%!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: Report99-02.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -o Report99-02.ps Report99-02.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.03.04:0637 %%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 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5066 y(with)i(termination.)g(Since)g Fo(U)10 b Fq(\()p Fn(R)p Fq(\))27 b(inherits)e(non-o)m(v)m(erlappingness)g(from)i(a)g (syn)m(tactically)g(deterministic)75 5179 y(CTRS)39 b Fn(R)p Fq(,)i(termination)e(of)i Fo(U)10 b Fq(\()p Fn(R)p Fq(\))41 b(and)f(quasi-reductivit)m(y)f(of)i Fn(R)g Fq(are)g(equiv)-5 b(alen)m(t)40 b(for)g(this)f(class)i(of)75 5292 y(CTRSs.)31 b(An)g(immediate)g(consequence)i(is)e(that)i(quasi-reductivit)m(y)d(is) h(mo)s(dular)f(for)i(non-o)m(v)m(erlapping)75 5405 y(syn)m(tactically) 26 b(deterministic)e(comp)s(osable)i(CTRSs.)f(Under)g(certain)h (\(natural\))g(conditions,)f(it)g(is)g(mo)s(d-)75 5518 y(ular)j(ev)m(en)i(for)f(hierarc)m(hical)e(com)m(binations)i(of)g (these)h(systems.)f(T)-8 b(o)30 b(the)f(b)s(est)g(of)g(our)g(kno)m (wledge,)g(these)1890 5767 y(2)p eop %%Page: 3 3 3 2 bop 75 324 a Fq(are)31 b(the)f(\014rst)g(non-trivial)e(mo)s (dularit)m(y)h(results)g(for)h(CTRSs)f(that)i(ha)m(v)m(e)g(extra)g(v)-5 b(ariables)29 b(on)i(the)f(righ)m(t-)75 437 y(hand)f(sides)g(of)i(the)g (rules.)216 551 y(There)d(is)e(another)j(imp)s(ortan)m(t)e(application) f(of)i(our)f(main)g(result.)g(It)h(has)g(already)f(b)s(een)g(men)m (tioned)75 664 y(that)41 b(a)g(w)m(ell-mo)s(ded)d(logic)j(program)f Fn(P)48 b Fq(is)39 b(terminating)g(whenev)m(er)h(it)g(can)h(b)s(e)f (transformed)f(in)m(to)i(a)75 776 y(quasi-reductiv)m(e)32 b(deterministic)f(CTRS)h Fo(R)1607 790 y Fj(P)1666 776 y Fq(.)h(Using)f(an)h(imp)s(erativ)m(e)f(pro)s(cedure,)h(Arts)g(and)f (Zan)m(tema)75 889 y([AZ95)q(,AZ96)q(])41 b(transformed)e(a)i(logic)g (program)f Fn(P)48 b Fq(directly)39 b(in)m(to)i(an)g Fk(unc)-5 b(onditional)42 b Fq(TRS)e(whic)m(h)f(in)75 1002 y(essence)c(coincides)e(with)f Fo(U)10 b Fq(\()p Fo(R)1167 1016 y Fj(P)1227 1002 y Fq(\).)34 b(They)g(pro)m(v)m(ed)g (that)g(single-redex)f(termination)g(of)h Fo(U)10 b Fq(\()p Fo(R)3343 1016 y Fj(P)3402 1002 y Fq(\))35 b(implies)75 1115 y(termination)c(of)i Fn(P)7 b Fq(.)33 b(Although)f Fo(U)10 b Fq(\()p Fo(R)1386 1129 y 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1629 y(quence)23 b(of)g Fo(s)507 1643 y Fg(j)543 1629 y Fo(\033)s Fn(!)689 1596 y Fj(\003)689 1656 y(R)784 1629 y Fo(t)817 1643 y Fg(j)853 1629 y Fo(\033)s Fq(,)g(where)f(1)k Fn(\024)f Fo(j)31 b Fn(\024)25 b Fo(i)p Fq(.)e(Since)f(\()p Fn(F)10 b Fo(;)15 b Fn(R)p Fq(\))23 b(is)f(quasi-reductiv)m(e,)g(it)g(follo)m (ws)g Fo(l)r(\033)28 b(>)3421 1643 y Fg(st)3509 1629 y Fo(s)3552 1643 y Fg(i)p Fl(+1)3670 1629 y Fo(\033)s Fq(.)75 1742 y(Finally)-8 b(,)29 b(w)m(e)i(obtain)f Fo(l)r(\033)e Fn(\037)d Fo(s)1067 1756 y Fg(i)p Fl(+1)1185 1742 y Fo(\033)33 b Fq(from)d Fo(l)r(\033)n(;)15 b(s)1647 1756 y Fg(i)p Fl(+1)1766 1742 y Fo(\033)28 b Fn(2)d(T)e Fq(\()p Fn(F)9 b Fo(;)15 b Fn(V)7 b Fq(\))q(.)75 1942 y(W)-8 b(e)35 b(do)f(not)g(kno)m(w)f(whether)h(the)g(con)m(v)m(erse)h(of)f(the)g (preceding)e(lemma)i(holds)e(true)i(as)g(w)m(ell.)f(Ho)m(w)m(ev)m(er,) 75 2055 y(quasi-decreasingness)25 b(is)h(b)s(etter)g(to)i(handle)d(b)s (ecause)h(it)g(has)g(t)m(w)m(o)i(adv)-5 b(an)m(tages)29 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Fq(\))26 b(=)f Fn(f)p Fo(f)10 b Fq(\()p Fo(x;)15 b(y)s(;)g(y)s Fq(\))26 b Fn(!)f Fo(U)10 b Fq(\()p Fo(x)18 b Fq(+)g Fo(y)s(;)d(x;)g(y)s Fq(\),)75 3419 y Fo(U)10 b Fq(\()p Fo(z)25 b Fq(+)20 b Fo(z)t(;)15 b(x;)g(y)s Fq(\))26 b Fn(!)f Fo(y)e Fq(+)d Fo(z)t Fn(g)p Fq(.)32 b(The)d(transformation)h(of)h(the)f(quic)m(ksort) g(system)h(is)1360 3597 y Fo(spl)r(it)p Fq(\()p Fo(x;)15 b Fq([)30 b(]\))24 b Fn(!)f Fq(\([)31 b(])p Fo(;)15 b Fq([)32 b(]\))1227 3710 y Fo(spl)r(it)p Fq(\()p Fo(x;)15 b(y)28 b Fq(:)d Fo(y)s(s)p Fq(\))e Fn(!)g Fo(U)1994 3677 y Fj(0)1984 3734 y Fl(1)2024 3710 y Fq(\()p Fo(spl)r(it)p Fq(\()p Fo(x;)15 b(y)s(s)p Fq(\))p Fo(;)g(x;)g(y)s(;)g(y)s(s)p Fq(\))1007 3823 y Fo(U)1079 3790 y Fj(0)1069 3847 y Fl(1)1108 3823 y Fq(\(\()p Fo(xs;)g(z)t(s)p Fq(\))p Fo(;)g(x;)g(y)s(;)g(y)s(s)p Fq(\))25 b Fn(!)e Fo(U)1994 3790 y Fj(0)1984 3847 y Fl(2)2024 3823 y Fq(\()p Fo(x)j Fn(\024)e Fo(y)s(;)15 b(x;)g(y)s(;)g(y)s(s;)g (xs;)g(z)t(s)p Fq(\))866 3936 y Fo(U)938 3903 y Fj(0)928 3960 y Fl(2)968 3936 y Fq(\()p Fo(tr)s(ue;)g(x;)g(y)s(;)g(y)s(s;)g(xs;) g(z)t(s)p Fq(\))24 b Fn(!)f Fq(\()p Fo(xs;)15 b(y)29 b Fq(:)c Fo(z)t(s)p Fq(\))1227 4049 y Fo(spl)r(it)p Fq(\()p Fo(x;)15 b(y)28 b Fq(:)d Fo(y)s(s)p Fq(\))e Fn(!)g Fo(U)1994 4016 y Fj(0)q(0)1984 4073 y Fl(1)2037 4049 y Fq(\()p Fo(spl)r(it)p Fq(\()p Fo(x;)15 b(y)s(s)p