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[Jason Gross <jasongross9 AT gmail.com>]

Thanks, but that's not quite what I'm looking for, unless I'm mis-understanding it; I'm looking for some way to reduce the size of my proof goals by eliminating unnecessary arguments (rather than reduce how much I have to type to define them, which seems to be what [Generalizable All Variables] does).  For example, it would be nice if I could not have to say that a natural transformation was of type

  NaturalTransformation : forall objC homC (C : @Category objC homC) objD homD (D : @Category objD homD), @Functor objC homC C objD homD D -> @Functor objC homC C objD homD D -> Type

and instead say that a NaturalTransformation was of type

  NaturalTransformation : forall (C : @Category _ _) (D : @Category _ _), @Functor C D -> @Functor C D -> Type

I realize that I can do 
  NaturalTransformation : forall `(F G : @Functor objC homC C objD homD D), Type

and while this is useful, it does not reduce the size of my proof goals (because it simply de-sugars to the first form that I typed).

I guess I'm looking for something like telling Coq that the internal representation of [NaturalTransformation] should be something like [forall `(F G : @Functor objC homC C objD homD D), Type], but that Coq shouldn't carry around the types of all the arguments to [Functor], or something like that.

-Jason

On Fri, Aug 3, 2012 at 2:55 AM, Arnaud Spiwack <aspiwack AT lix.polytechnique.fr> wrote:

Not exactly what you want. But close enough:

http://coq.inria.fr/doc/Reference-Manual004.html#@command88

On 2 August 2012 22:40, Jason Gross <jasongross9 AT gmail.com> wrote:

Hi,

I'm trying to encode some category theory in Coq, and have run in to problems with comma categories; I'm passing around objects and hom sets as arguments to everything, because I need to be able to deal with functors from the category of functors between two categories, and for this I'm making using of sort-polymorphism.  However, my sort-polymorphic comma category construction routinely gives me goals of ~300,000 terms (thank goodness for the ability to hide implicit arguments), and operating on these goals takes a long time.

Is there any support (either in the literature, in theory, in the works for Coq, or elsewhere) for talking about "partially applied types" without talking about their arguments, and having these types carry around their universe level?

For example, say I have

  Parameter Category : forall (obj : Type) (hom : obj -> obj -> Type), Type.

currently, I could have

  Parameter Functor : forall objA homA objB homB, @Category objA homA -> @Category objB homB.

I'd like to instead be able to do something like

  Parameter Functor : @Cateogry _ _ -> @Category _ _.

so that the [Functor] does not carry around the information of objects and hom-sets with it.  I can do this by saying something like

  Record Category' := { obj : Type; hom : obj -> obj -> Type; Cat : @Category obj hom }.

but this [Category'] is not universe polymorphic, and so any [Functor]s that I build with it all live at the same universe-level.

Thanks.

-Jason