I did some digging and found the first commit with an attempt to compu= te the number written by Guillaume Brunerie, Thierry Coquand and Simon Hube= r already in December 2014:

https://github.com= /simhu/cubical/commit/6e6278c6a626a9034789ab11cdd6bfb0bc8550be
This code is written for the predecessor of cubicaltt called "= cubical" and Thierry reminded me that it was based on the buggy regularity = evaluator which we eventually fixed in cubicaltt. I also think this cubical= code is what became Appendix B in Guillaume's thesis (or maybe Guillaume a= lready had a draft at the time, I don't remember). Anyway, Guillaume gave a= nice talk in 2017 with an overview of the attempts up to then and the prob= lems we had encountered:

https://guillaumebrun= erie.github.io/pdf/cubicalexperiments.pdf

This was= before Cubical Agda was invented and there are some tricks in Cubical Agda= that might make a difference compared to cubicaltt (in particular the "ghc= omp" trick which I learned about from Angiuli-Favonia-Harper and which elim= inates empty systems in hcomps). Now that we have a computation that actual= ly terminates I'm looking forward to seeing if any of these tricks actually= were necessary or if it was "just" a matter of simplifying the definition = of the number.

Best,
Anders

On = Monday, May 23, 2022 at 11:00:17 PM UTC+2 Anders M=C3=B6rtberg wrote:
Thanks Nicolai! And yes, our =CE=B2' is a different definition of the or= der of pi_4(S^3). In fact, the number =CE=B2 in the Summary file is not exa= ctly the same number as in Guillaume's Appendix B either for various reason= s. For instance, Guillaume only uses 1-HITs while we are quite liberal in u= sing higher HITs as they are not much harder to work with in Cubical Agda t= han 1-HITs. Also Guillaume of course defines everything with path-induction= while we use cubical primitives and the maps in appendix B are quite unnec= essarily complex from a cubical point of view (for instance, the equivalenc= e S^3 =3D S^1 * S^1 can be written quite directly in Cubical Agda while in = Book HoTT it's a bit more involved and Guillaume uses a chain of equivalenc= es to define it).

One could of course define t= he number exactly like Guillaume does and try to compute it, but I don't fi= nd that very interesting now that we have a much simpler definition which i= s fast to compute. However, we have come up with various other interesting= numbers that we can't get Cubical Agda to compute, so there's definitely r= oom to make cubical evaluation faster. Surprisingly enough though, one does= n't need to do this in order to get Cubical Agda to compute the order of pi= _4(S^3)   :-)

--
Anders

On Mon, May 23, 2022 at 10:23 PM Nicolai Kraus <nicola...@gmail.com> wrote:<= br>
Congratulations! It's great that this number finally computes in practice = and not just in theory, after all these years. :-)
And it's impressive = how short the new proof is! But this still doesn't mean that Cubical Agda p= asses the test that Guillaume formulates in Appendix B of his thesis, right= ? Because this test refers to the Brunerie number =CE=B2 (in the Summary.ag= da file you linked), and not to =CE=B2'.
In any case, that's a fa= ntastic result!
Best,
Nicolai

On Mon, May 23, 2022 at 8:30 PM Anders Mortberg <andersm...@gmail.com= > wrote:
We're very happy to announce that we have finally managed= to=20 compute the Brunerie number using Cubical Agda... and the result is -2!

The computation was made possible by a new direct synthetic proof that=20 pi_4(S^3) =3D Z/2Z by Axel Ljungstr=C3=B6m. This new proof involves a seri= es=20 of new Brunerie=20 numbers (i.e. numbers n : Z such that pi_4(S^3) =3D Z/nZ) and we got the= =20 one called =CE=B2' in the file above to reduce to -2 in just a= few seconds. With some work we then managed to prove that pi_4(S^3) =3D = Z / =CE=B2' Z, leading to a proof by normalization of the numb= er as conjectured in Brunerie's thesis.

Axel's new proof is very direct and completely=20 avoids chapters 4-6 in Brunerie's thesis (so no cohomology theory!), but it relies on chapters 1-3 to define the number. It also does not rely=20 on any special features of cubical type theory=20 and should be possible to formalize also in systems based on Book HoTT.=20 For a proof sketch as well as the formalization of the new proof in just=20 ~700 lines (not counting what is needed from chapters 1-3) see:
<= div>
=
So to summarize we now have both a new direct HoTT proof, not relying on=20 cubical computations, as well as a cubical proof by computation.
<= div>
Univalent regards,
A= nders and Axel

PS: the minus sign is a= ctually not very significant and we can get +2 by slightly modifying <= span>=CE=B2', but it's quite funny that we ended up getting -2 when = we finally got a definition which terminates!

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