mathlib-initiative/mathlib-tactics huggingface dataset:
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 (U \ s ∩ t)).Finite

Tactic:
inter_comm,

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.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:
inter_comm,

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
⊢ (U \ s ∩ insert z t).Finite
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
⊢ (U \ s ∩ t).Finite

Tactic:
inter_comm,

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
⊢ (t ∩ (U \ s)).Finite
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
⊢ (𝓝 x).ker = {x}

Tactic:
by
  simp [ker_nhds_eq_specializes]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
s : Set X
hs : s ∈ 𝓝 x
hsf : s.Finite
A : {x} ⊆ s
B : IsClosed[inst✝¹] (s \ {x})
C : (s \ {x})ᶜ ∈ 𝓝 x
D : {x} ∈ 𝓝 x
⊢ IsOpen[inst✝¹] {x}

Tactic:
rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
p : X
s : Set X
hs : Disjoint (𝓝[s] p) cofinite
⊢ 𝓝[s] p = 𝓟 ({x | x ⤳ p} ∩ s)

Tactic:
by
  apply le_antisymm
  · simpa using ⟨hs.eventually_nhdsWithin_specializes, self_mem_nhdsWithin⟩
  · rw [← inf_principal, nhdsWithin]
    gcongr
    rw [Filter.principal_le_iff]
    exact fun s hs x hx ↦ mem_of_mem_nhds (hx hs)

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
y : X
K : Set X
hK : IsCompact K
hy : ∀ x ∈ K, ¬Inseparable x y
this : Disjoint (𝓝 y) (𝓝ˢ K)
⊢ y ∉ closure[inst✝¹] K

Tactic:
simpa only [disjoint_iff, notMem_closure_iff_nhdsWithin_eq_bot]
    using! this.mono_right principal_le_nhdsSet

Goal state after:
Module: Mathlib.Topology.Clopen

Goal state before:
pp: case refine_2
X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ Disjoint (s ∩ a) (s ∩ b)

Tactic:
rw [disjoint_iff_inter_eq_empty, inter_comm s b, ← inter_assoc, hab, empty_inter]

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

Goal state before:
pp: case refl
α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
t : Set ι
s : ι → Set α
H : ∀ i ∈ t, IsPreconnected (s i)
K : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty ∧ i ∈ t) i j
R : ι → ι → Prop := fun i j ↦ (s i ∩ s j).Nonempty ∧ i ∈ t
i : ι
hi : i ∈ t
j : ι
hj : i ∈ t
⊢ ∃ p ⊆ t, i ∈ p ∧ i ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)

Tactic:
refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩
      rw [biUnion_singleton]
      exact H i hi

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

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
s : Set α
⊢ s = ∅ ∨ s = univ → IsClopen s

Tactic:
rintro (rfl | rfl)

Goal state after:
case inl
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
⊢ IsClopen ∅

case inr
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
⊢ IsClopen univ
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case inr
X : Type u_1
inst✝² : TopologicalSpace X
s t : Set X
inst✝¹ : LindelofSpace X
inst✝ : RegularSpace X
s_cl : IsClosed[inst✝²] s
t_cl : IsClosed[inst✝²] t
st_dis : Disjoint s t
nonempty_X : Nonempty X
this : ∀ (a : X), ∃ n, IsOpen[inst✝²] n ∧ Disjoint (closure[inst✝²] n) t ∧ (a ∈ s → a ∈ n)
⊢ _root_.HasSeparatingCover s t

Tactic:
choose u u_open u_dis u_nhds using this

Goal state after:
case inr
X : Type u_1
inst✝² : TopologicalSpace X
s t : Set X
inst✝¹ : LindelofSpace X
inst✝ : RegularSpace X
s_cl : IsClosed[inst✝²] s
t_cl : IsClosed[inst✝²] t
st_dis : Disjoint s t
nonempty_X : Nonempty X
u : X → Set X
u_open : ∀ (a : X), IsOpen[inst✝²] (u a)
u_dis : ∀ (a : X), Disjoint (closure[inst✝²] (u a)) t
u_nhds : ∀ a ∈ s, a ∈ u a
⊢ _root_.HasSeparatingCover s t
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝²] U
hKU : K ⊆ U
⊢ ∃ V, IsOpen[inst✝²] V ∧ K ⊆ V ∧ closure[inst✝²] V ⊆ U ∧ IsCompact (closure[inst✝²] V)

