Re: [Haskell-beginners] Enum for natural numbers

20 Dec 2009

      -------- Original-Nachricht --------
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Datum: Sun, 20 Dec 2009 14:59:45 +0100
Von: Daniel Fischer 
An: beginners@haskell.org
CC: kane96@gmx.de
Betreff: Re: [Haskell-beginners] Enum for natural numbers
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Am Sonntag 20 Dezember 2009 14:36:06 schrieb kane96@gmx.de:
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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?
what is P?

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?

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Re: [Haskell-beginners] Enum for natural numbers

kane96＠gmx.de