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589 517 V 598 517 V 116 w(Y)l(es)f([T)l(o)o(y87b)q(])p 959 517 V 89 w Fz(Y)-5 b(es)17 b FB(\(4.3.5\))p 1329 517 V 130 w Fz(Y)-5 b(es)17 b FB(\(4.3.5\))p 1657 517 V 88 w({)p 1985 517 V 0 518 1986 2 v -1 578 2 61 v 53 560 a(+)f(Left-Linearit)o(y)p 589 578 V 598 578 V 220 w(Y)l(es)f([R)-5 b(V80])p 959 578 V 129 w(Y)l(es)16 b([R)-5 b(V80)o(])p 1329 578 V 137 w(?)22 b(\(op)q(en\))p 1657 578 V 144 w(Y)l(es)15 b([R)-5 b(V80])p 1985 578 V 0 580 1986 2 v 0 590 V -1 650 2 61 v 25 632 a(T)l(ermination)p 589 650 V 598 650 V 335 w(No)33 b([T)l(o)o(y87a)q(])p 959 650 V 87 w(No)g([T)l(o)o(y87a)q (])p 1329 650 V 95 w(No)g([T)l(o)o(y87a)q(])p 1657 650 V 53 w(No)g(\(3.2.1\))p 1985 650 V 0 652 1986 2 v -1 712 2 61 v 53 694 a(+)16 b(La)o(y)o(er-Preserv)m(ation)p 589 712 V 598 712 V 116 w(Y)l(es)f([Rus87a)q(])p 959 712 V 90 w(Y)l(es)h([Gra93a)q(])p 1329 712 V 98 w Fz(Y)-5 b(es)17 b FB(\(5.3.8\))p 1657 712 V 88 w({)p 1985 712 V 0 714 1986 2 v -1 774 2 61 v 53 756 a(+)f(Non-Duplication)p 589 774 V 598 774 V 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b(T)l(ermination)p 589 1031 V 598 1031 V 175 w Fz(Y)-5 b(es)16 b FB(\(5.2.25\))p 959 1031 V 99 w Fz(No)33 b FB(\(5.3.2\))p 1329 1031 V 127 w Fz(No)g FB(\(5.3.2\))p 1657 1031 V 85 w(No)g(\(3.2.1\))p 1985 1031 V 0 1033 1986 2 v -1 1093 2 61 v 25 1075 a(Simplifying)13 b(Prop)q(ert)o(y)p 589 1093 V 598 1093 V 152 w(Y)l(es)i([K)o(O90])p 959 1093 V 122 w(Y)l(es)h([K)o(O92])p 1329 1093 V 129 w Fz(Y)-5 b(es)17 b FB(\(5.3.10\))p 1657 1093 V 64 w(No)33 b(\(3.2.1\))p 1985 1093 V 0 1095 1986 2 v -1 1155 2 61 v 25 1137 a(Innermost)15 b(T)l(ermination)p 589 1155 V 598 1155 V 103 w(Y)l(es)g([Gra93b)r(])p 959 1155 V 87 w(Y)l(es)h([Gra93b)q(])p 1329 1155 V 95 w Fz(Y)-5 b(es)17 b FB(\(3.4.12\))p 1657 1155 V 64 w(No)33 b(\(3.2.1\))p 1985 1155 V 0 1157 1986 2 v 0 1167 V -1 1227 2 61 v 25 1209 a(Completeness)p 589 1227 V 598 1227 V 306 w(No)g([T)l(o)o(y87a)q(])p 959 1227 V 87 w(No)g([T)l(o)o(y87a)q (])p 1329 1227 V 95 w(No)g([T)l(o)o(y87a)q(])p 1657 1227 V 53 w(No)g(\(3.2.1\))p 1985 1227 V 0 1228 1986 2 v 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y(In)g(order)h(to)g (co)q(de)g(certain)e(sp)q(ecial)h(subterms)g(b)o(y)g(v)m(ariables)g (and)h(to)g(cop)q(e)g(with)f(transparen)o(t)h(or)-59 2617 y(outer)d(rewrite)g(steps)g(using)h(non-left-linear)e(rules,)h (the)g(follo)o(wing)g(notation)h(is)f(con)o(v)o(enien)o(t.)-59 2748 y Fz(De\014nition)h(3.3.12)24 b FB(Let)c Fy(s)484 2755 y Fs(1)504 2748 y Fy(;)8 b(:)g(:)g(:)f(;)h(s)636 2755 y Fw(n)680 2748 y FB(and)21 b Fy(t)797 2755 y Fs(1)816 2748 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)943 2755 y Fw(n)986 2748 y FB(b)q(e)21 b(sequences)e(of)h(terms)f(from)g Fx(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))o(.)33 b(W)l(e)-59 2808 y(write)17 b Fy(s)90 2815 y Fs(1)109 2808 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)242 2815 y Fw(n)281 2808 y Fx(/)15 b Fy(t)353 2815 y Fs(1)372 2808 y Fy(;)8 b(:)g(:)g(:)g(;)g(t)500 2815 y Fw(n)540 2808 y FB(if)16 b Fy(s)608 2815 y Fw(i)638 2808 y FB(=)g Fy(s)715 2815 y Fw(j)750 2808 y FB(implies)f Fy(t)935 2815 y Fw(i)964 2808 y FB(=)g Fy(t)1035 2815 y Fw(j)1071 2808 y FB(for)i(all)g(1)f 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Fz(Pro)r(of:)21 b FB(Routine.)g Fq(2)-59 1764 y Fz(Prop)r(osition)d(3.3.18)23 b FB(If)16 b Fy(s)e Fx(!)547 1746 y Fr(\003)580 1764 y Fy(t)p FB(,)i(then)g Fy(r)q(ank)r FB(\()p Fy(s)p FB(\))e Fx(\025)f Fy(r)q(ank)r FB(\()p Fy(t)p FB(\).)-59 1845 y Fz(Pro)r(of:)21 b FB(W)l(e)16 b(sho)o(w)h(that)g Fy(r)q(ank)r FB(\()p Fy(s)p FB(\))d Fx(\025)f Fy(r)q(ank)r FB(\()p Fy(t)p FB(\))j(is)g(implied)e(b)o(y)i Fy(s)d Fx(!)h Fy(t)p FB(.)21 b(The)16 b(prop)q(osition)i(then)e(follo)o(ws)-59 1905 y(b)o(y)k(a)i(straigh)o(tforw)o(ard)g(induction)e(on)i(the)f (length)f(of)i Fy(s)g Fx(!)1099 1887 y Fr(\003)1140 1905 y Fy(t)p FB(.)35 b(So)22 b(assume)e Fy(s)i Fx(!)g Fy(t)p FB(.)35 b(W)l(e)21 b(pro)q(ceed)-59 1966 y(b)o(y)e(induction)h(on)h Fy(r)q(ank)r FB(\()p Fy(s)p FB(\).)32 b(If)20 b Fy(r)q(ank)r FB(\()p Fy(s)p FB(\))g(=)g(0,)h(then)f Fy(s)h Fx(2)f(T)13 b FB(\()p Fx(B)r Fy(;)8 b Fx(V)t FB(\))o(.)32 b(By)20 b(Lemma)d(3.3.14,)22 b Fy(t)d FB(is)h(also)-59 2026 y(transparen)o(t,)h (i.e.,)e Fy(r)q(ank)r 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b(sho)o(w)i(the)e(assertion)h(for)g Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))g Fx(2)f(A)1145 2825 y Fs(1)1172 2818 y Fx(])8 b(A)1253 2825 y Fs(2)1273 2818 y FB(.)20 b(Again,)15 b(the)f(case)h Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))f Fx(2)h(B)-59 2878 y FB(follo)o(ws)23 b(straigh)o(tforw)o(ardly) f(from)g(this.)40 b(Without)23 b(loss)g(of)g(generalit)o(y)f(let)f Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))26 b Fx(2)f(A)1695 2885 y Fs(1)1714 2878 y FB(.)41 b(Th)o(us,)p eop %%Page: 35 43 35 42 bop 0 -39 a Fv(3.3.)38 b(COMPOSABLE)16 b(SYSTEMS)1188 b FB(35)0 94 y Fy(s)14 b FB(=)g Fy(C)128 75 y Fw(b)144 94 y FB([)-8 b([)p Fy(s)187 101 y Fs(1)206 94 y Fy(;)8 b(:)g(:)g(:)f(;)h(s)338 101 y Fw(j)356 94 y Fy(;)g(:)g(:)g(:)g(;)g(s) 489 101 y Fw(n)512 94 y FB(])-8 b(],)14 b Fy(t)f FB(=)h Fy(C)682 75 y Fw(b)699 94 y FB([)p Fy(s)736 101 y Fs(1)755 94 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)882 101 y Fw(j)900 94 y Fy(;)g(:)g(:)g(:)f(;)h(s)1032 101 y Fw(n)1056 94 y FB(],)14 b(and)i Fy(s)1215 101 y Fw(j)1247 94 y Fx(!)e Fy(t)1329 101 y 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Fx(D)240 1472 y Fr(0)265 1493 y Fx(\022)e(D)k FB(b)q(ecause)e Fx(R)596 1472 y Fr(0)622 1493 y Fx(\022)d(R)q FB(.)-59 1553 y(\(1\))22 b Fx(\))f FB(\(2\):)31 b(Since)21 b(\()p Fx(F)t Fy(;)8 b Fx(R)p FB(\))22 b(and)f(\()p Fx(F)659 1532 y Fr(0)671 1553 y Fy(;)8 b Fx(R)735 1532 y Fr(0)747 1553 y FB(\))21 b(are)g(comp)q(osable,)h(w)o(e)f(ha)o(v)o(e)f Fx(D)c(\\)f(C)1480 1532 y Fr(0)1514 1553 y FB(=)22 b Fx(C)c(\\)d(D)1706 1532 y Fr(0)1740 1553 y FB(=)22 b Fx(;)f FB(as)-59 1613 y(w)o(ell)c(as)j Fx(f)p Fy(l)e Fx(!)g Fy(r)i Fx(2)e(R)366 1593 y Fr(0)391 1613 y Fx([)13 b(R)19 b(j)f Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))h Fx(2)f(D)781 1593 y Fr(0)805 1613 y Fx(\\)13 b(D)r(g)18 b(\022)g(R)1033 1593 y Fr(0)1058 1613 y Fx(\\)12 b(R)q FB(.)28 b(No)o(w)19 b Fx(C)1331 1593 y Fr(0)1361 1613 y Fx(\022)f(C)k FB(imm)o(ediatel)o(y) 16 b(follo)o(ws)-59 1674 y(from)21 b Fx(D)c(\\)e(C)194 1653 y Fr(0)229 1674 y FB(=)24 b Fx(;)p FB(.)38 b(Moreo)o(v)o(er,)22 b(it)g(follo)o(ws)g(from)f(the)g(ab)q(o)o(v)o(e)h(observ)m(ations)h(in) f(conjunction)g(with)-59 1734 y Fx(f)p Fy(l)c Fx(!)g Fy(r)i Fx(2)e(R)h(j)f Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))h Fx(2)f(D)504 1713 y Fr(0)515 1734 y Fx(g)g FB(=)g Fx(f)p Fy(l)h Fx(!)e Fy(r)j Fx(2)e(R)875 1713 y Fr(0)900 1734 y Fx([)13 b(R)18 b(j)h Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))f Fx(2)h(D)1289 1713 y Fr(0)1314 1734 y Fx(\\)13 b(D)q(g)18 b(\022)g(R)1541 1713 y Fr(0)1566 1734 y Fx(\\)13 b(R)18 b FB(=)g Fx(R)1770 1713 y Fr(0)1800 1734 y FB(that)-59 1794 y Fx(f)p Fy(l)c Fx(!)g Fy(r)h Fx(2)f(R)j(j)f Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))e Fx(2)g(D)474 1773 y Fr(0)485 1794 y Fx(g)g FB(=)g Fx(R)618 1773 y Fr(0)630 1794 y FB(.)-59 1854 y(\(2\))19 b Fx(\))f FB(\(1\):)27 b(W)l(e)18 b(ha)o(v)o(e)g Fx(C)423 1833 y Fr(0)448 1854 y Fx(\\)12 b(D)20 b FB(=)e Fx(;)h FB(b)q(ecause)g Fx(C)863 1833 y Fr(0)892 1854 y Fx(\022)f(C)s FB(.)29 b(Since)17 b Fx(D)1190 1833 y Fr(0)1220 1854 y Fx(\022)g(D)r FB(,)i(w)o(e)f(also)h(ha)o(v)o(e)f Fx(D)1678 1833 y Fr(0)1702 1854 y Fx(\\)13 b(C)21 b FB(=)d Fx(;)p FB(.)-59 1914 y(It)i(remains)e(to)j(sho)o(w)g Fx(f)p Fy(l)g Fx(!)f Fy(r)i Fx(2)f(R)643 1893 y Fr(0)669 1914 y Fx([)13 b(R)21 b(j)f Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))h Fx(2)g(D)1067 1893 y Fr(0)1093 1914 y Fx(\\)14 b(D)r(g)20 b(\022)g(R)1327 1893 y Fr(0)1352 1914 y Fx(\\)14 b(R)p FB(.)33 b(No)o(w)20 b(this)g(assertion)-59 1975 y(follo)o(ws)d(from)g Fx(R)261 1954 y Fr(0)289 1975 y FB(=)f Fx(f)p Fy(l)h Fx(!)f Fy(r)h Fx(2)g(R)g(j)h Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))e Fx(2)h(D)893 1954 y Fr(0)904 1975 y Fx(g)f FB(=)g Fx(f)p Fy(l)h Fx(!)f Fy(r)i Fx(2)e(R)1253 1954 y Fr(0)1277 1975 y Fx([)c(R)18 b(j)f Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))g Fx(2)f(D)1660 1954 y Fr(0)1684 1975 y Fx(\\)c(D)r(g)17 b FB(and)-59 2035 y Fx(R)-17 2014 y Fr(0)9 2035 y FB(=)c Fx(R)103 2014 y Fr(0)125 2035 y Fx(\\)f(R)p FB(.)21 b Fq(2)-59 2173 y Fz(Prop)r(osition)d (3.3.24)23 b FB(Let)17 b(\()p Fx(F)t Fy(;)8 b Fx(R)p FB(\))17 b(b)q(e)f(a)h(TRS.)1 2298 y(1.)24 b(The)16 b(set)g Fy(N)22 b FB(consists)16 b(of)h(pairwise)f(comp)q(osable)g(TRSs.)1 2403 y(2.)24 b Fy(N)c FB(is)15 b(closed)g(under)g(union,)h(i.e.,)d(if)i (\()p Fx(F)793 2410 y Fs(1)813 2403 y Fy(;)8 b Fx(R)877 2410 y Fs(1)897 2403 y FB(\))p Fy(;)g FB(\()p Fx(F)997 2410 y Fs(2)1017 2403 y Fy(;)g Fx(R)1081 2410 y Fs(2)1101 2403 y FB(\))14 b Fx(2)g Fy(N)5 b FB(,)15 b(then)g(\()p Fx(F)1424 2410 y Fs(1)1453 2403 y Fx([)9 b(F)1535 2410 y Fs(2)1555 2403 y Fy(;)f Fx(R)1619 2410 y Fs(1)1648 2403 y Fx([)h(R)1732 2410 y Fs(2)1752 2403 y FB(\))14 b Fx(2)g Fy(N)5 b FB(.)-59 2528 y Fz(Pro)r(of:)25 b FB(\(1\))18 b(Let)g(\()p Fx(F)347 2535 y Fs(1)366 2528 y Fy(;)8 b Fx(R)430 2535 y Fs(1)450 2528 y FB(\))p Fy(;)g FB(\()p Fx(F)551 2535 y Fs(2)570 2528 y Fy(;)g Fx(R)635 2535 y Fs(2)654 2528 y FB(\))17 b Fx(2)g Fy(N)5 b FB(,)18 b(i.e.,)e Fx(F)951 2535 y Fw(j)986 2528 y Fx(\022)h(F)5 b FB(,)18 b Fx(R)1157 2535 y Fw(j)1192 2528 y Fx(\022)e(R)q FB(,)h Fx(D)e(\\)d(C)1448 2535 y Fw(j)1483 2528 y 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2885 y Fs(1)1032 2878 y Fx(\\)d(R)1118 2885 y Fs(2)1138 2878 y FB(.)p eop %%Page: 37 45 37 44 bop 0 -39 a Fv(3.3.)38 b(COMPOSABLE)16 b(SYSTEMS)1188 b FB(37)0 94 y(\(i\))15 b(Since)f Fx(D)233 101 y Fs(1)267 94 y Fx(\022)f(D)k FB(and)f Fx(D)11 b(\\)e(C)588 101 y Fs(2)622 94 y FB(=)14 b Fx(;)p FB(,)h(it)g(follo)o(ws)g Fx(D)975 101 y Fs(1)1003 94 y Fx(\\)10 b(C)1075 101 y Fs(2)1108 94 y FB(=)k Fx(;)p FB(.)21 b Fx(D)1260 101 y Fs(2)1289 94 y Fx(\\)9 b(C)1360 101 y Fs(1)1394 94 y FB(=)k Fx(;)i FB(is)h(pro)o(v)o(ed)e(analogously)l(.)0 154 y(\(ii\))h(W)l(e)h(ha)o(v)o(e)g Fx(f)p Fy(l)e Fx(!)g Fy(r)h Fx(2)f(R)522 161 y Fs(1)553 154 y Fx([)d(R)639 161 y Fs(2)675 154 y Fx(j)17 b Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))d Fx(2)g(D)948 161 y Fs(1)978 154 y Fx(\\)e(D)1063 161 y Fs(2)1082 154 y Fx(g)i(\022)g(f)p Fy(l)g Fx(!)g Fy(r)h Fx(2)f(R)i(j)g Fy(r)q(oot)p FB(\()p Fy(l)q FB(\))f Fx(2)f(D)1706 161 y Fs(1)1737 154 y Fx(\\)e(D)1821 161 y Fs(2)1841 154 y Fx(g)i FB(=)0 214 y Fx(f)p Fy(l)g Fx(!)g Fy(r)h 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Fs(1)233 1293 y Fy(\033)r FB(\))22 b(b)q(e)g(a)g(critical)e(pair)h(obtained)h(b)o(y)g (an)g(o)o(v)o(erlap)f(of)h(the)f(rules)h Fy(l)1511 1300 y Fs(1)1553 1293 y Fx(!)h Fy(r)1648 1300 y Fs(1)1691 1293 y Fx(2)g(R)f FB(and)-59 1353 y Fy(l)-44 1360 y Fs(2)-8 1353 y Fx(!)17 b Fy(r)81 1360 y Fs(2)118 1353 y Fx(2)g(R)p FB(.)27 b(That)19 b(is,)f Fy(l)455 1360 y Fs(1)491 1353 y FB(=)f Fy(C)t FB([)p Fy(t)p FB(])g(with)h Fy(t)e Fx(62)h(V)22 b FB(suc)o(h)c(that)h Fy(t\033)f FB(=)f Fy(l)1252 1360 y Fs(2)1271 1353 y Fy(\033)r FB(,)h(where)g Fy(\033)h FB(is)f(a)h(most)e(general)-59 1413 y(uni\014er.)i(If)13 b(b)q(oth)g(rules)f(stem)g(from)f(the)i(same)e(system)h Fx(R)1005 1420 y Fw(j)1023 1413 y FB(,)h Fy(j)k Fx(2)d(f)p FB(1)p Fy(;)8 b FB(2)p Fx(g)p FB(,)13 b(then)g(\()p Fy(C)t FB([)p Fy(r)1483 1420 y Fs(2)1501 1413 y FB(])p Fy(\033)o(;)8 b(r)1586 1420 y Fs(1)1605 1413 y Fy(\033)r FB(\))14 b Fx(2)g Fy(C)t(P)7 b FB(\()p Fx(R)1853 1420 y Fw(j)1871 1413 y FB(\))-59 1473 y(and)14 b(w)o(e)f(are)h(done.)21 b(Supp)q(ose)14 b(that)g(the)g(rules)f(come)f(from)g(di\013eren)o(t)h (systems.)19 b(W.l.o.g.)h Fy(l)1629 1480 y Fs(1)1662 1473 y Fx(!)13 b Fy(r)1747 1480 y Fs(1)1781 1473 y Fx(2)h(R)1870 1480 y Fs(1)-59 1533 y FB(and)20 b Fy(l)54 1540 y Fs(2)92 1533 y Fx(!)f Fy(r)183 1540 y Fs(2)221 1533 y Fx(2)h(R)316 1540 y Fs(2)335 1533 y FB(.)31 b(Since)18 b Fy(t\033)i FB(=)f Fy(l)648 1540 y Fs(2)667 1533 y Fy(\033)i FB(and)f Fy(t;)8 b(l)869 1540 y Fs(2)906 1533 y Fx(62)20 b(V)t FB(,)f(w)o(e)g(deriv)o(e)e Fy(r)q(oot)p FB(\()p Fy(t)p FB(\))j(=)e Fy(r)q(oot)p FB(\()p Fy(l)1588 1540 y Fs(2)1608 1533 y FB(\).)31 b(Therefore,)-59 1594 y Fy(r)q(oot)p FB(\()p Fy(l)62 1601 y Fs(2)82 1594 y FB(\))14 b Fx(2)g(F)203 1601 y Fs(1)229 1594 y Fx(\\)7 b(D)309 1601 y Fs(2)343 1594 y FB(=)14 b Fx(D)434 1601 y Fs(1)461 1594 y Fx(\\)7 b(D)541 1601 y Fs(2)561 1594 y FB(.)20 b(Hence)13 b Fy(l)753 1601 y Fs(2)786 1594 y Fx(!)h Fy(r)872 1601 y Fs(2)906 1594 y Fx(2)g(R)995 1601 y Fs(1)1029 1594 y FB(and)h(again)g(\()p Fy(C)t FB([)p Fy(r)1344 1601 y Fs(2)1362 1594 y FB(])p Fy(\033)o(;)8 b(r)1447 1601 y Fs(1)1466 1594 y Fy(\033)r FB(\))14 b Fx(2)g Fy(C)t(P)7 b FB(\()p Fx(R)1714 1601 y Fs(1)1733 1594 y FB(\))14 b(whic)o(h)-59 1654 y(concludes)i(the)g (pro)q(of.)22 b Fq(2)-59 1805 y Fz(Lemma)16 b(3.4.3)23 b FB(Let)17 b Fx(R)h FB(b)q(e)f(the)g(hierarc)o(hical)f(com)o(bination) g(of)h(base)h Fx(R)1329 1812 y Fs(1)1366 1805 y FB(and)g(extension)f Fx(R)1721 1812 y Fs(2)1740 1805 y FB(.)25 b(Then)-59 1865 y Fx(R)-17 1872 y Fs(1)19 1865 y FB(and)17 b Fx(R)156 1872 y Fs(2)192 1865 y FB(are)f(orthogonal)i(to)f(eac)o(h)f(other.)-59 1946 y Fz(Pro)r(of:)24 b FB(By)18 b(De\014nition)f(3.1.5,)h Fx(C)585 1953 y Fs(1)617 1946 y Fx(\\)13 b(D)703 1953 y Fs(2)739 1946 y FB(=)k Fx(D)833 1953 y Fs(1)865 1946 y Fx(\\)c(D)951 1953 y Fs(2)987 1946 y FB(=)k Fx(;)g FB(and)i Fx(R)1223 1953 y Fs(2)1259 1946 y Fx(\032)e(T)c FB(\()p Fx(F)1414 1953 y Fs(2)1446 1946 y Fx(n)f(D)1523 1953 y Fs(1)1543 1946 y Fy(;)c Fx(V)s FB(\))13 b Fx(\002)f(T)g FB(\()p Fx(F)1781 1953 y Fs(2)1801 1946 y Fy(;)c Fx(V)s FB(\).)-59 2006 y(Since)18 b(no)h(de\014ned)f(function)h(sym)o(b)q(ol)e (from)g Fx(D)831 2013 y Fs(2)870 2006 y FB(app)q(ears)i(in)g Fx(R)1154 2013 y Fs(1)1174 2006 y FB(,)f(no)h(rule)f(from)g Fx(R)1536 2013 y Fs(2)1574 2006 y FB(o)o(v)o(erlaps)g(a)h(rule)-59 2066 y(from)14 b Fx(R)97 2073 y Fs(1)117 2066 y FB(.)21 b(On)15 b(the)g(other)h(hand,)f(a)h(rule)e(from)h Fx(R)869 2073 y Fs(1)904 2066 y FB(cannot)h(o)o(v)o(erlap)e(a)i(rule)e(from)g Fx(R)1524 2073 y Fs(2)1559 2066 y FB(b)q(ecause)h(de\014ned)-59 2126 y(function)h(sym)o(b)q(ols)f(from)g Fx(D)473 2133 y Fs(1)509 2126 y FB(do)i(not)g(app)q(ear)g(in)f(left-hand)g(sides)g (of)h(rules)e(from)h Fx(R)1536 2133 y Fs(2)1555 2126 y FB(.)22 b Fq(2)-59 2278 y Fz(Lemma)16 b(3.4.4)23 b FB(Let)e Fx(R)g FB(b)q(e)g(the)f(com)o(bined)f(system)g(of)i(comp)q (osable)f(term)f(rewriting)h(systems)g Fx(R)1870 2285 y Fs(1)-59 2338 y FB(and)i Fx(R)83 2345 y Fs(2)103 2338 y FB(.)36 b(If)20 b Fy(s)i Fx(2)h(T)13 b FB(\()p Fx(F)406 2345 y Fw(j)425 2338 y Fy(;)8 b Fx(V)s FB(\),)22 b Fy(j)j Fx(2)d(f)p FB(1)p Fy(;)8 b FB(2)p Fx(g)p FB(,)23 b(and)f Fy(s)g Fx(!)g Fy(t)f FB(b)o(y)f(an)i(application)f(of)g(some)f(rewrite) g(rule)-59 2398 y Fy(l)14 b Fx(!)g Fy(r)h Fx(2)f(R)p FB(,)i(then)g Fy(l)f Fx(!)f Fy(r)h Fx(2)f(R)521 2405 y Fw(j)555 2398 y FB(and)j Fy(t)d Fx(2)g(T)f FB(\()p Fx(F)828 2405 y Fw(j)847 2398 y Fy(;)8 b Fx(V)s FB(\).)-59 2479 y Fz(Pro)r(of:)21 b FB(Routine.)g Fq(2)-59 2630 y Fz(Prop)r(osition)d(3.4.5)1 2763 y FB(1.)24 b(Lo)q(cal)17 b(con\015uence)f(is)g(a)g(mo)q(dular)g(prop)q(ert)o(y)g(of)h(comp)q (osable)f(TRSs.)1 2878 y(2.)24 b(Lo)q(cal)17 b(con\015uence)f(is)g(a)g (mo)q(dular)g(prop)q(ert)o(y)g(of)h(hierarc)o(hical)d(com)o(binations.) p eop %%Page: 41 49 41 48 bop 0 -39 a Fv(3.4.)38 b(LOCAL)17 b(CONFLUENCE)f(AND)f(NORMALIZA) l(TION)678 b FB(41)0 94 y Fz(Pro)r(of:)22 b FB(\(1\))17 b(Let)g Fx(R)383 101 y Fs(1)419 94 y FB(and)h Fx(R)557 101 y Fs(2)593 94 y FB(b)q(e)f(t)o(w)o(o)g(comp)q(osable)f(TRSs,)g(and) i(let)e Fx(R)g FB(b)q(e)h(their)f(com)o(bined)f(system.)0 154 y(W)l(e)h(ha)o(v)o(e)g(to)i(sho)o(w)f(that)g Fx(R)g FB(is)f(lo)q(cally)g(con\015uen)o(t)h(if)f(and)h(only)g(if)f Fx(R)1294 161 y Fs(1)1331 154 y FB(and)h Fx(R)1468 161 y Fs(2)1505 154 y FB(are)g(lo)q(cally)f(con\015uen)o(t.)0 214 y(The)g(if)g(direction)f(follo)o(ws)h(imme)o(diately)d(from)i(the)g (com)o(bination)g(of)i(Prop)q(osition)g(3.4.1)f(and)h(Lemma)0 274 y(3.4.2.)33 b(Let)21 b(us)f(turn)h(to)f(the)g(only-if)g(direction.) 32 b(W.l.o.g.)g(w)o(e)20 b(sho)o(w)g(that)h(the)f(TRS)g Fx(R)1718 281 y Fs(1)1758 274 y FB(is)g(lo)q(cally)0 334 y(con\015uen)o(t,)d(or)h(equiv)m(alen)o(tly)l(,)d(that)j(eac)o(h)f (critical)f(pair)h(\()p Fy(s;)8 b(t)p FB(\))16 b Fx(2)g Fy(C)t(P)7 b FB(\()p Fx(R)1355 341 y Fs(1)1374 334 y FB(\))18 b(is)f(con)o(v)o(ergen)o(t.)24 b(Since)16 b Fx(R)i FB(is)0 394 y(lo)q(cally)d(con\015uen)o(t)h(and)g(\()p Fy(s;)8 b(t)p FB(\))14 b Fx(2)g Fy(C)t(P)7 b FB(\()p Fx(R)p FB(\),)15 b(there)g(is)h(a)g(term)f Fy(u)g FB(so)i(that)f Fy(s)h Fx(!)1434 376 y Fr(\003)1434 407 y(R)1482 394 y Fy(u)1526 376 y Fr(\003)1526 407 y(R)1555 394 y Fx( )f Fy(t)p FB(.)21 b(According)15 b(to)0 455 y(Lemma)f(3.4.4,)i(it)g(follo) o(ws)h(from)e Fy(s;)8 b(t)13 b Fx(2)i(T)d FB(\()p Fx(F)852 462 y Fs(1)872 455 y Fy(;)c Fx(V)s FB(\))17 b(that)f(ev)o(ery)f(term)g (in)h(the)g(con)o(v)o(ersion)g(is)g(blac)o(k)f(and)0 515 y(furthermore)g Fy(s)h Fx(!)359 497 y Fr(\003)359 527 y(R)389 532 y Fh(1)424 515 y Fy(u)468 497 y Fr(\003)468 527 y(R)498 532 y Fh(1)515 515 y Fx( )g Fy(t)p FB(.)21 b(Hence)15 b Fx(R)821 522 y Fs(1)857 515 y FB(is)h(lo)q(cally)g (con\015uen)o(t.)0 575 y(\(2\))h(Let)f Fx(R)g FB(b)q(e)h(the)f(hierarc) o(hical)e(com)o(bination)h(of)i(base)f Fx(R)1120 582 y Fs(1)1156 575 y FB(and)h(extension)f Fx(R)1509 582 y Fs(2)1529 575 y FB(.)21 b(Since)16 b Fx(R)1734 582 y Fs(1)1767 575 y Fx(?)e(R)1862 582 y Fs(2)0 635 y FB(according)k(to)h (Lemma)d(3.4.3,)i Fx(R)634 642 y Fs(1)672 635 y FB(and)g Fx(R)811 642 y Fs(2)848 635 y FB(are)h(esp)q(ecially)d(non-in)o (terfering)i(\(cf.)f(remark)g(after)h(Def-)0 695 y(inition)k(2.2.26\).) 40 b(Th)o(us)23 b(lo)q(cal)f(con\015uence)g(of)h Fx(R)955 702 y Fs(1)997 695 y FB(and)g Fx(R)1141 702 y Fs(2)1183 695 y FB(implies,)e(b)o(y)g(Prop)q(osition)j(3.4.1,)g(lo)q(cal)0 756 y(con\015uence)16 b(of)g(their)g(com)o(bined)e(system)h Fx(R)p FB(.)21 b Fq(2)73 894 y FB(The)16 b(follo)o(wing)f(example)e (sho)o(ws)j(that)g(a)g(lo)q(cally)f(con\015uen)o(t)g(TRS)g(can)h(b)q(e) g(divided)e(in)o(to)h(a)h(\(lo)q(cally)0 954 y(con\015uen)o(t\))g(base) h(system)d(and)j(an)g(extension)f(whic)o(h)f(is)h(not)h(lo)q(cally)e (con\015uen)o(t.)0 1092 y Fz(Example)h(3.4.6)24 b FB(Let)17 b Fx(R)501 1099 y Fs(1)535 1092 y FB(=)e Fx(f)p Fy(b)f Fx(!)h Fy(c)p Fx(g)h FB(and)i Fx(R)913 1099 y Fs(2)947 1092 y FB(=)d Fx(f)p Fy(a)f Fx(!)h Fy(b;)8 b(a)14 b Fx(!)g Fy(c)p Fx(g)p FB(.)23 b(Their)16 b(hierarc)o(hical)f(com)o(bina-)0 1153 y(tion)h(is)g(lo)q(cally)g(con\015uen)o(t)g(\(in)g(fact)g (complete\))e(but)i Fx(R)1042 1160 y Fs(2)1078 1153 y FB(is)g(not.)73 1291 y(Sev)o(eral)h(authors)h(\(Bergstra)g(et)g(al.)f ([BKM89)o(],)g(Drosten)i([Dro89)q(],)e(Kurihara)h(&)g(Ka)s(ji)f([KK88]) 1910 1273 y Fs(1)1930 1291 y FB(\))0 1351 y(ha)o(v)o(e)c(indep)q(enden) o(tly)g(observ)o(ed)h(that)g(normalization)f(is)h(a)h(mo)q(dular)e (prop)q(ert)o(y)h(of)h Fu(disjoint)f FB(TRSs.)21 b(All)0 1411 y(authors)e(but)e(Kurihara)h(&)f(Ka)s(ji)g(tak)o(e)g(the)g(follo)o (wing)g(approac)o(h:)24 b(Ev)o(ery)17 b(term)e Fy(t)h Fx(2)g(T)d FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))16 b(can)i(b)q(e)0 1471 y(rewritten)e(to)i(normal)e(form)g(reducing)h(la)o(y)o(er)f(b)o(y) h(la)o(y)o(er)e(in)i(a)h(b)q(ottom)f(up)g(fashion.)25 b(That)18 b(is,)f(\014rst)g(the)0 1532 y(b)q(ottom)i(la)o(y)o(er)f (\(the)h(\\innermost")g(blac)o(k)f(or)i(white)f(parts\))h(of)g Fy(t)f FB(is)g(reduced)f(to)i(normal)e(form,)h(then)0 1592 y(the)d(same)e(is)i(done)g(with)g(the)g(la)o(y)o(er)e(ab)q(o)o(v)o (e)i(the)f(b)q(ottom)h(la)o(y)o(er)f(and)h(so)h(on.)k(Ev)o(en)o(tually) 14 b(the)i(top)g(la)o(y)o(er)0 1652 y(is)f(reduced)g(to)h(normal)f (form;)f(the)i(term)d(obtained)j(is)g(a)g(normal)e(form)h(of)h Fy(t)p FB(.)k(W)l(e)c(will)e(sho)o(w)i(next)f(that)0 1712 y(this)h(metho)q(d)g(can)g(also)h(b)q(e)g(successfully)e(applied)g (for)i(comp)q(osable)f(systems.)k(The)c(di\013eren)o(t)g(metho)q(d)0 1772 y(of)h(Kurihara)g(&)f(Ka)s(ji)g(will)f(b)q(e)i(treated)f(in)g (Chapter)h(7.)22 b(In)17 b(the)f(sequel,)f(w)o(e)h(use)g(for)h(an)o(y)f (TRS)h(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))0 1833 y(the)16 b(notation)h Fy(N)5 b(F)i FB(\()p Fx(F)e Fy(;)j Fx(R)p FB(\))13 b(=)h Fy(N)5 b(F)i FB(\(\()p Fx(T)12 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))p Fy(;)g Fx(!)938 1840 y Fr(R)969 1833 y FB(\)\))14 b(=)g Fx(f)p Fy(t)f Fx(2)h(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))16 b Fx(j)g Fy(t)d Fx(2)h Fy(N)5 b(F)i FB(\()p Fx(!)1628 1840 y Fr(R)1660 1833 y FB(\))p Fx(g)p FB(.)0 1971 y Fz(Lemma)16 b(3.4.7)23 b FB(If)c(\()p Fx(F)445 1978 y Fs(1)465 1971 y Fy(;)8 b Fx(R)529 1978 y Fs(1)549 1971 y FB(\))19 b(and)i(\()p Fx(F)741 1978 y Fs(2)760 1971 y Fy(;)8 b Fx(R)824 1978 y Fs(2)844 1971 y FB(\))20 b(are)f(arbitrary)h(\(not)g(necessarily)f (comp)q(osable\))g(TRSs,)0 2031 y(and)e(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))30 b(=)g(\()p Fx(F)391 2038 y Fs(1)421 2031 y Fx([)12 b(F)502 2038 y Fs(2)521 2031 y Fy(;)c Fx(R)585 2038 y Fs(1)616 2031 y Fx([)j(R)702 2038 y Fs(2)722 2031 y FB(\),)16 b(then)g Fy(N)5 b(F)i FB(\()p Fx(F)e Fy(;)j Fx(R)p FB(\))13 b(=)h Fy(N)5 b(F)i FB(\()p Fx(F)e Fy(;)j Fx(R)1380 2038 y Fs(1)1399 2031 y FB(\))j Fx(\\)g Fy(N)5 b(F)i FB(\()p Fx(F)e Fy(;)j Fx(R)1680 2038 y Fs(2)1700 2031 y FB(\).)0 2112 y Fz(Pro)r(of:)21 b FB(This)c(follo)o(ws)f(imm)o (ediatel)o(y)d(from)j Fx(!)888 2119 y Fr(R)936 2112 y FB(=)g Fx(!)1040 2119 y Fr(R)1070 2124 y Fh(1)1087 2119 y Fr([R)1141 2124 y Fh(2)1177 2112 y FB(=)g Fx(!)1281 2119 y Fr(R)1311 2124 y 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y Fs(1)1209 1487 y Fy(;)8 b Fx(V)t FB(\))k(suc)o(h)f (that)h Fy(t)1521 1494 y Fs(1)1557 1487 y Fx(!)1607 1469 y Fr(\003)1607 1500 y(R)1637 1505 y Fh(1)1673 1487 y Fy(t)1691 1494 y Fs(3)1726 1469 y Fr(\003)1726 1500 y(R)1756 1505 y Fh(1)1773 1487 y Fx( )k Fy(t)1857 1494 y Fs(2)1876 1487 y FB(.)-59 1547 y(By)d(Lemma)e(3.3.14,)j(ev)o(ery)e(term)f(in)i (the)g(con)o(v)o(ersion)g(is)g(transparen)o(t)h(and)g(moreo)o(v)o(er)d Fy(t)1569 1554 y Fs(1)1604 1547 y Fx(!)1654 1529 y Fr(\003)1654 1560 y(S)1696 1547 y Fy(t)1714 1554 y Fs(3)1750 1529 y Fr(\003)1750 1560 y(S)1773 1547 y Fx( )16 b Fy(t)1857 1554 y Fs(2)1876 1547 y FB(.)-59 1608 y(\(2\))h(This)f(follo)o(ws)g (from)f(Lemma)f(3.4.8)j(in)f(conjunction)g(with)g(\(1\).)22 b Fq(2)-59 1742 y Fz(Theorem)16 b(3.5.2)24 b FB(Semi-comple)o(teness)13 b(is)k(a)f(mo)q(dular)g(prop)q(ert)o(y)g(of)h(comp)q(osable)e(TRSs.)