ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsOpen.exterior_subset_iff
lemma IsOpen.exterior_subset_iff (ht : IsOpen t) : exterior s ⊆ t ↔ s ⊆ t 
Arguments: {α : Type u_3} [TopologicalSpace α] {s t : Set α} (ht : IsOpen t)
Goal: exterior s ⊆ t ↔ s ⊆ t
-- exterior_mono
@[mono] lemma exterior_mono : Monotone (exterior : Set α → Set α) 
Arguments: {α : Type u_3} [TopologicalSpace α]
Goal: Monotone exterior
-- exterior_empty
@[simp] lemma exterior_empty : exterior (∅ : Set α) = ∅ 
Arguments: {α : Type u_3} [TopologicalSpace α]
Goal: exterior ∅ = ∅
-- exterior_univ
@[simp] lemma exterior_univ : exterior (univ : Set α) = univ 
Arguments: {α : Type u_3} [TopologicalSpace α]
Goal: exterior Set.univ = Set.univ
-- exterior_eq_empty
@[simp] lemma exterior_eq_empty : exterior s = ∅ ↔ s = ∅ 
Arguments: {α : Type u_3} [TopologicalSpace α] {s : Set α}
Goal: exterior s = ∅ ↔ s = ∅
-- isOpen_exterior
@[simp] lemma isOpen_exterior : IsOpen (exterior s) 
Arguments: {α : Type u_3} [TopologicalSpace α] {s : Set α} [AlexandrovDiscrete α]
Goal: IsOpen (exterior s)
-- exterior_mem_nhdsSet
lemma exterior_mem_nhdsSet : exterior s ∈ 𝓝ˢ s 
Arguments: {α : Type u_3} [TopologicalSpace α] {s : Set α} [AlexandrovDiscrete α]
Goal: exterior s ∈ 𝓝ˢ s
-- exterior_eq_iff_isOpen
@[simp] lemma exterior_eq_iff_isOpen : exterior s = s ↔ IsOpen s 
Arguments: {α : Type u_3} [TopologicalSpace α] {s : Set α} [AlexandrovDiscrete α]
Goal: exterior s = s ↔ IsOpen s
-- exterior_subset_iff_isOpen
@[simp] lemma exterior_subset_iff_isOpen : exterior s ⊆ s ↔ IsOpen s 
Arguments: {α : Type u_3} [TopologicalSpace α] {s : Set α} [AlexandrovDiscrete α]
Goal: exterior s ⊆ s ↔ IsOpen s
-- exterior_subset_iff
lemma exterior_subset_iff : exterior s ⊆ t ↔ ∃ U, IsOpen U ∧ s ⊆ U ∧ U ⊆ t 
Arguments: {α : Type u_3} [TopologicalSpace α] {s t : Set α} [AlexandrovDiscrete α]
Goal: exterior s ⊆ t ↔ ∃ U, IsOpen U ∧ s ⊆ U ∧ U ⊆ t
-- exterior_subset_iff_mem_nhdsSet
lemma exterior_subset_iff_mem_nhdsSet : exterior s ⊆ t ↔ t ∈ 𝓝ˢ s 
Arguments: {α : Type u_3} [TopologicalSpace α] {s t : Set α} [AlexandrovDiscrete α]
Goal: exterior s ⊆ t ↔ t ∈ 𝓝ˢ s
-- exterior_singleton_subset_iff_mem_nhds
lemma exterior_singleton_subset_iff_mem_nhds : exterior {a} ⊆ t ↔ t ∈ 𝓝 a 
Arguments: {α : Type u_3} [TopologicalSpace α] {t : Set α} {a : α} [AlexandrovDiscrete α]
Goal: exterior {a} ⊆ t ↔ t ∈ 𝓝 a
-- IsOpen.exterior_subset
lemma IsOpen.exterior_subset (ht : IsOpen t) : exterior s ⊆ t ↔ s ⊆ t 
Arguments: {α : Type u_3} [TopologicalSpace α] {s t : Set α} (ht : IsOpen t)
Goal: exterior s ⊆ t ↔ s ⊆ t
-- gc_exterior_interior
lemma gc_exterior_interior : GaloisConnection (exterior : Set α → Set α) interior 
Arguments: {α : Type u_3} [TopologicalSpace α] [AlexandrovDiscrete α]
Goal: GaloisConnection exterior interior
-- exterior_exterior
@[simp] lemma exterior_exterior (s : Set α) : exterior (exterior s) = exterior s 
Arguments: {α : Type u_3} [TopologicalSpace α] [AlexandrovDiscrete α] (s : Set α)
Goal: exterior (exterior s) = exterior s
-- exterior_union
@[simp] lemma exterior_union (s t : Set α) : exterior (s ∪ t) = exterior s ∪ exterior t 
Arguments: {α : Type u_3} [TopologicalSpace α] [AlexandrovDiscrete α] (s t : Set α)
Goal: exterior (s ∪ t) = exterior s ∪ exterior t
-- nhdsSet_exterior
@[simp] lemma nhdsSet_exterior (s : Set α) : 𝓝ˢ (exterior s) = 𝓝ˢ s 
Arguments: {α : Type u_3} [TopologicalSpace α] [AlexandrovDiscrete α] (s : Set α)
Goal: 𝓝ˢ (exterior s) = 𝓝ˢ s
-- principal_exterior
@[simp] lemma principal_exterior (s : Set α) : 𝓟 (exterior s) = 𝓝ˢ s 
Arguments: {α : Type u_3} [TopologicalSpace α] [AlexandrovDiscrete α] (s : Set α)
Goal: 𝓟 (exterior s) = 𝓝ˢ s
-- exterior_subset_exterior
@[simp] lemma exterior_subset_exterior : exterior s ⊆ exterior t ↔ 𝓝ˢ s ≤ 𝓝ˢ t 
Arguments: {α : Type u_3} [TopologicalSpace α] {s t : Set α} [AlexandrovDiscrete α]
Goal: exterior s ⊆ exterior t ↔ 𝓝ˢ s ≤ 𝓝ˢ t
-- specializes_iff_exterior_subset
lemma specializes_iff_exterior_subset : x ⤳ y ↔ exterior {x} ⊆ exterior {y} 
Arguments: {α : Type u_3} [TopologicalSpace α] {x y : α} [AlexandrovDiscrete α]
Goal: x ⤳ y ↔ exterior {x} ⊆ exterior {y}
-- isOpen_iff_forall_specializes
lemma isOpen_iff_forall_specializes : IsOpen s ↔ ∀ x y, x ⤳ y → y ∈ s → x ∈ s 
Arguments: {α : Type u_3} [TopologicalSpace α] {s : Set α} [AlexandrovDiscrete α]
Goal: IsOpen s ↔ ∀ (x y : α), x ⤳ y → y ∈ s → x ∈ s
-- Set.Finite.isCompact_exterior
lemma Set.Finite.isCompact_exterior (hs : s.Finite) : IsCompact (exterior s) 
Arguments: {α : Type u_3} [TopologicalSpace α] {s : Set α} (hs : s.Finite)
Goal: IsCompact (exterior s)
-- Inducing.alexandrovDiscrete
lemma Inducing.alexandrovDiscrete {f : β → α} (h : Inducing f) : AlexandrovDiscrete β 
Arguments: {α : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [AlexandrovDiscrete α] {f : β → α} (h : Inducing f)
Goal: AlexandrovDiscrete β
-- alexandrovDiscrete_coinduced
lemma alexandrovDiscrete_coinduced {β : Type*} {f : α → β} :
    @AlexandrovDiscrete β (coinduced f ‹_›) 
Arguments: {α : Type u_3} [TopologicalSpace α] [AlexandrovDiscrete α] {β : Type u_5} {f : α → β}
Goal: AlexandrovDiscrete β
-- AlexandrovDiscrete.sup
lemma AlexandrovDiscrete.sup {t₁ t₂ : TopologicalSpace α} (_ : @AlexandrovDiscrete α t₁)
    (_ : @AlexandrovDiscrete α t₂) :
    @AlexandrovDiscrete α (t₁ ⊔ t₂) 
Arguments: {α : Type u_3} {t₁ t₂ : TopologicalSpace α}
Goal: AlexandrovDiscrete α → AlexandrovDiscrete α → AlexandrovDiscrete α
-- alexandrovDiscrete_iSup
lemma alexandrovDiscrete_iSup {t : ι → TopologicalSpace α} (_ : ∀ i, @AlexandrovDiscrete α (t i)) :
    @AlexandrovDiscrete α (⨆ i, t i) 
Arguments: {ι : Sort u_1} {α : Type u_3} {t : ι → TopologicalSpace α}
Goal: (∀ (i : ι), AlexandrovDiscrete α) → AlexandrovDiscrete α
-- nhds_def
irreducible_def nhds (x : X) : Filter X 
Arguments: {X : Type u_3} [TopologicalSpace X] (x : X)
Goal: 𝓝 x = ⨅ s ∈ {s | x ∈ s ∧ IsOpen s}, 𝓟 s
-- CompactSpace.isCompact_univ
class CompactSpace : Prop where
    isCompact_univ : IsCompact (Set.univ : Set X)
Arguments: {X : Type u_1} [TopologicalSpace X] [self : CompactSpace X]
Goal: IsCompact Set.univ
-- NoncompactSpace.noncompact_univ
class NoncompactSpace : Prop where
    noncompact_univ : ¬IsCompact (Set.univ : Set X)
Arguments: {X : Type u_1} [TopologicalSpace X] [self : NoncompactSpace X]
Goal: ¬IsCompact Set.univ
-- WeaklyLocallyCompactSpace.exists_compact_mem_nhds
class WeaklyLocallyCompactSpace (X : Type*) [TopologicalSpace X] : Prop where
    exists_compact_mem_nhds (x : X) : ∃ s, IsCompact s ∧ s ∈ 𝓝 x
Arguments: {X : Type u_3} [TopologicalSpace X] [self : WeaklyLocallyCompactSpace X] (x : X)
Goal: ∃ s, IsCompact s ∧ s ∈ 𝓝 x
-- LocallyCompactSpace.local_compact_nhds
class LocallyCompactSpace (X : Type*) [TopologicalSpace X] : Prop where
    local_compact_nhds : ∀ (x : X), ∀ n ∈ 𝓝 x, ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s
Arguments: {X : Type u_3} [TopologicalSpace X] [self : LocallyCompactSpace X] (x : X) (n : Set X)
Goal: n ∈ 𝓝 x → ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s
-- LocallyCompactPair.exists_mem_nhds_isCompact_mapsTo
class LocallyCompactPair (X Y : Type*) [TopologicalSpace X] [TopologicalSpace Y] : Prop where
    exists_mem_nhds_isCompact_mapsTo : ∀ {f : X → Y} {x : X} {s : Set Y},
    Continuous f → s ∈ 𝓝 (f x) → ∃ K ∈ 𝓝 x, IsCompact K ∧ Set.MapsTo f K s
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [self : LocallyCompactPair X Y] {f : X → Y} {x : X} {s : Set Y}
Goal: Continuous f → s ∈ 𝓝 (f x) → ∃ K ∈ 𝓝 x, IsCompact K ∧ Set.MapsTo f K s
-- Filter.tendsto_nhds_atTop
protected theorem tendsto_nhds_atTop [NoMaxOrder X] : Tendsto 𝓝 (atTop : Filter X) (𝓝 atTop) 
Arguments: {X : Type u_2} [TopologicalSpace X] [PartialOrder X] [OrderTopology X] [NoMaxOrder X]
Goal: Filter.Tendsto 𝓝 Filter.atTop (𝓝 Filter.atTop)
-- Filter.tendsto_nhds_atBot
protected theorem tendsto_nhds_atBot [NoMinOrder X] : Tendsto 𝓝 (atBot : Filter X) (𝓝 atBot) 
Arguments: {X : Type u_2} [TopologicalSpace X] [PartialOrder X] [OrderTopology X] [NoMinOrder X]
Goal: Filter.Tendsto 𝓝 Filter.atBot (𝓝 Filter.atBot)
-- Filter.Tendsto.nhds_atTop
theorem Tendsto.nhds_atTop [NoMaxOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atTop) :
    Tendsto (𝓝 ∘ f) l (𝓝 atTop) 
Arguments: {α : Type u_1} {X : Type u_2} [TopologicalSpace X] [PartialOrder X] [OrderTopology X] [NoMaxOrder X] {f : α → X} {l : Filter α} (h : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (𝓝 ∘ f) l (𝓝 Filter.atTop)
-- Filter.Tendsto.nhds_atBot
theorem Tendsto.nhds_atBot [NoMinOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atBot) :
    Tendsto (𝓝 ∘ f) l (𝓝 atBot) 
Arguments: {α : Type u_1} {X : Type u_2} [TopologicalSpace X] [PartialOrder X] [OrderTopology X] [NoMinOrder X] {f : α → X} {l : Filter α} (h : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (𝓝 ∘ f) l (𝓝 Filter.atBot)
-- Filter.isBounded_le_map_of_bounded_range
@[deprecated isBoundedUnder_of (since := "2024-06-07")]
lemma Filter.isBounded_le_map_of_bounded_range {ι : Type*} (F : Filter ι) {f : ι → ℝ}
    (h : Bornology.IsBounded (Set.range f)) :
    (F.map f).IsBounded (· ≤ ·) 
Arguments: {ι : Type u_1} (F : Filter ι) {f : ι → ℝ} (h : Bornology.IsBounded (Set.range f))
Goal: Filter.IsBounded (fun x x_1 => x ≤ x_1) (Filter.map f F)
-- Filter.isBounded_ge_map_of_bounded_range
@[deprecated isBoundedUnder_of (since := "2024-06-07")]
lemma Filter.isBounded_ge_map_of_bounded_range {ι : Type*} (F : Filter ι) {f : ι → ℝ}
    (h : Bornology.IsBounded (Set.range f)) :
    (F.map f).IsBounded (· ≥ ·) 
Arguments: {ι : Type u_1} (F : Filter ι) {f : ι → ℝ} (h : Bornology.IsBounded (Set.range f))
Goal: Filter.IsBounded (fun x x_1 => x ≥ x_1) (Filter.map f F)
-- frequently_lt_nhds
lemma frequently_lt_nhds (a : α) [NeBot (𝓝[<] a)] : ∃ᶠ x in 𝓝 a, x < a 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] (a : α) [(𝓝[<] a).NeBot]
Goal: ∃ᶠ (x : α) in 𝓝 a, x < a
-- frequently_gt_nhds
lemma frequently_gt_nhds (a : α) [NeBot (𝓝[>] a)] : ∃ᶠ x in 𝓝 a, a < x 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] (a : α) [(𝓝[>] a).NeBot]
Goal: ∃ᶠ (x : α) in 𝓝 a, a < x
-- Filter.Eventually.exists_lt
theorem Filter.Eventually.exists_lt {a : α} [NeBot (𝓝[<] a)] {p : α → Prop}
    (h : ∀ᶠ x in 𝓝 a, p x) : ∃ b < a, p b 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] {a : α} [(𝓝[<] a).NeBot] {p : α → Prop} (h : ∀ᶠ (x : α) in 𝓝 a, p x)
Goal: ∃ b < a, p b
-- Filter.Eventually.exists_gt
theorem Filter.Eventually.exists_gt {a : α} [NeBot (𝓝[>] a)] {p : α → Prop}
    (h : ∀ᶠ x in 𝓝 a, p x) : ∃ b > a, p b 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] {a : α} [(𝓝[>] a).NeBot] {p : α → Prop} (h : ∀ᶠ (x : α) in 𝓝 a, p x)
Goal: ∃ b > a, p b
-- nhdsWithin_Ici_neBot
theorem nhdsWithin_Ici_neBot {a b : α} (H₂ : a ≤ b) : NeBot (𝓝[Ici a] b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] {a b : α} (H₂ : a ≤ b)
Goal: (𝓝[Set.Ici a] b).NeBot
-- nhdsWithin_Iic_neBot
theorem nhdsWithin_Iic_neBot {a b : α} (H : a ≤ b) : NeBot (𝓝[Iic b] a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] {a b : α} (H : a ≤ b)
Goal: (𝓝[Set.Iic b] a).NeBot
-- nhds_left'_le_nhds_ne
theorem nhds_left'_le_nhds_ne (a : α) : 𝓝[<] a ≤ 𝓝[≠] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] (a : α)
Goal: 𝓝[<] a ≤ 𝓝[≠] a
-- nhds_right'_le_nhds_ne
theorem nhds_right'_le_nhds_ne (a : α) : 𝓝[>] a ≤ 𝓝[≠] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] (a : α)
Goal: 𝓝[>] a ≤ 𝓝[≠] a
-- IsAntichain.interior_eq_empty
lemma IsAntichain.interior_eq_empty [∀ x : α, (𝓝[<] x).NeBot] {s : Set α}
    (hs : IsAntichain (· ≤ ·) s) : interior s = ∅ 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [∀ (x : α), (𝓝[<] x).NeBot] {s : Set α} (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: interior s = ∅
-- IsAntichain.interior_eq_empty'
lemma IsAntichain.interior_eq_empty' [∀ x : α, (𝓝[>] x).NeBot] {s : Set α}
    (hs : IsAntichain (· ≤ ·) s) : interior s = ∅ 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [∀ (x : α), (𝓝[>] x).NeBot] {s : Set α} (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: interior s = ∅
-- continuousWithinAt_Ioi_iff_Ici
theorem continuousWithinAt_Ioi_iff_Ici {a : α} {f : α → β} :
    ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [PartialOrder α] [TopologicalSpace β] {a : α} {f : α → β}
Goal: ContinuousWithinAt f (Set.Ioi a) a ↔ ContinuousWithinAt f (Set.Ici a) a
-- continuousWithinAt_Iio_iff_Iic
theorem continuousWithinAt_Iio_iff_Iic {a : α} {f : α → β} :
    ContinuousWithinAt f (Iio a) a ↔ ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [PartialOrder α] [TopologicalSpace β] {a : α} {f : α → β}
Goal: ContinuousWithinAt f (Set.Iio a) a ↔ ContinuousWithinAt f (Set.Iic a) a
-- nhds_left_sup_nhds_right
theorem nhds_left_sup_nhds_right (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] (a : α)
Goal: 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a
-- nhds_left'_sup_nhds_right
theorem nhds_left'_sup_nhds_right (a : α) : 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] (a : α)
Goal: 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a
-- nhds_left_sup_nhds_right'
theorem nhds_left_sup_nhds_right' (a : α) : 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] (a : α)
Goal: 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a
-- nhds_left'_sup_nhds_right'
theorem nhds_left'_sup_nhds_right' (a : α) : 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] (a : α)
Goal: 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a
-- nhdsWithin_right_sup_nhds_singleton
lemma nhdsWithin_right_sup_nhds_singleton (a : α) :
    𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] (a : α)
Goal: 𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a
-- continuousAt_iff_continuous_left_right
theorem continuousAt_iff_continuous_left_right {a : α} {f : α → β} :
    ContinuousAt f a ↔ ContinuousWithinAt f (Iic a) a ∧ ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β] {a : α} {f : α → β}
Goal: ContinuousAt f a ↔ ContinuousWithinAt f (Set.Iic a) a ∧ ContinuousWithinAt f (Set.Ici a) a
-- continuousAt_iff_continuous_left'_right'
theorem continuousAt_iff_continuous_left'_right' {a : α} {f : α → β} :
    ContinuousAt f a ↔ ContinuousWithinAt f (Iio a) a ∧ ContinuousWithinAt f (Ioi a) a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β] {a : α} {f : α → β}
Goal: ContinuousAt f a ↔ ContinuousWithinAt f (Set.Iio a) a ∧ ContinuousWithinAt f (Set.Ioi a) a
-- orderBornology_isBounded
@[simp] lemma orderBornology_isBounded : orderBornology.IsBounded s ↔ BddBelow s ∧ BddAbove s 
Arguments: {α : Type u_1} {s : Set α} [Lattice α] [Nonempty α]
Goal: Bornology.IsBounded s ↔ BddBelow s ∧ BddAbove s
-- IsOrderBornology.isBounded_iff_bddBelow_bddAbove
class IsOrderBornology : Prop where
  protected isBounded_iff_bddBelow_bddAbove (s : Set α) : IsBounded s ↔ BddBelow s ∧ BddAbove s
Arguments: {α : Type u_1} [Bornology α] [Preorder α] [self : IsOrderBornology α] (s : Set α)
Goal: Bornology.IsBounded s ↔ BddBelow s ∧ BddAbove s
-- isOrderBornology_iff_eq_orderBornology
lemma isOrderBornology_iff_eq_orderBornology [Lattice α] [Nonempty α] :
    IsOrderBornology α ↔ ‹Bornology α› = orderBornology 
Arguments: {α : Type u_1} [Bornology α] [Lattice α] [Nonempty α]
Goal: IsOrderBornology α ↔ inst✝ = orderBornology
-- isBounded_iff_bddBelow_bddAbove
lemma isBounded_iff_bddBelow_bddAbove : IsBounded s ↔ BddBelow s ∧ BddAbove s 
Arguments: {α : Type u_1} [Bornology α] {s : Set α} [Preorder α] [IsOrderBornology α]
Goal: Bornology.IsBounded s ↔ BddBelow s ∧ BddAbove s
-- Bornology.IsBounded.bddBelow
protected lemma Bornology.IsBounded.bddBelow (hs : IsBounded s) : BddBelow s 
Arguments: {α : Type u_1} [Bornology α] {s : Set α} [Preorder α] [IsOrderBornology α] (hs : Bornology.IsBounded s)
Goal: BddBelow s
-- Bornology.IsBounded.bddAbove
protected lemma Bornology.IsBounded.bddAbove (hs : IsBounded s) : BddAbove s 
Arguments: {α : Type u_1} [Bornology α] {s : Set α} [Preorder α] [IsOrderBornology α] (hs : Bornology.IsBounded s)
Goal: BddAbove s
-- BddBelow.isBounded
protected lemma BddBelow.isBounded (hs₀ : BddBelow s) (hs₁ : BddAbove s) : IsBounded s 
Arguments: {α : Type u_1} [Bornology α] {s : Set α} [Preorder α] [IsOrderBornology α] (hs₀ : BddBelow s) (hs₁ : BddAbove s)
Goal: Bornology.IsBounded s
-- BddAbove.isBounded
protected lemma BddAbove.isBounded (hs₀ : BddAbove s) (hs₁ : BddBelow s) : IsBounded s 
Arguments: {α : Type u_1} [Bornology α] {s : Set α} [Preorder α] [IsOrderBornology α] (hs₀ : BddAbove s) (hs₁ : BddBelow s)
Goal: Bornology.IsBounded s
-- BddBelow.isBounded_inter
lemma BddBelow.isBounded_inter (hs : BddBelow s) (ht : BddAbove t) : IsBounded (s ∩ t) 
Arguments: {α : Type u_1} [Bornology α] {s t : Set α} [Preorder α] [IsOrderBornology α] (hs : BddBelow s) (ht : BddAbove t)
Goal: Bornology.IsBounded (s ∩ t)
-- BddAbove.isBounded_inter
lemma BddAbove.isBounded_inter (hs : BddAbove s) (ht : BddBelow t) : IsBounded (s ∩ t) 
Arguments: {α : Type u_1} [Bornology α] {s t : Set α} [Preorder α] [IsOrderBornology α] (hs : BddAbove s) (ht : BddBelow t)
Goal: Bornology.IsBounded (s ∩ t)
-- Bornology.IsBounded.subset_Icc_sInf_sSup
protected lemma Bornology.IsBounded.subset_Icc_sInf_sSup (hs : IsBounded s) :
    s ⊆ Icc (sInf s) (sSup s) 
Arguments: {α : Type u_1} [Bornology α] [ConditionallyCompleteLattice α] [IsOrderBornology α] {s : Set α} (hs : Bornology.IsBounded s)
Goal: s ⊆ Set.Icc (sInf s) (sSup s)
-- ContinuousOrderHom.continuous_toFun
structure ContinuousOrderHom (α β : Type*) [Preorder α] [Preorder β] [TopologicalSpace α]
  [TopologicalSpace β] extends OrderHom α β where
  continuous_toFun : Continuous toFun
Arguments: {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] (self : α →Co β)
Goal: Continuous self.toFun
-- ContinuousOrderHomClass.map_monotone
class ContinuousOrderHomClass (F : Type*) (α β : outParam Type*) [Preorder α] [Preorder β]
    [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] extends
    ContinuousMapClass F α β : Prop where
  map_monotone (f : F) : Monotone f
Arguments: {F : Type u_6} {α : outParam (Type u_7)} {β : outParam (Type u_8)} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [self : ContinuousOrderHomClass F α β] (f : F)
Goal: Monotone ⇑f
-- ContinuousOrderHom.coe_toOrderHom
@[simp] theorem coe_toOrderHom (f : α →Co β) : ⇑f.toOrderHom = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : α →Co β)
Goal: ⇑f.toOrderHom = ⇑f
-- ContinuousOrderHom.toFun_eq_coe
theorem toFun_eq_coe {f : α →Co β} : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f : α →Co β}
Goal: f.toFun = ⇑f
-- ContinuousOrderHom.ext
@[ext]
theorem ext {f g : α →Co β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f g : α →Co β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- ContinuousOrderHom.coe_copy
@[simp]
theorem coe_copy (f : α →Co β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : α →Co β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- ContinuousOrderHom.copy_eq
theorem copy_eq (f : α →Co β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : α →Co β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- ContinuousOrderHom.coe_id
@[simp]
theorem coe_id : ⇑(ContinuousOrderHom.id α) = id 
Arguments: (α : Type u_2) [TopologicalSpace α] [Preorder α]
Goal: ⇑(ContinuousOrderHom.id α) = id
-- ContinuousOrderHom.id_apply
@[simp]
theorem id_apply (a : α) : ContinuousOrderHom.id α a = a 
Arguments: {α : Type u_2} [TopologicalSpace α] [Preorder α] (a : α)
Goal: (ContinuousOrderHom.id α) a = a
-- ContinuousOrderHom.coe_comp
@[simp]
theorem coe_comp (f : β →Co γ) (g : α →Co β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (f : β →Co γ) (g : α →Co β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- ContinuousOrderHom.comp_apply
@[simp]
theorem comp_apply (f : β →Co γ) (g : α →Co β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (f : β →Co γ) (g : α →Co β) (a : α)
Goal: (f.comp g) a = f (g a)
-- ContinuousOrderHom.comp_assoc
@[simp]
theorem comp_assoc (f : γ →Co δ) (g : β →Co γ) (h : α →Co β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] [TopologicalSpace δ] [Preorder δ] (f : γ →Co δ) (g : β →Co γ) (h : α →Co β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- ContinuousOrderHom.comp_id
@[simp]
theorem comp_id (f : α →Co β) : f.comp (ContinuousOrderHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : α →Co β)
Goal: f.comp (ContinuousOrderHom.id α) = f
-- ContinuousOrderHom.id_comp
@[simp]
theorem id_comp (f : α →Co β) : (ContinuousOrderHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : α →Co β)
Goal: (ContinuousOrderHom.id β).comp f = f
-- ContinuousOrderHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : β →Co γ} {f : α →Co β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] {g₁ g₂ : β →Co γ} {f : α →Co β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- ContinuousOrderHom.cancel_left
@[simp]
theorem cancel_left {g : β →Co γ} {f₁ f₂ : α →Co β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] {g : β →Co γ} {f₁ f₂ : α →Co β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- Topology.WithUpperSet.to_WithUpperSet_symm_eq
@[simp] lemma to_WithUpperSet_symm_eq : (@toUpperSet α).symm = ofUpperSet 
Arguments: {α : Type u_1}
Goal: Topology.WithUpperSet.toUpperSet.symm = Topology.WithUpperSet.ofUpperSet
-- Topology.WithUpperSet.of_WithUpperSet_symm_eq
@[simp] lemma of_WithUpperSet_symm_eq : (@ofUpperSet α).symm = toUpperSet 
Arguments: {α : Type u_1}
Goal: Topology.WithUpperSet.ofUpperSet.symm = Topology.WithUpperSet.toUpperSet
-- Topology.WithUpperSet.toUpperSet_ofUpperSet
@[simp] lemma toUpperSet_ofUpperSet (a : WithUpperSet α) : toUpperSet (ofUpperSet a) = a 
Arguments: {α : Type u_1} (a : Topology.WithUpperSet α)
Goal: Topology.WithUpperSet.toUpperSet (Topology.WithUpperSet.ofUpperSet a) = a
-- Topology.WithUpperSet.ofUpperSet_toUpperSet
@[simp] lemma ofUpperSet_toUpperSet (a : α) : ofUpperSet (toUpperSet a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: Topology.WithUpperSet.ofUpperSet (Topology.WithUpperSet.toUpperSet a) = a
-- Topology.WithUpperSet.toUpperSet_inj
lemma toUpperSet_inj {a b : α} : toUpperSet a = toUpperSet b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: Topology.WithUpperSet.toUpperSet a = Topology.WithUpperSet.toUpperSet b ↔ a = b
-- Topology.WithUpperSet.ofUpperSet_inj
lemma ofUpperSet_inj {a b : WithUpperSet α} : ofUpperSet a = ofUpperSet b ↔ a = b 
Arguments: {α : Type u_1} {a b : Topology.WithUpperSet α}
Goal: Topology.WithUpperSet.ofUpperSet a = Topology.WithUpperSet.ofUpperSet b ↔ a = b
-- Topology.WithUpperSet.ofUpperSet_le_iff
lemma ofUpperSet_le_iff {a b : WithUpperSet α} : ofUpperSet a ≤ ofUpperSet b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : Topology.WithUpperSet α}
Goal: Topology.WithUpperSet.ofUpperSet a ≤ Topology.WithUpperSet.ofUpperSet b ↔ a ≤ b