%!PS-Adobe-2.0 %%Creator: dvips 5.516 Copyright 1986, 1993 Radical Eye Software %%Title: CTRS.dvi %%CreationDate: Fri Jul 16 14:03:23 1993 %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o CTRS.ps CTRS.dvi %DVIPSSource: TeX output 1993.07.16:1359 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 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b(W)l(e)21 b(assume)g Fr(F)26 b Fu(to)c(b)q(e)g(the)f(disjoin)o(t)g(union)h(of)g(t)o(w)o(o)f(sets)h Fr(D)h Fu(and)f Fr(C)s Fu(.)38 b(The)0 1821 y(elemen)o(ts)15 b(of)i Fr(D)i Fu(and)f Fr(C)j Fu(are)c(called)f Fm(de\014ne)n(d)k(symb) n(ols)d Fu(and)h Fm(c)n(onstructors)p Fu(,)f(resp)q(ectiv)o(ely)l(.)22 b(Eac)o(h)17 b Fs(f)k Fr(2)16 b(F)0 1881 y Fu(is)h(asso)q(ciated)h (with)f(a)h(natural)f(n)o(um)o(b)q(er)f(denoting)h(its)g Fm(arity)p Fu(,)f(i.e.,)f(the)i(n)o(um)o(b)q(er)f(of)h('argumen)o(ts')e (it)i(is)0 1941 y(supp)q(osed)g(to)f(ha)o(v)o(e.)k(F)l(unction)c(sym)o (b)q(ols)f(of)h(arit)o(y)f(0)h(are)g(called)e Fm(c)n(onstants)p Fu(.)22 b(If)15 b Fr(F)1535 1948 y Fo(n)1574 1941 y Fu(denotes)h(the)g (set)g(of)0 2001 y(all)g(function)g(sym)o(b)q(ols)f(of)h(arit)o(y)g Fs(n)g Fu(then)g Fr(F)j Fu(=)896 1968 y Fi(S)880 2039 y Fo(n)p Fl(\025)p Fq(0)955 2001 y Fr(F)991 2008 y Fo(n)1014 2001 y Fu(.)0 2158 y Ft(De\014nition)f(2.7)24 b Fu(Let)e Fr(F)28 b Fu(b)q(e)23 b(a)g(signature)g(and)h(let)e Fr(V)k Fu(b)q(e)d(a)g(den)o(umerable)e(set)h(of)h Fm(variables)h Fu(with)0 2218 y Fr(F)16 b(\\)11 b(V)18 b Fu(=)13 b Fr(;)p Fu(.)21 b(The)c(set)f(of)g Fm(terms)h Fr(T)12 b Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))16 b(is)g(de\014ned)g(to)g(b)q(e)h(the)f (smallest)e(set)j(suc)o(h)f(that)60 2338 y(1.)24 b Fr(V)17 b(\022)d(T)f Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))60 2438 y(2.)24 b(if)16 b Fs(f)j Fr(2)14 b(F)21 b Fu(is)16 b(an)h Fs(n)p Fu(-ary)f(function)g(sym)o(b)q(ol)f(and)i Fs(t)1032 2445 y Fq(1)1052 2438 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)1179 2445 y Fo(n)1216 2438 y Fr(2)14 b(T)f Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o(,)122 2499 y(then)16 b Fs(f)5 b Fu(\()p Fs(t)299 2506 y Fq(1)319 2499 y Fs(;)j(:)g(:)g(:)f(;)h(t)446 2506 y Fo(n)469 2499 y Fu(\))14 b Fr(2)g(T)f Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o(.)0 2618 y(W)l(e)20 b(write)g Fs(f)25 b Fu(instead)20 b(of)h Fs(f)5 b Fu(\(\))20 b(for)h(ev)o(ery)e (constan)o(t)h Fs(f)5 b Fu(.)34 b(The)20 b(set)g(of)h(v)m(ariables)f(o) 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Fu(\))14 b(:=)g Fr(f)p Fs(x)f Fr(2)h(V)t(j)p Fs(\033)r Fu(\()p Fs(x)p Fu(\))e Fr(6\021)i Fs(x)p Fr(g)f Fu(of)g Fs(\033)h Fu(is)f(\014nite.)19 b(A)13 b Fm(substitution)h Fs(\033)h Fu(extends)d(uniquely)g(to)h(a)g(morphism)0 147 y Fs(\033)21 b Fu(:)d Fr(T)13 b Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o Fr(!T)13 b Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o(,)19 b(that)h(is)f(to)h(sa)o(y)l(,)g Fs(\033)r Fu(\()p Fs(f)5 b Fu(\()p Fs(t)951 154 y Fq(1)970 147 y Fs(;)j(:)g(:)g(:)f(;)h(t)1097 154 y Fo(n)1120 147 y Fu(\)\))19 b Fr(\021)g Fs(f)5 b Fu(\()p Fs(\033)r Fu(\()p Fs(t)1350 154 y Fq(1)1369 147 y Fu(\))p Fs(;)j(:)g(:)g(:)f(;)h(\033)r Fu(\()p Fs(t)1564 154 y Fo(n)1587 147 y Fu(\)\))19 b(for)h(eac)o(h)e Fs(n)p Fu(-ary)0 207 y(function)e(sym)o(b)q(ol)g Fs(f)22 b Fu(and)17 b(terms)e Fs(t)654 214 y Fq(1)673 207 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)800 214 y Fo(n)823 207 y Fu(.)23 b Fs(\033)r Fu(\()p Fs(t)p Fu(\))15 b(is)i(said)f(to)h(b)q(e)g(an)g (instance)f(of)h Fs(t)p Fu(.)22 b(W)l(e)16 b(will)g(also)h(use)0 267 y(the)f(notation)h Fs(t\033)g Fu(for)g Fs(\033)r Fu(\()p Fs(t)p Fu(\).)0 400 y Ft(De\014nition)h(2.9)24 b Fu(A)17 b Fm(c)n(ontext)i Fs(C)t Fu([)p Fs(;)8 b(:)g(:)g(:)e(;)i Fu(])17 b(is)g(an)h(elemen)o(t)d(of)j Fr(T)12 b Fu(\()p Fr(D)q Fs(;)c Fr(C)16 b([)c(f)p Fh(2)p Fr(g)p Fs(;)c Fr(V)t Fu(\))17 b(whic)o(h)g(con)o(tains)h(zero,)0 460 y(one)13 b(or)g(more)e(o)q(ccurrences)h(of)h(the)f(sp)q(ecial)g (constan)o(t)h Fh(2)p Fu(.)21 b(If)12 b Fs(C)t Fu([)p Fs(;)c(:)g(:)g(:)e(;)i Fu(])k(is)g(a)h(con)o(text)f(with)h Fs(n)f Fu(o)q(ccurrences)0 521 y(of)i Fh(2)g Fu(and)h Fs(t)215 528 y Fq(1)234 521 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)361 528 y Fo(n)398 521 y Fr(2)14 b(T)f Fu(\()p Fr(D)q Fs(;)8 b Fr(C)s Fs(;)g Fr(V)t Fu(\),)14 b(then)g Fs(C)t Fu([)p Fs(t)878 528 y Fq(1)896 521 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)1023 528 y Fo(n)1046 521 y Fu(])14 b(is)f(the)h(result)f(of)h(replacing)f (from)g(left)g(to)h(righ)o(t)0 581 y(the)i(o)q(ccurrences)f(of)i Fh(2)f Fu(with)g Fs(t)583 588 y Fq(1)602 581 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)729 588 y Fo(n)752 581 y Fu(.)22 b(A)15 b(con)o(text)g(with)h(exactly)f(one)h(o)q(ccurrence)f(of)i Fh(2)f Fu(is)g(denoted)0 641 y(b)o(y)h Fs(C)t Fu([)10 b(])o(.)25 b(A)17 b(term)f Fs(s)h Fu(is)g(a)h Fm(subterm)g Fu(of)f(a)h(term)e Fs(t)h Fu(if)f(there)h(exists)g(a)h(con)o(text)e Fs(C)t Fu([)10 b(])17 b(suc)o(h)g(that)h Fs(t)d Fr(\021)h Fs(C)t Fu([)p Fs(s)p Fu(])n(.)0 701 y(By)g(abuse)i(of)g(notation)g(w)o (e)e(write)h Fr(T)c Fu(\()p Fr(D)q Fs(;)8 b Fr(C)s Fs(;)g Fr(V)t Fu(\))17 b(for)g Fr(T)c Fu(\()p Fr(D)q Fs(;)8 b Fr(C)16 b([)11 b(f)p Fh(2)p Fr(g)p Fs(;)d Fr(V)t Fu(\),)17 b(in)o(terpreting)f Fh(2)h Fu(as)h(a)f(sp)q(ecial)0 761 y(constructor)g(whic)o(h)e(is)h(alw)o(a)o(ys)g(a)o(v)m(ailable)g(but)g (used)h(only)f(for)g(the)g(aforemen)o(tioned)f(purp)q(ose.)0 895 y Ft(De\014nition)j(2.10)24 b Fu(Let)16 b Fs(t)d Fr(2)h(T)f Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o(.)60 1016 y(1.)24 b(The)16 b(set)g Fr(P)t Fs(os)p Fu(\()p Fs(t)p Fu(\))g(of)h Fm(p)n(ositions)f Fu(in)g Fs(t)f Fu(is)h(de\014ned)h(inductiv)o(ely:)189 1117 y(-)24 b Fr(P)t Fs(os)p Fu(\()p Fs(t)p Fu(\))14 b(=)g Fr(f)p Fj(\003)o Fr(g)i Fu(if)g Fs(t)e Fr(2)g(V)t Fu(,)189 1198 y(-)24 b Fr(P)t Fs(os)p Fu(\()p Fs(t)p Fu(\))14 b(=)g Fr(f)p Fj(\003)o Fr(g)d([)h(f)p Fs(ip)p Fr(j)p Fs(p)i Fr(2)g(P)t Fs(os)p Fu(\()p Fs(t)857 1205 y Fo(i)871 1198 y Fu(\))i(and)h(1)d Fr(\024)f Fs(i)h Fr(\024)g Fs(n)p Fr(g)i Fu(if)g Fs(t)d Fr(\021)h Fs(f)5 b Fu(\()p Fs(t)1440 1205 y Fq(1)1459 1198 y Fs(;)j(:)g(:)g(:)g(;)g(t)1587 1205 y Fo(n)1610 1198 y Fu(\).)122 1299 y(A)17 b(p)q(osition)h Fs(p)g Fu(within)e(a)i(term)e Fs(t)h Fu(is)g(th)o(us)g(a)h(sequence)e(of)i (natural)g(n)o(um)o(b)q(ers)e(\(where)h Fj(\003)g Fu(denotes)122 1359 y(the)e(empt)o(y)f(sequence\))g(describing)h(the)h(path)g(from)e (the)i(ro)q(ot)g(of)g Fs(t)g Fu(to)g(the)f(ro)q(ot)i(of)f(the)f (subterm)122 1419 y(at)i(p)q(osition)f Fs(p)p Fu(,)h(denoted)f(b)o(y)g Fs(t)p Fr(j)706 1426 y Fo(p)725 1419 y Fu(,)g(where)431 1561 y Fs(t)p Fr(j)463 1568 y Fo(p)496 1561 y Fu(=)548 1488 y Fi(\()649 1530 y Fs(t)48 b Fu(if)k Fs(p)14 b Fu(=)g Fj(\003)602 1590 y Fs(t)620 1597 y Fo(i)633 1590 y Fr(j)647 1597 y Fo(q)715 1590 y Fu(if)52 b Fs(p)14 b Fu(=)g Fs(iq)j Fu(and)g Fs(t)c Fr(\021)h Fs(f)5 b Fu(\()p Fs(t)1187 1597 y Fq(1)1207 1590 y Fs(;)j(:)g(:)g(:)f(;)h(t)1334 1597 y Fo(n)1357 1590 y Fu(\))p Fs(;)g Fu(1)14 b Fr(\024)g Fs(i)f Fr(\024)h Fs(n:)60 1718 y Fu(2.)24 b(P)o(ositions)16 b(are)g(partially)f(ordered)h(b)o(y)g(the)g(so-called)f Fm(pr)n(e\014x)j(or)n(dering)e Fr(\024)p Fu(,)f(i.e.,)f Fs(p)g Fr(\024)g Fs(q)j Fu(if)f(there)f(is)122 1778 y(an)i Fs(r)h Fu(suc)o(h)e(that)i Fs(pr)e Fu(=)e Fs(q)r Fu(.)22 b(If)16 b Fs(p)f Fr(\024)g Fs(q)j Fu(and)f Fs(p)e Fr(6)p Fu(=)f Fs(q)r Fu(,)i(w)o(e)g(write)g Fs(p)f(<)g(q)r Fu(.)22 b(Tw)o(o)17 b(p)q(ositions)g Fs(p)g Fu(and)h Fs(q)g Fu(are)122 1838 y Fm(disjoint)e Fu(if)g(neither)f Fs(p)g Fr(\024)e Fs(q)18 b Fu(nor)f Fs(q)e Fr(\024)e Fs(p)p Fu(.)60 1940 y(3.)24 b(Let)16 b Fs(p)f Fr(2)f(P)t Fs(os)p Fu(\()p Fs(t)p Fu(\))i(and)g Fs(t)e Fr(2)g(T)f Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o(.)21 b(The)16 b(term)f Fs(s)p Fu([)p Fs(p)p Fr( )p Fs(t)p Fu(])g(is)h(de\014ned)g(as)h(follo)o(ws:) 189 2041 y(-)24 b Fs(s)p Fu([)p Fs(p)p Fr( )p Fs(t)p Fu(])13 b(=)h Fs(t)i Fu(if)f Fs(p)g Fu(=)f Fj(\003)o Fu(,)189 2122 y(-)24 b Fs(s)p Fu([)p Fs(p)p Fr( )p Fs(t)p Fu(])13 b(=)h Fs(f)5 b Fu(\()p Fs(s)508 2129 y Fq(1)528 2122 y Fs(;)j(:)g(:)g(:)f(;)h(s)660 2129 y Fo(i)674 2122 y Fu([)p Fs(q)r Fr( )o Fs(t)p Fu(])p Fs(;)g(:)g(:)g(:)f(;)h(s)925 2129 y Fo(n)948 2122 y Fu(\))16 b(if)g Fs(p)e Fu(=)g Fs(iq)r Fu(,)h(1)f Fr(\024)g Fs(i)f Fr(\024)h Fs(n)p Fu(,)h(and)i Fs(s)d Fr(\021)g Fs(f)5 b Fu(\()p Fs(s)1676 2129 y Fq(1)1696 2122 y Fs(;)j(:)g(:)g(:)f(;)h(s)1828 2129 y Fo(n)1851 2122 y Fu(\).)60 2223 y(4.)24 b(Let)16 b Fs(t)e Fr(2)g(T)f Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o(.)21 b(Then)16 b Fs(r)q(oot)p Fu(\()p Fs(t)p Fu(\))h(is)f(de\014ned)g(as:) 189 2324 y(-)24 b Fs(r)q(oot)p Fu(\()p Fs(t)p Fu(\))14 b(=)g Fs(t)i Fu(if)g Fs(t)d Fr(2)h(V)t Fu(,)189 2405 y(-)24 b Fs(r)q(oot)p Fu(\()p Fs(t)p Fu(\))14 b(=)g Fs(f)22 b Fu(if)16 b Fs(t)d Fr(\021)h Fs(f)5 b Fu(\()p Fs(t)679 2412 y Fq(1)698 2405 y Fs(;)j(:)g(:)g(:)g(;)g(t)826 2412 y Fo(n)849 2405 y Fu(\).)0 2538 y Ft(De\014nition)18 b(2.11)60 2659 y Fu(1.)24 b(A)16 b(relation)f Fs(>)8 b Fr(\022)14 b(T)f Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))i Fr(\002)g(T)j Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))15 b(has)i(the)e Fm(r)n(eplac)n(ement)k(pr)n(op)n(erty)1506 2641 y Fq(2)1541 2659 y Fu(if)d(it)f(is)h(closed)g(under)122 2719 y(con)o(texts,)f(i.e.,)f(if)i Fs(s)e(>)g(t)h Fu(then)i Fs(C)t Fu([)p Fs(s)p Fu(])12 b Fs(>)i(C)t Fu([)p Fs(t)p Fu(])g(for)j(all)e(con)o(texts)h Fs(C)t Fu([)f(].)p 0 2765 780 2 v 56 2796 a Fk(2)75 2811 y Fj(often)f(also)f(called)h (monotonicit)o(y)962 2936 y Fu(4)p eop %%Page: 5 5 5 4 bop 60 26 a Fu(2.)24 b Fs(>)14 b Fu(has)h(the)f Fm(ful)r(l)j (invarianc)n(e)g(pr)n(op)n(erty)848 8 y Fq(3)882 26 y 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936 y Fr(#)14 b Fs(t)1863 943 y Fo(n)1905 936 y Fu(to)0 996 y(a)19 b(term)e Fs(s)h Fr(\021)f Fs(C)t Fu([)p Fs(l)q(\033)r Fu(])f(can)j(not)g(b)q(e)g(c)o(hec)o(k)o(ed)d(b)o(y)i(recursiv)o(ely)e (reducing)i(the)g(terms)f Fs(s)1581 1003 y Fo(i)1596 996 y Fs(\033)j Fu(and)f Fs(t)1759 1003 y Fo(i)1772 996 y Fs(\033)i Fu(for)d(all)0 1056 y Fs(i)13 b Fr(2)i(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)f(;)h(n)p Fr(g)16 b Fu(if)g Fr(R)h Fu(is)f(a)g(semi-equational)f(CTRS's.)0 1183 y Ft(Example)h(3.7)24 b Fu(Let)14 b Fr(R)g Fu(=)g Fr(f)p Fs(a)f Fr(!)h Fs(b)f Fr(\()h Fs(g)i Fu(=)e Fs(h;)8 b(f)19 b Fr(!)13 b Fs(g)j Fr(\()e Fs(a)g Fu(=)f Fs(b;)8 b(f)19 b Fr(!)14 b Fs(h)g Fr(\()f Fs(a)h Fu(=)g Fs(b)p Fr(g)p Fu(.)19 b Fr(R)14 b Fu(is)g(simplifyi)o(ng)0 1243 y(as)j(is)f(sho)o(wn) h(b)o(y)f(the)g(simpli\014cation)e(ordering)i Fs(f)j(>)14 b(a)f(>)h(b)g(>)g(g)i(>)d(h)p Fu(.)22 b(Do)q(es)17 b Fs(a)f Fu(reduce)f(to)i Fs(b)p Fu(?)0 1370 y Ft(F)-5 b(rom)18 b(no)n(w)i(on)e(w)n(e)i(assume)d(eac)n(h)i(CTRS)g(to)g (consist)f(of)h(\014nitely)e(man)n(y)h(rules.)0 1497 y(Prop)r(osition)g(3.8)24 b Fu(The)13 b(follo)o(wing)g(implicati)o(ons) e(hold:)20 b Fr(R)13 b Fu(is)g(simplifying)d Fr(\))28 b(R)13 b Fu(is)g(reductiv)o(e)e Fr(\))28 b(R)0 1557 y Fu(is)16 b(decreasing.)0 1638 y Ft(Pro)r(of:)j Fu(It)12 b(is)h(easy)f(to)h(see)f(that)h Fr(R)g Fu(is)f(reductiv)o(e)e(and)k (decreasing)e(if)g Fr(R)g Fu(is)h(simplifying.)k(The)12 b(remaining)0 1698 y(implication)i(is)i(pro)o(v)o(ed)f(in)h([DOS88)q (].)21 b Fh(2)0 1825 y Ft(De\014nition)d(3.9)60 1941 y Fu(1.)24 b(Let)18 b Fs(l)226 1948 y Fq(1)245 1941 y Fr(!)p Fs(r)317 1948 y Fq(1)353 1941 y Fr(\()f Fs(s)443 1948 y Fq(1)463 1941 y Fr(#)o Fs(t)505 1948 y Fq(1)525 1941 y Fs(;)8 b(:)g(:)g(:)f(;)h(s)657 1948 y Fo(n)681 1941 y Fr(#)o Fs(t)723 1948 y Fo(n)764 1941 y Fu(and)19 b Fs(l)876 1948 y Fq(2)895 1941 y Fr(!)p Fs(r)967 1948 y Fq(2)1003 1941 y Fr(\()e Fs(u)1098 1948 y Fq(1)1117 1941 y Fr(#)p Fs(v)1166 1948 y Fq(1)1186 1941 y Fs(;)8 b(:)g(:)g(:)f(;)h(u)1323 1948 y Fo(m)1356 1941 y Fr(#)p Fs(v)1405 1948 y Fo(m)1456 1941 y Fu(b)q(e)18 b(renamed)e(v)o(ersions)i (of)122 2001 y(rewrite)13 b(rules)g(of)h Fr(R)g Fu(suc)o(h)f(that)h (they)g(ha)o(v)o(e)e(no)j(v)m(ariables)e(in)g(common.)19 b(Supp)q(ose)14 b Fs(l)1671 2008 y Fq(1)1705 2001 y Fr(\021)f Fs(C)t Fu([)p Fs(t)p Fu(])f(with)122 2062 y Fs(t)j Fr(62)g(V)20 b Fu(suc)o(h)d(that)g Fs(t)g Fu(and)h Fs(l)617 2069 y Fq(2)653 2062 y Fu(are)f(uni\014able,)f(i.e.,)f Fs(t\033)h Fr(\021)f Fs(l)1171 2069 y Fq(2)1190 2062 y Fs(\033)k Fu(for)e(a)g(most)f(general)h(uni\014er)1795 2044 y Fq(7)1814 2062 y Fs(\033)r Fu(.)23 b(W)l(e)122 2122 y(call)392 2242 y Fr(h)p Fs(C)t Fu([)p Fs(r)486 2249 y Fq(2)505 2242 y Fu(])p Fs(\033)o(;)8 b(r)590 2249 y Fq(1)609 2242 y Fs(\033)r Fr(i)13 b(\()h Fs(s)758 2249 y Fq(1)778 2242 y Fs(\033)r Fr(#)o Fs(t)850 2249 y Fq(1)869 2242 y Fs(\033)o(;)8 b(:)g(:)g(:)g(;)g(s)1029 2249 y Fo(n)1052 2242 y Fs(\033)r Fr(#)o Fs(t)1124 2249 y Fo(n)1147 2242 y Fs(\033)o(;)g(u)1224 2249 y Fq(1)1244 2242 y Fs(\033)r Fr(#)o Fs(v)1322 2249 y Fq(1)1341 2242 y Fs(\033)o(;)g(:)g(:)g(:)f(;)h(u)1505 2249 y Fo(m)1538 2242 y Fs(\033)r Fr(#)p Fs(v)1617 2249 y Fo(m)1650 2242 y Fs(\033)122 2326 y Fu(a)22 b Fm(c)n(onditional)h (critic)n(al)g(p)n(air)d Fu(of)i Fr(R)p Fu(.)37 b(If)21 b(the)g(t)o(w)o(o)h(rules)f(are)h(renamed)e(v)o(ersions)h(of)h(the)f (same)122 2386 y(rewrite)15 b(rule)h(of)g Fr(R)p Fu(,)g(w)o(e)g(do)h (not)g(consider)f(the)g(case)g Fs(C)t Fu([)f(])e Fr(\021)h Fh(2)p Fu(.)60 2486 y(2.)24 b(A)c(conditional)g(critical)f(pair)h Fr(h)p Fs(s;)8 b(t)p Fr(i)21 b(\()g Fs(s)922 2493 y Fq(1)941 2486 y Fr(#)p Fs(t)984 2493 y Fq(1)1003 2486 y Fs(;)8 b(:)g(:)g(:)g(;)g(s)1136 2493 y Fo(n)1159 2486 y Fr(#)p Fs(t)1202 2493 y Fo(n)1246 2486 y Fu(of)20 b Fr(R)h Fu(is)f Fm(c)n(onver)n(gent)i Fu(if)e Fs(s\033)r Fr(#)1791 2498 y Fl(R)1823 2486 y Fs(t\033)i Fu(for)122 2546 y(ev)o(ery)15 b(substitution)h Fs(\033)i Fu(with)e Fs(s)702 2553 y Fo(i)716 2546 y Fs(\033)r Fr(#)771 2558 y Fl(R)803 2546 y Fs(t)821 2553 y Fo(i)834 2546 y Fs(\033)i Fu(for)f(all)e Fs(i)f Fr(2)g(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(;)g(n)p Fr(g)p Fu(.)73 2673 y(The)25 b(same)f(de\014nition)g(applies)h(also,)i (m)o(utatis)d(m)o(utandis,)h(for)g(semi-equational)e(and)j(normal)0 2733 y(CTRS's.)p 0 2765 780 2 v 56 2796 a Fk(7)75 2811 y Fj(see)15 b(e.g.)j([DJ90)o(])962 2936 y Fu(8)p eop %%Page: 9 9 9 8 bop 0 26 a Ft(Lemma)16 b(3.10)24 b Fu(A)17 b(decreasing)g(join)g (CTRS)h(is)f(lo)q(cally)g(con\015uen)o(t)g(if)g(and)h(only)f(if)g(all)g (its)g(conditional)0 86 y(critical)11 b(pairs)j(are)f(con)o(v)o(ergen)o (t.)19 b(If)12 b(w)o(e)h(replace)f(join)h(with)g(semi-equational,)f (then)h(the)g(decreasingness)0 147 y(requiremen)o(t)g(ma)o(y)i(ev)o(en) g(b)q(e)h(dropp)q(ed.)0 228 y Ft(Pro)r(of:)21 b Fu(See)16 b([DOS88].)21 b Fh(2)0 346 y Ft(Lemma)16 b(3.11)24 b Fu(A)13 b(decreasing)g(\(reductiv)o(e,)f(simplifying\))f(join)j (\(normal,)f(semi-equational\))f(CTRS)i(is)0 407 y(con)o(v)o(ergen)o(t) h(if)h(and)g(only)g(if)g(all)g(its)g(conditional)g(critical)f(pairs)h (are)g(con)o(v)o(ergen)o(t.)0 487 y Ft(Pro)r(of:)27 b Fu(Reductiv)o(e)17 b(and)j(simplifying)d(systems)h(are)h(sp)q(ecial)g (decreasing)g(CTRS's.)30 b(These)19 b(systems)0 548 y(are)14 b(terminating.)k(The)c(assertion)g(follo)o(ws)g(for)g(join)f(systems)g (from)f(the)h(ab)q(o)o(v)o(e)h(lemma)d(in)i(conjunction)0 608 y(with)i(Newman's)e(Lemma.)18 b(Since)c(normal)g(systems)g(are)h (sp)q(ecial)f(join)h(systems,)e(w)o(e)i(ma)o(y)e(replace)h(join)0 668 y(with)h(normal.)20 b(The)15 b(ab)q(o)o(v)o(e)g(lemma)d(tells)j(us) g(that)h(the)f(replacemen)o(t)d(of)j(join)h(with)f(semi-equational)e (is)0 728 y(also)k(feasible.)j Fh(2)0 891 y Fn(3.3)70 b(A)22 b(Characterization)g(of)h(Simplifying)d(CTRS's)0 1004 y Fu(In)12 b(this)g(section)g(w)o(e)g(sho)o(w)h(that)g(a)f(CTRS's) h Fr(R)f Fu(is)g(simplifying)e(if)i(and)h(only)f(if)g(a)g(certain)g (rewrite)f(relation)0 1065 y(is)16 b(irre\015exiv)o(e.)j(This)d(is)g (the)g(k)o(ey)f(to)i(the)f(pro)q(of)h(of)g(the)f(main)f(theorem.)0 1183 y Ft(De\014nition)j(3.12)24 b Fu(Let)15 b Fr(R)h Fu(b)q(e)g(a)g(CTRS.)f(With)g Fr(R)h Fu(w)o(e)f(asso)q(ciate)i(the)e (follo)o(wing)g(unconditional)h(TRS)288 1279 y Fr(S)i Fu(:=)116 b Fr(f)p Fs(l)q Fr(!)o Fs(r)q Fr(j)p Fs(l)15 b Fr(!)e Fs(r)i Fr(\()f Fs(s)848 1286 y Fq(1)882 1279 y Fr(#)f Fs(t)938 1286 y Fq(1)958 1279 y Fs(;)8 b(:)g(:)g(:)f(;)h(s) 1090 1286 y Fo(n)1127 1279 y Fr(#)14 b Fs(t)1184 1286 y Fo(n)1221 1279 y Fr(2)g(Rg)429 1352 y([)42 b(f)p Fs(l)q Fr(!)o Fs(s)617 1359 y Fo(i)631 1352 y Fr(j)p Fs(l)14 b Fr(!)g Fs(r)h Fr(\()f Fs(s)862 1359 y Fq(1)895 1352 y Fr(#)g Fs(t)952 1359 y Fq(1)971 1352 y Fs(;)8 b(:)g(:)g(:)g(;)g(s) 1104 1359 y Fo(n)1141 1352 y Fr(#)14 b Fs(t)1198 1359 y Fo(n)1235 1352 y Fr(2)g(R)p Fu(;)8 b Fs(i)13 b Fr(2)h(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(;)g(n)p Fr(gg)429 1425 y([)42 b(f)p Fs(l)q Fr(!)o Fs(t)612 1432 y Fo(i)626 1425 y Fr(j)p Fs(l)14 b Fr(!)g Fs(r)h Fr(\()e Fs(s)856 1432 y Fq(1)890 1425 y Fr(#)h Fs(t)947 1432 y Fq(1)966 1425 y Fs(;)8 b(:)g(:)g(:)f(;)h(s)1098 1432 y Fo(n)1136 1425 y Fr(#)13 b Fs(t)1192 1432 y Fo(n)1229 1425 y Fr(2)h(R)p Fu(;)8 b Fs(i)14 b Fr(2)g(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(;)g(n)p Fr(gg)0 1543 y Ft(De\014nition)18 b(3.13)24 b Fu(W)l(e)16 b(are)g(also)h(in)o(terested)d(in)i(another)h(TRS)g(related)e(to)i(\()p Fr(F)5 b Fs(;)j Fr(R)p Fu(\):)383 1639 y Fr(F)424 1621 y Fo(e)456 1639 y Fu(=)14 b Fr(f)p Fs(f)5 b Fu(\()p Fs(x)609 1646 y Fq(1)628 1639 y Fs(;)j(:)g(:)g(:)g(;)g(x)766 1646 y Fo(n)789 1639 y Fu(\))p Fr(!)p Fs(x)886 1646 y Fo(i)899 1639 y Fr(j)p Fs(f)19 b Fr(2)c(F)1040 1646 y Fo(n)1063 1639 y Fu(;)8 b Fs(n)14 b Fr(2)g Fe(I)-7 b(N)p Fu(;)8 b Fs(i)14 b Fr(2)g(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(;)g(n)p Fr(gg)p Fs(;)0 1735 y Fu(that)22 b(is)f(to)g(sa)o(y)l(,)h Fs(s)p Fr(!)402 1717 y Fl(\003)402 1747 y(F)430 1738 y Fd(e)450 1735 y Fs(t)e Fu(if)h(and)h(only)f(if)g Fs(t)p Fr(\036)824 1762 y Fl(\030)856 1768 y Fo(emb)921 1735 y Fs(s)p Fu(,)h(where)f Fr(\036)1132 1762 y Fl(\030)1165 1768 y Fo(emb)1251 1735 y Fu(denotes)g(the)g(pure)g(homeomorphic)0 1810 y(em)o(b)q(edding)15 b(relation)g(\(cf.)21 b(for)c(instance)f ([DJ90]\).)0 1928 y Ft(Lemma)g(3.14)24 b Fu(Let)d Fr(S)k Fu(b)q(e)c(an)g(unconditional)g(TRS,)g(and)h(let)e Fr(!)1282 1935 y Fl(S)r Fo(e)1346 1928 y Fu(:=)h(\()p Fr(!)1488 1935 y Fl(S)1528 1928 y Fr([)15 b(!)1626 1935 y Fl(F)1654 1927 y Fd(e)1673 1928 y Fu(\).)35 b Fr(!)1791 1908 y Fq(+)1791 1941 y Fl(S)r Fo(e)1871 1928 y Fu(is)20 b(a)0 1989 y(simpli\014cation)14 b(ordering)i(if)g(it)g(is)g(irre\015exiv)o (e.)0 2070 y Ft(Pro)r(of:)21 b Fu(W)l(e)16 b(ha)o(v)o(e)g(to)g(sho)o(w) h(the)f(follo)o(wing)g(prop)q(erties)g(of)h Fr(!)1175 2049 y Fq(+)1175 2082 y Fl(S)r Fo(e)1233 2070 y Fu(:)0 2130 y(Irre\015exivit)o(y:)h(b)o(y)e(assumption.)0 2190 y(T)l(ransitivit)o(y:)k(b)o(y)c(de\014nition.)0 2250 y(Replacemen)o(t)d(prop)q(ert)o(y:)21 b(follo)o(ws)c(from)e(the)h (replacemen)o(t)d(prop)q(ert)o(y)j(of)h Fr(!)1444 2257 y Fl(S)1486 2250 y Fu(and)g Fr(!)1630 2257 y Fl(F)1658 2248 y Fd(e)1678 2250 y Fu(.)0 2310 y(F)l(ull)e(in)o(v)m(ariance)h (prop)q(ert)o(y:)21 b(follo)o(ws)16 b(from)f(the)h(full)f(in)o(v)m (ariance)h(prop)q(ert)o(y)g(of)g Fr(!)1526 2317 y Fl(S)1568 2310 y Fu(and)h Fr(!)1713 2317 y Fl(F)1741 2308 y Fd(e)1760 2310 y Fu(.)0 2371 y(Subterm)e(prop)q(ert)o(y:)21 b(ob)o(vious)16 b(b)q(ecause)h Fr(!)823 2378 y Fl(F)851 2369 y Fd(e)886 2371 y Fu(is)f(con)o(tained)g(in)g Fr(!)1263 2350 y Fq(+)1263 2383 y Fl(S)r Fo(e)1321 2371 y Fu(.)22 b Fh(2)0 2489 y Ft(Lemma)16 b(3.15)24 b Fr(!)373 2469 y Fq(+)373 2502 y Fl(S)r Fo(e)431 2489 y Fu(is)17 b(the)g(smallest)e(simpli\014cation)g (ordering)j(con)o(taining)f Fr(!)1538 2496 y Fl(S)1564 2489 y Fu(,)g(precisely:)k(if)c Fs(>)g Fu(is)0 2550 y(a)h (simpli\014cation)e(ordering)i(with)g Fr(!)701 2557 y Fl(S)744 2550 y Fr(\022)p Fs(>)p Fu(,)f(then)h Fr(!)1015 2529 y Fq(+)1015 2562 y Fl(S)r Fo(e)1090 2550 y Fr(\022)8 b Fs(>)18 b Fu(\(and)g(hence)f Fr(!)1495 2529 y Fq(+)1495 2562 y Fl(S)r Fo(e)1571 2550 y Fu(is)h(a)g(simpli\014cation)0 2610 y(ordering\).)0 2691 y Ft(Pro)r(of:)25 b Fu(Let)18 b Fs(s)g Fr(!)357 2698 y Fl(S)r Fo(e)416 2691 y Fs(t)p Fu(.)27 b(If)18 b Fs(s)p Fr(!)598 2698 y Fl(S)624 2691 y Fs(t)g Fu(then)g Fs(s)f(>)h(t)p Fu(.)26 b(Otherwise)18 b Fs(s)p Fr(!)1233 2698 y Fl(F)1261 2689 y Fd(e)1280 2691 y Fs(t)p Fu(,)g(i.e.,)f Fs(s)g Fr(\021)g Fs(C)t Fu([)p Fs(f)5 b Fu(\()p Fs(s)1645 2698 y Fq(1)1664 2691 y Fs(;)j(:)g(:)g(:)f(;)h(s)1796 2698 y Fo(n)1819 2691 y Fu(\)])18 b(and)0 2751 y Fs(t)f Fr(\021)g Fs(C)t Fu([)p Fs(s)167 2758 y Fo(i)180 2751 y Fu(],)h(where)g Fs(f)23 b Fr(2)17 b(F)502 2758 y Fo(n)526 2751 y Fu(.)27 b(No)o(w)18 b(the)g(assertion)h(follo)o(ws)f(from)g(the)g(fact)g(that)h Fs(>)f Fu(has)h(the)f(subterm)0 2811 y(prop)q(ert)o(y)e(and)h(the)f (replacemen)o(t)d(prop)q(ert)o(y)l(.)21 b Fh(2)962 2936 y Fu(9)p eop %%Page: 10 10 10 9 bop 0 26 a Ft(Prop)r(osition)18 b(3.16)24 b Fu(Let)16 b Fr(R)g Fu(b)q(e)g(a)g(CTRS)g(and)g Fr(S)k Fu(the)15 b(asso)q(ciated)i(TRS.)f(Then)f(the)h(follo)o(wing)f(state-)0 86 y(men)o(ts)g(are)h(equiv)m(alen)o(t:)60 209 y(1.)24 b Fr(R)16 b Fu(is)g(simplifying.)60 311 y(2.)24 b Fr(!)172 290 y Fq(+)172 323 y Fl(S)r Fo(e)230 311 y Fu(is)16 b(irre\015exiv)o (e.)60 412 y(3.)24 b Fr(!)172 392 y Fq(+)172 424 y Fl(S)r Fo(e)230 412 y Fu(is)16 b(terminating.)0 535 y Ft(Pro)r(of:)31 b Fu(2)p Fs(:)23 b Fr(\))f Fu(1)p Fs(:)h Fu(:)e(By)f(Lemma)g(3.14)i Fr(!)831 514 y Fq(+)831 547 y Fl(S)r Fo(e)889 535 y Fu(is)f(a)h (simpli\014cation)d(ordering.)37 b(Hence)20 b(the)h(assertion)0 595 y(follo)o(ws)16 b(with)g Fr(!)321 602 y Fl(S)361 595 y Fr(\022)d(!)463 574 y Fq(+)463 607 y Fl(S)r Fo(e)522 595 y Fu(.)0 655 y(1)p Fs(:)h Fr(\))f Fu(3)p Fs(:)h Fu(:)f(Let)g Fs(>)g Fu(b)q(e)g(a)h(simpli\014cation)c(ordering)k(sho)o(wing)f(that)h Fr(R)f Fu(is)g(simplifying.)k Fs(>)c Fu(is)g(w)o(ell-founded.)0 715 y(Since)i Fr(!)177 695 y Fq(+)177 727 y Fl(S)r Fo(e)250 715 y Fr(\022)p Fs(>)g Fu(b)o(y)h(Lemma)e(3.15,)j Fr(!)751 695 y Fq(+)751 727 y Fl(S)r Fo(e)809 715 y Fu(is)f(w)o(ell-founded)g (as)h(w)o(ell.)0 776 y(3)p Fs(:)d Fr(\))f Fu(2)p Fs(:)h Fu(:)i(Eviden)o(tly)l(,)e(if)i Fr(!)523 755 y Fq(+)523 788 y Fl(S)r Fo(e)581 776 y Fu(is)g(not)h(irre\015exiv)o(e)c(then)j Fr(!)1101 755 y Fq(+)1101 788 y Fl(S)r Fo(e)1160 776 y Fu(is)g(not)h(terminating.)i Fh(2)0 910 y Ft(Example)d(3.17)0 971 y Fr(R)42 978 y Fq(1)76 971 y Fu(=)e Fr(f)p Fs(F)7 b Fu(\(0)p Fs(;)h Fu(1)p Fs(;)g(x)p Fu(\))p Fr(!)o Fs(F)f Fu(\()p Fs(x;)h(x;)g(x)p Fu(\))p Fr(g)i Fu(is)i(eviden)o(tly)f (terminating)f(but)j(not)g(simplifying,)d(since)i Fr(!)1747 950 y Fq(+)1747 983 y Fl(S)1768 988 y Fc(1)1785 983 y Fo(e)1801 988 y Fc(1)1833 971 y Fu(is)g(not)0 1031 y(irre\015exiv)o(e:) 19 b(If)c Fs(t)f Fr(\021)f Fs(F)7 b Fu(\(0)p Fs(;)h Fu(1)p Fs(;)g Fu(1\))17 b(and)g Fs(s)c Fr(\021)h Fs(F)7 b Fu(\()p Fs(t;)h(t;)g(t)p Fu(\),)14 b(then)i Fs(s)d Fr(!)1169 1038 y Fl(F)1194 1043 y Fc(1)1212 1026 y Fd(e)1244 1031 y Fs(F)7 b Fu(\(0)p Fs(;)h(t;)g(t)p 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1607 y(Therefore,)13 b(w)o(e)g(state)h(the)f(resp)q(ectiv)o(e)f (de\014nitions)i(and)g(considerations)g(only)f(for)h(the)f(color)g (blac)o(k)g(\(the)0 1668 y(same)i(applies)h(cum)f(grano)i(coloris)f (for)h(the)f(color)g(white\).)0 1803 y Ft(De\014nition)i(4.2)24 b Fu(Let)19 b Fs(s)g Fu(b)q(e)g(a)g(top)g(blac)o(k)f(term.)28 b(There)19 b(is)f(a)i(uniquely)d(determined)g(blac)o(k)h(con)o(text)0 1863 y Fs(C)t Fu([)p Fs(;)8 b(:)g(:)g(:)e(;)i Fu(])20 b(suc)o(h)f(that)i Fs(s)f Fr(\021)g Fs(C)t Fu([)p Fs(s)596 1870 y Fq(1)614 1863 y Fs(;)8 b(:)g(:)g(:)g(;)g(s)747 1870 y Fo(n)770 1863 y Fu(])20 b(and)g Fs(r)q(oot)p Fu(\()p Fs(s)1031 1870 y Fo(i)1046 1863 y Fu(\))g Fr(2)g(D)1176 1870 y Fq(2)1216 1863 y Fu(for)h Fs(i)e Fr(2)i(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)f(n)p Fr(g)p Fu(.)32 b(W)l(e)20 b(denote)g(this)0 1923 y(b)o(y)d Fs(s)g Fr(\021)f Fs(C)t Fu([)-8 b([)p Fs(s)246 1930 y Fq(1)264 1923 y Fs(;)8 b(:)g(:)g(:)g(;)g(s)397 1930 y Fo(n)420 1923 y Fu(])-8 b(].)25 b(Moreo)o(v)o(er,)16 b(w)o(e)i(de\014ne)f(the)h(set)g Fs(S)s Fu(\()p Fs(s)p Fu(\))f(of)i(all)e Fm(princip)n(al)h Fu(subterms)e(of)j Fs(s)e Fu(to)i(b)q(e)0 1983 y Fs(S)s Fu(\()p Fs(s)p Fu(\))14 b(:=)f Fr(f)p Fs(s)221 1990 y Fq(1)241 1983 y Fs(;)8 b(:)g(:)g(:)f(;)h(s)373 1990 y Fo(n)396 1983 y Fr(g)p Fu(.)20 b(No)o(w)11 b(let)g Fs(s)g Fu(b)q(e)h(an)g(arbitrary)f(term.)18 b(There)11 b(is)h(a)f(\(p)q(ossibly)h(empt)o(y\))d(transparen)o(t)0 2043 y(con)o(text)j Fs(C)t Fu([)p Fs(;)c(:)g(:)g(:)f(;)h Fu(])k(suc)o(h)h(that)h Fs(s)g Fr(\021)f Fs(C)t Fu([)p Fs(s)733 2050 y Fq(1)752 2043 y Fs(;)8 b(:)g(:)g(:)f(;)h(s)884 2050 y Fo(n)908 2043 y Fu(])o(,)14 b(where)e Fs(r)q(oot)p Fu(\()p Fs(s)1215 2050 y Fo(i)1230 2043 y Fu(\))i Fr(2)g(D)1348 2050 y Fq(1)1373 2043 y Fr([)5 b(D)1449 2050 y Fq(2)1482 2043 y Fu(for)14 b Fs(i)f Fr(2)h(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(n)p Fr(g)p Fu(;)14 b(from)0 2104 y(no)o(w)j(on)h(suc)o(h)f(a)g(transparen)o (t)h(con)o(text)e(will)g(b)q(e)h(denoted)g(b)o(y)g Fs(C)1213 2085 y Fo(t)1227 2104 y Fu([)p Fs(;)8 b(:)g(:)g(:)f(;)h Fu(].)23 b(A)17 b(subterm)f Fs(t)h Fu(of)g Fs(s)g Fu(is)g(called)0 2164 y(an)i Fm(inner)g Fu(subterm)e(of)h Fs(s)g Fu(if)g(it)g(is)g(a)g (subterm)f(of)i(some)e(principal)g(subterm)g(of)i(an)f Fs(s)1603 2171 y Fo(j)1622 2164 y Fu(,)g Fs(j)i Fr(2)e(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)f(n)p Fr(g)p Fu(.)0 2224 y(Otherwise)16 b Fs(t)g Fu(is)h(said)g(to)g(b)q(e)g(an)g Fm(outer)g Fu(subterm)f(of)h Fs(s)p Fu(.)22 b(W)l(e)17 b(denote)g(the)f(set)h(of)g (all)f(outer)h(subterms)f(of)0 2284 y Fs(s)g Fu(b)o(y)g Fs(O)q Fu(\()p Fs(s)p Fu(\).)0 2419 y Ft(De\014nition)i(4.3)24 b Fu(Let)c Fs(s)h Fr(\021)g Fs(C)t Fu([)-8 b([)p Fs(s)635 2426 y Fq(1)653 2419 y Fs(;)8 b(:)g(:)g(:)g(;)g(s)786 2426 y Fo(n)809 2419 y Fu(])-8 b(])19 b(and)i Fs(s)h Fr(!)1042 2426 y Fl(R)1095 2419 y Fs(t)e Fu(b)o(y)g(an)h(application)f (of)h(a)g(rewrite)f(rule)g(of)0 2479 y Fr(R)p Fu(.)25 b(W)l(e)18 b(write)f Fs(s)f Fr(!)382 2461 y Fo(i)382 2492 y Fl(R)430 2479 y Fs(t)h 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Fq(+1)1706 2600 y Fs(;)g(:)g(:)g(:)g(;)g (s)1839 2607 y Fo(n)1862 2600 y Fu(])15 b(for)0 2660 y(some)g Fs(j)i Fr(2)d(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(n)p Fr(g)p Fu(.)21 b(Then)16 b(w)o(e)g(write)g Fs(s)e Fr(!)843 2642 y Fo(i)843 2672 y Fl(R)888 2660 y Fs(t)i Fu(if)g Fs(s)990 2667 y Fo(j)1022 2660 y Fr(!)1072 2642 y Fo(i)1072 2672 y Fl(R)1118 2660 y Fs(t)1136 2667 y Fo(j)1170 2660 y Fu(and)h Fs(s)d Fr(!)1352 2642 y Fo(o)1352 2672 y Fl(R)1398 2660 y Fs(t)h Fu(if)h Fs(s)1499 2667 y Fo(j)1531 2660 y Fr(!)1581 2642 y Fo(o)1581 2672 y Fl(R)1627 2660 y Fs(t)1645 2667 y Fo(j)1663 2660 y Fu(.)950 2936 y(11)p eop %%Page: 12 12 12 11 bop 0 26 a Ft(De\014nition)18 b(4.4)24 b Fu(Let)16 b Fs(s)g Fu(b)q(e)h(a)f(top)h(blac)o(k)e(or)i(top)g(white)e(term.)20 b(W)l(e)c(de\014ne)147 159 y Fs(r)q(ank)r Fu(\()p Fs(s)p Fu(\))d(=)378 86 y Fi(\()432 129 y Fu(1)620 b(,)16 b(if)g Fs(s)d Fr(2)i(T)d Fu(\()p Fr(D)1331 136 y Fq(1)1351 129 y Fs(;)c Fr(C)s Fs(;)g Fr(V)t Fu(\))j Fr([)g(T)i Fu(\()p Fr(D)1630 136 y Fq(2)1650 129 y Fs(;)8 b Fr(C)s Fs(;)g Fr(V)t Fu(\))432 189 y(1)k(+)f Fs(max)p Fr(f)p Fs(r)q(ank)r Fu(\()p Fs(s)786 196 y Fo(i)799 189 y Fu(\))d Fr(j)g Fu(1)14 b Fr(\024)g Fs(i)g Fr(\024)f Fs(n)p Fr(g)p Fu(,)j(if)g Fs(s)d Fr(\021)h Fs(C)t Fu([)-8 b([)p Fs(s)1322 196 y Fq(1)1340 189 y Fs(;)8 b(:)g(:)g(:)f(;)h(s)1472 196 y Fo(n)1496 189 y Fu(])-8 b(])0 292 y(No)o(w)16 b(let)g Fs(t)f Fu(b)q(e)i(a)f(top)h(transparen)o(t)g(term.)i(Then)e(w)o(e)f (de\014ne)314 425 y Fs(r)q(ank)r Fu(\()p Fs(t)p Fu(\))e(=)540 352 y Fi(\()595 394 y Fu(0)554 b(,)15 b(if)h Fs(t)e Fr(2)g(T)e Fu(\()p Fr(C)s Fs(;)c Fr(V)t Fu(\))595 454 y Fs(max)p Fr(f)p Fs(r)q(ank)r Fu(\()p Fs(t)859 461 y Fo(j)875 454 y Fu(\))g Fr(j)g Fu(1)15 b Fr(\024)f Fs(j)i Fr(\024)e Fs(m)p Fr(g)p Fu(,)h(if)h Fs(t)e Fr(\021)f Fs(C)1370 436 y Fo(t)1384 454 y Fu([)p Fs(t)1416 461 y Fq(1)1435 454 y Fs(;)8 b(:)g(:)g(:)g(;)g(t)1563 461 y Fo(m)1595 454 y Fu(])73 582 y(The)21 b(pro)q(ofs)h(of)g(the)e(next)h(lemm)o(ata)e (are)i(straigh)o(tforw)o(ard.)36 b Fr(R)22 b Fu(denotes)f(as)g(usual)h (a)f(com)o(bined)0 642 y(CTRS)c(and)g Fs(s;)8 b(t)15 b Fu(are)i(terms)d(from)h Fr(T)e Fu(\()p Fr(F)5 b Fs(;)j Fr(V)t Fu(\))o(.)0 771 y Ft(Lemma)16 b(4.5)24 b Fs(s)p Fr(!)368 778 y Fl(R)400 771 y Fs(t)13 b Fr(\))h Fs(r)q(ank)r Fu(\()p Fs(s)p Fu(\))f Fr(\025)h Fs(r)q(ank)r Fu(\()p Fs(t)p Fu(\))p Fs(:)0 899 y Ft(Lemma)i(4.6)24 b Fu(Let)12 b Fs(r)q(oot)p Fu(\()p Fs(s)p Fu(\))i Fr(2)g(D)625 906 y Fq(1)657 899 y Fu(and)f Fs(s)g Fr(\021)h Fs(s)860 906 y Fq(1)880 899 y Fr(!)930 906 y Fl(R)962 899 y Fs(s)985 906 y Fq(2)1004 899 y Fr(!)1054 906 y Fl(R)1095 899 y Fs(:)8 b(:)g(:)f Fr(!)1210 906 y Fl(R)1242 899 y Fs(s)1265 906 y Fo(n)1303 899 y Fr(\021)13 b Fs(s)f Fu(b)q(e)g(a)g(cyclic)e (sequence.)19 b(Then)0 959 y Fs(r)q(ank)r Fu(\()p Fs(s)147 966 y Fo(i)161 959 y Fu(\))14 b(=)g Fs(r)q(ank)r Fu(\()p Fs(s)p Fu(\))i(and)h Fs(r)q(oot)p Fu(\()p Fs(s)652 966 y Fo(i)667 959 y Fu(\))c Fr(2)i(D)785 966 y Fq(1)821 959 y Fu(for)h(ev)o(ery)f Fs(i)f Fr(2)g(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)f(n)p Fr(g)p Fu(.)0 1088 y Ft(Example)16 b(4.7)24 b Fu(Let)c Fr(R)460 1095 y Fq(1)501 1088 y Fu(=)g Fr(f)p Fs(F)7 b Fu(\()p Fo(C)r Fu(\()p Fs(x;)h(y)r Fu(\)\))19 b Fr(!)h Fo(C)s Fu(\()p Fs(F)7 b Fu(\()p Fs(x)p Fu(\))p Fs(;)h(y)r Fu(\))19 b Fr(\()h Fs(x)p Fr(#)o Fs(y)r Fr(g)g Fu(and)h Fr(R)1463 1095 y Fq(2)1503 1088 y Fu(=)f Fr(f)p Fs(g)r Fu(\()p Fs(x)p Fu(\))g Fr(!)h Fo(C)r Fu(\()p Fs(x;)8 b(x)p Fu(\))p Fr(g)p Fu(.)0 1148 y(Let)16 b(us)h(consider)f(the)g (reduction)g(sequence)126 1253 y Fs(s)149 1260 y Fq(1)183 1253 y Fr(\021)e Fs(F)7 b Fu(\()p Fo(C)q Fu(\()p Fs(g)r Fu(\()p Fs(A)p Fu(\))p Fs(;)h(g)r Fu(\()p Fs(A)p Fu(\)\)\))13 b Fr(!)662 1232 y Fo(o)662 1265 y Fl(R)708 1253 y Fs(s)731 1260 y Fq(2)765 1253 y Fr(\021)g Fo(C)s Fu(\()p Fs(F)7 b Fu(\()p Fs(g)r Fu(\()p Fs(A)p Fu(\)\))p Fs(;)h(g)r Fu(\()p Fs(A)p Fu(\)\))k Fr(!)1244 1232 y Fo(i)1244 1265 y Fl(R)1290 1253 y Fs(s)1313 1260 y Fq(3)1346 1253 y Fr(\021)i Fo(C)r Fu(\()p Fs(F)7 b Fu(\()p Fo(C)r Fu(\()p Fs(A;)h(A)p Fu(\)\))p Fs(;)g(g)r Fu(\()p Fs(A)p Fu(\)\))0 1357 y(W)l(e)16 b(ha)o(v)o(e)f Fs(r)q(ank)r Fu(\()p Fs(s)343 1364 y Fo(i)357 1357 y Fu(\))f(=)g(3)i(for)h Fs(i)c Fr(2)h(f)p Fu(1)p Fs(;)8 b Fu(2)p Fr(g)17 b Fu(and)f Fs(r)q(ank)r Fu(\()p Fs(s)1012 1364 y Fq(3)1032 1357 y Fu(\))e(=)g(2.)21 b Fs(S)s Fu(\()p Fs(s)1251 1364 y Fq(1)1271 1357 y Fu(\))14 b(=)g Fr(f)p Fs(g)r Fu(\()p Fs(A)p Fu(\))p Fr(g)h Fu(whereas)h Fs(S)s Fu(\()p Fs(s)1780 1364 y Fo(i)1794 1357 y Fu(\))g(is)g(not)0 1417 y(de\014ned)j(for)g Fs(i)f Fr(2)g(f)p Fu(2)p Fs(;)8 b Fu(3)p Fr(g)p Fu(.)30 b(F)l(urthermore,)17 b(w)o(e)i(ha)o(v)o(e)f (for)h(instance)g Fs(O)q Fu(\()p Fs(s)1337 1424 y Fq(1)1357 1417 y Fu(\))f(=)g Fr(f)p Fs(s)1498 1424 y Fq(1)1518 1417 y Fs(;)8 b Fo(C)r Fu(\()p Fs(g)r Fu(\()p Fs(A)p Fu(\))p Fs(;)g(g)r Fu(\()p Fs(A)p Fu(\)\))p Fr(g)18 b Fu(and)0 1478 y Fs(O)q Fu(\()p Fs(s)80 1485 y Fq(3)100 1478 y Fu(\))c(=)g Fr(f)p Fs(s)233 1485 y Fq(3)252 1478 y Fs(;)8 b(F)f Fu(\()p Fo(C)r Fu(\()p Fs(A;)h(A)p Fu(\)\))p Fs(;)g 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Fu(is)g(blac)o(k,)f Fs(\033)638 2808 y Fq(2)673 2801 y Fu(is)h(top)h(white,)e(and)i Fs(\033)1078 2808 y Fq(2)1111 2801 y Fr(/)d Fs(\017)p Fu(.)21 b Fh(2)950 2936 y Fu(12)p eop %%Page: 13 13 13 12 bop 0 26 a Fn(4.2)70 b(The)22 b(Main)h(Theorem)0 139 y Ft(Theorem)17 b(4.11)24 b Fu(Let)19 b Fr(R)494 146 y Fq(1)534 139 y Fu(and)h Fr(R)674 146 y Fq(2)713 139 y Fu(b)q(e)g(t)o(w)o(o)f(CTRS's)h(and)g(let)e Fr(R)i Fu(b)q(e)g(the)f(com)o(bined)e(CTRS)j(of)g Fr(R)1929 146 y Fq(1)0 200 y Fu(and)d Fr(R)137 207 y Fq(2)174 200 y Fu(with)f(shared)h(constructors.)23 b(Then)17 b Fr(R)g Fu(is)f(simplifying)e(if)i(and)h(only)g(if)f(b)q(oth)h Fr(R)1673 207 y Fq(1)1710 200 y Fu(and)g Fr(R)1847 207 y Fq(2)1884 200 y Fu(are)0 260 y(simplifying.)0 341 y Ft(Pro)r(of:)22 b Fu(Let)17 b Fr(S)t Fu(,)p Fr(S)340 348 y Fq(1)359 341 y Fu(,)f(and)h Fr(S)514 348 y Fq(2)551 341 y Fu(b)q(e)g(the)f(unconditional)h(TRS's)f(asso)q(ciated)i(to)f Fr(R)p Fu(,)f Fr(R)1563 348 y Fq(1)1583 341 y Fu(,)g(and)i Fr(R)1751 348 y Fq(2)1771 341 y Fu(,)e(resp)q(ec-)0 401 y(tiv)o(ely)l(.)j(W)l(e)d(ha)o(v)o(e)293 508 y Fr(!)343 515 y Fl(S)r Fo(e)399 508 y Fu(=)p Fr(!)487 516 y Fq(\()p Fl(S)522 521 y Fc(1)539 516 y Fo(e)555 521 y Fc(1)573 516 y Fq(\))p Fl([)p Fq(\()p Fl(S)646 521 y Fc(2)663 516 y Fo(e)679 521 y Fc(2)696 516 y Fq(\))712 508 y Fu(=)p Fr(!)800 515 y Fl(S)821 520 y Fc(1)839 515 y Fo(e)855 520 y Fc(1)888 508 y Fr([)e(!)985 515 y Fl(S)1006 520 y Fc(2)1024 515 y Fo(e)1040 520 y Fc(2)1059 508 y Fu(=)p Fr(!)1147 515 y Fl(S)1168 520 y Fc(1)1202 508 y Fr([)g(!)1299 515 y Fl(F)1327 504 y Fd(e)1324 527 y Fc(1)1360 508 y Fr([)g(!)1457 515 y Fl(S)1478 520 y Fc(2)1511 508 y Fr([)g(!)1608 515 y Fl(F)1636 504 y Fd(e)1633 527 y Fc(2)0 615 y Fu(The)19 b(only-if)f(part)i(is)e(trivial)g(so)i(let)e(us)h(consider)g(the)f(if)h (direction.)28 b(According)18 b(to)i(Lemma)c(3.14,)k(it)0 675 y(m)o(ust)f(b)q(e)h(sho)o(wn)h(that)f Fr(!)506 655 y Fq(+)506 688 y Fl(S)r Fo(e)569 675 y Fu(is)g(irre\015exiv)o(e.)30 b(Without)20 b(loss)h(of)f(generalit)o(y)f(\(Lemma)f(3.18\))j(w)o(e)e (ma)o(y)0 736 y(assume)f(that)h(the)f(set)g Fr(C)k Fu(of)d (constructors)g(con)o(tains)g(t)o(w)o(o)f(sym)o(b)q(ols)g(not)h(used)f (previously)l(,)g(namely)e(a)0 796 y(constructor)h Fo(C)r(ons)g Fu(of)g(arit)o(y)e(2)i(and)g(a)f(constan)o(t)h Fo(N)t(il)o Fu(.)22 b(Assuming)15 b(that)h Fr(!)1370 775 y Fq(+)1370 808 y Fl(S)r Fo(e)1428 796 y Fu(is)g(not)h(irre\015exiv)o(e,)c(w)o(e)j (will)0 856 y(deriv)o(e)f(a)h(con)o(tradiction.)21 b(W)l(e)16 b(assume)f(that)i(there)f(is)g(a)g(cyclic)f(sequence)685 963 y Fs(s)f Fr(\021)g Fs(s)798 970 y Fq(1)831 963 y Fr(!)881 970 y Fl(S)r Fo(e)937 963 y Fs(:)8 b(:)g(:)13 b Fr(!)1058 970 y Fl(S)r Fo(e)1114 963 y Fs(s)1137 970 y Fo(n)1174 963 y Fr(\021)h Fs(s;)0 1070 y(n)23 b(>)h Fu(1,)f(with)f Fs(s)315 1077 y Fo(i)352 1070 y Fr(2)h(T)13 b Fu(\()p Fr(D)505 1077 y Fq(1)536 1070 y Fr(])f(D)619 1077 y Fq(2)639 1070 y Fs(;)c Fr(C)s Fs(;)g Fr(V)s Fu(\))22 b(for)g(all)f Fs(i)i Fr(2)h(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(;)g(n)p Fr(g)p Fu(.)37 b(Without)22 b(loss)h(of)f(generalit)o(y)e(w)o(e)0 1130 y(ma)o(y)d(further)h(assume)f(that)i Fs(s)f Fu(is)g(top)h(blac)o (k,)e(and)i(that)f(rank\()p Fs(s)p Fu(\))g(=:)e Fs(k)k Fu(is)e(minimal,)d(i.e.,)i(there)g(is)h(no)0 1191 y(cyclic)e(sequence)i Fs(t)f Fr(\021)g Fs(t)450 1198 y Fq(1)487 1191 y Fr(!)537 1198 y Fl(S)r Fo(e)597 1191 y Fs(:)8 b(:)g(:)17 b Fr(!)722 1198 y Fl(S)r Fo(e)781 1191 y Fs(t)799 1198 y Fo(m)850 1191 y Fr(\021)g Fs(t;)8 b(m)17 b(>)h Fu(1,)h(with)f(rank\()p Fs(t)p Fu(\))f Fs(<)h(k)r Fu(.)28 b(Consequen)o(tly)l(,)18 b Fr(!)1891 1170 y Fq(+)1891 1203 y Fl(S)r Fo(e)0 1251 y Fu(is)k(irre\015exiv)o(e)d(on)j Fs(T)30 b Fu(=)23 b Fr(f)p Fs(t)p Fr(j)p Fs(r)q(ank)r Fu(\()p Fs(t)p Fu(\))f Fs(<)h(r)q(ank)r Fu(\()p Fs(s)p Fu(\))p Fr(g)f Fu(\(or)g(equiv)m(alen)o (tly)e Fr(!)1402 1230 y Fq(+)1402 1263 y Fl(S)r Fo(e)1482 1251 y Fu(is)i(terminating)e(on)i Fs(T)7 b Fu(\).)0 1311 y(According)16 b(to)g(Lemma)f(4.6)h(ev)o(ery)f Fs(s)696 1318 y Fo(i)710 1311 y Fs(;)8 b Fu(0)14 b Fr(\024)g Fs(i)f Fr(\024)h Fs(n)p Fu(,)i(is)g(top)h(blac)o(k)e(and)i(has)g(rank)g Fs(k)r Fu(.)k(The)c(pro)q(of)g(idea)f(is)0 1371 y(to)h(construct)f(a)h (cyclic)d(sequence)h(of)h(blac)o(k)g(terms)534 1478 y Fs(\032)p Fu(\()p Fs(s)p Fu(\))e Fr(\021)f Fs(\032)p Fu(\()p Fs(s)753 1485 y Fq(1)773 1478 y Fu(\))h Fr(!)856 1458 y Fq(+)856 1491 y Fl(S)877 1496 y Fc(1)894 1491 y Fo(e)910 1496 y Fc(1)944 1478 y Fs(:)8 b(:)g(:)13 b Fr(!)1065 1458 y Fq(+)1065 1491 y Fl(S)1086 1496 y Fc(1)1103 1491 y Fo(e)1119 1496 y Fc(1)1153 1478 y Fs(\032)p Fu(\()p Fs(s)1220 1485 y Fo(n)1244 1478 y Fu(\))g Fr(\021)h Fs(\032)p Fu(\()p Fs(s)p Fu(\))0 1585 y(yielding)j(a)h(con)o(tradiction)g(to)g (the)g(irre\015exivit)o(y)d(of)j Fr(!)1042 1565 y Fq(+)1042 1598 y Fl(S)1063 1603 y Fc(1)1081 1598 y Fo(e)1097 1603 y Fc(1)1116 1585 y Fu(.)27 b(First)18 b(w)o(e)f(transform)h(eac)o(h)g (term)e Fs(s)1830 1592 y Fo(j)1866 1585 y Fu(in)o(to)0 1646 y(a)k(blac)o(k)f(term)f Fs(\032)p Fu(\()p Fs(s)362 1653 y Fo(j)380 1646 y Fu(\))h(and)i(then)e(sho)o(w)h(that)g Fs(s)887 1653 y Fo(j)925 1646 y Fr(!)975 1653 y Fl(S)r Fo(e)1036 1646 y Fs(s)1059 1653 y Fo(j)r Fq(+1)1142 1646 y Fu(implies)d Fs(\032)p Fu(\()p Fs(s)1378 1653 y Fo(j)1397 1646 y Fu(\))i Fr(!)1485 1625 y Fq(+)1485 1658 y Fl(S)1506 1663 y Fc(1)1524 1658 y Fo(e)1540 1663 y Fc(1)1579 1646 y Fs(\032)p Fu(\()p Fs(s)1646 1653 y Fo(j)r Fq(+1)1709 1646 y Fu(\).)31 b(In)20 b(order)0 1706 y(to)g(de\014ne)g(the)f (transformation,)i(w)o(e)e(consider)h(the)f(sets)h Fs(S)j Fu(=)1212 1673 y Fi(S)1247 1686 y Fo(n)1247 1718 y(j)r Fq(=1)1319 1706 y Fs(S)s Fu(\()p Fs(s)1394 1713 y Fo(j)1412 1706 y Fu(\))d(and)g Fs(O)i Fu(=)1666 1673 y Fi(S)1700 1686 y Fo(n)1700 1718 y(j)r Fq(=1)1772 1706 y Fs(O)q Fu(\()p Fs(s)1852 1713 y Fo(j)1871 1706 y Fu(\))e(of)0 1766 y(all)c(principal)g(and)h(outer)f(subterms)g(o)q(ccurring)h(in)f (the)g(sequence.)21 b(Notice)16 b(that)h Fs(r)q(oot)p Fu(\()p Fs(t)p Fu(\))d Fr(2)h(D)1786 1773 y Fq(2)1823 1766 y Fu(for)h(all)0 1826 y Fs(t)d Fr(2)h Fs(S)k Fu(and)e Fs(r)q(oot)p Fu(\()p Fs(t)p Fu(\))e Fr(2)g(D)462 1833 y Fq(1)490 1826 y Fr([)8 b(C)k([)c(V)19 b Fu(for)c(all)g Fs(t)e Fr(2)h Fs(O)j Fu(as)e(w)o(ell)f(as)h Fs(r)q(ank)r Fu(\()p Fs(t)p Fu(\))f Fr(\024)f Fs(r)q(ank)r Fu(\()p Fs(s)p Fu(\))i(for)g(eac)o(h)g Fs(t)e Fr(2)h Fs(S)d Fr([)d Fs(O)q Fu(.)0 1886 y(F)l(urthermore,)14 b(for)j(ev)o(ery)d Fs(t)g Fr(2)g Fs(S)19 b Fu(let)0 1994 y(\001\()p Fs(t)p Fu(\))13 b(=)h Fr(f)p Fs(t)205 1973 y Fl(0)216 1994 y Fr(j9)p Fs(j)i Fr(2)e(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(;)g(n)p Fr(\000)p Fu(1)p Fr(g)16 b Fu(with)h Fs(s)768 2001 y Fo(j)800 1994 y Fr(\021)c Fs(C)t Fu([)-8 b([)p Fs(:)8 b(:)g(:)e(;)i(t;)g(:)g(:)g(:)o Fu(])-8 b(])12 b Fr(!)1176 1973 y Fo(i)1176 2006 y Fl(S)r Fo(e)1232 1994 y Fs(C)t Fu([)p Fs(:)c(:)g(:)f(;)h(t)1390 1973 y Fl(0)1401 1994 y Fs(;)g(:)g(:)g(:)o Fu(])13 b Fr(\021)h Fs(s)1583 2001 y Fo(j)r Fq(+1)1662 1994 y Fu(and)j Fs(t)d Fr(!)1839 2001 y Fl(S)r Fo(e)1895 1994 y Fs(t)1913 1973 y Fl(0)1924 1994 y Fr(g)0 2101 y Fu(b)q(e)24 b(the)f(set)h(of)g(successors)g(of)g Fs(t)p Fu(.)44 b(Note)23 b(that)h(\001\()p Fs(t)p Fu(\))f(is)h (\014nite)f(for)h(an)o(y)f Fs(t)k Fr(2)f Fs(S)s Fu(.)44 b(Then)24 b(w)o(e)f(de\014ne)0 2161 y Fs(\032)14 b Fu(:)f Fs(S)h Fr([)e Fs(O)q Fr(!T)h Fu(\()p Fr(D)340 2168 y Fq(1)360 2161 y Fs(;)8 b Fr(C)s Fs(;)g Fr(V)s Fu(\))17 b(recursiv)o(ely)c(b)o(y:)60 2281 y(1.)24 b Fs(\032)p Fu(\()p Fs(t)p Fu(\))13 b(=)h([)p Fs(\032)p Fu(\()p Fs(t)344 2263 y Fl(0)344 2293 y Fq(1)363 2281 y Fu(\))p Fs(;)8 b(:)g(:)g(:)g(;)g(\032)p Fu(\()p Fs(t)554 2263 y Fl(0)554 2293 y Fo(m)587 2281 y Fu(\)])620 2263 y Fq(8)655 2281 y Fu(if)16 b Fs(t)d Fr(2)h Fs(S)19 b Fu(and)e(\001\()p Fs(t)p Fu(\))c(=)h Fr(f)p Fs(t)1127 2263 y Fl(0)1127 2293 y Fq(1)1146 2281 y Fs(;)8 b(:)g(:)g(:)g(;)g(t)1274 2263 y Fl(0)1274 2293 y Fo(m)1306 2281 y Fr(g)p Fu(.)122 2341 y(Note)16 b(that)g Fs(t)363 2323 y Fl(0)363 2353 y Fq(1)383 2341 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)510 2323 y Fl(0)510 2353 y Fo(m)557 2341 y Fr(2)14 b Fs(S)g Fr([)d Fs(O)q Fu(.)60 2442 y(2.)24 b Fs(\032)p Fu(\()p Fs(t)p Fu(\))13 b(=)h Fs(F)7 b Fu(\()p Fs(\032)p Fu(\()p Fs(t)388 2449 y Fq(1)407 2442 y Fu(\))p Fs(;)h(:)g(:)g(:)f(;)h(\032)p Fu(\()p Fs(t)597 2449 y Fo(m)630 2442 y Fu(\)\))16 b(if)g Fs(t)e Fr(2)g Fs(O)k Fu(and)e Fs(t)e Fr(\021)f Fs(F)7 b Fu(\()p Fs(t)1117 2449 y Fq(1)1136 2442 y Fs(;)h(:)g(:)g(:)f(;)h(t) 1263 2449 y Fo(m)1296 2442 y Fu(\))1331 2424 y Fq(9)1367 2442 y Fu(.)122 2502 y(Again)16 b(observ)o(e)g(that)h Fs(t)562 2509 y Fq(1)581 2502 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)708 2509 y Fo(m)755 2502 y Fr(2)14 b Fs(S)g Fr([)d Fs(O)q Fu(.)0 2622 y(Here)19 b([)p Fs(t)151 2629 y Fq(1)169 2622 y Fs(;)8 b(:)g(:)g(:)g(;)g(t)297 2629 y Fo(m)330 2622 y Fu(])19 b(is)g(an)h(abbreviation)g(for)g(the)f(term)f Fo(C)r(ons)q Fu(\()p Fs(t)1177 2629 y Fq(1)1196 2622 y Fs(;)8 b Fo(C)r(ons)r Fu(\()p Fs(:)g(:)g(:)f(;)h Fo(C)r(ons)q Fu(\()p Fs(t)1526 2629 y Fo(m)1559 2622 y Fs(;)g Fo(N)t(il)o Fu(\))g Fs(:)g(:)g(:)p Fu(\)\))19 b(where)h([])0 2682 y(stands)d(for)g Fo(N)t(il)o Fu(.)p 0 2716 780 2 v 56 2746 a Fk(8)75 2761 y Fj(The)d(order)h(is)e(arbitrary)m(.)56 2796 y Fk(9)75 2811 y Fa(F)k Ff(2)11 b(D)190 2817 y Fk(1)218 2811 y Ff([)e(C)i([)e(V)s Fj(.)950 2936 y Fu(13)p eop %%Page: 14 14 14 13 bop 0 26 a Fu(This)16 b(de\014nition)g(is)g(illustrated)f(in)h (Example)f(4.12.)0 86 y(W)l(e)f(ha)o(v)o(e)g(to)h(sho)o(w)g(that)g(the) f(recursion)g(ev)o(en)o(tually)e(stops,)j(i.e.,)e Fs(\032)i Fu(is)f(w)o(ell-de\014ned)f(for)i(an)o(y)f Fs(t)g Fr(2)g Fs(S)c Fr([)d Fs(O)q Fu(.)0 147 y(First)17 b(w)o(e)h(pro)o(v)o(e)f (that)h(\(i\))f Fs(\032)h Fu(is)f(w)o(ell-de\014ned)g(for)h(an)o(y)f Fs(t)f Fr(2)h Fs(T)1154 129 y Fl(0)1181 147 y Fu(=)g Fr(f)p Fs(t)p Fr(j)p Fs(r)q(ank)r Fu(\()p Fs(t)p Fu(\))e Fs(<)i(r)q(ank)r Fu(\()p Fs(s)p Fu(\))p Fs(;)8 b(t)15 b Fr(2)i Fs(S)e Fr([)d Fs(O)q Fr(g)0 207 y Fu(and)i(afterw)o(ards)f (that)g(\(ii\))g Fs(\032)g Fu(is)f(w)o(ell-de\014ned)g(for)h(an)o(y)g Fs(t)h Fr(2)g Fs(O)1132 189 y Fl(0)1158 207 y Fu(=)g Fr(f)p Fs(t)p Fr(j)p Fs(r)q(ank)r Fu(\()p Fs(t)p Fu(\))e(=)i Fs(r)q(ank)r Fu(\()p Fs(s)p Fu(\))p Fs(;)8 b(t)14 b Fr(2)g Fs(S)7 b Fr([)t Fs(O)q Fr(g)15 b Fu(=)0 267 y Fr(f)p Fs(t)p Fr(j)p Fs(r)q(ank)r Fu(\()p Fs(t)p Fu(\))e(=)h Fs(r)q(ank)r Fu(\()p Fs(s)p Fu(\))p Fs(;)8 b(t)13 b Fr(2)h Fs(O)q Fr(g)g(\022)g Fs(O)q Fu(.)0 327 y(\(i\))i Fr(!)118 307 y Fq(+)118 339 y Fl(S)r Fo(e)192 327 y Fu(is)g(terminating)f(on)i Fs(T)611 309 y Fl(0)638 327 y Fu(since)e Fs(T)793 309 y Fl(0)818 327 y Fr(\022)f Fs(T)7 b Fu(.)20 b(Let)d Fr(!)1079 334 y Fo(ar)q(g)1151 327 y Fu(b)q(e)f(the)g(smallest)e(relation)i(on)h Fs(T)1771 309 y Fl(0)1798 327 y Fu(with)73 438 y Fr(\017)24 b Fs(t)13 b Fr(!)203 445 y Fo(ar)q(g)273 438 y Fs(t)291 420 y Fl(0)291 451 y Fo(j)325 438 y Fu(if)j Fs(t)d Fr(2)h Fs(S)19 b Fu(and)e Fs(t)610 420 y Fl(0)610 451 y Fo(j)642 438 y Fr(2)d Fu(\001\()p Fs(t)p Fu(\),)73 536 y Fr(\017)24 b Fs(t)13 b Fr(!)203 543 y Fo(ar)q(g)273 536 y Fs(t)291 543 y Fo(j)325 536 y Fu(if)j Fs(t)d Fr(2)h Fs(O)k Fu(and)f Fs(t)c Fr(\021)h Fs(F)7 b Fu(\()p Fs(t)758 543 y Fq(1)776 536 y Fs(;)h(:)g(:)g(:)g(;)g(t)904 543 y Fo(m)937 536 y Fu(\).)0 647 y(Notice)k(that)h Fs(t)g Fr(!)333 654 y Fo(ar)q(g)403 647 y Fs(u)f Fu(implies)f Fs(t)p Fr(!)674 626 y Fq(+)674 659 y Fl(S)r Fo(e)731 647 y Fs(u)p Fu(.)20 b(Th)o(us)13 b Fr(!)963 654 y Fo(ar)q(g)1032 647 y Fu(is)g(w)o (ell-founded.)19 b(F)l(or)13 b(if)f(there)g(w)o(ere)g(an)i(in\014nite)0 707 y(sequence)i Fs(t)221 714 y Fq(1)256 707 y Fr(!)306 714 y Fo(ar)q(g)377 707 y Fs(t)395 714 y Fq(2)429 707 y Fr(!)479 714 y Fo(ar)q(g)550 707 y Fs(:)8 b(:)g(:)17 b Fu(of)g(terms)f Fs(t)836 714 y Fo(j)869 707 y Fr(2)f Fs(T)953 689 y Fl(0)964 707 y Fu(,)i(it)g(w)o(ould)g(follo)o(w)g(that)h Fr(!)1487 687 y Fq(+)1487 719 y Fl(S)r Fo(e)1562 707 y Fu(is)f(not)h(terminating)0 767 y(on)f Fs(T)104 749 y Fl(0)115 767 y Fu(,)f(a)g(con)o(tradiction.)21 b(Consequen)o(tly)l(,) 15 b Fs(\032)h Fu(is)g(w)o(ell-de\014ned)f(on)i Fs(T)1274 749 y Fl(0)1285 767 y Fu(.)0 828 y(\(ii\))e(Assume)g(that)h Fs(\032)h Fu(is)f(not)g(w)o(ell-de\014ned)f(on)i Fs(O)917 810 y Fl(0)929 828 y Fu(,)f(that)g(is)g(to)h(sa)o(y)l(,)e Fr(!)1316 835 y Fo(ar)q(g)1388 828 y Fu(\(no)o(w)h(de\014ned)g(on)h Fs(O)1782 810 y Fl(0)1794 828 y Fu(\))f(is)g(not)0 888 y(w)o(ell-founded)k(on)h Fs(O)399 870 y Fl(0)411 888 y Fu(.)33 b(Hence)20 b(there)f(exists)h(an)h(in\014nite)e(sequence)h Fs(t)1341 895 y Fq(1)1381 888 y Fr(!)1431 895 y Fo(ar)q(g)1507 888 y Fs(t)1525 895 y Fq(2)1566 888 y Fr(!)1616 895 y Fo(ar)q(g)1692 888 y Fs(:)8 b(:)g(:)20 b Fu(of)g(terms)0 948 y Fs(t)18 955 y Fo(j)57 948 y Fr(2)i Fs(O)150 930 y Fl(0)162 948 y Fu(.)35 b(Since)20 b Fs(O)381 930 y Fl(0)415 948 y Fr(\022)h Fs(O)q Fu(,)h(it)e(follo)o(ws)h(that)g Fs(t)895 955 y Fq(1)935 948 y Fu(m)o(ust)e(con)o(tain)i(in\014nitely)e (man)o(y)g(function)i(sym)o(b)q(ols)0 1008 y(\(since)14 b(the)g(second)g(clause)g(of)h(the)f(de\014nition)g(of)h Fs(\032)f Fu(applies)g(in\014nitely)f(man)o(y)g(times\),)f(a)j(con)o (tradiction.)0 1068 y(Next)g(w)o(e)h(sho)o(w)h(that)g Fs(s)441 1075 y Fo(j)473 1068 y Fr(!)523 1075 y Fl(S)r Fo(e)579 1068 y Fs(s)602 1075 y Fo(j)r Fq(+1)681 1068 y Fu(implies)d Fs(\032)p Fu(\()p Fs(s)914 1075 y Fo(j)932 1068 y Fu(\))g Fr(!)1015 1048 y Fq(+)1015 1081 y Fl(S)1036 1086 y Fc(1)1053 1081 y Fo(e)1069 1086 y Fc(1)1103 1068 y Fs(\032)p Fu(\()p Fs(s)1170 1075 y Fo(j)r Fq(+1)1233 1068 y Fu(\).)22 b(W)l(e)15 b(consider)h(the)g(cases:)0 1129 y(\(i\))g Fs(s)91 1136 y Fo(j)123 1129 y Fr(!)173 1111 y Fo(o)173 1141 y Fl(S)r Fo(e)229 1129 y Fs(s)252 1136 y Fo(j)r Fq(+1)73 1240 y Fr(\017)24 b Fu(if)17 b Fs(s)191 1247 y Fo(j)226 1240 y Fr(!)276 1222 y Fo(o)276 1252 y Fl(S)297 1257 y Fc(1)333 1240 y Fs(s)356 1247 y Fo(j)r Fq(+1)437 1240 y Fu(then)h Fs(\032)p Fu(\()p Fs(s)617 1247 y Fo(j)635 1240 y Fu(\))f Fr(!)721 1247 y Fl(S)742 1252 y Fc(1)778 1240 y Fs(\032)p Fu(\()p Fs(s)845 1247 y Fo(j)r Fq(+1)908 1240 y Fu(\).)26 b(F)l(or)18 b(if)g Fs(s)1126 1247 y Fo(j)1160 1240 y Fr(\021)f Fs(C)t Fu([)-8 b([)p Fs(t)1293 1247 y Fq(1)1310 1240 y Fs(;)8 b(:)g(:)g(:)g(;)g(t)1438 1247 y Fo(m)1470 1240 y Fu(])-8 b(])17 b(then)h(w)o(e)f(ha)o(v)o(e)g Fs(s)1830 1247 y Fo(j)r Fq(+1)1910 1240 y Fr(\021)122 1300 y Fs(C)161 1282 y Fl(0)172 1300 y Fu([)p Fs(t)204 1307 y Fo(i)216 1312 y Fc(1)235 1300 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)362 1307 y Fo(i)374 1313 y Fd(l)388 1300 y Fu(])21 b(for)h(some)f(blac)o(k) g(con)o(text)g Fs(C)982 1282 y Fl(0)993 1300 y Fu([)p Fs(;)8 b(:)g(:)g(:)f(;)h Fu(])21 b(and)h(some)f Fs(i)1396 1307 y Fq(1)1416 1300 y Fs(;)8 b(:)g(:)g(:)f(;)h(i)1542 1307 y Fo(l)1578 1300 y Fr(2)23 b(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)g(;)g (m)p Fr(g)p Fu(.)37 b(It)122 1360 y(follo)o(ws)22 b(with)g Fs(\032)p Fu(\()p Fs(s)472 1367 y Fo(j)490 1360 y Fu(\))i Fr(\021)f Fs(C)t Fu([)p Fs(\032)p Fu(\()p Fs(t)710 1367 y Fq(1)728 1360 y Fu(\))p Fs(;)8 b(:)g(:)g(:)g(;)g(\032)p Fu(\()p Fs(t)919 1367 y Fo(m)952 1360 y Fu(\)])21 b(and)i Fs(\032)p Fu(\()p Fs(s)1174 1367 y Fo(j)r Fq(+1)1237 1360 y Fu(\))h Fr(\021)f Fs(C)1381 1342 y Fl(0)1392 1360 y Fu([)p Fs(\032)p Fu(\()p Fs(t)1468 1367 y Fo(i)1480 1372 y Fc(1)1499 1360 y Fu(\))p Fs(;)8 b(:)g(:)g(:)f(;)h(\032)p Fu(\()p Fs(t)1689 1367 y Fo(i)1701 1373 y Fd(l)1715 1360 y Fu(\)])22 b(that)g(the)122 1420 y(rule)15 b(that)i(reduced)f Fs(s)530 1427 y Fo(j)564 1420 y Fu(to)h Fs(s)647 1427 y Fo(j)r Fq(+1)726 1420 y Fu(also)g(reduces)f Fs(\032)p Fu(\()p Fs(s)1065 1427 y Fo(j)1083 1420 y Fu(\))h(to)f Fs(\032)p Fu(\()p Fs(s)1245 1427 y Fo(j)r Fq(+1)1309 1420 y Fu(\).)73 1518 y Fr(\017)24 b Fu(if)12 b Fs(s)186 1525 y Fo(j)218 1518 y Fr(!)268 1500 y Fo(o)268 1530 y Fl(F)296 1521 y Fd(e)329 1518 y Fs(s)352 1525 y Fo(j)r Fq(+1)428 1518 y Fu(then)h Fs(\032)p Fu(\()p Fs(s)603 1525 y Fo(j)621 1518 y Fu(\))p Fr(!)690 1525 y Fl(F)718 1514 y Fd(e)715 1537 y Fc(1)737 1518 y Fs(\032)p Fu(\()p Fs(s)804 1525 y Fo(j)r Fq(+1)867 1518 y Fu(\),)g(since)f(only)h(blac)o (k)e(sym)o(b)q(ols)h(are)h(in)o(v)o(olv)o(ed)d(in)i(the)h(outer)122 1578 y(reduction.)0 1689 y(\(ii\))i Fs(s)104 1696 y Fo(j)136 1689 y Fr(!)186 1671 y Fo(i)186 1702 y Fl(S)r Fo(e)242 1689 y Fs(s)265 1696 y Fo(j)r Fq(+1)0 1749 y Fu(Then)20 b Fs(s)154 1756 y Fo(j)193 1749 y Fr(\021)g Fs(C)t Fu([)-8 b([)p Fs(t)329 1756 y Fq(1)347 1749 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)474 1756 y Fo(m)507 1749 y Fu(])-8 b(])19 b(and)h Fs(s)667 1756 y Fo(j)r Fq(+1)751 1749 y Fr(\021)g Fs(C)t Fu([)p Fs(t)881 1756 y Fq(1)900 1749 y Fs(;)8 b(:)g(:)g(:)f(;)h(t)1027 1756 y Fo(l)p Fl(\000)p Fq(1)1085 1749 y Fs(;)g(t)1125 1731 y Fl(0)1125 1762 y Fo(l)1137 1749 y Fs(;)g(t)1177 1756 y Fo(l)p Fq(+1)1235 1749 y Fs(;)g(:)g(:)g(:)f(;)h(t)1362 1756 y Fo(m)1395 1749 y Fu(])20 b(for)g(some)f Fs(l)i Fr(2)g(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)f(;)h(m)p Fr(g)0 1810 y Fu(and)17 b(some)e(term)g Fs(t)352 1792 y Fl(0)352 1822 y Fo(l)364 1810 y Fu(,)h(where)g Fs(t)553 1817 y Fo(l)579 1810 y Fr(!)629 1817 y Fl(S)r Fo(e)685 1810 y Fs(t)703 1792 y Fl(0)703 1822 y Fo(l)716 1810 y Fu(.)21 b(F)l(urthermore,)67 1906 y Fs(\032)p Fu(\()p Fs(s)134 1913 y Fo(j)153 1906 y Fu(\))14 b Fr(\021)f Fs(C)t Fu([)p Fs(\032)p Fu(\()p Fs(t)353 1913 y Fq(1)372 1906 y Fu(\))p Fs(;)8 b(:)g(:)g(:)f(;)h(\032)p Fu(\()p Fs(t)562 1913 y Fo(m)595 1906 y Fu(\)])16 b(and)h Fs(\032)p Fu(\()p Fs(s)806 1913 y Fo(j)r Fq(+1)869 1906 y Fu(\))d Fr(\021)f Fs(C)t Fu([)p Fs(\032)p Fu(\()p Fs(t)1069 1913 y Fq(1)1088 1906 y Fu(\))p Fs(;)8 b(:)g(:)g(:)f(;)h(\032)p Fu(\()p Fs(t)1278 1913 y Fo(l)p Fl(\000)p Fq(1)1336 1906 y Fu(\))p Fs(;)g(\032)p Fu(\()p Fs(t)1439 1886 y Fl(0)1439 1919 y Fo(l)1451 1906 y Fu(\))p Fs(;)g(\032)p Fu(\()p Fs(t)1554 1913 y Fo(l)p Fq(+1)1612 1906 y Fu(\))p Fs(;)g(:)g(:)g(:)f(;)h(\032)p Fu(\()p Fs(t)1802 1913 y Fo(m)1835 1906 y Fu(\)])p Fs(:)0 2003 y Fu(Since)14 b Fs(t)144 1985 y Fl(0)144 2015 y Fo(l)170 2003 y Fr(2)g Fu(\001\()p Fs(t)295 2010 y Fo(l)307 2003 y Fu(\),)h(w)o(e)f(ha)o(v)o(e)g(b)o(y)g(de\014nition)g(of)h Fs(\032)g Fu(that)g Fs(\032)p Fu(\()p Fs(t)1077 2010 y Fo(l)1089 2003 y Fu(\))f(=)g([)p Fs(:)8 b(:)g(:)f(;)h(\032)p Fu(\()p Fs(t)1337 1985 y Fl(0)1337 2015 y Fo(l)1350 2003 y Fu(\))p Fs(;)g(:)g(:)g(:)o Fu(],)14 b(i.e.,)e Fs(\032)p Fu(\()p Fs(t)1642 1985 y Fl(0)1642 2015 y Fo(l)1655 2003 y Fu(\))j(is)f(a)h(subterm)0 2063 y(of)23 b Fs(\032)p Fu(\()p Fs(t)124 2070 y Fo(l)136 2063 y Fu(\).)39 b(Hence)21 b Fs(\032)p Fu(\()p Fs(t)421 2070 y Fo(l)434 2063 y Fu(\))j(=)g([)p Fs(:)8 b(:)g(:)f(;)h(\032)p Fu(\()p Fs(t)702 2045 y Fl(0)702 2075 y Fo(l)715 2063 y Fu(\))p Fs(;)g(:)g(:)g(:)o Fu(])p Fr(!)877 2042 y Fq(+)877 2075 y Fl(F)905 2064 y Fd(e)902 2087 y Fc(1)923 2063 y Fs(\032)p Fu(\()p Fs(t)985 2045 y Fl(0)985 2075 y Fo(l)998 2063 y Fu(\),)23 b(where)f(only)g(the)g (rules)g Fo(C)r(ons)q Fu(\()p Fs(x;)8 b(y)r Fu(\))23 b Fr(!)h Fs(x)e Fu(and)0 2128 y Fo(C)r(ons)q Fu(\()p Fs(x;)8 b(y)r Fu(\))13 b Fr(!)h Fs(y)k Fu(are)e(used.)0 2188 y(Therefore,)f Fs(\032)p Fu(\()p Fs(s)301 2195 y Fo(j)320 2188 y Fu(\))p Fr(!)389 2168 y Fq(+)389 2200 y Fl(F)417 2189 y Fd(e)414 2212 y Fc(1)436 2188 y Fs(\032)p Fu(\()p Fs(s)503 2195 y Fo(j)r Fq(+1)566 2188 y Fu(\).)0 2256 y(All)23 b(in)g(all,)i(w)o(e)f(ha)o(v)o(e)f(sho)o(wn)h(that)h Fs(\032)p Fu(\()p Fs(s)p Fu(\))i Fr(!)875 2236 y Fq(+)875 2269 y Fl(S)896 2274 y Fc(1)913 2269 y Fo(e)929 2274 y Fc(1)976 2256 y Fs(\032)p Fu(\()p Fs(s)p Fu(\),)e(a)g(con)o (tradiction)e(to)h(the)g(fact)g(that)g Fr(R)1872 2263 y Fq(1)1916 2256 y Fu(is)0 2317 y(simplifying)14 b(\(Prop)q(osition)j (3.16\).)22 b Fh(2)0 2436 y Ft(Example)16 b(4.12)24 b Fu(Let)17 b Fr(S)473 2443 y Fq(1)507 2436 y Fu(=)d Fr(f)p Fs(F)7 b Fu(\()p Fo(C)q Fu(\()p Fs(x)p Fu(\))p Fs(;)h(F)f Fu(\()p Fs(y)r(;)h Fo(D)p Fu(\()p Fs(z)r Fu(\)\)\))13 b Fr(!)h Fs(F)7 b Fu(\()p Fs(y)r(;)h(x)p Fu(\))p Fs(;)g(A)k Fr(!)i Fs(B)s Fr(g)i Fu(and)h Fr(S)1563 2443 y Fq(2)1597 2436 y Fu(=)d Fr(f)p Fs(g)r Fu(\()p Fs(x)p Fu(\))f Fr(!)h Fo(C)s Fu(\()p Fs(x)p Fu(\))p Fs(;)0 2497 y(g)r Fu(\()p Fs(x)p Fu(\))g Fr(!)f Fo(D)q Fu(\()p Fs(x)p Fu(\))p Fr(g)p Fu(.)21 b(Let)16 b(us)h(consider)f(the)g(reduction)g(sequence)365 2593 y Fs(s)388 2600 y Fq(1)421 2593 y Fr(\021)e Fs(F)7 b Fu(\()p Fs(g)r Fu(\()p Fs(A)p Fu(\))p Fs(;)h(F)f Fu(\()p Fs(b;)h(g)r Fu(\()p Fs(A)p Fu(\)\)\))38 b Fr(!)981 2575 y Fo(i)981 2605 y Fl(S)1049 2593 y Fs(s)1072 2600 y Fq(2)1105 2593 y Fr(\021)14 b Fs(F)7 b Fu(\()p Fs(g)r Fu(\()p Fs(B)s Fu(\))p Fs(;)h(F)f Fu(\()p Fs(b;)h(g)r Fu(\()p Fs(A)p Fu(\)\)\))931 2666 y Fr(!)981 2648 y Fo(i)981 2678 y Fl(S)1049 2666 y Fs(s)1072 2673 y Fq(3)1105 2666 y Fr(\021)14 b Fs(F)7 b Fu(\()p Fo(C)r Fu(\()p Fs(B)s Fu(\))p Fs(;)h(F)f Fu(\()p Fs(b;)h(g)r Fu(\()p Fs(A)p Fu(\)\)\))931 2738 y Fr(!)981 2720 y Fo(i)981 2751 y Fl(S)1049 2738 y Fs(s)1072 2745 y Fq(4)1105 2738 y Fr(\021)14 b Fs(F)7 b Fu(\()p Fo(C)r Fu(\()p Fs(B)s Fu(\))p Fs(;)h(F)f Fu(\()p Fs(b;)h Fo(D)o Fu(\()p Fs(A)p Fu(\)\)\))931 2811 y Fr(!)981 2793 y Fo(o)981 2823 y Fl(S)1049 2811 y Fs(s)1072 2818 y Fq(5)1105 2811 y Fr(\021)14 b Fs(F)7 b Fu(\()p Fs(b;)h(B)s Fu(\))950 2936 y(14)p eop %%Page: 15 15 15 14 bop 0 26 a Fs(S)25 b Fu(=)d Fr(f)p Fs(g)r Fu(\()p Fs(A)p Fu(\))p Fs(;)8 b(b;)g(g)r Fu(\()p Fs(B)s Fu(\))p Fr(g)p Fu(,)21 b(and)g Fs(O)j Fu(=)e Fr(f)p Fs(s)736 33 y Fq(1)756 26 y Fs(;)8 b(s)801 33 y Fq(2)820 26 y Fs(;)g(s)865 33 y Fq(3)885 26 y Fs(;)g(s)930 33 y Fq(4)949 26 y Fs(;)g(s)994 33 y Fq(5)1014 26 y Fs(;)g(F)f Fu(\()p Fs(b;)h(g)r Fu(\()p Fs(A)p Fu(\)\))p Fs(;)g Fo(C)q Fu(\()p Fs(B)s Fu(\))p Fs(;)g(B)s(;)g(F)f Fu(\()p Fs(b;)h Fu(\()p Fo(D)n Fu(\()p Fs(A)p Fu(\)\)\))p Fs(;)g Fo(D)p Fu(\()p Fs(A)p Fu(\))p Fs(;)g(A)p Fr(g)p Fu(.)0 86 y(Moreo)o(v)o(er,)15 b(w)o(e)g(ha)o(v)o(e)h(\001\()p Fs(b)p Fu(\))d(=)h Fr(fg)p Fu(,)h(\001\()p Fs(g)r Fu(\()p Fs(A)p Fu(\)\))e(=)h Fr(f)p Fs(g)r Fu(\()p Fs(B)s Fu(\))p Fs(;)8 b Fo(D)q Fu(\()p Fs(A)p Fu(\))p Fr(g)p Fu(,)15 b(and)i(\001\()p Fs(g)r Fu(\()p Fs(B)s Fu(\)\))c(=)g Fr(f)p Fo(C)s Fu(\()p Fs(B)s Fu(\))p Fr(g)p Fu(.)0 147 y(Therefore,)i Fs(\032)h Fu(is)f(completely)e (de\014ned)j(on)g Fs(S)d Fr([)d Fs(O)17 b Fu(b)o(y)f Fs(\032)p Fu(\()p Fs(b)p Fu(\))d(=)h([],)h Fs(\032)p Fu(\()p Fs(g)r Fu(\()p 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Fu(are)f(simplifying)e(and)j(con\015uen)o(t.)0 1858 y Ft(Pro)r(of:)j Fu(The)15 b(only-if)g(case)g(is)f(straigh)o (tforw)o(ard.)22 b(W)l(e)14 b(sho)o(w)i(the)e(if)h(direction.)20 b(By)14 b(Newman's)f(Lemma)0 1918 y(in)h(conjunction)h(with)g(the)f (Critical)g(P)o(air)h(Lemma)961 1900 y Fq(10)1013 1918 y Fu(it)g(su\016ces)f(to)h(sho)o(w)h(that)f(all)f(critical)f(pairs)i (of)h Fr(R)0 1978 y Fu(are)f(con)o(v)o(ergen)o(t.)20 b(Since)14 b Fr(R)511 1985 y Fq(1)546 1978 y Fu(and)i Fr(R)682 1985 y Fq(2)717 1978 y Fu(do)g(not)g(share)f(de\014ned)g (function)g(sym)o(b)q(ols,)f(rewrite)h(rules)f(of)i Fr(R)1929 1985 y Fq(1)0 2038 y Fu(do)i(not)f(o)o(v)o(erlap)g(with)g(rewrite)f (rules)h(of)h Fr(R)822 2045 y Fq(2)859 2038 y Fu(and)g(vice)d(v)o (ersa.)24 b(Hence)16 b(the)h(set)g(of)h(critical)e(pairs)h(of)h Fr(R)0 2098 y Fu(consists)e(of)g(the)f(critical)f(pairs)i(of)g Fr(R)695 2105 y Fq(1)730 2098 y Fu(and)h(those)f(of)f Fr(R)1048 2105 y Fq(2)1068 2098 y Fu(.)21 b(These)16 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Fs(s)k Fr(\021)f Fs(C)t Fu([)p Fs(l)q(\033)r Fu(])p Fs(;)8 b(t)24 b Fr(\021)j Fs(C)t Fu([)p Fs(r)q(\033)r Fu(])c(and)i Fs(s)1717 1181 y Fo(i)1731 1174 y Fs(\033)r Fr(#)1785 1153 y Fo(o)1785 1187 y Fq(1)1805 1174 y Fs(t)1823 1181 y Fo(i)1837 1174 y Fs(\033)h Fu(for)0 1235 y Fs(i)18 b Fu(=)g(1)p Fs(;)8 b(:)g(:)g(:)f(;)h(n)p Fu(.)29 b(Here)17 b(the)i(sup)q(erscript)f Fs(o)i Fu(in)e Fs(s)878 1242 y Fo(i)892 1235 y Fs(\033)r Fr(#)946 1213 y Fo(o)946 1247 y Fq(1)966 1235 y Fs(t)984 1242 y Fo(i)998 1235 y Fs(\033)i Fu(means)e(that)h Fs(s)1329 1242 y Fo(i)1343 1235 y Fs(\033)h Fu(and)f Fs(t)1506 1242 y Fo(i)1520 1235 y Fs(\033)h Fu(are)f(joinable)f(using)0 1295 y(only)g Fm(outer)g Fr(!)287 1302 y Fq(1)325 1295 y Fu(-reduction)g(steps.)27 b(The)18 b(relation)f Fr(!)1039 1302 y Fq(2)1077 1295 y Fu(is)h(de\014ned)g(analogously)l(.)28 b(The)18 b(union)g(of)g Fr(!)1929 1302 y Fq(1)0 1355 y Fu(and)f Fr(!)145 1362 y Fq(2)181 1355 y Fu(is)f(denoted)g(b)o(y)g Fr(!)531 1362 y Fq(1)p Fo(;)p Fq(2)579 1355 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b(it)g(can)g(b)q(e)g(sho)o(wn)0 86 y(as)i(in)g([Mid90a])f (that)h Fr(!)476 93 y Fl(S)497 98 y Fd(i)529 86 y Fu(and)g Fr(!)674 93 y Fo(i)705 86 y Fu(are)g(also)g(the)f(same)g(on)h Fr(T)c Fu(\()p Fr(D)1257 93 y Fq(1)1288 86 y Fr(])e(D)1370 93 y Fq(2)1391 86 y Fs(;)d Fr(C)s Fs(;)g Fr(V)s Fu(\))j Fr(\002)g(T)i Fu(\()p Fr(D)1675 93 y Fq(1)1706 86 y Fr(])f(D)1789 93 y Fq(2)1809 86 y Fs(;)c Fr(C)s Fs(;)g Fr(V)s Fu(\).)0 147 y(It)16 b(follo)o(ws)g(with)536 207 y Fr(!)586 214 y Fl(S)607 219 y Fc(1)624 214 y Fl([S)669 219 y Fc(2)702 207 y Fu(=)e Fr(!)804 214 y Fl(S)825 219 y Fc(1)856 207 y Fr([)d(!)950 214 y Fl(S)971 219 y Fc(2)1004 207 y Fu(=)j Fr(!)1106 214 y Fq(1)1137 207 y Fr([)d(!)1231 214 y Fq(2)1264 207 y Fu(=)j Fr(!)1366 214 y Fq(1)p Fo(;)p Fq(2)0 293 y Fu(that)j Fr(!)155 300 y Fq(1)p Fo(;)p Fq(2)219 293 y Fu(is)f(simplifying)e(and)i(con\015uen)o(t.)21 b Fh(2)0 427 y Ft(Lemma)16 b(4.21)24 b Fu(If)15 b Fs(s)p Fr(!)444 434 y Fl(R)476 427 y Fs(t)h Fu(then)g Fs(s)p Fr(#)669 439 y Fq(1)p 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b(Example)15 b(3.17\).)73 2751 y(Additionally)l(,)24 b(one)f(ma)o(y)f(ask)i(whether) g Fr(R)913 2758 y Fq(1)956 2751 y Fu(and)g Fr(R)1100 2758 y Fq(2)1144 2751 y Fu(ma)o(y)e(share)i(de\014ned)f(function)g(sym) o(b)q(ols.)0 2811 y(Esp)q(ecially)d(in)h(the)g(con)o(text)g(of)g ("parameterized")f(term)f(rewriting)i(systems)f(\(cf.)36 b([Dro89)q(])21 b(for)g(the)950 2936 y(17)p eop %%Page: 18 18 18 17 bop 0 26 a Fu(unconditional)21 b(case\))h(it)f(w)o(ould)h(b)q(e)f (nice)g(if)g(at)h(least)f(one)h(of)f(the)h(CTRS's)f(could)h(con)o(tain) f(de\014ned)0 86 y(function)16 b(sym)o(b)q(ols)g(of)h(the)f(other)h(in) g(its)f(conditions)h(\(but)g(not)g(vice)e(v)o(ersa\).)22 b(Unfortunately)l(,)16 b(there)g(is)0 147 y(no)i(a\016rmativ)o(e)d (answ)o(er)j(to)g(that)h(question,)e(not)h(ev)o(en)f(in)g(the)g (restricted)g(case)g(that)h(the)g(systems)e(do)0 207 y(not)i(share)h(constructors)f(in)g(the)g(left-)f(and)h(righ)o(t-hand)h (sides)f(of)g(rules)f(\(i.e.,)f(shared)j(sym)o(b)q(ols)e(o)q(ccur)0 267 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