mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝³ : Group G
inst✝² : MulAction G X
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace X
x✝ :
  ContinuousSMul G X ∧
    ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
      Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)
cont : ContinuousSMul G X
h :
  ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
    Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
β : Type v
inst✝² : TopologicalSpace β
inst✝¹ : T2Space β
inst✝ : CompactSpace β
g : α → β
hg : Continuous[inst✝³, inst✝²] g
a b : α
h : Relation.EqvGen (fun F G ↦ ∃ x, ↑F ≤ 𝓝 x ∧ ↑G ≤ 𝓝 x) (pure a) (pure b)
e : Ultrafilter.extend g ∘ pure = g
⊢ Ultrafilter.extend g (pure a) = Ultrafilter.extend g (pure b)

Tactic:
generalize (pure a : Ultrafilter α) = F at h

Goal state after:
α : Type u
inst✝³ : TopologicalSpace α
β : Type v
inst✝² : TopologicalSpace β
inst✝¹ : T2Space β
inst✝ : CompactSpace β
g : α → β
hg : Continuous[inst✝³, inst✝²] g
a b : α
e : Ultrafilter.extend g ∘ pure = g
F : Ultrafilter α
h : Relation.EqvGen (fun F G ↦ ∃ x, ↑F ≤ 𝓝 x ∧ ↑G ≤ 𝓝 x) F (pure b)
⊢ Ultrafilter.extend g F = Ultrafilter.extend g (pure b)
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
β : Type v
inst✝² : TopologicalSpace β
inst✝¹ : T2Space β
inst✝ : CompactSpace β
g : α → β
hg : Continuous[inst✝³, inst✝²] g
a b : α
h : preStoneCechUnit a = preStoneCechUnit b
⊢ g a = g b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ Function.Surjective T2Quotient.mk

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ DenseRange stoneCechUnit

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝³ : Group G
inst✝² : MulAction G X
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace X
x✝¹ :
  ContinuousSMul G X ∧
    ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
      Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)
cont : ContinuousSMul G X
h :
  ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
    Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)
𝒰 : Ultrafilter (G × X)
x✝ : X × X
x₁ x₂ : X
hxx : Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂))
g : G
hg1 : g • x₂ = x₁
hg2 : Tendsto fst (↑𝒰) (𝓝 g)
⊢ ((g, x₂).1 • (g, x₂).2, (g, x₂).2) = (x₁, x₂)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
β : Type v
inst✝¹ : TopologicalSpace β
inst✝ : T2Space β
g₁ g₂ : StoneCech α → β
h₁ : Continuous[_, inst✝¹] g₁
h₂ : Continuous[_, inst✝¹] g₂
h : g₁ ∘ stoneCechUnit = g₂ ∘ stoneCechUnit
⊢ g₁ = g₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z w : ℍ
r : ℝ
g : SL(2, ℝ)
⊢ Isometry fun x ↦ g • x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
β : Type v
inst✝² : TopologicalSpace β
inst✝¹ : T2Space β
g : α → β
hg : Continuous[inst✝³, inst✝²] g
inst✝ : CompactSpace β
⊢ stoneCechExtend hg ∘ stoneCechUnit = g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
β : Type v
inst✝² : TopologicalSpace β
inst✝¹ : T2Space β
inst✝ : CompactSpace β
a b : α
f : α → β
hcf : Continuous[inst✝³, inst✝²] f
h : stoneCechUnit a = stoneCechUnit b
⊢ f a = f b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝³ : Group G
inst✝² : MulAction G X
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace X
⊢ ProperSMul G X ↔
    ContinuousSMul G X ∧
      ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
        Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: X : TopCat
Y : CompHaus
⊢ Function.LeftInverse (fun f ↦ ofHom (fun x ↦ True) { toFun := stoneCechExtend ⋯, continuous_toFun := ⋯ }) fun f ↦
    TopCat.ofHom { toFun := ⇑(ConcreteCategory.hom f) ∘ stoneCechUnit, continuous_toFun := ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: X : TopCat
Y : CompHaus
⊢ Function.RightInverse (fun f ↦ ofHom (fun x ↦ True) { toFun := stoneCechExtend ⋯, continuous_toFun := ⋯ }) fun f ↦
    TopCat.ofHom { toFun := ⇑(ConcreteCategory.hom f) ∘ stoneCechUnit, continuous_toFun := ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
x₁ x₂ : X
⊢ ↑x₁ = ↑x₂ ↔ x₁ = x₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
s : Set X
⊢ (some '' s)ᶜ = some '' sᶜ ∪ {∞}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
x : OnePoint X
⊢ x ≠ ∞ ↔ ∃ y, ↑y = x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
x : OnePoint X
⊢ x ∉ range some ↔ x = ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
⊢ some ⁻¹' {∞} = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
⊢ (∞ ∈ univ → IsCompact (some ⁻¹' univ)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' univ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: J : Type v
inst✝ : SmallCategory J
F : J ⥤ CompHaus
FF : J ⥤ TopCat := F ⋙ compHausToTop
⊢ {u | ∀ {i j : J} (f : i ⟶ j), (ConcreteCategory.hom (F.map f)) (u i) = u j} =
    ⋂ i, ⋂ j, ⋂ f, {u | (ConcreteCategory.hom (F.map f)) (u i) = u j}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
⊢ ProperSMul G X ↔
    ContinuousSMul G X ∧
      ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
        Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, Tendsto fst (↑𝒰) (𝓝 g)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
s t : Set (OnePoint X)
⊢ (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' s) →
    (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t) →
      (∞ ∈ s ∩ t → IsCompact (some ⁻¹' (s ∩ t))ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' (s ∩ t))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)
this : IsOpen[inst✝] (some ⁻¹' ⋃₀ S)
⊢ (∞ ∈ ⋃₀ S → IsCompact (some ⁻¹' ⋃₀ S)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' ⋃₀ S)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)
⊢ (∞ ∈ ⋃₀ S → IsCompact (some ⁻¹' ⋃₀ S)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' ⋃₀ S)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
h : ∞ ∈ s
⊢ IsOpen[instTopologicalSpace] s ↔ IsCompact (some ⁻¹' s)ᶜ ∧ IsOpen[inst✝] (some ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
h : ∞ ∈ s
⊢ IsOpen[instTopologicalSpace] s ↔ IsClosed[inst✝] (some ⁻¹' s)ᶜ ∧ IsCompact (some ⁻¹' s)ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
h : ∞ ∉ s
⊢ IsOpen[instTopologicalSpace] s ↔ IsOpen[inst✝] (some ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
h : ∞ ∈ s
⊢ IsClosed[instTopologicalSpace] s ↔ IsClosed[inst✝] (some ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
h : ∞ ∉ s
⊢ IsClosed[instTopologicalSpace] s ↔ IsClosed[inst✝] (some ⁻¹' s) ∧ IsCompact (some ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsOpen[instTopologicalSpace] (some '' s) ↔ IsOpen[inst✝] s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: J : Type v
inst✝ : SmallCategory J
F : J ⥤ CompHaus
FF : J ⥤ TopCat := F ⋙ compHausToTop
⊢ CompactSpace ↑(TopCat.limitCone FF).pt

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsOpen[instTopologicalSpace] (some '' s)ᶜ ↔ IsClosed[inst✝] s ∧ IsCompact s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : ProperSMul G X
⊢ T2Space (Quotient (MulAction.orbitRel G X))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsClosed[instTopologicalSpace] (some '' s) ↔ IsClosed[inst✝] s ∧ IsCompact s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
h_proper : ProperSMul G X
inst✝ : T1Space G
f : X → G × X := fun x ↦ (1, x)
⊢ range f = {1} ×ˢ univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ IsClosed[instTopologicalSpace] {∞}

Tactic:
by

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

Goal state before:
pp: X : Type w
Y : Type x
inst✝ : TopologicalSpace X
t : TopologicalSpace Y
f : X → Y
h : ∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous[_, t] (f ∘ ⇑g)
⊢ Continuous[compactlyGenerated X, t] f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
x : X
h : (𝓝[≠] x).NeBot
⊢ (𝓝[≠] ↑x).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
h_proper : ProperSMul G X
inst✝ : T1Space G
f : X → G × X := fun x ↦ (1, x)
⊢ IsProperMap f

Tactic:
by

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

Goal state before:
pp: X : Type w
t : TopologicalSpace X
inst✝ : UCompactlyGeneratedSpace X
⊢ t = compactlyGenerated X

Tactic:
by

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

Goal state before:
pp: X✝ : Type w
Y : Type x
X : Type v
t : TopologicalSpace X
inst✝ : DiscreteTopology X
⊢ t ≤ compactlyGenerated X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: J : Type v
inst✝ : SmallCategory J
F : J ⥤ CompHaus
FF : J ⥤ TopCat := F ⋙ compHausToTop
⊢ ∀ ⦃X Y : J⦄ (f : X ⟶ Y),
    ((Functor.const J).obj
              { toTop := (TopCat.limitCone FF).pt, is_compact := ⋯, is_hausdorff := ⋯, prop := trivial }).map
          f ≫
        InducedCategory.homMk ((TopCat.limitCone FF).π.app Y) =
      InducedCategory.homMk ((TopCat.limitCone FF).π.app X) ≫ F.map f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
h_proper : ProperSMul G X
inst✝ : T1Space G
f : X → G × X := fun x ↦ (1, x)
proper_f : IsProperMap f
g : G × X → X × X := fun gx ↦ (gx.1 • gx.2, gx.2)
proper_g : IsProperMap g
⊢ g ∘ f = Function.diag

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
h_proper : ProperSMul G X
inst✝ : T1Space G
f : X → G × X := fun x ↦ (1, x)
proper_f : IsProperMap f
g : G × X → X × X := fun gx ↦ (gx.1 • gx.2, gx.2)
proper_g : IsProperMap g
this : g ∘ f = Function.diag
⊢ range (g ∘ f) = diagonal X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: J : Type v
inst✝ : SmallCategory J
F : J ⥤ CompHaus
FF : J ⥤ TopCat := F ⋙ compHausToTop
S : Limits.Cone F
m : S.pt ⟶ (limitCone F).pt
hm : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
j : J
⊢ m.hom ≫ (TopCat.limitCone FF).π.app j = (compHausToTop.mapCone S).π.app j

Tactic:
by

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

Goal state before:
pp: X : Type w
t : TopologicalSpace X
h :
  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),
    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous[t, _] f
this : Continuous[t, compactlyGenerated X] id
⊢ t ≤ compactlyGenerated X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ 𝓝[≠] ∞ = map some (coclosedCompact X)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
h_proper : ProperSMul G X
inst✝ : T1Space G
⊢ T2Space X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
inst✝ : NoncompactSpace X
⊢ (𝓝[≠] ∞).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁷ : Group G
inst✝⁶ : MulAction G X
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace X
H : Type u_3
inst✝³ : Group H
inst✝² : MulAction H X
inst✝¹ : TopologicalSpace H
inst✝ : ProperSMul G X
f : H →* G
f_clemb : IsClosedEmbedding ⇑f
f_compat : ∀ (h : H) (x : X), f h • x = h • x
h : IsProperMap (Prod.map ⇑f fun x ↦ x)
⊢ (fun hx ↦ (hx.1 • hx.2, hx.2)) = fun hx ↦ (f hx.1 • hx.2, hx.2)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: X Y : CompHaus
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X ↦ ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
D : Set ↑Y.toTop := {y}
hD : IsClosed D
⊢ Disjoint C D

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ 𝓝 ∞ = map some (coclosedCompact X) ⊔ pure ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ Tendsto some (coclosedCompact X) (𝓝 ∞)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ (𝓝 ∞).HasBasis (fun s ↦ IsClosed[inst✝] s ∧ IsCompact s) fun s ↦ some '' sᶜ ∪ {∞}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁷ : Group G
inst✝⁶ : MulAction G X
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace X
H : Type u_3
inst✝³ : Group H
inst✝² : MulAction H X
inst✝¹ : TopologicalSpace H
inst✝ : ProperSMul G X
f : H →* G
f_clemb : IsClosedEmbedding ⇑f
f_compat : ∀ (h : H) (x : X), f h • x = h • x
⊢ IsProperMap fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ comap some (𝓝 ∞) = coclosedCompact X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : IsTopologicalGroup G
x✝ : G × G
⊢ (fun gh ↦ (gh.1 * gh.2⁻¹, gh.2)) ((fun gh ↦ (gh.1 * gh.2, gh.2)) x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
f : Filter (OnePoint X)
⊢ f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed[inst✝] s → IsCompact s → some '' sᶜ ∪ {∞} ∈ f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : IsTopologicalGroup G
x✝ : G × G
⊢ (fun gh ↦ (gh.1 * gh.2, gh.2)) ((fun gh ↦ (gh.1 * gh.2⁻¹, gh.2)) x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
f : Ultrafilter (OnePoint X)
⊢ ↑f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed[inst✝] s → IsCompact s → some '' s ∉ f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
α : Type u_3
f : OnePoint X → α
l : Filter α
⊢ Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : IsTopologicalGroup G
⊢ IsProperMap fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
α : Type u_3
f : OnePoint X → α
l : Filter α
⊢ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l ↔
    ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t, IsClosed[inst✝] t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : IsTopologicalGroup G
x✝ : Gᵐᵒᵖ × G
⊢ (fun gh ↦ (op (gh.2⁻¹ * gh.1), gh.2)) ((fun gh ↦ (gh.2 * unop gh.1, gh.2)) x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : IsTopologicalGroup G
x✝ : G × G
⊢ (fun gh ↦ (gh.2 * unop gh.1, gh.2)) ((fun gh ↦ (op (gh.2⁻¹ * gh.1), gh.2)) x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
⊢ Tendsto (f ∘ some) (coclosedCompact X) (𝓝 (f ∞)) ↔
    ∀ s ∈ 𝓝 (f ∞), ∃ t, IsClosed[inst✝¹] t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: X Y : CompHaus
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X ↦ ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
D : Set ↑Y.toTop := {y}
hD : IsClosed D
hCD : Disjoint C D
φ : C((fun X ↦ ↑X.toTop) Y, ℝ)
hφ0 : Set.EqOn (⇑φ) 0 C
hφ1 : Set.EqOn (⇑φ) 1 D
hφ01 : ∀ (x : (fun X ↦ ↑X.toTop) Y), φ x ∈ Set.Icc 0 1
this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1))
this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1))
Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1))
g : Y ⟶ Z := ofHom (fun x ↦ True) { toFun := fun y' ↦ { down := ⟨φ y', ⋯⟩ }, continuous_toFun := ⋯ }
h : Y ⟶ Z := ofHom (fun x ↦ True) { toFun := fun x ↦ { down := ⟨0, ⋯⟩ }, continuous_toFun := ⋯ }
⊢ h = g

Tactic:
by

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

Goal state before:
pp: X : Type w
t : TopologicalSpace X
h :
  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),
    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous[t, _] f
⊢ t ≤ compactlyGenerated X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
x : X
⊢ ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
⊢ Continuous[instTopologicalSpace, inst✝] f ↔
    Tendsto (fun x ↦ f ↑x) (coclosedCompact X) (𝓝 (f ∞)) ∧ Continuous[inst✝¹, inst✝] fun x ↦ f ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : ProperSMul G X
t : Set X
Φ : G × X → X × X := fun gx ↦ (gx.1 • gx.2, gx.2)
Φ_proper : IsProperMap Φ
⊢ univ ×ˢ t = Φ ⁻¹' snd ⁻¹' t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y✝ : Type u_2
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
Y : Type u_3
inst✝ : TopologicalSpace Y
f : C(X, Y)
y : Y
h : Tendsto (⇑f) (coclosedCompact X) (𝓝 y)
⊢ Continuous[instTopologicalSpace, inst✝] fun x ↦ x.elim y ⇑f

Tactic:
by

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

Goal state before:
pp: X : Type w
Y : Type x
tX : TopologicalSpace X
tY : TopologicalSpace Y
inst✝ : UCompactlyGeneratedSpace X
f : X → Y
h : ∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous[_, tY] (f ∘ ⇑g)
⊢ Continuous[tX, tY] f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Y : Type u_3
inst✝¹ : TopologicalSpace Y
inst✝ : DiscreteTopology X
f : OnePoint X → Y
⊢ Continuous[instTopologicalSpace, inst✝¹] f ↔ Tendsto (fun x ↦ f ↑x) cofinite (𝓝 (f ∞))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : ProperSMul G X
t : Set X
⊢ IsProperMap fun gx ↦ (gx.1 • ↑gx.2, gx.2)

Tactic:
by

Goal state after: