I'm attempting to implement church numerals in Haskell, but I've hit a minor problem. Haskell complains of an infinite type with
Occurs check: cannot construct the infinite type: t = (t -> t1) -> (t1 -> t2) -> t2
when I try and do subtraction. I'm 99% positive that my lambda calculus is valid (although if it isn't, please tell me). What I want to know, is whether there is anything I can do to make haskell work with my functions.
module Church where
type (Church a) = ((a -> a) -> (a -> a))
makeChurch :: Int -> (Church a)
makeChurch 0 = \f -> \x -> x
makeChurch n = \f -> \x -> f (makeChurch (n-1) f x)
numChurch x = (x succ) 0
showChurch x = show $ numChurch x
succChurch = \n -> \f -> \x -> f (n f x)
multChurch = \f2 -> \x2 -> \f1 -> \x1 -> f2 (x2 f1) x1
powerChurch = \exp -> \n -> exp (multChurch n) (makeChurch 1)
predChurch = \n -> \f -> \x -> n (\g -> \h -> h (g f)) (\u -> x) (\u -> u)
subChurch = \m -> \n -> (n predChurch) m