<div dir="ltr"><div>I was aware that equation (1) is stronger than (2) without extensionality, but I'm curious why this is a problem, since it is an axiom of the system and doesn't require proof. Any v5 proof that starts with (2) can easily be rebuilt to use (1), while conversely for a v6 proof there are very few theorems that follow from (1) but not (2) in the absence of extensionality (mostly trivial restatements of (1) itself), and I expect that any proof that starts from (1) will immediately wrap it with appThm, upon which point it will again be equivalent to (2). </div>Mario </div><div class="gmail_extra"> <div class="gmail_quote">On Thu, Nov 6, 2014 at 1:31 AM, Joe Leslie-Hurd <<a href="mailto:joe@gilith.com" target="_blank">joe@gilith.com</a>> wrote: <blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Now that there is a release of the opentheory tool for processing version 6 articles, and I am sitting on a release of the standard theory library that uses version 6 articles, I'd like to finalize version 6 of the article file format standard, which is currently in draft mode: <a href="http://www.gilith.com/research/opentheory/article.html" target="_blank">http://www.gilith.com/research/opentheory/article.html</a> The commands that are new in version 6 are highlighted in yellow and the command with changed semantics (defineTypeOp) is highlighted in red. We have discussed the changes on this list, so there shouldn't be any surprises in store. Actually, there is one minor change that hasn't been discussed before. During implementation work on defineTypeOp I discovered that one of Mario's proposed theorems ⊦ p = (\r. rep (abs r) = r) (1) required the axiom of extensionality to be logically equivalent to the same theorem produced by version 5 of defineTypeOp: ⊦ p r = (rep (abs r) = r) (2) Although it would be theoretically possible to invoke the axiom of extensionality to convert between the two versions of defineTypeOp, I thought it was more elegant to reformulate theorem (1) by eta-expanding p: ⊦ (\r. rep (abs r) = r) = \r. p r (3) [I also flipped the LHS and RHS so that theorem (3) has a pleasing similarity to the other theorem produced by defineTypeOp.] Now theorem (3) is logically equivalent to theorem (2) without requiring the axiom of extensionality, and retains the desirable property of having no free term variables. Anyway, there are no other "brown bag updates" to the standard, and I'd like to finalize it so we can start using it for real work. Does anybody have any objections or last-minute additions? Cheers, Joe _______________________________________________ opentheory-users mailing list <a href="mailto:opentheory-users@gilith.com">opentheory-users@gilith.com</a> <a href="http://www.gilith.com/opentheory/mailing-list" target="_blank">http://www.gilith.com/opentheory/mailing-list</a> </blockquote></div> </div>