*To*: Andreas Lochbihler <andreas.lochbihler at inf.ethz.ch>, Lawrence Paulson <lp15 at cam.ac.uk>*Subject*: Re: [isabelle] extending well-founded partial orders to total well-founded orders*From*: Andrei Popescu <uuomul at yahoo.com>*Date*: Mon, 18 Feb 2013 17:40:31 -0800 (PST)*Cc*: Christian Sternagel <c.sternagel at gmail.com>, cl-isabelle-users at lists.cam.ac.uk*In-reply-to*: <5A47F103-8F19-47FF-8BA5-04F6DBF18F16@cam.ac.uk>

I had a look at theorem well_ordering from Zorn, and it is not clear how to adapt its proof to this case. The set of wellorders disjoint from W^-1, with the initial-segment relation, does satisfy the hypothesis of Zorn, yielding a maximal element R. But then it is not apparent whether W <= R. Or maybe use a stronger notion of consistency with W? I confess I did not try too hard though, being happily married with my proof which traverses W breadth-first. Regards, Andrei --- On Mon, 2/18/13, Lawrence Paulson <lp15 at cam.ac.uk> wrote: From: Lawrence Paulson <lp15 at cam.ac.uk> Subject: Re: [isabelle] extending well-founded partial orders to total well-founded orders To: "Andreas Lochbihler" <andreas.lochbihler at inf.ethz.ch> Cc: "Christian Sternagel" <c.sternagel at gmail.com>, "Andrei Popescu" <uuomul at yahoo.com>, cl-isabelle-users at lists.cam.ac.uk Date: Monday, February 18, 2013, 5:22 PM I see, it is a little more subtle than I thought, but I would guess that the proof of the well-ordering theorem itself can be modified to exhibit a well-ordering that is consistent with a given well-founded relation. As I recall, the well-ordering theorem is proved by Zorn's lemma and considers the set of all well orderings of subsets of a given set; one would need to modify the argument to restrict attention to well orderings that were consistent with W. Surely this theorem has a name and a proof can be found somewhere. Larry On 18 Feb 2013, at 14:55, Andreas Lochbihler <andreas.lochbihler at inf.ethz.ch> wrote: > Hi Larry, > > As far as I understand your suggestion, it does not even yield a ordering. For example, suppose that there are three elements a, b, c such that the well ordering <R obtained by the well-ordering theorems orders them as a <R b <R c. Now, consider the well-founded relation <W which orders c <W a and nothing else. Then, c <TO a as c<W a, and a <TO b and b <TO c as neither a <W b, b <W a, b <W c, nor c <W b. Hence, > > c <TO a <TO b <TO c > > which violates the ordering properties. Or am I misunderstanding something? > > Andreas > > On 02/18/2013 03:18 PM, Lawrence Paulson wrote: >> I still don't see what's wrong with the following approach: >> >> 1. Prove the well ordering theorem (maybe it has been proved already). >> >> 2. Obtain the desired total ordering as a lexicographic combination of the partial order with the total well ordering of your type >> >> [More specifically: given W a well founded relation and R a well ordering obtained by the well ordering theorem, define TO x y == W x y | (~ W x y & ~W y x & R x y)] >> >> Larry Paulson >> >> >> On 18 Feb 2013, at 06:32, Christian Sternagel <c.sternagel at gmail.com> wrote: >> >>> Dear Andrei, >>> >>> finally deadlines are over for the time being and I found your email again ;) >>> >>> On 01/19/2013 12:22 AM, Andrei Popescu wrote: >>>> My AFP formalization ordinals >>>> >>>> http://afp.sourceforge.net/entries/Ordinals_and_Cardinals.shtml >>> >>> I guess since Isabelle2013 this is now "~~/src/HOL/Cardinals/", right? >>> >>>> (hopefully) provides the necessary ingredients: Initial segments in >>>> Wellorder_Embedding, ordinal sum in theory Constructions_on_Wellorders, >>>> and a transfinite recursion combinator (a small adaptation of the >>>> wellfounded combinator) in theory Wellorder_Relation. >>> >>> Could you elaborate on the mentioned finite recursion combinator and how it is used? >>> >>> thanks in advance, >>> >>> chris >>> >>> >>> >> >>

**References**:**Re: [isabelle] extending well-founded partial orders to total well-founded orders***From:*Lawrence Paulson

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