Equality constraints (~): type-theory behind them

[Anthony Clayden]

On Fri, 7 Dec 2018 at 10:57 PM, Adam Gundry wrote:

>
> Regarding inference, the place that comes to mind is Vytiniotis et al.
> OutsideIn(X):
>

Thanks Adam for the references. Hmm That OutsideIn paper is formidable in
so many senses ...

I'm not sure I grok your response on my specific q; to adapt your example
slightly, without a signature

    foo2 x@(_: _) = x

GHC infers `foo2 :: [a] -> [a]`. That's not general enough. So I give a
signature with bare `b` as the return type, using the (~) to improve it:

    foo2 :: [a] ~ b => [a] -> b

... and GHC infers `foo2 :: [a] -> [a]`. Oh. Is that what I should expect?

I'm chiefly concerned about improving bare tyvars under a FunDep. So to
take the time-worn example, I want instances morally equivalent to

    data TTrue = TTrue;  data TFalse = TFalse

    class TEq a b r  | a b -> r  where tEq :: a -> b -> r

    instance TEq a a TTrue   where tEq _ _ = TTrue
    instance TEq a b TFalse  where tEq _ _ = TFalse

The FunDeps via CHRs 2006 paper tells me that first instance is supposed to
be as if

    instance (r ~ TTrue) => TEq a a r  where ...

but it isn't

    tEq 'a' 'a' :: TFalse      -- returns TFalse, using the first pair of
instances
    tEq 'a' 'a' :: TFalse      -- rejected, using the (~) instance:
'Couldn't match type TFalse with TTrue ...'

The 'happily' returning the 'wrong' answer in the first case has caused
consternation amongst beginners, and a few Trac tickets.

So what magic is happening with (~) that it can eagerly improve `foo2` but
semi-lazily hold back for `tEq`?

Of course I lied about the first-given two instances for TEq: instances are
not consistent with Functional Dependency. So I need to resort to overlaps
and a (~) and exploit GHC's bogus consistency check:

    instance {-# OVERLAPPABLE #-} (r ~ TFalse) => TEq a b r  where ...

> ... like automatically inferring where `typeCast` needs to be placed

Yes the non-automatic `TypeCast` version

    instance (TypeCast TTrue r) => TEq a a r  where
      tEq _ _ = TTrue         -- rejected 'Couldn't match expected type 'r'
with actual type 'TTrue' ...'

      tEq _ _ = typeCast TTrue    -- accepted

Even though `typeCast` is really `id` (after it's jumped through the
devious indirection hoops I talked about).

So what I'm trying to understand is not just "where" to place `typeCast`
but also "when" in inference it unmasks the unification.


AntC


> About your specific question, if a wanted constraint `a ~ b` can be
> solved by unifying `a` with `b`, then GHC is free to do so (in a sense,
> either parameter of (~) can be improved from the other). The real
> difference with `TypeCast` arises when local equality assumptions (given
> constraints) are involved, because then there may be wanted constraints
> that cannot be solved by unification but can be solved by exploiting the
> local assumptions.
>
> For example, consider this function:
>
>     foo :: forall a b . a ~ b => [a] -> [b]
>     foo x = x
>
> When checking the RHS of `foo`, a wanted constraint `[a] ~ [b]` arises
> (because `x` has type `[a]` and is used in a place where `[b]` is
> expected). Since `a` and `b` are universally quantified variables, this
> cannot be solved by unification. However, the local assumption (given
> constraint) `a ~ b` can be used to solve this wanted.
>
> This is rather like automatically inferring where `typeCast` needs to be
> placed (indeed, at the Core level there is a construct for casting by an
> equality proof, which plays much the same role).
>
> Hope this helps,
>
> Adam
>
>
> On 06/12/2018 12:01, Anthony Clayden wrote:
> > The (~) constraint syntax is enabled by either `GADTs` or `TypeFamilies`
> > language extension.
> >
> > GADTs/Assoc Data Types/Assoc Type Synonyms/Type Families arrived in a
> > series of papers 2005 to 2008, and IIRC the development wasn't finished
> > in full in GHC until after that. (Superclass constraints took up to
> > about 2010.)
> >
> > Suppose I wanted just the (~) parts of those implementations. Which
> > would be the best place to look? (The Users Guide doesn't give a
> > reference.) ICFP 2008 'Type Checking with Open Type Functions' shows
> > uses of (~) in user-written code, but doesn't explain it or motivate it
> > as a separate feature.
> >
> > My specific question is: long before (~), it was possible to write a
> > TypeCast class, with bidirectional FunDeps to improve each type
> > parameter from the other. But for the compiler to see the type
> > improvement at term level, typically you also need a typeCast method and
> > to explicitly wrap a term in it. If you just wrote a term of type `a` in
> > a place where `b` was wanted, the compiler would either fail to see your
> > `TypeCast a b`, or unify the two too eagerly, then infer an overall type
> > that wasn't general enough.
> >
> > (For that reason, the instance for TypeCast goes through some
> > devious/indirect other classes/instances or exploits separate
> > compilation, to hide what's going on from the compiler's enthusiasm.)
> >
> > But (~) does some sort of magic: it's a kinda typeclass but without a
> > method. How does it mesh with type improvement in the goldilocks zone:
> > neither too eager nor too lazy?
> >
> >
> > AntC
>
>
>
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