%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: final.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips -o final.ps final.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1998.08.17:1533 %%BeginProcSet: tex.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 0 0 -1 0 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y(results)d(also)h(hold)f(for)i(in\014nite)d (TRSs;)h(see)1924 3589 y([)1949 3597 y(Middeldorp)f(and)i(Zan)m(tema,)h (1997)3195 3589 y(])3222 3597 y(.)523 3709 y(Ho)m(w)m(ev)m(er,)i(there) f(are)f(n)m(umerous)f(relev)-5 b(an)m(t)31 b(TRSs)f(where)g (simpli\014cation)e(order-)382 3822 y(ings)42 b(fail)h(in)f(pro)m(ving) h(termination.)79 b(F)-8 b(or)44 b(that)g(purp)s(ose,)i(a)e(new)f(tec)m (hnique)g(for)382 3935 y(automated)29 b(termination)e(pro)s(ofs,)h (viz.)g(the)h(so-called)f Fs(dep)-5 b(endency)32 b(p)-5 b(air)39 b Fu(approac)m(h,)382 4048 y(w)m(as)27 b(dev)m(elop)s(ed)g(b)m (y)g(Arts)f(and)h(Giesl)1695 4040 y([)1720 4048 y(1997a;)j(1997b;)e (1997c;)h(1998)2738 4040 y(])2766 4048 y(.)39 b(Giv)m(en)27 b(a)g(TRS,)382 4161 y(this)42 b(approac)m(h)h(generates)i(a)e(set)g(of) h(constrain)m(ts)f(and)f(the)h(existence)h(of)f(a)g(w)m(ell-)382 4274 y(founded)32 b(\(quasi-\)ordering)f(satisfying)h(these)i (constrain)m(ts)f(is)f(su\016cien)m(t)g(for)h(termi-)382 4387 y(nation.)38 b(The)23 b(adv)-5 b(an)m(tage)26 b(is)c(that)j (standard)d(tec)m(hniques)i(can)g(often)g(generate)h(suc)m(h)e(a)382 4500 y(w)m(ell-founded)h(ordering)h(ev)m(en)j(if)d(a)i(direct)f (termination)f(pro)s(of)h(with)f(the)h(same)h(tec)m(h-)382 4613 y(niques)j(fails.)44 b(In)31 b(this)f(w)m(a)m(y)-8 b(,)34 b(simpli\014cation)29 b(orderings)h(can)i(no)m(w)g(b)s(e)f(used) g(to)i(pro)m(v)m(e)382 4726 y(termination)39 b(of)i(non-simply)d (terminating)h(TRSs.)71 b(Sev)m(eral)40 b(suc)m(h)h(systems)f(from)382 4839 y(di\013eren)m(t)f(areas)i(of)f(computer)g(science)g(\(including)d (man)m(y)j(c)m(hallenging)f(problems)382 4951 y(from)30 b(the)g(literature\))g(can)h(for)f(instance)g(b)s(e)g(found)f(in)2333 4943 y([)2358 4951 y(Arts)h(and)g(Giesl,)g(1997c)3214 4943 y(])3241 4951 y(.)523 5064 y(Th)m(us,)h(the)h(dep)s(endency)e (pair)g(approac)m(h)i(pushed)e(the)i(fron)m(tier)f(of)h(those)g(TRSs) 382 5177 y(whose)38 b(termination)f(is)h(pro)m(v)-5 b(able)38 b(automatically)g(a)h(lot)f(further.)64 b(No)m(w)39 b(the)g(class)382 5290 y(of)34 b(TRSs)e(where)h(automated)h(termination)f(pro)s(ofs)f (are)i(\(p)s(oten)m(tially\))f(feasible)f(are)382 5403 y(no)k(longer)f(just)h(the)g(simply)d(terminating)i(systems,)i(but)e (the)i Fs(DP-\(quasi\))f(simply)1854 5652 y Fu(2)p eop %%Page: 3 3 3 2 bop 382 548 a Fs(terminating)33 b Fu(systems,)25 b(i.e.)f(those)h(systems)f(whose)g(termination)e(can)i(b)s(e)g(v)m (eri\014ed)f(b)m(y)382 661 y(using)e(simpli\014cation)f(orderings)i(in) f(com)m(bination)h(with)g(dep)s(endency)g(pairs.)37 b(Hence,)382 774 y(a)46 b(natural)e(question)h(is)f(whether)h(the)h(curren)m(t)f (fron)m(tier)g(of)g(mo)s(dularit)m(y)f(can)h(b)s(e)382 887 y(pushed)32 b(further)h(as)i(w)m(ell)e(b)m(y)h(extending)g(the)g (mo)s(dularit)m(y)f(results)g(from)h(simple)e(to)382 1000 y(DP-\(quasi\))43 b(simple)f(termination.)77 b(In)43 b(this)f(pap)s(er,)k(w)m(e)d(will)e(sho)m(w)i(that)h(this)e(is)382 1112 y(indeed)31 b(p)s(ossible.)44 b(Th)m(us,)32 b(the)h(class)f(of)g (TRSs)g(whose)g(termination)f(can)i(b)s(e)e(pro)m(v)m(ed)382 1225 y(in)e(a)i(mo)s(dular)d(w)m(a)m(y)j(is)f(extended)g(considerably) -8 b(.)523 1338 y(The)35 b(pap)s(er)g(is)g(organized)g(as)h(follo)m (ws:)51 b(First)35 b(w)m(e)h(brie\015y)e(recall)h(the)h(basic)f(no-)382 1451 y(tions)i(of)g(the)h(com)m(bination)e(of)i(TRSs.)61 b(Sect.)38 b(3)g(con)m(tains)f(a)h(short)f(description)e(of)382 1564 y(the)h(dep)s(endency)f(pair)g(metho)s(d.)57 b(In)35 b(Sect.)i(4)f(w)m(e)h(in)m(tro)s(duce)e(the)h(concept)h(of)f(DP-)382 1677 y(\(quasi\))28 b(simple)e(termination)i(and)f(sho)m(w)i(in)e (Sect.)i(5)g(that)g(DP-quasi)f(simple)e(termi-)382 1790 y(nation)j(is)g(mo)s(dular)f(for)h(disjoin)m(t)g(unions.)38 b(Sect.)31 b(6)f(con)m(tains)g(similar)d(results)i(ab)s(out)382 1903 y(constructor-sharing)h(TRSs.)382 2188 y Fv(2)135 b(Basic)45 b(Notions)h(of)f(the)g(Union)g(of)g(TRSs)382 2391 y Fu(F)-8 b(or)32 b(an)g(in)m(tro)s(duction)e(to)i(term)g (rewriting)d(see)k(e.g.)2234 2383 y([)2259 2391 y(Dersho)m(witz)f(and)f (Jouannaud,)382 2504 y(1990;)26 b(Klop,)f(1992)1036 2496 y(])1063 2504 y(.)39 b(Let)25 b Ft(R)f Fu(b)s(e)g(a)h(TRS)f(o)m(v)m(er) i(the)e(signature)g Ft(F)9 b Fu(.)39 b(A)25 b(function)e(sym)m(b)s(ol) 382 2617 y Fo(f)43 b Ft(2)33 b(F)44 b Fu(is)35 b(called)f(a)i Fs(de\014ne)-5 b(d)38 b(symb)-5 b(ol)37 b Fu(if)d(there)i(is)e(a)i (rewrite)f(rule)f Fo(l)h Ft(!)e Fo(r)k Ft(2)c(R)i Fu(suc)m(h)382 2730 y(that)e Fo(f)39 b Fu(=)29 b Fo(r)s(oot)p Fu(\()p Fo(l)r Fu(\).)48 b(F)-8 b(unction)32 b(sym)m(b)s(ols)f(from)i Ft(F)42 b Fu(whic)m(h)31 b(are)i(not)h(de\014ned)d(sym)m(b)s(ols)382 2843 y(are)g(called)e Fs(c)-5 b(onstructors)p Fu(.)43 b(Th)m(us,)30 b(if)f(a)i(TRS)e(consists)h(of)h(the)f(follo)m(wing)f(t)m (w)m(o)j(rules)1391 3041 y Fo(f)10 b Fu(\(0)p Fo(;)15 b Fu(1)p Fo(;)g(x)p Fu(\))85 b Ft(!)e Fo(f)10 b Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(x;)g(x)p Fu(\))784 b(\(1\))1274 3179 y Fo(f)10 b Fu(\()p Fo(x;)15 b(y)s(;)g(s)p Fu(\()p Fo(z)t Fu(\)\))85 b Ft(!)e Fo(s)p Fu(\()p Fo(f)10 b Fu(\(0)p Fo(;)15 b Fu(1)p Fo(;)g(z)t Fu(\)\))p Fo(;)779 b Fu(\(2\))382 3377 y(then)30 b Fo(f)40 b Fu(is)29 b(the)i(only)e (de\014ned)g(sym)m(b)s(ol,)h(whereas)g(0,)h(1,)g(and)f Fo(s)f Fu(are)i(constructors.)523 3490 y(Let)40 b Ft(R)772 3504 y Fr(1)851 3490 y Fu(and)f Ft(R)1114 3504 y Fr(2)1194 3490 y Fu(b)s(e)g(TRSs)f(o)m(v)m(er)j(the)f(signatures)f Ft(F)2460 3504 y Fr(1)2539 3490 y Fu(and)g Ft(F)2790 3504 y Fr(2)2830 3490 y Fu(,)j(resp.)68 b(Their)382 3603 y Fs(c)-5 b(ombine)g(d)28 b(system)d Fu(is)f(their)f(union)g Ft(R)i Fu(=)g Ft(R)1879 3617 y Fr(1)1926 3603 y Ft([)8 b(R)2072 3617 y Fr(2)2136 3603 y Fu(o)m(v)m(er)25 b(the)g(signature)f Ft(F)34 b Fu(=)25 b Ft(F)3124 3617 y Fr(1)3172 3603 y Ft([)8 b(F)3306 3617 y Fr(2)3345 3603 y Fu(.)382 3716 y(Its)41 b(set)h(of)f(de\014ned)f(sym)m(b)s(ols)g(is)h Ft(D)46 b Fu(=)d Ft(D)1881 3730 y Fr(1)1948 3716 y Ft([)27 b(D)2109 3730 y Fr(2)2190 3716 y Fu(and)40 b(its)h(set)h(of)f (constructors)h(is)382 3829 y Ft(C)30 b Fu(=)25 b Ft(F)j(n)19 b(D)s Fu(,)30 b(where)f Ft(D)1173 3843 y Fn(i)1230 3829 y Fu(\()p Ft(C)1313 3843 y Fn(i)1342 3829 y Fu(\))h(denotes)f(the)h (de\014ned)e(sym)m(b)s(ols)h(\(constructors\))h(in)e Ft(R)3317 3843 y Fn(i)3345 3829 y Fu(.)448 4035 y(\(1\))46 b Ft(R)686 4049 y Fr(1)756 4035 y Fu(and)30 b Ft(R)1010 4049 y Fr(2)1080 4035 y Fu(are)g Fs(disjoint)i Fu(if)d Ft(F)1714 4049 y Fr(1)1774 4035 y Ft(\\)20 b(F)1929 4049 y Fr(2)1994 4035 y Fu(=)25 b Ft(;)p Fu(.)448 4221 y(\(2\))46 b Ft(R)686 4235 y Fr(1)756 4221 y Fu(and)30 b Ft(R)1010 4235 y Fr(2)1080 4221 y Fu(are)g Fs(c)-5 b(onstructor-sharing)33 b Fu(if)d Ft(F)2187 4235 y Fr(1)2247 4221 y Ft(\\)20 b(F)2402 4235 y Fr(2)2467 4221 y Fu(=)25 b Ft(C)2616 4235 y Fr(1)2675 4221 y Ft(\\)20 b(C)2809 4235 y Fr(2)2874 4221 y Fu(\()p Ft(\022)25 b(C)6 b Fu(\).)448 4406 y(\(3\))46 b Ft(R)686 4420 y Fr(1)755 4406 y Fu(and)28 b Ft(R)1007 4420 y Fr(2)1075 4406 y Fu(are)i Fs(c)-5 b(omp)g(osable)31 b Fu(if)d Ft(C)1821 4420 y Fr(1)1878 4406 y Ft(\\)17 b(D)2026 4420 y Fr(2)2091 4406 y Fu(=)25 b Ft(D)2257 4420 y Fr(1)2314 4406 y Ft(\\)17 b(C)2440 4420 y Fr(2)2505 4406 y Fu(=)25 b Ft(;)k Fu(and)f(b)s(oth)h(systems)609 4519 y(con)m(tain)37 b(all)f(rewrite)f(rules)h(that)h(de\014ne)f(a)h (de\014ned)e(sym)m(b)s(ol)h(whenev)m(er)g(that)609 4632 y(sym)m(b)s(ol)29 b(is)h(shared:)40 b Ft(f)p Fo(l)27 b Ft(!)e Fo(r)j Ft(2)d(R)30 b(j)h Fo(r)s(oot)p Fu(\()p Fo(l)r Fu(\))25 b Ft(2)g(D)2317 4646 y Fr(1)2377 4632 y Ft(\\)19 b(D)2530 4646 y Fr(2)2569 4632 y Ft(g)26 b(\022)f(R)2813 4646 y Fr(1)2873 4632 y Ft(\\)20 b(R)3031 4646 y Fr(2)3070 4632 y Fu(.)523 4839 y(W)-8 b(e)42 b(next)f(giv)m(e)g(a)g(brief)e(o)m (v)m(erview)i(of)g(the)g(basic)f(notions)g(of)g(disjoin)m(t)f(unions.) 382 4951 y(In)e(the)i(sequel)e(let)h Fo(t)g Ft(2)g(T)22 b Fu(\()p Ft(F)1427 4965 y Fr(1)1493 4951 y Ft([)j(F)1644 4965 y Fr(2)1684 4951 y Fo(;)15 b Ft(V)7 b Fu(\).)65 b(Let)38 b Fm(2)g Fu(b)s(e)g(a)g(sp)s(ecial)f(constan)m(t)i Ft(62)f(F)3245 4965 y Fr(1)3310 4951 y Ft([)382 5064 y(F)447 5078 y Fr(2)487 5064 y Fu(.)82 b(A)44 b Fs(c)-5 b(ontext)54 b Fo(C)d Fu(is)43 b(a)i(term)f(in)f Ft(T)23 b Fu(\()p Ft(F)1873 5078 y Fr(1)1942 5064 y Ft([)29 b(F)2097 5078 y Fr(2)2167 5064 y Ft([)g(f)p Fm(2)p Ft(g)p Fo(;)15 b Ft(V)7 b Fu(\))46 b(and)d Fo(C)7 b Fu([)p Fo(t)2919 5078 y Fr(1)2958 5064 y Fo(;)15 b(:)g(:)g(:)i(;)e(t)3193 5078 y Fn(n)3240 5064 y Fu(])45 b(is)382 5177 y(the)37 b(result)f(of)h(replacing)e(from)i(left)f(to)i(righ)m(t)e(the)i Fo(n)d Ft(\025)h Fu(0)h(o)s(ccurrences)g(of)g Fm(2)g Fu(with)382 5290 y Fo(t)415 5304 y Fr(1)454 5290 y Fo(;)15 b(:)g(:)g(:)i(;)e(t)689 5304 y Fn(n)736 5290 y Fu(.)57 b(W)-8 b(e)37 b(write)e Fo(t)g Fu(=)f Fo(C)7 b Fu([)-15 b([)p Fo(t)1531 5304 y Fr(1)1570 5290 y Fo(;)15 b(:)g(:)g(:)h(;)f(t) 1804 5304 y Fn(n)1852 5290 y Fu(])-15 b(])36 b(if)f Fo(C)41 b Ft(2)34 b(T)22 b Fu(\()p Ft(F)2385 5304 y Fn(i)2438 5290 y Ft([)i(f)p Fm(2)p Ft(g)p Fo(;)15 b Ft(V)7 b Fu(\),)39 b Fo(C)i Ft(6)p Fu(=)34 b Fm(2)p Fu(,)j(and)382 5403 y Fo(r)s(oot)p Fu(\()p Fo(t)615 5417 y Fr(1)654 5403 y Fu(\))p Fo(;)15 b(:)g(:)g(:)i(r)s(oot)p Fu(\()p Fo(t)1084 5417 y Fn(n)1131 5403 y Fu(\))43 b Ft(2)h(F)1379 5417 y Fr(3)p Fz(\000)p Fn(i)1539 5403 y Fu(for)d(some)g Fo(i)j Ft(2)f(f)p Fu(1)p Fo(;)15 b Fu(2)p Ft(g)p Fu(.)76 b(In)41 b(this)f(case,)45 b(the)d Fo(t)3172 5417 y Fn(j)3249 5403 y Fu(are)1854 5652 y(3)p eop %%Page: 4 4 4 3 bop 382 548 a Fu(the)29 b Fs(princip)-5 b(al)31 b Fu(subterms)c(of)j Fo(t)e Fu(and)h Fo(C)35 b Fu(is)28 b(the)h(topmost)g Ft(F)2401 562 y Fn(i)2430 548 y Fu(-homogeneous)h (part)f(of)g Fo(t)p Fu(,)382 661 y(denoted)k(b)m(y)g Fo(top)980 675 y Fn(i)1008 661 y Fu(\()p Fo(t)p Fu(\))g(\(whereas)h Fo(top)1650 675 y Fr(3)p Fz(\000)p Fn(i)1768 661 y Fu(\()p Fo(t)p Fu(\))f(is)f Fm(2)p Fu(\).)49 b(So)33 b(for)g(example,)h(if)e Ft(R)2994 675 y Fr(1)3066 661 y Fu(consists)382 774 y(of)e(the)h(rules) e(\(1\))i(and)f(\(2\),)i(and)e Ft(R)1609 788 y Fr(2)1678 774 y Fu(con)m(tains)h(the)g(rules)1583 956 y Fo(g)s Fu(\()p Fo(x;)15 b(y)s Fu(\))84 b Ft(!)f Fo(x)1106 b Fu(\(3\))1583 1094 y Fo(g)s Fu(\()p Fo(x;)15 b(y)s Fu(\))84 b Ft(!)f Fo(y)s(;)1085 b Fu(\(4\))382 1277 y(then)42 b Ft(R)678 1291 y Fr(1)761 1277 y Fu(and)g Ft(R)1027 1291 y Fr(2)1109 1277 y Fu(are)h(disjoin)m(t)e(and)i(a)g(term)g(lik)m (e)f Fo(f)10 b Fu(\()p Fo(g)s Fu(\(0)p Fo(;)15 b Fu(0\))p Fo(;)g(x;)g(g)s Fu(\()p Fo(y)s(;)g(y)s Fu(\)\))47 b(can)c(b)s(e)382 1390 y(written)31 b(as)i Fo(C)7 b Fu([)-15 b([)p Fo(g)s Fu(\(0)p Fo(;)15 b Fu(0\))p Fo(;)g(g)s Fu(\()p Fo(y)s(;)g(y)s Fu(\)])-15 b(],)37 b(where)32 b Fo(C)39 b Fu(is)31 b Fo(f)10 b Fu(\()p Fm(2)p Fo(;)15 b(x;)g Fm(2)p Fu(\).)47 b(Th)m(us)31 b Fo(top)2840 1404 y Fr(1)2879 1390 y Fu(\()p Fo(f)10 b Fu(\()p Fo(g)s Fu(\(0)p Fo(;)15 b Fu(0\))p Fo(;)g(x;)382 1503 y(g)s Fu(\()p Fo(y)s(;)g(y)s Fu(\)\)\))27 b(=)e Fo(f)10 b Fu(\()p Fm(2)p Fo(;)15 b(x;)g Fm(2)p Fu(\))31 b(and)f Fo(top)1551 1517 y Fr(2)1590 1503 y Fu(\()p Fo(f)10 b Fu(\()p Fo(g)s Fu(\(0)p Fo(;)15 b Fu(0\))p Fo(;)g(x;)g(g)s Fu(\()p 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b(Our)382 2207 y(mo)s(dularit)m(y)e (results)i(crucially)e(dep)s(end)h(on)h(the)h(fact)g(that)h Fo(s)f Ft(!)2736 2221 y Fz(R)2796 2230 y Fl(1)2831 2221 y Fz([R)2938 2230 y Fl(2)3019 2207 y Fo(t)f Fu(implies)382 2320 y Fo(r)s(ank)s Fu(\()p Fo(s)p Fu(\))25 b Ft(\025)g Fo(r)s(ank)s Fu(\()p Fo(t)p Fu(\))30 b(\(the)h(pro)s(of)f(is)f(straigh) m(tforw)m(ard)h(b)m(y)g(induction)e(on)j Fo(r)s(ank)s Fu(\()p Fo(s)p Fu(\)\).)523 2433 y(A)k(rewrite)f(step)h Fo(s)d Ft(!)1302 2447 y Fz(R)1362 2456 y Fl(1)1397 2447 y Fz([R)1504 2456 y Fl(2)1575 2433 y Fo(t)j Fu(is)f Fs(destructive)j (at)g(level)f(1)g Fu(if)d Fo(r)s(oot)p Fu(\()p Fo(s)p Fu(\))g Ft(2)f(F)3161 2447 y Fn(i)3224 2433 y Fu(and)382 2546 y Fo(r)s(oot)p Fu(\()p Fo(t)p Fu(\))26 b Ft(2)f(F)827 2560 y Fr(3)p Fz(\000)p Fn(i)976 2546 y Fu(for)31 b(some)g Fo(i)26 b Ft(2)f(f)p Fu(1)p Fo(;)15 b Fu(2)p Ft(g)p Fu(.)44 b(A)31 b(reduction)f(step)g Fo(s)c Ft(!)2632 2560 y Fz(R)2692 2569 y Fl(1)2727 2560 y Fz([R)2834 2569 y Fl(2)2898 2546 y Fo(t)31 b Fu(is)f Fs(destruc-)382 2659 y(tive)38 b(at)i(level)d Fo(m)25 b Fu(+)f(1)38 b(\(for)f(some)h Fo(m)f Ft(\025)f Fu(1\))i(if)e Fo(s)h Fu(=)f Fo(C)7 b Fu([)-15 b([)p Fo(s)2355 2673 y Fr(1)2394 2659 y Fo(;)15 b(:)g(:)g(:)i(;)e(s)2639 2673 y Fn(j)2675 2659 y Fo(;)g(:)g(:)g(:)i(;)e(s)2920 2673 y Fn(n)2967 2659 y Fu(])-15 b(])37 b Ft(!)3130 2673 y Fz(R)3190 2682 y Fl(1)3225 2673 y Fz([R)3332 2682 y Fl(2)382 2772 y Fo(C)7 b Fu([)p Fo(s)522 2786 y Fr(1)561 2772 y Fo(;)15 b(:)g(:)g(:)h(;)f(t)795 2786 y Fn(j)832 2772 y Fo(;)g(:)g(:)g(:)i(;)e(s)1077 2786 y Fn(n)1124 2772 y Fu(])38 b(=)f Fo(t)h Fu(with)f Fo(s)1624 2786 y Fn(j)1697 2772 y Ft(!)1788 2786 y Fz(R)1848 2795 y Fl(1)1883 2786 y Fz([R)1990 2795 y Fl(2)2067 2772 y Fo(t)2100 2786 y Fn(j)2174 2772 y Fu(destructiv)m(e)h(at)h(lev)m(el)e Fo(m)p Fu(.)63 b(Ob)m(vi-)382 2884 y(ously)-8 b(,)38 b(if)e(a)h(rewrite)f(step)h(is)f(destructiv)m(e,)j(then)e(the)g (rewrite)f(rule)g(applied)e(is)i(col-)382 2997 y(lapsing,)e(i.e.)h(the) g(righ)m(t-hand)e(side)h(of)g(the)h(rule)f(is)f(a)i(v)-5 b(ariable.)52 b(F)-8 b(or)36 b(example,)f(the)382 3110 y(rewrite)29 b(step)i Fo(f)10 b Fu(\()p Fo(g)s Fu(\(0)p Fo(;)15 b Fu(0\))p Fo(;)g(x;)g(g)s Fu(\()p Fo(y)s(;)g(y)s Fu(\)\))30 b Ft(!)25 b Fo(f)10 b Fu(\(0)p Fo(;)15 b(x;)g(g)s Fu(\()p Fo(y)s(;)g(y)s Fu(\)\))33 b(is)c(destructiv)m(e)i(at)g(lev)m (el)f(2.)523 3223 y(Finally)-8 b(,)28 b(w)m(e)j(recall)e(that)h(for)f (ev)m(ery)i(signature)e Ft(F)39 b Fu(the)30 b(TRS)e Ft(E)8 b Fo(mb)p Fu(\()p Ft(F)h Fu(\))31 b(\(whic)m(h)e(is)382 3336 y(imp)s(ortan)m(t)g(in)h(the)g(con)m(text)i(of)f(simple)d (termination\))i(is)f(de\014ned)g(b)m(y)552 3519 y Ft(E)8 b Fo(mb)p Fu(\()p Ft(F)h Fu(\))26 b(=)f Ft(f)p Fo(f)10 b Fu(\()p Fo(x)1180 3533 y Fr(1)1220 3519 y Fo(;)15 b(:)g(:)g(:)i(;)e (x)1474 3533 y Fn(n)1521 3519 y Fu(\))25 b Ft(!)h Fo(x)1750 3533 y Fn(i)1793 3519 