This doesn't seem right; as far as I can tell, composition for Glue =
types in CCHM preserves regularity and reduces to composition in A on phi.<=
div>

- Jasper Hugunin

On Sun, Sep 15, 2019 at 3:28 AM An=
ders Mortberg <anders.mortberg@math.su.se> wrote:

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". Thi= s 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 t= his is

not even possible!

Another approach would be to have weak Glue types that don't strictlyreduce 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, be= cause of a subtlety I had been forgetting.

>

> Since types are *Kan* cubical sets, we need that the Kan operations ag= ree as well as the sets.

> So part 1 could be thought of as (Glue [ phi |-> equivRefl A ] A, c= ompGlue) =3D (A, compA), and getting that the Kan operations to agree was/i= s 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&g= t; 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:<= br> >> 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 s= trict 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 f= ound to not satisfy regularity.

>> I don't remember seeing the regularity condition defined anywh= ere, but my understanding is that it requires that composition in a degener= ate line of types, with the system of constraints giving the sides of the b= ox 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 rea= sons 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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