data Nat = Zero | Succ Nat deriving (Show, Eq)

instance Num Nat where
  n + Zero = n
  n + (Succ m) = Succ (n + m)

  Zero - _ = Zero
  n - Zero = n
  (Succ n) - (Succ m) = n - m

  n * Zero = Zero
  n * (Succ m) = n + (n * m)

  negate = const Zero
  abs = id
  signum = id
  fromInteger i = if i > 0 then Succ (fromInteger (i-1)) else Zero
  
instance Ord Nat where
  Zero <= b = True
  _ <= Zero = False
  (Succ n) <= (Succ b) = n <= b

instance Enum Nat where
  toEnum = fromInteger . toInteger

  fromEnum Zero = 0
  fromEnum (Succ n) = 1 + (fromEnum n)

-- yet "(2::Nat) <= product [2..]" does not terminate.