Natural & Integer numbers definition

Number.hs

module Number where

data Nat = S Nat | Z
         deriving(Eq)

natIntoHaskellInt :: Nat -> Int
natIntoHaskellInt Z = 0
natIntoHaskellInt (S v) = natIntoHaskellInt v + 1

instance Show Nat where
  show = show . natIntoHaskellInt

data Sign = Pos | Neg
data Int' = Int' Sign Nat

instance Show Int' where
  show (Int' Pos integral) = show integral
  show (Int' Neg integral) = '-' : show integral

natFromInt :: Int -> Nat
natFromInt 0 = Z
natFromInt x | x < 0 = error "Cannot build natural number from the negative integer"
             | x > 0 = S (natFromInt $ x - 1)

fromInt :: Int -> Int'
fromInt 0 = Int' Pos Z
fromInt x | x > 0 = Int' Pos $ natFromInt x
          | x < 0 = Int' Neg $ natFromInt (-x)

add :: Nat -> Nat -> Nat
add  Z y    = y
add (S x) y = S (add x y)

add' :: Int' -> Int' -> Int'
add' (Int' Pos lhs) (Int' Pos rhs) = Int' Pos $ add lhs rhs
add' (Int' Neg lhs) (Int' Neg rhs) = Int' Neg $ add lhs rhs
add' (Int' lS lhs) (Int' rS rhs) =
  case lS of
    Pos -> subFrom lhs rhs
    Neg -> subFrom rhs lhs
  where subFrom :: Nat -> Nat -> Int'
        subFrom Z rhs           = Int' Neg rhs
        subFrom lhs Z           = Int' Pos lhs
        subFrom (S lhs) (S rhs) = subFrom lhs rhs