| Safe Haskell | Safe-Inferred |
|---|---|
| Language | Haskell98 |
Algebra.Units
- class C a => C a where
- isUnit :: a -> Bool
- stdAssociate, stdUnitInv, stdUnit :: a -> a
- 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
Class
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
isUnit, (stdUnit | stdUnitInv)
Standard implementations for instances
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.