ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsLUB.isGLB_of_tendsto
theorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :
    ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},
      AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ}
Goal: AntitoneOn f s → IsLUB s a → s.Nonempty → Filter.Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b
-- IsGLB.mem_upperBounds_of_tendsto
theorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
    {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)
    (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a) (hb : Filter.Tendsto f (𝓝[s] a) (𝓝 b))
Goal: b ∈ upperBounds (f '' s)
-- IsGLB.isLUB_of_tendsto
theorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :
    ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},
      AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ}
Goal: AntitoneOn f s → IsGLB s a → s.Nonempty → Filter.Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b
-- IsLUB.mem_of_isClosed
theorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)
    (sc : IsClosed s) : a ∈ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) (sc : IsClosed s)
Goal: a ∈ s
-- IsGLB.mem_of_isClosed
theorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)
    (sc : IsClosed s) : a ∈ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) (sc : IsClosed s)
Goal: a ∈ s
-- IsLUB.exists_seq_strictMono_tendsto_of_not_mem
theorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}
    [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :
    ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {t : Set α} {x : α} [(𝓝 x).IsCountablyGenerated] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty)
Goal: ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Filter.Tendsto u Filter.atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
-- IsLUB.exists_seq_monotone_tendsto
theorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]
    (htx : IsLUB t x) (ht : t.Nonempty) :
    ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {t : Set α} {x : α} [(𝓝 x).IsCountablyGenerated] (htx : IsLUB t x) (ht : t.Nonempty)
Goal: ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Filter.Tendsto u Filter.atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
-- exists_seq_strictMono_tendsto'
theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
    [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
    ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) 
Arguments: {α : Type u_4} [LinearOrder α] [TopologicalSpace α] [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x)
Goal: ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Set.Ioo y x) ∧ Filter.Tendsto u Filter.atTop (𝓝 x)
-- exists_seq_strictMono_tendsto
theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
    (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α] (x : α)
Goal: ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Filter.Tendsto u Filter.atTop (𝓝 x)
-- exists_seq_strictMono_tendsto_nhdsWithin
theorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]
    [FirstCountableTopology α] (x : α) :
    ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α] (x : α)
Goal: ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Filter.Tendsto u Filter.atTop (𝓝[<] x)
-- exists_seq_tendsto_sSup
theorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]
    [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
    (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S 
Arguments: {α : Type u_4} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty) (hS' : BddAbove S)
Goal: ∃ u, Monotone u ∧ Filter.Tendsto u Filter.atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S
-- IsGLB.exists_seq_strictAnti_tendsto_of_not_mem
theorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}
    [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :
    ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {t : Set α} {x : α} [(𝓝 x).IsCountablyGenerated] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty)
Goal: ∃ u, StrictAnti u ∧ (∀ (n : ℕ), x < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
-- IsGLB.exists_seq_antitone_tendsto
theorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]
    (htx : IsGLB t x) (ht : t.Nonempty) :
    ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {t : Set α} {x : α} [(𝓝 x).IsCountablyGenerated] (htx : IsGLB t x) (ht : t.Nonempty)
Goal: ∃ u, Antitone u ∧ (∀ (n : ℕ), x ≤ u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
-- exists_seq_strictAnti_tendsto'
theorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}
    (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [FirstCountableTopology α] {x y : α} (hy : x < y)
Goal: ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Set.Ioo x y) ∧ Filter.Tendsto u Filter.atTop (𝓝 x)
-- exists_seq_strictAnti_tendsto
theorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]
    (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α] (x : α)
Goal: ∃ u, StrictAnti u ∧ (∀ (n : ℕ), x < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 x)
-- exists_seq_strictAnti_tendsto_nhdsWithin
theorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]
    [FirstCountableTopology α] (x : α) :
    ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α] (x : α)
Goal: ∃ u, StrictAnti u ∧ (∀ (n : ℕ), x < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝[>] x)
-- exists_seq_strictAnti_strictMono_tendsto
theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
    {x y : α} (h : x < y) :
    ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
      (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [FirstCountableTopology α] {x y : α} (h : x < y)
Goal: ∃ u v,
  StrictAnti u ∧
    StrictMono v ∧
      (∀ (k : ℕ), u k ∈ Set.Ioo x y) ∧
        (∀ (l : ℕ), v l ∈ Set.Ioo x y) ∧
          (∀ (k l : ℕ), u k < v l) ∧ Filter.Tendsto u Filter.atTop (𝓝 x) ∧ Filter.Tendsto v Filter.atTop (𝓝 y)
-- exists_seq_tendsto_sInf
theorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]
    [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
    (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S 
Arguments: {α : Type u_4} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty) (hS' : BddBelow S)
Goal: ∃ u, Antitone u ∧ Filter.Tendsto u Filter.atTop (𝓝 (sInf S)) ∧ ∀ (n : ℕ), u n ∈ S
-- AlexDisc.coe_forgetToTop
@[simp] lemma coe_forgetToTop (X : AlexDisc) : ↥((forget₂ _ TopCat).obj X) = X 
Arguments: (X : AlexDisc)
Goal: ↑((CategoryTheory.forget₂ AlexDisc TopCat).obj X) = ↑X
-- AlexDisc.coe_of
@[simp] lemma coe_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] : ↥(of α) = α 
Arguments: (α : Type u_1) [TopologicalSpace α] [AlexandrovDiscrete α]
Goal: ↑(AlexDisc.of α) = α
-- AlexDisc.forgetToTop_of
@[simp] lemma forgetToTop_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] :
  (forget₂ AlexDisc TopCat).obj (of α) = TopCat.of α 
Arguments: (α : Type u_1) [TopologicalSpace α] [AlexandrovDiscrete α]
Goal: (CategoryTheory.forget₂ AlexDisc TopCat).obj (AlexDisc.of α) = TopCat.of α
-- AlexDisc.Iso.mk_inv
@[simps]
def Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β 
Arguments: {α β : AlexDisc} (e : ↑α ≃ₜ ↑β)
Goal: (AlexDisc.Iso.mk e).inv = e.symm.toContinuousMap
-- AlexDisc.Iso.mk_hom
@[simps]
def Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β 
Arguments: {α β : AlexDisc} (e : ↑α ≃ₜ ↑β)
Goal: (AlexDisc.Iso.mk e).hom = e.toContinuousMap
-- alexDiscEquivPreord_functor
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
Goal: alexDiscEquivPreord.functor = CategoryTheory.forget₂ AlexDisc TopCat ⋙ topToPreord
-- alexDiscEquivPreord_inverse_map
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
Goal: ∀ {X Y : Preord} (f : ↑X →o ↑Y), alexDiscEquivPreord.inverse.map f = Topology.WithUpperSet.map f
-- alexDiscEquivPreord_unitIso
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
Goal: alexDiscEquivPreord.unitIso =
  CategoryTheory.NatIso.ofComponents (fun X => AlexDisc.Iso.mk (id (homeoWithUpperSetTopologyorderIso ↑X)))
    @alexDiscEquivPreord.proof_4
-- alexDiscEquivPreord_counitIso
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
Goal: alexDiscEquivPreord.counitIso =
  CategoryTheory.NatIso.ofComponents (fun X => Preord.Iso.mk (id (orderIsoSpecializationWithUpperSetTopology ↑X).symm))
    @alexDiscEquivPreord.proof_5
-- alexDiscEquivPreord_inverse_obj
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
Arguments: (X : Preord)
Goal: alexDiscEquivPreord.inverse.obj X = AlexDisc.of (Topology.WithUpperSet ↑X)
-- Topology.IsLawson.topology_eq_lawson
class IsLawson : Prop where
  topology_eq_lawson : ‹TopologicalSpace α› = lawson α
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [self : Topology.IsLawson α]
Goal: inst✝ = Topology.lawson α
-- Topology.IsLawson.isTopologicalBasis
protected theorem isTopologicalBasis : TopologicalSpace.IsTopologicalBasis (lawsonBasis α) 
Arguments: (α : Type u_1) [Preorder α] [TopologicalSpace α] [Topology.IsLawson α]
Goal: TopologicalSpace.IsTopologicalBasis (Topology.IsLawson.lawsonBasis α)
-- Topology.WithLawson.to_Lawson_symm_eq
@[simp] lemma to_Lawson_symm_eq : (@toLawson α).symm = ofLawson 
Arguments: {α : Type u_1}
Goal: Topology.WithLawson.toLawson.symm = Topology.WithLawson.ofLawson
-- Topology.WithLawson.of_Lawson_symm_eq
@[simp] lemma of_Lawson_symm_eq : (@ofLawson α).symm = toLawson 
Arguments: {α : Type u_1}
Goal: Topology.WithLawson.ofLawson.symm = Topology.WithLawson.toLawson
-- Topology.WithLawson.toLawson_ofLawson
@[simp] lemma toLawson_ofLawson (a : WithLawson α) : toLawson (ofLawson a) = a 
Arguments: {α : Type u_1} (a : Topology.WithLawson α)
Goal: Topology.WithLawson.toLawson (Topology.WithLawson.ofLawson a) = a
-- Topology.WithLawson.ofLawson_toLawson
@[simp] lemma ofLawson_toLawson (a : α) : ofLawson (toLawson a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: Topology.WithLawson.ofLawson (Topology.WithLawson.toLawson a) = a
-- Topology.WithLawson.toLawson_inj
@[simp, nolint simpNF]
lemma toLawson_inj {a b : α} : toLawson a = toLawson b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: Topology.WithLawson.toLawson a = Topology.WithLawson.toLawson b ↔ a = b
-- Topology.WithLawson.ofLawson_inj
@[simp, nolint simpNF]
lemma ofLawson_inj {a b : WithLawson α} : ofLawson a = ofLawson b ↔ a = b 
Arguments: {α : Type u_1} {a b : Topology.WithLawson α}
Goal: Topology.WithLawson.ofLawson a = Topology.WithLawson.ofLawson b ↔ a = b
-- Topology.WithLawson.isOpen_preimage_ofLawson
theorem isOpen_preimage_ofLawson {S : Set α} :
    IsOpen (ofLawson ⁻¹' S) ↔ (lawson α).IsOpen S 
Arguments: {α : Type u_1} [Preorder α] {S : Set α}
Goal: IsOpen (⇑Topology.WithLawson.ofLawson ⁻¹' S) ↔ TopologicalSpace.IsOpen S
-- Topology.WithLawson.isClosed_preimage_ofLawson
theorem isClosed_preimage_ofLawson {S : Set α} :
    IsClosed (ofLawson ⁻¹' S) ↔ IsClosed[lawson α] S 
Arguments: {α : Type u_1} [Preorder α] {S : Set α}
Goal: IsClosed (⇑Topology.WithLawson.ofLawson ⁻¹' S) ↔ IsClosed S
-- Topology.WithLawson.isOpen_def
theorem isOpen_def {T : Set (WithLawson α)} :
    IsOpen T ↔ (lawson α).IsOpen (toLawson ⁻¹' T) 
Arguments: {α : Type u_1} [Preorder α] {T : Set (Topology.WithLawson α)}
Goal: IsOpen T ↔ TopologicalSpace.IsOpen (⇑Topology.WithLawson.toLawson ⁻¹' T)
-- Topology.lawson_le_scott
lemma lawson_le_scott : lawson α ≤ scott α 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.lawson α ≤ Topology.scott α
-- Topology.lawson_le_lower
lemma lawson_le_lower : lawson α ≤ lower α 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.lawson α ≤ Topology.lower α
-- Topology.scottHausdorff_le_lawson
lemma scottHausdorff_le_lawson : scottHausdorff α  ≤ lawson α 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.scottHausdorff α ≤ Topology.lawson α
-- Topology.lawsonClosed_of_scottClosed
lemma lawsonClosed_of_scottClosed (s : Set α) (h : IsClosed (WithScott.ofScott ⁻¹' s)) :
    IsClosed (WithLawson.ofLawson ⁻¹' s) 
Arguments: {α : Type u_1} [Preorder α] (s : Set α) (h : IsClosed (⇑Topology.WithScott.ofScott ⁻¹' s))
Goal: IsClosed (⇑Topology.WithLawson.ofLawson ⁻¹' s)
-- Topology.lawsonClosed_of_lowerClosed
lemma lawsonClosed_of_lowerClosed (s : Set α) (h : IsClosed (WithLower.ofLower ⁻¹' s)) :
    IsClosed (WithLawson.ofLawson ⁻¹' s) 
Arguments: {α : Type u_1} [Preorder α] (s : Set α) (h : IsClosed (⇑Topology.WithLower.ofLower ⁻¹' s))
Goal: IsClosed (⇑Topology.WithLawson.ofLawson ⁻¹' s)
-- Topology.lawsonOpen_iff_scottOpen_of_isUpperSet
lemma lawsonOpen_iff_scottOpen_of_isUpperSet {s : Set α} (h : IsUpperSet s) :
    IsOpen (WithLawson.ofLawson ⁻¹' s) ↔ IsOpen (WithScott.ofScott ⁻¹' s) 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (h : IsUpperSet s)
Goal: IsOpen (⇑Topology.WithLawson.ofLawson ⁻¹' s) ↔ IsOpen (⇑Topology.WithScott.ofScott ⁻¹' s)
-- Topology.isLawson_le_isScott
lemma isLawson_le_isScott : L ≤ S 
Arguments: {α : Type u_1} [Preorder α] (L S : TopologicalSpace α) [Topology.IsLawson α] [Topology.IsScott α]
Goal: L ≤ S
-- Topology.scottHausdorff_le_isLawson
lemma scottHausdorff_le_isLawson : scottHausdorff α ≤ L 
Arguments: {α : Type u_1} [Preorder α] (L : TopologicalSpace α) [Topology.IsLawson α]
Goal: Topology.scottHausdorff α ≤ L
-- Topology.lawsonOpen_iff_scottOpen_of_isUpperSet'
lemma lawsonOpen_iff_scottOpen_of_isUpperSet' (s : Set α) (h : IsUpperSet s) :
    IsOpen[L] s ↔ IsOpen[S] s 
Arguments: {α : Type u_1} [Preorder α] (L S : TopologicalSpace α) [Topology.IsLawson α] [Topology.IsScott α] (s : Set α) (h : IsUpperSet s)
Goal: IsOpen s ↔ IsOpen s
-- Topology.lawsonClosed_iff_scottClosed_of_isLowerSet
lemma lawsonClosed_iff_scottClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) :
    IsClosed[L] s ↔ IsClosed[S] s 
Arguments: {α : Type u_1} [Preorder α] (L S : TopologicalSpace α) [Topology.IsLawson α] [Topology.IsScott α] (s : Set α) (h : IsLowerSet s)
Goal: IsClosed s ↔ IsClosed s
-- Topology.lawsonClosed_iff_dirSupClosed_of_isLowerSet
lemma lawsonClosed_iff_dirSupClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) :
    IsClosed[L] s ↔ DirSupClosed s 
Arguments: {α : Type u_1} [Preorder α] (L S : TopologicalSpace α) [Topology.IsLawson α] [Topology.IsScott α] (s : Set α) (h : IsLowerSet s)
Goal: IsClosed s ↔ DirSupClosed s
-- Topology.IsLawson.singleton_isClosed
lemma singleton_isClosed (a : α) : IsClosed ({a} : Set α) 
Arguments: {α : Type u_1} [PartialOrder α] [TopologicalSpace α] [Topology.IsLawson α] (a : α)
Goal: IsClosed {a}
-- IsMaxOn.closure
protected theorem IsMaxOn.closure (h : IsMaxOn f s a) (hc : ContinuousOn f (closure s)) :
    IsMaxOn f (closure s) a 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Preorder Y] [OrderClosedTopology Y] {f : X → Y} {s : Set X} {a : X} (h : IsMaxOn f s a) (hc : ContinuousOn f (closure s))
Goal: IsMaxOn f (closure s) a
-- IsMinOn.closure
protected theorem IsMinOn.closure (h : IsMinOn f s a) (hc : ContinuousOn f (closure s)) :
    IsMinOn f (closure s) a 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Preorder Y] [OrderClosedTopology Y] {f : X → Y} {s : Set X} {a : X} (h : IsMinOn f s a) (hc : ContinuousOn f (closure s))
Goal: IsMinOn f (closure s) a
-- IsExtrOn.closure
protected theorem IsExtrOn.closure (h : IsExtrOn f s a) (hc : ContinuousOn f (closure s)) :
    IsExtrOn f (closure s) a 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Preorder Y] [OrderClosedTopology Y] {f : X → Y} {s : Set X} {a : X} (h : IsExtrOn f s a) (hc : ContinuousOn f (closure s))
Goal: IsExtrOn f (closure s) a
-- IsLocalMaxOn.closure
protected theorem IsLocalMaxOn.closure (h : IsLocalMaxOn f s a) (hc : ContinuousOn f (closure s)) :
    IsLocalMaxOn f (closure s) a 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Preorder Y] [OrderClosedTopology Y] {f : X → Y} {s : Set X} {a : X} (h : IsLocalMaxOn f s a) (hc : ContinuousOn f (closure s))
Goal: IsLocalMaxOn f (closure s) a
-- IsLocalMinOn.closure
protected theorem IsLocalMinOn.closure (h : IsLocalMinOn f s a) (hc : ContinuousOn f (closure s)) :
    IsLocalMinOn f (closure s) a 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Preorder Y] [OrderClosedTopology Y] {f : X → Y} {s : Set X} {a : X} (h : IsLocalMinOn f s a) (hc : ContinuousOn f (closure s))
Goal: IsLocalMinOn f (closure s) a
-- IsLocalExtrOn.closure
protected theorem IsLocalExtrOn.closure (h : IsLocalExtrOn f s a)
    (hc : ContinuousOn f (closure s)) : IsLocalExtrOn f (closure s) a 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Preorder Y] [OrderClosedTopology Y] {f : X → Y} {s : Set X} {a : X} (h : IsLocalExtrOn f s a) (hc : ContinuousOn f (closure s))
Goal: IsLocalExtrOn f (closure s) a
-- nhdsSet_Ioi
@[simp] theorem nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a : α}
Goal: 𝓝ˢ (Set.Ioi a) = 𝓟 (Set.Ioi a)
-- nhdsSet_Iio
@[simp] theorem nhdsSet_Iio : 𝓝ˢ (Iio a) = 𝓟 (Iio a) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a : α}
Goal: 𝓝ˢ (Set.Iio a) = 𝓟 (Set.Iio a)
-- nhdsSet_Ioo
@[simp] theorem nhdsSet_Ioo : 𝓝ˢ (Ioo a b) = 𝓟 (Ioo a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α}
Goal: 𝓝ˢ (Set.Ioo a b) = 𝓟 (Set.Ioo a b)
-- nhdsSet_Ici
theorem nhdsSet_Ici : 𝓝ˢ (Ici a) = 𝓝 a ⊔ 𝓟 (Ioi a) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a : α}
Goal: 𝓝ˢ (Set.Ici a) = 𝓝 a ⊔ 𝓟 (Set.Ioi a)
-- nhdsSet_Iic
theorem nhdsSet_Iic : 𝓝ˢ (Iic a) = 𝓝 a ⊔ 𝓟 (Iio a) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a : α}
Goal: 𝓝ˢ (Set.Iic a) = 𝓝 a ⊔ 𝓟 (Set.Iio a)
-- nhdsSet_Ico
theorem nhdsSet_Ico (h : a < b) : 𝓝ˢ (Ico a b) = 𝓝 a ⊔ 𝓟 (Ioo a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α} (h : a < b)
Goal: 𝓝ˢ (Set.Ico a b) = 𝓝 a ⊔ 𝓟 (Set.Ioo a b)
-- nhdsSet_Ioc
theorem nhdsSet_Ioc (h : a < b) : 𝓝ˢ (Ioc a b) = 𝓝 b ⊔ 𝓟 (Ioo a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α} (h : a < b)
Goal: 𝓝ˢ (Set.Ioc a b) = 𝓝 b ⊔ 𝓟 (Set.Ioo a b)
-- nhdsSet_Icc
theorem nhdsSet_Icc (h : a ≤ b) : 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α} (h : a ≤ b)
Goal: 𝓝ˢ (Set.Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Set.Ioo a b)
-- Ioi_mem_nhdsSet_Ici_iff
@[simp]
theorem Ioi_mem_nhdsSet_Ici_iff : Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α}
Goal: Set.Ioi a ∈ 𝓝ˢ (Set.Ici b) ↔ a < b
-- Ici_mem_nhdsSet_Ici
theorem Ici_mem_nhdsSet_Ici (h : a < b) : Ici a ∈ 𝓝ˢ (Ici b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α} (h : a < b)
Goal: Set.Ici a ∈ 𝓝ˢ (Set.Ici b)
-- Iio_mem_nhdsSet_Iic_iff
theorem Iio_mem_nhdsSet_Iic_iff : Iio b ∈ 𝓝ˢ (Iic a) ↔ a < b 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α}
Goal: Set.Iio b ∈ 𝓝ˢ (Set.Iic a) ↔ a < b
-- Iic_mem_nhdsSet_Iic
theorem Iic_mem_nhdsSet_Iic (h : a < b) : Iic b ∈ 𝓝ˢ (Iic a) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b : α} (h : a < b)
Goal: Set.Iic b ∈ 𝓝ˢ (Set.Iic a)
-- Ioi_mem_nhdsSet_Icc
theorem Ioi_mem_nhdsSet_Icc (h : a < b) : Ioi a ∈ 𝓝ˢ (Icc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : a < b)
Goal: Set.Ioi a ∈ 𝓝ˢ (Set.Icc b c)
-- Ici_mem_nhdsSet_Icc
theorem Ici_mem_nhdsSet_Icc (h : a < b) : Ici a ∈ 𝓝ˢ (Icc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : a < b)
Goal: Set.Ici a ∈ 𝓝ˢ (Set.Icc b c)
-- Iio_mem_nhdsSet_Icc
theorem Iio_mem_nhdsSet_Icc (h : b < c) : Iio c ∈ 𝓝ˢ (Icc a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : b < c)
Goal: Set.Iio c ∈ 𝓝ˢ (Set.Icc a b)
-- Iic_mem_nhdsSet_Icc
theorem Iic_mem_nhdsSet_Icc (h : b < c) : Iic c ∈ 𝓝ˢ (Icc a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : b < c)
Goal: Set.Iic c ∈ 𝓝ˢ (Set.Icc a b)
-- Ioo_mem_nhdsSet_Icc
theorem Ioo_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Icc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c < d)
Goal: Set.Ioo a d ∈ 𝓝ˢ (Set.Icc b c)
-- Ico_mem_nhdsSet_Icc
theorem Ico_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Icc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c < d)
Goal: Set.Ico a d ∈ 𝓝ˢ (Set.Icc b c)
-- Ioc_mem_nhdsSet_Icc
theorem Ioc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Icc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c < d)
Goal: Set.Ioc a d ∈ 𝓝ˢ (Set.Icc b c)
-- Icc_mem_nhdsSet_Icc
theorem Icc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Icc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c < d)
Goal: Set.Icc a d ∈ 𝓝ˢ (Set.Icc b c)