mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inl
X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
f : X → Y
s : Set X
inj : InjOn f s
sc : IsCompact s
fc : ∀ x ∈ s, ContinuousAt f x
loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u
x✝¹ : X × X
x : X
x✝ : (x, x) ∈ s ×ˢ s
hx : (x, x).1 ∈ s
hy : (x, x).2 ∈ s
⊢ ∀ᶠ (y : X × X) in 𝓝 (x, x), f y.1 = f y.2 → y.1 = y.2

pp: case inr
X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
f : X → Y
s : Set X
inj : InjOn f s
sc : IsCompact s
fc : ∀ x ∈ s, ContinuousAt f x
loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u
x✝¹ : X × X
x y : X
x✝ : (x, y) ∈ s ×ˢ s
hx : (x, y).1 ∈ s
hy : (x, y).2 ∈ s
hne : x ≠ y
⊢ ∀ᶠ (y : X × X) in 𝓝 (x, y), f y.1 = f y.2 → y.1 = y.2

Tactic:
·

Goal state after:
case inl
X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
f : X → Y
s : Set X
inj : InjOn f s
sc : IsCompact s
fc : ∀ x ∈ s, ContinuousAt f x
loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u
x✝¹ : X × X
x : X
x✝ : (x, x) ∈ s ×ˢ s
hx : (x, x).1 ∈ s
hy : (x, x).2 ∈ s
⊢ ∀ᶠ (y : X × X) in 𝓝 (x, x), f y.1 = f y.2 → y.1 = y.2
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case pos
X : Type u_1
inst✝ : TopologicalSpace X
s U : Set X
hs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ s))
hU : IsClosed[inst✝] U
x : X
hx : x ∈ (U \ s)ᶜ
h₁x : x ∈ U
a : X
ha : a ∈ {x}ᶜ → a ∉ U \ s
h₂a : a = x
⊢ a ∈ (U \ s)ᶜ

pp: case neg
X : Type u_1
inst✝ : TopologicalSpace X
s U : Set X
hs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ s))
hU : IsClosed[inst✝] U
x : X
hx : x ∈ (U \ s)ᶜ
h₁x : x ∈ U
a : X
ha : a ∈ {x}ᶜ → a ∉ U \ s
h₂a : ¬a = x
⊢ a ∈ (U \ s)ᶜ

Tactic:
·

Goal state after:
case pos
X : Type u_1
inst✝ : TopologicalSpace X
s U : Set X
hs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ s))
hU : IsClosed[inst✝] U
x : X
hx : x ∈ (U \ s)ᶜ
h₁x : x ∈ U
a : X
ha : a ∈ {x}ᶜ → a ∉ U \ s
h₂a : a = x
⊢ a ∈ (U \ s)ᶜ
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inr
X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
f : X → Y
s : Set X
inj : InjOn f s
sc : IsCompact s
fc : ∀ x ∈ s, ContinuousAt f x
loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u
x✝¹ : X × X
x y : X
x✝ : (x, y) ∈ s ×ˢ s
hx : (x, y).1 ∈ s
hy : (x, y).2 ∈ s
hne : x ≠ y
⊢ ∀ᶠ (y : X × X) in 𝓝 (x, y), f y.1 = f y.2 → y.1 = y.2

Tactic:
·

Goal state after:
case inr
X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
f : X → Y
s : Set X
inj : InjOn f s
sc : IsCompact s
fc : ∀ x ∈ s, ContinuousAt f x
loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u
x✝¹ : X × X
x y : X
x✝ : (x, y) ∈ s ×ˢ s
hx : (x, y).1 ∈ s
hy : (x, y).2 ∈ s
hne : x ≠ y
⊢ ∀ᶠ (y : X × X) in 𝓝 (x, y), f y.1 = f y.2 → y.1 = y.2
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case neg
X : Type u_1
inst✝ : TopologicalSpace X
s U : Set X
hs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ s))
hU : IsClosed[inst✝] U
x : X
hx : x ∈ (U \ s)ᶜ
h₁x : x ∈ U
a : X
ha : a ∈ {x}ᶜ → a ∉ U \ s
h₂a : ¬a = x
⊢ a ∈ (U \ s)ᶜ

