[Haskell-beginners] Enum for natural numbers

[Isaac Dupree]

kane96 at gmx.de wrote:
> -------- Original-Nachricht --------
>> Datum: Sun, 20 Dec 2009 21:28:21 +0100
>> Von: Daniel Fischer <daniel.is.fischer at web.de>
>> An: beginners at haskell.org
>> Betreff: Re: [Haskell-beginners] Enum for natural numbers
> 
>> Am Sonntag 20 Dezember 2009 21:06:36 schrieb kane96 at gmx.de:
>>>> 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?
>>> what is P?
>> Any element of Nat (p for Peano).
>>
>>> Now I read some short textes about it and think I know more or less what
>> I
>>> have to do for the exercise. But I don't know really how. Do you know
>> any
>>> examples for it, how it normally looks like?
>> Deniz Dogan posted a few links to recursion earlier today, if you look at
>> them, you should 
>> get the general idea.
>> As a further example,
>>
>> replicate :: Int -> a -> [a]
>> replicate n x
>>     | n <= 0    = []
>>     | otherwise = x:replicate (n-1) x
>>
>> may help.
> 
> my problem is that I don't know how to use Enum correctly and didn't find any helpfull example

Enum is used for things that are in sequence and can be numbered 
likewise.  For example, (if you consider Monday the first day of the week)

data Weekday = Monday | Tuesday | Wednesday | Thursday |
                    Friday | Saturday | Sunday

probably Monday would correspond to 0, Tuesday to 1, Wednesday to 2, and 
so on through Sunday corresponding to 6.

In your problem,
data Nat = Z | S Nat

Z corresponds to 0, (S Z) corresponds to 1, (S (S Z)) corresponds to 2, 
and so on forever.

Perhaps you can see this by seeing "Z" as "0" and "S" as "1 +".
Z <-> 0
(S Z) <-> (1 + 0)
(S (S Z)) <-> (1 + (1 + 0))
etc...

but you'll have to find a way to define this pattern with just a small 
number of cases, not a huge infinite number.

-Isaac