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Ft(!)g Fu(F)5 b Fw(\()p Fi(D)p Fw(\))p Ft(g)p Fw(,)17 b(w)o(e)g(obtain)i(an)f(example)g(whic)o(h)h(sho)o(ws)e(that)h(in)g(general)g (total)379 2432 y(orderings)d(are)e(not)g(su\016cien)o(t)h(to)f(sho)o(w)g (the)g(termination)j(of)c(the)h(com)o(bined)i(system.)p eop %%Page: 3 3 bop 245 194 a Fx(A)11 b Fm(substitution)h Fo(\033)g Fx(is)g(a)f(mapping)e (from)h Fq(V)15 b Fx(to)c Fq(T)f Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))12 b(suc)o(h)h(that)e Fq(f)p Fo(x)g Fq(2)g(V)17 b(j)12 b Fo(\033)q Fx(\()p Fo(x)p Fx(\))p Fq(6\021)p Fo(x)p Fq(g)183 244 y Fx(is)18 b(\014nite.)g(This)g(set)i(is)e(called)g(the)h Fm(domain)g Fx(of)f Fo(\033)i Fx(and)e(will)f(b)q(e)i(denoted)h(b)o(y)e Fq(D)q Fo(om)p Fx(\()p Fo(\033)q Fx(\))q(.)183 293 y(Occasionally)11 b(w)o(e)h(presen)o(t)i(a)d(substitution)h Fo(\033)h Fx(as)f Fq(f)p Fo(x)f Fq(7!)g Fo(\033)q Fx(\()p Fo(x)p Fx(\))i Fq(j)f Fo(x)f Fq(2)h(D)q Fo(om)p Fx(\()p Fo(\033)q Fx(\))q Fq(g)p Fx(.)f(The)h(sub-)183 343 y(stitution)f(with)g(empt)o(y)g(domain)e(will)h(b)q (e)i(denoted)h(b)o(y)e Fo(")p Fx(.)h(Substitutions)g(are)g(extended)h(to)183 393 y(morphisms)g(from)g Fq(T)d Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))16 b(to)f Fq(T)10 b Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\),)16 b(i.e.)e Fo(\033)q Fx(\()p Fo(f)t Fx(\()p Fo(t)1046 399 y Fp(1)1066 393 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)1174 399 y Fn(n)1196 393 y Fx(\)\))14 b Fq(\021)g Fo(f)t Fx(\()p Fo(\033)q Fx(\()p Fo(t)1384 399 y Fp(1)1404 393 y Fx(\))p Fo(;)7 b(:)g(:)g(:)e(;)i(\033)q Fx(\()p Fo(t)1569 399 y Fn(n)1591 393 y Fx(\)\))183 443 y(for)12 b(ev)o(ery)i Fo(n)p Fx(-ary)f(function)f(sym)o(b)q(ol)g Fo(f)17 b Fx(and)c(terms)g Fo(t)1015 449 y Fp(1)1034 443 y Fo(;)7 b(:)g(:)g(:)t(;)g (t)1141 449 y Fn(n)1164 443 y Fx(.)12 b(W)m(e)h(call)f Fo(\033)q Fx(\()p Fo(t)p Fx(\))h(an)g Fm(instanc)n(e)183 493 y Fx(of)g Fo(t)p Fx(.)g(W)m(e)h(frequen)o(tly)g(write)g Fo(t\033)h Fx(instead)f(of)g Fo(\033)q Fx(\()p Fo(t)p Fx(\).)245 554 y(In)9 b(order)h(to)g(describ)q(e)h (subterm)e(o)q(ccurrences)k(of)8 b(a)h(term,)g(w)o(e)g(in)o(tro)q(duce)i(the) f(notation-)183 604 y(ally)k(con)o(v)o(enien)o(t)j(notion)e(\\con)o(text")h (instead)g(of)g(the)g(more)f(precise)j(notion)d(\\p)q(osition")183 654 y(\(cf.)d([DJ90)o(]\).)g(Let)i Fh(2)e Fx(b)q(e)i(a)f(sp)q(ecial)g (constan)o(t)g(sym)o(b)q(ol.)e(A)h Fm(c)n(ontext)i Fo(C)s Fx([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fx(])k(is)i(a)g(term)f(in)183 703 y Fq(T)e Fx(\()p Fq(F)f([)c(f)p Fh(2)p Fq(g)p Fo(;)i Fq(V)s Fx(\).)j(If)h Fo(C)s Fx([)p Fo(;)c(:)g(:)g(:)t(;)g Fx(])k(is)g(a)g(con)o (text)i(with)e Fo(n)g Fx(o)q(ccurrences)k(of)c Fh(2)h Fx(and)f Fo(t)1408 709 y Fp(1)1427 703 y Fo(;)c(:)g(:)g(:)e(;)i(t)1535 709 y Fn(n)1568 703 y Fx(are)183 753 y(terms,)15 b(then)i Fo(C)s Fx([)p Fo(t)469 759 y Fp(1)486 753 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)594 759 y Fn(n)616 753 y Fx(])16 b(is)f(the)i(result)g(of)e(replacing)h(from)e (left)i(to)g(righ)o(t)f(the)i(o)q(ccur-)183 803 y(rences)c(of)f Fh(2)g Fx(with)f Fo(t)501 809 y Fp(1)520 803 y Fo(;)c(:)g(:)g(:)t(;)g(t)627 809 y Fn(n)650 803 y Fx(.)k(A)h(con)o(text)g(con)o(taining)f(precisely)i(one) f(o)q(ccurrence)j(of)c Fh(2)h Fx(is)183 853 y(denoted)j(b)o(y)e Fo(C)s Fx([)f(].)h(A)h(term)g Fo(s)g Fx(is)g(a)g Fm(subterm)g Fx(of)f(a)h(term)f Fo(t)h Fx(if)g(there)h(exists)g(a)f(con)o(text)g Fo(C)s Fx([)e(])183 903 y(suc)o(h)k(that)f Fo(t)f Fq(\021)h Fo(C)s Fx([)p Fo(s)p Fx(].)f(By)i(abuse)g(of)e(notation)h(w)o(e)h(write)f Fq(T)c Fx(\()p Fq(F)t Fo(;)c Fq(V)s Fx(\))16 b(for)f Fq(T)10 b Fx(\()p Fq(F)k([)c(f)p Fh(2)p Fq(g)p Fo(;)d Fq(V)s Fx(\),)183 952 y(in)o(terpreting)14 b Fh(2)h Fx(as)f(a)g(sp)q(ecial)g(constan)o(t)h (whic)o(h)f(is)h(alw)o(a)o(ys)e(a)o(v)n(ailable)f(but)i(used)h(only)f(for)183 1002 y(the)g(aforemen)o(tioned)f(purp)q(ose.)245 1064 y(Let)e Fq(!)358 1070 y Fk(R)399 1064 y Fx(b)q(e)g(a)g(relation)f(on)g(terms,)h (i.e.,)e Fq(!)929 1070 y Fk(R)959 1064 y Fq(\022)j(T)e Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))s Fq(\002)s(T)k Fx(\()p Fq(F)t Fo(;)c Fq(V)s Fx(\).)k(The)g(transitiv)o(e-)183 1113 y(re\015exiv)o(e)k(closure)h (of)f Fq(!)577 1119 y Fk(R)622 1113 y Fx(is)g(denoted)h(b)o(y)e Fq(!)923 1098 y Fk(\003)923 1125 y(R)953 1113 y Fx(.)h(If)g Fo(s)e Fq(!)1097 1098 y Fk(\003)1097 1125 y(R)1141 1113 y Fo(t)p Fx(,)h(w)o(e)i(sa)o(y)e(that)h Fo(s)h Fm(r)n(e)n(duc)n(es)f Fx(to)183 1163 y Fo(t)i Fx(and)f(w)o(e)i(call)e Fo(t)h Fx(a)f Fm(r)n(e)n(duct)h Fx(of)f Fo(s)p Fx(.)h(W)m(e)g(write)g Fo(s)g Fq( )997 1169 y Fk(R)1044 1163 y Fo(t)g Fx(if)f Fo(t)h Fq(!)1191 1169 y Fk(R)1238 1163 y Fo(s)p Fx(;)g(lik)o(ewise)f(for)h Fo(s)1534 1148 y Fk(\003)1534 1175 y(R)1556 1163 y Fq( )o Fo(t)p Fx(.)183 1213 y(The)12 b(transitiv)o(e)f(closure)h(of)f Fq(!)671 1219 y Fk(R)713 1213 y Fx(is)g(denoted)h(b)o(y)g Fq(!)1004 1195 y Fp(+)1004 1225 y Fk(R)1034 1213 y Fx(,)f(and)g Fq($)1177 1219 y Fk(R)1219 1213 y Fx(denotes)h(the)g(symmetric)183 1263 y(closure)i(of)g Fq(!)411 1269 y Fk(R)455 1263 y Fx(\(i.e.,)e Fq($)591 1269 y Fk(R)622 1263 y Fx(=)p Fq(!)696 1269 y Fk(R)738 1263 y Fq([)f( )819 1269 y Fk(R)849 1263 y Fx(\).)j(The)g(transitiv)o (e-re\015exiv)o(e)h(closure)f(of)g Fq($)1552 1269 y Fk(R)1596 1263 y Fx(is)183 1313 y(called)i Fm(c)n(onversion)g Fx(and)g(denoted)h(b)o(y) f(=)844 1319 y Fk(R)875 1313 y Fx(.)f(If)h Fo(s)g Fx(=)1013 1319 y Fk(R)1059 1313 y Fo(t)p Fx(,)g(then)g Fo(s)h Fx(and)f Fo(t)g Fx(are)h Fm(c)n(onvertible)p Fx(.)183 1363 y(Tw)o(o)c(terms)g Fo(t)406 1369 y Fp(1)425 1363 y Fo(;)7 b(t)459 1369 y Fp(2)490 1363 y Fx(are)14 b Fm(joinable)p Fx(,)g(denoted)g(b)o(y)g Fo(t)954 1369 y Fp(1)984 1363 y Fq(#)1005 1369 y Fk(R)1047 1363 y Fo(t)1062 1369 y Fp(2)1080 1363 y Fx(,)f(if)g(there)i(exists)f(a)g(term)f Fo(t)1512 1369 y Fp(3)1544 1363 y Fx(suc)o(h)183 1412 y(that)e Fo(t)285 1418 y Fp(1)315 1412 y Fq(!)357 1397 y Fk(\003)357 1424 y(R)399 1412 y Fo(t)414 1418 y Fp(3)452 1397 y Fk(\003)452 1424 y(R)473 1412 y Fq( )h Fo(t)542 1418 y Fp(2)561 1412 y Fx(.)f(Suc)o(h)h(a)f(term)g Fo(t)826 1418 y Fp(3)856 1412 y Fx(is)h(called)f(a)g Fm(c)n(ommon)j(r)n(e)n(duct)d Fx(of)g 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1767 y Fk(R)1473 1761 y Fo(s)e Fq(!)1547 1767 y Fk(R)1590 1761 y Fo(t)1605 1767 y Fp(2)183 1811 y Fx(w)o(e)h(ha)o(v)o(e)f Fo(t)354 1817 y Fp(1)385 1811 y Fq(#)406 1817 y Fk(R)447 1811 y Fo(t)462 1817 y Fp(2)481 1811 y Fx(.)g(If)h Fq(!)590 1817 y Fk(R)634 1811 y Fx(is)g(con\015uen)o(t)g(and)g(terminating,)e(it)h(is)h (called)g Fm(c)n(onver)n(gent)p Fx(.)245 1872 y(A)19 b Fm(term)g(r)n (ewriting)f(system)h Fx(\(TRS)g(for)g(short\))h(is)f(a)g(pair)g(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))19 b(consisting)g(of)g(a)183 1922 y(signature)12 b Fq(F)j Fx(and)d(a)f(set)i Fq(R)e(\032)h(T)e Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))e Fq(\002)g(T)11 b Fx(\()p Fq(F)t Fo(;)c Fq(V)s Fx(\))12 b(of)f Fm(r)n(ewrite)g(rules)g Fx(or)h Fm(r)n(e)n(duction)h(rules)p Fx(.)183 1972 y(Ev)o(ery)j(rewrite)g (rule)f(\()p Fo(l)q(;)7 b(r)q Fx(\))16 b(m)o(ust)e(satisfy)h(the)h(follo)o (wing)d(t)o(w)o(o)h(constrain)o(ts:)i(\(i\))f(the)h(left-)183 2022 y(hand)e(side)h Fo(l)h Fx(is)e(not)h(a)f(v)n(ariable,)f(and)i(\(ii\))f (v)n(ariables)g(o)q(ccurring)h(in)f(the)h(righ)o(t-hand)f(side)183 2071 y Fo(r)g Fx(also)e(o)q(ccur)i(in)f Fo(l)q Fx(.)g(Rewrite)g(rules)h(\()p Fo(l)q(;)7 b(r)q Fx(\))13 b(will)e(b)q(e)j(denoted)g(b)o(y)f Fo(l)g Fq(!)e Fo(r)q Fx(.)h(The)i(rewrite)g(rules)183 2121 y(of)i(a)h(TRS)f(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))17 b(de\014ne)h(a)f Fm(r)n(ewrite)e(r)n(elation)i Fq(!)1008 2127 y Fk(R)1055 2121 y Fx(on)f Fq(T)10 b Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))18 b(as)f(follo)o(ws:) e Fo(s)i Fq(!)1562 2127 y Fk(R)1609 2121 y Fo(t)183 2171 y Fx(if)f(there)j(exists)g(a)e(rewrite)h(rule)g Fo(l)h Fq(!)e Fo(r)i Fx(in)e Fq(R)p Fx(,)g(a)g(substitution)h Fo(\033)h Fx(and)e(a)g(con)o (text)h Fo(C)s Fx([)12 b(])183 2221 y(suc)o(h)k(that)f Fo(s)f Fq(\021)g Fo(C)s Fx([)p Fo(l)q(\033)q Fx(])h(and)g Fo(t)f Fq(\021)g Fo(C)s Fx([)p Fo(r)q(\033)q Fx(].)f(W)m(e)i(call)f Fo(s)h Fq(!)1068 2227 y Fk(R)1112 2221 y Fo(t)g Fx(a)g Fm(r)n(ewrite)f(step)h Fx(or)h Fm(r)n(e)n(duction)183 2271 y(step)p Fx(.)g(A)g(TRS)g(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))17 b(has)g(one)f(of)g(the)h(ab)q(o)o(v)o(e)g (prop)q(erties)h(if)d(its)i(rewrite)g(relation)f(has)183 2321 y(the)e(resp)q(ectiv)o(e)i(prop)q(ert)o(y)m(.)245 2382 y(A)f Fm(p)n(artial)g(or)n(dering)g Fo(>)g Fx(is)g(a)g(transitiv)o(e)g(and)g (irre\015exiv)o(e)h(relation)f(on)g Fq(T)10 b Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\).)15 b(If)g(a)183 2432 y(partial)e(ordering)g(is)h (terminating,)e(it)h(is)h(also)f(called)h Fm(wel)r(l-founde)n(d)p Fx(.)p eop %%Page: 4 4 bop 340 194 a Fr(3)56 b(Simplifying)16 b(Conditional)j(T)-5 b(erm)17 b(Rewriting)g(Systems)340 280 y Fg(De\014nition)t(1.)k Fx(A)12 b Fm(c)n(onditional)h(term)g(r)n(ewriting)e(system)g Fx(is)h(a)f(pair)h(\()p Fq(F)t Fo(;)7 b Fq(R)o Fx(\))12 b(consisting)g(of)340 330 y(a)i(signature)h Fq(F)j Fx(and)c(a)g(set)h Fq(R)g Fx(of)e Fm(c)n(onditional)j(r)n(ewrite)d(rules)p Fx(.)g(Eac)o(h)i(of)e(these)j(rules) f(is)f(of)340 380 y(the)j(form)d Fo(l)f Fq(!)e Fo(r)h Fq(\()f Fo(s)695 386 y Fp(1)726 380 y Fq(')h Fo(t)785 386 y Fp(1)804 380 y Fo(;)7 b(:)g(:)g(:)t(;)g(s)915 386 y Fn(n)949 380 y Fq(')12 b Fo(t)1008 386 y Fn(n)1047 380 y Fx(with)j Fo(l)q(;)7 b(r)o(;)g(s)1231 386 y Fp(1)1249 380 y Fo(;)g(:)g(:)g(:)e(;)i(s)1361 386 y Fn(n)1383 380 y Fo(;)g(t)1417 386 y Fp(1)1435 380 y Fo(;)g(:)g(:)g(:)e(;)i(t)1543 386 y Fn(n)1580 380 y Fq(2)15 b(T)10 b Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))q(,)340 429 y(where)k Fq(')f Fx(denotes)h(either)g Fq(#)e Fx(\(joinabilit)o(y\))e(or)j(=)g(\(con)o(v)o(ersion\))1322 414 y Fp(2)1341 429 y Fx(.)f Fo(s)1381 435 y Fp(1)1412 429 y Fq(')i Fo(t)1470 435 y Fp(1)1489 429 y Fo(;)c(:)g(:)g(:)e(;)i(s)1601 435 y Fn(n)1635 429 y Fq(')12 b Fo(t)1694 435 y Fn(n)1726 429 y Fx(are)340 479 y(the)i Fm(c)n(onditions)g Fx(of)f(the)h(rewrite)h(rule)f(.) f(If)g(a)g(rewrite)h(rule)g(has)g(no)f(conditions,)g(that)g(is)g(to)340 529 y(sa)o(y)m(,)c Fo(n)j Fx(=)g(0,)d(w)o(e)h(write)g Fo(l)j Fq(!)e Fo(r)q Fx(.)e(As)i(for)e(unconditional)g(TRSs,)g(w)o(e)i(imp)q(ose)e (t)o(w)o(o)g(restrictions)340 579 y(on)i(eac)o(h)g(rewrite)h(rule)f(in)f Fq(R)p Fx(:)g(the)i(left-hand)e(side)h Fo(l)h Fx(is)e(not)h(a)f(v)n(ariable,) f(and)i(ev)o(ery)g(v)n(ariable)340 629 y(whic)o(h)20 b(o)q(ccurs)h(in)f(the)g (righ)o(t-hand)f(side)i Fo(r)f Fx(also)g(o)q(ccurs)h(in)e Fo(l)q Fx(.)h(If)f Fq(')h Fx(denotes)i Fq(#)o Fx(,)e(then)340 679 y(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))13 b(is)f(said)h(to)f(b)q(e)i(a)e Fm(join)h Fx(CTRS.)e(The)i(rewrite)h(relation)e(asso)q(ciated)i(with)e(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))12 b(is)340 728 y(de\014ned)i(b)o(y:)e Fo(s)p Fq(!)611 734 y Fk(R)642 728 y Fo(t)h Fx(if)f(there)i(exists)g(a)e (rewrite)i(rule)f Fo(l)g Fq(!)e Fo(r)h Fq(\()f Fo(s)1362 734 y Fp(1)1393 728 y Fq(#)g Fo(t)1440 734 y Fp(1)1459 728 y Fo(;)c(:)g(:)g(:)e (;)i(s)1571 734 y Fn(n)1605 728 y Fq(#)k Fo(t)1652 734 y Fn(n)1687 728 y Fx(in)i Fq(R)p Fx(,)340 778 y(a)f(substitution)h Fo(\033)h Fx(and)e(a)g(con)o(text)h Fo(C)s Fx([)f(])g(suc)o(h)h(that)f Fo(s)g Fq(\021)g Fo(C)s Fx([)p Fo(l)q(\033)q Fx(])p Fo(;)7 b(t)j Fq(\021)i Fo(C)s Fx([)p Fo(r)q(\033)q Fx(])f(and)h Fo(s)1629 784 y Fn(j)1647 778 y Fo(\033)q Fq(#)1693 788 y Fk(R)1724 778 y Fo(t)1739 784 y Fn(j)1756 778 y Fo(\033)340 828 y Fx(for)20 b(all)e Fo(j)23 b Fq(2)e(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(n)p Fq(g)p Fx(.)19 b(If)g Fq(')h Fx(denotes)h(=,)e(then)h(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))20 b(is)f(called)g Fm(semi-e)n(quational)340 878 y Fx(CTRS,)13 b(and)h Fq(!)602 884 y Fk(R)646 878 y Fx(is)g(de\014ned)h (analogously)m(.)340 952 y(W)m(e)d(often)g(simply)f(write)h Fq(R)g Fx(if)g(the)h(underlying)e(signature)i Fq(F)j Fx(is)c(clear)h(from)d (the)j(con)o(text.)340 1002 y(Note)18 b(that)f(if)f(ev)o(ery)i(rewrite)f (rule)h(in)e(a)h(CTRS)f Fq(R)h Fx(has)g(no)g(conditions,)f(then)i Fq(R)f Fx(is)g(an)340 1052 y(unconditional)f(TRS)g(in)h(the)g(usual)g(sense.) h(A)f(least)g(\014xp)q(oin)o(t)f(c)o(haracterization)h(of)g(the)340 1102 y(asso)q(ciated)h(rewrite)f(relation)f(can)h(b)q(e)g(found)g(in)f ([Mid90a)n(].)g(With)g(regard)h(to)f(Prop)q(osi-)340 1152 y(tion)d(5)h(w)o(e) g(con\014ne)g(the)g(considerations)g(in)g(this)f(pap)q(er)i(to)e(\014nite)h (CTRSs.)f(Th)o(us,)h(in)f(the)340 1202 y(sequel)j(w)o(e)g(will)d(tacitly)i (assume)g(all)f(systems)h(considered)i(to)e(consist)h(of)e(\014nitely)h(man)o (y)340 1251 y(rules.)d(Exceptions)g(from)e(this)i(assumption)e(will)g(b)q(e)i (explicitly)e(indicated.)h(Some)f(consid-)340 1301 y(erations)15 b(concerning)f(in\014nite)g(TRSs)g(in)g([Ohl92a)n(])g(sho)o(w)g(that)g(the)g (restriction)h(to)f(\014nite)340 1351 y(systems)g(ma)o(y)e(b)q(e)j(w)o(eak)o (ened)g(at)e(some)g(places.)340 1426 y Fg(De\014nition)t(2.)37 b Fx(A)16 b Fm(r)n(ewrite)e(or)n(dering)h Fx(is)h(a)f(partial)g(ordering)g Fo(>)h Fx(on)g Fq(T)10 b Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))16 b(whic)o(h)f(is)340 1475 y Fm(close)n(d)j(under)f(c)n(ontexts)g Fx(\(i.e.,)e(if)h Fo(s)g(>)g(t)h Fx(then)g Fo(C)s Fx([)p Fo(s)p Fx(])e Fo(>)h(C)s Fx([)p Fo(t)p Fx(])f(for)h(all)f(con)o(texts)j Fo(C)s Fx([)11 b(]\))16 b(and)340 1525 y Fm(close)n(d)g(under)g (substitutions)e Fx(\(i.e.,)f(if)h Fo(s)f(>)f(t)j Fx(then)g Fo(s\033)f(>)f(t\033)j Fx(for)e(all)f(substitutions)i Fo(\033)q Fx(\).)f(A)340 1575 y Fm(simpli\014c)n(ation)h Fx(ordering)g Fo(>)g Fx(is)f(a)h(rewrite)h(ordering)f(p)q(ossessing)h(the)f Fm(subterm)g(pr)n(op)n(erty)p Fx(,)340 1625 y(i.e.,)e Fo(C)s Fx([)p Fo(t)p Fx(])d Fo(>)i(t)i Fx(for)f(all)g(con)o(texts)i Fo(C)s Fx([)c(])h Fq(6\021)f Fh(2)p Fx(.)340 1700 y Fg(Lemma)d(3.)21 b Fm(L)n(et)14 b Fo(>)h Fm(b)n(e)g(a)g(simpli\014c)n(ation)g(or)n(dering)f (on)i Fq(T)10 b Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))p Fm(.)356 1769 y(1.)21 b(Distinct)15 b(variables)f Fo(x;)7 b(y)13 b Fq(2)e(V)18 b Fm(ar)n(e)d(inc)n(omp)n(ar)n(able)g(w.r.t.)e Fo(>)p Fm(.)356 1819 y(2.)21 b(If)15 b Fo(s)d(>)f(t)p Fm(,)k(then)g Fo(V)10 b(ar)q Fx(\()p Fo(t)p Fx(\))h Fq(\022)h Fo(V)e(ar)q Fx(\()p Fo(s)p Fx(\))p Fm(.)340 1894 y(Pr)n(o)n(of.)20 b Fx(1.)15 b(Supp)q(ose)h Fo(x)e(>)g(y)q Fx(.)h(Then)h(it)f(follo)o(ws)f(that)h Fo(x\033)g(>)f(y)q (\033)j Fx(for)e(an)o(y)g(substitution)h Fo(\033)q Fx(.)340 1944 y(With)e Fo(\033)e Fx(=)g Fq(f)p Fo(x)f Fq(7!)g Fo(y)q Fx(;)c Fo(y)14 b Fq(7!)d Fo(x)p Fq(g)i Fx(this)h(implies)e Fo(y)h(>)f(x)p Fx(,)h(a)h(con)o(tradiction.)340 1993 y(2.)e(Let)h Fo(s)f(>)g(t)p Fx(.)f(Assume)i(that)f(there)i(is)e(a)g(v)n(ariable)f Fo(x)g Fq(2)g Fo(V)f(ar)q Fx(\()p Fo(t)p Fx(\))c Fq(n)g Fo(V)j(ar)q Fx(\()p Fo(s)p Fx(\).)k(Then)g Fo(t)e Fq(\021)h Fo(C)s Fx([)p Fo(x)p Fx(])340 2043 y(for)k(some)f(con)o(text)i Fo(C)s Fx([)12 b(].)j(With)g Fo(\033)i Fx(=)e Fq(f)p Fo(x)g Fq(7!)g Fo(s)p Fq(g)h Fx(it)g(follo)o(ws)e(that)i Fo(s)g Fq(\021)f Fo(s\033)i(>)f(t\033)g Fq(\021)g Fo(C)s Fx([)p Fo(s)p Fx(],)340 2093 y(con)o(tradicting)e(the)h (subterm)e(prop)q(ert)o(y)i(or)f(irre\015exivit)o(y)g(of)f Fo(>)p Fx(.)394 b Fq(u)-28 b(t)340 2168 y Fg(De\014nition)t(4)16 b(\(Kaplan\).)j Fx(A)c(CTRS)f(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))14 b(is)g Fm(simplifying)1377 2153 y Fp(3)1410 2168 y Fx(if)f(there)j(exists)f (a)f(sim-)340 2218 y(pli\014cation)f(ordering)h Fo(>)g Fx(on)f Fq(T)d Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\))14 b(with)g Fo(l)e(>)g(r)o(;)7 b(l)12 b(>)g(s)1248 2224 y Fn(j)1266 2218 y Fx(,)h(and)h Fo(l)e(>)g(t)1455 2224 y Fn(j)1473 2218 y Fx(,)h(for)g(eac)o(h)h(rewrite)340 2267 y(rule)h Fo(l)q Fq(!)o Fo(r)e Fq(\()e Fo(s)582 2273 y Fp(1)612 2267 y Fq(')h Fo(t)671 2273 y Fp(1)690 2267 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)802 2273 y Fn(n)836 2267 y Fq(')k Fo(t)894 2273 y Fn(n)931 2267 y Fx(in)i Fq(R)h Fx(and)g(eac)o(h)g Fo(j)g Fq(2)e(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(n)p Fq(g)p Fx(.)p 340 2298 237 2 v 349 2325 a Fs(2)379 2340 y Fw(W)m(e)13 b(restrict)h(ourselv) o(es)g(to)f(these)h(t)o(w)o(o)e(most)h(common)h(in)o(terpretations,)h(cf.)d (also)i([DOS88)q(].)349 2370 y Fs(3)379 2386 y Fw(F)m(or)f(uncondition)q(al)k (TRSs)d(Kurihara)h(and)f(Oh)o(uc)o(hi)h(\(see)f([K)o(O90b]\))f(in)o(tro)q (duced)j(the)e(notion)379 2432 y Ff(simply)e(terminating.)p eop %%Page: 5 5 bop 183 194 a Fx(Note)19 b(that)f(according)h(to)f(the)h(ab)q(o)o(v)o(e)f (lemma,)d(a)j(simplifying)d(CTRS)j(do)q(es)h(not)g(ha)o(v)o(e)183 244 y(extra-v)n(ariables)d(in)g(its)g(conditions,)g(i.e.,)f(the)i(v)n (ariables)f(app)q(earing)g(in)g(the)h(conditions)183 293 y(of)c(a)h(rule)g(m) o(ust)f(also)g(app)q(ear)h(in)g(the)g(left-hand)f(side)i(of)e(that)h(rule.) 245 343 y(The)i(next)g(prop)q(osition)e(sho)o(ws)i(that)g(simplifyi)o(ng)d (join)h(CTRSs)h(p)q(ossess)j(t)o(w)o(o)c(in)o(ter-)183 393 y(esting)20 b(prop)q(erties.)g(Unfortunately)m(,)f(simplifying)e (semi-equational)g(CTRSs)j(lac)o(k)f(the)183 443 y(second)c(prop)q(ert)o(y)f (as)g(is)g(demonstrated)g(in)f(the)i(example)e(b)q(elo)o(w.)183 519 y Fg(Prop)q(ositi)o(on)5 b(5.)21 b Fm(L)n(et)15 b(the)h(\014nite)f(CTRS)h Fq(R)g Fm(b)n(e)f(simplifying.)g(Then)g Fq(R)h Fm(is)f(terminating,)183 569 y(and)20 b(if)e(additional)r(ly)h Fq(R)h Fm(is)e(a)i(join)f(system,)g (then)g(the)g(applic)n(ability)g(of)g(a)g(c)n(onditional)183 618 y(r)n(ewrite)g(rule)i Fo(l)12 b Fq(!)f Fo(r)i Fq(\()e Fo(s)601 624 y Fp(1)631 618 y Fq(#)h Fo(t)679 624 y Fp(1)697 618 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)809 624 y Fn(n)843 618 y Fq(#)k Fo(t)890 624 y Fn(n)934 618 y Fm(to)21 b(a)h(term)e Fo(s)k Fq(\021)f Fo(C)s Fx([)p Fo(l)q(\033)q Fx(])d Fm(is)h(de)n(cidable)g(by)183 668 y(r)n(e)n(cursively)13 b(r)n(e)n(ducing)i(the)g(terms)f Fo(s)756 674 y Fn(j)774 668 y Fo(\033)i Fm(and)g Fo(t)910 674 y Fn(j)928 668 y Fo(\033)g Fm(for)e(al)r(l)g Fo(j)g Fq(2)e(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(n)p Fq(g)p Fm(.)183 744 y(Pr)n(o)n(of.)20 b Fq(R)f Fx(is)g(certainly)g(terminating)e(if)h(the)h(unconditional)f(TRS)h (obtained)f(from)f Fq(R)183 794 y Fx(b)o(y)c(remo)o(ving)f(the)i(conditions)f (from)f(the)j(rules)f(is)f(terminating.)f(But)i(\014nite)g(simplifying)183 844 y(unconditional)8 b(TRSs)i(are)g(terminating)e(as)i(w)o(as)g(\014rst)h (pro)o(v)o(ed)f(b)o(y)f(Dersho)o(witz)i(in)e([Der82].)183 894 y(F)m(or)k(the)i(second)g(statemen)o(t)e(w)o(e)i(refer)f(to)g([Kap87)o(].)580 b Fq(u)-28 b(t)183 976 y Fg(Example)7 b(6.)21 b Fx(Consider)15 b Fq(R)e Fx(=)g Fq(f)p Fo(a)g Fq(!)f Fo(b)h Fq(\()f Fo(g)i Fx(=)f Fo(h;)7 b(f)17 b Fq(!)c Fo(g)h Fq(\()e Fo(a)h Fx(=)g Fo(b;)7 b(f)17 b Fq(!)12 b Fo(h)h Fq(\()g Fo(a)f Fx(=)h Fo(b)p Fq(g)p Fx(.)183 1026 y Fq(R)i Fx(is)g(simplifyi)o(ng)d(as)j(is)g(sho)o(wn)g (b)o(y)g(the)h(simpli\014cation)c(ordering)j Fo(f)k(>)13 b(a)h(>)g(b)f(>)h(g) g(>)g(h)p Fx(.)183 1076 y(Do)q(es)g Fo(a)g Fx(reduce)h(to)f Fo(b)p Fx(?)183 1158 y(Next)g(w)o(e)f(sho)o(w)g(that)h(a)f(CTRS)g Fq(R)g Fx(is)g(simplifying)d(if)i(and)i(only)e(if)h(the)g(transitiv)o(e)h (closure)183 1208 y(of)j(a)g(certain)i(rewrite)g(relation)e(is)h (irre\015exiv)o(e.)g(This)f(c)o(haracterization)i(of)e(simplifying)183 1258 y(CTRSs)j(is)g(the)g(k)o(ey)g(to)g(the)h(pro)q(of)f(of)f(Theorem)h(23)f (and)h(do)q(es)h(not)f(dep)q(end)h(on)f(the)183 1308 y(in)o(terpretation)14 b(of)f Fq(')p Fx(.)183 1383 y Fg(De\014niti)o(on)5 b(7.)20 b Fx(Let)15 b Fq(R)f Fx(b)q(e)g(a)g(CTRS.)f(With)g Fq(R)h Fx(w)o(e)g(asso)q (ciate)h(the)f(follo)o(wing)e(TRS)310 1462 y Fq(S)j Fx(:=)51 b Fq(f)p Fo(l)q Fq(!)o Fo(r)14 b Fq(j)e Fo(l)h Fq(!)e Fo(r)h Fq(\()f Fo(s)758 1468 y Fp(1)789 1462 y Fq(')g Fo(t)847 1468 y Fp(1)866 1462 y Fo(;)c(:)g(:)g(:)e(;)i(s)978 1468 y Fn(n)1012 1462 y Fq(')12 b Fo(t)1071 1468 y Fn(n)1105 1462 y Fq(2)f(Rg)406 1524 y([)g(f)p Fo(l)q Fq(!)o Fo(s)539 1530 y Fn(i)566 1524 y Fq(j)h Fo(l)h Fq(!)e Fo(r)h Fq(\()f Fo(s)771 1530 y Fp(1)802 1524 y Fq(')h Fo(t)861 1530 y Fp(1)879 1524 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)991 1530 y Fn(n)1025 1524 y Fq(')12 b Fo(t)1084 1530 y Fn(n)1118 1524 y Fq(2)f(R)p Fx(;)19 b Fo(i)12 b Fq(2)g(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(n)p Fq(gg)406 1586 y([)k(f)p Fo(l)q Fq(!)o Fo(t)535 1592 y Fn(i)562 1586 y Fq(j)h Fo(l)g Fq(!)f Fo(r)i Fq(\()e Fo(s)767 1592 y Fp(1)798 1586 y Fq(')g Fo(t)856 1592 y Fp(1)875 1586 y Fo(;)c(:)g(:)g(:)e(;)i(s)987 1592 y Fn(n)1021 1586 y Fq(')12 b Fo(t)1080 1592 y Fn(n)1114 1586 y Fq(2)f(R)p Fx(;)19 b Fo(i)12 b Fq(2)f(f)p Fx(1)p Fo(;)c(:)g(:)g(:)t(;)g(n)p Fq(gg)p Fo(:)183 1664 y Fx(W)m(e)17 b(are)h(also)f(in)o(terested)i(in)e (another)h(unconditional)e(TRS)h(related)h(to)f(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))18 b(whic)o(h)183 1714 y(dep)q(ends)i(only)e(on)g(the)h (underlying)f(signature)h(\(and)g(th)o(us)g(ma)o(y)d(consist)j(of)f (in\014nitely)183 1764 y(man)o(y)12 b(rules)i(if)f(the)i(signature)f(is)g (in\014nite\):)355 1842 y Fq(F)389 1827 y Fn(e)419 1842 y Fx(:=)d Fq(f)p Fo(f)t Fx(\()p Fo(x)559 1848 y Fp(1)578 1842 y Fo(;)c(:)g(:)g(:)e(;)i (x)695 1848 y Fn(n)717 1842 y Fx(\))12 b Fq(!)g Fo(x)823 1848 y Fn(i)849 1842 y Fq(j)g Fo(f)17 b Fq(2)11 b(F)979 1848 y Fn(n)1001 1842 y Fx(;)19 b Fo(n)12 b Fq(2)f Fe(I)-6 b(N)p Fx(;)19 b Fo(i)12 b Fq(2)f(f)p Fx(1)p Fo(;)c(:)g(:)g(:)t(;)g(n)p Fq(gg)p Fo(:)183 1925 y Fg(Lemma)g(8.)21 b Fm(L)n(et)13 b Fx(\()p Fq(F)t Fo(;)7 b Fq(S)s Fx(\))13 b Fm(b)n(e)g(an)h(unc)n(onditional)h(TRS,)e(and)h(let)f Fx(\()p Fq(F)t Fo(;)7 b Fq(S)s Fo(e)p Fx(\))12 b(:=)f(\()p Fq(F)t Fo(;)c Fq(S)h([)e(F)1577 1910 y Fn(e)1595 1925 y Fx(\))p Fm(.)183 1974 y Fq(!)225 1957 y Fp(+)225 1987 y Fk(S)r Fn(e)294 1974 y Fm(is)15 b(a)g(simpli\014c)n(ation)f(or)n(dering)h(if)f(it)h(is)f(irr) n(e\015exive.)183 2050 y(Pr)n(o)n(of.)20 b Fq(!)355 2033 y Fp(+)355 2063 y Fk(S)r Fn(e)421 2050 y Fx(is)12 b(transitiv)o(e)g(b)o(y)g (de\014nition.)f(An)o(y)h(rewrite)h(relation)f(is)g(closed)g(under)h(con-)183 2100 y(texts)19 b(and)f(substitutions.)g(Hence)i(the)f(same)e(holds)h(for)g Fq(!)1173 2082 y Fp(+)1173 2112 y Fk(S)r Fn(e)1244 2100 y Fx(\(b)q(eing)h (the)g(transitiv)o(e)183 2150 y(closure)e(of)e(a)h(rewrite)h(relation\).)e (Finally)m(,)e Fq(!)928 2132 y Fp(+)928 2162 y Fk(S)r Fn(e)998 2150 y Fx(has)j(the)h(subterm)f(prop)q(ert)o(y)h(b)q(ecause)183 2200 y Fq(!)224 2206 y Fk(F)251 2198 y Fd(e)283 2200 y Fx(is)d(con)o(tained)g (in)f Fq(!)603 2182 y Fp(+)603 2212 y Fk(S)r Fn(e)657 2200 y Fx(.)g(Th)o(us,)h(the)g(assertion)h(follo)o(ws.)416 b Fq(u)-28 b(t)183 2282 y Fg(Lemma)7 b(9.)21 b Fm(L)n(et)15 b(the)h(unc)n(onditional)h (TRS)f Fx(\()p Fq(F)t Fo(;)7 b Fq(S)s Fx(\))15 b Fm(b)n(e)h(simplifying)f (w.r.t.)e(the)j(simpli\014-)183 2332 y(c)n(ation)c(or)n(dering)f Fo(>)p Fm(.)h(Then)g(\(i\))f Fq(!)727 2314 y Fp(+)727 2344 y Fk(S)r Fn(e)794 2332 y Fq(\022)c Fo(>)12 b Fm(and)h(\(ii\))e Fq(!)1068 2314 y Fp(+)1068 2344 y Fk(S)r Fn(e)1134 2332 y Fm(is)h(a)g (simpli\014c)n(ation)f(or)n(dering.)183 2382 y(In)k(other)g(wor)n(ds,)g Fq(!)518 2364 y Fp(+)518 2394 y Fk(S)r Fn(e)572 2382 y Fm(is)g(the)h(smal)r (lest)e(simpli\014c)n(ation)h(or)n(dering)g(showing)g(that)h Fx(\()p Fq(F)t Fo(;)7 b Fq(S)s Fx(\))183 2432 y Fm(is)14 b(simplifying.)p eop %%Page: 6 6 bop 340 194 a Fm(Pr)n(o)n(of.)20 b Fx(\(i\))c(Since)h Fo(>)f Fx(is)g(transitiv)o(e,)f(it)h(su\016ces)h(to)f(sho)o(w)g(that)g Fo(s)f Fq(!)1444 200 y Fk(S)r Fn(e)1499 194 y Fo(t)h Fx(implies)e Fo(s)i(>)f(t)p Fx(.)340 244 y(So)i(let)f Fo(s)h Fq(!)541 250 y Fk(S)r Fn(e)597 244 y Fo(t)p Fx(.)f(If)g Fo(s)p Fq(!)745 250 y Fk(S)782 244 y Fo(t)p Fx(,)g(then)h(clearly)g Fo(s)f(>)g(t)p Fx(.)g(If)h(otherwise)g Fo(s)p Fq(!)1477 250 y Fk(F)1504 241 y Fd(e)1522 244 y Fo(t)p Fx(,)f(that)h(is,)f Fo(s)g Fq(\021)340 293 y Fo(C)s Fx([)p Fo(f)t Fx(\()p Fo(s)444 299 y Fp(1)464 293 y Fo(;)7 b(:)g(:)g(:)t(;)g(s)575 299 y Fn(n)598 293 y Fx(\)])12 b(and)g Fo(t)f Fq(\021)h Fo(C)s Fx([)p Fo(s)851 299 y Fn(i)864 293 y Fx(])g(where)i Fo(f)i Fq(2)11 b(F)1112 299 y Fn(n)1134 293 y Fx(,)h(then)h(the)g(assertion)g(follo)o(ws)d(from)h(the)340 343 y(fact)j(that)g Fo(>)g Fx(has)g(the)h(subterm)f(prop)q(ert)o(y)g(and)g (is)g(closed)g(under)h(con)o(texts.)340 393 y(\(ii\))f(F)m(ollo)o(ws)e(from)g (Lemma)f(8)j(b)q(ecause)h(\(i\))f(implies)e(that)i Fq(!)1322 375 y Fp(+)1322 405 y Fk(S)r Fn(e)1389 393 y Fx(is)g(irre\015exiv)o(e.)135 b Fq(u)-28 b(t)340 485 y Fg(Prop)q(osition)5 b(10.)20 b Fm(L)n(et)13 b Fq(R)g Fm(b)n(e)g(a)g(CTRS)g(and)h Fq(S)i Fm(the)d(asso)n(ciate)n(d)g(TRS.) g(Then)g(the)g(fol)r(low-)340 535 y(ing)i(statements)g(ar)n(e)g(e)n (quivalent:)356 618 y(1.)21 b Fq(R)15 b Fm(is)g(simplifying.)356 668 y(2.)21 b Fq(S)d Fm(is)d(simplifying.)356 718 y(3.)21 b Fq(S)s Fo(e)15 b Fm(is)g(terminating.)356 768 y(4.)21 b Fq(!)453 750 y Fp(+)453 780 y Fk(S)r Fn(e)508 768 y Fm(is)14 b(irr)n(e\015exive.)340 860 y(Pr)n(o)n(of.)20 b Fx(1)p Fo(:)11 b Fq(,)g Fx(2)p Fo(:)g Fx(:)j(By)g(de\014nition)f(of)g Fq(S)s Fx(.)340 910 y(2)p Fo(:)22 b Fq(\))f Fx(3)p Fo(:)h Fx(:)d(Let)i Fo(>)f Fx(b)q(e)h(a)f(simpli\014cation)e (ordering)i(sho)o(wing)g(that)g Fq(S)j Fx(is)d(simplifying.)340 960 y(Assume)15 b(that)h Fq(S)s Fo(e)f Fx(is)g(not)g(terminating,)e(i.e.,)h (there)i(is)f(an)g(in\014nite)g(reduction)h(sequence)340 1009 y Fo(s)359 1015 y Fp(1)395 1009 y Fq(!)437 1015 y Fk(S)r Fn(e)494 1009 y Fo(s)513 1015 y Fp(2)549 1009 y Fq(!)591 1015 y Fk(S)r Fn(e)647 1009 y Fo(s)666 1015 y Fp(3)702 1009 y Fq(!)744 1015 y Fk(S)r Fn(e)801 1009 y Fo(:)7 b(:)g(:)n Fx(,)17 b(where)h Fo(s)1020 1015 y Fn(i)1051 1009 y Fq(2)f(T)10 b Fx(\()p Fq(F)t Fo(;)d Fq(V)s Fx(\).)17 b(According)g(to)g(Lemma)d(9)j(this)340 1059 y(implies)d Fo(s)502 1065 y Fp(1)535 1059 y Fo(>)g(s)600 1065 y Fp(2)633 1059 y Fo(>)h(s)699 1065 y Fp(3)732 1059 y Fo(>)f(:)7 b(:)g(:)o Fx(.)15 b(Note)g(that)h(Lemma)d(3)i(states)h(that)g Fq(V)s Fo(ar)q Fx(\()p Fo(s)1549 1065 y Fn(i)1564 1059 y Fx(\))e Fq(\022)g(V)s Fo(ar)q Fx(\()p Fo(s)1745 1065 y Fp(1)1765 1059 y Fx(\))340 1109 y(for)k(all)f Fo(i)i Fq(2)f Fo(I)-11 b(N)5 b Fx(.)18 b(Let)g Fq(F)736 1094 y Fk(0)766 1109 y Fx(b)q(e)h(the)f(set)h(of)f (function)g(sym)o(b)q(ols)e(o)q(ccurring)j(in)f Fq(S)s Fx(,)f(and)h(let)340 1159 y Fq(F)374 1144 y Fk(00)414 1159 y Fx(b)q(e)h(the)h(set)f(of)f(function) g(sym)o(b)q(ols)g(o)q(ccurring)h(in)f Fo(s)1264 1165 y Fp(1)1283 1159 y Fx(.)g(Then)h(eac)o(h)g Fo(s)1543 1165 y Fn(i)1576 1159 y Fx(\()p Fo(i)h Fq(2)f Fo(I)-11 b(N)5 b Fx(\))19 b(is)340 1209 y(an)14 b(elemen)o(t)f(of)f Fq(T)e Fx(\()p Fq(F)681 1188 y Fk(0)701 1209 y Fq([)e(F)771 1194 y Fk(00)792 1209 y Fo(;)f Fq(V)s Fo(ar)q Fx(\()p Fo(s)916 1215 y Fp(1)935 1209 y Fx(\)\).)13 b(This,)g(ho)o(w)o(ev)o(er,)h(con)o(tradicts)g(the)g(fact)f(that)h(the)340 1259 y(restriction)g(of)f Fo(>)g Fx(to)f Fq(T)f Fx(\()p Fq(F)763 1238 y Fk(0)782 1259 y Fq([)c(F)851 1243 y Fk(00)872 1259 y Fo(;)g Fq(V)s Fo(ar)q Fx(\()p Fo(s)996 1265 y Fp(1)1015 1259 y Fx(\)\))13 b(is)g(a)g(w)o(ell-founded)f(simpli\014cation)e(ordering)340 1308 y(\(since)15 b Fq(F)492 1293 y Fk(0)504 1308 y Fo(;)7 b Fq(F)557 1293 y Fk(00)577 1308 y Fx(,)14 b(and)g Fq(V)s Fo(ar)q Fx(\()p Fo(s)789 1314 y Fp(1)808 1308 y Fx(\))g(are)h(\014nite)f(sets,)h(cf.) e(also)g([Der82,)g(Ohl92a)o(]\).)340 1358 y(3)p Fo(:)e Fq(\))g Fx(4)p Fo(:)g Fx(:)i(Eviden)o(tly)m(,)g(if)g Fq(!)783 1340 y Fp(+)783 1370 y Fk(S)r Fn(e)837 1358 y Fx(is)g(not)h(irre\015exiv)o(e,)g (then)h Fq(S)s Fo(e)f Fx(is)g(not)g(terminating.)340 1408 y(4)p Fo(:)k Fq(\))f Fx(2)p Fo(:)g Fx(:)g(By)h(Lemma)d(8,)i Fq(!)842 1390 y Fp(+)842 1420 y Fk(S)r Fn(e)896 1408 y Fx(is)g(a)h(simpli\014cation)d (ordering.)i(Hence)i(the)g(assertion)340 1458 y(follo)o(ws)13 b(from)f Fq(!)616 1464 y Fk(S)652 1458 y Fq(\022)g(!)738 1440 y Fp(+)738 1470 y Fk(S)r Fn(e)792 1458 y Fx(.)950 b Fq(u)-28 b(t)340 1550 y Fx(The)14 b(follo)o(wing)c(example)i(w)o(as)h(\014rst)h(giv)o (en)f(b)o(y)f(T)m(o)o(y)o(ama)e(in)j([T)m(o)o(y87a)n(])f(for)h(the)h(purp)q (ose)g(of)340 1599 y(sho)o(wing)g(that)g(termination)e(is)h(not)h(mo)q (dular.)340 1683 y Fg(Example)8 b(11.)20 b Fq(R)646 1689 y Fp(1)685 1683 y Fx(=)g Fq(f)p Fo(F)6 b Fx(\(0)p Fo(;)h Fx(1)p Fo(;)g(x)p Fx(\))p Fq(!)m Fo(F)f Fx(\()p Fo(x;)h(x;)g(x)p Fx(\))p Fq(g)17 b Fx(is)h(eviden)o(tly)h(terminating)e(but)i(not)340 1733 y(simplifying,)8 b(since)k Fq(!)703 1715 y Fp(+)703 1745 y Fk(S)723 1749 y Fc(1)739 1745 y Fn(e)755 1749 y Fc(1)784 1733 y Fx(is)f(not)h(irre\015exiv)o(e:)f(If)g Fo(t)g Fq(\021)h Fo(F)6 b Fx(\(0)p Fo(;)h Fx(1)p Fo(;)g Fx(1\))i(and)i Fo(s)h Fq(\021)g Fo(F)6 b Fx(\()p Fo(t;)h(t;)g(t)p Fx(\),)j(then)340 1783 y(there)16 b(is)d(the)i(cyclic)f(reduction)h(sequence)h Fo(s)c Fq(!)1104 1789 y Fk(F)1128 1793 y Fc(1)1143 1780 y Fd(e)1173 1783 y Fo(F)6 b Fx(\(0)p Fo(;)h(t;)g(t)p Fx(\))j Fq(!)1379 1789 y Fk(F)1403 1793 y Fc(1)1418 1780 y Fd(e)1448 1783 y Fo(F)c Fx(\(0)p Fo(;)h Fx(1)p Fo(;)g(t)p Fx(\))j Fq(!)1660 1789 y Fk(S)1680 1793 y Fc(1)1709 1783 y Fo(s)p Fx(.)340 1832 y(The)k(TRS)f(\()p Fq(F)568 1838 y Fp(2)586 1832 y Fo(;)7 b Fq(R)640 1838 y Fp(2)659 1832 y Fx(\))k(=)h(\()p Fq(f)p Fo(g)q Fq(g)p Fo(;)7 b Fq(f)p Fo(g)q Fx(\()p Fo(x;)g(y)q Fx(\))p Fq(!)o Fo(x;)g(g)q Fx(\()p Fo(x;)g(y)q Fx(\))p Fq(!)p Fo(y)q Fq(g)p Fx(\))14 b(is)f(simplifyi)o(ng)d(b)q (ecause)15 b Fq(R)1721 1838 y Fp(2)1753 1832 y Fx(is)340 1882 y(terminating)e(and)g Fq(R)682 1888 y Fp(2)713 1882 y Fx(=)e Fq(S)781 1888 y Fp(2)812 1882 y Fx(=)g Fq(F)885 1888 y Fp(2)904 1864 y Fn(e)933 1882 y Fx(=)h Fq(S)1002 1888 y Fp(2)1021 1882 y Fo(e)1040 1888 y Fp(2)1059 1882 y Fx(.)340 1932 y(The)21 b(union)e(of)h(the)g(TRSs)g Fq(R)839 1938 y Fp(1)878 1932 y Fx(and)g Fq(R)1000 1938 y Fp(2)1039 1932 y Fx(is)g(not)g(terminating,)d(for)j (there)h(is)f(a)g(cyclic)340 1982 y(reduction)15 b(sequence)h(starting)e (with)f Fo(F)6 b Fx(\()p Fo(g)q Fx(\(0)p Fo(;)h Fx(1\))p Fo(;)g(g)q Fx(\(0)p Fo(;)g Fx(1\))p Fo(;)g(g)q Fx(\(0)p Fo(;)g Fx(1\)\).)340 2065 y(F)m(or)14 b(the)g(sak)o(e)g(of)f(simplicit)o(y)f(w)o(e)i(will)e(later) i(in)o(tro)q(duce)g(a)g(new)g(binary)f(constructor)j Fn(C)r(ons)340 2115 y Fx(and)e(a)g(constan)o(t)g Fn(N)s(il)q Fx(.)f(That)h(this)g(do)q(es)h (no)e(harm)g(is)h(sho)o(wn)f(in)h(the)g(next)h(lemma.)340 2199 y Fg(Lemma)8 b(12.)20 b Fm(L)n(et)f Fx(\()p Fq(F)t Fo(;)7 b Fq(S)s Fx(\))19 b Fm(b)n(e)h(an)f(unc)n(onditional)i(TRS.)e(Set)g Fq(F)1398 2184 y Fk(0)1429 2199 y Fx(:=)g Fq(F)d([)c(f)p Fn(C)r(ons)p Fo(;)7 b Fn(N)s(il)p Fq(g)p Fm(.)340 2249 y(Then)16 b Fq(S)s Fo(e)f Fm(is)g(terminating)f(if)g(and)i(only)f(if)f Fq(S)s 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642 y Fx(is)f(rewritten)h(to)e Fo(s)991 627 y Fk(0)991 653 y Fn(i)p Fp(+1)1058 642 y Fx(b)o(y)h(the)g(rule)g Fo(f)t Fx(\()p Fo(x;)c(y)q(;)g(z)r(;)g(:)g(:)g(:)f(;)h(z)r Fx(\))k Fq(!)g Fo(x)183 692 y Fx(\(resp.)i Fo(f)t Fx(\()p Fo(x;)7 b(y)q(;)g(z)r(;)g(:)g(:)g(:)e(;)i(z)r Fx(\))12 b Fq(!)f Fo(y)q Fx(\))i(if)f Fo(s)742 698 y Fn(i)769 692 y Fx(is)h(reduced)h(to)f Fo(s)1032 698 y Fn(i)p Fp(+1)1101 692 y Fx(using)f(the)i(rule)f Fn(C)r(ons)o Fx(\()p Fo(x;)7 b(y)q Fx(\))12 b Fq(!)f Fo(x)183 742 y Fx(\(resp.)j Fn(C)r(ons)p Fx(\()p Fo(x;)7 b(y)q Fx(\))12 b Fq(!)f Fo(y)q Fx(\).)j(This)g(con)o(tradicts)g(the)h(termination)d(of)h Fq(S)s Fo(e)p Fx(.)283 b Fq(u)-28 b(t)183 866 y Fr(4)56 b(Com)n(bined)18 b(Conditional)g(T)-5 b(erm)17 b(Rewriting)h(Systems)183 956 y Fx(In)9 b(this)g(section,)h(w)o(e)g(pro)o(v)o(e)f(that)h(the)g(prop)q(ert)o (y)g(to)f(b)q(e)h(simplifying)c(is)j(a)g(mo)q(dular)f(prop)q(ert)o(y)183 1006 y(\(Theorem)14 b(23\).)f(The)i(pro)q(of)f(dep)q(ends)i(on)e(Prop)q 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Fx(is)e(called)h(the)h Fm(c)n(ombine)n(d)h(CTRS)183 1738 y(of)f Fq(R)264 1744 y Fp(1)294 1738 y Fm(and)h Fq(R)407 1744 y Fp(2)438 1738 y Fm(with)e(shar)n(e)n(d)h(c)n(onstructors)g Fq(C)r Fx(.)e(T)m(o)g(b)q(e)h(able)g(to)g(distinguish)f(b)q(et)o(w)o(een)i (sym-)183 1788 y(b)q(ols)i(from)f(di\013eren)o(t)j(sets,)f(w)o(e)g(use)h (capitals)e Fo(F)q(;)7 b(G;)g(:)g(:)g(:)k Fx(for)k(function)f(sym)o(b)q(ols)f (from)g Fq(D)1593 1794 y Fp(1)1612 1788 y Fx(,)183 1837 y(small)e(letters)j Fo(f)r(;)7 b(g)q(;)g(:)g(:)g(:)k Fx(for)i(those)h(from)d Fq(D)859 1843 y Fp(2)878 1837 y Fx(,)i(and)g(small)d(capitals)j Fn(C)r Fo(;)7 b Fn(D)p Fo(;)g(:)g(:)g(:)k Fx(for)i(construc-)183 1887 y(tors.)h(As)g(con)o(v)o(en)o(tional)f Fo(x;)7 b(y)q(;)g(z)r(;)g(x)726 1893 y Fp(1)743 1887 y Fo(;)g(y)782 1893 y Fp(1)801 1887 y Fo(;)g(z)839 1893 y Fp(1)857 1887 y Fo(;)g(:)g(:)g(:)12 b Fx(will)g(denote)j (v)n(ariables.)183 1960 y Fg(De\014niti)o(on)5 b(13.)20 b Fx(Let)c Fo(t)f Fq(2)f(T)c Fx(\()p Fq(D)708 1966 y Fp(1)736 1960 y Fq([)f(D)805 1966 y Fp(2)823 1960 y Fo(;)e Fq(C)r Fo(;)g Fq(V)s Fx(\))q(.)15 b(W)m(e)g(color)g(eac)o(h)h(sym)o(b)q(ol)e(in)h Fo(t)p Fx(.)g(F)m(unction)183 2010 y(sym)o(b)q(ols)g(from)h Fq(D)478 2016 y Fp(1)513 2010 y Fx(are)i(colored)f(blac)o(k,)f(those)i(from)d Fq(D)1100 2016 y Fp(2)1136 2010 y Fx(white,)h(and)h(constructors)i(as)183 2060 y(w)o(ell)d(as)i(v)n(ariables)f(are)h(transparen)o(t.)g(If)f Fo(t)g Fx(do)q(es)i(not)e(con)o(tain)g(white)h(\(blac)o(k\))f(function)183 2110 y(sym)o(b)q(ols,)c(w)o(e)i(sp)q(eak)h(of)e(a)h Fm(black)h(\(white\))f (term)p Fx(.)f Fo(t)h Fx(is)g(said)g(to)g(b)q(e)h Fm(tr)n(ansp)n(ar)n(ent)e Fx(if)h(it)f(only)183 2159 y(con)o(tains)i(constructors)i(and)f(v)n (ariables.)e(Consequen)o(tly)m(,)h(a)g(transparen)o(t)i(term)d(ma)o(y)g(b)q (e)183 2209 y(regarded)h(as)f(blac)o(k)g(or)g(white,)g(this)g(is)g(con)o(v)o (enien)o(t)h(for)f(later)g(purp)q(oses.)h Fo(t)f Fx(is)g(called)g Fm(top)183 2259 y(black)g(\(top)g(white,)f(top)h(tr)n(ansp)n(ar)n(ent\))e Fx(if)g Fo(r)q(oot)p Fx(\()p Fo(t)p Fx(\))h(is)g(blac)o(k)g(\(white,)f (transparen)o(t\).)183 2332 y(Sev)o(eral)k(de\014nitions)f(and)h (considerations)g(are)h(symmetrical)c(in)i(the)i(colors)e(blac)o(k)h(and)183 2382 y(white.)c(Therefore,)h(w)o(e)g(state)g(the)h(resp)q(ectiv)o(e)g (de\014nitions)f(and)f(considerations)h(only)f(for)183 2432 y(the)h(color)g(blac)o(k)f(\(the)i(same)e(applies)g(cum)g(grano)h(coloris)f (for)h(the)g(color)g(white\).)p eop %%Page: 8 8 bop 340 194 a Fg(De\014nition)t(14.)21 b Fx(Let)13 b Fo(s)f Fx(b)q(e)g(a)g(top)g(blac)o(k)f(term.)g(There)i(is)f(a)f(uniquely)h (determined)g(blac)o(k)340 244 y(con)o(text)h Fo(C)s Fx([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fx(])k(suc)o(h)i(that)f Fo(s)g Fq(\021)g Fo(C)s Fx([)p Fo(s)965 250 y Fp(1)983 244 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1095 250 y Fn(n)1117 244 y Fx(])12 b(and)f Fo(r)q(oot)p Fx(\()p Fo(s)1329 250 y Fn(j)1348 244 y Fx(\))g Fq(2)g(D)1446 250 y Fp(2)1477 244 y Fx(for)h Fo(j)i Fq(2)d(f)p Fx(1)p Fo(;)c(:)g(:)g(:)t(n)p Fq(g)p Fx(.)340 293 y(W)m(e)18 b(denote)h(this)f(b)o(y)g Fo(s)h Fq(\021)g Fo(C)s Fx([)-7 b([)p Fo(s)859 299 y Fp(1)876 293 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)988 299 y Fn(n)1011 293 y Fx(])-7 b(].)16 b(Moreo)o(v)o(er,)j(w)o(e)f(de\014ne)h(the)g(set)f Fo(S)r Fx(\()p Fo(s)p Fx(\))i(of)e(all)340 343 y Fm(princip)n(al)f Fx(subterms)h(of)f Fo(s)g Fx(to)g(b)q(e)h Fo(S)r Fx(\()p Fo(s)p Fx(\))h(:=)e Fq(f)p Fo(s)1096 349 y Fp(1)1115 343 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1227 349 y Fn(n)1250 343 y Fq(g)p Fx(.)16 b(No)o(w)h(let)h Fo(s)f Fx(b)q(e)h(an)f(arbitrary)340 393 y(term.)f(There)i (is)e(a)h(\(p)q(ossibly)f(empt)o(y\))g(transparen)o(t)i(con)o(text)g Fo(C)s Fx([)p Fo(;)7 b(:)g(:)g(:)s(;)g Fx(])16 b(suc)o(h)i(that)e Fo(s)h Fq(\021)340 443 y Fo(C)s Fx([)p Fo(s)404 449 y Fp(1)423 443 y Fo(;)7 b(:)g(:)g(:)t(;)g(s)534 449 y Fn(n)557 443 y Fx(])o(,)18 b(where)h Fo(r)q(oot)p Fx(\()p Fo(s)832 449 y Fn(j)851 443 y Fx(\))g Fq(2)f(D)964 449 y Fp(1)995 443 y Fq([)11 b(D)1066 449 y Fp(2)1103 443 y Fx(for)18 b Fo(j)j 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Fx(b)o(y)g(an)f(application)g(of)g(a)g(rewrite)340 781 y(rule)16 b(of)e Fq(R)p Fx(.)g(W)m(e)h(write)g Fo(s)f Fq(!)789 766 y Fn(i)789 793 y Fk(R)832 781 y Fo(t)h Fx(if)f(the)h(rule)g(is)g(applied) f(in)h(one)g(of)f(the)h Fo(s)1512 787 y Fn(j)1545 781 y Fx(and)g(w)o(e)g (write)340 831 y Fo(s)f Fq(!)415 816 y Fn(o)415 843 y Fk(R)458 831 y Fo(t)g Fx(otherwise.)h(The)g(relation)f Fq(!)965 816 y Fn(i)965 843 y Fk(R)1010 831 y Fx(is)h(called)f Fm(inner)h Fx(reduction)g(and)f Fq(!)1590 816 y Fn(o)1590 843 y Fk(R)1635 831 y Fx(is)h(called)340 881 y Fm(outer)g Fx(reduction.)h(No)o(w)e(let)i Fo(s)e Fq(\021)p 882 848 V 13 w Fo(C)s Fx([)p Fo(s)946 887 y Fp(1)965 881 y Fo(;)7 b(:)g(:)g(:)t(;)g(s)1076 887 y Fn(n)1099 881 y Fx(])15 b(with)p 1221 848 V 14 w Fo(C)s Fx([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fx(])13 b Fq(6\021)h Fh(2)h Fx(and)g Fo(s)f Fq(!)1632 887 y Fk(R)1675 881 y Fo(t)p Fx(,)h(i.e.,)340 931 y Fo(t)d Fq(\021)p 411 898 V 12 w Fo(C)r Fx([)p Fo(s)474 937 y Fp(1)493 931 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)605 937 y Fn(j)r Fk(\000)p Fp(1)665 931 y Fo(;)g(t)699 937 y Fn(j)716 931 y Fo(;)g(s)754 937 y Fn(j)r Fp(+1)813 931 y Fo(;)g(:)g(:)g(:)e(;)i(s)925 937 y Fn(n)947 931 y Fx(])k(for)f(some)g Fo(j)k Fq(2)e(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(n)p Fq(g)p Fx(.)j(Then)h(w)o(e)g(write)h Fo(s)g Fq(!)1725 916 y Fn(i)1725 942 y Fk(R)1766 931 y Fo(t)340 981 y Fx(if)h Fo(s)397 987 y Fn(j)427 981 y Fq(!)469 966 y Fn(i)469 992 y Fk(R)511 981 y Fo(t)526 987 y Fn(j)557 981 y Fx(and)h Fo(s)e Fq(!)711 966 y Fn(o)711 992 y Fk(R)752 981 y Fo(t)i Fx(if)f Fo(s)838 987 y Fn(j)868 981 y Fq(!)910 966 y Fn(o)910 992 y Fk(R)951 981 y Fo(t)966 987 y Fn(j)984 981 y Fx(.)340 1070 y Fg(De\014nition)t(16.)21 b Fx(Let)14 b Fo(s)h Fx(b)q(e)f(a)g(top)g(blac)o(k)f(or)h(top)g(white)g(term.)f(W)m(e)g(de\014ne) 353 1177 y Fo(r)q(ank)q Fx(\()p Fo(s)p Fx(\))f(=)549 1118 y Fl(\032)586 1151 y Fx(1)559 b(,)14 b(if)f Fo(s)f Fq(2)f(T)f Fx(\()p Fq(D)1381 1157 y Fp(1)1400 1151 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\))i Fq([)g(T)h Fx(\()p Fq(D)1633 1157 y Fp(2)1652 1151 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\))586 1201 y(1)i(+)h Fo(max)p Fq(f)p Fo(r)q(ank)q Fx(\()p Fo(s)886 1207 y Fn(j)903 1201 y Fx(\))20 b Fq(j)f Fx(1)11 b Fq(\024)h Fo(j)i Fq(\024)e Fo(n)p Fq(g)p Fx(,)h(if)g Fo(s)f Fq(\021)g Fo(C)s Fx([)-7 b([)p Fo(s)1374 1207 y Fp(1)1391 1201 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1503 1207 y Fn(n)1526 1201 y Fx(])-7 b(])340 1283 y(No)o(w)14 b(let)g Fo(t)g Fx(b)q(e)g(a)g(top)g(transparen)o(t)h(term.)d(Then)j(w)o(e)f(de\014ne) 502 1391 y Fo(r)q(ank)q Fx(\()p Fo(t)p Fx(\))e(=)694 1332 y Fl(\032)731 1364 y Fx(0)495 b(,)13 b(if)g Fo(t)f Fq(2)f(T)f Fx(\()p Fq(C)r Fo(;)d Fq(V)s Fx(\))731 1416 y Fo(max)p Fq(f)p Fo(r)q(ank)q Fx(\()p Fo(t)955 1422 y Fn(j)972 1416 y Fx(\))20 b Fq(j)f Fx(1)11 b Fq(\024)h Fo(j)i Fq(\024)e Fo(m)p Fq(g)p Fx(,)h(if)g Fo(t)f Fq(\021)p 1381 1383 V 12 w Fo(C)r Fx([)p Fo(t)1440 1422 y Fp(1)1459 1416 y Fo(;)7 b(:)g(:)g(:)t(;)g(t)1566 1422 y Fn(m)1597 1416 y Fx(])340 1504 y(The)14 b(pro)q(ofs)g(of)e(the)i(next) g(lemmata)d(are)i(straigh)o(tforw)o(ard.)g Fq(R)g Fx(denotes)i(as)e(usual)h (a)f(com-)340 1553 y(bined)h(CTRS)g(and)g Fo(s;)7 b(t)13 b Fx(are)i(terms)e(from)f Fq(T)f Fx(\()p Fq(F)t Fo(;)c Fq(V)s Fx(\).)340 1635 y Fg(Lemma)h(17.)20 b Fo(s)p Fq(!)644 1641 y Fk(R)675 1635 y Fo(t)12 b Fq(\))f Fo(r)q(ank)q Fx(\()p Fo(s)p Fx(\))h Fq(\025)f Fo(r)q(ank)q Fx(\()p Fo(t)p Fx(\))p Fo(:)340 1725 y Fg(Lemma)d(18.)20 b Fm(L)n(et)14 b Fo(r)q(oot)p Fx(\()p Fo(s)p Fx(\))f Fq(2)e(D)864 1731 y Fp(1)897 1725 y Fm(and)k(let)e Fo(s)f Fq(\021)g Fo(s)1128 1731 y Fp(1)1147 1725 y Fq(!)1189 1731 y Fk(R)1219 1725 y Fo(s)1238 1731 y Fp(2)1257 1725 y Fq(!)1299 1731 y Fk(R)1336 1725 y Fo(:)7 b(:)g(:)f Fq(!)1433 1731 y Fk(R)1464 1725 y Fo(s)1483 1731 y Fn(n)1518 1725 y Fq(\021)11 b Fo(s)k Fm(b)n(e)f(a)h(cyclic)340 1774 y(r)n(e)n(duction)f(se)n(quenc)n(e.)f(Then)h Fo(r)q(ank)q Fx(\()p Fo(s)931 1780 y Fn(j)949 1774 y Fx(\))d(=)h Fo(r)q(ank)q Fx(\()p Fo(s)p Fx(\))i Fm(and)g Fo(r)q(oot)p Fx(\()p Fo(s)1364 1780 y Fn(j)1382 1774 y Fx(\))e Fq(2)f(D)1481 1780 y Fp(1)1513 1774 y Fm(for)i(every)g(index)340 1824 y Fo(j)h Fq(2)e(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(n)p Fq(g)p Fm(.)340 1914 y Fg(Example)h(19.)20 b Fx(Consider)f(the)h(CTRSs)e Fq(R)1049 1920 y Fp(1)1087 1914 y Fx(=)i Fq(f)p Fo(F)6 b Fx(\()p Fn(C)q Fx(\()p Fo(x;)h(y)q Fx(\)\))20 b Fq(!)f Fn(C)r Fx(\()p Fo(F)6 b Fx(\()p Fo(x)p Fx(\))p Fo(;)h(y)q Fx(\))19 b Fq(\()g Fo(x)p Fq(#)p Fo(y)q Fq(g)340 1964 y Fx(and)14 b Fq(R)456 1970 y Fp(2)486 1964 y Fx(=)e Fq(f)p Fo(g)q Fx(\()p Fo(x)p Fx(\))g Fq(!)f Fn(C)r Fx(\()p Fo(x;)c(x)p Fx(\))p Fq(g)13 b Fx(as)g(w)o(ell)h(as)g(the)g(reduction) h(sequence)340 2048 y Fo(s)359 2054 y Fp(1)390 2048 y Fq(\021)d Fo(F)6 b Fx(\()p Fn(C)q Fx(\()p Fo(g)q Fx(\()p Fo(A)p Fx(\))p Fo(;)h(g)q Fx(\()p Fo(A)p Fx(\)\)\))13 b Fq(!)798 2030 y Fn(o)798 2058 y Fk(R)840 2048 y Fo(s)859 2054 y Fp(2)889 2048 y Fq(\021)f Fn(C)r Fx(\()p Fo(F)6 b Fx(\()p Fo(g)q Fx(\()p Fo(A)p Fx(\)\))p Fo(;)h(g)q Fx(\()p Fo(A)p Fx(\)\))12 b Fq(!)1297 2030 y Fn(i)1297 2058 y Fk(R)1339 2048 y Fo(s)1358 2054 y Fp(3)1389 2048 y Fq(\021)f Fn(C)r Fx(\()p Fo(F)6 b Fx(\()p Fn(C)r Fx(\()p Fo(A;)h(A)p Fx(\)\))p Fo(;)g(g)q Fx(\()p Fo(A)p Fx(\)\))340 2131 y(W)m(e)14 b(ha)o(v)o(e)g Fo(r)q(ank)q Fx(\()p Fo(s)633 2137 y Fn(i)647 2131 y Fx(\))e(=)h(3)g(for)h Fo(i)e Fq(2)g(f)p Fx(1)p Fo(;)7 b Fx(2)p Fq(g)13 b Fx(and)h Fo(r)q(ank)q Fx(\()p Fo(s)1206 2137 y Fp(3)1225 2131 y Fx(\))e(=)g(2.)h Fo(S)r Fx(\()p Fo(s)1405 2137 y Fp(1)1425 2131 y Fx(\))f(=)h Fq(f)p Fo(g)q Fx(\()p Fo(A)p Fx(\))p Fq(g)h Fx(whereas)340 2181 y Fo(S)r Fx(\()p Fo(s)402 2187 y Fn(i)418 2181 y Fx(\))e(is)h(not)f(de\014ned)i(for)e Fo(i)g Fq(2)f(f)p Fx(2)p Fo(;)c Fx(3)p Fq(g)p Fx(.)j(F)m(urthermore,)i(e.g.)g Fo(O)q Fx(\()p Fo(s)1347 2187 y Fp(1)1366 2181 y Fx(\))f(=)h Fq(f)p Fo(s)1477 2187 y Fp(1)1496 2181 y Fo(;)7 b Fn(C)q Fx(\()p Fo(g)q Fx(\()p Fo(A)p Fx(\))p Fo(;)g(g)q Fx(\()p Fo(A)p Fx(\)\))p Fq(g)340 2231 y Fx(and)14 b Fo(O)q Fx(\()p Fo(s)489 2237 y Fp(3)508 2231 y Fx(\))e(=)g Fq(f)p Fo(s)620 2237 y Fp(3)638 2231 y Fo(;)7 b(F)f Fx(\()p Fn(C)q Fx(\()p Fo(A;)h(A)p Fx(\)\))p Fo(;)g Fn(C)r Fx(\()p Fo(A;)g(A)p Fx(\))p Fo(;)g(A;)g(g)q Fx(\()p Fo(A)p Fx(\))p Fq(g)p Fx(.)340 2321 y Fg(De\014nition)t(20.)21 b Fx(Let)12 b Fo(\033)h Fx(and)e Fo(\034)16 b Fx(b)q(e)c(substitutions.)g(W)m (e)f(write)g Fo(\033)i Fq(/)f Fo(\034)k Fx(if)11 b Fo(x\033)h Fq(\021)g Fo(y)q(\033)h Fx(implies)340 2370 y Fo(x\034)j Fq(\021)c Fo(y)q(\034)17 b Fx(for)11 b(all)f Fo(x;)d(y)13 b Fq(2)e(V)s Fx(.)h(The)g(notation)e Fo(\033)j Fq(!)1106 2355 y Fk(\003)1106 2382 y(R)1147 2370 y Fo(\034)k Fx(is)11 b(used)h(if)e Fo(x\033)j Fq(!)1451 2355 y Fk(\003)1451 2382 y(R)1493 2370 y Fo(x\034)i Fx(for)c(all)f Fo(x)i Fq(2)f(V)s Fx(.)340 2420 y(Ob)o(viously)m(,)i Fo(\033)f Fq(!)622 2405 y Fk(\003)622 2432 y(R)664 2420 y Fo(\034)19 b Fx(implies)12 b Fo(t\033)g Fq(!)935 2405 y Fk(\003)935 2432 y(R)977 2420 y Fo(t\034)19 b Fx(for)13 b(all)g Fo(t)e Fq(2)g(T)g Fx(\()p Fq(F)t Fo(;)c Fq(V)s Fx(\).)p eop %%Page: 9 9 bop 183 194 a Fg(De\014niti)o(on)5 b(21.)20 b Fx(A)h(substitution)f Fo(\033)h Fx(is)f(said)g(to)g(b)q(e)h Fm(black)f Fx(if)f Fo(x\033)i Fx(is)f(blac)o(k)g(for)g(ev)o(ery)183 244 y Fo(x)11 b Fq(2)g(D)q Fo(om)p Fx(\()p Fo(\033)q Fx(\))k(and)e(it)h(is)f(said)g(to)h(b)q(e)g Fm(top)h(black)e Fx(if)g Fo(x\033)i Fx(is)e(top)h(blac)o(k)f(for)g(all)f Fo(x)g Fq(2)f(D)q Fo(om)p Fx(\()p Fo(\033)q Fx(\))q(.)183 330 y Fg(Prop)q(ositi)o(on)5 b(22.)21 b Fm(Every)14 b(substitution)g Fo(\033)h Fm(c)n(an)f(b)n(e)h(de)n(c)n(omp)n(ose)n(d)f(into)h Fo(\033)1350 336 y Fp(2)1375 330 y Fq(\016)8 b Fo(\033)1428 336 y Fp(1)1460 330 y Fm(such)15 b(that)183 379 y Fo(\033)207 385 y Fp(1)240 379 y Fm(is)f(black)h(and)h Fo(\033)493 385 y Fp(2)526 379 y Fm(is)f(top)g(white)f(and)i Fo(\033)856 385 y Fp(2)886 379 y Fq(/)11 b Fo(\017)k Fm(\(r)n(e)n(c)n(al)r(l)f(that)h Fo(\017)f Fm(has)i(empty)f(domain\).)183 458 y(Pr)n(o)n(of.)20 b Fx(The)14 b(pro)q(of)g(can)g(b)q(e)g(found)g(in)f([Mid90a)n(].)622 b Fq(u)-28 b(t)183 579 y Fg(4.2)47 b(T)l(o)16 b(b)q(e)g(Simplif)o(yin)o(g)d (is)i(a)h(Mo)q(dular)f(Prop)q(ert)o(y)183 658 y(Theorem)6 b(23.)21 b Fm(L)n(et)15 b Fq(R)566 664 y Fp(1)599 658 y Fm(and)h Fq(R)715 664 y Fp(2)749 658 y Fm(b)n(e)e(two)h(CTRSs)g(and)g(let)g Fq(R)f Fm(b)n(e)h(the)g(c)n(ombine)n(d)h(CTRS)183 708 y(of)c Fq(R)264 714 y Fp(1)296 708 y Fm(and)i Fq(R)410 714 y Fp(2)441 708 y Fm(with)e(shar)n(e)n(d)h(c)n(onstructors.)f(Then)i Fq(R)f Fm(is)f (simplifying)g(if)g(and)i(only)f(if)f(b)n(oth)183 757 y Fq(R)218 763 y Fp(1)251 757 y Fm(and)k Fq(R)367 763 y Fp(2)401 757 y Fm(ar)n(e)e(simplifying.)183 836 y(Pr)n(o)n(of.)20 b Fx(Let)13 b Fq(S)s Fx(,)p Fq(S)451 842 y Fp(1)470 836 y Fx(,)f(and)h Fq(S)599 842 y Fp(2)630 836 y Fx(b)q(e)h(the)f(unconditional)f(TRSs)h(asso)q (ciated)g(with)g Fq(R)p Fx(,)g Fq(R)1514 842 y Fp(1)1532 836 y Fx(,)g(and)183 886 y Fq(R)218 892 y Fp(2)236 886 y Fx(,)h(resp)q(ectiv)o 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b(12\))j(w)o(e)g(ma)o(y)f(assume)g(that)h(the)h(set)g Fq(C)i Fx(of)c(constructors)j(con)o(tains)e(t)o(w)o(o)g(sym)o(b)q(ols)183 1198 y(not)17 b(used)i(previously)m(,)d(namely)g(a)h(constructor)i Fn(C)r(ons)e Fx(of)g(arit)o(y)g(2)g(and)h(a)f(constan)o(t)h Fn(N)s(il)q Fx(.)183 1248 y(Assuming)11 b(that)i Fq(!)503 1230 y Fp(+)503 1260 y Fk(S)r Fn(e)555 1248 y Fx(is)g(not)f(irre\015exiv)o(e,)h(w) o(e)g(will)e(deriv)o(e)i(a)g(con)o(tradiction.)f(W)m(e)g(assume)183 1297 y(that)i(there)h(is)f(a)f(cyclic)h(reduction)h(sequence)653 1378 y Fo(s)d Fq(\021)g Fo(s)747 1384 y Fp(1)778 1378 y Fq(!)820 1384 y Fk(S)r Fn(e)871 1378 y Fo(:)7 b(:)g(:)j Fq(!)973 1384 y Fk(S)r Fn(e)1024 1378 y Fo(s)1043 1384 y Fn(n)1078 1378 y Fq(\021)i Fo(s;)183 1460 y(n)j(>)i Fx(1,)e(with)i Fo(s)437 1466 y Fn(j)470 1460 y Fq(2)f(T)10 b Fx(\()p Fq(D)595 1466 y Fp(1)623 1460 y Fq([)f(D)692 1466 y Fp(2)711 1460 y Fo(;)e Fq(C)r Fo(;)g Fq(V)s Fx(\))16 b(for)h(all)e Fo(j)j Fq(2)e(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(n)p Fq(g)p Fx(.)15 b(Without)h(loss)h(of)e (gen-)183 1509 y(eralit)o(y)j(w)o(e)h(ma)o(y)e(further)j(assume)f(that)g Fo(s)g Fx(is)g(top)g(blac)o(k,)f(and)g(that)h(rank\()p Fo(s)p Fx(\))i(=:)f Fo(k)g Fx(is)183 1559 y(minim)o(al)o(,)8 b(i.e.,)i(there)j(is)f (no)f(cyclic)h(sequence)i Fo(t)d Fq(\021)h Fo(t)995 1565 y Fp(1)1025 1559 y Fq(!)1067 1565 y Fk(S)r Fn(e)1119 1559 y Fo(:)7 b(:)g(:)j Fq(!)1221 1565 y Fk(S)r Fn(e)1272 1559 y Fo(t)1287 1565 y Fn(m)1330 1559 y Fq(\021)i Fo(t;)7 b(m)k(>)h Fx(1,)f(with)183 1609 y(rank\()p Fo(t)p Fx(\))k Fo(<)g(k)q Fx(.)g(Consequen)o(tly)m(,)g Fq(!)733 1591 y Fp(+)733 1621 y Fk(S)r Fn(e)803 1609 y Fx(is)h(irre\015exiv)o (e)g(on)g Fo(T)k Fx(=)15 b Fq(f)p Fo(t)e Fq(j)f Fo(r)q(ank)q Fx(\()p Fo(t)p Fx(\))i Fo(<)h(r)q(ank)q Fx(\()p Fo(s)p Fx(\))p Fq(g)183 1659 y Fx(\(or)21 b(equiv)n(alen)o(tly)e Fq(!)536 1641 y Fp(+)536 1671 y Fk(S)r Fn(e)611 1659 y Fx(is)i(terminating)e(on)i Fo(T)6 b Fx(\).)20 b(According)i(to)f(Lemma)d(18,)i(ev)o(ery)183 1709 y Fo(s)202 1715 y Fn(j)220 1709 y Fo(;)7 b Fx(0)17 b Fq(\024)i Fo(j)i Fq(\024)d Fo(n)p Fx(,)g(is)f(top)h(blac)o(k)g(and)g(has)g(rank)g Fo(k)q Fx(.)f(The)h(pro)q(of)g(idea)g(is)f(to)h(construct)i(a)183 1758 y(cyclic)14 b(reduction)g(sequence)i(of)e(blac)o(k)f(terms)521 1840 y Fo(\032)p Fx(\()p Fo(s)p Fx(\))g Fq(\021)f Fo(\032)p Fx(\()p Fo(s)706 1846 y Fp(1)726 1840 y Fx(\))f Fq(!)795 1822 y Fp(+)795 1852 y Fk(S)815 1856 y Fc(1)831 1852 y Fn(e)847 1856 y Fc(1)877 1840 y Fo(:)c(:)g(:)j Fq(!)979 1822 y Fp(+)979 1852 y Fk(S)999 1856 y Fc(1)1015 1852 y Fn(e)1031 1856 y Fc(1)1061 1840 y Fo(\032)p Fx(\()p Fo(s)1117 1846 y Fn(n)1140 1840 y Fx(\))i Fq(\021)g Fo(\032)p Fx(\()p Fo(s)p Fx(\))183 1927 y(yielding)k(a)h (con)o(tradiction)g(to)g(the)g(irre\015exivit)o(y)g(of)g Fq(!)1082 1909 y Fp(+)1082 1939 y Fk(S)1102 1943 y Fc(1)1118 1939 y Fn(e)1134 1943 y Fc(1)1152 1927 y Fx(.)g(First)h(w)o(e)f(transform)f(eac)o(h)183 1977 y(term)21 b Fo(s)309 1983 y Fn(j)350 1977 y Fx(in)o(to)g(a)h(blac)o(k)g (term)g Fo(\032)p Fx(\()p Fo(s)766 1983 y Fn(j)785 1977 y Fx(\))g(and)g(then) h(sho)o(w)g(that)f Fo(s)1244 1983 y Fn(j)1288 1977 y Fq(!)1330 1983 y Fk(S)r Fn(e)1395 1977 y Fo(s)1414 1983 y Fn(j)r Fp(+1)1497 1977 y Fx(implies)183 2027 y Fo(\032)p Fx(\()p Fo(s)239 2033 y Fn(j)257 2027 y Fx(\))12 b Fq(!)327 2009 y Fp(+)327 2039 y Fk(S)347 2043 y Fc(1)363 2039 y Fn(e)379 2043 y Fc(1)409 2027 y Fo(\032)p Fx(\()p Fo(s)465 2033 y Fn(j)r Fp(+1)525 2027 y Fx(\).)h(In)h(order)h(to)e(de\014ne)i(the)f(transformation,)d(w)o(e)j (consider)h(the)f(sets)183 2077 y Fo(S)g Fx(=)266 2045 y Fl(S)301 2056 y Fn(n)301 2089 y(j)r Fp(=1)367 2077 y Fo(S)r Fx(\()p Fo(s)429 2083 y Fn(j)448 2077 y Fx(\))h(and)e Fo(O)g Fx(=)648 2045 y Fl(S)683 2056 y Fn(n)683 2089 y(j)r Fp(=1)749 2077 y Fo(O)q Fx(\()p Fo(s)817 2083 y Fn(j)835 2077 y Fx(\))h(of)g(all)f(principal)g (and)h(outer)g(subterms)h(o)q(ccur-)183 2126 y(ring)9 b(in)g(the)h(sequence.) h(Notice)f(that)f Fo(r)q(oot)p Fx(\()p Fo(t)p Fx(\))j Fq(2)f(D)970 2132 y Fp(2)998 2126 y Fx(for)e(all)g Fo(t)i Fq(2)g Fo(S)i Fx(and)c Fo(r)q(oot)p Fx(\()p Fo(t)p Fx(\))j Fq(2)f(D)1495 2132 y Fp(1)1514 2126 y Fq([C)s([V)183 2176 y Fx(for)j(all)f Fo(t)g Fq(2)g Fo(O)i Fx(as)g(w)o(ell)f(as)g Fo(r)q(ank)q Fx(\()p Fo(t)p Fx(\))f Fq(\024)g Fo(r)q(ank)q Fx(\()p Fo(s)p Fx(\))i(for)g(eac)o(h)g Fo(t)d Fq(2)h Fo(S)f Fq([)e Fo(O)q Fx(.)k(F)m(urthermore,)g(for)183 2226 y(ev)o(ery)g Fo(t)e Fq(2)f Fo(S)17 b Fx(let)183 2307 y Fo(\001)p Fx(\()p Fo(t)p Fx(\))11 b(=)h Fq(f)p Fo(t)356 2292 y Fk(0)380 2307 y Fq(j)g(9)p Fo(j)i Fq(2)d(f)p Fx(1)p Fo(;)c(:)g(:)g(:)t(;)g (n)i Fq(\000)g Fx(1)p Fq(g)14 b Fx(with)f Fo(s)875 2313 y Fn(j)905 2307 y Fq(\021)f Fo(C)s Fx([)-7 b([)p Fo(:)7 b(:)g(:)t(;)g(t;)g(:)g(:)g(:)m Fx(])-7 b(])11 b Fq(!)1223 2292 y Fn(i)1223 2319 y Fk(S)r Fn(e)1274 2307 y Fo(C)s Fx([)p Fo(:)c(:)g(:)e(;)i(t)1408 2292 y Fk(0)1419 2307 y Fo(;)g(:)g(:)g(:)n Fx(])k Fq(\021)h Fo(s)1572 2313 y Fn(j)r Fp(+1)859 2369 y Fx(and)h Fo(t)f Fq(!)1008 2375 y Fk(S)r Fn(e)1059 2369 y Fo(t)1074 2354 y Fk(0)1086 2369 y Fq(g)p eop %%Page: 10 10 bop 340 194 a Fx(b)q(e)18 b(the)g(set)h(of)e(successors)j(of)d Fo(t)p Fx(.)g(Note)g(that)h Fo(\001)p Fx(\()p Fo(t)p Fx(\))f(is)g(\014nite)h (for)f(an)o(y)g Fo(t)g Fq(2)g Fo(S)r Fx(.)h(Then)g(w)o(e)340 244 y(de\014ne)d Fo(\032)d Fx(:)f Fo(S)h Fq([)d Fo(O)q Fq(!T)h Fx(\()p Fq(D)746 250 y Fp(1)765 244 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\))14 b(recursiv)o(ely)g(b)o(y:)358 317 y(1.)20 b Fo(\032)p Fx(\()p Fo(t)p Fx(\))12 b(=)g([)p Fo(\032)p Fx(\()p Fo(t)599 302 y Fk(0)599 327 y Fp(1)618 317 y Fx(\))p Fo(;)7 b(:)g(:)g(:)e(;)i(\032)p Fx(\()p Fo(t)779 302 y Fk(0)779 327 y Fn(m)810 317 y Fx(\)])852 302 y Fp(4)884 317 y Fx(if)13 b Fo(t)f Fq(2)f Fo(S)17 b Fx(and)c Fo(\001)p Fx(\()p Fo(t)p Fx(\))f(=)g Fq(f)p Fo(t)1284 302 y Fk(0)1284 327 y Fp(1)1302 317 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)1410 302 y Fk(0)1410 327 y Fn(m)1441 317 y Fq(g)p Fx(.)411 367 y(Note)14 b(that)g Fo(t)616 352 y Fk(0)616 377 y Fp(1)635 367 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)743 352 y Fk(0)743 377 y 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y Fd(l)821 1639 y Fx(])17 b(for)h(some)e(blac)o(k)i(con)o(text)g Fo(C)1322 1624 y Fk(0)1333 1639 y Fx([)p Fo(;)7 b(:)g(:)g(:)e(;)i Fx(])16 b(and)i(some)f(indices)340 1689 y Fo(i)354 1695 y Fp(1)373 1689 y Fo(;)7 b(:)g(:)g(:)e(;)i(i)480 1695 y Fn(l)508 1689 y Fq(2)16 b(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(m)p Fq(g)p Fx(.)15 b(Clearly)m(,)g(it)h(follo)o(ws)f(from)f Fo(\032)p Fx(\()p Fo(s)1264 1695 y Fn(j)1283 1689 y Fx(\))i Fq(\021)g Fo(C)s Fx([)p Fo(\032)p Fx(\()p Fo(t)1460 1695 y Fp(1)1478 1689 y Fx(\))p Fo(;)7 b(:)g(:)g(:)e(;)i(\032)p Fx(\()p Fo(t)1639 1695 y Fn(m)1670 1689 y Fx(\)])16 b(and)340 1739 y Fo(\032)p Fx(\()p Fo(s)396 1745 y Fn(j)r Fp(+1)457 1739 y Fx(\))c Fq(\021)f Fo(C)561 1723 y Fk(0)573 1739 y Fx([)p Fo(\032)p Fx(\()p Fo(t)637 1745 y Fn(i)649 1749 y Fc(1)667 1739 y Fx(\))p Fo(;)c(:)g(:)g(:)e(;)i(\032)p Fx(\()p Fo(t)828 1745 y Fn(i)840 1749 y Fd(l)854 1739 y Fx(\)])12 b(that)g(the)h(rule)g(that)f(reduced)j Fo(s)1395 1745 y Fn(j)1425 1739 y Fx(to)d Fo(s)1493 1745 y Fn(j)r Fp(+1)1565 1739 y Fx(also)g(reduces) 340 1788 y Fo(\032)p Fx(\()p Fo(s)396 1794 y Fn(j)415 1788 y Fx(\))17 b(to)g Fo(\032)p Fx(\()p Fo(s)558 1794 y Fn(j)r Fp(+1)619 1788 y Fx(\).)g(If)g(otherwise)h Fo(s)916 1794 y Fn(j)951 1788 y Fq(!)993 1773 y Fn(o)993 1800 y Fk(F)1020 1791 y Fd(e)1055 1788 y Fo(s)1074 1794 y Fn(j)r Fp(+1)1134 1788 y Fx(,)e(then)i Fo(\032)p Fx(\()p Fo(s)1316 1794 y Fn(j)1335 1788 y Fx(\))p Fq(!)1393 1794 y Fk(F)1420 1785 y Fd(e)1417 1805 y Fc(1)1438 1788 y Fo(\032)p Fx(\()p Fo(s)1494 1794 y Fn(j)r Fp(+1)1554 1788 y Fx(\),)f(since)h(only)340 1838 y(sym)o(b)q(ols)13 b(from)f Fq(F)627 1844 y Fp(1)659 1838 y Fx(are)j(in)o(v)o(olv)o(ed)d(in)i (the)g(outer)h(reduction)f(step.)340 1888 y(\(ii\))g Fo(s)429 1894 y Fn(j)458 1888 y Fq(!)500 1873 y Fn(i)500 1899 y Fk(S)r Fn(e)551 1888 y Fo(s)570 1894 y Fn(j)r Fp(+1)340 1938 y Fx(Then)i Fo(s)469 1944 y Fn(j)501 1938 y Fq(\021)f Fo(C)s Fx([)-7 b([)p Fo(t)613 1944 y Fp(1)630 1938 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)738 1944 y Fn(m)769 1938 y Fx(])-7 b(])14 b(and)i Fo(s)902 1944 y Fn(j)r Fp(+1)976 1938 y Fq(\021)e Fo(C)s Fx([)p Fo(t)1082 1944 y Fp(1)1100 1938 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)1208 1944 y Fn(l)p Fk(\000)p Fp(1)1262 1938 y Fo(;)g(t)1296 1923 y Fk(0)1296 1950 y Fn(l)1308 1938 y Fo(;)g(t)1342 1944 y Fn(l)p Fp(+1)1397 1938 y Fo(;)g(:)g(:)g(:)t(;)g(t)1504 1944 y Fn(m)1535 1938 y Fx(])15 b(for)g(some)g(in-)340 1988 y(dex)j Fo(l)h Fq(2)f(f)p Fx(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(m)p Fq(g)17 b Fx(and)h(some)f(term)g Fo(t)1017 1973 y Fk(0)1017 1999 y Fn(l)1030 1988 y Fx(,)g(where)h Fo(t)1197 1994 y Fn(l)1228 1988 y Fq(!)1270 1994 y Fk(S)r Fn(e)1328 1988 y Fo(t)1343 1973 y Fk(0)1343 1999 y Fn(l)1355 1988 y Fx(.)f(F)m (urthermore,)h Fo(\032)p Fx(\()p Fo(s)1697 1994 y Fn(j)1715 1988 y Fx(\))g Fq(\021)340 2037 y Fo(C)s Fx([)p Fo(\032)p Fx(\()p Fo(t)437 2043 y Fp(1)456 2037 y Fx(\))p Fo(;)7 b(:)g(:)g(:)e(;)i(\032)p Fx(\()p Fo(t)617 2043 y Fn(m)648 2037 y Fx(\)])17 b(and)g Fo(\032)p Fx(\()p Fo(s)833 2043 y Fn(j)r Fp(+1)894 2037 y Fx(\))g Fq(\021)g Fo(C)s Fx([)p Fo(\032)p Fx(\()p Fo(t)1073 2043 y Fp(1)1092 2037 y Fx(\))p Fo(;)7 b(:)g(:)g(:)t(;)g(\032)p Fx(\()p Fo(t)1252 2043 y Fn(l)p Fk(\000)p Fp(1)1308 2037 y Fx(\))p Fo(;)g(\032)p Fx(\()p Fo(t)1395 2022 y Fk(0)1395 2049 y Fn(l)1408 2037 y Fx(\))p Fo(;)g(\032)p Fx(\()p Fo(t)1495 2043 y Fn(l)p Fp(+1)1550 2037 y Fx(\))p Fo(;)g(:)g(:)g(:)e(;)i(\032)p Fx(\()p Fo(t)1711 2043 y Fn(m)1742 2037 y Fx(\)])p Fo(:)340 2087 y Fx(Since)14 b Fo(t)463 2072 y Fk(0)463 2099 y Fn(l)488 2087 y Fq(2)d Fo(\001)p Fx(\()p Fo(t)593 2093 y Fn(l)605 2087 y Fx(\),)i(w)o(e)h(ha)o(v)o(e)f(b)o(y)h (de\014nition)f(of)f Fo(\032)i Fx(that)g Fo(\032)p Fx(\()p Fo(t)1267 2093 y Fn(l)1280 2087 y Fx(\))e(=)g([)p Fo(:)7 b(:)g(:)e(;)i(\032)p Fx(\()p Fo(t)1490 2072 y Fk(0)1490 2099 y Fn(l)1502 2087 y Fx(\))p Fo(;)g(:)g(:)g(:)n Fx(],)13 b(i.e.,)f Fo(\032)p Fx(\()p Fo(t)1752 2072 y Fk(0)1752 2099 y Fn(l)1765 2087 y Fx(\))340 2137 y(is)g(a)g(subterm)g(of)f Fo(\032)p Fx(\()p Fo(t)670 2143 y Fn(l)684 2137 y Fx(\).)g(Hence)j Fo(\032)p Fx(\()p Fo(t)897 2143 y Fn(l)910 2137 y Fx(\))e(=)g([)p Fo(:)7 b(:)g(:)e(;)i(\032)p Fx(\()p Fo(t)1120 2122 y Fk(0)1120 2149 y Fn(l)1133 2137 y Fx(\))p Fo(;)g(:)g(:)g(:)n Fx(])p Fq(!)1270 2119 y Fp(+)1270 2149 y Fk(F)1297 2139 y Fd(e)1294 2159 y Fc(1)1314 2137 y Fo(\032)p Fx(\()p Fo(t)1366 2122 y Fk(0)1366 2149 y Fn(l)1379 2137 y Fx(\),)12 b(where)h(only)e(the)i(rules)340 2199 y Fn(C)r(ons)p Fx(\()p Fo(x;)7 b(y)q Fx(\))12 b Fq(!)f Fo(x)i Fx(and)h Fn(C)r(ons)o Fx(\()p Fo(x;)7 b(y)q Fx(\))12 b Fq(!)f Fo(y)16 b Fx(are)e(used.)h (Therefore,)f Fo(\032)p Fx(\()p Fo(s)1404 2205 y Fn(j)1423 2199 y Fx(\))p Fq(!)1481 2181 y Fp(+)1481 2211 y Fk(F)1508 2201 y Fd(e)1505 2221 y Fc(1)1525 2199 y Fo(\032)p Fx(\()p Fo(s)1581 2205 y Fn(j)r Fp(+1)1642 2199 y Fx(\).)340 2261 y(All)g(in)g(all,)f (w)o(e)h(ha)o(v)o(e)g(sho)o(wn)h(that)f Fo(\032)p Fx(\()p Fo(s)p Fx(\))g Fq(!)1030 2243 y Fp(+)1030 2273 y Fk(S)1050 2277 y Fc(1)1066 2273 y Fn(e)1082 2277 y Fc(1)1112 2261 y Fo(\032)p Fx(\()p Fo(s)p Fx(\),)h(a)f(con)o(tradiction)h(to)f(the)h(fact)f(that)340 2311 y Fq(R)375 2317 y Fp(1)408 2311 y Fx(is)g(simplifying)c(\(Prop)q (osition)k(10\).)782 b Fq(u)-28 b(t)p 340 2343 237 2 v 349 2370 a Fs(4)379 2386 y Fw(The)13 b(order)g(is)h(arbitrary)g(but)g(\014xed.) 349 2416 y Fs(5)379 2432 y Fu(F)h Ft(2)c(D)486 2436 y Fs(1)511 2432 y Ft([)d(C)j([)d(V)s Fw(.)p eop %%Page: 11 11 bop 183 194 a Fg(Example)7 b(24.)21 b Fx(Consider)16 b Fq(S)654 200 y Fp(1)688 194 y Fx(=)f Fq(f)p Fo(F)6 b Fx(\()p Fn(C)r Fx(\()p Fo(x)p Fx(\))p Fo(;)h(F)f Fx(\()p Fo(y)q(;)h Fn(D)p Fx(\()p Fo(z)r Fx(\)\)\))16 b Fq(!)f Fo(F)6 b Fx(\()p Fo(y)q(;)h(x)p Fx(\))p Fo(;)g(A)14 b Fq(!)h Fo(B)r Fq(g)h Fx(as)g(w)o(ell)183 244 y(as)e Fq(S)259 250 y Fp(2)289 244 y Fx(=)e Fq(f)p Fo(g)q Fx(\()p Fo(x)p Fx(\))f Fq(!)g Fn(C)r Fx(\()p Fo(x)p Fx(\))p Fo(;)j(g)q Fx(\()p Fo(x)p Fx(\))d Fq(!)g Fn(D)q Fx(\()p Fo(x)p Fx(\))p Fq(g)p Fx(.)j(In)g(the)g(reduction)h(sequence)405 326 y Fo(s)424 332 y Fp(1)454 326 y Fq(\021)d Fo(F)6 b Fx(\()p Fo(g)q Fx(\()p Fo(A)p Fx(\))p Fo(;)h(F)f Fx(\()p Fo(b;)h(g)q Fx(\()p Fo(A)p Fx(\)\)\))k Fq(!)905 311 y Fn(i)905 338 y Fk(S)941 326 y Fo(s)960 332 y Fp(2)991 326 y Fq(\021)g Fo(F)6 b Fx(\()p Fo(g)q Fx(\()p Fo(B)r Fx(\))p Fo(;)h(F)f Fx(\()p Fo(b;)h(g)q Fx(\()p Fo(A)p Fx(\)\)\))863 389 y Fq(!)905 374 y Fn(i)905 400 y Fk(S)941 389 y Fo(s)960 395 y Fp(3)991 389 y Fq(\021)k Fo(F)6 b Fx(\()p Fn(C)r Fx(\()p Fo(B)r Fx(\))p Fo(;)h(F)f Fx(\()p Fo(b;)h(g)q Fx(\()p Fo(A)p Fx(\)\)\))863 451 y Fq(!)905 436 y Fn(i)905 462 y Fk(S)941 451 y Fo(s)960 457 y Fp(4)991 451 y Fq(\021)k Fo(F)6 b Fx(\()p Fn(C)r Fx(\()p Fo(B)r Fx(\))p Fo(;)h(F)f Fx(\()p Fo(b;)h Fn(D)p Fx(\()p Fo(A)p Fx(\)\)\))863 513 y Fq(!)905 498 y Fn(o)905 525 y Fk(S)941 513 y Fo(s)960 519 y Fp(5)991 513 y Fq(\021)k Fo(F)6 b Fx(\()p Fo(b;)h(B)r Fx(\))p Fo(;)183 596 y Fx(the)14 b(sets)h(of)e(all)f(outer)i(and)g(principal) f(subterms)g(o)q(ccurring)i(in)e(the)h(reduction)g(sequence)183 646 y(are)21 b(giv)o(en)f(b)o(y)g Fo(O)k Fx(=)f Fq(f)p Fo(s)589 652 y Fp(1)608 646 y Fo(;)7 b(s)646 652 y Fp(2)664 646 y Fo(;)g(s)702 652 y Fp(3)721 646 y Fo(;)g(s)759 652 y Fp(4)777 646 y Fo(;)g(s)815 652 y Fp(5)834 646 y Fo(;)g(F)f Fx(\()p Fo(b;)h(g)q Fx(\()p Fo(A)p Fx(\)\))p Fo(;)g Fn(C)p Fx(\()p Fo(B)r Fx(\))p Fo(;)g(B)r(;)g(F)f Fx(\()p Fo(b;)h Fx(\()p Fn(D)q Fx(\()p Fo(A)p Fx(\)\)\))p Fo(;)g Fn(D)q Fx(\()p Fo(A)p Fx(\))p Fo(;)g(A)p Fq(g)183 696 y Fx(and)15 b Fo(S)j Fx(=)d Fq(f)p Fo(g)q Fx(\()p Fo(A)p Fx(\))p Fo(;)7 b(b;)g(g)q Fx(\()p Fo(B)r Fx(\))p Fq(g)p Fx(,)15 b(resp)q(ectiv)o(ely)m(.)i (Moreo)o(v)o(er,)f(w)o(e)g(ha)o(v)o(e)g Fo(\001)p Fx(\()p Fo(b)p Fx(\))e(=)h Fq(fg)g Fx(as)h(w)o(ell)f(as)183 745 y Fo(\001)p Fx(\()p Fo(g)q Fx(\()p Fo(A)p Fx(\)\))d(=)g Fq(f)p Fo(g)q Fx(\()p Fo(B)r Fx(\))p Fo(;)7 b Fn(D)q Fx(\()p Fo(A)p Fx(\))p Fq(g)p Fx(,)14 b(and)f Fo(\001)p Fx(\()p Fo(g)q Fx(\()p Fo(B)r Fx(\)\))g(=)f Fq(f)p Fn(C)q Fx(\()p Fo(B)r Fx(\))p Fq(g)p Fx(.)183 795 y(Therefore,)k Fo(\032)f Fx(is)g(completely)f(de\014ned)i(b)o(y)f Fo(\032)p Fx(\()p Fo(b)p Fx(\))f(=)f([)g(],)h Fo(\032)p Fx(\()p Fo(g)q Fx(\()p Fo(B)r Fx(\)\))i(=)e([)p Fo(\032)p Fx(\()p Fn(C)r Fx(\()p Fo(B)r Fx(\)\)])g(=)f([)p Fn(C)r Fx(\()p Fo(B)r Fx(\)],)183 845 y(and)g Fo(\032)p Fx(\()p Fo(g)q Fx(\()p Fo(A)p Fx(\)\))g(=)f([)p Fo(\032)p Fx(\()p Fo(g)q Fx(\()p Fo(B)r Fx(\)\))p Fo(;)7 b(\032)p Fx(\()p Fn(D)s Fx(\()p Fo(A)p Fx(\)\)])k(=)h([[)p Fn(C)q Fx(\()p Fo(B)r Fx(\)])p Fo(;)7 b Fn(D)q Fx(\()p Fo(A)p Fx(\)].)183 895 y(As)14 b(in)f(the)i(ab)q(o)o(v)o(e)f(pro)q(of,)f(w)o(e)h(get)g(a)f (reduction)i(sequence)h(of)d(blac)o(k)h(terms)536 978 y Fo(\032)p Fx(\()p Fo(s)592 984 y Fp(1)612 978 y Fx(\))e Fq(\021)f Fo(F)6 b Fx(\([[)p Fn(C)q Fx(\()p Fo(B)r Fx(\)])p Fo(;)h Fn(D)q Fx(\()p Fo(A)p Fx(\)])p Fo(;)g(F)f Fx(\([)k(])p Fo(;)d Fx([[)p Fn(C)p Fx(\()p Fo(B)r Fx(\)])p Fo(;)g Fn(D)q Fx(\()p Fo(A)p Fx(\)]\)\))438 1040 y Fq(!)480 1022 y Fp(+)480 1052 y Fk(F)507 1042 y Fd(e)504 1062 y Fc(1)536 1040 y Fo(\032)p Fx(\()p Fo(s)592 1046 y Fp(2)612 1040 y Fx(\))12 b Fq(\021)f Fo(F)6 b Fx(\([)p Fn(C)q Fx(\()p Fo(B)r Fx(\)])p Fo(;)h(F)f Fx(\([)11 b(])p Fo(;)c Fx([[)p Fn(C)p Fx(\()p Fo(B)r Fx(\)])p Fo(;)g Fn(D)q Fx(\()p Fo(A)p Fx(\)]\)\))438 1114 y Fq(!)480 1097 y Fp(+)480 1127 y Fk(F)507 1117 y Fd(e)504 1137 y Fc(1)536 1114 y Fo(\032)p Fx(\()p Fo(s)592 1120 y Fp(3)612 1114 y Fx(\))12 b Fq(\021)f Fo(F)6 b Fx(\()p Fn(C)r Fx(\()p Fo(B)r Fx(\))p Fo(;)h(F)f Fx(\([)11 b(])p Fo(;)c Fx([[)p Fn(C)p Fx(\()p Fo(B)r Fx(\)])p Fo(;)g Fn(D)p Fx(\()p Fo(A)p Fx(\)]\)\))438 1189 y Fq(!)480 1171 y Fp(+)480 1201 y Fk(F)507 1191 y Fd(e)504 1211 y Fc(1)536 1189 y Fo(\032)p Fx(\()p Fo(s)592 1195 y Fp(4)612 1189 y Fx(\))12 b Fq(\021)f Fo(F)6 b Fx(\()p Fn(C)r Fx(\()p Fo(B)r Fx(\))p Fo(;)h(F)f Fx(\([)11 b(])p Fo(;)c Fn(D)p Fx(\()p Fo(A)p Fx(\)\)\))441 1259 y Fq(!)483 1265 y Fk(S)503 1269 y Fc(1)536 1259 y Fo(\032)p Fx(\()p Fo(s)592 1265 y Fp(5)612 1259 y Fx(\))12 b Fq(\021)f Fo(F)6 b Fx(\([)11 b(])p Fo(;)c(B)r Fx(\))183 1381 y Fg(4.3)47 b(Con\015uence)15 b(is)h(a)h(Mo)q(dular)e(Prop)q (ert)o(y)g(of)h(Simplif)o(yin)o(g)e(Join)h(CTRSs)183 1431 y(with)f(Shared)h (Constructors)183 1512 y Fx(Con\015uence)k(is)f(a)g(mo)q(dular)e(prop)q(ert)o (y)j(of)f(disjoin)o(t)f(TRSs)h(as)g(w)o(as)g(sho)o(wn)h(b)o(y)f(T)m(o)o(y)o (ama)183 1561 y(\(T)m(o)o(y)o(ama')o(s)12 b(Theorem,)h(see)j([T)m(o)o(y87b)n (]\).)d(Middeldorp)h(\([Mid90a)o(]\))g(extended)h(this)g(result)183 1611 y(to)g(disjoin)o(t)f(CTRSs.)h(Ho)o(w)o(ev)o(er,)h(con\015uence)h(is)e (not)h(mo)q(dular)d(if)i(the)h(TRSs)f(share)h(con-)183 1661 y(structors)i(as)f(w)o(as)f(p)q(oin)o(ted)g(out)h(b)o(y)f(Kurihara)h(and)f (Oh)o(uc)o(hi)g([K)o(O90a])g(using)g(the)h(next)183 1711 y(example.)183 1791 y Fg(Example)7 b(25.)21 b Fx(Let)14 b Fq(R)563 1797 y Fp(1)593 1791 y Fx(=)e Fq(f)p Fo(F)6 b Fx(\()p Fo(x;)h(x)p Fx(\))p Fq(!)n Fo(A;)g(F)f Fx(\()p Fo(x;)h Fn(C)p Fx(\()p Fo(x)p Fx(\)\))p Fq(!)p Fo(B)r Fq(g)14 b Fx(and)f(let)h Fq(R)1353 1797 y Fp(2)1383 1791 y Fx(=)e Fq(f)p Fo(a)p Fq(!)o Fn(C)r Fx(\()p Fo(a)p Fx(\))p Fq(g)p Fx(.)183 1841 y Fq(R)218 1847 y Fp(1)254 1841 y Fx(and)k Fq(R)372 1847 y Fp(2)408 1841 y Fx(are)i(con\015uen)o(t,)f(but)h Fq(R)789 1847 y Fp(1)819 1841 y Fq([)11 b(R)893 1847 y Fp(2)929 1841 y Fx(is)17 b(not)g(con\015uen)o(t)h (since)g Fo(F)6 b Fx(\()p Fo(a;)h(a)p Fx(\))16 b(has)h(t)o(w)o(o)183 1891 y(normal)11 b(forms)i Fo(A)h Fx(and)g Fo(B)r Fx(.)183 1972 y(F)m(ortunately)m(,)c(con\015uence)j(is)f(a)f(mo)q(dular)f(prop)q(ert)o (y)i(of)f Fm(simplifying)h(unc)n(onditional)h Fx(TRSs)183 2021 y(with)g(shared)i(constructors.)183 2102 y Fg(Corollary)6 b(26.)21 b Fm(L)n(et)14 b Fq(R)575 2108 y Fp(1)609 2102 y Fm(and)i Fq(R)725 2108 y Fp(2)758 2102 y Fm(b)n(e)f(two)g(TRSs)g(and)h(let)e Fq(R)h Fm(b)n(e)g(the)g(c)n(ombine)n(d)g(TRS)h(of)183 2152 y Fq(R)218 2158 y Fp(1)248 2152 y Fm(and)d Fq(R)361 2158 y Fp(2)392 2152 y Fm(with)f(shar)n(e)n(d)g(c)n(onstructors.)f(Then)h Fq(R)h Fm(is)e(simplifying)h(and)g(c)n(on\015uent)i(\(henc)n(e)183 2202 y(c)n(onver)n(gent\))h(if)f(and)i(only)f(if)f(b)n(oth)h Fq(R)779 2208 y Fp(1)813 2202 y Fm(and)h Fq(R)929 2208 y Fp(2)962 2202 y Fm(ar)n(e)f(simplifying)f(and)i(c)n(on\015uent.)183 2282 y(Pr)n(o)n(of.)k Fx(The)13 b(only-if)d(case)k(is)e(trivial.)e(W)m(e)i (sho)o(w)h(the)g(if)e(direction.)h(By)h(Theorem)f(23,)f Fq(R)i Fx(is)183 2332 y(simplifyi)o(ng)e(and)j(th)o(us)g(terminating.)d(Since)k (rewrite)f(rules)h(of)e Fq(R)1235 2338 y Fp(1)1267 2332 y Fx(do)h(not)g(o)o (v)o(erlap)f(with)183 2382 y(rewrite)j(rules)g(of)f Fq(R)510 2388 y Fp(2)544 2382 y Fx(and)g(vice)h(v)o(ersa,)g(the)g(set)g(of)f(critical) g(pairs)g(of)g Fq(R)h Fx(consists)g(of)f(the)183 2432 y(critical)d(pairs)g (of)g Fq(R)501 2438 y Fp(1)532 2432 y Fx(and)g(those)i(of)d Fq(R)799 2438 y Fp(2)818 2432 y Fx(.)h(Hence)i(the)f(assertion)g(follo)o(ws)e (from)g(Newman's)p eop %%Page: 12 12 bop 340 194 a Fx(Lemma)487 179 y Fp(6)518 194 y Fx(in)11 b(conjunction)h (with)g(the)h(Critical)e(P)o(air)h(Lemma)1332 179 y Fp(7)1363 194 y Fx(b)q(ecause)i Fq(R)1550 200 y Fp(1)1581 194 y Fx(and)e Fq(R)1695 200 y Fp(2)1726 194 y Fx(are)340 244 y(con\015uen)o(t.)1238 b Fq(u)-28 b(t)340 324 y Fx(In)12 b(the)g(con)o(text)g(of)f(simplifying)d (join)j(CTRSs)g(with)h(shared)g(constructors,)h(the)f(mo)q(dular-)340 373 y(it)o(y)g(of)f(con\015uence)j(is)e(not)h(so)f(easily)f(obtained;)h (although)f(a)h(Critical)f(P)o(air)h(Lemma)e(holds)340 423 y(for)16 b(simplifying)d(CTRSs)i(\(cf.)h([Kap87)o(]\))g(and)g(the)g(set)h(of) f(conditional)e(critical)i(pairs)g(of)340 473 y Fq(R)i Fx(consists)h(of)e (the)h(conditional)e(critical)h(pairs)h(of)f Fq(R)1228 479 y Fp(1)1264 473 y Fx(and)g(those)i(of)d Fq(R)1546 479 y Fp(2)1565 473 y Fx(,)h(w)o(e)h(cannot)340 523 y(argue)d(as)f(in)g(the)h(unconditional)e (case.)i(This)f(is)g(due)h(to)f(the)h(fact)f(that)h(w)o(e)f(ha)o(v)o(e)g(to)h (deal)340 573 y(with)h Fm(c)n(ontextual)g Fx(critical)f(pairs)h(\(see)h ([Kap87)o(]\))f(and)f(that)h(the)h(fundamen)o(tal)c(prop)q(ert)o(y)340 623 y(of)i(TRSs,)g(that)g(is,)g Fo(s)p Fq(!)723 629 y Fk(R)753 623 y Fo(t)g Fx(either)i(implies)c Fo(s)p Fq(!)1106 629 y Fk(R)1135 633 y Fc(1)1153 623 y Fo(t)i Fx(or)g Fo(s)p Fq(!)1296 629 y Fk(R)1325 633 y Fc(2)1343 623 y Fo(t)p Fx(,)g(do)q(es)h(not)f(hold)f(for)h (join)340 672 y(CTRSs)j(\(cf.)f([Mid90a)o(]\).)f(T)m(o)h(o)o(v)o(ercome)g (this)g(obstacle,)h(w)o(e)g(use)g(the)g(structure)i(of)d(the)340 722 y(pro)q(of)12 b(of)f(mo)q(dularit)o(y)e(of)i(con\015uence)j(for)d (disjoin)o(t)g(CTRSs)g(\([Mid90a)o(]\).)g(It)h(go)q(es)g(without)340 772 y(sa)o(ying)i(that)h(w)o(e)g(ha)o(v)o(e)g(to)g(mo)q(dify)d(parts)k(of)e (the)h(pro)q(of.)f(The)h(pro)q(ofs)g(of)g(some)f(lemma)o(ta)340 822 y(hold)j(cum)f(grano)h(salis,)f(whereas)j(some)d(in)o(tricate)i(tec)o (hnical)f(details)g(simplify)e(due)i(to)340 872 y(the)k(termination)d(prop)q (ert)o(y)i(of)f(the)i(com)o(bined)d(system.)h(The)h(pro)q(of)g(idea)f(is)h (to)f(con-)340 921 y(struct)d(t)o(w)o(o)e(rewrite)i(relations)e Fq(!)892 927 y Fp(1)925 921 y Fx(and)h Fq(!)1048 927 y Fp(2)1081 921 y Fx(suc)o(h)g(that)g(their)g(union)e(is)i(con\015uen)o(t,)f(and)340 971 y(reduction)i(in)f(the)h(com)o(bined)e(system)h Fq(R)g Fx(corresp)q(onds)i(to)f(joinabilit)o(y)c(with)j(resp)q(ect)i(to)340 1021 y Fq(!)382 1027 y Fp(1)405 1021 y Fq([)t(!)478 1027 y Fp(2)497 1021 y Fx(.)11 b(F)m(rom)e(these)k(t)o(w)o(o)e(prop)q(erties)i(the)f (mo)q(dularit)o(y)d(of)i(con\015uence)i(for)e(simplifying)340 1071 y(join)i(CTRSs)h(with)g(shared)g(constructors)i(is)e(easily)f(inferred.) 340 1151 y Fg(De\014nition)t(27.)21 b Fx(The)c(rewrite)g(relation)f Fq(!)1065 1157 y Fp(1)1100 1151 y Fx(is)g(de\014ned)h(as)g(follo)o(ws:)d Fo(s)p Fq(!)1554 1157 y Fp(1)1573 1151 y Fo(t)i Fx(if)g(there)h(is)340 1201 y(a)e(rewrite)h(rule)f Fo(l)e Fq(!)e Fo(r)i Fq(\()e Fo(s)784 1207 y Fp(1)814 1201 y Fq(#)h Fo(t)862 1207 y Fp(1)880 1201 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)992 1207 y Fn(n)1026 1201 y Fq(#)k Fo(t)1073 1207 y Fn(n)1111 1201 y Fx(in)j Fq(R)1195 1207 y Fp(1)1229 1201 y Fx(suc)o(h)i(that)f Fo(s)f Fq(\021)g Fo(C)s Fx([)p Fo(l)q(\033)q Fx(])p Fo(;)7 b(t)12 b Fq(\021)i Fo(C)s Fx([)p Fo(r)q(\033)q Fx(])340 1251 y(and)f Fo(s)439 1257 y Fn(i)454 1251 y Fo(\033)q Fq(#)500 1232 y Fn(o)500 1261 y Fp(1)518 1251 y Fo(t)533 1257 y Fn(i)547 1251 y Fo(\033)h Fx(for)f Fo(i)f Fx(=)g(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(n)p Fx(.)12 b(Here)j(the)e(sup)q (erscript)j Fo(o)d Fx(in)f Fo(s)1361 1257 y Fn(i)1376 1251 y Fo(\033)q Fq(#)1422 1232 y Fn(o)1422 1261 y Fp(1)1440 1251 y Fo(t)1455 1257 y Fn(i)1469 1251 y Fo(\033)i Fx(means)f(that)g Fo(s)1742 1257 y Fn(i)1756 1251 y Fo(\033)340 1300 y Fx(and)k Fo(t)439 1306 y Fn(i)452 1300 y Fo(\033)h Fx(are)f(joinable)e(using)h(only)g Fm(outer)g Fq(!)1080 1306 y Fp(1)1115 1300 y Fx(-reduction)h(steps.)g(The)g (relation)f Fq(!)1718 1306 y Fp(2)1753 1300 y Fx(is)340 1350 y(de\014ned)f(analogously)m(.)d(The)i(union)f(of)g Fq(!)1006 1356 y Fp(1)1038 1350 y Fx(and)h Fq(!)1161 1356 y Fp(2)1193 1350 y Fx(is)g(denoted)h(b)o(y)e Fq(!)1491 1356 y Fp(1)p Fn(;)p Fp(2)1536 1350 y Fx(.)340 1438 y Fg(Example)8 b(28.)20 b Fx(Let)d Fq(R)723 1444 y Fp(1)757 1438 y Fx(=)f Fq(f)p Fo(F)6 b Fx(\()p Fo(x;)h Fn(C)p Fx(\))p Fq(!)p Fo(G)p Fx(\()p Fo(x)p Fx(\))15 b Fq(\()f Fo(x)p Fq(#)p Fn(C)r Fq(g)i Fx(and)g Fq(R)1386 1444 y Fp(2)1420 1438 y Fx(=)f Fq(f)p Fo(a)p Fq(!)o Fn(C)r Fq(g)p Fx(.)h(W)m(e)f(ha)o(v)o(e)340 1488 y Fo(F)6 b Fx(\()p Fo(a;)h Fn(C)q Fx(\))p Fq(!)513 1494 y Fk(R)543 1488 y Fo(G)p Fx(\()p Fo(a)p Fx(\))16 b(but)g(neither)h Fo(F)6 b Fx(\()p Fo(a;)h Fn(C)q Fx(\))p Fq(!)1040 1494 y Fp(1)1058 1488 y Fo(G)p Fx(\()p Fo(a)p Fx(\))16 b(nor)g Fo(F)6 b Fx(\()p Fo(a;)h Fn(C)q Fx(\))p Fq(!)1410 1494 y Fp(2)1428 1488 y Fo(G)p Fx(\()p Fo(a)p Fx(\).)16 b(Ho)o(w)o(ev)o(er,)g(the)340 1538 y(terms)e(are)g(joinable)f(with)h(resp)q (ect)i(to)d Fq(!)1010 1544 y Fp(1)p Fn(;)p Fp(2)1055 1538 y Fx(:)g Fo(F)6 b Fx(\()p Fo(a;)h Fn(C)q Fx(\))p Fq(!)1253 1544 y Fp(2)1271 1538 y Fo(F)f Fx(\()p Fn(C)r Fo(;)h Fn(C)q Fx(\))p Fq(!)1448 1544 y Fp(1)1467 1538 y Fo(G)p Fx(\()p Fn(C)q Fx(\))12 b Fq( )1611 1544 y Fp(2)1640 1538 y Fo(G)p Fx(\()p Fo(a)p Fx(\))p Fo(:)340 1625 y Fg(Prop)q(osition)5 b(29.)20 b Fm(If)15 b Fo(s)p Fq(!)777 1631 y Fp(1)p Fn(;)p Fp(2)822 1625 y Fo(t)g Fm(then)g Fo(s)p Fq(!)1005 1631 y Fk(R)1036 1625 y Fo(t)p Fm(.)340 1706 y(Pr)n(o)n(of.)20 b Fx(T)m(rivial.)1150 b Fq(u)-28 b(t)340 1793 y Fg(Prop)q(osition)5 b(30.)20 b Fm(L)n(et)f Fo(s;)7 b(t)18 b Fm(b)n(e)g(black)h(terms)f(and)h(let)f Fo(\033)i Fm(b)n(e)e(a)h(top)f (white)g(substitution)340 1843 y(with)d Fo(s\033)q Fq(!)517 1828 y Fn(o)517 1853 y Fp(1)535 1843 y Fo(t\033)q Fm(.)g(If)g Fo(\034)k Fm(is)c(a)g(substitution)g(with)f Fo(\033)f Fq(/)f Fo(\034)19 b Fm(then)d Fo(s\034)5 b Fq(!)1373 1828 y Fn(o)1373 1853 y Fp(1)1392 1843 y Fo(t\034)g Fm(.)340 1923 y(Pr)n(o)n(of.)20 b Fx(Essen)o(tially)14 b(the)h(same)e(as)h(in)f([Mid90a)n(].)630 b Fq(u)-28 b(t)340 2011 y Fg(Lemma)8 b(31.)20 b Fm(The)15 b(r)n(elation)g Fq(!)862 2017 y Fp(1)p Fn(;)p Fp(2)922 2011 y Fm(is)f(c)n(onver)n(gent.)340 2091 y(Pr)n(o)n(of.)20 b Fx(W)m(e)14 b(de\014ne)h(t)o(w)o(o)e(unconditional)g (TRSs)h Fq(S)1140 2097 y Fp(1)1172 2091 y Fx(and)g Fq(S)1278 2097 y Fp(2)1311 2091 y Fx(b)o(y)546 2174 y Fq(S)571 2180 y Fn(i)596 2174 y Fx(=)e Fq(f)p Fo(u)f Fq(!)g Fo(v)j Fq(j)e Fo(u;)7 b(v)12 b Fq(2)g(T)e Fx(\()p Fq(D)1004 2186 y Fn(i)1018 2174 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\))p Fo(;)g(r)q(oot)p Fx(\()p Fo(u)p Fx(\))k Fq(62)g(C)16 b Fx(and)e Fo(u)p Fq(!)1508 2180 y Fn(i)1522 2174 y Fo(v)q Fq(g)p Fo(:)340 2256 y Fx(First)21 b(of)f(all)g(note)h(that)f(the)i Fq(S)866 2262 y Fn(i)900 2256 y Fx(are)f(in\014nite)f(in)h(general.)f(It)h(is)f(easy)h(to)g(sho)o(w)f(that) 340 2306 y(the)15 b(restrictions)h(of)e(the)g(relations)h Fq(!)958 2312 y Fk(S)978 2316 y Fd(i)994 2306 y Fo(;)7 b Fq(!)1054 2312 y Fn(i)1068 2306 y Fx(,)13 b(and)h Fq(!)1216 2312 y Fk(R)1245 2316 y Fd(i)1274 2306 y Fx(to)g Fq(T)c Fx(\()p Fq(D)1408 2318 y Fn(i)1421 2306 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\))j Fq(\002)g(T)g Fx(\()p Fq(D)1661 2318 y Fn(i)1675 2306 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\))p 340 2343 237 2 v 349 2370 a Fs(6)379 2386 y Fw(A)12 b(terminating)k(TRS)d(is)g(con\015uen)o(t)i(if)e(and)g(only)i(if)e (it)g(is)h(lo)q(cally)i(con\015uen)o(t.)349 2416 y Fs(7)379 2432 y Fw(A)c(TRS)i(is)f(lo)q(cally)j(con\015uen)o(t)e(if)f(and)h(only)g(if)f (all)i(its)e(critical)i(pairs)f(are)g(con)o(v)o(ergen)o(t.)p eop %%Page: 13 13 bop 183 194 a Fx(coincide)16 b(\(note)h(that)g Fq(R)583 200 y Fn(i)613 194 y Fx(is)g(con)o(v)o(ergen)o(t\).)g(In)f(particular,)g(the)h (TRS)f Fq(S)1355 200 y Fn(i)1385 194 y Fx(is)h(con)o(v)o(ergen)o(t)183 244 y(on)c Fq(T)d Fx(\()p Fq(D)323 256 y Fn(i)337 244 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\))i Fq(\002)h(T)g Fx(\()p Fq(D)576 256 y Fn(i)590 244 y Fo(;)d Fq(C)r Fo(;)g Fq(V)s Fx(\).)245 293 y(F)m(urthermore,)14 b(it)g(can)h(b)q(e)g(sho)o(wn)f(as)h(in)f([Mid90a)n (])g(that)h(the)g(rewrite)g(relations)f Fq(!)1588 299 y Fk(S)1608 303 y Fd(i)183 343 y Fx(and)c Fq(!)302 349 y Fn(i)327 343 y Fx(are)h(also)f(the)i(same)e(on)h Fq(T)f Fx(\()p Fq(D)779 349 y Fp(1)807 343 y Fq([)f(D)876 349 y Fp(2)894 343 y Fo(;)e Fq(C)r Fo(;)g Fq(V)s Fx(\))t Fq(\002)s(T)k Fx(\()p Fq(D)1121 349 y Fp(1)1149 343 y Fq([)d(D)1217 349 y Fp(2)1236 343 y Fo(;)f Fq(C)r Fo(;)g Fq(V)s Fx(\).)k(If)f(w)o(e)h(can)h(sho)o(w)183 393 y(that)i Fq(S)298 399 y Fp(1)326 393 y Fq([)8 b(S)387 399 y Fp(2)420 393 y Fx(is)14 b(con)o(v)o(ergen)o(t,)g(then)h(it)e(follo)o(ws)f (with)521 474 y Fq(!)562 480 y Fk(S)582 484 y Fc(1)599 480 y Fk([S)641 484 y Fc(2)671 474 y Fx(=)g Fq(!)757 480 y Fk(S)777 484 y Fc(1)805 474 y Fq([)d(!)883 480 y Fk(S)903 484 y Fc(2)933 474 y Fx(=)j Fq(!)1019 480 y Fp(1)1047 474 y Fq([)c(!)1125 480 y Fp(2)1155 474 y Fx(=)k Fq(!)1240 480 y Fp(1)p Fn(;)p Fp(2)183 555 y Fx(that)h Fq(!)314 561 y Fp(1)p Fn(;)p Fp(2)372 555 y Fx(is)g(also)g(con)o(v)o(ergen)o(t)h(on)f Fq(T)d Fx(\()p Fq(D)841 561 y Fp(1)869 555 y Fq([)f(D)938 561 y Fp(2)957 555 y Fo(;)e Fq(C)r Fo(;)g Fq(V)s Fx(\))h 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Fx(is)f(also)g(con\015uen)o(t.)42 b Fq(u)-28 b(t)183 840 y Fg(Lemma)7 b(32.)21 b Fm(If)14 b Fo(s)p Fq(!)530 846 y Fk(R)561 840 y Fo(t)h Fm(then)g Fo(s)p Fq(#)723 850 y Fp(1)p Fn(;)p Fp(2)768 840 y Fo(t)p Fm(.)183 919 y(Pr)n(o)n(of.)20 b Fx(The)13 b(same)g(as)g(in)g([Mid90a)n(].)f(The)i(pro)q(of)f(uses)h(the)g (next)f(prop)q(osition)g(\(the)h(pro)q(of)183 969 y(of)f(whic)o(h)h(is)f(tec) o(hnically)h(in)o(tricate)g(without)g(the)g(termination)e(prop)q(ert)o(y)j (of)e Fq(R)p Fx(\).)101 b Fq(u)-28 b(t)183 1055 y Fg(Prop)q(ositi)o(on)5 b(33.)21 b Fm(L)n(et)13 b Fo(s)604 1061 y Fp(1)623 1055 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)735 1061 y Fn(n)758 1055 y Fm(,)13 b Fo(t)799 1061 y Fp(1)818 1055 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)926 1061 y Fn(n)948 1055 y Fm(b)n(e)14 b(black)g(terms.)f(F)m(or)h(every)g (substitution)183 1104 y Fo(\033)22 b Fm(with)d Fo(s)343 1110 y Fn(i)358 1104 y Fo(\033)q Fq(#)404 1115 y Fp(1)p Fn(;)p Fp(2)449 1104 y Fo(t)464 1110 y Fn(i)478 1104 y Fo(\033)i Fm(for)g(every)f Fo(i)i 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Fg(Prop)q(ositi)o(on)5 b(34.)21 b Fm(The)14 b(r)n(elations)h Fx(=)801 1885 y Fk(R)846 1879 y Fm(and)h Fq(#)948 1889 y Fp(1)p Fn(;)p Fp(2)1008 1879 y Fm(c)n(oincide.)183 1958 y(Pr)n(o)n(of.)k Fx(This)13 b(is)h(a)g (consequence)i(of)d(Prop)q(osition)h(29)f(and)h(Lemmata)d(31)i(and)h(32.)96 b Fq(u)-28 b(t)183 2044 y Fg(Theorem)6 b(35.)21 b Fm(Con\015uenc)n(e)14 b(is)f(a)g(mo)n(dular)g(pr)n(op)n(erty)f(of)h(simplifying)f(join)g(CTRSs)h (with)183 2094 y(shar)n(e)n(d)h(c)n(onstructors.)183 2172 y(Pr)n(o)n(of.)20 b Fx(Let)c(\()p Fq(D)437 2178 y Fp(1)466 2172 y Fq([)11 b(C)r Fo(;)c Fq(R)583 2178 y Fp(1)601 2172 y Fx(\))16 b(and)g(\()p Fq(D)764 2178 y Fp(2)793 2172 y Fq([)10 b(C)r Fo(;)d Fq(R)909 2178 y Fp(2)928 2172 y Fx(\))16 b(b)q(e)g(t)o(w)o(o)g(simplifying)c(join)j (CTRSs)h(with)183 2222 y(shared)i(constructors.)h(W)m(e)e(ha)o(v)o(e)h(to)f (sho)o(w)h(that)f(the)i(com)o(bined)d(system)h Fq(R)h Fx(\(whic)o(h)g(is)183 2272 y(simplifyi)o(ng\))11 b(is)i(con\015uen)o(t)h(if)f(and)g(only)g(if)g(b)q (oth)g Fq(R)1009 2278 y Fp(1)1041 2272 y Fx(and)h Fq(R)1157 2278 y Fp(2)1189 2272 y Fx(are)g(con\015uen)o(t.)g(The)f(only-)183 2322 y(if)h(case)i(is)f(straigh)o(tforw)o(ard.)f(In)h(order)h(to)e(sho)o(w)h (the)h(other)g(implication)c(w)o(e)j(consider)h(a)183 2372 y(con)o(v)o(ersion)11 b Fo(t)397 2378 y Fp(1)423 2357 y Fk(\003)423 2383 y(R)444 2372 y Fq( )o Fo(s)h Fq(!)558 2357 y Fk(\003)558 2383 y(R)600 2372 y Fo(t)615 2378 y Fp(2)634 2372 y Fx(.)e(According)i(to)f (Prop)q(osition)g(34)f(w)o(e)i(ha)o(v)o(e)f Fo(t)1339 2378 y Fp(1)1358 2372 y Fq(#)1378 2382 y Fp(1)p Fn(;)p Fp(2)1423 2372 y Fo(t)1438 2378 y Fp(2)1468 2372 y Fx(and)g(th)o(us)183 2422 y(Prop)q(osition)i(29)g(implies)f Fo(t)617 2428 y Fp(1)636 2422 y Fq(#)657 2432 y Fk(R)687 2422 y Fo(t)702 2428 y Fp(2)721 2422 y Fx(.)863 b Fq(u)-28 b(t)p eop %%Page: 14 14 bop 340 194 a Fg(4.4)48 b(A)o(ttempts)14 b(to)h(w)o(eak)o(en)g(the)g (Premises)340 262 y Fx(Naturally)i(the)h(question)g(arises)h(whether)g (Theorem)e(23)g(also)g(holds)g(for)h(reductiv)o(e)g(or)340 312 y(decreasing)i(CTRSs)f(\(see)h([DOS88)o(]\).)e(The)h(answ)o(er)g(is)g (negativ)o(e)f(as)h(is)g(sho)o(wn)f(b)o(y)h(the)340 361 y("classical")13 b(coun)o(terexample)h(of)f(T)m(o)o(y)o(ama)e(\(cf.)j(Example)e(11\).)403 411 y(Additionally)m(,)h(one)j(ma)o(y)f(ask)h(whether)i Fq(R)1096 417 y Fp(1)1131 411 y Fx(and)e Fq(R)1249 417 y Fp(2)1284 411 y Fx(ma)o(y)e(share)j(de\014ned)h(function)340 461 y(sym)o(b)q(ols)d(\(sym)o (b)q(ols)g(that)h(do)g(o)q(ccur)i(at)e(the)g(ro)q(ot)h(p)q(osition)e(of)h (the)h(left-hand)e(side)i(of)f(a)340 511 y(rewrite)f(rule\).)e(Esp)q(ecially) g(in)f(the)i(con)o(text)g(of)e(parameterized)i(algebraic)e(sp)q (eci\014cations)340 561 y(it)18 b(w)o(ould)g(b)q(e)h(nice)g(if)e(at)i(least)f (one)h(of)f(the)h(CTRSs)f(could)g(con)o(tain)g(de\014ned)i(function)340 610 y(sym)o(b)q(ols)13 b(of)g(the)h(other)g(in)f(its)h(conditions)f(\(but)h (not)g(vice)g(v)o(ersa\).)f(Unfortunately)m(,)g(there)340 660 y(is)f(no)g(a\016rmativ)o(e)d(answ)o(er)k(to)e(that)h(question,)f(not)h(ev)o (en)h(in)e(the)h(restricted)i(case)f(that)f(the)340 710 y(systems)i(do)f(not) g(share)h(constructors)i(in)c(the)i(left-)f(and)g(righ)o(t-hand)g(sides)h(of) f(rules)h(\(i.e.,)340 760 y(shared)h(sym)o(b)q(ols)e(o)q(ccur)i(only)e(in)g (the)i(conditions\).)340 828 y Fg(Example)8 b(36.)20 b Fx(Let)f Fq(R)725 834 y Fp(1)762 828 y Fx(=)f Fq(f)p Fo(F)6 b Fx(\()p Fo(x;)h(x)p Fx(\))j Fq(!)h Fo(x)g Fq(\()g Fo(F)6 b Fx(\()p Fo(a;)h(b)p Fx(\))k Fq(#)g Fo(x)p Fq(g)17 b Fx(and)h Fq(R)1465 834 y Fp(2)1502 828 y Fx(=)g Fq(f)p Fo(a)12 b Fq(!)f Fo(b)p Fq(g)o Fx(.)18 b Fq(R)1763 834 y Fp(1)340 878 y Fx(and)c Fq(R)456 884 y Fp(2)489 878 y Fx(are)g(simplifying)c(but)15 b(the)f(union)f Fq(R)f Fx(=)g Fq(R)1159 884 y Fp(1)1187 878 y Fq([)d(R)1259 884 y Fp(2)1291 878 y Fx(is)14 b(not)g(\(see)h([Ohl92b)o(]\).)403 946 y(In)d(the)i(ab)q(o)o(v)o(e)e(example)g(there)i(is)f(a)f(\\mixed")f(term) h Fo(F)6 b Fx(\()p Fo(a;)h(b)p Fx(\))12 b(in)g(the)h(condition)g(of)f(the)340 996 y(only)i(rule)h(of)f Fq(R)599 1002 y Fp(1)617 996 y Fx(.)g(It)h(needs)h (some)d(further)j(in)o(v)o(estigations)d(if)h(the)h(union)f(\()p Fq(F)t Fo(;)7 b Fq(R)p Fx(\))14 b(of)g(t)o(w)o(o)340 1045 y(simplifying)d (CTRSs)j(\()p Fq(F)743 1051 y Fp(1)761 1045 y Fo(;)7 b Fq(R)815 1051 y Fp(1)833 1045 y Fx(\))14 b(and)g(\()p Fq(F)990 1051 y Fp(2)1009 1045 y Fo(;)7 b Fq(R)1063 1051 y Fp(2)1081 1045 y Fx(\))14 b(is)g(again)e(simplifying)f(pro)o(vided)j(that)366 1113 y Fg({)21 b Fq(F)441 1119 y Fp(1)476 1113 y Fx(is)c(the)h(disjoin)o(t)e (union)g(of)h Fq(F)950 1098 y Fk(0)946 1124 y Fp(1)981 1113 y Fx(and)g Fq(F)1095 1119 y Fp(2)1130 1113 y Fx(\(in)g(particular,)f Fq(R)1439 1119 y Fp(2)1475 1113 y Fx(do)q(es)i(not)f(con)o(tain)411 1163 y(an)o(y)c(sym)o(b)q(ol)f(from)h Fq(F)764 1148 y Fk(0)760 1174 y Fp(1)778 1163 y Fx(\),)h(and)366 1213 y Fg({)21 b Fx(if)c Fo(l)12 b Fq(!)f Fo(r)i Fq(\()e Fo(s)634 1219 y Fp(1)664 1213 y Fq(#)h Fo(t)712 1219 y Fp(1)730 1213 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)842 1219 y Fn(n)876 1213 y Fq(#)k Fo(t)923 1219 y Fn(n)963 1213 y Fx(is)18 b(a)f(rule)g(of)g Fq(R)1219 1219 y Fp(1)1255 1213 y Fx(then)h Fo(l)q(;)7 b(r)18 b Fq(2)f(T)10 b Fx(\()p Fq(F)1550 1192 y Fk(0)1550 1226 y Fp(1)1569 1213 y Fo(;)d Fq(V)s Fx(\))18 b(and)f(w)o(e)411 1263 y(ha)o(v)o(e)d Fo(s)526 1269 y Fn(i)540 1263 y Fo(;)7 b(t)574 1269 y Fn(i)599 1263 y Fq(2)k(T)f Fx(\()p Fq(F)721 1242 y Fk(0)721 1275 y Fp(1)740 1263 y Fo(;)d Fq(V)s Fx(\))14 b(or)g Fo(s)887 1269 y Fn(i)901 1263 y Fo(;)7 b(t)935 1269 y Fn(i)960 1263 y Fq(2)k(T)f Fx(\()p Fq(F)1082 1275 y Fp(2)1101 1263 y Fo(;)d Fq(V)s Fx(\))q(.)340 1331 y(Ho)o(w)o(ev)o(er,)13 b(w)o(e)g(note)g(that)g(ev)o(en)h(if)e(the)h(union)g(of)f(suc)o(h)h(t)o(w)o (o)g(simplifyi)o(ng)d(CTRSs)j(is)f(again)340 1381 y(simplifying,)6 b(con\015uence)12 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g(w)o(eak)o(ening)g(of)f(the)i(premises)f(leads)h(to)f(the)h(loss)f(of)f(the) i(mo)q(dularit)o(y)d(of)340 1834 y(at)h(least)g(one)g(of)f(these)i(t)o(w)o(o) f(prop)q(erties.)g(Because)i(of)d(Prop)q(osition)h(5,)f(w)o(e)h(ha)o(v)o(e)f (fo)q(cused)340 1884 y(on)i(\014nite)h(simplifyi)o(ng)c Fm(join)j Fx(systems.)g(Ho)o(w)o(ev)o(er,)g(con\015uence)i(seems)f(to)f(b)q(e)g(a)g(mo) q(dular)340 1934 y(prop)q(ert)o(y)j(of)f(simplifying)d(semi-equational)h (CTRSs,)i(to)q(o)g(\(this)g(m)o(ust)f(b)q(e)i(c)o(hec)o(k)o(ed)h(in)340 1983 y(detail)14 b(along)f(the)i(lines)g(of)e(the)i(pro)q(of)g(of)e(mo)q (dularit)o(y)f(of)i(con\015uence)i(for)e(disjoin)o(t)g(semi-)340 2033 y(equational)h(CTRSs)h(giv)o(en)f(in)g([Mid90b)o(]\).)g(Moreo)o(v)o(er,) g(w)o(e)h(p)q(oin)o(t)f(out)h(that)g(the)g(results)340 2083 y(obtained)11 b(for)g(join)f(CTRSs)g(also)h(hold)f(for)h Fm(normal)f Fx(CTRSs)h(\(since)h(ev)o(ery)g(normal)c(system)340 2133 y(can)14 b(b)q(e)h(view)o(ed)f(as)g(a)g(join)e(system,)i(cf.)f([Mid90a)o(]\).)340 2183 y(Recen)o(tly)20 b(Gramlic)o(h)d(\(cf.)i([Gra91)o(]\))g(obtained)g(sev)o (eral)h(su\016cien)o(t)g(conditions)g(for)f(the)340 2232 y(mo)q(dularit)o(y) 11 b(of)i(termination)f(of)g(\(\014nite)i(and)f(\014nitely)g(branc)o(hing,)g (resp)q(ectiv)o(ely\))i(uncon-)340 2282 y(ditional)g(term)h(rewriting)g (systems)h(\(p)q(ossibly)f(sharing)g(constructors\))i(and)f(of)e(disjoin)o(t) 340 2332 y(conditional)e(term)h(rewriting)h(systems.)f(The)h(careful)g (reader)g(will)e(observ)o(e)j(that)e(his)h(re-)340 2382 y(sults)g(and)e (those)i(giv)o(en)e(in)h(the)g(presen)o(t)i(pap)q(er)e(complemen)o(t)e(one)i (another.)p eop %%Page: 15 15 bop 183 194 a Fg(Ac)o(kno)o(wledgemen)o(t)o(s:)8 b Fx(The)j(author)g(is)f (grateful)g(to)h(Rob)q(ert)g(Giegeric)o(h,)f(Stefan)h(Kurtz,)183 244 y(J\177)-21 b(org)15 b(S)q(\177)-22 b(uggel,)15 b(and)g(Aart)h (Middeldorp)f(for)g(v)n(aluable)f(commen)o(ts)g(on)h(previous)h(v)o(ersions) 183 293 y(of)d(the)h(pap)q(er.)h(Thanks)f(also)f(go)g(to)h(Ank)o(e)g(Bo)q (dzin)g(for)g(t)o(yp)q(esetting)h(the)f(man)o(uscript.)183 423 y Fr(References)199 514 y Fw([Der82])21 b(N.)12 b(Dersho)o(witz.)33 b(Orderings)19 b(for)e(T)m(erm-Rewriting)i(Systems.)32 b Ff(The)n(or)n(etic)n (al)16 b(Com-)341 560 y(puter)c(Scienc)n(e)g Fv(17)p Ff(\(3\))p Fw(,)f(pages)j(279{301,)g(1982.)211 605 y([DJ90])21 b(N.)12 b(Dersho)o(witz)18 b(and)f(J.P)m(.)e(Jouannaud.)28 b(Rewrite)17 b(Systems.)28 b(In)16 b(L.)d(v)n(an)j(Leeu)o(w)o(en,)341 651 y(editor,)d Ff(Handb)n(o)n(ok)d(of)i(The)n(or)n(etic)n(al)f(Computer)h (Scienc)n(e,)f(V)m(ol.)g(B)p Fw(,)h(c)o(hapter)i(6.)e(North-)341 697 y(Holland,)j(1990.)211 742 y([DJ91])21 b(N.)12 b(Dersho)o(witz)i(and)g (J.P)m(.)d(Jouannaud.)19 b(Notations)14 b(for)f(Rewriting.)18 b Ff(Bul)r(letin)12 b(of)h(the)341 788 y(EA)m(TCS)h Fv(43)p Fw(,)e(pages)i(162{172,)g(F)m(ebruary)g(1991.)183 834 y([DOS88])21 b(N.)12 b(Dersho)o(witz,)31 b(M.)12 b(Ok)n(ada,)30 b(and)h(G.)13 b(Siv)n(akumar.)70 b(Canonical)32 b(Conditional)341 879 y(Rewrite)17 b(Systems.)29 b(In)16 b Ff(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)j(the)f(9th)g (Confer)n(enc)n(e)f(on)h(A)o(utomate)n(d)e(De-)341 925 y(duction)p Fw(,)g(pages)j(538{549.)h(Lecture)f(Notes)g(in)h(Computer)f(Science)h Fv(310)p Fw(,)f(Springer)341 971 y(V)m(erlag,)c(1988.)196 1016 y([Gra91])21 b(B.)12 b(Gramlic)o(h.)18 b(A)10 b(Structural)j(Analysis)f(of)f (Mo)q(dular)h(T)m(ermination)h(of)d(T)m(erm)g(Rewrit-)341 1062 y(ing)k(Systems.)k(SEKI)13 b(Rep)q(ort)g(SR-91-15,)h(Univ)o(ersit\177)-19 b(at)14 b(Kaiserslautern,)i(1991.)190 1108 y([Kap87])21 b(S.)12 b(Kaplan.)35 b(Simplifyin)q(g)21 b(Conditional)h(T)m(erm)17 b(Rewriting)k(Systems:)d(Uni\014cation,)341 1153 y(T)m(ermination)e(and)f (Con\015uence.)22 b Ff(Journal)14 b(of)g(Symb)n(olic)f(Computation)g Fv(4)p Ff(\(3\))p Fw(,)g(pages)341 1199 y(295{334,)h(1987.)201 1244 y([Klo90])21 b(J.W.)13 b(Klop.)21 b(T)m(erm)14 b(Rewriting)h(Systems.)21 b(Rep)q(ort)15 b(CS-R9073,)f(Cen)o(tre)g(for)g(Mathe-)341 1290 y(matics)g(and)f(Computer)h(Science,)g(1990.)183 1336 y([K)o(O90a])20 b(M.)12 b(Kurihara)21 b(and)f(A.)12 b(Oh)o(uc)o(hi.)38 b(Mo)q(dularit)o(y)22 b(of)d(Simple)i(T)m(ermination)h(of)d(T)m(erm)341 1381 y(Rewriting)c(Systems) f(with)g(Shared)g(Constructors.)19 b(Rep)q(ort)14 b(SF-36,)f(Hokk)n(aido)i (Uni-)341 1427 y(v)o(ersit)o(y)m(,)e(Sapp)q(oro,)i(1990.)183 1473 y([K)o(O90b])20 b(M.)12 b(Kurihara)21 b(and)f(A.)12 b(Oh)o(uc)o(hi.)37 b(Mo)q(dularit)o(y)22 b(of)d(Simple)i(T)m(ermination)g(of)e(T)m(erm)341 1518 y(Rewriting)c(Systems.)i Ff(Journal)12 b(of)i(IPS)f(Jap)n(an)f Fv(31)p Ff(\(5\))p Fw(,)g(pages)h(633{642,)h(1990.)183 1564 y([Mid90a])22 b(A.)11 b(Middeldorp)q(.)48 b(Con\015uence)23 b(of)g(the)f(Disjoin)o(t)j(Union)e(of)f(Conditional)k(T)m(erm)341 1610 y(Rewriting)20 b(Systems.)32 b(In)18 b Ff(Pr)n(o)n(c)n(e)n(e)n(dings)d (of)i(the)h(2nd)f(Internation)o(al)e(Workshop)h(on)341 1655 y(Conditional)c(and)j(T)m(yp)n(e)n(d)f(R)n(ewriting)g(Systems)p Fw(,)f(pages)j(295{306.)g(Lecture)f(Notes)g(in)341 1701 y(Computer)e(Science) i Fv(516)p Fw(,)e(Springer)h(V)m(erlag,)g(1990.)183 1747 y([Mid90b])22 b(A.)12 b(Middeldorp.)19 b Ff(Mo)n(dular)10 b(Pr)n(op)n(erties)g(of)h(T)m (erm)h(R)n(ewriting)f(Systems)p Fw(.)j(PhD)e(thesis,)341 1792 y(F)m(ree)h(Univ)o(ersit)o(y)i(Amsterdam,)e(1990.)183 1838 y([Ohl92a])21 b(E.)12 b(Ohlebusc)o(h.)34 b(A)17 b(Note)g(on)i(Simple)g(T)m (ermination)h(of)d(In\014nite)i(T)m(erm)f(Rewriting)341 1884 y(Systems.)g(Rep)q(ort)13 b(Nr.)g(7,)f(T)m(ec)o(hnisc)o(he)j(F)m(akult\177) -19 b(at,)13 b(Univ)o(ersit\177)-19 b(at)15 b(Bielefeld,)g(1992.)183 1929 y([Ohl92b])21 b(E.)13 b(Ohlebusc)o(h.)k(Com)o(binations)12 b(of)d(Simplifyi)q(ng)j(Conditional)h(T)m(erm)c(Rewriting)i(Sys-)341 1975 y(tems.)17 b(Rep)q(ort)d(Nr.)e(6,)g(T)m(ec)o(hnisc)o(he)j(F)m(akult\177) -19 b(at,)14 b(Univ)o(ersit\177)-19 b(at)14 b(Bielefeld,)h(1992.)196 2021 y([Rus87])21 b(M.)12 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b(On)17 b(the)h(Ch)o(urc)o (h-Rosser)g(Prop)q(ert)o(y)h(for)e(the)g(Direct)i(Sum)f(of)f(T)m(erm)341 2432 y(Rewriting)e(Systems.)i Ff(Journal)12 b(of)i(the)e(A)o(CM)i Fv(34)p Ff(\(1\))p Fw(,)e(pages)i(128{143,)g(1987.)p eop %%Page: 16 16 bop 340 194 a Fw(This)14 b(article)g(w)o(as)f(pro)q(cessed)i(using)f(the)f(L) 968 189 y Fa(a)986 194 y Fw(T)1008 205 y(E)1029 194 y(X)f(macro)i(pac)o(k)n (age)g(with)f(LLNCS)g(st)o(yle)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF