ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LocallyBoundedMapClass.comap_cobounded_le
class LocallyBoundedMapClass (F : Type*) (α β : outParam Type*) [Bornology α]
    [Bornology β] [FunLike F α β] : Prop where
    comap_cobounded_le (f : F) : (cobounded β).comap f ≤ cobounded α
Arguments: {F : Type u_6} {α : outParam (Type u_7)} {β : outParam (Type u_8)} [Bornology α] [Bornology β] [FunLike F α β] [self : LocallyBoundedMapClass F α β] (f : F)
Goal: Filter.comap (⇑f) (Bornology.cobounded β) ≤ Bornology.cobounded α
-- Bornology.IsBounded.image
theorem Bornology.IsBounded.image [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] (f : F)
    {s : Set α} (hs : IsBounded s) : IsBounded (f '' s) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] (f : F) {s : Set α} (hs : Bornology.IsBounded s)
Goal: Bornology.IsBounded (⇑f '' s)
-- LocallyBoundedMap.ext
@[ext]
theorem ext {f g : LocallyBoundedMap α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] {f g : LocallyBoundedMap α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- LocallyBoundedMap.coe_copy
@[simp]
theorem coe_copy (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- LocallyBoundedMap.copy_eq
theorem copy_eq (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- LocallyBoundedMap.coe_ofMapBounded
@[simp]
theorem coe_ofMapBounded (f : α → β) {h} : ⇑(ofMapBounded f h) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : α → β) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)}
Goal: ⇑(LocallyBoundedMap.ofMapBounded f h) = f
-- LocallyBoundedMap.ofMapBounded_apply
@[simp]
theorem ofMapBounded_apply (f : α → β) {h} (a : α) : ofMapBounded f h a = f a 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : α → β) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)} (a : α)
Goal: (LocallyBoundedMap.ofMapBounded f h) a = f a
-- LocallyBoundedMap.coe_id
@[simp]
theorem coe_id : ⇑(LocallyBoundedMap.id α) = id 
Arguments: (α : Type u_2) [Bornology α]
Goal: ⇑(LocallyBoundedMap.id α) = id
-- LocallyBoundedMap.id_apply
@[simp]
theorem id_apply (a : α) : LocallyBoundedMap.id α a = a 
Arguments: {α : Type u_2} [Bornology α] (a : α)
Goal: (LocallyBoundedMap.id α) a = a
-- LocallyBoundedMap.coe_comp
@[simp]
theorem coe_comp (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- LocallyBoundedMap.comp_apply
@[simp]
theorem comp_apply (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) (a : α) :
    f.comp g a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- LocallyBoundedMap.comp_assoc
@[simp]
theorem comp_assoc (f : LocallyBoundedMap γ δ) (g : LocallyBoundedMap β γ)
    (h : LocallyBoundedMap α 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} [Bornology α] [Bornology β] [Bornology γ] [Bornology δ] (f : LocallyBoundedMap γ δ) (g : LocallyBoundedMap β γ) (h : LocallyBoundedMap α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- LocallyBoundedMap.comp_id
@[simp]
theorem comp_id (f : LocallyBoundedMap α β) : f.comp (LocallyBoundedMap.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β)
Goal: f.comp (LocallyBoundedMap.id α) = f
-- LocallyBoundedMap.id_comp
@[simp]
theorem id_comp (f : LocallyBoundedMap α β) : (LocallyBoundedMap.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β)
Goal: (LocallyBoundedMap.id β).comp f = f
-- LocallyBoundedMap.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : LocallyBoundedMap β γ} {f : LocallyBoundedMap α 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} [Bornology α] [Bornology β] [Bornology γ] {g₁ g₂ : LocallyBoundedMap β γ} {f : LocallyBoundedMap α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- LocallyBoundedMap.cancel_left
@[simp]
theorem cancel_left {g : LocallyBoundedMap β γ} {f₁ f₂ : LocallyBoundedMap α 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} [Bornology α] [Bornology β] [Bornology γ] {g : LocallyBoundedMap β γ} {f₁ f₂ : LocallyBoundedMap α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- RestrictGenTopology.isOpen_iff
protected theorem isOpen_iff (hS : RestrictGenTopology S) :
    IsOpen t ↔ ∀ s ∈ S, IsOpen ((↑) ⁻¹' t : Set s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} {t : Set X} (hS : RestrictGenTopology S)
Goal: IsOpen t ↔ ∀ s ∈ S, IsOpen (Subtype.val ⁻¹' t)
-- RestrictGenTopology.isClosed_iff
protected theorem isClosed_iff (hS : RestrictGenTopology S) :
    IsClosed t ↔ ∀ s ∈ S, IsClosed ((↑) ⁻¹' t : Set s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} {t : Set X} (hS : RestrictGenTopology S)
Goal: IsClosed t ↔ ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
-- RestrictGenTopology.continuous_iff
protected theorem continuous_iff {Y : Type*} [TopologicalSpace Y] {f : X → Y}
    (hS : RestrictGenTopology S) :
    Continuous f ↔ ∀ s ∈ S, ContinuousOn f s 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} {Y : Type u_2} [TopologicalSpace Y] {f : X → Y} (hS : RestrictGenTopology S)
Goal: Continuous f ↔ ∀ s ∈ S, ContinuousOn f s
-- RestrictGenTopology.of_continuous_prop
theorem of_continuous_prop (h : ∀ f : X → Prop, (∀ s ∈ S, ContinuousOn f s) → Continuous f) :
    RestrictGenTopology S 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} (h : ∀ (f : X → Prop), (∀ s ∈ S, ContinuousOn f s) → Continuous f)
Goal: RestrictGenTopology S
-- RestrictGenTopology.of_isClosed
theorem of_isClosed (h : ∀ t : Set X, (∀ s ∈ S, IsClosed ((↑) ⁻¹' t : Set s)) → IsClosed t) :
    RestrictGenTopology S 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} (h : ∀ (t : Set X), (∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)) → IsClosed t)
Goal: RestrictGenTopology S
-- RestrictGenTopology.enlarge
protected theorem enlarge {T} (hS : RestrictGenTopology S) (hT : ∀ s ∈ S, ∃ t ∈ T, s ⊆ t) :
    RestrictGenTopology T 
Arguments: {X : Type u_1} [TopologicalSpace X] {S T : Set (Set X)} (hS : RestrictGenTopology S) (hT : ∀ s ∈ S, ∃ t ∈ T, s ⊆ t)
Goal: RestrictGenTopology T
-- RestrictGenTopology.mono
protected theorem mono {T} (hS : RestrictGenTopology S) (hT : S ⊆ T) : RestrictGenTopology T 
Arguments: {X : Type u_1} [TopologicalSpace X] {S T : Set (Set X)} (hS : RestrictGenTopology S) (hT : S ⊆ T)
Goal: RestrictGenTopology T
-- RestrictGenTopology.of_seq
lemma of_seq [SequentialSpace X]
    (h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S) :
    RestrictGenTopology S 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} [SequentialSpace X] (h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Filter.Tendsto u Filter.atTop (𝓝 x) → insert x (Set.range u) ∈ S)
Goal: RestrictGenTopology S
-- RestrictGenTopology.isCompact_of_seq
lemma isCompact_of_seq [SequentialSpace X] : RestrictGenTopology {K : Set X | IsCompact K} 
Arguments: {X : Type u_1} [TopologicalSpace X] [SequentialSpace X]
Goal: RestrictGenTopology {K | IsCompact K}
-- Locale.openOfElementHom_toFun
@[simps]
def openOfElementHom : FrameHom L (Set (PT L)) 
Arguments: (L : Type u_1) [CompleteLattice L] (u : L)
Goal: (Locale.openOfElementHom L) u = {x | x u}
-- Locale.PT.isOpen_iff
lemma isOpen_iff (U : Set (PT L)) : IsOpen U ↔ ∃ u : L, {x | x u} = U 
Arguments: (L : Type u_1) [CompleteLattice L] (U : Set (Locale.PT L))
Goal: IsOpen U ↔ ∃ u, {x | x u} = U
-- Locale.localePointOfSpacePoint_toFun
@[simps]
def localePointOfSpacePoint (x : X) : PT (Opens X) 
Arguments: (X : Type u_1) [TopologicalSpace X] (x : X)
Goal: ∀ (x_1 : TopologicalSpace.Opens X), (Locale.localePointOfSpacePoint X x) x_1 = (x ∈ x_1)
-- Set.ordConnectedComponent_mem_nhds
@[simp]
theorem ordConnectedComponent_mem_nhds : ordConnectedComponent s a ∈ 𝓝 a ↔ s ∈ 𝓝 a 
Arguments: {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {a : X} {s : Set X}
Goal: s.ordConnectedComponent a ∈ 𝓝 a ↔ s ∈ 𝓝 a
-- Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Ici
theorem compl_section_ordSeparatingSet_mem_nhdsWithin_Ici (hd : Disjoint s (closure t))
    (ha : a ∈ s) : (ordConnectedSection (ordSeparatingSet s t))ᶜ ∈ 𝓝[≥] a 
Arguments: {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {a : X} {s t : Set X} (hd : Disjoint s (closure t)) (ha : a ∈ s)
Goal: (s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝[≥] a
-- Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Iic
theorem compl_section_ordSeparatingSet_mem_nhdsWithin_Iic (hd : Disjoint s (closure t))
    (ha : a ∈ s) : (ordConnectedSection <| ordSeparatingSet s t)ᶜ ∈ 𝓝[≤] a 
Arguments: {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {a : X} {s t : Set X} (hd : Disjoint s (closure t)) (ha : a ∈ s)
Goal: (s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝[≤] a
-- Set.compl_section_ordSeparatingSet_mem_nhds
theorem compl_section_ordSeparatingSet_mem_nhds (hd : Disjoint s (closure t)) (ha : a ∈ s) :
    (ordConnectedSection <| ordSeparatingSet s t)ᶜ ∈ 𝓝 a 
Arguments: {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {a : X} {s t : Set X} (hd : Disjoint s (closure t)) (ha : a ∈ s)
Goal: (s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝 a
-- Set.ordT5Nhd_mem_nhdsSet
theorem ordT5Nhd_mem_nhdsSet (hd : Disjoint s (closure t)) : ordT5Nhd s t ∈ 𝓝ˢ s 
Arguments: {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {s t : Set X} (hd : Disjoint s (closure t))
Goal: s.ordT5Nhd t ∈ 𝓝ˢ s
-- Bornology.le_cofinite'
class Bornology (α : Type*) where
    cobounded' : Filter α
    le_cofinite' : cobounded' ≤ cofinite
Arguments: {α : Type u_4} [self : Bornology α]
Goal: Bornology.cobounded' ≤ Filter.cofinite
-- Bornology.le_cofinite
lemma Bornology.le_cofinite (α : Type*) [Bornology α] : cobounded α ≤ cofinite 
Arguments: (α : Type u_4) [Bornology α]
Goal: Bornology.cobounded α ≤ Filter.cofinite
-- Bornology.ext
@[ext]
lemma Bornology.ext (t t' : Bornology α)
    (h_cobounded : @Bornology.cobounded α t = @Bornology.cobounded α t') :
    t = t' 
Arguments: {α : Type u_2} (t t' : Bornology α) (h_cobounded : Bornology.cobounded α = Bornology.cobounded α)
Goal: t = t'
-- Bornology.ext_iff
lemma Bornology.ext_iff (t t' : Bornology α) :
    t = t' ↔ @Bornology.cobounded α t = @Bornology.cobounded α t' 
Arguments: {α : Type u_2} (t t' : Bornology α)
Goal: t = t' ↔ Bornology.cobounded α = Bornology.cobounded α
-- Bornology.ofBounded_cobounded
@[simps]
def Bornology.ofBounded {α : Type*} (B : Set (Set α))
    (empty_mem : ∅ ∈ B)
    (subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B)
    (union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B)
    (singleton_mem : ∀ x, {x} ∈ B) : Bornology α 
Arguments: {α : Type u_4} (B : Set (Set α)) (empty_mem : ∅ ∈ B) (subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B) (union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B) (singleton_mem : ∀ (x : α), {x} ∈ B)
Goal: Bornology.cobounded α = Filter.comk (fun x => x ∈ B) empty_mem subset_mem union_mem
-- Bornology.ofBounded'_cobounded
@[simps! cobounded]
def Bornology.ofBounded' {α : Type*} (B : Set (Set α))
    (empty_mem : ∅ ∈ B)
    (subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B)
    (union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B)
    (sUnion_univ : ⋃₀ B = univ) :
    Bornology α 
Arguments: {α : Type u_4} (B : Set (Set α)) (empty_mem : ∅ ∈ B) (subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B) (union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B) (sUnion_univ : ⋃₀ B = Set.univ)
Goal: Bornology.cobounded α = Filter.comk (fun x => x ∈ B) empty_mem subset_mem union_mem
-- Bornology.isCobounded_def
theorem isCobounded_def {s : Set α} : IsCobounded s ↔ s ∈ cobounded α 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α}, Bornology.IsCobounded s ↔ s ∈ Bornology.cobounded α
-- Bornology.isBounded_def
theorem isBounded_def {s : Set α} : IsBounded s ↔ sᶜ ∈ cobounded α 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded s ↔ sᶜ ∈ Bornology.cobounded α
-- Bornology.isBounded_compl_iff
@[simp]
theorem isBounded_compl_iff : IsBounded sᶜ ↔ IsCobounded s 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded sᶜ ↔ Bornology.IsCobounded s
-- Bornology.isCobounded_compl_iff
@[simp]
theorem isCobounded_compl_iff : IsCobounded sᶜ ↔ IsBounded s 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α}, Bornology.IsCobounded sᶜ ↔ Bornology.IsBounded s
-- Bornology.isBounded_empty
@[simp]
theorem isBounded_empty : IsBounded (∅ : Set α) 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α}, Bornology.IsBounded ∅
-- Bornology.nonempty_of_not_isBounded
theorem nonempty_of_not_isBounded (h : ¬IsBounded s) : s.Nonempty 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α}, ¬Bornology.IsBounded s → s.Nonempty
-- Bornology.isBounded_singleton
@[simp]
theorem isBounded_singleton : IsBounded ({x} : Set α) 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {x_1 : α}, Bornology.IsBounded {x_1}
-- Bornology.isBounded_iff_forall_mem
theorem isBounded_iff_forall_mem : IsBounded s ↔ ∀ x ∈ s, IsBounded s 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded s ↔ ∀ x_1 ∈ s, Bornology.IsBounded s
-- Bornology.isCobounded_univ
@[simp]
theorem isCobounded_univ : IsCobounded (univ : Set α) 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α}, Bornology.IsCobounded Set.univ
-- Bornology.isCobounded_inter
@[simp]
theorem isCobounded_inter : IsCobounded (s ∩ t) ↔ IsCobounded s ∧ IsCobounded t 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s t : Set α}, Bornology.IsCobounded (s ∩ t) ↔ Bornology.IsCobounded s ∧ Bornology.IsCobounded t
-- Bornology.IsCobounded.inter
theorem IsCobounded.inter (hs : IsCobounded s) (ht : IsCobounded t) : IsCobounded (s ∩ t) 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s t : Set α}, Bornology.IsCobounded s → Bornology.IsCobounded t → Bornology.IsCobounded (s ∩ t)
-- Bornology.isBounded_union
@[simp]
theorem isBounded_union : IsBounded (s ∪ t) ↔ IsBounded s ∧ IsBounded t 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s t : Set α}, Bornology.IsBounded (s ∪ t) ↔ Bornology.IsBounded s ∧ Bornology.IsBounded t
-- Bornology.IsBounded.union
theorem IsBounded.union (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ∪ t) 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s t : Set α}, Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s ∪ t)
-- Bornology.IsCobounded.superset
theorem IsCobounded.superset (hs : IsCobounded s) (ht : s ⊆ t) : IsCobounded t 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s t : Set α}, Bornology.IsCobounded s → s ⊆ t → Bornology.IsCobounded t
-- Bornology.IsBounded.subset
theorem IsBounded.subset (ht : IsBounded t) (hs : s ⊆ t) : IsBounded s 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s t : Set α}, Bornology.IsBounded t → s ⊆ t → Bornology.IsBounded s
-- Bornology.sUnion_bounded_univ
@[simp]
theorem sUnion_bounded_univ : ⋃₀ { s : Set α | IsBounded s } = univ 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α}, ⋃₀ {s | Bornology.IsBounded s} = Set.univ
-- Bornology.IsBounded.insert
theorem IsBounded.insert (h : IsBounded s) (x : α) : IsBounded (insert x s) 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded s → ∀ (x_1 : α), Bornology.IsBounded (insert x_1 s)
-- Bornology.isBounded_insert
@[simp]
theorem isBounded_insert : IsBounded (insert x s) ↔ IsBounded s 
Arguments: {α : Type u_2}
Goal: ∀ {x : Bornology α} {s : Set α} {x_1 : α}, Bornology.IsBounded (insert x_1 s) ↔ Bornology.IsBounded s
-- Bornology.comap_cobounded_le_iff
theorem comap_cobounded_le_iff [Bornology β] {f : α → β} :
    (cobounded β).comap f ≤ cobounded α ↔ ∀ ⦃s⦄, IsBounded s → IsBounded (f '' s) 
Arguments: {α : Type u_2} {β : Type u_3}
Goal: ∀ {x : Bornology α} [inst : Bornology β] {f : α → β},
  Filter.comap f (Bornology.cobounded β) ≤ Bornology.cobounded α ↔
    ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)
-- Bornology.ext_iff'
theorem ext_iff' {t t' : Bornology α} :
    t = t' ↔ ∀ s, s ∈ @cobounded α t ↔ s ∈ @cobounded α t' 
Arguments: {α : Type u_2} {t t' : Bornology α}
Goal: t = t' ↔ ∀ (s : Set α), s ∈ Bornology.cobounded α ↔ s ∈ Bornology.cobounded α
-- Bornology.ext_iff_isBounded
theorem ext_iff_isBounded {t t' : Bornology α} :
    t = t' ↔ ∀ s, @IsBounded α t s ↔ @IsBounded α t' s 
Arguments: {α : Type u_2} {t t' : Bornology α}
Goal: t = t' ↔ ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s
-- Bornology.isCobounded_ofBounded_iff
theorem isCobounded_ofBounded_iff (B : Set (Set α)) {empty_mem subset_mem union_mem sUnion_univ} :
    @IsCobounded _ (ofBounded B empty_mem subset_mem union_mem sUnion_univ) s ↔ sᶜ ∈ B 
Arguments: {α : Type u_2} {s : Set α} (B : Set (Set α)) {empty_mem : ∅ ∈ B} {subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B} {union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B} {sUnion_univ : ∀ (x : α), {x} ∈ B}
Goal: Bornology.IsCobounded s ↔ sᶜ ∈ B
-- Bornology.isBounded_ofBounded_iff
theorem isBounded_ofBounded_iff (B : Set (Set α)) {empty_mem subset_mem union_mem sUnion_univ} :
    @IsBounded _ (ofBounded B empty_mem subset_mem union_mem sUnion_univ) s ↔ s ∈ B 
Arguments: {α : Type u_2} {s : Set α} (B : Set (Set α)) {empty_mem : ∅ ∈ B} {subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B} {union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B} {sUnion_univ : ∀ (x : α), {x} ∈ B}
Goal: Bornology.IsBounded s ↔ s ∈ B
-- Bornology.isCobounded_biInter
theorem isCobounded_biInter {s : Set ι} {f : ι → Set α} (hs : s.Finite) :
    IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, IsCobounded (f i) 
Arguments: {ι : Type u_1} {α : Type u_2} [Bornology α] {s : Set ι} {f : ι → Set α} (hs : s.Finite)
Goal: Bornology.IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsCobounded (f i)
-- Bornology.isCobounded_biInter_finset
@[simp]
theorem isCobounded_biInter_finset (s : Finset ι) {f : ι → Set α} :
    IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, IsCobounded (f i) 
Arguments: {ι : Type u_1} {α : Type u_2} [Bornology α] (s : Finset ι) {f : ι → Set α}
Goal: Bornology.IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsCobounded (f i)
-- Bornology.isCobounded_iInter
@[simp]
theorem isCobounded_iInter [Finite ι] {f : ι → Set α} :
    IsCobounded (⋂ i, f i) ↔ ∀ i, IsCobounded (f i) 
Arguments: {ι : Type u_1} {α : Type u_2} [Bornology α] [Finite ι] {f : ι → Set α}
Goal: Bornology.IsCobounded (⋂ i, f i) ↔ ∀ (i : ι), Bornology.IsCobounded (f i)
-- Bornology.isCobounded_sInter
theorem isCobounded_sInter {S : Set (Set α)} (hs : S.Finite) :
    IsCobounded (⋂₀ S) ↔ ∀ s ∈ S, IsCobounded s 
Arguments: {α : Type u_2} [Bornology α] {S : Set (Set α)} (hs : S.Finite)
Goal: Bornology.IsCobounded (⋂₀ S) ↔ ∀ s ∈ S, Bornology.IsCobounded s
-- Bornology.isBounded_biUnion
theorem isBounded_biUnion {s : Set ι} {f : ι → Set α} (hs : s.Finite) :
    IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, IsBounded (f i) 
Arguments: {ι : Type u_1} {α : Type u_2} [Bornology α] {s : Set ι} {f : ι → Set α} (hs : s.Finite)
Goal: Bornology.IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsBounded (f i)
-- Bornology.isBounded_biUnion_finset
theorem isBounded_biUnion_finset (s : Finset ι) {f : ι → Set α} :
    IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, IsBounded (f i) 
Arguments: {ι : Type u_1} {α : Type u_2} [Bornology α] (s : Finset ι) {f : ι → Set α}
Goal: Bornology.IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsBounded (f i)
-- Bornology.isBounded_sUnion
theorem isBounded_sUnion {S : Set (Set α)} (hs : S.Finite) :
    IsBounded (⋃₀ S) ↔ ∀ s ∈ S, IsBounded s 
Arguments: {α : Type u_2} [Bornology α] {S : Set (Set α)} (hs : S.Finite)
Goal: Bornology.IsBounded (⋃₀ S) ↔ ∀ s ∈ S, Bornology.IsBounded s
-- Bornology.isBounded_iUnion
@[simp]
theorem isBounded_iUnion [Finite ι] {s : ι → Set α} :
    IsBounded (⋃ i, s i) ↔ ∀ i, IsBounded (s i) 
Arguments: {ι : Type u_1} {α : Type u_2} [Bornology α] [Finite ι] {s : ι → Set α}
Goal: Bornology.IsBounded (⋃ i, s i) ↔ ∀ (i : ι), Bornology.IsBounded (s i)
-- Bornology.eventually_ne_cobounded
lemma eventually_ne_cobounded (a : α) : ∀ᶠ x in cobounded α, x ≠ a 
Arguments: {α : Type u_2} [Bornology α] (a : α)
Goal: ∀ᶠ (x : α) in Bornology.cobounded α, x ≠ a
-- Filter.HasBasis.disjoint_cobounded_iff
theorem Filter.HasBasis.disjoint_cobounded_iff [Bornology α] {ι : Sort*} {p : ι → Prop}
    {s : ι → Set α} {l : Filter α} (h : l.HasBasis p s) :
    Disjoint l (cobounded α) ↔ ∃ i, p i ∧ IsBounded (s i) 
Arguments: {α : Type u_2} [Bornology α] {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} {l : Filter α} (h : l.HasBasis p s)
Goal: Disjoint l (Bornology.cobounded α) ↔ ∃ i, p i ∧ Bornology.IsBounded (s i)
-- Set.Finite.isBounded
theorem Set.Finite.isBounded [Bornology α] {s : Set α} (hs : s.Finite) : IsBounded s 
Arguments: {α : Type u_2} [Bornology α] {s : Set α} (hs : s.Finite)
Goal: Bornology.IsBounded s
-- Filter.Tendsto.eventually_ne_cobounded
nonrec lemma Filter.Tendsto.eventually_ne_cobounded [Bornology α] {f : β → α} {l : Filter β}
    (h : Tendsto f l (cobounded α)) (a : α) : ∀ᶠ x in l, f x ≠ a 
Arguments: {α : Type u_2} {β : Type u_3} [Bornology α] {f : β → α} {l : Filter β} (h : Filter.Tendsto f l (Bornology.cobounded α)) (a : α)
Goal: ∀ᶠ (x : β) in l, f x ≠ a
-- BoundedSpace.bounded_univ
class BoundedSpace (α : Type*) [Bornology α] : Prop where
    bounded_univ : Bornology.IsBounded (univ : Set α)
Arguments: {α : Type u_4} [Bornology α] [self : BoundedSpace α]
Goal: Bornology.IsBounded Set.univ
-- Bornology.isBounded_univ
theorem isBounded_univ : IsBounded (univ : Set α) ↔ BoundedSpace α 
Arguments: {α : Type u_2} [Bornology α]
Goal: Bornology.IsBounded Set.univ ↔ BoundedSpace α
-- Bornology.cobounded_eq_bot_iff
theorem cobounded_eq_bot_iff : cobounded α = ⊥ ↔ BoundedSpace α 
Arguments: {α : Type u_2} [Bornology α]
Goal: Bornology.cobounded α = ⊥ ↔ BoundedSpace α
-- Bornology.IsBounded.all
theorem IsBounded.all (s : Set α) : IsBounded s 
Arguments: {α : Type u_2} [Bornology α] [BoundedSpace α] (s : Set α)
Goal: Bornology.IsBounded s
-- Bornology.IsCobounded.all
theorem IsCobounded.all (s : Set α) : IsCobounded s 
Arguments: {α : Type u_2} [Bornology α] [BoundedSpace α] (s : Set α)
Goal: Bornology.IsCobounded s
-- Bornology.cobounded_eq_bot
@[simp]
theorem cobounded_eq_bot : cobounded α = ⊥ 
Arguments: (α : Type u_2) [Bornology α] [BoundedSpace α]
Goal: Bornology.cobounded α = ⊥
-- OrderDual.isCobounded_preimage_ofDual
@[simp] lemma isCobounded_preimage_ofDual {s : Set α} :
    IsCobounded (ofDual ⁻¹' s) ↔ IsCobounded s 
Arguments: {α : Type u_2} [Bornology α] {s : Set α}
Goal: Bornology.IsCobounded (⇑OrderDual.ofDual ⁻¹' s) ↔ Bornology.IsCobounded s
-- OrderDual.isCobounded_preimage_toDual
@[simp] lemma isCobounded_preimage_toDual {s : Set αᵒᵈ} :
    IsCobounded (toDual ⁻¹' s) ↔ IsCobounded s 
Arguments: {α : Type u_2} [Bornology α] {s : Set αᵒᵈ}
Goal: Bornology.IsCobounded (⇑OrderDual.toDual ⁻¹' s) ↔ Bornology.IsCobounded s
-- OrderDual.isBounded_preimage_ofDual
@[simp] lemma isBounded_preimage_ofDual {s : Set α} :
    IsBounded (ofDual ⁻¹' s) ↔ IsBounded s 
Arguments: {α : Type u_2} [Bornology α] {s : Set α}
Goal: Bornology.IsBounded (⇑OrderDual.ofDual ⁻¹' s) ↔ Bornology.IsBounded s
-- OrderDual.isBounded_preimage_toDual
@[simp] lemma isBounded_preimage_toDual {s : Set αᵒᵈ} :
    IsBounded (toDual ⁻¹' s) ↔ IsBounded s 
Arguments: {α : Type u_2} [Bornology α] {s : Set αᵒᵈ}
Goal: Bornology.IsBounded (⇑OrderDual.toDual ⁻¹' s) ↔ Bornology.IsBounded s