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1420 y Fv(and)g(its)h(extension)f Fs(R)3108 1432 y Fr(2)3169 1420 y Fv(is)h(ter-)515 1519 y(minating)c(if)h Fs(R)999 1531 y Fr(1)1057 1519 y Fv(is)g Fs(C)1179 1531 y Fq(E)1223 1519 y Fv(-terminating)f(and)g(there)g(is)g(no)g(in\014nite)h(c)n(hain) e(of)h Fs(R)2889 1531 y Fr(2)2927 1519 y Fv(-dep)r(endency)515 1619 y(pairs)i(o)n(v)n(er)f Fs(R)p Fv(.)36 b Fs(C)1062 1631 y Fq(E)1107 1619 y Fv(-termination)23 b(is)h(a)g(sligh)n(tly)g (stronger)e(notion)i(than)g(termination,)g(but)515 1719 y(w)n(e)g(will)i(sho)n(w)e(in)h(Section)g(3)f(that)i(virtually)e(all)h (automatic)f(termination)h(pro)r(ofs)f(b)n(y)h(the)515 1818 y(dep)r(endency)c(pair)g(metho)r(d)h(yield)f Fs(C)1659 1830 y Fq(E)1703 1818 y Fv(-termination.)35 b(Of)21 b(course,)g(it)h(w) n(ould)f(b)r(e)g(ev)n(en)g(more)515 1918 y(desirable)33 b(if)i(the)f(termination)g(pro)r(ofs)f(of)h(a)g(base)g(system)g Fs(R)2536 1930 y Fr(1)2607 1918 y Fv(and)g(its)h(extension)e Fs(R)3341 1930 y Fr(2)515 2017 y Fv(could)23 b(b)r(e)h(done)f(indep)r (enden)n(tly)h(of)f(eac)n(h)g(other)g(and)g(then)h(\(if)g(some)f (easily)f(c)n(hec)n(k)n(ed)h(syn-)515 2117 y(tactic)30 b(criteria)f(are)g(satis\014ed\))h(one)g(ma)n(y)f(conclude|without)i (an)n(y)e(extra)g(pro)r(of|that)515 2217 y(the)e(hierarc)n(hical)d(com) n(bination)i(of)g Fs(R)1736 2229 y Fr(1)1800 2217 y Fv(and)h Fs(R)2031 2229 y Fr(2)2095 2217 y Fv(is)f(also)g(terminating.)36 b(In)26 b(other)g(w)n(ords,)515 2316 y(for)e(hierarc)n(hical)f(com)n (binations)g(of)i(TRSs,)g(it)g(w)n(ould)f(b)r(e)h(desirable)f(to)h(kno) n(w)f(conditions)515 2416 y(under)g(whic)n(h)h(termination)f(is)g (preserv)n(ed.)35 b(Dersho)n(witz)23 b([Der95)o(])i(coined)f(the)h (name)g(\\hi-)515 2516 y(erarc)n(hical)d(termination")i(for)g(this)h (preserv)-5 b(ation,)24 b(while)g(other)g(authors)g(use)g(the)h(longer) 515 2615 y(term)30 b(\\mo)r(dularit)n(y)f(of)h(termination)g(for)g (hierarc)n(hical)e(com)n(binations".)43 b(By)30 b(using)g(Ur-)515 2715 y(bain's)24 b([Urb01)o(])h(result)f(ab)r(out)g(incremen)n(tal)g (termination)h(pro)r(ofs,)f(w)n(e)g(will)h(pro)n(v)n(e)e(a)h(new)515 2814 y(result)g(ab)r(out)h(hierarc)n(hical)d Fs(C)1464 2826 y Fq(E)1509 2814 y Fv(-termination)i(whic)n(h)g(is)h(more)f(p)r(o) n(w)n(erful)g(than)g(all)h(results)515 2914 y(kno)n(wn)h(so)h(far.)36 b(Moreo)n(v)n(er,)25 b(w)n(e)i(will)h(not)f(only)g(pro)n(vide)f(v)n (ery)g(simple)i(pro)r(ofs)e(of)i(already)515 3014 y(kno)n(wn)f(mo)r (dularit)n(y)g(results)g(on)g(simple)h(termination,)f(but)h(also)f (extend)h(these)g(results.)515 3113 y(Finally)-7 b(,)27 b(w)n(e)h(will)f(refute)h(a)f(conjecture)h(of)f(Krishna)f(Rao)h([KR94)o (].)639 3213 y(The)41 b(article)e(is)h(organized)f(as)g(follo)n(ws.)74 b(In)40 b(Section)h(2)e(w)n(e)h(will)h(review)e(the)i(ba-)515 3313 y(sics)36 b(of)g(the)h(dep)r(endency)g(pair)e(metho)r(d.)64 b(In)37 b(Section)f(3)g(it)h(will)g(b)r(e)g(sho)n(wn)e(that)i(vir-)515 3412 y(tually)c(all)h(automatic)f(termination)g(pro)r(ofs)g(b)n(y)h (the)g(dep)r(endency)g(pair)f(metho)r(d)h(yield)515 3512 y Fs(C)559 3524 y Fq(E)603 3512 y Fv(-termination.)k(Section)28 b(4)g(con)n(tains)f(an)h(alternativ)n(e)f(pro)r(of)g(of)h(Urbain's)g ([Urb01)o(])g(re-)515 3611 y(sult)38 b(ab)r(out)g(incremen)n(tal)f (termination)g(pro)r(ofs.)67 b(In)38 b(Section)g(5)g(w)n(e)f(will)h (presen)n(t)g(the)515 3711 y(mo)r(dularit)n(y)27 b(results.)515 3986 y Fw(2)134 b(Preliminaries)515 4167 y Fv(W)-7 b(e)31 b(assume)f(the)h(reader)e(to)i(b)r(e)g(familiar)f(with)h(the)g(basic)f (notions)h(of)f(term)h(rewriting.)515 4267 y(F)-7 b(or)27 b(an)g(in)n(tro)r(duction)g(to)h(term)f(rewriting)g(see)g(e.g.)g ([DJ90,)g(Klo92)n(,)h(BN98)o(].)515 4500 y Fp(2.1)112 b(Dep)s(endency)39 b(pairs)515 4653 y Fv(First,)27 b(w)n(e)g(brie\015y) g(recapitulate)f(the)i(dep)r(endency)g(pair)e(metho)r(d)i(whic)n(h)f (is)h(due)f(to)g(Arts)515 4752 y(and)g(Giesl)h([A)n(G00)o(].)639 4852 y(If)37 b Fo(f)9 b Fv(\()p Fo(s)852 4864 y Fr(1)889 4852 y Fo(;)14 b(:)g(:)g(:)g(;)g(s)1113 4864 y Fn(n)1158 4852 y Fv(\))37 b Fs(!)h Fo(C)6 b Fv([)p Fo(g)s Fv(\()p Fo(t)1541 4864 y Fr(1)1578 4852 y Fo(;)14 b(:)g(:)g(:)g(;)g(t)1793 4864 y Fn(m)1856 4852 y Fv(\)])37 b(is)f(a)g(rewrite)f(rule)h(of)g(a)g (TRS)g Fs(R)h Fv(and)f Fo(g)j Fv(is)515 4952 y(a)f(de\014ned)g(sym)n(b) r(ol,)j(then)e Fs(h)p Fo(F)12 b Fv(\()p Fo(s)1580 4964 y Fr(1)1617 4952 y Fo(;)i(:)g(:)g(:)g(;)g(s)1841 4964 y Fn(n)1886 4952 y Fv(\))p Fo(;)g(G)p Fv(\()p Fo(t)2082 4964 y Fr(1)2120 4952 y Fo(;)g(:)g(:)g(:)g(;)g(t)2335 4964 y Fn(m)2398 4952 y Fv(\))p Fs(i)39 b Fv(is)f(a)g Fm(dep)l(endency)j(p)l(air)48 b Fv(of)1926 5255 y(2)p eop %%Page: 3 3 3 2 bop 515 523 a Fs(R)p Fv(,)38 b(where)e Fo(F)48 b Fv(and)36 b Fo(G)g Fv(are)f(new)h(tuple)h(sym)n(b)r(ols.)62 b(A)36 b(sequence)g(of)g(dep)r(endency)g(pairs)515 623 y Fs(h)p Fo(s)586 635 y Fr(1)623 623 y Fo(;)14 b(t)690 635 y Fr(1)727 623 y Fs(ih)p Fo(s)830 635 y Fr(2)868 623 y Fo(;)g(t)935 635 y Fr(2)972 623 y Fs(i)g Fo(:)g(:)g(:)56 b Fv(of)41 b Fs(R)h Fv(is)g(an)f Fs(R)p Fv(-)p Fm(chain)50 b Fv(if)42 b(there)f(is)h(a)f(substitution)h Fo(\033)j Fv(suc)n(h)d(that)515 722 y Fo(t)545 734 y Fn(j)580 722 y Fo(\033)h Fs(!)753 692 y Fq(\003)753 745 y(R)853 722 y Fo(s)892 734 y Fn(j)s Fr(+1)1011 722 y Fo(\033)e Fv(holds)c(for)g(ev) n(ery)g(t)n(w)n(o)f(consecutiv)n(e)h(pairs)f Fs(h)p Fo(s)2592 734 y Fn(j)2628 722 y Fo(;)14 b(t)2695 734 y Fn(j)2729 722 y Fs(i)38 b Fv(and)g Fs(h)p Fo(s)3042 734 y Fn(j)s Fr(+1)3161 722 y Fo(;)14 b(t)3228 734 y Fn(j)s Fr(+1)3347 722 y Fs(i)515 822 y Fv(in)28 b(the)g(sequence.)37 b(F)-7 b(urthermore,)27 b(if)h(ev)n(ery)e(prop)r(er)h(subterm)h(of)g Fo(s)2656 834 y Fn(j)2691 822 y Fo(\033)j Fv(and)c Fo(t)2960 834 y Fn(j)2995 822 y Fo(\033)32 b Fv(is)27 b(termi-)515 922 y(nating)i(\(or)g(equiv)-5 b(alen)n(tly)e(,)29 b(ev)n(ery)f Fo(s)1650 934 y Fn(j)1685 922 y Fo(\033)33 b Fv(and)c Fo(t)1958 934 y Fn(j)1993 922 y Fo(\033)k Fv(is)d(terminating\),)g (then)f(the)h Fs(R)p Fv(-c)n(hain)f(is)515 1021 y(said)e(to)g(b)r(e)h Fm(minimal)p Fv(.)515 1195 y Fl(Theorem)i(2.1)41 b Fm(A)34 b(TRS)g Fs(R)h Fm(is)g(terminating)g(if)g(and)g(only)h(if)f(ther)l(e)g (is)g(no)g(in\014nite)f Fs(R)p Fm(-)515 1294 y(chain.)j(In)22 b(fact,)j(if)e Fs(R)f Fm(is)h(non-terminating,)h(then)e(ther)l(e)g(is)h (a)f(minimal)h(in\014nite)f Fs(R)p Fm(-chain.)639 1468 y Fv(A)27 b(pro)r(of)f(of)g(the)h(\014rst)f(statemen)n(t)h(in)g (Theorem)e(2.1)h(can)g(b)r(e)h(found)f(in)h([A)n(G00)o(].)37 b(As)26 b(a)515 1567 y(matter)18 b(of)h(fact,)h(that)f(pro)r(of)f(also) g(pro)n(v)n(es)f(the)i(second)f(statemen)n(t)g(as)g(a)h(careful)f(insp) r(ection)515 1667 y(sho)n(ws.)75 b(But)42 b(ho)n(w)e(do)h(w)n(e)f(pro)n (v)n(e)f(the)j(absence)e(of)h(in\014nite)g(c)n(hains?)77 b(As)41 b(sho)n(wn)f(in)515 1767 y([A)n(G00)o(],)e(Theorem)d(2.1)g(can) h(b)r(e)g(used)g(to)g(obtain)f(a)h(termination)f(criterion)g(that)h (can)515 1866 y(b)r(e)g(automated.)62 b(T)-7 b(o)35 b(this)i(end,)h(a)e (set)g(IN)g(of)g(inequalities,)i(eac)n(h)d(of)h(the)h(form)e Fo(s)i Fs(\025)g Fo(t)515 1966 y Fv(or)e Fo(s)j(>)f(t)p Fv(,)i(is)d(generated)f(and)h(to)g(pro)n(v)n(e)f(termination)h(of)g Fs(R)p Fv(,)j(one)d(has)g(to)g(sho)n(w)g(that)515 2065 y(these)26 b(inequalities)h(are)e(satis\014ed)h(b)n(y)h(some)f(pair)f (\()p Fk(\045)p Fo(;)14 b Fs(\037)p Fv(\))27 b(consisting)e(of)i(a)f (quasi-rewrite)515 2165 y(ordering)h Fk(\045)h Fv(\(a)g(quasi-ordering) e(whic)n(h)j(is)f(closed)g(under)g(substitutions)h(and)f(con)n(texts\)) 515 2265 y(and)f(a)g(strict)h(ordering)e Fs(\037)h Fv(with)h(the)g (prop)r(erties)639 2423 y Fs(\017)41 b(\037)28 b Fv(is)f(closed)g (under)g(substitutions)h(and)g(w)n(ell-founded,)639 2586 y Fs(\017)41 b Fk(\045)23 b Fs(\016)f(\037)14 b(\022)g(\037)27 b Fv(or)f Fs(\037)d(\016)f Fk(\045)14 b Fs(\022)g(\037)p Fv(.)515 2744 y(Suc)n(h)28 b(a)h(pair)e(is)i(called)f(a)g Fm(r)l(e)l(duction)j(p)l(air)39 b Fv([KNT99)n(].)h(Giv)n(en)29 b(a)f(quasi-rewrite)f(ordering)515 2844 y Fk(\045)p Fv(,)c(a)f(natural) f(candidate)h(for)g(the)h(corresp)r(onding)d(ordering)h Fs(\037)h Fv(is)g(the)h Fm(strict)30 b Fv(relation)21 b Fs(\037)3344 2814 y Fn(s)515 2944 y Fv(de\014ned)k(b)n(y)g Fo(t)e Fs(\037)1029 2913 y Fn(s)1088 2944 y Fo(u)i Fv(if)g(and)g(only)g (if)h Fo(t)d Fk(\045)g Fo(u)i Fv(and)g Fo(u)e Fs(6)p Fk(\045)f Fo(t)p Fv(.)36 b(Unfortunately)-7 b(,)26 b Fs(\037)2884 2913 y Fn(s)2945 2944 y Fv(is)f(in)g(general)515 3043 y(not)j(closed)g(under)g(substitutions;)h(see)f([GA)n(O01)n(].)40 b(Therefore,)27 b(one)h(usually)g(c)n(ho)r(oses)e Fs(\037)515 3143 y Fv(to)j(b)r(e)h(the)g(so-called)e Fm(stable-strict)38 b Fv(relation)28 b Fs(\037)2045 3113 y Fn(ss)2141 3143 y Fv(corresp)r(onding)g(to)h Fk(\045)p Fv(.)43 b(This)29 b(relation)515 3242 y(is)j(de\014ned)i(b)n(y)e Fo(t)g Fs(\037)1142 3212 y Fn(ss)1240 3242 y Fo(u)g Fv(if)i(and)e(only)h(if)g Fo(t\033)i Fk(\045)d Fo(u\033)k Fv(and)c Fo(u\033)j Fs(6)p Fk(\045)c Fo(t\033)37 b Fv(hold)32 b(for)h(all)f(ground)515 3342 y(substitutions)c Fo(\033)s Fv(;)g(see)f([GA)n(O01)o(])h(for)f (more)f(details.)515 3516 y Fl(Theorem)k(2.2)41 b Fm(L)l(et)30 b Fs(R)h Fm(b)l(e)f(a)h(TRS)f(over)i(the)f(signatur)l(e)f Fs(F)i Fv(=)24 b Fs(D)e(])d(C)35 b Fm(and)c(let)g Fo(T)12 b(up)3234 3528 y Fq(F)3314 3516 y Fv(=)515 3615 y Fs(f)p Fo(F)40 b Fs(j)28 b Fo(f)k Fs(2)23 b(D)r(g)28 b Fm(b)l(e)h(the)f(set)g (of)i(tuple)e(symb)l(ols.)39 b Fs(R)29 b Fm(is)f(terminating)h(if)g (and)g(only)g(ther)l(e)g(is)g(a)515 3715 y(r)l(e)l(duction)g(p)l(air)i Fv(\()p Fk(\045)p Fo(;)14 b Fs(\037)p Fv(\))30 b Fm(on)f Fs(T)22 b Fv(\()p Fs(F)k([)19 b Fo(T)12 b(up)1836 3727 y Fq(F)1892 3715 y Fo(;)i Fs(V)7 b Fv(\))30 b Fm(such)g(that)639 3873 y Fs(\017)41 b Fo(l)25 b Fk(\045)d Fo(r)33 b Fm(for)d(every)h (rule)f Fo(l)24 b Fs(!)f Fo(r)33 b Fm(in)d Fs(R)g Fm(and)639 4036 y Fs(\017)41 b Fo(s)23 b Fs(\037)g Fo(t)30 b Fm(for)g(every)h(dep) l(endency)g(p)l(air)g Fs(h)p Fo(s;)14 b(t)p Fs(i)p Fm(.)639 4209 y Fv(In)40 b(fact,)j(for)c(\014nite)h(systems)f(w)n(e)g(can)h(do)f (ev)n(en)g(b)r(etter.)73 b(This)40 b(is)f(b)r(ecause)h(some)515 4309 y(dep)r(endency)29 b(pairs)f(cannot)h(o)r(ccur)g(t)n(wice)g(in)g (an)n(y)f(c)n(hain)h(and)g(hence)g(they)g(need)h(not)f(b)r(e)515 4408 y(tak)n(en)f(in)n(to)h(accoun)n(t)g(in)g(a)g(pro)r(of)g(that)g(no) g(in\014nite)h(c)n(hain)f(exists.)41 b(In)30 b(order)d(to)j(iden)n (tify)515 4508 y(these)e(insigni\014can)n(t)g(dep)r(endency)h(pairs,)f (the)h(notion)f(of)g(dep)r(endency)h(graph)f(has)g(b)r(een)515 4608 y(in)n(tro)r(duced)f(b)n(y)g(Arts)h(and)f(Giesl)h([A)n(G00)o(].) 639 4707 y(The)h Fm(dep)l(endency)j(gr)l(aph)k Fv(of)29 b(a)f(TRS)h Fs(R)g Fv(is)g(the)g(directed)f(graph)g(whose)g(no)r(des)g (\(v)n(er-)515 4807 y(tices\))k(are)f(the)h(dep)r(endency)h(pairs)d(of) i Fs(R)h Fv(and)e(there)h(is)g(an)g(arc)f(\(directed)h(edge\))f(from) 515 4907 y Fs(h)p Fo(s;)14 b(t)p Fs(i)34 b Fv(to)f Fs(h)p Fo(u;)14 b(v)s Fs(i)34 b Fv(if)g Fs(h)p Fo(s;)14 b(t)p Fs(ih)p Fo(u;)g(v)s Fs(i)34 b Fv(is)f(a)g(c)n(hain.)54 b(If)34 b(w)n(e)f(restrict)g(ourselv)n(es)f(to)h Fm(\014nite)g Fv(TRSs,)515 5006 y(then)28 b(ev)n(ery)e(in\014nite)j(c)n(hain)e (corresp)r(onds)e(to)j(a)f(cycle)g(in)h(the)g(dep)r(endency)g(graph.) 1926 5255 y(3)p eop %%Page: 4 4 4 3 bop 639 523 a Fv(A)39 b(non-empt)n(y)g(set)f Fs(P)46 b Fv(of)38 b(dep)r(endency)i(pairs)d(is)i(called)f(a)h Fm(cycle)h Fv(if,)i(for)c(an)n(y)g(t)n(w)n(o)515 623 y(dep)r(endency)21 b(pairs)g Fs(h)p Fo(s;)14 b(t)p Fs(i)p Fo(;)g Fs(h)p Fo(u;)g(v)s Fs(i)23 b(2)g(P)7 b Fv(,)23 b(there)e(is)g(a)f(non-empt)n(y)h(path)g(from)g Fs(h)p Fo(s;)14 b(t)p Fs(i)22 b Fv(to)f Fs(h)p Fo(u;)14 b(v)s Fs(i)515 722 y Fv(and)32 b(from)g Fs(h)p Fo(u;)14 b(v)s Fs(i)33 b Fv(to)f Fs(h)p Fo(s;)14 b(t)p Fs(i)33 b Fv(in)g(the)f(dep)r (endency)h(graph)e(whic)n(h)i(tra)n(v)n(erses)d(dep)r(endency)515 822 y(pairs)22 b(from)h Fs(P)30 b Fv(only)-7 b(.)35 b(\(In)23 b(particular,)g(there)g(m)n(ust)g(also)f(b)r(e)i(a)f(path)g(from)g Fs(h)p Fo(s;)14 b(t)p Fs(i)23 b Fv(to)g(itself.\))515 1005 y Fl(Theorem)30 b(2.3)41 b Fm(A)26 b(\014nite)g(TRS)h Fs(R)g Fm(is)g(terminating)g(if)h(and)f(only)h(if)f(for)h(e)l(ach)g (cycle)g Fs(P)33 b Fm(in)515 1104 y(the)c(dep)l(endency)h(gr)l(aph)g (of)f Fs(R)g Fm(ther)l(e)g(exists)f(no)h(in\014nite)f Fs(R)p Fm(-chain)i(of)g(dep)l(endency)g(p)l(airs)515 1204 y(fr)l(om)g Fs(P)7 b Fm(.)639 1386 y Fv(A)35 b(pro)r(of)f(of)g (Theorem)g(2.3)f(and)i(the)g(subsequen)n(t)f(results)f(of)i(this)g (section)f(can)g(b)r(e)515 1486 y(found)27 b(in)g([GA)n(O01)n(].)37 b(Similar)26 b(to)h(Theorem)f(2.1,)g(Theorem)g(2.3)g(can)g(b)r(e)h (used)g(to)f(obtain)515 1586 y(a)h(termination)g(criterion)g(that)h (can)f(b)r(e)h(tested)g(automatically)-7 b(.)515 1768 y Fl(Theorem)30 b(2.4)41 b Fm(L)l(et)c Fs(R)h Fm(b)l(e)f(a)h(\014nite)f (TRS)g(over)h(the)g(signatur)l(e)f Fs(F)45 b Fv(=)37 b Fs(D)27 b(])d(C)42 b Fm(and)c(let)515 1868 y Fo(T)12 b(up)666 1880 y Fq(F)745 1868 y Fv(=)22 b Fs(f)p Fo(F)38 b Fs(j)26 b Fo(f)32 b Fs(2)23 b(D)r(g)j Fm(b)l(e)g(the)h(set)e(of)i (tuple)f(symb)l(ols.)39 b Fs(R)26 b Fm(is)g(terminating)h(if)g(and)f (only)h(if)515 1968 y(for)i(every)g(cycle)g Fs(P)35 b Fm(in)29 b(the)f(dep)l(endency)i(gr)l(aph)f(of)g Fs(R)g Fm(ther)l(e)f(is)h(a)g(r)l(e)l(duction)f(p)l(air)h Fv(\()p Fk(\045)p Fo(;)14 b Fs(\037)p Fv(\))515 2067 y Fm(on)30 b Fs(T)21 b Fv(\()p Fs(F)27 b([)18 b Fo(T)12 b(up)1043 2079 y Fq(F)1099 2067 y Fo(;)i Fs(V)7 b Fv(\))30 b Fm(such)g(that)639 2233 y Fs(\017)41 b Fo(l)25 b Fk(\045)d Fo(r)33 b Fm(for)d(every)h (rule)f Fo(l)24 b Fs(!)f Fo(r)33 b Fm(in)d Fs(R)p Fm(,)639 2399 y Fs(\017)41 b Fo(s)23 b Fk(\045)g Fo(t)30 b Fm(for)g(every)h(dep) l(endency)g(p)l(air)g Fs(h)p Fo(s;)14 b(t)p Fs(i)30 b Fm(in)g Fs(P)7 b Fm(,)639 2565 y Fs(\017)41 b Fo(s)23 b Fs(\037)g Fo(t)30 b Fm(for)g(at)g(le)l(ast)g(one)g(dep)l(endency)h(p) l(air)g Fs(h)p Fo(s;)14 b(t)p Fs(i)30 b Fm(in)g Fs(P)7 b Fm(.)639 2748 y Fv(W)-7 b(e)32 b(stress)e(that,)i(according)d(to)i (Theorem)f(2.4,)h(one)f(ma)n(y)h(use)f(di\013eren)n(t)h(reduction)515 2848 y(pairs)25 b(for)i(di\013eren)n(t)f(cycles)g(in)h(a)g(termination) f(pro)r(of)g(of)g(a)h Fm(\014nite)f Fv(TRS.)h(This)g(is)f(in)h(stark) 515 2947 y(con)n(trast)32 b(to)i(Theorem)f(2.2,)h(where)g(one)f (reduction)h(pair)f(has)g(to)h(satisfy)f(all)h(the)g(con-)515 3047 y(strain)n(ts)26 b(sim)n(ultaneously)-7 b(.)639 3147 y(The)33 b(primary)f(goal)f(of)i(the)g(dep)r(endency)g(pair)f (metho)r(d)i(is)e(to)h(use)g(standard)e(tec)n(h-)515 3246 y(niques)j(to)g(automatically)f(generate)g(a)h(suitable)g (reduction)g(pair)g(satisfying)f(the)i(con-)515 3346 y(strain)n(ts)26 b(of)h(Theorem)f(2.4.)36 b(In)27 b(order)f(to)h (facilitate)g(the)h(syn)n(thesis)e(of)h(a)g(suitable)g(reduc-)515 3445 y(tion)g(pair,)g(the)h(follo)n(wing)f(concept)g(w)n(as)g(in)n(tro) r(duced)g(b)n(y)h(Arts)f(and)g(Giesl)h([A)n(G00)o(].)639 3545 y(An)40 b Fm(ar)l(gument)g(\014ltering)g(system)f Fv(\(AFS\))h(o)n(v)n(er)e(a)g(signature)g Fs(F)48 b Fv(is)39 b(a)f(TRS)i(whose)515 3645 y(rewrite)31 b(rules)g(are)f(of)i(the)g (form)f Fo(f)9 b Fv(\()p Fo(x)1721 3657 y Fr(1)1758 3645 y Fo(;)14 b(:)g(:)g(:)g(;)g(x)1990 3657 y Fn(n)2036 3645 y Fv(\))30 b Fs(!)f Fo(r)35 b Fv(with)d Fo(f)38 b Fs(2)30 b(F)2707 3615 y Fr(\()p Fn(n)p Fr(\))2836 3645 y Fv(and)h(there)h(is)f (at)515 3744 y(most)24 b(one)f(suc)n(h)h(rule)g(for)g(ev)n(ery)f Fo(f)31 b Fs(2)24 b(F)8 b Fv(.)35 b(In)25 b(the)f(rule)g Fo(f)9 b Fv(\()p Fo(x)2359 3756 y Fr(1)2397 3744 y Fo(;)14 b(:)g(:)g(:)f(;)h(x)2628 3756 y Fn(n)2674 3744 y Fv(\))23 b Fs(!)g Fo(r)r Fv(,)j(the)e(v)-5 b(ariables)515 3844 y Fo(x)562 3856 y Fr(1)599 3844 y Fo(;)14 b(:)g(:)g(:)g(;)g(x)831 3856 y Fn(n)906 3844 y Fv(are)28 b(pairwise)f(distinct)j(and)f Fo(r)j Fv(is)d(either)f(one)h(of)g(these)g(v)-5 b(ariables)27 b(or)i(a)f(term)515 3944 y Fo(f)565 3913 y Fq(0)588 3944 y Fv(\()p Fo(y)661 3956 y Fr(1)698 3944 y Fo(;)14 b(:)g(:)g(:)f(;)h(y) 923 3956 y Fn(m)986 3944 y Fv(\),)23 b(where)d Fo(f)1347 3913 y Fq(0)1393 3944 y Fs(62)k(F)k Fv(is)21 b(a)g(fresh)f(function)i (sym)n(b)r(ol)e(of)h(arit)n(y)f Fo(m)h Fv(o)r(ccurring)e(only)515 4043 y(once)k(in)h(the)h(AFS)f(and)g Fo(y)1318 4055 y Fr(1)1355 4043 y Fo(;)14 b(:)g(:)g(:)g(;)g(y)1581 4055 y Fn(m)1667 4043 y Fv(are)23 b(pairwise)g(distinct)i(v)-5 b(ariables)22 b(out)i(of)g Fo(x)3041 4055 y Fr(1)3079 4043 y Fo(;)14 b(:)g(:)g(:)f(;)h(x)3310 4055 y Fn(n)3356 4043 y Fv(.)515 4143 y(The)32 b(use)g(of)g(an)g(appropriate)e(AFS)j (often)f(facilitates)g(a)g(termination)g(pro)r(of)f(b)n(y)h(either)515 4242 y(Theorem)27 b(2.2)f(or)h(Theorem)g(2.4.)639 4342 y(T)-7 b(o)24 b(estimate)f(whic)n(h)h(dep)r(endency)g(pairs)f(ma)n(y)g (o)r(ccur)g(consecutiv)n(ely)g(in)h(a)g(c)n(hain,)g(the)515 4442 y(estimated)34 b(dep)r(endency)g(graph)f(has)h(b)r(een)g(in)n(tro) r(duced;)j(cf.)d([A)n(G00].)56 b(W)-7 b(e)34 b(\014rst)g(recall)515 4541 y(the)26 b(needed)g(notions.)36 b Fz(CAP)p Fv(\()p Fo(t)p Fv(\))27 b(results)f(from)g(replacing)e(all)i(subterms)g(of)g Fo(t)g Fv(that)h(ha)n(v)n(e)d(a)515 4641 y(de\014ned)h(ro)r(ot)g(sym)n (b)r(ol)f(b)n(y)h(di\013eren)n(t)g(fresh)g(v)-5 b(ariables)24 b(and)h Fz(REN)p Fv(\()p Fo(t)p Fv(\))g(results)g(from)f(replac-)515 4741 y(ing)32 b(all)g(v)-5 b(ariables)32 b(in)g Fo(t)h Fv(b)n(y)f(di\013eren)n(t)h(fresh)f(v)-5 b(ariables.)51 b(Then,)34 b(in)e(order)g(to)g(determine)515 4840 y(whether)37 b Fs(h)p Fo(u;)14 b(v)s Fs(i)38 b Fv(can)f(follo)n(w)f Fs(h)p Fo(s;)14 b(t)p Fs(i)38 b Fv(in)g(a)e(c)n(hain,)k(one)d(c)n(hec)n (ks)f(whether)h Fz(REN)p Fv(\()p Fz(CAP)p Fv(\()p Fo(t)p Fv(\)\))515 4940 y(uni\014es)30 b(with)g Fo(u)p Fv(.)42 b(The)30 b(function)g Fz(REN)f Fv(is)h(needed)g(to)f(rename)g(m)n (ultiple)h(o)r(ccurrences)e(of)1926 5255 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fv(the)36 b(same)f(v)-5 b(ariable)35 b Fo(x)h Fv(in)h Fo(t)e Fv(b)r(ecause)h(t)n(w)n(o)f(di\013eren)n(t)h(o) r(ccurrences)e(of)i Fo(x\033)k Fv(in)c Fo(t\033)j Fv(could)515 623 y(reduce)27 b(to)g(di\013eren)n(t)h(terms.)639 722 y(The)k Fm(estimate)l(d)i(dep)l(endency)h(gr)l(aph)40 b Fv(of)32 b(a)f(TRS)h Fs(R)p Fv(,)i(denoted)d(b)n(y)h(EDG\()p Fs(R)p Fv(\),)i(is)e(the)515 822 y(directed)d(graph)e(whose)i(no)r(des) f(are)g(the)i(dep)r(endency)f(pairs)f(of)h Fs(R)g