ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AddCircle.exists_gcd_eq_one_of_isOfFinAddOrder
theorem exists_gcd_eq_one_of_isOfFinAddOrder {u : AddCircle p} (h : IsOfFinAddOrder u) :
    ∃ m : ℕ, m.gcd (addOrderOf u) = 1 ∧ m < addOrderOf u ∧ ↑((m : 𝕜) / addOrderOf u * p) = u 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {u : AddCircle p} (h : IsOfFinAddOrder u)
Goal: ∃ m, m.gcd (addOrderOf u) = 1 ∧ m < addOrderOf u ∧ ↑(↑m / ↑(addOrderOf u) * p) = u
-- AddCircle.card_addOrderOf_eq_totient
@[simp]
theorem card_addOrderOf_eq_totient {n : ℕ} :
    Nat.card { u : AddCircle p // addOrderOf u = n } = n.totient 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] {n : ℕ}
Goal: Nat.card { u // addOrderOf u = n } = n.totient
-- AddCircle.finite_setOf_add_order_eq
theorem finite_setOf_add_order_eq {n : ℕ} (hn : 0 < n) :
    { u : AddCircle p | addOrderOf u = n }.Finite 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] {n : ℕ} (hn : 0 < n)
Goal: {u | addOrderOf u = n}.Finite
-- AddCircle.equivIccQuot_comp_mk_eq_toIcoMod
theorem equivIccQuot_comp_mk_eq_toIcoMod :
    equivIccQuot p a ∘ Quotient.mk'' = fun x =>
      Quot.mk _ ⟨toIcoMod hp.out a x, Ico_subset_Icc_self <| toIcoMod_mem_Ico _ _ x⟩ 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p a : 𝕜) [hp : Fact (0 < p)] [Archimedean 𝕜]
Goal: ⇑(AddCircle.equivIccQuot p a) ∘ Quotient.mk'' = fun x => Quot.mk (AddCircle.EndpointIdent p a) ⟨toIcoMod ⋯ a x, ⋯⟩
-- AddCircle.equivIccQuot_comp_mk_eq_toIocMod
theorem equivIccQuot_comp_mk_eq_toIocMod :
    equivIccQuot p a ∘ Quotient.mk'' = fun x =>
      Quot.mk _ ⟨toIocMod hp.out a x, Ioc_subset_Icc_self <| toIocMod_mem_Ioc _ _ x⟩ 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p a : 𝕜) [hp : Fact (0 < p)] [Archimedean 𝕜]
Goal: ⇑(AddCircle.equivIccQuot p a) ∘ Quotient.mk'' = fun x => Quot.mk (AddCircle.EndpointIdent p a) ⟨toIocMod ⋯ a x, ⋯⟩
-- AddCircle.liftIco_eq_lift_Icc
theorem liftIco_eq_lift_Icc {f : 𝕜 → B} (h : f a = f (a + p)) :
    liftIco p a f =
      Quot.lift (restrict (Icc a <| a + p) f)
          (by
            rintro _ _ ⟨_⟩
            exact h) ∘
        equivIccQuot p a 
Arguments: {𝕜 : Type u_1} {B : Type u_2} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p a : 𝕜} [hp : Fact (0 < p)] [Archimedean 𝕜] {f : 𝕜 → B} (h : f a = f (a + p))
Goal: AddCircle.liftIco p a f = Quot.lift ((Set.Icc a (a + p)).restrict f) ⋯ ∘ ⇑(AddCircle.equivIccQuot p a)
-- AddCircle.liftIco_continuous
theorem liftIco_continuous [TopologicalSpace B] {f : 𝕜 → B} (hf : f a = f (a + p))
    (hc : ContinuousOn f <| Icc a (a + p)) : Continuous (liftIco p a f) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p a : 𝕜} [hp : Fact (0 < p)] [Archimedean 𝕜] [TopologicalSpace B] {f : 𝕜 → B} (hf : f a = f (a + p)) (hc : ContinuousOn f (Set.Icc a (a + p)))
Goal: Continuous (AddCircle.liftIco p a f)
-- AddCircle.liftIco_zero_coe_apply
theorem liftIco_zero_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ico 0 p) : liftIco p 0 f ↑x = f x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] [Archimedean 𝕜] {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Set.Ico 0 p)
Goal: AddCircle.liftIco p 0 f ↑x = f x
-- AddCircle.liftIco_zero_continuous
theorem liftIco_zero_continuous [TopologicalSpace B] {f : 𝕜 → B} (hf : f 0 = f p)
    (hc : ContinuousOn f <| Icc 0 p) : Continuous (liftIco p 0 f) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] [Archimedean 𝕜] [TopologicalSpace B] {f : 𝕜 → B} (hf : f 0 = f p) (hc : ContinuousOn f (Set.Icc 0 p))
Goal: Continuous (AddCircle.liftIco p 0 f)
-- ZMod.toAddCircle_intCast
lemma toAddCircle_intCast (j : ℤ) :
    toAddCircle (j : ZMod N) = ↑(j / N : ℝ) 
Arguments: {N : ℕ} [NeZero N] (j : ℤ)
Goal: ZMod.toAddCircle ↑j = ↑(↑j / ↑N)
-- ZMod.toAddCircle_natCast
lemma toAddCircle_natCast (j : ℕ) :
    toAddCircle (j : ZMod N) = ↑(j / N : ℝ) 
Arguments: {N : ℕ} [NeZero N] (j : ℕ)
Goal: ZMod.toAddCircle ↑j = ↑(↑j / ↑N)
-- ZMod.toAddCircle_apply
lemma toAddCircle_apply (j : ZMod N) :
    toAddCircle j = ↑(j.val / N : ℝ) 
Arguments: {N : ℕ} [NeZero N] (j : ZMod N)
Goal: ZMod.toAddCircle j = ↑(↑j.val / ↑N)
-- ZMod.toAddCircle_injective
lemma toAddCircle_injective : Function.Injective (toAddCircle : ZMod N → _) 
Arguments: (N : ℕ) [NeZero N]
Goal: Function.Injective ⇑ZMod.toAddCircle
-- ZMod.toAddCircle_inj
@[simp] lemma toAddCircle_inj {j k : ZMod N} : toAddCircle j = toAddCircle k ↔ j = k 
Arguments: {N : ℕ} [NeZero N] {j k : ZMod N}
Goal: ZMod.toAddCircle j = ZMod.toAddCircle k ↔ j = k
-- ZMod.toAddCircle_eq_zero
@[simp] lemma toAddCircle_eq_zero {j : ZMod N} : toAddCircle j = 0 ↔ j = 0 
Arguments: {N : ℕ} [NeZero N] {j : ZMod N}
Goal: ZMod.toAddCircle j = 0 ↔ j = 0
-- TopCat.adj₁_unit
@[simps! unit counit]
def adj₁ : discrete ⊣ forget TopCat.{u} 
Goal: TopCat.adj₁.unit = { app := fun X => id, naturality := TopCat.adj₁.proof_1 }
-- TopCat.adj₁_counit
@[simps! unit counit]
def adj₁ : discrete ⊣ forget TopCat.{u} 
Goal: TopCat.adj₁.counit = { app := fun X => { toFun := id, continuous_toFun := ⋯ }, naturality := TopCat.adj₁.proof_3 }
-- TopCat.adj₂_counit
@[simps! unit counit]
def adj₂ : forget TopCat.{u} ⊣ trivial 
Goal: TopCat.adj₂.counit = { app := fun X => id, naturality := TopCat.adj₂.proof_3 }
-- TopCat.adj₂_unit
@[simps! unit counit]
def adj₂ : forget TopCat.{u} ⊣ trivial 
Goal: TopCat.adj₂.unit = { app := fun X => { toFun := id, continuous_toFun := ⋯ }, naturality := TopCat.adj₂.proof_2 }
-- Sequential.is_sequential
structure Sequential where
    toTop : TopCat.{u}
    [is_sequential : SequentialSpace toTop]
Arguments: (self : Sequential)
Goal: SequentialSpace ↑self.toTop
-- Sequential.sequentialToTop_obj
@[simps!]
def sequentialToTop : Sequential.{u} ⥤ TopCat.{u} 
Arguments: (self : Sequential)
Goal: Sequential.sequentialToTop.obj self = self.toTop
-- Sequential.sequentialToTop_map
@[simps!]
def sequentialToTop : Sequential.{u} ⥤ TopCat.{u} 
Goal: ∀ {X Y : CategoryTheory.InducedCategory TopCat Sequential.toTop} (f : X ⟶ Y), Sequential.sequentialToTop.map f = f
-- Sequential.isoOfHomeo_inv
@[simps hom inv]
def isoOfHomeo {X Y : Sequential.{u}} (f : X ≃ₜ Y) : X ≅ Y 
Arguments: {X Y : Sequential} (f : ↑X.toTop ≃ₜ ↑Y.toTop)
Goal: (Sequential.isoOfHomeo f).inv = { toFun := ⇑f.symm, continuous_toFun := ⋯ }
-- Sequential.isoOfHomeo_hom
@[simps hom inv]
def isoOfHomeo {X Y : Sequential.{u}} (f : X ≃ₜ Y) : X ≅ Y 
Arguments: {X Y : Sequential} (f : ↑X.toTop ≃ₜ ↑Y.toTop)
Goal: (Sequential.isoOfHomeo f).hom = { toFun := ⇑f, continuous_toFun := ⋯ }
-- Sequential.homeoOfIso_symm_apply
@[simps]
def homeoOfIso {X Y : Sequential.{u}} (f : X ≅ Y) : X ≃ₜ Y 
Arguments: {X Y : Sequential} (f : X ≅ Y) (a : (CategoryTheory.forget Sequential).obj Y)
Goal: (Sequential.homeoOfIso f).symm a = f.inv a
-- Sequential.homeoOfIso_apply
@[simps]
def homeoOfIso {X Y : Sequential.{u}} (f : X ≅ Y) : X ≃ₜ Y 
Arguments: {X Y : Sequential} (f : X ≅ Y) (a : (CategoryTheory.forget Sequential).obj X)
Goal: (Sequential.homeoOfIso f) a = f.hom a
-- Sequential.isoEquivHomeo_apply
@[simps]
def isoEquivHomeo {X Y : Sequential.{u}} : (X ≅ Y) ≃ (X ≃ₜ Y) 
Arguments: {X Y : Sequential} (f : X ≅ Y)
Goal: Sequential.isoEquivHomeo f = Sequential.homeoOfIso f
-- Sequential.isoEquivHomeo_symm_apply
@[simps]
def isoEquivHomeo {X Y : Sequential.{u}} : (X ≅ Y) ≃ (X ≃ₜ Y) 
Arguments: {X Y : Sequential} (f : ↑X.toTop ≃ₜ ↑Y.toTop)
Goal: Sequential.isoEquivHomeo.symm f = Sequential.isoOfHomeo f
-- IsCompact.exists_clusterPt
lemma IsCompact.exists_clusterPt (hs : IsCompact s) {f : Filter X} [NeBot f] (hf : f ≤ 𝓟 s) :
    ∃ x ∈ s, ClusterPt x f 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) {f : Filter X} [f.NeBot] (hf : f ≤ 𝓟 s)
Goal: ∃ x ∈ s, ClusterPt x f
-- IsCompact.exists_mapClusterPt
lemma IsCompact.exists_mapClusterPt {ι : Type*} (hs : IsCompact s) {f : Filter ι} [NeBot f]
    {u : ι → X} (hf : Filter.map u f ≤ 𝓟 s) :
    ∃ x ∈ s, MapClusterPt x f u 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type u_2} (hs : IsCompact s) {f : Filter ι} [f.NeBot] {u : ι → X} (hf : Filter.map u f ≤ 𝓟 s)
Goal: ∃ x ∈ s, MapClusterPt x f u
-- IsCompact.compl_mem_sets
theorem IsCompact.compl_mem_sets (hs : IsCompact s) {f : Filter X} (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f) :
    sᶜ ∈ f 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) {f : Filter X} (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f)
Goal: sᶜ ∈ f
-- IsCompact.compl_mem_sets_of_nhdsWithin
theorem IsCompact.compl_mem_sets_of_nhdsWithin (hs : IsCompact s) {f : Filter X}
    (hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) {f : Filter X} (hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f)
Goal: sᶜ ∈ f
-- IsCompact.induction_on
@[elab_as_elim]
theorem IsCompact.induction_on (hs : IsCompact s) {p : Set X → Prop} (he : p ∅)
    (hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s) (hunion : ∀ ⦃s t⦄, p s → p t → p (s ∪ t))
    (hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) : p s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) {p : Set X → Prop} (he : p ∅) (hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s) (hunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)) (hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t)
Goal: p s
-- IsCompact.inter_right
theorem IsCompact.inter_right (hs : IsCompact s) (ht : IsClosed t) : IsCompact (s ∩ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsCompact s) (ht : IsClosed t)
Goal: IsCompact (s ∩ t)
-- IsCompact.inter_left
theorem IsCompact.inter_left (ht : IsCompact t) (hs : IsClosed s) : IsCompact (s ∩ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (ht : IsCompact t) (hs : IsClosed s)
Goal: IsCompact (s ∩ t)
-- IsCompact.diff
theorem IsCompact.diff (hs : IsCompact s) (ht : IsOpen t) : IsCompact (s \ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsCompact s) (ht : IsOpen t)
Goal: IsCompact (s \ t)
-- IsCompact.of_isClosed_subset
theorem IsCompact.of_isClosed_subset (hs : IsCompact s) (ht : IsClosed t) (h : t ⊆ s) :
    IsCompact t 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (h : t ⊆ s)
Goal: IsCompact t
-- IsCompact.image_of_continuousOn
theorem IsCompact.image_of_continuousOn {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s) :
    IsCompact (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s)
Goal: IsCompact (f '' s)
-- IsCompact.image
theorem IsCompact.image {f : X → Y} (hs : IsCompact s) (hf : Continuous f) : IsCompact (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hs : IsCompact s) (hf : Continuous f)
Goal: IsCompact (f '' s)
-- IsCompact.adherence_nhdset
theorem IsCompact.adherence_nhdset {f : Filter X} (hs : IsCompact s) (hf₂ : f ≤ 𝓟 s)
    (ht₁ : IsOpen t) (ht₂ : ∀ x ∈ s, ClusterPt x f → x ∈ t) : t ∈ f 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} {f : Filter X} (hs : IsCompact s) (hf₂ : f ≤ 𝓟 s) (ht₁ : IsOpen t) (ht₂ : ∀ x ∈ s, ClusterPt x f → x ∈ t)
Goal: t ∈ f
-- isCompact_iff_ultrafilter_le_nhds
theorem isCompact_iff_ultrafilter_le_nhds :
    IsCompact s ↔ ∀ f : Ultrafilter X, ↑f ≤ 𝓟 s → ∃ x ∈ s, ↑f ≤ 𝓝 x 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsCompact s ↔ ∀ (f : Ultrafilter X), ↑f ≤ 𝓟 s → ∃ x ∈ s, ↑f ≤ 𝓝 x
-- isCompact_iff_ultrafilter_le_nhds'
theorem isCompact_iff_ultrafilter_le_nhds' :
    IsCompact s ↔ ∀ f : Ultrafilter X, s ∈ f → ∃ x ∈ s, ↑f ≤ 𝓝 x 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsCompact s ↔ ∀ (f : Ultrafilter X), s ∈ f → ∃ x ∈ s, ↑f ≤ 𝓝 x
