File: Mathlib/Topology/Order/Bornology.lean

full_name: orderBornology
code: /-- Order-bornology on a nonempty lattice. The bounded sets are the sets that are bounded both above
and below. -/
def orderBornology : Bornology α 
goal: Bornology α

full_name: orderBornology_isBounded
code: @[simp] lemma orderBornology_isBounded : orderBornology.IsBounded s ↔ BddBelow s ∧ BddAbove s 
goal: Bornology.IsBounded s ↔ BddBelow s ∧ BddAbove s

full_name: IsOrderBornology
code: /-- Predicate for a preorder to be equipped with its order-bornology, namely for its bounded sets
to be the ones that are bounded both above and below. -/
class IsOrderBornology : Prop where
  protected isBounded_iff_bddBelow_bddAbove (s : Set α) : IsBounded s ↔ BddBelow s ∧ BddAbove s
goal: Prop

full_name: IsOrderBornology.isBounded_iff_bddBelow_bddAbove
code: /-- Predicate for a preorder to be equipped with its order-bornology, namely for its bounded sets
to be the ones that are bounded both above and below. -/
class IsOrderBornology : Prop where
  protected isBounded_iff_bddBelow_bddAbove (s : Set α) : IsBounded s ↔ BddBelow s ∧ BddAbove s
goal: Bornology.IsBounded s ↔ BddBelow s ∧ BddAbove s

full_name: isOrderBornology_iff_eq_orderBornology
code: lemma isOrderBornology_iff_eq_orderBornology [Lattice α] [Nonempty α] :
    IsOrderBornology α ↔ ‹Bornology α› = orderBornology 
goal: IsOrderBornology α ↔ inst✝ = orderBornology

full_name: isBounded_iff_bddBelow_bddAbove
code: lemma isBounded_iff_bddBelow_bddAbove : IsBounded s ↔ BddBelow s ∧ BddAbove s 
goal: Bornology.IsBounded s ↔ BddBelow s ∧ BddAbove s

full_name: Bornology.IsBounded.bddBelow
code: protected lemma Bornology.IsBounded.bddBelow (hs : IsBounded s) : BddBelow s 
goal: BddBelow s

full_name: Bornology.IsBounded.bddAbove
code: protected lemma Bornology.IsBounded.bddAbove (hs : IsBounded s) : BddAbove s 
goal: BddAbove s

full_name: BddBelow.isBounded
code: protected lemma BddBelow.isBounded (hs₀ : BddBelow s) (hs₁ : BddAbove s) : IsBounded s 
goal: Bornology.IsBounded s

full_name: BddAbove.isBounded
code: protected lemma BddAbove.isBounded (hs₀ : BddAbove s) (hs₁ : BddBelow s) : IsBounded s 
goal: Bornology.IsBounded s

full_name: BddBelow.isBounded_inter
code: lemma BddBelow.isBounded_inter (hs : BddBelow s) (ht : BddAbove t) : IsBounded (s ∩ t) 
goal: Bornology.IsBounded (s ∩ t)

full_name: BddAbove.isBounded_inter
code: lemma BddAbove.isBounded_inter (hs : BddAbove s) (ht : BddBelow t) : IsBounded (s ∩ t) 
goal: Bornology.IsBounded (s ∩ t)

full_name: OrderDual.instIsOrderBornology
code: instance OrderDual.instIsOrderBornology : IsOrderBornology αᵒᵈ 
goal: IsOrderBornology αᵒᵈ

full_name: Prod.instIsOrderBornology
code: instance Prod.instIsOrderBornology {β : Type*} [Preorder β] [Bornology β] [IsOrderBornology β] :
    IsOrderBornology (α × β) 
goal: IsOrderBornology (α × β)

full_name: Pi.instIsOrderBornology
code: instance Pi.instIsOrderBornology {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]
    [∀ i, Bornology (α i)] [∀ i, IsOrderBornology (α i)] : IsOrderBornology (∀ i, α i) 
goal: IsOrderBornology ((i : ι) → α i)

full_name: Bornology.IsBounded.subset_Icc_sInf_sSup
code: protected lemma Bornology.IsBounded.subset_Icc_sInf_sSup (hs : IsBounded s) :
    s ⊆ Icc (sInf s) (sSup s) 
goal: s ⊆ Set.Icc (sInf s) (sSup s)