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The document also points to an external source for a description of the formalization referenced in the bibliography.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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\nAbstract  \nThis theory contains a formalization of decreasing diagrams show  \ning that any locally decreasing abstract rewrite system is confluent.  \nWe consider the valley (van Oostrom, TCS 1994) and the conversion  \nversion (van Oostrom, RTA 2008) and closely follow the original proofs.  \nAs an application we prove Newman’s lemma.  \nA description of this formalization is available in [3] .  \nContents  \n1 Decreasing Diagrams 1  \n1.1 Valley Version . . . . . . . . . . . . . . . . . . . . . . . . . . . 1  \n1.1. 1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 1  \n1.1.2 Multisets . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.1.3 Lexicographic maximum measure . . . . . . . . . . . . 12  \n1.1.4 Labeled Rewriting . . . . . . . . . . . . . . . . . . . . 20  \n1.1.5 Application: Newman’s Lemma . . . . . . . . . . . . . 27  \n1.2 Conversion Version . . . . . . . . . . . . . . . . . . . . . . . . 29  \n1 Decreasing Diagrams  \ntheory Decreasing-Diagrams imports HOL􀀀Library:Multiset Abstract􀀀Rewriting:Abstract-Rewriting begin  \n1.1 Valley Version  \nThis section follows [1] .  \n1.1.1 Appendix  \ninteraction of multisets with sets  \ndefinition diff :: 0a multiset ) 0a set ) 0a multiset where diff M S = filter-mset (􀀕x: x  S) M  \ndefinition intersect :: 0a multiset ) 0a set ) 0a multiset where intersect M S = filter-mset (􀀕x: x 2 S) M  \nnotation  \ndiff (infixl ‹􀀀s› 800 ) and intersect (infixl ‹\\s› 800 )  \nlemma count-diff [simp]:  \ncount (M 􀀀s A) a = count M a 􀀃 of-bool (a  A)  \nby (simp add: diff-def )  \nlemma set-mset-diff [simp]:  \nset-mset (M 􀀀s A) = set-mset M 􀀀 Aby (auto simp add: diff-def )  \nlemma diff-eq-singleton-imp:  \nM 􀀀s A = f\\#a\\#g =) a 2 (set-mset M 􀀀 A) unfolding diff-def filter-mset-eq-conv by auto  \nlemma count-intersect [simp]:  \ncount (M \\s A) a = count M a 􀀃 of-bool (a 2 A)  \nby (simp add: intersect-def )  \nlemma set-mset-intersect [simp]:  \nset-mset (M \\s A) = set-mset M \\ Aby (auto simp add: intersect-def )  \nlemma diff-from-empty: f\\#g􀀀s S = f\\#g unfolding diff-def by auto  \nlemma diff-empty: M 􀀀s fg = M unfolding diff-def by (rule multiset-eqI ) simp  \nlemma submultiset-implies-subset: assumes M 􀀒\\# N shows set-mset M 􀀒 set-mset N  \nusing assms mset-subset-eqD by auto  \nlemma subset-implies-remove-empty: assumes set-mset M 􀀒 S shows M 􀀀s S = f\\#g  \nunfolding diff-def using assms by (induct M ) auto  \nlemma remove-empty-implies-subset: assumes M 􀀀s S = f\\#g shows set-mset M 􀀒 S proof  \nfix x assume A: x 2 set-mset M  \nhave x  set-mset (M 􀀀s S) using assms by auto thus x 2 S using A unfolding diff-def by auto qed  \nlemma lemmaA-3-8 : (M + N ) 􀀀s S = (M 􀀀s S) + (N 􀀀s S) unfolding diff-def by (rule multiset-eqI ) simp  \nlemma lemmaA-3-9 : (M 􀀀s S ) 􀀀s T = M 􀀀s (S [ T ) unfolding diff-def by (rule multiset-eqI ) simp  \nlemma lemmaA-3-10 : M = (M \\s S ) + (M 􀀀s S ) unfolding diff-def intersect-def by auto  \nlemma lemmaA-3-11 : (M 􀀀s T ) \\s S = (M \\s S ) 􀀀s T unfolding diff-def intersect-def by (rule multiset-eqI ) simp  \n1.1.2 Multisets  \nDefinition 2.5(1)  \ndefinition ds :: 0a rel ) 0a set ) 0a set where ds r S = fy : 9 x 2 S: (y ;x) 2 rg  \ndefinition dm :: 0a rel ) 0a multiset ) 0a set where dm r M = ds r (set-mset M )  \ndefinition dl :: 0a rel ) 0a list ) 0a set where dl r 􀀛 = ds r (set 􀀛)  \nnotation  \nds (infixl ‹\\#s› 900 ) and  \ndm (infixl ‹\\# m› 900 ) and  \ndl (infixl ‹\\#l› 900 ) missing but useful  \nlemma ds-ds-subseteq-ds: assumes t: trans r shows ds r (ds r S ) 􀀒 ds r S proof  \nfix x assume A: x 2 ds r (ds r S ) show x 2 ds r S proof 􀀀  \nfrom A obtain y z where (x ; y) 2 r and (y ;z) 2 r and mem: z 2 S unfolding ds-def by auto  \nthus ?thesis using mem t trans-def unfolding ds-def by fast qed  \nqed  \nfrom PhD thesis of van Oostrom  \nlemma ds-monotone: assumes S 􀀒 T shows ds r S 􀀒 ds r T using assms unfolding ds-def by auto  \nlemma subset-imp-ds-subset: assumes trans r and S 􀀒 ds r T shows ds r S 􀀒  \nds r T  \nusing assms ds-monotone ds-ds-subseteq-ds by blast Definition 2.5(2) ","cbCaitzzUfzMd7nQ","https://ap.wps.com/l/cbCaitzzUfzMd7nQ","pdf",440368,47,"English","# Decreasing Diagrams\n## Valley Version\n## Conversion Version","[{\"question\":\"What does the theory prove about locally decreasing abstract rewrite systems?\",\"answer\":\"It proves that any locally decreasing abstract rewrite system is confluent.\"},{\"question\":\"Which two versions of decreasing diagrams are considered?\",\"answer\":\"The theory considers the valley version and the conversion version, following van Oostrom’s original proofs.\"},{\"question\":\"What application result is proved using the formalization?\",\"answer\":\"Newman’s lemma is proved as an application of the decreasing-diagrams framework.\"}]","Decreasing-Diagrams - Formalization of Decreasing Diagrams | PDF",1792981743,118]