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3748 y(After)26 b(in)m(tro)s(ducing)f(required)h(preliminaries)d(on)i (orderings)h(in)f(Section)h(2,)f(in)g(Section)h(3)183 3869 y(a)32 b(re\014nemen)m(t)h(of)f(the)h(dep)s(endency)i(pair)c(tec)m (hnique)j(is)e(presen)m(ted)i(that)e(allo)m(ws)f FC(mo)-5 b(dular)183 3989 y FE(termination)40 b(pro)s(ofs)j(using)g(dep)s (endency)j(pairs.)c(In)i(other)f(w)m(ords,)h(no)m(w)g(sev)m(eral)g(w)m (ell-)183 4110 y(founded)24 b(relations)e(ma)m(y)h(b)s(e)h(used)g(in)f (the)h(termination)d(pro)s(of)h(of)h(one)h(TRS.)g(Applying)e(the)183 4230 y(dep)s(endency)36 b(pair)c(approac)m(h)i(in)f(the)h(prop)s(osed)g (mo)s(dular)e(w)m(a)m(y)j(cannot)f(complicate)d(the)183 4350 y(pro)s(of,)26 b(whereas)i(it)e(ma)m(y)h(allo)m(w)e(a)i (successful)i(application)24 b(where)k(the)g(original)23 b(tec)m(hnique)183 4471 y(failed.)35 b(Hence,)j(it)e(is)h(alw)m(a)m(ys) g(adv)-5 b(an)m(tageous,)38 b(and)f(often)g(more)f(p)s(o)m(w)m(erful,)h (to)g(tak)m(e)g(this)183 4591 y(mo)s(dular)30 b(approac)m(h)j(in)m(to)f 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b(to)e(kno)m(w)i(whether)f(a)g (large)e(TRS)i(has)g(a)f(certain)h(mo)s(dular)d(prop)s(ert)m(y)k FB(')p FE(,)e(then)h(this)183 5554 y(system)36 b(can)g(b)s(e)g(decomp)s (osed)g(in)m(to)f(small)e(subsystems)38 b(and)e(one)f(merely)g(has)h (to)g(c)m(hec)m(k)p eop %%Page: 3 3 3 2 bop 306 191 a FF(Giesl,)30 b(Arts,)h(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)f(Dep)s(endency)h(P)m(airs) 123 b(3)183 390 y FE(whether)39 b(eac)m(h)g(of)e(these)j(subsystems)g (has)e(prop)s(ert)m(y)h FB(')p FE(.)f(This)g(con)m(v)m(en)m(tional)g (notion)f(of)183 511 y(mo)s(dularit)m(y)26 b(is)j(inspired)g(b)m(y)h(a) f(w)m(ell-kno)m(wn)g(paradigm)e(in)h(computer)h(science;)i(programs)183 631 y(are)h(dev)m(elop)s(ed)h(in)e(small)f(mo)s(dules)h(that)h (together)g(form)f(the)h(whole)g(program.)f(In)h(prac-)183 751 y(tice)i(it)g(is)g(an)g(enormous)h(b)s(ene\014t)g(if)f(it)f (su\016ces)k(to)d(pro)m(v)m(e)i(a)f(prop)s(ert)m(y)g(of)f(a)g(mo)s (dule)g(just)183 872 y(once,)f(indep)s(enden)m(t)h(of)e(the)h(con)m (text)h(in)d(whic)m(h)j(the)f(mo)s(dule)e(is)h(used)i(afterw)m(ards.) 280 992 y(Clearly)-8 b(,)24 b(this)h(con)m(v)m(en)m(tional)g(notion)f (of)g(mo)s(dularit)m(y)f(can)i(also)g(b)s(e)g(applied)f(successfully) 183 1112 y(in)33 b(com)m(bination)f(with)i(the)g(original)d(dep)s (endency)37 b(pair)c(approac)m(h.)h(Ho)m(w)m(ev)m(er,)j(termina-)183 1233 y(tion)31 b(and)i(innermost)f(termination)f(are)i(not)f(mo)s (dular)f(prop)s(erties)i(for)f(arbitrary)g(TRSs.)183 1353 y(The)f(mo)s(dular)d(re\014nemen)m(t)k(of)e(the)h(dep)s(endency)i (pair)c(approac)m(h)i(in)m(tro)s(duced)f(in)g(Section)183 1474 y(3)40 b(is)f(applicable)g(to)h(n)m(umerous)h(TRSs)g(that)f(do)g (not)g(b)s(elong)f(to)h(one)h(of)f(the)g(restricted)183 1594 y(classes)33 b(where)h(con)m(v)m(en)m(tional)f(mo)s(dularit)m(y)d (results)j(are)f(applicable.)280 1714 y(T)-8 b(o)m(y)m(ama)22 b([1987])f(sho)m(w)m(ed)j(that)d(termination)e(is)j(not)f(ev)m(en)j(mo) s(dular)19 b(for)i(disjoin)m(t)g(unions,)183 1835 y(i.e.,)36 b(com)m(binations)e(of)i(TRSs)h(without)f(common)f(function)h(sym)m(b)s (ols.)g(So)g(the)g(question)183 1955 y(is)h(what)i(restrictions)e(ha)m (v)m(e)i(to)f(b)s(e)g(imp)s(osed)f(on)h(the)h(constituen)m(t)g(TRSs)f (so)h(that)e(their)183 2076 y(disjoin)m(t)44 b(union)g(is)h(again)e (terminating.)g(The)j(\014rst)g(results)f(w)m(ere)i(obtained)d(b)m(y)i (in)m(v)m(es-)183 2196 y(tigating)e(the)k(distribution)e(of)g (collapsing)f(rules)j(and)f(duplicating)e(rules)i(among)f(the)183 2316 y(TRSs;)36 b(see)g([Rusino)m(witc)m(h,)f(1987,)f(Middeldorp,)g (1989].)g(In)h([T)-8 b(o)m(y)m(ama)35 b(et)h(al.,)d(1995])h(it)g(is)183 2437 y(sho)m(wn)27 b(that)g(termination)d(is)i(mo)s(dular)e(for)i (con\015uen)m(t)i(and)f(left-linear)c(TRSs.)28 b(Ev)m(er)f(since)183 2557 y(an)k(abundance)i(of)e(mo)s(dularit)m(y)e(results)j(for)f (disjoin)m(t)f(unions,)i(constructor-sharing)f(sys-)183 2677 y(tems,)41 b(comp)s(osable)f(systems,)j(and)e(hierarc)m(hical)f (com)m(binations)f(has)j(b)s(een)g(published;)183 2798 y(see)33 b([Middeldorp,)f(1990,)g(Ohlebusc)m(h,)i(1994a,)e(Gramlic)m (h,)e(1996b])i(for)g(an)g(o)m(v)m(erview.)280 2918 y(Most)f(of)e(the)h (mo)s(dularit)m(y)e(results)i(are)g(often)g(not)g(applicable)d(in)j (practice.)f(F)-8 b(or)30 b(exam-)183 3039 y(ple,)k(collapsing)e(and)j (duplicating)e(rules)i(o)s(ccur)f(naturally)g(in)g(most)g(TRSs.)h(In)g (con)m(trast)183 3159 y(to)24 b(this,)h(since)h(most)f(standard)g (metho)s(ds)g(for)g FC(automate)-5 b(d)35 b FE(termination)23 b(pro)s(ofs)i(are)g(based)183 3279 y(on)e(syn)m(thesizing)h (simpli\014cation)c(orderings,)j(the)h(result)f(of)g(Kurihara)f(and)i (Oh)m(uc)m(hi)g([1992])183 3400 y(for)34 b(constructor-sharing)h (systems)i(is)e(of)f(practical)g(relev)-5 b(ance.)36 b(They)g(sho)m(w)m(ed)h(that)e(the)183 3520 y(constructor-sharing)28 b(com)m(bination)f(of)h(\014nite)h(simply)e(terminating)f(TRSs)k(is)e (again)g(sim-)183 3641 y(ply)35 b(terminating.)d(Their)k(result)f(w)m (as)h(extended)h(to)e(comp)s(osable)f(systems)j([Ohlebusc)m(h,)183 3761 y(1995])g(and)i(to)f(certain)g(hierarc)m(hical)e(com)m(binations)h ([Krishna)h(Rao,)g(1994].)f(Moreo)m(v)m(er,)183 3881 y(all)24 b(these)k(results)f(also)f(hold)f(for)h(in\014nite)g(TRSs;)h (see)h([Middeldorp)e(and)h(Zan)m(tema,)f(1997].)280 4002 y(Th)m(us,)42 b(if)d(one)h(has)g(a)g(metho)s(d)f(to)h(pro)m(v)m(e)h (simple)d(termination)g(of)h(a)h(TRS,)g(then)h(one)183 4122 y(can)30 b(use)g(this)g(metho)s(d)f(in)g(a)g(mo)s(dular)f(w)m(a)m (y)j(for)e(the)h(ab)s(o)m(v)m(e-men)m(tioned)g(classes)h(of)e(TRSs,)183 4242 y(whereas)e(an)e(arbitrary)g(metho)s(d)g(for)g(pro)m(ving)h (termination)d(cannot)j(b)s(e)g(used)g(in)f(this)h(w)m(a)m(y)-8 b(.)183 4363 y(Ho)m(w)m(ev)m(er,)29 b(simple)d(termination)f(is)h(a)h (considerably)g(restricted)h(form)e(of)h(termination.)d(As)183 4483 y(indicated)f(ab)s(o)m(v)m(e,)i(the)g(reason)f(for)g(the)g(dev)m (elopmen)m(t)h(of)f(the)h(dep)s(endency)h(pair)d(approac)m(h)183 4604 y(w)m(as)41 b(that)f(there)h(are)f(n)m(umerous)h(relev)-5 b(an)m(t)41 b(TRSs)g(for)f(whic)m(h)g(simpli\014cation)d(orderings)183 4724 y(fail)d(in)h(pro)m(ving)i(termination.)d(Th)m(us,)k(no)m(w)f (TRSs)g(for)f(whic)m(h)h(automated)e(termination)183 4844 y(pro)s(ofs)h(are)g(\(p)s(oten)m(tially\))e(feasible)h(are)i(no)f (longer)f(just)i(simply)e(terminating)e(systems,)183 4965 y(but)28 b FC(DP)i(\(quasi-\)simply)g(terminating)36 b FE(systems,)30 b(i.e.,)d(systems)j(whose)f(termination)d(can)183 5085 y(b)s(e)32 b(v)m(eri\014ed)g(b)m(y)h(using)e (\(quasi-\)simpli\014cation)d(orderings)j(in)g(com)m(bination)f(with)h (dep)s(en-)183 5205 y(dency)41 b(pairs.)e(Hence,)i(a)f(natural)e (question)i(is)g(whether)h(the)f(curren)m(t)h(state)f(of)f(the)h(art) 183 5326 y(of)c(mo)s(dularit)m(y)e(can)j(b)s(e)f(re\014ned)i(as)f(w)m (ell)e(b)m(y)j(extending)f(the)g(con)m(v)m(en)m(tional)f(mo)s(dularit)m (y)183 5446 y(results)29 b(from)e(simple)g(to)h(DP)g(\(quasi-\)simple)f (termination.)f(In)i(Section)h(4)f(w)m(e)h(sho)m(w)h(that)183 5567 y(this)g(is)h(indeed)h(p)s(ossible.)e(Th)m(us,)j(the)f(n)m(um)m(b) s(er)f(of)g(TRSs)h(for)f(whic)m(h)h(termination)c(can)k(b)s(e)p eop %%Page: 4 4 4 3 bop 306 191 a FF(Giesl,)30 b(Arts,)h(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)f(Dep)s(endency)h(P)m(airs) 123 b(4)183 390 y FE(pro)m(v)m(ed)32 b(in)f(a)g(mo)s(dular)e(w)m(a)m(y) j(is)f(extended)i(signi\014can)m(tly)-8 b(.)30 b(The)i(practical)e (consequence)k(of)183 511 y(this)d(result)g(is)g(that)g(if)f(one)i(has) f(pro)m(v)m(ed)i(termination)c(of)i(a)g(TRS)h(using)f(the)g(dep)s (endency)183 631 y(pair)k(approac)m(h,)j(then)f(adding)f(a)g(TRS)i(and) e(pro)m(ving)h(termination)d(of)j(the)g(new)g(com)m(bi-)183 751 y(nation)29 b(reduces)k(to)d(no)g(more)g(than)h(pro)m(ving)f (termination)e(of)i(the)h(added)g(TRS)g(with)f(the)183 872 y(dep)s(endency)35 b(pair)c(tec)m(hnique.)280 992 y(Subsequen)m(tly)-8 b(,)42 b(w)m(e)e(consider)f FC(innermost)48 b FE(termination,)37 b(i.e.,)i(the)h(requiremen)m(t)f(that)183 1112 y(all)26 b(reductions)k(where)g(only)e FC(innermost)37 b FE(redexes)31 b(are)e(rewritten)g(are)f(\014nite.)h(W)-8 b(e)29 b(dev)m(elop)183 1233 y(a)38 b(mo)s(dular)f(tec)m(hnique)k(for)d (innermost)g(termination)f(pro)s(ofs)h(using)h(dep)s(endency)i(pairs) 183 1353 y(in)31 b(Section)i(5.)280 1474 y(The)41 b(kno)m(wn)h(mo)s (dularit)m(y)37 b(results)k(for)e(innermost)h(termination)d(are)j(less) h(restrictiv)m(e)183 1594 y(than)27 b(those)g(for)g(termination.)d (Innermost)j(termination)e(is)h(mo)s(dular)f(for)h(disjoin)m(t)g (unions)183 1714 y(and)41 b(for)f(TRSs)h(with)g(shared)h(constructors)g ([Gramlic)m(h,)c(1995],)i(for)g(comp)s(osable)g(con-)183 1835 y(structor)d(systems)h([Middeldorp)f(and)g(T)-8 b(o)m(y)m(ama,)37 b(1993],)f(for)g(comp)s(osable)g(TRSs)i([Ohle-)183 1955 y(busc)m(h,)h(1995],)d(and)h(for)g(prop)s(er)g(extensions)h ([Krishna)f(Rao,)g(1995],)f(whic)m(h)i(are)f(sp)s(ecial)183 2076 y(hierarc)m(hical)23 b(com)m(binations.)f(As)j(innermost)f (termination)e(implies)g(termination)g(for)i(sev-)183 2196 y(eral)34 b(classes)i(of)f(TRSs)i([Gramlic)m(h,)c(1995,)h(1996a],) h(these)h(results)g(can)g(also)e(b)s(e)i(used)g(for)183 2316 y(termination)27 b(pro)s(ofs)i(of)g(suc)m(h)i(systems.)h(F)-8 b(or)28 b(example,)i(this)f(holds)g(for)g(lo)s(cally)e(con\015uen)m(t) 183 2437 y(o)m(v)m(erla)m(y)33 b(systems)h(\(and)f(in)e(particular)g (for)h(non-o)m(v)m(erlapping)g(TRSs\).)280 2557 y(In)42 b(Section)g(6)f(w)m(e)i(sho)m(w)g(that)f(the)g(mo)s(dular)d(dep)s (endency)45 b(pair)40 b(approac)m(h)j(leads)e(to)183 2677 y(new)d(mo)s(dularit)m(y)d(criteria)h(for)h(innermost)f (termination)f(\(whic)m(h)j(can)g(also)e(b)s(e)i(used)g(in-)183 2798 y(dep)s(enden)m(tly)32 b(of)f(the)g(dep)s(endency)j(pair)c(tec)m (hnique\).)i(Moreo)m(v)m(er,)h(w)m(e)f(demonstrate)f(that)183 2918 y(in)43 b(our)h(framew)m(ork)h(the)g(kno)m(wn)g(mo)s(dularit)m(y)d (results)j(for)f(innermost)f(termination)f(of)183 3039 y(comp)s(osable)31 b(TRSs)j(and)e(prop)s(er)h(extensions)h(are)e (obtained)g(as)h(easy)h(consequences.)280 3159 y(Preliminary)27 b(v)m(ersions)j(of)f(parts)g(of)g(this)g(article)f(app)s(eared)h(in)g ([Arts)h(and)f(Giesl,)f(1998])183 3279 y(and)k([Giesl)g(and)g(Ohlebusc) m(h,)i(2000].)183 3578 y FD(2.)68 b(Preliminaries)46 b(on)f(Orderings)183 3748 y FE(W)-8 b(e)38 b(assume)h(the)f(reader)h (to)e(b)s(e)i(familiar)33 b(with)38 b(the)h(basic)f(notions)f(of)h (term)f(rewriting.)183 3869 y(F)-8 b(or)26 b(an)h(in)m(tro)s(duction)f (to)h(term)f(rewriting)g(see)j(e.g.)e([Dersho)m(witz)h(and)f (Jouannaud,)h(1990,)183 3989 y(Klop,)g(1992,)h(Baader)g(and)h(Nipk)m(o) m(w,)g(1998].)f(W)-8 b(e)30 b(restrict)f(ourselv)m(es)i(to)e(\014nite)h (signatures)183 4110 y(con)m(taining)g(at)i(least)f(one)i(constan)m(t)f (\(i.e.,)g(w)m(e)h(assume)f(that)g(there)h(exist)f FC(gr)-5 b(ound)34 b(terms)8 b FE(\))183 4230 y(and)37 b(to)g(TRSs)h(with)f (\014nitely)f(man)m(y)h(rules.)h(In)f(the)h(follo)m(wing)c(w)m(e)39 b(in)m(tro)s(duce)e(the)g(bac)m(k-)183 4350 y(ground)28 b(material)d(on)j(orderings)g(whic)m(h)h(is)e(relev)-5 b(an)m(t)28 b(to)g(this)g(article.)f(A)h FC(r)-5 b(ewrite)31 b(or)-5 b(dering)183 4471 y FA(\037)37 b FE(o)m(v)m(er)g(a)g(set)g(of)f (terms)g FA(T)26 b FE(\()p FA(F)10 b FB(;)17 b FA(V)8 b FE(\))36 b(is)g(an)g(ordering)g(\(i.e.,)g(an)h(irre\015exiv)m(e)g (and)f(transitiv)m(e)183 4591 y(relation\))23 b(that)j(is)f (\(strongly\))g(monotonic)f(\(i.e.,)h FB(s)j FA(\037)g FB(t)e FE(implies)d FB(f)11 b FE(\()p FB(:)17 b(:)g(:)f(s)h(:)g(:)g(:)o FE(\))28 b FA(\037)g FB(f)11 b FE(\()p FB(:)17 b(:)g(:)f(t)h(:)g(:)g(:) o FE(\))183 4711 y(for)24 b(all)e(function)j(sym)m(b)s(ols)f FB(f)38 b FA(2)29 b(F)10 b FE(\))24 b(and)g(closed)h(under)h (substitutions)e(\(i.e.,)h FB(s)i FA(\037)h FB(t)d FE(implies)183 4832 y FB(s\033)36 b FA(\037)d FB(t\033)39 b FE(for)c(all)e (substitutions)j FB(\033)t FE(\).)f(A)g FC(simpli\014c)-5 b(ation)36 b(or)-5 b(dering)43 b FE(is)35 b(a)g(rewrite)h(ordering)183 4952 y(ha)m(ving)i(the)g(subterm)h(prop)s(ert)m(y)f(\(i.e.,)g FB(f)11 b FE(\()p FB(:)17 b(:)g(:)f(x)h(:)g(:)g(:)p FE(\))37 b FA(\037)g FB(x)i FE(for)e(all)f FB(f)48 b FA(2)38 b(F)10 b FE(\).)37 b(It)i(is)e(a)h(w)m(ell-)183 5073 y(kno)m(wn)31 b(consequence)j(of)29 b(Krusk)-5 b(al's)30 b(theorem)g(that)g(ev)m(ery) j(simpli\014cation)26 b(ordering)k(o)m(v)m(er)183 5193 y FA(T)25 b FE(\()p FA(F)10 b FB(;)17 b FA(V)8 b FE(\))40 b(is)g(w)m(ell)g(founded)h(pro)m(vided)g(that)g FA(F)50 b FE(is)40 b(\014nite.)2365 5157 y Fy(\003)2445 5193 y FE(It)h(is)f(also)f(w)m(ell)h(kno)m(wn)i(that)183 5313 y(simpli\014cation)29 b(orderings)j(satisfy)h(the)g(follo)m(wing)c (prop)s(ert)m(y)-8 b(.)294 5452 y Fx(\003)332 5482 y Fw(F)h(or)27 b(details)g(on)h(in\014nite)g(signatures)e(see)i ([Middeldorp)f(and)g(Zan)n(tema,)g(1997].)p eop %%Page: 5 5 5 4 bop 306 191 a FF(Giesl,)30 b(Arts,)h(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)f(Dep)s(endency)h(P)m(airs) 123 b(5)183 390 y FK(Lemma)38 b(2.1)f(\(V)-12 b(ariables)38 b(in)g(Inequalities)g(f)n(or)g(Simplifica)-7 b(tion)39 b(Orderings\):)183 511 y FC(L)-5 b(et)35 b FA(\037)g FC(b)-5 b(e)35 b(a)f(simpli\014c)-5 b(ation)34 b(or)-5 b(dering.)34 b(If)g FB(s)28 b FA(\037)g FB(t)p FC(,)35 b(then)f FA(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))28 b FA(\022)g(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))35 b FC(and)g FB(s)27 b FA(62)h(V)8 b FC(.)280 739 y FE(A)45 b(TRS)g FA(R)g FE(o)m(v)m(er)h(a)e(\014nite)g(signature)h FA(F)54 b FE(is)44 b(called)f FC(simply)j(terminating)53 b FE(if)43 b(its)h(ter-)183 859 y(mination)36 b(can)j(b)s(e)g(pro)m(v)m(en)i(b)m (y)f(a)f(simpli\014cation)c(ordering.)j(This)i(is)e(equiv)-5 b(alen)m(t)39 b(to)f(the)183 980 y(statemen)m(t)33 b(that)f(the)h(TRS)g FA(R)23 b([)f(E)9 b FB(mb)p FE(\()p FA(F)h FE(\))32 b(is)g (terminating,)f(where)453 1200 y FA(E)9 b FB(mb)p FE(\()p FA(F)h FE(\))28 b(=)f FA(f)p FB(f)11 b FE(\()p FB(x)1132 1215 y 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b(the)h(corresp)s(onding)f(ordering)g FA(\037)h FE(is)f(the)h FC(strict)54 b FE(relation)43 b FA(\037)2692 2953 y Fv(s)2773 2989 y FE(de\014ned)j(b)m(y)g FB(t)i FA(\037)3429 2953 y Fv(s)3514 2989 y FB(u)183 3109 y FE(if)e(and)i(only)g(if)e FB(t)54 b Fu(\045)g FB(u)47 b FE(and)h FB(u)54 b FA(6)p Fu(\045)g FB(t)p FE(.)48 b(Unfortunately)-8 b(,)47 b FA(\037)2447 3073 y Fv(s)2532 3109 y FE(is)h(in)f(general)g(not)h(closed)183 3230 y(under)27 b(substitutions)f(\(see)i(b)s(elo)m(w\).)e(Therefore,)i (to)e(determine)g(suitable)g(reduction)g(pairs)183 3350 y(automatically)-8 b(,)22 b(one)k(usually)g(c)m(ho)s(oses)h FA(\037)g FE(to)e(b)s(e)h(the)h(so-called)e FC(stable-strict)35 b FE(relation)24 b FA(\037)3500 3314 y Fv(ss)183 3470 y FE(corresp)s(onding)39 b(to)f(the)i(quasi-rewrite)e(ordering)h Fu(\045)p FE(.)g(W)-8 b(e)40 b(ha)m(v)m(e)g FB(t)f FA(\037)2797 3434 y Fv(ss)2906 3470 y FB(u)f FE(if)g(and)h(only)g(if)183 3591 y FB(t\033)34 b FA(\037)384 3555 y Fv(s)451 3591 y FB(u\033)k 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(stable-strict)183 5156 y(relation)d FA(\037)611 5120 y Fv(ss)706 5156 y FE(is)h(closed)h(under)h(substitutions)f(to)s(o,)f (whereas)i(this)e(is)h(not)f(necessarily)i(true)183 5276 y(for)32 b(the)h(strict)f(relation)f FA(\037)1190 5240 y Fv(s)1227 5276 y FE(.)280 5396 y(F)-8 b(or)30 b(example,)h(if)e FA(j)p Ft(0)p FA(j)e FE(=)h(0,)i FA(j)p Ft(s)p FE(\()p FB(t)p FE(\))p FA(j)e FE(=)f FA(j)p FB(t)p FA(j)19 b FE(+)f(1,)31 b(and)g FA(j)p Ft(f)6 b FE(\()p FB(t)p FE(\))p FA(j)27 b FE(=)h(2)p FA(j)p FB(t)p FA(j)i FE(for)g(all)f(ground)i (terms)g FB(t)p FE(,)183 5517 y(then)g(w)m(e)g(ha)m(v)m(e)h Ft(f)7 b FE(\()p FB(x)p FE(\))28 b Fu(\045)g FB(x)j FE(and)f FB(x)f FA(6)p Fu(\045)f Ft(f)6 b FE(\()p FB(x)p FE(\).)31 b(Hence,)h(this)e(implies)e Ft(f)7 b FE(\()p FB(x)p FE(\))28 b FA(\037)2860 5481 y Fv(s)2925 5517 y FB(x)p FE(.)j(Ho)m(w)m(ev)m(er,) i FA(\037)3533 5481 y Fv(s)p eop %%Page: 6 6 6 5 bop 306 191 a FF(Giesl,)30 b(Arts,)h(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)f(Dep)s(endency)h(P)m(airs) 123 b(6)183 390 y FE(is)31 b(not)g(closed)h(under)g(substitutions)f(b)s (ecause)i Ft(f)6 b FE(\()p Ft(0)p FE(\))28 b FA(\037)2220 354 y Fv(s)2285 390 y Ft(0)j FE(do)s(es)h(not)f(hold.)g(This)g(example) 183 511 y(also)36 b(demonstrates)h(that)g(in)g(general)f FA(\037)1736 474 y Fv(s)1809 511 y FA(\022)f(\037)1998 474 y Fv(ss)2105 511 y FE(is)i(not)g(true)g(b)s(ecause)i(for)d(the)i (stable-)183 631 y(strict)32 b(relation)f FA(\037)873 595 y Fv(ss)975 631 y FE(w)m(e)j(ha)m(v)m(e)g Ft(f)6 b FE(\()p FB(x)p FE(\))28 b FA(6\037)1616 595 y Fv(ss)1714 631 y FB(x)p FE(.)280 751 y(Moreo)m(v)m(er,)36 b(in)e(general)g FA(\037)1268 715 y Fv(ss)1369 751 y FA(\022)d Fu(\045)k FE(do)s(es)g(not)f(hold)g(either)g(\(hence,)i FA(\037)2898 715 y Fv(ss)2998 751 y FA(\022)c(\037)3184 715 y Fv(s)3255 751 y FE(is)i(false,)183 872 y(to)s(o\).)28 b(If)h FA(R)h FE(is)f(the)h(TRS)g(con)m(taining)e(only)g(the)i(rule)f Ft(f)7 b FE(\()p Ft(0)o FE(\))28 b FA(!)f Ft(0)i 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Fv(s)2103 2723 y FC(is)h(wel)5 b(l)34 b(founde)-5 b(d,)34 b(then)h FE(\()p Fu(\045)p FB(;)17 b FA(\037)3251 2687 y Fv(ss)3321 2723 y FE(\))34 b FC(is)h(a)g(r)-5 b(e)g(duction)34 b(p)-5 b(air)183 2951 y(Pr)g(o)g(of:)48 b FE(The)28 b(statemen)m(ts)f(\(i\))f(and)g(\(ii\))f(follo)m(w)g(from)g (the)i(re\015exivit)m(y)g(and)g(the)g(transitivit)m(y)183 3072 y(of)g Fu(\045)p FE(.)i(Statemen)m(ts)g(\(iii\),)c(\(iv\),)i(and)i (\(v\))f(are)g(direct)g(consequences)j(of)d(the)h(de\014nition.)e(F)-8 b(or)183 3192 y(\(vi\),)35 b(ev)m(ery)j(p)s(oten)m(tial)c(in\014nite)h (descending)j(sequence)g FB(t)2350 3207 y Fz(0)2423 3192 y FA(\037)2500 3156 y Fv(ss)2604 3192 y FB(t)2639 3207 y Fz(1)2712 3192 y FA(\037)2789 3156 y Fv(ss)2893 3192 y FB(:)17 b(:)g(:)35 b FE(w)m(ould)h(result)183 3313 y(in)e(an)h(in\014nite)e(descending)j(sequence)i FB(t)1707 3328 y Fz(0)1747 3313 y FB(\033)d FA(\037)1914 3276 y Fv(s)1983 3313 y FB(t)2018 3328 y Fz(1)2058 3313 y FB(\033)g FA(\037)2225 3276 y Fv(s)2294 3313 y FB(:)17 b(:)g(:)34 b FE(Statemen)m(ts)i(\(vii\))d(and)i(\(viii\))183 3433 y(follo)m(w)f(from)h(the)i(transitivit)m(y)e(and)i(stabilit)m(y)d(of)i Fu(\045)p FE(.)h(Statemen)m(t)g(\(ix\))f(follo)m(ws)f(from)g(\(i\),)183 3553 y(\(ii\),)30 b(\(iii\),)g(\(vi\))i(and)h(\(vii\))e(\(or)h (\(viii\)\).)1883 b Fs(2)280 3782 y FE(In)30 b(this)g(article,)e FA(\037)j FE(alw)m(a)m(ys)f(denotes)h(an)f(arbitrary)f(ordering)g(suc)m (h)j(that)d(\()p Fu(\045)p FB(;)17 b FA(\037)p FE(\))31 b(forms)183 3902 y(a)24 b(reduction)h(pair.)e(As)i(sho)m(wn)h(in)e (Lemma)f(2.2)h(\(ix\),)g(one)h(p)s(ossibilit)m(y)e(is)h(to)g(c)m(ho)s (ose)h FA(\037)g FE(to)g(b)s(e)183 4022 y(the)i(stable-strict)g (relation)e(corresp)s(onding)j(to)f(the)g(quasi-rewrite)g(relation)f Fu(\045)h FE(\(pro)m(vided)183 4143 y(that)35 b(it)g(is)h(w)m(ell)f (founded\).)h(Lemma)f(2.2)g(\(v\))h(indicates)g(that)f(this)h(c)m (hoice)g(is)g(at)f(least)h(as)183 4263 y(p)s(o)m(w)m(erful)c(as)h(c)m (ho)s(osing)f FA(\037)h 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b(2.3)f(\(V)-12 b(ariables)38 b(in)g(Strict)g (Inequalities\):)49 b FC(L)-5 b(et)37 b Fu(\045)i FC(b)-5 b(e)37 b(a)g(QSO)g(and)g(let)183 5334 y FE(\()p Fu(\045)p FB(;)17 b FA(\037)p FE(\))35 b FC(b)-5 b(e)35 b(a)f(r)-5 b(e)g(duction)35 b(p)-5 b(air.)34 b(If)h FB(s)27 b FA(\037)h FB(t)p FC(,)35 b(then)g FA(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))28 b FA(\022)g(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))35 b FC(and)f FB(s)28 b FA(62)g(V)8 b FC(.)p eop %%Page: 7 7 7 6 bop 306 191 a FF(Giesl,)30 b(Arts,)h(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)f(Dep)s(endency)h(P)m(airs) 123 b(7)183 390 y FC(Pr)-5 b(o)g(of:)48 b FE(Assume)37 b(that)f(there)h(is)e(a)h(v)-5 b(ariable)34 b FB(x)g FA(2)g(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))25 b FA(n)f(V)8 b FB(ar)s FE(\()p FB(s)p FE(\).)36 b(Then)i FB(t)33 b FE(=)h FB(C)7 b FE([)p FB(x)p FE(])36 b(for)183 511 y(some)g(con)m (text)i FB(C)7 b FE(.)36 b(With)g FB(\033)i FE(=)c FA(f)p FB(x)h FA(7!)f FB(s)p FA(g)i FE(it)f(follo)m(ws)g(that)h FB(s)f FE(=)f FB(s\033)k FA(\037)d FB(t\033)j FE(=)c FB(C)7 b FE([)p FB(s)p FE(].)37 b(Since)183 631 y FB(C)7 b FE([)p FB(s)p FE(])33 b Fu(\045)g FB(s)i FE(according)g(to)h(the)g (subterm)g(prop)s(ert)m(y)-8 b(,)36 b(w)m(e)h(obtain)d FB(s)f FA(\037)g FB(C)7 b FE([)p FB(s)p FE(])33 b Fu(\045)h FB(s)p FE(.)h(This)h(is)f(a)183 751 y(con)m(tradiction)j(to)h(the)h(w)m (ell-foundedness)g(of)f FA(\037)p FE(.)h(Th)m(us)h FA(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))40 b FA(\022)f(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))40 b(holds.)f(The)183 872 y(pro)s(of)31 b(of)h FB(s)c FA(62)g(V)41 b FE(is)32 b(just)h(as)g(straigh)m(tforw)m (ard.)1586 b Fs(2)183 1035 y FE(A)32 b(similar)e(prop)s(ert)m(y)j(ev)m (en)h(holds)e(for)g(non-strict)g(inequalities.)183 1199 y FK(Lemma)38 b(2.4)f(\(V)-12 b(ariables)38 b(in)g(Non-Strict)g (Inequalities\):)49 b FC(L)-5 b(et)25 b Fu(\045)g FC(b)-5 b(e)25 b(a)g(QSO)f(and)183 1319 y(let)51 b FE(\()p Fu(\045)p FB(;)17 b FA(\037)p FE(\))51 b FC(b)-5 b(e)51 b(a)g(r)-5 b(e)g(duction)51 b(p)-5 b(air)51 b(such)g(that)g FB(s)2055 1283 y Fy(0)2136 1319 y FA(\037)58 b FB(t)2306 1283 y Fy(0)2381 1319 y FC(for)51 b(some)f(terms)h FB(s)3153 1283 y Fy(0)3176 1319 y FB(;)17 b(t)3255 1283 y Fy(0)3330 1319 y FC(wher)-5 b(e)183 1440 y FA(V)8 b FB(ar)s FE(\()p FB(t)423 1404 y Fy(0)446 1440 y FE(\))28 b FA(6)p FE(=)f FA(;)p FC(.)35 b(If)f FB(s)28 b Fu(\045)g FB(t)p FC(,)35 b(then)g FA(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))28 b FA(\022)g(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))p FC(.)183 1603 y(Pr)-5 b(o)g(of:)48 b FE(First)39 b(of)g(all,)e FB(s)1066 1567 y Fy(0)1129 1603 y FA(\037)j FB(t)1281 1567 y Fy(0)1344 1603 y FE(implies)d FA(V)8 b FB(ar)s FE(\()p FB(t)1922 1567 y Fy(0)1945 1603 y FE(\))40 b FA(\022)f(V)8 b FB(ar)s FE(\()p FB(s)2390 1567 y Fy(0)2414 1603 y FE(\))39 b(according)g(to)g (Lemma)f(2.3.)183 1724 y(Without)33 b(loss)h(of)g(generalit)m(y)-8 b(,)34 b(w)m(e)h(assume)g(that)f FB(s)g FE(and)h FB(t)f FE(are)h(renamed)f(suc)m(h)i(that)e(they)183 1844 y(ha)m(v)m(e)f(no)f (v)-5 b(ariables)30 b(in)i(common)e(with)i FB(s)1716 1808 y Fy(0)1771 1844 y FE(or)f FB(t)1924 1808 y Fy(0)1948 1844 y FE(.)h(W)-8 b(e)32 b(sho)m(w)h FA(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))28 b FA(\022)g(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))32 b(indirectly)-8 b(.)183 1964 y(Supp)s(ose)27 b(that)g(there)g(is)g(a)f(v)-5 b(ariable)25 b FB(y)31 b FA(2)d(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))i FA(n)g(V)e FB(ar)s FE(\()p FB(s)p FE(\).)27 b(Since)g FA(V)8 b FB(ar)s FE(\()p FB(t)2889 1928 y Fy(0)2913 1964 y FE(\))27 b FA(6)p FE(=)h FA(;)p FE(,)e(there)i(is)e(a)183 2085 y(v)-5 b(ariable)24 b FB(x)k FA(2)g(V)8 b FB(ar)s FE(\()p FB(t)959 2049 y Fy(0)983 2085 y FE(\))28 b FA(\022)g(V)8 b FB(ar)s FE(\()p FB(s)1405 