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

Goal state before:
pp: case intro.refine_2
ι : Type u_5
X : ι → Type u_9
g : (a : ι) → Set (Set (X a))
inst✝ : Finite ι
hg : ∀ (a : ι), ⋃₀ g a = univ
val✝ : Fintype ι
t : (a : ι) → Set (X a)
i : Finset ι
ht : ∀ a ∈ i, t a ∈ g a
this : TopologicalSpace ((i : ι) → X i) := generateFrom {t | ∃ s, (∀ (a : ι), s a ∈ g a) ∧ t = univ.pi s}
f : (i : ι) → X i
hf : f ∈ (↑i).pi t
c : (a : ι) → Set (X a)
hcg : ∀ (a : ι), c a ∈ g a
hfc : ∀ (a : ι), f a ∈ c a
⊢ ∃ t_1 ⊆ (↑i).pi t, IsOpen[generateFrom {t | ∃ s, (∀ (a : ι), s a ∈ g a) ∧ t = univ.pi s}] t_1 ∧ f ∈ t_1

Tactic:
refine ⟨pi i t ∩ pi ((↑i)ᶜ : Set ι) c, inter_subset_left, ?_, ⟨hf, fun a _ => hfc a⟩⟩

Goal state after:
case intro.refine_2
ι : Type u_5
X : ι → Type u_9
g : (a : ι) → Set (Set (X a))
inst✝ : Finite ι
hg : ∀ (a : ι), ⋃₀ g a = univ
val✝ : Fintype ι
t : (a : ι) → Set (X a)
i : Finset ι
ht : ∀ a ∈ i, t a ∈ g a
this : TopologicalSpace ((i : ι) → X i) := generateFrom {t | ∃ s, (∀ (a : ι), s a ∈ g a) ∧ t = univ.pi s}
f : (i : ι) → X i
hf : f ∈ (↑i).pi t
c : (a : ι) → Set (X a)
hcg : ∀ (a : ι), c a ∈ g a
hfc : ∀ (a : ι), f a ∈ c a
⊢ IsOpen[generateFrom {t | ∃ s, (∀ (a : ι), s a ∈ g a) ∧ t = univ.pi s}] ((↑i).pi t ∩ (↑i)ᶜ.pi c)
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X✝ : Type u
Y✝ : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(ULift.{v, u} X, ULift.{u, v} Y)
⊢ Continuous[inst✝¹, inst✝] (ULift.down ∘ ⇑f ∘ ULift.up)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X✝ : Type u
Y✝ : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(X, Y)
⊢ Continuous[ULift.topologicalSpace, ULift.topologicalSpace] (ULift.up ∘ ⇑f ∘ ULift.down)

Tactic:
by fun_prop

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✝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)
⊢ SigmaCompactSpace ((i : ι) → X i)

Tactic:
by
  refine ⟨⟨fun n => Set.pi univ fun i => compactCovering (X i) n,
    fun n => isCompact_univ_pi fun i => isCompact_compactCovering (X i) _, ?_⟩⟩
  rw [iUnion_univ_pi_of_monotone]
  · simp only [iUnion_compactCovering, pi_univ]
  · exact fun i => compactCovering_subset (X i)

Goal state after:
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
⊢ ⋃ n, (fun n ↦ e ⁻¹' compactCovering X n) n = univ

Tactic:
rw [← preimage_iUnion, iUnion_compactCovering, preimage_univ]

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
f : Filter X
y : Y
hf : Disjoint f (cocompact X)
⊢ f ×ˢ 𝓝 y ≤ 𝓝ˢ (univ ×ˢ {y})

Tactic:
simpa using prod_nhdsSet_le_of_disjoint_cocompact isCompact_singleton hf

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s)

Tactic:
← comap_mk_nhdsSet_image,

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ map mk (comap mk (𝓝ˢ (mk '' s))) = 𝓝ˢ (mk '' s)
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
⊢ IsDiscrete S ↔ ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s

Tactic:
rw [isDiscrete_iff_forall_subset_exists_isOpen]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
⊢ (∀ t ⊆ S, ∃ u, IsOpen[inst✝] u ∧ u ∩ S = t) ↔ ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s
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

