Overlapping families (specificity/degrees of freedom)

[Richard Eisenberg]

On Apr 3, 2013, at 3:01 PM, Gabor Greif <ggreif at gmail.com> wrote:
> 
> What I want is a function that can e.g. create the minimal union of
> two Symbol singletons, possibly by consulting decidable (in)equality:
> 
> {{{ (sketch)
> type family MinUnion (a :: Symbol) (b :: Symbol) :: [Symbol]
> type instance where
>  MinUnion a a = '[a]
>  MinUnion a b = '[a, b]
> 
> minUnion :: DecEq a b -> Sing (a :: Symbol) -> Sing (b :: Symbol) ->
> Sing (MinUnion a b)
> minUnion (Right Refl) a b -> Cons a Nil
> minUnion (Left Refl) a b -> Cons a b
> }}}
> 

I think you're out of luck with this approach, I'm afraid. GHC does not know how to use something like (a :~: b -> Void) as evidence of inequality -- not by a long shot. The best thing I can think of is to use a "normal" promoted data type instead of Symbol, so you can write a recursive equality function and avoid "type instance where". I think, then, you'd be able to get it all to work out. "type instance where" and GADT pattern matching don't really play nicely with each other.

> 
>> 
>>> {-# LANGUAGE TemplateHaskell, DataKinds, PolyKinds, TypeFamilies, GADTs,
>>>             UndecidableInstances, FlexibleContexts, RankNTypes #-}
>>> 
>>> import Data.Singletons
>>> 
>>> $(singletons [d|
>>>  member :: Eq a => a -> [a] -> Bool
>>>  member _ [] = False
>>>  member x (h : t) = x == h || member x t
>>> 
>>>  intersect :: Eq a => [a] -> [a] -> [a]
>>>  intersect []      _ = []
>>>  intersect (h : t) b = if member h b then h : (intersect t b) else intersect t b
>>> |])
> 
> You appear to be able to lift `Eq a` up to the type level (and
> `member` too) as a type function. How do you compare Symbol? Also,will
> this give me an `intersect` function at the value level?

Well, sort of. I cheat a little. `Eq` is not promoted to the type level, but that's OK. There is a type family (:==:) that 'singletons' has for type-level Boolean equality. That type family only has instances for datatypes that have Eq instances, at the value level. There is no real need to promote the constraint. `member` and `intersect` are promoted to the type level to become `Member` and `Intersect`. They are also refined into functions that process singletons, called `sMember` and `sIntersect`. Equality is refined into a function over singletons called `%==%`, and the type class for singleton types with equality is called `SEq`.

One problem with all of this is that it doesn't really work with Symbols, because Symbols aren't recursively defined. Using unsafe features, we can squeeze out an instance of `SEq` (which would, in fact, be safe, but GHC wouldn't know it), but without inequality evidence, I don't think there's a way to get a branched type family instance (a "type instance where") to do the right thing. Somehow, to get that instance to trigger, you need to show GHC that the two types have different constructors used somewhere in their structure. Because Symbols have no structure, this isn't possible.

Richard