{-# LANGUAGE GeneralizedNewtypeDeriving #-} module Algebra.RingUtils ( module Prelude , AbelianGroup(..) , AbelianGroupZ(..) , Ring(..) , RingP(..) , Pair(..), select, onlyLeft, onlyRight , O(..) , sum , mulDefault , module Data.Pair ) where import qualified Prelude as P import Prelude hiding ( (+), (*), splitAt, sum ) import Control.Applicative import Data.Pair class AbelianGroup a where zero :: a (+) :: a -> a -> a instance AbelianGroup Int where zero = 0 (+) = (P.+) class AbelianGroup a => AbelianGroupZ a where isZero :: a -> Bool instance AbelianGroupZ Int where isZero x = x == 0 class AbelianGroupZ a => Ring a where (*) :: a -> a -> a class (AbelianGroupZ a) => RingP a where mul :: Bool -> a -> a -> Pair a -- mul _ x y = pure \$ x * y mulDefault x y = leftOf (mul False x y) onlyLeft x = x :/: [] onlyRight x = [] :/: x select p = if p then onlyRight else onlyLeft newtype O f g a = O {fromO :: f (g a)} deriving (AbelianGroup, AbelianGroupZ, Show) instance (Functor f,Functor g) => Functor (O f g) where fmap f (O x) = O (fmap (fmap f) x) instance AbelianGroup a => AbelianGroup (Pair a) where zero = (zero:/:zero) (a:/:b) + (x:/:y) = (a+x) :/: (b+y) instance AbelianGroupZ a => AbelianGroupZ (Pair a) where isZero (a:/:b) = isZero a && isZero b instance Ring Int where (*) = (P.*) infixl 7 * infixl 6 + sum :: AbelianGroup a => [a] -> a sum = foldr (+) zero instance AbelianGroup Bool where zero = False (+) = (||)