*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 11:58:34 +0200*In-reply-to*: <1430056276.6087.7.camel@weber>*References*: <1429885969.5902.26.camel@weber> <553C929B.208@in.tum.de> <1430056276.6087.7.camel@weber>*User-agent*: Mozilla/5.0 (X11; Linux x86_64; rv:31.0) Gecko/20100101 Thunderbird/31.6.0

Dear Tjark and Tobias,

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.

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 ...

[1] Alfons Geser. Relative Termination. PhD-Thesis. 1990.

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:*Christian Sternagel

**Re: [isabelle] Well-foundedness of Relational Composition***From:*Jeremy Dawson

**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

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