[Agda] Telescope syntax

[Liam O'Connor]

For the record, NuPRL’s approach is summarised here:

http://www.cs.cornell.edu/home/sfa/Nuprl/NuprlPrimitives/Xuniverse_doc.html

http://www.cs.cornell.edu/home/sfa/Nuprl/NuprlPrimitives/Xtype_functionality_sequents_doc.html

From what I can tell, NuPRLs approach is much the same as Agda, with universe polymorphism and a hierarchy of universes, although it distinguishes between Type and Prop, and from what I can tell it has cumulativity, unlike Agda.

Jon, did you mean “why does Agda not have cumulativity?”



-- 
Liam O'Connor
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On 27 November 2014 at 3:43:06 am, Jon Sterling (jon at jonmsterling.com) wrote:

Perhaps this is the wrong place to ask, but I am wondering if there is something I can read about why Agda does the universe hierarchy, which even with syntactic sugar, is a little bit tiring to use—as opposed to the approach used in Nuprl, and soon (I think), in Coq? My assumption has always been that nobody had got around to doing it, which is fine since it is difficult to implement, but I wanted to be sure I wasn't missing more foundational.
 
Thanks very much,
Jon
 
 
On Wed, Nov 26, 2014, at 07:07 AM, Jesper Cockx wrote:
I think this would be a very nice thing to reduce the verbosity of using universe polymorphism. A few remarks:
 
- If I understand correctly, a telsyntax statement would always contain exactly one visible argument plus any number of hidden and instance arguments before and after? This looks very similar to the idea Andreas proposed in Tallinn for changing the internal representation of hidden function types. So your proposal could be seen as a concrete syntax for this new internal representation.
 
- Do you want (A B C : Type) to be translated to {i : Level} (A : Set i) {j : Level} (B : Set j) {k : Level} (C : Set k), or to {i j k : Level} (A : Set i) (B : Set j) (C : Set k)? I usually write the latter, though the former is more consistent with the idea of grouping hidden arguments with a visible argument.
 
- There is a way to make sense of having Type and Group as the return type of a function: "f : ... -> Type" just stands for "f : ... -> Set _", and "g : ... -> Group0" stands for "g : ... -> Set", but using g also brings a term of type "GroupStr (g ...)" into scope for instance resolution. Then you could make (A : Set) a synonym for {i : Level} (A : Set i) instead of (A : Set0), so that functions that look non-level-polymorphic can actually be used at any level.
 
Cheers,
Jesper
 
On Wed, Nov 26, 2014 at 2:33 PM, Guillaume Brunerie <guillaume.brunerie at gmail.com> wrote:
Hello all,

As is well known, it’s currently a bit annoying to use universe
polymorphism in Agda because instead of writing

  f : (A B C : Type) -> …

you have to write

  f : {i j k : Level} (A : Set i) (B : Set j) (C : Set k) -> …

Would it be a good idea to make the first one a syntactic sugar for
the second one?
I’m thinking about adding a "telsyntax" keyword, such that you can
write for instance

  telsyntax {i : Level} (X : Set i) = (X : Type)

and then (A B C : Type) (in a telescope) would be automatically translated into

  {i : Level} (A : Set i) {j : Level} (B : Set j) {k : Level} (C : Set k)

And this is not just about universe management, but it would also be
very useful when using instance arguments. For instance if a group is
a type (of level 0, say) with a group structure (which will be found
by instance search), and you want to write a function taking three
groups as arguments you have to write

  g : (G H K : Set) {{_ : GroupStr G}} {{_ : GroupStr H}} {{_ :
GroupStr K}} -> …

But you could say instead

  telsyntax (G : Set) {{_ : GroupStr G}} = (G : Group0)
  g : (G H K : Group0) -> …

And of course you can combine the two, if groups can be at any
universe level then the following:

  telsyntax {i : Level} (G : Set i) {{_ : GroupStr G}} = (G : Group)
  g : (G H K : Group) -> …

would be a shorthand for

  g : {i j k : Level} {G : Set i} {H : Set j} {K : Set k} {{_ :
GroupStr G}} {{_ : GroupStr H}} {{_ : GroupStr K}} -> …

which is much less readable and annoying to write.

One drawback (in the case of universe levels) is that you don’t have
access to the level anymore, but I don’t think that would really be a
problem, and you still can make the levels explicit if you need to.
Another drawback is that when writing (A : Type) or (G : Group) in a
telescope, it makes it look like Type and Group are types, but it’s
not the case so it could be confusing (for instance you can’t end a
function with "-> Group"). If that’s indeed too confusing, maybe we
could use a different notation than a colon, to make it clear that
it’s just syntactic sugar (on the other hand, it looks nice with a
colon).

What do you think?

Guillaume
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