On Thursday 12 July 2007, Andrew Coppin wrote:
> Jonathan Cast wrote:
> > On Wednesday 11 July 2007, Chaddaï Fouché wrote:
> >> Is there something I misunderstood in the exchange ?
> >
> > Yeah. The reference to the "lazy natural type", which is:
> >
> > data Nat
> > = Zero
> >
> > | Succ Nat
> >
> > deriving (Eq, Ord, Read, Show)
> >
> > instance Num Nat where
> > fromInteger 0 = Zero
> > fromInteger (n + 1) = Succ (fromInteger n)
> > etc.
> >
> > then genericLength xn > n does exactly what Andrew wants, when n :: Nat.
>
> Wow.
>
> Show me a simple problem, and some Haskeller somewhere will find a
> completely unexpected way to solve it... LOL!
>
> OTOH, doesn't that just mean that Nat is itself a degenerate list, and
> genericList is just converting one list to another, and the Ord instance
> for Nat is doing the short-cut stuff?
Yes. Nat ~ [()], where ~ means `is isomorphic to'. But Nat is also the
obvious way to encode Peano arithmetic in Haskell, so this is a deep thought,
not a shallow one.
(Properly speaking, the set of total values of Nat is the set of natural
numbers + infinity (infinity = x where x = Succ x), and the set of total
lists of type [alpha] is
sum (n :: Nat). f :: {m :: Nat | m < n} -> alpha
where f and n are total. sum is a dependent sum, which is just a product, and
the only total function with co-domain () is const () (well, and (`seq` ()),
but they're equal on total arguments), so that type is just
sum (n :: Nat^inf). {const ()}
which is isomorphic to Nat^inf. But you can see that this is a deep thought,
not a shallow one. . .)
Jonathan Cast
http://sourceforge.net/projects/fid-core
http://sourceforge.net/projects/fid-emacs
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