[Haskell-cafe] Type arithmetic with ATs/TFs

[Robert Greayer]

What Ryan said, and here's an example of addition with ATs,
specifically (not thoroughly tested, but tested a little).  The
translation to TFs sans ATs is straightforward.

class Add a b where
    type SumType a b

instance Add Zero Zero where
    type SumType Zero Zero = Zero

instance Add (Succ a) Zero where
    type SumType (Succ a) Zero = Succ a

instance Add Zero (Succ a) where
    type SumType Zero (Succ a) = Succ a

instance Add (Succ a) (Succ b) where
    type SumType (Succ a) (Succ b) = Succ (Succ (SumType a b))


On Thu, Feb 11, 2010 at 4:10 PM, Andrew Coppin
<andrewcoppin at btinternet.com> wrote:
> 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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