-59 1823 y Fz(Pro)r(of:)23 b FB(Let)17 b(\()p Fx(F)259 1830 y Fs(1)278 1823 y Fy(;)8 b Fx(R)342 1830 y Fs(1)362 1823 y FB(\))17 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b(:)g(:)g(:)g(;)g(u)1511 522 y Fw(m)1544 515 y Fx(g)-17 b(g)21 b FB(in)g(whic)o(h)f(the)h Fy(u)1921 522 y Fw(i)1935 515 y FB(,)0 575 y Fy(i)14 b Fx(2)g(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(m)p Fx(g)p FB(,)15 b(are)i(top)g(white) f(normal)f(forms.)21 b(Cho)q(ose)d(v)m(ariables)e Fy(x)1355 582 y Fs(1)1375 575 y Fy(;)8 b(:)g(:)g(:)f(;)h(x)1512 582 y Fw(m)1561 575 y FB(not)17 b(o)q(ccurring)g(in)i(~)-27 b Fy(u)0 635 y FB(satisfying)15 b Fy(u)244 642 y Fs(1)263 635 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)400 642 y Fw(m)448 635 y Fx(1)14 b Fy(x)540 642 y Fs(1)560 635 y Fy(;)8 b(:)g(:)g(:)f(;)h(x) 697 642 y Fw(m)730 635 y FB(.)21 b(Since)901 623 y(~)890 635 y Fy(C)929 617 y Fw(b)946 635 y Fx(f)p Fy(x)999 642 y Fs(1)1018 635 y Fy(;)8 b(:)g(:)g(:)g(;)g(x)1156 642 y Fw(m)1188 635 y Fx(g)14 b(2)g(T)f FB(\()p Fx(F)1369 642 y Fs(1)1389 635 y Fy(;)8 b Fx(V)t FB(\))13 b(and)j(the)e(TRS)h(\()p Fx(F)1826 642 y Fs(1)1846 635 y Fy(;)8 b Fx(R)1910 642 y Fs(1)1930 635 y FB(\))0 695 y(is)14 b(semi-complete,)d(it)j(follo)o (ws)h(that)698 683 y(~)687 695 y Fy(C)726 677 y Fw(b)743 695 y Fx(f)p Fy(x)796 702 y Fs(1)815 695 y Fy(;)8 b(:)g(:)g(:)f(;)h(x) 952 702 y Fw(m)985 695 y Fx(g)15 b FB(rewrites)f(to)h(its)f(unique)g (\()p Fx(F)1549 702 y Fs(1)1569 695 y Fy(;)8 b Fx(R)1633 702 y Fs(1)1653 695 y FB(\))14 b(normal)g(form)11 743 y(^)0 756 y Fy(C)39 738 y Fw(b)56 756 y Fx(h)p Fy(x)103 763 y Fw(i)115 768 y Fh(1)134 756 y Fy(;)8 b(:)g(:)g(:)g(;)g(x)272 763 y Fw(i)284 769 y Fg(l)297 756 y Fx(i)p FB(.)20 b(W)l(e)12 b(set)j(^)-27 b Fy(u)14 b FB(=)606 743 y(^)595 756 y Fy(C)634 738 y Fw(b)651 756 y Fx(h)p Fy(u)698 763 y Fw(i)710 768 y Fh(1)729 756 y Fy(;)8 b(:)g(:)g(:)g(;)g(u)867 763 y Fw(i)879 769 y Fg(l)893 756 y Fx(i)p FB(.)20 b(It)11 b(is)h(easy)g(to)g(v)o(erify)f(that)k(^)-27 b Fy(u)14 b Fx(2)g Fy(N)5 b(F)i FB(\()p Fx(F)t Fy(;)h Fx(R)p FB(\).)20 b(Observ)o(e)0 816 y(that)g(~)-27 b Fy(u)13 b Fx(!)197 798 y Fr(\003)197 828 y(R)227 833 y Fh(1)263 816 y FB(^)-27 b Fy(u)16 b FB(and)h(hence)f Fy(u)d Fx(!)626 798 y Fr(\003)663 816 y FB(^)-27 b Fy(u)o FB(.)73 901 y(W)l(e)17 b(claim)e(that)403 892 y(^)396 901 y Fy(t)414 908 y Fs(1)449 901 y FB(=)503 892 y(^)503 901 y Fy(t)g FB(=)596 892 y(^)589 901 y Fy(t)607 908 y Fs(2)627 901 y FB(.)24 b(Let)18 b Fy(u)782 908 y Fs(1)817 901 y Fx(!)d Fy(u)910 908 y Fs(2)947 901 y FB(b)q(e)j(a)f(step)h(in)e(the)i(con)o(v)o(ersion)e Fy(t)1560 908 y Fs(1)1604 883 y Fr(\003)1621 901 y Fx( )g Fy(t)f Fx(!)1770 883 y Fr(\003)1805 901 y Fy(t)1823 908 y Fs(2)1843 901 y FB(.)24 b(W)l(e)0 962 y(sho)o(w)15 b(that)26 b(^)-36 b Fy(u)250 969 y Fs(1)283 962 y FB(=)26 b(^)-36 b Fy(u)363 969 y Fs(2)383 962 y FB(.)20 b(If)14 b Fy(r)q(ank)r FB(\()p Fy(u)616 969 y Fs(1)635 962 y FB(\))g Fy(<)g(k)r FB(,)g(then)g Fy(r)q(ank)r FB(\()p Fy(u)1036 969 y Fs(2)1055 962 y FB(\))g Fy(<)g(k)i FB(as)f(w)o(ell.)k(Hence)d(^)-27 b Fy(u)1527 969 y Fs(1)1560 962 y FB(=)14 b Fy(u)1640 969 y Fs(1)1659 962 y Fx(#)g FB(=)g Fy(u)1778 969 y Fs(2)1797 962 y Fx(#)g FB(=)j(^)-27 b Fy(u)1916 969 y Fs(2)1935 962 y FB(.)0 1022 y(If)17 b Fy(r)q(ank)r FB(\()p Fy(u)202 1029 y Fs(1)222 1022 y FB(\))f(=)h Fy(k)r FB(,)g(then)h Fy(u)511 1029 y Fs(1)548 1022 y FB(is)g(a)g(top)g(blac)o(k)f(or)h(top)g (transparen)o(t)h(term,)d(i.e.,)g Fy(u)1525 1029 y Fs(1)1560 1022 y FB(=)h Fy(C)1654 1004 y Fw(b)1650 1034 y Fs(1)1670 1022 y Fx(f)-17 b(f)q Fy(s)1727 1029 y Fs(1)1746 1022 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)1879 1029 y Fw(n)1902 1022 y Fx(g)-17 b(g)p FB(.)0 1082 y(Here)15 b(w)o(e)h(ha)o(v)o(e)f(the)h (follo)o(wing)g(sub)q(cases.)0 1202 y(\(a\))28 b(If)g Fy(u)179 1209 y Fs(1)215 1202 y Fx(!)265 1179 y Fw(t;o)265 1215 y Fr(A)293 1220 y Fh(1)328 1202 y Fy(u)356 1209 y Fs(2)376 1202 y FB(,)i(then)e Fy(u)571 1209 y Fs(2)619 1202 y FB(can)g(b)q(e)g(written)g(as)g Fy(u)1080 1209 y Fs(2)1134 1202 y FB(=)34 b Fy(C)1245 1184 y Fw(b)1241 1215 y Fs(2)1261 1202 y Fx(h)-8 b(h)p Fy(s)1314 1209 y Fw(i)1326 1214 y Fh(1)1346 1202 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)1479 1209 y Fw(i)1491 1213 y Fg(m)1522 1202 y Fx(i)-8 b(i)p FB(.)57 b(It)27 b(follo)o(ws)h(that)3 1263 y(~)-27 b Fy(u)28 1270 y Fs(1)74 1263 y FB(=)26 b Fy(C)177 1245 y Fw(b)173 1275 y Fs(1)193 1263 y Fx(f)p Fy(s)241 1270 y Fs(1)261 1263 y Fx(#)p Fy(;)8 b(:)g(:)g(:)f(;)h(s)418 1270 y Fw(n)441 1263 y Fx(#g)24 b FB(and)j(~)-27 b Fy(u)645 1270 y Fs(2)690 1263 y FB(=)26 b Fy(C)793 1245 y Fw(b)789 1275 y Fs(2)810 1263 y Fx(h)p Fy(s)852 1270 y Fw(i)864 1275 y Fh(1)884 1263 y Fx(#)p Fy(;)8 b(:)g(:)g(:)f(;)h(s)1041 1270 y Fw(i)1053 1276 y Fg(l)1067 1263 y Fx(#i)p FB(.)43 b(W)l(e)23 b(obtain)k(~)-27 b Fy(u)1446 1270 y Fs(1)1482 1263 y Fx(!)1532 1270 y Fr(R)1562 1275 y Fh(1)1601 1263 y FB(~)g Fy(u)1626 1270 y Fs(2)1669 1263 y FB(from)22 b(Lemma)0 1323 y(3.3.16.)g(Ev)o(ery)15 b Fy(s)323 1330 y Fw(j)341 1323 y Fx(#)h FB(has)h(a)g(represen)o(tation)f Fy(s)852 1330 y Fw(j)870 1323 y Fx(#)e FB(=)972 1310 y(\026)961 1323 y Fy(C)1000 1305 y Fw(b)996 1335 y(j)1016 1323 y Fx(h)-8 b(h)p Fy(u)1074 1299 y Fw(j)1074 1334 y Fs(1)1094 1323 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)1231 1305 y Fw(j)1231 1335 y(m)1262 1340 y Fg(j)1281 1323 y Fx(i)-8 b(i)p FB(.)21 b(Hence)391 1443 y(~)-27 b Fy(u)416 1450 y Fs(1)449 1443 y FB(=)14 b Fy(C)540 1422 y Fw(b)536 1455 y Fs(1)557 1443 y Fx(f)593 1430 y FB(\026)582 1443 y Fy(C)621 1422 y Fw(b)617 1455 y Fs(1)637 1443 y Fx(h)-8 b(h)p Fy(u)695 1422 y Fs(1)695 1455 y(1)715 1443 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)852 1422 y Fs(1)852 1455 y Fw(m)883 1460 y Fh(1)903 1443 y Fx(i)-8 b(i)p Fy(;)8 b(:)g(:)g(:)f(;)1053 1430 y FB(\026)1042 1443 y Fy(C)1081 1422 y Fw(b)1077 1455 y(n)1101 1443 y Fx(h)-8 b(h)p Fy(u)1159 1422 y Fw(n)1159 1455 y Fs(1)1182 1443 y Fy(;)8 b(:)g(:)g(:)g(;)g(u)1320 1422 y Fw(n)1320 1455 y(m)1351 1459 y Fg(n)1374 1443 y Fx(i)-8 b(ig)16 b(!)1495 1450 y Fr(R)1525 1455 y Fh(1)430 1558 y Fy(C)469 1538 y Fw(b)465 1571 y Fs(2)486 1558 y Fx(h)516 1546 y FB(\026)505 1558 y Fy(C)544 1538 y Fw(b)540 1571 y(i)552 1576 y Fh(1)571 1558 y Fx(h)-8 b(h)p Fy(u)629 1538 y Fw(i)641 1543 y Fh(1)629 1571 y Fs(1)661 1558 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)798 1538 y Fw(i)810 1543 y Fh(1)798 1571 y Fw(m)829 1576 y Fg(i)840 1583 y Fh(1)862 1558 y Fx(i)-8 b(i)p Fy(;)8 b(:)g(:)g(:)g(;)1012 1546 y FB(\026)1002 1558 y Fy(C)1041 1538 y Fw(b)1037 1571 y(i)1049 1577 y Fg(l)1062 1558 y Fx(h)-8 b(h)p Fy(u)1120 1535 y Fw(i)1132 1541 y Fg(l)1120 1569 y Fs(1)1147 1558 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)1284 1538 y Fw(i)1296 1544 y Fg(l)1284 1571 y Fw(m)1315 1576 y Fg(i)1326 1585 y(l)1342 1558 y Fx(i)-8 b(ii)15 b FB(=)i(~)-27 b Fy(u)1486 1565 y Fs(2)1505 1558 y Fy(:)0 1655 y FB(Cho)q(ose)20 b(fresh)e(v)m (ariables)g Fy(x)524 1637 y Fs(1)524 1667 y(1)543 1655 y Fy(;)8 b(:)g(:)g(:)g(;)g(x)681 1637 y Fw(n)681 1667 y(m)712 1671 y Fg(n)753 1655 y FB(satisfying)18 b Fy(u)1000 1637 y Fs(1)1000 1667 y(1)1020 1655 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)1157 1637 y Fw(n)1157 1667 y(m)1188 1671 y Fg(n)1230 1655 y Fx(1)18 b Fy(x)1326 1637 y Fs(1)1326 1667 y(1)1345 1655 y Fy(;)8 b(:)g(:)g(:)g(;)g(x)1483 1637 y Fw(n)1483 1667 y(m)1514 1671 y Fg(n)1555 1655 y FB(and)19 b(note)f(that)h(this)0 1715 y(implies)14 b Fy(u)194 1694 y Fw(i)206 1699 y Fh(1)194 1726 y Fs(1)225 1715 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)362 1697 y Fw(i)374 1703 y Fg(l)362 1728 y Fw(m)393 1733 y Fg(i)404 1741 y(l)437 1715 y Fx(1)16 b Fy(x)531 1694 y Fw(i)543 1699 y Fh(1)531 1726 y Fs(1)562 1715 y Fy(;)8 b(:)g(:)g(:)f(;)h(x)699 1697 y Fw(i)711 1703 y Fg(l)699 1728 y Fw(m)730 1733 y Fg(i)741 1741 y(l)757 1715 y FB(.)21 b(Another)16 b(application)g(of)h(Lemma)d(3.3.16)j(yields)3 1840 y Fy(C)42 1820 y Fw(b)38 1853 y Fs(1)59 1840 y Fx(f)95 1828 y FB(\026)84 1840 y Fy(C)123 1820 y Fw(b)119 1853 y Fs(1)140 1840 y Fx(h)p Fy(x)187 1820 y Fs(1)187 1853 y(1)206 1840 y Fy(;)8 b(:)g(:)g(:)g(;)g(x)344 1820 y Fs(1)344 1853 y Fw(m)375 1858 y Fh(1)394 1840 y Fx(i)p Fy(;)g(:)g(:)g(:)g(;)534 1828 y FB(\026)523 1840 y Fy(C)562 1820 y Fw(b)558 1853 y(n)581 1840 y Fx(h)p Fy(x)628 1820 y Fw(n)628 1853 y Fs(1)651 1840 y Fy(;)g(:)g(:)g(:)g(;)g(x)789 1820 y Fw(n)789 1853 y(m)820 1857 y Fg(n)843 1840 y Fx(ig)14 b(!)951 1847 y Fr(R)981 1852 y Fh(1)1014 1840 y Fy(C)1053 1820 y Fw(b)1049 1853 y Fs(2)1070 1840 y Fx(h)1100 1828 y FB(\026)1089 1840 y Fy(C)1128 1820 y Fw(b)1124 1853 y(i)1136 1858 y Fh(1)1156 1840 y Fx(h)p Fy(x)1203 1820 y Fw(i)1215 1825 y Fh(1)1203 1853 y Fs(1)1234 1840 y Fy(;)8 b(:)g(:)g(:)f(;)h(x)1371 1820 y Fw(i)1383 1825 y Fh(1)1371 1853 y Fw(m)1402 1858 y Fg(i)1413 1865 y Fh(1)1435 1840 y Fx(i)p Fy(;)g(:)g(:)g(:)g(;)1574 1828 y FB(\026)1564 1840 y Fy(C)1603 1820 y Fw(b)1599 1853 y(i)1611 1859 y Fg(l)1624 1840 y Fx(h)p Fy(x)1671 1817 y Fw(i)1683 1823 y Fg(l)1671 1851 y Fs(1)1698 1840 y Fy(;)g(:)g(:)g(:)f(;)h(x)1835 1820 y Fw(i)1847 1826 y Fg(l)1835 1853 y Fw(m)1866 1858 y Fg(i)1877 1866 y(l)1893 1840 y Fx(ii)p Fy(:)0 1958 y FB(Since)k(b)q(oth)i(terms)e(are)h (trivially)e(joinable,)i(they)g(reduce)f(to)i(the)f(same)f(unique)g(\() p Fx(F)1552 1965 y Fs(1)1572 1958 y Fy(;)c Fx(R)1636 1965 y Fs(1)1656 1958 y FB(\))13 b(normal)f(form)11 2005 y(^)0 2018 y Fy(C)39 2000 y Fw(b)56 2018 y Fx(h)p Fy(y)99 2025 y Fs(1)119 2018 y Fy(;)c(:)g(:)g(:)f(;)h(y)252 2025 y Fw(p)271 2018 y Fx(i)p FB(,)19 b(where)f Fy(y)490 2025 y Fs(1)510 2018 y Fy(;)8 b(:)g(:)g(:)f(;)h(y)643 2025 y Fw(p)680 2018 y Fx(2)17 b(f)p Fy(x)783 2000 y Fs(1)783 2030 y(1)803 2018 y Fy(;)8 b(:)g(:)g(:)f(;)h(x)940 2000 y Fw(n)940 2030 y(m)971 2034 y Fg(n)994 2018 y Fx(g)p FB(.)27 b(Hence)21 b(^)-28 b Fy(u)1235 2025 y Fs(1)1272 2018 y FB(=)18 b Fy(\033)r FB(\()1387 2005 y(^)1377 2018 y Fy(C)1416 2000 y Fw(b)1432 2018 y Fx(h)p Fy(y)1475 2025 y Fs(1)1495 2018 y Fy(;)8 b(:)g(:)g(:)f(;)h(y)1628 2025 y Fw(p)1647 2018 y Fx(i)p FB(\))18 b(=)i(^)-27 b Fy(u)1786 2025 y Fs(2)1824 2018 y FB(where)0 2084 y Fy(\033)15 b FB(=)f Fx(f)p Fy(x)148 2061 y Fw(j)148 2096 y(i)180 2084 y Fx(7!)f Fy(u)271 2061 y Fw(j)271 2096 y(i)306 2084 y Fx(j)j Fy(j)h Fx(2)d(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(n)p Fx(g)p Fy(;)g(i)14 b Fx(2)g(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(m)933 2091 y Fw(j)951 2084 y Fx(gg)p 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Fg(j)q Fj(\000)p Fh(1)929 1232 y Fy(;)g(y)975 1239 y Fs(1)994 1232 y Fy(;)g(:)g(:)g(:)f(;)h(y)1127 1239 y Fw(q)1146 1232 y Fy(;)g(x)1196 1209 y Fw(j)r Fs(+1)1196 1243 y(1)1259 1232 y Fy(;)g(:)g(:)g(:)f(;)h(x)1396 1214 y Fw(n)1396 1245 y(m)1427 1249 y Fg(n)1450 1232 y Fx(g)p FB(.)20 b(It)11 b(follo)o(ws)h(as)g(ab)q(o)o(v)o(e)-59 1293 y(that)20 b(^)-27 b Fy(u)75 1300 y Fs(1)108 1293 y FB(=)17 b(^)-27 b Fy(u)188 1300 y Fs(2)207 1293 y FB(.)-59 1413 y(All)19 b(in)i(all,)175 1404 y(^)174 1413 y Fy(t)g FB(=)279 1404 y(^)273 1413 y Fy(t)291 1420 y Fs(1)332 1413 y FB(=)398 1404 y(^)391 1413 y Fy(t)409 1420 y Fs(2)449 1413 y FB(is)g(a)g(common) e(reduct)h(of)h Fy(t)982 1420 y Fs(1)1022 1413 y FB(and)h Fy(t)1140 1420 y Fs(2)1159 1413 y FB(.)35 b(Moreo)o(v)o(er)19 b Fy(t)j Fx(!)1516 1395 y Fr(\003)1558 1404 y FB(^)1557 1413 y Fy(t)f Fx(2)h Fy(N)5 b(F)i FB(\()p Fx(F)t Fy(;)h Fx(R)p FB(\).)-59 1473 y(Therefore,)21 b Fy(t)f FB(is)h(con\015uen)o(t) f(and)i(normalizing.)885 1464 y(^)885 1473 y Fy(t)e 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b(If)19 b Fy(r)q(ank)r FB(\()p Fy(u)p FB(\))i Fy(<)g(k)r FB(,)g(then)f Fy(u)h FB(has)g(a)g(unique)e(normal)h(form)-59 1955 y Fy(u)p Fx(#)f FB(according)i(to)f(the)f(induction)h(h)o(yp)q (othesis)g(and)g(w)o(e)g(set)i(~)-27 b Fy(u)20 b FB(=)j(^)-27 b Fy(u)20 b FB(=)g Fy(u)p Fx(#)o FB(.)33 b(If)19 b Fy(r)q(ank)r FB(\()p Fy(u)p FB(\))h(=)g Fy(k)i FB(and)e Fy(u)-59 2015 y FB(is)f(top)h(blac)o(k)e(or)i(top)f(white,)g(then)g Fy(u)g FB(has)h(a)g(unique)f(normal)f(form)g Fy(u)p Fx(#)h FB(according)g(to)h(cases)f(\(i\))g(and)-59 2075 y(\(ii\).)30 b(Again,)19 b(w)o(e)g(set)j(~)-27 b Fy(u)19 b FB(=)j(^)-27 b Fy(u)19 b FB(=)g Fy(u)p Fx(#)p FB(.)31 b(If)18 b Fy(r)q(ank)r FB(\()p Fy(u)p FB(\))h(=)g Fy(k)j FB(and)e Fy(u)f FB(is)g(top)h (transparen)o(t,)g(then)f(it)g(can)h(b)q(e)-59 2135 y(written)f(as)h Fy(u)f FB(=)g Fy(C)321 2117 y Fw(t)335 2135 y FB([)-8 b([)o Fy(s)377 2142 y Fs(1)397 2135 y Fy(;)8 b(:)g(:)g(:)f(;)h(s)529 2142 y Fw(n)552 2135 y FB(])-8 b(])o(.)31 b(It)19 b(follo)o(ws)g(from)f (the)h(foregoing)i(that)e(ev)o(ery)f Fy(s)1517 2142 y Fw(j)1535 2135 y FB(,)i Fy(j)i Fx(2)e(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(n)p Fx(g)p FB(,)-59 2196 y(has)24 b(a)g(unique)f (normal)g(form)g Fy(s)568 2203 y Fw(j)586 2196 y Fx(#)p FB(.)44 b(The)23 b(result)g(of)h(replacing)g(eac)o(h)f Fy(s)1339 2203 y Fw(j)1381 2196 y FB(with)g(its)h(unique)f(normal)-59 2256 y(form)d(is)i(denoted)f(b)o(y)k(~)-27 b Fy(u)o FB(,)23 b(i.e.,)h(~)-27 b Fy(u)22 b FB(=)h Fy(C)692 2238 y Fw(t)706 2256 y FB([)p Fy(s)743 2263 y Fs(1)763 2256 y Fx(#)p Fy(;)8 b(:)g(:)g(:)f(;)h(s)920 2263 y Fw(n)943 2256 y Fx(#)p FB(].)37 b(Note)21 b(that)h Fy(u)h Fx(!)1368 2238 y Fr(\003)1413 2256 y FB(~)-27 b Fy(u)p FB(.)37 b(Moreo)o(v)o(er,)25 b(~)-27 b Fy(u)21 b FB(has)h(a)-59 2316 y(represen)o(tation)h(~)-27 b Fy(u)20 b FB(=)381 2303 y(~)370 2316 y Fy(C)409 2298 y Fw(t)424 2316 y Fx(f)-17 b(f)p Fy(u)485 2323 y Fs(1)504 2316 y Fy(;)8 b(:)g(:)g(:)g(;)g(u)642 2323 y Fw(m)675 2316 y Fx(g)-17 b(g)20 b FB(in)g(whic)o(h)f(the)h 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Fx(F)476 215 y Fs(1)495 208 y Fy(;)8 b Fx(R)559 215 y Fs(1)579 208 y FB(\))21 b(and)g(\()p Fx(F)773 215 y Fs(2)792 208 y Fy(;)8 b Fx(R)856 215 y Fs(2)876 208 y FB(\))21 b(are)g(assumed)f(to)h(b)q(e)g(con\015uen)o(t)g(comp)q(osable)f(TRSs.)0 268 y(First)c(w)o(e)g(sho)o(w)i(that)f(white)f(preserv)o(ed)f(terms)g (are)i(con\015uen)o(t)f(w.r.t.)g(the)g(com)o(bined)e(system)i(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\).)0 328 y(If)18 b Fx(!)101 335 y Fw(c)136 328 y FB(is)g(normalizing,)f(then)h(this)g(result)g(can) g(b)q(e)g(used)h(to)f(sho)o(w)h(con\015uence)f(of)g(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\).)27 b(The)18 b(next)0 388 y(prop)q(osition)f(states)g(that)g(mono)q(c)o(hrome)d(outer)i (reduction)g(is)g(con\015uen)o(t.)0 530 y Fz(Prop)r(osition)i(4.2.5)23 b FB(The)17 b(relations)f Fx(!)791 512 y Fw(t)805 530 y FB(,)g Fx(!)885 506 y Fw(t;o)885 542 y Fr(A)913 547 y Fh(1)933 530 y FB(,)f(and)i Fx(!)1107 506 y Fw(t;o)1107 542 y Fr(A)1135 547 y Fh(2)1171 530 y FB(are)f(con\015uen)o(t.)0 625 y 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b(:)g(:)g(:)g(;)g(m)p Fx(g)p FB(.)20 b(Ob)o(viously)l(,)15 b(the)h(set)g Fx(f)p Fy(t)1168 1980 y Fs(1)1187 1973 y Fy(;)8 b(:)g(:)g(:)g(;)g(t)1315 1980 y Fw(n)1338 1973 y Fx(g)16 b FB(represen)o(ts)g Fy(S)s FB(.)21 b Fq(2)73 2114 y FB(F)l(or)13 b(sho)o(wing)g (con\015uence)g(of)g(white)f(preserv)o(ed)g(terms,)f(con\015uence)h(of) i(preserv)o(ed)d(terms)g(is)i(needed.)0 2256 y Fz(Lemma)j(4.2.8)23 b FB(Preserv)o(ed)15 b(terms)g(are)h(con\015uen)o(t.)0 2337 y Fz(Pro)r(of:)36 b FB(W)l(e)24 b(sho)o(w)g(that)h(ev)o(ery)d (preserv)o(ed)h(term)f Fy(t)h FB(is)h(con\015uen)o(t)g(b)o(y)f (induction)g(on)i Fy(r)q(ank)r FB(\()p Fy(t)p FB(\).)43 b(If)0 2397 y Fy(r)q(ank)r FB(\()p Fy(t)p FB(\))18 b(=)g(0,)h(then)g Fy(t)e Fx(2)i(T)12 b FB(\()p Fx(B)r Fy(;)c Fx(V)t FB(\))18 b(and)h(the)g(assertion)g(follo)o(ws)g(from)e(the)i(con\015uence)f(of)h (\()p Fx(B)q Fy(;)8 b Fx(S)t FB(\))19 b(\(cf.)0 2457 y(Lemma)c(3.5.1\).)24 b(So)17 b(let)f Fy(r)q(ank)r FB(\()p Fy(t)p FB(\))f(=)g Fy(k)i(>)e 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FB(consists)d(of)h(con\015uen)o(t)f (terms)e(b)q(ecause)i(ev)o(ery)f(elemen)o(t)e(of)j Fy(S)j FB(is)d(preserv)o(ed)e(and)j(has)0 2878 y(rank)e(less)g(than)h Fy(k)r FB(.)29 b(It)19 b(follo)o(ws)g(from)f(Lemma)f(4.2.7)i(that)g Fy(S)j FB(can)e(b)q(e)f(represen)o(ted)f(b)o(y)g(a)i(set)1813 2866 y(^)1805 2878 y Fy(S)r FB(.)30 b(W)l(e)p eop %%Page: 52 60 52 59 bop -59 -39 a FB(52)1216 b Fv(CHAPTER)16 b(4.)38 b(CONFLUENCE)-59 94 y FB(write)23 b(~)-27 b Fy(u)21 b FB(for)g(the)g(term)e(obtained)i(from)f Fy(u)h FB(b)o(y)f(replacing)h (eac)o(h)f(white)h(principal)f(subterm)f(with)i(its)-59 154 y(represen)o(tativ)o(e.)e(Note)d(that)h Fy(u)c Fx(!)588 136 y Fr(\003)625 154 y FB(~)-27 b Fy(u)o FB(.)14 235 y(W)l(e)14 b(claim)f(that)329 225 y(~)328 235 y Fy(t)346 242 y Fs(1)18 b Fr(A)410 247 y Fh(1)394 208 y Fr(\003)e Fw(o)410 235 y Fx( )494 225 y FB(~)493 235 y Fy(t)560 208 y Fw(o)g Fr(\003)544 235 y Fx(!)593 242 y Fr(A)621 247 y Fh(1)658 225 y FB(~)657 235 y Fy(t)675 242 y Fs(2)694 235 y FB(.)21 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Fy(u)139 684 y Fs(1)172 677 y Fx(!)222 659 y Fw(i)250 677 y Fy(u)278 684 y Fs(2)297 677 y FB(,)g(w)o(e)g(ma)o(y)e(write)i Fy(u)653 684 y Fs(2)686 677 y FB(=)f Fy(C)777 659 y Fw(b)773 690 y Fs(1)793 677 y Fx(f)-17 b(f)p Fy(s)849 684 y Fs(1)869 677 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)1002 659 y Fr(0)1002 690 y Fw(j)1020 677 y Fy(;)g(:)g(:)g(:)f(;)h(s)1152 684 y Fw(n)1176 677 y Fx(g)-17 b(g)p FB(,)15 b(where)f Fy(s)1400 684 y Fw(j)1432 677 y Fx(!)g Fy(s)1519 659 y Fr(0)1519 690 y Fw(j)1537 677 y FB(.)21 b(Since)14 b Fy(s)1721 684 y Fw(j)1755 677 y FB(and)i Fy(s)1872 659 y Fr(0)1872 690 y Fw(j)63 738 y FB(are)g(trivially)e(joinable,)i(w)o(e)g(ha)o(v)o (e)h(^)-26 b Fy(s)732 745 y Fw(j)764 738 y FB(=)16 b(^)-26 b Fy(s)839 720 y Fr(0)839 750 y Fw(j)873 738 y FB(and)17 b(hence)i(~)-27 b Fy(u)1132 745 y Fs(1)1165 738 y FB(=)14 b Fy(C)1256 720 y Fw(b)1252 750 y Fs(1)1273 738 y Fx(f)r FB(^)-26 b Fy(s)1321 745 y Fs(1)1340 738 y Fy(;)8 b(:)g(:)g(:)g(;)h FB(^)-25 b Fy(s)1473 745 y Fw(j)1491 738 y Fy(;)8 b(:)g(:)g(:)f(;)j FB(^)-26 b Fy(s)1623 745 y Fw(n)1646 738 y Fx(g)14 b FB(=)j(~)-27 b Fy(u)1765 745 y Fs(2)1784 738 y FB(.)-59 852 y(This)19 b(pro)o(v)o(es)g(the)g(claim)430 843 y(~)429 852 y Fy(t)447 859 y Fs(1)f Fr(A)511 864 y Fh(1)494 826 y Fr(\003)e Fw(o)511 852 y Fx( )595 843 y FB(~)594 852 y Fy(t)661 826 y Fw(o)g Fr(\003)645 852 y Fx(!)694 859 y Fr(A)722 864 y Fh(1)758 843 y FB(~)758 852 y Fy(t)776 859 y Fs(2)795 852 y FB(.)31 b(Since)18 b Fx(!)1020 829 y Fw(t;o)1020 865 y Fr(A)1048 870 y Fh(1)1087 852 y FB(is)h(con\015uen) o(t)g(b)o(y)g(Prop)q(osition)h(4.2.5,)g(the)-59 913 y(terms)77 903 y(~)77 913 y Fy(t)95 920 y Fs(1)130 913 y FB(and)226 903 y(~)225 913 y Fy(t)243 920 y Fs(2)278 913 y FB(ha)o(v)o(e)c(a)g (common)e(reduct)779 903 y(~)778 913 y Fy(t)796 920 y Fs(3)815 913 y FB(,)i(whic)o(h)f(at)i(the)f(same)f(time)f(is)i(a)g (common)e(reduct)i(of)g Fy(t)1870 920 y Fs(1)-59 973 y FB(and)h Fy(t)54 980 y Fs(2)73 973 y FB(,)f(more)f(precisely)f Fy(t)442 980 y Fs(1)475 973 y 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b(sho)o(w)g(that)h(ev)o(ery)d (white)h(preserv)o(ed)g(term)f(is)i(con\015uen)o(t.)37 b(So)23 b(supp)q(ose)g Fy(t)e FB(is)h(a)g(white)-59 1588 y(preserv)o(ed)15 b(term)f(and)j(consider)e(a)i(con)o(v)o(ersion)e Fy(t)854 1595 y Fs(1)898 1570 y Fr(\003)915 1588 y Fx( )h Fy(t)d Fx(!)1062 1570 y Fr(\003)1096 1588 y Fy(t)1114 1595 y Fs(2)1133 1588 y FB(.)21 b(It)16 b(has)h(to)f(b)q(e)g(sho)o(wn)h (that)f(the)g(terms)-59 1648 y Fy(t)-41 1655 y Fs(1)-7 1648 y FB(and)g Fy(t)105 1655 y Fs(2)139 1648 y FB(are)f(joinable.)20 b(As)15 b(in)g(the)f(pro)q(of)i(of)g(Lemma)c(4.2.8,)j(let)f Fy(S)k FB(b)q(e)d(the)g(set)g(of)g(all)f(white)h(principal)-59 1708 y(subterms)21 b(o)q(ccurring)h(in)g(the)f(con)o(v)o(ersion.)38 b(Notice)21 b(that)h(if)g Fy(u)g FB(is)f(a)i(top)f(white)g(term)e(o)q (ccurring)i(in)-59 1768 y(the)c(con)o(v)o(ersion,)g(then)h Fy(u)f FB(itself)g(b)q(elongs)i(to)f Fy(S)s FB(.)28 b(By)18 b(Lemma)f(4.2.8,)i Fy(S)j FB(consists)d(of)g(con\015uen)o(t)f(terms)-59 1828 y(b)q(ecause)f(ev)o(ery)f(elemen)o(t)e(of)k Fy(S)i FB(is)d(preserv)o(ed.)23 b(Th)o(us)18 b(b)o(y)e(Lemma)f(4.2.7,)j Fy(S)i FB(can)d(b)q(e)h(represen)o(ted)e(b)o(y)g(a)-59 1889 y(set)25 1876 y(^)17 1889 y Fy(S)s FB(.)21 b(Recall)15 b(that)k(~)-27 b Fy(u)16 b FB(denotes)g(the)g(result)g(of)g(replacing)f (ev)o(ery)g(white)h(principal)f(subterm)f(in)i Fy(u)g FB(with)-59 1949 y(its)g(represen)o(tativ)o(e.)-59 2017 y(W)l(e)j(claim)d(that)269 2007 y(~)269 2017 y Fy(t)287 2024 y Fs(1)h Fr(A)350 2029 y Fh(1)334 1987 y Fr(\003)f Fw(t;o)346 2017 y Fx( )424 2007 y FB(~)424 2017 y Fy(t)474 1987 y Fw(t;o)g Fr(\003)478 2017 y Fx(!)530 2024 y Fr(A)558 2029 y Fh(1)594 2007 y FB(~)594 2017 y Fy(t)612 2024 y Fs(2)631 2017 y FB(.)29 b(Let)19 b Fy(u)792 2024 y Fs(1)830 2017 y Fx(!)g Fy(u)927 2024 y Fs(2)965 2017 y FB(b)q(e)g(a)h(step)e(in)h(the)g(con)o(v)o(ersion)f Fy(t)1588 2024 y Fs(1)1632 1999 y Fr(\003)1649 2017 y Fx( )e Fy(t)i Fx(!)1801 1999 y Fr(\003)1839 2017 y Fy(t)1857 2024 y Fs(2)1876 2017 y 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b(Unfortunately)l(,)18 b(con\015uence)h(of)h (\()p Fx(F)t Fy(;)8 b Fx(R)p FB(\))20 b(is)f(exactly)f(what)i(w)o(e)e (w)o(an)o(t)h(to)0 2638 y(pro)o(v)o(e.)h(So)15 b(w)o(e)g(are)g(going)h (to)f(sho)o(w)h(normalization)d(of)j Fx(!)1086 2645 y Fw(c)1103 2638 y FB(.)21 b(Since)14 b Fx(!)1314 2645 y Fw(c)1346 2638 y FB(is)h(not)g(closed)g(under)g(con)o(texts,)0 2698 y(w)o(e)d(cannot)i(apply)f(the)f(metho)q(d)g(of)i(Prop)q(osition)g (3.4.9.)20 b(Ho)o(w)o(ev)o(er,)11 b(innermost)h Fx(!)1538 2705 y Fw(c)1568 2698 y FB(deriv)m(ations)h(satisfy)0 2758 y(the)g(desired)g(closure)g(prop)q(ert)o(y)g(\(this)g(is)h(made)e (more)g(precise)g(in)h(the)g(next)g(lemma\).)18 b(This)13 b(observ)m(ation)0 2818 y(leads)k(to)h(a)f(simple)e(pro)q(of)k(of)e (innermost)f(normalization)g(of)i Fx(!)1209 2825 y Fw(c)1226 2818 y FB(.)24 b(Therefore,)17 b(w)o(e)g(actually)f(sho)o(w)i(the)0 2878 y(stronger)f(statemen)o(t)d(that)j Fx(!)572 2885 y Fw(c)606 2878 y FB(is)f(innermost)f(normalizing.)p eop %%Page: 58 66 58 65 bop -59 -39 a FB(58)1216 b Fv(CHAPTER)16 b(4.)38 b(CONFLUENCE)-59 94 y Fz(Lemma)16 b(4.3.10)23 b FB(Let)16 b Fy(s)g FB(b)q(e)f(a)h(top)h(white)e(term)f(and)i(let)f Fy(s)f Fx(!)1124 75 y Fr(\003)1124 106 y Fw(c)1157 94 y Fy(s)1180 75 y Fr(0)1207 94 y FB(b)q(e)i(an)g(innermost)f(deriv)m (ation)g(suc)o(h)-59 154 y(that)i Fy(s)70 136 y Fr(0)95 154 y Fx(2)d Fy(N)5 b(F)i FB(\()p Fx(!)294 161 y Fw(c)311 154 y FB(\).)21 b(Then)16 b Fy(C)531 136 y Fw(b)548 154 y FB([)p Fy(:)8 b(:)g(:)f(;)h(s;)g(:)g(:)g(:)o FB(])13 b Fx(!)828 136 y Fr(\003)828 166 y Fw(c)862 154 y Fy(C)901 136 y Fw(b)917 154 y FB([)p Fy(:)8 b(:)g(:)g(;)g(s)1042 136 y Fr(0)1053 154 y Fy(;)g(:)g(:)g(:)o FB(])16 b(for)g(an)o(y)h(blac) o(k)e(con)o(text)g Fy(C)1668 136 y Fw(b)1685 154 y FB([)p Fy(;)8 b(:)g(:)g(:)f(;)h FB(].)