mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsClosed[inst✝] s
h : s ⊆ t
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcov : ⋃ n, K n = t
⊢ s ∩ t = s

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsClosed[inst✝] s
h : s ⊆ t
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcov : ⋃ n, K n = t
⊢ s ∩ t = s
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hf : IsInducing f
L : ℕ → Set Y
hcomp : ∀ (n : ℕ), IsCompact (L n)
hcov : ⋃ n, L n = f '' s
n : ℕ
⊢ L n = L n

pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hf : IsInducing f
L : ℕ → Set Y
hcomp : ∀ (n : ℕ), IsCompact (L n)
hcov : ⋃ n, L n = f '' s
n : ℕ
⊢ L n ⊆ f '' s

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hf : IsInducing f
L : ℕ → Set Y
hcomp : ∀ (n : ℕ), IsCompact (L n)
hcov : ⋃ n, L n = f '' s
n : ℕ
⊢ L n ⊆ f '' s
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hf : IsInducing f
L : ℕ → Set Y
hcomp : ∀ (n : ℕ), IsCompact (L n)
hcov : ⋃ n, L n = f '' s
⊢ ⋃ n, f ⁻¹' L n ∩ s = ⋃ i, f ⁻¹' L i ∩ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hf : IsInducing f
L : ℕ → Set Y
hcomp : ∀ (n : ℕ), IsCompact (L n)
hcov : ⋃ n, L n = f '' s
⊢ f ⁻¹' f '' s ∩ s = f ⁻¹' f '' s ∩ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ (∃ K, (∀ (n : ℕ), IsCompact (K n)) ∧ ⋃ n, K n = univ) ↔ ∃ K, (∀ (n : ℕ), IsCompact (K n)) ∧ ⋃ n, K n = univ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsSigmaCompact s ↔ IsSigmaCompact s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
s✝ t : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology X
K : X → Set X
hKc : ∀ (x : X), IsCompact (K x)
hxK : ∀ (x : X), K x ∈ 𝓝 x
s : Set X
hsc : s.Countable
hsU : ⋃ x ∈ s, K x = univ
⊢ ⋃ a ∈ s, K a = univ

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
s✝ t : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology X
K : X → Set X
hKc : ∀ (x : X), IsCompact (K x)
hxK : ∀ (x : X), K x ∈ 𝓝 x
s : Set X
hsc : s.Countable
hsU : ⋃ x ∈ s, K x = univ
⊢ ⋃ a ∈ s, K a = univ
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : SigmaCompactSpace X
⊢ ⋃ x, ⋯.choose x = univ

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : SigmaCompactSpace X
⊢ ⋃ x, ⋯.choose x = univ
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y
s t : Set X✝
inst✝³ : SigmaCompactSpace X✝
inst✝² : Finite ι
X : ι → Type u_4
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), SigmaCompactSpace (X i)
⊢ (univ.pi fun i ↦ ⋃ j, compactCovering (X i) j) = univ

pp: X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y
s t : Set X✝
inst✝³ : SigmaCompactSpace X✝
inst✝² : Finite ι
X : ι → Type u_4
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), SigmaCompactSpace (X i)
⊢ ∀ (i : ι), Monotone (compactCovering (X i))

Tactic:
]

Goal state after:
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y
s t : Set X✝
inst✝³ : SigmaCompactSpace X✝
inst✝² : Finite ι
X : ι → Type u_4
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), SigmaCompactSpace (X i)
⊢ (univ.pi fun i ↦ ⋃ j, compactCovering (X i) j) = univ

X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y
s t : Set X✝
inst✝³ : SigmaCompactSpace X✝
inst✝² : Finite ι
X : ι → Type u_4
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), SigmaCompactSpace (X i)
⊢ ∀ (i : ι), Monotone (compactCovering (X i))
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
p : Set X → Prop
he : p ∅
hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s
hunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)
hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t
f : Filter X := comk p he ⋯ ⋯
this : sᶜ ∈ f
⊢ p sᶜᶜ

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
p : Set X → Prop
he : p ∅
hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s
hunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)
hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t
f : Filter X := comk p he ⋯ ⋯
this : sᶜ ∈ f
⊢ p sᶜᶜ
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : SecondCountableTopology β
hf : IsInducing f
⊢ SecondCountableTopology α

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : SecondCountableTopology β
hf : IsInducing f
⊢ SecondCountableTopology α
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : SigmaCompactSpace X
e : Y → X
he : IsClosedEmbedding e
⊢ univ = univ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsCompact s
hf : ContinuousOn f s
l : Filter Y
lne : l.NeBot
ls : l ≤ 𝓟 (f '' s)
this✝ : (comap f l ⊓ 𝓟 s).NeBot
x : X
hxs : x ∈ s
hx : ClusterPt x (comap f l ⊓ 𝓟 s)
this : (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)).NeBot
⊢ 𝓝 x ⊓ (comap f l ⊓ 𝓟 s) = 𝓝 x ⊓ 𝓟 s ⊓ comap f l

Tactic:
]

Goal state after:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsCompact s
hf : ContinuousOn f s
l : Filter Y
lne : l.NeBot
ls : l ≤ 𝓟 (f '' s)
this✝ : (comap f l ⊓ 𝓟 s).NeBot
x : X
hxs : x ∈ s
hx : ClusterPt x (comap f l ⊓ 𝓟 s)
this : (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)).NeBot
⊢ 𝓝 x ⊓ (comap f l ⊓ 𝓟 s) = 𝓝 x ⊓ 𝓟 s ⊓ comap f l
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
l : Filter X
s' : Set X
hmem : s ∈ l
h : ∀ x ∈ s, ClusterPt x l → x ∈ s'
f : Ultrafilter X
hf : ↑f ≤ l
x : X
hxs : x ∈ s
hx : ↑f ≤ 𝓝 x
⊢ 𝓝 x ≤ 𝓝ˢ s'

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
l : Filter X
s' : Set X
hmem : s ∈ l
h : ∀ x ∈ s, ClusterPt x l → x ∈ s'
f : Ultrafilter X
hf : ↑f ≤ l
x : X
hxs : x ∈ s
hx : ↑f ≤ 𝓝 x
⊢ 𝓝 x ≤ 𝓝ˢ s'
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
l : Filter X
y : X
hmem : s ∈ l
h : ∀ x ∈ s, ClusterPt x l → x = y
⊢ l ≤ 𝓝ˢ {y}

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
l : Filter X
y : X
hmem : s ∈ l
h : ∀ x ∈ s, ClusterPt x l → x = y
⊢ l ≤ 𝓝ˢ {y}
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
Y : Type u_2
l : Filter Y
y : X
f : Y → X
hs : IsCompact s
hmem : ∀ᶠ (x : Y) in l, f x ∈ s
h : ∀ x ∈ s, MapClusterPt x l f → x = y
⊢ Tendsto f l (𝓝ˢ {y})

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
Y : Type u_2
l : Filter Y
y : X
f : Y → X
hs : IsCompact s
hmem : ∀ᶠ (x : Y) in l, f x ∈ s
h : ∀ x ∈ s, MapClusterPt x l f → x = y
⊢ Tendsto f l (𝓝ˢ {y})
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
U : X → Set X
hU : ∀ x ∈ s, U x ∈ 𝓝 x
t : Finset ↑s
ht : ⋃ x ∈ t, U ↑x ∈ 𝓝ˢ s
⊢ ⋃ y ∈ t, U ↑y ∈ 𝓝ˢ s

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
U : X → Set X
hU : ∀ x ∈ s, U x ∈ 𝓝 x
t : Finset ↑s
ht : ⋃ x ∈ t, U ↑x ∈ 𝓝ˢ s
⊢ ⋃ y ∈ t, U ↑y ∈ 𝓝ˢ s
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
l : Filter X
hs : IsCompact s
H : ∀ x ∈ s, Disjoint (𝓝 x) l
U : X → Set X
hUl : ∀ x ∈ s, (U x)ᶜ ∈ l
hxU : ∀ x ∈ s, x ∈ U x
hUo : ∀ x ∈ s, IsOpen[inst✝] (U x)
t : Finset X
hts : ∀ x ∈ t, x ∈ s
hst : s ⊆ ⋃ x ∈ t, U x
⊢ ∀ i ∈ t, (U i)ᶜ ∈ l

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
l : Filter X
hs : IsCompact s
H : ∀ x ∈ s, Disjoint (𝓝 x) l
U : X → Set X
hUl : ∀ x ∈ s, (U x)ᶜ ∈ l
hxU : ∀ x ∈ s, x ∈ U x
hUo : ∀ x ∈ s, IsOpen[inst✝] (U x)
t : Finset X
hts : ∀ x ∈ t, x ∈ s
hst : s ⊆ ⋃ x ∈ t, U x
⊢ ∀ i ∈ t, (U i)ᶜ ∈ l
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
ι : Type v
hs : IsCompact s
t : ι → Set X
htc : ∀ (i : ι), IsClosed[inst✝] (t i)
hst : s ∩ ⋂ i, t i = ∅
⊢ s ∩ ⋂ i, t i = ∅

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
ι : Type v
hs : IsCompact s
t : ι → Set X
htc : ∀ (i : ι), IsClosed[inst✝] (t i)
hst : s ∩ ⋂ i, t i = ∅
⊢ s ∩ ⋂ i, t i = ∅
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
t : Set (Set X)
ht : ∀ a ∈ t, IsClosed[inst✝] a
h : ∀ a ⊆ t, a.Finite → (s ∩ ⋂₀ a).Nonempty
⊢ (s ∩ ⋂ i, ↑i).Nonempty

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
t : Set (Set X)
ht : ∀ a ∈ t, IsClosed[inst✝] a
h : ∀ a ⊆ t, a.Finite → (s ∩ ⋂₀ a).Nonempty
⊢ (s ∩ ⋂ i, ↑i).Nonempty
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
ι : Type v
hι : Nonempty ι
t : ι → Set X
htd : Directed (fun x1 x2 ↦ x1 ⊇ x2) t
htn : ∀ (i : ι), (t i).Nonempty
htc : ∀ (i : ι), IsCompact (t i)
htcl : ∀ (i : ι), IsClosed[inst✝] (t i)
i₀ : ι := hι.some
this : (⋂ i, t i).Nonempty
⊢ (⋂ i, t i).Nonempty

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
ι : Type v
hι : Nonempty ι
t : ι → Set X
htd : Directed (fun x1 x2 ↦ x1 ⊇ x2) t
htn : ∀ (i : ι), (t i).Nonempty
htc : ∀ (i : ι), IsCompact (t i)
htcl : ∀ (i : ι), IsClosed[inst✝] (t i)
i₀ : ι := hι.some
this : (⋂ i, t i).Nonempty
⊢ (⋂ i, t i).Nonempty
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hS : Nonempty ↑S
hSd : DirectedOn (fun x1 x2 ↦ x1 ⊇ x2) S
hSn : ∀ U ∈ S, U.Nonempty
hSc : ∀ U ∈ S, IsCompact U
hScl : ∀ U ∈ S, IsClosed[inst✝] U
⊢ (⋂ i, ↑i).Nonempty

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hS : Nonempty ↑S
hSd : DirectedOn (fun x1 x2 ↦ x1 ⊇ x2) S
hSn : ∀ U ∈ S, U.Nonempty
hSc : ∀ U ∈ S, IsCompact U
hScl : ∀ U ∈ S, IsClosed[inst✝] U
⊢ (⋂ i, ↑i).Nonempty
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case refine_2
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set X
b : Set ι
c : ι → Set X
hs : IsCompact s
hc₁ : ∀ (x : { a // a ∈ b }), IsOpen[inst✝] (c ↑x)
hc₂ : s ⊆ ⋃ x, c ↑x
d : Finset ↑b
hd : s ⊆ ⋃ i ∈ d, c ↑i
⊢ s ⊆ ⋃ y ∈ ↑d, c ↑y

Tactic:
]

Goal state after:
case refine_2
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set X
b : Set ι
c : ι → Set X
hs : IsCompact s
hc₁ : ∀ (x : { a // a ∈ b }), IsOpen[inst✝] (c ↑x)
hc₂ : s ⊆ ⋃ x, c ↑x
d : Finset ↑b
hd : s ⊆ ⋃ i ∈ d, c ↑i
⊢ s ⊆ ⋃ y ∈ ↑d, c ↑y
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen[inst✝] (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
t : Finset ↑s
ht : s ⊆ ⋃ i ∈ t, U i
⊢ s ⊆ ⋃ i ∈ t, (U i)ᶜᶜ

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen[inst✝] (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
t : Finset ↑s
ht : s ⊆ ⋃ i ∈ t, U i
⊢ s ⊆ ⋃ i ∈ t, (U i)ᶜᶜ
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
h : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed[inst✝] (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen[inst✝] (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
⊢ ∃ t, s ⊆ ⋃ i ∈ t, U i

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed[inst✝] (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen[inst✝] (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
⊢ ∃ t, s ⊆ ⋃ i ∈ t, U i
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
h : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed[inst✝] (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen[inst✝] (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
t : Finset ι✝
ht : s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅
⊢ s ⊓ ⋂ i ∈ t, (U i)ᶜ = ⊥

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed[inst✝] (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen[inst✝] (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
t : Finset ι✝
ht : s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅
⊢ s ⊓ ⋂ i ∈ t, (U i)ᶜ = ⊥
Module: Mathlib.Data.Set.Subset

Goal state before:
pp: α : Type u_2
A B : Set α
h : A ⊆ B
⊢ ∀ (a : α) (b : a ∈ A), ⟨a, b⟩ ∈ A ↓∩ B

Tactic:
]

Goal state after:
α : Type u_2
A B : Set α
h : A ⊆ B
⊢ ∀ (a : α) (b : a ∈ A), ⟨a, b⟩ ∈ A ↓∩ B
Module: Mathlib.Data.Set.Subset

Goal state before:
pp: α : Type u_2
A : Set α
S : Set (Set α)
⊢ A ↓∩ ⋃₀ S = ⋃ a ∈ S, A ↓∩ a

Tactic:
]

Goal state after:
α : Type u_2
A : Set α
S : Set (Set α)
⊢ A ↓∩ ⋃₀ S = ⋃ a ∈ S, A ↓∩ a
Module: Mathlib.Data.Set.Subset

Goal state before:
pp: α : Type u_2
A : Set α
S : Set (Set α)
⊢ A ↓∩ ⋂₀ S = ⋂ a ∈ S, A ↓∩ a

Tactic:
]

Goal state after:
α : Type u_2
A : Set α
S : Set (Set α)
⊢ A ↓∩ ⋂₀ S = ⋂ a ∈ S, A ↓∩ a
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set ι
f : ι → Set X
hs : s.Finite
hf : ∀ i ∈ s, IsCompact (f i)
l : Ultrafilter X
hl : ∃ i ∈ s, f i ∈ l
⊢ ∃ x ∈ ⋃ i ∈ s, f i, ↑l ≤ 𝓝 x

Tactic:
]

Goal state after:
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set ι
f : ι → Set X
hs : s.Finite
hf : ∀ i ∈ s, IsCompact (f i)
l : Ultrafilter X
hl : ∃ i ∈ s, f i ∈ l
⊢ ∃ x ∈ ⋃ i ∈ s, f i, ↑l ≤ 𝓝 x
Module: Mathlib.Data.Set.Subset

Goal state before:
pp: α : Type u_2
A : Set α
D : Set ↑A
⊢ A \ Subtype.val '' D = A \ Subtype.val '' D

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hf : S.Finite
hc : ∀ s ∈ S, IsCompact s
⊢ IsCompact (⋃ i ∈ S, i)

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hf : S.Finite
hc : ∀ s ∈ S, IsCompact s
⊢ IsCompact (⋃ i ∈ S, i)
Module: Mathlib.Data.Set.Subset

Goal state before:
pp: α : Type u_2
A : Set α
T : Set (Set ↑A)
⊢ ⋃₀ {x | ∃ a ∈ T, Subtype.val '' a = x} = ⋃₀ {x | ∃ B ∈ T, Subtype.val '' B = x}

Tactic:
]

Goal state after:
Module: Mathlib.Data.Set.Subset

Goal state before:
pp: α : Type u_2
A : Set α
T : Set (Set ↑A)
hT : T.Nonempty
⊢ ⋂ i ∈ T, Subtype.val '' i = ⋂ a ∈ T, Subtype.val '' a

Tactic:
]

Goal state after:
Module: Mathlib.Data.Set.Subset

Goal state before:
pp: α : Type u_2
A : Set α
D E : Set ↑A
⊢ D ⊆ E ↔ D ⊆ E

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hs : IsCompact s
ht : IsCompact t
⊢ IsCompact (⋃ b, bif b then s else t)

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hs : IsCompact s
ht : IsCompact t
⊢ IsCompact (⋃ b, bif b then s else t)
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
T : TopologicalSpace X
S : Set (Set X)
hTS : T = generateFrom S
s : Set X
h : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q
⊢ ∀ (f : Ultrafilter X), s ∈ f → ∃ x ∈ s, ↑f ≤ 𝓝 x

Tactic:
]

Goal state after:
X : Type u
T : TopologicalSpace X
S : Set (Set X)
hTS : T = generateFrom S
s : Set X
h : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q
⊢ ∀ (f : Ultrafilter X), s ∈ f → ∃ x ∈ s, ↑f ≤ 𝓝 x
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
f : α ≃o β
⊢ upperClosure (⇑f '' s) = (UpperSet.map f.symm).symm (upperClosure s)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
f : α ≃o β
⊢ upperClosure (⇑f '' s) = (UpperSet.map f.symm).symm (upperClosure s)
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : Set α
⊢ s ⊆ ↑⊤ ↔ s = ∅

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Preorder α
s : Set α
⊢ s ⊆ ↑⊤ ↔ s = ∅
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : Set α
h : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s
⊢ (↑(upperClosure s) ∩ ↑(lowerClosure s)).OrdConnected

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Preorder α
s : Set α
h : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s
⊢ (↑(upperClosure s) ∩ ↑(lowerClosure s)).OrdConnected
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : PartialOrder α
s : Set α
x : α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
h : Minimal (fun x ↦ x ∈ { carrier := {x | ∃ a ∈ s, a ≤ x}, upper' := ⋯ }) x
a : α
has : a ∈ s
hax : a ≤ x
⊢ a ∈ s

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
inst✝ : PartialOrder α
s : Set α
x : α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
h : Minimal (fun x ↦ x ∈ { carrier := {x | ∃ a ∈ s, a ≤ x}, upper' := ⋯ }) x
a : α
has : a ∈ s
hax : a ≤ x
⊢ a ∈ s
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : PartialOrder α
s : Set α
x : α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
h : x ∈ s
b : α
x✝ : (fun x ↦ x ∈ { carrier := {x | ∃ a ∈ s, a ≤ x}, upper' := ⋯ }) b
hbx : b ≤ x
a : α
has : a ∈ s
hab : a ≤ b
⊢ a ≤ b

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
inst✝ : PartialOrder α
s : Set α
x : α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
h : x ∈ s
b : α
x✝ : (fun x ↦ x ∈ { carrier := {x | ∃ a ∈ s, a ≤ x}, upper' := ⋯ }) b
hbx : b ≤ x
a : α
has : a ∈ s
hab : a ≤ b
⊢ a ≤ b
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
⊢ s.sdiff {a} = s.erase a

Tactic:
by
  simp [sdiff, erase]

Goal state after:
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
⊢ s.sdiff {a} = s ↔ a ∉ s

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
⊢ s.sdiff {a} = s ↔ a ∉ s
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : UpperSet α
t : Set α
hts : t ⊆ ↑s
hst : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : UpperSet α
t : Set α
hts : t ⊆ ↑s
hst : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t
⊢ upperClosure t ⊓ s.sdiff t = s

Tactic:
by rw [inf_comm, sdiff_inf_upperClosure hts hst]

Goal state after:
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
f : ι → X
x : X
this : TopologicalSpace ι := ⊥
hf : Tendsto f (cocompact ι) (𝓝 x)
h : DiscreteTopology ι
⊢ IsCompact (insert x (range f))

Tactic:
]

Goal state after:
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
f : ι → X
x : X
this : TopologicalSpace ι := ⊥
hf : Tendsto f (cocompact ι) (𝓝 x)
h : DiscreteTopology ι
⊢ IsCompact (insert x (range f))
Module: Mathlib.Order.UpperLower.CompleteLattice

Goal state before:
pp: α : Type u_1
ι : Sort u_4
inst✝ : LE α
a : α
f : ι → UpperSet α
⊢ a ∈ ⋂ i, ↑(f i) ↔ ∀ (i : ι), a ∈ f i

Tactic:
]

Goal state after:
α : Type u_1
ι : Sort u_4
inst✝ : LE α
a : α
f : ι → UpperSet α
⊢ a ∈ ⋂ i, ↑(f i) ↔ ∀ (i : ι), a ∈ f i
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
ha : a ∈ s
has : ∀ b ∈ s, b ≤ a → b = a
⊢ s = s

pp: case hts
α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
ha : a ∈ s
has : ∀ b ∈ s, b ≤ a → b = a
⊢ {a} ⊆ ↑s

pp: case hst
α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
ha : a ∈ s
has : ∀ b ∈ s, b ≤ a → b = a
⊢ ∀ b ∈ s, ∀ c ∈ {a}, b ≤ c → b ∈ {a}

Tactic:
]

Goal state after:
case hts
α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
ha : a ∈ s
has : ∀ b ∈ s, b ≤ a → b = a
⊢ {a} ⊆ ↑s

case hst
α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
ha : a ∈ s
has : ∀ b ∈ s, b ≤ a → b = a
⊢ ∀ b ∈ s, ∀ c ∈ {a}, b ≤ c → b ∈ {a}
Module: Mathlib.Order.UpperLower.CompleteLattice

Goal state before:
pp: α : Type u_1
ι : Sort u_4
inst✝ : LE α
a : α
f : ι → UpperSet α
⊢ a ∈ ⋃ i, ↑(f i) ↔ ∃ i, a ∈ f i

Tactic:
]

Goal state after:
α : Type u_1
ι : Sort u_4
inst✝ : LE α
a : α
f : ι → UpperSet α
⊢ a ∈ ⋃ i, ↑(f i) ↔ ∃ i, a ∈ f i
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : UpperSet α
a : α
ha : a ∈ s
has : ∀ b ∈ s, b ≤ a → b = a
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ IsCompact (closure[inst✝] s) ↔ IsCompact (closure[inst✝] s)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ t, IsCompact t ∧ tᶜ ⊆ sᶜ) ↔ ∃ t, IsCompact t ∧ s ⊆ t

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ t, IsCompact t ∧ tᶜ ⊆ sᶜ) ↔ ∃ t, IsCompact t ∧ s ⊆ t
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
s : Set α
x : α
inst✝ : DecidableEq α
y : β
⊢ Tendsto (update f x y) (𝓝[s \ {x}] x) (𝓝 y) ↔ Tendsto (update f x y) (𝓝[s \ {x}] x) (𝓝 y)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
x : α
inst✝ : DecidableEq α
y : β
⊢ Tendsto f (𝓝[univ \ {x}] x) (𝓝 y) ↔ Tendsto f (𝓝[univ \ {x}] x) (𝓝 y)

Tactic:
]

Goal state after:
Module: Mathlib.Order.UpperLower.CompleteLattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ≃o β
s : UpperSet α
b : β
⊢ b ∈ (map f.symm).symm s ↔ f.symm b ∈ s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ≃o β
s : UpperSet α
b : β
⊢ b ∈ (map f.symm).symm s ↔ f.symm b ∈ s