Fv(and)g(there)g(is)f (an)h(arc)515 922 y(from)e Fs(h)p Fo(s;)14 b(t)p Fs(i)28 b Fv(to)g Fs(h)p Fo(u;)14 b(v)s Fs(i)27 b Fv(if)35 b Fz(REN)p Fv(\()p Fz(CAP)p Fv(\()p Fo(t)p Fv(\)\))28 b(and)g Fo(u)f Fv(are)g(uni\014able.)515 1083 y Fl(Corollary)k(2.5)41 b Fm(A)33 b(\014nite)f(TRS)h Fs(R)g Fm(is)g(terminating)g(if)h(and)f (only)h(if)g(for)g(e)l(ach)f(cycle)h Fs(P)515 1183 y Fm(in)c Fv(EDG\()p Fs(R)p Fv(\))g Fm(ther)l(e)g(is)g(an)g(AFS)f Fs(A)h Fm(and)h(a)f(r)l(e)l(duction)f(p)l(air)i Fv(\()p Fk(\045)p Fo(;)14 b Fs(\037)p Fv(\))30 b Fm(such)f(that)639 1332 y Fs(\017)41 b Fo(l)25 b Fs(#)814 1344 y Fq(A)885 1332 y Fk(\045)d Fo(r)k Fs(#)1077 1344 y Fq(A)1164 1332 y Fm(for)k(al)t(l)h(rules)f Fo(l)24 b Fs(!)f Fo(r)33 b Fm(fr)l(om)d Fs(R)p Fm(,)639 1490 y Fs(\017)41 b Fo(s)23 b Fs(#)826 1502 y Fq(A)897 1490 y Fk(\045)g Fo(t)g Fs(#)1080 1502 y Fq(A)1166 1490 y Fm(for)31 b(al)t(l)g(dep)l(endency)g(p)l(airs)g Fs(h)p Fo(s;)14 b(t)p Fs(i)30 b Fm(fr)l(om)g Fs(P)7 b Fm(,)639 1648 y Fs(\017)41 b Fo(s)23 b Fs(#)826 1660 y Fq(A)897 1648 y Fs(\037)g Fo(t)g Fs(#)1080 1660 y Fq(A)1166 1648 y Fm(for)31 b(at)f(le)l(ast)g(one)g(dep)l(endency)h(p)l(air)g Fs(h)p Fo(s;)14 b(t)p Fs(i)30 b Fm(fr)l(om)g Fs(P)7 b Fm(.)639 1810 y Fv(Recall)24 b(that)g(a)f Fm(quasi-simpli\014c)l(ation) 28 b(or)l(dering)d Fv(\(QSO\))f Fk(\045)f Fv(is)h(a)f(quasi-rewrite)f (order-)515 1909 y(ing)k(p)r(ossessing)e(the)j(subterm)f(prop)r(ert)n (y)-7 b(.)35 b(Krusk)-5 b(al's)24 b(theorem)i(implies)g(that)h(\(the)f (strict)515 2009 y(and\))f(the)h(stable-strict)e(relation)g(corresp)r (onding)g(to)h(a)g(QSO)f(is)h(w)n(ell-founded)g(pro)n(vided)515 2109 y(that)40 b Fs(F)48 b Fv(is)40 b(\014nite.)74 b(Man)n(y)40 b(standard)f(orderings)f(\(lik)n(e)i(recursiv)n(e)e(path)i(orderings)e (or)515 2208 y(certain)32 b(p)r(olynomial)h(orderings\))e(are)h(in)i (fact)f(QSOs.)53 b(This)33 b(motiv)-5 b(ates)33 b(the)g(follo)n(wing) 515 2308 y(de\014nition)h([GA)n(O01)o(]:)49 b(A)35 b(\014nite)f(TRS)g Fs(R)h Fv(is)f(called)f(DP)h Fm(quasi-simply)j(terminating)42 b Fv(if)515 2408 y(its)32 b(termination)g(can)g(b)r(e)g(pro)n(v)n(en)f (b)n(y)h(means)g(of)g(Corollary)e(2.5)h(suc)n(h)h(that)g(the)h(reduc-) 515 2507 y(tion)i(pair)g(consists)g(of)g(a)g(QSO)g(and)g(its)h(corresp) r(onding)d(stable-strict)i(relation.)59 b(The)515 2607 y(in)n(ten)n(tion)24 b(of)g(this)h(de\014nition)f(is)g(to)g(co)n(v)n (er)f(most)h(of)g(the)h(TRSs)f(for)f(whic)n(h)i(an)f(automated)515 2706 y(termination)j(pro)r(of)g(using)g(dep)r(endency)h(pairs)f(is)g(p) r(oten)n(tially)h(feasible.)36 b(Note)28 b(that)g(the)515 2806 y(de\014nition)c(of)h(DP)f(quasi-simple)f(termination)h(uses)g (the)g(estimated)h(dep)r(endency)f(graph)515 2906 y(instead)32 b(of)g(the)g(dep)r(endency)g(graph.)49 b(This)32 b(is)g(b)r(ecause)g (the)g(dep)r(endency)h(graph)e(can-)515 3005 y(not)36 b(b)r(e)h(used)f(in)h(automated)f(termination)g(pro)r(ofs)g(\(it)h(is)f (undecidable)h(whether)f(t)n(w)n(o)515 3105 y(dep)r(endency)28 b(pairs)e(form)i(a)f(c)n(hain\).)515 3334 y Fp(2.2)112 b(Mo)s(dularit)m(y)515 3487 y Fv(Next,)35 b(w)n(e)f(summarize)f(the)h (basic)f(concepts)h(concerning)e(the)i(com)n(bination)f(of)h(TRSs.)515 3587 y(Let)h Fs(R)741 3599 y Fr(1)814 3587 y Fv(and)h Fs(R)1054 3599 y Fr(2)1127 3587 y Fv(b)r(e)f(TRSs)h(o)n(v)n(er)d(the)j (signatures)e Fs(F)2280 3599 y Fr(1)2352 3587 y Fv(and)i Fs(F)2582 3599 y Fr(2)2619 3587 y Fv(,)h(resp)r(ectiv)n(ely)-7 b(.)59 b(Their)515 3686 y Fm(c)l(ombine)l(d)28 b(system)j Fv(is)25 b(the)g(union)g Fs(R)f Fv(=)e Fs(R)1846 3698 y Fr(1)1897 3686 y Fs([)13 b(R)2035 3698 y Fr(2)2098 3686 y Fv(o)n(v)n(er)23 b(the)i(signature)f Fs(F)31 b Fv(=)23 b Fs(F)3010 3698 y Fr(1)3060 3686 y Fs([)13 b(F)3188 3698 y Fr(2)3225 3686 y Fv(.)36 b(Its)515 3786 y(set)27 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Fv(.)616 4807 y(4.)41 b Fm(Base)28 b(system)c Fs(R)1258 4819 y Fr(1)1319 4807 y Fv(and)g Fm(extension)g Fs(R)1909 4819 y Fr(2)1971 4807 y Fv(form)f(a)h Fm(hier)l(ar)l(chic)l(al)29 b(c)l(ombination)j Fv(if)24 b Fs(D)3275 4819 y Fr(1)3324 4807 y Fs(\\)722 4907 y(D)786 4919 y Fr(2)852 4907 y Fv(=)k Fs(C)989 4919 y Fr(1)1046 4907 y Fs(\\)21 b(D)1186 4919 y Fr(2)1252 4907 y Fv(=)28 b Fs(;)p Fv(.)46 b(So)30 b(de\014ned)h(sym)n(b)r(ols)f(of)h Fs(R)2351 4919 y Fr(1)2420 4907 y Fv(ma)n(y)f(o)r(ccur)g(as)g(constructors)722 5006 y(in)e Fs(R)889 5018 y Fr(2)927 5006 y Fv(,)f(but)i(not)e(vice)h(v)n (ersa.)1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 616 523 a Fv(5.)41 b Fs(R)29 b Fv(is)f(a)f Fm(hier)l(ar)l(chic) l(al)33 b(c)l(ombination)d Fv(of)e Fm(b)l(ase)i(system)e Fs(R)2499 535 y Fr(1)2565 523 y Fv(and)g Fm(extension)g Fs(R)3163 535 y Fr(2)3228 523 y Fm(with)722 623 y(c)l(ommon)d (subsystem)d Fs(R)1499 635 y Fr(0)1560 623 y Fv(=)h Fs(f)p Fo(l)h Fs(!)f Fo(r)i Fs(2)f(R)f(j)f Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))i Fs(2)g(D)2531 635 y Fr(1)2576 623 y Fs(\\)8 b(D)2703 635 y Fr(2)2740 623 y Fs(g)22 b Fv(if)h Fs(R)2945 635 y Fr(0)3005 623 y Fv(=)g Fs(R)3163 635 y Fr(1)3208 623 y Fs(\\)8 b(R)3341 635 y Fr(2)722 722 y Fv(and)28 b Fs(F)944 734 y Fr(1)999 722 y Fs(\\)19 b Fv(\()p Fs(D)1169 734 y Fr(2)1225 722 y Fs(n)f(D)1349 734 y Fr(1)1386 722 y Fv(\))24 b(=)e Fs(;)p Fv(.)639 864 y(Both)32 b(comp)r(osable)f (systems)h(and)g(hierarc)n(hical)e(com)n(binations)h(are)h(a)g(prop)r (er)f(gen-)515 963 y(eralization)k(of)i(constructor-sharing)d(systems)i (but)h(neither)g(one)g(is)g(a)f(generalization)515 1063 y(of)i(the)h(other.)69 b(In)39 b(practice,)h(hierarc)n(hical)d(com)n (binations)g(with)i(a)f(common)g(subsys-)515 1163 y(tem)28 b(are)f(most)h(imp)r(ortan)n(t)f(b)r(ecause)h(they)g(generalize)f(b)r (oth)h(comp)r(osable)f(systems)g(and)515 1262 y(hierarc)n(hical)e(com)n (binations.)639 1362 y(A)i(prop)r(ert)n(y)e Fs(P)33 b Fv(is)27 b Fm(mo)l(dular)i(for)g(disjoint)h(\(c)l(onstructor-sharing)f (etc.\))38 b(TRSs)26 b Fv(if,)i(for)515 1462 y(all)22 b(disjoin)n(t)g(\(constructor-sharing)d(etc.\))36 b(TRSs)22 b(\()p Fs(F)2183 1474 y Fr(1)2220 1462 y Fo(;)14 b Fs(R)2327 1474 y Fr(1)2364 1462 y Fv(\))23 b(and)f(\()p Fs(F)2667 1474 y Fr(2)2704 1462 y Fo(;)14 b Fs(R)2811 1474 y Fr(2)2849 1462 y Fv(\))22 b(that)h(ha)n(v)n(e)e(the)515 1561 y(prop)r(ert)n(y)26 b Fs(P)7 b Fv(,)28 b(their)f(union)h(\()p Fs(F)1491 1573 y Fr(1)1546 1561 y Fs([)19 b(F)1680 1573 y Fr(2)1717 1561 y Fo(;)14 b Fs(R)1824 1573 y Fr(1)1880 1561 y Fs([)19 b(R)2024 1573 y Fr(2)2061 1561 y Fv(\))28 b(also)f(has)g(prop)r(ert)n (y)f Fs(P)7 b Fv(.)639 1661 y(T)-7 b(o)n(y)n(ama)23 b([T)-7 b(o)n(y87)n(])25 b(sho)n(w)n(ed)f(that)h(termination)f(is)h(not)g(mo)r (dular)f(for)g(disjoin)n(t)h(unions.)515 1760 y(Th)n(us,)31 b(the)h(question)f(is)g(what)g(restrictions)e(ha)n(v)n(e)h(to)h(b)r(e)h (imp)r(osed)f(on)g(the)g(constituen)n(t)515 1860 y(TRSs)h(so)g(that)h (their)g(disjoin)n(t)g(union)f(is)h(again)e(terminating.)52 b(The)33 b(\014rst)f(results)g(w)n(ere)515 1960 y(obtained)k(b)n(y)g (in)n(v)n(estigating)f(the)i(distribution)g(of)f(collapsing)g(rules)g (and)g(duplicating)515 2059 y(rules)27 b(among)g(the)h(TRSs;)h(see)e ([Rus87)o(,)h(Mid89].)38 b(Ev)n(er)26 b(since)i(an)g(abundance)f(of)h (mo)r(du-)515 2159 y(larit)n(y)i(results)h(has)g(b)r(een)h(published;)i (see)e(the)f(dissertations)g(of)g(Middeldorp)g([Mid90],)515 2259 y(Ohlebusc)n(h)c([Ohl94a)n(],)h(and)f(Gramlic)n(h)g([Gra96)n(])h (for)f(an)h(o)n(v)n(erview.)639 2358 y(Ho)n(w)n(ev)n(er,)e(man)n(y)g (mo)r(dularit)n(y)h(results)f(are)g(rarely)g(applicable)h(in)g (practice.)36 b(F)-7 b(or)27 b(ex-)515 2458 y(ample,)i(collapsing)f (and)h(duplicating)g(rules)g(o)r(ccur)g(naturally)f(in)i(most)f(TRSs.) 41 b(In)30 b(con-)515 2557 y(trast)21 b(to)g(this,)i(since)f(most)f (standard)g(metho)r(ds)h(for)f(automated)g(termination)g(pro)r(ofs)g (are)515 2657 y(based)32 b(on)g(syn)n(thesizing)f(simpli\014cation)i (orderings,)f(Kurihara)e(and)i(Oh)n(uc)n(hi's)g([K)n(O92)n(])515 2757 y(result)19 b(is)h(of)g(practical)e(relev)-5 b(ance.)34 b(They)19 b(sho)n(w)n(ed)g(that)h(the)g(constructor-sharing)d(com)n (bi-)515 2856 y(nation)24 b(of)g(\014nite)h(simply)g(terminating)f (TRSs)g(is)h(again)e(simply)i(terminating.)35 b(Kurihara)515 2956 y(and)23 b(Oh)n(uc)n(hi's)h(result)f(w)n(as)g(extended)h(to)g (comp)r(osable)f(systems)g([Ohl95)o(])h(and)g(to)g(certain)515 3056 y(hierarc)n(hical)g(com)n(binations)i([KR94)o(].)36 b(Gramlic)n(h)26 b([Gra94)n(])h(sho)n(w)n(ed)f(that)h(simple)f(termi-) 515 3155 y(nation)35 b(is)g(also)f(mo)r(dular)h(for)g (constructor-sharing)d Fm(\014nitely)37 b(br)l(anching)f Fv(TRSs.)60 b(As)35 b(a)515 3255 y(matter)24 b(of)g(fact,)h(he)f(sho)n (w)n(ed)f(a)g(stronger)g(result,)h(namely)g(that)g Fs(C)2591 3267 y Fq(E)2636 3255 y Fv(-termination)f(is)h(mo)r(d-)515 3354 y(ular)j(for)g(these)h(systems.)36 b(The)28 b(coun)n(terexamples)e (in)i([Ohl94b)o(])g(sho)n(w)f(that)h(the)g(\014nitely)515 3454 y(branc)n(hing)e(requiremen)n(t)h(cannot)g(b)r(e)h(dropp)r(ed.)515 3723 y Fw(3)134 b(Pro)l(ving)46 b Fj(C)1285 3741 y Fs(E)1340 3723 y Fw(-termination)h(automatically)515 3905 y Fv(The)35 b(name)g Fs(C)966 3917 y Fq(E)1011 3905 y Fv(-termination)f(w)n(as)h (coined)g(in)g([Ohl94b)o(].)60 b(The)36 b(notion)e(w)n(as)h(originally) 515 4005 y(in)n(tro)r(duced)h(b)n(y)g(Gramlic)n(h)g([Gra94)n(])h(who)f (called)g(it)h(\\termination)e(preserving)g(under)515 4104 y(non-deterministic)27 b(collapses".)35 b Fs(C)1635 4116 y Fq(E)1680 4104 y Fv(-termination)27 b(is)g(formally)g(de\014ned) h(as)f(follo)n(ws.)515 4256 y Fl(De\014nition)k(3.1)40 b Fv(A)34 b(TRS)g Fs(R)g Fv(is)f(called)g Fs(C)1899 4268 y Fq(E)1944 4256 y Fv(-)p Fm(terminating)h Fv(if)g(the)g Fs(C)5 b Fm(ol)t(lapsing)35 b Fs(E)7 b Fm(xtende)l(d)515 4356 y Fv(term)33 b(rewriting)f(system)g Fs(R)23 b(])f(f)p Fo(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))33 b Fs(!)f Fo(x;)14 b(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))33 b Fs(!)f Fo(y)s Fs(g)g Fv(is)h(terminating,)515 4456 y(where)21 b Fo(C)6 b(ons)23 b Fv(is)f(some)g(binary)f(function)i(sym)n (b)r(ol)e(that)i(do)r(es)f(not)g(o)r(ccur)f(in)i(the)f(signature)515 4555 y(of)27 b Fs(R)p Fv(.)639 4707 y Fs(C)683 4719 y Fq(E)728 4707 y Fv(-termination)h(pla)n(ys)g(a)h(crucial)f(role)g(in)i (incremen)n(tal)e(termination)h(pro)r(ofs)f(as)g(w)n(e)515 4807 y(shall)i(see)h(in)g(Section)g(4.)47 b(It)31 b(is)g(kno)n(wn)f (that)i Fs(C)2030 4819 y Fq(E)2074 4807 y Fv(-termination)f(is)g (undecidable)g([Mid90)o(,)515 4907 y(Theorem)22 b(4.1.30].)34 b(On)23 b(the)h(other)f(hand,)h(the)g(next)g(prop)r(osition)e(states)h (some)g(su\016cien)n(t)515 5006 y(conditions)k(that)h(can)f(easily)g(b) r(e)h(c)n(hec)n(k)n(ed.)35 b(They)28 b(are)e(due)i(to)g(Gramlic)n(h)f ([Gra94)n(].)1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fl(Prop)s(osition)30 b(3.2)40 b Fm(L)l(et)30 b Fv(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\))30 b Fm(b)l(e)g(a)g (terminating)g(TRS.)613 689 y(1.)42 b(If)30 b Fs(R)h Fm(is)f(non-duplic)l(ating,)h(then)e(it)h(is)g Fs(C)2010 701 y Fq(E)2055 689 y Fm(-terminating.)613 855 y(2.)42 b(If)30 b Fs(R)h Fm(is)f(simply)h(terminating,)f(then)f Fs(R)19 b([)g(E)7 b Fo(mb)p Fv(\()p Fs(F)h Fv(\))30 b Fm(is)g Fs(C)2538 867 y Fq(E)2583 855 y Fm(-terminating.)639 1038 y Fv(T)-7 b(o)24 b(date,)g(ho)n(w)n(ev)n(er,)f(the)h(TRSs)g(for)f (whic)n(h)h(automated)f(termination)h(pro)r(ofs)f(are)f(p)r(o-)515 1137 y(ten)n(tially)30 b(feasible)h(are)f(no)h(longer)e(just)j(the)f (simply)g(terminating)f(systems,)i(i.e.,)f(those)515 1237 y(systems)26 b(whose)f(termination)i(can)f(b)r(e)g(v)n(eri\014ed)g (b)n(y)g(a)g(simpli\014cation)h(ordering,)e(but)i(the)515 1337 y Fm(DP)d(quasi-simply)i(terminating)j Fv(systems,)22 b(i.e.,)h(those)e(systems)g(whose)g(termination)g(can)515 1436 y(b)r(e)32 b(v)n(eri\014ed)f(b)n(y)h(using)f (quasi-simpli\014cation)g(orderings)f(in)i(com)n(bination)f(with)i(dep) r(en-)515 1536 y(dency)25 b(pairs.)36 b(Hence,)26 b(a)f(natural)g (question)h(is)f(whether)h(DP)g(quasi-simple)e(termination)515 1636 y(also)i(implies)i Fs(C)1007 1648 y Fq(E)1052 1636 y Fv(-termination.)36 b(The)27 b(next)h(lemma)g(sho)n(ws)e(that)i(this) g(is)g(the)f(case.)515 1818 y Fl(Lemma)i(3.3)41 b Fm(A)26 b(\014nite)f(TRS)h Fs(R)h Fm(is)f(DP)g(quasi-simply)i(terminating)e(if) h(and)g(only)g(if)g Fs(R)11 b(])515 1918 y(C)559 1930 y Fq(E)631 1918 y Fm(is)29 b(DP)f(quasi-simply)i(terminating.)38 b(In)28 b(p)l(articular,)i(DP)e(quasi-simple)i(termination)515 2017 y(implies)h Fs(C)843 2029 y Fq(E)888 2017 y Fm(-termination.)515 2217 y(Pr)l(o)l(of)72 b Fv(The)34 b(if-direction)g(is)h(ob)n(viously)e (true.)57 b(F)-7 b(or)34 b(the)h(other)f(direction,)i(let)f Fs(R)g Fv(b)r(e)f(a)515 2316 y(DP)26 b(quasi-simply)e(terminating)i (TRS.)g(That)g(is,)g(for)f(ev)n(ery)f(cycle)i Fs(P)32 b Fv(in)26 b(EDG\()p Fs(R)p Fv(\))h(there)515 2416 y(exists)g(an)h(AFS) h Fs(A)f Fv(and)g(a)g(QSO)g Fk(\045)f Fv(on)h Fs(T)21 b Fv(\()p Fs(F)1955 2386 y Fq(0)1979 2416 y Fo(;)14 b Fs(V)7 b Fv(\))28 b(suc)n(h)g(that)g Fk(\045)g Fv(and)g(its)g(corresp)r (onding)515 2516 y(stable-strict)f(relation)f(satisfy)h(the)h (constrain)n(ts)575 2682 y(\(a\))41 b Fo(l)25 b Fs(#)814 2694 y Fq(A)885 2682 y Fs(\025)d Fo(r)k Fs(#)1077 2694 y Fq(A)1162 2682 y Fv(for)h(eac)n(h)g(rule)g Fo(l)d Fs(!)f Fo(r)31 b Fv(from)c Fs(R)p Fv(,)570 2848 y(\(b\))42 b Fo(s)23 b Fs(#)826 2860 y Fq(A)897 2848 y Fs(\025)g Fo(t)g Fs(#)1080 2860 y Fq(A)1164 2848 y Fv(for)28 b(eac)n(h)e(dep)r(endency)i (pair)f Fs(h)p Fo(s;)14 b(t)p Fs(i)28 b Fv(from)f Fs(P)7 b Fv(,)579 3014 y(\(c\))42 b Fo(s)23 b Fs(#)826 3026 y Fq(A)897 3014 y Fo(>)g(t)g Fs(#)1080 3026 y Fq(A)1164 3014 y Fv(for)28 b(at)f(least)g(one)g(dep)r(endency)h(pair)f Fs(h)p Fo(s;)14 b(t)p Fs(i)28 b Fv(from)g Fs(P)7 b Fv(,)515 3180 y(where)25 b Fs(F)821 3150 y Fq(0)870 3180 y Fv(consists)g(of)g (all)h(function)g(sym)n(b)r(ols)f(o)r(ccurring)f(in)i(the)g(constrain)n (ts.)35 b(W)-7 b(e)26 b(ha)n(v)n(e)515 3279 y(to)c(sho)n(w)f(that)h Fs(R)7 b(])g(C)1168 3291 y Fq(E)1236 3279 y Fv(is)22 b(DP)g(quasi-simply)f(terminating.)35 b(First)22 b(of)g(all,)h(note)f (that)g(ev)n(ery)515 3379 y(cycle)27 b Fs(P)34 b Fv(in)28 b(EDG\()p Fs(R)19 b(])g(C)1333 3391 y Fq(E)1378 3379 y Fv(\))28 b(is)f(also)g(a)g(cycle)g(in)h(EDG\()p Fs(R)p Fv(\).)2401 3349 y Fr(1)639 3479 y Fv(Let)g(\012)g(b)r(e)g(the)g (function)g(whic)n(h)f(replaces)g(ev)n(ery)f(subterm)h Fo(C)6 b(ons)p Fv(\()p Fo(u)2825 3491 y Fr(1)2863 3479 y Fo(;)14 b(u)2948 3491 y Fr(2)2984 3479 y Fv(\))28 b(of)g(a)f(term)515 3578 y Fo(u)22 b Fs(2)i(T)d Fv(\()p Fs(F)830 3548 y Fq(0)862 3578 y Fs([)9 b(f)p Fo(C)d(ons)p Fs(g)p Fo(;)14 b Fs(V)7 b Fv(\))22 b(with)i(the)f(distinguished)f(v)-5 b(ariable)22 b Fo(z)t Fv(.)35 b(Let)23 b(the)g(quasi-ordering)515 3678 y Fk(\045)580 3648 y Fq(0)630 3678 y Fv(b)r(e)28 b(de\014ned)g(on)g Fs(T)21 b Fv(\()p Fs(F)1311 3648 y Fq(0)1353 3678 y Fs([)e(f)p Fo(C)6 b(ons)p Fs(g)p Fo(;)14 b Fs(V)7 b Fv(\))26 b(b)n(y)1336 3861 y Fo(u)d Fk(\045)1472 3826 y Fq(0)1517 3861 y Fo(v)31 b Fv(if)d(and)g(only)f(if)h(\012\()p Fo(u)p Fv(\))23 b Fk(\045)g Fv(\012\()p Fo(v)s Fv(\))p Fo(:)515 4043 y Fv(It)37 b(is)g(relativ)n(ely)f(simple)h(to)g(sho)n(w)g (that)g Fk(\045)1928 4013 y Fq(0)1988 4043 y Fv(is)g(a)g(quasi-rewrite) e(ordering,)j(i.e.,)i Fk(\045)3263 4013 y Fq(0)3323 4043 y Fv(is)515 4143 y(closed)g(under)h(con)n(texts)f(and)g(substitutions.) 77 b(The)41 b(quasi-ordering)d Fk(\045)2909 4113 y Fq(00)2992 4143 y Fv(de\014ned)j(b)n(y)515 4242 y Fk(\045)580 4212 y Fq(0)o(0)622 4242 y Fv(=)36 b(\()p Fk(\045)820 4212 y Fq(0)880 4242 y Fs([)h(!)1055 4254 y Fq(C)1091 4262 y Fi(E)1134 4242 y Fv(\))1166 4212 y Fq(\003)1240 4242 y Fv(inherits)f(closure)f(under)g(con)n(texts)g(and)h(substitutions)g (from)f Fk(\045)3356 4212 y Fq(0)515 4342 y Fv(and)28 b Fs(!)760 4354 y Fq(C)796 4362 y Fi(E)840 4342 y Fv(.)40 b(W)-7 b(e)29 b(claim)g(that)g Fk(\045)1516 4312 y Fq(00)1587 4342 y Fv(is)f(a)h(QSO)f(on)g Fs(T)21 b Fv(\()p Fs(F)2228 4312 y Fq(0)2271 4342 y Fs([)f(f)p Fo(C)6 b(ons)p Fs(g)p Fo(;)14 b Fs(V)7 b Fv(\).)39 b(F)-7 b(or)28 b(ev)n(ery)g Fo(n)p Fv(-ary)515 4442 y Fo(f)j Fs(2)24 b(F)734 4412 y Fq(0)785 4442 y Fv(and)j(1)c Fs(\024)f Fo(i)h Fs(\024)g Fo(n)p Fv(,)k(w)n(e)g(ha)n(v)n(e)1105 4624 y(\012\()p Fo(f)9 b Fv(\()p Fo(x)1326 4636 y Fr(1)1364 4624 y Fo(;)14 b(:)g(:)g(:)g(;)g(x)1596 4636 y Fn(n)1641 4624 y Fv(\)\))24 b(=)e Fo(f)9 b Fv(\()p Fo(x)1945 4636 y Fr(1)1983 4624 y Fo(;)14 b(:)g(:)g(:)g(;)g(x)2215 4636 y Fn(n)2260 4624 y Fv(\))23 b Fk(\045)g Fo(x)2450 4636 y Fn(i)2501 4624 y Fv(=)g(\012\()p Fo(x)2728 4636 y Fn(i)2756 4624 y Fv(\))p 515 4698 1146 4 v 607 4752 a Fh(1)642 4775 y Fu(This)d(is)f(not)i(true) g(if)e(w)n(e)i(consider)g(the)g(dep)r(endency)i(graph)d(instead)h(of)g (the)g(estimated)f(dep)r(endency)515 4854 y(graph,)j(as)h(the)h(system) e Fg(f)7 b Fu(\(0)p Fg(;)12 b Fu(1)p Fg(;)g(x)p Fu(\))19 b FB(!)g Fg(f)7 b Fu(\()p Fg(x;)12 b(x;)f(x)p Fu(\))24 b(sho)n(ws.)1926 5255 y Fv(7)p eop %%Page: 8 8 8 7 bop 515 523 a Fv(b)r(ecause)24 b Fk(\045)h Fv(is)g(a)g(QSO)f(on)h Fs(T)c Fv(\()p Fs(F)1536 493 y Fq(0)1560 523 y Fo(;)14 b Fs(V)7 b Fv(\).)36 b(Therefore,)24 b Fo(f)9 b Fv(\()p Fo(x)2272 535 y Fr(1)2310 523 y Fo(;)14 b(:)g(:)g(:)f(;)h(x)2541 535 y Fn(n)2587 523 y Fv(\))23 b Fk(\045)2707 493 y Fq(0)2753 523 y Fo(x)2800 535 y Fn(i)2853 523 y Fv(and)i(hence)g(also)515 623 y Fo(f)9 b Fv(\()p Fo(x)644 635 y Fr(1)681 623 y Fo(;)14 b(:)g(:)g(:)g(;)g(x)913 635 y Fn(n)958 623 y Fv(\))24 b Fk(\045)1079 593 y Fq(0)o(0)1144 623 y Fo(x)1191 635 y Fn(i)1219 623 y Fv(.)37 b(Moreo)n(v)n(er,)25 b(w)n(e)i(ha)n(v)n (e)1600 787 y Fo(C)6 b(ons)p Fv(\()p Fo(x)1873 799 y Fr(1)1911 787 y Fo(;)14 b(x)1995 799 y Fr(2)2033 787 y Fv(\))23 b Fk(\045)2153 753 y Fq(00)2218 787 y Fo(x)2265 799 y Fn(i)515 951 y Fv(b)r(ecause)37 b Fo(C)6 b(ons)p Fv(\()p Fo(x)1105 963 y Fr(1)1143 951 y Fo(;)14 b(x)1227 963 y Fr(2)1265 951 y Fv(\))39 b Fs(!)1419 963 y Fq(C)1455 971 y Fi(E)1539 951 y Fo(x)1586 963 y Fn(i)1614 951 y Fv(,)h(where)d Fo(i)i Fs(2)h(f)p Fv(1)p Fo(;)14 b Fv(2)p Fs(g)p Fv(.)64 b(Th)n(us,)40 b Fk(\045)2693 921 y Fq(00)2772 951 y Fv(has)d(the)h(subterm)515 1051 y(prop)r(ert)n(y)-7 b(.)57 b(In)36 b(other)e(w)n(ords,)h Fk(\045)1572 1021 y Fq(00)1649 1051 y Fv(is)g(a)g(QSO)f(on)g Fs(T)22 b Fv(\()p Fs(F)2316 1021 y Fq(0)2363 1051 y Fs([)h(f)p Fo(C)6 b(ons)p Fs(g)p Fo(;)14 b Fs(V)7 b Fv(\).)58 b(It)35 b(is)g(easy)f(to)515 1151 y(v)n(erify)21 b(that)i Fk(\045)980 1120 y Fq(00)1045 1151 y Fv(and)f(its)g(corresp)r(onding)f (stable-strict)g(relation)h(satisfy)g(the)g(constrain)n(ts)515 1250 y(\(a\){\(c\))k(with)h(the)g(same)f(AFS)h(as)f(ab)r(o)n(v)n(e)f (and)h(furthermore)g Fo(C)6 b(ons)p Fv(\()p Fo(x)2743 1262 y Fr(1)2781 1250 y Fo(;)14 b(x)2865 1262 y Fr(2)2902 1250 y Fv(\))24 b Fk(\045)3023 1220 y Fq(0)o(0)3088 1250 y Fo(x)3135 1262 y Fn(i)3189 1250 y Fv(holds)515 1350 y(true.)37 b(W)-7 b(e)28 b(conclude)f(that)h Fs(R)19 b(])f(C)1593 1362 y Fq(E)1666 1350 y Fv(is)27 b(DP)h(quasi-simply)e (terminating.)639 1449 y(Lastly)-7 b(,)26 b(DP)h(quasi-simple)e (termination)h(of)h Fs(R)16 b(])g(C)2272 1461 y Fq(E)2343 1449 y Fv(implies)27 b(that)f Fs(R)17 b(])f(C)3004 1461 y Fq(E)3075 1449 y Fv(is)26 b(termi-)515 1549 y(nating,)h(or)g(equiv)-5 b(alen)n(tly)e(,)27 b(that)h Fs(R)g Fv(is)g Fs(C)1781 1561 y Fq(E)1825 1549 y Fv(-terminating.)1013 b Fk(\003)639 1734 y Fv(It)28 b(is)g(an)f(immediate)h(consequence)e(of)i(the)g(mo)r (dularit)n(y)f(of)g Fs(C)2605 1746 y Fq(E)2650 1734 y Fv(-termination)g(in)h(con-)515 1833 y(junction)19 b(with)g(the)f (preceding)g(lemma)g(that)h(the)g(com)n(bination)e(of)i(t)n(w)n(o)e (\014nite)i(constructor-)515 1933 y(sharing)35 b(DP)h(quasi-simply)g (terminating)g(TRS)g(is)h Fs(C)2262 1945 y Fq(E)2306 1933 y Fv(-terminating.)63 b(In)36 b(fact,)j(it)e(w)n(as)515 2033 y(sho)n(wn)g(in)i([GA)n(O01)o(])f(that)h(DP)f(quasi-simple)f (termination)h(is)h(mo)r(dular)e(for)h(disjoin)n(t)515 2132 y(unions)32 b(but)h(not)f(for)g(constructor-sharing)d(systems.)51 b(The)32 b(latter)g(result)h(particularly)515 2232 y(implies)c(that)g Fs(C)1023 2244 y Fq(E)1068 2232 y Fv(-termination)f(do)r(es)h(not)g (imply)g(DP)g(quasi-simple)f(termination)h(\(oth-)515 2331 y(erwise)h(DP)h(quasi-simple)f(termination)g(w)n(ould)g(b)r(e)i (mo)r(dular)e(for)g(constructor-sharing)515 2431 y(systems\).)515 2702 y Fw(4)134 b(Pro)l(ving)46 b Fj(C)1285 2720 y Fs(E)1340 2702 y Fw(-termination)h(incremen)l(tally)515 2884 y Fv(The)19 b(results)g(in)g(this)h(section)f(are)f(due)i(to)f(Urbain)g ([Urb01)o(];)j(he)e(dev)n(elop)r(ed)f(tec)n(hniques)g(for)515 2984 y(pro)n(ving)e(termination)h(of)h(TRSs)g(incremen)n(tally)-7 b(.)33 b(Ho)n(w)n(ev)n(er,)18 b(w)n(e)h(will)g(pro)n(vide)e(alternativ) n(e)515 3084 y(pro)r(ofs)26 b(of)i(Urbain's)f(results.)36 b(W)-7 b(e)28 b(need)g(the)g(follo)n(wing)f(re\014ned)g(de\014nition)h (of)g(c)n(hains.)515 3248 y Fl(De\014nition)j(4.1)40 b Fv(Let)h(\()p Fs(F)1378 3260 y Fr(1)1415 3248 y Fo(;)14 b Fs(R)1522 3260 y Fr(1)1560 3248 y Fv(\))41 b(and)f(\()p Fs(F)1899 3260 y Fr(2)1936 3248 y Fo(;)14 b Fs(R)2043 3260 y Fr(2)2081 3248 y Fv(\))41 b(b)r(e)g(TRSs.)76 b(If)41 b(there)f(is)h(an)f(in\014nite)515 3348 y(sequence)27 b Fs(h)p Fo(s)930 3360 y Fr(1)968 3348 y Fo(;)14 b(t)1035 3360 y Fr(1)1072 3348 y Fs(ih)p Fo(s)1175 3360 y Fr(2)1213 3348 y Fo(;)g(t)1280 3360 y Fr(2)1317 3348 y Fs(i)g Fo(:)g(:)g(:)42 b Fv(of)28 b(dep)r(endency)h(pairs)e(from)h Fs(R)2521 3360 y Fr(1)2587 3348 y Fv(and)g(a)g(substitution)g Fo(\033)g Fv(:)515 3447 y Fs(V)h(!)24 b(T)d Fv(\()p Fs(F)860 3459 y Fr(1)914 3447 y Fs([)c(F)1046 3459 y Fr(2)1082 3447 y Fo(;)d Fs(V)7 b Fv(\))27 b(suc)n(h)f(that)h Fo(t)1631 3459 y Fn(j)1666 3447 y Fo(\033)g Fs(!)1823 3417 y Fq(\003)1823 3470 y(R)1880 3478 y Ff(1)1912 3470 y Fq([R)2014 3478 y Ff(2)2074 3447 y Fo(s)2113 3459 y Fn(j)s Fr(+1)2231 3447 y Fo(\033)k Fv(holds)26 b(for)g(all)g(consecutiv)n(e)g(pairs)515 3547 y Fs(h)p Fo(s)586 3559 y Fn(j)621 3547 y Fo(;)14 b(t)688 3559 y Fn(j)723 3547 y Fs(i)39 b Fv(and)f Fs(h)p Fo(s)1037 3559 y Fn(j)s Fr(+1)1156 3547 y Fo(;)14 b(t)1223 3559 y Fn(j)s Fr(+1)1342 3547 y Fs(i)39 b Fv(in)g(the)g(sequence,)i (then)e(the)g(sequence)f Fs(h)p Fo(s)2836 3559 y Fr(1)2873 3547 y Fo(;)14 b(t)2940 3559 y Fr(1)2977 3547 y Fs(ih)p Fo(s)3080 3559 y Fr(2)3118 3547 y Fo(;)g(t)3185 3559 y Fr(2)3222 3547 y Fs(i)g Fo(:)g(:)g(:)515 3646 y Fv(is)37 b(called)g(an)g Fm(in\014nite)h Fs(R)1343 3658 y Fr(1)1380 3646 y Fm(-chain)i(over)f Fs(R)1906 3658 y Fr(1)1969 3646 y Fs([)25 b(R)2119 3658 y Fr(2)2156 3646 y Fv(.)66 b(F)-7 b(urthermore,)39 b(if)f(ev)n(ery)d(\(prop)r(er\))515 3746 y(subterm)25 b Fo(u)g Fv(of)g Fo(s)1040 3758 y Fn(j)1075 3746 y Fo(\033)j Fv(and)d Fo(t)1339 3758 y Fn(j)1374 3746 y Fo(\033)k Fv(is)c(terminating)g(w.r.t.)g Fs(R)2270 3758 y Fr(1)2321 3746 y Fs([)14 b(R)2460 3758 y Fr(2)2498 3746 y Fv(,)26 b(then)f(the)h(c)n(hain)e(is)h(called)515 3846 y Fm(minimal)p Fv(.)639 4010 y(In)33 b(what)f(follo)n(ws,)g(w)n(e) g(will)h(extend)f(Gramlic)n(h's)f([Gra94)o(])h(transformation|whic)n(h) 515 4110 y(is)e(de\014ned)h(on)f(the)h(set)g Fs(T)21 b Fv(\()p Fs(F)8 b Fv(\))31 b(of)g(ground)e(terms|to)h(hierarc)n(hical) f(com)n(binations.)44 b(T)-7 b(o)515 4209 y(this)22 b(end,)i(w)n(e)d (assume)h(that)g(ev)n(ery)f(signature)g(has)g(at)h(least)g(one)g (constan)n(t)f Fo(c)h Fv(whic)n(h)g(is)g(not)515 4309 y(used)i(in)g(the)h(rewrite)e(rules.)35 b(Then)24 b(for)g(ev)n(ery)f (minimal)h(c)n(hain)g(that)g(uses)g(a)g(substitution)515 4408 y Fo(\033)j Fv(:)d Fs(V)30 b(!)24 b(T)d Fv(\()p Fs(F)8 b Fo(;)14 b Fs(V)7 b Fv(\),)29 b(there)f(is)f(also)g(a)h (minimal)g(c)n(hain)g(that)g(uses)g(a)f Fm(gr)l(ound)i Fv(substitution)515 4508 y Fo(\033)565 4478 y Fq(0)628 4508 y Fv(:)39 b Fs(V)46 b(!)39 b(T)21 b Fv(\()p Fs(F)8 b Fv(\).)66 b(The)38 b(substitution)g Fo(\033)1899 4478 y Fq(0)1960 4508 y Fv(is)f(obtained)g(from)g Fo(\033)j Fv(b)n(y)d(replacing)f(ev)n(ery)515 4608 y(v)-5 b(ariable)25 b(in)i Fo(x\033)j Fv(with)d(the)f(constan)n(t)g Fo(c)p Fv(.)36 b(T)-7 b(o)26 b(assume)g(that)g(ev)n(ery)f(signature)h(has)f (at)h(least)515 4707 y(one)d(constan)n(t)f Fo(c)h Fv(whic)n(h)g(is)h (not)f(used)g(in)h(the)f(rewrite)g(rules)f(en)n(tails)h(no)g(loss)f(of) h(generalit)n(y)-7 b(.)515 4807 y(In)26 b(fact,)g(as)f(in)h([Ohl94b)o (])g(one)g(can)f(use)h(a)g(distinguished)f(v)-5 b(ariable)25 b Fo(z)k Fv(instead)d(and)g(de\014ne)515 4907 y(the)i(transformation)e (on)h(the)h(set)g Fs(T)21 b Fv(\()p Fs(F)8 b Fo(;)14 b Fs(f)p Fo(z)t Fs(g)p Fv(\).)639 5006 y(T)-7 b(o)27 b(start,)h(there)f(is)g(the)h(follo)n(wing)f(c)n(haracterization)e(of)j Fs(C)2519 5018 y Fq(E)2563 5006 y Fv(-termination.)1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fl(Lemma)29 b(4.2)41 b Fm(A)30 b(TRS)g Fs(R)h Fm(is)g Fs(C)1541 535 y Fq(E)1586 523 y Fm(-terminating)f(if)i (and)f(only)g(if)h(ther)l(e)f(is)g(no)g(in\014nite)f Fs(R)p Fm(-)515 623 y(chain)g(over)f Fs(R)17 b(])g(C)1124 635 y Fq(E)1168 623 y Fm(.)39 b(In)28 b(fact,)i(if)g Fs(R)f Fm(is)g(not)f Fs(C)1983 635 y Fq(E)2028 623 y Fm(-terminating,)h(then)g(ther)l(e)g(is)g(a)g(minimal)515 722 y(in\014nite)g Fs(R)p Fm(-chain)i(over)g Fs(R)18 b(])h(C)1516 734 y Fq(E)1561 722 y Fm(.)515 922 y(Pr)l(o)l(of)45 b Fv(A)21 b(TRS)g Fs(R)g Fv(is)g Fs(C)1235 934 y Fq(E)1279 922 y Fv(-terminating)f(if)i(and)e(only)g(if)i Fs(R)5 b(])g(C)2400 934 y Fq(E)2465 922 y Fv(is)21 b(terminating.)34 b(Hence)21 b(the)515 1021 y(lemma)i(follo)n(ws)g(directly)h(from)f(the) h(fact)g(that)g(the)g(dep)r(endency)h(pairs)e(of)g Fs(R)h Fv(and)g Fs(R)11 b(])g(C)3334 1033 y Fq(E)515 1121 y Fv(coincide)30 b(\(rewrite)f(rules)h(from)g Fs(C)1597 1133 y Fq(E)1672 1121 y Fv(do)g(not)g(cause)f(dep)r(endency)i(pairs\))e (in)i(conjunction)515 1220 y(with)d(Theorem)f(2.1.)2129 b Fk(\003)639 1420 y Fv(In)27 b(the)f(follo)n(wing,)g(w)n(e)f(deal)h (with)h(a)f(hierarc)n(hical)e(com)n(bination)h(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\))27 b(of)f(the)h(base)515 1519 y(system)38 b(\()p Fs(F)893 1531 y Fr(1)931 1519 y Fo(;)14 b Fs(R)1038 1531 y Fr(1)1075 1519 y Fv(\))40 b(and)e(the)i(extension)e (\()p Fs(F)1944 1531 y Fr(2)1982 1519 y Fo(;)14 b Fs(R)2089 1531 y Fr(2)2126 1519 y Fv(\).)72 b(That)38 b(is)h(to)g(sa)n(y)-7 b(,)41 b Fs(R)f Fv(is)e(the)i(TRS)515 1619 y Fs(R)585 1631 y Fr(1)633 1619 y Fs([)11 b(R)769 1631 y Fr(2)831 1619 y Fv(o)n(v)n(er)21 b(the)k(signature)d Fs(F)31 b Fv(=)23 b Fs(F)1739 1631 y Fr(1)1786 1619 y Fs([)11 b(F)1912 1631 y Fr(2)1973 1619 y Fv(and)24 b Fs(F)2191 1631 y Fr(1)2238 1619 y Fs(\\)11 b(D)2368 1631 y Fr(2)2429 1619 y Fv(=)22 b Fs(;)p Fv(.)36 b(Moreo)n(v)n(er,)21 b(w)n(e)j(assume)515 1719 y(that)k(b)r(oth)g Fs(R)961 1731 y Fr(1)1026 1719 y Fv(and)f Fs(R)1257 1731 y Fr(2)1323 1719 y Fv(are)f Fm(\014nitely)k(br)l(anching)p Fv(.)639 1818 y(Let)35 b Fo(s)f Fs(2)h(T)21 b Fv(\()p Fs(F)8 b Fv(\))35 b(with)g Fo(r)r(oot)p Fv(\()p Fo(s)p Fv(\))i Fs(2)e(F)1826 1830 y Fr(1)1862 1818 y Fv(.)58 b(W)-7 b(e)35 b(use)f(the)h(notation)f Fo(s)g Fv(=)g Fo(C)6 b Fv([)-14 b([)p Fo(s)3040 1830 y Fr(1)3078 1818 y Fo(;)14 b(:)g(:)g(:)g(;)g(s)3302 1830 y Fn(n)3347 1818 y Fv(])-14 b(])515 1918 y(if)33 b Fo(C)6 b Fv([)p Fo(;)14 b(:)g(:)g(:)g(;)g Fv(])32 b Fs(2)g(T)21 b Fv(\()p Fs(F)1169 1930 y Fr(1)1206 1918 y Fv(\))33 b(and)g Fo(r)r(oot)p Fv(\()p Fo(s)1658 1930 y Fr(1)1697 1918 y Fv(\))p Fo(;)14 b(:)g(:)g(:)f(;)h(r)r(oot)p Fv(\()p Fo(s)2133 1930 y Fn(n)2180 1918 y Fv(\))32 b Fs(2)g(F)2391 1930 y Fr(2)2450 1918 y Fs(n)21 b(F)2573 1930 y Fr(1)2610 1918 y Fv(.)53 b(In)33 b(this)g(situation,)g(a)515 2017 y(reduction)h(step)h Fo(s)g Fs(!)1231 2029 y Fq(R)1328 2017 y Fo(t)f Fv(is)h(called)g Fm(inner)g Fv(\(denoted)g(b)n(y)f Fo(s)h Fs(!)2585 1987 y Fn(i)2585 2040 y Fq(R)2682 2017 y Fo(t)p Fv(\),)i(if)e(the)g(reduction)515 2117 y(tak)n(es)28 b(place)h(in)g(one)g(of)g(the)h(subterms)f Fo(s)1833 2129 y Fr(1)1870 2117 y Fo(;)14 b(:)g(:)g(:)g(;)g(s)2094 2129 y Fn(n)2139 2117 y Fv(.)42 b(Otherwise,)28 b(w)n(e)h(sp)r(eak)g (of)g(an)g Fm(outer)515 2217 y Fv(reduction)e(step)h(and)f(write)h Fo(s)22 b Fs(!)1578 2187 y Fn(o)1578 2240 y Fq(R)1663 2217 y Fo(t)p Fv(.)639 2316 y(F)-7 b(urthermore,)31 b Fs(T)1193 2328 y Fr(SN)1283 2316 y Fv(\()p Fs(F)8 b Fv(\))32 b(stands)e(for)g(the)i(set)f Fs(f)p Fo(t)d Fs(2)h(T)21 b Fv(\()p Fs(F)8 b Fv(\))31 b Fs(j)62 b Fv(ev)n(ery)30 b(subterm)h Fo(t)3197 2286 y Fq(0)3251 2316 y Fv(of)g Fo(t)515 2416 y Fv(with)d Fo(r)r(oot)p Fv(\()p Fo(t)915 2386 y Fq(0)940 2416 y Fv(\))23 b Fs(2)g(F)1133 2428 y Fr(2)1189 2416 y Fs(n)18 b(F)1309 2428 y Fr(1)1373 2416 y Fv(is)28 b(terminating)f Fs(g)p Fv(.)639 2516 y(The)36 b(transformation)e(\010)i(de\014ned)g(b)r(elo)n(w)f(extends)h (Gramlic)n(h's)f([Gra94)n(])h(transfor-)515 2615 y(mation)27 b(to)h(hierarc)n(hical)d(com)n(binations.)515 2798 y Fl(De\014nition)31 b(4.3)40 b Fv(De\014ne)28 b(\010)23 b(:)g Fs(T)1555 2810 y Fr(SN)1645 2798 y Fv(\()p Fs(F)8 b Fv(\))24 b Fs(!)f(T)e Fv(\()p Fs(F)2065 2810 y Fr(1)2121 2798 y Fs(])e(f)p Fo(C)6 b(ons;)14 b(N)9 b(il)r Fs(g)p Fv(\))26 b(b)n(y)639 2964 y Fs(\017)41 b Fv(\010\()p Fo(t)p Fv(\))24 b(=)f Fo(t)p Fv(,)k(if)h Fo(t)23 b Fs(2)h(T)d Fv(\()p Fs(F)1434 2976 y Fr(1)1471 2964 y Fv(\))639 3130 y Fs(\017)41 b Fv(\010\()p Fo(t)p Fv(\))24 b(=)f Fo(C)6 b Fv([\010\()p Fo(s)1207 3142 y Fr(1)1244 3130 y Fv(\))p Fo(;)14 b(:)g(:)g(:)g(;)g Fv(\010\()p Fo(s)1592 3142 y Fn(n)1637 3130 y Fv(\)],)28 b(if)h Fo(r)r(oot)p Fv(\()p Fo(t)p Fv(\))24 b Fs(2)g(F)2226 3142 y Fr(1)2290 3130 y Fv(and)k Fo(t)23 b Fv(=)g Fo(C)6 b Fv([)-14 b([)p Fo(t)2720 3142 y Fr(1)2757 3130 y Fo(;)14 b(:)g(:)g(:)g(;)g(t)2972 3142 y Fn(n)3017 3130 y Fv(])-14 b(])639 3296 y Fs(\017)41 b Fv(\010\()p Fo(t)p Fv(\))24 b(=)f Fo(S)5 b(or)r(t)p Fv(\(\010)1245 3266 y Fq(\003)1284 3296 y Fv(\(\001)1385 3266 y Fq(\003)1385 3317 y Fr(1)1423 3296 y Fv(\()p Fo(t)p Fv(\)\)\),)29 b(if)f Fo(r)r(oot)p Fv(\()p Fo(t)p Fv(\))d Fs(2)e(F)2115 3308 y Fr(2)2171 3296 y Fs(n)18 b(F)2291 3308 y Fr(1)515 3462 y Fv(with)1251 3645 y(\001)1320 3610 y Fq(\003)1320 3665 y Fr(1)1358 3645 y Fv(\()p Fo(t)p Fv(\))84 b(=)e Fs(f)p Fo(u)22 b Fs(2)i(T)d Fv(\()p Fs(F)8 b Fv(\))28 b Fs(j)g Fo(t)23 b Fs(!)2287 3610 y Fq(\003)2287 3665 y(R)2371 3645 y Fo(u;)14 b(r)r(oot)p Fv(\()p Fo(u)p Fv(\))24 b Fs(2)f(F)2879 3657 y Fr(1)2916 3645 y Fs(g)1200 3769 y Fv(\010)1260 3735 y Fq(\003)1298 3769 y Fv(\()p Fo(M)9 b Fv(\))84 b(=)e Fs(f)p Fv(\010\()p Fo(u)p Fv(\))27 b Fs(j)h Fo(u)23 b Fs(2)g Fo(M)9 b Fs(g)27 b Fv(for)g Fo(M)32 b Fs(\022)22 b(T)2655 3781 y Fr(SN)2745 3769 y Fv(\()p Fs(F)8 b Fv(\))794 3894 y Fo(S)d(or)r(t)p Fv(\()p Fs(f)p Fo(t)1063 3906 y Fr(1)1101 3894 y Fo(;)14 b(:)g(:)g(:)g(;)g(t) 1316 3906 y Fn(m)1379 3894 y Fs(g)p Fv(\))83 b(=)f Fs(h)p Fo(t)1745 3909 y Fn(\031)r Fr(\(1\))1875 3894 y Fo(;)14 b(:)g(:)g(:)g(;)g(t)2090 3909 y Fn(\031)r Fr(\()p Fn(m)p Fr(\))2246 3894 y Fs(i)1711 4018 y Fv(suc)n(h)27 b(that)h Fo(t)2108 4033 y Fn(\031)r Fr(\()p Fn(j)s Fr(\))2250 4018 y Fs(\037)13 b Fo(t)2358 4033 y Fn(\031)r Fr(\()p Fn(j)s Fr(+1\))2597 4018 y Fv(for)28 b(1)22 b Fs(\024)h Fo(j)28 b(<)22 b(m)858 4143 y Fs(h)p Fo(t)920 4158 y Fn(\031)r Fr(\(1\))1050 4143 y Fo(;)14 b(:)g(:)g(:)f(;)h(t)1264 4158 y Fn(\031)r Fr(\()p Fn(m)p Fr(\))1420 4143 y Fs(i)84 b Fv(=)e Fo(C)6 b(ons)p Fv(\()p Fo(t)1939 4158 y Fn(\031)r Fr(\(1\))2070 4143 y Fo(;)14 b(:)g(:)g(:)f(;)h(C)6 b(ons)p Fv(\()p Fo(t)2510 4158 y Fn(\031)r Fr(\()p Fn(m)p Fr(\))2667 4143 y Fo(;)14 b(N)9 b(il)r Fv(\))14 b Fo(:)g(:)g(:)e Fv(\))515 4325 y(where)27 b Fs(\037)g Fv(is)h(an)f(arbitrary)e(but)k (\014xed)e(total)h(ordering)d(on)j Fs(T)21 b Fv(\()p Fs(F)2552 4337 y Fr(1)2608 4325 y Fs(])e(f)p Fo(C)6 b(ons;)14 b(N)9 b(il)r Fs(g)p Fv(\).)639 4425 y(Note)30 b(that)f(if)h Fo(t)c Fs(2)g(T)1283 4437 y Fr(SN)1373 4425 y Fv(\()p Fs(F)8 b Fv(\),)31 b(then)e(\001)1818 4395 y Fq(\003)1818 4446 y Fr(1)1857 4425 y Fv(\()p Fo(t)p Fv(\))d Fs(\022)g(T)2113 4437 y Fr(SN)2203 4425 y Fv(\()p Fs(F)8 b Fv(\).)42 b(Th)n(us,)30 b(\010)2696 4395 y Fq(\003)2764 4425 y Fv(is)f(alw)n(a)n(ys)e(applied) 515 4525 y(to)g(a)g(subset)h(of)g Fs(T)1080 4537 y Fr(SN)1170 4525 y Fv(\()p Fs(F)8 b Fv(\).)1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fl(Example)30 b(4.4)40 b Fv(Consider)27 b(the)h(t)n(w)n(o)f(TRSs)1274 821 y Fs(R)1344 833 y Fr(1)1465 821 y Fv(=)1552 601 y Fe(8)1552 676 y(>)1552 701 y(>)1552 726 y(<)1552 875 y(>)1552 900 y(>)1552 925 y(:)1670 671 y Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(x)p Fv(\))83 b Fs(!)g Fo(f)9 b Fv(\()p Fo(x;)14 b(x;)g(x)p Fv(\))1668 771 y Fo(f)9 b Fv(\()p Fo(x;)14 b(y)s(;)g(z)t Fv(\))82 b Fs(!)h Fv(2)1947 870 y(0)g Fs(!)g Fv(2)1947 970 y(1)g Fs(!)g Fv(2)1274 1220 y Fs(R)1344 1232 y Fr(2)1465 1220 y Fv(=)1552 1103 y Fe(\032)1656 1169 y Fo(g)j Fs(!)d Fv(0)1656 1269 y Fo(g)j Fs(!)d Fv(1)515 1423 y(Then)31 b(\010\()p Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(g)s Fv(\)\))27 b(=)h Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g Fv(\010\()p Fo(g)s Fv(\)\))28 b(=)g Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(C)6 b(ons)p Fv(\(0)p Fo(;)14 b(C)6 b(ons)p Fv(\(1)p Fo(;)14 b(N)9 b(il)r Fv(\)\)\),)29 b(where)i Fs(\037)f Fv(is)515 1523 y(a)d(total)g(ordering)f(on)h Fs(T)22 b Fv(\()p Fs(F)1384 1535 y Fr(1)1439 1523 y Fs(])d(f)p Fo(C)6 b(ons;)14 b(N)9 b(il)r Fs(g)p Fv(\))26 b(with)i(0)23 b Fs(\037)f Fv(1.)515 1677 y Fl(Lemma)29 b(4.5)41 b Fm(L)l(et)29 b Fo(s)23 b Fs(2)g(T)1345 1689 y Fr(SN)1436 1677 y Fv(\()p Fs(F)8 b Fv(\))p Fm(.)613 1819 y(1.)42 b(If)30 b Fo(s)23 b Fs(2)h(T)d Fv(\()p Fs(F)1108 1831 y Fr(1)1145 1819 y Fv(\))30 b Fm(and)h Fo(s)23 b Fs(!)1514 1831 y Fq(R)1598 1819 y Fo(t)p Fm(,)30 b(then)g Fv(\010\()p Fo(s)p Fv(\))23 b Fs(!)2137 1831 y Fq(R)2194 1839 y Ff(1)2254 1819 y Fv(\010\()p Fo(t)p Fv(\))p Fm(.)613 1974 y(2.)42 b(If)30 b Fo(r)r(oot)p Fv(\()p Fo(s)p Fv(\))25 b Fs(2)f(F)1225 1986 y Fr(1)1291 1974 y Fm(and)31 b Fo(s)22 b Fv(=)h Fo(C)6 b Fv([)-14 b([)p Fo(s)1738 1986 y Fr(1)1776 1974 y Fo(;)14 b(:)g(:)g(:)f(;)h(s)1999 1986 y Fn(n)2044 1974 y Fv(])-14 b(])p Fm(,)31 b(then)753 2128 y(\(a\))42 b Fo(s)23 b Fs(!)1050 2098 y Fn(o)1050 2151 y Fq(R)1134 2128 y Fo(t)30 b Fm(implies)h Fv(\010\()p Fo(s)p Fv(\))24 b Fs(!)1748 2140 y Fq(R)1805 2148 y Ff(1)1864 2128 y Fv(\010\()p Fo(t)p Fv(\))p Fm(,)757 2250 y(\(b\))42 b Fo(s)23 b Fs(!)1050 2219 y Fn(i)1050 2272 y Fq(R)1134 2250 y Fo(t)30 b Fm(implies)h Fv(\010\()p Fo(s)p Fv(\))24 b Fs(!)1748 2219 y Fq(\003)1748 2272 y(C)1784 2280 y Fi(E)1850 2250 y Fv(\010\()p Fo(t)p Fv(\))p Fm(.)613 2404 y(3.)42 b(If)30 b Fo(r)r(oot)p Fv(\()p Fo(s)p Fv(\))25 b Fs(2)f(F)1225 2416 y Fr(2)1280 2404 y Fs(n)18 b(F)1400 2416 y Fr(1)1467 2404 y Fm(and)30 b Fo(s)23 b Fs(!)1773 2416 y Fq(R)1857 2404 y Fo(t)p Fm(,)30 b(then)g Fv(\010\()p Fo(s)p Fv(\))23 b Fs(!)2396 2374 y Fq(\003)2396 2427 y(C)2432 2435 y Fi(E)2499 2404 y Fv(\010\()p Fo(t)p Fv(\))p Fm(.)515 2580 y(Pr)l(o)l(of)58 b Fv(\(1\))28 b(T)-7 b(rivial)27 b(b)r(ecause)g(\010\()p Fo(s)p Fv(\))c(=)g Fo(s)28 b Fv(and)f(\010\()p Fo(t)p Fv(\))d(=)e Fo(t)p Fv(.)515 2680 y(\(2a\))35 b(W)-7 b(e)35 b(ha)n(v)n(e)f(either)h Fo(s)h Fv(=)f Fo(C)6 b Fv([)-14 b([)p Fo(s)1601 2692 y Fr(1)1639 2680 y Fo(;)14 b(:)g(:)g(:)g(;)g(s)1863 2692 y Fn(n)1908 2680 y Fv(])-14 b(])36 b Fs(!)2059 2649 y Fn(o)2059 2702 y Fq(R)2156 2680 y Fo(C)2221 2649 y Fq(0)2244 2680 y Fv([)-14 b([)p Fo(s)2315 2692 y Fn(i)2338 2700 y Ff(1)2375 2680 y Fo(;)14 b(:)g(:)g(:)g(;)g(s)2599 2692 y Fn(i)2622 2700 y Fd(m)2682 2680 y Fv(])-14 b(])35 b(with)h Fo(i)2975 2692 y Fr(1)3012 2680 y Fo(;)14 b(:)g(:)g(:)f(;)h(i)3225 2692 y Fn(m)3324 2680 y Fs(2)515 2779 y(f)p Fv(1)p Fo(;)g(:)g(:)g(:)e (;)i(n)p Fs(g)24 b Fv(or)g Fo(s)f Fv(=)g Fo(C)6 b Fv([)-14 b([)p Fo(s)1283 2791 y Fr(1)1321 2779 y Fo(;)14 b(:)g(:)g(:)f(;)h(s) 1544 2791 y Fn(n)1589 2779 y Fv(])-14 b(])24 b Fs(!)1728 2749 y Fn(o)1728 2802 y Fq(R)1812 2779 y Fo(s)1851 2791 y Fn(j)1909 2779 y Fv(=)e Fo(t)j Fv(for)g(some)f Fo(j)k Fs(2)23 b(f)p Fv(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(n)p Fs(g)p Fv(.)35 b(Applying)25 b(\010,)515 2879 y(w)n(e)k(obtain)h(\010\()p Fo(s)p Fv(\))d(=)f Fo(C)6 b Fv([\010\()p Fo(s)1400 2891 y Fr(1)1438 2879 y Fv(\))p Fo(;)14 b(:)g(:)g(:)g(;)g Fv(\010\()p Fo(s)1786 2891 y Fn(n)1831 2879 y Fv(\)])30 b(and)g(either)g(\010\()p Fo(t)p Fv(\))d(=)f Fo(C)2655 2849 y Fq(0)2679 2879 y Fv([\010\()p Fo(s)2833 2891 y Fn(i)2856 2899 y Ff(1)2893 2879 y Fv(\))p Fo(;)14 b(:)g(:)g(:)g(;)g Fv(\010\()p Fo(s)3241 2891 y Fn(i)3264 2899 y Fd(m)3324 2879 y Fv(\)])515 2978 y(or)27 b(\010\()p Fo(t)p Fv(\))e(=)g(\010\()p Fo(s)1017 2990 y Fn(j)1052 2978 y Fv(\).)40 b(Th)n(us)28 b(\010\()p Fo(s)p Fv(\))d Fs(!)1629 2990 y Fq(R)1715 2978 y Fv(\010\()p Fo(t)p Fv(\))k(b)n(y)f(the)h(same)f(rule)g(b)r (ecause)g Fo(s)2881 2990 y Fn(i)2933 2978 y Fv(=)c Fo(s)3061 2990 y Fn(j)3125 2978 y Fv(implies)515 3078 y(\010\()p Fo(s)646 3090 y Fn(i)674 3078 y Fv(\))f(=)g(\010\()p Fo(s)948 3090 y Fn(j)983 3078 y Fv(\))28 b(for)f(all)h(1)23 b Fs(\024)g Fo(i)g(<)f(j)29 b Fs(\024)23 b Fo(n)k Fv(\(i.e.,)h(ev)n(en) g(non-left-linear)e(rules)i(do)f(not)h(cause)515 3178 y(trouble\).)515 3277 y(\(2b\))j(Supp)r(ose)g Fo(s)e Fv(=)f Fo(C)6 b Fv([)-14 b([)p Fo(s)1324 3289 y Fr(1)1362 3277 y Fo(;)14 b(:)g(:)g(:)g(;)g(s)1586 3289 y Fn(j)1620 3277 y Fo(;)g(:)g(:)g(:)g(s)1807 3289 y Fn(n)1852 3277 y Fv(])-14 b(])29 b Fs(!)1996 3247 y Fn(i)1996 3300 y Fq(R)2086 3277 y Fo(C)6 b Fv([)p Fo(s)2213 3289 y Fr(1)2251 3277 y Fo(;)14 b(:)g(:)g(:)f(;)h(s)2474 3247 y Fq(0)2474 3299 y Fn(j)2509 3277 y Fo(;)g(:)g(:)g(:)g(s)2696 3289 y Fn(n)2741 3277 y Fv(])29 b(=)f Fo(t)p Fv(,)k(i.e.,)g Fo(s)3171 3289 y Fn(j)3235 3277 y Fs(!)3318 3289 y Fq(R)515 3390 y Fo(s)554 3360 y Fq(0)554 3412 y Fn(j)589 3390 y Fv(.)k(If)27 b Fo(r)r(oot)p Fv(\()p Fo(s)950 3360 y Fq(0)950 3412 y Fn(j)986 3390 y Fv(\))c Fs(2)h(F)1180 3402 y Fr(1)1217 3390 y Fv(,)i(then)h Fo(s)1493 3360 y Fq(0)1493 3412 y Fn(j)1551 3390 y Fs(2)c Fv(\001)1698 3360 y Fq(\003)1698 3411 y Fr(1)1737 3390 y Fv(\()p Fo(s)1808 3402 y Fn(j)1843 3390 y Fv(\))j(and)g(hence)g(\010\()p Fo(s)2421 3402 y Fn(j)2457 3390 y Fv(\))d Fs(!)2595 3355 y Fr(+)2595 3414 y Fq(C)2631 3422 y Fi(E)2698 3390 y Fv(\010\()p Fo(s)2829 3360 y Fq(0)2829 3412 y Fn(j)2864 3390 y Fv(\).)36 b(If)27 b(otherwise)515 3490 y Fo(r)r(oot)p Fv(\()p Fo(s)735 3460 y Fq(0)735 3511 y Fn(j)771 3490 y Fv(\))c Fs(2)h(F)965 3502 y Fr(2)1020 3490 y Fs(n)18 b(F)1140 3502 y Fr(1)1177 3490 y Fv(,)28 b(then)g(\001)1486 3460 y Fq(\003)1486 3510 y Fr(1)1524 3490 y Fv(\()p Fo(s)1595 3460 y Fq(0)1595 3511 y Fn(j)1631 3490 y Fv(\))23 b Fs(\022)g Fv(\001)1843 3460 y Fq(\003)1843 3510 y Fr(1)1881 3490 y Fv(\()p Fo(s)1952 3502 y Fn(j)1987 3490 y Fv(\))28 b(implies)g(that)g(\010\()p Fo(s)2640 3502 y Fn(j)2675 3490 y Fv(\))23 b Fs(!)2813 3460 y Fq(\003)2813 3513 y(C)2849 3521 y Fi(E)2916 3490 y Fv(\010\()p Fo(s)3047 3460 y Fq(0)3047 3511 y Fn(j)3082 3490 y Fv(\).)515 3589 y(\(3\))k(As)h(in)g(case)f(\(2b\).)2095 b Fk(\003)515 3765 y Fl(Lemma)29 b(4.6)41 b Fm(If)30 b Fo(s)23 b Fs(2)g(T)1289 3777 y Fr(SN)1379 3765 y Fv(\()p Fs(F)8 b Fv(\))31 b Fm(and)f Fo(s)23 b Fs(!)1848 3735 y Fq(\003)1848 3788 y(R)1932 3765 y Fo(t)p Fm(,)30 b(then)g Fv(\010\()p Fo(s)p Fv(\))23 b Fs(!)2471 3735 y Fq(\003)2471 3788 y(R)2528 3796 y Ff(1)2561 3788 y Fq(]C)2642 3796 y Fi(E)2708 3765 y Fv(\010\()p Fo(t)p Fv(\))p Fm(.)515 3941 y(Pr)l(o)l(of)59 b Fv(Using)28 b(Lemma)f(4.5,)h(this)g(follo)n(ws)f(b)n(y)g(induction)i (on)e(the)i(length)f(of)g(the)g(rewrite)515 4041 y(sequence)f Fo(s)c Fs(!)1004 4011 y Fq(\003)1004 4064 y(R)1088 4041 y Fo(t)p Fv(.)2173 b Fk(\003)639 4217 y Fv(Let)30 b Fs(R)860 4229 y Fr(1)927 4217 y Fv(and)f Fs(R)1160 4229 y Fr(2)1228 4217 y Fv(b)r(e)h(as)e(in)i(Example)f(4.4.)42 b(If)30 b(w)n(e)f(apply)g(the)h(transformation)e(\010)h(to)515 4317 y(the)k(deriv)-5 b(ation)32 b Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(g)s Fv(\))30 b Fs(!)1486 4329 y Fq(R)1543 4337 y Ff(1)1611 4317 y Fo(f)9 b Fv(\()p Fo(g)s(;)14 b(g)s(;)g(g)s Fv(\))30 b Fs(!)2041 4329 y Fq(R)2098 4337 y Ff(2)2166 4317 y Fo(f)9 b Fv(\(0)p Fo(;)14 b(g)s(;)g(g)s Fv(\))30 b Fs(!)2595 4329 y Fq(R)2652 4337 y Ff(2)2720 4317 y Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(g)s Fv(\),)32 b(then)h(w)n(e)515 4416 y(obtain)1652 4548 y Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(t)p Fv(\))22 b(=)g(\010\()p Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(g)s Fv(\)\))1392 4648 y Fs(!)1475 4660 y Fq(R)1532 4668 y Ff(1)1652 4648 y Fo(f)9 b Fv(\()p Fo(t;)14 b(t;)g(t)p Fv(\))23 b(=)f(\010\()p Fo(f)9 b Fv(\()p Fo(g)s(;)14 b(g)s(;)g(g)s Fv(\)\))1406 4748 y Fs(!)1489 4760 y Fq(C)1525 4768 y Fi(E)1652 4748 y Fo(f)9 b Fv(\(0)p Fo(;)14 b(t;)g(t)p Fv(\))22 b(=)h(\010\()p Fo(f)9 b Fv(\(0)p Fo(;)14 b(g)s(;)g(g)s Fv(\)\))1406 4849 y Fs(!)1489 4813 y Fr(+)1489 4873 y Fq(C)1525 4881 y Fi(E)1652 4849 y Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(t)p Fv(\))22 b(=)g(\010\()p Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(g)s Fv(\)\))515 5006 y(where)27 b Fo(t)c Fv(=)f(\010\()p Fo(g)s Fv(\))i(=)e Fo(C)6 b(ons)p Fv(\(0)p Fo(;)14 b(C)6 b(ons)p Fv(\(1)p Fo(;)14 b(N)9 b(il)r Fv(\)\).)1905 5255 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fl(Lemma)29 b(4.7)41 b Fm(If)30 b(ther)l(e)g(is)g (a)h(minimal)g(in\014nite)e Fs(R)2156 535 y Fr(1)2194 523 y Fm(-chain)i(over)f Fs(R)p Fm(,)h(then)f(ther)l(e)g(is)g(also)515 623 y(an)g(in\014nite)f Fs(R)990 635 y Fr(1)1028 623 y Fm(-chain)h(over)h Fs(R)1536 635 y Fr(1)1592 623 y Fs(])19 b(C)1710 635 y Fq(E)1754 623 y Fm(.)515 822 y(Pr)l(o)l(of)78 b Fv(Let)38 b Fs(h)p Fo(s)1015 834 y Fr(1)1052 822 y Fo(;)14 b(t)1119 834 y Fr(1)1156 822 y Fs(ih)p Fo(s)1259 834 y Fr(2)1297 822 y Fo(;)g(t)1364 834 y Fr(2)1401 822 y Fs(i)g Fo(:)g(:)g(:)52 b Fv(b)r(e)38 b(a)g(minimal)g(in\014nite)g Fs(R)2493 834 y Fr(1)2531 822 y Fv(-c)n(hain)f(o)n(v)n(er)f Fs(R)p Fv(,)41 b(that)d(is,)515 922 y(there)d(is)h(a)f(substitution)i Fo(\033)j Fv(:)c Fs(V)44 b(!)36 b(T)22 b Fv(\()p Fs(F)8 b Fv(\))36 b(suc)n(h)g(that)g Fo(t)2384 934 y Fn(j)2419 922 y Fo(\033)k Fs(!)2589 891 y Fq(\003)2589 944 y(R)2687 922 y Fo(s)2726 934 y Fn(j)s Fr(+1)2844 922 y Fo(\033)g Fv(holds)35 b(for)g(all)515 1021 y(consecutiv)n(e)23 b(pairs)h Fs(h)p Fo(s)1222 1033 y Fn(j)1257 1021 y Fo(;)14 b(t)1324 1033 y Fn(j)1359 1021 y Fs(i)25 b Fv(and)g Fs(h)p Fo(s)1646 1033 y Fn(j)s Fr(+1)1765 1021 y Fo(;)14 b(t)1832 1033 y Fn(j)s Fr(+1)1951 1021 y Fs(i)25 b Fv(in)g(the)g(sequence.)36 b(Since)25 b(the)g Fs(R)3042 1033 y Fr(1)3079 1021 y Fv(-c)n(hain)f(is)515 1121 y(minimal,)32 b(ev)n(ery)d(subterm)i(of)g Fo(s)1551 1133 y Fn(j)1586 1121 y Fo(\033)j Fv(and)d Fo(t)1862 1133 y Fn(j)1897 1121 y Fo(\033)s Fv(,)h(resp)r(ectiv)n(ely) -7 b(,)31 b(is)g(an)g(elemen)n(t)g(of)f Fs(T)3133 1133 y Fr(SN)3223 1121 y Fv(\()p Fs(F)8 b Fv(\).)515 1220 y(Therefore,)35 b(b)n(y)f(Lemma)g(4.6,)i(\010\()p Fo(t)1636 1232 y Fn(j)1671 1220 y Fo(\033)s Fv(\))g Fs(!)1872 1190 y Fq(\003)1872 1243 y(R)1929 1251 y Ff(1)1961 1243 y Fq(]C)2042 1251 y Fi(E)2120 1220 y Fv(\010\()p Fo(s)2251 1232 y Fn(j)s Fr(+1)2370 1220 y Fo(\033)s Fv(\))g(holds)e(for)g(all)g (consecutiv)n(e)515 1320 y(pairs)24 b Fs(h)p Fo(s)787 1332 y Fn(j)823 1320 y Fo(;)14 b(t)890 1332 y Fn(j)925 1320 y Fs(i)26 b Fv(and)f Fs(h)p Fo(s)1213 1332 y Fn(j)s Fr(+1)1332 1320 y Fo(;)14 b(t)1399 1332 y Fn(j)s Fr(+1)1518 1320 y Fs(i)26 b Fv(in)g(the)g(sequence.)36 b(In)26 b(other)f(w)n (ords,)g Fs(h)p Fo(s)2836 1332 y Fr(1)2873 1320 y Fo(;)14 b(t)2940 1332 y Fr(1)2977 1320 y Fs(ih)p Fo(s)3080 1332 y Fr(2)3118 1320 y Fo(;)g(t)3185 1332 y Fr(2)3222 1320 y Fs(i)g Fo(:)g(:)g(:)515 1420 y Fv(is)27 b(an)h Fs(R)784 1432 y Fr(1)821 1420 y Fv(-c)n(hain)f(o)n(v)n(er)f Fs(R)1316 1432 y Fr(1)1372 1420 y Fs(])19 b(C)1490 1432 y Fq(E)1534 1420 y Fv(.)1757 b Fk(\003)639 1619 y Fv(No)n(w)25 b(w)n(e)g(are)f(in)i (a)f(p)r(osition)g(to)g(pro)n(v)n(e)e(Urbain's)i([Urb01)o(])g(main)h (result)f(ab)r(out)g(incre-)515 1719 y(men)n(tal)i(termination)g(pro)r (ofs.)515 1901 y Fl(Theorem)j(4.8)41 b Fm(L)l(et)31 b Fs(R)i Fm(b)l(e)f(the)h(hier)l(ar)l(chic)l(al)i(c)l(ombination)e(of)g (two)g(\014nitely)f(br)l(anching)515 2001 y(TRSs)d Fs(R)815 2013 y Fr(1)882 2001 y Fm(and)i Fs(R)1114 2013 y Fr(2)1151 2001 y Fm(.)613 2167 y(1.)42 b Fs(R)g Fm(is)h(terminating)e(if)i(b)l (ase)f(system)g Fs(R)2035 2179 y Fr(1)2114 2167 y Fm(is)g Fs(C)2259 2179 y Fq(E)2304 2167 y Fm(-terminating)g(and)g(ther)l(e)g (is)g(no)722 2267 y(minimal)31 b(in\014nite)e Fs(R)1405 2279 y Fr(2)1443 2267 y Fm(-chain)i(over)f Fs(R)p Fm(.)613 2433 y(2.)42 b Fs(R)d Fm(is)h(terminating)e(if)i Fs(R)1549 2445 y Fr(1)1625 2433 y Fm(is)g Fs(C)1768 2445 y Fq(E)1812 2433 y Fm(-terminating)f(and)g(ther)l(e)g(is)g(a)g(r)l(e)l(duction)g(p) l(air)722 2532 y Fv(\()p Fk(\045)p Fo(;)14 b Fs(\037)p Fv(\))30 b Fm(such)f(that)822 2698 y Fs(\017)41 b Fo(l)24 b Fk(\045)f Fo(r)33 b Fm(for)d(al)t(l)h(rules)f Fo(l)24 b Fs(!)f Fo(r)33 b Fm(in)c Fs(R)h Fm(and)822 2831 y Fs(\017)41 b Fo(s)23 b Fs(\037)g Fo(t)29 b Fm(for)i(al)t(l)g(dep)l(endency)g(p)l (airs)g Fs(h)p Fo(s;)14 b(t)p Fs(i)30 b Fm(of)g Fs(R)2377 2843 y Fr(2)2415 2831 y Fm(.)515 3030 y(Pr)l(o)l(of)80 b Fv(\(1\))39 b(F)-7 b(or)38 b(a)h(pro)r(of)f(b)n(y)g(con)n (tradiction,)j(supp)r(ose)d(that)h Fs(R)k Fv(=)e Fs(R)2873 3042 y Fr(1)2937 3030 y Fs([)26 b(R)3088 3042 y Fr(2)3164 3030 y Fv(is)39 b(not)515 3130 y(terminating.)33 b(According)18 b(to)g(Theorem)g(2.1,)h(there)f(is)h(a)f(minimal)g(in\014nite)i Fs(R)p Fv(-c)n(hain)e(\(o)n(v)n(er)515 3230 y Fs(R)p Fv(\).)54 b(By)32 b(assumption,)i(there)f(is)g(no)g(minimal)g (in\014nite)h Fs(R)2411 3242 y Fr(2)2448 3230 y Fv(-c)n(hain)f(o)n(v)n (er)e Fs(R)p Fv(.)53 b(Therefore)515 3329 y(the)40 b(minimal)g (in\014nite)h Fs(R)p Fv(-c)n(hain)e(m)n(ust)h(con)n(tain)g(a)f(dep)r (endency)i(pair)e(from)g Fs(R)3159 3341 y Fr(1)3197 3329 y Fv(,)k(sa)n(y)515 3429 y Fs(h)p Fo(s)586 3441 y Fn(j)621 3429 y Fo(;)14 b(t)688 3441 y Fn(j)723 3429 y Fs(i)p Fv(.)36 b(Since)27 b(the)f(rewrite)f(rules)g(from)h Fs(R)1913 3441 y Fr(1)1976 3429 y Fv(do)g(not)g(con)n(tain)f(sym)n(b)r(ols)g (from)h Fs(D)3101 3441 y Fr(2)3138 3429 y Fv(,)g(ev)n(ery)515 3528 y(dep)r(endency)34 b(pair)f Fs(h)p Fo(s)1221 3540 y Fn(i)1249 3528 y Fo(;)14 b(t)1316 3540 y Fn(i)1343 3528 y Fs(i)34 b Fv(in)g(the)g(minimal)g(in\014nite)h Fs(R)p Fv(-c)n(hain)e(with)h Fo(i)f Fs(\025)g Fo(j)38 b Fv(m)n(ust)c(also)515 3628 y(originate)27 b(from)h(a)h(rewrite)f (rule)g(of)h Fs(R)1746 3640 y Fr(1)1784 3628 y Fv(.)40 b(In)29 b(other)f(w)n(ords,)g(there)g(is)h(a)f(minimal)h(in\014nite)515 3728 y Fs(R)585 3740 y Fr(1)622 3728 y Fv(-c)n(hain)j(o)n(v)n(er)e Fs(R)p Fv(.)52 b(According)31 b(to)h(Lemma)g(4.7,)h(there)f(is)g(also)f (an)h(in\014nite)h Fs(R)3122 3740 y Fr(1)3160 3728 y Fv(-c)n(hain)515 3827 y(o)n(v)n(er)h Fs(R)771 3839 y Fr(1)832 3827 y Fs(])24 b(C)955 3839 y Fq(E)1000 3827 y Fv(.)62 b(W)-7 b(e)36 b(conclude)f(from)h(Lemma)f(4.2)g(that)h Fs(R)2494 3839 y Fr(1)2568 3827 y Fv(is)f(not)h Fs(C)2859 3839 y Fq(E)2904 3827 y Fv(-terminating.)515 3927 y(This)27 b(con)n(tradiction)g(pro)n(v)n(es)e(statemen)n(t)j(\(1\).)515 4027 y(\(2\))f(F)-7 b(ollo)n(ws)27 b(directly)g(from)h(\(1\).)1743 b Fk(\003)639 4226 y Fv(F)-7 b(or)39 b(\014nite)h(TRSs,)j(Theorem)38 b(4.8)h(can)g(b)r(e)h(reform)n(ulated)e(suc)n(h)h(that)h(it)g(captures) 515 4325 y(cycles.)e(The)29 b(next)g(example)f(from)g([Gra94)n(])h (demonstrates)e(that)i(in)f(Theorem)g(4.8,)g(the)515 4425 y(presumption)g(\\there)g(is)h(no)f(in\014nite)i Fs(R)1805 4437 y Fr(2)1842 4425 y Fv(-c)n(hain)e(o)n(v)n(er)f Fs(R)2339 4437 y Fr(1)2396 4425 y Fs([)19 b(R)2540 4437 y Fr(2)2578 4425 y Fv(")28 b(cannot)g(b)r(e)h(w)n(eak)n(ened)515 4525 y(to)e(\\there)g(is)h(no)f(in\014nite)h Fs(R)1420 4537 y Fr(2)1458 4525 y Fv(-c)n(hain)f(o)n(v)n(er)e Fs(R)1952 4537 y Fr(2)1990 4525 y Fv(.")515 4707 y Fl(Example)30 b(4.9)40 b Fv(The)d(TRSs)g Fs(R)1559 4719 y Fr(1)1635 4707 y Fv(=)h Fs(f)p Fo(a)f Fs(!)i Fo(b)p Fs(g)c Fv(and)i Fs(R)2337 4719 y Fr(2)2413 4707 y Fv(=)h Fs(f)p Fo(h)p Fv(\()p Fo(x;)14 b(x)p Fv(\))39 b Fs(!)f Fo(h)p Fv(\()p Fo(a;)14 b(b)p Fv(\))p Fs(g)36 b Fv(are)515 4807 y(simply)h (terminating,)i(hence)f Fs(C)1562 4819 y Fq(E)1606 4807 y Fv(-terminating)f(according)e(to)i(Prop)r(osition)f(3.2.)65 b(But)515 4907 y(their)27 b(union)h(is)f(not)h(terminating:)37 b Fo(h)p Fv(\()p Fo(a;)14 b(b)p Fv(\))22 b Fs(!)1994 4919 y Fq(R)2051 4927 y Ff(1)2111 4907 y Fo(h)p Fv(\()p Fo(b;)14 b(b)p Fv(\))23 b Fs(!)2438 4919 y Fq(R)2495 4927 y Ff(2)2554 4907 y Fo(h)p Fv(\()p Fo(a;)14 b(b)p Fv(\).)1905 5255 y(11)p eop %%Page: 12 12 12 11 bop 639 523 a Fv(A)24 b(sligh)n(t)e(mo)r(di\014cation)h(of)g (Theorem)f(4.8)h(yields)f(a)h(metho)r(d)h(to)e(incremen)n(tally)h(pro)n (v)n(e)515 623 y Fs(C)559 635 y Fq(E)603 623 y Fv(-termination,)k(whic) n(h)h(is)f(also)g(due)h(to)f(Urbain)g([Urb01)o(].)515 794 y Fl(Corollary)k(4.10)41 b Fm(L)l(et)24 b Fs(R)h Fm(b)l(e)g(the)g(hier)l(ar)l(chic)l(al)j(c)l(ombination)e(of)f(two)g (\014nitely)g(br)l(anching)515 894 y(TRSs)k Fs(R)815 906 y Fr(1)882 894 y Fm(and)i Fs(R)1114 906 y Fr(2)1151 894 y Fm(.)613 1051 y(1.)42 b Fs(R)33 b Fm(is)f Fs(C)960 1063 y Fq(E)1004 1051 y Fm(-terminating)g(if)h(b)l(ase)f(system)g Fs(R)2094 1063 y Fr(1)2164 1051 y Fm(is)g Fs(C)2299 1063 y Fq(E)2343 1051 y Fm(-terminating)g(and)h(ther)l(e)f(is)g(no)722 1151 y(minimal)f(in\014nite)e Fs(R)1405 1163 y Fr(2)1443 1151 y Fm(-chain)i(over)f Fs(R)19 b(])g(C)2088 1163 y Fq(E)2132 1151 y Fm(.)613 1312 y(2.)42 b Fs(R)30 b Fm(is)f Fs(C)954 1324 y Fq(E)998 1312 y Fm(-terminating)g(if)h Fs(R)1628 1324 y Fr(1)1694 1312 y Fm(is)g Fs(C)1827 1324 y Fq(E)1871 1312 y Fm(-terminating)f(and)g(ther)l(e)g(is)h(a)f(r)l(e)l (duction)g(p)l(air)722 1412 y Fv(\()p Fk(\045)p Fo(;)14 b Fs(\037)p Fv(\))30 b Fm(such)f(that)822 1574 y Fs(\017)41 b Fo(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))24 b Fk(\045)f Fo(x)30 b Fm(and)g Fo(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))24 b Fk(\045)e Fo(y)s Fm(,)822 1702 y Fs(\017)41 b Fo(l)24 b Fk(\045)f Fo(r)33 b Fm(for)d(al)t(l)h(rules)f Fo(l)24 b Fs(!)f Fo(r)33 b Fm(in)c Fs(R)p Fm(,)822 1830 y Fs(\017)41 b Fo(s)23 b Fs(\037)g Fo(t)29 b Fm(for)i(al)t(l)g(dep)l (endency)g(p)l(airs)g Fs(h)p Fo(s;)14 b(t)p Fs(i)30 b Fm(of)g Fs(R)2377 1842 y Fr(2)2415 1830 y Fm(.)515 2021 y(Pr)l(o)l(of)58 b Fv(Apply)28 b(Theorem)f(4.8)g(with)h Fs(R)1758 2033 y Fr(2)1814 2021 y Fs(])19 b(C)1932 2033 y Fq(E)2004 2021 y Fv(instead)28 b(of)f Fs(R)2455 2033 y Fr(2)2493 2021 y Fv(.)798 b Fk(\003)639 2211 y Fv(W)-7 b(e)25 b(exemplify)f(the)g(usefulness)g(of)g(Corollary)e(4.10)g(b)n(y)i (an)g(example)f(from)h([GA)n(O01)o(].)515 2383 y Fl(Example)30 b(4.11)40 b Fv(Let)864 2649 y Fs(R)934 2661 y Fr(1)995 2649 y Fv(=)1083 2479 y Fe(8)1083 2554 y(<)1083 2703 y(:)1544 2549 y Fo(add)p Fv(\()p Fo(nil)r(;)14 b(y)s(s)p Fv(\))82 b Fs(!)h Fo(y)s(s)1452 2648 y(add)p Fv(\(0)23 b(:)g Fo(xs;)14 b(y)s(s)p Fv(\))83 b Fs(!)g Fo(add)p Fv(\()p Fo(xs;)14 b(y)s(s)p Fv(\))1198 2748 y Fo(add)p Fv(\()p Fo(s)p Fv(\()p Fo(n)p Fv(\))24 b(:)f Fo(xs;)14 b(m)23 b Fv(:)h Fo(y)s(s)p Fv(\))82 b Fs(!)h Fo(add)p Fv(\()p Fo(n)24 b Fv(:)f Fo(xs;)14 b(s)p Fv(\()p Fo(m)p Fv(\))24 b(:)f Fo(y)s(s)p Fv(\))515 2916 y(and)760 3117 y Fs(R)830 3129 y Fr(2)890 3117 y Fv(=)978 2999 y Fe(\032)1205 3066 y Fo(w)r(eig)s(ht)p Fv(\()p Fo(n)g Fv(:)g Fo(nil)r Fv(\))82 b Fs(!)h Fo(n)1082 3166 y(w)r(eig)s(ht)p Fv(\()p Fo(n)22 b Fv(:)h Fo(m)g Fv(:)h Fo(xs)p Fv(\))83 b Fs(!)g Fo(w)r(eig)s(ht)p Fv(\()p Fo(add)p Fv(\()p Fo(n)23 b Fv(:)h Fo(m)e Fv(:)i Fo(xs;)14 b Fv(0)22 b(:)i Fo(xs)p Fv(\)\))515 3338 y(where)31 b Fo(n)e Fv(:)g Fo(m)h Fv(:)f Fo(x)j Fv(abbreviates)e Fo(n)f Fv(:)g(\()p Fo(m)h Fv(:)f Fo(x)p Fv(\).)49 b(The)32 b(function)g Fo(add)g Fv(computes)f Fo(add)p Fv(\()p Fo(n)3289 3350 y Fr(0)3356 3338 y Fv(:)515 3449 y Fo(n)565 3461 y Fr(1)644 3449 y Fv(:)41 b Fs(\001)14 b(\001)g(\001)42 b Fv(:)g Fo(n)962 3461 y Fn(k)1044 3449 y Fv(:)g Fo(nil)r(;)14 b(m)41 b Fv(:)h Fo(y)s(s)p Fv(\))f(=)h(\()p Fo(m)26 b Fv(+)1915 3387 y Fe(P)2003 3407 y Fn(k)2003 3474 y(i)p Fr(=0)2128 3449 y Fo(n)2178 3461 y Fn(i)2206 3449 y Fv(\))42 b(:)g Fo(y)s(s)c Fv(and)h(the)g(function)h Fo(w)r(eig)s(ht)515 3564 y Fv(computes)24 b Fo(w)r(eig)s(ht)p Fv(\()p Fo(n)1213 3576 y Fr(0)1273 3564 y Fv(:)f Fo(n)1369 3576 y Fr(1)1429 3564 y Fv(:)g Fs(\001)14 b(\001)g(\001)23 b Fv(:)g Fo(n)1691 3576 y Fn(k)1755 3564 y Fv(:)g Fo(nil)r Fv(\))f(=)h Fo(n)2099 3576 y Fr(0)2147 3564 y Fv(+)2224 3502 y Fe(P)2311 3523 y Fn(k)2311 3589 y(i)p Fr(=1)2437 3564 y Fo(i)12 b Fs(\001)g Fo(n)2563 3576 y Fn(i)2589 3564 y Fv(.)36 b(In)24 b(order)f(to)h(sho)n(w)g Fs(C)3307 3576 y Fq(E)3351 3564 y Fv(-)515 3664 y(termination)g(of)h(the)h (hierarc)n(hical)c(com)n(bination)j(of)f(the)i(t)n(w)n(o)e(TRSs,)i (will)f(use)g(Corollary)515 3764 y(4.10.)50 b(Simple)33 b(termination)f(of)h Fs(R)1641 3776 y Fr(1)1711 3764 y Fv(can)f(b)r(e)h(sho)n(wn)e(b)n(y)i(the)f Fm(lp)l(o)i Fv(with)f(the)g(precedence)515 3863 y Fo(add)i Fs(\037)745 3875 y Fn(p)797 3863 y Fv(:)f(and)h Fo(add)g Fs(\037)1253 3875 y Fn(p)1325 3863 y Fo(s)p Fv(.)58 b(Therefore,)35 b Fs(R)1923 3875 y Fr(1)1995 3863 y Fv(is)g Fs(C)2130 3875 y Fq(E)2175 3863 y Fv(-terminating.)57 b(W)-7 b(e)35 b(ha)n(v)n(e)e(to)h(\014nd)h(a)515 3963 y(reduction)27 b(pair)g(\()p Fk(\045)p Fo(;)14 b Fs(\037)p Fv(\))27 b(satisfying)g(the)h(constrain)n(ts:)1230 4134 y Fo(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))84 b Fs(\025)f Fo(x)1230 4259 y(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))84 b Fs(\025)f Fo(y)1198 4384 y(add)p Fv(\()p Fo(nil)r(;)14 b(y)s(s)p Fv(\))82 b Fs(\025)h Fo(y)s(s)1106 4508 y(add)p Fv(\(0)23 b(:)g Fo(xs;)14 b(y)s(s)p Fv(\))83 b Fs(\025)g Fo(add)p Fv(\()p Fo(xs;)14 b(y)s(s)p Fv(\))852 4633 y Fo(add)p Fv(\()p Fo(s)p Fv(\()p Fo(n)p Fv(\))24 b(:)f Fo(xs;)14 b(m)23 b Fv(:)g Fo(y)s(s)p Fv(\))83 b Fs(\025)g Fo(add)p Fv(\()p Fo(n)23 b Fv(:)g Fo(xs;)14 b(s)p Fv(\()p Fo(m)p Fv(\))24 b(:)f Fo(y)s(s)p Fv(\))1079 4757 y Fo(w)r(eig)s(ht)p Fv(\()p Fo(n)g Fv(:)g Fo(nil)r Fv(\))82 b Fs(\025)h Fo(n)956 4882 y(w)r(eig)s(ht)p Fv(\()p Fo(n)22 b Fv(:)h Fo(m)g Fv(:)g Fo(xs)p Fv(\))84 b Fs(\025)f Fo(w)r(eig)s(ht)p Fv(\()p Fo(add)p Fv(\()p Fo(n)23 b Fv(:)g Fo(m)g Fv(:)g Fo(xs;)14 b Fv(0)23 b(:)g Fo(xs)p Fv(\)\))846 5006 y(WEIGHT\()p Fo(n)g Fv(:)g Fo(m)g Fv(:)g Fo(xs)p Fv(\))84 b Fo(>)f Fv(WEIGHT\()p Fo(add)p Fv(\()p Fo(n)24 b Fv(:)f Fo(m)g Fv(:)g Fo(xs;)14 b Fv(0)23 b(:)g Fo(xs)p Fv(\)\))1905 5255 y(12)p eop %%Page: 13 13 13 12 bop 515 523 a Fv(Note)42 b(that)h Fs(h)p Fv(WEIGHT\()p Fo(n)48 b Fv(:)g Fo(m)g Fv(:)g Fo(xs)p Fv(\))p Fo(;)14 b Fv(ADD)r(\()p Fo(n)48 b Fv(:)g Fo(m)f Fv(:)h Fo(xs;)14 b Fv(0)48 b(:)g Fo(xs)p Fv(\))p Fs(i)43 b Fv(is)g(not)f(a)g(de-)515 623 