numeric-prelude-0.4.3: An experimental alternative hierarchy of numeric type classes

Algebra.Units

Synopsis

# Class

class C a => C a where Source #

This class lets us deal with the units in a ring. isUnit tells whether an element is a unit. The other operations let us canonically write an element as a unit times another element. Two elements a, b of a ring R are _associates_ if a=b*u for a unit u. For an element a, we want to write it as a=b*u where b is an associate of a. The map (a->b) is called StandardAssociate by Gap, "unitCanonical" by Axiom, and "canAssoc" by DoCon. The map (a->u) is called "canInv" by DoCon and "unitNormal(x).unit" by Axiom.

The laws are

  stdAssociate x * stdUnit x === x
stdUnit x * stdUnitInv x === 1
isUnit u ==> stdAssociate x === stdAssociate (x*u)

Currently some algorithms assume

 stdAssociate(x*y) === stdAssociate x * stdAssociate y

Minimal definition: isUnit and (stdUnit or stdUnitInv) and optionally stdAssociate

Minimal complete definition

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # MethodsstdUnit :: T -> T Source # Integral a => C (T a) Source # MethodsisUnit :: T a -> Bool Source #stdAssociate :: T a -> T a Source #stdUnit :: T a -> T a Source #stdUnitInv :: T a -> T a Source # (C a, C a) => C (T a) Source # MethodsisUnit :: T a -> Bool Source #stdAssociate :: T a -> T a Source #stdUnit :: T a -> T a Source #stdUnitInv :: T a -> T a Source # (Ord a, C a) => C (T a) Source # MethodsisUnit :: T a -> Bool Source #stdAssociate :: T a -> T a Source #stdUnit :: T a -> T a Source #stdUnitInv :: T a -> T a Source # C a => C (T a) Source # MethodsisUnit :: T a -> Bool Source #stdAssociate :: T a -> T a Source #stdUnit :: T a -> T a Source #stdUnitInv :: T a -> T a Source #

isUnit :: C a => a -> Bool Source #

# Standard implementations for instances

intQuery :: (Integral a, C a) => a -> Bool Source #

intAssociate :: (Integral a, C a, C a) => a -> a Source #

intStandard :: (Integral a, C a, C a) => a -> a Source #

intStandardInverse :: (Integral a, C a, C a) => a -> a Source #

# Properties

propComposition :: (Eq a, C a) => a -> Bool Source #

propInverseUnit :: (Eq a, C a) => a -> Bool Source #

propUniqueAssociate :: (Eq a, C a) => a -> a -> Property Source #

propAssociateProduct :: (Eq a, C a) => a -> a -> Bool Source #

Currently some algorithms assume this property.