%!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%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, compressed %DVIPSSource: TeX output 1999.07.14:1046 %%BeginProcSet: texc.pro %! /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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b(to)h(the)h(approac)n(h)d(of)i(Marc)n(hiori)e([Mar96)n(],)j(w)n(e)e (will)i(sho)n(w)e(ho)n(w)g(ev)n(ery)g(de-)523 1247 y(terministic)27 b(CTRS)h Fm(R)g Fw(can)e(b)r(e)i(transformed)e(in)n(to)h(an)g (unconditional)g(TRS)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\))28 b(suc)n(h)523 1347 y(that)f(termination)f(of)h Fn(U)9 b Fw(\()p Fm(R)p Fw(\))27 b(implies)g(quasi-reductivit)n(y)e(of)i Fm(R)p Fw(.)g(This)g(result)f(is)h(in)n(terest-)523 1447 y(ing)c(b)r(ecause)f(standard)g(metho)r(ds)i(for)e(pro)n(ving)f (termination)i(of)g(TRSs)g(can)g(b)r(e)g(emplo)n(y)n(ed)523 1546 y(to)31 b(infer)g(quasi-reductivit)n(y)e(automatically;)h(see)h ([A)n(G97a)n(].)g(On)g(the)g(one)g(hand,)g(an)g(ex-)523 1646 y(ample)c(in)g([Mar95)n(])g(sho)n(ws)f(that)i(quasi-reductivit)n (y)d(of)i Fm(R)g Fw(do)r(es)g(not)g(imply)g(termination)523 1745 y(of)21 b Fn(U)9 b Fw(\()p Fm(R)p Fw(\))23 b(but)f(on)f(the)h (other)f(hand)g(the)h(main)f(theorem)g(of)h(this)g(pap)r(er)f(states)g (that)g(it)h(do)r(es)523 1845 y(imply)h Fp(innermost)f Fw(termination)f(of)i Fn(U)9 b Fw(\()p Fm(R)p Fw(\).)22 b(\(The)h(pro)r(of)f(of)g(this)g(theorem)g(is)g(non-trivial.\))523 1945 y(This)28 b(has)f(t)n(w)n(o)f(striking)h(consequences.)648 2046 y(Firstly)-7 b(,)30 b(Gramlic)n(h)g([Gra95)n(])h(sho)n(w)n(ed)f (that)h(for)f(non-o)n(v)n(erlapping)e(TRSs)i(innermost)523 2146 y(termination)19 b(coincides)f(with)i(termination.)f(Since)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\))20 b(inherits)f(non-o)n(v)n (erlappingness)523 2246 y(from)38 b(a)g(syn)n(tactically)f (deterministic)i(CTRS)f Fm(R)p Fw(,)h(termination)f(of)h Fn(U)9 b Fw(\()p Fm(R)p Fw(\))39 b(and)f(quasi-)523 2345 y(reductivit)n(y)e(of)h Fm(R)g Fw(are)f(equiv)-5 b(alen)n(t)36 b(for)g(this)h(class)f(of)g(CTRSs.)h(Consequen)n(tly)-7 b(,)36 b(quasi-)523 2445 y(reductivit)n(y)c(is)g(mo)r(dular)g(for)g (non-o)n(v)n(erlapping)d(syn)n(tactically)i(deterministic)i(comp)r(os-) 523 2544 y(able)27 b(CTRSs.)648 2646 y(Secondly)-7 b(,)38 b(Ganzinger)g(and)g(W)-7 b(aldmann)39 b([GW93])g(pro)n(v)n(ed)e(that)i (a)f(translation)g(of)523 2746 y(a)c(w)n(ell-mo)r(ded)h(logic)e (program)g Fm(P)41 b Fw(in)n(to)35 b(a)f(quasi-reductiv)n(e)f (deterministic)i(CTRS)g Fn(R)3349 2758 y Fl(P)523 2845 y Fw(yields)26 b(a)g(termination)f(pro)r(of)h(for)g Fm(P)7 b Fw(.)25 b(Using)h(an)g(imp)r(erativ)n(e)g(pro)r(cedure,)f(Arts)h(and) g(Zan-)523 2945 y(tema)e([AZ95)o(,AZ96)o(])f(transformed)g(a)g(logic)g (program)e Fm(P)30 b Fw(directly)23 b(in)n(to)h(an)f Fp(unc)l(onditional)523 3045 y Fw(TRS)31 b(\(whic)n(h)h(in)f(essence)g (coincides)f(with)i Fn(U)9 b Fw(\()p Fn(R)2094 3057 y Fl(P)2150 3045 y Fw(\)\))32 b(and)f(sho)n(w)n(ed)e(that)j(innermost)f (ter-)523 3144 y(mination)e(of)f(this)h(system)f(ensures)g(termination) g(of)h Fm(P)7 b Fw(.)28 b(Consequen)n(tly)-7 b(,)28 b(it)h(is)g(remark) n(ed)523 3244 y(in)39 b([AZ95)o(])g(that)g(the)h(suggested)d(metho)r(d) j(\\is)e(applicable)g(to)h(a)g(wider)f(class)g(of)h(logic)523 3343 y(programs")30 b(and)i(hence)g(it)h(is)f(\\stronger)f(than)h(the)h (other)f(results".)f(Although)i Fn(U)9 b Fw(\()p Fn(R)3317 3355 y Fl(P)3373 3343 y Fw(\))523 3443 y(is)31 b(not)f(necessarily)f (non-o)n(v)n(erlapping,)f(it)j(can)g(b)r(e)g(sho)n(wn)f(that)h(in)g (this)g(particular)e(case)523 3543 y(innermost)22 b(termination)g(and)g (termination)g(are)g(equiv)-5 b(alen)n(t.)22 b(A)g(consequence)g(is)g (the)h(sur-)523 3642 y(prising)29 b(fact)h(that)g(the)g(metho)r(ds)g (of)g(Ganzinger)f(&)g(W)-7 b(aldmann)30 b(and)g(Arts)g(&)f(Zan)n(tema) 523 3742 y(are)d(equally)g(p)r(o)n(w)n(erful,)h(in)g(the)g(sense)g (that)g(ev)n(ery)f(logic)g(program)f(whic)n(h)i(can)f(b)r(e)i(sho)n(wn) 523 3842 y(as)37 b(terminating)h(b)n(y)f(one)h(of)f(the)i(metho)r(ds)f (can)f(b)r(e)h(sho)n(wn)f(as)h(terminating)f(b)n(y)h(b)r(oth)523 3941 y(metho)r(ds.)24 b(Our)g(new)g(t)n(w)n(o-stage)e(transformation)h (approac)n(h)f(to)i(pro)n(ving)e(termination)i(of)523 4041 y(an)i(LR-w)n(ell-mo)r(ded)f(logic)g(program)f(can)h(b)r(e)i (automated)e(and)h(moreo)n(v)n(er)d(it)j(has)g(certain)523 4141 y(adv)-5 b(an)n(tages)34 b(o)n(v)n(er)g(the)i(direct)g (transformation.)e(These)h(adv)-5 b(an)n(tages)34 b(are)h(discussed)g (at)523 4240 y(the)28 b(end)g(of)f(the)h(pap)r(er.)523 4516 y Fq(2)112 b(Preliminaries)523 4725 y Fw(The)29 b(reader)f(is)h(assumed)g(to)g(b)r(e)g(familiar)g(with)h(the)f(basic)g (concepts)g(of)g(term)g(rewriting)523 4825 y(whic)n(h)e(can)f(for)h (instance)g(b)r(e)g(found)g(in)g(the)h(textb)r(o)r(ok)e(of)h(Baader)f (and)h(Nipk)n(o)n(w)f([BN98)o(].)523 4924 y(Here)h(w)n(e)h(will)f(only) h(recall)e(the)i(de\014nitions)g(whic)n(h)f(are)g(crucial)g(to)g(this)h (pap)r(er.)p eop %%Page: 3 3 3 2 bop 648 448 a Fw(In)33 b(a)f(CTRS)h(\()p Fm(F)8 b Fn(;)14 b Fm(R)p Fw(\))34 b(rules)e(ha)n(v)n(e)g(the)i(form)e Fn(l)h Fm(!)f Fn(r)j Fm(\()d Fn(s)2514 460 y Fk(1)2583 448 y Fw(=)g Fn(t)2710 460 y Fk(1)2747 448 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)2971 460 y Fj(k)3043 448 y Fw(=)32 b Fn(t)3170 460 y Fj(k)3243 448 y Fw(with)523 548 y Fn(l)r(;)14 b(r)n(;)g(s)698 560 y Fk(1)735 548 y Fn(;)g(:)g(:)g(:)f(;)h(s)958 560 y Fj(k)999 548 y Fw(,)30 b Fn(t)1082 560 y Fk(1)1120 548 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)1334 560 y Fj(k)1403 548 y Fm(2)28 b(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\).)31 b Fn(l)h Fw(ma)n(y)e(not)g(b)r(e)h(a)f(v)-5 b(ariable.)30 b(W)-7 b(e)30 b(frequen)n(tly)h(ab-)523 648 y(breviate)j(the)h(conditional)f(part)g(of)h(the)g(rule)f(b)n(y)h Fn(c)p Fw(.)g(If)g(a)f(rule)g(has)h(no)f(conditions,)g(w)n(e)523 747 y(write)23 b Fn(l)h Fm(!)g Fn(r)r Fw(,)g(demand)f(that)g Fm(V)7 b Fn(ar)r Fw(\()p Fn(r)r Fw(\))25 b Fm(\022)e(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\),)23 b(and)g(call)g Fn(l)i Fm(!)e Fn(r)j Fw(an)d(unconditional)f(rule.)523 847 y(The)27 b(=)g(sym)n(b)r(ol)g(in)h(the)g(conditions)f(can)g(b)r(e)g(in)n (terpreted)g(in)h(di\013eren)n(t)f(w)n(a)n(ys,)f(leading)h(to)523 946 y(di\013eren)n(t)j(rewrite)f(relations)f(asso)r(ciated)g(with)i Fm(R)p Fw(.)g(F)-7 b(or)29 b(instance,)h(in)g(a)f Fp(join)37 b Fw(CTRS)30 b(the)523 1046 y(=)d(sym)n(b)r(ol)h(stands)f(for)g (joinabilit)n(y)h(\()p Fm(#)1755 1058 y Fl(R)1816 1046 y Fw(\).)g(This)g(pap)r(er)f(deals)g(with)i Fp(oriente)l(d)37 b Fw(CTRSs)28 b(in)523 1146 y(whic)n(h)g(the)g(equalit)n(y)g(signs)f (are)g(in)n(terpreted)g(as)h(reac)n(habilit)n(y)e(\()p Fm(!)2659 1116 y Fl(\003)2659 1169 y(R)2720 1146 y Fw(\).)j(A)f Fp(normal)h Fw(CTRS)523 1245 y(\()p Fm(F)8 b Fn(;)14 b Fm(R)p Fw(\))35 b(is)f(an)h(orien)n(ted)e(CTRS)i(in)f(whic)n(h)h(the) f(rewrite)g(rules)g(are)f(sub)5 b(ject)35 b(to)f(the)h(ad-)523 1345 y(ditional)e(constrain)n(t)f(that)h(ev)n(ery)f Fn(t)1669 1357 y Fj(j)1737 1345 y Fw(is)h(a)g(ground)f(normal)g(form)h(with)g (resp)r(ect)g(to)g Fm(R)3338 1357 y Fj(u)3382 1345 y Fw(,)523 1445 y(where)27 b Fm(R)833 1457 y Fj(u)900 1445 y Fw(=)c Fm(f)p Fn(l)h Fm(!)f Fn(r)30 b Fm(j)e Fn(l)c Fm(!)f Fn(r)j Fm(\()d Fn(c)g Fm(2)h(Rg)p Fw(.)648 1551 y(F)-7 b(or)30 b(ev)n(ery)h(rule)g Fn(\032)e Fw(:)g Fn(l)i Fm(!)f Fn(r)i Fm(\()d Fn(c)p Fw(,)j(the)g(set)f(of)g(v)-5 b(ariables)31 b(o)r(ccurring)f(in)h Fn(\032)h Fw(is)f(denoted)523 1651 y(b)n(y)26 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(\032)p Fw(\))26 b(and)g(the)h(set)e(of)h(extra)f(v)-5 b(ariables)25 b(in)h Fn(\032)g Fw(is)g Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(\032)p Fw(\))24 b(=)f Fm(V)7 b Fn(ar)r Fw(\()p Fn(\032)p Fw(\))16 b Fm(n)e(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\).)26 b(A)h(1-)523 1751 y(CTRS)22 b(has)g(no)f(extra)g(v)-5 b(ariables,)21 b(a)h(2-CTRS)f(has)h(no)f(extra)g(v)-5 b(ariables)21 b(on)h(the)g(righ)n(t-hand)523 1850 y(sides)27 b(of)g(the)h(rules,)f(and)g(a)g(3-CTRS)g(ma)n(y)g(con)n(tain)g(extra)f (v)-5 b(ariables)26 b(on)h(the)h(righ)n(t-hand)523 1950 y(sides)23 b(of)g(the)h(rules)e(pro)n(vided)g(that)i(these)f(also)f(o)r (ccur)g(in)i(the)f(corresp)r(onding)f(conditional)523 2049 y(part)27 b(\(i.e.,)h Fm(V)7 b Fn(ar)r Fw(\()p Fn(r)r Fw(\))25 b Fm(\022)e(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))18 b Fm([)h(V)7 b Fn(ar)r Fw(\()p Fn(c)p Fw(\)\).)523 2350 y Fq(3)112 b(Quasi-Reductiv)m(e)37 b(Deterministic)d(3-CTRSs)523 2585 y Fw(First)27 b(of)g(all,)f(w)n(e)h(will)g(review)f(the)i (de\014nition)f(of)g(deterministic)g(systems)f(from)h([Gan91)o(].)523 2794 y Fi(De\014nition)k(1.)41 b Fp(A)n(n)36 b(oriente)l(d)h(3-CTRS)g Fm(R)g Fp(is)h(c)l(al)t(le)l(d)47 b Fw(deterministic)37 b Fp(if)g(\(after)h(appr)l(o-)523 2894 y(priately)45 b(changing)g(the)e(or)l(der)h(of)h(the)e(c)l(onditions)i(in)e(the)h(r)l (ewrite)g(rules\))f(for)h(every)523 2993 y Fn(l)33 b Fm(!)e Fn(r)j Fm(\()d Fn(s)919 3005 y Fk(1)987 2993 y Fm(!)h Fn(t)1132 3005 y Fk(1)1169 2993 y Fn(;)14 b(:)g(:)g(:)f(;)h(s) 1392 3005 y Fj(k)1464 2993 y Fm(!)31 b Fn(t)1608 3005 y Fj(k)1683 2993 y Fp(in)k Fm(R)f Fp(and)h(every)g Fw(1)c Fm(\024)g Fn(i)f Fm(\024)h Fn(k)s Fp(,)k(we)f(have)i Fm(V)7 b Fn(ar)r Fw(\()p Fn(s)3249 3005 y Fj(i)3277 2993 y Fw(\))31 b Fm(\022)523 3103 y(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))30 b Fm([)870 3040 y Fh(S)939 3061 y Fj(i)p Fl(\000)p Fk(1)939 3128 y Fj(j)s Fk(=1)1072 3103 y Fm(V)7 b Fn(ar)r Fw(\()p Fn(t)1275 3115 y Fj(j)1311 3103 y Fw(\))p Fp(.)46 b(In)f(the)g(fol)t(lowing,)k(we)c(wil)t(l)i(fr)l(e)l(quently)e (use)g(the)h(notation)523 3227 y Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)777 3239 y Fj(i)806 3227 y Fw(\))23 b(=)g Fm(V)7 b Fn(ar)r Fw(\()p Fn(t)1152 3239 y Fj(i)1180 3227 y Fw(\))19 b Fm(n)e Fw(\()p Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))20 b Fm([)1647 3165 y Fh(S)1717 3186 y Fj(i)p Fl(\000)p Fk(1)1717 3252 y Fj(j)s Fk(=1)1849 3227 y Fm(V)7 b Fn(ar)r Fw(\()p Fn(t)2052 3239 y Fj(j)2088 3227 y Fw(\)\))p Fp(.)648 3430 y Fw(The)29 b(rewrite)f(relation)g Fm(!)1492 3442 y Fl(R)1582 3430 y Fw(asso)r(ciated)g(with)h(an)g(orien)n(ted)f (deterministic)i(3-CTRS)523 3530 y Fm(R)j Fw(is)f(de\014ned)h(b)n(y:)f Fn(s)f Fm(!)1301 3542 y Fl(R)1393 3530 y Fn(t)h Fw(if)h(and)f(only)g (if)h(there)f(exists)g(a)g(rewrite)f(rule)h Fn(\032)f Fw(:)g Fn(l)h Fm(!)f Fn(r)j Fm(\()523 3629 y Fn(s)562 3641 y Fk(1)624 3629 y Fm(!)26 b Fn(t)763 3641 y Fk(1)800 3629 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)1024 3641 y Fj(k)1090 3629 y Fm(!)25 b Fn(t)1228 3641 y Fj(k)1298 3629 y Fw(in)k Fm(R)p Fw(,)g(a)g(substitution)g Fn(\033)g Fw(:)c Fm(V)7 b Fn(ar)r Fw(\()p Fn(\032)p Fw(\))26 b Fm(!)g(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\),)29 b(and)g(a)f(con)n(text) 523 3729 y Fn(C)6 b Fw([)14 b(])28 b(suc)n(h)f(that)h Fn(s)23 b Fw(=)g Fn(C)6 b Fw([)p Fn(l)r(\033)s Fw(])p Fn(;)14 b(t)23 b Fw(=)g Fn(C)6 b Fw([)p Fn(r)r(\033)s Fw(],)30 b(and)d Fn(s)2012 3741 y Fj(i)2040 3729 y Fn(\033)f Fm(!)2196 3699 y Fl(\003)2196 3752 y(R)2281 3729 y Fn(t)2311 3741 y Fj(i)2338 3729 y Fn(\033)31 b Fw(for)c(all)h(1)23 b Fm(\024)f Fn(i)h Fm(\024)g Fn(k)s Fw(.)k(W)-7 b(e)28 b(w)n(ould)523 3828 y(lik)n(e)g(to)g(stress)f(the)i(fact)g(that)f Fn(\033)k Fw(instan)n(tiates)c(ev)n(ery)f(v)-5 b(ariable)27 b(in)i Fn(\032)f Fw(and)g(not)g(only)g(those)523 3928 y(v)-5 b(ariables)23 b(o)r(ccurring)f(in)i Fn(l)r Fw(;)f(for)g(an)h (extra)f(v)-5 b(ariable)22 b Fn(x)p Fw(,)j Fn(x\033)i Fw(is)d(determined)g(as)f(follo)n(ws.)g(The)523 4028 y(conditions)h(are)f(ev)-5 b(aluated)24 b(from)g(left-to-righ)n(t.)f (Since)i Fn(s)2333 4040 y Fk(1)2394 4028 y Fw(con)n(tains)f(only)f(v)-5 b(ariables)23 b(from)523 4127 y Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\),)28 b(the)f(v)-5 b(ariables)26 b(in)h Fm(V)7 b Fn(ar)r Fw(\()p Fn(s)1600 4139 y Fk(1)1638 4127 y Fw(\))27 b(ha)n(v)n(e)f(a)h(binding.)g(Then)g Fn(s)2534 4139 y Fk(1)2572 4127 y Fn(\033)j Fw(is)d(rewritten)g(un)n(til)g Fn(t)3317 4139 y Fk(1)3355 4127 y Fn(\033)523 4227 y Fw(matc)n(hes)e(a)h(reduct.)g(The)g(term)g Fn(t)1586 4239 y Fk(1)1623 4227 y Fn(\033)k Fw(ma)n(y)25 b(con)n(tain)h(extra)f (v)-5 b(ariables)25 b(but)h(all)g(of)g(these)g(are)523 4327 y(b)r(ound)e(during)f(the)i(matc)n(h.)e(No)n(w)h Fn(s)1669 4339 y Fk(2)1730 4327 y Fw(con)n(tains)e(only)i(v)-5 b(ariables)22 b(whic)n(h)i(already)e(o)r(ccurred)523 4426 y(to)34 b(its)h(left)g(\(in)g Fn(l)h Fw(and)e Fn(t)1301 4438 y Fk(1)1338 4426 y Fw(\))h(and)f(are)f(th)n(us)i(b)r(ound.)g(The)f (instan)n(tiated)g(term)h Fn(s)3082 4438 y Fk(2)3153 4426 y Fw(is)f(then)523 4526 y(reduced)25 b(un)n(til)i(the)f (\(partially\))f(instan)n(tiated)h(term)f Fn(t)2247 4538 y Fk(2)2310 4526 y Fw(matc)n(hes)g(a)h(reduct)f(and)h(so)f(on.)h(If)523 4625 y(all)h(the)h(conditions)f(are)f(satis\014ed,)h(then)h(all)f(v)-5 b(ariables)26 b(in)h(the)h(conditions)f(are)f(b)r(ound)i(in)523 4725 y(the)j(pro)r(cess)e(of)i(ev)-5 b(aluating)30 b(the)h(conditions.) f(Hence)h(the)f(reduct)h(of)f Fn(l)r(\033)k Fw(is)c(w)n(ell-de\014ned) 523 4825 y(\(but)d(in)g(general)e(not)i(unique\))g(b)r(ecause)f Fn(r)j Fw(con)n(tains)d(only)g(v)-5 b(ariables)25 b(whic)n(h)i(also)e (app)r(ear)523 4924 y(in)j(the)g(conditions.)p eop %%Page: 4 4 4 3 bop 648 448 a Fw(The)32 b(prop)r(er)f(subterm)g(relation)g(is)h (denoted)g(b)n(y)g Fg(B)p Fw(.)f(The)h(next)g(de\014nition)h(is)f (based)523 548 y(on)d(the)g(w)n(ell-kno)n(wn)f(fact)h(that)g(if)h Fm(\037)f Fw(is)g(a)f(w)n(ell-founded)h(partial)f(order)g(whic)n(h)h (is)g(closed)523 648 y(under)e(con)n(texts,)g(then)i(the)f(order)e Fm(\037)1724 660 y Fj(st)1784 648 y Fw(=)d(\()p Fm(\037)g([)g Fg(B)p Fw(\))2167 617 y Fk(+)2250 648 y Fw(is)k(also)g(w)n (ell-founded.)523 821 y Fi(De\014nition)k(2.)41 b Fp(A)25 b(deterministic)i(3-CTRS)f Fw(\()p Fm(F)8 b Fn(;)14 b Fm(R)p Fw(\))26 b Fp(is)g(c)l(al)t(le)l(d)36 b Fw(quasi-reductiv)n(e)24 b Fp(if)j(ther)l(e)523 921 y(is)38 b(an)g(extension)g Fm(F)1191 890 y Fl(0)1252 921 y Fp(of)h(the)f(signatur)l(e)f Fm(F)46 b Fp(\(so)38 b Fm(F)46 b(\022)37 b(F)2399 890 y Fl(0)2422 921 y Fp(\))h(and)h(a)f(r)l(e)l(duction)g(or)l(der)g Fm(\037)523 1020 y Fp(on)e Fm(T)21 b Fw(\()p Fm(F)814 990 y Fl(0)838 1020 y Fn(;)14 b Fm(V)7 b Fw(\))36 b Fp(which,)i(for)f (every)g(rule)f Fn(l)g Fm(!)e Fn(r)k Fm(\()c Fn(s)2215 1032 y Fk(1)2287 1020 y Fm(!)g Fn(t)2434 1032 y Fk(1)2472 1020 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2695 1032 y Fj(k)2770 1020 y Fm(!)35 b Fn(t)2918 1032 y Fj(k)2993 1020 y Fm(2)g(R)p Fp(,)i(every)523 1120 y(substitution)29 b Fn(\033)12 b Fw(:)28 b Fm(V)i(!)23 b(T)e Fw(\()p Fm(F)1434 1090 y Fl(0)1458 1120 y Fn(;)14 b Fm(V)7 b Fw(\))p Fp(,)30 b(and)g(every)h Fw(0)22 b Fm(\024)h Fn(i)g(<)f(k)33 b Fp(satis\014es:)555 1285 y(1.)42 b(if)31 b Fn(s)784 1297 y Fj(j)819 1285 y Fn(\033)26 b Fm(\027)d Fn(t)1010 1297 y Fj(j)1045 1285 y Fn(\033)33 b Fp(for)d(every)h Fw(1)23 b Fm(\024)f Fn(j)28 b Fm(\024)23 b Fn(i)p Fp(,)30 b(then)f Fn(l)r(\033)d Fm(\037)2212 1297 y Fj(st)2295 1285 y Fn(s)2334 1297 y Fj(i)p Fk(+1)2446 1285 y Fn(\033)s Fp(,)555 1384 y(2.)42 b(if)31 b Fn(s)784 1396 y Fj(j)819 1384 y Fn(\033)26 b Fm(\027)d Fn(t)1010 1396 y Fj(j)1045 1384 y Fn(\033)33 b Fp(for)d(every)h Fw(1)23 b Fm(\024)f Fn(j)28 b Fm(\024)23 b Fn(k)s Fp(,)30 b(then)g Fn(l)r(\033)25 b Fm(\037)e Fn(r)r(\033)s Fp(.)523 1557 y Fw(Quasi-reductiv)n(e)30 b(deterministic)i(3-CTRSs)f(w)n(ere)f(in)n(tro)r(duced)i(b)n(y)f (Ganzinger)g([Gan91)o(,)523 1657 y(Def.)d(4.2])f(without)h(men)n (tioning)f(that)h(the)g(original)e(signature)g(can)h(b)r(e)h(extended.) g(This,)523 1757 y(ho)n(w)n(ev)n(er,)c(is)i(crucial)f(b)r(ecause)h (otherwise)f(Prop)r(ositions)f(4.3)h(and)h(4.4)f(in)h([Gan91)o(])g(w)n (ould)523 1856 y(b)r(e)g(incorrect.)f(F)-7 b(or)25 b(instance,)g ([Gan91)o(,)h(Prop.)f(4.3])g(states)g(the)h(follo)n(wing)f(su\016cien)n (t)h(con-)523 1956 y(dition)34 b(for)e(quasi-reductivit)n(y)g(\(cf.)i ([BG89)o(]\):)g(Let)f Fm(F)2259 1926 y Fl(0)2316 1956 y Fw(b)r(e)g(an)g(enric)n(hmen)n(t)g(of)g(the)h(orig-)523 2056 y(inal)g(signature)g Fm(F)42 b Fw(suc)n(h)35 b(that)f(the)h(order) f Fm(\037)g Fw(can)g(b)r(e)h(extended)g(to)f(a)g(reduction)g(order)523 2155 y(o)n(v)n(er)h Fm(T)22 b Fw(\()p Fm(F)877 2125 y Fl(0)900 2155 y Fn(;)14 b Fm(V)7 b Fw(\).)37 b(A)h(deterministic)f (rule)g Fn(l)j Fm(!)f Fn(r)j Fm(\()d Fn(s)2298 2167 y Fk(1)2374 2155 y Fm(!)g Fn(t)2526 2167 y Fk(1)2564 2155 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2787 2167 y Fj(k)2867 2155 y Fm(!)39 b Fn(t)3019 2167 y Fj(k)3097 2155 y Fw(is)e(quasi-)523 2255 y(reductiv)n(e)26 b(if)h(there)g(exists)f(a)h(sequence)f Fn(h)1857 2267 y Fj(i)1884 2255 y Fw(\()p Fn(x)p Fw(\))i(of)f(terms)f (in)h Fm(T)22 b Fw(\()p Fm(F)2610 2225 y Fl(0)2633 2255 y Fn(;)14 b Fm(V)7 b Fw(\),)27 b Fn(x)c Fm(2)h(V)7 b Fw(,)26 b(suc)n(h)h(that)523 2354 y Fn(l)e Fm(\037)d Fn(h)708 2366 y Fk(1)745 2354 y Fw(\()p Fn(s)816 2366 y Fk(1)854 2354 y Fw(\))p Fn(;)14 b(h)971 2366 y Fj(i)998 2354 y Fw(\()p Fn(t)1060 2366 y Fj(i)1088 2354 y Fw(\))24 b Fm(\027)e Fn(h)1279 2366 y Fj(i)p Fk(+1)1391 2354 y Fw(\()p Fn(s)1462 2366 y Fj(i)p Fk(+1)1574 2354 y Fw(\),)28 b(1)22 b Fm(\024)h Fn(i)g(<)f(k)s Fw(,)28 b(and)f Fn(h)2254 2366 y Fj(k)2295 2354 y Fw(\()p Fn(t)2357 2366 y Fj(k)2398 2354 y Fw(\))c Fm(\027)g Fn(r)r Fw(.)648 2454 y(In)39 b(order)g(to)g(sho)n(w)g(that)h([Gan91)o(,)g(Prop.)e(4.3])h(w)n(ould)g (b)r(e)h(incorrect)f(if)h(signature)523 2554 y(extensions)27 b(are)g(not)g(allo)n(w)n(ed,)f(consider)h(the)h(1-CTRS)f(from)g([DOS88) o(])1436 2835 y Fm(R)d Fw(=)1617 2665 y Fh(8)1617 2739 y(<)1617 2889 y(:)1817 2735 y Fn(b)f Fm(!)g Fn(c)1703 2834 y(f)9 b Fw(\()p Fn(b)p Fw(\))23 b Fm(!)g Fn(f)9 b Fw(\()p Fn(a)p Fw(\))1809 2934 y Fn(a)23 b Fm(!)g Fn(c)145 b Fm(\()23 b Fn(b)g Fm(!)g Fn(c)523 3116 y Fw(o)n(v)n(er)g(the)i (signature)f Fm(F)31 b Fw(=)23 b Fm(f)p Fn(a;)14 b(b;)g(c;)g(f)9 b Fm(g)p Fw(.)22 b(Note)j(that)g(no)g(reduction)f(order)g(on)g Fm(T)d Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))25 b(can)523 3216 y(pro)n(v)n(e)34 b(quasi-reductivit)n(y)h(of)g Fm(R)i Fw(b)r(ecause)e(no)h(partial)f(order)f Fm(\037)i Fw(on)f Fm(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))36 b(whic)n(h)g(is)523 3315 y(closed)d(under)g(con)n(texts)f(and)h(has)g Fn(f)9 b Fw(\()p Fn(b)p Fw(\))32 b Fm(\037)g Fn(f)9 b Fw(\()p Fn(a)p