On Sat, Aug 10, 2019 at 5:25 AM Valery Isaev <valery.isaev@gmail.com> wrote:
> The document is slightly outdated. We do not have the rule iso A B (λx ⇒ x) (λx ⇒ x) idp idp i ⇒β A in the actual implementation since univalence is true even without it. This rule has another problem. It seems that the theory as presented in the document introduces a quasi-equivalence between A = B and Equiv A B, which means that there are some true statements which are not provable in it.
I don't understand. By "quasi-equivalence" do you mean an incoherent
equivalence (what the book calls a map with a quasi-inverse)? If so,
then every quasi-equivalence can of course be promoted to a strong
equivalence.
Yes, a quasi-equivalence is a function together with its quasi-inverse. The problem is that we've got some terms in the theory of which we know nothing about. It's the same as if I just add a new type Magic without any additional rules. Then we cannot prove anything about it and the resulting theory won't be equivalent to the original one.
However, as I said, I'm more worried about the fourth rule coe_{λ k ⇒
iso A B f g p q k} a right ⇒β f a. That's the one that I have trouble
seeing how to interpret in a model category. Can you say anything
about that?
I don't remember well, but I think the idea is that you need to prove that there is a trivial cofibration Eq(A,B) -> F(U^I,A,B), where the first object is the object of equivalences between A and B and the second object is the fiber of U^I over A and B. The fact that this map is a weak equivalence is just the univalence axiom. The problem is to show that it is a cofibration and whether this is true or not depends on the definition of Eq(A,B). I don't actually remember whether I finished this proof.
> If you can prove that some \data or \record satisfies isSet (or, more generally, that it is an n-type), then you can put this proof in \use \level function corresponding to this definition and it will be put in the corresponding universe.
What does it mean for it to be "put in" the corresponding universe?
I mean F(A,p) will have type \Set0 instead of \Type0 that it would have without the \use \level annotation.
The documentation for \use \level makes it sound as though the
definition *itself*, rather than something equivalent to it, ends up
in the corresponding universe.
Yes, F(A,p) itself has type \Set0, but A still has type \Type0.
How is the equivalence between A and
F(A,p) accessed inside the proof assistant?
Since F(A,p) is the usual (inductive) data type, you can do everything you can do with other data types. In particular, since it has only one constructor with one parameter A, it is easy to proof that it is equivalent to A.