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

[Christian Sternagel]

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.)
>