In CCHM, my understanding is that composition for Glue [ phi |-> (T, f) ] A is actually regular, and reduces to composition in T on (forall i. phi), assuming composition is regular for A and T.
The regularity does seem to fail in ABCFHL, as you pointed out Anders, but is perhaps repairable.
What Carlo told me was that the problem is getting composition for Glue [ phi |-> (T, f) ] A which is regular, reduces to composition in T on (forall i. phi), *and* reduces to composition in A when (T, f) = (A, equivRefl).
We get the first two but not the third in CCHM.
The problem as Carlo explained it was that given a square of types i, j |- A and a point a : A (i0) (j0), we can prove Path (coe^i A o coe^j A) (coe^j A o coe^i A) (coercing around the square in right then up vs up then right), and (assuming regularity for A), we can take this to be degenerate when A is degenerate in i, or give a different path which is degenerate when A is degenerate in j, but constructing a path which is degenerate when A is degenerate in i or j is elusive.
I have an idea for how to add new terms/operations (a sort of generalization of comp) such that such a path is constructible, but I haven't yet checked that there is a univalent universe with these more general operations.
I also don't understand the negative results from Andrew well enough to tell if they rule out such an approach.
The idea is to add terms witnessing that for every square x, y |- A x y we have Path (coe^i (A i i)) ((coe^i (A i 0)) o (coe^i (A 1 i))) which is degenerate when A is degenerate in x OR y, plus similar terms for higher cubes, since once you need dimensions one and two you probably need dimension n, and a similar thing for hcomp, or you can do coe and hcomp together in just comp.
Composing two of these paths then gives the needed proof that coercing around the square either way is equal, in a degenerate fashion if A is degenerate in x or y.
Best regards,
- Jasper Hugunin