ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TopologicalSpace.NonemptyCompacts.ext
@[ext]
protected theorem ext {s t : NonemptyCompacts α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : TopologicalSpace.NonemptyCompacts α} (h : ↑s = ↑t)
Goal: s = t
-- TopologicalSpace.NonemptyCompacts.coe_mk
@[simp]
theorem coe_mk (s : Compacts α) (h) : (mk s h : Set α) = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α) (h : s.carrier.Nonempty)
Goal: ↑{ toCompacts := s, nonempty' := h } = ↑s
-- TopologicalSpace.NonemptyCompacts.carrier_eq_coe
theorem carrier_eq_coe (s : NonemptyCompacts α) : s.carrier = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.NonemptyCompacts α)
Goal: s.carrier = ↑s
-- TopologicalSpace.NonemptyCompacts.coe_toCompacts
@[simp]
theorem coe_toCompacts (s : NonemptyCompacts α) : (s.toCompacts : Set α) = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.NonemptyCompacts α)
Goal: ↑s.toCompacts = ↑s
-- TopologicalSpace.NonemptyCompacts.coe_sup
@[simp]
theorem coe_sup (s t : NonemptyCompacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t 
Arguments: {α : Type u_1} [TopologicalSpace α] (s t : TopologicalSpace.NonemptyCompacts α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- TopologicalSpace.NonemptyCompacts.coe_top
@[simp]
theorem coe_top [CompactSpace α] [Nonempty α] : (↑(⊤ : NonemptyCompacts α) : Set α) = univ 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α] [Nonempty α]
Goal: ↑⊤ = Set.univ
-- TopologicalSpace.NonemptyCompacts.coe_prod
@[simp]
theorem coe_prod (K : NonemptyCompacts α) (L : NonemptyCompacts β) :
    (K.prod L : Set (α × β)) = (K : Set α) ×ˢ (L : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (K : TopologicalSpace.NonemptyCompacts α) (L : TopologicalSpace.NonemptyCompacts β)
Goal: ↑(K.prod L) = ↑K ×ˢ ↑L
-- TopologicalSpace.PositiveCompacts.interior_nonempty'
structure PositiveCompacts (α : Type*) [TopologicalSpace α] extends Compacts α where
  interior_nonempty' : (interior carrier).Nonempty
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.PositiveCompacts α)
Goal: (interior self.carrier).Nonempty
-- TopologicalSpace.PositiveCompacts.isCompact
protected theorem isCompact (s : PositiveCompacts α) : IsCompact (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.PositiveCompacts α)
Goal: IsCompact ↑s
-- TopologicalSpace.PositiveCompacts.interior_nonempty
theorem interior_nonempty (s : PositiveCompacts α) : (interior (s : Set α)).Nonempty 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.PositiveCompacts α)
Goal: (interior ↑s).Nonempty
-- TopologicalSpace.PositiveCompacts.nonempty
protected theorem nonempty (s : PositiveCompacts α) : (s : Set α).Nonempty 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.PositiveCompacts α)
Goal: (↑s).Nonempty
-- TopologicalSpace.PositiveCompacts.ext
@[ext]
protected theorem ext {s t : PositiveCompacts α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : TopologicalSpace.PositiveCompacts α} (h : ↑s = ↑t)
Goal: s = t
-- TopologicalSpace.PositiveCompacts.coe_mk
@[simp]
theorem coe_mk (s : Compacts α) (h) : (mk s h : Set α) = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α) (h : (interior s.carrier).Nonempty)
Goal: ↑{ toCompacts := s, interior_nonempty' := h } = ↑s
-- TopologicalSpace.PositiveCompacts.carrier_eq_coe
theorem carrier_eq_coe (s : PositiveCompacts α) : s.carrier = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.PositiveCompacts α)
Goal: s.carrier = ↑s
-- TopologicalSpace.PositiveCompacts.coe_toCompacts
@[simp]
theorem coe_toCompacts (s : PositiveCompacts α) : (s.toCompacts : Set α) = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.PositiveCompacts α)
Goal: ↑s.toCompacts = ↑s
-- TopologicalSpace.PositiveCompacts.coe_sup
