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

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : HereditarilyLindelofSpace X
ι : Type u_2
C : ι → Set X
h : ∀ (i : ι), IsClosed[inst✝¹] (C i)
⊢ ∃ t, t.Countable ∧ ⋃ i ∈ t, (C i)ᶜ = ⋃ i, (C i)ᶜ

Tactic:
exact eq_open_union_countable (fun i ↦ (C i)ᶜ) (fun i ↦ (h i).isOpen_compl)

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

Goal state before:
pp: case mp
α : Type u
inst✝ : TopologicalSpace α
h : TotallyDisconnectedSpace α
x : α
⊢ (connectedComponent x).Subsingleton

Tactic:
apply h.1

Goal state after:
case mp.a
α : Type u
inst✝ : TopologicalSpace α
h : TotallyDisconnectedSpace α
x : α
⊢ connectedComponent x ⊆ univ

case mp.a
α : Type u
inst✝ : TopologicalSpace α
h : TotallyDisconnectedSpace α
x : α
⊢ IsPreconnected (connectedComponent x)
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsGδ s → ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ s = ⋂ n, f n

Tactic:
rintro ⟨T, hT, T_count, rfl⟩

Goal state after:
case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
T : Set (Set X)
hT : ∀ t ∈ T, IsOpen[inst✝] t
T_count : T.Countable
⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ T = ⋂ n, f n
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_1.inr
X : Type u_1
inst✝ : TopologicalSpace X
T : Set (Set X)
hT✝ : ∀ t ∈ T, IsOpen[inst✝] t
T_count : T.Countable
hT : T.Nonempty
⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ T = ⋂ n, f n

Tactic:
obtain ⟨f, hf⟩ : ∃ (f : ℕ → Set X), T = range f := Countable.exists_eq_range T_count hT
      exact ⟨f, by simp_all, by simp [hf]⟩

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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsNowhereDense s ↔ ∀ x ∈ s, closure[inst✝] s ∉ 𝓝 x

Tactic:
simp [isNowhereDense_iff_disjoint, disjoint_iff_inter_eq_empty, eq_empty_iff_forall_notMem,
    mem_interior_iff_mem_nhds]

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
⊢ Disjoint (f ⁻¹' T₁) (f ⁻¹' T₂)

Tactic:
T₁_u,

Goal state after:
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
⊢ Disjoint (f ⁻¹' t ∩ u) (f ⁻¹' T₂)
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
h : Disjoint (connectedComponent a) (connectedComponent b)
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v

Tactic:
connectedComponent_eq_iInter_isClopen b,

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
h : Disjoint (connectedComponent a) (⋂ s, ↑s)
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Homeomorph.Lemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set Y
h : X ≃ₜ Y
⊢ IsPreconnected (⇑h.symm '' s) ↔ IsPreconnected s

Tactic:
isPreconnected_image

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set Y
h : X ≃ₜ Y
⊢ IsPreconnected s ↔ IsPreconnected s
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: β : Type v
u v : UniformSpace β
F : Filter β
⊢ Cauchy F ↔ Cauchy F ∧ Cauchy F

Tactic:
unfold Cauchy
  rw [inf_uniformity (u := u), le_inf_iff, and_and_left]

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

Goal state before:
pp: α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : UniformSpace β
f : Filter α
m : α → β
hf : Cauchy f
s : Set α
hm : UniformContinuousOn m s
hfs : f ≤ 𝓟 s
⊢ Cauchy (Filter.comap Subtype.val f)

Tactic:
exact hf.comap' (fun _ x ↦ x) (comap_coe_neBot_of_le_principal (h := hf.1) hfs)

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

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
s : Set α
⊢ IsComplete s ↔ ∀ (l : Ultrafilter α), Cauchy ↑l → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x

Tactic:
refine ⟨fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => ?_⟩

Goal state after:
α : Type u
uniformSpace : UniformSpace α
s : Set α
H : ∀ (l : Ultrafilter α), Cauchy ↑l → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x
l : Filter α
hl : Cauchy l
hls : l ≤ 𝓟 s
⊢ ∃ x ∈ s, ClusterPt x l
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:
by
  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
f : α → β
u₁ : UniformSpace α
u₂ : Set (UniformSpace β)
⊢ Tendsto (fun x ↦ (f x.1, f x.2)) (𝓤 α) (𝓤 β) ↔ ∀ u ∈ u₂, Tendsto (fun x ↦ (f x.1, f x.2)) (𝓤 α) (𝓤 β)

Tactic:
rw [sInf_eq_iInf', iInf_uniformity, tendsto_iInf, SetCoe.forall]

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

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
S : Set (Filter α)
hS : S.Finite
⊢ (sSup S).TotallyBounded ↔ ∀ f ∈ S, f.TotallyBounded

Tactic:
sSup_eq_iSup,

Goal state after:
α : Type u
uniformSpace : UniformSpace α
S : Set (Filter α)
hS : S.Finite
⊢ (⨆ a ∈ S, a).TotallyBounded ↔ ∀ f ∈ S, f.TotallyBounded
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
s : Set β
⊢ ContinuousOn f s ↔ ∀ b ∈ s, Tendsto (fun x ↦ (f x, f b)) (𝓝[s] b) (𝓤 α)

Tactic:
simp [ContinuousOn, continuousWithinAt_iff'_left]

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] (mulSupport (g ∘ f)) ⊆ f ⁻¹' closure[inst✝] (mulSupport g)

Tactic:
mulSupport_comp_eq_preimage

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.UniformSpace.Cauchy

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

Tactic:
have := fun n => (tC n).to_subtype

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

Goal state before:
pp: X : Type u_1
R : Type u_8
ι : Type u_9
inst✝¹ : TopologicalSpace X
U : ι → Set X
inst✝ : One R
f : ι → X → R
hlf : LocallyFinite fun i ↦ mulSupport (f i)
hso : ∀ (i : ι), mulTSupport (f i) ⊆ U i
ho : ∀ (i : ι), IsOpen[inst✝¹] (U i)
x : X
n : Set X
hn : n ∈ 𝓝 x
hnf : {i | ((fun i ↦ mulSupport (f i)) i ∩ n).Nonempty}.Finite
is : Finset ι := {i ∈ hnf.toFinset | x ∈ U i}
js : Finset ι := {j ∈ hnf.toFinset | x ∉ U j}
z : X
hzn : z ∈ n
hz : ∀ (i : ι), (mulSupport (f i) ∩ n).Nonempty → x ∉ U i → z ∈ (mulTSupport (f i))ᶜ
this : (mulSupport fun i ↦ f i z) ⊆ ↑hnf.toFinset
⊢ (mulSupport fun i ↦ f i z) ⊆ ↑is

Tactic:
refine hnf.toFinset.subset_coe_filter_of_subset_forall _ this fun i hi => ?_

Goal state after:
X : Type u_1
R : Type u_8
ι : Type u_9
inst✝¹ : TopologicalSpace X
U : ι → Set X
inst✝ : One R
f : ι → X → R
hlf : LocallyFinite fun i ↦ mulSupport (f i)
hso : ∀ (i : ι), mulTSupport (f i) ⊆ U i
ho : ∀ (i : ι), IsOpen[inst✝¹] (U i)
x : X
n : Set X
hn : n ∈ 𝓝 x
hnf : {i | ((fun i ↦ mulSupport (f i)) i ∩ n).Nonempty}.Finite
is : Finset ι := {i ∈ hnf.toFinset | x ∈ U i}
js : Finset ι := {j ∈ hnf.toFinset | x ∉ U j}
z : X
hzn : z ∈ n
hz : ∀ (i : ι), (mulSupport (f i) ∩ n).Nonempty → x ∉ U i → z ∈ (mulTSupport (f i))ᶜ
this : (mulSupport fun i ↦ f i z) ⊆ ↑hnf.toFinset
i : ι
hi : i ∈ mulSupport fun i ↦ f i z
⊢ x ∈ U i
Module: Mathlib.Topology.UniformSpace.LocallyUniformConvergence

Goal state before:
pp: case mpr
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
⊢ (∀ x ∈ s,
      ∀ u ∈ 𝓤 β,
        ∃ pa,
          (∀ᶠ (x : ι) in p, pa x) ∧
            ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u) →
    TendstoLocallyUniformlyOn F f p s