2049 y Fy(0)1428 2085 y FE(\).)27 b(Let)f FB(\033)32 b FE(=)27 b FA(f)p FB(x)h FA(7!)g FB(s)p FA(g)e FE(and)g FB(\033)2503 2049 y Fy(0)2554 2085 y FE(=)i FA(f)p FB(x)g FA(7!)f FB(t;)17 b(y)31 b FA(7!)c FB(s)3249 2049 y Fy(0)3272 2085 y FB(\033)t FA(g)p FE(.)f(W)-8 b(e)183 2205 y(ha)m(v)m(e)32 b(\(a\))f FB(s)608 2169 y Fy(0)631 2205 y FB(\033)h FA(\037)c FB(t)858 2169 y Fy(0)882 2205 y FB(\033)35 b FE(b)s(ecause)e FB(s)1378 2169 y Fy(0)1428 2205 y FA(\037)c FB(t)1569 2169 y Fy(0)1623 2205 y FE(and)j FA(\037)f FE(is)g(closed)h(under)g(substitutions,)f (\(b\))g FB(t)3382 2169 y Fy(0)3406 2205 y FB(\033)g Fu(\045)183 2326 y FB(t)218 2289 y Fy(0)241 2326 y FB(\033)300 2289 y Fy(0)357 2326 y FE(b)s(ecause)j FB(s)29 b Fu(\045)g FB(t)k FE(and)h Fu(\045)f FE(is)g(w)m(eakly)h(monotonic,)e(and)h(\(c\)) h FB(t)2573 2289 y Fy(0)2596 2326 y FB(\033)2655 2289 y Fy(0)2707 2326 y Fu(\045)29 b FB(y)t(\033)2924 2289 y Fy(0)2976 2326 y FE(=)f FB(s)3126 2289 y Fy(0)3149 2326 y FB(\033)38 b FE(b)s(ecause)183 2446 y FB(y)g FA(2)d FB(t)405 2410 y Fy(0)428 2446 y FE(,)i Fu(\045)h FE(has)f(the)g(w)m (eak)h(subterm)f(prop)s(ert)m(y)-8 b(,)38 b(and)f Fu(\045)g FE(is)f(closed)h(under)h(substitutions.)183 2566 y(In)32 b(summary)-8 b(,)31 b FB(s)794 2530 y Fy(0)817 2566 y FB(\033)h FA(\037)c FB(t)1044 2530 y Fy(0)1067 2566 y FB(\033)k Fu(\045)c FB(t)1294 2530 y Fy(0)1318 2566 y FB(\033)1377 2530 y Fy(0)1428 2566 y Fu(\045)g FB(s)1579 2530 y Fy(0)1602 2566 y FB(\033)36 b FE(is)31 b(a)h(con)m(tradiction)e (to)i(the)g(w)m(ell-foundedness)g(of)183 2687 y FA(\037)p FE(.)3209 b Fs(2)280 2850 y FE(Examples)27 b(of)f(simpli\014cation)e (orderings)i(and)h(QSOs)g(include)f(path)h(orderings)f(lik)m(e)g(the) 183 2971 y(lexicographic)33 b(path)h(ordering)g(\(LPO\))h([Kamin)d(and) j(L)m(\023)-46 b(evy,)36 b(1980],)e(the)h(recursiv)m(e)h(path)183 3091 y(ordering)30 b(\(RPO\))i([Dersho)m(witz,)g(1987,)f(Stein)m(bac)m (h,)h(1995,)f(F)-8 b(erreira,)31 b(1995],)g(the)h(Kn)m(uth-)183 3211 y(Bendix)26 b(ordering)e(\(KBO\))i([Kn)m(uth)g(and)g(Bendix,)g (1970,)e(Dic)m(k)i(et)f(al.,)g(1990,)g(Koro)m(vin)g(and)183 3332 y(V)-8 b(oronk)m(o)m(v,)37 b(2001],)f(etc.)i(P)m(olynomial)c (orderings,)i(ho)m(w)m(ev)m(er,)j(are)e(not)f(QSOs)h(in)f(general.)183 3452 y(F)-8 b(or)26 b(instance,)i(if)e(the)i(constan)m(t)g Ft(0)f FE(is)g(asso)s(ciated)g(with)g(the)g(n)m(um)m(b)s(er)h(0,)f Ft(s)q FE(\()p FB(x)p FE(\))g(is)g(asso)s(ciated)183 3573 y(with)41 b FB(x)28 b FE(+)h(1,)41 b(and)g Ft(f)7 b FE(\()p FB(x;)17 b(y)t FE(\))41 b(is)g(asso)s(ciated)g(with)g(the)h (m)m(ultiplication)37 b(of)k FB(x)h FE(and)g FB(y)t FE(,)e(then)183 3693 y(this)i(p)s(olynomial)d(ordering)j(do)s(es)h(not)f(satisfy)h(the) g(subterm)g(prop)s(ert)m(y)g(\(for)f(example,)183 3813 y Ft(f)6 b FE(\()p Ft(s)q FE(\()p Ft(0)o FE(\))p FB(;)17 b Ft(0)p FE(\))27 b Fu(\045)h Ft(s)q FE(\()p Ft(0)p FE(\))j(do)s(es)h (not)f(hold\).)g(Ho)m(w)m(ev)m(er,)i(the)f(follo)m(wing)d(lemma)g(sho)m (ws)k(that)e(if)f(the)183 3934 y(p)s(olynomial)f(ordering)i(resp)s (ects)k(some)d(restrictions,)g(then)i(it)d(is)h(indeed)h(a)f(QSO.)183 4097 y FK(Lemma)38 b(2.5)f(\(Pol)-7 b(ynomial)40 b(Orderings)e(as)f (QSOs\):)49 b FC(L)-5 b(et)28 b Fu(\045)h FC(b)-5 b(e)28 b(a)g(p)-5 b(olynomial)27 b(or-)183 4218 y(dering)40 b(wher)-5 b(e)41 b(every)h(function)f(symb)-5 b(ol)41 b(is)g(asso)-5 b(ciate)g(d)40 b(with)i(a)f(p)-5 b(olynomial)40 b(c)-5 b(ontaining)183 4338 y(only)34 b(non-ne)-5 b(gative)34 b(c)-5 b(o)g(e\016cients.)299 4492 y FA(\017)48 b FC(If)29 b(every)g(function)f(symb)-5 b(ol)29 b FB(f)11 b FE(\()p FB(x)1586 4507 y Fz(1)1625 4492 y FB(;)17 b(:)g(:)g(:)f(;)h(x)1899 4507 y Fv(n)1946 4492 y FE(\))29 b FC(is)g(asso)-5 b(ciate)g(d)28 b(with)g(a)h(p)-5 b(olynomial)28 b(which)397 4612 y(c)-5 b(ontains)37 b(a)g(\(non-mixe)-5 b(d\))35 b(monomial)g(of)i(the)g(form) g FB(m)2487 4627 y Fv(i)2532 4612 y FB(x)2587 4568 y Fv(k)2624 4578 y Fr(i)2587 4638 y Fv(i)2692 4612 y FC(\(with)g FB(m)3031 4627 y Fv(i)3059 4612 y FB(;)17 b(k)3154 4627 y Fv(i)3214 4612 y FA(\025)32 b FE(1)p FC(\))37 b(for)397 4733 y(every)e FB(i)28 b FE(=)g(1)p FB(;)17 b(:)g(:)g(:)e(;)i(n)p FC(,)35 b(then)g Fu(\045)g FC(is)f(a)h(QSO.)299 4878 y FA(\017)48 b FC(If)35 b(every)f(function)h(symb)-5 b(ol)35 b FB(f)11 b FE(\()p FB(x)1610 4893 y Fz(1)1649 4878 y FB(;)17 b(:)g(:)g(:)f(;)h(x)1923 4893 y Fv(n)1970 4878 y FE(\))35 b FC(is)f(asso)-5 b(ciate)g(d)34 b(with)h(a)g(p)-5 b(olynomial)34 b(c)-5 b(on-)397 4999 y(taining)43 b(al)5 b(l)43 b(variables)f FB(x)1353 5014 y Fz(1)1393 4999 y FB(;)17 b(:)g(:)g(:)f(;)h(x)1667 5014 y Fv(n)1758 4999 y FC(and)43 b(if)g(every)g(c)-5 b(onstant)43 b(is)g(asso)-5 b(ciate)g(d)43 b(with)g(a)397 5119 y(numb)-5 b(er)35 b FB(>)27 b FE(0)p FC(,)35 b(then)f Fu(\045)i FC(is)e(a)h(QSO.)183 5283 y(Pr)-5 b(o)g(of:)48 b FE(Straigh)m(tforw)m(ard.)2311 b Fs(2)280 5446 y FE(In)35 b(fact,)f(the)g(conditions)g(in)f(Lemma)g (2.5)g(also)h(en)m(tail)e(\(strong\))i(monotonicit)m(y)f(of)g(the)183 5567 y(strict)f(and)h(stable-strict)e(relations)g(corresp)s(onding)i (to)f(the)h(p)s(olynomial)c(ordering.)p eop %%Page: 8 8 8 7 bop 306 191 a FF(Giesl,)30 b(Arts,)h(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)f(Dep)s(endency)h(P)m(airs) 123 b(8)183 390 y FD(3.)68 b(Mo)t(dular)44 b(T)-11 b(ermination)46 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b(of)i(a)h(TRS)g(in)f(a)h(mo)s(dular)e(w)m (a)m(y)-8 b(,)35 b(b)s(ecause)g(absence)h(of)d(in\014nite)g(c)m(hains)h (can)h(b)s(e)183 4648 y(pro)m(v)m(ed)f(separately)f(for)f(ev)m(ery)i (cycle.)183 4867 y FK(Theorem)k(3.3)f(\(Modular)i(Termina)-7 b(tion)39 b(Criterion\):)49 b FC(A)k(TRS)g FA(R)g FC(is)g(termi-)183 4987 y(nating)40 b(if)h(and)f(only)h(if)g(for)f(e)-5 b(ach)40 b(cycle)h FA(P)49 b FC(in)41 b(the)g(dep)-5 b(endency)40 b(gr)-5 b(aph)40 b(ther)-5 b(e)41 b(exists)f(no)183 5107 y(in\014nite)34 b FA(R)p FC(-chain)g(of)h(dep)-5 b(endency)33 b(p)-5 b(airs)35 b(fr)-5 b(om)34 b FA(P)8 b FC(.)183 5326 y(Pr)-5 b(o)g(of:)48 b FE(The)25 b(only-if)c(direction) i(is)g(a)g(direct)h(consequence)i(of)e(Theorem)g(3.1.)f(F)-8 b(or)23 b(the)h(other)183 5446 y(direction,)30 b(supp)s(ose)i(that)f FA(R)h FE(is)f(not)g(terminating.)d(Then)33 b(b)m(y)f(Theorem)f(3.1)g (there)h(exists)183 5567 y(an)42 b(in\014nite)g FA(R)p FE(-c)m(hain.)h(As)g(w)m(e)h(only)e(regard)h(\014nite)f(TRSs)i FA(R)p FE(,)f(there)h(are)e(only)h(\014nitely)p eop %%Page: 10 10 10 9 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(10)183 390 y FE(man)m(y)33 b(dep)s(endency)j(pairs)d(and)h (hence,)h(one)f(dep)s(endency)i(pair)c(o)s(ccurs)j(in\014nitely)d(man)m (y)183 511 y(times)27 b(in)h(the)h(c)m(hain)f(\(up)g(to)g(renaming)f (of)h(the)h(v)-5 b(ariables\).)27 b(Th)m(us,)j(the)f(in\014nite)e(c)m (hain)h(has)183 631 y(the)33 b(form)846 751 y FB(:)17 b(:)g(:)f(;)h(s\032)1117 766 y Fz(1)1184 751 y FA(!)27 b FB(t\032)1396 766 y Fz(1)1436 751 y FB(;)17 b(:)g(:)g(:)f(;)h(s\032) 1751 766 y Fz(2)1818 751 y FA(!)28 b FB(t\032)2031 766 y Fz(2)2071 751 y FB(;)17 b(:)g(:)g(:)e(;)i(s\032)2385 766 y Fz(3)2452 751 y FA(!)28 b FB(t\032)2665 766 y Fz(3)2705 751 y FB(;)17 b(:)g(:)g(:)f(;)183 926 y FE(where)39 b FB(\032)520 941 y Fz(1)559 926 y FB(;)17 b(\032)653 941 y Fz(2)693 926 y FB(;)g(\032)787 941 y Fz(3)826 926 y FB(;)g(:)g(:)g(:)37 b FE(are)h(renamings.)e(Hence,)j(the)f(tail)e FB(s\032)2458 941 y Fz(1)2525 926 y FA(!)28 b FB(t\032)2738 941 y Fz(1)2778 926 y FB(;)17 b(:)g(:)g(:)e(;)i(s\032)3092 941 y Fz(2)3159 926 y FA(!)28 b FB(t\032)3372 941 y Fz(2)3412 926 y FB(;)17 b(:)g(:)g(:)183 1046 y FE(is)34 b(an)h(in\014nite)f FA(R)p FE(-c)m(hain)g(whic)m(h)i(consists)f(of)g(dep)s(endency)i(pairs) d(from)g(one)h(cycle)g(in)f(the)183 1166 y(dep)s(endency)h(graph)d (only)-8 b(.)2305 b Fs(2)280 1395 y FE(According)40 b(to)g(the)g(ab)s (o)m(v)m(e)h(theorem,)f(in)f(our)h(example)f(w)m(e)i(can)g(separate)f (the)h(pro)s(of)183 1515 y(that)34 b(there)h(is)e(no)h(in\014nite)f(c)m (hain)i(consisting)e(of)h(the)g(dep)s(endency)j(pair)c FA(f)p FE(\(2\))p FA(g)h FE(from)e(the)183 1636 y(corresp)s(onding)g (pro)s(of)g(for)g(the)h(dep)s(endency)i(pair)d FA(f)p FE(\(3\))p FA(g)p FE(.)280 1756 y(One)j(should)g(remark)f(that)g(for)h (the)g(soundness)i(of)d(this)g(theorem)g(one)h(indeed)g(has)g(to)183 1876 y(regard)42 b FC(al)5 b(l)52 b FE(cycles,)44 b(not)e(just)h(the)g (minimal)38 b(ones)43 b(\(i.e.,)g(not)f(just)h(those)g(cycles)g(whic)m (h)183 1997 y(con)m(tain)d(no)g(other)g(cycles)i(as)e(prop)s(er)g (subsets\).)j(F)-8 b(or)39 b(a)h(coun)m(terexample)h(to)f(illustrate) 183 2117 y(this)32 b(fact)g(see)i([Giesl)d(and)i(Arts,)g(2001,)f(p.)g (50].)280 2237 y(Note)40 b(that)f(in)g(standard)h(graph)f(terminology) -8 b(,)37 b(a)j(path)f FB(v)2494 2252 y Fz(0)2573 2237 y FA(\))h FB(v)2760 2252 y Fz(1)2839 2237 y FA(\))f FB(:)17 b(:)g(:)39 b FA(\))g FB(v)3318 2252 y Fv(k)3400 2237 y FE(in)g(a)183 2358 y(directed)g(graph)f(forms)g(a)h(cycle)g(if)f FB(v)1598 2373 y Fz(0)1676 2358 y FE(=)g FB(v)1837 2373 y Fv(k)1918 2358 y FE(and)h FB(k)j FA(\025)c FE(1.)h(In)g(our)g(con)m (text)h(w)m(e)f(iden)m(tify)183 2478 y(cycles)d(with)e(the)i FC(set)44 b FE(of)34 b(elemen)m(ts)i(that)e(o)s(ccur)h(in)g(it,)f (i.e.,)g(w)m(e)i(call)d FA(f)p FB(v)2868 2493 y Fz(0)2908 2478 y FB(;)17 b(v)2999 2493 y Fz(1)3038 2478 y FB(;)g(:)g(:)g(:)f(;)h (v)3304 2493 y Fv(k)r Fy(\000)p Fz(1)3437 2478 y FA(g)34 b FE(a)183 2599 y(cycle.)d(Since)g(a)g(set)h(nev)m(er)g(con)m(tains)f (m)m(ultiple)e(o)s(ccurrences)k(of)d(an)h(elemen)m(t,)g(this)f(results) 183 2719 y(in)e(sev)m(eral)i(cycling)e(paths)i(b)s(eing)e(iden)m (ti\014ed)h(with)g(the)g(same)g(set.)h(Moreo)m(v)m(er,)h(for)d(a)h (\014nite)183 2839 y(TRS)h(w)m(e)h(only)f(ha)m(v)m(e)h(\014nitely)e (man)m(y)h(cycles,)i(since)e(the)g(n)m(um)m(b)s(er)h(of)e(dep)s (endency)k(pairs)c(is)183 2960 y(\014nite,)j(to)s(o.)183 3222 y Fq(3.2.)53 b(Chec)m(king)35 b(the)g(Mo)s(dular)g(T)-9 b(ermination)34 b(Criterion)g(Automatically)183 3392 y FE(F)-8 b(or)31 b(an)h(automatic)e(approac)m(h)i(the)g(de\014nition)f (of)h(a)f(dep)s(endency)k(graph)c(is)h(impractical,)183 3513 y(since)39 b(it)f(is)g(in)g(general)g(undecidable)h(whether)h(t)m (w)m(o)f(dep)s(endency)j(pairs)c(form)f(a)i(c)m(hain.)183 3633 y(Ho)m(w)m(ev)m(er,)j(in)c(order)i(to)f(obtain)f(a)i(sound)g(tec)m (hnique)h(for)e(termination)e(pro)s(ofs,)i(w)m(e)i(can)183 3753 y(safely)46 b(use)i(an)m(y)g(appro)m(ximation)d(of)h(the)h(dep)s (endency)j(graph)c(that)h(preserv)m(es)j(all)44 b(its)183 3874 y(cycles.)25 b(T)-8 b(o)24 b(estimate)f(whic)m(h)h(dep)s(endency)j (pairs)c(ma)m(y)h(o)s(ccur)g(consecutiv)m(e,)i(the)e FC(estimate)-5 b(d)183 3994 y(dep)g(endency)41 b(gr)-5 b(aph)49 b FE(has)42 b(b)s(een)g(in)m(tro)s(duced,)g(cf.)f([Arts)h(and) g(Giesl,)e(2000].)h(Let)g FK(cap)p FE(\()p FB(t)p FE(\))183 4115 y(result)22 b(from)e(replacing)h(all)f(subterms)j(of)e FB(t)h FE(that)g(ha)m(v)m(e)h(a)f(de\014ned)h(ro)s(ot)f(sym)m(b)s(ol)f (b)m(y)i(di\013eren)m(t)183 4235 y(fresh)30 b(v)-5 b(ariables)29 b(and)h(let)g FK(ren)p FE(\()p FB(t)p FE(\))g(result)g(from)f (replacing)f(all)g(v)-5 b(ariables)29 b(in)g FB(t)i FE(b)m(y)g (di\013eren)m(t)183 4355 y(fresh)i(v)-5 b(ariables.)32 b(Then,)i(to)e(determine)h(whether)h FB(v)d FA(!)c FB(w)36 b FE(can)d(follo)m(w)e FB(s)c FA(!)h FB(t)k FE(in)g(a)h(c)m(hain,)183 4476 y(w)m(e)h(c)m(hec)m(k)i(whether)f FK(ren)o FE(\()p FK(cap)p FE(\()p FB(t)p FE(\)\))e(uni\014es)h(with)f FB(v)t FE(.)g(So)h(w)m(e)g(ha)m(v)m(e)h FK(ren)o FE(\()p FK(cap)p FE(\()p Ft(F)p FE(\()p FB(y)t(;)17 b(y)t FE(\)\)\))27 b(=)183 4596 y FK(ren)o FE(\()p Ft(F)q FE(\()p FB(y)t(;)17 b(y)t FE(\)\))52 b(=)j Ft(F)q FE(\()p FB(y)1040 4611 y Fz(1)1079 4596 y FB(;)17 b(y)1171 4611 y Fz(2)1209 4596 y FE(\))49 b(and)f FK(ren)p FE(\()p FK(cap)p FE(\()p Ft(F)p FE(\()p FB(x;)17 b Ft(s)p FE(\()p Ft(f)7 b FE(\()p FB(y)t(;)17 b(y)t FE(\)\)\)\)\))52 b(=)j FK(ren)p FE(\()p Ft(F)p FE(\()p FB(x;)17 b Ft(s)q FE(\()p FB(z)t FE(\)\)\))55 b(=)183 4716 y Ft(F)p FE(\()p FB(x)331 4731 y Fz(1)371 4716 y FB(;)17 b Ft(s)p FE(\()p FB(z)535 4731 y Fz(1)575 4716 y FE(\)\).)43 b(Hence,)i(\(1\))e(can)h(nev)m(er)h(follo)m(w)d (itself)g(in)h(a)g(c)m(hain,)g(b)s(ecause)i Ft(F)p FE(\()p FB(x)3250 4731 y Fz(1)3290 4716 y FB(;)17 b Ft(s)p FE(\()p FB(z)3454 4731 y Fz(1)3494 4716 y FE(\)\))183 4837 y(do)s(es)33 b(not)f(unify)g(with)h Ft(F)p FE(\()p FB(x;)17 b Ft(c)p FE(\()p FB(y)t FE(\)\).)183 5065 y FK(Definition)38 b(3.4)f(\(Estima)-7 b(ted)38 b(Dependency)e(Graph\):)49 b FC(The)e FE(estimated)e(dep)s (en-)183 5186 y(dency)38 b(graph)f FC(of)f(a)h(TRS)f FA(R)i FC(is)e(the)h(dir)-5 b(e)g(cte)g(d)37 b(gr)-5 b(aph)36 b(whose)g(no)-5 b(des)36 b(ar)-5 b(e)36 b(the)h(dep)-5 b(endency)183 5306 y(p)g(airs)35 b(and)f(ther)-5 b(e)36 b(is)f(an)g(ar)-5 b(c)35 b(fr)-5 b(om)35 b FB(s)28 b FA(!)f FB(t)35 b FC(to)h FB(v)31 b FA(!)d FB(w)38 b FC(i\013)d FK(ren)o FE(\()p FK(cap)p FE(\()p FB(t)p FE(\)\))g FC(and)g FB(v)k FC(ar)-5 b(e)35 b(uni\014-)183 5426 y(able.)p eop %%Page: 11 11 11 10 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(11)280 390 y FE(In)40 b(our)f(example,)g(the)h(estimated)f(dep)s (endency)j(graph)d(is)g(the)h(same)f(as)h(the)f(dep)s(en-)183 511 y(dency)h(graph)f(giv)m(en)g(in)f(Figure)g(1.)g(F)-8 b(or)38 b(an)h(automation)e(of)h(the)h(mo)s(dular)e(criterion)g(of)183 631 y(Theorem)26 b(3.3,)h(w)m(e)g(use)h(this)e(estimated)g(dep)s (endency)j(graph.)d(Indeed,)i(Theorem)f(3.3)f(also)183 751 y(holds)k(for)g(the)h(estimated)f(dep)s(endency)k(graph)c(instead)h (of)f(the)h(dep)s(endency)j(graph,)c(b)s(e-)183 872 y(cause)c(all)d (dep)s(endency)28 b(pairs)d(on)g(a)g(cycle)h(in)e(the)i(dep)s(endency)i (graph)d(are)g(also)f(on)h(a)g(cycle)183 992 y(in)33 b(its)h(estimation.)e(The)k(only-if)c(direction)h(of)h(Theorem)g(3.3)g (holds)g(an)m(yw)m(a)m(y)j(regardless)183 1112 y(of)g(the)i(estimation) d(used,)k(since)f(whenev)m(er)i(a)d(TRS)g(is)g(terminating,)e(then)j (there)g(is)f(no)183 1233 y(in\014nite)31 b(c)m(hain)i(\(Theorem)f (3.1\).)280 1353 y(T)-8 b(o)46 b(c)m(hec)m(k)h(the)f(criterion)e(of)h (Theorem)g(3.3)g(automatically)-8 b(,)42 b(for)j(eac)m(h)h(cycle)g FA(P)8 b FE(,)46 b(w)m(e)183 1474 y(generate)30 b(a)f(set)i(of)e (inequalities)f(suc)m(h)j(that)e(the)i(existence)g(of)e(reduction)h (pairs)f(\()p Fu(\045)3309 1489 y Fy(P)3368 1474 y FB(;)p FA(\037)3472 1489 y Fy(P)3532 1474 y FE(\))183 1594 y(satisfying)22 b(these)j(inequalities)c(is)i(su\016cien)m(t)h(for)f(the)h(absence)h (of)e(in\014nite)f(c)m(hains.)i(F)-8 b(or)22 b(that)183 1714 y(purp)s(ose)i(w)m(e)g(ha)m(v)m(e)g(to)f(ensure)i(that)d(the)i (dep)s(endency)i(pairs)c(from)g FA(P)32 b FE(are)23 b(decreasing)g (w.r.t.)183 1835 y Fu(\045)260 1850 y Fy(P)319 1835 y FE(.)f(More)g(precisely)-8 b(,)22 b(for)f(an)m(y)i(sequence)h(of)d(dep) s(endency)k(pairs)c FB(s)2616 1850 y Fz(1)2683 1835 y FA(!)27 b FB(t)2845 1850 y Fz(1)2885 1835 y FB(;)17 b(s)2975 1850 y Fz(2)3042 1835 y FA(!)27 b FB(t)3204 1850 y Fz(2)3244 1835 y FB(;)17 b(s)3334 1850 y Fz(3)3400 1835 y FA(!)28 b FB(t)3563 1850 y Fz(3)3619 1835 y FB(:)17 b(:)g(:)183 1955 y FE(from)31 b FA(P)41 b FE(and)33 b(for)f(an)m(y)h(substitution)f FB(\033)37 b FE(with)32 b FB(t)1938 1970 y Fv(j)1974 1955 y FB(\033)21 b FA(!)2150 1919 y Fy(\003)2150 1980 y(R)2231 1955 y FB(s)2277 1970 y Fv(j)t Fz(+1)2403 1955 y FB(\033)37 b FE(\(for)32 b(all)e FB(j)6 b FE(\))33 b(w)m(e)g(demand)804 2175 y FB(s)850 2190 y Fz(1)890 2175 y FB(\033)f Fu(\045)1054 2190 y Fy(P)1141 2175 y FB(t)1176 2190 y Fz(1)1215 2175 y FB(\033)g Fu(\045)1379 2190 y Fy(P)1466 2175 y FB(s)1512 2190 y Fz(2)1552 2175 y FB(\033)f Fu(\045)1715 2190 y Fy(P)1803 2175 y FB(t)1838 2190 y Fz(2)1877 2175 y FB(\033)h Fu(\045)2041 2190 y Fy(P)2128 2175 y FB(s)2174 2190 y Fz(3)2214 2175 y FB(\033)f Fu(\045)2377 2190 y Fy(P)2464 2175 y FB(t)2499 2190 y Fz(3)2539 2175 y FB(\033)h Fu(\045)2703 2190 y Fy(P)2790 2175 y FB(:)17 b(:)g(:)f(;)183 2395 y FE(and)30 b(for)f(at)g(least)h (one)g FB(s)e FA(!)f FB(t)j FE(in)f FA(P)38 b FE(w)m(e)31 b(demand)f(the)g FC(strict)40 b FE(inequalit)m(y)29 b FB(s\033)i FA(\037)3109 2410 y Fy(P)3197 2395 y FB(t\033)t FE(.)f(Then)183 2516 y(there)e(exists)g(no)f(c)m(hain)g(of)g(dep)s (endency)j(pairs)c(from)g FA(P)36 b FE(whic)m(h)28 b(tra)m(v)m(erses)h (all)c(dep)s(endency)183 2636 y(pairs)32 b(in)f FA(P)41 b FE(in\014nitely)32 b(man)m(y)g(times.)280 2756 y(Since)40 b Fu(\045)619 2771 y Fy(P)718 2756 y FE(is)g(closed)g(under)g (substitutions)g(and)g(w)m(eakly)h(monotonic,)d(to)h(guaran)m(tee)183 2877 y FB(t)218 2892 y Fv(j)254 2877 y FB(\033)32 b Fu(\045)418 2892 y Fy(P)505 2877 y FB(s)551 2892 y Fv(j)t Fz(+1)678 2877 y FB(\033)j FE(whenev)m(er)f FB(t)1230 2892 y Fv(j)1267 2877 y FB(\033)20 b FA(!)1442 2841 y Fy(\003)1442 2902 y(R)1523 2877 y FB(s)1569 2892 y Fv(j)t Fz(+1)1695 2877 y FB(\033)36 b FE(holds,)30 b(it)h(is)f(su\016cien)m(t)j(to)d(demand)i FB(l)d Fu(\045)3288 2892 y Fy(P)3376 2877 y FB(r)k FE(for)183 2997 y(all)g(rules)j FB(l)19 b FA(!)d FB(r)39 b FE(of)c(the)h(TRS.)h (Moreo)m(v)m(er,)g FB(s)1854 3012 y Fv(j)1924 2997 y Fu(\045)2001 3012 y Fy(P)2094 2997 y FB(t)2129 3012 y Fv(j)2201 2997 y FE(and)f FB(s)2440 3012 y Fv(j)2510 2997 y FA(\037)2587 3012 y Fy(P)2680 2997 y FB(t)2715 3012 y Fv(j)2787 2997 y FE(ensure)i FB(s)3141 3012 y Fv(j)3177 2997 y FB(\033)f Fu(\045)3346 3012 y Fy(P)3439 2997 y FB(t)3474 3012 y Fv(j)3511 2997 y FB(\033)183 3117 y FE(and)32 b FB(s)418 3132 y Fv(j)455 3117 y FB(\033)f FA(\037)618 3132 y Fy(P)706 3117 y FB(t)741 3132 y Fv(j)777 3117 y FB(\033)t FE(,)i(resp)s(ectiv)m(ely)-8 b(,)34 b(for)e(all)e(substitutions)j FB(\033)t FE(.)280 3238 y(Because)k(rewrite)e(rules)f(and)i(dep)s(endency)h(pairs)d(are)h(just) g(pairs)g(of)f(terms,)h(w)m(e)h(write)183 3358 y FA(R)22 b([)h(P)48 b(\022)39 b Fu(\045)688 3373 y Fy(P)787 3358 y FE(as)h(a)f(shorthand)h(for)f FB(l)j Fu(\045)1774 3373 y Fy(P)1873 3358 y FB(r)g FE(for)c(ev)m(ery)k(rewrite)d(rule)g FB(l)19 b FA(!)d FB(r)42 b FE(in)d FA(R)h FE(and)183 3479 y(ev)m(ery)28 b(dep)s(endency)h(pair)c FB(l)30 b FA(!)d FB(r)i FE(from)c FA(P)8 b FE(.)27 b(Moreo)m(v)m(er,)h FA(P)d(\\)k(\037)2482 3494 y Fy(P)2569 3479 y FA(6)p FE(=)e FA(;)f FE(denotes)i(that)e FB(l)k FA(\037)3436 3494 y Fy(P)3523 3479 y FB(r)183 3599 y FE(holds)i(for)g(at)g(least)g (one)h(dep)s(endency)i(pair)d FB(l)e FA(!)d FB(r)35 b FE(from)d FA(P)8 b FE(.)183 3827 y FK(Theorem)38 b(3.5)f(\(Modular)i (Termina)-7 b(tion)39 b(Pr)n(oofs\):)49 b FC(A)30 b(TRS)g FA(R)h FC(is)f(terminating)183 3948 y(if)41 b(and)f(only)h(if)g(for)g (e)-5 b(ach)41 b(cycle)g FA(P)50 b FC(in)41 b(the)g(\(estimate)-5 b(d\))40 b(dep)-5 b(endency)40 b(gr)-5 b(aph)41 b(ther)-5 b(e)41 b(is)g(a)183 4068 y(r)-5 b(e)g(duction)34 b(p)-5 b(air)35 b FE(\()p Fu(\045)923 4083 y Fy(P)982 4068 y FB(;)17 b FA(\037)1103 4083 y Fy(P)1162 4068 y FE(\))35 b FC(such)g(that)230 4271 y(\(a\))f FA(R)23 b([)f(P)37 b(\022)28 b Fu(\045)877 4286 y Fy(P)971 4271 y FC(and)230 4392 y(\(b\))39 b FA(P)25 b(\\)j(\037)659 4407 y Fy(P)747 4392 y FA(6)p FE(=)f FA(;)p FC(.)183 4620 y(Pr)-5 b(o)g(of:)48 b FE(F)-8 b(or)35 b(the)h(if)f(direction,)f(supp)s(ose)j(that)f(there)g (exists)h(an)e(in\014nite)g FA(R)p FE(-c)m(hain)h(of)f(de-)183 4741 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Fz(2)2572 1219 y FB(\033)f FA(\037)2735 1234 y Fy(P)2822 1219 y FB(t\032)2907 1234 y Fz(2)2947 1219 y FB(\033)h Fu(\045)3111 1234 y Fy(P)3198 1219 y FB(:)17 b(:)g(:)183 1422 y FE(But)44 b(this)g(is)g(a)g(con)m(tradiction)g(to)g(the)g(w)m (ell-foundedness)i(of)e FA(\037)2665 1437 y Fy(P)2724 1422 y FE(.)h(Hence,)h(no)e(in\014nite)183 1542 y(c)m(hain)30 b(of)h(dep)s(endency)i(pairs)d(from)g FA(P)39 b FE(exists)32 b(and)e(b)m(y)i(Theorem)f(3.3,)g FA(R)g FE(is)f(terminating.)280 1663 y(F)-8 b(or)24 b(the)h(only-if)e(direction)h(w)m(e)i(refer)f(to)f ([Arts)i(and)f(Giesl,)e(2000,)h(Theorem)h(7],)g(where)h(it)183 1783 y(is)e(sho)m(wn)j(that)d(termination)f(of)h FA(R)i FE(ev)m(en)h(implies)22 b(termination)h(of)h FA(R)7 b([)g FK(DP)p FE(\()p FA(R)p FE(\).)25 b(A)g(simple)183 1903 y(alternativ)m(e)38 b(pro)s(of)h(for)g(this)g(statemen)m(t)h(using)g(t) m(yping)f(can)h(b)s(e)g(found)g(in)e([Middeldorp)183 2024 y(and)32 b(Ohsaki,)h(2000].)2518 b Fs(2)280 2232 y FE(W)-8 b(e)41 b(already)f(men)m(tioned)f(that)h(for)g(Theorem)h(3.3) e(\(and)i(hence,)g(also)f(for)f(the)i(ab)s(o)m(v)m(e)183 2352 y(theorem\))k(considering)h(just)g(the)g(minimal)c(cycles)47 b(w)m(ould)f(b)s(e)g(unsound.)h(In)g(fact,)e(for)183 2473 y(Theorem)g(3.5)g(it)g(w)m(ould)g(also)g(b)s(e)g(unsound)i(just)f (to)f(consider)g FC(maximal)55 b FE(cycles)46 b(\(i.e.,)183 2593 y(those)39 b(cycles)h(whic)m(h)f(are)g(not)f(con)m(tained)h(in)f (an)m(y)i(other)e(cycle\).)i(The)f(problem)f(is)g(that)183 2714 y(it)c(is)i(not)g(su\016cien)m(t)h(if)d(just)j(one)f(dep)s (endency)i(pair)d(of)g(eac)m(h)i(maximal)c(cycle)j(is)g(strictly)183 2834 y(decreasing.)27 b(F)-8 b(or)26 b(a)g(coun)m(terexample)i(to)e (illustrate)f(this)h(fact)g(see)i([Giesl)e(and)g(Arts,)i(2001,)183 2954 y(p.)k(51].)h(Th)m(us,)h(it)d(is)i(crucial)e(to)h(consider)h FC(al)5 b(l)42 b FE(cycles)34 b FA(P)41 b FE(for)32 b(Theorem)h(3.5.) 280 3075 y(With)c(the)g(ab)s(o)m(v)m(e)h(theorem,)e(termination)f(of)h 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Ft(f)36 b FE(is)28 b(a)g(new)183 5205 y(unary)j(function)g (sym)m(b)s(ol\).)f(So)h(instead)h(of)e(\(4\))h(w)m(e)h(obtain)e(the)i (inequalit)m(y)e Ft(f)6 b FE(\()p FB(x)p FE(\))28 b Fu(\045)3307 5220 y Fz(1)3375 5205 y Ft(f)7 b FE(\()p FB(x)p FE(\).)183 5326 y(By)31 b(comparing)e(the)i(terms)f(resulting)g(from)f(this)h (replacemen)m(t)h(\(instead)f(of)g(the)h(original)183 5446 y(terms\))40 b(w)m(e)h(can)g(tak)m(e)g(adv)-5 b(an)m(tage)40 b(of)g(the)h(fact)f(that)g Ft(f)47 b FE(do)s(es)41 b(not)f(ha)m(v)m(e)i (to)e(b)s(e)g(strongly)183 5567 y(monotonic)31 b(in)g(its)i(second)h (argumen)m(t.)e(No)m(w)h(the)g(inequalities)e(resulting)h(from)f(\(4\)) h(-)g(\(6\))p eop %%Page: 13 13 13 12 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(13)183 390 y FE(are)29 b(satis\014ed)h(b)m(y)g(the)g (lexicographic)e(path)i(ordering)e(\(LPO\))i(where)g(subterms)h(are)e (com-)183 511 y(pared)42 b(righ)m(t-to-left)e(\(i.e.,)i Fu(\045)1308 526 y Fz(1)1390 511 y FE(is)g(c)m(hosen)i(to)e(b)s(e)h Fu(\045)2169 526 y Fv(LP)10 b(O)2374 511 y FE(and)43 b FA(\037)2651 526 y Fz(1)2733 511 y FE(is)f(c)m(hosen)i(to)e(b)s(e)g (the)183 631 y(\(stable-\)strict)33 b(relation)g FA(\037)1235 646 y Fv(LP)10 b(O)1397 631 y FE(\).)35 b(F)-8 b(or)33 b(the)i(inequalities)e(\(7\))h(-)g(\(9\))g(w)m(e)i(again)d(delete)h (the)183 751 y(second)40 b(argumen)m(t)f(of)g Ft(f)6 b FE(.)40 b(Then)g(these)g(inequalities)e(are)h(also)f(satis\014ed)i(b) m(y)g(LPO)f(\(with)183 872 y(the)30 b(precedence)j Ft(F)28 b FB(>)f Ft(s)q FB(;)33 b Ft(F)28 b FB(>)g Ft(c)p FE(\),)i(but)g(this)g (time)f(subterms)i(are)f(compared)g(left-to-righ)m(t.)183 992 y(Hence,)44 b(termination)c(of)i(the)h(TRS)g(under)h(consideration) e(is)g(pro)m(v)m(ed.)i(Note)f(that)f(this)183 1112 y(TRS)e(is)f(not)h (simply)e(terminating.)f(So)j(in)f(the)h(dep)s(endency)i(pair)d (approac)m(h,)h(simpli\014-)183 1233 y(cation)34 b(orderings)h(lik)m(e) 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Fz(1)2092 3478 y FB(;)17 b(:)g(:)g(:)e(;)i (t)2345 3493 y Fv(n)2392 3478 y FE(\))35 b FC(and)f FB(\031)t FE(\()p FB(f)11 b FE(\))28 b(=)f FB(i)p FC(,)733 3599 y FB(f)11 b FE(\()p FB(\031)t FE(\()p FB(t)962 3614 y Fv(i)986 3623 y Fm(1)1025 3599 y FE(\))p FB(;)17 b(:)g(:)g(:)f(;)h (\031)t FE(\()p FB(t)1414 3614 y Fv(i)1438 3622 y Fr(m)1500 3599 y FE(\)\))83 b FC(if)35 b FB(t)28 b FE(=)f FB(f)11 b FE(\()p FB(t)2052 3614 y Fz(1)2092 3599 y FB(;)17 b(:)g(:)g(:)e(;)i (t)2345 3614 y Fv(n)2392 3599 y FE(\))35 b FC(and)f FB(\031)t FE(\()p FB(f)11 b FE(\))28 b(=)f([)p FB(i)3039 3614 y Fz(1)3079 3599 y FB(;)17 b(:)g(:)g(:)f(;)h(i)3331 3614 y Fv(m)3397 3599 y FE(])p FC(.)280 3826 y FE(As)28 b(pro)m(v)m(ed)h(b)m (y)g(Arts)f(and)f(Giesl)f([2000],)h(in)g(order)g(to)h(\014nd)g(a)f (reduction)g(pair)g(satisfying)183 3946 y(a)j(particular)f(set)i(of)f (inequalities,)f(one)i(ma)m(y)f(\014rst)h(apply)g(an)f(argumen)m(t)g (\014ltering)f(for)h(the)183 4067 y(signature)g FA(F)686 4031 y Fv(])748 4067 y 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b(gr)-5 b(aph)25 b(ther)-5 b(e)26 b(is)g(an)f(ar)-5 b(gument)26 b(\014ltering)g FB(\031)2553 5012 y Fy(P)2638 4997 y FC(for)g FA(F)2867 4960 y Fv(])2924 4997 y FC(and)f(a)h(r)-5 b(e)g(duction)183 5117 y(p)g(air)34 b FE(\()p Fu(\045)498 5132 y Fy(P)558 5117 y FB(;)17 b FA(\037)679 5132 y Fy(P)738 5117 y FE(\))35 b FC(such)f(that)230 5320 y(\(a\))g FB(\031)449 5335 y Fy(P)509 5320 y FE(\()p FA(R)22 b([)h(P)8 b FE(\))28 b FA(\022)g Fu(\045)1067 5335 y Fy(P)1161 5320 y FC(and)230 5441 y(\(b\))39 b FB(\031)449 5456 y Fy(P)509 5441 y FE(\()p FA(P)8 b FE(\))17 b FA(\\)28 b(\037)850 5456 y Fy(P)937 5441 y FA(6)p FE(=)f FA(;)p FC(.)p eop %%Page: 14 14 14 13 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(14)280 390 y FE(Note)43 b(that)g(there)h(exist)f(only)g (\014nitely)f(man)m(y)h(p)s(ossibilities)d(for)i(the)i(c)m(hoice)f(of)f (suc)m(h)183 511 y(argumen)m(t)31 b(\014lterings.)g(Therefore)i(in)e (principle,)f(all)g(these)j(p)s(ossibilities)c(can)j(b)s(e)g(c)m(hec)m (k)m(ed)183 631 y(automatically)-8 b(.)25 b(Hence,)30 b(b)m(y)g(com)m(bining)d(the)i(generation)f(of)g(a)h(suitable)e (argumen)m(t)i(\014lter-)183 751 y(ing)35 b(with)i(w)m(ell-kno)m(wn)g (automatic)d(tec)m(hniques)39 b(for)d(the)h(syn)m(thesis)i(of)d(\()p FC(str)-5 b(ongly)45 b FE(mono-)183 872 y(tonic\))33 b(simpli\014cation)d(orderings,)k(no)m(w)h(the)f(searc)m(h)h(for)f(a)f FC(we)-5 b(akly)42 b FE(monotonic)33 b(ordering)183 992 y(satisfying)25 b(the)h(constrain)m(ts)g(can)h(b)s(e)f(automated.)f(As) h(men)m(tioned)g(b)s(efore,)g(in)f(a)h(reduction)183 1112 y(pair)31 b(\()p Fu(\045)p FB(;)17 b FA(\037)p FE(\))33 b(one)g(usually)f(c)m(ho)s(oses)i FA(\037)f FE(to)f(b)s(e)h(the)g (stable-strict)e(relation)g(corresp)s(onding)183 1233 y(to)h(the)h(quasi-ordering)f Fu(\045)p FE(.)h(By)g(using)g(the)g (estimated)f(dep)s(endency)j(graph,)e(this)f(results)183 1353 y(in)c(a)h(fully)e(automatic)g(termination)g(pro)s(of)h(of)h(our)g (TRS,)g(whereas)i(a)d(direct)h(termination)183 1474 y(pro)s(of)35 b(with)g(simpli\014cation)e(orderings)i(w)m(as)i(not)f(p)s(ossible.)f (So)h(Criterion)f(3.7)g(allo)m(ws)g(us)183 1594 y(to)44 b(use)i FC(di\013er)-5 b(ent)54 b FE(quasi-orderings)44 b(resp.)i(reduction)f(pairs)g(to)f(pro)m(v)m(e)i(the)g(absence)g(of)183 1714 y(c)m(hains)34 b(for)g(di\013eren)m(t)h(cycles.)h(In)e(our)h (example)f(this)g(is)g(essen)m(tial,)g(b)s(ecause)i(there)f(exists)183 1835 y(no)f(reduction)f(pair)g(with)h(a)g(quasi-simpli\014cation)c (ordering)j(satisfying)g FC(al)5 b(l)44 b FE(inequalities)183 1955 y(\(4\))36 b(-)g(\(9\))g(\(not)g(ev)m(en)i(after)f(elimination)32 b(of)k(argumen)m(ts\).)h(The)g(reason)g(is)f(that)h(\(9\))f(and)183 2076 y(\(6\))c(en)m(tail)385 2296 y Ft(F)p FE(\()p Ft(s)q FE(\()p FB(x)p FE(\))p FB(;)17 b Ft(s)p FE(\()p FB(x)p FE(\)\))28 b FA(\037)1002 2311 y Fz(2)1070 2296 y Ft(F)p FE(\()p FB(x;)17 b Ft(s)q FE(\()p Ft(c)p FE(\()p Ft(s)p FE(\()p FB(x)p FE(\)\)\)\))28 b FA(!)1829 2315 y Fy(E)6 b Fv(mb)p Fz(\()p Fy(F)2048 2296 y Fr(])2078 2315 y Fz(\))2137 2296 y Ft(F)q FE(\()p FB(x;)17 b Ft(c)p FE(\()p Ft(s)q FE(\()p FB(x)p FE(\)\)\))27 b FA(\037)2760 2311 y Fz(1)2828 2296 y Ft(F)p FE(\()p Ft(s)q FE(\()p FB(x)p FE(\))p FB(;)17 b Ft(s)p FE(\()p FB(x)p FE(\)\))p FB(:)183 2516 y FE(Hence,)30 b(without)f(our)g(mo)s(dularit)m(y)e(result,)j(an)f(automated)f (termination)f(pro)s(of)i(with)f(the)183 2636 y(dep)s(endency)35 b(pair)c(approac)m(h)i(fails.)280 2756 y(In)f(order)f(to)g(syn)m (thesize)i(suitable)d(reduction)i(pairs,)e(the)i(argumen)m(t)f (\014lterings)f(should)183 2877 y(b)s(e)39 b(c)m(hosen)j(in)c(a)i(w)m (a)m(y)g(suc)m(h)h(that)f(for)f(all)e(resulting)i(inequalities)e(the)j (v)-5 b(ariables)39 b(in)f(the)183 2997 y(righ)m(t-hand)c(side)h(also)g (o)s(ccur)g(in)g(the)h(left-hand)e(side.)h(Then)i(the)e(resulting)g (inequalities)183 3117 y(could)44 b(b)s(e)g(transformed)h(in)m(to)e(a)h (TRS)h(as)g(w)m(ell)f(and)g(for)g(pro)m(ving)g(termination)e(of)i(the) 183 3238 y(original)h(TRS)k(it)f(w)m(ould)g(b)s(e)h(su\016cien)m(t)h (to)e(pro)m(v)m(e)i(termination)c(of)i(the)h(transformed)183 3358 y(TRSs)33 b(for)f(all)f(cycles.)183 3587 y FK(Criterion)38 b(3.8)f(\(Termina)-7 b(tion)40 b(Criterion)e(by)f(Transf)n(orma)-7 b(tion\):)51 b FC(A)26 b(TRS)e FA(R)183 3707 y FC(over)40 b(a)h(signatur)-5 b(e)40 b FA(F)51 b FC(is)40 b(terminating)g(if)h(and) f(only)h(if)f(for)h(e)-5 b(ach)40 b(cycle)h FA(P)49 b FC(in)40 b(the)h(\(esti-)183 3827 y(mate)-5 b(d\))47 b(dep)-5 b(endency)48 b(gr)-5 b(aph)47 b(ther)-5 b(e)49 b(is)f(an)g(ar)-5 b(gument)48 b(\014ltering)g FB(\031)2732 3842 y Fy(P)2840 3827 y FC(for)g FA(F)3091 3791 y Fv(])3170 3827 y FC(such)h(that)183 3948 y FB(\031)238 3963 y Fy(P)297 3948 y FE(\()p FA(R)23 b([)f(P)8 b FE(\))35 b FC(is)g(a)g(terminating)f (TRS)g(and)h(such)f(that)i FB(\031)2295 3963 y Fy(P)2354 3948 y FE(\()p FA(P)8 b FE(\))28 b FA(6)p FE(=)g FA(;)p FC(.)280 4176 y FE(This)22 b(criterion)e(is)i(su\016cien)m(t)h(for)e (termination,)e(since)j(one)g(ma)m(y)g(c)m(ho)s(ose)g(\()p FA(!)3076 4140 y Fy(\003)3076 4206 y Fv(\031)3117 4217 y Fk(P)3169 4206 y Fz(\()p Fy(R[P)6 b Fz(\))3390 4176 y FB(;)p FA(!)3517 4135 y Fz(+)3517 4207 y Fv(\031)3558 4218 y Fk(P)3609 4207 y Fz(\()p Fy(P)g Fz(\))3723 4176 y FE(\))183 4296 y(as)36 b(the)g(reduction)f(pairs)g(in)g(Criterion)g (3.7.)g(It)h(is)f(also)g(necessary)j(for)d(termination,)e(b)s(e-)183 4417 y(cause)39 b(due)f(to)g([Arts)g(and)h(Giesl,)d(2000,)h(Theorem)h (7],)g(termination)e(of)h FA(R)i FE(implies)c(ter-)183 4537 y(mination)30 b(of)j(all)e FA(R)23 b([)f(P)42 b FE(\(and)33 b(hence,)i(of)d FB(\031)1842 4552 y Fy(P)1902 4537 y FE(\()p FA(R)22 b([)h(P)8 b FE(\),)34 b(if)e FB(\031)2456 4552 y Fy(P)2515 4537 y FE(\()p FB(f)11 b FE(\))28 b(=)h([1)p FB(;)17 b(:)g(:)g(:)e(;)i(n)p FE(])33 b(for)g(ev)m(ery)183 4658 y FB(f)38 b FA(2)28 b(F)42 b FE(with)32 b(arit)m(y)g FB(n)p FE(,)h(i.e.,)f(if)g FB(\031)1379 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(orderings)h(searc)m(h)g(for)f(total)f(orderings.)i(Ho)m(w-)183 4359 y(ev)m(er,)j(w)m(e)g(concen)m(trate)g(on)f(simpli\014cation)c (orderings)k(or)f(quasi-simpli\014cation)d(orderings)183 4480 y([Dersho)m(witz,)31 b(1987,)f(Stein)m(bac)m(h,)i(1995,)e (Middeldorp)g(and)h(Zan)m(tema,)f(1997])g(b)s(ecause)i(all)183 4600 y(TRSs)41 b(that)g(are)f(totally)f(terminating)f(ha)m(v)m(e)k(b)s (een)g(sho)m(wn)g(to)e(b)s(e)h(simply)e(terminating)183 4721 y([Zan)m(tema,)31 b(1994])h(and)h(b)s(ecause)g(simple)e (termination)f(has)j(a)f(nice)h(mo)s(dular)d(b)s(eha)m(viour,)183 4841 y(whereas)k(mo)s(dularit)m(y)c(of)i(total)f(termination)f(is)i (still)f(an)h(op)s(en)h(problem.)280 4961 y(No)m(w)i(w)m(e)h(formally) 31 b(de\014ne)36 b(the)f(notion)e(of)h(DP)h(quasi-simple)d(termination) g(whic)m(h)j(re-)183 5082 y(sults)42 b(from)f(restricting)h(ourselv)m (es)h(to)f(QSOs)h(when)h(using)e(the)g(dep)s(endency)j(pair)c(ap-)183 5202 y(proac)m(h)32 b(\(i.e.,)f(when)i(using)f(Criterion)e(3.7\).)i (The)h(motiv)-5 b(ation)28 b(for)k(this)f(notion)g(is)g(that)h(it)183 5323 y(con)m(tains)f(all)e(TRSs)k(where)f(termination)d(can)j(b)s(e)g (pro)m(v)m(ed)h FC(automatic)-5 b(al)5 b(ly)40 b FE(in)30 b(the)i(follo)m(w-)183 5443 y(ing)c(w)m(a)m(y:)i(First,)e(the)i (constrain)m(ts)f(describ)s(ed)h(in)f(Theorem)g(3.5)g(are)g(generated)h (using)e(the)183 5563 y(estimated)37 b(dep)s(endency)k(graph,)d(whic)m (h)h(can)g(b)s(e)f(determined)g(mec)m(hanically)-8 b(.)37 b(Then)i(an)p eop %%Page: 17 17 17 16 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(17)183 390 y FE(argumen)m(t)34 b(\014ltering)f(is)h(applied)g(to) g(eliminate)e(argumen)m(ts)i(of)h(function)f(sym)m(b)s(ols)g(\(or)g(to) 183 511 y(replace)c(functions)h(b)m(y)h(their)e(argumen)m(ts\))h(as)g (in)f(Criterion)g(3.7,)g(and)h(\014nally)f(a)g(standard)183 631 y(tec)m(hnique)d(is)f(used)h(to)e(generate)i(a)f(QSO)f Fu(\045)i FE(suc)m(h)g(that)f(a)g(reduction)g(pair)f(\()p Fu(\045)p FB(;)17 b FA(\037)p FE(\))26 b(satis\014es)183 751 y(the)37 b(resulting)f(constrain)m(ts.)h(F)-8 b(or)36 b(example,)g FA(\037)i FE(can)f(b)s(e)g(c)m(hosen)h(to)f(b)s(e)g(the)g (stable-strict)183 872 y(relation)30 b(corresp)s(onding)j(to)f Fu(\045)p FE(.)183 1100 y FK(Definition)38 b(4.1)f(\(DP)h(quasi-simple) f(termina)-7 b(tion\):)50 b FC(A)38 b(TRS)f FA(R)h FC(over)f(a)h (signa-)183 1220 y(tur)-5 b(e)37 b FA(F)45 b FC(is)36 b(c)-5 b(al)5 b(le)-5 b(d)36 b FE(DP)e(quasi-simply)e(terminating)i FC(if)i(and)g(only)g(if)g(for)g(e)-5 b(ach)35 b(cycle)h FA(P)45 b FC(in)183 1341 y(the)31 b(estimate)-5 b(d)32 b(dep)-5 b(endency)30 b(gr)-5 b(aph)31 b(ther)-5 b(e)32 b(exists)g(an)f(ar)-5 b(gument)32 b(\014ltering)f FB(\031)3027 1356 y Fy(P)3118 1341 y FC(for)h FA(F)3353 1305 y Fv(])3416 1341 y FC(and)183 1461 y(a)i(r)-5 b(e)g(duction)35 b(p)-5 b(air)34 b FE(\()p Fu(\045)1007 1476 y Fy(P)1067 1461 y FB(;)17 b FA(\037)1188 1476 y Fy(P)1247 1461 y FE(\))35 b FC(with)g(a)f(QSO)g Fu(\045)1932 1476 y Fy(P)2027 1461 y FC(such)h(that)230 1665 y(\(a\))f FB(\031)449 1680 y Fy(P)509 1665 y FE(\()p FA(R)22 b([)h(P)8 b FE(\))28 b FA(\022)g Fu(\045)1067 1680 y Fy(P)1161 1665 y FC(and)230 1785 y(\(b\))39 b FB(\031)449 1800 y Fy(P)509 1785 y FE(\()p FA(P)8 b FE(\))17 b FA(\\)28 b(\037)850 1800 y Fy(P)937 1785 y FA(6)p FE(=)f FA(;)p FC(.)280 2013 y FE(De\014nition)36 b(4.1)h(captures)h(the)g(TRSs)g(for)f(whic)m(h)h (an)f(automated)g(termination)e(pro)s(of)183 2134 y(using)46 b(dep)s(endency)j(pairs)e(with)f(the)h(estimated)f(dep)s(endency)k (graph)2918 2098 y Fy(y)3000 2134 y FE(is)c(p)s(oten)m(tially)183 2254 y(feasible)31 b(\(since)i(virtually)d(all)g(quasi-orderings)i (that)g(can)h(b)s(e)f(generated)h(are)g(QSOs\).)g(In)183 2374 y(fact,)21 b(there)i(are)f(n)m(umerous)g(DP)g(quasi-simply)e (terminating)g(TRSs)i(whic)m(h)h(are)f(not)f(simply)183 2495 y(terminating;)i(cf.)i(e.g.)h(the)g(collection)d(b)m(y)j(Arts)g (and)f(Giesl)f([2001].)h(This)g(observ)-5 b(ation)25 b(mo-)183 2615 y(tiv)-5 b(ated)30 b(the)i(dev)m(elopmen)m(t)g(of)f(the) h(dep)s(endency)h(pair)e(approac)m(h)g(and)h(it)e(also)h(motiv)-5 b(ated)183 2736 y(the)40 b(w)m(ork)h(of)f(the)g(presen)m(t)i(section,)f (as)f(our)g(aim)e(is)i(to)g(extend)h(w)m(ell-kno)m(wn)f(mo)s(dular-)183 2856 y(it)m(y)31 b(results)h(for)f(simple)f(termination)f(to)i(DP)h (quasi-simple)d(termination.)h(F)-8 b(or)30 b(instance,)183 2976 y(the)k(TRS)g(from)f(Section)h(3)g(is)f(ob)m(viously)h(DP)f (quasi-simply)g(terminating,)e(b)s(ecause)k(the)183 3097 y(resulting)29 b(constrain)m(ts)h(are)g(satis\014ed)g(b)m(y)h(LPO)f (\(whic)m(h)h(is)e(a)h(quasi-simpli\014cation)c(order-)183 3217 y(ing\).)38 b(Similarly)-8 b(,)34 b(for)39 b(the)g(TRS)g FA(R)1505 3232 y Fz(1)1584 3217 y FE(=)f FA(f)p FE(\(10\))p FB(;)17 b FE(\(11\))p FA(g)37 b FE(from)h(Section)g(4.1)h(w)m(e)h (obtain)d(the)183 3337 y(follo)m(wing)30 b(dep)s(endency)35 b(pairs)1350 3557 y Ft(F)q FE(\()p Ft(0)o FB(;)17 b Ft(1)p FB(;)g(x)p FE(\))99 b FA(!)h Ft(F)p FE(\()p Ft(s)p FE(\()p FB(x)p FE(\))p FB(;)17 b(x;)g(x)p FE(\))879 b(\(14\))1234 3703 y Ft(F)p FE(\()p FB(x;)17 b(y)t(;)g Ft(s)o FE(\()p FB(z)t FE(\)\))100 b FA(!)g Ft(F)p FE(\()p Ft(0)p FB(;)17 b Ft(1)o FB(;)g(z)t FE(\))p FB(:)984 b FE(\(15\))183 3923 y(Our)43 b(estimation)f(tec)m(hnique)j(determines)f(that)g(the)g (\014rst)g(dep)s(endency)j(pair)42 b(\(14\))i(can)183 4043 y(nev)m(er)31 b(follo)m(w)d(itself)h(in)g(a)g(c)m(hain,)h(b)s (ecause)h Ft(F)q FE(\()p Ft(s)p FE(\()p FB(x)2006 4058 y Fz(1)2046 4043 y FE(\))p FB(;)17 b(x)2183 4058 y Fz(2)2222 4043 y FB(;)g(x)2321 4058 y Fz(3)2361 4043 y FE(\))p FB(\033)32 b FA(!)2586 4007 y Fy(\003)2586 4068 y(R)2646 4077 y Fm(1)2712 4043 y Ft(F)p FE(\()p Ft(0)p FB(;)17 b Ft(1)o FB(;)g(x)3045 4058 y Fz(4)3085 4043 y FE(\))p FB(\033)34 b FE(do)s(es)c(not)183 4164 y(hold)e(for)g(an)m(y)i (substitution)e FB(\033)t FE(.)h(So)g(in)f(our)h(example,)f(the)h (estimated)g(dep)s(endency)i(graph)183 4284 y(con)m(tains)g(an)g(arc)g (from)f(\(14\))h(to)g(\(15\))g(and)g(arcs)h(from)e(\(15\))g(to)h (\(14\))g(and)g(to)g(itself.)f(Th)m(us,)183 4404 y(the)j(only)g(cycles) h(in)e(our)h(example)g(are)g FA(f)p FE(\(15\))p FA(g)f FE(and)h FA(f)p FE(\(14\))p FB(;)17 b FE(\(15\))p FA(g)p FE(.)31 b(Hence,)k(according)d(to)183 4525 y(Theorem)39 b(3.5,)g(to)h(pro)m(v)m(e)g(the)g(absence)h(of)e(in\014nite)g(c)m (hains)g(from)g(the)g(cycle)h FA(f)p FE(\(15\))p FA(g)f FE(w)m(e)183 4645 y(ha)m(v)m(e)34 b(to)e(\014nd)h(a)f(reduction)h(pair) e(satisfying)1406 4865 y Ft(f)7 b FE(\()p Ft(0)o FB(;)17 b Ft(1)p FB(;)g(x)p FE(\))83 b Fu(\045)g Ft(f)7 b FE(\()p Ft(s)p FE(\()p FB(x)p FE(\))p FB(;)17 b(x;)g(x)p FE(\))1290 5010 y Ft(f)6 b FE(\()p FB(x;)17 b(y)t(;)g Ft(s)o FE(\()p FB(z)t FE(\)\))84 b Fu(\045)f Ft(s)q FE(\()p Ft(f)6 b FE(\()p Ft(0)p FB(;)17 b Ft(1)p FB(;)g(z)t FE(\)\))1271 5156 y Ft(F)p FE(\()p FB(x;)g(y)t(;)g Ft(s)o FE(\()p FB(z)t FE(\)\))84 b FA(\037)f Ft(F)q FE(\()p Ft(0)p FB(;)17 b Ft(1)o FB(;)g(z)t FE(\))p FB(:)297 5300 y Fx(y)332 5330 y Fw(Note)33 b(that)g(the)h(notion)e(of)h(DP)g(quasi-simple)f (termination)h(and)g(therefore)f(also)g(our)g(mo)r(dularit)n(y)183 5430 y(results)37 b(dep)r(end)h(on)f(the)h(estimation)f(of)h(the)f(dep) r(endency)h(graph.)f(Th)n(us,)g(for)g(other)g(appro)n(ximation)183 5529 y(tec)n(hniques)27 b(one)g(w)n(ould)g(ha)n(v)n(e)g(to)g(in)n(v)n (estigate)f(the)i(resulting)f(mo)r(dularit)n(y)g(prop)r(erties)g (separately)-7 b(.)p eop %%Page: 18 18 18 17 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(18)183 390 y FE(By)39 b(using)g(the)h(argumen)m(t)e(\014ltering)g (that)h(maps)g Ft(f)45 b FE(to)39 b(its)f(third)h(argumen)m(t,)g(these) h(con-)183 511 y(strain)m(ts)f(are)g(satis\014ed)g(b)m(y)h(RPO)f(with)g (the)h(precedence)h Ft(s)f FB(>)e Ft(0)h FE(and)g Ft(s)g FB(>)g Ft(1)p FE(.)g(Similarly)-8 b(,)183 631 y(\(b)m(y)34 b(eliminating)c(the)k(\014rst)h(t)m(w)m(o)f(argumen)m(ts)g(of)g Ft(F)p FE(\))g(one)g(can)g(also)f(pro)m(v)m(e)j(the)e(absence)h(of)183 751 y(in\014nite)g(c)m(hains)h(from)e(the)j(cycle)f FA(f)p FE(\(14\))p FB(;)17 b FE(\(15\))p FA(g)p FE(.)34 b(Hence,)k (termination)33 b(of)j(the)g(TRS)g(con-)183 872 y(sisting)28 b(of)h(the)h(rules)g(\(10\))f(and)g(\(11\))g(is)h(pro)m(v)m(ed)h(and)e (\(as)h(RPO)f(is)h(a)f(quasi-simpli\014cation)183 992 y(ordering\),)i(it)h(is)g(DP)g(quasi-simply)f(terminating.)280 1112 y(In)h(this)g(article,)f(w)m(e)h(imp)s(ose)f(a)h(minor)e (restriction)h(on)h(the)g(argumen)m(t)g(\014lterings)f(used,)183 1233 y(viz.)g(for)g(all)f(cycles)j FA(P)40 b FE(w)m(e)32 b(restrict)g(ourselv)m(es)h(to)e(argumen)m(t)g(\014lterings)g FB(\031)2933 1248 y Fy(P)3024 1233 y FE(suc)m(h)i(that)e(for)183 1353 y(all)h(rules)i FB(s)c FA(!)g FB(t)35 b FE(in)e FB(\031)1001 1368 y Fy(P)1060 1353 y FE(\()p FA(R)24 b([)g(P)8 b FE(\))34 b(b)s(oth)g FA(V)8 b FB(ar)s FE(\()p FB(t)p FE(\))31 b FA(\022)g(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))34 b(and)g FB(s)c FA(62)h(V)8 b FE(.)35 b(This)f(restriction)183 1474 y(ensures)g(that)d(the)i(rules)f FB(\031)1191 1489 y Fy(P)1250 1474 y FE(\()p FA(R)22 b([)f(P)8 b FE(\))32 b(from)f(Criterion)g(3.8)g(indeed)i(form)e(a)g(term)h(rewrit-)183 1594 y(ing)i(system.)j(According)e(to)h(Lemma)e(2.4,)h(if)g(there)h(is) g(a)f(quasi-simpli\014cation)d(ordering)183 1714 y(satisfying)41 b(the)i(constrain)m(ts)g(in)e(Criterion)h(3.7)g(\(i.e.,)g(in)f (De\014nition)g(4.1\))h(and)g(if)g(these)183 1835 y(constrain)m(ts)32 b(include)g(at)g(least)g(one)g(strict)g(inequalit)m(y)g(with)g(v)-5 b(ariables)31 b(in)g(its)h(righ)m(t-hand)183 1955 y(side,)k(then)h FA(V)8 b FB(ar)s FE(\()p FB(\031)t FE(\()p FB(r)s FE(\)\))34 b FA(\022)g(V)8 b FB(ar)s FE(\()p FB(\031)t FE(\()p FB(l)r FE(\)\))37 b(is)f(alw)m(a)m(ys)h(satis\014ed)f(for)g(all)f FB(l)h FA(!)e FB(r)39 b FE(in)c FA(R)26 b([)f(P)8 b FE(.)37 b(In)183 2076 y(other)32 b(w)m(ords,)i(the)f(restriction)f(is)g(not)g (v)m(ery)i(sev)m(ere.)280 2196 y(In)h(fact,)f(in)g(the)h(pro)s(of)f(of) g(mo)s(dularit)m(y)e(of)i(DP)g(quasi-simple)f(termination)f(it)h(is)h (su\016-)183 2316 y(cien)m(t)f(to)g(kno)m(w)i(that)e(for)g(ev)m(ery)i (cycle)f(of)f(a)h(DP)f(quasi-simply)e(terminating)g(TRS)j(there)183 2437 y(is)e(at)g(least)g FC(one)39 b FE(argumen)m(t)33 b(\014ltering)e(satisfying)g(the)i(minor)e(restriction)h(and)h(a)f (suitable)183 2557 y(QSO)40 b(that)h(pro)m(v)m(e)h(termination.)d(Ho)m (w)m(ev)m(er,)k(it)d(is)g(an)h(op)s(en)g(problem)f(whether)i(for)f(ev-) 183 2677 y(ery)30 b(DP)f(quasi-simply)f(terminating)f(TRS)j(suc)m(h)h (an)f(argumen)m(t)f(\014ltering)f(and)i(a)f(suitable)183 2798 y(QSO)36 b(alw)m(a)m(ys)g(exist.)h(Nev)m(ertheless,)h(ev)m(en)g (if)d(there)i(w)m(ere)g(a)f(coun)m(terexample,)h(then)f(the)183 2918 y(QSO)h(satisfying)f(the)i(constrain)m(ts)f(m)m(ust)h(ful\014ll)c FB(s)i Fu(\045)g FB(C)7 b FE([)p FB(y)t FE(])35 b Fu(\045)h FB(y)41 b FE(for)36 b(some)h(term)g FB(s)g FE(with)183 3039 y FB(y)42 b FA(62)d(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))40 b(or)e FB(x)i Fu(\045)g FB(t)f FE(for)g(a)g(term)g FB(t)g FA(6)p FE(=)g FB(x)p FE(.)h(Clearly)-8 b(,)38 b(this)h(is)g(imp)s(ossible)e(for)h(path)i(or-)183 3159 y(derings)31 b(lik)m(e)f(LPO)h(or)g(RPO.)g(Hence,)h(whenev)m(er)i(the)d (constrain)m(ts)g(of)g(De\014nition)e(4.1)i(are)183 3279 y(satis\014ed)f(b)m(y)h(suc)m(h)h(a)e(path)g(ordering,)f(then)i(the)f (restriction)g(on)g(the)g(argumen)m(t)g(\014lterings)183 3400 y(is)37 b(ful\014lled)f(an)m(yw)m(a)m(y)-8 b(.)40 b(A)d(constrain)m(t)h(of)f(the)i(form)d FB(s)h Fu(\045)g FB(y)k FE(with)c FB(y)j FA(62)d(V)8 b FB(ar)s FE(\()p FB(s)p FE(\))38 b(cannot)g(b)s(e)183 3520 y(satis\014ed)28 b(b)m(y)h(p)s(olynomial)c(orderings)j(either)f(unless)i(terms)f(are)g (only)g(mapp)s(ed)g(to)g(\014nitely)183 3641 y(man)m(y)34 b(di\013eren)m(t)h(n)m(um)m(b)s(ers.)h(Th)m(us,)g(the)f(question)h (whether)g(DP)e(quasi-simple)f(termina-)183 3761 y(tion)27 b(w)m(ould)h(also)f(b)s(e)i(mo)s(dular)d(without)i(the)g(ab)s(o)m(v)m (e)h(restriction)f(is)g(not)g(so)g(imp)s(ortan)m(t)e(for)183 3881 y(practical)31 b(termination)f(pro)s(ofs.)280 4002 y(A)i(straigh)m(tforw)m(ard)e(w)m(a)m(y)j(to)e(generate)h(a)f(QSO)g FA(\027)h FE(from)e(a)h(simpli\014cation)d(ordering)i FA(\037)183 4122 y FE(is)i(to)h(de\014ne)h FB(t)29 b FA(\027)g FB(u)j FE(if)g(and)h(only)g(if)f FB(t)d FA(\037)f FB(u)33 b FE(or)g FB(t)28 b FE(=)h FB(u)p FE(,)j(where)j(=)d(is)h(syn)m (tactic)h(equalit)m(y)-8 b(.)33 b(In)183 4242 y(the)c(follo)m(wing,)e (w)m(e)j(denote)g(the)g(re\015exiv)m(e)h(closure)e(of)g(a)g(relation)f (b)m(y)i(underlining,)d(i.e.,)i FA(\027)183 4363 y FE(denotes)h(the)g (re\015exiv)m(e)h(closure)e(of)g FA(\037)p FE(.)h(By)f(restricting)g (ourselv)m(es)h(to)f(this)g(class)h(of)e(QSOs,)183 4483 y(w)m(e)33 b(obtain)f(the)h(notion)e(of)h(DP)h(simple)e(termination.) 