ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- tendsto_of_monotone
theorem tendsto_of_monotone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
    [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Monotone f) :
    Tendsto f atTop atTop ∨ ∃ l, Tendsto f atTop (𝓝 l) 
Arguments: {ι : Type u_3} {α : Type u_4} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Monotone f)
Goal: Filter.Tendsto f Filter.atTop Filter.atTop ∨ ∃ l, Filter.Tendsto f Filter.atTop (𝓝 l)
-- tendsto_of_antitone
theorem tendsto_of_antitone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
    [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Antitone f) :
    Tendsto f atTop atBot ∨ ∃ l, Tendsto f atTop (𝓝 l) 
Arguments: {ι : Type u_3} {α : Type u_4} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Antitone f)
Goal: Filter.Tendsto f Filter.atTop Filter.atBot ∨ ∃ l, Filter.Tendsto f Filter.atTop (𝓝 l)
-- tendsto_iff_tendsto_subseq_of_monotone
theorem tendsto_iff_tendsto_subseq_of_monotone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
    [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
    [NoMaxOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Monotone f)
    (hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) 
Arguments: {ι₁ : Type u_3} {ι₂ : Type u_4} {α : Type u_5} [SemilatticeSup ι₁] [Preorder ι₂] [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [NoMaxOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Monotone f) (hg : Filter.Tendsto φ Filter.atTop Filter.atTop)
Goal: Filter.Tendsto f Filter.atTop (𝓝 l) ↔ Filter.Tendsto (f ∘ φ) Filter.atTop (𝓝 l)
-- tendsto_iff_tendsto_subseq_of_antitone
theorem tendsto_iff_tendsto_subseq_of_antitone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
    [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
    [NoMinOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Antitone f)
    (hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) 
Arguments: {ι₁ : Type u_3} {ι₂ : Type u_4} {α : Type u_5} [SemilatticeSup ι₁] [Preorder ι₂] [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [NoMinOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Antitone f) (hg : Filter.Tendsto φ Filter.atTop Filter.atTop)
Goal: Filter.Tendsto f Filter.atTop (𝓝 l) ↔ Filter.Tendsto (f ∘ φ) Filter.atTop (𝓝 l)
-- Monotone.ge_of_tendsto
theorem Monotone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) :
    f b ≤ a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (b : β)
Goal: f b ≤ a
-- Monotone.le_of_tendsto
theorem Monotone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) :
    a ≤ f b 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f) (ha : Filter.Tendsto f Filter.atBot (𝓝 a)) (b : β)
Goal: a ≤ f b
-- Antitone.le_of_tendsto
theorem Antitone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) :
    a ≤ f b 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (b : β)
Goal: a ≤ f b
-- Antitone.ge_of_tendsto
theorem Antitone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) :
    f b ≤ a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f) (ha : Filter.Tendsto f Filter.atBot (𝓝 a)) (b : β)
Goal: f b ≤ a
-- isLUB_of_tendsto_atTop
theorem isLUB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f)
    (ha : Tendsto f atTop (𝓝 a)) : IsLUB (Set.range f) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: IsLUB (Set.range f) a
-- isGLB_of_tendsto_atBot
theorem isGLB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f)
    (ha : Tendsto f atBot (𝓝 a)) : IsGLB (Set.range f) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f) (ha : Filter.Tendsto f Filter.atBot (𝓝 a))
Goal: IsGLB (Set.range f) a
-- isLUB_of_tendsto_atBot
theorem isLUB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f)
    (ha : Tendsto f atBot (𝓝 a)) : IsLUB (Set.range f) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f) (ha : Filter.Tendsto f Filter.atBot (𝓝 a))
Goal: IsLUB (Set.range f) a
-- isGLB_of_tendsto_atTop
theorem isGLB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f)
    (ha : Tendsto f atTop (𝓝 a)) : IsGLB (Set.range f) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: IsGLB (Set.range f) a
-- iSup_eq_of_tendsto
theorem iSup_eq_of_tendsto {α β} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) :
    Tendsto f atTop (𝓝 a) → iSup f = a 
Arguments: {α : Type u_3} {β : Type u_4} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f)
Goal: Filter.Tendsto f Filter.atTop (𝓝 a) → iSup f = a
-- iInf_eq_of_tendsto
theorem iInf_eq_of_tendsto {α} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) :
    Tendsto f atTop (𝓝 a) → iInf f = a 
Arguments: {β : Type u_2} {α : Type u_3} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f)
Goal: Filter.Tendsto f Filter.atTop (𝓝 a) → iInf f = a
-- iSup_eq_iSup_subseq_of_monotone
theorem iSup_eq_iSup_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
    (hφ : Tendsto φ l atTop) : ⨆ i, f i = ⨆ i, f (φ i) 
Arguments: {ι₁ : Type u_3} {ι₂ : Type u_4} {α : Type u_5} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f) (hφ : Filter.Tendsto φ l Filter.atTop)
Goal: ⨆ i, f i = ⨆ i, f (φ i)
-- iSup_eq_iSup_subseq_of_antitone
theorem iSup_eq_iSup_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
    (hφ : Tendsto φ l atBot) : ⨆ i, f i = ⨆ i, f (φ i) 
Arguments: {ι₁ : Type u_3} {ι₂ : Type u_4} {α : Type u_5} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f) (hφ : Filter.Tendsto φ l Filter.atBot)
Goal: ⨆ i, f i = ⨆ i, f (φ i)
-- iInf_eq_iInf_subseq_of_monotone
theorem iInf_eq_iInf_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
    (hφ : Tendsto φ l atBot) : ⨅ i, f i = ⨅ i, f (φ i) 
Arguments: {ι₁ : Type u_3} {ι₂ : Type u_4} {α : Type u_5} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f) (hφ : Filter.Tendsto φ l Filter.atBot)
Goal: ⨅ i, f i = ⨅ i, f (φ i)
-- iInf_eq_iInf_subseq_of_antitone
theorem iInf_eq_iInf_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
    (hφ : Tendsto φ l atTop) : ⨅ i, f i = ⨅ i, f (φ i) 
Arguments: {ι₁ : Type u_3} {ι₂ : Type u_4} {α : Type u_5} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f) (hφ : Filter.Tendsto φ l Filter.atTop)
Goal: ⨅ i, f i = ⨅ i, f (φ i)
-- ClopenUpperSet.upper'
structure ClopenUpperSet (α : Type*) [TopologicalSpace α] [LE α] extends Clopens α where
  upper' : IsUpperSet carrier
Arguments: {α : Type u_3} [TopologicalSpace α] [LE α] (self : ClopenUpperSet α)
Goal: IsUpperSet self.carrier
-- ClopenUpperSet.upper
theorem upper (s : ClopenUpperSet α) : IsUpperSet (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α] (s : ClopenUpperSet α)
Goal: IsUpperSet ↑s
-- ClopenUpperSet.isClopen
theorem isClopen (s : ClopenUpperSet α) : IsClopen (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α] (s : ClopenUpperSet α)
Goal: IsClopen ↑s
-- ClopenUpperSet.toUpperSet_coe
@[simps]
def toUpperSet (s : ClopenUpperSet α) : UpperSet α 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α] (s : ClopenUpperSet α)
Goal: ↑s.toUpperSet = ↑s
-- ClopenUpperSet.ext
@[ext]
protected theorem ext {s t : ClopenUpperSet α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α] {s t : ClopenUpperSet α} (h : ↑s = ↑t)
Goal: s = t
-- ClopenUpperSet.coe_mk
@[simp]
theorem coe_mk (s : Clopens α) (h) : (mk s h : Set α) = s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α] (s : TopologicalSpace.Clopens α) (h : IsUpperSet s.carrier)
Goal: ↑{ toClopens := s, upper' := h } = ↑s
-- ClopenUpperSet.coe_sup
@[simp]
theorem coe_sup (s t : ClopenUpperSet α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α] (s t : ClopenUpperSet α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- ClopenUpperSet.coe_inf
@[simp]
theorem coe_inf (s t : ClopenUpperSet α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α] (s t : ClopenUpperSet α)
Goal: ↑(s ⊓ t) = ↑s ∩ ↑t
-- ClopenUpperSet.coe_top
@[simp]
theorem coe_top : (↑(⊤ : ClopenUpperSet α) : Set α) = univ 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α]
Goal: ↑⊤ = Set.univ
-- ClopenUpperSet.coe_bot
@[simp]
theorem coe_bot : (↑(⊥ : ClopenUpperSet α) : Set α) = ∅ 
Arguments: {α : Type u_1} [TopologicalSpace α] [LE α]
Goal: ↑⊥ = ∅
-- Topology.WithLower.to_WithLower_symm_eq
@[simp] lemma to_WithLower_symm_eq : (@toLower α).symm = ofLower 
Arguments: {α : Type u_1}
Goal: Topology.WithLower.toLower.symm = Topology.WithLower.ofLower
-- Topology.WithLower.of_WithLower_symm_eq
@[simp] lemma of_WithLower_symm_eq : (@ofLower α).symm = toLower 
Arguments: {α : Type u_1}
Goal: Topology.WithLower.ofLower.symm = Topology.WithLower.toLower
-- Topology.WithLower.toLower_ofLower
@[simp] lemma toLower_ofLower (a : WithLower α) : toLower (ofLower a) = a 
Arguments: {α : Type u_1} (a : Topology.WithLower α)
Goal: Topology.WithLower.toLower (Topology.WithLower.ofLower a) = a
-- Topology.WithLower.ofLower_toLower
@[simp] lemma ofLower_toLower (a : α) : ofLower (toLower a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: Topology.WithLower.ofLower (Topology.WithLower.toLower a) = a
-- Topology.WithLower.toLower_inj
lemma toLower_inj {a b : α} : toLower a = toLower b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: Topology.WithLower.toLower a = Topology.WithLower.toLower b ↔ a = b
-- Topology.WithLower.ofLower_inj
theorem ofLower_inj {a b : WithLower α} : ofLower a = ofLower b ↔ a = b 
Arguments: {α : Type u_1} {a b : Topology.WithLower α}
Goal: Topology.WithLower.ofLower a = Topology.WithLower.ofLower b ↔ a = b
-- Topology.WithLower.isOpen_preimage_ofLower
lemma isOpen_preimage_ofLower : IsOpen (ofLower ⁻¹' s) ↔ (lower α).IsOpen s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: IsOpen (⇑Topology.WithLower.ofLower ⁻¹' s) ↔ TopologicalSpace.IsOpen s
-- Topology.WithLower.isOpen_def
lemma isOpen_def (T : Set (WithLower α)) : IsOpen T ↔ (lower α).IsOpen (WithLower.toLower ⁻¹' T) 
Arguments: {α : Type u_1} [Preorder α] (T : Set (Topology.WithLower α))
Goal: IsOpen T ↔ TopologicalSpace.IsOpen (⇑Topology.WithLower.toLower ⁻¹' T)
-- Topology.WithUpper.to_WithUpper_symm_eq
@[simp] lemma to_WithUpper_symm_eq {α} : (@toUpper α).symm = ofUpper 
Arguments: {α : Type u_1}
Goal: Topology.WithUpper.toUpper.symm = Topology.WithUpper.ofUpper
-- Topology.WithUpper.of_WithUpper_symm_eq
@[simp] lemma of_WithUpper_symm_eq : (@ofUpper α).symm = toUpper 
Arguments: {α : Type u_1}
Goal: Topology.WithUpper.ofUpper.symm = Topology.WithUpper.toUpper
-- Topology.WithUpper.toUpper_ofUpper
@[simp] lemma toUpper_ofUpper (a : WithUpper α) : toUpper (ofUpper a) = a 
Arguments: {α : Type u_1} (a : Topology.WithUpper α)
Goal: Topology.WithUpper.toUpper (Topology.WithUpper.ofUpper a) = a
-- Topology.WithUpper.ofUpper_toUpper
@[simp] lemma ofUpper_toUpper (a : α) : ofUpper (toUpper a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: Topology.WithUpper.ofUpper (Topology.WithUpper.toUpper a) = a
-- Topology.WithUpper.toUpper_inj
lemma toUpper_inj {a b : α} : toUpper a = toUpper b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: Topology.WithUpper.toUpper a = Topology.WithUpper.toUpper b ↔ a = b
-- Topology.WithUpper.ofUpper_inj
lemma ofUpper_inj {a b : WithUpper α} : ofUpper a = ofUpper b ↔ a = b 
Arguments: {α : Type u_1} {a b : Topology.WithUpper α}
Goal: Topology.WithUpper.ofUpper a = Topology.WithUpper.ofUpper b ↔ a = b
-- Topology.WithUpper.isOpen_preimage_ofUpper
lemma isOpen_preimage_ofUpper : IsOpen (ofUpper ⁻¹' s) ↔ (upper α).IsOpen s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: IsOpen (⇑Topology.WithUpper.ofUpper ⁻¹' s) ↔ TopologicalSpace.IsOpen s
-- Topology.WithUpper.isOpen_def
lemma isOpen_def {s : Set (WithUpper α)} : IsOpen s ↔ (upper α).IsOpen (toUpper ⁻¹' s) 
Arguments: {α : Type u_1} [Preorder α] {s : Set (Topology.WithUpper α)}
Goal: IsOpen s ↔ TopologicalSpace.IsOpen (⇑Topology.WithUpper.toUpper ⁻¹' s)
-- Topology.IsLower.topology_eq_lowerTopology
class IsLower (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
  topology_eq_lowerTopology : t = lower α
Arguments: {α : Type u_1} [t : TopologicalSpace α] [Preorder α] [self : Topology.IsLower α]
Goal: t = Topology.lower α
-- Topology.IsUpper.topology_eq_upperTopology
class IsUpper (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
  topology_eq_upperTopology : t = upper α
Arguments: {α : Type u_1} [t : TopologicalSpace α] [Preorder α] [self : Topology.IsUpper α]
Goal: t = Topology.upper α
-- Topology.IsLower.topology_eq
lemma topology_eq : ‹_› = lower α 
Arguments: (α : Type u_1) [Preorder α] [TopologicalSpace α] [Topology.IsLower α]
Goal: inst✝ = Topology.lower α
-- Topology.IsLower.isOpen_iff_generate_Ici_compl
theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Ici a)ᶜ = t } s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α] {s : Set α}
Goal: IsOpen s ↔ TopologicalSpace.GenerateOpen {t | ∃ a, (Set.Ici a)ᶜ = t} s
-- Topology.IsLower.isClosed_upperClosure
theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α] {s : Set α} (h : s.Finite)
Goal: IsClosed ↑(upperClosure s)
-- Topology.IsLower.isLowerSet_of_isOpen
theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α] {s : Set α} (h : IsOpen s)
Goal: IsLowerSet s
-- Topology.IsLower.isUpperSet_of_isClosed
theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α] {s : Set α} (h : IsClosed s)
Goal: IsUpperSet s
-- Topology.IsLower.closure_singleton
@[simp]
theorem closure_singleton (a : α) : closure {a} = Ici a 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α] (a : α)
Goal: closure {a} = Set.Ici a
-- Topology.IsLower.isTopologicalBasis
protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α]
Goal: TopologicalSpace.IsTopologicalBasis (Topology.IsLower.lowerBasis α)
-- Topology.IsLower.continuous_iff_Ici
lemma continuous_iff_Ici [TopologicalSpace β] {f : β → α} :
    Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Ici a)) 
Arguments: {α : Type u_2} {β : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsLower α] [TopologicalSpace β] {f : β → α}
Goal: Continuous f ↔ ∀ (a : α), IsClosed (f ⁻¹' Set.Ici a)
-- Topology.IsUpper.topology_eq
lemma topology_eq : ‹_› = upper α 
Arguments: (α : Type u_1) [Preorder α] [TopologicalSpace α] [Topology.IsUpper α]
Goal: inst✝ = Topology.upper α
-- Topology.IsUpper.isOpen_iff_generate_Iic_compl
theorem isOpen_iff_generate_Iic_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Iic a)ᶜ = t } s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α] {s : Set α}
Goal: IsOpen s ↔ TopologicalSpace.GenerateOpen {t | ∃ a, (Set.Iic a)ᶜ = t} s
-- Topology.IsUpper.isClosed_lowerClosure
theorem isClosed_lowerClosure (h : s.Finite) : IsClosed (lowerClosure s : Set α) 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α] {s : Set α} (h : s.Finite)
Goal: IsClosed ↑(lowerClosure s)
-- Topology.IsUpper.isUpperSet_of_isOpen
theorem isUpperSet_of_isOpen (h : IsOpen s) : IsUpperSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α] {s : Set α} (h : IsOpen s)
Goal: IsUpperSet s
-- Topology.IsUpper.isLowerSet_of_isClosed
theorem isLowerSet_of_isClosed (h : IsClosed s) : IsLowerSet s 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α] {s : Set α} (h : IsClosed s)
Goal: IsLowerSet s
-- Topology.IsUpper.closure_singleton
@[simp]
theorem closure_singleton (a : α) : closure {a} = Iic a 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α] (a : α)
Goal: closure {a} = Set.Iic a
-- Topology.IsUpper.isTopologicalBasis
protected theorem isTopologicalBasis : IsTopologicalBasis (upperBasis α) 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α]
Goal: TopologicalSpace.IsTopologicalBasis (Topology.IsUpper.upperBasis α)
-- Topology.IsUpper.continuous_iff_Iic
lemma continuous_iff_Iic [TopologicalSpace β] {f : β → α} :
    Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Iic a)) 
Arguments: {α : Type u_2} {β : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α] [TopologicalSpace β] {f : β → α}
Goal: Continuous f ↔ ∀ (a : α), IsClosed (f ⁻¹' Set.Iic a)
-- Topology.IsUpper.continuous_of_Iic
@[deprecated (since := "2023-12-24")]
lemma continuous_of_Iic [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Iic a))) :
    Continuous f 
Arguments: {α : Type u_2} {β : Type u_1} [Preorder α] [TopologicalSpace α] [Topology.IsUpper α] [TopologicalSpace β] {f : β → α} (h : ∀ (a : α), IsClosed (f ⁻¹' Set.Iic a))
Goal: Continuous f
-- sInfHom.continuous
protected lemma _root_.sInfHom.continuous (f : sInfHom α β) : Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [Topology.IsLower α] [TopologicalSpace β] [Topology.IsLower β] (f : sInfHom α β)
Goal: Continuous ⇑f
-- sSupHom.continuous
protected lemma _root_.sSupHom.continuous (f : sSupHom α β) : Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [Topology.IsUpper α] [TopologicalSpace β] [Topology.IsUpper β] (f : sSupHom α β)
Goal: Continuous ⇑f
-- Topology.isUpper_orderDual
lemma isUpper_orderDual [Preorder α] [TopologicalSpace α] : IsUpper αᵒᵈ ↔ IsLower α 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α]
Goal: Topology.IsUpper αᵒᵈ ↔ Topology.IsLower α
-- Topology.isLower_orderDual
lemma isLower_orderDual [Preorder α] [TopologicalSpace α] : IsLower αᵒᵈ ↔ IsUpper α 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α]
Goal: Topology.IsLower αᵒᵈ ↔ Topology.IsUpper α
-- Absorbs.empty
protected lemma empty : Absorbs M s ∅ 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α}
Goal: Absorbs M s ∅
-- Absorbs.eventually
protected lemma eventually (h : Absorbs M s t) : ∀ᶠ a in cobounded M, t ⊆ a • s 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s t : Set α} (h : Absorbs M s t)
Goal: ∀ᶠ (a : M) in Bornology.cobounded M, t ⊆ a • s
-- Absorbs.of_boundedSpace
@[simp] lemma of_boundedSpace [BoundedSpace M] : Absorbs M s t 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s t : Set α} [BoundedSpace M]
Goal: Absorbs M s t
-- Absorbs.mono_left
lemma mono_left (h : Absorbs M s₁ t) (hs : s₁ ⊆ s₂) : Absorbs M s₂ t 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s₁ s₂ t : Set α} (h : Absorbs M s₁ t) (hs : s₁ ⊆ s₂)
Goal: Absorbs M s₂ t
-- Absorbs.mono_right
lemma mono_right (h : Absorbs M s t₁) (ht : t₂ ⊆ t₁) : Absorbs M s t₂ 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s t₁ t₂ : Set α} (h : Absorbs M s t₁) (ht : t₂ ⊆ t₁)
Goal: Absorbs M s t₂
-- Absorbs.mono
lemma mono (h : Absorbs M s₁ t₁) (hs : s₁ ⊆ s₂) (ht : t₂ ⊆ t₁) : Absorbs M s₂ t₂ 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s₁ s₂ t₁ t₂ : Set α} (h : Absorbs M s₁ t₁) (hs : s₁ ⊆ s₂) (ht : t₂ ⊆ t₁)
Goal: Absorbs M s₂ t₂
-- absorbs_union
@[simp]
lemma _root_.absorbs_union : Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂ 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s t₁ t₂ : Set α}
Goal: Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂
-- Absorbs.union
protected lemma union (h₁ : Absorbs M s t₁) (h₂ : Absorbs M s t₂) : Absorbs M s (t₁ ∪ t₂) 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s t₁ t₂ : Set α} (h₁ : Absorbs M s t₁) (h₂ : Absorbs M s t₂)
Goal: Absorbs M s (t₁ ∪ t₂)
-- Set.Finite.absorbs_sUnion
lemma _root_.Set.Finite.absorbs_sUnion {T : Set (Set α)} (hT : T.Finite) :
    Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α} {T : Set (Set α)} (hT : T.Finite)
Goal: Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t
-- Absorbs.sUnion
protected lemma sUnion (hT : T.Finite) (hs : ∀ t ∈ T, Absorbs M s t) :
    Absorbs M s (⋃₀ T) 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α} {T : Set (Set α)} (hT : T.Finite) (hs : ∀ t ∈ T, Absorbs M s t)
Goal: Absorbs M s (⋃₀ T)
-- absorbs_iUnion
@[simp]
lemma _root_.absorbs_iUnion {ι : Sort*} [Finite ι] {t : ι → Set α} :
    Absorbs M s (⋃ i, t i) ↔ ∀ i, Absorbs M s (t i) 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α} {ι : Sort u_3} [Finite ι] {t : ι → Set α}
Goal: Absorbs M s (⋃ i, t i) ↔ ∀ (i : ι), Absorbs M s (t i)
-- Set.Finite.absorbs_biUnion
lemma _root_.Set.Finite.absorbs_biUnion {ι : Type*} {t : ι → Set α} {I : Set ι} (hI : I.Finite) :
    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α} {ι : Type u_3} {t : ι → Set α} {I : Set ι} (hI : I.Finite)
Goal: Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i)