(Note that well-foundedness of the given partial order is irrelevant. So the actual lemma is: every partial order can be extended to a total well-order.)
Now, I'll check whether this is also applicable in my setting (with explicit domains).
Thanks for all your hints and suggestions! (I'm still interested in the alternative proof ;))
cheers chris On 02/19/2013 12:22 AM, Lawrence Paulson wrote:
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.shtmlI 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