[Haskell-cafe] Type arithmetic with ATs/TFs
Andrew Coppin
andrewcoppin at btinternet.com
Thu Feb 11 16:10:34 EST 2010
Andrew Coppin wrote:
> OK, so I sat down today and tried this, but I can't figure out how.
>
> There are various examples of type-level arithmetic around the place.
> For example,
>
> http://www.haskell.org/haskellwiki/Type_arithmetic
>
> (This is THE first hit on Google, by the way. Haskell is apparently
> THAT popular!) But this does type arithmetic using functional
> dependencies; what I'm trying to figure out is how to do that with
> associated types.
>
> Any hints?
Several people have now replied to this, both on and off-list. But all
the replies use type families, not associated types.
Now type families are something I don't yet comprehend. (Perhaps the
replies will help... I haven't studied them yet.) What I understand is
that ATs allow you to write things like
class Container c where
type Element c :: *
...
And now you can explicitly talk about the kind of element a container
can hold, rather than relying on the type constructor having a
particular kind or something. So the above works for containers that can
hold *anything* (such as lists), containers which can only hold *one*
thing (e.g., ByteString), and containers which can hold only certain
things (e.g., Set).
...which is great. But I can't see a way to use this for type
arithmetic. Possibly because I don't have a dramatically solid mental
model of exactly how it works. You'd *think* that something like
class Add x y where
type Sum x y :: *
instance Add x y => Add (Succ x) y where
type Sum (Succ x) y = Succ (Sum x y)
ought to work, but apparently not.
As to what type families - type declarations outside of a class - end up
meaning, I haven't the vaguest idea. The Wiki page makes it sound
increadibly complicated...
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