## Stream: sphere eversion

### Topic: Chapter 3 coordination

#### Patrick Massot (Mar 03 2022 at 14:00):

We are ready to start working on Chapter 3. This is the chapter where all the manifold theory is concentrated. On paper in the real world this chapter would be three lines long, saying the local proofs obviously globalize (I'm lying a bit because there is also half a page about getting parametricity for free). But mathlib lacks a lot of infrastructure about vector bundles.

#### Patrick Massot (Mar 03 2022 at 14:01):

Discussions with Sébastien and Heather gives the following first targets:

• topological vector bundle of continuous linear maps from one topological vector bundle to another.
• pull-back bundles
• induced has_groupoid structure on a topological vector bundle over a base which has_groupoid

#### Patrick Massot (Mar 03 2022 at 14:02):

The first two points are not there even for topological vector bundles over topological spaces. Then the third point is needed to get smooth structures everywhere.

#### Patrick Massot (Mar 03 2022 at 14:02):

Once these are done, we'll need to upgrade the tangent bundle to a smooth vector bundle, and then define $J^1(M, N)$ for smooth manifolds $M$ and $N$.

#### Patrick Massot (Mar 03 2022 at 14:06):

An independent goal is:

#### Patrick Massot (Mar 03 2022 at 14:07):

Anyone who wants to tackle one of those four accessible points is very welcome to say it here.

#### Heather Macbeth (Mar 03 2022 at 14:10):

Patrick Massot said:

Discussions with Sébastien and Heather gives the following first targets:

• topological vector bundle of continuous linear maps from one topological vector bundle to another.

I think this one, in particular, should develop quite similarly to the corresponding construction for the direct sum of two vector bundles, which I have nearly finished: branch#vb-direct-sum

#### Heather Macbeth (Mar 03 2022 at 14:11):

So that can serve as a model (or I myself can do that, after I finish the direct sum).

#### Patrick Massot (Mar 03 2022 at 14:13):

I have another independent target to add:

#### Heather Macbeth (Mar 03 2022 at 14:13):

Patrick Massot said:

• induced has_groupoid structure on a topological vector bundle over a base which has_groupoid

For this one, it's first necessarily to define the correct groupoid. Given a structure groupoid G over a model space H, you want to define the groupoid of local homeomorphisms of F \times H whose sources and targets contain only full fibres of F, and which are linear in each fibre, and which cover G.

#### Oliver Nash (Aug 10 2022 at 12:05):

Yesterday I returned to working on Sphere Eversion after a break and I would be grateful for help clarifying:

• Who is working on what
• What has been done so far, in particular I am not clear on the status of each the following:

I note that:
@Patrick Massot said:

Heather Macbeth already claimed Lemma 3.2

https://leanprover.zulipchat.com/#narrow/stream/303200-sphere-eversion/topic/Lemma.203.2E6/near/290324201

@Yury G. Kudryashov said:

I'm behind the schedule on Lemma 3.6 (waiting for CI on #15490 to merge it and PR the actual proof)

In the meantime, here is Lemma 3.10: #15580

Do we have all definitions for L 3.23?

@Floris van Doorn said:

I am now continuing with Definition 3.20 and Lemma 3.21

https://leanprover.zulipchat.com/#narrow/stream/303200-sphere-eversion/topic/Lemma.203.2E6/near/290357122

#### Oliver Nash (Aug 10 2022 at 12:09):

It seems to me that the Blueprint should be updated since:

#### Oliver Nash (Aug 10 2022 at 12:10):

Finally I would be very grateful if we could permanently fix the issue with the stale dependency graph. I swear the URL has changed more than once. We have both: bad link and good link.

#### Oliver Nash (Aug 10 2022 at 12:12):

cc also @Heather Macbeth

#### Floris van Doorn (Aug 10 2022 at 12:48):

I restarted to work on Lemma 3.21 again today. It's mostly done, I think.

#### Floris van Doorn (Aug 10 2022 at 12:49):

The dependency graph issue is the reason I'm not updating the blueprint myself anymore: my installation generates the dependency graph at the bad link...

#### Floris van Doorn (Aug 10 2022 at 12:51):

There is one more thing that needs to be fixed in the dependency graph: I removed the edge from one_jet_extension_prop to holonomic_section, but one_jet_extension_prop needs an edge to some other vertex.

#### Patrick Massot (Aug 10 2022 at 13:02):

I agree that clarification is much needed.
Oliver Nash said:

Patrick Massot said:

Heather Macbeth already claimed Lemma 3.2

https://leanprover.zulipchat.com/#narrow/stream/303200-sphere-eversion/topic/Lemma.203.2E6/near/290324201

I think Heather unclaimed this to work on constructing the family of formal immersions used in the actual sphere eversion case.

#### Patrick Massot (Aug 10 2022 at 13:06):

I'm confused about Lemma 3.6. I thought that Yury did it but I'm not 100% sure

#### Oliver Nash (Aug 10 2022 at 13:07):

IIUC he had a proof but may not have PR'd it as he waited for some other work to get merged in Mathlib.

#### Patrick Massot (Aug 10 2022 at 13:08):

Regarding the blueprint, I know it's currently a mess. In providence I tried to update it but couldn't build on my laptop. I'll try again (but I'm still on my laptop)

#### Oliver Nash (Aug 10 2022 at 13:09):

Thanks. If it helps, that revolting commutative diagram is now gone!

#### Oliver Nash (Aug 10 2022 at 13:10):

There's no emergency btw and it might not be the best use of your time to do it from your laptop. I just wanted a bit more clarity on where we stand.

#### Patrick Massot (Aug 10 2022 at 13:20):

I can explain the dep_graph url issue. The problem is that the blueprint in this project isn't built in CI, it is built locally and pushed to the repository. Hence anyone who has an outdate leanblueprint plugin for plastex will generate the dep graph in the old way. The reason why there is an old way and a new way it that LTE became too big to have a single dep graph and I arranged that each part of a blueprint can have a dep graph.

#### Oliver Nash (Aug 10 2022 at 13:22):

Sounds like it's Johan's fault then ;-)

#### Johan Commelin (Aug 10 2022 at 13:36):

I'll buy you a beer next time we meet, if that helps...

#### Floris van Doorn (Aug 10 2022 at 14:46):

I finished and pushed Lemma 3.21

#### Floris van Doorn (Aug 10 2022 at 14:54):

Shall I take a look at Lemma 3.26 next?

#### Patrick Massot (Aug 10 2022 at 15:21):

Yes, that would be great.

#### Patrick Massot (Aug 10 2022 at 15:27):

Note that the numbering was shifted because I removed an intermediate lemma.

#### Patrick Massot (Aug 10 2022 at 19:38):

Patrick Massot said:

I'm confused about Lemma 3.6. I thought that Yury did it but I'm not 100% sure

I managed to find it, Yury indeed did it in #15609.

#### Patrick Massot (Aug 10 2022 at 19:51):

Let me try to summarize the current status, knowing that some numbering changed today:

#### Oliver Nash (Aug 10 2022 at 20:26):

Thanks, this is really helpful!

#### Floris van Doorn (Aug 12 2022 at 10:25):

lem:ample_iff_loc turned out to be easy. I'll take a look at the parametricity lemma next.

#### Oliver Nash (Aug 15 2022 at 07:34):

I've finished various other non-Sphere-Eversion bits and pieces so while I'm waiting for #15980 to land in Mathlib I'm going to start on lem:updating this morning.

#### Oliver Nash (Aug 16 2022 at 18:22):

Today I pushed proofs of lem:nice_atlas and lem:updating (after adding the definition def:update yesterday).

#### Oliver Nash (Aug 16 2022 at 18:24):

I also changed the informal statement of lem:updating in the blueprint slightly (the main point being that I dropped the second compact set L from the input since it did nothing).

#### Patrick Massot (Aug 16 2022 at 19:20):

Amazing, thank you very much!

#### Johan Commelin (Aug 17 2022 at 06:03):

Cool! Looks like you're now rapidly converging!

#### Oliver Nash (Aug 17 2022 at 08:30):

I'll start work on lem:localisation_stability later this morning.

#### Oliver Nash (Aug 17 2022 at 10:51):

Happily lem:localisation_stability turned out to be trivial: now done!

#### Oliver Nash (Aug 17 2022 at 10:52):

I did have to tweak the def:localisation_data and lem:ex_localisation but I think all correct now.

#### Johan Commelin (Aug 17 2022 at 11:24):

So what is even left?

#### Johan Commelin (Aug 17 2022 at 11:25):

It sounds to me like half of the non-green things at https://leanprover-community.github.io/sphere-eversion/blueprint/dep_graph_document.html are actually green already?

#### Oliver Nash (Aug 17 2022 at 11:53):

I think that there are five nodes (four lemmas and a definition) which would turn green if we updated the dependency graph now. I believe the following is what remains:

1. lem:param_trick [I think Floris has claimed this]
2. lem:param_for_free
3. lem:ample_parameter
4. lem:improve_htpy_loc
5. lem:open_ample_immersion
6. thm:open_ample
7. thm:sphere_eversion

#### Patrick Massot (Aug 17 2022 at 12:49):

A lot of the lemmas that remained were indeed meant to be easy (including lem:localisation_stability), but they still require some work and it's very nice to have them done. And lem:updating still had some content.

#### Patrick Massot (Aug 17 2022 at 12:50):

There are also sorries in the project currently, and I don't really know what is their status.

#### Oliver Nash (Aug 17 2022 at 12:52):

Yeah I picked off a few easy ones. I'm going to take a day or so to do a little Mathlib work. When I resume on this project, I'll start by surveying the sorrys not captured in the dependency graph.

#### Patrick Massot (Aug 17 2022 at 12:53):

I'm very happy this is moving. I still think we can finish before the end of summer.

#### Oliver Nash (Aug 17 2022 at 12:53):

We can definitely finish soon. When does the Summer end?

#### Patrick Massot (Aug 17 2022 at 12:53):

I'll answer this question when we'll be done

#### Oliver Nash (Aug 23 2022 at 13:49):

Patrick Massot said:

There are also sorries in the project currently, and I don't really know what is their status.

I looked over these earlier today. Excluding sorrys in the unused folder we currently have the following sorry counts:

src/to_mathlib/analysis/normed_space/operator_norm.lean:1
src/to_mathlib/geometry/manifold/misc_manifold.lean:1
src/global/relation.lean:8
src/global/one_jet_sec.lean:1
src/global/one_jet_bundle.lean:4
src/global/immersion.lean:5
src/global/gromov.lean:2


Some will definitely not be needed and for others there is some uncertainty about whether they will ultimately be needed. Also at least three are already captured in our dependency graph.

#### Oliver Nash (Aug 23 2022 at 13:50):

After this I convinced myself that the best thing I could do would be to start work on the lemma lem:improve_htpy_loc so I have just pushed a commit adding its formal statement.

#### Oliver Nash (Aug 23 2022 at 13:52):

I fixed an apparent typo in the Blueprint as part of that commit. The statement of this lemma is quite complicated so hopefully I got it right. I'll start working on the proof later this afternoon so I suppose I'll find out as I work on that over the next day or two.

#### Oliver Nash (Aug 23 2022 at 13:54):

I think the easiest remaining node in the dependency graph might be lem:open_ample_immersion if anyone is interested in a target.

#### Patrick Massot (Aug 23 2022 at 19:15):

Thank you very much Oliver. I'm sorry I see this so late.

#### Patrick Massot (Aug 23 2022 at 19:18):

The story of lem:improve_htpy_loc is that I thought that Chapter 2 was finished but then when I tried to add LaTeX details in Chapter 3 I realized what I wrote in Chapter 2 was not strong enough. So if you aren't too far into that lemma it would probably make more sense to postpone proving it until we make sure we can actually use it in the main theorem.

#### Oliver Nash (Aug 23 2022 at 21:21):

OK no problem! I'll pick something else tomorrow morning.

#### Patrick Massot (Aug 23 2022 at 21:22):

Picking the main theorem would be nice for instance :wink:

#### Oliver Nash (Aug 23 2022 at 21:23):

Indeed! Let's see.

#### Oliver Nash (Aug 25 2022 at 12:50):

I started working on lem:open_ample_immersion this morning. I hope to finish later today.

#### Oliver Nash (Aug 25 2022 at 12:52):

If anyone is looking for a useful self-contained lemma then I think closing this sorry would make a good target. It is the statement that the immersion relation for smooth maps between manifolds is open.

#### Oliver Nash (Aug 25 2022 at 12:53):

(It's just a globalisation of the fact that the set of injective maps is open in the space of linear maps $Hom(E, F)$.)

#### Patrick Massot (Aug 25 2022 at 14:41):

Thanks Oliver. Here is a global update:

• Oliver is working on lem:open_ample_immersion
• Floris will soon state a more general version of Gromov theorem than what is currently in the Lean code base. He needs to include the relative version and C⁰-close version
• Floris will work on parametricity for free (this is a set of three nodes in the graph: param_trick, param_for_free and ample_parameter)
• Patrick will sorry the homotopy of formal immersions for sphere eversion, waiting to hear from Heather's rotations and deduce existence of sphere eversion from the statement of Gromov's theorem
• Nobody should work on proving improve_htpy_loc until we made sure it's enough to prove Gromov's theorem
• Gromov's theorem isn't assigned yet, but note that is can be cut in half: use parametricity for free to get rid of parameters and do the actual work.

#### Oliver Nash (Aug 25 2022 at 16:33):

I just pushed a proof of lem:open_ample_immersion. In fact, following a suggestion from the blueprint, I really just took the existing proof of lem:open_ample_immersion_loc, split it up a bit and used that. I then also removed lem:open_ample_immersion_loc entirely.

#### Patrick Massot (Aug 26 2022 at 08:42):

I just bumped mathlib.

#### Oliver Nash (Aug 26 2022 at 08:49):

Today I'm going to work on proving that the (global) immersion relation is open. (We have that it is ample as of yesterday.)

Very good.

#### Patrick Massot (Aug 26 2022 at 08:50):

I'm working in global/immersion.lean right now.

#### Johan Commelin (Aug 26 2022 at 08:51):

Is it worth it to regenerate the dependency graph for enthusiastic onlookers like me?

#### Oliver Nash (Aug 26 2022 at 08:52):

Sadly it's basically up to date.

#### Patrick Massot (Aug 26 2022 at 08:52):

The graph is up to date

#### Patrick Massot (Aug 26 2022 at 08:53):

You're confused because Oliver wrote yesterday he proved lem:open_ample_immersion but that was a typo, he proved half of it (the non-trivial half on paper).

Aha

#### Oliver Nash (Aug 26 2022 at 08:54):

That's right. I'm just about to push a commit which will clarify this.

#### Patrick Massot (Aug 26 2022 at 08:55):

Actually I can't proceed further until Floris pushes his fix to Gromov's theorem statement.

#### Patrick Massot (Aug 26 2022 at 08:56):

So I'll wait for Oliver's push then push my thing in the same file and do something else until Floris pushes

#### Oliver Nash (Aug 26 2022 at 08:56):

I should push in about nine minutes. Just waiting on a build after pulling the Mathlib bump.

#### Oliver Nash (Aug 26 2022 at 09:05):

OK the build was slow but I have now pushed.

#### Patrick Massot (Aug 26 2022 at 09:06):

That was expected, the bump changed files very low in the project hierarchy because the main change was Yury replacing with_top Nat by enat everywhere in calculus.

#### Patrick Massot (Aug 26 2022 at 09:07):

I pushed a tiny commit and I'll wait for Floris now.

#### Floris van Doorn (Aug 26 2022 at 12:18):

I pushed the fix. Sorry for the delay.

#### Patrick Massot (Aug 26 2022 at 12:20):

No problem. I did some admin in the mean time.

#### Patrick Massot (Aug 26 2022 at 12:45):

@Floris van Doorn do we agree that you now work on parametricity for free?

#### Floris van Doorn (Aug 26 2022 at 12:46):

Yes, I'm working on that and param trick now!

Great!

#### Patrick Massot (Aug 26 2022 at 13:42):

Can I propose that we try to use variable names indicating their type? In places like https://github.com/leanprover-community/sphere-eversion/blob/master/src/global/relation.lean#L150 I find it confusing that we have three manifolds $M$, $P$ and $X$ and lots of ∀ (s : P) (x : M), especially the x : M when the next manifold is called X

#### Floris van Doorn (Aug 26 2022 at 13:46):

I understand.
It is a bit tricky to do this consistently, since we sometimes want the codomain manifold to be a topological_space and sometimes to be a metric_space, which means we probably want to use different letters for them.
I have been trying (since yesterday) to use s, t and x for the variable names following the conventions in chapters 1 & 2:

• x is an element in the (domain) space we're considering
• s is a parameter
• t is the variable indicating how far you are along a homotopy

Would you be happy/happier if I replace X by M'' or M₂?

#### Patrick Massot (Aug 26 2022 at 13:47):

I never claimed the blueprint was consistent :wink:

#### Patrick Massot (Aug 26 2022 at 13:48):

In fact it would be even better to make the union of blueprint and Lean code consistent.

#### Patrick Massot (Aug 29 2022 at 13:12):

@Floris van Doorn could you push as soon as you have definitions and statements for the parametricity related stuff? It would allow me to start working on Gromov's theorem.

#### Floris van Doorn (Aug 29 2022 at 13:31):

What definitions do you need? $\Psi$ is bundle_snd, the section $\overline F$ is F.uncurry (where F is a family_one_jet_sec) and $\mathcal{R}^P$ is R.relativize IP P

#### Floris van Doorn (Aug 29 2022 at 13:34):

Feel free to rename any of these definitions

#### Patrick Massot (Aug 29 2022 at 13:42):

I think we have enough definitions indeed. So we need statements.

#### Floris van Doorn (Aug 29 2022 at 13:45):

Those are also already there. See blueprint. (ample_parameter in the blueprint is rel_mfld.ample.relativize.)

#### Patrick Massot (Aug 29 2022 at 13:46):

I'm sorry, I've doing administration while waiting for answers instead of looking at the project code. I guess I no longer have any excuse to do administration instead of working on the project.

#### Floris van Doorn (Aug 29 2022 at 13:46):

I guess I haven't stated param_for_free yet. If you already need that, I can try to prioritize it.

#### Patrick Massot (Aug 29 2022 at 13:47):

Oh but the graph isn't updated. Stated stuff should have green outlines.

#### Patrick Massot (Aug 29 2022 at 13:47):

Yes, prioritizing stating param_for_free would be great

#### Patrick Massot (Aug 29 2022 at 13:48):

I'll try to update the graph in the mean time.

#### Floris van Doorn (Aug 29 2022 at 13:49):

You're right, I didn't update the graph yet. I'll prioritize it.

#### Patrick Massot (Aug 29 2022 at 13:55):

Compiling the blueprint reveals that param_for_free is stated as rel_mfld.relativize_satisfies_h_principle!

#### Patrick Massot (Aug 29 2022 at 13:56):

Actually the graph currently claims that everything is stated! Of course it doesn't mean we won't add 50 lemmas...

#### Floris van Doorn (Aug 29 2022 at 13:56):

Yes, I realized that I had already (tried) to state it. I accidentally stated it as an iff though, but I should have stated it as follows. I will push

/-- This might need some additional assumptions or other modifications. -/
lemma rel_mfld.satisfies_h_principle.satisfies_h_principle_with
(R : rel_mfld I M IX X) (C₁ : set P) (C₂ : set M)
(ε : M → ℝ) (h : (R.relativize IP P).satisfies_h_principle (C₁ ×ˢ C₂) (λ x, ε x.2)) :
R.satisfies_h_principle_with IP C₁ C₂ ε :=
sorry


#### Patrick Massot (Aug 29 2022 at 13:56):

Working on Gromov's theorem will allow to test all those statements.

#### Patrick Massot (Aug 29 2022 at 13:57):

Tell me when you push and I'll recompile the blueprint

#### Floris van Doorn (Aug 29 2022 at 13:59):

currently compiling to check if I broke anything...

#### Patrick Massot (Aug 29 2022 at 14:05):

Damn, I think I messed up in 04cb6032f9. I asked Yury to prove a lemma from the blueprint and he proved something that looked simpler. It was the middle of my vacations so I didn't check and assumed my version was overcomplicated without a purpose. But now that I reading back the proof of Gromov's theorem I remember why my complicated version was needed.

#### Patrick Massot (Aug 29 2022 at 14:06):

The issue is that assumptions in Yury's version are too strong.

#### Patrick Massot (Aug 29 2022 at 14:06):

And I was too quickly to happily believe I overcomplicated things.

#### Patrick Massot (Aug 29 2022 at 14:07):

This shouldn't be too bad to fix.

#### Floris van Doorn (Aug 29 2022 at 14:07):

Oh, that is very unfortunate...

I pushed, btw

#### Johan Commelin (Aug 29 2022 at 14:09):

Btw, in the current online version of the blueprint, I noticed that https://leanprover-community.github.io/sphere-eversion/blueprint/chap-global.html#def:convenient_indexing is disjoint from the rest of the graph.

#### Patrick Massot (Aug 29 2022 at 14:10):

That's very much related to the story I was just telling

#### Patrick Massot (Aug 29 2022 at 14:12):

I compiled the blueprint and pushed.

#### Oliver Nash (Aug 29 2022 at 14:14):

Did you compile against an unpushed commit? The lean links in the dependency graph reference 4a1d60ccffec5b40c83aadfc5f779fe74147997b which seems to be missing.

#### Patrick Massot (Aug 29 2022 at 14:15):

Oh, I see what happened :sad:

I'll fix it

#### Oliver Nash (Aug 29 2022 at 14:15):

Graph is looking good though!

#### Patrick Massot (Aug 29 2022 at 14:25):

TODO: find a way to hide the (is_closed_empty : is_closed (∅ : set empty)) I just added to the project.

#### Patrick Massot (Aug 29 2022 at 14:31):

@Floris van Doorn do you confirm we don't have:

lemma rel_mfld.is_open_relativize
{E : Type u_1} {H : Type u_2} {M : Type u_3} {EP : Type u_7}
{HP : Type u_8} {P : Type u_9} {EX : Type u_10} {HX : Type u_11}
{X : Type u_12}
[normed_space ℝ E]
[topological_space H]
{I : model_with_corners ℝ E H}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
[normed_space ℝ EP]
[topological_space HP]
{IP : model_with_corners ℝ EP HP}
[topological_space P]
[charted_space HP P]
[smooth_manifold_with_corners IP P]
[normed_space ℝ EX]
[topological_space HX]
{IX : model_with_corners ℝ EX HX}
[topological_space X]
[charted_space HX X]
[smooth_manifold_with_corners IX X]
(R : rel_mfld I M IX X)
(h2 : is_open R) :
is_open (rel_mfld.relativize IP P R)


#### Patrick Massot (Aug 29 2022 at 14:36):

I pushed anyway, with this statement sorried.

#### Floris van Doorn (Aug 29 2022 at 14:37):

That is correct.
It would require continuity of bundle_snd, which shouldn't be too hard (but probably we should prove smoothness first).

#### Floris van Doorn (Aug 29 2022 at 14:37):

Feel free to add a bunch of sorry'd lemmas like that, they should be easy to knock down.

#### Patrick Massot (Aug 29 2022 at 14:38):

I can confirm that the properties stated for relativize (including the one I just wrote) are enough to deduce the parametric h-principle from the non-parametric one.

#### Patrick Massot (Aug 29 2022 at 14:40):

It's becoming harder to avoid confronting the last big proof. Luckily I need to go away for a little while (my family will return from vacations tonight and I need to prepare a bit).

#### Patrick Massot (Aug 29 2022 at 15:31):

Patrick Massot said:

TODO: find a way to hide the (is_closed_empty : is_closed (∅ : set empty)) I just added to the project.

This is hilarious. Sometimes Lean goes too far in allowing us to prove things without thinking. Lean was asking me to prove some subset of the parameter space was closed while deducing parametric Gromov from the non-parametric one. So I thought: there is no parameter here so this must be the empty set of the empty type. But actually this was the copy-paste typo that we noticed the other day in the statement but nobody fixed. So that closed set wasn't used at all and Lean would have been happy with any closed set in any topological space! And my vague semi-conscious rationalization wasn't even consistent: I should have written unit instead of empty since the non-parametric situation correspond to a parameter space which is unit, not empty...

Ha!

#### Oliver Nash (Aug 29 2022 at 15:31):

All is well anyway.

#### Patrick Massot (Aug 29 2022 at 15:32):

Now I need to find and kill anyone who saw that, in order to protect my academic reputation.

#### Oliver Nash (Aug 29 2022 at 15:32):

You just needed an initial object for any old category.

#### Oliver Nash (Aug 29 2022 at 15:33):

Your reputation is very safe IMHO.

#### Patrick Massot (Aug 29 2022 at 15:33):

My only consolation is that I saw the issue as soon as put begin/end for non-parametric Gromov and looked at the tactic state.

#### Patrick Massot (Aug 29 2022 at 15:34):

Speaking of tactic state, I can't wait for the magic tricks Lean 4 will use to turn

E: Type u_1
_inst_2: normed_space ℝ E
H: Type u_2
_inst_3: topological_space H
I: model_with_corners ℝ E H
M: Type u_3
_inst_4: topological_space M
_inst_5: charted_space H M
_inst_6: smooth_manifold_with_corners I M


into

M : manifold


#### Patrick Massot (Aug 29 2022 at 15:41):

And now Lean seeks revenge and asks me to prove X is non-empty.

#### Patrick Massot (Aug 29 2022 at 15:43):

I'm pretty sure the blueprint version of std_localisation_data does not have any non-emptyness assumption, but I guess Lean may be right anyway.

#### Oliver Nash (Aug 29 2022 at 15:45):

This could be my fault. I did this for convenience.

#### Patrick Massot (Aug 29 2022 at 15:45):

@Oliver Nash could you have a look at global/localization_data.lean and figure out whether those nonempty assumptions are really needed? There are also several linting errors in that file.

#### Patrick Massot (Aug 29 2022 at 15:46):

I don't mind proving a preliminary lemma saying the $h$-principle holds for empty manifold (or additing that assumption everywhere if it doesn't, who knows?) but I'd rather avoid it if possible.

#### Oliver Nash (Aug 29 2022 at 15:46):

Our comments crossed. I'll take a look. I'm very close to turning an ellipse green though. Would you like me to look now?

#### Patrick Massot (Aug 29 2022 at 15:46):

There is no emergency, I can add the assumption to Gromov's theorem for now.

#### Oliver Nash (Aug 29 2022 at 15:47):

OK I'll take it up as soon as I've finished my goal. If that means excising it from the statement of Gromov's theorem, then I'll do that too.

#### Patrick Massot (Aug 29 2022 at 15:50):

I'm editing Gromov's theorem so please don't touch it yet. If you could simply investigate whether we can remove nonempty from the localisation file it would be great.

#### Patrick Massot (Aug 29 2022 at 15:51):

I also noticed that Gromov's theorem misses finite dimensionality assumptions. This didn't occur to me because manifold implies finite dimension in my mind.

#### Oliver Nash (Aug 29 2022 at 15:52):

I'm wrestling with some annoying linear algebra and I doubt I'll get to this nonempty M stuff for at least an hour.

No problem

#### Patrick Massot (Aug 29 2022 at 15:53):

I'm adding assumptions to the theorem for now

#### Patrick Massot (Aug 29 2022 at 15:59):

There are lots of assumptions to add anyway because all the localization stuff assume finite dimensional manifolds without boundary.

#### Patrick Massot (Aug 29 2022 at 16:00):

And this totally makes sense, this is the default meaning of manifold and this should be enough for us.

#### Patrick Massot (Aug 29 2022 at 16:00):

And of course the blueprint is totally inconsistent here.

#### Oliver Nash (Aug 29 2022 at 16:54):

Patrick Massot said:

You're confused because Oliver wrote yesterday he proved lem:open_ample_immersion but that was a typo, he proved half of it (the non-trivial half on paper).

lem:open_ample_immersion is now genuinely sorry-free.

#### Oliver Nash (Aug 29 2022 at 16:56):

I'm being called to dinner now. I'll tidy up the nonempty M issue this evening if I can find some time or else I'll take it up tomorrow morning.

Great!

#### Floris van Doorn (Aug 30 2022 at 17:23):

I just finished the lemma param_trick. By far the hardest part was showing that $\bar F$ is smooth.

Great!

#### Patrick Massot (Aug 30 2022 at 17:50):

It's very sad to read that obvious smoothness proofs are still so hard, but very good to have this lemma!

#### Patrick Massot (Aug 30 2022 at 17:51):

I'm currently fighting off-by-one discrepancies when piecing various lemmas together...

#### Patrick Massot (Aug 30 2022 at 17:51):

There is no problem at all but it's very time consuming and annoying

#### Floris van Doorn (Aug 30 2022 at 19:19):

The main problem was the library building part of it; I had to prove that the derivative (as a continuous linear map) of a smooth map between manifolds is smooth in the point where you take the derivative. And then I had to generalize that to partial derivatives. I think it makes more sense that it is hard if you have to do library building as a part of it.

#### Floris van Doorn (Aug 30 2022 at 19:20):

I also finished ample_parameter, so the 4 hour old blueprint is 3 lemmas behind :grinning_face_with_smiling_eyes:

Wonderful!

#### Patrick Massot (Aug 30 2022 at 20:05):

Are those 3 lemmas fully sorry-free or do they still depend on some library building sorries?

#### Floris van Doorn (Aug 30 2022 at 20:06):

The 2 lemmas I pushed are fully sorry-free, I checked with #print axioms.

#### Patrick Massot (Aug 30 2022 at 20:06):

Do you have a clear view of where the current sorries are used?

#### Patrick Massot (Aug 30 2022 at 20:07):

If some are not used then it would be nice to comment them out.

#### Floris van Doorn (Aug 30 2022 at 20:11):

I've tried to put a comment near most of my recent sorry's when I think we're likely not using them (and also replacing sorry by admit so we can count by word search). I can go over the sorry's tomorrow to see which ones we are likely not using.
I believe there are not much sorry's in the files I've been recently using.

#### Floris van Doorn (Aug 30 2022 at 20:12):

I think htpy_one_jet_sec.unlocalize in global/relation is yours? Will we need that?

#### Patrick Massot (Aug 30 2022 at 20:15):

About unlocalize, we should find out pretty soon

#### Patrick Massot (Aug 30 2022 at 20:15):

But clearly if it is useful then it will be at the very end.

#### Patrick Massot (Aug 30 2022 at 20:19):

What about one_jet_bundle_model_space_chart_at and one_jet_bundle_model_space_homeomorph?

#### Patrick Massot (Aug 30 2022 at 20:20):

They are probably stuff I wrote at the very beginning of Chapter 3 and maybe there are n longer relevant

#### Patrick Massot (Aug 30 2022 at 20:21):

As unlocalize, they were meant to use allow using Chapter 2 in Chapter 3.

#### Floris van Doorn (Aug 30 2022 at 20:30):

Yeah, I haven't touched them, and I don't to what extend we need those to glue chapter 2 to chapter 3.

Ok, let's wait.

#### Patrick Massot (Aug 30 2022 at 20:33):

In the mean time the graph got updated: https://leanprover-community.github.io/sphere-eversion/blueprint/dep_graph_document.html

#### Patrick Massot (Aug 30 2022 at 20:33):

It's starting to look good. Too bad nice_atlas went in the wrong direction.

#### Patrick Massot (Aug 30 2022 at 21:04):

I'm done for today. I just push https://github.com/leanprover-community/sphere-eversion/commit/e7e165e59b366466e620d5ad42c513664adb6712 which features actually starting the main big proof.

#### Patrick Massot (Aug 30 2022 at 21:06):

Things are going pretty smoothly so far, but it's always painful to have a local context with more than 60 items.

#### Patrick Massot (Aug 30 2022 at 21:07):

(at least there are only 23 numbered instances, which is way less than the maximum reached when deducing the parametric result from the unparametric one).

#### Oliver Nash (Aug 30 2022 at 21:27):

Wow, amazing progress! Congratulations @Floris van Doorn

#### Oliver Nash (Aug 30 2022 at 21:29):

I’ll fix nice_atlas tomorrow. It’s rather messy because I’ve pushed for the quickest way to cater for the two changes but there are absolutely no mathematical issues so I’m confident I can do it tomorrow.

#### Patrick Massot (Aug 31 2022 at 08:56):

I just pushed https://github.com/leanprover-community/sphere-eversion/commit/ba3ab0478213c8d20f3d4829dff54f61ac8217c5 to help handling the very large proof states at the end of the project.

#### Johan Commelin (Aug 31 2022 at 08:58):

Nice! Would have been useful in LTE as well :smiley:

#### Patrick Massot (Aug 31 2022 at 08:58):

This is a customized version of Ed's interactive_expr file from mathlib. It makes sure goals are printed before the local context so that you don't have to scroll to see the goal. And it adds options for permanent tactic state filtering (hacking the trace option framework as usual)

#### Patrick Massot (Aug 31 2022 at 08:59):

All this as been possible in Lean 3 forever, but people were too lazy to dive into this (admittedly rather intimidating) file.

#### Patrick Massot (Aug 31 2022 at 09:01):

Let me clarify: the big file I added in that commit is a file taken from mathlib with something like 15 added lines

#### Yaël Dillies (Aug 31 2022 at 09:02):

Filtering, as in this?

#### Johan Commelin (Aug 31 2022 at 09:03):

Those aren't persistent, for me.

#### Patrick Massot (Aug 31 2022 at 09:03):

Yaël, what you show isn't persistent.

#### Yaël Dillies (Aug 31 2022 at 09:04):

Ah, so it's being reset when you touch the file?

#### Patrick Massot (Aug 31 2022 at 09:04):

It's much worse, it's reset when you move to the next line of your proof.

#### Yaël Dillies (Aug 31 2022 at 09:04):

Patrick Massot said:

All this as been possible in Lean 3 forever, but people were too lazy to dive into this (admittedly rather intimidating) file.

Also, #15959 :stuck_out_tongue_closed_eyes:

#### Yaël Dillies (Aug 31 2022 at 09:05):

What I really want is being able to inspect error states. This would have been handy countless times.

#### Patrick Massot (Aug 31 2022 at 09:07):

Now I need to un-train my brain which is asking my right hand to go to the mouse scroll button every time I finish typing a line.

#### Floris van Doorn (Aug 31 2022 at 09:21):

Oh, the new tactic state view looks very useful!

#### Floris van Doorn (Aug 31 2022 at 13:37):

FYI: I'm currently reformulating satisfies_h_principle_with, I'm replacing the {C₁ : set P} {C₂ : set M} with a single set {C : set (P × M)}. The previous phrasing was pretty unnatural.

#### Patrick Massot (Aug 31 2022 at 13:47):

I guess I'll be able to adapt.

#### Floris van Doorn (Aug 31 2022 at 13:48):

I fixed all current proofs, which was trivial.

#### Patrick Massot (Aug 31 2022 at 13:53):

Did you push? This will surely create conflicts

I just pushed

#### Floris van Doorn (Aug 31 2022 at 13:54):

I'm happy to resolve all conflicts if you push a compiling development to a branch (now or later).

#### Patrick Massot (Aug 31 2022 at 13:57):

I was talking nonsense, this doesn't affect at all the proof I'm working on since I'm working on the non-parametric case.

#### Patrick Massot (Aug 31 2022 at 13:57):

Sorry about the noise.

#### Patrick Massot (Aug 31 2022 at 13:59):

Does param_for_free still seem doable?

#### Patrick Massot (Aug 31 2022 at 13:59):

For casual observers, let me point out the blueprint proof of that lemma is "This obviously follows from lemma 3.22. "

#### Floris van Doorn (Aug 31 2022 at 14:22):

It's doable, but of course, Lemma 3.22 is only part of the work (though hopefully more than half of it).

#### Floris van Doorn (Aug 31 2022 at 14:23):

The extra work comes from transforming the homotopy on the product back to a homotopy with parameters.

#### Patrick Massot (Aug 31 2022 at 14:24):

Isn't that (un)curryfication?

yeah

#### Floris van Doorn (Aug 31 2022 at 14:25):

but of course, if you have extra structure, (un)currying might be non-trivial. Lemma 3.22 is uncurrying, and now I have to curry back.

#### Floris van Doorn (Aug 31 2022 at 14:31):

@Patrick Massot: shall I change all equalities in rel_mfld.satisfies_h_principle and rel_mfld.satisfies_h_principle_with to equalities of elements in J¹(M, M') (instead of equalities of formal solutions / 1-jet sections)? That might simplify some things.

#### Patrick Massot (Aug 31 2022 at 14:49):

Indeed that will probably simplify things a bit.

#### Floris van Doorn (Aug 31 2022 at 14:49):

ok, I'll go ahead

#### Floris van Doorn (Aug 31 2022 at 14:56):

I pushed. For the non-parametric version only the first Prop-field changed.

#### Patrick Massot (Aug 31 2022 at 15:01):

Did you check the immersion file is still ok?

It seems ok

#### Patrick Massot (Aug 31 2022 at 15:41):

I just pushed after reaching what was my goal for this morning.

#### Patrick Massot (Aug 31 2022 at 15:43):

I did the last paragraph of the main theorem:

Now that the inductive construction is completed, we apply Lemma 3.9 to make sure our sequence $F_n$ is locally ultimately constant, hence it converges pointwise to a smooth homotopy relative to $A$ and ending at a holonomic section of $\mathcal{R}$.

which is slightly less than 100 lines in Lean.

#### Patrick Massot (Aug 31 2022 at 15:44):

But I had to make some change to what I assumed from the inductive construction. One change was clearly correcting a typo but another one has some content, I'll need to think a bit.

#### Patrick Massot (Aug 31 2022 at 15:45):

But I only have 12 minutes now so I'll rather remove a tiny sorry I just introduced.

#### Patrick Massot (Aug 31 2022 at 15:52):

Ok I proved it and added it to to_mathlib/topology/misc.lean which is now 644 lines long... :innocent:

#### Floris van Doorn (Aug 31 2022 at 15:55):

Note: docs#set.bUnion_mono exists.

#### Floris van Doorn (Aug 31 2022 at 15:56):

And I'm doing even worse: I have a few tiny topology lemmas currently polluting the relation file :see_no_evil:

#### Patrick Massot (Aug 31 2022 at 15:57):

Oh, I'm also polluting.

lemma univ_prod_inter_univ_prod {α β : Type*} {s t : set β} :
(univ : set α) ×ˢ s ∩ (univ : set α) ×ˢ t = (univ : set α) ×ˢ (s ∩ t) :=
begin
ext ⟨a, b⟩,
simp
end


is really properly a part of differential topology.

#### Floris van Doorn (Aug 31 2022 at 15:58):

that checks out ^^

#### Patrick Massot (Aug 31 2022 at 16:00):

I pushed and I'll move to admin and then go back home. I don't know yet whether I'll have time for the project tonight.

#### Oliver Nash (Sep 01 2022 at 16:02):

Patrick Massot said:

Oliver Nash could you have a look at global/localization_data.lean and figure out whether those nonempty assumptions are really needed? There are also several linting errors in that file.

I finally had time to check this. Unfortunately these nonempty instances are necessary because index_type always has at least one element. Thus for example nice_atlas demands that there is at least one embedding of F into M but F always contains 0 and so is non-empty and thus M must be also.

#### Oliver Nash (Sep 01 2022 at 16:04):

The way to fix this would be to fix this deficiency in index_type but given that you have already suffered one round of off-by-one-ness pain for indexing, I doubt it's worth it for now.

#### Johan Commelin (Sep 01 2022 at 16:07):

How hard would it be to prove the main theorem assuming is_empty M? Can't be that hard, I hope?

#### Patrick Massot (Sep 01 2022 at 16:16):

Of course in the worse case scenario we can simply start the proof with a case disjunction according to whether M is empty or not, but it feels a bit silly.

#### Patrick Massot (Sep 01 2022 at 16:17):

Oliver Nash said:

The way to fix this would be to fix this deficiency in index_type but given that you have already suffered one round of off-by-one-ness pain for indexing, I doubt it's worth it for now.

Guess what? Today I suffered back and partly reverted yesterday's change of mind.

#### Patrick Massot (Sep 01 2022 at 16:17):

Life is so much easier on paper or blackboard.

#### Patrick Massot (Sep 01 2022 at 16:18):

Today had a lot more admin and off-by-one suffering than I hoped for, but I'm back to a stable situation (I just pushed). Now I hope the inductive construction is really setup.

#### Patrick Massot (Sep 01 2022 at 16:20):

I need to go back home now (and I'm too hungry to Lean anyway), but I'll have at least one more hour tonight.

#### Johan Commelin (Sep 01 2022 at 16:21):

It's only 18:20!!

#### Patrick Massot (Sep 01 2022 at 16:23):

I have 25 minutes biking and then I need to cook

#### Floris van Doorn (Sep 01 2022 at 18:51):

Today was a very long day... But I did finish param_for_free and updated the blueprint.
Now I also need to go home :-)

#### Patrick Massot (Sep 01 2022 at 19:04):

Great! Thank you very much for your perseverance!

#### Patrick Massot (Sep 01 2022 at 19:08):

Only three dependency graph nodes to go! One is @Heather Macbeth rotation project, one is chapter two coming back for an encore and the last one is Gromov's theorem for open ample relations (aka the whole project, but we are pretty close).

#### Patrick Massot (Sep 01 2022 at 19:59):

Nooooo, the definition of updating is also given in the restricted context of smooth_manifold_with_corners 𝓘(𝕜, EX) X

:cry:

#### Patrick Massot (Sep 01 2022 at 20:00):

We really need to put a huge warning in the docstring of docs#model_with_corners_self and the corresponding module docstring.

#### Patrick Massot (Sep 01 2022 at 20:11):

Ha Ha! Generalizing in math is so easy!

#### Patrick Massot (Sep 01 2022 at 20:12):

It even has some content, the new definition and its companion 40 lines long lemma now work with manifolds with boundaries and corners.

#### Floris van Doorn (Sep 01 2022 at 20:18):

I recommend Oliver to do more copy-pasting. Copy-pasting variable lists (or lemma statements) of similar situations is a good way to get the formulation of a particular concept right.
I expected that [smooth_manifold_with_corners 𝓘(𝕜, E)] would not occur in mathlib at all, but actually I am wrong: @Heather Macbeth uses [smooth_manifold_with_corners 𝓘(ℂ, E) M] once in file#geometry/manifold/complex, which likely should also be generalized.

#### Oliver Nash (Sep 01 2022 at 20:25):

I agree this is a good tip (which I will follow) and I did formalise the statement of nice_update_of_eq_outside_compact (which was fortunately easy to generalise) but in my defence, I did not formalise the original statement of nice_atlas.

#### Oliver Nash (Sep 01 2022 at 20:26):

Super job on param_for_free btw!

#### Patrick Massot (Sep 01 2022 at 20:27):

@Floris van Doorn this docstring is outdated, right?

#### Oliver Nash (Sep 01 2022 at 20:27):

I'm being told I must help with packing now. See you all in a week!

#### Patrick Massot (Sep 01 2022 at 20:28):

Have good vacations!

#### Floris van Doorn (Sep 01 2022 at 20:30):

@Oliver Nash That's fair; and if you copied/mimicked these variables from the nice_atlas formulation, then you cannot be faulted for that at all. Good vacations!

#### Floris van Doorn (Sep 01 2022 at 20:30):

Patrick Massot said:

Floris van Doorn this docstring is outdated, right?

Yes, that is correct

#### Patrick Massot (Sep 01 2022 at 20:31):

I'd like to make clear I wasn't "faulting" anyone :smile:

#### Floris van Doorn (Sep 01 2022 at 20:31):

(although mathematically it might still be correct, since 100% of a construction is still a part of a construction :smile: )

#### Floris van Doorn (Sep 01 2022 at 20:32):

I think you're missing a negation in your last message?

#### Patrick Massot (Sep 01 2022 at 20:33):

And the dependency graph makes it very clear that we were expecting impedance mismatch exactly here. This node has 10 in-going edges!

#### Patrick Massot (Sep 01 2022 at 20:46):

I pushed a commit that will create conflict so please pull. I split off parametricity for free from the relation file.

#### Patrick Massot (Sep 01 2022 at 21:14):

I reach the first point where I see rel_loc.jet_sec EM EX in the main proof!

#### Patrick Massot (Sep 01 2022 at 21:14):

This mean the central object of Chapter 2 is entering the main proof of Chapter 3.

#### Patrick Massot (Sep 01 2022 at 21:16):

However it seems that global/relation.lean misses a 1-parameter version of one_jet_sec.localize. I mean it seem we don't have htpy_one_jet_sec.localize (or, more to the point htpy_formal_sol.localize).

#### Patrick Massot (Sep 01 2022 at 21:17):

@Floris van Doorn in case you need a new project...

#### Patrick Massot (Sep 01 2022 at 21:18):

The funny thing is htpy_one_jet_sec.unlocalize is there.

#### Patrick Massot (Sep 01 2022 at 21:34):

I also bumped mathlib. It means we now have access to @Kyle Miller's awesome work on dot notation. This will allow to get rid of a lot of to_one_jet_section etc.

#### Patrick Massot (Sep 01 2022 at 21:34):

And I'm done for today.

#### Floris van Doorn (Sep 05 2022 at 15:48):

I proved 4 sorry's in localisation, 1 sorry in one-jet_bundle (smoothness of $\Psi$) and 1 in relation (smoothness of transfer $\psi_{g,h}$)

Great!

#### Patrick Massot (Sep 05 2022 at 15:58):

I've been besieged by students all day :sad:

#### Patrick Massot (Sep 05 2022 at 15:59):

I wrote some statements but the situation still isn't clarified

#### Patrick Massot (Sep 05 2022 at 15:59):

I'll try to work more tonight.

#### Patrick Massot (Sep 05 2022 at 16:08):

If you don't know what to do, I think we need tu upgrade transfer to a smooth open embedding.

#### Floris van Doorn (Sep 05 2022 at 20:08):

There's still a bunch of sorry's I can attack. I started looking into the computation of $J^1(V,V')$ when $V$ and $V'$ are vector spaces, and I will continue with that tomorrow.

#### Floris van Doorn (Sep 05 2022 at 20:18):

But I'll look into that after that, if you are (reasonably) certain that we will use it.

#### Patrick Massot (Sep 05 2022 at 22:39):

The story is that on paper I thought it was enough to discuss updating maps from $M$ to $N$ using embeddings $h : X \hookrightarrow M$ and $g : Y \hookrightarrow N$. The reasoning is that a section $F : V \to J^1(V, W)$ is a map from the manifold $V$ to the manifold $J^1(V, W)$ with some extra condition. But now I think this trick isn't usable in a formal context. We need a clean API around modifying sections of jet bundles. The thing is that $h : X \hookrightarrow M$ and $g : Y \hookrightarrow N$ give rise to $\psi : J^1(X, Y) \hookrightarrow J^1(M, N)$ (this is the transfer map) and we can use the updating API with the pair $(h, \psi)$ but this requires an API registering that sections becomes sections and a convenient smoothness condition.

#### Patrick Massot (Sep 05 2022 at 22:41):

I've started stating stuff, but many statement miss some side-conditions. At least in global/relation.lean I think all sorries that require additional conditions are clearly marked.

Last updated: Dec 20 2023 at 11:08 UTC