Re: [HoTT] New theorem prover Arend is released

[Michael Shulman <shulman@sandiego.edu>]

On Sun, Aug 11, 2019 at 6:47 AM Valery Isaev <valery.isaev@gmail.com> wrote:
> 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.

Yes, in a straightforward approach that's what you would need to
prove, or more precisely that the map Eq -> U^I is a trivial
cofibration in the slice over UxU (since the lifting has to be done in
the universal case rather than just over each global element).
Possibly you could get away with less, since you're only asking to
recover the action of the given equivalence as a function (rather than
the entire equivalence data).  Perhaps a clever choice of fibration
structure in the equivalence extension property would suffice.

> 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.

Okay, it sounds like you're saying that the datatype defined with
\use\level is F(A,p) and the analogous one defined without \use\level
is A, while both have their usual rules but are not identical.  In
this case you have somewhat *more* than an equivalence between F(A,p)
and A, at least if all datatypes have their usual definitional
computation rules, since an arbitrary type "F(A,p)" that is only
*equivalent* to a datatype will only inherit a typal computation rule.
It seems plausible though that one could construct universes of
n-types that are strictly closed under a given datatype construction
that preserves n-types (at least, assuming one can do the same for
arbitrary universes).

However, the documentation also suggests that \use\level can be
applied to plain definitions (\func).  How is F(A,p) distinguished
from A in this case?

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