```
From: Kenji Maillard <chocobodralliam@gmail.com>
To: HomotopyTypeTheory@googlegroups.com
Subject: Re: [HoTT] Injective types
Date: Fri, 3 May 2019 15:25:49 +0200
Message-ID: <5782d183-f880-9a85-78bf-49c86687473f@gmail.com> (raw)
In-Reply-To: <CAOvivQy0A3mUJ8904+PjwAzTpmx1J1A3Le9jmdjJWG7ZSvz3FA@mail.gmail.com>
I think the paper "A calculus of constructions with explicit subtyping"
by Ali Assaf that can be found at https://hal.inria.fr/hal-01097401 is
a relevant reference for coercive lifts between Tarski style universes.
K.
On 03/05/2019 13:45, Michael Shulman wrote:
> I am more optimistic: I think there's a good chance we could do better
> than both Coq and Agda. Of course sometimes when we try new things
> they flop; but we can't make progress without trying new things.
>
> For one thing, I don't think we should conflate typical ambiguity with
> cumulativity. It just so happens that Coq has both and Agda has
> neither, but in principle I see no reason they have to go together. I
> see typical ambiguity as basically syntactic sugar or abuse of
> notation, analogous to the use of implicit arguments: the reader or
> proof assistant is tasked (if they feel like it) to go through and
> insert level parameters as needed, accumulating constraints on these
> parameters according to how the instance are used and thereby
> "elaborating" a typically ambiguous development to a fully precise one
> with (polymorphic) universe levels. Usually this will be possible,
> but occasionally if the writer was careless there may be a universe
> inconsistency.
>
> It seems to me that this could be done in both a cumulative and a
> non-cumulative system. True, in a non-cumulative system we get
> different constraints, e.g. if we ever write "A=B" then it must be
> that A and B live in the same universe, whereas in a cumulative system
> we could be looser about such constraints. But your evidence (and
> that of other universe-polymorphic users of Agda) suggests that such
> constraints arising from non-cumulativity are not in practice a
> problem. In fact, the unique assignment of levels in a non-cumulative
> system suggests that the universe inference algorithm in a
> hypothetical typically-ambiguous non-cumulative proof assistant would
> probably be *simpler*, and less error-prone, than that of Coq. So I
> don't see any argument here against typical ambiguity, as long as
> there is the *option* to be unambiguous when necessary (which again,
> even Coq now supports).
>
> In particular, note that when a development is formalized in a
> typically ambiguous proof assistant, it's not necessary for the
> universe levels to be written in the source code, or even thought
> about by the author, in order for the interested reader -- or even the
> author themselves! -- to learn about what the universe constraints
> are. They only have to compile the code, in particular running it
> through the universe checker/elaborator, and then inspect the
> resulting universe levels/constraints. I've done this in present-day
> Coq myself, although the proliferation of universe parameters makes
> the output hard to undertsand; I expect it would only be easier in a
> hypothetical typically-ambiguous non-cumulative proof assistant. So
> it seems to me that it should be possible to be "fair to the reader",
> as you say, and still retain (some of) the advantages of typical
> ambiguity.
>
> I also think there's a good chance we can retain some cumulativity
> without losing the benefits of non-cumulativity, by using a
> Tarski-style lifting coercion as I sketched in my last email. (Isn't
> this in the literature somewhere? I didn't think I'd made it up.) I
> agree that it's rare to need this, but neither is it unheard-of; so if
> we can make it more convenient to use with no drawback, why not? (Of
> course, cumulativity is also trickier to model semantically, but
> probably possible.)
>
>
>
>
>
> On Thu, May 2, 2019 at 1:46 PM <escardo.martin@gmail.com> wrote:
>>
>>
>> On Wednesday, 1 May 2019 17:55:50 UTC+1, Michael Shulman wrote
>>>
>>> Yes, this is a good point in favor of Agda-style non-cumulative
>>> Russell universes over Coq-style cumulative Russell universes.
>>>
>>> But isn't there a middle ground with Tarski universes?
>>
>> It would be nice to have such a middle ground, particularly because formulating universe assumptions in each single definition and theorem is unfamiliar in mathematical practice, and so "typical ambiguity" (pretending there is only one universe) is potentially a good idea for many (or even most) examples. But not in the paper I advertised in this thread.
>>
>> Here I post an example when Giraud did precisely that, namely to assume two arbitrary universes U and V, explaining why this is needed in that level of generality after the formulation of a theorem and its proof, given to me by Thierry Coquand:
>>
>> https://www.cs.bham.ac.uk/~mhe/giraud-universes.pdf (photo of one page of a book).
>>
>> The book is “Cohmologie non abelienne” (1971, https://www.springer.com/gp/book/9783540053071).
>>
>>> Suppose we
>>> have explicit lifting operators Lift : U_i -> U_{i+1}, so that as in
>>> Agda we have unique small polymorphic level assignments. But then
>>> suppose we have *definitional* equalities El(Lift(A)) == El(A). Then
>>> on the (rare) occasions when we do have to explicitly lift types from
>>> one universe to another,
>>
>> I can confirm from a 26k line Agda development (with comments and repeated blank lines removed in this counting of the number of lines) that not once did I need to embed a universe into a larger universe, except when I wanted to state the theorem that any universe is a retract of any larger universe if one assumes the propositional resizing axiom (any proposition in a universe U has an equivalent copy in any universe V). So I would say that such situations are *extremely rare* in practice.
>>
>>> we would get stronger cumulativity behavior.
>>> (And I could imagine a proof assistant that implements this and sugars
>>> away the El to look like Russell universes to the user.)
>>
>> At the moment we can choose cumulativity (Coq) or non-cumulativity (Agda), and there is no system that combines the virtues of Coq and Agda regarding universe handling. (And I fear that such a system would potentially multiply the vices of both. :-) )
>>
>> M.
>>
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```

next prev parent reply indexThread overview:11+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-04-30 23:05 escardo.martin 2019-05-01 2:50 ` Michael Shulman 2019-05-01 6:25 ` escardo.martin 2019-05-01 16:55 ` Michael Shulman 2019-05-02 20:46 ` escardo.martin 2019-05-03 11:45 ` Michael Shulman2019-05-03 13:25 ` Kenji Maillard [this message]2019-05-03 18:23 ` Thierry Coquand 2019-05-07 12:42 ` Nils Anders Danielsson 2019-05-07 13:51 ` Andreas Nuyts 2019-05-07 22:06 ` Martín Hötzel Escardó

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