*To*: cl-isabelle-users at lists.cam.ac.uk*Subject*: Re: [isabelle] Well-foundedness of Relational Composition*From*: Christian Sternagel <c.sternagel at gmail.com>*Date*: Mon, 27 Apr 2015 13:04:17 +0200*In-reply-to*: <553E084A.3020303@gmail.com>*References*: <1429885969.5902.26.camel@weber> <553C929B.208@in.tum.de> <1430056276.6087.7.camel@weber> <553E084A.3020303@gmail.com>*User-agent*: Mozilla/5.0 (X11; Linux x86_64; rv:31.0) Gecko/20100101 Thunderbird/31.6.0

Okay, here is a little bit more:

lemma qc_wf_relto_iff: assumes "R O S â S O (R â S)â*" -- âR quasi-commutes over Sâ shows "wf (Sâ* O R O Sâ*) â wf R" proof assume "wf (Sâ* O R O Sâ*)" moreover have "R â Sâ* O R O Sâ*" by auto ultimately show "wf R" using wf_subset by auto next assume "wf R" then show "wf (Sâ* O R O Sâ*)" sorry qed It can be used to proof "Tjark's Lemma" :) corollary assumes "wf R" and "R O S â S O R" shows "wf (S O R)" proof - have "R O S â S O (R â S)â*" using âR O S â S O Râ by auto then have "wf (Sâ* O R O Sâ*)" using âwf Râ by (simp add: qc_wf_relto_iff) moreover have "S O R â Sâ* O R O Sâ*" by auto ultimately show ?thesis using wf_subset by auto qed Depending on who you cite the "subset assumption" could be called: - R commutes over S (BD86)

- ... hope this helps chris On 04/27/2015 11:58 AM, Christian Sternagel wrote:

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

**Follow-Ups**:**Re: [isabelle] Well-foundedness of Relational Composition***From:*Tjark Weber

**References**:**[isabelle] Well-foundedness of Relational Composition***From:*Tjark Weber

**Re: [isabelle] Well-foundedness of Relational Composition***From:*Tobias Nipkow

**Re: [isabelle] Well-foundedness of Relational Composition***From:*Tjark Weber

**Re: [isabelle] Well-foundedness of Relational Composition***From:*Christian Sternagel

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