-- Peano numbers --------------------------------------------------------------

data Pe = Zero                  -- 0
        | S Pe                  -- Pe + 1
        deriving (Show)

-- basic arithmetic operations ------------------------------------------------

add :: Pe -> Pe -> Pe
add Zero y = y                  -- 0 + y = y
add (S x) y = S (add x y)       -- (x + 1) + y = (x + y) + 1

mul :: Pe -> Pe -> Pe
mul Zero y = Zero               -- 0 * y = 0
mul (S x) y = add (mul x y) y   -- (x + 1) * y = (x * y) + y

sub :: Pe -> Pe -> Pe
sub Zero y = Zero               -- 0 - y = 0
sub x Zero = x                  -- x - 0 = x
sub (S x) (S y) = sub x y       -- (x + 1) - (y + 1) = x - y

-- input/output ---------------------------------------------------------------

toInt :: Pe -> Int
toInt Zero = 0
toInt (S x) = 1 + toInt x

toPe :: Int -> Pe
toPe 0 = Zero
toPe x = S (toPe (x - 1))

-- ----------------------------------------------------------------------------

main = do

    let zero = Zero
    let one  = S(Zero)
    let two  = S(S(Zero))

    print $ (show . toInt $ zero) ++ " := " ++ (show zero)
    print $ (show . toInt $ one)  ++ " := " ++ (show one)
    print $ (show . toInt $ two)  ++ " := " ++ (show two)

    print $ "3 := " ++ (show $ S(S(S(Zero))))
    print $ "4 := " ++ (show $ add two two)
    print $ "5 := " ++ (show $ add (mul two two) one)
    print $ "6 := " ++ (show $ toPe (3 + 3))
    print $ "7 := " ++ (show $ sub (mul (mul two two) two) one)
    print $ "8 := " ++ (show $ mul (mul two two) two)
    print $ "9 := " ++ (show $ toPe (3 * 3))