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

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
p : ℕ → Prop
U : ℕ → SetRel α α
H : (𝓤 α).HasBasis p U
xs : ℕ → α
u : ℕ → ℕ
⊢ ∀ (i : ℕ), p i → ∃ t, t.Finite ∧ interUnionBalls xs u U ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U i}

Tactic:
]

Goal state after:
α : Type u
uniformSpace : UniformSpace α
p : ℕ → Prop
U : ℕ → SetRel α α
H : (𝓤 α).HasBasis p U
xs : ℕ → α
u : ℕ → ℕ
⊢ ∀ (i : ℕ), p i → ∃ t, t.Finite ∧ interUnionBalls xs u U ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U i}
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
hf : IsUniformInducing f
hsurj : Surjective f
⊢ IsComplete univ ↔ IsComplete univ

Tactic:
]

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
f : α → β
s : Set α
l : Filter ι
inst✝ : l.IsCountablyGenerated
h : ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformlyOn (fun n ↦ F (u n)) f atTop s
⊢ ∀ (x : ℕ → ι × α), Tendsto x atTop (l ×ˢ 𝓟 s) → Tendsto ((fun q ↦ (f q.2, F q.1 q.2)) ∘ x) atTop (𝓤 β)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
l : Filter ι
inst✝ : l.IsCountablyGenerated
h : ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformlyOn (fun n ↦ F (u n)) f atTop s
⊢ ∀ (x : ℕ → ι × α), Tendsto x atTop (l ×ˢ 𝓟 s) → Tendsto ((fun q ↦ (f q.2, F q.1 q.2)) ∘ x) atTop (𝓤 β)
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u
inst✝ : UniformSpace α
s : Set α
⊢ IsComplete s ↔ IsComplete s

Tactic:
]

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
f : α → β
s : Set α
l : Filter ι
inst✝ : l.IsCountablyGenerated
h : ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformlyOn (fun n ↦ F (u n)) f atTop s
u : ℕ → ι × α
hu : Tendsto (fun n ↦ (u n).1) atTop l ∧ Tendsto (fun n ↦ (u n).2) atTop (𝓟 s)
⊢ Tendsto ((fun q ↦ (f q.2, F q.1 q.2)) ∘ u) atTop (𝓤 β)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
l : Filter ι
inst✝ : l.IsCountablyGenerated
h : ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformlyOn (fun n ↦ F (u n)) f atTop s
u : ℕ → ι × α
hu : Tendsto (fun n ↦ (u n).1) atTop l ∧ Tendsto (fun n ↦ (u n).2) atTop (𝓟 s)
⊢ Tendsto ((fun q ↦ (f q.2, F q.1 q.2)) ∘ u) atTop (𝓤 β)
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
f : α → β
s : Set α
l : Filter ι
inst✝ : l.IsCountablyGenerated
u : ℕ → ι × α
h : Tendsto (fun q ↦ (f q.2, F (u q.1).1 q.2)) (atTop ×ˢ 𝓟 s) (𝓤 β)
hu : Tendsto (fun n ↦ (u n).1) atTop l ∧ Tendsto (fun n ↦ (u n).2) atTop (𝓟 s)
⊢ Tendsto ((fun q ↦ (f q.2, F q.1 q.2)) ∘ u) atTop (𝓤 β)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
l : Filter ι
inst✝ : l.IsCountablyGenerated
u : ℕ → ι × α
h : Tendsto (fun q ↦ (f q.2, F (u q.1).1 q.2)) (atTop ×ˢ 𝓟 s) (𝓤 β)
hu : Tendsto (fun n ↦ (u n).1) atTop l ∧ Tendsto (fun n ↦ (u n).2) atTop (𝓟 s)
⊢ Tendsto ((fun q ↦ (f q.2, F q.1 q.2)) ∘ u) atTop (𝓤 β)
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
p : ℕ → Prop
U : ℕ → SetRel α α
H : (𝓤 α).HasBasis p U
inst✝ : CompleteSpace α
xs : ℕ → α
u : ℕ → ℕ
⊢ TotallyBounded (closure[uniformSpace.toTopologicalSpace] (interUnionBalls xs u U)) ∧
    IsComplete (closure[uniformSpace.toTopologicalSpace] (interUnionBalls xs u U))

Tactic:
]

