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 an OrderBot
• Set.Ici.boundedOrder, within an OrderTop

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]

abbrev Set.Ico.orderBot {α : Type u_1} [PartialOrder α] {a b : α} (h : a < b) : OrderBot ↑(Set.Ico a b)

Ico a b has a bottom element whenever a < b.

Equations

• Set.Ico.orderBot h = ⋯.orderBot

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]

abbrev Set.Ioc.orderTop {α : Type u_1} [PartialOrder α] {a b : α} (h : a < b) : OrderTop ↑(Set.Ioc a b)

Ioc a b has a top element whenever a < b.

Equations

• Set.Ioc.orderTop h = ⋯.orderTop

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]

theorem Set.Iic.coe_inf {α : Type u_1} {a : α} [SemilatticeInf α] {x y : ↑(Set.Iic a)} : ↑(x ⊓ y) = ↑x ⊓ ↑y

instance Set.Iic.semilatticeSup {α : Type u_1} {a : α} [SemilatticeSup α] : SemilatticeSup ↑(Set.Iic a)

Equations

• Set.Iic.semilatticeSup = Subtype.semilatticeSup ⋯

@[simp]

theorem Set.Iic.coe_sup {α : Type u_1} {a : α} [SemilatticeSup α] {x y : ↑(Set.Iic a)} : ↑(x ⊔ y) = ↑x ⊔ ↑y

instance Set.Iic.instLatticeElem {α : Type u_1} {a : α} [Lattice α] : Lattice ↑(Set.Iic a)

Equations

• Set.Iic.instLatticeElem = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯

instance Set.Iic.orderTop {α : Type u_1} {a : α} [Preorder α] : OrderTop ↑(Set.Iic a)

Equations

• Set.Iic.orderTop = OrderTop.mk ⋯

@[simp]

theorem Set.Iic.coe_top {α : Type u_1} {a : α} [Preorder α] : ↑⊤ = a

theorem Set.Iic.eq_top_iff {α : Type u_1} {a : α} [Preorder α] {x : ↑(Set.Iic a)} : x = ⊤ ↔ ↑x = a

instance Set.Iic.orderBot {α : Type u_1} {a : α} [Preorder α] [OrderBot α] : OrderBot ↑(Set.Iic a)

Equations

• Set.Iic.orderBot = OrderBot.mk ⋯

@[simp]

theorem Set.Iic.coe_bot {α : Type u_1} {a : α} [Preorder α] [OrderBot α] : ↑⊥ = ⊥

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)} : Disjoint x y ↔ Disjoint ↑x ↑y

theorem Set.Iic.codisjoint_iff {α : Type u_1} {a : α} [SemilatticeSup α] {x y : ↑(Set.Iic a)} : Codisjoint x y ↔ ↑x ⊔ ↑y = a

theorem Set.Iic.isCompl_iff {α : Type u_1} {a : α} [Lattice α] [BoundedOrder α] {x y : ↑(Set.Iic a)} : IsCompl x y ↔ Disjoint ↑x ↑y ∧ ↑x ⊔ ↑y = a

theorem Set.Iic.complementedLattice_iff {α : Type u_1} {a : α} [Lattice α] [BoundedOrder α] : ComplementedLattice ↑(Set.Iic a) ↔ ∀ b ≤ a, ∃ c ≤ a, b ⊓ c = ⊥ ∧ b ⊔ c = 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 ⋯

instance Set.Ici.lattice {α : Type u_1} [Lattice α] {a : α} : Lattice ↑(Set.Ici a)

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 ⋯

instance Set.Ici.orderBot {α : Type u_1} [Preorder α] {a : α} : OrderBot ↑(Set.Ici a)

Equations

• Set.Ici.orderBot = OrderBot.mk ⋯

@[simp]

theorem Set.Ici.coe_bot {α : Type u_1} [Preorder α] {a : α} : ↑⊥ = a

instance Set.Ici.orderTop {α : Type u_1} [Preorder α] [OrderTop α] {a : α} : OrderTop ↑(Set.Ici a)

Equations

• Set.Ici.orderTop = OrderTop.mk ⋯

@[simp]

theorem Set.Ici.coe_top {α : Type u_1} [Preorder α] [OrderTop α] {a : α} : ↑⊤ = ⊤

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} {a b : α} [SemilatticeInf α] : SemilatticeInf ↑(Set.Icc a b)

Equations

• Set.Icc.semilatticeInf = Subtype.semilatticeInf ⋯

instance Set.Icc.semilatticeSup {α : Type u_1} {a b : α} [SemilatticeSup α] : SemilatticeSup ↑(Set.Icc a b)

Equations

• Set.Icc.semilatticeSup = Subtype.semilatticeSup ⋯

instance Set.Icc.lattice {α : Type u_1} {a b : α} [Lattice α] : Lattice ↑(Set.Icc a b)

Equations

• Set.Icc.lattice = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯

instance Set.Icc.instOrderBotElem {α : Type u_1} {a b : α} [Preorder α] [Fact (a ≤ b)] : OrderBot ↑(Set.Icc a b)

Icc a b has a bottom element whenever a ≤ b.

Equations

• Set.Icc.instOrderBotElem = ⋯.orderBot

instance Set.Icc.instOrderTopElem {α : Type u_1} {a b : α} [Preorder α] [Fact (a ≤ b)] : OrderTop ↑(Set.Icc a b)

Icc a b has a top element whenever a ≤ b.

Equations

• Set.Icc.instOrderTopElem = ⋯.orderTop

instance Set.Icc.instBoundedOrderElem {α : Type u_1} {a b : α} [Preorder α] [Fact (a ≤ b)] : BoundedOrder ↑(Set.Icc a b)

Icc a b is a BoundedOrder whenever a ≤ b.

Equations

• Set.Icc.instBoundedOrderElem = BoundedOrder.mk