[Haskell-cafe] Re: having fun with GADT's

[Reiner Pope]

Anatoly Yakovenko <aeyakovenko <at> gmail.com> writes:

> 
> I dont remember where i saw it, but i think someone had an example of
> a list whose type is the maximum element in the list.  I've been
> trying to reproduce that with GADT's.
> 
> data One = One
> data Two = Two
> data Three = Three
> 
> data MaxList t where
>    Elem1 :: MaxList One
>    Elem2 :: MaxList Two
>    ML1Cons1 :: MaxList One -> MaxList One -> MaxList One
>    ML1Cons2 :: MaxList One -> MaxList Two -> MaxList Two
>    ML2Cons1 :: MaxList Two -> MaxList One -> MaxList Two
>    ML2Cons2 :: MaxList Two -> MaxList Two -> MaxList Two
> 
> a = ML2Cons2 Elem2 $ ML2Cons1 Elem2 $ ML1Cons1 Elem1 $ Elem1
> 
> so one problem is the tedium of defining a cons for each possible
> combination.  The other problem is that i cant define a usefull tail
> that makes any sense.
> 
> mlTail :: MaxList Two -> MaxList t
> mlTail (ML2Cons2 a b) = b
> mlTail (ML2Cons1 a b) = b
> 
> this one doesn't work, and probably because there is nothing that i
> could do with the return value.

Your problem in this example is that the t in "MaxList t" is universally
quantified when it needs to be existentially quantified. The following
definition encodes the existential quantification as a rank-2 type:

mlTail :: MaxList n -> (forall t. MaxList t -> a) -> a
mlTail (ML1Cons1 h t) f = f t
mlTail (ML1Cons2 h t) f = f t
mlTail (ML2Cons1 h t) f = f t
mlTail (ML2Cons2 h t) f = f t

It works with the rest of your code unmodified.

> 
> mlTail :: MaxList Two -> MaxList Two
> mlTail (ML2Cons2 a b) = b
> mlTail (ML2Cons1 a b) = b  --wont compile because b is a MaxList One
> 
> will only work for lists that only contain Two's, which is not what i
> want either.  So is this somehow possible?

This example here suggests that you are happy merely with a (not necessarily
tight) upper bound on the list elements. The following code solves your problem
in this case, using only type unification and not fundeps or TFs:

data Nat a where
    Z :: Nat a
    S :: Nat a -> Nat (S a)

data Z
data S a

n00 = Z
n01 = S n00
n02 = S n01
n03 = S n02
n04 = S n03

data MaxList t where
   Nil :: MaxList a
   Cons :: Nat a -> MaxList a -> MaxList a

a = Cons n02 $ Cons n02 $ Cons n01 $ Nil
--- ":t a" gives "forall a. MaxList (S (S a))" which tells you exactly 
--- what you want: elements are at least 2.

mlTail :: forall t. MaxList t -> MaxList t
mlTail (Cons h t) = t
--- unfortunately, you lose information here if the first
--- element is larger than the rest.

Reiner