Tactic:
·

Goal state after:
case neg
X : Type u_1
inst✝ : TopologicalSpace X
s U : Set X
hs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ s))
hU : IsClosed[inst✝] U
x : X
hx : x ∈ (U \ s)ᶜ
h₁x : x ∈ U
a : X
ha : a ∈ {x}ᶜ → a ∉ U \ s
h₂a : ¬a = x
⊢ a ∈ (U \ s)ᶜ
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case neg
X : Type u_1
inst✝ : TopologicalSpace X
s U : Set X
hs : s ∈ codiscreteWithin U
hU : IsClosed[inst✝] U
x : X
hx : x ∈ (U \ s)ᶜ
h₁x : x ∉ U
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ (U \ s)ᶜ

Tactic:
·

Goal state after:
case neg
X : Type u_1
inst✝ : TopologicalSpace X
s U : Set X
hs : s ∈ codiscreteWithin U
hU : IsClosed[inst✝] U
x : X
hx : x ∈ (U \ s)ᶜ
h₁x : x ∉ U
⊢ ∀ᶠ (y : X) in 𝓝 x, y ∈ (U \ s)ᶜ
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inl.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x y : X
h : Sum.inl x ≠ Sum.inl y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inl x ∈ u ∧ Sum.inl y ∈ v ∧ Disjoint u v

pp: case inl.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : X
y : Y
h : Sum.inl x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inl x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v

pp: case inr.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : Y
y : X
h : Sum.inr x ≠ Sum.inl y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inl y ∈ v ∧ Disjoint u v

pp: case inr.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x y : Y
h : Sum.inr x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case inl.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x y : X
h : Sum.inl x ≠ Sum.inl y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inl x ∈ u ∧ Sum.inl y ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Y → X
h : IsOpenMap f
s : Set X
hne : (f ⁻¹' s).Nonempty
hs : IsPreirreducible s
hs' : IsClosed[inst✝¹] s
⊢ (s ∩ range f).Nonempty

pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Y → X
h : IsOpenMap f
s : Set X
hne : (f ⁻¹' s).Nonempty
hs : IsPreirreducible s
hs' : IsClosed[inst✝¹] s
⊢ closure[inst✝¹] (s ∩ range f) ⊆ closure[inst✝¹] (f '' f ⁻¹' s)

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Y → X
h : IsOpenMap f
s : Set X
hne : (f ⁻¹' s).Nonempty
hs : IsPreirreducible s
hs' : IsClosed[inst✝¹] s
⊢ (s ∩ range f).Nonempty
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inl.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : X
y : Y
h : Sum.inl x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inl x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v

pp: case inr.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : Y
y : X
h : Sum.inr x ≠ Sum.inl y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inl y ∈ v ∧ Disjoint u v

pp: case inr.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x y : Y
h : Sum.inr x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case inl.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : X
y : Y
h : Sum.inl x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inl x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Y → X
h : IsOpenMap f
s : Set X
hne : (f ⁻¹' s).Nonempty
hs : IsPreirreducible s
hs' : IsClosed[inst✝¹] s
⊢ closure[inst✝¹] (s ∩ range f) ⊆ closure[inst✝¹] (f '' f ⁻¹' s)

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Y → X
h : IsOpenMap f
s : Set X
hne : (f ⁻¹' s).Nonempty
hs : IsPreirreducible s
hs' : IsClosed[inst✝¹] s
⊢ closure[inst✝¹] (s ∩ range f) ⊆ closure[inst✝¹] (f '' f ⁻¹' s)
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inr.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : Y
y : X
h : Sum.inr x ≠ Sum.inl y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inl y ∈ v ∧ Disjoint u v

pp: case inr.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x y : Y
h : Sum.inr x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case inr.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : Y
y : X
h : Sum.inr x ≠ Sum.inl y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inl y ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inr.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x y : Y
h : Sum.inr x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case inr.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x y : Y
h : Sum.inr x ≠ Sum.inr y
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inl
X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
ι : Type u_4
X : ι → Type u_3
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (a : ι), T2Space (X a)
i : ι
x y : X i
ne : ⟨i, x⟩ ≠ ⟨i, y⟩
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSigma] u ∧ IsOpen[instTopologicalSpaceSigma] v ∧ ⟨i, x⟩ ∈ u ∧ ⟨i, y⟩ ∈ v ∧ Disjoint u v

