ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.isLindelof_biUnion
theorem Finset.isLindelof_biUnion (s : Finset ι) {f : ι → Set X} (hf : ∀ i ∈ s, IsLindelof (f i)) :
    IsLindelof (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] (s : Finset ι) {f : ι → Set X} (hf : ∀ i ∈ s, IsLindelof (f i))
Goal: IsLindelof (⋃ i ∈ s, f i)
-- isLindelof_accumulate
theorem isLindelof_accumulate {K : ℕ → Set X} (hK : ∀ n, IsLindelof (K n)) (n : ℕ) :
    IsLindelof (Accumulate K n) 
Arguments: {X : Type u} [TopologicalSpace X] {K : ℕ → Set X} (hK : ∀ (n : ℕ), IsLindelof (K n)) (n : ℕ)
Goal: IsLindelof (Set.Accumulate K n)
-- Set.Countable.isLindelof_sUnion
theorem Set.Countable.isLindelof_sUnion {S : Set (Set X)} (hf : S.Countable)
    (hc : ∀ s ∈ S, IsLindelof s) : IsLindelof (⋃₀ S) 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Countable) (hc : ∀ s ∈ S, IsLindelof s)
Goal: IsLindelof (⋃₀ S)
-- Set.Finite.isLindelof_sUnion
theorem Set.Finite.isLindelof_sUnion {S : Set (Set X)} (hf : S.Finite)
    (hc : ∀ s ∈ S, IsLindelof s) : IsLindelof (⋃₀ S) 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Finite) (hc : ∀ s ∈ S, IsLindelof s)
Goal: IsLindelof (⋃₀ S)
-- isLindelof_iUnion
theorem isLindelof_iUnion {ι : Sort*} {f : ι → Set X} [Countable ι] (h : ∀ i, IsLindelof (f i)) :
    IsLindelof (⋃ i, f i) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Sort u_2} {f : ι → Set X} [Countable ι] (h : ∀ (i : ι), IsLindelof (f i))
Goal: IsLindelof (⋃ i, f i)
-- Set.Countable.isLindelof
theorem Set.Countable.isLindelof (hs : s.Countable) : IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Countable)
Goal: IsLindelof s
-- Set.Finite.isLindelof
theorem Set.Finite.isLindelof (hs : s.Finite) : IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Finite)
Goal: IsLindelof s
-- IsLindelof.countable_of_discrete
theorem IsLindelof.countable_of_discrete [DiscreteTopology X] (hs : IsLindelof s) :
    s.Countable 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [DiscreteTopology X] (hs : IsLindelof s)
Goal: s.Countable
-- isLindelof_iff_countable
theorem isLindelof_iff_countable [DiscreteTopology X] : IsLindelof s ↔ s.Countable 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [DiscreteTopology X]
Goal: IsLindelof s ↔ s.Countable
-- IsLindelof.union
theorem IsLindelof.union (hs : IsLindelof s) (ht : IsLindelof t) : IsLindelof (s ∪ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsLindelof s) (ht : IsLindelof t)
Goal: IsLindelof (s ∪ t)
-- IsLindelof.insert
protected theorem IsLindelof.insert (hs : IsLindelof s) (a) : IsLindelof (insert a s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (a : X)
Goal: IsLindelof (insert a s)
-- isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis
theorem isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis (b : ι → Set X)
    (hb : IsTopologicalBasis (Set.range b)) (hb' : ∀ i, IsLindelof (b i)) (U : Set X) :
    IsLindelof U ∧ IsOpen U ↔ ∃ s : Set ι, s.Countable ∧ U = ⋃ i ∈ s, b i 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] (b : ι → Set X) (hb : TopologicalSpace.IsTopologicalBasis (Set.range b)) (hb' : ∀ (i : ι), IsLindelof (b i)) (U : Set X)
Goal: IsLindelof U ∧ IsOpen U ↔ ∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i
-- hasBasis_coLindelof
theorem hasBasis_coLindelof : (coLindelof X).HasBasis IsLindelof compl 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.coLindelof X).HasBasis IsLindelof compl
-- mem_coLindelof
theorem mem_coLindelof : s ∈ coLindelof X ↔ ∃ t, IsLindelof t ∧ tᶜ ⊆ s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.coLindelof X ↔ ∃ t, IsLindelof t ∧ tᶜ ⊆ s
-- mem_coLindelof'
theorem mem_coLindelof' : s ∈ coLindelof X ↔ ∃ t, IsLindelof t ∧ sᶜ ⊆ t 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.coLindelof X ↔ ∃ t, IsLindelof t ∧ sᶜ ⊆ t
-- IsLindelof.compl_mem_coLindelof
theorem _root_.IsLindelof.compl_mem_coLindelof (hs : IsLindelof s) : sᶜ ∈ coLindelof X 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s)
Goal: sᶜ ∈ Filter.coLindelof X
-- coLindelof_le_cofinite
theorem coLindelof_le_cofinite : coLindelof X ≤ cofinite 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Filter.coLindelof X ≤ Filter.cofinite
-- Tendsto.isLindelof_insert_range_of_coLindelof
theorem Tendsto.isLindelof_insert_range_of_coLindelof {f : X → Y} {y}
    (hf : Tendsto f (coLindelof X) (𝓝 y)) (hfc : Continuous f) :
    IsLindelof (insert y (range f)) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {y : Y} (hf : Filter.Tendsto f (Filter.coLindelof X) (𝓝 y)) (hfc : Continuous f)
Goal: IsLindelof (insert y (Set.range f))
-- hasBasis_coclosedLindelof
theorem hasBasis_coclosedLindelof :
    (Filter.coclosedLindelof X).HasBasis (fun s => IsClosed s ∧ IsLindelof s) compl 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.coclosedLindelof X).HasBasis (fun s => IsClosed s ∧ IsLindelof s) compl
-- mem_coclosedLindelof
theorem mem_coclosedLindelof : s ∈ coclosedLindelof X ↔
    ∃ t, IsClosed t ∧ IsLindelof t ∧ tᶜ ⊆ s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.coclosedLindelof X ↔ ∃ t, IsClosed t ∧ IsLindelof t ∧ tᶜ ⊆ s
-- mem_coclosed_Lindelof'
theorem mem_coclosed_Lindelof' : s ∈ coclosedLindelof X ↔
    ∃ t, IsClosed t ∧ IsLindelof t ∧ sᶜ ⊆ t 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.coclosedLindelof X ↔ ∃ t, IsClosed t ∧ IsLindelof t ∧ sᶜ ⊆ t
-- coLindelof_le_coclosedLindelof
theorem coLindelof_le_coclosedLindelof : coLindelof X ≤ coclosedLindelof X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Filter.coLindelof X ≤ Filter.coclosedLindelof X
-- IsLindeof.compl_mem_coclosedLindelof_of_isClosed
theorem IsLindeof.compl_mem_coclosedLindelof_of_isClosed (hs : IsLindelof s) (hs' : IsClosed s) :
    sᶜ ∈ Filter.coclosedLindelof X 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (hs' : IsClosed s)
Goal: sᶜ ∈ Filter.coclosedLindelof X
-- LindelofSpace.isLindelof_univ
class LindelofSpace (X : Type*) [TopologicalSpace X] : Prop where
    isLindelof_univ : IsLindelof (univ : Set X)
Arguments: {X : Type u_2} [TopologicalSpace X] [self : LindelofSpace X]
Goal: IsLindelof Set.univ
-- isLindelof_univ_iff
theorem isLindelof_univ_iff : IsLindelof (univ : Set X) ↔ LindelofSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsLindelof Set.univ ↔ LindelofSpace X
-- isLindelof_univ
theorem isLindelof_univ [h : LindelofSpace X] : IsLindelof (univ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] [h : LindelofSpace X]
Goal: IsLindelof Set.univ
-- cluster_point_of_Lindelof
theorem cluster_point_of_Lindelof [LindelofSpace X] (f : Filter X) [NeBot f]
    [CountableInterFilter f] : ∃ x, ClusterPt x f 
