Re: [isabelle] Well-foundedness of Relational Composition



Dear Tjark and Tobias,

This is my first impression: I've definitely seen this fact applied (although implicitly) and think it counts as, as they say, "folklore".

Especially so, since the paper "proof" is trivial ;):

Assume: RS RS RS RS RS RS ...
Rearrange into: R SR SR SR SR SR ...
Apply "RS <= SR": R RS RS RS RS RS ...
Repeat this process to obtain: R R R R R R ...
Contradiction.

I checked IsaFoR by hypersearch but didn't find anything that looks alike immediately. That is, I'm not aware of any formalized proof.


After investigating a little further, I found some relevant literature. Commutation has been investigated by Rosen [3], but as far as I can tell on a short glance only w.r.t. confluence. Bachmair and Dershowitz [2] pronounce "RS <= SR" as "R *commutes over* S" (whereas "commutation" without "over" would result in a "diamond" diagram). And Geser [1] (which was the reference I started from) at least comments about something similar to your lemma on page 38:

  If R is Noetherian, it is more convenient to check for strict
  local commutation than for quasi-commutation: They are
  equivalent, and checking for strict local commutation is
  less complex ...

But also here I did not find a proof by skimming through. Maybe a closer look would reveal something.

[1] Alfons Geser. Relative Termination. PhD-Thesis. 1990.
[2] Leo Bachmair, Nachum Dershowitz. Commutation, Transformation, and Termination. CADE. 1986. http://dx.doi.org/10.1007/3-540-16780-3_76 [3] Barry K. Rosen. Tree-Manipulating Systems and Church-Rosser Theorems. J. ACM (JACM) 20(1):160-187 (1973). http://doi.acm.org/10.1145/321738.321750

cheers

chris

On 04/26/2015 03:51 PM, Tjark Weber wrote:
Tobias,

On Sun, 2015-04-26 at 09:24 +0200, Tobias Nipkow wrote:
I'll be happy to include it in HOL/Wellfounded.thy if you do me the favour of
getting rid of metis - it is not available at that theory yet.

Certainly. A proof that uses only simp and blast is attached.

Maybe somebody can also think of a more telling name suffix than "_compatible".

This was inspired by the existing lemma wf_union_compatible. Of course,
it is a terrible name.

Best,
Tjark

===== 8< =====

lemma wf_relcomp_compatible:
   assumes "wf R" and "R O S \<subseteq> S O R"
   shows "wf (S O R)"
proof (rule wfI_pf)
   fix A assume A: "A \<subseteq> (S O R) `` A"
   {
     fix n have "(S ^^ n) `` A \<subseteq> R `` (S ^^ Suc n) `` A"
     proof (induction n)
       case 0 show ?case
         using A by (simp add: relcomp_Image)
     next
       case (Suc n)
       then have "S `` (S ^^ n) `` A \<subseteq> S `` R `` (S ^^ Suc n)
`` A"
         by (simp add: Image_mono)
       also have "\<dots> \<subseteq> R `` S `` (S ^^ Suc n) `` A"
         using assms(2) by (simp add: Image_mono O_assoc
relcomp_Image[symmetric] relcomp_mono)
       finally show ?case
         by (simp add: relcomp_Image)
     qed
   }
   then have "(\<Union>n. (S ^^ n) `` A) \<subseteq> R `` (\<Union>n. (S
^^ n) `` A)"
     by blast
   then have "(\<Union>n. (S ^^ n) `` A) = {}"
     using assms(1) by (simp only: wfE_pf)
   then show "A = {}"
     using relpow.simps(1) by blast
qed







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