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





class C a => C a whereSource

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


isUnit :: a -> BoolSource

stdAssociate :: a -> aSource

stdUnitInv :: a -> aSource

stdUnit :: a -> aSource


C Int 
C Int8 
C Int16 
C Int32 
C Int64 
C Integer 
C T 
(Ord a, C a) => C (T a) 
(C a, C a) => C (T a) 

Standard implementations for instances

intQuery :: (Integral a, C a) => a -> BoolSource

intAssociate :: (Integral a, C a, C a) => a -> aSource

intStandard :: (Integral a, C a, C a) => a -> aSource

intStandardInverse :: (Integral a, C a, C a) => a -> aSource


propComposition :: (Eq a, C a) => a -> BoolSource

propInverseUnit :: (Eq a, C a) => a -> BoolSource

propUniqueAssociate :: (Eq a, C a) => a -> a -> PropertySource

propAssociateProduct :: (Eq a, C a) => a -> a -> BoolSource