Tactic:
rcases loc x hx with ⟨u, hu, hf⟩

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
u : Set X
hu : u ∈ 𝓝 x
hf : InjOn f u
⊢ ∀ᶠ (y : X × X) in 𝓝 (x, x), f y.1 = f y.2 → y.1 = y.2
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: 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
⊢ ∀ᶠ (y : X × X) in 𝓝 (x, y), f y.1 = f y.2 → y.1 = y.2

Tactic:
by
    rcases eq_or_ne x y with rfl | hne
    · rcases loc x hx with ⟨u, hu, hf⟩
      exact Filter.mem_of_superset (prod_mem_nhds hu hu) <| forall_prod_set.2 hf
    · suffices ∀ᶠ z in 𝓝 (x, y), f z.1 ≠ f z.2 from this.mono fun _ hne h ↦ absurd h hne
      refine (fc x hx).prodMap' (fc y hy) <| isClosed_diagonal.isOpen_compl.mem_nhds ?_
      exact inj.ne hx hy hne

Goal state after:
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
s : Set X
hxy : x ≠ y
t : Set X
ht : t ∈ 𝓝[s] x
o : Set X
ho : IsOpen[inst✝¹] o
hxo : x ∈ o
host : o ∩ s ⊆ t
⊢ o \ {y} ∩ insert y s ⊆ t

Tactic:
inter_insert_of_notMem <| notMem_sdiff_of_mem (mem_singleton y)

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
s : Set X
hxy : x ≠ y
t : Set X
ht : t ∈ 𝓝[s] x
o : Set X
ho : IsOpen[inst✝¹] o
hxo : x ∈ o
host : o ∩ s ⊆ t
⊢ o \ {y} ∩ s ⊆ t
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
x y : X
h : ∀ᶠ (x : X) in 𝓝[{y}ᶜ] x, x ∈ s ↔ x ∈ t
⊢ (∀ᶠ (x_1 : X) in 𝓝[≠] x, x_1 ∈ s ∪ {x} ↔ x_1 ∈ t ∪ {x}) ∧ ∀ᶠ (x_1 : X) in 𝓝[{x}] x, x_1 ∈ s ∪ {x} ↔ x_1 ∈ t ∪ {x}

Tactic:
nhdsWithin_singleton,

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
x y : X
h : ∀ᶠ (x : X) in 𝓝[{y}ᶜ] x, x ∈ s ↔ x ∈ t
⊢ (∀ᶠ (x_1 : X) in 𝓝[≠] x, x_1 ∈ s ∪ {x} ↔ x_1 ∈ t ∪ {x}) ∧ ∀ᶠ (x_1 : X) in pure x, x_1 ∈ s ∪ {x} ↔ x_1 ∈ t ∪ {x}
Module: Mathlib.Topology.Separation.Hausdorff

Goal state before:
pp: case mp
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : X
f g : X → Y
hf : ContinuousAt f x
hg : ContinuousAt g x
hfg : f x ≠ g x
⊢ ∀ᶠ (x : X) in 𝓝 x, f x ≠ g x

Tactic:
obtain ⟨Uf, Ug, h₁U, h₂U, h₃U, h₄U, h₅U⟩ := t2_separation hfg

Goal state after:
case mp
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : X
f g : X → Y
hf : ContinuousAt f x
hg : ContinuousAt g x
hfg : f x ≠ g x
Uf Ug : Set Y
h₁U : IsOpen[inst✝¹] Uf
h₂U : IsOpen[inst✝¹] Ug
h₃U : f x ∈ Uf
h₄U : g x ∈ Ug
h₅U : Disjoint Uf Ug
⊢ ∀ᶠ (x : X) in 𝓝 x, f x ≠ g x
Module: Mathlib.Topology.DiscreteSubset

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

Tactic:
use t \ (t ∩ (U \ s)), nhdsNE_of_nhdsNE_sdiff_finite (mem_nhdsWithin_of_mem_nhds h₁t) h₂t

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

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
u v : Set α
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hs : s ⊆ u ∪ v
z : α
zs : z ∈ s
zu : z ∈ u
y : α
ys : y ∈ s
yv : y ∈ v
⊢ (s ∩ (u ∩ v)).Nonempty

Tactic:
have xs : x ∈ s := by
    rcases H y ys with ⟨t, ts, xt, -, -⟩
    exact ts xt

