ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsCompact.eventually_forall_of_forall_eventually
theorem IsCompact.eventually_forall_of_forall_eventually {x₀ : X} {K : Set Y} (hK : IsCompact K)
    {P : X → Y → Prop} (hP : ∀ y ∈ K, ∀ᶠ z : X × Y in 𝓝 (x₀, y), P z.1 z.2) :
    ∀ᶠ x in 𝓝 x₀, ∀ y ∈ K, P x y 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ : X} {K : Set Y} (hK : IsCompact K) {P : X → Y → Prop} (hP : ∀ y ∈ K, ∀ᶠ (z : X × Y) in 𝓝 (x₀, y), P z.1 z.2)
Goal: ∀ᶠ (x : X) in 𝓝 x₀, ∀ y ∈ K, P x y
-- isCompact_empty
@[simp]
theorem isCompact_empty : IsCompact (∅ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsCompact ∅
-- isCompact_singleton
@[simp]
theorem isCompact_singleton {x : X} : IsCompact ({x} : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] {x : X}
Goal: IsCompact {x}
-- Set.Subsingleton.isCompact
theorem Set.Subsingleton.isCompact (hs : s.Subsingleton) : IsCompact s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Subsingleton)
Goal: IsCompact s
-- Set.Finite.isCompact_biUnion
theorem Set.Finite.isCompact_biUnion {s : Set ι} {f : ι → Set X} (hs : s.Finite)
    (hf : ∀ i ∈ s, IsCompact (f i)) : IsCompact (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set ι} {f : ι → Set X} (hs : s.Finite) (hf : ∀ i ∈ s, IsCompact (f i))
Goal: IsCompact (⋃ i ∈ s, f i)
-- Finset.isCompact_biUnion
theorem Finset.isCompact_biUnion (s : Finset ι) {f : ι → Set X} (hf : ∀ i ∈ s, IsCompact (f i)) :
    IsCompact (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] (s : Finset ι) {f : ι → Set X} (hf : ∀ i ∈ s, IsCompact (f i))
Goal: IsCompact (⋃ i ∈ s, f i)
-- isCompact_accumulate
theorem isCompact_accumulate {K : ℕ → Set X} (hK : ∀ n, IsCompact (K n)) (n : ℕ) :
    IsCompact (Accumulate K n) 
Arguments: {X : Type u} [TopologicalSpace X] {K : ℕ → Set X} (hK : ∀ (n : ℕ), IsCompact (K n)) (n : ℕ)
Goal: IsCompact (Set.Accumulate K n)
-- Set.Finite.isCompact_sUnion
theorem Set.Finite.isCompact_sUnion {S : Set (Set X)} (hf : S.Finite) (hc : ∀ s ∈ S, IsCompact s) :
    IsCompact (⋃₀ S) 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Finite) (hc : ∀ s ∈ S, IsCompact s)
Goal: IsCompact (⋃₀ S)
-- isCompact_iUnion
theorem isCompact_iUnion {ι : Sort*} {f : ι → Set X} [Finite ι] (h : ∀ i, IsCompact (f i)) :
    IsCompact (⋃ i, f i) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Sort u_2} {f : ι → Set X} [Finite ι] (h : ∀ (i : ι), IsCompact (f i))
Goal: IsCompact (⋃ i, f i)
-- Set.Finite.isCompact
theorem Set.Finite.isCompact (hs : s.Finite) : IsCompact s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Finite)
Goal: IsCompact s
-- IsCompact.finite_of_discrete
theorem IsCompact.finite_of_discrete [DiscreteTopology X] (hs : IsCompact s) : s.Finite 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [DiscreteTopology X] (hs : IsCompact s)
Goal: s.Finite
-- isCompact_iff_finite
theorem isCompact_iff_finite [DiscreteTopology X] : IsCompact s ↔ s.Finite 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [DiscreteTopology X]
Goal: IsCompact s ↔ s.Finite
-- IsCompact.union
theorem IsCompact.union (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s ∪ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t)
Goal: IsCompact (s ∪ t)
-- IsCompact.insert
protected theorem IsCompact.insert (hs : IsCompact s) (a) : IsCompact (insert a s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (a : X)
Goal: IsCompact (insert a s)
-- exists_subset_nhds_of_isCompact'
theorem exists_subset_nhds_of_isCompact' [Nonempty ι] {V : ι → Set X}
    (hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) (hV_closed : ∀ i, IsClosed (V i))
    {U : Set X} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] [Nonempty ι] {V : ι → Set X} (hV : Directed (fun x x_1 => x ⊇ x_1) V) (hV_cpct : ∀ (i : ι), IsCompact (V i)) (hV_closed : ∀ (i : ι), IsClosed (V i)) {U : Set X} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x)
Goal: ∃ i, V i ⊆ U
-- eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open
lemma eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open (b : ι → Set X)
    (hb : IsTopologicalBasis (Set.range b)) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U) :
    ∃ s : Set ι, s.Finite ∧ 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)) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U)
Goal: ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
-- eq_sUnion_finset_of_isTopologicalBasis_of_isCompact_open
lemma eq_sUnion_finset_of_isTopologicalBasis_of_isCompact_open (b : Set (Set X))
    (hb : IsTopologicalBasis b) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U) :
    ∃ s : Finset b, U = s.toSet.sUnion 
Arguments: {X : Type u} [TopologicalSpace X] (b : Set (Set X)) (hb : TopologicalSpace.IsTopologicalBasis b) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U)
Goal: ∃ s, U = ⋃₀ (Subtype.val '' ↑s)
-- isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis
theorem isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis (b : ι → Set X)
    (hb : IsTopologicalBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i)) (U : Set X) :
    IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ 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), IsCompact (b i)) (U : Set X)
Goal: IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
-- Filter.hasBasis_cocompact
theorem hasBasis_cocompact : (cocompact X).HasBasis IsCompact compl 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.cocompact X).HasBasis IsCompact compl
-- Filter.mem_cocompact
theorem mem_cocompact : s ∈ cocompact X ↔ ∃ t, IsCompact t ∧ tᶜ ⊆ s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.cocompact X ↔ ∃ t, IsCompact t ∧ tᶜ ⊆ s
-- Filter.mem_cocompact'
theorem mem_cocompact' : s ∈ cocompact X ↔ ∃ t, IsCompact t ∧ sᶜ ⊆ t 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.cocompact X ↔ ∃ t, IsCompact t ∧ sᶜ ⊆ t
-- IsCompact.compl_mem_cocompact
theorem _root_.IsCompact.compl_mem_cocompact (hs : IsCompact s) : sᶜ ∈ Filter.cocompact X 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s)
Goal: sᶜ ∈ Filter.cocompact X
-- Filter.cocompact_le_cofinite
theorem cocompact_le_cofinite : cocompact X ≤ cofinite 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Filter.cocompact X ≤ Filter.cofinite
-- Filter.cocompact_eq_cofinite
theorem cocompact_eq_cofinite (X : Type*) [TopologicalSpace X] [DiscreteTopology X] :
    cocompact X = cofinite 
Arguments: (X : Type u_2) [TopologicalSpace X] [DiscreteTopology X]
Goal: Filter.cocompact X = Filter.cofinite
-- Filter.disjoint_cocompact_left
theorem disjoint_cocompact_left (f : Filter X) :
    Disjoint (Filter.cocompact X) f ↔ ∃ K ∈ f, IsCompact K 
Arguments: {X : Type u} [TopologicalSpace X] (f : Filter X)
Goal: Disjoint (Filter.cocompact X) f ↔ ∃ K ∈ f, IsCompact K
-- Filter.disjoint_cocompact_right
theorem disjoint_cocompact_right (f : Filter X) :
    Disjoint f (Filter.cocompact X) ↔ ∃ K ∈ f, IsCompact K 
Arguments: {X : Type u} [TopologicalSpace X] (f : Filter X)
Goal: Disjoint f (Filter.cocompact X) ↔ ∃ K ∈ f, IsCompact K
-- Nat.cocompact_eq
@[deprecated "see `cocompact_eq_atTop` with `import Mathlib.Topology.Instances.Nat`"
  (since := "2024-02-07")]
theorem _root_.Nat.cocompact_eq : cocompact ℕ = atTop 
Goal: Filter.cocompact ℕ = Filter.atTop
-- Filter.Tendsto.isCompact_insert_range_of_cocompact
theorem Tendsto.isCompact_insert_range_of_cocompact {f : X → Y} {y}
    (hf : Tendsto f (cocompact X) (𝓝 y)) (hfc : Continuous f) : IsCompact (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.cocompact X) (𝓝 y)) (hfc : Continuous f)
Goal: IsCompact (insert y (Set.range f))
-- Filter.Tendsto.isCompact_insert_range_of_cofinite
theorem Tendsto.isCompact_insert_range_of_cofinite {f : ι → X} {x} (hf : Tendsto f cofinite (𝓝 x)) :
    IsCompact (insert x (range f)) 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] {f : ι → X} {x : X} (hf : Filter.Tendsto f Filter.cofinite (𝓝 x))
Goal: IsCompact (insert x (Set.range f))
-- Filter.Tendsto.isCompact_insert_range
theorem Tendsto.isCompact_insert_range {f : ℕ → X} {x} (hf : Tendsto f atTop (𝓝 x)) :
    IsCompact (insert x (range f)) 
Arguments: {X : Type u} [TopologicalSpace X] {f : ℕ → X} {x : X} (hf : Filter.Tendsto f Filter.atTop (𝓝 x))
Goal: IsCompact (insert x (Set.range f))
-- Filter.hasBasis_coclosedCompact
theorem hasBasis_coclosedCompact :
    (Filter.coclosedCompact X).HasBasis (fun s => IsClosed s ∧ IsCompact s) compl 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.coclosedCompact X).HasBasis (fun s => IsClosed s ∧ IsCompact s) compl
-- Filter.mem_coclosedCompact_iff
theorem mem_coclosedCompact_iff :
    s ∈ coclosedCompact X ↔ IsCompact (closure sᶜ) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.coclosedCompact X ↔ IsCompact (closure sᶜ)
-- Filter.mem_coclosedCompact
@[deprecated mem_coclosedCompact_iff (since := "2024-02-16")]
theorem mem_coclosedCompact : s ∈ coclosedCompact X ↔ ∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.coclosedCompact X ↔ ∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ s
-- Filter.mem_coclosed_compact'
@[deprecated mem_coclosedCompact_iff (since := "2024-02-16")]
theorem mem_coclosed_compact' : s ∈ coclosedCompact X ↔ ∃ t, IsClosed t ∧ IsCompact t ∧ sᶜ ⊆ t 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: s ∈ Filter.coclosedCompact X ↔ ∃ t, IsClosed t ∧ IsCompact t ∧ sᶜ ⊆ t
-- Filter.compl_mem_coclosedCompact
theorem compl_mem_coclosedCompact : sᶜ ∈ coclosedCompact X ↔ IsCompact (closure s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: sᶜ ∈ Filter.coclosedCompact X ↔ IsCompact (closure s)
-- Filter.cocompact_le_coclosedCompact
theorem cocompact_le_coclosedCompact : cocompact X ≤ coclosedCompact X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Filter.cocompact X ≤ Filter.coclosedCompact X
-- IsCompact.compl_mem_coclosedCompact_of_isClosed
theorem IsCompact.compl_mem_coclosedCompact_of_isClosed (hs : IsCompact s) (hs' : IsClosed s) :
    sᶜ ∈ Filter.coclosedCompact X 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (hs' : IsClosed s)
Goal: sᶜ ∈ Filter.coclosedCompact X
-- Bornology.inCompact.isBounded_iff
theorem inCompact.isBounded_iff : @IsBounded _ (inCompact X) s ↔ ∃ t, IsCompact t ∧ s ⊆ t 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: Bornology.IsBounded s ↔ ∃ t, IsCompact t ∧ s ⊆ t
-- IsCompact.nhdsSet_prod_eq
theorem IsCompact.nhdsSet_prod_eq {t : Set Y} (hs : IsCompact s) (ht : IsCompact t) :
    𝓝ˢ (s ×ˢ t) = 𝓝ˢ s ×ˢ 𝓝ˢ t 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {t : Set Y} (hs : IsCompact s) (ht : IsCompact t)
Goal: 𝓝ˢ (s ×ˢ t) = 𝓝ˢ s ×ˢ 𝓝ˢ t
-- nhdsSet_prod_le_of_disjoint_cocompact
theorem nhdsSet_prod_le_of_disjoint_cocompact {f : Filter Y} (hs : IsCompact s)
    (hf : Disjoint f (Filter.cocompact Y)) :
    𝓝ˢ s ×ˢ f ≤ 𝓝ˢ (s ×ˢ Set.univ) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : Filter Y} (hs : IsCompact s) (hf : Disjoint f (Filter.cocompact Y))
Goal: 𝓝ˢ s ×ˢ f ≤ 𝓝ˢ (s ×ˢ Set.univ)
-- prod_nhdsSet_le_of_disjoint_cocompact
theorem prod_nhdsSet_le_of_disjoint_cocompact {f : Filter X} (ht : IsCompact t)
    (hf : Disjoint f (Filter.cocompact X)) :
    f ×ˢ 𝓝ˢ t ≤ 𝓝ˢ (Set.univ ×ˢ t) 
Arguments: {X : Type u} [TopologicalSpace X] {t : Set X} {f : Filter X} (ht : IsCompact t) (hf : Disjoint f (Filter.cocompact X))
Goal: f ×ˢ 𝓝ˢ t ≤ 𝓝ˢ (Set.univ ×ˢ t)
-- generalized_tube_lemma
theorem generalized_tube_lemma (hs : IsCompact s) {t : Set Y} (ht : IsCompact t)
    {n : Set (X × Y)} (hn : IsOpen n) (hp : s ×ˢ t ⊆ n) :
    ∃ (u : Set X) (v : Set Y), IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (hs : IsCompact s) {t : Set Y} (ht : IsCompact t) {n : Set (X × Y)} (hn : IsOpen n) (hp : s ×ˢ t ⊆ n)
Goal: ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n
-- isCompact_univ_iff
theorem isCompact_univ_iff : IsCompact (univ : Set X) ↔ CompactSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsCompact Set.univ ↔ CompactSpace X
-- isCompact_univ
theorem isCompact_univ [h : CompactSpace X] : IsCompact (univ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] [h : CompactSpace X]
Goal: IsCompact Set.univ
-- exists_clusterPt_of_compactSpace
theorem exists_clusterPt_of_compactSpace [CompactSpace X] (f : Filter X) [NeBot f] :
    ∃ x, ClusterPt x f 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] (f : Filter X) [f.NeBot]
Goal: ∃ x, ClusterPt x f
-- Ultrafilter.le_nhds_lim
nonrec theorem Ultrafilter.le_nhds_lim [CompactSpace X] (F : Ultrafilter X) : ↑F ≤ 𝓝 F.lim 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] (F : Ultrafilter X)
Goal: ↑F ≤ 𝓝 F.lim
-- CompactSpace.elim_nhds_subcover
theorem CompactSpace.elim_nhds_subcover [CompactSpace X] (U : X → Set X) (hU : ∀ x, U x ∈ 𝓝 x) :
    ∃ t : Finset X, ⋃ x ∈ t, U x = ⊤ 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] (U : X → Set X) (hU : ∀ (x : X), U x ∈ 𝓝 x)
Goal: ∃ t, ⋃ x ∈ t, U x = ⊤
-- compactSpace_of_finite_subfamily_closed
theorem compactSpace_of_finite_subfamily_closed
    (h : ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → ⋂ i, t i = ∅ →
      ∃ u : Finset ι, ⋂ i ∈ u, t i = ∅) :
    CompactSpace X 
Arguments: {X : Type u} [TopologicalSpace X] (h : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed (t i)) → ⋂ i, t i = ∅ → ∃ u, ⋂ i ∈ u, t i = ∅)
Goal: CompactSpace X
-- IsClosed.isCompact
theorem IsClosed.isCompact [CompactSpace X] (h : IsClosed s) : IsCompact s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [CompactSpace X] (h : IsClosed s)
Goal: IsCompact s
-- le_nhds_of_unique_clusterPt
lemma le_nhds_of_unique_clusterPt [CompactSpace X] {l : Filter X} {y : X}
    (h : ∀ x, ClusterPt x l → x = y) : l ≤ 𝓝 y 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] {l : Filter X} {y : X} (h : ∀ (x : X), ClusterPt x l → x = y)
Goal: l ≤ 𝓝 y
-- tendsto_nhds_of_unique_mapClusterPt
lemma tendsto_nhds_of_unique_mapClusterPt [CompactSpace X] {l : Filter Y} {y : X} {f : Y → X}
    (h : ∀ x, MapClusterPt x l f → x = y) :
    Tendsto f l (𝓝 y) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [CompactSpace X] {l : Filter Y} {y : X} {f : Y → X} (h : ∀ (x : X), MapClusterPt x l f → x = y)
Goal: Filter.Tendsto f l (𝓝 y)
-- noncompact_univ
lemma noncompact_univ (X : Type*) [TopologicalSpace X] [NoncompactSpace X] :
    ¬IsCompact (univ : Set X) 
Arguments: (X : Type u_2) [TopologicalSpace X] [NoncompactSpace X]
Goal: ¬IsCompact Set.univ
-- IsCompact.ne_univ
theorem IsCompact.ne_univ [NoncompactSpace X] (hs : IsCompact s) : s ≠ univ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [NoncompactSpace X] (hs : IsCompact s)
Goal: s ≠ Set.univ
-- Filter.cocompact_eq_bot
@[simp]
theorem Filter.cocompact_eq_bot [CompactSpace X] : Filter.cocompact X = ⊥ 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X]
Goal: Filter.cocompact X = ⊥
-- noncompactSpace_of_neBot
theorem noncompactSpace_of_neBot (_ : NeBot (Filter.cocompact X)) : NoncompactSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.cocompact X).NeBot → NoncompactSpace X
-- Filter.cocompact_neBot_iff
theorem Filter.cocompact_neBot_iff : NeBot (Filter.cocompact X) ↔ NoncompactSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: (Filter.cocompact X).NeBot ↔ NoncompactSpace X
-- not_compactSpace_iff
theorem not_compactSpace_iff : ¬CompactSpace X ↔ NoncompactSpace X 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: ¬CompactSpace X ↔ NoncompactSpace X
-- finite_of_compact_of_discrete
theorem finite_of_compact_of_discrete [CompactSpace X] [DiscreteTopology X] : Finite X 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] [DiscreteTopology X]
Goal: Finite X
-- Set.Infinite.exists_accPt_cofinite_inf_principal_of_subset_isCompact
lemma Set.Infinite.exists_accPt_cofinite_inf_principal_of_subset_isCompact
    {K : Set X} (hs : s.Infinite) (hK : IsCompact K) (hsub : s ⊆ K) :
    ∃ x ∈ K, AccPt x (cofinite ⊓ 𝓟 s) 
Arguments: {X : Type u} [TopologicalSpace X] {s K : Set X} (hs : s.Infinite) (hK : IsCompact K) (hsub : s ⊆ K)
Goal: ∃ x ∈ K, AccPt x (Filter.cofinite ⊓ 𝓟 s)
-- Set.Infinite.exists_accPt_of_subset_isCompact
lemma Set.Infinite.exists_accPt_of_subset_isCompact {K : Set X} (hs : s.Infinite)
    (hK : IsCompact K) (hsub : s ⊆ K) : ∃ x ∈ K, AccPt x (𝓟 s) 
Arguments: {X : Type u} [TopologicalSpace X] {s K : Set X} (hs : s.Infinite) (hK : IsCompact K) (hsub : s ⊆ K)
Goal: ∃ x ∈ K, AccPt x (𝓟 s)
-- Set.Infinite.exists_accPt_cofinite_inf_principal
lemma Set.Infinite.exists_accPt_cofinite_inf_principal [CompactSpace X] (hs : s.Infinite) :
    ∃ x, AccPt x (cofinite ⊓ 𝓟 s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [CompactSpace X] (hs : s.Infinite)
Goal: ∃ x, AccPt x (Filter.cofinite ⊓ 𝓟 s)
-- Set.Infinite.exists_accPt_principal
lemma Set.Infinite.exists_accPt_principal [CompactSpace X] (hs : s.Infinite) : ∃ x, AccPt x (𝓟 s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} [CompactSpace X] (hs : s.Infinite)
Goal: ∃ x, AccPt x (𝓟 s)
-- exists_nhds_ne_neBot
theorem exists_nhds_ne_neBot (X : Type*) [TopologicalSpace X] [CompactSpace X] [Infinite X] :
    ∃ z : X, (𝓝[≠] z).NeBot 
Arguments: (X : Type u_2) [TopologicalSpace X] [CompactSpace X] [Infinite X]
Goal: ∃ z, (𝓝[≠] z).NeBot
-- finite_cover_nhds_interior
theorem finite_cover_nhds_interior [CompactSpace X] {U : X → Set X} (hU : ∀ x, U x ∈ 𝓝 x) :
    ∃ t : Finset X, ⋃ x ∈ t, interior (U x) = univ 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] {U : X → Set X} (hU : ∀ (x : X), U x ∈ 𝓝 x)
Goal: ∃ t, ⋃ x ∈ t, interior (U x) = Set.univ
-- finite_cover_nhds
theorem finite_cover_nhds [CompactSpace X] {U : X → Set X} (hU : ∀ x, U x ∈ 𝓝 x) :
    ∃ t : Finset X, ⋃ x ∈ t, U x = univ 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] {U : X → Set X} (hU : ∀ (x : X), U x ∈ 𝓝 x)
Goal: ∃ t, ⋃ x ∈ t, U x = Set.univ
-- LocallyFinite.finite_nonempty_of_compact
theorem LocallyFinite.finite_nonempty_of_compact [CompactSpace X] {f : ι → Set X}
    (hf : LocallyFinite f) : { i | (f i).Nonempty }.Finite 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] [CompactSpace X] {f : ι → Set X} (hf : LocallyFinite f)
Goal: {i | (f i).Nonempty}.Finite
-- LocallyFinite.finite_of_compact
theorem LocallyFinite.finite_of_compact [CompactSpace X] {f : ι → Set X}
    (hf : LocallyFinite f) (hne : ∀ i, (f i).Nonempty) : (univ : Set ι).Finite 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] [CompactSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hne : ∀ (i : ι), (f i).Nonempty)
Goal: Set.univ.Finite
-- Filter.comap_cocompact_le
theorem Filter.comap_cocompact_le {f : X → Y} (hf : Continuous f) :
    (Filter.cocompact Y).comap f ≤ Filter.cocompact X 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f)
Goal: Filter.comap f (Filter.cocompact Y) ≤ Filter.cocompact X
-- isCompact_range
theorem isCompact_range [CompactSpace X] {f : X → Y} (hf : Continuous f) : IsCompact (range f) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace X] {f : X → Y} (hf : Continuous f)
Goal: IsCompact (Set.range f)
-- isCompact_diagonal
theorem isCompact_diagonal [CompactSpace X] : IsCompact (diagonal X) 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X]
Goal: IsCompact (Set.diagonal X)
-- isClosedMap_snd_of_compactSpace
theorem isClosedMap_snd_of_compactSpace [CompactSpace X] :
    IsClosedMap (Prod.snd : X × Y → Y) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace X]
Goal: IsClosedMap Prod.snd
-- isClosedMap_fst_of_compactSpace
theorem isClosedMap_fst_of_compactSpace [CompactSpace Y] : IsClosedMap (Prod.fst : X × Y → X) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace Y]
Goal: IsClosedMap Prod.fst
-- exists_subset_nhds_of_compactSpace
theorem exists_subset_nhds_of_compactSpace [CompactSpace X] [Nonempty ι]
    {V : ι → Set X} (hV : Directed (· ⊇ ·) V) (hV_closed : ∀ i, IsClosed (V i)) {U : Set X}
    (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] [CompactSpace X] [Nonempty ι] {V : ι → Set X} (hV : Directed (fun x x_1 => x ⊇ x_1) V) (hV_closed : ∀ (i : ι), IsClosed (V i)) {U : Set X} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x)
Goal: ∃ i, V i ⊆ U
-- Inducing.isCompact_iff
theorem Inducing.isCompact_iff {f : X → Y} (hf : Inducing f) :
    IsCompact s ↔ IsCompact (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Inducing f)
Goal: IsCompact s ↔ IsCompact (f '' s)
-- Embedding.isCompact_iff
theorem Embedding.isCompact_iff {f : X → Y} (hf : Embedding f) :
    IsCompact s ↔ IsCompact (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Embedding f)
Goal: IsCompact s ↔ IsCompact (f '' s)
-- Inducing.isCompact_preimage
theorem Inducing.isCompact_preimage {f : X → Y} (hf : Inducing f) (hf' : IsClosed (range f))
    {K : Set Y} (hK : IsCompact K) : IsCompact (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 : IsCompact K)
Goal: IsCompact (f ⁻¹' K)
-- Inducing.isCompact_preimage_iff
lemma Inducing.isCompact_preimage_iff {f : X → Y} (hf : Inducing f) {K : Set Y}
    (Kf : K ⊆ range f) : IsCompact (f ⁻¹' K) ↔ IsCompact K 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Inducing f) {K : Set Y} (Kf : K ⊆ Set.range f)
Goal: IsCompact (f ⁻¹' K) ↔ IsCompact K
-- Inducing.isCompact_preimage'
lemma Inducing.isCompact_preimage' {f : X → Y} (hf : Inducing f) {K : Set Y}
    (hK : IsCompact K) (Kf : K ⊆ range f) : IsCompact (f ⁻¹' K) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Inducing f) {K : Set Y} (hK : IsCompact K) (Kf : K ⊆ Set.range f)
Goal: IsCompact (f ⁻¹' K)
-- ClosedEmbedding.isCompact_preimage
theorem ClosedEmbedding.isCompact_preimage {f : X → Y} (hf : ClosedEmbedding f)
    {K : Set Y} (hK : IsCompact K) : IsCompact (f ⁻¹' K) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : ClosedEmbedding f) {K : Set Y} (hK : IsCompact K)
Goal: IsCompact (f ⁻¹' K)
-- ClosedEmbedding.tendsto_cocompact
theorem ClosedEmbedding.tendsto_cocompact {f : X → Y} (hf : ClosedEmbedding f) :
    Tendsto f (Filter.cocompact X) (Filter.cocompact Y) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : ClosedEmbedding f)
Goal: Filter.Tendsto f (Filter.cocompact X) (Filter.cocompact Y)
-- Subtype.isCompact_iff
theorem Subtype.isCompact_iff {p : X → Prop} {s : Set { x // p x }} :
    IsCompact s ↔ IsCompact ((↑) '' s : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] {p : X → Prop} {s : Set { x // p x }}
Goal: IsCompact s ↔ IsCompact (Subtype.val '' s)
-- isCompact_iff_isCompact_univ
theorem isCompact_iff_isCompact_univ : IsCompact s ↔ IsCompact (univ : Set s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsCompact s ↔ IsCompact Set.univ
-- isCompact_iff_compactSpace
theorem isCompact_iff_compactSpace : IsCompact s ↔ CompactSpace s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsCompact s ↔ CompactSpace ↑s
-- IsCompact.finite
theorem IsCompact.finite (hs : IsCompact s) (hs' : DiscreteTopology s) : s.Finite 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (hs' : DiscreteTopology ↑s)
Goal: s.Finite