Tactic:
· rintro h u hu x hx
    obtain ⟨pa, hpa, pb, hpb, h⟩ := h x hx u hu
    exact ⟨{a | pb a}, hpb, eventually_of_mem hpa fun i hi y hy => h hi hy⟩

Goal state after:
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
F : ι → β → α
h : UniformEquicontinuous F
⊢ UniformEquicontinuous Subtype.val

Tactic:
rw [← comp_rangeSplitting F]

Goal state after:
ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
F : ι → β → α
h : UniformEquicontinuous F
⊢ UniformEquicontinuous (F ∘ rangeSplitting F)
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
F : ι → β → α
S : Set β
h : UniformEquicontinuousOn F S
⊢ UniformEquicontinuousOn Subtype.val S

Tactic:
rw [← comp_rangeSplitting F]

Goal state after:
ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
F : ι → β → α
S : Set β
h : UniformEquicontinuousOn F S
⊢ UniformEquicontinuousOn (F ∘ rangeSplitting F) S
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
ι : Sort u_5
p : ι → Prop
V : ι → Set (β × β)
h : (𝓤 β).HasBasis p V
f : α →ᵤ[𝔖] β
⊢ 𝓝 f = ⨅ s ∈ 𝔖, ⨅ i, ⨅ (_ : p i), 𝓟 {g | ∀ x ∈ s, ((toFun 𝔖) f x, (toFun 𝔖) g x) ∈ V i}

Tactic:
simp_rw [nhds_eq_comap_uniformity, UniformOnFun.uniformity_eq_of_basis _ _ h, comap_iInf,
    comap_principal, UniformOnFun.gen, preimage_ofPred_eq]

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

Goal state before:
pp: case mpr
α : Type u_2
inst✝ : DivisionMonoid α
f : Filter α
⊢ (∃ a, f = pure a ∧ IsUnit a) → IsUnit f

Tactic:
rintro ⟨a, rfl, ha⟩

Goal state after:
case mpr
α : Type u_2
inst✝ : DivisionMonoid α
a : α
ha : IsUnit a
⊢ IsUnit (pure a)
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✝³ : SMul α β
inst✝² : SMul α γ
inst✝¹ : SMul β γ
inst✝ : IsScalarTower α β γ
a : α
f : Filter β
g : Filter γ
⊢ (a • f) • g = a • f • g

Tactic:
refine (map_map₂_distrib_left fun _ _ => ?_).symm
    exact (smul_assoc a _ _).symm

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

Goal state before:
pp: 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
⊢ uniformSpace α β 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)

Tactic:
have : ∀ S ∈ 𝔖, 𝒰(S, β, _) = .comap (· ∘ ψ₁ S) 𝒰(_, β, _) ⊓ .comap (· ∘ ψ₂ S) 𝒰(_, β, _) := by
    refine fun S hS ↦ UniformFun.uniformSpace_eq_inf_precomp_of_cover β _ _ ?_
    simpa only [← univ_subset_iff, ψ₁, ψ₂, range_restrictPreimage, ← preimage_union,
      ← image_subset_iff, image_univ, Subtype.range_val] using h_cover S hS

Goal state after:
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)
⊢ uniformSpace α β 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)
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' Ici a = Iic (a / c)

Tactic:
simpa only [mul_comm] using preimage_mul_const_Ici_of_neg a h

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

Goal state before:
pp: Γ : Type u_4
inst✝³ : Group Γ
T : Type u_5
inst✝² : TopologicalSpace T
inst✝¹ : MulAction Γ T
inst✝ : ContinuousConstSMul Γ T
U : Set T
hU : IsOpen[inst✝²] U
⊢ IsOpen[inst✝²] (⋃ g, (fun x ↦ g • x) '' U)

Tactic:
exact isOpen_iUnion fun γ => isOpenMap_smul γ U hU

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
f_cont : Continuous[inst✝¹, inst✝] f
f_inj : Injective f
⊢ IsProperMap f ↔ IsClosedMap f

Tactic:
by
  refine ⟨fun h ↦ h.isClosedMap, fun h ↦ ?_⟩
  rw [isProperMap_iff_isClosedMap_and_compact_fibers]
  exact ⟨f_cont, h, fun y ↦ (subsingleton_singleton.preimage f_inj).isCompact⟩

