Church Booleans in Haskell

ChurchBool.hs

{-# LANGUAGE FlexibleInstances, Rank2Types, TypeFamilies #-}

module ChurchBool where

type CB = forall a. a -> a -> a

instance a ~ Bool => Show (a -> a -> a) where
    show cb = if cb True False then "cTrue" else "cFalse"

cTrue, cFalse :: CB
cTrue a _ = a
cFalse _ b = b

cNot :: CB -> CB
cNot m a b = m b a

cAnd :: CB -> CB -> CB
cAnd a b = a b a

cNand :: CB -> CB -> CB
cNand a b x y = a b a y x

cOr :: CB -> CB -> CB
cOr a b = a a b

cNor :: CB -> CB -> CB
cNor a b x y = a a b y x

cXor :: CB -> CB -> CB
cXor a b x y = a (b y x) (b x y)

cXnor :: CB -> CB -> CB
cXnor a b x y = a (b x y) (b y x)