Thanks Gino, I'm going to have a look. In addition to my post on the discussion group that you mention, I also began to do in my mathbox what you described, i.e., adding enough DV conditions to the technical lemmas so as to later remove some ax-13 usages, but less systematically. Some theorems made it to the main sections, but most remain in my mathbox, in the section "20.15.4.12 Removing dependencies on ax-13 (and ax-11)", which also has a section head comment to explain the principles. The plan was to move to Main only the ones that had the greatest benefits, but since there was no clear criterion, this kind of stalled.
Benoît On Tuesday, January 2, 2024 at 12:52:10 AM UTC+1 [email protected] wrote: > On Mon, Jan 1, 2024 at 6:43 PM Jim Kingdon <[email protected]> wrote: > >> One (historical?) note: some of what we have now is the result of >> experimentation in the opposite direction - trying to figure out whether a >> logical system can be built without distinct variable constraints at all (I >> think there is some reference to this in some comments or web pages). I >> think the verdict was that it was possible but so cumbersome as to be >> impractical (because all the distinctor antecedents need to be carried >> along until the point where that variable is no longer in use, I think). >> But perhaps I'm not summarizing that quite right - like I say this isn't a >> topic I've been hugely engaged with. >> > > No it's worse than that: even after the variable is done being used you > still can't discharge the distinctor, it permanently sticks around. > Effectively all proofs end up saying "provided there are at least N > variables in the metalogic, the following theorem holds" because you > actually can't prove anything about whether variables exist in this > setting. This admits models where e.g. the object logic only has three > variables v0 v1 v2 and so you can't write any expression containing four or > more forall or exists quantifiers without being degenerate in some way, so > the undischargeable assumptions that pile up are saying that the object > logic is at least nondegenerate enough to perform the proof in question. > > >> On 1/1/24 13:38, Gino Giotto wrote: >> >> In the last few days, I've been working on reducing the usage of ax-13, >> aiming at getting the highest result with the minimum amount of changes. >> The results of my findings are committed in my repository branch >> https://github.com/GinoGiotto/set.mm/tree/ax-13-complete, which is based >> on a version of set.mm dating back to December 13, 2023. >> >> This research was primarily motivated by curiosity. I read this email >> from Benoit >> https://groups.google.com/g/metamath/c/1wi1s6qBYqY/m/FPkPsd5oAwAJ. He >> described how most theorems use technical lemmas with dummy variables and I >> became interested in checking the real extent of this. The good news is >> that ax-13 can be erased almost everywhere. The bad news is that I needed >> 129 lemmas to accomplish this task, which is higher than the final >> estimates provided in that conversation (100 seemed to be the upper bound). >> >> The approach I pursued goes as follows: Starting from >> https://us.metamath.org/mpeuni/ax13w.html, I proved all theorems in the >> ax-13 section by adding the necessary dv conditions. Then I continued to >> the "Uniqueness and unique existence" section and the first few set theory >> sections until usage of proofs with dummy variables became prevalent. >> Distinguishing between the theorems that require additional dv conditions >> from those that don't isn't straightforward, so at first I simply proved >> them all and later I pruned away those that didn't necessitate additional >> dv conditions. >> >> This process resulted in more than 300 additional lemmas, which I later >> pruned again by eliminating unused and already existing ones. This job >> ultimately reduced the number to 129 additional lemmas, which I believe >> cannot be lowered further unless proof lenghtenings are introduced. >> >> In the meantime, I conducted multiple minimization sessions with the new >> lemmas using the /MAY_GROW option. This option allows to replace steps even >> when the proof length increases. In most of my minimizations, the overall >> proof shape and length remained unchanged as I replaced theorems with >> identical versions with more dv conditions. >> >> All and only my 129 additional lemmas have a *(Contributed by Gino >> Giotto, 30-Dec-2023.) *tag, so this information can be used to >> distinguish them from the other theorems in the database. >> >> I adopted the naming convention of adding a *w suffix to the original >> theorem names. The reason I did not use a *v suffix is because it would >> have resulted in 17 naming collisions. Since all the pre-exisiting versions >> have more dv conditions than mine, they would have to be renamed with *vv, >> which would have increased the amount of changes in the commit. Also I >> believe it makes sense to name them as *w since they all originated from >> ax13w (even tho after shortening their proofs they don't use ax13w >> anymore). So in the end, by using a *w suffix, no naming collision was >> generated. >> >> But enough rambling, let's get to the numbers: >> As of commit >> https://github.com/metamath/set.mm/tree/5228c50ed1c4f3e7c41dd0d5fe49c91f5c7725c8 >> dating >> back to December 13, 2023, ax-13 was used by 32,347 out of 41,652 theorems, >> covering 77.66% of the entire database. As of January 1, 2024, this >> percentage is at 77.64%, so it hasn't changed much since then. >> >> In my branch https://github.com/GinoGiotto/set.mm/tree/ax-13-complete, >> thanks to the lemmas I added and the minimizations I performed, ax-13 is >> used by only 819 theorems out of 41,781, which is just 1.96% of the entire >> database. If we exclude OLD/ALT versions then the number of theorems that >> use ax-13 goes down to around 700. The majority of these remaining theorems >> are found in the ax-13 section, in the "Alternate definition of >> substitution" section, and within mathboxes. Many of those 700 theorems >> could drop ax-13 by adding dv conditions directly to them, but I believe >> that would be considered cheating (I only did this for 2 or 3 theorems >> where adding a new version didn't seem worth it, also they didn't affect >> the dv conditions of later theorems). >> >> It's possible to check these numbers with *metamath-knife set.mm >> <http://set.mm> --stmt-use use.txt ax-13 *which shows what theorems in >> set.mm use ax-13. A comparison between axiom usage of my branch >> https://github.com/GinoGiotto/set.mm/tree/ax-13-complete and the base >> branch >> https://github.com/GinoGiotto/set.mm/tree/5228c50ed1c4f3e7c41dd0d5fe49c91f5c7725c8 >> shows >> the result of my minimizations. The command metamath-knife set.mm -X >> ax.txt can be used to check that other axioms haven't been added, >> however it's better to find a way to ignore ax-13 for this, otherwise >> you're going to be overwhelmed by the amount of changes from it. So far >> I've not yet seen axioms that have been mistakenly introduced (on the >> contrary there are a few theorems with a reduced usage of ax-10, ax-11 and >> ax-12). >> >> Unfortunately, despite my efforts to make as few changes as possible, the >> commit on my branch6fc6153 >> <https://github.com/metamath/set.mm/commit/6fc6153e0e4df76bf73bfeef76151d5354bc972c>still >> >> looks huge, with about 48,000 changed lines. Most of these changed lines >> are the result of the minimization process and rewrapping (the proof >> changes themselves are often very minor, in reality it's the rewrapping the >> skews everything). I didn't find ways to lower this number down without >> tradeoffs. >> >> This result (aka the mentioned branch in my fork) can be used in >> different ways, one could use it as a simple consultation for future axiom >> minimizations, or maybe it can be converted into a proper PR series. The >> latter would require some non-trivial work of systematization, so probably >> it would be better to discuss about it first. >> >> Regards >> >> Gino >> -- >> You received this message because you are subscribed to the Google Groups >> "Metamath" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/metamath/84b387ba-298f-4252-8c70-821e3a23d372n%40googlegroups.com >> >> <https://groups.google.com/d/msgid/metamath/84b387ba-298f-4252-8c70-821e3a23d372n%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> >> -- >> You received this message because you are subscribed to the Google Groups >> "Metamath" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/metamath/4d7b01cb-fca5-4713-9dad-308941403db8%40panix.com >> >> <https://groups.google.com/d/msgid/metamath/4d7b01cb-fca5-4713-9dad-308941403db8%40panix.com?utm_medium=email&utm_source=footer> >> . >> > -- You received this message because you are subscribed to the Google Groups "Metamath" group. 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