Integral.hs

{-# LANGUAGE NoImplicitPrelude #-}

module Precursor.Numeric.Integral
  ( Integral(..)
  , even
  , odd
  )where

import           Data.Int                   (Int, Int16, Int32, Int64, Int8)
import           Precursor.Algebra.Semiring
import           Prelude                    (even, odd)
import qualified Prelude

-- | An integral domain. Members of this class must be 'Semiring's with
-- commutative '*'.
--
-- * @a '*' b = b '*' a@
-- * @(a '//' b) '*' b '+' (a '%' b) = a@
-- * @(a'+'k'*'b) '%' b = a '%' b@
-- * @'zero' '%' b = 'zero'@
class Semiring a => Integral a where
  {-# MINIMAL divMod | ((//), (%)) #-}
  -- | The divisor and modulo
  divMod :: a -> a -> (a, a)
  infixl 7 //
  -- | Integer division
  (//) :: a -> a -> a
  infixl 7 %
  -- | Modulo
  (%)  :: a -> a -> a
  x // y = let (d,_) = divMod x y in d
  x %  y = let (_,m) = divMod x y in m
  divMod x y = (x % y, x // y)

instance Integral Int where divMod = Prelude.divMod
instance Integral Int8 where divMod = Prelude.divMod
instance Integral Int16 where divMod = Prelude.divMod
instance Integral Int32 where divMod = Prelude.divMod
instance Integral Int64 where divMod = Prelude.divMod
instance Integral Prelude.Integer where divMod = Prelude.divMod