Intervals in Lattices #
In this file, we provide instances of lattice structures on intervals within lattices. Some of them depend on the order of the endpoints of the interval, and thus are not made global instances. These are probably not all of the lattice instances that could be placed on these intervals, but more can be added easily along the same lines when needed.
Main definitions #
In the following, *
can represent either c
, o
, or i
.
Set.Ic*.orderBot
Set.Ii*.semillaticeInf
Set.I*c.orderTop
Set.I*c.semillaticeInf
Set.I**.lattice
Set.Iic.boundedOrder
, within anOrderBot
Set.Ici.boundedOrder
, within anOrderTop
instance
Set.Ico.semilatticeInf
{α : Type u_1}
[SemilatticeInf α]
{a b : α}
:
SemilatticeInf ↑(Set.Ico a b)
Equations
- Set.Ico.semilatticeInf = Subtype.semilatticeInf ⋯
@[reducible, inline]
Ico a b
has a bottom element whenever a < b
.
Equations
- Set.Ico.orderBot h = ⋯.orderBot
Instances For
instance
Set.Iio.semilatticeInf
{α : Type u_1}
[SemilatticeInf α]
{a : α}
:
SemilatticeInf ↑(Set.Iio a)
Equations
- Set.Iio.semilatticeInf = Subtype.semilatticeInf ⋯
instance
Set.Ioc.semilatticeSup
{α : Type u_1}
[SemilatticeSup α]
{a b : α}
:
SemilatticeSup ↑(Set.Ioc a b)
Equations
- Set.Ioc.semilatticeSup = Subtype.semilatticeSup ⋯
@[reducible, inline]
Ioc a b
has a top element whenever a < b
.
Equations
- Set.Ioc.orderTop h = ⋯.orderTop
Instances For
instance
Set.Ioi.semilatticeSup
{α : Type u_1}
[SemilatticeSup α]
{a : α}
:
SemilatticeSup ↑(Set.Ioi a)
Equations
- Set.Ioi.semilatticeSup = Subtype.semilatticeSup ⋯
instance
Set.Iic.semilatticeInf
{α : Type u_1}
{a : α}
[SemilatticeInf α]
:
SemilatticeInf ↑(Set.Iic a)
Equations
- Set.Iic.semilatticeInf = Subtype.semilatticeInf ⋯
@[simp]
instance
Set.Iic.semilatticeSup
{α : Type u_1}
{a : α}
[SemilatticeSup α]
:
SemilatticeSup ↑(Set.Iic a)
Equations
- Set.Iic.semilatticeSup = Subtype.semilatticeSup ⋯
@[simp]
Equations
- Set.Iic.instLatticeElem = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯
Equations
- Set.Iic.orderTop = OrderTop.mk ⋯
Equations
- Set.Iic.orderBot = OrderBot.mk ⋯
instance
Set.Iic.instBoundedOrderElemOfOrderBot
{α : Type u_1}
{a : α}
[Preorder α]
[OrderBot α]
:
BoundedOrder ↑(Set.Iic a)
Equations
- Set.Iic.instBoundedOrderElemOfOrderBot = BoundedOrder.mk
theorem
Set.Iic.disjoint_iff
{α : Type u_1}
{a : α}
[SemilatticeInf α]
[OrderBot α]
{x y : ↑(Set.Iic a)}
:
theorem
Set.Iic.codisjoint_iff
{α : Type u_1}
{a : α}
[SemilatticeSup α]
{x y : ↑(Set.Iic a)}
:
Codisjoint x y ↔ ↑x ⊔ ↑y = a
instance
Set.Ici.semilatticeInf
{α : Type u_1}
[SemilatticeInf α]
{a : α}
:
SemilatticeInf ↑(Set.Ici a)
Equations
- Set.Ici.semilatticeInf = Subtype.semilatticeInf ⋯
instance
Set.Ici.semilatticeSup
{α : Type u_1}
[SemilatticeSup α]
{a : α}
:
SemilatticeSup ↑(Set.Ici a)
Equations
- Set.Ici.semilatticeSup = Subtype.semilatticeSup ⋯
Equations
- Set.Ici.lattice = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯
instance
Set.Ici.distribLattice
{α : Type u_1}
[DistribLattice α]
{a : α}
:
DistribLattice ↑(Set.Ici a)
Equations
- Set.Ici.distribLattice = DistribLattice.mk ⋯
Equations
- Set.Ici.orderBot = OrderBot.mk ⋯
Equations
- Set.Ici.orderTop = OrderTop.mk ⋯
instance
Set.Ici.boundedOrder
{α : Type u_1}
[Preorder α]
[OrderTop α]
{a : α}
:
BoundedOrder ↑(Set.Ici a)
Equations
- Set.Ici.boundedOrder = BoundedOrder.mk
instance
Set.Icc.semilatticeInf
{α : Type u_1}
[SemilatticeInf α]
{a b : α}
:
SemilatticeInf ↑(Set.Icc a b)
Equations
- Set.Icc.semilatticeInf = Subtype.semilatticeInf ⋯
instance
Set.Icc.semilatticeSup
{α : Type u_1}
[SemilatticeSup α]
{a b : α}
:
SemilatticeSup ↑(Set.Icc a b)
Equations
- Set.Icc.semilatticeSup = Subtype.semilatticeSup ⋯
Equations
- Set.Icc.lattice = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯
@[reducible, inline]
Icc a b
has a bottom element whenever a ≤ b
.
Equations
- Set.Icc.orderBot h = ⋯.orderBot
Instances For
@[reducible, inline]
Icc a b
has a top element whenever a ≤ b
.
Equations
- Set.Icc.orderTop h = ⋯.orderTop
Instances For
@[reducible, inline]
abbrev
Set.Icc.boundedOrder
{α : Type u_1}
[Preorder α]
{a b : α}
(h : a ≤ b)
:
BoundedOrder ↑(Set.Icc a b)
Icc a b
is a BoundedOrder
whenever a ≤ b
.
Equations
- Set.Icc.boundedOrder h = BoundedOrder.mk