y Ft(j)46 b Fo(f)34 b Ft(2)25 b(F)9 b Fu(,)31 b Fo(f)40 b Fu(is)29 b Fo(n)p Fu(-ary)h(and)g(1)c Ft(\024)f Fo(i)g Ft(\024)g Fo(n)p Ft(g)p Fo(:)382 3802 y Fv(3)135 b(Dep)t(endency)44 b(P)l(airs)382 4005 y Fu(In)29 b(the)i(dep)s(endency)e(pair)g(approac)m(h)h(of)h(Arts)f(and)g(Giesl) 2439 3997 y([)2464 4005 y(1997a;)j(1997c;)g(1998)3205 3997 y(])3262 4005 y(for)382 4118 y(sho)m(wing)j(termination,)j(if)d Fo(f)10 b Fu(\()p Fo(s)1488 4132 y Fr(1)1527 4118 y Fo(;)15 b(:::;)g(s)1725 4132 y Fn(n)1773 4118 y Fu(\))38 b(rewrites)e(to)i Fo(C)7 b Fu([)p Fo(g)s Fu(\()p Fo(t)2526 4132 y Fr(1)2566 4118 y Fo(;)15 b(:::;)g(t)2754 4132 y Fn(m)2822 4118 y Fu(\)])38 b(\(where)g Fo(g)j Fu(is)382 4231 y(a)36 b(de\014ned)e(sym)m(b)s(ol\),)i(then)f(one)h(has)f(to)i(compare)f(the)f (argumen)m(t)h(tuples)e Fo(s)3085 4245 y Fr(1)3125 4231 y Fo(;)15 b(:::;)g(s)3323 4245 y Fn(n)382 4344 y Fu(and)29 b Fo(t)591 4358 y Fr(1)630 4344 y Fo(;)15 b(:::;)g(t)818 4358 y Fn(m)886 4344 y Fu(.)41 b(T)-8 b(o)30 b(a)m(v)m(oid)g(the)g (handling)d(of)j Fs(tuples)p Fu(,)g(a)g(new)g Fs(tuple)i(symb)-5 b(ol)40 b Fo(F)f Ft(62)25 b(F)39 b Fu(is)382 4456 y(in)m(tro)s(duced)23 b(for)h(ev)m(ery)i(de\014ned)d(sym)m(b)s(ol)h Fo(f)10 b Fu(.)38 b(Instead)24 b(of)h(comparing)f Fs(tuples)p Fu(,)j(no)m(w)d(the)382 4569 y Fs(terms)34 b Fo(F)13 b Fu(\()p Fo(s)786 4583 y Fr(1)826 4569 y Fo(;)i(:::;)g(s)1024 4583 y Fn(n)1072 4569 y Fu(\))27 b(and)e Fo(G)p Fu(\()p Fo(t)1446 4583 y Fr(1)1485 4569 y Fo(;)15 b(:::;)g(t)1673 4583 y Fn(m)1741 4569 y Fu(\))27 b(are)f(compared.)40 b(Th)m(us,)26 b(to)h(ease)g(readabilit)m(y)382 4682 y(w)m(e)e(assume)g (that)h(the)f(signature)f Ft(F)34 b Fu(consists)25 b(of)g(lo)m(w)m(er)g (case)h(sym)m(b)s(ols)e(only)g(and)g(that)382 4795 y(tuple)29 b(sym)m(b)s(ols)g(are)i(denoted)f(b)m(y)h(the)f(corresp)s(onding)f(upp) s(er)f(case)j(sym)m(b)s(ols.)382 4986 y Fk(De\014nition)k(1)g(\(Dep)s (endency)h(P)m(air\))45 b Fs(If)d Fo(f)10 b Fu(\()p Fo(s)2109 5000 y Fr(1)2148 4986 y Fo(;)15 b(:::;)g(s)2346 5000 y Fn(n)2394 4986 y Fu(\))44 b Ft(!)f Fo(C)7 b Fu([)p Fo(g)s Fu(\()p Fo(t)2818 5000 y Fr(1)2858 4986 y Fo(;)15 b(:::;)g(t)3046 5000 y Fn(m)3114 4986 y Fu(\)])43 b Fs(is)f(a)382 5099 y(rule)34 b(of)g(a)h(TRS)g Ft(R)f Fs(and)h Fo(g)i Fs(is)d(a)h(de\014ne)-5 b(d)35 b(symb)-5 b(ol,)36 b(then)f Ft(h)p Fo(F)13 b Fu(\()p Fo(s)2544 5113 y Fr(1)2583 5099 y Fo(;)i(:::;)g(s)2781 5113 y Fn(n)2830 5099 y Fu(\))p Fo(;)g(G)p Fu(\()p Fo(t)3045 5113 y Fr(1)3084 5099 y Fo(;)g(:::t)3232 5113 y Fn(m)3300 5099 y Fu(\))p Ft(i)382 5212 y Fs(is)32 b(a)40 b Fu(dep)s(endency)29 b(pair)j Fs(of)h Ft(R)p Fs(.)382 5403 y Fu(So)d(for)g(the)h(TRS)e Ft(R)1094 5417 y Fr(1)1159 5403 y Fu(=)c Ft(f)p Fu(\(1\))p Fo(;)15 b Fu(\(2\))p Ft(g)34 b Fu(w)m(e)d(obtain)f(the)g(follo)m(wing)f (dep)s(endency)g(pairs)1854 5652 y(4)p eop %%Page: 5 5 5 4 bop 1470 548 a Ft(h)p Fo(F)13 b Fu(\(0)p Fo(;)i Fu(1)p Fo(;)g(x)p Fu(\))p Fo(;)g(F)e Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\))p Fo(;)i(x;)g(x)p Fu(\))p Ft(i)822 b Fu(\(5\))1470 686 y Ft(h)p Fo(F)13 b Fu(\()p Fo(x;)i(y)s(;)g(s)p Fu(\()p Fo(z)t Fu(\)\))p Fo(;)g(F)e Fu(\(0)p Fo(;)i Fu(1)p Fo(;)g(z)t Fu(\))p Ft(i)p Fo(:)813 b Fu(\(6\))523 841 y(T)-8 b(o)38 b(trace)g(those)g(subterms)e(whic)m(h)g(ma)m(y)i(start)g(new)f (reductions,)h(w)m(e)g(examine)382 953 y(sp)s(ecial)g(sequences)h(of)g (dep)s(endency)f(pairs,)i(so-called)f Fs(chains)p Fu(.)68 b(In)39 b(the)g(follo)m(wing,)382 1066 y(w)m(e)c(consider)f (substitutions)e(whose)j(domains)f(ma)m(y)h(b)s(e)g(in\014nite)d(and)i (assume)h(that)382 1179 y(di\013eren)m(t)30 b(\(o)s(ccurrences)g(of)7 b(\))31 b(dep)s(endency)e(pairs)g(ha)m(v)m(e)j(disjoin)m(t)d(sets)h(of) h(v)-5 b(ariables.)382 1342 y Fk(De\014nition)35 b(2)g(\(Chain\))45 b Fs(A)37 b(se)-5 b(quenc)g(e)37 b(of)h(dep)-5 b(endency)38 b(p)-5 b(airs)40 b Ft(h)p Fo(s)2740 1356 y Fr(1)2779 1342 y Fo(;)15 b(t)2852 1356 y Fr(1)2892 1342 y Ft(ih)p Fo(s)3005 1356 y Fr(2)3044 1342 y Fo(;)g(t)3117 1356 y Fr(2)3157 1342 y Ft(i)p Fo(:::)39 b Fs(is)382 1455 y(an)h Ft(R)p Fs(-)p Fu(c)m(hain)f Fs(if)h(ther)-5 b(e)40 b(is)g(a)g(substitution)h Fo(\033)i Fs(such)d(that)h Fo(t)2431 1469 y Fn(j)2467 1455 y Fo(\033)h Ft(!)2652 1422 y Fz(\003)2652 1482 y(R)2754 1455 y Fo(s)2797 1469 y Fn(j)t Fr(+1)2924 1455 y Fo(\033)g Fs(holds)g(for)382 1568 y(every)32 b(two)i(c)-5 b(onse)g(cutive)33 b(p)-5 b(airs)34 b Ft(h)p Fo(s)1568 1582 y Fn(j)1605 1568 y Fo(;)15 b(t)1678 1582 y Fn(j)1714 1568 y Ft(i)33 b Fs(and)h Ft(h)p Fo(s)2037 1582 y Fn(j)t Fr(+1)2163 1568 y Fo(;)15 b(t)2236 1582 y Fn(j)t Fr(+1)2363 1568 y Ft(i)33 b Fs(in)g(the)g(se)-5 b(quenc)g(e.)382 1731 y Fu(F)d(or)31 b(instance,)f(in)f(our)h(example)g (w)m(e)h(ha)m(v)m(e)h(the)e(c)m(hain)392 1886 y Fj(h)p Fi(F)11 b Fp(\(0)p Fi(;)i Fp(1)p Fi(;)g(x)700 1894 y Fl(1)735 1886 y Fp(\))p Fi(;)g(F)e Fp(\()p Fi(s)p Fp(\()p Fi(x)999 1894 y Fl(1)1032 1886 y Fp(\))p Fi(;)i(x)1140 1894 y Fl(1)1174 1886 y Fi(;)g(x)1252 1894 y Fl(1)1286 1886 y Fp(\))p Fj(i)64 b(h)p Fi(F)11 b Fp(\()p Fi(x)1574 1894 y Fl(2)1608 1886 y Fi(;)i(y)1680 1894 y Fl(2)1714 1886 y Fi(;)g(s)p Fp(\()p Fi(z)1850 1894 y Fl(2)1884 1886 y Fp(\)\))p Fi(;)g(F)e Fp(\(0)p Fi(;)i Fp(1)p Fi(;)h(z)2249 1894 y Fl(2)2283 1886 y Fp(\))p Fj(i)64 b(h)p Fi(F)11 b Fp(\(0)p Fi(;)i Fp(1)p Fi(;)g(x)2715 1894 y Fl(3)2749 1886 y Fp(\))p Fi(;)g(F)e Fp(\()p Fi(s)p Fp(\()p Fi(x)3013 1894 y Fl(3)3047 1886 y Fp(\))p Fi(;)i(x)3155 1894 y Fl(3)3189 1886 y Fi(;)g(x)3267 1894 y Fl(3)3301 1886 y Fp(\))p Fj(i)382 2041 y Fu(b)s(ecause)25 b(with)g Fo(\033)j Fu(=)d Ft(f)p Fo(x)1189 2055 y Fr(1)1254 2041 y Ft(7!)g Fo(s)p Fu(\()p Fo(x)1500 2055 y Fr(3)1540 2041 y Fu(\))p Fo(;)15 b(x)1667 2055 y Fr(2)1732 2041 y Ft(7!)25 b Fo(s)p Fu(\()p Fo(s)p Fu(\()p Fo(x)2056 2055 y Fr(3)2095 2041 y Fu(\)\))p Fo(;)15 b(y)2250 2055 y Fr(2)2316 2041 y Ft(7!)25 b Fo(s)p Fu(\()p Fo(x)2562 2055 y Fr(3)2601 2041 y Fu(\))p Fo(;)15 b(z)2718 2055 y Fr(2)2784 2041 y Ft(7!)25 b Fo(x)2952 2055 y Fr(3)2992 2041 y Ft(g)g Fu(w)m(e)h(ha)m(v)m(e)382 2154 y Fo(F)13 b Fu(\()p Fo(s)p Fu(\()p Fo(x)618 2168 y Fr(1)658 2154 y Fu(\))p Fo(;)i(x)785 2168 y Fr(1)825 2154 y Fo(;)g(x)917 2168 y Fr(1)957 2154 y Fu(\))p Fo(\033)38 b Ft(!)1173 2121 y Fz(\003)1173 2181 y(R)1233 2190 y Fl(1)1307 2154 y Fo(F)13 b Fu(\()p Fo(x)1465 2168 y Fr(2)1505 2154 y Fo(;)i(y)1590 2168 y Fr(2)1630 2154 y Fo(;)g(s)p Fu(\()p Fo(z)1790 2168 y Fr(2)1830 2154 y Fu(\)\))p Fo(\033)40 b Fu(and)c Fo(F)13 b Fu(\(0)p Fo(;)i Fu(1)p Fo(;)g(z)2493 2168 y Fr(2)2535 2154 y Fu(\))p Fo(\033)38 b Ft(!)2751 2121 y Fz(\003)2751 2181 y(R)2811 2190 y Fl(1)2885 2154 y Fo(F)13 b Fu(\(0)p Fo(;)i Fu(1)p Fo(;)g(x)3213 2168 y Fr(3)3255 2154 y Fu(\))p Fo(\033)s Fu(.)382 2267 y(In)29 b(fact,)i(ev)m(ery)f(\014nite)e (alternating)h(sequence)h(of)g(\(5\))h(and)e(\(6\))h(is)f(a)h(c)m (hain.)40 b(Arts)29 b(and)382 2380 y(Giesl)608 2372 y([)633 2380 y(1997a;)i(1997c)1134 2372 y(])1190 2380 y(pro)m(v)m(ed)e(that)g (the)f(absence)h(of)f Fs(in\014nite)35 b Fu(c)m(hains)28 b(is)f(a)i(su\016cien)m(t)382 2493 y(and)h(necessary)g(criterion)g(for) g(termination.)382 2656 y Fk(Theorem)k(3)h(\(T)-9 b(ermination)34 b(Criterion\))44 b Fs(A)g(TRS)g Ft(R)h Fs(is)f(terminating)h(if)f(and) 382 2769 y(only)33 b(if)g(ther)-5 b(e)33 b(exists)g(no)g(in\014nite)g Ft(R)p Fs(-chain.)523 2932 y Fu(Note)28 b(that)f(the)g(\014rst)e(dep)s (endency)g(pair)h(\(5\))h(can)g(nev)m(er)g(follo)m(w)e(itself)h(in)f(a) i(c)m(hain,)382 3045 y(b)s(ecause)33 b Fo(F)13 b Fu(\()p Fo(s)p Fu(\()p Fo(x)957 3059 y Fr(1)997 3045 y Fu(\))p Fo(;)i(x)1124 3059 y Fr(1)1164 3045 y Fo(;)g(x)1256 3059 y Fr(1)1296 3045 y Fu(\))p Fo(\033)33 b Ft(!)1507 3012 y Fz(\003)1507 3071 y(R)1567 3080 y Fl(1)1635 3045 y Fo(F)13 b Fu(\(0)p Fo(;)i Fu(1)p Fo(;)g(x)1963 3059 y Fr(2)2005 3045 y Fu(\))p Fo(\033)36 b Fu(do)s(es)d(not)g(hold)f(for)h (an)m(y)g(substitu-)382 3157 y(tion)40 b Fo(\033)s Fu(.)71 b(T)-8 b(o)41 b(estimate)g(whic)m(h)e(dep)s(endency)g(pairs)g(ma)m(y)i (o)s(ccur)g(consecutiv)m(e)g(in)e(a)382 3270 y(c)m(hain,)j(the)e Fs(estimate)-5 b(d)43 b(dep)-5 b(endency)42 b(gr)-5 b(aph)49 b Fu(has)39 b(b)s(een)h(in)m(tro)s(duced,)g(cf.)h(Arts)e(and)382 3383 y(Giesl)616 3375 y([)641 3383 y(1997a;)f(1997c;)h(1998)1393 3375 y(])1420 3383 y(.)57 b(W)-8 b(e)37 b(\014rst)e(recall)g(the)h (needed)g(notions.)56 b Fx(CAP)p Fu(\()p Fo(t)p Fu(\))36 b(re-)382 3496 y(sults)g(from)h(replacing)f(all)g(subterms)h(of)g Fo(t)g Fu(that)h(ha)m(v)m(e)h(a)f(de\014ned)e(ro)s(ot)i(sym)m(b)s(ol)e (b)m(y)382 3609 y(di\013eren)m(t)30 b(fresh)g(v)-5 b(ariables)30 b(and)g Fx(REN)p Fu(\()p Fo(t)p Fu(\))i(results)d(from)i(replacing)f (all)g(v)-5 b(ariables)29 b(in)h Fo(t)382 3722 y Fu(b)m(y)g(di\013eren) m(t)g(fresh)g(v)-5 b(ariables.)40 b(Then,)30 b(in)f(order)h(to)i (determine)d(whether)h Ft(h)p Fo(u;)15 b(v)s Ft(i)32 b Fu(can)382 3835 y(follo)m(w)e Ft(h)p Fo(s;)15 b(t)p Ft(i)31 b Fu(in)f(a)h(c)m(hain,)g(w)m(e)g(c)m(hec)m(k)i(whether)d Fx(REN)p Fu(\()p Fx(CAP)p Fu(\()p Fo(t)p Fu(\)\))i(uni\014es)c(with)i Fo(u)p Fu(.)42 b(The)382 3948 y(function)25 b Fx(REN)h Fu(is)f(needed)h(to)h(rename)g(m)m(ultiple)d(o)s(ccurrences)i(of)h(the) f(same)h(v)-5 b(ariable)382 4061 y Fo(x)37 b Fu(in)f Fo(t)h Fu(b)s(ecause)h(when)e(instan)m(tiated)h(with)f Fo(\033)s Fu(,)k(t)m(w)m(o)f(o)s(ccurrences)e(of)g Fo(x\033)k Fu(could)36 b(re-)382 4174 y(duce)25 b(to)h(di\013eren)m(t)f(terms.)39 b(So)25 b(in)f(our)h(example,)h(the)g(estimated)f(dep)s(endency)f (graph)382 4287 y(con)m(tains)30 b(an)h(arc)f(from)g(\(5\))i(to)f (\(6\))g(and)f(arcs)h(from)f(\(6\))h(to)g(\(5\))h(and)e(to)h(itself.) 382 4450 y Fk(De\014nition)k(4)g(\(Estimated)e(Dep)s(endency)j(Graph\)) 45 b Fs(The)53 b Fu(estimated)45 b(dep)s(en-)382 4563 y(dency)27 b(graph)j Fs(is)g(the)g(dir)-5 b(e)g(cte)g(d)32 b(gr)-5 b(aph)32 b(whose)f(no)-5 b(des)31 b(ar)-5 b(e)31 b(the)g(dep)-5 b(endency)31 b(p)-5 b(airs)32 b(and)382 4675 y(ther)-5 b(e)33 b(is)g(an)g(ar)-5 b(c)34 b(fr)-5 b(om)33 b Ft(h)p Fo(s;)15 b(t)p Ft(i)34 b Fs(to)f Ft(h)p Fo(u;)15 b(v)s Ft(i)34 b Fs(if)51 b Fw(REN)p Fu(\()p Fw(CAP)p Fu(\()p Fo(t)p Fu(\)\))34 b Fs(and)g Fo(u)e Fs(ar)-5 b(e)34 b(uni\014able.)523 4839 y Fu(A)c(set)h Ft(P)38 b Fu(of)31 b(dep)s(endency)e(pairs)g(is)g(called)h(a)g Fs(cycle)37 b Fu(if)30 b(for)g(an)m(y)g(t)m(w)m(o)i(dep)s(endency)382 4951 y(pairs)d Ft(h)p Fo(s;)15 b(t)p Ft(i)p Fo(;)g Ft(h)p Fo(u;)g(v)s Ft(i)29 b(2)c(P)39 b Fu(there)31 b(is)f(a)h(path)g(from)f Ft(h)p Fo(s;)15 b(t)p Ft(i)31 b Fu(to)h Ft(h)p Fo(u;)15 b(v)s Ft(i)32 b Fu(and)e(from)h Ft(h)p Fo(u;)15 b(v)s Ft(i)32 b Fu(to)382 5064 y Ft(h)p Fo(s;)15 b(t)p Ft(i)33 b Fu(in)d(the)j(estimated)f(dep)s(endency)f(graph)h(whic)m(h)f(tra)m(v) m(erses)i(dep)s(endency)e(pairs)382 5177 y(from)f Ft(P)39 b Fu(only)-8 b(.)41 b(\(In)30 b(particular,)g(there)h(m)m(ust)g(also)f (b)s(e)g(a)i(path)e(from)g Ft(h)p Fo(s;)15 b(t)p Ft(i)32 b Fu(to)f(itself.\))382 5290 y(Th)m(us,)c(the)f(only)g(non-empt)m(y)h (cycles)g(in)f(our)g(example)g(are)h Ft(f)p Fu(\(6\))p Ft(g)i Fu(and)d Ft(f)p Fu(\(5\))p Fo(;)15 b Fu(\(6\))p Ft(g)p Fu(.)43 b(In)382 5403 y(the)27 b(remainder)f(of)h(the)h(pap)s (er,)f(w)m(e)h(alw)m(a)m(ys)g(restrict)f(ourselv)m(es)g(to)h(\014nite)e (TRSs)g(\(and)1854 5652 y(5)p eop %%Page: 6 6 6 5 bop 382 548 a Fu(to)34 b(\014nite)f(signatures\).)49 b(Then)33 b(an)m(y)g(in\014nite)f(c)m(hain)h(corresp)s(onds)f(to)i(a)g (cycle,)h(i.e.)e(it)382 661 y(su\016ces)26 b(to)h(pro)m(v)m(e)h(that)f (there)g(is)e(no)h(in\014nite)f(c)m(hain)h(of)g(dep)s(endency)g(pairs)f Fs(fr)-5 b(om)30 b(any)382 774 y(cycle)p Fu(,)g(cf.)754 766 y([)779 774 y(Arts)h(and)e(Giesl,)h(1998)1595 766 y(])1622 774 y(.)523 887 y(F)-8 b(or)25 b(an)g(automation)g(of)g(this)e (criterion,)i(w)m(e)g(generate)h(a)f(set)g(of)g(inequalities)d(suc)m(h) 382 1000 y(that)k(the)g(existence)h(of)f(a)g(w)m(ell-founded)e Fs(quasi-or)-5 b(dering)35 b Fu(satisfying)24 b(these)j(inequali-)382 1112 y(ties)g(is)e(su\016cien)m(t)i(for)f(the)h(absence)h(of)f (in\014nite)d(c)m(hains.)39 b(As)27 b(usual,)f(a)h Fs(quasi-or)-5 b(dering)382 1225 y Fh(\045)38 b Fu(is)f(a)i(re\015exiv)m(e)f(and)g (transitiv)m(e)f(relation.)64 b(The)38 b(corresp)s(onding)e Fs(strict)47 b Fu(relation)382 1338 y Ft(\037)453 1305 y Fn(s)528 1338 y Fu(is)38 b(de\014ned)g(as)i Fo(t)f Ft(\037)1214 1305 y Fn(s)1290 1338 y Fo(u)g Fu(i\013)g Fo(t)g Fh(\045)h Fo(u)f Fu(and)f Fo(u)i Ft(6)p Fh(\045)f Fo(t)p Fu(.)67 b(Moreo)m(v)m(er,)43 b(w)m(e)d(also)f(de\014ne)f(a)382 1451 y(corresp)s(onding)29 b Fs(stable-strict)41 b Fu(relation)31 b Ft(\037)1872 1418 y Fn(ss)1973 1451 y Fu(as)h Fo(t)27 b Ft(\037)2217 1418 y Fn(ss)2313 1451 y Fo(u)32 b Fu(i\013)e Fo(t\033)h Ft(\037)2693 1418 y Fn(s)2757 1451 y Fo(u\033)j Fu(holds)c(for)i(all)382 1564 y(ground)d(substitutions)f Fo(\033)s Fu(,)j(where)f(a)g Fs(gr)-5 b(ound)41 b Fu(substitution)29 b(is)g(a)i(substitution)d(map-)382 1677 y(ping)22 b(all)g(v)-5 b(ariables)22 b(to)j(ground)d(terms.)38 b(In)23 b(other)h(w)m(ords,)h (for)e(all)f(those)i(substitutions)382 1790 y Fo(\033)33 b Fu(w)m(e)e(m)m(ust)g(ha)m(v)m(e)g Fo(t\033)d Fh(\045)d Fo(u\033)34 b Fu(and)c Fo(u\033)e Ft(6)p Fh(\045)d Fo(t\033)s Fu(.)523 1903 y(F)-8 b(or)26 b(instance,)h(man)m(y)f(useful)e (quasi-orderings)f(are)j(constructed)g(b)m(y)g(using)e(map-)382 2016 y(pings)k Ft(j)p Fo(:)p Ft(j)j Fu(from)f(the)g(set)h(of)f(ground)f (terms)h(to)h(a)f(w)m(ell-founded)e(set)j(lik)m(e)e(the)i(natural)382 2129 y(n)m(um)m(b)s(ers)36 b(I)-15 b(N,)40 b(cf.)e(e.g.)1213 2121 y([)1238 2129 y(Lankford,)h(1979,)j(\\p)s(olynomial)35 b(orderings")2843 2121 y(])2868 2129 y(.)63 b(Then)37 b Fh(\045)h Fu(is)382 2242 y(de\014ned)f(as)i Fo(t)f Fh(\045)h Fo(u)f Fu(i\013)g Ft(j)p Fo(t\033)s Ft(j)h(\025)1459 2256 y Fr(I)-12 b(N)1568 2242 y Ft(j)p Fo(u\033)s Ft(j)39 b Fu(holds)e(for)h(all)f(ground)h(substitutions)e Fo(\033)s Fu(.)65 b(A)382 2355 y(natural)32 b(w)m(a)m(y)h(to)h(de\014ne)e(a)h (corresp)s(onding)d(irre\015exiv)m(e)i(ordering)f Ft(\037)h Fu(is)g(to)h(let)g Fo(t)c Ft(\037)g Fo(u)382 2467 y Fu(hold)h(i\013)h Ft(j)p Fo(t\033)s Ft(j)c Fo(>)931 2481 y Fr(I)-12 b(N)1029 2467 y Ft(j)p Fo(u\033)s Ft(j)32 b Fu(for)f(all)f(ground)g (substitutions)f Fo(\033)s Fu(.)44 b(Ho)m(w)m(ev)m(er,)34 b(no)m(w)e Ft(\037)f Fu(is)f(not)382 2580 y(the)38 b(corresp)s(onding)f (strict)h(relation,)h(but)f(the)g(corresp)s(onding)f(stable-strict)h (rela-)382 2693 y(tion)31 b(of)g Fh(\045)p Fu(.)44 b(Th)m(us,)31 b(the)g(irre\015exiv)m(e)g(relation)f(in)m(tuitiv)m(ely)g(asso)s (ciated)h(with)g(a)g(quasi-)382 2806 y(ordering)24 b(is)g(often)i(the)f (stable-strict)g(one)h(instead)e(of)i(the)f(strict)g(one.)40 b(In)24 b(particular,)382 2919 y(if)h(the)h(quasi-ordering)e Fh(\045)h Fu(is)g(stable)h(under)e(substitutions,)h(then)h(the)g (corresp)s(onding)382 3032 y(stable-strict)36 b(relation)g Ft(\037)1300 2999 y Fn(ss)1405 3032 y Fu(is)g(stable)g(under)f (substitutions)f(to)s(o,)39 b(whereas)d(this)g(is)382 3145 y(not)31 b(necessarily)e(true)h(for)g(the)h(strict)f(relation)f Ft(\037)2127 3112 y Fn(s)2164 3145 y Fu(.)523 3258 y(F)-8 b(or)39 b(example,)i(if)d Ft(j)p Fo(a)p Ft(j)h Fu(=)g(0,)i Ft(j)p Fo(f)1606 3272 y Fr(1)1645 3258 y Fu(\()p Fo(t)p Fu(\))p Ft(j)f Fu(=)f Ft(j)p Fo(t)p Ft(j)p Fu(,)i(and)d Ft(j)p Fo(f)2327 3272 y Fr(2)2366 3258 y Fu(\()p Fo(t)p Fu(\))p Ft(j)i Fu(=)e(2)p Ft(j)p Fo(t)p Ft(j)i Fu(for)e(all)g(ground) 382 3371 y(terms)30 b Fo(t)p Fu(,)h(then)f(w)m(e)h(ha)m(v)m(e)h Fo(f)1320 3385 y Fr(2)1359 3371 y Fu(\()p Fo(x)p Fu(\))26 b Fh(\045)g Fo(f)1649 3385 y Fr(1)1688 3371 y Fu(\()p Fo(x)p Fu(\))31 b(and)f Fo(f)2063 3385 y Fr(1)2102 3371 y Fu(\()p Fo(x)p Fu(\))c Ft(6)p Fh(\045)f Fo(f)2391 3385 y Fr(2)2430 3371 y Fu(\()p Fo(x)p Fu(\).)42 b(Hence,)32 b(this)d(implies)382 3484 y Fo(f)427 3498 y Fr(2)466 3484 y Fu(\()p Fo(x)p Fu(\))50 b Ft(\037)709 3451 y Fn(s)796 3484 y Fo(f)841 3498 y Fr(1)880 3484 y Fu(\()p Fo(x)p Fu(\).)85 b(Ho)m(w)m(ev)m(er,)51 b Ft(\037)1593 3451 y Fn(s)1675 3484 y Fu(is)44 b(not)h(stable)g(under)f(substitutions)f(b) s(ecause)382 3597 y Fo(f)427 3611 y Fr(2)466 3597 y Fu(\()p Fo(a)p Fu(\))51 b Ft(\037)706 3564 y Fn(s)793 3597 y Fo(f)838 3611 y Fr(1)877 3597 y Fu(\()p Fo(a)p Fu(\))46 b(do)s(es)f(not)h(hold.)84 b(This)44 b(example)h(also)g(demonstrates)h (that)g(in)382 3709 y(general)d Ft(\037)779 3676 y Fn(s)816 3709 y Ft(\022\037)958 3676 y Fn(ss)1069 3709 y Fu(is)g(not)g(true)g(b) s(ecause)g(for)g(the)h(stable-strict)f(relation)f Ft(\037)3155 3676 y Fn(ss)3267 3709 y Fu(w)m(e)382 3822 y(ha)m(v)m(e)31 b Fo(f)636 3836 y Fr(2)675 3822 y Fu(\()p Fo(x)p Fu(\))26 b Ft(6\037)894 3789 y Fn(ss)989 3822 y Fo(f)1034 3836 y Fr(1)1073 3822 y Fu(\()p Fo(x)p Fu(\).)523 3935 y(Moreo)m(v)m(er,)g (in)21 b(general)h Ft(\037)1415 3902 y Fn(ss)1484 3935 y Ft(\022)p Fh(\045)g Fu(do)s(es)f(not)i(hold)d(either)i(\(hence,)i Ft(\037)2818 3902 y Fn(ss)2887 3935 y Ft(\022\037)3029 3902 y Fn(s)3088 3935 y Fu(is)d(false,)382 4048 y(to)s(o\).)41 b(If)27 b Ft(R)h Fu(is)f(the)h(TRS)f(con)m(taining)h(only)f(the)h(rule) e Fo(h)p Fu(\()p Fo(a)p Fu(\))h Ft(!)e Fo(a)j Fu(and)f Fh(\045)h Fu(is)f(de\014ned)f(as)382 4161 y Ft(!)473 4128 y Fz(\003)473 4188 y(R)537 4161 y Fu(,)31 b(then)f(w)m(e)h(ha)m(v) m(e)g Fo(h)p Fu(\()p Fo(x)p Fu(\))c Ft(\037)1415 4128 y Fn(ss)1509 4161 y Fo(x)p Fu(,)j(but)g Fo(h)p Fu(\()p Fo(x)p Fu(\))c Ft(6)p Fh(\045)f Fo(x)p Fu(.)523 4274 y(The)38 b(follo)m(wing)f(lemma)g(states)j(some)e(straigh)m(tforw)m (ard)g(prop)s(erties)f(of)i(stable-)382 4387 y(strict)29 b(relations,)f(where)h(in)e(the)i(follo)m(wing)f(w)m(e)h(alw)m(a)m(ys)h (assume)e(that)i(our)e(signature)382 4500 y(con)m(tains)i(at)i(least)e (one)h(constan)m(t)h(\(i.e.)e(that)h(there)g(exist)f(ground)f(terms\).) 382 4669 y Fk(Lemma)k(5)i(\(Prop)s(erties)g(of)g(Stable-Strict)f (Relations\))46 b Fs(L)-5 b(et)57 b Fh(\045)g Fs(b)-5 b(e)32 b(a)g(quasi-)382 4782 y(or)-5 b(dering)34 b(that)g(is)e(stable)i (under)f(substitutions.)42 b(Then)33 b(we)g(have)382 4951 y(\(i\))102 b Ft(\037)657 4918 y Fn(ss)759 4951 y Fs(is)33 b(irr)-5 b(e\015exive)382 5064 y(\(ii\))74 b Ft(\037)657 5031 y Fn(ss)759 5064 y Fs(is)33 b(tr)-5 b(ansitive)382 5177 y(\(iii\))46 b Ft(\037)657 5144 y Fn(ss)759 5177 y Fs(is)33 b(stable)g(under)g(substitutions)382 5290 y(\(iv\))60 b(if)33 b Ft(\037)746 5257 y Fn(s)815 5290 y Fs(is)f(stable)h(under)g(substitutions,)h(then)f Ft(\037)2253 5257 y Fn(s)2289 5290 y Ft(\022\037)2431 5257 y Fn(ss)382 5403 y Fs(\(v\))88 b(if)33 b Ft(\037)746 5370 y Fn(s)815 5403 y Fs(is)f(wel)5 b(l)33 b(founde)-5 b(d,)34 b(then)f Ft(\037)1736 5370 y Fn(ss)1838 5403 y Fs(is)f(wel)5 b(l)33 b(founde)-5 b(d,)34 b(to)-5 b(o)1854 5652 y Fu(6)p eop %%Page: 7 7 7 6 bop 382 548 a Fs(\(vi\))60 b Fo(s)25 b Fh(\045)g Fo(t)g Ft(\037)879 515 y Fn(ss)974 548 y Fo(u)32 b Fs(implies)i Fo(s)24 b Ft(\037)1508 515 y Fn(ss)1603 548 y Fo(u)382 661 y Fs(\(vii\))32 b Fo(s)25 b Ft(\037)725 628 y Fn(ss)820 661 y Fo(t)g Fh(\045)g Fo(u)32 b Fs(implies)i Fo(s)24 b Ft(\037)1508 628 y Fn(ss)1603 661 y Fo(u)p Fs(.)382 825 y(Pr)-5 b(o)g(of.)43 b Fu(The)30 b(conjectures)h(\(i\))g(and)f (\(ii\))g(follo)m(w)g(from)h(the)f(re\015exivit)m(y)g(and)h(the)f (tran-)382 938 y(sitivit)m(y)25 b(of)h Fh(\045)p Fu(.)39 b(Conjectures)26 b(\(iii\))e(and)h(\(iv\))h(are)h(direct)e (consequences)i(of)f(the)g(de\014ni-)382 1051 y(tion.)45 b(F)-8 b(or)33 b(\(v\),)g(ev)m(ery)g(p)s(oten)m(tial)f(in\014nite)d (descending)i(sequence)i Fo(t)2761 1065 y Fr(0)2828 1051 y Ft(\037)2899 1018 y Fn(ss)2996 1051 y Fo(t)3029 1065 y Fr(1)3096 1051 y Ft(\037)3167 1018 y Fn(ss)3265 1051 y Fo(:)15 b(:)g(:)382 1164 y Fu(w)m(ould)35 b(result)g(in)g(an)h (in\014nite)e(descending)h(sequence)h Fo(t)2351 1178 y Fr(0)2391 1164 y Fo(\033)i Ft(\037)2552 1131 y Fn(s)2623 1164 y Fo(t)2656 1178 y Fr(1)2695 1164 y Fo(\033)h Ft(\037)2857 1131 y Fn(s)2928 1164 y Fo(:)15 b(:)g(:)37 b Fu(Conjec-)382 1277 y(tures)30 b(\(vi\))g(and)g(\(vii\))f(follo)m(w)h(from)g(the)h (transitivit)m(y)e(and)g(stabilit)m(y)g(of)i Fh(\045)p Fu(.)305 b Fm(2)523 1449 y Fu(In)30 b(the)i(follo)m(wing,)e(instead)g (of)i(the)f(corresp)s(onding)e(strict)i(relations)f(w)m(e)i(alw)m(a)m (ys)382 1562 y(consider)f(the)i(corresp)s(onding)d(stable-strict)i (relations)g(of)g(quasi-orderings)f Fh(\045)p Fu(.)46 b(F)-8 b(or)382 1675 y(the)35 b(sak)m(e)i(of)e(brevit)m(y)-8 b(,)37 b(w)m(e)e(write)g Ft(\037)g Fu(instead)f(of)i Ft(\037)2169 1642 y Fn(ss)2238 1675 y Fu(,)h(i.e.)e(in)f(this)g(pap)s (er)g Ft(\037)h Fu(alw)m(a)m(ys)382 1788 y(denotes)e(the)g (stable-strict)g(relation)f(corresp)s(onding)f(to)j Fh(\045)p Fu(.)47 b(Analogously)-8 b(,)34 b(w)m(e)f(will)382 1900 y(call)c(a)h(quasi-ordering)d Fs(wel)5 b(l-founde)-5 b(d)40 b Fu(if)29 b(the)g(corresp)s(onding)f(stable-strict)h(relation) 382 2013 y(is)g(w)m(ell)h(founded.)523 2126 y(The)20 b(follo)m(wing)f(theorem)i(is)f(from)1705 2118 y([)1730 2126 y(Arts)h(and)f(Giesl,)i(1998)2519 2118 y(])2546 2126 y(,)h(where)d(instead)g(of)g(the)382 2239 y(strict)37 b(relation)g(corresp)s(onding)e(to)j(the)g(quasi-ordering)d(w)m(e)j(no) m(w)g(use)f(the)h(stable-)382 2352 y(strict)g(relation.)64 b(Note)40 b(that)f(the)g(presen)m(t)f(form)m(ulation)f(of)i(Thm.)f(6)g (with)g(stable-)382 2465 y(strict)33 b(relations)e(is)h(more)h(p)s(o)m (w)m(erful)f(than)g(the)h(form)m(ulation)f(with)g(strict)g(relations.) 382 2578 y(T)-8 b(o)32 b(use)g(the)g(strict)f(relation)h Ft(\037)1480 2545 y Fn(s)1548 2578 y Fu(of)g(a)g(quasi-ordering)e(in)h (Thm.)g(6,)h Ft(\037)2856 2545 y Fn(s)2925 2578 y Fu(w)m(ould)e(ha)m(v) m(e)382 2691 y(to)38 b(b)s(e)f(stable)h(under)e(substitutions;)j(cf.) 1876 2683 y([)1901 2691 y(Arts)f(and)f(Giesl,)i(1998,)i(Thm.)c(6)3107 2683 y(])3133 2691 y(.)62 b(But)382 2804 y(then)30 b(b)m(y)h(Lemma)g(5) g(\(iv\),)g Fo(s)25 b Ft(\037)1456 2771 y Fn(s)1518 2804 y Fo(t)31 b Fu(alw)m(a)m(ys)g(implies)d Fo(s)d Ft(\037)2321 2771 y Fn(ss)2416 2804 y Fo(t)p Fu(.)41 b(Hence,)32 b(all)e(constrain)m (ts)382 2917 y(satis\014ed)22 b(b)m(y)h Ft(\037)914 2884 y Fn(s)974 2917 y Fu(are)g(satis\014ed)f(b)m(y)h(the)h(corresp)s (onding)d(stable-strict)i(relation)f Ft(\037)3197 2884 y Fn(ss)3289 2917 y Fu(as)382 3030 y(w)m(ell.)39 b(Using)28 b(Lemma)g(5)h(\(vii\),)g(the)f(pro)s(of)g(for)g(the)h(if-part)f(of)h (this)e(sligh)m(tly)g(mo)s(di\014ed)382 3142 y(theorem)g(is)f(iden)m (tical)g(to)i(the)f(corresp)s(onding)e(one)i(in)2291 3134 y([)2316 3142 y(Arts)g(and)g(Giesl,)g(1998)3123 3134 y(])3150 3142 y(.)39 b(The)382 3255 y(pro)s(of)30 b(for)g(the)g(only-if-part)f(can)i(b)s(e)f(found)f(in)2047 3247 y([)2072 3255 y(Arts)h(and)g(Giesl,)g(1997c)2928 3247 y(])2955 3255 y(.)382 3440 y Fk(Theorem)k(6)h(\(Dep)s(endency)g(P) m(air)h(Approac)m(h\))46 b Fs(A)25 b(TRS)h Ft(R)g Fs(is)g(terminating)h (i\013)382 3553 y(for)35 b(e)-5 b(ach)35 b(cycle)f Ft(P)43 b Fs(in)34 b(the)h(estimate)-5 b(d)36 b(dep)-5 b(endency)36 b(gr)-5 b(aph)36 b(ther)-5 b(e)35 b(is)g(a)f(wel)5 b(l-founde)-5 b(d)382 3666 y(we)g(akly)34 b(monotonic)g(quasi-or)-5 b(dering)34 b Fh(\045)e Fs(stable)h(under)g(substitutions)h(such)e (that)429 3830 y Ft(\017)h Fo(l)27 b Fh(\045)e Fo(r)35 b Fs(for)f(al)5 b(l)33 b(rules)f Fo(l)c Ft(!)d Fo(r)35 b Fs(in)d Ft(R)p Fs(,)429 3943 y Ft(\017)h Fo(s)25 b Fh(\045)g Fo(t)32 b Fs(for)i(al)5 b(l)33 b(dep)-5 b(endency)34 b(p)-5 b(airs)34 b Ft(h)p Fo(s;)15 b(t)p Ft(i)33 b Fs(fr)-5 b(om)34 b Ft(P)7 b Fs(,)429 4056 y Ft(\017)33 b Fo(s)25 b Ft(\037)g Fo(t)32 b Fs(for)i(at)f(le)-5 b(ast)33 b(one)g(dep)-5 b(endency)34 b(p)-5 b(air)34 b Ft(h)p Fo(s;)15 b(t)p Ft(i)33 b Fs(fr)-5 b(om)34 b Ft(P)7 b Fs(.)523 4240 y Fu(Th)m(us,)42 b(to)f(pro)m(v)m(e)g(the)f(absence)h(of)g(in\014nite)c (c)m(hains)j(from)g(the)g(cycle)h Ft(f)p Fu(\(6\))p Ft(g)h Fu(w)m(e)382 4353 y(ha)m(v)m(e)31 b(to)h(\014nd)c(a)j(quasi-ordering)e (satisfying)1276 4530 y Fo(F)13 b Fu(\()p Fo(x;)i(y)s(;)g(s)p Fu(\()p Fo(z)t Fu(\)\))85 b Ft(\037)e Fo(F)13 b Fu(\(0)p Fo(;)i Fu(1)p Fo(;)g(z)t Fu(\))903 b(\(7\))1409 4667 y Fo(f)10 b Fu(\(0)p Fo(;)15 b Fu(1)p Fo(;)g(x)p Fu(\))85 b Fh(\045)e Fo(f)10 b Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(x;)g(x)p Fu(\))786 b(\(8\))1293 4805 y Fo(f)10 b Fu(\()p Fo(x;)15 b(y)s(;)g(s)p Fu(\()p Fo(z)t Fu(\)\))84 b Fh(\045)f Fo(s)p Fu(\()p Fo(f)10 b Fu(\(0)p Fo(;)15 b Fu(1)p Fo(;)g(z)t Fu(\)\))p Fo(:)781 b Fu(\(9\))382 5087 y Fv(4)135 b(DP-\(quasi\))45 b(simple)h(termination)382 5290 y Fu(As)31 b(men)m(tioned,)f(our)h(aim)f(is)g(to)h(use)g(standard) f(tec)m(hniques)g(to)i(generate)g(a)f(suitable)382 5403 y(quasi-ordering)c(satisfying)g(the)i(constrain)m(ts)g(of)g(Thm.)f(6.) 40 b(Ho)m(w)m(ev)m(er,)32 b(most)d(existing)1854 5652 y(7)p eop %%Page: 8 8 8 7 bop 382 548 a Fu(metho)s(ds)32 b(generate)j(orderings)d(whic)m(h)g (are)h Fs(str)-5 b(ongly)43 b Fu(monotonic,)34 b(whereas)f(for)g(the) 382 661 y(dep)s(endency)e(pair)h(approac)m(h)h(w)m(e)h(only)e(need)g(a) i Fs(we)-5 b(akly)42 b Fu(monotonic)33 b(ordering.)47 b(F)-8 b(or)382 774 y(that)31 b(reason,)g(b)s(efore)f(syn)m(thesizing)f (a)h(suitable)f(ordering,)h(some)g(of)h(the)f(argumen)m(ts)382 887 y(of)c(the)h(function)d(sym)m(b)s(ols)h(can)i(b)s(e)e(eliminated,)h (cf.)g(Arts)g(and)g(Giesl)2794 879 y([)2819 887 y(1997a;)j(1997c)3318 879 y(])3345 887 y(.)382 1000 y(F)-8 b(or)44 b(instance,)i(one)e(ma)m (y)f(eliminate)f(the)i(\014rst)e(t)m(w)m(o)j(argumen)m(ts)e(of)h(the)f (function)382 1112 y(sym)m(b)s(ol)38 b Fo(f)10 b Fu(.)69 b(Then)39 b(ev)m(ery)h(term)g Fo(f)10 b Fu(\()p Fo(t)1697 1126 y Fr(1)1736 1112 y Fo(;)15 b(t)1809 1126 y Fr(2)1849 1112 y Fo(;)g(t)1922 1126 y Fr(3)1962 1112 y Fu(\))40 b(in)e(the)i(inequalities)e(is)g(replaced)i(b)m(y)382 1225 y Fo(f)437 1192 y Fz(0)459 1225 y Fu(\()p Fo(t)527 1239 y Fr(3)567 1225 y Fu(\),)k(where)c Fo(f)999 1192 y Fz(0)1063 1225 y Fu(is)f(a)i(new)g(function)e(sym)m(b)s(ol.)71 b(So)40 b(instead)g(of)h(\(8\))h(and)e(\(9\))i(w)m(e)382 1338 y(w)m(ould)35 b(obtain)h(the)h(inequalities)d Fo(f)1644 1305 y Fz(0)1667 1338 y Fu(\()p Fo(x)p Fu(\))i Fh(\045)g Fo(f)1987 1305 y Fz(0)2009 1338 y Fu(\()p Fo(x)p Fu(\))i(and)e Fo(f)2407 1305 y Fz(0)2429 1338 y Fu(\()p Fo(s)p Fu(\()p Fo(z)t Fu(\)\))h Fh(\045)f Fo(s)p Fu(\()p Fo(f)2935 1305 y Fz(0)2957 1338 y Fu(\()p Fo(z)t Fu(\)\).)61 b(No)m(w)382 1451 y(the)36 b(resulting)e(constrain)m(ts)i(are)g(satis\014ed)f(b)m(y) h(the)g(recursiv)m(e)f(path)g(ordering)g(\(rp)s(o\))382 1564 y(with)c(the)h(precedence)h Fo(f)1266 1531 y Fz(0)1310 1564 y Fm(\003)21 b Fo(s)g Fm(\003)h Fu(0)f Fm(\003)h Fu(1.)46 b(Similarly)-8 b(,)30 b(\(b)m(y)j(eliminating)c(the)k(\014rst) e(t)m(w)m(o)382 1677 y(argumen)m(ts)i(of)f Fo(F)13 b Fu(\))33 b(one)g(can)f(also)h(pro)m(v)m(e)g(the)f(absence)h(of)g (in\014nite)d(c)m(hains)i(from)f(the)382 1790 y(cycle)k Ft(f)p Fu(\(5\))p Fo(;)15 b Fu(\(6\))p Ft(g)p Fu(.)57 b(Hence,)37 b(termination)c(of)i(the)g(TRS)e(consisting)h(of)h(the)f (rules)g(\(1\))382 1903 y(and)c(\(2\))h(is)f(pro)m(v)m(ed.)42 b(Note)31 b(that)g(this)f(TRS)f(is)h(not)h(simply)d(terminating.)40 b(So)30 b(in)f(the)382 2016 y(dep)s(endency)e(pair)g(approac)m(h,)i (simpli\014cation)c(orderings)h(lik)m(e)i(the)g(rp)s(o)f(can)i(b)s(e)e (used)382 2129 y(to)k(pro)m(v)m(e)g(termination)e(of)i(TRSs)e(where)h (their)f(direct)h(application)f(w)m(ould)g(fail.)523 2242 y(Apart)c(from)g(eliminating)d(argumen)m(ts)j(of)g(function)f(sym) m(b)s(ols,)h(another)g(p)s(ossibil-)382 2355 y(it)m(y)30 b(is)g(to)h(replace)f(functions)f(b)m(y)i(one)g(of)f(their)g(argumen)m (ts.)41 b(So)30 b(instead)g(of)h(deleting)382 2467 y(the)h(\014rst)g(t) m(w)m(o)i(argumen)m(ts)e(of)h Fo(f)10 b Fu(,)32 b(one)g(could)g (replace)g(all)f(terms)h Fo(f)10 b Fu(\()p Fo(t)2795 2481 y Fr(1)2834 2467 y Fo(;)15 b(t)2907 2481 y Fr(2)2947 2467 y Fo(;)g(t)3020 2481 y Fr(3)3060 2467 y Fu(\))32 b(b)m(y)g Fo(f)10 b Fu('s)382 2580 y(third)26 b(argumen)m(t)i Fo(t)1044 2594 y Fr(3)1083 2580 y Fu(.)40 b(Then)27 b(the)h(resulting)e (inequalities)g(are)i(again)g(satis\014ed)f(b)m(y)g(the)382 2693 y(rp)s(o.)38 b(T)-8 b(o)28 b(p)s(erform)d(this)h(elimination)e(of) j(argumen)m(ts)g(resp.)f(of)h(function)e(sym)m(b)s(ols)g(the)382 2806 y(follo)m(wing)k(concept)i(w)m(as)g(in)m(tro)s(duced)e(in)1835 2798 y([)1860 2806 y(Arts)h(and)g(Giesl,)g(1997c)2716 2798 y(])2743 2806 y(.)382 3011 y Fk(De\014nition)35 b(7)g(\(AFS\))45 b Fs(A)n(n)g Fu(argumen)m(t)h(\014ltering)d(system) 2495 2978 y Fr(1)2580 3011 y Fs(\(AFS\))k(over)f Ft(F)55 b Fs(is)46 b(a)382 3124 y(TRS)33 b(whose)h(r)-5 b(ewrite)33 b(rules)g(ar)-5 b(e)34 b(of)f(the)g(form)1525 3320 y Fo(f)10 b Fu(\()p Fo(x)1667 3334 y Fr(1)1706 3320 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)1960 3334 y Fn(n)2007 3320 y Fu(\))26 b Ft(!)f Fo(r)382 3516 y Fs(with)44 b Fo(f)55 b Ft(2)45 b(F)53 b Fs(and)44 b(ther)-5 b(e)45 b(is)e(at)i(most)f(one)g(such)g (rule)g(for)g(every)f Fo(f)55 b Ft(2)45 b(F)9 b Fs(.)75 b(Her)-5 b(e)382 3629 y Fo(x)434 3643 y Fr(1)473 3629 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)727 3643 y Fn(n)798 3629 y Fs(ar)-5 b(e)24 b(p)-5 b(airwise)25 b(distinct)f(variables)g(and)h Fo(r)g Fs(is)f(either)g(one)f(of)h(these)g(variables)382 3742 y(or)35 b(it)g(is)g(a)g(term)g Fo(f)1049 3709 y Fz(0)1072 3742 y Fu(\()p Fo(y)1152 3756 y Fr(1)1191 3742 y Fo(;)15 b(:)g(:)g(:)i(;)e(y)1438 3756 y Fn(m)1504 3742 y Fu(\))p Fs(,)36 b(wher)-5 b(e)36 b Fo(f)1917 3709 y Fz(0)1968 3742 y Ft(62)29 b(F)44 b Fs(is)35 b(a)g(fr)-5 b(esh)36 b(function)e(symb)-5 b(ol)37 b(and)382 3855 y Fo(y)427 3869 y Fr(1)466 3855 y Fo(;)15 b(:)g(:)g(:)i(;)e(y)713 3869 y Fn(m)812 3855 y Fs(ar)-5 b(e)33 b(p)-5 b(airwise)34 b(distinct)g(variables)f(out)g(of)g Fo(x)2340 3869 y Fr(1)2379 3855 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)2633 3869 y Fn(n)2680 3855 y Fs(.)523 4060 y Fu(As)36 b(pro)m(v)m(ed)g(in)1076 4052 y([)1101 4060 y(Arts)g(and)g(Giesl,)h(1997c)1976 4052 y(])2003 4060 y(,)h(in)c(order)i(to)h(\014nd)d(a)j(quasi-ordering) 382 4173 y(satisfying)28 b(a)j(particular)d(set)i(of)g(inequalities,)e (one)i(ma)m(y)h(\014rst)e(normalize)g(the)h(terms)382 4286 y(in)25 b(the)i(inequalities)d(with)h(resp)s(ect)i(to)g(an)g(AFS)g (\(where)f(the)h(AFS)f(ma)m(y)i(also)e(con)m(tain)382 4399 y(rules)37 b(for)i(the)g(tuple)f(sym)m(b)s(ols\).)65 b(Subsequen)m(tly)-8 b(,)40 b(one)f(only)f(has)h(to)h(\014nd)d(a)i (quasi-)382 4512 y(ordering)28 b(that)j(satis\014es)e(these)i(mo)s (di\014ed)c(inequalities.)38 b(Hence,)31 b(b)m(y)f(com)m(bining)e(the) 382 4624 y(syn)m(thesis)34 b(of)g(a)h(suitable)e(AFS)i(with)e(w)m (ell-kno)m(wn)h(tec)m(hniques)g(for)g(the)h(generation)382 4737 y(of)26 b(\()p Fs(str)-5 b(ongly)35 b Fu(monotonic\))26 b(simpli\014cation)c(orderings,)k(no)m(w)f(the)h(searc)m(h)g(for)g(a)f Fs(we)-5 b(akly)382 4850 y Fu(monotonic)30 b(ordering)f(satisfying)g (the)i(constrain)m(ts)f(can)h(b)s(e)f(automated.)523 4963 y(In)k(this)f(pap)s(er,)i(w)m(e)h(imp)s(ose)d(a)i(\(minor\))f (restriction)2381 4930 y Fr(2)2454 4963 y Fu(on)g(the)h(AFSs)g(used,)g (viz.)382 5076 y(w)m(e)c(restrict)f(ourselv)m(es)g(to)h(AFSs)f Ft(A)g Fu(suc)m(h)g(that)p 382 5135 1196 4 v 486 5189 a Fl(1)520 5220 y Fp(AFSs)25 b(are)i(a)f(sp)r(ecial)h(form)e(of)i (recursiv)n(e)e(program)h(sc)n(hemes)2359 5214 y([)2380 5220 y(Courcelle,)i(1990;)g(Klop,)e(1992)3301 5214 y(])3323 5220 y(.)486 5280 y Fl(2)520 5312 y Fp(This)e(restriction)f(is)h(not)e (v)n(ery)g(sev)n(ere.)34 b(If)23 b(there)f(exists)i(a)f (quasi-simpli\014cation)g(ordering)g(satis-)382 5403 y(fying)c(the)f(constrain)n(ts)h(in)f(Thm.)g(6)h(and)f(if)h(these)g (constrain)n(ts)g(include)f(at)h(least)g(one)g(strict)g(inequalit)n(y) 1854 5652 y Fu(8)p eop %%Page: 9 9 9 8 bop 382 548 a Ft(\017)31 b Fg(V)7 b Ff(ar)r Fx(\()p Ff(r)26 b Fg(#)736 560 y Fe(A)793 548 y Fx(\))d Fg(\022)g(V)7 b Ff(ar)r Fx(\()p Ff(l)25 b Fg(#)1201 560 y Fe(A)1258 548 y Fx(\))j(and)g Ff(l)c Fg(#)1571 560 y Fe(A)1628 548 y Fg(62)f(V)35 b Fx(or)27 b Ff(l)d Fg(#)1985 560 y Fe(A)2042 548 y Fx(=)f Ff(r)i Fg(#)2234 560 y Fe(A)2319 548 y Fx(for)i(all)g(rules)g Ff(l)e Fg(!)e Ff(r)30 b Fx(in)e Fg(R)382 661 y Ft(\017)j Fg(V)7 b Ff(ar)r Fx(\()p Ff(t)23 b Fg(#)726 673 y Fe(A)783 661 y Fx(\))h Fg(\022)e(V)7 b Ff(ar)r Fx(\()p Ff(s)24 b Fg(#)1204 673 y Fe(A)1261 661 y Fx(\))k(and)f Ff(s)c Fg(#)1586 673 y Fe(A)1643 661 y Fg(62)h(V)34 b Fx(or)27 b Ff(s)c Fg(#)2013 673 y Fe(A)2070 661 y Fx(=)f Ff(t)h Fg(#)2252 673 y Fe(A)2337 661 y Fx(for)k(all)g Fg(h)p Ff(s;)14 b(t)p Fg(i)28 b Fx(in)g(cycles)523 823 y Fu(As)38 b(already)f(men)m(tioned,)i(most)f (metho)s(ds)f(for)h(the)g(automated)g(generation)g(of)382 936 y(w)m(ell-founded)d(orderings)h(construct)i(simpli\014cation)c (orderings)i(or)i(quasi-simpli\014-)382 1049 y(cation)d(orderings)1059 1041 y([)1084 1049 y(Dersho)m(witz,)i(1987;)f(Stein)m(bac)m(h,)h(1995;) f(Middeldorp)d(and)h(Zan-)382 1162 y(tema,)h(1997)818 1154 y(])845 1162 y(.)49 b(Here)34 b(w)m(e)g(use)f(the)g(follo)m(wing)f (de\014nition)f(of)2493 1154 y([)2519 1162 y(Middeldorp)f(and)j(Zan-) 382 1275 y(tema,)e(1997)814 1267 y(])841 1275 y(:)41 b(A)30 b Fs(simpli\014c)-5 b(ation)35 b(or)-5 b(dering)32 b Fu(is)d(an)h(ordering)f(\(i.e.)i(an)f(irre\015exiv)m(e)f(and)382 1388 y(transitiv)m(e)h(relation\))h(that)g(is)f(monotonic)h(\(closed)h (under)d(con)m(texts\),)k(closed)e(under)382 1501 y(substitutions,)25 b(and)h(p)s(ossesses)g(the)h(subterm)e(prop)s(ert)m(y)-8 b(.)39 b(It)27 b(is)e(a)i(w)m(ell-kno)m(wn)f(conse-)382 1614 y(quence)34 b(of)f(Krusk)-5 b(al's)32 b(theorem)i(that)g(ev)m(ery) g(simpli\014cation)d(ordering)h(on)h Ft(T)23 b Fu(\()p Ft(F)9 b Fo(;)15 b Ft(V)7 b Fu(\))382 1727 y(is)29 b(w)m(ell)h(founded) f(pro)m(vided)g(that)i Ft(F)39 b Fu(is)30 b(\014nite.)1999 1694 y Fr(3)523 1840 y Fu(Analogously)-8 b(,)52 b(a)c Fs(quasi-simpli\014c)-5 b(ation)51 b(or)-5 b(dering)49 b Fu(\(qso\))g(is)d(a)j(quasi-ordering)382 1952 y(whic)m(h)23 b(is)g(\(w)m(eakly\))i(monotonic,)h(closed)e(under)e(substitutions,)h (and)h(has)g(the)g(\(w)m(eak\))382 2065 y(subterm)43 b(prop)s(ert)m(y)-8 b(.)82 b(Since)43 b(w)m(e)i(restrict)f(ourselv)m (es)g(to)h(\014nite)e(signatures,)k(ev)m(ery)382 2178 y(quasi-simpli\014cation)21 b(ordering)j(\(more)h(precisely)-8 b(,)26 b(the)f(corresp)s(onding)e(stable-strict)382 2291 y(relation\))30 b(is)f(w)m(ell)h(founded,)f(to)s(o.)523 2404 y(Examples)40 b(of)i(simpli\014cation)c(orderings)i(and)g(qso's)i (include)d(path)i(orderings)382 2517 y(lik)m(e)33 b(the)g(rp)s(o,)h (the)f(lexicographic)g(path)g(ordering)f(\(lp)s(o\),)i(etc.)2638 2509 y([)2663 2517 y(Dersho)m(witz,)h(1987;)382 2630 y(Stein)m(bac)m(h,)h(1995)1007 2622 y(])1035 2630 y(.)55 b(P)m(olynomial)34 b(orderings,)h(ho)m(w)m(ev)m(er,)j(are)e(not)g (qso's)f(in)f(general.)382 2743 y(F)-8 b(or)28 b(instance,)h(if)d(the)i (constan)m(t)h(0)f(is)f(asso)s(ciated)h(with)f(the)h(n)m(um)m(b)s(er)e (0,)j Fo(s)p Fu(\()p Fo(x)p Fu(\))f(is)f(asso-)382 2856 y(ciated)i(with)e Fo(x)17 b Fu(+)g(1,)30 b(and)e Fo(f)10 b Fu(\()p Fo(x;)15 b(y)s Fu(\))29 b(is)e(asso)s(ciated)j(with)d(the)i (m)m(ultiplication)d(of)j Fo(x)f Fu(and)382 2969 y Fo(y)s Fu(,)i(then)g(this)e(p)s(olynomial)g(ordering)g(do)s(es)i(not)g (satisfy)f(the)h(subterm)f(prop)s(ert)m(y)h(\(for)382 3082 y(example,)39 b Fo(f)10 b Fu(\()p Fo(s)p Fu(\(0\))p Fo(;)15 b Fu(0\))38 b Fh(\045)e Fo(s)p Fu(\(0\))i(do)s(es)f(not)g (hold\).)60 b(Ho)m(w)m(ev)m(er,)41 b(the)c(follo)m(wing)f(lemma)382 3194 y(sho)m(ws)d(that)h(if)e(the)h(p)s(olynomial)e(ordering)g(resp)s (ects)i(some)h(restrictions,)f(then)g(it)g(is)382 3307 y(indeed)c(a)i(qso.)382 3490 y Fk(Lemma)i(8)i(\(P)m(olynomial)g (Orderings)g(as)g(qso's\))46 b Fs(L)-5 b(et)33 b Fh(\045)f Fs(b)-5 b(e)33 b(a)g(p)-5 b(olynomial)35 b(or-)382 3603 y(dering)c(wher)-5 b(e)31 b(every)g(function)f(symb)-5 b(ol)32 b(is)f(asso)-5 b(ciate)g(d)33 b(with)e(a)g(p)-5 b(olynomial)34 b(c)-5 b(ontain-)382 3716 y(ing)32 b(only)i(non-ne)-5 b(gative)33 b(c)-5 b(o)g(e\016cients.)518 3878 y Ft(\017)46 b Fs(If)32 b(every)h(function)f(symb)-5 b(ol)34 b Fo(f)10 b Fu(\()p Fo(x)1739 3892 y Fr(1)1778 3878 y Fo(;)15 b(:)g(:)g(:)i(;)e (x)2032 3892 y Fn(n)2079 3878 y Fu(\))33 b Fs(is)f(asso)-5 b(ciate)g(d)35 b(with)f(a)f(p)-5 b(olynomial)609 3991 y(c)g(ontaining)37 b(al)5 b(l)36 b(variables)h Fo(x)1622 4005 y Fr(1)1661 3991 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)1915 4005 y Fn(n)1998 3991 y Fs(and)36 b(if)f(every)h(c)-5 b(onstant)37 b(is)f(asso)-5 b(ciate)g(d)609 4104 y(with)34 b(a)f(numb)-5 b(er)33 b Fo(>)25 b Fu(0)p Fs(,)33 b(then)g Fh(\045)f Fs(is)g(a)h(qso.)518 4281 y Ft(\017)46 b Fs(If)32 b(every)h(function)f(symb)-5 b(ol)34 b Fo(f)10 b Fu(\()p Fo(x)1739 4295 y Fr(1)1778 4281 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)2032 4295 y Fn(n)2079 4281 y Fu(\))33 b Fs(is)f(asso)-5 b(ciate)g(d)35 b(with)f(a)f(p)-5 b(olynomial)609 4394 y(which)26 b(c)-5 b(ontains)27 b(a)f(\(non-mixe)-5 b(d\))27 b(monomial)g(of)f(the)g(form) g Fo(m)15 b(x)2809 4361 y Fn(k)2809 4420 y(i)2877 4394 y Fs(\(with)26 b Fo(m;)15 b(k)29 b Ft(\025)609 4507 y Fu(1)p Fs(\))k(for)h(every)e Fo(i)26 b Fu(=)f(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(n)p Fs(,)32 b(then)h Fh(\045)f Fs(is)h(a)g(qso.)382 4690 y(Pr)-5 b(o)g(of.)42 b Fu(Straigh)m(tforw)m(ard.)1991 b Fm(2)523 4862 y Fu(In)33 b(fact,)i(whenev)m(er)e(p)s(olynomial)e (orderings)h(can)h(b)s(e)g(used)f(in)g(connection)i(with)382 4975 y(the)27 b(dep)s(endency)e(pair)h(approac)m(h,)i(one)f(can)g (usually)d(apply)i(a)h(p)s(olynomial)d(ordering)p 382 5044 1196 4 v 382 5129 a Fp(with)37 b(v)l(ariables)g(in)g(its)g(righ)n (t-hand)f(side,)k(then)c Fj(V)6 b Fi(ar)r Fp(\()p Fi(r)41 b Fj(#)2155 5137 y Fq(A)2209 5129 y Fp(\))f Fj(\022)f(V)6 b Fi(ar)r Fp(\()p Fi(l)41 b Fj(#)2641 5137 y Fq(A)2695 5129 y Fp(\))36 b(and)g Fj(V)6 b Fi(ar)r Fp(\()p Fi(t)40 b Fj(#)3188 5137 y Fq(A)3241 5129 y Fp(\))g Fj(\022)382 5220 y(V)6 b Fi(ar)r Fp(\()p Fi(s)26 b Fj(#)643 5228 y Fq(A)697 5220 y Fp(\))j(are)g(alw)n(a)n(ys)h(satis\014ed,)h(b)r (ecause)e(otherwise)h(the)f(constrain)n(ts)h(w)n(ould)f(imply)e Fi(t)g Fj(\037)f Fi(x)j Fp(for)382 5312 y(some)c(term)g Fi(t)g Fp(with)h Fi(x)21 b Fj(62)h(V)6 b Fi(ar)r Fp(\()p Fi(t)p Fp(\).)486 5371 y Fl(3)520 5403 y Fp(F)-6 b(or)26 b(details)h(on)e(in\014nite)g(signatures)i(see)1756 5397 y([)1778 5403 y(Middeldorp)f(and)f(Zan)n(tema,)g(1997)2830 5397 y(])2853 5403 y(.)1854 5652 y Fu(9)p eop %%Page: 10 10 10 9 bop 382 548 a Fu(whic)m(h)24 b(satis\014es)h(one)h(of)f(the)h(ab)s (o)m(v)m(e)h(conditions.)37 b(By)26 b(restricting)f(ourselv)m(es)g(to)h (qso's,)382 661 y(w)m(e)32 b(obtain)g(the)g(follo)m(wing)e(restricted)i (notion)g(of)g(termination.)44 b(Again,)33 b Ft(\037)e Fu(denotes)382 774 y(the)f(stable-strict)h(relation)e(corresp)s(onding) g(to)i Fh(\045)p Fu(.)382 982 y Fk(De\014nition)k(9)g(\(DP-quasi)h (simple)e(termination\))44 b Fs(A)51 b(TRS)i Ft(R)f Fs(is)g Fu(DP-quasi)382 1095 y(simply)41 b(terminating)j Fs(i\013)g(for)h (every)f(non-empty)i(cycle)e Ft(P)52 b Fs(in)44 b(the)h(estimate)-5 b(d)46 b(de-)382 1208 y(p)-5 b(endency)34 b(gr)-5 b(aph)34 b(ther)-5 b(e)33 b(exists)g(an)g(AFS)g Ft(A)f Fs(and)i(a)f(qso)g Fh(\045)f Fs(such)h(that)429 1392 y(\(a\))g Fo(l)28 b Ft(#)682 1406 y Fz(A)758 1392 y Fh(\045)d Fo(r)j Ft(#)968 1406 y Fz(A)1061 1392 y Fs(for)33 b(al)5 b(l)33 b(rules)g Fo(l)27 b Ft(!)f Fo(r)35 b Fs(in)d Ft(R)p Fs(,)429 1505 y(\(b\))37 b Fo(s)25 b Ft(#)695 1519 y Fz(A)772 1505 y Fh(\045)f Fo(t)i Ft(#)971 1519 y Fz(A)1064 1505 y Fs(for)33 b(al)5 b(l)33 b(dep)-5 b(endency)34 b(p)-5 b(airs)35 b Ft(h)p Fo(s;)15 b(t)p Ft(i)33 b Fs(fr)-5 b(om)33 b Ft(P)7 b Fs(,)429 1618 y(\(c\))37 b Fo(s)25 b Ft(#)695 1632 y Fz(A)772 1618 y Ft(\037)f Fo(t)i Ft(#)971 1632 y Fz(A)1064 1618 y Fs(for)33 b(at)g(le)-5 b(ast)34 b(one)f(dep)-5 b(endency)34 b(p)-5 b(air)34 b Ft(h)p Fo(s;)15 b(t)p Ft(i)33 b Fs(fr)-5 b(om)34 b Ft(P)7 b Fs(.)523 1827 y Fu(De\014nition)36 b(9)i(captures)f(all)f(TRSs)h(where)f(an)i (automated)g(termination)e(pro)s(of)382 1940 y(using)c(dep)s(endency)h (pairs)f(is)h(p)s(oten)m(tially)g(feasible.)50 b(In)33 b(fact,)j(there)e(are)h(n)m(umerous)382 2053 y(DP-quasi)j(simply)e (terminating)i(TRSs)f(whic)m(h)g(are)i(not)g(simply)d(terminating;)42 b(cf.)382 2166 y(e.g.)d(the)g(collection)e(in)1246 2158 y([)1271 2166 y(Arts)h(and)g(Giesl,)h(1997c)2152 2158 y(])2179 2166 y(.)64 b(This)36 b(observ)-5 b(ation)38 b(motiv)-5 b(ated)382 2278 y(the)28 b(dev)m(elopmen)m(t)f(of)h(the)g (dep)s(endency)e(pair)g(approac)m(h)h(and)g(it)g(also)h(motiv)-5 b(ated)27 b(the)382 2391 y(presen)m(t)43 b(w)m(ork,)j(as)d(our)f(aim)g (is)g(to)h(extend)g(w)m(ell-kno)m(wn)f(mo)s(dularit)m(y)e(results)i (for)382 2504 y(simple)28 b(termination)h(to)j(DP-quasi)e(simple)e (termination.)523 2617 y(A)c(straigh)m(tforw)m(ard)g(w)m(a)m(y)i(to)f (generate)g(a)g(qso)g Ft(\027)e Fu(from)h(a)h(simpli\014cation)c (ordering)382 2730 y Ft(\037)30 b Fu(is)g(to)i(de\014ne)e Fo(t)c Ft(\027)g Fo(u)31 b Fu(i\013)f Fo(t)c Ft(\037)g Fo(u)31 b Fu(or)g Fo(t)26 b Fu(=)g Fo(u)p Fu(,)31 b(where)f(=)h(is)f (syn)m(tactic)i(equalit)m(y)-8 b(.)42 b(In)30 b(the)382 2843 y(follo)m(wing,)d(w)m(e)i(denote)g(the)f(re\015exiv)m(e)g(closure) g(of)g(a)h(relation)e(b)m(y)h(underlining,)d(i.e.)j Ft(\027)382 2956 y Fu(denotes)35 b(the)h(re\015exiv)m(e)f(closure)f(of)h Ft(\037)p Fu(.)55 b(By)35 b(restricting)f(ourselv)m(es)h(to)h(this)e (class)g(of)382 3069 y(qso's,)d(w)m(e)g(obtain)f(the)g(notion)g(of)g (DP-simple)f(termination.)382 3277 y Fk(De\014nition)35 b(10)g(\(DP-simple)g(termination\))44 b Fs(A)25 b(TRS)i Ft(R)f Fs(is)g Fu(DP-simply)21 b(termi-)382 3390 y(nating)39 b Fs(i\013)e(for)i(every)e(non-empty)i(cycle)f Ft(P)46 b Fs(in)38 b(the)g(estimate)-5 b(d)40 b(dep)-5 b(endency)39 b(gr)-5 b(aph)382 3503 y(ther)g(e)33 b(is)g(an)g(AFS)g Ft(A)f Fs(and)h(a)g(simpli\014c)-5 b(ation)35 b(or)-5 b(dering)34 b Ft(\037)f Fs(such)f(that)429 3688 y(\(a\))h Fo(l)28 b Ft(#)682 3702 y Fz(A)758 3688 y Ft(\027)d Fo(r)j Ft(#)968 3702 y Fz(A)1061 3688 y Fs(for)33 b(al)5 b(l)33 b(rules)g Fo(l)27 b Ft(!)f Fo(r)35 b Fs(in)d Ft(R)p Fs(,)429 3800 y(\(b\))37 b Fo(s)25 b Ft(#)695 3814 y Fz(A)772 3800 y Ft(\027)f Fo(t)i Ft(#)971 3814 y Fz(A)1064 3800 y Fs(for)33 b(al)5 b(l)33 b(dep)-5 b(endency)34 b(p)-5 b(airs)35 b Ft(h)p Fo(s;)15 b(t)p Ft(i)33 b Fs(fr)-5 b(om)33 b Ft(P)7 b Fs(,)429 3913 y(\(c\))37 b Fo(s)25 b Ft(#)695 3927 y Fz(A)772 3913 y Ft(\037)f Fo(t)i Ft(#)971 3927 y Fz(A)1064 3913 y Fs(for)33 b(at)g(le)-5 b(ast)34 b(one)f(dep)-5 b(endency)34 b(p)-5 b(air)34 b Ft(h)p Fo(s;)15 b(t)p Ft(i)33 b Fs(fr)-5 b(om)34 b Ft(P)7 b Fs(.)382 4122 y Fu(Note)32 b(that)f(\(a\))g(-)g(\(c\))g(are)g(equiv)-5 b(alen)m(t)30 b(to)h(simple)d(termination)h(of)i(the)f(TRS)400 4319 y Ft(S)455 4333 y Fz(P)562 4319 y Fu(=)47 b Ft(f)p Fo(l)28 b Ft(#)825 4333 y Fz(A)886 4319 y Ft(!)d Fo(r)j Ft(#)1116 4333 y Fz(A)1202 4319 y Ft(j)51 b Fo(l)27 b Ft(!)e Fo(r)j Ft(2)d(R)30 b Fu(and)g Fo(l)d Ft(#)1986 4333 y Fz(A)2072 4319 y Ft(6)p Fu(=)e Fo(r)j Ft(#)2282 4333 y Fz(A)2343 4319 y Ft(g)e([)680 4432 y(f)p Fo(s)f Ft(#)838 4446 y Fz(A)899 4432 y Ft(!)g Fo(t)h Ft(#)1119 4446 y Fz(A)1205 4432 y Ft(j)51 b(h)p Fo(s;)15 b(t)p Ft(i)30 b Fu(is)g(a)h(dep)s(endency)e(pair)g(from)h Ft(P)38 b Fu(and)29 b Fo(s)c Ft(#)2952 4446 y Fz(A)3038 4432 y Ft(6)p Fu(=)g Fo(t)g Ft(#)3237 4446 y Fz(A)3299 4432 y Ft(g)382 4630 y Fu(pro)m(vided)f(that)j Fo(s)e Ft(#)1057 4644 y Fz(A)1143 4630 y Ft(6)p Fu(=)g Fo(t)g Ft(#)1342 4644 y Fz(A)1429 4630 y Fu(holds)f(for)i(at)g(least)h(one)f(dep)s (endency)e(pair)h Ft(h)p Fo(s;)15 b(t)p Ft(i)41 b(2)f(P)7 b Fu(.)523 4743 y(It)27 b(turns)e(out)i(that)h(most)f(of)g(the)g (examples)f(in)2166 4735 y([)2192 4743 y(Arts)g(and)g(Giesl,)h(1997c) 3037 4735 y(])3091 4743 y(are)g(not)382 4856 y(only)39 b(DP-quasi)g(simply)e(terminating)i(but)g(ev)m(en)h(DP-simply)e (terminating.)67 b(The)382 4969 y(follo)m(wing)29 b(lemma)h (illustrates)e(the)j(connections)f(b)s(et)m(w)m(een)h(the)g(di\013eren) m(t)f(notions.)382 5177 y Fk(Lemma)j(11)i(\(Characterizing)g (DP-\(quasi\))g(simple)f(termination\))382 5290 y Fs(simple)e (termination)i Ft(\))d Fs(DP-simple)g(termination)j Ft(\))d Fs(DP-quasi)g(simple)h(termina-)382 5403 y(tion)h Ft(\))g Fs(termination)1831 5652 y Fu(10)p eop %%Page: 11 11 11 10 bop 382 548 a Fs(Pr)-5 b(o)g(of.)42 b Fu(The)30 b(second)h(implication)d(holds)h(as)i Ft(\037)f Fu(is)g(stable)g(under) f(substitutions)f(and)382 661 y(therefore)h Ft(\037)g Fu(is)f(con)m(tained)h(in)f(the)h(stable-strict)g(relation)f(of)h Ft(\027)p Fu(,)h(cf.)f(Lemma)g(5)h(\(iv\).)382 774 y(The)40 b(last)g(implication)e(follo)m(ws)h(from)h(Thm.)f(6)i(b)m(y)f(using)f (the)h(quasi-ordering)f Ft(\027)3348 741 y Fz(0)382 887 y Fu(where)30 b Fo(u)25 b Ft(\027)793 854 y Fz(0)841 887 y Fo(v)34 b Fu(holds)29 b(i\013)g Fo(u)d Ft(#)1388 901 y Fz(A)1449 887 y Ft(\027)f Fo(v)j Ft(#)1662 901 y Fz(A)1723 887 y Fu(.)523 1000 y(It)i(remains)f(to)h(sho)m(w)g(the)g (\014rst)f(implication.)38 b(Let)30 b Ft(R)g Fu(b)s(e)f(a)h(simply)e (terminating)382 1112 y(TRS)c(o)m(v)m(er)i(the)f(signature)f Ft(F)34 b Fu(=)25 b Ft(C)14 b([)9 b(D)27 b Fu(and)d(let)h Ft(T)e Fo(up)2211 1126 y Fz(F)2297 1112 y Fu(=)i Ft(f)p Fo(F)38 b Ft(j)25 b Fo(f)35 b Ft(2)24 b(D)s(g)h Fu(b)s(e)f(the)h(set)g (of)382 1225 y(tuple)i(sym)m(b)s(ols.)38 b(If)28 b Ft(R)g Fu(is)f(simply)e(terminating,)j(then)g(there)g(exists)g(a)g (simpli\014cation)382 1338 y(ordering)h Ft(\037)h Fu(suc)m(h)g(that)h Fo(l)c Ft(\037)e Fo(r)33 b Fu(holds)c(for)h(all)f(rules)g Fo(l)e Ft(!)f Fo(r)32 b Fu(of)f Ft(R)p Fu(.)523 1451 y(Let)24 b(\012)g(b)s(e)f(the)h(function)e(whic)m(h)h(in)f(a)i(term)g Fo(s)h Ft(2)g(T)e Fu(\()p Ft(F)16 b([)7 b(T)23 b Fo(up)2600 1465 y Fz(F)2661 1451 y Fo(;)15 b Ft(V)7 b Fu(\))24 b(replaces)g(ev)m (ery)382 1564 y(tuple)j(sym)m(b)s(ol)g Fo(F)41 b Fu(with)27 b(its)h(corresp)s(onding)e(function)h(sym)m(b)s(ol)g Fo(f)35 b Ft(2)25 b(F)9 b Fu(.)40 b(Then)27 b Ft(\037)h Fu(can)382 1677 y(b)s(e)h(extended)i(to)g(a)f(simpli\014cation)d (ordering)i Ft(\037)2069 1644 y Fz(0)2122 1677 y Fu(on)h Ft(T)23 b Fu(\()p Ft(F)29 b([)20 b(T)i Fo(up)2701 1691 y Fz(F)2762 1677 y Fo(;)15 b Ft(V)7 b Fu(\))31 b(b)m(y)g(de\014ning)382 1790 y Fo(t)g Ft(\037)517 1757 y Fz(0)572 1790 y Fo(u)j Fu(i\013)g(\012\()p Fo(t)p Fu(\))d Ft(\037)h Fu(\012\()p Fo(u)p Fu(\))i(holds.)51 b(W)-8 b(e)35 b(claim)f(that)g(the)h (simpli\014cation)c(ordering)i Ft(\037)3348 1757 y Fz(0)382 1903 y Fu(satis\014es)d(the)g(constrain)m(ts)h(\(a\))g(-)g(\(c\))g(of)g (Def.)g(10)g(without)e(applying)g(an)h(AFS.)523 2016 y(Ob)m(viously)-8 b(,)37 b Fo(l)g Ft(\037)1111 1983 y Fz(0)1169 2016 y Fo(r)i Fu(holds)c(for)i(all)e(rules)g Fo(l)j Ft(!)d Fo(r)k Fu(of)e Ft(R)p Fu(.)59 b(Th)m(us)35 b Ft(\037)2844 1983 y Fz(0)2903 2016 y Fu(satis\014es)h(the)382 2129 y(constrain)m(t)43 b(\(a\).)78 b(Moreo)m(v)m(er,)49 b(for)42 b(ev)m(ery)h(dep)s(endency)f(pair)f Ft(h)p Fo(s;)15 b(t)p Ft(i)43 b Fu(w)m(e)g(ha)m(v)m(e)h Fo(s)i Ft(\037)3348 2096 y Fz(0)382 2242 y Fo(t)p Fu(.)71 b(The)40 b(reason)h(is)f(that)h (eac)m(h)h(dep)s(endency)d(pair)g Ft(h)p Fo(F)13 b Fu(\()p Fo(s)2412 2256 y Fr(1)2452 2242 y Fo(;)i(:)g(:)g(:)i(;)e(s)2697 2256 y Fn(n)2744 2242 y Fu(\))p Fo(;)g(G)p Fu(\()p Fo(t)2959 2256 y Fr(1)2999 2242 y Fo(;)g(:)g(:)g(:)h(;)f(t)3233 2256 y Fn(m)3300 2242 y Fu(\))p Ft(i)382 2355 y Fu(originates)28 b(from)g(a)h(rule)e Fo(f)10 b Fu(\()p Fo(s)1395 2369 y Fr(1)1434 2355 y Fo(;)15 b(:)g(:)g(:)h(;)f(s)1678 2369 y Fn(n)1725 2355 y Fu(\))26 b Ft(!)f Fo(C)7 b Fu([)p Fo(g)s Fu(\()p Fo(t)2113 2369 y Fr(1)2153 2355 y Fo(;)15 b(:)g(:)g(:)i(;)e(t)2388 2369 y Fn(m)2454 2355 y Fu(\)])29 b(in)f Ft(R)p Fu(.)40 b(Th)m(us,)28 b Fo(f)10 b Fu(\()p Fo(:)15 b(:)g(:)p Fu(\))26 b Ft(\037)382 2467 y Fo(C)7 b Fu([)p Fo(g)s Fu(\()p Fo(:)15 b(:)g(:)r Fu(\)])42 b(implies)d Fo(f)10 b Fu(\()p Fo(:)15 b(:)g(:)q Fu(\))44 b Ft(\037)g Fo(g)s Fu(\()p Fo(:)15 b(:)g(:)r Fu(\))42 b(whic)m(h)f(in)f(turn)h (implies)e Fo(F)13 b Fu(\()p Fo(:)i(:)g(:)q Fu(\))45 b Ft(\037)3030 2434 y Fz(0)3097 2467 y Fo(G)p Fu(\()p Fo(:)15 b(:)g(:)q Fu(\).)382 2580 y(Hence,)31 b Ft(\037)748 2547 y Fz(0)802 2580 y Fu(also)f(satis\014es)g(the)g(constrain)m(ts)h (\(b\))f(and)g(\(c\))h(of)g(Def.)g(10.)483 b Fm(2)523 2752 y Fu(The)34 b(follo)m(wing)f(examples)h(sho)m(w)g(that)h(none)f (of)g(the)h(con)m(v)m(erse)h(implications)31 b(of)382 2865 y(Lemma)f(11)i(holds.)382 3078 y Fk(Example)i(12)46 b Fs(The)35 b(system)i Ft(f)p Fo(f)10 b Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\))31 b Ft(!)g Fo(f)10 b Fu(\()p Fo(c)p Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\)\))p Ft(g)37 b Fs(is)e(DP-simply)h(terminat-)382 3191 y(ing)g(as)i(the)f(only)g(dep) -5 b(endency)39 b(p)-5 b(air)38 b(on)f(a)g(cycle)g(is)f Ft(h)p Fo(F)13 b Fu(\()p Fo(f)d Fu(\()p Fo(x)p Fu(\)\))p Fo(;)15 b(F)e Fu(\()p Fo(x)p Fu(\))p Ft(i)p Fs(.)57 b(Henc)-5 b(e,)37 b(the)382 3304 y(r)-5 b(esulting)35 b(c)-5 b(onstr)g(aints)37 b(ar)-5 b(e)35 b(satis\014e)-5 b(d)36 b(by)e(the)h(rp)-5 b(o)36 b(if)e(one)h(uses)f(the)h(AFS)f Fo(c)p Fu(\()p Fo(x)p Fu(\))29 b Ft(!)g Fo(x)p Fs(.)382 3416 y(However,)k(this)g(TRS)g (is)f(not)i(simply)f(terminating.)43 b(The)33 b(TRS)424 3613 y Fo(f)10 b Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\))83 b Ft(!)g Fo(f)10 b Fu(\()p Fo(c)p Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\)\))276 b Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\))86 b Ft(!)d Fo(x)259 b(g)s Fu(\()p Fo(c)p Fu(\(0\)\))85 b Ft(!)e Fo(g)s Fu(\()p Fo(d)p Fu(\(1\)\))424 3726 y Fo(f)10 b Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\))83 b Ft(!)g Fo(f)10 b Fu(\()p Fo(d)p Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\)\))260 b Fo(g)s Fu(\()p Fo(d)p Fu(\()p Fo(x)p Fu(\)\))86 b Ft(!)d Fo(x)259 b(g)s Fu(\()p Fo(c)p Fu(\(1\)\))85 b Ft(!)e Fo(g)s Fu(\()p Fo(d)p Fu(\(0\)\))382 3922 y Fs(is)32 b(DP-quasi)f(simply)j(terminating)f(as)g(c)-5 b(an)33 b(b)-5 b(e)32 b(pr)-5 b(ove)g(d)34 b(in)e(a)h(similar)g(way)g (using)f(the)382 4035 y(AFS)j(with)h(the)f(rules)g Fo(c)p Fu(\()p Fo(x)p Fu(\))c Ft(!)e Fo(x)35 b Fs(and)h Fo(d)p Fu(\()p Fo(x)p Fu(\))30 b Ft(!)g Fo(x)k Fs(and)i(the)g(rp)-5 b(o)36 b(wher)-5 b(e)36 b Fu(0)g Fs(and)f Fu(1)h Fs(ar)-5 b(e)382 4148 y(e)g(qual)34 b(in)g(the)g(pr)-5 b(e)g(c)g(e)g(denc)g(e.) 47 b(However,)34 b(it)f(is)h(not)g(DP-simply)g(terminating,)h(b)-5 b(e)g(c)g(ause)382 4261 y(due)46 b(to)h(the)g(\014rst)g(four)g(rules,)j (the)d(AFS)f(must)h(r)-5 b(e)g(duc)g(e)47 b Fo(c)p Fu(\()p Fo(x)p Fu(\))g Fs(and)h Fo(d)p Fu(\()p Fo(x)p Fu(\))f Fs(to)g(their)382 4374 y(ar)-5 b(guments.)43 b(But)33 b(then)g Fo(g)s Fu(\(0\))27 b Ft(\025)e Fo(g)s Fu(\(1\))34 b Fs(and)f Fo(g)s Fu(\(1\))28 b Ft(\025)c Fo(g)s Fu(\(0\))35 b Fs(le)-5 b(ad)34 b(to)f(a)g(c)-5 b(ontr)g(adiction.)523 4487 y(Final)5 b(ly,)35 b(the)g(system)f Ft(f)p Fo(f)10 b Fu(\(0)p Fo(;)15 b Fu(1)p Fo(;)g(x)p Fu(\))30 b Ft(!)e Fo(f)10 b Fu(\()p Fo(x;)15 b(x;)g(x)p Fu(\))p Ft(g)35 b Fs(is)f(terminating)h(but)e(not)i(DP-)382 4600 y(quasi)e(simply)g (terminating.)1913 b Fm(2)382 4886 y Fv(5)135 b(Com)l(bining)45 b(Disjoin)l(t)i(Systems)382 5089 y Fu(In)39 b(this)f(section)i(w)m(e)g (sho)m(w)f(that)i(DP-quasi)e(simple)e(termination)i(is)f(mo)s(dular)g (for)382 5202 y(disjoin)m(t)29 b(TRSs.)39 b(F)-8 b(or)31 b(the)g(pro)s(of,)f(w)m(e)h(need)f(the)g(follo)m(wing)f(lemma.)1831 5652 y(11)p eop %%Page: 12 12 12 11 bop 382 548 a Fk(Lemma)33 b(13)i(\(T)-9 b(ransforming)34 b(Reduction)i(Sequences\))46 b Fs(L)-5 b(et)41 b Ft(R)2871 562 y Fr(1)2951 548 y Fs(and)g Ft(R)3212 562 y Fr(2)3292 548 y Fs(b)-5 b(e)382 661 y(two)30 b(TRSs)g(over)g(disjoint)g(signatur) -5 b(es)30 b Ft(F)1809 675 y Fr(1)1878 661 y Fs(and)g Ft(F)2116 675 y Fr(2)2156 661 y Fs(,)g(r)-5 b(esp)g(e)g(ctively.)42 b(F)-7 b(urthermor)i(e,)32 b(let)382 774 y Ft(R)g Fu(=)f Ft(R)670 788 y Fr(1)732 774 y Ft([)23 b(R)893 788 y Fr(2)969 774 y Fs(b)-5 b(e)36 b(their)g(union.)53 b(If)36 b Fo(u;)15 b(v)40 b Fs(ar)-5 b(e)37 b(terms)g(over)g(the)f(signatur)-5 b(e)38 b Ft(F)3121 788 y Fr(1)3196 774 y Fs(such)382 887 y(that)f Fo(u)31 b Ft(!)745 901 y Fz(R)805 910 y Fl(1)875 887 y Fo(v)39 b Fs(and)e Fo(v)s(\033)e Ft(!)1363 854 y Fz(\003)1363 914 y(R)1458 887 y Fo(u\033)40 b Fs(hold)d(for)f(a)h (gr)-5 b(ound)37 b(substitution)f Fo(\033)f Fu(:)c Ft(V)7 b Fo(ar)s Fu(\()p Fo(u)p Fu(\))32 b Ft(!)382 1000 y(T)23 b Fu(\()p Ft(F)555 1014 y Fr(1)619 1000 y Ft([)h(F)769 1014 y Fr(2)809 1000 y Fu(\))p Fs(,)40 b(then)f(ther)-5 b(e)39 b(is)g(also)h(a)e(gr)-5 b(ound)40 b(substitution)f Fo(\034)46 b Fu(:)36 b Ft(V)7 b Fo(ar)s Fu(\()p Fo(u)p Fu(\))37 b Ft(!)f(T)22 b Fu(\()p Ft(F)3295 1014 y Fr(1)3335 1000 y Fu(\))382 1112 y Fs(such)33 b(that)h Fo(u\034)h Ft(!)993 1126 y Fz(R)1053 1135 y Fl(1)1117 1112 y Fo(v)s(\034)h Ft(!)1331 1080 y Fz(\003)1331 1144 y(R)1391 1153 y Fl(1)1425 1144 y Fz([E)6 b Fn(mb)p Fr(\()p Fz(F)1684 1153 y Fl(1)1720 1144 y Fr(\))1777 1112 y Fo(u\034)k Fs(.)382 1321 y(Pr)-5 b(o)g(of.)42 b Fu(Clearly)-8 b(,)29 b(all)h(terms)g(in)f(the)h(cyclic)g (deriv)-5 b(ation)1372 1522 y Fo(D)28 b Fu(:)117 b Fo(u\033)28 b Ft(!)1840 1536 y Fz(R)1900 1545 y Fl(1)1965 1522 y Fo(v)s(\033)g Ft(!)2183 1484 y Fz(\003)2183 1544 y(R)2273 1522 y Fo(u\033)382 1722 y Fu(ha)m(v)m(e)i(the)e(same)h(rank.)40 b(Since)27 b(the)i(ro)s(ot)f(sym)m(b)s(ol)g(of)g Fo(u)h Fu(is)e(in)g Ft(F)2547 1736 y Fr(1)2587 1722 y Fu(,)i(the)g(ro)s(ot)f (sym)m(b)s(ol)f(of)382 1835 y(ev)m(ery)i(term)e(in)g(the)h(reduction)f (sequence)h Fo(D)i Fu(is)d(also)h(in)f Ft(F)2411 1849 y Fr(1)2478 1835 y Fu(\(reduction)g(steps)h(whic)m(h)382 1948 y(are)j(destructiv)m(e)f(at)h(lev)m(el)f(1)h(w)m(ould)e(decrease)j (the)e(rank\).)523 2061 y(Supp)s(ose)20 b(\014rst)h(that)h(ev)m(ery)h (function)d(sym)m(b)s(ol)h(in)f Ft(F)2282 2075 y Fr(1)2343 2061 y Fu(has)i(arit)m(y)g Ft(\024)j Fu(1.)38 b(In)21 b(this)f(case,)382 2174 y(ev)m(ery)35 b(reduction)f(step)g(in)g Fo(D)j Fu(whic)m(h)c(is)h(destructiv)m(e)h(at)g(lev)m(el)f(2)h (strictly)f(decreases)382 2287 y(the)29 b(rank.)40 b(Consequen)m(tly)-8 b(,)29 b(there)g(is)f(no)h(reduction)f(step)h(of)g(this)f(kind)f(in)h Fo(D)s Fu(.)40 b(Hence)1128 2487 y Fo(top)1251 2501 y Fr(1)1290 2487 y Fu(\()p Fo(u\033)s Fu(\))27 b Ft(!)1585 2501 y Fz(R)1645 2510 y Fl(1)1709 2487 y Fo(top)1832 2501 y Fr(1)1871 2487 y Fu(\()p Fo(v)s(\033)s Fu(\))f Ft(!)2160 2449 y Fz(\003)2160 2509 y(R)2220 2518 y Fl(1)2284 2487 y Fo(top)2407 2501 y Fr(1)2446 2487 y Fu(\()p Fo(u\033)s Fu(\))382 2687 y(is)c(an)g Ft(R)661 2701 y Fr(1)701 2687 y Fu(-reduction)f(sequence)i(of)g(ground)f(terms)g(o)m(v)m(er)i Ft(F)2390 2701 y Fr(1)2430 2687 y Fu(.)38 b(Let)23 b Ft(V)7 b Fo(ar)s Fu(\()p Fo(u)p Fu(\))26 b(=)f Ft(f)p Fo(x)3144 2701 y Fr(1)3184 2687 y Fo(;)15 b(:)g(:)g(:)h(;)382 2800 y(x)434 2814 y Fn(n)481 2800 y Ft(g)33 b Fu(and)f(recall)g Ft(V)7 b Fo(ar)s Fu(\()p Fo(v)s Fu(\))30 b Ft(\022)f(V)7 b Fo(ar)s Fu(\()p Fo(u)p Fu(\).)49 b(In)32 b(this)f(case,)k(w)m(e)e (de\014ne)f(the)h(substitution)e Fo(\034)382 2913 y Fu(b)m(y)f Fo(\034)35 b Fu(=)25 b Ft(f)p Fo(x)776 2927 y Fn(i)830 2913 y Ft(7!)g Fo(top)1069 2927 y Fr(1)1108 2913 y Fu(\()p Fo(x)1195 2927 y Fn(i)1224 2913 y Fo(\033)s Fu(\))31 b Ft(j)f Fu(1)c Ft(\024)f Fo(i)h Ft(\024)f Fo(n)p Ft(g)30 b Fu(and)g(indeed)796 3114 y Fo(u\034)35 b Fu(=)25 b Fo(top)1142 3128 y Fr(1)1181 3114 y Fu(\()p Fo(u\033)s Fu(\))h Ft(!)1475 3128 y Fz(R)1535 3137 y Fl(1)1599 3114 y Fo(top)1722 3128 y Fr(1)1761 3114 y Fu(\()p Fo(v)s(\033)s Fu(\))h(=)e Fo(v)s(\034)35 b Ft(!)2269 3076 y Fz(\003)2269 3136 y(R)2329 3145 y Fl(1)2394 3114 y Fo(top)2517 3128 y Fr(1)2556 3114 y Fu(\()p Fo(u\033)s Fu(\))26 b(=)f Fo(u\034)382 3314 y Fu(is)k(the)i(reduction)e(sequence)i(w)m(e)g(are)g (lo)s(oking)e(for.)523 3427 y(Supp)s(ose)37 b(otherwise)h(that)i(there) 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(v)n(er)h(the)f(signature)i Fj(F)27 b Fp(then)19 b Fj(!)1920 4723 y Fq(\003)1920 4773 y(R[E)t Fd(mb)p Fl(\()p Fq(F)5 b Fl(\))2254 4755 y Fp(is)20 b(the)f(smallest)i(qso)f(con)n(taining)g Fj(!)3314 4763 y Fq(R)382 4846 y Fp(\(that)28 b(is,)h(if)g Fc(\045)f Fp(is)g(a)h(qso)f(with)g Fj(!)1380 4854 y Fq(R)1437 4846 y Fj(\022)p Fc(\045)p Fp(,)g(then)g Fj(!)1861 4814 y Fq(\003)1861 4864 y(R[E)t Fd(mb)p Fl(\()p Fq(F)5 b Fl(\))2175 4846 y Fj(\022)p Fc(\045)p Fp(\).)41 b(Note)28 b(ho)n(w)n(ev)n(er,)h(that)f(the)f Fb(strict)382 4937 y Fp(part)21 b(of)h(a)g(qso)g Fj(!)895 4906 y Fq(\003)895 4956 y(R[E)t Fd(mb)p Fl(\()p Fq(F)5 b Fl(\))1230 4937 y Fp(is)22 b(not)g(necessarily)g(closed)h(under)d(substitutions.)33 b(Hence,)22 b(without)g(the)382 5029 y(extension)h(of)h(the)e(dep)r (endency)g(pair)h(approac)n(h)g(in)g(Thm.)f(6)h(and)g(Def.)g(9)h(to)f Fb(stable-strict)33 b Fp(relations,)382 5120 y(suc)n(h)20 b(a)h(qso)f(cannot)h(b)r(e)f(used)g(for)h(termination)f(pro)r(ofs)i (with)f(dep)r(endency)e(pairs.)33 b(As)20 b(this)h(extension)382 5211 y(leads)32 b(to)g(a)g(more)f(p)r(o)n(w)n(erful)h(criterion,)i(w)n (e)e(did)f(not)h(in)n(v)n(estigate)g(whether)f(Thm.)g(14)h(w)n(ould)g (also)382 5303 y(hold)26 b(for)g(a)g(form)n(ulation)g(of)h(Def.)f(9)g (with)g(strict)g(instead)g(of)g(stable-strict)h(relations.)1831 5652 y Fu(13)p eop %%Page: 14 14 14 13 bop 523 548 a Fu(Conditions)31 b(\(a\))k(and)e(\(b\))h(are)g(ob)m (viously)e(satis\014ed)h(as)h(for)f(all)g Fo(l)f Ft(!)f Fo(r)i Ft(2)d(R)3194 562 y Fr(2)3267 548 y Fu(w)m(e)382 661 y(ha)m(v)m(e)23 b Fo(l)k Ft(#)682 675 y Fz(A)743 661 y Fu(=)e Fo(l)f Fu(and)d Fo(r)28 b Ft(#)1172 675 y Fz(A)1233 661 y Fu(=)d Fo(r)s Fu(.)37 b(Hence,)25 b(w)m(e)d(only)f (ha)m(v)m(e)j(to)e(sho)m(w)g(conjecture)h(\(c\).)39 b(Since)382 774 y Fh(\045)28 b Fu(is)g(the)g(qso)h(used)f(for)g(the)h(DP-quasi)f (simple)f(termination)g(pro)s(of)h(of)h Ft(R)2966 788 y Fr(1)3005 774 y Fu(,)g(w)m(e)g(ha)m(v)m(e)382 887 y Ft(!)473 854 y Fz(\003)473 914 y(S)516 923 y Fl(1)569 887 y Ft(\022)15 b Fh(\045)p Fu(.)58 b(Let)37 b Ft(h)p Fo(s;)15 b(t)p Ft(i)37 b Fu(b)s(e)e(a)i(dep)s(endency)e(pair)g(from)h Ft(P)43 b Fu(suc)m(h)36 b(that)h Fo(s)e Ft(#)2970 901 y Fz(A)3066 887 y Ft(\037)g Fo(t)f Ft(#)3284 901 y Fz(A)3345 887 y Fu(.)382 1000 y(Supp)s(ose)21 b(that)i(there)h(exists)e(a)h (ground)f(substitution)f Fo(\033)28 b Fu(:)e Ft(V)7 b Fo(ar)s Fu(\()p Fo(s)25 b Ft(#)2691 1014 y Fz(A)2752 1000 y Fu(\))g Ft(!)h(T)c Fu(\()p Ft(F)3111 964 y Fz(0)3111 1022 y Fr(1)3156 1000 y Ft([)5 b(F)3296 1014 y Fr(2)3335 1000 y Fu(\))382 1112 y(suc)m(h)30 b(that)h Fo(t)25 b Ft(#)887 1126 y Fz(A)974 1112 y Fo(\033)j Ft(!)1145 1080 y Fz(\003)1145 1141 y(R)1205 1122 y Fq(0)1257 1112 y Fo(s)d Ft(#)1370 1126 y Fz(A)1457 1112 y Fo(\033)s Fu(.)41 b(By)31 b(Lemma)f(13,)i(this)d(implies)f(the)j(existence)g(of)f(a)382 1225 y(ground)j(substitution)e Fo(\034)41 b Fu(:)32 b Ft(V)7 b Fo(ar)s Fu(\()p Fo(s)31 b Ft(#)1650 1239 y Fz(A)1711 1225 y Fu(\))g Ft(!)g(T)23 b Fu(\()p Ft(F)2081 1190 y Fz(0)2081 1248 y Fr(1)2121 1225 y Fu(\))34 b(suc)m(h)g(that)g Fo(t)d Ft(#)2708 1239 y Fz(A)2800 1225 y Fo(\034)41 b Ft(!)2972 1192 y Fz(\003)2972 1252 y(S)3015 1261 y Fl(1)3085 1225 y Fo(s)30 b Ft(#)3203 1239 y Fz(A)3295 1225 y Fo(\034)10 b Fu(.)382 1338 y(This,)34 b(ho)m(w)m(ev)m(er,)i(w)m(ould)d(imply)f Fo(t)g Ft(#)1636 1352 y Fz(A)1728 1338 y Fo(\034)42 b Fh(\045)31 b Fo(s)g Ft(#)2031 1352 y Fz(A)2124 1338 y Fo(\034)44 b Fu(and)34 b(con)m(tradicts)h(the)f(fact)h(that)382 1451 y Fo(s)26 b Ft(#)496 1465 y Fz(A)583 1451 y Ft(\037)g Fo(t)g Ft(#)784 1465 y Fz(A)845 1451 y Fu(.)43 b(Th)m(us,)30 b Fo(t)c Ft(#)1273 1465 y Fz(A)1361 1451 y Fo(\033)j Ft(6!)1533 1418 y Fz(\003)1533 1479 y(R)1593 1460 y Fq(0)1646 1451 y Fo(s)d Ft(#)1760 1465 y Fz(A)1847 1451 y Fo(\033)35 b Fu(holds)29 b(for)i(all)f(ground)g(substitutions)e Fo(\033)s Fu(.)382 1564 y(This)g(pro)m(v)m(es)k(conjecture)f(\(c\).) 1860 b Fm(2)523 1736 y Fu(Th)m(us,)23 b(if)f Ft(R)925 1750 y Fr(1)987 1736 y Fu(is)f(the)i(TRS)e(consisting)g(of)i(the)g (rules)e(\(1\))i(and)f(\(2\))i(and)e Ft(R)2983 1750 y Fr(2)3044 1736 y Fu(con)m(tains)382 1849 y(the)k(rules)e(\(3\))j(and)e (\(4\),)j(then)d(this)g(theorem)h(allo)m(ws)f(us)g(to)i(conclude)e (termination)f(of)382 1962 y(their)34 b(com)m(bination)g(b)s(ecause)h (b)s(oth)f(systems)h(are)g(DP-quasi)f(simply)f(terminating.)382 2075 y(This)27 b(example)i(cannot)h(b)s(e)f(handled)e(b)m(y)j(an)m(y)f (of)h(the)f(previous)f(mo)s(dularit)m(y)g(results.)382 2188 y(Note)k(also)e(that)h(in)e(this)g(example,)h(mo)s(dularit)m(y)e (of)j(termination)e(is)g(far)h(from)g(b)s(eing)382 2301 y(trivial)j(b)s(ecause)i(if)f Ft(R)1163 2315 y Fr(1)1202 2301 y Fu('s)h(rule)f Fo(f)10 b Fu(\(0)p Fo(;)15 b Fu(1)p Fo(;)g(x)p Fu(\))34 b Ft(!)f Fo(f)10 b Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(x;)g(x)p Fu(\))36 b(w)m(ould)e(b)s(e)g (just)g(sligh)m(tly)382 2414 y(c)m(hanged)g(to)h Fo(f)10 b Fu(\(0)p Fo(;)15 b Fu(1)p Fo(;)g(x)p Fu(\))32 b Ft(!)f Fo(f)10 b Fu(\()p Fo(x;)15 b(x;)g(x)p Fu(\),)35 b(then)f Ft(R)2061 2428 y Fr(1)2134 2414 y Fu(w)m(ould)e(still)g(b)s(e)h (terminating,)g(but)382 2526 y(the)d(union)f(with)g Ft(R)1075 2540 y Fr(2)1145 2526 y Fu(w)m(ould)g(not)i(terminate)f(an)m(y)h(more,) g(cf.)2534 2518 y([)2559 2526 y(T)-8 b(o)m(y)m(ama,)33 b(1987a)3153 2518 y(])3181 2526 y(.)523 2639 y(DP-quasi)g(simply)f (terminating)g(systems)i(o)s(ccur)g(frequen)m(tly)f(in)f(practice.)52 b(F)-8 b(or)382 2752 y(example,)30 b(the)h(t)m(w)m(o)g(TRSs)395 2977 y Fg(R)465 2989 y Fa(1)526 2977 y Fx(:)117 b Ff(x)19 b Fg(\000)f Fx(0)35 b Fg(!)h Ff(x)457 3102 y(s)p Fx(\()p Ff(x)p Fx(\))19 b Fg(\000)f Ff(s)p Fx(\()p Ff(y)s Fx(\))36 b Fg(!)g Ff(x)19 b Fg(\000)f Ff(y)526 3226 y(q)s Fx(\(0)p Ff(;)c(s)p Fx(\()p Ff(y)s Fx(\)\))36 b Fg(!)g Fx(0)417 3351 y Ff(q)s Fx(\()p Ff(s)p Fx(\()p Ff(x)p Fx(\))p Ff(;)14 b(s)p Fx(\()p Ff(y)s Fx(\)\))37 b Fg(!)f Ff(s)p Fx(\()p Ff(q)s Fx(\()p Ff(x)20 b Fg(\000)e Ff(y)s(;)c(s)p Fx(\()p Ff(y)s Fx(\)\)\))1503 2977 y Fg(R)1573 2989 y Fa(2)1633 2977 y Fx(:)118 b Ff(app)p Fx(\()p Ff(nil)r(;)14 b(k)s Fx(\))40 b Fg(!)h Ff(k)1793 3102 y(app)p Fx(\()p Ff(l)r(;)14 b(nil)r Fx(\))40 b Fg(!)h Ff(l)1736 3226 y(app)p Fx(\()p Ff(x)23 b Fx(:)g Ff(l)r(;)14 b(k)s Fx(\))41 b Fg(!)g Ff(x)24 b Fx(:)f Ff(app)p Fx(\()p Ff(l)r(;)14 b(k)s Fx(\))1708 3351 y Ff(sum)p Fx(\()p Ff(x)23 b Fx(:)g Ff(nil)r Fx(\))41 b Fg(!)g Ff(x)24 b Fx(:)f Ff(nil)1609 3475 y(sum)p Fx(\()p Ff(x)g Fx(:)g(\()p Ff(y)j Fx(:)d Ff(l)r Fx(\)\))42 b Fg(!)f Ff(sum)p Fx(\(\()p Ff(x)19 b Fx(+)f Ff(y)s Fx(\))23 b(:)g Ff(l)r Fx(\))1334 3600 y Ff(sum)p Fx(\()p Ff(app)p Fx(\()p Ff(l)r(;)14 b(x)22 b Fx(:)i(\()p Ff(y)i Fx(:)d Ff(k)s Fx(\)\)\))42 b Fg(!)f Ff(sum)p Fx(\()p Ff(app)p Fx(\()p Ff(l)r(;)14 b(sum)p Fx(\()p Ff(x)22 b Fx(:)h(\()p Ff(y)j Fx(:)e Ff(k)s Fx(\)\)\)\))382 3796 y Fu(are)37 b(not)g(simply)d(terminating,)j(but)f(they)h(are)g(b)s(oth)f(DP-quasi)g (simply)f(terminat-)382 3909 y(ing,)g(cf.)g(Arts)g(and)f(Giesl)1316 3901 y([)1342 3909 y(1997c;)j(1997a)1849 3901 y(])1876 3909 y(.)54 b(Hence,)37 b(Thm.)d(14)i(no)m(w)f(also)g(allo)m(ws)f(to) 382 4021 y(conclude)c(DP-quasi)g(simple)e(termination)h(of)i(their)e (union.)382 4308 y Fv(6)135 b(Com)l(bining)45 b(Constructor-Sharing)g (Systems)382 4511 y Fu(It)d(ma)m(y)g(b)s(e)f(a)h(bit)e(surprising)e (that)43 b(Thm.)d(14)j(cannot)f(b)s(e)f(directly)f(extended)i(to)382 4624 y(constructor-sharing)36 b(TRSs)f(\(ev)m(en)j(if)d(w)m(e)j (disallo)m(w)c(the)j(use)f(of)h(AFSs\).)60 b(In)35 b(other)382 4737 y(w)m(ords,)c(there)h(are)f(constructor-sharing)g(TRSs)f Ft(R)2183 4751 y Fr(1)2254 4737 y Fu(and)h Ft(R)2509 4751 y Fr(2)2580 4737 y Fu(whic)m(h)f(are)h(b)s(oth)g(DP-)382 4850 y(quasi)38 b(simply)f(terminating,)j(but)f(their)f(union)g Ft(R)i Fu(=)f Ft(R)2425 4864 y Fr(1)2491 4850 y Ft([)26 b(R)2655 4864 y Fr(2)2733 4850 y Fu(is)39 b(not)g(DP-quasi)382 4962 y(simply)28 b(terminating.)1831 5652 y(14)p eop %%Page: 15 15 15 14 bop 382 548 a Fk(Example)34 b(15)46 b Fs(Consider)33 b(the)g(fol)5 b(lowing)34 b(TRSs:)610 723 y Ft(R)687 737 y Fr(1)752 723 y Fu(:)119 b Fo(f)10 b Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\))85 b Ft(!)d Fo(f)10 b Fu(\()p Fo(x)p Fu(\))897 861 y Fo(f)g Fu(\()p Fo(b)p Fu(\()p Fo(x)p Fu(\)\))84 b Ft(!)e Fo(x)2005 723 y Ft(R)2082 737 y Fr(2)2147 723 y Fu(:)120 b Fo(g)s Fu(\()p Fo(d)p Fu(\()p Fo(x)p Fu(\)\))85 b Ft(!)e Fo(g)s Fu(\()p Fo(x)p Fu(\))2265 861 y Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\))i Ft(!)e Fo(c)p Fu(\()p Fo(g)s Fu(\()p Fo(b)p Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\)\))523 1026 y Fs(The)39 b(TRS)g Ft(R)1013 1040 y Fr(1)1092 1026 y Fs(is)g(simply)h (terminating)g(and)g Ft(R)2248 1040 y Fr(2)2326 1026 y Fs(is)f(DP-quasi)f(simply)i(termi-)382 1139 y(nating.)68 b(The)41 b(latter)h(c)-5 b(an)42 b(b)-5 b(e)41 b(shown)i(without)f(any) g(AFS)f(by)g(using)g(a)h(p)-5 b(olynomial)382 1252 y(or)g(dering)42 b(which)g(maps)g Fo(c)p Fs(,)g Fo(b)p Fs(,)h Fo(g)s Fs(,)g(and)e Fo(G)f Fs(to)h(the)g(identity)h(and)f(which)h(maps)g Fo(d)p Fu(\()p Fo(x)p Fu(\))382 1365 y Fs(to)h Fo(x)27 b Fu(+)g(1)p Fs(.)71 b(However,)44 b(their)f(union)g(is)f(not)h (DP-quasi)e(simply)i(terminating.)72 b(As)382 1478 y Ft(h)p Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\))p Fo(;)i(F)e Fu(\()p Fo(x)p Fu(\))p Ft(i)45 b Fs(r)-5 b(epr)g(esents)44 b(a)e(cycle)f(in)h(the)g(estimate)-5 b(d)43 b(dep)-5 b(endency)43 b(gr)-5 b(aph)43 b(one)382 1591 y(would)34 b(have)f(to)g(\014nd)g(a)g(qso)g(satisfying)1344 1791 y Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\))85 b Ft(\037)d Fo(F)13 b Fu(\()p Fo(x)p Fu(\))1132 b(\(10\))1361 1929 y Fo(f)10 b Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\))84 b Fh(\045)e Fo(f)10 b Fu(\()p Fo(x)p Fu(\))1148 b(\(11\))1361 2066 y Fo(f)10 b Fu(\()p Fo(b)p Fu(\()p Fo(x)p Fu(\)\))84 b Fh(\045)e Fo(x)1273 b Fu(\(12\))1360 2204 y Fo(g)s Fu(\()p Fo(d)p Fu(\()p Fo(x)p Fu(\)\))86 b Fh(\045)c Fo(g)s Fu(\()p Fo(x)p Fu(\))1157 b(\(13\))1368 2342 y Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\))86 b Fh(\045)c Fo(c)p Fu(\()p Fo(g)s Fu(\()p Fo(b)p Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\)\))p Fo(:)805 b Fu(\(14\))382 2542 y Fs(With)27 b(empty)h(AFS,)g(no)f(qso)h(satis\014es)g(\(10\))g(-)f (\(14\),)i(sinc)-5 b(e)28 b(otherwise)g(we)g(would)g(have)551 2739 y Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\)\))88 b Ft(\037)82 b Fo(F)13 b Fu(\()p Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\))339 b Fs(due)32 b(to)h(\(10\))1166 2852 y Fh(\045)82 b Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(g)s Fu(\()p Fo(b)p Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\)\)\))121 b Fs(due)32 b(to)h(\(14\))1166 2965 y Fh(\045)82 b Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\)\))230 b Fs(due)32 b(to)h(the)g(subterm)g(pr)-5 b(op)g(erty.)382 3162 y(By)38 b(\(10\),)j(the)d(AFS)g(c)-5 b(annot)40 b(c)-5 b(ontain)40 b(any)e Fo(c)p Fs(-rules.)59 b(If)38 b(the)g(ar)-5 b(gument)40 b(of)e Fo(b)g Fs(would)382 3275 y(b)-5 b(e)34 b(eliminate)-5 b(d,)37 b(then)e(\(12\))h(would)f(b) -5 b(e)35 b(tr)-5 b(ansforme)g(d)38 b(into)d Fo(f)10 b Fu(\()p Fo(b)2567 3242 y Fz(0)2590 3275 y Fu(\))29 b Fh(\045)g Fo(x)p Fs(.)47 b(But)35 b(as)g(ther)-5 b(e)382 3388 y(exists)40 b(the)g(strict)h(ine)-5 b(quality)40 b(\(10\))h(with)g(a)f(variable)g(in)g(its)g(right-hand)h(side,)h(this) 382 3501 y(r)-5 b(esults)39 b(in)g(the)g(c)-5 b(ontr)g(adiction)41 b Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(f)d Fu(\()p Fo(b)1811 3468 y Fz(0)1835 3501 y Fu(\)\)\))37 b Ft(\037)f Fo(x)p Fs(.)59 b(Similarly,)42 b(the)d(ar)-5 b(gument)39 b(of)g Fo(g)382 3614 y Fs(c)-5 b(annot)27 b(b)-5 b(e)25 b(eliminate)-5 b(d)27 b(either,)g(sinc)-5 b(e)26 b Fo(g)1754 3581 y Fz(0)1803 3614 y Fh(\045)f Fo(c)p Fu(\()p Fo(g)2019 3581 y Fz(0)2044 3614 y Fu(\))g Fs(would)i(b)-5 b(e)25 b(a)h(c)-5 b(ontr)g(adiction)29 b(to)d(\(10\).)523 3727 y(Thus,)i(the)f(only)g(p) -5 b(ossible)27 b(rules)g(in)f(the)h(AFS)f(ar)-5 b(e)27 b Fo(f)10 b Fs(-)25 b(and)i Fo(d)p Fs(-rules)g(and)g(rules)f(that)382 3840 y(map)i Fo(b)e Fs(or)i Fo(g)i Fs(to)d(its)g(ar)-5 b(gument.)41 b(But)26 b(then)h(we)g(would)h(again)f(obtain)h Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\)\))29 b Ft(\037)382 3953 y Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\)\))38 b Fs(or)c Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\))30 b Ft(\037)d Fo(F)13 b Fu(\()p Fo(c)p Fu(\()p Fo(c)p Fu(\()p Fo(x)p Fu(\)\)\))36 b Fs(as)f(ab)-5 b(ove.)45 b(Henc)-5 b(e,)33 b(the)h(TRS)g(inde)-5 b(e)g(d)382 4066 y(is)32 b(not)i(DP-quasi)d (simply)j(terminating.)1492 b Fm(2)523 4274 y Fu(Th)m(us,)36 b(in)d(the)j(remainder)d(of)j(the)f(section)g(w)m(e)h(will)c(restrict)j (ourselv)m(es)g(to)h(DP-)382 4387 y(simple)24 b(termination)g(instead)h (of)h(DP-quasi)f(simple)f(termination.)38 b(Without)26 b(apply-)382 4500 y(ing)33 b(AFSs,)i(DP-simple)d(termination)h(of)h (the)h(TRS)e Ft(R)2320 4514 y Fr(2)2393 4500 y Fu(cannot)h(b)s(e)g(pro) m(v)m(ed.)52 b(Ho)m(w-)382 4613 y(ev)m(er,)25 b(if)c(one)i(uses)e(for)h (example)g(the)h(AFS)f Fo(b)p Fu(\()p Fo(x)p Fu(\))k Ft(!)f Fo(b)2181 4580 y Fz(0)2205 4613 y Fu(,)f(then)e(its)f(resulting) g(constrain)m(ts)382 4726 y(are)34 b(again)g(satis\014ed)f(b)m(y)h(a)h (simpli\014cation)30 b(ordering.)50 b(Th)m(us,)34 b(to)h(ac)m(hiev)m(e) g(mo)s(dular-)382 4839 y(it)m(y)42 b(results)e(for)h (constructor-sharing)g(TRSs,)j(w)m(e)e(ha)m(v)m(e)h(to)f(restrict)g (ourselv)m(es)f(to)382 4951 y(systems)30 b(where)g(the)h(AFSs)f(con)m (tain)h(no)f(rules)f(for)h(shared)g(sym)m(b)s(ols)f(lik)m(e)g Fo(b)p Fu(.)523 5064 y(But)38 b(w)m(e)g(need)f(another)g(requiremen)m (t)g(to)h(ensure)f(mo)s(dularit)m(y)-8 b(.)60 b(F)-8 b(or)39 b(example,)382 5177 y(let)h(us)f(eliminate)g(the)h(\014rst)g (rule)e Fo(g)s Fu(\()p Fo(d)p Fu(\()p Fo(x)p Fu(\)\))44 b Ft(!)d Fo(g)s Fu(\()p Fo(x)p Fu(\))h(from)d Ft(R)2580 5191 y Fr(2)2620 5177 y Fu(.)70 b(No)m(w)40 b(there)h(is)e(no)382 5290 y(non-empt)m(y)f(cycle)g(in)f(the)h(estimated)g(dep)s(endency)f (graph)g(of)h Ft(R)2732 5304 y Fr(2)2810 5290 y Fu(an)m(y)g(more)g(and) 382 5403 y(hence)31 b(w)m(e)g(obtain)f(no)h(constrain)m(ts)g(at)h(all)d (for)i Ft(R)2098 5417 y Fr(2)2137 5403 y Fu(.)42 b(Th)m(us,)30 b(DP-simple)g(termination)1831 5652 y(15)p eop %%Page: 16 16 16 15 bop 382 548 a Fu(of)23 b Ft(R)555 562 y Fr(2)618 548 y Fu(can)h(no)m(w)f(ev)m(en)h(b)s(e)f(pro)m(v)m(ed)g(without)f (using)g(AFSs,)j(but)e(the)g(com)m(bined)f(system)382 661 y Ft(R)459 675 y Fr(1)513 661 y Ft([)15 b(R)666 675 y Fr(2)733 661 y Fu(is)27 b(still)e(not)j(DP-simply)e(terminating.)39 b(Here,)29 b(the)f(problem)e(is)h(due)g(to)h(the)382 774 y(fact)k(that)g(TRSs)e(without)g(non-empt)m(y)i(cycles)f(are)h (DP-simply)d(terminating,)i(ev)m(en)382 887 y(if)e(there)i(is)e(no)i (simpli\014cation)c(ordering)i Ft(\037)h Fu(suc)m(h)g(that)h Fo(l)c Ft(\027)e Fo(r)33 b Fu(holds)c(for)h(their)g(rules.)382 1000 y(T)-8 b(o)39 b(exclude)g(suc)m(h)f(TRSs)g(w)m(e)h(will)d(demand)i (that)i(the)f(constrain)m(t)g(\(a\))h(of)f(Def.)h(10)382 1112 y(should)28 b(also)j(b)s(e)e(satis\014ed)h(for)g(the)h Fs(empty)39 b Fu(cycle)31 b Ft(P)7 b Fu(.)2217 1080 y Fr(7)523 1225 y Fu(Nonetheless,)29 b(the)g(follo)m(wing)e(example)h (sho)m(ws)h(that)g(this)e(restriction)h(is)g(not)g(y)m(et)382 1338 y(su\016cien)m(t)42 b(for)h(obtaining)f(a)i(mo)s(dularit)m(y)d (result)h(for)h(DP-simple)f(termination)f(of)382 1451 y(constructor-sharing)30 b(systems.)382 1660 y Fk(Example)k(16)46 b Fs(L)-5 b(et)33 b Ft(R)1198 1674 y Fr(1)1270 1660 y Fs(c)-5 b(onsist)33 b(of)g(the)g(rules)759 1836 y Fo(g)s Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\)\))85 b Ft(!)e Fo(g)s Fu(\()p Fo(x)p Fu(\))759 1974 y Fo(g)s Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\)\))i Ft(!)e Fo(x)2189 1836 y(g)s Fu(\(0\))i Ft(!)e Fo(g)s Fu(\(1\))2182 1974 y Fo(f)10 b Fu(\(0\))83 b Ft(!)g Fo(g)s Fu(\()p Fo(f)10 b Fu(\()p Fo(s)p Fu(\(0\)\)\))382 2144 y Fs(and)29 b(let)f Ft(R)754 2158 y Fr(2)822 2144 y Fs(c)-5 b(onsist)29 b(of)f(the)h(rule)f Fo(h)p Fu(\(1\))f Ft(!)e Fo(h)p Fu(\(0\))p Fs(.)41 b(T)-7 b(o)29 b(pr)-5 b(ove)29 b(DP-simple)f(termination)382 2257 y(of)k Ft(R)565 2271 y Fr(1)635 2257 y Fs(we)g(have)g(to)g(use)f (an)h(AFS)f(c)-5 b(ontaining)32 b(the)g(rule)g Fo(g)s Fu(\()p Fo(x)p Fu(\))26 b Ft(!)f Fo(x)p Fs(.)42 b(This,)32 b(however,)382 2370 y(would)46 b(imply)g Fu(0)i Ft(\037)g Fu(1)d Fs(which)h(is)f(a)h(c)-5 b(ontr)g(adiction)48 b(to)d Fo(h)p Fu(\(1\))50 b Ft(\027)d Fo(h)p Fu(\(0\))p Fs(.)81 b(Thus,)48 b(the)382 2482 y(c)-5 b(ombination)34 b(of)f(b)-5 b(oth)34 b(systems)g(is)e(not)h(DP-simply)g(terminating.) 579 b Fm(2)523 2692 y Fu(So)25 b(w)m(e)h(also)f(ha)m(v)m(e)h(to)g (ensure)f(that)h(an)f(application)e(of)j(the)f(AFS)g(to)h(the)g (resulting)382 2805 y(inequalities)g(do)s(es)i(not)g(transform)g (left-hand)f(sides)g(whic)m(h)g(had)h(a)h(non-shared)e(ro)s(ot)382 2917 y(sym)m(b)s(ol)22 b(lik)m(e)g Fo(g)27 b Fu(in)m(to)c(terms)g(with) f(a)h(shared)f(ro)s(ot)i(sym)m(b)s(ol)e(\(lik)m(e)g(the)i(former)e (construc-)382 3030 y(tor)33 b(0\).)636 2997 y Fr(8)723 3030 y Fu(The)f(restrictions)f(needed)h(are)h(captured)f(b)m(y)g(the)h (notion)e(of)i Fo(C)7 b Fs(-r)-5 b(estricte)g(d)382 3143 y Fu(DP-simple)29 b(termination.)382 3352 y Fk(De\014nition)35 b(17)g(\()p Fo(C)7 b Fk(-restricted)34 b(DP-simple)h(termination\))44 b Fs(L)-5 b(et)24 b Ft(R)g Fs(b)-5 b(e)24 b(a)g(TRS)382 3465 y(over)33 b Ft(F)42 b Fs(and)33 b(let)g Fo(C)f Ft(\022)25 b(F)9 b Fs(.)42 b Ft(R)32 b Fs(is)h Fo(C)7 b Fu(-restricted)32 b Fs(DP-simply)h(terminating)g(i\013)382 3650 y(\(1\))g(F)-7 b(or)34 b(every)e(cycle)h Ft(P)40 b Fs(\(including)33 b Ft(P)g Fu(=)25 b Ft(;)p Fs(\),)33 b(ther)-5 b(e)33 b(is)g(an)g(AFS)f Ft(A)h Fs(such)f(that)577 3859 y Ft(S)54 b Fu(=)48 b Ft(f)p Fo(l)27 b Ft(#)949 3873 y Fz(A)1010 3859 y Ft(!)e Fo(r)j Ft(#)1240 3873 y Fz(A)1327 3859 y Ft(j)50 b Fo(l)27 b Ft(!)f Fo(r)h Ft(2)e(R)33 b Fs(and)g Fo(l)28 b Ft(#)2113 3873 y Fz(A)2189 3859 y Ft(6)p Fu(=)d Fo(r)i Ft(#)2398 3873 y Fz(A)2460 3859 y Ft(g)32 b([)805 3972 y(f)p Fo(s)25 b Ft(#)963 3986 y Fz(A)1024 3972 y Ft(!)g Fo(t)g Ft(#)1243 3986 y Fz(A)1330 3972 y Ft(j)50 b(h)p Fo(s;)15 b(t)p Ft(i)33 b Fs(is)g(a)g(dep)-5 b(endency)34 b(p)-5 b(air)34 b(in)e Ft(P)41 b Fs(and)33 b Fo(s)25 b Ft(#)2973 3986 y Fz(A)3049 3972 y Ft(6)p Fu(=)g Fo(t)g Ft(#)3248 3986 y Fz(A)3309 3972 y Ft(g)535 4181 y Fs(is)41 b(simply)h(terminating)f(and)h(such)f(that)h Fo(s)d Ft(#)2157 4195 y Fz(A)2234 4181 y Ft(6)p Fu(=)g Fo(t)h Ft(#)2462 4195 y Fz(A)2563 4181 y Fs(for)i(some)f Ft(h)p Fo(s;)15 b(t)p Ft(i)56 b(2)39 b(P)535 4294 y Fs(if)k Ft(P)52 b(6)p Fu(=)43 b Ft(;)p Fs(.)74 b(\(Equivalently,)45 b(some)f(simpli\014c)-5 b(ation)45 b(or)-5 b(dering)45 b Ft(\037)e Fs(satis\014es)g(the)535 4407 y(c)-5 b(onstr)g(aints)32 b(\(a\),)f(\(b\),)f(and)h(\(c\))f(of)f (Def.)g(10)i(for)f(every)f(cycle,)h(wher)-5 b(e)31 b(\(a\))f(is)g(also) 535 4520 y(r)-5 b(e)g(quir)g(e)g(d)34 b(for)g Ft(P)f Fu(=)25 b Ft(;)p Fs(\).)382 4622 y(\(2\))33 b(F)-7 b(or)34 b(every)e(rule)h Fo(u)25 b Ft(#)1252 4636 y Fz(A)1329 4622 y Ft(!)g Fo(v)j Ft(#)1562 4636 y Fz(A)1656 4622 y Fs(in)k Ft(S)7 b Fs(:)42 b(if)32 b Fo(r)s(oot)p Fu(\()p Fo(u)p Fu(\))26 b Ft(62)f Fo(C)7 b Fs(,)31 b(then)j Fo(r)s(oot)p Fu(\()p Fo(u)25 b Ft(#)3042 4636 y Fz(A)3103 4622 y Fu(\))g Ft(62)g Fo(C)7 b Fs(.)382 4735 y(\(3\))33 b(None)g(of)g(the)g(AFSs)f(c) -5 b(ontains)34 b(a)f(rule)g(for)g(a)g(function)g(symb)-5 b(ol)34 b Fo(f)h Ft(2)25 b Fo(C)7 b Fs(.)523 4944 y Fu(The)36 b(follo)m(wing)g(theorem)h(sho)m(ws)f(that)i(under)d(this)h Fo(C)7 b Fu(-restriction,)37 b(DP-simple)382 5057 y(termination)29 b(is)h(mo)s(dular)e(for)i(constructor-sharing)g(TRSs.)p 382 5135 1196 4 v 486 5189 a Fl(7)520 5220 y Fp(In)25 b(fact,)i(without)f(this)g(additional)h(requiremen)n(t)d(DP-simple)h (termination)g(is)h(not)g(ev)n(en)f(mo)r(d-)382 5312 y(ular)h(for)g Fb(disjoint)33 b Fp(com)n(binations.)i(F)-6 b(or)25 b(reasons)i(of)g(space,)f(a)g(coun)n(terexample)f(is)h (omitted.)486 5371 y Fl(8)520 5403 y Fp(If)g(all)h(AFSs)e(used)g(are)h (non-collapsing,)i(then)d(this)g(requiremen)n(t)f(is)j(alw)n(a)n(ys)f (ful\014lled.)1831 5652 y Fu(16)p eop %%Page: 17 17 17 16 bop 382 548 a Fk(Theorem)34 b(18)h(\(Mo)s(dularit)m(y)g(of)g Fo(C)7 b Fk(-restricted)35 b(DP-simple)f(termination\))382 661 y Fs(L)-5 b(et)30 b Ft(R)613 675 y Fr(1)682 661 y Fs(and)h Ft(R)933 675 y Fr(2)1002 661 y Fs(b)-5 b(e)30 b(c)-5 b(onstructor-sharing)32 b(TRSs)f(over)f(the)g(signatur)-5 b(es)31 b Ft(F)2995 675 y Fr(1)3064 661 y Fs(and)g Ft(F)3303 675 y Fr(2)3343 661 y Fs(,)382 774 y(r)-5 b(esp)g(e)g(ctively.)43 b(If)30 b Ft(F)1052 788 y Fr(1)1109 774 y Ft(\\)16 b(F)1251 788 y Fr(2)1316 774 y Ft(\022)25 b Fo(C)7 b Fs(,)30 b(then)h(their)h(c) -5 b(ombine)g(d)32 b(system)g Ft(R)25 b Fu(=)g Ft(R)2922 788 y Fr(1)2978 774 y Ft([)16 b(R)3132 788 y Fr(2)3202 774 y Fs(over)382 887 y(the)28 b(signatur)-5 b(e)29 b Ft(F)35 b Fu(=)24 b Ft(F)1175 901 y Fr(1)1225 887 y Ft([)10 b(F)1361 901 y Fr(2)1428 887 y Fs(is)28 b Fo(C)7 b Fs(-r)-5 b(estricte)g(d)29 b(DP-simply)e(terminating)i(if)f(and)h(only)382 1000 y(if)j(b)-5 b(oth)34 b Ft(R)740 1014 y Fr(1)812 1000 y Fs(and)g Ft(R)1066 1014 y Fr(2)1138 1000 y Fs(ar)-5 b(e)33 b Fo(C)7 b Fs(-r)-5 b(estricte)g(d)33 b(DP-simply)g (terminating.)382 1196 y(Pr)-5 b(o)g(of.)47 b Fu(The)31 b(only-if)f(direction)h(is)g(trivial.)44 b(F)-8 b(or)33 b(the)f(if)f(direction,)g(let)h Ft(P)40 b Fu(b)s(e)31 b(a)h(cycle)382 1308 y(in)f(the)h(estimated)h(dep)s(endency)e(graph)h (of)g Ft(R)g Fu(\(where)g Ft(P)40 b Fu(ma)m(y)33 b(also)g(b)s(e)e(empt) m(y\).)48 b(It)382 1421 y(is)32 b(not)g(di\016cult)f(to)j(pro)m(v)m(e)f (that)g Ft(P)41 b Fu(is)31 b(also)i(a)g(cycle)g(in)e(the)i(estimated)g (dep)s(endency)382 1534 y(graph)23 b(of)i Ft(R)808 1548 y Fr(1)871 1534 y Fu(or)f(in)f(the)i(estimated)f(dep)s(endency)f(graph) g(of)i Ft(R)2550 1548 y Fr(2)2613 1534 y Fu(b)s(ecause)f Ft(R)3020 1548 y Fr(1)3084 1534 y Fu(and)f Ft(R)3331 1548 y Fr(2)382 1647 y Fu(are)34 b(constructor-sharing.)51 b(Without)34 b(loss)g(of)g(generalit)m(y)g(let)g Ft(P)42 b Fu(b)s(e)33 b(a)h(cycle)h(in)d(the)382 1760 y(estimated)f(dep)s (endency)f(graph)g(of)h Ft(R)1732 1774 y Fr(1)1771 1760 y Fu(.)43 b(W)-8 b(e)32 b(ha)m(v)m(e)g(to)g(sho)m(w)e(that)i(there)f (is)f(an)h(AFS)382 1873 y Ft(A)h Fu(suc)m(h)g(that)h(the)f(corresp)s (onding)e(TRS)i Ft(S)39 b Fu(is)31 b(simply)f(terminating)h(and)h (moreo)m(v)m(er)382 1986 y(conditions)d(\(2\))i(and)f(\(3\))h(of)g (Def.)g(17)g(are)g(satis\014ed.)523 2099 y(Since)g Ft(R)839 2113 y Fr(1)910 2099 y Fu(is)f Fo(C)7 b Fu(-restricted)32 b(DP-simply)d(terminating)i(and)g Ft(P)39 b Fu(is)31 b(a)h(cycle)g(in)e(the)382 2212 y(estimated)h(dep)s(endency)d(graph)i (of)h Ft(R)1730 2226 y Fr(1)1769 2212 y Fu(,)g(there)f(is)g(an)g(AFS)g Ft(A)2556 2226 y Fr(1)2626 2212 y Fu(suc)m(h)g(that)h(the)f(TRS)400 2391 y Ft(S)455 2405 y Fr(1)542 2391 y Fu(=)47 b Ft(f)p Fo(l)28 b Ft(#)805 2405 y Fz(A)861 2414 y Fl(1)901 2391 y Ft(!)d Fo(r)j Ft(#)1131 2405 y Fz(A)1187 2414 y Fl(1)1251 2391 y Ft(j)51 b Fo(l)27 b Ft(!)e Fo(r)j Ft(2)d(R)1729 2405 y Fr(1)1799 2391 y Fu(and)30 b Fo(l)d Ft(#)2075 2405 y Fz(A)2131 2414 y Fl(1)2196 2391 y Ft(6)p Fu(=)e Fo(r)i Ft(#)2405 2405 y Fz(A)2461 2414 y Fl(1)2501 2391 y Ft(g)k([)660 2504 y(f)p Fo(s)26 b Ft(#)819 2518 y Fz(A)875 2527 y Fl(1)914 2504 y Ft(!)f Fo(t)g Ft(#)1133 2518 y Fz(A)1189 2527 y Fl(1)1254 2504 y Ft(j)51 b(h)p Fo(s;)15 b(t)p Ft(i)31 b Fu(is)e(a)i(dep)s(endency)e(pair)g(in)g Ft(P)38 b Fu(and)30 b Fo(s)25 b Ft(#)2893 2518 y Fz(A)2949 2527 y Fl(1)3014 2504 y Ft(6)p Fu(=)g Fo(t)g Ft(#)3213 2518 y Fz(A)3269 2527 y Fl(1)3308 2504 y Ft(g)382 2684 y Fu(is)38 b(simply)g(terminating)g(and)h(suc)m(h)g(that)h Fo(s)g Ft(#)2021 2698 y Fz(A)2077 2707 y Fl(1)2132 2684 y Ft(6)p Fu(=)g Fo(t)h Ft(#)2362 2698 y Fz(A)2418 2707 y Fl(1)2497 2684 y Fu(holds)d(for)h(at)i(least)e(one)382 2797 y(dep)s(endency)26 b(pair)g Ft(h)p Fo(s;)15 b(t)p Ft(i)28 b Fu(from)f Ft(P)35 b Fu(if)26 b Ft(P)33 b(6)p Fu(=)25 b Ft(;)p Fu(.)40 b(On)27 b(the)g(other)h(hand,)f(there)h(is)e (an)i(AFS)382 2910 y Ft(A)455 2924 y Fr(2)524 2910 y Fu(suc)m(h)i(that)h(the)g(TRS)830 3098 y Ft(S)885 3112 y Fr(2)950 3098 y Fu(=)25 b Ft(f)p Fo(l)j Ft(#)1191 3112 y Fz(A)1247 3121 y Fl(2)1286 3098 y Ft(!)d Fo(r)j Ft(#)1516 3112 y Fz(A)1572 3121 y Fl(2)1637 3098 y Ft(j)51 b Fo(l)27 b Ft(!)e Fo(r)j Ft(2)d(R)2115 3112 y Fr(2)2185 3098 y Fu(and)k Fo(l)f Ft(#)2461 3112 y Fz(A)2517 3121 y Fl(2)2571 3098 y Ft(6)p Fu(=)d Fo(r)j Ft(#)2781 3112 y Fz(A)2837 3121 y Fl(2)2877 3098 y Ft(g)382 3286 y Fu(is)40 b(simply)f (terminating)h(b)s(ecause)h Ft(R)1716 3300 y Fr(2)1797 3286 y Fu(is)f Fo(C)7 b Fu(-restricted)41 b(DP-simply)f(terminating.) 382 3398 y(\(Here,)31 b Ft(A)730 3412 y Fr(2)800 3398 y Fu(is)e(the)i(AFS)f(corresp)s(onding)e(to)k(the)e(empt)m(y)h (cycle.\))523 3511 y(Let)41 b Ft(F)770 3478 y Fz(0)761 3537 y Fn(i)834 3511 y Fu(b)s(e)e(the)h(set)h(of)g(all)e(function)g (sym)m(b)s(ols)f(in)h Ft(S)2428 3525 y Fn(i)2497 3511 y Fu(for)h Fo(i)i Ft(2)f(f)p Fu(1)p Fo(;)15 b Fu(2)p Ft(g)42 b Fu(and)e(let)382 3624 y Ft(A)25 b Fu(=)g Ft(A)649 3638 y Fr(1)707 3624 y Ft([)18 b(A)859 3638 y Fr(2)898 3624 y Fu(.)41 b(Since)28 b Ft(R)1277 3638 y Fr(1)1347 3624 y Fu(and)h Ft(R)1600 3638 y Fr(2)1669 3624 y Fu(are)h Fo(C)7 b Fu(-restricted)29 b(DP-simply)f(terminating,)h Ft(A)382 3737 y Fu(do)s(es)i(not)g(con)m(tain)h(rules)e(for)h(elemen)m (ts)g(of)h Fo(C)7 b Fu(.)42 b(Consequen)m(tly)-8 b(,)32 b(it)e(follo)m(ws)h(for)g(ev)m(ery)382 3850 y(term)24 b Fo(u)i Ft(2)e(T)f Fu(\()p Ft(F)929 3864 y Fn(i)958 3850 y Fo(;)15 b Ft(V)7 b Fu(\))25 b(that)g Fo(u)g Ft(#)1434 3864 y Fz(A)1520 3850 y Fu(=)g Fo(u)h Ft(#)1739 3864 y Fz(A)1795 3874 y Fd(i)1826 3850 y Fu(.)39 b(So)24 b(in)f(particular,) h Fo(s)h Ft(#)2663 3864 y Fz(A)2719 3873 y Fl(1)2774 3850 y Ft(6)p Fu(=)g Fo(t)g Ft(#)2973 3864 y Fz(A)3029 3873 y Fl(1)3092 3850 y Fu(implies)382 3963 y Fo(s)g Ft(#)495 3977 y Fz(A)571 3963 y Ft(6)p Fu(=)g Fo(t)g Ft(#)770 3977 y Fz(A)831 3963 y Fu(.)41 b(Moreo)m(v)m(er,)32 b(b)m(y)f(requiremen)m(ts)e(\(2\))j(and)d(\(3\),)j(the)f(TRSs)423 4148 y Ft(S)478 4162 y Fr(1)601 4148 y Fu(=)83 b Ft(f)p Fo(l)27 b Ft(#)899 4162 y Fz(A)960 4148 y Ft(!)e Fo(r)j Ft(#)1190 4162 y Fz(A)1277 4148 y Ft(j)50 b Fo(l)27 b Ft(!)f Fo(r)h Ft(2)e(R)1754 4162 y Fr(1)1824 4148 y Fu(and)30 b Fo(l)d Ft(#)2100 4162 y Fz(A)2186 4148 y Ft(6)p Fu(=)e Fo(r)j Ft(#)2396 4162 y Fz(A)2457 4148 y Ft(g)j([)755 4261 y(f)p Fo(s)25 b Ft(#)913 4275 y Fz(A)974 4261 y Ft(!)g Fo(t)g Ft(#)1193 4275 y Fz(A)1280 4261 y Ft(j)50 b(h)p Fo(s;)15 b(t)p Ft(i)31 b Fu(is)f(a)g(dep)s(endency)f(pair)g(in)g Ft(P)38 b Fu(and)30 b Fo(s)25 b Ft(#)2918 4275 y Fz(A)3004 4261 y Ft(6)p Fu(=)g Fo(t)g Ft(#)3203 4275 y Fz(A)3264 4261 y Ft(g)p Fo(;)423 4487 y Ft(S)478 4501 y Fr(2)601 4487 y Fu(=)83 b Ft(f)p Fo(l)27 b Ft(#)899 4501 y Fz(A)960 4487 y Ft(!)e Fo(r)j Ft(#)1190 4501 y Fz(A)1277 4487 y Ft(j)50 b Fo(l)27 b Ft(!)f Fo(r)h Ft(2)e(R)1754 4501 y Fr(2)1824 4487 y Fu(and)30 b Fo(l)d Ft(#)2100 4501 y Fz(A)2176 4487 y Ft(6)p Fu(=)e Fo(r)j Ft(#)2386 4501 y Fz(A)2447 4487 y Ft(g)382 4667 y Fu(are)j(also)f (constructor-sharing.)40 b(This)29 b(is)h(b)s(ecause)g(a)h(former)f (constructor)h(can)g(only)382 4779 y(b)s(e)39 b(a)h(de\014ned)f(sym)m (b)s(ol)f(in)h Ft(S)1414 4793 y Fr(1)1493 4779 y Fu(or)g Ft(S)1668 4793 y Fr(2)1748 4779 y Fu(if)f(it)i(is)e(not)i(in)f Fo(C)7 b Fu(,)42 b(i.e.)d(if)g(it)g(is)g(not)h(shared.)382 4892 y(Th)m(us)d(b)m(y)753 4884 y([)779 4892 y(Kurihara)e(and)j(Oh)m (uc)m(hi,)g(1992,)k(Thm.)37 b(3.10)2369 4884 y(])2396 4892 y(,)j(their)c(com)m(bined)h(system)382 5005 y Ft(S)32 b Fu(=)25 b Ft(S)620 5019 y Fr(1)680 5005 y Ft([)19 b(S)815 5019 y Fr(2)885 5005 y Fu(is)30 b(also)g(simply)e(terminating.)523 5118 y(Since)c(the)i(AFS)f Ft(A)g Fu(and)f(the)i(TRS)e Ft(S)32 b Fu(ob)m(viously)24 b(satisfy)h(the)g(conditions)f(\(2\))i (and)382 5231 y(\(3\))31 b(of)g(Def.)g(17,)h Ft(R)e Fu(is)f Fo(C)7 b Fu(-restricted)30 b(DP-simply)f(terminating.)682 b Fm(2)382 5403 y Fu(F)-8 b(or)31 b(example,)f(let)h Ft(R)1135 5417 y Fr(2)1204 5403 y Fu(b)s(e)f(the)h(\(simply)d (terminating\))i(TRS)1831 5652 y(17)p eop %%Page: 18 18 18 17 bop 1527 712 a Fo(x)20 b Fu(+)g(0)83 b Ft(!)g Fo(x)1411 849 y(x)20 b Fu(+)g Fo(s)p Fu(\()p Fo(y)s Fu(\))83 b Ft(!)g Fo(s)p Fu(\()p Fo(x)20 b Fu(+)g Fo(y)s Fu(\))1296 987 y Fo(x)g Fu(+)g(\()p Fo(y)k Fu(+)c Fo(z)t Fu(\))83 b Ft(!)g Fu(\()p Fo(x)21 b Fu(+)f Fo(y)s Fu(\))g(+)g Fo(z)t(:)382 1151 y Fu(As)31 b(the)g(TRS)f Ft(R)965 1165 y Fr(1)1035 1151 y Fu(for)h(subtraction)f(and)g(division)e(from)i(the)h (end)f(of)h(Sect.)h(5)f(is)f(DP-)382 1264 y(simply)24 b(terminating,)j(no)m(w)f(Thm.)g(18)h(allo)m(ws)f(us)g(to)i(conclude)e (DP-simple)f(termina-)382 1377 y(tion)g(of)i(the)f(com)m(bined)f (system)h Ft(R)1592 1391 y Fr(1)1643 1377 y Ft([)11 b(R)1792 1391 y Fr(2)1831 1377 y Fu(,)28 b(since)d(these)h(t)m(w)m(o)i(TRSs)d (are)h(constructor-)382 1489 y(sharing)j(\(they)i(b)s(oth)e(con)m(tain) i(the)g(same)f(constructors)h(0)g(and)f Fo(s)p Fu(\).)523 1602 y(There)42 b(are)g(ev)m(en)h(TRSs)e Ft(R)1518 1616 y Fr(1)1586 1602 y Ft([)27 b(R)1751 1616 y Fr(2)1833 1602 y Fu(where)41 b(DP-simple)g(termination)g(of)h(b)s(oth)382 1715 y Ft(R)459 1729 y Fr(1)532 1715 y Fu(and)33 b Ft(R)789 1729 y Fr(2)862 1715 y Fu(can)h(b)s(e)f(pro)m(v)m(ed)h(with)f(a)h (standard)f(tec)m(hnique)g(lik)m(e)g(the)h(lp)s(o,)g(whereas)382 1828 y(suc)m(h)28 b(standard)f(orderings)g(fail)g(if)h(one)g(w)m(an)m (ts)h(to)g(pro)m(v)m(e)g(DP-simple)e(termination)g(of)382 1941 y(their)i(union)f(directly)-8 b(.)40 b(Hence,)31 b(for)e(suc)m(h)h(examples)f(our)h(result)e(enables)i(automatic)382 2054 y(termination)f(pro)s(ofs)h(whic)m(h)f(w)m(ere)i(not)f(p)s (ossible)e(b)s(efore.)382 2226 y Fk(Example)34 b(19)46 b Fs(L)-5 b(et)33 b Ft(R)1198 2240 y Fr(1)1270 2226 y Fs(b)-5 b(e)32 b(the)h(TRS)1259 2390 y Fo(f)10 b Fu(\()p Fo(c)p Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(y)s Fu(\)\))85 b Ft(!)e Fo(f)10 b Fu(\()p Fo(c)p Fu(\()p Fo(x;)15 b(s)p Fu(\()p Fo(y)s Fu(\)\)\))1446 2527 y Fo(f)10 b Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\))83 b Ft(!)g Fo(f)10 b Fu(\()p Fo(d)p Fu(\()p Fo(f)g Fu(\()p Fo(x)p Fu(\)\)\))1571 2665 y Fo(f)g Fu(\()p Fo(x)p Fu(\))83 b Ft(!)g Fo(x)382 2829 y Fs(and)33 b(let)g Ft(R)763 2843 y Fr(2)835 2829 y Fs(c)-5 b(onsist)34 b(of)f(the)g(rule)g Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(x;)15 b(s)p Fu(\()p Fo(y)s Fu(\)\)\))28 b Ft(!)d Fo(g)s Fu(\()p Fo(c)p Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(y)s Fu(\)\))p Fs(.)523 2942 y Ft(R)600 2956 y Fr(1)686 2942 y Fs(is)46 b(DP-simply)g(terminating)h (\(using)g(the)f(AFS)g Fo(d)p Fu(\()p Fo(x)p Fu(\))51 b Ft(!)f Fo(x)c Fs(and)i(the)e(lp)-5 b(o)382 3055 y(c)g(omp)g(aring)33 b(left-to-right\),)e(but)f(it)g(is)g(not)h(simply)h(terminating.)42 b Ft(R)2746 3069 y Fr(2)2815 3055 y Fs(is)31 b(even)e(simply)382 3167 y(terminating)46 b(as)g(c)-5 b(an)46 b(b)-5 b(e)45 b(shown)i(with)f(the)f(lp)-5 b(o)47 b(c)-5 b(omp)g(aring)47 b(right-to-left.)80 b(Thus,)382 3280 y(DP-simple)32 b(termination)i(of) f(b)-5 b(oth)34 b(systems)g(c)-5 b(an)33 b(b)-5 b(e)33 b(veri\014e)-5 b(d)33 b(by)f(the)h(lp)-5 b(o.)523 3393 y(By)39 b(Thm.)h(18)g(their)g(union)f(is)g(also)i(DP-simply)e (terminating.)62 b(However,)41 b(the)382 3506 y(c)-5 b(onstr)g(aints)37 b(for)e(the)f(cycle)g Ft(fh)p Fo(G)p Fu(\()p Fo(c)p Fu(\()p Fo(x;)15 b(s)p Fu(\()p Fo(y)s Fu(\)\)\))p Fo(;)g(G)p Fu(\()p Fo(c)p Fu(\()p Fo(s)p Fu(\()p Fo(x)p Fu(\))p Fo(;)g(y)s Fu(\)\))p Ft(ig)40 b Fs(ar)-5 b(e)35 b(not)g(satis\014e)-5 b(d)36 b(by)382 3619 y(any)h(lp)-5 b(o)37 b(\(nor)g(by)f(any)h(rp)-5 b(o)38 b(nor)f(by)f(any)g(p)-5 b(olynomial)40 b(or)-5 b(dering\).)53 b(Thus,)38 b(ther)-5 b(e)37 b(ar)-5 b(e)382 3732 y(inde)g(e)g(d)32 b(TRSs)f(wher)-5 b(e)31 b(termination)h(of)f (the)g(subsystems)g(c)-5 b(an)31 b(b)-5 b(e)31 b(shown)h(with)f(dep)-5 b(en-)382 3845 y(dency)34 b(p)-5 b(airs)36 b(and)f(the)f(lp)-5 b(o,)36 b(but)e(\(without)h(our)g(mo)-5 b(dularity)37 b(r)-5 b(esult\))35 b(termination)g(of)382 3958 y(their)e(union)g(c)-5 b(annot)34 b(b)-5 b(e)33 b(pr)-5 b(ove)g(d)34 b(with)g(dep)-5 b(endency)34 b(p)-5 b(airs)34 b(and)f(the)g(lp)-5 b(o.)387 b Fm(2)523 4130 y Fu(W)-8 b(e)37 b(stress)f(that)g(Thm.)f(18)i(can)f(b) s(e)f(extended)h(to)h(comp)s(osable)e(systems.)57 b(The)382 4243 y(pro)s(of)44 b(is)f(almost)i(iden)m(tical)e(b)s(ecause)i(simple)d (termination)h(is)h(also)g(mo)s(dular)f(for)382 4356 y(comp)s(osable)32 b(systems)1211 4348 y([)1236 4356 y(Ohlebusc)m(h,)f(1995,)k(Thm.)c(5.16)2355 4348 y(])2382 4356 y(.)46 b(Only)31 b(the)h(pro)s(of)f(that)i Ft(S)3331 4370 y Fr(1)382 4468 y Fu(and)25 b Ft(S)609 4482 y Fr(2)675 4468 y Fu(are)h(indeed)f(comp)s(osable)h(needs)f(an)h(extra)h (e\013ort.)40 b(Therefore,)27 b Fo(C)7 b Fu(-restricted)382 4581 y(DP-simple)41 b(termination)h(is)g(ev)m(en)i(mo)s(dular)d(for)i (com)m(binations)f(of)h(TRSs)f(whic)m(h)382 4694 y(ha)m(v)m(e)27 b(common)g(de\014ned)e(sym)m(b)s(ols)f(pro)m(vided)h(that)i(b)s(oth)e (systems)h(con)m(tain)h(the)f(same)382 4807 y(rules)j(for)h(shared)g (de\014ned)f(sym)m(b)s(ols.)382 5087 y Fv(7)135 b(Conclusion)382 5290 y Fu(W)-8 b(e)33 b(ha)m(v)m(e)g(sho)m(wn)e(that)h(the)g(existing)f (mo)s(dularit)m(y)e(results)i(for)g(simple)f(termination)382 5403 y(of)j(disjoin)m(t)e(unions)g(can)i(b)s(e)f(extended)h(to)g (DP-quasi)g(simple)e(termination.)46 b(Under)1831 5652 y(18)p eop %%Page: 19 19 19 18 bop 382 548 a Fu(certain)39 b(restrictions)f(a)h(similar)e(mo)s (dularit)m(y)g(result)h(also)h(holds)f(for)h(constructor-)382 661 y(sharing)33 b(\(and)g(ev)m(en)i(comp)s(osable\))f(com)m (binations.)51 b(F)-8 b(uture)34 b(w)m(ork)g(will)e(include)g(an)382 774 y(examination)37 b(of)h(whether)f(suc)m(h)g(an)h(extension)f(is)g (also)g(p)s(ossible)f(for)h(hierarc)m(hical)382 887 y(com)m(binations.) k(In)31 b(summary)-8 b(,)30 b(the)h(progress)g(in)f(automated)i (termination)d(pro)m(ving)382 1000 y(whic)m(h)g(w)m(as)h(made)g(p)s (ossible)e(b)m(y)i(the)g(dev)m(elopmen)m(t)g(of)g(dep)s(endency)f (pairs)g(no)m(w)h(also)382 1112 y(has)k(a)h(coun)m(terpart)g(in)e(the)i (area)g(of)g(mo)s(dularit)m(y)-8 b(.)51 b(Dep)s(endency)34 b(pairs)f(enable)h(au-)382 1225 y(tomated)j(termination)d(pro)s(ofs)h (of)h(non-simply)d(terminating)i(TRSs)f(and)h(no)m(w)h(our)382 1338 y(results)29 b(allo)m(w)h(to)h(p)s(erform)e(them)h(in)f(a)i(mo)s (dular)d(w)m(a)m(y)-8 b(.)523 1451 y(The)29 b(presen)m(t)h(w)m(ork)g (completes)g(the)g(results)f(in)2216 1443 y([)2242 1451 y(Arts)g(and)h(Giesl,)f(1998)3056 1443 y(])3083 1451 y(,)h(where)382 1564 y(the)j(theory)h(of)f(dep)s(endency)f(pairs)g(w)m (as)h(re\014ned)g(in)f(a)h(mo)s(dular)e(w)m(a)m(y)k(\(cf.)f(Thm.)e(6\)) 382 1677 y(and)h(where)g(sev)m(eral)i(new)e(mo)s(dularit)m(y)f (criteria)h(for)h Fs(innermost)43 b Fu(termination)33 b(w)m(ere)382 1790 y(presen)m(ted.)39 b(Ho)m(w)m(ev)m(er,)27 b(these)e(criteria)e(are)i(only)e(applicable)f(for)i(termination)g(pro) s(ofs)382 1903 y(of)k(lo)s(cally)f(con\015uen)m(t)h(o)m(v)m(erla)m(y)i (systems)e(\(and)g(in)e(particular,)i(non-o)m(v)m(erlapping)f(sys-)382 2016 y(tems\))649 2008 y([)674 2016 y(Gramlic)m(h,)48 b(1995)1295 2008 y(])1322 2016 y(.)83 b(But)45 b(in)e(practice)i(there) g(are)g(man)m(y)g(cases)h(in)d(whic)m(h)382 2129 y(innermost)32 b(termination)g(is)g(not)h(su\016cien)m(t)g(for)g(termination.)48 b(Up)32 b(to)i(no)m(w,)g(due)f(to)382 2242 y(missing)i(mo)s(dularit)m (y)h(results,)i(the)g(adv)-5 b(an)m(tages)39 b(of)f(dep)s(endency)e (pairs)g(could)g(not)382 2355 y(b)s(e)26 b(fully)e(exploited)h(for)h (these)h(systems.)40 b(So)26 b(compared)g(to)h(previous)e(w)m(ork)i(on) f(mo)s(d-)382 2467 y(ularit)m(y)-8 b(,)35 b(the)h(mo)s(dularit)m(y)c (criteria)j(dev)m(elop)s(ed)f(in)f(the)j(presen)m(t)e(pap)s(er)g (represen)m(t)h(a)382 2580 y(signi\014can)m(t)29 b(extension.)382 2774 y Fy(Ac)m(kno)m(wledgemen)m(ts.)44 b Fx(W)-7 b(e)31 b(thank)f(Aart)h(Middeldorp)f(for)g(man)n(y)g(helpful)h(remarks)e(and) 382 2874 y(hin)n(ts.)42 b(This)29 b(w)n(ork)f(w)n(as)h(partially)f (supp)r(orted)h(b)n(y)g(the)h(DF)n(G)g(under)f(gran)n(t)f(W)-7 b(a)29 b(652/7-2)d(as)382 2974 y(part)h(of)h(the)g(fo)r(cus)f(program)f (\\Deduktion".)382 3248 y Fv(References)382 3423 y Fx([)405 3430 y(Arts)h(and)h(Giesl,)g(1997a)1196 3423 y(])1262 3430 y(T.)h(Arts)g(and)g(J.)g(Giesl.)42 b(Automatically)29 b(Pro)n(ving)e(T)-7 b(ermination)473 3530 y(Where)35 b(Simpli\014cation)g(Orderings)e(F)-7 b(ail.)58 b(In)35 b Fw(Pr)l(o)l(c.)i(T)-6 b(APSOFT)36 b('97)p Fx(,)i(pages)c(261{272.)473 3629 y(LNCS)28 b(1214,)e(1997.)382 3772 y([)405 3779 y(Arts)h(and)h(Giesl,)g(1997b)1200 3772 y(])1266 3779 y(T.)38 b(Arts)h(and)f(J.)f(Giesl.)69 b(Pro)n(ving)36 b(Innermost)h(Normalisation)473 3878 y(Automatically.)g(In)28 b Fw(Pr)l(o)l(c.)i(R)-6 b(T)g(A)29 b('97)p Fx(,)g(pages)e(157{171.)d (LNCS)k(1232,)e(1997.)382 4021 y([)405 4028 y(Arts)h(and)h(Giesl,)g (1997c)1191 4021 y(])1257 4028 y(T.)45 b(Arts)h(and)f(J.)g(Giesl.)89 b(T)-7 b(ermination)45 b(of)g(T)-7 b(erm)45 b(Rewriting)473 4127 y(Using)35 b(Dep)r(endency)h(P)n(airs.)56 b(T)-7 b(ec)n(hnical)34 b(Rep)r(ort)h(IBN)g(97/46,)f(TU)h(Darmstadt,)i(1997.) 473 4227 y(h)n (ttp://www.inferenzsysteme.informatik.tu-darmstadt.de/~rep)r (orts/notes/ibn-)473 4327 y(97-46.ps.)26 b(Revised)h(v)n(ersion)f(to)i (app)r(ear)e(in)i Fw(The)l(or)l(etic)l(al)k(Computer)e(Scienc)l(e)p Fx(.)382 4469 y([)405 4476 y(Arts)d(and)h(Giesl,)g(1998)1154 4469 y(])1220 4476 y(T.)f(Arts)g(and)g(J.)g(Giesl.)35 b(Mo)r(dularit)n(y)27 b(of)g(T)-7 b(ermination)26 b(Using)h(De-)473 4576 y(p)r(endency)h(P)n(airs.)35 b(In)28 b Fw(Pr)l(o)l(c.)j(R)-6 b(T)g(A)28 b('98)p Fx(,)h(pages)e(226{240.)d(LNCS)k(1379,)e(1998.)382 4718 y([)405 4725 y(Courcelle,)h(1990)961 4718 y(])1027 4725 y(B.)g(Courcelle.)34 b(Recursiv)n(e)26 b(Applicativ)n(e)g(Program) e(Sc)n(hemes.)35 b(In)27 b(J.)g(v)-5 b(an)473 4825 y(Leeu)n(w)n(en,)37 b(editor,)g Fw(Handb)l(o)l(ok)h(of)f(The)l(or)l(etic)l(al)i(Computer)e (Scienc)l(e)p Fx(,)h(v)n(olume)d(B,)g(pages)473 4924 y(459{492.)25 b(North-Holland,)i(1990.)382 5067 y([)405 5074 y(Dersho)n(witz)g(and)g(Jouannaud,)g(1990)1614 5067 y(])1680 5074 y(N.)36 b(Dersho)n(witz)e(and)g(J.-P)-7 b(.)34 b(Jouannaud.)58 b(Rewrite)473 5174 y(Systems.)35 b(In)27 b(J.)f(v)-5 b(an)27 b(Leeu)n(w)n(en,)f(editor,)g Fw(Handb)l(o)l(ok)k(of)g(The)l(or)l(etic)l(al)g(Computer)f(Scienc)l(e)p Fx(,)473 5273 y(v)n(olume)e(B,)h(pages)e(243{320.)f(North-Holland,)i (1990.)1831 5652 y Fu(19)p eop %%Page: 20 20 20 19 bop 382 541 a Fx([)405 548 y(Dersho)n(witz,)27 b(1987)1027 541 y(])1093 548 y(N.)h(Dersho)n(witz.)36 b(T)-7 b(ermination)26 b(of)i(Rewriting.)36 b Fw(Journal)29 b(of)i(Symb)l(olic)473 648 y(Computation)p Fx(,)e(3:69{115,)24 b(1987.)382 774 y([)405 781 y(Gramlic)n(h,)j(1994)959 774 y(])1025 781 y(B.)g(Gramlic)n(h.)34 b(Generalized)26 b(Su\016cien)n(t)h(Conditions)f(for)g(Mo)r(dular)g(T)-7 b(er-)473 880 y(mination)38 b(of)f(Rewriting.)66 b Fw(Applic)l(able)41 b(A)n(lgebr)l(a)f(in)f(Engine)l(ering,)j(Commutation)d(and)473 980 y(Computing)p Fx(,)29 b(5:131{158,)24 b(1994.)382 1106 y([)405 1113 y(Gramlic)n(h,)j(1995)959 1106 y(])1025 1113 y(B.)21 b(Gramlic)n(h.)k(Abstract)c(Relations)f(Bet)n(w)n(een)h (Restricted)f(T)-7 b(ermination)473 1213 y(and)20 b(Con\015uence)f (Prop)r(erties)f(of)h(Rewrite)h(System.)j Fw(F)-6 b(und.)22 b(Informatic)l(ae)p Fx(,)h(24:3{23,)c(1995.)382 1339 y([)405 1346 y(Gramlic)n(h,)27 b(1996)959 1339 y(])1025 1346 y(B.)34 b(Gramlic)n(h.)56 b Fw(T)-6 b(ermination)36 b(and)g(Con\015uenc)l(e)g(Pr)l(op)l(erties)g(of)h(Struc-)473 1446 y(tur)l(e)l(d)29 b(R)l(ewrite)h(Systems)p Fx(.)36 b(PhD)28 b(thesis,)g(Univ)n(ersit\177)-42 b(at)27 b(Kaiserslautern,)e (German)n(y)-7 b(,)27 b(1996.)382 1572 y([)405 1579 y(Klop,)g(1992)799 1572 y(])866 1579 y(J.)j(W.)h(Klop.)45 b(T)-7 b(erm)30 b(Rewriting)h(Systems.)45 b(In)31 b(S.)g(Abramsky)-7 b(,)30 b(D.)h(M.)g(Gab-)473 1679 y(ba)n(y)-7 b(,)36 b(and)f(T.)g(S.)g (E.)f(Maibaum,)i(editors,)g Fw(Handb)l(o)l(ok)h(of)g(L)l(o)l(gic)g(in)g (Computer)f(Scienc)l(e)p Fx(,)473 1778 y(v)n(olume)27 b(2,)h(pages)e(1{116.)f(Oxford)i(Univ)n(ersit)n(y)g(Press,)f(New)i(Y)-7 b(ork,)27 b(1992.)382 1904 y([)405 1911 y(Krishna)f(Rao,)h(1994)1082 1904 y(])1149 1911 y(M.)35 b(R.)h(K.)f(Krishna)e(Rao.)59 b(Simple)36 b(T)-7 b(ermination)35 b(of)g(Hierarc)n(hical)473 2011 y(Com)n(binations)27 b(of)g(T)-7 b(erm)27 b(Rewriting)g(Systems.) 37 b(In)27 b Fw(Pr)l(o)l(c.)k(ST)-6 b(A)n(CS)29 b('94)p Fx(,)g(pages)d(203{223.)473 2111 y(LNCS)i(789,)f(1994.)382 2237 y([)405 2244 y(Kurihara)f(and)h(Oh)n(uc)n(hi,)g(1992)1399 2237 y(])1466 2244 y(M.)34 b(Kurihara)e(and)i(A.)g(Oh)n(uc)n(hi.)55 b(Mo)r(dularit)n(y)33 b(of)h(Simple)473 2343 y(T)-7 b(ermination)29 b(of)g(T)-7 b(erm)28 b(Rewriting)h(Systems)g(with)g(Shared)g (Constructors.)39 b Fw(The)l(or)l(etic)l(al)473 2443 y(Computer)31 b(Scienc)l(e)p Fx(,)d(103:273{282,)23 b(1992.)382 2569 y([)405 2576 y(Lankford,)k(1979)953 2569 y(])1019 2576 y(D.)c(S.)g(Lankford.)28 b(On)23 b(Pro)n(ving)d(T)-7 b(erm)23 b(Rewriting)f(Systems)h(are)f(No)r(ethe-)473 2676 y(rian.)37 b(T)-7 b(ec)n(hnical)27 b(Rep)r(ort)g(MTP-3,)f (Louisiana)h(T)-7 b(ec)n(h.)27 b(Univ.,)h(Ruston,)g(LA,)g(1979.)382 2802 y([)405 2809 y(Middeldorp)f(and)h(Zan)n(tema,)e(1997)1546 2802 y(])1613 2809 y(A.)32 b(Middeldorp)g(and)g(H.)g(Zan)n(tema.)49 b(Simple)32 b(T)-7 b(ermi-)473 2909 y(nation)28 b(of)f(Rewrite)h (Systems.)36 b Fw(The)l(or)l(etic)l(al)c(Computer)e(Scienc)l(e)p Fx(,)e(175:127{158,)23 b(1997.)382 3035 y([)405 3042 y(Middeldorp,)k(1989)1041 3035 y(])1108 3042 y(A.)j(Middeldorp.)44 b(A)30 b(Su\016cien)n(t)g(Condition)g(for)f(the)i(T)-7 b(ermination)29 b(of)473 3142 y(the)24 b(Direct)f(Sum)g(of)g(T)-7 b(erm)23 b(Rewriting)f(Systems.)30 b(In)23 b Fw(Pr)l(o)l(c.)j(LICS)g ('89,)i Fx(p.)23 b(396-401,)e(1989.)382 3268 y([)405 3275 y(Middeldorp,)27 b(1990)1041 3268 y(])1108 3275 y(A.)19 b(Middeldorp.)k Fw(Mo)l(dular)g(Pr)l(op)l(erties)g(of)f(T)-6 b(erm)22 b(R)l(ewriting)g(Systems)p Fx(.)473 3374 y(PhD)28 b(thesis,)g(V)-7 b(rije)28 b(Univ)n(ersiteit)f(te)h(Amsterdam,)f(1990.) 382 3501 y([)405 3508 y(Ohlebusc)n(h,)g(1994a)1042 3501 y(])1108 3508 y(E.)21 b(Ohlebusc)n(h.)k Fw(Mo)l(dular)g(Pr)l(op)l (erties)g(of)g(Comp)l(osable)g(T)-6 b(erm)24 b(R)l(ewrit-)473 3607 y(ing)31 b(Systems)p Fx(.)36 b(PhD)28 b(thesis,)f(Univ)n (ersit\177)-42 b(at)27 b(Bielefeld,)h(1994.)382 3733 y([)405 3740 y(Ohlebusc)n(h,)f(1994b)1046 3733 y(])1113 3740 y(E.)42 b(Ohlebusc)n(h.)79 b(On)42 b(the)h(Mo)r(dularit)n(y)e(of)h (T)-7 b(ermination)42 b(of)g(T)-7 b(erm)473 3840 y(Rewriting)28 b(Systems.)36 b Fw(The)l(or)l(etic)l(al)c(Computer)e(Scienc)l(e)p Fx(,)e(136:333{360,)23 b(1994.)382 3966 y([)405 3973 y(Ohlebusc)n(h,)k(1995)1000 3966 y(])1067 3973 y(E.)22 b(Ohlebusc)n(h.)30 b(Mo)r(dular)22 b(Prop)r(erties)g(of)h(Comp)r (osable)f(T)-7 b(erm)23 b(Rewrit-)473 4073 y(ing)28 b(Systems.)37 b Fw(Journal)29 b(of)i(Symb)l(olic)g(Computation)p Fx(,)d(20:1{41,)d (1995.)382 4199 y([)405 4206 y(Rusino)n(witc)n(h,)i(1987)1068 4199 y(])1135 4206 y(M.)j(Rusino)n(witc)n(h.)44 b(On)30 b(T)-7 b(ermination)29 b(of)h(the)h(Direct)f(Sum)h(of)f(T)-7 b(erm)473 4306 y(Rewriting)28 b(Systems.)36 b Fw(Information)31 b(Pr)l(o)l(c)l(essing)f(L)l(etters)p Fx(,)e(26:65{70,)c(1987.)382 4432 y([)405 4439 y(Stein)n(bac)n(h,)j(1995)974 4432 y(])1041 4439 y(J.)g(Stein)n(bac)n(h.)37 b(Simpli\014cation)27 b(Orderings:)36 b(History)26 b(of)i(Results.)37 b Fw(F)-6 b(un-)473 4538 y(damenta)31 b(Informatic)l(ae)p Fx(,)e(24:47{87,)24 b(1995.)382 4665 y([)405 4672 y(T)-7 b(o)n(y)n(ama,)26 b(1987a)952 4665 y(])1018 4672 y(Y.)g(T)-7 b(o)n(y)n(ama.)32 b(Coun)n(terexamples)24 b(to)i(T)-7 b(ermination)25 b(for)g(the)h (Direct)g(Sum)473 4771 y(of)i(T)-7 b(erm)27 b(Rewriting)h(Systems.)36 b Fw(Information)31 b(Pr)l(o)l(c)l(essing)f(L)l(etters)p Fx(,)d(25:141{143,)d(1987.)382 4897 y([)405 4904 y(T)-7 b(o)n(y)n(ama,)26 b(1987b)956 4897 y(])1022 4904 y(Y.)k(T)-7 b(o)n(y)n(ama.)40 b(On)29 b(the)g(Ch)n(urc)n(h-Rosser)e(Prop)r(ert)n(y) h(for)g(the)i(Direct)f(Sum)473 5004 y(of)f(T)-7 b(erm)27 b(Rewriting)h(Systems.)36 b Fw(Journal)30 b(of)g(the)g(A)n(CM)p Fx(,)f(34:128{143,)23 b(1987.)382 5130 y([)405 5137 y(T)-7 b(o)n(y)n(ama)26 b Fw(et)j(al.)p Fx(,)g(1995)1122 5130 y(])1188 5137 y(Y.)f(T)-7 b(o)n(y)n(ama,)25 b(J.)h(W.)h(Klop,)g(and)f (H.)h(P)-7 b(.)27 b(Barendregt.)33 b(T)-7 b(ermination)473 5237 y(for)23 b(the)g(Direct)g(Sum)h(of)f(Left-Linear)f(T)-7 b(erm)22 b(Rewriting)h(Systems.)29 b Fw(Journal)c(of)i(the)e(A)n(CM)p Fx(,)473 5337 y(42:1275{1304,)e(1995.)1831 5652 y Fu(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF