|class C a => C a where|
|intQuery :: (Integral a, C a) => a -> Bool|
|intAssociate :: (Integral a, C a, C a) => a -> a|
|intStandard :: (Integral a, C a, C a) => a -> a|
|intStandardInverse :: (Integral a, C a, C a) => a -> a|
|propComposition :: (Eq a, C a) => a -> Bool|
|propInverseUnit :: (Eq a, C a) => a -> Bool|
|propUniqueAssociate :: (Eq a, C a) => a -> a -> Property|
|propAssociateProduct :: (Eq a, C a) => a -> a -> Bool|
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
isUnit and (stdUnit or stdUnitInv) and optionally stdAssociate
|Standard implementations for instances
|Produced by Haddock version 2.4.2|