[Haskell-beginners] Enum for natural numbers
Daniel Fischer
daniel.is.fischer at web.de
Sun Dec 20 08:59:45 EST 2009
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?
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