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y(Lemma)e(states)j(that)g(termination)d(and)j(lo)q(cal)f (con\015uence)g(of)g Fn(!)g Fv(imply)e(con\015uence.)962 3024 y(5)p eop %%Page: 6 6 6 5 bop 73 115 a Fv(A)20 b Fi(term)i(r)n(ewriting)g(system)f Fv(\(TRS)g(for)g(short\))g(is)f(a)i(pair)e(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))21 b(consisting)g(of)g(a)g(signature)g Fn(F)0 175 y Fv(and)g(a)f(set)g Fn(R)g(\032)g(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))k Fn(\002)i(T)e Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))20 b(of)g Fi(r)n(ewrite)g(rules)h Fv(or)f Fi(r)n(e)n(duction)h(rules)p Fv(.)33 b(Ev)o(ery)19 b(rewrite)g(rule)0 235 y(\()p Fl(l)q(;)8 b(r)q Fv(\))20 b(m)o(ust)f(satisfy)h(the)h(follo)o(wing)f(t)o(w)o(o)g(constrain)o(ts:) 30 b(\(i\))19 b(the)i(left-hand)f(side)g Fl(l)h Fv(is)f(not)h(a)g(v)m (ariable,)0 295 y(and)c(\(ii\))f(v)m(ariables)h(o)q(ccurring)g(in)f (the)h(righ)o(t-hand)g(side)g Fl(r)h Fv(also)f(o)q(ccur)g(in)g Fl(l)q Fv(.)22 b(Rewrite)16 b(rules)g(\()p Fl(l)q(;)8 b(r)q Fv(\))17 b(will)0 356 y(b)q(e)i(denoted)g(b)o(y)f Fl(l)h 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Fv(\))14 b(=)g Fn(h)p Fl(t)617 2367 y Fh(\031)q Fm(\(1\))685 2359 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)813 2367 y Fh(\031)q Fm(\()p Fh(n)p Fm(\))884 2359 y Fn(i)17 b Fv(suc)o(h)f(that)h Fl(t)1154 2367 y Fh(\031)q Fm(\()p Fh(j)r Fm(\))1229 2359 y Fn(\037)8 b Fl(t)1294 2367 y Fh(\031)q Fm(\()p Fh(j)r Fm(+1\))1422 2359 y Fv(for)17 b(1)d Fn(\024)g Fl(j)j(<)d(n)p Fv(.)21 b(Note)16 b(that)0 2425 y(the)j(sets)g(to)h(b)q(e)f(sorted)h(are)f(\014nite)g(and)h(th)o (us)f(the)g(sets)g Fl(L)1105 2407 y Fh(d)1105 2438 y(n)1148 2425 y Fv(and)h Fl(L)1285 2398 y Fm(\026)1279 2407 y Fh(d)1279 2438 y(n)1322 2425 y Fv(are)f(w)o(ell-de\014ned.)29 b(As)19 b(w)o(e)g(shall)0 2485 y(see)h(later,)g(the)g(sorting)g(pro)q (cess)h(is)f(necessary)g(in)f(order)h(to)h(cop)q(e)f(with)g (non-left-linear)f(rules:)29 b(The)0 2546 y(succession)15 b(of)g(the)g(listed)g(elemen)o(ts)d(of)j(a)h(set)f(has)h(to)f(b)q(e)h (uniquely)d(determined.)19 b(Again,)14 b(w)o(e)h(suppress)0 2606 y(the)h(argumen)o(t)f Fl(s)h 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b(B)1167 2907 y Fm(1)1187 2900 y Fn(i)950 3024 y Fv(12)p eop %%Page: 13 13 13 12 bop 579 115 a Fl(L)612 94 y Fm(2)612 127 y(2)646 115 y Fv(=)42 b Fn(h)p Fl(L)778 97 y Fm(1)778 127 y(1)798 115 y Fl(;)8 b(L)853 97 y Fm(2)853 127 y(1)873 115 y Fn(i)41 b Fl(L)966 94 y Fm(1)966 127 y(2)1000 115 y Fv(=)14 b Fn(h)p Fl(H)t Fv(\()p Fl(L)1167 94 y Fm(2)1167 127 y(1)1187 115 y Fv(\))p Fn(i)579 187 y Fl(L)612 167 y Fm(2)612 200 y(3)646 187 y Fv(=)42 b Fn(h)p Fl(L)778 169 y Fm(1)778 200 y(2)798 187 y Fl(;)8 b(L)853 169 y Fm(2)853 200 y(2)873 187 y Fn(i)41 b Fl(L)966 167 y Fm(1)966 200 y(3)1000 187 y Fv(=)14 b Fn(h)p Fl(G)p Fv(\()p Fl(L)1161 167 y Fm(2)1161 200 y(2)1182 187 y Fv(\))p Fn(i)579 260 y Fl(L)612 240 y Fm(2)612 272 y(4)646 260 y Fv(=)42 b Fn(h)p Fl(L)778 242 y Fm(1)778 272 y(3)798 260 y Fl(;)8 b(L)853 242 y Fm(2)853 272 y(3)873 260 y Fn(i)41 b Fl(L)966 240 y Fm(1)966 272 y(4)1000 260 y Fv(=)14 b Fn(h)j(i)579 333 y Fl(L)612 312 y Fm(2)612 345 y(5)646 333 y Fv(=)97 b Fn(h)17 b(i)97 b 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2366 y(m)p Fr(\000)p Fm(1)885 2354 y Fv(])16 b Fn(j)g Fl(t)e Fv(=)g Fl(C)t Fv([)-8 b([)p Fl(t)1106 2361 y Fm(1)1124 2354 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1251 2361 y Fh(n)1274 2354 y Fv(])-8 b(])13 b Fn(2)i Fl(S)1385 2361 y Fh(d)1405 2354 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(r)q(ank)r Fv(\()p Fl(t)p Fv(\))14 b(=)g Fl(m)p Fn(g)p Fv(\).)22 b(Since)0 2426 y Fl(u)d Fn(2)h Fl(S)130 2433 y Fh(d)150 2426 y Fv(\()p Fl(s)p Fv(\))f(and)h Fl(r)q(ank)r Fv(\()p Fl(u)p Fv(\))f(=)h Fl(m)p Fv(,)f(it)g(follo)o(ws)g(that)h Fl(C)1015 2408 y Fr(0)1026 2426 y Fv([)p Fl(L)1079 2399 y Fm(\026)1073 2408 y Fh(d)1073 2438 y(m)p Fr(\000)p Fm(1)1152 2426 y Fl(;)8 b(:)g(:)g(:)f(;)h(L)1300 2399 y Fm(\026)1294 2408 y Fh(d)1294 2438 y(m)p Fr(\000)p Fm(1)1372 2426 y Fv(])19 b(o)q(ccurs)h(in)f Fl(L)1652 2408 y Fh(d)1652 2438 y(m)1686 2426 y Fv(.)31 b(Therefore,)0 2491 y Fl(L)33 2473 y Fh(d)33 2504 y(m)93 2491 y Fn(!)143 2471 y Fm(+)143 2504 y Fr(C)162 2510 y Fb(E)210 2491 y Fl(C)249 2473 y Fr(0)260 2491 y Fv([)p Fl(L)313 2465 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Fh(d)1870 2631 y Fv(\()p Fl(u)p Fv(\).)0 2691 y Ff(2)950 3024 y Fv(13)p eop %%Page: 14 14 14 13 bop 0 115 a Fk(Example)16 b(3.12)24 b Fv(\(Example)15 b(3.8)h(con)o(tin)o(ued\))g(Consider)g(the)g(reduction)g(sequence)189 220 y Fl(D)f Fv(:)f Fl(s)f Fv(=)h Fl(s)383 227 y Fm(1)417 220 y Fn(!)467 227 y Fr(R)497 232 y Fj(1)530 220 y Fl(s)553 227 y Fm(2)586 220 y Fn(!)636 227 y Fr(R)666 232 y Fj(2)700 220 y Fl(s)723 227 y Fm(3)756 220 y Fn(!)806 227 y Fr(R)836 232 y Fj(1)869 220 y Fl(s)892 227 y Fm(4)926 220 y Fn(!)976 227 y Fr(R)1006 232 y Fj(1)1039 220 y Fl(s)1062 227 y Fm(5)1095 220 y Fn(!)1145 227 y Fr(R)1175 232 y Fj(1)1209 220 y Fl(s)1232 227 y Fm(6)1265 220 y Fn(!)1315 227 y Fr(R)1345 232 y Fj(2)1378 220 y Fl(s)1401 227 y Fm(7)1435 220 y Fn(!)1485 227 y Fr(R)1515 232 y Fj(1)1548 220 y Fl(s)1571 227 y Fm(8)1604 220 y Fn(!)1654 227 y Fr(R)1684 232 y Fj(2)1718 220 y Fl(s)1741 227 y Fm(9)0 325 y Fv(where)i Fl(s)p Fv(,)g Fn(R)236 332 y Fm(1)256 325 y Fv(,)f(and)i Fn(R)422 332 y Fm(2)458 325 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Fl(l)k Fv(=)e(0)f(is)f(trivial,)g(so)h(consider)f Fl(t)i Fn(!)950 1939 y Fh(l)950 1969 y Fr(R)980 1974 y Fj(1)998 1969 y Fr(]R)1052 1974 y Fj(2)1096 1957 y Fl(t)1114 1939 y Fr(00)1161 1957 y Fn(!)1211 1964 y Fr(R)1241 1969 y Fj(1)1258 1964 y Fr(]R)1312 1969 y Fj(2)1357 1957 y Fl(t)1375 1939 y Fr(0)1386 1957 y Fv(.)42 b(By)23 b(the)g(inner)f (induction)0 2017 y(h)o(yp)q(othesis)c(\(on)g Fl(l)q Fv(\))f(\010)417 2024 y Fh(d)438 2017 y Fv(\()p Fl(t)p Fv(\))f Fn(!)560 1999 y Fr(\003)560 2029 y(R)590 2035 y Fc(d)608 2029 y Fr(]C)651 2035 y Fb(E)689 2017 y Fv(\010)724 2024 y Fh(d)745 2017 y Fv(\()p Fl(t)782 1999 y Fr(00)803 2017 y Fv(\).)25 b(It)18 b(remains)e(to)i(b)q(e)g(sho)o(wn)g(that)h (\010)1520 2024 y Fh(d)1540 2017 y Fv(\()p Fl(t)1577 1999 y Fr(00)1598 2017 y Fv(\))d Fn(!)1683 1999 y Fr(\003)1683 2029 y(R)1713 2035 y Fc(d)1731 2029 y Fr(]C)1774 2035 y Fb(E)1813 2017 y Fv(\010)1848 2024 y Fh(d)1868 2017 y Fv(\()p Fl(t)1905 1999 y Fr(0)1916 2017 y Fv(\).)0 2077 y(The)g(case)g 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