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b(:)g(:)g(:)e(;)i(t)1293 2038 y Fk(n)1315 2032 y Fs(])14 b(is)g(the)g(result)h(of)e(replacing) 340 2082 y(from)e(left)i(to)f(righ)o(t)g(the)h(o)q(ccurrences)j(of)c Fj(2)h Fs(with)f Fm(t)1147 2088 y Fl(1)1165 2082 y Fm(;)7 b(:)g(:)g(:)e(;)i(t)1273 2088 y Fk(n)1295 2082 y Fs(.)12 b(A)h(con)o(text)g(con)o(taining)f(pre-)340 2132 y(cisely)g(one)g(o)q (ccurrence)j(of)c Fj(2)h Fs(is)f(denoted)i(b)o(y)f Fm(C)s Fs([)f(].)g(A)h(term)f Fm(t)h Fs(is)f(a)h Fo(subterm)f Fs(of)g(a)h(term)f Fm(s)h Fs(if)340 2182 y(there)h(exists)f(a)f(con)o (text)h Fm(C)s Fs([)g(])f(suc)o(h)h(that)f Fm(s)h Fs(=)g Fm(C)s Fs([)p Fm(t)p Fs(].)d(A)j(subterm)f Fm(t)g Fs(of)g Fm(s)h Fs(is)f Fo(pr)n(op)n(er)p Fs(,)f(denoted)340 2232 y(b)o(y)17 b Fm(s)p 444 2234 3 25 v 20 w(>)11 b(t)p Fs(,)16 b(if)g Fm(s)h Fn(6)p Fs(=)f Fm(t)p Fs(.)g(By)h(abuse)g(of)f(notation)g (w)o(e)h(write)f Fn(T)11 b Fs(\()p Fn(F)t Fm(;)c Fn(V)s Fs(\))17 b(for)f Fn(T)10 b Fs(\()p Fn(F)15 b([)c(f)p Fj(2)p Fn(g)p Fm(;)c Fn(V)s Fs(\),)340 2282 y(in)o(terpreting)15 b Fj(2)f Fs(as)h(a)f(sp)q(ecial)g(constan)o(t)h(whic)o(h)f(is)g(alw)o (a)o(ys)f(a)o(v)n(ailable)g(but)h(used)h(only)f(for)340 2331 y(the)h(aforemen)o(tioned)e(purp)q(ose.)403 2382 y(Let)20 b Fn(!)f Fs(b)q(e)h(a)f(binary)g(relation)g(on)g(terms,)g (i.e.,)f Fn(!)12 b(\022)21 b(T)10 b Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))14 b Fn(\002)f(T)d Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))q(.)19 b(The)340 2432 y(re\015exiv)o(e)h(transitiv)o(e)e (closure)i(of)e Fn(!)g Fs(is)h(denoted)g(b)o(y)g Fn(!)1266 2417 y Fi(\003)1284 2432 y Fs(.)f(If)h Fm(s)h Fn(!)1442 2417 y Fi(\003)1480 2432 y Fm(t)p Fs(,)e(w)o(e)h(sa)o(y)f(that)h Fm(s)p eop %%Page: 3 3 3 2 bop 183 194 a Fo(r)n(e)n(duc)n(es)19 b Fs(to)h Fm(t)f Fs(and)h(w)o(e)g(call)f Fm(t)g Fs(a)h Fo(r)n(e)n(duct)f Fs(of)g Fm(s)p Fs(.)h(W)m(e)f(write)h Fm(s)h Fn( )g Fm(t)f Fs(if)f Fm(t)i Fn(!)f Fm(s)p Fs(;)g(lik)o(ewise)183 244 y(for)15 b Fm(s)288 228 y Fi(\003)305 244 y Fn( )e Fm(t)p Fs(.)i(The)g(transitiv)o(e)g(closure)h(of)f Fn(!)g 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941 y Fs(with)12 b Fm(t)1093 947 y Fl(1)1123 941 y Fn( )f Fm(s)h Fn(!)f Fm(t)1275 947 y Fl(2)1307 941 y Fs(w)o(e)h(ha)o(v)o(e)h Fm(t)1476 947 y Fl(1)1506 941 y Fn(#)e Fm(t)1553 947 y Fl(2)1572 941 y Fs(.)h(If)183 991 y Fn(!)h Fs(is)g(con\015uen)o(t)h(and)g(terminating,)d(it)i(is)h (called)f Fo(c)n(omplete)g Fs(or)h Fo(c)n(onver)n(gent)p Fs(.)f(The)h(famous)183 1041 y(Newman's)8 b(Lemma)g(states)j(that)f (termination)e(and)i(lo)q(cal)f(con\015uence)j(imply)7 b(con\015uence.)183 1090 y(If)i Fn(!)h Fs(is)g(con\015uen)o(t)h(and)f (normalizing,)d(then)k(it)f(is)g(called)g Fo(semi-c)n(omplete)p Fs(.)f(Sometimes)f(this)183 1140 y(prop)q(ert)o(y)15 b(is)f(called)f Fo(unique)j(normalization)e Fs(b)q(ecause)i(it)e(is)g (equiv)n(alen)o(t)f(to)h(the)h(prop)q(ert)o(y)183 1190 y(that)f(ev)o(ery)g(term)g(has)g(a)f(unique)h(normal)e(form.)245 1263 y(A)19 b Fo(term)g(r)n(ewriting)e(system)i Fs(\(TRS\))f(is)h(a)g (pair)f(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))19 b(consisting)f(of)h(a)f (signature)183 1313 y Fn(F)h Fs(and)d(a)f(set)i Fn(R)d(\032)h(T)10 b Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))k Fn(\002)g(T)f Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))16 b(of)f Fo(r)n(ewrite)g(rules)g Fs(or)h Fo(r)n(e)n(duction)g(rules)p Fs(.)f(Ev)o(ery)183 1363 y(rewrite)k(rule)f(\()p Fm(l)q(;)7 b(r)q Fs(\))18 b(m)o(ust)f(satisfy)h(the)h(follo)o(wing)c(t)o(w)o(o)j(constrain)o(ts:) g(\(i\))g(the)h(left-hand)183 1412 y(side)14 b Fm(l)h Fs(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)i(righ)o(t-hand)e(side)h Fm(r)g Fs(also)183 1462 y(o)q(ccur)i(in)g Fm(l)q Fs(.)f(Rewrite)h(rules)g(\()p Fm(l)q(;)7 b(r)q Fs(\))16 b(will)e(b)q(e)i(denoted)h(b)o(y)e Fm(l)h Fn(!)e Fm(r)q Fs(.)h(An)h(instance)g(of)g(a)f(left-)183 1512 y(hand)h(side)h(of)e(a)i(rewrite)g(rule)g(is)f(a)g Fo(r)n(e)n(dex)g Fs(\(reducible)h(expression\).)h(The)f(rewrite)g (rules)183 1562 y(of)d(a)g(TRS)g(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))14 b(de\014ne)i(a)e Fo(r)n(ewrite)g(r)n(elation)g Fn(!)989 1568 y Fi(R)1033 1562 y Fs(on)h Fn(T)10 b Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))15 b(as)g(follo)o(ws:)d Fm(s)h Fn(!)1527 1568 y Fi(R)1570 1562 y Fm(t)h Fs(if)183 1612 y(there)h(exists)h(a)e(rewrite)h(rule)g Fm(l)f Fn(!)e Fm(r)k Fs(in)e Fn(R)p Fs(,)g(a)g(substitution)h Fm(\033)g Fs(and)g(a)f(con)o(text)h Fm(C)s Fs([)d(])i(suc)o(h)183 1661 y(that)h Fm(s)e Fs(=)g Fm(C)s Fs([)p Fm(l)q(\033)q Fs(])h(and)h Fm(t)e Fs(=)g Fm(C)s Fs([)p Fm(r)q(\033)q Fs(].)g(W)m(e)i(sa)o(y)f(that)h Fm(s)h Fs(rewrites)g(to)e Fm(t)h Fs(b)o(y)g Fo(c)n(ontr)n(acting)f Fs(redex)183 1711 y Fm(l)q(\033)q Fs(.)i(W)m(e)f(call)g Fm(s)h Fn(!)477 1717 y Fi(R)522 1711 y Fm(t)g Fs(a)g Fo(r)n(ewrite)f(step)h Fs(or)g Fo(r)n(e)n(duction)h(step)p Fs(.)e(A)h(TRS)g(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))16 b(has)g(one)g(of)183 1761 y(the)f(ab)q(o)o(v)o(e)f(prop)q(erties)i(\(e.g.)e(termination\))f (if)h(its)h(rewrite)g(relation)g(has)f(the)i(resp)q(ectiv)o(e)183 1811 y(prop)q(ert)o(y)m(.)g(Let)h(\()p Fn(F)t Fm(;)7 b Fn(R)o Fs(\))17 b(b)q(e)g(an)f(arbitrary)g(TRS.)f(A)i(function)f(sym) o(b)q(ol)e Fm(f)20 b Fn(2)15 b(F)21 b Fs(is)16 b(called)183 1861 y(a)i Fo(de\014ne)n(d)i(symb)n(ol)e Fs(if)f(there)i(is)g(a)e (rewrite)j(rule)e Fm(l)i Fn(!)e Fm(r)i Fn(2)e(R)h Fs(suc)o(h)g(that)f Fm(f)23 b Fs(=)c Fm(r)q(oot)p Fs(\()p Fm(l)q Fs(\).)183 1910 y(F)m(unction)11 b(sym)o(b)q(ols)g(from)f Fn(F)17 b Fs(whic)o(h)12 b(are)g(not)g(de\014ned)h(sym)o(b)q(ols)e(are)i (called)e Fo(c)n(onstructors)p Fs(.)183 1960 y(The)j(set)g(of)f(normal) f(forms)g(of)h(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))14 b(will)e(also)h(b)q (e)h(denoted)h(b)o(y)e Fm(N)5 b(F)h Fs(\()p Fn(F)t Fm(;)h Fn(R)p Fs(\).)12 b(W)m(e)h(often)183 2010 y(simply)f(write)j Fn(R)f Fs(instead)h(of)f(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))14 b(if)g(there)i(is)e(no)g(am)o(biguit)o(y)e(ab)q(out)i(the)h (underlying)183 2060 y(signature)e Fn(F)t Fs(.)f(A)g(rewrite)i(rule)f Fm(l)g Fn(!)e Fm(r)i Fs(of)f(a)h(TRS)f Fn(R)g Fs(is)h Fo(c)n(ol)r(lapsing)g Fs(if)e Fm(r)j Fs(is)e(a)h(v)n(ariable,)e(and)183 2110 y Fo(duplic)n(ating)j Fs(if)g Fm(r)h Fs(con)o(tains)g(more)e(o)q (ccurrences)18 b(of)c(some)f(v)n(ariable)h(than)g Fm(l)q Fs(.)g(A)h(TRS)f Fn(R)h Fs(is)183 2160 y Fo(non-duplic)n(ating)h Fs(\(non-collapsing,)d(resp)q(ectiv)o(ely\))k(if)d(it)h(do)q(es)h(not)f (con)o(tain)g(duplicating)183 2209 y(\(collapsing,)d(resp)q(ectiv)o (ely\))j(rewrite)g(rules.)245 2282 y(In)k(a)f Fo(join)h(c)n(onditional) h(term)f(r)n(ewriting)e(system)i Fs(\(CTRS)f(for)g(short\))i(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\),)18 b(the)183 2332 y(rewrite)13 b(rules)g(of)f Fn(R)g Fs(ha)o(v)o(e)g(the)h(form)d Fm(l)j Fn(!)e Fm(r)i Fn(\()e Fm(s)955 2338 y Fl(1)985 2332 y Fn(#)h Fm(t)1033 2338 y Fl(1)1051 2332 y Fm(;)7 b(:)g(:)g(:)e(;)i(s) 1163 2338 y Fk(n)1197 2332 y Fn(#)k Fm(t)1244 2338 y Fk(n)1279 2332 y Fs(with)h Fm(l)q(;)7 b(r)o(;)g(s)1460 2338 y Fl(1)1478 2332 y Fm(;)g(:)g(:)g(:)e(;)i(s)1590 2338 y Fk(n)1612 2332 y Fs(,)183 2382 y Fm(t)198 2388 y Fl(1)216 2382 y Fm(;)g(:)g(:)g(:)e(;)i(t)324 2388 y Fk(n)358 2382 y Fn(2)k(T)f Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))q(.)k Fm(s)586 2388 y Fl(1)618 2382 y Fn(#)h Fm(t)666 2388 y Fl(1)685 2382 y Fm(;)7 b(:)g(:)g(:)t(;)g(s)796 2388 y Fk(n)831 2382 y Fn(#)13 b Fm(t)880 2388 y Fk(n)914 2382 y Fs(are)g(the)g Fo(c)n(onditions)g Fs(of)f(the)h(rewrite)g(rule.) f(If)183 2432 y(a)e(rewrite)i(rule)g(has)f(no)f(conditions,)h(w)o(e)g (write)g Fm(l)i Fn(!)e Fm(r)q Fs(.)f(W)m(e)h(imp)q(ose)f(the)h(same)f (restrictions)p eop %%Page: 4 4 4 3 bop 340 194 a Fs(on)14 b(conditional)e(rewrite)i(rules)g(as)g(on)f (unconditional)g(rewrite)h(rules.)g(That)f(is,)g(w)o(e)h(allo)o(w)340 244 y Fo(extr)n(a)f(variables)d Fs(in)h(the)h(conditions)f(but)g(not)g (on)g(righ)o(t-hand)g(sides)h(of)e(rewrite)i(rules.)g(The)340 293 y(rewrite)18 b(relation)e(asso)q(ciated)i(with)f(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))16 b(is)h(de\014ned)h(b)o(y:)e Fm(s)h Fn(!)1415 299 y Fi(R)1461 293 y Fm(t)g Fs(if)f(there)i(exists)g (a)340 343 y(rewrite)f(rule)f Fm(l)d Fn(!)e Fm(r)h Fn(\()f Fm(s)749 349 y Fl(1)780 343 y Fn(#)g Fm(t)827 349 y Fl(1)846 343 y Fm(;)c(:)g(:)g(:)t(;)g(s)957 349 y Fk(n)991 343 y Fn(#)12 b Fm(t)1039 349 y Fk(n)1077 343 y Fs(in)j Fn(R)p Fs(,)g(a)h(substitution)f Fm(\033)h Fs(:)e Fn(V)k(!)c(T)c Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))q(,)340 393 y(and)17 b(a)g(con)o(text)g Fm(C)s Fs([)12 b(])k(suc)o(h)i(that)f Fm(s)g Fs(=)g Fm(C)s Fs([)p Fm(l)q(\033)q Fs(])p Fm(;)7 b(t)15 b Fs(=)i Fm(C)s Fs([)p Fm(r)q(\033)q Fs(],)e(and)i Fm(s)1397 399 y Fk(j)1414 393 y Fm(\033)d Fn(#)1473 403 y Fi(R)1516 393 y Fm(t)1531 399 y Fk(j)1548 393 y Fm(\033)k Fs(for)f(all)f Fm(j)j Fn(2)340 443 y(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)e(;)i(n)p Fn(g)p Fs(.)12 b(F)m(or)i(ev)o(ery)g(CTRS)g Fn(R)p Fs(,)f(w)o(e)h(inductiv)o(ely)g(de\014ne)h(TRSs)e Fn(R)1461 449 y Fk(i)1474 443 y Fs(,)h Fm(i)e Fn(2)f Fs(I)-7 b(N,)13 b(b)o(y:)580 525 y Fn(R)615 531 y Fl(0)646 525 y Fs(=)e Fn(f)p Fm(l)i Fn(!)e Fm(r)i Fn(j)f Fm(l)h Fn(!)e Fm(r)h Fn(2)g(Rg)543 575 y(R)578 581 y Fk(i)p Fl(+1)646 575 y Fs(=)f Fn(f)p Fm(l)q(\033)i Fn(!)e Fm(r)q(\033)j Fn(j)e Fm(l)g Fn(!)f Fm(r)i Fn(\()e Fm(s)1076 581 y Fl(1)1106 575 y Fn(#)h Fm(t)1154 581 y Fl(1)1172 575 y Fm(;)7 b(:)g(:)g(:)e(;)i (s)1284 581 y Fk(n)1318 575 y Fn(#)k Fm(t)1365 581 y Fk(n)1400 575 y Fn(2)g(R)j Fs(and)953 624 y Fm(s)972 630 y Fk(j)990 624 y Fm(\033)f Fn(#)1048 635 y Fi(R)1077 639 y Fh(i)1104 624 y Fm(t)1119 630 y Fk(j)1137 624 y Fm(\033)i Fs(for)e(all)g Fm(j)h Fn(2)d(f)p Fs(1)p Fm(;)c(:)g(:)g(:)t(;) g(n)p Fn(gg)p Fm(:)340 705 y Fs(Note)15 b(that)f Fn(R)566 711 y Fk(i)592 705 y Fn(\022)e(R)671 711 y Fk(i)p Fl(+1)741 705 y Fs(for)i(all)e Fm(i)h Fn(2)e Fs(I)-7 b(N.)14 b(F)m(urthermore,)f Fm(s)g Fn(!)1319 711 y Fi(R)1360 705 y Fm(t)h Fs(if)g(and)g(only)f(if)g Fm(s)f Fn(!)1711 711 y Fi(R)1740 715 y Fh(i)1766 705 y Fm(t)340 755 y Fs(for)h(some)f Fm(i)f Fn(2)h Fs(I)-7 b(N.)12 b(The)h Fo(depth)h Fs(of)e(a)g(rewrite)i(step)g Fm(s)e Fn(!)1207 761 y Fi(R)1248 755 y Fm(t)h Fs(is)g(de\014ned)g(to)g (b)q(e)g(the)h(minim)o(al)340 805 y Fm(i)g Fs(with)e Fm(s)g Fn(!)534 811 y Fi(R)563 815 y Fh(i)589 805 y Fm(t)p Fs(.)h(Depths)h(of)e(reduction)i(sequences)h Fm(s)d Fn(!)1261 790 y Fi(\003)1261 816 y(R)1303 805 y Fm(t)p Fs(,)h(con)o(v)o(ersions)g Fm(s)f Fn($)1633 790 y Fi(\003)1633 816 y(R)1675 805 y Fm(t)p Fs(,)g(and)340 854 y(v)n(alleys)j Fm(s)e Fn(#)529 865 y Fi(R)572 854 y Fm(t)j Fs(are)g(de\014ned)g(analogously)m(.)d(All) i(notions)h(de\014ned)g(previously)g(for)f(TRSs)340 904 y(extend)g(to)f(CTRSs.)403 954 y(A)c Fo(p)n(artial)g(or)n(dering)g Fs(\()p Fm(A;)d(>)p Fs(\))j(is)g(a)g(pair)g(consisting)g(of)f(a)h(set)h Fm(A)f Fs(and)g(a)g(binary)g(irre\015exiv)o(e)340 1004 y(and)k(transitiv)o(e)f(relation)g Fm(>)g Fs(on)g Fm(A)p Fs(.)g(A)g(partial)g(ordering)g(is)g(called)g Fo(wel)r(l-founde)n(d)g Fs(if)g(there)340 1054 y(are)i(no)e(in\014nite)h(sequences)i Fm(a)818 1060 y Fl(1)848 1054 y Fm(>)c(a)914 1060 y Fl(2)944 1054 y Fm(>)g(a)1010 1060 y Fl(3)1040 1054 y Fm(>)g(:)7 b(:)g(:)12 b Fs(of)h(elemen)o(ts)h(from)e Fm(A)p Fs(.)h(A)h Fo(multiset)f Fs(is)h(a)340 1104 y(collection)g(in)g(whic)o(h)h(elemen) o(ts)f(are)h(allo)o(w)o(ed)e(to)h(o)q(ccur)i(more)d(than)h(once.)h(If)f Fm(A)g Fs(is)h(a)f(set,)340 1153 y(then)h(the)f(set)h(of)e(all)f (\014nite)i(m)o(ultisets)f(o)o(v)o(er)g Fm(A)h Fs(is)g(denoted)g(b)o(y) g Fn(M)p Fs(\()p Fm(A)p Fs(\).)f(The)h Fo(multiset)g(ex-)340 1203 y(tension)f Fs(of)e(a)g(partial)g(ordering)h(\()p Fm(A;)7 b(>)p Fs(\))12 b(is)f(the)i(partial)d(ordering)i(\()p Fn(M)p Fs(\()p Fm(A)p Fs(\))p Fm(;)7 b(>)1562 1188 y Fk(mul)1624 1203 y Fs(\))12 b(de\014ned)340 1253 y(as)h(follo)o(ws:)e Fm(M)577 1259 y Fl(1)608 1253 y Fm(>)640 1238 y Fk(mul)713 1253 y Fm(M)753 1259 y Fl(2)785 1253 y Fs(if)h Fm(M)862 1259 y Fl(2)893 1253 y Fs(=)f(\()p Fm(M)992 1259 y Fl(1)1019 1253 y Fn(n)c Fm(X)s Fs(\))h Fn([)f Fm(Y)22 b Fs(for)12 b(some)g(m)o(ultisets)g Fm(X)q(;)7 b(Y)21 b Fn(2)11 b(M)p Fs(\()p Fm(A)p Fs(\))340 1303 y(that)16 b(satisfy)f(\(i\))h Fn(;)e(6)p Fs(=)h Fm(X)j Fn(\022)d Fm(M)844 1309 y Fl(1)878 1303 y Fs(and)h(\(ii\))f(for)g(all)g Fm(y)h Fn(2)e Fm(Y)25 b Fs(there)17 b(exists)f(an)g Fm(x)e Fn(2)g Fm(X)20 b Fs(suc)o(h)340 1353 y(that)15 b Fm(x)e(>)h(y)q Fs(.)h(Dersho)o(witz)h (and)f(Manna)f([DM79)o(])g(pro)o(v)o(ed)h(that)g(the)h(m)o(ultiset)e (extension)340 1402 y(of)g(a)f(w)o(ell-founded)g(partial)g(ordering)h (is)g(a)g(w)o(ell-founded)f(partial)g(ordering.)340 1534 y Fp(3)56 b(Basic)19 b(Notions)f(of)h(Constructor-Sharing)f(CTRSs)340 1631 y Fg(De\014nition)12 b(3.1)21 b Fs(Tw)o(o)k(CTRSs)g(\()p Fn(F)951 1637 y Fl(1)969 1631 y Fm(;)7 b Fn(R)1023 1637 y Fl(1)1042 1631 y Fs(\))25 b(and)g(\()p Fn(F)1221 1637 y Fl(2)1240 1631 y Fm(;)7 b Fn(R)1294 1637 y Fl(2)1312 1631 y Fs(\))25 b(are)h(called)f Fo(c)n(onstructor-)340 1681 y(sharing)14 b Fs(if)f(they)i(share)f(at)g(most)f(constructors,)i (more)e(precisely)i(if)474 1763 y Fn(F)504 1769 y Fl(1)532 1763 y Fn(\\)9 b(F)599 1769 y Fl(2)626 1763 y Fn(\\)g(f)p Fm(r)q(oot)p Fs(\()p Fm(l)q Fs(\))k Fn(j)f Fm(l)h Fn(!)e Fm(r)h Fn(\()f Fm(s)1022 1769 y Fl(1)1053 1763 y Fn(#)g Fm(t)1100 1769 y Fl(1)1119 1763 y Fm(;)c(:)g(:)g(:)e(;)i(s)1231 1769 y Fk(n)1265 1763 y Fn(#)k Fm(t)1312 1769 y Fk(n)1346 1763 y Fn(2)g(R)1420 1769 y Fl(1)1448 1763 y Fn([)e(R)1520 1769 y Fl(2)1539 1763 y Fn(g)i Fs(=)h Fn(;)p Fm(:)340 1846 y Fs(In)18 b(this)g(case,)h(their)f(union)g(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))18 b(=)g(\()p Fn(F)1042 1852 y Fl(1)1073 1846 y Fn([)12 b(F)1143 1852 y Fl(2)1161 1846 y Fm(;)7 b Fn(R)1215 1852 y Fl(1)1245 1846 y Fn([)12 b(R)1320 1852 y Fl(2)1339 1846 y Fs(\))18 b(is)g(called)f(the)i Fo(c)n(ombine)n(d)340 1896 y(CTRS)14 b(of)g Fs(\()p Fn(F)561 1902 y Fl(1)579 1896 y Fm(;)7 b Fn(R)633 1902 y Fl(1)651 1896 y Fs(\))14 b Fo(and)h Fs(\()p Fn(F)807 1902 y Fl(2)825 1896 y Fm(;)7 b Fn(R)879 1902 y Fl(2)898 1896 y Fs(\))13 b Fo(with)g(shar)n(e)n(d)h(c)n(onstructors)f Fn(C)r Fs(,)f(where)i Fn(C)g Fs(=)e Fn(F)1658 1902 y Fl(1)1683 1896 y Fn(\\)7 b(F)1751 1902 y Fl(2)1770 1896 y Fs(.)340 1945 y(F)m(urthermore,)j(w)o (e)h(de\014ne)h Fn(D)796 1951 y Fl(1)826 1945 y Fs(=)g Fn(F)900 1951 y Fl(1)921 1945 y Fn(n)s(C)r Fs(,)e Fn(D)1023 1951 y Fl(2)1053 1945 y Fs(=)i Fn(F)1127 1951 y Fl(2)1148 1945 y Fn(n)s(C)r Fs(,)e(and)h Fn(D)h Fs(=)g Fn(D)1416 1951 y Fl(1)1438 1945 y Fn([)s(D)1501 1951 y Fl(2)1519 1945 y Fs(.)e(A)h(prop)q(ert)o(y)g Fn(P)340 1995 y Fs(of)i(CTRSs)h(is)g (called)f Fo(mo)n(dular)h(for)h(c)n(onstructor-sharing)f(CTRSs)f Fs(if)g(for)h(all)e(CTRSs)i Fn(R)1763 2001 y Fl(1)340 2045 y Fs(and)f Fn(R)455 2051 y Fl(2)486 2045 y Fs(whic)o(h)g(share)h (at)e(most)g(constructors,)i(their)f(union)f Fn(R)1355 2051 y Fl(1)1381 2045 y Fn([)7 b(R)1451 2051 y Fl(2)1482 2045 y Fs(has)13 b(the)g(prop)q(ert)o(y)340 2095 y Fn(P)18 b Fs(if)13 b(and)g(only)g(if)g(b)q(oth)h Fn(R)767 2101 y Fl(1)800 2095 y Fs(and)g Fn(R)916 2101 y Fl(2)948 2095 y Fs(ha)o(v)o(e)g(the)h(prop)q(ert)o(y)f Fn(P)s Fs(.)403 2183 y(If)g(\()p Fn(F)495 2189 y Fl(1)514 2183 y Fm(;)7 b Fn(R)568 2189 y Fl(1)586 2183 y Fs(\))16 b(and)e(\()p Fn(F)750 2189 y Fl(2)768 2183 y Fm(;)7 b Fn(R)822 2189 y Fl(2)840 2183 y Fs(\))16 b(are)f(constructor-sharing)i(CTRSs,)d(then) i(\()p Fn(F)t Fm(;)7 b Fn(R)1665 2189 y Fl(1)1683 2183 y Fs(\))15 b(and)340 2232 y(\()p Fn(F)t Fm(;)7 b Fn(R)444 2238 y Fl(2)463 2232 y Fs(\))13 b(are)h(also)f(CTRSs,)g(where)i Fn(F)g Fs(=)d Fn(F)1042 2238 y Fl(1)1069 2232 y Fn([)c(F)1139 2238 y Fl(2)1158 2232 y Fs(.)13 b(In)g(order)h(to)g(a)o(v)o(oid)e (misunderstand-)340 2282 y(ings,)f(w)o(e)h(write)g Fn(\))640 2288 y Fi(R)669 2292 y Fh(i)695 2282 y Fs(for)g(the)g(rewrite)h (relation)e(asso)q(ciated)h(with)g(\()p Fn(F)1447 2288 y Fk(i)1461 2282 y Fm(;)7 b Fn(R)1515 2288 y Fk(i)1528 2282 y Fs(\))12 b(and)g Fn(!)1677 2288 y Fi(R)1706 2292 y Fh(i)1732 2282 y Fs(for)340 2332 y(the)k(rewrite)f(relation)g(asso)q (ciated)g(with)f(\()p Fn(F)5 b Fm(;)i Fn(R)1105 2338 y Fk(i)1119 2332 y Fs(\),)14 b(where)i Fm(i)d Fn(2)g(f)p Fs(1)p Fm(;)7 b Fs(2)p Fn(g)p Fs(.)13 b(If)h Fm(s;)7 b(t)13 b Fn(2)f(T)f Fs(\()p Fn(F)1704 2338 y Fk(i)1718 2332 y Fm(;)c Fn(V)s Fs(\))340 2382 y(and)20 b Fm(s)h Fn(\))509 2388 y Fi(R)538 2392 y Fh(i)574 2382 y Fm(t)p Fs(,)e(then)h(w)o(e)g(clearly)g(ha)o(v)o(e)f Fm(s)j Fn(!)1110 2388 y Fi(R)1139 2392 y Fh(i)1174 2382 y Fm(t)p Fs(.)d(A)h(priori,)f (it)g(is)g(not)h(clear)g(at)g(all)340 2432 y(whether)13 b(the)g(con)o(v)o(erse)g(is)e(also)g(true.)h(F)m(or,)f(if)g Fm(s)h Fn(!)1140 2438 y Fi(R)1169 2442 y Fh(i)1194 2432 y Fm(t)p Fs(,)f(then)i(there)g(exists)f(a)f(rewrite)i(rule)p eop %%Page: 5 5 5 4 bop 183 194 a Fm(l)12 b Fn(!)f Fm(r)i Fn(\()e Fm(s)364 200 y Fl(1)394 194 y Fn(#)h Fm(t)442 200 y Fl(1)460 194 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)572 200 y Fk(n)606 194 y Fn(#)12 b Fm(t)654 200 y Fk(n)686 194 y Fs(in)e Fn(R)766 200 y Fk(i)780 194 y Fs(,)g(a)g(substitution)g Fm(\033)j Fs(:)e Fn(V)k(!)c(T)g Fs(\()p Fn(F)t Fm(;)c Fn(V)s Fs(\),)j(and)g(a)g (con)o(text)183 244 y Fm(C)s Fs([)h(])j(suc)o(h)i(that)e Fm(s)g Fs(=)f Fm(C)s Fs([)p Fm(l)q(\033)q Fs(])p Fm(;)7 b(t)k 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b Fn(f)p Fm(a)g Fn(!)f Fk(C)r Fn(g)p Fs(.)c(W)m(e)f(ha)o(v)o(e)h Fm(F)6 b Fs(\()p Fm(a;)h Fk(C)q Fs(\))14 b Fn(!)925 1969 y Fi(R)969 1963 y Fm(G)p Fs(\()p Fm(a)p Fs(\))24 b(but)h(neither)g Fm(F)6 b Fs(\()p Fm(a;)h Fk(C)q Fs(\))14 b Fn(!)1504 1969 y Fl(1)1537 1963 y Fm(G)p Fs(\()p Fm(a)p Fs(\))183 2013 y(nor)j Fm(F)6 b Fs(\()p Fm(a;)h Fk(C)q Fs(\))14 b Fn(!)446 2019 y Fl(2)479 2013 y Fm(G)p Fs(\()p Fm(a)p Fs(\).)i(Ho)o(w)o(ev)o(er,)h(the)h(terms)e(are)i(joinable)e(with)h (resp)q(ect)i(to)31 b Fn(!)1553 2019 y Fl(1)p Fk(;)p Fl(2)1612 2013 y Fs(:)183 2063 y Fm(F)6 b Fs(\()p Fm(a;)h Fk(C)p Fs(\))14 b Fn(!)369 2069 y Fl(2)401 2063 y Fm(F)6 b Fs(\()p Fk(C)r Fm(;)h Fk(C)q Fs(\))14 b Fn(!)592 2069 y Fl(1)624 2063 y Fm(G)p Fs(\()p Fk(C)r Fs(\))d Fn( )768 2069 y Fl(2)798 2063 y Fm(G)p Fs(\()p Fm(a)p Fs(\))p Fm(:)183 2166 y Fg(Lemma)k(4.3)21 b Fs(If)13 b Fm(s)h Fn(!)550 2172 y Fl(1)p Fk(;)p Fl(2)609 2166 y Fm(t)p Fs(,)f(then)i Fm(s)f Fn(!)819 2172 y Fi(R)863 2166 y Fm(t)p Fs(.)183 2218 y Fg(Pro)q(of:)f Fs(T)m(rivial.)e Fj(2)183 2321 y Fg(Lemma)k(4.4)21 b Fs(Let)15 b Fm(s)g Fs(b)q(e)h(a)f(blac)o(k)f(term)h(and)f(let)h Fm(\033)h Fs(b)q(e)g(a)f(top)g(white)g(substitution)g(suc)o(h)183 2371 y(that)f Fm(s\033)f Fn(!)371 2356 y Fk(o)371 2381 y Fl(1)400 2371 y Fm(t)p Fs(.)h(Then)g(there)h(is)f(a)g(blac)o(k)f (term)g Fm(u)h Fs(suc)o(h)h(that)f Fm(t)d Fs(=)h Fm(u\033)q Fs(.)183 2424 y Fg(Pro)q(of:)h Fs(Straigh)o(tforw)o(ard.)f Fj(2)p eop %%Page: 8 8 8 7 bop 340 194 a Fg(Lemma)16 b(4.5)k Fs(Let)15 b Fm(s;)7 b(t)13 b Fs(b)q(e)i(blac)o(k)e(terms)h(and)f(let)h Fm(\033)h Fs(b)q(e)g(a)e(top)h(white)g(substitution)g(with)340 244 y Fm(s\033)g Fn(!)439 228 y Fk(o)439 254 y Fl(1)468 244 y Fm(t\033)q Fs(.)g(If)f Fm(\034)19 b Fs(is)14 b(a)f(substitution)h (with)g Fm(\033)f Fn(/)f Fm(\034)5 b Fs(,)13 b(then)h Fm(s\034)j Fn(!)1333 228 y Fk(o)1333 254 y Fl(1)1363 244 y Fm(t\034)5 b Fs(.)340 293 y Fg(Pro)q(of:)20 b Fs(The)h(lemma)c (is)k(pro)o(v)o(ed)f(b)o(y)g(induction)h(on)f(the)h(depth)g(of)f Fm(s\033)k Fn(!)1597 278 y Fk(o)1597 304 y Fl(1)1638 293 y Fm(t\033)q Fs(.)c(The)340 343 y(case)e(of)e(zero)i(depth)f(is)g (straigh)o(tforw)o(ard.)f(Let)h(the)g(depth)h(of)e Fm(s\033)i Fn(!)1469 328 y Fk(o)1469 353 y Fl(1)1504 343 y Fm(t\033)f Fs(equal)g Fm(d)11 b Fs(+)g(1,)340 393 y Fm(d)16 b Fn(\025)f Fs(0.)h(There)h(is)g(a)f(con)o(text)h Fm(C)s Fs([)11 b(],)16 b(a)g(substitution)g Fm(\032)g Fs(:)f Fn(V)20 b(!)15 b(T)10 b Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\),)16 b(and)g(a)g(rewrite)340 443 y(rule)h Fm(l)h Fn(!)d Fm(r)i Fn(\()f Fm(s)626 449 y Fl(1)657 443 y Fn(#)c Fm(t)705 449 y Fl(1)724 443 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)836 449 y Fk(n)871 443 y Fn(#)12 b Fm(t)919 449 y Fk(n)958 443 y Fs(in)k Fn(R)1044 449 y Fl(1)1080 443 y Fs(suc)o(h)h(that)g Fm(s\033)g Fs(=)g Fm(C)s Fs([)p Fm(l)q(\032)p Fs(],)e Fm(t\033)i Fs(=)g Fm(C)s Fs([)p Fm(r)q(\032)p Fs(])e(and)340 493 y Fm(s)359 499 y Fk(j)377 493 y Fm(\032)e Fn(#)432 474 y Fk(o)432 503 y Fl(1)463 493 y Fm(t)478 499 y Fk(j)496 493 y Fm(\032)d Fs(is)g(of)g(depth)h Fn(\024)h Fm(d)d Fs(for)h(ev)o(ery)h Fm(j)j Fn(2)d(f)p Fs(1)p Fm(;)c(:)g(:)g(:)e(;)i(n)p Fn(g)p Fs(.)i(According)h(to)g(Prop)q(osition)g(3.12,)340 542 y Fm(\032)18 b Fs(has)e(a)h(decomp)q(osition)e Fm(\032)i Fs(=)g Fm(\032)877 548 y Fl(2)907 542 y Fn(\016)11 b Fm(\032)960 548 y Fl(1)996 542 y Fs(suc)o(h)17 b(that)g Fm(\032)1206 548 y Fl(1)1242 542 y Fs(is)f(blac)o(k,)g Fm(\032)1430 548 y Fl(2)1466 542 y Fs(is)g(top)h(white,)f(and)340 592 y Fm(\032)361 598 y Fl(2)393 592 y Fn(/)d Fm(\017)p Fs(.)h(W)m(e)h(de\014ne)g(a)g(substitution)f Fm(\032)962 577 y Fi(0)989 592 y Fs(b)o(y)h Fm(\032)1069 577 y Fi(0)1081 592 y Fs(\()p Fm(x)p Fs(\))d(=)i Fm(\034)5 b Fs(\()p Fm(y)q Fs(\))15 b(for)f(ev)o(ery)i Fm(x)c Fn(2)h(D)q Fm(om)p Fs(\()p Fm(\032)1664 598 y Fl(2)1684 592 y Fs(\))h(and)340 642 y Fm(y)g Fn(2)d(D)q Fm(om)p Fs(\()p Fm(\033)q Fs(\))k(satisfying)e Fm(\032)780 648 y Fl(2)799 642 y Fs(\()p Fm(x)p Fs(\))f(=)g Fm(\033)q Fs(\()p Fm(y)q Fs(\).)i Fm(\032)1036 627 y Fi(0)1062 642 y Fs(is)g(w)o(ell-de\014ned)g(b)q(ecause)i Fm(\033)d Fn(/)f Fm(\034)5 b Fs(.)13 b(It)h(follo)o(ws)340 692 y(from)f Fm(\032)460 698 y Fl(2)492 692 y Fn(/)g Fm(\017)h Fs(and)h Fm(\017)d Fn(/)h Fm(\032)745 677 y Fi(0)772 692 y Fs(that)h Fm(\032)883 698 y Fl(2)915 692 y Fn(/)f Fm(\032)981 677 y Fi(0)993 692 y Fs(.)h(By)h(Lemma)d(4.4,)h (for)h(an)o(y)g Fm(j)h Fn(2)e(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)t(;)g(n)p Fn(g)p Fs(,)13 b(w)o(e)340 742 y(ma)o(y)f(write)358 824 y Fm(\032)379 830 y Fl(2)398 824 y Fs(\()p Fm(\032)435 830 y Fl(1)454 824 y Fs(\()p Fm(s)489 830 y Fk(j)507 824 y Fs(\)\))g(=)g Fm(\032)616 830 y Fl(2)635 824 y Fs(\()p Fm(u)675 830 y Fl(1)693 824 y Fs(\))g Fn(!)763 807 y Fk(o)763 834 y Fl(1)793 824 y Fm(:)7 b(:)g(:)j Fn(!)895 807 y Fk(o)895 834 y Fl(1)924 824 y Fm(\032)945 830 y Fl(2)964 824 y Fs(\()p Fm(u)1004 830 y Fk(k)1025 824 y Fs(\))h(=)h Fm(\032)1117 830 y Fl(2)1136 824 y Fs(\()p Fm(v)1172 830 y Fk(l)1185 824 y Fs(\))1214 807 y Fk(o)1214 834 y Fl(1)1230 824 y Fn( )f Fm(:)c(:)g(:)1351 807 y Fk(o)1351 834 y Fl(1)1367 824 y Fn( )k Fm(\032)1441 830 y Fl(2)1460 824 y Fs(\()p Fm(v)1496 830 y Fl(1)1515 824 y Fs(\))h(=)g Fm(\032)1608 830 y Fl(2)1627 824 y Fs(\()p Fm(\032)1664 830 y Fl(1)1683 824 y Fs(\()p Fm(t)1714 830 y Fk(j)1732 824 y Fs(\)\))340 907 y(for)k(some)e(blac)o(k)h(terms)h Fm(u)763 913 y Fl(1)781 907 y Fm(;)7 b(:)g(:)g(:)e(;)i(u)898 913 y Fk(k)918 907 y Fm(;)g(v)957 913 y Fl(1)975 907 y Fm(;)g(:)g(:)g(:)e(;)i(v)1088 913 y Fk(l)1100 907 y Fs(.)15 b(No)o(w)g(rep)q(eated)j(application)c(of)h(the)h(in-)340 957 y(duction)e(h)o(yp)q(othesis)h(yields)379 1039 y Fm(\032)400 1022 y Fi(0)412 1039 y Fs(\()p Fm(\032)449 1045 y Fl(1)468 1039 y Fs(\()p Fm(s)503 1045 y Fk(j)521 1039 y Fs(\)\))d(=)g Fm(\032)630 1022 y Fi(0)642 1039 y Fs(\()p Fm(u)682 1045 y Fl(1)700 1039 y Fs(\))g Fn(!)770 1022 y Fk(o)770 1049 y Fl(1)800 1039 y Fm(:)7 b(:)g(:)j Fn(!)902 1022 y Fk(o)902 1049 y Fl(1)931 1039 y Fm(\032)952 1022 y Fi(0)964 1039 y Fs(\()p Fm(u)1004 1045 y Fk(k)1025 1039 y Fs(\))h(=)h Fm(\032)1117 1022 y Fi(0)1129 1039 y Fs(\()p Fm(v)1165 1045 y Fk(l)1178 1039 y Fs(\))1207 1022 y Fk(o)1207 1049 y Fl(1)1223 1039 y Fn( )f Fm(:)c(:)g(:)1344 1022 y Fk(o)1344 1049 y Fl(1)1360 1039 y Fn( )k Fm(\032)1434 1022 y Fi(0)1447 1039 y Fs(\()p Fm(v)1483 1045 y Fl(1)1501 1039 y Fs(\))h(=)g Fm(\032)1594 1022 y Fi(0)1606 1039 y Fs(\()p Fm(\032)1643 1045 y Fl(1)1662 1039 y Fs(\()p Fm(t)1693 1045 y Fk(j)1711 1039 y Fs(\)\))340 1127 y(Th)o(us)21 b Fm(\032)473 1112 y Fi(0)486 1127 y Fs(\()p Fm(\032)523 1133 y Fl(1)542 1127 y Fs(\()p Fm(l)q Fs(\)\))i Fn(!)668 1112 y Fk(o)668 1137 y Fl(1)710 1127 y Fm(\032)731 1112 y Fi(0)743 1127 y Fs(\()p Fm(\032)780 1133 y Fl(1)799 1127 y Fs(\()p Fm(r)q Fs(\)\).)e(Let)990 1117 y(^)981 1127 y Fm(C)r Fs([)12 b(])21 b(b)q(e)g(the)h(con)o(text)f(obtained)g (from)e Fm(C)s Fs([)11 b(])21 b(b)o(y)340 1177 y(replacing)f(ev)o(ery)g (white)f(principal)g(subterm)g(whic)o(h)g(m)o(ust)g(b)q(e)h(of)e(the)i (form)e Fm(\033)q Fs(\()p Fm(x)p Fs(\))i(for)340 1227 y(some)c(v)n(ariable)g Fm(x)h Fn(2)f(D)q Fm(om)p Fs(\()p Fm(\033)q Fs(\))i(b)o(y)f(the)h(corresp)q(onding)g Fm(\034)5 b Fs(\()p Fm(x)p Fs(\).)16 b(\(This)h(is)g(a)f(sligh)o(t)h(abuse)340 1277 y(of)h(notation)f(b)q(ecause)729 1266 y(^)719 1277 y Fm(C)s Fs([)12 b(])17 b(con)o(tains)h(in)g(general)g(more)f(than)h (one)g(o)q(ccurrence)i(of)e Fj(2)p Fs(.\))340 1326 y(It)e(is)e(fairly)g (simple)g(to)h(v)o(erify)f(that)h Fm(s\034)k Fs(=)1040 1316 y(^)1031 1326 y Fm(C)s Fs([)p Fm(\032)1097 1311 y Fi(0)1108 1326 y Fs(\()p Fm(\032)1145 1332 y Fl(1)1165 1326 y Fs(\()p Fm(l)q Fs(\)\)])c(and)g Fm(t\034)j Fs(=)1441 1316 y(^)1432 1326 y Fm(C)s Fs([)p Fm(\032)1498 1311 y Fi(0)1509 1326 y Fs(\()p Fm(\032)1546 1332 y Fl(1)1565 1326 y Fs(\()p Fm(r)q Fs(\)\)].)d(Hence)340 1376 y Fm(s\034)i Fn(!)436 1361 y Fk(o)436 1387 y Fl(1)466 1376 y Fm(t\034)5 b Fs(.)13 b Fj(2)340 1464 y Fg(Lemma)j(4.6)k Fs(The)15 b(restriction)g(of)e Fn(!)965 1470 y Fl(1)997 1464 y Fs(to)h Fn(T)c Fs(\()p Fn(F)1127 1470 y Fl(1)1145 1464 y Fm(;)d Fn(V)s Fs(\))j Fn(\002)f(T)i Fs(\()p Fn(F)1339 1470 y Fl(1)1357 1464 y Fm(;)c Fn(V)s Fs(\))14 b(and)g Fn(\))1557 1470 y Fi(R)1586 1474 y Ff(1)1617 1464 y Fs(coincide.)340 1514 y Fg(Pro)q(of:)f Fs(\\)p Fn(\023)p Fs(")h(T)m(rivial.)340 1564 y(\\)p Fn(\022)p Fs(")f(Let)g Fm(s;)7 b(t)12 b Fn(2)f(T)f Fs(\()p Fn(F)683 1570 y Fl(1)702 1564 y Fm(;)d Fn(V)s Fs(\))13 b(with)f Fm(s)g Fn(!)944 1570 y Fl(1)974 1564 y Fm(t)p Fs(.)g(In)h(order)g(to)g(sho)o(w)g(that)g Fm(s)f Fn(\))1485 1570 y Fi(R)1514 1574 y Ff(1)1542 1564 y Fm(t)p Fs(,)h(w)o(e)g(pro)q(ceed)340 1614 y(b)o(y)d(induction)g(on)g(the)g (depth)h(of)f Fm(s)i Fn(!)927 1599 y Fk(o)927 1624 y Fl(1)956 1614 y Fm(t)p Fs(.)e(The)g(case)h(of)f(zero)h(depth)g(is)e (straigh)o(tforw)o(ard.)h(So)340 1663 y(supp)q(ose)k(that)e(the)g (depth)h(of)f Fm(s)g Fn(!)887 1648 y Fk(o)887 1674 y Fl(1)916 1663 y Fm(t)g Fs(equals)g Fm(d)6 b Fs(+)g(1,)11 b Fm(d)g Fn(\025)h Fs(0.)f(Then)h(there)i(exists)f(a)e(rewrite)340 1713 y(rule)17 b Fm(l)12 b Fn(!)f Fm(r)i Fn(\()e Fm(s)607 1719 y Fl(1)637 1713 y Fn(#)h Fm(t)685 1719 y Fl(1)703 1713 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)815 1719 y Fk(n)849 1713 y Fn(#)k Fm(t)896 1719 y Fk(n)935 1713 y Fs(in)k Fn(R)1020 1719 y Fl(1)1039 1713 y Fs(,)h(a)f(substitution)i Fm(\033)f Fs(:)e Fn(V)19 b(!)c(T)10 b Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\),)16 b(and)g(a)340 1763 y(con)o(text)11 b Fm(C)s Fs([)g(])e(suc)o(h)h(that)g Fm(s)i Fs(=)g Fm(C)s Fs([)p Fm(l)q(\033)q Fs(])p Fm(;)7 b(t)j Fs(=)i Fm(C)s Fs([)p Fm(r)q(\033)q Fs(],)7 b(and)j Fm(s)1211 1769 y Fk(j)1229 1763 y Fm(\033)j Fn(#)1287 1745 y Fk(o)1287 1773 y Fl(1)1318 1763 y Fm(t)1333 1769 y Fk(j)1351 1763 y Fm(\033)d Fs(with)f(depth)i Fn(\024)g Fm(d)f Fs(for)f Fm(j)14 b Fn(2)340 1813 y(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)e(;)i(n)p Fn(g)p Fs(.)j(According)h(to)g(Prop)q(osition)g(3.12,)f Fm(\033)i Fs(can)f(b)q(e)h(decomp)q(osed)g(in)o(to)e Fm(\033)1601 1819 y Fl(2)1623 1813 y Fn(\016)t Fm(\033)1672 1819 y Fl(1)1702 1813 y Fs(suc)o(h)340 1863 y(that)j Fm(\033)453 1869 y Fl(1)483 1863 y Fs(is)f(blac)o(k,)f Fm(\033)665 1869 y Fl(2)696 1863 y Fs(is)h(top)g(white,)f(and)h Fm(\033)1033 1869 y Fl(2)1063 1863 y Fn(/)g Fm(\017)p Fs(.)f(Induction)i(on)f(the)g(n)o(um)o(b)q(er)g(of)f(rewrite)340 1913 y(steps)k(in)f Fm(s)512 1919 y Fk(j)530 1913 y Fm(\033)f Fn(#)588 1894 y Fk(o)588 1923 y Fl(1)619 1913 y Fm(t)634 1919 y Fk(j)652 1913 y Fm(\033)h Fs(in)g(com)o(bination)d(with)i(Lemma) e(4.5)i(yields)g Fm(\017)p Fs(\()p Fm(\033)1457 1919 y Fl(1)1475 1913 y Fs(\()p Fm(s)1510 1919 y Fk(j)1529 1913 y Fs(\)\))f Fn(#)1594 1894 y Fk(o)1594 1923 y Fl(1)1625 1913 y Fm(\017)p Fs(\()p Fm(\033)1682 1919 y Fl(1)1700 1913 y Fs(\()p Fm(t)1731 1919 y Fk(j)1749 1913 y Fs(\)\))340 1962 y(for)j Fm(j)i Fn(2)c(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)t(;)g(n)p Fn(g)p Fs(.)14 b(Since)i(ev)o(ery)g(term)f(in)g(the)g(con)o(v)o(ersion) h Fm(\033)1358 1968 y Fl(1)1376 1962 y Fs(\()p Fm(s)1411 1968 y Fk(j)1429 1962 y Fs(\))d Fn(#)1479 1944 y Fk(o)1479 1973 y Fl(1)1510 1962 y Fm(\033)1534 1968 y Fl(1)1552 1962 y Fs(\()p Fm(t)1583 1968 y Fk(j)1601 1962 y Fs(\))i(is)g(blac)o (k,)340 2012 y(w)o(e)d(obtain)e Fm(\033)549 2018 y Fl(1)568 2012 y Fs(\()p Fm(s)603 2018 y Fk(j)621 2012 y Fs(\))h Fn(+)673 2018 y Fi(R)702 2022 y Ff(1)732 2012 y Fm(\033)756 2018 y Fl(1)774 2012 y Fs(\()p Fm(t)805 2018 y Fk(j)823 2012 y Fs(\))g(b)o(y)g(rep)q(eated)i(application)d(of)h(the)g (induction)g(h)o(yp)q(othesis.)340 2062 y(Consequen)o(tly)m(,)j(w)o(e)g (ha)o(v)o(e)f Fm(\033)786 2068 y Fl(1)804 2062 y Fs(\()p Fm(l)q Fs(\))f Fn(\))903 2068 y Fi(R)932 2072 y Ff(1)961 2062 y Fm(\033)985 2068 y Fl(1)1004 2062 y Fs(\()p Fm(r)q Fs(\).)h(No)o(w)g Fm(s)f Fn(\))1248 2068 y Fi(R)1277 2072 y Ff(1)1306 2062 y Fm(t)i Fs(follo)o(ws)e(from)g Fm(s)g Fs(=)f Fm(C)s Fs([)p Fm(l)q(\033)q Fs(])g(=)340 2112 y Fm(C)s Fs([)p Fm(l)q(\033)422 2118 y Fl(1)440 2112 y Fs(])i(and)h Fm(t)e Fs(=)g Fm(C)s Fs([)p Fm(r)q(\033)q Fs(])e(=)i Fm(C)s Fs([)p Fm(r)q(\033)862 2118 y Fl(1)879 2112 y Fs(])h(b)q(ecause)j Fm(s)e Fs(and)g Fm(t)g Fs(are)g(blac)o(k.)f Fj(2)340 2200 y Fg(Prop)q(osition)f(4.7)21 b Fs(If)27 b(\()p Fn(F)785 2206 y Fl(1)803 2200 y Fm(;)7 b Fn(R)857 2206 y Fl(1)876 2200 y Fs(\))27 b(and)f(\()p Fn(F)1063 2206 y Fl(2)1081 2200 y Fm(;)7 b Fn(R)1135 2206 y Fl(2)1153 2200 y Fs(\))27 b(are)h(semi-complete)d(constructor-)340 2250 y(sharing)20 b(conditional)e(term)g(rewriting)i(systems,)f(then)h (the)g(relation)33 b Fn(!)1561 2256 y Fl(1)p Fk(;)p Fl(2)1639 2250 y Fs(is)20 b(semi-)340 2299 y(complete.)340 2349 y Fg(Pro)q(of:)13 b Fs(W)m(e)h(de\014ne)h(t)o(w)o(o)e(unconditional)g (TRSs)h(\()p Fn(F)1178 2355 y Fl(1)1196 2349 y Fm(;)7 b Fn(S)1240 2355 y Fl(1)1259 2349 y Fs(\))14 b(and)f(\()p Fn(F)1419 2355 y Fl(2)1438 2349 y Fm(;)7 b Fn(S)1482 2355 y Fl(2)1500 2349 y Fs(\))14 b(b)o(y)493 2432 y Fn(S)518 2438 y Fk(i)544 2432 y Fs(=)e Fn(f)p Fm(u)f Fn(!)g Fm(v)j Fn(j)e Fm(u;)7 b(v)12 b Fn(2)f(T)f Fs(\()p Fn(F)948 2438 y Fk(i)962 2432 y Fm(;)d Fn(V)s Fs(\))p Fm(;)g(r)q(oot)p Fs(\()p Fm(u)p Fs(\))12 b Fn(62)f(F)1256 2438 y Fl(1)1284 2432 y Fn(\\)e(F)1355 2438 y Fl(2)1387 2432 y Fs(and)14 b Fm(u)f Fn(!)1547 2438 y Fk(i)1575 2432 y Fm(v)q Fn(g)p Fm(:)p eop %%Page: 9 9 9 8 bop 183 194 a Fs(First)17 b(of)g(all)f(note)i(that)g(\()p Fn(F)639 200 y Fl(1)657 194 y Fm(;)7 b Fn(S)701 200 y Fl(1)720 194 y Fs(\))17 b(and)g(\()p Fn(F)887 200 y Fl(2)906 194 y Fm(;)7 b Fn(S)950 200 y Fl(2)968 194 y Fs(\))18 b(are)g(constructor-sharing)g(TRSs.)f(By)183 244 y(Lemma)8 b(4.6,)i(the)i(restriction)g(of)25 b Fn(!)770 250 y Fk(i)809 244 y Fs(to)11 b Fn(T)f Fs(\()p Fn(F)930 250 y Fk(i)944 244 y Fm(;)d Fn(V)s Fs(\))e Fn(\002)t(T)10 b Fs(\()p Fn(F)1121 250 y Fk(i)1136 244 y Fm(;)d Fn(V)s Fs(\))k(and)h Fn(\))1331 250 y Fi(R)1360 254 y Fh(i)1385 244 y Fs(coincide.)f(It)h (is)183 293 y(easy)e(to)g(sho)o(w)g(that)24 b Fn(!)558 299 y Fi(S)578 303 y Fh(i)618 293 y Fs(and)10 b(the)g(restriction)h(of) 24 b Fn(!)1056 299 y Fk(i)1094 293 y Fs(to)10 b Fn(T)g Fs(\()p Fn(F)1214 299 y Fk(i)1228 293 y Fm(;)d Fn(V)s Fs(\))r Fn(\002)r(T)k Fs(\()p Fn(F)1401 299 y Fk(i)1415 293 y Fm(;)c Fn(V)s Fs(\))k(are)f(also)183 343 y(the)k(same.)e(Hence)29 b Fn(!)548 349 y Fi(S)568 353 y Fh(i)611 343 y Fs(and)13 b Fn(\))733 349 y Fi(R)762 353 y Fh(i)790 343 y Fs(coincide)h(on)f Fn(T)d Fs(\()p Fn(F)1079 349 y Fk(i)1094 343 y Fm(;)d Fn(V)s Fs(\))i Fn(\002)g(T)h Fs(\()p Fn(F)1280 349 y Fk(i)1294 343 y Fm(;)d Fn(V)s Fs(\).)14 b(In)f(particular,)183 393 y(the)j(TRS)e(\()p Fn(F)399 399 y Fk(i)413 393 y Fm(;)7 b Fn(S)457 399 y Fk(i)470 393 y Fs(\))16 b(is)f(semi-complete)e (b)q(ecause)k(\()p Fn(F)1016 399 y Fk(i)1030 393 y Fm(;)7 b Fn(R)1083 399 y Fk(i)1097 393 y Fs(\))15 b(is)h(semi-complete.)d(It)i (follo)o(ws)183 443 y(from)10 b(the)k(mo)q(dularit)o(y)c(of)i 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b Fm(t)654 748 y Fk(n)692 742 y Fs(in)j Fn(R)777 748 y Fl(1)795 742 y Fs(,)g(a)h(substitution)f Fm(\033)h Fs(:)d Fn(V)19 b(!)13 b(T)d Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))q(,)15 b(and)g(a)g(con-)183 791 y(text)f Fm(C)s Fs([)e(])h(suc)o(h)i(that)f Fm(s)e Fs(=)g Fm(C)s Fs([)p Fm(l)q(\033)q Fs(])p Fm(;)7 b(t)j Fs(=)i Fm(C)s Fs([)p Fm(r)q(\033)q Fs(],)g(and)i Fm(s)1019 797 y Fk(j)1037 791 y Fm(\033)g Fn(#)1096 773 y Fk(o)1096 802 y Fl(1)1127 791 y Fm(t)1142 797 y Fk(j)1159 791 y Fm(\033)h Fs(for)f Fm(j)g Fn(2)d(f)p Fs(1)p Fm(;)c(:)g(:)g(:)e(;)i(n)p Fn(g)p Fs(.)12 b(Note)183 841 y(that)j(particularly)f Fm(l)q(\033)g Fn(!)593 847 y Fl(1)625 841 y Fm(r)q(\033)q Fs(.)g(According)h(to)g (Prop)q(osition)g(3.12,)e Fm(\033)j Fs(has)f(a)g(decomp)q(osi-)183 891 y(tion)c Fm(\033)i Fs(=)f Fm(\033)371 897 y Fl(2)394 891 y Fn(\016)5 b Fm(\033)444 897 y Fl(1)475 891 y Fs(suc)o(h)12 b(that)g Fm(\033)678 897 y Fl(1)709 891 y Fs(is)f(blac)o(k,)g Fm(\033)890 897 y Fl(2)921 891 y Fs(is)g(top)h(white,)g(and)g Fm(\033)1258 897 y Fl(2)1287 891 y Fn(/)g Fm(\017)p Fs(.)f(No)o(w)h(w)o (e)g(apply)183 941 y(Lemma)g(4.5:)h Fm(\033)435 947 y Fl(1)453 941 y Fs(\()p Fm(l)q Fs(\))j(and)e Fm(\033)619 947 y Fl(1)638 941 y Fs(\()p Fm(r)q Fs(\))h(are)g(blac)o(k)f(terms)h (and)f Fm(\033)1106 947 y Fl(2)1139 941 y Fs(is)h(a)g(top)f(white)h (substitution)183 991 y(with)f Fm(\033)302 997 y Fl(2)320 991 y Fs(\()p Fm(\033)360 997 y Fl(1)379 991 y Fs(\()p Fm(l)q Fs(\)\))f Fn(!)495 997 y Fl(1)526 991 y Fm(\033)550 997 y Fl(2)569 991 y Fs(\()p Fm(\033)609 997 y Fl(1)627 991 y Fs(\()p Fm(r)q Fs(\)\))i(and)g Fm(\017)f Fs(is)h(a)f (substitution)h(with)g Fm(\033)1253 997 y Fl(2)1284 991 y Fn(/)e Fm(\017)p Fs(.)h(Consequen)o(tly)m(,)183 1041 y(w)o(e)g(obtain)g Fm(\033)398 1047 y Fl(1)416 1041 y Fs(\()p Fm(l)q Fs(\))e(=)h Fm(\017)p Fs(\()p Fm(\033)575 1047 y Fl(1)593 1041 y Fs(\()p Fm(l)q Fs(\)\))g Fn(!)709 1047 y Fl(1)739 1041 y Fm(\017)p Fs(\()p Fm(\033)796 1047 y Fl(1)814 1041 y Fs(\()p Fm(r)q Fs(\)\))g(=)f Fm(\033)963 1047 y Fl(1)981 1041 y Fs(\()p Fm(r)q Fs(\).)i(Since)h Fm(\033)1192 1047 y Fl(1)1210 1041 y Fs(\()p Fm(l)q Fs(\))g(and)f Fm(\033)1375 1047 y Fl(1)1393 1041 y Fs(\()p Fm(r)q Fs(\))g(are)h(blac) o(k)183 1090 y(terms)h(and)h Fm(r)q(oot)p Fs(\()p Fm(\033)500 1096 y Fl(1)518 1090 y Fs(\()p Fm(l)q Fs(\)\))g(=)g Fm(r)q(oot)p Fs(\()p Fm(l)q Fs(\))g Fn(62)f(F)860 1096 y Fl(1)889 1090 y Fn(\\)11 b(F)962 1096 y Fl(2)981 1090 y Fs(,)16 b(it)g(follo)o(ws)f(that)i Fm(\033)1309 1096 y Fl(1)1328 1090 y Fs(\()p Fm(l)q Fs(\))f Fn(!)g Fm(\033)1471 1096 y Fl(1)1489 1090 y Fs(\()p Fm(r)q Fs(\))h(is)g(a)183 1140 y(rewrite)e(rule)f(of)f Fn(S)482 1146 y Fl(1)501 1140 y Fs(.)g(Th)o(us)h Fm(s)e Fs(=)g Fm(C)s Fs([)p Fm(\033)775 1146 y Fl(2)793 1140 y Fs(\()p Fm(\033)833 1146 y Fl(1)851 1140 y Fs(\()p Fm(l)q Fs(\)\)])g Fn(!)978 1146 y Fi(S)998 1150 y Ff(1)1028 1140 y Fm(C)s Fs([)p Fm(\033)1097 1146 y Fl(2)1114 1140 y Fs(\()p Fm(\033)1154 1146 y Fl(1)1173 1140 y Fs(\()p Fm(r)q Fs(\)\)])f(=)h Fm(t)p Fs(.)245 1190 y(With)h(the)i(ab)q(o)o(v)o(e)e(results,)i(it)e(further)i(follo)o (ws)d(from)452 1273 y Fn(!)493 1279 y Fi(S)513 1283 y Ff(1)530 1279 y Fi([S)572 1283 y Ff(2)616 1273 y Fs(=)26 b Fn(!)715 1279 y Fi(S)735 1283 y Ff(1)777 1273 y Fn([)d(!)869 1279 y Fi(S)889 1283 y Ff(2)933 1273 y Fs(=)j Fn(!)1033 1279 y Fl(1)1074 1273 y Fn([)d(!)1166 1279 y Fl(2)1210 1273 y Fs(=)j Fn(!)1310 1279 y Fl(1)p Fk(;)p Fl(2)183 1357 y Fs(that)h Fn(!)328 1363 y Fl(1)p Fk(;)p Fl(2)401 1357 y Fs(is)13 b(semi-complete)f(on)i Fn(T)c Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))j Fn(\002)g(T)g Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\).)14 b Fj(2)183 1446 y Fg(De\014niti)o(on)f(4.8)20 b Fs(If)c Fn(!)573 1452 y Fl(1)p Fk(;)p Fl(2)633 1446 y Fs(is)g(semi-complete,)d(then)k(ev)o(ery)f(term)g Fm(t)f Fs(has)h(a)g(unique)g(nor-)183 1496 y(mal)f(form)h(w.r.t.)g Fn(!)525 1502 y Fl(1)p Fk(;)p Fl(2)570 1496 y Fs(.)g(In)i(the)g (sequel,)g(this)f(normal)f(form)f(will)h(b)q(e)i(denoted)h(b)o(y)e Fm(t)1577 1481 y Fi(!)1612 1496 y 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1492 y Fm(;)7 b Fn(R)760 1498 y Fl(2)778 1492 y Fs(\))12 b(b)q(e)h(CTRSs)f(with)g(shared)h (constructors.)h(W)m(e)e(ha)o(v)o(e)183 1542 y(to)h(sho)o(w)h(that)g (their)g(com)o(bined)e(system)h(\()p Fn(F)5 b Fm(;)i Fn(R)p Fs(\))13 b(is)h(semi-complete)d(if)i(and)h(only)f(if)f(b)q(oth) 183 1592 y(\()p Fn(F)229 1598 y Fl(1)247 1592 y Fm(;)7 b Fn(R)301 1598 y Fl(1)320 1592 y Fs(\))i(and)h(\()p Fn(F)468 1598 y Fl(2)486 1592 y Fm(;)d Fn(R)540 1598 y Fl(2)558 1592 y Fs(\))j(are)g(semi-complete.)d(In)j(order)g(to)f(sho) o(w)h(the)g(if)f(case,)h(w)o(e)g(consider)183 1641 y(a)j(con)o(v)o (ersion)g Fm(t)433 1647 y Fl(1)473 1626 y Fi(\003)473 1653 y(R)501 1641 y Fn( )26 b Fm(s)12 b Fn(!)642 1626 y Fi(\003)642 1653 y(R)683 1641 y Fm(t)698 1647 y Fl(2)717 1641 y Fs(.)h(According)g(to)h(Prop)q(osition)e(4.12)g(w)o(e)i(ha)o(v)o (e)f Fm(t)1469 1647 y Fl(1)1500 1641 y Fn(#)1521 1652 y Fl(1)p Fk(;)p Fl(2)1578 1641 y Fm(t)1593 1647 y Fl(2)1612 1641 y Fs(.)183 1691 y(Since)j Fn(!)335 1697 y Fl(1)p Fk(;)p Fl(2)395 1691 y Fs(is)g(semi-complete,)d Fm(t)737 1697 y Fl(1)771 1691 y Fn(!)813 1676 y Fi(\003)813 1702 y Fl(1)p Fk(;)p Fl(2)872 1691 y Fm(t)887 1697 y Fl(3)921 1691 y Fs(and)j Fm(t)1019 1697 y Fl(2)1052 1691 y Fn(!)1094 1676 y Fi(\003)1094 1702 y Fl(1)p Fk(;)p Fl(2)1153 1691 y Fm(t)1168 1697 y Fl(3)1187 1691 y Fs(,)f(where)i Fm(t)1351 1697 y Fl(3)1385 1691 y Fs(is)f(the)h(unique)183 1741 y(normal)d(form)g(of)h Fm(s)p Fs(,)i Fm(t)538 1747 y Fl(1)556 1741 y Fs(,)f(and)g Fm(t)682 1747 y Fl(2)700 1741 y Fs(.)g(No)o(w)g(Lemma)d(4.3)i(implies)f Fm(t)1202 1747 y Fl(1)1235 1741 y Fn(!)1277 1726 y Fi(\003)1277 1752 y(R)1321 1741 y Fm(t)1336 1747 y Fl(3)1375 1726 y Fi(\003)1375 1752 y(R)1403 1741 y Fn( )26 b Fm(t)1486 1747 y Fl(2)1505 1741 y Fs(.)15 b(Th)o(us)183 1791 y(\()p Fn(F)t Fm(;)7 b Fn(R)o Fs(\))15 b(is)g(con\015uen)o(t.)g(It)f(remains)g (to)h(sho)o(w)f(normalization)e(of)i Fn(!)1261 1797 y Fi(R)1291 1791 y Fs(.)g(Let)i Fm(s)d Fn(2)f(T)f Fs(\()p Fn(F)t Fm(;)c Fn(V)s Fs(\).)183 1841 y(Since)18 b Fn(!)337 1847 y Fl(1)p Fk(;)p Fl(2)400 1841 y Fs(is)g(normalizing,)c Fm(s)19 b Fn(!)767 1826 y Fi(\003)767 1851 y Fl(1)p Fk(;)p Fl(2)830 1841 y Fm(t)f Fs(for)g(some)f Fm(t)i Fn(2)f Fm(N)5 b(F)h Fs(\()p Fn(!)1248 1847 y Fl(1)p Fk(;)p Fl(2)1292 1841 y Fs(\).)17 b(By)i(Lemma)c(4.3,)183 1891 y Fm(s)g Fn(!)259 1875 y Fi(\003)259 1902 y(R)303 1891 y Fm(t)p Fs(.)g(It)h(follo)o(ws)e(from)g(Lemma)f(4.13)h(that)i Fm(t)f Fn(2)f Fm(N)5 b(F)h Fs(\()p Fn(F)s Fm(;)h Fn(R)p Fs(\).)15 b(Hence)i(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))16 b(is)f(also)183 1940 y(normalizing.)6 b(This)j(all)f(pro)o(v)o(es)i (that)g(\()p Fn(F)t Fm(;)d Fn(R)o Fs(\))j(is)f(semi-complete.)e(The)j (only-if)e(case)i(follo)o(ws)183 1990 y(straigh)o(tforw)o(ardly)i(from) h(Lemma)e(4.15.)h Fj(2)183 2082 y Fg(Lemma)j(4.15)21 b Fs(Let)d(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))17 b(b)q(e)h(the)g(com)o (bined)f(system)g(of)g(t)o(w)o(o)g(constructor-sharing)183 2132 y(CTRSs)c(\()p Fn(F)372 2138 y Fl(1)391 2132 y Fm(;)7 b Fn(R)445 2138 y Fl(1)463 2132 y Fs(\))14 b(and)f(\()p Fn(F)619 2138 y Fl(2)638 2132 y Fm(;)7 b Fn(R)692 2138 y Fl(2)710 2132 y Fs(\))14 b(suc)o(h)g(that)g(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\))13 b(is)h(semi-complete.)d(If)i Fm(s)h Fs(is)g(a)f(blac)o(k)183 2182 y(term)g(and)h Fm(s)e Fn(!)436 2188 y Fi(R)477 2182 y Fm(t)p Fs(,)i(then)g Fm(s)e Fn(\))685 2188 y Fi(R)714 2192 y Ff(1)743 2182 y Fm(t)p Fs(.)183 2232 y Fg(Pro)q(of:)g Fs(W)m(e)g(sho)o(w)h(the)g (follo)o(wing)e(stronger)j(claim,)c(where)k(the)f(rewrite)h(relation)e (asso)q(ci-)183 2282 y(ated)i(with)f(\()p Fn(F)420 2288 y Fl(1)447 2282 y Fn([)c(f)p Fj(2)p Fn(g)p Fm(;)e Fn(R)610 2288 y Fl(1)629 2282 y Fs(\))14 b(is)g(also)f(denoted)i(b)o(y)f Fn(\))1041 2288 y Fi(R)1070 2292 y Ff(1)1087 2282 y Fs(.)183 2332 y Fg(Claim:)h Fs(If)h Fm(s)g Fs(is)g(a)g(blac)o(k)f(term)h(and)g Fm(\033)h Fs(is)f(a)g(top)g(white)g Fn(!)1146 2338 y Fi(R)1192 2332 y Fs(normalized)e(substitution)183 2382 y(suc)o(h)k(that)g Fm(s\033)i Fn(!)479 2388 y Fi(R)527 2382 y Fm(t\033)q Fs(,)e(then)g Fm(s\033)739 2367 y Fd(2)783 2382 y Fn(\))825 2388 y Fi(R)854 2392 y Ff(1)890 2382 y Fm(t\033)930 2367 y Fd(2)955 2382 y Fs(,)f(where)i Fm(\033)1133 2367 y Fd(2)1177 2382 y Fs(denotes)g(the)f(substitution) 183 2432 y Fn(f)p Fm(x)11 b Fn(7!)g Fj(2)h Fn(j)g Fm(x)f Fn(2)h(D)q Fm(om)p Fs(\()p Fm(\033)q Fs(\))q Fn(g)p Fs(.)p eop %%Page: 12 12 12 11 bop 340 194 a Fs(Since)11 b Fn(R)f Fs(is)g(semi-complete,)e(ev)o (ery)j(term)e Fm(t)h Fs(has)g(a)g(unique)g(normal)e(form)g Fm(t)p Fn(#)i Fs(w.r.t.)f Fn(R)p Fs(.)h(F)m(ur-)340 244 y(thermore,)g(for)g(an)o(y)f(substitution)h Fm(\033)q Fs(,)g(let)g Fm(\033)q Fn(#)g Fs(denote)h(the)g(substitution)f Fn(f)p Fm(x)h Fn(7!)g Fm(\033)q Fs(\()p Fm(x)p Fs(\))p Fn(#)h(j)g Fm(x)g Fn(2)340 293 y(D)q Fm(om)p Fs(\()p Fm(\033)q Fs(\))r Fn(g)p Fs(.)19 b(The)h(claim)e(is)i(pro)o(v)o(ed)g(b) o(y)f(induction)h(on)g(the)g(depth)h(of)e Fm(s\033)k Fn(!)e Fm(t\033)q Fs(.)f(The)340 343 y(case)f(of)f(zero)g(depth)h(is)f (straigh)o(tforw)o(ard.)e(Let)j(the)f(depth)h(of)e Fm(s\033)j Fn(!)e Fm(t\033)h Fs(equal)e Fm(d)12 b Fs(+)g(1,)340 393 y Fm(d)k Fn(\025)f Fs(0.)h(There)h(is)g(a)f(con)o(text)h Fm(C)s Fs([)11 b(],)16 b(a)g(substitution)g Fm(\032)g Fs(:)f Fn(V)20 b(!)15 b(T)10 b Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\),)16 b(and)g(a)g(rewrite)340 443 y(rule)h Fm(l)h Fn(!)d Fm(r)i Fn(\()f Fm(s)626 449 y Fl(1)657 443 y Fn(#)c Fm(t)705 449 y Fl(1)724 443 y Fm(;)7 b(:)g(:)g(:)e(;)i(s) 836 449 y Fk(n)871 443 y Fn(#)12 b Fm(t)919 449 y Fk(n)958 443 y Fs(in)k Fn(R)1044 449 y Fl(1)1080 443 y Fs(suc)o(h)h(that)g Fm(s\033)g Fs(=)g Fm(C)s Fs([)p Fm(l)q(\032)p Fs(],)e Fm(t\033)i Fs(=)g Fm(C)s Fs([)p Fm(r)q(\032)p Fs(])e(and)340 493 y Fm(s)359 499 y Fk(j)377 493 y Fm(\032)e Fn(#)f Fm(t)459 499 y Fk(j)477 493 y Fm(\032)i Fs(is)f(of)g(depth)h Fn(\024)e Fm(d)h Fs(for)g(ev)o(ery)h Fm(j)g Fn(2)d(f)p Fs(1)p Fm(;)c(:)g(:)g(:)t(;)g(n)p Fn(g)p Fs(.)12 b(By)i(Prop)q(osition) f(3.12,)e Fm(\032)j Fs(can)g(b)q(e)340 542 y(decomp)q(osed)g(in)o(to)e Fm(\032)676 548 y Fl(2)704 542 y Fn(\016)7 b Fm(\032)753 548 y Fl(1)786 542 y Fs(suc)o(h)14 b(that)f Fm(\032)989 548 y Fl(1)1022 542 y Fs(is)g(blac)o(k)g(and)g Fm(\032)1272 548 y Fl(2)1304 542 y Fs(is)g(top)h(white.)f(Note)g(that)h(for)340 592 y(ev)o(ery)e Fm(x)f Fn(2)h(D)q Fm(om)p Fs(\()p Fm(\032)648 598 y Fl(2)668 592 y Fs(\))s Fn(\\)s(V)s Fm(ar)q Fs(\()p Fm(l)q(\032)838 598 y Fl(1)858 592 y Fs(\),)f(w)o(e)g(ha)o(v)o(e)f Fm(\032)1068 598 y Fl(2)1087 592 y Fs(\()p Fm(x)p Fs(\))i Fn(2)f Fm(N)5 b(F)h Fs(\()p Fn(!)p Fs(\).)k(Nev)o(ertheless,)j(w)o(e)e (do)f(not)340 642 y(ha)o(v)o(e)15 b Fm(\032)458 648 y Fl(2)477 642 y Fs(\()p Fm(x)p Fs(\))d Fn(2)h Fm(N)5 b(F)h Fs(\()p Fn(!)p Fs(\))13 b(in)h(general)h(b)q(ecause)h(of)e(p)q(ossible) h(extra)g(v)n(ariables.)e(Since)i Fn(!)f Fs(is)340 692 y(semi-complete,)e Fm(\032)643 698 y Fl(2)674 692 y Fn(!)716 677 y Fi(\003)746 692 y Fm(\032)767 698 y Fl(2)786 692 y Fn(#)p Fs(.)h(Th)o(us)h Fm(\032)958 698 y Fl(2)977 692 y Fn(#)p Fs(\()p Fm(\032)1035 698 y Fl(1)1054 692 y Fs(\()p Fm(s)1089 698 y Fk(j)1107 692 y Fs(\)\))1173 677 y Fi(\003)1189 692 y Fn( )g Fm(s)1264 698 y Fk(j)1282 692 y Fm(\032)f Fn(#)f Fm(t)1364 698 y Fk(j)1381 692 y Fm(\032)g Fn(!)1456 677 y Fi(\003)1487 692 y Fm(\032)1508 698 y Fl(2)1527 692 y Fn(#)o Fs(\()p Fm(\032)1584 698 y Fl(1)1604 692 y Fs(\()p Fm(t)1635 698 y Fk(j)1652 692 y Fs(\)\).)i(The)340 742 y(con\015uence)k(of)e Fn(!)g Fs(guaran)o(tees)h Fm(\032)881 748 y Fl(2)900 742 y Fn(#)p Fs(\()p Fm(\032)958 748 y Fl(1)977 742 y Fs(\()p Fm(s)1012 748 y Fk(j)1030 742 y Fs(\)\))c Fn(#)24 b Fm(\032)1141 748 y Fl(2)1160 742 y Fn(#)p Fs(\()p Fm(\032)1218 748 y Fl(1)1237 742 y Fs(\()p Fm(t)1268 748 y Fk(j)1286 742 y Fs(\)\))16 b(for)g(ev)o(ery)h Fm(j)h Fn(2)d(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)e(;)i(n)p Fn(g)p Fs(.)340 791 y(By)16 b(Prop)q(osition)f(3.12,)f Fm(\032)754 797 y Fl(2)773 791 y Fn(#)h Fs(can)h(b)q(e)g(decomp)q(osed)g(in)o(to)e Fm(\032)1285 797 y Fl(4)1315 791 y Fn(\016)c Fm(\032)1367 797 y Fl(3)1401 791 y Fs(suc)o(h)16 b(that)g Fm(\032)1609 797 y Fl(3)1643 791 y Fs(is)g(blac)o(k)340 841 y(and)d Fm(\032)441 847 y Fl(4)473 841 y Fs(is)g(top)g(white.)g(Eviden)o(tly)m (,)e Fm(\032)927 847 y Fl(3)946 841 y Fs(\()p Fm(\032)983 847 y Fl(1)1003 841 y Fs(\()p Fm(s)1038 847 y Fk(j)1056 841 y Fs(\)\))i(and)g Fm(\032)1202 847 y Fl(3)1221 841 y Fs(\()p Fm(\032)1258 847 y Fl(1)1277 841 y Fs(\()p Fm(t)1308 847 y Fk(j)1326 841 y Fs(\)\))g(are)g(blac)o(k)g(terms)f(and) h Fm(\032)1762 847 y Fl(4)340 891 y Fs(is)f(a)g(top)g(white)g Fn(!)g Fs(normalized)e(substitution.)i(Rep)q(eated)h(application)e(of)g (the)i(induction)340 941 y(h)o(yp)q(othesis)21 b(yields)e Fm(\032)694 926 y Fd(2)694 951 y Fl(4)720 941 y Fs(\()p Fm(\032)757 947 y Fl(3)776 941 y Fs(\()p Fm(\032)813 947 y Fl(1)833 941 y Fs(\()p Fm(s)868 947 y Fk(j)886 941 y Fs(\)\)\))34 b Fn(+)993 947 y Fi(R)1022 951 y Ff(1)1074 941 y Fm(\032)1095 926 y Fd(2)1095 951 y Fl(4)1120 941 y Fs(\()p Fm(\032)1157 947 y Fl(3)1177 941 y Fs(\()p Fm(\032)1214 947 y Fl(1)1233 941 y Fs(\()p Fm(t)1264 947 y Fk(j)1281 941 y Fs(\)\)\).)20 b(W)m(e)f(obtain)g(as)h(a)f(conse-) 340 991 y(quence)f(that)e Fm(\032)593 976 y Fd(2)593 1001 y Fl(4)619 991 y Fs(\()p Fm(\032)656 997 y Fl(3)676 991 y Fs(\()p Fm(\032)713 997 y Fl(1)732 991 y Fs(\()p Fm(l)q Fs(\)\)\))g Fn(\))867 997 y Fi(R)896 1001 y Ff(1)929 991 y Fm(\032)950 976 y Fd(2)950 1001 y Fl(4)976 991 y Fs(\()p Fm(\032)1013 997 y Fl(3)1032 991 y Fs(\()p Fm(\032)1069 997 y Fl(1)1088 991 y Fs(\()p Fm(r)q Fs(\)\)\).)h(Clearly) m(,)e Fm(s\033)i Fs(=)f Fm(C)s Fs([)p Fm(\032)1530 997 y Fl(2)1548 991 y Fn(#)p Fs(\()p Fm(\032)1606 997 y Fl(1)1625 991 y Fs(\()p Fm(l)q Fs(\)\)])g(and)340 1041 y Fm(t\033)e Fs(=)f Fm(C)s Fs([)p Fm(\032)504 1047 y Fl(2)522 1041 y Fn(#)p Fs(\()p Fm(\032)580 1047 y Fl(1)599 1041 y Fs(\()p Fm(r)q Fs(\)\)])h(b)q(ecause)i Fm(\032)868 1047 y Fl(2)887 1041 y Fs(\()p Fm(x)p Fs(\))d Fn(2)f Fm(N)5 b(F)h Fs(\()p Fn(!)p Fs(\))13 b(for)h(ev)o(ery)h Fm(x)e Fn(2)f(D)q Fm(om)p Fs(\()p Fm(\032)1531 1047 y Fl(2)1551 1041 y Fs(\))d Fn(\\)h(V)s Fm(ar)q Fs(\()p Fm(l)q(\032)1734 1047 y Fl(1)1754 1041 y Fs(\).)340 1090 y(Let)420 1080 y(^)411 1090 y Fm(C)s Fs([)i(])d(b)q(e)i(the)g(con)o(text)f(obtained)g (from)f Fm(C)s Fs([)i(])f(b)o(y)g(replacing)g(ev)o(ery)g(white)h (principal)e(sub-)340 1140 y(term)j(whic)o(h)f(m)o(ust)g(b)q(e)i(of)e (the)h(form)e Fm(\033)q Fs(\()p Fm(x)p Fs(\))i(for)g(some)f(v)n (ariable)f Fm(x)i Fn(2)f(D)q Fm(om)p Fs(\()p Fm(\033)q Fs(\))i(with)e Fj(2)p Fs(.)h(It)f(is)340 1190 y(fairly)i(simple)g(to)h (v)o(erify)f(that)h Fm(s\033)879 1175 y Fd(2)917 1190 y Fs(=)971 1179 y(^)961 1190 y Fm(C)s Fs([)p Fm(\032)1027 1175 y Fd(2)1027 1200 y Fl(4)1052 1190 y Fs(\()p Fm(\032)1089 1196 y Fl(3)1108 1190 y Fs(\()p Fm(\032)1145 1196 y Fl(1)1165 1190 y Fs(\()p Fm(l)q Fs(\)\)\)])g(and)g Fm(t\033)1389 1175 y Fd(2)1426 1190 y Fs(=)1480 1179 y(^)1470 1190 y Fm(C)s Fs([)p Fm(\032)1536 1175 y Fd(2)1536 1200 y Fl(4)1561 1190 y Fs(\()p Fm(\032)1598 1196 y Fl(3)1618 1190 y Fs(\()p Fm(\032)1655 1196 y Fl(1)1674 1190 y Fs(\()p Fm(r)q Fs(\)\)\)].)340 1240 y(Th)o(us)h Fm(s\033)490 1225 y Fd(2)527 1240 y Fn(\))569 1246 y Fi(R)598 1250 y Ff(1)627 1240 y Fm(t\033)667 1225 y Fd(2)693 1240 y Fs(.)e(This)h(pro)o(v)o(es)g(the)h(claim.)c Fj(2)403 1318 y Fs(It)19 b(has)h(b)q(een)h(sho)o(wn)f(b)o(y)f(Middeldorp)g ([Mid90)o(,)g(Mid93)o(])g(that)h(con\015uence)h(is)f(also)340 1368 y(mo)q(dular)12 b(for)h(semi-equational)f(disjoin)o(t)g(CTRSs.)h (Analogously)m(,)e(it)i(should)h(b)q(e)g(p)q(ossible)340 1418 y(to)g(pro)o(v)o(e)g(that)g(Theorem)f(4.14)g(also)g(holds)h(true)h (for)e(semi-equational)f(CTRSs.)340 1531 y Fg(4.2)48 b(T)l(ermination)340 1604 y Fs(In)15 b(con)o(trast)g(to)f(the)h (unconditional)e(case)j(\(see)f([Rus87)o(]\),)f(termination)f(is)h(not) g(mo)q(dular)340 1654 y(for)g(non-duplicating)e(disjoin)o(t)h(CTRSs.)g (This)h(is)g(witnessed)h(b)o(y)f(the)g(follo)o(wing)e(example)340 1704 y(whic)o(h)i(stems)g(from)e([Mid93)o(].)340 1782 y Fg(Example)k(4.16)k Fs(Consider)14 b(the)h(CTRSs)700 1854 y Fn(R)735 1860 y Fl(1)765 1854 y Fs(=)809 1820 y Fe(\010)839 1854 y Fm(F)6 b Fs(\()p Fm(x)p Fs(\))11 b Fn(!)g Fm(F)6 b Fs(\()p Fm(x)p Fs(\))11 b Fn(\()24 b Fm(x)12 b Fn(#)g Fm(A;)19 b(x)12 b Fn(#)g Fm(B)1398 1820 y Fe(\011)700 1979 y Fn(R)735 1985 y Fl(2)765 1979 y Fs(=)809 1920 y Fe(\032)846 1954 y Fm(g)q Fs(\()p Fm(x;)7 b(y)q Fs(\))12 b Fn(!)f Fm(x)846 2003 y(g)q Fs(\()p Fm(x;)c(y)q Fs(\))12 b Fn(!)f Fm(y)340 2076 y Fs(Both)i(systems)f(are)h (terminating)d(and)j(non-duplicating)d(but)j(their)f(com)o(bined)f (system)h(is)340 2126 y(non-terminating.)403 2204 y(Note)e(that)f(the)h (CTRS)f Fn(R)810 2210 y Fl(2)838 2204 y Fs(in)g(Example)f(4.16)g(is)h (not)h(con\015uen)o(t.)g(Middeldorp)f([Mid90)o(,)340 2254 y(Mid93])k(has)h(sho)o(wn)g(that)g(this)g(is)f(essen)o(tial.)h(He) h(pro)o(v)o(ed)f(the)g(follo)o(wing)e(theorem.)340 2332 y Fg(Theorem)j(4.17)21 b Fs(If)13 b Fn(R)725 2338 y Fl(1)757 2332 y Fs(and)g Fn(R)872 2338 y Fl(2)904 2332 y Fs(are)h(terminating)e (disjoin)o(t)g(CTRSs,)h(then)h(their)g(com-)340 2382 y(bined)k(system)f Fn(R)g Fs(=)g Fn(R)734 2388 y Fl(1)764 2382 y Fn([)11 b(R)838 2388 y Fl(2)874 2382 y Fs(is)17 b(terminating)e(pro)o(vided)i(that)g(one)h(of)e(the)i(follo)o(wing)340 2432 y(conditions)c(is)g(satis\014ed:)p eop %%Page: 13 13 13 12 bop 200 194 a Fs(1.)20 b(Neither)15 b Fn(R)437 200 y Fl(1)470 194 y Fs(nor)f Fn(R)579 200 y Fl(2)611 194 y Fs(con)o(tain)g(collapsing)e(rules.)200 244 y(2.)20 b(Both)14 b(systems)g(are)h(con\015uen)o(t)f(and)g(non-duplicating.)200 294 y(3.)20 b(Both)10 b(systems)g(are)g(con\015uen)o(t)h(and)e(one)h (of)f(the)i(systems)f(con)o(tains)f(neither)i(collapsing)253 343 y(nor)j(duplicating)f(rules.)245 435 y(F)m(urthermore,)h(he)g (conjectured)i(that)f(the)g(disjoin)o(t)e(union)h(of)f(t)o(w)o(o)h (terminating)f(join)183 485 y(CTRSs)f(is)h(terminating)e(if)i(one)g(of) f(them)g(con)o(tains)h(neither)h(collapsing)e(nor)h(duplicating)183 535 y(rules)e(and)g(the)h(other)f(is)g(con\015uen)o(t.)g(The)h(next)f (example)f(dispro)o(v)o(es)h(this)g(conjecture.)h(The)183 585 y(function)f(sym)o(b)q(ols)f(ha)o(v)o(e)i(b)q(een)g(c)o(hosen)h(in) e(resem)o(blance)h(to)g(other)g(kno)o(wn)f(coun)o(terexam-)183 634 y(ples.)183 726 y Fg(Example)k(4.18)20 b Fs(Let)371 860 y Fn(R)406 866 y Fl(1)436 860 y Fs(=)480 774 y Fe(8)480 812 y(<)480 887 y(:)530 815 y Fs(0)168 b(1)525 911 y Fm(A)625 910 y Fs(2)713 911 y Fm(B)p 540 873 2 48 v 541 873 a Fc(?)564 867 y(@)570 873 y(@)-42 b(R)664 867 y(\000)659 873 y(\000)g(\011)p 729 873 V 29 w(?)889 860 y Fm(F)6 b Fs(\()p Fm(x)p Fs(\))11 b Fn(!)g Fm(F)6 b Fs(\()p Fm(x)p Fs(\))11 b Fn(\()g Fm(x)h Fn(#)h Fm(A;)19 b(x)12 b Fn(#)g Fm(B)183 993 y Fs(and)692 1058 y Fn(R)727 1064 y Fl(2)758 1058 y Fs(=)802 999 y Fe(\032)838 1033 y Fm(g)d Fs(\()p Fm(x;)e(y)q(;)g(y)q Fs(\))12 b Fn(!)f Fm(x)838 1082 y(g)e Fs(\()p Fm(y)q(;)e(y)q(;)g(x)p Fs(\))12 b Fn(!)f Fm(x:)183 1157 y Fs(Clearly)m(,)e Fn(R)367 1163 y Fl(1)397 1157 y Fs(is)i(non-collapsing,)e(non-duplicating,)g(and)i(terminating)f (\(there)i(is)f(no)g(term)183 1206 y Fm(t)j Fn(2)f(T)e Fs(\()p Fn(F)333 1212 y Fl(1)351 1206 y Fm(;)c Fn(V)s Fs(\))16 b(whic)o(h)f(rewrites)i(to)e(b)q(oth)h Fm(A)f Fs(and)h Fm(B)r Fs(\).)f(Note)h(that)f Fn(R)1295 1212 y Fl(1)1329 1206 y Fs(is)g(not)g(con\015uen)o(t.)183 1256 y(Moreo)o(v)o(er,)e(the)i(CTRS)e Fn(R)610 1262 y Fl(2)643 1256 y Fs(is)g(eviden)o(tly)h(terminating)e(and)h(con\015uen)o (t.)h(Ho)o(w)o(ev)o(er,)g(their)183 1306 y(disjoin)o(t)e(union)i Fn(R)d Fs(=)h Fn(R)571 1312 y Fl(1)599 1306 y Fn([)d(R)671 1312 y Fl(2)704 1306 y Fs(is)14 b(not)f(terminating.)f(Since)327 1392 y Fm(B)375 1398 y Fi(R)403 1392 y Fn( )h Fs(1)493 1398 y Fi(R)521 1392 y Fn( )h Fm(g)q Fs(\(0)p Fm(;)7 b Fs(0)p Fm(;)g Fs(1\))j Fn(!)783 1398 y Fi(R)825 1392 y Fm(g)q Fs(\(0)p Fm(;)d Fs(2)p Fm(;)g Fs(1\))j Fn(!)1031 1398 y Fi(R)1072 1392 y Fm(g)q Fs(\(0)p Fm(;)d Fs(2)p Fm(;)g Fs(2\))k Fn(!)1279 1398 y Fi(R)1320 1392 y Fs(0)h Fn(!)1395 1398 y Fi(R)1436 1392 y Fm(A;)183 1478 y Fs(there)j(is)f(the) g(cyclic)g(reduction)h("sequence")g Fm(F)6 b Fs(\()p Fm(g)q Fs(\(0)p Fm(;)h Fs(0)p Fm(;)g Fs(1\)\))13 b Fn(!)1188 1484 y Fi(R)1232 1478 y Fm(F)6 b Fs(\()p Fm(g)q Fs(\(0)p Fm(;)h Fs(0)p Fm(;)g Fs(1\)\).)245 1570 y(It)14 b(will)f(next)i(b)q(e)g (sho)o(wn)f(that)h(the)g(ab)q(o)o(v)o(e)f(theorem)g(also)f(holds,)h(m)o (utatis)f(m)o(utandis,)183 1619 y(in)18 b(the)h(presence)j(of)c(shared) i(constructors.)g(W)m(e)e(p)q(oin)o(t)h(out)f(that)h(our)g(pro)q(of)g (\(though)183 1669 y(based)14 b(on)g(the)g(ideas)g(of)g([Mid93)n(]\))g (is)g(considerably)g(simpler)e(than)i(that)g(of)f([Mid93)o(].)245 1719 y(As)20 b(in)g(the)g(previous)h(subsection,)g(let)f(\()p Fn(F)951 1725 y Fl(1)969 1719 y Fm(;)7 b Fn(R)1023 1725 y Fl(1)1042 1719 y Fs(\))20 b(and)f(\()p Fn(F)1215 1725 y Fl(2)1233 1719 y Fm(;)7 b Fn(R)1287 1725 y Fl(2)1305 1719 y Fs(\))20 b(b)q(e)h(constructor-)183 1769 y(sharing)16 b(join)f(CTRSs.)g(It)h(is)g(not)g(di\016cult)g(to)g(v)o(erify)g(that)g (the)g(CTRS)g(\()p Fn(F)1405 1775 y Fk(i)1429 1769 y Fn([)10 b(f)p Fj(2)p Fn(g)p Fm(;)d Fn(R)1594 1775 y Fk(i)1607 1769 y Fs(\))183 1819 y(is)15 b(terminating)f(if)h(and)g(only)g(if)f (\()p Fn(F)753 1825 y Fk(i)767 1819 y Fm(;)7 b Fn(R)820 1825 y Fk(i)834 1819 y Fs(\))16 b(is)f(terminating.)f(Again,)g(w)o(e)i (also)f(denote)h(the)183 1869 y(rewrite)f(relation)e(asso)q(ciated)i (with)e(\()p Fn(F)813 1875 y Fk(i)836 1869 y Fn([)8 b(f)p Fj(2)p Fn(g)p Fm(;)f Fn(R)999 1875 y Fk(i)1012 1869 y Fs(\))14 b(b)o(y)g Fn(\))1142 1875 y Fi(R)1171 1879 y Fh(i)1199 1869 y Fs(\(b)o(y)g(abuse)g(of)g(notation\).)183 1960 y Fg(Prop)q(ositi)o(on)f(4.19)20 b Fs(Let)15 b(\()p Fn(F)671 1966 y Fl(2)689 1960 y Fm(;)7 b Fn(R)743 1966 y Fl(2)762 1960 y Fs(\))14 b(b)q(e)g(la)o(y)o(er-preserving.)200 2044 y(1.)20 b(If)14 b Fm(s)e Fn(!)368 2029 y Fk(o)397 2044 y Fm(t)i Fs(b)o(y)g(some)f(rule)h(from)e Fn(R)804 2050 y Fl(1)823 2044 y Fs(,)h(then)i Fm(top)999 2029 y Fk(b)1015 2044 y Fs(\()p Fm(s)p Fs(\))e Fn(\))1121 2050 y Fi(R)1150 2054 y Ff(1)1178 2044 y Fm(top)1234 2029 y Fk(b)1251 2044 y Fs(\()p Fm(t)p Fs(\).)200 2094 y(2.)20 b(If)14 b Fm(s)e Fn(!)368 2079 y Fk(o)397 2094 y Fm(t)i Fs(b)o(y)g(some)f(rule)h(from)e Fn(R)804 2100 y Fl(2)837 2094 y Fs(or)h Fm(s)f Fn(!)960 2079 y Fk(i)985 2094 y Fm(t)p Fs(,)i(then)g Fm(top)1176 2079 y Fk(b)1193 2094 y Fs(\()p Fm(s)p Fs(\))e(=)g Fm(top)1356 2079 y Fk(b)1373 2094 y Fs(\()p Fm(t)p Fs(\).)183 2177 y Fg(Pro)q(of:)e Fs(W)m(e)h(pro)q(ceed)h(b)o(y)f(induction)g(on)g(the)h(depth)f(of)g Fm(s)h Fn(!)f Fm(t)p Fs(.)g(The)g(case)h(of)f(zero)h(depth)g(is)183 2227 y(straigh)o(tforw)o(ard.)f(So)i(supp)q(ose)h(that)f(the)g(depth)h (of)e Fm(s)g Fn(!)f Fm(t)i Fs(equals)g Fm(d)6 b Fs(+)i(1,)k Fm(d)f Fn(\025)h Fs(0.)g(The)h(in-)183 2277 y(duction)e(h)o(yp)q (othesis)g(co)o(v)o(ers)h(the)g(statemen)o(t)f(that)g Fm(u)g Fn(!)g Fm(v)i Fs(implies)c Fm(top)1313 2262 y Fk(b)1330 2277 y Fs(\()p Fm(u)p Fs(\))i Fn(\))1439 2262 y Fi(\003)1439 2288 y(R)1468 2292 y Ff(1)1497 2277 y Fm(top)1553 2262 y Fk(b)1570 2277 y Fs(\()p Fm(v)q Fs(\))183 2327 y(whenev)o(er)k Fm(u)c Fn(!)g Fm(v)16 b Fs(is)d(of)h(depth)g(less) h(than)f(or)g(equal)f(to)h Fm(d)p Fs(.)183 2376 y(\(1\))23 b(If)f Fm(s)27 b Fn(!)397 2361 y Fk(o)442 2376 y Fm(t)22 b Fs(b)o(y)h(some)f(rule)h(from)e Fn(R)894 2382 y Fl(1)912 2376 y Fs(,)i(then)g Fm(s)k Fs(=)g Fm(C)1188 2361 y Fk(b)1204 2376 y Fn(f)-14 b(f)p Fm(u)1256 2382 y Fl(1)1274 2376 y Fm(;)7 b(:)g(:)g(:)e(;)i(u)1391 2382 y Fk(p)1410 2376 y Fn(g)-14 b(g)22 b Fs(and)h Fm(t)j Fs(=)192 2421 y(^)183 2432 y Fm(C)216 2417 y Fk(b)232 2432 y Fn(h)-7 b(h)p Fm(u)281 2438 y Fk(i)293 2442 y Ff(1)311 2432 y Fm(;)7 b(:)g(:)g(:)e(;)i(u)428 2438 y Fk(i)440 2442 y Fh(q)457 2432 y Fn(i)-7 b(i)q Fs(,)11 b(where)i Fm(i)638 2438 y Fl(1)657 2432 y Fm(;)7 b(:)g(:)g(:)e(;)i(i)764 2438 y Fk(q)793 2432 y Fn(2)k(f)p Fs(1)p Fm(;)c(:)g(:)g(:)e(;)i(p)p Fn(g)p Fs(.)j(Moreo)o(v)o(er,)i(there)h(is)e(a)h(con)o(text)g Fm(C)s Fs([)g(],)p eop %%Page: 14 14 14 13 bop 340 194 a Fs(a)20 b(substitution)f Fm(\033)i Fs(and)e(a)h(rewrite)g(rule)g Fm(l)i Fn(!)f Fm(r)g Fn(\()g Fm(s)1243 200 y Fl(1)1274 194 y Fn(#)13 b Fm(t)1323 200 y Fl(1)1341 194 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)1453 200 y Fk(n)1488 194 y Fn(#)12 b Fm(t)1536 200 y Fk(n)1580 194 y Fn(2)20 b(R)1663 200 y Fl(1)1702 194 y Fs(suc)o(h)340 244 y(that)f Fm(s)i Fs(=)f Fm(C)s Fs([)p Fm(l)q(\033)q Fs(],)d Fm(t)j Fs(=)h Fm(C)s Fs([)p Fm(r)q(\033)q Fs(])c(and)i Fm(s)963 250 y Fk(j)981 244 y Fm(\033)14 b Fn(#)e Fm(t)1067 250 y Fk(j)1084 244 y Fm(\033)20 b Fs(is)f(of)f(depth)i(less)g(than)f (or)f(equal)h(to)g Fm(d)340 293 y Fs(for)e(ev)o(ery)g Fm(j)i Fn(2)d(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)t(;)g(n)p Fn(g)p Fs(.)16 b(W)m(e)g(\014rst)h(sho)o(w)g(that)g Fm(top)1224 278 y Fk(b)1241 293 y Fs(\()p Fm(s)1276 299 y Fk(j)1294 293 y Fm(\033)q Fs(\))f Fn(+)1376 299 y Fi(R)1405 303 y Ff(1)1440 293 y Fm(top)1496 278 y Fk(b)1512 293 y Fs(\()p Fm(t)1543 299 y Fk(j)1561 293 y Fm(\033)q Fs(\))h(for)g(ev)o(ery)340 343 y Fm(j)d Fn(2)e(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)t(;)g(n)p Fn(g)p Fs(.)j(Fix)g Fm(j)r Fs(.)h(Let)h Fm(w)g Fs(b)q(e)f(the)h(common) c(reduct)k(of)f Fm(s)1315 349 y Fk(j)1333 343 y Fm(\033)h Fs(and)f Fm(t)1462 349 y Fk(j)1479 343 y Fm(\033)q Fs(.)g(Clearly)m(,)f (it)g(suf-)340 393 y(\014ces)17 b(to)e(sho)o(w)h(that)f Fm(top)736 378 y Fk(b)753 393 y Fs(\()p Fm(s)788 399 y Fk(j)806 393 y Fm(\033)q Fs(\))g Fn(\))904 378 y Fi(\003)904 404 y(R)933 408 y Ff(1)964 393 y Fm(top)1020 378 y Fk(b)1037 393 y Fs(\()p Fm(w)q Fs(\))g(and)h Fm(top)1254 378 y Fk(b)1270 393 y Fs(\()p Fm(t)1301 399 y Fk(j)1319 393 y Fm(\033)q Fs(\))e Fn(\))1416 378 y Fi(\003)1416 404 y(R)1445 408 y Ff(1)1477 393 y Fm(top)1533 378 y Fk(b)1549 393 y Fs(\()p Fm(w)q Fs(\).)i(W.l.o.g.)340 443 y(w)o(e)10 b(consider)h(only)e(the)i(former)d(claim.)g(The)i(claim)e(is)h(pro)o(v) o(ed)h(b)o(y)g(induction)f(on)h(the)g(length)340 493 y(of)k Fm(s)407 499 y Fk(j)425 493 y Fm(\033)f Fn(!)504 478 y Fi(\003)534 493 y Fm(w)q Fs(.)g(The)i(case)g(of)e(zero)i(length)f (is)g(trivial,)e(so)i(let)g Fm(s)1333 499 y Fk(j)1351 493 y Fm(\033)f Fn(!)e Fm(v)j Fn(!)1517 478 y Fk(l)1541 493 y Fm(w)g Fs(with)g Fm(l)f Fn(\025)f Fs(0.)340 542 y(The)17 b(induction)e(h)o(yp)q(othesis)i(\(on)f Fm(l)q Fs(\))g(yields)g Fm(top)1117 527 y Fk(b)1134 542 y Fs(\()p Fm(v)q Fs(\))g Fn(\))1245 527 y Fi(\003)1245 554 y(R)1274 558 y Ff(1)1306 542 y Fm(top)1362 527 y Fk(b)1379 542 y Fs(\()p Fm(w)q Fs(\).)f(F)m(urthermore,)h(the)340 597 y(induction)11 b(h)o(yp)q(othesis)h(\(on)f Fm(d)p Fs(\))g(yields)g Fm(top)1014 582 y Fk(b)1031 597 y Fs(\()p Fm(s)1066 603 y Fk(j)1084 597 y Fm(\033)q Fs(\))h Fn(\))1179 582 y Fi(\003)1179 609 y(R)1208 613 y Ff(1)1237 597 y Fm(top)1293 582 y Fk(b)1310 597 y Fs(\()p Fm(v)q Fs(\).)f(This)g(pro)o(v)o(es)h (the)g(claim.)340 652 y(Th)o(us)f Fm(top)498 637 y Fk(b)515 652 y Fs(\()p Fm(w)q Fs(\))g(is)g(a)f(common)e(reduct)k(of)e Fm(top)1048 637 y Fk(b)1065 652 y Fs(\()p Fm(s)1100 658 y Fk(j)1118 652 y Fm(\033)q Fs(\))h(and)g Fm(top)1304 637 y Fk(b)1320 652 y Fs(\()p Fm(t)1351 658 y Fk(j)1369 652 y Fm(\033)q Fs(\))g(w.r.t.)f Fn(\))1571 658 y Fi(R)1600 662 y Ff(1)1617 652 y Fs(.)g(Accord-)340 702 y(ing)k(to)h(Prop)q (osition)f(3.12,)f Fm(\033)g Fs(=)g Fm(\033)892 708 y Fl(2)920 702 y Fn(\016)d Fm(\033)975 708 y Fl(1)993 702 y Fs(,)k(where)i Fm(\033)1164 708 y Fl(1)1197 702 y Fs(is)e(a)g(blac)o (k)g(substitution)h(and)g Fm(\033)1721 708 y Fl(2)1753 702 y Fs(is)340 752 y(top)j(white.)g(Recall)g(that)g Fm(\033)796 737 y Fd(2)795 762 y Fl(2)840 752 y Fs(denotes)h(the)g (substitution)f Fn(f)p Fm(x)g Fn(7!)g Fj(2)13 b Fn(j)f Fm(x)18 b Fn(2)h(D)q Fm(om)p Fs(\()p Fm(\033)1714 758 y Fl(2)1733 752 y Fs(\))p Fn(g)p Fs(.)340 802 y(It)j(is)g(clear)h(that) f Fm(top)705 787 y Fk(b)721 802 y Fs(\()p Fm(s)756 808 y Fk(j)774 802 y Fm(\033)q Fs(\))k(=)f Fm(\033)923 787 y Fd(2)922 812 y Fl(2)949 802 y Fs(\()p Fm(\033)989 808 y Fl(1)1007 802 y Fs(\()p Fm(s)1042 808 y Fk(j)1060 802 y Fs(\)\))e(and)e Fm(top)1259 787 y Fk(b)1276 802 y Fs(\()p Fm(t)1307 808 y Fk(j)1325 802 y Fm(\033)q Fs(\))k(=)g Fm(\033)1473 787 y Fd(2)1472 812 y Fl(2)1499 802 y Fs(\()p Fm(\033)1539 808 y Fl(1)1557 802 y Fs(\()p Fm(t)1588 808 y Fk(j)1606 802 y Fs(\)\).)d(Hence)340 857 y Fm(\033)365 842 y Fd(2)364 867 y Fl(2)391 857 y Fs(\()p Fm(\033)431 863 y Fl(1)449 857 y Fs(\()p Fm(s)484 863 y Fk(j)503 857 y Fs(\)\))14 b Fn(+)574 863 y Fi(R)603 867 y Ff(1)635 857 y Fm(\033)660 842 y Fd(2)659 867 y Fl(2)686 857 y Fs(\()p Fm(\033)726 863 y Fl(1)744 857 y Fs(\()p Fm(t)775 863 y Fk(j)793 857 y Fs(\)\))h(and)g(th)o(us)h Fm(\033)1040 842 y Fd(2)1039 867 y Fl(2)1066 857 y Fs(\()p Fm(\033)1106 863 y Fl(1)1124 857 y Fs(\()p Fm(l)q Fs(\)\))f Fn(\))1242 863 y Fi(R)1271 867 y Ff(1)1302 857 y Fm(\033)1327 842 y Fd(2)1326 867 y Fl(2)1353 857 y Fs(\()p Fm(\033)1393 863 y Fl(1)1411 857 y Fs(\()p Fm(r)q Fs(\)\).)g(Let)1592 847 y(^)1582 857 y Fm(C)s Fs([)d(])j(b)q(e)h(the)340 907 y(con)o(text)j(obtained)f(from)f Fm(C)s Fs([)12 b(])17 b(b)o(y)h(replacing)g(all)f(white)i(principal)e(subterms)i(with)f Fj(2)p Fs(.)340 957 y(No)o(w)c(\(1\))g(follo)o(ws)e(from)g Fm(top)792 942 y Fk(b)809 957 y Fs(\()p Fm(s)p Fs(\))g(=)925 946 y(^)916 957 y Fm(C)s Fs([)p Fm(\033)986 942 y Fd(2)985 967 y Fl(2)1011 957 y Fs(\()p Fm(\033)1051 963 y Fl(1)1069 957 y Fs(\()p Fm(l)q Fs(\)\)])i(and)g Fm(top)1293 942 y Fk(b)1310 957 y Fs(\()p Fm(t)p Fs(\))d(=)1422 946 y(^)1412 957 y Fm(C)s Fs([)p Fm(\033)1482 942 y Fd(2)1481 967 y Fl(2)1507 957 y Fs(\()p Fm(\033)1547 963 y Fl(1)1566 957 y Fs(\()p Fm(r)q Fs(\)\)].)340 1007 y(\(2\))17 b(Let)g Fm(s)f Fn(!)564 992 y Fk(o)598 1007 y Fm(t)h Fs(b)o(y)f(some)g(rule)h (from)e Fn(R)1019 1013 y Fl(2)1054 1007 y Fs(or)i Fm(s)f Fn(!)1185 992 y Fk(i)1214 1007 y Fm(t)p Fs(.)g(Since)i Fn(R)1404 1013 y Fl(2)1439 1007 y Fs(is)e(la)o(y)o(er-preserving,)340 1056 y(w)o(e)d(ma)o(y)d(write)j Fm(s)f Fs(=)g Fm(C)701 1041 y Fk(b)717 1056 y Fn(h)-7 b(h)p Fm(u)766 1062 y Fl(1)785 1056 y Fm(;)7 b(:)g(:)g(:)t(;)g(u)901 1062 y Fk(j)918 1056 y Fm(;)g(:)g(:)g(:)e(;)i(u)1035 1062 y Fk(p)1053 1056 y Fn(i)-7 b(i)13 b Fs(and)f Fm(t)f Fs(=)h Fm(C)1273 1041 y Fk(b)1290 1056 y Fn(h)-7 b(h)p Fm(u)1339 1062 y Fl(1)1357 1056 y Fm(;)7 b(:)g(:)g(:)e(;)i(u)1474 1041 y Fi(0)1474 1067 y Fk(j)1491 1056 y Fm(;)g(:)g(:)g(:)e(;)i(u)1608 1062 y Fk(p)1626 1056 y Fn(i)-7 b(i)p Fs(,)12 b(where)340 1112 y Fm(u)364 1118 y Fk(j)393 1112 y Fn(!)f Fm(u)470 1097 y Fi(0)470 1123 y Fk(j)487 1112 y Fs(.)j(Hence)h Fm(top)692 1097 y Fk(b)709 1112 y Fs(\()p Fm(s)p Fs(\))d(=)g Fm(top)872 1097 y Fk(b)889 1112 y Fs(\()p Fm(t)p Fs(\).)h Fj(2)403 1202 y Fs(In)h(the)h(preceding)g(prop)q(osition,)f(the)g (assumption)f(that)i(\()p Fn(F)1377 1208 y Fl(2)1396 1202 y Fm(;)7 b Fn(R)1449 1208 y Fl(2)1468 1202 y Fs(\))14 b(has)h(to)f(b)q(e)h(la)o(y)o(er-)340 1252 y(preserving)g(cannot)f(b)q (e)h(dropp)q(ed,)f(as)g(is)f(witnessed)j(b)o(y)d(the)i(next)f(example)f (\(cf.)g([Mid90)o(,)340 1302 y(Mid93]\).)340 1391 y Fg(Example)j(4.20)k Fs(Let)f Fn(R)757 1397 y Fl(1)794 1391 y Fs(=)g Fn(f)p Fm(F)6 b Fs(\()p Fm(x)p Fs(\))17 b Fn(!)h Fm(G)p Fs(\()p Fm(x)p Fs(\))g Fn(\()g Fm(x)12 b Fn(#)h Fm(A)p Fn(g)k Fs(and)h Fn(R)1459 1397 y Fl(2)1496 1391 y Fs(=)g Fn(f)p Fm(h)p Fs(\()p Fm(x)p Fs(\))g Fn(!)g Fm(x)p Fn(g)p Fs(.)340 1441 y(Then)e Fm(F)6 b Fs(\()p Fm(h)p Fs(\()p Fm(A)p Fs(\)\))14 b Fn(!)658 1426 y Fk(o)690 1441 y Fm(G)p Fs(\()p Fm(h)p Fs(\()p Fm(A)p Fs(\)\))h(b)o(y)g(the)h(only)f(rule)g(of)g Fn(R)1250 1447 y Fl(1)1284 1441 y Fs(but)h Fm(top)1418 1426 y Fk(b)1434 1441 y Fs(\()p Fm(F)6 b Fs(\()p Fm(h)p Fs(\()p Fm(a)p Fs(\)\)\)\))14 b(=)h Fm(F)6 b Fs(\()p Fj(2)p Fs(\))340 1491 y(is)14 b(a)g(normal)e(form)g(w.r.t.)g Fn(\))808 1497 y Fi(R)837 1501 y Ff(1)855 1491 y Fs(.)403 1581 y(Our)e(next)h(goal)e(is)h(to)g(sho)o(w)g(an)g(analogous)f (statemen)o(t)h(to)g(Prop)q(osition)g(4.19)f(\(1\))h(with-)340 1630 y(out)16 b(the)g(la)o(y)o(er-preservingness)h(requiremen)o(t)e(on) h(\()p Fn(F)1207 1636 y Fl(2)1225 1630 y Fm(;)7 b Fn(R)1279 1636 y Fl(2)1298 1630 y Fs(\))15 b(but)h(under)g(the)g(additional)340 1680 y(assumption)d(that)h Fn(!)692 1686 y Fl(1)p Fk(;)p Fl(2)750 1680 y Fs(is)g(semi-complete.)340 1770 y Fg(De\014nition)e (4.21)21 b Fs(Let)k(the)g(rewrite)g(relation)f Fn(!)1192 1776 y Fl(1)p Fk(;)p Fl(2)1261 1770 y Fs(b)q(e)h(semi-complete.)d(F)m (or)i Fm(t)29 b Fs(=)340 1820 y Fm(C)373 1805 y Fk(b)390 1820 y Fn(h)-7 b(h)p Fm(t)430 1826 y Fl(1)449 1820 y Fm(;)7 b(:)g(:)g(:)t(;)g(t)556 1826 y Fk(m)587 1820 y Fn(i)-7 b(i)q Fs(,)13 b(w)o(e)h(de\014ne)h Fm(top)875 1805 y Fk(b)875 1830 y Fi(!)910 1820 y Fs(\()p Fm(t)p Fs(\))g(b)o(y:)769 1904 y Fm(top)825 1886 y Fk(b)825 1914 y Fi(!)860 1904 y Fs(\()p Fm(t)p Fs(\))d(=)f Fm(top)1018 1886 y Fk(b)1035 1904 y Fs(\()p Fm(C)1084 1886 y Fk(b)1101 1904 y Fn(h)p Fm(t)1132 1886 y Fi(!)1132 1914 y Fl(1)1167 1904 y Fm(;)c(:)g(:)g(:)e(;)i(t)1275 1886 y Fi(!)1275 1914 y Fk(m)1309 1904 y Fn(i)p Fs(\))p Fm(:)403 1993 y Fs(In)12 b(other)g(w)o(ords,)g(\014rst)h(the)g(white)f(principal)f (subterms)h(in)g Fm(t)g Fs(are)g(replaced)h(with)f(their)340 2043 y(unique)h Fn(!)516 2049 y Fl(1)p Fk(;)p Fl(2)574 2043 y Fs(normal)e(form,)g(and)h(then)i(the)g(topmost)e(blac)o(k)g (homogeneous)g(part)h(of)g(the)340 2093 y(term)h(obtained)f(is)h(tak)o (en.)340 2183 y Fg(Lemma)i(4.22)k Fs(Let)14 b Fn(!)731 2189 y Fl(1)p Fk(;)p Fl(2)789 2183 y Fs(b)q(e)g(semi-complete.)d(If)i Fm(s)h Fs(and)f Fm(t)h Fs(are)f(blac)o(k)g(terms)h(and)f Fm(\033)h Fs(is)g(a)340 2232 y(top)h(white)g(substitution)g(suc)o(h)h (that)e Fm(s\033)h Fn(!)1046 2217 y Fk(o)1077 2232 y Fm(t\033)h Fs(b)o(y)f(some)f(rewrite)h(rule)g(from)f Fn(R)1656 2238 y Fl(1)1674 2232 y Fs(,)h(then)340 2282 y Fm(\033)365 2267 y Fi(!)401 2282 y Fs(\()p Fm(s)p Fs(\))d Fn(!)506 2267 y Fk(o)506 2293 y Fl(1)536 2282 y Fm(\033)561 2267 y Fi(!)596 2282 y Fs(\()p Fm(t)p Fs(\).)340 2332 y Fg(Pro)q(of:)k Fs(There)h(is)f(a)g(con)o(text)h Fm(C)s Fs([)12 b(],)j(a)h(substitution)h Fm(\032)f Fs(:)f Fn(V)k(!)c(T)10 b Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))17 b(and)f(a)g(rewrite)340 2382 y(rule)21 b Fm(l)k Fn(!)d Fm(r)i Fn(\()f Fm(s)658 2388 y Fl(1)690 2382 y Fn(#)12 b Fm(t)738 2388 y Fl(1)756 2382 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)868 2388 y Fk(n)903 2382 y Fn(#)12 b Fm(t)951 2388 y Fk(n)997 2382 y Fn(2)23 b(R)1083 2388 y Fl(1)1122 2382 y Fs(suc)o(h)f(that)f Fm(s\033)k Fs(=)e Fm(C)s Fs([)p Fm(l)q(\032)p Fs(],)d Fm(t\033)k Fs(=)f Fm(C)s Fs([)p Fm(r)q(\032)p Fs(])340 2432 y(and)c Fm(s)445 2438 y Fk(j)463 2432 y Fm(\032)12 b Fn(#)h Fm(t)545 2438 y Fk(j)562 2432 y Fm(\032)19 b Fs(for)f Fm(j)k Fn(2)c(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)e(;)i(n)p Fn(g)p Fs(.)17 b(Fix)h Fm(j)r Fs(.)g(F)m(rom)f(Prop)q(osition)h(4.12)f (w)o(e)h(kno)o(w)g(that)p eop %%Page: 15 15 15 14 bop 183 194 a Fm(s)202 200 y Fk(j)220 194 y Fm(\032)13 b Fn(#)274 204 y Fl(1)p Fk(;)p Fl(2)332 194 y Fm(t)347 200 y Fk(j)364 194 y Fm(\032)p Fs(.)k(According)h(to)f(Prop)q(osition)g (3.12,)f Fm(\032)h Fs(can)h(b)q(e)g(decomp)q(osed)f(in)o(to)g Fm(\032)1521 200 y Fl(2)1551 194 y Fn(\016)12 b Fm(\032)1605 200 y Fl(1)183 244 y Fs(suc)o(h)h(that)g Fm(\032)385 250 y Fl(1)417 244 y Fs(is)f(blac)o(k)g(and)h Fm(\032)665 250 y Fl(2)697 244 y Fs(is)f(top)h(white.)f(Since)i Fn(!)1083 250 y Fl(1)p Fk(;)p Fl(2)1140 244 y Fs(is)e(semi-complete,)f(it)h (follo)o(ws)183 293 y(as)g(in)g(the)i(pro)q(of)e(of)g(Lemma)e(4.9)h (that)i Fm(\032)825 278 y Fi(!)825 304 y Fl(2)861 293 y Fs(\()p Fm(\032)898 299 y Fl(1)917 293 y Fs(\()p Fm(s)952 299 y Fk(j)970 293 y Fs(\)\))g Fn(#)1035 303 y Fl(1)p Fk(;)p Fl(2)1093 293 y Fm(\032)1114 278 y Fi(!)1114 304 y Fl(2)1149 293 y Fs(\()p Fm(\032)1186 299 y Fl(1)1206 293 y Fs(\()p Fm(t)1237 299 y Fk(j)1254 293 y Fs(\)\).)g(Applying)e (Lemma)183 343 y(4.10)f(to)h(the)h(blac)o(k)f(terms)g Fm(\032)625 349 y Fl(1)644 343 y Fs(\()p Fm(s)679 349 y Fl(1)698 343 y Fs(\))p Fm(;)c(:)g(:)g(:)e(;)i(\032)828 349 y Fl(1)847 343 y Fs(\()p Fm(s)882 349 y Fk(n)905 343 y Fs(\))p Fm(;)g(\032)961 349 y Fl(1)980 343 y Fs(\()p Fm(t)1011 349 y Fl(1)1029 343 y Fs(\))p Fm(;)g(:)g(:)g(:)e(;)i(\032) 1159 349 y Fl(1)1178 343 y Fs(\()p Fm(t)1209 349 y Fk(n)1232 343 y Fs(\))k(and)g(the)h(substitution)183 393 y Fm(\032)204 378 y Fi(!)204 403 y Fl(2)257 393 y Fs(yields)18 b Fm(\032)399 378 y Fi(!)399 403 y Fl(2)435 393 y Fs(\()p Fm(\032)472 399 y Fl(1)491 393 y Fs(\()p Fm(s)526 399 y Fk(j)544 393 y Fs(\)\))13 b Fn(#)610 374 y Fk(o)610 403 y Fl(1)641 393 y Fm(\032)662 378 y Fi(!)662 403 y Fl(2)697 393 y Fs(\()p Fm(\032)734 399 y Fl(1)754 393 y Fs(\()p Fm(t)785 399 y Fk(j)802 393 y Fs(\)\).)18 b(Therefore,)h Fm(\032)1089 378 y Fi(!)1089 403 y Fl(2)1125 393 y Fs(\()p Fm(\032)1162 399 y Fl(1)1181 393 y Fs(\()p Fm(l)q Fs(\)\))g Fn(!)1303 378 y Fk(o)1303 403 y Fl(1)1352 393 y Fm(\032)1373 378 y Fi(!)1373 403 y Fl(2)1409 393 y Fs(\()p Fm(\032)1446 399 y Fl(1)1465 393 y Fs(\()p Fm(r)q Fs(\)\).)f(Let)192 438 y(^)183 448 y Fm(C)r Fs([)12 b(])h(b)q(e)g(the)h(con)o(text)g (obtained)e(from)g Fm(C)s Fs([)f(])i(b)o(y)f(replacing)h(all)f(white)h (principal)f(subterms)183 498 y(with)j(their)i(resp)q(ectiv)o(e)h Fn(!)618 504 y Fl(1)p Fk(;)p Fl(2)679 498 y Fs(normal)c(form.)g(It)i (is)g(clear)g(that)g Fm(\033)1243 483 y Fi(!)1279 498 y Fs(\()p Fm(s)p Fs(\))g(=)1403 488 y(^)1393 498 y Fm(C)s Fs([)p Fm(\032)1459 483 y Fi(!)1459 508 y Fl(2)1494 498 y Fs(\()p Fm(\032)1531 504 y Fl(1)1550 498 y Fs(\()p Fm(l)q Fs(\)\)])183 553 y(and)d Fm(\033)288 538 y Fi(!)324 553 y Fs(\()p Fm(t)p Fs(\))f(=)436 543 y(^)426 553 y Fm(C)s Fs([)p Fm(\032)492 538 y Fi(!)492 564 y Fl(2)527 553 y Fs(\()p Fm(\032)564 559 y Fl(1)583 553 y Fs(\()p Fm(r)q Fs(\)\)].)i(Th)o(us)g Fm(\033)819 538 y Fi(!)854 553 y Fs(\()p Fm(s)p Fs(\))e Fn(!)959 538 y Fk(o)959 564 y Fl(1)989 553 y Fm(\033)1014 538 y Fi(!)1049 553 y Fs(\()p Fm(t)p Fs(\).)i Fj(2)183 656 y Fg(Prop)q(ositi)o(on)f(4.23)20 b Fs(Let)13 b Fn(!)661 662 y Fl(1)p Fk(;)p Fl(2)718 656 y Fs(b)q(e)g(semi-complete.)d(If)h Fm(s)h Fn(!)1165 641 y Fk(o)1195 656 y Fm(t)g Fs(b)o(y)g(some)f(rule)i(from)d Fn(R)1593 662 y Fl(1)1612 656 y Fs(,)183 706 y(then)k Fm(top)333 691 y Fk(b)333 717 y Fi(!)368 706 y Fs(\()p Fm(s)p Fs(\))f Fn(\))474 712 y Fi(R)503 716 y Ff(1)531 706 y Fm(top)587 691 y Fk(b)587 717 y Fi(!)623 706 y Fs(\()p Fm(t)p Fs(\).)183 759 y Fg(Pro)q(of:)23 b Fs(W)m(e)i(ma)o(y)d (write)j Fm(s)30 b Fs(=)g Fm(C)780 744 y Fk(b)797 759 y Fn(f)-14 b(f)o Fm(s)843 765 y Fl(1)862 759 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)974 765 y Fk(n)997 759 y Fn(g)-14 b(g)24 b Fs(and)g Fm(t)30 b Fs(=)1256 748 y(^)1247 759 y Fm(C)1280 744 y Fk(b)1296 759 y Fn(h)-7 b(h)p Fm(s)1340 765 y Fk(i)1352 769 y Ff(1)1371 759 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)1483 765 y Fk(i)1495 769 y Fh(m)1524 759 y Fn(i)-7 b(i)25 b Fs(for)183 809 y(some)c(blac)o(k)h(con)o(texts)i Fm(C)618 794 y Fk(b)634 809 y Fn(f)p Fm(;)7 b(:)g(:)g(:)e(;)i Fn(g)p Fs(,)811 798 y(^)802 809 y Fm(C)835 794 y Fk(b)851 809 y Fn(h)p Fm(;)g(:)g(:)g(:)e(;)i Fn(i)p Fs(,)22 b(and)g Fm(i)1113 815 y Fl(1)1132 809 y Fm(;)7 b(:)g(:)g(:)e(;)i(i)1239 815 y Fk(m)1296 809 y Fn(2)26 b(f)p Fs(1)p Fm(;)7 b(:)g(:)g(:)t(;)g(n)p Fn(g)p Fs(.)21 b(Let)183 859 y Fm(x)207 865 y Fl(1)225 859 y Fm(;)7 b(:)g(:)g(:)e(;)i(x)342 865 y Fk(n)381 859 y Fs(b)q(e)19 b(distinct)e(fresh)i(v)n(ariables)e(and)g(de\014ne)i Fm(\033)g Fs(=)f Fn(f)p Fm(x)1222 865 y Fk(j)1257 859 y Fn(7!)g Fm(s)1336 865 y Fk(j)1366 859 y Fn(j)12 b Fs(1)18 b Fn(\024)g Fm(j)i Fn(\024)e Fm(n)p Fn(g)p Fs(,)183 914 y Fm(s)202 899 y Fi(0)227 914 y Fs(=)13 b Fm(C)305 899 y Fk(b)321 914 y Fn(f)p Fm(x)366 920 y Fl(1)384 914 y Fm(;)7 b(:)g(:)g(:)e(;)i(x)501 920 y Fk(n)523 914 y Fn(g)p Fs(,)14 b(and)g Fm(t)666 899 y Fi(0)691 914 y Fs(=)745 903 y(^)736 914 y Fm(C)769 899 y Fk(b)785 914 y Fn(h)p Fm(x)825 920 y Fk(i)837 924 y Ff(1)855 914 y Fm(;)7 b(:)g(:)g(:)e(;)i (x)972 920 y Fk(i)984 924 y Fh(m)1012 914 y Fn(i)q Fs(.)14 b(Since)h Fm(\033)h Fs(is)e(top)h(white,)f(w)o(e)h(obtain)183 964 y Fm(\033)208 949 y Fi(!)243 964 y Fs(\()p Fm(s)278 949 y Fi(0)290 964 y Fs(\))k Fn(!)367 949 y Fk(o)367 974 y Fl(1)404 964 y Fm(\033)429 949 y Fi(!)464 964 y Fs(\()p Fm(t)495 949 y Fi(0)507 964 y Fs(\))f(b)o(y)g(Lemma)d(4.22.)i (According)h(to)g(Prop)q(osition)g(3.12,)e Fm(\033)1471 949 y Fi(!)1525 964 y Fs(has)i(a)183 1014 y(decomp)q(osition)c Fm(\033)483 999 y Fi(!)533 1014 y Fs(=)h Fm(\033)604 1020 y Fl(2)633 1014 y Fn(\016)10 b Fm(\033)688 1020 y Fl(1)707 1014 y Fs(,)15 b(where)i Fm(\033)880 1020 y Fl(1)914 1014 y Fs(is)e(blac)o(k)h(and)f Fm(\033)1174 1020 y Fl(2)1208 1014 y Fs(is)h(top)g(white.)f(It)h(follo)o(ws)183 1063 y(from)10 b(Lemma)g(4.5)i(that)g Fm(\033)605 1048 y Fd(2)604 1074 y Fl(2)631 1063 y Fs(\()p Fm(\033)671 1069 y Fl(1)690 1063 y Fs(\()p Fm(s)725 1048 y Fi(0)737 1063 y Fs(\)\))g Fn(!)823 1048 y Fk(o)823 1074 y Fl(1)852 1063 y Fm(\033)877 1048 y Fd(2)876 1074 y Fl(2)903 1063 y Fs(\()p Fm(\033)943 1069 y Fl(1)961 1063 y Fs(\()p Fm(t)992 1048 y Fi(0)1004 1063 y Fs(\)\))h(b)q(ecause)h Fm(\033)1225 1069 y Fl(2)1255 1063 y Fn(/)e Fm(\033)1324 1048 y Fd(2)1323 1074 y Fl(2)1350 1063 y Fs(.)g(T)m(o)f(v)o(erify)h (that)183 1113 y Fm(\033)208 1098 y Fd(2)207 1123 y Fl(2)233 1113 y Fs(\()p Fm(\033)273 1119 y Fl(1)292 1113 y Fs(\()p Fm(s)327 1098 y Fi(0)339 1113 y Fs(\)\))i Fn(\))427 1119 y Fi(R)456 1123 y Ff(1)486 1113 y Fm(\033)511 1098 y Fd(2)510 1123 y Fl(2)537 1113 y Fs(\()p Fm(\033)577 1119 y Fl(1)596 1113 y Fs(\()p Fm(t)627 1098 y Fi(0)638 1113 y Fs(\)\))i(is)e(relativ)o(ely)h(simple.)e(No)o(w)h Fm(top)1204 1098 y Fk(b)1204 1123 y Fi(!)1240 1113 y Fs(\()p Fm(s)p Fs(\))g Fn(\))1347 1119 y Fi(R)1376 1123 y Ff(1)1407 1113 y Fm(top)1463 1098 y Fk(b)1463 1123 y Fi(!)1498 1113 y Fs(\()p Fm(t)p Fs(\))h(is)g(a)183 1163 y(consequence)h(of)183 1266 y Fm(top)239 1249 y Fk(b)239 1276 y Fi(!)274 1266 y Fs(\()p Fm(s)p Fs(\))c(=)g Fm(top)437 1249 y Fk(b)454 1266 y Fs(\()p Fm(C)503 1249 y Fk(b)519 1266 y Fn(f)p Fm(s)559 1249 y Fi(!)559 1276 y Fl(1)594 1266 y Fm(;)7 b(:)g(:)g(:)e(;)i(s)706 1249 y Fi(!)706 1276 y Fk(n)741 1266 y Fn(g)p Fs(\))12 b(=)g Fm(top)890 1249 y Fk(b)906 1266 y Fs(\()p Fm(\033)947 1249 y Fi(!)983 1266 y Fs(\()p Fm(s)1018 1249 y Fi(0)1030 1266 y Fs(\)\))g(=)g Fm(top)1174 1249 y Fk(b)1190 1266 y Fs(\()p Fm(\033)1230 1272 y Fl(2)1249 1266 y Fs(\()p Fm(\033)1289 1272 y Fl(1)1307 1266 y Fs(\()p Fm(s)1342 1249 y Fi(0)1354 1266 y Fs(\)\)\))g(=)g Fm(\033)1483 1249 y Fd(2)1482 1276 y Fl(2)1509 1266 y Fs(\()p Fm(\033)1549 1272 y Fl(1)1567 1266 y Fs(\()p Fm(s)1602 1249 y Fi(0)1615 1266 y Fs(\)\))183 1368 y(and)h Fm(top)319 1353 y Fk(b)319 1379 y Fi(!)355 1368 y Fs(\()p Fm(t)p Fs(\))e(=)h Fm(\033)482 1353 y Fd(2)481 1379 y Fl(2)508 1368 y Fs(\()p Fm(\033)548 1374 y Fl(1)566 1368 y Fs(\()p Fm(t)597 1353 y Fi(0)609 1368 y Fs(\)\).)i Fj(2)245 1471 y Fs(The)j(ab)q(o)o(v)o(e)g (preparatory)h(considerations)f(and)g(the)g(follo)o(wing)e(t)o(w)o(o)h (lemmata)e(\(the)183 1521 y(pro)q(ofs)d(of)f(whic)o(h)h(are)h (omitted\))e(pa)o(v)o(e)g(the)i(w)o(a)o(y)f(for)f(the)i(main)d(results) j(of)f(this)g(subsection.)183 1627 y Fg(Lemma)k(4.24)21 b Fs(If)14 b Fn(!)g Fs(is)g(w)o(ell-founded)g(on)h(a)f(set)i Fm(T)i Fn(\022)13 b(T)d Fs(\()p Fn(F)t Fm(;)d Fn(V)s Fs(\))q(,)14 b(then)h Fm(>)7 b Fs(=)14 b(\()p Fn(!)e([)p 1551 1629 3 25 v 9 w Fm(>)p Fs(\))1596 1612 y Fl(+)183 1677 y Fs(is)h(a)h(w)o(ell-founded)f(partial)g(ordering)h(on)g Fm(T)6 b Fs(.)13 b Fj(2)183 1780 y Fg(Lemma)i(4.25)21 b Fs(Let)c Fm(s)h Fn(!)614 1765 y Fk(o)649 1780 y Fm(t)f Fs(b)o(y)g(an)g(application)f(of)g(a)h(non-duplicating)f(rewrite)i (rule)183 1830 y(from)12 b Fn(R)316 1836 y Fl(1)335 1830 y Fs(.)h(Then)h Fm(S)495 1815 y Fk(w)493 1841 y(P)523 1830 y Fs(\()p Fm(t)p Fs(\))e Fn(\022)g Fm(S)653 1815 y Fk(w)651 1841 y(P)680 1830 y Fs(\()p Fm(s)p Fs(\).)j Fj(2)183 1933 y Fg(Theorem)f(4.26)21 b Fs(Let)11 b Fn(R)597 1939 y Fl(1)626 1933 y Fs(and)f Fn(R)738 1939 y Fl(2)767 1933 y Fs(b)q(e)h(terminating)d(constructor-sharing)k(CTRSs)e(suc)o(h) 183 1983 y(that)i(their)g(com)o(bined)f(system)h Fn(R)g Fs(=)g Fn(R)814 1989 y Fl(1)838 1983 y Fn([)6 b(R)907 1989 y Fl(2)938 1983 y Fs(is)12 b(not)g(terminating.)e(Then)i(the)h (follo)o(wing)183 2032 y(statemen)o(ts)h(hold)f(\(where)i Fm(d;)668 2021 y Fs(\026)661 2032 y Fm(d)10 b Fn(2)i(f)p Fs(1)p Fm(;)7 b Fs(2)p Fn(g)12 b Fs(with)i Fm(d)d Fn(6)p Fs(=)1027 2021 y(\026)1019 2032 y Fm(d)p Fs(\):)200 2127 y(1.)20 b(There)f(exists)f(an)g(in\014nite)f Fn(R)h Fs(rewrite)g(deriv) n(ation)f Fm(D)i Fs(:)e Fm(s)1198 2133 y Fl(1)1235 2127 y Fn(!)g Fm(s)1313 2133 y Fl(2)1350 2127 y Fn(!)g Fm(s)1428 2133 y Fl(3)1465 2127 y Fn(!)g Fm(:)7 b(:)g(:)16 b Fs(of)253 2177 y(minima)o(l)e(rank)k(suc)o(h)g(that)g Fm(D)h Fs(con)o(tains)f (in\014nitely)f(man)o(y)f Fm(s)1243 2183 y Fk(j)1278 2177 y Fn(!)1320 2162 y Fk(o)1356 2177 y Fm(s)1375 2183 y Fk(j)r Fl(+1)1453 2177 y Fs(reduction)253 2227 y(steps)f(where)g Fm(s)496 2233 y Fk(j)528 2227 y Fs(reduces)h(to)e Fm(s)746 2233 y Fk(j)r Fl(+1)820 2227 y Fs(b)o(y)f(some)g(rule)i(from)d Fn(R)1198 2233 y Fk(d)1217 2227 y Fs(.)200 2279 y(2.)20 b Fn(R)294 2281 y Fl(\026)288 2289 y Fk(d)322 2279 y Fs(is)13 b(not)h(la)o(y)o(er-preserving.)200 2332 y(3.)20 b(If)c(b)q(oth)g(systems)g(are)g(con\015uen)o(t,)h(then)f Fm(D)i Fs(con)o(tains)d(in\014nitely)h(man)o(y)e(duplicating)253 2382 y Fm(s)272 2388 y Fk(j)303 2382 y Fn(!)345 2367 y Fk(o)376 2382 y Fm(s)395 2388 y Fk(j)r Fl(+1)470 2382 y Fs(reduction)h(steps)i(suc)o(h)e(that)g Fm(s)965 2388 y Fk(j)998 2382 y Fs(reduces)h(to)f Fm(s)1217 2388 y Fk(j)r Fl(+1)1292 2382 y Fs(b)o(y)f(some)g(rule)h(from)253 2432 y Fn(R)288 2438 y Fk(d)308 2432 y Fs(.)p eop %%Page: 16 16 16 15 bop 340 194 a Fg(Pro)q(of:)17 b Fs(First)i(of)e(all,)f(the)j (rank)f(of)f(a)h(deriv)n(ation)f Fm(D)j Fs(:)d Fm(s)1284 200 y Fl(1)1322 194 y Fn(!)h Fm(s)1401 200 y Fl(2)1438 194 y Fn(!)g Fm(s)1517 200 y Fl(3)1554 194 y Fn(!)g Fm(:)7 b(:)g(:)16 b Fs(is)i(de-)340 244 y(\014ned)f(to)f(b)q(e)h Fm(r)q(ank)q Fs(\()p Fm(D)q Fs(\))f(=)g Fm(r)q(ank)q Fs(\()p Fm(s)902 250 y Fl(1)921 244 y Fs(\).)g(Let)h Fm(D)h Fs(b)q(e)f(an)f(in\014nite)g Fn(R)g Fs(rewrite)h(deriv)n(ation)f (of)340 293 y(minima)o(l)e(rank,)j(sa)o(y)h Fm(r)q(ank)q Fs(\()p Fm(D)q Fs(\))g(=)g Fm(k)q Fs(.)f(Then)h Fm(r)q(ank)q Fs(\()p Fm(s)1206 299 y Fk(j)1223 293 y Fs(\))g(=)g Fm(r)q(ank)q Fs(\()p Fm(D)q Fs(\))g(for)f(all)f(indices)i Fm(j)r Fs(.)340 343 y(Moreo)o(v)o(er,)f Fn(!)579 349 y Fi(R)626 343 y Fs(is)f(terminating)f(on)h Fn(T)992 328 y Fk()f Fs(=)f(\()p Fn(!)628 1428 y Fi(R)669 1422 y Fn([)p 710 1424 3 25 v 9 w Fm(>)p Fs(\))754 1406 y Fl(+)782 1422 y Fs(.)h(According)h(to)f(Lemma)e(4.24,) h(\()p Fn(T)1350 1406 y Fk()p Fs(\))13 b(is)h(a)f(w)o(ell-founded)340 1471 y(ordering.)g(Let)h(\()p Fn(M)p Fs(\()p Fn(T)704 1456 y Fk()818 1456 y Fk(mul)879 1471 y Fs(\))14 b(denote)g(its)f(w)o (ell-founded)g(m)o(ultiset)e(extension.)j(Note)340 1521 y(that)j Fm(S)460 1506 y Fk(w)458 1533 y(P)488 1521 y Fs(\()p Fm(s)523 1527 y Fk(j)541 1521 y Fs(\))f Fn(2)f(M)p Fs(\()p Fn(T)715 1506 y Fk()1237 1986 y Fk(mul)1310 2001 y Fm(S)1337 1986 y Fk(w)1335 2013 y(P)1365 2001 y Fs(\()p Fm(s)1400 2007 y Fk(j)r Fl(+1)1460 2001 y Fs(\).)366 2051 y Fg({)21 b Fs(If)14 b Fm(s)472 2057 y Fk(j)501 2051 y Fn(!)d Fm(s)573 2057 y Fk(j)r Fl(+1)647 2051 y Fs(b)o(y)j(some)f(rule)h(from)e Fn(R)1026 2057 y Fl(2)1044 2051 y Fs(,)i(then)g(there)i(is)d(a)h(white) g(principal)f(subterm)411 2101 y Fm(u)22 b Fn(2)f Fm(S)533 2086 y Fk(w)531 2112 y(P)561 2101 y Fs(\()p Fm(s)596 2107 y Fk(j)614 2101 y Fs(\))f(suc)o(h)h(that)f Fm(u)i Fn(!)f Fm(v)h Fs(for)e(some)f Fm(v)q Fs(,)h(i.e.,)f Fm(s)1334 2107 y Fk(j)1374 2101 y Fs(=)j Fm(C)1461 2086 y Fk(b)1477 2101 y Fs([)-7 b([)p Fm(;)7 b(:)g(:)g(:)t(;)g(u;)g(:)g(:)g(:)t(;)g Fs(])-7 b(])21 b Fn(!)411 2151 y Fm(C)444 2136 y Fk(b)460 2151 y Fs([)p Fm(;)7 b(:)g(:)g(:)e(;)i(v)q(;)g(:)g(:)g(:)t(;)g Fs(])k(=)h Fm(s)764 2157 y Fk(j)r Fl(+1)824 2151 y Fs(.)h(Th)o(us)h(w)o 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Fk(j)r Fl(+1)931 2300 y Fs(\).)340 2382 y(W)m(e)18 b(conclude)g(from)e(the)i(w)o(ell-foundedness)g(of)f(\()p Fn(M)p Fs(\()p Fn(T)1253 2367 y Fk()1367 2367 y Fk(mul)1429 2382 y Fs(\))17 b(that)h(only)f(a)g (\014nite)340 2432 y(n)o(um)o(b)q(er)c(of)g(inner)h(reduction)g(steps)h (as)e(w)o(ell)g(as)h(reduction)g(steps)h(using)e(a)g(rule)h(from)e Fn(R)1763 2438 y Fl(2)p eop %%Page: 17 17 17 16 bop 183 194 a Fs(o)q(ccur)16 b(in)f Fm(D)q Fs(.)h(W.l.o.g.)c(w)o (e)j(ma)o(y)f(supp)q(ose)j(that)e(there)i(are)f(no)f(reduction)h(steps) h(of)d(that)183 244 y(kind)i(in)h Fm(D)q Fs(.)g(Consequen)o(tly)m(,)f (for)h(all)f Fm(j)j Fn(2)d Fs(I)-7 b(N)q(,)16 b(w)o(e)h(ha)o(v)o(e)g Fm(s)1121 250 y Fk(j)1156 244 y Fn(!)1198 228 y Fk(o)1233 244 y Fm(s)1252 250 y Fk(j)r Fl(+1)1329 244 y Fs(b)o(y)f(some)h (rewrite)183 293 y(rule)h(from)e Fn(R)407 299 y Fl(1)425 293 y Fs(.)i(No)o(w)f Fn(!)595 299 y Fl(1)p Fk(;)p Fl(2)657 293 y Fs(is)h(semi-complete)e(b)q(ecause)j(\()p Fn(F)1184 299 y Fl(1)1203 293 y Fm(;)7 b 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b(of)d(Corollary)f(4.27.)g(\(2\))i(and)f(\(3\))h(follo)o(w)183 1226 y(from)12 b(Theorem)h(4.14)g(in)g(conjunction)h(with)g(Corollary)e (4.27.)g Fj(2)245 1302 y Fs(Theorem)h(4.17)f(is)i(also)f(true)i(for)e (semi-equational)e(CTRSs,)i(see)i([Mid90)o(].)e(Again,)f(it)183 1352 y(is)g(plausible)f(that)i(Corollaries)e(4.27)g(and)h(4.28)g(also)f (hold)h(for)g(semi-equational)e(CTRSs.)183 1474 y Fp(5)56 b(Conclusions)183 1562 y Fs(W)m(e)17 b(ha)o(v)o(e)g(sho)o(wn)g(that)h (semi-completeness)f(is)g(a)g(mo)q(dular)f(prop)q(ert)o(y)i(of)f (constructor-)183 1612 y(sharing)9 b(CTRSs)g(and)g(that)g(completeness) h(is)f(mo)q(dular)f(for)h(non-duplicating)f(constructor-)183 1661 y(sharing)20 b(CTRSs.)g(It)h(is)f(de\014nitely)h(w)o(orth)o(while) f(to)g(try)h(to)f(extend)i(these)g(results)g(to)183 1711 y(more)12 b(general)h(kinds)g(of)f(com)o(binations)f(of)h(CTRSs.)h(In)g (particular,)f(the)i(question)f(arises)183 1761 y(whether)19 b(the)g(ab)q(o)o(v)o(e)e(results)i(also)f(hold)f(for)h(comp)q(osable)e 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2286 y(<)628 2361 y(>)628 2374 y(>)628 2386 y(>)628 2398 y(>)628 2411 y(:)671 2234 y Fs(0)g(+)h Fm(x)149 b Fn(!)11 b Fm(x)671 2284 y(S)r Fs(\()p Fm(x)p Fs(\))f(+)g Fm(y)90 b Fn(!)11 b Fm(S)r Fs(\()p Fm(x)f Fs(+)f Fm(y)q Fs(\))671 2334 y Fm(f)t(ib)p Fs(\(0\))136 b Fn(!)11 b Fs(0)671 2384 y Fm(f)t(ib)p Fs(\()p Fm(S)r Fs(\(0\)\))77 b Fn(!)11 b Fm(S)r Fs(\(0\))671 2433 y Fm(f)t(ib)p Fs(\()p Fm(S)r Fs(\()p Fm(S)r Fs(\()p Fm(x)p Fs(\)\)\))k Fn(!)c Fm(f)t(ib)p Fs(\()p Fm(S)r Fs(\()p Fm(x)p Fs(\)\))g(+)f Fm(f)t(ib)p Fs(\()p Fm(x)p Fs(\))p eop %%Page: 18 18 18 17 bop 340 194 a Fn(R)376 200 y Fl(1)412 194 y Fs(and)18 b Fn(R)533 200 y Fl(2)569 194 y Fs(share)i(the)e(constructors)j(0)d (and)g Fm(S)j Fs(as)d(w)o(ell)g(as)g(the)h(de\014ned)g(sym)o(b)q(ol)e (+.)340 244 y Fn(R)376 250 y Fl(1)414 244 y Fs(and)i Fn(R)536 250 y Fl(2)574 244 y Fs(are)h(comp)q(osable)f(since)h(the)h(t) o(w)o(o)e(de\014ning)h(rewrite)g(rules)h(for)e(+)h(o)q(ccur)340 293 y(in)c(b)q(oth)g(of)g(them.)f(Recen)o(tly)m(,)g(w)o(e)i(ha)o(v)o(e) e(sho)o(wn)i(that)f(the)h(ab)q(o)o(v)o(e)e(results)j(also)d(hold)h(for) 340 343 y(comp)q(osable)11 b(unconditional)g(TRSs)h({)g(see)h([Ohl94a)o (].)e(This)h(giv)o(es)g(reason)h(to)f(exp)q(ect)h(that)340 393 y(the)19 b(aforemen)o(tioned)d(question)i(has)f(an)h(answ)o(er)g (in)f(the)h(a\016rmativ)o(e.)d(Note,)i(ho)o(w)o(ev)o(er,)340 443 y(that)12 b(the)g(pro)q(ofs)g(presen)o(ted)h(in)e(this)h(pap)q(er)g (do)g(not)f(carry)h(o)o(v)o(er)g(to)f(comp)q(osable)g(systems.)340 493 y(Moreo)o(v)o(er,)18 b(w)o(e)f(p)q(oin)o(t)g(out)h(that)f(there)i (are)f(t)o(w)o(o)f(closely)g(related)h(results)g(obtained)g(b)o(y)340 542 y(Middeldorp)h([Mid94)n(].)f(He)h(pro)o(v)o(ed)g(that)f (semi-completeness)g(and)h(completeness)g(are)340 592 y(mo)q(dular)10 b(for)h(comp)q(osable)g(conditional)f(constructor)j (systems)f(without)f(extra)h(v)n(ariables)340 642 y(\(it)h(is)f(y)o(et) h(unkno)o(wn)g(if)f(the)h(same)f(is)g(true)i(when)f(extra)g(v)n 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b(is)f(semi-complete)f(accord-)183 244 y(ing)f(to)h(Theorem)f(4.14.)f (Th)o(us)j(ev)o(ery)f(term)g Fm(t)e Fn(2)g(T)g Fs(\()p Fn(F)t Fm(;)c Fn(V)s Fs(\))13 b(has)g(a)g(unique)g(normal)e(form)g Fm(t)p Fn(#)183 293 y Fs(with)16 b(resp)q(ect)i(to)e(\()p Fn(F)t Fm(;)7 b Fn(R)p Fs(\).)16 b(No)o(w)g(if)f(in)h(addition)f(ev)o (ery)i Fn(R)1126 299 y Fk(j)1160 293 y Fs(is)f(\014nite,)g(then,)g(due) h(to)f(the)183 343 y(decreasingness)j(of)d(the)i(constituen)o(t)f (systems,)g Fm(t)p Fn(#)g Fs(is)f(computable)g(b)o(y)h(computing)e(the) 183 393 y(unique)f(normal)f(form)g(of)i Fm(t)f Fs(w.r.t.)g(the)h(mo)q (dular)e(reduction)j(relation.)e(Rigorous)g(pro)q(ofs)183 443 y(of)f(these)i(facts)f(can)h(b)q(e)f(found)g(in)f([Ohl94a)o(].)183 542 y Fg(Ac)o(kno)o(wledgemen)o(t)o(s:)7 b Fs(The)i(author)g(thanks)h (Aart)g(Middeldorp)f(for)g(discussions)h(ab)q(out)183 592 y(the)k(subtleties)h(of)e(CTRSs.)183 729 y Fp(References)183 827 y Fr([Der95])33 b(N.)12 b(Dersho)o(witz.)18 b(Hierarc)o(hical)d(T)m (ermination.)k(1995.)e(This)d(v)o(olume.)183 874 y([DJ90])45 b(N.)12 b(Dersho)o(witz)e(and)f(J.-P)m(.)f(Jouannaud.)17 b(Rewrite)9 b(Systems.)16 b(In)9 b(L.)j(v)n(an)e(Leeu)o(w)o(en,)e(ed-) 337 919 y(itor,)j Fb(Handb)n(o)n(ok)e(of)j(The)n(or)n(etic)n(al)e (Computer)h(Scienc)n(e)p Fr(,)d(v)o(olume)14 b(B,)d(c)o(hapter)j(6.)d (North-)337 965 y(Holland,)j(1990.)183 1011 y([DM79])30 b(N.)12 b(Dersho)o(witz)h(and)f(Z.)h(Manna.)k(Pro)o(ving)d(T)m (ermination)f(with)g(Multiset)g(Orderings.)337 1057 y Fb(Communic)n(ations)e(of)i(the)g(A)o(CM)h Fq(22)p Fb(\(8\))p Fr(,)d(pages)j(465{476,)g(1979.)183 1103 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)337 1149 y(Rewrite)17 b(Systems.)30 b(In)17 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)d (of)j(the)f(9th)h(Confer)n(enc)n(e)e(on)h(A)o(utomate)n(d)f(De-)337 1194 y(duction)p Fr(,)f(pages)k(538{549.)g(Lecture)g(Notes)f(in)h (Computer)g(Science)h Fq(310)p Fr(,)e(Springer)337 1240 y(V)m(erlag,)c(1988.)183 1286 y([FJ94])49 b(M.)12 b(F)m(ern\023)-19 b(andez)12 b(and)f(J.-P)m(.)f(Jouannaud.)18 b(Mo)q(dular)12 b(T)m(ermination)h(of)d(T)m(erm)h(Rewriting)337 1332 y(Systems)i(Revisited.)20 b(In)13 b Fb(ADT)g(Workshop)p Fr(,)e(1994.)183 1378 y([Gra93])30 b(B.)12 b(Gramlic)o(h.)25 b(Su\016cien)o(t)17 b(Conditions)g(for)e(Mo)q(dular)i(T)m(ermination)g (of)d(Conditional)337 1424 y(T)m(erm)g(Rewriting)i(Systems.)22 b(In)15 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)i(the)h(3r)n(d)f (Internationa)o(l)e(Workshop)337 1469 y(on)h(Conditional)d(T)m(erm)15 b(R)n(ewriting)d(Systems)g(1992)p Fr(,)f(pages)i(128{142.)h(Lecture)f (Notes)337 1515 y(in)h(Computer)f(Science)h Fq(656)p Fr(,)f(Springer)i(V)m(erlag,)e(1993.)183 1561 y([Gra94a])21 b(B.)12 b(Gramlic)o(h.)21 b(Abstract)15 b(Relations)h(b)q(et)o(w)o(een) e(Restricted)h(T)m(ermination)h(and)f(Con-)337 1607 y(\015uence)g(Prop) q(erties)h(of)f(Rewrite)g(Systems.)22 b Fb(F)m(undamenta)12 b(Informatic)n(ae)p Fr(,)g(1994.)22 b(T)m(o)337 1652 y(app)q(ear.)183 1699 y([Gra94b])f(B.)12 b(Gramlic)o(h.)21 b(Generalized)16 b(Su\016cien)o(t)f(Conditions)h(for)e(Mo)q(dular)h(T)m (ermination)h(of)337 1744 y(Rewriting.)k Fb(Applic)n(able)10 b(A)o(lgebr)n(a)j(in)g(Engine)n(ering,)e(Communic)n(ation)h(and)h (Comput-)337 1790 y(ing)f Fq(5)p Fr(,)h(pages)h(131{158,)g(1994.)183 1836 y([Klo92])35 b(J.W.)20 b(Klop.)41 b(T)m(erm)20 b(Rewriting)i (Systems.)41 b(In)20 b(S.)13 b(Abramsky)m(,)21 b(D.)12 b(Gabba)o(y)m(,)22 b(and)337 1882 y(T.)12 b(Maibaum,)22 b(editors,)g Fb(Handb)n(o)n(ok)d(of)h(L)n(o)n(gic)g(in)h(Computer)g (Scienc)n(e)p Fr(,)d(v)o(olume)c(2,)337 1927 y(pages)f(1{116.)h(Oxford) f(Univ)o(ersit)o(y)i(Press,)e(1992.)183 1974 y([K)o(O91])35 b(M.)12 b(Kurihara)j(and)f(A.)e(Oh)o(uc)o(hi.)19 b(Mo)q(dular)c(T)m (erm)e(Rewriting)i(Systems)f(with)g(Shared)337 2019 y(Constructors.)j Fb(Journal)c(of)g(Information)e(Pr)n(o)n(c)n(essing)g Fq(14)p Fb(\(3\),)i(IPS)g(of)g(Jap)n(an)p Fr(,)e(pages)337 2065 y(357{358,)j(1991.)183 2111 y([K)o(O92])35 b(M.)12 b(Kurihara)21 b(and)g(A.)12 b(Oh)o(uc)o(hi.)39 b(Mo)q(dularit)o(y)22 b(of)e(Simple)i(T)m(ermination)f(of)f(T)m(erm)337 2157 y(Rewriting)15 b(Systems)e(with)h(Shared)g(Constructors.)k Fb(The)n(or)n(etic)n(al)12 b(Computer)h(Scienc)n(e)337 2202 y Fq(103)p Fr(,)g(pages)g(273{282,)h(1992.)183 2249 y([K)o(O94])35 b(M.)12 b(Kurihara)20 b(and)f(A.)12 b(Oh)o(uc)o(hi.)35 b(T)m(ermination)21 b(of)d(Com)o(bination)j(of)e(Comp)q(osable)337 2294 y(T)m(erm)12 b(Rewriting)j(Systems.)k(In)13 b Fb(Pr)n(o)n(c)n(e)n (e)n(dings)e(of)i(the)g(7th)g(A)o(ustr)n(alian)e(Joint)i(Confer-)337 2340 y(enc)n(e)f(on)h(A)o(rti\014cial)e(Intel)r(ligenc)n(e)o Fr(,)f(1994.)17 b(T)m(o)c(app)q(ear.)183 2386 y([KR93])36 b(M.R.K.)16 b(Krishna)e(Rao.)28 b(Completeness)18 b(of)f(Hierarc)o (hical)i(Com)o(binations)g(of)e(T)m(erm)337 2432 y(Rewriting)12 b(Systems.)k(In)11 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)j(the)f(13th)g (Confer)n(enc)n(e)f(on)i(the)f(F)m(oundations)p eop %%Page: 20 20 20 19 bop 494 194 a Fb(of)19 b(Softwar)n(e)e(T)m(e)n(chnolo)n(gy)f(and) i(The)n(or)n(etic)n(al)e(Computer)i(Scienc)n(e)p Fr(,)e(pages)j (125{139.)494 239 y(Lecture)14 b(Notes)f(in)h(Computer)f(Science)h Fq(761)p Fr(,)f(Springer)i(V)m(erlag,)e(1993.)340 285 y([KR94])36 b(M.R.K.)16 b(Krishna)e(Rao.)28 b(Simple)19 b(T)m(ermination)f(of)e(Hierarc)o(hical)j(Com)o(binations)g(of)494 331 y(T)m(erm)14 b(Rewriting)h(Systems.)k(In)14 b Fb(Pr)n(o)n(c)n(e)n (e)n(dings)d(of)j(the)f(International)e(Symp)n(osium)h(on)494 376 y(The)n(or)n(etic)n(al)i(Asp)n(e)n(cts)g(of)h(Computer)g(Softwar)n (e)p Fr(,)e(pages)j(203{223.)g(Lecture)g(Notes)f(in)494 422 y(Computer)f(Science)g Fq(789)p Fr(,)f(Springer)i(V)m(erlag,)e (1994.)340 468 y([Mar95])25 b(M.)13 b(Marc)o(hiori.)24 b(Mo)q(dularit)o(y)18 b(of)c(Completeness)j(Revisited.)24 b(In)15 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)h(the)494 513 y(6th)e(Internationa)o(l)d(Confer)n(enc)n(e)i(on)g(R)n(ewriting)g (T)m(e)n(chniques)f(and)h(Applic)n(ations)o Fr(.)d(Lec-)494 559 y(ture)14 b(Notes)f(in)g(Computer)h(Science)g(,)f(Springer)h(V)m (erlag,)g(1995.)j(T)m(o)c(app)q(ear.)340 605 y([Mid89])28 b(A.)12 b(Middeldorp)q(.)18 b(A)11 b(Su\016cien)o(t)h(Condition)i(for)c (the)h(T)m(ermination)i(of)e(the)g(Direct)h(Sum)494 650 y(of)j(T)m(erm)f(Rewriting)j(Systems.)23 b(In)15 b Fb(Pr)n(o)n(c)n(e)n (e)n(dings)e(of)h(the)h(4th)f(IEEE)i(Symp)n(osium)d(on)494 696 y(L)n(o)n(gic)g(in)g(Computer)g(Scienc)n(e)p Fr(,)d(pages)k (396{401,)f(1989.)340 742 y([Mid90])28 b(A.)12 b(Middeldorp)q(.)20 b Fb(Mo)n(dular)12 b(Pr)n(op)n(erties)g(of)h(T)m(erm)i(R)n(ewriting)d (Systems)p Fr(.)k(PhD)d(thesis,)494 787 y(V)m(rije)g(Univ)o(ersiteit)j (te)c(Amsterdam,)h(1990.)340 833 y([Mid93])28 b(A.)12 b(Middeldorp)q(.)19 b(Mo)q(dular)14 b(Prop)q(erties)f(of)f(Conditional) j(T)m(erm)d(Rewriting)i(Systems.)494 879 y Fb(Information)e(and)g (Computation)g Fq(104)p Fb(\(1\))p Fr(,)f(pages)j(110{158,)g(1993.)340 924 y([Mid94])28 b(A.)12 b(Middeldorp)q(.)31 b(Completeness)18 b(of)f(Com)o(binations)i(of)e(Conditional)i(Constructor)494 970 y(Systems.)f Fb(Journal)12 b(of)h(Symb)n(olic)f(Computation)f Fq(17)p Fr(,)i(pages)h(3{21,)f(1994.)340 1016 y([MT93])31 b(A.)12 b(Middeldorp)17 b(and)d(Y.)e(T)m(o)o(y)o(ama.)18 b(Completeness)d(of)f(Com)o(binations)i(of)d(Construc-)494 1061 y(tor)g(Systems.)18 b Fb(Journal)12 b(of)h(Symb)n(olic)f (Computation)f Fq(15)p Fb(\(3\))p Fr(,)h(pages)i(331{348,)g(1993.)340 1107 y([Ohl93])33 b(E.)13 b(Ohlebusc)o(h.)18 b(Com)o(binations)13 b(of)e(Simplifyin)q(g)j(Conditional)g(T)m(erm)c(Rewriting)j(Sys-)494 1153 y(tems.)k(In)11 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)j(the)f(3r)n (d)g(International)e(Workshop)h(on)h(Conditional)f(T)m(erm)494 1198 y(R)n(ewriting)k(Systems)e(1992)p Fr(,)g(pages)i(113{127.)h (Lecture)f(Notes)f(in)i(Computer)f(Science)494 1244 y Fq(656)p Fr(,)f(Springer)i(V)m(erlag,)e(1993.)340 1290 y([Ohl94a])22 b(E.)12 b(Ohlebusc)o(h.)25 b Fb(Mo)n(dular)14 b(Pr)n(op)n(erties)g(of)h(Comp)n(osable)f(T)m(erm)i(R)n(ewriting)e (Systems)p Fr(.)494 1335 y(PhD)g(thesis,)g(Univ)o(ersit\177)-19 b(at)15 b(Bielefeld,)g(1994.)340 1381 y([Ohl94b])22 b(E.)12 b(Ohlebusc)o(h.)42 b(On)21 b(the)g(Mo)q(dularit)o(y)i(of)e (Con\015uence)h(of)f(Constructor-Sharing)494 1427 y(T)m(erm)c (Rewriting)j(Systems.)31 b(In)18 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)i (the)g(19th)f(Col)r(lo)n(quium)h(on)g(T)m(r)n(e)n(es)494 1472 y(in)12 b(A)o(lgebr)n(a)e(and)h(Pr)n(o)n(gr)n(amming)p Fr(,)f(pages)i(261{275.)g(Lecture)f(Notes)h(in)g(Computer)f(Sci-)494 1518 y(ence)j Fq(787)p Fr(,)f(Springer)h(V)m(erlag,)g(1994.)340 1564 y([Ohl94c])22 b(E.)12 b(Ohlebusc)o(h.)17 b(On)10 b(the)f(Mo)q(dularit)o(y)j(of)d(T)m(ermination)h(of)f(T)m(erm)g (Rewriting)i(Systems.)494 1609 y Fb(The)n(or)n(etic)n(al)h(Computer)h (Scienc)n(e)e Fq(136)p Fr(,)i(pages)h(333{360,)f(1994.)340 1655 y([Rus87])30 b(M.)13 b(Rusino)o(witc)o(h.)20 b(On)14 b(T)m(ermination)h(of)e(the)h(Direct)g(Sum)g(of)g(T)m(erm)f(Rewriting)i (Sys-)494 1701 y(tems.)i Fb(Information)12 b(Pr)n(o)n(c)n(essing)f(L)n (etters)h Fq(26)p Fr(,)h(pages)h(65{70,)f(1987.)340 1746 y([TKB89])21 b(Y.)12 b(T)m(o)o(y)o(ama,)17 b(J.W.)f(Klop,)i(and)g(H.P)m (.)e(Barendregt.)31 b(T)m(ermination)19 b(for)e(the)g(Direct)494 1792 y(Sum)c(of)e(Left-Linear)h(T)m(erm)g(Rewriting)h(Systems.)k(In)12 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)j(the)g(3r)n(d)g(Inter-)494 1838 y(national)j(Confer)n(enc)n(e)g(on)h(R)n(ewriting)g(T)m(e)n (chniques)e(and)i(Applic)n(ation)o(s)p Fr(,)d(pages)18 b(477{)494 1883 y(491.)c(Lecture)f(Notes)g(in)h(Computer)f(Science)h Fq(355)p Fr(,)f(Springer)i(V)m(erlag,)e(1989.)340 1929 y([T)m(o)o(y87a])21 b(Y.)12 b(T)m(o)o(y)o(ama.)27 b(Coun)o(terexamples) 19 b(to)d(T)m(ermination)i(for)f(the)f(Direct)i(Sum)f(of)f(T)m(erm)494 1975 y(Rewriting)f(Systems.)i Fb(Information)11 b(Pr)n(o)n(c)n(essing)g (L)n(etters)h Fq(25)p Fr(,)h(pages)g(141{143,)h(1987.)340 2020 y([T)m(o)o(y87b])21 b(Y.)12 b(T)m(o)o(y)o(ama.)30 b(On)18 b(the)f(Ch)o(urc)o(h-Rosser)i(Prop)q(ert)o(y)f(for)g(the)f (Direct)i(Sum)e(of)h(T)m(erm)494 2066 y(Rewriting)d(Systems.)j Fb(Journal)12 b(of)h(the)g(A)o(CM)h Fq(34)p Fb(\(1\))p Fr(,)d(pages)j(128{143,)g(1987.)340 2420 y(This)g(article)g(w)o(as)f (pro)q(cessed)i(using)f(the)f(L)968 2415 y Fa(a)985 2420 y Fr(T)1007 2432 y(E)1028 2420 y(X)g(macro)g(pac)o(k)n(age)h(with)f (LLNCS)g(st)o(yle)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF