%!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: ipl.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -o ipl.ps ipl.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.02.25:0943 %%BeginProcSet: texc.pro %! /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 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2075 y(sho)n(w)n(ed)26 b(the)i(follo)n(wing)e(theorem:)36 b(If)28 b Fv(R)g Fx(is)f(simply)g(terminating,)g(then)h Fv(R)18 b([)h(E)7 b Ft(mb)p Fx(\()p Fv(F)h Fx(\))27 b(is)515 2175 y Fv(C)559 2187 y Fu(E)603 2175 y Fx(-terminating.)1079 2145 y Fs(1)1152 2175 y Fx(T)-7 b(o)25 b(date,)g(ho)n(w)n(ev)n(er,)f (the)h(TRSs)g(for)g(whic)n(h)g(automated)f(termination)515 2275 y(pro)r(ofs)30 b(are)g(\(p)r(oten)n(tially\))h(feasible)f(are)g (no)h(longer)e(just)i(the)h(simply)f(terminating)f(sys-)515 2374 y(tems,)i(but)g(the)f Fy(DP)i(quasi-simply)h(terminating)39 b Fx(systems,)31 b(i.e.,)i(those)d(systems)h(whose)515 2474 y(termination)18 b(can)h(b)r(e)g(v)n(eri\014ed)f(b)n(y)h (quasi-simpli\014cation)e(orderings)g(in)j(com)n(bination)e(with)515 2573 y(dep)r(endency)28 b(pairs.)639 2673 y(As)g(already)f(men)n (tioned,)h(the)g(de\014nitions)g(of)g(the)g(follo)n(wing)f(notions)g (can)g(b)r(e)i(found)515 2773 y(in)34 b([A)n(G00)o(,)g(GA)n(O02)o(].)56 b(A)34 b(\014nite)h(TRS)f Fv(R)h 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Fv(#)1126 3382 y Fu(A)1211 3370 y Fx(for)k(at)g(least)g(one)g(dep)r(endency)h(pair)f Fv(h)p Ft(s;)14 b(t)p Fv(i)28 b Fx(in)g Fv(P)7 b Fx(,)515 3470 y(where)27 b Fv(F)823 3440 y Fu(0)874 3470 y Fx(consists)f(of)i (all)f(function)i(sym)n(b)r(ols)d(o)r(ccurring)h(in)g(the)h(constrain)n (ts.)639 3570 y(Recall)g(that)g(a)f(QSO)h(is)g(a)f(quasi-rewrite)f (ordering)g(p)r(ossessing)h(the)h(subterm)g(prop-)515 3669 y(ert)n(y)-7 b(.)38 b(Krusk)-5 b(al's)27 b(theorem)h(implies)g (that)h(the)f(stable-strict)g(relation)f(corresp)r(onding)f(to)515 3769 y(a)h(QSO)g(is)g(w)n(ell-founded)h(pro)n(vided)e(the)i(underlying) f(signature)g(is)g(\014nite.)639 3869 y(A)g(natural)f(question)g(is)g (whether)h(DP)f(quasi-simple)g(termination)g(also)f(implies)i Fv(C)3307 3881 y Fu(E)3351 3869 y Fx(-)515 3968 y(termination.)36 b(The)28 b(next)g(lemma)f(sho)n(ws)g(that)h(this)f(is)h(indeed)g(the)g (case.)515 4151 y Fm(Lemma)h(2.1)41 b Fy(A)26 b(\014nite)f(TRS)h Fv(R)h 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Fv(R)1522 3067 y Fs(1)1560 3055 y Fx(\))41 b(and)f(\()p Fv(F)1899 3067 y Fs(2)1936 3055 y Ft(;)14 b Fv(R)2043 3067 y Fs(2)2081 3055 y Fx(\))41 b(b)r(e)g(TRSs.)76 b(If)41 b(there)f(is)h(an)f(in\014nite)515 3155 y(sequence)27 b Fv(h)p Ft(s)930 3167 y Fs(1)968 3155 y Ft(;)14 b(t)1035 3167 y Fs(1)1072 3155 y Fv(ih)p Ft(s)1175 3167 y Fs(2)1213 3155 y Ft(;)g(t)1280 3167 y Fs(2)1317 3155 y Fv(i)g Ft(:)g(:)g(:)42 b Fx(of)28 b(dep)r(endency)h(pairs)e(from)h Fv(R)2521 3167 y Fs(1)2587 3155 y Fx(and)g(a)g(substitution)g Ft(\033)g Fx(:)515 3254 y Fv(V)h(!)24 b(T)d Fx(\()p Fv(F)860 3266 y Fs(1)914 3254 y Fv([)c(F)1046 3266 y Fs(2)1082 3254 y Ft(;)d Fv(V)7 b Fx(\))27 b(suc)n(h)f(that)h Ft(t)1631 3266 y Fh(j)1666 3254 y Ft(\033)g Fv(!)1823 3224 y Fu(\003)1823 3277 y(R)1880 3285 y Fg(1)1912 3277 y Fu([R)2014 3285 y Fg(2)2074 3254 y Ft(s)2113 3266 y Fh(j)s Fs(+1)2231 3254 y Ft(\033)k Fx(holds)26 b(for)g(all)g(consecutiv)n(e)g(pairs)515 3354 y Fv(h)p Ft(s)586 3366 y Fh(j)621 3354 y Ft(;)14 b(t)688 3366 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1149 y Fs(1)2757 1137 y Ft(;)14 b(:)g(:)g(:)g(;)g(t)2972 1149 y Fh(n)3017 1137 y Fx(])-14 b(])639 1303 y Fv(\017)41 b Fx(\010\()p Ft(t)p Fx(\))24 b(=)f Ft(S)5 b(or)r(t)p Fx(\(\010)1245 1273 y Fu(\003)1284 1303 y Fx(\(\001)1385 1273 y Fu(\003)1385 1324 y Fs(1)1423 1303 y Fx(\()p Ft(t)p Fx(\)\)\),)29 b(if)f Ft(r)r(oot)p Fx(\()p Ft(t)p Fx(\))d Fv(2)e(F)2115 1315 y Fs(2)2171 1303 y Fv(n)18 b(F)2291 1315 y Fs(1)1062 1553 y Fx(with)28 b(\001)1320 1518 y Fu(\003)1320 1573 y Fs(1)1358 1553 y Fx(\()p Ft(t)p Fx(\))84 b(=)e Fv(f)p Ft(u)22 b Fv(2)i(T)d Fx(\()p Fv(F)8 b Fx(\))28 b Fv(j)g Ft(t)23 b Fv(!)2287 1518 y Fu(\003)2287 1573 y(R)2371 1553 y Ft(u;)14 b(r)r(oot)p Fx(\()p Ft(u)p Fx(\))24 b Fv(2)f(F)2879 1565 y Fs(1)2916 1553 y Fv(g)1200 1677 y Fx(\010)1260 1643 y Fu(\003)1298 1677 y Fx(\()p Ft(M)9 b Fx(\))84 b(=)e Fv(f)p Fx(\010\()p Ft(u)p Fx(\))27 b Fv(j)h Ft(u)23 b Fv(2)g Ft(M)9 b Fv(g)27 b Fx(for)g Ft(M)32 b Fv(\022)22 b(T)2655 1689 y Fs(SN)2745 1677 y Fx(\()p Fv(F)8 b Fx(\))794 1802 y Ft(S)d(or)r(t)p Fx(\()p Fv(f)p Ft(t)1063 1814 y Fs(1)1101 1802 y Ft(;)14 b(:)g(:)g(:)g(;)g(t)1316 1814 y Fh(m)1379 1802 y Fv(g)p Fx(\))83 b(=)f Fv(h)p Ft(t)1745 1817 y Fh(\031)r Fs(\(1\))1875 1802 y Ft(;)14 b(:)g(:)g(:)g(;)g(t)2090 1817 y Fh(\031)r Fs(\()p Fh(m)p Fs(\))2246 1802 y Fv(i)1711 1926 y Fx(suc)n(h)27 b(that)h Ft(t)2108 1941 y Fh(\031)r Fs(\()p Fh(j)s Fs(\))2250 1926 y Fv(\037)13 b Ft(t)2358 1941 y Fh(\031)r Fs(\()p Fh(j)s Fs(+1\))2597 1926 y Fx(for)28 b(1)22 b Fv(\024)h Ft(j)28 b(<)22 b(m)858 2051 y Fv(h)p Ft(t)920 2066 y Fh(\031)r Fs(\(1\))1050 2051 y Ft(;)14 b(:)g(:)g(:)f(;)h(t)1264 2066 y Fh(\031)r Fs(\()p Fh(m)p Fs(\))1420 2051 y Fv(i)84 b Fx(=)e Ft(C)6 b(ons)p Fx(\()p Ft(t)1939 2066 y Fh(\031)r Fs(\(1\))2070 2051 y Ft(;)14 b(:)g(:)g(:)f(;)h(C)6 b(ons)p Fx(\()p Ft(t)2510 2066 y Fh(\031)r Fs(\()p Fh(m)p Fs(\))2667 2051 y Ft(;)14 b(N)9 b(il)r Fx(\))14 b Ft(:)g(:)g(:)e Fx(\))515 2233 y(where)27 b Fv(\037)g Fx(is)h(an)f(arbitrary)e(but)k (\014xed)e(total)h(ordering)d(on)j Fv(T)21 b Fx(\()p Fv(F)2552 2245 y Fs(1)2608 2233 y Fv(])e(f)p Ft(C)6 b(ons;)14 b(N)9 b(il)r Fv(g)p Fx(\).)639 2333 y(Note)30 b(that)f(if)h Ft(t)c Fv(2)g(T)1283 2345 y Fs(SN)1373 2333 y Fx(\()p Fv(F)8 b Fx(\),)31 b(then)e(\001)1818 2303 y Fu(\003)1818 2354 y Fs(1)1857 2333 y Fx(\()p Ft(t)p Fx(\))d Fv(\022)g(T)2113 2345 y Fs(SN)2203 2333 y Fx(\()p Fv(F)8 b Fx(\).)42 b(Th)n(us,)30 b(\010)2696 2303 y Fu(\003)2764 2333 y Fx(is)f(alw)n(a)n(ys)e(applied) 515 2433 y(to)g(a)g(subset)h(of)g Fv(T)1080 2445 y Fs(SN)1170 2433 y Fx(\()p Fv(F)8 b Fx(\).)515 2615 y Fm(Lemma)29 b(3.3)41 b Fy(If)30 b Ft(s)23 b Fv(2)g(T)1289 2627 y Fs(SN)1379 2615 y Fx(\()p Fv(F)8 b Fx(\))31 b Fy(and)f Ft(s)23 b Fv(!)1848 2585 y Fu(\003)1848 2638 y(R)1932 2615 y Ft(t)p Fy(,)30 b(then)g Fx(\010\()p Ft(s)p Fx(\))23 b Fv(!)2471 2585 y Fu(\003)2471 2638 y(R)2528 2646 y Fg(1)2561 2638 y Fu(]C)2642 2646 y Fi(E)2708 2615 y Fx(\010\()p Ft(t)p Fx(\))p Fy(.)639 2798 y Fx(With)29 b(the)g(preceding)f(lemma)g (\(a)g(pro)r(of)g(of)g(whic)n(h)h(can)f(b)r(e)g(found)h(in)g([Ohl01)n (]\),)g(it)g(is)515 2897 y(not)d(di\016cult)i(to)e(pro)n(v)n(e)f(the)i (follo)n(wing)e(result)h(of)h(Urbain)f([Urb01a)o(];)h(see)f([Ohl01)o(]) g(for)g(an)515 2997 y(alternativ)n(e)g(pro)r(of.)515 3180 y Fm(Theorem)k(3.4)41 b Fy(L)l(et)31 b Fv(R)i Fy(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 3279 y(TRSs)22 b Fv(R)808 3291 y Fs(1)868 3279 y Fy(and)h Fv(R)1092 3291 y Fs(2)1130 3279 y Fy(.)36 b Fv(R)23 b Fy(is)g(terminating)f(\()p Fv(C)1887 3291 y Fu(E)1932 3279 y Fy(-terminating,)i(r)l(esp)l(e)l(ctively\))g(if)f(b) l(ase)g(system)515 3379 y Fv(R)585 3391 y Fs(1)656 3379 y Fy(is)34 b Fv(C)793 3391 y Fu(E)837 3379 y Fy(-terminating)g(and)g (ther)l(e)g(is)f(no)h(minimal)h(in\014nite)e Fv(R)2604 3391 y Fs(2)2642 3379 y Fy(-chain)h(over)g Fv(R)g Fy(\(over)515 3479 y Fv(R)19 b(])f(C)721 3491 y Fu(E)766 3479 y Fy(,)30 b(r)l(esp)l(e)l(ctively\).)515 3753 y Fw(4)134 b(Hierarc)l(hical)47 b Fo(C)1534 3771 y Fv(E)1589 3753 y Fw(-termination)515 3935 y Fx(In)25 b(this)g(section,)g(w)n(e)f(pro)n(vide)g(a)g(new)h(mo)r (dularit)n(y)f(result)h(ab)r(out)f(hierarc)n(hical)f(com)n(bina-)515 4035 y(tions)k(of)h(TRSs.)37 b(W)-7 b(e)28 b(start)f(with)h(a)f(result) g(due)h(to)f(Gramlic)n(h)g([Gra94)o(].)515 4217 y Fm(Prop)s(osition)j (4.1)40 b Fv(C)1238 4229 y Fu(E)1283 4217 y Fy(-termination)h(is)h(a)g (mo)l(dular)g(pr)l(op)l(erty)g(of)h(\014nitely)e(br)l(anching)515 4317 y(c)l(onstructor-sharing)30 b(TRSs.)639 4500 y Fx(Next,)i(w)n(e)f (will)g(generalize)e(this)i(mo)r(dularit)n(y)f(result)g(to)h(certain)f (hierarc)n(hical)f(com-)515 4599 y(binations,)37 b(viz.)e(prop)r(er)f (extensions.)59 b(In)35 b(order)f(to)h(review)g(the)g(de\014nition)h (of)f(prop)r(er)515 4699 y(extensions,)24 b(due)h(to)f(Krishna)f(Rao)h ([KR94)n(,)h(KR95)o(],)g(w)n(e)f(ha)n(v)n(e)f(to)h(in)n(tro)r(duce)g (the)h(dep)r(en-)515 4798 y(dency)i(relation)g Ff(\004)1109 4813 y Fh(d)1148 4798 y Fx(.)639 4898 y(F)-7 b(or)23 b(a)h(TRS)f(\()p Fv(F)8 b Ft(;)14 b Fv(R)p Fx(\),)26 b(the)e(dep)r(endency)g(relation)f Ff(\004)2268 4913 y Fh(d)2330 4898 y Fx(is)h(the)g(smallest)f(quasi-ordering)515 4998 y(satisfying)41 b(the)i(condition)f Ft(f)57 b Ff(\004)1586 5013 y Fh(d)1672 4998 y Ft(g)45 b Fx(whenev)n(er)c(there)i(is)f(a)g (rewrite)f(rule)h Ft(f)9 b Fx(\()p Ft(:)14 b(:)g(:)g Fx(\))48 b Fv(!)1926 5255 y Fx(4)p eop %%Page: 5 5 5 4 bop 515 523 a Ft(C)6 b Fx([)p Ft(g)s Fx(\()p Ft(:)14 b(:)g(:)g Fx(\)])42 b Fv(2)g(R)d Fx(with)g Ft(g)44 b Fv(2)e(D)r Fx(.)71 b(So)39 b Ft(f)50 b Ff(\004)1903 538 y Fh(d)1983 523 y Ft(g)42 b Fx(holds)c(if)h(the)g(function)h Ft(f)47 b Fx(dep)r(ends)39 b(on)515 623 y(the)28 b(de\014nition)g(of)f (the)h(function)g Ft(g)s Fx(.)639 722 y(Let)41 b(\()p Fv(F)8 b Ft(;)14 b Fv(R)p Fx(\))41 b(b)r(e)g(the)f(hierarc)n(hical)e (com)n(bination)i(of)g(base)g(system)g(\()p Fv(F)2991 734 y Fs(1)3028 722 y Ft(;)14 b Fv(R)3135 734 y Fs(1)3173 722 y Fx(\))40 b(and)515 822 y(extension)32 b(\()p Fv(F)979 834 y Fs(2)1017 822 y Ft(;)14 b Fv(R)1124 834 y Fs(2)1161 822 y Fx(\).)54 b(The)34 b(de\014ned)f(sym)n(b)r(ols)g Fv(D)2125 834 y Fs(2)2195 822 y Fx(of)g Fv(R)2365 834 y Fs(2)2436 822 y Fx(are)f(split)i(in)n(to)f(t)n(w)n(o)f(sets)h Fv(D)3341 792 y Fs(1)3339 843 y(2)515 922 y Fx(and)27 b Fv(D)742 891 y Fs(2)740 942 y(2)780 922 y Fx(,)g(where)g Fv(D)1136 891 y Fs(1)1134 942 y(2)1202 922 y Fx(con)n(tains)f(all)h (de\014ned)h(sym)n(b)r(ols)f(of)g Fv(D)2403 934 y Fs(2)2468 922 y Fx(that)h(dep)r(end)g(on)g(a)f(de\014ned)515 1021 y(sym)n(b)r(ol)20 b(of)h Fv(R)950 1033 y Fs(1)988 1021 y Fx(,)h(i.e.,)h Fv(D)1251 991 y Fs(1)1249 1042 y(2)1312 1021 y Fx(=)f Fv(f)p Ft(f)g Fv(j)14 b Ft(f)32 b Fv(2)23 b(D)1756 1033 y Fs(2)1793 1021 y Ft(;)14 b(f)32 b Ff(\004)1955 1036 y Fh(d)2017 1021 y Ft(g)e Fx(for)d(some)g Ft(g)f Fv(2)d(D)2630 1033 y Fs(1)2667 1021 y Fv(g)e Fx(and)g Fv(D)2951 991 y Fs(2)2949 1042 y(2)3011 1021 y Fx(=)i Fv(D)3163 1033 y Fs(2)3206 1021 y Fv(n)5 b(D)3319 991 y Fs(1)3317 1042 y(2)3356 1021 y Fx(.)639 1121 y Fv(R)709 1133 y Fs(2)782 1121 y Fx(is)36 b(a)f Fy(pr)l(op)l(er)j(extension)d Fx(of)g Fv(R)1760 1133 y Fs(1)1834 1121 y Fx(if)h(ev)n(ery)e(rewrite)g (rule)i Ft(l)h Fv(!)f Ft(r)j Fv(2)e(R)3028 1133 y Fs(2)3101 1121 y Fx(satis\014es)515 1220 y(the)j(follo)n(wing)f(condition:)62 b(If)40 b Ft(t)h Fx(is)f(a)f(subterm)h(of)g Ft(r)j Fx(suc)n(h)d(that)h Ft(r)r(oot)p Fx(\()p Ft(t)p Fx(\))k Fv(2)f(D)3167 1190 y Fs(1)3165 1241 y(2)3245 1220 y Fx(and)515 1320 y Ft(r)r(oot)p Fx(\()p Ft(t)p Fx(\))d Ff(\004)851 1335 y Fh(d)929 1320 y Ft(r)r(oot)p Fx(\()p Ft(l)r Fx(\),)g(then)c Ft(t)h Fx(do)r(es)f(not)g(con)n(tain)f(a)h(function)h(sym)n(b)r(ol)f(from)g Fv(D)3133 1332 y Fs(1)3195 1320 y Fv([)25 b(D)3341 1290 y Fs(1)3339 1341 y(2)515 1420 y Fx(strictly)i(b)r(elo)n(w)g(its)h(ro)r (ot.)639 1519 y Fv(R)709 1531 y Fs(2)781 1519 y Fx(is)33 b(a)g Fy(r)l(estricte)l(d)j(pr)l(op)l(er)g(extension)e Fx(of)f Fv(R)2118 1531 y Fs(1)2190 1519 y Fx(if)h(it)g(is)f(a)h(prop)r (er)f(extension)g(of)g Fv(R)3341 1531 y Fs(1)515 1619 y Fx(suc)n(h)23 b(that)h(no)f Fy(left-hand)k(side)e Fx(of)e(the)h (rewrite)f(rules)g(of)g Fv(R)2353 1631 y Fs(2)2415 1619 y Fx(con)n(tains)f(a)h(function)i(sym)n(b)r(ol)515 1719 y(from)i Fv(D)775 1731 y Fs(1)831 1719 y Fv([)19 b(D)971 1688 y Fs(1)969 1739 y(2)1036 1719 y Fx(strictly)27 b(b)r(elo)n(w)g (its)h(ro)r(ot.)515 1901 y Fm(Example)i(4.2)40 b Fv(R)1142 1913 y Fu(\003)1209 1901 y Fx(is)27 b(a)g(restricted)g(prop)r(er)g (extension)g(of)h Fv(R)2529 1913 y Fs(+)2584 1901 y Fx(.)1270 2132 y Fv(R)1340 2144 y Fs(+)1479 2132 y Fx(=)1567 2015 y Fe(\032)1780 2082 y Fx(0)18 b(+)g Ft(y)86 b Fv(!)d Ft(y)1670 2181 y(s)p Fx(\()p Ft(x)p Fx(\))20 b(+)e Ft(y)86 b Fv(!)d Ft(s)p Fx(\()p Ft(x)19 b Fx(+)f Ft(y)s Fx(\))1287 2431 y Fv(R)1357 2443 y Fu(\003)1479 2431 y Fx(=)1567 2314 y Fe(\032)1780 2381 y Fx(0)g Fv(\003)g Ft(y)85 b Fv(!)f Fx(0)1670 2480 y Ft(s)p Fx(\()p Ft(x)p Fx(\))20 b Fv(\003)e Ft(y)85 b Fv(!)f Fx(\()p Ft(x)19 b Fv(\003)f Ft(y)s Fx(\))g(+)g Ft(y)639 2659 y Fx(No)n(w)27 b(w)n(e)h(are)e(in)i(a) f(p)r(osition)h(to)f(state)h(our)e(new)i(mo)r(dularit)n(y)f(result.)515 2842 y Fm(Theorem)j(4.3)41 b Fy(L)l(et)35 b Fv(R)1303 2854 y Fs(1)1376 2842 y Fy(and)h Fv(R)1613 2854 y Fs(2)1686 2842 y Fy(b)l(e)g(\014nite)f Fv(C)2058 2854 y Fu(E)2103 2842 y Fy(-terminating)g(TRSs.)56 b(If)36 b Fv(R)3034 2854 y Fs(2)3107 2842 y Fy(is)g(a)g(r)l(e-)515 2941 y(stricte)l(d)30 b(pr)l(op)l(er)g(extension)g(of)g Fv(R)1600 2953 y Fs(1)1638 2941 y Fy(,)g(then)g Fv(R)1948 2953 y Fs(1)2004 2941 y Fv([)18 b(R)2147 2953 y Fs(2)2215 2941 y Fy(is)30 b Fv(C)2348 2953 y Fu(E)2392 2941 y Fy(-terminating.)515 3141 y(Pr)l(o)l(of)61 b Fx(Let)29 b Fv(R)988 3110 y Fs(1)988 3161 y(2)1051 3141 y Fx(=)24 b Fv(f)p Ft(l)j Fv(!)e Ft(r)j Fv(2)d(R)1557 3153 y Fs(2)1609 3141 y Fv(j)14 b Ft(r)r(oot)p Fx(\()p Ft(l)r Fx(\))26 b Fv(2)g(D)2059 3110 y Fs(1)2057 3161 y(2)2096 3141 y Fv(g)j Fx(and)f Fv(R)2399 3110 y Fs(2)2399 3161 y(2)2462 3141 y Fx(=)d Fv(f)p Ft(l)h Fv(!)f Ft(r)j Fv(2)e(R)2969 3153 y Fs(2)3020 3141 y Fv(j)14 b Ft(r)r(oot)p Fx(\()p Ft(l)r Fx(\))27 b Fv(2)515 3240 y(D)581 3210 y Fs(2)579 3261 y(2)618 3240 y Fv(g)p Fx(.)52 b(The)33 b(TRS)g Fv(R)1181 3210 y Fs(2)1181 3261 y(2)1252 3240 y Fx(do)r(es)f(not)h(con)n(tain)f(function)h(sym)n(b)r(ols)f(from) h Fv(D)2813 3252 y Fs(1)2872 3240 y Fv([)22 b(D)3015 3210 y Fs(1)3013 3261 y(2)3053 3240 y Fx(.)52 b(This)33 b(is)515 3340 y(b)r(ecause)22 b(for)h(all)f Ft(l)j Fv(!)e Ft(r)j Fv(2)d(R)1417 3310 y Fs(2)1417 3360 y(2)1478 3340 y Fx(neither)g Ft(l)h Fx(\(b)r(ecause)f Fv(R)2209 3352 y Fs(2)2270 3340 y Fx(is)g(a)f(restricted)g(prop)r(er)h(extension)515 3439 y(of)29 b Fv(R)681 3451 y Fs(1)719 3439 y Fx(\))h(nor)e Ft(r)33 b Fx(con)n(tains)28 b(function)j(sym)n(b)r(ols)d(from)i Fv(D)2236 3451 y Fs(1)2293 3439 y Fv([)20 b(D)2434 3409 y Fs(1)2432 3460 y(2)2501 3439 y Fx(\(if)30 b Ft(r)j Fx(w)n(ould)c(con)n(tain)f(suc)n(h)515 3539 y(a)k(function)i(sym)n(b)r (ol,)g(then)f Ft(r)r(oot)p Fx(\()p Ft(l)r Fx(\))i(w)n(ould)d(b)r(e)i (an)e(elemen)n(t)i(of)f Fv(D)2668 3509 y Fs(1)2666 3560 y(2)2705 3539 y Fx(\).)54 b(In)33 b(other)f(w)n(ords,)515 3639 y Fv(R)585 3651 y Fs(1)661 3639 y Fx(and)38 b Fv(R)903 3609 y Fs(2)903 3659 y(2)978 3639 y Fx(are)g(constructor-sharing.)65 b(Because)37 b(b)r(oth)i(are)e Fv(C)2650 3651 y Fu(E)2694 3639 y Fx(-terminating,)k(so)c(is)515 3738 y(their)29 b(union)f(b)n(y)h(Prop)r(osition)e(4.1.)40 b(In)29 b(order)f(to)g(sho)n (w)g Fv(C)2358 3750 y Fu(E)2403 3738 y Fx(-termination)g(of)h Fv(R)3050 3750 y Fs(1)3107 3738 y Fv([)20 b(R)3252 3750 y Fs(2)3314 3738 y Fx(=)515 3838 y(\()p Fv(R)617 3850 y Fs(1)675 3838 y Fv([)h(R)821 3808 y Fs(2)821 3859 y(2)859 3838 y Fx(\))g Fv([)g(R)1058 3808 y Fs(1)1058 3859 y(2)1096 3838 y Fx(,)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 Fv(R)3122 3808 y Fs(1)3122 3859 y(2)3160 3838 y Fx(-c)n(hain)515 3938 y(o)n(v)n(er)c(\()p Fv(R)796 3950 y Fs(1)854 3938 y Fv([)20 b(R)999 3950 y Fs(2)1036 3938 y Fx(\))g Fv(])g(C)1207 3950 y Fu(E)1252 3938 y Fx(;)30 b(see)f(Theorem)f(3.4.)41 b(Supp)r(ose,)30 b(on)f(the)h(con)n(trary)-7 b(,)28 b(that)h(there)g(is)515 4037 y(a)k(minimal)i(in\014nite)f Fv(R)1277 4007 y Fs(1)1277 4058 y(2)1315 4037 y Fx(-c)n(hain)f Fv(h)p Ft(s)1639 4049 y Fs(1)1677 4037 y Ft(;)14 b(t)1744 4049 y Fs(1)1781 4037 y Fv(ih)p Ft(s)1884 4049 y Fs(2)1921 4037 y Ft(;)g(t)1988 4049 y Fs(2)2026 4037 y Fv(i)g Ft(:)g(:)g(:)48 b Fx(o)n(v)n(er)32 b(\()p Fv(R)2503 4049 y Fs(1)2563 4037 y Fv([)23 b(R)2711 4049 y Fs(2)2749 4037 y Fx(\))g Fv(])g(C)2926 4049 y Fu(E)2971 4037 y Fx(,)35 b(i.e.,)h(there)515 4137 y(exists)41 b(a)f(substitution)i Ft(\033)49 b Fx(:)d Fv(V)52 b(!)46 b(T)21 b Fx(\()p Fv(F)36 b(])28 b(f)p Ft(C)6 b(ons)p Fv(g)p Fx(\))41 b(suc)n(h)g(that)g Ft(t)2765 4149 y Fh(j)2800 4137 y Ft(\033)49 b Fv(!)2979 4107 y Fu(\003)2979 4164 y Fs(\()p Fu(R)3062 4172 y Fg(1)3095 4164 y Fu([R)3197 4172 y Fg(2)3229 4164 y Fs(\))p Fu(]C)3336 4172 y Fi(E)515 4249 y Ft(s)554 4261 y Fh(j)s Fs(+1)673 4249 y Ft(\033)38 b Fx(holds)d(for)f(all)h(consecutiv)n(e)f(pairs)g Fv(h)p Ft(s)1968 4261 y Fh(j)2003 4249 y Ft(;)14 b(t)2070 4261 y Fh(j)2105 4249 y Fv(i)36 b Fx(and)e Fv(h)p Ft(s)2412 4261 y Fh(j)s Fs(+1)2532 4249 y Ft(;)14 b(t)2599 4261 y Fh(j)s Fs(+1)2718 4249 y Fv(i)35 b Fx(in)g(the)h(sequence.)515 4348 y(Because)31 b(w)n(e)h(consider)g(\014nite)h(TRSs,)g(w)n(e)g(ma)n (y)e(assume)h(that)h(ev)n(ery)e(dep)r(endency)i(pair)515 4448 y(in)39 b Fv(h)p Ft(s)694 4460 y Fs(1)732 4448 y Ft(;)14 b(t)799 4460 y Fs(1)836 4448 y Fv(ih)p Ft(s)939 4460 y Fs(2)976 4448 y Ft(;)g(t)1043 4460 y Fs(2)1080 4448 y Fv(i)g Ft(:)g(:)g(:)53 b Fx(b)r(elongs)39 b(to)g(a)f(cycle)h Fv(P)46 b Fx(that)39 b(consists)f(solely)h(of)g(dep)r(endency)515 4548 y(pairs)f(from)g Fv(R)1007 4517 y Fs(1)1007 4568 y(2)1045 4548 y Fx(.)71 b(Let)40 b Fv(h)p Ft(F)12 b Fx(\()p Ft(u)1477 4560 y Fs(1)1514 4548 y Ft(;)i(:)g(:)g(:)f(;)h(u)1746 4560 y Fh(n)1791 4548 y Fx(\))p Ft(;)g(G)p 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y Ft(f)k Fx(b)r(ecause)38 b Fv(P)45 b Fx(is)38 b(a)g(cycle.)69 b(Because)38 b Fv(R)2082 4858 y Fs(2)2158 4846 y Fx(is)g(a)g(prop)r(er) g(extension)f(of)i Fv(R)3162 4858 y Fs(1)3199 4846 y Fx(,)j(the)515 4946 y(terms)32 b Ft(v)791 4958 y Fs(1)829 4946 y Ft(;)14 b(:)g(:)g(:)f(;)h(v)1053 4958 y Fh(m)1149 4946 y Fx(do)32 b(not)h(con)n(tain)f(sym)n(b)r(ols)g(from)g Fv(D)2307 4958 y Fs(1)2366 4946 y Fv([)22 b(D)2509 4916 y Fs(1)2507 4967 y(2)2547 4946 y Fx(.)52 b(The)33 b(same)f(is)g(true)h (for)1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Ft(u)563 535 y Fs(1)600 523 y Ft(;)14 b(:)g(:)g(:)f(;)h(u)832 535 y Fh(n)903 523 y Fx(b)r(ecause)25 b Fv(R)1278 535 y Fs(2)1342 523 y Fx(is)h(a)f(restricted)g(prop)r(er)g (extension)h(of)g Fv(R)2652 535 y Fs(1)2689 523 y Fx(.)37 b(All)26 b(in)g(all,)g(none)g(of)515 623 y(the)i(subterms)f(of)h(the)g (dep)r(endency)g(pairs)e(in)i Fv(P)34 b Fx(con)n(tains)27 b(a)g(sym)n(b)r(ol)g(from)h Fv(D)3036 635 y Fs(1)3091 623 y Fv([)19 b(D)3231 593 y Fs(1)3229 643 y(2)3269 623 y Fx(.)639 722 y(Since)32 b Fv(R)930 692 y Fs(2)930 743 y(2)1000 722 y Fx(do)r(es)f(not)h(con)n(tain)f(function)h(sym)n(b)r (ols)f(from)g Fv(D)2554 734 y Fs(1)2613 722 y Fv([)21 b(D)2755 692 y Fs(1)2753 743 y(2)2793 722 y Fx(,)33 b(the)f(TRS)g Fv(R)3265 734 y Fs(1)3324 722 y Fv([)515 822 y(R)585 792 y Fs(1)585 843 y(2)645 822 y Fv([)23 b(C)767 834 y Fu(E)845 822 y Fx(can)34 b(b)r(e)g(view)n(ed)f(as)h(an)f(extension)h (of)f Fv(R)2175 792 y Fs(2)2175 843 y(2)2213 822 y Fx(.)56 b(Because)32 b(the)j(in\014nite)f Fv(R)3122 792 y Fs(1)3122 843 y(2)3160 822 y Fx(-c)n(hain)515 922 y(is)f(minimal,)i(\(i.e.,)g Ft(u)e Fx(is)g(terminating)g(w.r.t.)g Fv(R)2078 934 y Fs(1)2138 922 y Fv([)22 b(R)2285 934 y Fs(2)2345 922 y Fv([)h(C)2467 934 y Fu(E)2544 922 y Fx(for)33 b(ev)n(ery)f(subterm)h Ft(u)g Fx(of)515 1021 y(some)i Ft(s)770 1033 y Fh(j)805 1021 y Ft(\033)k Fx(and)d Ft(t)1091 1033 y Fh(j)1126 1021 y Ft(\033)s Fx(\),)j(w)n(e)d(ma)n(y)f(apply)g(Lemma)h(3.3)f(to)h (ev)n(ery)f Ft(t)2640 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Fx(\)\))84 b Fv(!)f Ft(s)p Fx(\()p Ft(q)s(uot)p Fx(\()p Ft(x)19 b Fv(\000)f Ft(y)s(;)c(s)p Fx(\()p Ft(y)s Fx(\)\)\))105 b Ft(l)r(eng)s(th)p Fx(\()p Ft(nil)r Fx(\))e Fv(!)86 b Fx(0)833 4316 y Ft(l)r(eng)s(th)p Fx(\()p Ft(x)22 b Fx(:)h Ft(xs)p Fx(\))84 b Fv(!)f Ft(s)p Fx(\()p Ft(l)r(eng)s(th)p Fx(\()p Ft(xs)p Fx(\)\))340 b Ft(hd)p Fx(\()p Ft(x)24 b Fx(:)f Ft(xs)p Fx(\))129 b Fv(!)83 b Ft(x)1186 4426 y(av)s(g)s Fx(\()p Ft(xs)p Fx(\))24 b Fv(!)f Ft(q)s(uot)p Fx(\()p Ft(hd)p Fx(\()p Ft(sum)p Fx(\()p Ft(xs)p Fx(\)\))p Ft(;)14 b(l)r(eng)s(th)p Fx(\()p Ft(xs)p Fx(\)\))515 4592 y(By)36 b(using)g(an)h(appropriate)e (AFS)i(and)g Fy(rp)l(o)p Fx(,)i(it)f(is)e(not)h(di\016cult)g(to)g(sho)n (w)f(that)h Fv(R)3249 4604 y Fs(2)3323 4592 y Fx(is)515 4692 y(DP)h(quasi-simply)g(terminating;)43 b(see)38 b([A)n(G00)o(,)k (Example)37 b(9].)69 b(Therefore,)40 b Fv(R)3092 4704 y Fs(2)3168 4692 y Fx(is)f Fv(C)3307 4704 y Fu(E)3351 4692 y Fx(-)515 4791 y(terminating)32 b(as)g(w)n(ell.)52 b(Note)32 b(that)h(neither)g Fv(R)2042 4803 y Fs(1)2112 4791 y Fx(nor)f Fv(R)2335 4803 y Fs(2)2405 4791 y Fx(is)g(simply)h (terminating.)52 b(W)-7 b(e)515 4891 y(ha)n(v)n(e)28 b Fv(D)774 4861 y Fs(1)772 4912 y(2)837 4891 y Fx(=)e Fv(f)p Ft(av)s(g)s Fv(g)p Fx(,)j(whereas)f(all)h(other)f(sym)n(b)r(ols) h(of)g Fv(D)2324 4903 y Fs(2)2391 4891 y Fx(b)r(elong)g(to)g Fv(D)2827 4861 y Fs(2)2825 4912 y(2)2865 4891 y Fx(.)42 b(Because)28 b Ft(av)s(g)515 4991 y Fx(do)r(es)g(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 Fv(R)2551 5003 y Fs(2)2618 4991 y Fx(is)f(a)h(prop)r(er)f(extension) 1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fx(of)31 b Fv(R)683 535 y Fs(1)721 523 y Fx(.)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 Ft(sum)p Fx(,)g(and)g Ft(av)s(g)i Fx(app)r(ears)515 623 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 Fv(R)2840 635 y Fs(2)2878 623 y Fx(,)g Fv(R)2999 635 y Fs(2)3064 623 y Fx(is)g(ev)n(en)f(a)515 722 y(restricted)h(prop)r(er) g(extension)h(of)g Fv(R)1688 734 y Fs(1)1725 722 y Fx(.)42 b(Because)28 b Fv(R)2179 734 y Fs(2)2245 722 y Fx(is)h(a)g(restricted)f (prop)r(er)h(extension)515 822 y(of)e Fv(R)679 834 y Fs(1)717 822 y Fx(,)h(it)g(follo)n(ws)e(from)h(Theorem)g(4.3)g(that)h Fv(R)2054 834 y Fs(1)2110 822 y Fv([)19 b(R)2254 834 y Fs(2)2319 822 y Fx(is)28 b Fv(C)2447 834 y Fu(E)2491 822 y Fx(-terminating.)639 1005 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 1104 y(results)27 b(is)g(applicable;)g(see)h(Section)f(6.)639 1204 y(The)e(follo)n(wing)e(example)h(tak)n(en)g(from)h([Gra94)n(])g (demonstrates)e(wh)n(y)h(a)g(sym)n(b)r(ol)g(from)515 1303 y Fv(D)579 1315 y Fs(1)635 1303 y Fv([)c(D)776 1273 y Fs(1)774 1324 y(2)842 1303 y Fx(ma)n(y)28 b(not)h(o)r(ccur)f(b)r(elo) n(w)h(a)f Fv(D)1770 1273 y Fs(1)1768 1324 y(2)1808 1303 y Fx(-sym)n(b)r(ol)g(on)g(the)h(righ)n(t-hand)f(side)h(of)f(a)h (rewrite)515 1403 y(rule)36 b(from)g Fv(R)965 1415 y Fs(2)1002 1403 y Fx(:)55 b(The)36 b(TRSs)g Fv(R)1565 1415 y Fs(1)1641 1403 y Fx(=)h Fv(f)p Ft(a)g Fv(!)g Ft(b)p Fv(g)f Fx(and)g Fv(R)2340 1415 y Fs(2)2415 1403 y Fx(=)h Fv(f)p Ft(h)p Fx(\()p Ft(x;)14 b(x)p Fx(\))39 b Fv(!)e Ft(h)p Fx(\()p Ft(a;)14 b(b)p Fx(\))p Fv(g)36 b Fx(are)515 1503 y(simply)27 b(terminating)h(\(hence)f Fv(C)1539 1515 y Fu(E)1584 1503 y Fx(-terminating\),)g(but)h(their)g(union)f(is)h (not)f(terminating,)515 1602 y(as)g(the)h(deriv)-5 b(ation)27 b Ft(h)p Fx(\()p Ft(a;)14 b(b)p Fx(\))23 b Fv(!)1485 1614 y Fu(R)1542 1622 y Fg(1)1601 1602 y Ft(h)p Fx(\()p Ft(b;)14 b(b)p Fx(\))23 b Fv(!)1928 1614 y Fu(R)1985 1622 y Fg(2)2044 1602 y Ft(h)p Fx(\()p Ft(a;)14 b(b)p Fx(\))28 b(sho)n(ws.)639 1702 y(The)d(requiremen)n(t)e(that)i(function) f(sym)n(b)r(ols)g(from)g Fv(D)2334 1714 y Fs(1)2383 1702 y Fv([)12 b(D)2516 1672 y Fs(1)2514 1723 y(2)2578 1702 y Fx(ma)n(y)24 b(not)g(o)r(ccur)f(at)i(non-)515 1802 y(ro)r(ot)j(p)r(ositions)g(on)g(the)h(left-hand)g(sides)f(of)h(rewrite) f(rules)g(from)g Fv(R)2704 1814 y Fs(2)2771 1802 y Fx(is)g(also)g (necessary)-7 b(.)515 1901 y(This)27 b(is)h(sho)n(wn)f(b)n(y)g(the)h (follo)n(wing)e(example.)515 2084 y Fm(Example)k(4.5)40 b Fx(The)28 b(term)f(rewriting)g(system)g Fv(R)2142 2096 y Fs(1)2203 2084 y Fx(=)22 b Fv(f)p Ft(g)s Fx(\()p Ft(x;)14 b(y)s Fx(\))23 b Fv(!)g Ft(x)p Fv(g)k Fx(is)h(ob)n(viously)e Fv(C)3307 2096 y Fu(E)3351 2084 y Fx(-)515 2183 y(terminating.)36 b(It)28 b(can)f(b)r(e)h(sho)n(wn)f(that)h(the)g(same)f(is)h(true)f(for) g(the)h(TRS)g(\(see)g([Ohl01)n(]\):)1178 2545 y Fv(R)1248 2557 y Fs(2)1309 2545 y Fx(=)1397 2374 y Fe(8)1397 2449 y(<)1397 2599 y(:)1541 2444 y Ft(f)9 b Fx(\(0)p Ft(;)14 b Fx(1)p Ft(;)g(x)p Fx(\))83 b Fv(!)g Ft(f)9 b Fx(\()p Ft(h)p Fx(\()p Ft(x)p Fx(\))p Ft(;)14 b(h)p Fx(\()p Ft(x)p Fx(\))p Ft(;)g(x)p Fx(\))1706 2544 y Ft(h)p Fx(\(0\))83 b Fv(!)g Fx(0)1512 2643 y Ft(h)p Fx(\()p Ft(g)s Fx(\()p Ft(x;)14 b(y)s Fx(\)\))84 b Fv(!)f Ft(y)515 2822 y Fx(Ho)n(w)n(ev)n (er,)25 b(the)j(hierarc)n(hical)e(com)n(bination)h(of)g Fv(R)2098 2834 y Fs(1)2164 2822 y Fx(and)g Fv(R)2395 2834 y Fs(2)2460 2822 y Fx(is)h(not)f(terminating:)980 2999 y Ft(f)9 b Fx(\(0)p Ft(;)14 b Fx(1)p Ft(;)g(g)s Fx(\(0)p Ft(;)g Fx(1\)\))81 b Fv(!)1644 3011 y Fu(R)1701 3019 y Fg(2)1820 2999 y Ft(f)9 b Fx(\()p Ft(h)p Fx(\()p Ft(g)s Fx(\(0)p Ft(;)14 b Fx(1\)\))p Ft(;)g(h)p Fx(\()p Ft(g)s Fx(\(0)p Ft(;)g Fx(1\)\))p Ft(;)g(g)s Fx(\(0)p Ft(;)g Fx(1\)\))1561 3099 y Fv(!)1644 3111 y Fu(R)1701 3119 y Fg(1)1820 3099 y Ft(f)9 b Fx(\()p Ft(h)p Fx(\(0\))p Ft(;)14 b(h)p Fx(\()p Ft(g)s Fx(\(0)p Ft(;)g Fx(1\)\))p Ft(;)g(g)s Fx(\(0)p Ft(;)g Fx(1\)\))1561 3201 y Fv(!)1644 3165 y Fs(+)1644 3225 y Fu(R)1701 3233 y Fg(2)1820 3201 y Ft(f)9 b Fx(\(0)p Ft(;)14 b Fx(1)p Ft(;)g(g)s Fx(\(0)p Ft(;)g Fx(1\)\))639 3382 y(Note)28 b(that)f(the)h(mo)r(dularit)n(y)f (of)g(simple)g(termination)g(for)g(restricted)g(prop)r(er)f(exten-)515 3482 y(sions)j(do)r(es)g(not)h(directly)f(follo)n(w)g(from)g(Theorem)g (4.3.)42 b(This)30 b(is)g(b)r(ecause)f(the)h(fact)g(that)515 3581 y Fv(R)585 3593 y Fs(2)652 3581 y Fx(is)f(a)g(prop)r(er)f (extension)h(of)g Fv(R)1610 3593 y Fs(1)1677 3581 y Fx(do)r(es)g(not)h (imply)f(that)h Fv(R)2502 3593 y Fs(2)2559 3581 y Fv([)20 b(E)7 b Ft(mb)p Fx(\()p Fv(F)2886 3593 y Fs(2)2923 3581 y Fx(\))30 b(is)f(a)g(prop)r(er)515 3681 y(extension)h(of)i Fv(R)1054 3693 y Fs(1)1112 3681 y Fv([)21 b(E)7 b Ft(mb)p Fx(\()p Fv(F)1440 3693 y Fs(1)1477 3681 y Fx(\))32 b(\(the)g(systems)e Fv(E)7 b Ft(mb)p Fx(\()p Fv(F)2283 3693 y Fs(1)2320 3681 y Fx(\))32 b(and)f Fv(E)7 b Ft(mb)p Fx(\()p Fv(F)2801 3693 y Fs(2)2838 3681 y Fx(\))31 b(usually)g(share)515 3781 y(de\014ned)25 b(sym)n(b)r(ols)g(whic)n(h)g(is)g(forbidden)g(in)h (hierarc)n(hical)d(com)n(binations\).)35 b(Nev)n(ertheless,)515 3880 y(it)29 b(will)g(b)r(e)g(sho)n(wn)f(in)h(the)h(next)f(section)f (that)h(simple)g(termination)g(is)g(indeed)g(mo)r(dular)515 3980 y(for)e(restricted)g(prop)r(er)f(extensions.)515 4254 y Fw(5)134 b(Hierarc)l(hical)47 b(simple)f(termination)515 4436 y Fx(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 4536 y(ularit)n(y)k(of)h(simple)g(termination.)64 b(F)-7 b(urthermore,)38 b(w)n(e)f(will)g(dispro)n(v)n(e)e(a)i(conjecture)f(of) 515 4636 y(Krishna)26 b(Rao)h([KR94)o(].)37 b(Again,)27 b(w)n(e)g(start)g(with)h(a)f(lemma)h(due)g(to)f(Gramlic)n(h)g([Gra94)n (].)515 4818 y Fm(Lemma)i(5.1)41 b Fy(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 4918 y(c)l(onstructor-sharing)30 b(TRSs.)1926 5255 y Fx(7)p eop %%Page: 8 8 8 7 bop 639 523 a Fx(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)515 623 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.)515 805 y Fm(Theorem)k(5.2)41 b Fy(L)l(et)25 b Fx(\()p Fv(F)1315 817 y Fs(1)1353 805 y Ft(;)14 b Fv(R)1460 817 y Fs(1)1497 805 y Fx(\))26 b Fy(and)h Fx(\()p Fv(F)1805 817 y Fs(2)1842 805 y Ft(;)14 b Fv(R)1949 817 y Fs(2)1987 805 y Fx(\))26 b Fy(b)l(e)g(\014nite)g(simply)h(terminating)f(TRSs.)38 b(If)515 905 y Fv(R)585 917 y Fs(2)657 905 y Fy(is)d(a)f(r)l(estricte)l (d)h(pr)l(op)l(er)g(extension)f(of)i Fv(R)2001 917 y Fs(1)2038 905 y Fy(,)g(then)f(their)g(hier)l(ar)l(chic)l(al)i(c)l (ombination)515 1005 y Fx(\()p Fv(F)8 b Ft(;)14 b Fv(R)p Fx(\))24 b(=)e(\()p Fv(F)957 1017 y Fs(1)1013 1005 y Fv([)d(F)1147 1017 y Fs(2)1184 1005 y Ft(;)14 b Fv(R)1291 1017 y Fs(1)1346 1005 y Fv([)19 b(R)1490 1017 y Fs(2)1528 1005 y Fx(\))30 b Fy(is)g(simply)h(terminating.)515 1204 y(Pr)l(o)l(of)61 b Fx(Let)29 b Fv(R)988 1174 y Fs(1)988 1224 y(2)1051 1204 y Fx(=)24 b Fv(f)p Ft(l)j Fv(!)e Ft(r)j 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1503 y Fx(are)33 b(constructor-sharing.)53 b(Because)33 b(b)r(oth)i(are)e(simply)h(terminating,)h(so)e(is)h(their)515 1602 y(union)20 b(according)e(to)i(Lemma)f(5.1.)34 b(In)20 b(order)f(to)g(sho)n(w)g(termination)h(of)g Fv(R)s([)s(E)7 b Ft(mb)p Fx(\()p Fv(F)h Fx(\),)23 b(it)d(is)515 1702 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 Fv(R)2438 1672 y Fs(1)2438 1723 y(2)2476 1702 y Fx(-c)n(hain)e(o)n(v)n(er)f Fv(R)19 b([)g(E)7 b Ft(mb)p Fx(\()p Fv(F)h Fx(\);)515 1802 y(see)36 b(Theorem)g(3.4.)65 b(Supp)r(ose,)39 b(on)e(the)g(con)n(trary)-7 b(,)38 b(that)f(there)g (is)g(a)f(minimal)h(in\014nite)515 1901 y Fv(R)585 1871 y Fs(1)585 1922 y(2)622 1901 y Fx(-c)n(hain)e Fv(h)p Ft(s)948 1913 y Fs(1)986 1901 y Ft(;)14 b(t)1053 1913 y Fs(1)1090 1901 y Fv(ih)p Ft(s)1193 1913 y Fs(2)1231 1901 y Ft(;)g(t)1298 1913 y Fs(2)1335 1901 y Fv(i)g Ft(:)g(:)g(:)50 b Fx(o)n(v)n(er)34 b Fv(R)24 b([)h(E)7 b Ft(mb)p Fx(\()p Fv(F)h Fx(\),)38 b(i.e.,)h(there)c(exists)h(a)g(substitution)515 2001 y Ft(\033)d Fx(:)c Fv(V)36 b(!)30 b(T)21 b Fx(\()p Fv(F)8 b Fx(\))32 b(suc)n(h)f(that)h Ft(t)1482 2013 y Fh(j)1517 2001 y Ft(\033)h Fv(!)1680 1971 y Fu(\003)1680 2028 y(R[E)5 b Fh(mb)p Fs(\()p Fu(F)h Fs(\))2049 2001 y Ft(s)2088 2013 y Fh(j)s Fs(+1)2207 2001 y Ft(\033)35 b Fx(holds)c(for)g(all)h(consecutiv)n(e)e(pairs)515 2113 y Fv(h)p Ft(s)586 2125 y Fh(j)621 2113 y Ft(;)14 b(t)688 2125 y Fh(j)723 2113 y Fv(i)32 b Fx(and)g Fv(h)p Ft(s)1024 2125 y Fh(j)s Fs(+1)1144 2113 y Ft(;)14 b(t)1211 2125 y Fh(j)s Fs(+1)1330 2113 y Fv(i)32 b Fx(in)h(the)f(sequence.)51 b(Because)31 b(w)n(e)h(consider)f(\014nite)i(TRSs,)g(w)n(e)515 2212 y(ma)n(y)26 b(assume)g(that)h(ev)n(ery)e(dep)r(endency)i(pair)f (in)h Fv(h)p Ft(s)2165 2224 y Fs(1)2203 2212 y Ft(;)14 b(t)2270 2224 y Fs(1)2307 2212 y Fv(ih)p Ft(s)2410 2224 y Fs(2)2448 2212 y Ft(;)g(t)2515 2224 y Fs(2)2552 2212 y Fv(i)g Ft(:)g(:)g(:)40 b Fx(b)r(elongs)27 b(to)f(a)g(cycle)515 2312 y Fv(P)k Fx(that)24 b(consists)f(solely)f(of)i(dep)r(endency)g (pairs)e(from)i Fv(R)2306 2282 y Fs(1)2306 2333 y(2)2343 2312 y Fx(.)36 b(As)24 b(in)f(the)h(pro)r(of)f(of)h(Theorem)515 2412 y(4.3,)42 b(one)d(can)g(sho)n(w)g(that)g(none)h(of)f(the)h (subterms)g(of)f(the)h(dep)r(endency)g(pairs)f(in)g Fv(P)515 2511 y Fx(con)n(tains)26 b(a)i(sym)n(b)r(ol)f(from)g Fv(D)1454 2523 y Fs(1)1510 2511 y Fv([)18 b(D)1649 2481 y Fs(1)1647 2532 y(2)1687 2511 y Fx(.)639 2611 y(Let)34 b Fv(F)862 2581 y Fu(0)919 2611 y Fx(=)f(\()p Fv(F)1109 2623 y Fs(1)1169 2611 y Fv([)23 b(F)1307 2623 y Fs(2)1344 2611 y Fx(\))g Fv(n)f(F)1531 2581 y Fs(2)1523 2631 y(2)1568 2611 y Fx(.)56 b(Because)32 b Fv(R)2040 2581 y Fs(2)2040 2631 y(2)2112 2611 y Fx(do)r(es)h(not)h(con)n(tain)f(function)i(sym)n (b)r(ols)515 2710 y(from)30 b Fv(D)778 2722 y Fs(1)836 2710 y Fv([)21 b(D)978 2680 y Fs(1)976 2731 y(2)1016 2710 y Fx(,)32 b(the)f(TRS)g Fv(R)1485 2722 y Fs(1)1544 2710 y Fv([)21 b(R)1690 2680 y Fs(1)1690 2731 y(2)1748 2710 y Fv([)g(E)7 b Ft(mb)p Fx(\()p Fv(F)2084 2680 y Fu(0)2107 2710 y Fx(\))31 b(can)g(b)r(e)g(view)n(ed)g(as)f(an)h (extension)f(of)515 2810 y Fv(R)585 2780 y Fs(2)585 2831 y(2)633 2810 y Fv([)11 b(E)c Ft(mb)p Fx(\()p Fv(F)959 2780 y Fs(2)951 2831 y(2)997 2810 y Fx(\).)36 b(Because)22 b(the)j(in\014nite)f Fv(R)1888 2780 y Fs(1)1888 2831 y(2)1926 2810 y Fx(-c)n(hain)f(is)g(minimal,)i(w)n(e)f(ma)n(y)f(apply)g (Lemma)515 2910 y(3.3)j(to)i(ev)n(ery)1181 3092 y Ft(t)1211 3104 y Fh(j)1246 3092 y Ft(\033)f Fv(!)1403 3058 y Fu(\003)1403 3118 y Fs(\()p Fu(R)1486 3098 y Fg(2)1486 3136 y(2)1518 3118 y Fu([E)5 b Fh(mb)p Fs(\()p Fu(F)1772 3098 y Fg(2)1766 3136 y(2)1803 3118 y Fs(\)\))p Fu([)p Fs(\()p Fu(R)1983 3126 y Fg(1)2015 3118 y Fu([R)2117 3098 y Fg(1)2117 3136 y(2)2150 3118 y Fu([E)g Fh(mb)p Fs(\()p Fu(F)2404 3101 y Fi(0)2425 3118 y Fs(\)\))2504 3092 y Ft(s)2543 3104 y Fh(j)s Fs(+1)2662 3092 y Ft(\033)515 3290 y Fx(and)35 b(obtain)h(\010\()p Ft(t)1073 3302 y Fh(j)1108 3290 y Ft(\033)s Fx(\))h Fv(!)1310 3260 y Fu(\003)1310 3320 y Fs(\()p Fu(R)1393 3300 y Fg(2)1393 3338 y(2)1426 3320 y Fu([E)5 b Fh(mb)p Fs(\()p Fu(F)1680 3300 y Fg(2)1674 3338 y(2)1711 3320 y Fs(\)\))p Fu(]C)1844 3328 y Fi(E)1924 3290 y Fx(\010\()p Ft(s)2055 3302 y Fh(j)s Fs(+1)2174 3290 y Ft(\033)s Fx(\).)62 b(In)36 b(other)f(w)n(ords,)i(the)f(cycle)f Fv(P)515 3414 y Fx(admits)22 b(an)g(in\014nite)i Fv(R)1239 3383 y Fs(1)1239 3434 y(2)1276 3414 y Fx(-c)n(hain)e(o)n(v)n(er)f(\()p Fv(R)1793 3383 y Fs(2)1793 3434 y(2)1839 3414 y Fv([)8 b(E)f Ft(mb)p Fx(\()p Fv(F)2162 3383 y Fs(2)2154 3434 y(2)2200 3414 y Fx(\)\))h Fv(])g(C)2379 3426 y Fu(E)2425 3414 y Fx(.)35 b(Therefore,)22 b Fv(R)2948 3426 y Fs(2)2994 3414 y Fv([)8 b(E)f Ft(mb)p Fx(\()p Fv(F)3309 3426 y Fs(2)3347 3414 y Fx(\))515 3513 y(is)24 b(not)g Fv(C)783 3525 y Fu(E)828 3513 y Fx(-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 Fv(R)3318 3525 y Fs(2)3356 3513 y Fx(.)515 3613 y Fn(\003)639 3812 y Fx(Theorem)d(5.2)h(can)f(also)g(b)r(e)i(extended)f(to)g (restricted)f(prop)r(er)g(extension)h(with)g(a)g(com-)515 3912 y(mon)27 b(subsystem;)f(see)h([Ohl01)o(])g(for)f(details.)37 b(It)27 b(should)f(b)r(e)i(p)r(oin)n(ted)f(out)g(that)g(Theorem)515 4011 y(5.2)c(is)h(not)g(applicable)g(to)g(Example)g(4.4)f(b)r(ecause)h (the)g(TRSs)h(are)e(not)h(simply)g(terminat-)515 4111 y(ing.)42 b(Krishna)28 b(Rao)h(conjectured)g(in)h([KR94)o(])f(that)h (Theorem)f(5.2)f(can)i(b)r(e)f(extended)h(to)515 4211 y(\014nitely)e(branc)n(hing)e(TRSs,)i(but)g(Example)f(5.3)g(dispro)n(v) n(es)e(his)j(conjecture.)515 4393 y Fm(Example)i(5.3)40 b Fx(The)55 b(term)f(rewriting)f(systems)g Fv(R)2281 4405 y Fs(1)2386 4393 y Fx(=)67 b Fv(f)p Ft(a)f Fv(!)h Ft(b)p Fv(g)54 b Fx(and)g Fv(R)3210 4405 y Fs(2)3314 4393 y Fx(=)515 4493 y Fv(f)p Ft(h)605 4505 y Fh(j)639 4493 y Fx(\()p Ft(x;)14 b(x)p Fx(\))47 b Fv(!)g Ft(h)1059 4505 y Fh(j)s Fs(+1)1177 4493 y Fx(\()p Ft(a;)14 b(b)p Fx(\))g Fv(j)g Ft(j)51 b Fv(2)46 b Fc(N)t Fv(g)h Fx(are)41 b(\014nitely)h(branc)n(hing)e(and)h(simply)g(terminat-)515 4592 y(ing)29 b(\(hence)i Fv(C)965 4604 y Fu(E)1009 4592 y Fx(-terminating\).)44 b Fv(R)1631 4604 y Fs(2)1698 4592 y Fx(is)30 b(a)f(restricted)h(prop)r(er)f(extension)g(of)h Fv(R)3032 4604 y Fs(1)3099 4592 y Fx(b)r(ecause)515 4692 y Ft(h)563 4704 y Fh(j)s Fs(+1)708 4692 y Fv(6)p Ff(\004)760 4707 y Fh(d)825 4692 y Ft(h)873 4704 y Fh(j)938 4692 y Fx(\(so)f Ft(a)e Fv(2)g(D)1291 4704 y Fs(1)1358 4692 y Fx(is)i(allo)n(w)n(ed)g(b)r(elo)n(w)g Ft(h)2026 4704 y Fh(j)s Fs(+1)2172 4692 y Fv(2)d(D)2319 4662 y Fs(1)2317 4713 y(2)2357 4692 y Fx(\).)44 b(Ho)n(w)n(ev)n(er,)28 b(the)i(hierarc)n(hical)515 4792 y(com)n(bination)d(of)g Fv(R)1152 4804 y Fs(1)1217 4792 y Fx(and)h Fv(R)1449 4804 y Fs(2)1514 4792 y Fx(is)g(not)f(terminating:)685 4974 y Ft(h)733 4986 y Fs(1)770 4974 y Fx(\()p Ft(a;)14 b(b)p Fx(\))23 b Fv(!)1057 4986 y Fu(R)1114 4994 y Fg(1)1174 4974 y Ft(h)1222 4986 y Fs(1)1259 4974 y Fx(\()p Ft(b;)14 b(b)p Fx(\))22 b Fv(!)1537 4986 y Fu(R)1594 4994 y Fg(2)1654 4974 y Ft(h)1702 4986 y Fs(2)1739 4974 y Fx(\()p Ft(a;)14 b(b)p Fx(\))23 b Fv(!)2026 4986 y Fu(R)2083 4994 y Fg(1)2143 4974 y Ft(h)2191 4986 y Fs(2)2228 4974 y Fx(\()p Ft(b;)14 b(b)p Fx(\))23 b Fv(!)2507 4986 y Fu(R)2564 4994 y Fg(2)2623 4974 y Ft(h)2671 4986 y Fs(3)2708 4974 y Fx(\()p Ft(a;)14 b(b)p Fx(\))23 b Fv(!)2995 4986 y Fu(R)3052 4994 y Fg(1)3112 4974 y Ft(:)14 b(:)g(:)1926 5255 y Fx(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fw(6)134 b(Related)47 b(w)l(ork)515 705 y Fx(Krishna)25 b(Rao)h([KR95)o(])h(and)f(Giesl)h(et)g(al.)g([GA)n (O02)n(])g(pro)n(v)n(ed)f(strong)f(results)h(concerning)515 805 y(hierarc)n(hical)31 b(innermost)i(termination.)53 b(Using)33 b(these)g(results,)h(it)g(is)f(p)r(ossible)g(to)g(infer)515 904 y(innermost)25 b(termination)g(of)g(the)h(com)n(bined)f(system)g Fv(R)h Fx(in)g(Example)f(4.4.)35 b(If)26 b Fv(R)g Fx(w)n(ere)e(an)515 1004 y(o)n(v)n(erla)n(y)g(system,)j(one)g(could)g(further)h(conclude)e (that)i(it)g(is)f(terminating;)g(see)g(Gramlic)n(h)515 1103 y([Gra95)n(].)50 b(Note,)33 b(ho)n(w)n(ev)n(er,)e(that)h Fv(R)h Fx(is)e(not)h(an)g(o)n(v)n(erla)n(y)d(system.)50 b(Th)n(us,)33 b(these)f(results)515 1203 y(cannot)27 b(sho)n(w)g(hierarc)n(hical)e(termination)i(of)h(systems)f(lik)n(e)g (that)h(of)g(Example)e(4.4.)639 1303 y(Apart)35 b(from)g(the)g(results) g(on)g(hierarc)n(hical)e(innermost)h(termination,)j(Giesl)e(et)g(al.) 515 1402 y([GA)n(O02)o(])18 b(also)g(pro)n(v)n(ed)f(that)i(DP)g (quasi-simple)e(termination)i(is)f(mo)r(dular)g(under)h(disjoin)n(t)515 1502 y(union,)30 b(but)h(this)f(and)g(related)f(results)h(in)g([GA)n (O02)o(])g(are)f(con\014ned)g(to)h(disjoin)n(t)g(unions,)515 1602 y(constructor-sharing)18 b(systems,)23 b(or)e(comp)r(osable)g (systems.)34 b(Th)n(us,)23 b(these)f(results)g(cannot)515 1701 y(deal)27 b(with)h(hierarc)n(hical)e(termination.)639 1801 y(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 1900 y(sho)n(witz)c ([Der95)o(].)76 b(In)41 b(his)g(theorems,)i(the)f(system)e Fv(R)2357 1912 y Fs(2)2435 1900 y Fx(m)n(ust)h(b)r(e)h Fy(\015at)p Fx(,)i(i.e.,)g(in)d(an)n(y)515 2000 y(left-)30 b(and)g(righ)n(t-hand)f(side)h(of)h(a)e(rule)h(from)g Fv(R)2041 2012 y Fs(2)2079 2000 y Fx(,)h(no)f(path)g(from)g(the)g(ro)r (ot)g(sym)n(b)r(ol)g(has)515 2100 y(more)h(than)h(one)f Ft(D)1150 2112 y Fs(2)1219 2100 y Fx(sym)n(b)r(ol)g(along)g(it.)50 b(Clearly)-7 b(,)32 b(the)g(system)g Fv(R)2668 2112 y Fs(2)2737 2100 y Fx(of)g(Example)f(4.4)g(is)515 2199 y(not)24 b(\015at)g(b)r(ecause)f(the)i(righ)n(t-hand)d(side)i(of)g(the) g(last)g(rule)g(con)n(tains)f(nested)h Ft(D)3006 2211 y Fs(2)3067 2199 y Fx(sym)n(b)r(ols.)639 2299 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 2399 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 2498 y(the)22 b(argumen)n(ts)e(of)i(recursiv)n(e)e(calls)h(in)h Fv(R)1820 2510 y Fs(2)1880 2498 y Fx(decrease)e(w.r.t.)i(the)g(subterm) g(relation)e(\(com-)515 2598 y(pared)e(as)h(m)n(ultisets)h(or)e (lexicographically\).)32 b(Hence,)22 b(their)d(result)g(is)h(also)e (not)i(applicable)515 2697 y(to)27 b(Example)g(4.4.)515 2958 y Fw(References)515 3123 y Fb([A)n(G00])102 b(T.)22 b(Arts)g(and)f(J.)h(Giesl.)30 b(T)-6 b(ermination)21 b(of)h(term)f(rewriting)i(using)f(dep)r(endency)e(pairs.)851 3215 y Fa(The)l(or)l(etic)l(al)29 b(Computer)g(Scienc)l(e)p Fb(,)e(236:133{178,)j(2000.)515 3336 y([Der95])95 b(N.)25 b(Dersho)n(witz.)33 b(Hierarc)n(hical)26 b(termination.)33 b(In)24 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)f(the)g(4th)g(Interna-)851 3427 y(tional)i(Workshop)h(on)f(Conditional)g(and)g(T)-6 b(yp)l(e)l(d)30 b(R)l(ewriting)g(Systems)p Fb(,)f(v)n(olume)d(968)851 3518 y(of)d Fa(L)l(e)l(ctur)l(e)j(Notes)f(in)e(Computer)j(Scienc)l(e)p Fb(,)d(pages)g(89{105,)i(Berlin,)f(1995.)f(Springer-)851 3610 y(V)-6 b(erlag.)515 3730 y([FJ95])129 b(M.)25 b(F)-6 b(ern\023)-38 b(andez)24 b(and)g(J.-P)-6 b(.)24 b(Jouannaud.)32 b(Mo)r(dular)26 b(termination)d(of)i(term)f(rewriting)851 3822 y(systems)g(revisited.)32 b(In)23 b Fa(R)l(e)l(c)l(ent)k(T)-6 b(r)l(ends)28 b(in)d(Data)i(T)-6 b(yp)l(e)27 b(Sp)l(e)l(ci\014c)l (ation)g(\(Pr)l(o)l(c)l(e)l(e)l(dings)851 3913 y(of)20 b(the)h(10th)h(Workshop)f(on)g(Sp)l(e)l(ci\014c)l(ation)h(of)e(A)n (bstr)l(act)i(Data)f(T)-6 b(yp)l(es\))p Fb(,)22 b(v)n(olume)16 b(906)j(of)851 4004 y Fa(L)l(e)l(ctur)l(e)31 b(Notes)f(in)f(Computer)h (Scienc)l(e)p Fb(,)e(pages)g(255{272,)j(Berlin,)d(1995.)h(Springer-)851 4096 y(V)-6 b(erlag.)515 4217 y([GA)n(O02])42 b(J.)24 b(Giesl,)i(T.)e(Arts,)f(and)g(E.)h(Ohlebusc)n(h.)30 b(Mo)r(dular)24 b(termination)f(pro)r(ofs)h(for)h(rewrit-)851 4308 y(ing)33 b(using)h(dep)r(endency)d(pairs.)57 b Fa(Journal)35 b(of)f(Symb)l(olic) g(Computation)p Fb(,)i(2002.)58 b(T)-6 b(o)851 4399 y(app)r(ear.)515 4520 y([Gra94])90 b(B.)31 b(Gramlic)n(h.)50 b(Generalized)31 b(su\016cien)n(t)g(conditions)g(for)h(mo)r(dular)e(termination)g(of)851 4611 y(rewriting.)j Fa(Applic)l(able)26 b(A)n(lgebr)l(a)h(in)f(Engine)l (ering,)g(Communic)l(ation)g(and)h(Comput-)851 4703 y(ing)p Fb(,)f(5:131{158,)j(1994.)515 4824 y([Gra95])90 b(B.)35 b(Gramlic)n(h.)59 b(Abstract)34 b(relations)h(b)r(et)n(w)n(een)f (restricted)h(termination)f(and)f(con-)851 4915 y(\015uence)f(prop)r (erties)h(of)h(rewrite)f(systems.)54 b Fa(F)-6 b(undamenta)36 b(Informatic)l(ae)p Fb(,)f(24:3{23,)851 5006 y(1995.)1926 5255 y Fx(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fb([KR94])101 b(M.R.K.)34 b(Krishna)f(Rao.)56 b(Simple)32 b(termination)h(of)h(hierarc)n(hical)g(com)n(binations)f (of)851 614 y(term)23 b(rewriting)j(systems.)31 b(In)23 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)29 b(of)d(the)h(2nd)f(International)i (Symp)l(osium)851 706 y(on)i(The)l(or)l(etic)l(al)g(Asp)l(e)l(cts)i(of) d(Computer)h(Softwar)l(e)p Fb(,)g(v)n(olume)c(789)j(of)f Fa(L)l(e)l(ctur)l(e)k(Notes)851 797 y(in)27 b(Computer)i(Scienc)l(e)p Fb(,)e(pages)f(203{223,)j(Berlin,)e(1994.)g(Springer-V)-6 b(erlag.)515 922 y([KR95])101 b(M.R.K.)26 b(Krishna)f(Rao.)33 b(Mo)r(dular)26 b(pro)r(ofs)g(for)g(completeness)f(of)h(hierarc)n (hical)g(term)851 1013 y(rewriting)h(systems.)34 b Fa(The)l(or)l(etic)l (al)29 b(Computer)g(Scienc)l(e)p Fb(,)e(151:487{512,)j(1995.)515 1137 y([Mid01])84 b(A.)38 b(Middeldorp.)71 b(Appro)n(ximating)36 b(dep)r(endency)h(graphs)h(using)g(tree)g(automata)851 1229 y(tec)n(hniques.)45 b(In)29 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e (the)g(International)i(Joint)e(Confer)l(enc)l(e)h(on)g(A)n(uto-)851 1320 y(mate)l(d)37 b(R)l(e)l(asoning)p Fb(,)i(v)n(olume)33 b(2083)k(of)e Fa(L)l(e)l(ctur)l(e)j(Notes)g(in)d(A)n(rti\014cial)h (Intel)t(ligenc)l(e)p Fb(,)851 1411 y(pages)27 b(593{610,)h(Berlin,)f (2001.)h(Springer-V)-6 b(erlag.)515 1536 y([Ohl01])94 b(E.)21 b(Ohlebusc)n(h.)26 b(Hierarc)n(hical)c(termination)e (revisited.)27 b(Rep)r(ort)20 b(01-05,)j(F)-6 b(orsc)n(h)n(ungs-)851 1627 y(b)r(eric)n(h)n(te)28 b(der)h(T)-6 b(ec)n(hnisc)n(hen)28 b(F)-6 b(akult\177)-38 b(at,)29 b(Abteilung)f(Informationstec)n(hnik,)h (Univ)n(er-)851 1719 y(sit\177)-38 b(at)26 b(Bielefeld,)i(2001.)515 1843 y([Urb01a])49 b(X.)37 b(Urbain.)67 b(Automated)36 b(incremen)n(tal)g(termination)h(pro)r(ofs)h(for)g(hierarc)n(hically) 851 1934 y(de\014ned)32 b(term)g(rewriting)h(systems.)55 b(In)32 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)37 b(of)c(the)i(International)g (Joint)851 2026 y(Confer)l(enc)l(e)c(on)g(A)n(utomate)l(d)g(R)l(e)l (asoning)p Fb(,)g(v)n(olume)c(2083)j(of)f Fa(L)l(e)l(ctur)l(e)j(Notes)f (in)e(A)n(r-)851 2117 y(ti\014cial)e(Intel)t(ligenc)l(e)p Fb(,)g(pages)f(485{499,)j(Berlin,)e(2001.)g(Springer-V)-6 b(erlag.)515 2242 y([Urb01b])44 b(Xa)n(vier)28 b(Urbain.)39 b Fa(Appr)l(o)l(che)31 b(incr)n(\023)-37 b(ementale)31 b(des)f(pr)l(euves)h(automatiques)g(de)f(termi-)851 2333 y(naison)p Fb(.)35 b(PhD)25 b(thesis,)i(Univ)n(ersit)n(\023)-36 b(e)25 b(P)n(aris-Sud,)h(2001.)1905 5255 y Fx(10)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF