[Haskell-cafe] Reverse unification question

[Ryan Ingram]

there's no y.z that fulfills that requirement.  Lets rewrite in a
System-F-style language with data types:

Given

read (/\) as forall, \\ as "big lambda", and @ as "type-level
application" which eliminates big-lambda.

data Category (~>) = CategoryDict
    { id :: (/\a. a ~> a)
    , (.) :: (/\a b c. (b ~> c) -> (a ~> b) -> (a ~> c)
    }

-- this gives
(.) :: /\((~>) :: * -> * -> *).
       Category (~>) ->
           /\a b c.
           (b ~> c) -> (a ~> b) -> (a ~> c)

Givens:
  C :: * -> * -> *
  A :: *
  dictC :: Category C
  y :: (/\r. C r (A -> r))
  x :: unknown

We want to find the type of x0 such that
   \\r. (.) C dictC r (A -> r) r x0 (y @r) :: /\r. C r r

given r :: *
   (.) C dictC r (A -> r) r :: (C (A -> r) r) -> (C r (A -> r)) -> C r r

So clearly x0 has type (C (A -> r) r)
However, our input is parametric in r, which is mentioned in x0's
type, so we need to pass that parameter in.  Therefore, we end up
with:

(x :: /\r . C (A -> r) r)

Similar logic will show you that there is no z such that y.z :: /\r. C
r r exists, since y.z must have type (C a (A -> r)) for some type a.

  -- ryan


On Mon, Aug 2, 2010 at 11:44 AM, Martijn van Steenbergen
<martijn at van.steenbergen.nl> wrote:
> Dear café,
>
> Given:
>>
>> instance Category C
>> y :: forall r. C r (A -> r)
>
> I am looking for the types of x and z such that:
>>
>> x . y :: forall r. C r r
>> y . z :: forall r. C r r
>
> Can you help me find such types? I suspect only one of them exists.
>
> Less importantly, at least to me at this moment: how do I solve problems
> like these in general?
>
> Thank you,
>
> Martijn.
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