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Fp(t)d(>)h(u)j Fs(w)m(e)g(ha)m(v)m(e)212 4739 y Fp(f)10 b Fs(\()p Fp(:)15 b(:)g(:)h(;)f(t;)g(:)g(:)g(:)i Fs(\))25 b Fp(>)g(f)10 b Fs(\()p Fp(:)15 b(:)g(:)h(;)f(u;)g(:)g(:)g(:)i Fs(\).)40 b(A)28 b(TRS)e Fq(R)h Fs(o)m(v)m(er)i Fq(F)36 b Fs(and)27 b(an)g(order)g Fp(>)g Fs(on)g Fq(T)22 b Fs(\()p Fq(F)9 b Fs(\))29 b(are)e(called)g Fo(c)-5 b(om-)212 4852 y(p)g(atible)35 b Fs(if)25 b Fp(t)g(>)g(u)i Fs(for)f(all)f(rewrite)h(steps)g Fp(t)f Fq(!)1746 4866 y FA(R)1836 4852 y Fp(u)p Fs(.)39 b(F)-8 b(or)27 b(compatibilit)m(y)e(with)g(a)i(monotonic)g(order)f(it) 212 4965 y(su\016ces)j(to)h(c)m(hec)m(k)h(that)e Fp(l)r(\033)g(>)c(r)s (\033)32 b Fs(for)d(all)f(rules)f Fp(l)g Fq(!)f Fp(r)31 b Fs(in)d Fq(R)h Fs(and)g(all)f(ground)g(substitutions)f Fp(\033)s Fs(.)40 b(It)212 5078 y(is)26 b(w)m(ell-kno)m(wn)f(that)i(a)g (TRS)e(is)h(terminating)f(if)h(and)g(only)f(if)h(it)g(is)f(compatible)h (with)f(a)i(monotonic)212 5191 y(w)m(ell-founded)34 b(order.)56 b(An)35 b Fq(F)9 b Fs(-algebra)37 b(consists)e(of)h(a)g(set)g Fp(A)g Fs(and)f(for)h(ev)m(ery)g Fp(f)44 b Fq(2)33 b(F)45 b Fs(a)37 b(function)1920 5462 y(3)p eop %%Page: 4 4 4 3 bop 212 337 a Fp(f)257 351 y Fm(A)324 337 y Fs(:)30 b Fp(A)447 304 y Fm(n)520 337 y Fq(!)25 b Fp(A)p Fs(,)i(where)f Fp(n)g Fs(is)f(the)h(arit)m(y)h(of)f Fp(f)10 b Fs(.)39 b(A)26 b Fo(monotone)35 b Fq(F)9 b Fs(-algebra)27 b(\()p Fp(A;)15 b(>)p Fs(\))27 b(is)e(an)h Fq(F)9 b Fs(-algebra)27 b Fp(A)212 450 y Fs(for)h(whic)m(h)e(the)i(underlying)d(set)k(is)e(pro) m(vided)f(with)h(an)g(order)h Fp(>)f Fs(suc)m(h)h(that)g(ev)m(ery)h (algebra)f(op)s(era-)212 562 y(tion)e(is)g(monotonic)h(in)e(all)g(of)i (its)f(argumen)m(ts.)40 b(More)27 b(precisely)-8 b(,)27 b(for)f(all)g Fp(f)34 b Fq(2)25 b(F)36 b Fs(and)26 b Fp(a;)15 b(b)26 b Fq(2)f Fp(A)i Fs(with)212 675 y Fp(a)32 b(>)g(b)j Fs(w)m(e)g(ha)m(v)m(e)h Fp(f)866 689 y Fm(A)922 675 y Fs(\()p Fp(:)15 b(:)g(:)i(;)e(a;)g(:)g(:)g(:)i Fs(\))33 b Fp(>)f(f)1545 689 y Fm(A)1601 675 y Fs(\()p Fp(:)15 b(:)g(:)i(;)e(b;)g(:)g(:)g(:)i Fs(\).)53 b(A)35 b(monotone)h Fq(F)9 b Fs(-algebra)35 b(\()p Fp(A;)15 b(>)p Fs(\))35 b(is)f(called)212 788 y Fo(wel)5 b(l-founde)-5 b(d)39 b Fs(if)26 b Fp(>)i Fs(is)f(a)h(w)m(ell-founded)e(order.)40 b(Ev)m(ery)28 b(monotone)h Fq(F)9 b Fs(-algebra)28 b(\()p Fp(A;)15 b(>)p Fs(\))29 b(induces)d(an)212 901 y(order)33 b Fp(>)524 915 y Fm(A)614 901 y Fs(on)h(the)f(set)h(of)g(terms)f Fq(T)23 b Fs(\()p Fq(F)9 b Fp(;)15 b Fq(X)e Fs(\))35 b(as)f(follo)m(ws:)46 b Fp(t)30 b(>)2370 915 y Fm(A)2457 901 y Fp(u)j Fs(if)g(and)g(only)f(if)h([)p Fp(\013)p Fs(]\()p Fp(t)p Fs(\))e Fp(>)g Fs([)p Fp(\013)p Fs(]\()p Fp(u)p Fs(\))212 1014 y(for)i(all)e(assignmen)m(ts)i Fp(\013)10 b Fs(:)32 b Fq(X)42 b(!)29 b Fp(A)p Fs(.)48 b(Here)33 b([)p Fp(\013)p Fs(])h(denotes)f(the)g(homomorphic)f (extension)g(of)h Fp(\013)p Fs(,)h(i.e.,)212 1127 y([)p Fp(\013)p Fs(]\()p Fp(x)p Fs(\))40 b(=)e Fp(\013)p Fs(\()p Fp(x)p Fs(\))i(for)e Fp(x)g Fq(2)g(X)51 b Fs(and)38 b([)p Fp(\013)p Fs(]\()p Fp(f)10 b Fs(\()p Fp(t)1714 1141 y Fn(1)1754 1127 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)1989 1141 y Fm(n)2036 1127 y Fs(\)\))39 b(=)f Fp(f)2299 1141 y Fm(A)2356 1127 y Fs(\([)p Fp(\013)p Fs(]\()p Fp(t)2567 1141 y Fn(1)2608 1127 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(\013)p Fs(]\()p Fp(t)3021 1141 y Fm(n)3069 1127 y Fs(\)\))39 b(for)f(all)f Fp(n)p Fs(-ary)212 1240 y Fp(f)k Fq(2)31 b(F)44 b Fs(and)33 b Fp(t)712 1254 y Fn(1)751 1240 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)986 1254 y Fm(n)1065 1240 y Fq(2)31 b(T)23 b Fs(\()p Fq(F)9 b Fp(;)15 b Fq(X)e Fs(\))q(.)52 b(A)35 b(TRS)e Fq(R)h Fs(and)g(a)g (monotone)h(algebra)f(\()p Fp(A;)15 b(>)p Fs(\))35 b(are)g(called)212 1353 y Fo(c)-5 b(omp)g(atible)42 b Fs(if)32 b Fq(R)i Fs(and)e Fp(>)1110 1367 y Fm(A)1200 1353 y Fs(are)i(compatible.)48 b(It)33 b(is)f(w)m(ell-kno)m(wn)h(that)g(a)h(TRS)e(is)g(terminating)g (if)212 1466 y(and)h(only)h(if)f(it)g(is)g(compatible)g(with)g(a)h(w)m (ell-founded)e(monotone)j(algebra.)52 b(The)33 b(set)i(of)f(rewrite)212 1579 y(rules)e Fp(f)10 b Fs(\()p Fp(x)575 1593 y Fn(1)614 1579 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)867 1593 y Fm(n)915 1579 y Fs(\))30 b Fq(!)g Fp(x)1153 1593 y Fm(i)1214 1579 y Fs(for)j(all)f Fp(f)39 b Fq(2)29 b(F)42 b Fs(and)33 b(all)f Fp(i)e Fs(=)g(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)p Fs(,)34 b(where)e Fp(n)e Fl(>)f Fs(1)34 b(is)e(the)h(arit)m(y)g(of)212 1692 y Fp(f)10 b Fs(,)28 b(is)f(denoted)i(b)m(y)f Fq(E)8 b Fs(m)m(b\()p Fq(F)h Fs(\),)30 b(or)e(simply)e(b)m(y)i Fq(E)8 b Fs(m)m(b)28 b(when)f(the)h(signature)g Fq(F)37 b Fs(can)29 b(b)s(e)f(inferred)e(from)212 1804 y(the)31 b(con)m(text.)353 1917 y(The)g(prop)s(erties)e(in)h(the)h(termination)f (hierarc)m(h)m(y)h(are)g(de\014ned)f(as)h(follo)m(ws.)42 b(A)31 b(TRS)f(is)g(called)212 2030 y Fo(terminating)46 b Fs(if)36 b(it)g(do)s(es)g(not)h(allo)m(w)g(an)f(in\014nite)f(rewrite) h(sequence.)60 b(A)37 b(TRS)f Fq(R)g Fs(o)m(v)m(er)i(a)g(signa-)212 2143 y(ture)32 b Fq(F)42 b Fs(is)32 b(called)f Fo(simply)36 b(terminating)42 b Fs(if)31 b Fq(R)22 b([)f(E)8 b Fs(m)m(b\()p Fq(F)h Fs(\))33 b(is)f(terminating,)g(or,)h(equiv)-5 b(alen)m(tly)31 b(\(b)m(y)212 2256 y(Krusk)-5 b(al's)26 b(T)-8 b(ree)29 b(Theorem)e([14)r(]\),)i Fq(R)15 b([)g(E)8 b Fs(m)m(b\()p Fq(F)h Fs(\))29 b(has)f(no)f(cycle.)41 b(A)28 b(w)m(ell-kno)m(wn)f(su\016cien)m(t)g(condi-)212 2369 y(tion)32 b(for)f(simple)g(termination)f(of)j(terminating)e(TRSs)f (is)i Fo(length-pr)-5 b(eservingness)p Fs(,)34 b(whic)m(h)c(means)212 2482 y(that)40 b Fq(j)p Fp(l)r(\033)s Fq(j)i Fs(=)e Fq(j)p Fp(r)s(\033)s Fq(j)g Fs(for)f(all)g(rules)f Fp(l)k Fq(!)f Fp(r)h Fs(and)d(all)g(ground)f(substitutions)f Fp(\033)s Fs(.)69 b(Here)40 b Fq(j)p Fp(t)p Fq(j)g Fs(denotes)212 2595 y(the)33 b(n)m(um)m(b)s(er)e(of)i(\(o)s(ccurrences)g(of)7 b(\))33 b(function)e(sym)m(b)s(ols)g(in)g Fp(t)p Fs(.)47 b(Note)34 b(that)f(length-preservingness)212 2708 y(is)40 b(equiv)-5 b(alen)m(t)40 b(to)h(the)g(com)m(bination)f(of)g(v)-5 b(ariable-preservingness)38 b(and)i(the)h(requiremen)m(t)f(that)212 2821 y Fq(j)p Fp(l)r Fq(j)33 b Fs(=)g Fq(j)p Fp(r)s Fq(j)i Fs(for)g(all)f(rules)f Fp(l)i Fq(!)e Fp(r)s Fs(.)54 b(A)35 b(TRS)f(o)m(v)m(er)j(a)e(signature)f Fq(F)45 b Fs(is)34 b(called)g Fo(total)5 b(ly)38 b(terminating)44 b Fs(if)212 2934 y(it)29 b(is)f(compatible)g(with)f(a)j(monotonic)f(w)m (ell-founded)e(total)i(order)g(on)g Fq(T)22 b Fs(\()p Fq(F)9 b Fs(\))q(,)30 b(or,)f(equiv)-5 b(alen)m(tly)d(,)29 b(it)212 3046 y(is)j(compatible)g(with)f Fp(>)1053 3060 y Fm(A)1143 3046 y Fs(for)h(some)h(w)m(ell-founded)e(monotone)i Fq(F)9 b Fs(-algebra)34 b(\()p Fp(A;)15 b(>)p Fs(\))33 b(in)f(whic)m(h)f(the)212 3159 y(order)i Fp(>)g Fs(is)g(total.)51 b(A)33 b(TRS)g(o)m(v)m(er)h(a)g(signature)f Fq(F)43 b Fs(is)32 b(called)h Fp(!)s Fo(-terminating)42 b Fs(if)32 b(it)h(is)g(compatible)212 3272 y(with)26 b(some)j(w)m(ell-founded)c (monotone)k Fq(F)9 b Fs(-algebra)28 b(\()p Fp(A;)15 b(>)p Fs(\))29 b(suc)m(h)e(that)h Fp(A)g Fs(is)f(a)h(subset)f(of)h(the)f(set) i Fk(N)212 3385 y Fs(of)e(natural)e(n)m(um)m(b)s(ers)g(and)h Fp(>)g Fs(is)g(the)g(restriction)g(of)g(the)h(usual)e(order)h(on)g Fk(N)39 b Fs(to)27 b Fp(A)p Fs(.)40 b(If,)27 b(in)e(addition,)212 3498 y(ev)m(ery)35 b(in)m(terpretation)e(function)f Fp(f)1442 3512 y Fm(A)1533 3498 y Fs(is)h(a)h(p)s(olynomial,)e(w)m(e)i(sa)m(y)h (that)f(the)g(TRS)f(is)g Fo(p)-5 b(olynomial)5 b(ly)212 3611 y(terminating)p Fs(.)57 b(Our)35 b(de\014nitions)e(of)j Fp(!)s Fs(-termination)e(and)h(p)s(olynomial)e(termination)h(di\013er)h (from)212 3724 y(the)g(ones)f(in)f([25)q(])i(in)e(that)h(w)m(e)h(allo)m (w)f(an)g(arbitrary)f(subset)h(of)g Fk(N)47 b Fs(as)35 b(carrier)e(of)i(the)f(compatible)212 3837 y(algebra.)40 b(F)-8 b(or)29 b Fp(!)s Fs(-termination)e(this)g(mak)m(es)i(no)g (di\013erence:)39 b(If)27 b Fp(A)i Fs(is)e(an)h(in\014nite)e(subset)h (of)i Fk(N)41 b Fs(then)212 3950 y(there)36 b(exists)f(exactly)i(one)f (monotonic)g(bijection)e Fp(\036)10 b Fs(:)33 b Fp(A)h Fq(!)g Fk(N)7 b Fs(.)62 b(Let)37 b Fp( )i Fs(b)s(e)c(its)g (\(monotonic\))h(in-)212 4063 y(v)m(erse)31 b(and)f(de\014ne)g Fp(f)925 4077 y Fj(N)976 4063 y Fs(\()p Fp(x)1063 4077 y Fn(1)1103 4063 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)1357 4077 y Fm(n)1404 4063 y Fs(\))26 b(=)f Fp(\036)p Fs(\()p Fp(f)1695 4077 y Fm(A)1752 4063 y Fs(\()p Fp( )s Fs(\()p Fp(x)1936 4077 y Fn(1)1976 4063 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e( )s Fs(\()p Fp(x)2362 4077 y Fm(n)2411 4063 y Fs(\)\)\))31 b(for)f(ev)m(ery)h(function)f(sym)m(b)s(ol)f Fp(f)10 b Fs(.)212 4176 y(In)32 b(this)g(w)m(a)m(y)i(w)m(e)g(obtain)e(a)i (monotone)g(algebra)f(with)f(carrier)g Fk(N)7 b Fs(.)54 b(This)32 b(construction)g(preserv)m(es)212 4288 y(all)i(compatibilit)m (y)e(requiremen)m(ts,)j(hence)g(b)s(oth)f(de\014nitions)e(of)j Fp(!)s Fs(-termination)e(yield)g(the)i(same)212 4401 y(class)30 b(of)f(TRSs.)40 b(\(F)-8 b(or)31 b(p)s(olynomial)c (termination)h(this)h(is)g(unclear.\))40 b(A)29 b(TRS)g Fq(R)h Fs(is)f(called)g Fo(lo)-5 b(oping)212 4514 y Fs(if)32 b(it)h(admits)f(a)h(rewrite)f(sequence)i Fp(t)29 b Fq(!)1615 4476 y Fn(+)1615 4543 y FA(R)1709 4514 y Fp(C)7 b Fs([)p Fp(t\033)s Fs(])33 b(for)g(some)g(term)g Fp(t)p Fs(,)h(some)f(con)m (text)i Fp(C)k Fs(and)33 b(some)212 4627 y(substitution)j Fp(\033)s Fs(.)66 b(A)38 b(TRS)g Fq(R)g Fs(is)g(called)f Fo(cyclic)44 b Fs(if)37 b(it)h(admits)g(a)h(rewrite)f(sequence)g Fp(t)h Fq(!)3390 4589 y Fn(+)3390 4656 y FA(R)3493 4627 y Fp(t)f Fs(for)212 4740 y(some)28 b(term)f Fp(t)p Fs(.)39 b(A)28 b(TRS)e Fq(R)h Fs(o)m(v)m(er)h(a)g(signature)e Fq(F)37 b Fs(is)26 b(called)g Fo(self-emb)-5 b(e)g(dding)36 b Fs(if)27 b(it)f(admits)h(a)g(rewrite)212 4853 y(sequence)40 b Fp(t)g Fq(!)762 4815 y Fn(+)762 4882 y FA(R)867 4853 y Fp(u)g Fq(!)1050 4820 y FA(\003)1050 4884 y(E)6 b Fn(m)n(b)o(\()p FA(F)h Fn(\))1346 4853 y Fp(t)39 b Fs(for)g(some)h(terms)g Fp(t)f Fs(and)g Fp(u)p Fs(.)68 b(Recen)m(t)41 b(in)m(v)m(estigations)e (of)h(these)212 4977 y(notions)30 b(include)e([2)q(,)i(6)q(,)g(7)q(,)g (16)q(,)h(17)q(,)f(21)q(,)h(24)q(,)f(27)q(].)353 5090 y(V)-8 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Fs(\))p Fp(;)p 1930 1712 V 15 w Fs($)q(\()p Fp(z)t Fs(\))p Fp(;)15 b Fe(LHS)q Fs(\))564 1960 y Fq(!)655 1979 y Fn(\(I)r(I\))877 1960 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b Fs(1)p Fp(;)g Fs(0)p Fp(;)g(\013)1306 1974 y Fm(i)1330 1983 y Fd(2)1387 1960 y Fp(:)g(:)g(:)h(\013)1566 1974 y Fm(i)1590 1986 y Fc(k)1633 1960 y Fs(\()p Fp(t)1701 1974 y Fn(1)1740 1960 y Fs(\))p Fp(;)p 1815 1881 167 4 v 15 w Fs($)p Fp(\013)1918 1974 y Fm(i)1942 1983 y Fd(1)1982 1960 y Fs(\()p Fp(x)p Fs(\))p Fp(;)f(\014)2195 1974 y Fm(i)2219 1983 y Fd(2)2275 1960 y Fp(:)g(:)g(:)h(\014)2447 1974 y Fm(i)2471 1986 y Fc(k)2514 1960 y Fs(\()p Fp(t)2582 1974 y Fn(2)2621 1960 y Fs(\))p Fp(;)p 2696 1881 160 4 v 15 w Fs($)p Fp(\014)2792 1974 y Fm(i)2816 1983 y Fd(1)2857 1960 y Fs(\()p Fp(z)t Fs(\))p Fp(;)f Fe(LHS)q Fs(\))564 2129 y Fq(!)655 2096 y FA(\003)655 2161 y Fn(\(I)r(I\))877 2129 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b Fs(1)p Fp(;)g Fs(0)p Fp(;)g(t)1281 2143 y Fn(1)1323 2129 y Fp(;)p 1363 2050 98 4 v 15 w Fs($)p Fp(\015)6 b Fs(\()p 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Fp(;)15 b(:)g(:)g(:)i(;)e(x)925 4607 y Fm(k)968 4592 y Fs(\))25 b(=)1124 4436 y Ff(\()1197 4528 y Fs(10)c Fq(\001)g Fe(co)s(de)q Fs(\()p Fp(x)1617 4542 y Fn(1)1677 4528 y Fs(+)f Fq(\001)15 b(\001)g(\001)21 b Fs(+)f Fp(x)2037 4543 y Fm(k)2079 4528 y Fs(\))92 b(if)29 b Fp(k)f(>)d Fs(0)1197 4664 y(10)919 b(if)29 b Fp(k)f Fs(=)d(0)353 4874 y(Before)32 b(w)m(e)e(can)h(extend)f(the)h(in)m(terpretation)f(to) h Fp(A)p Fs(,)f(w)m(e)h(need)f(a)h(few)f(further)f(auxiliary)f(func-) 212 4987 y(tions,)i(some)h(of)f(whic)m(h)f(dep)s(end)g(on)h(the)h(in)m (terpretation)f(of)g(terms)h(o)m(v)m(er)g Fq(F)2845 5001 y FA(U)2921 4987 y Fq(n)20 b(f)p Fp(A)p Fq(g)p Fs(.)353 5100 y(F)-8 b(or)32 b(the)g(treatmen)m(t)h(of)e(the)g(last)h(argumen)m (t)f(of)h Fp(A)f Fs(w)m(e)h(\014rst)e(de\014ne)h(t)m(w)m(o)h(unary)f (functions)e Fp(\036)3611 5114 y FA(Q)212 5213 y Fs(and)38 b Fp( )456 5227 y FA(Q)556 5213 y Fs(whic)m(h)e(dep)s(end)h(on)h(the)g (TRS)f Fq(Q)p Fs(.)64 b(F)-8 b(unction)38 b Fp(\036)2271 5227 y FA(Q)2371 5213 y Fs(estimates)g(the)g(gro)m(wth)h(of)f(the)g (last)1897 5462 y(13)p eop %%Page: 14 14 14 13 bop 212 337 a Fs(argumen)m(t)38 b(of)g Fp(A)g Fs(caused)g(b)m(y)g (an)g(application)e(of)i(a)h(t)m(yp)s(e)f(\(I)s(I)s(I\))f(rewrite)g (rule)f(of)i Fq(U)3190 304 y FA(0)3214 337 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\):)57 b(F)-8 b(or)212 450 y(ev)m(ery)41 b(upp)s(er)c(b)s(ound)h Fp(x)i Fs(of)g(the)g(in)m(terpretations)f(of)h Fe(LHS)p Fs(,)i Fp(\036)2414 464 y FA(Q)2476 450 y Fs(\()p Fp(x)p Fs(\))e(is)f(an)h(upp)s(er)d(b)s(ound)h(of)i(the)212 562 y(corresp)s(onding)26 b(in)m(terpretations)h(of)h Fe(RHS)o Fs(,)h(for)e(all)g Fe(LHS)f Fq(!)f Fe(RHS)g Fq(2)f(Q)p Fs(.)40 b(F)-8 b(unction)28 b Fp(\036)3155 576 y FA(Q)3227 562 y Fs(:)j Fk(N)3342 576 y Fn(+)3432 562 y Fq(!)26 b Fk(N)3608 576 y Fn(+)212 675 y Fs(is)k(de\014ned)f(as)h (follo)m(ws:)439 878 y Fp(\036)493 892 y FA(Q)555 878 y Fs(\()p Fp(x)p Fs(\))c(=)f(max)46 b Fq(f)p Fp(x)p Fq(g)21 b([)f(f)p Fs([)p Fp(\022)s Fs(]\()p Fe(RHS)p Fs(\))25 b Fq(j)h Fe(LHS)f Fq(!)h Fe(RHS)e Fq(2)h(Q)31 b Fs(and)f([)p Fp(\022)s Fs(]\()p Fe(LHS)p Fs(\))25 b Fl(6)g Fp(x)p Fq(g)212 1080 y Fs(Since)d Fe(LHS)i Fs(and)f Fe(RHS)f Fs(only)h(con)m(tain)h(function)e(sym)m(b)s(ols)g(in)g(the)h(signature) g(of)g Fq(Q)p Fs(,)j(w)m(e)d(can)h(compute)212 1193 y([)p Fp(\022)s Fs(]\()p Fe(LHS)p Fs(\))39 b(and)f([)p Fp(\022)s Fs(]\()p Fe(RHS)p Fs(\))g(for)h(ev)m(ery)g(assignmen)m(t)f Fp(\022)j Fs(of)d(p)s(ositiv)m(e)g(in)m(tegers)h(for)f(the)g(v)-5 b(ariables)38 b(in)212 1306 y Fe(LHS)25 b Fq(!)h Fe(RHS)o Fs(.)41 b(Here)31 b([)p Fp(\022)s Fs(]\()p Fp(t)p Fs(\))f(for)h Fp(t)25 b Fq(2)f(T)f Fs(\()p Fq(F)1657 1320 y FA(Q)1720 1306 y Fp(;)15 b Fq(X)e Fs(\))31 b(is)e(inductiv)m(ely)f(de\014ned)h (as)i(follo)m(ws:)439 1586 y([)p Fp(\022)s Fs(]\()p Fp(t)p Fs(\))26 b(=)760 1430 y Ff(\()833 1522 y Fs([)p Fp(f)10 b Fs(]\([)p Fp(\022)s Fs(]\()p Fp(t)1137 1536 y Fn(1)1177 1522 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(\022)s Fs(]\()p Fp(t)1578 1537 y Fm(k)1620 1522 y Fs(\)\))92 b(if)29 b Fp(t)c Fs(=)g Fp(f)10 b Fs(\()p Fp(t)2142 1536 y Fn(1)2181 1522 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)2416 1537 y Fm(k)2459 1522 y Fs(\))31 b(with)e Fp(k)f Fl(>)d Fs(0)833 1658 y Fp(\022)s Fs(\()p Fp(t)p Fs(\))800 b(if)29 b Fp(t)c Fq(2)g(X)212 1860 y Fs(Because)30 b(the)g(in)m(terpretation)e(of)h(ev)m (ery)h(function)e(sym)m(b)s(ol)f(in)h Fq(Q)h Fs(is)f(strictly)g (monotonic)h(in)f(all)g(its)212 1973 y(argumen)m(ts|\(an)c(immediate)e (consequence)i(of)g(the)g(strict)f(monotonicit)m(y)g(of)h Fq(\016)f Fs(and)g Fe(co)s(de)q Fs(\)|there)212 2086 y(can)k(only)f(b)s(e)h(a)g(\014nite)f(n)m(um)m(b)s(er)f(of)i(assignmen) m(ts)g Fp(\022)i Fs(suc)m(h)d(that)i([)p Fp(\022)s Fs(]\()p Fe(LHS)p Fs(\))e Fl(6)f Fp(x)h Fs(for)h(a)g(giv)m(en)g Fp(x)f Fq(2)e Fk(N)3583 2100 y Fn(+)3648 2086 y Fs(.)212 2199 y(Hence)37 b(the)e(maxim)m(um)g(is)f(formed)h(o)m(v)m(er)i(a)f (\014nite)f(computable)g(set)h(and)f(th)m(us)g(the)h(function)e Fp(\036)3611 2213 y FA(Q)212 2312 y Fs(is)h(w)m(ell-de\014ned.)55 b(Note)37 b(that)g(for)e(ev)m(ery)i(ground)e(instance)g Fp(t)g Fq(!)f Fp(u)i Fs(of)g(a)g(rewrite)f(rule)g(in)f Fq(Q)p Fs(,)k(w)m(e)212 2425 y(ha)m(v)m(e)33 b Fp(\036)477 2439 y FA(Q)538 2425 y Fs(\([)p Fp(t)p Fs(]\))28 b Fl(>)e Fs([)p Fp(u)p Fs(].)44 b(It)31 b(is)g(easy)g(to)h(c)m(hec)m(k)h(that)f Fp(\036)1989 2439 y FA(Q)2082 2425 y Fs(is)e(monotonic)h(and)g(that)h Fp(\036)3049 2439 y FA(Q)3111 2425 y Fs(\()p Fp(x)p Fs(\))27 b Fl(>)f Fp(x)32 b Fs(holds.)212 2538 y(F)-8 b(unction)42 b Fp( )663 2552 y FA(Q)735 2538 y Fs(:)34 b Fk(N)41 b Fq(\002)28 b Fk(N)1047 2552 y Fn(+)1156 2538 y Fq(!)45 b Fk(N)1352 2552 y Fn(+)1459 2538 y Fs(is)c(used)g(to)i(comp)s(ensate)g (a)f(p)s(oten)m(tial)g(increase)g(of)g(the)g(last)212 2651 y(argumen)m(t)28 b(of)g Fp(A)g Fs(along)f(an)h(application)e(of)h (a)h(t)m(yp)s(e)g(\(I)s(I)s(I\))f(rewrite)g(rule)f(of)i Fq(U)2872 2618 y FA(0)2895 2651 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\).)41 b(It)28 b(is)f(de\014ned)212 2763 y(inductiv)m(ely)h(as)j(follo)m(ws:)602 2966 y Fp( )661 2980 y FA(Q)723 2966 y Fs(\(0)p Fp(;)15 b(y)s Fs(\))27 b(=)e Fp(y)e Fs(+)d(1)439 3104 y Fp( )498 3118 y FA(Q)560 3104 y Fs(\()p Fp(x)h Fs(+)f(1)p Fp(;)15 b(y)s Fs(\))26 b(=)f Fp(y)e Fs(+)d Fp( )1267 3118 y FA(Q)1329 3104 y Fs(\()p Fp(x;)15 b(\036)1510 3118 y FA(Q)1573 3104 y Fs(\()p Fp(y)s Fs(\)\))212 3306 y(One)30 b(easily)g(v)m(eri\014es)f (that)i Fp( )1214 3320 y FA(Q)1307 3306 y Fs(is)e(strictly)g(monotonic) i(in)e(b)s(oth)h(argumen)m(ts.)353 3419 y(One)41 b(more)g(auxiliary)d (function)i(is)g(needed)g(b)s(efore)h(w)m(e)g(can)h(de\014ne)e(the)h (in)m(terpretation)f(of)212 3532 y(function)25 b(sym)m(b)s(ol)f Fp(A)p Fs(.)39 b(The)26 b(in)m(terpretation)f(de\014ned)f(ab)s(o)m(v)m (e)j(has)f(the)g(prop)s(ert)m(y)f(that)i(for)e(a)h(ground)212 3645 y(term)45 b Fp(t)f Fs(not)h(con)m(taining)f(an)m(y)h Fp(A)f Fs(sym)m(b)s(ol,)k(the)c(top)h(part)g(of)f Fp(t)h Fs(that)g(consists)f(of)g(sym)m(b)s(ols)g(in)212 3758 y Fq(f)p Fe(cons)q Fp(;)p 465 3706 168 4 v 15 w Fe(cons)q Fp(;)15 b(";)g Fs($)p Fp(;)g Fs(0)p Fp(;)g Fs(1)p Fq(g)50 b Fs(can)45 b(b)s(e)g(extracted)i(from)e([)p Fp(t)p Fs(].)86 b(A)46 b(suitable)e(extraction)i(function)e Fp(\031)k Fs(is)212 3871 y(de\014ned)29 b(next.)212 4081 y Fb(De\014nition)35 b(5.5)46 b Fs(The)30 b(function)e Fp(\031)13 b Fs(:)31 b Fk(N)1601 4095 y Fn(+)1692 4081 y Fq(!)25 b(T)e Fs(\()p Fq(F)9 b Fs(\))30 b(from)g(p)s(ositiv)m(e)e(in)m(tegers)i(to)h(ground)e (terms)g(is)212 4194 y(inductiv)m(ely)f(de\014ned)h(as)i(follo)m(ws:) 439 4807 y Fp(\031)s Fs(\()p Fp(x)p Fs(\))26 b(=)738 4320 y Ff(8)738 4402 y(>)738 4429 y(>)738 4457 y(>)738 4484 y(>)738 4511 y(>)738 4539 y(>)738 4566 y(>)738 4593 y(>)738 4620 y(>)738 4648 y(>)738 4675 y(>)738 4702 y(<)738 4866 y(>)738 4893 y(>)738 4920 y(>)738 4948 y(>)738 4975 y(>)738 5002 y(>)738 5029 y(>)738 5057 y(>)738 5084 y(>)738 5111 y(>)738 5139 y(>)738 5166 y(:)819 4405 y Fp(")674 b Fs(if)29 b Fp(x)c Fs(=)g(1)819 4540 y($)671 b(if)29 b Fp(x)c Fs(=)g(2)819 4676 y(0)671 b(if)29 b Fp(x)c Fs(=)g(3)819 4811 y(1)671 b(if)29 b Fp(x)c Fs(=)g(4)819 4947 y Fe(cons)q Fs(\()p Fp(\031)s Fs(\()p Fp(y)s Fs(\))p Fp(;)15 b(\031)s Fs(\()p Fp(z)t Fs(\)\))94 b(if)29 b Fp(x)c Fs(=)g Fp(z)g Fq(\016)20 b Fs(5)h Fq(\016)f Fp(y)k Fq(\016)c Fs(6)31 b(with)e Fp(y)f(>)d Fs(0)31 b(w)m(ell-balanced)p 819 5031 V 819 5082 a Fe(cons)q Fs(\()p Fp(\031)s Fs(\()p Fp(y)s Fs(\))p Fp(;)15 b(\031)s Fs(\()p Fp(z)t Fs(\)\))94 b(if)29 b Fp(x)c Fs(=)g(5)c Fq(\016)f Fp(y)j Fq(\016)e Fs(6)g Fq(\016)f Fp(z)25 b Fq(\016)20 b Fs(7)26 b Fq(")g Fs(\(2)21 b(+)f Fp(`)p Fs(\()p Fp(y)s Fs(\)\))31 b(with)e Fp(y)f(>)d Fs(0)31 b(w)m(ell-balanced)819 5218 y Fp(A)p Fs(\()p Fp(";)15 b(:)g(:)g(:)j(;)d(")p Fs(\))291 b(otherwise)1897 5462 y(14)p eop %%Page: 15 15 15 14 bop 212 337 a Fs(Here)32 b(for)f(w)m(ell-de\014nedness)e(w)m(e)j (require)e(that)i(the)g(digits)d(5)j(\(\\left)g(paren)m(thesis"\))f (and)g(6)h(\(\\righ)m(t)212 450 y(paren)m(thesis"\))k(form)g(a)h(w)m (ell-balanced)e(sequence)i(in)e(the)i(decimal)e(represen)m(tation)h(of) h Fp(y)i Fs(in)c(the)212 562 y(\014fth)25 b(and)g(sixth)g(clause)h(of)g (the)g(de\014nition.)36 b(F)-8 b(ormally)g(,)27 b(the)f(decimal)f (represen)m(tation)h(of)g(a)g(natural)212 675 y(n)m(um)m(b)s(er)j(is)h (w)m(ell-balanced)f(if)g(it)h(is)f(generated)j(b)m(y)e(the)h(con)m (text-free)h(grammar)486 872 y Fp(S)88 b Fq(!)83 b Fs(5)p Fp(S)5 b Fs(6)26 b Fq(j)f Fp(S)5 b(S)30 b Fq(j)c Fp(T)38 b Fq(j)25 b Fs(56)481 985 y Fp(T)96 b Fq(!)83 b Fs(0)25 b Fq(j)h Fs(1)g Fq(j)f Fs(2)h Fq(j)f Fs(3)h Fq(j)f Fs(4)h Fq(j)g Fs(7)f Fq(j)h Fs(8)g Fq(j)f Fs(9)212 1181 y(with)30 b(start)h(sym)m(b)s(ol)f Fp(S)5 b Fs(.)42 b(F)-8 b(or)32 b(example,)f(123)h(and)e(1556576236)36 b(are)31 b(w)m(ell-balanced)f(n) m(um)m(b)s(ers)f(but)212 1294 y(65)i(is)f(not.)353 1507 y(The)20 b(premise)f(on)h Fp(y)j Fs(in)c(the)i(\014fth)e(and)h(sixth)f (clause)h(of)g(the)h(de\014nition)d(ensures)h(w)m(ell-de\014nedness)212 1620 y(of)31 b(the)f(de\014nition)e(of)j Fp(\031)s Fs(.)212 1832 y Fb(Lemma)i(5.6)46 b Fo(The)33 b(function)f Fp(\031)k Fo(is)d(wel)5 b(l-de\014ne)-5 b(d.)212 1965 y(Pr)g(o)g(of)52 b Fs(First)28 b(w)m(e)h(sho)m(w)f(that)h(there)g(is)f(no)g(am)m(biguit) m(y)g(in)g(the)g(\014fth)g(clause)g(of)h(the)g(de\014nition)d(of)j Fp(\031)s Fs(.)212 2078 y(Supp)s(ose)24 b(to)j(the)f(con)m(trary)g (that)h(there)f(exists)f Fp(x)h Fq(2)e Fk(N)2072 2092 y Fn(+)2163 2078 y Fs(suc)m(h)h(that)i Fp(x)e Fs(=)g Fp(z)15 b Fq(\016)c Fs(5)g Fq(\016)g Fp(y)j Fq(\016)d Fs(6)29 b(=)24 b Fp(z)3284 2045 y FA(0)3319 2078 y Fq(\016)11 b Fs(5)g Fq(\016)g Fp(y)3535 2045 y FA(0)3571 2078 y Fq(\016)g Fs(6)212 2190 y(with)34 b(di\013eren)m(t)g(w)m(ell-balanced)g Fp(y)s(;)15 b(y)1491 2157 y FA(0)1547 2190 y Fp(>)32 b Fs(0.)54 b(Without)35 b(loss)f(of)h(generalit)m(y)-8 b(,)37 b(assume)e(that)g Fp(y)h(>)c(y)3625 2157 y FA(0)3648 2190 y Fs(.)212 2303 y(Then)h Fp(y)h Fs(=)d Fp(z)680 2270 y FA(00)745 2303 y Fq(\016)23 b Fs(5)g Fq(\016)g Fp(y)997 2270 y FA(0)1054 2303 y Fs(for)34 b(some)g Fp(z)1474 2270 y FA(0)q(0)1548 2303 y Fq(2)d Fk(N)6 b Fs(.)58 b(Ho)m(w)m(ev)m (er,)37 b(since)c Fp(y)2452 2270 y FA(0)2509 2303 y Fs(is)g(w)m (ell-balanced,)h Fp(y)i Fs(cannot)f(b)s(e)212 2416 y(w)m(ell-balanced) 27 b(\(as)h(there)g(is)f(no)h(6)g(in)f Fp(y)1585 2383 y FA(0)1636 2416 y Fs(that)h(corresp)s(onds)f(to)h(the)h(displa)m(y)m (ed)d(5)i(in)f Fp(y)s Fs(\).)40 b(A)28 b(similar)212 2529 y(argumen)m(t)j(sho)m(ws)f(that)h(there)g(is)e(no)h(am)m(biguit)m (y)g(in)f(the)i(sixth)e(clause)h(of)h(the)f(de\014nition)e(of)j Fp(\031)s Fs(.)45 b Fl(\003)353 2742 y Fs(F)-8 b(or)31 b(instance,)439 2946 y Fp(\031)s Fs(\(156655635626\))h(=)25 b Fp(\031)s Fs(\(1566)d Fq(\016)f Fs(5)g Fq(\016)f Fs(563562)j Fq(\016)e Fs(6\))26 b(=)f Fe(cons)q Fs(\()p Fp(\031)s Fs(\(563562\))p Fp(;)15 b(\031)s Fs(\(1566\)\))1136 3084 y(=)25 b Fe(cons)q Fs(\()p Fp(A)p Fs(\()p Fp(";)15 b(:)g(:)g(:)i(;)e(") p Fs(\))p Fp(;)g(A)p Fs(\()p Fp(";)g(:)g(:)g(:)k(;)c(")p Fs(\)\))212 3288 y(and)30 b Fp(\031)s Fs(\(5462777536\))g(=)25 b Fe(cons)q Fs(\()p Fp(\031)s Fs(\(3\))p Fp(;)15 b(\031)s Fs(\(5462777\)\))32 b(=)25 b Fe(cons)p Fs(\(0)p Fp(;)p 2393 3237 168 4 v 15 w Fe(cons)r Fs(\(1)p Fp(;)15 b Fs($\)\).)353 3401 y(W)-8 b(e)32 b(need)e(t)m(w)m(o)i(more)e(de\014nitions:)521 3605 y Fe(revc)q Fs(\()p Fp(x;)15 b(y)s Fs(\))26 b(=)f Fp(x)c Fq(\016)f Fp(y)j Fq(\016)e Fs(7)26 b Fq(")f Fp(`)p Fs(\()p Fp(x)p Fs(\))439 3743 y Fe(b)s(ound)p Fs(\()p Fp(x;)15 b(y)s Fs(\))26 b(=)f(6)p Fp(`)p Fs(\()p Fp(x)p Fs(\))d(+)d(3)p Fp(`)p Fs(\()p Fp(y)s Fs(\))212 3947 y(The)k(function)g Fe(revc)h Fs(\(for)g(\\rev)m(erse)h (concatenation"\))h(is)c(strictly)h(monotonic)h(and)f Fe(b)s(ound)g Fs(is)g(mono-)212 4060 y(tonic)30 b(in)f(b)s(oth)h (argumen)m(ts.)212 4273 y Fb(De\014nition)35 b(5.7)46 b Fs(The)30 b(in)m(terpretation)g([)p Fp(A)p Fs(])h(is)e(de\014ned)h (as)g(follo)m(ws:)439 4477 y([)p Fp(A)p Fs(]\()p Fp(x)644 4491 y Fn(1)685 4477 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)938 4491 y Fn(2)p Fm(n)p Fn(+15)1146 4477 y Fs(\))26 b(=)f Fe(co)s(de)q Fs(\()p Fp( )1573 4491 y FA(Q)1635 4477 y Fs(\()p Fp(D)s Fs(\()p Fp(x)1835 4491 y Fn(1)1875 4477 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)2129 4491 y Fn(2)p Fm(n)p Fn(+14)2337 4477 y Fs(\))p Fp(;)g(x)2464 4491 y Fn(2)p Fm(n)p Fn(+15)2672 4477 y Fs(\)\))21 b Fq(\016)g Fs(8)212 4681 y(Here)31 b Fp(D)s Fs(\()p Fp(x)592 4695 y Fn(1)632 4681 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)885 4695 y Fn(2)p Fm(n)p Fn(+14)1093 4681 y Fs(\))31 b(denotes)g(the)f(expression)439 4814 y Ff( )511 4855 y Fn(2)p Fm(n)p Fn(+8)537 4883 y Ff(Y)538 5078 y Fm(i)p Fn(=1)695 4969 y Fp(\037)p Fs(\()p Fp(x)839 4983 y Fm(i)892 4969 y Fq(\025)25 b Fs([)p Fp(V)1066 4983 y Fm(i)1095 4969 y Fs(]\))1155 4814 y Ff(!)1248 4969 y Fq(\001)20 b Fp(E)5 b Fs(\()p Fp(x)1452 4983 y Fn(1)1492 4969 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)1746 4983 y Fn(2)p Fm(n)p Fn(+14)1954 4969 y Fs(\))20 b(+)2100 4814 y Ff( )2172 4969 y Fs(1)h Fq(\000)2329 4855 y Fn(2)p Fm(n)p Fn(+8)2355 4883 y Ff(Y)2356 5078 y Fm(i)p Fn(=1)2512 4969 y Fp(\037)p Fs(\()p Fp(x)2656 4983 y Fm(i)2710 4969 y Fq(\025)k Fs([)p Fp(V)2884 4983 y Fm(i)2912 4969 y Fs(]\))2972 4814 y Ff(!)3065 4969 y Fq(\001)3111 4814 y Ff( )3182 4855 y Fn(2)p Fm(n)p Fn(+14)3219 4883 y Ff(X)3227 5078 y Fm(i)p Fn(=1)3401 4969 y Fp(x)3453 4983 y Fm(i)3481 4814 y Ff(!)1897 5462 y Fs(15)p eop %%Page: 16 16 16 15 bop 212 337 a Fp(\037)10 b Fs(:)31 b Fq(f)p Fe(false)p Fp(;)15 b Fe(true)p Fq(g)26 b(!)f(f)p Fs(0)p Fp(;)15 b Fs(1)p Fq(g)22 b Fs(is)e(de\014ned)f(b)m(y)h Fp(\037)p Fs(\()p Fe(false)p Fs(\))25 b(=)g(0)c(and)f Fp(\037)p Fs(\()p Fe(true)p Fs(\))26 b(=)f(1,)e(and)c Fp(E)5 b Fs(\()p Fp(x)3138 351 y Fn(1)3178 337 y Fp(;)15 b(:)g(:)g(:)i(;)e(x) 3432 351 y Fn(2)p Fm(n)p Fn(+14)3640 337 y Fs(\))212 450 y(denotes)31 b(the)f(expression)439 647 y Fe(len)p Fs(\()p Fp(\031)s Fs(\()p Fp(x)725 661 y Fn(2)p Fm(n)p Fn(+9)899 647 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e(\031)s Fs(\()p Fp(x)1278 661 y Fn(2)p Fm(n)p Fn(+14)1486 647 y Fs(\)\))21 b(+)f Fe(b)s(ound)p Fs(\()p Fp(x)1991 661 y Fn(2)p Fm(n)p Fn(+11)2199 647 y Fp(;)15 b(x)2291 661 y Fn(2)p Fm(n)p Fn(+12)2499 647 y Fs(\))20 b Fq(\001)h Fe(facto)m(r)r Fs(\()p Fp(x)2907 661 y Fn(1)2947 647 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)3200 661 y Fn(2)p Fm(n)p Fn(+14)3408 647 y Fs(\))212 845 y(where)30 b Fe(facto)m(r)r Fs(\()p Fp(x)782 859 y Fn(1)822 845 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)1075 859 y Fn(2)p Fm(n)p Fn(+14)1283 845 y Fs(\))31 b(is)e(an)i (abbreviation)e(for)439 1127 y(2\()p Fp(x)571 1141 y Fn(2)p Fm(n)p Fn(+9)765 1127 y Fs(+)20 b Fp(x)908 1141 y Fn(2)p Fm(n)p Fn(+10)1115 1127 y Fs(\))h(+)f Fe(revc)q Fs(\()p Fp(x)1503 1141 y Fn(2)p Fm(n)p Fn(+11)1711 1127 y Fp(;)15 b(x)1803 1141 y Fn(2)p Fm(n)p Fn(+12)2011 1127 y Fs(\))20 b(+)g Fe(revc)q Fs(\()p Fp(x)2398 1141 y Fn(2)p Fm(n)p Fn(+13)2606 1127 y Fp(;)15 b(x)2698 1141 y Fn(2)p Fm(n)p Fn(+14)2906 1127 y Fs(\))21 b(+)3053 1013 y Fn(2)p Fm(n)p Fn(+8)3071 1041 y Ff(X)3080 1236 y Fm(i)p Fn(=1)3236 1127 y Fp(x)3288 1141 y Fm(i)353 1415 y Fs(Note)34 b(that)f Fp(D)s Fs(\()p Fp(x)939 1429 y Fn(1)978 1415 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)1232 1429 y Fn(2)p Fm(n)p Fn(+14)1440 1415 y Fs(\))29 b(=)f Fp(E)5 b Fs(\()p Fp(x)1762 1429 y Fn(1)1802 1415 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)2056 1429 y Fn(2)p Fm(n)p Fn(+14)2264 1415 y Fs(\))32 b(if)f Fp(x)2468 1429 y Fm(i)2525 1415 y Fl(>)d Fs([)p Fp(V)2702 1429 y Fm(i)2730 1415 y Fs(])33 b(for)f(all)f(1)e Fl(6)f Fp(i)h Fl(6)f Fs(2)p Fp(n)21 b Fs(+)h(8.)212 1528 y(If)36 b(there)h(is)f(at)i (least)f(one)g(1)f Fl(6)g Fp(i)g Fl(6)f Fs(2)p Fp(n)25 b Fs(+)f(8)37 b(suc)m(h)g(that)g Fp(x)2286 1542 y Fm(i)2350 1528 y Fp(<)e Fs([)p Fp(V)2534 1542 y Fm(i)2563 1528 y Fs(],)k(then)d Fp(D)s Fs(\()p Fp(x)3030 1542 y Fn(1)3070 1528 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)3323 1542 y Fn(2)p Fm(n)p Fn(+14)3531 1528 y Fs(\))37 b(=)212 1572 y Ff(P)308 1599 y Fn(2)p Fm(n)p Fn(+14)308 1668 y Fm(i)p Fn(=1)531 1641 y Fp(x)583 1655 y Fm(i)611 1641 y Fs(.)212 1847 y Fb(Lemma)c(5.8)46 b Fo(F)-7 b(or)29 b(every)f(gr)-5 b(ound)29 b(term)g Fp(t)f Fo(the)g(de)-5 b(cimal)30 b(r)-5 b(epr)g(esentation)31 b(of)d(its)g(interpr)-5 b(etation)31 b Fs([)p Fp(t)p Fs(])212 1960 y Fo(has)j(a)f(wel)5 b(l-b)-5 b(alanc)g(e)g(d)34 b(se)-5 b(quenc)g(e)33 b(of)g Fs(5)g Fo(and)g Fs(6)g Fo(digits.)212 2092 y(Pr)-5 b(o)g(of)52 b Fs(Easy)30 b(induction)c(on)j(the)h(structure)e(of)i Fp(t)p Fs(.)40 b(If)29 b Fp(t)c Fq(2)f(f)p Fp(";)15 b Fs($)p Fp(;)g Fs(0)p Fp(;)g Fs(1)p Fq(g)33 b Fs(then)c(the)h(decimal)e (represen-)212 2205 y(tation)34 b(of)f([)p Fp(t)p Fs(])h(do)s(es)f(not) g(con)m(tain)h(an)m(y)f(5)h(or)f(6.)50 b(If)33 b Fp(t)c Fs(=)h Fe(cons)q Fs(\()p Fp(t)2382 2219 y Fn(1)2422 2205 y Fp(;)15 b(t)2495 2219 y Fn(2)2534 2205 y Fs(\))34 b(or)f Fp(t)d Fs(=)p 2881 2154 168 4 v 30 w Fe(cons)p Fs(\()p Fp(t)3116 2219 y Fn(1)3156 2205 y Fp(;)15 b(t)3229 2219 y Fn(2)3269 2205 y Fs(\))33 b(then)g(the)212 2318 y(result)27 b(follo)m(ws)g(from)h(the)g(induction)e(h)m(yp)s(othesis.)39 b(If)28 b Fp(t)d Fs(=)g Fp(A)p Fs(\()p Fp(t)2390 2332 y Fn(1)2429 2318 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)2664 2332 y Fn(2)p Fm(n)p Fn(+15)2872 2318 y Fs(\))28 b(or)h Fp(t)c Fs(=)g Fp(f)10 b Fs(\()p Fp(t)3322 2332 y Fn(1)3360 2318 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)3595 2333 y Fm(k)3638 2318 y Fs(\))212 2431 y(with)32 b Fp(f)38 b Fq(2)29 b(F)660 2445 y FA(Q)755 2431 y Fs(then)k(the)g(decimal)f(represen)m(tation)h (of)g([)p Fp(t)p Fs(])g(do)s(es)f(not)i(con)m(tain)f(an)m(y)g(5)g(or)g (6)g(b)m(y)g(the)212 2544 y(de\014nition)28 b(of)j(the)f(function)f Fe(co)s(de)q Fs(.)2170 b Fl(\003)353 2750 y Fs(The)30 b(next)h(lemma)f(states)h(that)g Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))h(is)d(su\016cien)m(tly)g(close)i(to)g Fp(t)p Fs(.)212 2956 y Fb(De\014nition)k(5.9)46 b Fs(Let)40 b Fq(\030)f Fs(b)s(e)g(the)h(smallest)e(congruence)j(on)e(ground)f (terms)i(suc)m(h)f(that)h Fp(t)g Fq(\030)g Fp(t)3650 2923 y FA(0)212 3069 y Fs(holds)27 b(if)g(ro)s(ot)q(\()p Fp(t)p Fs(\))p Fp(;)15 b Fs(ro)s(ot)q(\()p Fp(t)1070 3036 y FA(0)1094 3069 y Fs(\))25 b Fq(2)g(f)p Fp(A)p Fq(g)18 b([)d(F)1557 3083 y FA(Q)1620 3069 y Fs(.)40 b(In)27 b(other)i(w)m(ords,)f Fp(t)e Fq(\030)f Fp(t)2504 3036 y FA(0)2555 3069 y Fs(if)i(the)i(top)g(parts)f(consisting)f(of)212 3182 y(sym)m(b)s(ols)21 b(in)h Fq(f)p Fp(";)15 b Fs($)p Fp(;)g Fs(0)p Fp(;)g Fs(1)p Fp(;)g Fe(cons)t Fp(;)p 1243 3131 V 15 w Fe(cons)q Fq(g)23 b Fs(in)f Fp(t)g Fs(and)g Fp(t)1835 3149 y FA(0)1881 3182 y Fs(coincide.)37 b(Let)23 b(us)f(call)g(a)h(term)g(o)m(v)m(er)h(the)f(restricted)212 3295 y(signature)g Fq(f)p Fp(";)15 b Fs($)p Fp(;)g Fs(0)p Fp(;)g Fs(1)p Fp(;)g Fe(cons)t Fp(;)p 1193 3244 V 15 w Fe(cons)q Fq(g)24 b Fo(pur)-5 b(e)p Fs(.)39 b(W)-8 b(e)25 b(extend)e Fq(\030)g Fs(to)i(sequences)e(of)h(terms)f(comp)s (onen)m(t)m(wise.)353 3501 y(F)-8 b(or)28 b(instance,)f(if)f Fp(f)34 b Fq(2)25 b(F)1199 3515 y FA(Q)1288 3501 y Fs(then)h Fe(cons)q Fs(\(0)p Fp(;)15 b Fe(cons)q Fs(\(1)p Fp(;)g(A)p Fs(\()p Fp(:)g(:)g(:)j Fs(\)\)\))27 b Fq(\030)e Fe(cons)p Fs(\(0)p Fp(;)15 b Fe(cons)r Fs(\(1)p Fp(;)g(f)10 b Fs(\()p Fp(:)15 b(:)g(:)i Fs(\)\)\))28 b(for)e(all)212 3614 y(sequences)31 b(of)f(argumen)m(ts)h(of)f Fp(A)h Fs(and)f Fp(f)10 b Fs(.)40 b(On)29 b(the)i(other)f(hand,)p 2468 3563 V 30 w Fe(cons)q Fs(\(0)p Fp(;)15 b(")p Fs(\))27 b Fq(6\030)p 2956 3563 V 25 w Fe(cons)p Fs(\(1)p Fp(;)15 b(")p Fs(\).)212 3820 y Fb(Lemma)33 b(5.10)46 b Fo(If)f Fp(t)f Fo(is)h(a)g(gr)-5 b(ound)46 b(term)g(then)f Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))k Fq(\030)e Fp(t)p Fo(.)78 b(In)45 b(addition,)k(if)c Fp(t)f Fo(is)h(pur)-5 b(e)46 b(then)212 3933 y Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))26 b(=)f Fp(t)p Fo(.)212 4066 y(Pr)-5 b(o)g(of)56 b Fs(Let)33 b Fp(t)f Fs(b)s(e)g(a)h(ground)f(term.)47 b(W)-8 b(e)34 b(pro)m(v)m(e)g(that)f Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))d Fq(\030)f Fp(t)j Fs(b)m(y)h(induction)d(on)j(the)g (structure)212 4179 y(of)28 b Fp(t)p Fs(.)40 b(The)28 b(base)g(case)h(is)e(an)h(immediate)f(consequence)i(of)f(the)g (de\014nitions)e(of)i([)g(])h(and)e Fp(\031)s Fs(.)40 b(Supp)s(ose)212 4292 y Fp(t)25 b Fs(=)g Fe(cons)q Fs(\()p Fp(t)602 4306 y Fn(1)641 4292 y Fp(;)15 b(t)714 4306 y Fn(2)754 4292 y Fs(\).)40 b(W)-8 b(e)28 b(ha)m(v)m(e)g([)p Fp(t)p Fs(])e(=)f([)p Fp(t)1476 4306 y Fn(2)1515 4292 y Fs(])13 b Fq(\016)g Fs(5)g Fq(\016)g Fs([)p Fp(t)1785 4306 y Fn(1)1828 4292 y Fs(])g Fq(\016)g Fs(6.)42 b(According)27 b(to)h(Lemma)f(5.8)h(the)f(subsequence)212 4405 y(of)42 b(the)g(digits)e(5)i(and)f(6)h(in)f([)p Fp(t)1292 4419 y Fn(1)1331 4405 y Fs(])h(is)f(w)m(ell-balanced.)73 b(Hence)43 b Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))i(=)f Fe(cons)p Fs(\()p Fp(\031)s Fs(\([)p Fp(t)3133 4419 y Fn(1)3174 4405 y Fs(]\))p Fp(;)15 b(\031)s Fs(\([)p Fp(t)3422 4419 y Fn(2)3463 4405 y Fs(]\)\))45 b Fq(\030)212 4517 y Fe(cons)q Fs(\()p Fp(t)448 4531 y Fn(1)487 4517 y Fp(;)15 b(t)560 4531 y Fn(2)600 4517 y Fs(\))27 b(b)m(y)f(the)g(de\014nition)f(of)h Fp(\031)j Fs(and)d(the)h(induction)d(h)m(yp)s(othesis.)38 b(The)26 b(case)h Fp(t)e Fs(=)p 3250 4466 V 25 w Fe(cons)q Fs(\()p Fp(t)3486 4531 y Fn(1)3525 4517 y Fp(;)15 b(t)3598 4531 y Fn(2)3638 4517 y Fs(\))212 4630 y(is)25 b(just)g(as)h(easy)-8 b(.)40 b(If)26 b(ro)s(ot\()p Fp(t)p Fs(\))g(=)f Fp(A)h Fs(or)f(ro)s(ot)q(\()p Fp(t)p Fs(\))h Fq(2)e(F)1927 4644 y FA(Q)2015 4630 y Fs(then)i(the)g(decimal)e(represen)m(tation)i(of)g ([)p Fp(t)p Fs(])g(ends)212 4743 y(with)j(the)i(digit)e(8)i(or)f(0.)41 b(Hence)31 b Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))c(=)e Fp(A)p Fs(\()p Fp(";)15 b(:)g(:)g(:)i(;)e(")p Fs(\))32 b(and)e(th)m(us)g Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))c Fq(\030)f Fp(t)30 b Fs(b)m(y)g(the)h(de\014nition)d(of)212 4856 y Fq(\030)p Fs(.)353 4969 y(T)-8 b(o)36 b(conclude)g(the)f(latter)i(statemen)m(t,)i (according)c(to)i(the)f(former)f(statemen)m(t)j(and)d(the)h(de\014-)212 5082 y(nition)31 b(of)i Fq(\030)f Fs(it)g(is)f(su\016cien)m(t)h(to)h (sho)m(w)g(that)g Fp(\031)s Fs(\([)p Fp(t)p Fs(]\))h(is)d(pure)h (whenev)m(er)g Fp(t)g Fs(is)g(pure.)46 b(This)31 b(is)g(easily)212 5195 y(pro)m(v)m(ed)g(b)m(y)f(induction)e(on)i(the)h(structure)f(of)g Fp(t)p Fs(,)h(similar)c(to)32 b(the)e(ab)s(o)m(v)m(e)i(pro)s(of.)643 b Fl(\003)1897 5462 y Fs(16)p eop %%Page: 17 17 17 16 bop 212 337 a Fb(Lemma)33 b(5.11)46 b Fo(If)33 b Fp(W)k Fq(\030)25 b Fp(W)1228 304 y FA(0)1284 337 y Fo(then)33 b Fe(len)p Fs(\()p Fp(W)13 b Fs(\))25 b(=)g Fe(len)p Fs(\()p Fp(W)2128 304 y FA(0)2151 337 y Fs(\))p Fo(.)212 469 y(Pr)-5 b(o)g(of)62 b Fs(Let)40 b Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(t)p Fs(\))42 b Fq(!)e Fp(A)p Fs(\()p Fp(V)1377 483 y Fn(1)1417 469 y Fp(;)15 b(W)1543 483 y Fn(1)1583 469 y Fp(;)g(t)1656 483 y Fn(1)1696 469 y Fs(\))39 b(and)g Fp(W)53 b Fq(\030)40 b Fp(W)2305 436 y FA(0)2328 469 y Fs(.)68 b(Due)39 b(to)h(the)g(fact)g(that)g(the)g (argu-)212 582 y(men)m(ts)30 b(of)g(the)f(left-hand)g(side)f(of)i(the)g (rewrite)f(rules)f(in)g Fq(U)2257 549 y FA(0)2280 582 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))31 b(are)f(terms)f(o)m(v)m(er)i (the)f(signature)212 695 y Fq(f)p Fp(";)15 b Fs($)p Fp(;)g Fs(0)p Fp(;)g Fs(1)p Fp(;)g Fe(cons)5 b Fp(;)p 806 644 168 4 v 15 w Fe(cons)q Fq(g)p Fs(,)36 b(it)f(follo)m(ws)f(that)i Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)1981 662 y FA(0)2004 695 y Fp(;)g(t)p Fs(\))36 b(also)f(matc)m(hes)h(the)f(left-hand)f (side.)53 b(If)35 b(w)m(e)212 808 y(apply)f(the)h(same)h(rewrite)f (rule,)g(w)m(e)h(obtain)e Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)2114 775 y FA(0)2138 808 y Fp(;)g(t)p Fs(\))34 b Fq(!)f Fp(A)p Fs(\()p Fp(V)2581 775 y FA(0)2560 832 y Fn(1)2604 808 y Fp(;)15 b(W)2743 775 y FA(0)2730 832 y Fn(1)2770 808 y Fp(;)g(t)2843 822 y Fn(1)2883 808 y Fs(\))35 b(with)f Fp(V)3218 822 y Fn(1)3291 808 y Fq(\030)f Fp(V)3468 775 y FA(0)3448 832 y Fn(1)3527 808 y Fs(and)212 921 y Fp(W)298 935 y Fn(1)368 921 y Fq(\030)d Fp(W)568 888 y FA(0)555 945 y Fn(1)595 921 y Fs(.)50 b(If)33 b(one)h(of)g Fp(V)1094 935 y Fn(1)1133 921 y Fs(,)h Fp(V)1266 888 y FA(0)1246 945 y Fn(1)1323 921 y Fs(equals)e Fp(V)54 b Fs(then)33 b(b)s(oth)g(are)h(equal)f(to)h Fp(V)21 b Fs(.)50 b(F)-8 b(rom)34 b(this)f(observ)-5 b(ation)212 1034 y(w)m(e)31 b(easily)e(obtain)h Fe(len)p Fs(\()p 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1455 y Fs(\))26 b Fl(>)f Fe(b)s(ound)p Fs(\()p Fp(y)1743 1469 y Fn(2)p Fm(n)p Fn(+11)1950 1455 y Fp(;)15 b(y)2035 1469 y Fn(2)p Fm(n)p Fn(+12)2243 1455 y Fs(\))212 1651 y(b)m(y)30 b(the)h(monotonicit)m(y)f(of)h Fe(b)s(ound)p Fs(.)40 b(Using)30 b(Lemma)h(5.12,)h(it)e(follo)m(ws)f(that)439 1847 y Fp(E)5 b Fs(\()p Fp(x)598 1861 y Fn(1)638 1847 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)892 1861 y Fn(2)p Fm(n)p Fn(+14)1100 1847 y Fs(\))465 1985 y(=)25 b Fe(len)p Fs(\()p Fp(\031)s Fs(\()p Fp(x)847 1999 y Fn(2)p Fm(n)p Fn(+9)1020 1985 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e(\031)s Fs(\()p Fp(x)1399 1999 y Fn(2)p Fm(n)p Fn(+14)1607 1985 y Fs(\)\))21 b(+)f Fe(b)s(ound)p Fs(\()p Fp(x)2112 1999 y Fn(2)p Fm(n)p Fn(+11)2320 1985 y Fp(;)15 b(x)2412 1999 y Fn(2)p Fm(n)p Fn(+12)2620 1985 y Fs(\))21 b Fq(\001)f Fe(facto)m(r)r Fs(\()p Fp(x)3028 1999 y Fn(1)3068 1985 y Fp(;)15 b(:)g(:)g(:)h(;)f(x) 3321 1999 y Fn(2)p Fm(n)p Fn(+14)3529 1985 y Fs(\))465 2122 y Fl(>)25 b Fe(b)s(ound)o Fs(\()p Fp(x)883 2136 y Fn(2)p Fm(n)p 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1046 y Fn(2)p Fm(n)p Fn(+9)899 1032 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e (\031)s Fs(\()p Fp(x)1278 1046 y Fn(2)p Fm(n)p Fn(+14)1486 1032 y Fs(\)\))21 b(+)f Fe(b)s(ound)p Fs(\()p Fp(x)1991 1046 y Fn(2)p Fm(n)p Fn(+11)2199 1032 y Fp(;)15 b(x)2291 1046 y Fn(2)p Fm(n)p Fn(+12)2499 1032 y Fs(\))20 b Fq(\001)h Fe(facto)m(r)r Fs(\()p Fp(x)2907 1046 y Fn(1)2947 1032 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)3200 1046 y Fn(2)p Fm(n)p Fn(+14)3408 1032 y Fs(\))212 1216 y(and)439 1462 y Fp(D)s Fs(\()p Fp(y)597 1476 y Fn(1)637 1462 y Fp(;)g(:)g(:)g(:)h(;)f(y)883 1476 y Fn(2)p Fm(n)p Fn(+14)1091 1462 y Fs(\))25 b(=)1247 1348 y Fn(2)p Fm(n)p Fn(+14)1284 1375 y Ff(X)1292 1571 y Fm(i)p Fn(=1)1466 1462 y Fp(y)1511 1476 y Fm(i)212 1728 y Fs(Since)d Fe(revc)q Fs(\()p Fp(x)683 1742 y Fn(2)p Fm(n)p Fn(+11)891 1728 y Fp(;)15 b(x)983 1742 y Fn(2)p Fm(n)p Fn(+12)1191 1728 y Fs(\))26 b Fl(>)f Fp(x)1400 1742 y Fn(2)p Fm(n)p Fn(+11)1613 1728 y Fs(+)6 b Fp(x)1742 1742 y Fn(2)p Fm(n)p Fn(+12)1972 1728 y Fs(and)22 b Fe(revc)q Fs(\()p Fp(x)2382 1742 y Fn(2)p Fm(n)p Fn(+13)2590 1728 y Fp(;)15 b(x)2682 1742 y Fn(2)p Fm(n)p Fn(+14)2890 1728 y Fs(\))26 b Fl(>)f Fp(x)3099 1742 y Fn(2)p Fm(n)p Fn(+13)3312 1728 y Fs(+)6 b Fp(x)3441 1742 y Fn(2)p Fm(n)p Fn(+14)3648 1728 y Fs(,)212 1841 y(it)30 b(follo)m(ws)f(that)439 2086 y Fe(facto)m(r)r Fs(\()p Fp(x)746 2100 y Fn(1)786 2086 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)1040 2100 y Fn(2)p Fm(n)p Fn(+14)1248 2086 y Fs(\))25 b Fl(>)1404 1972 y Fn(2)p Fm(n)p Fn(+14)1440 1999 y Ff(X)1449 2195 y Fm(i)p Fn(=1)1623 2086 y Fp(x)1675 2100 y Fm(i)212 2352 y Fs(and)36 b(th)m(us)h Fp(D)s Fs(\()p Fp(x)767 2366 y Fn(1)807 2352 y Fp(;)15 b(:)g(:)g(:)h(;)f(x)1060 2366 y Fn(2)p Fm(n)p Fn(+14)1268 2352 y Fs(\))37 b Fp(>)f(D)s Fs(\()p Fp(y)1605 2366 y Fn(1)1644 2352 y Fp(;)15 b(:)g(:)g(:)i(;)e(y)1891 2366 y Fn(2)p Fm(n)p Fn(+14)2098 2352 y Fs(\).)61 b(The)36 b(desired)g(result)g(no)m(w)h(follo)m(ws)f(from)212 2465 y(the)31 b(strict)f(monotonicit)m(y)g(of)h Fp( )1320 2479 y FA(Q)1382 2465 y Fs(,)f Fe(co)s(de)q Fs(,)h(and)e Fq(\016)p Fs(.)1688 b Fl(\003)353 2657 y Fs(In)22 b(the)g(\014nal)f (part)h(of)g(this)f(section)i(w)m(e)f(will)e(mak)m(e)j(go)s(o)s(d)f(on) g(our)g(claim)f(that)h(the)h(in)m(terpretation)212 2770 y([)32 b(])g(is)e(capable)i(of)f(orien)m(ting)g(all)f(ground)h (instances)g(of)h(the)f(rewrite)g(rules)f(in)h Fq(U)3028 2737 y FA(0)3051 2770 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))32 b(from)g(left)212 2883 y(to)f(righ)m(t.)41 b(W)-8 b(e)31 b(need)f(a)h(few)f(preliminary)d(results,)i(concerning)h(the)h (in)m(terpla)m(y)e(of)i Fe(revc)g Fs(and)f([)g(].)212 3076 y Fb(Lemma)j(5.14)46 b Fo(L)-5 b(et)38 b Fp(\013)f Fo(b)-5 b(e)37 b(a)h(se)-5 b(quenc)g(e)37 b(of)g(gr)-5 b(ound)39 b(terms.)56 b(F)-7 b(or)38 b(al)5 b(l)38 b(gr)-5 b(ound)38 b(terms)g Fp(s)f Fo(and)h Fp(t)f Fo(we)212 3189 y(have)c Fe(revc)q Fs(\([)p Fp(\013)p Fs(\()p Fp(s)p Fs(\)])p Fp(;)15 b Fs([)p Fp(t)p Fs(]\))28 b(=)d Fe(revc)q Fs(\([)p Fp(s)p Fs(])p Fp(;)15 b Fs([)p 1460 3139 59 4 v Fp(\013)q Fs(\()p Fp(t)p Fs(\)]\))p Fo(.)212 3321 y(Pr)-5 b(o)g(of)55 b Fs(By)33 b(induction)d(on)i(the)h(length)e(of)i Fp(\013)p Fs(.)47 b(If)32 b Fp(\013)h Fs(is)e(the)i(empt)m(y)f (sequence,)i(then)e(the)g(lemma)g(is)212 3434 y(trivially)c(true.)40 b(Let)31 b Fp(\013)26 b Fs(=)f Fp(u\014)5 b Fs(.)41 b(Then)439 3618 y Fe(revc)q Fs(\([)p Fp(\013)p Fs(\()p Fp(s)p Fs(\)])p Fp(;)15 b Fs([)p Fp(t)p Fs(]\))28 b(=)d Fe(revc)q Fs(\([)p Fe(cons)q Fs(\()p Fp(u;)15 b(\014)5 b Fs(\()p Fp(s)p Fs(\)\)])p Fp(;)15 b Fs([)p Fp(t)p Fs(]\))1035 3756 y(=)25 b Fe(revc)q Fs(\([)p Fe(cons)q Fs(]\([)p Fp(u)p Fs(])p Fp(;)15 b Fs([)p Fp(\014)5 b Fs(\()p Fp(s)p Fs(\)]\))p Fp(;)15 b Fs([)p Fp(t)p Fs(]\))1035 3894 y(=)25 b Fe(revc)q Fs(\([)p Fp(\014)5 b Fs(\()p Fp(s)p Fs(\)])22 b Fq(\016)e Fs(5)h Fq(\016)f Fs([)p Fp(u)p Fs(])h Fq(\016)g Fs(6)p Fp(;)15 b Fs([)p Fp(t)p Fs(]\))1035 4032 y(=)25 b([)p Fp(\014)5 b Fs(\()p Fp(s)p Fs(\)])21 b Fq(\016)g Fs(5)g Fq(\016)f Fs([)p Fp(u)p Fs(])h Fq(\016)g Fs(6)f Fq(\016)h Fs([)p Fp(t)p Fs(])g Fq(\016)f Fs(7)26 b Fq(")g Fp(`)p Fs(\([)p Fp(\014)5 b Fs(\()p Fp(s)p Fs(\)])21 b Fq(\016)g Fs(5)f Fq(\016)h Fs([)p Fp(u)p Fs(])g Fq(\016)g Fs(6\))1035 4170 y(=)k Fe(revc)q Fs(\([)p Fp(\014)5 b Fs(\()p Fp(s)p Fs(\)])p Fp(;)15 b Fs(5)22 b Fq(\016)f Fs([)p Fp(u)p Fs(])g Fq(\016)f Fs(6)h Fq(\016)g Fs([)p Fp(t)p Fs(])f Fq(\016)h Fs(7)26 b Fq(")f Fp(`)p Fs(\(5)c Fq(\016)g Fs([)p Fp(u)p Fs(])g Fq(\016)f Fs(6\))q(\))1035 4307 y(=)25 b Fe(revc)q Fs(\([)p Fp(\014)5 b Fs(\()p Fp(s)p Fs(\)])p Fp(;)15 b Fs([)p 1604 4256 168 4 v Fe(cons)s Fs(]\([)p Fp(u)p Fs(])p Fp(;)g Fs([)p Fp(t)p Fs(]\)\))1035 4445 y(=)25 b Fe(revc)q Fs(\([)p Fp(\014)5 b Fs(\()p Fp(s)p Fs(\)])p Fp(;)15 b Fs([)p 1604 4394 V Fe(cons)s Fs(\()p Fp(u;)g(t)p Fs(\)]\))1035 4594 y(=)25 b Fe(revc)q Fs(\([)p Fp(s)p Fs(])p Fp(;)15 b Fs([)p 1478 4520 57 4 v Fp(\014)6 b Fs(\()p 1570 4542 168 4 v Fe(cons)q Fs(\()p Fp(u;)15 b(t)p Fs(\)\)]\))1529 b(\(7\))1035 4742 y(=)25 b Fe(revc)q Fs(\([)p Fp(s)p Fs(])p Fp(;)15 b Fs([)p 1478 4668 57 4 v Fp(\014)6 b Fs(\()p 1570 4692 53 4 v Fp(u)q Fs(\()p Fp(t)p Fs(\)\)]\))1035 4890 y(=)25 b Fe(revc)q Fs(\([)p Fp(s)p Fs(])p Fp(;)15 b Fs([)p 1478 4816 109 4 v Fp(u\014)6 b Fs(\()p Fp(t)p Fs(\)]\))1035 5028 y(=)25 b Fe(revc)q Fs(\([)p Fp(s)p Fs(])p Fp(;)15 b Fs([)p 1478 4978 59 4 v Fp(\013)r Fs(\()p Fp(t)p Fs(\)]\))212 5213 y(Here)31 b(\(7\))g(follo)m(ws)f(from)g(the)g(induction)e(h)m(yp)s (othesis.)1512 b Fl(\003)1897 5462 y Fs(19)p eop %%Page: 20 20 20 19 bop 212 337 a Fb(Lemma)33 b(5.15)46 b Fo(L)-5 b(et)31 b Fp(s)p Fo(,)g Fp(t)p Fo(,)g(and)h Fp(u)f Fo(b)-5 b(e)31 b(gr)-5 b(ound)32 b(terms)g(and)g Fp(x)f Fo(a)g(p)-5 b(ositive)32 b(inte)-5 b(ger.)41 b(If)31 b Fs([)p Fp(s)p Fs(])26 b Fl(>)f Fs([)p Fp(t)p Fs(])31 b Fo(then)439 500 y Fe(revc)q Fs(\([)p Fe(cons)q Fs(\()p Fp(s;)15 b(u)p Fs(\)])p Fp(;)g(x)p Fs(\))22 b(+)e([)p Fp(t)p Fs(])26 b Fl(>)f Fe(revc)q Fs(\([)p Fe(cons)q Fs(\()p Fp(t;)15 b(u)p Fs(\)])p Fp(;)g(x)p Fs(\))22 b(+)e([)p Fp(s)p Fs(])439 638 y Fe(revc)q Fs(\()p Fp(x;)15 b Fs([)p 745 587 168 4 v Fe(cons)r Fs(\()p Fp(s;)g(u)p Fs(\)]\))21 b(+)f([)p Fp(t)p Fs(])26 b Fl(>)f Fe(revc)q Fs(\()p Fp(x;)15 b Fs([)p 1802 587 V Fe(cons)q Fs(\()p Fp(t;)g(u)p Fs(\)]\))22 b(+)e([)p Fp(s)p Fs(])212 802 y Fo(Mor)-5 b(e)g(over,)34 b(if)e Fs([)p Fp(s)p Fs(])25 b Fp(>)g Fs([)p Fp(t)p Fs(])33 b Fo(then)g(b)-5 b(oth)34 b(ine)-5 b(qualities)33 b(ar)-5 b(e)34 b(strict.)212 934 y(Pr)-5 b(o)g(of)53 b Fs(The)30 b(\014rst)g(statemen)m(t)i(is)e(obtained)f(as)i(follo)m(ws:)439 1098 y Fe(revc)q Fs(\([)p Fe(cons)q Fs(\()p Fp(s;)15 b(u)p Fs(\)])p Fp(;)g(x)p Fs(\))22 b(+)e([)p Fp(t)p Fs(])26 b(=)f([)p Fe(cons)q Fs(\()p Fp(s;)15 b(u)p Fs(\)])21 b Fq(\016)g Fp(x)f Fq(\016)g Fs(7)26 b Fq(")g Fp(`)p Fs(\([)p Fe(cons)q Fs(\()p Fp(s;)15 b(u)p Fs(\)]\))22 b(+)e([)p Fp(t)p Fs(])1400 1236 y(=)25 b([)p Fp(u)p Fs(])c Fq(\016)f Fs(5)h Fq(\016)g Fs([)p Fp(s)p Fs(])f Fq(\016)h Fs(6)f Fq(\016)h Fp(x)f Fq(\016)h Fs(7)k Fq(")h Fp(`)p Fs(\([)p Fe(cons)q Fs(\()p Fp(s;)15 b(u)p Fs(\)]\))22 b(+)e([)p Fp(t)p Fs(])1400 1374 y Fl(>)25 b Fs([)p Fp(u)p Fs(])c Fq(\016)f Fs(5)h Fq(\016)g Fs([)p Fp(t)p Fs(])f Fq(\016)h Fs(6)f Fq(\016)h Fp(x)f Fq(\016)h Fs(7)26 b Fq(")f Fp(`)p Fs(\([)p Fe(cons)q Fs(\()p Fp(s;)15 b(u)p Fs(\)]\))22 b(+)e([)p Fp(s)p Fs(])425 b(\(8\))1400 1511 y Fl(>)25 b Fs([)p Fp(u)p Fs(])c Fq(\016)f Fs(5)h Fq(\016)g Fs([)p Fp(t)p Fs(])f Fq(\016)h Fs(6)f Fq(\016)h Fp(x)f Fq(\016)h Fs(7)26 b Fq(")f Fp(`)p Fs(\([)p Fe(cons)q Fs(\()p Fp(t;)15 b(u)p Fs(\)]\))22 b(+)e([)p Fp(s)p Fs(])435 b(\(9\))1400 1649 y(=)25 b([)p Fe(cons)q Fs(\()p Fp(t;)15 b(u)p Fs(\)])21 b Fq(\016)g Fp(x)f Fq(\016)h Fs(7)k Fq(")h Fp(`)p Fs(\([)p Fe(cons)q Fs(\()p Fp(t;)15 b(u)p Fs(\)]\))22 b(+)e([)p Fp(s)p Fs(])1400 1787 y(=)25 b Fe(revc)q Fs(\([)p Fe(cons)q Fs(\()p Fp(t;)15 b(u)p Fs(\)])p Fp(;)g(x)p Fs(\))22 b(+)e([)p Fp(s)p Fs(])212 1951 y(Here)36 b(\(8\))g(follo)m(ws) f(from)f(the)i(fact)g(that,)i(for)d(all)f Fp(x;)15 b(y)s(;)g(z)38 b Fq(2)33 b Fk(N)2370 1965 y Fn(+)2435 1951 y Fs(,)j Fp(x)24 b Fq(\016)g Fp(y)i Fq(\016)e Fp(z)k Fs(+)23 b Fp(y)2993 1918 y FA(0)3050 1951 y Fl(>)33 b Fp(x)23 b Fq(\016)h Fp(y)3346 1918 y FA(0)3393 1951 y Fq(\016)g Fp(z)j Fs(+)c Fp(y)212 2064 y Fs(whenev)m(er)k Fp(y)h Fl(>)d Fp(y)825 2031 y FA(0)875 2064 y Fs(and)h(\(9\))i(follo)m(ws)e (from)h(the)g(monotonicit)m(y)g(of)g Fp(`)p Fs(,)h Fq(\016)p Fs(,)g(and)f(7)2906 2031 y FA(\016)p Fn(\()p FA(\001)p Fn(\))3020 2064 y Fs(.)40 b(If)26 b([)p Fp(s)p Fs(])f Fp(>)g Fs([)p Fp(t)p Fs(])j(then)212 2177 y(\(9\))k(b)s(ecomes)e (strict,)h(and)e(so)439 2340 y Fe(revc)q Fs(\([)p Fe(cons)q Fs(\()p Fp(s;)15 b(u)p Fs(\)])p Fp(;)g(x)p Fs(\))22 b(+)e([)p Fp(t)p Fs(])26 b Fp(>)f Fe(revc)q Fs(\([)p Fe(cons)q Fs(\()p Fp(t;)15 b(u)p Fs(\)])p Fp(;)g(x)p Fs(\))22 b(+)e([)p Fp(s)p Fs(])212 2504 y(The)30 b(other)h(t)m(w)m(o)g(statemen)m(ts)h (are)f(obtained)f(in)f(a)i(similar)c(fashion.)1024 b Fl(\003)353 2676 y Fs(The)30 b(last)g(preliminary)d(result)i(is)h(a)h (v)-5 b(arian)m(t)30 b(of)h(the)f(previous)f(lemma.)212 2848 y Fb(Lemma)k(5.16)46 b Fo(L)-5 b(et)42 b Fp(\013)g Fo(and)h Fp(\014)k Fo(b)-5 b(e)41 b(se)-5 b(quenc)g(es)42 b(of)g(gr)-5 b(ound)42 b(terms,)j Fp(t)c Fo(a)h(gr)-5 b(ound)43 b(term,)i(and)d Fp(x)g Fo(a)212 2961 y(p)-5 b(ositive)34 b(inte)-5 b(ger.)42 b(If)32 b Fs([)p Fp(\013)p Fs(\()p Fp(")p Fs(\)])c Fl(>)c Fs([)p Fp(\014)5 b Fs(\()p Fp(")p Fs(\)])35 b Fo(then)439 3124 y Fe(revc)q Fs(\([)p Fp(\013)p Fs(\()p Fp(t)p Fs(\)])p Fp(;)15 b(x)p Fs(\))23 b(+)d([)p Fp(\014)5 b Fs(\()p Fp(")p Fs(\)])27 b Fl(>)e Fe(revc)q Fs(\([)p Fp(\014)5 b Fs(\()p Fp(t)p Fs(\)])p Fp(;)15 b(x)p Fs(\))22 b(+)e([)p Fp(\013)p Fs(\()p Fp(")p Fs(\)])212 3288 y Fo(Mor)-5 b(e)g(over,)34 b(if)e Fs([)p Fp(\013)p Fs(\()p Fp(")p Fs(\)])27 b Fp(>)e Fs([)p Fp(\014)5 b Fs(\()p Fp(")p Fs(\)])35 b Fo(then)e(the)g(ine)-5 b(quality)33 b(is)g(strict.)212 3421 y(Pr)-5 b(o)g(of)53 b Fs(W)-8 b(rite)31 b Fp(\013)26 b Fs(=)f Fp(s)951 3435 y Fn(1)1005 3421 y Fp(:)15 b(:)g(:)h(s)1169 3436 y Fm(k)1242 3421 y Fs(and)30 b Fp(\014)g Fs(=)25 b Fp(t)1629 3435 y Fn(1)1683 3421 y Fp(:)15 b(:)g(:)i(t)1838 3436 y Fm(l)1864 3421 y Fs(.)40 b(W)-8 b(e)32 b(ha)m(v)m(e)439 3584 y([)p Fp(\013)p Fs(\()p Fp(")p Fs(\)])c(=)d(1)20 b Fq(\016)h Fs(5)g Fq(\016)f Fs([)p Fp(s)1113 3599 y Fm(k)1156 3584 y Fs(])g Fq(\016)h Fs(6)15 b Fq(\001)g(\001)g(\001)22 b(\016)e Fs(5)h Fq(\016)g Fs([)p Fp(s)1719 3598 y Fn(1)1758 3584 y Fs(])g Fq(\016)f Fs(6)687 3722 y Fl(>)25 b Fs(1)20 b Fq(\016)h Fs(5)g Fq(\016)f Fs([)p Fp(t)1103 3737 y Fm(l)1129 3722 y Fs(])h Fq(\016)f Fs(6)15 b Fq(\001)g(\001)g(\001)22 b(\016)f Fs(5)g Fq(\016)f Fs([)p Fp(t)1682 3736 y Fn(1)1722 3722 y Fs(])g Fq(\016)h Fs(6)26 b(=)f([)p Fp(\014)5 b Fs(\()p Fp(")p Fs(\)])212 3886 y(and)30 b(th)m(us)439 4049 y Fp(a)c Fs(=)f(5)20 b Fq(\016)h Fs([)p Fp(s)808 4064 y Fm(k)850 4049 y Fs(])g Fq(\016)g Fs(6)15 b Fq(\001)g(\001)g(\001)21 b(\016)g Fs(5)g Fq(\016)f Fs([)p Fp(s)1413 4063 y Fn(1)1453 4049 y Fs(])g Fq(\016)h Fs(6)26 b Fl(>)f Fs(5)20 b Fq(\016)h Fs([)p Fp(t)1920 4064 y Fm(l)1946 4049 y Fs(])f Fq(\016)h Fs(6)15 b Fq(\001)g(\001)g(\001)22 b(\016)f Fs(5)f Fq(\016)h Fs([)p Fp(t)2499 4063 y Fn(1)2538 4049 y Fs(])g Fq(\016)f Fs(6)26 b(=)f Fp(b)212 4213 y Fs(whic)m(h)j(implies)e([)p Fp(\013)p Fs(\()p Fp(t)p Fs(\)])g(=)f([)p Fp(t)p Fs(])18 b Fq(\016)f Fp(a)26 b Fl(>)f Fs([)p Fp(t)p Fs(])17 b Fq(\016)g Fp(b)26 b Fs(=)f([)p Fp(\014)5 b Fs(\()p Fp(t)p Fs(\)].)41 b(No)m(w)30 b(the)f(desired)e(inequalit)m(y)g(is)h(obtained) g(as)212 4326 y(in)h(the)i(pro)s(of)e(of)i(Lemma)f(5.15:)439 4490 y Fe(revc)q Fs(\([)p Fp(\013)p Fs(\()p Fp(t)p Fs(\)])p Fp(;)15 b(x)p Fs(\))23 b(+)d([)p Fp(\014)5 b Fs(\()p Fp(")p Fs(\)])27 b(=)e([)p Fp(\013)p Fs(\()p Fp(t)p Fs(\)])d Fq(\016)e Fp(x)g Fq(\016)h Fs(7)26 b Fq(")g Fp(`)p Fs(\([)p Fp(\013)p Fs(\()p Fp(t)p Fs(\)]\))c(+)e([)p Fp(\014)5 b Fs(\()p Fp(")p Fs(\)])1325 4627 y(=)25 b([)p Fp(t)p Fs(])20 b Fq(\016)h Fp(a)f Fq(\016)h Fp(x)f Fq(\016)h Fs(7)26 b Fq(")f Fp(`)p Fs(\([)p Fp(\013)p Fs(\()p Fp(t)p Fs(\)]\))e(+)c(1)i Fq(\016)g Fp(b)1325 4765 y Fl(>)k Fs([)p Fp(t)p Fs(])20 b Fq(\016)h Fp(b)f Fq(\016)h Fp(x)f Fq(\016)h Fs(7)k Fq(")h Fp(`)p Fs(\([)p Fp(\013)p Fs(\()p Fp(t)p Fs(\)]\))d(+)c(1)i Fq(\016)g Fp(a)906 b Fs(\(10\))1325 4903 y Fl(>)25 b Fs([)p Fp(t)p Fs(])20 b Fq(\016)h Fp(b)f Fq(\016)h Fp(x)f Fq(\016)h Fs(7)k Fq(")h Fp(`)p Fs(\([)p Fp(\014)5 b Fs(\()p Fp(t)p Fs(\)]\))22 b(+)e(1)h Fq(\016)f Fp(a)1325 5041 y Fs(=)25 b([)p Fp(\014)5 b Fs(\()p Fp(t)p Fs(\)])21 b Fq(\016)g Fp(x)f Fq(\016)h Fs(7)26 b Fq(")f Fp(`)p Fs(\([)p Fp(\014)5 b Fs(\()p Fp(t)p Fs(\)]\))22 b(+)e([)p Fp(\013)p Fs(\()p Fp(")p Fs(\)])1325 5179 y(=)25 b Fe(revc)q Fs(\([)p Fp(\014)5 b Fs(\()p Fp(t)p Fs(\)])p Fp(;)15 b(x)p Fs(\))22 b(+)e([)p Fp(\013)p Fs(\()p Fp(")p Fs(\)])1897 5462 y(20)p eop %%Page: 21 21 21 20 bop 212 337 a Fs(Here)27 b(\(10\))g(follo)m(ws)e(from)h(the)g (fact)h(that,)g(for)f(all)f Fp(x;)15 b(y)s(;)g(z)30 b Fq(2)25 b Fk(N)2314 351 y Fn(+)2379 337 y Fs(,)i Fp(x)11 b Fq(\016)g Fp(y)k Fq(\016)c Fp(z)17 b Fs(+)11 b(1)g Fq(\016)g Fp(y)2967 304 y FA(0)3017 337 y Fl(>)25 b Fp(x)11 b Fq(\016)g Fp(y)3280 304 y FA(0)3316 337 y Fq(\016)g Fp(z)16 b Fs(+)11 b(1)g Fq(\016)g Fp(y)212 450 y Fs(whenev)m(er)26 b Fp(y)i Fl(>)d Fp(y)824 417 y FA(0)847 450 y Fs(.)39 b(Note)28 b(that)e(the)h(second)f(inequalit)m(y)e(in)h(the)h(ab)s(o)m (v)m(e)h(deriv)-5 b(ation)25 b(b)s(ecomes)i(strict)212 562 y(if)i([)p Fp(\013)p Fs(\()p Fp(")p Fs(\)])f Fp(>)d Fs([)p Fp(\014)5 b Fs(\()p Fp(")p Fs(\)].)2721 b Fl(\003)212 758 y Fb(Theorem)34 b(5.17)46 b Fo(F)-7 b(or)34 b(every)e(gr)-5 b(ound)34 b(instanc)-5 b(e)439 945 y Fp(l)r(\033)29 b Fs(=)c Fp(A)p Fs(\()p Fp(t)781 959 y Fn(1)821 945 y Fp(;)15 b(:)g(:)g(:)h(;)f(t)1055 959 y Fn(2)p Fm(n)p Fn(+15)1263 945 y Fs(\))26 b Fq(!)f Fp(A)p Fs(\()p Fp(u)1595 959 y Fn(1)1635 945 y Fp(;)15 b(:)g(:)g(:)h(;)f(u)1888 959 y Fn(2)p Fm(n)p Fn(+15)2097 945 y Fs(\))25 b(=)g Fp(r)s(\033)212 1132 y Fo(of)33 b(a)g(r)-5 b(ewrite)34 b(rule)e Fp(l)27 b Fq(!)f Fp(r)35 b Fo(in)d Fq(U)1313 1099 y FA(0)1336 1132 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))34 b Fo(we)f(have)g Fs([)p Fp(l)r(\033)s Fs(])26 b Fp(>)f Fs([)p Fp(r)s(\033)s Fs(])p Fo(.)212 1264 y(Pr)-5 b(o)g(of)58 b Fs(First)35 b(w)m(e)g(consider)g(the)g(case)i(that)e([)p Fp(u)1842 1278 y Fm(i)1871 1264 y Fs(])f Fl(>)f Fs([)p Fp(t)2092 1278 y Fm(i)2120 1264 y Fs(])j(for)f(all)f(1)g Fl(6)f Fp(i)h Fl(6)f Fs(2)p Fp(n)24 b Fs(+)f(8.)56 b(By)36 b(de\014nition)212 1377 y(of)i Fq(U)389 1344 y FA(0)412 1377 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))38 b(w)m(e)g(ha)m(v)m(e)h Fp(t)1086 1391 y Fn(1)1125 1377 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)1360 1391 y Fn(2)p Fm(n)p Fn(+8)1569 1377 y Fs(=)37 b Fp(V)20 b Fs(,)39 b Fp(t)1847 1391 y Fn(2)p Fm(n)p Fn(+15)2092 1377 y Fs(=)d Fe(LHS)p Fp(\033)s Fs(,)k(and)d(either)f Fp(u)2983 1391 y Fn(2)p Fm(n)p Fn(+15)3228 1377 y Fs(=)h Fe(LHS)p Fp(\033)j Fs(or)212 1490 y Fp(u)264 1504 y Fn(2)p Fm(n)p Fn(+15)497 1490 y Fs(=)25 b Fe(RHS)767 1504 y Fm(j)803 1490 y Fp(\033)k Fs(for)d(some)h(1)e Fl(6)g Fp(j)31 b Fl(6)25 b Fp(m)p Fs(.)39 b(P)m(ositiv)m(e)26 b(in)m(tegers)h Fp(E)2455 1504 y Fn(1)2520 1490 y Fs(and)f Fp(F)2751 1504 y Fn(1)2816 1490 y Fs(are)h(de\014ned)e(as)h(follo)m (ws:)439 1677 y Fp(E)506 1691 y Fn(1)571 1677 y Fs(=)f([)p Fp(t)725 1691 y Fn(1)765 1677 y Fs(])20 b(+)g([)p Fp(t)959 1691 y Fn(2)999 1677 y Fs(])g(+)g(2\([)p Fp(t)1273 1691 y Fn(2)p Fm(n)p Fn(+9)1446 1677 y Fs(])h(+)f([)p Fp(t)1641 1691 y Fn(2)p Fm(n)p Fn(+10)1848 1677 y Fs(]\))448 1815 y Fp(F)506 1829 y Fn(1)571 1815 y Fs(=)25 b([)p Fp(u)744 1829 y Fn(1)784 1815 y Fs(])20 b(+)g([)p Fp(u)997 1829 y Fn(2)1037 1815 y Fs(])g(+)g(2\([)p Fp(u)1330 1829 y Fn(2)p Fm(n)p Fn(+9)1504 1815 y Fs(])g(+)g([)p Fp(u)1717 1829 y Fn(2)p Fm(n)p Fn(+10)1925 1815 y Fs(]\))212 2002 y(W)-8 b(e)32 b(claim)d(that)439 2189 y Fp(E)506 2203 y Fn(1)571 2189 y Fl(>)c Fp(F)725 2203 y Fn(1)3512 2189 y Fs(\(11\))212 2376 y(W)-8 b(e)33 b(claim)d(moreo)m(v)m(er)j(that,)f (if)e([)p Fp(u)1390 2390 y Fm(i)1419 2376 y Fs(])d Fp(>)f Fs([)p Fp(t)1626 2390 y Fm(i)1654 2376 y Fs(])32 b(for)f(some)g 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Fn(1)895 5041 y Fp(:)15 b(:)g(:)h(w)1081 5055 y Fm(\026)1128 5041 y Fs(\()p Fp(w)r Fs(\))p Fp(\033)s Fs(])p Fp(;)f Fs([)p 1410 4971 46 4 v(1)r(\()p Fp(x)p Fs(\))p Fp(\033)s Fs(]\))22 b(+)e([)p Fp(x)1884 5055 y Fn(1)1924 5041 y Fp(\033)s Fs(])465 5179 y(=)25 b Fe(revc)p Fs(\([1)p Fp(w)884 5193 y Fn(1)940 5179 y Fp(:)15 b(:)g(:)h(w)1126 5193 y Fm(\026)1173 5179 y Fs(\()p Fp(w)r Fs(\))p Fp(\033)s Fs(])p Fp(;)f Fs([)p Fp(x\033)s Fs(]\))23 b(+)d([)p Fp(x)1813 5193 y Fn(1)1853 5179 y Fp(\033)s Fs(])1897 5462 y(22)p eop %%Page: 23 23 23 22 bop 212 337 a Fs(and)30 b(th)m(us)g Fp(E)656 351 y Fn(2)716 337 y Fq(\000)20 b Fp(F)865 351 y Fn(2)930 337 y Fl(>)25 b Fs(0.)41 b(Moreo)m(v)m(er,)32 b(if)e([)p Fp(u)1722 351 y Fn(4)1762 337 y Fs(])25 b Fp(>)g Fs([)p Fp(t)1966 351 y Fn(4)2006 337 y Fs(])30 b(then)g Fp(E)2335 351 y Fn(2)2395 337 y Fq(\000)20 b Fp(F)2544 351 y Fn(2)2609 337 y Fp(>)25 b Fs(0.)353 450 y(This)33 b(concludes)h(the)h(pro)s(of)f (of)g(claim)g(\(12\))r(.)53 b(P)m(ositiv)m(e)35 b(in)m(tegers)g Fp(E)2731 464 y Fn(3)2805 450 y Fs(and)f Fp(F)3044 464 y Fn(3)3119 450 y Fs(are)h(de\014ned)e(as)212 562 y(follo)m(ws:)439 828 y Fp(E)506 842 y Fn(3)571 828 y Fs(=)25 b Fe(revc)q Fs(\([)p Fp(t)914 842 y Fn(2)p Fm(n)p Fn(+13)1122 828 y Fs(])p Fp(;)15 b Fs([)p Fp(t)1245 842 y Fn(2)p Fm(n)p Fn(+14)1454 828 y Fs(]\))20 b(+)1647 714 y Fn(2)p Fm(n)p Fn(+8)1666 741 y Ff(X)1625 937 y Fm(i)p Fn(=)p Fm(n)p Fn(+6)1837 828 y Fs([)p Fp(t)1895 842 y Fm(i)1923 828 y Fs(])448 1151 y Fp(F)506 1165 y Fn(3)571 1151 y Fs(=)25 b Fe(revc)q Fs(\([)p Fp(u)933 1165 y Fn(2)p Fm(n)p Fn(+13)1141 1151 y Fs(])p Fp(;)15 b Fs([)p Fp(u)1283 1165 y Fn(2)p Fm(n)p Fn(+14)1492 1151 y Fs(]\))21 b(+)1685 1038 y Fn(2)p Fm(n)p Fn(+8)1704 1065 y Ff(X)1664 1260 y Fm(i)p Fn(=)p Fm(n)p Fn(+6)1876 1151 y Fs([)p Fp(u)1953 1165 y Fm(i)1981 1151 y Fs(])212 1439 y(W)-8 b(e)32 b(claim)d(that)439 1643 y Fp(E)506 1657 y Fn(3)571 1643 y Fl(>)c Fp(F)725 1657 y Fn(3)3512 1643 y Fs(\(13\))212 1848 y(Moreo)m(v)m(er,)39 b(if)c([)p Fp(u)809 1862 y Fm(i)837 1848 y Fs(])f Fp(>)f Fs([)p Fp(t)1058 1862 y Fm(i)1086 1848 y Fs(])j(for)f(some)h Fp(n)23 b Fs(+)g(6)35 b Fl(6)e Fp(i)h Fl(6)f Fs(2)p Fp(n)24 b Fs(+)f(8)36 b(then)f Fp(E)2627 1862 y Fn(3)2700 1848 y Fp(>)f(F)2863 1862 y Fn(3)2902 1848 y Fs(.)56 b(The)35 b(pro)s(of)g(of)g(this)212 1961 y(claim)30 b(is)f(v)m(ery)i(similar)c (to)32 b(the)e(pro)s(of)g(of)37 b(\(12\))32 b(and)e(hence)h(omitted.) 353 2074 y(F)-8 b(rom)31 b(\(11\))r(,)f(\(12\))r(,)g(and)g(\(13\))i(w)m (e)f(immediately)e(obtain)439 2278 y Fe(facto)m(r)r Fs(\([)p Fp(t)752 2292 y Fn(1)792 2278 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(t)1077 2292 y Fn(2)p Fm(n)p Fn(+14)1285 2278 y Fs(]\))26 b(=)f Fp(E)1534 2292 y Fn(1)1594 2278 y Fs(+)20 b Fp(E)1752 2292 y Fn(2)1811 2278 y Fs(+)g Fp(E)1969 2292 y Fn(3)1371 2416 y Fl(>)25 b Fp(F)1525 2430 y Fn(1)1585 2416 y Fs(+)20 b Fp(F)1734 2430 y Fn(2)1794 2416 y Fs(+)g Fp(F)1943 2430 y Fn(3)2008 2416 y Fs(=)25 b Fe(facto)m(r)r Fs(\([)p Fp(u)2436 2430 y Fn(1)2476 2416 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(u)2780 2430 y Fn(2)p Fm(n)p Fn(+14)2989 2416 y Fs(]\))463 b(\(14\))212 2620 y(and)30 b(if)f(additionally)f([)p Fp(u)1052 2634 y Fm(i)1080 2620 y Fs(])e Fp(>)f Fs([)p Fp(t)1285 2634 y Fm(i)1313 2620 y Fs(])30 b(for)h(some)f(1)c Fl(6)f Fp(i)h Fl(6)f Fs(2)p Fp(n)20 b Fs(+)g(8)30 b(then)439 2824 y Fe(facto)m(r)r Fs(\([)p Fp(t)752 2838 y Fn(1)792 2824 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(t)1077 2838 y Fn(2)p Fm(n)p Fn(+14)1285 2824 y Fs(]\))26 b Fp(>)f Fe(facto)m(r)r Fs(\([)p Fp(u)1799 2838 y Fn(1)1839 2824 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(u)2143 2838 y Fn(2)p Fm(n)p Fn(+14)2351 2824 y Fs(]\))1101 b(\(15\))212 3028 y(>F)-8 b(rom)31 b(\(12\))h(w)m(e)f(easily)e(obtain)439 3233 y Fe(revc)q Fs(\([)p Fp(t)686 3247 y Fn(2)p Fm(n)p Fn(+11)894 3233 y Fs(])p Fp(;)15 b Fs([)p Fp(t)1017 3247 y Fn(2)p Fm(n)p Fn(+12)1226 3233 y Fs(]\))26 b Fl(>)f Fe(revc)p Fs(\([)p Fp(u)1673 3247 y Fn(2)p Fm(n)p Fn(+11)1882 3233 y Fs(])p Fp(;)15 b Fs([)p Fp(u)2024 3247 y Fn(2)p Fm(n)p Fn(+12)2232 3233 y Fs(]\))212 3437 y(Hence)439 3641 y(2)p Fp(`)p Fs(\([)p Fp(t)615 3655 y Fn(2)p Fm(n)p Fn(+11)824 3641 y Fs(]\))21 b(+)f Fp(`)p Fs(\([)p Fp(t)1127 3655 y Fn(2)p Fm(n)p Fn(+12)1335 3641 y Fs(]\))25 b Fl(>)g Fs(2)p Fp(`)p Fs(\([)p Fp(u)1711 3655 y Fn(2)p Fm(n)p Fn(+11)1920 3641 y Fs(]\))c(+)f Fp(`)p Fs(\([)p Fp(u)2242 3655 y Fn(2)p Fm(n)p Fn(+12)2450 3641 y Fs(]\))212 3845 y(b)m(y)29 b(the)h(monotonicit)m(y)f(of)g Fp(`)g Fs(and)g(the)g(fact)i (that)e Fp(`)p Fs(\()p Fe(revc)q Fs(\()p Fp(x;)15 b(y)s Fs(\)\))27 b(=)e(2)p Fp(`)p Fs(\()p Fp(x)p Fs(\))19 b(+)e Fp(`)p Fs(\()p Fp(y)s Fs(\))30 b(for)f(all)f Fp(x;)15 b(y)29 b Fq(2)24 b Fk(N)3583 3859 y Fn(+)3648 3845 y Fs(.)212 3958 y(Therefore)439 4162 y Fe(b)s(ound)p Fs(\([)p Fp(t)768 4176 y Fn(2)p Fm(n)p Fn(+11)976 4162 y Fs(])p Fp(;)15 b Fs([)p Fp(t)1099 4176 y Fn(2)p Fm(n)p Fn(+12)1308 4162 y Fs(]\))26 b Fl(>)f Fe(b)s(ound)p Fs(\([)p Fp(u)1838 4176 y Fn(2)p Fm(n)p Fn(+11)2046 4162 y Fs(])p Fp(;)15 b Fs([)p Fp(u)2188 4176 y Fn(2)p Fm(n)p Fn(+12)2397 4162 y Fs(]\))1055 b(\(16\))212 4367 y(No)m(w)37 b(if)e([)p Fp(u)591 4381 y Fm(i)620 4367 y Fs(])g Fp(>)g Fs([)p Fp(t)844 4381 y Fm(i)872 4367 y Fs(])h(for)g(some)h(1)e Fl(6)g Fp(i)g Fl(6)g Fs(2)p Fp(n)24 b Fs(+)g(8)36 b(then)g(the)h (statemen)m(t)h(of)e(the)h(theorem)f(follo)m(ws)212 4480 y(from)29 b(\(15\))j(and)d(\(16\))i(as)f(in)f(the)h(pro)s(of)f(of)h (Lemma)g(5.13.)42 b(Otherwise,)28 b(w)m(e)j(ha)m(v)m(e)g([)p Fp(u)3123 4494 y Fm(i)3151 4480 y Fs(])26 b(=)f([)p Fp(t)3356 4494 y Fm(i)3384 4480 y Fs(])30 b(for)f(all)212 4592 y(1)g Fl(6)e Fp(i)i Fl(6)e Fs(2)p Fp(n)22 b Fs(+)f(8.)46 b(>F)-8 b(rom)32 b(the)h(\014rst)e(part)h(of)g(Lemma)g(5.10)i(w)m(e)e (obtain)g Fp(u)2799 4606 y Fm(i)2855 4592 y Fq(\030)c Fp(\031)s Fs(\([)p Fp(u)3121 4606 y Fm(i)3150 4592 y Fs(]\))h(=)e Fp(\031)s Fs(\([)p Fp(t)3485 4606 y Fm(i)3514 4592 y Fs(]\))i Fq(\030)212 4705 y Fp(t)245 4719 y Fm(i)303 4705 y Fs(=)h Fp(V)457 4719 y Fm(i)485 4705 y Fs(.)50 b(Since)32 b Fp(V)853 4719 y Fm(i)914 4705 y Fs(is)h(a)g(pure)g(ground) f(term,)i(the)g(second)f(part)g(yields)f Fp(\031)s Fs(\([)p Fp(u)2937 4719 y Fm(i)2965 4705 y Fs(]\))f(=)f Fp(V)3210 4719 y Fm(i)3271 4705 y Fs(and)j(hence)212 4818 y Fp(u)264 4832 y Fm(i)318 4818 y Fs(=)25 b Fp(V)467 4832 y Fm(i)525 4818 y Fs(b)m(y)30 b(the)h(de\014nition)d(of)i Fq(\030)p Fs(.)41 b(Hence)31 b Fp(r)s(\033)d Fs(=)d Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(u)2195 4832 y Fn(2)p Fm(n)p Fn(+9)2368 4818 y Fp(;)g(:)g(:)g(:)i(;)e(u)2622 4832 y Fn(2)p Fm(n)p Fn(+15)2830 4818 y Fs(\))31 b(and)f(therefore)439 5022 y Fe(len)p Fs(\()p Fp(t)616 5036 y Fn(2)p Fm(n)p Fn(+9)789 5022 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)1024 5036 y Fn(2)p Fm(n)p Fn(+14)1232 5022 y Fs(\))25 b Fp(>)g Fe(len)p Fs(\()p Fp(u)1584 5036 y Fn(2)p Fm(n)p Fn(+9)1757 5022 y Fp(;)15 b(:)g(:)g(:)i(;)e(u)2011 5036 y Fn(2)p Fm(n)p Fn(+14)2219 5022 y Fs(\))1258 b(\(17\))1897 5462 y(23)p eop %%Page: 24 24 24 23 bop 212 337 a Fs(b)m(y)30 b(the)h(de\014nition)d(of)j Fe(len)p Fs(.)40 b(F)-8 b(rom)31 b(\(14\))r(,)f(\(16\))r(,)g(and)g (\(17\))i(w)m(e)f(obtain)439 518 y Fp(D)s Fs(\([)p Fp(t)610 532 y Fn(1)650 518 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(t)935 532 y Fn(2)p Fm(n)p Fn(+14)1143 518 y Fs(]\))26 b(=)f Fp(E)5 b Fs(\([)p Fp(t)1490 532 y Fn(1)1530 518 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(t)1815 532 y Fn(2)p Fm(n)p Fn(+14)2023 518 y Fs(]\))1229 656 y Fp(>)25 b(E)5 b Fs(\([)p Fp(u)1509 670 y Fn(1)1549 656 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(u)1853 670 y Fn(2)p Fm(n)p Fn(+14)2062 656 y Fs(]\))25 b(=)g Fp(D)s Fs(\([)p Fp(u)2433 670 y Fn(1)2473 656 y Fs(])p Fp(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fp(u)2777 670 y Fn(2)p Fm(n)p Fn(+14)2986 656 y Fs(]\))212 838 y(With)30 b(help)f(of)h(the)h (strict)f(monotonicit)m(y)h(of)f(the)h(v)-5 b(arious)29 b(functions,)g(w)m(e)i(no)m(w)g(obtain)439 1020 y([)p Fp(l)r(\033)s Fs(])26 b(=)f Fe(co)s(de)q Fs(\()p Fp( )965 1034 y FA(Q)1027 1020 y Fs(\()p Fp(D)s Fs(\([)p 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Fm(i)1827 5087 y Fs(])1897 5462 y(24)p eop %%Page: 25 25 25 24 bop 212 337 a Fs(Note)32 b(that)f Fe(facto)m(r)r Fs(\([)p Fp(t)942 351 y Fn(1)982 337 y Fs(])p Fp(;)15 b(:)g(:)g(:)h(;)f Fs([)p Fp(t)1266 351 y Fn(2)p Fm(n)p Fn(+14)1475 337 y Fs(]\))26 b(=)e Fp(E)1723 351 y Fn(1)1783 337 y Fs(+)c Fp(E)1941 351 y Fn(2)2001 337 y Fs(+)g Fp(E)2159 351 y Fn(3)2229 337 y Fs(and)439 511 y Fn(2)p Fm(n)p Fn(+14)475 538 y Ff(X)484 734 y Fm(i)p Fn(=1)643 625 y Fs([)p Fp(u)720 639 y Fm(i)748 625 y Fs(])26 b(=)f Fp(F)966 587 y FA(0)953 647 y Fn(1)1013 625 y Fs(+)20 b Fp(F)1175 587 y FA(0)1162 647 y Fn(2)1222 625 y Fs(+)g Fp(F)1384 587 y FA(0)1371 647 y Fn(3)212 912 y Fs(In)30 b(order)h(to)h(sho)m(w)g(\(18\))h(it)d(is)g(su\016cien)m(t)h(to)h(sho)m (w)f(that)g Fp(E)2265 926 y Fn(1)2331 912 y Fp(>)26 b(F)2499 879 y FA(0)2486 937 y Fn(1)2526 912 y Fs(,)32 b Fp(E)2650 926 y Fn(2)2716 912 y Fp(>)26 b(F)2884 879 y FA(0)2871 937 y Fn(2)2911 912 y Fs(,)31 b(and)g Fp(E)3212 926 y Fn(3)3278 912 y Fp(>)26 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5091 y(b)s(et)m(w)m(een)39 b(the)g(sequences)g(of)g(terms)g Fp(t)1564 5105 y Fm(n)p Fn(+6)1701 5091 y Fp(;)15 b(:)g(:)g(:)h(;)f(t) 1935 5105 y Fn(2)p Fm(n)p Fn(+8)2146 5091 y Fs(and)39 b Fp(u)2384 5105 y Fm(n)p Fn(+6)2521 5091 y Fp(;)15 b(:)g(:)g(:)h(;)f (u)2774 5105 y Fn(2)p Fm(n)p Fn(+8)2986 5091 y Fs(is)38 b(the)g(third)f(term)212 5204 y(and)g(either)g(the)h(\014rst)f(or)h (second)g(term.)63 b(W)-8 b(e)39 b(consider)d(here)i(the)g(latter)g (\(so)g(a)g(rule)f Fp(r)3332 5218 y Fm(n)p Fn(+1+)p Fm(m)p Fn(+)p Fm(j)1897 5462 y Fs(25)p eop %%Page: 26 26 26 25 bop 212 337 a Fs(with)34 b(1)f Fl(6)f Fp(j)38 b Fl(6)32 b Fp(m)j Fs(is)e(used\);)k(the)e(former)f(is)g(pro)m(v)m(ed)h (in)f(exactly)h(the)g(same)g(w)m(a)m(y)-8 b(.)55 b(So)35 b Fp(t)3330 351 y Fm(n)p Fn(+7)3499 337 y Fs(=)e(1,)212 450 y Fp(t)245 464 y Fm(n)p Fn(+8)407 450 y Fs(=)25 b($,)31 b Fp(u)656 464 y Fm(n)p Fn(+7)819 450 y Fs(=)25 b Fp(z)957 464 y Fn(1)996 450 y Fp(\033)s Fs(,)31 b(and)f Fp(u)1336 464 y Fm(n)p Fn(+8)1498 450 y Fs(=)25 b Fp(y)1639 464 y Fn(1)1678 450 y Fp(\033)s Fs(.)41 b(Hence)439 654 y Fp(E)506 668 y Fn(3)566 654 y Fq(\000)20 b Fp(F)728 616 y FA(0)715 676 y Fn(3)780 654 y Fs(=)60 b Fe(revc)q Fs(\([)p Fp(y)1170 668 y Fn(1)1225 654 y Fp(:)15 b(:)g(:)h(y)1391 668 y Fm(\026)1437 654 y Fs(\()p Fp(y)s Fs(\))p Fp(\033)s Fs(])p Fp(;)f Fs([)p 1700 604 82 4 v Fp(z)1742 668 y Fn(1)1784 654 y Fs(\()p Fp(z)t Fs(\))p Fp(\033)s Fs(]\))22 b(+)e([1])h(+)f([$])h Fq(\000)876 792 y Fs(\([1$)p Fp(y)1071 806 y Fn(2)1127 792 y Fp(:)15 b(:)g(:)h(y)1293 806 y Fm(\026)1339 792 y Fs(\()p Fp(y)s Fs(\))p Fp(\033)s Fs(])21 b(+)f([)p Fp(z)t(\033)s Fs(])h(+)f([)p Fp(z)1979 806 y Fn(1)2019 792 y Fp(\033)s Fs(])h(+)f([)p Fp(y)2281 806 y Fn(1)2320 792 y Fp(\033)s Fs(]\))212 996 y(If)30 b(b)s(oth)g([)p Fp(y)588 1010 y Fn(1)627 996 y Fp(\033)s Fs(])c Fl(>)f Fs([$])31 b(and)f([)p Fp(z)1199 1010 y Fn(1)1239 996 y Fp(\033)s Fs(])25 b Fl(>)g Fs([1])32 b(then)e(w)m(e)g(obtain)439 1200 y Fe(revc)q Fs(\([)p Fp(y)698 1214 y Fn(1)753 1200 y Fp(:)15 b(:)g(:)h(y)919 1214 y Fm(\026)965 1200 y Fs(\()p Fp(y)s Fs(\))p Fp(\033)s Fs(])p Fp(;)f Fs([)p 1228 1150 V Fp(z)1270 1214 y Fn(1)1312 1200 y Fs(\()p Fp(z)t Fs(\))p Fp(\033)s Fs(]\))22 b(+)e([1])h(+)f([$]) 465 1338 y Fl(>)25 b Fe(revc)p Fs(\([$)p Fp(y)864 1352 y Fn(2)920 1338 y Fp(:)15 b(:)g(:)h(y)1086 1352 y Fm(\026)1132 1338 y Fs(\()p Fp(y)s Fs(\))p Fp(\033)s Fs(])p Fp(;)f Fs([)p 1395 1288 V Fp(z)1437 1352 y Fn(1)1478 1338 y Fs(\()p Fp(z)t Fs(\))p Fp(\033)s Fs(]\))22 b(+)e([1])h(+)f([)p Fp(y)2099 1352 y Fn(1)2139 1338 y Fp(\033)s Fs(])465 1476 y Fl(>)25 b Fe(revc)p Fs(\([$)p Fp(y)864 1490 y Fn(2)920 1476 y Fp(:)15 b(:)g(:)h(y)1086 1490 y Fm(\026)1132 1476 y Fs(\()p Fp(y)s Fs(\))p Fp(\033)s Fs(])p Fp(;)f Fs([)p 1395 1406 46 4 v(1)r(\()p Fp(z)t Fs(\))p Fp(\033)s Fs(]\))22 b(+)e([)p Fp(z)1853 1490 y Fn(1)1893 1476 y Fp(\033)s Fs(])h(+)f([)p Fp(y)2155 1490 y Fn(1)2194 1476 y Fp(\033)s Fs(])465 1614 y(=)25 b Fe(revc)p Fs(\([1$)p Fp(y)909 1628 y Fn(2)965 1614 y Fp(:)15 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Fp(A)p Fs(\()p Fp(V)e(;)15 b(W)m(;)g(d)p Fs(\))26 b(according)e(to)h(Lemma)f(4.2)h(and)e(th)m(us)h Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(d)p Fs(\))28 b Fq(!)2499 3573 y Fn(+)2499 3645 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)2699 3654 y Fd(1)2732 3645 y Fn(\))2789 3612 y Fp(C)i Fs([)p Fp(A)p Fs(\()p Fp(V)e(;)15 b(W)m(;)g(d)p Fs(\)])26 b(for)e(some)212 3738 y(con)m(text)37 b Fp(C)42 b Fs(b)m(y)36 b(Lemma)f(6.4,)k(hence)c Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1816 3752 y Fn(1)1856 3738 y Fs(\))36 b(is)f(lo)s(oping.)54 b(On)35 b(the)h(other)f(hand,)h(if)f Fp(P)48 b Fs(has)35 b(no)212 3851 y(solution)29 b(then)h Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1035 3865 y Fn(1)1075 3851 y Fs(\))31 b(is)e Fp(!)s Fs(-terminating)h(b)m(y)g(Lemma)g(6.3)i(and)e(th)m(us)g (also)g(non-lo)s(oping.)135 b Fl(\003)212 4137 y Fr(8)g(SN)45 b Fi(\))f Fr(NL)212 4340 y Fs(Before)36 b(de\014ning)e(the)h(TRS)f Fq(Q)1303 4354 y Fn(2)1378 4340 y Fs(used)g(for)h(the)h(relativ)m(e)f (undecidabilit)m(y)d(of)j(SN)g Fq(\))g 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b(o)g(of)46 b Fs(Let)23 b Fp(t)i Fq(!)772 436 y Fn(+)857 469 y Fp(C)7 b Fs([)p Fp(t\033)s Fs(])22 b(b)s(e)g(an)m(y)h(lo)s(op.)37 b(W)-8 b(e)24 b(sho)m(w)e(b)m(y)h(induction)d(on)i(the)h(structure)f(of)h Fp(t)f Fs(that)h(there)212 582 y(exists)f(a)g(lo)s(op)f(\(not)h (necessarily)f(starting)h(at)h Fp(t)p Fs(\))f(whic)m(h)e(con)m(tains)i (a)h(ro)s(ot)f(rewrite)f(step.)38 b(In)21 b(the)h(base)212 695 y(case)k(\()p Fp(t)g Fs(is)e(a)i(constan)m(t\))h(the)f(\014rst)e (step)h(of)h(the)f(giv)m(en)h(lo)s(op)e(m)m(ust)h(tak)m(e)i(place)f(at) g(the)f(ro)s(ot)h(p)s(osition.)212 808 y(F)-8 b(or)32 b(the)g(induction)c(step,)k(supp)s(ose)e Fp(t)d Fs(=)f Fp(f)10 b Fs(\()p Fp(t)1780 822 y Fn(1)1819 808 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)2054 823 y Fm(k)2097 808 y Fs(\))31 b(with)f Fp(k)g Fl(>)c Fs(1)32 b(and)f(no)g(step)g(in)f Fp(t)d Fq(!)3378 775 y Fn(+)3463 808 y Fp(C)7 b Fs([)p Fp(t\033)s Fs(])212 921 y(tak)m(es)28 b(place)f(at)g(the)g(ro)s(ot)g(p) s(osition.)37 b(So)26 b Fp(C)7 b Fs([)p Fp(t\033)s Fs(])26 b(=)f Fp(f)10 b Fs(\()p Fp(u)2097 935 y Fn(1)2136 921 y Fp(;)15 b(:)g(:)g(:)h(;)f(u)2389 936 y Fm(k)2433 921 y Fs(\))26 b(with)g Fp(t)2731 935 y Fm(i)2784 921 y Fq(!)2875 888 y FA(\003)2939 921 y Fp(u)2991 935 y Fm(i)3046 921 y Fs(for)h(all)e(1)h Fl(6)f Fp(i)g Fl(6)g Fp(k)212 1034 y Fs(and)g Fp(t)417 1048 y Fm(j)479 1034 y Fq(!)570 1001 y Fn(+)654 1034 y Fp(u)706 1048 y Fm(j)768 1034 y Fs(for)g(at)i(least)e (one)h(1)g Fl(6)f Fp(j)31 b Fl(6)25 b Fp(k)s Fs(.)39 b(If)25 b(the)h(con)m(text)h Fp(C)32 b Fs(is)25 b(empt)m(y)h(then)f Fp(u)3090 1048 y Fm(j)3152 1034 y Fs(=)g Fp(t)3281 1048 y Fm(j)3317 1034 y Fp(\033)k Fs(and)c(w)m(e)212 1147 y(obtain)32 b(the)g(desired)e(lo)s(op)h(from)h(the)g(induction)e(h)m (yp)s(othesis)g(\(applied)g(to)j Fp(t)2897 1161 y Fm(j)2934 1147 y Fs(\).)45 b(Otherwise)31 b(there)212 1259 y(exist)e(a)g(con)m (text)i Fp(C)894 1226 y FA(0)946 1259 y Fs(and)d(an)h(index)f(1)e Fl(6)f Fp(l)i Fl(6)e Fp(k)32 b Fs(suc)m(h)d(that)g Fp(C)j Fs(=)25 b Fp(f)10 b Fs(\()p Fp(v)2612 1273 y Fn(1)2651 1259 y Fp(;)15 b(:)g(:)g(:)i(;)e(v)2897 1274 y Fm(l)q FA(\000)p Fn(1)3013 1259 y Fp(;)g(C)3125 1226 y FA(0)3149 1259 y Fp(;)g(v)3233 1274 y Fm(l)q Fn(+1)3349 1259 y Fp(;)g(:)g(:)g(:)i(;)e(v)3595 1274 y Fm(k)3638 1259 y Fs(\))212 1372 y(and)35 b Fp(t)427 1387 y Fm(l)486 1372 y Fq(!)577 1339 y FA(\003)650 1372 y Fp(C)722 1339 y FA(0)745 1372 y Fs([)p Fp(t\033)s Fs(])f(=)f Fp(C)1093 1339 y FA(0)1116 1372 y Fs([)p Fp(f)10 b Fs(\()p Fp(:)15 b(:)g(:)h(;)f(t)1425 1387 y Fm(l)1452 1372 y Fp(\033)n(;)g(:)g(:)g(:)h Fs(\)].)57 b(Since)34 b Fp(t)2080 1387 y Fm(l)2139 1372 y Fq(6)p Fs(=)g Fp(C)2316 1339 y FA(0)2339 1372 y Fs([)p Fp(f)10 b Fs(\()p Fp(:)15 b(:)g(:)h(;)f(t)2648 1387 y Fm(l)2674 1372 y Fp(\033)n(;)g(:)g(:)g(:)i Fs(\)])36 b(w)m(e)f(can)h(apply)e(the)212 1485 y(induction)29 b(h)m(yp)s(othesis) g(to)j Fp(t)1210 1500 y Fm(l)1236 1485 y Fs(,)f(yielding)e(a)i(lo)s(op) g(whic)m(h)e(con)m(tains)j(a)f(ro)s(ot)g(rewrite)g(step.)42 b(No)m(w)32 b(the)212 1598 y(result)d(follo)m(ws)h(from)g(the)g (preceding)g(lemma.)1770 b Fl(\003)353 1793 y Fs(Let)439 2011 y Fq(Q)513 2025 y Fn(2)578 2011 y Fs(=)674 1882 y Ff(\032)784 1954 y Fp(f)10 b Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(x)p Fs(\))p Fp(;)g(y)s Fs(\))85 b Fq(!)e Fp(f)10 b Fs(\()p Fp(d;)15 b(x;)g(b)p Fs(\()p Fp(y)s Fs(\)\))784 2067 y Fp(f)10 b Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(x)p Fs(\))p Fp(;)g(y)s Fs(\))85 b Fq(!)e Fp(f)10 b Fs(\()p Fp(x;)15 b(y)s(;)g(b)p Fs(\()p Fp(b)p Fs(\()p Fp(d)p Fs(\)\)\))212 2259 y Fb(Lemma)33 b(8.3)i(\()p Fs([27)q(])p Fb(\))45 b Fo(The)33 b(TRS)g Fq(Q)1494 2273 y Fn(2)1566 2259 y Fo(is)g(non-lo)-5 b(oping)35 b(and)e(not)g(terminating.)212 2392 y(Pr)-5 b(o)g(of)56 b Fs(First)32 b(w)m(e)h(sho)m(w)g(that)g Fq(Q)1339 2406 y Fn(2)1411 2392 y Fs(is)f(not)h(terminating.)47 b(De\014ne)33 b(terms)f Fp(t)2779 2406 y Fm(i)2837 2392 y Fs(=)c Fp(f)10 b Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(d)p Fs(\))p Fp(;)g(b)3348 2359 y Fm(i)3378 2392 y Fs(\()p Fp(d)p Fs(\)\))34 b(for)212 2504 y(all)i Fp(i)g Fl(>)f Fs(1.)60 b(W)-8 b(e)37 b(ha)m(v)m(e)h Fp(t)1059 2518 y Fm(i)1123 2504 y Fq(!)1214 2471 y Fn(+)1308 2504 y Fp(t)1341 2518 y Fm(i)p Fn(+1)1496 2504 y Fs(b)m(y)f(one)f(application) f(of)i(the)g(second)g(rewrite)e(rule)h(follo)m(w)m(ed)212 2617 y(b)m(y)41 b Fp(i)28 b Fq(\000)g Fs(1)42 b(applications)d(of)j (the)g(\014rst)e(rewrite)h(rule.)73 b(Hence)42 b Fq(Q)2517 2631 y Fn(2)2599 2617 y Fs(admits)e(the)i(in\014nite)d(rewrite)212 2730 y(sequence)34 b Fp(t)625 2744 y Fn(1)696 2730 y Fq(!)787 2697 y Fn(+)877 2730 y Fp(t)910 2744 y Fn(2)980 2730 y Fq(!)1071 2697 y Fn(+)1161 2730 y Fp(t)1194 2744 y Fn(3)1264 2730 y Fq(!)1355 2697 y Fn(+)1445 2730 y Fq(\001)15 b(\001)g(\001)h Fs(.)51 b(Next)35 b(w)m(e)f(sho)m(w)g(that)h Fq(Q)2509 2744 y Fn(2)2582 2730 y Fs(is)e(non-lo)s(oping.)49 b(F)-8 b(or)35 b(a)f(pro)s(of)212 2843 y(b)m(y)d(con)m(tradiction)g (supp)s(ose)e(that)j Fq(Q)1507 2857 y Fn(2)1578 2843 y Fs(is)e(lo)s(oping.)41 b(According)30 b(to)i(Lemma)f(8.2)h(there)g(m) m(ust)f(b)s(e)f(a)212 2956 y(lo)s(op)c(that)i(starts)f(with)f(a)h(ro)s (ot)g(rewrite)f(step,)i(whic)m(h)e(is)g(only)g(p)s(ossible)e(if)i(the)h (lo)s(op)g(is)f(of)h(the)g(form)439 3143 y Fp(f)10 b Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(t)723 3157 y Fn(1)763 3143 y Fs(\))p Fp(;)g(t)871 3157 y Fn(2)911 3143 y Fs(\))26 b Fq(!)1063 3105 y Fn(+)1147 3143 y Fp(C)7 b Fs([)p Fp(f)j Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(t)1528 3157 y Fn(1)1568 3143 y Fp(\033)s Fs(\))p Fp(;)g(t)1731 3157 y Fn(2)1771 3143 y Fp(\033)s Fs(\)])1626 b(\(19\))212 3329 y(Since)39 b(rewrite)h(steps)g(do)g(not)h(c)m(hange)g(the)f(n)m(um)m(b)s(er)f(of)i Fp(f)49 b Fs(sym)m(b)s(ols,)42 b(the)e(con)m(text)i Fp(C)47 b Fs(m)m(ust)40 b(b)s(e)212 3442 y(empt)m(y)-8 b(.)42 b(Moreo)m(v)m(er,)33 b(the)e(substitution)d Fp(\033)34 b Fs(do)s(es)c(not)h(assign)f(terms)h(that)g(con)m(tain)g(an)m(y)g Fp(f)40 b Fs(sym)m(b)s(ols)212 3555 y(to)32 b(the)f(v)-5 b(ariables)30 b(in)f Fp(t)998 3569 y Fn(1)1068 3555 y Fs(and)i Fp(t)1279 3569 y Fn(2)1318 3555 y Fs(.)42 b(W)-8 b(e)32 b(ma)m(y)g(assume)f(that)g Fp(t)2285 3569 y Fn(1)2355 3555 y Fs(and)g Fp(t)2566 3569 y Fn(2)2636 3555 y Fs(do)g(not)g(con)m (tain)g Fp(f)40 b Fs(sym)m(b)s(ols.)212 3668 y(\(If)27 b(they)g(do,)g(w)m(e)g(replace)g(their)f(outermost)h Fp(f)36 b Fs(sym)m(b)s(ols)26 b(b)m(y)g(a)h(fresh)f(v)-5 b(ariable,)27 b(resulting)e(in)g(a)i(lo)s(op)212 3781 y(that)k(has)e(the)h(desired)e(prop)s(ert)m(y)-8 b(.\))41 b(Consequen)m(tly)-8 b(,)30 b(all)f(rewrite)g(steps)g(in)g(\(19\))i (tak)m(e)h(place)e(at)g(the)212 3894 y(ro)s(ot)37 b(p)s(osition.)57 b(W)-8 b(e)38 b(claim)e(that)h Fp(t)1461 3908 y Fn(1)1537 3894 y Fs(and)f Fp(t)1753 3908 y Fn(2)1828 3894 y Fs(are)h(ground)f (terms.)59 b(First)36 b(note)h(that)g(the)g(second)212 4007 y(rule)d(of)h Fq(Q)581 4021 y Fn(2)655 4007 y Fs(m)m(ust)g(b)s(e)g (used)f(in)f(\(19\))k(as)e(the)g(\014rst)f(rule)g(constitutes)h(a)h (terminating)d(\(and)i(hence)212 4119 y(non-lo)s(oping\))29 b(TRS.)h(Hence)h(w)m(e)g(can)f(render)g(\(19\))i(as)e(follo)m(ws:)439 4306 y Fp(f)10 b Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(t)723 4320 y Fn(1)763 4306 y Fs(\))p Fp(;)g(t)871 4320 y Fn(2)911 4306 y Fs(\))26 b Fq(!)1063 4268 y FA(\003)1128 4306 y Fp(f)10 b Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(u)1431 4320 y Fn(1)1471 4306 y Fs(\))p Fp(;)g(u)1598 4320 y Fn(2)1638 4306 y Fs(\))26 b Fq(!)f Fp(f)10 b Fs(\()p Fp(u)1957 4320 y Fn(1)1996 4306 y Fp(;)15 b(u)2088 4320 y Fn(2)2128 4306 y Fp(;)g(b)p Fs(\()p Fp(b)p Fs(\()p Fp(d)p Fs(\)\)\))28 b Fq(!)2587 4268 y FA(\003)2651 4306 y Fp(f)10 b Fs(\()p Fp(d;)15 b(b)p Fs(\()p Fp(t)2935 4320 y Fn(1)2975 4306 y Fp(\033)s Fs(\))p Fp(;)g(t)3138 4320 y Fn(2)3178 4306 y Fp(\033)s Fs(\))244 b(\(20\))212 4501 y(suc)m(h)40 b(that)h Fp(t)667 4515 y Fn(1)748 4501 y Fs(=)g Fp(b)899 4468 y Fm(k)942 4501 y Fs(\()p Fp(u)1029 4515 y Fn(1)1068 4501 y Fs(\))g(and)e Fp(b)1369 4468 y Fm(k)1412 4501 y Fs(\()p Fp(t)1480 4515 y Fn(2)1520 4501 y Fs(\))j(=)f Fp(u)1761 4515 y Fn(2)1841 4501 y Fs(for)f(some)g Fp(k)45 b Fl(>)c Fs(0.)71 b(Because)41 b(of)g(the)f(form)g(of)g(the)212 4614 y(left-hand)28 b(sides)f(of)i(the)g(rules)f(of)h Fq(Q)1463 4628 y Fn(2)1531 4614 y Fs(w)m(e)g(m)m(ust)g(ha)m(v)m(e)h Fp(u)2147 4628 y Fn(1)2211 4614 y Fs(=)25 b Fp(d)k Fs(and)g(hence)f Fp(t)2842 4628 y Fn(1)2907 4614 y Fs(=)d Fp(b)3042 4581 y Fm(k)3085 4614 y Fs(\()p Fp(d)p Fs(\))k(is)f(a)h(ground)212 4727 y(term.)41 b(Rep)s(eating)30 b(the)h(same)f(reasoning)g(for)g(the) h(lo)s(op)439 4913 y Fp(f)10 b Fs(\()p Fp(d;)15 b(u)668 4927 y Fn(2)708 4913 y Fp(;)g(b)p Fs(\()p Fp(b)p Fs(\()p Fp(d)p Fs(\)\)\))28 b Fq(!)1167 4876 y Fn(+)1251 4913 y Fp(f)10 b Fs(\()p Fp(d;)15 b(u)1480 4927 y Fn(2)1520 4913 y Fp(\033)n(;)g(b)p Fs(\()p Fp(b)p Fs(\()p Fp(d)p Fs(\)\)\))1602 b(\(21\))212 5100 y(whose)26 b(existence)h(is)f(guaran)m (teed)h(b)m(y)f(\(the)h(pro)s(of)f(of)7 b(\))27 b(Lemma)f(8.1)i(sho)m (ws)e(that)h Fp(u)3061 5114 y Fn(2)3100 5100 y Fs(,)h(and)d(th)m(us)h (also)212 5213 y Fp(t)245 5227 y Fn(2)284 5213 y Fs(,)32 b(is)f(a)h(ground)f(term.)44 b(Hence)33 b Fp(t)1382 5227 y Fn(1)1421 5213 y Fp(\033)d Fs(=)e Fp(t)1635 5227 y Fn(1)1674 5213 y Fs(,)k Fp(t)1764 5227 y Fn(2)1803 5213 y Fp(\033)f Fs(=)c Fp(t)2017 5227 y Fn(2)2056 5213 y Fs(,)32 b(and)f(th)m(us)g(\(19\))j(is)c(actually)h(a)h(cycle.)45 b(Since)1897 5462 y(30)p eop %%Page: 31 31 31 30 bop 212 337 a Fs(applications)25 b(of)j(the)f(second)g(rewrite)g (rule)f(increase)h(the)g(size)g(of)h(terms)f(whereas)g(applications)e (of)212 450 y(the)33 b(\014rst)e(rewrite)h(rule)f(do)h(not)g(c)m(hange) i(the)e(size)g(of)h(terms,)g(only)e(the)h(\014rst)g(rewrite)f(rule)g (can)i(b)s(e)212 562 y(used.)53 b(Ho)m(w)m(ev)m(er,)39 b(w)m(e)c(ha)m(v)m(e)h(already)e(observ)m(ed)h(that)h(the)f(\014rst)f (rule)f(constitutes)i(a)g(terminating)212 675 y(\(and)30 b(th)m(us)g(acyclic\))h(TRS.)f(Therefore)g Fq(Q)1680 689 y Fn(2)1750 675 y Fs(is)f(non-lo)s(oping.)1268 b Fl(\003)353 875 y Fs(W)-8 b(e)29 b(w)m(an)m(t)g(to)f(sho)m(w)g(that)g Fq(S)7 b 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2071 y Fs(\)\))26 b(=)f Fp(g)s Fs(\()p Fp( )s Fs(\()p Fp(t)1145 2085 y Fn(1)1186 2071 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e( )s Fs(\()p Fp(t)1553 2086 y Fm(k)1596 2071 y Fs(\)\))30 b(for)e(all)g(function)f(sym)m(b)s(ols)g Fp(g)32 b Fs(di\013eren)m(t)d (from)f Fp(A)h Fs(and)f Fp(B)5 b Fs(,)212 2184 y(and)30 b Fp( )s Fs(\()p Fp(x)p Fs(\))c(=)f Fp(x)30 b Fs(for)h(all)e(v)-5 b(ariables)29 b Fp(x)p Fs(.)41 b(Here)30 b Fp(z)35 b Fs(is)29 b(a)i(designated)f(fresh)g(v)-5 b(ariable.)353 2384 y(The)37 b(purp)s(ose)f(of)i(these)h(mappings)c(is)i(to)i (simplify)34 b(the)k(structure)f(of)h Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))39 b(rewrite)e(se-)212 2497 y(quences)30 b(b)m(y)h(replacing)e(all)g(descendan)m(ts)i(of)f (non-outermost)h Fp(A)g Fs(sym)m(b)s(ols)d(b)m(y)j(the)f(v)-5 b(ariable)29 b Fp(z)t Fs(.)212 2696 y Fb(Lemma)k(8.5)46 b Fo(If)23 b Fp(t)i Fq(!)997 2658 y Fn(+)997 2729 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)p Fn(\))1253 2696 y Fp(u)23 b Fo(with)h Fs(ro)s(ot)q(\()p Fp(t)p Fs(\))h(=)g Fp(A)f Fo(c)-5 b(ontains)25 b(a)e(r)-5 b(o)g(ot)26 b(r)-5 b(ewrite)24 b(step)g(then)g Fp(\036)p Fs(\()p Fp(t)p Fs(\))i Fq(!)3536 2658 y Fn(+)3536 2729 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)p Fn(\))212 2823 y Fp(\036)p Fs(\()p Fp(u)p Fs(\))p Fo(.)40 b(Mor)-5 b(e)g(over,)27 b(if)e Fp(v)j Fo(is)c(a)i(maximal)g(subterm)f (of)g Fp(u)g Fo(with)g(r)-5 b(o)g(ot)27 b(symb)-5 b(ol)26 b Fp(A)f Fo(then)g Fp(\036)p Fs(\()p Fp(v)s Fs(\))h Fo(is)f(a)g (subterm)212 2935 y(of)33 b Fp(\036)p Fs(\()p Fp(u)p Fs(\))g Fo(and)h Fp(\036)p Fs(\()p Fp(t)p Fs(\))26 b Fq(!)979 2897 y Fn(+)979 2969 y FA(U)7 b Fn(\()p Fm(P)r(;)p FA(Q)p Fn(\))1238 2935 y Fp(\036)p Fs(\()p Fp(v)s Fs(\))p Fo(.)212 3083 y(Pr)-5 b(o)g(of)49 b Fs(It)26 b(is)e(easy)j(to)f(see)g (that)g(for)g(ev)m(ery)g(step)g Fp(t)1878 3050 y FA(0)1926 3083 y Fq(!)2017 3101 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)2217 3110 y Fd(2)2251 3101 y Fn(\))2307 3083 y Fp(u)2359 3050 y FA(0)2408 3083 y Fs(in)25 b(the)g(giv)m(en)h(rewrite)f(sequence)h(w)m (e)212 3196 y(get)e Fp(\036)p Fs(\()p Fp(t)478 3163 y FA(0)502 3196 y Fs(\))i Fq(!)654 3214 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)854 3223 y Fd(2)887 3214 y Fn(\))944 3196 y Fp(\036)p Fs(\()p Fp(u)1085 3163 y FA(0)1109 3196 y Fs(\))23 b(if)g(the)g(con)m(tracted)i(redex)e(is)f(outermost)i(and)f Fp(\036)p Fs(\()p Fp(t)2870 3163 y FA(0)2894 3196 y Fs(\))i(=)g Fp(\036)p Fs(\()p Fp(u)3191 3163 y FA(0)3215 3196 y Fs(\))f(otherwise.) 212 3318 y(Hence)36 b Fp(\036)p Fs(\()p Fp(t)p Fs(\))d Fq(!)768 3279 y Fn(+)768 3351 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)p Fn(\))1031 3318 y Fp(\036)p Fs(\()p Fp(u)p Fs(\).)55 b(The)35 b(term)g Fp(u)f Fs(can)i(b)s(e)e(\(uniquely\))f(written)g(as)j Fp(C)7 b Fs([)p Fp(v)3111 3332 y Fn(1)3150 3318 y Fp(;)15 b(:)g(:)g(:)h(;)f(v)3395 3333 y Fm(k)3439 3318 y Fs(])35 b(suc)m(h)212 3430 y(that)f Fp(C)39 b Fs(is)32 b(a)i(con)m(text)g (consisting)e(of)h Fp(B)38 b Fs(sym)m(b)s(ols)31 b(and)i(ev)m(ery)h Fp(v)2466 3444 y Fm(i)2527 3430 y Fs(starts)f(with)f(an)h Fp(A)g Fs(sym)m(b)s(ol.)47 b(So)212 3543 y Fp(v)256 3557 y Fn(1)296 3543 y Fs(,)37 b Fp(:)15 b(:)g(:)h Fs(,)38 b Fp(v)586 3558 y Fm(k)664 3543 y Fs(are)f(the)f(maximal)f(subterms)f (of)i Fp(u)g Fs(with)f(ro)s(ot)h(sym)m(b)s(ol)f Fp(A)p Fs(.)57 b(By)37 b(de\014nition)c Fp(\036)p Fs(\()p Fp(u)p Fs(\))j(=)212 3656 y Fp(C)7 b Fs([)p Fp(\036)p Fs(\()p Fp(v)442 3670 y Fn(1)482 3656 y Fs(\))p Fp(;)15 b(:)g(:)g(:)i(;)e(\036) p Fs(\()p Fp(v)852 3671 y Fm(k)895 3656 y Fs(\)].)69 b(A)40 b(straigh)m(tforw)m(ard)g(induction)d(on)i(the)h(length)f(of)h Fp(t)g Fq(!)3085 3623 y FA(\003)3085 3688 y(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)p Fn(\))3356 3656 y Fp(u)40 b Fs(yields)212 3781 y Fp(\036)p Fs(\()p Fp(t)p Fs(\))26 b Fq(!)486 3748 y FA(\003)486 3812 y(U)7 b Fn(\()p Fm(P)r(;)p FA(Q)p Fn(\))745 3781 y Fp(\036)p Fs(\()p Fp(v)878 3795 y Fm(i)907 3781 y Fs(\))30 b(for)g(ev)m(ery)i(1)25 b Fl(6)g Fp(i)h Fl(6)f Fp(k)s Fs(.)1857 b Fl(\003)212 3992 y Fb(Lemma)33 b(8.6)46 b Fo(If)32 b Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1133 4006 y Fn(2)1174 3992 y Fs(\))32 b Fo(is)h(non-lo)-5 b(oping)35 b(then)e Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)2306 4006 y Fn(2)2346 3992 y Fs(\))33 b Fo(is)f(non-lo)-5 b(oping.)212 4124 y(Pr)g(o)g(of)50 b Fs(Supp)s(ose)25 b(on)i(the)g(con)m(trary)g(that)g Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1926 4138 y Fn(2)1967 4124 y Fs(\))27 b(is)f(lo)s(oping.)37 b(According)27 b(to)g(Lemma)g(8.2)h(there)212 4237 y(exists)36 b(a)h(lo)s(op)e Fp(t)g Fq(!)915 4199 y Fn(+)915 4270 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)1115 4279 y Fd(2)1148 4270 y Fn(\))1215 4237 y Fp(C)i Fs([)p Fp(t\033)s Fs(])36 b(that)h(starts)g(with)e(a)h(ro)s(ot)h(rewrite)e(step.)59 b(This)34 b(implies)g(that)212 4363 y(ro)s(ot\()p Fp(t)p Fs(\))j(=)g Fp(A)p Fs(.)61 b(Since)36 b(the)h(maxim)m(um)f(nesting)g (of)i Fp(A)f Fs(sym)m(b)s(ols)e(do)s(es)i(not)h(c)m(hange)g(b)m(y)f Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)3598 4377 y Fn(2)3638 4363 y Fs(\))212 4476 y(rewrite)32 b(steps,)h(there)f(cannot)h(b)s(e)f Fp(A)h Fs(sym)m(b)s(ols)e(ab)s(o)m(v)m(e)i(the)g(p)s(osition)e(of)h (the)h(hole)f(in)f(the)h(con)m(text)212 4589 y Fp(C)7 b Fs(.)47 b(In)32 b(other)g(w)m(ords,)h Fp(t\033)j Fs(is)31 b(a)i(maximal)f(subterm)f(of)i Fp(C)7 b Fs([)p Fp(t\033)s Fs(])33 b(with)e(ro)s(ot)i(sym)m(b)s(ol)e Fp(A)p Fs(.)47 b(Lemma)33 b(8.5)212 4702 y(yields)439 4893 y Fp(\036)p Fs(\()p Fp(t)p Fs(\))26 b Fq(!)713 4855 y Fn(+)713 4926 y FA(U)7 b Fn(\()p Fm(P)r(;)p FA(Q)916 4935 y Fd(2)950 4926 y Fn(\))1007 4893 y Fp(\036)p Fs(\()p Fp(t\033)s Fs(\))2293 b(\(22\))212 5100 y(Note)36 b(that)f Fp(\036)p Fs(\()p Fp(t\033)s Fs(\))f(=)e Fp(\036)p Fs(\()p Fp(t)p Fs(\))p Fp(\033)1198 5067 y FA(0)1257 5100 y Fs(where)i(the)h (substitution)d Fp(\033)2250 5067 y FA(0)2308 5100 y Fs(is)i(de\014ned)f(as)i(the)g(comp)s(osition)e(of)i Fp(\033)212 5213 y Fs(and)30 b Fp( )s Fs(.)41 b(Hence)31 b(\(22\))h(is)e(a)g(lo)s(op,)g(whic)m(h)f(con)m(tradicts)i(the)g (assumption.)870 b Fl(\003)1897 5462 y Fs(31)p eop %%Page: 32 32 32 31 bop 212 337 a Fb(Lemma)33 b(8.7)46 b Fo(The)33 b(TRS)g Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1435 351 y Fn(2)1475 337 y Fs(\))33 b Fo(is)f(non-lo)-5 b(oping)35 b(for)e(every)g(PCP)e(instanc)-5 b(e)34 b Fp(P)13 b Fo(.)212 469 y(Pr)-5 b(o)g(of)52 b Fs(Assume)28 b Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1081 483 y Fn(2)1121 469 y Fs(\))29 b(admits)f(a)h(lo)s(op.)39 b(F)-8 b(rom)29 b(Lemma)g(8.6)g(w)m(e)g (obtain)f(a)h(lo)s(op,)g Fp(t)c Fq(!)3354 436 y Fn(+)3438 469 y Fp(C)7 b Fs([)p Fp(t\033)s Fs(],)212 582 y(in)27 b Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)592 596 y Fn(2)633 582 y Fs(\).)40 b(According)28 b(to)h(Lemma)g(8.2)h(w)m(e)f(ma)m(y)g (assume)f(that)h(this)f(lo)s(op)g(starts)g(with)g(a)h(ro)s(ot-)212 695 y(rewrite)d(step.)39 b(This)24 b(is)i(only)f(p)s(ossible)f(if)h Fp(t)h Fs(is)f(a)i(redex)f(and)f(hence)i(w)m(e)g(ma)m(y)f(write)g Fp(t)f Fs(=)g Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(t)3588 662 y FA(0)3613 695 y Fs(\).)212 808 y(The)34 b(linear)g(in)m (terpretation)g Fp(\036)h Fs(is)f(de\014ned)f(b)m(y)i Fp(\036)p Fs(\()p Fp(b)p Fs(\()p Fp(t)p Fs(\)\))f(=)f Fp(\036)p Fs(\()p Fp(t)p Fs(\))i(and)f Fp(\036)p Fs(\()p Fp(g)s Fs(\()p Fp(t)2857 822 y Fn(1)2898 808 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)3133 823 y Fm(k)3176 808 y Fs(\)\))33 b(=)f Fp(\036)p Fs(\()p Fp(t)3504 822 y Fn(1)3544 808 y Fs(\))24 b(+)212 921 y Fq(\001)15 b(\001)g(\001)29 b Fs(+)e Fp(\036)p Fs(\()p Fp(t)566 936 y Fm(k)609 921 y Fs(\))g(+)g(1)42 b(for)f(ev)m(ery)h(other)f(function)f(sym)m(b)s(ol)g Fp(g)45 b Fs(of)c(arit)m(y)g Fp(k)s Fs(.)73 b(Clearly)-8 b(,)43 b Fp(s)g Fq(!)3296 939 y FA(U)7 b Fn(\()p Fm(P)r(;)p FA(Q)3499 948 y Fd(2)3533 939 y Fn(\))3607 921 y Fp(s)3650 888 y FA(0)212 1034 y Fs(implies)27 b Fp(\036)p Fs(\()p Fp(s)p Fs(\))f(=)f Fp(\036)p Fs(\()p Fp(s)941 1001 y FA(0)964 1034 y Fs(\))30 b(for)g(all)e(terms)i Fp(s)f Fs(and)g Fp(s)1837 1001 y FA(0)1860 1034 y Fs(,)h(hence)g Fp(C)36 b Fs(consists)29 b(of)h Fp(b)g Fs(sym)m(b)s(ols)e(only)-8 b(.)40 b(Another)212 1147 y(linear)f(in)m(terpretation)h Fp( )k Fs(is)c(de\014ned)f(b)m(y)h Fp( )s Fs(\()p Fp(b)p Fs(\()p Fp(t)p Fs(\)\))k(=)e Fp( )s Fs(\()p Fp(t)p Fs(\))28 b(+)f(1)41 b(and)f Fp( )s Fs(\()p Fp(g)s Fs(\()p Fp(t)2934 1161 y Fn(1)2975 1147 y Fp(;)15 b(:)g(:)g(:)i(;)e(t)3210 1162 y Fm(k)3253 1147 y Fs(\)\))43 b(=)f(0)e(for)212 1259 y(ev)m(ery)45 b(other)g(function)e(sym)m(b)s(ol)g Fp(g)48 b Fs(of)c(arit)m(y)h Fp(k)s Fs(.)82 b(F)-8 b(or)45 b(all)f(terms)g Fp(s)g Fs(and)f Fp(s)2915 1226 y FA(0)2938 1259 y Fs(,)48 b(if)c Fp(s)k Fq(!)3291 1278 y FA(U)7 b Fn(\()p Fm(P)r(;)p FA(Q)3494 1287 y Fd(2)3527 1278 y Fn(\))3607 1259 y Fp(s)3650 1226 y FA(0)212 1372 y Fs(then)34 b Fp( )s Fs(\()p Fp(s)p Fs(\))e(=)f Fp( )s Fs(\()p Fp(s)872 1339 y FA(0)896 1372 y Fs(\),)k(hence)f Fp(C)40 b Fs(is)33 b(empt)m(y)-8 b(.)53 b(W)-8 b(e)35 b(conclude)e(that)i(the)f(lo)s(op)f(m)m(ust)h(b)s(e)f(of)i(the)f(form) 212 1485 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(t)569 1452 y FA(0)594 1485 y Fs(\))26 b Fq(!)746 1452 y Fn(+)831 1485 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)e(\033)n(;)i(t)1254 1452 y FA(0)1278 1485 y Fp(\033)s Fs(\).)42 b(Since)29 b Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(t)2029 1452 y FA(0)2054 1485 y Fs(\))26 b Fq(!)2206 1447 y Fn(+)2206 1518 y FA(U)2250 1527 y Fu(\000)2302 1518 y Fn(\()p Fm(P)r(;)p FA(Q)p Fn(\))2511 1485 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)e(\033)n(;)i(t)2934 1452 y FA(0)2958 1485 y Fp(\033)s Fs(\))31 b(con)m(tradicts)g(the)212 1622 y(\()p Fp(!)s Fs(-\)termination)37 b(of)f Fq(U)1041 1636 y FA(\000)1100 1622 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))38 b(\(Corollary)e(5.20\),)k (w)m(e)e(obtain)e Fp(t)2550 1589 y FA(0)2609 1622 y Fq(!)2700 1583 y Fn(+)2700 1650 y FA(Q)2758 1659 y Fd(2)2832 1622 y Fp(t)2865 1589 y FA(0)2888 1622 y Fp(\033)k Fs(from)c(Lemma)h(4.3.) 212 1735 y(This)29 b(is)g(imp)s(ossible)e(as)k Fq(Q)1140 1749 y Fn(2)1210 1735 y Fs(is)e(non-lo)s(oping)g(\(Lemma)h(8.3\).)1270 b Fl(\003)212 1947 y Fb(Lemma)33 b(8.8)46 b Fo(The)33 b(TRS)g Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1435 1961 y Fn(2)1475 1947 y Fs(\))33 b Fo(is)f(terminating)i(if)e(and)i(only)f (if)g Fp(P)45 b Fo(admits)34 b(no)f(solution.)212 2080 y(Pr)-5 b(o)g(of)54 b Fs(Supp)s(ose)29 b Fp(P)45 b Fs(has)30 b(a)i(solution.)42 b(F)-8 b(rom)31 b(\(the)h(pro)s(of)e(of)7 b(\))32 b(Lemma)f(8.3)i(w)m(e)e(kno)m(w)g(there)h(exists)212 2192 y(an)g(in\014nite)d(rewrite)i(sequence)h Fp(t)1370 2206 y Fn(1)1437 2192 y Fq(!)1528 2206 y FA(Q)1586 2215 y Fd(2)1652 2192 y Fp(t)1685 2206 y Fn(2)1752 2192 y Fq(!)1843 2206 y FA(Q)1901 2215 y Fd(2)1967 2192 y Fp(t)2000 2206 y Fn(3)2066 2192 y Fq(!)2157 2206 y FA(Q)2215 2215 y Fd(2)2281 2192 y Fq(\001)15 b(\001)g(\001)48 b Fs(in)31 b(whic)m(h)f(all)h(steps)g(tak)m(e)j(place)e(at)212 2305 y(the)e(ro)s(ot)f(p)s(osition.)39 b(According)29 b(to)h(Lemmata)g(4.2)h (and)d(6.4)j(this)d(sequence)i(can)f(b)s(e)g(transformed)212 2418 y(in)m(to)h(an)h(in\014nite)d(rewrite)h(sequence)i(in)e Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1894 2432 y Fn(2)1934 2418 y Fs(\):)439 2623 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(t)796 2637 y Fn(1)837 2623 y Fs(\))26 b Fq(!)989 2585 y Fn(+)1073 2623 y Fp(C)1138 2637 y Fn(1)1178 2623 y Fs([)p Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(t)1560 2637 y Fn(2)1601 2623 y Fs(\)])26 b Fq(!)1778 2585 y Fn(+)1862 2623 y Fp(C)1927 2637 y Fn(1)1966 2623 y Fs([)p Fp(C)2056 2637 y Fn(2)2096 2623 y Fs([)p Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(t)2478 2637 y Fn(3)2519 2623 y Fs(\)]])26 b Fq(!)g(\001)15 b(\001)g(\001)212 2827 y Fs(Note)33 b(that)f(for)f(the)h(applicabilit)m(y)c(of)k(Lemma)f(4.2)i (it)e(is)g(essen)m(tial)g(that)h(all)e(steps)i(in)e(the)i(in\014nite) 212 2940 y Fq(Q)286 2954 y Fn(2)326 2940 y Fs(-rewrite)k(sequence)h (tak)m(e)h(place)e(at)i(the)e(ro)s(ot)h(p)s(osition.)57 b(Con)m(v)m(ersely)-8 b(,)39 b(if)d Fp(P)49 b Fs(has)36 b(no)h(solution)212 3053 y(then)30 b Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)691 3067 y Fn(2)731 3053 y Fs(\))31 b(is)e Fp(!)s Fs(-terminating)h(b)m(y)g(Lemma)h(6.3)g(and)f(therefore)h (also)f(terminating.)300 b Fl(\003)212 3339 y Fr(9)135 b(NSE)45 b Fi(\))f Fr(SN)212 3542 y Fs(Let)31 b Fq(Q)449 3556 y Fn(3)514 3542 y Fs(=)25 b Fq(f)p Fp(f)10 b Fs(\()p Fp(d)p Fs(\))26 b Fq(!)f Fp(f)10 b Fs(\()p Fp(g)s Fs(\()p Fp(d)p Fs(\)\))p Fq(g)p Fs(.)43 b(This)28 b(TRS)i(is)f(terminating)g (and)h(self-em)m(b)s(edding.)212 3754 y Fb(Lemma)j(9.1)46 b Fo(The)33 b(TRS)g Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1435 3768 y Fn(3)1475 3754 y Fs(\))33 b Fo(is)f(terminating)i(for)f(every)g (PCP)f(instanc)-5 b(e)33 b Fp(P)13 b Fo(.)212 3887 y(Pr)-5 b(o)g(of)46 b Fs(According)21 b(to)i(Lemma)f(6.1\(1\))j(it)c(su\016ces) h(to)h(sho)m(w)f(that)g Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)2732 3901 y Fn(3)2773 3887 y Fs(\))22 b(is)f(terminating.)37 b(There)212 4000 y(are)43 b(sev)m(eral)f(w)m(a)m(ys)i(to)f(ac)m(hiev)m (e)g(this.)76 b(W)-8 b(e)43 b(use)f(t)m(yp)s(e)g(in)m(tro)s(duction)f (\([25)q(]\),)46 b(whic)m(h)41 b(is)h(p)s(ossible)212 4113 y(since)30 b Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)711 4127 y Fn(3)751 4113 y Fs(\))31 b(lac)m(ks)g(collapsing)d(\(and)i (duplicating\))f(rules.)39 b(Hence)31 b(w)m(e)g(ma)m(y)g(assume)f(that) h(the)212 4226 y(function)f(sym)m(b)s(ols)f(come)k(from)d(a)i(man)m (y-sorted)f(signature)g(suc)m(h)f(that)i(the)f(left)g(and)f(righ)m (t-hand)212 4339 y(side)g(of)i(an)m(y)f(rewrite)f(rule)g(are)i(w)m (ell-t)m(yp)s(ed)e(and)g(of)i(the)f(same)h(t)m(yp)s(e.)43 b(W)-8 b(e)32 b(use)f(t)m(w)m(o)i(sorts)e(1)g(and)g(2)212 4451 y(with)f Fp(A)i Fs(of)g(t)m(yp)s(e)f(1)22 b Fq(\002)f(\001)15 b(\001)g(\001)22 b(\002)e Fs(1)28 b Fq(!)f Fs(2)32 b(and)f(all)f(other) i(function)e(sym)m(b)s(ols)h(of)g(t)m(yp)s(e)h(1)21 b Fq(\002)g(\001)15 b(\001)g(\001)22 b(\002)f Fs(1)28 b Fq(!)f Fs(1.)212 4564 y(T)-8 b(erms)29 b(of)h(t)m(yp)s(e)f(1)h(are)g (in)e(normal)g(form.)40 b(So)29 b(if)g Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)2165 4578 y Fn(3)2205 4564 y Fs(\))30 b(is)e(not)i(terminating)e(then)h(there)h(exists)212 4677 y(an)h(in\014nite)e(rewrite)h(sequence)h(consisting)f(of)h(terms)g (of)g(t)m(yp)s(e)g(2.)43 b(Hence)32 b(all)e(steps)h(tak)m(e)i(place)e (at)212 4790 y(the)e(ro)s(ot)g(p)s(osition.)39 b(Since)27 b(terms)i(of)g(t)m(yp)s(e)g(2)g(do)g(not)g(con)m(tain)g(o)s(ccurrences) g(of)g Fp(A)g Fs(b)s(elo)m(w)f(the)h(ro)s(ot)212 4903 y(p)s(osition,)h(Lemma)h(4.3)h(applies.)41 b(Because)33 b Fq(U)1804 4917 y FA(\000)1863 4903 y Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))32 b(is)e(\(simply\))g(terminating)g(\(Lemma)h(5.20\),) 212 5016 y(w)m(e)g(obtain)f(an)g(in\014nite)e(rewrite)i(sequence)g(in)g Fq(Q)1929 5030 y Fn(3)1968 5016 y Fs(,)h(con)m(tradicting)f(its)g (termination.)408 b Fl(\003)1897 5462 y Fs(32)p eop %%Page: 33 33 33 32 bop 212 337 a Fb(Lemma)33 b(9.2)46 b Fo(The)31 b(TRS)h Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1432 351 y Fn(3)1472 337 y Fs(\))32 b Fo(is)f(non-self-emb)-5 b(e)g(dding)33 b(if)e(and)h(only)g(if)f Fp(P)45 b Fo(admits)32 b(no)g(solu-)212 450 y(tion.)212 582 y(Pr)-5 b(o)g(of)53 b Fs(If)30 b Fp(P)44 b Fs(admits)29 b(a)i(solution)e(then)h(w)m(e)h (obtain)439 783 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d)p Fs(\)\))28 b Fq(!)1089 745 y Fn(+)1089 816 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)1289 825 y Fd(3)1323 816 y Fn(\))1379 783 y Fp(C)i Fs([)p Fp(A)p Fs(\()p Fp(V)e(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(g)s Fs(\()p Fp(d)p Fs(\)\)\)])212 998 y(from)23 b(Lemmata)i(4.2)g(and)e(6.4.)40 b(Since)22 b Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d)p Fs(\)\))26 b(is)d(em)m(b)s(edded)g(in)f Fp(C)7 b Fs([)p Fp(A)p Fs(\()p Fp(V)e(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(g)s Fs(\()p Fp(d)p Fs(\)\)\)],)29 b(this)212 1111 y(sho)m(ws)j(that)h Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)946 1125 y Fn(3)987 1111 y Fs(\))32 b(is)g(self-em)m(b)s (edding.)45 b(Con)m(v)m(ersely)-8 b(,)33 b(if)f Fp(P)45 b Fs(has)32 b(no)h(solution)e(then)h Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)3598 1125 y Fn(3)3638 1111 y Fs(\))212 1224 y(is)30 b Fp(!)s Fs(-terminating)f(and)h(th)m(us)g(non-self-em)m (b)s(edding)e(b)m(y)i(Lemma)g(6.3.)951 b Fl(\003)212 1510 y Fr(10)135 b(ST)44 b Fi(\))h Fr(NSE)212 1713 y Fs(Before)23 b(de\014ning)d(the)i(TRS)f Fq(Q)1250 1727 y Fn(4)1312 1713 y Fs(used)g(for)h(the)g(relativ)m(e)g(undecidabilit)m (y)d(of)j(ST)f Fq(\))h Fs(NSE,)g(w)m(e)g(presen)m(t)212 1825 y(a)31 b(simple)d(fact)j(ab)s(out)g(non-self-em)m(b)s(eddingness.) 212 2035 y Fb(Lemma)i(10.1)46 b Fo(If)40 b(a)h(TRS)g Fq(R)f Fo(is)g(self-emb)-5 b(e)g(dding)42 b(then)e(it)h(admits)h(a)e(r) -5 b(ewrite)42 b(se)-5 b(quenc)g(e)40 b Fp(t)f Fq(!)3609 1997 y Fn(+)3609 2064 y FA(R)212 2148 y Fp(u)25 b Fq(!)380 2115 y FA(\003)380 2176 y(E)6 b Fn(m)n(b)549 2148 y Fp(t)32 b Fo(such)h(that)h(the)f(subse)-5 b(quenc)g(e)32 b(fr)-5 b(om)34 b Fp(t)e Fo(to)h Fp(u)g Fo(c)-5 b(ontains)34 b(a)f(r)-5 b(o)g(ot)35 b(r)-5 b(ewrite)33 b(step.)212 2280 y(Pr)-5 b(o)g(of)62 b Fs(Similar)36 b(to)k(the)g(pro)s(of)f(of)g (Lemma)h(8.2,)j(but)c(note)h(that)g(w)m(e)g(cannot)g(pro)m(v)m(e)g (that)g(there)212 2393 y(exists)35 b(a)h(rewrite)f(sequence)h Fp(t)e Fq(!)1403 2355 y Fn(+)1403 2422 y FA(R)1501 2393 y Fp(u)g Fq(!)1678 2360 y FA(\003)1678 2421 y(E)6 b Fn(m)n(b)1855 2393 y Fp(t)36 b Fs(that)g(starts)g(with)e(ro)s(ot)i(rewrite)f(step:)51 b(consider)212 2506 y(the)31 b(TRS)e Fq(f)p Fp(f)10 b Fs(\()p Fp(a)p Fs(\))26 b Fq(!)f Fp(f)10 b Fs(\()p Fp(g)s Fs(\()p Fp(h)p Fs(\()p Fp(b)p Fs(\)\)\))p Fp(;)15 b(g)s Fs(\()p Fp(b)p Fs(\))29 b Fq(!)d Fp(a)p Fq(g)p Fs(.)1800 b Fl(\003)353 2716 y Fs(W)-8 b(e)32 b(de\014ne)439 2948 y Fq(Q)513 2962 y Fn(4)578 2948 y Fs(=)674 2820 y Ff(\032)784 2891 y Fp(f)10 b Fs(\()p Fp(d;)15 b(e;)g(x)p Fs(\))84 b Fq(!)f Fp(f)10 b Fs(\()p Fp(x;)15 b(g)s Fs(\()p Fp(e)p Fs(\))p Fp(;)g(e)p Fs(\))784 3004 y Fp(f)10 b Fs(\()p Fp(d;)15 b(e;)g(x)p Fs(\))84 b Fq(!)f Fp(f)10 b Fs(\()p Fp(g)s Fs(\()p Fp(d)p Fs(\))p Fp(;)15 b(x;)g(d)p Fs(\))212 3206 y Fb(Lemma)33 b(10.2)46 b Fo(The)33 b(TRS)g Fq(Q)1289 3220 y Fn(4)1361 3206 y Fo(is)g(non-self-emb)-5 b(e)g(dding.)212 3339 y(Pr)g(o)g(of)67 b Fs(If)44 b Fq(Q)668 3353 y Fn(4)753 3339 y Fs(is)g(self-em)m(b)s(edding)f(then)h(b)m(y)h(Lemma)g(10.1)i (there)e(exists)f(a)i(rewrite)e(sequence)212 3452 y Fp(t)36 b Fq(!)372 3414 y Fn(+)372 3480 y FA(Q)430 3489 y Fd(4)504 3452 y Fp(u)g Fq(!)683 3419 y FA(\003)683 3479 y(E)6 b Fn(m)n(b)861 3452 y Fp(t)37 b Fs(suc)m(h)f(that)i(the)e(subsequence)g (from)h Fp(t)f Fs(to)h Fp(u)g Fs(con)m(tains)g(a)g(ro)s(ot)g(rewrite)f (step.)212 3565 y(By)g(the)f(form)g(of)g(the)g(rules)f(in)g Fq(Q)1418 3579 y Fn(4)1458 3565 y Fs(,)i(the)f(latter)h(condition)e (requires)f(that)j(the)g(\014rst)e(step)h(in)f(the)212 3678 y(subsequence)h(from)g Fp(t)g Fs(to)h Fp(u)g Fs(is)e(a)i(ro)s(ot)g (rewrite)f(step.)56 b(Moreo)m(v)m(er,)39 b(later)c(steps)h(m)m(ust)f (tak)m(e)i(place)212 3790 y(b)s(elo)m(w)d(the)h(ro)s(ot.)55 b(It)35 b(follo)m(ws)f(that)i Fp(t)c Fs(=)h Fp(f)10 b Fs(\()p Fp(d;)15 b(e;)g(s)p Fs(\),)37 b Fp(s)c Fq(!)2222 3758 y FA(\003)2222 3817 y(Q)2280 3826 y Fd(4)2351 3790 y Fp(s)2394 3758 y FA(0)2452 3790 y Fs(and)h(either)g Fp(u)f Fs(=)g Fp(f)10 b Fs(\()p Fp(s)3217 3758 y FA(0)3239 3790 y Fp(;)15 b(g)s Fs(\()p Fp(e)p Fs(\))p Fp(;)g(e)p Fs(\))38 b(or)212 3903 y Fp(u)25 b Fs(=)g Fp(f)10 b Fs(\()p Fp(g)s Fs(\()p Fp(d)p Fs(\))p Fp(;)15 b(s)721 3870 y FA(0)746 3903 y Fp(;)g(d)p Fs(\).)40 b(But)26 b(then)f Fp(t)h Fs(can)g(only)f(b)s(e)g(em)m(b)s(edded)f(in)h Fp(u)g Fs(if)g Fp(s)g Fs(=)g Fp(e)h Fs(and)f Fp(s)2975 3870 y FA(0)3023 3903 y Fq(!)3114 3870 y FA(\003)3114 3931 y(E)6 b Fn(m)n(b)3283 3903 y Fp(d)25 b Fs(or)h Fp(s)f Fs(=)g Fp(d)212 4016 y Fs(and)30 b Fp(s)432 3983 y FA(0)480 4016 y Fq(!)571 3983 y FA(\003)571 4044 y(E)6 b Fn(m)n(b)740 4016 y Fp(e)p Fs(.)41 b(Note)32 b(that)f Fp(t)f Fs(cannot)h(b)s(e)f(em) m(b)s(edded)f(in)h Fp(s)2329 3983 y FA(0)2382 4016 y Fs(as)h Fp(t)f Fs(con)m(tains)g(one)h(more)g Fp(f)40 b Fs(sym)m(b)s(ol)212 4129 y(than)30 b Fp(s)467 4096 y FA(0)490 4129 y Fs(.)41 b(Ho)m(w)m(ev)m(er,)32 b(b)s(oth)e(cases)h (con)m(tradict)g Fp(s)25 b Fq(!)1981 4096 y FA(\003)1981 4156 y(Q)2039 4165 y Fd(4)2103 4129 y Fp(s)2146 4096 y FA(0)2169 4129 y Fs(.)40 b(Hence)32 b Fq(Q)2579 4143 y Fn(4)2649 4129 y Fs(is)d(non-self-em)m(b)s(edding.)70 b Fl(\003)212 4339 y Fb(Lemma)33 b(10.3)46 b Fo(The)33 b(TRS)g Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1487 4353 y Fn(4)1527 4339 y Fs(\))33 b Fo(is)g(non-self-emb)-5 b(e)g(dding)33 b(for)g(every)g(PCP)f(instanc)-5 b(e)33 b Fp(P)13 b Fo(.)212 4471 y(Pr)-5 b(o)g(of)44 b Fs(Supp)s(ose)19 b(on)h(the)g(con)m(trary)h(that)g Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)1888 4485 y Fn(4)1928 4471 y Fs(\))21 b(is)e(self-em)m(b)s (edding.)35 b(According)20 b(to)h(Lemma)g(10.1)212 4584 y(there)31 b(exists)f(a)g(rewrite)g(sequence)439 4785 y Fp(t)25 b Fq(!)588 4747 y Fn(+)588 4819 y FA(S)5 b Fn(\()p Fm(P)r(;)p FA(Q)788 4828 y Fd(4)822 4819 y Fn(\))879 4785 y Fp(u)25 b Fq(!)1047 4748 y FA(\003)1047 4809 y(E)6 b Fn(m)n(b)1215 4785 y Fp(t)2264 b Fs(\(23\))212 4987 y(suc)m(h)31 b(that)g(its)f(\014rst)h(part)f(con)m(tains)i(a)f(step)g (at)g(the)h(ro)s(ot)f(p)s(osition.)40 b(By)31 b(the)g(form)g(of)g(the)g (rules)f(in)212 5100 y Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)484 5114 y Fn(4)524 5100 y Fs(\))25 b(this)e(implies)e(that)k Fp(t)f Fs(is)f(a)i(redex,)g(so)f(w)m(e)h(ma)m(y)g(write)e Fp(t)i Fs(=)g Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d;)15 b(e;)g(t)3145 5067 y FA(0)3171 5100 y Fs(\)\).)40 b(The)23 b(term)212 5213 y Fp(u)g Fs(can)g(b)s(e)g (written)f(\(cf.)i(the)f(pro)s(of)f(of)i(Lemma)f(8.5\))h(as)g Fp(C)7 b Fs([)p Fp(v)2239 5227 y Fn(1)2278 5213 y Fp(;)15 b(:)g(:)g(:)h(;)f(v)2523 5228 y Fm(k)2566 5213 y Fs(])24 b(suc)m(h)e(that)i Fp(C)30 b Fs(is)22 b(a)h(non-empt)m(y)1897 5462 y(33)p eop %%Page: 34 34 34 33 bop 212 337 a Fs(con)m(text)38 b(consisting)d(of)i Fp(B)k Fs(sym)m(b)s(ols)35 b(and)g(ev)m(ery)j Fp(v)2013 351 y Fm(i)2077 337 y Fs(starts)f(with)e(an)h Fp(A)g Fs(sym)m(b)s(ol.)58 b(W)-8 b(e)37 b(can)g(rear-)212 450 y(range)28 b(the)g(second)g(part)f(of)35 b(\(23\))29 b(in)m(to)f Fp(u)d Fq(!)1743 411 y Fn(+)1743 479 y FA(E)6 b Fn(m)n(b)1911 450 y Fp(v)1955 464 y Fm(i)2009 450 y Fq(!)2100 417 y 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b(a)g(solution.)59 b(W)-8 b(e)38 b(sho)m(w)e(that)i Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)3500 5084 y Fn(5)3540 5070 y Fs(\))37 b(is)212 5183 y(not)32 b(totally)g(terminating.)45 b(According)32 b(to)h(Lemma)f(6.1\(3\))i(this)d(is)g(equiv)-5 b(alen)m(t)32 b(to)h(sho)m(wing)e(that)1897 5462 y(34)p eop %%Page: 35 35 35 34 bop 212 337 a Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)488 351 y Fn(5)528 337 y Fs(\))23 b(is)e(not)i(totally)f(terminating.)37 b(Supp)s(ose)20 b(on)i(the)h(con)m(trary)g(that)f Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)3068 351 y Fn(5)3109 337 y Fs(\))22 b(is)f(totally)i(ter-)212 450 y(minating.)39 b(So)28 b(there)h(exists)g(a)g(compatible)f(total)i(reduction)d(order)i Fp(>)p Fs(.)39 b(Because)31 b(b)m(y)d(Lemma)h(4.2)212 562 y(b)s(oth)i Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d;)15 b(e)p Fs(\)\))31 b Fq(!)1163 529 y Fn(+)1249 562 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(e;)15 b(e)p Fs(\)\))35 b(and)c Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d;)15 b(e)p Fs(\)\))31 b Fq(!)2805 529 y Fn(+)2891 562 y Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d;)15 b(d)p Fs(\)\),)36 b(w)m(e)212 675 y(ha)m(v)m(e)29 b Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d;)15 b(e)p Fs(\)\))29 b Fp(>)c(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(e;)15 b(e)p Fs(\)\))31 b(and)d Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d;)15 b(e)p Fs(\)\))28 b Fp(>)d(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()p Fp(d;)15 b(d)p Fs(\)\))31 b(b)m(y)d(com-)212 788 y(patibilit)m(y)-8 b(.)62 b(By)38 b(the)h(truncation)e(rule)g(for)h (total)h(reduction)e(orders)g(\([25)r(,)j(Prop)s(osition)c(9]\))j(one) 212 901 y(ma)m(y)32 b(remo)m(v)m(e)g(a)f(con)m(text)i Fp(C)k Fs(from)31 b(an)f(inequation)g Fp(C)7 b Fs([)p Fp(t)p Fs(])26 b Fp(>)g(C)7 b Fs([)p Fp(t)2421 868 y FA(0)2443 901 y Fs(].)43 b(By)31 b(remo)m(ving)g Fp(A)p Fs(\()p Fp(V)5 b(;)15 b(W)m(;)g(f)10 b Fs(\()32 b Fp(;)15 b(e)p Fs(\)\))212 1014 y(and)30 b Fp(A)p 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4862 y Fs(is)f(not)212 4975 y Fp(!)s Fs(-terminating.)2810 b Fl(\003)1897 5462 y Fs(35)p eop %%Page: 36 36 36 35 bop 212 337 a Fr(13)135 b(PT)44 b Fi(\))h Fg(!)t Fr(T)212 540 y Fs(A)m(t)30 b(presen)m(t)e(it)g(is)g(unkno)m(wn)f(as)i (to)g(whether)f(p)s(olynomial)d(termination)j(is)f(an)i(undecidable)d (prop-)212 652 y(ert)m(y)e(of)g(\()p Fp(!)s Fs(-terminating\))f(TRSs.) 37 b(The)23 b(follo)m(wing)f(TRS)g(is)h Fp(!)s Fs(-terminating)f(but)h (not)h(p)s(olynomially)212 765 y(terminating)29 b(\(Zan)m(tema)j([25)q (,)f(Prop)s(osition)d(10]\):)481 922 y Fp(f)10 b Fs(\()p Fp(g)s Fs(\()p Fp(h)p Fs(\()p Fp(x)p Fs(\)\)\))85 b Fq(!)e Fp(g)s Fs(\()p Fp(f)10 b Fs(\()p Fp(h)p Fs(\()p Fp(g)s Fs(\()p Fp(x)p Fs(\)\)\)\))212 1079 y(So)43 b(the)h(p)s(olynomially)c (terminating)i(TRSs)g(form)h(a)g(prop)s(er)f(sub)s(class)f(of)j(the)f Fp(!)s Fs(-terminating)212 1192 y(TRSs.)52 b(W)-8 b(e)36 b(conjecture)f(that)g(the)g(implication)d(PT)i Fq(\))h Fp(!)s 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Fb(Theorem)j(14.2)46 b Fo(A)n(n)32 b(ortho)-5 b(gonal)35 b(and)e(non-er)-5 b(asing)34 b(TRS)e(is)g(we)-5 b(akly)34 b(normalizing)f(if)f(and)h (only)212 4169 y(if)f(it)h(is)g(str)-5 b(ongly)34 b(normalizing.)2282 b Fl(\003)353 4334 y Fs(The)22 b(implication)e(SN)j Fq(\))f Fs(WN)i(is)d(kno)m(wn)h(to)i(b)s(e)e(strict)g(ev)m(en)i(for)e(one-rule) g(TRSs.)37 b(An)23 b(example)212 4447 y(of)31 b(a)f(w)m(eakly)h(but)f (not)g(strongly)g(normalizing)e(one-rule)i(TRS)f(is)h(the)g(system)439 4604 y Fq(Q)513 4618 y Fn(8)578 4604 y Fs(=)25 b Fq(f)p Fp(f)10 b Fs(\()p Fp(a;)15 b(f)10 b Fs(\()p Fp(x;)15 b(y)s Fs(\)\))27 b Fq(!)e Fp(f)10 b Fs(\()p Fp(x;)15 b(f)10 b Fs(\()p Fp(x;)15 b(f)10 b Fs(\()p Fp(b;)15 b(b)p Fs(\)\)\))p Fq(g)212 4761 y Fs(of)44 b(Akk)m(erman)g(\(see)h([13)q (]\).)82 b(Note)45 b(that)g(this)d(TRS)h(is)g(neither)g(orthogonal)h (nor)g(non-erasing.)212 4874 y(Ho)m(w)m(ev)m(er,)51 b Fq(Q)696 4888 y Fn(8)781 4874 y Fs(cannot)46 b(b)s(e)e(used)g(either:) 70 b(Since)44 b Fq(U)9 b Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))46 b(and)f Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)p Fs(\))46 b(only)e(sim)m(ulate)g Fo(r)-5 b(o)g(ot)212 4987 y Fs(rewrite)30 b(sequences)g(in)f Fq(Q)i Fs(\(for)f(PCP)g (instances)f Fp(P)44 b Fs(that)31 b(admit)e(a)i(solution\))e(and)h(ro)s (ot)g(rewriting)212 5100 y(in)35 b Fq(Q)398 5114 y Fn(8)473 5100 y Fs(is)g(not)h(w)m(eakly)g(normalizing,)g(viz.)g Fp(f)10 b Fs(\()p Fp(a;)15 b(f)10 b Fs(\()p Fp(a;)15 b(f)10 b Fs(\()p Fp(b;)15 b(b)p Fs(\)\)\))35 b Fq(!)g Fp(f)10 b Fs(\()p Fp(a;)15 b(f)10 b Fs(\()p Fp(a;)15 b(f)10 b Fs(\()p Fp(b;)15 b(b)p Fs(\)\)\),)39 b(it)c(follo)m(ws)212 5213 y(that)c Fq(S)7 b Fs(\()p Fp(P)s(;)15 b Fq(Q)681 5227 y Fn(8)721 5213 y Fs(\))31 b(is)e(not)i(w)m(eakly)f(normalizing)f (for)h(solv)-5 b(able)29 b Fp(P)13 b Fs(.)1897 5462 y(36)p eop %%Page: 37 37 37 36 bop 212 337 a Fr(Conclusion)212 540 y Fs(The)30 b(results)f(pro)m(v)m(ed)i(in)e(this)g(pap)s(er)g(are)i(summarized)e(b) s(elo)m(w.)212 726 y Fb(Theorem)34 b(14.3)46 b Fo(F)-7 b(or)39 b(the)f(fol)5 b(lowing)38 b(implic)-5 b(ations)40 b Fp(X)h Fq(\))34 b Fp(Y)20 b Fo(,)39 b Fp(X)45 b Fo(is)37 b(an)h(unde)-5 b(cidable)38 b(pr)-5 b(op)g(erty)212 839 y(of)33 b(one-rule)g(TRSs)g(that)h(satisfy)f(pr)-5 b(op)g(erty)35 b Fp(Y)20 b Fo(:)481 1017 y Fp(!)s Fo(T)83 b Fq(\))g Fo(TT)g Fq(\))g Fo(ST)g Fq(\))g Fo(NSE)f Fq(\))h Fo(SN)g Fq(\))g Fo(NL)f Fq(\))h Fo(A)n(C)212 1195 y(In)33 b(addition,)h(SN)e (is)h(an)g(unde)-5 b(cidable)34 b(pr)-5 b(op)g(erty)35 b(of)e(TRSs)g(that)h(satisfy)f(WN.)654 b Fl(\003)353 1381 y Fs(W)-8 b(e)27 b(already)e(iden)m(ti\014ed)e(the)j(follo)m(wing) d(op)s(en)i(problems:)37 b(Is)25 b(the)g(implication)e(SN)i Fq(\))g Fs(WN)h(rel-)212 1494 y(ativ)m(e)g(undecidable)c(for)i Fo(one-rule)32 b Fs(TRSs?)37 b(Is)24 b(the)h(implication)d(PT)i Fq(\))h Fp(!)s Fs(T)f(relativ)m(e)h(undecidable?)212 1607 y(Another)34 b(problem)d(is)i(whether)g(the)h(results)e(obtained)h (in)f(this)g(pap)s(er)h(can)g(b)s(e)g(strengthened)h(to)212 1720 y(string)29 b(rewriting)g(systems.)353 1833 y(In)36 b(part)h(2)f(of)h(this)f(pap)s(er)f([8)q(])i(w)m(e)g(presen)m(t)f (relativ)m(e)h(undecidabilit)m(y)c(results)j(for)g(prop)s(erties)212 1946 y(related)30 b(to)h(con\015uence.)212 2228 y Fr(Ac)l(kno)l (wledgemen)l(ts)212 2431 y Fs(The)39 b(detailed)f(commen)m(ts)i(b)m(y)f (the)g(anon)m(ymous)g(referees)h(help)s(ed)d(to)j(shorten)e(the)i(pap)s (er.)66 b(Al-)212 2544 y(fons)35 b(Geser)h(is)e(partially)f(supp)s (orted)h(b)m(y)h(gran)m(t)h Fa(Ku)47 b(966/3-1)34 b Fs(of)h(the)h (Deutsc)m(he)g(F)-8 b(orsc)m(h)m(ungsge-)212 2657 y(meinsc)m(haft)37 b(\(DF)m(G\))j(within)34 b(the)k(Sc)m(h)m(w)m(erpunkt)e(Deduktion)h(at) h(the)f(Univ)m(ersit)m(y)g(of)g(T)s(\177)-48 b(ubingen.)212 2770 y(Aart)27 b(Middeldorp)c(is)i(partially)g(supp)s(orted)f(b)m(y)i (the)g(Gran)m(t-in-Aid)f(for)h(Scien)m(ti\014c)f(Researc)m(h)j(C\(2\)) 212 2883 y(11680338)34 b(of)d(the)f(Ministry)f(of)h(Education,)g (Science,)g(Sp)s(orts)g(and)f(Culture)g(of)h(Japan.)212 3165 y Fr(References)258 3368 y Fs([1])46 b(F.)36 b(Baader)h(and)e(T.)g (Nipk)m(o)m(w.)57 b Fo(T)-7 b(erm)38 b(R)-5 b(ewriting)38 b(and)h(A)n(l)5 b(l)37 b(That)p Fs(.)57 b(Cam)m(bridge)35 b(Univ)m(ersit)m(y)399 3481 y(Press,)30 b(1998.)258 3660 y([2])46 b(G.)c(Bonfan)m(te,)k(A.)c(Cic)m(hon,)i(J.-Y.)e(Marion,)i(and) d(H.)h(T)-8 b(ouzet.)75 b(Complexit)m(y)40 b(classes)i(and)399 3773 y(rewrite)29 b(systems)g(with)g(p)s(olynomial)e(in)m (terpretation.)39 b(In)28 b Fo(Pr)-5 b(o)g(c)g(e)g(e)g(dings)34 b(of)f(the)f(11th)h(A)n(nnual)399 3886 y(Confer)-5 b(enc)g(e)34 b(of)g(the)f(Eur)-5 b(op)g(e)g(an)36 b(Asso)-5 b(ciation)35 b(for)f(Computer)g(Scienc)-5 b(e)33 b(L)-5 b(o)g(gic)p Fs(,)33 b(v)m(olume)e(1584)399 3998 y(of)21 b Fo(L)-5 b(e)g(ctur)g(e)25 b(Notes)f(in)g(Computer)h(Scienc)-5 b(e)p Fs(,)23 b(pages)f(372{384,)k(Berlin,)21 b(1998.)i(Springer-V)-8 b(erlag.)258 4177 y([3])46 b(M.)32 b(Dauc)m(het.)44 b(Sim)m(ulation)29 b(of)j(Turing)d(mac)m(hines)h(b)m(y)i(a)f(regular)g(rewrite)f(rule.)42 b Fo(The)-5 b(or)g(etic)g(al)399 4290 y(Computer)34 b(Scienc)-5 b(e)p Fs(,)30 b(103\(2\):409{420,)37 b(1992.)258 4469 y([4])46 b(N.)33 b(Dersho)m(witz)g(and)g(J.-P)-8 b(.)33 b(Jouannaud.)47 b(Rewrite)32 b(systems.)49 b(In)32 b(J.)g(v)-5 b(an)33 b(Leeu)m(w)m(en,)i(editor,)399 4582 y Fo(Handb)-5 b(o)g(ok)47 b(of)f(The)-5 b(or)g(etic)g(al)47 b(Computer)g(Scienc)-5 b(e)p Fs(,)48 b(v)m(olume)43 b(B,)i(pages)g(243{320.)i(Elsevier,)399 4695 y(1990.)258 4874 y([5])f(N.)40 b(Dersho)m(witz,)j(J.-P)-8 b(.)41 b(Jouannaud,)g(and)e(J.W.)h(Klop.)68 b(Problems)38 b(in)h(rewriting)f(I)s(I)s(I.)67 b(In)399 4987 y Fo(Pr)-5 b(o)g(c)g(e)g(e)g(dings)37 b(of)e(the)g(6th)h(International)h(Confer)-5 b(enc)g(e)35 b(on)h(R)-5 b(ewriting)35 b(T)-7 b(e)i(chniques)35 b(and)h(Ap-)399 5100 y(plic)-5 b(ations)p Fs(,)26 b(v)m(olume)c(914)h (of)f Fo(L)-5 b(e)g(ctur)g(e)26 b(Notes)f(in)f(Computer)j(Scienc)-5 b(e)p Fs(,)23 b(pages)g(457{471,)k(Berlin,)399 5213 y(1995.)32 b(Springer-V)-8 b(erlag.)1897 5462 y(37)p eop %%Page: 38 38 38 37 bop 258 337 a Fs([6])46 b(M.)f(F)-8 b(erreira)45 b(and)f(H.)h(Zan)m(tema.)84 b(T)-8 b(otal)46 b(termination)d(of)i(term) g(rewriting.)81 b Fo(Applic)-5 b(able)399 450 y(A)n(lgebr)g(a)33 b(in)f(Engine)-5 b(ering,)32 b(Communic)-5 b(ation)35 b(and)f(Computing)p Fs(,)d(7\(2\):133{162,)36 b(1996.)258 634 y([7])46 b(A.)27 b(Geser.)35 b(Omega-termination)27 b(is)f(undecidable)e(for)j(totally)g(terminating)e(term)i(rewriting)399 746 y(systems.)41 b Fo(Journal)33 b(of)g(Symb)-5 b(olic)34 b(Computation)p Fs(,)f(23:399{411,)i(1997.)258 930 y([8])46 b(A.)32 b(Geser,)i(A.)e(Middeldorp,)e(E.)j(Ohlebusc)m(h,)d(and)i(H.)g (Zan)m(tema.)47 b(Relativ)m(e)33 b(undecidabilit)m(y)399 1043 y(in)24 b(term)i(rewriting,)f(part)g(2:)39 b(The)25 b(con\015uence)h(hierarc)m(h)m(y)-8 b(.)33 b Fo(Information)d(and)g (Computation)p Fs(.)399 1156 y(To)g(app)s(ear.)258 1340 y([9])46 b(A.)32 b(Geser,)i(A.)e(Middeldorp,)e(E.)j(Ohlebusc)m(h,)d (and)i(H.)g(Zan)m(tema.)47 b(Relativ)m(e)33 b(undecidabilit)m(y)399 1453 y(in)g(term)h(rewriting.)51 b(In)33 b Fo(Pr)-5 b(o)g(c)g(e)g(e)g (dings)39 b(of)d(the)h(10th)h(A)n(nnual)e(Confer)-5 b(enc)g(e)37 b(of)g(the)f(Eur)-5 b(op)g(e)g(an)399 1566 y(Asso)g(ciation)27 b(for)g(Computer)g(Scienc)-5 b(e)25 b(L)-5 b(o)g(gic)p Fs(,)26 b(v)m(olume)d(1258)h(of)g Fo(L)-5 b(e)g(ctur)g(e)26 b(Notes)g(in)g(Computer)399 1679 y(Scienc)-5 b(e)p Fs(,)30 b(pages)h(150{166,)j(Berlin,)29 b(1997.)j(Springer-V)-8 b(erlag.)212 1863 y([10])47 b(A.)32 b(Geser,)i(A.)e(Middeldorp,)e(E.)j (Ohlebusc)m(h,)d(and)i(H.)g(Zan)m(tema.)47 b(Relativ)m(e)33 b(undecidabilit)m(y)399 1976 y(in)40 b(the)i(termination)e(hierarc)m(h) m(y)h(of)h(single)e(rewrite)h(rules.)73 b(In)40 b Fo(Pr)-5 b(o)g(c)g(e)g(e)g(dings)45 b(of)f(the)f(22nd)399 2089 y(Col)5 b(lo)-5 b(quium)38 b(on)g(T)-7 b(r)i(e)g(es)39 b(in)e(A)n(lgebr)-5 b(a)37 b(and)i(Pr)-5 b(o)g(gr)g(amming)p Fs(,)39 b(v)m(olume)d(1214)h(of)f Fo(L)-5 b(e)g(ctur)g(e)38 b(Notes)399 2202 y(in)32 b(Computer)i(Scienc)-5 b(e)p Fs(,)31 b(pages)g(237{248,)i(Berlin,)c(1997.)k(Springer-V)-8 b(erlag.)212 2386 y([11])47 b(G.)37 b(Huet)h(and)e(D.S.)i(Lankford.)59 b(On)36 b(the)h(uniform)e(halting)h(problem)f(for)i(term)g(rewriting) 399 2499 y(systems.)k(Rapp)s(ort)29 b(Lab)s(oria)h(283,)i(INRIA,)e (1978.)212 2683 y([12])47 b(G.)39 b(Huet)h(and)e(D.C.)h(Opp)s(en.)64 b(Equations)38 b(and)h(rewrite)f(rules:)56 b(A)39 b(surv)m(ey)-8 b(.)66 b(In)38 b(R.)h(Bo)s(ok,)399 2796 y(editor,)29 b Fo(F)-7 b(ormal)34 b(L)-5 b(anguage)32 b(The)-5 b(ory:)42 b(Persp)-5 b(e)g(ctives)32 b(and)h(Op)-5 b(en)31 b(Pr)-5 b(oblems)p Fs(,)31 b(pages)f(349{405.)399 2908 y(Academic)h(Press,)f (1980.)212 3092 y([13])47 b(J.W.)27 b(Klop.)34 b(T)-8 b(erm)26 b(rewriting)f(systems.)35 b(In)26 b(S.)g(Abramski,)g(D.)h (Gabba)m(y)-8 b(,)29 b(and)d(T.)h(Maibaum,)399 3205 y(editors,)40 b Fo(Handb)-5 b(o)g(ok)41 b(of)f(L)-5 b(o)g(gic)40 b(in)g(Computer)h (Scienc)-5 b(e)p Fs(,)40 b(v)m(olume)e(2,)i(pages)f(1{116.)h(Oxford)399 3318 y(Univ)m(ersit)m(y)30 b(Press,)g(1992.)212 3502 y([14])47 b(J.B.)41 b(Krusk)-5 b(al.)70 b(W)-8 b(ell-quasi-ordering,)42 b(the)f(tree)h(theorem,)i(and)c(Vazson)m(yi's)h(conjecture.)399 3615 y Fo(T)-7 b(r)i(ansactions)35 b(of)e(the)g(A)n(meric)-5 b(an)33 b(Mathematic)-5 b(al)35 b(So)-5 b(ciety)p Fs(,)31 b(95:210{225,)k(1960.)212 3799 y([15])47 b(P)-8 b(.)38 b(Lescanne.)62 b(On)37 b(termination)f(of)i(one)f(rule)g(rewrite)f (systems.)62 b Fo(The)-5 b(or)g(etic)g(al)42 b(Computer)399 3912 y(Scienc)-5 b(e)p Fs(,)30 b(132:395{401,)36 b(1994.)212 4096 y([16])47 b(U.)34 b(Martin.)51 b(New)34 b(directions)f(for)h(syn)m (tactic)h(termination)d(orderings.)50 b Fo(Pr)-5 b(o)g(gr)g(ess)38 b(in)e(Com-)399 4209 y(puter)d(Scienc)-5 b(e)32 b(and)i(Applie)-5 b(d)34 b(L)-5 b(o)g(gic)p Fs(,)31 b(15:209{224,)k(1998.)212 4393 y([17])47 b(U.)42 b(Martin)f(and)h(E.)f(Scott.)76 b(The)41 b(order)h(t)m(yp)s(es)f(of)h(termination)f(orderings)f(on)i (monadic)399 4506 y(terms,)31 b(strings)e(and)g(m)m(ultisets.)40 b Fo(Journal)34 b(of)f(Symb)-5 b(olic)33 b(L)-5 b(o)g(gic)p Fs(,)32 b(62\(2\):624{635,)k(1997.)212 4690 y([18])47 b(Y.)c(Matiy)m(asevic)m(h)h(and)e(G.)i(Senizergues.)76 b(Decision)43 b(problems)e(for)h(semi-Th)m(ue)g(systems)399 4803 y(with)30 b(a)i(few)f(rules.)43 b(In)31 b Fo(Pr)-5 b(o)g(c)g(e)g(e)g(dings)36 b(of)e(the)g(11th)h(IEEE)e(A)n(nnual)g(Symp) -5 b(osium)36 b(on)e(L)-5 b(o)g(gic)34 b(in)399 4916 y(Computer)g(Scienc)-5 b(e)p Fs(,)30 b(pages)h(523{531,)j(1996.)212 5100 y([19])47 b(A.)26 b(Middeldorp)e(and)h(B.)i(Gramlic)m(h.)32 b(Simple)24 b(termination)h(is)g(di\016cult.)32 b Fo(Applic)-5 b(able)29 b(A)n(lgebr)-5 b(a)399 5213 y(in)32 b(Engine)-5 b(ering,)33 b(Communic)-5 b(ation)34 b(and)g(Computing)p Fs(,)e(6\(2\):115{128,)j(1995.)1897 5462 y(38)p eop %%Page: 39 39 39 38 bop 212 337 a Fs([20])47 b(A.)41 b(Middeldorp,)g(H.)g(Ohsaki,)h (and)e(H.)h(Zan)m(tema.)72 b(T)-8 b(ransforming)39 b(termination)g(b)m (y)i(self-)399 450 y(lab)s(elling.)53 b(In)35 b Fo(Pr)-5 b(o)g(c)g(e)g(e)g(dings)40 b(of)d(the)h(13th)h(International)h(Confer) -5 b(enc)g(e)39 b(on)f(A)n(utomate)-5 b(d)38 b(De-)399 562 y(duction)p Fs(,)47 b(v)m(olume)c(1104)i(of)e Fo(L)-5 b(e)g(ctur)g(e)45 b(Notes)f(in)h(A)n(rti\014cial)f(Intel)5 b(ligenc)-5 b(e)p Fs(,)46 b(pages)e(373{386,)399 675 y(Berlin,)29 b(1996.)j(Springer-V)-8 b(erlag.)212 863 y([21])47 b(A.)25 b(Middeldorp)d(and)i(H.)i(Zan)m(tema.)32 b(Simple)22 b(termination)i(of)h(rewrite)f(systems.)32 b Fo(The)-5 b(or)g(etic)g(al)399 976 y(Computer)34 b(Scienc)-5 b(e)p Fs(,)30 b(175\(1\):127{158,)37 b(1997.)212 1164 y([22])47 b(D.)33 b(Plaisted.)45 b(The)32 b(undecidabilit)m(y)c(of)33 b(self-em)m(b)s(edding)d(for)i(term)g(rewriting)e(systems.)47 b Fo(In-)399 1276 y(formation)35 b(Pr)-5 b(o)g(c)g(essing)33 b(L)-5 b(etters)p Fs(,)32 b(20:61{64,)i(1985.)212 1464 y([23])47 b(E.)34 b(P)m(ost.)52 b(A)35 b(v)-5 b(arian)m(t)33 b(of)i(a)f(recursiv)m(ely)f(unsolv)-5 b(able)32 b(problem.)50 b Fo(Bul)5 b(letin)35 b(of)i(the)f(A)n(meric)-5 b(an)399 1577 y(Mathematic)g(al)35 b(So)-5 b(ciety)p Fs(,)31 b(52:264{268,)k (1946.)212 1765 y([24])47 b(H.)41 b(T)-8 b(ouzet.)71 b(A)40 b(complex)g(example)g(of)g(a)h(simplifying)36 b(rewrite)k(system.)70 b(In)39 b Fo(Pr)-5 b(o)g(c)g(e)g(e)g(dings)399 1878 y(of)37 b(the)f(25th)i(International)h(Col)5 b(lo)-5 b(quium)37 b(on)g(A)n(utomata,)h(L)-5 b(anguages)37 b(and)g(Pr)-5 b(o)g(gr)g(amming)p Fs(,)399 1990 y(v)m(olume)36 b(1443)h(of)f Fo(L)-5 b(e)g(ctur)g(e)38 b(Notes)g(in)g(Computer)h(Scienc)-5 b(e)p Fs(,)37 b(pages)f(507{517,)41 b(Berlin,)35 b(1998.)399 2103 y(Springer-V)-8 b(erlag.)212 2291 y([25])47 b(H.)34 b(Zan)m(tema.)51 b(T)-8 b(ermination)32 b(of)i(term)f(rewriting:)45 b(In)m(terpretation)34 b(and)f(t)m(yp)s(e)h(elimination.)399 2404 y Fo(Journal)g(of)e(Symb)-5 b(olic)34 b(Computation)p Fs(,)f(17:23{50,)h(1994.)212 2592 y([26])47 b(H.)21 b(Zan)m(tema.)k(T) -8 b(otal)21 b(termination)e(of)i(term)f(rewriting)f(is)g(undecidable.) j Fo(Journal)i(of)g(Symb)-5 b(olic)399 2704 y(Computation)p Fs(,)33 b(20:43{60,)h(1995.)212 2892 y([27])47 b(H.)31 b(Zan)m(tema)h(and)f(A.)g(Geser.)43 b(Non-lo)s(oping)30 b(rewriting.)41 b(T)-8 b(ec)m(hnical)30 b(Rep)s(ort)h(UU-CS-1996-)399 3005 y(03,)g(Utrec)m(h)m(t)h(Univ)m(ersit)m(y)-8 b(,)31 b(1996.)1897 5462 y(39)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF