mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure[inst✝] ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure[inst✝] ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure[inst✝] = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure[inst✝] = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed[inst✝] t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed[inst✝] t ∧ t ⊆ s) →
    ∀ (s : Set X), ∀ x ∉ closure[inst✝] s, Disjoint (𝓝ˢ s) (𝓝 x)
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure[inst✝] s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure[inst✝] s,
      ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed[inst✝] t ∧ t ⊆ s, ∀ (x : X), (𝓝 x).lift' closure[inst✝] ≤ 𝓝 x,
      ∀ (x : X), (𝓝 x).lift' closure[inst✝] = 𝓝 x].TFAE

Tactic:
tfae_have 2 → 3 := by
    refine fun H a s => ⟨fun hd has => mem_closure_iff_nhds_ne_bot.mp has ?_, H s a⟩
    exact (hd.symm.mono_right <| @principal_le_nhdsSet _ _ s).eq_bot

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure[inst✝] ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure[inst✝] ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure[inst✝] = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure[inst✝] = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed[inst✝] t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed[inst✝] t ∧ t ⊆ s) →
    ∀ (s : Set X), ∀ x ∉ closure[inst✝] s, Disjoint (𝓝ˢ s) (𝓝 x)
tfae_2_to_3 :
  (∀ (s : Set X), ∀ x ∉ closure[inst✝] s, Disjoint (𝓝ˢ s) (𝓝 x)) →
    ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure[inst✝] s
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure[inst✝] s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure[inst✝] s,
      ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed[inst✝] t ∧ t ⊆ s, ∀ (x : X), (𝓝 x).lift' closure[inst✝] ≤ 𝓝 x,
      ∀ (x : X), (𝓝 x).lift' closure[inst✝] = 𝓝 x].TFAE
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case sUnion
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (Set X)
h : inst✝ = generateFrom s
h' : ∀ t ∈ s, ∀ a ∈ t, Disjoint (𝓝ˢ tᶜ) (𝓝 a)
a : X
t : Set X
S : Set (Set X)
a✝ : ∀ s_1 ∈ S, GenerateOpen s s_1
ih : ∀ s ∈ S, a ∈ s → Disjoint (𝓝ˢ sᶜ) (𝓝 a)
ha : a ∈ ⋃₀ S
⊢ Disjoint (𝓝ˢ (⋃₀ S)ᶜ) (𝓝 a)

Tactic:
obtain ⟨t, ht, ha⟩ := ha

Goal state after:
case sUnion
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (Set X)
h : inst✝ = generateFrom s
h' : ∀ t ∈ s, ∀ a ∈ t, Disjoint (𝓝ˢ tᶜ) (𝓝 a)
a : X
t✝ : Set X
S : Set (Set X)
a✝ : ∀ s_1 ∈ S, GenerateOpen s s_1
ih : ∀ s ∈ S, a ∈ s → Disjoint (𝓝ˢ sᶜ) (𝓝 a)
t : Set X
ht : t ∈ S
ha : a ∈ t
⊢ Disjoint (𝓝ˢ (⋃₀ S)ᶜ) (𝓝 a)
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this : IsPreconnected (Subtype.val ⁻¹' s)
⊢ ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s

Tactic:
mem_preimage

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this : IsPreconnected (Subtype.val ⁻¹' s)
⊢ ↑⟨x, ⋯⟩ ∈ s
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
u s : Set X
h : s ∈ 𝓝ˢ u
hu : IsClosed[inst✝¹] u
o : Set X
ho_open : IsOpen[inst✝¹] o
huo : u ⊆ o
hos : o ⊆ s
⊢ ∃ t ∈ 𝓝ˢ u, IsClosed[inst✝¹] t ∧ t ⊆ s

Tactic:
obtain ⟨v, hv_open, huv, hcvo⟩ := normal_exists_closure_subset hu ho_open huo

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
u s : Set X
h : s ∈ 𝓝ˢ u
hu : IsClosed[inst✝¹] u
o : Set X
ho_open : IsOpen[inst✝¹] o
huo : u ⊆ o
hos : o ⊆ s
v : Set X
hv_open : IsOpen[inst✝¹] v
huv : u ⊆ v
hcvo : closure[inst✝¹] v ⊆ o
⊢ ∃ t ∈ 𝓝ˢ u, IsClosed[inst✝¹] t ∧ t ⊆ s
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hf : S.Countable
hc : ∀ s ∈ S, IsLindelof s
⊢ IsLindelof (⋃₀ S)

Tactic:
rw [sUnion_eq_biUnion]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hf : S.Countable
hc : ∀ s ∈ S, IsLindelof s
⊢ IsLindelof (⋃ i ∈ S, i)
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hf : S.Finite
hc : ∀ s ∈ S, IsLindelof s
⊢ IsLindelof (⋃₀ S)

Tactic:
rw [sUnion_eq_biUnion]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
S : Set (Set X)
hf : S.Finite
hc : ∀ s ∈ S, IsLindelof s
⊢ IsLindelof (⋃ i ∈ S, i)
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : HereditarilyLindelofSpace X
s : Set X
⊢ IsLindelof s

Tactic:
apply HereditarilyLindelofSpace.isHereditarilyLindelof_univ
  exact subset_univ s

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

Goal state before:
pp: X : Type u_1
ι' : Sort u_4
inst✝¹ : TopologicalSpace X
inst✝ : Countable ι'
f : ι' → Set X
hs : ∀ (i : ι'), IsMeagre (f i)
⊢ ⋂ i, (f i)ᶜ ∈ residual X

Tactic:
exact countable_iInter_mem.mpr hs

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

Goal state before:
pp: case refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed[inst✝²] (⋂ s, ↑s)
a b : Set X
ha : IsClosed[inst✝²] a
hb : IsClosed[inst✝²] b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen[inst✝²] u
hv : IsOpen[inst✝²] v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
H1 : Disjoint (u ∪ v)ᶜ (⋂ i, ↑i) → ∃ x_1, ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ x_1, ↑i).Nonempty
si : Finset { s // IsClopen s ∧ x ∈ s }
H2 : ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ si, ↑i).Nonempty
⊢ ⋂ U ∈ si, ↑U ⊆ u ∪ v

Tactic:
← disjoint_compl_left_iff_subset,

Goal state after:
case refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed[inst✝²] (⋂ s, ↑s)
a b : Set X
ha : IsClosed[inst✝²] a
hb : IsClosed[inst✝²] b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen[inst✝²] u
hv : IsOpen[inst✝²] v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
H1 : Disjoint (u ∪ v)ᶜ (⋂ i, ↑i) → ∃ x_1, ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ x_1, ↑i).Nonempty
si : Finset { s // IsClopen s ∧ x ∈ s }
H2 : ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ si, ↑i).Nonempty
⊢ Disjoint (u ∪ v)ᶜ (⋂ U ∈ si, ↑U)
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case inl
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 ∧ f ⁻¹' t ∩ u ⊆ f ⁻¹' T₁
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : t ⊆ T₁
⊢ f ⁻¹' t ⊆ u

Tactic:
suffices f ⁻¹' t ⊆ f ⁻¹' T₁
      from (this.trans T₁_u.1).trans inter_subset_right

Goal state after:
case inl
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 ∧ f ⁻¹' t ∩ u ⊆ f ⁻¹' T₁
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : t ⊆ T₁
⊢ f ⁻¹' t ⊆ f ⁻¹' T₁
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Set α
hclopen : ∀ (i : ι), IsClopen (U i)
hdisj : Pairwise (Disjoint on U)
hunion : ⋃ i, U i = univ
i : ι
x : α
hx : x ∈ U i
⊢ (equivOfIsClopen hclopen hdisj hunion) (mk x) = ⟨i, mk ⟨x, hx⟩⟩

Tactic:
apply (equivOfIsClopen hclopen hdisj hunion).symm.injective
  simp

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

Goal state before:
pp: X : Type u_1
Y✝ : Type u_2
W : Type u_3
Z : Type u_4
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y✝
inst✝⁵ : TopologicalSpace W
inst✝⁴ : TopologicalSpace Z
X' : Type u_5
Y' : Type u_6
inst✝³ : TopologicalSpace X'
inst✝² : TopologicalSpace Y'
ι : Type u_7
inst✝¹ : DecidableEq ι
i : ι
Y : ι → Type u_8
inst✝ : (j : ι) → TopologicalSpace (Y j)
j : ι
⊢ Continuous[instTopologicalSpaceProd, inst✝ j] fun a ↦ if h : j = i then ⋯ ▸ a.1 else a.2 ⟨j, h⟩

Tactic:
split_ifs with h

Goal state after:
case pos
X : Type u_1
Y✝ : Type u_2
W : Type u_3
Z : Type u_4
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y✝
inst✝⁵ : TopologicalSpace W
inst✝⁴ : TopologicalSpace Z
X' : Type u_5
Y' : Type u_6
inst✝³ : TopologicalSpace X'
inst✝² : TopologicalSpace Y'
ι : Type u_7
inst✝¹ : DecidableEq ι
i : ι
Y : ι → Type u_8
inst✝ : (j : ι) → TopologicalSpace (Y j)
j : ι
h : j = i
⊢ Continuous[instTopologicalSpaceProd, inst✝ j] fun a ↦ ⋯ ▸ a.1

case neg
X : Type u_1
Y✝ : Type u_2
W : Type u_3
Z : Type u_4
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y✝
inst✝⁵ : TopologicalSpace W
inst✝⁴ : TopologicalSpace Z
X' : Type u_5
Y' : Type u_6
inst✝³ : TopologicalSpace X'
inst✝² : TopologicalSpace Y'
ι : Type u_7
inst✝¹ : DecidableEq ι
i : ι
Y : ι → Type u_8
inst✝ : (j : ι) → TopologicalSpace (Y j)
j : ι
h : ¬j = i
⊢ Continuous[instTopologicalSpaceProd, inst✝ j] fun a ↦ a.2 ⟨j, h⟩
Module: Mathlib.Topology.Homeomorph.Lemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
f : X → Y
inst✝¹ : CompactSpace X
inst✝ : T2Space Y
hf : Continuous[inst✝³, inst✝²] f
⊢ IsClosedMap f ∧ Bijective f ↔ Bijective f

Tactic:
eq_true hf.isClosedMap,

Goal state after:
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
f : X → Y
inst✝¹ : CompactSpace X
inst✝ : T2Space Y
hf : Continuous[inst✝³, inst✝²] f
⊢ True ∧ Bijective f ↔ Bijective f
Module: Mathlib.Topology.UniformSpace.Basic

Goal state before:
pp: α : Type ua
β : Type ub
γ : Type uc
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
s : SetRel α β
t : SetRel β γ
hs : IsOpen[instTopologicalSpaceProd] s
ht : IsOpen[instTopologicalSpaceProd] t
⊢ s ○ t = ⋃ b, (fun p ↦ (p.1, b)) ⁻¹' s ∩ (fun p ↦ (b, p.2)) ⁻¹' t

Tactic:
ext ⟨_, _⟩

Goal state after:
α : Type ua
β : Type ub
γ : Type uc
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
s : SetRel α β
t : SetRel β γ
hs : IsOpen[instTopologicalSpaceProd] s
ht : IsOpen[instTopologicalSpaceProd] t
fst✝ : α
snd✝ : γ
⊢ (fst✝, snd✝) ∈ s ○ t ↔ (fst✝, snd✝) ∈ ⋃ b, (fun p ↦ (p.1, b)) ⁻¹' s ∩ (fun p ↦ (b, p.2)) ⁻¹' t
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:
exact h.comp ((hu.comp tendsto_fst).prodMk tendsto_snd)

Goal state after:
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, ∃ V ∈ 𝓤 α, ball x (V ○ V) ⊆ U i

Tactic:
obtain ⟨i, hi⟩ := mem_iUnion.1 (hcover hx)
    rw [← (hopen i).mem_nhds_iff, nhds_eq_comap_uniformity, ← lift'_comp_uniformity] at hi
    exact ⟨i, (((basis_sets _).lift' <| monotone_id.relComp monotone_id).comap _).mem_iff.1 hi⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
p : Filter ι
x✝³ : α
x✝² : x✝³ ∈ s
x✝¹ : Set (β × β)
x✝ : x✝¹ ∈ 𝓤 β
⊢ (∃ t ∈ 𝓝[s] x✝³, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ x✝¹) ↔
    ∃ t ∈ 𝓝[s] x✝³, ∀ᶠ (x : ι) in p, ∀ y ∈ t, (x, y) ∈ (fun y ↦ (f y.2, F y.1 y.2)) ⁻¹' x✝¹

Tactic:
mem_preimage

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
S : Set (Set α)
hS : ∀ s ∈ S, IsOpen[inst✝¹] s
h : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s
⊢ TendstoLocallyUniformlyOn F f p (⋃₀ S)

Tactic:
rw [sUnion_eq_biUnion]

Goal state after:
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
p : Filter ι
S : Set (Set α)
hS : ∀ s ∈ S, IsOpen[inst✝¹] s
h : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s
⊢ TendstoLocallyUniformlyOn F f p (⋃ i ∈ S, i)
Module: Mathlib.Topology.UniformSpace.HeineCantor

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
ι : Type u_4
F : ι → β → α
inst✝ : CompactSpace β
h : Equicontinuous F
⊢ UniformEquicontinuous F

Tactic:
equicontinuous_iff_continuous

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
ι : Type u_4
F : ι → β → α
inst✝ : CompactSpace β
h : Continuous[inst✝¹.toTopologicalSpace, UniformFun.topologicalSpace ι α] (⇑UniformFun.ofFun ∘ Function.swap F)
⊢ UniformEquicontinuous F
Module: Mathlib.Algebra.Order.Group.MinMax

Goal state before:
pp: α : Type u_1
inst✝² : CommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedMonoid α
a b c : α
⊢ max (a / b) (a / c) = a / min b c

Tactic:
by
  simp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv']

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

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
⊢ comap (⇑e) atTop = atTop

Tactic:
by
  simp [atTop, ← e.surjective.iInf_comp]

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

Goal state before:
pp: α : Type u_1
β : Type u_3
f : α → β → α
b : β
h : ∀ (a : α), f a b = a
l : Filter α
⊢ map₂ f l (pure b) = l

Tactic:
rw [map₂_pure_right, funext h, map_id']

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
hA : EquicontinuousOn (u ∘ Subtype.val) S
hu : Continuous[tY, Pi.topologicalSpace] (S.domRestrict ∘ u)
x : X
hx : x ∈ S
⊢ Continuous[tY, uα.toTopologicalSpace] (eval x ∘ u)

Tactic:
by exact continuous_apply ⟨x, hx⟩ |>.comp hu

Goal state after:
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 : Tendsto f (𝓝[s] x) (𝓝 z)
hs : ∀ y ∈ s, Tendsto (fun x ↦ F x y) l (𝓝 (f y))
hx : x ∈ closure[tX] s
⊢ Tendsto (fun x_1 ↦ F x_1 x) l (𝓝 z)

Tactic:
rw [(nhds_basis_uniformity (𝓤 α).basis_sets).tendsto_right_iff] at hf ⊢

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
⊢ ∀ i ∈ 𝓤 α, ∀ᶠ (x_1 : ι) in l, F x_1 x ∈ {y | (y, z) ∈ id i}
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: case e_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : UniformSpace γ
f : γ → β
hf : 𝓤 γ = 𝓤 γ
⊢ UniformSpace.comap (⇑(ofFun 𝔖) ∘ (fun x ↦ f ∘ x) ∘ ⇑(toFun 𝔖)) (uniformSpace α β 𝔖) = uniformSpace α γ 𝔖

Tactic:
rw [← UniformSpace.ext hf, UniformOnFun.comap_eq]

Goal state after:
case e_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : UniformSpace γ
f : γ → β
hf : 𝓤 γ = 𝓤 γ
⊢ UniformSpace.comap (⇑(ofFun 𝔖) ∘ (fun x ↦ f ∘ x) ∘ ⇑(toFun 𝔖)) (uniformSpace α β 𝔖) =
    UniformSpace.comap (fun x ↦ f ∘ x) (uniformSpace α β 𝔖)
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 d : α
h : [[c, d]] ⊆ [[a, b]]
⊢ |d - c| ≤ |b - a|

Tactic:
rw [← max_sub_min_eq_abs, ← max_sub_min_eq_abs]

Goal state after:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c d : α
h : [[c, d]] ⊆ [[a, b]]
⊢ max c d - min c d ≤ max a b - min a b
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
h : IsCoherentWith 𝔖
f : α →ᵤ[𝔖] β
u : Filter (α →ᵤ[𝔖] β)
x✝ : u.NeBot
hu : u ≤ 𝓟 {f | Continuous[inst✝, inst✝¹.toTopologicalSpace] ((toFun 𝔖) f)}
huf : ∀ s ∈ 𝔖, TendstoUniformlyOn (⇑(toFun 𝔖) ∘ id) ((toFun 𝔖) f) u s
s : Set α
hs : s ∈ 𝔖
⊢ ContinuousOn ((toFun 𝔖) f) s

Tactic:
exact (huf s hs).continuousOn <| Eventually.frequently <| hu fun _ ↦ Continuous.continuousOn

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

Goal state before:
pp: G₀ : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
ha : 0 ≤ a
hbc : b ≤ c
⊢ (fun x ↦ a * x) '' Icc b c = Icc (a * b) (a * c)

Tactic:
by
  rcases ha.eq_or_lt with rfl | ha
  · simp [(nonempty_Icc.2 hbc).image_const]
  · exact image_mul_left_Icc' ha b c

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : C(β, γ)
g₁ g₂ : C(α, β)
hf : Injective ⇑f
h : f.comp g₁ = f.comp g₂
a : α
⊢ f (g₁ a) = f (g₂ a)

Tactic:
by rw [← comp_apply, h, comp_apply]

Goal state after:
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 b c : α
⊢ (fun x ↦ a * x) '' [[b, c]] = [[a * b, a * c]]

Tactic:
by
  simpa only [mul_comm] using image_mul_const_uIcc a b c

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hf : Continuous[inst✝², inst✝¹] f
hg : Continuous[inst✝¹, inst✝] g
hgf : IsProperMap (g ∘ f)
f_surj : Surjective f
ℱ : Filter Y
z : Z
h : MapClusterPt z (map f (comap f ℱ)) g
⊢ ∃ x, g x = z ∧ ClusterPt x ℱ

Tactic:
← mapClusterPt_comp

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hf : Continuous[inst✝², inst✝¹] f
hg : Continuous[inst✝¹, inst✝] g
hgf : IsProperMap (g ∘ f)
f_surj : Surjective f
ℱ : Filter Y
z : Z
h : MapClusterPt z (comap f ℱ) (g ∘ f)
⊢ ∃ x, g x = z ∧ ClusterPt x ℱ
Module: Mathlib.Topology.Algebra.Monoid

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
l : Filter (M × M)
hl : Disjoint l (cocompact (M × M))
this : (𝓝 0).coprod (𝓝 0) ⊓ l ≤ 𝓝 0 ×ˢ map Prod.snd l ⊔ map Prod.fst l ×ˢ 𝓝 0
⊢ Tendsto (fun x ↦ x.1 * x.2) (map Prod.fst l ×ˢ 𝓝 0) (𝓝 0)

Tactic:
apply tendsto_mul_prod_nhds_zero_of_disjoint_cocompact
    exact disjoint_map_cocompact continuous_fst hl

Goal state after:
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:
simp_rw [nhds_prod_eq, tendsto_prod_iff'] at hyw

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 (fun n ↦ (Prod.map f g n).1) (↑𝒰) (𝓝 y) ∧ Tendsto (fun n ↦ (Prod.map f g n).2) (↑𝒰) (𝓝 w)
⊢ ∃ x, Prod.map f g x = (y, w) ∧ ↑𝒰 ≤ 𝓝 x
Module: Mathlib.Topology.Algebra.Monoid

Goal state before:
pp: M : Type u_3
inst✝ : Mul M
t₁ t₂ : TopologicalSpace M
h₁ : ContinuousMul M
h₂ : ContinuousMul M
⊢ ContinuousMul M

Tactic:
by
  rw [inf_eq_iInf]
  refine continuousMul_iInf fun b => ?_
  cases b <;> assumption

Goal state after:
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
dr : β × β → β := fun xy ↦ xy.2 * xy.1⁻¹
dl : β × β → β := fun xy ↦ xy.1⁻¹ * xy.2
conj : β × β → β := fun gx ↦ gx.1 * gx.2 * gx.1⁻¹
φ : β × β ≃ β × β := (Equiv.refl β).prodShear fun b ↦ Equiv.symm (Equiv.mulLeft b)
conj_φ : conj ∘ ⇑φ = dr
snd_φ : snd ∘ ⇑φ = dl
⊢ comap dr (𝓝 1) = comap dl (𝓝 1)

Tactic:
← uniformity_eq_comap_inv_mul_nhds_one,

Goal state after:
β : Type u_2
inst✝³ : UniformSpace β
inst✝² : Group β
inst✝¹ : IsLeftUniformGroup β
inst✝ : IsRightUniformGroup β
dr : β × β → β := fun xy ↦ xy.2 * xy.1⁻¹
dl : β × β → β := fun xy ↦ xy.1⁻¹ * xy.2
conj : β × β → β := fun gx ↦ gx.1 * gx.2 * gx.1⁻¹
φ : β × β ≃ β × β := (Equiv.refl β).prodShear fun b ↦ Equiv.symm (Equiv.mulLeft b)
conj_φ : conj ∘ ⇑φ = dr
snd_φ : snd ∘ ⇑φ = dl
⊢ comap dr (𝓝 1) = 𝓤 β
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
G₀ : Type u_3
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : TopologicalSpace G₀
inst✝³ : ContinuousInv₀ G₀
inst✝² : ContinuousMul G₀
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
a : α
f g : α → G₀
h : α → G₀ → β
hf : ContinuousAt f a
hg : ContinuousAt g a
hh : g a ≠ 0 → ContinuousAt ↿h (a, f a / g a)
h2h : g a = 0 → Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))
hga : g a = 0
⊢ Tendsto (↿h ∘ fun x ↦ (x, f x / g x)) (𝓝 a) (𝓝 (↿h (a, 0)))

Tactic:
exact (h2h hga).comp (continuousAt_id.tendsto.prodMk tendsto_top)

Goal state after:
Module: Mathlib.Order.OrdContinuous

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
f : α → β
hf : LeftOrdContinuous f
h : Injective f
x y : α
⊢ f x ≤ f y ↔ x ≤ y

Tactic:
simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]

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

Goal state before:
pp: case pos
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}
H : t.Countable
x : α
hx : x ∈ {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥}
h'x : IsTop x
⊢ x ∈ t ∪ {x | IsTop x}

Tactic:
simp [h'x]

Goal state after: