Re: [isabelle] Well-foundedness of Relational Composition

I am not aware of this lemma from the literature, but maybe our friends in Innsbruck are?

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


On 24/04/2015 16:32, Tjark Weber wrote:
The attached lemma about well-foundedness of relational composition
featured prominently in a recent termination proof of mine. If the
lemma is not available already, I would like to propose it for
inclusion in HOL/Wellfounded.thy (or some derived theory).

I discovered the lemma and proof myself, but I suspect that the result
is well-known. Pointers to the literature would be appreciated.


===== 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)
       case (Suc n)
       then have "S `` (S ^^ n) `` A \<subseteq> S `` R `` (S ^^ Suc n)
`` A"
         by (metis Image_mono subsetCI)
       also have "\<dots> \<subseteq> R `` S `` (S ^^ Suc n) `` A"
         using assms(2) by (metis (no_types, hide_lams) Image_mono
order_refl relcomp_Image)
       finally show ?case
         by (metis relcomp_Image relpow.simps(2))
   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 (metis wfE_pf)
   then show "A = {}"
     using relpow.simps(1) by blast

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