Fq(\))p Fo(;)g(x;)g(y)s(;)g(y) s(s)p Fq(\))994 4161 y Fo(U)1066 4128 y Fj(00)1056 4186 y Fl(1)1108 4161 y Fq(\(\()p Fo(xs;)g(z)t(s)p Fq(\))p Fo(;)g(x;)g(y)s(;)g(y)s(s)p Fq(\))25 b Fn(!)e Fo(U)1994 4128 y Fj(0)q(0)1984 4186 y Fl(2)2037 4161 y Fq(\()p Fo(x)i Fn(\024)g Fo(y)s(;)15 b(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t(s)p Fq(\))808 4274 y Fo(U)880 4241 y Fj(00)870 4299 y Fl(2)922 4274 y Fq(\()p Fo(f)10 b(al)r(se;)15 b(x;)g(y)s(;)g(y)s(s;)g(xs;)g(z)t (s)p Fq(\))24 b Fn(!)f Fq(\()p Fo(y)29 b Fq(:)c Fo(xs;)15 b(z)t(s)p Fq(\))1427 4387 y Fo(q)s(sor)s(t)p Fq(\([)29 b(]\))24 b Fn(!)f Fq([)31 b(])1285 4500 y Fo(q)s(sor)s(t)p Fq(\()p Fo(x)25 b Fq(:)g Fo(xs)p Fq(\))e Fn(!)g Fo(U)1994 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Fo(\033)36 b Fq(yields)30 b Fo(s)1735 4779 y Fl(1)1774 4765 y Fo(\033)1826 4779 y Fj(r)1918 4765 y Fn(!)2009 4732 y Fj(\003)2009 4792 y(R)2102 4765 y Fo(t)2135 4779 y Fl(1)2175 4765 y Fo(\033)2227 4779 y Fj(r)2349 4713 y Fg(i)g Fj(\003)2333 4765 y Fn(!)2415 4783 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))2618 4765 y Fo(t)2651 4779 y Fl(1)2690 4765 y Fo(\033)36 b Fq(b)m(y)c(Lemma)h(26.)g(It)g (follo)m(ws)75 4878 y Fo(l)r(\033)156 4892 y Fj(r)248 4878 y Fn(\037)d Fo(s)392 4892 y Fl(2)431 4878 y Fo(\033)483 4892 y Fj(r)578 4878 y Fq(b)s(ecause)j Fn(R)g Fq(is)f(quasi-decreasing) g(w.r.t.)h Fn(\037)g Fq(and)f(th)m(us)h(w)m(e)g(ma)m(y)h(con)m(tin)m (ue)f(as)g(ab)s(o)m(v)m(e.)h(All)75 4991 y(in)29 b(all,)h Fo(D)j Fq(m)m(ust)d(ha)m(v)m(e)i(the)e(form)176 5182 y Fo(l)r(\033)257 5196 y Fj(r)376 5130 y Fg(i)g Fj(\003)360 5182 y Fn(!)442 5200 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))641 5182 y Fo(l)r(\033)784 5130 y Fg(i)750 5182 y Fn(!)841 5200 y Fg(U)g Fl(\()p Fj(R)p Fl(\))1041 5182 y Fo(U)1113 5137 y Fg(\032)1103 5208 y Fl(1)1153 5182 y Fq(\()p Fo(s)1231 5196 y Fl(1)1270 5182 y Fo(;)15 b Fn(V)7 b Fo(ar)s Fq(\()p Fo(l)r Fq(\))q(\))p Fo(\033)1711 5130 y Fg(i)31 b Fj(\003)1695 5182 y Fn(!)1777 5200 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))1977 5182 y Fo(U)2049 5137 y Fg(\032)2039 5215 y Fj(j)p Fg(\032)p Fj(j)2118 5182 y Fq(\()p Fo(t)2186 5200 y Fj(j)p Fg(\032)p Fj(j)2266 5182 y Fo(;)15 b Fn(V)7 b Fo(ar)s Fq(\()p Fo(l)r Fq(\))q Fo(;)15 b(:)g(:)g(:)h Fq(\))p Fo(\033)2872 5130 y Fg(i)2838 5182 y Fn(!)2929 5200 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))3128 5182 y Fo(r)s(\033)3286 5130 y Fg(i)3252 5182 y Fn(!)3343 5200 y Fg(U)g Fl(\()p Fj(R)p Fl(\))3543 5182 y Fo(:)15 b(:)g(:)75 5405 y Fq(Hence)29 b Fo(l)r(\033)424 5419 y Fj(r)512 5405 y Fn(!)603 5419 y Fj(R)692 5405 y Fo(r)s(\033)788 5419 y Fj(r)906 5353 y Fg(i)i Fj(\003)890 5405 y Fn(!)972 5423 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))1172 5405 y Fo(r)s(\033)30 b Fq(b)m(y)e(Lemma)g(26.)h(W)-8 b(e)29 b(conclude)e(that)h(there)g(is)f(an)h(in\014nite)d(innermost)75 5518 y Fo(U)10 b Fq(\()p Fn(R)p Fq(\)-deriv)-5 b(ation)30 b(starting)g(from)g Fo(r)s(\033)1402 5532 y Fj(r)1495 5518 y Fq(whic)m(h)f(is)g(imp)s(ossible)e(b)s(ecause)k Fo(s)25 b Fq(=)f Fo(l)r(\033)2870 5532 y Fj(r)2959 5518 y Fn(\037)h Fo(r)s(\033)3151 5532 y Fj(r)3213 5518 y Fq(.)1867 5767 y(13)p eop %%Page: 14 14 14 13 bop 75 324 a Ff(Theorem)34 b(28.)46 b Fk(If)e Fn(R)g Fk(is)g(a)h(quasi-de)-5 b(cr)g(e)g(asing)45 b(deterministic)g(3-CTRS,)g (then)g Fo(U)10 b Fq(\()p Fn(R)p Fq(\))44 b Fk(is)h(innermost)75 437 y(terminating.)75 651 y(Pr)-5 b(o)g(of.)47 b Fq(W)-8 b(e)32 b(pro)m(v)m(e)h(b)m(y)e(structural)f(induction)f(that)j(there)g (is)e(no)h(in\014nite)e(innermost)h Fo(U)10 b Fq(\()p Fn(R)q Fq(\)-deriv)-5 b(ation)75 764 y(starting)27 b(from)g(a)g(term)h Fo(s)c Fn(2)h(T)e Fq(\()p Fn(F)1246 729 y Fj(0)1269 764 y Fo(;)15 b Fn(V)8 b Fq(\).)27 b(If)g(\(a\))h Fo(s)f Fq(is)f(a)i(v)-5 b(ariable,)26 b(then)h(it)g(is)f(in)g(normal)g(form.)h (If)g(\(b\))h Fo(s)e Fq(is)h(a)75 877 y(constan)m(t,)j(then)e Fo(s)d Fn(2)g(F)37 b Fq(and)28 b(the)g(claim)g(follo)m(ws)f(from)h (Lemma)h(27.)g(Supp)s(ose)d(\(c\))k Fo(s)25 b Fq(=)g Fo(U)3202 833 y Fg(\032)3192 904 y(i)3242 877 y Fq(\()p Fo(u)3329 891 y Fl(1)3369 877 y Fo(;)15 b(:)g(:)g(:)h(;)f(u)3622 891 y Fg(m)3689 877 y Fq(\).)75 990 y(Ev)m(ery)31 b(in\014nite)d (innermost)h Fo(U)10 b Fq(\()p Fn(R)p Fq(\)-deriv)-5 b(ation)30 b(starting)g(from)g Fo(s)g Fq(m)m(ust)g(ha)m(v)m(e)i(the)e (form)569 1225 y Fo(D)e Fq(:)d Fo(s)g Fq(=)g Fo(U)958 1181 y Fg(\032)948 1252 y(i)998 1225 y Fq(\()p Fo(u)1085 1239 y Fl(1)1125 1225 y Fo(;)15 b(:)g(:)g(:)i(;)e(u)1379 1239 y Fg(m)1446 1225 y Fq(\))1535 1173 y Fg(i)30 b Fj(\003)1519 1225 y Fn(!)1601 1243 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))1800 1225 y Fo(U)1872 1181 y Fg(\032)1862 1252 y(i)1913 1225 y Fq(\()p Fo(v)1992 1239 y Fl(1)2031 1225 y Fo(;)15 