Re: [isabelle] Term rewriting systems

I am not aware of any support for confluence proofs other than abstract
relation-algebraic theorems.

Have you proved confluence by hand yet? By some standard method, like
absence of critical pairs? If not, do that first, by reducing it to some
standard variety of rewrite system. As you correctly note, list patterns
of the form xs @ [t] @ ys could be a problem. Either you have to do
rewriting modulo associativity of @ or add associativity of @ in both
dirctions as a rewrite rule, which loses termination but may preserve


Peter Lammich schrieb:
> 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

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