Hi Jasper,

Indeed, the problem is to construct an algorithm for comp (or even
coe/transp) for Glue that reduces to the one of A when phi is true
while still preserving regularity. It was pointed out independently by
Sattler and Orton around 2016 that one can factor out this step in our
algorithm in a separate lemma that is now called "alignment". This is
Thm 6.13 in Orton-Pitts and discussed in a paragraph in the end of
section 2.11 of ABCFHL. Unless I'm misremembering this is exactly
where regularity for comp for Glue types break down. In this step we
do an additional comp/hcomp that inserts an additional forall i. phi
face making the comp/coe irregular.

One could imagine there being a way to modify the algorithm to avoid
this, maybe by inlining the alignment step... But despite considerable
efforts no one has been able to figure this out and I think Swan's
recent paper (https://arxiv.org/abs/1808.00920v3) shows that this is
not even possible!

Another approach would be to have weak Glue types that don't strictly
reduce to A when phi is true, but this causes problems for the
composition in the universe and would be weird for cubical type
theory...

In light of Swan's negative results I think we need a completely new
approach if we ever hope to solve this problem. Luckily for you Andrew
Swan is starting as a postdoc over in Baker Hall in October, so he can
explain his counterexamples to you in person.

Best,
Anders

On Sun, Sep 15, 2019 at 7:57 AM Jasper Hugunin
<jasperh@cs.washington.edu> wrote:
>
> Offline, Carlo Angiuli showed me that the difficulty was in part 1, because of a subtlety I had been forgetting.
>
> Since types are *Kan* cubical sets, we need that the Kan operations agree as well as the sets.
> So part 1 could be thought of as (Glue [ phi |-> equivRefl A ] A, compGlue) = (A, compA), and getting that the Kan operations to agree was/is difficult.
> (Now that I know what the answer is, it is clear that this was already explained in the initial discussion.)
>
> Humbly,
> - Jasper Hugunin
>
> On Fri, Sep 13, 2019 at 2:10 AM Jasper Hugunin <jasperh@cs.washington.edu> wrote:
>>
>> Hello all,
>>
>> I've been trying to understand better why composition for the universe does not satisfy regularity.
>> Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |-> equiv^i E ] A, I would expect regularity to follow from two parts:
>> 1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of regularity condition for the Glue type constructor itself)
>> 2. That equiv^i (refl A) reduces to equivRefl A
>> I'm curious as to which (or both) of these parts was the issue, or if regularity for the universe was supposed to follow from a different argument.
>>
>> Context:
>> I've been studying and using CCHM cubical type theory recently, and often finding myself wishing that J computed strictly.
>> If I understand correctly, early implementations of ctt did have strict J for Path types, and this was justified by a "regularity" condition on the composition operation, but as discussed in this thread on the HoTT mailing list, the definition of composition for the universe was found to not satisfy regularity.
>> I don't remember seeing the regularity condition defined anywhere, but my understanding is that it requires that composition in a degenerate line of types, with the system of constraints giving the sides of the box also degenerate in that direction, reduces to just the bottom of the box. This seems to be closed under the usual type formers, plus Glue, but not the universe with computation defined as in the CCHM paper (for trivial reasons and non-trivial reasons; it gets stuck at the start with Glue [ phi |-> equiv^i refl ] A not reducing to anything).
>>
>> Best regards,
>> - Jasper Hugunin
>
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