add vertex/edge access theorems to iset.mm #5139
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avekens
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Reviewed up to "GRAPH THEORY". Only minor remarks. I will review section about graph theory later.
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Graph Theory reviewed, still only minor remarks about wording
It's fantastic to see how graph theory becomes intuitonized. Hopefully, the definition of graphs either as ordered pairs or as extensible structures was the most difficult part, and the rest can be done more easily.
This is the syntax and df-edgf . Copied without change from set.mm.
Stated as in set.mm. The proof is the set.mm proof with small changes to use strnfvnd
This is the syntaxes , df-vtx , and df-iedg . Copied without change from set.mm.
This is vtxval from set.mm with a set existence condition added. The proof is taken from a portion of the set.mm proof with changes for set existence.
This is iedgval from set.mm with a set existence condition added. The proof is taken from a portion of the set.mm proof, modified for differences in set existence.
Stated as in set.mm. The proof is based on the set.mm proof but needs to be significantly longer for differences in set existence theorems.
Stated as in set.mm. The proof is the set.mm proof with a very small change for set existence.
Stated as in set.mm. The proof is the set.mm proof with a small change for differences in set existence.
Although this is similar to various existing theorems, we don't seem to have quite this form until now.
Stated as in set.mm. The proof is based on the iset.mm proof of en1 .
Stated as in set.mm. The proof is the set.mm proof with small changes in various places.
This is structgrssvtxlem from set.mm with changes to how we specify that a set has at least two elements. The proof is the set.mm proof with several small changes.
Stated as in set.mm. The proof is the set.mm proof with small changes.
Stated as in set.mm. The proof is the set.mm proof with several small changes.
This is struct2grstr from set.mm with set existence conditions added. The proof is the set.mm proof with small changes.
Stated as in set.mm. The proof is the set.mm proof with small changes.
Stated as in set.mm. The proof is the set.mm proof with one small change.
This is grstructd from set.mm with a change to how we specify that a set has at least two elements. The proof is the set.mm proof with small changes.
This is grstructeld from set.mm with changes to how we specify that a set has at least two elements. The proof is the set.mm proof with small changes.
Update text for basprssdmsets
Add ax-coll to the $j usage. Remove ax-pow from the comment because it is used via breng and brdom2g , respectively.
Change "unordered pair" to "proper unordered pair" (meaning the two elements of the pair are unequal). Suggested by avekens .
Change "pair" to "unordered pair". As suggested by avekens .
Per "iset.mm versus set.mm names" at https://us.metamath.org/ileuni/conventions.html , the name structvtxvallem isn't available. So struct2slots2dom perhaps is a less awkward name.
Seems like the obvious name by analogy to struct2slots2dom .
(this is just to undo some odd ordering which happened during rebasing - the desired order is the order in set.mm) also, update mmil.html entries for structvtxvallem and structgrssvtxlem
I've only gotten slightly further than what is in this pull request, so I'm not too sure. Do we assume that graphs are finite throughout? Sets not known to be finite are one place where excluded middle can potentially creep in but to be honest I have been taking things one step at a time rather than studying the set.mm treatment enough to see what happens later on. |
No, in general, graphs can be infinite. There is a special definition for finite graphs (~df-fusgr), but only for undirected simple graphs. So let's wait until we get problems with potentially infinite graphs... |
Good to know. We'll see what it looks like when we get there. (What is necessary/possible will depend on the details). |
This is the most straightforward thing to do in iset.mm in the sense that the axiom is called ax-coll and I guess in principle would be stronger than zfrep6 although currently we only use ax-coll to prove zfrep6 (and a handful of theorems not used further). There isn't anything going on differently than in the set.mm comments which say "Axiom of Replacement".
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This ended up a lot longer than I expected, but there's always a lot of machinery involved in extensible structures, and in this case the definition (unaltered from set.mm) allows either an extensible structure or an ordered pair of vertexes and edges.
Most of the discouraged theorems are also discouraged in set.mm, although I did re-intuitionize
2strstrto reflect changes in set.mm since the first one, and that accounts for a few of the entries.One of the biggest differences from set.mm is that
2o ~<_ Ais well behaved, in iset.mm, for saying a set has at least two elements (https://us.metamath.org/mpeuni/rex2dom.html is provable and is added here) whereas2 <_ ( # ` A )is not (at least with current theorems,#works on finite sets or on infinite sets, but not on arbitrary sets).