183 4711 y FK(Definition)38 b(4.2)f(\(DP)h(simple)f(termina)-7 b(tion\):)50 b FC(A)29 b(TRS)g FA(R)h FC(over)f(a)g(signatur)-5 b(e)29 b FA(F)39 b FC(is)183 4832 y(c)-5 b(al)5 b(le)-5 b(d)41 b FE(DP)f(simply)f(terminating)g FC(if)j(and)f(only)h(if)g(for)f (e)-5 b(ach)42 b(cycle)f FA(P)51 b FC(in)41 b(the)h(estimate)-5 b(d)183 4952 y(dep)g(endency)36 b(gr)-5 b(aph)37 b(ther)-5 b(e)38 b(is)g(an)f(ar)-5 b(gument)38 b(\014ltering)f FB(\031)2318 4967 y Fy(P)2415 4952 y FC(for)h FA(F)2656 4916 y Fv(])2725 4952 y FC(and)f(a)h(simpli\014c)-5 b(ation)183 5073 y(or)g(dering)34 b FA(\037)646 5088 y Fy(P)740 5073 y FC(such)h(that)230 5276 y(\(a\))f FB(\031)449 5291 y Fy(P)509 5276 y FE(\()p FA(R)22 b([)h(P)8 b FE(\))28 b FA(\022)g(\027)1067 5291 y Fy(P)1161 5276 y FC(and)230 5396 y(\(b\))39 b FB(\031)449 5411 y Fy(P)509 5396 y FE(\()p FA(P)8 b FE(\))17 b FA(\\)28 b(\037)850 5411 y Fy(P)937 5396 y FA(6)p FE(=)f FA(;)p FC(.)p eop %%Page: 19 19 19 18 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(19)280 390 y FE(Note)38 b(that)g(whenev)m(er)j(there)e(exist)f (argumen)m(t)g(\014lterings)f(and)i(simpli\014cation)34 b(order-)183 511 y(ings)47 b(satisfying)h(the)g(constrain)m(ts)h(\(a\)) f(and)g(\(b\))h(of)f(De\014nition)e(4.2,)i(then)h(the)g(minor)183 631 y(restriction)35 b(on)i(the)g(argumen)m(t)f(\014lterings)f(is)h (satis\014ed)h(according)f(to)h(Lemma)e(2.1.)h(Due)183 751 y(to)31 b(that)h(lemma,)e(there)j(is)e(the)i(follo)m(wing)c (alternativ)m(e)i(c)m(haracterization)g(for)h(DP)f(simple)183 872 y(termination)f(\(whic)m(h)j(uses)h(Criterion)d(3.8)h(instead)h(of) f(Criterion)f(3.7\).)183 1092 y FK(Cor)n(ollar)-7 b(y)39 b(4.3)e(\(Al)-7 b(terna)g(tive)39 b(Chara)n(cteriza)-7 b(tion)38 b(of)h(DP)e(simple)g(termina)-7 b(tion\):)183 1213 y FC(A)38 b(TRS)g FA(R)g FC(over)f(a)h(signatur)-5 b(e)38 b FA(F)47 b FC(is)38 b(DP)g(simply)f(terminating)g(if)h(and)f (only)h(if)g(for)g(e)-5 b(ach)183 1333 y(cycle)33 b FA(P)43 b FC(in)34 b(the)g(estimate)-5 b(d)33 b(dep)-5 b(endency)33 b(gr)-5 b(aph)33 b(ther)-5 b(e)34 b(is)g(an)f(ar)-5 b(gument)34 b(\014ltering)g FB(\031)3356 1348 y Fy(P)3449 1333 y FC(for)183 1454 y FA(F)265 1417 y Fv(])331 1454 y FC(such)g(that)i FB(\031)807 1469 y Fy(P)866 1454 y FE(\()p FA(R)22 b([)h(P)8 b FE(\))35 b FC(is)g(a)g(simply)f(terminating)g(TRS.)280 1674 y FE(F)-8 b(or)25 b(instance,)h(b)s(oth)f(the)h(TRS)f(from)f (Section)i(3)f(and)g FA(R)2351 1689 y Fz(1)2419 1674 y FE(=)i FA(f)p FE(\(10\))p FB(;)17 b FE(\(11\))p FA(g)24 b FE(from)g(Section)183 1795 y(4.1)46 b(are)h(already)g(DP)g(simply)f (terminating,)e(b)s(ecause)49 b(for)d(their)h(termination)d(pro)s(ofs) 183 1915 y(w)m(e)39 b(ma)m(y)f(use)h(quasi-simpli\014cation)c (orderings)j(in)f(whic)m(h)i(only)f(syn)m(tactically)g(iden)m(tical)183 2035 y(terms)c(are)h(considered)g(to)g(b)s(e)g(equiv)-5 b(alen)m(t.)34 b(Moreo)m(v)m(er,)j(it)c(also)h(turns)h(out)g(that)f (most)g(of)183 2156 y(the)27 b(examples)g(in)g([Arts)g(and)g(Giesl,)f (2001])g(are)i(not)f(only)f(DP)h(quasi-simply)e(terminating)183 2276 y(but)e(ev)m(en)h(DP)f(simply)e(terminating.)g(The)j(follo)m(wing) c(lemma)h(illustrates)g(the)i(connections)183 2397 y(b)s(et)m(w)m(een) 34 b(the)f(di\013eren)m(t)g(notions.)183 2617 y FK(Lemma)38 b(4.4)f(\(Chara)n(cterizing)h(DP)g(\(quasi-\)simple)f(termina)-7 b(tion\):)50 b FC(The)25 b(fol-)183 2738 y(lowing)37 b(implic)-5 b(ations)38 b(hold:)g(simple)g(termination)g FA(\))g FC(DP)h(simple)f(termination)g FA(\))g FC(DP)183 2858 y(quasi-simple)33 b(termination)h FA(\))h FC(termination.)183 3079 y(Pr)-5 b(o)g(of:)48 b FE(The)24 b(second)f(implication)c(holds)j (as)h FA(\037)g FE(is)f(closed)g(under)i(substitutions)e(and)h(there-) 183 3199 y(fore)32 b(\()p FA(\027)p FB(;)17 b FA(\037)p FE(\))33 b(is)f(a)g(reduction)h(pair.)f(The)h(last)f(implication)c (follo)m(ws)k(from)f(Criterion)g(3.7.)280 3319 y(It)i(remains)g(to)g (sho)m(w)h(the)g(\014rst)f(implication.)d(Let)j FA(R)h FE(b)s(e)f(a)g(simply)f(terminating)f(TRS)183 3440 y(o)m(v)m(er)23 b(the)h(signature)e FA(F)9 b FE(.)23 b(If)g FA(R)g FE(is)f(simply)f (terminating,)g(then)i(there)g(exists)h(a)e(simpli\014cation)183 3560 y(ordering)31 b FA(\037)i FE(suc)m(h)h(that)f FB(l)d FA(\037)e FB(r)35 b FE(holds)d(for)g(all)f(rules)h FB(l)e FA(!)e FB(r)35 b FE(of)d FA(R)p FE(.)280 3681 y(Let)42 b(\012)h(b)s(e)f(the)h(function)e(whic)m(h)i(replaces)g(ev)m(ery)h (tuple)d(sym)m(b)s(ol)h FB(f)2884 3644 y Fv(])2957 3681 y FE(in)f(a)h(term)g FB(s)i FA(2)183 3801 y(T)25 b FE(\()p FA(F)382 3765 y Fv(])413 3801 y FB(;)17 b FA(V)8 b FE(\))31 b(b)m(y)g(its)g(corresp)s(onding)f(function)g(sym)m(b)s(ol)g FB(f)39 b FA(2)28 b(F)9 b FE(.)31 b(Then)h FA(\037)f FE(can)g(b)s(e)g(extended)183 3921 y(to)43 b(a)h(simpli\014cation)c (ordering)j FA(\037)1492 3885 y Fy(0)1559 3921 y FE(on)h FA(T)26 b FE(\()p FA(F)1906 3885 y Fv(])1937 3921 y FB(;)17 b FA(V)8 b FE(\))43 b(b)m(y)i(de\014ning)f FB(t)j FA(\037)2817 3885 y Fy(0)2887 3921 y FB(u)c FE(if)g(and)h(only)f(if)183 4042 y(\012\()p FB(t)p FE(\))i FA(\037)g FE(\012\()p FB(u)p FE(\))d(holds.)h(W)-8 b(e)43 b(claim)d(that)i(the)h (simpli\014cation)c(ordering)j FA(\037)2999 4006 y Fy(0)3065 4042 y FE(satis\014es)h(the)183 4162 y(constrain)m(ts)30 b(\(a\))f(and)h(\(b\))f(of)h(De\014nition)d(4.2)j(without)f(applying)f (an)i(argumen)m(t)f(\014ltering.)280 4283 y(Ob)m(viously)-8 b(,)46 b FB(l)54 b FA(\037)931 4246 y Fy(0)1006 4283 y FB(r)49 b FE(holds)d(for)g(all)e(rules)j FB(l)53 b FA(!)e FB(r)e FE(of)d FA(R)p FE(.)h(Moreo)m(v)m(er,)h(for)e(ev)m(ery)i (de-)183 4403 y(p)s(endency)40 b(pair)d FB(s)28 b FA(!)f FB(t)38 b FE(w)m(e)h(ha)m(v)m(e)h FB(s)d FA(\037)1642 4367 y Fy(0)1703 4403 y FB(t)p FE(.)i(The)g(reason)f(is)g(that)g(eac)m (h)h(dep)s(endency)i(pair)183 4523 y FB(f)242 4487 y Fv(])273 4523 y FE(\()p FB(s)357 4538 y Fz(1)396 4523 y FB(;)17 b(:)g(:)g(:)f(;)h(s)661 4538 y Fv(n)708 4523 y FE(\))27 b FA(!)h FB(g)952 4487 y Fv(])983 4523 y FE(\()p FB(t)1056 4538 y Fz(1)1095 4523 y FB(;)17 b(:)g(:)g(:)f(;)h(t)1349 4538 y Fv(m)1415 4523 y FE(\))22 b(originates)e(from)h(a)g(rule)g FB(f)11 b FE(\()p FB(s)2526 4538 y Fz(1)2565 4523 y FB(;)17 b(:)g(:)g(:)f(;)h(s)2830 4538 y Fv(n)2877 4523 y FE(\))27 b FA(!)h FB(C)7 b FE([)p FB(g)t FE(\()p FB(t)3298 4538 y Fz(1)3337 4523 y FB(;)17 b(:)g(:)g(:)e(;)i(t)3590 4538 y Fv(m)3657 4523 y FE(\)])183 4644 y(in)34 b FA(R)p FE(.)h(Th)m(us,)h FB(f)11 b FE(\()p FB(:)17 b(:)g(:)o FE(\))31 b FA(\037)h FB(C)7 b FE([)p FB(g)t FE(\()p FB(:)17 b(:)g(:)o FE(\)])35 b(implies)d FB(f)11 b FE(\()p FB(:)17 b(:)g(:)o FE(\))31 b FA(\037)h FB(g)t FE(\()p FB(:)17 b(:)g(:)o FE(\))34 b(b)m(y)i(the)f(subterm)g(prop)s(ert)m(y)183 4764 y(whic)m(h)f(in)f (turn)h(implies)d FB(f)1187 4728 y Fv(])1219 4764 y FE(\()p FB(:)17 b(:)g(:)o FE(\))30 b FA(\037)1516 4728 y Fy(0)1569 4764 y FB(g)1620 4728 y Fv(])1651 4764 y FE(\()p FB(:)17 b(:)g(:)p FE(\).)33 b(Hence,)j FA(\037)2298 4728 y Fy(0)2355 4764 y FE(satis\014es)f(b)s(oth)e(constrain)m(ts)h(\(a\))183 4884 y(and)e(\(b\))h(of)f(De\014nition)f(4.2.)2244 b Fs(2)183 5105 y 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b(gument.)35 b(However,)f(this)h(TRS)f(is)h(not)g(simply)f (terminating.)g(The)h(TRS)390 684 y Ft(f)7 b FE(\()p Ft(f)f FE(\()p FB(x)p FE(\)\))83 b FA(!)g Ft(f)7 b FE(\()p Ft(c)p FE(\()p Ft(f)f FE(\()p FB(x)p FE(\)\)\))280 b Ft(g)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))83 b FA(!)g FB(x)266 b Ft(g)q FE(\()p Ft(c)q FE(\()p Ft(0)o FE(\)\))83 b FA(!)g Ft(g)q FE(\()p Ft(d)p FE(\()p Ft(1)p FE(\)\))390 804 y Ft(f)7 b FE(\()p Ft(f)f FE(\()p FB(x)p FE(\)\))83 b FA(!)g Ft(f)7 b FE(\()p Ft(d)p FE(\()p Ft(f)f FE(\()p FB(x)p FE(\)\)\))266 b Ft(g)q FE(\()p Ft(d)p FE(\()p FB(x)p FE(\)\))83 b FA(!)g FB(x)266 b Ft(g)q FE(\()p Ft(c)q FE(\()p Ft(1)o FE(\)\))83 b FA(!)g Ft(g)q FE(\()p Ft(d)p FE(\()p Ft(0)p FE(\)\))183 974 y FC(is)48 b(DP)g(quasi-simply)g (terminating)g(as)g(c)-5 b(an)48 b(b)-5 b(e)48 b(pr)-5 b(ove)g(d)48 b(in)g(a)h(similar)e(way)i(using)f(the)183 1094 y(ar)-5 b(gument)28 b(\014ltering)h(which)f(maps)g Ft(c)h FC(and)f Ft(d)h FC(to)g(their)g(ar)-5 b(guments,)29 b(and)f(RPO)h(wher)-5 b(e)28 b Ft(0)h FC(and)183 1215 y Ft(1)i FC(ar)-5 b(e)32 b(e)-5 b(qual)31 b(in)g(the)h(pr)-5 b(e)g(c)g(e)g(denc)g(e.)30 b(However,)h(it)h(is)g(not)f(DP)g(simply)h (terminating,)f(b)-5 b(e)g(c)g(ause)183 1335 y(due)39 b(to)g(the)g(\014rst)h(four)f(rules,)g(the)g(ar)-5 b(gument)39 b(\014ltering)g(must)g(r)-5 b(e)g(duc)g(e)39 b Ft(c)p FE(\()p FB(x)p FE(\))h FC(and)e Ft(d)p FE(\()p FB(x)p FE(\))i FC(to)183 1455 y(their)35 b(ar)-5 b(guments.)34 b(But)h(then)g Ft(g)q FE(\()p Ft(0)p FE(\))27 b FA(\027)h Ft(g)q FE(\()p Ft(1)p FE(\))35 b FC(and)f Ft(g)q FE(\()p Ft(1)p FE(\))28 b FA(\027)g Ft(g)q FE(\()p Ft(0)p FE(\))35 b FC(le)-5 b(ad)34 b(to)h(a)g(c)-5 b(ontr)g(adiction.)280 1576 y(Final)5 b(ly,)37 b(the)h(system)g FA(f)p Ft(f)7 b FE(\()p Ft(0)o FB(;)17 b Ft(1)p FB(;)g(x)p FE(\))33 b FA(!)g Ft(f)7 b FE(\()p FB(x;)17 b(x;)g(x)p FE(\))p FA(g)38 b FC(is)g(terminating)f(but)i(not)f(DP)f(quasi-)183 1696 y(simply)45 b(terminating.)g(The)h(r)-5 b(e)g(ason)45 b(is)h(that)g FA(f)p Ft(F)p FE(\()p Ft(0)p FB(;)17 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b(F)-8 b(or)32 b(the)h(pro)s(of,)f(w)m(e)h(need)h(the)f(follo)m(wing)d(lemma.)183 4965 y FK(Lemma)38 b(4.6)f(\(Transf)n(orming)j(Reduction)d (Sequences\):)48 b FC(L)-5 b(et)25 b FA(R)2977 4980 y Fz(1)3042 4965 y FC(and)f FA(R)3305 4980 y Fz(2)3370 4965 y FC(b)-5 b(e)24 b(two)183 5085 y(TRSs)44 b(over)h(disjoint)g (signatur)-5 b(es)45 b FA(F)1593 5100 y Fz(1)1677 5085 y FC(and)f FA(F)1948 5100 y Fz(2)1988 5085 y FC(,)h(r)-5 b(esp)g(e)g(ctively.)44 b(F)-7 b(urthermor)i(e,)44 b(let)i FA(R)h FE(=)183 5205 y FA(R)267 5220 y Fz(1)315 5205 y FA([)8 b(R)473 5220 y Fz(2)542 5205 y FC(b)-5 b(e)29 b(their)f(union.)g(If)g FB(u;)17 b(v)32 b FC(ar)-5 b(e)29 b(terms)f(over)g(the)h(signatur)-5 b(e)29 b FA(F)2734 5220 y Fz(1)2801 5205 y FC(such)g(that)g FB(u)e FA(!)3393 5220 y Fy(R)3453 5229 y Fm(1)3519 5205 y FB(v)183 5326 y FC(and)38 b FB(v)t(\033)i FA(!)622 5290 y Fy(\003)622 5351 y(R)722 5326 y FB(u\033)j FC(hold)c(for)g(a)g(gr)-5 b(ound)39 b(substitution)h FB(\033)g FE(:)c FA(V)8 b FB(ar)s FE(\()p 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546 y Fy(\003)2208 611 y(R)2300 587 y FB(u\033)183 783 y FE(ha)m(v)m(e)44 b(the)f(same)f(rank.)h(Since)g(the)g(ro)s(ot)f(sym)m (b)s(ol)g(of)g FB(u)h FE(is)f(in)g FA(F)2642 798 y Fz(1)2681 783 y FE(,)h(the)g(ro)s(ot)e(sym)m(b)s(ol)h(of)183 903 y(ev)m(ery)33 b(term)e(in)g(the)g(reduction)h(sequence)i FB(D)g FE(is)d(also)g(in)f FA(F)2377 918 y Fz(1)2448 903 y FE(\(reduction)h(steps)i(whic)m(h)f(are)183 1024 y(destructiv)m(e)i(at)e(lev)m(el)g(1)g(w)m(ould)h(decrease)h(the)f (rank\).)280 1144 y(Supp)s(ose)46 b(\014rst)f(that)f(ev)m(ery)i (function)f(sym)m(b)s(ol)e(in)h FA(F)2318 1159 y Fz(1)2402 1144 y FE(has)h(arit)m(y)f FA(\024)k FE(1.)c(Then)i(ev)m(ery)183 1265 y(reduction)38 b(step)h(in)e FB(D)k FE(whic)m(h)e(is)f(destructiv) m(e)h(at)f(lev)m(el)g(2)g(strictly)f(decreases)k(the)d(rank.)183 1385 y(Consequen)m(tly)-8 b(,)34 b(there)g(is)e(no)g(reduction)h(step)g (of)f(this)h(kind)f(in)g FB(D)s FE(.)g(Hence)1069 1581 y(top)1210 1605 y Fz(1)1249 1581 y FE(\()p FB(u\033)t FE(\))27 b FA(!)1567 1596 y Fy(R)1627 1605 y Fm(1)1693 1581 y FE(top)1834 1605 y Fz(1)1874 1581 y FE(\()p FB(v)t(\033)t FE(\))g FA(!)2187 1540 y Fy(\003)2187 1606 y(R)2247 1615 y Fm(1)2313 1581 y FE(top)2454 1605 y Fz(1)2494 1581 y FE(\()p FB(u\033)t FE(\))183 1778 y(is)47 b(an)i FA(R)532 1793 y Fz(1)572 1778 y FE(-reduction)e(sequence)k(of)d(ground)g(terms)h (o)m(v)m(er)g FA(F)2530 1793 y Fz(1)2602 1778 y FA([)33 b(f)p Fs(2)p FA(g)p FE(.)49 b(Let)f FA(V)8 b FB(ar)s FE(\()p FB(u)p FE(\))54 b(=)183 1898 y FA(f)p FB(x)288 1913 y Fz(1)327 1898 y FB(;)17 b(:)g(:)g(:)f(;)h(x)601 1913 y Fv(n)648 1898 y FA(g)28 b FE(and)g(recall)f FA(V)8 b FB(ar)s FE(\()p FB(v)t FE(\))27 b FA(\022)i(V)8 b FB(ar)s FE(\()p FB(u)p FE(\).)27 b(In)i(this)e(case,)i(w)m(e)h(de\014ne)f(the)f (substitution)183 2019 y FB(\034)46 b FE(b)m(y)37 b FB(\034)44 b FE(=)32 b FA(f)p FB(x)709 2034 y Fv(i)770 2019 y FA(7!)g FE(top)1043 2042 y Fz(1)1082 2019 y FE(\()p FB(x)1175 2034 y Fv(i)1204 2019 y FB(\033)t FE(\))1301 1983 y Fy(0)1359 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FK(Theorem)38 b(4.7)f(\(Modularity)i(of)f(DP)f(quasi-simple)g(termina)-7 b(tion\):)50 b FC(L)-5 b(et)47 b FA(R)3530 750 y Fz(1)183 856 y FC(and)29 b FA(R)451 871 y Fz(2)521 856 y FC(b)-5 b(e)30 b(two)g(TRSs)g(over)g(disjoint)f(signatur)-5 b(es)30 b FA(F)2163 871 y Fz(1)2232 856 y FC(and)g FA(F)2489 871 y Fz(2)2528 856 y FC(,)g(r)-5 b(esp)g(e)g(ctively.)30 b(Then)f(their)183 976 y(union)36 b FA(R)30 b FE(=)h FA(R)766 991 y Fz(1)829 976 y FA([)23 b(R)1002 991 y Fz(2)1079 976 y FC(is)36 b(DP)g(quasi-simply)f(terminating)h(if)g(and)g (only)g(if)g(b)-5 b(oth)36 b FA(R)3339 991 y Fz(1)3416 976 y FC(and)183 1097 y FA(R)267 1112 y Fz(2)341 1097 y FC(ar)-5 b(e)35 b(DP)g(quasi-simply)e(terminating.)183 1321 y(Pr)-5 b(o)g(of:)48 b FE(The)29 b(only-if)d(direction)i(is)f (trivial.)f(F)-8 b(or)27 b(the)i(if)e(direction,)g(let)h FA(P)37 b FE(b)s(e)28 b(a)g(cycle)h(in)e(the)183 1442 y(estimated)j(dep)s(endency)j(graph)e(of)f FA(R)p FE(.)h(Since)g FA(R)2018 1457 y Fz(1)2089 1442 y FE(and)g FA(R)2361 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y(only)c(ha)m(v)m(e)i(to)e(sho)m(w)230 3643 y(\(a\))g FB(\031)t FE(\()p FA(R)22 b([)h(P)8 b FE(\))28 b FA(\022)g(!)1027 3607 y Fy(\003)1027 3670 y(R)1087 3651 y Fk(0)1146 3643 y FE(and)230 3764 y(\(b\))f(there)33 b(exists)h(a)e(dep)s(endency)j(pair)d FB(s)27 b FA(!)h FB(t)k FE(from)g FA(P)41 b FE(suc)m(h)34 b(that)387 3884 y FB(\031)t FE(\()p FB(t)p FE(\))p FB(\033)e FA(6!)744 3848 y Fy(\003)744 3911 y(R)804 3892 y Fk(0)858 3884 y FB(\031)t FE(\()p FB(s)p FE(\))p FB(\033)k FE(holds)c(for)g(all)f (ground)h(substitutions)h FB(\033)t FE(.)280 4062 y(Condition)f(\(a\))h (is)g(ob)m(viously)g(satis\014ed,)h(since)f(for)g(all)e FB(l)g FA(!)e FB(r)i FA(2)e(R)2796 4077 y Fz(2)2869 4062 y FE(w)m(e)35 b(ha)m(v)m(e)f FB(\031)t FE(\()p FB(l)r FE(\))29 b(=)g FB(l)183 4182 y FE(and)40 b FB(\031)t FE(\()p FB(r)s FE(\))g(=)h FB(r)h FE(and)f(for)e(all)g FB(l)k FA(!)d FB(r)j FE(in)c FA(R)1807 4197 y Fz(1)1874 4182 y FA([)28 b(P)49 b FE(either)40 b FB(\031)t FE(\()p FB(l)r FE(\))g(=)h FB(\031)t FE(\()p 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Fx(R)1396 5567 y Fi(\022)36 b Fh(\045)p Fw(,)27 b(then)h Fi(!)1884 5536 y Fx(\003)1884 5593 y(R[E)5 b Fj(mb)p FI(\()p Fx(F)h FI(\))2261 5567 y Fi(\022)37 b Fh(\045)o Fw(\).)p eop %%Page: 23 23 23 22 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(23)183 390 y FE(w)m(ould)29 b(ha)m(v)m(e)i FB(\031)t FE(\()p FB(s)p FE(\))p FB(\034)38 b FA(\037)29 b FB(\031)t FE(\()p FB(t)p FE(\))p FB(\034)39 b Fu(\045)28 b FB(\031)t FE(\()p FB(s)p FE(\))p FB(\034)39 b FA(\037)28 b FB(:)17 b(:)g(:)29 b FE(whic)m(h)h(con)m(tradicts)f(the)h(w)m(ell-foundedness) 183 511 y(of)36 b FA(\037)p FE(.)h(Th)m(us,)i FB(\031)t FE(\()p FB(t)p FE(\))p FB(\033)f FA(6!)1081 474 y Fy(\003)1081 537 y(R)1141 518 y Fk(0)1202 511 y FB(\031)t FE(\()p FB(s)p FE(\))p FB(\033)j FE(holds)36 b(for)g(all)f(ground)i (substitutions)f FB(\033)t FE(.)h(This)g(pro)m(v)m(es)183 631 y(conjecture)29 b(\(b\).)f(Finally)-8 b(,)25 b(note)j(that,)g (since)g FB(\031)t FE(\()p FA(R)13 b([)g(P)8 b FE(\))28 b(is)f(a)h(TRS,)g(the)h(minor)d(restriction)183 751 y(on)32 b(the)h(argumen)m(t)f(\014lterings)g(holds)g(for)g(this)g FB(\031)t FE(.)1492 b Fs(2)280 980 y FE(Th)m(us,)35 b(if)e FA(R)730 995 y Fz(1)803 980 y FE(is)g(the)h(TRS)g(consisting)e(of)h (the)h(rules)g(\(10\))f(and)g(\(11\))g(and)h FA(R)3147 995 y Fz(2)3220 980 y FE(con)m(tains)183 1100 y(the)g(rules)g(\(12\))g (and)g(\(13\),)f(then)i(this)f(theorem)g(allo)m(ws)f(us)h(to)g (conclude)h(termination)c(of)183 1220 y(their)g(com)m(bination)f(b)s (ecause)k(b)s(oth)e(systems)h(are)g(DP)f(quasi-simply)e(terminating.)g (This)183 1341 y(example)46 b(cannot)h(b)s(e)g(handled)f(b)m(y)i(an)m (y)f(of)g(the)g(previous)g(mo)s(dularit)m(y)e(results.)i(Note)183 1461 y(also)42 b(that)h(in)f(this)h(example,)f(mo)s(dularit)m(y)f(of)i (termination)d(is)j(far)f(from)g(b)s(eing)h(trivial)183 1582 y(b)s(ecause)d(if)e FA(R)730 1597 y Fz(1)769 1582 y FE('s)i(rule)e Ft(f)7 b FE(\()p Ft(0)p FB(;)17 b Ft(1)o FB(;)g(x)p FE(\))39 b FA(!)f Ft(f)6 b FE(\()p Ft(s)q FE(\()p FB(x)p FE(\))p FB(;)17 b(x;)g(x)p FE(\))39 b(w)m(ould)g(b)s(e)g (just)g(sligh)m(tly)e(c)m(hanged)j(to)183 1702 y Ft(f)6 b FE(\()p Ft(0)p FB(;)17 b Ft(1)o FB(;)g(x)p FE(\))41 b FA(!)f Ft(f)7 b FE(\()p FB(x;)17 b(x;)g(x)p FE(\),)41 b(then)g FA(R)1464 1717 y Fz(1)1544 1702 y FE(w)m(ould)f(still)d(b)s(e) k(terminating,)d(but)i(the)h(union)e(with)183 1822 y FA(R)267 1837 y Fz(2)343 1822 y FE(w)m(ould)d(not)g(terminate)f(an)m(y) i(more,)f(cf.)h([T)-8 b(o)m(y)m(ama,)36 b(1987].)g(It)g(is)g(in)m (teresting)g(to)g(note)183 1943 y(that)27 b(Theorem)g(4.7)g(pro)m (vides)h(an)f(elegan)m(t)g(pro)s(of)f(of)h(the)h(fact)f(that)g Ft(f)7 b FE(\()p Ft(0)o FB(;)17 b Ft(1)p FB(;)g(x)p FE(\))27 b FA(!)h Ft(f)6 b FE(\()p FB(x;)17 b(x;)g(x)p FE(\))183 2063 y(is)31 b(not)h(DP)g(quasi-simply)e(terminating)g(b)s(ecause)j FA(R)2165 2078 y Fz(2)2237 2063 y FE(is)e(DP)h(quasi-simply)e (terminating)183 2183 y(but)i(its)h(union)e(with)i Ft(f)6 b FE(\()p Ft(0)p FB(;)17 b Ft(1)o FB(;)g(x)p FE(\))28 b FA(!)f Ft(f)7 b FE(\()p FB(x;)17 b(x;)g(x)p FE(\))33 b(is)f(non-terminating.)280 2304 y(>F)-8 b(rom)42 b(the)i(pro)s(of)f (it)f(is)h(clear)g(that)g(the)h(mo)s(dularit)m(y)d(result)i(of)g (Theorem)h(4.7)f(also)183 2424 y(holds)31 b(if)f(in)g(the)i (de\014nition)f(of)f(DP)i(quasi-simple)d(termination)g(w)m(e)j(\014x)g (the)g(ordering)e FA(\037)3510 2439 y Fy(P)183 2545 y FE(to)38 b(b)s(e)h(the)g(stable-strict)f(relation)e(corresp)s(onding)j (to)f(the)h(QSO)g Fu(\045)2782 2560 y Fy(P)2841 2545 y FE(.)g(In)g(other)g(w)m(ords,)183 2665 y(the)c(termination)e(pro)s (of)i(of)g FA(R)1343 2680 y Fz(1)1407 2665 y FA([)24 b(R)1581 2680 y Fz(2)1656 2665 y FE(also)34 b(succeeds)39 b(with)34 b(reduction)i(pairs)f(consisting)183 2785 y(of)d(a)g(QSO)g (and)h(its)f(asso)s(ciated)h(stable-strict)e(relation.)280 2906 y(One)g(should)f(remark)g(that)h(a)f(further)g(extension)i(of)e (the)g(mo)s(dularit)m(y)f(result)h(in)g(Theo-)183 3026 y(rem)25 b(4.7)h(b)s(ey)m(ond)i(the)f(class)f(of)g(DP)g(quasi-simply)e (terminating)g(systems)k(is)e(not)g(straigh)m(t-)183 3147 y(forw)m(ard.)j(F)-8 b(or)29 b(example,)g(if)f(one)i(w)m(ould)f (de\014ne)h(DP)g(quasi-simple)d(termination)g(b)m(y)j(using)183 3267 y(the)38 b(real)f(dep)s(endency)j(graph)e(instead)g(of)f(the)i (estimated)e(graph,)h(then)g(this)g(notion)e(of)183 3387 y(termination)29 b(w)m(ould)i(no)h(longer)e(b)s(e)i(mo)s(dular)d(for)i (disjoin)m(t)g(systems.)i(The)f(previous)g(sys-)183 3508 y(tem)43 b(w)m(ould)i(serv)m(e)h(as)e(a)g(coun)m(terexample,)h(since)f (in)g(the)g(real)g(dep)s(endency)i(graph)e(of)183 3628 y Ft(f)6 b FE(\()p Ft(0)p FB(;)17 b Ft(1)o FB(;)g(x)p FE(\))37 b FA(!)e Ft(f)7 b FE(\()p FB(x;)17 b(x;)g(x)p FE(\))38 b(there)g(is)f(no)h(cycle.)g(Hence,)h(it)d(w)m(ould)i(dep)s (end)g(on)g(the)g(rules)f(of)183 3748 y FA(R)267 3763 y Fz(2)351 3748 y FE(whether)46 b(dep)s(endency)i(pairs)c(of)g FA(R)1739 3763 y Fz(1)1824 3748 y FE(form)f(a)i(cycle.)g(The)h(same)f (problem)e(o)s(ccurs)183 3869 y(with)c(the)g(recen)m(t)i(tec)m(hnique)g (of)e([Middeldorp,)f(2001])h(where)h(dep)s(endency)i(graphs)e(are)183 3989 y(appro)m(ximated)31 b(using)i(tree)g(automata)e(tec)m(hniques.) 280 4110 y(DP)39 b(quasi-simply)e(terminating)g(systems)j(o)s(ccur)f (frequen)m(tly)h(in)e(practice.)h(Consider)183 4230 y(the)j(follo)m (wing)d(t)m(w)m(o)k(TRSs)g(where)g Ft(nil)f FE(denotes)h(the)g(empt)m (y)f(list)f(and)h FB(x)i FE(:)g FB(l)g FE(represen)m(ts)183 4350 y(the)36 b(insertion)g(of)g(a)g(n)m(um)m(b)s(er)g FB(x)h FE(in)m(to)f(a)g(list)f FB(l)r FE(.)h(Here)h Ft(sum)q FE(\()p FB(l)r FE(\))f(computes)h(a)f(singleton)f(list)183 4471 y(con)m(taining)c(the)i(sum)f(of)g(all)f(elemen)m(ts)i(in)f(the)h (list)e FB(l)r FE(.)332 4724 y Fg(R)409 4738 y Fz(1)474 4724 y FF(:)120 b Ff(x)20 b Fg(\000)g Fe(0)83 b Fg(!)g Ff(x)405 4862 y Fe(s)p FF(\()p Ff(x)p FF(\))21 b Fg(\000)f Fe(s)p FF(\()p Ff(y)s FF(\))83 b Fg(!)g Ff(x)20 b Fg(\000)g Ff(y)345 5000 y Fe(quot)p FF(\()p Fe(0)q Ff(;)15 b Fe(s)p FF(\()p Ff(y)s FF(\)\))84 b Fg(!)f Fe(0)233 5138 y(quot)p FF(\()p Fe(s)p FF(\()p Ff(x)p FF(\))p Ff(;)15 b Fe(s)q FF(\()p Ff(y)s FF(\)\))84 b Fg(!)f Fe(s)p FF(\()p Fe(quot)q FF(\()p Ff(x)20 b Fg(\000)g Ff(y)s(;)15 b Fe(s)p FF(\()p Ff(y)s FF(\)\)\))1737 4724 y Fg(R)1814 4738 y Fz(2)1879 4724 y FF(:)120 b Fe(app)p FF(\()p Fe(nil)o Ff(;)15 b(k)s FF(\))84 b Fg(!)f Ff(k)2045 4862 y Fe(app)p FF(\()p Ff(l)r(;)15 b Fe(nil)p FF(\))83 b Fg(!)g Ff(l)1958 5000 y Fe(app)p FF(\()p Ff(x)25 b FF(:)h Ff(l)r(;)15 b(k)s FF(\))83 b Fg(!)g Ff(x)26 b FF(:)f Fe(app)p FF(\()p Ff(l)r(;)15 b(k)s FF(\))1971 5138 y Fe(sum)p FF(\()p Ff(x)25 b FF(:)h Fe(nil)o FF(\))83 b Fg(!)g Ff(x)26 b FF(:)f Fe(nil)1838 5276 y(sum)p FF(\()p Ff(x)g FF(:)h(\()p Ff(y)i FF(:)e Ff(l)r FF(\)\))83 b Fg(!)g Fe(sum)p FF(\(\()p Ff(x)21 b FF(+)f Ff(y)s FF(\))26 b(:)f Ff(l)r FF(\))1539 5414 y Fe(sum)p FF(\()p Fe(app)p FF(\()p Ff(l)r(;)15 b(x)26 b FF(:)f(\()p Ff(y)k FF(:)c Ff(k)s FF(\)\)\))84 b Fg(!)f Fe(sum)p FF(\()p Fe(app)p FF(\()p Ff(l)r(;)15 b Fe(sum)q FF(\()p Ff(x)26 b FF(:)f(\()p Ff(y)k FF(:)c Ff(k)s FF(\)\)\)\))p eop %%Page: 24 24 24 23 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(24)183 390 y FE(Both)24 b(TRSs)h(ab)s(o)m(v)m(e)f(are)g(not)g (simply)f(terminating,)f(but)i(they)h(are)f(b)s(oth)g(DP)g (quasi-simply)183 511 y(terminating,)33 b(cf.)k([Arts)g(and)f(Giesl,)f (2000].)g(Hence,)j(Theorem)e(4.7)g(no)m(w)h(also)e(allo)m(ws)g(to)183 631 y(conclude)e(DP)f(quasi-simple)f(termination)f(of)i(their)g(union.) 183 892 y Fq(4.4.)53 b(Com)m(bining)34 b(Constructor-Sharing)h(and)g (Comp)s(osable)f(Systems)183 1062 y FE(It)45 b(ma)m(y)g(b)s(e)g(a)g (bit)g(surprising)f(that)h(Theorem)h(4.7)f(cannot)g(b)s(e)g(directly)g (extended)i(to)183 1182 y(constructor-sharing)33 b(TRSs;)h(ev)m(en)g (if)e(w)m(e)j(disallo)m(w)c(the)j(use)g(of)e(argumen)m(t)h (\014lterings.)g(In)183 1303 y(other)d(w)m(ords,)i(there)f(are)f (constructor-sharing)g(TRSs)i FA(R)2352 1318 y Fz(1)2422 1303 y FE(and)e FA(R)2693 1318 y Fz(2)2764 1303 y FE(whic)m(h)h(are)f (b)s(oth)g(DP)183 1423 y(quasi-simply)g(terminating,)g(but)j(their)f (union)g FA(R)c FE(=)f FA(R)2287 1438 y Fz(1)2349 1423 y FA([)22 b(R)2521 1438 y Fz(2)2593 1423 y FE(is)32 b(not)h(DP)f (quasi-simply)183 1544 y(terminating.)183 1745 y FK(Example)37 b(4.8:)49 b FC(Consider)33 b(the)i(fol)5 b(lowing)34 b(TRSs:)360 2014 y FA(R)444 2029 y Fz(1)512 2014 y FE(:)122 b Ft(f)6 b FE(\()p Ft(c)q FE(\()p FB(x)p FE(\)\))83 b FA(!)g Ft(f)6 b FE(\()p FB(x)p FE(\))654 2160 y Ft(f)h FE(\()p Ft(b)p FE(\()p FB(x)p FE(\)\))83 b FA(!)g FB(x)1767 2014 y FA(R)1852 2029 y Fz(2)1919 2014 y FE(:)122 b Ft(g)q FE(\()p Ft(d)p FE(\()p FB(x)p FE(\)\))83 b FA(!)g Ft(g)q FE(\()p FB(x)p FE(\))2075 2160 y Ft(g)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))g FA(!)g Ft(c)p FE(\()p Ft(g)r FE(\()p Ft(b)o FE(\()p Ft(c)q FE(\()p FB(x)p FE(\)\)\)\))280 2339 y FA(R)364 2354 y Fz(1)439 2339 y FC(and)34 b FA(R)712 2354 y Fz(2)786 2339 y FC(ar)-5 b(e)35 b(DP)f(quasi-simply)g (terminating.)g(\()p FA(R)2367 2354 y Fz(1)2441 2339 y FC(is)g(even)g(simply)h(terminating)183 2459 y(and)k FA(R)461 2474 y Fz(2)540 2459 y FC(is)g(alr)-5 b(e)g(ady)40 b(DP)f(simply)g(terminating)g(as)g(c)-5 b(an)39 b(b)-5 b(e)40 b(shown)e(using)i(the)f(ar)-5 b(gument)183 2579 y(\014ltering)36 b FB(\031)t FE(\()p Ft(b)p FE(\))c(=)f([)17 b(])37 b FC(and)g(RPO.)g(A)n(lternatively,)g(DP)f(quasi-simple)g (termination)g(of)h FA(R)3530 2594 y Fz(2)183 2700 y FC(c)-5 b(an)29 b(even)f(b)-5 b(e)29 b(shown)g(without)h(any)f(ar)-5 b(gument)29 b(\014ltering)g(by)h(using)f(a)g(p)-5 b(olynomial)29 b(or)-5 b(dering)183 2820 y(which)25 b(maps)h Ft(c)p FC(,)g Ft(b)p FC(,)g Ft(g)r FC(,)g(and)f Ft(G)i FC(to)f(the)h(identity) f(and)g(which)f(maps)h Ft(d)p FE(\()p FB(x)p FE(\))h FC(to)f FB(x)s FE(+)s(1)p FC(.\))g(However,)183 2940 y(the)32 b(union)f(of)h FA(R)811 2955 y Fz(1)883 2940 y FC(and)f FA(R)1153 2955 y Fz(2)1225 2940 y FC(is)g(not)h(DP)g (quasi-simply)f(terminating.)g(As)h Ft(F)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))c FA(!)f Ft(F)q FE(\()p FB(x)p FE(\))183 3061 y FC(r)-5 b(epr)g(esents)39 b(a)g(cycle)g(in)h(the)f (estimate)-5 b(d)39 b(dep)-5 b(endency)38 b(gr)-5 b(aph)39 b(one)g(would)h(have)e(to)i(\014nd)f(a)183 3181 y(QSO)34 b(satisfying)1348 3399 y Ft(f)7 b FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))83 b Fu(\045)h Ft(f)6 b FE(\()p FB(x)p FE(\))1351 b(\(16\))1342 3544 y Ft(f)6 b FE(\()p Ft(b)p FE(\()p FB(x)p FE(\)\))83 b Fu(\045)h FB(x)1463 b FE(\(17\))1328 3689 y Ft(g)q FE(\()p Ft(d)p FE(\()p FB(x)p FE(\)\))83 b Fu(\045)h Ft(g)q FE(\()p FB(x)p FE(\))1337 b(\(18\))1335 3835 y Ft(g)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))83 b Fu(\045)h Ft(c)p FE(\()p Ft(g)q FE(\()p Ft(b)p FE(\()p Ft(c)q FE(\()p FB(x)p FE(\)\)\)\))972 b(\(19\))1329 3980 y Ft(F)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))83 b FA(\037)h Ft(F)p FE(\()p FB(x)p FE(\))p FB(:)1305 b FE(\(20\))183 4198 y FC(Without)47 b(ar)-5 b(gument)46 b(\014ltering,)g(no)g(QSO)g (satis\014es)g(\(16\))g(-)g(\(20\),)g(sinc)-5 b(e)46 b(otherwise)f(we)183 4318 y(would)34 b(have)478 4508 y Ft(F)p FE(\()p Ft(c)q FE(\()p Ft(g)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\)\)\))83 b FA(\037)h Ft(F)p FE(\()p Ft(g)q FE(\()p Ft(c)q FE(\()p FB(x)p FE(\)\)\))363 b FC(due)35 b(to)g(\(20\))1112 4628 y Fu(\045)84 b Ft(F)p FE(\()p Ft(c)p FE(\()p Ft(g)r FE(\()p Ft(b)p FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\)\)\)\))118 b FC(due)35 b(to)g(\(19\))1112 4748 y Fu(\045)84 b Ft(F)p FE(\()p Ft(c)p FE(\()p Ft(g)r FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\)\)\))244 b FC(due)35 b(to)g(the)g(subterm)f(pr)-5 b(op)g(erty.)183 4965 y(By)42 b(\(20\),)f(the)i(ar)-5 b(gument)42 b(\014ltering)f(c)-5 b(an)42 b(only)g(map)g Ft(c)g FC(to)h FE([1])p FC(,)f(i.e.,)g FB(\031)t FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))f(=)h Ft(c)p FE(\()p FB(x)p FE(\))p FC(.)h(If)183 5085 y FB(\031)t FE(\()p Ft(b)p FE(\))30 b(=)h([)17 b(])37 b FC(then)f(\(17\))g(would)g (b)-5 b(e)37 b(tr)-5 b(ansforme)g(d)35 b(into)i Ft(f)7 b FE(\()p Ft(b)p FE(\))30 b Fu(\045)i FB(x)p FC(.)37 b(But)g(as)f(ther)-5 b(e)37 b(exists)f(the)183 5205 y(strict)44 b(ine)-5 b(quality)44 b(\(20\))g(with)f(a)h(variable)f(in)h(its)g (right-hand)g(side,)f(this)h(r)-5 b(esults)44 b(in)g(the)183 5326 y(c)-5 b(ontr)g(adiction)37 b Ft(F)q FE(\()p Ft(c)p FE(\()p Ft(f)7 b FE(\()p Ft(b)p FE(\)\)\))34 b FA(\037)h Ft(F)q FE(\()p Ft(f)6 b FE(\()p Ft(b)p FE(\)\))35 b Fu(\045)g Ft(F)p FE(\()p FB(x)p FE(\))p FC(.)k(Similarly,)f(the)h(ar)-5 b(gument)38 b(of)h Ft(g)g FC(c)-5 b(annot)183 5446 y(b)g(e)34 b(eliminate)-5 b(d)34 b(either,)h(sinc)-5 b(e)34 b Ft(g)28 b Fu(\045)h Ft(c)p FE(\()p Ft(g)q FE(\))35 b FC(would)f(b)-5 b(e)35 b(a)g(c)-5 b(ontr)g(adiction)34 b(to)h(\(20\).)280 5567 y(Thus,)25 b(the)g(only)f(p)-5 b(ossible)24 b(ar)-5 b(gument)25 b(\014ltering)f(maps)g Ft(b)h FC(or)g Ft(g)h FC(to)f(its)g(ar)-5 b(gument.)25 b(But)g(then)p eop %%Page: 25 25 25 24 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(25)183 390 y FC(we)40 b(would)h(again)f(obtain)g Ft(F)q FE(\()p Ft(c)p FE(\()p Ft(g)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\)\)\))f FA(\037)h(\016)e Fu(\045)i Ft(F)p FE(\()p Ft(c)p FE(\()p Ft(g)q FE(\()p Ft(c)q FE(\()p FB(x)p FE(\)\)\)\))h FC(or)f Ft(F)q FE(\()p Ft(c)p FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\)\))f FA(\037)h(\016)e Fu(\045)183 511 y Ft(F)p FE(\()p Ft(c)p FE(\()p Ft(c)q FE(\()p FB(x)p FE(\)\)\))31 b FC(as)g(ab)-5 b(ove.)31 b(Henc)-5 b(e,)31 b(the)h(TRS)f(inde)-5 b(e)g(d)30 b(is)h(not)h(DP)f(quasi-simply)g (terminating.)280 714 y FE(Th)m(us,)h(in)d(order)i(to)f(obtain)f(a)h (mo)s(dularit)m(y)e(result)i(for)f(constructor-sharing)h(com)m(bina-) 183 834 y(tions)d(w)m(e)h(ha)m(v)m(e)h(to)e(exclude)h(TRSs)h(lik)m(e)d FA(R)1753 849 y Fz(2)1793 834 y FE(.)i(Note)f(that)h(without)f (applying)f(an)h(argumen)m(t)183 955 y(\014ltering,)41 b(DP)h(simple)g(termination)e(of)i(the)h(TRS)g FA(R)2244 970 y Fz(2)2327 955 y FE(cannot)f(b)s(e)h(pro)m(v)m(ed)h(\(while)e(DP) 183 1075 y(quasi-simple)31 b(termination)f(can)j(b)s(e)g(sho)m(wn)i (without)d(using)h(an)m(y)g(argumen)m(t)g(\014ltering)e(at)183 1196 y(all\).)k(Th)m(us,)k(w)m(e)f(will)d(imp)s(ose)h(t)m(w)m(o)i (restrictions:)e(\(a\))h(In)h(the)f(remainder)f(of)h(the)h(section)183 1316 y(w)m(e)f(will)c(restrict)j(ourselv)m(es)i(to)d(DP)h(simple)e (termination)g(instead)i(of)f(DP)h(quasi-simple)183 1436 y(termination)24 b(and)j(\(b\))g(w)m(e)h(ha)m(v)m(e)g(to)f(restrict)g (ourselv)m(es)h(to)e(systems)j(where)f(the)f(argumen)m(t)183 1557 y(\014ltering)k(do)s(es)i(not)f(eliminate)e(argumen)m(ts)j(for)f (shared)h(sym)m(b)s(ols)f(lik)m(e)g Ft(b)p FE(.)280 1677 y(But)39 b(w)m(e)g(need)h(another)e(requiremen)m(t)h(to)f(ensure)i(mo)s (dularit)m(y)-8 b(.)36 b(F)-8 b(or)38 b(example,)g(let)f(us)183 1797 y(remo)m(v)m(e)45 b(the)h(\014rst)f(rule)g Ft(g)q FE(\()p Ft(d)p FE(\()p FB(x)p FE(\)\))k FA(!)g Ft(g)q FE(\()p FB(x)p FE(\))c(from)f FA(R)2188 1812 y Fz(2)2228 1797 y FE(.)h(No)m(w)g(there)h(is)f(no)g(cycle)g(in)f(the)183 1918 y(estimated)26 b(dep)s(endency)j(graph)e(of)f FA(R)1608 1933 y Fz(2)1674 1918 y FE(an)m(y)i(more)e(and)h(hence)h(w)m(e)g (obtain)e(no)g(constrain)m(ts)183 2038 y(at)d(all)e(for)i FA(R)644 2053 y Fz(2)683 2038 y FE(.)h(Th)m(us,)h(DP)f(simple)e (termination)f(of)i FA(R)2167 2053 y Fz(2)2230 2038 y FE(can)h(no)m(w)g(ev)m(en)i(b)s(e)d(pro)m(v)m(ed)i(without)183 2159 y(using)e(argumen)m(t)g(\014lterings,)g(but)h(the)g(com)m(bined)f (system)h FA(R)2431 2174 y Fz(1)2474 2159 y FA([)t(R)2628 2174 y Fz(2)2691 2159 y FE(is)f(still)f(not)h(DP)g(simply)183 2279 y(terminating.)31 b(Here,)j(the)g(problem)e(is)h(due)h(to)f(the)h (fact)f(that)g(TRSs)h(without)f(cycles)i(are)183 2399 y(DP)h(simply)g(terminating,)f(ev)m(en)j(if)e(there)i(is)e(no)h (simpli\014cation)d(ordering)i FA(\037)i FE(suc)m(h)g(that)183 2520 y FB(l)53 b FA(\027)e FB(r)e FE(holds)c(for)h(their)g(rules.)g(T) -8 b(o)46 b(exclude)h(suc)m(h)h(TRSs)f(w)m(e)g(will)c(demand)k(that)e (the)183 2640 y(constrain)m(t)d(\(a\))g(of)g(De\014nition)f(4.2)h (\(i.e.,)g FB(\031)t FE(\()p FB(l)r FE(\))j FA(\027)g FB(\031)t FE(\()p FB(r)s FE(\))d(for)g(all)e(rules\))j(should)f(also)g (b)s(e)183 2760 y(satis\014ed)32 b(ev)m(en)i(if)c(there)j(do)s(es)f (not)g(exist)g(an)m(y)h(cycle)f FA(P)8 b FE(.)33 b(Th)m(us,)g(in)e(the) i(follo)m(wing)c(w)m(e)k(also)183 2881 y(tak)m(e)g(the)g FC(empty)41 b FE(cycle)33 b FA(P)41 b FE(in)m(to)32 b(accoun)m(t.)280 3001 y(With)41 b(this)g(additional)e(requiremen)m(t,)j(DP)g(simple)e (termination)f(is)i(at)g(least)g(mo)s(du-)183 3122 y(lar)35 b(for)h FC(disjoint)46 b FE(com)m(binations)1412 3085 y Fy({)1457 3122 y FE(,)37 b(whereas)h(without)f(this)f(requiremen)m (t,)h(Theorem)g(4.7)183 3242 y(w)m(ould)c(not)h(hold)f(for)h(DP)f (simple)g(termination)e(instead)j(of)f(DP)h(quasi-simple)e(termina-)183 3362 y(tion.)39 b(As)i(a)f(coun)m(terexample)h(consider)g(the)g(TRS)f FA(R)2251 3377 y Fz(1)2331 3362 y FE(with)g(the)g(rule)g Ft(f)7 b FE(\()p Ft(s)p FE(\()p FB(x)p FE(\)\))41 b FA(!)g Ft(f)7 b FE(\()p FB(x)p FE(\))183 3483 y(and)32 b(the)h(TRS)g FA(R)854 3498 y Fz(2)926 3483 y FE(with)f(the)h(rules)484 3690 y Ft(g)q FE(\()p Ft(0)p FE(\))83 b FA(!)g Ft(g)q FE(\()p Ft(c)p FE(\()p Ft(0)p FE(\)\))277 b Ft(g)q FE(\()p Ft(c)p FE(\()p FB(x)p FE(\)\))83 b FA(!)g FB(x)264 b Ft(g)q FE(\()p Ft(c)p FE(\()p Ft(0)p FE(\)\))83 b FA(!)g Ft(g)q FE(\()p Ft(d)p FE(\()p Ft(1)o FE(\)\))484 3810 y Ft(g)q FE(\()p Ft(0)p FE(\))g FA(!)g Ft(g)q FE(\()p Ft(d)p FE(\()p Ft(0)o FE(\)\))264 b Ft(g)q FE(\()p Ft(d)p FE(\()p FB(x)p FE(\)\))83 b FA(!)g FB(x)264 b Ft(g)q FE(\()p Ft(c)p FE(\()p Ft(1)p FE(\)\))83 b FA(!)g Ft(g)q FE(\()p Ft(d)p FE(\()p Ft(0)o FE(\)\))p FB(:)183 4024 y FA(R)267 4039 y Fz(1)349 4024 y FE(is)43 b(ev)m(en)h(simply)e (terminating.)e FA(R)1666 4039 y Fz(2)1748 4024 y FE(is)i(DP)h(simply)e (terminating,)g(but)i(the)g(reason)183 4144 y(is)c(just)g(that)h(there) g(do)s(es)g(not)f(exist)h(an)m(y)g(cycle)g(in)e(its)h(estimated)g(dep)s (endency)j(graph.)183 4264 y(Ho)m(w)m(ev)m(er,)26 b(when)f(com)m (bining)d FA(R)1384 4279 y Fz(1)1447 4264 y FE(and)i FA(R)1713 4279 y Fz(2)1752 4264 y FE(,)g(their)f(union)g(has)h(a)g (cycle)g(and)g(hence,)h(one)f(no)m(w)183 4385 y(also)35 b(has)i(to)f(demand)h FB(\031)t FE(\()p FB(l)r FE(\))d FA(\027)h FB(\031)t FE(\()p FB(r)s FE(\))g(for)h(the)h(rules)g(of)f FA(R)2347 4400 y Fz(2)2387 4385 y FE(.)g(Ho)m(w)m(ev)m(er,)j(for)d(all) e(argumen)m(t)183 4505 y(\014lterings)j FB(\031)t FE(,)i(this)f(is)g (not)h(ful\014lled)d(b)m(y)k(an)m(y)f(QSO)g(whose)h(equiv)-5 b(alence)39 b(relation)d(is)j(just)183 4625 y(syn)m(tactic)g(equalit)m (y)-8 b(.)39 b(So)f(their)g(union)h(is)f(not)g(DP)h(simply)e (terminating,)g(but)i(of)f(course)183 4746 y(due)33 b(to)f(Theorem)h (4.7)f(it)g(is)g(DP)g(quasi-simply)f(terminating.)280 4866 y(Nonetheless,)h(the)f(follo)m(wing)d(example)i(sho)m(ws)i(that)e (this)g(restriction)g(is)g(not)g(y)m(et)h(su\016-)183 4987 y(cien)m(t)26 b(for)g(obtaining)f(a)h(mo)s(dularit)m(y)e(result)j (for)f(DP)g(simple)f(termination)f(of)i FC(c)-5 b(onstructor-)183 5107 y(sharing)40 b FE(systems.)287 5246 y Fx({)332 5276 y Fw(This)31 b(can)g(b)r(e)h(pro)n(v)n(ed)d(similar)i(to)g(Theorem)f (4.7)h(using)f(the)i(simpli\014cation)f(ordering)f Fi(!)3243 5240 y FI(+)3243 5300 y Fx(R)3300 5284 y Fd(0)3358 5276 y Fw(where)183 5375 y(instead)i(of)h(condition)f(\(b\))h(in)g(this)g (pro)r(of)f(one)g(only)h(has)f(to)g(sho)n(w)g(that)h Fc(\031)s Fw(\()p Fc(s)p Fw(\))f Fi(6)p Fw(=)f Fc(\031)s Fw(\()p Fc(t)p Fw(\))i(holds)g(for)f(some)183 5475 y(dep)r(endency)c (pair)e Fc(s)d Fi(!)g Fc(t)28 b Fw(from)f Fi(P)34 b Fw(\(this)28 b(follo)n(ws)f(immediately)g(from)g(DP)h(simple)f(termination)h(of)f Fi(R)3477 5487 y FI(1)3515 5475 y Fw(\).)p eop %%Page: 26 26 26 25 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(26)183 390 y FK(Example)37 b(4.9:)49 b FC(L)-5 b(et)35 b FA(R)1128 405 y Fz(1)1203 390 y FC(c)-5 b(onsist)34 b(of)g(the)h(rules)594 612 y Ft(g)q FE(\()p Ft(s)q FE(\()p FB(x)p FE(\)\))83 b FA(!)g Ft(g)q FE(\()p FB(x)p FE(\))594 758 y Ft(g)q FE(\()p Ft(s)q FE(\()p FB(x)p FE(\)\))g FA(!)g FB(x)2150 612 y Ft(g)q FE(\()p Ft(0)p FE(\))g FA(!)f Ft(g)r FE(\()p Ft(1)o FE(\))2163 758 y Ft(f)7 b FE(\()p Ft(0)p FE(\))83 b FA(!)f Ft(g)r FE(\()p Ft(f)6 b FE(\()p Ft(s)p FE(\()p Ft(0)p FE(\)\)\))183 937 y FC(and)44 b(let)g FA(R)613 952 y Fz(2)697 937 y FC(c)-5 b(onsist)44 b(of)g(the)h(rule)f Ft(h)p FE(\()p Ft(1)p FE(\))i FA(!)f Ft(h)p FE(\()p Ft(0)p FE(\))p FC(.)f(T)-7 b(o)44 b(pr)-5 b(ove)44 b(DP)g(simple)g(termination)183 1057 y(of)d FA(R)388 1072 y Fz(1)469 1057 y FC(we)g(have)g(to)h(use)f(an)h(ar)-5 b(gument)41 b(\014ltering)g(mapping)f Ft(f)48 b FC(to)42 b FE([)17 b(])41 b FC(and)g Ft(g)i FC(to)f FE(1)p FC(.)f(This,)183 1177 y(however,)36 b(would)i(imply)f Ft(0)c FA(\037)h Ft(1)j FC(which)g(is)h(a)g(c)-5 b(ontr)g(adiction)37 b(to)h Ft(h)p FE(\()p Ft(1)o FE(\))33 b FA(\027)h Ft(h)p FE(\()p Ft(0)p FE(\))p FC(.)j(Thus,)h(the)183 1298 y(c)-5 b(ombination)33 b(of)i(b)-5 b(oth)34 b(systems)h(is)g(not)g(DP)f (simply)g(terminating.)280 1513 y FE(So)44 b(w)m(e)i(also)d(ha)m(v)m(e) i(to)f(ensure)i(that)e(an)g(application)d(of)j(the)h(argumen)m(t)f (\014ltering)e(to)183 1633 y(the)e(resulting)e(inequalities)f(do)s(es)j (not)g(transform)e(left-hand)h(sides)h(whic)m(h)g(had)f(a)g(non-)183 1754 y(shared)32 b(ro)s(ot)f(sym)m(b)s(ol)f(lik)m(e)h Ft(g)i FE(in)m(to)e(terms)g(with)g(a)g(shared)i(ro)s(ot)d(sym)m(b)s(ol) h(\(lik)m(e)g(the)h(former)183 1874 y(constructor)h Ft(0)o FE(\).)811 1838 y Fy(k)883 1874 y FE(F)-8 b(or)32 b(that)g(reason)g(w)m (e)i(ha)m(v)m(e)f(to)f(demand)g(the)h(follo)m(wing)c(compatibilit)m(y) 183 1995 y(requiremen)m(t)41 b(for)g(all)e(argumen)m(t)i(\014lterings)f (used,)j(where)f FA(G)48 b FE(m)m(ust)41 b(con)m(tain)g(all)e(shared) 183 2115 y(function)32 b(sym)m(b)s(ols.)183 2330 y FK(Definition)38 b(4.10)f(\()p FA(G)6 b FK(-Comp)-7 b(a)g(tibility\):)51 b FC(L)-5 b(et)38 b FA(R)g FC(b)-5 b(e)37 b(a)g(TRS)g(over)g(the)g (signatur)-5 b(e)37 b FA(F)183 2451 y FC(and)g(let)h FA(G)44 b FC(b)-5 b(e)38 b(a)g(signatur)-5 b(e.)37 b(A)n(n)h(ar)-5 b(gument)38 b(\014ltering)f FB(\031)42 b FC(for)c FA(F)47 b FC(is)38 b FA(G)6 b FE(-compatible)36 b FC(for)i FA(R)183 2571 y FC(if)c(and)h(only)f(if)219 2731 y(\(a\))48 b FB(\031)t FE(\()p FB(f)11 b FE(\))27 b(=)h([1)p FB(;)17 b(:)g(:)g(:)f(;)h(n)p FE(])35 b FC(for)f(every)h FB(f)j FA(2)28 b(F)k(\\)23 b(G)6 b FC(,)35 b(wher)-5 b(e)34 b FB(n)h FC(is)g(the)f(arity)i(of)e FB(f)397 2852 y FC(\(i.e.,)g FB(\031)39 b FC(do)-5 b(es)34 b(not)h(\014lter)g(ar)-5 b(guments)35 b(for)f(function)h(symb)-5 b(ols)34 b(fr)-5 b(om)34 b FA(G)6 b FC(\).)224 3010 y(\(b\))48 b(F)-7 b(or)34 b(every)h(rule)g FB(l)30 b FA(!)d FB(r)k FA(2)d(R)p FC(:)35 b(if)f FE(ro)s(ot\()p FB(l)r FE(\))27 b FA(62)h(G)6 b FC(,)35 b(then)g FE(ro)s(ot)o(\()p FB(\031)t FE(\()p FB(l)r FE(\)\))28 b FA(62)g(G)6 b FC(.)280 3226 y FE(The)28 b(restriction)d(to)h FA(G)6 b FE(-compatible)25 b(argumen)m(t)h (\014lterings)f(ensures)k(that)d(sym)m(b)s(ols)g(from)183 3346 y FA(F)e(\\)32 b(G)j FE(are)29 b(not)g(c)m(hanged)h(and)f (furthermore)f(constructors)j(from)c FA(F)d(\\)32 b(G)j FE(are)29 b(not)g(turned)183 3467 y(in)m(to)d(de\014ned)j(sym)m(b)s (ols)f(after)f(application)e(of)i(the)h(argumen)m(t)f(\014ltering.)g (In)g(the)h(follo)m(wing,)183 3587 y(for)35 b(an)m(y)h(TRS)f FA(R)h FE(o)m(v)m(er)h(the)f(signature)f FA(F)45 b FE(let)34 b FA(C)1994 3602 y Fv(\031)2077 3587 y FE(b)s(e)i(the)g(set)g(of)f (constructors)i(of)e FB(\031)t FE(\()p FA(R)p FE(\),)183 3707 y(and)d(let)g FA(D)590 3722 y Fv(\031)670 3707 y FE(b)s(e)g(the)h(set)h(of)e(de\014ned)i(sym)m(b)s(ols)e(in)g FB(\031)t FE(\()p FA(R)p FE(\).)183 3923 y FK(Lemma)38 b(4.11)f(\(Pr)n(oper)-7 b(ties)37 b(of)i(Appl)-7 b(ying)37 b FA(G)6 b FK(-Comp)-7 b(a)g(tible)38 b(Ar)n(gument)g(Fil)-7 b(terings\):)183 4043 y FC(L)i(et)32 b FA(R)g FC(b)-5 b(e)31 b(a)g(TRS)g(over)g(the)h(signatur)-5 b(e)31 b FA(F)38 b FE(=)27 b FA(C)21 b([)15 b(D)35 b FC(and)c(let)g FB(\031)36 b FC(b)-5 b(e)31 b(an)g(ar)-5 b(gument)31 b(\014ltering)183 4164 y(for)j FA(F)45 b FC(that)35 b(is)g FA(G)6 b FC(-c)-5 b(omp)g(atible)33 b(for)i FA(R)p FC(.)g(Then)f(the)h (fol)5 b(lowing)34 b(statements)g(hold:)239 4324 y(\(i\))48 b(F)-7 b(or)34 b(every)h(rule)g FB(l)30 b FA(!)d FB(r)k FA(2)d(R)p FC(:)35 b(if)f FE(ro)s(ot\()p FB(l)r FE(\))27 b FA(2)h(G)6 b FC(,)35 b(then)g FE(ro)s(ot)o(\()p FB(\031)t FE(\()p FB(l)r FE(\)\))28 b(=)f(ro)s(ot)o(\()p FB(l)r FE(\))p FC(.)209 4483 y(\(ii\))48 b(F)-7 b(or)34 b(every)h(rule)g FB(l)30 b FA(!)d FB(r)k FA(2)d(R)p FC(:)35 b(if)f FE(ro)s(ot\()p FB(\031)t FE(\()p FB(l)r FE(\)\))27 b FA(2)h(G)6 b FC(,)35 b(then)g FE(ro)s(ot)o(\()p FB(\031)t FE(\()p FB(l)r FE(\)\))28 b(=)f(ro)s(ot)o(\()p FB(l)r FE(\))p FC(.)179 4641 y(\(iii\))48 b FA(G)29 b(\\)22 b(D)653 4656 y Fv(\031)727 4641 y FA(\022)29 b(G)f(\\)23 b(D)r FC(.)183 4857 y(Pr)-5 b(o)g(of:)246 5017 y FE(\(i\))47 b(Immediate)31 b(consequence)36 b(of)c(De\014nition) f(4.10)h(\(a\).)218 5176 y(\(ii\))47 b(It)37 b(follo)m(ws)e(from)g (De\014nition)g(4.10)h(\(b\))g(that)h(ro)s(ot)o(\()p FB(l)r FE(\))d FA(2)h(G)6 b FE(.)37 b(Hence)h(\(ii\))c(is)i(a)g(conse-) 397 5296 y(quence)f(of)d(\(i\).)294 5437 y Fx(k)332 5467 y Fw(If)f(the)f(argumen)n(t)g(\014ltering)f(is)i(non-collapsing)d (\(i.e.,)j Fc(\031)s Fw(\()p Fc(f)9 b Fw(\))28 b Fi(6)p Fw(=)f Fc(i)j Fw(for)f(all)h(de\014ned)h(sym)n(b)r(ols)e Fc(f)9 b Fw(\),)31 b(then)183 5567 y(this)d(requiremen)n(t)e(is)i(alw)n (a)n(ys)d(ful\014lled.)p eop %%Page: 27 27 27 26 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(27)191 390 y FE(\(iii\))46 b(F)-8 b(ollo)m(ws)31 b(directly)h(from)g(\(ii\).)3496 667 y Fs(2)280 850 y FE(The)d(follo)m(wing)c(lemma)h(is)h(crucial)g(to)g(our)h(mo)s(dularit) m(y)d(result,)j(b)s(ecause)i(it)c(states)j(that)183 970 y(if)43 b FA(R)369 985 y Fz(1)453 970 y FE(and)i FA(R)739 985 y Fz(2)823 970 y FE(are)g(constructor-sharing,)g(then)g(applying)f (an)g(argumen)m(t)h(\014ltering)e FB(\031)183 1091 y FE(will)35 b(also)h(result)i(in)e(constructor-sharing)i(TRSs)g FB(\031)t FE(\()p FA(R)2283 1106 y Fz(1)2323 1091 y FE(\))f(and)h FB(\031)t FE(\()p FA(R)2774 1106 y Fz(2)2813 1091 y FE(\))g(pro)m (vided)g(that)f FB(\031)183 1211 y FE(is)e(compatible)f(with)h(the)h (set)h(of)e(all)f(shared)i(sym)m(b)s(ols.)g(In)g(fact,)g(this)f(result) h(ev)m(en)h(holds)183 1331 y(for)32 b FC(c)-5 b(omp)g(osable)38 b FE(TRSs)33 b(instead)g(of)f(constructor-sharing)g(ones.)183 1515 y FK(Lemma)38 b(4.12)f(\()p FA(G)6 b FK(-comp)-7 b(a)g(tible)39 b(Ar)n(gument)e(Fil)-7 b(terings)38 b(Maint)-7 b(ain)38 b(Composability\):)183 1635 y FC(L)-5 b(et)32 b FA(R)433 1650 y Fz(1)505 1635 y FC(and)f FA(R)776 1650 y Fz(2)848 1635 y FC(b)-5 b(e)32 b(c)-5 b(omp)g(osable)30 b(TRSs)i(over)g(the)g(signatur)-5 b(es)32 b FA(F)2634 1650 y Fz(1)2706 1635 y FC(and)f FA(F)2964 1650 y Fz(2)3003 1635 y FC(,)i(r)-5 b(esp)g(e)g(ctively.)183 1755 y(If)30 b FA(F)353 1770 y Fz(1)415 1755 y FA(\\)22 b(F)575 1770 y Fz(2)642 1755 y FA(\022)28 b(G)37 b FC(and)31 b(if)g FB(\031)k FC(is)c(an)g(ar)-5 b(gument)31 b(\014ltering)g(for)g FA(F)2456 1770 y Fz(1)2509 1755 y FA([)14 b(F)2661 1770 y Fz(2)2732 1755 y FC(that)32 b(is)f FA(G)6 b FC(-c)-5 b(omp)g(atible)183 1876 y(for)34 b FA(R)423 1891 y Fz(1)497 1876 y FC(and)g(for)h FA(R)926 1891 y Fz(2)966 1876 y FC(,)g(then)f FB(\031)t FE(\()p FA(R)1429 1891 y Fz(1)1468 1876 y FE(\))h FC(and)f FB(\031)t FE(\()p FA(R)1911 1891 y Fz(2)1951 1876 y FE(\))h FC(ar)-5 b(e)34 b(also)h(c)-5 b(omp)g(osable.)183 2059 y(Pr)g(o)g(of:)48 b FE(W)-8 b(e)31 b(pro)m(v)m(e)g(the)g(follo)m(wing)d(claims)g(\(where)k(\(B\))e (and)h(\(C\))f(imply)f(that)h FB(\031)t FE(\()p FA(R)3305 2074 y Fz(1)3345 2059 y FE(\))g(and)183 2179 y FB(\031)t FE(\()p FA(R)364 2194 y Fz(2)403 2179 y FE(\))j(are)f(comp)s(osable\):) 199 2335 y(\(A\))49 b FA(f)p FB(l)30 b FA(!)d FB(r)k FA(2)d FB(\031)t FE(\()p FA(R)983 2350 y Fz(1)1022 2335 y FE(\))23 b FA([)f FB(\031)t FE(\()p FA(R)1352 2350 y Fz(2)1392 2335 y FE(\))27 b FA(j)h FE(ro)s(ot)o(\()p FB(l)r FE(\))g FA(2)g(D)1997 2350 y Fz(1)2059 2335 y FA(\\)23 b(D)2227 2350 y Fz(2)2267 2335 y FA(g)k(\022)h FB(\031)t FE(\()p FA(R)2630 2350 y Fz(1)2670 2335 y FE(\))22 b FA(\\)h FB(\031)t FE(\()p FA(R)3000 2350 y Fz(2)3039 2335 y FE(\))204 2486 y(\(B\))48 b FA(f)p FB(l)30 b FA(!)d FB(r)k FA(2)d FB(\031)t FE(\()p FA(R)983 2501 y Fz(1)1022 2486 y FE(\))23 b FA([)f FB(\031)t FE(\()p FA(R)1352 2501 y Fz(2)1392 2486 y FE(\))27 b FA(j)h FE(ro)s(ot)o(\()p FB(l)r FE(\))g FA(2)g(D)1997 2501 y Fz(1)2037 2505 y Fv(\031)2106 2486 y FA(\\)23 b(D)2274 2501 y Fz(2)2314 2505 y Fv(\031)2361 2486 y FA(g)k(\022)h FB(\031)t FE(\()p FA(R)2725 2501 y Fz(1)2764 2486 y FE(\))22 b FA(\\)h FB(\031)t FE(\()p FA(R)3094 2501 y Fz(2)3133 2486 y FE(\))202 2637 y(\(C\))49 b FA(C)449 2652 y Fz(1)489 2656 y Fv(\031)558 2637 y FA(\\)23 b(D)724 2652 y Fz(2)763 2656 y Fv(\031)838 2637 y FE(=)k FA(D)1018 2652 y Fz(1)1058 2656 y Fv(\031)1127 2637 y FA(\\)c(C)1268 2652 y Fz(2)1307 2656 y Fv(\031)1382 2637 y FE(=)28 b FA(;)280 2793 y FE(\(A\))35 b(If)f FB(l)f FA(!)d FB(r)k FA(2)d FB(\031)t FE(\()p FA(R)1111 2808 y Fz(1)1151 2793 y FE(\))23 b FA([)h FB(\031)t FE(\()p FA(R)1483 2808 y Fz(2)1523 2793 y FE(\),)34 b(then)h(w)m(e)h(ha)m(v)m (e)f FB(l)e FE(=)e FB(\031)t FE(\()p FB(u)p FE(\))j(and)g FB(r)g FE(=)c FB(\031)t FE(\()p FB(v)t FE(\))k(for)g(some)183 2913 y FB(u)i FA(!)g FB(v)k FA(2)d(R)685 2928 y Fz(1)750 2913 y FA([)26 b(R)927 2928 y Fz(2)966 2913 y FE(.)38 b(Note)g(that)f(ro)s(ot)o(\()p FB(\031)t FE(\()p FB(u)p FE(\)\))f FA(2)h(D)2150 2928 y Fz(1)2215 2913 y FA(\\)26 b(D)2386 2928 y Fz(2)2462 2913 y FA(\022)37 b(G)44 b FE(implies)35 b(ro)s(ot)o(\()p FB(\031)t FE(\()p FB(u)p FE(\)\))h(=)183 3034 y(ro)s(ot)o(\()p FB(u)p FE(\))48 b(b)m(y)i(Lemma)d(4.11)h(\(ii\).)f(As)i(ro)s(ot)o(\()p FB(u)p FE(\))55 b FA(2)h(D)2207 3049 y Fz(1)2280 3034 y FA(\\)33 b(D)2459 3049 y Fz(2)2547 3034 y FE(and)49 b(as)g FA(R)2973 3049 y Fz(1)3061 3034 y FE(and)g FA(R)3352 3049 y Fz(2)3440 3034 y FE(are)183 3154 y(comp)s(osable,)43 b(this)h(implies)d FB(u)47 b FA(!)g FB(v)52 b FA(2)c(R)1836 3169 y Fz(1)1905 3154 y FA(\\)g(R)2103 3169 y Fz(2)2142 3154 y FE(.)d(It)f(follo)m(ws)f(that)h FB(\031)t FE(\()p FB(u)p FE(\))i FA(!)i FB(\031)t FE(\()p FB(v)t FE(\))f FA(2)183 3274 y FB(\031)t FE(\()p FA(R)364 3289 y Fz(1)403 3274 y FE(\))22 b FA(\\)h FB(\031)t FE(\()p FA(R)733 3289 y Fz(2)773 3274 y FE(\))32 b(b)s(ecause)i FB(\031)t FE(\()p FB(u)p FE(\))27 b FA(!)g FB(\031)t FE(\()p FB(v)t FE(\))h FA(2)g FB(\031)t FE(\()p FA(R)2038 3289 y Fz(1)2077 3274 y FE(\))22 b FA([)h FB(\031)t FE(\()p FA(R)2407 3289 y Fz(2)2447 3274 y FE(\))32 b(implies)e FB(\031)t FE(\()p FB(u)p FE(\))d FA(6)p FE(=)h FB(\031)t FE(\()p FB(v)t FE(\).)280 3395 y(\(B\))44 b(If)g FB(f)57 b FE(=)47 b(ro)s(ot)o(\()p FB(l)r FE(\))g FA(2)g(D)1326 3410 y Fz(1)1365 3414 y Fv(\031)1442 3395 y FA(\\)g(D)1632 3410 y Fz(2)1671 3414 y Fv(\031)1718 3395 y FE(,)d(then)h(a)e(function)h (sym)m(b)s(ol)f FB(f)54 b FE(\(with)43 b(p)s(ossibly)183 3515 y(di\013eren)m(t)37 b(arit)m(y\))g(o)s(ccurs)h(in)f FA(F)1345 3530 y Fz(1)1410 3515 y FA(\\)26 b(F)1574 3530 y Fz(2)1649 3515 y FA(\022)36 b(G)44 b FE(b)m(y)38 b(the)g (de\014nition)f(of)g(argumen)m(t)g(\014lterings.)183 3635 y(But)k(then)h(due)g(to)e(De\014nition)g(4.10)g(\(a\),)h FB(f)53 b FA(2)43 b(F)2088 3650 y Fz(1)2155 3635 y FA(\\)28 b(F)2321 3650 y Fz(2)2403 3635 y FA(\022)43 b(G)k FE(has)41 b(the)h(same)f(arit)m(y)g(as)183 3756 y FB(f)49 b FA(2)39 b(D)462 3771 y Fz(1)501 3775 y Fv(\031)575 3756 y FA(\\)k(D)761 3771 y Fz(2)801 3775 y Fv(\031)848 3756 y FE(.)c(Hence,)h FB(f)49 b FA(2)39 b(D)1519 3771 y Fz(1)1585 3756 y FA(\\)k(D)1774 3771 y Fz(2)1852 3756 y FE(follo)m(ws)38 b(from)g(Lemma)f(4.11)i (\(iii\).)d(No)m(w)j(the)183 3876 y(claim)30 b(is)i(implied)e(b)m(y)j (\(A\).)280 3997 y(\(C\))e(If)f(there)h(w)m(ere)h(an)e FB(f)39 b FA(2)28 b(C)1388 4012 y Fz(1)1428 4016 y Fv(\031)1493 3997 y FA(\\)34 b(D)1670 4012 y Fz(2)1710 4016 y Fv(\031)1757 3997 y FE(,)c(then)h(similar)c(to)j(the)h(argumen)m(tation)e(in)h (\(B\),)183 4117 y(w)m(e)40 b(w)m(ould)g(ha)m(v)m(e)h FB(f)51 b FA(2)40 b(F)1134 4132 y Fz(1)1200 4117 y FA(\\)k(F)1382 4132 y Fz(2)1461 4117 y FA(\022)d(G)46 b FE(b)m(y)40 b(the)h(de\014nition)d(of)i(argumen)m(t)f(\014lterings)g(and)183 4237 y(since)j FB(\031)j FE(is)c FA(G)6 b FE(-compatible.)40 b(This)i(implies)d FB(f)54 b FA(2)43 b(F)2126 4252 y Fz(1)2194 4237 y FA(\\)i(D)2382 4252 y Fz(2)2463 4237 y FE(according)c(to)g(Lemma)g(4.11)183 4358 y(\(iii\).)30 b(W)-8 b(e)34 b(kno)m(w)g FA(C)876 4373 y Fz(1)939 4358 y FA(\\)39 b(D)1121 4373 y Fz(2)1190 4358 y FE(=)28 b FA(;)33 b FE(b)s(ecause)i FA(R)1823 4373 y Fz(1)1896 4358 y FE(and)f 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b(restrictions)d(needed)j(for)e(the)h (desired)g(mo)s(dularit)m(y)d(result)i(are)h(captured)g(b)m(y)g(the)183 5022 y(notion)e(of)h FA(G)6 b FC(-r)-5 b(estricte)g(d)43 b FE(DP)33 b(simple)e(termination.)183 5205 y FK(Definition)38 b(4.13)f(\()p FA(G)6 b FK(-restricted)37 b(DP)g(simple)g(termina)-7 b(tion\):)50 b FC(A)25 b(TRS)g FA(R)g FC(over)183 5326 y(a)k(signatur)-5 b(e)28 b FA(F)39 b FC(is)29 b(c)-5 b(al)5 b(le)-5 b(d)28 b FA(G)6 b FE(-restricted)29 b FC(DP)g(simply)f(terminating)h(if)g(and)f(only)h(if)g(for)g(e)-5 b(ach)183 5446 y(cycle)30 b FA(P)39 b FC(in)30 b(the)g(estimate)-5 b(d)30 b(dep)-5 b(endency)29 b(gr)-5 b(aph)30 b(of)g FA(R)g FC(\(including)g(the)37 b FE(empt)m(y)31 b FC(one\))f(ther)-5 b(e)183 5567 y(is)34 b(an)h(ar)-5 b(gument)34 b(\014ltering)h FB(\031)1274 5582 y Fy(P)1368 5567 y FC(for)g FA(F)1606 5530 y Fv(])1672 5567 y FC(that)g(is)g FA(G)6 b FC(-c)-5 b(omp)g(atible)33 b(for)i FA(R)23 b([)f(P)43 b FC(such)35 b(that)p eop %%Page: 28 28 28 27 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(28)299 390 y FA(\017)48 b FB(\031)452 405 y Fy(P)512 390 y FE(\()p FA(R)22 b([)h(P)8 b FE(\))35 b FC(is)g(a)f(simply)h (terminating)f(TRS)g(and)299 552 y FA(\017)48 b FB(\031)452 567 y Fy(P)512 552 y FE(\()p FA(P)8 b FE(\))28 b FA(6)p FE(=)f FA(;)35 b FC(whenever)f FA(P)i(6)p FE(=)27 b FA(;)p FC(.)280 780 y FE(So)32 b(ob)m(viously)-8 b(,)31 b FA(G)6 b FE(-restricted)32 b(DP)g(simple)e(termination)f(implies)g(DP)j (simple)e(termina-)183 901 y(tion,)22 b(cf.)i(Corollary)e(4.3.)i(The)h (follo)m(wing)c(theorem)i(sho)m(ws)j(that)d(under)i(this)e FA(G)6 b FE(-restriction,)183 1021 y(DP)30 b(simple)e(termination)g(is) i(mo)s(dular)e(for)h(constructor-sharing)h(and)g(ev)m(en)i(for)e(comp)s (os-)183 1142 y(able)i(TRSs.)183 1370 y FK(Theorem)38 b(4.14)f(\(Modularity)i(of)f FA(G)6 b FK(-restricted)36 b(DP)h(simple)g(termina)-7 b(tion\):)183 1490 y FC(L)i(et)32 b FA(R)433 1505 y Fz(1)505 1490 y FC(and)f FA(R)776 1505 y Fz(2)848 1490 y FC(b)-5 b(e)32 b(c)-5 b(omp)g(osable)30 b(TRSs)i(over)g(the)g(signatur)-5 b(es)32 b FA(F)2634 1505 y Fz(1)2706 1490 y FC(and)f FA(F)2964 1505 y Fz(2)3003 1490 y FC(,)i(r)-5 b(esp)g(e)g(ctively.)183 1611 y(If)40 b FA(F)363 1626 y Fz(1)429 1611 y FA(\\)27 b(F)594 1626 y Fz(2)672 1611 y FA(\022)39 b(G)6 b FC(,)41 b(then)g(their)g(c)-5 b(ombine)g(d)40 b(system)h FA(R)e FE(=)g FA(R)2463 1626 y Fz(1)2530 1611 y FA([)27 b(R)2707 1626 y Fz(2)2787 1611 y FC(is)41 b FA(G)6 b FC(-r)-5 b(estricte)g(d)41 b(DP)183 1731 y(simply)h(terminating)h(if)g(and)g(only)g(if)g(b)-5 b(oth)44 b FA(R)1959 1746 y Fz(1)2042 1731 y FC(and)f FA(R)2324 1746 y Fz(2)2407 1731 y FC(ar)-5 b(e)43 b FA(G)6 b FC(-r)-5 b(estricte)g(d)43 b(DP)g(simply)183 1851 y(terminating.)183 2080 y(Pr)-5 b(o)g(of:)48 b FE(The)29 b(only-if)d(direction)i(is)f (trivial.)f(F)-8 b(or)27 b(the)i(if)e(direction,)g(let)h FA(P)37 b FE(b)s(e)28 b(a)g(cycle)h(in)e(the)183 2200 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Fz(1)3413 4006 y FE(and)183 4126 y FB(\031)238 4141 y Fz(2)313 4126 y FE(do)d(not)e(mo)s(dify)g(function)h(sym)m(b)s(ols)f(from)g FA(F)2028 4141 y Fz(1)2092 4126 y FA(\\)25 b(F)2255 4141 y Fz(2)2328 4126 y FA(\022)34 b(G)6 b FE(.\))36 b(Clearly)-8 b(,)35 b FB(\031)t FE(\()p FA(P)8 b FE(\))34 b(=)g FB(\031)3377 4141 y Fz(1)3417 4126 y FE(\()p FA(P)8 b FE(\))183 4247 y(and)38 b(th)m(us,)i FB(\031)t FE(\()p FA(P)8 b FE(\))37 b FA(6)p FE(=)h FA(;)g FE(if)f FA(P)46 b(6)p FE(=)38 b FA(;)p FE(.)g(Moreo)m(v)m(er,)i(ob)m(viously)e FB(\031)k FE(is)c FA(G)6 b FE(-compatible)36 b(for)i(b)s(oth)183 4367 y FA(S)243 4382 y Fz(1)325 4367 y FE(and)k FA(S)584 4382 y Fz(2)666 4367 y FE(and)h(hence,)h(also)d(for)h FA(S)7 b FE(.)44 b(Then)f(b)m(y)h(Lemma)d(4.12,)g FB(\031)t FE(\()p FA(S)2843 4382 y Fz(1)2883 4367 y FE(\))h(and)h FB(\031)t FE(\()p FA(S)3320 4382 y Fz(2)3360 4367 y FE(\))f(are)183 4487 y(comp)s(osable,)27 b(since)i FA(S)1024 4502 y Fz(1)1092 4487 y FE(and)g FA(S)1338 4502 y Fz(2)1406 4487 y 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FE(\))p eop %%Page: 29 29 29 28 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(29)183 390 y FE(and)35 b(moreo)m(v)m(er,)i(w)m(e)f(also)f(add)h (the)g(rule)f(\()p FB(x)24 b FA(\000)h FB(y)t FE(\))f FA(\000)g FB(z)38 b FA(!)32 b FB(x)25 b FA(\000)g FE(\()p FB(y)i FE(+)d FB(z)t FE(\))36 b(to)g FA(R)3107 405 y Fz(1)3147 390 y FE(.)f(No)m(w)h(the)183 511 y(resulting)f(TRSs)i(are)g (comp)s(osable,)e(since)i(they)g(b)s(oth)g(con)m(tain)f(the)g(same)h (constructors)183 631 y Ft(0)f FE(and)g Ft(s)h FE(and)f(they)i(also)d (share)i(the)g(de\014ned)h(sym)m(b)s(ol)d(+,)i(but)f(b)s(oth)g(TRSs)i (con)m(tain)d(the)183 751 y(same)48 b(+-rules.)g(As)h(b)s(oth)f(TRSs)h (are)g FA(f)p Ft(0)o FB(;)17 b Ft(s)p FB(;)g FE(+)p FA(g)p FE(-restricted)48 b(DP)g(simply)f(terminating,)183 872 y(Theorem)37 b(4.14)g(allo)m(ws)f(us)i(to)f(conclude)g FA(f)p Ft(0)p FB(;)17 b Ft(s)p FB(;)g FE(+)p FA(g)p FE(-restricted)37 b(DP)g(simple)e(termination)183 992 y(of)d(the)h(com)m(bined)f(system.) 280 1112 y(There)j(are)g(ev)m(en)g(TRSs)g FA(R)1304 1127 y Fz(1)1367 1112 y FA([)24 b(R)1541 1127 y Fz(2)1614 1112 y FE(where)36 b(DP)e(simple)e(termination)g(of)h(b)s(oth)h FA(R)3339 1127 y Fz(1)3413 1112 y FE(and)183 1233 y FA(R)267 1248 y Fz(2)343 1233 y FE(can)j(b)s(e)g(pro)m(v)m(ed)h(with)e(a)g (standard)h(tec)m(hnique)h(lik)m(e)e(LPO,)g(whereas)i(suc)m(h)g (standard)183 1353 y(orderings)25 b(fail)e(if)h(one)i(w)m(an)m(ts)g(to) f(pro)m(v)m(e)i(DP)e(simple)f(termination)f(of)i(their)g(union)g (directly)-8 b(.)183 1474 y(Hence,)30 b(for)e(suc)m(h)i(examples)f(our) g(result)g(enables)g(automatic)e(termination)f(pro)s(ofs)i(whic)m(h)183 1594 y(w)m(ere)34 b(not)e(p)s(ossible)g(b)s(efore.)183 1761 y FK(Example)37 b(4.15:)49 b FC(L)-5 b(et)35 b FA(R)1183 1776 y Fz(1)1258 1761 y FC(b)-5 b(e)34 b(the)h(TRS)1248 1928 y Ft(f)7 b FE(\()p Ft(c)p FE(\()p Ft(s)p FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t FE(\)\))82 b FA(!)h Ft(f)7 b FE(\()p Ft(c)p FE(\()p FB(x;)17 b Ft(s)p FE(\()p FB(y)t FE(\)\)\))1463 2073 y Ft(f)7 b FE(\()p Ft(f)f FE(\()p FB(x)p FE(\)\))83 b FA(!)g Ft(f)7 b FE(\()p Ft(d)p FE(\()p Ft(f)f FE(\()p FB(x)p FE(\)\)\))1576 2218 y Ft(f)g FE(\()p FB(x)p FE(\))83 b FA(!)g FB(x)183 2385 y FC(and)34 b(let)h FA(R)593 2400 y Fz(2)668 2385 y FC(c)-5 b(onsist)34 b(of)h(the)f(rule)h Ft(g)r FE(\()p Ft(c)p FE(\()p FB(x;)17 b Ft(s)p FE(\()p FB(y)t FE(\)\)\))27 b FA(!)g Ft(g)q FE(\()p Ft(c)q FE(\()p Ft(s)p FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t FE(\)\))p FC(.)280 2505 y FA(R)364 2520 y Fz(1)435 2505 y FC(is)30 b(DP)g(simply)g(terminating)g(\(using)g(the)h(ar)-5 b(gument)30 b(\014ltering)h FB(\031)t FE(\()p Ft(d)o FE(\))d(=)g([)17 b(])30 b FC(and)g(LPO)183 2625 y(c)-5 b(omp)g(aring)39 b(subterms)i(left-to-right\),)f(but)i(it)f(is)f(not)h(simply)f (terminating.)g FA(R)3189 2640 y Fz(2)3270 2625 y FC(is)h(even)183 2746 y(simply)33 b(terminating)h(as)g(c)-5 b(an)34 b(b)-5 b(e)34 b(shown)f(with)i(LPO)f(c)-5 b(omp)g(aring)33 b(subterms)h (right-to-left.)183 2866 y(Thus,)g(DP)h(simple)f(termination)g(of)g(b) -5 b(oth)35 b(systems)g(c)-5 b(an)34 b(b)-5 b(e)35 b(veri\014e)-5 b(d)34 b(by)h(LPO.)280 2986 y(By)45 b(The)-5 b(or)g(em)44 b(4.14)h(their)g(union)f(is)h(also)g(DP)f(simply)h(terminating.)f (However,)h(the)183 3107 y(c)-5 b(onstr)g(aints)30 b(for)g(the)g(cycle) g FA(f)p Ft(G)p FE(\()p Ft(c)p FE(\()p FB(x;)17 b Ft(s)q FE(\()p FB(y)t FE(\)\)\))26 b FA(!)i Ft(G)p FE(\()p Ft(c)p FE(\()p Ft(s)p FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t FE(\)\))o FA(g)30 b FC(ar)-5 b(e)31 b(not)f(satis\014e)-5 b(d)30 b(by)g(LPO)183 3227 y(\(nor)45 b(by)h(RPO)g(nor)g(by)g(any)g(p)-5 b(olynomial)44 b(or)-5 b(dering\).)45 b(Thus,)h(ther)-5 b(e)46 b(ar)-5 b(e)46 b(inde)-5 b(e)g(d)44 b(TRSs)183 3348 y(wher)-5 b(e)42 b(termination)h(of)f(the)i(subsystems)f(c)-5 b(an)42 b(b)-5 b(e)43 b(shown)f(with)h(dep)-5 b(endency)42 b(p)-5 b(airs)43 b(and)183 3468 y(LPO,)38 b(but)h(\(without)g(our)g(mo) -5 b(dularity)38 b(r)-5 b(esult\))39 b(termination)f(of)g(their)h (union)f(c)-5 b(annot)38 b(b)-5 b(e)183 3588 y(pr)g(ove)g(d)34 b(with)g(dep)-5 b(endency)34 b(p)-5 b(airs)34 b(and)h(LPO.)183 3864 y FD(5.)68 b(Mo)t(dular)56 b(Innermost)h(T)-11 b(ermination)57 b(Pro)t(ofs)g(With)g(Dep)t(en-)355 4014 y(dency)44 b(P)l(airs)183 4176 y FE(Arts)37 b(and)g(Giesl)e([2000])i(sho)m(w)m(ed)h(that)f(the)h (dep)s(endency)h(pair)d(approac)m(h)h(can)g(b)s(e)g(mo)s(d-)183 4297 y(i\014ed)44 b(in)f(order)i(to)f(v)m(erify)g FC(innermost)53 b FE(termination.)42 b(Unlik)m(e)i(previous)h(metho)s(ds,)f(this)183 4417 y(tec)m(hnique)30 b(can)f(ev)m(en)i(pro)m(v)m(e)f(innermost)f (termination)d(of)j(non-terminating)d(systems)k(au-)183 4537 y(tomatically)-8 b(.)31 b(Similar)g(to)k(the)g(mo)s(dular)d (approac)m(h)k(for)e(termination)e(in)i(Section)h(3,)f(this)183 4658 y(tec)m(hnique)h(for)f(innermost)g(termination)e(pro)s(ofs)i(can)h (also)e(b)s(e)i(used)h(in)d(a)h(mo)s(dular)f(w)m(a)m(y)-8 b(.)183 4778 y(As)33 b(an)f(example)g(consider)h(the)g(follo)m(wing)d (TRS:)658 4977 y Ft(f)7 b FE(\()p FB(x;)17 b Ft(c)p FE(\()p FB(x)p FE(\))p FB(;)g Ft(c)q FE(\()p FB(y)t FE(\)\))98 b FA(!)h Ft(f)7 b FE(\()p FB(y)t(;)17 b(y)t(;)g Ft(f)t FE(\()p FB(y)t(;)g(x;)g(y)t FE(\)\))789 5122 y Ft(f)7 b FE(\()p Ft(s)p FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t(;)g(z)t FE(\))99 b FA(!)g Ft(f)7 b FE(\()p FB(x;)17 b Ft(s)p FE(\()p Ft(c)q FE(\()p FB(y)t FE(\)\))p FB(;)g Ft(c)o FE(\()p FB(z)t FE(\)\))777 5267 y Ft(f)7 b FE(\()p Ft(c)p FE(\()p FB(x)p FE(\))p FB(;)17 b(x;)g(y)t FE(\))99 b FA(!)g Ft(c)q FE(\()p FB(y)t FE(\))2501 4977 y Ft(g)q FE(\()p FB(x;)17 b(y)t FE(\))99 b FA(!)h FB(x)2501 5122 y Ft(g)q FE(\()p FB(x;)17 b(y)t FE(\))99 b FA(!)h FB(y)183 5446 y FE(By)32 b(applying)f(the)i(\014rst)g Ft(f)6 b FE(-rule)32 b(to)g Ft(f)6 b FE(\()p FB(x;)17 b Ft(c)q FE(\()p FB(x)p FE(\))p FB(;)g Ft(c)p FE(\()p Ft(g)q FE(\()p FB(x;)g Ft(c)p FE(\()p FB(x)p FE(\)\)\)\),)33 b(w)m(e)g(obtain)e(an)h (in\014nite)g(\(cy-)183 5567 y(cling\))k(reduction.)j(Ho)m(w)m(ev)m (er,)h(it)e(is)g(not)g(an)g(innermost)f(reduction,)i(b)s(ecause)g(this) f(term)p eop %%Page: 30 30 30 29 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(30)183 390 y FE(con)m(tains)38 b(a)g(redex)i Ft(g)q FE(\()p FB(:)17 b(:)g(:)o FE(\))38 b(as)h(a)f(prop)s(er)g(subterm.)h (It)f(turns)h(out)g(that)f(the)h(TRS)f(is)g(not)183 511 y(terminating,)30 b(but)j(it)e(is)h(innermost)g(terminating.)280 631 y(T)-8 b(o)41 b(dev)m(elop)g(a)f(criterion)g(for)g(innermost)g (termination)e(similar)f(to)j(the)h(termination)183 751 y(criterion)c(of)h(Section)g(3,)g(the)h(notion)e(of)h(c)m(hains)h(has)g (to)f(b)s(e)h(restricted.)g(A)f(sequence)j(of)183 872 y(dep)s(endency)33 b(pairs)d FB(s)995 887 y Fz(1)1062 872 y FA(!)e FB(t)1225 887 y Fz(1)1264 872 y FE(,)j FB(s)1368 887 y Fz(2)1435 872 y FA(!)d FB(t)1598 887 y Fz(2)1637 872 y FE(,)j FB(:)17 b(:)g(:)31 b FE(is)f(an)h FC(innermost)h FA(R)p FC(-chain)38 b FE(if)30 b(there)h(exists)h(a)183 1011 y(substitution)g FB(\033)37 b FE(suc)m(h)d(that)f(all)d FB(s)1431 1026 y Fv(j)1468 1011 y FB(\033)37 b FE(are)c(in)f(normal)e (form)i(and)h FB(t)2623 1026 y Fv(j)2660 1011 y FB(\033)2788 955 y Fb(i)2747 1011 y FA(!)2841 975 y Fy(\003)2841 1037 y(R)2933 1011 y FB(s)2979 1026 y Fv(j)t Fz(+1)3106 1011 y FB(\033)k FE(holds)32 b(for)183 1159 y(ev)m(ery)j(t)m(w)m(o)g (consecutiv)m(e)g(pairs)f FB(s)1430 1174 y Fv(j)1494 1159 y FA(!)27 b FB(t)1656 1174 y Fv(j)1727 1159 y FE(and)33 b FB(s)1963 1174 y Fv(j)t Fz(+1)2118 1159 y FA(!)27 b FB(t)2280 1174 y Fv(j)t Fz(+1)2441 1159 y FE(in)33 b(the)h(sequence.)j (Here,)d(`)3485 1103 y Fb(i)3443 1159 y FA(!)p FE(')183 1280 y(denotes)f(innermost)f(reductions.)280 1400 y(Of)40 b(course,)h(ev)m(ery)h(innermost)e(c)m(hain)g(is)f(also)g(a)h(c)m (hain,)g(but)h(not)f(vice)g(v)m(ersa.)i(In)e(our)183 1521 y(example,)32 b(w)m(e)h(ha)m(v)m(e)h(the)f(follo)m(wing)d(dep)s (endency)35 b(pairs.)1054 1734 y Ft(F)p FE(\()p FB(x;)17 b Ft(c)q FE(\()p FB(x)p FE(\))p FB(;)g Ft(c)p FE(\()p FB(y)t FE(\)\))99 b FA(!)g Ft(F)p FE(\()p FB(y)t(;)17 b(y)t(;)g Ft(f)5 b FE(\()p FB(y)t(;)17 b(x;)g(y)t FE(\)\))744 b(\(21\))1054 1880 y Ft(F)p FE(\()p FB(x;)17 b Ft(c)q FE(\()p FB(x)p FE(\))p FB(;)g Ft(c)p FE(\()p FB(y)t FE(\)\))99 b FA(!)g Ft(F)p FE(\()p FB(y)t(;)17 b(x;)g(y)t FE(\))1047 b(\(22\))1185 2025 y Ft(F)p FE(\()p Ft(s)q FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t(;)g(z)t FE(\))99 b FA(!)g Ft(F)p FE(\()p FB(x;)17 b Ft(s)q FE(\()p Ft(c)p FE(\()p FB(y)t FE(\)\))p FB(;)g Ft(c)o FE(\()p FB(z)t FE(\)\))699 b(\(23\))280 2239 y(The)48 b(in\014nite)e(sequence)j(consisting)d(of)g(the)h(dep)s (endency)i(pair)d(\(21\))g(is)g(an)h(in\014nite)183 2359 y(c)m(hain,)38 b(but)g(no)h FC(innermost)47 b FE(c)m(hain,)38 b(b)s(ecause)i Ft(F)p FE(\()p FB(y)2072 2374 y Fz(1)2111 2359 y FB(;)17 b(y)2203 2374 y Fz(1)2242 2359 y FB(;)g Ft(f)6 b FE(\()p FB(y)2408 2374 y Fz(1)2447 2359 y FB(;)17 b(x)2546 2374 y Fz(1)2585 2359 y FB(;)g(y)2677 2374 y Fz(1)2716 2359 y FE(\)\))p FB(\033)42 b FE(can)d(only)f(reduce)183 2480 y(to)d Ft(F)p FE(\()p FB(x)453 2495 y Fz(2)493 2480 y FB(;)17 b Ft(c)p FE(\()p FB(x)673 2495 y Fz(2)713 2480 y FE(\))p FB(;)g Ft(c)p FE(\()p FB(y)924 2495 y Fz(2)963 2480 y FE(\)\))p FB(\033)39 b FE(for)34 b(substitutions)h FB(\033)40 b FE(where)c FB(y)2295 2495 y Fz(1)2334 2480 y FB(\033)j FE(is)c(not)g(a)g(normal)e(form.)h(Arts)183 2600 y(and)22 b(Giesl)f([2000])g(pro)m(v)m(ed)j(that)e(the)h(absence)h (of)e(in\014nite)f(innermost)g(c)m(hains)i(is)f(a)g(su\016cien)m(t)183 2720 y(and)32 b(necessary)j(criterion)c(for)h(innermost)g(termination.) 183 2942 y FK(Theorem)38 b(5.1)f(\(Innermost)i(Termina)-7 b(tion)39 b(Criterion\):)49 b FC(A)38 b(TRS)g FA(R)g FC(is)g(inner-)183 3062 y(most)c(terminating)h(if)f(and)g(only)h(if)g (ther)-5 b(e)35 b(exists)f(no)h(in\014nite)f(innermost)g FA(R)p FC(-chain.)183 3283 y FE(Analogous)d(to)i(Section)f(3,)g(the)h (notion)f(of)g(a)g(graph)h(is)f(de\014ned)i(for)e(innermost)g(c)m (hains.)183 3481 y FK(Definition)38 b(5.2)f(\(Innermost)i(dependency)d (graph\):)49 b FC(The)24 b FE(innermost)d(dep)s(endency)183 3602 y(graph)39 b FC(of)h(a)f(TRS)h FA(R)g FC(is)g(the)f(dir)-5 b(e)g(cte)g(d)40 b(gr)-5 b(aph)39 b(whose)g(no)-5 b(des)39 b(ar)-5 b(e)39 b(the)h(dep)-5 b(endency)38 b(p)-5 b(airs)183 3722 y(and)28 b(ther)-5 b(e)29 b(is)f(an)h(ar)-5 b(c)29 b(fr)-5 b(om)28 b FB(s)g FA(!)f FB(t)i FC(to)g FB(v)j FA(!)27 b FB(w)k FC(i\013)e FB(s)e FA(!)h FB(t)p FC(,)h FB(v)i FA(!)c FB(w)32 b FC(is)c(an)h(innermost)f(chain.)280 3920 y FE(F)-8 b(or)26 b(the)h(purp)s(ose)h(of)e(automation)e(w)m(e)k (again)e(need)h(an)g(estimation,)e(since)i(in)f(general)g(it)183 4041 y(is)33 b(undecidable)h(whether)i(t)m(w)m(o)f(dep)s(endency)h (pairs)e(form)f(an)h(innermost)f(c)m(hain.)h(T)-8 b(o)34 b(this)183 4161 y(end,)f(w)m(e)g(again)e(replace)h(subterms)h(in)e FB(t)h 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FC(such)g(that)g FB(s\026)g FC(and)g FB(v)t(\026)f FC(ar)-5 b(e)183 5105 y(normal)34 b(forms.)280 5326 y FE(In)h(the)f(estimated)g(innermost)f(dep)s(endency)k(graph)d(of)g(our) g(example,)f(there)i(are)f(arcs)183 5446 y(from)j(\(22\))i(to)f(eac)m (h)i(dep)s(endency)i(pair,)c(from)f(\(21\))h(to)h(\(23\),)f(and)i(from) d(\(23\))h(to)h(itself.)183 5567 y(Ho)m(w)m(ev)m(er,)27 b(there)f(is)f(no)g(arc)g(from)f(\(21\))g(to)h(itself,)f(b)s(ecause)j FK(cap)o FE(\()p Ft(F)p FE(\()p FB(y)2708 5582 y Fz(1)2747 5567 y FB(;)17 b(y)2839 5582 y Fz(1)2878 5567 y FB(;)g Ft(f)6 b FE(\()p FB(y)3044 5582 y Fz(1)3083 5567 y FB(;)17 b(x)3182 5582 y Fz(1)3222 5567 y FB(;)g(y)3314 5582 y Fz(1)3353 5567 y FE(\)\)\))27 b(=)p eop %%Page: 31 31 31 30 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(31)183 390 y Ft(F)p FE(\()p FB(y)324 405 y Fz(1)363 390 y FB(;)17 b(y)455 405 y Fz(1)494 390 y FB(;)g(z)t FE(\))47 b(do)s(es)h(not)f(unify)f(with)h Ft(F)q FE(\()p FB(x)1744 405 y Fz(2)1783 390 y FB(;)17 b Ft(c)p FE(\()p FB(x)1963 405 y Fz(2)2003 390 y FE(\))p FB(;)g Ft(c)p FE(\()p FB(y)2214 405 y Fz(2)2253 390 y FE(\)\).)47 b(Hence,)i(the)e (only)g(cycles)h(are)183 511 y FA(f)p FE(\(22\))p FA(g)28 b FE(and)h FA(f)p FE(\(23\))p FA(g)p FE(.)g(In)h(fact,)f(in)f(this)h (example)g(the)h(estimated)f(innermost)f(dep)s(endency)183 631 y(graph)c(coincides)g(with)g(the)h(\(real\))e(innermost)h(dep)s (endency)j(graph.)d(Similar)d(to)j(Theorem)183 751 y(3.3)i(one)h(can)h (sho)m(w)g(that)e(it)g(su\016ces)j(to)e(pro)m(v)m(e)h(the)f(absence)i (of)d(in\014nite)g(innermost)h(c)m(hains)183 872 y(separately)33 b(for)f(ev)m(ery)i(cycle.)183 1100 y FK(Theorem)k(5.4)f(\(Modular)i (Innermost)f(Termina)-7 b(tion)39 b(Criterion\):)49 b FC(A)25 b(TRS)g FA(R)183 1220 y FC(is)41 b(innermost)g(terminating)h (if)g(and)f(only)h(if)g(for)f(e)-5 b(ach)42 b(cycle)f FA(P)51 b FC(in)41 b(the)h(innermost)f(de-)183 1341 y(p)-5 b(endency)31 b(gr)-5 b(aph)31 b(ther)-5 b(e)32 b(is)f(no)h(in\014nite)f (innermost)g FA(R)p FC(-chain)g(of)h(dep)-5 b(endency)30 b(p)-5 b(airs)31 b(fr)-5 b(om)183 1461 y FA(P)8 b FC(.)183 1689 y(Pr)-5 b(o)g(of:)48 b FE(The)27 b(pro)s(of)f(is)f(absolutely)h (analogous)f(to)h(the)h(pro)s(of)e(of)h(Theorem)g(3.3:)g(If)h FA(R)f FE(is)g(not)183 1810 y(innermost)35 b(terminating,)f(then)j(b)m (y)h(Theorem)e(5.1)h(there)g(exists)g(an)f(in\014nite)g(innermost)183 1930 y(c)m(hain)c(and)h(its)f(tail)e(corresp)s(onds)k(to)e(a)h(cycle)g (in)f(the)h(innermost)e(dep)s(endency)k(graph.)41 b Fs(2)280 2159 y FE(T)-8 b(o)32 b(pro)m(v)m(e)h(innermost)e(termination)f(in)h(a) g(mo)s(dular)f(w)m(a)m(y)-8 b(,)33 b(w)m(e)g(again)e(generate)h(a)g (set)g(of)183 2279 y(inequalities)25 b(for)i(ev)m(ery)j(cycle)e FA(P)36 b FE(and)28 b(searc)m(h)g(for)f(a)h(reduction)f(pair)g(\()p Fu(\045)2863 2294 y Fy(P)2922 2279 y FB(;)17 b FA(\037)3043 2294 y Fy(P)3102 2279 y FE(\))28 b(satisfying)183 2399 y(them.)22 b(Ho)m(w)m(ev)m(er,)j(to)e(ensure)h FB(t\033)32 b Fu(\045)1462 2414 y Fy(P)1549 2399 y FB(v)t(\033)27 b FE(whenev)m(er)e FB(t\033)i FE(reduces)d(to)f FB(v)t(\033)t FE(,)f(no)m(w)i(it)d(is)i(su\016cien)m(t)183 2520 y(to)37 b(require)h FB(l)h Fu(\045)785 2535 y Fy(P)881 2520 y FB(r)h FE(only)d(for)g(those)i(rules)e(that)h(are)g FC(usable)44 b FE(in)37 b(a)h(reduction)f(of)h FB(t\033)j FE(\(for)183 2640 y FC(normal)h FE(substitutions)32 b FB(\033)t FE(\).)183 2868 y FK(Definition)38 b(5.5)f(\(Usable)h(R)m(ules\):)48 b FC(L)-5 b(et)53 b FA(R)h FC(b)-5 b(e)52 b(a)h(TRS.)g(F)-7 b(or)52 b(any)h(symb)-5 b(ol)52 b FB(f)64 b FC(let)183 2989 y(R)n(ules)413 3004 y Fy(R)478 2989 y FE(\()p FB(f)11 b FE(\))53 b(=)g FA(f)p FB(l)19 b FA(!)d FB(r)56 b FA(2)e(R)17 b(j)g FE(ro)s(ot)o(\()p FB(l)r FE(\))53 b(=)h FB(f)11 b FA(g)p FC(.)48 b(F)-7 b(or)48 b(any)h(term)g(we)f(de\014ne)g(the)h FE(usable)183 3109 y(rules)p FC(:)299 3271 y FA(\017)f(U)459 3286 y Fy(R)524 3271 y FE(\()p FB(x)p FE(\))28 b(=)g FA(;)p FC(,)299 3433 y FA(\017)48 b(U)459 3448 y Fy(R)524 3433 y FE(\()p FB(f)11 b FE(\()p FB(t)694 3448 y Fz(1)733 3433 y FB(;)17 b(:)g(:)g(:)f(;)h(t)987 3448 y Fv(n)1034 3433 y FE(\)\))27 b(=)h FC(R)n(ules)1472 3448 y Fy(R)1536 3433 y FE(\()p FB(f)11 b FE(\))49 b FA([)1837 3358 y Fl(S)1920 3462 y Fv(l)q Fy(!)p Fv(r)r Fy(2)p Fv(Rul)q(es)2276 3473 y Fk(R)2331 3462 y Fz(\()p Fv(f)7 b Fz(\))2448 3433 y FA(U)2510 3448 y Fy(R)2570 3429 y Fk(0)2597 3433 y FE(\()p FB(r)s FE(\))50 b FA([)2886 3358 y Fl(S)2969 3385 y Fv(n)2969 3462 y(j)t Fz(=1)3112 3433 y FA(U)3174 3448 y Fy(R)3234 3429 y Fk(0)3261 3433 y FE(\()p FB(t)3334 3448 y Fv(j)3371 3433 y FE(\))p FC(,)183 3607 y(wher)-5 b(e)44 b FA(R)553 3565 y Fy(0)622 3607 y FE(=)j FA(R)30 b(n)g FC(R)n(ules)1169 3622 y Fy(R)1233 3607 y FE(\()p FB(f)11 b FE(\))p FC(.)45 b(Mor)-5 b(e)g(over,)45 b(for)f(any)i(set)f FA(P)53 b FC(of)45 b(dep)-5 b(endency)44 b(p)-5 b(airs)44 b(we)183 3728 y(de\014ne)34 b FA(U)529 3743 y Fy(R)593 3728 y FE(\()p FA(P)8 b FE(\))28 b(=)878 3653 y Fl(S)961 3757 y Fv(s)p Fy(!)p Fv(t)p Fy(2P)1229 3728 y FA(U)1291 3743 y Fy(R)1356 3728 y FE(\()p FB(t)p FE(\))p FC(.)280 3956 y FE(So)23 b(w)m(e)i(ha)m(v)m(e)f FA(U)818 3971 y Fy(R)883 3956 y FE(\()p Ft(F)p FE(\()p FB(y)t(;)17 b(y)t(;)g Ft(f)t FE(\()p FB(y)t(;)g(x;)g(y)t FE(\)\)\))26 b(=)i(Rules)2003 3971 y Fy(R)2068 3956 y FE(\()p Ft(f)6 b FE(\))24 b(and)f FA(U)2446 3971 y Fy(R)2511 3956 y FE(\()p FA(f)p FE(\(22\))p FA(g)p FE(\))j(=)i FA(U)3053 3971 y Fy(R)3117 3956 y FE(\()p FA(f)p FE(\(23\))p FA(g)p FE(\))f(=)183 4076 y FA(;)p FE(,)43 b(i.e.,)g(there)h(are)g(no)f (usable)h(rules)f(for)g(the)h(cycles.)h(Note)f(that)f(Rules)2993 4091 y Fy(R)3057 4076 y FE(\()p FB(f)11 b FE(\))46 b(=)g FA(;)d FE(for)183 4197 y(an)m(y)k(constructor)h FB(f)11 b FE(.)46 b(No)m(w)i(our)f(theorem)f(for)g(automatic)2454 4160 y Fy(\003\003)2574 4197 y FE(modular)f(v)m(eri\014cation)h(of)183 4317 y(innermost)31 b(termination)g(can)h(b)s(e)h(pro)m(v)m(ed)h (analogously)d(to)h(Theorem)h(3.5.)183 4545 y FK(Theorem)38 b(5.6)f(\(Modular)i(Innermost)f(Termina)-7 b(tion)39 b(Pr)n(oofs\):)49 b FC(A)25 b(TRS)g FA(R)g FC(is)183 4666 y(innermost)37 b(terminating)i(if)f(for)h(e)-5 b(ach)38 b(cycle)g FA(P)47 b FC(in)39 b(the)f(\(estimate)-5 b(d\))38 b(innermost)g(dep)-5 b(en-)183 4786 y(dency)34 b(gr)-5 b(aph)34 b(ther)-5 b(e)35 b(is)g(a)g(r)-5 b(e)g(duction)34 b(p)-5 b(air)35 b FE(\()p Fu(\045)1897 4801 y Fy(P)1956 4786 y FB(;)17 b FA(\037)2077 4801 y Fy(P)2136 4786 y FE(\))35 b FC(such)g(that)230 4989 y(\(a\))f FA(U)456 5004 y Fy(R)521 4989 y FE(\()p FA(P)8 b FE(\))23 b FA([)f(P)36 b(\022)28 b Fu(\045)1072 5004 y Fy(P)1167 4989 y FC(and)230 5110 y(\(b\))39 b FA(P)25 b(\\)j(\037)659 5125 y Fy(P)747 5110 y FA(6)p FE(=)f FA(;)p FC(.)260 5265 y Fx(\003\003)332 5295 y Fw(Detailed)d(explanations)f(and)h(additional)g(re\014nemen)n (ts)f(for)h(the)g(automated)f(c)n(hec)n(king)g(of)h(the)h(inner-)183 5395 y(most)i(termination)g(criterion)g(can)g(b)r(e)h(found)g(in)g 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FB(:)183 1460 y FE(F)-8 b(or)37 b FA(\037)440 1475 y Fz(1)518 1460 y FE(w)m(e)i(ma)m(y)f(use)h(LPO)f(comparing)f(subterms)i(righ)m (t-to-left)c(and)j(for)g FA(\037)3164 1475 y Fz(2)3242 1460 y FE(w)m(e)h(ma)m(y)183 1580 y(use)29 b(LPO)g(comparing)d (subterms)j(left-to-righ)m(t.)d(Hence,)k(innermost)d(termination)f(of)i (this)183 1701 y(example)h(can)h(easily)f(b)s(e)h(pro)m(v)m(ed)h (automatically)-8 b(.)26 b(Without)k(our)f(mo)s(dularit)m(y)f(result,)h (the)183 1821 y(ab)s(o)m(v)m(e)37 b(innermost)e(termination)f(pro)s(of) i(w)m(ould)g(not)g(b)s(e)h(p)s(ossible,)e(b)s(ecause)j(there)f(exists) 183 1941 y(no)f(simpli\014cation)e(ordering)i(satisfying)g FC(b)-5 b(oth)44 b FE(inequalities)35 b(\(not)h(ev)m(en)j(after)d (argumen)m(t)183 2062 y(\014ltering\).)280 2182 y(Note)e(that)g(unlik)m (e)g(Theorem)g(3.5,)g(the)h(rev)m(erse)h(direction)d(of)g(Theorem)i (5.6)f(do)s(es)g(not)183 2303 y(hold,)c(i.e.,)h(this)f(criterion)g(is)g 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FE(b)s(e)h(the)g(system)g(consisting)f(of)183 3399 y(the)h(t)m(w)m(o)g Ft(w)m(eight)q FE(-rules.)g(Then)h(these)g(t)m (w)m(o)f(systems)i(form)c(a)i(hierarc)m(hical)e(com)m(bination,)183 3520 y(where)e Ft(add)g FE(is)f(a)g(de\014ned)i(sym)m(b)s(ol)e(of)g FA(R)1689 3535 y Fz(1)1760 3520 y FE(and)h(a)f(constructor)i(of)e FA(R)2742 3535 y Fz(2)2782 3520 y FE(.)280 3640 y(Note)24 b(that)g(tuple)f(sym)m(b)s(ols)h(from)e(dep)s(endency)k(pairs)e(of)f FA(R)2474 3655 y Fz(1)2538 3640 y FE(do)g(not)h(o)s(ccur)g(in)f (left-hand)183 3761 y(sides)k(of)f FA(R)601 3776 y Fz(2)640 3761 y FE(-dep)s(endency)j(pairs.)d(Hence,)j(a)d(cycle)h(in)g(the)g (innermost)f(dep)s(endency)j(graph)183 3881 y(either)42 b(consists)h(of)f FA(R)1043 3896 y Fz(1)1083 3881 y FE(-dep)s(endency)i (pairs)e(or)g(of)g FA(R)2239 3896 y Fz(2)2278 3881 y FE(-dep)s(endency)j(pairs)d(only)-8 b(.)41 b(So)i(in)183 4001 y(our)29 b(example,)f(ev)m(ery)j(cycle)f(either)f(con)m(tains)g (just)g Ft(ADD)q FE(-)g(or)f(just)i Ft(WEIGHT)o FE(-dep)s(endency)183 4122 y(pairs.)i(Th)m(us,)i(w)m(e)g(obtain)d(the)i(follo)m(wing)d (corollary)-8 b(.)2142 4086 y Fy(yy)183 4315 y FK(Cor)n(ollar)h(y)39 b(6.1)e(\(Innermost)i(Termina)-7 b(tion)39 b(f)n(or)f(Hierar)n(chical)g (Combina)-7 b(tions\):)183 4436 y FC(L)i(et)35 b FA(R)g FC(b)-5 b(e)35 b(the)g(hier)-5 b(ar)g(chic)g(al)33 b(c)-5 b(ombination)34 b(of)g FA(R)2022 4451 y Fz(1)2096 4436 y FC(and)g FA(R)2370 4451 y Fz(2)2409 4436 y FC(.)219 4593 y(\(a\))48 b FA(R)32 b FC(is)f(innermost)g(terminating)g(if)g(and) f(only)i(if)f FA(R)2259 4608 y Fz(1)2330 4593 y FC(is)g(innermost)g (terminating)g(and)397 4714 y(ther)-5 b(e)35 b(exists)g(no)f (in\014nite)g(innermost)g FA(R)q FC(-chain)g(of)g FA(R)2427 4729 y Fz(2)2466 4714 y FC(-dep)-5 b(endency)34 b(p)-5 b(airs.)224 4867 y(\(b\))48 b FA(R)e FC(is)f(innermost)g(terminating)g (if)g FA(R)1844 4882 y Fz(1)1929 4867 y FC(is)g(innermost)g (terminating)g(and)g(if)g(ther)-5 b(e)397 4987 y(exists)34 b(an)g(ar)-5 b(gument)34 b(\014ltering)g FB(\031)k FC(and)33 b(a)h(r)-5 b(e)g(duction)34 b(p)-5 b(air)34 b FE(\()p Fu(\045)p FB(;)17 b FA(\037)p FE(\))35 b FC(such)f(that)g(for)g(al)5 b(l)397 5108 y(dep)-5 b(endency)34 b(p)-5 b(airs)34 b FB(s)28 b FA(!)f FB(t)35 b FC(of)g FA(R)1623 5123 y Fz(2)267 5238 y Fx(yy)332 5268 y Fw(Of)d(course,)f(an)h(ob)n(vious)e (re\014nemen)n(t)i(of)g(Corollary)d(6.1)j(\(b\))g(is)g(to)g(regard)e (the)j(di\013eren)n(t)f(cycles)f(of)183 5367 y Fi(R)253 5379 y FI(2)290 5367 y Fw(-dep)r(endency)c(pairs)e(in)h Fi(R)p Fw('s)h(\(estimated\))g(innermost)e(dep)r(endency)i(graph)e (separately)-7 b(.)25 b(Moreo)n(v)n(er,)e(a)183 5467 y(v)-5 b(arian)n(t)19 b(of)h(Corollary)e(6.1)h(also)g(holds)h(for)g Fi(C)1563 5479 y Fx(E)1608 5467 y Fw(-termination)f(instead)h(of)h (innermost)e(termination)h([Urbain,)183 5567 y(2001].)p eop %%Page: 34 34 34 33 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(34)481 390 y FA(\017)49 b FB(\031)t FE(\()p FB(l)r FE(\))27 b Fu(\045)i FB(\031)t FE(\()p FB(r)s FE(\))34 b FC(for)h(al)5 b(l)34 b(rules)h FB(l)19 b FA(!)d FB(r)37 b FC(in)e FA(U)2055 405 y Fy(R)2120 390 y FE(\()p FB(t)p FE(\))g FC(and)481 531 y FA(\017)49 b FB(\031)t FE(\()p FB(s)p FE(\))27 b FA(\037)h FB(\031)t FE(\()p FB(t)p FE(\))p FC(.)183 760 y(Pr)-5 b(o)g(of:)48 b FE(The)h(corollary)e(is)g (a)h(direct)g(consequence)k(of)47 b(Theorems)i(5.4)f(and)h(5.6,)e (since)183 880 y(ev)m(ery)d(cycle)f(consists)g(of)f FA(R)1276 895 y Fz(1)1315 880 y FE(-)g(or)g(of)g FA(R)1725 895 y Fz(2)1764 880 y FE(-dep)s(endency)j(pairs)d(only)g(and)g(since)h(for) f(an)m(y)183 1000 y(dep)s(endency)31 b(pair)e FB(s)e FA(!)h FB(t)h FE(of)g FA(R)1365 1015 y Fz(1)1434 1000 y FE(the)h(only)e(rules)i(that)f(can)g(b)s(e)h(used)g(to)f(reduce)i(a)e (normal)183 1121 y(instan)m(tiation)h(of)i FB(t)h FE(are)f(the)h(rules) g(from)e FA(R)1813 1136 y Fz(1)1885 1121 y FE(\(i.e.,)h FA(U)2169 1136 y Fy(R)2234 1121 y FE(\()p FB(t)p FE(\))27 b FA(\022)i(R)2562 1136 y Fz(1)2601 1121 y FE(\).)830 b Fs(2)280 1349 y FE(\(Innermost\))27 b(termination)e(of)h(the)i Ft(add)o FE(-system)g(\()p FA(R)2227 1364 y Fz(1)2267 1349 y FE(\))f(is)f(easily)g(pro)m(v)m(ed)j(\(e.g.,)e(b)m(y)h(LPO)183 1469 y(with)40 b(the)h(precedence)i Ft(add)e FB(>)17 b FE(:)41 b(and)f Ft(add)h FB(>)h Ft(s)p FE(\).)f(F)-8 b(or)40 b(the)h Ft(w)m(eight)q FE(-subsystem)h(\()p FA(R)3341 1484 y Fz(2)3380 1469 y FE(\))f(w)m(e)183 1590 y(obtain)d(the)h(follo)m (wing)e(constrain)m(ts.)j(\(Note)f(that)g Fe(WEIGHT)p FF(\()p Ff(:)15 b(:)g(:)q FF(\))26 b Fg(!)f Fe(ADD)o FF(\()p Ff(:)15 b(:)g(:)r FF(\))39 b FE(is)g(no)g(de-)183 1710 y(p)s(endency)34 b(pair)e(of)g FA(R)1012 1725 y Fz(2)1052 1710 y FE(,)g(since)h Ft(add)28 b FA(62)g(D)1698 1725 y Fz(2)1738 1710 y FE(.\))599 1930 y FB(\031)t FE(\()p Ft(add)o FE(\()p Ft(s)q FE(\()p FB(n)p FE(\))f(:)h FB(x;)17 b(m)28 b FE(:)g FB(y)t FE(\)\))82 b Fu(\045)i FB(\031)t FE(\()p Ft(add)o FE(\()p FB(n)28 b FE(:)g FB(x;)17 b Ft(s)p FE(\()p FB(m)p FE(\))28 b(:)g FB(y)t FE(\)\))889 2076 y FB(\031)t FE(\()p Ft(add)p FE(\()p Ft(0)f FE(:)h FB(x;)17 b(y)t FE(\)\))82 b Fu(\045)i FB(\031)t FE(\()p Ft(add)o FE(\()p FB(x;)17 b(y)t FE(\)\))980 2221 y FB(\031)t FE(\()p Ft(add)o FE(\()p Ft(nil)p FB(;)g(y)t FE(\)\))82 b Fu(\045)i FB(\031)t FE(\()p FB(y)t FE(\))577 2366 y FB(\031)t FE(\()p Ft(WEIGHT)o FE(\()p FB(n)28 b FE(:)g FB(m)g FE(:)g FB(x)p FE(\)\))83 b FA(\037)h FB(\031)t FE(\()p Ft(WEIGHT)o FE(\()p Ft(add)o FE(\()p FB(n)28 b FE(:)g FB(m)g FE(:)g FB(x;)17 b Ft(0)27 b FE(:)h FB(x)p FE(\)\)\))183 2586 y(By)k(c)m(ho)s(osing)g(the)h(argumen)m(t)e (\014ltering)g FB(\031)t FE(\()p Ft(add)p FE(\))c(=)h FB(\031)t FE(\(:\))f(=)h([2],)k(the)g(inequalities)f(are)h(also)183 2707 y(satis\014ed)h(b)m(y)g(LPO,)g(but)g(no)m(w)g(w)m(e)h(ha)m(v)m(e)g (to)e(use)h(the)g(precedence)j(:)17 b FB(>)27 b Ft(add)p FE(.)280 2827 y(In)d(this)g(w)m(a)m(y)-8 b(,)24 b(innermost)f (termination)f(of)h(this)g(non-simply)f(terminating)f(example)j(can)183 2947 y(b)s(e)37 b(pro)m(v)m(ed)i(automatically)-8 b(.)33 b(Moreo)m(v)m(er,)39 b(as)e(the)h(system)g(is)f(non-o)m(v)m(erlapping,) f(this)h(also)183 3068 y(pro)m(v)m(es)e(its)e(termination.)e(A)j (criterion)e(lik)m(e)h(Corollary)f(6.1)h(can)h(also)e(b)s(e)i(form)m (ulated)e(for)183 3188 y(termination)j(instead)j(of)f(innermost)g (termination,)e(b)s(ecause)k(in)e(the)i(termination)c(case)183 3308 y(there)41 b(cannot)f(b)s(e)h(a)f(cycle)h(consisting)e(of)h(dep)s (endency)j(pairs)c(from)h(b)s(oth)g FA(R)3169 3323 y Fz(1)3249 3308 y FE(and)g FA(R)3531 3323 y Fz(2)183 3429 y FE(either.)24 b(But)h(in)f(con)m(trast)h(to)f(the)h(innermost)f (termination)e(case,)k(rules)e(of)h FA(R)3020 3444 y Fz(2)3084 3429 y FE(can)g(b)s(e)g(used)183 3549 y(to)40 b(reduce)i(instan)m(tiated)d(righ)m(t-hand)h(sides)h(of)f FA(R)2108 3564 y Fz(1)2147 3549 y FE(-dep)s(endency)j(pairs)c(\(as)i(w) m(e)h(cannot)183 3670 y(restrict)34 b(ourselv)m(es)h(to)f(normal)e (substitutions)h(then\).)i(Hence,)g(to)f(pro)m(v)m(e)h(the)g(absence)g (of)183 3790 y(in\014nite)28 b FA(R)594 3805 y Fz(1)634 3790 y FE(-c)m(hains)h(w)m(e)h(ha)m(v)m(e)h(to)d(use)j(a)e (quasi-ordering)f(where)i(the)g(rules)f(of)g FA(R)3179 3805 y Fz(2)3248 3790 y FE(are)g(also)183 3910 y(w)m(eakly)24 b(decreasing.)h(Therefore,)g(the)f(constrain)m(ts)g(for)f(the)h (termination)e(pro)s(of)h(of)g(the)h Ft(add)183 4031 y FE(and)35 b Ft(w)m(eight)q FE(-example)f(\(according)h(to)g(Section)g (3\))g(are)h(not)f(satis\014ed)h(b)m(y)g(an)m(y)g(reduction)183 4151 y(pair)21 b(with)g(a)h(quasi-simpli\014cation)c(ordering)j (amenable)g(to)h(automation)e([Arts)i(and)g(Giesl,)183 4271 y(2001],)27 b(whereas)j(the)e(constrain)m(ts)h(for)e FC(innermost)37 b FE(termination)26 b(are)i(ful\014lled)e(b)m(y)j(suc)m (h)h(an)183 4392 y(ordering.)c(Hence,)i(for)e(non-o)m(v)m(erlapping)f (systems,)j(it)e(is)g(alw)m(a)m(ys)h(adv)-5 b(an)m(tageous)27 b(to)f(v)m(erify)183 4512 y(termination)k(b)m(y)j(pro)m(ving)g FC(innermost)41 b FE(termination)30 b(only)-8 b(.)183 4732 y FC(6.1.2.)49 b(Splitting)34 b(into)h(Subsystems)183 4902 y FE(The)d(mo)s(dularit)m(y)c(results)k(for)e(innermost)h (termination)d(presen)m(ted)34 b(so)d(far)f(w)m(ere)j(all)c(used)183 5023 y(in)h(the)h(con)m(text)h(of)f(dep)s(endency)i(pairs.)e(Ho)m(w)m (ev)m(er)i(as)e(already)f(men)m(tioned,)h(the)g(classical)183 5143 y(approac)m(h)d(to)g(mo)s(dularit)m(y)e(is)i(to)g(split)f(a)g(TRS) i(in)m(to)e(subsystems)k(and)d(to)g(pro)m(v)m(e)i(their)d(\(in-)183 5264 y(nermost\))34 b(termination)e(separately)-8 b(.)35 b(The)g(follo)m(wing)d(corollary)g(of)i(Theorem)h(5.4)f(sho)m(ws)183 5384 y(that)e(the)h(consideration)f(of)g(cycles)i(in)e(the)h(innermost) e(dep)s(endency)36 b(graph)c(can)h(also)f(b)s(e)183 5504 y(used)e(to)e(decomp)s(ose)i(a)f(TRS)g(in)m(to)f(mo)s(dular)f (subsystems.)32 b(\(Similarly)-8 b(,)24 b(the)30 b(cycles)g(of)e(the)p eop %%Page: 35 35 35 34 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(35)183 390 y FE(estimated)30 b(innermost)g(dep)s(endency)j(graph) e(ma)m(y)f(b)s(e)h(used)h(as)f(w)m(ell)f(for)h(this)f(decomp)s(osi-)183 511 y(tion.\))280 631 y(In)e(the)g(follo)m(wing,)d(let)h FA(O)s FE(\()p FA(P)8 b FE(\))28 b(denote)g(the)g FC(origin)34 b FE(of)27 b(the)h(dep)s(endency)i(pairs)d(in)f FA(P)8 b FE(,)28 b(i.e.,)183 751 y FA(O)s FE(\()p FA(P)8 b FE(\))39 b(is)g(a)h(set)g(of)f(those)h(rules)f(where)i(the)f(dep)s(endency)i (pairs)c(of)h FA(P)48 b FE(stem)40 b(from.)e(If)h(a)183 872 y(dep)s(endency)d(pair)d(of)g FA(P)43 b FE(ma)m(y)33 b(stem)h(from)f FC(sever)-5 b(al)43 b FE(rules,)34 b(then)h(it)d(is)i (su\016cien)m(t)h(if)e FA(O)s FE(\()p FA(P)8 b FE(\))183 992 y(just)31 b(con)m(tains)f(one)h(of)g(them.)f(So)h(for)f(the)h (example)f(of)h(Section)f(5)h(w)m(e)g(ha)m(v)m(e)h FA(O)s FE(\()p FA(f)p FE(\(22\))p FA(g)p FE(\))27 b(=)183 1112 y FA(f)p Ft(f)6 b FE(\()p FB(x;)17 b Ft(c)p FE(\()p FB(x)p FE(\))p FB(;)g Ft(c)q FE(\()p FB(y)t FE(\)\))g FA(!)e Ft(f)6 b FE(\()p FB(y)t(;)17 b(y)t(;)g Ft(f)t FE(\()p FB(y)t(;)g(x;)g(y)t FE(\)\))p FA(g)j FE(and)i FA(O)s FE(\()p FA(f)p FE(\(23\))p FA(g)p FE(\))k(=)i FA(f)p Ft(f)6 b FE(\()p Ft(s)q FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t(;)g(z)t FE(\))g FA(!)e Ft(f)7 b FE(\()p FB(x;)17 b Ft(s)p FE(\()p Ft(c)p FE(\()p FB(y)t FE(\)\))p FB(;)g Ft(c)o FE(\()p FB(z)t FE(\)\))p FA(g)p FE(.)183 1301 y FK(Cor)n(ollar)-7 b(y)39 b(6.2)e(\(Modularity)i(f)n(or)f (Subsystems\):)48 b FC(L)-5 b(et)25 b FA(R)g FC(b)-5 b(e)25 b(a)g(TRS,)f(let)h FA(P)3430 1316 y Fz(1)3470 1301 y FB(;)17 b(:)g(:)g(:)f(;)h FA(P)3758 1316 y Fv(n)183 1422 y FC(b)-5 b(e)25 b(the)h(cycles)f(in)h(its)g(\(estimate)-5 b(d\))24 b(innermost)h(dep)-5 b(endency)25 b(gr)-5 b(aph,)25 b(and)g(let)h FA(R)3098 1437 y Fv(j)3161 1422 y FC(b)-5 b(e)25 b(subsys-)183 1542 y(tems)32 b(of)h FA(R)g FC(such)f(that)i FA(U)1121 1557 y Fy(R)1185 1542 y FE(\()p FA(P)1292 1557 y Fv(j)1329 1542 y FE(\))17 b FA([)h(O)s FE(\()p FA(P)1657 1557 y Fv(j)1694 1542 y FE(\))28 b FA(\022)g(R)1949 1557 y Fv(j)2019 1542 y FC(\(for)k(al)5 b(l)32 b FB(j)i FA(2)28 b(f)p FE(1)p FB(;)17 b Fv(:)11 b(:)g(:)p FB(;)17 b(n)p FA(g)p FC(\).)32 b(If)g FA(R)3181 1557 y Fz(1)3220 1542 y FB(;)17 b(:)g(:)g(:)f(;)h FA(R)3523 1557 y Fv(n)183 1663 y FC(ar)-5 b(e)34 b(innermost)g(terminating,)g(then)h FA(R)g FC(is)g(also)f(innermost)g(terminating.)183 1852 y(Pr)-5 b(o)g(of:)48 b FE(As)24 b FA(P)692 1867 y Fv(j)752 1852 y FE(is)f(a)g(cycle,)h(ev)m(ery)i(dep)s(endency)g(pair)c(from)g FA(P)2424 1867 y Fv(j)2484 1852 y FE(is)h(an)g FA(R)2784 1867 y Fv(j)2820 1852 y FE(-dep)s(endency)j(pair.)183 1972 y(\(In)36 b(order)g(to)f(see)i(this,)f(let)f FB(f)1311 1936 y Fv(])1342 1972 y FE(\()l FB(~)-45 b(s)p FE(\))28 b FA(!)f FB(g)1670 1936 y Fv(])1701 1972 y FE(\()1733 1954 y FB(~)1739 1972 y(t)p FE(\))36 b(b)s(e)g(an)g FA(R)p FE(-dep)s(endency)i(pair)d(in)g FA(P)3164 1987 y Fv(j)3200 1972 y FE(.)h(Here,)d FB(~)-45 b(s)183 2092 y FE(and)368 2074 y FB(~)374 2092 y(t)34 b FE(denote)h FC(tuples)42 b FE(of)34 b(terms)g FB(s)1477 2107 y Fz(1)1516 2092 y FB(;)17 b(:)g(:)g(:)f(;)h(s)1781 2107 y Fv(n)1862 2092 y FE(and)34 b FB(t)2088 2107 y Fz(1)2127 2092 y FB(;)17 b(:)g(:)g(:)f(;)h(t)2381 2107 y Fv(m)2448 2092 y FE(,)34 b(resp)s(ectiv)m(ely)-8 b(.)35 b(Clearly)-8 b(,)33 b FB(g)k FE(is)183 2213 y(a)g(de\014ned)h(sym)m(b)s(ol)f(of)f FA(R)1148 2228 y Fv(j)1221 2213 y FE(b)s(ecause)j(there)f(is)e(also)h (a)g(dep)s(endency)i(pair)d FB(g)3021 2177 y Fv(])3052 2213 y FE(\()l FB(~)-45 b(v)t FE(\))28 b FA(!)f FB(h)3390 2177 y Fv(])3422 2213 y FE(\()12 b FB(~)-61 b(w)r FE(\))183 2333 y(in)42 b FA(P)376 2348 y Fv(j)412 2333 y FE(.)h(Hence,)i(since)e FB(g)j FE(is)c(a)h(de\014ned)h(sym)m(b)s(ol)e(of)g FA(R)2247 2348 y Fv(j)2284 2333 y FE(,)h FB(f)2413 2297 y Fv(])2444 2333 y FE(\()l FB(~)-45 b(s)p FE(\))28 b FA(!)f FB(g)2772 2297 y Fv(])2803 2333 y FE(\()2835 2315 y FB(~)2841 2333 y(t)p FE(\))43 b(is)f(also)g(an)g FA(R)3501 2348 y Fv(j)3537 2333 y FE(-)183 2454 y(dep)s(endency)36 b(pair.\))d(Th)m(us,)i(ev)m (ery)h(innermost)c FA(R)p FE(-c)m(hain)i(of)f(dep)s(endency)k(pairs)c (from)f FA(P)3533 2469 y Fv(j)183 2574 y FE(is)47 b(also)g(an)h (innermost)f FA(R)1214 2589 y Fv(j)1250 2574 y FE(-c)m(hain.)h(No)m(w)h (the)f(corollary)e(is)i(a)f(direct)h(consequence)j(of)183 2694 y(Theorem)33 b(5.4.)2749 b Fs(2)280 2883 y FE(F)-8 b(or)31 b(instance,)g(in)g(the)h(example)e(of)h(Section)g(5)g(w)m(e)i (only)e(ha)m(v)m(e)h(t)m(w)m(o)g(cycles,)h(viz.)e FA(f)p FE(\(22\))p FA(g)183 3004 y FE(and)36 b FA(f)p FE(\(23\))p FA(g)p FE(.)g(As)i(these)f(dep)s(endency)j(pairs)c(ha)m(v)m(e)i(no)f (de\014ned)h(sym)m(b)s(ols)e(in)g(their)g(righ)m(t-)183 3124 y(hand)f(sides,)g(their)f(sets)i(of)f(usable)g(rules)f(are)h(empt) m(y)-8 b(.)35 b(Hence,)i(to)d(pro)m(v)m(e)i(innermost)e(ter-)183 3244 y(mination)24 b(of)i(the)i(whole)f(system,)h(b)m(y)f(Corollary)f (6.2)g(it)g(su\016ces)j(to)e(pro)m(v)m(e)h(innermost)e(ter-)183 3365 y(mination)36 b(of)j(the)h(t)m(w)m(o)g(one-rule)e(subsystems)k Ft(f)6 b FE(\()p FB(x;)17 b Ft(c)q FE(\()p FB(x)p FE(\))p FB(;)g Ft(c)p FE(\()p FB(y)t FE(\)\))g FA(!)e Ft(f)6 b FE(\()p FB(y)t(;)17 b(y)t(;)g Ft(f)5 b FE(\()p FB(y)t(;)17 b(x;)g(y)t FE(\)\))37 b(and)183 3485 y Ft(f)6 b FE(\()p Ft(s)q FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t(;)g(z)t FE(\))g FA(!)e Ft(f)7 b FE(\()p FB(x;)17 b Ft(s)p FE(\()p Ft(c)p FE(\()p FB(y)t FE(\)\))p FB(;)g Ft(c)o FE(\()p FB(z)t FE(\)\).)280 3606 y(In)38 b(fact,)f(b)s(oth)f(subsystems)k(are)d(ev)m (en)h(terminating)d(as)i(can)h(easily)e(b)s(e)h(pro)m(v)m(ed)i(auto-) 183 3726 y(matically)-8 b(.)27 b(F)-8 b(or)29 b(the)i(\014rst)g(system) g(one)g(can)f(use)h(a)f(p)s(olynomial)d(in)m(terpretation)i(mapping)183 3846 y Ft(f)6 b FE(\()p FB(x;)17 b(y)t(;)g(z)t FE(\))34 b(to)f FB(x)23 b FE(+)g FB(y)j FE(+)d FB(z)39 b FE(and)34 b Ft(c)p FE(\()p FB(x)p FE(\))g(to)g(5)p FB(x)23 b FE(+)g(1)33 b([Lankford,)h(1979].)f(Metho)s(ds)i(for)e(the)h(au-)183 3967 y(tomated)28 b(generation)h(of)g(p)s(olynomial)d(orderings)j(ha)m (v)m(e)i(for)e(instance)h(b)s(een)g(dev)m(elop)s(ed)h(in)183 4087 y([Stein)m(bac)m(h,)h(1994,)e(Giesl,)f(1995].)i(F)-8 b(or)30 b(the)h(second)h(system)g(one)g(can)f(use)h(LPO)f(with)f(the) 183 4207 y(precedence)35 b Ft(f)f FB(>)28 b Ft(s)33 b FE(and)f Ft(f)i FB(>)28 b Ft(c)p FE(.)280 4328 y(Hence,)34 b(the)g(mo)s(dularit)m(y)c(criterion)h(of)i(Corollary)e(6.2)h(allo)m (ws)g(the)h(use)g(of)g(w)m(ell-kno)m(wn)183 4448 y(simpli\014cation)21 b(orderings)k(for)g(innermost)f(termination)e(pro)s(ofs)j(of)g (non-terminating)d(sys-)183 4569 y(tems,)38 b(b)s(ecause)h(it)d(guaran) m(tees)j(that)f(innermost)f(termination)e(of)i(the)h(t)m(w)m(o)h (simply)d(ter-)183 4689 y(minating)26 b(subsystems)32 b(is)d(su\016cien)m(t)h(for)f(innermost)g(termination)e(of)i(the)g (original)d(TRS.)280 4809 y(A)35 b(similar)c(splitting)h(is)i(also)g(p) s(ossible)g(for)g(the)h(example)f(in)g(Section)g(3.)g(Ev)m(en)i(b)s (etter,)183 4930 y(if)31 b(w)m(e)j(mo)s(dify)d(the)i(TRS)g(in)m(to)e(a) i(non-o)m(v)m(erlapping)e(one)1272 5115 y Ft(f)6 b FE(\()p FB(x;)17 b Ft(c)q FE(\()p FB(y)t FE(\)\))98 b FA(!)i Ft(f)6 b FE(\()p FB(x;)17 b Ft(s)q FE(\()p Ft(f)6 b FE(\()p FB(y)t(;)17 b(y)t FE(\)\)\))1164 5261 y Ft(f)7 b FE(\()p Ft(s)p FE(\()p FB(x)p FE(\))p FB(;)17 b Ft(s)q FE(\()p FB(y)t FE(\)\))98 b FA(!)i Ft(f)6 b FE(\()p FB(x;)17 b Ft(s)q FE(\()p Ft(c)p FE(\()p Ft(s)p FE(\()p FB(y)t FE(\)\)\)\))p FB(;)183 5446 y FE(then)31 b(Corollary)d(6.2)i(allo)m(ws) g(to)g(conclude)h(termination)d(of)i(the)g(whole)h(system)g(from)e (ter-)183 5567 y(mination)h(of)j(the)h(t)m(w)m(o)g(one-rule)f (subsystems.)j(Innermost)d(termination)e(of)i(the)h(original)p eop %%Page: 36 36 36 35 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(36)183 390 y FE(example)24 b(resp.)i(termination)d(of)i(the)g(ab) s(o)m(v)m(e)h(mo)s(di\014ed)e(example)g(can)i(b)s(e)f(pro)m(v)m(ed)i(b) m(y)f(LPO,)183 511 y(but)34 b(for)f(the)i(\014rst)f(rule)g(one)g(needs) i(the)e(precedence)j Ft(c)31 b FB(>)f Ft(s)k FE(and)g Ft(c)d FB(>)f Ft(f)6 b FE(,)34 b(whereas)i(for)d(the)183 631 y(second)h(rule)e(the)h(precedence)i Ft(f)f FB(>)28 b Ft(s)k FE(and)h Ft(f)h FB(>)28 b Ft(c)33 b FE(is)f(required.)280 751 y(Note)40 b(that)f(the)g(rev)m(erse)j(direction)c(of)g(the)i (corollary)e(do)s(es)h(not)g(hold.)g(Consider)g(the)183 872 y(TRS)30 b(\(24\))e(-)i(\(27\))e(from)h(the)h(end)g(of)f(Section)g (5)h(again.)e(The)i(only)f(cycle)i(of)e(its)g(innermost)183 992 y(dep)s(endency)24 b(graph)d(is)h FA(f)p Ft(F)p FE(\()p Ft(a)p FE(\()p FB(x)p FE(\))p FB(;)17 b(y)t FE(\))26 b FA(!)i Ft(G)p FE(\()p FB(x;)17 b(y)t FE(\))o FB(;)g Ft(G)p FE(\()p FB(x;)g(y)t FE(\))26 b FA(!)i Ft(H)p FE(\()p FB(x;)17 b(y)t FE(\))o FB(;)g Ft(H)p FE(\()p Ft(0)p FB(;)g(y)t FE(\))26 b FA(!)i Ft(F)p FE(\()p FB(y)t(;)17 b(y)t FE(\))n FA(g)p FE(.)183 1112 y(Since)33 b(this)h(cycle)g(do)s(es)g(not)f(ha)m (v)m(e)i(an)m(y)f(usable)g(rules,)g(Corollary)e(6.2)h(states)h(that)g (inner-)183 1233 y(most)c(termination)e(of)i(the)i(subsystem)g (consisting)e(of)h(the)g(\014rst)g(three)h(rules)e(is)h(su\016cien)m(t) 183 1353 y(for)41 b(innermost)g(termination)f(of)h(the)h(whole)g(TRS.)h (Ho)m(w)m(ev)m(er,)h(the)e(con)m(v)m(erse)j(do)s(es)d(not)183 1474 y(hold,)36 b(since)i(the)g(whole)g(system)g(is)f(innermost)g (terminating,)e(whereas)k(the)f(subsystem)183 1594 y(consisting)29 b(of)g(the)i(\014rst)f(three)h(rules)f(is)g(not.)f(\(The)i(term)f Ft(f)6 b FE(\()p Ft(a)p FE(\()p Ft(0)p FE(\))p FB(;)17 b Ft(a)o FE(\()p Ft(0)p FE(\)\))30 b(starts)g(an)g(in\014nite)183 1714 y(innermost)h(reduction.\))183 1977 y Fq(6.2.)53 b(Comparison)34 b(with)g(Related)h(W)-9 b(ork)183 2147 y FE(No)m(w)32 b(w)m(e)h(sho)m(w)g(that)f(in)f(the)h(case)h(of)e (\014nite)h(TRSs,)h(existing)e(mo)s(dularit)m(y)f(results)i(for)f(in-) 183 2267 y(nermost)i(termination)d(are)j(obtained)g(as)g(easy)h (consequences)j(of)32 b(our)h(criteria)f(and)h(that)183 2388 y(our)38 b(criteria)e(extend)k(previously)f(dev)m(elop)s(ed)g (results.)g(Section)f(6.2.1)f(fo)s(cuses)j(on)e(com-)183 2508 y(p)s(osable)33 b(TRSs)h(and)g(Section)g(6.2.2)f(giv)m(es)h(a)f (comparison)g(with)g(results)h(on)f(hierarc)m(hical)183 2628 y(com)m(binations.)183 2848 y FC(6.2.1.)49 b(Shar)-5 b(e)g(d)34 b(Constructors)g(and)h(Comp)-5 b(osable)33 b(R)-5 b(ewrite)35 b(Systems)183 3019 y FE(By)j(the)h(framew)m(ork)f (of)g(the)g(previous)h(sections)f(w)m(e)i(can)e(easily)f(pro)m(v)m(e)j (that)e(innermost)183 3139 y(termination)32 b(is)i(mo)s(dular)f(for)i (comp)s(osable)e(TRSs)j([Ohlebusc)m(h,)g(1995])f(and)g(hence)h(also)183 3259 y(for)f(TRSs)h(with)g(disjoin)m(t)e(sets)j(of)e(de\014ned)i(sym)m (b)s(ols)f(and)f(shared)i(constructors)g([Gram-)183 3380 y(lic)m(h,)i(1995].)g(In)i(fact,)f(Corollary)f(6.2)g(immediately)e (implies)2512 3344 y Fy(zz)2617 3380 y FE(the)k(follo)m(wing)c(result)j (of)183 3500 y(Ohlebusc)m(h)33 b([1995].)183 3729 y FK(Theorem)38 b(6.3)f(\(Modularity)i(f)n(or)f(Composable)g(TRSs\):)48 b FC(L)-5 b(et)43 b FA(R)3041 3744 y Fz(1)3123 3729 y FC(and)e FA(R)3404 3744 y Fz(2)3485 3729 y FC(b)-5 b(e)183 3849 y(c)g(omp)g(osable)27 b(TRSs.)i(If)g FA(R)1156 3864 y Fz(1)1225 3849 y FC(and)g FA(R)1493 3864 y Fz(2)1562 3849 y FC(ar)-5 b(e)29 b(innermost)f(terminating,)h(then)g FA(R)3028 3864 y Fz(1)3077 3849 y FA([)10 b(R)3238 3864 y Fz(2)3306 3849 y FC(is)30 b(also)183 3969 y(innermost)k(terminating.) 183 4198 y(Pr)-5 b(o)g(of:)48 b FE(Let)29 b FB(f)718 4161 y Fv(])749 4198 y FE(\()l FB(~)-45 b(s)p FE(\))28 b FA(!)f FB(g)1077 4161 y Fv(])1108 4198 y FE(\()1140 4179 y FB(~)1146 4198 y(t)p FE(\))i(b)s(e)g(a)g(dep)s(endency)i(pair)d (of)g FA(R)2370 4213 y Fz(1)2425 4198 y FA([)15 b(R)2590 4213 y Fz(2)2630 4198 y FE(.)28 b(If)h FB(f)39 b FA(2)28 b(D)3039 4213 y Fz(1)3079 4198 y FE(,)h(then)g(there)183 4318 y(exists)37 b(a)e(rule)h FB(f)11 b FE(\()830 4300 y FB(~)836 4318 y(t)p FE(\))17 b FA(!)f FB(C)7 b FE([)p FB(g)t FE(\()1229 4300 y FB(~)1235 4318 y(t)o FE(\)])36 b(in)g FA(R)1572 4333 y Fz(1)1612 4318 y FE(.)g(\(This)g(rule)g(cannot) g(b)s(e)h(from)e FA(R)2916 4333 y Fz(2)2980 4318 y FA(n)24 b(R)3139 4333 y Fz(1)3178 4318 y FE(,)37 b(b)s(ecause)183 4438 y FA(R)267 4453 y Fz(1)337 4438 y FE(and)31 b FA(R)610 4453 y Fz(2)680 4438 y FE(are)g(comp)s(osable.\))f(Hence,)i FB(g)f FA(2)d(D)1995 4453 y Fz(1)2034 4438 y FE(,)j(b)s(ecause)i (constructors)f(of)e FA(R)3198 4453 y Fz(1)3268 4438 y FE(are)h(not)183 4559 y(de\014ned)41 b(sym)m(b)s(ols)e(of)g FA(R)1108 4574 y Fz(2)1147 4559 y FE(.)h(Similarly)-8 b(,)35 b FB(f)51 b FA(2)40 b(D)1939 4574 y Fz(2)2018 4559 y FE(implies)d FB(g)43 b FA(2)d(D)2631 4574 y Fz(2)2671 4559 y FE(.)f(So)h(an)m(y)g(dep)s(endency)183 4679 y(pair)31 b(of)h FA(R)578 4694 y Fz(1)640 4679 y FA([)23 b(R)813 4694 y Fz(2)885 4679 y FE(is)32 b(an)h FA(R)1203 4694 y Fz(1)1243 4679 y FE(-dep)s(endency)h(pair)e(or)g(an)h FA(R)2346 4694 y Fz(2)2385 4679 y FE(-dep)s(endency)i(pair.)280 4800 y(Moreo)m(v)m(er,)j(there)e(can)g(only)g(b)s(e)g(an)f(arc)h(from)f FB(f)2124 4763 y Fv(])2155 4800 y FE(\()l FB(~)-45 b(s)p FE(\))28 b FA(!)f FB(g)2483 4763 y Fv(])2514 4800 y FE(\()2546 4781 y FB(~)2552 4800 y(t)p FE(\))36 b(to)f(a)h(dep)s(endency)i(pair) 183 4920 y(of)k(the)i(form)e FB(g)775 4884 y Fv(])806 4920 y FE(\()l FB(~)-45 b(v)t FE(\))27 b FA(!)g FB(h)1143 4884 y Fv(])1175 4920 y FE(\()12 b FB(~)-61 b(w)s FE(\).)43 b(Hence,)h(if)e FB(f)1880 4884 y Fv(])1912 4920 y FE(\()l FB(~)-45 b(s)p FE(\))27 b FA(!)h FB(g)2240 4884 y Fv(])2270 4920 y FE(\()2302 4902 y FB(~)2308 4920 y(t)q FE(\))43 b(is)f(an)i FA(R)2764 4935 y Fv(j)2801 4920 y FE(-dep)s(endency)h (pair,)183 5040 y(then)d FB(g)k FA(2)d(D)695 5055 y Fv(j)773 5040 y FE(and)e(therefore,)h FB(g)1467 5004 y Fv(])1498 5040 y FE(\()l FB(~)-45 b(v)t FE(\))28 b FA(!)f FB(h)1836 5004 y Fv(])1868 5040 y FE(\()12 b FB(~)-61 b(w)r FE(\))41 b(is)g(also)g(an)g FA(R)2597 5055 y Fv(j)2634 5040 y FE(-dep)s(endency)i(pair)d(\(for)183 5161 y FB(j)47 b FA(2)41 b(f)p FE(1)p FB(;)17 b FE(2)p FA(g)p FE(\).)39 b(So)h(ev)m(ery)i(cycle)f FA(P)49 b FE(in)40 b(the)g(innermost)g(dep)s (endency)j(graph)d(of)g FA(R)3285 5176 y Fz(1)3352 5161 y FA([)28 b(R)3531 5176 y Fz(2)183 5281 y FE(either)k(consists)h(of)f FA(R)1014 5296 y Fz(1)1053 5281 y FE(-dep)s(endency)j(pairs)d(or)g(of)g FA(R)2171 5296 y Fz(2)2210 5281 y FE(-dep)s(endency)j(pairs)d(only)-8 b(.)267 5420 y Fx(zz)332 5450 y Fw(A)30 b(direct)g(pro)r(of)f(of)g (Theorem)g(6.3)g(is)h(not)f(to)r(o)h(di\016cult)g(either,)g(but)g(our)f (alternativ)n(e)g(pro)r(of)g(serv)n(es)183 5549 y(to)e(illustrate)g (the)h(connections)f(b)r(et)n(w)n(een)h(our)f(criteria)f(and)h (existing)h(mo)r(dularit)n(y)e(results.)p eop %%Page: 37 37 37 36 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(37)280 390 y FE(If)47 b(a)g(cycle)h FA(P)56 b FE(only)46 b(con)m(tains)h FA(R)1579 405 y Fz(1)1618 390 y FE(-dep)s(endency)j (pairs,)c(then)i FA(R)2798 405 y Fz(1)2885 390 y FE(is)f(a)g(sup)s (erset)h(of)183 511 y FA(U)245 526 y Fy(R)305 535 y Fm(1)340 526 y Fy([R)447 535 y Fm(2)486 511 y FE(\()p FA(P)8 b FE(\))t FA([)t(O)t FE(\()p FA(P)g FE(\),)24 b(as)g(the)g(de\014ned)h (sym)m(b)s(ols)f(of)f FA(R)2147 526 y Fz(2)2191 511 y FA(n)t(R)2329 526 y Fz(1)2392 511 y FE(do)h(not)g(o)s(ccur)g(as)g (constructors)183 631 y(in)39 b FA(R)389 646 y Fz(1)428 631 y FE(.)h(Similarly)-8 b(,)36 b(for)k(a)g(cycle)h FA(P)48 b FE(of)40 b FA(R)1751 646 y Fz(2)1791 631 y FE(-dep)s(endency)i(pairs,)e(w)m(e)h(ha)m(v)m(e)h FA(U)3082 646 y Fy(R)3142 655 y Fm(1)3177 646 y Fy([R)3284 655 y Fm(2)3323 631 y FE(\()p FA(P)8 b FE(\))28 b FA([)183 751 y(O)s FE(\()p FA(P)8 b FE(\))40 b FA(\022)g(R)660 766 y Fz(2)699 751 y FE(.)g(Hence)h(b)m(y)g(Corollary)d(6.2,)h FA(R)1920 766 y Fz(1)1986 751 y FA([)27 b(R)2164 766 y Fz(2)2243 751 y FE(is)39 b(innermost)g(terminating)e(if)i FA(R)3531 766 y Fz(1)183 872 y FE(and)32 b FA(R)457 887 y Fz(2)529 872 y FE(are)g(innermost)g(terminating.)1821 b Fs(2)280 1100 y FE(Note)33 b(that)f(our)g(results)g(extend)i(mo)s (dularit)m(y)c(to)i(a)g(m)m(uc)m(h)h(larger)e(class)h(of)g(TRSs,)i (e.g.,)183 1220 y(they)f(also)e(allo)m(w)g(a)g(splitting)f(in)m(to)i (non-comp)s(osable)e(subsystems)35 b(whic)m(h)e(share)f(de\014ned)183 1341 y(sym)m(b)s(ols)g(as)h(demonstrated)g(in)e(Section)i(6.1.2.)183 1561 y FC(6.2.2.)49 b(Pr)-5 b(op)g(er)34 b(Extensions)183 1731 y FE(Krishna)f(Rao)g([1995])g(pro)m(v)m(ed)i(that)e(innermost)g (termination)e(is)i(mo)s(dular)f(for)h(\(general-)183 1851 y(ized\))28 b FC(pr)-5 b(op)g(er)39 b FE(extensions)30 b(whic)m(h)f(are)g(a)f(certain)h(kind)f(of)g(hierarc)m(hical)f(com)m (binations.)h(In)183 1972 y(this)i(section)g(w)m(e)h(sho)m(w)g(that)f (for)g(\014nite)g(TRSs)h(this)f(is)g(also)f(a)h(direct)g(consequence)j (of)d(our)183 2092 y(results.)280 2213 y(F)-8 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3615 y Fz(2)262 3682 y(2)329 3657 y FE(=)28 b FA(;)p FE(.)g(This)f(example)h(is)f(not)h (a)f(prop)s(er)h(extension,)h(b)s(ecause)g(there)f(is)f(a)h Ft(w)m(eight)q FE(-rule)f(in)183 3778 y(whic)m(h)j(the)h FA(D)705 3793 y Fz(1)744 3778 y FE(-sym)m(b)s(ol)e Ft(add)h FE(o)s(ccurs)h(b)s(elo)m(w)f(the)h FA(D)2104 3736 y Fz(1)2104 3802 y(2)2143 3778 y FE(-sym)m(b)s(ol)e Ft(w)m(eight)q FE(.)h(Th)m(us,)i(in)e(a)g(prop)s(er)183 3898 y(extension)35 b(functions)g(dep)s(ending)g(on)g FA(R)1732 3913 y Fz(1)1806 3898 y FE(are)g(nev)m(er)h(called)e(within)g(a)g(recursiv)m(e)i(call)d (of)183 4018 y FA(R)267 4033 y Fz(2)307 4018 y FE(-functions.)g(As)i (an)f(example)g(for)f(a)h(prop)s(er)h(extension)g(consider)f(the)h (TRSs)g FA(R)3339 4033 y Fz(1)3413 4018 y FE(and)183 4139 y FA(R)267 4154 y Fz(2)339 4139 y FE(from)c(the)i(end)h(of)e (Section)g(4.3)g(again,)g(where)h FA(R)2185 4154 y Fz(2)2257 4139 y FE(is)f(extended)i(b)m(y)g(the)f(rule)1079 4359 y Ft(avg)p FE(\()p FB(l)r FE(\))56 b FA(!)f Ft(quot)p 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b(rise)g(to)g(additional)e(dep)s(endency)32 b(pairs\).)c(In)i(fact,)f (its)g(innermost)g(termination)e(also)183 5181 y(follo)m(ws)33 b(directly)h(from)g(Corollary)f(6.1)h(\(b\),)h(since)g(the)g(original)c (TRS)k FA(R)2959 5196 y Fz(2)3034 5181 y FE(and)g(the)g Ft(avg)p FE(-)183 5301 y(rule)28 b(form)g(a)h(hierarc)m(hical)e(com)m (bination.)h(Corollaries)e(6.1)j(and)g(6.2)g(imply)e(the)j(follo)m (wing)183 5421 y(result)d(of)h([Krishna)f(Rao,)h(1995])f(whic)m(h)h(in) f(turn)h(ensures)i(that)e(the)g(union)f(of)h FA(R)3182 5436 y Fz(1)3249 5421 y FE(and)g(the)183 5542 y(extended)34 b(system)g FA(R)1005 5557 y Fz(2)1077 5542 y FE(in)e(our)g(example)g (is)h(innermost)e(terminating.)p eop %%Page: 38 38 38 37 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(38)183 390 y FK(Theorem)38 b(6.4)f(\(Modularity)i(f)n(or)f(Pr)n (oper)f(Extensions\):)48 b FC(L)-5 b(et)25 b FA(R)3076 405 y Fz(2)3140 390 y FC(b)-5 b(e)25 b(a)f(pr)-5 b(op)g(er)183 511 y(extension)34 b(of)g FA(R)813 526 y Fz(1)852 511 y FC(.)h(The)g(TRS)f FA(R)1432 526 y Fz(1)1493 511 y FA([)23 b(R)1666 526 y Fz(2)1741 511 y FC(is)35 b(innermost)f (terminating)g(if)h FA(R)3016 526 y Fz(1)3091 511 y FC(and)f FA(R)3365 526 y Fz(2)3439 511 y FC(ar)-5 b(e)183 631 y(innermost)34 b(terminating.)183 859 y(Pr)-5 b(o)g(of:)48 b FE(As)43 b(in)f(the)h(pro)s(of)f(of)g(Corollary)f(6.1,)i(since)g FA(R)2299 874 y Fz(1)2381 859 y FE(and)g FA(R)2666 874 y Fz(2)2748 859 y FE(form)e(a)h(hierarc)m(hical)183 980 y(com)m(bination,)25 b(ev)m(ery)30 b(cycle)e(in)f(the)i(innermost)e (dep)s(endency)j(graph)d(of)h FA(R)2961 995 y Fz(1)3013 980 y FA([)12 b(R)3176 995 y Fz(2)3243 980 y FE(consists)183 1100 y(solely)26 b(of)h FA(R)636 1115 y Fz(1)676 1100 y FE(-dep)s(endency)i(pairs)d(or)h(of)g FA(R)1771 1115 y Fz(2)1811 1100 y FE(-dep)s(endency)i(pairs.)d(If)h(a)g(cycle)h FA(P)36 b FE(consists)27 b(of)183 1220 y(dep)s(endency)33 b(pairs)d(of)g FA(R)1142 1235 y Fz(1)1182 1220 y FE(,)g(w)m(e)i(ha)m(v) m(e)g FA(U)1666 1235 y Fy(R)1726 1244 y Fm(1)1761 1235 y Fy([R)1868 1244 y Fm(2)1907 1220 y FE(\()p FA(P)8 b FE(\))18 b FA([)h(O)s FE(\()p FA(P)8 b FE(\))28 b FA(\022)g(R)2615 1235 y Fz(1)2655 1220 y FE(,)i(b)s(ecause)i(dep)s(endency)183 1341 y(pairs)g(of)g FA(R)617 1356 y Fz(1)689 1341 y FE(do)h(not)f(con)m (tain)g(an)m(y)h(de\014ned)h(sym)m(b)s(ols)f(of)f FA(R)2430 1356 y Fz(2)2470 1341 y FE(.)280 1461 y(Otherwise,)44 b(the)f(cycle)g FA(P)52 b FE(consists)43 b(of)g FA(R)1904 1476 y Fz(2)1943 1461 y FE(-dep)s(endency)i(pairs.)e(If)g FB(f)2961 1425 y Fv(])2992 1461 y FE(\()l FB(~)-45 b(s)p FE(\))27 b FA(!)h FB(g)3320 1425 y Fv(])3350 1461 y FE(\()3382 1443 y FB(~)3388 1461 y(t)q FE(\))42 b(is)183 1582 y(an)34 b FA(R)405 1597 y Fz(2)444 1582 y FE(-dep)s(endency)j(pair)d(in)g FA(P)8 b FE(,)36 b(then)f(there)h(exists)f(a)g(rule)f FB(f)11 b FE(\()l FB(~)-45 b(s)p FE(\))17 b FA(!)f FB(C)7 b FE([)p FB(g)t FE(\()2998 1563 y FB(~)3004 1582 y(t)o FE(\)])35 b(in)f FA(R)3339 1597 y Fz(2)3413 1582 y FE(and)183 1702 y FB(f)5 b(;)17 b(g)31 b FA(2)d(D)531 1717 y Fz(2)571 1702 y FE(.)k(In)h(addition,)e(w)m(e)j(ha)m(v)m(e)g FB(f)k Fa(\004)1674 1717 y Fv(d)1742 1702 y FB(g)e FE(and)d FB(g)e Fa(\004)2147 1717 y Fv(d)2216 1702 y FB(f)43 b FE(\(as)32 b FA(P)41 b FE(is)33 b(a)f(cycle\).)280 1822 y(If)41 b FB(g)k FA(2)e(D)666 1781 y Fz(2)666 1847 y(2)706 1822 y FE(,)e(then)g FB(f)52 b FE(also)40 b(b)s(elongs)g(to)h FA(D)1873 1781 y Fz(2)1873 1847 y(2)1913 1822 y FE(,)g(hence)h(no)f (de\014ned)h(sym)m(b)s(ol)e(of)h FA(D)3289 1837 y Fz(1)3357 1822 y FA([)28 b(D)3531 1781 y Fz(1)3531 1847 y(2)183 1943 y FE(o)s(ccurs)33 b(in)591 1924 y FB(~)597 1943 y(t)p FE(.)f(Otherwise,)h(if)e FB(g)g FA(2)d(D)1520 1901 y Fz(1)1520 1967 y(2)1559 1943 y FE(,)33 b(then)f(b)m(y)h(de\014nition) f(of)f(a)h(prop)s(er)g(extension)h(again)183 2063 y(all)c(de\014ned)j (sym)m(b)s(ols)f(in)1128 2045 y FB(~)1134 2063 y(t)h FE(are)f(from)e FA(D)1670 2021 y Fz(2)1670 2088 y(2)1709 2063 y FE(.)j(Th)m(us,)g(in)f(b)s(oth)f(cases,)j(all)c(de\014ned)k(sym) m(b)s(ols)d(of)183 2183 y FA(U)245 2198 y Fy(R)305 2207 y Fm(1)340 2198 y Fy([R)447 2207 y Fm(2)486 2183 y FE(\()p FB(g)575 2147 y Fv(])606 2183 y FE(\()638 2165 y FB(~)644 2183 y(t)p FE(\)\))i(b)s(elong)g(to)g FA(D)1298 2142 y Fz(2)1298 2208 y(2)1337 2183 y FE(.)h(Hence,)h FA(U)1776 2198 y Fy(R)1836 2207 y Fm(1)1871 2198 y Fy([R)1978 2207 y Fm(2)2017 2183 y FE(\()p FB(g)2106 2147 y Fv(])2137 2183 y FE(\()2169 2165 y FB(~)2175 2183 y(t)p FE(\)\))e(is)g(a)h (subsystem)h(of)e FA(R)3164 2198 y Fz(2)3203 2183 y FE(.)280 2304 y(So)d(for)g(an)m(y)g(cycle)h FA(P)37 b FE(of)29 b FA(R)1274 2319 y Fz(2)1314 2304 y FE(-dep)s(endency)i(pairs,)d(w)m(e) j(ha)m(v)m(e)f FA(U)2560 2319 y Fy(R)2620 2328 y Fm(1)2655 2319 y Fy([R)2762 2328 y Fm(2)2801 2304 y FE(\()p FA(P)8 b FE(\))15 b FA([)g(O)t FE(\()p FA(P)8 b FE(\))28 b FA(\022)g(R)3503 2319 y Fz(2)3543 2304 y FE(.)183 2424 y(Hence,)33 b(b)m(y)g(Corollary)d 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y Fz(1)1980 5008 y(2)2027 4983 y FA(n)7 b(D)2163 4941 y Fy(0)2212 4983 y FE(\(since)25 b FB(g)31 b FA(2)d(D)2733 4941 y Fy(0)2733 5008 y Fz(2)2772 4983 y FE(\).)e(By)f(the)h(de\014nition)183 5103 y(of)i(generalized)g (prop)s(er)h(extensions,)1585 5085 y FB(~)1591 5103 y(t)g FE(again)e(con)m(tains)i(no)g(sym)m(b)s(ols)f(of)h FA(D)2977 5062 y Fy(0)2977 5128 y Fz(1)3031 5103 y FA([)14 b(D)3191 5062 y Fz(1)3191 5128 y(2)3231 5103 y FE(,)29 b(i.e.,)f(all)183 5224 y(de\014ned)39 b(sym)m(b)s(ols)e(in)1014 5205 y FB(~)1020 5224 y(t)h FE(are)g(from)e FA(D)1576 5182 y Fz(2)1576 5248 y(2)1615 5224 y FE(.)i(Hence,)h(w)m(e)g(obtain)d FA(U)2521 5239 y Fy(R)2581 5248 y Fm(1)2616 5239 y Fy([R)2723 5248 y Fm(2)2762 5224 y FE(\()p FA(P)8 b FE(\))26 b FA([)g(O)s FE(\()p FA(P)8 b FE(\))37 b FA(\022)g(R)3503 5239 y Fz(2)3543 5224 y FE(.)183 5344 y(Therefore,)g(innermost)e(termination)f(of)i FA(R)1848 5359 y Fz(1)1924 5344 y FE(and)g FA(R)2201 5359 y Fz(2)2277 5344 y FE(implies)e(innermost)h(termination)183 5465 y(of)d FA(R)378 5480 y Fz(1)440 5465 y FA([)22 b(R)613 5480 y Fz(2)685 5465 y FE(b)m(y)33 b(Corollary)e(6.2.)2093 b Fs(2)p eop %%Page: 40 40 40 39 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(40)280 390 y FE(T)-8 b(o)37 b(summarize,)f(w)m(e)i(ha)m(v)m(e)h (sho)m(wn)f(that)f(our)g(results)g(\(in)g(particular,)e(Corollary)g (6.2\))183 511 y(directly)46 b(imply)e(sev)m(eral)j(mo)s(dularit)m(y)e (results)i(for)e(innermost)h(termination)e(from)h(the)183 631 y(literature.)40 b(On)j(the)f(other)h(hand,)f(our)g(mo)s(dularit)m (y)e(results)j(signi\014can)m(tly)e(extend)i(the)183 751 y(class)32 b(of)g(TRSs)g(where)i(innermost)d(termination)f(can)i(b) s(e)g(pro)m(v)m(ed)i(in)d(a)h(mo)s(dular)e(w)m(a)m(y)-8 b(.)34 b(In)183 872 y(other)d(w)m(ords,)i(they)g(can)f(handle)f(man)m (y)g(systems)j(where)e(all)e(previously)i(kno)m(wn)g(criteria)183 992 y(for)g(mo)s(dularit)m(y)e(of)i(innermost)g(termination)e(fail.)280 1112 y(F)-8 b(or)42 b(example,)g(w)m(e)h(can)g(deal)f(with)g(com)m (binations)e(whic)m(h)j(are)g(neither)f(comp)s(osable)183 1233 y(nor)d(hierarc)m(hical)g(com)m(binations)f(\(nor)h(generalized)h (prop)s(er)f(extensions\))i(as)g(sho)m(wn)g(in)183 1353 y(Section)36 b(6.1.2.)g(This)g(is)g(not)g(p)s(ossible)g(with)g(an)m(y)h (of)f(the)h(previous)f(mo)s(dularit)m(y)e(results.)183 1474 y(Moreo)m(v)m(er,)g(in)d(con)m(trast)i(to)f([Krishna)g(Rao,)f (1995],)h(our)g(results)h(are)f(also)f(applicable)f(for)183 1594 y(hierarc)m(hical)41 b(com)m(binations)g(in)h(whic)m(h)i FA(R)1818 1609 y Fz(2)1900 1594 y FE(con)m(tains)f(de\014ned)i(sym)m(b) s(ols)d(of)h FA(R)3228 1609 y Fz(1)3310 1594 y FE(in)f(the)183 1714 y(argumen)m(ts)d(of)g(its)g(recursiv)m(e)h(calls,)e(cf.)i(the)f Ft(add)g FE(and)h Ft(w)m(eight)q FE(-example.)e(Suc)m(h)i(systems)183 1835 y(o)s(ccur)32 b(frequen)m(tly)i(in)e(practice.)280 1955 y(Another)51 b(mo)s(dularit)m(y)d(criterion)h(for)h(hierarc)m (hical)f(com)m(binations)g(is)h(due)h(to)f(Der-)183 2076 y(sho)m(witz)25 b([1994].)f(There,)i(o)s(ccurrences)g(of)e FA(D)1825 2091 y Fz(1)1864 2076 y FE(-sym)m(b)s(ols)g(in)g(recursiv)m (e)i(calls)d(of)i FA(D)3158 2091 y Fz(2)3198 2076 y FE(-sym)m(b)s(ols) 183 2196 y(are)37 b(allo)m(w)m(ed,)f(but)h(only)f(if)g FA(R)1308 2211 y Fz(2)1384 2196 y FE(is)g FC(oblivious)44 b FE(of)37 b(the)g FA(R)2270 2211 y Fz(1)2309 2196 y FE(-rules,)g(i.e.,)f(termination)f(of)h FA(R)3531 2211 y Fz(2)183 2316 y FE(m)m(ust)k(not)g(dep)s(end)i(on)e(the)h FA(R)1363 2331 y Fz(1)1403 2316 y FE(-rules.)f(Ho)m(w)m(ev)m(er,)j (this)d(criterion)f(is)h(not)g(applicable)e(to)183 2437 y(systems)27 b(lik)m(e)f(the)g Ft(add)g FE(and)g Ft(w)m(eight)q FE(-example,)f(b)s(ecause)j(termination)23 b(of)j(the)g Ft(w)m(eight)q FE(-rules)183 2557 y(of)32 b(course)h(dep)s(ends)i(on)d (the)h(result)g(of)f Ft(add)o FE(\()p FB(n)c FE(:)g FB(m)g FE(:)g FB(x;)17 b Ft(0)27 b FE(:)h FB(x)p FE(\).)280 2677 y(An)34 b(alternativ)m(e)g(mo)s(dularit)m(y)d(result)j(for)g (hierarc)m(hical)e(com)m(binations)h(w)m(as)i(presen)m(ted)183 2798 y(b)m(y)j(F)-8 b(ern\023)-49 b(andez)37 b(and)h(Jouannaud)f ([1995].)f(Ho)m(w)m(ev)m(er,)k(their)c(result)h(is)g(restricted)g(to)g (sys-)183 2918 y(tems)g(where)i(the)g(argumen)m(ts)f(of)f(recursiv)m(e) i(calls)e(in)g FA(R)2316 2933 y Fz(2)2393 2918 y FE(decrease)j(w.r.t.)e (the)h(subterm)183 3039 y(relation)33 b(\(compared)i(as)g(m)m(ultisets) f(or)h(lexicographically\).)d(Hence,)k(their)f(result)f(is)h(not)183 3159 y(applicable)42 b(to)i(the)g Ft(add)g FE(and)g Ft(w)m(eight)q FE(-example)f(either)h(\(and)g(also)f(not)h(to)g(most)g(other)183 3279 y(systems)e(where)h FA(R)928 3294 y Fz(2)1009 3279 y FE(is)d(not)h(simply)f(terminating\),)f(whereas)k(our)e(mo)s(dularit) m(y)d(results)183 3400 y(are)32 b(often)h(successful)h(in)e(these)i (examples.)183 3699 y FD(7.)68 b(Conclusion)183 3869 y FE(In)39 b(this)f(article)f(w)m(e)j(in)m(tro)s(duced)f(a)f (re\014nemen)m(t)i(of)e(the)h(dep)s(endency)i(pair)d(approac)m(h)h(in) 183 3989 y(order)j(to)f(p)s(erform)g(termination)f(and)i(innermost)f (termination)f(pro)s(ofs)h(in)g(a)h(mo)s(dular)183 4110 y(w)m(a)m(y)-8 b(.)35 b(This)g(re\014nemen)m(t)g(allo)m(ws)f(automated) f(termination)f(and)j(innermost)e(termination)183 4230 y(pro)s(ofs)44 b(for)h(man)m(y)g(TRSs)h(for)e(whic)m(h)i(suc)m(h)g(pro) s(ofs)f(w)m(ere)h(not)f(p)s(ossible)g(b)s(efore.)g(F)-8 b(or)44 b(a)183 4350 y(collection)30 b(of)i(suc)m(h)i(examples)f(see)h ([Arts)f(and)f(Giesl,)g(2001].)280 4471 y(Using)22 b(our)g(mo)s(dular)d (re\014nemen)m(t)24 b(of)d(the)h(dep)s(endency)j(pair)c(framew)m(ork,)h (w)m(e)h(dev)m(elop)s(ed)183 4591 y(sev)m(eral)34 b(new)g(mo)s(dularit) m(y)d(criteria)g(whic)m(h)j(extend)h(previous)e(results)h(for)e(mo)s (dularit)m(y)f(of)183 4711 y(innermost)39 b(termination.)f(Within)h (this)h(framew)m(ork,)g(w)m(e)h(also)e(obtain)g(easy)j(pro)s(ofs)d(for) 183 4832 y(existing)32 b(mo)s(dularit)m(y)e(theorems.)280 4952 y(Ho)m(w)m(ev)m(er,)39 b(criteria)c(for)h FC(innermost)45 b FE(termination)34 b(are)i(only)g(applicable)f(for)h(termina-)183 5073 y(tion)25 b(pro)s(ofs)h(of)g(certain)f(restricted)i(TRSs)g (\(e.g.,)g(lo)s(cally)c(con\015uen)m(t)28 b(o)m(v)m(erla)m(y)f(systems) h(and)183 5193 y(in)35 b(particular,)f(non-o)m(v)m(erlapping)h(systems) i([Gramlic)m(h,)c(1995]\).)i(But)h(in)f(practice)h(there)183 5313 y(are)30 b(man)m(y)h(cases)h(in)e(whic)m(h)h(innermost)f (termination)e(is)i(not)g(su\016cien)m(t)i(for)e(termination.)280 5434 y(Th)m(us,)h(to)d(fully)g(exploit)f(the)j(adv)-5 b(an)m(tages)29 b(of)f(dep)s(endency)k(pairs)c(for)g(these)i(systems)h (as)183 5554 y(w)m(ell,)d(w)m(e)i(sho)m(w)m(ed)h(that)e(the)g(w)m (ell-kno)m(wn)g(mo)s(dularit)m(y)e(result)i(for)f(simple)g(termination) e(of)p eop %%Page: 41 41 41 40 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(41)183 390 y FE(disjoin)m(t)35 b(unions)i(can)g(b)s(e)g(extended) h(to)f(DP)f(quasi-simple)f(termination.)f(F)-8 b(urthermore,)183 511 y FA(G)6 b FE(-restricted)30 b(DP)f(simple)g(termination)e(is)j(ev) m(en)h(mo)s(dular)d(for)h(constructor-sharing)h(and)183 631 y(comp)s(osable)h(systems.)280 751 y(T)-8 b(o)23 b(conclude,)g([Arts)h(and)f(Giesl,)e(2000])h(presen)m(ted)j(the)e(dep)s (endency)i(pair)d(tec)m(hnique)i(to)183 872 y(p)s(erform)31 b(automated)h(termination)f(and)i(innermost)f(termination)e(pro)s(ofs.) i(Ho)m(w)m(ev)m(er,)k(in)183 992 y(that)30 b(article)e(dep)s(endency)33 b(pairs)c(w)m(ere)j(not)e(used)h(in)f(a)f(mo)s(dular)g(w)m(a)m(y)i(and) f(th)m(us)h(one)g(had)183 1112 y(to)g(pro)m(v)m(e)h(termination)d(of)i (a)g(TRS)g(at)g(once)h(\(i.e.,)f(without)g(b)s(eing)g(able)f(to)h (decomp)s(ose)h(it)183 1233 y(in)m(to)c(subsystems)k(and)d(to)g(use)i (sev)m(eral)f(di\013eren)m(t)f(orderings)g(for)g(its)g(termination)e (pro)s(of)7 b(\).)183 1353 y(In)41 b(particular,)e(whenev)m(er)k(a)e (TRS)g(w)m(as)h(constructed)g(b)m(y)g(com)m(bining)d(sev)m(eral)i (systems)183 1474 y(whose)i(termination)c(had)j(b)s(een)g(pro)m(v)m(ed) h(b)s(efore,)f(then)h(the)f(whole)f(termination)f(pro)s(of)183 1594 y(had)32 b(to)h(b)s(e)f(re-done.)280 1714 y(Therefore,)49 b(the)g(presen)m(t)h(article)c(dev)m(elops)j(the)g(ideas)f(of)f([Arts)i (and)f(Giesl,)f(2000])183 1835 y(further)h(in)e(a)i(signi\014can)m(t)f (w)m(a)m(y)-8 b(.)48 b(The)h(progress)f(in)f(automated)g(termination)e (pro)m(ving)183 1955 y(whic)m(h)33 b(w)m(as)h(made)e(p)s(ossible)g(b)m (y)i(the)f(dev)m(elopmen)m(t)g(of)g(dep)s(endency)i(pairs)d(no)m(w)i (also)d(has)183 2076 y(a)h(coun)m(terpart)i(in)e(the)h(area)f(of)h(mo)s (dularit)m(y)-8 b(.)30 b(With)i(dep)s(endency)k(pairs)c(one)h(can)g (obtain)183 2196 y(automated)44 b(termination)f(pro)s(ofs)i(of)h (non-simply)d(terminating)g(TRSs)k(and)e(with)g(the)183 2316 y(results)31 b(of)g(the)h(presen)m(t)h(article)c(one)j(can)f(p)s (erform)f(them)h(in)g(a)g(mo)s(dular)e(w)m(a)m(y)-8 b(.)32 b(In)g(fact,)f(it)183 2437 y(is)g(this)g(mo)s(dularit)m(y)f(whic)m(h)i (mak)m(es)g(an)g(application)d(of)i(dep)s(endency)k(pairs)c(to)g(large) g(and)183 2557 y(realistic)c(systems)k(p)s(ossible;)d(see)j([Giesl)d (and)h(Arts,)h(2001])e(for)h(an)g(industrial)f(case)i(study)-8 b(.)183 2677 y(Compared)30 b(to)f(previous)i(w)m(ork)g(on)f(mo)s (dularit)m(y)-8 b(,)28 b(the)i(mo)s(dularit)m(y)e(criteria)h(dev)m (elop)s(ed)i(in)183 2798 y(this)h(article)f(represen)m(t)j(a)f(substan) m(tial)f(extension.)183 3060 y FG(Ac)m(kno)m(wledgemen)m(ts.)23 b FE(W)-8 b(e)25 b(thank)g(Aart)f(Middeldorp)g(for)g(man)m(y)h(helpful) e(remarks)i(and)183 3180 y(hin)m(ts.)32 b(J)s(\177)-51 b(urgen)33 b(Giesl)e(w)m(as)i(supp)s(orted)h(b)m(y)f(the)g(DF)m(G)f (under)h(gran)m(t)f(GI)h(274/4-1.)183 3479 y FD(References)183 3649 y FE(T.)39 b(Arts.)62 b(System)40 b(description:)e(The)i(dep)s (endency)i(pair)37 b(metho)s(d.)62 b(In)39 b FC(Pr)-5 b(o)g(c)g(e)g(e)g(dings)39 b(of)280 3770 y(the)34 b(11th)g (International)e(Confer)-5 b(enc)g(e)33 b(on)g(R)-5 b(ewriting)33 b(T)-7 b(e)i(chniques)33 b(and)g(Applic)-5 b(ations,)280 3890 y(R)e(T)g(A-00)p FE(,)34 b(v)m(olume)h(1833)f(of)h FC(L)-5 b(e)g(ctur)g(e)37 b(Notes)h(in)f(Computer)g(Scienc)-5 b(e)p FE(,)34 b(pages)h(261{264,)280 4011 y(Norwic)m(h,)e(England,)f (2000.)g(Springer)g(V)-8 b(erlag,)32 b(Berlin.)183 4214 y(T.)43 b(Arts)g(and)g(J.)f(Giesl.)72 b(Mo)s(dularit)m(y)42 b(of)g(termination)e(using)j(dep)s(endency)i(pairs.)73 b(In)280 4334 y FC(Pr)-5 b(o)g(c)g(e)g(e)g(dings)33 b(of)h(the)g(9th)h (International)e(Confer)-5 b(enc)g(e)32 b(on)i(R)-5 b(ewriting)34 b(T)-7 b(e)i(chniques)33 b(and)280 4455 y(Applic)-5 b(ations,)45 b(R)-7 b(T)g(A-98)p FE(,)44 b(v)m(olume)g(1379)g(of)g FC(L)-5 b(e)g(ctur)g(e)46 b(Notes)h(in)e(Computer)h(Scienc)-5 b(e)p FE(,)280 4575 y(pages)33 b(226{240,)e(Tsukuba,)k(Japan,)d(1998.)g (Springer)g(V)-8 b(erlag,)32 b(Berlin.)183 4779 y(T.)42 b(Arts)g(and)g(J.)f(Giesl.)70 b(T)-8 b(ermination)39 b(of)i(term)g(rewriting)g(using)g(dep)s(endency)j(pairs.)280 4899 y FC(The)-5 b(or)g(etic)g(al)34 b(Computer)h(Scienc)-5 b(e)p FE(,)31 b(236:133{178,)g(2000.)183 5102 y(T.)38 b(Arts)f(and)h(J.)f(Giesl.)57 b(A)37 b(collection)f(of)h(examples)g (for)g(termination)e(of)i(term)g(rewrit-)280 5223 y(ing)d(using)h(dep)s (endency)j(pairs.)50 b(T)-8 b(ec)m(hnical)35 b(Rep)s(ort)g(AIB)g (2001-09,)f(R)-11 b(WTH)36 b(Aac)m(hen,)280 5343 y(German)m(y)-8 b(,)33 b(2001.)42 b(h)m(ttp://aib.informatik.rwth-aac)m(hen.de.)p eop %%Page: 42 42 42 41 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(42)183 390 y FE(F.)29 b(Baader)g(and)g(T.)h(Nipk)m(o)m(w.)38 b FC(T)-7 b(erm)31 b(R)-5 b(ewriting)31 b(and)h(A)n(l)5 b(l)31 b(That)p FE(.)38 b(Cam)m(bridge)28 b(Univ)m(ersit)m(y)280 511 y(Press,)34 b(1998.)183 714 y(CiME)e(2,)h(1999.)42 b(Av)-5 b(ailable)30 b(at)j(h)m(ttp://cime.lri.fr.)183 917 y(N.)f(Dersho)m(witz.)45 b(A)32 b(note)h(on)g(simpli\014cation)c (orderings.)43 b FC(Information)34 b(Pr)-5 b(o)g(c)g(essing)34 b(L)-5 b(et-)280 1038 y(ters)p FE(,)33 b(9\(5\):212{215,)e(1979.)183 1241 y(N.)36 b(Dersho)m(witz.)56 b(T)-8 b(ermination)35 b(of)h(rewriting.)54 b FC(Journal)38 b(of)g(Symb)-5 b(olic)38 b(Computation)p FE(,)e(3)280 1362 y(\(1-2\):69{116,)30 b(1987.)183 1565 y(N.)e(Dersho)m(witz.)37 b(Hierarc)m(hical)26 b(termination.)34 b(In)29 b FC(Pr)-5 b(o)g(c)g(e)g(e)g(dings)30 b(of)g(the)h(4th)g(International)280 1685 y(Workshop)45 b(on)g(Conditional)f(and)g(T)-7 b(yp)i(e)g(d)45 b(R)-5 b(ewriting)45 b(Systems,)g(CTRS-94)p FE(,)d(v)m(olume)280 1806 y(968)e(of)g FC(L)-5 b(e)g(ctur)g(e)42 b(Notes)h(in)f(Computer)f (Scienc)-5 b(e)p FE(,)40 b(pages)h(89{105,)e(Jerusalem,)h(Israel,)280 1926 y(1994.)32 b(Springer)g(V)-8 b(erlag,)32 b(Berlin.)183 2130 y(N.)25 b(Dersho)m(witz)g(and)f(C.)i(Ho)s(ot.)j(Natural)24 b(termination.)k FC(The)-5 b(or)g(etic)g(al)27 b(Computer)g(Scienc)-5 b(e)p FE(,)280 2250 y(142\(2\):179{207,)30 b(1995.)183 2453 y(N.)39 b(Dersho)m(witz)g(and)h(J.-P)-8 b(.)39 b(Jouannaud.)63 b(Rewrite)39 b(systems.)64 b(In)40 b FC(F)-7 b(ormal)39 b(Mo)-5 b(dels)41 b(and)280 2574 y(Semantics)p FE(,)47 b(v)m(olume)g(B)h(of)f FC(Handb)-5 b(o)g(ok)48 b(of)g(The)-5 b(or)g(etic)g(al)48 b(Computer)h(Scienc)-5 b(e)p FE(,)46 b(pages)280 2694 y(243{320.)31 b(North-Holland,)g(1990.)183 2897 y(J.)42 b(Dic)m(k,)g(J.)g(Kalm)m(us,)f(and)h(U.)g(Martin.)71 b(Automating)41 b(the)h(Kn)m(uth-Bendix)h(ordering.)280 3018 y FC(A)-5 b(cta)36 b(Informatic)-5 b(a)p FE(,)31 b(28:95{119,)g(1990.)183 3221 y(M.)47 b(F)-8 b(ern\023)-49 b(andez)48 b(and)g(J.-P)-8 b(.)47 b(Jouannaud.)87 b(Mo)s(dular)46 b(termination)f(of)i(term)f(rewriting)280 3342 y(systems)f(revisited.) 76 b(In)43 b FC(Pr)-5 b(o)g(c)g(e)g(e)g(dings)44 b(of)h(the)g(10th)f (Workshop)g(on)h(Sp)-5 b(e)g(ci\014c)g(ation)43 b(of)280 3462 y(A)n(bstr)-5 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1196 y(F)-7 b(r)i(ontiers)30 b(of)g(Combining)f(Systems,)h(F)-7 b(r)i(oCoS)o(-98)p FE(,)27 b(v)m(olume)g(7)h(of)g FC(Studies)i(in)g(L) -5 b(o)g(gic)31 b(and)280 1316 y(Computation)p FE(,)39 b(pages)i(141{160,)d(Amsterdam,)h(The)i(Netherlands,)f(2000.)f(Researc) m(h)280 1436 y(Studies)33 b(Press,)h(John)f(Wiley)f(&)g(Sons.)183 1640 y(B.)24 b(Gramlic)m(h.)k(Generalized)23 b(su\016cien)m(t)j (conditions)d(for)h(mo)s(dular)e(termination)g(of)i(rewrit-)280 1760 y(ing.)61 b FC(Applic)-5 b(able)39 b(A)n(lgebr)-5 b(a)40 b(in)g(Engine)-5 b(ering,)39 b(Communic)-5 b(ation,)39 b(and)h(Computing)p FE(,)e(5:)280 1880 y(131{158,)31 b(1994.)183 2084 y(B.)k(Gramlic)m(h.)48 b(Abstract)35 b(relations)f(b)s(et)m(w)m(een)j(restricted)e(termination)e(and)i (con\015uence)280 2204 y(prop)s(erties)e(of)f(rewrite)g(systems.)45 b FC(F)-7 b(undamenta)33 b(Informatic)-5 b(ae)p FE(,)31 b(24:3{23,)h(1995.)183 2408 y(B.)e(Gramlic)m(h.)38 b(On)31 b(pro)m(ving)f(termination)f(b)m(y)i(innermost)f(termination.)38 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y(S.)40 b(Kamin)f(and)h(J.-J.)g(L)m(\023)-46 b(evy)-8 b(.)68 b(Tw)m(o)41 b(generalizations)e(of)h(the)g(recursiv)m(e)i(path)f (ordering.)280 3740 y(Departmen)m(t)32 b(of)h(Computer)f(Science,)i (Univ)m(ersit)m(y)f(of)f(Illinois,)e(IL,)j(USA,)g(1980.)183 3944 y(J.)c(W.)h(Klop.)36 b(T)-8 b(erm)30 b(rewriting)e(systems.)39 b(In)30 b FC(Backgr)-5 b(ound:)30 b(Computational)h(Structur)-5 b(es)p FE(,)280 4064 y(v)m(olume)45 b(2)g(of)g FC(Handb)-5 b(o)g(ok)46 b(of)g(L)-5 b(o)g(gic)46 b(in)g(Computer)h(Scienc)-5 b(e)p FE(,)44 b(pages)i(1{116.)e(Oxford)280 4184 y(Univ)m(ersit)m(y)33 b(Press,)i(New)e(Y)-8 b(ork,)33 b(1992.)183 4388 y(D.)f(E.)h(Kn)m(uth)g (and)g(P)-8 b(.)33 b(B.)g(Bendix.)44 b(Simple)31 b(w)m(ord)j(problems)e (in)g(univ)m(ersal)g(algebras.)44 b(In)280 4508 y FC(Computational)28 b(Pr)-5 b(oblems)29 b(in)f(A)n(bstr)-5 b(act)29 b(A)n(lgebr)-5 b(a)p FE(,)26 b(pages)h(263{297.)d(P)m(ergamon)i(Press,)280 4628 y(1970.)183 4832 y(K.)36 b(Koro)m(vin)g(and)g(A.)h(V)-8 b(oronk)m(o)m(v.)56 b(V)-8 b(erifying)35 b(orien)m(tabilit)m(y)f(of)i (rewrite)h(rules)f(using)g(the)280 4952 y(Kn)m(uth-Bendix)k(order.)64 b(In)40 b FC(Pr)-5 b(o)g(c)g(e)g(e)g(dings)40 b(of)h(the)g(5th)h (International)e(Confer)-5 b(enc)g(e)39 b(on)280 5073 y(R)-5 b(ewriting)26 b(T)-7 b(e)i(chniques)25 b(and)h(Applic)-5 b(ations,)26 b(R)-7 b(T)g(A-01)p FE(,)23 b(v)m(olume)g(2051)f(of)i FC(L)-5 b(e)g(ctur)g(e)27 b(Notes)280 5193 y(in)e(Computer)g(Scienc)-5 b(e)p FE(,)21 b(pages)i(137{153,)d(Utrec)m(h)m(t,)k(The)f(Netherlands,) g(2001.)e(Springer)280 5313 y(V)-8 b(erlag,)32 b(Berlin.)p eop %%Page: 44 44 44 43 bop 284 191 a FF(Giesl,)29 b(Arts,)i(Ohlebusc)m(h:)d(Mo)s(dular)h (T)-8 b(ermination)29 b(Pro)s(ofs)h(Using)g(Dep)s(endency)g(P)m(airs) 100 b(44)183 390 y FE(M.)30 b(R.)g(K.)g(Krishna)g(Rao.)39 b(Simple)28 b(termination)g(of)i(hierarc)m(hical)e(com)m(binations)h (of)g(term)280 511 y(rewriting)40 b(systems.)69 b(In)41 b FC(Pr)-5 b(o)g(c)g(e)g(e)g(dings)41 b(of)h(the)g(Symp)-5 b(osium)42 b(on)g(The)-5 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