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 --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.
