Constraint implication

[Edward Kmett]

Just a few old observations I've made from implementing these things into
type systems of my own:

1.) An internal hom for the category of constraints is admissible:

(|-) :: Constraint -> Constraint -> Constraint

models entailment, and effectively brings into scope a local rule, but
global instance resolution isn't lost if this is only produced from
existing instances.

To your point, this is analogous to the (:-) :: Constraint -> Constraint ->
* external hom for the category of constraints provided by my constraints
package, but it is internal, with all of the appropriate CCC operations.

2.) Quantification over constraints is also admissible.

Neither one compromises the "thinness" of the category of constraints that
provides us global coherence of instance resolution

In this case the property that if D is thin, so is [C,D]. forall here can
quantify over any other kind you want.

Unfortunately, neither implemented in Haskell.

You'd need both of them to be able to talk about constraints like (forall
a. Eq a |- Eq (f a)).

Together could eliminate, morally, the entire Data.Functor.Classes mess.
(That said, as implemented those classes are a bit stronger than the
quantified form)

3.) Constraint also admits a sum type, (\/) but it acts more like a least
upper bound than an either.

It is much easier to talk about in the category of constraints using the
first part above.

Given (p |- r, q |- r), (p \/ q) |- r and vice versa.

The key to keeping the category of constraints thin is that you can't case
analyze on it, its more like if you look at, say, Ord [a] \/ Eq a you can
get to any constraint that you could get to given the intersection of both,
not using the particulars of either. e.g. its morally as powerful as Eq [a]
in this case.

Getting this stuff into GHC is the tricky part!

-Edward

On Wed, Dec 27, 2017 at 2:50 AM, Ryan Reich <ryan.reich at gmail.com> wrote:

> The Constraint kind appears to lack an interface to an important
> capability that is already part of the type checker: constraint
> implication.  Namely, the ability to provide a witness for the statement
> "constraint c1 implies constraint c2" or, more importantly, "for all a,
> constraint (c1 a) implies constraint (c2 a)", where c1 and c2 are now
> constraint-valued type functions (and possibly even for constraint
> functions with multiple parameters).  It seems to me that this can follow
> the pattern of the "magic" Coercible type class and the (non-magic)
> Coercion data type; it provides the programmer with an analogous value to
> this example that can be obtained in apparently no other way.
>
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