Fw(\))34 b(can)e(ha)n(v)n(e)g Fn(a)h Fm(\037)f Fn(b)p Fw(.)h(Ho)n(w)n(ev)n(er,)e([Gan91)o(,)523 3415 y(Prop.)26 b(4.3])h(is)h(applicable)f(with)h Fm(F)1640 3385 y Fl(0)1686 3415 y Fw(=)23 b Fm(F)j([)19 b(f)p Fn(h)p Fm(g)26 b Fw(and)i Fm(\037)23 b Fw(=)f Fm(!)2512 3380 y Fk(+)2512 3439 y Fl(R)2569 3423 y Ff(0)2596 3415 y Fw(,)28 b(where)1590 3750 y Fm(R)1660 3716 y Fl(0)1706 3750 y Fw(=)1794 3530 y Fh(8)1794 3604 y(>)1794 3629 y(>)1794 3654 y(<)1794 3804 y(>)1794 3829 y(>)1794 3854 y(:)1994 3600 y Fn(b)23 b Fm(!)g Fn(c)1879 3699 y(f)9 b Fw(\()p Fn(b)p Fw(\))24 b Fm(!)f Fn(f)9 b Fw(\()p Fn(a)p Fw(\))1985 3799 y Fn(a)24 b Fm(!)f Fn(h)p Fw(\()p Fn(b)p Fw(\))1881 3898 y Fn(h)p Fw(\()p Fn(c)p Fw(\))h Fm(!)f Fn(c)523 4081 y Fw(The)28 b(relation)e Fm(\037)i Fw(is)f(a)g(reduction) g(order)g(on)g Fm(T)21 b Fw(\()p Fm(F)2111 4051 y Fl(0)2135 4081 y Fn(;)14 b Fm(V)7 b Fw(\))27 b(b)r(ecause)h Fm(R)2667 4051 y Fl(0)2718 4081 y Fw(is)f(terminating.)648 4181 y(Next,)h(w)n(e)f(will)h(in)n(tro)r(duce)f(the)h(new)f(notion)h(of)f (quasi-decreasingness.)523 4338 y Fi(De\014nition)k(3.)41 b Fp(A)21 b(deterministic)i(3-CTRS)e Fw(\()p Fm(F)8 b Fn(;)14 b Fm(R)p Fw(\))23 b Fp(is)f(c)l(al)t(le)l(d)h Fw(quasi-decreasing)18 b Fp(if)23 b(ther)l(e)523 4437 y(is)30 b(a)g(wel)t(l-founde)l(d)i(p)l(artial)f(or)l(der)f Fm(\037)f Fp(on)h Fm(T)22 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))29 b Fp(satisfying:)555 4602 y(1.)42 b Fm(!)747 4614 y Fl(R)831 4602 y Fm(\022)23 b(\037)p Fp(,)555 4702 y(2.)42 b Fm(\037)29 b Fp(has)i(the)f(subterm)f(pr)l(op)l (erty)i(\(henc)l(e)f Fm(\037)22 b Fw(=)h Fm(\037)2188 4714 y Fj(st)2248 4702 y Fp(\),)555 4801 y(3.)42 b(for)32 b(every)g(rule)e Fn(l)d Fm(!)e Fn(r)j Fm(\()d Fn(s)1560 4813 y Fk(1)1622 4801 y Fm(!)g Fn(t)1760 4813 y Fk(1)1797 4801 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)2021 4813 y Fj(k)2086 4801 y Fm(!)25 b Fn(t)2224 4813 y Fj(k)2290 4801 y Fm(2)h(R)p Fp(,)31 b(every)h(substitution)e Fn(\033)s Fp(,)h(and)664 4901 y Fw(0)23 b Fm(\024)f Fn(i)h(<)g(k)s Fp(:)30 b(if)g Fn(s)1176 4913 y Fj(j)1211 4901 y Fn(\033)s Fm(!)1345 4870 y Fl(\003)1345 4923 y(R)1406 4901 y Fn(t)1436 4913 y Fj(j)1471 4901 y Fn(\033)j Fp(for)d(every)h Fw(1)23 b Fm(\024)f Fn(j)28 b Fm(\024)23 b Fn(i)p Fp(,)30 b(then)f Fn(l)r(\033)d Fm(\037)d Fn(s)2700 4913 y Fj(i)p Fk(+1)2811 4901 y Fn(\033)s Fp(.)p eop %%Page: 5 5 5 4 bop 523 448 a Fi(Lemma)30 b(4.)40 b Fp(Every)31 b(quasi-r)l(e)l (ductive)f(deterministic)h(3-CTRS)f(is)g(quasi-de)l(cr)l(e)l(asing.)523 664 y(Pr)l(o)l(of.)43 b Fw(If)33 b(\()p Fm(F)8 b Fn(;)14 b Fm(R)p Fw(\))33 b(is)e(quasi-reductiv)n(e,)g(then)h(there)g(is)g(an)g (extension)g Fm(F)2891 634 y Fl(0)2946 664 y Fw(of)g(the)g(signa-)523 763 y(ture)g Fm(F)41 b Fw(and)32 b(a)h(reduction)f(order)f Fn(>)h Fw(on)g Fm(T)22 b Fw(\()p Fm(F)2025 733 y Fl(0)2048 763 y Fn(;)14 b Fm(V)7 b Fw(\))33 b(whic)n(h)f(meets)h(the)g (requiremen)n(ts)e(of)523 863 y(De\014nition)h(2.)f(It)h(follo)n(ws)f (that)g(the)h(relation)f Fm(!)2102 875 y Fl(R)2194 863 y Fw(on)h Fm(T)21 b Fw(\()p Fm(F)2480 833 y Fl(0)2503 863 y Fn(;)14 b Fm(V)7 b Fw(\))32 b(is)f(a)g(subset)h(of)f Fn(>)p Fw(;)g(see)523 963 y([Gan91)o(,)j(Lemma)g(4.5].)f(Let)h Fm(\037)g Fw(denote)g(the)g(restriction)f(of)h Fn(>)2575 975 y Fj(st)2670 963 y Fw(to)f Fm(T)22 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\).)34 b(Clearly)-7 b(,)523 1062 y Fm(\037)40 b Fw(is)h(a)f(w)n(ell-founded)g(order)g(on)g Fm(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))41 b(suc)n(h)f(that)h Fm(!)2473 1074 y Fl(R)2579 1062 y Fm(\022)14 b(\037)40 b Fw(and)g Fg(B)14 b Fm(\022)g(\037)p Fw(.)39 b(Th)n(us)523 1162 y Fm(R)i Fw(is)f(quasi-decreasing)d(w.r.t.)j Fm(\037)g Fw(pro)n(vided)f(that)h(for)g(ev)n(ery)e(rule)i Fn(l)45 b Fm(!)f Fn(r)j Fm(\()d Fn(s)3241 1174 y Fk(1)3322 1162 y Fm(!)523 1262 y Fn(t)553 1274 y Fk(1)590 1262 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)814 1274 y Fj(k)878 1262 y Fm(!)23 b Fn(t)1014 1274 y Fj(k)1078 1262 y Fm(2)g(R)p Fw(,)h(ev)n(ery)e(substitution)h Fn(\033)12 b Fw(:)29 b Fm(V)h(!)23 b(T)e Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\),)24 b(and)f(0)f Fm(\024)h Fn(i)f(<)h(k)j Fw(w)n(e)d(ha)n(v)n (e:)523 1361 y Fn(s)562 1373 y Fj(j)597 1361 y Fn(\033)s Fm(!)730 1331 y Fl(\003)730 1384 y(R)792 1361 y Fn(t)822 1373 y Fj(j)857 1361 y Fn(\033)37 b Fw(for)d(ev)n(ery)f(1)h Fm(\024)g Fn(j)39 b Fm(\024)34 b Fn(i)g Fw(implies)h Fn(l)r(\033)i Fm(\037)d Fn(s)2249 1373 y Fj(i)p Fk(+1)2361 1361 y Fn(\033)s Fw(.)h(Ob)n(viously)e Fn(s)2905 1373 y Fj(j)2940 1361 y Fn(\033)k Fm(\025)d Fn(t)3153 1373 y Fj(j)3188 1361 y Fn(\033)k Fw(is)c(a)523 1461 y(consequence)29 b(of)i Fn(s)1131 1473 y Fj(j)1165 1461 y Fn(\033)s Fm(!)1299 1431 y Fl(\003)1299 1484 y(R)1360 1461 y Fn(t)1390 1473 y Fj(j)1425 1461 y Fn(\033)s Fw(,)g(where)f(1)d Fm(\024)g Fn(j)32 b Fm(\024)27 b Fn(i)p Fw(.)j(Since)h(\()p Fm(F)8 b Fn(;)14 b Fm(R)p Fw(\))31 b(is)f(quasi-reductiv)n(e,)f(it)523 1560 y(follo)n(ws)e Fn(l)r(\033)f(>)961 1572 y Fj(st)1044 1560 y Fn(s)1083 1572 y Fj(i)p Fk(+1)1194 1560 y Fn(\033)s Fw(.)i(Finally)-7 b(,)28 b(w)n(e)f(obtain)h Fn(l)r(\033)e Fm(\037)c Fn(s)2201 1572 y Fj(i)p Fk(+1)2313 1560 y Fn(\033)31 b Fw(from)c Fn(l)r(\033)n(;)14 b(s)2735 1572 y Fj(i)p Fk(+1)2847 1560 y Fn(\033)26 b Fm(2)e(T)d Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\).)648 1776 y(W)-7 b(e)36 b(do)g(not)h(kno)n (w)e(whether)h(or)g(not)g(quasi-decreasingness)d(implies)k (quasi-reduc-)523 1875 y(tivit)n(y)-7 b(.)23 b(It)g(is)g(fa)n(v)n (orable,)d(ho)n(w)n(ev)n(er,)h(to)i(handle)f(quasi-decreasingness)e(b)r (ecause)i(it)i(has)e(t)n(w)n(o)523 1975 y(adv)-5 b(an)n(tages)20 b(o)n(v)n(er)g(quasi-reductivit)n(y:)h(\(i\))h(It)g(do)r(es)g(not)g (dep)r(end)g(on)g(signature)f(extensions)523 2075 y(and)f(\(ii\))g(in)h (requiremen)n(t)e(\(3\))h(of)f(De\014nition)i(3,)f Fn(l)r(\033)25 b Fm(\037)2191 2087 y Fj(st)2275 2075 y Fn(s)2314 2087 y Fj(i)p Fk(+1)2425 2075 y Fn(\033)f Fw(m)n(ust)c(hold)f(only)h(if)g Fn(s)3152 2087 y Fj(j)3187 2075 y Fn(\033)27 b Fm(!)3344 2045 y Fl(\003)3344 2098 y(R)523 2174 y Fn(t)553 2186 y Fj(j)588 2174 y Fn(\033)k Fw(whereas)26 b(it)i(m)n(ust)g(hold)g(for)f (all)g Fn(s)1735 2186 y Fj(j)1770 2174 y Fn(\033)f Fm(\027)d Fn(t)1961 2186 y Fj(j)1996 2174 y Fn(\033)31 b Fw(according)26 b(to)h(De\014nition)i(2\(1\).)648 2284 y(Finite)j(quasi-reductiv)n(e)e (deterministic)i(3-CTRSs)f(ha)n(v)n(e)g(a)g(terminating)h(and)g(com-) 523 2384 y(putable)25 b(rewrite)f(relation)g([Gan91)n(,ALS94])h(and)f (the)h(same)f(holds)h(for)f(quasi-decreasing)523 2483 y(systems.)18 b(Our)g(next)g(goal)f(is)i(to)f(pro)n(vide)f(a)h (systematic)g(w)n(a)n(y)f(of)h(sho)n(wing)f(quasi-reductivit)n(y)-7 b(.)523 2583 y(T)g(o)27 b(this)h(end,)f(w)n(e)g(transform)f(ev)n(ery)g (deterministic)i(3-CTRS)f Fm(R)g Fw(in)n(to)g(an)g(unconditional)523 2683 y(TRS)35 b Fn(U)9 b Fw(\()p Fm(R)p Fw(\).)36 b(F)-7 b(or)35 b(normal)f(1-CTRSs,)g(a)h(similar)g(transformation)e(w)n(as)h (already)g(giv)n(en)523 2782 y(in)28 b([BK86)n(,)g(Def.)g(2.5.1].)e (Marc)n(hiori)g([Mar96)n(,Mar95)n(])i(studied)f(suc)n(h)h (transformations)d(of)523 2882 y(1-CTRSs)i(\(whic)n(h)h(he)f(called)h Fp(unr)l(avelings)p Fw(\))g(in)g(detail.)523 3081 y Fi(De\014nition)j (5.)41 b Fp(L)l(et)22 b Fm(R)i Fp(b)l(e)f(a)g(deterministic)h(3-CTRS)g (over)f(the)h(signatur)l(e)e Fm(F)8 b Fp(.)24 b(F)-6 b(or)23 b(every)523 3180 y(r)l(ewrite)40 b(rule)f Fn(\032)h Fw(:)g Fn(l)i Fm(!)e Fn(r)j Fm(\()e Fn(c)f Fm(2)g(R)p Fp(,)g(let)g Fm(j)p Fn(\032)p Fm(j)f Fp(denote)h(the)f(numb)l(er)f(of)j (c)l(onditions)f(in)523 3280 y Fn(\032)p Fp(.)i(In)f(the)h(tr)l (ansformation,)h(we)f(ne)l(e)l(d)f Fm(j)p Fn(\032)p Fm(j)h Fp(fr)l(esh)g(function)g(symb)l(ols)g Fn(U)2903 3240 y Fj(\032)2894 3302 y Fk(1)2941 3280 y Fn(;)14 b(:)g(:)g(:)g(;)g(U)3192 3240 y Fj(\032)3183 3308 y Fl(j)p Fj(\032)p Fl(j)3302 3280 y Fp(for)523 3393 y(every)29 b(c)l(onditional)h(rule)f Fn(\032)23 b Fm(2)g(R)p Fp(.)29 b(Mor)l(e)l(over,)i(by)d(abuse)h(of)g (notation,)h Fm(V)7 b Fn(ar)30 b Fp(\(r)l(esp.)f Fm(E)7 b(V)g Fn(ar)r Fp(\))523 3493 y(denotes)39 b(a)f(function)g(which)i (assigns)f(the)f(se)l(quenc)l(e)g(of)h(the)f(variables)j(\(in)d(some)g (\014xe)l(d)523 3593 y(or)l(der\))c(in)f(the)h(set)f Fm(V)7 b Fn(ar)r Fw(\()p Fn(t)p Fw(\))34 b Fp(\(r)l(esp.)g Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)p Fw(\))p Fp(;)35 b(cf.)f(Def.)g(1\)) g(to)f(a)h(term)f Fn(t)p Fp(.)g(We)h(tr)l(ansform)523 3692 y Fn(\032)h Fw(:)h Fn(l)h Fm(!)e Fn(r)j Fm(\()e Fn(s)1073 3704 y Fk(1)1145 3692 y Fm(!)g Fn(t)1294 3704 y Fk(1)1331 3692 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)1554 3707 y Fl(j)p Fj(\032)p Fl(j)1668 3692 y Fm(!)35 b Fn(t)1816 3707 y Fl(j)p Fj(\032)p Fl(j)1931 3692 y Fp(into)h(a)h(set)f Fn(U)9 b Fw(\()p Fn(\032)p Fw(\))37 b Fp(of)h Fm(j)p Fn(\032)p Fm(j)23 b Fw(+)g(1)36 b Fp(unc)l(onditional)523 3792 y(r)l(ewrite)30 b(rules)g(as)g(fol)t(lows:)1719 3990 y Fn(l)24 b Fm(!)g Fn(U)1941 3950 y Fj(\032)1932 4012 y Fk(1)1979 3990 y Fw(\()p Fn(s)2050 4002 y Fk(1)2087 3990 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))1240 4089 y Fn(U)1306 4049 y Fj(\032)1297 4111 y Fk(1)1344 4089 y Fw(\()p Fn(t)1406 4101 y Fk(1)1444 4089 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))23 b Fm(!)h Fn(U)1941 4049 y Fj(\032)1932 4111 y Fk(2)1979 4089 y Fw(\()p Fn(s)2050 4101 y Fk(2)2087 4089 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)2647 4101 y Fk(1)2686 4089 y Fw(\)\))879 4189 y Fn(U)945 4149 y Fj(\032)936 4211 y Fk(2)983 4189 y Fw(\()p Fn(t)1045 4201 y Fk(2)1082 4189 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)1642 4201 y Fk(1)1681 4189 y Fw(\)\))23 b Fm(!)h Fn(U)1941 4149 y Fj(\032)1932 4211 y Fk(3)1979 4189 y Fw(\()p Fn(s)2050 4201 y Fk(3)2087 4189 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)2647 4201 y Fk(1)2686 4189 y Fw(\))p Fn(;)14 b Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)3009 4201 y Fk(2)3047 4189 y Fw(\)\))1768 4289 y Fn(:)14 b(:)g(:)817 4388 y(U)883 4348 y Fj(\032)874 4416 y Fl(j)p Fj(\032)p Fl(j)951 4388 y Fw(\()p Fn(t)1013 4403 y Fl(j)p Fj(\032)p Fl(j)1092 4388 y Fn(;)g Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)1652 4400 y Fk(1)1690 4388 y Fw(\))p Fn(;)23 b(:)14 b(:)g(:)g(;)g Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)2170 4403 y Fl(j)p Fj(\032)p Fl(j\000)p Fk(1)2334 4388 y Fw(\)\))24 b Fm(!)f Fn(r)523 4605 y Fp(Sinc)l(e)38 b Fm(R)g Fp(is)g(deterministic,)h(the)f(system)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\))38 b(=)2259 4543 y Fh(S)2328 4630 y Fj(\032)p Fl(2R)2482 4605 y Fn(U)9 b Fw(\()p Fn(\032)p Fw(\))38 b Fp(is)g(an)g(unc)l(onditional)523 4715 y(TRS)25 b(over)i(the)f(extende)l(d)g(signatur)l(e)f Fm(F)1784 4684 y Fl(0)1831 4715 y Fw(=)d Fm(F)c([)2061 4652 y Fh(S)2130 4739 y Fj(\032)p Fl(2R)p Fj(;)p Fk(1)p Fl(\024)p Fj(i)p Fl(\024j)p Fj(\032)p Fl(j)2525 4715 y Fm(f)p Fn(U)2633 4675 y Fj(\032)2624 4738 y(i)2671 4715 y Fm(g)25 b Fp(\(that)h(is,)g Fm(V)7 b Fn(ar)r Fw(\()p Fn(r)3260 4684 y Fl(0)3285 4715 y Fw(\))23 b Fm(\022)523 4825 y(V)7 b Fn(ar)r Fw(\()p Fn(l)723 4795 y Fl(0)746 4825 y Fw(\))23 b Fp(holds)g(for)g(every)g(r)l (ewrite)f(rule)g Fn(l)1802 4795 y Fl(0)1848 4825 y Fm(!)h Fn(r)1993 4795 y Fl(0)2040 4825 y Fm(2)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\))p Fp(\).)23 b(In)f(the)g(fol)t(lowing,)j(the)d(symb)l(ols) 523 4924 y(fr)l(om)30 b Fm(F)787 4894 y Fl(0)829 4924 y Fm(n)18 b(F)38 b Fp(ar)l(e)30 b(c)l(al)t(le)l(d)h Fn(U)9 b Fp(-symb)l(ols.)p eop %%Page: 6 6 6 5 bop 523 448 a Fw(F)-7 b(or)27 b(example,)g(the)h(transformation)e (of)i(the)g Fp(quicksort)g Fw(system)f(is)1473 639 y Fn(spl)r(it)p Fw(\()p Fn(x;)14 b Fw([)g(]\))23 b Fm(!)g Fw(\([)14 b(])p Fn(;)g Fw([)g(]\))1337 738 y Fn(spl)r(it)p Fw(\()p Fn(x;)g(y)25 b Fw(:)f Fn(y)s(s)p Fw(\))f Fm(!)g Fn(U)2043 708 y Fl(0)2034 759 y Fk(1)2071 738 y Fw(\()p Fn(spl)r(it)p Fw(\()p Fn(x;)14 b(y)s(s)p Fw(\))p Fn(;)g(x;)g(y)s(;)g(y) s(s)p Fw(\))1135 838 y Fn(U)1201 808 y Fl(0)1192 858 y Fk(1)1229 838 y Fw(\(\()p Fn(xs;)g(z)t(s)p Fw(\))p Fn(;)g(x;)g(y)s(;)g(y)s(s)p Fw(\))24 b Fm(!)f Fn(U)2043 808 y Fl(0)2034 858 y Fk(2)2071 838 y Fw(\()p Fn(x)h Fm(\024)e Fn(y)s(;)14 b(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t(s)p Fw(\))1007 937 y Fn(U)1073 907 y Fl(0)1064 958 y Fk(2)1101 937 y Fw(\()p Fn(tr)r(ue;)g(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t(s)p Fw(\))23 b Fm(!)g Fw(\()p Fn(xs;)14 b(y)26 b Fw(:)d Fn(z)t(s)p Fw(\))1337 1037 y Fn(spl)r(it)p Fw(\()p Fn(x;)14 b(y)25 b Fw(:)f Fn(y)s(s)p Fw(\))f Fm(!)g Fn(U)2043 1007 y Fl(00)2034 1058 y Fk(1)2085 1037 y Fw(\()p Fn(spl)r(it)p Fw(\()p Fn(x;)14 b(y)s(s)p Fw(\))p Fn(;)g(x;)g(y)s(;)g(y)s(s)p Fw(\))1121 1137 y Fn(U)1187 1107 y Fl(00)1178 1157 y Fk(1)1229 1137 y Fw(\(\()p Fn(xs;)g(z)t(s)p Fw(\))p Fn(;)g(x;)g(y)s(;)g (y)s(s)p Fw(\))24 b Fm(!)f Fn(U)2043 1107 y Fl(00)2034 1157 y Fk(2)2085 1137 y Fw(\()p Fn(x)h Fm(\024)f Fn(y)s(;)14 b(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t(s)p Fw(\))951 1236 y Fn(U)1017 1206 y Fl(00)1008 1257 y Fk(2)1060 1236 y Fw(\()p Fn(f)9 b(al)r(se;)14 b(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t(s)p Fw(\))21 b Fm(!)i Fw(\()p Fn(y)j Fw(:)d Fn(xs;)14 b(z)t(s)p Fw(\))1534 1336 y Fn(q)s(sor)r(t)p Fw(\([)g(]\))25 b Fm(!)e Fw([)14 b(])1391 1436 y Fn(q)s(sor)r(t)p Fw(\()p Fn(x)25 b Fw(:)e Fn(xs)p Fw(\))h Fm(!)f Fn(U)2043 1405 y Fl(000)2034 1456 y Fk(1)2104 1436 y Fw(\()p Fn(spl)r(it)p Fw(\()p Fn(x;)14 b(xs)p Fw(\))p Fn(;)g(x;)g(xs)p Fw(\))1183 1535 y Fn(U)1249 1505 y Fl(000)1240 1556 y Fk(1)1310 1535 y Fw(\(\()p Fn(y)s(s;)g(z)t(s)p Fw(\))p Fn(;)g(x;)g(xs)p Fw(\))24 b Fm(!)f Fn(q)s(sor)r(t)p Fw(\()p Fn(y)s(s)p Fw(\))c(+)-23 b(+)23 b(\()p Fn(x)g Fw(:)h Fn(q)s(sor)r(t)p Fw(\()p Fn(z)t(s)p Fw(\)\))523 1732 y(If)19 b(functions)h(are)e(sp)r (eci\014ed)h(via)f(distinct)i(cases)e(as)g(in)h(the)h Fp(split)f Fw(function)h(\(that)f(is,)g(the)h(left-)523 1831 y(hand)26 b(sides)f(of)h(t)n(w)n(o)f(or)g(more)h(rules)f(and)h(a)f (pre\014x)h(of)f(the)i(sequences)e(of)h(their)f(conditions)523 1931 y(coincide\),)j(then)g(the)g(transformation)e(can)h(b)r(e)h (\\optimized")f(as)g(follo)n(ws:)1387 2126 y Fn(spl)r(it)p Fw(\()p Fn(x;)14 b(y)25 b Fw(:)e Fn(y)s(s)p Fw(\))g Fm(!)h Fn(U)2093 2096 y Fl(0)2084 2146 y Fk(1)2120 2126 y Fw(\()p Fn(spl)r(it)p Fw(\()p Fn(x;)14 b(y)s(s)p Fw(\))p Fn(;)g(x;)g(y)s(;)g(y) s(s)p Fw(\))1185 2225 y Fn(U)1251 2195 y Fl(0)1242 2246 y Fk(1)1279 2225 y Fw(\(\()p Fn(xs;)g(z)t(s)p Fw(\))p Fn(;)g(x;)g(y)s(;)g(y)s(s)p Fw(\))23 b Fm(!)h Fn(U)2093 2195 y Fl(0)2084 2246 y Fk(2)2120 2225 y Fw(\()p Fn(x)g Fm(\024)f Fn(y)s(;)14 b(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t(s)p Fw(\))1057 2325 y Fn(U)1123 2295 y Fl(0)1114 2346 y Fk(2)1151 2325 y Fw(\()p Fn(tr)r(ue;)g(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t(s)p Fw(\))22 b Fm(!)i Fw(\()p Fn(xs;)14 b(y)26 b Fw(:)d Fn(z)t(s)p Fw(\))1015 2425 y Fn(U)1081 2394 y Fl(0)1072 2445 y Fk(2)1109 2425 y Fw(\()p Fn(f)9 b(al)r(se;)14 b(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t (s)p Fw(\))21 b Fm(!)j Fw(\()p Fn(y)i Fw(:)d Fn(xs;)14 b(z)t(s)p Fw(\))523 2616 y(W)-7 b(e)24 b(omit)g(the)g(formal)e (de\014nition)i(of)g(this)g(ob)n(vious)e(optimization)h(of)h(the)g (transformation)523 2716 y Fn(U)9 b Fw(.)38 b(It)g(turns)g(out)g(that)g (termination)g(of)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\))38 b(is)g(a)g(su\016cien)n(t)g(but)g(not)g(a)g(necessary)523 2816 y(condition)27 b(for)g(quasi-reductivit)n(y)g(of)g Fm(R)p Fw(.)523 2999 y Fi(Prop)s(osition)j(6.)41 b Fp(If)30 b Fn(U)9 b Fw(\()p Fm(R)p Fw(\))31 b Fp(is)f(terminating,)g(then)g Fm(R)g Fp(is)g(quasi-r)l(e)l(ductive.)523 3206 y(Pr)l(o)l(of.)43 b Fw(Let)32 b Fm(\037)d Fw(=)h Fm(!)1208 3171 y Fk(+)1208 3234 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1404 3206 y Fw(and)31 b(note)h(that)g Fm(\037)f Fw(is)g(a)g(reduction)h(order)e(on)h Fm(T)22 b Fw(\()p Fm(F)3079 3176 y Fl(0)3102 3206 y Fn(;)14 b Fm(V)7 b Fw(\).)32 b(F)-7 b(or)523 3315 y(ev)n(ery)25 b(rule)h Fn(l)e Fm(!)f Fn(r)j Fm(\()d Fn(s)1269 3327 y Fk(1)1329 3315 y Fm(!)g Fn(t)1465 3327 y Fk(1)1503 3315 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)1726 3327 y Fj(k)1790 3315 y Fm(!)23 b Fn(t)1926 3327 y Fj(k)1993 3315 y Fw(in)j Fm(R)p Fw(,)h(w)n(e)f(sho)n(w)f(that)h Fn(s)2750 3327 y Fj(j)2785 3315 y Fn(\033)h Fm(\027)22 b Fn(t)2976 3327 y Fj(j)3011 3315 y Fn(\033)30 b Fw(for)25 b(ev)n(ery)523 3415 y(1)e Fm(\024)f Fn(j)28 b Fm(\024)23 b Fn(i)g(<)f(k)31 b Fw(implies)d Fn(l)r(\033)d Fm(\037)1484 3427 y Fj(st)1568 3415 y Fn(s)1607 3427 y Fj(i)p Fk(+1)1718 3415 y Fn(\033)s Fw(.)j(W)-7 b(e)28 b(ha)n(v)n(e)f(the)h(follo)n(wing)e(deriv)-5 b(ation)997 3608 y Fn(l)r(\033)26 b Fm(!)1180 3623 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1367 3608 y Fn(U)1433 3568 y Fj(\032)1424 3630 y Fk(1)1472 3608 y Fw(\()p Fn(s)1543 3620 y Fk(1)1580 3608 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)1097 3708 y Fm(!)1180 3677 y Fl(\003)1180 3734 y Fj(U)f Fk(\()p Fl(R)p Fk(\))1367 3708 y Fn(U)1433 3668 y Fj(\032)1424 3730 y Fk(1)1472 3708 y Fw(\()p Fn(t)1534 3720 y Fk(1)1571 3708 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)1097 3819 y Fm(!)1180 3834 y Fj(U)f Fk(\()p Fl(R)p Fk(\))1367 3819 y Fn(U)1433 3779 y Fj(\032)1424 3841 y Fk(2)1472 3819 y Fw(\()p Fn(s)1543 3831 y Fk(2)1580 3819 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)2140 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4253 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1246 4226 y Fn(t)1276 4238 y Fj(j)1311 4226 y Fn(\033)s Fw(.)26 b(Th)n(us)g Fn(l)r(\033)f Fm(\037)1783 4238 y Fj(st)1867 4226 y Fn(s)1906 4238 y Fj(i)p Fk(+1)2017 4226 y Fn(\033)s Fw(.)h(Requiremen)n(t)g (\(2\))f(of)h(De\014nition)g(2)f(can)523 4326 y(b)r(e)j(sho)n(wn)f (similarly)-7 b(.)523 4526 y(A)21 b(similar)f(result)g(for)g(normal)g (1-CTRSs)f(and)i(a)f(surprising)f(coun)n(terexample)g(to)i(the)g(con-) 523 4625 y(v)n(erse)i(of)h(Prop)r(osition)e(6)i(can)g(b)r(e)g(found)h (in)f([Mar96)n(,Mar95)o(].)g(In)g(our)g Fp(quicksort)g Fw(example,)523 4725 y(termination)31 b(of)h(the)f(transformed)g (system)g(can)g(b)r(e)h(sho)n(wn)f(automatically)f(b)n(y)h(the)h(de-) 523 4825 y(p)r(endency)c(pair)g(metho)r(d)g(of)g(Arts)g(and)g(Giesl)g ([A)n(G97a)o(].)g(Th)n(us)g(the)h Fp(quicksort)f Fw(system)g(is)523 4924 y(quasi-reductiv)n(e.)p eop %%Page: 7 7 7 6 bop 523 448 a Fq(4)112 b(The)38 b(Main)f(Theorem)523 656 y Fw(Although)25 b(quasi-decreasingness)c(of)j(a)g(deterministic)h (3-CTRS)f Fm(R)g Fw(do)r(es)g(not)h(imply)g(ter-)523 756 y(mination)37 b(of)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\),)38 b(it)f(do)r(es)g(imply)g(innermost)f(termination)g(of)h Fn(U)9 b Fw(\()p Fm(R)p Fw(\).)38 b(This)e(en)n(tire)523 855 y(section)27 b(is)h(dedicated)f(to)h(a)f(pro)r(of)g(of)h(this)f (non-trivial)g(fact.)523 1038 y Fi(De\014nition)k(7.)41 b Fp(L)l(et)j Fm(R)h Fp(b)l(e)g(a)h(deterministic)g(3-CTRS)f(and)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\))46 b Fp(its)e(tr)l(ansforme)l(d)523 1138 y(TRS.)30 b(We)g(de\014ne)g(a)g(tr)l(ansformation)g Fm(r)9 b Fw(:)28 b Fm(T)22 b Fw(\()p Fm(F)2072 1108 y Fl(0)2095 1138 y Fn(;)14 b Fm(V)7 b Fw(\))23 b Fm(!)g(T)e Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))30 b Fp(by)885 1427 y Fm(r)p Fw(\()p Fn(t)p Fw(\))24 b(=)1159 1257 y Fh(8)1159 1332 y(<)1159 1481 y(:)1245 1327 y Fn(t)731 b Fp(if)31 b Fn(t)23 b Fm(2)g(V)i([)19 b(F)2436 1297 y Fk(\(0\))1245 1431 y Fn(f)9 b Fw(\()p Fm(r)p Fw(\()p Fn(t)1458 1443 y Fk(1)1495 1431 y Fw(\))p Fn(;)14 b(:)g(:)g(:)g(;)g Fm(r)p Fw(\()p Fn(t)1843 1443 y Fj(n)1889 1431 y Fw(\)\))53 b Fp(if)31 b Fn(t)23 b Fw(=)g Fn(f)9 b Fw(\()p Fn(t)2340 1443 y Fk(1)2377 1431 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)2591 1443 y Fj(n)2636 1431 y Fw(\))p Fn(;)g(f)32 b Fm(2)24 b(F)2925 1400 y Fk(\()p Fj(n)p Fk(\))1245 1530 y Fn(l)r(\034)698 b Fp(if)31 b Fn(t)23 b Fw(=)g Fn(U)2294 1490 y Fj(\032)2285 1553 y(i)2332 1530 y Fw(\()p Fn(u;)14 b(t)2479 1542 y Fk(1)2516 1530 y Fn(;)g(:)g(:)g(:)f(;)h(t)2730 1542 y Fj(n)2775 1530 y Fn(;)g(:)g(:)g(:)g Fw(\))523 1717 y Fp(wher)l(e)35 b Fn(l)g Fp(is)g(the)f(left-hand)h(side)g(of)g(the)g (rule)f Fn(\032)p Fp(,)g Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))32 b(=)e Fn(x)2508 1729 y Fk(1)2546 1717 y Fn(;)14 b(:)g(:)g(:)f(;)h(x)2777 1729 y Fj(n)2857 1717 y Fp(\(that)34 b(is)h(to)f(say,)523 1817 y Fn(U)589 1777 y Fj(\032)580 1840 y(i)627 1817 y Fw(\()p Fn(u;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))23 b(=)g Fn(U)1333 1777 y Fj(\032)1324 1840 y(i)1371 1817 y Fw(\()p Fn(u;)14 b(x)1535 1829 y Fk(1)1573 1817 y Fn(;)g(:)g(:)g(:)f (;)h(x)1804 1829 y Fj(n)1850 1817 y Fn(;)g(:)g(:)g(:)f Fw(\))p Fp(\),)30 b(and)f Fn(\034)19 b Fw(:)27 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))24 b Fm(!)f(T)e Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))29 b Fp(is)g(de\014ne)l(d)523 1916 y(by)h Fn(\034)9 b Fw(\()p Fn(x)755 1928 y Fj(j)792 1916 y Fw(\))23 b(=)g Fm(r)p Fw(\()p Fn(t)1066 1928 y Fj(j)1101 1916 y Fw(\))p Fp(.)31 b(F)-6 b(or)29 b(any)i Fn(\033)s Fp(,)f(we)g(de\014ne)g Fn(\033)2020 1928 y Fl(r)2110 1916 y Fp(by)g Fn(x\033)2312 1928 y Fl(r)2396 1916 y Fw(=)22 b Fm(r)p Fw(\()p Fn(x\033)s Fw(\))p Fp(.)648 2098 y Fw(F)-7 b(or)31 b(instance,)h(if)h Fm(R)d Fw(=)h Fm(f)p Fn(\032)1514 2110 y Fk(1)1581 2098 y Fw(:)f Fn(f)9 b Fw(\()p Fn(x;)14 b(y)s Fw(\))31 b Fm(!)g Fn(z)j Fm(\()c Fn(x)h Fm(!)g Fn(z)t(;)14 b(\032)2522 2110 y Fk(2)2589 2098 y Fw(:)30 b Fn(a)h Fm(!)f Fn(b)g Fm(\()h Fn(g)s Fw(\()p Fn(c)p Fw(\))g Fm(!)f Fn(d)p Fm(g)p Fw(,)523 2197 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Fk(1)2289 2297 y Fw(\()p Fn(a;)g(b;)g(U)2541 2257 y Fj(\032)2575 2265 y Fe(2)2532 2319 y Fk(1)2612 2297 y Fw(\()p Fn(d)p Fw(\)\)\)\),)20 b(w)n(e)e(ha)n(v)n(e)g Fm(r)p Fw(\()p Fn(t)p Fw(\))23 b(=)523 2396 y Fn(g)s Fw(\()p Fn(f)9 b Fw(\()p Fn(a;)14 b(f)9 b Fw(\()p Fn(b;)14 b(a)p Fw(\)\)\).)648 2521 y(Informally)-7 b(,)33 b(in)h(a)f(deriv)-5 b(ation)34 b Fn(D)h Fw(:)f Fn(s)1910 2474 y Fj(i)28 b Fl(\003)1897 2521 y Fm(!)1972 2536 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2170 2521 y Fn(t)p Fw(,)34 b(where)g Fn(s)f Fm(2)h(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\),)34 b Fm(r)g Fw(replaces)523 2621 y(ev)n(ery)26 b(maximal)h Fn(U)9 b Fw(-subterm)26 b Fn(u)h Fw(\(a)g Fn(U)9 b Fw(-)p Fp(subterm)26 b Fw(is)h(a)g(subterm)g(with)h(a)e Fn(U)9 b Fw(-sym)n(b)r(ol)27 b(at)g(the)523 2720 y(ro)r(ot\))g(b)n(y)g Fm(r)p Fw(\()p Fn(u)p Fw(\))d Fm(2)f(T)f Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))27 b({)g(the)h(term)g(whic)n(h)g(created)e Fn(u)i Fw(in)f Fn(D)r Fw(.)523 2926 y Fi(Lemma)j(8.)40 b Fp(L)l(et)30 b Fn(s)23 b Fm(2)g(T)e Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))p Fp(.)31 b(If)f Fn(s)1788 2879 y Fj(i)h Fl(\003)1776 2926 y Fm(!)1853 2941 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2041 2926 y Fn(t)p Fp(,)30 b(then)f Fn(s)23 b Fm(!)2455 2896 y Fl(\003)2455 2949 y(R)2540 2926 y Fm(r)p Fw(\()p Fn(t)p Fw(\))2756 2879 y Fj(i)30 b Fl(\003)2743 2926 y Fm(!)2821 2941 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3008 2926 y Fn(t)p Fp(.)523 3139 y(Pr)l(o)l(of.)43 b Fw(By)d(induction)i(on)e(the)h(length)g Fn(`)f Fw(of)g(the)h(deriv)-5 b(ation)40 b Fn(D)47 b Fw(:)e Fn(s)2896 3092 y Fj(i)29 b Fl(\003)2884 3139 y Fm(!)2959 3154 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3168 3139 y Fn(t)p Fw(.)41 b(The)523 3273 y(base)d(case)g Fn(`)i Fw(=)h(0)d(is)h(ob)n(viously)e(true.)h(So)g (supp)r(ose)h Fn(D)12 b Fw(:)30 b Fn(s)2541 3226 y Fj(i)e(`)2525 3273 y Fm(!)2597 3288 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2803 3273 y Fn(t)2833 3243 y Fl(0)2927 3226 y Fj(i)2898 3273 y Fm(!)2981 3288 y Fj(U)g Fk(\()p Fl(R)p Fk(\))3186 3273 y Fn(t)p Fw(.)39 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3683 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2840 3668 y Fn(U)2906 3628 y Fj(\032)2897 3690 y Fk(1)2945 3668 y Fw(\()p Fn(s)3016 3680 y Fk(1)3053 3668 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)523 3800 y Fw(or)38 b Fn(U)702 3760 y Fj(\032)693 3823 y(i)740 3800 y Fw(\()p Fn(t)802 3812 y Fj(i)830 3800 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)1401 3753 y Fj(i)1371 3800 y Fm(!)1454 3815 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1660 3800 y Fn(U)1726 3760 y Fj(\032)1717 3823 y(i)p Fk(+1)1829 3800 y Fw(\()p Fn(s)1900 3812 y Fj(i)p Fk(+1)2012 3800 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)42 b Fw(imply)d(that)g(ev)n(ery)f(argu-)523 3899 y(men)n(t)43 b(in)g(the)g(terms)f Fn(U)1330 3859 y Fj(\032)1321 3921 y Fk(1)1368 3899 y Fw(\()p Fn(s)1439 3911 y Fk(1)1477 3899 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)46 b Fw(and)c Fn(U)2113 3859 y Fj(\032)2104 3922 y(i)p Fk(+1)2216 3899 y Fw(\()p Fn(s)2287 3911 y Fj(i)p Fk(+1)2399 3899 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)46 b Fw(is)c(in)h(normal)523 3999 y(form)32 b(except)g(for)f (\(p)r(ossibly\))h(the)h(\014rst)f(one.)f(This)h(means)g(that)g(the)h (redex)e(con)n(tracted)523 4122 y(in)d Fn(t)650 4092 y Fl(0)726 4075 y Fj(i)696 4122 y Fm(!)779 4137 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))967 4122 y Fn(t)27 b Fw(is)h(a)f(subterm)h(of)f (the)h(\014rst)g(argumen)n(t)e(of)i(some)f Fn(U)9 b Fw(-subterm)27 b(of)g Fn(t)3125 4092 y Fl(0)3148 4122 y Fw(.)h(So)1394 4340 y Fm(r)p Fw(\()p Fn(t)p Fw(\))c(=)f Fm(r)p Fw(\()p Fn(t)1800 4306 y Fl(0)1824 4340 y Fw(\))1907 4293 y Fj(i)28 b Fl(\003)1894 4340 y Fm(!)1969 4355 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2157 4340 y Fn(t)2187 4306 y Fl(0)2263 4293 y Fj(i)2233 4340 y Fm(!)2316 4355 y Fj(U)g Fk(\()p Fl(R)p Fk(\))2504 4340 y Fn(t)523 4526 y Fw(b)r(ecause)21 b(the)h (de\014nition)g(of)f Fm(r)p Fw(\()p Fn(t)p Fw(\))h(is)g(indep)r(enden)n (t)g(of)f(the)h(\014rst)f(argumen)n(t)g(of)g Fn(U)9 b Fw(-subterms)523 4625 y(of)20 b Fn(t)p Fw(.)g(Next,)h(w)n(e)f(assume)f (that)i(there)f(is)g(no)g Fn(U)9 b Fw(-sym)n(b)r(ol)19 b(ab)r(o)n(v)n(e)g(the)h(con)n(tracted)f(redex.)h(Note)523 4725 y(that)33 b(if)h Fn(t)820 4695 y Fl(0)875 4725 y Fw(=)e Fn(C)6 b Fw([)p Fn(l)1087 4695 y Fl(0)1110 4725 y Fn(\033)s Fw(])33 b Fm(2)f(T)21 b Fw(\()p Fm(F)1469 4695 y Fl(0)1493 4725 y Fn(;)14 b Fm(V)7 b Fw(\))33 b(with)g Fn(l)1874 4695 y Fl(0)1929 4725 y Fm(!)f Fn(r)2083 4695 y Fl(0)2139 4725 y Fm(2)h Fn(U)9 b Fw(\()p Fm(R)p Fw(\))34 b(and)f(there)f(is)h(no)g Fn(U)9 b Fw(-sym)n(b)r(ol)523 4825 y(ab)r(o)n(v)n(e)29 b Fn(l)788 4795 y Fl(0)811 4825 y Fn(\033)34 b Fw(in)d Fn(C)6 b Fw(,)31 b(then)g Fm(r)p Fw(\()p Fn(t)1434 4795 y Fl(0)1458 4825 y Fw(\))d(=)g Fm(r)p Fw(\()p Fn(C)6 b Fw(\)[)p Fm(r)p Fw(\()p Fn(l)1960 4795 y Fl(0)1984 4825 y Fn(\033)s Fw(\)].)32 b(Th)n(us,)e(it)i (su\016ces)e(to)g(sho)n(w)g(the)h(claim)523 4924 y(for)c(the)h(follo)n (wing)f(cases:)p eop %%Page: 8 8 8 7 bop 558 459 a Fw(1.)41 b Fn(t)694 429 y Fl(0)741 459 y Fw(=)22 b Fn(l)r(\033)958 412 y Fj(i)928 459 y Fm(!)1011 474 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1199 459 y Fn(r)r(\033)27 b Fw(=)22 b Fn(t)p Fw(,)28 b(where)f Fn(l)e Fm(!)e Fn(r)j Fm(2)d(R)p Fw(.)558 590 y(2.)41 b Fn(t)694 560 y Fl(0)741 590 y Fw(=)22 b Fn(l)r(\033)958 543 y Fj(i)928 590 y Fm(!)1011 605 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1199 590 y Fn(U)1265 550 y Fj(\032)1256 612 y Fk(1)1303 590 y Fw(\()p Fn(s)1374 602 y Fk(1)1411 590 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)27 b Fw(=)c Fn(t)p Fw(.)558 721 y(3.)41 b Fn(t)694 691 y Fl(0)741 721 y Fw(=)22 b Fn(U)894 681 y Fj(\032)885 744 y(i)932 721 y Fw(\()p Fn(t)994 733 y Fj(i)1022 721 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)1575 674 y Fj(i)1545 721 y Fm(!)1628 736 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1816 721 y Fn(U)1882 681 y Fj(\032)1873 744 y(i)p Fk(+1)1984 721 y Fw(\()p Fn(s)2055 733 y Fj(i)p Fk(+1)2167 721 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)26 b Fw(=)d Fn(t)p Fw(.)558 853 y(4.)41 b Fn(t)694 823 y Fl(0)741 853 y Fw(=)22 b Fn(U)894 813 y Fj(\032)885 881 y Fl(j)p Fj(\032)p Fl(j)963 853 y Fw(\()p Fn(t)1025 868 y Fl(j)p Fj(\032)p Fl(j)1103 853 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)1656 806 y Fj(i)1626 853 y Fm(!)1709 868 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1896 853 y Fn(r)r(\033)28 b Fw(=)22 b Fn(t)p Fw(.)523 1030 y(The)33 b(\014rst)h(three)f(cases)f(are)g(easy)h(to)g(pro)n(v)n(e.)f (In)h(case)g(\(4\),)g(w)n(e)g(ha)n(v)n(e)f Fm(r)p Fw(\()p Fn(t)2911 1000 y Fl(0)2935 1030 y Fw(\))h(=)f Fm(r)p Fw(\()p Fn(l)r(\033)s Fw(\))h(=)523 1153 y Fn(l)r(\033)597 1165 y Fl(r)656 1153 y Fw(.)e(By)f(the)h(inductiv)n(e)g(h)n(yp)r (othesis,)f Fm(r)p Fw(\()p Fn(t)1918 1123 y Fl(0)1942 1153 y Fw(\))e(=)g Fn(l)r(\033)2169 1165 y Fl(r)2284 1106 y Fj(i)g Fl(\003)2271 1153 y Fm(!)2347 1168 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2539 1153 y Fn(t)2569 1123 y Fl(0)2623 1153 y Fw(holds)30 b(true.)h(Moreo)n(v)n(er,)523 1285 y Fm(r)p Fw(\()p Fn(t)p Fw(\))25 b(=)e Fn(r)r(\033)885 1297 y Fl(r)946 1285 y Fw(.)29 b(Th)n(us)f(one)f(m)n(ust)i(sho)n(w)e (that)h Fn(l)r(\033)2028 1297 y Fl(r)2112 1285 y Fm(!)2195 1255 y Fl(\003)2195 1308 y(R)2280 1285 y Fn(r)r(\033)2366 1297 y Fl(r)2479 1238 y Fj(i)g Fl(\003)2466 1285 y Fm(!)2541 1300 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2730 1285 y Fn(r)r(\033)s Fw(.)29 b(The)g(deriv)-5 b(ation)523 1385 y Fn(D)30 b Fw(m)n(ust)d(con)n(tain)g(a)h(subsequence)769 1583 y Fn(D)840 1553 y Fl(0)887 1583 y Fw(:)23 b Fn(l)r(\033)1007 1595 y Fl(r)1117 1536 y Fj(i)28 b Fl(\003)1104 1583 y Fm(!)1180 1598 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1367 1583 y Fn(l)r(\033)1497 1536 y Fj(i)1467 1583 y Fm(!)1550 1598 y Fj(U)g Fk(\()p Fl(R)p Fk(\))1738 1583 y Fn(U)1804 1543 y Fj(\032)1795 1605 y Fk(1)1842 1583 y Fw(\()p Fn(s)1913 1595 y Fk(1)1950 1583 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)2353 1536 y Fj(i)28 b Fl(\003)2340 1583 y Fm(!)2416 1598 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2603 1583 y Fn(U)2669 1543 y Fj(\032)2660 1605 y Fk(1)2707 1583 y Fw(\()p Fn(t)2769 1595 y Fk(1)2807 1583 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)1119 1660 y Fj(i)1089 1707 y Fm(!)1172 1722 y Fj(U)f Fk(\()p Fl(R)p Fk(\))1360 1707 y Fn(U)1426 1667 y Fj(\032)1417 1729 y Fk(2)1464 1707 y Fw(\()p Fn(s)1535 1719 y Fk(2)1573 1707 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b Fm(E)7 b(V)g Fn(ar)r Fw(\()p Fn(t)2133 1719 y Fk(1)2171 1707 y Fw(\)\))p Fn(\033)2337 1660 y Fj(i)28 b Fl(\003)2324 1707 y Fm(!)2399 1722 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2587 1707 y Fn(:)14 b(:)g(:)1117 1783 y Fj(i)28 b Fl(\003)1104 1830 y Fm(!)1180 1845 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1367 1830 y Fn(U)1433 1791 y Fj(\032)1424 1859 y Fl(j)p Fj(\032)p Fl(j)1502 1830 y Fw(\()p Fn(t)1564 1845 y Fl(j)p Fj(\032)p Fl(j)1642 1830 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p 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Fj(i)28 b Fl(\003)3165 2278 y Fm(!)3240 2293 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))523 2410 y Fn(s)562 2422 y Fk(1)599 2410 y Fn(\033)718 2363 y Fj(i)28 b Fl(\003)705 2410 y Fm(!)781 2425 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))986 2410 y Fn(t)1016 2422 y Fk(1)1053 2410 y Fn(\033)s Fw(.)39 b(Since)g Fn(l)r(eng)s(th)p Fw(\()p Fn(D)1731 2422 y Fk(1)1766 2410 y Fw(\))i Fn(<)g(l)r(eng)s(th)p Fw(\()p Fn(D)r Fw(\),)c(w)n(e)h(infer)g(from)g(the)h(inductiv)n(e)523 2541 y(h)n(yp)r(othesis)21 b(that)i Fn(s)1140 2553 y Fk(1)1177 2541 y Fn(\033)1224 2553 y Fl(r)1307 2541 y Fm(!)1390 2511 y Fl(\003)1390 2564 y(R)1474 2541 y Fm(r)p Fw(\()p Fn(t)1605 2553 y Fk(1)1643 2541 y Fn(\033)s Fw(\))h(=)e Fn(t)1866 2553 y Fk(1)1903 2541 y Fn(\033)1950 2553 y Fl(r)2061 2494 y Fj(i)28 b Fl(\003)2048 2541 y Fm(!)2124 2556 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2311 2541 y Fn(t)2341 2553 y Fk(1)2379 2541 y Fn(\033)s Fw(.)22 b(In)g(particular,)f Fn(x\033)3067 2553 y Fl(r)3178 2494 y Fj(i)28 b Fl(\003)3165 2541 y Fm(!)3240 2556 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))523 2641 y Fn(x\033)38 b Fw(is)d(true)f(for)g(ev)n(ery)f(v)-5 b(ariable)33 b Fn(x)i Fm(2)g(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))23 b Fm([)g(V)7 b Fn(ar)r Fw(\()p Fn(t)2314 2653 y Fk(1)2352 2641 y Fw(\).)35 b(By)f(con)n(tin)n(uing)g(along)f(these) 523 2741 y(lines,)28 b(w)n(e)f(deriv)n(e)1484 2939 y Fn(s)1523 2951 y Fj(j)1558 2939 y Fn(\033)1605 2951 y Fl(r)1688 2939 y Fm(!)1771 2905 y Fl(\003)1771 2959 y(R)1856 2939 y Fn(t)1886 2951 y Fj(j)1920 2939 y Fn(\033)1967 2951 y Fl(r)2078 2892 y Fj(i)h Fl(\003)2066 2939 y Fm(!)2141 2954 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2328 2939 y Fn(t)2358 2951 y Fj(j)2393 2939 y Fn(\033)523 3151 y Fw(for)32 b(all)g Fn(j)j Fm(2)c(f)p Fw(1)p Fn(;)14 b(:)g(:)g(:)f(;)h Fm(j)p Fn(\032)p Fm(jg)p Fw(.)32 b(Consequen)n(tly)-7 b(,)31 b Fn(l)r(\033)1993 3163 y Fl(r)2083 3151 y Fm(!)2166 3163 y Fl(R)2258 3151 y Fn(r)r(\033)2344 3163 y Fl(r)2405 3151 y Fw(.)h(Finally)-7 b(,)33 b Fn(r)r(\033)2850 3163 y Fl(r)2969 3104 y Fj(i)28 b Fl(\003)2956 3151 y Fm(!)3031 3166 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3227 3151 y Fn(r)r(\033)36 b Fw(is)523 3282 y(a)h(consequence)f(of)h(the)g(fact)g(that)h Fn(x\033)1793 3294 y Fl(r)1920 3235 y Fj(i)28 b Fl(\003)1907 3282 y Fm(!)1982 3297 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2185 3282 y Fn(x\033)41 b Fw(holds)c(for)f(ev)n(ery)g Fn(x)j Fm(2)h(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))25 b Fm([)523 3343 y Fh(S)592 3364 y Fl(j)p Fj(\032)p Fl(j)592 3430 y Fj(j)s Fk(=1)725 3405 y Fm(V)7 b Fn(ar)r Fw(\()p Fn(t)928 3417 y Fj(j)964 3405 y Fw(\))28 b(in)g(conjunction)f(with)h Fm(V)7 b Fn(ar)r Fw(\()p Fn(r)r Fw(\))25 b Fm(\022)e(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))18 b Fm([)2440 3343 y Fh(S)2509 3364 y Fl(j)p Fj(\032)p Fl(j)2509 3430 y Fj(j)s Fk(=1)2642 3405 y Fm(V)7 b Fn(ar)r Fw(\()p Fn(t)2845 3417 y Fj(j)2881 3405 y Fw(\).)523 3577 y Fi(Lemma)30 b(9.)40 b Fp(If)35 b Fm(R)f Fp(is)h(a)g(quasi-de)l(cr)l(e)l(asing)g (deterministic)g(3-CTRS,)g(then)f(ther)l(e)g(is)h(no)523 3677 y(in\014nite)29 b(innermost)h Fn(U)9 b Fw(\()p Fm(R)p Fw(\))p Fp(-derivation)32 b(starting)e(fr)l(om)g(a)g(term)f Fn(s)23 b Fm(2)h(T)d Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))p Fp(.)523 3848 y(Pr)l(o)l(of.)43 b Fw(Let)30 b Fm(R)g Fw(b)r(e)g(quasi-decreasing)c(w.r.t.)k Fm(\037)p Fw(.)f(W)-7 b(e)30 b(sho)n(w)e(b)n(y)h(w)n(ell-founded)h(induction)523 3948 y(on)21 b Fm(\037)f Fw(that)i(ev)n(ery)d(innermost)i Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)21 b(starting)f(from)g Fn(s)j Fm(2)h(T)d Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))21 b(is)g(\014nite.)h(If)523 4047 y Fn(s)h Fm(!)668 4012 y Fk(+)668 4072 y Fl(R)752 4047 y Fn(u)g Fm(2)g(T)f Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\),)20 b(then)g Fn(s)j Fm(\037)f Fn(u)e Fw(b)r(ecause)f Fm(!)2018 4059 y Fl(R)2093 4047 y Fm(\022)14 b(\037)19 b Fw(and)g(it)i(follo)n(ws)d (from)i(the)g(inductiv)n(e)523 4147 y(h)n(yp)r(othesis)30 b(that)h(ev)n(ery)f(innermost)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)30 b(starting)g(from)h Fn(u)c Fm(2)i(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))31 b(is)523 4246 y(\014nite.)g(Moreo)n(v)n(er,)e(if)i Fn(t)g Fw(is)g(a)f(prop)r(er)g(subterm)h(of)f Fn(s)h Fw(and)g Fn(t)d Fm(!)2521 4216 y Fl(\003)2521 4269 y(R)2611 4246 y Fn(u)g Fm(2)h(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\),)31 b(then)g(w)n(e)523 4346 y(infer)e Fn(s)c Fm(\037)f Fn(t)h Fm(\027)g Fn(u)j Fw(b)r(ecause)g Fm(\037)g Fw(has)h(the)g(subterm)f(prop)r(ert)n(y)g(and)h Fm(!)2696 4358 y Fl(R)2771 4346 y Fm(\022)14 b(\037)p Fw(.)27 b(Hence)i(ev)n(ery) 523 4446 y(innermost)e Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)27 b(starting)g(from)g Fn(t)h Fw(or)f Fn(u)g Fw(is)h(also)e(\014nite.)648 4545 y(F)-7 b(or)39 b(a)g(pro)r(of)g(b)n (y)h(con)n(tradiction,)e(supp)r(ose)h(that)h(there)g(is)g(an)f (in\014nite)h(innermost)523 4645 y Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)31 b Fn(D)i Fw(starting)d(from)h Fn(s)p Fw(.)g(According)f(to)h(the)g(ab)r(o)n(v)n(e,)f Fn(D)j Fw(m)n(ust)e(b)r(e)h(of)f(the)523 4745 y(form)766 4924 y Fn(D)25 b Fw(:)e Fn(s)996 4877 y Fj(i)28 b Fl(\003)983 4924 y Fm(!)1058 4939 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1246 4924 y Fn(f)j Fw(\()p Fn(v)1368 4936 y Fk(1)1405 4924 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)1630 4936 y Fj(m)1693 4924 y Fw(\))24 b(=)e Fn(l)r(\033)1966 4877 y Fj(i)1936 4924 y Fm(!)2019 4939 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2207 4924 y Fn(U)2273 4884 y Fj(\032)2264 4946 y Fk(1)2311 4924 y Fw(\()p Fn(s)2382 4936 y Fk(1)2419 4924 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)2824 4877 y Fj(i)2794 4924 y Fm(!)2877 4939 y Fj(U)f Fk(\()p Fl(R)p Fk(\))3065 4924 y Fn(:)14 b(:)g(:)p eop %%Page: 9 9 9 8 bop 523 448 a Fw(where)37 b(ev)n(ery)f Fn(v)1042 460 y Fj(j)1117 448 y Fm(2)j Fn(N)9 b(F)j Fw(\()p Fm(!)1467 463 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1632 448 y Fw(\),)38 b(i.e.,)f(ev)n(ery)f Fn(v)2160 460 y Fj(j)2233 448 y Fw(is)h(a)g(normal)g(form)g(w.r.t.)g Fm(!)3217 463 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3382 448 y Fw(.)523 580 y(Since)32 b Fn(u)792 592 y Fj(j)885 533 y(i)c Fl(\003)872 580 y Fm(!)947 595 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1142 580 y Fn(v)1182 592 y Fj(j)1217 580 y Fw(,)32 b(w)n(e)g(deriv)n(e)f (from)h(Lemma)g(8)f(that)h Fn(u)2455 592 y Fj(j)2520 580 y Fm(!)2603 550 y Fl(\003)2603 603 y(R)2695 580 y Fm(r)p Fw(\()p Fn(v)2836 592 y Fj(j)2872 580 y Fw(\))2962 533 y Fj(i)c Fl(\003)2949 580 y Fm(!)3024 595 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3219 580 y Fn(v)3259 592 y Fj(j)3294 580 y Fw(.)33 b(If)523 689 y Fn(u)571 701 y Fj(j)629 689 y Fm(!)712 654 y Fk(+)712 714 y Fl(R)796 689 y Fm(r)p Fw(\()p Fn(v)937 701 y Fj(j)973 689 y Fw(\),)28 b(then)g(the)g(deriv)-5 b(ation)564 881 y Fn(s)23 b Fm(!)709 845 y Fk(+)709 905 y Fl(R)794 881 y Fn(f)9 b Fw(\()p Fn(u)924 893 y Fk(1)960 881 y Fn(;)14 b(:)g(:)g(:)g(;)g(u)1193 893 y Fj(j)s Fl(\000)p Fk(1)1312 881 y Fn(;)g Fm(r)p Fw(\()p Fn(v)1490 893 y Fj(j)1526 881 y Fw(\))p Fn(;)g(u)1643 893 y Fj(j)s Fk(+1)1762 881 y Fn(;)g(:)g(:)g(:)f(u)1957 893 y Fj(m)2020 881 y Fw(\))23 b(=)g Fn(t)2244 834 y Fj(i)28 b Fl(\003)2231 881 y Fm(!)2306 896 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2494 881 y Fn(f)j Fw(\()p Fn(v)2616 893 y Fk(1)2653 881 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)2878 893 y Fj(m)2941 881 y Fw(\))3026 834 y Fj(i)2996 881 y Fm(!)3079 896 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3267 881 y Fn(:)14 b(:)g(:)523 1055 y Fw(is)38 b(in\014nite)h(whic)n(h)f(is)g(imp)r(ossible)g(b)r (ecause)g Fm(!)2067 1019 y Fk(+)2067 1079 y Fl(R)2128 1055 y Fm(\022\037)f Fw(and)h(hence)g(there)g(is)g(no)g(in\014nite)523 1155 y(innermost)26 b Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)26 b(starting)g(from)g Fn(t)p Fw(.)h(Th)n(us)f Fn(u)2368 1167 y Fj(j)2425 1155 y Fw(=)d Fm(r)p Fw(\()p Fn(v)2654 1167 y Fj(j)2690 1155 y Fw(\))k(for)e(ev)n(ery)h Fn(j)i Fm(2)23 b Fd(N)t Fw(.)33 b(It)523 1254 y(is)26 b(not)g(di\016cult)g(to)g(c)n(hec)n(k)f(that)h Fn(s)d Fw(=)f Fn(l)r(\033)1776 1266 y Fl(r)1862 1254 y Fw(follo)n(ws)j(as)g(a) g(consequence.)g(Therefore,)g Fn(D)j Fw(has)523 1354 y(the)g(form)939 1518 y Fn(D)d Fw(:)e Fn(s)g Fw(=)g Fn(l)r(\033)1303 1530 y Fl(r)1413 1471 y Fj(i)28 b Fl(\003)1400 1518 y Fm(!)1476 1533 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1663 1518 y Fn(l)r(\033)1793 1471 y Fj(i)1763 1518 y Fm(!)1846 1533 y Fj(U)g Fk(\()p Fl(R)p Fk(\))2034 1518 y Fn(U)2100 1479 y Fj(\032)2091 1541 y Fk(1)2138 1518 y Fw(\()p Fn(s)2209 1530 y Fk(1)2246 1518 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)2651 1471 y Fj(i)2621 1518 y Fm(!)2704 1533 y Fj(U)f Fk(\()p Fl(R)p Fk(\))2892 1518 y Fn(:)14 b(:)g(:)523 1683 y Fw(The)26 b(v)-5 b(alidit)n(y)26 b(of)g(the)h(inequalit)n(y)f Fn(l)r(\033)1681 1695 y Fl(r)1763 1683 y Fm(\037)d Fn(s)1890 1695 y Fk(1)1927 1683 y Fn(\033)1974 1695 y Fl(r)2060 1683 y Fw(is)j(a)g(consequence)f (of)h(the)h(fact)f(that)h Fm(R)f Fw(is)523 1783 y(quasi-decreasing.)h (Hence)i(there)g(is)h(no)f(in\014nite)h(innermost)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)29 b(starting)523 1905 y(from)36 b Fn(s)767 1917 y Fk(1)805 1905 y Fn(\033)852 1917 y Fl(r)912 1905 y Fw(.)g(Since)h Fn(s)1236 1917 y Fk(1)1274 1905 y Fn(\033)1321 1917 y Fl(r)1446 1858 y Fj(i)29 b Fl(\003)1434 1905 y Fm(!)1509 1920 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1712 1905 y Fn(s)1751 1917 y Fk(1)1788 1905 y Fn(\033)s Fw(,)37 b(ev)n(ery)f(in\014nite)h(innermost)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)523 2005 y(starting)27 b(from)g Fn(s)1068 2017 y Fk(1)1105 2005 y Fn(\033)k Fw(m)n(ust)d(b)r(e)g(\014nite.)g(The)g(deriv)-5 b(ation)27 b Fn(D)j Fw(th)n(us)d(lo)r(oks)g(lik)n(e)581 2188 y Fn(s)c Fw(=)g Fn(l)r(\033)805 2200 y Fl(r)915 2141 y Fj(i)28 b Fl(\003)902 2188 y Fm(!)977 2203 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1165 2188 y Fn(l)r(\033)1294 2141 y Fj(i)1265 2188 y Fm(!)1348 2203 y Fj(U)g Fk(\()p Fl(R)p Fk(\))1535 2188 y Fn(U)1601 2148 y Fj(\032)1592 2210 y Fk(1)1640 2188 y Fw(\()p Fn(s)1711 2200 y Fk(1)1748 2188 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)2151 2141 y Fj(i)28 b Fl(\003)2138 2188 y Fm(!)2213 2203 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2401 2188 y Fn(U)2467 2148 y Fj(\032)2458 2210 y Fk(1)2505 2188 y Fw(\()p Fn(t)2567 2200 y Fk(1)2605 2188 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)3009 2141 y Fj(i)2980 2188 y Fm(!)3063 2203 y Fj(U)f Fk(\()p Fl(R)p Fk(\))3250 2188 y Fn(:)14 b(:)g(:)523 2385 y Fw(No)n(w)28 b Fn(s)752 2397 y Fk(1)790 2385 y Fn(\033)837 2397 y Fl(r)950 2338 y Fj(i)g Fl(\003)937 2385 y Fm(!)1012 2400 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1202 2385 y Fn(s)1241 2397 y Fk(1)1278 2385 y Fn(\033)1382 2338 y Fj(i)28 b Fl(\003)1369 2385 y Fm(!)1444 2400 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1634 2385 y Fn(t)1664 2397 y Fk(1)1701 2385 y Fn(\033)33 b Fw(yields)28 b Fn(s)2054 2397 y Fk(1)2091 2385 y Fn(\033)2138 2397 y Fl(r)2224 2385 y Fm(!)2307 2355 y Fl(\003)2307 2408 y(R)2393 2385 y Fn(t)2423 2397 y Fk(1)2461 2385 y Fn(\033)2508 2397 y Fl(r)2621 2338 y Fj(i)g Fl(\003)2608 2385 y Fm(!)2683 2400 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2873 2385 y Fn(t)2903 2397 y Fk(1)2940 2385 y Fn(\033)33 b Fw(b)n(y)28 b(Lemma)523 2485 y(8.)35 b(It)g(follo)n(ws)g Fn(l)r(\033)1075 2497 y Fl(r)1170 2485 y Fm(\037)g Fn(s)1309 2497 y Fk(2)1346 2485 y Fn(\033)1393 2497 y Fl(r)1489 2485 y Fw(b)r(ecause)f Fm(R)i Fw(is)f(quasi-decreasing)e(w.r.t.)i Fm(\037)g Fw(and)g(w)n(e)g(ma)n(y)523 2584 y(con)n(tin)n(ue)27 b(with)h(the)g(ab)r(o)n(v)n(e)e(reasoning.)g(All)i(in)g(all,)f Fn(D)j Fw(m)n(ust)e(ha)n(v)n(e)e(the)i(form)531 2767 y Fn(l)r(\033)605 2779 y Fl(r)715 2720 y Fj(i)h Fk(+)711 2767 y Fm(!)795 2782 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))982 2767 y Fn(U)1048 2727 y Fj(\032)1039 2790 y Fk(1)1087 2767 y Fw(\()p Fn(s)1158 2779 y Fk(1)1195 2767 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\)\))p Fn(\033)1598 2720 y Fj(i)28 b Fl(\003)1585 2767 y Fm(!)1660 2782 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1848 2767 y Fn(U)1914 2727 y Fj(\032)1905 2796 y Fl(j)p Fj(\032)p Fl(j)1983 2767 y Fw(\()p Fn(t)2045 2782 y Fl(j)p Fj(\032)p Fl(j)2123 2767 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)2675 2720 y Fj(i)2646 2767 y Fm(!)2729 2782 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2916 2767 y Fn(r)r(\033)3059 2720 y Fj(i)3029 2767 y Fm(!)3112 2782 y Fj(U)g Fk(\()p Fl(R)p Fk(\))3300 2767 y Fn(:)14 b(:)g(:)523 2977 y Fw(Hence)20 b Fn(l)r(\033)836 2989 y Fl(r)918 2977 y Fm(!)1001 2989 y Fl(R)1086 2977 y Fn(r)r(\033)1172 2989 y Fl(r)1283 2930 y Fj(i)28 b Fl(\003)1270 2977 y Fm(!)1346 2992 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1533 2977 y Fn(r)r(\033)24 b Fw(b)n(y)c(Lemma)f(8.)g(W)-7 b(e)20 b(conclude)g(that)g(there)f(is)h(an)f(in\014nite)523 3077 y(innermost)31 b Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)30 b(starting)g(from)h Fn(r)r(\033)2135 3089 y Fl(r)2227 3077 y Fw(whic)n(h)g(is)f(imp)r(ossible)h(b)r(ecause)g Fn(s)d Fw(=)523 3176 y Fn(l)r(\033)597 3188 y Fl(r)680 3176 y Fm(\037)22 b Fn(r)r(\033)853 3188 y Fl(r)914 3176 y Fw(.)523 3333 y Fi(Theorem)30 b(10.)41 b Fp(If)33 b Fm(R)g Fp(is)g(a)g(quasi-de)l(cr)l(e)l(asing)g(deterministic)h(3-CTRS,) f(then)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\))34 b Fp(is)523 3432 y(innermost)c(terminating.)523 3589 y(Pr)l(o)l(of.)43 b Fw(W)-7 b(e)44 b(pro)n(v)n(e)d(b)n(y)i(structural)f(induction)h(that) g(there)g(is)g(no)g(in\014nite)g(innermost)523 3688 y Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)31 b(starting)f(from)g (a)h(term)f Fn(s)f Fm(2)g(T)21 b Fw(\()p Fm(F)2249 3658 y Fl(0)2272 3688 y Fn(;)14 b Fm(V)7 b Fw(\).)31 b(If)h(\(a\))f Fn(s)f Fw(is)h(a)g(v)-5 b(ariable,)29 b(then)523 3788 y(it)34 b(is)g(in)h(normal)d(form.)i(If)h(\(b\))f Fn(s)g Fw(is)g(a)g(constan)n(t,)f(then)h Fn(s)g Fm(2)g(F)42 b Fw(and)34 b(the)g(claim)g(follo)n(ws)523 3888 y(from)22 b(Lemma)h(9.)f(Supp)r(ose)h(\(c\))g Fn(s)g Fw(=)g Fn(U)1754 3848 y Fj(\032)1745 3911 y(i)1792 3888 y Fw(\()p Fn(u)1872 3900 y Fk(1)1909 3888 y Fn(;)14 b(:)g(:)g(:)g(;)g(u)2142 3900 y Fj(m)2205 3888 y Fw(\).)23 b(Ev)n(ery)e(in\014nite)i(innermost)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-)523 3987 y(deriv)-5 b(ation)27 b(starting)g(from)g Fn(s)h Fw(m)n(ust)f(ha)n(v)n(e)g(the)h(form)725 4170 y Fn(D)d Fw(:)e Fn(s)g Fw(=)g Fn(U)1081 4130 y Fj(\032)1072 4193 y(i)1119 4170 y Fw(\()p Fn(u)1199 4182 y Fk(1)1236 4170 y Fn(;)14 b(:)g(:)g(:)g(;)g(u)1469 4182 y Fj(m)1531 4170 y Fw(\))1614 4123 y Fj(i)29 b Fl(\003)1602 4170 y Fm(!)1677 4185 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1864 4170 y Fn(U)1930 4130 y Fj(\032)1921 4193 y(i)1969 4170 y Fw(\()p Fn(v)2041 4182 y Fk(1)2079 4170 y Fn(;)14 b(:)g(:)g(:)f(;)h (v)2303 4182 y Fj(m)2366 4170 y Fw(\))24 b(=)e Fn(U)2575 4130 y Fj(\032)2566 4193 y(i)2614 4170 y Fw(\()p Fn(t)2676 4182 y Fj(i)2704 4170 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)f Fw(\))p Fn(\033)1616 4247 y Fj(i)1587 4294 y Fm(!)1670 4309 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1857 4294 y Fn(U)1923 4254 y Fj(\032)1914 4317 y(i)p Fk(+1)2026 4294 y Fw(\()p Fn(s)2097 4306 y Fj(i)p Fk(+1)2209 4294 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)f Fw(\))p Fn(\033)2761 4247 y Fj(i)2731 4294 y Fm(!)2814 4309 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3002 4294 y Fn(:)14 b(:)g(:)523 4460 y Fw(where)27 b Fn(v)803 4472 y Fj(j)861 4460 y Fm(2)c Fn(N)9 b(F)j Fw(\()p Fm(!)1195 4475 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1360 4460 y Fw(\))27 b(for)g(ev)n(ery)f(1)d Fm(\024)f Fn(j)28 b Fm(\024)23 b Fn(m)p Fw(.)k(If)h Fn(s)2312 4472 y Fj(i)p Fk(+1)2451 4460 y Fw(is)f(a)g(v)-5 b(ariable,)26 b(then)h Fn(s)3164 4472 y Fj(i)p Fk(+1)3276 4460 y Fn(\033)g Fm(2)523 4559 y Fn(N)9 b(F)j Fw(\()p Fm(!)779 4574 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))944 4559 y Fw(\).)40 b(Otherwise)f Fn(r)r(oot)p Fw(\()p Fn(s)1664 4571 y Fj(i)p Fk(+1)1778 4559 y Fw(\))44 b Fm(2)g(F)k Fw(and)40 b(b)n(y)g(case)f(\(d\))i(there)f (is)g(no)g(in\014nite)523 4659 y(innermost)23 b Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)23 b(starting)g(from)g Fn(s)2058 4671 y Fj(i)p Fk(+1)2170 4659 y Fn(\033)s Fw(.)h(A)g(rep)r (etition)f(of)h(these)f(argumen)n(ts)523 4758 y(sho)n(ws)j(that)i Fn(D)i Fw(has)d(the)h(form)620 4923 y Fn(D)d Fw(:)e Fn(s)g Fw(=)g Fn(U)976 4883 y Fj(\032)967 4946 y(i)1014 4923 y Fw(\()p Fn(u)1094 4935 y Fk(1)1131 4923 y Fn(;)14 b(:)g(:)g(:)g(;)g (u)1364 4935 y Fj(m)1426 4923 y Fw(\))1509 4876 y Fj(i)28 b Fl(\003)1496 4923 y Fm(!)1572 4938 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1759 4923 y Fn(U)1825 4883 y Fj(\032)1816 4951 y Fl(j)p Fj(\032)p Fl(j)1894 4923 y Fw(\()p Fn(t)1956 4938 y Fl(j)p Fj(\032)p Fl(j)2034 4923 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\))p Fn(\033)2587 4876 y Fj(i)2557 4923 y Fm(!)2640 4938 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2827 4923 y Fn(r)r(\033)2970 4876 y Fj(i)2941 4923 y Fm(!)3024 4938 y Fj(U)g Fk(\()p Fl(R)p Fk(\))3211 4923 y Fn(:)14 b(:)g(:)p eop %%Page: 10 10 10 9 bop 523 448 a Fw(Again,)28 b(if)g Fn(r)j Fw(is)d(a)g(v)-5 b(ariable,)27 b(then)h Fn(r)r(\033)g Fm(2)d Fn(N)9 b(F)j Fw(\()p Fm(!)2061 463 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2225 448 y Fw(\).)29 b(Otherwise)e Fn(r)r(oot)p Fw(\()p Fn(r)r Fw(\))f Fm(2)e(F)36 b Fw(and)28 b(b)n(y)523 548 y(case)f(\(d\))h(there) g(is)f(no)g(in\014nite)i(innermost)e Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)27 b(starting)g(from)g Fn(r)r(\033)s Fw(.)523 648 y(\(d\))h(Supp)r(ose)g Fn(s)23 b Fw(=)g Fn(f)9 b Fw(\()p Fn(u)1267 660 y Fk(1)1304 648 y Fn(;)14 b(:)g(:)g(:)f(;)h(u)1536 660 y Fj(m)1599 648 y Fw(\),)28 b(where)f Fn(f)32 b Fm(2)23 b(F)36 b Fw(and)27 b Fn(u)2378 660 y Fj(j)2436 648 y Fm(2)c(T)f Fw(\()p Fm(F)2681 617 y Fl(0)2704 648 y Fn(;)14 b Fm(V)7 b Fw(\).)28 b(F)-7 b(or)27 b(a)g(pro)r(of)g(b)n(y)523 747 y(con)n(tradiction,)f (supp)r(ose)i(that)g(there)f(is)g(an)h(in\014nite)g(innermost)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)916 928 y Fn(D)25 b Fw(:)f Fn(s)f Fw(=)f Fn(f)9 b Fw(\()p Fn(u)1336 940 y Fk(1)1373 928 y Fn(;)14 b(:)g(:)g(:)f(;)h(u)1605 940 y Fj(m)1668 928 y Fw(\))1751 880 y Fj(i)28 b Fl(\003)1738 928 y Fm(!)1813 943 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2001 928 y Fn(f)j Fw(\()p Fn(v)2123 940 y Fk(1)2160 928 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)2385 940 y Fj(m)2448 928 y Fw(\))23 b(=)g Fn(t)2674 880 y Fj(i)2644 928 y Fm(!)2727 943 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2915 928 y Fn(:)14 b(:)g(:)523 1085 y Fw(where)26 b Fn(v)802 1097 y Fj(j)860 1085 y Fm(2)d Fn(N)9 b(F)j Fw(\()p Fm(!)1194 1100 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1359 1085 y Fw(\).)26 b(If)h Fn(t)c Fm(2)g(T)f Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\),)26 b(then)g(the)h(assertion)d(follo)n(ws)i(from)f(Lemma)523 1184 y(9.)e(Otherwise,)f Fn(t)h Fw(con)n(tains)f Fn(U)9 b Fw(-subterms)22 b(but)h(all)g(of)g(them)h(are)e(in)h(normal)f(form.)g (W)-7 b(e)24 b(ma)n(y)523 1284 y(write)19 b Fn(t)k Fw(=)g Fn(C)6 b Fw([)p Fn(w)1015 1296 y Fk(1)1053 1284 y Fn(;)14 b(:)g(:)g(:)f(;)h(w)1296 1296 y Fj(n)1342 1284 y Fw(],)20 b(where)e Fn(C)30 b Fm(2)23 b(T)e Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))20 b(and)f(ev)n(ery)f(term)i Fn(w)2733 1296 y Fj(i)2780 1284 y Fw(is)f(a)g(\(maximal\))h Fn(U)9 b Fw(-)523 1384 y(subterm)29 b(of)f Fn(t)p Fw(.)h(In)f(order)g(to)g (cop)r(e)g(with)h(non-left-linear)f(rules,)g(w)n(e)g(ha)n(v)n(e)g(to)g (distinguish)523 1483 y(b)r(et)n(w)n(een)40 b(those)g Fn(U)9 b Fw(-subterms)39 b(of)h Fn(t)g Fw(whic)n(h)h(can)e(b)r(e)i (created)e(b)n(y)h(a)g(term)g(without)h Fn(U)9 b Fw(-)523 1583 y(sym)n(b)r(ols)34 b(and)g(those)h(whic)n(h)f(can't.)h(F)-7 b(or)34 b(the)h(sak)n(e)e(of)i(simplicit)n(y)-7 b(,)34 b(let)h(us)g(assume)f(that)523 1701 y(for)c(ev)n(ery)f(1)f Fm(\024)f Fn(j)33 b Fm(\024)27 b Fn(i)k Fw(there)f(is)g(a)g(term)h Fn(w)1892 1671 y Fl(0)1890 1723 y Fj(j)1953 1701 y Fm(2)e(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))31 b(suc)n(h)f(that)h Fn(w)2795 1671 y Fl(0)2793 1723 y Fj(j)2884 1654 y(i)d Fk(+)2879 1701 y Fm(!)2963 1716 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3155 1701 y Fn(w)3214 1713 y Fj(j)3280 1701 y Fw(but)523 1801 y(for)35 b(ev)n(ery)f Fn(i)i(<)f(j)41 b Fm(\024)36 b Fn(n)f Fw(no)h(suc)n(h)f(term)g(exists.)g(Then)h(there)f(is)g(an)g (in\014nite)h(innermost)523 1900 y Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation)652 2081 y Fn(D)723 2046 y Fl(0)769 2081 y Fw(:)23 b Fn(s)854 2046 y Fl(0)900 2081 y Fw(=)g Fn(C)6 b Fw([)p Fn(w)1137 2046 y Fl(0)1135 2101 y Fk(1)1173 2081 y Fn(;)14 b(:)g(:)g(:)g(;)g(w)1419 2046 y Fl(0)1417 2101 y Fj(i)1445 2081 y Fn(;)g(w)1541 2093 y Fj(i)p Fk(+1)1653 2081 y Fn(;)g(:)g(:)g(:)f(;)h(w)1896 2093 y Fj(n)1942 2081 y Fw(])2016 2034 y Fj(i)28 b Fl(\003)2003 2081 y Fm(!)2078 2096 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2266 2081 y Fn(f)j Fw(\()p Fn(v)2388 2093 y Fk(1)2425 2081 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)2650 2093 y Fj(m)2713 2081 y Fw(\))23 b(=)g Fn(t)2939 2034 y Fj(i)2909 2081 y Fm(!)2992 2096 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))3179 2081 y Fn(:)14 b(:)g(:)523 2238 y Fw(In)31 b(order)f(to)h(get)f(rid)h(of)g(the)g(remaining)f Fn(U)9 b Fw(-subterms)30 b Fn(w)2377 2250 y Fj(i)p Fk(+1)2489 2238 y Fn(;)14 b(:)g(:)g(:)g(;)g(w)2733 2250 y Fj(n)2778 2238 y Fw(,)31 b(w)n(e)g(mark)3146 2208 y Fk(1)3213 2238 y Fw(ev)n(ery)523 2337 y Fn(w)582 2349 y Fj(j)617 2337 y Fw(,)j Fn(i)d(<)h(j)37 b Fm(\024)31 b Fn(n)p Fw(,)i(and)g(c)n(ho)r (ose)e(fresh)i(v)-5 b(ariables)32 b Fn(x)2143 2349 y Fj(i)p Fk(+1)2255 2337 y Fn(;)14 b(:)g(:)g(:)g(;)g(x)2487 2349 y Fj(n)2565 2337 y Fw(with)34 b(the)f(prop)r(ert)n(y)f(that)523 2437 y Fn(x)570 2449 y Fj(j)631 2437 y Fw(=)25 b Fn(x)768 2449 y Fj(k)838 2437 y Fw(if)30 b(and)f(only)f(if)i Fn(w)1399 2449 y Fj(j)1459 2437 y Fw(=)25 b Fn(w)1608 2449 y Fj(k)1679 2437 y Fw(for)j Fn(i)d(<)g(j)31 b(<)25 b(k)j Fm(\024)d Fn(n)p Fw(.)k(Let)g Fn(\011)38 b Fw(b)r(e)29 b(the)h(transformation)523 2537 y(whic)n(h)c(replaces)f(ev)n(ery)g(mark)n(ed)g(o)r(ccurrence)g(of) i Fn(w)2149 2549 y Fj(j)2211 2537 y Fw(in)f(a)g(term)g(with)h Fn(x)2806 2549 y Fj(j)2841 2537 y Fw(.)g(W)-7 b(e)27 b(claim)f(that)523 2636 y(the)i(transformed)e(deriv)-5 b(ation)27 b Fn(\011)9 b Fw(\()p Fn(D)1683 2606 y Fl(0)1707 2636 y Fw(\))559 2817 y Fn(\011)g Fw(\()p Fn(s)690 2782 y Fl(0)713 2817 y Fw(\))23 b(=)g Fn(C)6 b Fw([)p Fn(w)1005 2782 y Fl(0)1003 2837 y Fk(1)1041 2817 y Fn(;)14 b(:)g(:)g(:)g(;)g(w) 1287 2782 y Fl(0)1285 2837 y Fj(i)1313 2817 y Fn(;)g(x)1397 2829 y Fj(i)p Fk(+1)1509 2817 y Fn(;)g(:)g(:)g(:)f(;)h(x)1740 2829 y Fj(n)1786 2817 y Fw(])1860 2770 y Fj(i)28 b Fl(\003)1847 2817 y Fm(!)1922 2832 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2110 2817 y Fn(\011)j Fw(\()p Fn(f)g Fw(\()p Fn(v)2324 2829 y Fk(1)2361 2817 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)2586 2829 y Fj(m)2649 2817 y Fw(\)\))24 b(=)e Fn(\011)9 b Fw(\()p Fn(t)p Fw(\))3032 2770 y Fj(i)3002 2817 y Fm(!)3085 2832 y Fj(U)d Fk(\()p Fl(R)p Fk(\))3273 2817 y Fn(:)14 b(:)g(:)523 2974 y Fw(is)41 b(an)g(in\014nite)h(innermost)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\)-deriv)-5 b(ation.)41 b(Since)h Fn(\011)9 b Fw(\()p Fn(s)2463 2944 y Fl(0)2486 2974 y Fw(\))46 b Fm(2)h(T)21 b Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\),)42 b(this)f(yields)523 3097 y(the)34 b(desired)f(con)n (tradiction)f(to)h(Lemma)g(9.)g(Let)h Fn(C)6 b Fw([)p Fn(l)r(\033)s Fw(])2384 3049 y Fj(i)2354 3097 y Fm(!)2437 3112 y Fj(U)g Fk(\()p Fl(R)p Fk(\))2634 3097 y Fn(C)g Fw([)p Fn(r)r(\033)s Fw(])36 b(b)r(e)d(a)g(reduction)523 3196 y(step)j(in)h Fn(D)884 3166 y Fl(0)907 3196 y Fw(,)f(where)g Fn(l)i Fm(!)f Fn(r)j Fm(2)e Fn(U)9 b Fw(\()p Fm(R)p