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Fu(\(b)o(y)0 1432 y(construction,)f(where)g(1)g Fs(\024)h Fr(i)f(<)g(k)i Fs(\024)e Fr(j)781 1441 y Fq(1)816 1432 y Fs(\000)14 b Fu(1\))22 b(and)g(no)g Fr(t)1124 1441 y Fl(i)1160 1432 y Fs(\024)1198 1441 y Fl(emb)1285 1432 y Fr(s)1308 1441 y Fl(k)1351 1432 y Fu(\(otherwise)f Fr(t)1610 1441 y Fl(i)1646 1432 y Fs(\024)1684 1441 y Fl(emb)1771 1432 y Fr(t)1789 1441 y Fl(j)1803 1447 y Fk(k)1846 1432 y Fu(since)0 1492 y Fr(s)23 1501 y Fl(k)60 1492 y Fs(\024)98 1501 y Fl(emb)178 1492 y Fr(t)196 1501 y Fl(j)210 1507 y Fk(k)231 1492 y Fu(,)c(where)g Fr(i)e(<)g(j)509 1501 y Fq(1)544 1492 y Fs(\024)g Fr(j)617 1501 y Fl(k)639 1492 y Fu(\))i(it)g(follo)o(ws)g(that)h Fr(t)1011 1501 y Fq(1)1030 1492 y Fr(;)8 b(t)1070 1501 y Fq(2)1089 1492 y Fr(;)g(:)g(:)g(:)f(;)h(t)1216 1501 y Fl(j)1230 1506 y Fe(1)1247 1501 y Fp(\000)p Fq(1)1294 1492 y Fr(;)g(s)1339 1501 y Fq(1)1358 1492 y Fr(;)g(s)1403 1501 y Fq(2)1423 1492 y Fr(;)g(:)g(:)g(:)16 b Fu(is)h(a)g(bad)h(sequence)f (of)0 1552 y(elemen)o(ts)12 b(of)j Fs(T)e Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\),)13 b(con)o(tradicting)h(the)h(minim)o(alit)n(y)d(prop)q (ert)o(y)i(of)h(the)f(constructed)h(bad)g(sequence)0 1613 y Fr(t)18 1621 y Fq(1)37 1613 y Fr(;)8 b(t)77 1621 y Fq(2)96 1613 y Fr(;)g(:)g(:)g(:)o Fu(.)21 b(This)c(pro)o(v)o(es)e(the)h(claim.)0 1673 y(Notice)f(that)i(b)o(y)f(Higman's)e(Lemma)g(\()p Fr(S)768 1654 y Fl()f(t)j Fu(then)f Fr(C)t Fu([)p Fr(s)p Fu(])e Fr(>)g(C)t Fu([)p Fr(t)p Fu(])h(for)i(all)f(con)o(texts)g Fr(C)t Fu([)g(]\))g(and)h Fj(close)n(d)h(under)h(substitutions)0 1026 y Fu(\(i.e.,)14 b(if)i Fr(s)d(>)h(t)i Fu(then)g Fr(s\033)f(>)e(t\033)k Fu(for)g(all)f (substitutions)g Fr(\033)r Fu(\).)21 b(A)16 b Fj(simpli\014c)n(ation)h Fu(ordering)f(\()p Fs(T)d Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))o Fr(;)g(<)p Fu(\))16 b(is)0 1086 y(a)h(rewrite)e(ordering)h(p)q(ossessing)i (the)e Fj(subterm)i(pr)n(op)n(erty)p Fu(,)c(i.e.,)g Fr(C)t Fu([)p Fr(t)p Fu(])e Fr(>)h(t)j Fu(for)h(all)e(con)o(texts)h Fr(C)t Fu([)f(])e Fs(6\021)h Fi(2)p Fu(.)0 1217 y Fh(Lemma)i(4.2)24 b Fu(Let)16 b(\()p Fs(T)d Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))o Fr(;)g(<)p Fu(\))16 b(b)q(e)h(a)f(simpli\014cation)e(ordering.)60 1336 y(1.)24 b(Distinct)15 b(v)m(ariables)i Fr(x;)8 b(y)14 b Fs(2)f(V)20 b Fu(are)c(incomparable)f(w.r.t.)21 b(\()p Fs(T)13 b Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))o Fr(;)g(<)p Fu(\).)60 1436 y(2.)24 b(If)16 b Fr(s)d(>)h(t)p Fu(,)h(then)h Fr(V)11 b(ar)q Fu(\()p Fr(t)p Fu(\))j Fs(\022)f Fr(V)e(ar)q Fu(\()p Fr(s)p Fu(\).)0 1556 y Fh(Pro)r(of:)30 b Fu(1.)36 b(Supp)q(ose)22 b Fr(x)f(>)g(y)r Fu(.)35 b(Then)20 b(it)h(follo)o(ws)g(that)g Fr(x\033)h(>)g(y)r(\033)g Fu(for)f(an)o(y)f(substitution)i Fr(\033)r Fu(.)34 b(With)0 1616 y Fr(\033)15 b Fu(=)f Fs(f)p Fr(x)f Fs(7!)h Fr(y)r Fu(;)8 b Fr(y)14 b Fs(7!)f Fr(x)p Fs(g)k Fu(this)f(implies)d Fr(y)i(>)f(x)p Fu(,)h(a)i(con)o(tradiction.)0 1676 y(2.)23 b(Let)17 b Fr(s)d(>)h(t)p Fu(.)22 b(Assume)15 b(that)j(there)e(is)g(a)i(v)m(ariable)e Fr(x)e Fs(2)g Fr(V)d(ar)q Fu(\()p Fr(t)p Fu(\))g Fs(n)h Fr(V)f(ar)q Fu(\()p Fr(s)p Fu(\).)23 b(Then)16 b Fr(t)e Fs(\021)h Fr(C)t Fu([)p Fr(x)p Fu(])g(for)i(some)0 1736 y(con)o(text)j Fr(C)t Fu([)g(].)34 b(With)21 b Fr(\033)h Fu(=)f Fs(f)p Fr(x)h Fs(7!)e Fr(s)p Fs(g)i Fu(it)e(follo)o(ws)h(that)g Fr(s)g Fs(\021)g Fr(s\033)h(>)g(t\033)g Fs(\021)f Fr(C)t Fu([)p Fr(s)p Fu(],)f(con)o(tradicting)g(the)0 1796 y(subterm)15 b(prop)q(ert)o(y)h (or)h(irre\015exivit)o(y)c(of)j(\()p Fs(T)d Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))p Fr(;)g(<)p Fu(\).)21 b Fi(2)0 1927 y Fh(Lemma)16 b(4.3)24 b Fu(Let)18 b Fs(F)24 b Fu(b)q(e)18 b(a)h(signature,)g Fs(V)j Fu(b)q(e)d(a)g(coun)o(table)f(set)g(of)h(v)m (ariables)f(and)h(let)f(\()p Fs(T)13 b Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))p Fr(;)g(<)p Fu(\))0 1987 y(b)q(e)19 b(a)g(simpli\014cation)d (ordering.)29 b(Moreo)o(v)o(er,)18 b(let)g Fs(F)978 1969 y Fp(0)1008 1987 y Fu(and)h Fs(V)1139 1969 y Fp(0)1169 1987 y Fu(b)q(e)g(subsets)g(of)g Fs(F)24 b Fu(and)19 b Fs(V)t Fu(,)g(resp)q(ectiv)o (ely)l(.)0 2048 y(Then)d(the)g(restriction)g(of)g Fr(<)g Fu(to)h Fs(T)c Fu(\()p Fs(F)712 2029 y Fp(0)724 2048 y Fr(;)8 b Fs(V)780 2029 y Fp(0)791 2048 y Fu(\),)16 b(that)g(is)g(\()p Fs(T)d Fu(\()p Fs(F)1111 2029 y Fp(0)1123 2048 y Fr(;)8 b Fs(V)1179 2029 y Fp(0)1190 2048 y Fu(\))p Fr(;)g(<)p Fu(\),)15 b(is)h(a)h (simpli\014cation)d(ordering.)0 2178 y Fh(Pro)r(of:)21 b Fu(T)l(rivial.)f Fi(2)0 2309 y Fh(Prop)r(osition)e(4.4)24 b Fu(Let)11 b Fs(F)16 b Fu(b)q(e)11 b(a)g(signature,)h Fs(V)j Fu(b)q(e)c(a)h(coun)o(table)e(set)h (of)g(v)m(ariables)g(and)h(let)e(\()p Fs(T)j Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))p Fr(;)g(<)p Fu(\))0 2369 y(b)q(e)16 b(a)g(simpli\014cation)e(ordering.)21 b(Moreo)o(v)o(er,)14 b(let)h Fs(F)955 2350 y Fp(0)982 2369 y Fu(b)q(e)h(a)g(\014nite)g(subset)g (of)g Fs(F)5 b Fu(.)21 b(Then)16 b(\()p Fs(T)d Fu(\()p Fs(F)1736 2350 y Fp(0)1747 2369 y Fr(;)8 b Fs(V)t Fu(\))p Fr(;)g(<)p Fu(\))15 b(is)0 2429 y(a)i(w)o(ell-founded)e(simpli\014cation)f(ordering.)0 2510 y Fh(Pro)r(of:)20 b 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Fu(\()p Fr(t)308 2699 y Fq(1)327 2691 y Fu(\)\).)23 b(W)l(e)17 b(de\014ne)g Fr(s)e Fs(\024)g Fr(t)h Fu(if)h Fr(s)e(<)g(t)i Fu(or)g Fr(s)e Fs(\021)g Fr(t)p Fu(.)23 b(\()p Fs(T)13 b Fu(\()p Fs(F)1249 2672 y Fp(0)1261 2691 y Fr(;)8 b Fs(V)t Fr(ar)q Fu(\()p Fr(t)1403 2699 y Fq(1)1421 2691 y Fu(\)\))p Fr(;)g Fs(\024)p Fu(\))17 b(is)g(a)g(quasi-ordering)0 2751 y(whic)o(h)f(is)g(additionally)f (an)o(ti-symmetric)o(.)0 2811 y(W)l(e)c(further)h(kno)o(w)g(that)g(b)o(y)f (Krusk)m(al's)g(Theorem)g(\()p Fs(T)i Fu(\()p Fs(F)1053 2787 y Fp(0)1064 2811 y Fr(;)8 b Fs(V)t Fr(ar)q Fu(\()p Fr(t)1206 2820 y Fq(1)1225 2811 y Fu(\)\))p Fr(;)g Fs(\024)1323 2787 y Fl(pur)q(e)1323 2824 y(emb)1396 2811 y Fu(\))k(is)f(a)h(wqo.)20 b(Since)11 b Fr(<)h Fu(has)g(the)962 2936 y(7)p eop %%Page: 8 8 bop 0 26 a Fu(subterm)16 b(prop)q(ert)o(y)l(,)h(\()p Fs(T)c Fu(\()p Fs(F)518 2 y Fp(0)530 26 y Fr(;)8 b Fs(V)t Fr(ar)q Fu(\()p Fr(t)672 35 y Fq(1)690 26 y Fu(\)\))p Fr(;)g Fs(\024)p Fu(\))17 b(is)g(an)h(extension)e(of)i(\()p 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Fh(Pro)r(of:)k Fu(The)15 b(pro)q(of)i(dep)q(ends)f(on)g(Lemma)e(4.6)i (and)g(can)f(b)q(e)h(found)g(in)g([K)o(O90b],)e(cf.)21 b(also)16 b([Ohl92].)21 b Fi(2)p 0 2760 780 2 v 56 2791 a Fo(7)75 2806 y Fn(Constructors)14 b(are)e(sym)o(b)q(ols)f(that)h(do)g(not)g(o)q(ccur)i(at) e(the)h(ro)q(ot)f(p)q(osition)f(of)h(the)h(left-hand)f(side)g(of)g(an)o(y)g (rewrite)h(rule.)950 2936 y Fu(11)p eop %%Page: 12 12 bop 0 26 a Fu(In)15 b(the)g(pro)q(of)i(referred)d(to,)h(a)h(new)g (constructor)f Fr(E)k Fu(of)c(v)m(ariable)g(arit)o(y)g(is)g(used)h(to)f (\\store)h(\(recursiv)o(ely\))0 86 y(descendan)o(ts)f(of)g(so-called)f (principal)g(aliens".)21 b(In)14 b(our)i(setting)e(\(where)h(function)f(sym)o (b)q(ols)g(ha)o(v)o(e)g(\014xed)0 147 y(arit)o(y\))e(this)h(can)g(b)q(e)g (mimi)o(c)o(k)n(ed)d(b)o(y)i(in)o(tro)q(ducing)h(a)g(new)g(binary)g (constructor)g Fl(C)r(ons)i Fu(and)e(a)g(new)g(constan)o(t)0 207 y Fl(N)t(il)o Fu(.)35 b(\(W)l(riting)20 b Fl(C)r(ons)r Fu(\()p Fr(t)425 216 y Fq(1)444 207 y Fr(;)8 b 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Fp(R[F)378 447 y Fa(0)389 444 y Fk(ar)q(g)21 b Fu(is)16 b(irre\015exiv)o(e.)0 525 y Fh(Pro)r(of:)21 b Fu(The)16 b(if)g(direction)f(is)h(trivial.)k(W)l(e)c(sho)o(w)h(the)f(only-if)g(case)g (b)o(y)g(considering)g(t)o(w)o(o)g(cases.)0 585 y(Case)i(1:)24 b Fs(F)e Fu(con)o(tains)17 b(at)h(most)f(constan)o(ts)h(and)f(function)h(sym) o(b)q(ols)e(of)h(arit)o(y)g(1.)25 b(Then)17 b Fs(R)12 b([)f(F)1805 566 y Fl(ar)q(g)1878 585 y Fu(has)0 645 y(no)23 b(duplicating)e(rule)g (\(i.e.,)h(no)g(righ)o(t-hand)h(side)e(of)i(a)f(rule)f(con)o(tains)h(more)f (o)q(ccurrences)h(of)g(some)0 705 y(v)m(ariable)e(than)h(the)f(left-hand)h (side)f(of)h(that)g(rule\).)33 b(The)20 b(same)g(holds)h(for)f Fs(R)14 b([)f(F)1607 687 y Fp(0)1619 681 y Fl(ar)q(g)1675 705 y Fu(.)34 b(Let)20 b Fs(j)p Fr(s)p Fs(j)1865 714 y Fl(C)r(ons)0 765 y Fu(denote)14 b(the)f(n)o(um)o(b)q(er)f(of)j Fl(C)r(ons)g Fu(op)q(erators)g(in)f Fr(s)p Fu(.)20 b(W)l(e)14 b(pro)o(v)o(e)f(b)o(y)g (induction)g(on)i Fr(m)e Fu(=)g Fs(j)p Fr(s)p Fs(j)1600 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Fs(\025)h(j)p Fr(s)p Fs(j)1809 1496 y Fl(C)r(ons)1892 1488 y Fu(,)h(a)0 1548 y(con)o(tradiction.)28 b(Th)o(us)19 b Fr(s)e Fs(\021)h Fr(s)568 1557 y Fq(1)605 1548 y Fs(!)654 1557 y Fp(R[F)736 1547 y Fk(ar)q(g)806 1548 y Fr(s)829 1557 y Fq(2)866 1548 y Fs(!)915 1557 y Fp(R[F)997 1547 y Fk(ar)q(g)1067 1548 y Fr(:)8 b(:)g(:)16 b Fs(!)1190 1557 y Fp(R[F)1272 1547 y Fk(ar)q(g)1342 1548 y Fr(s)1365 1557 y Fl(n)1406 1548 y Fs(\021)i Fr(s)g Fu(where)h Fr(s)1670 1557 y Fl(i)1701 1548 y Fs(2)e(T)c Fu(\()p Fs(F)1849 1529 y Fp(0)1861 1548 y Fr(;)8 b Fs(V)s Fu(\).)0 1608 y(Moreo)o(v)o(er,)14 b Fr(s)250 1617 y Fl(j)282 1608 y Fs(\021)g Fr(C)369 1617 y Fl(j)387 1608 y Fu([)p Fl(C)r(ons)p Fu(\()p Fr(t)519 1617 y Fq(1)539 1608 y Fr(;)8 b(t)579 1617 y Fq(2)598 1608 y Fu(\)],)15 b(where)g Fr(C)835 1617 y Fl(j)853 1608 y Fu([)10 b(])j Fs(2)g(T)g Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))15 b(for)h(ev)o(ery)f Fr(j)h Fs(2)d(f)p Fu(1)p Fr(;)8 b(:)g(:)g(:)g(n)p Fs(g)p Fu(.)21 b(Hence)15 b(there)g(is)0 1668 y(a)j(cyclic)e(reduction)h(sequence)g Fr(C)t Fu([)p Fr(z)r Fu(])e Fs(\021)h Fr(C)796 1677 y Fq(1)815 1668 y Fu([)p Fr(z)r Fu(])g Fs(!)933 1677 y Fp(R[F)1015 1668 y Fk(ar)q(g)21 b Fr(C)1118 1677 y Fq(2)1138 1668 y Fu([)p Fr(z)r Fu(])15 b Fs(!)1255 1677 y Fp(R[F)1337 1668 y Fk(ar)q(g)21 b Fr(:)8 b(:)g(:)15 b Fs(!)1527 1677 y Fp(R[F)1609 1668 y Fk(ar)q(g)22 b Fr(C)1713 1677 y Fl(n)1736 1668 y Fu([)p Fr(z)r Fu(])15 b Fs(\021)h Fr(C)t Fu([)p Fr(z)r Fu(])0 1728 y(of)k(terms)e Fr(C)233 1737 y Fl(j)251 1728 y Fu([)g(])h Fs(2)f(T)13 b Fu(\()p Fs(F)5 b Fr(;)j Fs(V)t Fu(\))o(,)20 b(where)f Fr(z)j Fu(is)d(a)h(fresh)f(v)m(ariable.)30 b(This)20 b(con)o(tradicts)f(the)g(irre\015exivit)o(y)e(of)0 1789 y Fs(!)49 1768 y Fq(+)49 1801 y Fp(R[F)131 1791 y Fk(ar)q(g)183 1789 y Fu(.)0 1849 y(Case)g(2:)23 b Fs(F)f Fu(con)o(tains)16 b(at)i(least)e(one)h (function)g(sym)o(b)q(ol)e Fr(f)22 b Fu(of)17 b(arit)o(y)f Fr(n)f Fs(\025)f Fu(2.)23 b(Supp)q(ose)18 b Fs(!)1667 1828 y Fq(+)1667 1862 y Fp(R[F)1749 1853 y Fa(0)1760 1849 y Fk(ar)q(g)k Fu(is)16 b(not)0 1909 y(irre\015exiv)o(e,)f(i.e.,)h(there)i(is)f(a)i(cyclic)c (sequence)i Fr(s)f Fs(\021)h Fr(s)1009 1918 y Fq(1)1045 1909 y Fs(!)1094 1918 y Fp(R[F)1176 1908 y Fa(0)1187 1905 y Fk(ar)q(g)1255 1909 y Fr(s)1278 1918 y Fq(2)1314 1909 y Fs(!)1363 1918 y Fp(R[F)1445 1908 y Fa(0)1456 1905 y Fk(ar)q(g)1525 1909 y Fr(:)8 b(:)g(:)15 b Fs(!)1647 1918 y Fp(R[F)1729 1908 y Fa(0)1740 1905 y Fk(ar)q(g)1809 1909 y Fr(s)1832 1918 y Fl(n)1872 1909 y Fs(\021)h Fr(s)0 1969 y Fu(of)i(terms)e Fr(s)217 1978 y Fl(i)248 1969 y Fs(2)f(T)e Fu(\()p Fs(F)394 1951 y Fp(0)406 1969 y Fr(;)8 b Fs(V)s Fu(\).)26 b(In)18 b(ev)o(ery)e Fr(s)736 1978 y Fl(j)772 1969 y Fu(replace)h(eac)o(h)h Fl(C)r(ons)q Fu(\()p Fr(t)1170 1978 y Fq(1)1189 1969 y Fr(;)8 b(t)1229 1978 y Fq(2)1248 1969 y Fu(\))18 b(with)g Fr(f)5 b Fu(\()p Fr(t)1464 1978 y Fq(1)1483 1969 y Fr(;)j(t)1523 1978 y Fq(2)1542 1969 y Fr(;)g(z)r(;)g(:)g(:)g(:)f(;)h(z)r Fu(\))17 b(and)h(eac)o(h)0 2029 y Fl(N)t(il)d Fu(with)g 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Fj(Pr)n(o)n(c.)g(Cam-)207 1460 y(bridge)j(Philos.) g(So)n(c.)g Fh(59)p Fj(\(4\))p Fu(,)d(pages)i(833{835,)i(1963.)28 1562 y([Ohl92])24 b(E.)15 b(Ohlebusc)o(h.)k(Com)o(binations)c(of)h(Simplifyi) o(ng)d(Conditional)j(T)l(erm)e(Rewriting)h(Systems.)207 1622 y(In)c Fj(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)i(the)g(3r)n(d)e(International)j (Workshop)e(on)h(Conditional)g(and)g(T)l(yp)n(e)n(d)f(R)n(ewrit-)207 1683 y(ing)24 b(Systems)e(\(to)h(app)n(e)n(ar\))p Fu(.)d(Lecture)h(Notes)h (in)f(Computer)g(Science)f(,)j(Springer)f(V)l(erlag,)207 1743 y(1992.)28 1844 y([Ros82])i(J.)16 b(Rosenstein.)21 b Fj(Line)n(ar)c(Or)n (derings)p Fu(.)k(Academic)14 b(Press,)i(1982.)28 1946 y([Sho67])25 b(J.R.)16 b(Sho)q(en\014eld.)21 b Fj(Mathematic)n(al)d(L)n(o)n(gic)p Fu(.)i(Addison-W)l(esley)l(,)15 b(1967.)3 2048 y([T)l(o)o(y87a])24 b(Y.)11 b(T)l(o)o(y)o(ama.)g(Coun)o(terexamples)f(to)h(T)l(ermination)f(for)h (the)g(Direct)f(Sum)g(of)i(T)l(erm)e(Rewriting)207 2108 y(Systems.)20 b 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