BooleanAlgebra.hs

{-# LANGUAGE DerivingVia, TypeApplications #-}
import Prelude hiding ((&&), (||), not)
import qualified Prelude ((&&), (||), not)

class BooleanAlgebra a where
    (&&) :: a -> a -> a
    (||) :: a -> a -> a
    not :: a -> a
    true :: a
    false :: a

instance BooleanAlgebra Bool where
    (&&) = (Prelude.&&)
    (||) = (Prelude.||)
    not = Prelude.not
    true = True
    false = False

newtype BooleanRing a = BooleanRing a
    deriving (Read, Show, Eq, Ord, Bounded)
    deriving (Enum, BooleanAlgebra) via a

-- Every boolean algebra is a ring in a canonical way.
instance BooleanAlgebra a => Num (BooleanRing a) where
    x + y = (x || y) && not (x && y)
    x * y = x && y
    negate = id
    abs = id
    signum = const 1
    fromInteger n = case n `mod` 2 of
        0 -> false
        1 -> true

type Z2 = BooleanRing Bool

main :: IO ()
main = do
    putStrLn "> 42 * true == 42"
    print (42 * true @Z2 == 42)