Re: [isabelle] Term rewriting systems



Peter Lammich wrote:
Hi all,

I wonder whether there is any tool support (best Isabelle compatible tool support) to reason about term rewriting systems of the following form:

I have a term rewriting system defined over terms of the form:

datatype 'm action = ENTER "'m" | LEAVE "'m" | USE "'m set"
datatype 'm tree = NIL | NODE "'m action" "'m tree list" "'m tree"

the rules are like this:

1) NODE (ENTER m) c1 (NODE (LEAVE m) c2 t) -> NODE (USE m) (c1 at c2) t 2) NODE (ENTER m) c1 (NODE (USE m') c2 (NODE (LEAVE m) c3 t)) -> NODE (USE (m\<union>m')) (c1 at c2@c3) t 3) NODE (USE u1) c1 (NODE (USE u2) c2 t) -> NODE (USE (u1\<union>u2)) (c1 at c2) t 4) NODE (USE u) (c1 at NIL#c2) t -> NODE (USE u) (c1 at c2) t 5) NODE (USE u) (c1@(NODE (USE u1) cs1 ts1)@c2@(NODE (USE u2) cs2 ts2)@c3) t -> NODE (USE u) (c1@(NODE (USE (u1\<union>u2)) cs1 ts1)#c2 at cs2@ts2#c3) t 6) NODE (USE u1) (c1@(NODE (USE u2) cs ts)#c2) NIL -> NODE (USE (u1\<union>u2)) (c1 at cs@ts#c2)

My rewriting strategy is rewriting of an arbitrary subterm. Currently I define an inductive predicate step ("->") by the rules 1-6 and additionally the inductive rules:
7) t -> t'  ==>  NODE a c t                    -> NODE a c t'
8) t -> t'  ==>  NODE a (c1 at t#c2) x   -> NODE a (c1 at t'#c2) x

I want to show termination and confluence of this system. Termination is easy, as the size of the tree decreases in any step, hence I easily get
 lemma "wfP (step^--)"

By Newman's lemma (A version is in HOL/Lambda/Commutation.thy), it suffices to show local confluence, I even think my system above has the diamond property.

Is there any standard approach to term rewriting systems in Isabelle or are there some other tools out there, to show confluence (and termination) as automatic as possible? Are there any suggestions on how to show local confluence of such a system in Isabelle (as automatic as possible)?

The main problem seems to be the non-constructor patterns (like c1 at t#c2) I use in my rewriting rules (4,5,6,8)

Thanks for any suggestions/comments in advance, best
 Peter



Peter,

I have a lot of proofs relating to some theorems for proving rewrite systems terminating. There are a number of examples of its application to specific rewrite rule systems.

The model of term which is rewritten is (n-ary function symbol) (list of n subterms),
so rewriting a subterm naturally is rewriting f (c1 at t#c2) to f (c1 at t'#c2).

The work is focussed on proving general results rather than applying them to particular systems, but you may find it useful. At the moment it runs in Isabelle 2005, I started trying to update it to Isabelle 2007 but the incompatibilities are significant.

It's at http://users.rsise.anu.edu.au/~jeremy/isabelle/2005/snabs/
(also uses http://users.rsise.anu.edu.au/~jeremy/isabelle/2005/gen/)

Jeremy










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