y(p)r(endency)c(pair)f(of)h Fs(R)1250 635 y Fr(2)1325 623 y Fv(b)r(ecause)g Fo(add)i Fs(62)h(D)1973 635 y Fr(2)2010 623 y Fv(.)67 b(After)39 b(normalization)d(with)i(the)g(AFS)515 722 y Fs(A)e Fv(=)f Fs(f)p Fo(add)p Fv(\()p Fo(x;)14 b(y)s Fv(\))36 b Fs(!)g Fo(add)1366 692 y Fq(0)1389 722 y Fv(\()p Fo(y)s Fv(\))p Fo(;)14 b(x)37 b Fv(:)e Fo(xs)h Fs(!)g Fv(:)1940 692 y Fq(0)1971 722 y Fv(\()p Fo(xs)p Fv(\))p Fs(g)p Fv(,)i(the)e(inequalities)e(are)h(satis\014ed)f(b)n(y) 515 822 y(the)26 b Fm(lp)l(o)h Fv(based)e(on)g(the)i(precedence)e(:) 1709 792 y Fq(0)1746 822 y Fs(\037)1811 834 y Fn(p)1872 822 y Fo(add)2002 792 y Fq(0)2026 822 y Fv(.)36 b(Th)n(us,)26 b Fs(R)2387 834 y Fr(1)2439 822 y Fs([)16 b(R)2580 834 y Fr(2)2643 822 y Fv(is)26 b Fs(C)2769 834 y Fq(E)2813 822 y Fv(-terminating)f(b)n(y)515 922 y(Corollary)g(4.10.)515 1192 y Fw(5)134 b(Hierarc)l(hical)47 b Fj(C)1534 1209 y Fs(E)1589 1192 y Fw(-termination)515 1373 y Fv(In)31 b(the)h(follo)n(wing,)g(w)n(e)f(pro)n(vide)f(new)h(mo)r(dularit)n(y)g (results)g(ab)r(out)g(hierarc)n(hical)f(com)n(bi-)515 1473 y(nations)d(of)g(TRSs)h(as)f(w)n(ell)g(as)g(elegan)n(t)g(simple)h (pro)r(ofs)e(of)i(already)e(kno)n(wn)h(results.)639 1573 y(By)32 b(means)g(of)g(the)h(tec)n(hniques)f(dev)n(elop)r(ed)f(so)h (far,)h(w)n(e)f(are)f(able)h(to)g(giv)n(e)f(a)h(simple)515 1672 y(pro)r(of)j(of)h(the)h(follo)n(wing)e(result)h(o)n(wing)f(to)h (Kurihara)e(and)i(Oh)n(uc)n(hi)f([K)n(O95)n(];)40 b(see)c(also)515 1772 y([Urb01)o(].)515 1928 y Fl(Prop)s(osition)30 b(5.1)40 b Fs(C)1238 1940 y Fq(E)1283 1928 y Fm(-termination)h(is)h(a)g(mo)l (dular)g(pr)l(op)l(erty)g(of)h(\014nitely)e(br)l(anching)515 2028 y(c)l(omp)l(osable)31 b(TRSs.)515 2206 y(Pr)l(o)l(of)65 b Fv(Let)31 b Fs(R)994 2218 y Fr(1)1063 2206 y Fv(and)f Fs(R)1297 2218 y Fr(2)1366 2206 y Fv(b)r(e)h Fs(C)1526 2218 y Fq(E)1571 2206 y Fv(-terminating,)g(\014nitely)h(branc)n(hing,)e (and)h(comp)r(osable.)515 2305 y(De\014ne)1619 2440 y Fs(R)1689 2452 y Fr(0)1810 2440 y Fv(=)83 b Fs(R)2028 2452 y Fr(1)2084 2440 y Fs(\\)18 b(R)2227 2452 y Fr(2)1619 2539 y Fs(R)1689 2509 y Fq(0)1689 2560 y Fr(1)1810 2539 y Fv(=)83 b Fs(R)2028 2551 y Fr(1)2084 2539 y Fs(n)18 b(R)2214 2551 y Fr(0)1619 2639 y Fs(R)1689 2609 y Fq(0)1689 2660 y Fr(2)1810 2639 y Fv(=)83 b Fs(R)2028 2651 y Fr(2)2084 2639 y Fs(n)18 b(R)2214 2651 y Fr(0)2251 2639 y Fo(:)515 2796 y Fv(Note)39 b(that)h Fs(R)989 2808 y Fr(1)1052 2796 y Fs([)27 b(R)1204 2808 y Fr(2)1284 2796 y Fv(=)42 b Fs(R)1461 2808 y Fr(1)1525 2796 y Fs([)26 b(R)1676 2766 y Fq(0)1676 2817 y Fr(2)1756 2796 y Fv(=)42 b Fs(R)1933 2766 y Fq(0)1933 2817 y Fr(1)1997 2796 y Fs([)27 b(R)2149 2808 y Fr(2)2186 2796 y Fv(.)72 b(F)-7 b(or)39 b(a)g(pro)r(of)f(b)n(y)h (con)n(tradiction,)515 2896 y(supp)r(ose)28 b(that)h Fs(R)1079 2908 y Fr(1)1136 2896 y Fs([)20 b(R)1281 2908 y Fr(2)1347 2896 y Fv(is)29 b(not)g Fs(C)1625 2908 y Fq(E)1669 2896 y Fv(-terminating.)40 b(According)28 b(to)h(Theorem)f (4.2,)g(there)515 2995 y(is)23 b(a)g(minimal)h(in\014nite)h Fs(R)1326 3007 y Fr(1)1374 2995 y Fs([)10 b(R)1509 3007 y Fr(2)1547 2995 y Fv(-c)n(hain)23 b(o)n(v)n(er)f(\()p Fs(R)2066 3007 y Fr(1)2114 2995 y Fs([)10 b(R)2249 3007 y Fr(2)2288 2995 y Fv(\))g Fs(])g(C)2439 3007 y Fq(E)2485 2995 y Fv(.)35 b(If)25 b(there)e(w)n(ere)g(a)g(minimal)515 3095 y(in\014nite)c Fs(R)857 3107 y Fr(1)895 3095 y Fv(-c)n(hain)e(o)n (v)n(er)g(\()p Fs(R)1403 3107 y Fr(1)1441 3095 y Fs([R)1566 3107 y Fr(2)1604 3095 y Fv(\))p Fs(]C)1735 3107 y Fq(E)1780 3095 y Fv(,)k(then)e(an)f(application)g(of)g(Lemma)g(4.7)g(to)g Fs(R)3189 3107 y Fr(1)3245 3095 y Fv(and)515 3195 y Fs(R)585 3165 y Fq(0)585 3215 y Fr(2)651 3195 y Fv(w)n(ould)29 b(yield)g(an)g(in\014nite)g Fs(R)1566 3207 y Fr(1)1604 3195 y Fv(-c)n(hain)f(o)n(v)n(er)f Fs(R)2101 3207 y Fr(1)2158 3195 y Fs(])20 b(C)2282 3165 y Fq(0)2277 3218 y(E)2321 3195 y Fv(,)30 b(where)e Fs(C)2664 3165 y Fq(0)2659 3218 y(E)2729 3195 y Fv(=)d Fs(f)p Fo(C)6 b(ons)3055 3165 y Fq(0)3078 3195 y Fv(\()p Fo(x;)14 b(y)s Fv(\))26 b Fs(!)515 3294 y Fo(x;)36 b(C)6 b(ons)815 3264 y Fq(0)839 3294 y Fv(\()p Fo(x;)14 b(y)s Fv(\))37 b Fs(!)g Fo(y)s Fs(g)e Fv(\(b)r(ecause)h(w)n(e)f(ha)n(v)n(e)g(to)h(use)g(a)f(fresh)h (function)g(sym)n(b)r(ol)f Fo(C)6 b(ons)3355 3264 y Fq(0)515 3394 y Fv(instead)32 b(of)h(the)g(already)f(used)g(sym)n(b)r(ol)h Fo(C)6 b(ons)p Fv(\).)53 b(This)33 b(yields)f(a)h(con)n(tradiction)e (to)i(the)515 3494 y Fs(C)559 3506 y Fq(E)603 3494 y Fv(-termination)22 b(of)g Fs(R)1237 3506 y Fr(1)1274 3494 y Fv(.)35 b(Analogously)-7 b(,)22 b(there)g(cannot)f(b)r(e)i(a)e (minimal)i(in\014nite)f Fs(R)3122 3506 y Fr(2)3160 3494 y Fv(-c)n(hain)515 3593 y(o)n(v)n(er)31 b(\()p Fs(R)800 3605 y Fr(1)860 3593 y Fs([)22 b(R)1007 3605 y Fr(2)1045 3593 y Fv(\))g Fs(])h(C)1221 3605 y Fq(E)1265 3593 y Fv(.)54 b(It)33 b(follo)n(ws)f(that)h(the)h(in\014nite)g Fs(R)2406 3605 y Fr(1)2465 3593 y Fs([)23 b(R)2613 3605 y Fr(2)2650 3593 y Fv(-c)n(hain)33 b(m)n(ust)g(con)n(tain)515 3693 y(dep)r(endency)28 b(pairs)g(from)g(b)r(oth)g Fs(R)1635 3663 y Fq(0)1635 3713 y Fr(1)1701 3693 y Fv(and)h Fs(R)1934 3663 y Fq(0)1934 3713 y Fr(2)1971 3693 y Fv(.)39 b(Ho)n(w)n(ev)n(er,)27 b(it)h(is)h(easy)e(to)h(see)g(that)h(this)f(is)515 3792 y(imp)r(ossible.)2399 b Fk(\003)639 3971 y Fv(In)28 b(fact,)f(the)h(ab) r(o)n(v)n(e)e(mo)r(dularit)n(y)g(result)h(can)g(b)r(e)g(generalized)f (to)h(certain)g(hierarc)n(hi-)515 4070 y(cal)32 b(com)n(binations.)52 b(First)33 b(of)g(all,)h(the)f(hierarc)n(hical)e(com)n(bination)h(of)h Fs(R)2903 4082 y Fr(1)2973 4070 y Fv(=)e Fs(f)p Fo(a)h Fs(!)g Fo(b)p Fs(g)515 4170 y Fv(and)i Fs(R)753 4182 y Fr(2)825 4170 y Fv(=)h Fs(f)p Fo(f)9 b Fv(\()p Fo(b)p Fv(\))34 b Fs(!)g Fo(f)9 b Fv(\()p Fo(a)p Fv(\))p Fs(g)35 b Fv(sho)n(ws)e(that)i(neither)f(simple)h(nor)f Fs(C)2688 4182 y Fq(E)2732 4170 y Fv(-termination)g(is)h(in)515 4269 y(general)28 b(mo)r(dular)h(for)g(hierarc)n(hical)f(com)n (binations.)42 b(On)29 b(the)h(other)f(hand,)i(the)f(hierar-)515 4369 y(c)n(hical)23 b(com)n(bination)h(of)g(the)h(base)e(system)h Fs(R)1968 4381 y Fr(+)2048 4369 y Fv(and)g(the)h(extension)f Fs(R)2780 4381 y Fq(\003)2843 4369 y Fv(is)g(again)f(simply)515 4469 y(terminating:)1270 4670 y Fs(R)1340 4682 y Fr(+)1479 4670 y Fv(=)1567 4553 y Fe(\032)1780 4619 y Fv(0)18 b(+)g Fo(y)86 b Fs(!)d Fo(y)1670 4719 y(s)p Fv(\()p Fo(x)p Fv(\))20 b(+)e Fo(y)86 b Fs(!)d Fo(s)p Fv(\()p Fo(x)19 b Fv(+)f Fo(y)s Fv(\))1287 4969 y Fs(R)1357 4981 y Fq(\003)1479 4969 y Fv(=)1567 4852 y Fe(\032)1780 4918 y Fv(0)g Fs(\003)g Fo(y)85 b Fs(!)f Fv(0)1670 5018 y Fo(s)p Fv(\()p Fo(x)p Fv(\))20 b Fs(\003)e Fo(y)85 b Fs(!)f Fv(\()p Fo(x)19 b Fs(\003)f Fo(y)s Fv(\))g(+)g Fo(y)1905 5255 y Fv(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fv(What)24 b(is)g(the)g(di\013erence)g(b)r(et)n(w)n (een)g(these)g(examples?)35 b(That)24 b(is,)h(wh)n(y)f(is)g(the)g (hierarc)n(hical)515 623 y(com)n(bination)34 b(of)h Fs(R)1167 635 y Fr(+)1257 623 y Fv(and)g Fs(R)1496 635 y Fq(\003)1570 623 y Fv(so)f(b)r(enign)h(and)g(that)g(of)g Fs(R)2484 635 y Fr(1)2557 623 y Fv(and)g Fs(R)2796 635 y Fr(2)2868 623 y Fv(so)g(malignan)n(t?)515 722 y(In)i(essence,)h(this)f(is)f(due)h (to)g(the)g(fact)g(that)g(on)f(the)h(righ)n(t-hand)e(side)i(\()p Fo(x)25 b Fs(\003)f Fo(y)s Fv(\))h(+)f Fo(y)39 b Fv(of)515 822 y(the)h(second)f(rule)h(from)f Fs(R)1407 834 y Fq(\003)1486 822 y Fv(the)h(function)h(sym)n(b)r(ol)e(+)h(from)f Fs(D)2652 834 y Fr(+)2747 822 y Fv(o)r(ccurs)g(ab)r(o)n(v)n(e)g(the)515 922 y(function)33 b(sym)n(b)r(ol)f Fs(\003)h Fv(from)f Fs(D)1474 934 y Fq(\003)1512 922 y Fv(,)i(whereas)e(on)g(the)h(righ)n (t-hand)f(side)h Fo(f)9 b Fv(\()p Fo(a)p Fv(\))32 b(the)h(function)515 1021 y(sym)n(b)r(ol)c Fo(a)g Fv(from)g Fs(D)1136 1033 y Fr(1)1203 1021 y Fv(o)r(ccurs)g(b)r(elo)n(w)g(the)h(function)g(sym)n (b)r(ol)f Fo(f)38 b Fv(from)29 b Fs(D)2798 1033 y Fr(2)2836 1021 y Fv(.)42 b(This)30 b(fact)g(has)515 1121 y(b)r(een)20 b(observ)n(ed)e(indep)r(enden)n(tly)i(and)f(con)n(temp)r(oraneously)f (b)n(y)h(Dersho)n(witz)g([Der95)o(])g(and)515 1220 y(Krishna)32 b(Rao)h([KR95)n(])h(and)f(lead)g(to)h(the)g(notion)f(of)g(prop)r(er)g (extensions;)i(cf.)f(Krishna)515 1320 y(Rao)c([KR94)n(,)h(KR95)o(].)47 b(In)31 b(order)e(to)i(review)f(the)h(de\014nition)g(of)g(prop)r(er)f (extensions,)g(w)n(e)515 1420 y(ha)n(v)n(e)c(to)i(in)n(tro)r(duce)f (the)h(dep)r(endency)g(relation)e Fc(\004)2128 1435 y Fn(d)2167 1420 y Fv(.)515 1602 y Fl(De\014nition)31 b(5.2)40 b Fv(F)-7 b(or)29 b(a)f(TRS)i(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\),)30 b(the)g(dep)r(endency)f(relation)f Fc(\004)2790 1617 y Fn(d)2858 1602 y Fv(is)h(the)g(smallest)515 1702 y(quasi-ordering)e(satisfying)j(the)h(condition)f Fo(f)36 b Fc(\004)2075 1717 y Fn(d)2142 1702 y Fo(g)d Fv(whenev)n(er)c(there)h (is)h(a)f(rewrite)f(rule)515 1802 y Fo(f)9 b Fv(\()p Fo(:)14 b(:)g(:)f Fv(\))25 b Fs(!)f Fo(C)6 b Fv([)p Fo(g)s Fv(\()p Fo(:)14 b(:)g(:)g Fv(\)])25 b Fs(2)f(R)29 b Fv(with)g Fo(g)e Fs(2)d(D)r Fv(.)40 b(So)28 b Fo(f)33 b Fc(\004)2109 1817 y Fn(d)2172 1802 y Fo(g)e Fv(holds)d(if)h(the)g(function)g Fo(f)37 b Fv(dep)r(ends)515 1901 y(on)27 b(the)h(de\014nition)g(of)g (the)g(function)g Fo(g)s Fv(.)515 2084 y Fl(De\014nition)j(5.3)40 b Fv(Let)45 b(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\))46 b(b)r(e)f(the)g(hierarc)n(hical)e(com)n(bination)g(of)i(base)f(system) 515 2183 y(\()p Fs(F)607 2195 y Fr(1)644 2183 y Fo(;)14 b Fs(R)751 2195 y Fr(1)788 2183 y Fv(\))31 b(and)f(extension)g(\()p Fs(F)1477 2195 y Fr(2)1515 2183 y Fo(;)14 b Fs(R)1622 2195 y Fr(2)1659 2183 y Fv(\).)46 b(The)30 b(de\014ned)h(sym)n(b)r(ols) f Fs(D)2606 2195 y Fr(2)2674 2183 y Fv(of)g Fs(R)2841 2195 y 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2860 y Fr(1)1834 2848 y Fv(if)h(ev)n(ery)e(rewrite)g(rule)i Fo(l)h Fs(!)f Fo(r)j Fs(2)e(R)3028 2860 y Fr(2)3101 2848 y Fv(satis\014es)515 2947 y(the)j(follo)n(wing)f(condition:)62 b(If)40 b Fo(t)h Fv(is)f(a)f(subterm)h(of)g Fo(r)j Fv(suc)n(h)d(that)h Fo(r)r(oot)p Fv(\()p Fo(t)p Fv(\))k Fs(2)f(D)3167 2917 y Fr(1)3165 2968 y(2)3245 2947 y Fv(and)515 3047 y Fo(r)r(oot)p Fv(\()p Fo(t)p Fv(\))d Fc(\004)851 3062 y Fn(d)929 3047 y Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\),)g(then)c Fo(t)h Fv(do)r(es)f(not)g(con)n(tain)f(a)h(function)h(sym)n(b)r(ol)f(from)g Fs(D)3133 3059 y Fr(1)3195 3047 y Fs([)25 b(D)3341 3017 y Fr(1)3339 3068 y(2)515 3147 y Fv(strictly)i(b)r(elo)n(w)g(its)h(ro)r (ot.)639 3246 y Fs(R)709 3258 y Fr(2)781 3246 y Fv(is)33 b(a)g Fm(r)l(estricte)l(d)j(pr)l(op)l(er)g(extension)e Fv(of)f Fs(R)2118 3258 y Fr(1)2190 3246 y Fv(if)h(it)g(is)f(a)h(prop)r (er)f(extension)g(of)g Fs(R)3341 3258 y Fr(1)515 3346 y Fv(suc)n(h)23 b(that)h(no)f Fm(left-hand)k(side)e Fv(of)e(the)h 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Fo(l)r Fv(\))42 b Fs(2)f(D)3277 4578 y Fr(2)3275 4628 y(2)3314 4608 y Fs(g)p Fv(.)515 4707 y(The)27 b(TRS)g Fs(R)949 4677 y Fr(2)949 4728 y(2)1013 4707 y Fv(do)r(es)g(not)f(con)n (tain)g(function)i(sym)n(b)r(ols)e(from)g Fs(D)2538 4719 y Fr(1)2593 4707 y Fs([)17 b(D)2731 4677 y Fr(1)2729 4728 y(2)2768 4707 y Fv(.)37 b(This)27 b(is)f(b)r(ecause)515 4807 y(for)35 b(all)h Fo(l)i Fs(!)g Fo(r)h Fs(2)f(R)1197 4777 y Fr(2)1197 4828 y(2)1271 4807 y Fv(neither)e Fo(l)h Fv(\(b)r(ecause)f Fs(R)2041 4819 y Fr(2)2115 4807 y Fv(is)g(a)f (restricted)g(prop)r(er)h(extension)f(of)515 4907 y Fs(R)585 4919 y Fr(1)622 4907 y Fv(\))40 b(nor)e Fo(r)k Fv(do)r(es)d(con)n(tain) g(function)h(sym)n(b)r(ols)e(from)h Fs(D)2373 4919 y Fr(1)2436 4907 y Fs([)27 b(D)2584 4876 y Fr(1)2582 4927 y(2)2661 4907 y Fv(\(if)40 b Fo(r)i Fv(w)n(ould)d(con)n(tain)515 5006 y(suc)n(h)e(a)h(function)g(sym)n(b)r(ol,)i(then)f Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))g(w)n(ould)e(b)r(e)h(an)g(elemen)n(t) g(of)g Fs(D)2915 4976 y Fr(1)2913 5027 y(2)2953 5006 y Fv(\).)68 b(In)38 b(other)1905 5255 y(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fv(w)n(ords,)24 b Fs(R)844 535 y Fr(1)906 523 y Fv(and)g Fs(R)1134 493 y Fr(2)1134 544 y(2)1196 523 y Fv(are)g(constructor-sharing.)32 b(Since)25 b(b)r(oth)g(are)e Fs(C)2679 535 y Fq(E)2724 523 y Fv(-terminating,)h (so)g(is)515 623 y(their)29 b(union)f(b)n(y)h(Prop)r(osition)e(5.1.)40 b(In)29 b(order)f(to)g(sho)n(w)g Fs(C)2358 635 y Fq(E)2403 623 y Fv(-termination)g(of)h Fs(R)3050 635 y Fr(1)3107 623 y Fs([)20 b(R)3252 635 y Fr(2)3314 623 y Fv(=)515 722 y(\()p Fs(R)617 734 y Fr(1)675 722 y Fs([)h(R)821 692 y Fr(2)821 743 y(2)859 722 y Fv(\))g Fs([)g(R)1058 692 y Fr(1)1058 743 y(2)1096 722 y Fv(,)32 b(it)f(su\016ces)g(to)g(sho) n(w)f(that)h(there)g(is)g(no)f(minimal)i(in\014nite)f Fs(R)3122 692 y Fr(1)3122 743 y(2)3160 722 y Fv(-c)n(hain)515 822 y(o)n(v)n(er)h(\()p Fs(R)801 834 y Fr(1)861 822 y Fs([)23 b(R)1009 834 y Fr(2)1047 822 y Fv(\))f Fs(])h(C)1223 834 y Fq(E)1268 822 y Fv(;)37 b(see)c(Corollary)e(4.10.)54 b(Supp)r(ose)34 b(on)f(the)i(con)n(trary)c(that)j(there)515 922 y(is)27 b(a)f(minimal)h(in\014nite)h Fs(R)1339 891 y Fr(1)1339 942 y(2)1376 922 y Fv(-c)n(hain)e Fs(h)p Fo(s)1693 934 y Fr(1)1731 922 y Fo(;)14 b(t)1798 934 y Fr(1)1835 922 y Fs(ih)p Fo(s)1938 934 y Fr(2)1976 922 y Fo(;)g(t)2043 934 y Fr(2)2080 922 y Fs(i)g Fo(:)g(:)g(:)41 b Fv(o)n(v)n(er)25 b(\()p Fs(R)2543 934 y Fr(1)2598 922 y Fs([)17 b(R)2740 934 y Fr(2)2778 922 y Fv(\))g Fs(])g(C)2943 934 y Fq(E)2988 922 y Fv(,)27 b(i.e.,)g(there)515 1021 y(exists)41 b(a)f(substitution)i Fo(\033)49 b Fv(:)d Fs(V)52 b(!)46 b(T)21 b Fv(\()p Fs(F)36 b(])28 b(f)p Fo(C)6 b(ons)p Fs(g)p Fv(\))41 b(suc)n(h)g(that)g Fo(t)2765 1033 y Fn(j)2800 1021 y Fo(\033)49 b Fs(!)2979 991 y Fq(\003)2979 1048 y Fr(\()p Fq(R)3062 1056 y Ff(1)3095 1048 y Fq([R)3197 1056 y Ff(2)3229 1048 y Fr(\))p Fq(]C)3336 1056 y Fi(E)515 1133 y Fo(s)554 1145 y Fn(j)s Fr(+1)673 1133 y Fo(\033)38 b Fv(holds)d(for)f(all)h(consecutiv)n(e)f(pairs)g Fs(h)p Fo(s)1968 1145 y Fn(j)2003 1133 y Fo(;)14 b(t)2070 1145 y Fn(j)2105 1133 y Fs(i)36 b Fv(and)e Fs(h)p Fo(s)2412 1145 y Fn(j)s Fr(+1)2532 1133 y Fo(;)14 b(t)2599 1145 y Fn(j)s Fr(+1)2718 1133 y Fs(i)35 b Fv(in)g(the)h(sequence.)515 1233 y(Since)c(w)n(e)g(consider)g(\014nite)h(TRSs,)g(w)n(e)f(ma)n(y)g (assume)g(that)g(ev)n(ery)f(dep)r(endency)i(pair)f(in)515 1332 y Fs(h)p Fo(s)586 1344 y Fr(1)623 1332 y Fo(;)14 b(t)690 1344 y Fr(1)727 1332 y Fs(ih)p Fo(s)830 1344 y Fr(2)868 1332 y Fo(;)g(t)935 1344 y Fr(2)972 1332 y Fs(i)g Fo(:)g(:)g(:)38 b Fv(b)r(elongs)24 b(to)g(a)f(cycle)h Fs(P)31 b Fv(whic)n(h)24 b(consists)f(solely)g(of)i(dep)r(endency)f (pairs)515 1432 y(from)38 b Fs(R)792 1402 y Fr(1)792 1452 y(2)830 1432 y Fv(.)71 b(Let)39 b Fs(h)p Fo(F)12 b Fv(\()p Fo(u)1261 1444 y Fr(1)1299 1432 y Fo(;)i(:)g(:)g(:)f(;)h(u) 1531 1444 y Fn(n)1576 1432 y Fv(\))p Fo(;)g(G)p Fv(\()p Fo(t)1772 1444 y Fr(1)1810 1432 y Fo(;)g(:)g(:)g(:)g(;)g(t)2025 1444 y Fn(m)2087 1432 y Fv(\))p Fs(i)40 b Fv(b)r(e)g(one)e(of)h(the)h (dep)r(endency)f(pairs)515 1532 y(in)34 b Fs(P)7 b Fv(.)55 b(Supp)r(ose)34 b Fs(h)p Fo(F)12 b Fv(\()p Fo(u)1270 1544 y Fr(1)1307 1532 y Fo(;)i(:)g(:)g(:)g(;)g(u)1540 1544 y Fn(n)1585 1532 y Fv(\))p Fo(;)g(G)p Fv(\()p Fo(v)1791 1544 y Fr(1)1829 1532 y Fo(;)g(:)g(:)g(:)g(;)g(v)2054 1544 y Fn(m)2117 1532 y Fv(\))p Fs(i)34 b Fv(originates)e(from)i(the)g (rewrite)f(rule)515 1631 y Fo(f)9 b Fv(\()p Fo(u)645 1643 y Fr(1)681 1631 y Fo(;)14 b(:)g(:)g(:)g(;)g(u)914 1643 y Fn(n)959 1631 y Fv(\))33 b Fs(!)g Fo(C)6 b Fv([)p Fo(g)s Fv(\()p Fo(v)1343 1643 y Fr(1)1380 1631 y Fo(;)14 b(:)g(:)g(:)g(;)g(v)1605 1643 y Fn(m)1668 1631 y Fv(\)])34 b(of)g Fs(R)1928 1601 y Fr(1)1928 1652 y(2)1965 1631 y Fv(.)55 b(Clearly)-7 b(,)34 b(w)n(e)f(ha)n(v)n(e)g Fo(f)41 b Fc(\004)2815 1646 y Fn(d)2887 1631 y Fo(g)36 b Fv(and)d Fo(g)j Fc(\004)3258 1646 y Fn(d)3329 1631 y Fo(f)515 1731 y Fv(b)r(ecause)23 b Fs(P)30 b Fv(is)24 b(a)f(cycle.)35 b(Since)24 b Fs(R)1570 1743 y Fr(2)1632 1731 y Fv(is)f(a)g(prop)r(er)g(extension)g(of)h Fs(R)2562 1743 y Fr(1)2600 1731 y Fv(,)g(the)g(terms)g Fo(v)3054 1743 y Fr(1)3091 1731 y Fo(;)14 b(:)g(:)g(:)g(;)g(v)3316 1743 y Fn(m)515 1830 y Fv(do)19 b(not)h(con)n(tain)e(sym)n(b)r(ols)h (from)h Fs(D)1608 1842 y Fr(1)1647 1830 y Fs([)r(D)1770 1800 y Fr(1)1768 1851 y(2)1809 1830 y Fv(.)34 b(The)20 b(same)f(is)g(true)h(for)f Fo(u)2639 1842 y Fr(1)2675 1830 y Fo(;)14 b(:)g(:)g(:)g(;)g(u)2908 1842 y Fn(n)2972 1830 y Fv(b)r(ecause)19 b Fs(R)3341 1842 y Fr(2)515 1930 y Fv(is)26 b(a)f(restricted)h(prop)r(er)f(extension)h(of)g Fs(R)1826 1942 y Fr(1)1863 1930 y Fv(.)37 b(All)27 b(in)f(all,)g(none)g (of)g(the)h(\(prop)r(er\))e(subterms)515 2030 y(of)i(the)h(dep)r (endency)g(pairs)f(in)h Fs(P)34 b Fv(con)n(tains)27 b(a)g(sym)n(b)r(ol) g(from)g Fs(D)2536 2042 y Fr(1)2592 2030 y Fs([)19 b(D)2732 2000 y Fr(1)2730 2050 y(2)2769 2030 y Fv(.)639 2129 y(Since)32 b Fs(R)930 2099 y Fr(2)930 2150 y(2)1000 2129 y Fv(do)r(es)f(not)h(con) n(tain)f(function)h(sym)n(b)r(ols)f(from)g Fs(D)2554 2141 y Fr(1)2613 2129 y Fs([)21 b(D)2755 2099 y Fr(1)2753 2150 y(2)2793 2129 y Fv(,)33 b(the)f(TRS)g Fs(R)3265 2141 y Fr(1)3324 2129 y Fs([)515 2229 y(R)585 2199 y Fr(1)585 2250 y(2)645 2229 y Fs([)23 b(C)767 2241 y Fq(E)845 2229 y Fv(can)34 b(b)r(e)g(view)n(ed)f(as)h(an)f(extension)h(of)f Fs(R)2175 2199 y Fr(2)2175 2250 y(2)2213 2229 y Fv(.)56 b(Because)32 b(the)j(in\014nite)f Fs(R)3122 2199 y Fr(1)3122 2250 y(2)3160 2229 y Fv(-c)n(hain)515 2329 y(is)f(minimal,)i(\(i.e.,)g Fo(u)e Fv(is)g(terminating)g(w.r.t.)g Fs(R)2078 2341 y Fr(1)2138 2329 y Fs([)22 b(R)2285 2341 y Fr(2)2345 2329 y Fs([)h(C)2467 2341 y Fq(E)2544 2329 y Fv(for)33 b(ev)n(ery)f(subterm)h Fo(u)g Fv(of)515 2428 y(some)i Fo(s)770 2440 y Fn(j)805 2428 y Fo(\033)k Fv(and)d Fo(t)1091 2440 y Fn(j)1126 2428 y Fo(\033)s Fv(\),)j(w)n(e)d(ma)n(y)f(apply)g (Lemma)h(4.6)f(to)h(ev)n(ery)f Fo(t)2640 2440 y Fn(j)2675 2428 y Fo(\033)40 b Fs(!)2845 2398 y Fq(\003)2845 2458 y(R)2902 2438 y Ff(2)2902 2476 y(2)2934 2458 y Fq([)p Fr(\()p Fq(R)3062 2466 y Ff(1)3095 2458 y Fq([R)3197 2438 y Ff(1)3197 2476 y(2)3229 2458 y Fq([C)3310 2466 y Fi(E)3349 2458 y Fr(\))515 2547 y Fo(s)554 2559 y Fn(j)s Fr(+1)673 2547 y Fo(\033)s Fv(.)77 b(This)41 b(yields)g(\010\()p Fo(t)1395 2559 y Fn(j)1430 2547 y Fo(\033)s Fv(\))47 b Fs(!)1642 2517 y Fq(\003)1642 2577 y(R)1699 2557 y Ff(2)1699 2595 y(2)1731 2577 y Fq(]C)1815 2557 y Fi(0)1812 2596 y(E)1901 2547 y Fv(\010\()p Fo(s)2032 2559 y Fn(j)s Fr(+1)2151 2547 y Fo(\033)s Fv(\),)e(where)c Fs(C)2604 2517 y Fq(0)2599 2570 y(E)2689 2547 y Fv(=)k Fs(f)p Fo(C)6 b(ons)3035 2517 y Fq(0)3058 2547 y Fv(\()p Fo(x;)14 b(y)s Fv(\))46 b Fs(!)515 2667 y Fo(x;)14 b(C)6 b(ons)793 2637 y Fq(0)817 2667 y Fv(\()p Fo(x;)14 b(y)s Fv(\))36 b Fs(!)g Fo(y)s Fs(g)e Fv(b)r(ecause)h(w)n(e)g(ha)n(v)n(e)f(to)h(use)g(a)g (fresh)g(sym)n(b)r(ol)f Fo(C)6 b(ons)2959 2637 y Fq(0)3018 2667 y Fv(instead)35 b(of)515 2767 y(the)30 b(already)f(used)h(sym)n(b) r(ol)f Fo(C)6 b(ons)p Fv(.)44 b(In)30 b(other)g(w)n(ords,)f(the)h (cycle)g Fs(P)37 b Fv(admits)30 b(an)f(in\014nite)515 2866 y Fs(R)585 2836 y Fr(1)585 2887 y(2)622 2866 y Fv(-c)n(hain)e(o)n (v)n(er)f Fs(R)1117 2836 y Fr(2)1117 2887 y(2)1173 2866 y Fs(])19 b(C)1296 2836 y Fq(0)1291 2889 y(E)1335 2866 y Fv(.)37 b(This)28 b(con)n(tradicts)e Fs(C)2056 2878 y Fq(E)2101 2866 y Fv(-termination)h(of)g Fs(R)2745 2878 y Fr(2)2783 2866 y Fv(.)508 b Fk(\003)639 3066 y Fv(With)41 b(the)f(aid)f(of)h(Theorem)f(5.4)g(w)n(e)g(are)g(able)g(to)h(infer)f Fs(C)2620 3078 y Fq(E)2665 3066 y Fv(-termination)g(of)h(the)515 3165 y(follo)n(wing)26 b(hierarc)n(hical)g(com)n(bination.)515 3348 y Fl(Example)k(5.5)40 b Fv(The)28 b(TRS)g Fs(R)1508 3360 y Fr(1)1556 3525 y Fv(0)18 b(+)g Fo(y)86 b Fs(!)d Fo(y)1447 3625 y(s)p Fv(\()p Fo(x)p Fv(\))19 b(+)f Fo(y)86 b Fs(!)d Fo(s)p Fv(\()p Fo(x)19 b Fv(+)f Fo(y)s Fv(\))1408 3724 y Fo(nil)h Fv(+)-23 b(+)22 b Fo(y)s(s)83 b Fs(!)g Fo(y)s(s)1404 3824 y(xs)19 b Fv(+)-23 b(+)22 b Fo(nil)84 b Fs(!)f Fo(xs)1245 3924 y Fv(\()p Fo(x)24 b Fv(:)f Fo(xs)p Fv(\))c(+)-23 b(+)22 b Fo(y)s(s)83 b Fs(!)g Fo(x)23 b Fv(:)g(\()p Fo(xs)c Fv(+)-23 b(+)23 b Fo(y)s(s)p Fv(\))1297 4023 y Fo(sum)p Fv(\()p Fo(x)g Fv(:)h Fo(nil)r Fv(\))82 b Fs(!)h Fo(x)23 b Fv(:)g Fo(nil)1203 4123 y(sum)p Fv(\()p Fo(x)g Fv(:)g Fo(y)j Fv(:)d Fo(xs)p Fv(\))84 b Fs(!)f Fo(sum)p Fv(\(\()p Fo(x)19 b Fv(+)f Fo(y)s Fv(\))23 b(:)g Fo(xs)p Fv(\))908 4222 y Fo(sum)p Fv(\()p Fo(xs)c Fv(+)-23 b(+)22 b(\()p Fo(x)i Fv(:)f Fo(y)i Fv(:)e Fo(y)s(s)p Fv(\)\))84 b Fs(!)f Fo(sum)p Fv(\()p Fo(xs)18 b Fv(+)-23 b(+)22 b Fo(sum)p Fv(\()p Fo(x)h Fv(:)h Fo(y)h Fv(:)e Fo(y)s(s)p Fv(\)\))515 4406 y(stems)k(from)h([A)n(G97a)n(];)g(cf.)g([A) n(G97b)o(,)g(Example)f(5.1.17].)35 b(The)28 b(function)g Fo(sum)p Fv(\()p Fo(xs)p Fv(\))g(com-)515 4506 y(putes)f(the)h(sum)g (of)f(all)g(n)n(um)n(b)r(ers)g(in)g(the)h(list)g Fo(xs)g Fv(\(e.g.)f Fo(sum)g Fv(applied)g(to)g(the)h(list)g([1)p Fo(;)14 b Fv(2)p Fo(;)g Fv(3])1905 5255 y(15)p eop %%Page: 16 16 16 15 bop 515 523 a Fv(returns)27 b([6]\).)37 b(The)27 b(p)r(olynomial)g(in)n(terpretation)1289 683 y(0)1331 695 y Fb(N)1460 683 y Fv(=)83 b(0)565 b Fo(nil)2319 695 y Fb(N)2448 683 y Fv(=)82 b(0)1180 783 y Fo(s)1219 795 y Fb(N)1265 783 y Fv(\()p Fo(x)p Fv(\))i(=)f Fo(x)18 b Fv(+)g(1)317 b Fo(x)24 b Fv(:)2209 795 y Fb(N)2278 783 y Fo(xs)84 b Fv(=)e Fo(xs)19 b Fv(+)f(1)1138 882 y Fo(x)h Fv(+)1269 894 y Fb(N)1333 882 y Fo(y)86 b Fv(=)d Fo(x)18 b Fv(+)g Fo(y)205 b(xs)19 b Fv(+)-23 b(+)2214 894 y Fb(N)2282 882 y Fo(y)s(s)83 b Fv(=)f Fo(xs)19 b Fv(+)f Fo(y)s(s)g Fv(+)g(1)1021 982 y Fo(sum)1181 994 y Fb(N)1226 982 y Fv(\()p Fo(xs)p Fv(\))84 b(=)f(1)259 b(PLUS)2126 994 y Fb(N)2172 982 y Fv(\()p Fo(x;)14 b(y)s Fv(\))84 b(=)e Fo(x)885 1082 y Fv(APP)1060 1094 y Fb(N)1106 1082 y Fv(\()p Fo(xs;)14 b(y)s(s)p Fv(\))84 b(=)f Fo(xs)289 b Fv(SUM)2168 1094 y Fb(N)2214 1082 y Fv(\()p Fo(xs)p Fv(\))84 b(=)e Fo(xs)944 1181 y(C)6 b(ons)1138 1193 y Fb(N)1184 1181 y Fv(\()p Fo(x;)14 b(y)s Fv(\))84 b(=)f Fo(x)18 b Fv(+)g Fo(y)515 1343 y Fv(sho)n(ws)37 b Fs(C)808 1355 y Fq(E)853 1343 y Fv(-termination)h(of)g Fs(R)1519 1355 y Fr(1)1557 1343 y Fv(.)69 b(\()p Fs(R)1751 1355 y Fr(1)1828 1343 y Fv(is)38 b(actually)g(DP-quasi)f(simply)i (terminating.\))515 1442 y(No)n(w)24 b(w)n(e)f(w)n(an)n(t)h(to)g (extend)h(this)f(system)g(suc)n(h)g(that)h(one)f(can)g(also)f(compute)h (the)h(a)n(v)n(erage)515 1542 y(of)i(the)h(n)n(um)n(b)r(ers)f(in)h(a)f (list.)38 b(The)27 b(TRS)h Fs(R)1860 1554 y Fr(2)1925 1542 y Fv(do)r(es)g(this:)1255 1694 y(0)18 b Fs(\000)g Fo(s)p Fv(\()p Fo(y)s Fv(\))83 b Fs(!)g Fv(0)1355 1793 y Fo(x)19 b Fs(\000)f Fv(0)82 b Fs(!)h Fo(x)1146 1893 y(s)p Fv(\()p Fo(x)p Fv(\))19 b Fs(\000)f Fo(s)p Fv(\()p Fo(y)s Fv(\))83 b Fs(!)g Fo(x)19 b Fs(\000)f Fo(y)1097 1993 y(q)s(uot)p Fv(\(0)p Fo(;)c(s)p Fv(\()p Fo(y)s Fv(\)\))83 b Fs(!)g Fv(0)988 2092 y Fo(q)s(uot)p Fv(\()p Fo(s)p Fv(\()p Fo(x)p Fv(\))p Fo(;)14 b(s)p Fv(\()p Fo(y)s Fv(\)\))84 b Fs(!)f Fo(s)p Fv(\()p Fo(q)s(uot)p Fv(\()p Fo(x)20 b Fs(\000)e Fo(y)s(;)c(s)p Fv(\()p Fo(y)s Fv(\)\)\))1141 2192 y Fo(l)r(eng)s(th)p Fv(\()p Fo(nil)r Fv(\))80 b Fs(!)j Fv(0)1042 2292 y Fo(l)r(eng)s(th)p Fv(\()p Fo(x)23 b Fv(:)g Fo(xs)p Fv(\))83 b Fs(!)g Fo(s)p Fv(\()p Fo(l)r(eng)s(th)p Fv(\()p Fo(xs)p Fv(\)\))1187 2391 y Fo(hd)p Fv(\()p Fo(x)24 b Fv(:)f Fo(xs)p Fv(\))83 b Fs(!)g Fo(x)1265 2491 y(av)s(g)s Fv(\()p Fo(xs)p Fv(\))g Fs(!)g Fo(q)s(uot)p Fv(\()p Fo(hd)p Fv(\()p Fo(sum)p Fv(\()p Fo(xs)p Fv(\)\))p Fo(;)14 b(l)r(eng)s(th)p Fv(\()p Fo(xs)p Fv(\)\))515 2648 y(By)36 b(using)g(an)g(appropriate)e (AFS)j(and)f(rp)r(o,)i(it)f(is)f(not)g(di\016cult)h(to)f(sho)n(w)g (that)g Fs(R)3249 2660 y Fr(2)3323 2648 y Fv(is)515 2747 y(DP-quasi)h(simply)i(terminating;)k(see)38 b([A)n(G00)o(,)k(Example)37 b(9].)69 b(Therefore,)40 b Fs(R)3092 2759 y Fr(2)3168 2747 y Fv(is)f Fs(C)3307 2759 y Fq(E)3351 2747 y Fv(-)515 2847 y(terminating)32 b(as)g(w)n(ell.)52 b(Note)32 b(that)h(neither)g Fs(R)2042 2859 y Fr(1)2112 2847 y Fv(nor)f Fs(R)2335 2859 y Fr(2)2405 2847 y Fv(is)g(simply)h(terminating.)52 b(W)-7 b(e)515 2947 y(ha)n(v)n(e)34 b Fs(D)780 2917 y Fr(1)778 2967 y(2)853 2947 y Fv(=)h Fs(f)p Fo(av)s(g)s Fs(g)p Fv(,)h(whereas)d(all)i(other)g(sym)n(b)r(ols)f(of)h Fs(D)2385 2959 y Fr(2)2457 2947 y Fv(b)r(elong)g(to)g Fs(D)2905 2917 y Fr(2)2903 2967 y(2)2943 2947 y Fv(.)59 b(Since)35 b Fo(av)s(g)515 3046 y Fv(do)r(es)28 b(not)h(o)r(ccur)f(on)h (a)f(righ)n(t-hand)g(side)g(of)h(a)f(rewrite)g(rule,)h Fs(R)2551 3058 y Fr(2)2618 3046 y Fv(is)f(a)h(prop)r(er)f(extension)515 3146 y(of)j Fs(R)683 3158 y Fr(1)721 3146 y Fv(.)49 b(F)-7 b(urthermore,)32 b(since)f(none)h(of)f(the)h(sym)n(b)r(ols)f(+,)i(+)-28 b(+,)32 b Fo(sum)p Fv(,)g(and)g Fo(av)s(g)i Fv(app)r(ears)515 3246 y(in)28 b(a)f(prop)r(er)g(subterm)h(of)g(a)g(left-hand)g(side)g (of)f(the)i(rewrite)e(rules)g(of)h Fs(R)2840 3258 y Fr(2)2878 3246 y Fv(,)g Fs(R)2999 3258 y Fr(2)3064 3246 y Fv(is)g(ev)n(en)f(a)515 3345 y(restricted)h(prop)r(er)h(extension)g(of)g Fs(R)1689 3357 y Fr(1)1727 3345 y Fv(.)42 b(Since)30 b Fs(R)2081 3357 y Fr(2)2148 3345 y Fv(is)f(a)g(restricted)g(prop)r(er)f(extension) h(of)515 3445 y Fs(R)585 3457 y Fr(1)622 3445 y Fv(,)f(it)g(follo)n(ws) f(from)g(Theorem)g(5.4)f(that)i Fs(R)1959 3457 y Fr(1)2015 3445 y Fs([)19 b(R)2159 3457 y Fr(2)2224 3445 y Fv(is)28 b Fs(C)2352 3457 y Fq(E)2397 3445 y Fv(-terminating.)639 3606 y(W)-7 b(e)42 b(stress)f(that)g(in)h(the)g(preceding)f(example)g (none)g(of)g(the)h(kno)n(wn)f(mo)r(dularit)n(y)515 3705 y(results)29 b(is)h(applicable.)44 b(On)30 b(the)h(other)e(hand,)i (Theorem)e(5.4)h(cannot)f(b)r(e)i(used)f(to)g(infer)515 3805 y Fs(C)559 3817 y Fq(E)603 3805 y Fv(-termination)e(of)g(the)g (hierarc)n(hical)e(com)n(bination)h(of)h(the)h(t)n(w)n(o)e(TRSs)h(from) g(Example)515 3905 y(4.11.)35 b(This)28 b(is)f(b)r(ecause)h(there)f Fs(R)1585 3917 y Fr(2)1650 3905 y Fv(is)h(a)f(not)g(prop)r(er)g (extension)g(of)h Fs(R)2748 3917 y Fr(1)2785 3905 y Fv(.)639 4004 y(Example)23 b(4.9)g(demonstrates)g(wh)n(y)h(a)f(sym)n(b)r(ol)h (from)f Fs(D)2389 4016 y Fr(1)2437 4004 y Fs([)11 b(D)2569 3974 y Fr(1)2567 4025 y(2)2631 4004 y Fv(ma)n(y)23 b(not)h(o)r(ccur)f (b)r(elo)n(w)515 4104 y(a)h Fs(D)647 4074 y Fr(1)645 4124 y(2)685 4104 y Fv(-sym)n(b)r(ol)g(on)g(the)i(righ)n(t-hand)d(side) i(of)g(a)f(rewrite)g(rule)h(from)f Fs(R)2680 4116 y Fr(2)2718 4104 y Fv(.)36 b(The)25 b(requiremen)n(t)515 4203 y(that)f(function)h (sym)n(b)r(ols)f(from)g Fs(D)1584 4215 y Fr(1)1634 4203 y Fs([)12 b(D)1767 4173 y Fr(1)1765 4224 y(2)1829 4203 y Fv(ma)n(y)24 b(not)g(o)r(ccur)g(at)g(non-ro)r(ot)g(p)r(ositions)g(on) g(the)515 4303 y(left-hand)k(sides)g(of)g(rewrite)f(rules)g(from)h Fs(R)1911 4315 y Fr(2)1977 4303 y Fv(is)g(also)e(necessary)-7 b(.)37 b(This)28 b(is)g(sho)n(wn)f(b)n(y)h(the)515 4403 y(follo)n(wing)e(example.)515 4564 y Fl(Example)k(5.6)40 b Fv(The)f(term)g(rewriting)f(system)h Fs(R)2188 4576 y Fr(1)2267 4564 y Fv(=)j Fs(f)p Fo(g)s Fv(\()p Fo(x;)14 b(y)s Fv(\))41 b Fs(!)h Fo(x)p Fs(g)d Fv(is)g(ob)n(viously)515 4663 y Fs(C)559 4675 y Fq(E)603 4663 y Fv(-terminating.)e(W)-7 b(e)27 b(sho)n(w)g(that)h(the)g(same)f(is)h(true)f(for)g(the)h(TRS)1178 4919 y Fs(R)1248 4931 y Fr(2)1309 4919 y Fv(=)1397 4749 y Fe(8)1397 4824 y(<)1397 4973 y(:)1541 4819 y Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(x)p Fv(\))83 b Fs(!)g Fo(f)9 b Fv(\()p Fo(h)p Fv(\()p Fo(x)p Fv(\))p Fo(;)14 b(h)p Fv(\()p Fo(x)p Fv(\))p Fo(;)g(x)p Fv(\))1706 4918 y Fo(h)p Fv(\(0\))83 b Fs(!)g Fv(0)1512 5018 y Fo(h)p Fv(\()p Fo(g)s Fv(\()p Fo(x;)14 b(y)s Fv(\)\))84 b Fs(!)f Fo(y)1905 5255 y Fv(16)p eop %%Page: 17 17 17 16 bop 515 523 a Fv(or)48 b(equiv)-5 b(alen)n(tly)48 b(that)h Fs(R)1391 535 y Fr(2)1462 523 y Fs(])33 b(C)1594 535 y Fq(E)1687 523 y Fv(is)49 b(terminating.)100 b(There)49 b(is)f(only)h(one)f(cycle)h(in)515 623 y(EDG\()p Fs(R)802 635 y Fr(2)863 623 y Fs(])24 b(C)986 635 y Fq(E)1031 623 y Fv(\),)38 b(viz.)d Fs(fh)p Fo(F)12 b Fv(\(0)p Fo(;)i Fv(1)p Fo(;)g(x)p Fv(\))p Fo(;)g(F)e Fv(\()p Fo(h)p Fv(\()p Fo(x)p Fv(\))p Fo(;)i(h)p Fv(\()p Fo(x)p Fv(\))p Fo(;)g(x)p Fv(\))p Fs(ig)p Fv(.)63 b(After)36 b(normalization)e(with)515 722 y(the)28 b(AFS)g Fs(f)p Fo(f)9 b Fv(\()p Fo(x;)14 b(y)s(;)g(z)t Fv(\))22 b Fs(!)h Fo(f)1390 692 y Fq(0)1413 722 y Fo(;)14 b(h)p Fv(\()p Fo(x)p Fv(\))24 b Fs(!)f Fo(x;)14 b(g)s Fv(\()p Fo(x;)g(y)s Fv(\))23 b Fs(!)g Fo(y)s Fs(g)k Fv(the)h(resulting)f(constrain)n(ts)f(are)1465 902 y Fo(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))84 b Fs(\025)f Fo(x)1465 1001 y(C)6 b(ons)p Fv(\()p Fo(x;)14 b(y)s Fv(\))84 b Fs(\025)f Fo(y)1779 1101 y(f)1829 1071 y Fq(0)1935 1101 y Fs(\025)g Fo(f)2133 1071 y Fq(0)1811 1201 y Fv(0)f Fs(\025)h Fv(0)1808 1300 y Fo(y)j Fs(\025)d Fo(y)1518 1400 y(F)12 b Fv(\(0)p Fo(;)i Fv(1)p Fo(;)g(x)p Fv(\))83 b Fo(>)g(F)12 b Fv(\()p Fo(x;)i(x;)g(x)p Fv(\))515 1576 y(In)36 b(order)e(to)i(construct)f(a)g(reduction)h(pair)f (satisfying)g(the)h(inequalities,)i(w)n(e)d(will)h(use)515 1676 y(algebras;)h(see)e(e.g.)g([Ohl01)o(,)j(Section)d(6].)60 b(As)36 b(algebra)e Fs(A)h Fv(w)n(e)h(c)n(ho)r(ose)e(the)i(carrier)d (set)515 1775 y Fo(A)38 b Fv(=)f Fs(f\016)p Fo(;)14 b Fs(\017)p Fo(;)g Fk(\006)p Fo(;)g Fk(\007)p Fs(g)35 b Fv(together)h(with)h(the)g(in)n(terpretation)e(functions)i(0)2774 1787 y Fq(A)2869 1775 y Fv(=)h Fs(\016)p Fv(,)g(1)3117 1787 y Fq(A)3212 1775 y Fv(=)f Fs(\017)p Fv(,)515 1875 y Fo(f)565 1845 y Fq(0)556 1898 y(A)636 1875 y Fv(=)23 b Fk(\007)p Fv(,)28 b Fo(C)6 b(ons)1024 1887 y Fq(A)1081 1875 y Fv(\()p Fo(a)1157 1887 y Fr(1)1195 1875 y Fo(;)14 b(a)1276 1887 y Fr(2)1313 1875 y Fv(\))23 b(=)g Fk(\006)28 b Fv(for)f(all)g Fo(a)1825 1887 y Fr(1)1862 1875 y Fo(;)14 b(a)1943 1887 y Fr(2)2003 1875 y Fs(2)24 b Fo(A)p Fv(,)k(and)1132 2107 y Fo(F)1185 2119 y Fq(A)1243 2107 y Fv(\()p Fo(a)1319 2119 y Fr(1)1357 2107 y Fo(;)14 b(a)1438 2119 y Fr(2)1475 2107 y Fo(;)g(a)1556 2119 y Fr(3)1593 2107 y Fv(\))23 b(=)1736 1990 y Fe(\032)1840 2057 y Fk(\006)83 b Fv(if)28 b Fo(a)2098 2069 y Fr(1)2158 2057 y Fv(=)23 b Fs(\016)k Fv(and)g Fo(a)2520 2069 y Fr(2)2581 2057 y Fv(=)22 b Fs(\017)1840 2156 y Fk(\007)83 b Fv(otherwise.)515 2336 y(F)-7 b(urthermore,)24 b(w)n(e)h(de\014ne)h(a)e(quasi-ordering)f Fk(\045)i Fv(on)g Fo(A)g Fv(b)n(y)g Fk(\006)e(\045)g(\007)p Fv(.)37 b(It)25 b(is)g(fairly)g(simple)g(to)515 2435 y(pro)n(v)n(e)g(that)i(\()p Fs(A)p Fo(;)14 b Fk(\045)p Fv(\))27 b(is)g(a)f(w)n(ell-founded)h(w)n(eakly)e(monotone)i(algebra)e (and)h(that)h Fk(\045)3161 2447 y Fq(A)3245 2435 y Fv(and)515 2535 y(its)f(corresp)r(onding)d(stable-strict)i(relation)g(satisfy)g (the)h(constrain)n(ts.)35 b(Hence)26 b Fs(R)3087 2547 y Fr(2)3139 2535 y Fs(])15 b(C)3253 2547 y Fq(E)3323 2535 y Fv(is)515 2634 y(terminating)30 b(b)n(y)g(Corollary)e(2.5.)45 b(But)31 b(the)g(hierarc)n(hical)e(com)n(bination)g(of)i Fs(R)3039 2646 y Fr(1)3107 2634 y Fv(and)f Fs(R)3341 2646 y Fr(2)515 2734 y Fv(is)d(not)h(terminating:)980 2909 y Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(g)s Fv(\(0)p Fo(;)g Fv(1\)\))81 b Fs(!)1644 2921 y Fq(R)1701 2929 y Ff(2)1820 2909 y Fo(f)9 b Fv(\()p Fo(h)p Fv(\()p Fo(g)s Fv(\(0)p Fo(;)14 b Fv(1\)\))p Fo(;)g(h)p Fv(\()p Fo(g)s Fv(\(0)p Fo(;)g Fv(1\)\))p Fo(;)g(g)s Fv(\(0)p Fo(;)g Fv(1\)\))1561 3008 y Fs(!)1644 3020 y Fq(R)1701 3028 y Ff(1)1820 3008 y Fo(f)9 b Fv(\()p Fo(h)p Fv(\(0\))p Fo(;)14 b(h)p Fv(\()p Fo(g)s Fv(\(0)p Fo(;)g Fv(1\)\))p Fo(;)g(g)s Fv(\(0)p Fo(;)g Fv(1\)\))1561 3110 y Fs(!)1644 3074 y Fr(+)1644 3134 y Fq(R)1701 3142 y Ff(2)1820 3110 y Fo(f)9 b Fv(\(0)p Fo(;)14 b Fv(1)p Fo(;)g(g)s Fv(\(0)p Fo(;)g Fv(1\)\))639 3289 y(F)-7 b(urthermore,)23 b(the)f(next)h (example)f(illustrates)f(that)i(Theorem)e(5.4)h(do)r(es)f(not)i(remain) 515 3389 y(v)-5 b(alid)27 b(if)h(one)g(considers)e(\014nitely)i(branc)n (hing)e(systems)h(instead)h(of)f(\014nite)i(systems.)515 3569 y Fl(Example)h(5.7)40 b Fv(The)55 b(term)f(rewriting)f(systems)g Fs(R)2281 3581 y Fr(1)2386 3569 y Fv(=)67 b Fs(f)p Fo(a)f Fs(!)h Fo(b)p Fs(g)54 b Fv(and)g Fs(R)3210 3581 y Fr(2)3314 3569 y Fv(=)515 3669 y Fs(f)p Fo(h)605 3681 y Fn(j)639 3669 y Fv(\()p Fo(x;)14 b(x)p Fv(\))47 b Fs(!)g Fo(h)1059 3681 y Fn(j)s Fr(+1)1177 3669 y Fv(\()p Fo(a;)14 b(b)p Fv(\))g Fs(j)g Fo(j)51 b Fs(2)46 b Fa(N)t Fs(g)h Fv(are)41 b(\014nitely)h(branc)n(hing)e(and)h(simply)g(terminat-)515 3768 y(ing.)54 b(According)33 b(to)g(Prop)r(osition)f(3.2,)i(b)r(oth)g (systems)f(are)f Fs(C)2532 3780 y Fq(E)2577 3768 y Fv(-terminating.)54 b Fs(R)3177 3780 y Fr(2)3248 3768 y Fv(is)33 b(a)515 3868 y(restricted)j(prop)r(er)f(extension)i(of)f Fs(R)1718 3880 y Fr(1)1793 3868 y Fv(b)r(ecause)g Fo(h)2157 3880 y Fn(j)s Fr(+1)2314 3868 y Fs(6)p Fc(\004)2366 3883 y Fn(d)2443 3868 y Fo(h)2491 3880 y Fn(j)2562 3868 y Fv(\(so)g Fo(a)i Fs(2)h(D)2945 3880 y Fr(1)3019 3868 y Fv(is)d(allo)n(w)n(ed)515 3967 y(b)r(elo)n(w)28 b Fo(h)799 3979 y Fn(j)s Fr(+1)942 3967 y Fs(2)d(D)1088 3937 y Fr(1)1086 3988 y(2)1126 3967 y Fv(\).)40 b(Ho)n(w)n(ev)n(er,)27 b(the)i(hierarc)n(hical)e(com)n (bination)g(of)i Fs(R)2810 3979 y Fr(1)2876 3967 y Fv(and)f Fs(R)3108 3979 y Fr(2)3175 3967 y Fv(is)g(not)515 4067 y(terminating:)685 4247 y Fo(h)733 4259 y Fr(1)770 4247 y Fv(\()p Fo(a;)14 b(b)p Fv(\))23 b Fs(!)1057 4259 y Fq(R)1114 4267 y Ff(1)1174 4247 y Fo(h)1222 4259 y Fr(1)1259 4247 y Fv(\()p Fo(b;)14 b(b)p Fv(\))22 b Fs(!)1537 4259 y Fq(R)1594 4267 y Ff(2)1654 4247 y Fo(h)1702 4259 y Fr(2)1739 4247 y Fv(\()p Fo(a;)14 b(b)p Fv(\))23 b Fs(!)2026 4259 y Fq(R)2083 4267 y Ff(1)2143 4247 y Fo(h)2191 4259 y Fr(2)2228 4247 y Fv(\()p Fo(b;)14 b(b)p Fv(\))23 b Fs(!)2507 4259 y Fq(R)2564 4267 y Ff(2)2623 4247 y Fo(h)2671 4259 y Fr(3)2708 4247 y Fv(\()p Fo(a;)14 b(b)p Fv(\))23 b Fs(!)2995 4259 y Fq(R)3052 4267 y Ff(1)3112 4247 y Fo(:)14 b(:)g(:)639 4428 y Fv(As)19 b(hierarc)n(hical)e(com)n (binations)h(with)h(a)f(common)h(subsystem)f(generalize)f(b)r(oth)j (com-)515 4527 y(p)r(osable)e(systems)h(and)g(hierarc)n(hical)f(com)n (binations,)i(prop)r(er)e(extension)h(with)h(a)f(common)515 4627 y(subsystem)32 b(extend)g(b)r(oth)h(comp)r(osable)e(systems)h(and) g(prop)r(er)g(extensions.)50 b(F)-7 b(ormally)g(,)515 4726 y(these)27 b(systems)g(are)g(de\014ned)h(as)f(follo)n(ws.)515 4907 y Fl(De\014nition)k(5.8)40 b Fv(Let)35 b(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\))36 b(b)r(e)g(a)f(hierarc)n(hical)e(com)n (bination)h(of)h(the)h(base)e(system)515 5006 y(\()p Fs(F)607 5018 y Fr(1)644 5006 y Fo(;)14 b Fs(R)751 5018 y Fr(1)788 5006 y Fv(\))25 b(and)f(the)g(extension)g(\()p Fs(F)1598 5018 y Fr(2)1635 5006 y Fo(;)14 b Fs(R)1742 5018 y Fr(2)1780 5006 y Fv(\))24 b(with)h(a)e(common)h(subsystem)g(\()p Fs(F)2905 5018 y Fr(0)2942 5006 y Fo(;)14 b Fs(R)3049 5018 y Fr(0)3086 5006 y Fv(\),)26 b(where)1905 5255 y(17)p eop %%Page: 18 18 18 17 bop 515 523 a Fs(F)575 535 y Fr(0)636 523 y Fv(=)25 b Fs(f)p Fo(f)36 b Fs(j)29 b Fo(f)36 b Fv(o)r(ccurs)27 b(in)h Fs(R)1398 535 y Fr(0)1436 523 y Fs(g)p Fv(.)39 b(De\014ne)29 b Fs(D)1862 535 y Fr(0)1924 523 y Fv(=)c Fs(f)p Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))k Fs(j)g Fo(l)d Fs(!)f Fo(r)i Fs(2)e(R)2750 535 y Fr(0)2788 523 y Fs(g)p Fv(,)j Fs(D)2947 493 y Fq(0)2945 544 y Fr(1)3007 523 y Fv(=)d Fs(D)3161 535 y Fr(1)3217 523 y Fs(n)19 b(D)3342 535 y Fr(0)515 631 y Fv(and)k Fs(D)738 601 y Fq(0)736 652 y Fr(2)796 631 y Fv(=)g Fs(D)948 643 y Fr(2)996 631 y Fs(n)10 b(D)1112 643 y Fr(0)1149 631 y Fv(.)35 b(No)n(w)23 b Fs(D)1458 601 y Fq(0)1456 652 y Fr(2)1517 631 y Fv(is)h(split)f(in)h (t)n(w)n(o)f(sets)g Fs(D)2247 601 y Fq(0)2271 590 y Fr(1)2271 652 y(2)2332 631 y Fv(and)g Fs(D)2555 601 y Fq(0)2579 590 y Fr(2)2579 652 y(2)2616 631 y Fv(,)h(where)f Fs(D)2965 601 y Fq(0)2989 590 y Fr(1)2989 652 y(2)3049 631 y Fv(=)g Fs(f)p Fo(f)9 b Fs(j)p Fo(f)31 b Fs(2)515 740 y(D)581 710 y Fq(0)579 760 y Fr(2)616 740 y Fo(;)14 b(f)41 b Fc(\004)787 755 y Fn(d)858 740 y Fo(g)30 b Fv(for)d(some)g Fo(g)35 b Fs(2)e(D)1492 710 y Fq(0)1490 760 y Fr(1)1527 740 y Fs(g)g Fv(and)g Fs(D)1835 710 y Fq(0)1858 698 y Fr(2)1858 760 y(2)1928 740 y Fv(=)f Fs(D)2091 710 y Fq(0)2089 760 y Fr(2)2148 740 y Fs(n)22 b(D)2278 710 y Fq(0)2302 698 y Fr(1)2302 760 y(2)2339 740 y Fv(.)53 b(The)34 b(TRS)f Fs(R)2862 752 y Fr(2)2933 740 y Fv(is)g(a)g(called)f(a)515 839 y Fm(pr)l(op)l(er)37 b(extension)k Fv(of)35 b Fs(R)1329 851 y Fr(1)1401 839 y Fm(with)i(a)g(c)l(ommon)f(subsystem)f Fs(R)2468 851 y Fr(0)2540 839 y Fv(if)g(ev)n(ery)f(rewrite)g(rule)g(of) 515 939 y Fo(l)g Fs(!)e Fo(r)k Fs(2)d(R)920 951 y Fr(2)991 939 y Fv(satis\014es)f(the)i(follo)n(wing)e(condition:)48 b(If)34 b Fo(t)f Fv(is)g(a)g(subterm)g(of)h Fo(r)i Fv(suc)n(h)d(that) 515 1039 y Fo(r)r(oot)p Fv(\()p Fo(t)p Fv(\))e Fs(2)f(D)940 1009 y Fq(0)964 997 y Fr(1)964 1059 y(2)1033 1039 y Fv(and)h Fo(r)r(oot)p Fv(\()p Fo(t)p Fv(\))h Fc(\004)1524 1054 y Fn(d)1593 1039 y Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\),)i(then)e Fo(t)f Fv(con)n(tains)g(no)g(function)h(sym)n(b)r(ol)f(from)515 1147 y Fs(D)581 1117 y Fq(0)579 1168 y Fr(1)634 1147 y Fs([)19 b(D)774 1117 y Fq(0)798 1105 y Fr(1)798 1168 y(2)863 1147 y Fv(strictly)27 b(b)r(elo)n(w)g(its)h(ro)r(ot.)639 1247 y Fs(R)709 1259 y Fr(2)775 1247 y Fv(is)f(a)g Fm(r)l(estricte)l(d) j(pr)l(op)l(er)h(extension)c Fv(of)h Fs(R)2077 1259 y Fr(1)2142 1247 y Fm(with)j(a)f(c)l(ommon)g(subsystem)d Fs(R)3182 1259 y Fr(0)3247 1247 y Fv(if)i(it)515 1346 y(is)22 b(a)g(prop)r(er)f(extension)h(of)g Fs(R)1438 1358 y Fr(1)1498 1346 y Fv(with)g(a)g(common)g(subsystem)g Fs(R)2537 1358 y Fr(0)2597 1346 y Fv(suc)n(h)f(that)i(no)f Fm(left-hand)515 1446 y(side)j Fv(of)f(the)h(rewrite)e(rules)h(of)g Fs(R)1545 1458 y Fr(2)1607 1446 y Fv(con)n(tains)g(a)f(function)i(sym)n (b)r(ol)f(from)g Fs(D)2857 1416 y Fq(0)2855 1467 y Fr(1)2904 1446 y Fs([)12 b(D)3037 1416 y Fq(0)3061 1404 y Fr(1)3061 1467 y(2)3122 1446 y Fv(strictly)515 1546 y(b)r(elo)n(w)27 b(its)h(ro)r(ot.)639 1728 y(It)i(should)f(b)r(e)g(p)r(oin)n(ted)h(out)f (that)g(prop)r(er)g(extensions)f(with)i(a)f(common)f(subsystem)515 1828 y(w)n(ere)e(in)n(tro)r(duced)i(as)f Fm(gener)l(alize)l(d)k(pr)l (op)l(er)g(extension)i Fv(in)28 b([KR95)o(].)515 2011 y Fl(Example)i(5.9)40 b Fv(Let)32 b Fs(R)1295 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y Fr(2)832 4555 y Fv(\))g Fs(])g(C)977 4567 y Fq(E)1022 4555 y Fv(,)23 b(i.e.,)h(there)d(exists)h(a)g(substitution)g Fo(\033)27 b Fv(:)c Fs(V)29 b(!)24 b(T)d Fv(\()p Fs(F)15 b(])7 b(f)p Fo(C)f(ons)p Fs(g)p Fv(\))22 b(suc)n(h)g(that)515 4655 y Fo(t)545 4667 y Fn(j)580 4655 y Fo(\033)33 b Fs(!)743 4624 y Fq(\003)743 4681 y Fr(\()p Fq(R)826 4689 y Ff(1)858 4681 y Fq([R)960 4689 y Ff(2)992 4681 y Fr(\))p Fq(]C)1099 4689 y Fi(E)1172 4655 y Fo(s)1211 4667 y Fn(j)s Fr(+1)1330 4655 y Fo(\033)i Fv(holds)c(for)g(all)g(consecutiv)n(e)g(pairs)f Fs(h)p Fo(s)2604 4667 y Fn(j)2640 4655 y Fo(;)14 b(t)2707 4667 y Fn(j)2741 4655 y Fs(i)32 b Fv(and)g Fs(h)p Fo(s)3042 4667 y Fn(j)s Fr(+1)3161 4655 y Fo(;)14 b(t)3228 4667 y Fn(j)s Fr(+1)3347 4655 y Fs(i)515 4754 y Fv(in)39 b(the)g(sequence.) 71 b(Since)39 b(w)n(e)g(consider)f(\014nite)h(TRSs,)j(w)n(e)d(ma)n(y)f (assume)g(that)h(ev)n(ery)515 4854 y(dep)r(endency)27 b(pair)f(in)h Fs(h)p Fo(s)1303 4866 y Fr(1)1341 4854 y Fo(;)14 b(t)1408 4866 y Fr(1)1445 4854 y Fs(ih)p Fo(s)1548 4866 y Fr(2)1586 4854 y Fo(;)g(t)1653 4866 y Fr(2)1690 4854 y Fs(i)g Fo(:)g(:)g(:)41 b Fv(b)r(elongs)26 b(to)h(a)f(cycle)g Fs(P)34 b Fv(whic)n(h)27 b(consists)f(solely)515 4962 y(of)19 b(dep)r(endency)h(pairs)e(from)h Fs(R)1498 4932 y Fq(0)1522 4920 y Fr(1)1522 4983 y(2)1559 4962 y Fv(.)34 b(As)20 b(in)f(the)h(pro)r(of)f(of)g(Theorem)f(5.4,)j(one)d(can)h(sho)n (w)g(that)1905 5255 y(18)p eop %%Page: 19 19 19 18 bop 515 523 a Fv(ev)n(ery)31 b(\(prop)r(er\))h(subterm)g(of)g(a)g (dep)r(endency)h(pair)f(in)h Fs(P)39 b Fv(do)r(es)32 b(not)g(con)n(tain)g(a)g(sym)n(b)r(ol)515 631 y(from)27 b Fs(D)777 601 y Fq(0)775 652 y Fr(1)831 631 y Fs([)19 b(D)971 601 y Fq(0)994 590 y Fr(1)994 652 y(2)1031 631 y Fv(.)639 740 y(Since)k(b)r(oth)g Fs(R)1112 752 y Fr(0)1172 740 y Fv(and)f Fs(R)1398 710 y Fq(0)1422 698 y Fr(2)1422 760 y(2)1482 740 y Fv(do)g(not)h(con)n(tain)e(function)j(sym)n(b)r(ols) d(from)h Fs(D)2911 710 y Fq(0)2909 760 y Fr(1)2955 740 y Fs([)8 b(D)3084 710 y Fq(0)3108 698 y Fr(1)3108 760 y(2)3168 740 y Fv(\(if)23 b Fs(R)3341 752 y Fr(0)515 839 y Fv(w)n(ould)e(con)n(tain)g(a)g(function)h(sym)n(b)r(ol)f Fo(f)32 b Fs(2)23 b(D)1915 809 y Fq(0)1913 860 y Fr(1)1951 839 y Fv(,)g(then)f(the)g(rewrite)f(rules)g(de\014ning)g Fo(f)31 b Fv(w)n(ould)515 948 y(b)r(e)25 b(in)g Fs(R)789 960 y Fr(0)826 948 y Fv(,)h(i.e.,)f Fo(f)32 b Fs(2)23 b(D)1244 960 y Fr(0)1282 948 y Fv(\),)i(the)h(TRS)e Fs(R)1764 917 y Fq(0)1764 968 y Fr(1)1815 948 y Fs([)13 b(R)1953 917 y Fq(0)1976 906 y Fr(1)1976 968 y(2)2026 948 y Fs([)g(C)2138 960 y Fq(E)2208 948 y Fv(can)24 b(b)r(e)h(view)n(ed)f(as)g(an)h (extension)f(of)515 1056 y Fs(R)585 1068 y Fr(0)640 1056 y Fs([)17 b(R)782 1026 y Fq(0)806 1014 y Fr(2)806 1076 y(2)843 1056 y Fv(.)37 b(Because)26 b(the)h(in\014nite)h Fs(R)1713 1026 y Fq(0)1737 1014 y Fr(1)1737 1076 y(2)1774 1056 y Fv(-c)n(hain)e(is)h(minimal,)g(w)n(e)g(ma)n(y)f(apply)h(Lemma)g (4.6)515 1156 y(to)33 b(ev)n(ery)g Fo(t)878 1168 y Fn(j)913 1156 y Fo(\033)k Fs(!)1080 1125 y Fq(\003)1080 1186 y Fr(\()p Fq(R)1163 1194 y Ff(0)1195 1186 y Fq([R)1297 1169 y Fi(0)1319 1165 y Ff(2)1319 1204 y(2)1352 1186 y Fr(\))p Fq([)p Fr(\()p Fq(R)1506 1165 y Fi(0)1506 1204 y Ff(1)1538 1186 y Fq([R)1640 1169 y Fi(0)1662 1165 y Ff(1)1662 1204 y(2)1694 1186 y Fq([C)1775 1194 y Fi(E)1815 1186 y Fr(\))1878 1156 y Fo(s)1917 1168 y Fn(j)s Fr(+1)2036 1156 y Fo(\033)s Fv(.)56 b(This)34 b(yields)f(\010\()p Fo(t)2722 1168 y Fn(j)2758 1156 y Fo(\033)s Fv(\))h Fs(!)2957 1125 y Fq(\003)2957 1186 y Fr(\()p Fq(R)3040 1194 y Ff(0)3072 1186 y Fq([R)3174 1169 y Fi(0)3196 1165 y Ff(2)3196 1204 y(2)3229 1186 y Fr(\))p Fq(]C)3339 1165 y Fi(0)3336 1205 y(E)515 1276 y Fv(\010\()p Fo(s)646 1288 y Fn(j)s Fr(+1)765 1276 y Fo(\033)s Fv(\),)j(where)e Fs(C)1204 1246 y Fq(0)1199 1299 y(E)1278 1276 y Fv(denotes)g(the)g(system)f Fs(f)p Fo(C)6 b(ons)2254 1246 y Fq(0)2277 1276 y Fv(\()p Fo(x;)14 b(y)s Fv(\))36 b Fs(!)f Fo(x;)14 b(C)6 b(ons)2901 1246 y Fq(0)2925 1276 y Fv(\()p Fo(x;)14 b(y)s Fv(\))36 b Fs(!)f Fo(y)s Fs(g)p Fv(.)515 1387 y(In)29 b(other)f(w)n(ords,)g(the)h (cycle)g Fs(P)35 b Fv(admits)29 b(an)g(in\014nite)g Fs(R)2288 1357 y Fq(0)2312 1345 y Fr(1)2312 1408 y(2)2349 1387 y Fv(-c)n(hain)f(o)n(v)n(er)f(\()p Fs(R)2878 1399 y Fr(0)2936 1387 y Fs([)19 b(R)3080 1357 y Fq(0)3104 1345 y Fr(2)3104 1408 y(2)3141 1387 y Fv(\))h Fs(])f(C)3316 1357 y Fq(0)3311 1410 y(E)3356 1387 y Fv(.)515 1487 y(Since)26 b(this)h(is)f(an)f (in\014nite)i Fs(R)1436 1499 y Fr(2)1474 1487 y Fv(-c)n(hain)e(o)n(v)n (er)g Fs(R)1966 1499 y Fr(2)2019 1487 y Fs(])16 b(C)2139 1457 y Fq(0)2134 1510 y(E)2178 1487 y Fv(,)27 b(w)n(e)f(deriv)n(e)f(a)h (con)n(tradiction)f(to)h(the)515 1587 y Fs(C)559 1599 y Fq(E)603 1587 y Fv(-termination)h(of)h Fs(R)1248 1599 y Fr(2)1285 1587 y Fv(.)2006 b Fk(\003)639 1786 y Fv(By)26 b(means)g(of)g(Theorem)g(5.10,)f(one)h(can)g(for)g(example)g(infer)g Fs(C)2647 1798 y Fq(E)2692 1786 y Fv(-termination)f(of)i(the)515 1885 y(hierarc)n(hical)e(com)n(bination)i(of)h(the)g(t)n(w)n(o)e(TRSs)i (from)f(Example)g(5.9.)639 1985 y(The)h(next)g(corollary)d(follo)n(ws)h (easily)h(from)g(the)h(preceding)f(results.)515 2168 y Fl(Corollary)k(5.11)76 b Fm(T)-6 b(ermination)37 b(is)f(mo)l(dular)g (for)h(\014nite)e(non-duplic)l(ating)i(r)l(estricte)l(d)515 2267 y(pr)l(op)l(er)30 b(extensions)g(\(with)g(a)g(c)l(ommon)g (subsystem\).)515 2467 y(Pr)l(o)l(of)52 b Fv(Clearly)-7 b(,)24 b(the)h(union)g(of)f(t)n(w)n(o)g(non-duplicating)g(TRSs)g(is)g (again)g(non-duplicating.)515 2566 y(Hence)31 b(the)h(assertions)e (follo)n(w)g(from)h(Theorem)g(5.4)f(\(Theorem)h(5.10,)g(resp)r(ectiv)n (ely\))g(in)515 2666 y(conjunction)c(with)i(Prop)r(osition)c(3.2.)1584 b Fk(\003)639 2865 y Fv(Ho)n(w)n(ev)n(er,)29 b(the)h(mo)r(dularit)n(y)f (of)h(simple)g(termination)f(for)g(restricted)g(prop)r(er)g(exten-)515 2965 y(sions)k(\(with)j(a)e(common)g(subsystem\))g(do)r(es)g(not)g (directly)h(follo)n(w)e(from)h(Theorem)g(5.4)515 3064 y(\(Theorem)39 b(5.10,)i(resp)r(ectiv)n(ely\).)73 b(This)40 b(is)f(b)r(ecause)h(the)g(fact)g(that)g Fs(R)2887 3076 y Fr(2)2964 3064 y Fv(is)g(a)f(prop)r(er)515 3164 y(extension)d(of)g Fs(R)1064 3176 y Fr(1)1138 3164 y Fv(do)r(es)h(not)f(imply)h(that)g Fs(R)1992 3176 y Fr(2)2054 3164 y Fs([)24 b(E)7 b Fo(mb)p Fv(\()p Fs(F)2385 3176 y Fr(2)2422 3164 y Fv(\))37 b(is)f(a)h(prop)r (er)e(extension)h(of)515 3264 y Fs(R)585 3276 y Fr(1)640 3264 y Fs([)18 b(E)7 b Fo(mb)p Fv(\()p Fs(F)965 3276 y Fr(1)1002 3264 y Fv(\).)37 b(Nev)n(ertheless,)26 b(it)i(will)g(b)r(e) f(sho)n(wn)g(b)r(elo)n(w)f(that)i(simple)f(termination)g(is)515 3363 y(indeed)h(mo)r(dular)f(for)g(restricted)g(prop)r(er)f(extensions) h(\(with)i(a)e(common)g(subsystem\).)515 3638 y Fw(6)134 b(Hierarc)l(hical)47 b(simple)f(termination)515 3820 y Fv(In)31 b(this)h(section)f(w)n(e)g(will)h(pro)n(vide)e(simple)i(pro) r(ofs)e(of)h(kno)n(wn)g(results)g(ab)r(out)g(the)h(mo)r(d-)515 3919 y(ularit)n(y)g(of)i(simple)f(termination,)h(and)g(extend)f(these)h (results.)53 b(F)-7 b(urthermore,)34 b(w)n(e)f(will)515 4019 y(dispro)n(v)n(e)e(a)h(conjecture)g(of)h(Krishna)e(Rao)h([KR94)o (].)52 b(W)-7 b(e)33 b(start)f(with)h(a)g(lemma)f(due)h(to)515 4118 y(Gramlic)n(h)27 b([Gra96)n(].)515 4301 y Fl(Lemma)i(6.1)41 b Fm(Simple)j(termination)f(is)g(a)h(mo)l(dular)f(pr)l(op)l(erty)h(of)g (\014nitely)f(br)l(anching)515 4401 y(c)l(omp)l(osable)31 b(TRSs.)515 4600 y(Pr)l(o)l(of)55 b Fv(Let)27 b(\()p Fs(F)1002 4612 y Fr(1)1039 4600 y Fo(;)14 b Fs(R)1146 4612 y Fr(1)1183 4600 y Fv(\))27 b(and)f(\()p Fs(F)1494 4612 y Fr(2)1531 4600 y Fo(;)14 b Fs(R)1638 4612 y Fr(2)1675 4600 y Fv(\))27 b(b)r(e)f(simply)h(terminating)e(comp)r(osable)g(TRSs.) 37 b(W)-7 b(e)515 4700 y(ha)n(v)n(e)18 b(to)h(sho)n(w)g(that)h(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\))24 b(=)e(\()p Fs(F)1603 4712 y Fr(1)1643 4700 y Fs([)r(F)1760 4712 y Fr(2)1797 4700 y Fo(;)14 b Fs(R)1904 4712 y Fr(1)1944 4700 y Fs([)r(R)2071 4712 y Fr(2)2109 4700 y Fv(\))20 b(is)g(also)e(simply)i(terminating.)34 b(Simple)515 4799 y(termination)26 b(implies)h(that)g Fs(R)1496 4811 y Fr(1)1551 4799 y Fs([)17 b(E)7 b Fo(mb)p Fv(\()p Fs(F)1875 4811 y Fr(1)1912 4799 y Fv(\))28 b(and)e Fs(R)2202 4811 y Fr(2)2257 4799 y Fs([)17 b(E)7 b Fo(mb)p Fv(\()p Fs(F)2581 4811 y Fr(2)2618 4799 y Fv(\))27 b(are)f Fs(C)2859 4811 y Fq(E)2904 4799 y Fv(-terminating;)515 4899 y(see)d(Prop)r(osition)f(3.2.)35 b(Moreo)n(v)n(er,)22 b(it)i(is)g(fairly)f(easy)f(to)i(see)f(that)h(these)g(systems)f(are)g (also)1905 5255 y(19)p eop %%Page: 20 20 20 19 bop 515 523 a Fv(comp)r(osable.)53 b(Th)n(us,)34 b(an)f(application)g(of)h(Prop)r(osition)d(5.1)i(yields)g Fs(C)2782 535 y Fq(E)2826 523 y Fv(-termination)g(of)515 623 y Fs(R)19 b([)f(E)7 b Fo(mb)p Fv(\()p Fs(F)h Fv(\).)38 b(In)27 b(other)g(w)n(ords,)g Fs(R)h Fv(is)f(simply)h(terminating.)807 b Fk(\003)639 807 y Fv(The)33 b(next)g(theorem)f(constitutes)g(the)h (main)g(result)f(from)h([KR94)n(].)52 b(The)33 b(reader)e(is)515 907 y(in)n(vited)c(to)h(compare)e(our)h(pro)r(of)g(with)h(the)g(long)f (and)h(in)n(v)n(olv)n(ed)e(original)g(one.)515 1071 y Fl(Theorem)k(6.2)41 b Fm(L)l(et)25 b Fv(\()p Fs(F)1315 1083 y Fr(1)1353 1071 y Fo(;)14 b Fs(R)1460 1083 y Fr(1)1497 1071 y Fv(\))26 b Fm(and)h Fv(\()p Fs(F)1805 1083 y Fr(2)1842 1071 y Fo(;)14 b Fs(R)1949 1083 y Fr(2)1987 1071 y Fv(\))26 b Fm(b)l(e)g(\014nite)g(simply)h(terminating)f(TRSs.)38 b(If)515 1170 y Fs(R)585 1182 y Fr(2)657 1170 y Fm(is)d(a)f(r)l (estricte)l(d)h(pr)l(op)l(er)g(extension)f(of)i Fs(R)2001 1182 y Fr(1)2038 1170 y Fm(,)g(then)f(their)g(hier)l(ar)l(chic)l(al)i (c)l(ombination)515 1270 y Fv(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\))24 b(=)e(\()p Fs(F)957 1282 y Fr(1)1013 1270 y Fs([)d(F)1147 1282 y Fr(2)1184 1270 y Fo(;)14 b Fs(R)1291 1282 y Fr(1)1346 1270 y Fs([)19 b(R)1490 1282 y Fr(2)1528 1270 y Fv(\))30 b Fm(is)g(simply)h(terminating.)515 1454 y(Pr)l(o)l(of)79 b Fv(Let)38 b Fs(R)1015 1424 y Fr(1)1015 1475 y(2)1094 1454 y Fv(=)i Fs(f)p Fo(l)i Fs(!)f Fo(r)16 b Fs(j)e Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))42 b Fs(2)f(D)1966 1424 y Fr(1)1964 1475 y(2)2004 1454 y Fs(g)d Fv(and)g Fs(R)2326 1424 y Fr(2)2326 1475 y(2)2404 1454 y Fv(=)j Fs(f)p Fo(l)g Fs(!)g Fo(r)17 b Fs(j)d Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))42 b Fs(2)f(D)3277 1424 y Fr(2)3275 1475 y(2)3314 1454 y Fs(g)p Fv(.)515 1554 y(F)-7 b(urthermore,)39 b(let)g Fs(F)1231 1524 y Fr(2)1223 1574 y(2)1308 1554 y Fv(=)h Fs(f)p Fo(f)22 b Fs(j)14 b Fo(f)36 b Fv(o)r(ccurs)27 b(in)h Fs(R)2056 1524 y Fr(2)2056 1574 y(2)2093 1554 y Fs(g)p Fv(.)68 b(The)38 b(TRS)g Fs(R)2682 1524 y Fr(2)2682 1574 y(2)2758 1554 y Fv(do)r(es)g(not)g(con)n(tain)515 1653 y(function)31 b(sym)n(b)r(ols)f(from)g Fs(D)1426 1665 y Fr(1)1483 1653 y Fs([)21 b(D)1625 1623 y Fr(1)1623 1674 y(2)1663 1653 y Fv(;)32 b(cf.)f(pro)r(of)f(of)g(Theorem)g(5.4.)45 b(In)31 b(other)e(w)n(ords,)h Fs(R)3341 1665 y Fr(1)515 1753 y Fv(and)e Fs(R)747 1723 y Fr(2)747 1774 y(2)814 1753 y Fv(are)g(constructor-sharing.)38 b(Since)29 b(b)r(oth)g(are)f (simply)h(terminating,)g(so)f(is)h(their)515 1853 y(union)20 b(according)e(to)i(Lemma)f(6.1.)34 b(In)20 b(order)f(to)g(sho)n(w)g (termination)h(of)g Fs(R)s([)s(E)7 b Fo(mb)p Fv(\()p Fs(F)h Fv(\),)23 b(it)d(is)515 1952 y(su\016cien)n(t)28 b(to)f(sho)n(w)g(that)h(there)g(is)g(no)f(minimal)h(in\014nite)h Fs(R)2438 1922 y Fr(1)2438 1973 y(2)2476 1952 y Fv(-c)n(hain)e(o)n(v)n (er)f Fs(R)19 b([)g(E)7 b Fo(mb)p Fv(\()p Fs(F)h Fv(\);)515 2052 y(see)39 b(Theorem)h(4.8.)73 b(Supp)r(ose)41 b(on)e(the)i(con)n (trary)d(that)i(there)g(is)g(a)g(minimal)g(in\014nite)515 2151 y Fs(R)585 2121 y Fr(1)585 2172 y(2)622 2151 y Fv(-c)n(hain)35 b Fs(h)p Fo(s)948 2163 y Fr(1)986 2151 y Fo(;)14 b(t)1053 2163 y Fr(1)1090 2151 y Fs(ih)p Fo(s)1193 2163 y Fr(2)1231 2151 y Fo(;)g(t)1298 2163 y Fr(2)1335 2151 y Fs(i)g Fo(:)g(:)g(:)50 b Fv(o)n(v)n(er)34 b Fs(R)24 b([)h(E)7 b Fo(mb)p Fv(\()p Fs(F)h Fv(\),)38 b(i.e.,)h(there)c(exists)h(a)g(substitution)515 2251 y Fo(\033)d Fv(:)c Fs(V)36 b(!)30 b(T)21 b Fv(\()p Fs(F)8 b Fv(\))32 b(suc)n(h)f(that)h Fo(t)1482 2263 y Fn(j)1517 2251 y Fo(\033)h Fs(!)1680 2221 y Fq(\003)1680 2278 y(R[E)5 b Fn(mb)p Fr(\()p Fq(F)h Fr(\))2049 2251 y Fo(s)2088 2263 y Fn(j)s Fr(+1)2207 2251 y Fo(\033)35 b Fv(holds)c(for)g(all)h(consecutiv)n(e)e(pairs)515 2363 y Fs(h)p Fo(s)586 2375 y Fn(j)621 2363 y Fo(;)14 b(t)688 2375 y Fn(j)723 2363 y Fs(i)26 b Fv(and)g Fs(h)p Fo(s)1012 2375 y Fn(j)s Fr(+1)1131 2363 y Fo(;)14 b(t)1198 2375 y Fn(j)s Fr(+1)1317 2363 y Fs(i)27 b Fv(in)f(the)g(sequence.)36 b(Since)26 b(w)n(e)g(consider)f(\014nite)i(TRSs,)f(w)n(e)g(ma)n(y)515 2463 y(assume)34 b(that)h(ev)n(ery)e(dep)r(endency)i(pair)f(in)h Fs(h)p Fo(s)2034 2475 y Fr(1)2072 2463 y Fo(;)14 b(t)2139 2475 y Fr(1)2176 2463 y Fs(ih)p Fo(s)2279 2475 y Fr(2)2316 2463 y Fo(;)g(t)2383 2475 y Fr(2)2421 2463 y Fs(i)g Fo(:)g(:)g(:)48 b Fv(b)r(elongs)34 b(to)h(a)f(cycle)g Fs(P)515 2562 y Fv(whic)n(h)26 b(consists)f(solely)g(of)h(dep)r(endency)h(pairs)e(from) g Fs(R)2292 2532 y Fr(1)2292 2583 y(2)2330 2562 y Fv(.)37 b(As)26 b(in)g(the)g(pro)r(of)g(of)g(Theorem)515 2662 y(5.4,)j(one)h(can)f(sho)n(w)g(that)h(none)g(of)g(the)g(\(prop)r(er\))f (subterms)h(of)g(the)g(dep)r(endency)g(pairs)515 2761 y(in)e Fs(P)34 b Fv(con)n(tains)26 b(a)i(sym)n(b)r(ol)f(from)g Fs(D)1643 2773 y Fr(1)1699 2761 y Fs([)19 b(D)1839 2731 y Fr(1)1837 2782 y(2)1876 2761 y Fv(.)639 2861 y(Let)41 b Fs(F)869 2831 y Fq(0)936 2861 y Fv(=)i(\()p Fs(F)1136 2873 y Fr(1)1200 2861 y Fs([)27 b(F)1342 2873 y Fr(2)1379 2861 y Fv(\))g Fs(n)g(F)1575 2831 y Fr(2)1567 2882 y(2)1612 2861 y Fv(.)75 b(Since)40 b Fs(R)2009 2831 y Fr(2)2009 2882 y(2)2087 2861 y Fv(do)r(es)g(not)g(con)n(tain)f(function)i(sym)n (b)r(ols)515 2961 y(from)30 b Fs(D)778 2973 y Fr(1)836 2961 y Fs([)21 b(D)978 2931 y Fr(1)976 2981 y(2)1016 2961 y Fv(,)32 b(the)f(TRS)g Fs(R)1485 2973 y Fr(1)1544 2961 y Fs([)21 b(R)1690 2931 y Fr(1)1690 2981 y(2)1748 2961 y Fs([)g(E)7 b Fo(mb)p Fv(\()p Fs(F)2084 2931 y Fq(0)2107 2961 y Fv(\))31 b(can)g(b)r(e)g(view)n(ed)g(as)f(an)h (extension)f(of)515 3060 y Fs(R)585 3030 y Fr(2)585 3081 y(2)643 3060 y Fs([)22 b(E)7 b Fo(mb)p Fv(\()p Fs(F)980 3030 y Fr(2)972 3081 y(2)1017 3060 y Fv(\).)48 b(Since)32 b(the)g(in\014nite)g Fs(R)1843 3030 y Fr(1)1843 3081 y(2)1880 3060 y Fv(-c)n(hain)f(is)g(minimal,)i(w)n(e)e(ma)n(y)f(apply)h (Lemma)515 3160 y(4.6)26 b(to)i(ev)n(ery)1181 3324 y Fo(t)1211 3336 y Fn(j)1246 3324 y Fo(\033)f Fs(!)1403 3290 y Fq(\003)1403 3350 y Fr(\()p Fq(R)1486 3330 y Ff(2)1486 3368 y(2)1518 3350 y Fq([E)5 b Fn(mb)p Fr(\()p Fq(F)1772 3330 y Ff(2)1766 3368 y(2)1803 3350 y Fr(\)\))p Fq([)p Fr(\()p Fq(R)1983 3358 y Ff(1)2015 3350 y Fq([R)2117 3330 y Ff(1)2117 3368 y(2)2150 3350 y Fq([E)g Fn(mb)p Fr(\()p Fq(F)2404 3333 y Fi(0)2425 3350 y Fr(\)\))2504 3324 y Fo(s)2543 3336 y Fn(j)s Fr(+1)2662 3324 y Fo(\033)515 3503 y Fv(and)35 b(obtain)h(\010\()p Fo(t)1073 3515 y Fn(j)1108 3503 y Fo(\033)s Fv(\))h Fs(!)1310 3472 y Fq(\003)1310 3533 y Fr(\()p Fq(R)1393 3512 y Ff(2)1393 3551 y(2)1426 3533 y Fq([E)5 b Fn(mb)p Fr(\()p Fq(F)1680 3512 y Ff(2)1674 3551 y(2)1711 3533 y Fr(\)\))p Fq(]C)1844 3541 y Fi(E)1924 3503 y Fv(\010\()p Fo(s)2055 3515 y Fn(j)s Fr(+1)2174 3503 y Fo(\033)s Fv(\).)62 b(In)36 b(other)f(w)n(ords,)i(the)f(cycle)f Fs(P)515 3626 y Fv(admits)22 b(an)g(in\014nite)i Fs(R)1239 3596 y Fr(1)1239 3647 y(2)1276 3626 y Fv(-c)n(hain)e(o)n(v)n(er)f(\()p Fs(R)1793 3596 y Fr(2)1793 3647 y(2)1839 3626 y Fs([)8 b(E)f Fo(mb)p Fv(\()p Fs(F)2162 3596 y Fr(2)2154 3647 y(2)2200 3626 y Fv(\)\))h Fs(])g(C)2379 3638 y Fq(E)2425 3626 y Fv(.)35 b(Therefore,)22 b Fs(R)2948 3638 y Fr(2)2994 3626 y Fs([)8 b(E)f Fo(mb)p Fv(\()p Fs(F)3309 3638 y Fr(2)3347 3626 y Fv(\))515 3726 y(is)24 b(not)g Fs(C)783 3738 y Fq(E)828 3726 y Fv(-terminating.)35 b(This,)25 b(ho)n(w)n(ev)n(er,)e(con)n(tradicts)g(the)i(simple)g(termination)f(of) g Fs(R)3318 3738 y Fr(2)3356 3726 y Fv(;)515 3826 y(see)j(Prop)r (osition)f(3.2.)2088 b Fk(\003)639 4010 y Fv(Theorem)36 b(6.2)g(is)g(applicable)g(to)h(the)g(hierarc)n(hical)d(com)n(bination)i (of)h Fs(R)3008 4022 y Fr(+)3100 4010 y Fv(and)f Fs(R)3340 4022 y Fq(\003)515 4110 y Fv(but)d(neither)f(to)g(Example)f(5.5)h(\(b)r (ecause)g(the)h(systems)e(are)h(not)g(simply)g(terminating\))515 4209 y(nor)d(to)h(Example)f(4.11)g(\(since)h Fs(R)1602 4221 y Fr(2)1669 4209 y Fv(is)g(not)g(a)g(prop)r(er)f(extension)g(of)h Fs(R)2781 4221 y Fr(1)2819 4209 y Fv(\).)44 b(Krishna)29 b(Rao)515 4309 y(conjectured)g(in)g([KR94)o(])h(that)f(Theorem)g(6.2)f (can)h(b)r(e)h(extended)f(to)h(\014nitely)f(branc)n(hing)515 4408 y(TRSs,)21 b(but)e(Example)g(5.7)f(dispro)n(v)n(es)f(his)i (conjecture.)34 b(Moreo)n(v)n(er,)18 b(no)h(example)g(is)g(kno)n(wn)515 4508 y(whic)n(h)32 b(sho)n(ws)g(the)h(necessit)n(y)e(of)i(the)g (condition)f(that)h(no)f(left-hand)h(side)f(of)h(a)f(rewrite)515 4608 y(rule)37 b(from)g Fs(R)967 4620 y Fr(2)1042 4608 y Fv(ma)n(y)g(con)n(tain)g(function)h(sym)n(b)r(ols)f(from)g Fs(D)2467 4620 y Fr(1)2530 4608 y Fs([)25 b(D)2676 4578 y Fr(1)2674 4628 y(2)2751 4608 y Fv(strictly)38 b(b)r(elo)n(w)f(its)515 4707 y(ro)r(ot.)61 b(It)36 b(should)g(b)r(e)g(p)r(oin)n(ted)g(out)g (that)g(Example)g(5.6)f(sho)n(ws)f(the)j(necessit)n(y)e(of)h(this)515 4807 y(condition)28 b(for)g Fs(C)1052 4819 y Fq(E)1096 4807 y Fv(-termination)g(but)h(not)f(for)g(simple)g(termination)g(\(b)r (ecause)g(there)g Fs(R)3341 4819 y Fr(2)515 4907 y Fv(is)33 b(not)h(simply)f(terminating\).)55 b(Therefore,)34 b(it)f(is)h(still)g (an)f(op)r(en)g(question)h(whether)f(or)515 5006 y(not)27 b(Theorem)g(6.2)g(is)g(also)g(v)-5 b(alid)27 b(for)h(non-restricted)e (prop)r(er)h(extensions.)1905 5255 y(20)p eop %%Page: 21 21 21 20 bop 639 523 a Fv(Next,)24 b(w)n(e)d(extend)h(Theorem)f(6.2)g(to)h (restricted)f(prop)r(er)g(extensions)g(with)h(a)g(common)515 623 y(subsystem.)515 803 y Fl(Theorem)30 b(6.3)41 b Fm(L)l(et)25 b Fv(\()p Fs(F)1315 815 y Fr(1)1353 803 y Fo(;)14 b Fs(R)1460 815 y Fr(1)1497 803 y Fv(\))26 b Fm(and)h Fv(\()p Fs(F)1805 815 y Fr(2)1842 803 y Fo(;)14 b Fs(R)1949 815 y Fr(2)1987 803 y Fv(\))26 b Fm(b)l(e)g(\014nite)g(simply)h(terminating)f(TRSs.)38 b(If)515 902 y Fv(\()p Fs(F)607 914 y Fr(2)644 902 y Fo(;)14 b Fs(R)751 914 y Fr(2)788 902 y Fv(\))31 b Fm(is)f(a)h(r)l (estricte)l(d)f(pr)l(op)l(er)i(extension)d(of)i Fv(\()p Fs(F)2191 914 y Fr(1)2229 902 y Fo(;)14 b Fs(R)2336 914 y Fr(1)2373 902 y Fv(\))31 b Fm(with)f(a)h(c)l(ommon)f(subsystem)515 1002 y Fv(\()p Fs(F)607 1014 y Fr(0)644 1002 y Fo(;)14 b Fs(R)751 1014 y Fr(0)788 1002 y Fv(\))p Fm(,)31 b(then)e Fv(\()p Fs(F)8 b Fo(;)14 b Fs(R)p Fv(\))24 b(=)f(\()p Fs(F)1503 1014 y Fr(1)1559 1002 y Fs([)18 b(F)1692 1014 y Fr(2)1729 1002 y Fo(;)c Fs(R)1836 1014 y Fr(1)1892 1002 y Fs([)19 b(R)2036 1014 y Fr(2)2074 1002 y Fv(\))30 b Fm(also)g(is)g(simply)h(terminating.)515 1208 y(Pr)l(o)l(of)69 b Fv(Let)33 b Fs(R)1000 1178 y Fq(0)1000 1228 y Fr(1)1069 1208 y Fv(=)f Fs(f)p Fo(l)h Fs(!)f Fo(r)i Fs(2)f(R)1610 1220 y Fr(1)1661 1208 y Fs(j)14 b Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))33 b Fs(2)g(D)2125 1178 y Fq(0)2123 1228 y Fr(1)2160 1208 y Fs(g)p Fv(,)h Fs(R)2329 1178 y Fq(0)2352 1166 y Fr(1)2352 1228 y(2)2422 1208 y Fv(=)d Fs(f)p Fo(l)i Fs(!)f Fo(r)j Fs(2)d(R)2962 1220 y Fr(2)3014 1208 y Fs(j)14 b Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))33 b Fs(2)515 1316 y(D)581 1286 y Fq(0)604 1274 y Fr(1)604 1337 y(2)642 1316 y Fs(g)p Fv(,)39 b(and)f Fs(R)988 1286 y Fq(0)1011 1274 y Fr(2)1011 1337 y(2)1088 1316 y Fv(=)h Fs(f)p Fo(l)i Fs(!)f Fo(r)i Fs(2)e(R)1667 1328 y Fr(2)1719 1316 y Fs(j)14 b Fo(r)r(oot)p Fv(\()p Fo(l)r Fv(\))40 b Fs(2)g(D)2197 1286 y Fq(0)2221 1274 y Fr(2)2221 1337 y(2)2258 1316 y Fs(g)p Fv(.)67 b(Th)n(us,)39 b Fs(R)2705 1328 y Fr(1)2783 1316 y Fv(=)g Fs(R)2957 1328 y Fr(0)3020 1316 y Fs([)25 b(R)3170 1286 y Fq(0)3170 1337 y Fr(1)3245 1316 y Fv(and)515 1425 y Fs(R)585 1437 y Fr(2)648 1425 y Fv(=)g Fs(R)808 1437 y Fr(0)865 1425 y Fs([)19 b(R)1009 1394 y Fq(0)1033 1383 y Fr(1)1033 1445 y(2)1090 1425 y Fs([)g(R)1234 1394 y Fq(0)1258 1383 y Fr(2)1258 1445 y(2)1295 1425 y Fv(.)41 b(F)-7 b(urthermore,)29 b(let)g Fs(F)2055 1394 y Fq(0)2078 1383 y Fr(2)2078 1445 y(2)2141 1425 y Fv(=)c Fs(f)p Fo(f)d Fs(j)14 b Fo(f)36 b Fv(o)r(ccurs)26 b(in)i Fs(R)2873 1394 y Fq(0)2897 1383 y Fr(2)2897 1445 y(2)2934 1425 y Fs(g)p Fv(.)41 b(The)29 b(TRS)515 1533 y Fs(R)585 1503 y Fq(0)608 1491 y Fr(2)608 1553 y(2)674 1533 y Fv(do)r(es)f(not)g(con)n (tain)g(function)h(sym)n(b)r(ols)f(from)g Fs(D)2211 1503 y Fq(0)2209 1554 y Fr(1)2265 1533 y Fs([)19 b(D)2405 1503 y Fq(0)2429 1491 y Fr(1)2429 1553 y(2)2466 1533 y Fv(,)29 b(hence)f Fs(R)2819 1545 y Fr(1)2885 1533 y Fv(and)g Fs(R)3117 1545 y Fr(0)3174 1533 y Fs([)19 b(R)3318 1503 y Fq(0)3342 1491 y Fr(2)3342 1553 y(2)515 1641 y Fv(are)27 b(comp)r(osable.)38 b(Since)28 b(b)r(oth)h(are)e(simply)i (terminating,)f(so)g(is)g(their)g(union)g Fs(R)3117 1653 y Fr(1)3174 1641 y Fs([)19 b(R)3318 1611 y Fq(0)3342 1599 y Fr(2)3342 1662 y(2)515 1741 y Fv(b)n(y)27 b(Lemma)g(6.1.)639 1840 y(In)44 b(order)e(to)h(sho)n(w)g(simple)g(termination)g(of)h Fs(R)49 b Fv(=)g(\()p Fs(R)2519 1852 y Fr(1)2586 1840 y Fs([)29 b(R)2740 1810 y Fq(0)2764 1799 y Fr(2)2764 1861 y(2)2801 1840 y Fv(\))h Fs([)f(R)3017 1810 y Fq(0)3041 1799 y Fr(1)3041 1861 y(2)3078 1840 y Fv(,)47 b(it)d(suf-)515 1949 y(\014ces)39 b(to)h(sho)n(w)e(that)i(there)g(is)f(no)g(minimal)h (in\014nite)h Fs(R)2376 1919 y Fq(0)2399 1907 y Fr(1)2399 1969 y(2)2436 1949 y Fv(-c)n(hain)e(o)n(v)n(er)f Fs(R)27 b([)g(E)7 b Fo(mb)p Fv(\()p Fs(F)h Fv(\);)515 2048 y(see)39 b(Theorem)h(4.8.)73 b(Supp)r(ose)41 b(on)e(the)i(con)n(trary)d(that)i (there)g(is)g(a)g(minimal)g(in\014nite)515 2148 y Fs(R)585 2118 y Fq(0)608 2106 y Fr(1)608 2169 y(2)646 2148 y Fv(-c)n(hain)k Fs(h)p Fo(s)981 2160 y Fr(1)1019 2148 y Fo(;)14 b(t)1086 2160 y Fr(1)1123 2148 y Fs(ih)p Fo(s)1226 2160 y Fr(2)1263 2148 y Fo(;)g(t)1330 2160 y Fr(2)1368 2148 y Fs(i)g Fo(:)g(:)g(:)59 b Fv(o)n(v)n(er)43 b Fs(R)31 b([)g(E)7 b Fo(mb)p Fv(\()p Fs(F)h Fv(\),)50 b(i.e.,)g(there)45 b(exists)g(a)g(substitu-)515 2248 y(tion)39 b Fo(\033)46 b Fv(:)c Fs(V)49 b(!)43 b(T)21 b Fv(\()p Fs(F)8 b Fv(\))40 b(suc)n(h)f(that)g Fo(t)1739 2260 y Fn(j)1774 2248 y Fo(\033)46 b Fs(!)1950 2218 y Fq(\003)1950 2274 y(R[E)5 b Fn(mb)p Fr(\()p Fq(F)h Fr(\))2332 2248 y Fo(s)2371 2260 y Fn(j)s Fr(+1)2490 2248 y Fo(\033)43 b Fv(holds)c(for)f(all)h(consecu-)515 2360 y(tiv)n(e)h(pairs)g Fs(h)p Fo(s)977 2372 y Fn(j)1012 2360 y Fo(;)14 b(t)1079 2372 y Fn(j)1114 2360 y Fs(i)41 b Fv(and)g Fs(h)p Fo(s)1433 2372 y Fn(j)s Fr(+1)1552 2360 y Fo(;)14 b(t)1619 2372 y Fn(j)s Fr(+1)1738 2360 y Fs(i)41 b Fv(in)g(the)g(sequence.)76 b(Since)41 b(w)n(e)f(consider)g(\014nite)515 2459 y(TRSs,)46 b(w)n(e)c(ma)n(y)f(assume)h(that)g(ev)n(ery)g(dep)r(endency)g(pair)g (in)g Fs(h)p Fo(s)2680 2471 y Fr(1)2718 2459 y Fo(;)14 b(t)2785 2471 y Fr(1)2822 2459 y Fs(ih)p Fo(s)2925 2471 y Fr(2)2963 2459 y Fo(;)g(t)3030 2471 y Fr(2)3067 2459 y Fs(i)g Fo(:)g(:)g(:)56 b Fv(b)r(e-)515 2568 y(longs)38 b(to)i(a)f(cycle)g Fs(P)46 b Fv(whic)n(h)40 b(consists)e(solely)h(of)h (dep)r(endency)g(pairs)e(from)h Fs(R)3127 2537 y Fq(0)3151 2526 y Fr(1)3151 2588 y(2)3188 2568 y Fv(.)73 b(As)515 2667 y(in)41 b(the)h(pro)r(of)f(of)g(Theorem)g(5.4,)j(one)d(can)g(sho)n (w)f(that)i(none)f(of)h(the)f(\(prop)r(er\))g(sub-)515 2776 y(terms)d(of)h(the)h(dep)r(endency)f(pairs)f(in)h Fs(P)46 b Fv(con)n(tains)38 b(a)g(sym)n(b)r(ol)h(from)f Fs(D)2894 2745 y Fq(0)2892 2796 y Fr(1)2956 2776 y Fs([)26 b(D)3103 2745 y Fq(0)3127 2734 y Fr(1)3127 2796 y(2)3164 2776 y Fv(.)71 b(Let)515 2884 y Fs(F)583 2854 y Fq(0)632 2884 y Fv(=)26 b Fs(F)i(n)19 b Fv(\()p Fs(F)964 2896 y Fr(0)1021 2884 y Fs([)h(F)1164 2854 y Fq(0)1188 2842 y Fr(2)1188 2904 y(2)1225 2884 y Fv(\).)43 b(Since)30 b(b)r(oth)g Fs(R)1810 2896 y Fr(0)1877 2884 y Fv(and)f Fs(R)2110 2854 y Fq(0)2134 2842 y Fr(2)2134 2904 y(2)2201 2884 y Fv(do)g(not)h(con)n(tain)f(function)h(sym)n(b)r(ols)515 2992 y(from)d Fs(D)777 2962 y Fq(0)775 3013 y Fr(1)831 2992 y Fs([)19 b(D)971 2962 y Fq(0)995 2950 y Fr(1)995 3013 y(2)1032 2992 y Fv(,)28 b(the)g(TRS)h Fs(R)1492 2962 y Fq(0)1492 3013 y Fr(1)1548 2992 y Fs([)19 b(R)1692 2962 y Fq(0)1715 2950 y Fr(1)1715 3013 y(2)1771 2992 y Fs([)g(E)7 b Fo(mb)p Fv(\()p Fs(F)2105 2962 y Fq(0)2129 2992 y Fv(\))28 b(can)f(b)r(e)i(view)n(ed)e(as)g(an)h(extension)f(of) 515 3101 y Fs(R)585 3113 y Fr(0)642 3101 y Fs([)19 b(R)786 3071 y Fq(0)810 3059 y Fr(2)810 3121 y(2)866 3101 y Fs([)h(E)7 b Fo(mb)p Fv(\()p Fs(F)1193 3113 y Fr(0)1249 3101 y Fs([)19 b(F)1391 3071 y Fq(0)1415 3059 y Fr(2)1415 3121 y(2)1452 3101 y Fv(\).)40 b(Because)28 b(the)h(in\014nite)h Fs(R)2363 3071 y Fq(0)2386 3059 y Fr(1)2386 3121 y(2)2424 3101 y Fv(-c)n(hain)e(is)g(minimal,)h(w)n(e)g(ma)n(y)515 3200 y(apply)h(Lemma)h(4.6)f(to)h(ev)n(ery)f Fo(t)1541 3212 y Fn(j)1576 3200 y Fo(\033)i Fs(!)1738 3170 y Fq(\003)1738 3230 y Fr(\()p Fq(R)1821 3238 y Ff(0)1853 3230 y Fq([R)1955 3214 y Fi(0)1977 3210 y Ff(2)1977 3248 y(2)2010 3230 y Fq([E)5 b Fn(mb)p Fr(\()p Fq(F)2258 3238 y Ff(0)2289 3230 y Fq([F)2388 3214 y Fi(0)2410 3210 y Ff(2)2410 3248 y(2)2442 3230 y Fr(\)\))p Fq([)p Fr(\()p Fq(R)2622 3210 y Fi(0)2622 3248 y Ff(1)2654 3230 y Fq([R)2756 3214 y Fi(0)2778 3210 y Ff(1)2778 3248 y(2)2811 3230 y Fq([E)g Fn(mb)p Fr(\()p Fq(F)3065 3214 y Fi(0)3086 3230 y Fr(\)\))3171 3200 y Fo(s)3210 3212 y Fn(j)s Fr(+1)3329 3200 y Fo(\033)515 3319 y Fv(and)22 b(obtain)g(\010\()p Fo(t)1046 3331 y Fn(j)1081 3319 y Fo(\033)s Fv(\))i Fs(!)1270 3289 y Fq(\003)1270 3349 y Fr(\()p Fq(R)1353 3357 y Ff(0)1386 3349 y Fq([R)1488 3332 y Fi(0)1510 3329 y Ff(2)1510 3367 y(2)1542 3349 y Fq([E)5 b Fn(mb)p Fr(\()p Fq(F)1790 3357 y Ff(0)1822 3349 y Fq([F)1921 3332 y Fi(0)1942 3329 y Ff(2)1942 3367 y(2)1975 3349 y Fr(\)\))p Fq(]C)2108 3357 y Fi(E)2174 3319 y Fv(\010\()p Fo(s)2305 3331 y Fn(j)s Fr(+1)2424 3319 y Fo(\033)s Fv(\).)36 b(In)23 b(other)f(w)n(ords,)g(the)h(cy-)515 3454 y(cle)j Fs(P)32 b Fv(admits)26 b(an)g(in\014nite)h Fs(R)1463 3424 y Fq(0)1487 3413 y Fr(1)1487 3475 y(2)1524 3454 y 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(union,)g(but)h(this)f(and)g(related)f(results)h(in)g([GA)n(O01)o(])g (cannot)f(deal)515 5006 y(with)f(hierarc)n(hical)d(termination.)1905 5255 y(21)p eop %%Page: 22 22 22 21 bop 639 523 a Fv(Other)44 b(mo)r(dularit)n(y)g(criteria)f(for)h (hierarc)n(hical)e(com)n(binations)h(are)h(due)g(to)g(Der-)515 623 y(sho)n(witz)33 b([Der95)o(].)58 b(His)34 b(theorems)g(deal)g(with) h Fm(\015at)f Fv(systems,)i(systems)e(in)g(whic)n(h)h(b)r(oth)515 722 y(sides)30 b(of)h(rewrite)e(rules)h(do)h(not)f(con)n(tain)g(nested) h(de\014ned)g(sym)n(b)r(ols.)45 b(That's)31 b(wh)n(y)f(they)515 822 y(are)c(also)h(not)g(applicable)g(to)h(systems)f(lik)n(e)g(that)h (of)g(Example)e(5.5.)639 922 y(Another)35 b(mo)r(dularit)n(y)f(result)h (for)f(hierarc)n(hical)f(com)n(binations)h(w)n(as)g(presen)n(ted)g(b)n (y)515 1021 y(F)-7 b(ern\023)-42 b(andez)32 b(and)g(Jouannaud)g([FJ95)o (].)52 b(Their)32 b(result)g(is)h(restricted)f(to)g(systems)h(where)515 1121 y(the)22 b(argumen)n(ts)e(of)i(recursiv)n(e)e(calls)h(in)h Fs(R)1820 1133 y Fr(2)1880 1121 y Fv(decrease)e(w.r.t.)i(the)g(subterm) g(relation)e(\(com-)515 1220 y(pared)e(as)h(m)n(ultisets)h(or)e (lexicographically\).)32 b(Hence,)22 b(their)d(result)g(is)h(also)e (not)i(applicable)515 1320 y(to)27 b(Example)g(5.5.)515 1595 y Fw(References)515 1776 y Fv([A)n(G97a])63 b(T.)30 b(Arts)f(and)g(J.)h(Giesl.)42 b(Automatically)29 b(pro)n(ving)f (termination)h(where)g(sim-)875 1876 y(pli\014cation)j(orderings)e (fail.)51 b(In)32 b Fm(The)l(ory)j(and)g(Pr)l(actic)l(e)f(of)h(Softwar) l(e)g(Devel-)875 1976 y(opment)g(\(Pr)l(o)l(c)l(e)l(e)l(dings)h(of)g (the)f(22nd)h(International)g(Col)t(lo)l(quium)g(on)f(T)-6 b(r)l(e)l(es)875 2075 y(in)27 b(A)n(lgebr)l(a)h(and)g(Pr)l(o)l(gr)l (amming\))p Fv(,)e(v)n(olume)f(1214)e(of)i Fm(L)l(e)l(ctur)l(e)h(Notes) h(in)h(Com-)875 2175 y(puter)i(Scienc)l(e)p Fv(,)e(pages)e(261{272,)e (Berlin,)k(1997.)d(Springer-V)-7 b(erlag.)515 2341 y([A)n(G97b])59 b(T.)39 b(Arts)f(and)g(J.)g(Giesl.)69 b(T)-7 b(ermination)38 b(of)g(term)g(rewriting)g(using)g(dep)r(en-)875 2441 y(dency)33 b(pairs.)53 b(T)-7 b(ec)n(hnical)33 b(Rep)r(ort)g(IBN)g (97/46,)g(Darmstadt)g(Univ)n(ersit)n(y)f(of)875 2540 y(T)-7 b(ec)n(hnology)g(,)26 b(1997.)515 2706 y([A)n(G00])105 b(T.)23 b(Arts)f(and)h(J.)f(Giesl.)29 b(T)-7 b(ermination)22 b(of)g(term)h(rewriting)e(using)i(dep)r(endency)875 2806 y(pairs.)36 b Fm(The)l(or)l(etic)l(al)31 b(Computer)g(Scienc)l(e)p Fv(,)d(236:133{178,)23 b(2000.)515 2972 y([BN98])109 b(F.)27 b(Baader)d(and)i(T.)h(Nipk)n(o)n(w.)33 b Fm(T)-6 b(erm)29 b(R)l(ewriting)f(and)h(A)n(l)t(l)g(That)p Fv(.)35 b(Cam)n(bridge)875 3072 y(Univ)n(ersit)n(y)27 b(Press,)f(1998.)515 3238 y([Der95])97 b(N.)23 b(Dersho)n(witz.)k(Hierarc)n(hical)21 b(termination.)28 b(In)23 b Fm(Pr)l(o)l(c)l(e)l(e)l(dings)i(of)h(the)f (4th)h(In-)875 3337 y(ternational)35 b(Workshop)h(on)e(Conditional)i (and)e(T)-6 b(yp)l(e)l(d)35 b(R)l(ewriting)f(Systems)p Fv(,)875 3437 y(v)n(olume)43 b(968)f(of)h Fm(L)l(e)l(ctur)l(e)h(Notes)g (in)g(Computer)h(Scienc)l(e)p Fv(,)j(pages)42 b(89{105,)875 3537 y(Berlin,)27 b(1995.)f(Springer-V)-7 b(erlag.)515 3703 y([DJ90])124 b(N.)41 b(Dersho)n(witz)g(and)f(J.-P)-7 b(.)40 b(Jouannaud.)76 b(Rewrite)40 b(systems.)76 b(In)41 b(J.)g(v)-5 b(an)875 3802 y(Leeu)n(w)n(en,)33 b(editor,)g Fm(F)-6 b(ormal)35 b(Mo)l(dels)h(and)f(Semantics)p Fv(,)e(v)n(olume)f (B)h(of)f Fm(Hand-)875 3902 y(b)l(o)l(ok)49 b(of)f(The)l(or)l(etic)l (al)i(Computer)e(Scienc)l(e)p Fv(,)53 b(c)n(hapter)46 b(6,)52 b(pages)46 b(243{320.)875 4001 y(Elsevier|The)26 b(MIT)i(Press,)e(1990.)515 4167 y([FJ95])133 b(M.)39 b(F)-7 b(ern\023)-42 b(andez)38 b(and)h(J.-P)-7 b(.)38 b(Jouannaud.)69 b(Mo)r(dular)38 b(termination)g(of)h(term)875 4267 y(rewriting)33 b(systems)g(revisited.)54 b(In)34 b Fm(R)l(e)l(c)l(ent)f(T)-6 b(r)l(ends)36 b(in)f(Data)g(T)-6 b(yp)l(e)36 b(Sp)l(e)l(ci\014-)875 4367 y(c)l(ation)28 b(\(Pr)l(o)l(c)l(e)l(e)l(dings)g(of)g(the)f(10th)h(Workshop)h(on)e(Sp)l (e)l(ci\014c)l(ation)h(of)g(A)n(bstr)l(act)875 4466 y(Data)d(T)-6 b(yp)l(es\))p Fv(,)24 b(v)n(olume)d(906)f(of)i Fm(L)l(e)l(ctur)l(e)i (Notes)g(in)h(Computer)g(Scienc)l(e)p Fv(,)e(pages)875 4566 y(255{272,)h(Berlin,)k(1995.)d(Springer-V)-7 b(erlag.)515 4732 y([GA)n(O01])40 b(J.)29 b(Giesl,)g(T.)g(Arts,)g(and)f(E.)h (Ohlebusc)n(h.)39 b(Mo)r(dular)28 b(termination)h(pro)r(ofs)f(for)875 4832 y(rewriting)f(using)g(dep)r(endency)i(pairs.)36 b Fm(Journal)30 b(of)h(Symb)l(olic)g(Computation)p Fv(,)875 4931 y(2001.)k(T)-7 b(o)27 b(app)r(ear.)1905 5255 y(22)p eop %%Page: 23 23 23 22 bop 515 523 a Fv([Gra94])90 b(B.)31 b(Gramlic)n(h.)48 b(Generalized)31 b(su\016cien)n(t)g(conditions)g(for)g(mo)r(dular)g (termina-)875 623 y(tion)c(of)g(rewriting.)35 b Fm(Applic)l(able)c(A)n (lgebr)l(a)f(in)f(Engine)l(ering,)h(Communic)l(ation)875 722 y(and)g(Computing)p Fv(,)f(5:131{158,)24 b(1994.)515 888 y([Gra95])90 b(B.)32 b(Gramlic)n(h.)50 b(Abstract)32 b(relations)f(b)r(et)n(w)n(een)h(restricted)f(termination)h(and)875 988 y(con\015uence)g(prop)r(erties)f(of)h(rewrite)g(systems.)50 b Fm(F)-6 b(undamenta)33 b(Informatic)l(ae)p Fv(,)875 1088 y(24:3{23,)25 b(1995.)515 1254 y([Gra96])90 b(B.)35 b(Gramlic)n(h.)58 b Fm(T)-6 b(ermination)38 b(and)f(Con\015uenc)l(e)g (Pr)l(op)l(erties)g(of)h(Structur)l(e)l(d)875 1353 y(R)l(ewrite)30 b(Systems)p Fv(.)36 b(PhD)28 b(thesis,)f(Univ)n(ersit\177)-42 b(at)27 b(Kaiserslautern,)f(1996.)515 1519 y([Klo92])100 b(J.W.)37 b(Klop.)64 b(T)-7 b(erm)37 b(rewriting)f(systems.)64 b(In)37 b(S.)g(Abramsky)-7 b(,)38 b(D.)g(Gabba)n(y)-7 b(,)875 1619 y(and)35 b(T.)h(Maibaum,)h(editors,)g Fm(Handb)l(o)l(ok)h (of)f(L)l(o)l(gic)h(in)f(Computer)h(Scienc)l(e)p Fv(,)875 1719 y(v)n(olume)27 b(2,)g(pages)g(1{116.)e(Oxford)i(Univ)n(ersit)n(y)f (Press,)g(1992.)515 1885 y([KNT99])43 b(K.)20 b(Kusak)-5 b(ari,)20 b(M.)h(Nak)-5 b(am)n(ura,)20 b(and)g(Y.)h(T)-7 b(o)n(y)n(ama.)23 b(Argumen)n(t)d(\014ltering)g(trans-)875 1984 y(formation.)71 b(In)39 b Fm(Principles)j(and)f(Pr)l(actic)l(e)g 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b(term)f(rewriting)g (systems.)63 b(In)37 b Fm(Pr)l(o)l(c)l(e)l(e)l(dings)i(of)g(the)f(2nd)h (International)875 3279 y(Symp)l(osium)26 b(on)g(The)l(or)l(etic)l(al)i (Asp)l(e)l(cts)d(of)i(Computer)f(Softwar)l(e)p Fv(,)f(v)n(olume)d(789) 875 3379 y(of)33 b Fm(L)l(e)l(ctur)l(e)h(Notes)g(in)h(Computer)g (Scienc)l(e)p Fv(,)f(pages)e(203{223,)f(Berlin,)j(1994.)875 3479 y(Springer-V)-7 b(erlag.)515 3645 y([KR95])104 b(M.R.K.)23 b(Krishna)f(Rao.)29 b(Mo)r(dular)22 b(pro)r(ofs)h(for)f(completeness)h (of)g(hierarc)n(hical)875 3744 y(term)30 b(rewriting)e(systems.)42 b Fm(The)l(or)l(etic)l(al)33 b(Computer)f(Scienc)l(e)p Fv(,)f(151:487{512,)875 3844 y(1995.)515 4010 y([Mid89])85 b(A.)33 b(Middeldorp.)49 b(A)33 b(su\016cien)n(t)f(condition)g(for)f (the)i(termination)f(of)g(the)g(di-)875 4110 y(rect)f(sum)h(of)f(term)g (rewriting)f(systems.)48 b(In)31 b Fm(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)g (the)f(4th)h(IEEE)875 4209 y(Symp)l(osium)c(on)g(L)l(o)l(gic)h(in)e (Computer)i(Scienc)l(e)p 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Fv(,)e(20:1{41,)c(1995.)515 789 y([Ohl01])96 b(E.)19 b(Ohlebusc)n(h.)j(T)-7 b(ermination)18 b(of)h(logic)f(programs:)30 b(T)-7 b(ransformational)17 b(meth-)875 888 y(o)r(ds)23 b(revisited.)29 b Fm(Applic)l(able)f(A)n (lgebr)l(a)e(in)g(Engine)l(ering,)i(Communic)l(ation)e(and)875 988 y(Computing)p Fv(,)i(12:73{116,)c(2001.)515 1154 y([Rus87])90 b(M.)29 b(Rusino)n(witc)n(h.)39 b(On)29 b(termination)f(of)h(the)g(direct)f(sum)h(of)f(term)h(rewriting)875 1254 y(systems.)36 b Fm(Information)31 b(Pr)l(o)l(c)l(essing)f(L)l (etters)p Fv(,)e(26:65{70,)c(1987.)515 1420 y([T)-7 b(o)n(y87])93 b(Y.)36 b(T)-7 b(o)n(y)n(ama.)60 b(Coun)n(terexamples)35 b(to)h(termination)f(for)h(the)g(direct)g(sum)g(of)875 1519 y(term)30 b(rewriting)g(systems.)44 b Fm(Information)33 b(Pr)l(o)l(c)l(essing)g(L)l(etters)p Fv(,)d(25:141{143,)875 1619 y(1987.)515 1785 y([Urb01])89 b(X.)30 b(Urbain.)44 b(Automated)30 b(incremen)n(tal)f(termination)h(pro)r(ofs)f(for)g (hierarc)n(hi-)875 1885 y(cally)40 b(de\014ned)g(term)g(rewriting)g (systems.)74 b(In)40 b Fm(Pr)l(o)l(c)l(e)l(e)l(dings)i(of)g(the)g (Inter-)875 1984 y(national)35 b(Joint)e(Confer)l(enc)l(e)i(on)e(A)n (utomate)l(d)h(R)l(e)l(asoning)p Fv(,)f(v)n(olume)e(2083)f(of)875 2084 y Fm(L)l(e)l(ctur)l(e)j(Notes)g(in)h(A)n(rti\014cial)g(Intel)t (ligenc)l(e)p Fv(,)g(pages)d(485{499,)e(Berlin,)k(2001.)875 2183 y(Springer-V)-7 b(erlag.)1905 5255 y(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF