lattices-2.2: Fine-grained library for constructing and manipulating lattices
Copyright(C) 2010-2015 Maximilian Bolingbroke 2015-2019 Oleg Grenrus
LicenseBSD-3-Clause (see the file LICENSE)
MaintainerOleg Grenrus <oleg.grenrus@iki.fi>
Safe HaskellSafe
LanguageHaskell2010

Algebra.PartialOrd

Description

 
Synopsis

Partial orderings

class Eq a => PartialOrd a where Source #

A partial ordering on sets (http://en.wikipedia.org/wiki/Partially_ordered_set) is a set equipped with a binary relation, leq, that obeys the following laws

Reflexive:     a `leq` a
Antisymmetric: a `leq` b && b `leq` a ==> a == b
Transitive:    a `leq` b && b `leq` c ==> a `leq` c

Two elements of the set are said to be comparable when they are are ordered with respect to the leq relation. So

comparable a b ==> a `leq` b || b `leq` a

If comparable always returns true then the relation leq defines a total ordering (and an Ord instance may be defined). Any Ord instance is trivially an instance of PartialOrd. Ordered provides a convenient wrapper to satisfy PartialOrd given Ord.

As an example consider the partial ordering on sets induced by set inclusion. Then for sets a and b,

a `leq` b

is true when a is a subset of b. Two sets are comparable if one is a subset of the other. Concretely

a = {1, 2, 3}
b = {1, 3, 4}
c = {1, 2}

a `leq` a = True
a `leq` b = False
a `leq` c = False
b `leq` a = False
b `leq` b = True
b `leq` c = False
c `leq` a = True
c `leq` b = False
c `leq` c = True

comparable a b = False
comparable a c = True
comparable b c = False

Minimal complete definition

leq

Methods

leq :: a -> a -> Bool Source #

The relation that induces the partial ordering

comparable :: a -> a -> Bool Source #

Whether two elements are ordered with respect to the relation. A default implementation is given by

comparable x y = leq x y || leq y x

Instances

Instances details
PartialOrd All Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: All -> All -> Bool Source #

comparable :: All -> All -> Bool Source #

PartialOrd Any Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: Any -> Any -> Bool Source #

comparable :: Any -> Any -> Bool Source #

PartialOrd Void Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: Void -> Void -> Bool Source #

comparable :: Void -> Void -> Bool Source #

PartialOrd IntSet Source # 
Instance details

Defined in Algebra.PartialOrd

PartialOrd M2 Source # 
Instance details

Defined in Algebra.Lattice.M2

Methods

leq :: M2 -> M2 -> Bool Source #

comparable :: M2 -> M2 -> Bool Source #

PartialOrd M3 Source # 
Instance details

Defined in Algebra.Lattice.M3

Methods

leq :: M3 -> M3 -> Bool Source #

comparable :: M3 -> M3 -> Bool Source #

PartialOrd N5 Source # 
Instance details

Defined in Algebra.Lattice.N5

Methods

leq :: N5 -> N5 -> Bool Source #

comparable :: N5 -> N5 -> Bool Source #

PartialOrd ZeroHalfOne Source # 
Instance details

Defined in Algebra.Lattice.ZeroHalfOne

PartialOrd () Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: () -> () -> Bool Source #

comparable :: () -> () -> Bool Source #

PartialOrd Bool Source #

Since: 2

Instance details

Defined in Algebra.PartialOrd

Methods

leq :: Bool -> Bool -> Bool Source #

comparable :: Bool -> Bool -> Bool Source #

(PartialOrd v, Finite v) => PartialOrd (Endo v) Source # 
Instance details

Defined in Algebra.PartialOrd.Instances

Methods

leq :: Endo v -> Endo v -> Bool Source #

comparable :: Endo v -> Endo v -> Bool Source #

PartialOrd v => PartialOrd (IntMap v) Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: IntMap v -> IntMap v -> Bool Source #

comparable :: IntMap v -> IntMap v -> Bool Source #

Ord a => PartialOrd (Set a) Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: Set a -> Set a -> Bool Source #

comparable :: Set a -> Set a -> Bool Source #

Ord a => PartialOrd (Free a) Source # 
Instance details

Defined in Algebra.Heyting.Free

Methods

leq :: Free a -> Free a -> Bool Source #

comparable :: Free a -> Free a -> Bool Source #

(Eq a, Lattice a) => PartialOrd (Join a) Source # 
Instance details

Defined in Algebra.Lattice

Methods

leq :: Join a -> Join a -> Bool Source #

comparable :: Join a -> Join a -> Bool Source #

(Eq a, Lattice a) => PartialOrd (Meet a) Source # 
Instance details

Defined in Algebra.Lattice

Methods

leq :: Meet a -> Meet a -> Bool Source #

comparable :: Meet a -> Meet a -> Bool Source #

(Eq a, Integral a) => PartialOrd (Divisibility a) Source # 
Instance details

Defined in Algebra.Lattice.Divisibility

PartialOrd a => PartialOrd (Dropped a) Source # 
Instance details

Defined in Algebra.Lattice.Dropped

Methods

leq :: Dropped a -> Dropped a -> Bool Source #

comparable :: Dropped a -> Dropped a -> Bool Source #

Ord a => PartialOrd (Free a) Source # 
Instance details

Defined in Algebra.Lattice.Free

Methods

leq :: Free a -> Free a -> Bool Source #

comparable :: Free a -> Free a -> Bool Source #

PartialOrd a => PartialOrd (Levitated a) Source # 
Instance details

Defined in Algebra.Lattice.Levitated

PartialOrd a => PartialOrd (Lifted a) Source # 
Instance details

Defined in Algebra.Lattice.Lifted

Methods

leq :: Lifted a -> Lifted a -> Bool Source #

comparable :: Lifted a -> Lifted a -> Bool Source #

PartialOrd a => PartialOrd (Op a) Source # 
Instance details

Defined in Algebra.Lattice.Op

Methods

leq :: Op a -> Op a -> Bool Source #

comparable :: Op a -> Op a -> Bool Source #

Ord a => PartialOrd (Ordered a) Source # 
Instance details

Defined in Algebra.Lattice.Ordered

Methods

leq :: Ordered a -> Ordered a -> Bool Source #

comparable :: Ordered a -> Ordered a -> Bool Source #

Eq a => PartialOrd (Wide a) Source # 
Instance details

Defined in Algebra.Lattice.Wide

Methods

leq :: Wide a -> Wide a -> Bool Source #

comparable :: Wide a -> Wide a -> Bool Source #

(Eq k, Hashable k) => PartialOrd (HashSet k) Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: HashSet k -> HashSet k -> Bool Source #

comparable :: HashSet k -> HashSet k -> Bool Source #

Eq a => PartialOrd [a] Source #

leq = isSequenceOf.

Instance details

Defined in Algebra.PartialOrd

Methods

leq :: [a] -> [a] -> Bool Source #

comparable :: [a] -> [a] -> Bool Source #

(PartialOrd a, PartialOrd b) => PartialOrd (Either a b) Source #

Ordinal sum.

Since: 2.1

Instance details

Defined in Algebra.PartialOrd

Methods

leq :: Either a b -> Either a b -> Bool Source #

comparable :: Either a b -> Either a b -> Bool Source #

(Ord k, PartialOrd v) => PartialOrd (Map k v) Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: Map k v -> Map k v -> Bool Source #

comparable :: Map k v -> Map k v -> Bool Source #

(PartialOrd k, PartialOrd v) => PartialOrd (Lexicographic k v) Source # 
Instance details

Defined in Algebra.Lattice.Lexicographic

(Eq k, Hashable k, PartialOrd v) => PartialOrd (HashMap k v) Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: HashMap k v -> HashMap k v -> Bool Source #

comparable :: HashMap k v -> HashMap k v -> Bool Source #

(PartialOrd v, Finite k) => PartialOrd (k -> v) Source #

Eq (k -> v) is from Eq

Instance details

Defined in Algebra.PartialOrd.Instances

Methods

leq :: (k -> v) -> (k -> v) -> Bool Source #

comparable :: (k -> v) -> (k -> v) -> Bool Source #

(PartialOrd a, PartialOrd b) => PartialOrd (a, b) Source # 
Instance details

Defined in Algebra.PartialOrd

Methods

leq :: (a, b) -> (a, b) -> Bool Source #

comparable :: (a, b) -> (a, b) -> Bool Source #

partialOrdEq :: PartialOrd a => a -> a -> Bool Source #

The equality relation induced by the partial-order structure. It satisfies the laws of an equivalence relation: Reflexive: a == a Symmetric: a == b ==> b == a Transitive: a == b && b == c ==> a == c

Fixed points of chains in partial orders

lfpFrom :: PartialOrd a => a -> (a -> a) -> a Source #

Least point of a partially ordered monotone function. Checks that the function is monotone.

unsafeLfpFrom :: Eq a => a -> (a -> a) -> a Source #

Least point of a partially ordered monotone function. Does not checks that the function is monotone.

gfpFrom :: PartialOrd a => a -> (a -> a) -> a Source #

Greatest fixed point of a partially ordered antinone function. Checks that the function is antinone.

unsafeGfpFrom :: Eq a => a -> (a -> a) -> a Source #

Greatest fixed point of a partially ordered antinone function. Does not check that the function is antinone.