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
u v : Set α
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hs : s ⊆ u ∪ v
z : α
zs : z ∈ s
zu : z ∈ u
y : α
ys : y ∈ s
yv : y ∈ v
xs : x ∈ s
⊢ (s ∩ (u ∩ v)).Nonempty
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
x : α
c : Set (Set α)
H1 : ∀ s ∈ c, x ∈ s
H2 : ∀ s ∈ c, IsPreconnected s
⊢ IsPreconnected (⋃₀ c)

Tactic:
apply isPreconnected_of_forall x

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
x : α
c : Set (Set α)
H1 : ∀ s ∈ c, x ∈ s
H2 : ∀ s ∈ c, IsPreconnected s
⊢ ∀ y ∈ ⋃₀ c, ∃ t ⊆ ⋃₀ c, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
Module: Mathlib.Order.Filter.CountableInter

Goal state before:
pp: ι : Sort u_1
α✝ : Type u_2
β✝ : Type u_3
l✝ : Filter α✝
inst✝² : CountableInterFilter l✝
α : Type u_4
β : Type u_5
l : Filter α
m : Filter β
inst✝¹ : CountableInterFilter l
inst✝ : CountableInterFilter m
S : Set (Set (α × β))
Sct : S.Countable
hS : ∀ s ∈ S, s ∈ l.curry m
⊢ ∀ᶠ (x : α) in l, ∀ i ∈ S, ∀ᶠ (x_1 : β) in m, (x, x_1) ∈ i

Tactic:
eventually_countable_ball (p := fun _ _ _ => ∀ᶠ (_ : β) in m, _)  Sct,

Goal state after:
ι : Sort u_1
α✝ : Type u_2
β✝ : Type u_3
l✝ : Filter α✝
inst✝² : CountableInterFilter l✝
α : Type u_4
β : Type u_5
l : Filter α
m : Filter β
inst✝¹ : CountableInterFilter l
inst✝ : CountableInterFilter m
S : Set (Set (α × β))
Sct : S.Countable
hS : ∀ s ∈ S, s ∈ l.curry m
⊢ ∀ i ∈ S, ∀ᶠ (x : α) in l, ∀ᶠ (x_1 : β) in m, (x, x_1) ∈ i
Module: Mathlib.Order.Filter.CountableInter

Goal state before:
pp: case mp.sInter
α : Type u_2
g : Set (Set α)
s : Set α
S : Set (Set α)
Sct : S.Countable
a✝ : ∀ s ∈ S, CountableGenerateSets g s
T : (s : Set α) → s ∈ S → Set (Set α)
Tg : ∀ (s : Set α) (a : s ∈ S), T s a ⊆ g
Tct : ∀ (s : Set α) (a : s ∈ S), (T s a).Countable
hT : ∀ (s : Set α) (a : s ∈ S), ⋂₀ T s a ⊆ s
⊢ ⋂₀ ⋃ s, ⋃ (H : s ∈ S), T s H ⊆ ⋂₀ S

Tactic:
apply subset_sInter

Goal state after:
case mp.sInter
α : Type u_2
g : Set (Set α)
s : Set α
S : Set (Set α)
Sct : S.Countable
a✝ : ∀ s ∈ S, CountableGenerateSets g s
T : (s : Set α) → s ∈ S → Set (Set α)
Tg : ∀ (s : Set α) (a : s ∈ S), T s a ⊆ g
Tct : ∀ (s : Set α) (a : s ∈ S), (T s a).Countable
hT : ∀ (s : Set α) (a : s ∈ S), ⋂₀ T s a ⊆ s
⊢ ∀ t' ∈ S, ⋂₀ ⋃ s, ⋃ (H : s ∈ S), T s H ⊆ t'
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsLindelof s
p : Set X → Prop
hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s
hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S)
hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t
f : Filter X := ofCountableUnion {t | p t} hcountable_union ⋯
⊢ p s

Tactic:
have : sᶜ ∈ f := hs.compl_mem_sets_of_nhdsWithin (by simpa [f] using! hnhds)

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsLindelof s
p : Set X → Prop
hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s
hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S)
hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t
f : Filter X := ofCountableUnion {t | p t} hcountable_union ⋯
this : sᶜ ∈ f
⊢ p s
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsLindelof s
U : X → Set X
hU : ∀ x ∈ s, U x ∈ 𝓝 x
⊢ ∃ t, t.Countable ∧ (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x

