





Synopsis 

class C a => C a where  div :: a > a > a  mod :: a > a > a  divMod :: a > a > (a, a) 
  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 mod b picks a canonical element
of the equivalence class of a 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.
Minimal definition: divMod or (div and mod)
  Methods      divMod :: a > a > (a, a)  Source 

  Instances  


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.






safeDiv :: (C a, C a) => a > a > a  Source 

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 


Properties


















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