pp: case inr
X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
ι : Type u_4
X : ι → Type u_3
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (a : ι), T2Space (X a)
i : ι
x : X i
j : ι
y : X j
ne : ⟨i, x⟩ ≠ ⟨j, y⟩
h : i ≠ j
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSigma] u ∧ IsOpen[instTopologicalSpaceSigma] v ∧ ⟨i, x⟩ ∈ u ∧ ⟨j, y⟩ ∈ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case inl
X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
ι : Type u_4
X : ι → Type u_3
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (a : ι), T2Space (X a)
i : ι
x y : X i
ne : ⟨i, x⟩ ≠ ⟨i, y⟩
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSigma] u ∧ IsOpen[instTopologicalSpaceSigma] v ∧ ⟨i, x⟩ ∈ u ∧ ⟨i, y⟩ ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case inr
X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
ι : Type u_4
X : ι → Type u_3
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (a : ι), T2Space (X a)
i : ι
x : X i
j : ι
y : X j
ne : ⟨i, x⟩ ≠ ⟨j, y⟩
h : i ≠ j
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSigma] u ∧ IsOpen[instTopologicalSpaceSigma] v ∧ ⟨i, x⟩ ∈ u ∧ ⟨j, y⟩ ∈ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case inr
X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
ι : Type u_4
X : ι → Type u_3
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (a : ι), T2Space (X a)
i : ι
x : X i
j : ι
y : X j
ne : ⟨i, x⟩ ≠ ⟨j, y⟩
h : i ≠ j
⊢ ∃ u v,
    IsOpen[instTopologicalSpaceSigma] u ∧ IsOpen[instTopologicalSpaceSigma] v ∧ ⟨i, x⟩ ∈ u ∧ ⟨j, y⟩ ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case h.left
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
U s : Set X
hs : Finite ↑s
t : Set X
ht : IsOpen[inst✝¹] t
h₁ts : x ∈ t
h₂ts : t ∩ {x}ᶜ ⊆ U
⊢ IsOpen[inst✝¹] (t \ (s \ {x}))

pp: case h.right
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
U s : Set X
hs : Finite ↑s
t : Set X
ht : IsOpen[inst✝¹] t
h₁ts : x ∈ t
h₂ts : t ∩ {x}ᶜ ⊆ U
⊢ x ∈ t \ (s \ {x}) ∧ t \ (s \ {x}) ∩ {x}ᶜ ⊆ U \ s

Tactic:
·

Goal state after:
case h.left
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
U s : Set X
hs : Finite ↑s
t : Set X
ht : IsOpen[inst✝¹] t
h₁ts : x ∈ t
h₂ts : t ∩ {x}ᶜ ⊆ U
⊢ IsOpen[inst✝¹] (t \ (s \ {x}))
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case h.right
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
U s : Set X
hs : Finite ↑s
t : Set X
ht : IsOpen[inst✝¹] t
h₁ts : x ∈ t
h₂ts : t ∩ {x}ᶜ ⊆ U
⊢ x ∈ t \ (s \ {x}) ∧ t \ (s \ {x}) ∩ {x}ᶜ ⊆ U \ s

Tactic:
·

Goal state after:
case h.right
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
U s : Set X
hs : Finite ↑s
t : Set X
ht : IsOpen[inst✝¹] t
h₁ts : x ∈ t
h₂ts : t ∩ {x}ᶜ ⊆ U
⊢ x ∈ t \ (s \ {x}) ∧ t \ (s \ {x}) ∩ {x}ᶜ ⊆ U \ s
Module: Mathlib.Order.SuccPred.Relation

Goal state before:
pp: case refine_1
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
hnm : n ≤ m
⊢ (∀ i ∈ Ico n n, r i (succ i)) → ReflTransGen r n n

pp: case refine_2
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
hnm : n ≤ m
⊢ ∀ (n_1 : α),
    n ≤ n_1 →
      ((∀ i ∈ Ico n n_1, r i (succ i)) → ReflTransGen r n n_1) →
        (∀ i ∈ Ico n (succ n_1), r i (succ i)) → ReflTransGen r n (succ n_1)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
hnm : n ≤ m
⊢ (∀ i ∈ Ico n n, r i (succ i)) → ReflTransGen r n n
Module: Mathlib.Order.SuccPred.Relation

Goal state before:
pp: case refine_2
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
hnm : n ≤ m
⊢ ∀ (n_1 : α),
    n ≤ n_1 →
      ((∀ i ∈ Ico n n_1, r i (succ i)) → ReflTransGen r n n_1) →
        (∀ i ∈ Ico n (succ n_1), r i (succ i)) → ReflTransGen r n (succ n_1)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
hnm : n ≤ m
⊢ ∀ (n_1 : α),
    n ≤ n_1 →
      ((∀ i ∈ Ico n n_1, r i (succ i)) → ReflTransGen r n n_1) →
        (∀ i ∈ Ico n (succ n_1), r i (succ i)) → ReflTransGen r n (succ n_1)
Module: Mathlib.Order.SuccPred.Relation

Goal state before:
pp: case refine_2.inl
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m✝ : α
hnm✝ : n ≤ m✝
m : α
hnm : n ≤ m
ih : (∀ i ∈ Ico n m, r i (succ i)) → ReflTransGen r n m
h : ∀ i ∈ Ico n (succ m), r i (succ i)
this : ReflTransGen r n m
hm : m = succ m
⊢ ReflTransGen r n (succ m)

pp: case refine_2.inr
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m✝ : α
hnm✝ : n ≤ m✝
m : α
hnm : n ≤ m
ih : (∀ i ∈ Ico n m, r i (succ i)) → ReflTransGen r n m
h : ∀ i ∈ Ico n (succ m), r i (succ i)
this : ReflTransGen r n m
hm : m < succ m
⊢ ReflTransGen r n (succ m)

Tactic:
·

Goal state after:
case refine_2.inl
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m✝ : α
hnm✝ : n ≤ m✝
m : α
hnm : n ≤ m
ih : (∀ i ∈ Ico n m, r i (succ i)) → ReflTransGen r n m
h : ∀ i ∈ Ico n (succ m), r i (succ i)
this : ReflTransGen r n m
hm : m = succ m
⊢ ReflTransGen r n (succ m)
Module: Mathlib.Order.SuccPred.Relation

Goal state before:
pp: case inl
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
r : α → α → Prop
m : α
inst✝ : Std.Refl r
h1 : ∀ i ∈ Ico m m, r i (succ i)
h2 : ∀ i ∈ Ico m m, r (succ i) i
⊢ TransGen r m m

pp: case inr
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
r : α → α → Prop
n m : α
inst✝ : Std.Refl r
h1 : ∀ i ∈ Ico n m, r i (succ i)
h2 : ∀ i ∈ Ico m n, r (succ i) i
hmn : m ≠ n
⊢ TransGen r n m

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
r : α → α → Prop
m : α
inst✝ : Std.Refl r
h1 : ∀ i ∈ Ico m m, r i (succ i)
h2 : ∀ i ∈ Ico m m, r (succ i) i
⊢ TransGen r m m
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case mp
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
⊢ (∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅) → ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ (U \ s)).Finite

pp: case mpr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
⊢ (∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ (U \ s)).Finite) → ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅

Tactic:
·

Goal state after:
case mp
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
⊢ (∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅) → ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ (U \ s)).Finite
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed[inst✝] {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed[inst✝] {x}) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
tfae_1_to_4 : T1Space X → Continuous[inst✝, _] ⇑CofiniteTopology.of
tfae_4_to_2 : Continuous[inst✝, _] ⇑CofiniteTopology.of → ∀ (x : X), IsClosed[inst✝] {x}
tfae_2_iff_10 : (∀ (x : X), IsClosed[inst✝] {x}) ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y
tfae_10_iff_11 : (∀ ⦃x y : X⦄, x ⤳ y → x = y) ↔ T0Space X ∧ R0Space X
⊢ [T1Space X, ∀ (x : X), IsClosed[inst✝] {x}, ∀ (x : X), IsOpen[inst✝] {x}ᶜ, Continuous[inst✝, _] ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y, T0Space X ∧ R0Space X].TFAE

Tactic:
tfae_finish

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case pos
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
h : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅
z : X
h₁z : z ∈ U
t : Set X
h₁t : t ∈ 𝓝[≠] z
h₂t : t ∩ (U \ s) = ∅
hz : z ∈ U \ s
⊢ (insert z t ∩ (U \ s)).Finite

pp: case neg
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
h : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅
z : X
h₁z : z ∈ U
t : Set X
h₁t : t ∈ 𝓝[≠] z
h₂t : t ∩ (U \ s) = ∅
hz : z ∉ U \ s
⊢ (insert z t ∩ (U \ s)).Finite

Tactic:
·

Goal state after:
case pos
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
h : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅
z : X
h₁z : z ∈ U
t : Set X
h₁t : t ∈ 𝓝[≠] z
h₂t : t ∩ (U \ s) = ∅
hz : z ∈ U \ s
⊢ (insert z t ∩ (U \ s)).Finite
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
H : ∀ (b : X), b ≠ x → b ∈ s → ContinuousWithinAt (update f x y) s b
z : X
hz : z ∈ s \ {x}
⊢ ContinuousWithinAt f (s \ {x}) z

pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
H : ContinuousOn f (s \ {x})
z : X
hzx : z ≠ x
hzs : z ∈ s
⊢ ContinuousWithinAt (update f x y) s z

pp: case refine_3
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
x✝ : x ∈ s
⊢ ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y)

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
H : ∀ (b : X), b ≠ x → b ∈ s → ContinuousWithinAt (update f x y) s b
z : X
hz : z ∈ s \ {x}
⊢ ContinuousWithinAt f (s \ {x}) z
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
H : ContinuousOn f (s \ {x})
z : X
hzx : z ≠ x
hzs : z ∈ s
⊢ ContinuousWithinAt (update f x y) s z

pp: case refine_3
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
x✝ : x ∈ s
⊢ ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y)

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
H : ContinuousOn f (s \ {x})
z : X
hzx : z ≠ x
hzs : z ∈ s
⊢ ContinuousWithinAt (update f x y) s z
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case refine_3
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
x✝ : x ∈ s
⊢ ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y)

Tactic:
·

Goal state after:
case refine_3
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
x✝ : x ∈ s
⊢ ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y)
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
hs : ∅.Subsingleton
⊢ IsClosed[inst✝¹] ∅

pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
hs : {x}.Subsingleton
⊢ IsClosed[inst✝¹] {x}

Tactic:
·

Goal state after:
case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
hs : ∅.Subsingleton
⊢ IsClosed[inst✝¹] ∅
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
hs : {x}.Subsingleton
⊢ IsClosed[inst✝¹] {x}

Tactic:
·

Goal state after:
case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
hs : {x}.Subsingleton
⊢ IsClosed[inst✝¹] {x}
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case mp
ι : Type u_4
X : ι → Type u_5
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), T2Space (X i)
s : Set ((i : ι) → X i)
H : IsCompact s
⊢ IsClosed[topologicalSpace] s ∧ ∀ (i : ι), IsCompact (eval i '' s)

pp: case mpr
ι : Type u_4
X : ι → Type u_5
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), T2Space (X i)
s : Set ((i : ι) → X i)
H : IsClosed[topologicalSpace] s ∧ ∀ (i : ι), IsCompact (eval i '' s)
⊢ IsCompact s

Tactic:
·

Goal state after:
case mp
ι : Type u_4
X : ι → Type u_5
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), T2Space (X i)
s : Set ((i : ι) → X i)
H : IsCompact s
⊢ IsClosed[topologicalSpace] s ∧ ∀ (i : ι), IsCompact (eval i '' s)
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case neg
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
h : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅
z : X
h₁z : z ∈ U
t : Set X
h₁t : t ∈ 𝓝[≠] z
h₂t : t ∩ (U \ s) = ∅
hz : z ∉ U \ s
⊢ (insert z t ∩ (U \ s)).Finite

Tactic:
·

Goal state after:
case neg
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
h : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅
z : X
h₁z : z ∈ U
t : Set X
h₁t : t ∈ 𝓝[≠] z
h₂t : t ∩ (U \ s) = ∅
hz : z ∉ U \ s
⊢ (insert z t ∩ (U \ s)).Finite