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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : UniformSpace α
inst✝³ : UniformSpace β
γ : Type u_3
inst✝² : UniformSpace γ
inst✝¹ : CompleteSpace β
inst✝ : CompleteSpace γ
i : α → β
f : α → γ
hid : IsDenseInducing i
hi : IsUniformInducing i
h : IsUniformInducing f
sf : α → SeparationQuotient γ := SeparationQuotient.mk ∘ f
this : CompleteSpace ↑(closure (range sf))
ff : α → ↑(closure (range sf)) := inclusion ⋯ ∘ rangeFactorization sf
hgu : IsUniformInducing ff
hgd : DenseRange ff
hg : IsDenseInducing ff
fwd : β → ↑(closure (range sf)) := hid.extend ff
hfwd : UniformContinuous fwd
hg' : UniformContinuous (hg.extend i)
key : SeparationQuotient.mk ∘ hg.extend i ∘ fwd = SeparationQuotient.mk
hfu : IsUniformInducing fwd
hrr : range (SeparationQuotient.mk ∘ hid.extend f) ⊆ closure (range (SeparationQuotient.mk ∘ f))
⊢ IsUniformInducing (hid.extend f)

Tactic:
suffices Subtype.val ∘ fwd = SeparationQuotient.mk ∘ hid.extend f by
    rw [← SeparationQuotient.isUniformInducing_mk.of_comp_iff, ← this]
    exact (isUniformInducing_val _).comp hfu

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : UniformSpace α
inst✝³ : UniformSpace β
γ : Type u_3
inst✝² : UniformSpace γ
inst✝¹ : CompleteSpace β
inst✝ : CompleteSpace γ
i : α → β
f : α → γ
hid : IsDenseInducing i
hi : IsUniformInducing i
h : IsUniformInducing f
sf : α → SeparationQuotient γ := SeparationQuotient.mk ∘ f
this : CompleteSpace ↑(closure (range sf))
ff : α → ↑(closure (range sf)) := inclusion ⋯ ∘ rangeFactorization sf
hgu : IsUniformInducing ff
hgd : DenseRange ff
hg : IsDenseInducing ff
fwd : β → ↑(closure (range sf)) := hid.extend ff
hfwd : UniformContinuous fwd
hg' : UniformContinuous (hg.extend i)
key : SeparationQuotient.mk ∘ hg.extend i ∘ fwd = SeparationQuotient.mk
hfu : IsUniformInducing fwd
hrr : range (SeparationQuotient.mk ∘ hid.extend f) ⊆ closure (range (SeparationQuotient.mk ∘ f))
⊢ Subtype.val ∘ fwd = SeparationQuotient.mk ∘ hid.extend f
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
f : Filter α
g : Filter β
ι : Type u_11
ι' : Type u_12
p : ι → Prop
q : ι' → Prop
s : ι → Set α
t : ι' → Set β
m : α → β → γ
hf : f.HasBasis p s
hg : g.HasBasis q t
⊢ (map₂ m f g).HasBasis (fun i ↦ p i.1 ∧ q i.2) fun i ↦ image2 m (s i.1) (t i.2)

Tactic:
simpa only [← map_prod_eq_map₂, ← image_prod] using (hf.prod hg).map _

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s : Set α
p : Filter ι
inst✝ : UniformSpace β
𝔖 : Set (Set α)
h : univ ∈ 𝔖
⊢ ∀ i ∈ 𝓤 β,
    ∀ᶠ (x : (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)) in 𝓤 (α →ᵤ[𝔖] β),
      (UniformFun.ofFun ((toFun 𝔖) x.1), UniformFun.ofFun ((toFun 𝔖) x.2)) ∈ UniformFun.gen α β i

Tactic:
intro U

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s : Set α
p : Filter ι
inst✝ : UniformSpace β
𝔖 : Set (Set α)
h : univ ∈ 𝔖
U : Set (β × β)
⊢ U ∈ 𝓤 β →
    ∀ᶠ (x : (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)) in 𝓤 (α →ᵤ[𝔖] β),
      (UniformFun.ofFun ((toFun 𝔖) x.1), UniformFun.ofFun ((toFun 𝔖) x.2)) ∈ UniformFun.gen α β U
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
β : Type u_8
tX : TopologicalSpace X
uα : UniformSpace α
uβ : UniformSpace β
F : ι → X → α
u : α → β
hu : IsUniformInducing u
⊢ Equicontinuous F ↔ Equicontinuous ((fun x ↦ u ∘ x) ∘ F)

Tactic:
by
  congrm ∀ x, ?_
  rw [hu.equicontinuousAt_iff]

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

Goal state before:
pp: X : Type u_3
Y : Type u_5
α : Type u_6
tX : TopologicalSpace X
tY : TopologicalSpace Y
uα : UniformSpace α
A : Set Y
u : Y → X → α
S : Set X
x₀ : X
hA : EquicontinuousWithinAt (u ∘ Subtype.val) S x₀
hu₁ : Continuous[tY, Pi.topologicalSpace] (S.domRestrict ∘ u)
hu₂ : Continuous[tY, uα.toTopologicalSpace] (eval x₀ ∘ u)
⊢ EquicontinuousWithinAt (u ∘ Subtype.val) S x₀

Tactic:
intro U

Goal state after:
X : Type u_3
Y : Type u_5
α : Type u_6
tX : TopologicalSpace X
tY : TopologicalSpace Y
uα : UniformSpace α
A : Set Y
u : Y → X → α
S : Set X
x₀ : X
hA : EquicontinuousWithinAt (u ∘ Subtype.val) S x₀
hu₁ : Continuous[tY, Pi.topologicalSpace] (S.domRestrict ∘ u)
hu₂ : Continuous[tY, uα.toTopologicalSpace] (eval x₀ ∘ u)
U : Set (α × α)
⊢ U ∈ 𝓤 α → ∀ᶠ (x : X) in 𝓝[S] x₀, ∀ (i : ↑(closure[tY] A)), ((u ∘ Subtype.val) i x₀, (u ∘ Subtype.val) i x) ∈ U
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: Y : Type u_5
α : Type u_6
β : Type u_8
tY : TopologicalSpace Y
uα : UniformSpace α
uβ : UniformSpace β
A : Set Y
u : Y → β → α
S : Set β
hA : UniformEquicontinuousOn (u ∘ Subtype.val) S
hu : Continuous[tY, Pi.topologicalSpace] (S.domRestrict ∘ u)
⊢ UniformEquicontinuousOn (u ∘ Subtype.val) S

Tactic:
intro U

Goal state after:
Y : Type u_5
α : Type u_6
β : Type u_8
tY : TopologicalSpace Y
uα : UniformSpace α
uβ : UniformSpace β
A : Set Y
u : Y → β → α
S : Set β
hA : UniformEquicontinuousOn (u ∘ Subtype.val) S
hu : Continuous[tY, Pi.topologicalSpace] (S.domRestrict ∘ u)
U : Set (α × α)
⊢ U ∈ 𝓤 α →
    ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S),
      ∀ (i : ↑(closure[tY] A)), ((u ∘ Subtype.val) i xy.1, (u ∘ Subtype.val) i xy.2) ∈ U
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 ↦ F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x ↦ F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
⊢ ContinuousWithinAt f S x₀

Tactic:
intro U

Goal state after:
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 ↦ F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x ↦ F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
⊢ U ∈ 𝓝 (f x₀) → U ∈ map f (𝓝[S] x₀)
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
l : Filter ι
inst✝ : l.NeBot
F : ι → β → α
f : β → α
S : Set β
h₁ : ∀ x ∈ S, Tendsto (fun x_1 ↦ F x_1 x) l (𝓝 (f x))
h₂ : UniformEquicontinuousOn F S
⊢ UniformContinuousOn f S

Tactic:
intro U

Goal state after:
ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
l : Filter ι
inst✝ : l.NeBot
F : ι → β → α
f : β → α
S : Set β
h₁ : ∀ x ∈ S, Tendsto (fun x_1 ↦ F x_1 x) l (𝓝 (f x))
h₂ : UniformEquicontinuousOn F S
U : Set (α × α)
⊢ U ∈ 𝓤 α → U ∈ map (fun x ↦ (f x.1, f x.2)) (𝓤 β ⊓ 𝓟 (S ×ˢ S))
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
F : ι → X → α
f : X → α
s : Set X
x : X
z : α
hF : EquicontinuousAt F x
hf : ∀ i ∈ 𝓤 α, ∀ᶠ (x : X) in 𝓝[s] x, f x ∈ {y | (y, z) ∈ id i}
hs : ∀ y ∈ s, Tendsto (fun x ↦ F x y) l (𝓝 (f y))
hx : x ∈ closure[tX] s
⊢ ∀ i ∈ 𝓤 α, ∀ᶠ (x_1 : ι) in l, F x_1 x ∈ {y | (y, z) ∈ id i}

Tactic:
intro U

Goal state after:
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
F : ι → X → α
f : X → α
s : Set X
x : X
z : α
hF : EquicontinuousAt F x
hf : ∀ i ∈ 𝓤 α, ∀ᶠ (x : X) in 𝓝[s] x, f x ∈ {y | (y, z) ∈ id i}
hs : ∀ y ∈ s, Tendsto (fun x ↦ F x y) l (𝓝 (f y))
hx : x ∈ closure[tX] s
U : Set (α × α)
⊢ U ∈ 𝓤 α → ∀ᶠ (x_1 : ι) in l, F x_1 x ∈ {y | (y, z) ∈ id U}
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : TopologicalSpace α
f : α →ᵤ[𝔖] β
x : α
h𝔖 : ∃ V ∈ 𝔖, V ∈ 𝓝 x
hc : ContinuousAt ((toFun 𝔖) f) x
⊢ ∀ s ∈ 𝓤 β,
    ∀ᶠ (a : (α →ᵤ[𝔖] β) × α) in 𝓝 (f, x),
      (Prod.mk ((toFun 𝔖) (f, x).1 (f, x).2) ∘ fun fx ↦ (toFun 𝔖) fx.1 fx.2) a ∈ SetRel.comp s s

Tactic:
intro U

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : TopologicalSpace α
f : α →ᵤ[𝔖] β
x : α
h𝔖 : ∃ V ∈ 𝔖, V ∈ 𝓝 x
hc : ContinuousAt ((toFun 𝔖) f) x
U : Set (β × β)
⊢ U ∈ 𝓤 β →
    ∀ᶠ (a : (α →ᵤ[𝔖] β) × α) in 𝓝 (f, x),
      (Prod.mk ((toFun 𝔖) (f, x).1 (f, x).2) ∘ fun fx ↦ (toFun 𝔖) fx.1 fx.2) a ∈ SetRel.comp U U
Module: Mathlib.Order.Filter.Pointwise

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Monoid γ
inst✝ : MulAction γ β
m : α → β
c : γ
f : Filter α
g : Filter β
hc : IsUnit c
⊢ Tendsto (c • m) f (c • g) ↔ Tendsto m f g

Tactic:
rcases hc.exists_left_inv with ⟨d, hd⟩

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Monoid γ
inst✝ : MulAction γ β
m : α → β
c : γ
f : Filter α
g : Filter β
hc : IsUnit c
d : γ
hd : d * c = 1
⊢ Tendsto (c • m) f (c • g) ↔ Tendsto m f g
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a c : α
h : c < 0
⊢ (fun x ↦ c * x) ⁻¹' Iio a = Ioi (a / c)

Tactic:
simpa only [mul_comm] using preimage_mul_const_Iio_of_neg a h

Goal state after:
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}
y : α
⊢ IsClosed[inst✝⁴] {y}

Tactic:
rcases MulAction.exists_smul_eq G x y with ⟨g, rfl⟩

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] {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}
y : α
⊢ IsOpen[inst✝⁴] {y}

Tactic:
rcases MulAction.exists_smul_eq G x y with ⟨g, rfl⟩

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] {g • x}
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
F : Set G
hF : IsClosed[inst✝²] F
⊢ F * closure[inst✝²] {1} = closure[inst✝²] F * closure[inst✝²] {1}

Tactic:
rw [hF.closure_eq]

Goal state after:
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 : Subgroup G
hS : IsDiscrete ↑S
U : Set G
hU : U ∈ 𝓝 1
hUinv : U⁻¹ = U
h : ∀ g ∈ S, ((fun x ↦ g * x) '' U ∩ U).Nonempty → g = 1
g : G
hgS : g ∈ S
hgU : ((fun x ↦ x * g) '' U ∩ U).Nonempty
⊢ ((fun x ↦ g⁻¹ * x) '' U ∩ U).Nonempty

Tactic:
rwa [Set.nonempty_image_mulLeft_inv_inter_iff, hUinv]

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: A : Type u_1
inst✝⁶ : Group A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousMul A
B : Type u_2
inst✝³ : Group B
inst✝² : TopologicalSpace B
F : Type u_3
inst✝¹ : FunLike F A B
inst✝ : MonoidHomClass F A B
φ : F
hφ : IsQuotientMap ⇑φ
⊢ IsOpenMap ⇑φ

Tactic:
intro U

Goal state after:
A : Type u_1
inst✝⁶ : Group A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousMul A
B : Type u_2
inst✝³ : Group B
inst✝² : TopologicalSpace B
F : Type u_3
inst✝¹ : FunLike F A B
inst✝ : MonoidHomClass F A B
φ : F
hφ : IsQuotientMap ⇑φ
U : Set A
⊢ IsOpen[inst✝⁵] U → IsOpen[inst✝²] (⇑φ '' U)
Module: Mathlib.Topology.Algebra.Group.Basic

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

Tactic:
by
  refine IsTopologicalGroup.of_nhds_one' hmul hinv hleft fun x₀ => ?_
  replace hconj : ∀ x₀ : G, map (x₀ * · * x₀⁻¹) (𝓝 1) = 𝓝 1 :=
    fun x₀ => map_eq_of_inverse (x₀⁻¹ * · * x₀⁻¹⁻¹) (by ext; simp [mul_assoc]) (hconj _) (hconj _)
  rw [← hconj x₀]
  simpa [Function.comp_def] using hleft _

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

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝¹.toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
x₀ : α
y₁ : δ
Nx : Filter α := 𝓝 x₀
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
lim1 : Tendsto (fun a ↦ (a.2 - a.1, y₁)) (comap (⇑e) Nx ×ˢ comap (⇑e) Nx) (𝓝 (0, y₁))
lim2 : Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, y₁)) (𝓝 0)
lim :
  ∀ W ∈ 𝓝 0,
    ∃ U ∈ comap (⇑e) Nx, ∀ (x x' : β), x ∈ U → x' ∈ U → ((fun p ↦ (φ p.1) p.2) ∘ fun a ↦ (a.2 - a.1, y₁)) (x, x') ∈ W
⊢ ∃ U₂ ∈ comap (⇑e) (𝓝 x₀), ∀ (a b : β), a ∈ U₂ → b ∈ U₂ → (φ (b - a)) y₁ ∈ W'

Tactic:
exact lim W' W'_nhds

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

Goal state before:
pp: case left
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : TopologicalSpace δ
inst✝⁵ : AddCommGroup δ
inst✝⁴ : UniformSpace G
inst✝³ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝⁴.toTopologicalSpace] fun p ↦ (φ p.1) p.2
inst✝² : IsUniformAddGroup G
inst✝¹ : T0Space G
inst✝ : CompleteSpace G
x₀ : α
y₀ : γ
⊢ ∀ t ∈ 𝓝 (x₀, y₀), ∃ a, Prod.map (⇑e) (⇑f) a ∈ t

Tactic:
intro U

Goal state after:
case left
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : TopologicalSpace δ
inst✝⁵ : AddCommGroup δ
inst✝⁴ : UniformSpace G
inst✝³ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝⁴.toTopologicalSpace] fun p ↦ (φ p.1) p.2
inst✝² : IsUniformAddGroup G
inst✝¹ : T0Space G
inst✝ : CompleteSpace G
x₀ : α
y₀ : γ
U : Set (α × γ)
⊢ U ∈ 𝓝 (x₀, y₀) → ∃ a, Prod.map (⇑e) (⇑f) a ∈ U
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: case inr.inr
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : Dense (range fun x ↦ a ^ x)
ha₀ : a ≤ 1
this : ∀ {a : G}, Dense (range fun x ↦ a ^ x) → 1 < a → (range fun x ↦ a ^ x) = univ
hlt : a < 1
H : (range fun x ↦ a⁻¹ ^ x) = range fun x ↦ a ^ x
⊢ (range fun x ↦ a⁻¹ ^ x) = univ

Tactic:
apply this <;> simpa only [H, one_lt_inv']

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
⊢ (𝓝[<] 0)⁻¹ = atBot

Tactic:
← inv_atBot₀,

Goal state after:
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
⊢ atBot⁻¹⁻¹ = atBot
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
β : Type v
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : TopologicalSpace β
f : β → α
⊢ Continuous[inst✝, ts] f ↔ ∀ (a : α), IsOpen[inst✝] (f ⁻¹' Ioi a) ∧ IsOpen[inst✝] (f ⁻¹' Iio a)

Tactic:
by
  simp_rw +instances [OrderTopology.topology_eq_generate_intervals, continuous_generateFrom_iff]
  aesop

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
a : α
b b' : β
hb' : b' ∈ {b}
⊢ const α b (const β a b') = b'

Tactic:
by rw [eq_of_mem_singleton hb', const_apply]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
a : AddCircle p
⊢ ∃ b ∈ Ico 0 p, ↑b = a

Tactic:
let b := QuotientAddGroup.equivIcoMod hp.out 0 a

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
a : AddCircle p
b : ↑(Ico 0 (0 + p)) := (QuotientAddGroup.equivIcoMod ⋯ 0) a
⊢ ∃ b ∈ Ico 0 p, ↑b = a
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1.h
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
m : ℕ
⊢ ?refine_1.h.b * addOrderOf ↑(↑m / ↑n * p) = n

pp: case refine_1.h.b
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
m : ℕ
⊢ ℕ

Tactic:
exact gcd_mul_addOrderOf_div_eq p m h

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
hS : S.Exact
A : C
x₂ : A ⟶ S.X₂
hx₂ : x₂ ≫ S.g = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f

Tactic:
by
  rw [ShortComplex.exact_iff_exact_up_to_refinements] at hS
  exact hS x₂ hx₂

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
A : C
x₂ x₂' : A ⟶ S.X₂
hx₂ : x₂ ≫ S.g = 0
hx₂' : x₂' ≫ S.g = 0
this :
  S.liftCycles x₂ hx₂ ≫ S.homologyπ = S.liftCycles x₂' hx₂' ≫ S.homologyπ ↔ S.liftCycles (x₂ - x₂') ⋯ ≫ S.homologyπ = 0
⊢ S.liftCycles x₂ hx₂ ≫ S.homologyπ = S.liftCycles x₂' hx₂' ≫ S.homologyπ ↔
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = π ≫ x₂' + x₁ ≫ S.f

Tactic:
by
    simp [this, S.liftCycles_comp_homologyπ_eq_zero_iff_up_to_refinements,
      sub_eq_iff_eq_add']

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.SnakeLemma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
⊢ Epi S.L₂.op.g

Tactic:
dsimp; infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.SnakeLemma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
⊢ Mono S.L₁.op.f

Tactic:
dsimp; infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
f : J → C
inst✝⁷ : HasProduct f
D : Type t
inst✝⁶ : Category.{r, t} D
FD : D → D → Type u_1
DD : D → Type (max w r)
inst✝⁵ : (X Y : D) → FunLike (FD X Y) (DD X) (DD Y)
inst✝⁴ : ConcreteCategory D FD
F : C ⥤ D
inst✝³ : PreservesLimit (Discrete.functor f) F
inst✝² : HasProduct fun j ↦ F.obj (f j)
inst✝¹ : PreservesLimitsOfShape WalkingCospan (forget D)
inst✝ : PreservesLimit (Discrete.functor fun b ↦ F.obj (f b)) (forget D)
x y : ToType (F.obj (∏ᶜ f))
h : ∀ (i : J), (hom (F.map (Pi.π f i))) x = (hom (F.map (Pi.π f i))) y
j : J
⊢ (hom (piComparison F f ≫ limit.π (Discrete.functor fun b ↦ F.obj (f b)) { as := j })) x =
    (hom (piComparison F f ≫ limit.π (Discrete.functor fun b ↦ F.obj (f b)) { as := j })) y

Tactic:
piComparison_comp_π

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
f : J → C
inst✝⁷ : HasProduct f
D : Type t
inst✝⁶ : Category.{r, t} D
FD : D → D → Type u_1
DD : D → Type (max w r)
inst✝⁵ : (X Y : D) → FunLike (FD X Y) (DD X) (DD Y)
inst✝⁴ : ConcreteCategory D FD
F : C ⥤ D
inst✝³ : PreservesLimit (Discrete.functor f) F
inst✝² : HasProduct fun j ↦ F.obj (f j)
inst✝¹ : PreservesLimitsOfShape WalkingCospan (forget D)
inst✝ : PreservesLimit (Discrete.functor fun b ↦ F.obj (f b)) (forget D)
x y : ToType (F.obj (∏ᶜ f))
h : ∀ (i : J), (hom (F.map (Pi.π f i))) x = (hom (F.map (Pi.π f i))) y
j : J
⊢ (hom (F.map (Pi.π f j))) x = (hom (F.map (Pi.π f j))) y
Module: Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type s
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
J : MulticospanShape
I : MulticospanIndex J C
x : { x // ∀ (i : J.R), (hom (I.fst i)) (x (J.fst i)) = (hom (I.snd i)) (x (J.snd i)) }
⊢ (fun x ↦ ⟨fun x_1 ↦ ↑x (WalkingMulticospan.left x_1), ⋯⟩)
      ((fun x ↦
          ⟨fun j ↦
            match j with
            | WalkingMulticospan.left a => ↑x a
            | WalkingMulticospan.right b => (hom (I.fst b)) (↑x (J.fst b)),
            ⋯⟩)
        x) =
    x

Tactic:
ext i

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type s
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
J : MulticospanShape
I : MulticospanIndex J C
x : { x // ∀ (i : J.R), (hom (I.fst i)) (x (J.fst i)) = (hom (I.snd i)) (x (J.snd i)) }
i : J.L
⊢ ↑((fun x ↦ ⟨fun x_1 ↦ ↑x (WalkingMulticospan.left x_1), ⋯⟩)
          ((fun x ↦
              ⟨fun j ↦
                match j with
                | WalkingMulticospan.left a => ↑x a
                | WalkingMulticospan.right b => (hom (I.fst b)) (↑x (J.fst b)),
                ⋯⟩)
            x))
      i =
    ↑x i
Module: Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer

Goal state before:
pp: case left.right
J : MultispanShape
l : J.L
r' : J.R
T₁ : Type := { u // J.fst l = r' }
T₂ : Type := { u // J.snd l = r' }
⊢ Small.{t, max w w'} (left l ⟶ right r')

Tactic:
let f : T₁ ⊕ T₂ → (left l ⟶ right r') :=
        Sum.elim (fun ⟨_, h⟩ ↦ by subst h; exact Hom.fst l)
          (fun ⟨_, h⟩ ↦ by subst h; exact Hom.snd l)

Goal state after:
case left.right
J : MultispanShape
l : J.L
r' : J.R
T₁ : Type := { u // J.fst l = r' }
T₂ : Type := { u // J.snd l = r' }
f : T₁ ⊕ T₂ → (left l ⟶ right r') :=
  Sum.elim
    (fun x ↦
      match x with
      | ⟨val, h⟩ =>
        (h ▸
            let T₁ := { u // J.fst l = J.fst l };
            let T₂ := { u // J.snd l = J.fst l };
            fun x ↦ Hom.fst l)
          x)
    fun x ↦
    match x with
    | ⟨val, h⟩ =>
      (h ▸
          let T₁ := { u // J.fst l = J.snd l };
          let T₂ := { u // J.snd l = J.snd l };
          fun x ↦ Hom.snd l)
        x
⊢ Small.{t, max w w'} (left l ⟶ right r')
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ComposableArrows C 0
φ φ' : F ⟶ G
h : app' φ 0 homMk₀._proof_1 = app' φ' 0 homMk₀._proof_1
⊢ φ = φ'

Tactic:
ext i

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ComposableArrows C 0
φ φ' : F ⟶ G
h : app' φ 0 homMk₀._proof_1 = app' φ' 0 homMk₀._proof_1
i : Fin (0 + 1)
⊢ φ.app i = φ'.app i
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ComposableArrows C 1
φ φ' : F ⟶ G
h₀ : app' φ 0 homMk₁._proof_4 = app' φ' 0 homMk₁._proof_4
h₁ : app' φ 1 homMk₁._proof_5 = app' φ' 1 homMk₁._proof_5
⊢ φ = φ'

Tactic:
ext i

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ComposableArrows C 1
φ φ' : F ⟶ G
h₀ : app' φ 0 homMk₁._proof_4 = app' φ' 0 homMk₁._proof_4
h₁ : app' φ 1 homMk₁._proof_5 = app' φ' 1 homMk₁._proof_5
i : Fin (1 + 1)
⊢ φ.app i = φ'.app i