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

Goal state before:
pp: M : Type u_1
s : Set M
inst✝³ : Mul M
inst✝² : TopologicalSpace M
inst✝¹ : SeparatelyContinuousMul M
inst✝ : T2Space M
⊢ IsClosed[inst✝²] s.centralizer

Tactic:
by
  rw [centralizer, ofPred_forall]
  refine isClosed_sInter ?_
  rintro - ⟨m, ht, rfl⟩
  refine isClosed_imp (by simp) <| isClosed_eq ?_ ?_
  all_goals fun_prop

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

Goal state before:
pp: G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
N : Subgroup G
⊢ DiscreteTopology (G ⧸ N) ↔ IsOpen[inst✝²] ↑N

Tactic:
rw [← QuotientGroup.preimage_mk_one, MulAction.IsPretransitive.discreteTopology_iff G (mk 1),
      isOpen_coinduced]
  rfl

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

Goal state before:
pp: G : Type w
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
H : Type u_1
inst✝⁴ : Monoid H
inst✝³ : TopologicalSpace H
inst✝² : ContinuousConstSMul H H
F : Type u_2
inst✝¹ : FunLike F G H
inst✝ : MonoidHomClass F G H
f : F
h : 𝓝 1 ≤ map (⇑f) (𝓝 1)
x : G
this : map (fun x_1 ↦ f x * x_1) (𝓝 1) = 𝓝 (f x)
⊢ 𝓝 (f x) ≤ map (⇑f) (𝓝 x)

Tactic:
rw [← map_mul_left_nhds_one x, Filter.map_map, Function.comp_def, ← this]

Goal state after:
G : Type w
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
H : Type u_1
inst✝⁴ : Monoid H
inst✝³ : TopologicalSpace H
inst✝² : ContinuousConstSMul H H
F : Type u_2
inst✝¹ : FunLike F G H
inst✝ : MonoidHomClass F G H
f : F
h : 𝓝 1 ≤ map (⇑f) (𝓝 1)
x : G
this : map (fun x_1 ↦ f x * x_1) (𝓝 1) = 𝓝 (f x)
⊢ map (fun x_1 ↦ f x * x_1) (𝓝 1) ≤ map (fun x_1 ↦ f (x * x_1)) (𝓝 1)
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: case refine_2
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
ψ : (β × β) × β × β → β :=
  fun x ↦
    match x with
    | ((x₁, x₂), y₁, y₂) => x₁⁻¹ * x₂ * (y₂⁻¹ * y₁)
g : (β × β) × β × β → β :=
  fun x ↦
    match x with
    | ((x₁, x₂), y₁, y₂) => x₁
φ_ψ_conj : φ = g * ψ * g⁻¹
⊢ Tendsto (fun x ↦ x.2.2⁻¹ * x.2.1) (𝓤 β ×ˢ 𝓤 β) (𝓝 1)

Tactic:
rw [uniformity_eq_comap_inv_mul_nhds_one_swapped]
        exact tendsto_comap.comp tendsto_snd

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
n : ℤ
hn : 0 < n
⊢ Tendsto (fun x ↦ x ^ n) atTop atTop

Tactic:
lift n to ℕ using hn.le

Goal state after:
α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
n : ℕ
hn : 0 < ↑n
⊢ Tendsto (fun x ↦ x ^ ↑n) atTop atTop
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k : α
hk : 0 < k
b : α
x✝ : 1 ≤ b
⊢ b ≤ k * b / k

Tactic:
by rw [mul_div_assoc, mul_div_cancel₀]; exact ne_of_gt hk

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

Goal state before:
pp: A : Type u_2
E : Type u_6
inst✝⁴ : Monoid A
inst✝³ : TopologicalSpace A
inst✝² : CommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousMul E
f : A →ₜ* E
a : A
⊢ (f ^ 0) a = f a ^ 0

Tactic:
pow_zero,

Goal state after:
A : Type u_2
E : Type u_6
inst✝⁴ : Monoid A
inst✝³ : TopologicalSpace A
inst✝² : CommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousMul E
f : A →ₜ* E
a : A
⊢ 1 a = f a ^ 0
Module: Mathlib.Topology.Algebra.ContinuousMonoidHom

Goal state before:
pp: A : Type u_2
E : Type u_6
inst✝⁴ : Monoid A
inst✝³ : TopologicalSpace A
inst✝² : CommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousMul E
f : A →ₜ* E
a : A
⊢ 1 a = f a ^ 0

Tactic:
pow_zero,

Goal state after:
A : Type u_2
E : Type u_6
inst✝⁴ : Monoid A
inst✝³ : TopologicalSpace A
inst✝² : CommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousMul E
f : A →ₜ* E
a : A
⊢ 1 a = 1
Module: Mathlib.Order.Filter.AtTopBot.CompleteLattice

Goal state before:
pp: α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLinearOrder γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
g : α → β
hg : Tendsto g l atTop
hb : BddAbove (range f)
⊢ ⨆ a, f (g a) = ⨆ b, f b

Tactic:
exact hf.ciSup_comp_tendsto_atTop hb hg

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

Goal state before:
pp: case inl
α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : ContinuousMul 𝕜
hS : IsPreconnected S
hf : ContinuousOn f S
hg : ContinuousOn g S
hsq : EqOn (f ^ 2) (g ^ 2) S
hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0
y : α
hy : y ∈ S
hy' : f y = g y
x : α
hx : x ∈ S
h : EqOn f g S
⊢ f x = g x

Tactic:
exact h hx

Goal state after:
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : x i₀ ≤ m
⊢ (univ.pi fun i ↦ Ioc (update x i₀ m i) (y i)) = {z | m < z i₀} ∩ univ.pi fun i ↦ Ioc (x i) (y i)

Tactic:
have : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀) := by
    rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, ← inter_assoc,
      inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]
  simp_rw [univ_pi_update i₀ _ _ fun i z ↦ Ioc z (y i), ← pi_inter_compl ({i₀} : Set ι),
    singleton_pi', ← inter_assoc, this]
  rfl

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

Goal state before:
pp: α : Type u_1
s t : Set α
Ixx : α → α → Set α
⊢ Tendsto (fun p ↦ Ixx p.1 p.2) (𝓟 s ×ˢ 𝓟 s) (𝓟 t).smallSets ↔ ∀ x ∈ s, ∀ y ∈ s, Ixx x y ⊆ t

Tactic:
by
    simp only [smallSets_principal, prod_principal_principal, tendsto_principal_principal,
      forall_prod_set, mem_powerset_iff]

Goal state after:
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
y : α → α
hy : ∀ x ∈ t, y x > x
h'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a ≠ b
⊢ Disjoint (Ioo a (y a)) (Ioo b (y b))

Tactic:
wlog! H : a < b generalizing a b with h

Goal state after:
case inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
y : α → α
hy : ∀ x ∈ t, y x > x
h'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a ≠ b
h : ∀ ⦃a : α⦄, a ∈ t → ∀ ⦃b : α⦄, b ∈ t → a ≠ b → a < b → Disjoint (Ioo a (y a)) (Ioo b (y b))
H : b ≤ a
⊢ Disjoint (Ioo a (y a)) (Ioo b (y b))

α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
y : α → α
hy : ∀ x ∈ t, y x > x
h'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a ≠ b
H : a < b
⊢ Disjoint (Ioo a (y a)) (Ioo b (y b))
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : ∀ (x : α), ↑e x = ↑e' x
hsymm : ∀ (x : β), ↑e.symm x = ↑e'.symm x
hs : e.source = e'.source
A : ↑e = ↑e'
B : ↑e.symm = ↑e'.symm
I : ↑e '' e.source = e.target
⊢ e = e'

Tactic:
have I' : e' '' e'.source = e'.target := e'.image_source_eq_target

Goal state after:
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : ∀ (x : α), ↑e x = ↑e' x
hsymm : ∀ (x : β), ↑e.symm x = ↑e'.symm x
hs : e.source = e'.source
A : ↑e = ↑e'
B : ↑e.symm = ↑e'.symm
I : ↑e '' e.source = e.target
I' : ↑e' '' e'.source = e'.target
⊢ e = e'