-59 235 y Fz(Pro)r(of:)25 b FB(Clearly)l(,)17 b(if)g(all)h(terms)e(in)i Fy(s)f Fx(!)701 217 y Fr(\003)701 247 y Fw(c)737 235 y Fy(s)760 217 y Fr(0)790 235 y FB(are)h(top)g(white,)g(then)f(the)h(lemma)d (holds.)27 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b(].)24 b(Since)16 b Fy(r)q(ank)r FB(\()p Fy(t)1047 1701 y Fw(j)1065 1694 y FB(\))f Fy(<)h(r)q(ank)r FB(\()p Fy(t)p FB(\),)h(it)g(follo)o(ws)g(from)f(the)h(induc-)-59 1755 y(tion)g(h)o(yp)q(othesis)f(that,)h(for)g(ev)o(ery)e Fy(j)i Fx(2)e(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(,)16 b(there)g(exists)g(an)h(innermost)e(deriv)m(ation)i Fy(t)1738 1762 y Fw(j)1770 1755 y Fx(!)1820 1737 y Fr(\003)1820 1767 y Fw(c)1854 1755 y Fy(t)1872 1737 y Fr(0)1872 1767 y Fw(j)-59 1815 y FB(suc)o(h)h(that)i Fy(t)180 1797 y Fr(0)180 1827 y Fw(j)216 1815 y Fx(2)e Fy(N)5 b(F)i FB(\()p Fx(!)419 1822 y Fw(c)436 1815 y FB(\).)28 b(According)18 b(to)h(Lemma)e(4.3.10,)j Fy(t)d FB(=)h Fy(C)1258 1797 y Fw(b)1275 1815 y FB([)-8 b([)p Fy(t)1313 1822 y Fs(1)1331 1815 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)1458 1822 y Fw(n)1481 1815 y FB(])-8 b(])17 b Fx(!)1568 1797 y Fr(\003)1568 1827 y Fw(c)1606 1815 y Fy(C)1645 1797 y Fw(b)1661 1815 y FB([)p Fy(t)1693 1797 y Fr(0)1693 1827 y Fs(1)1712 1815 y Fy(;)8 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Fy(D)h FB(:)d Fy(s)830 2168 y Fs(1)864 2161 y Fx(!)g Fy(s)950 2168 y Fs(2)984 2161 y Fx(!)g Fy(s)1070 2168 y Fs(3)1104 2161 y Fx(!)h Fy(:)8 b(:)g(:)0 2276 y FB(As)18 b(in)h(the)f(pro)q(of)i(of)f(Prop)q(osition)h(5.2.6,)f(w)o (e)f(ma)o(y)f(assume)h(that)h Fy(D)i FB(is)d(an)i(in\014nite)d Fx(R)i FB(deriv)m(ation)g(of)0 2337 y(minim)o(al)12 b(rank,)j(i.e.,)e (an)o(y)i Fx(R)g FB(deriv)m(ation)g(of)g(smaller)e(rank)i(is)g (\014nite.)20 b(Let)15 b Fy(r)q(ank)r FB(\()p Fy(D)q FB(\))g(=)f Fy(k)i Fx(2)e Fl(I)-7 b(N)p FB(.)21 b(By)14 b(our)0 2397 y(assumptions,)f(it)g(follo)o(ws)f(that)i(for)f(all)f (indices)g Fy(j)s FB(,)h Fy(r)q(ank)r FB(\()p Fy(s)1082 2404 y Fw(j)1101 2397 y FB(\))h(=)f Fy(r)q(ank)r FB(\()p Fy(D)q FB(\))h(and)g Fy(r)q(oot)p FB(\()p Fy(s)1604 2404 y Fw(j)1623 2397 y FB(\))g Fx(2)g(F)1739 2404 y Fw(d)1772 2397 y FB(for)f(some)0 2457 y Fy(d)h Fx(2)g(f)p FB(1)p Fy(;)8 b FB(2)p Fx(g)13 b FB(\(i.e.,)e(the)h(terms)f(all)h(ha)o(v)o(e)g 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y Fr(0)856 1630 y Fx(6)p FB(=)f Fy(t)h FB(there)g(is)g(a)h Fy(j)h Fx(2)d(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g (n)p Fx(g)15 b FB(suc)o(h)g(that)h Fy(t)1697 1612 y Fr(0)1722 1630 y Fx(2)f Fy(S)s FB(\()p Fy(t)1840 1637 y Fw(j)1857 1630 y FB(\).)-59 1690 y(Let)34 1677 y(\026)25 1690 y Fy(d)g Fx(2)f(f)p FB(1)p Fy(;)8 b FB(2)p Fx(g)d(n)g(f)p Fy(d)p Fx(g)p FB(.)21 b(Then)14 b(b)o(y)f(Lemma)e(5.2.10)j(there)f(is)g (a)h Fy(u)1110 1672 y Fr(0)1135 1690 y Fx(2)g Fy(S)1221 1663 y Fs(\026)1215 1672 y Fw(d)1235 1690 y FB(\()p Fy(u)p FB(\))g Fx(\022)f Fy(S)1406 1663 y Fs(\026)1400 1672 y Fw(d)1420 1690 y FB(\()p Fy(s)p FB(\))h(suc)o(h)f(that)h Fy(u)1733 1672 y Fr(0)1758 1690 y Fx(!)1808 1672 y Fr(\003)1808 1702 y(R)1854 1690 y Fy(t)1872 1697 y Fw(j)-59 1750 y FB(b)q(ecause)j Fy(r)q(oot)p FB(\()p Fy(t)246 1757 y Fw(j)265 1750 y FB(\))f Fx(2)f(F)390 1752 y Fs(\026)384 1761 y Fw(d)404 1750 y FB(.)24 b(Clearly)l(,)16 b Fy(r)q(ank)r FB(\()p Fy(u)774 1732 y Fr(0)786 1750 y FB(\))f Fy(<)h(r)q(ank)r FB(\()p 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2548 y FB(if)g(it)f(is)h(clear)g(from)f(the)h(con)o(text.) 0 2677 y Fz(Lemma)g(5.2.19)23 b FB(Let)16 b Fy(s)g FB(b)q(e)g(an)g (elemen)o(t)d(of)j Fx(T)d FB(\()p Fx(F)t Fy(;)8 b Fx(f)p Fy(z)r Fx(g)p FB(\))16 b(with)f Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))g Fx(2)f(F)1447 2684 y Fw(d)1467 2677 y FB(.)21 b(If)15 b Fy(u)f Fx(2)g Fy(S)1672 2659 y Fw(d)1692 2677 y FB(\()p Fy(s)p FB(\),)i(then)f(w)o(e)0 2737 y(ha)o(v)o(e)g Fy(L)145 2719 y Fw(d)145 2751 y(r)q(ank)q Fs(\()p Fw(u)p Fs(\))286 2737 y Fx(!)336 2717 y Fs(+)336 2749 y Fr(C)355 2755 y Fj(E)391 2737 y FB(\010)426 2744 y Fw(d)446 2737 y FB(\()p Fy(u)p FB(\).)0 2818 y Fz(Pro)r(of:)31 b FB(Let)21 b Fy(r)q(ank)r FB(\()p Fy(u)p FB(\))h(=)g Fy(m)p FB(.)35 b(Clearly)l(,)21 b Fy(m)h FB(=)g Fy( )r FB(\()p Fy(u)p FB(\))f Fx(\024)h Fy(r)q(ank)r FB(\()p Fy(s)p FB(\).)36 b(W)l(e)21 b(will)f(sho)o(w)i(the)f(lemm)o(a)e(for)0 2878 y Fy(m)d(>)h FB(1,)h(the)f(case)h Fy(m)e FB(=)h(1)h(is)g(obtained) g(b)o(y)f(similar)f(argumen)o(ts.)25 b(Since)17 b Fy(m)f(>)h 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161 y Fw(j)1087 154 y Fx(6)p FB(=)e Fy(t)1160 136 y Fr(00)1180 154 y FB(,)h(it)g(follo)o(ws)f(from)g(Lemma)f(5.2.14)i(that)122 214 y Fy( )r FB(\()p Fy(t)193 221 y Fw(j)210 214 y FB(\))24 b Fy(<)g( )r FB(\()p Fy(t)386 196 y Fr(00)406 214 y FB(\))e(and)h(th)o (us)f Fy( )r FB(\()p Fy(t)732 221 y Fw(j)749 214 y FB(\))i Fy(<)f( )r FB(\()p Fy(t)p FB(\))e(\(recall)g(that)i Fy( )r FB(\()p Fy(t)1302 196 y Fr(00)1322 214 y FB(\))h Fx(\024)f Fy( )r FB(\()p Fy(t)p FB(\))e(b)o(y)h(Lemma)e(5.2.13\).)122 274 y(The)15 b(outer)g(induction)f(h)o(yp)q(othesis)h(yields)e(\010)971 281 y Fw(d)992 274 y FB(\()p Fy(t)1029 281 y Fw(j)1047 274 y FB(\))g Fx(!)1129 256 y Fr(\003)1129 286 y(R)1159 292 y Fg(d)1177 286 y Fr(]C)1220 292 y Fj(E)1256 274 y FB(\010)1291 281 y Fw(d)1312 274 y FB(\()p Fy(t)1349 256 y Fr(0)1349 286 y Fw(j)1367 274 y FB(\).)20 b(Therefore,)14 b(it)h(follo)o(ws)f(that)122 334 y(\010)157 341 y Fw(d)177 334 y FB(\()p Fy(t)214 316 y Fr(00)235 334 y FB(\))i(=)f Fy(C)t FB([\010)411 341 y Fw(d)430 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Fr(0)1379 407 y Fw(j)1397 394 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h FB(\010)1560 401 y Fw(d)1581 394 y FB(\()p Fy(t)1618 401 y Fw(n)1641 394 y FB(\)].)39 b(It)22 b(is)h(quite)122 455 y(simple)10 b(to)j(v)o(erify)e(its)h(v)m(alidit)o(y)f(if)h Fy(r)q(oot)p FB(\()p Fy(t)858 437 y Fr(0)858 467 y Fw(j)877 455 y FB(\))i Fx(2)g(F)999 457 y Fs(\026)993 465 y Fw(d)1025 455 y FB(or)f Fy(t)1099 437 y Fr(0)1099 467 y Fw(j)1131 455 y Fx(2)h(T)f FB(\()p Fx(F)1278 462 y Fw(d)1298 455 y Fy(;)8 b Fx(V)s FB(\).)20 b(So,)14 b(supp)q(ose)f Fy(r)q(oot)p FB(\()p Fy(t)1789 437 y Fr(0)1789 467 y Fw(j)1808 455 y FB(\))h Fx(2)g(F)1929 462 y Fw(d)122 515 y FB(and)j Fy(t)235 497 y Fr(0)235 527 y Fw(j)266 515 y FB(=)d Fy(C)357 497 y Fr(0)368 515 y FB([)-8 b([)o Fy(u)415 522 y Fs(1)435 515 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)572 522 y Fw(m)605 515 y FB(])-8 b(])o(.)21 b(Set)16 b Fy(C)782 497 y Fr(00)803 515 y FB([)p Fy(;)8 b(:)g(:)g(:)f(;)h FB(])13 b(=)h Fy(C)t FB([)p Fy(;)8 b(:)g(:)g(:)e(;)i(C)1205 497 y Fr(0)1216 515 y FB([)p Fy(;)g(:)g(:)g(:)f(;)h FB(])p Fy(;)g(:)g(:)g(:)f(;)h FB(].)20 b(It)c(follo)o(ws)216 617 y(\010)251 624 y Fw(d)271 617 y FB(\()p Fy(t)308 599 y Fr(0)319 617 y FB(\))42 b(=)14 b Fy(C)471 599 y Fr(00)491 617 y FB([\010)540 624 y Fw(d)560 617 y FB(\()p Fy(t)597 624 y Fs(1)617 617 y FB(\))p Fy(;)8 b(:)g(:)g(:)f(;)h FB(\010)780 624 y Fw(d)800 617 y FB(\()p Fy(t)837 624 y Fw(j)r Fr(\000)p Fs(1)900 617 y FB(\))p Fy(;)g FB(\010)976 624 y Fw(d)996 617 y FB(\()p Fy(u)1043 624 y Fs(1)1063 617 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h FB(\010)1226 624 y Fw(d)1247 617 y FB(\()p Fy(u)1294 624 y Fw(m)1327 617 y FB(\))p Fy(;)g FB(\010)1403 624 y Fw(d)1423 617 y FB(\()p Fy(t)1460 624 y Fw(j)r Fs(+1)1523 617 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h FB(\010)1686 624 y Fw(d)1707 617 y FB(\()p Fy(t)1744 624 y Fw(n)1767 617 y FB(\)])380 678 y(=)14 b Fy(C)t FB([\010)520 685 y Fw(d)539 678 y FB(\()p Fy(t)576 685 y Fs(1)595 678 y FB(\))p Fy(;)8 b(:)g(:)g(:)g(;)g FB(\010)759 685 y Fw(d)779 678 y FB(\()p Fy(t)816 685 y Fw(j)r Fr(\000)p Fs(1)879 678 y FB(\))p Fy(;)g(C)959 660 y Fr(0)970 678 y FB([\010)1019 685 y Fw(d)1039 678 y FB(\()p Fy(u)1086 685 y Fs(1)1105 678 y FB(\))p Fy(;)g(:)g(:)g(:)g(;)g FB(\010)1269 685 y Fw(d)1289 678 y FB(\()p Fy(u)1336 685 y Fw(m)1369 678 y FB(\)])p Fy(;)g FB(\010)1459 685 y Fw(d)1479 678 y FB(\()p Fy(t)1516 685 y Fw(j)r Fs(+1)1579 678 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h FB(\010)1742 685 y Fw(d)1763 678 y FB(\()p Fy(t)1800 685 y Fw(n)1823 678 y FB(\)])380 738 y(=)14 b Fy(C)t FB([\010)520 745 y Fw(d)539 738 y FB(\()p Fy(t)576 745 y Fs(1)595 738 y FB(\))p Fy(;)8 b(:)g(:)g(:)g(;)g FB(\010)759 745 y Fw(d)779 738 y FB(\()p Fy(t)816 720 y Fr(0)816 750 y Fw(j)834 738 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h FB(\010)997 745 y Fw(d)1018 738 y FB(\()p Fy(t)1055 745 y Fw(n)1078 738 y FB(\)])p Fy(:)0 855 y FB(\(ii\))15 b Fy(r)q(oot)p FB(\()p Fy(t)205 837 y Fr(00)227 855 y FB(\))f Fx(2)g(F)349 857 y Fs(\026)343 865 y Fw(d)73 967 y Fx(\017)24 b FB(If)16 b Fy(t)189 949 y Fr(0)214 967 y FB(=)e Fy(z)r FB(,)h(then)h(the)g(assertion)h (follo)o(ws)f(straigh)o(tforw)o(ardly)l(.)73 1065 y Fx(\017)24 b FB(If)17 b(also)h Fy(r)q(oot)p FB(\()p Fy(t)395 1047 y Fr(0)407 1065 y FB(\))e Fx(2)g(F)533 1067 y Fs(\026)527 1076 y Fw(d)547 1065 y FB(,)i(then)f(b)o(y)g(Lemma)f(5.2.17)i(\010)1113 1072 y Fw(d)1133 1065 y FB(\()p Fy(t)1170 1047 y Fr(00)1191 1065 y FB(\))e(=)g Fy(L)1319 1038 y Fs(\026)1313 1047 y Fw(d)1313 1078 y( )q Fs(\()p Fw(t)1364 1069 y Fj(00)1384 1078 y Fs(\))1400 1065 y FB(\()p Fy(s)p FB(\))g Fx(!)1527 1047 y Fr(\003)1527 1077 y(C)1546 1083 y Fj(E)1584 1065 y Fy(L)1623 1038 y Fs(\026)1617 1047 y Fw(d)1617 1078 y( )q Fs(\()p Fw(t)1668 1069 y Fj(0)1679 1078 y Fs(\))1695 1065 y FB(\()p Fy(s)p FB(\))g(=)g(\010)1861 1072 y Fw(d)1882 1065 y FB(\()p Fy(t)1919 1047 y Fr(0)1930 1065 y FB(\))122 1125 y(b)q(ecause)g Fy( )r FB(\()p Fy(t)373 1107 y Fr(0)384 1125 y FB(\))e Fx(\024)g Fy( )r FB(\()p Fy(t)541 1107 y Fr(00)561 1125 y FB(\).)73 1223 y Fx(\017)24 b FB(If)e(otherwise)g Fy(r)q(oot)p FB(\()p Fy(t)524 1205 y Fr(0)536 1223 y FB(\))j Fx(2)g(F)674 1230 y Fw(d)694 1223 y FB(,)f(then)e Fy(t)867 1205 y Fr(00)912 1223 y FB(=)j Fy(C)t FB([)-8 b([)p Fy(t)1052 1230 y Fs(1)1070 1223 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)1197 1230 y Fw(n)1220 1223 y FB(])-8 b(])23 b Fx(!)1313 1230 y Fr(R)1348 1233 y Fh(\026)1343 1240 y Fg(d)1388 1223 y Fy(t)1406 1205 y Fr(0)1440 1223 y FB(where)f Fy(t)1605 1205 y Fr(0)1641 1223 y FB(=)j Fy(t)1722 1230 y Fw(j)1762 1223 y FB(for)e(some)122 1290 y Fy(j)18 b Fx(2)e(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(.)23 b(Since)17 b Fy( )r FB(\()p Fy(t)659 1272 y Fr(00)679 1290 y FB(\))e Fx(\024)h Fy( )r FB(\()p Fy(t)p FB(\))e Fx(6)p FB(=)h Fx(1)p FB(,)i(there)g(is)g(an)g Fy(s)1273 1272 y Fr(0)1300 1290 y Fx(2)f Fy(S)1388 1263 y Fs(\026)1382 1272 y Fw(d)1402 1290 y FB(\()p Fy(s)p FB(\))h(with)g Fy(r)q(ank)r FB(\()p Fy(s)1739 1272 y Fr(0)1751 1290 y FB(\))e(=)h Fy( )r FB(\()p Fy(t)1910 1272 y Fr(00)1930 1290 y FB(\))122 1351 y(suc)o(h)23 b(that)h Fy(s)375 1332 y Fr(0)413 1351 y Fx(!)463 1332 y Fr(\003)463 1363 y(R)493 1368 y Fh(1)510 1363 y Fr(]R)564 1368 y Fh(2)609 1351 y Fy(t)627 1332 y Fr(00)648 1351 y FB(.)43 b(No)o(w)24 b(\010)859 1358 y Fw(d)879 1351 y FB(\()p Fy(s)921 1332 y Fr(0)933 1351 y FB(\))i(=)g Fy(L)1081 1324 y Fs(\026)1075 1332 y Fw(d)1075 1364 y(r)q(ank)q Fs(\()p Fw(s)1181 1355 y Fj(0)1193 1364 y Fs(\))1208 1351 y FB(\()p Fy(s)p FB(\))h(=)f Fy(L)1399 1324 y Fs(\026)1393 1332 y Fw(d)1393 1364 y( )q Fs(\()p Fw(t)1444 1355 y Fj(00)1464 1364 y Fs(\))1480 1351 y FB(\()p Fy(s)p FB(\))g(=)g(\010)1666 1358 y Fw(d)1686 1351 y FB(\()p Fy(t)1723 1332 y Fr(00)1744 1351 y FB(\).)43 b(Hence)122 1411 y(it)22 b(remains)g(to)h(sho)o(w)h(\010)595 1418 y Fw(d)615 1411 y FB(\()p Fy(s)657 1393 y Fr(0)669 1411 y FB(\))h Fx(!)763 1393 y Fr(\003)763 1423 y(R)793 1429 y Fg(d)811 1423 y Fr(]C)854 1429 y Fj(E)902 1411 y FB(\010)937 1418 y Fw(d)957 1411 y FB(\()p Fy(t)994 1393 y Fr(0)1005 1411 y FB(\).)42 b(Since)22 b Fy(s)1237 1393 y Fr(0)1274 1411 y Fx(!)1324 1393 y Fr(\003)1324 1423 y(R)1354 1428 y Fh(1)1371 1423 y Fr(]R)1425 1428 y Fh(2)1470 1411 y Fy(t)1488 1393 y Fr(0)1499 1411 y FB(,)i Fy(r)q(oot)p FB(\()p Fy(t)1661 1393 y Fr(0)1673 1411 y FB(\))h Fx(2)h(F)1812 1418 y Fw(d)1832 1411 y FB(,)e(and)122 1471 y Fy(r)q(oot)p FB(\()p Fy(s)251 1453 y Fr(0)263 1471 y FB(\))c Fx(2)g(F)397 1473 y Fs(\026)391 1482 y Fw(d)411 1471 y FB(,)g(there)f(exists)g(a)h Fy(u)f Fx(2)h Fy(S)888 1453 y Fw(d)908 1471 y FB(\()p Fy(s)950 1453 y Fr(0)962 1471 y FB(\))f Fx(\022)h Fy(S)1092 1453 y Fw(d)1112 1471 y FB(\()p Fy(s)p FB(\))f(suc)o(h)h(that)g Fy(u)f Fx(!)1512 1453 y Fr(\003)1512 1483 y(R)1542 1488 y Fh(1)1559 1483 y Fr(]R)1613 1488 y Fh(2)1652 1471 y Fy(t)1670 1453 y Fr(0)1681 1471 y FB(.)31 b(Ob)o(viously)l(,)122 1531 y Fy( )r FB(\()p Fy(u)p FB(\))18 b(=)g Fy(r)q(ank)r FB(\()p Fy(u)p FB(\))h Fy(<)f(r)q(ank)r FB(\()p Fy(s)689 1513 y Fr(0)701 1531 y FB(\))g(=)h Fy( )r FB(\()p Fy(t)866 1513 y Fr(00)886 1531 y FB(\))g Fx(\024)f Fy( )r FB(\()p Fy(t)p FB(\).)28 b(Consequen)o(tly)19 b(it)f(follo)o(ws)h(from)f(the)h (outer)122 1591 y(induction)i(h)o(yp)q(othesis)g(that)h(\010)735 1598 y Fw(d)755 1591 y FB(\()p Fy(u)p FB(\))h Fx(!)894 1573 y Fr(\003)894 1604 y(R)924 1610 y Fg(d)942 1604 y Fr(]C)985 1610 y Fj(E)1029 1591 y FB(\010)1064 1598 y Fw(d)1085 1591 y FB(\()p Fy(t)1122 1573 y Fr(0)1133 1591 y FB(\).)36 b(Ev)o(en)o(tually)l(,)21 b(w)o(e)g(ha)o(v)o(e)f (\(cf.)h(Lemmata)122 1651 y(5.2.17)c(and)g(5.2.19\))273 1754 y(\010)308 1761 y Fw(d)328 1754 y FB(\()p Fy(t)365 1733 y Fr(00)386 1754 y FB(\))d(=)f(\010)505 1761 y Fw(d)526 1754 y FB(\()p Fy(s)568 1733 y Fr(0)579 1754 y FB(\))h(=)g Fy(L)703 1725 y Fs(\026)697 1733 y Fw(d)697 1766 y(r)q(ank)q Fs(\()p Fw(s)803 1757 y Fj(0)815 1766 y Fs(\))831 1754 y FB(\()p Fy(s)p FB(\))f Fx(!)955 1733 y Fs(+)955 1766 y Fr(C)974 1772 y Fj(E)1011 1754 y Fy(L)1044 1733 y Fw(d)1044 1766 y(r)q(ank)q Fs(\()p Fw(u)p Fs(\))1170 1754 y FB(\()p Fy(s)p FB(\))h Fx(!)1295 1733 y Fs(+)1295 1766 y Fr(C)1314 1772 y Fj(E)1350 1754 y FB(\010)1385 1761 y Fw(d)1405 1754 y FB(\()p Fy(u)p FB(\))g Fx(!)1535 1733 y Fr(\003)1535 1766 y(R)1565 1772 y Fg(d)1583 1766 y Fr(]C)1626 1772 y Fj(E)1662 1754 y FB(\010)1697 1761 y Fw(d)1717 1754 y FB(\()p Fy(t)1754 1733 y Fr(0)1766 1754 y FB(\))p Fy(:)0 1871 y Fq(2)0 1992 y Fz(Theorem)j(5.2.22)23 b FB(Let)15 b Fx(R)533 1999 y Fs(1)568 1992 y FB(and)h Fx(R)704 1999 y Fs(2)738 1992 y FB(b)q(e)f(t)o(w)o(o)g(disjoin)o(t)g(terminating)e (TRSs)i(suc)o(h)g(that)g(their)f(disjoin)o(t)0 2052 y(union)h Fx(R)f FB(=)g Fx(R)284 2059 y Fs(1)312 2052 y Fx(])8 b(R)395 2059 y Fs(2)430 2052 y FB(is)15 b(non-terminating.)k(Then)c Fx(R)1024 2059 y Fs(1)1059 2052 y FB(is)g(not)g Fx(C)1218 2059 y Fr(E)1242 2052 y FB(-terminating)e(and)j Fx(R)1658 2059 y Fs(2)1692 2052 y FB(is)f(collapsing)0 2112 y(or)i(vice)e(v)o (ersa.)0 2193 y Fz(Pro)r(of:)21 b FB(Let)680 2253 y Fy(D)15 b FB(:)f Fy(s)g FB(=)f Fy(s)874 2260 y Fs(1)908 2253 y Fx(!)h Fy(s)995 2260 y Fs(2)1028 2253 y Fx(!)g Fy(s)1115 2260 y Fs(3)1148 2253 y Fx(!)g Fy(:)8 b(:)g(:)0 2335 y FB(b)q(e)19 b(an)g(in\014nite)e Fx(R)348 2342 y Fs(1)381 2335 y Fx(])c(R)469 2342 y Fs(2)507 2335 y FB(deriv)m(ation)19 b(of)f(minimal)d(rank,)k(i.e.,)e(an)o(y)i Fx(R)1349 2342 y Fs(1)1381 2335 y Fx(])13 b(R)1469 2342 y Fs(2)1508 2335 y FB(deriv)m(ation)18 b(of)h(smaller)0 2395 y(rank)e(is)f (\014nite.)k(Let)d Fy(r)q(ank)r FB(\()p Fy(D)q FB(\))e(=)f Fy(k)r FB(.)21 b(Hence)15 b Fy(r)q(ank)r FB(\()p Fy(s)998 2402 y Fw(j)1016 2395 y FB(\))f(=)g Fy(r)q(ank)r FB(\()p Fy(D)q FB(\))j(for)g(all)f(indices)f Fy(j)s FB(.)21 b Fy(r)q(oot)p FB(\()p Fy(s)1792 2402 y Fs(1)1813 2395 y FB(\))14 b Fx(2)g(F)1929 2402 y Fw(d)0 2455 y FB(for)22 b(some)f Fy(d)j Fx(2)g(f)p FB(1)p Fy(;)8 b FB(2)p Fx(g)p FB(.)38 b(It)22 b(follo)o(ws)g(that)g Fy(r)q(oot)p FB(\()p Fy(s)951 2462 y Fw(j)970 2455 y FB(\))h Fx(2)h(F)1105 2462 y Fw(d)1147 2455 y FB(for)e(an)o(y)g Fy(j)s FB(.)38 b(In)22 b(particular,)g(there)g(is)f(no)0 2516 y(reduction)15 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Fy(S)730 1915 y Fw(b)727 1946 y Fs(1)747 1933 y FB(\()p Fy(s)p FB(\).)20 b(No)o(w)15 b(w)o(e)f(infer)h Fy(S)1167 1915 y Fw(w)1164 1946 y Fs(2)1195 1933 y FB(\()p Fy(t)p FB(\))e Fx(\022)h Fy(S)1350 1915 y Fw(w)1347 1946 y Fs(2)1378 1933 y FB(\()p Fy(s)p FB(\))h(from)e Fy(S)1600 1915 y Fw(b)1597 1946 y Fs(1)1617 1933 y FB(\()p Fy(t)p FB(\))h Fx(\022)f Fy(S)1772 1915 y Fw(b)1769 1946 y Fs(1)1789 1933 y FB(\()p Fy(s)p FB(\).)21 b(All)0 1993 y(in)c(all,)g(it)g(follo)o (ws)g Fy(S)385 1975 y Fw(w)382 2006 y(P)413 1993 y FB(\()p Fy(t)p FB(\))f(=)g Fy(S)572 1975 y Fw(w)569 2006 y Fs(1)600 1993 y FB(\()p Fy(t)p FB(\))11 b Fx([)h Fy(S)745 1975 y Fw(w)742 2006 y Fs(2)774 1993 y FB(\()p Fy(t)p FB(\))j Fx(\022)h Fy(S)933 1975 y Fw(w)930 2006 y Fs(1)961 1993 y FB(\()p Fy(s)p FB(\))c Fx([)g Fy(S)1112 1975 y Fw(w)1109 2006 y Fs(2)1140 1993 y FB(\()p Fy(s)p FB(\))k(=)f Fy(S)1303 1975 y Fw(w)1300 2006 y(P)1332 1993 y FB(\()p Fy(s)p FB(\).)24 b(The)18 b(remaining)e(inclusion)0 2054 y Fy(S)33 2036 y 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y(hold,)e(then)g(statemen)o(t)f(\(ii\))g(m)o(ust)g(hold.)0 2590 y(\(i\))i(There)h(is)f(an)i(in\014nite)e Fx(R)h FB(deriv)m(ation)f Fy(D)j FB(starting)e(from)f(a)h(non-top-transparen)o (t,)i(sa)o(y)e(top)g(blac)o(k,)0 2650 y(term)d(suc)o(h)h(that:)60 2774 y(1.)24 b(There)16 b(is)g(no)h(top)f(white)g(term)f(in)h Fy(D)q FB(.)60 2878 y(2.)24 b(There)16 b(are)g(in\014nitely)f(man)o(y) 31 b Fx(!)748 2855 y Fw(t;o)748 2890 y Fr(A)776 2895 y Fh(1)828 2878 y FB(reduction)15 b(steps)i(in)f Fy(D)q FB(.)p eop %%Page: 82 90 82 89 bop -59 -39 a FB(82)1200 b Fv(CHAPTER)16 b(5.)38 b(TERMINA)l(TION)1 94 y FB(3.)24 b(There)18 b(are)h(in\014nitely)e(man) o(y)g Fx(!)682 101 y Fr(R)712 106 y Fh(2)749 94 y FB(reduction)h(steps) h(in)f Fy(D)j FB(whic)o(h)d(are)g(destructiv)o(e)f(at)i(lev)o(el)d(1)63 154 y(or)g(lev)o(el)f(2.)1 259 y(4.)24 b(There)16 b(are)g(in\014nitely) f(man)o(y)f(duplicating)32 b Fx(!)943 235 y Fw(t;o)943 271 y Fr(A)971 276 y Fh(1)1023 259 y FB(reduction)16 b(steps)h(in)e Fy(D)q FB(.)-59 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(only)e(if)h(\()p Fx(F)1472 1889 y Fs(1)1492 1882 y Fy(;)8 b Fx(R)1556 1889 y Fs(1)1576 1882 y FB(\))17 b(and)h(\()p Fx(F)1767 1889 y Fs(2)1787 1882 y Fy(;)8 b Fx(R)1851 1889 y Fs(2)1871 1882 y FB(\))-59 1943 y(are)16 b(simplifying.)-59 2063 y(\\only-if)s(":)36 b(Let)23 b(\()p Fx(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))o Fy(;)g(>)p FB(\))23 b(b)q(e)h(a)f (simpli\014cation)e(ordering)j(with)39 b Fx(!)1381 2070 y Fr(R)1455 2063 y Fx(\022)23 b Fy(>)p FB(.)42 b(It)23 b(is)g(not)h(to)q(o)-59 2123 y(di\016cult)12 b(to)j(pro)o(v)o(e)e(that) h(\()p Fx(T)f FB(\()p Fx(F)521 2130 y Fw(j)539 2123 y Fy(;)8 b Fx(V)t FB(\))o Fy(;)22 b(>)6 b Fx(j)708 2131 y Fr(T)j Fs(\()p Fr(F)767 2141 y Fg(j)784 2131 y Fw(;)p Fr(V)s Fs(\))834 2123 y FB(\))14 b(is)f(a)i(simpli\014cation)c (ordering)k(and)f(that)g(furthermore)-59 2188 y Fx(!)-9 2195 y Fr(R)21 2200 y Fg(j)45 2188 y Fx(j)59 2195 y Fr(T)9 b Fs(\()p Fr(F)118 2206 y Fg(j)135 2195 y Fw(;)p Fr(V)s Fs(\))215 2188 y Fx(\022)16 b Fy(>)6 b Fx(j)328 2195 y Fr(T)i Fs(\()p Fr(F)387 2206 y Fg(j)403 2195 y Fw(;)p Fr(V)s Fs(\))453 2188 y FB(.)21 b(In)16 b(other)h(w)o(ords,)f(\()p Fx(F)884 2195 y Fw(j)902 2188 y Fy(;)8 b Fx(R)966 2195 y Fw(j)985 2188 y FB(\))16 b(is)g(simplifying.)-59 2308 y(\\if)s(":)22 b(First)15 b(of)h(all,)f(note)h(that)g Fx(R)566 2315 y Fs(1)596 2308 y Fx([)10 b(F)680 2284 y Fw(ar)q(g)675 2319 y Fs(1)751 2308 y FB(and)17 b Fx(R)888 2315 y Fs(2)918 2308 y Fx([)10 b(F)1002 2284 y Fw(ar)q(g)997 2319 y Fs(2)1073 2308 y FB(are)16 b(comp)q(osable)f(systems.)20 b(According)15 b(to)-59 2368 y(Lemma)h(5.1.7,)j(it)f(m)o(ust)f(b)q(e)i (sho)o(wn)g(that)36 b Fx(!)818 2348 y Fs(+)818 2380 y Fr(R[F)900 2371 y Fg(ar)q(g)987 2368 y FB(is)18 b(irre\015exiv)o(e.)25 b(Assuming)18 b(that)35 b Fx(!)1688 2348 y Fs(+)1688 2380 y Fr(R[F)1770 2371 y Fg(ar)q(g)1857 2368 y FB(is)-59 2428 y(not)17 b(irre\015exiv)o(e,)c(w)o(e)j(will)f(deriv)o(e)f(a)j(con) o(tradiction.)k(So)16 b(supp)q(ose)i(that)f(there)e(is)h(a)h(cyclic)d (deriv)m(ation)467 2533 y Fy(D)h FB(:)46 b Fy(t)14 b FB(=)f Fy(t)683 2540 y Fs(1)719 2533 y Fx(!)769 2540 y Fr(R[F)851 2531 y Fg(ar)q(g)928 2533 y Fy(:)8 b(:)g(:)24 b Fx(!)1060 2540 y Fr(R[F)1142 2531 y Fg(ar)q(g)1226 2533 y Fy(t)1244 2540 y Fw(n)1281 2533 y FB(=)14 b Fy(t;)-59 2638 y(n)k(>)g FB(1,)i(of)f(terms)e Fy(t)316 2645 y Fw(j)352 2638 y Fx(2)h(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))o(,)19 b Fy(j)j Fx(2)c(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(.)28 b(W.l.o.g.)g(w)o(e)19 b(ma)o(y)e(assume)h(that)h Fy(z)i FB(is)d(the)h(only)-59 2698 y(v)m(ariable)g(o)q(ccurring)h(in)g Fy(D)q FB(.)32 b(W)l(e)20 b(ma)o(y)e(further)h(assume)g(that)i Fy(r)q(ank)r FB(\()p Fy(t)p FB(\))e(=)h Fy(k)i FB(is)d(minimal,)e (i.e.,)h(there)-59 2758 y(is)i(no)h(cyclic)e(deriv)m(ation)h Fy(s)h FB(=)g Fy(s)564 2765 y Fs(1)600 2758 y Fx(!)650 2765 y Fr(R[F)732 2756 y Fg(ar)q(g)809 2758 y Fy(:)8 b(:)g(:)24 b Fx(!)941 2765 y Fr(R[F)1023 2756 y Fg(ar)q(g)1112 2758 y Fy(s)1135 2765 y Fw(m)1189 2758 y FB(=)d Fy(s;)29 b(m)21 b(>)g FB(1,)g(with)g(rank\()p Fy(s)p FB(\))g Fy(<)g(k)r FB(.)-59 2818 y(Consequen)o(tly)l(,)31 b Fx(!)319 2798 y Fs(+)319 2830 y Fr(R[F)401 2821 y Fg(ar)q(g)486 2818 y FB(is)16 b(irre\015exiv)o(e)d(on)k Fx(T)866 2800 y Fw()e FB(1)j(b)o(y)f (Lemma)e(5.3.9.)p eop %%Page: 85 93 85 92 bop 0 -39 a Fv(5.3.)38 b(COMPOSABLE)16 b(SYSTEMS)1188 b FB(85)0 94 y(Case)21 b(\(i\):)30 b Fy(t)20 b FB(is)g(top)h(blac)o(k.) 34 b(Ob)o(viously)l(,)20 b(ev)o(ery)f(term)g(in)i Fy(D)h FB(m)o(ust)d(ha)o(v)o(e)h(rank)h Fy(k)r FB(.)34 b(Therefore,)21 b(eac)o(h)0 154 y(term)15 b(in)g Fy(D)k FB(is)d(either)f(top)h(blac)o (k)g(or)g(top)h(transparen)o(t.)22 b(Let)288 287 y Fx(T)309 294 y Fw(D)355 287 y FB(=)14 b Fx(f)p Fy(s)f Fx(2)h(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(f)p Fy(z)r Fx(g)p FB(\))16 b Fx(j)g Fy(s)g FB(is)g(a)h(subterm)e(of)h(a)h(term)d(o)q(ccurring)j (in)f Fy(D)q Fx(g)p Fy(:)0 420 y FB(Note)21 b(that)i Fx(T)256 427 y Fw(D)310 420 y FB(is)e(\014nite.)37 b(Similar)20 b(to)i(v)m(arious)g(preceding)f(pro)q(ofs,)j(the)e(pro)q(of)h(idea)e (is)h(to)g(de\014ne)f(a)0 481 y(transformation)16 b(function)g(\010)558 462 y Fw(D)558 493 y(b)604 481 y FB(:)e Fx(T)653 488 y Fw(D)699 481 y Fx(!)f(T)g FB(\()p Fx(F)857 488 y Fs(1)888 481 y Fx(])e(f)p Fy(C)t(ons)p Fx(g)p Fy(;)d Fx(f)p Fy(z)r Fx(g)p FB(\))16 b(suc)o(h)g(that)170 614 y(\010)205 593 y Fw(D)205 626 y(b)237 614 y FB(\()p Fy(D)q FB(\))f(:)46 b(\010)426 593 y Fw(D)426 626 y(b)458 614 y FB(\()p Fy(t)p FB(\))13 b(=)h(\010)614 593 y Fw(D)614 626 y(b)646 614 y FB(\()p Fy(t)683 621 y Fs(1)703 614 y FB(\))i Fx(!)788 596 y Fr(\003)788 629 y Fs(\()p Fr(R)832 634 y Fh(1)849 629 y Fr([F)901 613 y Fg(ar)q(g)898 641 y Fh(1)951 629 y Fs(\))p Fr(]C)1008 635 y Fj(E)1054 614 y Fy(:)8 b(:)g(:)24 b Fx(!)1186 596 y Fr(\003)1186 629 y Fs(\()p Fr(R)1230 634 y Fh(1)1247 629 y Fr([F)1299 613 y Fg(ar)q(g)1296 641 y Fh(1)1349 629 y Fs(\))p Fr(]C)1406 635 y Fj(E)1444 614 y FB(\010)1479 593 y Fw(D)1479 626 y(b)1512 614 y FB(\()p Fy(t)1549 621 y Fw(n)1572 614 y FB(\))13 b(=)h(\010)1691 593 y Fw(D)1691 626 y(b)1723 614 y FB(\()p Fy(t)p FB(\))0 759 y(is)j(a)g(non-empt)o(y)f(cyclic)f(deriv)m(ation)h(of)i(terms)d (from)h Fx(T)d FB(\()p Fx(F)1104 766 y Fs(1)1135 759 y Fx(])f(f)p Fy(C)t(ons)p Fx(g)p Fy(;)c Fx(f)p Fy(z)r Fx(g)p FB(\).)22 b(Since)33 b Fx(!)1691 738 y Fs(+)1691 775 y(\()p Fr(R)1735 780 y Fh(1)1752 775 y Fr([F)1804 759 y Fg(ar)q(g)1801 786 y Fh(1)1854 775 y Fs(\))p Fr(]C)1911 781 y Fj(E)0 831 y FB(is)18 b(irre\015exiv)o(e)d(on)j Fx(T)13 b FB(\()p Fx(F)445 838 y Fs(1)477 831 y Fx(])f(f)p Fy(C)t(ons)p Fx(g)p Fy(;)c Fx(f)p Fy(z)r Fx(g)p FB(\))17 b(if)h(and)g(only)g(if)33 b Fx(!)1183 811 y Fs(+)1183 848 y Fr(R)1213 853 y Fh(1)1230 848 y Fr([F)1282 831 y Fg(ar)q(g)1279 859 y Fh(1)1369 831 y FB(is)18 b(irre\015exiv)o(e)d (on)j Fx(T)13 b FB(\()p Fx(F)1814 838 y Fs(1)1833 831 y Fy(;)8 b Fx(f)p Fy(z)r Fx(g)p FB(\))0 904 y(\(cf.)13 b(Lemma)f(5.1.14\),)j(this)f(con)o(tradicts)g(the)g(irre\015exivit)o(y) d(of)31 b Fx(!)1229 883 y Fs(+)1229 920 y Fr(R)1259 925 y Fh(1)1276 920 y Fr([F)1328 904 y Fg(ar)q(g)1325 932 y Fh(1)1397 904 y FB(.)20 b(In)14 b(order)g(to)h(de\014ne)f(\010)1847 886 y Fw(D)1847 916 y(b)1893 904 y FB(w)o(e)0 970 y(need)e(the)g(follo) o(wing)g(de\014nitions.)20 b(The)13 b Fu(inner)h(subterm)h(o)n(c)n (curr)n(enc)n(es)e(of)h Fy(D)h FB(are)d(those)h(terms)e(whic)o(h)h(are) 0 1031 y(subterms)k(of)h(a)g(white)f(principal)f(subterm)h(o)q (ccurring)h(in)f Fy(D)q FB(.)23 b(The)17 b(others)g(are)g(called)e Fu(outer)j(subterm)0 1091 y(o)n(c)n(curr)n(enc)n(es)c(of)f Fy(D)q FB(.)21 b(Let)13 b Fy(O)507 1073 y Fw(D)506 1103 y(b)552 1091 y FB(denote)f(the)h(set)f(of)h(all)f(outer)g(subterm)f(o)q (ccurrences)h(of)h Fy(D)q FB(.)21 b(F)l(urthermore,)0 1151 y(let)14 b Fy(S)102 1133 y Fw(w)99 1163 y(P)130 1151 y FB(\()p Fy(D)q FB(\))i(denote)f(the)g(set)g(of)h(all)e(white)h (principal)f(subterms)g(app)q(earing)i(in)f Fy(D)q FB(.)21 b(Observ)o(e)14 b(that)i(b)q(oth)0 1211 y(sets)24 b(are)f(\014nite)g (and)h(that)g(ev)o(ery)e(elemen)o(t)f(of)j Fy(S)958 1193 y Fw(w)955 1223 y(P)986 1211 y FB(\()p Fy(D)q FB(\))g(has)h(a)f(rank)f (less)h(than)g Fy(k)r FB(.)43 b(Moreo)o(v)o(er,)24 b(for)0 1271 y Fy(s)c Fx(2)f Fy(S)128 1253 y Fw(w)125 1284 y(P)157 1271 y FB(\()p Fy(D)q FB(\),)h(w)o(e)g(de\014ne)f(\001)531 1253 y Fw(D)531 1284 y(b)563 1271 y FB(\()p Fy(s)p FB(\))g(=)h Fx(f)p Fy(u)f Fx(2)h Fy(O)864 1253 y Fw(D)863 1284 y(b)916 1271 y Fx(j)g Fy(s)c Fx(!)1039 1251 y Fs(+)1039 1283 y Fr(R[F)1121 1274 y Fg(ar)q(g)1209 1271 y Fy(u)p Fx(g)p FB(.)31 b(It)19 b(is)h(imp)q(ortan)o(t)e(to)i(notice)f(that)0 1331 y(\001)41 1313 y Fw(D)41 1344 y(b)73 1331 y FB(\()p Fy(s)p FB(\))d(is)g(\014nite)g(for)g(an)o(y)g Fy(s)e Fx(2)g Fy(S)607 1313 y Fw(w)604 1344 y(P)635 1331 y FB(\()p Fy(D)q FB(\).)22 b(Let)17 b Fx(\037)f FB(b)q(e)g(a)h(total)f(ordering)h (on)f Fx(T)d FB(\()p Fx(F)j(])11 b(f)p Fy(C)t(ons)p Fx(g)p Fy(;)d Fx(f)p Fy(z)r Fx(g)p FB(\).)21 b(Let)282 1524 y(\010)317 1503 y Fw(D)317 1536 y(b)349 1524 y FB(\()p Fy(s)p FB(\))14 b(=)476 1424 y Ft(8)476 1462 y(>)476 1474 y(<)476 1549 y(>)476 1561 y(:)533 1463 y Fy(C)572 1445 y Fw(b)589 1463 y Fx(f)p FB(\010)649 1445 y Fw(D)649 1476 y(b)681 1463 y FB(\()p Fy(s)723 1470 y Fs(1)743 1463 y FB(\))p Fy(;)8 b(:)g(:)g(:)f(;)h FB(\010)906 1445 y Fw(D)906 1476 y(b)938 1463 y FB(\()p Fy(s)980 1470 y Fw(m)1013 1463 y FB(\))p Fx(g)121 b FB(if)15 b Fy(s)f FB(=)g Fy(C)1350 1445 y Fw(b)1367 1463 y Fx(f)-17 b(f)p Fy(s)1423 1470 y Fs(1)1442 1463 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)1575 1470 y Fw(m)1608 1463 y Fx(g)-17 b(g)533 1584 y Fy(S)s(or)q(t)p FB(\()p Fx(f)p FB(\010)709 1566 y Fw(D)709 1596 y(b)742 1584 y FB(\()p Fy(u)p FB(\))16 b Fx(j)g Fy(u)d Fx(2)h FB(\001)983 1566 y Fw(D)983 1596 y(b)1015 1584 y FB(\()p Fy(s)p FB(\))p Fx(g)p FB(\))58 b(if)15 b Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))g Fx(2)f(A)1472 1591 y Fs(2)0 1716 y FB(where)20 b Fy(S)s(or)q(t)p FB(\()p Fx(f)p Fy(t)304 1723 y Fs(1)323 1716 y Fy(;)8 b(:)g(:)g(:)g(;)g(t)451 1723 y Fw(n)474 1716 y Fx(g)p FB(\))20 b(=)h Fx(h)p Fy(t)634 1724 y Fw(\031)q Fs(\(1\))703 1716 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)830 1724 y Fw(\031)q Fs(\()p Fw(n)p Fs(\))902 1716 y Fx(i)21 b FB(suc)o(h)f(that)h Fy(t)1184 1724 y Fw(\031)q Fs(\()p Fw(j)r Fs(\))1259 1716 y Fx(\037)8 b Fy(t)1324 1724 y Fw(\031)q Fs(\()p Fw(j)r Fs(+1\))1456 1716 y FB(for)20 b(1)i Fx(\024)e Fy(j)k(<)d(n)p FB(.)33 b(Recall)0 1777 y(that)17 b Fx(h)p Fy(t)143 1784 y Fw(\031)q Fs(\(1\))211 1777 y Fy(;)8 b(:)g(:)g(:)g(;)g(t)339 1784 y Fw(\031)q Fs(\()p Fw(n)p Fs(\))411 1777 y Fx(i)16 b FB(stands)h(for)g(the)f(term) e Fy(C)t(ons)p FB(\()p Fy(t)1024 1784 y Fw(\031)q Fs(\(1\))1093 1777 y Fy(;)8 b(C)t(ons)p FB(\()p Fy(t)1266 1784 y Fw(\031)q Fs(\(2\))1334 1777 y Fy(;)g(:)g(:)g(:)f(;)h(C)t(ons)p FB(\()p Fy(t)1594 1784 y Fw(\031)q Fs(\()p Fw(n)p Fs(\))1666 1777 y Fy(;)g(z)r FB(\))g Fy(:)g(:)g(:)o FB(\)\).)0 1837 y(The)16 b(de\014nition)g(of)h(\010)408 1819 y Fw(D)408 1849 y(b)440 1837 y FB(\()p Fy(s)p FB(\))f(is)g(illustrated)f(in)h (Example)f(5.3.11.)0 1897 y(Note)h(that)h Fy(S)s(or)q(t)p FB(\()p Fx(f)p FB(\010)400 1879 y Fw(D)400 1909 y(b)432 1897 y FB(\()p Fy(u)p FB(\))f Fx(j)g Fy(u)e Fx(2)g FB(\001)674 1879 y Fw(D)674 1909 y(b)705 1897 y FB(\()p Fy(s)p FB(\))p Fx(g)p FB(\))g(=)g Fy(z)k FB(if)e Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))e Fx(2)g(A)1211 1904 y Fs(2)1247 1897 y FB(and)j(\001)1383 1879 y Fw(D)1383 1909 y(b)1414 1897 y FB(\()p Fy(s)p FB(\))d(=)g Fx(;)p FB(.)0 1957 y(It)i(is)g(easy)g(to)h (v)o(erify)d(that)j(the)f(transformation)g(function)g(\010)1152 1939 y Fw(D)1152 1969 y(b)1201 1957 y FB(is)g(w)o(ell-de\014ned.)0 2078 y(W)l(e)24 b(sho)o(w)g(next)f(that)i Fy(t)469 2085 y Fw(j)503 2078 y Fx(!)553 2085 y Fr(R[F)635 2076 y Fg(ar)q(g)727 2078 y Fy(t)745 2085 y Fw(j)r Fs(+1)832 2078 y FB(implies)c(\010)1040 2059 y Fw(D)1040 2090 y(b)1072 2078 y FB(\()p Fy(t)1109 2085 y Fw(j)1127 2078 y FB(\))c Fx(!)1213 2059 y Fr(\003)1213 2093 y Fs(\()p Fr(R)1257 2098 y Fh(1)1273 2093 y Fr([F)1325 2077 y Fg(ar)q(g)1322 2104 y Fh(1)1376 2093 y Fs(\))p Fr(]C)1433 2099 y Fj(E)1471 2078 y FB(\010)1506 2059 y Fw(D)1506 2090 y(b)1538 2078 y FB(\()p Fy(t)1575 2085 y Fw(j)r Fs(+1)1638 2078 y FB(\),)25 b(using)f Fx(!)g FB(as)0 2143 y(a)17 b(shorthand)g(for)33 b Fx(!)412 2150 y Fr(R[F)494 2141 y Fg(ar)q(g)576 2143 y FB(=)17 b Fx(!)681 2153 y Fs(\()p Fr(R)725 2158 y Fh(1)741 2153 y Fr([F)793 2137 y Fg(ar)q(g)790 2164 y Fh(1)844 2153 y Fs(\))873 2143 y Fx([)d(!)970 2153 y Fs(\()p Fr(R)1014 2158 y Fh(2)1031 2153 y Fr([F)1083 2137 y Fg(ar)q(g)1080 2164 y Fh(2)1133 2153 y Fs(\))1149 2143 y FB(.)22 b(There)15 b(are)i(the)f(follo)o(wing) g(cases.)0 2263 y(\(a\))g(If)f Fy(t)144 2270 y Fw(j)178 2263 y Fx(!)228 2240 y Fw(t;o)228 2276 y Fr(A)256 2281 y Fh(1)292 2263 y Fy(t)310 2270 y Fw(j)r Fs(+1)388 2263 y FB(b)o(y)g(some)g(rule)f Fy(l)h Fx(!)e Fy(r)q FB(,)j(then)f(w)o(e)g (ha)o(v)o(e)g Fy(l)g Fx(!)e Fy(r)i Fx(2)f(R)1331 2270 y Fs(1)1362 2263 y Fx([)e(F)1448 2240 y Fw(ar)q(g)1443 2274 y Fs(1)1503 2263 y FB(,)j Fy(t)1550 2270 y Fw(j)1582 2263 y FB(=)f Fy(C)1673 2245 y Fw(b)1689 2263 y FB([)-8 b([)o Fy(s)1731 2270 y Fs(1)1751 2263 y Fy(;)8 b(:)g(:)g(:)f(;)h(s)1883 2270 y Fw(m)1916 2263 y FB(])-8 b(])o(,)0 2332 y(and)17 b Fy(t)113 2339 y Fw(j)r Fs(+1)191 2332 y FB(=)255 2319 y(^)244 2332 y Fy(C)283 2314 y Fw(b)299 2332 y Fx(h)-8 b(h)q Fy(s)353 2339 y Fw(i)365 2344 y Fh(1)384 2332 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)517 2339 y Fw(i)529 2345 y Fg(l)542 2332 y Fx(i)-8 b(i)q FB(.)23 b(Applying)16 b(\010)857 2314 y Fw(D)857 2344 y(b)889 2332 y FB(,)g(w)o(e)h(obtain)g(\010)1179 2314 y Fw(D)1179 2344 y(b)1211 2332 y FB(\()p Fy(t)1248 2339 y Fw(j)1266 2332 y FB(\))e(=)f Fy(C)1391 2314 y Fw(b)1408 2332 y FB([\010)1457 2314 y Fw(D)1457 2344 y(b)1489 2332 y FB(\()p Fy(s)1531 2339 y Fs(1)1550 2332 y FB(\))p Fy(;)8 b(:)g(:)g(:)g(;)g FB(\010)1714 2314 y Fw(D)1714 2344 y(b)1746 2332 y FB(\()p Fy(s)1788 2339 y Fw(m)1821 2332 y FB(\)])16 b(and)0 2397 y(\010)35 2379 y Fw(D)35 2409 y(b)67 2397 y 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Fy(s)1009 1440 y Fr(0)1021 1458 y FB(\))c(=)h Fy(q)r FB(.)0 1575 y Fz(Pro)r(of:)21 b FB(The)16 b(\014rst)g(t)o(w)o(o)f (statemen)o(ts)g(are)g(immediate)d(consequences)j(of)h(Lemma)e(6.1.17.) 22 b(F)l(rom)14 b(\(2\))i(it)0 1645 y(follo)o(ws)h(that)h(there)f(is)g (a)h(deriv)m(ation)f Fy(s)e FB(=)h Fy(C)t FB([)p Fy(u)p FB(])934 1618 y Fw(im)g Fr(\003)939 1645 y Fx(!)38 b Fy(s)1050 1627 y Fr(0)1077 1645 y FB(=)16 b Fy(C)t FB([)p Fy(s)p Fx(#)o FB(])1277 1618 y Fw(im)h Fr(\003)1283 1645 y Fx(!)37 b Fy(s)p Fx(#)p FB(.)25 b(The)17 b(equalit)o(y)f Fx(})p FB(\()p Fy(s)1826 1627 y Fr(0)1837 1645 y FB(\))g(=)f Fy(q)0 1705 y FB(remains)22 b(to)i(b)q(e)f(pro)o(v)o(en.)42 b Fx(})p FB(\()p Fy(s)609 1687 y Fr(0)621 1705 y FB(\))23 b(is)g(the)h(p)q(osition)g(from)e Fx(S)t(P)t Fy(os)p FB(\()p Fy(s)1287 1687 y Fr(0)1299 1705 y FB(\))h(whic)o(h)g (satis\014es)h(the)f(equation)0 1774 y Fx(})p FB(\()p Fy(s)p FB(\))c(=)h Fx(r)207 1781 y Fw(D)239 1774 y FB(\()p Fx(})p FB(\()p Fy(s)339 1756 y Fr(0)350 1774 y FB(\)\),)g(where)g Fy(D)h FB(denotes)f(the)g(sub)q(deriv)m(ation)g Fy(s)1259 1747 y Fw(im)c Fr(\003)1264 1774 y Fx(!)38 b Fy(s)1375 1756 y Fr(0)1387 1774 y FB(.)31 b(Since)19 b Fy(r)q(oot)p FB(\()p Fy(u)p FB(\))h Fx(2)g(F)1830 1781 y Fw(d)1870 1774 y FB(and)0 1843 y Fy(u)60 1817 y Fw(im)d Fr(\003)66 1843 y Fx(!)38 b Fy(s)p Fx(#)p FB(,)17 b(Lemma)f(6.1.23)i(yields)f Fx(r)720 1850 y Fw(D)750 1841 y Fj(0)763 1843 y FB(\()p Fm(\003)p FB(\))f(=)h Fm(\003)o FB(,)h(where)g Fy(D)1145 1825 y Fr(0)1174 1843 y FB(:)e Fy(u)1264 1817 y Fw(im)g Fr(\003)1270 1843 y Fx(!)37 b Fy(s)p Fx(#)p FB(.)26 b(F)l(rom)17 b(this,)g(it)h(follo)o(ws)f(b)o(y)0 1904 y(rep)q(eated)i(application)g (of)g(Lemma)e(6.1.19)j(that)f Fx(r)974 1911 y Fw(D)1006 1904 y FB(\()p Fy(q)r FB(\))f(=)h Fy(q)h FB(=)e Fx(})p FB(\()p Fy(s)p FB(\).)30 b(Since)18 b Fy(q)j FB(satis\014es)e(the)g(ab) q(o)o(v)o(e)0 1964 y(equation,)d(w)o(e)g(infer)f Fx(})p FB(\()p Fy(s)478 1946 y Fr(0)489 1964 y FB(\))f(=)g Fy(q)r FB(.)21 b Fq(2)0 2091 y Fz(Lemma)16 b(6.1.29)23 b FB(Let)c Fx(R)g FB(b)q(e)h(complete,)c Fy(s)j Fx(2)f(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))18 b(with)h Fy(r)q(oot)p FB(\()p Fy(s)p Fx(#)q FB(\))f Fx(2)h(F)1495 2098 y Fw(d)1515 2091 y FB(,)g Fy(d)g Fx(2)f(f)p FB(1)p Fy(;)8 b FB(2)p Fx(g)p FB(,)20 b(and)g(let)0 2160 y Fy(s)14 b FB(=)g Fy(C)t FB([)p Fy(u)p FB(])g(with)h Fy(r)q(oot)p FB(\()p Fy(u)p FB(\))f Fx(2)g(F)563 2167 y Fw(d)583 2160 y FB(,)i(where)f Fy(C)t FB([)g(])p Fx(j)849 2167 y Fw(q)881 2160 y FB(=)f Fq(2)p FB(.)21 b(No)o(w)15 b(if)g Fy(s)f FB(=)g Fy(C)t FB([)p Fy(u)p FB(])1374 2134 y Fw(im)i Fr(\003)1379 2160 y Fx(!)38 b Fy(s)1490 2142 y Fr(0)1516 2160 y FB(=)13 b Fy(C)t FB([)p Fy(s)p Fx(#)o FB(])1713 2134 y Fw(im)k Fr(\003)1719 2160 y Fx(!)38 b Fy(s)p Fx(#)15 b FB(and)0 2221 y Fx(})p FB(\()p Fy(s)81 2203 y Fr(0)92 2221 y FB(\))f(=)g Fy(q)r FB(,)h(then)h Fx(})p FB(\()p Fy(s)p FB(\))e(=)f Fy(q)r FB(.)0 2311 y Fz(Pro)r(of:)21 b Fx(})p FB(\()p Fy(s)p FB(\))14 b(=)f Fx(r)368 2318 y Fw(D)400 2311 y FB(\()p Fx(})p FB(\()p Fy(s)500 2293 y Fr(0)511 2311 y FB(\)\))h(=)g Fx(r)645 2318 y Fw(D)677 2311 y FB(\()p Fy(q)r FB(\))f(=)h Fy(q)r FB(,)h(where)h Fy(D)g FB(:)d Fy(s)1136 2284 y Fw(im)k Fr(\003)1142 2311 y Fx(!)38 b Fy(s)1253 2293 y Fr(0)1264 2311 y FB(.)22 b Fq(2)0 2438 y Fz(Lemma)16 b(6.1.30)23 b FB(Let)c Fx(R)g FB(b)q(e)h(complete,)c Fy(s)j Fx(2)f(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))18 b(with)h Fy(r)q(oot)p FB(\()p Fy(s)p Fx(#)q FB(\))f Fx(2)h(F)1495 2445 y Fw(d)1515 2438 y FB(,)g Fy(d)g Fx(2)f(f)p FB(1)p Fy(;)8 b FB(2)p Fx(g)p FB(,)20 b(and)g(let)0 2507 y Fy(s)g FB(=)g Fy(C)t FB([)p Fy(u)p FB(])e(with)i Fy(r)q(oot)p FB(\()p Fy(u)p FB(\))g Fx(2)g(F)597 2509 y Fs(\026)591 2518 y Fw(d)611 2507 y FB(,)h(where)e Fy(C)t FB([)g(])p Fx(j)890 2514 y Fw(q)928 2507 y FB(=)h Fq(2)p FB(.)33 b(No)o(w)19 b(if)h Fy(s)f FB(=)h Fy(C)t FB([)p Fy(u)p FB(])1459 2481 y Fw(im)d Fr(\003)1465 2507 y Fx(!)38 b Fy(t)19 b FB(=)h Fy(C)t FB([)p Fy(s)p Fx(#)o FB(])1794 2481 y Fw(im)c Fr(\003)1800 2507 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FB(It)i(is)g(simple)e(to)i(sho)o(w)h Fx(r)476 2873 y Fw(s)492 2864 y Fj(0)503 2873 y Fr(!)538 2864 y Fj(\003)557 2873 y Fw(t)571 2866 y FB(\()p Fy(q)r FB(\))d(=)f Fy(q)r(:q)760 2848 y Fr(0)770 2866 y FB(.)22 b(It)15 b(follo)o(ws)i Fx(r)1061 2873 y Fw(s)p Fr(!)1112 2864 y Fj(\003)1130 2873 y Fw(s)1146 2864 y Fj(0)1159 2866 y FB(\()p Fy(q)r(:q)1240 2848 y Fr(0)1250 2866 y FB(\))d(=)g Fy(q)r(:q)1397 2848 y Fr(0)1423 2866 y FB(b)o(y)i(Lemma)e (6.1.29.)22 b Fq(2)p eop %%Page: 102 110 102 109 bop -59 -39 a FB(102)1132 b Fv(CHAPTER)16 b(6.)38 b(COMPLETENESS)-59 94 y Fz(Prop)r(osition)18 b(6.1.31)23 b FB(Let)d Fx(R)h FB(b)q(e)f(complete)e(and)i(let)f Fy(s)i Fx(2)f(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))19 b(with)h Fy(r)q(oot)p FB(\()p Fy(s)p Fx(#)q FB(\))g Fx(2)g(F)1712 101 y Fw(d)1732 94 y FB(.)33 b(Then)-59 154 y Fy(s)14 b Fx(!)28 136 y Fr(\003)61 154 y Fy(s)p Fx(j)98 161 y Fr(\005)p Fs(\()p Fw(s)p Fs(\))161 154 y FB(,)i(i.e.,)e Fy(s)j FB(reduces)e(to)i(its)f(essen)o(tial)f(subterm.)-59 235 y Fz(Pro)r(of:)28 b FB(Let)20 b Fy(q)i FB(=)e Fx(})p FB(\()p Fy(s)p FB(\))g(and)g Fy(s)p Fx(j)569 242 y Fw(q)608 235 y FB(=)g Fy(u)p FB(.)32 b(W)l(e)20 b(ma)o(y)e(write)h Fy(s)i FB(=)f Fy(C)t FB([)p Fy(u)p FB(])e(with)i(an)g(appropriate)h (con)o(text)-59 305 y Fy(C)t FB([)c(].)24 b(Clearly)l(,)17 b Fy(s)f FB(=)g Fy(C)t FB([)p Fy(u)p FB(])463 279 y Fw(im)g Fr(\003)468 305 y Fx(!)38 b Fy(s)579 287 y Fr(0)607 305 y FB(=)16 b Fy(C)t FB([)p Fy(s)p Fx(#)o FB(])807 279 y Fw(im)h Fr(\003)813 305 y Fx(!)37 b Fy(s)p Fx(#)p FB(,)18 b(By)f(Lemma)e(6.1.28,)j Fx(})p FB(\()p Fy(s)1470 287 y Fr(0)1482 305 y FB(\))e(=)g Fy(q)r FB(.)24 b(Consider)18 b(an)-59 375 y(arbitrary)h(innermost)f(deriv)m(ation)h Fy(D)i FB(:)e Fy(s)731 357 y Fr(0)761 375 y FB(=)g Fy(t)836 382 y Fs(1)875 348 y Fw(im)872 375 y Fx(!)d Fy(t)956 382 y Fs(2)995 348 y Fw(im)992 375 y Fx(!)24 b Fy(:)8 b(:)g(:)1151 348 y Fw(im)1148 375 y Fx(!)16 b Fy(t)1232 382 y Fw(l)1264 348 y(im)1261 375 y Fx(!)g Fy(t)1345 382 y Fw(l)p Fs(+1)1422 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Fy(t)549 816 y Fw(j)583 809 y FB(=)f Fy(t)655 816 y Fw(j)673 809 y FB([)p Fy(q)709 816 y Fw(j)743 809 y Fx( )g Fy(s)p Fx(#)p FB(])h(for)h(ev)o(ery)e Fy(j)j Fx(2)e(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(l)13 b FB(+)f(1)p Fx(g)p FB(.)26 b(Let)18 b Fy(t)1615 816 y Fw(j)1646 809 y Fx(\000)-8 b(!)1644 816 y Fg(p;l)p Fj(!)p Fg(r)1741 809 y Fy(t)1759 816 y Fw(j)r Fs(+1)1840 809 y FB(b)q(e)-59 869 y(innermost.)19 b(Note)14 b(that)h(the)f(equalit)o(y)f Fy(t)691 876 y Fw(j)709 869 y FB([)p Fy(q)745 876 y Fw(j)776 869 y Fx( )h Fy(s)p Fx(#)o FB(])g Fx(\000)-9 b(!)912 876 y Fg(p;l)p Fj(!)p Fg(r)1009 869 y Fy(t)1027 876 y Fw(j)r Fs(+1)1090 869 y FB([)p Fy(o)14 b Fx( )g Fy(s)p Fx(#)i(j)g Fy(o)e Fx(2)g FB(\001)1424 876 y Fw(t)1437 881 y Fg(j)1453 876 y Fr(!)p Fw(t)1501 881 y Fg(j)q Fh(+1)1558 869 y FB(\()p Fy(q)1599 876 y Fw(j)1617 869 y FB(\)])g(holds)h(true.)-59 929 y(W)l(e)i(pro)q(ceed)g(b)o(y)g(case)g(analysis.)25 b(W)l(e)17 b(ha)o(v)o(e)g(to)g(consider)g(the)g(cases)h(\(i\))f Fy(p)f Fx(?)f Fy(q)1430 936 y Fw(j)1466 929 y FB(or)i(\(ii\))g Fy(p)f(<)f(q)1724 936 y Fw(j)1760 929 y FB(\(other)-59 989 y(cases)h(cannot)h(o)q(ccur)g(since)e Fy(t)492 996 y Fw(j)510 989 y Fx(j)524 996 y Fw(q)540 1001 y Fg(j)572 989 y FB(=)f Fy(s)p Fx(#)f(2)h Fy(N)5 b(F)i FB(\()p Fx(F)e Fy(;)j Fx(R)p FB(\)\).)-59 1049 y(\(i\))16 b Fy(p)e Fx(?)g Fy(q)122 1056 y Fw(j)139 1049 y FB(:)22 b(Then)16 b Fy(q)324 1056 y Fw(j)356 1049 y FB(=)e Fy(q)430 1056 y Fw(j)r Fs(+1)509 1049 y FB(and)j(furthermore)d Fy(t)891 1056 y Fw(j)909 1049 y FB([)p Fy(q)945 1056 y Fw(j)976 1049 y Fx( )g Fy(u)p FB(])f Fx(\000)-8 b(!)1093 1056 y Fg(p;l)p Fj(!)p Fg(r)1190 1049 y Fy(t)1208 1056 y Fw(j)r Fs(+1)1271 1049 y FB([)p Fy(q)1307 1056 y Fw(j)r Fs(+1)1383 1049 y Fx( )14 b Fy(u)p FB(])o(.)-59 1110 y(\(ii\))h(It)h(follo)o(ws)g(from) f(the)h(ab)q(o)o(v)o(e)h(considerations)f(and)h Fy(q)1007 1117 y Fw(j)r Fs(+1)1084 1110 y Fx(2)d FB(\001)1172 1117 y Fw(t)1185 1122 y Fg(j)1200 1117 y Fr(!)p Fw(t)1248 1122 y Fg(j)q Fh(+1)1305 1110 y FB(\()p Fy(q)1346 1117 y Fw(j)1364 1110 y FB(\))i(that)203 1210 y Fy(t)221 1217 y Fw(j)239 1210 y FB([)p Fy(q)275 1217 y Fw(j)306 1210 y Fx( )e Fy(u)p FB(])40 b Fx(\000)-8 b(!)450 1217 y Fg(p;l)p Fj(!)p Fg(r)575 1210 y Fy(t)593 1217 y Fw(j)r Fs(+1)656 1210 y FB([)p Fy(o)14 b Fx( )f Fy(u)j Fx(j)g Fy(o)f Fx(2)f FB(\001)970 1217 y Fw(t)983 1222 y Fg(j)998 1217 y Fr(!)p Fw(t)1046 1222 y Fg(j)q Fh(+1)1103 1210 y FB(\()p Fy(q)1144 1217 y Fw(j)1162 1210 y FB(\)])474 1270 y(=)63 b Fy(t)593 1277 y Fw(j)r Fs(+1)656 1270 y FB([)p Fy(q)692 1277 y Fw(j)r Fs(+1)768 1270 y Fx( )14 b Fy(u)p FB(][)p Fy(o)f Fx( )h Fy(u)i Fx(j)g Fy(o)e Fx(2)g FB(\001)1187 1277 y Fw(t)1200 1282 y Fg(j)1216 1277 y Fr(!)p Fw(t)1264 1282 y Fg(j)q Fh(+1)1320 1270 y FB(\()p Fy(q)1361 1277 y Fw(j)1379 1270 y FB(\))p Fy(;)8 b(o)14 b Fx(6)p FB(=)g Fy(q)1531 1277 y Fw(j)r Fs(+1)1594 1270 y FB(])458 1330 y Fx(!)508 1312 y Fr(\003)575 1330 y Fy(t)593 1337 y Fw(j)r Fs(+1)656 1330 y FB([)p Fy(q)692 1337 y Fw(j)r Fs(+1)768 1330 y Fx( )g Fy(u)p FB(][)p Fy(o)f Fx( )h Fy(s)p Fx(#)i(j)g Fy(o)e Fx(2)g FB(\001)1207 1337 y Fw(t)1220 1342 y Fg(j)1236 1337 y Fr(!)p Fw(t)1284 1342 y Fg(j)q Fh(+1)1341 1330 y FB(\()p Fy(q)1382 1337 y Fw(j)1399 1330 y FB(\))p Fy(;)8 b(o)15 b Fx(6)p FB(=)e Fy(q)1551 1337 y Fw(j)r Fs(+1)1614 1330 y FB(])474 1390 y(=)63 b Fy(t)593 1397 y Fw(j)r Fs(+1)656 1390 y FB([)p Fy(q)692 1397 y Fw(j)r Fs(+1)768 1390 y Fx( )14 b Fy(u)p FB(])p Fy(:)-59 1471 y Fq(2)14 1590 y FB(W)l(e)i(ha)o(v)o(e)g(seen)g(that)h (if)f Fy(s)510 1564 y Fw(im)507 1590 y Fx(!)g Fy(t)p FB(,)g(then)g Fx(})p FB(\()p Fy(s)p FB(\))e(=)g Fx(r)928 1597 y Fw(s)p Fr(!)p Fw(t)994 1590 y FB(\()p Fx(})p FB(\()p Fy(t)p FB(\))o(\).)22 b(The)17 b(follo)o(wing)f(example)e(sho)o(ws)j (that)-59 1651 y(this)c(equalit)o(y)e(do)q(es)i(not)g(hold)g(if)f(the)h (rewrite)f(step)g(under)h(consideration)g(is)f(not)h(innermost.)19 b(Ho)o(w)o(ev)o(er,)-59 1711 y(it)12 b(will)f(b)q(e)i(sho)o(wn)g(that)g (if)f Fy(s)h Fx(!)h Fy(t)p FB(,)f(then)f(the)g(essen)o(tial)f(subterm)h (of)g Fy(s)h FB(reduces)f(to)g(the)h(essen)o(tial)e(subterm)-59 1771 y(of)16 b Fy(t)p FB(.)-59 1890 y Fz(Example)g(6.1.32)24 b FB(Let)c Fx(R)474 1897 y Fs(1)514 1890 y FB(=)h Fx(f)p Fy(F)7 b FB(\()p Fy(x)p FB(\))20 b Fx(!)h Fy(x)p Fx(g)f FB(and)h Fx(R)1008 1897 y Fs(2)1048 1890 y FB(=)g Fx(f)p Fy(g)r FB(\()p Fy(x)p FB(\))g Fx(!)g Fy(x)p Fx(g)p FB(.)33 b(F)l(urthermore,)19 b(consider)-59 1951 y Fy(s)h FB(=)g Fy(F)7 b FB(\()p Fy(g)r FB(\()p Fy(A)p FB(\)\))20 b Fx(!)g Fy(t)g FB(=)g Fy(g)r FB(\()p Fy(A)p FB(\))g Fx(!)g Fy(A)g FB(=)g Fy(s)p Fx(#)p FB(.)32 b(Note)20 b(that)g Fy(r)q(oot)p FB(\()p Fy(s)p Fx(#)q FB(\))g Fx(2)h(F)1324 1958 y Fs(1)1363 1951 y FB(and)g Fx(})p FB(\()p Fy(t)p FB(\))f(=)g(1.)33 b(W)l(e)20 b(ha)o(v)o(e)-59 2011 y Fx(r)-29 2018 y Fw(s)p Fr(!)p Fw(t)37 2011 y FB(\()p Fx(})p FB(\()p Fy(t)p FB(\))o(\))14 b(=)g Fx(r)265 2018 y Fw(s)p Fr(!)p Fw(t)331 2011 y FB(\(1\))g(=)g(1)p Fy(:)p FB(1)j(whereas)f Fx(})p FB(\()p Fy(s)p FB(\))e(=)f Fm(\003)k FB(according)f(to)h(Lemma)d(6.1.23.)-59 2130 y Fz(Theorem)i(6.1.33)24 b FB(Let)17 b Fx(R)g FB(b)q(e)g(complete,)d Fy(s)h Fx(2)g(T)e FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))16 b(with)g Fy(r)q(oot)p FB(\()p Fy(s)p Fx(#)q FB(\))f Fx(2)g(F)1447 2137 y Fw(d)1467 2130 y FB(,)h(and)i Fy(s)c Fx(\000)-9 b(!)1627 2137 y Fg(p;l)p Fj(!)p Fg(r)1724 2130 y Fy(t)p FB(.)23 b(Then)-59 2191 y Fy(s)p Fx(j)-22 2198 y Fr(\005)p Fs(\()p Fw(s)p Fs(\))55 2191 y Fx(!)105 2173 y Fr(\003)138 2191 y Fy(t)p Fx(j)170 2198 y Fr(\005)p Fs(\()p Fw(t)p Fs(\))230 2191 y FB(.)e(That)c(is,)f(the)g(essen)o(tial)f(subterm)g(of) h Fy(t)g FB(is)g(a)h(reduct)f(of)g(the)g(essen)o(tial)f(subterm)g(of)i Fy(s)p FB(.)-59 2272 y Fz(Pro)r(of:)k FB(Case)c(\(i\):)k Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))14 b Fx(2)g(F)568 2279 y Fw(d)604 2272 y FB(and)j Fy(r)q(oot)p FB(\()p Fy(t)p FB(\))e Fx(2)f(F)944 2279 y Fw(d)965 2272 y FB(.)-59 2332 y(By)h(Lemma)g(6.1.23,)h Fx(})p FB(\()p Fy(s)p FB(\))e(=)f Fm(\003)j FB(and)h Fx(})p FB(\()p Fy(t)p FB(\))c(=)h Fm(\003)p FB(.)21 b(Hence)15 b(the)h(assertion)h(follo)o(ws)f(from)f Fy(s)f Fx(!)g Fy(t)p FB(.)-59 2392 y(Case)j(\(ii\))e Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))g Fx(2)f(F)391 2394 y Fs(\026)385 2403 y Fw(d)421 2392 y FB(and)j Fy(r)q(oot)p FB(\()p Fy(t)p FB(\))d Fx(2)g(F)761 2399 y Fw(d)781 2392 y FB(.)-59 2452 y(Then)19 b Fy(s)13 b FB(=)h Fy(C)t FB([)-8 b([)o Fy(s)240 2459 y Fs(1)259 2452 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)392 2459 y Fw(j)r Fr(\000)p Fs(1)455 2452 y Fy(;)g(s)500 2459 y Fw(j)518 2452 y Fy(;)g(s)563 2459 y Fw(j)r Fs(+1)626 2452 y Fy(;)g(:)g(:)g(:)f(;)h(s)758 2459 y Fw(n)782 2452 y FB(])-8 b(])12 b Fx(\000)-8 b(!)824 2459 y Fg(l)p Fj(!)p Fg(r)909 2452 y Fy(s)932 2459 y Fw(j)964 2452 y FB(=)14 b Fy(t)p FB(.)28 b(Let)19 b Fy(s)p Fx(j)1203 2459 y Fw(q)1239 2452 y FB(=)f Fy(s)1318 2459 y Fw(j)1354 2452 y FB(=)g Fy(t)p FB(.)28 b(W)l(e)18 b(sho)o(w)i Fy(q)f FB(=)f Fx(})p FB(\()p Fy(s)p FB(\))o(.)-59 2523 y(Clearly)l(,)e Fy(s)g FB(=)f Fy(C)t FB([)-8 b([)n Fy(s)293 2530 y Fs(1)313 2523 y Fy(;)8 b(:)g(:)g(:)f(;)h(s)445 2530 y Fw(j)r Fr(\000)p Fs(1)509 2523 y Fy(;)g(s)554 2530 y Fw(j)572 2523 y Fy(;)g(s)617 2530 y Fw(j)r Fs(+1)680 2523 y Fy(;)g(:)g(:)g(:)f(;)h(s)812 2530 y Fw(n)835 2523 y FB(])-8 b(])887 2496 y Fw(im)16 b Fr(\003)892 2523 y Fx(!)38 b Fy(C)t FB([)-8 b([)n Fy(s)1060 2530 y Fs(1)1080 2523 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)1213 2530 y Fw(j)r Fr(\000)p Fs(1)1276 2523 y Fy(;)g(s)1321 2530 y Fw(j)1339 2523 y Fx(#)p Fy(;)g(s)1409 2530 y Fw(j)r Fs(+1)1472 2523 y Fy(;)g(:)g(:)g(:)f(;)h(s)1604 2530 y Fw(n)1628 2523 y FB(])-8 b(])14 b(=)h Fy(s)1738 2505 y Fr(0)1750 2523 y FB(.)24 b(Note)-59 2583 y(that)c Fy(s)73 2590 y Fw(j)91 2583 y Fx(#)g FB(=)g Fy(t)p Fx(#)f FB(=)h Fy(s)p Fx(#)p FB(.)32 b(According)19 b(to)h(Lemma)d(6.1.29,)k(it)f (su\016ces)f(to)h(sho)o(w)g(that)h Fx(})p FB(\()p Fy(s)1642 2565 y Fr(0)1653 2583 y FB(\))f(=)f Fy(q)r FB(.)32 b(Let)-57 2643 y(^)-26 b Fy(s)18 b FB(=)g Fy(C)t FB([)p Fy(s)114 2650 y Fs(1)132 2643 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)265 2650 y Fw(j)r Fr(\000)p Fs(1)328 2643 y Fy(;)g(x;)g(s)423 2650 y Fw(j)r Fs(+1)486 2643 y Fy(;)g(:)g(:)g(:)f(;)h(s)618 2650 y Fw(n)641 2643 y FB(].)28 b(Since)18 b Fx(R)h FB(is)f (left-linear,)g(w)o(e)g(also)h(ha)o(v)o(e)h(^)-26 b Fy(s)14 b Fx(\000)-9 b(!)1512 2650 y Fg(l)p Fj(!)p Fg(r)1598 2643 y Fy(x)18 b FB(\(cf.)g(Lemma)-59 2703 y(3.3.16\).)24 b(Th)o(us,)17 b Fy(x)g FB(is)g(the)g(unique)f(normal)g(form)g(of)k(^) -26 b Fy(s)17 b FB(b)q(ecause)g Fx(R)h FB(is)e(complete.)22 b(F)l(or)17 b(an)o(y)g(deriv)m(ation)-59 2773 y Fy(D)-18 2755 y Fr(0)8 2773 y FB(:)32 b(^)-26 b Fy(s)107 2746 y Fw(im)17 b Fr(\003)113 2773 y Fx(!)38 b Fy(x)p FB(,)15 b(it)h(follo)o(ws)g(that)h Fx(r)603 2780 y Fw(D)633 2771 y Fj(0)646 2773 y FB(\()p Fm(\003)p FB(\))c(=)h Fy(q)k FB(b)o(y)e(Prop)q(osition)h(6.1.10.)22 b(Consider)-59 2878 y Fy(s)-36 2858 y Fr(0)-11 2878 y FB(=)14 b Fy(C)t FB([)-8 b([)n Fy(s)121 2885 y Fs(1)141 2878 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)274 2885 y Fw(j)r Fr(\000)p Fs(1)337 2878 y Fy(;)g(s)382 2885 y Fw(j)400 2878 y Fx(#)p Fy(;)g(s)470 2885 y Fw(j)r Fs(+1)533 2878 y Fy(;)g(:)g(:)g(:)f(;)h(s)665 2885 y Fw(n)689 2878 y FB(])-8 b(])740 2852 y Fw(im)16 b Fr(\003)745 2878 y Fx(!)38 b Fy(s)856 2858 y Fr(00)891 2878 y FB(=)14 b Fy(C)t FB([)p Fy(s)1019 2885 y Fs(1)1038 2878 y Fx(#)p Fy(;)8 b(:)g(:)g(:)f(;)h(s)1195 2885 y Fw(j)r Fr(\000)p Fs(1)1258 2878 y Fx(#)p Fy(;)g(s)1328 2885 y Fw(j)1346 2878 y Fx(#)p Fy(;)g(s)1416 2885 y Fw(j)r Fs(+1)1479 2878 y Fx(#)p Fy(;)g(:)g(:)g(:)g(;)g(s)1637 2885 y Fw(n)1660 2878 y Fx(#)p FB(])1731 2852 y Fw(im)16 b Fr(\003)1736 2878 y Fx(!)38 b Fy(s)1847 2885 y Fw(j)1866 2878 y Fx(#)o Fy(:)p eop %%Page: 103 111 103 110 bop 0 -39 a Fv(6.1.)38 b(DISJOINT)15 b(SYSTEMS)1281 b FB(103)0 94 y(W)l(e)20 b(ma)o(y)e(write)h Fy(s)348 75 y Fr(00)389 94 y FB(=)458 81 y(^)447 94 y Fy(C)t FB([)-8 b([)o Fy(t)523 101 y Fs(1)542 94 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)669 101 y Fw(m)702 94 y FB(])-8 b(])o(,)20 b(where)g Fy(t)918 101 y Fw(i)951 94 y FB(=)g Fy(s)1032 101 y Fw(j)1051 94 y Fx(#)f FB(for)h(some)f Fy(i)h Fx(2)g(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(m)p Fx(g)p FB(.)31 b(Rep)q(eated)20 b(ap-)0 154 y(plication)d(of)h(Lemma)e(6.1.18)i(yields)f Fx(})p FB(\()p Fy(s)797 136 y Fr(00)818 154 y FB(\))f(=)h Fy(q)r FB(.)25 b(It)17 b(follo)o(ws)h Fx(})p FB(\()p Fy(s)1268 136 y Fr(0)1279 154 y FB(\))f(=)f Fy(q)j FB(b)q(ecause)f(in)g (the)f(deriv)m(ation)0 223 y Fy(s)23 205 y Fr(0)67 196 y Fw(im)g Fr(\003)73 223 y Fx(!)38 b Fy(s)184 205 y Fr(00)219 223 y FB(all)14 b(rewrite)f(p)q(ositions)j(are)e(disjoin)o(t)g(to)h Fy(q)r FB(.)20 b(Hence,)13 b(w)o(e)h(ha)o(v)o(e)f Fy(s)p Fx(j)1393 231 y Fr(\005)p Fs(\()p Fw(s)p Fs(\))1470 223 y FB(=)h Fy(t)g Fx(!)1604 205 y Fr(\003)1637 223 y Fy(t)g FB(=)f Fy(t)p Fx(j)1752 230 y Fs(\003)1792 223 y FB(=)h Fy(t)p Fx(j)1876 231 y Fr(\005)p Fs(\()p Fw(t)p Fs(\))1935 223 y FB(.)0 283 y(Case)j(\(iii\))e Fy(r)q(oot)p FB(\()p Fy(s)p FB(\))f Fx(2)g(F)462 290 y Fw(d)499 283 y FB(and)i Fy(r)q(oot)p FB(\()p Fy(t)p 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FB(are)c(the)f(essen)o(tial)f(subterms) h(of)h Fy(s)f FB(and)h Fy(t)p FB(,)f(resp)q(ectiv)o(ely)l(.)19 b(W)l(e)14 b(kno)o(w)h(from)e(Theorem)g(6.1.33)i(that)122 1816 y Fy(u)21 b Fx(!)221 1798 y Fr(\003)261 1816 y Fy(v)r FB(.)34 b(Rep)q(eated)20 b(application)g(of)h(the)g(induction)f(h)o(yp) q(othesis)g(yields)g(\010)1602 1823 y Fw(d)1622 1816 y FB(\()p Fy(u)p FB(\))h Fx(!)1759 1798 y Fr(\003)1759 1829 y(R)1789 1835 y Fg(d)1830 1816 y FB(\010)1865 1823 y Fw(d)1886 1816 y FB(\()p Fy(v)r FB(\))122 1876 y(b)q(ecause)16 b Fy(r)q(ank)r FB(\()p Fy(u)p FB(\))e Fy(<)g(r)q(ank)r FB(\()p Fy(s)p FB(\))g(=)f Fy(l)q FB(.)73 1979 y Fx(\017)24 b FB(If)d Fy(r)q(oot)p FB(\()p Fy(s)p Fx(#)q FB(\))i(=)g Fy(r)q(oot)p FB(\()p Fy(t)p Fx(#)p FB(\))g Fx(62)g(F)722 1986 y Fw(d)742 1979 y FB(,)g(then)e(\010)930 1986 y Fw(d)950 1979 y FB(\()p Fy(s)p FB(\))i(=)g Fy(pil)q(e)p FB(\(\010)1229 1986 y Fw(d)1249 1979 y FB(\()p Fy(t)1286 1986 y Fs(1)1305 1979 y FB(\))p Fy(;)8 b(:)g(:)g(:)g(;)g FB(\010)1469 1986 y 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Fy(;)8 b(x)1245 730 y Fs(1)1264 723 y Fy(;)g(x;)g(y)r(;)g(z)r FB(\))-59 859 y(Moreo)o(v)o(er,)15 b(consider)g(the)h(TRS)h(consisting)f(of)h Fx(F)874 866 y Fs(2)907 859 y FB(=)d Fx(f)p Fy(f)s(;)8 b(h)p Fx(g)16 b FB(and)424 1055 y Fx(R)466 1062 y Fs(2)500 1055 y FB(=)552 955 y Ft(8)552 993 y(>)552 1005 y(<)552 1080 y(>)552 1092 y(:)609 994 y Fy(f)5 b FB(\()p Fy(x;)j(y)r FB(\))42 b Fx(!)f Fy(h)p FB(\()p Fy(x;)8 b(y)r FB(\))609 1055 y Fy(f)d FB(\()p Fy(x;)j(y)r FB(\))42 b Fx(!)f Fy(h)p FB(\()p Fy(y)r(;)8 b(x)p FB(\))609 1115 y Fy(h)p FB(\()p Fy(x;)g(x)p FB(\))41 b Fx(!)g Fy(x)-59 1251 y FB(Both)16 b(TRSs)h(are)f(v)m(ariable-preserving)f(and)i(\()p Fx(F)848 1258 y Fs(2)868 1251 y Fy(;)8 b Fx(R)932 1258 y Fs(2)952 1251 y FB(\))16 b(is)g(clearly)e(terminating.)20 b(W)l(e)c(pro)o(v)o(e) f(next)g(that)-59 1311 y(the)h(TRS)g(\()p Fx(F)199 1318 y Fs(1)219 1311 y Fy(;)8 b Fx(R)283 1318 y Fs(1)303 1311 y FB(\))16 b(is)g(terminating)e(\(Prop)q(osition)k(6.1.44\).)-59 1500 y Fz(Lemma)e(6.1.43)23 b FB(F)l(or)16 b(all)g Fy(s)485 1507 y Fs(1)505 1500 y Fy(;)8 b(s)550 1507 y Fs(2)569 1500 y Fy(;)g(s)614 1507 y Fs(3)634 1500 y Fy(;)g(w)691 1507 y Fs(1)710 1500 y Fy(;)g(w)767 1507 y Fs(2)787 1500 y Fy(;)g(w)844 1507 y Fs(3)877 1500 y Fx(2)14 b(T)f FB(\()p Fx(F)1019 1507 y Fs(1)1038 1500 y Fy(;)8 b Fx(V)t FB(\))16 b(w)o(e)g(ha)o(v)o(e)144 1637 y Fy(E)180 1644 y Fs(1)200 1637 y FB(\()p Fy(H)259 1644 y Fs(1)279 1637 y Fy(;)8 b(H)341 1644 y Fs(2)361 1637 y Fy(;)g(s)406 1644 y Fs(1)425 1637 y Fy(;)g(s)470 1644 y Fs(2)490 1637 y Fy(;)g(s)535 1644 y Fs(3)554 1637 y FB(\))14 b Fx(!)637 1644 y Fr(R)667 1649 y Fh(1)700 1637 y Fy(E)736 1644 y Fs(2)756 1637 y FB(\()p Fy(s)798 1644 y Fs(1)818 1637 y Fy(;)8 b(s)863 1644 y Fs(1)882 1637 y Fy(;)g(s)927 1644 y Fs(2)947 1637 y Fy(;)g(s)992 1644 y Fs(3)1011 1637 y Fy(;)g(s)1056 1644 y Fs(3)1076 1637 y FB(\))14 b Fx(6!)1159 1616 y Fs(+)1159 1649 y Fr(R)1189 1654 y Fh(1)1222 1637 y Fy(E)1258 1644 y Fs(1)1278 1637 y FB(\()p Fy(H)1337 1644 y Fs(1)1357 1637 y Fy(;)8 b(H)1419 1644 y Fs(2)1439 1637 y Fy(;)g(w)1496 1644 y Fs(1)1515 1637 y Fy(;)g(w)1572 1644 y Fs(2)1592 1637 y Fy(;)g(w)1649 1644 y Fs(3)1668 1637 y FB(\))-59 1794 y Fz(Pro)r(of:)30 b FB(W)l(e)20 b(pro)o(v)o(e)g(the)h(lemm)o(a)d (b)o(y)i(con)o(tradiction.)34 b(Therefore,)21 b(assume)f(that)h(there)f (exist)g(terms)-59 1854 y Fy(s)-36 1861 y Fs(1)-16 1854 y Fy(;)8 b(s)29 1861 y Fs(2)48 1854 y Fy(;)g(s)93 1861 y Fs(3)113 1854 y FB(,)p Fy(w)162 1861 y Fs(1)181 1854 y Fy(;)g(w)238 1861 y Fs(2)257 1854 y Fy(;)g(w)314 1861 y Fs(3)348 1854 y Fx(2)14 b(T)e FB(\()p Fx(F)489 1861 y Fs(1)509 1854 y Fy(;)c Fx(V)t FB(\))16 b(suc)o(h)g(that)144 1991 y Fy(E)180 1998 y Fs(1)200 1991 y FB(\()p Fy(H)259 1998 y Fs(1)279 1991 y Fy(;)8 b(H)341 1998 y Fs(2)361 1991 y Fy(;)g(s)406 1998 y Fs(1)425 1991 y Fy(;)g(s)470 1998 y Fs(2)490 1991 y Fy(;)g(s)535 1998 y Fs(3)554 1991 y FB(\))14 b Fx(!)637 1998 y Fr(R)667 2003 y Fh(1)700 1991 y Fy(E)736 1998 y Fs(2)756 1991 y FB(\()p Fy(s)798 1998 y Fs(1)818 1991 y Fy(;)8 b(s)863 1998 y Fs(1)882 1991 y Fy(;)g(s)927 1998 y Fs(2)947 1991 y Fy(;)g(s)992 1998 y Fs(3)1011 1991 y Fy(;)g(s)1056 1998 y Fs(3)1076 1991 y FB(\))14 b Fx(!)1159 1971 y Fs(+)1159 2004 y Fr(R)1189 2009 y Fh(1)1222 1991 y Fy(E)1258 1998 y Fs(1)1278 1991 y FB(\()p Fy(H)1337 1998 y Fs(1)1357 1991 y Fy(;)8 b(H)1419 1998 y Fs(2)1439 1991 y Fy(;)g(w)1496 1998 y Fs(1)1515 1991 y Fy(;)g(w)1572 1998 y Fs(2)1592 1991 y Fy(;)g(w)1649 1998 y Fs(3)1668 1991 y FB(\))-59 2128 y(Then)16 b(there)g(m)o(ust)f(b) q(e)h(terms)f Fy(t)534 2135 y Fs(1)553 2128 y Fy(;)8 b(t)593 2135 y Fs(2)612 2128 y Fy(;)g(t)652 2135 y Fs(3)672 2128 y Fy(;)g(u)722 2135 y Fs(1)741 2128 y Fy(;)g(:)g(:)g(:)f(;)h(u)878 2135 y Fs(7)898 2128 y Fy(;)g(v)944 2135 y Fs(1)963 2128 y Fy(;)g(v)1009 2135 y Fs(2)1028 2128 y Fy(;)g(v)1074 2135 y Fs(3)1093 2128 y Fy(;)g(v)1139 2135 y Fs(4)1172 2128 y Fx(2)14 b(T)f FB(\()p Fx(F)1314 2135 y Fs(1)1333 2128 y Fy(;)8 b Fx(V)t FB(\))16 b(suc)o(h)g(that)435 2263 y Fy(E)471 2270 y Fs(1)491 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1035 y FB(and)i Fy(u)515 1042 y Fs(4)548 1035 y FB(=)e Fy(G)638 1042 y Fs(2)658 1035 y FB(,)i(w)o(e)f(ha)o(v)o(e)g(either)f Fy(v)1032 1042 y Fs(1)1065 1035 y FB(=)g Fy(H)1157 1042 y Fs(1)1193 1035 y FB(or)i Fy(v)1276 1042 y Fs(1)1309 1035 y FB(=)e Fy(H)1401 1042 y Fs(2)1421 1035 y FB(.)21 b(In)15 b(an)o(y)h(case)f Fy(v)1734 1042 y Fs(1)1769 1035 y FB(do)q(es)h(not)0 1095 y(rewrite)f(to)i(b)q(oth)g Fy(H)380 1102 y Fs(1)416 1095 y FB(and)g Fy(H)551 1102 y Fs(2)571 1095 y FB(.)22 b(This)16 b(con)o(tradiction)g(concludes)f(the)h(pro)q(of.)23 b Fq(2)0 1243 y Fz(Prop)r(osition)18 b(6.1.44)23 b FB(The)17 b(TRS)f(\()p Fx(F)745 1250 y Fs(1)765 1243 y Fy(;)8 b Fx(R)829 1250 y Fs(1)848 1243 y FB(\))17 b(is)f(terminating.)0 1324 y Fz(Pro)r(of:)22 b FB(W)l(e)16 b(will)g(pro)o(v)o(e)f(that)i(ev)o (ery)e(reduction)i(sequence)e(starting)i(from)f(some)f(term)g Fy(s)f Fx(2)h(T)e FB(\()p Fx(F)1854 1331 y Fs(1)1873 1324 y Fy(;)8 b Fx(V)t FB(\))0 1384 y(m)o(ust)22 b(b)q(e)h(\014nite.)42 b(T)l(o)24 b(this)f(end)g(w)o(e)g(use)g(induction)g(on)g(the)g (structure)g(of)h Fy(s)p FB(.)42 b(If)22 b Fy(s)i FB(is)f(a)g(constan)o (t)0 1445 y(or)e(a)g(v)m(ariable,)g(then)f(the)g(assertion)h(is)f(ob)o (viously)g(true.)34 b(So)21 b(let)f Fy(s)h FB(=)g Fy(F)7 b FB(\()p Fy(t)1456 1452 y Fs(1)1474 1445 y Fy(;)h(:)g(:)g(:)g(;)g(t) 1602 1452 y Fw(n)1625 1445 y FB(\))20 b(with)h Fy(F)27 b Fx(2)21 b(F)1929 1452 y Fs(1)0 1505 y FB(and)d(terms)e Fy(t)251 1512 y Fw(j)285 1505 y Fx(2)h(T)12 b FB(\()p Fx(F)429 1512 y Fs(1)449 1505 y Fy(;)c Fx(V)t FB(\).)25 b(By)16 b(induction)i(h)o(yp)q(othesis,)f(ev)o(ery)f(reduction)h (sequence)g(starting)h(from)0 1565 y Fy(t)18 1572 y Fw(j)36 1565 y Fy(;)8 b(j)17 b Fx(2)d(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(n)p Fx(g)p FB(,)16 b(is)g(\014nite.)21 b(Supp)q(ose)c(that)g(there)e(is)h (an)h(in\014nite)e(deriv)m(ation)647 1682 y Fy(s)f FB(=)g Fy(s)759 1689 y Fs(1)792 1682 y Fx(!)842 1689 y Fr(R)872 1694 y Fh(1)905 1682 y Fy(s)928 1689 y Fs(2)962 1682 y Fx(!)1012 1689 y Fr(R)1042 1694 y Fh(1)1075 1682 y Fy(s)1098 1689 y Fs(3)1131 1682 y Fx(!)1181 1689 y Fr(R)1211 1694 y Fh(1)1245 1682 y Fy(:)8 b(:)g(:)0 1799 y FB(It)18 b(is)f(easily)g(sho)o(wn)i(that)g(there)e(m)o(ust)g(exist)g(terms)f Fy(u)1030 1806 y Fs(1)1050 1799 y Fy(;)8 b(u)1100 1806 y Fs(2)1119 1799 y Fy(;)g(u)1169 1806 y Fs(3)1188 1799 y Fy(;)g(v)1234 1806 y Fs(1)1254 1799 y Fy(;)g(v)1300 1806 y Fs(2)1319 1799 y Fy(;)g(v)1365 1806 y Fs(3)1402 1799 y FB(and)18 b(indices)f Fy(j;)8 b(k)19 b Fx(2)e Fl(I)-7 b(N)19 b FB(suc)o(h)0 1859 y(that)g Fy(s)131 1866 y Fw(j)168 1859 y FB(=)f Fy(E)260 1866 y Fs(1)280 1859 y FB(\()p Fy(H)339 1866 y Fs(1)359 1859 y Fy(;)8 b(H)421 1866 y Fs(2)441 1859 y Fy(;)g(u)491 1866 y Fs(1)510 1859 y Fy(;)g(u)560 1866 y Fs(2)579 1859 y Fy(;)g(u)629 1866 y Fs(3)649 1859 y FB(\))p Fy(;)g(s)713 1866 y Fw(j)r Fs(+1)794 1859 y FB(=)18 b Fy(E)886 1866 y Fs(2)906 1859 y FB(\()p Fy(u)953 1866 y Fs(1)972 1859 y Fy(;)8 b(u)1022 1866 y Fs(1)1042 1859 y Fy(;)g(u)1092 1866 y Fs(2)1111 1859 y Fy(;)g(u)1161 1866 y Fs(3)1180 1859 y Fy(;)g(u)1230 1866 y Fs(3)1250 1859 y FB(\),)19 b Fy(s)1325 1866 y Fw(k)1364 1859 y FB(=)f Fy(E)1456 1866 y Fs(1)1476 1859 y FB(\()p Fy(H)1535 1866 y Fs(1)1555 1859 y Fy(;)8 b(H)1617 1866 y Fs(2)1637 1859 y Fy(;)g(v)1683 1866 y Fs(1)1702 1859 y Fy(;)g(v)1748 1866 y Fs(2)1767 1859 y Fy(;)g(v)1813 1866 y Fs(3)1832 1859 y FB(\))19 b(and)0 1919 y Fy(s)23 1926 y Fw(j)55 1919 y Fx(!)105 1926 y Fr(R)135 1931 y Fh(1)168 1919 y Fy(s)191 1926 y Fw(j)r Fs(+1)268 1919 y Fx(!)318 1898 y Fs(+)318 1931 y Fr(R)348 1936 y Fh(1)382 1919 y Fy(s)405 1926 y Fw(k)426 1919 y FB(.)i(This,)16 b(ho)o(w)o(ev)o(er,)e(con)o(tradicts)i(Lemma)e(6.1.43.)22 b Fq(2)73 2067 y FB(The)f(com)o(bined)f(system)f Fx(R)k FB(=)f Fx(R)735 2074 y Fs(1)770 2067 y Fx(])14 b(R)859 2074 y Fs(2)900 2067 y FB(of)22 b(the)f(systems)f(de\014ned)h(in)g (Example)e(6.1.42)j(is)f(not)0 2128 y(terminating.)f(W)l(e)c(ha)o(v)o (e)f(the)h(cyclic)e(reduction)i(sequence)466 2240 y Fy(E)502 2247 y 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b(is)g(done)g(via)g(the)g(transformation)g(function)g(\010)g(as)h (follo)o(ws.)41 b(In)-59 274 y(Lemma)12 b(6.2.7,)j(it)e(will)g(b)q(e)i (sho)o(wn)g(that)f(there)g(is)g(an)h(in\014nite)e Fx(R)i FB(deriv)m(ation)f Fy(D)h FB(:)28 b Fy(s)1495 281 y Fs(1)1529 274 y Fx(!)13 b Fy(s)1615 281 y Fs(2)1649 274 y Fx(!)g Fy(s)1735 281 y Fs(3)1769 274 y Fx(!)h Fy(:)8 b(:)g(:)-59 334 y FB(suc)o(h)21 b(that,)h(in)e(ev)o(ery)f(reduction)i(step)g Fy(s)728 341 y Fw(j)768 334 y Fx(!)g Fy(s)862 341 y Fw(j)r Fs(+1)925 334 y FB(,)h(the)f(con)o(tracted)f(redex)g(is)h(a)g(subterm)f (of)h(some)-59 394 y(term)f Fy(u)91 401 y Fw(j)133 394 y FB(=)k Fy(F)7 b FB(\()p Fy(t)271 401 y Fs(1)289 394 y Fy(;)h(:)g(:)g(:)g(;)g(t)417 401 y Fw(n)440 394 y FB(\),)23 b(where)f Fy(F)30 b Fx(2)24 b(F)803 374 y Fw(n)826 394 y FB(,)f Fy(u)891 401 y Fw(j)933 394 y Fx(62)h(T)1017 401 y Fw(c)1034 394 y FB(,)g(and)e Fy(t)1190 401 y Fw(i)1228 394 y Fx(2)i(T)1312 401 y Fw(c)1351 394 y FB(for)f(ev)o(ery)d Fy(i)k Fx(2)g(f)p FB(1)p Fy(;)8 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FB(=)e Fy(C)780 1649 y Fw(j)r Fs(+1)843 1642 y FB([)p Fy(u)885 1649 y Fw(j)r Fs(+1)948 1642 y FB(])f Fx(!)h Fy(s)1062 1649 y Fw(j)r Fs(+2)1141 1642 y FB(tak)o(es)i(place)g(in)g Fy(u)1476 1649 y Fw(j)r Fs(+1)1539 1642 y FB(.)-59 1767 y Fz(Pro)r(of:)28 b FB(Since)18 b Fx(R)i FB(is)f(non-terminating,)g (there)g(exists)g(an)h(in\014nite)e(deriv)m(ation)h(starting)h(from)f (some)-59 1828 y(term)d Fy(s)82 1835 y Fs(1)101 1828 y FB(;)i(in)f(particular,)g Fy(s)454 1835 y Fs(1)490 1828 y Fx(62)f(T)560 1835 y Fw(f)583 1828 y FB(.)25 b(Hence)17 b(there)g(is)g(a)h(minim)o(al)d(subterm)h(o)q(ccurrence)h Fy(u)1641 1835 y Fs(1)1678 1828 y FB(of)h Fy(s)1758 1835 y Fs(1)1795 1828 y FB(with)-59 1888 y Fy(u)-31 1895 y Fs(1)8 1888 y Fx(62)h(T)81 1895 y Fw(f)104 1888 y FB(.)31 b(That)20 b(is,)g Fy(u)369 1895 y Fs(1)408 1888 y FB(=)f Fy(F)7 b FB(\()p Fy(t)541 1895 y Fs(1)559 1888 y Fy(;)h(:)g(:)g(:)g(;)g (t)687 1895 y Fw(n)710 1888 y FB(\),)20 b(where)f Fy(F)25 b Fx(2)20 b(F)1058 1867 y Fw(n)1101 1888 y FB(and)g Fy(t)1217 1895 y Fw(i)1250 1888 y Fx(2)f(T)1323 1895 y Fw(f)1366 1888 y FB(for)g(ev)o(ery)f Fy(i)h Fx(2)h(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(n)p Fx(g)p FB(.)-59 1948 y(Let)17 b Fy(C)64 1955 y Fs(1)83 1948 y FB([)f(])g(b)q(e)h(the)f(con)o(text)g (suc)o(h)g(that)h Fy(s)707 1955 y Fs(1)741 1948 y FB(=)d Fy(C)828 1955 y Fs(1)848 1948 y FB([)p Fy(u)890 1955 y Fs(1)909 1948 y FB(].)21 b(Since)16 b Fy(u)1114 1955 y Fs(1)1148 1948 y Fx(62)e(T)1216 1955 y Fw(f)1239 1948 y FB(,)i(there)g(is)h(an)g(in\014nite)e(deriv)m(ation)-59 2008 y(starting)e(from)f Fy(u)259 2015 y Fs(1)279 2008 y FB(.)20 b(Let)13 b Fy(u)425 2015 y Fs(1)458 2008 y Fx(!)h Fy(u)550 1990 y Fr(0)550 2021 y Fs(1)582 2008 y FB(b)q(e)f(the)g(\014rst)g(reduction)g(step)g(in)f(it.)20 b(Note)13 b Fy(u)1400 1990 y Fr(0)1400 2021 y Fs(1)1433 2008 y Fx(62)h(T)1501 2015 y Fw(f)1524 2008 y FB(.)20 b(Set)13 b Fy(s)1662 2015 y Fs(2)1695 2008 y FB(=)h Fy(C)1782 2015 y Fs(1)1802 2008 y FB([)p Fy(u)1844 1990 y Fr(0)1844 2021 y Fs(1)1863 2008 y FB(].)-59 2068 y(No)o(w)f(there)g(are)g(t)o(w)o (o)h(p)q(ossibilities.)19 b(Either)13 b(\(i\))g(a)g(prop)q(er)h (subterm)e(of)i Fy(u)1289 2075 y Fs(1)1322 2068 y FB(w)o(as)g(con)o (tracted)f(in)g Fy(u)1732 2075 y Fs(1)1765 2068 y Fx(!)h Fy(u)1857 2050 y Fr(0)1857 2081 y Fs(1)1876 2068 y FB(.)-59 2129 y(Then)19 b Fy(u)99 2111 y Fr(0)99 2141 y Fs(1)137 2129 y FB(=)g Fy(F)7 b FB(\()p Fy(t)270 2136 y Fs(1)289 2129 y Fy(;)h(:)g(:)g(:)f(;)h(t)416 2111 y Fr(0)416 2141 y Fw(l)428 2129 y Fy(;)g(:)g(:)g(:)g(;)g(t)556 2136 y Fw(n)579 2129 y FB(\),)19 b(where)g Fy(t)793 2136 y Fw(l)824 2129 y Fx(!)g Fy(t)911 2111 y Fr(0)911 2141 y Fw(l)942 2129 y FB(for)h(some)e Fy(l)h Fx(2)g(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(.)29 b(In)19 b(this)g(case)g(w)o(e)g (set)-59 2189 y Fy(C)-24 2196 y Fs(2)-4 2189 y FB([)14 b(])g(=)f Fy(C)138 2196 y Fs(1)158 2189 y FB([)i(])g(as)g(w)o(ell)f(as) i Fy(u)459 2196 y Fs(2)492 2189 y FB(=)e Fy(u)572 2171 y Fr(0)572 2201 y Fs(1)592 2189 y FB(.)21 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Fw(im)16 b Fr(\003)1365 2818 y Fx(!)1464 2806 y FB(^)1453 2818 y Fy(C)t Fx(f)p Fy(t)1535 2825 y Fs(1)1554 2818 y Fx(#)p Fy(;)8 b(:)g(:)g(:)f(;)h(t)1706 2825 y Fw(n)1729 2818 y Fx(#g)14 b FB(=)f(\010\()p Fy(t)p FB(\).)0 2878 y Fq(2)p eop %%Page: 118 126 118 125 bop -59 -39 a FB(118)1132 b Fv(CHAPTER)16 b(6.)38 b(COMPLETENESS)-59 94 y Fz(Prop)r(osition)18 b(6.2.9)23 b FB(Let)16 b(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))16 b(b)q(e)g(a)h(lo)q (cally)e(con\015uen)o(t)g(o)o(v)o(erla)o(y)g(TRS.)h(If)f(there)g(is)h (an)h(in\014nite)e Fx(R)-59 154 y FB(deriv)m(ation,)g(then)i(there)e (is)h(also)h(an)g(in\014nite)e(innermost)g Fx(R)h FB(deriv)m(ation.)-59 235 y Fz(Pro)r(of:)25 b FB(According)17 b(to)h(Lemma)e(6.2.7,)i(there)g (is)g(an)g(in\014nite)f(deriv)m(ation)h Fy(D)h FB(:)d Fy(s)1477 242 y Fs(1)1514 235 y Fx(!)g Fy(s)1603 242 y Fs(2)1640 235 y Fx(!)g Fy(s)1729 242 y Fs(3)1766 235 y Fx(!)h Fy(:)8 b(:)g(:)-59 295 y FB(suc)o(h)16 b(that)h(for)f(ev)o (ery)f(rewrite)g(step)h Fy(s)651 302 y Fw(j)683 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FB(is)g(top)i(white,)e(and)h Fy(\032)400 1645 y Fs(2)434 1638 y Fx(/)d Fy(\017)p FB(.)21 b(W)l(e)15 b(de\014ne)h(a)g(substitution)g Fy(\032)1101 1620 y Fr(0)1128 1638 y FB(b)o(y)f Fy(\032)1220 1620 y Fr(0)1232 1638 y FB(\()p Fy(x)p FB(\))f(=)f Fy(\034)6 b FB(\()p Fy(y)r FB(\))15 b(for)h(ev)o(ery)e Fy(x)g Fx(2)g(D)q Fy(om)p FB(\()p Fy(\032)1909 1645 y Fs(2)1930 1638 y FB(\))0 1698 y(and)k Fy(y)g Fx(2)f(D)q Fy(om)p FB(\()p Fy(\033)r FB(\))h(satisfying)g Fy(\032)623 1705 y Fs(2)642 1698 y FB(\()p Fy(x)p FB(\))e(=)h Fy(\033)r FB(\()p Fy(y)r FB(\).)24 b Fy(\032)936 1680 y Fr(0)966 1698 y FB(is)17 b(w)o(ell-de\014ned)g(b)q(ecause)h Fy(\033)g Fx(/)e Fy(\034)6 b FB(.)25 b(It)17 b(follo)o(ws)h(from)0 1759 y Fy(\032)25 1766 y Fs(2)59 1759 y Fx(/)13 b Fy(\017)j FB(and)h Fy(\017)d Fx(/)f Fy(\032)353 1741 y Fr(0)381 1759 y FB(that)k Fy(\032)512 1766 y Fs(2)546 1759 y Fx(/)c Fy(\032)623 1741 y Fr(0)635 1759 y FB(.)21 b(By)16 b(Lemma)e(8.3.9,)i(for)g(an)o(y)h Fy(j)f Fx(2)e(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(,)16 b(w)o(e)g(ma)o(y)e(write)163 1864 y Fy(\032)188 1871 y Fs(2)207 1864 y FB(\()p Fy(\032)251 1871 y Fs(1)271 1864 y FB(\()p Fy(s)313 1871 y Fw(j)331 1864 y FB(\)\))g(=)g Fy(\032)460 1871 y Fs(2)480 1864 y FB(\()p Fy(u)527 1871 y Fs(1)546 1864 y FB(\))g Fx(!)629 1844 y Fw(o)629 1877 y Fs(1)662 1864 y Fy(:)8 b(:)g(:)14 b Fx(!)784 1844 y Fw(o)784 1877 y Fs(1)817 1864 y Fy(\032)842 1871 y Fs(2)862 1864 y FB(\()p Fy(u)909 1871 y Fw(k)930 1864 y FB(\))g(=)f Fy(\032)1039 1871 y Fs(2)1059 1864 y FB(\()p Fy(v)1102 1871 y Fw(l)1115 1864 y FB(\))1150 1844 y Fw(o)1150 1877 y Fs(1)1167 1864 y Fx( )h Fy(:)8 b(:)g(:)1312 1844 y Fw(o)1312 1877 y Fs(1)1329 1864 y Fx( )14 b Fy(\032)1418 1871 y Fs(2)1438 1864 y FB(\()p Fy(v)1481 1871 y Fs(1)1500 1864 y FB(\))g(=)g Fy(\032)1610 1871 y Fs(2)1630 1864 y FB(\()p Fy(\032)1674 1871 y Fs(1)1694 1864 y FB(\()p Fy(t)1731 1871 y Fw(j)1748 1864 y FB(\)\))0 1970 y(for)20 b(some)g(blac)o(k)f(terms)f Fy(u)503 1977 y Fs(1)523 1970 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)660 1977 y Fw(k)681 1970 y Fy(;)g(v)727 1977 y Fs(1)746 1970 y Fy(;)g(:)g(:)g(:)g(;)g(v)880 1977 y Fw(l)892 1970 y FB(.)33 b(No)o(w)20 b(rep)q(eated)g(application) g(of)h(the)e(induction)h(h)o(y-)0 2030 y(p)q(othesis)d(yields)187 2136 y Fy(\032)212 2115 y Fr(0)224 2136 y FB(\()p Fy(\032)268 2143 y Fs(1)287 2136 y FB(\()p Fy(s)329 2143 y Fw(j)348 2136 y FB(\)\))c(=)h Fy(\032)476 2115 y Fr(0)488 2136 y FB(\()p Fy(u)535 2143 y Fs(1)554 2136 y FB(\))g Fx(!)637 2115 y Fw(o)637 2148 y Fs(1)671 2136 y Fy(:)8 b(:)g(:)13 b Fx(!)792 2115 y Fw(o)792 2148 y Fs(1)825 2136 y Fy(\032)850 2115 y Fr(0)862 2136 y FB(\()p Fy(u)909 2143 y Fw(k)930 2136 y FB(\))h(=)f Fy(\032)1039 2115 y Fr(0)1051 2136 y FB(\()p Fy(v)1094 2143 y Fw(l)1107 2136 y FB(\))1142 2115 y Fw(o)1142 2148 y Fs(1)1159 2136 y Fx( )h Fy(:)8 b(:)g(:)1304 2115 y Fw(o)1304 2148 y Fs(1)1321 2136 y Fx( )14 b Fy(\032)1410 2115 y Fr(0)1422 2136 y FB(\()p Fy(v)1465 2143 y Fs(1)1484 2136 y FB(\))g(=)g Fy(\032)1594 2115 y Fr(0)1605 2136 y FB(\()p Fy(\032)1649 2143 y Fs(1)1669 2136 y FB(\()p Fy(t)1706 2143 y Fw(j)1724 2136 y FB(\)\))0 2246 y(Th)o(us)k Fy(\032)150 2228 y Fr(0)162 2246 y FB(\()p Fy(\032)206 2253 y Fs(1)226 2246 y FB(\()p Fy(l)q FB(\)\))d Fx(!)364 2228 y Fw(o)364 2259 y Fs(1)400 2246 y Fy(\032)425 2228 y Fr(0)437 2246 y FB(\()p Fy(\032)481 2253 y Fs(1)501 2246 y FB(\()p Fy(r)q FB(\)\).)25 b(Let)720 2234 y(^)709 2246 y Fy(C)t FB([)17 b(])g(b)q(e)h(the)f(con)o(text)g(obtained)h(from) e Fy(C)t FB([)h(])g(b)o(y)g(replacing)g(ev)o(ery)0 2306 y(white)g(principal)f(subterm)h(whic)o(h)f(m)o(ust)g(b)q(e)i(of)g(the)f (form)f Fy(\033)r FB(\()p Fy(x)p FB(\))h(for)g(some)g(v)m(ariable)g Fy(x)e Fx(2)i(D)q Fy(om)p FB(\()p Fy(\033)r FB(\))g(b)o(y)0 2367 y(the)f(corresp)q(onding)i Fy(\034)6 b FB(\()p Fy(x)p FB(\).)22 b(\(This)17 b(is)f(a)h(sligh)o(t)f(abuse)i(of)f(notation)g(b) q(ecause)1460 2354 y(^)1449 2367 y Fy(C)t FB([)f(])g(con)o(tains)h(in)f (general)0 2427 y(more)21 b(that)h(one)g(o)q(ccurrence)g(of)g Fq(2)p FB(.\))38 b(It)22 b(is)g(fairly)f(simple)f(to)i(v)o(erify)e (that)j Fy(s\034)29 b FB(=)1621 2414 y(^)1609 2427 y Fy(C)t FB([)p Fy(\032)1687 2409 y Fr(0)1698 2427 y FB(\()p Fy(\032)1742 2434 y Fs(1)1762 2427 y FB(\()p Fy(l)q FB(\)\)])21 b(and)0 2487 y Fy(t\034)e FB(=)121 2474 y(^)110 2487 y Fy(C)s FB([)p Fy(\032)187 2469 y Fr(0)199 2487 y FB(\()p Fy(\032)243 2494 y Fs(1)263 2487 y FB(\()p Fy(r)q FB(\)\)].)i(Hence)15 b Fy(s\034)k Fx(!)650 2469 y Fw(o)650 2499 y Fs(1)684 2487 y Fy(t\034)6 b FB(.)20 b Fq(2)0 2617 y Fz(Lemma)c(8.3.11)23 b FB(The)16 b(restriction)g(of)g Fx(!)805 2624 y Fs(1)841 2617 y FB(to)h Fx(T)12 b FB(\()p Fx(F)995 2624 y Fs(1)1015 2617 y Fy(;)c Fx(V)t FB(\))i Fx(\002)h(T)i FB(\()p Fx(F)1246 2624 y Fs(1)1266 2617 y Fy(;)8 b Fx(V)t FB(\))15 b(and)i Fx(\))1502 2624 y Fr(R)1532 2629 y Fh(1)1568 2617 y FB(coincide.)0 2698 y Fz(Pro)r(of:)k FB(\\)p Fx(\023)p FB(")c(T)l(rivial.)0 2758 y(\\)p Fx(\022)p FB(")c(Let)g Fy(s;)8 b(t)13 b Fx(2)h(T)f FB(\()p Fx(F)402 2765 y Fs(1)421 2758 y Fy(;)8 b Fx(V)t FB(\))13 b(with)f Fy(s)i Fx(!)704 2765 y Fs(1)737 2758 y Fy(t)p FB(.)20 b(In)12 b(order)h(to)g(sho)o(w)g(that)g Fy(s)h Fx(\))1332 2765 y Fr(R)1362 2770 y Fh(1)1395 2758 y Fy(t)p FB(,)e(w)o(e)h(pro)q(ceed)f(b)o(y)g(induction)0 2818 y(on)i(the)f(depth)g(of)h Fy(s)f Fx(!)420 2800 y Fw(o)420 2830 y Fs(1)454 2818 y Fy(t)p FB(.)20 b(The)13 b(case)g(of)h(zero)f(depth)g(is)g(straigh)o(tforw)o(ard.)21 b(So)14 b(supp)q(ose)g(that)g(the)f(depth)0 2878 y(of)18 b Fy(s)f Fx(!)147 2860 y Fw(o)147 2891 y Fs(1)183 2878 y Fy(t)g FB(equals)h Fy(d)12 b FB(+)g(1,)19 b Fy(d)d Fx(\025)h FB(0.)26 b(Then)18 b(there)f(exists)h(a)g(rewrite)f(rule)g Fy(l)d Fx(!)g Fy(r)h Fx(\()f Fy(s)1590 2885 y Fs(1)1623 2878 y Fx(#)g Fy(t)1680 2885 y Fs(1)1699 2878 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)1832 2885 y Fw(n)1869 2878 y Fx(#)14 b Fy(t)1926 2885 y Fw(n)p eop %%Page: 138 146 138 145 bop -59 -39 a FB(138)451 b Fv(CHAPTER)16 b(8.)38 b(CONDITIONAL)15 b(TERM)h(REWRITING)g(SYSTEMS)-59 94 y FB(in)h Fx(R)41 101 y Fs(1)61 94 y FB(,)h(a)g(substitution)g Fy(\033)h FB(:)d Fx(V)k(!)c(T)d FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))o(,)18 b(and)g(a)h(con)o(text)e Fy(C)t FB([)f(])i(suc)o(h) f(that)i Fy(s)d FB(=)h Fy(C)t FB([)p Fy(l)q(\033)r FB(])p Fy(;)8 b(t)13 b FB(=)k Fy(C)t FB([)p Fy(r)q(\033)r FB(],)-59 154 y(and)e Fy(s)57 161 y Fw(j)76 154 y Fy(\033)h Fx(#)145 132 y Fw(o)145 166 y Fs(1)180 154 y Fy(t)198 161 y Fw(j)216 154 y Fy(\033)g FB(with)f(depth)g Fx(\024)e Fy(d)j FB(for)f Fy(j)i Fx(2)d(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(n)p Fx(g)p FB(.)21 b(According)14 b(to)i(Prop)q(osition)g(3.3.20,)f Fy(\033)i FB(can)e(b)q(e)-59 214 y(decomp)q(osed)g(in)o(to)g Fy(\033)339 221 y Fs(2)368 214 y Fx(\016)9 b Fy(\033)430 221 y Fs(1)465 214 y FB(suc)o(h)15 b(that)h Fy(\033)707 221 y Fs(1)742 214 y FB(is)f(blac)o(k,)f Fy(\033)958 221 y Fs(2)993 214 y FB(is)h(top)h(white,)f(and)h Fy(\033)1395 221 y Fs(2)1428 214 y Fx(/)d Fy(\017)p FB(.)21 b(Induction)15 b(on)h(the)-59 274 y(n)o(um)o(b)q(er)9 b(of)j(rewrite)e(steps)h(in)g Fy(s)516 281 y Fw(j)534 274 y Fy(\033)i Fx(#)600 253 y Fw(o)600 286 y Fs(1)631 274 y Fy(t)649 281 y Fw(j)667 274 y Fy(\033)g FB(in)d(com)o(bination)g(with)h(Lemma)e(8.3.10)j (yields)e Fy(\033)1603 281 y Fs(1)1622 274 y FB(\()p Fy(s)1664 281 y Fw(j)1683 274 y FB(\))h Fx(#)1738 253 y Fw(o)1738 286 y Fs(1)1769 274 y Fy(\033)1797 281 y Fs(1)1816 274 y FB(\()p Fy(t)1853 281 y Fw(j)1871 274 y FB(\))-59 334 y(for)19 b Fy(j)j Fx(2)c(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(.)29 b(Since)18 b(ev)o(ery)g(term)f(in) i(the)f(con)o(v)o(ersion)g Fy(\033)1162 341 y Fs(1)1182 334 y FB(\()p Fy(s)1224 341 y Fw(j)1242 334 y FB(\))h Fx(#)1305 313 y Fw(o)1305 347 y Fs(1)1343 334 y Fy(\033)1371 341 y Fs(1)1391 334 y FB(\()p Fy(t)1428 341 y Fw(j)1446 334 y FB(\))g(is)f(blac)o(k,)h(w)o(e)f(obtain)-59 394 y Fy(\033)-31 401 y Fs(1)-12 394 y FB(\()p Fy(s)30 401 y Fw(j)49 394 y FB(\))k Fx(+)120 401 y Fr(R)150 406 y Fh(1)193 394 y Fy(\033)221 401 y Fs(1)240 394 y FB(\()p Fy(t)277 401 y Fw(j)295 394 y FB(\))f(b)o(y)g(rep)q(eated)g (application)h(of)f(the)g(induction)g(h)o(yp)q(othesis.)37 b(Consequen)o(tly)l(,)21 b(w)o(e)-59 455 y(ha)o(v)o(e)13 b Fy(\033)79 462 y Fs(1)99 455 y FB(\()p Fy(l)q FB(\))g Fx(\))216 462 y Fr(R)246 467 y Fh(1)279 455 y Fy(\033)307 462 y Fs(1)327 455 y FB(\()p Fy(r)q FB(\).)21 b(No)o(w)14 b Fy(s)g Fx(\))619 462 y Fr(R)649 467 y Fh(1)682 455 y Fy(t)g FB(follo)o(ws)g(from)f Fy(s)h FB(=)g Fy(C)t FB([)p Fy(l)q(\033)r FB(])d(=)j Fy(C)t FB([)p Fy(l)q(\033)1347 462 y Fs(1)1365 455 y FB(])g(and)h Fy(t)f FB(=)f Fy(C)t FB([)p Fy(r)q(\033)r FB(])g(=)g Fy(C)t FB([)p Fy(r)q(\033)1857 462 y Fs(1)1876 455 y FB(])-59 515 y(b)q(ecause)j Fy(s)h FB(and)f Fy(t)g FB(are)h(blac)o(k.)j Fq(2)-59 641 y Fz(Prop)r(osition)e (8.3.12)23 b FB(If)c(\()p Fx(F)523 648 y Fs(1)543 641 y Fy(;)8 b Fx(R)607 648 y Fs(1)626 641 y FB(\))20 b(and)f(\()p Fx(F)822 648 y Fs(2)842 641 y Fy(;)8 b Fx(R)906 648 y Fs(2)926 641 y FB(\))19 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bop 0 -39 a Fv(8.3.)38 b(CONSTR)o(UCTOR-SHARING)14 b(SYSTEMS)883 b FB(141)0 94 y(Lemma)14 b(8.3.8,)j Fy(s)d Fx(!)392 75 y Fr(\003)392 106 y(R)438 94 y Fy(t)p FB(.)22 b(It)16 b(follo)o(ws)g(from)f(Lemma)g(8.3.18)i(\(2\))g(that)f Fy(t)e Fx(2)h Fy(N)5 b(F)i FB(\()p Fx(F)t Fy(;)h Fx(R)p FB(\).)22 b(Hence)15 b(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))0 154 y(is)18 b(also)h(normalizing.)27 b(This)18 b(all)g(pro)o(v)o(es)g (that)h(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))19 b(is)f(semi-comple)o(te.) 25 b(The)18 b(only-if)g(case)h(follo)o(ws)0 214 y(straigh)o(tforw)o (ardly)d(from)g(Lemma)e(8.3.20.)21 b Fq(2)0 394 y Fz(Lemma)16 b(8.3.20)23 b FB(Let)i(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))25 b(b)q(e)g(the)f(com)o(bined)f(system)g(of)i(t)o(w)o(o)g (constructor-sharing)h(CTRSs)0 454 y(\()p Fx(F)55 461 y Fs(1)74 454 y Fy(;)8 b Fx(R)138 461 y Fs(1)158 454 y FB(\))18 b(and)g(\()p Fx(F)346 461 y Fs(2)365 454 y Fy(;)8 b Fx(R)429 461 y Fs(2)449 454 y FB(\))18 b(suc)o(h)f(that)h(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))17 b(is)h(semi-compl)o(ete.)k(If)17 b Fy(s)h FB(is)f(a)h(blac)o(k)f(term)e(and)k Fy(s)d Fx(!)1870 461 y Fr(R)1918 454 y Fy(t)p FB(,)0 514 y(then)g Fy(s)e Fx(\))198 521 y Fr(R)228 526 y Fh(1)261 514 y Fy(t)p FB(.)0 595 y Fz(Pro)r(of:)29 b FB(W)l(e)20 b(sho)o(w)g(the)g(follo)o (wing)g(stronger)h(claim,)d(where)i(the)g(rewrite)f(relation)h(asso)q (ciated)h(with)0 655 y(\()p Fx(F)60 662 y Fs(1)91 655 y Fx([)11 b(f)p Fq(2)p Fx(g)p Fy(;)d Fx(R)286 662 y Fs(1)306 655 y FB(\))16 b(is)g(also)h(denoted)f(b)o(y)g Fx(\))790 662 y Fr(R)820 667 y Fh(1)839 655 y FB(.)0 736 y Fz(Claim:)28 b FB(If)20 b Fy(s)h FB(is)f(a)h(blac)o(k)e(term)g(and)i Fy(\033)h FB(is)f(a)f(top)h(white)f Fx(!)1163 743 y Fr(R)1216 736 y FB(normalized)e(substitution)j(suc)o(h)f(that)0 797 y Fy(s\033)15 b Fx(!)116 804 y Fr(R)162 797 y Fy(t\033)r FB(,)g(then)h Fy(s\033)403 779 y Fc(2)444 797 y Fx(\))494 804 y Fr(R)524 809 y Fh(1)557 797 y Fy(t\033)605 779 y Fc(2)631 797 y FB(,)g(where)g Fy(\033)832 779 y Fc(2)875 797 y 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Fs(1)1258 1866 y FB(\()p Fy(s)1300 1873 y Fw(j)1318 1866 y FB(\)\))j(and)g Fy(top)1533 1848 y Fw(b)1550 1866 y FB(\()p Fy(t)1587 1873 y Fw(j)1605 1866 y Fy(\033)r FB(\))c(=)h Fy(\033)1749 1848 y Fc(2)1747 1878 y Fs(2)1776 1866 y FB(\()p Fy(\033)1823 1873 y Fs(1)1842 1866 y FB(\()p Fy(t)1879 1873 y Fw(j)1897 1866 y FB(\)\).)0 1931 y(Hence)f Fy(\033)173 1913 y Fc(2)171 1943 y Fs(2)200 1931 y FB(\()p Fy(\033)247 1938 y Fs(1)267 1931 y FB(\()p Fy(s)309 1938 y Fw(j)327 1931 y FB(\)\))h Fx(+)409 1938 y Fr(R)439 1943 y Fh(1)472 1931 y Fy(\033)502 1913 y Fc(2)500 1943 y Fs(2)529 1931 y FB(\()p Fy(\033)576 1938 y Fs(1)596 1931 y FB(\()p Fy(t)633 1938 y Fw(j)651 1931 y FB(\)\))g(and)i(th)o(us) e Fy(\033)932 1913 y Fc(2)930 1943 y Fs(2)959 1931 y FB(\()p Fy(\033)1006 1938 y Fs(1)1026 1931 y FB(\()p Fy(l)q FB(\)\))f Fx(\))1162 1938 y Fr(R)1192 1943 y Fh(1)1225 1931 y Fy(\033)1255 1913 y Fc(2)1253 1943 y Fs(2)1282 1931 y FB(\()p Fy(\033)1329 1938 y Fs(1)1348 1931 y FB(\()p Fy(r)q FB(\)\).)21 b(Let)1560 1918 y(^)1549 1931 y Fy(C)t FB([)14 b(])g(b)q(e)h(the)g(con)o(text)0 1991 y(obtained)i(from)f Fy(C)t FB([)g(])h(b)o(y)f(replacing)h(all)f(white)h(principal)f (subterms)g(with)h Fq(2)p FB(.)24 b(No)o(w)17 b(\(1\))g(follo)o(ws)g (from)0 2051 y Fy(top)65 2033 y Fw(b)83 2051 y FB(\()p Fy(s)p FB(\))c(=)220 2039 y(^)209 2051 y Fy(C)t FB([)p Fy(\033)292 2033 y Fc(2)290 2064 y Fs(2)318 2051 y FB(\()p Fy(\033)365 2058 y Fs(1)385 2051 y FB(\()p Fy(l)q FB(\)\)])i(and)i Fy(top)647 2033 y Fw(b)664 2051 y FB(\()p Fy(t)p FB(\))d(=)797 2039 y(^)786 2051 y Fy(C)s FB([)p Fy(\033)868 2033 y Fc(2)866 2064 y Fs(2)894 2051 y FB(\()p Fy(\033)941 2058 y Fs(1)961 2051 y FB(\()p Fy(r)q FB(\)\)].)0 2112 y(\(2\))22 b(Let)h Fy(s)g Fx(!)274 2093 y Fw(o)317 2112 y Fy(t)f FB(b)o(y)f(some)g(rule)h(from)e Fx(R)825 2119 y Fs(2)866 2112 y FB(or)j Fy(s)h Fx(!)1029 2093 y Fw(i)1066 2112 y Fy(t)p FB(.)38 b(Since)22 b Fx(R)1312 2119 y Fs(2)1354 2112 y FB(is)f(la)o(y)o(er-preserving,)h(w)o(e)f(ma)o(y)0 2172 y(write)f Fy(s)i FB(=)f Fy(C)272 2154 y Fw(b)289 2172 y Fx(h)-8 b(h)q Fy(u)348 2179 y Fs(1)367 2172 y Fy(;)8 b(:)g(:)g(:)f(;)h(u)504 2179 y Fw(j)522 2172 y Fy(;)g(:)g(:)g(:)g(;)g(u)660 2179 y Fw(p)679 2172 y Fx(i)-8 b(i)21 b FB(and)h Fy(t)f FB(=)h Fy(C)968 2154 y Fw(b)984 2172 y Fx(h)-8 b(h)q Fy(u)1043 2179 y Fs(1)1062 2172 y Fy(;)8 b(:)g(:)g(:)g(;)g(u)1200 2154 y Fr(0)1200 2184 y Fw(j)1218 2172 y Fy(;)g(:)g(:)g(:)f(;)h(u)1355 2179 y Fw(p)1374 2172 y Fx(i)-8 b(i)q FB(,)21 b(where)g Fy(u)1614 2179 y Fw(j)1654 2172 y Fx(!)g Fy(u)1753 2154 y Fr(0)1753 2184 y Fw(j)1771 2172 y FB(.)35 b(Hence)0 2232 y Fy(top)65 2214 y Fw(b)83 2232 y FB(\()p Fy(s)p FB(\))13 b(=)h Fy(top)274 2214 y Fw(b)292 2232 y FB(\()p Fy(t)p FB(\).)21 b Fq(2)73 2347 y FB(In)f(the)g(preceding)g(prop)q(osition,)i(the)e(assumption)g (that)h(\()p Fx(F)1244 2354 y Fs(2)1264 2347 y Fy(;)8 b Fx(R)1328 2354 y Fs(2)1348 2347 y FB(\))20 b(has)h(to)g(b)q(e)g(la)o (y)o(er-preserving)0 2407 y(cannot)c(b)q(e)f(dropp)q(ed,)h(as)g(is)f 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y(la)o(y)o(er-preservingness)12 b(requiremen)n(t)f(on)i(\()p Fx(F)829 2825 y Fs(2)849 2818 y Fy(;)8 b Fx(R)913 2825 y Fs(2)933 2818 y FB(\))13 b(but)g(under)g(the)g(additional)g (assumption)g(that)h Fx(!)1902 2825 y Fs(1)p Fw(;)p Fs(2)0 2878 y FB(is)i(semi-comple)o(te.)p eop %%Page: 144 152 144 151 bop -59 -39 a FB(144)451 b Fv(CHAPTER)16 b(8.)38 b(CONDITIONAL)15 b(TERM)h(REWRITING)g(SYSTEMS)-59 94 y Fz(De\014nition)h(8.3.26)24 b FB(Let)15 b(the)g(rewrite)f(relation)g Fx(!)930 101 y Fs(1)p Fw(;)p Fs(2)992 94 y FB(b)q(e)h(semi-complet)o (e.)j(F)l(or)d Fy(t)e FB(=)h Fy(C)1601 75 y Fw(b)1618 94 y Fx(h)-8 b(h)p Fy(t)1666 101 y Fs(1)1686 94 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)1813 101 y Fw(m)1846 94 y Fx(i)-8 b(i)p FB(,)-59 154 y(w)o(e)16 b(de\014ne)g Fy(top)219 136 y Fw(b)219 166 y Fr(!)257 154 y FB(\()p Fy(t)p FB(\))f(b)o(y:)586 214 y Fy(top)651 193 y Fw(b)651 226 y Fr(!)689 214 y FB(\()p Fy(t)p FB(\))e(=)h Fy(top)875 193 y Fw(b)893 214 y FB(\()p Fy(C)951 193 y Fw(b)968 214 y Fx(h)p Fy(t)1005 193 y Fr(!)1005 226 y Fs(1)1042 214 y Fy(;)8 b(:)g(:)g(:)f(;)h(t)1169 193 y Fr(!)1169 226 y Fw(m)1206 214 y Fx(i)q FB(\))14 341 y(In)13 b(other)h(w)o(ords,)g(\014rst)f(the)g(white)g(principal)f (subterms)h(in)g Fy(t)g FB(are)g(replaced)f(with)i(their)e(unique)h Fx(!)1843 348 y Fs(1)p Fw(;)p Fs(2)-59 401 y FB(normal)i(form,)g(and)i (then)f(the)g(topmost)g(blac)o(k)f(homogeneous)i(part)f(of)h(the)f (term)e(obtained)j(is)f(tak)o(en.)-59 528 y Fz(Lemma)g(8.3.27)23 b FB(Let)e Fx(!)449 535 y Fs(1)p Fw(;)p Fs(2)517 528 y FB(b)q(e)g(semi-comple)o(te.)32 b(If)21 b Fy(s;)8 b(t)20 b FB(are)h(blac)o(k)f(terms)f(and)j Fy(\033)g FB(is)f(a)g(top)g(white) -59 588 y(substitution)16 b(suc)o(h)g(that)h Fy(s\033)e Fx(!)544 570 y Fw(o)577 588 y Fy(t\033)j FB(b)o(y)d(some)h(rule)f(from) g Fx(R)1086 595 y Fs(1)1105 588 y FB(,)h(then)g Fy(\033)1276 570 y Fr(!)1313 588 y FB(\()p Fy(s)p FB(\))e Fx(!)1438 570 y Fw(o)1438 601 y Fs(1)1471 588 y Fy(\033)1501 570 y Fr(!)1538 588 y FB(\()p Fy(t)p FB(\).)-59 669 y Fz(Pro)r(of:)21 b FB(There)16 b(is)g(a)h(con)o(text)e Fy(C)t FB([)g(],)h(a)g (substitution)h Fy(\032)d FB(:)f Fx(V)18 b(!)13 b(T)g FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))15 b(and)i(a)g(rewrite)e(rule)h Fy(l)e Fx(!)g Fy(r)h Fx(\()-59 730 y Fy(s)-36 737 y Fs(1)1 730 y Fx(#)i Fy(t)61 737 y Fs(1)80 730 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)213 737 y Fw(n)253 730 y Fx(#)17 b Fy(t)313 737 y Fw(n)352 730 y Fx(2)e(R)442 737 y Fs(1)479 730 y FB(suc)o(h)i(that)h Fy(s\033)e FB(=)g Fy(C)t FB([)p Fy(l)q(\032)p FB(],)f Fy(t\033)h FB(=)g Fy(C)t FB([)p Fy(r)q(\032)p FB(])g(and)i Fy(s)1321 737 y Fw(j)1339 730 y Fy(\032)f Fx(#)g Fy(t)1441 737 y Fw(j)1459 730 y Fy(\032)g FB(for)h Fy(j)g Fx(2)e(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(n)p Fx(g)p FB(.)-59 790 y(Fix)13 b Fy(j)s FB(.)20 b(F)l(rom)13 b(Prop)q(osition)i(8.3.17)f(w)o (e)f(kno)o(w)h(that)g Fy(s)927 797 y Fw(j)946 790 y Fy(\032)g Fx(#)1010 802 y Fs(1)p Fw(;)p Fs(2)1070 790 y Fy(t)1088 797 y Fw(j)1106 790 y Fy(\032)p FB(.)21 b(According)13 b(to)h(Prop)q(osition)h(3.3.20,)g Fy(\032)-59 850 y FB(can)g(b)q(e)g (decomp)q(osed)f(in)o(to)h Fy(\032)488 857 y Fs(2)516 850 y Fx(\016)8 b Fy(\032)574 857 y Fs(1)609 850 y FB(suc)o(h)15 b(that)g Fy(\032)847 857 y Fs(1)882 850 y FB(is)f(blac)o(k)g(and)i Fy(\032)1174 857 y Fs(2)1209 850 y FB(is)e(top)i(white.)k(Since)14 b Fx(!)1669 857 y Fs(1)p Fw(;)p Fs(2)1731 850 y FB(is)g(semi-)-59 910 y(complete,)c(it)h(follo)o(ws)g(as)h(in)f(the)h(pro)q(of)g(of)g (Lemma)d(8.3.14)j(that)g Fy(\032)1147 892 y Fr(!)1147 922 y Fs(2)1185 910 y FB(\()p Fy(\032)1229 917 y Fs(1)1249 910 y FB(\()p Fy(s)1291 917 y Fw(j)1309 910 y FB(\)\))f Fx(#)1383 922 y Fs(1)p Fw(;)p Fs(2)1442 910 y Fy(\032)1467 892 y Fr(!)1467 922 y Fs(2)1504 910 y FB(\()p Fy(\032)1548 917 y Fs(1)1568 910 y FB(\()p Fy(t)1605 917 y Fw(j)1623 910 y FB(\)\).)20 b(Applying)-59 970 y(Lemma)e(8.3.15)j(to)g(the)f (blac)o(k)g(terms)f Fy(\032)713 977 y Fs(1)733 970 y FB(\()p Fy(s)775 977 y Fs(1)794 970 y FB(\))p Fy(;)8 b(:)g(:)g(:)g(;)g(\032)948 977 y Fs(1)967 970 y FB(\()p Fy(s)1009 977 y Fw(n)1033 970 y FB(\))p Fy(;)g(\032)1099 977 y Fs(1)1118 970 y FB(\()p Fy(t)1155 977 y Fs(1)1175 970 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h(\032)1328 977 y Fs(1)1348 970 y FB(\()p Fy(t)1385 977 y Fw(n)1408 970 y FB(\))20 b(and)h(the)g(substitution)-59 1031 y Fy(\032)-34 1012 y Fr(!)-34 1043 y Fs(2)23 1031 y FB(yields)e Fy(\032)189 1012 y Fr(!)189 1043 y Fs(2)226 1031 y FB(\()p Fy(\032)270 1038 y Fs(1)290 1031 y FB(\()p Fy(s)332 1038 y Fw(j)350 1031 y FB(\)\))h Fx(#)433 1009 y Fw(o)433 1043 y Fs(1)472 1031 y Fy(\032)497 1012 y Fr(!)497 1043 y Fs(2)535 1031 y FB(\()p Fy(\032)579 1038 y Fs(1)599 1031 y FB(\()p Fy(t)636 1038 y Fw(j)654 1031 y FB(\)\).)31 b(Therefore,)20 b Fy(\032)1001 1012 y Fr(!)1001 1043 y Fs(2)1039 1031 y FB(\()p Fy(\032)1083 1038 y Fs(1)1102 1031 y FB(\()p Fy(l)q FB(\)\))g Fx(!)1245 1012 y Fw(o)1245 1043 y Fs(1)1304 1031 y Fy(\032)1329 1012 y Fr(!)1329 1043 y Fs(2)1366 1031 y FB(\()p Fy(\032)1410 1038 y Fs(1)1430 1031 y FB(\()p Fy(r)q FB(\)\).)32 b(Let)1658 1018 y(^)1647 1031 y Fy(C)t FB([)19 b(])g(b)q(e)h(the)-59 1091 y(con)o(text)e(obtained)h(from)f Fy(C)t FB([)g(])g(b)o(y)h(replacing)f(all)g(white)h(principal)f (subterms)g(with)g(their)g(resp)q(ectiv)o(e)-59 1151 y Fx(!)-9 1158 y Fs(1)p Fw(;)p Fs(2)56 1151 y FB(normal)f(form.)26 b(It)18 b(is)g(clear)g(that)g Fy(\033)725 1133 y Fr(!)762 1151 y FB(\()p Fy(s)p FB(\))f(=)906 1138 y(^)895 1151 y Fy(C)t FB([)p Fy(\032)973 1133 y Fr(!)973 1163 y Fs(2)1010 1151 y FB(\()p Fy(\032)1054 1158 y Fs(1)1074 1151 y FB(\()p Fy(l)q FB(\)\)])g(and)i Fy(\033)1305 1133 y Fr(!)1341 1151 y FB(\()p Fy(t)p FB(\))e(=)1480 1138 y(^)1469 1151 y Fy(C)t FB([)p Fy(\032)1547 1133 y Fr(!)1547 1163 y Fs(2)1584 1151 y FB(\()p Fy(\032)1628 1158 y Fs(1)1648 1151 y FB(\()p Fy(r)q FB(\)\)].)26 b(Th)o(us)-59 1211 y Fy(\033)-29 1193 y Fr(!)8 1211 y FB(\()p Fy(s)p FB(\))14 b Fx(!)133 1193 y Fw(o)133 1223 y Fs(1)166 1211 y Fy(\033)196 1193 y Fr(!)233 1211 y FB(\()p Fy(t)p FB(\).)21 b Fq(2)-59 1338 y Fz(Prop)r(osition)d(8.3.28)23 b FB(Let)f Fx(!)554 1345 y Fs(1)p Fw(;)p Fs(2)622 1338 y FB(b)q(e)f(semi-complete)o(.)34 b(If)21 b Fy(s)h Fx(!)1194 1320 y Fw(o)1235 1338 y Fy(t)f FB(b)o(y)g(some)f(rule)h(from)f Fx(R)1739 1345 y Fs(1)1759 1338 y FB(,)i(then)-59 1398 y Fy(top)6 1380 y Fw(b)6 1411 y Fr(!)44 1398 y FB(\()p Fy(s)p FB(\))14 b Fx(\))169 1405 y Fr(R)199 1410 y Fh(1)232 1398 y Fy(top)297 1380 y Fw(b)297 1411 y Fr(!)335 1398 y FB(\()p Fy(t)p FB(\).)-59 1479 y Fz(Pro)r(of:)21 b FB(W)l(e)15 b(ma)o(y)f(write)h Fy(s)f FB(=)g Fy(C)554 1461 y Fw(b)571 1479 y Fx(f)-17 b(f)p Fy(s)627 1486 y Fs(1)647 1479 y Fy(;)8 b(:)g(:)g(:)f(;)h(s)779 1486 y Fw(n)802 1479 y Fx(g)-17 b(g)16 b FB(and)g Fy(t)e FB(=)1040 1467 y(^)1029 1479 y Fy(C)1068 1461 y Fw(b)1084 1479 y Fx(h)-8 b(h)q Fy(s)1138 1486 y Fw(i)1150 1491 y Fh(1)1169 1479 y Fy(;)8 b(:)g(:)g(:)g(;)g(s)1302 1486 y Fw(i)1314 1490 y Fg(m)1345 1479 y Fx(i)-8 b(i)16 b FB(for)g(some)f(blac)o(k)f(con)o(texts)-59 1540 y Fy(C)-20 1521 y Fw(b)-3 1540 y Fx(f)p Fy(;)8 b(:)g(:)g(:)f(;)h Fx(g)p FB(,)198 1527 y(^)187 1540 y Fy(C)226 1521 y Fw(b)242 1540 y Fx(h)q Fy(;)g(:)g(:)g(:)f(;)h Fx(i)p FB(,)17 b(and)h Fy(i)534 1547 y Fs(1)553 1540 y Fy(;)8 b(:)g(:)g(:)g(;)g(i)680 1547 y Fw(m)728 1540 y Fx(2)15 b(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n) p Fx(g)p FB(.)24 b(Let)17 b Fy(x)1143 1547 y Fs(1)1163 1540 y Fy(;)8 b(:)g(:)g(:)f(;)h(x)1300 1547 y Fw(n)1340 1540 y FB(b)q(e)18 b(distinct)e(fresh)h(v)m(ariables)-59 1600 y(and)g(de\014ne)f Fy(\033)g FB(=)e Fx(f)p Fy(x)326 1607 y Fw(j)358 1600 y Fx(7!)g Fy(s)445 1607 y Fw(j)479 1600 y Fx(j)i FB(1)f Fx(\024)f Fy(j)j Fx(\024)d Fy(n)p Fx(g)p FB(,)i Fy(s)798 1582 y Fr(0)823 1600 y FB(=)e Fy(C)914 1582 y Fw(b)931 1600 y Fx(f)p Fy(x)984 1607 y Fs(1)1003 1600 y Fy(;)8 b(:)g(:)g(:)g(;)g(x)1141 1607 y Fw(n)1164 1600 y Fx(g)p FB(,)16 b(and)h Fy(t)1332 1582 y Fr(0)1357 1600 y FB(=)1420 1587 y(^)1409 1600 y Fy(C)1448 1582 y Fw(b)1465 1600 y Fx(h)p Fy(x)1512 1607 y Fw(i)1524 1612 y Fh(1)1543 1600 y Fy(;)8 b(:)g(:)g(:)g(;)g(x)1681 1607 y Fw(i)1693 1611 y Fg(m)1724 1600 y Fx(i)p FB(.)22 b(Since)-59 1660 y Fy(\033)g FB(is)e(top)h(white,)f(w)o(e)g(obtain)h Fy(\033)548 1642 y Fr(!)584 1660 y FB(\()p Fy(s)626 1642 y Fr(0)638 1660 y FB(\))g Fx(!)728 1642 y Fw(o)728 1672 y Fs(1)768 1660 y Fy(\033)798 1642 y Fr(!)835 1660 y FB(\()p Fy(t)872 1642 y Fr(0)883 1660 y FB(\))f(b)o(y)g(Lemma)e (8.3.27.)34 b(According)20 b(to)h(Prop)q(osition)-59 1720 y(3.3.20,)h Fy(\033)131 1702 y Fr(!)188 1720 y FB(has)f(a)g (decomp)q(osition)f Fy(\033)680 1702 y Fr(!)738 1720 y FB(=)h Fy(\033)825 1727 y Fs(2)858 1720 y Fx(\016)14 b Fy(\033)925 1727 y Fs(1)944 1720 y FB(,)22 b(where)e Fy(\033)1153 1727 y Fs(1)1193 1720 y FB(is)g(blac)o(k)f(and)j Fy(\033)1505 1727 y Fs(2)1544 1720 y FB(is)f(top)g(white.)33 b(It)-59 1780 y(follo)o(ws)18 b(from)f(Lemma)e(8.3.10)k(that)f Fy(\033)675 1762 y Fc(2)673 1793 y Fs(2)702 1780 y FB(\()p Fy(\033)749 1787 y Fs(1)769 1780 y FB(\()p Fy(s)811 1762 y Fr(0)822 1780 y FB(\)\))f Fx(!)927 1762 y Fw(o)927 1793 y Fs(1)963 1780 y Fy(\033)993 1762 y Fc(2)991 1793 y Fs(2)1020 1780 y FB(\()p Fy(\033)1067 1787 y Fs(1)1087 1780 y FB(\()p Fy(t)1124 1762 y Fr(0)1135 1780 y FB(\)\))h(b)q(ecause)g Fy(\033)1401 1787 y Fs(2)1437 1780 y Fx(/)e Fy(\033)1522 1762 y Fc(2)1520 1793 y Fs(2)1549 1780 y FB(.)27 b(T)l(o)18 b(v)o(erify)e(that)-59 1840 y Fy(\033)-29 1822 y Fc(2)-31 1853 y Fs(2)-2 1840 y FB(\()p Fy(\033)45 1847 y Fs(1)64 1840 y FB(\()p Fy(s)106 1822 y Fr(0)118 1840 y FB(\)\))g Fx(\))222 1847 y Fr(R)252 1852 y Fh(1)287 1840 y Fy(\033)317 1822 y Fc(2)315 1853 y Fs(2)344 1840 y FB(\()p Fy(\033)391 1847 y Fs(1)411 1840 y FB(\()p Fy(t)448 1822 y Fr(0)459 1840 y FB(\)\))i(is)f(relativ)o(ely)e(simple.)23 b(No)o(w)18 b Fy(top)1132 1822 y Fw(b)1132 1853 y Fr(!)1170 1840 y FB(\()p Fy(s)p FB(\))e Fx(\))1297 1847 y Fr(R)1327 1852 y Fh(1)1362 1840 y Fy(top)1427 1822 y Fw(b)1427 1853 y Fr(!)1465 1840 y FB(\()p Fy(t)p FB(\))h(is)h(a)g(consequence)-59 1901 y(of)74 2004 y Fy(top)139 1983 y 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(imme)o(diately)d(from)i(the)h(ab)q(o)o(v)o(e)g(claim)e(that)644 1806 y Fy(D)i FB(:)29 b Fy(s)766 1813 y Fs(1)800 1806 y Fx(\025)839 1813 y Fs(2)872 1806 y Fy(s)895 1813 y Fs(2)929 1806 y Fx(\025)968 1813 y Fs(2)1001 1806 y Fy(s)1024 1813 y Fs(3)1057 1806 y Fx(\025)1096 1813 y Fs(2)1130 1806 y Fy(:)8 b(:)g(:)-59 1909 y FB(If)22 b(there)g(w)o(ere)g(only)g (\014nitely)f(man)o(y)g Fy(s)691 1916 y Fw(j)734 1909 y Fy(>)772 1916 y Fs(2)817 1909 y Fy(s)840 1916 y Fw(j)r Fs(+1)925 1909 y FB(steps)i(in)f Fy(D)q FB(,)j(then)d(there)g(w)o(ould) h(b)q(e)g(an)g(in\014nite)-59 1970 y(subsequence)626 2030 y Fy(D)667 2009 y Fr(0)693 2030 y FB(:)30 b Fy(s)760 2037 y Fw(l)p 799 2032 V 795 2030 a Fy(>)11 b(s)867 2037 y Fw(l)p Fs(+1)p 951 2032 V 947 2030 a Fy(>)g(s)1019 2037 y Fw(l)p Fs(+2)p 1102 2032 V 1098 2030 a Fy(>)g(:)d(:)g(:)-59 2114 y FB(in)17 b(con)o(trast)h(to)g(the)f(w)o(ell-foundedness)h(of)p 780 2116 V 28 w Fy(>)p FB(.)25 b(Hence)17 b(there)g(are)g(in\014nitely) f(man)o(y)g Fy(s)1571 2121 y 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Fr(0)1710 2818 y Fy(;)8 b Fx(V)s FB(\))24 b(is)f(a)63 2878 y(substitution,)16 b(then,)g(b)o(y)f (Lemma)f(8.3.40,)j Fy(l)q(\033)e(>)966 2885 y Fs(3)999 2878 y Fy(s)1022 2885 y Fw(j)1041 2878 y Fy(\033)i FB(and)g Fy(l)q(\033)e(>)1278 2885 y Fs(3)1312 2878 y Fy(t)1330 2885 y Fw(j)1348 2878 y Fy(\033)i FB(for)g(all)f Fy(j)g Fx(2)e(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(.)p eop %%Page: 151 159 151 158 bop 0 -39 a Fv(8.4.)38 b(COMPOSABLE)16 b(SYSTEMS)1164 b FB(151)60 94 y(4.)24 b Fy(>)160 101 y Fs(3)196 94 y FB(has)17 b(the)f(subterm)f(prop)q(ert)o(y)h(b)o(y)g(de\014nition.)0 225 y(In)g(other)g(w)o(ords,)h(\()p Fx(F)401 232 y Fs(1)432 225 y Fx(])11 b(F)517 204 y Fr(0)529 225 y Fy(;)d Fx(R)593 232 y Fs(1)613 225 y FB(\))16 b(is)g(decreasing)g(w.r.t.)f Fy(>)1101 232 y Fs(3)1121 225 y FB(.)21 b Fq(2)0 374 y Fz(Prop)r(osition)d(8.3.43)23 b FB(Let)29 b(\()p Fx(F)630 381 y Fs(1)650 374 y Fy(;)8 b Fx(R)714 381 y Fs(1)733 374 y FB(\))p Fy(;)g(:)g(:)g(:)g(;)g FB(\()p Fx(F)915 381 y Fw(n)939 374 y Fy(;)g Fx(R)1003 381 y Fw(n)1026 374 y FB(\))29 b(b)q(e)g(pairwise)f(constructor-sharing)i(CTRSs)0 434 y(consisting)20 b(of)g(\014nitely)f(man)o(y)f(rewrite)h(rules.)32 b(If)19 b(the)h(systems)f(are)h(decreasing,)g(then)f(the)h(function)0 494 y Fy(n)-8 b(f)19 b FB(:)13 b Fx(T)g FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))13 b Fx(!)h(S)t FB(\()p Fx(T)e FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\)\))16 b(de\014ned)g(b)o(y)499 611 y Fy(n)-8 b(f)5 b FB(\()p Fy(s)p FB(\))14 b(=)g Fx(f)p Fy(t)f Fx(2)h(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))15 b Fx(j)h Fy(s)e Fq(;)1087 591 y Fr(\003)1120 611 y Fy(t;)24 b(t)14 b Fx(2)g Fy(N)5 b(F)i FB(\()p Fq(;)p FB(\))p Fx(g)0 728 y FB(is)16 b(computable.)0 809 y Fz(Pro)r(of:)j FB(By)12 b(Theorem)f(8.3.32,)i Fq(;)f FB(is)g(terminating.)18 b(The)12 b(computabilit)o(y)e(of)j(the)f(function)g Fy(n)-8 b(f)17 b FB(is)12 b(sho)o(wn)0 869 y(b)o(y)k(induction)g(on)g(the)g(w)o (ell-founded)g(partial)g(ordering)g(\()p Fx(T)d FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))o Fy(;)g Fq(;)1336 851 y Fs(+)1365 869 y FB(\).)22 b(Let)16 b Fy(s)e Fx(2)g(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))o(,)16 b(and)g(let)141 986 y Fy(n)-8 b(f)p 178 986 15 2 v 192 993 a Fw(j)210 986 y FB(\()p Fy(s)p FB(\))14 b(=)g Fx(f)p Fy(t)f Fx(2)h(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))15 b Fx(j)h Fy(s)e Fx(!)748 966 y Fs(+)748 999 y Fr(R)778 1004 y Fg(j)810 986 y Fy(t;)24 b(t)13 b Fx(2)h Fy(N)5 b(F)i FB(\()p Fx(F)e Fy(;)j Fx(R)1151 993 y Fw(j)1169 986 y FB(\))p Fx(g)14 b FB(=)g Fx(f)p Fy(t)f Fx(2)h(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))15 b Fx(j)h Fy(s)e Fq(;)1690 993 y Fr(R)1720 998 y Fg(j)1752 986 y Fy(t)p Fx(g)p Fy(:)0 1103 y FB(Note)j(that)h(the) f(CTRSs)h(\()p Fx(F)5 b Fy(;)j Fx(R)605 1110 y Fs(1)625 1103 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h FB(\()p Fx(F)d Fy(;)j Fx(R)877 1110 y Fw(n)900 1103 y FB(\))18 b(are)f(decreasing)g(b) o(y)g(Prop)q(osition)i(8.3.42.)25 b(Th)o(us,)18 b(for)0 1164 y(an)o(y)h Fy(j)j Fx(2)d(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(,)19 b(the)g(\014nite)g(set)g Fy(n)-8 b(f)p 766 1164 V 780 1171 a Fw(j)799 1164 y FB(\()p Fy(s)p FB(\))19 b(is)g(computable)f(according)h(to)h(Prop)q(osition)g(8.1.13.) 31 b(If)0 1224 y Fy(n)-8 b(f)p 37 1224 V 51 1231 a Fw(j)69 1224 y FB(\()p Fy(s)p FB(\))23 b(is)g(empt)o(y)e(for)j(all)e Fy(j)29 b Fx(2)c(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)g(;)g(n)p Fx(g)p FB(,)24 b(then)f Fy(s)j Fx(2)g Fy(N)5 b(F)i FB(\()p Fx(F)t Fy(;)h Fx(R)1311 1231 y Fw(j)1329 1224 y FB(\))23 b(for)h(all)e Fy(j)29 b Fx(2)d(f)p FB(1)p Fy(;)8 b(:)g(:)g(:)f(;)h(n)p Fx(g)23 b FB(and)0 1284 y(th)o(us)f Fy(s)h Fx(2)g Fy(N)5 b(F)i FB(\()p Fq(;)p FB(\).)37 b(In)21 b(this)h(case)g Fy(n)-8 b(f)5 b FB(\()p Fy(s)p FB(\))23 b(=)g Fx(f)p Fy(s)p Fx(g)e FB(and)i(w)o(e)e(are)h(done.)37 b(Otherwise,)22 b(the)g(\014nite)f(set)0 1344 y Fy(R)p FB(\()p Fy(s)p FB(\))c(=)170 1311 y Ft(S)204 1324 y Fw(n)204 1356 y(j)r Fs(=1)276 1344 y Fy(n)-8 b(f)p 313 1344 V 327 1351 a Fw(j)345 1344 y FB(\()p Fy(s)p FB(\))18 b(of)h(all)e(one)h(step)g (reducts)g(of)g Fy(s)g FB(w.r.t.)f Fq(;)g FB(is)h(not)h(empt)o(y)l(.)24 b(Let)18 b Fy(t)e Fx(2)h Fy(R)p FB(\()p Fy(s)p FB(\).)27 b(Since)0 1404 y Fy(s)14 b Fq(;)f Fy(t)p FB(,)h(it)f(follo)o(ws)g(from) f(the)i(induction)f(h)o(yp)q(othesis)g(that)h(the)f(\014nite)g(set)h Fy(n)-8 b(f)5 b FB(\()p Fy(t)p FB(\))13 b(is)g(computable.)19 b(Hence)0 1465 y(the)d(\014nite)g(set)g Fy(n)-8 b(f)5 b FB(\()p Fy(s)p FB(\))14 b(=)461 1431 y Ft(S)496 1475 y Fw(t)p Fr(2)p Fw(R)p Fs(\()p Fw(s)p Fs(\))613 1465 y Fy(n)-8 b(f)5 b FB(\()p Fy(t)p FB(\))16 b(is)g(computable.)k Fq(2)0 1613 y Fz(Theorem)d(8.3.44)23 b FB(Let)d(\()p Fx(F)556 1620 y Fs(1)576 1613 y Fy(;)8 b Fx(R)640 1620 y Fs(1)659 1613 y FB(\))p Fy(;)g(:)g(:)g(:)g(;)g FB(\()p Fx(F)841 1620 y Fw(n)865 1613 y Fy(;)g Fx(R)929 1620 y Fw(n)952 1613 y FB(\))20 b(b)q(e)g(pairwise)g(constructor-sharing)h (CTRSs)f(con-)0 1674 y(sisting)j(of)g(\014nitely)e(man)o(y)g(rewrite)h (rules.)40 b(If)23 b(the)f(systems)g(are)g(decreasing)h(and)g (con\015uen)o(t,)h(then)0 1734 y(their)19 b(com)o(bined)f(system)g(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))19 b(is)h(semi-comple)o(te)d(and)j(the)f (unique)g(normal)g(form)g Fy(s)p Fx(#)g FB(of)h(a)h(term)0 1794 y Fy(s)14 b Fx(2)g(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))16 b(with)g(resp)q(ect)g(to)h(\()p Fx(F)t Fy(;)8 b Fx(R)p FB(\))17 b(is)f(computable)f(b)o(y)h(computing)f(the)h(normal) g(form)f(of)i Fy(s)f FB(with)0 1854 y(resp)q(ect)g(to)h Fq(;)p FB(.)0 1935 y Fz(Pro)r(of:)j FB(Since)15 b(the)f(CTRSs)i(\()p Fx(F)609 1942 y Fs(1)628 1935 y Fy(;)8 b Fx(R)692 1942 y Fs(1)712 1935 y FB(\))p Fy(;)g(:)g(:)g(:)f(;)h FB(\()p Fx(F)894 1942 y Fw(n)918 1935 y Fy(;)g Fx(R)982 1942 y Fw(n)1005 1935 y FB(\))15 b(are)g(decreasing,)f(they)h(are)g (particularly)f(termi-)0 1995 y(nating.)21 b(Hence)13 b(they)h(are)g(complete.)19 b(Semi-compl)o(etene)o(ss)12 b(of)j(\()p Fx(F)5 b Fy(;)j Fx(R)p FB(\))14 b(is)g(a)h(consequence)e (of)i(Theorem)0 2055 y(8.3.19.)33 b(It)19 b(remains)g(to)h(pro)o(v)o(e) f(the)g(computabilit)o(y)f(of)i(the)f(function)h(whic)o(h)f(computes)g (the)h(unique)0 2116 y(normal)14 b(form)f Fy(s)p Fx(#)h(2)g Fy(N)5 b(F)i FB(\()p Fx(F)t Fy(;)h Fx(R)p FB(\))15 b(of)g(a)g(giv)o(en) f(term)f Fy(s)h Fx(2)g(T)e FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\).)20 b(According)14 b(to)h(Theorem)f(8.3.35,)h Fq(;)0 2176 y FB(is)j(complete.)23 b(Moreo)o(v)o(er,)17 b(b)o(y)g(Prop)q (osition)i(8.3.43,)f(the)g(unique)f(normal)g(form)g Fy(s)1570 2137 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 1570 2137 a 18 x Fq(;)1570 2187 y currentpoint grestore moveto 1570 2187 a 1588 2176 a FB(of)h Fy(s)g FB(with)f(resp)q(ect)0 2236 y(to)g Fq(;)f FB(is)g(computable.)k(By)15 b(Corollary)i(8.3.36,)f Fy(s)p Fx(#)e FB(=)f Fy(s)1058 2197 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 1058 2197 a 18 x Fq(;)1058 2247 y currentpoint grestore moveto 1058 2247 a 1074 2236 a FB(whic)o(h)j(concludes)f (the)h(pro)q(of.)23 b Fq(2)0 2435 y FC(8.4)83 b(Comp)r(osable)27 b(Systems)0 2582 y Fo(8.4.1)70 b(The)22 b(Simplifying)e(Prop)r(ert)n(y) 0 2698 y FB(In)15 b(the)f(preceding)h(section,)f(w)o(e)h(ha)o(v)o(e)f (seen)g(that)i(decreasing)f(CTRSs)g(b)q(eha)o(v)o(e)g(\\nicely")f (w.r.t.)g(com)o(bi-)0 2758 y(nations)k(with)f(shared)h(constructors.)26 b(The)17 b(ob)s(jectiv)o(e)f(of)h(this)h(section)f(is)g(to)h(pro)o(v)o (e)e(that)i(the)f(related)0 2818 y(simplifying)10 b(prop)q(ert)o(y)i (is)g(mo)q(dular)g(ev)o(en)f(for)h(comp)q(osable)g(CTRSs.)21 b(This)12 b(will)f(b)q(e)h(done)h(b)o(y)f(a)g(straigh)o(t-)0 2878 y(forw)o(ard)17 b(reduction)f(to)g(Theorem)f(5.3.10.)p eop %%Page: 152 160 152 159 bop -59 -39 a FB(152)451 b Fv(CHAPTER)16 b(8.)38 b(CONDITIONAL)15 b(TERM)h(REWRITING)g(SYSTEMS)-59 94 y Fz(Lemma)g(8.4.1)23 b FB(Let)d(\()p Fx(F)429 101 y Fs(1)449 94 y Fy(;)8 b Fx(R)513 101 y Fs(1)533 94 y FB(\))19 b(and)h(\()p Fx(F)729 101 y Fs(2)748 94 y Fy(;)8 b Fx(R)812 101 y Fs(2)832 94 y FB(\))19 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Fy(;)g(:)g(:)g(:)17 b FB(is)h(a)h(bad)f(sequence)f(of)i(elemen)o(ts)c (from)i Fy(S)k FB(where)d Fy(s)1759 781 y Fw(k)1780 774 y FB(,)g Fy(k)h(>)e FB(1,)0 834 y(is)22 b(an)g(imme)o(diate)d(prop)q (er)j(subterm)f(of)h(some)e Fy(t)935 841 y Fw(j)949 847 y Fg(k)992 834 y FB(suc)o(h)i(that)g Fy(j)1239 841 y Fs(1)1282 834 y Fy(<)h(j)1363 841 y Fw(k)1385 834 y FB(.)37 b(Since)21 b(no)h Fy(t)1660 841 y Fw(i)1697 834 y Fb(-)1729 841 y Fw(emb)1818 834 y Fy(t)1836 841 y Fw(k)1878 834 y FB(\(b)o(y)0 894 y(construction,)h(where)e(1)i Fx(\024)g Fy(i)g(<)g(k)i Fx(\024)e Fy(j)788 901 y Fs(1)822 894 y Fx(\000)15 b FB(1\))22 b(and)g(no)g Fy(t)1132 901 y Fw(i)1169 894 y Fb(-)1201 901 y Fw(emb)1289 894 y Fy(s)1312 901 y Fw(k)1355 894 y FB(\(otherwise)f Fy(t)1614 901 y Fw(i)1651 894 y Fb(-)1683 901 y Fw(emb)1771 894 y Fy(t)1789 901 y Fw(j)1803 907 y Fg(k)1846 894 y FB(since)0 954 y Fy(s)23 961 y Fw(k)65 954 y Fb(-)97 961 y Fw(emb)182 954 y Fy(t)200 961 y Fw(j)214 967 y Fg(k)235 954 y FB(,)g(where)f Fy(i)g(<)g(j)530 961 y Fs(1)570 954 y Fx(\024)h Fy(j)650 961 y Fw(k)671 954 y FB(\))f(it)g(follo)o(ws)g(that)g Fy(t)1054 961 y Fs(1)1074 954 y Fy(;)8 b(t)1114 961 y Fs(2)1133 954 y Fy(;)g(:)g(:)g(:)f(;)h(t)1260 961 y Fw(j)1274 966 y Fh(1)1291 961 y Fr(\000)p Fs(1)1338 954 y Fy(;)g(s)1383 961 y Fs(1)1403 954 y Fy(;)g(s)1448 961 y Fs(2)1468 954 y Fy(;)g(:)g(:)g(:)19 b FB(is)h(a)g(bad)h(sequence)0 1014 y(of)h(elemen)o(ts)e(from)h Fx(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\),)22 b(con)o(tradicting)g(the)f(minimali)o(t)o(y)e (prop)q(ert)o(y)j(of)g(the)g(constructed)g(bad)0 1075 y(sequence)15 b Fy(t)220 1082 y Fs(1)239 1075 y Fy(;)8 b(t)279 1082 y Fs(2)299 1075 y Fy(;)g(:)g(:)g(:)o FB(.)21 b(This)16 b(pro)o(v)o(es)g(the)g(claim.)0 1135 y(Notice)f(that)i(b)o(y) f(Higman's)e(Lemma)g(\()p Fy(S)768 1117 y Fw()p FB(\))16 b(b)q(e)g(a)h(simpli\014cation)d(ordering.)60 2878 y(1.)24 b(Distinct)15 b(v)m(ariables)i Fy(x;)8 b(y)15 b Fx(2)f(V)20 b FB(are)c(incomparable)f(w.r.t.)g(\()p Fx(T)d FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))p Fy(;)g(>)p FB(\).)p eop %%Page: 168 176 168 175 bop -59 -39 a FB(168)805 b Fv(APPENDIX)16 b(A.)31 b(KR)o(USKAL'S)15 b(TREE)i(THEOREM)1 94 y FB(2.)24 b(If)16 b Fy(s)d(>)h(t)p FB(,)i(then)g Fx(V)t Fy(ar)q FB(\()p Fy(t)p FB(\))d Fx(\022)g(V)t Fy(ar)q FB(\()p Fy(s)p FB(\).)-59 210 y Fz(Pro)r(of:)21 b FB(\(1\))15 b(Supp)q(ose)h Fy(x)e(>)g(y)r FB(.)20 b(Then,)15 b(for)g(an)o(y)g(substitution)h Fy(\033)r FB(,)e(it)h(follo)o(ws)g(that)g Fy(x\033)g(>)f(y)r(\033)i FB(b)q(ecause)g Fy(>)-59 271 y FB(is)f(closed)g(under)g(substitutions.) 22 b(With)15 b Fy(\033)g FB(=)f Fx(f)p Fy(x)f Fx(7!)h Fy(y)r(;)23 b(y)15 b Fx(7!)f Fy(x)p Fx(g)h FB(this)g(implies)e Fy(y)i(>)f(x)p FB(,)h(a)g(con)o(tradiction)-59 331 y(to)i Fy(x)c(>)h(y)r FB(.)-59 391 y(\(2\))j(Let)f Fy(s)e(>)g(t)p FB(.)21 b(Assume)15 b(that)i(there)e(is)i(a)f(v)m(ariable)g Fy(x)e Fx(2)g(V)t Fy(ar)q FB(\()p Fy(t)p FB(\))d Fx(n)g(V)t Fy(ar)q FB(\()p Fy(s)p FB(\))o(.)22 b(Then)16 b Fy(t)e FB(=)g Fy(C)t FB([)p Fy(x)p FB(])g(for)j(some)-59 451 y(con)o(text)j Fy(C)t FB([)f(].)34 b(With)20 b Fy(\033)j FB(=)e Fx(f)p Fy(x)g Fx(7!)g Fy(s)p Fx(g)f FB(it)g(follo)o(ws)h(that)g Fy(s)g FB(=)g Fy(s\033)i(>)e(t\033)h FB(=)f Fy(C)t FB([)p Fy(s)p FB(],)f(con)o(tradicting)g(the)-59 511 y(subterm)15 b(prop)q(ert)o(y)h(or)h(irre\015exivit)o(y)c(of)j(\()p Fx(T)d FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))o Fy(;)g(>)p FB(\).)21 b Fq(2)-59 639 y Fz(Lemma)16 b(A.2.5)23 b FB(Let)c(\()p Fx(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))p Fy(;)g(>)p FB(\))17 b(b)q(e)h(a)h(simpli\014cation)d(ordering.)27 b(Moreo)o(v)o(er,)17 b(let)h Fx(F)1649 621 y Fr(0)1678 639 y FB(and)h Fx(V)1810 621 y Fr(0)1840 639 y FB(b)q(e)-59 699 y(subsets)d(of)f Fx(F)20 b FB(and)c Fx(V)t FB(,)f(resp)q(ectiv)o (ely)l(.)j(Then)d(\()p Fx(T)e FB(\()p Fx(F)902 681 y Fr(0)914 699 y Fy(;)8 b Fx(V)971 681 y Fr(0)982 699 y FB(\))p Fy(;)g(>)p FB(\),)14 b(the)h(restriction)g(of)g Fy(>)g FB(to)h Fx(T)d FB(\()p Fx(F)1690 681 y Fr(0)1701 699 y Fy(;)8 b Fx(V)1758 681 y Fr(0)1769 699 y FB(\),)15 b(is)g(a)-59 759 y(simpli\014cation)f(ordering.)-59 840 y Fz(Pro)r(of:)21 b FB(T)l(rivial.)f Fq(2)-59 967 y Fz(Prop)r(osition)e (A.2.6)24 b FB(Let)19 b(\()p Fx(T)13 b FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))p Fy(;)g(>)p FB(\))18 b(b)q(e)i(a)g (simpli\014cation)d(ordering.)31 b(Moreo)o(v)o(er,)18 b(let)h Fx(F)1765 949 y Fr(0)1796 967 y FB(b)q(e)g(a)-59 1028 y(\014nite)d(subset)g(of)h Fx(F)5 b FB(.)21 b(Then)16 b(\()p Fx(T)d FB(\()p Fx(F)593 1010 y Fr(0)604 1028 y Fy(;)8 b Fx(V)t FB(\))p Fy(;)g(>)p FB(\))16 b(is)g(a)g(w)o(ell-founded) g(simpli\014cation)e(ordering.)-59 1109 y Fz(Pro)r(of:)27 b FB(Supp)q(ose)20 b(that)g(\()p Fx(T)13 b FB(\()p Fx(F)542 1091 y Fr(0)553 1109 y Fy(;)8 b Fx(V)t FB(\))p Fy(;)g(>)p FB(\))18 b(is)h(not)h(w)o(ell-founded,)f(that)g(is)g(to)h(sa)o(y)l(,)f (there)g(is)g(an)h(in\014nite)-59 1169 y(sequence)15 b Fy(t)161 1176 y Fs(1)194 1169 y Fy(>)f(t)264 1176 y Fs(2)297 1169 y Fy(>)g(t)367 1176 y Fs(3)400 1169 y Fy(>)g(:)8 b(:)g(:)15 b FB(of)h(elemen)o(ts)d(from)i Fx(T)e FB(\()p Fx(F)993 1145 y Fr(0)1005 1169 y Fy(;)8 b Fx(V)t FB(\).)21 b(According)15 b(to)h(Lemma)e(A.2.4,)h(w)o(e)h(ha)o(v)o(e)-59 1229 y Fx(V)t Fy(ar)q FB(\()p Fy(t)62 1236 y Fw(i)75 1229 y FB(\))j Fx(\022)f(V)t Fy(ar)q FB(\()p Fy(t)291 1236 y Fs(1)310 1229 y FB(\))g(for)i(ev)o(ery)d Fy(i)h Fx(2)h Fl(I)-7 b(N)p FB(.)30 b(Hence)17 b(there)i(is)f(an)i(in\014nite) e(sequence)f Fy(t)1514 1236 y Fs(1)1552 1229 y Fy(>)h(t)1626 1236 y Fs(2)1664 1229 y Fy(>)g(t)1738 1236 y Fs(3)1776 1229 y Fy(>)h(:)8 b(:)g(:)-59 1289 y FB(of)25 b(elemen)o(ts)e(from)h Fx(T)13 b FB(\()p Fx(F)438 1265 y Fr(0)449 1289 y Fy(;)8 b Fx(V)t Fy(ar)q FB(\()p Fy(t)592 1296 y Fs(1)611 1289 y FB(\)\).)48 b(\()p Fx(T)13 b FB(\()p Fx(F)830 1271 y Fr(0)841 1289 y Fy(;)8 b Fx(V)t Fy(ar)q FB(\()p Fy(t)984 1296 y Fs(1)1003 1289 y FB(\)\))p Fy(;)g Fx(\025)p FB(\))25 b(is)g(a)g(re\015exiv)o(e)e(partial)i(ordering)h(\(cf.)-59 1349 y(Lemma)13 b(2.1.17\).)22 b(W)l(e)15 b(further)h(kno)o(w)g(from)e 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o(ell.)i(In)d(particular,)f(there)g(is)h(no)h(in\014nite)d(sequence)h Fy(t)1232 1537 y Fs(1)1265 1530 y Fy(>)f(t)1335 1537 y Fs(2)1368 1530 y Fy(>)g(t)1438 1537 y Fs(3)1471 1530 y Fy(>)g(:)8 b(:)g(:)15 b FB(of)h(elemen)o(ts)d(of)-59 1590 y Fx(T)g FB(\()p Fx(F)40 1566 y Fr(0)52 1590 y Fy(;)8 b Fx(V)t Fy(ar)q FB(\()p Fy(t)195 1597 y Fs(1)214 1590 y FB(\)\))252 1572 y Fs(1)271 1590 y FB(,)13 b(a)f(con)o(tradiction)f (to)h(the)g(assumption.)19 b(Therefore,)12 b(\()p Fx(T)h FB(\()p Fx(F)1386 1572 y Fr(0)1397 1590 y Fy(;)8 b Fx(V)t FB(\))p Fy(;)g(>)p FB(\))j(is)h(w)o(ell-founded.)-59 1650 y Fq(2)14 1778 y FB(It)h(has)h(already)g(b)q(een)f(sho)o(wn)h(in)f (Example)f(5.1.16)i(that)g(the)f(\014niteness)g(assumption)g(on)h Fx(F)1721 1760 y Fr(0)1746 1778 y FB(cannot)-59 1838 y(b)q(e)k(dropp)q(ed.)27 b(Note)18 b(that)g(the)g(in\014nit)o(y)e(of)j (the)e(set)h(of)g(v)m(ariables)g(do)q(es)h(no)f(harm,)f(since)g(only)h (\014nitely)-59 1898 y(man)o(y)d(v)m(ariables)h(can)g(o)q(ccur)h(in)f (an)o(y)g(sequence)f Fy(t)866 1905 y Fs(1)899 1898 y Fy(>)f(t)969 1905 y Fs(2)1002 1898 y Fy(>)g(t)1072 1905 y Fs(3)1105 1898 y Fy(>)g(:)8 b(:)g(:)o FB(.)14 1979 y(No)o(w)16 b(w)o(e)g(are)g(able)g(to)h(pro)o(v)o(e)e(Theorem)g (5.1.17.)-59 2106 y Fz(Theorem)h(A.2.7)24 b FB(Let)15 b Fx(!)f FB(b)q(e)g(a)h(binary)f(relation)g(on)g Fx(T)f FB(\()p Fx(F)5 b Fy(;)j Fx(V)t FB(\))13 b(whic)o(h)h(is)g(con)o(tained) g(in)g(some)f(simpli-)-59 2167 y(\014cation)g(ordering)f Fy(>)p FB(.)20 b(If,)13 b(for)f(ev)o(ery)f(reduction)h(sequence,)g(the) h(set)f(of)h(all)f(function)g(sym)o(b)q(ols)g(o)q(ccurring)-59 2227 y(in)k(the)g(\(terms)f(in)g(that\))i(reduction)f(sequence)f(is)h (\014nite,)f(then)h Fx(!)g FB(is)h(terminating.)-59 2308 y Fz(Pro)r(of:)23 b FB(Supp)q(ose)18 b(that)f Fx(!)g FB(is)g(not)g(terminating,)e(i.e.,)g(there)h(exists)h(an)g(in\014nite)f (reduction)h(sequence)-59 2368 y Fy(t)-41 2375 y Fs(1)0 2368 y Fx(!)22 b Fy(t)90 2375 y Fs(2)131 2368 y Fx(!)g Fy(t)221 2375 y Fs(3)262 2368 y 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2549 y Fy(;)8 b Fx(V)t FB(\))p Fy(;)g(>)p FB(\))16 b(\(see)g(Prop)q(osition)h(A.2.6\).)k Fq(2)14 2676 y FB(W)l(e)16 b(conclude)g(this)g(app)q(endix)g(b)o(y)g(men)o (tioning)f(that)h(Krusk)m(al's)h(Theorem)e(has)i(b)q(een)f(generalized) -59 2736 y(b)o(y)g(Puel)f([Pue89)q(].)p -59 2781 780 2 v -3 2813 a Fn(1)16 2829 y Fm(According)j(to)h(Lemma)c(2.1.17)h(the)j (asso)q(ciated)g(partial)f(ordering)g(of)g(\()p Fk(T)10 b Fm(\()p Fk(F)1285 2808 y Fa(0)1297 2829 y Fe(;)d Fk(V)s Fe(ar)q Fm(\()p Fe(t)1417 2835 y Fn(1)1436 2829 y Fm(\)\))p Fe(;)g Fk(\025)p Fm(\))18 b(coincides)h(with)f(the)-59 2878 y(original)12 b(simpli\014cation)f(ordering.)p eop %%Page: 169 177 169 176 bop 0 446 a FE(Biblio)q(graph)m(y)0 697 y FB([Bac89])98 b(L.)16 b(Bac)o(hmair.)j Fu(Canonic)n(al)f(Equational)h(Pr)n(o)n(ofs)p FB(.)g(Birkh\177)-24 b(auser,)15 b(Boston,)i(1989.)0 807 y([Bar84])101 b(H.P)l(.)17 b(Barendregt.)26 b Fu(The)20 b(L)n(amb)n(da)d(Calculus:)27 b(Its)20 b(Syntax)g(and)f(Semantics)p FB(.)28 b(Studies)18 b(in)255 867 y(Logic)e(and)h(the)f(F)l(oundations) i(of)e(Mathematics.)e(North-Holland,)i(1984.)23 b(2nd)16 b(edition.)0 977 y([Bar90])101 b(F.)20 b(Barbanera.)36 b(Adding)21 b(Algebraic)f(Rewriting)h(to)g(the)g(Calculus)g(of)g (Constructions:)255 1037 y(Strong)16 b(Normalization)e(Preserv)o(ed.)k (In)d Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)h(the)g(2nd)g(International)h (Work-)255 1097 y(shop)13 b(on)h(Conditional)g(and)g(T)l(yp)n(e)n(d)f (R)n(ewriting)g(Systems)p FB(,)g(pages)g(260{271.)h(Lecture)e(Notes)255 1158 y(in)k(Computer)f(Science)g Fz(516)p FB(,)g(Berlin:)20 b(Springer)c(V)l(erlag,)f(1990.)0 1267 y([Bar92])101 b(H.P)l(.)13 b(Barendregt.)19 b(Lam)o(b)q(da)c(Calculi)e(with)i(T)o(yp) q(es.)k(In)14 b(S.)h(Abramsky)l(,)d(D.)j(Gabba)o(y)l(,)g(and)255 1328 y(T.)h(Maibaum,)f(editors,)i Fu(Handb)n(o)n(ok)h(of)f(L)n(o)n(gic) h(in)g(Computer)f(Scienc)n(e)p FB(,)i(v)o(olume)14 b(2,)j(pages)255 1388 y(117{309.)h(Oxford)e(Univ)o(ersit)o(y)e(Press,)i(1992.)0 1498 y([BD86])107 b(L.)22 b(Bac)o(hmair)e(and)j(N.)e(Dersho)o(witz.)38 b(Comm)o(utation,)21 b(T)l(ransformation,)j(and)f(T)l(ermi-)255 1558 y(nation.)29 b(In)19 b Fu(8th)h(International)h(Confer)n(enc)n(e)f (on)h(A)o(utomate)n(d)e(De)n(duction)p FB(,)h(pages)g(5{20.)255 1618 y(Lecture)15 b(Notes)i(in)e(Computer)h(Science)e Fz(230)p FB(,)i(Berlin:)k(Springer)c(V)l(erlag,)f(1986.)0 1728 y([BF93])112 b(F.)15 b(Barbanera)i(and)g(M.)e(F)l(ern\023)-24 b(andez.)21 b(Mo)q(dularit)o(y)16 b(of)g(T)l(ermination)f(and)i (Con\015uence)f(in)255 1788 y(Com)o(binations)g(of)i(Rewrite)e(Systems) g(with)i Fy(\025)1137 1795 y Fw(!)1162 1788 y FB(.)25 b(In)17 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)h(the)h(20th)g(Interna-) 255 1848 y(tional)e(Col)r(lo)n(quium)g(on)g(A)o(utomata,)g(L)n (anguages)g(and)f(Pr)n(o)n(gr)n(amming)p FB(.)d(Lecture)i(Notes)g(in) 255 1908 y(Computer)g(Science)g Fz(700)p FB(,)h(Berlin:)j(Springer)d(V) l(erlag,)g(1993.)0 2018 y([BGF94])74 b(F.)20 b(Barbanera,)j(Geuv)o (ers,)e(and)h(M.)e(F)l(ern\023)-24 b(andez.)35 b(Mo)q(dularit)o(y)21 b(of)g(Strong)h(Normaliza-)255 2078 y(tion)e(and)g(Con\015uence)g(in)g (the)g(Algebraic)f Fy(\025)p FB(-cub)q(e.)32 b(In)20 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)h(the)g(9th)g(IEEE)255 2138 y(Symp)n(osium)16 b(on)i(L)n(o)n(gic)f(in)g(Computer)h(Scienc)n(e) p FB(,)f(1994.)23 b(T)l(o)17 b(app)q(ear.)0 2248 y([Bid81])103 b(M.)21 b(Bidoit.)39 b(Une)22 b(M)o(\023)-23 b(etho)q(de)23 b(de)f(Pr)o(\023)-23 b(esen)o(tation)23 b(de)f(T)o(yp)q(es)g (Abstraits:)34 b(Applications.)255 2308 y(Th)o(\023)-23 b(ese,)15 b(Univ)o(ersit)o(\023)-23 b(e)14 b(de)i(P)o(aris-Sud,)h(Orsa) o(y)l(,)e(F)l(rance,)h(1981.)0 2418 y([BK86])106 b(J.A.)15 b(Bergstra)i(and)h(J.W.)e(Klop.)23 b(Conditional)17 b(Rewrite)f(Rules:) 21 b(Con\015uence)c(and)h(T)l(er-)255 2478 y(mination.)42 b Fu(Journal)25 b(of)f(Computer)g(and)h(System)g(Scienc)n(es)h Fz(32)p Fu(\(3\))p FB(,)f(pages)g(323{362,)255 2538 y(1986.)0 2648 y([BKM89])61 b(J.A.)14 b(Bergstra,)h(J.W.)g(Klop,)g(and)h(A.)e (Middeldorp.)19 b Fu(T)l(ermherschrijfsystemen)p FB(.)h(Klu)o(w)o(er) 255 2708 y(Bedrijfsw)o(etensc)o(happ)q(en,)14 b(Dev)o(en)o(ter,)g (1989.)0 2818 y([BM84])99 b(J.A.)13 b(Bergstra)i(and)h(J.-J.Ch.)e(Mey)o (er.)k(On)d(Sp)q(ecifying)f(Sets)h(of)g(In)o(tegers.)j Fu(Elektr)n(onische)255 2878 y(Informationsver)n(arb)n(eitung)g(und)f (Kyb)n(ernetik)i Fz(20)f Fu(\(10/11\))p FB(,)d(pages)i(531{541,)h (1984.)938 3003 y(169)p eop %%Page: 170 178 170 177 bop -59 -39 a FB(170)1487 b Fv(BIBLIOGRAPHY)-59 94 y FB([BT88])109 b(V.)13 b(Breazu-T)l(annen.)k(Com)o(bining)c (Algebra)g(and)i(Higher)e(Order)h(T)o(yp)q(es.)j(In)d Fu(Pr)n(o)n(c)n(e)n(e)n(dings)196 154 y(of)j(the)h(3r)n(d)e(IEEE)h (Symp)n(osium)g(on)h(L)n(o)n(gic)e(in)i(Computer)g(Scienc)n(e)p FB(,)f(pages)g(82{90,)h(1988.)-59 254 y([BTG89])71 b(V.)17 b(Breazu-T)l(annen)h(and)h(J.)f(Gallier.)25 b(P)o(olymorphic)16 b(Rewriting)i(Conserv)o(es)g(Algebraic)196 314 y(Strong)g (Normalization)d(and)j(Con\015uence.)23 b(In)17 b Fu(Pr)n(o)n(c)n(e)n 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(Computer)h(Scienc)n(e)i Fz(126)p Fu(\(1\))p FB(,)d(pages)g(31{52,)196 975 y(1994.)-59 1075 y([CF58])112 b(H.B.)14 b(Curry)i(and)h(R.)f(F)l (eys.)k Fu(Combinatory)d(L)n(o)n(gic)p FB(,)f(v)o(olume)e(1.)21 b(North-Holland,)16 b(1958.)-59 1175 y([Ch)o(u41])91 b(A.)21 b(Ch)o(urc)o(h.)39 b Fu(The)23 b(Calculi)h(of)f(L)n(amb)n(da)f (Conversion)p FB(.)40 b(Princeton)22 b(Univ)o(ersit)o(y)e(Press,)196 1235 y(1941.)-59 1335 y([Cou90])93 b(B.)17 b(Courcelle.)28 b(Graph)19 b(Rewriting:)26 b(An)18 b(Algebraic)g(and)h(Logic)h(Approac) o(h.)28 b(In)18 b(L.)h(v)m(an)196 1396 y(Leeu)o(w)o(en,)h(editor,)g Fu(Handb)n(o)n(ok)h(of)g(The)n(or)n(etic)n(al)g(Computer)g(Scienc)n(e)p FB(,)h(v)o(olume)c(B,)i(c)o(hap-)196 1456 y(ter)15 b(5.)i (North-Holland,)e(1990.)-59 1556 y([Dau92])91 b(M.)15 b(Dauc)o(het.)23 b(Sim)o(ulation)14 b(of)j(T)l(uring)g(Mac)o(hines)f(b) o(y)g(a)h(Regular)g(Rewrite)e(Rule.)22 b Fu(The)n(o-)196 1616 y(r)n(etic)n(al)17 b(Computer)g(Scienc)n(e)j Fz(103)p Fu(\(2\))p FB(,)15 b(pages)i(409{420,)i(1992.)-59 1716 y([DCK94])69 b(R.)17 b(Di)g(Cosmo)h(and)g(D.)f(Kesner.)25 b(Com)o(bining)16 b(First)i(Order)f(Algebraic)f(Rewriting)h(Sys-)196 1776 y(tems,)j(Recursion)h(and)g(Extensional)g(Lam)o(b)q(da)h(Calculi.) 34 b(In)21 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)g(the)i(21st)196 1836 y(International)18 b(Col)r(lo)n(quium)h(on)f(A)o(utomata,)f(L)n (anguages)h(and)f(Pr)n(o)n(gr)n(amming)p FB(,)e(1994.)22 b(T)l(o)196 1897 y(app)q(ear.)-59 1997 y([Der79])101 b(N.)21 b(Dersho)o(witz.)39 b(A)22 b(Note)g(on)g(Simpli\014cation)e (Orderings.)40 b Fu(Information)22 b(Pr)n(o)n(c)n(essing)196 2057 y(L)n(etters)17 b Fz(9)p Fu(\(5\))p FB(,)e(pages)i(212{215,)i (1979.)-59 2157 y([Der81])101 b(N.)14 b(Dersho)o(witz.)20 b(T)l(ermination)14 b(of)i(Linear)f(Rewriting)g(Systems)f (\(preliminary)f(v)o(ersion\).)196 2217 y(In)g Fu(Pr)n(o)n(c)n(e)n(e)n (dings)h(of)g(the)i(8th)f(International)h(Col)r(lo)n(quium)g(on)g(A)o (utomata,)f(L)n(anguages)h(and)196 2277 y(Pr)n(o)n(gr)n(amming)p FB(,)g(pages)j(448{458.)i(Lecture)d(Notes)g(in)g(Computer)f(Science)f Fz(115)p FB(,)j(Berlin:)196 2338 y(Springer)d(V)l(erlag,)f(1981.)-59 2438 y([Der82])101 b(N.)18 b(Dersho)o(witz.)30 b(Orderings)19 b(for)h(T)l(erm-Rewriting)d(Systems.)29 b Fu(The)n(or)n(etic)n(al)20 b(Computer)196 2498 y(Scienc)n(e)f Fz(17)p Fu(\(3\))p FB(,)d(pages)h(279{301,)h(1982.)-59 2598 y([Der87])101 b(N.)20 b(Dersho)o(witz.)37 b(T)l(ermination)20 b(of)i(Rewriting.)37 b Fu(Journal)22 b(of)h(Symb)n(olic)g(Computation)196 2658 y Fz(3)p Fu(\(1\))p FB(,)15 b(pages)i(69{116,)h(1987.)-59 2758 y([Der93])101 b(N.)20 b(Dersho)o(witz.)34 b(Hierarc)o(hical)18 b(T)l(ermination.)34 b(Draft,)21 b(Dept.)g(of)g(Computer)f(Science,)196 2818 y(Hebrew)14 b(Univ)o(ersit)o(y)l(,)f(Jerusalem)h(91904,)j(Israel,) d(1993.)22 b(Revised)14 b(v)o(ersion)h(to)h(app)q(ear)h(in:)196 2878 y(4th)f(In)o(ternational)g(W)l(orkshop)h(on)g(Conditional)f(T)l (erm)f(Rewriting)h(Systems.)p eop %%Page: 171 179 171 178 bop 0 -39 a Fv(BIBLIOGRAPHY)1484 b FB(171)0 94 y([DJ90])117 b(N.)12 b(Dersho)o(witz)h(and)g(J.P)l(.)g(Jouannaud.)k (Rewrite)12 b(Systems.)j(In)e(L.)g(v)m(an)g(Leeu)o(w)o(en,)g(editor,) 255 154 y Fu(Handb)n(o)n(ok)h(of)f(The)n(or)n(etic)n(al)h(Computer)g (Scienc)n(e)p FB(,)g(v)o(olume)d(B,)g(c)o(hapter)h(6.)g(North-Holland,) 255 214 y(1990.)0 322 y([DJK93])79 b(N.)17 b(Dersho)o(witz,)i(J.P)l(.)e (Jouannaud,)k(and)e(J.W.)f(Klop.)27 b(More)19 b(Problems)e(in)h (Rewriting.)255 382 y(In)c Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)h(the)g (5th)g(International)i(Confer)n(enc)n(e)e(on)h(R)n(ewriting)f(T)l(e)n (chniques)i(and)255 442 y(Applic)n(ations)p FB(,)j(pages)h(468{487.)h (Lecture)d(Notes)g(in)g(Computer)g(Science)f Fz(690)p FB(,)i(Berlin:)255 502 y(Springer)16 b(V)l(erlag,)f(1993.)0 610 y([DM79])97 b(N.)20 b(Dersho)o(witz)g(and)h(Z.)g(Manna.)35 b(Pro)o(ving)21 b(T)l(ermination)e(with)i(Multiset)e(Orderings.)255 670 y Fu(Communic)n(ations)e(of)h(the)f(A)o(CM)g Fz(22)p Fu(\(8\))p FB(,)f(pages)h(465{476,)i(1979.)0 778 y([DO90])104 b(N.)12 b(Dersho)o(witz)h(and)i(M.)d(Ok)m(ada.)18 b(A)13 b(Rationale)g(for)h(Conditional)g(Equational)g(Program-)255 838 y(ming.)20 b Fu(The)n(or)n(etic)n(al)c(Computer)i(Scienc)n(e)h Fz(75)p FB(,)d(pages)h(111{138,)h(1990.)0 946 y([DOS88])77 b(N.)18 b(Dersho)o(witz,)i(M.)f(Ok)m(ada,)i(and)f(G.)f(Siv)m(akumar.)30 b(Canonical)20 b(Conditional)g(Rewrite)255 1006 y(Systems.)j(In)18 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)i(the)g(9th)g(Confer)n(enc)n(e)g (on)g(A)o(utomate)n(d)g(De)n(duction)p FB(,)f(pages)255 1066 y(538{549.)d(Lecture)e(Notes)g(in)g(Computer)f(Science)f Fz(310)p FB(,)j(Berlin:)k(Springer)13 b(V)l(erlag,)g(1988.)0 1174 y([DP85])109 b(N.)21 b(Dersho)o(witz)g(and)i(D.A.)d(Plaisted.)38 b(Logic)22 b(Programming)e(cum)h(Applicativ)o(e)e(Pro-)255 1234 y(gramming.)38 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)22 b(of)h(the)h(2nd)f(IEEE)g(Symp)n(osium)g(on)g(L)n(o)n(gic)g(Pr)n(o)n (gr)n(amming)p FB(,)255 1294 y(pages)17 b(54{66,)g(1985.)0 1402 y([DP87])109 b(N.)21 b(Dersho)o(witz)h(and)h(D.A.)f(Plaisted.)39 b(Equational)23 b(Programming.)38 b(In)23 b(J.E.)e(Ha)o(y)o(es,)255 1462 y(D.)16 b(Mic)o(hie,)f(and)j(J.)e(Ric)o(hards,)h(editors,)f Fu(Machine)j(Intel)r(ligenc)o(e)i Fz(11)p FB(,)c(pages)g(21{56.)i(Ox-) 255 1522 y(ford)d(Univ)o(ersit)o(y)e(Press,)i(1987.)0 1630 y([Dro89])99 b(K.)23 b(Drosten.)46 b Fu(T)l(ermersetzungssysteme)p FB(.)h(Informatik-F)l(ac)o(h)o(b)q(eric)n(h)o(te)21 b Fz(210)p FB(,)26 b(Springer)255 1690 y(V)l(erlag,)15 b(1989.)0 1798 y([EM85])101 b(H.)19 b(Ehrig)h(and)g(B.)f(Mahr.)32 b Fu(F)l(undamentals)23 b(of)d(A)o(lgebr)n(aic)i(Sp)n(e)n(ci\014c)n (ations)f(I:)g(Equations)255 1858 y(and)c(Initial)h(Semantics)p FB(.)23 b(Springer)16 b(V)l(erlag,)f(1985.)0 1966 y([Gal91])103 b(J.)17 b(Gallier.)23 b(What's)17 b(so)h(Sp)q(ecial)f(ab)q(out)h(Krusk) m(al's)g(Theorem)e(and)h(the)g(Ordinal)g(\000)1844 1973 y Fs(0)1865 1966 y FB(?)24 b(A)255 2026 y(Surv)o(ey)15 b(of)h(some)f(Results)h(in)g(Pro)q(of)h(Theory.)k Fu(A)o(nnals)d(of)g (Pur)n(e)f(and)g(Applie)n(d)h(L)n(o)n(gic)e Fz(53)p FB(,)255 2086 y(pages)h(199{260,)h(1991.)0 2194 y([Ges90])100 b(A.)15 b(Geser.)21 b Fu(R)n(elative)d(T)l(ermination)p FB(.)k(PhD)17 b(thesis,)e(Univ)o(ersit\177)-24 b(at)15 b(P)o(assau,)i(1990.)0 2302 y([Geu89])92 b(O.)11 b(Geup)q(el.)k(Ov)o (erlap)c(Closures)h(and)h(T)l(ermination)e(of)h(T)l(erm)f(Rewriting)g (Systems.)i(T)l(ec)o(h-)255 2362 y(nical)i(Rep)q(ort)i(MIP-8922,)g (Univ)o(ersit\177)-24 b(at)14 b(P)o(assau,)j(1989.)0 2470 y([GG87])103 b(H.)21 b(Ganzinger)h(and)h(R.)e(Giegeric)o(h.)38 b(A)21 b(Note)h(on)h(T)l(ermination)d(in)i(Com)o(binations)g(of)255 2530 y(Heterogeneous)c(T)l(erm)g(Rewriting)g(Systems.)27 b Fu(Bul)r(letin)c(of)d(the)g(Eur)n(op)n(e)n(an)f(Asso)n(ciation)255 2590 y(for)d(The)n(or)n(etic)n(al)h(Computer)h(Scienc)n(e)h Fz(31)p FB(,)d(pages)h(22{28,)h(1987.)0 2698 y([Gra92a])74 b(B.)23 b(Gramlic)o(h.)44 b(Generalized)23 b(Su\016cien)o(t)g (Conditions)j(for)e(Mo)q(dular)h(T)l(ermination)f(of)255 2758 y(Rewriting.)32 b(In)20 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)h (the)h(Thir)n(d)e(International)i(Confer)n(enc)n(e)g(on)g(A)o(lgebr)n (aic)255 2818 y(and)c(L)n(o)n(gic)f(Pr)n(o)n(gr)n(amming)p FB(,)e(pages)i(53{68.)h(Lecture)e(Notes)h(in)f(Computer)g(Science)f Fz(632)p FB(,)255 2878 y(Berlin:)k(Springer)d(V)l(erlag,)g(1992.)p eop %%Page: 172 180 172 179 bop -59 -39 a FB(172)1487 b Fv(BIBLIOGRAPHY)-59 94 y FB([Gra92b])71 b(B.)16 b(Gramlic)o(h.)k(Relating)d(Innermost,)e(W) l(eak,)h(Uniform)f(and)j(Mo)q(dular)f(T)l(ermination)e(of)196 154 y(T)l(erm)f(Rewriting)h(Systems.)20 b(In)c Fu(Confer)n(enc)n(e)i (on)f(L)n(o)n(gic)g(Pr)n(o)n(gr)n(amming)e(and)j(A)o(utomate)n(d)196 214 y(R)n(e)n(asoning)p FB(,)c(pages)i(285{296.)i(Lecture)c(Notes)h(in) g(Arti\014cial)f(In)o(telligence)e Fz(624)p FB(,)j(Springer)196 274 y(V)l(erlag,)g(1992.)-59 386 y([Gra93a])74 b(B.)23 b(Gramlic)o(h.)44 b(Generalized)23 b(Su\016cien)o(t)g(Conditions)i(for) g(Mo)q(dular)g(T)l(ermination)e(of)196 446 y(Rewriting.)e Fu(Applic)n(able)e(A)o(lgebr)n(a)g(in)f(Engine)n(ering,)i(Communic)n (ation)e(and)g(Computing)p FB(,)196 506 y(1993.)k(Extended)16 b(v)o(ersion)g(of)g([Gra92a)r(],)f(to)i(app)q(ear.)-59 618 y([Gra93b])71 b(B.)16 b(Gramlic)o(h.)k(Relating)d(Innermost,)e(W)l (eak,)h(Uniform)f(and)j(Mo)q(dular)f(T)l(ermination)e(of)196 678 y(T)l(erm)k(Rewriting)g(Systems.)33 b(SEKI)20 b(Rep)q(ort)h (SR-93-09,)i(Univ)o(ersit\177)-24 b(at)18 b(Kaiserslautern,)196 738 y(1993.)k(Extended)16 b(v)o(ersion)g(of)g([Gra92b)q(].)-59 850 y([Gra93c])76 b(B.)10 b(Gramlic)o(h.)i(Su\016cien)o(t)e(Conditions) i(for)f(Mo)q(dular)i(T)l(ermination)d(of)h(Conditional)h(T)l(erm)196 910 y(Rewriting)h(Systems.)j(In)e Fu(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)h (the)h(3r)n(d)e(International)j(Workshop)e(on)g(Condi-)196 970 y(tional)g(T)l(erm)f(R)n(ewriting)h(Systems)f(1992)p FB(,)f(pages)h(128{142.)h(Lecture)d(Notes)h(in)g(Computer)196 1031 y(Science)h Fz(656)p FB(,)i(Berlin:)k(Springer)c(V)l(erlag,)f (1993.)-59 1142 y([GTW78])56 b(J.A.)12 b(Goguen,)j(J.W.)e(Thatc)o(her,) h(and)g(E.G.)g(W)l(agner.)k(An)13 b(Initial)g(Algebra)h(Approac)o(h)f (to)196 1202 y(the)i(Sp)q(eci\014cation,)g(Correctness)g(and)i(Implem)o (e)o(n)o(tations)c(of)j(Abstract)g(Data)g(T)o(yp)q(es.)k(In)196 1263 y(R.)15 b(Y)l(eh,)g(editor,)h Fu(Curr)n(ent)h(T)l(r)n(ends)g(in)g (Pr)n(o)n(gr)n(amming)f(Metho)n(dolo)n(gy)p FB(,)f(v)o(olume)f(IV,)h (pages)196 1323 y(80{149.)j(Pren)o(tice)c(Hall,)h(1978.)-59 1434 y([Hig52])104 b(G.)15 b(Higman.)j(Ordering)d(b)o(y)g(Divisibilit)o (y)d(in)j(Abstract)g(Algebras.)20 b Fu(Pr)n(o)n(c.)15 b(L)n(ondon)i(Math-)196 1495 y(ematic)n(al)h(So)n(ciety)g Fz(2)p Fu(\(7\))p FB(,)d(pages)i(326{336,)i(1952.)-59 1606 y([Hin64])101 b(R.)19 b(Hindley)l(.)32 b Fu(The)21 b(Chur)n(ch-R)n(osser)f(Pr)n(op)n(erty)g(and)h(a)g(R)n(esult)h(in)f (Combinatory)g(L)n(o)n(gic)p FB(.)196 1667 y(PhD)16 b(thesis,)g(Univ)o (ersit)o(y)e(of)i(New)o(castle-up)q(on-T)o(yne,)g(1964.)-59 1778 y([Hin74])101 b(R.)13 b(Hindley)l(.)i(An)f(Abstract)f(Ch)o(urc)o (h-Rosser)h(Theorem,)f(P)o(art)h(I)q(I:)f(applications.)k Fu(Journal)196 1838 y(of)g(Symb)n(olic)h(L)n(o)n(gic)f Fz(39)p FB(,)e(pages)i(1{21,)h(1974.)-59 1950 y([HL78])112 b(G.)10 b(Huet)h(and)g(D.S.)g(Lankford.)j(On)d(the)f(Uniform)g(Halting) g(Problem)g(for)h(T)l(erm)e(Rewriting)196 2010 y(Systems.)19 b(Rapp)q(ort)f(lab)q(oria)f(283,)g(IRIA,)d(1978.)-59 2122 y([HM90])97 b(B.)15 b(Ho)o(w)o(ard)i(and)g(J.)g(Mitc)o(hell.)j(Op) q(erational)d(and)g(Axiomatic)d(Seman)o(tics)h(for)i(PCF.)23 b(In)196 2182 y Fu(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)h(the)g(LISP)g(and)h (F)l(unctional)h(Pr)n(o)n(gr)n(amming)c(Confer)n(enc)n(e)p FB(,)j(pages)f(298{)196 2242 y(306.)c(A)o(CM,)e(1990.)-59 2354 y([HO80])104 b(G.)16 b(Huet)h(and)g(D.C.)g(Opp)q(en.)24 b(Equations)17 b(and)h(Rewrite)e(Rules:)22 b(A)17 b(Surv)o(ey.)22 b(In)17 b Fu(F)l(ormal)196 2414 y(L)n(anguage)24 b(The)n(ory)e(Persp)n (e)n(ctives)i(and)g(Op)n(en)g(Pr)n(oblems)p FB(,)h(pages)e(349{405.)i (Academic)196 2474 y(Press,)16 b(1980.)-59 2586 y([Hue80])93 b(G.)16 b(Huet.)21 b(Con\015uen)o(t)c(Reductions:)22 b(Abstract)16 b(Prop)q(erties)h(and)g(Applications)e(to)i(T)l(erm)196 2646 y(Rewriting)e(Systems.)20 b Fu(Journal)d(of)h(the)g(A)o(CM)f Fz(27)p Fu(\(4\))p FB(,)e(pages)i(797{821,)i(1980.)-59 2758 y([JO91])116 b(J.-P)l(.)16 b(Jouannaud)i(and)f(M.)f(Ok)m(ada.)24 b(A)16 b(Computation)g(Mo)q(del)h(for)g(Executable)e(Higher-)196 2818 y(Order)h(Algebraic)f(Sp)q(eci\014cation)i(Languages.)24 b(In)16 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)h(the)g(6th)g(IEEE)f(Sym-) 196 2878 y(p)n(osium)f(on)i(L)n(o)n(gic)f(in)g(Computer)h(Scienc)n(e)p FB(,)f(pages)g(350{361,)i(1991.)p eop %%Page: 173 181 173 180 bop 0 -39 a Fv(BIBLIOGRAPHY)1484 b FB(173)0 94 y([JW86])104 b(J.-P)l(.)12 b(Jouannaud)j(and)f(B.)e(W)l(aldmann.)j (Reductiv)o(e)c(Conditional)j(T)l(erm)d(Rewriting)h(Sys-)255 154 y(tems.)k(In)d Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)i(the)f(3r)n(d)f (IFIP)h(Working)h(Confer)n(enc)n(e)g(on)f(F)l(ormal)g(Descripton)255 214 y(of)i(Pr)n(o)n(gr)n(amming)f(Conc)n(epts)p FB(,)g(pages)h (223{244,)i(1986.)0 318 y([Kah94])90 b(S.)17 b(Kahrs.)27 b(Con\015uence)18 b(of)g(Curried)g(T)l(erm-Rewriting)d(Systems.)25 b(Univ)o(ersit)o(y)15 b(of)k(Edin-)255 378 y(burgh,)d(submitted,)e (1994.)0 483 y([Kap84])90 b(S.)21 b(Kaplan.)37 b(Conditional)21 b(Rewrite)g(Rules.)36 b Fu(The)n(or)n(etic)n(al)21 b(Computer)h(Scienc) n(e)i Fz(33)p Fu(\(2\))p FB(,)255 543 y(pages)17 b(175{193,)h(1984.)0 647 y([Kap87])90 b(S.)16 b(Kaplan.)25 b(Simplifying)14 b(Conditional)k(T)l(erm)d(Rewriting)i(Systems:)k(Uni\014cation,)c(T)l (er-)255 707 y(mination)9 b(and)j(Con\015uence.)h Fu(Journal)g(of)f (Symb)n(olic)h(Computation)g Fz(4)p Fu(\(3\))p FB(,)f(pages)g(295{334,) 255 768 y(1987.)0 872 y([KB70])106 b(D.E.)20 b(Kn)o(uth)g(and)h(P)l(.)f (Bendix.)32 b(Simple)18 b(W)l(ord)j(Problems)e(in)h(Univ)o(ersal)f (Algebra.)33 b(In)255 932 y(J.)19 b(Leec)o(h,)h(editor,)g Fu(Computational)i(Pr)n(oblems)f(in)g(A)o(bstr)n(act)g(A)o(lgebr)n(a)p FB(,)h(pages)f(263{297.)255 992 y(P)o(ergamon)15 b(Press,)h(1970.)0 1097 y([KK88])103 b(M.)14 b(Kurihara)h(and)g(I.)f(Ka)s(ji.)19 b(Mo)q(dular)c(T)l(erm)e(Rewriting)i(Systems:)k(T)l(ermination,)13 b(Con-)255 1157 y(\015uence)i(and)i(Strategies.)k(IEICE)16 b(T)l(ec)o(h.)f(Rep)q(ort)i(COMP88)g(\(143\),)g(pages)g(57-66,)h(1988.) 0 1261 y([KK90])103 b(M.)10 b(Kurihara)i(and)h(I.)d(Ka)s(ji.)j(Mo)q (dular)f(T)l(erm)e(Rewriting)h(Systems)f(and)i(the)g(T)l(ermination.) 255 1321 y Fu(Information)17 b(Pr)n(o)n(c)n(essing)g(L)n(etters)g Fz(34)p FB(,)f(pages)h(1{4,)g(1990.)0 1426 y([KKSV93])39 b(J.R.)19 b(Kenna)o(w)o(a)o(y)l(,)i(J.W.)f(Klop,)i(M.R.)d(Sleep,)i(and) g(F.-J.)f(de)g(V)l(ries.)34 b(Comparing)20 b(Cur-)255 1486 y(ried)d(and)h(Uncurried)f(Rewriting.)25 b(Rep)q(ort)19 b(CS-R9350,)h(Cen)o(tre)d(for)h(Mathematics)e(and)255 1546 y(Computer)f(Science,)f(Amsterdam,)f(1993.)0 1650 y([KL80])111 b(S.)10 b(Kamin)g(and)h(J.-J.)g(L)o(\023)-23 b(evy)l(.)12 b(A)o(ttempts)d(for)i(Generalizing)f(the)g(Recursiv)o(e)f (P)o(ath)j(Ordering.)255 1711 y(Unpublished)j(note,)h(Dept.)g(of)g (Computer)g(Science,)e(Univ)o(ersit)o(y)g(of)i(Illinois,)e(1980.)0 1815 y([Klo80])103 b(J.W.)24 b(Klop.)48 b(Com)o(binatory)24 b(Reduction)h(Systems.)46 b(Mathematical)23 b(Cen)o(tre)h(T)l(racts)255 1875 y(V)l(ol.127,)16 b(Cen)o(tre)f(for)i(Mathematics)d(and)j(Computer) e(Science,)g(Amsterdam,)e(1980.)0 1980 y([Klo87])103 b(J.W.)22 b(Klop.)40 b(T)l(erm)21 b(Rewriting)h(Systems:)32 b(A)23 b(T)l(utorial.)40 b Fu(Bul)r(letin)26 b(of)d(the)h(Eur)n(op)n(e) n(an)255 2040 y(Asso)n(ciation)17 b(for)g(The)n(or)n(etic)n(al)g (Computer)g(Scienc)n(e)j Fz(32)p FB(,)c(pages)h(143{182,)h(1987.)0 2144 y([Klo92])103 b(J.W.)27 b(Klop.)58 b(T)l(erm)27 b(Rewriting)h(Systems.)56 b(In)28 b(S.)g(Abramsky)l(,)h(D.)g(Gabba)o(y) l(,)i(and)255 2204 y(T.)16 b(Maibaum,)f(editors,)i Fu(Handb)n(o)n(ok)h 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FB(,)g(pages)255 2878 y(357{358,)c(1991.)p eop %%Page: 174 182 174 181 bop -59 -39 a FB(174)1487 b Fv(BIBLIOGRAPHY)-59 94 y FB([K)o(O92])104 b(M.)13 b(Kurihara)h(and)g(A.)f(Oh)o(uc)o(hi.)i (Mo)q(dularit)o(y)f(of)g(Simple)d(T)l(ermination)h(of)i(T)l(erm)e (Rewrit-)196 154 y(ing)f(Systems)g(with)g(Shared)h(Constructors.)j Fu(The)n(or)n(etic)n(al)e(Computer)g(Scienc)n(e)i Fz(103)p FB(,)d(pages)196 214 y(273{282,)18 b(1992.)-59 321 y([K)o(O93])104 b(M.)20 b(Kurihara)h(and)g(A.)f(Oh)o(uc)o(hi.)33 b(Noncop)o(ying)20 b(T)l(erm)f(Rewriting)h(and)h(Mo)q(dularit)o(y)f(of)196 381 y(T)l(ermination.)f(T)l(ec)o(hnical)c(rep)q(ort,)h(Hokk)m(aido)h (Univ)o(ersit)o(y)l(,)c(Sapp)q(oro,)k(1993.)-59 487 y([KR93a])81 b(M.R.K.)24 b(Krishna)j(Rao.)53 b(Completeness)25 b(of)h(Hierarc)o (hical)f(Com)o(binations)g(of)i(T)l(erm)196 548 y(Rewriting)13 b(Systems.)j(In)d Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)h(the)h(13th)f (Confer)n(enc)n(e)h(on)f(the)h(F)l(oundations)g(of)196 608 y(Softwar)n(e)g(T)l(e)n(chnolo)n(gy)i(and)e(The)n(or)n(etic)n(al)g (Computer)h(Scienc)n(e)p FB(,)g(pages)f(125{139.)h(Lecture)196 668 y(Notes)f(in)g(Computer)f(Science)g Fz(761)p FB(,)h(Berlin:)j (Springer)d(V)l(erlag,)f(1993.)-59 775 y([KR93b])78 b(M.R.K.)21 b(Krishna)i(Rao.)43 b Fu(T)l(ermination)24 b(Char)n(acteristics)g(of)g (L)n(o)n(gic)f(Pr)n(o)n(gr)n(ams)p FB(.)39 b(PhD)196 835 y(thesis,)15 b(T)l(ata)i(Institute)f(of)g(F)l(undamen)o(tal)f (Researc)o(h,)g(Bom)o(ba)o(y)l(,)f(1993.)-59 942 y([KR94])105 b(M.R.K.)11 b(Krishna)j(Rao.)j(Simple)11 b(T)l(ermination)g(of)j (Hierarc)o(hical)d(Com)o(binations)i(of)g(T)l(erm)196 1002 y(Rewriting)i(Systems.)20 b(In)c Fu(Pr)n(o)n(c)n(e)n(e)n(dings)g (of)i(the)g(International)h(Symp)n(osium)d(on)i(The)n(or)n(et-)196 1062 y(ic)n(al)h(Asp)n(e)n(cts)g(of)f(Computer)h(Softwar)n(e)p FB(,)f(pages)h(203{223.)h(Lecture)d(Notes)h(in)g(Computer)196 1122 y(Science)c Fz(789)p FB(,)i(Berlin:)k(Springer)c(V)l(erlag,)f (1994.)-59 1229 y([Kru60])95 b(J.)13 b(Krusk)m(al.)18 b(W)l(ell-Quasi-Ordering,)13 b(the)h(T)l(ree)f(Theorem,)g(and)i(V)l (azson)o(yi's)e(Conjecture.)196 1289 y Fu(T)l(r)n(ansactions)k(of)g (the)h(A)o(meric)n(an)g(Mathematic)n(al)f(So)n(ciety)h Fz(95)p FB(,)e(pages)h(210{225,)i(1960.)-59 1396 y([Kru72])95 b(J.)15 b(Krusk)m(al.)21 b(The)16 b(Theory)g(of)g(W)l (ell-Quasi-Ordering:)k(A)c(F)l(requen)o(tly)d(Disco)o(v)o(ered)i(Con-) 196 1456 y(cept.)20 b Fu(J.)d(Combinatorial)h(The)n(ory)e(Ser.)i(A)f Fz(13)p Fu(\(3\))p FB(,)f(pages)h(297{305,)i(1972.)-59 1563 y([KV90])104 b(J.W.)14 b(Klop)i(and)g(R.)f(de)g(V)l(rijer.)j (Extended)d(T)l(erm)f(Rewriting)h(Systems.)k(In)c Fu(Pr)n(o)n(c)n(e)n (e)n(dings)196 1623 y(of)e(the)h(2nd)g(International)i(Workshop)c(on)j (Conditional)f(and)g(T)l(yp)n(e)n(d)f(R)n(ewriting)h(Systems)p FB(,)196 1683 y(pages)k(26{50.)h(Lecture)e(Notes)h(in)f(Computer)f (Science)h Fz(516)p FB(,)g(Berlin:)22 b(Springer)c(V)l(erlag,)196 1743 y(1990.)-59 1850 y([Mar94a])67 b(M.)21 b(Marc)o(hiori.)39 b(Mo)q(dularit)o(y)21 b(of)i(Completeness)e(Revisited.)38 b(T)l(ec)o(hnical)21 b(Rep)q(ort)h(No.)196 1910 y(6,)d(Departmen)o(t)f (of)h(Pure)g(and)g(Applied)f(Mathematics,)f(Univ)o(ersit)o(y)g(of)i(P)o (ado)o(v)m(a,)h(Italy)l(,)196 1970 y(1994.)-59 2077 y([Mar94b])64 b(M.)22 b(Marc)o(hiori.)39 b(Mo)q(dularit)o(y)23 b(of)g(UN)935 2059 y Fr(!)994 2077 y FB(for)g(Left-Linear)g(T)l(erm)e(Rewriting)i (Systems.)196 2137 y(Rep)q(ort)15 b(CS-R9433,)i(Cen)o(tre)e(for)g (Mathematics)f(and)h(Computer)g(Science,)e(Amsterdam,)196 2197 y(1994.)-59 2304 y([M)o(\023)-23 b(et83])93 b(Y.)22 b(M)o(\023)-23 b(etevier.)42 b(Ab)q(out)24 b(the)g(Rewriting)f(Systems) f(Pro)q(duced)i(b)o(y)f(the)h(Kn)o(uth-Bendix)196 2364 y(Completion)9 b(Algorithm.)i Fu(Information)h(Pr)n(o)n(c)n(essing)g(L) n(etters)h Fz(16)p Fu(\(1\))p FB(,)e(pages)h(31{34,)i(1983.)-59 2471 y([MG93])96 b(A.)13 b(Middeldorp)g(and)i(B.)e(Gramlic)o(h.)j (Simple)c(T)l(ermination)g(is)i(Di\016cult.)j(In)d Fu(Pr)n(o)n(c)n(e)n (e)n(dings)196 2531 y(of)k(the)i(5th)f(International)h(Confer)n(enc)n (e)g(on)f(R)n(ewriting)h(T)l(e)n(chniques)g(and)g(Applic)n(ations)p FB(,)196 2591 y(pages)13 b(228{242.)i(Lecture)d(Notes)h(in)f(Computer)f (Science)h Fz(690)p FB(,)g(Berlin:)18 b(Springer)13 b(V)l(erlag,)196 2651 y(1993.)-59 2758 y([Mid89a])69 b(A.)13 b(Middeldorp.)18 b(A)c(Su\016cien)o(t)f(Condition)i(for)g(the)g(T)l(ermination)e(of)i (the)f(Direct)g(Sum)f(of)196 2818 y(T)l(erm)j(Rewriting)h(Systems.)24 b(In)17 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)h(4th)g(IEEE)g(Symp) n(osium)e(on)j(L)n(o)n(gic)196 2878 y(in)d(Computer)h(Scienc)n(e)p FB(,)f(pages)g(396{401,)i(1989.)p eop %%Page: 175 183 175 182 bop 0 -39 a Fv(BIBLIOGRAPHY)1484 b FB(175)0 94 y([Mid89b])66 b(A.)15 b(Middeldorp.)22 b(Mo)q(dular)17 b(Asp)q(ects)f(of)h(Prop)q(erties)g(of)g(T)l(erm)e(Rewriting)h(Systems) f(Re-)255 154 y(lated)j(to)g(Normal)f(F)l(orms.)27 b(In)18 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)h(the)h(3r)n(d)f(International)h (Confer)n(enc)n(e)h(on)255 214 y(R)n(ewriting)e(T)l(e)n(chniques)i(and) e(Applic)n(ations)p FB(,)g(pages)g(263{277.)h(Lecture)e(Notes)g(in)f (Com-)255 274 y(puter)f(Science)e Fz(355)p FB(,)i(Berlin:)k(Springer)c (V)l(erlag,)f(1989.)0 386 y([Mid90a])69 b(A.)12 b(Middeldorp.)k (Con\015uence)e(of)f(the)h(Disjoin)o(t)f(Union)g(of)h(Conditional)g(T)l (erm)e(Rewriting)255 446 y(Systems.)19 b(In)c Fu(Pr)n(o)n(c)n(e)n(e)n (dings)h(of)h(the)h(2nd)f(International)h(Workshop)e(on)i(Conditional)g (and)255 506 y(T)l(yp)n(e)n(d)j(R)n(ewriting)h(Systems)p FB(,)g(pages)g(295{306.)i(Lecture)c(Notes)h(in)g(Computer)f(Science)255 566 y Fz(516)p FB(,)c(Berlin:)j(Springer)d(V)l(erlag,)g(1990.)0 678 y([Mid90b])66 b(A.)20 b(Middeldorp.)35 b Fu(Mo)n(dular)21 b(Pr)n(op)n(erties)f(of)i(T)l(erm)g(R)n(ewriting)g(Systems)p FB(.)36 b(PhD)22 b(thesis,)255 738 y(V)l(rije)15 b(Univ)o(ersiteit)e (te)j(Amsterdam,)d(1990.)0 850 y([Mid93a])69 b(A.)14 b(Middeldorp.)20 b(Completeness)14 b(of)h(Com)o(binations)g(of)h (Conditional)g(Constructor)g(Sys-)255 910 y(tems.)30 b(In)19 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)h(the)g(3r)n(d)e (International)j(Workshop)e(on)h(Conditional)h(T)l(erm)255 970 y(R)n(ewriting)i(Systems)h(1992)p FB(,)f(pages)h(148{154.)h (Lecture)d(Notes)g(in)h(Computer)e(Science)255 1031 y Fz(656)p FB(,)f(Berlin:)27 b(Springer)21 b(V)l(erlag,)f(1993.)35 b(Revised)19 b(v)o(ersion)h(to)g(app)q(ear)i(in:)29 b(Journal)21 b(of)255 1091 y(Sym)o(b)q(olic)14 b(Computation.)0 1202 y([Mid93b])66 b(A.)20 b(Middeldorp.)36 b(Mo)q(dular)23 b(Prop)q(erties)e(of)h(Conditional)g(T)l(erm)e(Rewriting)h(Systems.)255 1263 y Fu(Information)c(and)h(Computation)f Fz(104)p Fu(\(1\))p FB(,)f(pages)h(110{158,)h(1993.)0 1374 y([MM82])89 b(A.)18 b(Martelli)g(and)i(U.)e(Mon)o(tanari.)30 b(An)19 b(E\016cien)o(t)f(Uni\014cation)h(Algorithm.)28 b Fu(T)l(r)n(ans.)20 b(on)255 1434 y(Pr)n(o)n(gr)n(amming)15 b(L)n(anguages)j(and)g(Systems) g Fz(4)p Fu(\(2\))p FB(,)e(pages)h(258{282,)h(1982.)0 1546 y([MT93])99 b(A.)17 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b(Systems)g(1992)p FB(,)f(pages)g(113{127.)i (Lecture)d(Notes)g(in)g(Computer)f(Science)g Fz(656)p FB(,)255 2878 y(Berlin:)19 b(Springer)d(V)l(erlag,)g(1993.)p eop %%Page: 176 184 176 183 bop -59 -39 a FB(176)1487 b Fv(BIBLIOGRAPHY)-59 94 y FB([Ohl93c])78 b(E.)18 b(Ohlebusc)o(h.)26 b(On)19 b(the)f(Mo)q(dularit)o(y)g(of)g(T)l(ermination)f(of)i(T)l(erm)e (Rewriting)h(Systems.)196 154 y(Rep)q(ort)g(Nr.)e(11,)j(Univ)o (ersit\177)-24 b(at)15 b(Bielefeld,)g(1993.)27 b(Revised)16 b(v)o(ersion)h(to)h(app)q(ear)h(in)e(Theo-)196 214 y(retical)e (Computer)g(Science.)-59 314 y([Ohl93d])73 b(E.)20 b(Ohlebusc)o(h.)35 b(T)l(ermination)20 b(is)h(not)g(Mo)q(dular)h(for)f(Con\015uen)o(t)h(V) l(ariable-Preserving)196 374 y(T)l(erm)14 b(Rewriting)i(Systems,)e (1993.)23 b(T)l(o)17 b(app)q(ear)g(in)f(Information)f(Pro)q(cessing)i (Letters.)-59 474 y([Ohl94a])76 b(E.)22 b(Ohlebusc)o(h.)40 b(Mo)q(dular)23 b(Prop)q(erties)g(of)g(Constructor-Sharing)h (Conditional)f(T)l(erm)196 534 y(Rewriting)12 b(Systems.)i(In)e Fu(4th)j(International)g(Workshop)f(on)g(Conditional)i(T)l(erm)e(R)n (ewrit-)196 595 y(ing)k(Systems)p FB(,)e(1994.)23 b(T)l(o)17 b(app)q(ear.)-59 695 y([Ohl94b])73 b(E.)19 b(Ohlebusc)o(h.)28 b(On)20 b(the)f(Mo)q(dularit)o(y)f(of)i(Con\015uence)f(of)g (Constructor-Sharing)i(T)l(erm)196 755 y(Rewriting)g(Systems.)36 b(In)22 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)h(the)h(19th)g(Col)r(lo)n (quium)g(on)g(T)l(r)n(e)n(es)f(in)h(A)o(lge-)196 815 y(br)n(a)16 b(and)i(Pr)n(o)n(gr)n(amming)p FB(,)c(pages)j(261{275.)i (Lecture)d(Notes)g(in)g(Computer)f(Science)g Fz(787)p FB(,)196 875 y(Berlin:)k(Springer)d(V)l(erlag,)g(1994.)-59 975 y([Ok)m(a89])94 b(M.)14 b(Ok)m(ada.)21 b(Strong)c(Normalizabilit)n (y)12 b(for)k(the)f(Com)o(bined)f(System)g(of)i(the)f(Pure)h(T)o(yp)q (ed)196 1035 y Fy(\025)p FB(-Calculus)h(and)h(an)f(Arbitrary)g(Con)o(v) o(ergen)o(t)f(T)l(erm)f(Rewrite)h(System.)22 b(In)17 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)196 1096 y(of)g(the)h(International)h (Symp)n(osium)d(on)i(Symb)n(olic)g(and)g(A)o(lgebr)n(aic)g(Computation) p FB(,)e(1989.)-59 1196 y([Oos94])98 b(V.)20 b(v)m(an)h(Oostrom.)35 b(Con\015uence)21 b(b)o(y)f(Decreasing)h(Diagrams.)35 b Fu(The)n(or)n(etic)n(al)21 b(Computer)196 1256 y(Scienc)n(e)e Fz(126)p Fu(\(2\))p FB(,)d(pages)h(259{280,)h(1994.)-59 1356 y([PJ87])121 b(S.L.)15 b(P)o(eyton)g(Jones.)20 b Fu(The)d(Implementation)h(of)e(F)l(unctional)j(Pr)n(o)n(gr)n(amming)c (L)n(anguages)p FB(.)196 1416 y(Pren)o(tice)f(Hall,)h(1987.)-59 1516 y([Plu90])105 b(D.)15 b(Plump.)20 b(Implem)o(en)n(ting)13 b(T)l(erm)i(Rewriting)g(b)o(y)h(Graph)g(Reduction:)21 b(T)l(ermination)14 b(of)196 1576 y(Com)o(bined)c(Systems.)k(In)e Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)h(the)g(2nd)h(International)g (Workshop)e(on)h(Condi-)196 1636 y(tional)i(and)g(T)l(yp)n(e)n(d)f(R)n (ewriting)h(Systems)p FB(,)f(pages)g(307{317.)h(Lecture)e(Notes)h(in)f (Computer)196 1697 y(Science)g Fz(516)p FB(,)i(Berlin:)k(Springer)c(V)l (erlag,)f(1990.)-59 1797 y([Plu93])105 b(D.)20 b(Plump.)34 b(Collapsed)22 b(T)l(ree)e(Rewriting:)30 b(Completeness,)20 b(Con\015uence,)i(and)g(Mo)q(du-)196 1857 y(larit)o(y.)27 b(In)18 b Fu(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)h(3r)n(d)f (International)i(Workshop)d(on)i(Conditional)h(T)l(erm)196 1917 y(R)n(ewriting)15 b(Systems)h(1992)p FB(,)d(pages)i(97{112.)h (Lecture)d(Notes)h(in)g(Computer)f(Science)f Fz(656)p FB(,)196 1977 y(Berlin:)19 b(Springer)d(V)l(erlag,)g(1993.)-59 2077 y([P)o(ol92])109 b(J.)18 b(v)m(an)h(de)f(P)o(ol.)28 b(Mo)q(dularit)o(y)18 b(in)h(Man)o(y-Sorted)f(T)l(erm)f(Rewriting)h (Systems.)27 b(Master's)196 2137 y(thesis,)15 b(Utrec)o(h)o(t)g(Univ)o (ersit)o(y,)e(1992.)-59 2237 y([Pue89])97 b(L.)23 b(Puel.)42 b(Using)23 b(Una)o(v)o(oidable)f(Sets)h(of)h(T)l(rees)f(to)g (Generalize)f(Krusk)m(al's)i(Theorem.)196 2298 y Fu(Journal)17 b(of)g(Symb)n(olic)h(Computation)g Fz(8)p FB(,)e(pages)h(335{382,)h (1989.)-59 2398 y([Rob65])92 b(J.A.)19 b(Robinson.)35 b(A)20 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