@[simp]
theorem coe_sup (s t : PositiveCompacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t 
Arguments: {α : Type u_1} [TopologicalSpace α] (s t : TopologicalSpace.PositiveCompacts α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- TopologicalSpace.PositiveCompacts.coe_top
@[simp]
theorem coe_top [CompactSpace α] [Nonempty α] : (↑(⊤ : PositiveCompacts α) : Set α) = univ 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α] [Nonempty α]
Goal: ↑⊤ = Set.univ
-- TopologicalSpace.PositiveCompacts.coe_map
@[simp, norm_cast]
theorem coe_map {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : PositiveCompacts α) :
    (s.map f hf hf' : Set β) = f '' s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : TopologicalSpace.PositiveCompacts α)
Goal: ↑(TopologicalSpace.PositiveCompacts.map f hf hf' s) = f '' ↑s
-- TopologicalSpace.PositiveCompacts.map_id
@[simp]
theorem map_id (K : PositiveCompacts α) : K.map id continuous_id IsOpenMap.id = K 
Arguments: {α : Type u_1} [TopologicalSpace α] (K : TopologicalSpace.PositiveCompacts α)
Goal: TopologicalSpace.PositiveCompacts.map id ⋯ ⋯ K = K
-- TopologicalSpace.PositiveCompacts.map_comp
theorem map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f)
    (hg' : IsOpenMap g) (K : PositiveCompacts α) :
    K.map (f ∘ g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf' 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f) (hg' : IsOpenMap g) (K : TopologicalSpace.PositiveCompacts α)
Goal: TopologicalSpace.PositiveCompacts.map (f ∘ g) ⋯ ⋯ K =
  TopologicalSpace.PositiveCompacts.map f hf hf' (TopologicalSpace.PositiveCompacts.map g hg hg' K)
-- exists_positiveCompacts_subset
theorem _root_.exists_positiveCompacts_subset [LocallyCompactSpace α] {U : Set α} (ho : IsOpen U)
    (hn : U.Nonempty) : ∃ K : PositiveCompacts α, ↑K ⊆ U 
Arguments: {α : Type u_1} [TopologicalSpace α] [LocallyCompactSpace α] {U : Set α} (ho : IsOpen U) (hn : U.Nonempty)
Goal: ∃ K, ↑K ⊆ U
-- IsOpen.exists_positiveCompacts_closure_subset
theorem _root_.IsOpen.exists_positiveCompacts_closure_subset [R1Space α] [LocallyCompactSpace α]
    {U : Set α} (ho : IsOpen U) (hn : U.Nonempty) : ∃ K : PositiveCompacts α, closure ↑K ⊆ U 
Arguments: {α : Type u_1} [TopologicalSpace α] [R1Space α] [LocallyCompactSpace α] {U : Set α} (ho : IsOpen U) (hn : U.Nonempty)
Goal: ∃ K, closure ↑K ⊆ U
-- TopologicalSpace.PositiveCompacts.coe_prod
@[simp]
theorem coe_prod (K : PositiveCompacts α) (L : PositiveCompacts β) :
    (K.prod L : Set (α × β)) = (K : Set α) ×ˢ (L : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (K : TopologicalSpace.PositiveCompacts α) (L : TopologicalSpace.PositiveCompacts β)
Goal: ↑(K.prod L) = ↑K ×ˢ ↑L
-- TopologicalSpace.CompactOpens.isOpen'
structure CompactOpens (α : Type*) [TopologicalSpace α] extends Compacts α where
  isOpen' : IsOpen carrier
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.CompactOpens α)
Goal: IsOpen self.carrier
-- TopologicalSpace.CompactOpens.isCompact
protected theorem isCompact (s : CompactOpens α) : IsCompact (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.CompactOpens α)
Goal: IsCompact ↑s
-- TopologicalSpace.CompactOpens.isOpen
protected theorem isOpen (s : CompactOpens α) : IsOpen (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.CompactOpens α)
Goal: IsOpen ↑s
-- TopologicalSpace.CompactOpens.toOpens_coe
@[simps]
def toOpens (s : CompactOpens α) : Opens α 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.CompactOpens α)
Goal: ↑s.toOpens = ↑s
-- TopologicalSpace.CompactOpens.toClopens_coe
@[simps]
def toClopens [T2Space α] (s : CompactOpens α) : Clopens α 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] (s : TopologicalSpace.CompactOpens α)
Goal: ↑s.toClopens = ↑s
-- TopologicalSpace.CompactOpens.ext
@[ext]
protected theorem ext {s t : CompactOpens α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : TopologicalSpace.CompactOpens α} (h : ↑s = ↑t)
Goal: s = t
-- TopologicalSpace.CompactOpens.coe_mk
@[simp]
theorem coe_mk (s : Compacts α) (h) : (mk s h : Set α) = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α) (h : IsOpen s.carrier)
Goal: ↑{ toCompacts := s, isOpen' := h } = ↑s
-- TopologicalSpace.CompactOpens.coe_sup
@[simp, norm_cast] lemma coe_sup (s t : CompactOpens α) : ↑(s ⊔ t) = (s ∪ t : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] (s t : TopologicalSpace.CompactOpens α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- TopologicalSpace.CompactOpens.coe_bot
@[simp, norm_cast] lemma coe_bot : ↑(⊥ : CompactOpens α) = (∅ : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: ↑⊥ = ∅
-- TopologicalSpace.CompactOpens.coe_inf
@[simp, norm_cast] lemma coe_inf (s t : CompactOpens α) : ↑(s ⊓ t) = (s ∩ t : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [QuasiSeparatedSpace α] (s t : TopologicalSpace.CompactOpens α)
Goal: ↑(s ⊓ t) = ↑s ∩ ↑t
-- TopologicalSpace.CompactOpens.coe_sdiff
@[simp, norm_cast] lemma coe_sdiff (s t : CompactOpens α) : ↑(s \ t) = (s \ t : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] (s t : TopologicalSpace.CompactOpens α)
Goal: ↑(s \ t) = ↑s \ ↑t
-- TopologicalSpace.CompactOpens.coe_top
@[simp, norm_cast] lemma coe_top : ↑(⊤ : CompactOpens α) = (univ : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α]
Goal: ↑⊤ = Set.univ
-- TopologicalSpace.CompactOpens.coe_compl
@[simp, norm_cast] lemma coe_compl (s : CompactOpens α) : ↑sᶜ = (sᶜ : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α] [T2Space α] (s : TopologicalSpace.CompactOpens α)
Goal: ↑sᶜ = (↑s)ᶜ
-- TopologicalSpace.CompactOpens.coe_himp
@[simp, norm_cast] lemma coe_himp (s t : CompactOpens α) : ↑(s ⇨ t) = (s ⇨ t : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α] [T2Space α] (s t : TopologicalSpace.CompactOpens α)
Goal: ↑(s ⇨ t) = ↑s ⇨ ↑t
-- TopologicalSpace.CompactOpens.map_toCompacts
@[simps toCompacts]
def map (f : α → β) (hf : Continuous f) (hf' : IsOpenMap f) (s : CompactOpens α) : CompactOpens β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (hf : Continuous f) (hf' : IsOpenMap f) (s : TopologicalSpace.CompactOpens α)
Goal: (TopologicalSpace.CompactOpens.map f hf hf' s).toCompacts = TopologicalSpace.Compacts.map f hf s.toCompacts
-- TopologicalSpace.CompactOpens.coe_map
@[simp, norm_cast]
theorem coe_map {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : CompactOpens α) :
    (s.map f hf hf' : Set β) = f '' s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : TopologicalSpace.CompactOpens α)
Goal: ↑(TopologicalSpace.CompactOpens.map f hf hf' s) = f '' ↑s
-- TopologicalSpace.CompactOpens.map_id
@[simp]
theorem map_id (K : CompactOpens α) : K.map id continuous_id IsOpenMap.id = K 
Arguments: {α : Type u_1} [TopologicalSpace α] (K : TopologicalSpace.CompactOpens α)
Goal: TopologicalSpace.CompactOpens.map id ⋯ ⋯ K = K
-- TopologicalSpace.CompactOpens.map_comp
theorem map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f)
    (hg' : IsOpenMap g) (K : CompactOpens α) :
    K.map (f ∘ g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf' 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f) (hg' : IsOpenMap g) (K : TopologicalSpace.CompactOpens α)
Goal: TopologicalSpace.CompactOpens.map (f ∘ g) ⋯ ⋯ K =
  TopologicalSpace.CompactOpens.map f hf hf' (TopologicalSpace.CompactOpens.map g hg hg' K)
-- TopologicalSpace.CompactOpens.coe_prod
@[simp]
theorem coe_prod (K : CompactOpens α) (L : CompactOpens β) :
    (K.prod L : Set (α × β)) = (K : Set α) ×ˢ (L : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (K : TopologicalSpace.CompactOpens α) (L : TopologicalSpace.CompactOpens β)
Goal: ↑(K.prod L) = ↑K ×ˢ ↑L
-- isOpen_mk
lemma isOpen_mk {p h₁ h₂ h₃} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s 
Arguments: {X : Type u} {s : Set X} {p : Set X → Prop} {h₁ : p Set.univ} {h₂ : ∀ (s t : Set X), p s → p t → p (s ∩ t)} {h₃ : ∀ (s : Set (Set X)), (∀ t ∈ s, p t) → p (⋃₀ s)}
Goal: IsOpen s ↔ p s
-- TopologicalSpace.ext
@[ext]
protected theorem TopologicalSpace.ext :
    ∀ {f g : TopologicalSpace X}, IsOpen[f] = IsOpen[g] → f = g
  
Arguments: {X : Type u} {f g : TopologicalSpace X}
Goal: IsOpen = IsOpen → f = g
-- TopologicalSpace.ext_iff
protected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace X} :
    t = t' ↔ ∀ s, IsOpen[t] s ↔ IsOpen[t'] s 
Arguments: {X : Type u} {t t' : TopologicalSpace X}
Goal: t = t' ↔ ∀ (s : Set X), IsOpen s ↔ IsOpen s
-- isOpen_fold
theorem isOpen_fold {t : TopologicalSpace X} : t.IsOpen s = IsOpen[t] s 
Arguments: {X : Type u} {s : Set X} {t : TopologicalSpace X}
Goal: TopologicalSpace.IsOpen s = IsOpen s
-- isOpen_iUnion
theorem isOpen_iUnion {f : ι → Set X} (h : ∀ i, IsOpen (f i)) : IsOpen (⋃ i, f i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] {f : ι → Set X} (h : ∀ (i : ι), IsOpen (f i))
Goal: IsOpen (⋃ i, f i)
-- isOpen_biUnion
theorem isOpen_biUnion {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
    IsOpen (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i))
Goal: IsOpen (⋃ i ∈ s, f i)
-- IsOpen.union
theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∪ s₂) 
Arguments: {X : Type u} {s₁ s₂ : Set X} [TopologicalSpace X] (h₁ : IsOpen s₁) (h₂ : IsOpen s₂)
Goal: IsOpen (s₁ ∪ s₂)
-- isOpen_iff_of_cover
lemma isOpen_iff_of_cover {f : α → Set X} (ho : ∀ i, IsOpen (f i)) (hU : (⋃ i, f i) = univ) :
    IsOpen s ↔ ∀ i, IsOpen (f i ∩ s) 
Arguments: {X : Type u} {α : Type u_1} {s : Set X} [TopologicalSpace X] {f : α → Set X} (ho : ∀ (i : α), IsOpen (f i)) (hU : ⋃ i, f i = Set.univ)
Goal: IsOpen s ↔ ∀ (i : α), IsOpen (f i ∩ s)
-- isOpen_empty
@[simp] theorem isOpen_empty : IsOpen (∅ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsOpen ∅
-- Set.Finite.isOpen_sInter
theorem Set.Finite.isOpen_sInter {s : Set (Set X)} (hs : s.Finite) :
    (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set (Set X)} (hs : s.Finite)
Goal: (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s)
-- Set.Finite.isOpen_biInter
theorem Set.Finite.isOpen_biInter {s : Set α} {f : α → Set X} (hs : s.Finite)
    (h : ∀ i ∈ s, IsOpen (f i)) :
    IsOpen (⋂ i ∈ s, f i) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] {s : Set α} {f : α → Set X} (hs : s.Finite) (h : ∀ i ∈ s, IsOpen (f i))
Goal: IsOpen (⋂ i ∈ s, f i)
-- isOpen_iInter_of_finite
theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsOpen (s i)) :
    IsOpen (⋂ i, s i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] [Finite ι] {s : ι → Set X} (h : ∀ (i : ι), IsOpen (s i))
Goal: IsOpen (⋂ i, s i)
-- isOpen_biInter_finset
theorem isOpen_biInter_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
    IsOpen (⋂ i ∈ s, f i) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i))
Goal: IsOpen (⋂ i ∈ s, f i)
-- isOpen_const
@[simp] -- Porting note: added `simp`
theorem isOpen_const {p : Prop} : IsOpen { _x : X | p } 
Arguments: {X : Type u} [TopologicalSpace X] {p : Prop}
Goal: IsOpen {_x | p}
-- IsOpen.and
theorem IsOpen.and : IsOpen { x | p₁ x } → IsOpen { x | p₂ x } → IsOpen { x | p₁ x ∧ p₂ x } 
Arguments: {X : Type u} {p₁ p₂ : X → Prop} [TopologicalSpace X]
Goal: IsOpen {x | p₁ x} → IsOpen {x | p₂ x} → IsOpen {x | p₁ x ∧ p₂ x}
-- isOpen_compl_iff
@[simp] theorem isOpen_compl_iff : IsOpen sᶜ ↔ IsClosed s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsOpen sᶜ ↔ IsClosed s
-- TopologicalSpace.ext_iff_isClosed
theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
    t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s 
Arguments: {X : Type u} {t₁ t₂ : TopologicalSpace X}
Goal: t₁ = t₂ ↔ ∀ (s : Set X), IsClosed s ↔ IsClosed s
-- isClosed_const
theorem isClosed_const {p : Prop} : IsClosed { _x : X | p } 
Arguments: {X : Type u} [TopologicalSpace X] {p : Prop}
Goal: IsClosed {_x | p}
-- isClosed_empty
@[simp] theorem isClosed_empty : IsClosed (∅ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosed ∅
-- isClosed_univ
@[simp] theorem isClosed_univ : IsClosed (univ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosed Set.univ
-- IsClosed.union
theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) 
Arguments: {X : Type u} {s₁ s₂ : Set X} [TopologicalSpace X]
Goal: IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂)
-- isClosed_sInter
theorem isClosed_sInter {s : Set (Set X)} : (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set (Set X)}
Goal: (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s)
-- isClosed_iInter
theorem isClosed_iInter {f : ι → Set X} (h : ∀ i, IsClosed (f i)) : IsClosed (⋂ i, f i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] {f : ι → Set X} (h : ∀ (i : ι), IsClosed (f i))
Goal: IsClosed (⋂ i, f i)
-- isClosed_biInter
theorem isClosed_biInter {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i)) :
    IsClosed (⋂ i ∈ s, f i) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i))
Goal: IsClosed (⋂ i ∈ s, f i)
-- isClosed_compl_iff
@[simp]
theorem isClosed_compl_iff {s : Set X} : IsClosed sᶜ ↔ IsOpen s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsClosed sᶜ ↔ IsOpen s
-- IsOpen.sdiff
theorem IsOpen.sdiff (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h₁ : IsOpen s) (h₂ : IsClosed t)
Goal: IsOpen (s \ t)
-- IsClosed.inter
theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed (s₁ ∩ s₂) 
Arguments: {X : Type u} {s₁ s₂ : Set X} [TopologicalSpace X] (h₁ : IsClosed s₁) (h₂ : IsClosed s₂)
Goal: IsClosed (s₁ ∩ s₂)
-- IsClosed.sdiff
theorem IsClosed.sdiff (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h₁ : IsClosed s) (h₂ : IsOpen t)
Goal: IsClosed (s \ t)
-- Set.Finite.isClosed_biUnion
theorem Set.Finite.isClosed_biUnion {s : Set α} {f : α → Set X} (hs : s.Finite)
    (h : ∀ i ∈ s, IsClosed (f i)) :
    IsClosed (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] {s : Set α} {f : α → Set X} (hs : s.Finite) (h : ∀ i ∈ s, IsClosed (f i))
Goal: IsClosed (⋃ i ∈ s, f i)
-- isClosed_biUnion_finset
lemma isClosed_biUnion_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i)) :
    IsClosed (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i))
Goal: IsClosed (⋃ i ∈ s, f i)
-- isClosed_iUnion_of_finite
theorem isClosed_iUnion_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsClosed (s i)) :
    IsClosed (⋃ i, s i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] [Finite ι] {s : ι → Set X} (h : ∀ (i : ι), IsClosed (s i))
Goal: IsClosed (⋃ i, s i)
-- isClosed_imp
theorem isClosed_imp {p q : X → Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :
    IsClosed { x | p x → q x } 
Arguments: {X : Type u} [TopologicalSpace X] {p q : X → Prop} (hp : IsOpen {x | p x}) (hq : IsClosed {x | q x})
Goal: IsClosed {x | p x → q x}
-- IsClosed.not
theorem IsClosed.not : IsClosed { a | p a } → IsOpen { a | ¬p a } 
Arguments: {X : Type u} {p : X → Prop} [TopologicalSpace X]
Goal: IsClosed {a | p a} → IsOpen {a | ¬p a}
-- mem_interior
theorem mem_interior : x ∈ interior s ↔ ∃ t ⊆ s, IsOpen t ∧ x ∈ t 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ interior s ↔ ∃ t ⊆ s, IsOpen t ∧ x ∈ t
-- isOpen_interior
@[simp]
theorem isOpen_interior : IsOpen (interior s) 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsOpen (interior s)
-- interior_subset
theorem interior_subset : interior s ⊆ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior s ⊆ s
-- interior_maximal
theorem interior_maximal (h₁ : t ⊆ s) (h₂ : IsOpen t) : t ⊆ interior s 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h₁ : t ⊆ s) (h₂ : IsOpen t)
Goal: t ⊆ interior s
-- IsOpen.interior_eq
theorem IsOpen.interior_eq (h : IsOpen s) : interior s = s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (h : IsOpen s)
Goal: interior s = s
-- interior_eq_iff_isOpen
theorem interior_eq_iff_isOpen : interior s = s ↔ IsOpen s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior s = s ↔ IsOpen s
-- subset_interior_iff_isOpen
theorem subset_interior_iff_isOpen : s ⊆ interior s ↔ IsOpen s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: s ⊆ interior s ↔ IsOpen s
-- IsOpen.subset_interior_iff
theorem IsOpen.subset_interior_iff (h₁ : IsOpen s) : s ⊆ interior t ↔ s ⊆ t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h₁ : IsOpen s)
Goal: s ⊆ interior t ↔ s ⊆ t
-- subset_interior_iff
theorem subset_interior_iff : t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U ∧ U ⊆ s 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X]
Goal: t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U ∧ U ⊆ s
-- interior_subset_iff
lemma interior_subset_iff : interior s ⊆ t ↔ ∀ U, IsOpen U → U ⊆ s → U ⊆ t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X]
Goal: interior s ⊆ t ↔ ∀ (U : Set X), IsOpen U → U ⊆ s → U ⊆ t
-- interior_mono
@[mono, gcongr]
theorem interior_mono (h : s ⊆ t) : interior s ⊆ interior t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h : s ⊆ t)
Goal: interior s ⊆ interior t
-- interior_empty
@[simp]
theorem interior_empty : interior (∅ : Set X) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: interior ∅ = ∅
-- interior_univ
@[simp]
theorem interior_univ : interior (univ : Set X) = univ 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: interior Set.univ = Set.univ
-- interior_eq_univ
@[simp]
theorem interior_eq_univ : interior s = univ ↔ s = univ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior s = Set.univ ↔ s = Set.univ
-- interior_interior
@[simp]
theorem interior_interior : interior (interior s) = interior s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior (interior s) = interior s
-- interior_inter
@[simp]
theorem interior_inter : interior (s ∩ t) = interior s ∩ interior t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X]
Goal: interior (s ∩ t) = interior s ∩ interior t
-- Set.Finite.interior_biInter
theorem Set.Finite.interior_biInter {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set X) :
    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type u_3} {s : Set ι} (hs : s.Finite) (f : ι → Set X)
Goal: interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i)
-- Set.Finite.interior_sInter
theorem Set.Finite.interior_sInter {S : Set (Set X)} (hS : S.Finite) :
    interior (⋂₀ S) = ⋂ s ∈ S, interior s 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite)
Goal: interior (⋂₀ S) = ⋂ s ∈ S, interior s
-- Finset.interior_iInter
@[simp]
theorem Finset.interior_iInter {ι : Type*} (s : Finset ι) (f : ι → Set X) :
    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type u_3} (s : Finset ι) (f : ι → Set X)
Goal: interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i)
-- interior_iInter_of_finite
@[simp]
theorem interior_iInter_of_finite [Finite ι] (f : ι → Set X) :
    interior (⋂ i, f i) = ⋂ i, interior (f i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] [Finite ι] (f : ι → Set X)
Goal: interior (⋂ i, f i) = ⋂ i, interior (f i)