- class C a => C a where
- divModZero :: (C a, C a) => a -> a -> (a, a)
- divides :: (C a, C a) => a -> a -> Bool
- sameResidueClass :: (C a, C a) => a -> a -> a -> Bool
- safeDiv :: (C a, C a) => a -> a -> a
- even :: (C a, C a) => a -> Bool
- odd :: (C a, C a) => a -> Bool
- decomposeVarPositional :: (C a, C a) => [a] -> a -> [a]
- decomposeVarPositionalInf :: C a => [a] -> a -> [a]
- propInverse :: (Eq a, C a, C a) => a -> a -> Property
- propMultipleDiv :: (Eq a, C a, C a) => a -> a -> Property
- propMultipleMod :: (Eq a, C a, C a) => a -> a -> Property
- propProjectAddition :: (Eq a, C a, C a) => a -> a -> a -> Property
- propProjectMultiplication :: (Eq a, C a, C a) => a -> a -> a -> Property
- propUniqueRepresentative :: (Eq a, C a, C a) => a -> a -> a -> Property
- propZeroRepresentative :: (Eq a, C a, C a) => a -> Property
- propSameResidueClass :: (Eq a, C a, C a) => a -> a -> a -> Property
Class
IntegralDomain
corresponds to a commutative ring,
where a
picks a canonical element
of the equivalence class of mod
ba
in the ideal generated by b
.
div
and mod
satisfy the laws
a * b === b * a (a `div` b) * b + (a `mod` b) === a (a+k*b) `mod` b === a `mod` b 0 `mod` b === 0
Typical examples of IntegralDomain
include integers and
polynomials over a field.
Note that for a field, there is a canonical instance
defined by the above rules; e.g.,
instance IntegralDomain.C Rational where divMod a b = if isZero b then (undefined,a) else (a\/b,0)
It shall be noted, that div
, mod
, divMod
have a parameter order
which is unfortunate for partial application.
But it is adapted to mathematical conventions,
where the operators are used in infix notation.
Derived functions
divModZero :: (C a, C a) => a -> a -> (a, a)Source
Allows division by zero. If the divisor is zero, then the divident is returned as remainder.
sameResidueClass :: (C a, C a) => a -> a -> a -> BoolSource
safeDiv :: (C a, C a) => a -> a -> aSource
Returns the result of the division, if divisible. Otherwise undefined.
Algorithms
decomposeVarPositional :: (C a, C a) => [a] -> a -> [a]Source
decomposeVarPositional [b0,b1,b2,...] x
decomposes x
into a positional representation with mixed bases
x0 + b0*(x1 + b1*(x2 + b2*x3))
E.g. decomposeVarPositional (repeat 10) 123 == [3,2,1]
decomposeVarPositionalInf :: C a => [a] -> a -> [a]Source