ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Topology.WithUpperSet.toUpperSet_le_iff
lemma toUpperSet_le_iff {a b : α} : toUpperSet a ≤ toUpperSet b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Topology.WithUpperSet.toUpperSet a ≤ Topology.WithUpperSet.toUpperSet b ↔ a ≤ b
-- Topology.WithLowerSet.to_WithLowerSet_symm_eq
@[simp] lemma to_WithLowerSet_symm_eq : (@toLowerSet α).symm = ofLowerSet 
Arguments: {α : Type u_1}
Goal: Topology.WithLowerSet.toLowerSet.symm = Topology.WithLowerSet.ofLowerSet
-- Topology.WithLowerSet.of_WithLowerSet_symm_eq
@[simp] lemma of_WithLowerSet_symm_eq : (@ofLowerSet α).symm = toLowerSet 
Arguments: {α : Type u_1}
Goal: Topology.WithLowerSet.ofLowerSet.symm = Topology.WithLowerSet.toLowerSet
-- Topology.WithLowerSet.toLowerSet_ofLowerSet
@[simp] lemma toLowerSet_ofLowerSet (a : WithLowerSet α) : toLowerSet (ofLowerSet a) = a 
Arguments: {α : Type u_1} (a : Topology.WithLowerSet α)
Goal: Topology.WithLowerSet.toLowerSet (Topology.WithLowerSet.ofLowerSet a) = a
-- Topology.WithLowerSet.ofLowerSet_toLowerSet
@[simp] lemma ofLowerSet_toLowerSet (a : α) : ofLowerSet (toLowerSet a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: Topology.WithLowerSet.ofLowerSet (Topology.WithLowerSet.toLowerSet a) = a
-- Topology.WithLowerSet.toLowerSet_inj
lemma toLowerSet_inj {a b : α} : toLowerSet a = toLowerSet b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: Topology.WithLowerSet.toLowerSet a = Topology.WithLowerSet.toLowerSet b ↔ a = b
-- Topology.WithLowerSet.ofLowerSet_inj
lemma ofLowerSet_inj {a b : WithLowerSet α} : ofLowerSet a = ofLowerSet b ↔ a = b 
Arguments: {α : Type u_1} {a b : Topology.WithLowerSet α}
Goal: Topology.WithLowerSet.ofLowerSet a = Topology.WithLowerSet.ofLowerSet b ↔ a = b
-- Topology.WithLowerSet.ofLowerSet_le_iff
lemma ofLowerSet_le_iff {a b : WithLowerSet α} : ofLowerSet a ≤ ofLowerSet b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : Topology.WithLowerSet α}
Goal: Topology.WithLowerSet.ofLowerSet a ≤ Topology.WithLowerSet.ofLowerSet b ↔ a ≤ b
-- Topology.WithLowerSet.toLowerSet_le_iff
lemma toLowerSet_le_iff {a b : α} : toLowerSet a ≤ toLowerSet b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Topology.WithLowerSet.toLowerSet a ≤ Topology.WithLowerSet.toLowerSet b ↔ a ≤ b
-- Topology.IsUpperSet.topology_eq_upperSetTopology
protected class IsUpperSet (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
  topology_eq_upperSetTopology : t = upperSet α
Arguments: {α : Type u_4} [t : TopologicalSpace α] [Preorder α] [self : Topology.IsUpperSet α]
Goal: t = Topology.upperSet α
-- Topology.IsLowerSet.topology_eq_lowerSetTopology
protected class IsLowerSet (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
  topology_eq_lowerSetTopology : t = lowerSet α
Arguments: {α : Type u_4} [t : TopologicalSpace α] [Preorder α] [self : Topology.IsLowerSet α]
Goal: t = Topology.lowerSet α
-- Topology.IsUpperSet.topology_eq
lemma topology_eq : ‹_› = upperSet α 
Arguments: (α : Type u_1) [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α]
Goal: inst✝ = Topology.upperSet α
-- Topology.IsUpperSet.isOpen_iff_isUpperSet
lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] {s : Set α}
Goal: IsOpen s ↔ IsUpperSet s
-- Topology.IsUpperSet.isClosed_iff_isLower
lemma isClosed_iff_isLower : IsClosed s ↔ IsLowerSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] {s : Set α}
Goal: IsClosed s ↔ IsLowerSet s
-- Topology.IsUpperSet.closure_eq_lowerClosure
lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] {s : Set α}
Goal: closure s = ↑(lowerClosure s)
-- Topology.IsUpperSet.closure_singleton
@[simp] lemma closure_singleton {a : α} : closure {a} = Iic a 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] {a : α}
Goal: closure {a} = Set.Iic a
-- Topology.IsUpperSet.monotone_iff_continuous
protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
    [Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β}
Goal: Monotone f ↔ Continuous f
-- Topology.IsUpperSet.monotone_to_upperTopology_continuous
lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
    [Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [Topology.IsUpperSet α] [Topology.IsUpper β] {f : α → β} (hf : Monotone f)
Goal: Continuous f
-- Topology.IsUpperSet.upperSet_le_upper
lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
    [@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ 
Arguments: {α : Type u_1} [Preorder α] {t₁ t₂ : TopologicalSpace α} [Topology.IsUpperSet α] [Topology.IsUpper α]
Goal: t₁ ≤ t₂
-- Topology.IsLowerSet.topology_eq
lemma topology_eq : ‹_› = lowerSet α 
Arguments: (α : Type u_1) [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α]
Goal: inst✝ = Topology.lowerSet α
-- Topology.IsLowerSet.isOpen_iff_isLowerSet
lemma isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] {s : Set α}
Goal: IsOpen s ↔ IsLowerSet s
-- Topology.IsLowerSet.isClosed_iff_isUpper
lemma isClosed_iff_isUpper : IsClosed s ↔ IsUpperSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] {s : Set α}
Goal: IsClosed s ↔ IsUpperSet s
-- Topology.IsLowerSet.closure_eq_upperClosure
lemma closure_eq_upperClosure {s : Set α} : closure s = upperClosure s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] {s : Set α}
Goal: closure s = ↑(upperClosure s)
-- Topology.IsLowerSet.closure_singleton
@[simp] lemma closure_singleton {a : α} : closure {a} = Ici a 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] {a : α}
Goal: closure {a} = Set.Ici a
-- Topology.IsLowerSet.monotone_iff_continuous
protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
    [Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β}
Goal: Monotone f ↔ Continuous f
-- Topology.IsLowerSet.monotone_to_lowerTopology_continuous
lemma monotone_to_lowerTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
    [Topology.IsLowerSet α] [IsLower β] {f : α → β} (hf : Monotone f) : Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [Topology.IsLowerSet α] [Topology.IsLower β] {f : α → β} (hf : Monotone f)
Goal: Continuous f
-- Topology.IsLowerSet.lowerSet_le_lower
lemma lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
    [@IsLower α t₂ _] : t₁ ≤ t₂ 
Arguments: {α : Type u_1} [Preorder α] {t₁ t₂ : TopologicalSpace α} [Topology.IsLowerSet α] [Topology.IsLower α]
Goal: t₁ ≤ t₂
-- Topology.isUpperSet_orderDual
lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
    Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α]
Goal: Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α
-- Topology.isLowerSet_orderDual
lemma isLowerSet_orderDual [Preorder α] [TopologicalSpace α] :
    Topology.IsLowerSet αᵒᵈ ↔ Topology.IsUpperSet α 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α]
Goal: Topology.IsLowerSet αᵒᵈ ↔ Topology.IsUpperSet α
-- Topology.WithUpperSet.map_id
@[simp] lemma map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.WithUpperSet.map OrderHom.id = ContinuousMap.id (Topology.WithUpperSet α)
-- Topology.WithUpperSet.map_comp
@[simp] lemma map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] (g : β →o γ) (f : α →o β)
Goal: Topology.WithUpperSet.map (g.comp f) = (Topology.WithUpperSet.map g).comp (Topology.WithUpperSet.map f)
-- Topology.WithUpperSet.toUpperSet_specializes_toUpperSet
@[simp] lemma toUpperSet_specializes_toUpperSet {a b : α} :
    toUpperSet a ⤳ toUpperSet b ↔ b ≤ a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Topology.WithUpperSet.toUpperSet a ⤳ Topology.WithUpperSet.toUpperSet b ↔ b ≤ a
-- Topology.WithUpperSet.ofUpperSet_le_ofUpperSet
@[simp] lemma ofUpperSet_le_ofUpperSet {a b : WithUpperSet α} :
    ofUpperSet a ≤ ofUpperSet b ↔ b ⤳ a 
Arguments: {α : Type u_1} [Preorder α] {a b : Topology.WithUpperSet α}
Goal: Topology.WithUpperSet.ofUpperSet a ≤ Topology.WithUpperSet.ofUpperSet b ↔ b ⤳ a
-- Topology.WithUpperSet.isUpperSet_toUpperSet_preimage
@[simp] lemma isUpperSet_toUpperSet_preimage {s : Set (WithUpperSet α)} :
    IsUpperSet (toUpperSet ⁻¹' s) ↔ IsOpen s 
Arguments: {α : Type u_1} [Preorder α] {s : Set (Topology.WithUpperSet α)}
Goal: IsUpperSet (⇑Topology.WithUpperSet.toUpperSet ⁻¹' s) ↔ IsOpen s
-- Topology.WithUpperSet.isOpen_ofUpperSet_preimage
@[simp] lemma isOpen_ofUpperSet_preimage {s : Set α} :
    IsOpen (ofUpperSet ⁻¹' s) ↔ IsUpperSet s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: IsOpen (⇑Topology.WithUpperSet.ofUpperSet ⁻¹' s) ↔ IsUpperSet s
-- Topology.WithLowerSet.map_id
@[simp] lemma map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.WithLowerSet.map OrderHom.id = ContinuousMap.id (Topology.WithLowerSet α)
-- Topology.WithLowerSet.map_comp
@[simp] lemma map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] (g : β →o γ) (f : α →o β)
Goal: Topology.WithLowerSet.map (g.comp f) = (Topology.WithLowerSet.map g).comp (Topology.WithLowerSet.map f)
-- Topology.WithLowerSet.toLowerSet_specializes_toLowerSet
@[simp] lemma toLowerSet_specializes_toLowerSet {a b : α} :
  toLowerSet a ⤳ toLowerSet b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Topology.WithLowerSet.toLowerSet a ⤳ Topology.WithLowerSet.toLowerSet b ↔ a ≤ b
-- Topology.WithLowerSet.ofLowerSet_le_ofLowerSet
@[simp] lemma ofLowerSet_le_ofLowerSet {a b : WithLowerSet α} :
    ofLowerSet a ≤ ofLowerSet b ↔ a ⤳ b 
Arguments: {α : Type u_1} [Preorder α] {a b : Topology.WithLowerSet α}
Goal: Topology.WithLowerSet.ofLowerSet a ≤ Topology.WithLowerSet.ofLowerSet b ↔ a ⤳ b
-- Topology.WithLowerSet.isLowerSet_toLowerSet_preimage
@[simp] lemma isLowerSet_toLowerSet_preimage {s : Set (WithLowerSet α)} :
    IsLowerSet (toLowerSet ⁻¹' s) ↔ IsOpen s 
Arguments: {α : Type u_1} [Preorder α] {s : Set (Topology.WithLowerSet α)}
Goal: IsLowerSet (⇑Topology.WithLowerSet.toLowerSet ⁻¹' s) ↔ IsOpen s
-- Topology.WithLowerSet.isOpen_ofLowerSet_preimage
@[simp] lemma isOpen_ofLowerSet_preimage {s : Set α} :
    IsOpen (ofLowerSet ⁻¹' s) ↔ IsLowerSet s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: IsOpen (⇑Topology.WithLowerSet.ofLowerSet ⁻¹' s) ↔ IsLowerSet s
-- dirSupInacc_compl
@[simp] lemma dirSupInacc_compl : DirSupInacc sᶜ ↔ DirSupClosed s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: DirSupInacc sᶜ ↔ DirSupClosed s
-- dirSupClosed_compl
@[simp] lemma dirSupClosed_compl : DirSupClosed sᶜ ↔ DirSupInacc s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: DirSupClosed sᶜ ↔ DirSupInacc s
-- DirSupClosed.inter
lemma DirSupClosed.inter (hs : DirSupClosed s) (ht : DirSupClosed t) : DirSupClosed (s ∩ t) 
Arguments: {α : Type u_1} [Preorder α] {s t : Set α} (hs : DirSupClosed s) (ht : DirSupClosed t)
Goal: DirSupClosed (s ∩ t)
-- DirSupInacc.union
lemma DirSupInacc.union (hs : DirSupInacc s) (ht : DirSupInacc t) : DirSupInacc (s ∪ t) 
Arguments: {α : Type u_1} [Preorder α] {s t : Set α} (hs : DirSupInacc s) (ht : DirSupInacc t)
Goal: DirSupInacc (s ∪ t)
-- IsUpperSet.dirSupClosed
lemma IsUpperSet.dirSupClosed (hs : IsUpperSet s) : DirSupClosed s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (hs : IsUpperSet s)
Goal: DirSupClosed s
-- IsLowerSet.dirSupInacc
lemma IsLowerSet.dirSupInacc (hs : IsLowerSet s) : DirSupInacc s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (hs : IsLowerSet s)
Goal: DirSupInacc s
-- dirSupClosed_Iic
lemma dirSupClosed_Iic (a : α) : DirSupClosed (Iic a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: DirSupClosed (Set.Iic a)
-- dirSupInacc_iff_forall_sSup
lemma dirSupInacc_iff_forall_sSup :
    DirSupInacc s ↔ ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → sSup d ∈ s → (d ∩ s).Nonempty 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: DirSupInacc s ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (fun x x_1 => x ≤ x_1) d → sSup d ∈ s → (d ∩ s).Nonempty
-- dirSupClosed_iff_forall_sSup
lemma dirSupClosed_iff_forall_sSup :
    DirSupClosed s ↔ ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → d ⊆ s → sSup d ∈ s 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: DirSupClosed s ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (fun x x_1 => x ≤ x_1) d → d ⊆ s → sSup d ∈ s
-- Topology.IsScottHausdorff.topology_eq_scottHausdorff
class IsScottHausdorff : Prop where
  topology_eq_scottHausdorff : ‹TopologicalSpace α› = scottHausdorff α
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [self : Topology.IsScottHausdorff α]
Goal: inst✝ = Topology.scottHausdorff α
-- Topology.IsScottHausdorff.topology_eq
lemma topology_eq : ‹_› = scottHausdorff α 
Arguments: (α : Type u_1) [Preorder α] [TopologicalSpace α] [Topology.IsScottHausdorff α]
Goal: inst✝ = Topology.scottHausdorff α
-- Topology.IsScottHausdorff.isOpen_iff
lemma isOpen_iff :
    IsOpen s ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a : α⦄, IsLUB d a →
      a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScottHausdorff α] {s : Set α}
Goal: IsOpen s ↔
  ∀ ⦃d : Set α⦄,
    d.Nonempty → DirectedOn (fun x x_1 => x ≤ x_1) d → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s → ∃ b ∈ d, Set.Ici b ∩ d ⊆ s
-- Topology.IsScottHausdorff.isOpen_of_isLowerSet
lemma isOpen_of_isLowerSet (h : IsLowerSet s) : IsOpen s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScottHausdorff α] {s : Set α} (h : IsLowerSet s)
Goal: IsOpen s
-- Topology.IsScottHausdorff.isClosed_of_isUpperSet
lemma isClosed_of_isUpperSet (h : IsUpperSet s) : IsClosed s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScottHausdorff α] {s : Set α} (h : IsUpperSet s)
Goal: IsClosed s
-- Topology.IsScottHausdorff.dirSupInacc_of_isOpen
lemma dirSupInacc_of_isOpen (h : IsOpen s) : DirSupInacc s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScottHausdorff α] {s : Set α} (h : IsOpen s)
Goal: DirSupInacc s
-- Topology.IsScottHausdorff.dirSupClosed_of_isClosed
lemma dirSupClosed_of_isClosed (h : IsClosed s) : DirSupClosed s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScottHausdorff α] {s : Set α} (h : IsClosed s)
Goal: DirSupClosed s
-- Topology.upperSet_le_scott
lemma upperSet_le_scott : upperSet α ≤ scott α 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.upperSet α ≤ Topology.scott α
-- Topology.scottHausdorff_le_scott
lemma scottHausdorff_le_scott : scottHausdorff α ≤ scott α 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.scottHausdorff α ≤ Topology.scott α
-- Topology.IsScott.topology_eq_scott
class IsScott : Prop where
  topology_eq_scott : ‹TopologicalSpace α› = scott α
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [self : Topology.IsScott α]
Goal: inst✝ = Topology.scott α
-- Topology.IsScott.topology_eq
lemma topology_eq : ‹_› = scott α 
Arguments: (α : Type u_1) [Preorder α] [TopologicalSpace α] [Topology.IsScott α]
Goal: inst✝ = Topology.scott α
-- Topology.IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open
lemma isOpen_iff_isUpperSet_and_scottHausdorff_open :
    IsOpen s ↔ IsUpperSet s ∧ IsOpen[scottHausdorff α] s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {s : Set α}
Goal: IsOpen s ↔ IsUpperSet s ∧ IsOpen s
-- Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInacc
lemma isOpen_iff_isUpperSet_and_dirSupInacc : IsOpen s ↔ IsUpperSet s ∧ DirSupInacc s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {s : Set α}
Goal: IsOpen s ↔ IsUpperSet s ∧ DirSupInacc s
-- Topology.IsScott.isClosed_iff_isLowerSet_and_dirSupClosed
lemma isClosed_iff_isLowerSet_and_dirSupClosed : IsClosed s ↔ IsLowerSet s ∧ DirSupClosed s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {s : Set α}
Goal: IsClosed s ↔ IsLowerSet s ∧ DirSupClosed s
-- Topology.IsScott.isUpperSet_of_isOpen
lemma isUpperSet_of_isOpen : IsOpen s → IsUpperSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {s : Set α}
Goal: IsOpen s → IsUpperSet s
-- Topology.IsScott.isLowerSet_of_isClosed
lemma isLowerSet_of_isClosed : IsClosed s → IsLowerSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {s : Set α}
Goal: IsClosed s → IsLowerSet s
-- Topology.IsScott.dirSupClosed_of_isClosed
lemma dirSupClosed_of_isClosed : IsClosed s → DirSupClosed s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {s : Set α}
Goal: IsClosed s → DirSupClosed s
-- Topology.IsScott.lowerClosure_subset_closure
lemma lowerClosure_subset_closure : ↑(lowerClosure s) ⊆ closure s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {s : Set α}
Goal: ↑(lowerClosure s) ⊆ closure s
-- Topology.IsScott.isClosed_Iic
lemma isClosed_Iic : IsClosed (Iic a) 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {a : α}
Goal: IsClosed (Set.Iic a)
-- Topology.IsScott.closure_singleton
@[simp] lemma closure_singleton : closure {a} = Iic a 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] {a : α}
Goal: closure {a} = Set.Iic a
-- Topology.IsScott.monotone_of_continuous
lemma monotone_of_continuous (hf : Continuous f) : Monotone f 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] [Preorder β] [TopologicalSpace β] [Topology.IsScott β] {f : α → β} (hf : Continuous f)
Goal: Monotone f
-- Topology.IsScott.scottContinuous_iff_continuous
@[simp] lemma scottContinuous_iff_continuous : ScottContinuous f ↔ Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [TopologicalSpace α] [Topology.IsScott α] [Preorder β] [TopologicalSpace β] [Topology.IsScott β] {f : α → β}
Goal: ScottContinuous f ↔ Continuous f
-- Topology.IsScott.isOpen_iff_Iic_compl_or_univ
lemma isOpen_iff_Iic_compl_or_univ (U : Set α) :
    IsOpen U ↔ (∃ (a : α), U = (Iic a)ᶜ) ∨ U = univ 
Arguments: {α : Type u_1} [CompleteLinearOrder α] [TopologicalSpace α] [Topology.IsScott α] (U : Set α)
Goal: IsOpen U ↔ (∃ a, U = (Set.Iic a)ᶜ) ∨ U = Set.univ
-- Topology.WithScott.toScott_symm_eq
@[simp] lemma toScott_symm_eq : (@toScott α).symm = ofScott 
Arguments: {α : Type u_1}
Goal: Topology.WithScott.toScott.symm = Topology.WithScott.ofScott
-- Topology.WithScott.ofScott_symm_eq
@[simp] lemma ofScott_symm_eq : (@ofScott α).symm = toScott 
Arguments: {α : Type u_1}
Goal: Topology.WithScott.ofScott.symm = Topology.WithScott.toScott
-- Topology.WithScott.toScott_ofScott
@[simp] lemma toScott_ofScott (a : WithScott α) : toScott (ofScott a) = a 
Arguments: {α : Type u_1} (a : Topology.WithScott α)
Goal: Topology.WithScott.toScott (Topology.WithScott.ofScott a) = a
-- Topology.WithScott.ofScott_toScott
@[simp] lemma ofScott_toScott (a : α) : ofScott (toScott a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: Topology.WithScott.ofScott (Topology.WithScott.toScott a) = a
-- Topology.WithScott.toScott_inj
@[simp, nolint simpNF]
lemma toScott_inj {a b : α} : toScott a = toScott b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: Topology.WithScott.toScott a = Topology.WithScott.toScott b ↔ a = b
-- Topology.WithScott.ofScott_inj
@[simp, nolint simpNF]
lemma ofScott_inj {a b : WithScott α} : ofScott a = ofScott b ↔ a = b 
Arguments: {α : Type u_1} {a b : Topology.WithScott α}
Goal: Topology.WithScott.ofScott a = Topology.WithScott.ofScott b ↔ a = b
-- Topology.WithScott.isOpen_iff_isUpperSet_and_scottHausdorff_open'
lemma isOpen_iff_isUpperSet_and_scottHausdorff_open' {u : Set α} :
    IsOpen (WithScott.ofScott ⁻¹' u) ↔ IsUpperSet u ∧ (scottHausdorff α).IsOpen u 
Arguments: {α : Type u_1} [Preorder α] {u : Set α}
Goal: IsOpen (⇑Topology.WithScott.ofScott ⁻¹' u) ↔ IsUpperSet u ∧ TopologicalSpace.IsOpen u
-- Topology.scottHausdorff_le_lower
lemma scottHausdorff_le_lower : scottHausdorff α ≤ lower α 
Arguments: {α : Type u_1} [Preorder α]
Goal: Topology.scottHausdorff α ≤ Topology.lower α
-- Topology.IsScott.scottHausdorff_le
lemma IsScott.scottHausdorff_le [IsScott α] : scottHausdorff α ≤ ‹TopologicalSpace α› 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsScott α]
Goal: Topology.scottHausdorff α ≤ inst✝
-- Topology.IsLower.scottHausdorff_le
lemma IsLower.scottHausdorff_le [IsLower α] : scottHausdorff α ≤ ‹TopologicalSpace α› 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α]
Goal: Topology.scottHausdorff α ≤ inst✝
-- subset_tsupport
@[to_additive]
theorem subset_mulTSupport (f : X → α) : mulSupport f ⊆ mulTSupport f 
Arguments: {X : Type u_1} {α : Type u_2} [Zero α] [TopologicalSpace X] (f : X → α)
Goal: Function.support f ⊆ tsupport f
-- subset_mulTSupport
@[to_additive]
theorem subset_mulTSupport (f : X → α) : mulSupport f ⊆ mulTSupport f 
Arguments: {X : Type u_1} {α : Type u_2} [One α] [TopologicalSpace X] (f : X → α)
Goal: Function.mulSupport f ⊆ mulTSupport f
-- isClosed_mulTSupport
@[to_additive]
theorem isClosed_mulTSupport (f : X → α) : IsClosed (mulTSupport f) 
Arguments: {X : Type u_1} {α : Type u_2} [One α] [TopologicalSpace X] (f : X → α)
Goal: IsClosed (mulTSupport f)
-- isClosed_tsupport
@[to_additive]
theorem isClosed_mulTSupport (f : X → α) : IsClosed (mulTSupport f) 
Arguments: {X : Type u_1} {α : Type u_2} [Zero α] [TopologicalSpace X] (f : X → α)
Goal: IsClosed (tsupport f)
-- mulTSupport_eq_empty_iff
@[to_additive]
theorem mulTSupport_eq_empty_iff {f : X → α} : mulTSupport f = ∅ ↔ f = 1 
Arguments: {X : Type u_1} {α : Type u_2} [One α] [TopologicalSpace X] {f : X → α}
Goal: mulTSupport f = ∅ ↔ f = 1
-- tsupport_eq_empty_iff
@[to_additive]
theorem mulTSupport_eq_empty_iff {f : X → α} : mulTSupport f = ∅ ↔ f = 1 
Arguments: {X : Type u_1} {α : Type u_2} [Zero α] [TopologicalSpace X] {f : X → α}
Goal: tsupport f = ∅ ↔ f = 0
-- image_eq_one_of_nmem_mulTSupport
@[to_additive]
theorem image_eq_one_of_nmem_mulTSupport {f : X → α} {x : X} (hx : x ∉ mulTSupport f) : f x = 1 
Arguments: {X : Type u_1} {α : Type u_2} [One α] [TopologicalSpace X] {f : X → α} {x : X} (hx : x ∉ mulTSupport f)
Goal: f x = 1
-- image_eq_zero_of_nmem_tsupport
@[to_additive]
theorem image_eq_one_of_nmem_mulTSupport {f : X → α} {x : X} (hx : x ∉ mulTSupport f) : f x = 1 
Arguments: {X : Type u_1} {α : Type u_2} [Zero α] [TopologicalSpace X] {f : X → α} {x : X} (hx : x ∉ tsupport f)
Goal: f x = 0
-- range_subset_insert_image_mulTSupport
@[to_additive]
theorem range_subset_insert_image_mulTSupport (f : X → α) :
    range f ⊆ insert 1 (f '' mulTSupport f) 
Arguments: {X : Type u_1} {α : Type u_2} [One α] [TopologicalSpace X] (f : X → α)
Goal: Set.range f ⊆ insert 1 (f '' mulTSupport f)
-- range_subset_insert_image_tsupport
@[to_additive]
theorem range_subset_insert_image_mulTSupport (f : X → α) :
    range f ⊆ insert 1 (f '' mulTSupport f) 
Arguments: {X : Type u_1} {α : Type u_2} [Zero α] [TopologicalSpace X] (f : X → α)
Goal: Set.range f ⊆ insert 0 (f '' tsupport f)