Tactic:
rcases exists_compact_closed_between hK hU hKU with ⟨L, L_compact, L_closed, KL, LU⟩

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝²] U
hKU : K ⊆ U
L : Set X
L_compact : IsCompact L
L_closed : IsClosed[inst✝²] L
KL : K ⊆ interior L
LU : L ⊆ U
⊢ ∃ V, IsOpen[inst✝²] V ∧ K ⊆ V ∧ closure[inst✝²] V ⊆ U ∧ IsCompact (closure[inst✝²] V)
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case mp
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
Y : Type u
f : Y → Set X
h₁ : IsLindelof (⋃ i, f i)
h₂ : IsOpen[inst✝] (⋃ i, f i)
f' : Y → ι
hf' : ∀ (i : Y), b (f' i) = f i
this : b ∘ f' = f
⊢ ∃ s, s.Countable ∧ ⋃ i, f i = ⋃ i ∈ s, b i

Tactic:
subst this

Goal state after:
case mp
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
Y : Type u
f' : Y → ι
h₁ : IsLindelof (⋃ i, (b ∘ f') i)
h₂ : IsOpen[inst✝] (⋃ i, (b ∘ f') i)
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
⊢ ∃ s, s.Countable ∧ ⋃ i, (b ∘ f') i = ⋃ i ∈ s, b i
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: case mpr.inr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
hα : Nonempty α
⊢ PreconnectedSpace α

Tactic:
exact ⟨by rw [← h (Classical.choice hα)]; exact isPreconnected_connectedComponent⟩

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
i : α → β
inst✝ : TopologicalSpace γ
di : IsDenseInducing i
f : α → γ
a : α
hf : ContinuousAt f a
⊢ Tendsto f (𝓝 a) (𝓝 (di.extend f (i a)))

Tactic:
rw [IsDenseInducing.extend, ← di.nhds_eq_comap]
  exact tendsto_nhds_limUnder ⟨_, hf⟩

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : LindelofSpace X
hf : Continuous[inst✝², inst✝¹] f
hsur : Function.Surjective f
⊢ IsLindelof univ

Tactic:
← Set.image_univ_of_surjective hsur

Goal state after:
X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : LindelofSpace X
hf : Continuous[inst✝², inst✝¹] f
hsur : Function.Surjective f
⊢ IsLindelof (f '' univ)
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : LindelofSpace X
hf : Continuous[inst✝², inst✝¹] f
hsur : Function.Surjective f
⊢ IsLindelof univ

Tactic:
rw [← Set.image_univ_of_surjective hsur]
    exact IsLindelof.image (isLindelof_univ_iff.mpr ‹_›) hf

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
⊢ T ∈ 𝓝 (f x) ↔ ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T

Tactic:
refine ⟨fun hT => ?_, fun hT => ?_⟩

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
hT : T ∈ 𝓝 (f x)
⊢ ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T

case refine_2
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
hT : ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T
⊢ T ∈ 𝓝 (f x)
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ IsEmpty (ConnectedComponents α) ↔ IsEmpty α

Tactic:
by
  rw [← not_iff_not, not_isEmpty_iff, not_isEmpty_iff, nonempty_iff_nonempty]

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
W : Type u_3
Z : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace W
inst✝² : TopologicalSpace Z
X' : Type u_5
Y' : Type u_6
inst✝¹ : TopologicalSpace X'
inst✝ : TopologicalSpace Y'
ι : Type u_7
ι' : Type u_8
i✝ : ι ⊕ ι'
i : ι'
⊢ Continuous[instTopologicalSpaceProd, inst✝⁵] fun a ↦ (Equiv.sumArrowEquivProdArrow ι ι' X).invFun a (Sum.inr i)

Tactic:
apply (continuous_apply _).comp' continuous_snd

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
p : Filter ι
g : ι → β
b : β
hg : Tendsto g p (𝓝 b)
p' : Filter α
⊢ TendstoUniformlyOnFilter (fun n x ↦ g n) (fun x ↦ b) p p'

Tactic:
simpa only [nhds_eq_comap_uniformity, tendsto_comap_iff] using! hg.comp (tendsto_fst (g := p'))

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

Goal state before:
pp: α : Type ua
u₁ u₂ : UniformSpace α
h : u₁ ≤ u₂
a : α
⊢ 𝓝 a ≤ 𝓝 a

Tactic:
rw [@nhds_eq_uniformity α u₁ a, @nhds_eq_uniformity α u₂ a]; exact lift'_mono h le_rfl

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

Goal state before:
pp: α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
h : UniformContinuous (s.domRestrict f)
⊢ Continuous[instTopologicalSpaceSubtype, inst✝.toTopologicalSpace] (s.domRestrict f)

Tactic:
exact h.continuous

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

Goal state before:
pp: α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
s : SetRel β β
f : α → α → β
hf : Tendsto (fun x ↦ (f x.1.1 x.1.2, f x.2.1 x.2.2)) (𝓤 (α × α)) (𝓤 β)
hs : s ∈ 𝓤 β
⊢ ∃ u ∈ 𝓤 α, ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s

Tactic:
uniformity_prod_eq_prod,

Goal state after:
α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
s : SetRel β β
f : α → α → β
hf : Tendsto (fun x ↦ (f x.1.1 x.1.2, f x.2.1 x.2.2)) (map (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) (𝓤 α ×ˢ 𝓤 α)) (𝓤 β)
hs : s ∈ 𝓤 β
⊢ ∃ u ∈ 𝓤 α, ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s
Module: Mathlib.Topology.UniformSpace.Separation

Goal state before:
pp: α : Type u
inst✝ : UniformSpace α
U : SetRel α α
hU : U ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] U
x✝¹ : α × α
x₁ x₂ : α
x✝ : (x₁, x₂) ∈ Prod.map mk mk ⁻¹' Prod.map mk mk '' id U
y₁ y₂ : α
hyU : (y₁, y₂) ∈ id U
hxy : Prod.map mk mk (y₁, y₂) = Prod.map mk mk (x₁, x₂)
⊢ (x₁, x₂) ∈ id U

Tactic:
simp only [Prod.map, Prod.ext_iff, mk_eq_mk] at hxy

Goal state after:
α : Type u
inst✝ : UniformSpace α
U : SetRel α α
hU : U ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] U
x✝¹ : α × α
x₁ x₂ : α
x✝ : (x₁, x₂) ∈ Prod.map mk mk ⁻¹' Prod.map mk mk '' id U
y₁ y₂ : α
hyU : (y₁, y₂) ∈ id U
hxy : Inseparable y₁ x₁ ∧ Inseparable y₂ x₂
⊢ (x₁, x₂) ∈ id U
Module: Mathlib.Topology.UniformSpace.Basic

Goal state before:
pp: α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
f : β → α
b : β
s : Set β
⊢ ContinuousWithinAt f s b ↔ Tendsto (fun x ↦ (f b, f x)) (𝓝[s] b) (𝓤 α)

Tactic:
by
  rw [ContinuousWithinAt, tendsto_nhds_right]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
s : Set α
p : Filter ι
ι' : Type u_5
α' : Type u_6
β' : Type u_7
inst✝ : UniformSpace β'
F' : ι' → α' → β'
p' : Filter ι'
s' : Set α'
h : UniformCauchySeqOn F p s
h' : UniformCauchySeqOn F' p' s'
u : Set ((β × β') × β × β')
hu : u ∈ 𝓤 (β × β')
⊢ ∀ᶠ (m : (ι × ι') × ι × ι') in (p ×ˢ p') ×ˢ p ×ˢ p',
    ∀ x ∈ s ×ˢ s', ((fun i ↦ Prod.map (F i.1) (F' i.2)) m.1 x, (fun i ↦ Prod.map (F i.1) (F' i.2)) m.2 x) ∈ u

