Nat in Haskell

Nat.hs

data Nat where
  O :: Nat
  S :: Nat -> Nat

instance Eq Nat where
  (==) O O = True
  (==) O (S _) = False
  (==) (S _) O = False
  (==) (S n) (S m) = n == m

instance Num Nat where
  (+) n O = n
  (+) O m = m
  (+) n (S m) = S n + m
 
  (-) n O = n
  (-) O _ = O
  (-) (S n) (S m) = n - m

  (*) _ O = O
  (*) O _ = O
  (*) n (S m) = n + n * m

  abs n = n

  signum O = O
  signum (S _) = S O

  fromInteger i = if i <= 0 then O else S (fromInteger (i - 1))

instance Show Nat where
  show O = "O"
  show (S O) = "S O"
  show (S n) = "S (" ++ show n ++ ")"

fix :: ((a -> b) -> a -> b) -> a -> b
fix f x = f (fix f) x

fact :: Nat -> Nat
fact = fix $ \fact n ->
  case n of
    O -> S O
    S n' -> n * fact n'

main = putStrLn (show $ fact (fromInteger 3))