Fw(\).)37 b(The)f(reduction)f(step)i(cannot)e(tak)n(e)g(place)h(in)g(a)523 3296 y(mark)n(ed)20 b(subterm)i(b)r(ecause)f(ev)n(ery)f(mark)n(ed)h (subterm)g(is)h(in)g(normal)e(form,)i(so)f Fn(\011)9 b Fw(\()p Fn(C)d Fw([)p Fn(l)r(\033)s Fw(]\))23 b(=)523 3395 y Fn(\011)9 b Fw(\()p Fn(C)d Fw(\)[)p Fn(\011)j Fw(\()p Fn(l)r(\033)s Fw(\)].)37 b(Moreo)n(v)n(er,)32 b(w)n(e)j(ha)n(v)n(e)f Fn(\011)9 b Fw(\()p Fn(l)r(\033)s Fw(\))37 b(=)e Fn(l)r(\033)2155 3407 y Fj(\011)2208 3395 y Fw(,)g(where)g Fn(\033)2561 3407 y Fj(\011)2649 3395 y Fw(is)g(de\014ned)h(b)n(y)f Fn(x\033)3251 3407 y Fj(\011)3340 3395 y Fw(=)523 3495 y Fn(\011)9 b Fw(\()p Fn(x\033)s Fw(\))29 b(b)r(ecause)e(no)g(prop)r(er)g(subterm)g(of)h Fn(l)h Fw(con)n(tains)d(a)i Fn(U)9 b Fw(-sym)n(b)r(ol.)26 b(It)i(is)f(th)n(us)h(su\016cien)n(t)523 3618 y(to)g(sho)n(w)e Fn(l)r(\033)904 3630 y Fj(\011)1009 3571 y(i)980 3618 y Fm(!)1063 3633 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1250 3618 y Fn(r)r(\033)1336 3630 y Fj(\011)1418 3618 y Fw(b)r(ecause)27 b(this)h(yields)1007 3807 y Fn(\011)9 b Fw(\()p Fn(C)d Fw([)p Fn(l)r(\033)s Fw(]\))23 b(=)g Fn(\011)9 b Fw(\()p Fn(C)d Fw(\)[)p Fn(l)r(\033)1716 3819 y Fj(\011)1769 3807 y Fw(])1845 3760 y Fj(i)1815 3807 y Fm(!)1898 3822 y Fj(U)g Fk(\()p Fl(R)p Fk(\))2086 3807 y Fn(\011)j Fw(\()p Fn(C)d Fw(\)[)p Fn(r)r(\033)2384 3819 y Fj(\011)2438 3807 y Fw(])24 b(=)e Fn(\011)9 b Fw(\()p Fn(C)d Fw([)p Fn(r)r(\033)s Fw(]\))p Fn(:)523 3964 y Fw(W)-7 b(e)20 b(\014rst)g(sho)n(w)f Fn(l)r(\033)1094 3976 y Fj(\011)1146 3964 y Fm(!)1229 3979 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1394 3964 y Fn(r)r(\033)1480 3976 y Fj(\011)1534 3964 y Fw(.)20 b(Since)g Fn(l)h Fw(ma)n(y)e(b)r(e)i(non-left-linear,)e(w)n(e)g(ha)n(v) n(e)g(to)h(sho)n(w)f(that)523 4064 y Fn(x\033)27 b Fw(=)c Fn(y)s(\033)29 b Fw(implies)d Fn(x\033)1226 4076 y Fj(\011)1302 4064 y Fw(=)d Fn(y)s(\033)1481 4076 y Fj(\011)1560 4064 y Fw(for)i(ev)n(ery)g(pair)g Fn(x;)14 b(y)30 b Fw(of)c(v)-5 b(ariables)24 b(from)i Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\).)27 b(So)e(sup-)523 4163 y(p)r(ose)g(that)g Fn(x\033)k Fw(con)n(tains)c(a)f(mark)n(ed)g Fn(U)9 b Fw(-subterm,)25 b(sa)n(y)f(at)h(p)r(osition)g Fn(p)p Fw(.)g(Then)h Fn(y)s(\033)i Fw(con)n(tains)523 4263 y(the)f(same)g(subterm)g Fn(u)f Fw(at)h(p)r(osition)g Fn(p)p Fw(.)g(Since)g Fn(u)g Fw(hasn't)f(b)r(een) i(created)e(in)h(the)h(deriv)-5 b(ation)523 4381 y Fn(\011)9 b Fw(\()p Fn(D)686 4351 y Fl(0)710 4381 y Fw(\))21 b({)g(there)g(is)g (no)g Fn(u)1266 4351 y Fl(0)1312 4381 y Fm(2)i(T)e Fw(\()p Fm(F)8 b Fn(;)14 b Fm(V)7 b Fw(\))22 b(with)f Fn(u)1935 4351 y Fl(0)2009 4334 y Fj(i)28 b Fk(+)2005 4381 y Fm(!)2088 4396 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2276 4381 y Fn(u)21 b Fw({)f(it)i(is)f(also)f(mark)n(ed.)g(Therefore,)523 4481 y Fn(l)r(\033)597 4493 y Fj(\011)649 4481 y Fm(!)732 4496 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))897 4481 y Fn(r)r(\033)983 4493 y Fj(\011)1037 4481 y Fw(.)35 b(F)-7 b(urthermore,)33 b(b)n(y)i(the)g(c)n(hoice)f(of)h(the)g(fresh)g(v)-5 b(ariables)34 b Fn(x)2993 4493 y Fj(i)p Fk(+1)3105 4481 y Fn(;)14 b(:)g(:)g(:)f(;)h (x)3336 4493 y Fj(n)3382 4481 y Fw(,)523 4581 y Fn(x\033)617 4593 y Fj(\011)696 4581 y Fw(=)25 b Fn(y)s(\033)877 4593 y Fj(\011)958 4581 y Fw(also)j(implies)h Fn(x\033)g Fw(=)c Fn(y)s(\033)s Fw(.)k(Consequen)n(tly)-7 b(,)28 b Fn(l)r(\033)2374 4593 y Fj(\011)2427 4581 y Fm(!)2510 4596 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))2674 4581 y Fn(r)r(\033)2760 4593 y Fj(\011)2843 4581 y Fw(is)29 b(an)f(innermost)523 4680 y(reduction)19 b(step)h(\(for)g(otherwise)f Fn(l)r(\033)s Fm(!)1726 4695 y Fj(U)6 b Fk(\()p Fl(R)p Fk(\))1890 4680 y Fn(r)r(\033)24 b Fw(w)n(ouldn't)c(b)r(e)g(innermost\).)g(This)g (concludes)523 4780 y(the)28 b(pro)r(of.)p 523 4839 473 4 v 540 4893 a Fc(1)600 4924 y Fv(F)-6 b(or)26 b(instance)g(b)n(y)e (underlining)i(the)f(ro)r(ot)i(sym)n(b)r(ol.)p eop %%Page: 11 11 11 10 bop 523 448 a Fq(5)112 b(Application)35 b(1:)j(Mo)s(dularit)m(y) 523 655 y Fw(F)-7 b(rom)27 b(the)h(previous)f(sections,)g(w)n(e)g(kno)n (w)g(that)h(the)g(follo)n(wing)e(implications)h(hold.)675 835 y Fn(U)9 b Fw(\()p Fm(R)p Fw(\))29 b(is)e(terminating)51 b Fm(\))23 b(R)28 b Fw(is)f(quasi-reductiv)n(e)49 b Fm(\))23 b(R)28 b Fw(is)g(quasi-decreasing)1463 935 y Fm(\))23 b Fn(U)9 b Fw(\()p Fm(R)p Fw(\))28 b(is)g(innermost)f(terminating)o Fn(:)523 1121 y Fw(Gramlic)n(h)22 b([Gra95)o(,)h(Thm.)g(3.23])f(sho)n (w)n(ed)f(that)j(a)e(non-o)n(v)n(erlapping)e(TRS)k(is)e(terminating)523 1221 y(if)i(and)f(only)g(if)h(it)f(is)g(innermost)g(terminating.)g(Th)n (us,)g(if)h Fn(U)9 b Fw(\()p Fm(R)p Fw(\))24 b(is)f(non-o)n(v)n (erlapping,)d(then)523 1320 y(the)35 b(ab)r(o)n(v)n(e)f(implications)h (are)f(in)h(fact)g(equiv)-5 b(alences.)34 b(Ho)n(w)n(ev)n(er,)f(non-o)n (v)n(erlappingness)523 1420 y(of)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\))34 b(is)e(not)h(implied)g(b)n(y)f(non-o)n(v)n(erlappingness)d(of) j Fm(R)p Fw(.)h(F)-7 b(or)32 b(example,)g(the)h(system)523 1520 y Fm(R)c Fw(=)f Fm(f)p Fn(a)g Fm(!)h Fn(b)f Fm(\()h Fn(b)f Fm(!)h Fn(a)p Fm(g)h Fw(is)h(non-o)n(v)n(erlapping)d(but)k Fn(U)9 b Fw(\()p Fm(R)p Fw(\))29 b(=)g Fm(f)p Fn(a)f Fm(!)g Fn(U)9 b Fw(\()p Fn(b)p Fw(\))p Fn(;)14 b(U)9 b Fw(\()p Fn(a)p Fw(\))29 b Fm(!)g Fn(b)p Fm(g)523 1619 y Fw(is)h(not.)g(The)g(situation)g(is)g(di\013eren)n(t)g(for)g(syn)n (tactically)f(deterministic)h(3-CTRSs)f(whic)n(h)523 1719 y(will)f(b)r(e)g(considered)e(next.)523 1883 y Fi(De\014nition)31 b(11.)40 b Fp(A)27 b(deterministic)h(3-CTRS)f Fm(R)g Fp(is)g(c)l(al)t(le)l(d)h Fw(syn)n(tactically)c(deterministic)523 1982 y Fp(if,)32 b(for)g(every)g Fn(l)27 b Fm(!)e Fn(r)j Fm(\()d Fn(s)1357 1994 y Fk(1)1420 1982 y Fm(!)g Fn(t)1558 1994 y Fk(1)1595 1982 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)1819 1994 y Fj(k)1885 1982 y Fm(!)25 b Fn(t)2023 1994 y Fj(k)2095 1982 y Fp(in)31 b Fm(R)p Fp(,)h(every)g(term)e Fn(t)2775 1994 y Fj(i)2834 1982 y Fp(is)h(a)h(c)l(onstructor)523 2082 y(term)692 2052 y Fk(2)759 2082 y Fp(or)e(a)g(gr)l(ound)g Fm(R)1283 2094 y Fj(u)1326 2082 y Fp(-normal)h(form,)f(wher)l(e)h Fm(R)2172 2094 y Fj(u)2239 2082 y Fw(=)22 b Fm(f)p Fn(l)i Fm(!)f Fn(r)33 b Fm(j)d Fn(l)24 b Fm(!)f Fn(r)j Fm(\()d Fn(c)g Fm(2)h(Rg)p Fp(.)523 2244 y Fw(Syn)n(tactically)18 b(deterministic)g(CTRSs)h(are)e(a)h(natural)g(generalization)e(of)i (normal)g(CTRSs.)523 2344 y(The)28 b Fp(quicksort)h Fw(system)f(sho)n (ws)e(that)j(these)f(systems)f(arise)g(quite)h(naturally)-7 b(.)28 b(As)g(a)f(mat-)523 2443 y(ter)e(of)g(fact,)g(ev)n(ery)e(syn)n (tactically)h(deterministic)h(CTRS)h(is)e(strongly)g(deterministic)h (\(see)523 2543 y([ALS94)o(])33 b(for)g(a)f(de\014nition\))h(but)h(in)f (con)n(trast)e(to)i(the)g(latter)g(prop)r(ert)n(y)-7 b(,)31 b(it)j(is)e(decidable)523 2643 y(whether)38 b(a)g(CTRS)h(is)f (syn)n(tactically)f(deterministic)i(or)e(not.)h(Strongly)g (deterministic)523 2742 y(systems)32 b(are)f(in)n(teresting)h(b)r (ecause)g(of)g(the)h(critical)f(pair)g(lemma)g(that)h(holds)f(for)f (them)523 2842 y([ALS94)o(,)d(Thm.)g(4.1].)648 2943 y(According)20 b(to)i(the)g(next)g(lemma,)g(for)f(non-o)n(v)n(erlapping)e(syn)n (tactically)i(deterministic)523 3043 y(3-CTRSs)27 b(all)g(of)h(the)g (ab)r(o)n(v)n(e)e(implications)h(are)g(equiv)-5 b(alences.)523 3206 y Fi(Lemma)30 b(12.)40 b Fp(The)35 b(tr)l(ansforme)l(d)f(system)f Fn(U)9 b Fw(\()p Fm(R)p Fw(\))35 b Fp(of)f(a)g(syntactic)l(al)t(ly)h (deterministic)g(3-)523 3306 y(CTRS)30 b Fm(R)g Fp(is)g (non-overlapping)i(if)e Fm(R)h Fp(is)f(non-overlapping.)523 3488 y(Pr)l(o)l(of.)43 b Fw(Let)27 b Fn(l)956 3500 y Fk(1)1016 3488 y Fm(!)c Fn(r)1159 3500 y Fk(1)1223 3488 y Fw(and)j Fn(l)1408 3500 y Fk(2)1469 3488 y Fm(!)d Fn(r)1612 3500 y Fk(2)1676 3488 y Fw(b)r(e)k(renamed)f(v)n(ersions)e(of)j (rewrite)e(rules)h(from)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\))523 3587 y(suc)n(h)30 b(that)g(they)h(don't)f(ha)n(v)n(e)f(v)-5 b(ariables)29 b(in)i(common.)e(If)i Fn(l)2413 3599 y Fk(1)2480 3587 y Fw(and)f Fn(l)2669 3599 y Fk(2)2737 3587 y Fw(are)f(left-hand)h(sides)523 3687 y(of)25 b(rules)f(from)h Fm(R)p Fw(,)g(then)g(they)h(cannot)e(o)n(v)n(erlap)f(b)r(ecause)h Fm(R)h Fw(is)g(non-o)n(v)n(erlapping.)d(If)k(b)r(oth)523 3786 y Fn(l)548 3798 y Fk(1)609 3786 y Fw(and)e Fn(l)792 3798 y Fk(2)853 3786 y Fw(ha)n(v)n(e)f(a)h Fn(U)9 b Fw(-sym)n(b)r(ol)23 b(at)h(their)g(ro)r(ot,)f(then)i(they)f(cannot)g(o)n(v)n(erlap)d (either)j(b)r(ecause)523 3886 y(of)37 b(the)h(shap)r(e)f(of)h(the)f Fn(U)9 b Fw(-rules)37 b(\()p Fn(U)9 b Fw(-sym)n(b)r(ols)36 b(only)h(o)r(ccur)g(at)g(ro)r(ot)f(p)r(ositions)h(and)h(the)523 3986 y(ro)r(ot)25 b(sym)n(b)r(ols)h(of)g(t)n(w)n(o)f(di\013eren)n(t)h Fn(U)9 b Fw(-rules)25 b(cannot)h(b)r(e)g(the)g(same\).)g(Th)n(us)g(let) g Fn(l)3017 3998 y Fk(1)3081 3986 y Fw(b)r(e)g(a)g(left-)523 4085 y(hand)32 b(side)h(of)f(a)g(rule)g(from)g Fm(R)h Fw(and)f Fn(l)1746 4097 y Fk(2)1815 4085 y Fw(=)e Fn(U)1976 4045 y Fj(\032)1967 4108 y(i)2015 4085 y Fw(\()p Fn(t)2077 4097 y Fj(i)2104 4085 y Fn(;)14 b Fm(V)7 b Fn(ar)r Fw(\()p Fn(l)r Fw(\))p Fn(;)14 b(:)g(:)g(:)g Fw(\).)33 b(F)-7 b(or)32 b(an)g(indirect)h(pro)r(of,)523 4185 y(supp)r(ose)g(that)h Fn(l)1052 4197 y Fk(1)1123 4185 y Fw(and)f Fn(l)1315 4197 y Fk(2)1386 4185 y Fw(do)g(o)n(v)n(erlap.)f(Ob)n(viously)-7 b(,)32 b Fn(l)2262 4197 y Fk(1)2333 4185 y Fw(m)n(ust)h(o)n(v)n(erlap)f (with)i(a)f(subterm)523 4285 y(of)g Fn(t)653 4297 y Fj(i)681 4285 y Fw(.)g(This,)g(ho)n(w)n(ev)n(er,)e(is)i(imp)r(ossible)f(b)r (ecause)h Fn(t)2143 4297 y Fj(i)2204 4285 y Fw(is)g(either)f(a)h (constructor)e(term)i(or)f(a)523 4384 y(ground)27 b Fm(R)875 4396 y Fj(u)918 4384 y Fw(-normal)g(form.)523 4564 y(Lemma)e(12)f(can)h (b)r(e)h(re\014ned)f(to)g(demand)g(only)g(exactly)f(what)i(is)f (required)f(b)n(y)h(the)h(pro)r(of.)523 4664 y(F)-7 b(or)32 b(instance,)h(the)g(3-CTRS)g Fm(R)g Fw(need)g(not)g(b)r(e)h(syn)n (tactically)e(deterministic;)h(it)g(is)g(suf-)523 4764 y(\014cien)n(t)j(to)h(demand)f(that)h(no)e(left-hand)i(side)f Fn(l)2085 4776 y Fk(1)2159 4764 y Fw(of)g(a)g(rule)g(from)g Fm(R)g Fw(o)n(v)n(erlaps)e(a)i(term)p 523 4839 473 4 v 540 4893 a Fc(2)600 4924 y Fv(A)25 b Fb(c)l(onstructor)31 b(term)26 b Fv(is)g(a)g(term)f(without)h(de\014ned)f(sym)n(b)r(ols.)p eop %%Page: 12 12 12 11 bop 523 448 a Fn(t)553 460 y Fj(i)618 448 y Fw(of)37 b(another)f(rule)h Fn(l)1237 460 y Fk(2)1313 448 y Fm(!)i Fn(r)1472 460 y Fk(2)1549 448 y Fm(\()g Fn(s)1710 460 y Fk(1)1787 448 y Fm(!)g Fn(t)1939 460 y Fk(1)1976 448 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2199 460 y Fj(k)2279 448 y Fm(!)39 b Fn(t)2431 460 y Fj(k)2509 448 y Fw(from)e Fm(R)p Fw(.)h(Moreo)n(v)n(er,)c(non-)523 548 y(o)n(v)n(erlappingness)k (of)i Fm(R)h Fw(can)f(sligh)n(tly)g(b)r(e)h(w)n(eak)n(ened)f(in)g(the)h (preceding)f(lemma.)h(F)-7 b(or)523 648 y(instance,)37 b(w)n(e)h(ma)n(y)e(allo)n(w)h(infeasible)g(critical)g(pairs)g(as)g(in)h (the)g Fp(quicksort)g Fw(system)f(b)n(y)523 747 y(using)27 b(the)h(\\optimized")f Fn(U)9 b Fw(-transformation)25 b(explained)j(earlier.)648 849 y(No)n(w)g(w)n(e)i(are)e(in)i(a)f(p)r (osition)g(to)g(pro)n(v)n(e)f(a)h(nice)g(mo)r(dularit)n(y)g(result)g (for)g(CTRSs)g(with)523 949 y Fp(extr)l(a)f(variables)j(on)d(the)h (right-hand)h(sides)d Fw(of)f(the)h(rules.)f(The)g(reader)f(is)h (assumed)f(to)i(b)r(e)523 1048 y(familiar)k(with)i(the)f(concepts)g(of) g(the)g(\014eld)h(of)f(mo)r(dularit)n(y)-7 b(.)31 b(Details)h(and)g (references)f(to)523 1148 y(the)26 b(literature)g(can)f(b)r(e)i(found)f (e.g.)g(in)g([Ohl95)o(].)g(Let)g Fm(R)g Fw(b)r(e)h(a)e(CTRS)i(o)n(v)n (er)d(the)i(signature)523 1248 y Fm(F)8 b Fw(.)29 b(A)g(function)g(sym) n(b)r(ol)f Fn(f)34 b Fm(2)25 b(F)37 b Fw(is)28 b(called)h(a)f Fp(de\014ne)l(d)j(symb)l(ol)f Fw(if)f(there)f(is)h(a)f(rewrite)g(rule) 523 1347 y Fn(l)39 b Fm(!)f Fn(r)j Fm(\()d Fn(c)g Fm(2)g(R)f Fw(suc)n(h)f(that)h Fn(f)46 b Fw(=)37 b Fn(r)r(oot)p Fw(\()p Fn(l)r Fw(\).)h(F)-7 b(unction)37 b(sym)n(b)r(ols)f(from)g Fm(F)45 b Fw(whic)n(h)36 b(are)523 1447 y(not)30 b(de\014ned)g(sym)n(b) r(ols)g(are)f(called)g Fp(c)l(onstructors)p Fw(.)h(If)g Fm(R)2309 1459 y Fk(1)2377 1447 y Fw(and)g Fm(R)2611 1459 y Fk(2)2678 1447 y Fw(are)f(CTRSs)h(o)n(v)n(er)f(the)523 1546 y(signatures)34 b Fm(F)984 1558 y Fk(1)1056 1546 y Fw(and)h Fm(F)1285 1558 y Fk(2)1322 1546 y Fw(,)g(resp)r(ectiv)n(ely) -7 b(,)35 b(then)h(their)f Fp(c)l(ombine)l(d)j(system)d Fw(is)g(their)g(union)523 1646 y Fm(R)j Fw(=)e Fm(R)802 1658 y Fk(1)864 1646 y Fm([)24 b(R)1013 1658 y Fk(2)1087 1646 y Fw(o)n(v)n(er)34 b(the)j(signature)e Fm(F)45 b Fw(=)37 b Fm(F)2061 1658 y Fk(1)2122 1646 y Fm([)24 b(F)2261 1658 y Fk(2)2298 1646 y Fw(.)36 b(Its)g(set)g(of)g(de\014ned)h(sym)n(b) r(ols)e(is)523 1746 y Fm(D)25 b Fw(=)e Fm(D)764 1758 y Fk(1)816 1746 y Fm([)15 b(D)950 1758 y Fk(2)1014 1746 y Fw(and)25 b(its)h(set)g(of)g(constructors)e(is)i Fm(C)h Fw(=)c Fm(F)g(n)14 b(D)r Fw(,)26 b(where)g Fm(D)2776 1758 y Fj(i)2829 1746 y Fw(\()p Fm(C)2905 1758 y Fj(i)2933 1746 y Fw(\))g(denotes)g(the)523 1845 y(de\014ned)i(sym)n(b)r(ols)f (\(constructors\))f(in)i Fm(R)1829 1857 y Fj(i)1857 1845 y Fw(.)517 2039 y(\(1\))41 b Fm(R)734 2051 y Fk(1)799 2039 y Fw(and)28 b Fm(R)1031 2051 y Fk(2)1096 2039 y Fw(are)f Fp(disjoint)i Fw(if)f Fm(F)1668 2051 y Fk(1)1723 2039 y Fm(\\)19 b(F)1857 2051 y Fk(2)1917 2039 y Fw(=)k Fm(;)p Fw(.)517 2141 y(\(2\))41 b Fm(R)734 2153 y Fk(1)799 2141 y Fw(and)28 b Fm(R)1031 2153 y Fk(2)1096 2141 y Fw(are)f Fp(c)l(onstructor-sharing)h Fw(if)g Fm(F)2100 2153 y Fk(1)2155 2141 y Fm(\\)19 b(F)2289 2153 y Fk(2)2349 2141 y Fw(=)k Fm(C)2481 2153 y Fk(1)2536 2141 y Fm(\\)c(C)2654 2153 y Fk(2)2714 2141 y Fw(\()p Fm(\022)k(C)5 b Fw(\).)517 2243 y(\(3\))41 b Fm(R)734 2255 y Fk(1)810 2243 y Fw(and)d Fm(R)1052 2255 y Fk(2)1127 2243 y Fw(are)f Fp(c)l(omp)l(osable)j Fw(if)f Fm(C)1844 2255 y Fk(1)1906 2243 y Fm(\\)26 b(D)2051 2255 y Fk(2)2128 2243 y Fw(=)40 b Fm(D)2297 2255 y Fk(1)2360 2243 y Fm(\\)26 b(C)2485 2255 y Fk(2)2562 2243 y Fw(=)40 b Fm(;)d Fw(and)h(b)r(oth)g(systems)664 2342 y(con)n(tain)26 b(all)h(rewrite)f(rules)g(that)h(de\014ne)g(a)g(de\014ned)g(sym)n(b)r (ol)f(whenev)n(er)g(that)h(sym)n(b)r(ol)664 2442 y(is)h(shared,)e(that) i(is)g(to)f(sa)n(y)-7 b(,)27 b Fm(f)p Fn(l)d Fm(!)f Fn(r)j Fm(\()d Fn(c)g Fm(2)g(R)28 b(j)g Fn(r)r(oot)p Fw(\()p Fn(l)r Fw(\))c Fm(2)g(D)2618 2454 y Fk(1)2673 2442 y Fm(\\)19 b(D)2811 2454 y Fk(2)2848 2442 y Fm(g)k(\022)g(R)3071 2454 y Fk(1)3127 2442 y Fm(\\)18 b(R)3270 2454 y Fk(2)3308 2442 y Fw(.)523 2634 y(A)26 b(prop)r(ert)n(y)f Fm(P)32 b Fw(is)26 b Fp(mo)l(dular)g Fw(for)g(a)f(certain)g(class)g(of)g(CTRSs) h(if,)h(for)e(all)g(CTRSs)h(\()p Fm(F)3191 2646 y Fk(1)3228 2634 y Fn(;)14 b Fm(R)3335 2646 y Fk(1)3373 2634 y Fw(\))523 2733 y(and)29 b(\()p Fm(F)778 2745 y Fk(2)815 2733 y Fn(;)14 b Fm(R)922 2745 y Fk(2)960 2733 y Fw(\))29 b(b)r(elonging)g(to) g(that)g(class)f(and)h(ha)n(ving)f(prop)r(ert)n(y)g Fm(P)7 b Fw(,)29 b(their)g(union)g(\()p Fm(F)3293 2745 y Fk(1)3350 2733 y Fm([)523 2833 y(F)583 2845 y Fk(2)620 2833 y Fn(;)14 b Fm(R)727 2845 y Fk(1)783 2833 y Fm([)19 b(R)927 2845 y Fk(2)964 2833 y Fw(\))28 b(also)f(b)r(elongs)g(to)g(that)h(class)f (and)g(has)g(the)h(prop)r(ert)n(y)f Fm(P)7 b Fw(.)523 3000 y Fi(Prop)s(osition)30 b(13.)41 b Fp(L)l(et)36 b Fm(R)1423 3012 y Fk(1)1496 3000 y Fp(and)h Fm(R)1734 3012 y Fk(2)1808 3000 y Fp(b)l(e)f(quasi-r)l(e)l(ductive)h(\(quasi-de)l (cr)l(e)l(asing,)h(r)l(esp)l(e)l(c-)523 3099 y(tively\))24 b(deterministic)g(3-CTRSs.)f(Their)i(c)l(ombine)l(d)f(system)e Fm(R)2558 3111 y Fk(1)2600 3099 y Fm([)t(R)2729 3111 y Fk(2)2789 3099 y Fp(is)h(quasi-r)l(e)l(ductive)523 3199 y(if)555 3374 y(1.)42 b Fn(U)9 b Fw(\()p Fm(R)832 3386 y Fk(1)870 3374 y Fw(\))22 b Fp(and)h Fn(U)9 b Fw(\()p Fm(R)1246 3386 y Fk(2)1284 3374 y Fw(\))22 b Fp(b)l(elong)h(to)f(a)h (class)g(of)g(TRSs)f(for)h(which)g(innermost)g(termination)664 3473 y(is)30 b(mo)l(dular,)h(and)555 3575 y(2.)42 b Fn(U)9 b Fw(\()p Fm(R)832 3587 y Fk(1)888 3575 y Fm([)19 b(R)1032 3587 y Fk(2)1070 3575 y Fw(\))30 b Fp(is)g(non-overlapping.)523 3761 y(Pr)l(o)l(of.)43 b Fw(Since)19 b Fm(R)1061 3773 y Fk(1)1117 3761 y Fw(and)g Fm(R)1340 3773 y Fk(2)1396 3761 y Fw(are)f(quasi-reductiv)n(e,)f(they)h(are)g(quasi-decreasing)e (b)n(y)i(Lemma)523 3860 y(4.)26 b(Th)n(us)g(the)h(transformed)e(TRSs)h Fn(U)9 b Fw(\()p Fm(R)1822 3872 y Fk(1)1860 3860 y Fw(\))26 b(and)g Fn(U)9 b Fw(\()p Fm(R)2246 3872 y Fk(2)2284 3860 y Fw(\))26 b(are)g(innermost)f(terminating)h(b)n(y)523 3960 y(Theorem)34 b(10.)g(Their)g(com)n(bination)g Fn(U)9 b Fw(\()p Fm(R)1903 3972 y Fk(1)1940 3960 y Fw(\))24 b Fm([)f Fn(U)9 b Fw(\()p Fm(R)2242 3972 y Fk(2)2280 3960 y Fw(\))35 b(=)g Fn(U)9 b Fw(\()p Fm(R)2615 3972 y Fk(1)2675 3960 y Fm([)24 b(R)2824 3972 y Fk(2)2861 3960 y Fw(\))35 b(is)g(also)f(inner-)523 4060 y(most)i(terminating)f(b) r(ecause)g(innermost)g(termination)h(is)f(mo)r(dular.)h(Since)f (innermost)523 4159 y(termination)27 b(and)g(termination)g(coincide)f (for)h(non-o)n(v)n(erlapping)d(systems,)j Fn(U)9 b Fw(\()p Fm(R)3137 4171 y Fk(1)3192 4159 y Fm([)18 b(R)3335 4171 y Fk(2)3373 4159 y Fw(\))523 4259 y(is)28 b(terminating.)f(No)n(w)g (the)h(assertion)e(follo)n(ws)h(from)g(Prop)r(osition)f(6.)523 4442 y Fi(Theorem)k(14.)41 b Fp(Quasi-r)l(e)l(ductivity)29 b(\(quasi-de)l(cr)l(e)l(asingness,)i(r)l(esp)l(e)l(ctively\))g(is)e(mo) l(dular)523 4542 y(for)i(non-overlapping)g(syntactic)l(al)t(ly)h (deterministic)f(c)l(omp)l(osable)g(3-CTRSs.)523 4725 y(Pr)l(o)l(of.)43 b Fw(Let)34 b Fm(R)1008 4737 y Fk(1)1080 4725 y Fw(and)g Fm(R)1318 4737 y Fk(2)1390 4725 y Fw(b)r(e)g (quasi-reductiv)n(e)e(non-o)n(v)n(erlapping)g(syn)n(tactically)h (deter-)523 4825 y(ministic)38 b(comp)r(osable)f(3-CTRSs.)g(It)h(is)g (relativ)n(ely)f(easy)f(to)i(v)n(erify)f(that)h Fn(U)9 b Fw(\()p Fm(R)3163 4837 y Fk(1)3201 4825 y Fw(\))38 b(and)523 4924 y Fn(U)9 b Fw(\()p Fm(R)691 4936 y Fk(2)729 4924 y Fw(\))18 b(are)g(comp)r(osable)f(since)i Fm(R)1610 4936 y Fk(1)1666 4924 y Fw(and)f Fm(R)1888 4936 y Fk(2)1944 4924 y Fw(are)g(comp)r(osable)f(\(note)i(that)f(the)h Fn(U)9 b Fw(-sym)n(b)r(ols)p eop %%Page: 13 13 13 12 bop 523 448 a Fn(U)589 408 y Fj(\032)580 471 y(i)656 448 y Fw(used)29 b(in)g(the)h(transformation)d Fn(U)9 b Fw(\()p Fn(\032)p Fw(\))29 b(are)f(mark)n(ed)g(with)h(the)g(rule)g Fn(\032)p Fw(\).)g(Note)g(that)g(in-)523 548 y(nermost)34 b(termination)f(is)h(mo)r(dular)g(for)g(comp)r(osable)f(TRSs;)h(see)g ([Ohl95)o(].)g(According)523 648 y(to)j(Lemma)f(12,)g(the)h(TRSs)g Fn(U)9 b Fw(\()p Fm(R)1639 660 y Fk(1)1676 648 y Fw(\))38 b(and)e Fn(U)9 b Fw(\()p Fm(R)2084 660 y Fk(2)2122 648 y Fw(\))37 b(are)f(non-o)n(v)n(erlapping.)d(The)k(system)523 747 y Fn(U)9 b Fw(\()p Fm(R)691 759 y Fk(1)753 747 y Fm([)26 b(R)904 759 y Fk(2)941 747 y Fw(\))39 b(=)g Fn(U)9 b Fw(\()p Fm(R)1284 759 y Fk(1)1322 747 y Fw(\))25 b Fm([)g Fn(U)9 b Fw(\()p Fm(R)1627 759 y Fk(2)1665 747 y Fw(\))37 b(is)g(non-o)n(v)n(erlapping)d(as)j(w)n(ell)g(b)r(ecause)f (the)i(union)523 847 y(of)c(t)n(w)n(o)e(non-o)n(v)n(erlapping)f(comp)r (osable)i(TRSs)g(is)h(again)e(non-o)n(v)n(erlapping.)f(Hence)j(the)523 946 y(com)n(bined)h(system)f Fm(R)1254 958 y Fk(1)1315 946 y Fm([)24 b(R)1464 958 y Fk(2)1537 946 y Fw(is)35 b(quasi-reductiv)n(e)e(b)n(y)i(Prop)r(osition)e(13.)h(F)-7 b(urthermore,)523 1046 y Fm(R)593 1058 y Fk(1)649 1046 y Fm([)19 b(R)793 1058 y Fk(2)858 1046 y Fw(is)28 b(ob)n(viously)e (non-o)n(v)n(erlapping)f(and)i(syn)n(tactically)f(deterministic.)523 1184 y(As)h(already)e(men)n(tioned,)i(Lemma)f(12)g(can)g(b)r(e)h (generalized)e(as)h(follo)n(ws:)g(If)h(functions)g(are)523 1284 y(de\014ned)g(b)n(y)f(distinct)i(cases,)d(then)i(the)g (\\optimized")f(transformation)f(describ)r(ed)h(earlier)523 1384 y(still)37 b(yields)f(a)h(non-o)n(v)n(erlapping)c(transformed)j (system)g Fn(U)9 b Fw(\()p Fm(R)p Fw(\).)37 b(Clearly)-7 b(,)36 b(Theorem)g(14)523 1483 y(remains)29 b(v)-5 b(alid)30 b(for)f(these)h(systems.)f(F)-7 b(or)29 b(example,)h(this)g (generalized)e(v)n(ersion)g(of)i(Theo-)523 1583 y(rem)24 b(14)g(can)g(b)r(e)h(applied)g(to)g(the)g(function)g Fn(q)s(uor)r(em)g Fw(whic)n(h)f(computes)h(the)g(quotien)n(t)f(and)523 1683 y(the)k(remainder)f(of)g Fn(m)h Fw(and)f Fn(n)1221 1808 y Fw(0)18 b Fm(\000)g Fn(s)p Fw(\()p Fn(y)s Fw(\))24 b Fm(!)f Fw(0)1321 1907 y Fn(x)c Fm(\000)f Fw(0)23 b Fm(!)g Fn(x)1112 2007 y(s)p Fw(\()p Fn(x)p Fw(\))c Fm(\000)f Fn(s)p Fw(\()p Fn(y)s Fw(\))24 b Fm(!)f Fn(x)c Fm(\000)f Fn(y)1312 2106 y(x)23 b(<)g Fw(0)g Fm(!)g Fn(f)9 b(al)r(se)1208 2206 y Fw(0)23 b Fn(<)g(s)p Fw(\()p Fn(x)p Fw(\))h Fm(!)f Fn(tr)r(ue)1103 2306 y(s)p Fw(\()p Fn(x)p Fw(\))h Fn(<)e(s)p Fw(\()p Fn(y)s Fw(\))i Fm(!)f Fn(x)g(<)g(y)942 2405 y(q)s(uor)r(em)p Fw(\(0)p Fn(;)14 b(s)p Fw(\()p Fn(y)s Fw(\)\))24 b Fm(!)f(h)p Fw(0)p Fn(;)14 b Fw(0)p Fm(i)833 2505 y Fn(q)s(uor)r(em)p Fw(\()p Fn(s)p Fw(\()p Fn(x)p Fw(\))p Fn(;)g(s)p Fw(\()p Fn(y)s Fw(\)\))25 b Fm(!)e(h)p Fw(0)p Fn(;)14 b(s)p Fw(\()p Fn(x)p Fw(\))p Fm(i)24 b(\()f Fn(x)h(<)f(y)i Fm(!)e Fn(tr)r(ue)833 2605 y(q)s(uor)r(em)p Fw(\()p Fn(s)p Fw(\()p Fn(x)p Fw(\))p Fn(;)14 b(s)p Fw(\()p Fn(y)s Fw(\)\))25 b Fm(!)e(h)p Fn(s)p Fw(\()p Fn(q)s Fw(\))p Fn(;)14 b(r)r Fm(i)34 b(\()23 b Fn(x)h(<)f(y)i Fm(!)e Fn(f)9 b(al)r(se;)2064 2704 y(q)s(uor)r(em)p Fw(\()p Fn(x)20 b Fm(\000)e Fn(y)s(;)c(s)p Fw(\()p Fn(y)s Fw(\)\))23 b Fm(!)g(h)p Fn(q)s(;)14 b(r)r Fm(i)523 2847 y Fw(and)24 b(the)g(function)g Fn(f)9 b(il)r(ter)25 b Fw(whic)n(h)e(\014lters)h(all)f(elemen)n(ts)h(out)f(of)h(a)f(list)h(of) g(natural)f(n)n(um)n(b)r(ers)523 2947 y(that)28 b(ha)n(v)n(e)e (remainder)h Fn(r)j Fw(when)e(divided)g(b)n(y)f Fn(n)950 3088 y Fw(0)18 b Fm(\000)g Fn(s)p Fw(\()p Fn(y)s Fw(\))23 b Fm(!)g Fw(0)1050 3187 y Fn(x)18 b Fm(\000)g Fw(0)23 b Fm(!)g Fn(x)840 3287 y(s)p Fw(\()p Fn(x)p Fw(\))d Fm(\000)e Fn(s)p Fw(\()p Fn(y)s Fw(\))23 b Fm(!)g Fn(x)c Fm(\000)f Fn(y)1044 3387 y Fw(0)23 b Fm(\024)f Fn(y)k Fm(!)d Fn(tr)r(ue)937 3486 y(s)p Fw(\()p Fn(x)p Fw(\))h Fm(\024)e Fw(0)h Fm(!)g Fn(f)9 b(al)r(se)831 3586 y(s)p Fw(\()p Fn(x)p Fw(\))24 b Fm(\024)f Fn(s)p Fw(\()p Fn(y)s Fw(\))g Fm(!)g Fn(x)h Fm(\024)f Fn(y)977 3686 y(eq)s Fw(\(0)p Fn(;)14 b Fw(0\))22 b Fm(!)h Fn(tr)r(ue)867 3785 y(eq)s Fw(\()p Fn(s)p Fw(\()p Fn(x)p Fw(\))p Fn(;)14 b Fw(0\))24 b Fm(!)f Fn(f)9 b(al)r(se)871 3885 y(eq)s Fw(\(0)p Fn(;)14 b(s)p Fw(\()p Fn(y)s Fw(\)\))23 b Fm(!)g Fn(f)9 b(al)r(se)762 3984 y(eq)s Fw(\()p Fn(s)p Fw(\()p Fn(x)p Fw(\))p Fn(;)14 b(s)p Fw(\()p Fn(y)s Fw(\)\))24 b Fm(!)f Fn(eq)s Fw(\()p Fn(x;)14 b(y)s Fw(\))897 4084 y Fn(mod)p Fw(\(0)p Fn(;)g(y)s Fw(\))23 b Fm(!)g Fw(0)790 4184 y Fn(mod)p Fw(\()p Fn(s)p Fw(\()p Fn(x)p Fw(\))p Fn(;)14 b Fw(0\))24 b Fm(!)f Fw(0)684 4283 y Fn(mod)p Fw(\()p Fn(s)p Fw(\()p Fn(x)p Fw(\))p Fn(;)14 b(s)p Fw(\()p Fn(y)s Fw(\)\))25 b Fm(!)e Fn(mod)p Fw(\()p Fn(x)d Fm(\000)e Fn(y)s(;)c(s)p Fw(\()p Fn(y)s Fw(\)\))65 b Fm(\()23 b Fn(y)j Fm(\024)d Fn(x)g Fm(!)g Fn(tr)r(ue)684 4383 y(mod)p Fw(\()p Fn(s)p Fw(\()p Fn(x)p Fw(\))p Fn(;)14 b(s)p Fw(\()p Fn(y)s Fw(\)\))25 b Fm(!)e Fn(s)p Fw(\()p Fn(x)p Fw(\))513 b Fm(\()23 b Fn(y)j Fm(\024)d Fn(x)g Fm(!)g Fn(f)9 b(al)r(se)699 4483 y(f)g(il)r(ter)r Fw(\()p Fn(n;)14 b(r)n(;)g(nil)r Fw(\))21 b Fm(!)i Fn(nil)601 4582 y(f)9 b(il)r(ter)r Fw(\()p Fn(n;)14 b(r)n(;)g(x)22 b Fw(:)h Fn(xs)p Fw(\))h Fm(!)f Fn(x)h Fw(:)f Fn(f)9 b(il)r(ter)r Fw(\()p Fn(n;)14 b(r)n(;)g(xs)p Fw(\))23 b Fm(\()g Fn(mod)p Fw(\()p Fn(x;)14 b(n)p Fw(\))24 b Fm(!)f Fn(r)2661 4552 y Fl(0)2685 4582 y Fn(;)14 b(eq)s Fw(\()p Fn(r)n(;)g(r)2944 4552 y Fl(0)2968 4582 y Fw(\))24 b Fm(!)f Fn(tr)r(ue)601 4682 y(f)9 b(il)r(ter)r Fw(\()p Fn(n;)14 b(r)n(;)g(x)22 b Fw(:)h Fn(xs)p Fw(\))h Fm(!)f Fn(f)9 b(il)r(ter)r Fw(\()p Fn(n;)14 b(r)n(;)g(xs)p Fw(\))140 b Fm(\()23 b Fn(mod)p Fw(\()p Fn(x;)14 b(n)p Fw(\))24 b Fm(!)f Fn(r)2661 4652 y Fl(0)2685 4682 y Fn(;)14 b(eq)s Fw(\()p Fn(r)n(;)g(r)2944 4652 y Fl(0)2968 4682 y Fw(\))24 b Fm(!)f Fn(f)9 b(al)r(se)523 4825 y Fw(Both)24 b(systems)g(are)f(syn)n(tactically)g(deterministic)i(comp)r(osable)e (3-CTRSs)g(whic)n(h)i(can)f(b)r(e)523 4924 y(sho)n(wn)e(quasi-reductiv) n(e)f(b)n(y)h(Prop)r(osition)f(6)h(in)h(conjunction)g(with)g(the)g(dep) r(endency)g(pair)p eop %%Page: 14 14 14 13 bop 523 448 a Fw(tec)n(hnique.)32 b(Hence)f(w)n(e)h(can)f (conclude)g(from)g(the)h(generalized)f(v)n(ersion)f(of)h(Theorem)g(14) 523 548 y(that)d(their)f(com)n(bined)h(system)f(is)h(also)e (quasi-reductiv)n(e.)523 840 y Fq(6)112 b(Application)35 b(2:)j(W)-9 b(ell-Mo)s(ded)37 b(Logic)f(Programs)523 1066 y Fw(Next)20 b(w)n(e)g(will)f(sho)n(w)g(ho)n(w)g(our)g(results)h (can)f(b)r(e)h(used)g(to)f(sho)n(w)g(termination)h(of)f(w)n(ell-mo)r (ded)523 1165 y(logic)30 b(programs.)e(W)-7 b(e)32 b(assume)e(that)h (the)g(reader)e(is)i(familiar)f(with)h(logic)f(programming)523 1265 y(and)d(SLD)i(deriv)-5 b(ations)26 b(and)i(will)g(only)f(review)g (the)h(follo)n(wing)e(notions.)648 1370 y(If)k Fn(P)42 b Fw(is)30 b(a)g(predicate)f(sym)n(b)r(ol)h(and)f Fn(t)1828 1382 y Fk(1)1866 1370 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)2080 1382 y Fj(n)2155 1370 y Fw(are)30 b(terms,)f(then)i Fn(P)12 b Fw(\()p Fn(t)2872 1382 y Fk(1)2909 1370 y Fn(;)i(:)g(:)g(:)g(;)g(t) 3124 1382 y Fj(n)3169 1370 y Fw(\))30 b(is)g(an)523 1469 y Fp(atom)p Fw(.)k(A)g Fp(Horn-clause)40 b Fw(is)34 b(a)f(form)n(ula)g (of)g(the)h(form)g Fn(A)f Fm( )g Fn(B)2532 1481 y Fk(1)2569 1469 y Fn(;)14 b(:)g(:)g(:)g(;)g(B)2817 1481 y Fj(m)2914 1469 y Fw(where)33 b Fn(m)g Fm(\025)f Fw(0)523 1569 y(and)g Fn(A;)14 b(B)851 1581 y Fj(i)910 1569 y Fw(are)31 b(atoms.)g(A)h Fp(lo)l(gic)i(pr)l(o)l(gr)l(am)39 b Fm(P)g Fw(is)31 b(a)g(set)h(of)g (Horn-clauses.)e(A)i Fp(query)39 b Fw(is)31 b(a)523 1669 y(form)n(ula)c(of)g(the)h(form)f Fm( )d Fn(B)1431 1681 y Fk(1)1468 1669 y Fn(;)14 b(:)g(:)g(:)f(;)h(B)1715 1681 y Fj(m)1806 1669 y Fw(where)27 b Fn(m)c Fm(\025)g Fw(1)k(and)g Fn(B)2523 1681 y Fj(i)2579 1669 y Fw(are)f(atoms.)648 1773 y(A)18 b(logic)g(program)e Fm(P)25 b Fw(is)18 b Fp(mo)l(de)l(d)28 b Fw(if)19 b(for)f(eac)n(h)g(o)r(ccurrence)f(of)h(an) g(atom)g Fn(A)23 b Fw(=)g Fn(P)12 b Fw(\()p Fn(t)3106 1785 y Fk(1)3143 1773 y Fn(;)i(:)g(:)g(:)g(;)g(t)3358 1785 y Fj(n)3403 1773 y Fw(\))523 1873 y(there)33 b(is)f(a)h(function)g Fn(m)1307 1885 y Fj(A)1393 1873 y Fw(:)f Fm(f)p Fw(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(n)p Fm(g)31 b(!)h(f)p Fn(in;)14 b(out)p Fm(g)p Fw(.)31 b(If)i Fn(m)2487 1885 y Fj(A)2541 1873 y Fw(\()p Fn(i)p Fw(\))f(=)g Fn(in)g Fw(\()p Fn(m)2979 1885 y Fj(A)3033 1873 y Fw(\()p Fn(i)p Fw(\))g(=)g Fn(out)p Fw(\))523 1973 y(then)23 b(p)r(osition)f Fn(i)h Fw(is)f(called)g(an)h Fp(input)h(p)l(osition)j(\(output)d(p)l(osition\))j Fw(of)22 b Fn(A)p Fw(.)h(A)g(v)-5 b(ariable)21 b Fn(x)j Fw(o)r(c-)523 2072 y(curs)e(in)g(an)g(input)i(\(output\))f(p)r(osition)f(in)h Fn(A)f Fw(if)h Fn(x)h Fm(2)f(V)7 b Fn(ar)r Fw(\()p Fn(t)2353 2084 y Fj(i)2381 2072 y Fw(\))23 b(for)f(some)g Fn(i)g Fw(with)h Fn(m)3069 2084 y Fj(A)3123 2072 y Fw(\()p Fn(i)p Fw(\))g(=)f Fn(in)523 2172 y Fw(\()p Fn(m)628 2184 y Fj(A)682 2172 y Fw(\()p Fn(i)p Fw(\))i(=)e Fn(out)p Fw(\).)648 2277 y(Here)33 b(only)h(left-to-righ)n(t)f(SLD-deriv)-5 b(ations)33 b(will)h(b)r(e)g(considered.)f(In)h(these)g(deriv)-5 b(a-)523 2377 y(tions)37 b(it)h(is)f(alw)n(a)n(ys)e(the)j(leftmost)g (literal)f(of)g(a)g(query)f(that)i(is)f(selected)g(for)g(the)h(next)523 2476 y(resolution)28 b(step.)h(Moreo)n(v)n(er,)d(w)n(e)i(will)h (restrict)g(our)f(atten)n(tion)g(to)h(LR-w)n(ell-mo)r(ded)f(pro-)523 2576 y(grams.)523 2755 y Fi(De\014nition)j(15.)72 b Fp(1.)43 b(L)l(et)26 b Fn(C)k Fw(=)22 b Fn(A)h Fm( )h Fn(B)1845 2767 y Fk(1)1882 2755 y Fn(;)14 b(:)g(:)g(:)f(;)h(B)2129 2767 y Fj(m)2219 2755 y Fp(b)l(e)28 b(a)f(clause)h(and)f Fn(x)d Fm(2)f(V)7 b Fn(ar)r Fw(\()p Fn(C)f Fw(\))p Fp(.)29 b(The)664 2854 y(he)l(ad)35 b Fn(A)f Fp(of)h Fn(C)40 b Fp(is)34 b(c)l(al)t(le)l(d)h(a)41 b Fw(pro)r(ducer)31 b(\(consumer\))i Fp(of)i Fn(x)p Fp(,)f(if)h Fn(x)f Fp(o)l(c)l(curs)g (in)f(an)h(input)664 2954 y(\(output\))26 b(p)l(osition)i(of)f Fn(A)p Fp(.)h(The)g(b)l(o)l(dy)f(atom)g Fn(B)2124 2966 y Fj(j)2186 2954 y Fp(is)g(c)l(al)t(le)l(d)h(a)g(pr)l(o)l(duc)l(er)f (\(c)l(onsumer\))f(of)664 3053 y Fn(x)p Fp(,)31 b(if)f Fn(x)h Fp(o)l(c)l(curs)e(in)h(an)g(output)f(\(input\))g(p)l(osition)i (of)f Fn(B)2414 3065 y Fj(j)2449 3053 y Fp(.)555 3158 y(2.)42 b(The)36 b(clause)f Fn(B)1155 3170 y Fk(0)1224 3158 y Fm( )d Fn(B)1402 3170 y Fk(1)1439 3158 y Fn(;)14 b(:)g(:)g(:)g(;)g(B)1687 3170 y Fj(m)1784 3158 y Fp(is)35 b(c)l(al)t(le)l(d)45 b Fw(LR-w)n(ell-mo)r(ded)p Fp(,)34 b(if)h(every)h(variable)g Fn(x)664 3258 y Fp(in)c(the)h(clause)f(has)h (a)g(pr)l(o)l(duc)l(er)f Fn(B)1785 3270 y Fj(i)1845 3258 y Fw(\(0)27 b Fm(\024)g Fn(i)g Fm(\024)g Fn(m)p Fw(\))32 b Fp(and)h Fn(i)26 b(<)h(j)37 b Fp(for)c(every)g(c)l(onsumer)664 3358 y Fn(B)727 3370 y Fj(j)796 3358 y Fw(\(1)f Fm(\024)f Fn(j)36 b Fm(\024)31 b Fn(m)p Fw(\))j Fp(of)i Fn(x)e Fp(in)h(the)f(b)l(o)l(dy)h(of)h(the)e(clause.)h(A)f(lo)l(gic)i(pr)l(o)l (gr)l(am)f Fm(P)41 b Fp(is)g Fw(LR-)664 3457 y(w)n(ell-mo)r(ded)29 b Fp(if)i(every)g(clause)f(in)g Fm(P)36 b Fp(is)30 b(LR-wel)t(l-mo)l (de)l(d.)555 3562 y(3.)42 b(A)25 b(query)h Fm( )d Fn(B)1142 3574 y Fk(1)1179 3562 y Fn(;)14 b(:)g(:)g(:)g(;)g(B)1427 3574 y Fj(m)1515 3562 y Fp(is)26 b(LR-wel)t(l-mo)l(de)l(d)g(if)h(every) f(variable)h Fn(x)f Fp(in)g(the)f(query)h(has)664 3662 y(a)k(pr)l(o)l(duc)l(er)h Fn(B)1135 3674 y Fj(i)1192 3662 y Fp(such)f(that)f(for)i(every)g(c)l(onsumer)e Fn(B)2338 3674 y Fj(j)2403 3662 y Fp(of)h Fn(x)g Fp(we)g(have)h Fn(i)23 b(<)g(j)5 b Fp(.)523 3835 y Fw(By)32 b(this)h(de\014nition,)g (if)g Fm( )e Fn(B)1480 3847 y Fk(1)1517 3835 y Fn(;)14 b(:)g(:)g(:)g(;)g(B)1765 3847 y Fj(m)1860 3835 y Fw(is)33 b(LR-w)n(ell-mo)r(ded)e(and)i Fn(B)2762 3847 y Fk(1)2830 3835 y Fw(=)e Fn(P)12 b Fw(\()p Fn(t)3053 3847 y Fk(1)3090 3835 y Fn(;)i(:)g(:)g(:)g(;)g(t)3305 3847 y Fj(n)3350 3835 y Fw(\),)523 3935 y(then,)28 b(for)f(all)h(input)g(p)r(ositions)f Fn(i)g Fw(of)h Fn(B)1762 3947 y Fk(1)1799 3935 y Fw(,)g Fn(t)1880 3947 y Fj(i)1935 3935 y Fw(is)g(a)f(ground)f(term.)648 4040 y(W)-7 b(e)29 b(transform)e(ev)n(ery)h(LR-w)n(ell-mo)r(ded)f (logic)h(program)f Fm(P)35 b Fw(in)n(to)29 b(a)f(deterministic)h(3-)523 4139 y(CTRS)k Fn(R)846 4151 y Fl(P)934 4139 y Fw(as)f(in)h([GW93)o (,ALS94].)f(F)-7 b(or)32 b(ev)n(ery)f(atom)h Fn(A)g Fw(=)e Fn(P)12 b Fw(\()p Fn(t)2655 4151 y Fk(1)2693 4139 y Fn(;)i(:)g(:)g(:)f (;)h(t)2907 4151 y Fj(n)2952 4139 y Fw(\))33 b(with)g(input)523 4239 y(p)r(ositions)k Fn(i)911 4251 y Fk(1)948 4239 y Fn(;)14 b(:)g(:)g(:)g(;)g(i)1162 4251 y Fj(k)1240 4239 y Fw(and)38 b(output)g(p)r(ositions)g Fn(i)2083 4251 y Fj(k)q Fk(+1)2207 4239 y Fn(;)14 b(:)g(:)g(:)g(;)g(i)2421 4251 y Fj(n)2503 4239 y Fw(there)38 b(are)f(t)n(w)n(o)g(new)h(func-)523 4339 y(tion)32 b(sym)n(b)r(ols)g Fn(P)1073 4351 y Fj(in)1174 4339 y Fw(and)g Fn(P)1393 4351 y Fj(out)1527 4339 y Fw(and)g(w)n(e)g (de\014ne)g Fn(\032)2107 4351 y Fj(in)2176 4339 y Fw(\()p Fn(A)p Fw(\))f(=)f Fn(P)2481 4351 y Fj(in)2550 4339 y Fw(\()p Fn(t)2612 4351 y Fj(i)2635 4359 y Fe(1)2673 4339 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)2887 4351 y Fj(i)2910 4360 y Fa(k)2951 4339 y Fw(\),)33 b Fn(\032)3082 4351 y Fj(out)3183 4339 y Fw(\()p Fn(A)p Fw(\))e(=)523 4438 y Fn(P)576 4450 y Fj(out)678 4438 y Fw(\()p Fn(t)740 4450 y Fj(i)763 4459 y Fa(k)q Fe(+1)875 4438 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)1089 4450 y Fj(i)1112 4458 y Fa(n)1158 4438 y Fw(\),)30 b(and)g Fn(\032)p Fw(\()p Fn(A)p Fw(\))d(=)g Fn(\032)1738 4450 y Fj(in)1806 4438 y Fw(\()p Fn(A)p Fw(\))h Fm(!)f Fn(\032)2113 4450 y Fj(out)2214 4438 y Fw(\()p Fn(A)p Fw(\).)k(The)f(transformation) e Fn(\032)p Fw(\()p Fn(C)6 b Fw(\))31 b(of)523 4538 y(a)c(clause)g Fn(C)i Fw(=)23 b Fn(A)g Fm( )h Fn(B)1268 4550 y Fk(1)1305 4538 y Fn(;)14 b(:)g(:)g(:)f(;)h(B)1552 4550 y Fj(m)1643 4538 y Fw(is)27 b(de\014ned)h(to)g(b)r(e)g(the)g(rule)1502 4731 y Fn(\032)p Fw(\()p Fn(A)p Fw(\))23 b Fm(\()h Fn(\032)p Fw(\()p Fn(B)1939 4743 y Fk(1)1976 4731 y Fw(\))p Fn(;)14 b(:)g(:)g(:)g(;)g(\032)p Fw(\()p Fn(B)2331 4743 y Fj(m)2394 4731 y Fw(\))523 4924 y(and)27 b(with)i(ev)n(ery)d(logic)h(program)e Fm(P)34 b Fw(w)n(e)28 b(asso)r(ciate)e Fn(R)2246 4936 y Fl(P)2325 4924 y Fw(=)d Fm(f)p Fn(\032)p Fw(\()p Fn(C)6 b Fw(\))28 b Fm(j)f Fn(C)34 b Fw(in)28 b Fm(P)7 b(g)p Fw(.)p eop %%Page: 15 15 15 14 bop 648 448 a Fw(Note)39 b(that)h Fn(R)1115 460 y Fl(P)1210 448 y Fw(is)g(a)f(syn)n(tactically)f(deterministic)i (3-CTRS)f(o)n(v)n(er)e(the)j(signature)523 548 y Fn(F)576 560 y Fl(P)680 548 y Fw(=)48 b Fn(F)858 518 y Fj(P)846 571 y Fl(P)941 548 y Fm([)29 b Fn(F)1090 518 y Fj(T)1078 571 y Fl(P)1142 548 y Fw(,)43 b(where)e Fn(F)1527 518 y Fj(P)1515 571 y Fl(P)1631 548 y Fw(=)47 b Fm(f)p Fn(P)1838 560 y Fj(in)1906 548 y Fn(;)14 b(P)1996 560 y Fj(out)2112 548 y Fm(j)g Fn(P)39 b Fw(is)28 b(a)f(predicate)g(in)h Fm(P)7 b(g)41 b Fw(and)h Fn(F)3240 518 y Fj(T)3228 571 y Fl(P)3340 548 y Fw(=)523 648 y Fm(f)p Fn(f)22 b Fm(j)14 b Fn(f)36 b Fw(o)r(ccurs)27 b(in)h(a)f(term)g(of)h(an)f(atom)g(in)h Fm(P)7 b(g)p Fw(.)648 747 y(As)22 b(an)g(example,)g(consider)f(the)i (logic)e(program)f Fm(P)29 b Fw(whic)n(h)22 b(implemen)n(ts)h(the)f Fp(quicksort)523 847 y Fw(algorithm:)623 1018 y Fn(q)s(sor)r(t)p Fw(\([)14 b(])p Fn(;)g Fw([)g(]\))24 b Fm( )623 1117 y Fn(q)s(sor)r(t)p Fw(\()p Fn(x)h Fw(:)e Fn(l)r(;)14 b(s)p Fw(\))22 b Fm( )i Fn(spl)r(it)p Fw(\()p Fn(l)r(;)14 b(x;)g(l)1597 1129 y Fk(1)1632 1117 y Fn(;)g(l)1694 1129 y Fk(2)1731 1117 y Fw(\))p Fn(;)g(q)s(sor)r(t)p Fw(\()p Fn(l)2045 1129 y Fk(1)2084 1117 y Fn(;)g(s)2160 1129 y Fk(1)2197 1117 y Fw(\))p Fn(;)g(q)s(sor)r(t)p Fw(\()p Fn(l)2511 1129 y Fk(2)2549 1117 y Fn(;)g(s)2625 1129 y Fk(2)2662 1117 y Fw(\))p Fn(;)g(app)p Fw(\()p Fn(s)2930 1129 y Fk(1)2967 1117 y Fn(;)g(x)24 b Fw(:)f Fn(s)3160 1129 y Fk(2)3197 1117 y Fn(;)14 b(s)p Fw(\))623 1317 y Fn(spl)r(it)p Fw(\([)g(])p Fn(;)g(x;)g Fw([)g(])p Fn(;)g Fw([)g(]\))22 b Fm( )623 1416 y Fn(spl)r(it)p Fw(\()p Fn(x)h Fw(:)g Fn(l)r(;)14 b(y)s(;)g(x)22 b Fw(:)h Fn(l)1223 1428 y Fk(1)1260 1416 y Fn(;)14 b(l)1322 1428 y Fk(2)1359 1416 y Fw(\))23 b Fm( )h Fn(l)r(ess)p Fw(\()p Fn(x;)14 b(y)s Fw(\))p Fn(;)g(spl)r(it)p Fw(\()p Fn(l)r(;)g(y)s(;)g(l)2263 1428 y Fk(1)2297 1416 y Fn(;)g(l)2359 1428 y Fk(2)2396 1416 y Fw(\))623 1516 y Fn(spl)r(it)p 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Fj(in)1125 2159 y Fw(\()p Fn(x)24 b Fw(:)f Fn(l)r Fw(\))g Fm(!)g Fn(q)s(sor)r(t)1650 2171 y Fj(out)1752 2159 y Fw(\()p Fn(s)p Fw(\))45 b Fm(\()23 b Fn(spl)r(it)2173 2171 y Fj(in)2241 2159 y Fw(\()p Fn(l)r(;)14 b(x)p Fw(\))23 b Fm(!)g Fn(spl)r(it)2712 2171 y Fj(out)2812 2159 y Fw(\()p Fn(l)2869 2171 y Fk(1)2906 2159 y Fn(;)14 b(l)2968 2171 y Fk(2)3005 2159 y Fw(\))p Fn(;)2006 2259 y(q)s(sor)r(t)2194 2271 y Fj(in)2264 2259 y Fw(\()p Fn(l)2321 2271 y Fk(1)2358 2259 y Fw(\))24 b Fm(!)f Fn(q)s(sor)r(t)2708 2271 y Fj(out)2810 2259 y Fw(\()p Fn(s)2881 2271 y Fk(1)2918 2259 y Fw(\))p Fn(;)2006 2358 y(q)s(sor)r(t)2194 2370 y Fj(in)2264 2358 y Fw(\()p Fn(l)2321 2370 y Fk(2)2358 2358 y Fw(\))h Fm(!)f Fn(q)s(sor)r(t)2708 2370 y Fj(out)2810 2358 y Fw(\()p Fn(s)2881 2370 y Fk(2)2918 2358 y Fw(\))p Fn(;)2006 2458 y(app)2134 2470 y Fj(in)2202 2458 y Fw(\()p Fn(s)2273 2470 y Fk(1)2311 2458 y Fn(;)14 b(x)23 b Fw(:)g Fn(s)2503 2470 y Fk(2)2541 2458 y Fw(\))g Fm(!)g Fn(app)2830 2470 y Fj(out)2931 2458 y Fw(\()p Fn(s)p Fw(\))834 2719 y Fn(spl)r(it)1001 2731 y Fj(in)1069 2719 y Fw(\([)14 b(])p Fn(;)g(x)p Fw(\))24 b Fm(!)f Fn(spl)r(it)1574 2731 y Fj(out)1674 2719 y Fw(\([)14 b(])p Fn(;)g Fw([)g(]\))755 2819 y Fn(spl)r(it)922 2831 y Fj(in)989 2819 y Fw(\()p Fn(x)24 b Fw(:)f Fn(l)r(;)14 b(y)s Fw(\))23 b Fm(!)g Fn(spl)r(it)1574 2831 y Fj(out)1674 2819 y Fw(\()p Fn(x)h Fw(:)f Fn(l)1848 2831 y Fk(1)1885 2819 y Fn(;)14 b(l)1947 2831 y Fk(2)1984 2819 y Fw(\))23 b Fm(\()h Fn(l)r(ess)2290 2831 y Fj(in)2357 2819 y Fw(\()p Fn(x;)14 b(y)s Fw(\))24 b Fm(!)f Fn(l)r(ess)2823 2831 y Fj(out)2923 2819 y Fn(;)2146 2918 y(spl)r(it)2313 2930 y Fj(in)2380 2918 y Fw(\()p Fn(l)r(;)14 b(y)s Fw(\))23 b Fm(!)g Fn(spl)r(it)2848 2930 y Fj(out)2948 2918 y Fw(\()p Fn(l)3005 2930 y Fk(1)3042 2918 y Fn(;)14 b(l)3104 2930 y Fk(2)3141 2918 y Fw(\))755 3018 y Fn(spl)r(it)922 3030 y Fj(in)989 3018 y Fw(\()p Fn(x)24 b Fw(:)f Fn(l)r(;)14 b(y)s Fw(\))23 b Fm(!)g Fn(spl)r(it)1574 3030 y Fj(out)1674 3018 y Fw(\()p Fn(l)1731 3030 y Fk(1)1768 3018 y Fn(;)14 b(x)24 b Fw(:)f Fn(l)1947 3030 y Fk(2)1984 3018 y Fw(\))g Fm(\()h Fn(g)s(eq)2265 3030 y Fj(in)2333 3018 y Fw(\()p Fn(x;)14 b(y)s Fw(\))23 b Fm(!)g Fn(g)s(eq)2773 3030 y Fj(out)2874 3018 y Fn(;)2146 3117 y(spl)r(it)2313 3129 y Fj(in)2380 3117 y Fw(\()p Fn(l)r(;)14 b(y)s Fw(\))23 b Fm(!)g Fn(spl)r(it)2848 3129 y Fj(out)2948 3117 y Fw(\()p Fn(l)3005 3129 y Fk(1)3042 3117 y Fn(;)14 b(l)3104 3129 y Fk(2)3141 3117 y Fw(\))523 3300 y(Note)26 b(that)g(eac)n(h)f(rule)g(of)h Fn(R)1405 3312 y Fl(P)1487 3300 y Fw(has)f(the)i(form)e Fn(P)2034 3270 y Fk(0)2022 3322 y Fj(in)2091 3300 y Fw(\()p Fn(u)2171 3270 y Fk(0)2171 3321 y(1)2208 3300 y Fn(;)14 b(:)g(:)g(:)g(;)g(u)2441 3270 y Fk(0)2441 3320 y Fj(m)2500 3328 y Fe(0)2535 3300 y Fw(\))24 b Fm(!)f Fn(P)2762 3270 y Fk(0)2750 3320 y Fj(out)2851 3300 y Fw(\()p Fn(v)2926 3270 y Fk(0)2923 3321 y(1)2964 3300 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)3192 3270 y Fk(0)3189 3320 y Fj(n)3230 3328 y Fe(0)3267 3300 y Fw(\))23 b Fm(\()523 3400 y Fn(P)588 3369 y Fk(1)576 3421 y Fj(in)645 3400 y Fw(\()p Fn(u)725 3369 y Fk(1)725 3420 y(1)762 3400 y Fn(;)14 b(:)g(:)g(:)g(;)g(u)995 3369 y Fk(1)995 3420 y Fj(m)1054 3428 y Fe(1)1090 3400 y Fw(\))23 b Fm(!)g Fn(P)1316 3369 y Fk(1)1304 3420 y Fj(out)1406 3400 y Fw(\()p Fn(v)1481 3369 y Fk(1)1478 3420 y(1)1518 3400 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)1746 3369 y Fk(1)1743 3420 y Fj(n)1784 3428 y Fe(1)1821 3400 y Fw(\))p Fn(;)k(:)c(:)g(:)g(;)g (P)2107 3369 y Fj(k)2095 3421 y(in)2164 3400 y Fw(\()p Fn(u)2244 3369 y Fj(k)2244 3420 y Fk(1)2285 3400 y Fn(;)g(:)g(:)g(:)f (;)h(u)2517 3369 y Fj(k)2517 3420 y(m)2576 3429 y Fa(k)2616 3400 y Fw(\))23 b Fm(!)g Fn(P)2842 3369 y Fj(k)2830 3420 y(out)2932 3400 y Fw(\()p Fn(v)3007 3369 y Fj(k)3004 3420 y Fk(1)3048 3400 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)3276 3369 y Fj(k)3273 3420 y(n)3314 3429 y Fa(k)3355 3400 y Fw(\),)523 3516 y(where)30 b Fn(u)814 3476 y Fj(j)814 3539 y(i)848 3516 y Fn(;)14 b(v)928 3476 y Fj(j)925 3539 y(i)991 3516 y Fm(2)28 b(T)21 b Fw(\()p Fn(F)1237 3485 y Fj(T)1225 3538 y Fl(P)1290 3516 y Fn(;)14 b Fm(V)7 b Fw(\).)30 b(In)h(particular,)e(ev)n(ery)g Fn(l)g Fm(!)e Fn(r)k Fm(\()c Fn(s)2589 3528 y Fk(1)2654 3516 y Fm(!)h Fn(t)2795 3528 y Fk(1)2832 3516 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)3056 3528 y Fj(k)3124 3516 y Fm(!)28 b Fn(t)3265 3528 y Fj(k)3336 3516 y Fw(in)523 3615 y Fn(R)586 3627 y Fl(P)668 3615 y Fw(satis\014es:)c(\(1\))i(The)f(ro)r(ot)g(sym)n(b)r(ol)f(is)i(the)f (only)g(de\014ned)h(sym)n(b)r(ol)f(in)h Fn(l)g Fw(and)g Fn(s)3064 3627 y Fj(i)3091 3615 y Fw(,)g(and)f(\(2\))523 3715 y Fn(r)30 b Fw(and)e Fn(t)782 3727 y Fj(i)837 3715 y Fw(are)f(constructor)f(terms.)523 3883 y Fi(De\014nition)31 b(16.)40 b Fp(A)n(n)35 b(LR-wel)t(l-mo)l(de)l(d)i(lo)l(gic)f(pr)l(o)l (gr)l(am)g Fm(P)43 b Fp(is)f Fw(\(uniquely\))36 b(terminating)523 3982 y Fp(if,)41 b(for)f(every)g(LR-wel)t(l-mo)l(de)l(d)g(query)g Fm( )g Fn(A)p Fp(,)g(every)g(left-to-right)g(SLD-derivation)h(is)523 4082 y(terminating)35 b(\(and)h(every)f(left-to-right)h(SLD-r)l (efutation)e(c)l(omputes)h(the)g(same)h(answer)523 4182 y(substitution\).)523 4350 y Fw(Ganzinger)e(and)h(W)-7 b(aldmann)35 b([GW93)o(,)g(Thm.)h(14])e(sho)n(w)n(ed)f(that)j (quasi-reductivit)n(y)d(of)523 4449 y Fn(R)586 4461 y Fl(P)678 4449 y Fw(pro)n(v)n(es)h(termination)h(of)h(an)f(LR-w)n (ell-mo)r(ded)g(logic)g(program)f Fm(P)7 b Fw(.)35 b(Av)n(enhaus)g(and) 523 4549 y(Lor)-9 b(\023)-32 b(\020a-S\023)-42 b(aenz)37 b([ALS94,)j(Thm.)h(5.1])f(pro)n(v)n(ed)f(that)h(unique)h(termination)f (follo)n(ws)f(from)523 4648 y(quasi-reductivit)n(y)26 b(and)i(the)g(joinabilit)n(y)f(of)h(all)f(conditional)g(critical)g (pairs.)648 4748 y(By)g(means)g(of)h(the)g(follo)n(wing)e(implications) 726 4924 y Fn(U)9 b Fw(\()p Fn(R)887 4936 y Fl(P)943 4924 y Fw(\))28 b(is)f(terminating)51 b Fm(\))23 b Fn(R)1731 4936 y Fl(P)1814 4924 y Fw(is)28 b(quasi-reductiv)n(e)49 b Fm(\))23 b(P)34 b Fw(is)28 b(terminating)p eop %%Page: 16 16 16 15 bop 523 448 a Fw(it)35 b(su\016ces)e(to)h(sho)n(w)f(termination)h (of)g Fn(U)9 b Fw(\()p Fn(R)1942 460 y Fl(P)1998 448 y Fw(\))35 b(in)f(order)f(to)h(pro)n(v)n(e)e(termination)i(of)g(the)523 548 y Fp(quicksort)27 b Fw(program.)d(T)-7 b(ermination)26 b(of)g Fn(U)9 b Fw(\()p Fn(R)1953 560 y Fl(P)2009 548 y Fw(\))26 b(can)g(automatically)f(b)r(e)i(sho)n(wn)e(with)i(the)523 648 y(dep)r(endency)32 b(pair)f(tec)n(hnique;)g(see)h([A)n(G97a)n(].)g (Th)n(us)f(the)h Fp(quicksort)g Fw(program)e(is)h(termi-)523 747 y(nating.)c(Since)g(ev)n(ery)f(conditional)h(critical)g(pair)f(in)i Fn(R)2275 759 y Fl(P)2358 747 y Fw(is)f(infeasible)g(\(hence)h (joinable\),)523 847 y(the)g(logic)f(program)e(is)j(also)e(uniquely)i (terminating.)648 946 y(Arts)k(and)g(Zan)n(tema)f([AZ95)o(,AZ96)o(])h (stated)g(an)g(imp)r(erativ)n(e)g(pro)r(cedure)2998 916 y Fk(3)3066 946 y Fw(whic)n(h)g(di-)523 1046 y(rectly)f(transforms)g(a) g(logic)g(program)e Fm(P)38 b Fw(in)n(to)32 b(an)f Fp(unc)l(onditional) i Fw(TRS.)f(The)g(TRS)g(ob-)523 1146 y(tained)39 b(b)n(y)f(this)h(imp)r (erativ)n(e)f(pro)r(cedure)1870 1116 y Fk(4)1945 1146 y Fw(is)g(essen)n(tially)g(the)h(same)f(as)g Fn(U)9 b Fw(\()p Fn(R)3095 1158 y Fl(P)3151 1146 y Fw(\).)39 b(Arts)523 1245 y(and)33 b(Zan)n(tema)e(sho)n(w)n(ed)h(that)h(single-redex)e (termination)i(of)f Fn(U)9 b Fw(\()p Fn(R)2693 1257 y Fl(P)2749 1245 y Fw(\))34 b(su\016ces)e(to)h(pro)n(v)n(e)523 1345 y(termination)d(of)h Fm(P)7 b Fw(;)30 b(see)g([AZ95)o(,)g(Thm.)h (4.8])f(and)g([Art97,)g(Thm.)h(8.2.9].)e(W)-7 b(e)31 b(recall)f(the)523 1445 y(de\014nition:)523 1585 y Fi(De\014nition)h (17.)40 b Fp(A)29 b(r)l(e)l(duction)f(step)h Fn(s)23 b Fm(!)g Fn(t)28 b Fp(is)h(c)l(al)t(le)l(d)h(a)f Fw(single-redex)e Fp(r)l(e)l(duction)h(step)h(if)523 1684 y Fn(s)i Fp(c)l(ontains)h (exactly)g(one)f(r)l(e)l(dex.)h(If)g(a)g(term)f(do)l(es)h(not)f(have)h (exactly)g(one)g(r)l(e)l(dex,)g(then)f(it)523 1784 y(is)c(in)f Fw(single-redex)c(normal)h(form)p Fp(.)k(A)f Fw(single-redex)c(deriv)-5 b(ation)26 b Fp(is)g(a)h(r)l(e)l(duction)f(se)l(quenc)l(e)523 1883 y(c)l(onsisting)k(solely)h(of)g(single-r)l(e)l(dex)f(r)l(e)l (duction)f(steps.)i(A)e(TRS)g Fm(R)h Fp(is)g(c)l(al)t(le)l(d)i Fw(single-redex)523 1983 y(terminating)d Fp(if)i(al)t(l)g(single-r)l(e) l(dex)f(derivations)h(ar)l(e)f(\014nite.)523 2123 y Fw(Note)k(that)g (innermost)g(termination)g(implies)g(single-redex)e(termination.)i(Due) h(to)f(the)523 2223 y(follo)n(wing)27 b(hierarc)n(h)n(y)e(\(note)j (that)g Fn(U)9 b Fw(\()p Fn(R)1794 2235 y Fl(P)1850 2223 y Fw(\))28 b(ma)n(y)f(b)r(e)h(o)n(v)n(erlapping\))614 2382 y Fn(U)9 b Fw(\()p Fn(R)775 2394 y Fl(P)831 2382 y Fw(\))28 b(is)f(terminating)51 b Fm(\))23 b Fn(R)1619 2394 y Fl(P)1703 2382 y Fw(is)k(quasi-reductiv)n(e)49 b Fm(\))23 b Fn(R)2553 2394 y Fl(P)2637 2382 y Fw(is)k (quasi-decreasing)614 2482 y Fm(\))c Fn(U)9 b Fw(\()p Fn(R)881 2494 y Fl(P)937 2482 y Fw(\))28 b(is)g(innermost)f (terminating)22 b Fm(\))h Fn(U)9 b Fw(\()p Fn(R)2183 2494 y Fl(P)2239 2482 y Fw(\))28 b(is)g(single-redex)e(terminating)523 2639 y(it)f(seems)f(that)h(the)f(metho)r(d)h(of)g(Arts)f(and)g(Zan)n (tema)g(is)g(more)g(p)r(o)n(w)n(erful)g(than)g(Ganzinger)523 2738 y(and)33 b(W)-7 b(aldmann's,)33 b(in)g(the)h(sense)e(that)i(more)e (logic)g(programs)f(can)h(b)r(e)i(pro)n(v)n(en)d(termi-)523 2838 y(nating)26 b(b)n(y)g(the)g(former)f(metho)r(d.)i(Theorem)e(19,)g (ho)n(w)n(ev)n(er,)g(implies)h(that)g(b)r(oth)h(metho)r(ds)523 2938 y(are)g Fp(e)l(qual)t(ly)j(p)l(owerful)p Fw(.)f(T)-7 b(o)28 b(pro)n(v)n(e)e(it,)i(w)n(e)f(need)h(the)g(follo)n(wing)e (lemma.)523 3078 y Fi(Lemma)k(18.)40 b Fp(Supp)l(ose)i Fn(U)9 b Fw(\()p Fn(R)1514 3090 y Fl(P)1570 3078 y Fw(\))41 b Fp(is)h(single-r)l(e)l(dex)f(terminating.)h(If)f(every)h Fn(v)3059 3090 y Fj(j)3136 3078 y Fp(in)f Fn(u)i Fw(=)523 3177 y Fn(P)576 3189 y Fj(in)645 3177 y Fw(\()p Fn(v)717 3189 y Fk(1)755 3177 y Fn(;)14 b(:)g(:)g(:)f(;)h(v)979 3189 y Fj(m)1043 3177 y Fw(\))39 b Fp(is)h(a)g(term)f(in)h Fm(T)21 b Fw(\()p Fn(F)1778 3147 y Fj(T)1766 3200 y Fl(P)1830 3177 y Fn(;)14 b Fm(V)7 b Fw(\))p Fp(,)40 b(then)f(every)i Fn(U)9 b Fw(\()p Fn(R)2607 3189 y Fl(P)2663 3177 y Fw(\))p Fp(-derivation)41 b(starting)523 3277 y(fr)l(om)30 b Fn(u)g Fp(is)g(\014nite.)523 3429 y(Pr)l(o)l(of.)43 b Fw(It)31 b(is)g(fairly)f(easy)g(to)g(see)h(that)g(ev)n(ery)e Fn(U)9 b Fw(\()p Fn(R)2177 3441 y Fl(P)2233 3429 y Fw(\)-deriv)-5 b(ation)30 b(starting)g(from)g Fn(u)h Fw(is)f(a)523 3529 y(single-redex)c(deriv)-5 b(ation.)27 b(Th)n(us)g(it)h(is)g(\014nite.) 523 3681 y Fi(Theorem)i(19.)41 b Fp(If)30 b Fn(U)9 b Fw(\()p Fn(R)1340 3693 y Fl(P)1396 3681 y Fw(\))30 b Fp(is)g(single-r)l(e)l(dex)g(terminating,)h(then)e(it)h(is)g (terminating.)523 3833 y(Pr)l(o)l(of.)43 b Fw(By)18 b(structural)g (induction)h(on)f Fn(u)p Fw(,)g(it)g(will)h(b)r(e)g(sho)n(wn)e(that)i (ev)n(ery)e Fn(U)9 b Fw(\()p Fn(R)2943 3845 y Fl(P)2999 3833 y Fw(\)-deriv)-5 b(ation)523 3933 y(starting)23 b(from)g Fn(u)g Fw(is)h(\014nite.)g(If)g Fn(u)f Fw(is)h(a)f(v)-5 b(ariable,)22 b(then)j(it)f(is)f(in)h(normal)f(form.)g(If)h Fn(u)f Fw(is)h(a)f(con-)523 4032 y(stan)n(t,)h(then)h(it)g(is)f(either) g(in)h(normal)e(form)h(or)g(Lemma)g(18)f(applies.)i(If)f Fn(u)g Fw(has)g(a)g Fn(U)9 b Fw(-sym)n(b)r(ol)523 4132 y(at)36 b(its)h(ro)r(ot,)f(then)h(the)g(assertion)e(follo)n(ws)g(as)h (in)h(the)g(pro)r(of)f(of)g(case)g(\(c\))h(in)g(Theorem)523 4231 y(10.)h(So)g(supp)r(ose)f Fn(u)k Fw(=)f Fn(P)1363 4243 y Fj(in)1432 4231 y Fw(\()p Fn(v)1504 4243 y Fk(1)1542 4231 y Fn(;)14 b(:)g(:)g(:)g(;)g(v)1767 4243 y Fj(m)1830 4231 y Fw(\),)39 b(where)f(at)g(least)g(one)f(of)i(the)f Fn(v)2953 4243 y Fj(j)3027 4231 y Fw(con)n(tains)f(a)523 4331 y(function)31 b(sym)n(b)r(ol)f Fn(f)36 b Fm(62)27 b Fn(F)1362 4301 y Fj(T)1350 4354 y Fl(P)1415 4331 y Fw(.)j(By)g(the)g(inductiv)n(e)h(h)n(yp)r(othesis,)f(ev)n(ery)f Fn(U)9 b Fw(\()p Fn(R)2927 4343 y Fl(P)2983 4331 y Fw(\)-deriv)-5 b(ation)523 4431 y(starting)21 b(from)h Fn(v)1058 4443 y Fj(j)1115 4431 y Fw(is)g(\014nite.)h(F)-7 b(or)21 b(a)g(pro)r(of)h(b) n(y)g(con)n(tradiction)e(supp)r(ose)i(there)g(is)g(an)f(in\014nite)523 4530 y Fn(U)9 b Fw(\()p Fn(R)684 4542 y Fl(P)740 4530 y Fw(\)-deriv)-5 b(ation)1445 4691 y Fn(D)26 b Fw(:)d Fn(u)f Fw(=)h Fn(u)1792 4703 y Fk(0)1852 4691 y Fm(!)g Fn(u)2006 4703 y Fk(1)2066 4691 y Fm(!)g Fn(u)2220 4703 y Fk(2)2280 4691 y Fm(!)g Fn(:)14 b(:)g(:)p 523 4748 473 4 v 540 4801 a Fc(3)600 4833 y Fv(There)26 b(is)g(a)g(\015a)n(w)g (in)g(the)f(pro)r(cedure)h(whic)n(h)f(has)h(b)r(een)g(corrected)g(in)f ([Art97)q(,)h(Def.)g(8.2.2].)540 4893 y Fc(4)600 4924 y Fv(More)g(precisely)-6 b(,)27 b(b)n(y)d(the)i(imp)r(erativ)n(e)f(pro) r(cedure)g(in)h([Art97,)g(Def.)h(8.2.2].)p eop %%Page: 17 17 17 16 bop 523 448 a Fw(The)30 b(pro)r(of)f(idea)g(is)h(to)f(eliminate)h (all)f(\\aliens")f(in)i(the)g(terms)g Fn(u)2602 460 y Fj(j)2666 448 y Fw(b)n(y)f(a)g(function)i Fn(\011)38 b Fw(and)523 548 y(to)c(sho)n(w)g(that)h(the)g(deriv)-5 b(ation)34 b Fn(\011)9 b Fw(\()p Fn(D)r Fw(\))35 b(is)g(still)f (in\014nite.)i(In)e Fn(u)p Fw(,)h(ev)n(ery)e(maximal)h(prop)r(er)523 648 y(subterm)j Fn(t)h Fw(with)g(a)f Fn(U)9 b Fw(-sym)n(b)r(ol)36 b(or)h(a)g(function)h(sym)n(b)r(ol)f(from)g Fn(F)2680 617 y Fj(P)2668 670 y Fl(P)2772 648 y Fw(at)h(the)f(ro)r(ot)g(is)g(an) 523 747 y(alien.)g(Informally)-7 b(,)36 b(ev)n(ery)f(\\descendan)n(t")g (of)i(an)g(alien)f(from)g Fn(u)h Fw(is)f(an)h(alien)f(in)h Fn(u)3234 759 y Fj(j)3269 747 y Fw(.)g(In)523 847 y(order)32 b(to)i(formally)e(de\014ne)i(what)g(an)f(alien)g(is,)h(w)n(e)f(need)h (the)g(t)n(w)n(o)f(functions)h Fn(\010)g Fw(and)f Fn(\011)523 946 y Fw(de\014ned)28 b(b)r(elo)n(w.)907 1205 y Fn(\010)p Fw(\()p Fn(t)p Fw(\))c(=)1168 1035 y Fh(8)1168 1110 y(<)1168 1259 y(:)1253 1105 y Fn(t)702 b Fw(if)28 b Fn(t)23 b Fm(2)h(V)7 b Fn(;)1253 1204 y(f)i Fw(\()p Fn(\010)p Fw(\()p Fn(t)1452 1216 y Fk(1)1490 1204 y Fw(\))p Fn(;)14 b(:)g(:)g(:)g(;)g (\010)p Fw(\()p Fn(t)1824 1216 y Fj(n)1870 1204 y Fw(\)\))51 b(if)28 b Fn(t)23 b Fw(=)g Fn(f)9 b Fw(\()p Fn(t)2314 1216 y Fk(1)2351 1204 y Fn(;)14 b(:)g(:)g(:)g(;)g(t)2566 1216 y Fj(n)2611 1204 y Fw(\))p Fn(;)42 b(f)31 b Fm(2)24 b Fn(F)2924 1174 y Fj(T)2912 1227 y Fl(P)2976 1204 y Fn(;)1253 1304 y(z)693 b Fw(otherwise)523 1484 y(where)31 b Fn(z)k Fw(is)c(a)g(fresh)h(v)-5 b(ariable)30 b(\(i.e.,)i(it)g(do)r (es)f(not)h(o)r(ccur)f(in)h Fn(t)p Fw(\).)g(So)f(if)h Fn(t)d Fw(=)h Fn(C)6 b Fw([)p Fn(t)3062 1496 y Fk(1)3099 1484 y Fn(;)14 b(:)g(:)g(:)g(;)g(t)3314 1496 y Fj(n)3359 1484 y Fw(],)523 1584 y(where)30 b Fn(C)35 b Fm(2)29 b(T)21 b Fw(\()p Fn(F)1107 1554 y Fj(T)1095 1607 y Fl(P)1160 1584 y Fn(;)14 b Fm(V)7 b Fw(\))31 b(and)f(either)h Fn(r)r(oot)p Fw(\()p Fn(t)1932 1596 y Fj(j)1969 1584 y Fw(\))g(is)g(a)f Fn(U)9 b Fw(-sym)n(b)r(ol)30 b(or)g Fn(r)r(oot)p Fw(\()p Fn(t)2888 1596 y Fj(j)2924 1584 y Fw(\))f Fm(2)g Fn(F)3134 1554 y Fj(P)3122 1607 y Fl(P)3189 1584 y Fw(,)i(then)523 1684 y Fn(\010)p Fw(\()p Fn(t)p Fw(\))24 b(=)f Fn(C)6 b Fw([)p Fn(z)t(;)14 b(:)g(:)g(:)f(;)h(z)t Fw(].)775 1913 y Fn(\011)9 b Fw(\()p Fn(t)p Fw(\))24 b(=)1041 1796 y Fh(\032)1115 1863 y Fn(p)p Fw(\()p Fn(\010)p Fw(\()p Fn(t)1306 1875 y Fk(1)1343 1863 y Fw(\))p Fn(;)14 b(:)g(:)g(:)g(;)g (\010)p Fw(\()p Fn(t)1677 1875 y Fj(n)1723 1863 y Fw(\)\))343 b(if)28 b Fn(t)23 b Fw(=)g Fn(p)p Fw(\()p Fn(t)2451 1875 y Fk(1)2488 1863 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)2702 1875 y Fj(n)2747 1863 y Fw(\))p Fn(;)42 b(p)23 b Fm(2)h Fn(F)3053 1833 y Fj(P)3041 1886 y Fl(P)3108 1863 y Fn(;)1115 1962 y(U)1181 1922 y Fj(\032)1172 1985 y(i)1219 1962 y Fw(\()p Fn(\011)9 b Fw(\()p Fn(t)1373 1974 y Fk(1)1411 1962 y Fw(\))p Fn(;)14 b(\010)p Fw(\()p Fn(t)1597 1974 y Fk(2)1635 1962 y Fw(\))p Fn(;)g(:)g(:)g(:)g(;)g(\010)p Fw(\()p Fn(t)1969 1974 y Fj(n)2014 1962 y Fw(\)\))52 b(if)28 b Fn(t)23 b Fw(=)g Fn(U)2413 1922 y Fj(\032)2404 1985 y(i)2451 1962 y Fw(\()p Fn(t)2513 1974 y Fk(1)2550 1962 y Fn(;)14 b(:)g(:)g(:)g(;)g(t)2765 1974 y Fj(n)2810 1962 y Fw(\))p Fn(:)523 2138 y Fw(Note)30 b(that)g Fn(\011)39 b Fw(is)30 b(only)f(partially)g(de\014ned.)i(Supp)r(ose)f Fn(t)g Fw(has)f(only)g Fn(U)9 b Fw(-sym)n(b)r(ols)29 b(ab)r(o)n(v)n(e)g(the)523 2238 y(leftmost)39 b(outermost)f(function)i (sym)n(b)r(ol)e(whic)n(h)h(is)f(in)h Fn(F)2407 2208 y Fj(P)2395 2261 y Fl(P)2463 2238 y Fw(.)f Fn(\011)48 b Fw(do)r(es)39 b(not)f(mo)r(dify)i(this)523 2338 y(sym)n(b)r(ol)31 b(or)f(an)n(y)g(of)h(the)g Fn(U)9 b Fw(-sym)n(b)r(ols)30 b(ab)r(o)n(v)n(e)g(it.)h(All)h(other)e(function)i(sym)n(b)r(ols)e(in)i Fn(t)f Fw(that)523 2437 y(are)c(not)g(in)h Fn(F)971 2407 y Fj(T)959 2460 y Fl(P)1051 2437 y Fw(are)f(then)h(replaced)e(with)j (the)e(v)-5 b(ariable)27 b Fn(z)k Fw(b)n(y)c(the)h(function)g Fn(\010)p Fw(.)648 2537 y(No)n(w)j(the)i(aliens)e(in)h Fn(u)1373 2549 y Fj(j)1440 2537 y Fw(are)f(those)h(subterms)f(of)h Fn(u)2312 2549 y Fj(j)2379 2537 y Fw(whic)n(h)g(are)f(replaced)g(with)i (the)523 2636 y(fresh)26 b(v)-5 b(ariable)26 b Fn(z)k Fw(when)c Fn(\011)36 b Fw(is)26 b(applied)h(to)f Fn(u)1925 2648 y Fj(j)1960 2636 y Fw(.)g(W)-7 b(e)27 b(write)f Fn(u)2410 2648 y Fj(j)2468 2636 y Fm(!)2551 2648 y Fj(a)2614 2636 y Fn(u)2662 2648 y Fj(j)s Fk(+1)2807 2636 y Fw(if)h(the)g(con)n (tracted)523 2736 y(redex)22 b(is)h(a)f(subterm)h(of)g(an)f(alien)h(in) g Fn(u)1738 2748 y Fj(j)1795 2736 y Fw(and)g Fn(u)2000 2748 y Fj(j)2057 2736 y Fm(!)2140 2748 y Fj(na)2244 2736 y Fn(u)2292 2748 y Fj(j)s Fk(+1)2434 2736 y Fw(otherwise.)f(It)h(is)f (not)h(di\016cult)523 2836 y(to)f(sho)n(w)f(that)h(for)g(ev)n(ery)f (alien)g Fn(t)h Fw(in)h Fn(u)1713 2848 y Fj(j)1769 2836 y Fw(there)f(is)g(an)g(alien)g Fn(s)g Fw(in)g Fn(u)f Fw(suc)n(h)h(that)g Fn(s)h Fm(!)3079 2806 y Fl(\003)3141 2836 y Fn(t)p Fw(.)f(Since)523 2935 y(ev)n(ery)i Fn(U)9 b Fw(\()p Fn(R)901 2947 y Fl(P)957 2935 y Fw(\)-deriv)-5 b(ation)25 b(starting)f(from)h Fn(s)g Fw(is)h(\014nite,)g(the)f (relation)g Fm(!)2811 2947 y Fj(a)2876 2935 y Fw(is)g(terminating.)523 3035 y(Consequen)n(tly)-7 b(,)39 b Fn(D)i Fw(m)n(ust)f(con)n(tain)f (in\014nitely)h(man)n(y)e Fm(!)2379 3047 y Fj(na)2500 3035 y Fw(steps.)i(It)f(is)h(ob)n(vious)e(that)523 3135 y Fn(u)571 3147 y Fj(j)630 3135 y Fm(!)713 3147 y Fj(a)778 3135 y Fn(u)826 3147 y Fj(j)s Fk(+1)973 3135 y Fw(implies)29 b Fn(\011)9 b Fw(\()p Fn(u)1396 3147 y Fj(j)1431 3135 y Fw(\))25 b(=)f Fn(\011)9 b Fw(\()p Fn(u)1717 3147 y Fj(j)s Fk(+1)1836 3135 y Fw(\).)29 b(Moreo)n(v)n(er,)d(if)j Fn(u)2434 3147 y Fj(j)2493 3135 y Fm(!)2576 3147 y Fj(na)2682 3135 y Fn(u)2730 3147 y Fj(j)s Fk(+1)2848 3135 y Fw(,)g(then)g Fn(\011)9 b Fw(\()p Fn(u)3230 3147 y Fj(j)3265 3135 y Fw(\))25 b Fm(!)523 3234 y Fn(\011)9 b Fw(\()p Fn(u)663 3246 y Fj(j)s Fk(+1)782 3234 y Fw(\).)20 b(This)g(fact)g(can)f(b)r(e)h (pro)n(v)n(en)e(as)h(follo)n(ws.)g(W)-7 b(rite)20 b Fn(u)2359 3246 y Fj(j)2417 3234 y Fw(=)i Fn(C)6 b Fw([)p Fn(l)r(\033)s Fw(])24 b Fm(!)2799 3246 y Fj(na)2903 3234 y Fn(C)6 b Fw([)p Fn(r)r(\033)s Fw(])25 b(=)d Fn(u)3263 3246 y Fj(j)s Fk(+1)3382 3234 y Fw(,)523 3334 y(where)36 b Fn(l)k Fm(!)f Fn(r)i Fm(2)e Fn(U)9 b Fw(\()p Fn(R)1292 3346 y Fl(P)1348 3334 y Fw(\).)38 b(By)e(the)i(form)e(of)h(the)h(rewrite)e(rules)g(in)h Fn(U)9 b Fw(\()p Fn(R)2961 3346 y Fl(P)3017 3334 y Fw(\),)38 b(w)n(e)e(ha)n(v)n(e)523 3433 y Fn(\011)9 b Fw(\()p Fn(l)r(\033)s Fw(\))28 b(=)f Fn(l)r(\034)9 b Fw(,)30 b(where)f Fn(\034)40 b Fw(is)30 b(de\014ned)h(b)n(y)f Fn(x\034)37 b Fw(=)27 b Fn(\010)p Fw(\()p Fn(x\033)s Fw(\).)32 b(It)e(is)g(fairly)g(simple)g (to)g(pro)n(v)n(e)f(that)523 3533 y Fn(l)r(\034)k Fm(!)23 b Fn(r)r(\034)34 b Fw(=)23 b Fn(\011)9 b Fw(\()p Fn(r)r(\033)s Fw(\))30 b(whic)n(h)e(yields)f Fn(\011)9 b Fw(\()p Fn(u)1776 3545 y Fj(j)1811 3533 y Fw(\))24 b(=)f Fn(\011)9 b Fw(\()p Fn(C)d Fw(\)[)p Fn(\011)j Fw(\()p Fn(l)r(\033)s Fw(\)])25 b Fm(!)e Fn(\011)9 b Fw(\()p Fn(C)d Fw(\)[)p Fn(\011)j Fw(\()p Fn(r)r(\033)s Fw(\)])27 b(=)c Fn(u)3133 3545 y Fj(j)s Fk(+1)3251 3533 y Fw(.)28 b(By)523 3633 y(putting)g(all)g(the) g(facts)f(together,)g(w)n(e)g(conclude)g(that)h(the)g Fn(U)9 b Fw(\()p Fn(R)2571 3645 y Fl(P)2627 3633 y Fw(\)-deriv)-5 b(ation)1077 3812 y Fn(\011)9 b Fw(\()p Fn(D)r Fw(\))24 b(:)f Fn(\011)9 b Fw(\()p Fn(u)p Fw(\))23 b(=)f Fn(\011)9 b Fw(\()p Fn(u)1764 3824 y Fk(0)1801 3812 y Fw(\))24 b Fm(!)1940 3778 y Fl(\003)2001 3812 y Fn(\011)9 b Fw(\()p Fn(u)2141 3824 y Fk(1)2178 3812 y Fw(\))23 b Fm(!)2316 3778 y Fl(\003)2378 3812 y Fn(\011)9 b Fw(\()p Fn(u)2518 3824 y Fk(2)2555 3812 y Fw(\))23 b Fm(!)2693 3778 y Fl(\003)2754 3812 y Fn(:)14 b(:)g(:)523 3992 y Fw(is)37 b(in\014nite.)i(F)-7 b(urthermore,)36 b Fn(\011)9 b Fw(\()p Fn(u)p Fw(\))40 b(=)f Fn(P)1814 4004 y Fj(in)1883 3992 y Fw(\()p Fn(\010)p Fw(\()p Fn(v)2042 4004 y Fk(1)2081 3992 y Fw(\))p Fn(;)14 b(:)g(:)g(:)f(;)h(\010)p Fw(\()p Fn(v)2424 4004 y Fj(m)2488 3992 y Fw(\)\))38 b(and)g(ev)n(ery)e Fn(\010)p Fw(\()p Fn(v)3118 4004 y Fj(j)3154 3992 y Fw(\))i(is)f(an)523 4092 y(elemen)n(t)28 b(of)f Fm(T)22 b Fw(\()p Fn(F)1088 4062 y Fj(T)1076 4115 y Fl(P)1140 4092 y Fn(;)14 b Fm(V)7 b Fw(\).)28 b(This,)f(ho)n(w)n(ev)n(er,)f(con)n(tradicts)g(Lemma)i(18.) 523 4263 y(W)-7 b(e)24 b(ha)n(v)n(e)e(seen)i(that)g(the)g(termination)f (pro)r(of)g(tec)n(hnique)g(of)h(Ganzinger)e(and)i(W)-7 b(aldmann)523 4363 y(is)25 b(as)g(p)r(o)n(w)n(erful)g(as)g(that)g(of)h (Arts)f(and)g(Zan)n(tema.)g(All)h(in)g(all,)f(w)n(e)g(suggest)f(that)i (one)f(tak)n(es)523 4462 y(the)j(b)r(est)g(of)g(b)r(oth)g(w)n(orlds:) 558 4625 y(1.)41 b(The)36 b(t)n(w)n(o-stage)d(transformation)h (consisting)g(of)h(the)h(phases)f(\(i\))h(translation)e(of)h(a)664 4725 y(logic)25 b(program)f Fm(P)33 b Fw(in)n(to)26 b(a)f (deterministic)i(3-CTRS)e Fn(R)2395 4737 y Fl(P)2477 4725 y Fw(and)h(\(ii\))h(translation)e(of)h Fn(R)3349 4737 y Fl(P)664 4825 y Fw(in)n(to)39 b(an)f(unconditional)g(TRS)h Fn(U)9 b Fw(\()p Fn(R)1869 4837 y Fl(P)1926 4825 y Fw(\))39 b(is)f(m)n(uc)n(h)h(easier)e(to)i(grasp)e(than)i(a)f(direct)664 4924 y(transformation)26 b(via)h(an)g(imp)r(erativ)n(e)h(pro)r(cedure.) e(It)i(is)g(th)n(us)f(preferable.)p eop %%Page: 18 18 18 17 bop 558 448 a Fw(2.)41 b(Unique)36 b(termination)f(of)h Fm(P)42 b Fw(can)35 b(b)r(e)h(pro)n(v)n(en)e(on)h(the)h(lev)n(el)f(of)h (3-CTRSs)e(whereas)664 548 y(there)f(is)g(no)h(similar)e(metho)r(d)i (for)f(TRSs.)g(So)g(the)h(t)n(w)n(o-stage)d(transformation)h(has)664 648 y(another)27 b(adv)-5 b(an)n(tage)26 b(o)n(v)n(er)g(the)i(direct)f (transformation.)558 744 y(3.)41 b(W)-7 b(e)24 b(ha)n(v)n(e)f(seen)g (that)h(in)g(b)r(oth)g(approac)n(hes)e(to)h(pro)n(ving)f(termination)i (of)f(an)h(LR-w)n(ell-)664 844 y(mo)r(ded)32 b(logic)f(program)f Fm(P)38 b Fw(it)32 b(b)r(oils)g(do)n(wn)f(to)g(pro)n(ving)f(innermost)i (termination)f(of)664 943 y(the)36 b(unconditional)f(TRS)g Fn(U)9 b Fw(\()p Fn(R)1707 955 y Fl(P)1763 943 y Fw(\).)36 b(Since)g(the)f(latter)g(can)g(b)r(e)h(automated)f(b)n(y)g(the)664 1043 y(dep)r(endency)d(pair)f(tec)n(hnique)h([A)n(G97b)o(],)g(the)g (whole)g(metho)r(d)g(can)f(b)r(e)h(automated;)664 1143 y(cf.)c([Art97)o(].)523 1296 y(It)j(should)g(b)r(e)g(p)r(oin)n(ted)g (out)g(that)g(the)g(metho)r(ds)g(don't)g(yield)g(a)f(complete)h (criterion)f(for)523 1396 y(pro)n(ving)c(termination)h(of)h(LR-w)n (ell-mo)r(ded)f(logic)f(programs;)g(see)h([GW93].)523 1649 y Fq(7)112 b(Related)37 b(W)-9 b(ork)523 1836 y Fw(As)25 b(already)f(men)n(tioned,)h(the)g(idea)g(of)g(transforming)f (conditional)g(rewrite)h(systems)f(in)n(to)523 1936 y(unconditional)k (ones)f(dates)h(bac)n(k)f(to)h(the)h(w)n(ork)d(of)i(Bergstra)e(and)i (Klop)g([BK86)n(],)h(where)523 2035 y(suc)n(h)24 b(transformations)f(w) n(ere)h(used)h(as)f(a)g(heuristic)g(to)r(ol)h(to)f(construct)h(coun)n (terexamples)523 2135 y(to)35 b(con\015uence)f(of)h(certain)f(classes)f (of)i(CTRSs.)g(Bergstra)e(and)h(Klop)g(did)h(not)g(explore)523 2234 y(the)f(formal)f(asp)r(ects)h(of)g(these)f(transformations.)f (This)i(w)n(as)f(done)h(ten)g(y)n(ears)e(later)h(b)n(y)523 2334 y(Marc)n(hiori)i([Mar96)n(].)i(He)h(sho)n(w)n(ed)e(that)h(some)f (parts)g(of)h(the)h(theory)e(of)h(CTRSs)g(\\can)523 2434 y(b)r(e)30 b Fp(automatic)l(al)t(ly)i Fw(reco)n(v)n(ered)c(from)h(the)i (theory)e(of)h(TRSs".)g(In)g([Mar97)n(],)g(the)h(limits)f(of)523 2533 y(transformational)25 b(approac)n(hes)f(are)h(discussed.)i(Other)f (transformations)e(can)j(b)r(e)g(found)523 2633 y(in)42 b([GM88)o(])f(and)h([Hin95)o(].)g(The)f(applicabilit)n(y)g(of)g(Gio)n (v)-5 b(anetti)41 b(and)g(Moiso's)g([GM88)o(])523 2733 y(transformation)31 b(is)i(rather)f(limited)i(b)r(ecause)f(it)g(is)g (designed)g(to)f(preserv)n(e)g(\\the)h(equiv-)523 2832 y(alence)39 b(relation)f(induced)i(on)f(terms".)f(Hin)n(termeier)h ([Hin95])g(pro)n(vides)f(a)h(t)n(w)n(o-phase)523 2932 y(transformation)c(\(based)h(on)g(order-sorted)e(rewriting\))i(from)g (the)h(class)e(of)i(decreasing)523 3031 y(and)27 b(ground)g(con\015uen) n(t)h(CTRSs)f(to)h(TRSs.)648 3131 y(Man)n(y)j(metho)r(ds)h(ha)n(v)n(e)f (b)r(een)i(prop)r(osed)e(to)h(pro)n(v)n(e)e(termination)i(of)g(logic)f (programs)523 3231 y(and)37 b(w)n(e)f(will)i(not)f(attempt)g(to)g (review)f(all)h(of)g(these)g(here.)f(Instead,)h(w)n(e)g(refer)f(to)h (the)523 3330 y(o)n(v)n(erview)k(article)i(of)h(de)f(Sc)n(hrey)n(e)f (and)i(Decorte)f([SD94)o(].)h(More)e(recen)n(t)h(approac)n(hes)523 3430 y(to)36 b(pro)n(ving)f(termination)i(of)f(logic)g(programs)e(are)i (discussed)g(in)g(Krishna)g(Rao)f(et)i(al.)523 3530 y([KRKS98)o(].)28 b(W)-7 b(e)28 b(will)h(only)f(brie\015y)g(commen)n(t)g(on)f(other)h (transformational)e(tec)n(hniques.)523 3629 y(T)-7 b(o)37 b(the)h(b)r(est)g(of)g(our)e(kno)n(wledge,)h(the)h(\014rst)f (termination)g(pro)r(of)g(tec)n(hnique)h(for)f(w)n(ell-)523 3729 y(mo)r(ded)26 b(logic)f(programs)f(whic)n(h)i(uses)g(a)f (transformation)f(of)i(logic)g(programs)d(in)n(to)j(TRSs)523 3828 y(w)n(as)g(describ)r(ed)g(b)n(y)h(Krishna)f(Rao)g(et)h(al.)f ([KRKS92)o(].)h(Ho)n(w)n(ev)n(er,)e(Ganzinger)g(and)i(W)-7 b(ald-)523 3928 y(mann's)25 b(metho)r(d)g(is)f(not)h(only)f (conceptually)g(easier)g(but)h(it)g(is)g(also)e(able)i(to)f(pro)n(v)n (e)f(termi-)523 4028 y(nation)28 b(of)g(logic)g(programs)d(for)j(whic)n (h)g(the)h(metho)r(d)g(in)f([KRKS92)o(])g(fails.)g(An)h(approac)n(h)523 4127 y(similar)k(to)g(that)g(of)g(Arts)h(and)f(Zan)n(tema)f(w)n(as)g (suggested)g(b)n(y)h(Aguzzi)g(and)g(Mo)r(digliani)523 4227 y([AM93)o(].)k(In)g(con)n(trast)e(to)i(the)f(other)g(tec)n (hniques,)h(ho)n(w)n(ev)n(er,)d(their)j(metho)r(d)g(do)r(es)f(not)523 4327 y(require)24 b(an)n(y)h(prior)g(information)f(ab)r(out)i(mo)r(des) f(of)h(predicates)e(b)r(ecause)i(these)f(are)g(com-)523 4426 y(puted)36 b(during)e(the)h(transformation)e(according)h(to)g(a)h (giv)n(en)f(query)-7 b(.)34 b(The)h(transforma-)523 4526 y(tional)30 b(approac)n(h)e(of)i(Marc)n(hiori)e([Mar94)n(])i(is)g (rather)f(complex)h(but)h(it)f(can)g(handle)g(logic)523 4625 y(programs)16 b(for)i(whic)n(h)h(the)g(metho)r(ds)g(describ)r(ed)f (ab)r(o)n(v)n(e)f(fail.)i(It)g(is)f(sound)h(and)f(complete)h(for)523 4725 y(t)n(w)n(o)k(sub)r(classes)f(of)i(the)g(class)f(of)g(w)n(ell-mo)r (ded)g(logic)g(programs,)e(viz.)j(simply)f(w)n(ell-mo)r(ded)523 4825 y(and)34 b(\015atly)g(w)n(ell-mo)r(ded)g(programs.)e(Lastly)-7 b(,)34 b(another)f(translation)g(of)h(logic)f(programs)523 4924 y(in)n(to)27 b(conditional)g(rewrite)g(systems)g(can)g(b)r(e)h (found)g(in)g(v)-5 b(an)28 b(Raamsdonk)e([Raa97)n(].)p eop %%Page: 19 19 19 18 bop 523 548 a Fi(Ac)m(kno)m(wledgmen)m(t:)38 b Fw(I)h(thank)g(Mic)n(hael)g(Han)n(us)f(for)h(the)g(\(email\))h (discussion)e(whic)n(h)523 648 y(led)33 b(to)f(the)h(dev)n(elopmen)n(t) f(of)h(the)g(transformation)e Fn(U)9 b Fw(.)33 b(He)g(also)e(p)r(oin)n (ted)i(out)g(that)g(the)523 747 y(elimination)24 b(of)g(lo)r(cal)f (de\014nitions)h(via)f(program)f(transformations)f(is)j(w)n(ell-kno)n (wn)f(in)h(the)523 847 y(implemen)n(tation)k(of)f(functional)h (programming)d(languages;)h(see)h([Joh85)o(].)523 1106 y Fq(References)523 1289 y Fv([A)n(G97a])86 b(T.)18 b(Arts)f(and)g(J.)h (Giesl.)23 b(Automatically)17 b(pro)n(ving)g(termination)g(where)h (simpli\014cation)881 1380 y(orderings)23 b(fail.)30 b(In)22 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(the)h(22nd)g (International)g(Col)t(lo)l(quium)f(on)g(T)-6 b(r)l(e)l(es)881 1472 y(in)34 b(A)n(lgebr)l(a)i(and)f(Pr)l(o)l(gr)l(amming)p Fv(,)f(v)n(olume)e(1214)j(of)f Fb(L)l(e)l(ctur)l(e)j(Notes)f(in)e (Computer)881 1563 y(Scienc)l(e)p Fv(,)27 b(pages)g(261{272,)h(Berlin,) f(1997.)h(Springer-V)-6 b(erlag.)523 1653 y([A)n(G97b])81 b(T.)33 b(Arts)f(and)g(J.)g(Giesl.)55 b(Pro)n(ving)32 b(innermost)g(normalisation)h(automatically)-6 b(.)54 b(In)881 1744 y Fb(Pr)l(o)l(c)l(e)l(e)l(dings)40 b(of)d(the)h(8th)g (International)h(Confer)l(enc)l(e)g(on)e(R)l(ewriting)h(T)-6 b(e)l(chniques)881 1835 y(and)25 b(Applic)l(ations)p Fv(,)f(v)n(olume)e(1232)i(of)f Fb(L)l(e)l(ctur)l(e)k(Notes)f(in)e (Computer)i(Scienc)l(e)p Fv(,)e(pages)881 1927 y(157{172,)29 b(Berlin,)e(1997.)g(Springer-V)-6 b(erlag.)523 2016 y([ALS94])91 b(J.)30 b(Av)n(enhaus)e(and)g(C.)i(Lor)-9 b(\023)-30 b(\020a-S\023)-38 b(aenz.)45 b(On)28 b(conditional)i(rewrite)g(systems) f(with)g(ex-)881 2108 y(tra)34 b(v)l(ariables)h(and)f(deterministic)f (logic)j(programs.)59 b(In)33 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e (the)h(5th)881 2199 y(International)26 b(Confer)l(enc)l(e)h(on)e(L)l(o) l(gic)g(Pr)l(o)l(gr)l(amming)h(and)g(A)n(utomate)l(d)g(R)l(e)l(asoning) p Fv(,)881 2290 y(v)n(olume)20 b(822)h(of)h Fb(L)l(e)l(ctur)l(e)i (Notes)g(in)f(A)n(rti\014cial)g(Intel)t(ligenc)l(e)p Fv(,)e(pages)g(215{229,)j(Berlin,)881 2382 y(1994.)k(Springer-V)-6 b(erlag.)523 2471 y([AM93])112 b(G.)35 b(Aguzzi)g(and)f(U.)g(Mo)r (digliani.)63 b(Pro)n(ving)34 b(termination)g(of)h(logic)h(programs)f (b)n(y)881 2563 y(transforming)c(them)e(in)n(to)h(equiv)l(alen)n(t)g (term)f(rewriting)i(systems.)48 b(In)30 b Fb(Pr)l(o)l(c)l(e)l(e)l (dings)881 2654 y(of)21 b(the)h(13th)g(Confer)l(enc)l(e)h(on)e(the)h(F) -6 b(oundations)23 b(of)d(Softwar)l(e)j(T)-6 b(e)l(chnolo)l(gy)22 b(and)g(The)l(o-)881 2745 y(r)l(etic)l(al)h(Computer)h(Scienc)l(e)p Fv(,)d(v)n(olume)d(761)j(of)g Fb(L)l(e)l(ctur)l(e)j(Notes)g(in)d (Computer)j(Scienc)l(e)p Fv(,)881 2837 y(pages)j(114{124,)i(Berlin,)d (1993.)i(Springer-V)-6 b(erlag.)523 2926 y([Art97])122 b(T.)27 b(Arts.)36 b Fb(A)n(utomatic)l(al)t(ly)29 b(Pr)l(oving)g(T)-6 b(ermination)29 b(and)f(Innermost)h(Normalisation)881 3018 y(of)e(T)-6 b(erm)28 b(R)l(ewriting)g(Systems)p Fv(.)36 b(PhD)25 b(thesis,)i(Utrec)n(h)n(t)e(Univ)n(ersit)n(y,)g(1997.) 523 3107 y([AZ95])135 b(T.)30 b(Arts)e(and)g(H.)g(Zan)n(tema.)42 b(T)-6 b(ermination)29 b(of)g(logic)h(programs)f(via)f(lab)r(elled)i (term)881 3199 y(rewrite)h(systems.)47 b(In)29 b Fb(Pr)l(o)l(c)l(e)l(e) l(dings)34 b(of)d(Computing)i(Scienc)l(e)f(in)f(the)h(Netherlands)p Fv(,)881 3290 y(pages)27 b(22{34,)h(1995.)523 3380 y([AZ96])135 b(T.)23 b(Arts)e(and)g(H.)h(Zan)n(tema.)27 b(T)-6 b(ermination)21 b(of)i(logic)g(programs)f(using)f(seman)n(tic)h(uni-)881 3471 y(\014cation.)27 b(In)20 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)25 b(of)d(the)i(5th)f(International)h(Workshop)g(on)f(L)l(o)l(gic)h(Pr)l (o)l(gr)l(am)881 3563 y(Synthesis)33 b(and)e(T)-6 b(r)l(ansformation)p Fv(,)31 b(v)n(olume)d(1048)j(of)f Fb(L)l(e)l(ctur)l(e)j(Notes)f(in)f (Computer)881 3654 y(Scienc)l(e)p Fv(,)c(pages)g(219{233,)h(Berlin,)f (1996.)h(Springer-V)-6 b(erlag.)523 3744 y([BG89])126 b(H.)18 b(Bertling)i(and)e(H.)g(Ganzinger.)23 b(Completion-time)17 b(optimization)i(of)g(rewrite-time)881 3835 y(goal)27 b(solving.)35 b(In)24 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)30 b(of)d(the)h(3r)l(d)f(International)i(Confer)l(enc)l(e)f(on)g(R)l (ewrit-)881 3926 y(ing)f(T)-6 b(e)l(chniques)28 b(and)g(Applic)l (ations)p Fv(,)e(v)n(olume)d(355)j(of)g Fb(L)l(e)l(ctur)l(e)j(Notes)f (in)f(Computer)881 4018 y(Scienc)l(e)p Fv(,)g(pages)g(45{58,)h(Berlin,) e(1989.)i(Springer-V)-6 b(erlag.)523 4107 y([BK86])126 b(J.A.)22 b(Bergstra)g(and)e(J.W.)i(Klop.)27 b(Conditional)22 b(rewrite)g(rules:)f(Con\015uence)g(and)f(ter-)881 4199 y(mination.)34 b Fb(Journal)29 b(of)e(Computer)i(and)f(System)g(Scienc) l(es)p Fv(,)g(32\(3\):323{362,)i(1986.)523 4288 y([BN98])128 b(F.)28 b(Baader)g(and)f(T.)h(Nipk)n(o)n(w.)38 b Fb(T)-6 b(erm)29 b(R)l(ewriting)h(and)f(A)n(l)t(l)f(That)p Fv(.)40 b(Cam)n(bridge)27 b(Uni-)881 4380 y(v)n(ersit)n(y)e(Press,)i(1998.)523 4469 y([DOS88])78 b(N.)22 b(Dersho)n(witz,)h(M.)f(Ok)l(ada,)g(and)g(G.) g(Siv)l(akumar.)k(Canonical)e(conditional)f(rewrite)881 4561 y(systems.)64 b(In)35 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)d(the)h (9th)h(Confer)l(enc)l(e)g(on)f(A)n(utomate)l(d)h(De)l(duction)p Fv(,)881 4652 y(v)n(olume)30 b(310)i(of)g Fb(L)l(e)l(ctur)l(e)i(Notes)g (in)d(Computer)j(Scienc)l(e)p Fv(,)e(pages)g(538{549,)i(Berlin,)881 4743 y(1988.)28 b(Springer-V)-6 b(erlag.)523 4833 y([Gan91])99 b(H.)27 b(Ganzinger.)38 b(Order-sorted)26 b(completion:)h(The)g(man)n (y-sorted)e(w)n(a)n(y)-6 b(.)36 b Fb(The)l(or)l(etic)l(al)881 4924 y(Computer)29 b(Scienc)l(e)p Fv(,)e(89:3{32,)i(1991.)p eop %%Page: 20 20 20 19 bop 523 448 a Fv([GM88])110 b(E.)32 b(Gio)n(v)l(annetti)g(and)f (C.)i(Moiso.)54 b(Notes)31 b(on)h(the)f(elimination)h(of)h(conditions.) 53 b(In)881 540 y Fb(Pr)l(o)l(c)l(e)l(e)l(dings)39 b(of)d(the)h(1st)g (International)h(Workshop)f(on)g(Conditional)f(and)h(T)-6 b(yp)l(e)l(d)881 631 y(R)l(ewriting)24 b(Systems)p Fv(,)f(v)n(olume)c (308)j(of)g Fb(L)l(e)l(ctur)l(e)j(Notes)g(in)e(Computer)h(Scienc)l(e)p Fv(,)e(pages)881 722 y(91{97,)28 b(Berlin,)f(1988.)h(Springer-V)-6 b(erlag.)523 809 y([Gra95])112 b(B.)35 b(Gramlic)n(h.)58 b(Abstract)33 b(relations)i(b)r(et)n(w)n(een)f(restricted)g (termination)f(and)g(con-)881 900 y(\015uence)f(prop)r(erties)h(of)g (rewrite)g(systems.)54 b Fb(F)-6 b(undamenta)35 b(Informatic)l(ae)p Fv(,)e(24:3{23,)881 991 y(1995.)523 1077 y([GW93])101 b(H.)32 b(Ganzinger)g(and)f(U.)h(W)-6 b(aldmann.)50 b(T)-6 b(ermination)32 b(pro)r(ofs)h(of)f(w)n(ell-mo)r(ded)f(logic)881 1169 y(programs)e(via)g(conditional)h(rewrite)g(systems.)43 b(In)28 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)33 b(of)d(the)h(3r)l(d)h (Inter-)881 1260 y(national)c(Workshop)h(on)f(Conditional)f(T)-6 b(erm)28 b(R)l(ewriting)g(Systems)p Fv(,)g(v)n(olume)c(656)j(of)881 1351 y Fb(L)l(e)l(ctur)l(e)k(Notes)f(in)e(Computer)i(Scienc)l(e)p Fv(,)e(pages)f(113{127,)j(Berlin,)e(1993.)g(Springer-)881 1443 y(V)-6 b(erlag.)523 1529 y([Han94])101 b(M.)20 b(Han)n(us.)i(The)d (in)n(tegration)h(of)g(functions)f(in)n(to)g(logic)h(programming:)e(F) -6 b(rom)18 b(theory)881 1620 y(to)26 b(practice.)35 b Fb(The)28 b(Journal)g(of)g(L)l(o)l(gic)g(Pr)l(o)l(gr)l(amming)p Fv(,)f(19&20:583{628,)j(1994.)523 1707 y([Hin95])118 b(C.)28 b(Hin)n(termeier.)37 b(Ho)n(w)26 b(to)h(transform)g(canonical)h (decreasing)g(CTRSs)f(in)n(to)g(equiv-)881 1798 y(alen)n(t)e(canonical) h(TRSs.)33 b(In)24 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)29 b(of)e(the)g(4r)l(d)h(International)g(Workshop)g(on)881 1889 y(Conditional)j(and)g(T)-6 b(yp)l(e)l(d)31 b(R)l(ewriting)g (Systems)p Fv(,)g(v)n(olume)d(968)i(of)g Fb(L)l(e)l(ctur)l(e)i(Notes)g (in)881 1981 y(Computer)d(Scienc)l(e)p Fv(,)e(pages)f(186{205,)j (Berlin,)e(1995.)h(Springer-V)-6 b(erlag.)523 2067 y([Joh85])120 b(T.)22 b(Johnsson.)28 b(Lam)n(b)r(da)21 b(lifting:)i(T)-6 b(ransforming)22 b(programs)f(to)h(recursiv)n(e)f(functions.)881 2158 y(In)26 b Fb(F)-6 b(unctional)29 b(Pr)l(o)l(gr)l(amming)g(L)l (anguages)h(and)f(Computer)g(A)n(r)l(chite)l(ctur)l(e)p Fv(,)h(v)n(olume)881 2249 y(201)39 b(of)f Fb(L)l(e)l(ctur)l(e)j(Notes)e (in)f(Computer)i(Scienc)l(e)p Fv(,)f(pages)f(190{203,)i(Berlin,)f (1985.)881 2341 y(Springer-V)-6 b(erlag.)523 2427 y([KRKS92])42 b(M.R.K.)29 b(Krishna)g(Rao,)h(D.)f(Kapur,)f(and)h(R.K.)g(Sh)n(y)n (amasundar.)42 b(A)29 b(transforma-)881 2518 y(tional)h(metho)r(dology) e(for)h(pro)n(ving)f(termination)g(of)h(logic)h(programs.)43 b(In)27 b Fb(Pr)l(o)l(c)l(e)l(e)l(d-)881 2610 y(ings)e(of)e(the)i(5th)g (Workshop)h(on)e(Computer)h(Scienc)l(e)g(L)l(o)l(gic)p Fv(,)e(v)n(olume)e(626)i(of)g Fb(L)l(e)l(ctur)l(e)881 2701 y(Notes)29 b(in)e(Computer)i(Scienc)l(e)p Fv(,)e(pages)f(213{226,) j(Berlin,)e(1992.)g(Springer-V)-6 b(erlag.)523 2787 y([KRKS98])42 b(M.R.K.)18 b(Krishna)g(Rao,)h(D.)f(Kapur,)g(and)f(R.K.)h(Sh)n(y)n (amasundar.)i(T)-6 b(ransformational)881 2879 y(metho)r(dology)18 b(for)g(pro)n(ving)g(termination)f(of)h(logic)i(programs.)h Fb(The)g(Journal)g(of)e(L)l(o)l(gic)881 2970 y(Pr)l(o)l(gr)l(amming)p Fv(,)27 b(34\(1\):1{42,)i(1998.)523 3056 y([Mar94])102 b(M.)33 b(Marc)n(hiori.)55 b(Logic)33 b(programs)f(as)h(term)e (rewriting)i(systems.)53 b(In)32 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)881 3147 y(of)h(the)h(4th)g(International)g(Confer)l(enc)l(e)h(on)e(A)n (lgebr)l(aic)h(and)g(L)l(o)l(gic)g(Pr)l(o)l(gr)l(amming)p Fv(,)881 3239 y(v)n(olume)c(850)i(of)g Fb(L)l(e)l(ctur)l(e)i(Notes)g (in)d(Computer)j(Scienc)l(e)p Fv(,)e(pages)g(223{241,)i(Berlin,)881 3330 y(1994.)28 b(Springer-V)-6 b(erlag.)523 3416 y([Mar95])102 b(M.)27 b(Marc)n(hiori.)38 b(Unra)n(v)n(elings)26 b(and)g(ultra-prop)r (erties.)37 b(T)-6 b(ec)n(hnical)26 b(Rep)r(ort)g(8,)h(Dept.)881 3508 y(of)g(Pure)e(and)h(Applied)f(Mathematics,)h(Univ)n(ersit)n(y)f (of)h(P)n(ado)n(v)l(a,)g(Italy)-6 b(,)26 b(1995.)523 3594 y([Mar96])102 b(M.)31 b(Marc)n(hiori.)48 b(Unra)n(v)n(elings)30 b(and)f(ultra-prop)r(erties.)47 b(In)29 b Fb(Pr)l(o)l(c)l(e)l(e)l (dings)34 b(of)d(the)h(5th)881 3685 y(International)24 b(Confer)l(enc)l(e)f(on)f(A)n(lgebr)l(aic)h(and)g(L)l(o)l(gic)g(Pr)l(o) l(gr)l(amming)p Fv(,)e(v)n(olume)d(1139)881 3777 y(of)h Fb(L)l(e)l(ctur)l(e)j(Notes)f(in)f(Computer)h(Scienc)l(e)p Fv(,)e(pages)g(107{121,)h(Berlin,)f(1996.)g(Springer-)881 3868 y(V)-6 b(erlag.)523 3954 y([Mar97])102 b(M.)29 b(Marc)n(hiori.)44 b(On)27 b(the)h(expressiv)n(e)h(p)r(o)n(w)n(er)f(of)h(rewriting.)44 b(In)27 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)32 b(of)e(the)881 4045 y(17th)35 b(International)g(Confer)l(enc)l(e)g(on)f(the)h(F)-6 b(oundations)35 b(of)e(Softwar)l(e)i(T)-6 b(e)l(chnolo)l(gy)881 4137 y(and)32 b(The)l(or)l(etic)l(al)g(Computer)g(Scienc)l(e)p Fv(,)e(v)n(olume)e(1346)j(of)f Fb(L)l(e)l(ctur)l(e)j(Notes)f(in)f(Com-) 881 4228 y(puter)e(Scienc)l(e)p Fv(,)e(pages)g(88{102,)h(Berlin,)f (1997.)g(Springer-V)-6 b(erlag.)523 4314 y([Ohl95])116 b(E.)32 b(Ohlebusc)n(h.)50 b(Mo)r(dular)32 b(prop)r(erties)g(of)g(comp) r(osable)g(term)e(rewriting)i(systems.)881 4406 y Fb(Journal)d(of)e (Symb)l(olic)g(Computation)p Fv(,)h(20:1{41,)g(1995.)523 4492 y([Raa97])107 b(F.)32 b(v)l(an)e(Raamsdonk.)50 b(T)-6 b(ranslating)33 b(logic)g(programs)e(in)n(to)g(conditional)i(rewriting) 881 4583 y(systems.)g(In)25 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(the) g(14th)h(International)h(Confer)l(enc)l(e)f(on)f(L)l(o)l(gic)h(Pr)l(o-) 881 4675 y(gr)l(amming)p Fv(,)f(pages)f(168{182.)j(MIT)d(Press,)h (1997.)523 4761 y([SD94])138 b(D.)30 b(de)g(Sc)n(hrey)n(e)f(and)h(S.)f (Decorte.)48 b(T)-6 b(ermination)30 b(of)h(logic)g(programs:)g(The)f (nev)n(er-)881 4852 y(ending)c(story)-6 b(.)34 b Fb(The)28 b(Journal)g(of)f(L)l(o)l(gic)h(Pr)l(o)l(gr)l(amming)p Fv(,)f(19&20:199{260,)k(1994.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF