[Haskell-beginners] Enum for natural numbers

[Daniel Fischer]

Am Sonntag 20 Dezember 2009 14:36:06 schrieb kane96 at gmx.de:
> Maybe I didn't understand the exercise if have to do. It says:
> "Write the instance Enum Nat where toEnum is defined as a total function
> that returns Z for negative integers. Some examples: *Main> toEnum (-1) ::
> Nat
> Z
> *Main> toEnum 0 :: Nat
> Z
> *Main> toEnum 1 :: Nat
> S Z
> *Main> fromEnum (S (S Z))
> 2
>
> so I did:
> data Nat = Z | S Nat deriving (Eq,Ord,Show)
> instance Enum Nat where
> toEnum x|x > 0 = S Z
>
>         |otherwise = Z
>
> somehow it looks really wrong. Do you understand want I have to do and how
> it should look like?
>

Your task is to write an Enum instance for Nat.

class Enum a where
  succ :: a -> a
  pred :: a -> a
  toEnum :: Int -> a
  fromEnum :: a -> Int
  enumFrom :: a -> [a]
  enumFromThen :: a -> a -> [a]
  enumFromTo :: a -> a -> [a]
  enumFromThenTo :: a -> a -> a -> [a]

So you have to write functions

succ :: Nat -> Nat      -- that's very easy
pred :: Nat -> Nat      -- easy too, except you have to decide whether to throw an error 
on pred Z or have pred Z = Z (which would fit with the required toEnum)

toEnum :: Int -> Nat
-- that has the additional requirement that it should return Z for negative input

and so on.

The first thing is to understand the Nat datatype. That models the so-called Peano numbers 
(or, put another way, it's built to mirror the Peano axioms for the natural numbers).
Z corresponds to 0
S Z corresponds to 1, which is the successor of 0
S (S Z) corresponds to 2, which is the successor of 1 ...

For non-negative n, toEnum n should be the Peano number corresponding to n and for 
negative n, toEnum n should be Z (to avoid exceptions).
So,

toEnum n
    | n < 0 = Z
toEnum 0 = Z
toEnum n = ?    -- here, we know n > 0

fromEnum should be the correspondnece the other way round, so

fromEnum Z = 0
fromEnum (S p) = ?      -- which Int corresponds to the successor of p?