Goal state after:
α : Type u
uniformSpace : UniformSpace α
p : ℕ → Prop
U : ℕ → SetRel α α
H : (𝓤 α).HasBasis p U
inst✝ : CompleteSpace α
xs : ℕ → α
u : ℕ → ℕ
⊢ TotallyBounded (closure[uniformSpace.toTopologicalSpace] (interUnionBalls xs u U)) ∧
    IsComplete (closure[uniformSpace.toTopologicalSpace] (interUnionBalls xs u U))
Module: Mathlib.Topology.UniformSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
l : Filter ι
h : Tendsto (fun q ↦ (f q.2, F q.1 q.2)) (l ×ˢ 𝓟 s) (𝓤 β)
u : ℕ → ι
hu : Tendsto u atTop l
⊢ Tendsto (fun q ↦ (f q.2, F (u q.1) q.2)) (atTop ×ˢ 𝓟 s) (𝓤 β)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
l : Filter ι
h : Tendsto (fun q ↦ (f q.2, F q.1 q.2)) (l ×ˢ 𝓟 s) (𝓤 β)
u : ℕ → ι
hu : Tendsto u atTop l
⊢ Tendsto (fun q ↦ (f q.2, F (u q.1) q.2)) (atTop ×ˢ 𝓟 s) (𝓤 β)
Module: Mathlib.Topology.UniformSpace.CompleteSeparated

Goal state before:
pp: α : Type u_1
inst✝¹ : UniformSpace α
inst✝ : T0Space α
s : Set α
h : IsComplete s
a : α
ha : ClusterPt a (𝓟 s)
f : Filter α := 𝓝[s] a
this : Cauchy f
y : α
ys : y ∈ s
fy : f ≤ 𝓝 y
⊢ y ∈ s

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : UniformSpace α
inst✝ : T0Space α
s : Set α
h : IsComplete s
a : α
ha : ClusterPt a (𝓟 s)
f : Filter α := 𝓝[s] a
this : Cauchy f
y : α
ys : y ∈ s
fy : f ≤ 𝓝 y
⊢ y ∈ s
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
inst✝ : UniformSpace α
K : Set α
ι : Sort u_2
U : ι → Set α
hK : IsCompact K
hopen : ∀ (i : ι), IsOpen[inst✝.toTopologicalSpace] (U i)
hcover : K ⊆ ⋃ i, U i
x : α
hx : x ∈ K
i : ι
hi : U i ∈ Filter.comap (Prod.mk x) ((𝓤 α).lift' fun s ↦ s ○ s)
⊢ ∃ i, ∃ V ∈ 𝓤 α, ball x (V ○ V) ⊆ U i

Tactic:
]

Goal state after:
α : Type ua
inst✝ : UniformSpace α
K : Set α
ι : Sort u_2
U : ι → Set α
hK : IsCompact K
hopen : ∀ (i : ι), IsOpen[inst✝.toTopologicalSpace] (U i)
hcover : K ⊆ ⋃ i, U i
x : α
hx : x ∈ K
i : ι
hi : U i ∈ Filter.comap (Prod.mk x) ((𝓤 α).lift' fun s ↦ s ○ s)
⊢ ∃ i, ∃ V ∈ 𝓤 α, ball x (V ○ V) ⊆ U i
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝⁴ : UniformSpace α
inst✝³ : UniformSpace β
inst✝² : UniformSpace γ
f : α → β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
⊢ IsComplete (range Sum.inl ∪ range Sum.inr)

Tactic:
]

Goal state after:
α : Type u
β : Type v
γ : Type w
inst✝⁴ : UniformSpace α
inst✝³ : UniformSpace β
inst✝² : UniformSpace γ
f : α → β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
⊢ IsComplete (range Sum.inl ∪ range Sum.inr)
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
f : α → β
p : Filter ι
p' : Filter α
inst✝ : p.NeBot
L : ι → β
ℓ : β
h1 : TendstoUniformlyOnFilter F f p p'
h2 : ∀ᶠ (i : ι) in p, Tendsto (F i) p' (𝓝 (L i))
h3 : Tendsto L p (𝓝 ℓ)
⊢ Tendsto (fun x ↦ (f x, ℓ)) p' (𝓤 β)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
inst✝ : p.NeBot
L : ι → β
ℓ : β
h1 : TendstoUniformlyOnFilter F f p p'
h2 : ∀ᶠ (i : ι) in p, Tendsto (F i) p' (𝓝 (L i))
h3 : Tendsto L p (𝓝 ℓ)
⊢ Tendsto (fun x ↦ (f x, ℓ)) p' (𝓤 β)
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u
β : Type v
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
inst✝ : DiscreteUniformity β
f : α → β
hf : IsUniformEmbedding f
⊢ DiscreteUniformity α

