ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.tendsto_pure_self
protected theorem tendsto_pure_self (l : Filter X) :
    Tendsto (pure : X → Filter X) l (𝓝 l) 
Arguments: {X : Type u_4} (l : Filter X)
Goal: Filter.Tendsto pure l (𝓝 l)
-- Filter.HasBasis.nhds'
theorem HasBasis.nhds' {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : HasBasis l p s) :
    HasBasis (𝓝 l) p fun i => { l' | s i ∈ l' } 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: (𝓝 l).HasBasis p fun i => {l' | s i ∈ l'}
-- Filter.mem_nhds_iff
protected theorem mem_nhds_iff {l : Filter α} {S : Set (Filter α)} :
    S ∈ 𝓝 l ↔ ∃ t ∈ l, Iic (𝓟 t) ⊆ S 
Arguments: {α : Type u_2} {l : Filter α} {S : Set (Filter α)}
Goal: S ∈ 𝓝 l ↔ ∃ t ∈ l, Set.Iic (𝓟 t) ⊆ S
-- Filter.mem_nhds_iff'
theorem mem_nhds_iff' {l : Filter α} {S : Set (Filter α)} :
    S ∈ 𝓝 l ↔ ∃ t ∈ l, ∀ ⦃l' : Filter α⦄, t ∈ l' → l' ∈ S 
Arguments: {α : Type u_2} {l : Filter α} {S : Set (Filter α)}
Goal: S ∈ 𝓝 l ↔ ∃ t ∈ l, ∀ ⦃l' : Filter α⦄, t ∈ l' → l' ∈ S
-- Filter.nhds_bot
@[simp]
theorem nhds_bot : 𝓝 (⊥ : Filter α) = pure ⊥ 
Arguments: {α : Type u_2}
Goal: 𝓝 ⊥ = pure ⊥
-- Filter.nhds_top
@[simp]
theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ 
Arguments: {α : Type u_2}
Goal: 𝓝 ⊤ = ⊤
-- Filter.nhds_principal
@[simp]
theorem nhds_principal (s : Set α) : 𝓝 (𝓟 s) = 𝓟 (Iic (𝓟 s)) 
Arguments: {α : Type u_2} (s : Set α)
Goal: 𝓝 (𝓟 s) = 𝓟 (Set.Iic (𝓟 s))
-- Filter.nhds_pure
@[simp]
theorem nhds_pure (x : α) : 𝓝 (pure x : Filter α) = 𝓟 {⊥, pure x} 
Arguments: {α : Type u_2} (x : α)
Goal: 𝓝 (pure x) = 𝓟 {⊥, pure x}
-- Filter.nhds_iInf
@[simp]
theorem nhds_iInf (f : ι → Filter α) : 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} (f : ι → Filter α)
Goal: 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i)
-- Filter.nhds_inf
@[simp]
theorem nhds_inf (l₁ l₂ : Filter α) : 𝓝 (l₁ ⊓ l₂) = 𝓝 l₁ ⊓ 𝓝 l₂ 
Arguments: {α : Type u_2} (l₁ l₂ : Filter α)
Goal: 𝓝 (l₁ ⊓ l₂) = 𝓝 l₁ ⊓ 𝓝 l₂
-- Filter.monotone_nhds
theorem monotone_nhds : Monotone (𝓝 : Filter α → Filter (Filter α)) 
Arguments: {α : Type u_2}
Goal: Monotone 𝓝
-- Filter.sInter_nhds
theorem sInter_nhds (l : Filter α) : ⋂₀ { s | s ∈ 𝓝 l } = Iic l 
Arguments: {α : Type u_2} (l : Filter α)
Goal: ⋂₀ {s | s ∈ 𝓝 l} = Set.Iic l
-- Filter.nhds_mono
@[simp]
theorem nhds_mono {l₁ l₂ : Filter α} : 𝓝 l₁ ≤ 𝓝 l₂ ↔ l₁ ≤ l₂ 
Arguments: {α : Type u_2} {l₁ l₂ : Filter α}
Goal: 𝓝 l₁ ≤ 𝓝 l₂ ↔ l₁ ≤ l₂
-- Filter.mem_interior
protected theorem mem_interior {s : Set (Filter α)} {l : Filter α} :
    l ∈ interior s ↔ ∃ t ∈ l, Iic (𝓟 t) ⊆ s 
Arguments: {α : Type u_2} {s : Set (Filter α)} {l : Filter α}
Goal: l ∈ interior s ↔ ∃ t ∈ l, Set.Iic (𝓟 t) ⊆ s
-- Filter.mem_closure
protected theorem mem_closure {s : Set (Filter α)} {l : Filter α} :
    l ∈ closure s ↔ ∀ t ∈ l, ∃ l' ∈ s, t ∈ l' 
Arguments: {α : Type u_2} {s : Set (Filter α)} {l : Filter α}
Goal: l ∈ closure s ↔ ∀ t ∈ l, ∃ l' ∈ s, t ∈ l'
-- Filter.closure_singleton
@[simp]
protected theorem closure_singleton (l : Filter α) : closure {l} = Ici l 
Arguments: {α : Type u_2} (l : Filter α)
Goal: closure {l} = Set.Ici l
-- Filter.specializes_iff_le
@[simp]
theorem specializes_iff_le {l₁ l₂ : Filter α} : l₁ ⤳ l₂ ↔ l₁ ≤ l₂ 
Arguments: {α : Type u_2} {l₁ l₂ : Filter α}
Goal: l₁ ⤳ l₂ ↔ l₁ ≤ l₂
-- Filter.nhds_atTop
theorem nhds_atTop [Preorder α] : 𝓝 atTop = ⨅ x : α, 𝓟 (Iic (𝓟 (Ici x))) 
Arguments: {α : Type u_2} [Preorder α]
Goal: 𝓝 Filter.atTop = ⨅ x, 𝓟 (Set.Iic (𝓟 (Set.Ici x)))
-- Filter.tendsto_nhds_atTop_iff
protected theorem tendsto_nhds_atTop_iff [Preorder β] {l : Filter α} {f : α → Filter β} :
    Tendsto f l (𝓝 atTop) ↔ ∀ y, ∀ᶠ a in l, Ici y ∈ f a 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder β] {l : Filter α} {f : α → Filter β}
Goal: Filter.Tendsto f l (𝓝 Filter.atTop) ↔ ∀ (y : β), ∀ᶠ (a : α) in l, Set.Ici y ∈ f a
-- Filter.nhds_atBot
theorem nhds_atBot [Preorder α] : 𝓝 atBot = ⨅ x : α, 𝓟 (Iic (𝓟 (Iic x))) 
Arguments: {α : Type u_2} [Preorder α]
Goal: 𝓝 Filter.atBot = ⨅ x, 𝓟 (Set.Iic (𝓟 (Set.Iic x)))
-- Filter.tendsto_nhds_atBot_iff
protected theorem tendsto_nhds_atBot_iff [Preorder β] {l : Filter α} {f : α → Filter β} :
    Tendsto f l (𝓝 atBot) ↔ ∀ y, ∀ᶠ a in l, Iic y ∈ f a 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder β] {l : Filter α} {f : α → Filter β}
Goal: Filter.Tendsto f l (𝓝 Filter.atBot) ↔ ∀ (y : β), ∀ᶠ (a : α) in l, Set.Iic y ∈ f a
-- Filter.nhds_nhds
theorem nhds_nhds (x : X) :
    𝓝 (𝓝 x) = ⨅ (s : Set X) (_ : IsOpen s) (_ : x ∈ s), 𝓟 (Iic (𝓟 s)) 
Arguments: {X : Type u_4} [TopologicalSpace X] (x : X)
Goal: 𝓝 (𝓝 x) = ⨅ s, ⨅ (_ : IsOpen s), ⨅ (_ : x ∈ s), 𝓟 (Set.Iic (𝓟 s))
-- Filter.inducing_nhds
theorem inducing_nhds : Inducing (𝓝 : X → Filter X) 
Arguments: {X : Type u_4} [TopologicalSpace X]
Goal: Inducing 𝓝
-- Filter.continuous_nhds
@[continuity]
theorem continuous_nhds : Continuous (𝓝 : X → Filter X) 
Arguments: {X : Type u_4} [TopologicalSpace X]
Goal: Continuous 𝓝
-- Filter.Tendsto.nhds
protected theorem Tendsto.nhds {f : α → X} {l : Filter α} {x : X} (h : Tendsto f l (𝓝 x)) :
    Tendsto (𝓝 ∘ f) l (𝓝 (𝓝 x)) 
Arguments: {α : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : α → X} {l : Filter α} {x : X} (h : Filter.Tendsto f l (𝓝 x))
Goal: Filter.Tendsto (𝓝 ∘ f) l (𝓝 (𝓝 x))
-- ContinuousWithinAt.nhds
protected nonrec theorem ContinuousWithinAt.nhds (h : ContinuousWithinAt f s x) :
    ContinuousWithinAt (𝓝 ∘ f) s x 
Arguments: {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {s : Set X} (h : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (𝓝 ∘ f) s x
-- ContinuousAt.nhds
protected nonrec theorem ContinuousAt.nhds (h : ContinuousAt f x) : ContinuousAt (𝓝 ∘ f) x 
Arguments: {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} (h : ContinuousAt f x)
Goal: ContinuousAt (𝓝 ∘ f) x
-- ContinuousOn.nhds
protected nonrec theorem ContinuousOn.nhds (h : ContinuousOn f s) : ContinuousOn (𝓝 ∘ f) s 
Arguments: {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (h : ContinuousOn f s)
Goal: ContinuousOn (𝓝 ∘ f) s
-- Continuous.nhds
protected nonrec theorem Continuous.nhds (h : Continuous f) : Continuous (𝓝 ∘ f) 
Arguments: {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : Continuous f)
Goal: Continuous (𝓝 ∘ f)
-- IsOpen.isGδ
theorem IsOpen.isGδ {s : Set X} (h : IsOpen s) : IsGδ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : IsOpen s)
Goal: IsGδ s
-- IsGδ.empty
@[simp]
protected theorem IsGδ.empty : IsGδ (∅ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsGδ ∅
-- IsGδ.univ
@[simp]
protected theorem IsGδ.univ : IsGδ (univ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsGδ Set.univ
-- IsGδ.biInter_of_isOpen
theorem IsGδ.biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set X}
    (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) 
Arguments: {X : Type u_1} {ι : Type u_3} [TopologicalSpace X] {I : Set ι} (hI : I.Countable) {f : ι → Set X} (hf : ∀ i ∈ I, IsOpen (f i))
Goal: IsGδ (⋂ i ∈ I, f i)
-- IsGδ.iInter_of_isOpen
theorem IsGδ.iInter_of_isOpen [Countable ι'] {f : ι' → Set X} (hf : ∀ i, IsOpen (f i)) :
    IsGδ (⋂ i, f i) 
Arguments: {X : Type u_1} {ι' : Sort u_4} [TopologicalSpace X] [Countable ι'] {f : ι' → Set X} (hf : ∀ (i : ι'), IsOpen (f i))
Goal: IsGδ (⋂ i, f i)
-- isGδ_iff_eq_iInter_nat
lemma isGδ_iff_eq_iInter_nat {s : Set X} :
    IsGδ s ↔ ∃ (f : ℕ → Set X), (∀ n, IsOpen (f n)) ∧ s = ⋂ n, f n 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsGδ s ↔ ∃ f, (∀ (n : ℕ), IsOpen (f n)) ∧ s = ⋂ n, f n
-- IsGδ.iInter
protected theorem IsGδ.iInter [Countable ι'] {s : ι' → Set X} (hs : ∀ i, IsGδ (s i)) :
    IsGδ (⋂ i, s i) 
Arguments: {X : Type u_1} {ι' : Sort u_4} [TopologicalSpace X] [Countable ι'] {s : ι' → Set X} (hs : ∀ (i : ι'), IsGδ (s i))
Goal: IsGδ (⋂ i, s i)
-- IsGδ.biInter
theorem IsGδ.biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set X}
    (ht : ∀ (i) (hi : i ∈ s), IsGδ (t i hi)) : IsGδ (⋂ i ∈ s, t i ‹_›) 
Arguments: {X : Type u_1} {ι : Type u_3} [TopologicalSpace X] {s : Set ι} (hs : s.Countable) {t : (i : ι) → i ∈ s → Set X} (ht : ∀ (i : ι) (hi : i ∈ s), IsGδ (t i hi))
Goal: IsGδ (⋂ i, ⋂ (h : i ∈ s), t i h)
-- IsGδ.sInter
theorem IsGδ.sInter {S : Set (Set X)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable)
Goal: IsGδ (⋂₀ S)
-- IsGδ.inter
theorem IsGδ.inter {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} (hs : IsGδ s) (ht : IsGδ t)
Goal: IsGδ (s ∩ t)
-- IsGδ.union
theorem IsGδ.union {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} (hs : IsGδ s) (ht : IsGδ t)
Goal: IsGδ (s ∪ t)
-- IsGδ.sUnion
theorem IsGδ.sUnion {S : Set (Set X)} (hS : S.Finite) (h : ∀ s ∈ S, IsGδ s) : IsGδ (⋃₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite) (h : ∀ s ∈ S, IsGδ s)
Goal: IsGδ (⋃₀ S)
-- IsGδ.biUnion
theorem IsGδ.biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
    IsGδ (⋃ i ∈ s, f i) 
Arguments: {X : Type u_1} {ι : Type u_3} [TopologicalSpace X] {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i))
Goal: IsGδ (⋃ i ∈ s, f i)
-- IsGδ.iUnion
theorem IsGδ.iUnion [Finite ι'] {f : ι' → Set X} (h : ∀ i, IsGδ (f i)) : IsGδ (⋃ i, f i) 
Arguments: {X : Type u_1} {ι' : Sort u_4} [TopologicalSpace X] [Finite ι'] {f : ι' → Set X} (h : ∀ (i : ι'), IsGδ (f i))
Goal: IsGδ (⋃ i, f i)
-- IsClosed.isGδ
theorem IsClosed.isGδ {X : Type*} [UniformSpace X] [IsCountablyGenerated (𝓤 X)] {s : Set X}
    (hs : IsClosed s) : IsGδ s 
Arguments: {X : Type u_5} [UniformSpace X] [(𝓤 X).IsCountablyGenerated] {s : Set X} (hs : IsClosed s)
Goal: IsGδ s
-- IsGδ.setOf_continuousAt
theorem IsGδ.setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (𝓤 Y)] (f : X → Y) :
    IsGδ { x | ContinuousAt f x } 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [UniformSpace Y] [(𝓤 Y).IsCountablyGenerated] (f : X → Y)
Goal: IsGδ {x | ContinuousAt f x}
-- residual_of_dense_open
theorem residual_of_dense_open {s : Set X} (ho : IsOpen s) (hd : Dense s) : s ∈ residual X 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (ho : IsOpen s) (hd : Dense s)
Goal: s ∈ residual X
-- residual_of_dense_Gδ
theorem residual_of_dense_Gδ {s : Set X} (ho : IsGδ s) (hd : Dense s) : s ∈ residual X 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (ho : IsGδ s) (hd : Dense s)
Goal: s ∈ residual X
-- mem_residual_iff
theorem mem_residual_iff {s : Set X} :
    s ∈ residual X ↔
      ∃ S : Set (Set X), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: s ∈ residual X ↔ ∃ S, (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s
-- isNowhereDense_empty
@[simp]
lemma isNowhereDense_empty : IsNowhereDense (∅ : Set X) 
Arguments: {X : Type u_5} [TopologicalSpace X]
Goal: IsNowhereDense ∅
-- IsClosed.isNowhereDense_iff
lemma IsClosed.isNowhereDense_iff {s : Set X} (hs : IsClosed s) :
    IsNowhereDense s ↔ interior s = ∅ 
Arguments: {X : Type u_5} [TopologicalSpace X] {s : Set X} (hs : IsClosed s)
Goal: IsNowhereDense s ↔ interior s = ∅
-- IsNowhereDense.closure
protected lemma IsNowhereDense.closure {s : Set X} (hs : IsNowhereDense s) :
    IsNowhereDense (closure s) 
Arguments: {X : Type u_5} [TopologicalSpace X] {s : Set X} (hs : IsNowhereDense s)
Goal: IsNowhereDense (closure s)
-- IsNowhereDense.subset_of_closed_isNowhereDense
lemma IsNowhereDense.subset_of_closed_isNowhereDense {s : Set X} (hs : IsNowhereDense s) :
    ∃ t : Set X, s ⊆ t ∧ IsNowhereDense t ∧ IsClosed t 
Arguments: {X : Type u_5} [TopologicalSpace X] {s : Set X} (hs : IsNowhereDense s)
Goal: ∃ t, s ⊆ t ∧ IsNowhereDense t ∧ IsClosed t
-- isClosed_isNowhereDense_iff_compl
lemma isClosed_isNowhereDense_iff_compl {s : Set X} :
    IsClosed s ∧ IsNowhereDense s ↔ IsOpen sᶜ ∧ Dense sᶜ 
Arguments: {X : Type u_5} [TopologicalSpace X] {s : Set X}
Goal: IsClosed s ∧ IsNowhereDense s ↔ IsOpen sᶜ ∧ Dense sᶜ
-- meagre_empty
lemma meagre_empty : IsMeagre (∅ : Set X) 
Arguments: {X : Type u_5} [TopologicalSpace X]
Goal: IsMeagre ∅
-- IsMeagre.mono
lemma IsMeagre.mono {s t : Set X} (hs : IsMeagre s) (hts : t ⊆ s) : IsMeagre t 
Arguments: {X : Type u_5} [TopologicalSpace X] {s t : Set X} (hs : IsMeagre s) (hts : t ⊆ s)
Goal: IsMeagre t
-- IsMeagre.inter
lemma IsMeagre.inter {s t : Set X} (hs : IsMeagre s) : IsMeagre (s ∩ t) 
Arguments: {X : Type u_5} [TopologicalSpace X] {s t : Set X} (hs : IsMeagre s)
Goal: IsMeagre (s ∩ t)
-- isMeagre_iUnion
lemma isMeagre_iUnion {s : ℕ → Set X} (hs : ∀ n, IsMeagre (s n)) : IsMeagre (⋃ n, s n) 
Arguments: {X : Type u_5} [TopologicalSpace X] {s : ℕ → Set X} (hs : ∀ (n : ℕ), IsMeagre (s n))
Goal: IsMeagre (⋃ n, s n)
-- isMeagre_iff_countable_union_isNowhereDense
lemma isMeagre_iff_countable_union_isNowhereDense {s : Set X} :
    IsMeagre s ↔ ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S 
Arguments: {X : Type u_5} [TopologicalSpace X] {s : Set X}
Goal: IsMeagre s ↔ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
-- TopologicalSpace.Opens.is_open'
structure Opens where
    carrier : Set α
    is_open' : IsOpen carrier
Arguments: {α : Type u_2} [TopologicalSpace α] (self : TopologicalSpace.Opens α)
Goal: IsOpen self.carrier
-- TopologicalSpace.Opens.forall
theorem «forall» {p : Opens α → Prop} : (∀ U, p U) ↔ ∀ (U : Set α) (hU : IsOpen U), p ⟨U, hU⟩ 
Arguments: {α : Type u_2} [TopologicalSpace α] {p : TopologicalSpace.Opens α → Prop}
Goal: (∀ (U : TopologicalSpace.Opens α), p U) ↔ ∀ (U : Set α) (hU : IsOpen U), p { carrier := U, is_open' := hU }
-- TopologicalSpace.Opens.carrier_eq_coe
@[simp] theorem carrier_eq_coe (U : Opens α) : U.1 = ↑U 
Arguments: {α : Type u_2} [TopologicalSpace α] (U : TopologicalSpace.Opens α)
Goal: U.carrier = ↑U
-- TopologicalSpace.Opens.coe_mk
@[simp]
theorem coe_mk {U : Set α} {hU : IsOpen U} : ↑(⟨U, hU⟩ : Opens α) = U 
Arguments: {α : Type u_2} [TopologicalSpace α] {U : Set α} {hU : IsOpen U}
Goal: ↑{ carrier := U, is_open' := hU } = U
-- TopologicalSpace.Opens.mem_mk
@[simp]
theorem mem_mk {x : α} {U : Set α} {h : IsOpen U} : x ∈ mk U h ↔ x ∈ U 
Arguments: {α : Type u_2} [TopologicalSpace α] {x : α} {U : Set α} {h : IsOpen U}
Goal: x ∈ { carrier := U, is_open' := h } ↔ x ∈ U
-- TopologicalSpace.Opens.nonempty_coeSort
protected theorem nonempty_coeSort {U : Opens α} : Nonempty U ↔ (U : Set α).Nonempty 
Arguments: {α : Type u_2} [TopologicalSpace α] {U : TopologicalSpace.Opens α}
Goal: Nonempty ↥U ↔ (↑U).Nonempty
-- TopologicalSpace.Opens.nonempty_coe
protected theorem nonempty_coe {U : Opens α} : (U : Set α).Nonempty ↔ ∃ x, x ∈ U 
Arguments: {α : Type u_2} [TopologicalSpace α] {U : TopologicalSpace.Opens α}
Goal: (↑U).Nonempty ↔ ∃ x, x ∈ U
-- TopologicalSpace.Opens.ext
@[ext] -- Porting note (#11215): TODO: replace with `∀ x, x ∈ U ↔ x ∈ V`
theorem ext {U V : Opens α} (h : (U : Set α) = V) : U = V 
Arguments: {α : Type u_2} [TopologicalSpace α] {U V : TopologicalSpace.Opens α} (h : ↑U = ↑V)
Goal: U = V
-- TopologicalSpace.Opens.coe_inj
theorem coe_inj {U V : Opens α} : (U : Set α) = V ↔ U = V 
Arguments: {α : Type u_2} [TopologicalSpace α] {U V : TopologicalSpace.Opens α}
Goal: ↑U = ↑V ↔ U = V
-- TopologicalSpace.Opens.isOpen
protected theorem isOpen (U : Opens α) : IsOpen (U : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (U : TopologicalSpace.Opens α)
Goal: IsOpen ↑U
-- TopologicalSpace.Opens.mk_coe
@[simp] theorem mk_coe (U : Opens α) : mk (↑U) U.isOpen = U 
Arguments: {α : Type u_2} [TopologicalSpace α] (U : TopologicalSpace.Opens α)
Goal: { carrier := ↑U, is_open' := ⋯ } = U
-- TopologicalSpace.Opens.gc
theorem gc : GaloisConnection ((↑) : Opens α → Set α) interior 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: GaloisConnection SetLike.coe TopologicalSpace.Opens.interior
-- TopologicalSpace.Opens.mk_inf_mk
@[simp]
theorem mk_inf_mk {U V : Set α} {hU : IsOpen U} {hV : IsOpen V} :
    (⟨U, hU⟩ ⊓ ⟨V, hV⟩ : Opens α) = ⟨U ⊓ V, IsOpen.inter hU hV⟩ 
Arguments: {α : Type u_2} [TopologicalSpace α] {U V : Set α} {hU : IsOpen U} {hV : IsOpen V}
Goal: { carrier := U, is_open' := hU } ⊓ { carrier := V, is_open' := hV } = { carrier := U ⊓ V, is_open' := ⋯ }
-- TopologicalSpace.Opens.coe_inf
@[simp, norm_cast]
theorem coe_inf (s t : Opens α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Opens α)
Goal: ↑(s ⊓ t) = ↑s ∩ ↑t
-- TopologicalSpace.Opens.coe_sup
@[simp, norm_cast]
theorem coe_sup (s t : Opens α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Opens α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- TopologicalSpace.Opens.coe_bot
@[simp, norm_cast]
theorem coe_bot : ((⊥ : Opens α) : Set α) = ∅ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: ↑⊥ = ∅
-- TopologicalSpace.Opens.mk_empty
@[simp] theorem mk_empty : (⟨∅, isOpen_empty⟩ : Opens α) = ⊥ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: { carrier := ∅, is_open' := ⋯ } = ⊥
-- TopologicalSpace.Opens.coe_eq_empty
@[simp, norm_cast]
theorem coe_eq_empty {U : Opens α} : (U : Set α) = ∅ ↔ U = ⊥ 
Arguments: {α : Type u_2} [TopologicalSpace α] {U : TopologicalSpace.Opens α}
Goal: ↑U = ∅ ↔ U = ⊥
-- TopologicalSpace.Opens.mem_top
@[simp]
lemma mem_top (x : α) : x ∈ (⊤ : Opens α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (x : α)
Goal: x ∈ ⊤
-- TopologicalSpace.Opens.coe_top
@[simp, norm_cast]
theorem coe_top : ((⊤ : Opens α) : Set α) = Set.univ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: ↑⊤ = Set.univ
-- TopologicalSpace.Opens.mk_univ
@[simp] theorem mk_univ : (⟨univ, isOpen_univ⟩ : Opens α) = ⊤ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: { carrier := Set.univ, is_open' := ⋯ } = ⊤
-- TopologicalSpace.Opens.coe_eq_univ
@[simp, norm_cast]
theorem coe_eq_univ {U : Opens α} : (U : Set α) = univ ↔ U = ⊤ 
Arguments: {α : Type u_2} [TopologicalSpace α] {U : TopologicalSpace.Opens α}
Goal: ↑U = Set.univ ↔ U = ⊤
-- TopologicalSpace.Opens.coe_sSup
@[simp, norm_cast]
theorem coe_sSup {S : Set (Opens α)} : (↑(sSup S) : Set α) = ⋃ i ∈ S, ↑i 
Arguments: {α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Opens α)}
Goal: ↑(sSup S) = ⋃ i ∈ S, ↑i
-- TopologicalSpace.Opens.coe_finset_sup
@[simp, norm_cast]
theorem coe_finset_sup (f : ι → Opens α) (s : Finset ι) : (↑(s.sup f) : Set α) = s.sup ((↑) ∘ f) 
Arguments: {ι : Type u_1} {α : Type u_2} [TopologicalSpace α] (f : ι → TopologicalSpace.Opens α) (s : Finset ι)
Goal: ↑(s.sup f) = s.sup (SetLike.coe ∘ f)
-- TopologicalSpace.Opens.coe_finset_inf
@[simp, norm_cast]
theorem coe_finset_inf (f : ι → Opens α) (s : Finset ι) : (↑(s.inf f) : Set α) = s.inf ((↑) ∘ f) 
Arguments: {ι : Type u_1} {α : Type u_2} [TopologicalSpace α] (f : ι → TopologicalSpace.Opens α) (s : Finset ι)
Goal: ↑(s.inf f) = s.inf (SetLike.coe ∘ f)
-- TopologicalSpace.Opens.coe_iSup
@[simp, norm_cast]
theorem coe_iSup {ι} (s : ι → Opens α) : ((⨆ i, s i : Opens α) : Set α) = ⋃ i, s i 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_5} (s : ι → TopologicalSpace.Opens α)
Goal: ↑(⨆ i, s i) = ⋃ i, ↑(s i)
-- TopologicalSpace.Opens.iSup_def
theorem iSup_def {ι} (s : ι → Opens α) : ⨆ i, s i = ⟨⋃ i, s i, isOpen_iUnion fun i => (s i).2⟩ 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_5} (s : ι → TopologicalSpace.Opens α)
Goal: ⨆ i, s i = { carrier := ⋃ i, ↑(s i), is_open' := ⋯ }
-- TopologicalSpace.Opens.iSup_mk
@[simp]
theorem iSup_mk {ι} (s : ι → Set α) (h : ∀ i, IsOpen (s i)) :
    (⨆ i, ⟨s i, h i⟩ : Opens α) = ⟨⋃ i, s i, isOpen_iUnion h⟩ 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_5} (s : ι → Set α) (h : ∀ (i : ι), IsOpen (s i))
Goal: ⨆ i, { carrier := s i, is_open' := ⋯ } = { carrier := ⋃ i, s i, is_open' := ⋯ }
-- TopologicalSpace.Opens.mem_iSup
@[simp]
theorem mem_iSup {ι} {x : α} {s : ι → Opens α} : x ∈ iSup s ↔ ∃ i, x ∈ s i 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_5} {x : α} {s : ι → TopologicalSpace.Opens α}
Goal: x ∈ iSup s ↔ ∃ i, x ∈ s i
-- TopologicalSpace.Opens.mem_sSup
@[simp]
theorem mem_sSup {Us : Set (Opens α)} {x : α} : x ∈ sSup Us ↔ ∃ u ∈ Us, x ∈ u 
Arguments: {α : Type u_2} [TopologicalSpace α] {Us : Set (TopologicalSpace.Opens α)} {x : α}
Goal: x ∈ sSup Us ↔ ∃ u ∈ Us, x ∈ u
-- TopologicalSpace.Opens.openEmbedding'
theorem openEmbedding' (U : Opens α) : OpenEmbedding (Subtype.val : U → α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (U : TopologicalSpace.Opens α)
Goal: OpenEmbedding Subtype.val
-- TopologicalSpace.Opens.openEmbedding_of_le
theorem openEmbedding_of_le {U V : Opens α} (i : U ≤ V) :
    OpenEmbedding (Set.inclusion <| SetLike.coe_subset_coe.2 i) 
Arguments: {α : Type u_2} [TopologicalSpace α] {U V : TopologicalSpace.Opens α} (i : U ≤ V)
Goal: OpenEmbedding (Set.inclusion ⋯)
-- TopologicalSpace.Opens.not_nonempty_iff_eq_bot
theorem not_nonempty_iff_eq_bot (U : Opens α) : ¬Set.Nonempty (U : Set α) ↔ U = ⊥ 
Arguments: {α : Type u_2} [TopologicalSpace α] (U : TopologicalSpace.Opens α)
Goal: ¬(↑U).Nonempty ↔ U = ⊥
-- TopologicalSpace.Opens.ne_bot_iff_nonempty
theorem ne_bot_iff_nonempty (U : Opens α) : U ≠ ⊥ ↔ Set.Nonempty (U : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (U : TopologicalSpace.Opens α)
Goal: U ≠ ⊥ ↔ (↑U).Nonempty
-- TopologicalSpace.Opens.eq_bot_or_top
theorem eq_bot_or_top {α} [t : TopologicalSpace α] (h : t = ⊤) (U : Opens α) : U = ⊥ ∨ U = ⊤ 
Arguments: {α : Type u_5} [t : TopologicalSpace α] (h : t = ⊤) (U : TopologicalSpace.Opens α)
Goal: U = ⊥ ∨ U = ⊤
-- TopologicalSpace.Opens.isBasis_iff_nbhd
theorem isBasis_iff_nbhd {B : Set (Opens α)} :
    IsBasis B ↔ ∀ {U : Opens α} {x}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U 
Arguments: {α : Type u_2} [TopologicalSpace α] {B : Set (TopologicalSpace.Opens α)}
Goal: TopologicalSpace.Opens.IsBasis B ↔ ∀ {U : TopologicalSpace.Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
-- TopologicalSpace.Opens.isBasis_iff_cover
theorem isBasis_iff_cover {B : Set (Opens α)} :
    IsBasis B ↔ ∀ U : Opens α, ∃ Us, Us ⊆ B ∧ U = sSup Us 
Arguments: {α : Type u_2} [TopologicalSpace α] {B : Set (TopologicalSpace.Opens α)}
Goal: TopologicalSpace.Opens.IsBasis B ↔ ∀ (U : TopologicalSpace.Opens α), ∃ Us ⊆ B, U = sSup Us
-- TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion
theorem IsBasis.isCompact_open_iff_eq_finite_iUnion {ι : Type*} (b : ι → Opens α)
    (hb : IsBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i : Set α)) (U : Set α) :
    IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Type u_5} (b : ι → TopologicalSpace.Opens α) (hb : TopologicalSpace.Opens.IsBasis (Set.range b)) (hb' : ∀ (i : ι), IsCompact ↑(b i)) (U : Set α)
Goal: IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, ↑(b i)
-- TopologicalSpace.Opens.IsBasis.le_iff
lemma IsBasis.le_iff {α} {t₁ t₂ : TopologicalSpace α}
    {Us : Set (Opens α)} (hUs : @IsBasis α t₂ Us) :
    t₁ ≤ t₂ ↔ ∀ U ∈ Us, IsOpen[t₁] U 
Arguments: {α : Type u_5} {t₁ t₂ : TopologicalSpace α} {Us : Set (TopologicalSpace.Opens α)} (hUs : TopologicalSpace.Opens.IsBasis Us)
Goal: t₁ ≤ t₂ ↔ ∀ U ∈ Us, IsOpen ↑U
-- TopologicalSpace.Opens.isCompactElement_iff
@[simp]
theorem isCompactElement_iff (s : Opens α) :
    CompleteLattice.IsCompactElement s ↔ IsCompact (s : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Opens α)
Goal: CompleteLattice.IsCompactElement s ↔ IsCompact ↑s
-- TopologicalSpace.Opens.comap_id
@[simp]
theorem comap_id : comap (ContinuousMap.id α) = FrameHom.id _ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: TopologicalSpace.Opens.comap (ContinuousMap.id α) = FrameHom.id (TopologicalSpace.Opens α)
-- TopologicalSpace.Opens.comap_mono
theorem comap_mono (f : C(α, β)) {s t : Opens β} (h : s ≤ t) : comap f s ≤ comap f t 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) {s t : TopologicalSpace.Opens β} (h : s ≤ t)
Goal: (TopologicalSpace.Opens.comap f) s ≤ (TopologicalSpace.Opens.comap f) t