Arguments: {X : Type u} [TopologicalSpace X] [LindelofSpace X] (f : Filter X) [f.NeBot] [CountableInterFilter f]
Goal: ∃ x, ClusterPt x f
-- LindelofSpace.elim_nhds_subcover
theorem LindelofSpace.elim_nhds_subcover [LindelofSpace X] (U : X → Set X) (hU : ∀ x, U x ∈ 𝓝 x) :
    ∃ t : Set X, t.Countable ∧ ⋃ x ∈ t, U x = univ 
Arguments: {X : Type u} [TopologicalSpace X] [LindelofSpace X] (U : X → Set X) (hU : ∀ (x : X), U x ∈ 𝓝 x)
Goal: ∃ t, t.Countable ∧ ⋃ x ∈ t, U x = Set.univ
-- lindelofSpace_of_countable_subfamily_closed
theorem lindelofSpace_of_countable_subfamily_closed
    (h : ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → ⋂ i, t i = ∅ →
      ∃ u : Set ι, u.Countable ∧ ⋂ i ∈ u, t i = ∅) :
    LindelofSpace X 
Arguments: {X : Type u} [TopologicalSpace X] (h : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed (t i)) → ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ ⋂ i ∈ u, t i = ∅)
Goal: LindelofSpace X
-- IsClosed.isLindelof
theorem IsClosed.isLindelof [LindelofSpace X] (h : IsClosed s) : IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [LindelofSpace X] (h : IsClosed s)
Goal: IsLindelof s
-- IsCompact.isLindelof
theorem IsCompact.isLindelof (hs : IsCompact s) :
    IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s)
Goal: IsLindelof s
-- IsSigmaCompact.isLindelof
theorem IsSigmaCompact.isLindelof (hs : IsSigmaCompact s) :
    IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsSigmaCompact s)
Goal: IsLindelof s
-- NonLindelofSpace.nonLindelof_univ
class NonLindelofSpace (X : Type*) [TopologicalSpace X] : Prop where
    nonLindelof_univ : ¬IsLindelof (univ : Set X)
Arguments: {X : Type u_2} [TopologicalSpace X] [self : NonLindelofSpace X]
Goal: ¬IsLindelof Set.univ
-- nonLindelof_univ
lemma nonLindelof_univ (X : Type*) [TopologicalSpace X] [NonLindelofSpace X] :
    ¬IsLindelof (univ : Set X) 
Arguments: (X : Type u_2) [TopologicalSpace X] [NonLindelofSpace X]
Goal: ¬IsLindelof Set.univ
-- IsLindelof.ne_univ
theorem IsLindelof.ne_univ [NonLindelofSpace X] (hs : IsLindelof s) : s ≠ univ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [NonLindelofSpace X] (hs : IsLindelof s)
Goal: s ≠ Set.univ
-- Filter.coLindelof_eq_bot
@[simp]
theorem Filter.coLindelof_eq_bot [LindelofSpace X] : Filter.coLindelof X = ⊥ 
Arguments: {X : Type u} [TopologicalSpace X] [LindelofSpace X]
Goal: Filter.coLindelof X = ⊥
-- nonLindelofSpace_of_neBot
theorem nonLindelofSpace_of_neBot (_ : NeBot (Filter.coLindelof X)) : NonLindelofSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.coLindelof X).NeBot → NonLindelofSpace X
-- Filter.coLindelof_neBot_iff
theorem Filter.coLindelof_neBot_iff : NeBot (Filter.coLindelof X) ↔ NonLindelofSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.coLindelof X).NeBot ↔ NonLindelofSpace X
-- not_LindelofSpace_iff
theorem not_LindelofSpace_iff : ¬LindelofSpace X ↔ NonLindelofSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: ¬LindelofSpace X ↔ NonLindelofSpace X
-- countable_of_Lindelof_of_discrete
theorem countable_of_Lindelof_of_discrete [LindelofSpace X] [DiscreteTopology X] : Countable X 
Arguments: {X : Type u} [TopologicalSpace X] [LindelofSpace X] [DiscreteTopology X]
Goal: Countable X
-- countable_cover_nhds_interior
theorem countable_cover_nhds_interior [LindelofSpace X] {U : X → Set X} (hU : ∀ x, U x ∈ 𝓝 x) :
    ∃ t : Set X, t.Countable ∧ ⋃ x ∈ t, interior (U x) = univ 
Arguments: {X : Type u} [TopologicalSpace X] [LindelofSpace X] {U : X → Set X} (hU : ∀ (x : X), U x ∈ 𝓝 x)
Goal: ∃ t, t.Countable ∧ ⋃ x ∈ t, interior (U x) = Set.univ
-- countable_cover_nhds
theorem countable_cover_nhds [LindelofSpace X] {U : X → Set X} (hU : ∀ x, U x ∈ 𝓝 x) :
    ∃ t : Set X, t.Countable ∧ ⋃ x ∈ t, U x = univ 
Arguments: {X : Type u} [TopologicalSpace X] [LindelofSpace X] {U : X → Set X} (hU : ∀ (x : X), U x ∈ 𝓝 x)
Goal: ∃ t, t.Countable ∧ ⋃ x ∈ t, U x = Set.univ
-- Filter.comap_coLindelof_le
theorem Filter.comap_coLindelof_le {f : X → Y} (hf : Continuous f) :
    (Filter.coLindelof Y).comap f ≤ Filter.coLindelof X 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f)
Goal: Filter.comap f (Filter.coLindelof Y) ≤ Filter.coLindelof X
-- isLindelof_range
theorem isLindelof_range [LindelofSpace X] {f : X → Y} (hf : Continuous f) :
    IsLindelof (range f) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] [LindelofSpace X] {f : X → Y} (hf : Continuous f)
Goal: IsLindelof (Set.range f)
-- isLindelof_diagonal
theorem isLindelof_diagonal [LindelofSpace X] : IsLindelof (diagonal X) 
Arguments: {X : Type u} [TopologicalSpace X] [LindelofSpace X]
Goal: IsLindelof (Set.diagonal X)
-- Inducing.isLindelof_iff
theorem Inducing.isLindelof_iff {f : X → Y} (hf : Inducing f) :
    IsLindelof s ↔ IsLindelof (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Inducing f)
Goal: IsLindelof s ↔ IsLindelof (f '' s)
-- Embedding.isLindelof_iff
theorem Embedding.isLindelof_iff {f : X → Y} (hf : Embedding f) :
    IsLindelof s ↔ IsLindelof (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Embedding f)
Goal: IsLindelof s ↔ IsLindelof (f '' s)
-- Inducing.isLindelof_preimage
theorem Inducing.isLindelof_preimage {f : X → Y} (hf : Inducing f) (hf' : IsClosed (range f))
    {K : Set Y} (hK : IsLindelof K) : IsLindelof (f ⁻¹' K) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Inducing f) (hf' : IsClosed (Set.range f)) {K : Set Y} (hK : IsLindelof K)
Goal: IsLindelof (f ⁻¹' K)
-- ClosedEmbedding.isLindelof_preimage
theorem ClosedEmbedding.isLindelof_preimage {f : X → Y} (hf : ClosedEmbedding f)
    {K : Set Y} (hK : IsLindelof K) : IsLindelof (f ⁻¹' K) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : ClosedEmbedding f) {K : Set Y} (hK : IsLindelof K)
Goal: IsLindelof (f ⁻¹' K)
-- ClosedEmbedding.tendsto_coLindelof
theorem ClosedEmbedding.tendsto_coLindelof {f : X → Y} (hf : ClosedEmbedding f) :
    Tendsto f (Filter.coLindelof X) (Filter.coLindelof Y) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : ClosedEmbedding f)
Goal: Filter.Tendsto f (Filter.coLindelof X) (Filter.coLindelof Y)
-- Subtype.isLindelof_iff
theorem Subtype.isLindelof_iff {p : X → Prop} {s : Set { x // p x }} :
    IsLindelof s ↔ IsLindelof ((↑) '' s : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] {p : X → Prop} {s : Set { x // p x }}
Goal: IsLindelof s ↔ IsLindelof (Subtype.val '' s)
-- isLindelof_iff_isLindelof_univ
theorem isLindelof_iff_isLindelof_univ : IsLindelof s ↔ IsLindelof (univ : Set s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsLindelof s ↔ IsLindelof Set.univ
-- isLindelof_iff_LindelofSpace
theorem isLindelof_iff_LindelofSpace : IsLindelof s ↔ LindelofSpace s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsLindelof s ↔ LindelofSpace ↑s
-- IsLindelof.of_coe
lemma IsLindelof.of_coe [LindelofSpace s] : IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [LindelofSpace ↑s]
Goal: IsLindelof s
-- IsLindelof.countable
theorem IsLindelof.countable (hs : IsLindelof s) (hs' : DiscreteTopology s) : s.Countable 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (hs' : DiscreteTopology ↑s)
Goal: s.Countable
-- ClosedEmbedding.nonLindelofSpace
protected theorem ClosedEmbedding.nonLindelofSpace [NonLindelofSpace X] {f : X → Y}
    (hf : ClosedEmbedding f) : NonLindelofSpace Y 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] [NonLindelofSpace X] {f : X → Y} (hf : ClosedEmbedding f)
Goal: NonLindelofSpace Y
-- ClosedEmbedding.LindelofSpace
protected theorem ClosedEmbedding.LindelofSpace [h : LindelofSpace Y] {f : X → Y}
    (hf : ClosedEmbedding f) : LindelofSpace X 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] [h : LindelofSpace Y] {f : X → Y} (hf : ClosedEmbedding f)
Goal: LindelofSpace X
-- LindelofSpace.of_continuous_surjective
theorem LindelofSpace.of_continuous_surjective {f : X → Y} [LindelofSpace X] (hf : Continuous f)
    (hsur : Function.Surjective f) : LindelofSpace Y 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [LindelofSpace X] (hf : Continuous f) (hsur : Function.Surjective f)
Goal: LindelofSpace Y
-- HereditarilyLindelofSpace.isHereditarilyLindelof_univ
class HereditarilyLindelofSpace (X : Type*) [TopologicalSpace X] : Prop where
    isHereditarilyLindelof_univ : IsHereditarilyLindelof (univ : Set X)
Arguments: {X : Type u_2} [TopologicalSpace X] [self : HereditarilyLindelofSpace X]
Goal: IsHereditarilyLindelof Set.univ
-- IsHereditarilyLindelof.isLindelof_subset
lemma IsHereditarilyLindelof.isLindelof_subset (hs : IsHereditarilyLindelof s) (ht : t ⊆ s) :
    IsLindelof t 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsHereditarilyLindelof s) (ht : t ⊆ s)
Goal: IsLindelof t
-- IsHereditarilyLindelof.isLindelof
lemma IsHereditarilyLindelof.isLindelof (hs : IsHereditarilyLindelof s) :
    IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsHereditarilyLindelof s)
Goal: IsLindelof s
-- HereditarilyLindelof_LindelofSets
theorem HereditarilyLindelof_LindelofSets [HereditarilyLindelofSpace X] (s : Set X) :
    IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] [HereditarilyLindelofSpace X] (s : Set X)
Goal: IsLindelof s
-- eq_open_union_countable
lemma eq_open_union_countable [HereditarilyLindelofSpace X] {ι : Type u} (U : ι → Set X)
    (h : ∀ i, IsOpen (U i)) : ∃ t : Set ι, t.Countable ∧ ⋃ i∈t, U i = ⋃ i, U i 
Arguments: {X : Type u} [TopologicalSpace X] [HereditarilyLindelofSpace X] {ι : Type u} (U : ι → Set X) (h : ∀ (i : ι), IsOpen (U i))
Goal: ∃ t, t.Countable ∧ ⋃ i ∈ t, U i = ⋃ i, U i
-- OnePoint.forall
protected lemma «forall» {p : OnePoint X → Prop} :
    (∀ (x : OnePoint X), p x) ↔ p ∞ ∧ ∀ (x : X), p x 
Arguments: {X : Type u_1} {p : OnePoint X → Prop}
Goal: (∀ (x : OnePoint X), p x) ↔ p OnePoint.infty ∧ ∀ (x : X), p ↑x
-- OnePoint.exists
protected lemma «exists» {p : OnePoint X → Prop} :
    (∃ x, p x) ↔ p ∞ ∨ ∃ (x : X), p x 
Arguments: {X : Type u_1} {p : OnePoint X → Prop}
Goal: (∃ x, p x) ↔ p OnePoint.infty ∨ ∃ x, p ↑x
-- OnePoint.coe_injective
theorem coe_injective : Function.Injective ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1}
Goal: Function.Injective OnePoint.some
-- OnePoint.coe_eq_coe
@[norm_cast]
theorem coe_eq_coe {x y : X} : (x : OnePoint X) = y ↔ x = y 
Arguments: {X : Type u_1} {x y : X}
Goal: ↑x = ↑y ↔ x = y
-- OnePoint.coe_ne_infty
@[simp]
theorem coe_ne_infty (x : X) : (x : OnePoint X) ≠ ∞ 
Arguments: {X : Type u_1} (x : X)
Goal: ↑x ≠ OnePoint.infty
-- OnePoint.infty_ne_coe
@[simp]
theorem infty_ne_coe (x : X) : ∞ ≠ (x : OnePoint X) 
Arguments: {X : Type u_1} (x : X)
Goal: OnePoint.infty ≠ ↑x
-- OnePoint.isCompl_range_coe_infty
theorem isCompl_range_coe_infty : IsCompl (range ((↑) : X → OnePoint X)) {∞} 
Arguments: {X : Type u_1}
Goal: IsCompl (Set.range OnePoint.some) {OnePoint.infty}
-- OnePoint.range_coe_union_infty
theorem range_coe_union_infty : range ((↑) : X → OnePoint X) ∪ {∞} = univ 
Arguments: {X : Type u_1}
Goal: Set.range OnePoint.some ∪ {OnePoint.infty} = Set.univ
-- OnePoint.insert_infty_range_coe
@[simp]
theorem insert_infty_range_coe : insert ∞ (range (@some X)) = univ 
Arguments: {X : Type u_1}
Goal: insert OnePoint.infty (Set.range OnePoint.some) = Set.univ
-- OnePoint.range_coe_inter_infty
@[simp]
theorem range_coe_inter_infty : range ((↑) : X → OnePoint X) ∩ {∞} = ∅ 
Arguments: {X : Type u_1}
Goal: Set.range OnePoint.some ∩ {OnePoint.infty} = ∅
-- OnePoint.compl_range_coe
@[simp]
theorem compl_range_coe : (range ((↑) : X → OnePoint X))ᶜ = {∞} 
Arguments: {X : Type u_1}
Goal: (Set.range OnePoint.some)ᶜ = {OnePoint.infty}
-- OnePoint.compl_infty
theorem compl_infty : ({∞}ᶜ : Set (OnePoint X)) = range ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1}
Goal: {OnePoint.infty}ᶜ = Set.range OnePoint.some
-- OnePoint.compl_image_coe
theorem compl_image_coe (s : Set X) : ((↑) '' s : Set (OnePoint X))ᶜ = (↑) '' sᶜ ∪ {∞} 
Arguments: {X : Type u_1} (s : Set X)
Goal: (OnePoint.some '' s)ᶜ = OnePoint.some '' sᶜ ∪ {OnePoint.infty}
-- OnePoint.ne_infty_iff_exists
theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x 
Arguments: {X : Type u_1} {x : OnePoint X}
Goal: x ≠ OnePoint.infty ↔ ∃ y, ↑y = x
-- OnePoint.not_mem_range_coe_iff
theorem not_mem_range_coe_iff {x : OnePoint X} : x ∉ range some ↔ x = ∞ 
Arguments: {X : Type u_1} {x : OnePoint X}
Goal: x ∉ Set.range OnePoint.some ↔ x = OnePoint.infty
-- OnePoint.infty_not_mem_range_coe
theorem infty_not_mem_range_coe : ∞ ∉ range ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1}
Goal: OnePoint.infty ∉ Set.range OnePoint.some
-- OnePoint.infty_not_mem_image_coe
theorem infty_not_mem_image_coe {s : Set X} : ∞ ∉ ((↑) : X → OnePoint X) '' s 
Arguments: {X : Type u_1} {s : Set X}
Goal: OnePoint.infty ∉ OnePoint.some '' s
-- OnePoint.coe_preimage_infty
@[simp]
theorem coe_preimage_infty : ((↑) : X → OnePoint X) ⁻¹' {∞} = ∅ 
Arguments: {X : Type u_1}
Goal: OnePoint.some ⁻¹' {OnePoint.infty} = ∅
-- OnePoint.isOpen_def
theorem isOpen_def :
    IsOpen s ↔ (∞ ∈ s → IsCompact ((↑) ⁻¹' s : Set X)ᶜ) ∧ IsOpen ((↑) ⁻¹' s : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set (OnePoint X)}
Goal: IsOpen s ↔ (OnePoint.infty ∈ s → IsCompact (OnePoint.some ⁻¹' s)ᶜ) ∧ IsOpen (OnePoint.some ⁻¹' s)
-- OnePoint.isOpen_iff_of_mem'
theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
    IsOpen s ↔ IsCompact ((↑) ⁻¹' s : Set X)ᶜ ∧ IsOpen ((↑) ⁻¹' s : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set (OnePoint X)} (h : OnePoint.infty ∈ s)
Goal: IsOpen s ↔ IsCompact (OnePoint.some ⁻¹' s)ᶜ ∧ IsOpen (OnePoint.some ⁻¹' s)
-- OnePoint.isOpen_iff_of_mem
theorem isOpen_iff_of_mem (h : ∞ ∈ s) :
    IsOpen s ↔ IsClosed ((↑) ⁻¹' s : Set X)ᶜ ∧ IsCompact ((↑) ⁻¹' s : Set X)ᶜ 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set (OnePoint X)} (h : OnePoint.infty ∈ s)
Goal: IsOpen s ↔ IsClosed (OnePoint.some ⁻¹' s)ᶜ ∧ IsCompact (OnePoint.some ⁻¹' s)ᶜ
-- OnePoint.isOpen_iff_of_not_mem
theorem isOpen_iff_of_not_mem (h : ∞ ∉ s) : IsOpen s ↔ IsOpen ((↑) ⁻¹' s : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set (OnePoint X)} (h : OnePoint.infty ∉ s)
Goal: IsOpen s ↔ IsOpen (OnePoint.some ⁻¹' s)
-- OnePoint.isClosed_iff_of_mem
theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set (OnePoint X)} (h : OnePoint.infty ∈ s)
Goal: IsClosed s ↔ IsClosed (OnePoint.some ⁻¹' s)
-- OnePoint.isClosed_iff_of_not_mem
theorem isClosed_iff_of_not_mem (h : ∞ ∉ s) :
    IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) ∧ IsCompact ((↑) ⁻¹' s : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set (OnePoint X)} (h : OnePoint.infty ∉ s)
Goal: IsClosed s ↔ IsClosed (OnePoint.some ⁻¹' s) ∧ IsCompact (OnePoint.some ⁻¹' s)
-- OnePoint.isOpen_image_coe
@[simp]
theorem isOpen_image_coe {s : Set X} : IsOpen ((↑) '' s : Set (OnePoint X)) ↔ IsOpen s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsOpen (OnePoint.some '' s) ↔ IsOpen s
-- OnePoint.isOpen_compl_image_coe
theorem isOpen_compl_image_coe {s : Set X} :
    IsOpen ((↑) '' s : Set (OnePoint X))ᶜ ↔ IsClosed s ∧ IsCompact s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsOpen (OnePoint.some '' s)ᶜ ↔ IsClosed s ∧ IsCompact s
-- OnePoint.isClosed_image_coe
@[simp]
theorem isClosed_image_coe {s : Set X} :
    IsClosed ((↑) '' s : Set (OnePoint X)) ↔ IsClosed s ∧ IsCompact s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsClosed (OnePoint.some '' s) ↔ IsClosed s ∧ IsCompact s
-- OnePoint.infty_mem_opensOfCompl
theorem infty_mem_opensOfCompl {s : Set X} (h₁ : IsClosed s) (h₂ : IsCompact s) :
    ∞ ∈ opensOfCompl s h₁ h₂ 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (h₁ : IsClosed s) (h₂ : IsCompact s)
Goal: OnePoint.infty ∈ OnePoint.opensOfCompl s h₁ h₂
-- OnePoint.continuous_coe
@[continuity]
theorem continuous_coe : Continuous ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Continuous OnePoint.some
-- OnePoint.isOpenMap_coe
theorem isOpenMap_coe : IsOpenMap ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsOpenMap OnePoint.some
-- OnePoint.openEmbedding_coe
theorem openEmbedding_coe : OpenEmbedding ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: OpenEmbedding OnePoint.some
-- OnePoint.isOpen_range_coe
theorem isOpen_range_coe : IsOpen (range ((↑) : X → OnePoint X)) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsOpen (Set.range OnePoint.some)
-- OnePoint.isClosed_infty
theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsClosed {OnePoint.infty}
-- OnePoint.nhds_coe_eq
theorem nhds_coe_eq (x : X) : 𝓝 ↑x = map ((↑) : X → OnePoint X) (𝓝 x) 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X)
Goal: 𝓝 ↑x = Filter.map OnePoint.some (𝓝 x)