Tactic:
simp_rw [discreteUniformity_iff_eq_principal_setRelId, ← hf.comap_uniformity,
    DiscreteUniformity.eq_principal_setRelId, comap_principal, SetRel.id, preimage_ofPred_eq,
    hf.injective.eq_iff]

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
f : α → β
p : Filter ι
p' : Filter α
inst✝ : p.NeBot
L : ι → β
ℓ : β
h1 : TendstoUniformlyOnFilter F f p p'
h2 : ∀ᶠ (i : ι) in p, Tendsto (F i) p' (𝓝 (L i))
h3 : Tendsto L p (𝓝 ℓ)
s : Set (β × β)
hs : s ∈ 𝓤 β
⊢ ∀ᶠ (x : α) in p', (f x, ℓ) ∈ s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
inst✝ : p.NeBot
L : ι → β
ℓ : β
h1 : TendstoUniformlyOnFilter F f p p'
h2 : ∀ᶠ (i : ι) in p, Tendsto (F i) p' (𝓝 (L i))
h3 : Tendsto L p (𝓝 ℓ)
s : Set (β × β)
hs : s ∈ 𝓤 β
⊢ ∀ᶠ (x : α) in p', (f x, ℓ) ∈ s
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace γ
p : α → Prop
e : α → β
f : α → γ
b : β
s : Set α
hf : UniformContinuous fun x ↦ f ↑x
he : IsUniformEmbedding e
hd : ∀ (x : β), x ∈ closure[inst✝².toTopologicalSpace] (range e)
hb : closure[inst✝².toTopologicalSpace] (e '' s) ∈ 𝓝 b
hs : IsClosed[inst✝³.toTopologicalSpace] s
hp : ∀ x ∈ s, p x
de : IsDenseEmbedding e
de' : IsDenseEmbedding (IsDenseEmbedding.subtypeEmb p e)
ue' : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e)
this : b ∈ closure[inst✝².toTopologicalSpace] (e '' {x | p x})
c : γ
hc : Tendsto (f ∘ Subtype.val) (comap Subtype.val (comap e (𝓝 b))) (𝓝 c)
⊢ {x | p x} ∈ comap e (𝓝 b)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace γ
p : α → Prop
e : α → β
f : α → γ
b : β
s : Set α
hf : UniformContinuous fun x ↦ f ↑x
he : IsUniformEmbedding e
hd : ∀ (x : β), x ∈ closure[inst✝².toTopologicalSpace] (range e)
hb : closure[inst✝².toTopologicalSpace] (e '' s) ∈ 𝓝 b
hs : IsClosed[inst✝³.toTopologicalSpace] s
hp : ∀ x ∈ s, p x
de : IsDenseEmbedding e
de' : IsDenseEmbedding (IsDenseEmbedding.subtypeEmb p e)
ue' : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e)
this : b ∈ closure[inst✝².toTopologicalSpace] (e '' {x | p x})
c : γ
hc : Tendsto (f ∘ Subtype.val) (comap Subtype.val (comap e (𝓝 b))) (𝓝 c)
⊢ {x | p x} ∈ comap e (𝓝 b)
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace γ
p : α → Prop
e : α → β
f : α → γ
b : β
s : Set α
hf : UniformContinuous fun x ↦ f ↑x
he : IsUniformEmbedding e
hd : ∀ (x : β), x ∈ closure[inst✝².toTopologicalSpace] (range e)
hb : closure[inst✝².toTopologicalSpace] (e '' s) ∈ 𝓝 b
hs : IsClosed[inst✝³.toTopologicalSpace] s
hp : ∀ x ∈ s, p x
de : IsDenseEmbedding e
de' : IsDenseEmbedding (IsDenseEmbedding.subtypeEmb p e)
ue' : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e)
this : b ∈ closure[inst✝².toTopologicalSpace] (e '' {x | p x})
c : γ
hc : Tendsto (f ∘ Subtype.val) (comap Subtype.val (comap e (𝓝 b))) (𝓝 c)
⊢ s ⊆ {x | p x}

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace γ
p : α → Prop
e : α → β
f : α → γ
b : β
s : Set α
hf : UniformContinuous fun x ↦ f ↑x
he : IsUniformEmbedding e
hd : ∀ (x : β), x ∈ closure[inst✝².toTopologicalSpace] (range e)
hb : closure[inst✝².toTopologicalSpace] (e '' s) ∈ 𝓝 b
hs : IsClosed[inst✝³.toTopologicalSpace] s
hp : ∀ x ∈ s, p x
de : IsDenseEmbedding e
de' : IsDenseEmbedding (IsDenseEmbedding.subtypeEmb p e)
ue' : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e)
this : b ∈ closure[inst✝².toTopologicalSpace] (e '' {x | p x})
c : γ
hc : Tendsto (f ∘ Subtype.val) (comap Subtype.val (comap e (𝓝 b))) (𝓝 c)
⊢ s ⊆ {x | p x}
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
inst✝ : UniformSpace α
K : Set α
S : Set (Set α)
hK : IsCompact K
hopen : ∀ s ∈ S, IsOpen[inst✝.toTopologicalSpace] s
hcover : K ⊆ ⋃ i, ↑i
⊢ ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, ball x V ⊆ s

Tactic:
]

Goal state after:
α : Type ua
inst✝ : UniformSpace α
K : Set α
S : Set (Set α)
hK : IsCompact K
hopen : ∀ s ∈ S, IsOpen[inst✝.toTopologicalSpace] s
hcover : K ⊆ ⋃ i, ↑i
⊢ ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, ball x V ⊆ s
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
⊢ f = 1 ↔ f = 1

Tactic:
]

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
⊢ ∅ = ∅

Tactic:
]

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

Goal state before:
pp: X : Type u_1
α : Type u_2
β : Type u_4
inst✝² : One α
inst✝¹ : TopologicalSpace X
inst✝ : One β
g : α → β
hg : ∀ {x : α}, g x = 1 ↔ x = 1
f : X → α
⊢ closure[inst✝¹] (mulSupport f) = closure[inst✝¹] (mulSupport f)

Tactic:
]

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

Goal state before:
pp: X : Type u_1
α : Type u_2
β : Type u_4
inst✝² : One α
inst✝¹ : TopologicalSpace X
inst✝ : One β
g : α → β
f : X → α
hg : ∀ {x : α}, x ∈ range f → (g x = 1 ↔ x = 1)
⊢ closure[inst✝¹] (mulSupport f) = closure[inst✝¹] (mulSupport f)

Tactic:
]

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

Goal state before:
pp: X : Type u_1
α : Type u_2
inst✝² : One α
inst✝¹ : TopologicalSpace X
Y : Type u_9
inst✝ : TopologicalSpace Y
g : Y → α
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
⊢ closure[inst✝¹] (f ⁻¹' mulSupport g) ⊆ f ⁻¹' closure[inst✝] (mulSupport g)

Tactic:
]

Goal state after:
X : Type u_1
α : Type u_2
inst✝² : One α
inst✝¹ : TopologicalSpace X
Y : Type u_9
inst✝ : TopologicalSpace Y
g : Y → α
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
⊢ closure[inst✝¹] (f ⁻¹' mulSupport g) ⊆ f ⁻¹' closure[inst✝] (mulSupport g)
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: X : Type u_1
α : Type u_2
inst✝² : One α
inst✝¹ : TopologicalSpace X
Y : Type u_9
inst✝ : TopologicalSpace Y
g : Y → α
f : X ≃ₜ Y
⊢ closure[inst✝¹] (⇑f ⁻¹' mulSupport g) = closure[inst✝¹] (⇑f ⁻¹' mulSupport g)

Tactic:
]

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 '' mulSupport f)

Tactic:
]

Goal state after:
X : Type u_1
α : Type u_2
inst✝¹ : One α
inst✝ : TopologicalSpace X
f : X → α
⊢ range f ⊆ insert 1 (f '' mulSupport f)
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
inst✝ : UniformSpace α
A B : Set α
hA : IsCompact A
hB : IsClosed[inst✝.toTopologicalSpace] B
h : Disjoint A B
this : ∃ i ∈ 𝓤 α, ⋃ x ∈ A, ball x (id i) ⊆ Bᶜ
⊢ ∃ V ∈ 𝓤 α, Disjoint (⋃ x ∈ A, ball x V) (⋃ x ∈ B, ball x V)

Tactic:
]

Goal state after:
α : Type ua
inst✝ : UniformSpace α
A B : Set α
hA : IsCompact A
hB : IsClosed[inst✝.toTopologicalSpace] B
h : Disjoint A B
this : ∃ i ∈ 𝓤 α, ⋃ x ∈ A, ball x (id i) ⊆ Bᶜ
⊢ ∃ V ∈ 𝓤 α, Disjoint (⋃ x ∈ A, ball x V) (⋃ x ∈ B, ball x V)
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
e : β → α
h_e : IsUniformInducing e
h_dense : DenseRange e
f : β → γ
h_f : UniformContinuous f
inst✝ : CompleteSpace γ
d : Set (γ × γ)
hd : d ∈ 𝓤 γ
s : Set (γ × γ)
hs : s ∈ 𝓤 γ
hs_comp : s ○ (s ○ s) ⊆ d
h_pnt : ∀ {a : α} {m : Set α}, m ∈ 𝓝 a → ∃ c ∈ f '' e ⁻¹' m, (c, ⋯.extend f a) ∈ s ∧ (⋯.extend f a, c) ∈ s
this : Prod.map f f ⁻¹' s ∈ comap (fun x ↦ (e x.1, e x.2)) (𝓤 α)
⊢ Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
e : β → α
h_e : IsUniformInducing e
h_dense : DenseRange e
f : β → γ
h_f : UniformContinuous f
inst✝ : CompleteSpace γ
d : Set (γ × γ)
hd : d ∈ 𝓤 γ
s : Set (γ × γ)
hs : s ∈ 𝓤 γ
hs_comp : s ○ (s ○ s) ⊆ d
h_pnt : ∀ {a : α} {m : Set α}, m ∈ 𝓝 a → ∃ c ∈ f '' e ⁻¹' m, (c, ⋯.extend f a) ∈ s ∧ (⋯.extend f a, c) ∈ s
this : Prod.map f f ⁻¹' s ∈ comap (fun x ↦ (e x.1, e x.2)) (𝓤 α)
⊢ Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
inst✝ : UniformSpace α
A B : Set α
hA : IsCompact A
hB : IsClosed[inst✝.toTopologicalSpace] B
h : Disjoint A B
U : Set (α × α)
hU : U ∈ 𝓤 α
hUAB : ⋃ x ∈ A, ball x (id U) ⊆ Bᶜ
V : Set (α × α)
hV : V ∈ 𝓤 α
hVsymm : SetRel.IsSymm V
hVU : V ○ V ⊆ U
x a : α
ha : a ∈ A
hxa : a ∈ ball x V
b : α
hb : b ∈ B
hxb : b ∈ ball x V
⊢ False

Tactic:
]

Goal state after:
α : Type ua
inst✝ : UniformSpace α
A B : Set α
hA : IsCompact A
hB : IsClosed[inst✝.toTopologicalSpace] B
h : Disjoint A B
U : Set (α × α)
hU : U ∈ 𝓤 α
hUAB : ⋃ x ∈ A, ball x (id U) ⊆ Bᶜ
V : Set (α × α)
hV : V ∈ 𝓤 α
hVsymm : SetRel.IsSymm V
hVU : V ○ V ⊆ U
x a : α
ha : a ∈ A
hxa : a ∈ ball x V
b : α
hb : b ∈ B
hxb : b ∈ ball x V
⊢ False
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : One β
f : α → β
K : Set α
hK : IsCompact K
h'K : IsClosed[inst✝¹] K
hfK : ∀ x ∉ K, f x = 1
⊢ mulTSupport f ⊆ closure[inst✝¹] K

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : One β
f : α → β
K : Set α
hK : IsCompact K
h'K : IsClosed[inst✝¹] K
hfK : ∀ x ∉ K, f x = 1
⊢ mulTSupport f ⊆ closure[inst✝¹] K
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : (𝓤 α).IsCountablyGenerated
a : α
⊢ (Filter.comap (Prod.mk a) (𝓤 α)).IsCountablyGenerated

Tactic:
]

Goal state after:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : (𝓤 α).IsCountablyGenerated
a : α
⊢ (Filter.comap (Prod.mk a) (𝓤 α)).IsCountablyGenerated
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
inst✝¹ : UniformSpace α
inst✝ : CompactSpace α
⊢ (Filter.comap (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) (𝓤 α ×ˢ 𝓤 α)).HasBasis (fun U ↦ U ∈ 𝓤 α) fun U ↦
    (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' U ×ˢ U

Tactic:
]

Goal state after:
α : Type ua
inst✝¹ : UniformSpace α
inst✝ : CompactSpace α
⊢ (Filter.comap (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) (𝓤 α ×ˢ 𝓤 α)).HasBasis (fun U ↦ U ∈ 𝓤 α) fun U ↦
    (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' U ×ˢ U
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: case inl
α : Type u_2
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : One β
f : α → β
K : Set α
inst✝ : TopologicalSpace β
hf : Continuous[inst✝², inst✝] f
hk : IsCompact K
h'f : mulSupport f ⊆ K
h2 : range f = f '' K
⊢ IsCompact (f '' K)

Tactic:
]

Goal state after:
case inl
α : Type u_2
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : One β
f : α → β
K : Set α
inst✝ : TopologicalSpace β
hf : Continuous[inst✝², inst✝] f
hk : IsCompact K
h'f : mulSupport f ⊆ K
h2 : range f = f '' K
⊢ IsCompact (f '' K)
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: case inr
α : Type u_2
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : One β
f : α → β
K : Set α
inst✝ : TopologicalSpace β
hf : Continuous[inst✝², inst✝] f
hk : IsCompact K
h'f : mulSupport f ⊆ K
h2 : range f = insert 1 (f '' K)
⊢ IsCompact (insert 1 (f '' K))

Tactic:
]

Goal state after:
case inr
α : Type u_2
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : One β
f : α → β
K : Set α
inst✝ : TopologicalSpace β
hf : Continuous[inst✝², inst✝] f
hk : IsCompact K
h'f : mulSupport f ⊆ K
h2 : range f = insert 1 (f '' K)
⊢ IsCompact (insert 1 (f '' K))
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: γ : Type uc
t : TopologicalSpace γ
inst✝ : CompactSpace γ
u u' : UniformSpace γ
h : u.toTopologicalSpace = t
h' : u'.toTopologicalSpace = t
⊢ CompactSpace γ

Tactic:
]

Goal state after:
γ : Type uc
t : TopologicalSpace γ
inst✝ : CompactSpace γ
u u' : UniformSpace γ
h : u.toTopologicalSpace = t
h' : u'.toTopologicalSpace = t
⊢ CompactSpace γ
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: γ : Type uc
t : TopologicalSpace γ
inst✝ : CompactSpace γ
u u' : UniformSpace γ
h : u.toTopologicalSpace = t
h' : u'.toTopologicalSpace = t
this : CompactSpace γ
⊢ CompactSpace γ

Tactic:
]

Goal state after:
γ : Type uc
t : TopologicalSpace γ
inst✝ : CompactSpace γ
u u' : UniformSpace γ
h : u.toTopologicalSpace = t
h' : u'.toTopologicalSpace = t
this : CompactSpace γ
⊢ CompactSpace γ
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: γ : Type uc
t : TopologicalSpace γ
inst✝ : CompactSpace γ
u u' : UniformSpace γ
h : u.toTopologicalSpace = t
h' : u'.toTopologicalSpace = t
this✝ : CompactSpace γ
this : CompactSpace γ
⊢ ⨆ x, 𝓝 (x, x) = ⨆ x, 𝓝 (x, x)

Tactic:
]

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

Goal state before:
pp: α : Type ua
β : Type ub
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : SetRel α β
hs : ∀ x ∈ sᶜ, ∀ᶠ (y : α × β) in nhds x, y ∈ sᶜ
t : Set β
ht : IsCompact t
⊢ ∀ x ∈ (s.preimage t)ᶜ, ∀ᶠ (y : α) in nhds x, y ∈ (s.preimage t)ᶜ

Tactic:
]

Goal state after:
α : Type ua
β : Type ub
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : SetRel α β
hs : ∀ x ∈ sᶜ, ∀ᶠ (y : α × β) in nhds x, y ∈ sᶜ
t : Set β
ht : IsCompact t
⊢ ∀ x ∈ (s.preimage t)ᶜ, ∀ᶠ (y : α) in nhds x, y ∈ (s.preimage t)ᶜ
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
α' : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace α'
inst✝ : One β
f : α → β
hf : HasCompactMulSupport f
g : α' → α
hg : IsClosedEmbedding g
⊢ IsCompact (closure[inst✝¹] (g ⁻¹' mulSupport f))

Tactic:
]

Goal state after:
α : Type u_2
α' : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace α'
inst✝ : One β
f : α → β
hf : HasCompactMulSupport f
g : α' → α
hg : IsClosedEmbedding g
⊢ IsCompact (closure[inst✝¹] (g ⁻¹' mulSupport f))
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
α' : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace α'
inst✝ : One β
f : α → β
hf : HasCompactMulSupport f
g : α' → α
hg : IsClosedEmbedding g
⊢ g ⁻¹' closure[inst✝²] (g '' g ⁻¹' mulSupport f) ⊆ g ⁻¹' mulTSupport f

Tactic:
]

Goal state after:
α : Type u_2
α' : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace α'
inst✝ : One β
f : α → β
hf : HasCompactMulSupport f
g : α' → α
hg : IsClosedEmbedding g
⊢ g ⁻¹' closure[inst✝²] (g '' g ⁻¹' mulSupport f) ⊆ g ⁻¹' mulTSupport f
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
β : Type u_4
γ : Type u_5
δ : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : One β
inst✝¹ : One γ
inst✝ : One δ
f : α → β
f₂ : α → γ
m : β → γ → δ
hf : f =ᶠ[coclosedCompact α] 1
hf₂ : f₂ =ᶠ[coclosedCompact α] 1
hm : m 1 1 = 1
⊢ (fun x ↦ m (f x) (f₂ x)) =ᶠ[coclosedCompact α] 1

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_4
γ : Type u_5
δ : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : One β
inst✝¹ : One γ
inst✝ : One δ
f : α → β
f₂ : α → γ
m : β → γ → δ
hf : f =ᶠ[coclosedCompact α] 1
hf₂ : f₂ =ᶠ[coclosedCompact α] 1
hm : m 1 1 = 1
⊢ (fun x ↦ m (f x) (f₂ x)) =ᶠ[coclosedCompact α] 1
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α' : Type u_3
β : Type u_4
inst✝³ : TopologicalSpace α'
inst✝² : One β
inst✝¹ : T2Space α'
inst✝ : TopologicalSpace β
U : Set α'
hU : IsOpen[inst✝³] U
f : ↑U → β
cont : Continuous[instTopologicalSpaceSubtype, inst✝] f
supp : HasCompactMulSupport f
x : α'
h : x ∈ mulTSupport (extend Subtype.val f 1)
⊢ ContinuousAt f ⟨x, ⋯⟩

Tactic:
]

Goal state after:
α' : Type u_3
β : Type u_4
inst✝³ : TopologicalSpace α'
inst✝² : One β
inst✝¹ : T2Space α'
inst✝ : TopologicalSpace β
U : Set α'
hU : IsOpen[inst✝³] U
f : ↑U → β
cont : Continuous[instTopologicalSpaceSubtype, inst✝] f
supp : HasCompactMulSupport f
x : α'
h : x ∈ mulTSupport (extend Subtype.val f 1)
⊢ ContinuousAt f ⟨x, ⋯⟩
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
γ : Type u_5
inst✝² : TopologicalSpace α
inst✝¹ : One γ
f : α → γ
inst✝ : TopologicalSpace γ
h : HasCompactMulSupport f
N : Set γ
hN : N ∈ 𝓝 1
⊢ ∃ t, IsCompact t ∧ (f ⁻¹' N)ᶜ ⊆ t

Tactic:
]

Goal state after:
α : Type u_2
γ : Type u_5
inst✝² : TopologicalSpace α
inst✝¹ : One γ
f : α → γ
inst✝ : TopologicalSpace γ
h : HasCompactMulSupport f
N : Set γ
hN : N ∈ 𝓝 1
⊢ ∃ t, IsCompact t ∧ (f ⁻¹' N)ᶜ ⊆ t