-- IsCompact.le_nhds_of_unique_clusterPt
lemma IsCompact.le_nhds_of_unique_clusterPt (hs : IsCompact s) {l : Filter X} {y : X}
    (hmem : s ∈ l) (h : ∀ x ∈ s, ClusterPt x l → x = y) : l ≤ 𝓝 y 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) {l : Filter X} {y : X} (hmem : s ∈ l) (h : ∀ x ∈ s, ClusterPt x l → x = y)
Goal: l ≤ 𝓝 y
-- IsCompact.tendsto_nhds_of_unique_mapClusterPt
lemma IsCompact.tendsto_nhds_of_unique_mapClusterPt {l : Filter Y} {y : X} {f : Y → X}
    (hs : IsCompact s) (hmem : ∀ᶠ x in l, f x ∈ s) (h : ∀ x ∈ s, MapClusterPt x l f → x = y) :
    Tendsto f l (𝓝 y) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] {s : Set X} {l : Filter Y} {y : X} {f : Y → X} (hs : IsCompact s) (hmem : ∀ᶠ (x : Y) in l, f x ∈ s) (h : ∀ x ∈ s, MapClusterPt x l f → x = y)
Goal: Filter.Tendsto f l (𝓝 y)
-- IsCompact.elim_directed_cover
theorem IsCompact.elim_directed_cover {ι : Type v} [hι : Nonempty ι] (hs : IsCompact s)
    (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) (hdU : Directed (· ⊆ ·) U) :
    ∃ i, s ⊆ U i 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} [hι : Nonempty ι] (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ (i : ι), IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) (hdU : Directed (fun x x_1 => x ⊆ x_1) U)
Goal: ∃ i, s ⊆ U i
-- IsCompact.elim_finite_subcover
theorem IsCompact.elim_finite_subcover {ι : Type v} (hs : IsCompact s) (U : ι → Set X)
    (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ (i : ι), IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i)
Goal: ∃ t, s ⊆ ⋃ i ∈ t, U i
-- IsCompact.elim_nhds_subcover_nhdsSet'
lemma IsCompact.elim_nhds_subcover_nhdsSet' (hs : IsCompact s) (U : ∀ x ∈ s, Set X)
    (hU : ∀ x hx, U x hx ∈ 𝓝 x) : ∃ t : Finset s, (⋃ x ∈ t, U x.1 x.2) ∈ 𝓝ˢ s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (U : (x : X) → x ∈ s → Set X) (hU : ∀ (x : X) (hx : x ∈ s), U x hx ∈ 𝓝 x)
Goal: ∃ t, ⋃ x ∈ t, U ↑x ⋯ ∈ 𝓝ˢ s
-- IsCompact.elim_nhds_subcover_nhdsSet
lemma IsCompact.elim_nhds_subcover_nhdsSet (hs : IsCompact s) {U : X → Set X}
    (hU : ∀ x ∈ s, U x ∈ 𝓝 x) : ∃ t : Finset X, (∀ x ∈ t, x ∈ s) ∧ (⋃ x ∈ t, U x) ∈ 𝓝ˢ s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) {U : X → Set X} (hU : ∀ x ∈ s, U x ∈ 𝓝 x)
Goal: ∃ t, (∀ x ∈ t, x ∈ s) ∧ ⋃ x ∈ t, U x ∈ 𝓝ˢ s
-- IsCompact.elim_nhds_subcover'
theorem IsCompact.elim_nhds_subcover' (hs : IsCompact s) (U : ∀ x ∈ s, Set X)
    (hU : ∀ x (hx : x ∈ s), U x ‹x ∈ s› ∈ 𝓝 x) : ∃ t : Finset s, s ⊆ ⋃ x ∈ t, U (x : s) x.2 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (U : (x : X) → x ∈ s → Set X) (hU : ∀ (x : X) (hx : x ∈ s), U x hx ∈ 𝓝 x)
Goal: ∃ t, s ⊆ ⋃ x ∈ t, U ↑x ⋯
-- IsCompact.elim_nhds_subcover
theorem IsCompact.elim_nhds_subcover (hs : IsCompact s) (U : X → Set X) (hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