Tactic:
uniformity_prod_eq_prod,

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
s : Set α
p : Filter ι
ι' : Type u_5
α' : Type u_6
β' : Type u_7
inst✝ : UniformSpace β'
F' : ι' → α' → β'
p' : Filter ι'
s' : Set α'
h : UniformCauchySeqOn F p s
h' : UniformCauchySeqOn F' p' s'
u : Set ((β × β') × β × β')
hu : u ∈ map (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) (𝓤 β ×ˢ 𝓤 β')
⊢ ∀ᶠ (m : (ι × ι') × ι × ι') in (p ×ˢ p') ×ˢ p ×ˢ p',
    ∀ x ∈ s ×ˢ s', ((fun i ↦ Prod.map (F i.1) (F' i.2)) m.1 x, (fun i ↦ Prod.map (F i.1) (F' i.2)) m.2 x) ∈ u
Module: Mathlib.Topology.UniformSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
s : Set α
p : Filter ι
ι' : Type u_5
α' : Type u_6
β' : Type u_7
inst✝ : UniformSpace β'
F' : ι' → α' → β'
p' : Filter ι'
s' : Set α'
h : UniformCauchySeqOn F p s
h' : UniformCauchySeqOn F' p' s'
u : Set ((β × β') × β × β')
v : Set (β × β)
hv : v ∈ 𝓤 β
w : Set (β' × β')
hw : w ∈ 𝓤 β'
hvw : v ×ˢ w ⊆ (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' u
⊢ ∀ᶠ (m : (ι × ι') × ι × ι') in (p ×ˢ p') ×ˢ p ×ˢ p',
    ∀ (a : α) (b : α'), a ∈ s → b ∈ s' → ((F m.1.1 a, F' m.1.2 b), F m.2.1 a, F' m.2.2 b) ∈ u

Tactic:
← Set.image_subset_iff

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
s : Set α
p : Filter ι
ι' : Type u_5
α' : Type u_6
β' : Type u_7
inst✝ : UniformSpace β'
F' : ι' → α' → β'
p' : Filter ι'
s' : Set α'
h : UniformCauchySeqOn F p s
h' : UniformCauchySeqOn F' p' s'
u : Set ((β × β') × β × β')
v : Set (β × β)
hv : v ∈ 𝓤 β
w : Set (β' × β')
hw : w ∈ 𝓤 β'
hvw : (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) '' v ×ˢ w ⊆ u
⊢ ∀ᶠ (m : (ι × ι') × ι × ι') in (p ×ˢ p') ×ˢ p ×ˢ p',
    ∀ (a : α) (b : α'), a ∈ s → b ∈ s' → ((F m.1.1 a, F' m.1.2 b), F m.2.1 a, F' m.2.2 b) ∈ u
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
ι : Sort u_1
inst✝ : UniformSpace α
K : Set α
p : ι → Prop
V : ι → Set (α × α)
hbasis : (𝓤 α).HasBasis p V
hK : IsCompact K
U : Set α
H : U ∈ 𝓝ˢ K
⊢ K ⊆ ⋃ x, interior U

Tactic:
simpa only [iUnion_const, subset_interior_iff_mem_nhdsSet] using H

Goal state after:
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: X : Type u_1
α : Type u_2
inst✝¹ : One α
inst✝ : TopologicalSpace X
f : X → α
⊢ range f ⊆ insert 1 (f '' mulTSupport f)

Tactic:
grw [← subset_mulTSupport f]; exact range_subset_insert_image_mulSupport f

Goal state after:
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : One β
f : α → β
inst✝ : R1Space α
⊢ (∃ K, IsCompact K ∧ ∀ x ∉ K, f x = 1) ↔ HasCompactMulSupport f

Tactic:
simp_rw [← notMem_mulSupport, ← mem_compl_iff, ← subset_def, compl_subset_compl,
    hasCompactMulSupport_def, exists_isCompact_superset_iff]

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u
U : (i : ι) → UniformSpace (α i)
⊢ uniformSpace α = ⨅ i, UniformSpace.comap (eval i) (U i)

Tactic:
by
  ext : 1; rfl

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
p : Filter ι
⊢ TendstoLocallyUniformlyOn F f p s ↔ TendstoLocallyUniformly (fun i x ↦ F i ↑x) (f ∘ Subtype.val) p

Tactic:
by
  simp only [tendstoLocallyUniformly_iff_forall_tendsto, Subtype.forall', tendsto_map'_iff,
    tendstoLocallyUniformlyOn_iff_forall_tendsto, ← map_nhds_subtype_val, prod_map_right]; rfl

Goal state after:
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
F : ι → α → β
f : α → β
p : Filter ι
inst✝ : LocallyCompactSpace α
⊢ TendstoLocallyUniformly F f p ↔ ∀ (K : Set α), IsCompact K → TendstoUniformlyOn F f p K

Tactic:
by
  simp only [← tendstoLocallyUniformlyOn_univ,
    tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_univ, Set.subset_univ, forall_true_left]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
𝓐 : Filter ((α →ᵤ β) × (α →ᵤ β))
𝓕 : Filter (β × β)
⊢ 𝓐 ≤ (fun 𝓕 ↦ UniformFun.filter α β 𝓕) 𝓕 ↔ (fun 𝓐 ↦ map (UniformFun.phi α β) (𝓐 ×ˢ ⊤)) 𝓐 ≤ 𝓕

Tactic:
calc
    𝓐 ≤ UniformFun.filter α β 𝓕 ↔ (UniformFun.basis α β 𝓕).sets ⊆ 𝓐.sets := by
      rw [UniformFun.filter, ← FilterBasis.generate, le_generate_iff]
    _ ↔ ∀ U ∈ 𝓕, UniformFun.gen α β U ∈ 𝓐 := image_subset_iff
    _ ↔ ∀ U ∈ 𝓕,
          { uv | ∀ x, (uv, x) ∈ { t : ((α →ᵤ β) × (α →ᵤ β)) × α | (t.1.1 t.2, t.1.2 t.2) ∈ U } } ∈
            𝓐 :=
      Iff.rfl
    _ ↔ ∀ U ∈ 𝓕,
          { uvx : ((α →ᵤ β) × (α →ᵤ β)) × α | (uvx.1.1 uvx.2, uvx.1.2 uvx.2) ∈ U } ∈
            𝓐 ×ˢ (⊤ : Filter α) :=
      forall₂_congr fun U _hU => mem_prod_top.symm
    _ ↔ lowerAdjoint 𝓐 ≤ 𝓕 := Iff.rfl

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

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
δ₁ : Type u_6
δ₂ : Type u_7
φ₁ : δ₁ → α
φ₂ : δ₂ → α
h_cover : range φ₁ ∪ range φ₂ = univ
U : Set (β × β)
hU : U ∈ 𝓤 β
x✝ : (α →ᵤ β) × (α →ᵤ β)
x : α
f g : α →ᵤ β
hfg :
  (f, g) ∈
    (fun p ↦ ((⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toFun) p.1, (⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toFun) p.2)) ⁻¹'
        UniformFun.gen δ₁ β (U, U).1 ∩
      (fun p ↦ ((⇑ofFun ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑toFun) p.1, (⇑ofFun ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑toFun) p.2)) ⁻¹'
        UniformFun.gen δ₂ β (U, U).2
⊢ (toFun (f, g).1 x, toFun (f, g).2 x) ∈ U

Tactic:
rcases h_cover.ge <| mem_univ x with (⟨y, rfl⟩ | ⟨y, rfl⟩)

Goal state after:
case refine_3.inl
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
δ₁ : Type u_6
δ₂ : Type u_7
φ₁ : δ₁ → α
φ₂ : δ₂ → α
h_cover : range φ₁ ∪ range φ₂ = univ
U : Set (β × β)
hU : U ∈ 𝓤 β
x✝ : (α →ᵤ β) × (α →ᵤ β)
f g : α →ᵤ β
hfg :
  (f, g) ∈
    (fun p ↦ ((⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toFun) p.1, (⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toFun) p.2)) ⁻¹'
        UniformFun.gen δ₁ β (U, U).1 ∩
      (fun p ↦ ((⇑ofFun ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑toFun) p.1, (⇑ofFun ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑toFun) p.2)) ⁻¹'
        UniformFun.gen δ₂ β (U, U).2
y : δ₁
⊢ (toFun (f, g).1 (φ₁ y), toFun (f, g).2 (φ₁ y)) ∈ U

case refine_3.inr
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
δ₁ : Type u_6
δ₂ : Type u_7
φ₁ : δ₁ → α
φ₂ : δ₂ → α
h_cover : range φ₁ ∪ range φ₂ = univ
U : Set (β × β)
hU : U ∈ 𝓤 β
x✝ : (α →ᵤ β) × (α →ᵤ β)
f g : α →ᵤ β
hfg :
  (f, g) ∈
    (fun p ↦ ((⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toFun) p.1, (⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toFun) p.2)) ⁻¹'
        UniformFun.gen δ₁ β (U, U).1 ∩
      (fun p ↦ ((⇑ofFun ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑toFun) p.1, (⇑ofFun ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑toFun) p.2)) ⁻¹'
        UniformFun.gen δ₂ β (U, U).2
y : δ₂
⊢ (toFun (f, g).1 (φ₂ y), toFun (f, g).2 (φ₂ y)) ∈ U
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set α
inst✝ : UniformSpace β
𝔖 : Set (Set α)
f : α →ᵤ[𝔖] β
hs : s ∈ 𝔖
V : Set (β × β)
hV : V ∈ 𝓤 β
⊢ {g | ∀ x ∈ s, ((toFun 𝔖) f x, (toFun 𝔖) g x) ∈ V} ∈ 𝓝 f

Tactic:
by
  rw [UniformOnFun.nhds_eq]
  apply_rules [mem_iInf_of_mem, mem_principal_self]

Goal state after:
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b c d : α
⊢ Ico a b * Ioc c d ⊆ Ioo (a * c) (b * d)

Tactic:
by
  have := mulLeftMono_of_mulLeftStrictMono α
  have := mulRightMono_of_mulRightStrictMono α
  rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
  exact ⟨mul_lt_mul_of_le_of_lt hya hzc, mul_lt_mul_of_lt_of_le hyb hzd⟩

Goal state after:
Module: Mathlib.Topology.Algebra.MulAction

Goal state before:
pp: G : Type u_1
P : Type u_2
inst✝⁵ : AddGroup G
inst✝⁴ : AddTorsor G P
inst✝³ : TopologicalSpace G
inst✝² : PreconnectedSpace G
inst✝¹ : TopologicalSpace P
inst✝ : ContinuousVAdd G P
⊢ Set.univ = ⇑(Equiv.vaddConst (Classical.arbitrary P)) '' Set.univ

Tactic:
rw [Set.image_univ, Equiv.range_eq_univ]

Goal state after:
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: case h.inl
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
hb✝ : a ≠ 0
U V : Set α
hV : V = (fun x ↦ a * x) ⁻¹' U
aU : α
haU : U = Ioi aU
hb : a < 0
⊢ (fun x ↦ a * x) ⁻¹' Ioi aU = Iio (a⁻¹ * aU)

Tactic:
rw [Set.preimage_const_mul_Ioi_of_neg _ hb, div_eq_inv_mul]

Goal state after:
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: case inl.inr
G₀ : Type u_4
inst✝⁵ : GroupWithZero G₀
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
s : Set E
hs : s.Nonempty
⊢ closure[inst✝²] 0 = 0 • closure[inst✝²] s

Tactic:
zero_smul_set hs.closure

Goal state after:
case inl.inr
G₀ : Type u_4
inst✝⁵ : GroupWithZero G₀
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
s : Set E
hs : s.Nonempty
⊢ closure[inst✝²] 0 = 0
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : InvolutiveInv G
inst✝ : ContinuousInv G
a : G
⊢ (𝓝[≠] a)⁻¹ = 𝓝[≠] a⁻¹

Tactic:
by
  convert! (Homeomorph.inv G).isEmbedding.map_nhdsWithin_eq .. using 2
  simp

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
ι : Type u_3
f g : ι → α × α
l : Filter ι
hf : Tendsto f l (𝓤 α)
hg : Tendsto g l (𝓤 α)
⊢ Tendsto (f * g⁻¹) l (𝓤 α)

Tactic:
exact hf.uniformity_mul hg.uniformity_inv

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : UniformSpace α
inst✝² : Group α
inst✝¹ : IsUniformGroup α
inst✝ : UniformSpace β
f : β → α
hf : UniformContinuous f
n : ℕ
⊢ UniformContinuous fun x ↦ f x ^ ↑n

Tactic:
simp_rw [zpow_natCast]
    exact hf.pow_const _

Goal state after: