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

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s : Set α
p : Filter ι
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : CompleteSpace β
h✝ : Nonempty β
F : Filter (α →ᵤ[𝔖] β)
hF : Cauchy F
this : F.NeBot
g : α → β
hg : ∀ x ∈ ⋃₀ 𝔖, Tendsto (fun x_1 ↦ (toFun 𝔖) x_1 x) F (𝓝 (g x))
⊢ F ≤ 𝓝 ((ofFun 𝔖) g)

Tactic:
[

Goal state after:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s : Set α
p : Filter ι
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : CompleteSpace β
h✝ : Nonempty β
F : Filter (α →ᵤ[𝔖] β)
hF : Cauchy F
this : F.NeBot
g : α → β
hg : ∀ x ∈ ⋃₀ 𝔖, Tendsto (fun x_1 ↦ (toFun 𝔖) x_1 x) F (𝓝 (g x))
⊢ F ≤ 𝓝 ((ofFun 𝔖) g)
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝⁴ : CommMonoid α
inst✝³ : LinearOrder α
inst✝² : IsOrderedMonoid α
inst✝¹ : MulLeftReflectLE α
inst✝ : ExistsMulOfLE α
a b c : α
⊢ a • Icc b c = Icc (a * b) (a * c)

Tactic:
by
  ext x
  constructor
  · rintro ⟨y, ⟨hby, hyc⟩, rfl⟩
    dsimp
    constructor <;> gcongr
  · rintro ⟨habx, hxac⟩
    obtain ⟨y, hy, rfl⟩ := exists_one_le_mul_of_le habx
    refine ⟨b * y, ⟨le_mul_of_one_le_right' hy, ?_⟩, (mul_assoc ..).symm⟩
    rwa [mul_assoc, mul_le_mul_iff_left] at hxac

Goal state after:
Module: Mathlib.Order.Filter.Pointwise

Goal state before:
pp: α : Type u_2
inst✝ : Monoid α
f : Filter α
hf : 1 ≤ f
s : Set α
⊢ s ∈ f * ⊤ → s ∈ ⊤

Tactic:
simp only [mem_mul, mem_top, exists_eq_left]

Goal state after:
α : Type u_2
inst✝ : Monoid α
f : Filter α
hf : 1 ≤ f
s : Set α
⊢ (∃ t₁ ∈ f, t₁ * univ ⊆ s) → s = univ
Module: Mathlib.Order.Filter.Pointwise

Goal state before:
pp: α : Type u_2
inst✝ : Monoid α
f : Filter α
hf : 1 ≤ f
s : Set α
⊢ s ∈ ⊤ * f → s ∈ ⊤

Tactic:
simp only [mem_mul, mem_top, exists_eq_left]

Goal state after:
α : Type u_2
inst✝ : Monoid α
f : Filter α
hf : 1 ≤ f
s : Set α
⊢ (∃ t₂ ∈ f, univ * t₂ ⊆ s) → s = univ
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c : α
⊢ (fun x ↦ a - x) ⁻¹' [[b, c]] = [[a - b, a - c]]

Tactic:
[

Goal state after:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c : α
⊢ (fun x ↦ a - x) ⁻¹' [[b, c]] = [[a - b, a - c]]
Module: Mathlib.Order.Filter.Pointwise

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
ε : Type u_6
inst✝¹ : Monoid α
inst✝ : MulAction α β
f : Filter β
⊢ map (fun x2 ↦ 1 • x2) f = f

Tactic:
[

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
ε : Type u_6
inst✝¹ : Monoid α
inst✝ : MulAction α β
f : Filter β
⊢ map (fun x2 ↦ 1 • x2) f = f
Module: Mathlib.Order.Filter.Pointwise

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝² : Zero α
inst✝¹ : Zero β
inst✝ : SMulWithZero α β
g : Filter β
hg : g.NeBot
s : Set β
hs : s ∈ g
⊢ (fun x ↦ 0 • x) '' s ∈ 0

Tactic:
[

Goal state after:
α : Type u_2
β : Type u_3
inst✝² : Zero α
inst✝¹ : Zero β
inst✝ : SMulWithZero α β
g : Filter β
hg : g.NeBot
s : Set β
hs : s ∈ g
⊢ (fun x ↦ 0 • x) '' s ∈ 0
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
δ₁ : Type u_5
δ₂ : Type u_6
φ₁ : δ₁ → α
φ₂ : δ₂ → α
𝔗₁ : Set (Set δ₁)
𝔗₂ : Set (Set δ₂)
h_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖
h_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖
h_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁
h_preimage₂ : MapsTo (fun x ↦ φ₂ ⁻¹' x) 𝔖 𝔗₂
h_cover : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂
ψ₁ : (S : Set α) → ↑(φ₁ ⁻¹' S) → ↑S := fun S ↦ S.restrictPreimage φ₁
ψ₂ : (S : Set α) → ↑(φ₂ ⁻¹' S) → ↑S := fun S ↦ S.restrictPreimage φ₂
this :
  ∀ S ∈ 𝔖,
    UniformFun.uniformSpace (↑S) β =
      UniformSpace.comap (fun x ↦ x ∘ ψ₁ S) (UniformFun.uniformSpace (↑(φ₁ ⁻¹' S)) β) ⊓
        UniformSpace.comap (fun x ↦ x ∘ ψ₂ S) (UniformFun.uniformSpace (↑(φ₂ ⁻¹' S)) β)
S : Set α
hS : S ∈ 𝔖
⊢ UniformSpace.comap (⇑(ofFun 𝔗₁) ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑(toFun 𝔖)) (uniformSpace δ₁ β 𝔗₁) ⊓
      UniformSpace.comap (⇑(ofFun 𝔗₂) ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑(toFun 𝔖)) (uniformSpace δ₂ β 𝔗₂) ≤
    UniformSpace.comap (⇑UniformFun.ofFun ∘ S.domRestrict ∘ ⇑(toFun 𝔖)) (UniformFun.uniformSpace (↑S) β)

Tactic:
[

Goal state after:
case refine_3
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
δ₁ : Type u_5
δ₂ : Type u_6
φ₁ : δ₁ → α
φ₂ : δ₂ → α
𝔗₁ : Set (Set δ₁)
𝔗₂ : Set (Set δ₂)
h_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖
h_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖
h_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁
h_preimage₂ : MapsTo (fun x ↦ φ₂ ⁻¹' x) 𝔖 𝔗₂
h_cover : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂
ψ₁ : (S : Set α) → ↑(φ₁ ⁻¹' S) → ↑S := fun S ↦ S.restrictPreimage φ₁
ψ₂ : (S : Set α) → ↑(φ₂ ⁻¹' S) → ↑S := fun S ↦ S.restrictPreimage φ₂
this :
  ∀ S ∈ 𝔖,
    UniformFun.uniformSpace (↑S) β =
      UniformSpace.comap (fun x ↦ x ∘ ψ₁ S) (UniformFun.uniformSpace (↑(φ₁ ⁻¹' S)) β) ⊓
        UniformSpace.comap (fun x ↦ x ∘ ψ₂ S) (UniformFun.uniformSpace (↑(φ₂ ⁻¹' S)) β)
S : Set α
hS : S ∈ 𝔖
⊢ UniformSpace.comap (⇑(ofFun 𝔗₁) ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑(toFun 𝔖)) (uniformSpace δ₁ β 𝔗₁) ⊓
      UniformSpace.comap (⇑(ofFun 𝔗₂) ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑(toFun 𝔖)) (uniformSpace δ₂ β 𝔗₂) ≤
    UniformSpace.comap (⇑UniformFun.ofFun ∘ S.domRestrict ∘ ⇑(toFun 𝔖)) (UniformFun.uniformSpace (↑S) β)
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
𝔖 : Set (Set α)
δ : ι → Type u_5
φ : (i : ι) → δ i → α
𝔗 : (i : ι) → Set (Set (δ i))
h_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖
h_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)
h_cover : ∀ S ∈ 𝔖, ∃ I, I.Finite ∧ S ⊆ ⋃ i ∈ I, range (φ i)
ψ : (S : Set α) → (i : ι) → ↑(φ i ⁻¹' S) → ↑S := fun S i ↦ S.restrictPreimage (φ i)
this :
  ∀ S ∈ 𝔖,
    UniformFun.uniformSpace (↑S) β =
      ⨅ i, UniformSpace.comap (fun x ↦ x ∘ ψ S i) (UniformFun.uniformSpace (↑(φ i ⁻¹' S)) β)
S : Set α
hS : S ∈ 𝔖
⊢ ⨅ i, UniformSpace.comap (⇑(ofFun (𝔗 i)) ∘ (fun x ↦ x ∘ φ i) ∘ ⇑(toFun 𝔖)) (uniformSpace (δ i) β (𝔗 i)) ≤
    UniformSpace.comap (⇑UniformFun.ofFun ∘ S.domRestrict ∘ ⇑(toFun 𝔖)) (UniformFun.uniformSpace (↑S) β)