    ∃ t : Finset X, (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (U : X → Set X) (hU : ∀ x ∈ s, U x ∈ 𝓝 x)
Goal: ∃ t, (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x
-- IsCompact.disjoint_nhdsSet_left
theorem IsCompact.disjoint_nhdsSet_left {l : Filter X} (hs : IsCompact s) :
    Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {l : Filter X} (hs : IsCompact s)
Goal: Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l
-- IsCompact.disjoint_nhdsSet_right
theorem IsCompact.disjoint_nhdsSet_right {l : Filter X} (hs : IsCompact s) :
    Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {l : Filter X} (hs : IsCompact s)
Goal: Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x)
-- IsCompact.elim_directed_family_closed
theorem IsCompact.elim_directed_family_closed {ι : Type v} [hι : Nonempty ι] (hs : IsCompact s)
    (t : ι → Set X) (htc : ∀ i, IsClosed (t i)) (hst : (s ∩ ⋂ i, t i) = ∅)
    (hdt : Directed (· ⊇ ·) t) : ∃ i : ι, s ∩ t i = ∅ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} [hι : Nonempty ι] (hs : IsCompact s) (t : ι → Set X) (htc : ∀ (i : ι), IsClosed (t i)) (hst : s ∩ ⋂ i, t i = ∅) (hdt : Directed (fun x x_1 => x ⊇ x_1) t)
Goal: ∃ i, s ∩ t i = ∅
-- IsCompact.elim_finite_subfamily_closed
theorem IsCompact.elim_finite_subfamily_closed {ι : Type v} (hs : IsCompact s)
    (t : ι → Set X) (htc : ∀ i, IsClosed (t i)) (hst : (s ∩ ⋂ i, t i) = ∅) :
    ∃ u : Finset ι, (s ∩ ⋂ i ∈ u, t i) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsCompact s) (t : ι → Set X) (htc : ∀ (i : ι), IsClosed (t i)) (hst : s ∩ ⋂ i, t i = ∅)
Goal: ∃ u, s ∩ ⋂ i ∈ u, t i = ∅
-- LocallyFinite.finite_nonempty_inter_compact
theorem LocallyFinite.finite_nonempty_inter_compact {f : ι → Set X}
    (hf : LocallyFinite f) (hs : IsCompact s) : { i | (f i ∩ s).Nonempty }.Finite 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set X} {f : ι → Set X} (hf : LocallyFinite f) (hs : IsCompact s)
Goal: {i | (f i ∩ s).Nonempty}.Finite
-- IsCompact.inter_iInter_nonempty
theorem IsCompact.inter_iInter_nonempty {ι : Type v} (hs : IsCompact s) (t : ι → Set X)
    (htc : ∀ i, IsClosed (t i)) (hst : ∀ u : Finset ι, (s ∩ ⋂ i ∈ u, t i).Nonempty) :
    (s ∩ ⋂ i, t i).Nonempty 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsCompact s) (t : ι → Set X) (htc : ∀ (i : ι), IsClosed (t i)) (hst : ∀ (u : Finset ι), (s ∩ ⋂ i ∈ u, t i).Nonempty)
Goal: (s ∩ ⋂ i, t i).Nonempty
-- IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed
theorem IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed
    {ι : Type v} [hι : Nonempty ι] (t : ι → Set X) (htd : Directed (· ⊇ ·) t)
    (htn : ∀ i, (t i).Nonempty) (htc : ∀ i, IsCompact (t i)) (htcl : ∀ i, IsClosed (t i)) :
    (⋂ i, t i).Nonempty 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type v} [hι : Nonempty ι] (t : ι → Set X) (htd : Directed (fun x x_1 => x ⊇ x_1) t) (htn : ∀ (i : ι), (t i).Nonempty) (htc : ∀ (i : ι), IsCompact (t i)) (htcl : ∀ (i : ι), IsClosed (t i))
Goal: (⋂ i, t i).Nonempty
-- IsCompact.nonempty_sInter_of_directed_nonempty_isCompact_isClosed
theorem IsCompact.nonempty_sInter_of_directed_nonempty_isCompact_isClosed
    {S : Set (Set X)} [hS : Nonempty S] (hSd : DirectedOn (· ⊇ ·) S) (hSn : ∀ U ∈ S, U.Nonempty)
    (hSc : ∀ U ∈ S, IsCompact U) (hScl : ∀ U ∈ S, IsClosed U) : (⋂₀ S).Nonempty 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set (Set X)} [hS : Nonempty ↑S] (hSd : DirectedOn (fun x x_1 => x ⊇ x_1) S) (hSn : ∀ U ∈ S, U.Nonempty) (hSc : ∀ U ∈ S, IsCompact U) (hScl : ∀ U ∈ S, IsClosed U)
Goal: (⋂₀ S).Nonempty
-- IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed
theorem IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed (t : ℕ → Set X)
    (htd : ∀ i, t (i + 1) ⊆ t i) (htn : ∀ i, (t i).Nonempty) (ht0 : IsCompact (t 0))
    (htcl : ∀ i, IsClosed (t i)) : (⋂ i, t i).Nonempty 
Arguments: {X : Type u} [TopologicalSpace X] (t : ℕ → Set X) (htd : ∀ (i : ℕ), t (i + 1) ⊆ t i) (htn : ∀ (i : ℕ), (t i).Nonempty) (ht0 : IsCompact (t 0)) (htcl : ∀ (i : ℕ), IsClosed (t i))
Goal: (⋂ i, t i).Nonempty
-- IsCompact.elim_finite_subcover_image
theorem IsCompact.elim_finite_subcover_image {b : Set ι} {c : ι → Set X} (hs : IsCompact s)
    (hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i) :
    ∃ b', b' ⊆ b ∧ Set.Finite b' ∧ s ⊆ ⋃ i ∈ b', c i 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set X} {b : Set ι} {c : ι → Set X} (hs : IsCompact s) (hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i)
Goal: ∃ b' ⊆ b, b'.Finite ∧ s ⊆ ⋃ i ∈ b', c i
-- isCompact_of_finite_subcover
theorem isCompact_of_finite_subcover
    (h : ∀ {ι : Type u} (U : ι → Set X), (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) →
      ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i) :
    IsCompact s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (h : ∀ {ι : Type u} (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, s ⊆ ⋃ i ∈ t, U i)
Goal: IsCompact s
-- isCompact_of_finite_subfamily_closed
theorem isCompact_of_finite_subfamily_closed
    (h : ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → (s ∩ ⋂ i, t i) = ∅ →
      ∃ u : Finset ι, (s ∩ ⋂ i ∈ u, t i) = ∅) :
    IsCompact s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (h : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, s ∩ ⋂ i ∈ u, t i = ∅)
Goal: IsCompact s
-- isCompact_iff_finite_subcover
theorem isCompact_iff_finite_subcover :
    IsCompact s ↔ ∀ {ι : Type u} (U : ι → Set X),
      (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) → ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsCompact s ↔ ∀ {ι : Type u} (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, s ⊆ ⋃ i ∈ t, U i
-- isCompact_iff_finite_subfamily_closed
theorem isCompact_iff_finite_subfamily_closed :
    IsCompact s ↔ ∀ {ι : Type u} (t : ι → Set X),
      (∀ i, IsClosed (t i)) → (s ∩ ⋂ i, t i) = ∅ → ∃ u : Finset ι, (s ∩ ⋂ i ∈ u, t i) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsCompact s ↔ ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, s ∩ ⋂ i ∈ u, t i = ∅
-- IsCompact.mem_nhdsSet_prod_of_forall
theorem IsCompact.mem_nhdsSet_prod_of_forall {K : Set X} {l : Filter Y} {s : Set (X × Y)}
    (hK : IsCompact K) (hs : ∀ x ∈ K, s ∈ 𝓝 x ×ˢ l) : s ∈ (𝓝ˢ K) ×ˢ l 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] {K : Set X} {l : Filter Y} {s : Set (X × Y)} (hK : IsCompact K) (hs : ∀ x ∈ K, s ∈ 𝓝 x ×ˢ l)
Goal: s ∈ 𝓝ˢ K ×ˢ l
-- IsCompact.nhdsSet_prod_eq_biSup
theorem IsCompact.nhdsSet_prod_eq_biSup {K : Set X} (hK : IsCompact K) (l : Filter Y) :
    (𝓝ˢ K) ×ˢ l = ⨆ x ∈ K, 𝓝 x ×ˢ l 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] {K : Set X} (hK : IsCompact K) (l : Filter Y)
Goal: 𝓝ˢ K ×ˢ l = ⨆ x ∈ K, 𝓝 x ×ˢ l
-- IsCompact.prod_nhdsSet_eq_biSup
theorem IsCompact.prod_nhdsSet_eq_biSup {K : Set Y} (hK : IsCompact K) (l : Filter X) :
    l ×ˢ (𝓝ˢ K) = ⨆ y ∈ K, l ×ˢ 𝓝 y 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace Y] {K : Set Y} (hK : IsCompact K) (l : Filter X)
Goal: l ×ˢ 𝓝ˢ K = ⨆ y ∈ K, l ×ˢ 𝓝 y
-- IsCompact.mem_prod_nhdsSet_of_forall
theorem IsCompact.mem_prod_nhdsSet_of_forall {K : Set Y} {l : Filter X} {s : Set (X × Y)}
    (hK : IsCompact K) (hs : ∀ y ∈ K, s ∈ l ×ˢ 𝓝 y) : s ∈ l ×ˢ 𝓝ˢ K 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace Y] {K : Set Y} {l : Filter X} {s : Set (X × Y)} (hK : IsCompact K) (hs : ∀ y ∈ K, s ∈ l ×ˢ 𝓝 y)
Goal: s ∈ l ×ˢ 𝓝ˢ K
-- IsCompact.nhdsSet_inf_eq_biSup
theorem IsCompact.nhdsSet_inf_eq_biSup {K : Set X} (hK : IsCompact K) (l : Filter X) :
    (𝓝ˢ K) ⊓ l = ⨆ x ∈ K, 𝓝 x ⊓ l 
Arguments: {X : Type u} [TopologicalSpace X] {K : Set X} (hK : IsCompact K) (l : Filter X)
Goal: 𝓝ˢ K ⊓ l = ⨆ x ∈ K, 𝓝 x ⊓ l
-- IsCompact.inf_nhdsSet_eq_biSup
theorem IsCompact.inf_nhdsSet_eq_biSup {K : Set X} (hK : IsCompact K) (l : Filter X) :
    l ⊓ (𝓝ˢ K) = ⨆ x ∈ K, l ⊓ 𝓝 x 
Arguments: {X : Type u} [TopologicalSpace X] {K : Set X} (hK : IsCompact K) (l : Filter X)
Goal: l ⊓ 𝓝ˢ K = ⨆ x ∈ K, l ⊓ 𝓝 x
-- IsCompact.mem_nhdsSet_inf_of_forall
theorem IsCompact.mem_nhdsSet_inf_of_forall {K : Set X} {l : Filter X} {s : Set X}
    (hK : IsCompact K) (hs : ∀ x ∈ K, s ∈ 𝓝 x ⊓ l) : s ∈ (𝓝ˢ K) ⊓ l 
Arguments: {X : Type u} [TopologicalSpace X] {K : Set X} {l : Filter X} {s : Set X} (hK : IsCompact K) (hs : ∀ x ∈ K, s ∈ 𝓝 x ⊓ l)
Goal: s ∈ 𝓝ˢ K ⊓ l
-- IsCompact.mem_inf_nhdsSet_of_forall
theorem IsCompact.mem_inf_nhdsSet_of_forall {K : Set X} {l : Filter X} {s : Set X}
    (hK : IsCompact K) (hs : ∀ y ∈ K, s ∈ l ⊓ 𝓝 y) : s ∈ l ⊓ 𝓝ˢ K 
Arguments: {X : Type u} [TopologicalSpace X] {K : Set X} {l : Filter X} {s : Set X} (hK : IsCompact K) (hs : ∀ y ∈ K, s ∈ l ⊓ 𝓝 y)
Goal: s ∈ l ⊓ 𝓝ˢ K