Tactic:
by
  let ⟨t, ⟨htc, htsub⟩⟩ := hs.elim_nhds_subcover' (fun x _ ↦ U x) hU
  refine ⟨↑t, Countable.image htc Subtype.val, ?_⟩
  constructor
  · intro _
    simp only [mem_image, Subtype.exists, exists_and_right, exists_eq_right, forall_exists_index]
    tauto
  · have : ⋃ x ∈ t, U ↑x = ⋃ x ∈ Subtype.val '' t, U x := biUnion_image.symm
    rwa [← this]

Goal state after:
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : Set α
t : Set β
hs : IsPreconnected s
ht : IsPreconnected t
a₁ : α
b₁ : β
ha₁ : (a₁, b₁).1 ∈ s
hb₁ : (a₁, b₁).2 ∈ t
a₂ : α
b₂ : β
ha₂ : (a₂, b₂).1 ∈ s
hb₂ : (a₂, b₂).2 ∈ t
⊢ ContinuousOn (Prod.mk a₁) t

Tactic:
by fun_prop

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

Goal state before:
pp: case mpr
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
U : Set X
⊢ (∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i) → IsLindelof U ∧ IsOpen[inst✝] U

Tactic:
rintro ⟨s, hs, rfl⟩
    constructor
    · exact hs.isLindelof_biUnion fun i _ => hb' i
    · exact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _)

Goal state after:
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ s = ⋂ n, f n) → IsGδ s

Tactic:
rintro ⟨f, hf, rfl⟩
    exact .iInter_of_isOpen hf

Goal state after:
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: α : Type u
β : Type v
ι : Type u_1
X✝ : ι → Type u_2
inst✝⁴ : TopologicalSpace α
s t u v : Set α
X : Type u_3
Y : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TotallyDisconnectedSpace Y
inst✝ : ConnectedSpace X
y : Y
⊢ Continuous[inst✝³, inst✝²] fun x ↦ y

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case h_fibers
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
x y : α
⊢ IsConnected (connectedComponent y)

Tactic:
exact isConnected_connectedComponent

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case ht'
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : β
⊢ IsConnected (connectedComponent t)

Tactic:
exact isConnected_connectedComponent

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case refine_1
α : Type u
β : Type v
ι✝ : Type u_1
X : ι✝ → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
ι : Type u_3
U : ι → Set α
hclopen : ∀ (i : ι), IsClopen (U i)
hdisj : Pairwise (Disjoint on U)
hunion : ⋃ i, U i = univ
hconn : ∀ (i : ι), IsPreconnected (U i)
heq : ∀ {x : α} {i : ι} (hx : x ∈ U i), Subtype.val '' connectedComponent ⟨x, hx⟩ = connectedComponent x
x✝¹ x✝ : (i : ι) × ConnectedComponents ↑(U i)
i j : ι
x : ↑(U i)
y : ↑(U j)
hxy :
  (fun x ↦
        match x with
        | ⟨i, x⟩ => Quotient.lift (mk ∘ Subtype.val) ⋯ x)
      ⟨i, mk x⟩ =
    (fun x ↦
        match x with
        | ⟨i, x⟩ => Quotient.lift (mk ∘ Subtype.val) ⋯ x)
      ⟨j, mk y⟩
⊢ ⟨i, mk x⟩ = ⟨j, mk y⟩

Tactic:
replace hxy : ConnectedComponents.mk x.val = ConnectedComponents.mk y.val := hxy

Goal state after:
case refine_1
α : Type u
β : Type v
ι✝ : Type u_1
X : ι✝ → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
ι : Type u_3
U : ι → Set α
hclopen : ∀ (i : ι), IsClopen (U i)
hdisj : Pairwise (Disjoint on U)
hunion : ⋃ i, U i = univ
hconn : ∀ (i : ι), IsPreconnected (U i)
heq : ∀ {x : α} {i : ι} (hx : x ∈ U i), Subtype.val '' connectedComponent ⟨x, hx⟩ = connectedComponent x
x✝¹ x✝ : (i : ι) × ConnectedComponents ↑(U i)
i j : ι
x : ↑(U i)
y : ↑(U j)
hxy : mk ↑x = mk ↑y
⊢ ⟨i, mk x⟩ = ⟨j, mk y⟩
Module: Mathlib.Data.Rel.Separated

Goal state before:
pp: X : Type u_1
R : SetRel X X
s : Set X
x : X
inst✝ : R.IsSymm
⊢ R.IsSeparated (insert x s) ↔ R.IsSeparated s ∧ ∀ (y : X), y ∈ s → (x, y) ∈ R → x = y

Tactic:
by
  have : Std.Symm fun x y ↦ ¬(x, y) ∈ R := { symm _ _ := mt R.symm }
  simpa [not_imp_not, IsSeparated] using pairwise_insert_of_symm (r := fun x y ↦ ¬(x, y) ∈ R)

Goal state after:
Module: Mathlib.Topology.Homeomorph.Lemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace X
inst✝ : T2Space Y
f : X ≃ Y
hf : Continuous[inst✝³, inst✝²] ⇑f
⊢ Continuous[inst✝², inst✝³] ⇑f.symm

Tactic:
by
  rw [continuous_iff_isClosed]
  intro C hC
  have hC' : IsClosed (f '' C) := (hC.isCompact.image hf).isClosed
  rwa [Equiv.image_eq_preimage_symm] at hC'

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ univ.Nonempty ∧ PreconnectedSpace α ↔ Nonempty α ∧ ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ

Tactic:
preconnectedSpace_iff_clopen,

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
⊢ (univ.Nonempty ∧ ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ) ↔
    Nonempty α ∧ ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ
Module: Mathlib.Topology.UniformSpace.Basic

Goal state before:
pp: α : Type ua
β : Type ub
γ : Type uc
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
s : SetRel α β
t : SetRel β γ
hs : IsOpen[instTopologicalSpaceProd] s
ht : IsOpen[instTopologicalSpaceProd] t
a : β
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun p ↦ (p.1, a)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.UniformSpace.Basic

Goal state before:
pp: α : Type ua
β : Type ub
γ : Type uc
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
s : SetRel α β
t : SetRel β γ
hs : IsOpen[instTopologicalSpaceProd] s
ht : IsOpen[instTopologicalSpaceProd] t
a : β
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun p ↦ (a, p.2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.UniformSpace.Defs

Goal state before:
pp: case mp
β : Type ub
V W M : SetRel β β
inst✝ : W.IsSymm
x y z : β
hzy : (z, y) ∈ W
w : β
hpw : (x, w) ∈ V
hwz : (w, z) ∈ M
⊢ (ball (x, y).1 V ×ˢ ball (x, y).2 W ∩ M).Nonempty

Tactic:
exact ⟨(w, z), ⟨hpw, by rwa [mem_ball_symmetry]⟩, hwz⟩

Goal state after:
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
ι : Sort u_1
s : ι → Set α
hs : ∀ (i : ι), IsComplete (s i)
U : SetRel α α
hU : U ∈ 𝓤 α
hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j
S : Set α := ⋃ i, s i
l : Filter α
hl : Cauchy l
⊢ l ≤ 𝓟 S → ∃ x ∈ S, l ≤ 𝓝 x

Tactic:
hls

Goal state after:
α : Type u
uniformSpace : UniformSpace α
ι : Sort u_1
s : ι → Set α
hs : ∀ (i : ι), IsComplete (s i)
U : SetRel α α
hU : U ∈ 𝓤 α
hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j
S : Set α := ⋃ i, s i
l : Filter α
hl : Cauchy l
hls : l ≤ 𝓟 S
⊢ ∃ x ∈ S, l ≤ 𝓝 x
Module: Mathlib.Topology.UniformSpace.Separation

Goal state before:
pp: α : Type u
inst✝¹ : UniformSpace α
ι : Type u_1
inst✝ : LinearOrder ι
B : ι → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
s t : Set α
hSt : Disjoint (closure[inst✝¹.toTopologicalSpace] s) t
hsT : Disjoint s (closure[inst✝¹.toTopologicalSpace] t)
S : Bool → Set α := fun b ↦ Bool.casesOn b s t
U : (b : Bool) → ↑(S b) → ι
hU : ∀ (b : Bool) (x : ↑(S b)), Disjoint (ball (↑x) ((B (U b x)).comp (B (U b x)).inv)) (S !b)
hUS : ∀ (b : Bool), ⋃ x, ball (↑x) (B (U b x)) ∈ 𝓝ˢ (S b)
b : Bool
x : ↑(S b)
y : ↑(S !b)
hxy : U b x ≤ U (!b) y
⊢ Disjoint (ball (↑x) (B (U b x))) (ball (↑y) (B (U (!b) y)))

Tactic:
rw [Set.disjoint_iff]

Goal state after:
α : Type u
inst✝¹ : UniformSpace α
ι : Type u_1
inst✝ : LinearOrder ι
B : ι → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
s t : Set α
hSt : Disjoint (closure[inst✝¹.toTopologicalSpace] s) t
hsT : Disjoint s (closure[inst✝¹.toTopologicalSpace] t)
S : Bool → Set α := fun b ↦ Bool.casesOn b s t
U : (b : Bool) → ↑(S b) → ι
hU : ∀ (b : Bool) (x : ↑(S b)), Disjoint (ball (↑x) ((B (U b x)).comp (B (U b x)).inv)) (S !b)
hUS : ∀ (b : Bool), ⋃ x, ball (↑x) (B (U b x)) ∈ 𝓝ˢ (S b)
b : Bool
x : ↑(S b)
y : ↑(S !b)
hxy : U b x ≤ U (!b) y
⊢ ball (↑x) (B (U b x)) ∩ ball (↑y) (B (U (!b) y)) ⊆ ∅
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
inst✝ : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right✝ : ∀ (n : ℕ), (B n).IsSymm
hs : ∀ (n : ℕ), ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x (B n)
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t

Tactic:
choose t tC ht using hs

Goal state after:
α : Type u
uniformSpace : UniformSpace α
inst✝ : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right✝ : ∀ (n : ℕ), (B n).IsSymm
t : ℕ → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
Module: Mathlib.Topology.UniformSpace.Pi

Goal state before:
pp: ι : Type u_4
X : Type u_5
u : ι → UniformSpace X
hu : ∀ (i : ι), CompleteSpace X
ht : ∃ t, T2Space X ∧ ∀ (i : ι), (u i).toTopologicalSpace ≤ t
⊢ CompleteSpace X

Tactic:
nontriviality X

Goal state after:
ι : Type u_4
X : Type u_5
u : ι → UniformSpace X
hu : ∀ (i : ι), CompleteSpace X
ht : ∃ t, T2Space X ∧ ∀ (i : ι), (u i).toTopologicalSpace ≤ t
a✝ : Nontrivial X
⊢ CompleteSpace X
Module: Mathlib.Topology.UniformSpace.LocallyUniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
s : Set α
inst✝ : LocallyCompactSpace α
G : ι → α → β
g : α → β
p : Filter ι
hs : IsOpen[inst✝²] s
⊢ [TendstoLocallyUniformlyOn G g p s, ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn G g p K,
      ∀ x ∈ s, ∃ v ∈ 𝓝[s] x, TendstoUniformlyOn G g p v].TFAE

Tactic:
tfae_have 1 → 2
  | h, K, hK1, hK2 =>
    (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK2).mp (h.mono hK1)

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
s : Set α
inst✝ : LocallyCompactSpace α
G : ι → α → β
g : α → β
p : Filter ι
hs : IsOpen[inst✝²] s
tfae_1_to_2 : TendstoLocallyUniformlyOn G g p s → ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn G g p K
⊢ [TendstoLocallyUniformlyOn G g p s, ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn G g p K,
      ∀ x ∈ s, ∃ v ∈ 𝓝[s] x, TendstoUniformlyOn G g p v].TFAE
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
inst✝ : Finite ι
F : ι → β → α
⊢ UniformEquicontinuous F ↔ ∀ (i : ι), UniformContinuous (F i)

Tactic:
simp only [UniformEquicontinuous, eventually_all, @forall_comm _ ι]

Goal state after:
ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
inst✝ : Finite ι
F : ι → β → α
⊢ (∀ (b : ι), ∀ a ∈ 𝓤 α, ∀ᶠ (x : β × β) in 𝓤 β, (F b x.1, F b x.2) ∈ a) ↔ ∀ (i : ι), UniformContinuous (F i)
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: case mpr
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
S : Set X
x₀ : X
hx₀ : x₀ ∈ S
H : ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∀ᶠ (x : X) in 𝓝[S] x₀, ∀ (i : ι), (F i x₀, F i x) ∈ U

Tactic:
rcases H U hU with ⟨V, hV, hVU⟩
    filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhdsWithin hx₀ hV) x hx i

Goal state after: