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* [HoTT] The Brunerie number is -2
@ 2022-05-23 19:30 Anders Mortberg
2022-05-23 19:38 ` Steve Awodey
2022-05-23 20:22 ` Nicolai Kraus
0 siblings, 2 replies; 6+ messages in thread
From: Anders Mortberg @ 2022-05-23 19:30 UTC (permalink / raw)
To: Homotopy Type Theory; +Cc: Axel Ljungström

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We're very happy to announce that we have finally managed to compute the
Brunerie number using Cubical Agda... and the result is -2!

https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda#L129

The computation was made possible by a new direct synthetic proof that
pi_4(S^3) = Z/2Z by Axel Ljungström. This new proof involves a series of
new Brunerie numbers (i.e. numbers n : Z such that pi_4(S^3) = Z/nZ) and we
got the one called β' 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) = Z / β' Z,
leading to a proof by normalization of the number as conjectured in
Brunerie's thesis.

Axel's new proof is very direct and completely 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 on any special features of
cubical type theory and should be possible to formalize also in systems
based on Book HoTT. For a proof sketch as well as the formalization of the
new proof in just ~700 lines (not counting what is needed from chapters
1-3) see:

https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/DirectProof.agda

So to summarize we now have both a new direct HoTT proof, not relying on
cubical computations, as well as a cubical proof by computation.

Univalent regards,
Anders and Axel

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

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* Re: [HoTT] The Brunerie number is -2
2022-05-23 19:30 [HoTT] The Brunerie number is -2 Anders Mortberg
@ 2022-05-23 19:38 ` Steve Awodey
2022-05-23 20:22 ` Nicolai Kraus
1 sibling, 0 replies; 6+ messages in thread
From: Steve Awodey @ 2022-05-23 19:38 UTC (permalink / raw)
To: Anders Mortberg; +Cc: Homotopy Type Theory, Axel Ljungström

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Congratulations to Team A^2!
Great work - and a real milestone.
Best wishes,
Steve

> On May 23, 2022, at 21:30, Anders Mortberg <andersmortberg@gmail.com> wrote:
>
> ﻿
> We're very happy to announce that we have finally managed to compute the Brunerie number using Cubical Agda... and the result is -2!
>
> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda#L129
>
> The computation was made possible by a new direct synthetic proof that pi_4(S^3) = Z/2Z by Axel Ljungström. This new proof involves a series of new Brunerie numbers (i.e. numbers n : Z such that pi_4(S^3) = Z/nZ) and we got the one called β' 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) = Z / β' Z, leading to a proof by normalization of the number as conjectured in Brunerie's thesis.
>
> Axel's new proof is very direct and completely 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 on any special features of cubical type theory and should be possible to formalize also in systems based on Book HoTT. For a proof sketch as well as the formalization of the new proof in just ~700 lines (not counting what is needed from chapters 1-3) see:
>
> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/DirectProof.agda
>
> So to summarize we now have both a new direct HoTT proof, not relying on cubical computations, as well as a cubical proof by computation.
>
> Univalent regards,
> Anders and Axel
>
> PS: the minus sign is actually not very significant and we can get +2 by slightly modifying β', but it's quite funny that we ended up getting -2 when we finally got a definition which terminates!
> --
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
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* Re: [HoTT] The Brunerie number is -2
2022-05-23 19:30 [HoTT] The Brunerie number is -2 Anders Mortberg
2022-05-23 19:38 ` Steve Awodey
@ 2022-05-23 20:22 ` Nicolai Kraus
2022-05-23 20:59   ` Anders Mortberg
From: Nicolai Kraus @ 2022-05-23 20:22 UTC (permalink / raw)
To: Anders Mortberg; +Cc: Homotopy Type Theory, Axel Ljungström

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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 passes the test that Guillaume formulates in Appendix B
of his thesis, right? Because this test refers to the Brunerie number β (in
the Summary.agda file you linked), and not to β'.
In any case, that's a fantastic result!
Best,
Nicolai

On Mon, May 23, 2022 at 8:30 PM Anders Mortberg <andersmortberg@gmail.com>
wrote:

> We're very happy to announce that we have finally managed to compute the
> Brunerie number using Cubical Agda... and the result is -2!
>
>
> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda#L129
>
> The computation was made possible by a new direct synthetic proof that
> pi_4(S^3) = Z/2Z by Axel Ljungström. This new proof involves a series of
> new Brunerie numbers (i.e. numbers n : Z such that pi_4(S^3) = Z/nZ) and we
> got the one called β' 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) = Z / β' Z,
> leading to a proof by normalization of the number as conjectured in
> Brunerie's thesis.
>
> Axel's new proof is very direct and completely 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 on any special features of
> cubical type theory and should be possible to formalize also in systems
> based on Book HoTT. For a proof sketch as well as the formalization of the
> new proof in just ~700 lines (not counting what is needed from chapters
> 1-3) see:
>
>
> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/DirectProof.agda
>
> So to summarize we now have both a new direct HoTT proof, not relying on
> cubical computations, as well as a cubical proof by computation.
>
> Univalent regards,
> Anders and Axel
>
> PS: the minus sign is actually not very significant and we can get +2 by
> slightly modifying β', but it's quite funny that we ended up getting -2
> when we finally got a definition which terminates!
>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> To view this discussion on the web visit
> .
>

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* Re: [HoTT] The Brunerie number is -2
2022-05-23 20:22 ` Nicolai Kraus
@ 2022-05-23 20:59   ` Anders Mortberg
2022-05-24  9:46     ` Anders Mörtberg
From: Anders Mortberg @ 2022-05-23 20:59 UTC (permalink / raw)
To: Nicolai Kraus; +Cc: Homotopy Type Theory, Axel Ljungström

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Thanks Nicolai! And yes, our β' is a different definition of the order of
pi_4(S^3). In fact, the number β in the Summary file is not exactly the
same number as in Guillaume's Appendix B either for various reasons. For
instance, Guillaume only uses 1-HITs while we are quite liberal in using
higher HITs as they are not much harder to work with in Cubical Agda than
1-HITs. Also Guillaume of course defines everything with path-induction
while we use cubical primitives and the maps in appendix B are quite
unnecessarily complex from a cubical point of view (for instance, the
equivalence S^3 = 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
equivalences to define it).

One could of course define the number exactly like Guillaume does and try
to compute it, but I don't find that very interesting now that we have a
much simpler definition which is 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 room to make cubical evaluation faster.
Surprisingly enough though, one doesn'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 <nicolai.kraus@gmail.com>
wrote:

> 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 passes the test that Guillaume formulates in
> Appendix B of his thesis, right? Because this test refers to the Brunerie
> number β (in the Summary.agda file you linked), and not to β'.
> In any case, that's a fantastic result!
> Best,
> Nicolai
>
>
>
>
>
>
>
> On Mon, May 23, 2022 at 8:30 PM Anders Mortberg <andersmortberg@gmail.com>
> wrote:
>
>> We're very happy to announce that we have finally managed to compute the
>> Brunerie number using Cubical Agda... and the result is -2!
>>
>>
>> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda#L129
>>
>> The computation was made possible by a new direct synthetic proof that
>> pi_4(S^3) = Z/2Z by Axel Ljungström. This new proof involves a series of
>> new Brunerie numbers (i.e. numbers n : Z such that pi_4(S^3) = Z/nZ) and we
>> got the one called β' 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) = Z / β' Z,
>> leading to a proof by normalization of the number as conjectured in
>> Brunerie's thesis.
>>
>> Axel's new proof is very direct and completely 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 on any special features of
>> cubical type theory and should be possible to formalize also in systems
>> based on Book HoTT. For a proof sketch as well as the formalization of the
>> new proof in just ~700 lines (not counting what is needed from chapters
>> 1-3) see:
>>
>>
>> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/DirectProof.agda
>>
>> So to summarize we now have both a new direct HoTT proof, not relying on
>> cubical computations, as well as a cubical proof by computation.
>>
>> Univalent regards,
>> Anders and Axel
>>
>> PS: the minus sign is actually not very significant and we can get +2 by
>> slightly modifying β', but it's quite funny that we ended up getting -2
>> when we finally got a definition which terminates!
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "Homotopy Type Theory" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> To view this discussion on the web visit
>> .
>>
>

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* Re: [HoTT] The Brunerie number is -2
2022-05-23 20:59   ` Anders Mortberg
@ 2022-05-24  9:46     ` Anders Mörtberg
2022-05-24  9:49       ` Anders Mörtberg
From: Anders Mörtberg @ 2022-05-24  9:46 UTC (permalink / raw)
To: Homotopy Type Theory

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I did some digging and found the first commit with an attempt to compute
the number written by Guillaume Brunerie, Thierry Coquand and Simon Huber

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 already
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 problems

https://guillaumebrunerie.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 "ghcomp" trick which I learned about from
Angiuli-Favonia-Harper and which eliminates empty systems in hcomps). Now
that we have a computation that actually 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 Mörtberg wrote:

> Thanks Nicolai! And yes, our β' is a different definition of the order of
> pi_4(S^3). In fact, the number β in the Summary file is not exactly the
> same number as in Guillaume's Appendix B either for various reasons. For
> instance, Guillaume only uses 1-HITs while we are quite liberal in using
> higher HITs as they are not much harder to work with in Cubical Agda than
> 1-HITs. Also Guillaume of course defines everything with path-induction
> while we use cubical primitives and the maps in appendix B are quite
> unnecessarily complex from a cubical point of view (for instance, the
> equivalence S^3 = 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
> equivalences to define it).
>
> One could of course define the number exactly like Guillaume does and try
> to compute it, but I don't find that very interesting now that we have a
> much simpler definition which is 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 room to make cubical evaluation faster.
> Surprisingly enough though, one doesn'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:
>
>> 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 passes the test that Guillaume formulates in
>> Appendix B of his thesis, right? Because this test refers to the Brunerie
>> number β (in the Summary.agda file you linked), and not to β'.
>> In any case, that's a fantastic 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 compute the
>>> Brunerie number using Cubical Agda... and the result is -2!
>>>
>>>
>>> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda#L129
>>>
>>> The computation was made possible by a new direct synthetic proof that
>>> pi_4(S^3) = Z/2Z by Axel Ljungström. This new proof involves a series of
>>> new Brunerie numbers (i.e. numbers n : Z such that pi_4(S^3) = Z/nZ) and we
>>> got the one called β' 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) = Z / β'
>>> Z, leading to a proof by normalization of the number as conjectured in
>>> Brunerie's thesis.
>>>
>>> Axel's new proof is very direct and completely 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 on any special features of
>>> cubical type theory and should be possible to formalize also in systems
>>> based on Book HoTT. For a proof sketch as well as the formalization of the
>>> new proof in just ~700 lines (not counting what is needed from chapters
>>> 1-3) see:
>>>
>>>
>>> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/DirectProof.agda
>>>
>>> So to summarize we now have both a new direct HoTT proof, not relying on
>>> cubical computations, as well as a cubical proof by computation.
>>>
>>> Univalent regards,
>>> Anders and Axel
>>>
>>> PS: the minus sign is actually not very significant and we can get +2 by
>>> slightly modifying β', but it's quite funny that we ended up getting -2
>>> when we finally got a definition which terminates!
>>>
>>> --
>>> You received this message because you are subscribed to the Google
>>> Groups "Homotopy Type Theory" group.
>>> To unsubscribe from this group and stop receiving emails from it, send
>>> To view this discussion on the web visit
>>> .
>>>
>>

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* Re: [HoTT] The Brunerie number is -2
2022-05-24  9:46     ` Anders Mörtberg
@ 2022-05-24  9:49       ` Anders Mörtberg
0 siblings, 0 replies; 6+ messages in thread
From: Anders Mörtberg @ 2022-05-24  9:49 UTC (permalink / raw)
To: Homotopy Type Theory

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>
>
> This was before Cubical Agda was invented
>
>
Oops, I meant to say that this was around the time Andrea Vezzosi was
implementing Cubical Agda (which Guillaume mentions in the slides).

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