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stringlengths 7
101
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stringlengths 1
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stringlengths 6
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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | by_cases yt : y ∈ t | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
⊢ y ∈ f u | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ f u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
⊢ y ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [mem_union, mem_setOf, eventually_nhdsWithin_iff, ← hf] | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ f u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ u ∨ y ∈ s ∧ y ∈ t ∧ ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ f u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | right | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ u ∨ y ∈ s ∧ y ∈ t ∧ ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | case pos.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ s ∧ y ∈ t ∧ ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ u ∨ y ∈ s ∧ y ∈ t ∧ ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | use m, yt, n | case pos.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ s ∧ y ∈ t ∧ ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∈ t
⊢ y ∈ s ∧ y ∈ t ∧ ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact mono _ (n.self_of_nhds yt) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m✝ : x ∈ f u
xu : x ∉ u
xt : x ∈ t
n✝ : x ∈ s ∧ ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ u
y : X
n : ∀ᶠ (x : X) in 𝓝 y, x ∈ tᶜ → x ∈ u
m : y ∈ s
yt : y ∉ t
⊢ y ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rw [← hf] | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ f u | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact (o.eventually_mem xu).mp (eventually_of_forall fun q m ↦ subset_union_left _ _ m) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | contrapose xu | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xu : x ∉ u
xt : x ∉ t
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ f u | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xt : x ∉ t
xu : ¬∀ᶠ (y : X) in 𝓝 x, y ∈ f u
⊢ ¬x ∉ u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xu : x ∉ u
xt : x ∉ t
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | clear xu | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xt : x ∉ t
xu : ¬∀ᶠ (y : X) in 𝓝 x, y ∈ f u
⊢ ¬x ∉ u | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xt : x ∉ t
⊢ ¬x ∉ u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xt : x ∉ t
xu : ¬∀ᶠ (y : X) in 𝓝 x, y ∈ f u
⊢ ¬x ∉ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [mem_union, mem_setOf, xt, false_and_iff, and_false_iff, or_false_iff, ← hf] at m | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xt : x ∉ t
⊢ ¬x ∉ u | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
xt : x ∉ t
m : x ∈ u
⊢ ¬x ∉ u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
m : x ∈ f u
xt : x ∉ t
⊢ ¬x ∉ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [not_not] | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
xt : x ∉ t
m : x ∈ u
⊢ ¬x ∉ u | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
xt : x ∉ t
m : x ∈ u
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
xt : x ∉ t
m : x ∈ u
⊢ ¬x ∉ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact m | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
xt : x ∉ t
m : x ∈ u
⊢ x ∈ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
u : Set X
o : IsOpen u
x : X
xt : x ∉ t
m : x ∈ u
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | intro x u c m xt cn cu | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
⊢ ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ f u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
⊢ ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rw [← hf] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ f u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ f u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | right | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) u | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | use m, xt | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u} | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ x ∈ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [Filter.eventually_iff, setOf_mem_eq] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ u ∈ 𝓝[tᶜ] x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact Filter.mem_of_superset cn cu | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ u ∈ 𝓝[tᶜ] x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
x : X
u c : Set X
m : x ∈ s
xt : x ∈ t
cn : c ∈ 𝓝[tᶜ] x
cu : c ⊆ u
⊢ u ∈ 𝓝[tᶜ] x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | intro x m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
⊢ s ⊆ f u ∪ f v | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
⊢ s ⊆ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | by_cases xt : x ∉ t | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
⊢ x ∈ f u ∪ f v | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∉ t
⊢ x ∈ f u ∪ f v
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : ¬x ∉ t
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact union_subset_union (mono _) (mono _) (suv (mem_diff_of_mem m xt)) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∉ t
⊢ x ∈ f u ∪ f v
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : ¬x ∉ t
⊢ x ∈ f u ∪ f v | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : ¬x ∉ t
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∉ t
⊢ x ∈ f u ∪ f v
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : ¬x ∉ t
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [not_not] at xt | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : ¬x ∉ t
⊢ x ∈ f u ∪ f v | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : ¬x ∉ t
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rcases ts.loc x s xt (so.mem_nhds m) with ⟨c, cst, cn, cp⟩ | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
⊢ x ∈ f u ∪ f v | case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have d := inter_subset_inter_left (u ∩ v) cst | case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
⊢ x ∈ f u ∪ f v | case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) ⊆ s \ t ∩ (u ∩ v)
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rw [duv, subset_empty_iff] at d | case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) ⊆ s \ t ∩ (u ∩ v)
⊢ x ∈ f u ∪ f v | case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) ⊆ s \ t ∩ (u ∩ v)
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | cases' isPreconnected_iff_subset_of_disjoint.mp cp u v uo vo (_root_.trans cst suv) d with cu cv | case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
⊢ x ∈ f u ∪ f v | case neg.intro.intro.intro.inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cu : c ⊆ u
⊢ x ∈ f u ∪ f v
case neg.intro.intro.intro.inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cv : c ⊆ v
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact subset_union_left _ _ (mem m xt cn cu) | case neg.intro.intro.intro.inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cu : c ⊆ u
⊢ x ∈ f u ∪ f v
case neg.intro.intro.intro.inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cv : c ⊆ v
⊢ x ∈ f u ∪ f v | case neg.intro.intro.intro.inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cv : c ⊆ v
⊢ x ∈ f u ∪ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cu : c ⊆ u
⊢ x ∈ f u ∪ f v
case neg.intro.intro.intro.inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cv : c ⊆ v
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact subset_union_right _ _ (mem m xt cn cv) | case neg.intro.intro.intro.inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cv : c ⊆ v
⊢ x ∈ f u ∪ f v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
x : X
m : x ∈ s
xt : x ∈ t
c : Set X
cst : c ⊆ s \ t
cn : c ∈ 𝓝[tᶜ] x
cp : IsPreconnected c
d : c ∩ (u ∩ v) = ∅
cv : c ⊆ v
⊢ x ∈ f u ∪ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | intro u x m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
⊢ ∀ {u : Set X}, f u \ t ⊆ u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : x ∈ f u \ t
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
⊢ ∀ {u : Set X}, f u \ t ⊆ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [mem_diff, mem_union, mem_setOf, ← hf] at m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : x ∈ f u \ t
⊢ x ∈ u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : (x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u) ∧ x ∉ t
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : x ∈ f u \ t
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [m.2, false_and_iff, and_false_iff, or_false_iff, not_false_iff, and_true_iff] at m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : (x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u) ∧ x ∉ t
⊢ x ∈ u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : x ∈ u
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : (x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u) ∧ x ∉ t
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : x ∈ u
⊢ x ∈ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : x ∈ u
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | intro x u o m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
⊢ ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ f u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
⊢ ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [mem_union, mem_setOf, ← hf] at m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ f u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ f u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | cases' m with xu m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact (o.eventually_mem xu).filter_mono nhdsWithin_le_nhds | case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact m.2.2 | case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | contrapose duv | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ s ∩ (f u ∩ f v) = ∅ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : ¬s ∩ (f u ∩ f v) = ∅
⊢ ¬s \ t ∩ (u ∩ v) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ s ∩ (f u ∩ f v) = ∅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [← ne_eq, ← nonempty_iff_ne_empty] at duv ⊢ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : ¬s ∩ (f u ∩ f v) = ∅
⊢ ¬s \ t ∩ (u ∩ v) = ∅ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : (s ∩ (f u ∩ f v)).Nonempty
⊢ (s \ t ∩ (u ∩ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : ¬s ∩ (f u ∩ f v) = ∅
⊢ ¬s \ t ∩ (u ∩ v) = ∅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rcases duv with ⟨x, m⟩ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : (s ∩ (f u ∩ f v)).Nonempty
⊢ (s \ t ∩ (u ∩ v)).Nonempty | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∩ (f u ∩ f v)
⊢ (s \ t ∩ (u ∩ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : (s ∩ (f u ∩ f v)).Nonempty
⊢ (s \ t ∩ (u ∩ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [mem_inter_iff] at m | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∩ (f u ∩ f v)
⊢ (s \ t ∩ (u ∩ v)).Nonempty | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
⊢ (s \ t ∩ (u ∩ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∩ (f u ∩ f v)
⊢ (s \ t ∩ (u ∩ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have b := ((so.eventually_mem m.1).filter_mono nhdsWithin_le_nhds).and
((fnon uo m.2.1).and (fnon vo m.2.2)) | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
⊢ (s \ t ∩ (u ∩ v)).Nonempty | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝[tᶜ] x, x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [eventually_nhdsWithin_iff] at b | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝[tᶜ] x, x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝[tᶜ] x, x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rcases eventually_nhds_iff.mp b with ⟨n, h, no, xn⟩ | case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty | case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
⊢ (s \ t ∩ (u ∩ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rcases ts.dense.exists_mem_open no ⟨_, xn⟩ with ⟨y, yt, yn⟩ | case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
⊢ (s \ t ∩ (u ∩ v)).Nonempty | case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ (s \ t ∩ (u ∩ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
⊢ (s \ t ∩ (u ∩ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | use y | case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ (s \ t ∩ (u ∩ v)).Nonempty | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ y ∈ s \ t ∩ (u ∩ v) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ (s \ t ∩ (u ∩ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [mem_inter_iff, mem_diff, ← mem_compl_iff] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ y ∈ s \ t ∩ (u ∩ v) | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ (y ∈ s ∧ y ∈ tᶜ) ∧ y ∈ u ∧ y ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ y ∈ s \ t ∩ (u ∩ v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | specialize h y yn yt | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ (y ∈ s ∧ y ∈ tᶜ) ∧ y ∈ u ∧ y ∈ v | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
h : y ∈ s ∧ y ∈ u ∧ y ∈ v
⊢ (y ∈ s ∧ y ∈ tᶜ) ∧ y ∈ u ∧ y ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
⊢ (y ∈ s ∧ y ∈ tᶜ) ∧ y ∈ u ∧ y ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact ⟨⟨h.1,yt⟩,h.2.1,h.2.2⟩ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
h : y ∈ s ∧ y ∈ u ∧ y ∈ v
⊢ (y ∈ s ∧ y ∈ tᶜ) ∧ y ∈ u ∧ y ∈ v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
no : IsOpen n
xn : x ∈ n
y : X
yt : y ∈ tᶜ
yn : y ∈ n
h : y ∈ s ∧ y ∈ u ∧ y ∈ v
⊢ (y ∈ s ∧ y ∈ tᶜ) ∧ y ∈ u ∧ y ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.empty | [172, 1] | [174, 96] | simp only [compl_empty, dense_univ] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
⊢ Dense ∅ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
⊢ Dense ∅ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.empty | [172, 1] | [174, 96] | simp only [mem_empty_iff_false, IsEmpty.forall_iff, forall_const, imp_true_iff] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
⊢ ∀ (x : X) (u : Set X), x ∈ ∅ → u ∈ 𝓝 x → ∃ c ⊆ u \ ∅, c ∈ 𝓝[∅ᶜ] x ∧ IsPreconnected c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
⊢ ∀ (x : X) (u : Set X), x ∈ ∅ → u ∈ 𝓝 x → ∃ c ⊆ u \ ∅, c ∈ 𝓝[∅ᶜ] x ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | by_cases rp : r ≤ 0 | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
⊢ IsPreconnected (ball a r \ {a}) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [Metric.ball_eq_empty.mpr rp, empty_diff] | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected ∅
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | exact isPreconnected_empty | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected ∅
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected ∅
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [not_le] at rp | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | rw [e] | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (ball a r \ {a}) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected ((fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | apply IsPreconnected.image | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected ((fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ) | case neg.H
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (Ioo 0 r ×ˢ univ)
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected ((fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | exact isPreconnected_Ioo.prod isPreconnected_univ | case neg.H
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (Ioo 0 r ×ˢ univ)
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.H
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (Ioo 0 r ×ˢ univ)
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | apply Continuous.continuousOn | case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | case neg.hf.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ Continuous fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | continuity | case neg.hf.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ Continuous fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hf.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ Continuous fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | apply Set.ext | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ∀ (x : ℂ), x ∈ ball a r \ {a} ↔ x ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | intro z | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ∀ (x : ℂ), x ∈ ball a r \ {a} ↔ x ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ z ∈ ball a r \ {a} ↔ z ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ∀ (x : ℂ), x ∈ ball a r \ {a} ↔ x ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [mem_diff, mem_ball, Complex.dist_eq, mem_singleton_iff, mem_image, Prod.exists,
mem_prod_eq, mem_Ioo, mem_univ, and_true_iff] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ z ∈ ball a r \ {a} ↔ z ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a ↔ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ z ∈ ball a r \ {a} ↔ z ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | constructor | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a ↔ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a → ∃ a_2 b, (0 < a_2 ∧ a_2 < r) ∧ a + ↑a_2 * (↑b * Complex.I).exp = z
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ (∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z) → Complex.abs (z - a) < r ∧ ¬z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a ↔ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | intro ⟨zr, za⟩ | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a → ∃ a_2 b, (0 < a_2 ∧ a_2 < r) ∧ a + ↑a_2 * (↑b * Complex.I).exp = z | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a → ∃ a_2 b, (0 < a_2 ∧ a_2 < r) ∧ a + ↑a_2 * (↑b * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | use abs (z - a), Complex.arg (z - a) | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ (0 < Complex.abs (z - a) ∧ Complex.abs (z - a) < r) ∧ a + ↑(Complex.abs (z - a)) * (↑(z - a).arg * Complex.I).exp = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [AbsoluteValue.pos_iff, Ne, Complex.abs_mul_exp_arg_mul_I,
add_sub_cancel, eq_self_iff_true, sub_eq_zero, za, zr, not_false_iff, and_true_iff] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ (0 < Complex.abs (z - a) ∧ Complex.abs (z - a) < r) ∧ a + ↑(Complex.abs (z - a)) * (↑(z - a).arg * Complex.I).exp = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ (0 < Complex.abs (z - a) ∧ Complex.abs (z - a) < r) ∧ a + ↑(Complex.abs (z - a)) * (↑(z - a).arg * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | intro ⟨s, t, ⟨s0, sr⟩, e⟩ | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ (∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z) → Complex.abs (z - a) < r ∧ ¬z = a | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ Complex.abs (z - a) < r ∧ ¬z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ (∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z) → Complex.abs (z - a) < r ∧ ¬z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [← e, add_sub_cancel_left, Complex.abs.map_mul, Complex.abs_ofReal, abs_of_pos s0,
Complex.abs_exp_ofReal_mul_I, mul_one, sr, true_and_iff, add_right_eq_self, mul_eq_zero,
Complex.exp_ne_zero, or_false_iff, Complex.ofReal_eq_zero] | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ Complex.abs (z - a) < r ∧ ¬z = a | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ ¬s = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ Complex.abs (z - a) < r ∧ ¬z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | exact s0.ne' | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ ¬s = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ ¬s = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rw [dense_iff_inter_open] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ Dense {a}ᶜ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (U : Set ℂ), IsOpen U → U.Nonempty → (U ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ Dense {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | intro u uo ⟨z, m⟩ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (U : Set ℂ), IsOpen U → U.Nonempty → (U ∩ {a}ᶜ).Nonempty | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (U : Set ℂ), IsOpen U → U.Nonempty → (U ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | by_cases za : z ≠ a | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : ¬z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [not_not] at za | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : ¬z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : ¬z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rw [za] at m | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : a ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | clear za z | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : a ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : a ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rcases Metric.isOpen_iff.mp uo a m with ⟨r, rp, rs⟩ | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | case neg.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use a + r / 2 | case neg.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ (u ∩ {a}ᶜ).Nonempty | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u ∩ {a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [mem_inter_iff, mem_compl_iff, mem_singleton_iff, add_right_eq_self,
div_eq_zero_iff, Complex.ofReal_eq_zero, bit0_eq_zero, one_ne_zero, or_false_iff,
rp.ne', not_false_iff, and_true_iff, false_or, two_ne_zero] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u ∩ {a}ᶜ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u ∩ {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | apply rs | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u | case h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ ball a r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [mem_ball, dist_self_add_left, Complex.norm_eq_abs, map_div₀, Complex.abs_ofReal,
Complex.abs_two, abs_of_pos rp, half_lt_self rp] | case h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ ball a r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ ball a r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use z | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ u ∩ {a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use m | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ u ∩ {a}ᶜ | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ {a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ u ∩ {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | exact za | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ {a}ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | intro z u m n | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (x : ℂ) (u : Set ℂ), x ∈ {a} → u ∈ 𝓝 x → ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] x ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z ∈ {a}
n : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (x : ℂ) (u : Set ℂ), x ∈ {a} → u ∈ 𝓝 x → ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] x ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [mem_singleton_iff] at m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z ∈ {a}
n : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
n : u ∈ 𝓝 z
m : z = a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z ∈ {a}
n : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [m] at n ⊢ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
n : u ∈ 𝓝 z
m : z = a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z = a
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
n : u ∈ 𝓝 z
m : z = a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | clear m z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z = a
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z = a
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rcases Metric.mem_nhds_iff.mp n with ⟨r, rp, rs⟩ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | case intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use ball a r \ {a} | case intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ⊆ u \ {a} ∧ ball a r \ {a} ∈ 𝓝[≠] a ∧ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | refine ⟨diff_subset_diff_left rs, ?_, IsPreconnected.ball_diff_center⟩ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ⊆ u \ {a} ∧ ball a r \ {a} ∈ 𝓝[≠] a ∧ IsPreconnected (ball a r \ {a}) | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ∈ 𝓝[≠] a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ⊆ u \ {a} ∧ ball a r \ {a} ∈ 𝓝[≠] a ∧ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | exact diff_mem_nhdsWithin_compl (Metric.ball_mem_nhds _ rp) _ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ∈ 𝓝[≠] a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ∈ 𝓝[≠] a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | apply Nonseparating.complexManifold | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ Nonseparating {a} | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ Nonseparating {a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | intro z | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | by_cases az : a ∈ (extChartAt I z).source | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | convert Complex.nonseparating_singleton (extChartAt I z a) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = {↑(extChartAt I z) a} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | simp only [eq_singleton_iff_unique_mem, mem_inter_iff, PartialEquiv.map_source _ az, true_and_iff,
mem_preimage, mem_singleton_iff, PartialEquiv.left_inv _ az, eq_self_iff_true] | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = {↑(extChartAt I z) a} | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ ∀ (x : ℂ), x ∈ (extChartAt I z).target ∧ ↑(extChartAt I z).symm x = a → x = ↑(extChartAt I z) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = {↑(extChartAt I z) a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | intro x ⟨m, e⟩ | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ ∀ (x : ℂ), x ∈ (extChartAt I z).target ∧ ↑(extChartAt I z).symm x = a → x = ↑(extChartAt I z) a | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
e : ↑(extChartAt I z).symm x = a
⊢ x = ↑(extChartAt I z) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ ∀ (x : ℂ), x ∈ (extChartAt I z).target ∧ ↑(extChartAt I z).symm x = a → x = ↑(extChartAt I z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | simp only [← e, PartialEquiv.right_inv _ m] | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
e : ↑(extChartAt I z).symm x = a
⊢ x = ↑(extChartAt I z) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
e : ↑(extChartAt I z).symm x = a
⊢ x = ↑(extChartAt I z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | convert Nonseparating.empty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
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