[isabelle-dev] Zorn's lemma, the well-ordering-theorem, and extending well-founded relations to (total) well-orders

Christian Sternagel c.sternagel at gmail.com
Wed Feb 27 13:14:29 CET 2013

Dear Larry

please note that my proposal is not just about a split of the existing 
theory Zorn.thy, but also about a modernization of part of it (which I 
think makes it easier to understand, but I might be wrong... could be 
that the main purpose of this experiment was just to make me understand 
the formalized proofs ;)) as well as adding new facts (the 
order-extension principle). So please consider it, even if no split is done.

Nevertheless. Separating facts that are about the subset relation from 
the more general version of Zorn's lemma would make sense for at least 
one purpose: reusing the former in developments that use a different 
definition of partial order (and that are "incompatible" with the latter).

As to the point that a split would make examination of past versions 
more difficult. How do you mean? True, it would be hard to compare a 
version that comes somewhere after the split with one somewhere before 
the split (via plain diff), but how often does that happen? Isn't the 
typical use-case comparison of successive changesets?



On 02/27/2013 08:49 PM, Lawrence Paulson wrote:
> I don't see the point of splitting Zorn into multiple files. It isn't especially large. Such a change really has nothing to do with the question of locating proved results, and it would make it harder to examine past versions.
> Larry
> On 27 Feb 2013, at 05:57, Christian Sternagel <c.sternagel at gmail.com> wrote:
>> Dear all,
>> in the meanwhile I had a close look at the existing Zorn.thy (mostly to understand the proof myself) and came up with the following proposal:
>> see
>> https://bitbucket.org/csternagel/zorns-lemma-and-the-well-ordering-theorem/
>> for the related hg repository (from which you will hopefully merge into the Isabelle repo ;)).
>> I propose the following changes to ~~/src/HOL/Cardinals and ~~/src/HOL/Library.
>> 1) Make facts about the ordinal sum available in a separate theory, to avoid too early dependency on the old ~~/src/HOL/Library/Zorn. This is a prerequisite to make the remainder of my proposal work. (see Ordinal_Sum.thy)
>> 2) Split the current Zorn.thy into three separate parts.
>>   - Zorn_Subset.thy
>>     Here we are only concerned with the special case of Zorn's lemma for the subset relation. This constitutes a modernized version of the old Zorn.thy, employing locales for structuring (cf. Andrei's rel locale in ~~/src/HOL/Cardinals; I find this kind of structuring very convenient) and only Isar proofs (some of the old apply scripts were very brittle, e.g., using auto or simp as initial proof steps). Hopefully it is also easier to understand than the old scripts (or maybe it is just because I spend so much time with the proofs ;)).
>>   - Zorn.thy
>>     The general version of Zorn's lemma for arbitrary partial orders.
>>   - Well_Ordering_Theorem.thy
>>     The well-ordering-theorem. It seems important enough to give it it's own theory. Moreover, in the previous setup it seemed to be easily overlooked (not even some Isabelle veterans knew whether it was already formalized).
>> 3) Add a formalization of the well-order extension theorem to ~~/src/HOL/Library. (see Well_Order_Extension.thy)
>> In My_Zorn.thy it is illustrated that the new structure is more versatile than the old one. It is, e.g,. very easy to combine it with my alternative definitions of partial orders (po_on from AFP/Well_Quasi_Orders/Restricted_Predicates).
>> cheers
>> chris
>> On 02/21/2013 01:58 PM, Christian Sternagel wrote:
>>> Dear all,
>>> how about adding Andrei's proof (discussed on isbelle-users) to
>>> HOL/Library (or somewhere else)? I polished the latest version (see
>>> attachment).
>>> cheers
>>> chris
>>> PS: In case you are wondering: "Why is he interested in these results?"
>>> Here is my original motivation: In term rewriting, termination tools
>>> employ simplification orders (like the Knuth-Bendix order, the
>>> lexicographic path order, ...) whose definition is often based on a
>>> given well-partial-order as precedence. However, termination tools
>>> typically use well-founded partial orders as precedences. Thus we need
>>> to be able to extend a given well-founded (partial order) relation to a
>>> well-partial-order when we want to apply the theoretical results about
>>> simplification orders to proofs that are generated by termination tools.
>>> (Since every total well-order is also a well-partial-order, this is
>>> possible by the above mentioned results.)
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