Tactic:
[

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
𝔖 : Set (Set α)
δ : ι → Type u_5
φ : (i : ι) → δ i → α
𝔗 : (i : ι) → Set (Set (δ i))
h_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖
h_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)
h_cover : ∀ S ∈ 𝔖, ∃ I, I.Finite ∧ S ⊆ ⋃ i ∈ I, range (φ i)
ψ : (S : Set α) → (i : ι) → ↑(φ i ⁻¹' S) → ↑S := fun S i ↦ S.restrictPreimage (φ i)
this :
  ∀ S ∈ 𝔖,
    UniformFun.uniformSpace (↑S) β =
      ⨅ i, UniformSpace.comap (fun x ↦ x ∘ ψ S i) (UniformFun.uniformSpace (↑(φ i ⁻¹' S)) β)
S : Set α
hS : S ∈ 𝔖
⊢ ⨅ i, UniformSpace.comap (⇑(ofFun (𝔗 i)) ∘ (fun x ↦ x ∘ φ i) ∘ ⇑(toFun 𝔖)) (uniformSpace (δ i) β (𝔗 i)) ≤
    UniformSpace.comap (⇑UniformFun.ofFun ∘ S.domRestrict ∘ ⇑(toFun 𝔖)) (UniformFun.uniformSpace (↑S) β)
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: α : Type u_2
G : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul G α
s : Set G
t : Set α
ht : IsOpen[inst✝³] t
⊢ IsOpen[inst✝³] (s • t)

Tactic:
← iUnion_smul_set

Goal state after:
α : Type u_2
G : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul G α
s : Set G
t : Set α
ht : IsOpen[inst✝³] t
⊢ IsOpen[inst✝³] (⋃ a ∈ s, a • t)
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: α : Type u_2
G : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : ContinuousConstSMul G α
x : α
inst✝ : IsPretransitive G α
hx : IsClosed[inst✝⁴] {x}
g : G
⊢ IsClosed[inst✝⁴] {g • x}

Tactic:
← image_singleton,

Goal state after:
α : Type u_2
G : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : ContinuousConstSMul G α
x : α
inst✝ : IsPretransitive G α
hx : IsClosed[inst✝⁴] {x}
g : G
⊢ IsClosed[inst✝⁴] (HSMul.hSMul g '' {x})
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: α : Type u_2
G : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : ContinuousConstSMul G α
x : α
inst✝ : IsPretransitive G α
hx : IsOpen[inst✝⁴] {x}
g : G
⊢ IsOpen[inst✝⁴] {g • x}

Tactic:
← image_singleton,

Goal state after:
α : Type u_2
G : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : ContinuousConstSMul G α
x : α
inst✝ : IsPretransitive G α
hx : IsOpen[inst✝⁴] {x}
g : G
⊢ IsOpen[inst✝⁴] (HSMul.hSMul g '' {x})
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: case 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
c : G₀
s : Set E
hc : c ≠ 0
⊢ closure[inst✝²] (c • s) = c • closure[inst✝²] s

Tactic:
· exact closure_smul₀' hc s

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

Goal state before:
pp: Γ : Type u_4
T : Type u_5
inst✝² : TopologicalSpace T
inst✝¹ : SMul Γ T
inst✝ : ProperlyDiscontinuousSMul Γ T
x : T
⊢ {γ | γ • x = x}.Finite

Tactic:
[

Goal state after:
Γ : Type u_4
T : Type u_5
inst✝² : TopologicalSpace T
inst✝¹ : SMul Γ T
inst✝ : ProperlyDiscontinuousSMul Γ T
x : T
⊢ {γ | γ • x = x}.Finite
Module: Mathlib.Topology.Algebra.IsUniformGroup.Constructions

Goal state before:
pp: G : Type u_1
H : Type u_2
hom : Type u_3
inst✝⁶ : Group G
inst✝⁵ : Group H
inst✝⁴ : UniformSpace G
inst✝³ : UniformSpace H
inst✝² : IsUniformGroup H
inst✝¹ : FunLike hom G H
inst✝ : MonoidHomClass hom G H
f : hom
hf : IsUniformInducing ⇑f
⊢ UniformContinuous fun p ↦ p.1 / p.2

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
hom : Type u_3
inst✝⁶ : Group G
inst✝⁵ : Group H
inst✝⁴ : UniformSpace G
inst✝³ : UniformSpace H
inst✝² : IsUniformGroup H
inst✝¹ : FunLike hom G H
inst✝ : MonoidHomClass hom G H
f : hom
hf : IsUniformInducing ⇑f
⊢ UniformContinuous fun p ↦ p.1 / p.2
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: case mp
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous[inst✝¹, inst✝] f
H : ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
𝒰 : Ultrafilter X
y : Y
hY : ↑(Ultrafilter.map f 𝒰) ≤ 𝓝 y
⊢ ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x

Tactic:
[

Goal state after:
case mp
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous[inst✝¹, inst✝] f
H : ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
𝒰 : Ultrafilter X
y : Y
hY : ↑(Ultrafilter.map f 𝒰) ≤ 𝓝 y
⊢ ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: case mpr
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous[inst✝¹, inst✝] f
H : ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
⊢ ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ

Tactic:
[

Goal state after:
case mpr
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous[inst✝¹, inst✝] f
H : ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
⊢ ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
Module: Mathlib.Topology.Algebra.Monoid

Goal state before:
pp: M : Type u
inst✝¹ : CommMonoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
x₀ : M
⊢ 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)

Tactic:
[

Goal state after:
M : Type u
inst✝¹ : CommMonoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
x₀ : M
⊢ 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : IsProperMap f
hg : IsProperMap g
⊢ IsProperMap (Prod.map f g)

Tactic:
[

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : IsProperMap f
hg : IsProperMap g
⊢ IsProperMap (Prod.map f g)
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: case right
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : Continuous[inst✝³, inst✝²] f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
hg : Continuous[inst✝¹, inst✝] g ∧ ∀ ⦃𝒰 : Ultrafilter Z⦄ ⦃y : W⦄, Tendsto g (↑𝒰) (𝓝 y) → ∃ x, g x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter (X × Z)
y : Y
w : W
hyw : Tendsto (Prod.map f g) (↑𝒰) (𝓝 (y, w))
⊢ ∃ x, Prod.map f g x = (y, w) ∧ ↑𝒰 ≤ 𝓝 x

Tactic:
[

Goal state after:
case right
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : Continuous[inst✝³, inst✝²] f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
hg : Continuous[inst✝¹, inst✝] g ∧ ∀ ⦃𝒰 : Ultrafilter Z⦄ ⦃y : W⦄, Tendsto g (↑𝒰) (𝓝 y) → ∃ x, g x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter (X × Z)
y : Y
w : W
hyw : Tendsto (Prod.map f g) (↑𝒰) (𝓝 (y, w))
⊢ ∃ x, Prod.map f g x = (y, w) ∧ ↑𝒰 ≤ 𝓝 x
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: ι : Type u_5
X : ι → Type u_6
Y : ι → Type u_7
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : (i : ι) → TopologicalSpace (Y i)
f : (i : ι) → X i → Y i
h : ∀ (i : ι), IsProperMap (f i)
⊢ IsProperMap fun x i ↦ f i (x i)

Tactic:
[

Goal state after:
ι : Type u_5
X : ι → Type u_6
Y : ι → Type u_7
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : (i : ι) → TopologicalSpace (Y i)
f : (i : ι) → X i → Y i
h : ∀ (i : ι), IsProperMap (f i)
⊢ IsProperMap fun x i ↦ f i (x i)
Module: Mathlib.Topology.Algebra.Monoid

Goal state before:
pp: M : Type u_3
inst✝⁴ : GroupWithZero M
inst✝³ : TopologicalSpace M
inst✝² : ContinuousMul M
inst✝¹ : CompactSpace M
inst✝ : T1Space M
U : Set M
hU : IsOpen[inst✝³] U
h0U : 0 ∈ U
h1U : 1 ∉ U
W : Set M
hW : W ∈ 𝓝 0
hW' : univ * W ⊆ U
H : ¬∃ x, x ≠ 0 ∧ x ∈ W
⊢ W ⊆ {0}

Tactic:
by simpa [not_imp_not] using H

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : T1Space Y
⊢ IsProperMap f ↔ Continuous[inst✝², inst✝¹] f ∧ IsClosedMap f ∧ Tendsto f (cocompact X) cofinite

Tactic:
[

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : T1Space Y
⊢ IsProperMap f ↔ Continuous[inst✝², inst✝¹] f ∧ IsClosedMap f ∧ Tendsto f (cocompact X) cofinite
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: α : Type u
β : Type v
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : Group α
inst✝² : MulAction α β
inst✝¹ : ContinuousInv α
inst✝ : ContinuousSMul α β
s : Set α
t : Set β
ht : IsClosed[inst✝⁴] t
hs : IsCompact s
Φ : ↑s × β ≃ₜ ↑s × β :=
  { toFun := fun gx ↦ (gx.1, ↑gx.1 • gx.2), invFun := fun gx ↦ (gx.1, (↑gx.1)⁻¹ • gx.2), left_inv := ⋯, right_inv := ⋯,
    continuous_toFun := ⋯, continuous_invFun := ⋯ }
this : s • t = snd ∘ ⇑Φ '' snd ⁻¹' t
⊢ IsClosed[inst✝⁴] (s • t)

Tactic:
this

Goal state after:
α : Type u
β : Type v
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : Group α
inst✝² : MulAction α β
inst✝¹ : ContinuousInv α
inst✝ : ContinuousSMul α β
s : Set α
t : Set β
ht : IsClosed[inst✝⁴] t
hs : IsCompact s
Φ : ↑s × β ≃ₜ ↑s × β :=
  { toFun := fun gx ↦ (gx.1, ↑gx.1 • gx.2), invFun := fun gx ↦ (gx.1, (↑gx.1)⁻¹ • gx.2), left_inv := ⋯, right_inv := ⋯,
    continuous_toFun := ⋯, continuous_invFun := ⋯ }
this : s • t = snd ∘ ⇑Φ '' snd ⁻¹' t
⊢ IsClosed[inst✝⁴] (snd ∘ ⇑Φ '' snd ⁻¹' t)
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
s s' : Set G
hs₀ : s ∈ 𝓝 1
h_symm : ∀ x ∈ s, x⁻¹ ∈ s
y : G
hy : y ∈ closure[inst✝²] s'
⊢ y ∈ s * s'

Tactic:
obtain ⟨_, ⟨b, hb, rfl⟩, hc⟩ :=
    mem_closure_iff_nhds.mp hy ((· * y) '' s)
      (by simpa using (isOpenMap_mul_right y).image_mem_nhds hs₀)

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
s s' : Set G
hs₀ : s ∈ 𝓝 1
h_symm : ∀ x ∈ s, x⁻¹ ∈ s
y : G
hy : y ∈ closure[inst✝²] s'
b : G
hb : b ∈ s
hc : (fun x ↦ x * y) b ∈ s'
⊢ y ∈ s * s'
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
hs : IsOpen[inst✝²] s
t : Set G
⊢ s / closure[inst✝²] t = s / t

Tactic:
[

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
hs : IsOpen[inst✝²] s
t : Set G
⊢ s / closure[inst✝²] t = s / t
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
t : Set G
ht : IsOpen[inst✝²] t
s : Set G
⊢ closure[inst✝²] s / t = s / t

Tactic:
[

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
t : Set G
ht : IsOpen[inst✝²] t
s : Set G
⊢ closure[inst✝²] s / t = s / t
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: Gᵣ : Type u_3
inst✝² : UniformSpace Gᵣ
inst✝¹ : Group Gᵣ
inst✝ : IsRightUniformGroup Gᵣ
⊢ 𝓤 Gᵣ = comap (fun x ↦ x.2 / x.1) (𝓝 1)

Tactic:
[

Goal state after:
Gᵣ : Type u_3
inst✝² : UniformSpace Gᵣ
inst✝¹ : Group Gᵣ
inst✝ : IsRightUniformGroup Gᵣ
⊢ 𝓤 Gᵣ = comap (fun x ↦ x.2 / x.1) (𝓝 1)
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: case cons
α : Type u_3
β : Type u_4
ι : Type u_5
inst✝³ : UniformSpace α
inst✝² : CommGroup α
inst✝¹ : IsUniformGroup α
inst✝ : UniformSpace β
f : ι → β → α
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, UniformContinuous (f i)) → UniformContinuous fun x ↦ ∏ i ∈ s, f i x
h : ∀ i ∈ cons a s ha, UniformContinuous (f i)
⊢ UniformContinuous fun x ↦ ∏ i ∈ cons a s ha, f i x

Tactic:
[

Goal state after:
case cons
α : Type u_3
β : Type u_4
ι : Type u_5
inst✝³ : UniformSpace α
inst✝² : CommGroup α
inst✝¹ : IsUniformGroup α
inst✝ : UniformSpace β
f : ι → β → α
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, UniformContinuous (f i)) → UniformContinuous fun x ↦ ∏ i ∈ s, f i x
h : ∀ i ∈ cons a s ha, UniformContinuous (f i)
⊢ UniformContinuous fun x ↦ ∏ i ∈ cons a s ha, f i x
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
⊢ UniformContinuous fun x ↦ f x ^ 0

Tactic:
[

Goal state after:
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
⊢ UniformContinuous fun x ↦ f x ^ 0
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 + 1)

Tactic:
[

Goal state after:
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 + 1)
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:
[

Goal state after:
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
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 ^ Int.negSucc n

Tactic:
[

Goal state after:
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 ^ Int.negSucc n
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
ι : Sort u_1
p : ι → Prop
s : ι → Set G
hb : (𝓝 1).HasBasis p s
x : G
⊢ (Filter.comap (fun x_1 ↦ x_1 * x⁻¹) (𝓝 1)).HasBasis p fun i ↦ {y | y / x ∈ s i}

Tactic:
[

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
ι : Sort u_1
p : ι → Prop
s : ι → Set G
hb : (𝓝 1).HasBasis p s
x : G
⊢ (Filter.comap (fun x_1 ↦ x_1 * x⁻¹) (𝓝 1)).HasBasis p fun i ↦ {y | y / x ∈ s i}
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x : G
s : Set G
⊢ (∀ i ∈ 𝓝 1, ∃ y ∈ s, y ∈ {y | y / x ∈ id i}) ↔ ∀ U ∈ 𝓝 1, ∃ y ∈ s, y / x ∈ U

Tactic:
[

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x : G
s : Set G
⊢ (∀ i ∈ 𝓝 1, ∃ y ∈ s, y ∈ {y | y / x ∈ id i}) ↔ ∀ U ∈ 𝓝 1, ∃ y ∈ s, y / x ∈ U
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
α : Type u
inst✝⁵ : TopologicalSpace G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : TopologicalSpace α
inst✝¹ : Zero α
inst✝ : T1Space α
f : G → α
k : Set G
hk : IsCompact k
hf : support f ⊆ k
h'f : Continuous[inst✝⁵, inst✝²] f
h : ¬f = 0
⊢ LocallyCompactSpace G

Tactic:
[

Goal state after:
G : Type w
α : Type u
inst✝⁵ : TopologicalSpace G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : TopologicalSpace α
inst✝¹ : Zero α
inst✝ : T1Space α
f : G → α
k : Set G
hk : IsCompact k
hf : support f ⊆ k
h'f : Continuous[inst✝⁵, inst✝²] f
h : ¬f = 0
⊢ LocallyCompactSpace G
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: β : Type u_2
inst✝³ : UniformSpace β
inst✝² : Group β
inst✝¹ : IsLeftUniformGroup β
inst✝ : IsRightUniformGroup β
⊢ UniformContinuous fun p ↦ p.1 / p.2

Tactic:
let φ : (β × β) × (β × β) → β := fun ⟨⟨x₁, x₂⟩, ⟨y₁, y₂⟩⟩ ↦ x₂ * y₂⁻¹ * y₁ * x₁⁻¹

Goal state after:
β : Type u_2
inst✝³ : UniformSpace β
inst✝² : Group β
inst✝¹ : IsLeftUniformGroup β
inst✝ : IsRightUniformGroup β
φ : (β × β) × β × β → β :=
  fun x ↦
    match x with
    | ((x₁, x₂), y₁, y₂) => x₂ * y₂⁻¹ * y₁ * x₁⁻¹
⊢ UniformContinuous fun p ↦ p.1 / p.2
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
α : Type u
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
b c : G
f : α → G
l : Filter α
h : Tendsto (fun k ↦ b / f k) l (𝓝 (b / c))
⊢ Tendsto f l (𝓝 c)

Tactic:
convert! h.inv.mul_const b with k

Goal state after:
case e'_3
G : Type w
α : Type u
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
b c : G
f : α → G
l : Filter α
h : Tendsto (fun k ↦ b / f k) l (𝓝 (b / c))
k : α
⊢ f k = (b / f k)⁻¹ * b

case e'_5
G : Type w
α : Type u
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
b c : G
f : α → G
l : Filter α
h : Tendsto (fun k ↦ b / f k) l (𝓝 (b / c))
⊢ c = (b / c)⁻¹ * b