[Haskell-cafe] Re: type-level integers using type families

[Benedikt Huber]

Peter Gavin schrieb:
> Has anyone else tried implementing type-level integers using type families?
> 
> I tried using a couple of other type level arithmetic libraries 
> (including type-level on Hackage) and they felt a bit clumsy to use.  I 
> started looking at type families and realized I could pretty much build 
> an entire Scheme-like language based on them.
> 
> In short, I've got addition, subtraction, & multiplication working after 
> just a days worth of hacking. I'm going to post the darcs archive 
> sometime, sooner if anyone's interested.
> 
> I really like the type-families based approach to this, it's a lot 
> easier to understand, and you can think about things functionally 
> instead of relationally.  (Switching back and forth between Prolog-ish 
> thinking and Haskell gets old quick.) Plus you can do type arithmetic 
> directly in place, instead of using type classes everywhere.

I tried to rewrite Alfonso Acosta's type-level library (the one on 
hackage) using type-families too, because, right, they are much nicer to 
use than MPTCs.

It is a straigtforward translation, I've uploaded it to

http://code.haskell.org/~bhuber/type-level-tf-0.2.1.tar.gz

now (relevant files: src/Data/TypeLevel/{Bool.hs,Num/Ops.hs}).

I've also added a type-level ackermann to torture ghc a little bit ;),
but left out logarithms. Plus, I did very little testing.

> One thing that I'd like to be able to do is lazy unification on type instances, so that things like
> 
> data True
> data False
> 
> type family Cond x y z
> type instance Cond True y z = y
> type instance Cond False y z = z 

I'm not sure if this is what you had in mind, but I also found that e.g.

 > type instance Mod x y = Cond (y :>: x) x  (Mod (Sub x y) y)

won't terminate, as

 > Mod D0 D1 ==> Cond True D0 (Mod (Sub D0 D1) D0) ==> <loop>

rather than

 > Mod D0 D1 ==> Cond True D0 ? ==> D0

For Mod, I used the following (usual) encoding:

 > type family Mod' x y x_gt_y
 > type instance Mod' x y False = x
 > type instance Mod' x y True  = Mod' (Sub x y) y ((Sub x y) :>=: y)
 > type family Mod x y
 > type instance Mod x y = Mod' x y (x :>=: y)

There are few other things you cannot do with type families, and some 
for which you need type classes anyway (Alfonso pointed me to 
http://www.haskell.org/pipermail/haskell-cafe/2008-February/039489.html 
). Here's what I found:


1) One cannot define type equality (unless total type families become
available), i.e. use the overlapping instances trick:
> instance Eq e e  True
> instance Eq e e' False

Consequently, all type-level functions which depend on type equality
(see HList) need to be encoded using type classes.

2) One cannot use superclass contexts to derive instances e.g. to define
> instance Succ (s,s') =>  Pred (s',s)

In constrast, when using MPTC + FD, one can establish more than one TL 
function in one definition

 > class Succ x x' | x -> x', x' -> x

3) Not sure if this is a problem in general, but I think you cannot 
restrict the set of type family instances easily.

E.g., if you have an instance

 > type instance Mod10 (x :* D0) = D0

then you also have

 > Mod10 (FooBar :* D0) ~ D0

What would be nice is something like

 > type instance (IsPos x) => Mod10 (x :* D0) = D0

though

 > type family AssertThen b x
 > type instance AssertThen True x = x
 > type instance Mod10 (x :* D0) = AssertThen (IsPos x) D0

seems to work as well.

4) Not really a limitation, but if you want to use instance methods of
Nat or Bool (e.g. toBool) on the callee site, you have to provide
context that the type level functions are closed w.r.t. to the type class:

> test_1a :: forall a b b1 b2 b3.
>           (b1 ~ And a b,
>            b2 ~ Not (Or a b),
>            b3 ~ Or b1 b2,
>            Bool b3) => a -> b -> Prelude.Bool
> test_1a a b = toBool (undefined :: b3)

Actually, I think the 'a ~ b' syntax is very nice.

I'm really looking forward to type families.

best regards,

benedikt