numeric-prelude-0.4.2: 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

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

# 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.