b(:)g(:)g(:)i(;)e (v)2277 1239 y Fg(m)2344 1225 y Fq(\))26 b(=)f Fo(U)2573 1181 y Fg(\032)2563 1252 y(i)2613 1225 y Fq(\()p Fo(t)2681 1239 y Fg(i)2709 1225 y Fo(;)15 b Fn(V)7 b Fo(ar)s Fq(\()p Fo(l)r Fq(\))q Fo(;)15 b(:)g(:)g(:)i Fq(\))p Fo(\033)1538 1313 y Fg(i)1505 1365 y Fn(!)1595 1383 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))1795 1365 y Fo(U)1867 1320 y Fg(\032)1857 1392 y(i)p Fl(+1)1975 1365 y Fq(\()p Fo(s)2053 1379 y Fg(i)p Fl(+1)2171 1365 y Fo(;)15 b Fn(V)7 b Fo(ar)s Fq(\()p Fo(l)r Fq(\))q Fo(;)15 b(:)g(:)g(:)i Fq(\))p Fo(\033)2777 1313 y Fg(i)2744 1365 y Fn(!)2835 1383 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))3034 1365 y Fo(:)15 b(:)g(:)75 1575 y Fq(where)29 b Fo(v)381 1589 y Fg(j)443 1575 y Fn(2)c Fo(N)10 b(F)j Fq(\()p Fn(!)809 1594 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))983 1575 y Fq(\))30 b(for)f(ev)m(ery)h(1)c Fn(\024)f Fo(j)31 b Fn(\024)25 b Fo(m)p Fq(.)k(If)g Fo(s)2023 1589 y Fg(i)p Fl(+1)2171 1575 y Fq(is)f(a)i(v)-5 b(ariable,)29 b(then)g Fo(s)2951 1589 y Fg(i)p Fl(+1)3069 1575 y Fo(\033)f Fn(2)d Fo(N)10 b(F)j Fq(\()p Fn(!)3515 1594 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))3689 1575 y Fq(\).)75 1688 y(Otherwise)40 b Fo(r)s(oot)p Fq(\()p Fo(s)759 1702 y Fg(i)p Fl(+1)877 1688 y Fq(\))j Fn(2)g(F)51 b Fq(and)41 b(b)m(y)g(case)h(\(d\))g(there)f (is)f(no)h(in\014nite)e(innermost)h Fo(U)10 b Fq(\()p Fn(R)q Fq(\)-deriv)-5 b(ation)75 1801 y(starting)30 b(from)g Fo(s)672 1815 y Fg(i)p Fl(+1)790 1801 y Fo(\033)s Fq(.)h(A)f(rep)s (etition)f(of)i(these)g(argumen)m(ts)f(sho)m(ws)g(that)h Fo(D)i Fq(has)e(the)f(form)460 2034 y Fo(D)e Fq(:)d Fo(s)g Fq(=)g Fo(U)849 1990 y Fg(\032)839 2061 y(i)889 2034 y Fq(\()p Fo(u)976 2048 y Fl(1)1016 2034 y Fo(;)15 b(:)g(:)g(:)i(;)e(u) 1270 2048 y Fg(m)1337 2034 y Fq(\))1428 1982 y Fg(i)30 b Fj(\003)1412 2034 y Fn(!)1494 2052 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))1693 2034 y Fo(U)1765 1990 y Fg(\032)1755 2067 y Fj(j)p Fg(\032)p Fj(j)1835 2034 y Fq(\()p Fo(t)1903 2052 y Fj(j)p Fg(\032)p Fj(j)1982 2034 y Fo(;)15 b Fn(V)7 b Fo(ar)s Fq(\()p Fo(l)r Fq(\))q Fo(;)15 b(:)g(:)g(:)i Fq(\))p Fo(\033)2588 1982 y Fg(i)2555 2034 y Fn(!)2646 2052 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))2845 2034 y Fo(r)s(\033)3002 1982 y Fg(i)2969 2034 y Fn(!)3060 2052 y Fg(U)g Fl(\()p Fj(R)p Fl(\))3259 2034 y Fo(:)15 b(:)g(:)75 2259 y Fq(Again,)27 b(if)g Fo(r)j Fq(is)c(a)i(v)-5 b(ariable,)26 b(then)h Fo(r)s(\033)h Fn(2)d Fo(N)10 b(F)j Fq(\()p Fn(!)1732 2278 y Fg(U)7 b Fl(\()p Fj(R)p Fl(\))1906 2259 y Fq(\).)28 b(Otherwise)e Fo(r)s(oot)p Fq(\()p Fo(r)s Fq(\))f Fn(2)g(F)37 b Fq(and)27 b(b)m(y)g(case)h(\(d\))g(there)75 2372 y(is)h(no)i (in\014nite)d(innermost)h Fo(U)10 b Fq(\()p Fn(R)p Fq(\)-deriv)-5 b(ation)30 b(starting)g(from)g Fo(r)s(\033)s Fq(.)75 2485 y(\(d\))c(Supp)s(ose)e Fo(s)h Fq(=)g Fo(f)10 b Fq(\()p Fo(u)880 2499 y Fl(1)919 2485 y Fo(;)15 b(:)g(:)g(:)i(;)e(u)1173 2499 y Fg(m)1240 2485 y Fq(\),)26 b(where)f Fo(f)35 b Fn(2)25 b(F)35 b Fq(and)25 b Fo(u)2074 2499 y Fg(j)2136 2485 y Fn(2)g(T)d Fq(\()p Fn(F)2404 2449 y Fj(0)2427 2485 y Fo(;)15 b Fn(V)8 b Fq(\).)26 b(F)-8 b(or)27 b(a)f(pro)s(of)f(b)m (y)h(con)m(tradiction,)75 2598 y(supp)s(ose)j(that)i(there)f(is)g(an)g (in\014nite)e(innermost)h Fo(U)10 b Fq(\()p Fn(R)q Fq(\)-deriv)-5 b(ation)776 2833 y 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y(side)c(of)i(a)g(condition)e(to)i(normal)e(from)h (and)g(then)g(c)m(hec)m(k)i(whether)d(the)i(righ)m(t-hand)e(side)g(of)i (the)f(condition)75 3811 y(matc)m(hes)30 b(the)e(normal)g(form.)g(This) f(is)g(of)i(course)g(not)g(true)f(for)g Fn(R)p Fq(:)h(If)f(w)m(e)h(are) g(in)m(terested)f(in)g(the)g(reducts)75 3924 y(of)k Fo(f)10 b Fq(\()p Fo(s)313 3891 y Fl(5)352 3924 y Fq(\(0\))p Fo(;)15 b(s)p Fq(\(0\))p Fo(;)g(s)p Fq(\(0\)\))36 b(and)c(w)m(e)g (\014rst)g(reduce)g Fo(s)1771 3891 y Fl(5)1810 3924 y Fq(\(0\))22 b(+)f Fo(s)p Fq(\(0\))33 b(to)g(its)f(normal)f(form)g Fo(s)3041 3891 y Fl(6)3080 3924 y Fq(\(0\),)j(then)e(w)m(e)g(\014nd)75 4037 y(that)c Fo(z)18 b Fq(+)c Fo(z)31 b Fq(do)s(es)c(not)h(matc)m(h)g Fo(s)1163 4004 y Fl(6)1202 4037 y Fq(\(0\).)h(This)c(phenomenon)h (arises)h(b)s(ecause)g(the)h(righ)m(t-hand)e(side)g Fo(z)19 b Fq(+)14 b Fo(z)31 b Fq(of)75 4150 y(the)24 b(condition)f(con)m(tains) i(a)f(de\014ned)f(sym)m(b)s(ol)g(\(and)h(consequen)m(tly)-8 b(,)25 b(ev)m(en)g(for)f(normalized)3176 4117 y Fl(3)3238 4150 y Fq(substitutions)75 4263 y Fo(\034)10 b Fq(,)34 b(the)g(term)g Fo(z)t(\034)f Fq(+)22 b Fo(z)t(\034)44 b Fq(is)33 b(not)h(necessarily)f(a)h(normal)f(form\).)h(In)f(order)g (to)i(exclude)e(suc)m(h)g(a)i(b)s(eha)m(vior,)75 4376 y(Av)m(enhaus)30 b(and)g(Lor)-10 b(\023)-35 b(\020a-S\023)-45 b(aenz)30 b([ALS94)q(])h(in)m(tro)s(duced)d(the)j(class)f(of)h (strongly)e(deterministic)g(3-CTRSs.)75 4544 y Ff(De\014nition)35 b(31.)46 b Fq(Let)31 b Fn(R)f Fq(b)s(e)g(a)h(deterministic)d(3-CTRS.) 100 4709 y(1.)46 b(A)29 b(term)g Fo(t)f Fq(is)g(called)f Fk(str)-5 b(ongly)33 b(irr)-5 b(e)g(ducible)30 b Fq(w.r.t.)e Fn(R)h Fq(if)f Fo(t\033)j Fq(is)d(a)h(normal)e(form)i(for)f(ev)m(ery)h (normalized)216 4822 y(substitution)f Fo(\033)s Fq(.)100 4932 y(2.)46 b Fn(R)41 b Fq(is)e(called)h Fk(str)-5 b(ongly)44 b(deterministic)d Fq(if,)f(for)g(ev)m(ery)h(rule)f Fo(l)27 b Fn(!)e Fo(r)j Fn(\()d Fo(s)2719 4946 y Fl(1)2783 4932 y Fn(!)g Fo(t)2932 4946 y Fl(1)2972 4932 y Fo(;)15 b(:)g(:)g(:)31 b(;)15 b(s)3231 4947 y Fg(k)3299 4932 y Fn(!)26 b Fo(t)3449 4947 y Fg(k)3531 4932 y Fq(in)40 b Fn(R)p Fq(,)216 5044 y(ev)m(ery)31 b(term)g Fo(t)707 5058 y Fg(i)765 5044 y Fq(is)e(strongly)h(irreducible.)100 5155 y(3.)46 b Fn(R)37 b Fq(is)f(called)g Fk(syntactic)-5 b(al)5 b(ly)40 b(deterministic)e Fq(if,)e(for)g(ev)m(ery)i(rule)d Fo(l)27 b Fn(!)e Fo(r)j Fn(\()d Fo(s)2865 5169 y Fl(1)2930 5155 y Fn(!)g Fo(t)3079 5169 y Fl(1)3118 5155 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(s)3378 5170 y Fg(k)3446 5155 y Fn(!)25 b Fo(t)3595 5170 y Fg(k)3674 5155 y Fq(in)216 5267 y Fn(R)p Fq(,)31 b(ev)m(ery)g(term)f Fo(t)839 5281 y Fg(i)898 5267 y Fq(is)f(a)i (constructor)g(term)1733 5234 y Fl(4)1802 5267 y Fq(or)g(a)g(ground)e Fn(R)2375 5281 y Fg(u)2420 5267 y Fq(-normal)h(form.)p 75 5341 473 4 v 92 5395 a Fd(3)152 5426 y Fv(A)25 b(substitution)g Fs(\033)k Fv(is)d Fa(normalize)l(d)g Fv(if)h Fs(x\033)g Fv(is)f(a)g(normal)g(form)f(for)i(ev)n(ery)e(v)l(ariable)h Fs(x)p Fv(.)92 5486 y Fd(4)152 5518 y Fv(A)f Fa(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.) 1867 5767 y Fq(15)p eop %%Page: 16 16 16 15 bop 75 324 a Fq(Note)24 b(that)f(it)f(is)f(undecidable)f(whether) i(or)g(not)h(a)f(term)h Fo(t)f Fq(is)f(strongly)h(irreducible)d(w.r.t.) k Fn(R)p Fq(;)f(cf.)h([ALS94].)75 437 y(But)28 b(there)f(is)g(an)g(imp) s(ortan)m(t)g(su\016cien)m(t)g(syn)m(tactic)h(criterion:)f(if)f Fo(t)2381 451 y Fg(i)2437 437 y Fq(is)g(a)i(constructor)g(term)g(or)f (a)h(ground)75 550 y Fn(R)152 564 y Fg(u)197 550 y Fq(-normal)e(form,)h (then)g(it)g(is)f(strongly)g(irreducible.)e(Th)m(us)i(ev)m(ery)i(syn)m (tactically)f(deterministic)e(CTRS)75 663 y(is)42 b(strongly)g (deterministic.)e(Strongly)i(deterministic)e(systems)j(ha)m(v)m(e)h (further)d(nice)h(prop)s(erties.)f(F)-8 b(or)75 775 y(example,)44 b(a)h(critical)e(pair)g(lemma)h(holds)f(for)h(them.)h(This)d(has)i(b)s (een)g(pro)m(v)m(en)h(for)f(quasi-reductiv)m(e)75 888 y(systems)30 b(in)f([ALS94)q(,)i(Thm.)e(4.1])j(and)e(the)g(pro)s(of)g (also)g(applies)f(to)i(quasi-decreasing)e(systems.)75 1045 y Ff(Theorem)34 b(32.)46 b Fk(Every)g(quasi-de)-5 b(cr)g(e)g(asing)48 b(str)-5 b(ongly)48 b(deterministic)f(3-CTRS)g (with)g(joinable)g(c)-5 b(ondi-)75 1158 y(tional)34 b(critic)-5 b(al)33 b(p)-5 b(airs)35 b(is)d(c)-5 b(on\015uent.)75 1315 y Fq(The)29 b(follo)m(wing)g(lemma)h(states)h(that)f(the)g (transformed)g(system)g(of)g(a)g Fk(syntactic)-5 b(al)5 b(ly)32 b Fq(deterministic)c(sys-)75 1428 y(tem)j(inherits)d(non-o)m(v) m(erlappingness.)75 1585 y Ff(Lemma)33 b(33.)46 b Fk(The)c(tr)-5 b(ansforme)g(d)46 b(system)d Fo(U)10 b Fq(\()p Fn(R)q Fq(\))42 b Fk(of)h(a)g(syntactic)-5 b(al)5 b(ly)44 b(deterministic)g (3-CTRS)f Fn(R)f Fk(is)75 1697 y(non-overlapping)35 b(if)d Fn(R)g Fk(is)h(non-overlapping.)75 1873 y(Pr)-5 b(o)g(of.)47 b Fq(Routine.)75 2050 y(On)25 b(the)h(one)g(hand,)f(non-o)m(v)m (erlappingness)f(of)i Fn(R)f Fq(can)h(sligh)m(tly)e(b)s(e)h(w)m(eak)m 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y(transformed)29 b(TRS)g(is)g(terminating.)g(It)h(th)m(us)f(natural)g (to)h(conjecture)h(that)f(a)h(left-linear)d(syn)m(tactically)75 3864 y(deterministic)e(3-CTRS)h Fn(R)i Fq(is)e(quasi-decreasing)g(if)g (and)h(only)f(if)g(the)h(TRS)f Fo(U)10 b Fq(\()p Fn(R)q Fq(\))28 b(is)f(terminating.)g(The)75 3976 y(follo)m(wing)i(example)h (sho)m(ws)g(that)h(this)e(is)h(not)g(the)h(case.)75 4133 y Fk(Example)j(36.)46 b Fq(Let)31 b Fn(R)f Fq(con)m(tain)h(the)f(rules) 1279 4296 y Fo(g)s Fq(\()p Fo(x)p Fq(\))25 b Fn(!)e Fo(k)s Fq(\()p Fo(y)s Fq(\))217 b Fn(\()23 b Fo(h)p Fq(\()p Fo(x)p Fq(\))j Fn(!)f Fo(d;)15 b(h)p Fq(\()p Fo(x)p Fq(\))27 b Fn(!)e Fo(c)p Fq(\()p Fo(y)s Fq(\))1278 4409 y Fo(h)p Fq(\()p Fo(d)p Fq(\))g Fn(!)e Fo(c)p Fq(\()p Fo(a)p Fq(\))1278 4522 y Fo(h)p Fq(\()p Fo(d)p Fq(\))i Fn(!)e Fo(c)p Fq(\()p Fo(b)p Fq(\))862 4634 y Fo(f)10 b Fq(\()p Fo(k)s Fq(\()p Fo(a)p Fq(\))p Fo(;)15 b(k)s Fq(\()p Fo(b)p Fq(\))p Fo(;)g(x)p Fq(\))26 b Fn(!)d Fo(f)10 b Fq(\()p Fo(x;)15 b(x;)g(x)p Fq(\))75 4797 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Fq(con)m(tains)d Fo(c)f Fn(2)e(D)675 3579 y Fl(1)672 3637 y(2)749 3612 y Fq(with)i Fo(c)f Fn(\027)1104 3627 y Fg(d)1178 3612 y Fo(r)s(oot)p Fq(\()p Fo(l)r Fq(\).)i(If)g Fn(R)1675 3626 y Fl(1)1737 3612 y Fn([)23 b(R)1898 3626 y Fl(3)1973 3612 y Fq(w)m(ere)35 b(quasi-reductiv)m(e)f(w.r.t.)i(an)f(order)f Fn(\037)p Fq(,)h(then)75 3725 y Fo(c)26 b Fn(\037)211 3739 y Fg(st)298 3725 y Fo(c)k Fq(w)m(ould)g(hold)f(\(but)h(this)f(con)m(tradicts)i(the) f(irre\015exivit)m(y)f(of)h Fn(\037)2498 3739 y Fg(st)2560 3725 y Fq(\).)216 3838 y(Finally)-8 b(,)32 b Fn(R)623 3852 y Fl(1)695 3838 y Fq(and)h Fn(R)952 3852 y Fl(4)1021 3838 y Fq(=)d Fn(f)p Fo(c)g Fn(!)g Fo(d)g Fn(\()g Fo(e)g Fn(!)g Fo(f)5 b(;)15 b(f)39 b Fn(!)29 b Fo(g)34 b Fn(\()c Fo(a)f Fn(!)h Fo(b)p Fn(g)k Fq(are)f(strongly)g(deterministic)e(but)75 3951 y Fn(R)152 3965 y Fl(1)212 3951 y Fn([)19 b(R)369 3965 y Fl(4)439 3951 y Fq(is)30 b(not.)75 4128 y(Note)35 b(that)g Fn(R)576 4142 y Fl(2)650 4128 y Fq(is)e(in)g(fact)i(a)f 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Fo(U)137 451 y Fg(n)184 437 y Fq(\()p Fn(R)p Fq(\))k(has)f(b)s(een)f(giv)m(en:)h(F)-8 b(or)30 b(ev)m(ery)g Fo(\032)25 b Fq(:)h Fo(l)h Fn(!)e Fo(r)j Fn(\()d Fo(s)1918 451 y Fl(1)1982 437 y Fn(!)g Fo(t)2131 451 y Fl(1)2171 437 y Fo(;)15 b(:)g(:)g(:)31 b(;)15 b(s)2430 452 y Fg(k)2498 437 y Fn(!)25 b Fo(t)2647 452 y Fg(k)2715 437 y Fn(2)g(R)p Fq(,)k Fo(U)2994 451 y Fg(n)3041 437 y Fq(\()p Fo(\032)p Fq(\))h(consists)f(of)g(the)75 550 y(t)m(w)m(o)j(rules)794 737 y Fo(l)27 b Fn(!)e Fo(U)1036 700 y Fg(\032)1077 737 y Fq(\()p Fo(s)1155 751 y Fl(1)1194 737 y Fo(;)15 b(:)g(:)g(:)i(;)e(s)1439 752 y Fg(k)1481 737 y Fo(;)g Fn(V)7 b Fo(ar)s Fq(\()p Fo(l)r Fq(\))q(\))31 b(and)f Fo(U)2091 700 y Fg(\032)2131 737 y Fq(\()p Fo(t)2199 751 y Fl(1)2238 737 y Fo(;)15 b(:)g(:)g(:)i(;)e(t)2473 752 y Fg(k)2516 737 y Fo(;)g Fn(V)7 b Fo(ar)s Fq(\()p Fo(l)r Fq(\))q(\))25 b Fn(!)g Fo(r)75 925 y Fq(No)m(w)f(the)f(pro)s(of) f(is)g(almost)h(v)m(erbatim)g(the)g(same)h(as)f(that)h(of)f(Prop)s (osition)e(40.)j Fo(U)2844 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y([AZ95)q(,)d(Thm.)g(4.8])i(and)d([Art97)r(,)h(Thm.)g(8.2.9].)75 3679 y Ff(De\014nition)35 b(51.)46 b Fq(A)36 b(reduction)e(step)h Fo(s)e Fn(!)g Fo(t)i Fq(is)f(called)h(a)g Fk(single-r)-5 b(e)g(dex)36 b Fq(reduction)f(step)g(if)f Fo(s)h Fq(con)m(tains)75 3792 y(exactly)g(one)f(redex.)g(If)g(a)g(term)g(do)s(es)g(not)g(ha)m(v) m(e)h(exactly)g(one)f(redex,)g(then)g(it)f(is)g(in)g Fk(single-r)-5 b(e)g(dex)37 b(nor-)75 3905 y(mal)g(form)p Fq(.)f(A)f Fk(single-r)-5 b(e)g(dex)37 b(derivation)f Fq(is)d(a)i(reduction)f(sequence)g(consisting)g(solely)f(of)i (single-redex)75 4018 y(reduction)27 b(steps.)i(A)g(TRS)e Fn(R)i Fq(is)e(called)h Fk(single-r)-5 b(e)g(dex)32 b(terminating)d Fq(if)f(all)f(single-redex)h(deriv)-5 b(ations)27 b(are)75 4131 y(\014nite.)j(\(Note)i(that)f(innermost)e(termination)g(implies)e (single-redex)j(termination.\))75 4309 y Ff(Theorem)k(52.)46 b Fk(If)34 b Fn(P)42 b Fk(is)34 b(an)h(LR-wel)5 b(l-mo)-5 b(de)g(d)36 b(lo)-5 b(gic)35 b(pr)-5 b(o)g(gr)g(am)37 b(such)e(that)g Fo(U)10 b Fq(\()p Fo(R)2888 4323 y Fj(P)2947 4309 y Fq(\))35 b Fk(is)f(single-r)-5 b(e)g(dex)35 b(ter-)75 4422 y(minating,)e(then)g Fn(P)41 b Fk(is)32 b(terminating.)75 4599 y Fq(Because)26 b(of)e(the)g(follo)m(wing)f(hierarc)m(h)m(y)h (\(cf.)h(Corollary)e(29)i(and)e(note)i(that)g Fo(U)10 b Fq(\()p Fo(R)2830 4613 y Fj(P)2889 4599 y Fq(\))25 b(ma)m(y)g(b)s(e)e(o)m(v)m(erlapping\))437 4769 y Fo(U)10 b Fq(\()p Fo(R)613 4783 y Fj(P)672 4769 y Fq(\))31 b(is)e(terminating) 54 b Fn(\))26 b Fo(R)1535 4783 y Fj(P)1624 4769 y Fq(is)j (quasi-reductiv)m(e)55 b Fn(\))25 b Fo(R)2555 4783 y Fj(P)2645 4769 y Fq(is)k(quasi-decreasing)437 4882 y Fn(\))c Fo(U)10 b Fq(\()p Fo(R)729 4896 y Fj(P)788 4882 y Fq(\))31 b(is)e(innermost)g(terminating)24 b Fn(\))i Fo(U)10 b Fq(\()p Fo(R)2153 4896 y Fj(P)2212 4882 y Fq(\))31 b(is)e(single-redex)g(terminating)75 5046 y(it)j(seems)g(that)h(the)g (metho)s(d)f(of)g(Arts)h(and)e(Zan)m(tema)j(is)d(more)i(p)s(o)m(w)m (erful)e(than)h(Ganzinger)g(and)g(W)-8 b(ald-)75 5159 y(mann's,)43 b(in)e(the)j(sense)f(that)h(more)f(logic)g(programs)g(can) g(b)s(e)g(pro)m(v)m(en)g(terminating)f(b)m(y)h(the)g(former)75 5272 y(metho)s(d.)30 b(Theorem)g(54,)i(ho)m(w)m(ev)m(er,)g(implies)27 b(that)k(b)s(oth)f(metho)s(ds)f(are)i(equally)e(p)s(o)m(w)m(erful.)p 75 5341 473 4 v 92 5395 a Fd(5)152 5426 y Fv(There)d(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].)92 5486 y Fd(6)152 5518 y Fv(More)g(precisely)-6 b(,)27 b(the)e(TRS)g(obtained)h(b)n(y)e (the)i(imp)r(erativ)n(e)f(pro)r(cedure)g(in)h([Art97)q(,)g(Def.)g (8.2.2].)1867 5767 y Fq(23)p eop %%Page: 24 24 24 23 bop 75 324 a Ff(Lemma)33 b(53.)46 b Fk(Supp)-5 b(ose)37 b Fo(U)10 b Fq(\()p Fo(R)1150 338 y Fj(P)1209 324 y Fq(\))36 b Fk(is)g(single-r)-5 b(e)g(dex)36 b(terminating.)g(If)g (every)f Fo(v)2783 338 y Fg(j)2855 324 y Fk(in)h Fo(u)30 b Fq(=)h Fo(P)3212 338 y Fg(in)3283 324 y Fq(\()p Fo(v)3362 338 y Fl(1)3402 324 y Fo(;)15 b(:)g(:)g(:)i(;)e(v)3648 338 y Fg(m)3715 324 y Fq(\))75 437 y Fk(is)33 b(a)g(term)g(in)f Fn(T)23 b Fq(\()p Fo(F)759 404 y Fg(T)746 464 y Fj(P)814 437 y Fo(;)15 b Fn(V)8 b Fq(\))p Fk(,)33 b(then)g(every)f Fo(U)10 b Fq(\()p Fo(R)1632 451 y Fj(P)1692 437 y Fq(\))p Fk(-derivation)34 b(starting)f(fr)-5 b(om)34 b Fo(u)f Fk(is)f(\014nite.)75 623 y(Pr)-5 b(o)g(of.)47 b Fq(It)35 b(is)f(fairly)g(easy)i(to)g(see)g(that)f(ev)m(ery)h Fo(U)10 b Fq(\()p Fo(R)1897 637 y Fj(P)1957 623 y Fq(\)-deriv)-5 b(ation)34 b(starting)h(from)g Fo(u)g Fq(is)f(a)i(single-redex)75 736 y(deriv)-5 b(ation.)29 b(Th)m(us)h(it)f(is)h(\014nite.)75 923 y Ff(Theorem)k(54.)46 b Fk(If)32 b Fo(U)10 b Fq(\()p Fo(R)969 937 y Fj(P)1029 923 y Fq(\))33 b Fk(is)f(single-r)-5 b(e)g(dex)34 b(terminating,)f(then)g(it)g(is)g(terminating.)75 1109 y(Pr)-5 b(o)g(of.)47 b Fq(By)35 b(structural)f(induction)e(on)j Fo(u)p Fq(,)g(it)f(will)e(b)s(e)i(sho)m(wn)h(that)g(ev)m(ery)h Fo(U)10 b Fq(\()p Fo(R)2885 1123 y Fj(P)2944 1109 y Fq(\)-deriv)-5 b(ation)34 b(starting)75 1222 y(from)27 b Fo(u)g Fq(is)f(\014nite.)h (If)g Fo(u)g Fq(is)f(a)i(v)-5 b(ariable,)26 b(then)h(it)g(is)f(in)g (normal)g(form.)i(If)e Fo(u)i Fq(is)e(a)i(constan)m(t,)g(then)f(it)g (is)f(either)75 1335 y(in)39 b(normal)g(form)g(or)h(Lemma)g(53)h (applies.)e(If)g Fo(u)h Fq(has)g(a)g Fo(U)10 b Fq(-sym)m(b)s(ol)40 b(at)g(its)g(ro)s(ot,)g(then)g(the)g(assertion)75 1448 y(follo)m(ws)32 b(as)i(in)e(the)h(pro)s(of)g(of)g(case)i(\(c\))f(in)e (Theorem)h(28.)i(So)e(supp)s(ose)f Fo(u)e Fq(=)g Fo(P)2807 1462 y Fg(in)2878 1448 y Fq(\()p Fo(v)2957 1462 y Fl(1)2997 1448 y Fo(;)15 b(:)g(:)g(:)h(;)f(v)3242 1462 y Fg(m)3310 1448 y Fq(\),)33 b(where)g(at)75 1561 y(least)k(one)f(of)g(the)h Fo(v)782 1575 y Fg(j)855 1561 y Fq(con)m(tains)f(a)h(function)e(sym)m (b)s(ol)g Fo(f)44 b Fn(62)35 b Fo(F)2234 1528 y Fg(T)2221 1588 y Fj(P)2289 1561 y Fq(.)h(By)h(the)f(inductiv)m(e)f(h)m(yp)s (othesis,)g(ev)m(ery)75 1674 y Fo(U)10 b Fq(\()p Fo(R)251 1688 y Fj(P)310 1674 y Fq(\)-deriv)-5 b(ation)31 b(starting)g(from)g Fo(v)1404 1688 y Fg(j)1472 1674 y Fq(is)g(\014nite.)f(F)-8 b(or)33 b(a)e(pro)s(of)g(b)m(y)h(con)m(tradiction)f(supp)s(ose)f(there) i(is)e(an)75 1787 y(in\014nite)e Fo(U)10 b Fq(\()p Fo(R)559 1801 y Fj(P)618 1787 y Fq(\)-deriv)-5 b(ation)1347 1970 y Fo(D)28 b Fq(:)d Fo(u)h Fq(=)f Fo(u)1726 1984 y Fl(0)1790 1970 y Fn(!)g Fo(u)1958 1984 y Fl(1)2023 1970 y Fn(!)g Fo(u)2191 1984 y Fl(2)2256 1970 y Fn(!)g Fo(:)15 b(:)g(:)75 2153 y Fq(The)31 b(pro)s(of)g(idea)g(is)g(to)h(eliminate)e(all)h (\\aliens")g(in)f(the)i(terms)f Fo(u)2339 2167 y Fg(j)2408 2153 y Fq(b)m(y)g(a)h(function)e Fo(\011)42 b Fq(and)31 b(to)h(sho)m(w)f(that)75 2266 y(the)i(deriv)-5 b(ation)32 b Fo(\011)10 b Fq(\()p Fo(D)s Fq(\))33 b(is)f(still)e(in\014nite.)h(In) h Fo(u)p Fq(,)i(ev)m(ery)f(maximal)f(prop)s(er)f(subterm)h Fo(t)h Fq(with)f(a)h Fo(U)10 b Fq(-sym)m(b)s(ol)75 2379 y(or)34 b(a)h(function)e(sym)m(b)s(ol)h(from)g Fo(F)1236 2346 y Fg(P)1223 2406 y Fj(P)1329 2379 y 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5518 y Fq(\))26 b Fn(!)2708 5480 y Fj(\003)2772 5518 y Fo(:)15 b(:)g(:)1867 5767 y Fq(24)p eop %%Page: 25 25 25 24 bop 75 324 a Fq(is)31 b(an)g(in\014nite)f Fo(U)10 b Fq(\()p Fo(R)781 338 y Fj(P)840 324 y Fq(\)-deriv)-5 b(ation.)32 b(F)-8 b(urthermore,)32 b Fo(\011)10 b Fq(\()p Fo(u)p Fq(\))28 b(=)f Fo(P)2286 338 y Fg(in)2357 324 y Fq(\()p Fo(\010)p Fq(\()p Fo(v)2532 338 y Fl(1)2572 324 y Fq(\))p Fo(;)15 b(:)g(:)g(:)i(;)e(\010)p Fq(\()p Fo(v)2949 338 y Fg(m)3016 324 y Fq(\)\))32 b(and)g(ev)m(ery)g Fo(\010)p Fq(\()p Fo(v)3678 338 y Fg(j)3715 324 y Fq(\))75 437 y(is)d(an)i(elemen)m(t)g(of)f Fn(T)23 b Fq(\()p Fo(F)911 404 y Fg(T)898 464 y Fj(P)966 437 y Fo(;)15 b Fn(V)8 b Fq(\).)31 b(This,)d(ho)m(w)m(ev)m(er,)k(con)m(tradicts)f(Lemma)g(53.) 75 647 y Fk(Example)j(55.)46 b Fq(Consider)28 b(the)j(system)g Fo(R)1517 661 y Fj(P)1606 647 y Fq(from)f(Example)f(47)j(and)d(the)i Fo(U)10 b Fq(\()p Fo(R)2829 661 y Fj(P)2888 647 y Fq(\)-deriv)-5 b(ation)490 853 y Fo(q)s(s)577 867 y Fg(in)648 853 y 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b(b)s(oils)f(do)m(wn)h(to)i(pro)m(ving)e(innermost)f (termination)h(of)h(the)g(unconditional)d(TRS)216 3339 y Fo(U)10 b Fq(\()p Fo(R)q Fq(\()p Fn(P)d Fq(\)\).)32 b(Since)c(the)i(latter)f(can)h(b)s(e)f(automated)i(b)m(y)e(the)h(dep)s (endency)e(pair)g(metho)s(d)h(of)g(Arts)h(and)216 3452 y(Giesl)g([A)m(G97b)q(],)h(the)g(whole)e(metho)s(d)h(can)h(b)s(e)e (automated;)j(cf.)f([Art97)q(].)75 3635 y(It)25 b(should)e(b)s(e)h(p)s (oin)m(ted)f(out)i(that)h(none)e(of)h(the)g(ab)s(o)m(v)m(e-men)m (tioned)h(metho)s(ds)e(yield)f(a)i(complete)g(criterion)75 3748 y(for)30 b(pro)m(ving)g(termination)f(of)h(LR-w)m(ell-mo)s(ded)f (logic)h(programs;)h(see)g([GW93)r(].)75 3974 y Ff(Ac)m(kno)m(wledgmen) m(t:)c Fq(I)g(thank)f(Mic)m(hael)h(Han)m(us)g(for)g(the)g(\(email\))g (discussion)d(whic)m(h)i(led)g(to)i(the)f(dev)m(el-)75 4087 y(opmen)m(t)i(of)g(the)f(transformation)g Fo(U)10 b Fq(.)29 b(He)g(also)g(p)s(oin)m(ted)e(out)i(that)g(the)g(elimination) d(of)j(lo)s(cal)e(de\014nitions)75 4200 y(via)g(program)g (transformations)g(is)f(w)m(ell-kno)m(wn)g(in)g(the)i(implemen)m (tation)e(of)h(functional)f(programming)75 4313 y(languages;)31 b(see)g([Joh85)q(].)75 4586 y Fr(References)75 4784 y Fv([A)n(G97a])47 b(T.)31 b(Arts)g(and)f(J.)i(Giesl.)50 b(Automatically)31 b(Pro)n(ving)g(T)-6 b(ermination)30 b(where)h(Simpli\014cation)g(Orderings)g(F)-6 b(ail.)50 b(In)377 4875 y Fa(Pr)l(o)l(c)l(e)l(e)l(dings)30 b(of)d(the)h(22nd)h (International)f(Col)t(lo)l(quium)f(on)h(T)-6 b(r)l(e)l(es)29 b(in)e(A)n(lgebr)l(a)i(and)f(Pr)l(o)l(gr)l(amming)p Fv(,)f(pages)f (261{)377 4967 y(273.)h(LNCS)e Fu(1214)p Fv(,)i(1997.)75 5060 y([A)n(G97b])46 b(T.)33 b(Arts)f(and)g(J.)g(Giesl.)55 b(Pro)n(ving)33 b(Innermost)d(Normalisation)j(Automatically.)54 b(In)31 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)36 b(of)d(the)h(8th)377 5151 y(International)k(Confer)l(enc)l(e)h(on)e(R)l(ewriting)h(T)-6 b(e)l(chniques)38 b(and)g(Applic)l(ations)p Fv(,)f(pages)g(157{172.)i (LNCS)d Fu(1232)p Fv(,)377 5242 y(1997.)75 5335 y([ALS94])46 b(J.)27 b(Av)n(enhaus)e(and)g(C.)i(Lor)-9 b(\023)-30 b(\020a-S\023)-38 b(aenz.)37 b(On)25 b(Conditional)j(Rewrite)e(Systems) f(with)h(Extra)g(V)-6 b(ariables)27 b(and)f(Deter-)377 5426 y(ministic)h(Logic)i(Programs.)41 b(In)27 b Fa(Pr)l(o)l(c)l(e)l(e) l(dings)k(of)e(the)h(5th)h(International)f(Confer)l(enc)l(e)h(on)e(L)l (o)l(gic)h(Pr)l(o)l(gr)l(amming)377 5518 y(and)e(A)n(utomate)l(d)h(R)l (e)l(asoning)p Fv(,)e(pages)g(215{229.)i(LNAI)24 b Fu(822)p Fv(,)j(1994.)1867 5767 y Fq(25)p eop %%Page: 26 26 26 25 bop 75 324 a Fv([Art97])66 b(T.)30 b(Arts.)44 b Fa(A)n(utomatic)l(al)t(ly)32 b(Pr)l(oving)f(T)-6 b(ermination)31 b(and)g(Innermost)h(Normalisation)f(of)g(T)-6 b(erm)31 b(R)l(ewriting)g(Sys-)377 415 y(tems)p Fv(.)k(PhD)25 b(thesis,)i(Utrec)n(h)n(t)e(Univ)n(ersit)n(y)-6 b(,)24 b(1997.)75 506 y([AZ95])79 b(T.)32 b(Arts)e(and)h(H.)f(Zan)n(tema.)50 b(T)-6 b(ermination)30 b(of)i(Logic)g(Programs)g(via)f(Lab)r(elled)g(T) -6 b(erm)31 b(Rewrite)g(Systems.)49 b(In)377 598 y Fa(Pr)l(o)l(c)l(e)l (e)l(dings)30 b(of)d(Computing)h(Scienc)l(e)g(in)f(the)i(Netherlands)p Fv(,)f(pages)e(22{34,)i(1995.)75 689 y([AZ96])79 b(T.)25 b(Arts)f(and)h(H.)f(Zan)n(tema.)32 b(T)-6 b(ermination)24 b(of)h(Logic)h(Programs)f(using)g(Seman)n(tic)e(Uni\014cation.)33 b(In)24 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)377 780 y(of)h(the)g(Fifth)g (International)h(Workshop)h(on)e(L)l(o)l(gic)g(Pr)l(o)l(gr)l(am)i (Synthesis)f(and)g(T)-6 b(r)l(ansformation)p Fv(,)24 b(pages)g(219{233.)377 872 y(LNCS)h Fu(1048)p Fv(,)i(1996.)75 963 y([BK86])70 b(J.A.)30 b(Bergstra)g(and)f(J.W.)g(Klop.)45 b(Conditional)30 b(Rewrite)f(Rules:)h(Con\015uence)e(and)h(T)-6 b(ermination.)44 b Fa(Journal)31 b(of)377 1054 y(Computer)e(and)f (System)g(Scienc)l(es)h Fu(32)p Fa(\(3\))p Fv(,)e(pages)g(323{362,)i (1986.)75 1146 y([BN98])72 b(F.)26 b(Baader)h(and)e(T.)h(Nipk)n(o)n(w.) 34 b Fa(T)-6 b(erm)28 b(R)l(ewriting)g(and)g(A)n(l)t(l)f(That)p Fv(.)35 b(Cam)n(bridge)26 b(Univ)n(ersit)n(y)e(Press,)j(1998.)75 1237 y([Gan91])47 b(H.)26 b(Ganzinger.)37 b(Order-Sorted)26 b(Completion:)g(The)h(Man)n(y-Sorted)e(W)-6 b(a)n(y.)36 b Fa(The)l(or)l(etic)l(al)29 b(Computer)g(Scienc)l(e)g Fu(89)p Fv(,)377 1328 y(pages)d(3{32,)i(1991.)75 1420 y([Gie98])69 b(J.)26 b(Giesl,)i(Ma)n(y)d(1998.)36 b(P)n(ersonal)27 b(comm)n(unication.)75 1511 y([Gra95])56 b(B.)41 b(Gramlic)n(h.)77 b(Abstract)40 b(Relations)g(b)r(et)n(w)n(een)g(Restricted)g(T)-6 b(ermination)40 b(and)g(Con\015uence)g(Prop)r(erties)h(of)377 1602 y(Rewrite)26 b(Systems.)33 b Fa(F)-6 b(undamenta)29 b(Informatic)l(ae)f Fu(24)p Fv(,)f(pages)f(3{23,)i(1995.)75 1694 y([Gra96])56 b(B.)33 b(Gramlic)n(h.)55 b Fa(T)-6 b(ermination)34 b(and)g(Con\015uenc)l(e)h(Pr)l(op)l(erties)h(of)e (Structur)l(e)l(d)i(R)l(ewrite)f(Systems)p Fv(.)56 b(PhD)32 b(thesis,)377 1785 y(Univ)n(ersit\177)-38 b(at)25 b(Kaiserslautern,)j (1996.)75 1876 y([GW93])47 b(H.)22 b(Ganzinger)i(and)e(U.)g(W)-6 b(aldmann.)28 b(T)-6 b(ermination)22 b(Pro)r(ofs)i(of)f(W)-6 b(ell-Mo)r(ded)23 b(Logic)h(Programs)f(via)f(Conditional)377 1968 y(Rewrite)28 b(Systems.)37 b(In)27 b Fa(Pr)l(o)l(c)l(e)l(e)l (dings)k(of)d(the)i(3r)l(d)g(International)g(Workshop)g(on)f (Conditional)g(T)-6 b(erm)29 b(R)l(ewriting)377 2059 y(Systems)p Fv(,)e(pages)g(113{127.)i(LNCS)c Fu(656)p Fv(,)i(1993.)75 2150 y([Han94])47 b(M.)32 b(Han)n(us.)53 b(The)32 b(In)n(tegration)h(of)f(F)-6 b(unctions)32 b(in)n(to)g(Logic)h (Programming:)f(F)-6 b(rom)32 b(Theory)g(to)g(Practice.)55 b Fa(The)377 2242 y(Journal)28 b(of)f(L)l(o)l(gic)h(Pr)l(o)l(gr)l (amming)h Fu(19,20)p Fv(,)d(pages)h(583{628,)i(1994.)75 2333 y([Joh85])64 b(T.)31 b(Johnsson.)49 b(Lam)n(b)r(da)30 b(Lifting:)h(T)-6 b(ransforming)32 b(Programs)f(to)f(Recursiv)n(e)g(F) -6 b(unctions.)48 b(In)30 b Fa(F)-6 b(unctional)33 b(Pr)l(o-)377 2424 y(gr)l(amming)28 b(L)l(anguages)h(and)f(Computer)h(A)n(r)l(chite)l (ctur)l(e)p Fv(,)g(pages)d(190{203.)j(LNCS)c Fu(201)p Fv(,)i(1985.)75 2516 y([K)n(O92])66 b(M.)34 b(Kurihara)f(and)f(A.)h(Oh) n(uc)n(hi.)55 b(Mo)r(dularit)n(y)33 b(of)h(Simple)e(T)-6 b(ermination)33 b(of)g(T)-6 b(erm)32 b(Rewriting)i(Systems)e(with)377 2607 y(Shared)25 b(Constructors.)36 b Fa(The)l(or)l(etic)l(al)29 b(Computer)f(Scienc)l(e)h Fu(103)p Fv(,)d(pages)h(273{282,)i(1992.)75 2698 y([KR95])67 b(M.R.K.)34 b(Krishna)f(Rao.)58 b(Mo)r(dular)34 b(Pro)r(ofs)h(for)f(Completeness)g(of)g(Hierarc)n(hical)g(T)-6 b(erm)33 b(Rewriting)h(Systems.)377 2790 y Fa(The)l(or)l(etic)l(al)29 b(Computer)g(Scienc)l(e)f Fu(151)p Fv(,)f(pages)f(487{512,)j(1995.)75 2881 y([LS93])93 b(C.)29 b(Lor)-9 b(\023)-30 b(\020a-S\023)-38 b(aenz.)45 b Fa(A)30 b(The)l(or)l(etic)l(al)h(F)-6 b(r)l(amework)32 b(for)e(R)l(e)l(asoning)h(ab)l(out)h(Pr)l(o)l(gr)l(am)f(Construction)i (Base)l(d)e(on)f(Ex-)377 2972 y(tensions)f(of)e(R)l(ewrite)h(Systems)p Fv(.)36 b(PhD)26 b(thesis,)g(Univ)n(ersit\177)-38 b(at)26 b(Kaiserslautern,)h(German)n(y)-6 b(,)24 b(1993.)75 3064 y([Mar95])48 b(M.)34 b(Marc)n(hiori.)58 b(Unra)n(v)n(elings)33 b(and)g(Ultra-Prop)r(erties.)58 b(T)-6 b(ec)n(hnical)34 b(Rep)r(ort)f(8,)h(Dept.)f(of)g(Pure)h(and)f(Applied)377 3155 y(Mathematics,)26 b(Univ)n(ersit)n(y)f(of)h(P)n(ado)n(v)l(a,)h (Italy)-6 b(,)25 b(1995.)75 3246 y([Mar96])48 b(M.)27 b(Marc)n(hiori.)38 b(Unra)n(v)n(elings)27 b(and)f(Ultra-Prop)r(erties.) 38 b(In)26 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)31 b(of)d(the)h(5th)g (International)g(Confer)l(enc)l(e)377 3337 y(on)f(A)n(lgebr)l(aic)g (and)g(L)l(o)l(gic)g(Pr)l(o)l(gr)l(amming)p Fv(,)f(pages)g(107{121.)h (LNCS)e Fu(1139)p Fv(,)g(1996.)75 3429 y([Mid90])50 b(A.)32 b(Middeldorp.)56 b Fa(Mo)l(dular)35 b(Pr)l(op)l(erties)h(of)d(T)-6 b(erm)35 b(R)l(ewriting)f(Systems)p Fv(.)57 b(PhD)32 b(thesis,)i(V)-6 b(rije)32 b(Univ)n(ersiteit)h(te)377 3520 y(Amsterdam,)24 b(1990.)75 3611 y([Mid93])50 b(A.)25 b(Middeldorp.)33 b(Mo)r(dular)26 b(Prop)r(erties)g(of)g(Conditional)h (T)-6 b(erm)24 b(Rewriting)i(Systems.)32 b Fa(Information)26 b(and)i(Com-)377 3703 y(putation)h Fu(104)p Fa(\(1\))p Fv(,)e(pages)g(110{158,)i(1993.)75 3794 y([MZ97])67 b(A.)35 b(Middeldorp)g(and)g(H.)g(Zan)n(tema.)61 b(Simple)34 b(T)-6 b(ermination)35 b(of)h(Rewrite)f(Systems.)61 b Fa(The)l(or)l(etic)l(al)38 b(Computer)377 3885 y(Scienc)l(e)28 b Fu(175)p Fa(\(1\))p Fv(,)g(pages)e(127{158,)j(1997.)75 3977 y([Ohl94])60 b(E.)28 b(Ohlebusc)n(h.)39 b Fa(Mo)l(dular)29 b(Pr)l(op)l(erties)j(of)c(Comp)l(osable)j(T)-6 b(erm)29 b(R)l(ewriting)h(Systems)p Fv(.)41 b(PhD)27 b(thesis,)h(Univ)n (ersit\177)-38 b(at)377 4068 y(Bielefeld,)28 b(1994.)75 4159 y([Ohl95])60 b(E.)34 b(Ohlebusc)n(h.)57 b(Mo)r(dular)35 b(Prop)r(erties)f(of)h(Comp)r(osable)f(T)-6 b(erm)33 b(Rewriting)h(Systems.)57 b Fa(Journal)35 b(of)g(Symb)l(olic)377 4251 y(Computation)29 b Fu(20)p Fv(,)d(pages)g(1{41,)i(1995.)75 4342 y([SMI95])47 b(T.)25 b(Suzuki,)g(A.)g(Middeldorp,)g(and)g(T.)h (Ida.)33 b(Lev)n(el-Con\015uence)25 b(of)h(Conditional)g(Rewrite)g (Systems)e(with)h(Extra)377 4433 y(V)-6 b(ariables)38 b(in)f(Righ)n(t-Hand)f(Sides.)69 b(In)37 b Fa(Pr)l(o)l(c)l(e)l(e)l (dings)j(of)e(the)h(6th)g(International)g(Confer)l(enc)l(e)h(on)e(R)l (ewriting)377 4525 y(T)-6 b(e)l(chniques)29 b(and)f(Applic)l(ations)p Fv(,)f(pages)f(179{193.)j(LNCS)c Fu(914)p Fv(,)i(1995.)75 4616 y([T)-6 b(o)n(y87])58 b(Y.)25 b(T)-6 b(o)n(y)n(ama.)32 b(Coun)n(terexamples)25 b(to)g(T)-6 b(ermination)24 b(for)i(the)f (Direct)g(Sum)e(of)i(T)-6 b(erm)24 b(Rewriting)i(Systems.)32 b Fa(Infor-)377 4707 y(mation)c(Pr)l(o)l(c)l(essing)h(L)l(etters)h Fu(25)p Fv(,)c(pages)h(141{143,)i(1987.)75 4799 y([Zan94])56 b(H.)38 b(Zan)n(tema.)69 b(T)-6 b(ermination)38 b(of)g(T)-6 b(erm)37 b(Rewriting:)i(In)n(terpretation)e(and)h(T)n(yp)r(e)f (Elimination.)71 b Fa(Journal)39 b(of)377 4890 y(Symb)l(olic)27 b(Computation)i Fu(17)p Fv(,)d(pages)h(23{50,)h(1994.)1867 5767 y Fq(26)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF