%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -o paper.ps paper.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1997.06.18:1415 %%BeginProcSet: texc.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 round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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y(sound)g(and)h(complete)e(implemen)o(tation)e(\(w.r.t.)j(the)h (computation)e(of)g(normal)g(forms\))p eop %%Page: 2 2 2 1 bop 340 194 a Fw(of)18 b(prop)q(erly)h(orien)o(ted)f(orthogonal)g (CTRSs)g(with)g(strict)h(equalit)o(y)e(\(called)h(functional)340 244 y(CTRSs)f(in)f(what)h(follo)o(ws\).)e(Note)i(that)g(the)g(ab)q(o)o (v)o(e)g(CTRS)f(is)g(not)h(prop)q(erly)g(orien)o(ted.)340 293 y(Aside)c(from)d(the)j(men)o(tioned)e(reasons,)i(our)f(new)g (result)h(is)f(in)o(teresting)h(in)e(its)h(o)o(wn)g(righ)o(t{)340 343 y(simply)g(b)q(ecause)k(term)d(rewriting)h(is)g(usually)f(implem)o (en)o(ted)f(b)o(y)i(graph)g(rewriting.)403 394 y(The)i(remainder)e(of)h (the)h(pap)q(er)g(is)g(organized)f(as)h(follo)o(ws.)d(In)i(the)h(next)g (section,)g(w)o(e)340 444 y(recapitulate)j(the)g(basics)g(of)f (conditional)f(term)h(rewriting.)g(In)h(Section)g(3,)e(functional)340 493 y(CTRSs)c(are)g(in)o(tro)q(duced.)g(Unfortunately)m(,)e(functional) h(CTRSs)g(do)h(not)f(satisfy)g(the)h(par-)340 543 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(graphs)h(corresp)q(ond)i(to)d(w)o(ell-mark)o(ed)f(terms.)h(The)h (\014rst)g(part)g(of)f(Section)340 942 y(4)k(is)f(a)g(mirror)g(image)e (of)i(Section)h(3:)f(it)g(is)g(sho)o(wn)h(that)g(the)g(graph)f(rewrite) i(relation)e(of)340 992 y(a)e(functional)g(CTRS)f(has)i(almost)d(the)j (same)e(prop)q(erties)j(as)f(the)f(term)g(rewrite)h(relation.)340 1041 y(With)c(the)h(aid)f(of)g(the)i(aforemen)o(tioned)d(results)j(w)o (e)e(\014nally)g(ac)o(hiev)o(e)h(our)f(principal)g(result:)340 1091 y(graph)15 b(rewriting)h(is)f(a)f(sound)i(and)f(complete)f (implemen)o(tation)e(of)j(functional)f(CTRSs.)340 1141 y(It)j(should)f(b)q(e)i(p)q(ossible)f(to)f(sho)o(w)h(that)f(this)h (result)g(also)f(holds)h(for)f Fr(almost)h(ortho)n(gonal)340 1191 y Fw(systems.)d(F)m(or)f(space)i(reasons,)g(sev)o(eral)f(pro)q (ofs)g(are)g(omitted.)340 1330 y Fs(2)56 b(Preliminaries)340 1434 y Fw(The)16 b(reader)f(is)g(assumed)f(to)h(b)q(e)g(familia)o(r)d (with)j(the)g(basic)g(concepts)h(of)e(term)g(rewriting)340 1484 y(\(whic)o(h)e(can)g(b)q(e)g(found)f(in)g(the)h(surv)o(eys)g(of)f (Dersho)o(witz)h(&)g(Jouannaud)f([DJ90)o(])g(and)h(Klop)340 1533 y([Klo92)o(],)h(for)h(instance\).)g(Here,)h(w)o(e)g(merely)e (recall)h(less)h(common)c(de\014nitions)j(and)g(some)340 1583 y(basic)g(facts)h(concerning)f(conditional)f(term)g(rewriting.)403 1634 y(Let)19 b(\()p Fq(F)t Fo(;)7 b Fq(R)o Fw(\))19 b(b)q(e)g(a)f(term)g(rewriting)g(system)g(\(TRS\).)g(A)g(function)g (sym)o(b)q(ol)f Fo(f)24 b Fq(2)18 b(F)340 1684 y Fw(is)g(called)g(a)g Fr(de\014ne)n(d)h(symb)n(ol)f Fw(if)f(there)j(is)d(a)h(rewrite)h(rule)f Fo(l)i Fq(!)e Fo(r)h Fq(2)f(R)g Fw(suc)o(h)h(that)f Fo(l)h Fw(=)340 1733 y Fo(f)t Fw(\()p Fo(t)395 1739 y Fm(1)415 1733 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)523 1739 y Fl(k)543 1733 y Fw(\))12 b(for)h(some)f(terms)g Fo(t)866 1739 y Fm(1)885 1733 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)993 1739 y Fl(k)1013 1733 y Fw(,)12 b(otherwise)i(it)e(is)h(called)f Fr(c)n(onstructor)p Fw(.)g(The)h(set)340 1783 y(of)f(de\014ned)i(sym)o (b)q(ols)d(is)h(denoted)h(b)o(y)g Fq(F)967 1789 y Fn(D)1008 1783 y Fw(while)f Fq(F)1145 1789 y Fn(C)1178 1783 y Fw(stands)h(for)g (the)g(set)g(of)f(constructors.)340 1833 y(A)i Fr(c)n(onstructor)g (term)e Fw(is)i(a)f(term)f(built)h(from)f(constructors)j(and)e(v)n (ariables)g(only)m(.)e(A)j(non-)340 1883 y(o)o(v)o(erlapping)h (left-linear)g(TRS)h(is)g(called)f Fr(ortho)n(gonal)p Fw(.)h(In)g(an)g Fr(almost)g(ortho)n(gonal)g Fw(TRS,)340 1933 y(the)f(non-o)o(v)o(erlapping)d(restriction)j(is)f(relaxed)g(b)o (y)g(allo)o(wing)d(trivial)i(o)o(v)o(erla)o(ys.)403 1983 y(In)g(a)h(CTRS)f(\()p Fq(F)t Fo(;)7 b Fq(R)p Fw(\),)14 b(rewrite)g(rules)h(ha)o(v)o(e)f(the)g(form)e Fo(l)h Fq(!)e Fo(r)h Fq(\()f Fo(s)1448 1989 y Fm(1)1479 1983 y Fw(=)h Fo(t)1538 1989 y Fm(1)1556 1983 y Fo(;)7 b(:)g(:)g(:)e(;)i(s) 1668 1989 y Fl(n)1702 1983 y Fw(=)12 b Fo(t)1761 1989 y Fl(k)340 2033 y Fw(with)18 b Fo(l)q(;)7 b(r)o(;)g(s)527 2039 y Fm(1)545 2033 y Fo(;)g(:)g(:)g(:)e(;)i(s)657 2039 y Fl(k)677 2033 y Fw(,)18 b Fo(t)722 2039 y Fm(1)741 2033 y Fo(;)7 b(:)g(:)g(:)t(;)g(t)848 2039 y Fl(k)887 2033 y Fq(2)18 b(T)11 b Fw(\()p Fq(F)t Fo(;)c Fq(V)s Fw(\).)18 b(W)m(e)g(frequen)o(tly)g(abbreviate)h(the)g(condi-)340 2083 y(tional)14 b(part)i(of)e(the)i(rewrite)g(rule{the)g(\(p)q (ossibly)f(empt)o(y\))f(sequence)k Fo(s)1491 2089 y Fm(1)1524 2083 y Fw(=)c Fo(t)1585 2089 y Fm(1)1603 2083 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1715 2089 y Fl(k)1749 2083 y Fw(=)340 2133 y Fo(t)355 2139 y Fl(k)376 2133 y Fw({b)o(y)12 b Fo(c)p Fw(.)g(If)h(a)f(rewrite)i(rule)f(has)g(no)f(conditions,)g(w)o(e) h(write)g Fo(l)g Fq(!)e Fo(r)q Fw(.)h(It)h(is)g(alw)o(a)o(ys)e (required)340 2183 y(that)h Fo(l)g Fw(is)f(not)h(a)f(v)n(ariable.)e (The)j(=)g(sym)o(b)q(ol)d(in)i(the)h(conditions)f(can)g(b)q(e)h(in)o (terpreted)h(in)e(dif-)340 2232 y(feren)o(t)k(w)o(a)o(ys,)d(leading)g (to)i(di\013eren)o(t)g(rewrite)g(relations)f(asso)q(ciated)h(with)f Fq(R)q Fw(.)f(This)i(pap)q(er)340 2282 y(deals)g(with)f Fr(oriente)n(d)h Fw(CTRSs)f(in)h(whic)o(h)f(the)h(equalit)o(y)f(signs)h (are)g(in)o(terpreted)h(as)f(reac)o(h-)340 2332 y(abilit)o(y)g(\()p Fq(!)529 2317 y Fn(\003)529 2344 y(R)559 2332 y Fw(\).)g(F)m(ormally)m (,)d(the)16 b(rewrite)g(relation)e(asso)q(ciated)i(with)f(an)f(orien)o (ted)i(CTRS)340 2382 y(\()p Fq(F)t Fo(;)7 b Fq(R)p Fw(\))13 b(is)f(the)h(smallest)e(relation)h(that)g(satis\014es:)h Fo(s)f Fq(!)1216 2388 y Fn(R)1258 2382 y Fo(t)g Fw(if)g(there)h(exists) h(a)e(rewrite)h(rule)340 2432 y Fo(l)g Fq(!)e Fo(r)i Fq(\()e Fo(c)k Fw(in)h Fq(R)p Fw(,)f(a)g(substitution)h Fo(\033)g Fw(:)e Fq(V)19 b(!)14 b(T)c Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\),)16 b(and)f(a)h(con)o(text)g Fo(C)s Fw([)c(])j(suc)o(h)h(that)p eop %%Page: 3 3 3 2 bop 183 194 a Fo(s)17 b Fw(=)h Fo(C)s Fw([)p Fo(l)q(\033)q Fw(])p Fo(;)7 b(t)16 b Fw(=)h Fo(C)s Fw([)p Fo(r)q(\033)q Fw(],)f(and)h Fo(s)696 200 y Fl(i)710 194 y Fo(\033)i Fq(!)795 179 y Fn(\003)795 205 y(R)842 194 y Fo(t)857 200 y Fl(i)871 194 y Fo(\033)f Fw(for)f(all)f Fo(s)1060 200 y Fl(i)1092 194 y Fw(=)h Fo(t)1156 200 y Fl(i)1187 194 y Fw(in)g Fo(c)p Fw(.)g(F)m(or)g(ev)o(ery)h(orien)o(ted)183 244 y(CTRS)13 b Fq(R)p Fw(,)h(w)o(e)g(inductiv)o(ely)f(de\014ne)i(TRSs) f Fq(R)914 250 y Fl(n)937 244 y Fw(,)f Fo(n)e Fq(2)h Fk(I)-6 b(N)p Fw(,)13 b(b)o(y:)307 318 y Fq(R)342 324 y Fm(0)373 318 y Fw(=)e Fq(;)261 368 y(R)296 374 y Fl(n)p Fm(+1)373 368 y Fw(=)g Fq(f)p Fo(l)q(\033)i Fq(!)e Fo(r)q(\033)j Fq(j)e Fo(l)g Fq(!)g Fo(r)g Fq(\()f Fo(c)g Fq(2)h(R)i Fw(and)f Fo(s\033)26 b Fq(!)1093 353 y Fn(\003)1093 380 y(R)1122 384 y Fj(n)1167 368 y Fo(t\033)16 b Fw(for)d(all)g Fo(s)f Fw(=)g Fo(t)h Fw(in)h Fo(c)p Fq(g)p Fo(:)183 445 y Fw(Note)20 b(that)h Fq(R)421 451 y Fl(n)466 445 y Fq(\022)i(R)556 451 y Fl(n)p Fm(+1)641 445 y Fw(for)d(all)f Fo(n)j Fq(2)g Fk(I)-6 b(N)q Fw(.)20 b(F)m(urthermore,)f Fo(s)k Fq(!)1282 451 y Fn(R)1335 445 y Fo(t)d Fw(if)g(and)g(only)f(if)183 494 y Fo(s)12 b Fq(!)256 500 y Fn(R)285 504 y Fj(n)318 494 y Fo(t)g Fw(for)f(some)h Fo(n)f Fq(2)g Fk(I)-6 b(N)q Fw(.)11 b(The)i Fr(depth)g Fw(of)e(a)h(rewrite)h(step)g Fo(s)f Fq(!)1212 500 y Fn(R)1254 494 y Fo(t)g Fw(is)g(de\014ned)h(to)f (b)q(e)h(the)183 544 y(minim)n(um)c Fo(n)k Fw(with)g Fo(s)f Fq(!)573 550 y Fn(R)602 554 y Fj(n)635 544 y Fo(t)p Fw(.)g(A)i(CTRS)e Fq(R)i Fw(is)f(called)g Fr(level-c)n(on\015uent)h Fw(if)e(ev)o(ery)i(TRS)f Fq(R)1601 550 y Fl(n)183 594 y Fw(is)j(con\015uen)o(t.)g(As)h(in)e([MH94)o(],)h(rewrite)h(rules)f Fo(l)d Fq(!)e Fo(r)i Fq(\()e Fo(c)16 b Fw(will)e(b)q(e)j(classi\014ed)g (according)183 644 y(to)c(the)i(distribution)e(of)h(v)n(ariables)f (among)e Fo(l)q Fw(,)j Fo(r)q Fw(,)f(and)h Fo(c)p Fw(,)f(as)h(follo)o (ws:)627 717 y(t)o(yp)q(e)p 711 732 2 50 v 109 w(requiremen)o(t)p 621 733 565 2 v 656 768 a(1)p 711 783 2 50 v 41 w Fq(V)s Fo(ar)q Fw(\()p Fo(r)q Fw(\))c Fq([)f(V)s Fo(ar)q Fw(\()p Fo(c)p Fw(\))j Fq(\022)g(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))656 818 y(2)p 711 833 V 41 w Fq(V)s Fo(ar)q Fw(\()p Fo(r)q Fw(\))g Fq(\022)g(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))656 868 y(3)p 711 883 V 41 w Fq(V)s Fo(ar)q Fw(\()p Fo(r)q Fw(\))g Fq(\022)g(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))f Fq([)d(V)s Fo(ar)q Fw(\()p Fo(c)p Fw(\))656 918 y(4)p 711 933 V 41 w(no)14 b(restrictions)183 991 y(An)19 b Fo(n)p Fw(-CTRS)f(con)o(tains)h(only)g(rewrite)h(rules)f(of)g(t)o(yp)q (e)g Fo(n)p Fw(.)g(F)m(or)g(ev)o(ery)h(rule)f Fo(l)13 b Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(,)183 1041 y(w)o(e)19 b(de\014ne)i Fq(E)t(V)s Fo(ar)q Fw(\()p Fo(l)13 b Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(\))21 b(=)g Fq(V)s Fo(ar)q Fw(\()p Fo(r)q Fw(\))13 b Fq([)g(V)s Fo(ar)q Fw(\()p Fo(c)p Fw(\))h Fq(n)e(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))q(.)19 b(An)h Fr(extr)n(a)f(variable)g Fw(in)183 1091 y Fo(l)12 b Fq(!)f Fo(r)i Fq(\()e Fo(c)20 b Fw(is)f(a)h(v)n(ariable)f Fo(x)i Fq(2)g(E)t(V)s Fo(ar)q Fw(\()p Fo(l)13 b Fq(!)e Fo(r)h Fq(\()g Fo(c)o Fw(\))q(.)19 b(Th)o(us)h(a)g(1-CTRS)f(has)g(no)h(extra) 183 1141 y(v)n(ariables,)15 b(a)i(2-CTRS)f(has)g(no)h(extra)g(v)n (ariables)f(in)h(righ)o(t-hand)f(sides)h(of)f(rules,)h(and)g(a)183 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1450 y Fm(1)468 1439 y Fo(c)g Fw(and)g Fo(b)f Fq(!)648 1424 y Fn(\003)648 1450 y Fm(1)678 1439 y Fo(c)p Fw(.)h(Staples')f(result)i(states)g(that) f(a)g(compatible)e(re\014nemen)o(t)183 1489 y Fq(A)216 1495 y Fm(2)251 1489 y Fw(of)17 b Fq(A)335 1495 y Fm(1)370 1489 y Fw(is)g(con\015uen)o(t)h(if)e(and)h(only)f(if)g Fq(A)890 1495 y Fm(1)926 1489 y Fw(is)h(con\015uen)o(t.)g(In)g(fact,)f (w)o(e)i(also)e(need)i(the)183 1539 y(follo)o(wing)12 b(generalization)j(of)g(this)g(result.)g(Let)h Fq(A)994 1545 y Fm(1)1027 1539 y Fw(=)e(\()p Fo(A;)7 b Fq(!)1181 1545 y Fm(1)1199 1539 y Fw(\))15 b(and)g Fq(A)1345 1545 y Fm(2)1378 1539 y Fw(=)f(\()p Fo(A;)7 b Fq(!)1532 1545 y Fm(2)1550 1539 y Fw(\))15 b(b)q(e)183 1589 y(ARSs.)i(Let)h Fq(\030)h Fw(b)q(e)f(an)g(equiv)n(alence)g(relation)f(on)h Fo(A)g Fw(suc)o(h)g(that)g Fq(!)1287 1595 y Fm(1)1305 1589 y Fq(\022!)1379 1574 y Fn(\003)1379 1599 y Fm(2)1416 1589 y Fw(and,)f(for)h(all)183 1639 y Fo(a)11 b Fq(!)258 1624 y Fn(\003)258 1649 y Fm(2)288 1639 y Fo(b)p Fw(,)h(there)i(are)g Fo(c;)7 b(d)j Fq(2)h Fo(A)i Fw(suc)o(h)h(that)f Fo(a)e Fq(!)913 1624 y Fn(\003)913 1649 y Fm(1)943 1639 y Fo(c)p Fw(,)i Fo(b)e Fq(!)1057 1624 y Fn(\003)1057 1649 y Fm(1)1087 1639 y Fo(d)p Fw(,)h(and)h Fo(c)e Fq(\030)h Fo(d)p Fw(.)g(Let)h Fo(i)f Fq(2)f(f)p Fw(1)p Fo(;)c Fw(2)p Fq(g)p Fw(.)k(If)183 1689 y Fq(A)216 1695 y Fl(i)244 1689 y Fw(is)j(con\015uen)o(t)h(mo)q (dulo)d Fq(\030)j Fw(\(i.e.,)e(for)h(all)f Fo(c)908 1673 y Fn(\003)908 1699 y Fl(i)925 1689 y Fq( )f Fo(a)g Fq(\030)g Fo(b)g Fq(!)1129 1673 y Fn(\003)1129 1699 y Fl(i)1160 1689 y Fo(d)p Fw(,)i(there)h(are)g Fo(e;)7 b(f)17 b Fq(2)11 b Fo(A)k Fw(suc)o(h)183 1738 y(that)h Fo(c)f Fq(!)350 1723 y Fn(\003)350 1749 y Fl(i)384 1738 y Fo(e)h Fq(\030)f Fo(f)507 1723 y Fn(\003)507 1749 y Fl(i)527 1738 y Fq( )g Fo(d)p Fw(\))h(and,)f(for)h(all)f Fo(a)g Fq(\030)h Fo(b)f Fq(!)1018 1723 y Fn(\003)1018 1749 y Fm(3)p Fn(\000)p Fl(i)1089 1738 y Fo(c)p Fw(,)h(there)h(is)f(a)g Fo(d)f Fq(2)g Fo(A)i Fw(suc)o(h)f(that)183 1788 y Fo(a)11 b Fq(!)258 1773 y Fn(\003)258 1799 y Fm(3)p Fn(\000)p Fl(i)325 1788 y Fo(d)h Fq(\030)f Fo(c)p Fw(,)j(then)g Fq(A)573 1794 y Fm(3)p Fn(\000)p Fl(i)644 1788 y Fw(is)f(con\015uen)o(t)i(mo)q (dulo)d Fq(\030)p Fw(.)183 1915 y Fs(3)56 b(F)-5 b(unctional)18 b(CTRSs)183 2007 y Fw(The)13 b(unconditional)e(TRS)i(obtained)f(from)f (a)i(CTRS)f Fq(R)h Fw(b)o(y)g(omitting)d(the)k(conditions)e(in)183 2057 y(its)k(rewrite)i(rules)f(is)g(denoted)g(b)o(y)g Fq(R)789 2063 y Fl(u)811 2057 y Fw(.)823 2042 y Fm(1)858 2057 y Fw(F)m(or)f(a)g(CTRS)g Fq(R)q Fw(,)g(notions)g(lik)o(e)g (left-linearit)o(y)m(,)183 2107 y(\(almost\))g(orthogonalit)o(y)m(,)g (and)i(constructor)i(term)e(are)g(de\014ned)i(via)d(the)i(TRS)f Fq(R)1541 2113 y Fl(u)1563 2107 y Fw(.)f(A)183 2157 y Fr(normal)c Fw(CTRS)h Fq(R)g Fw(is)f(an)h(orien)o(ted)g(CTRS)f(whose)i (rules)f Fo(l)f Fq(!)e Fo(r)h Fq(\()f Fo(s)1290 2163 y Fm(1)1321 2157 y Fw(=)h Fo(t)1380 2163 y Fm(1)1399 2157 y Fo(;)7 b(:)g(:)g(:)t(;)g(s)1510 2163 y Fl(n)1544 2157 y Fw(=)12 b Fo(t)1603 2163 y Fl(k)183 2206 y Fw(are)k(sub)r(ject)h (to)f(the)g(additional)e(constrain)o(t)j(that)f(ev)o(ery)g Fo(t)1140 2212 y Fl(j)1173 2206 y Fw(is)g(a)g(ground)g(normal)d(form) 183 2256 y(with)18 b(resp)q(ect)j(to)d Fq(R)519 2262 y Fl(u)541 2256 y Fw(.)g(Orthogonal)f(normal)g(2-CTRS)g(satisfy)i(the)g (so-called)f(parallel)183 2306 y(mo)o(v)o(es)11 b(lemma.)e(Th)o(us)k (they)h(are)f(\(lev)o(el-\))f(con\015uen)o(t.)i(W)m(e)e(next)h (brie\015y)g(recall)g(this)g(result)183 2356 y(ac)o(hiev)o(ed)h(b)o(y)f (Bergstra)j(and)d(Klop)h([BK86];)f(see)i(also)e(Suzuki)h Fr(et)h(al.)e Fw([SMI95)o(].)p 183 2389 237 2 v 191 2416 a Ft(1)221 2432 y Fv(Note)g(that)g Fi(R)430 2436 y Fh(u)463 2432 y Fv(ma)o(y)g(con)o(tain)i(rules)f Fg(l)d Fi(!)g Fg(r)k Fv(with)e Fi(V)s Fg(ar)q Fv(\()p Fg(r)q Fv(\))e Fi(6\022)g(V)s Fg(ar)q Fv(\()p Fg(l)q Fv(\).)p eop %%Page: 4 4 4 3 bop 340 194 a Ff(De\014nition)t(1.)21 b Fw(Let)d Fo(A)e Fw(:)g Fo(s)h Fq(!)844 201 y Fm([)p Fl(p;l)p Fn(!)p Fl(r)q Fn(\()p Fl(c)p Fm(])1016 194 y Fo(t)g Fw(b)q(e)h(a)e(rewrite)i (step)g(in)e(a)h(CTRS)f Fq(R)h Fw(and)g(let)340 244 y Fo(q)c Fq(2)e(P)s Fo(os)p Fw(\()p Fo(s)p Fw(\))r(.)i(The)h(set)h Fo(q)q Fq(n)p Fo(A)f Fw(of)f Fr(desc)n(endants)i Fw(of)f Fo(q)g Fw(in)g Fo(t)g Fw(is)f(de\014ned)i(b)o(y:)441 374 y Fo(q)q Fq(n)p Fo(A)d Fw(=)569 289 y Fp(8)569 326 y(<)569 401 y(:)611 324 y Fq(f)p Fo(q)q Fq(g)379 b Fw(if)13 b Fo(q)f(<)g(p)i Fw(or)f Fo(q)h Fq(k)e Fo(p;)611 374 y Fq(f)p Fo(p)d Fq(\001)g Fo(p)704 380 y Fm(3)732 374 y Fq(\001)f Fo(p)773 380 y Fm(2)804 374 y Fq(j)k Fo(r)847 381 y Fn(j)p Fl(p)874 385 y Fe(3)904 374 y Fw(=)g Fo(l)960 381 y Fn(j)p Fl(p)987 385 y Fe(1)1005 374 y Fq(g)26 b Fw(if)13 b Fo(q)f Fw(=)g Fo(p)d Fq(\001)g Fo(p)1237 380 y Fm(1)1265 374 y Fq(\001)f Fo(p)1306 380 y Fm(2)1339 374 y Fw(with)13 b Fo(p)1454 380 y Fm(1)1484 374 y Fq(2)e(V)s(P)5 b Fo(os)p Fw(\()p Fo(l)q Fw(\))611 423 y Fq(;)420 b Fw(otherwise.)340 505 y(If)15 b Fo(Q)e Fq(\022)h(P)s Fo(os)p Fw(\()p Fo(s)p Fw(\))q(,)h(then)h Fo(Q)p Fq(n)p Fo(A)e Fw(denotes)i(the)g(set)1110 473 y Fp(S)1144 517 y Fl(q)q Fn(2)p Fl(Q)1218 505 y Fo(q)q Fq(n)p Fo(A)p Fw(.)f(The)g(notion)f(of)h(descendan)o(t)340 554 y(is)f(extended)i(to)d(rewrite)i(sequences)i(in)c(the)i(ob)o(vious) e(w)o(a)o(y)m(.)340 640 y Ff(De\014nition)t(2.)21 b Fw(Let)15 b Fq(R)f Fw(b)q(e)h(a)f(CTRS.)g(W)m(e)f(write)i Fo(s)e Fq(k)-30 b(!)1212 646 y Fn(R)1241 650 y Fj(n)1262 640 y Fo(t)14 b Fw(if)g Fo(t)g Fw(can)g(b)q(e)h(obtained)f(from)f Fo(s)340 690 y Fw(b)o(y)k(con)o(tracting)g(a)g(set)g(of)g(pairwise)f (disjoin)o(t)g(redexes)j(in)d Fo(s)i Fw(b)o(y)e Fq(R)1427 696 y Fl(n)1449 690 y Fw(.)g(W)m(e)h(write)g Fo(s)c Fq(k)-30 b(!)o Fo(t)17 b Fw(if)340 740 y Fo(s)c Fq(k)-30 b(!)405 746 y Fn(R)434 750 y Fj(n)456 740 y Fo(t)14 b Fw(for)h(some)f Fo(n)f Fq(2)g Fk(I)-6 b(N)p Fw(.)15 b(The)g(minim)n(um)c(suc)o(h)16 b Fo(n)e Fw(is)h(called)g(the)g(depth)h(of)e Fo(s)f Fq(k)-30 b(!)o Fo(t)p Fw(.)15 b(The)340 789 y(relation)26 b Fq(k)-30 b(!)13 b Fw(is)h(called)g Fr(p)n(ar)n(al)r(lel)f(r)n(ewriting)p Fw(.)340 875 y(The)h(parallel)f(mo)o(v)o(es)f(lemma)e(for)k(orthogonal) e(normal)f(2-CTRS)i(no)o(w)g(reads)i(as)e(follo)o(ws.)340 954 y Ff(Lemma)8 b(3.)21 b Fr(L)n(et)16 b Fq(R)h Fr(b)n(e)g(an)g(ortho) n(gonal)h(normal)e(2-CTRS.)h(If)f Fo(t)d Fq(k)-30 b(!)1434 960 y Fn(R)1463 964 y Fj(m)1491 954 y Fo(t)1506 960 y Fm(1)1542 954 y Fr(and)17 b Fo(t)c Fq(k)-30 b(!)1685 960 y Fn(R)1714 964 y Fj(n)1735 954 y Fo(t)1750 960 y Fm(2)1769 954 y Fr(,)340 1003 y(then)15 b(ther)n(e)e(is)h(a)g(term)g Fo(t)727 1009 y Fm(3)759 1003 y Fr(such)h(that)f Fo(t)952 1009 y Fm(1)983 1003 y Fq(k)-30 b(!)1016 1009 y Fn(R)1045 1013 y Fj(n)1066 1003 y Fo(t)1081 1009 y Fm(3)1114 1003 y Fr(and)15 b Fo(t)1209 1009 y Fm(2)1240 1003 y Fq(k)-30 b(!)1273 1009 y Fn(R)1302 1013 y Fj(m)1330 1003 y Fo(t)1345 1009 y Fm(3)1364 1003 y Fr(.)14 b(Mor)n(e)n(over,)f(the)h(r)n(e)n (dexes)340 1053 y(c)n(ontr)n(acte)n(d)h(in)g Fo(t)604 1059 y Fm(1)635 1053 y Fq(k)-30 b(!)668 1059 y Fn(R)697 1063 y Fj(n)719 1053 y Fo(t)734 1059 y Fm(3)767 1053 y Fr(\()p Fo(t)799 1059 y Fm(2)830 1053 y Fq(k)g(!)863 1059 y Fn(R)892 1063 y Fj(m)920 1053 y Fo(t)935 1059 y Fm(3)954 1053 y Fr(\))15 b(ar)n(e)g(the)g(desc)n(endants)h(in)f Fo(t)1420 1059 y Fm(1)1453 1053 y Fr(\()p Fo(t)1485 1059 y Fm(2)1504 1053 y Fr(\))g(of)f(the)h(r)n(e)n(dexes)340 1103 y(c)n(ontr)n(acte)n(d)g(in)g Fo(t)e Fq(k)-30 b(!)650 1109 y Fn(R)679 1113 y Fj(n)700 1103 y Fo(t)715 1109 y Fm(2)748 1103 y Fr(\()p Fo(t)13 b Fq(k)-30 b(!)826 1109 y Fn(R)855 1113 y Fj(m)883 1103 y Fo(t)898 1109 y Fm(1)917 1103 y Fr(\).)340 1181 y Fw(In)14 b(this)g(pap)q(er,)g(w)o (e)g(are)h(in)o(terested)g(in)f(CTRSs)g(with)f(strict)i(equalit)o(y)m (.)d(In)i(suc)o(h)h(systems,)340 1231 y(the)f(equalit)o(y)f(signs)g(in) g(the)h(conditional)e(part)h(of)g(a)g(rule)h(are)f(in)o(terpreted)i(as) f(\\the)g(terms)340 1281 y(are)h(reducible)f(to)g(a)g(same)f(ground)h (constructor)h(term)e(in)h Fq(R)p Fw(".)340 1360 y Ff(De\014nition)t (4.)21 b Fw(A)16 b(3-CTRS)g Fr(with)g(strict)g(e)n(quality)g Fw(is)g(a)g(3-CTRS)f(\()p Fq(F)g(])10 b(F)1559 1366 y Fl(eq)1593 1360 y Fo(;)d Fq(R)k(])f(R)1731 1366 y Fl(eq)1765 1360 y Fw(\))340 1409 y(whic)o(h)k(satis\014es)h(\()p Fq(])f Fw(denotes)h(the)f(disjoin)o(t)f(union)g(of)h(sets\):)358 1488 y(1.)20 b Fq(F)445 1494 y Fl(eq)491 1488 y Fw(=)11 b Fq(f)p Fw(==)q Fo(;)c Fq(^)p Fo(;)g(tr)q(ue;)g(f)t(al)q(se)p Fq(g)p Fw(,)13 b(where)i Fq(^)e Fw(is)h(assumed)f(to)h(b)q(e)h(righ)o (t-asso)q(ciativ)o(e.)358 1538 y(2.)20 b(The)14 b(TRS)g Fq(R)629 1544 y Fl(eq)677 1538 y Fw(consists)h(of)e(the)h(rules:)875 1619 y Fo(c)p Fw(==)p Fo(c)e Fq(!)f Fo(tr)q(ue)207 b Fw(for)13 b(all)g(constan)o(ts)i Fo(c)c Fq(2)g(F)1728 1625 y Fn(C)455 1669 y Fo(c)p Fw(\()p Fo(x)513 1675 y Fm(1)532 1669 y Fo(;)c(:)g(:)g(:)t(;)g(x)648 1675 y Fl(n)670 1669 y Fw(\)==)q Fo(c)p Fw(\()p Fo(y)805 1675 y Fm(1)824 1669 y Fo(;)g(:)g(:)g(:)e(;)i(y)937 1675 y Fl(n)959 1669 y Fw(\))12 b Fq(!)1040 1638 y Fp(V)1075 1648 y Fl(n)1075 1682 y(i)p Fm(=1)1131 1669 y Fw(\()p Fo(x)1171 1675 y Fl(i)1184 1669 y Fw(==)q Fo(y)1269 1675 y Fl(i)1283 1669 y Fw(\))26 b(for)13 b(all)g Fo(n)p Fw(-ary)g Fo(c)f Fq(2)f(F)1656 1675 y Fn(C)443 1719 y Fo(c)p Fw(\()p Fo(x)501 1725 y Fm(1)519 1719 y Fo(;)c(:)g(:)g(:)e(;)i(x)636 1725 y Fl(n)658 1719 y Fw(\)==)p Fo(d)p Fw(\()p Fo(y)796 1725 y Fm(1)815 1719 y Fo(;)g(:)g(:)g(:)e(;)i(y)928 1725 y Fl(m)959 1719 y Fw(\))12 b Fq(!)f Fo(f)t(al)q(se)188 b Fw(for)13 b(all)g Fo(c;)7 b(d)j Fq(2)i(F)1585 1725 y Fn(C)1606 1719 y Fo(;)7 b(c)k Fq(6)p Fw(=)h Fo(d)828 1769 y(tr)q(ue)d Fq(^)g Fo(x)i Fq(!)g Fo(x)807 1819 y(f)t(al)q(se)g Fq(^)e Fo(x)i Fq(!)g Fo(f)t(al)q(se)358 1900 y Fw(3.)20 b(Ev)o(ery)d(rule)g(in)g Fq(R)g Fw(has)g(the)g(form)e Fo(l)j Fq(!)e Fo(r)h Fq(\()f Fw(\()p Fo(s)1192 1906 y Fm(1)1211 1900 y Fw(==)p Fo(t)1290 1906 y Fm(1)1309 1900 y Fw(\))h(=)f Fo(tr)q(ue;)7 b(:)g(:)g(:)e(;)i Fw(\()p Fo(s)1596 1906 y Fl(k)1617 1900 y Fw(==)p Fo(t)1696 1906 y Fl(k)1717 1900 y Fw(\))16 b(=)411 1950 y Fo(tr)q(ue;)d Fw(where)i Fo(l)q(;)7 b(r)o(;)g(s)722 1956 y Fm(1)740 1950 y Fo(;)g(:)g(:)g(:)e(;)i(s)852 1956 y Fl(k)872 1950 y Fo(;)g(t)906 1956 y Fm(1)924 1950 y Fo(;)g(:)g(:)g(:)e(;)i(t)1032 1956 y Fl(k)1063 1950 y Fq(2)12 b(T)e Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\).)358 1999 y(4.)20 b(The)d(rewrite)h(relation)e Fq(!)840 2005 y Fn(R)887 1999 y Fw(is)g(de\014ned)i(b)o(y)e Fq(!)1179 2005 y Fn(R)1225 1999 y Fw(=)1274 1968 y Fp(S)1308 2012 y Fl(n)p Fn(\025)p Fm(0)1380 1999 y Fq(!)1422 2005 y Fn(R)1451 2009 y Fj(n)1473 1999 y Fw(,)g(where)i Fq(!)1666 2005 y Fn(R)1695 2009 y Fe(0)1712 1999 y Fw(=)f Fq(;)411 2049 y Fw(and,)g(for)h Fo(n)g(>)g Fw(0,)g Fo(s)g Fq(!)798 2055 y Fn(R)827 2059 y Fj(n)867 2049 y Fo(t)g Fw(if)f(and)h(only)f(if)g (there)i(is)f(a)f(rule)i Fo(l)12 b Fq(!)f Fo(r)i Fq(\()e Fo(c)18 b Fw(in)f Fq(R)p Fw(,)h(a)411 2099 y(substitution)e Fo(\033)f Fw(:)f Fq(V)k(!)c(T)d Fw(\()p Fq(F)t Fo(;)c Fq(V)s Fw(\))16 b(with)f Fq(D)q Fw(\()p Fo(\033)q Fw(\))g(=)g Fq(V)s Fo(ar)q Fw(\()p Fo(l)e Fq(!)e Fo(r)i Fq(\()e Fo(c)p Fw(\),)k(and)g(a)h(con)o(text)411 2149 y Fo(C)s Fw([)c(])g(suc)o(h)h (that)g Fo(s)f Fw(=)g Fo(C)s Fw([)p Fo(l)q(\033)q Fw(],)f Fo(t)h Fw(=)g Fo(C)s Fw([)p Fo(r)q(\033)q Fw(],)f(and)h(\()p Fo(s)1176 2155 y Fl(i)1191 2149 y Fo(\033)q Fw(==)q Fo(t)1296 2155 y Fl(i)1309 2149 y Fo(\033)q Fw(\))g Fq(!)1404 2134 y Fn(\003)1404 2160 y(R)1433 2164 y Fj(n)p Fd(\000)p Fe(1)1489 2160 y Fn([R)1540 2164 y Fj(eq)1584 2149 y Fo(tr)q(ue)h Fw(for)g(all)411 2204 y(\()p Fo(s)446 2210 y Fl(i)460 2204 y Fw(==)q Fo(t)540 2210 y Fl(i)554 2204 y Fw(\))e(=)h Fo(tr)q(ue)i Fw(in)f Fo(c)p Fw(.)340 2282 y(In)j(the)g(sequel,)g(w)o(e)g(simply)e(write)i(a)f(rewrite)i(rule)f (as)g Fo(l)g Fq(!)e Fo(r)h Fq(\()f Fo(s)1420 2288 y Fm(1)1440 2282 y Fw(==)p Fo(t)1519 2288 y Fm(1)1538 2282 y Fo(;)7 b(:)g(:)g(:)t(;)g(s)1649 2288 y Fl(k)1670 2282 y Fw(==)p Fo(t)1749 2288 y Fl(k)1770 2282 y Fw(.)340 2332 y(When)18 b(w)o(e)h(sp)q(eak)f(of)f(a)h(CTRS)f Fq(R)h Fw(with)g(strict)g(equalit) o(y)m(,)e(w)o(e)j(tacitly)e(assume)g(that)h Fq(R)340 2382 y Fw(is)i(extended)i(with)e Fq(R)707 2388 y Fl(eq)741 2382 y Fw(.)f(The)i(next)f(prop)q(osition)g(sho)o(ws)g(that)g(the)h (de\014nition)f(ab)q(o)o(v)o(e)340 2432 y(sp)q(eci\014es)c(the)f (desired)g(b)q(eha)o(vior)e(of)h(a)f(CTRS)h(with)f(strict)i(equalit)o (y;)d(cf.)i([AEH94)o(].)p eop %%Page: 5 5 5 4 bop 183 194 a Ff(Prop)q(ositi)o(on)5 b(5.)21 b Fr(L)n(et)d Fq(R)h Fr(b)n(e)f(a)h(3-CTRS)f(with)g(strict)g(e)n(quality.)g(The)g (fol)r(lowing)g(state-)183 244 y(ments)d(ar)n(e)f(e)n(quivalent)h(for)g (al)r(l)f(terms)g Fo(s)h Fr(and)h Fo(t)f Fr(in)g Fq(T)10 b Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\))q Fr(:)199 327 y(1.)20 b Fo(s)p Fw(==)q Fo(t)15 b Fr(is)g(r)n(e)n(ducible)f(to)h Fo(tr)q(ue)g Fr(in)g Fq(R)9 b(])g(R)893 333 y Fl(eq)927 327 y Fr(.)199 377 y(2.)20 b Fo(s)c Fr(and)f Fo(t)g Fr(ar)n(e)g(r)n(e)n (ducible)f(to)h(a)g(same)g(gr)n(ound)g(c)n(onstructor)g(term)f(in)h Fq(R)p Fr(.)183 468 y Ff(De\014niti)o(on)5 b(6.)20 b Fw(A)c(3-CTRS)e Fq(R)i Fw(with)f(strict)h(equalit)o(y)e(is)h(called)g Fr(pr)n(op)n(erly)h(oriente)n(d)f Fw(\(cf.)183 518 y([SMI95)o(]\))e (if,)g(for)g(all)g Fo(l)g Fq(!)e Fo(r)h Fq(\()f Fo(s)705 524 y Fm(1)724 518 y Fw(==)q Fo(t)804 524 y Fm(1)823 518 y Fo(;)c(:)g(:)g(:)t(;)g(s)934 524 y Fl(k)955 518 y Fw(==)p Fo(t)1034 524 y Fl(k)1069 518 y Fw(in)13 b Fq(R)p Fw(,)200 601 y(1.)20 b(ev)o(ery)15 b Fo(t)378 607 y Fl(j)409 601 y Fw(is)f(a)g(linear)f(constructor)i(term,)e(and)200 657 y(2.)20 b Fq(V)s Fo(ar)q Fw(\()p Fo(s)358 663 y Fl(i)373 657 y Fw(\))12 b Fq(\022)g(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))e Fq([)607 626 y Fp(S)642 636 y Fl(i)p Fn(\000)p Fm(1)642 670 y Fl(j)r Fm(=1)708 657 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)809 663 y Fl(j)827 657 y Fw(\))q(.)183 747 y(A)f Fr(functional)h Fw(CTRS)e(is)h(a)g(prop)q(erly)h(orien)o(ted)g (orthogonal)e(3-CTRS)g(with)h(strict)h(equalit)o(y)m(.)1628 732 y Fm(2)183 838 y Fw(The)k(functional)f(CTRS)g(b)q(elo)o(w)h (computes)g(the)g(Fib)q(onacci)g(n)o(um)o(b)q(ers)f(\(cf.)h([Klo92)o (]\).)296 998 y Fq(R)331 1004 y Fl(f)s(ib)390 998 y Fw(=)434 888 y Fp(8)434 926 y(>)434 938 y(>)434 951 y(<)434 1025 y(>)434 1038 y(>)434 1050 y(:)477 923 y Fo(pl)q(us)p Fw(\(0)p Fo(;)7 b(x)p Fw(\))64 b Fq(!)11 b Fo(x)477 973 y(pl)q(us)p Fw(\()p Fo(s)p Fw(\()p Fo(x)p Fw(\))p Fo(;)c(y)q Fw(\))13 b Fq(!)e Fo(s)p Fw(\()p Fo(pl)q(us)p Fw(\()p Fo(x;)c(y)q Fw(\)\))477 1023 y Fo(f)t(ib)p Fw(\(0\))128 b Fq(!)11 b Fo(pair)q Fw(\(0)p Fo(;)c(s)p Fw(\(0\)\))477 1073 y Fo(f)t(ib)p Fw(\()p Fo(s)p Fw(\()p Fo(x)p Fw(\)\))74 b Fq(!)11 b Fo(pair)q Fw(\()p Fo(z)r(;)c(pl)q(us)p Fw(\()p Fo(y)q(;)g(z)r Fw(\)\))12 b Fq(\()f Fo(f)t(ib)p Fw(\()p Fo(x)p Fw(\)==)r Fo(pair)q Fw(\()p Fo(y)q(;)c(z)r Fw(\))183 1156 y(Unfortunately)m(,)18 b(functional)g(CTRSs)g(do)h(not)g(satisfy)g (the)g(parallel)f(mo)o(v)o(es)g(lemma)d(as)183 1206 y(witnessed)g(b)o (y)f(the)g(follo)o(wing)d(simple)i(example)f(tak)o(en)i(from)e ([SMI95].)660 1342 y Fq(R)g Fw(=)751 1256 y Fp(8)751 1294 y(<)751 1368 y(:)793 1291 y Fo(f)t Fw(\()p Fo(x)p Fw(\))h Fq(!)e Fo(y)i Fq(\()e Fo(x)p Fw(==)q Fo(y)793 1341 y(a)h Fq(!)f Fo(b)793 1391 y(b)h Fq(!)f Fo(c)183 1477 y Fw(Then)j Fo(f)t Fw(\()p Fo(a)p Fw(\))g Fq(k)-30 b(!)415 1489 y Fn(R)444 1493 y Fe(2)462 1477 y Fo(a)13 b Fw(and)h Fo(f)t Fw(\()p Fo(a)p Fw(\))g Fq(k)-30 b(!)702 1489 y Fn(R)731 1493 y Fe(2)749 1477 y Fo(c)13 b Fw(but)i(not)e Fo(a)g Fq(k)-30 b(!)997 1489 y Fn(R)1026 1493 y Fe(2)1044 1477 y Fo(c)p Fw(.)183 1576 y(W)m(e)21 b(next)g(in)o(tro)q(duce)h(a)f (sp)q(ecial)h("deterministic")e(rewrite)i(relation)f Fq(!)1387 1585 y Fn(R)1416 1577 y Fj(d)1470 1576 y Fw(whic)o(h)g(is)183 1626 y(closely)11 b(related)h(to)g Fq(!)540 1632 y Fn(R)581 1626 y Fw(\(the)h(only)d(di\013erence)k(is)d(that)h(in)f Fq(!)1154 1635 y Fn(R)1183 1626 y Fj(d)29 b Fw(extra)12 b(v)n(ariables)e(cannot)183 1676 y(b)q(e)k(instan)o(tiated)g(b)o(y)f (non)h(ground)g(constructors)i(terms\).)183 1759 y Ff(De\014niti)o(on)5 b(7.)20 b Fw(Let)c Fq(R)e Fw(b)q(e)i(a)e(functional)g(CTRS.)f(Let)j Fq(!)1126 1769 y Fn(R)1155 1759 y Fj(d)1155 1779 y Fe(0)1173 1759 y Fw(=)d Fq(;)i Fw(and)f(for)g Fo(n)f(>)g Fw(0)i(de\014ne)183 1809 y Fo(s)p Fq(!)244 1818 y Fn(R)273 1809 y Fj(d)273 1826 y(n)295 1809 y Fo(t)k Fw(if)f(and)h(only)g(if)f(there)j(exists)f (a)f(rewrite)h(rule)g Fo(l)13 b Fq(!)e Fo(r)h Fq(\()f Fo(c)19 b Fw(in)g Fq(R)p Fw(,)g(a)g(substitu-)183 1859 y(tion)g Fo(\033)k Fw(:)f Fq(V)j(!)d(T)10 b Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\),)20 b(and)g(a)f(con)o(text)i Fo(C)s Fw([)12 b(])19 b(suc)o(h)i(that)f Fo(s)j Fw(=)f Fo(C)s Fw([)p Fo(l)q(\033)q Fw(])p Fo(;)7 b(t)20 b Fw(=)i Fo(C)s Fw([)p Fo(r)q(\033)q Fw(],)183 1909 y(\()p Fo(s)218 1915 y Fl(i)232 1909 y Fo(\033)q Fw(==)q Fo(t)337 1915 y Fl(i)351 1909 y Fo(\033)q Fw(\))12 b Fq(!)446 1894 y Fn(\003)446 1925 y(R)475 1915 y Fj(d)475 1935 y(n)p Fd(\000)p Fe(1)531 1925 y Fn([R)582 1929 y Fj(eq)626 1909 y Fo(tr)q(ue)g Fw(for)g(all)g Fo(s)854 1915 y Fl(i)868 1909 y Fw(==)q Fo(t)948 1915 y Fl(i)974 1909 y Fw(in)g Fo(c)p Fw(,)g(and)h Fo(x\033)g Fw(is)g(a)f(ground)h(constructor)183 1974 y(term)d(for)i(ev)o(ery)g(extra)g(v)n(ariable)e Fo(x)h Fw(in)g Fo(l)i Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(.)g(Finally)m(,)e (de\014ne)j Fq(!)1296 1982 y Fn(R)1325 1974 y Fj(d)29 b Fw(=)1413 1943 y Fp(S)1448 1986 y Fl(n)p Fn(\025)p Fm(0)1520 1974 y Fq(!)1562 1982 y Fn(R)1591 1974 y Fj(d)1591 1991 y(n)1612 1974 y Fw(.)183 2057 y(It)19 b(is)h(easy)g(to)f(pro)o(v)o (e)g(that)h Fo(s)p Fq(!)708 2066 y Fn(R)737 2057 y Fj(d)737 2074 y(n)759 2057 y Fo(t)f Fw(implies)f Fo(s)p Fq(!)1001 2063 y Fn(R)1030 2067 y Fj(n)1052 2057 y Fo(t)h Fw(but)h(not)f(vice)h (v)o(ersa.)g(The)g(\014rst)183 2107 y(statemen)o(t)12 b(of)g(the)h(next)g(lemma)c(sho)o(ws)k(that)g Fq(!)963 2115 y Fn(R)992 2107 y Fj(d)30 b Fw(is)12 b(deterministic)g(in)g(the)h (sense)h(that)183 2157 y(the)i(con)o(tractum)f(of)g(a)g(redex)i(is)e (uniquely)g(determined.)g(F)m(urthermore,)g(in)g(con)o(trast)i(to)183 2207 y Fq(!)225 2213 y Fn(R)255 2207 y Fw(,)d(the)i(relation)e Fq(!)549 2215 y Fn(R)578 2207 y Fj(d)32 b Fw(satis\014es)16 b(the)g(parallel)d(mo)o(v)o(es)h(lemma.)d(Because)17 b(of)d(the)i(\014rst)183 2256 y(statemen)o(t)f(of)g(Lemma)d(8,)j(there) i(is)e(a)g(strong)h(resem)o(blance)f(b)q(et)o(w)o(een)i(the)f(pro)q(of) f(of)g(the)183 2306 y(second)g(statemen)o(t)e(and)h(that)g(of)f(Lemma)f (3)h(\(giv)o(en)h(in)f([BK86]\).)p 183 2343 237 2 v 191 2370 a Ft(2)221 2386 y Fv(Note)c(that)g(Han)o(us)h([Han95])f(de\014nes) h(this)g(notion)h(di\013eren)o(tly)m(.)g(He)e(calls)h(a)g(CTRS)f Fi(R)f Fv(functional)221 2432 y(if)13 b(\(1\))g Fi(R)g Fv(is)g(normal,)h(\(2\))f Fi(R)638 2436 y Fh(u)671 2432 y Fv(is)g(almost)h(orthogonal,)h(and)f(\(3\))f Fi(!)1211 2436 y Fc(R)1252 2432 y Fv(is)h(con\015uen)o(t.)p eop %%Page: 6 6 6 5 bop 340 194 a Ff(Lemma)8 b(8.)21 b Fr(L)n(et)14 b Fq(R)h Fr(b)n(e)g(functional.)g(F)m(or)g(al)r(l)f Fo(m;)7 b(n)k Fq(2)h Fk(I)-6 b(N)p Fr(,)15 b(the)f(fol)r(lowing)h(holds:)356 284 y(1.)21 b(If)16 b Fo(s)g Fw(=)f Fo(l)550 290 y Fm(1)569 284 y Fo(\033)593 290 y Fm(1)611 284 y Fq(!)653 292 y Fn(R)682 284 y Fj(d)682 301 y(m)711 284 y Fo(r)730 290 y Fm(1)748 284 y Fo(\033)772 290 y Fm(1)807 284 y Fr(and)j Fo(s)d Fw(=)g Fo(l)983 290 y Fm(2)1002 284 y Fo(\033)1026 290 y Fm(2)1045 284 y Fq(!)1087 292 y Fn(R)1116 284 y Fj(d)1116 301 y(n)1137 284 y Fo(r)1156 290 y Fm(2)1174 284 y Fo(\033)1198 290 y Fm(2)1217 284 y Fr(,)h(then)h(the)g(applie)n (d)g(r)n(ewrite)e(rules)411 334 y(c)n(oincide)g(and)h Fo(\033)677 340 y Fm(1)707 334 y Fw(=)11 b Fo(\033)774 340 y Fm(2)793 334 y Fr(.)j(In)h(other)g(wor)n(ds,)f(if)g Fo(s)h Fr(is)g(a)g(r)n(e)n(dex,)f(then)i(ther)n(e)e(is)h(a)g(unique)411 384 y(r)n(ewrite)e(rule)g Fo(l)g Fq(!)e Fo(r)i Fq(\()e Fo(c)g Fq(2)g(R)k Fr(and)g(a)g(unique)g(substitution)f Fo(\033)f Fw(:)e Fq(V)16 b(!)11 b(T)f Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\))15 b Fr(such)411 433 y(that)g Fo(s)d Fw(=)g Fo(l)q(\033)q Fq(!)651 442 y Fn(R)680 434 y Fj(d)713 433 y Fo(r)q(\033)q Fr(.)356 485 y(2.)21 b(If)e Fo(t)13 b Fq(k)-30 b(!)520 493 y Fn(R)549 485 y Fj(d)549 502 y(m)577 485 y Fo(t)592 491 y Fm(1)630 485 y Fr(and)21 b Fo(t)12 b Fq(k)-30 b(!)776 493 y Fn(R)805 485 y Fj(d)805 502 y(n)826 485 y Fo(t)841 491 y Fm(2)860 485 y Fr(,)19 b(then)h(ther)n(e)f(is)h(a)f(term)g Fo(t)1306 491 y Fm(3)1344 485 y Fr(such)i(that)e Fo(t)1548 491 y Fm(1)1579 485 y Fq(k)-30 b(!)1612 493 y Fn(R)1641 485 y Fj(d)1641 502 y(n)1662 485 y Fo(t)1677 491 y Fm(3)1716 485 y Fr(and)411 535 y Fo(t)426 541 y Fm(2)457 535 y Fq(k)g(!)490 543 y Fn(R)519 535 y Fj(d)519 552 y(m)547 535 y Fo(t)562 541 y Fm(3)581 535 y Fr(.)14 b(Mor)n(e)n(over,)g(the)g(r)n(e)n(dexes)h (c)n(ontr)n(acte)n(d)f(in)h Fo(t)1277 541 y Fm(1)1308 535 y Fq(k)-30 b(!)1341 543 y Fn(R)1370 535 y Fj(d)1370 552 y(n)1391 535 y Fo(t)1406 541 y Fm(3)1439 535 y Fr(\()p Fo(t)1471 541 y Fm(2)1502 535 y Fq(k)g(!)1535 543 y Fn(R)1564 535 y Fj(d)1564 552 y(m)1592 535 y Fo(t)1607 541 y Fm(3)1626 535 y Fr(\))14 b(ar)n(e)g(the)411 585 y(desc)n(endants)i(in)f Fo(t)705 591 y Fm(1)738 585 y Fr(\()p Fo(t)770 591 y Fm(2)789 585 y Fr(\))g(of)g(the)g(r)n(e)n(dexes)f(c)n(ontr)n(acte)n(d)h (in)g Fo(t)e Fq(k)-30 b(!)1391 593 y Fn(R)1420 585 y Fj(d)1420 601 y(n)1441 585 y Fo(t)1456 591 y Fm(2)1489 585 y Fr(\()p Fo(t)13 b Fq(k)-30 b(!)1567 593 y Fn(R)1596 585 y Fj(d)1596 601 y(m)1624 585 y Fo(t)1639 591 y Fm(1)1658 585 y Fr(\).)340 685 y(Pr)n(o)n(of.)20 b Fw(The)e(pro)q(of)f(pro)q (ceeds)j(b)o(y)d(induction)g(on)g Fo(m)12 b Fw(+)g Fo(n)p Fw(.)17 b(The)g(base)h(case)h Fo(m)12 b Fw(+)g Fo(n)17 b Fw(=)h(0)340 735 y(holds)d(v)n(acuously)m(.)f(Supp)q(ose)j(the)f (lemma)c(holds)j(for)g(all)f Fo(m)1292 720 y Fn(0)1319 735 y Fw(and)h Fo(n)1426 720 y Fn(0)1453 735 y Fw(with)g Fo(m)1585 720 y Fn(0)1608 735 y Fw(+)10 b Fo(n)1675 720 y Fn(0)1701 735 y Fo(<)k(k)q Fw(.)340 785 y(In)i(the)h(induction)e (step,)i(w)o(e)f(ha)o(v)o(e)g(to)g(pro)o(v)o(e)g(that)g(the)g(lemma)d (holds)j(for)g(all)e Fo(m)j Fw(and)e Fo(n)340 834 y Fw(with)h Fo(m)11 b Fw(+)g Fo(n)k Fw(=)g Fo(k)q Fw(.)h(Observ)o(e)h(that)f(the)h (inductiv)o(e)f(h)o(yp)q(othesis)g(implies)e(the)j(v)n(alidit)o(y)d(of) 340 884 y(the)h(diagrams)d(in)h(Figure)h(1,)f(where)i Fo(m)971 869 y Fn(0)993 884 y Fw(+)9 b Fo(n)1059 869 y Fn(0)1082 884 y Fo(<)j(k)q Fw(.)427 1034 y Fg(:)508 1011 y Fi(R)541 995 y Fh(d)541 1020 y(n)560 1013 y Fd(0)p 596 1056 2 48 v 606 1056 V 710 1034 a Fg(:)444 1162 y Fi(R)477 1146 y Fh(d)477 1171 y(m)504 1164 y Fd(0)p 408 1208 48 2 v 408 1198 V 728 1162 a Fi(R)761 1146 y Fh(d)761 1171 y(m)788 1164 y Fd(0)p 692 1208 V 692 1198 V 427 1317 a Fg(:)508 1294 y Fi(R)541 1278 y Fh(d)541 1304 y(n)560 1297 y Fd(0)p 596 1339 2 48 v 606 1339 V 710 1317 a Fg(:)p 456 1033 237 2 v 650 1032 a Fb(-)p 431 1292 2 237 v 432 1292 a(?)p 715 1065 2 10 v 715 1084 V 715 1103 V 715 1122 V 715 1141 V 715 1160 V 715 1178 V 715 1197 V 715 1216 V 715 1235 V 715 1254 V 715 1273 V 715 1292 V 715 1292 2 24 v 241 w(?)p 456 1316 10 2 v 474 1316 V 493 1316 V 512 1316 V 531 1316 V 550 1316 V 569 1316 V 588 1316 V 607 1316 V 626 1316 V 645 1316 V 663 1316 V 668 1316 24 2 v 650 1315 a(-)890 1034 y Fg(:)972 1011 y Fi(R)1005 995 y Fh(d)1005 1020 y(n)1024 1013 y Fd(0)p 1060 1056 2 48 v 1069 1056 V 1174 1034 a Fg(:)908 1162 y Fi(R)941 1146 y Fh(d)941 1171 y(m)968 1164 y Fd(0)p 872 1208 48 2 v 872 1198 V 908 1291 a Fi(\003)1191 1162 y(R)1224 1146 y Fh(d)1224 1171 y(m)1251 1164 y Fd(0)p 1155 1208 V 1155 1198 V 1191 1291 a Fi(\003)890 1317 y Fg(:)972 1294 y Fi(R)1005 1278 y Fh(d)1005 1304 y(n)1024 1297 y Fd(0)p 1060 1339 2 48 v 1069 1339 V 1174 1317 a Fg(:)p 919 1033 237 2 v 1114 1032 a Fb(-)p 895 1292 2 237 v 896 1292 a(?)p 1178 1065 2 10 v 1178 1084 V 1178 1103 V 1178 1122 V 1178 1141 V 1178 1160 V 1178 1178 V 1178 1197 V 1178 1216 V 1178 1235 V 1178 1254 V 1178 1273 V 1178 1292 V 1178 1292 2 24 v 241 w(?)p 919 1316 10 2 v 938 1316 V 957 1316 V 976 1316 V 995 1316 V 1014 1316 V 1033 1316 V 1051 1316 V 1070 1316 V 1089 1316 V 1108 1316 V 1127 1316 V 1132 1316 24 2 v 1114 1315 a(-)1354 1034 y Fg(:)1435 1011 y Fi(R)1468 995 y Fh(d)1468 1020 y(n)1487 1013 y Fd(0)p 1524 1056 2 48 v 1533 1056 V 1637 1034 a Fg(:)1595 1017 y Fi(\003)1372 1162 y(R)1405 1146 y Fh(d)1405 1171 y(m)1432 1164 y Fd(0)p 1335 1208 48 2 v 1335 1198 V 1372 1291 a Fi(\003)1655 1162 y(R)1688 1146 y Fh(d)1688 1171 y(m)1715 1164 y Fd(0)p 1619 1208 V 1619 1198 V 1655 1291 a Fi(\003)1354 1317 y Fg(:)1435 1294 y Fi(R)1468 1278 y Fh(d)1468 1304 y(n)1487 1297 y Fd(0)p 1524 1339 2 48 v 1533 1339 V 1595 1300 a Fi(\003)1637 1317 y Fg(:)p 1383 1033 237 2 v 1577 1032 a Fb(-)p 1358 1292 2 237 v 1359 1292 a(?)p 1642 1065 2 10 v 1642 1084 V 1642 1103 V 1642 1122 V 1642 1141 V 1642 1160 V 1642 1178 V 1642 1197 V 1642 1216 V 1642 1235 V 1642 1254 V 1642 1273 V 1642 1292 V 1642 1292 2 24 v 242 w(?)p 1383 1316 10 2 v 1402 1316 V 1421 1316 V 1439 1316 V 1458 1316 V 1477 1316 V 1496 1316 V 1515 1316 V 1534 1316 V 1553 1316 V 1572 1316 V 1591 1316 V 1595 1316 24 2 v 1577 1315 a(-)996 1444 y Fu(Fig.)7 b(1.)340 1591 y Fw(\(1\))20 b(Supp)q(ose)i Fo(s)g Fw(=)g Fo(l)690 1597 y Fm(1)709 1591 y Fo(\033)733 1597 y Fm(1)751 1591 y Fq(!)793 1600 y Fn(R)822 1591 y Fj(d)822 1608 y(m)851 1591 y Fo(r)870 1597 y Fm(1)888 1591 y Fo(\033)912 1597 y Fm(1)951 1591 y Fw(and)e Fo(s)i Fw(=)g Fo(l)1145 1597 y Fm(2)1164 1591 y Fo(\033)1188 1597 y Fm(2)1207 1591 y Fq(!)1249 1600 y Fn(R)1278 1591 y Fj(d)1278 1608 y(n)1299 1591 y Fo(r)1318 1597 y Fm(2)1336 1591 y Fo(\033)1360 1597 y Fm(2)1379 1591 y Fw(.)d(Since)i Fq(R)1560 1597 y Fl(u)1602 1591 y Fw(is)f(orthog-)340 1641 y(onal,)e(the)i(applied)f (rewrite)h(rules)g(coincide)f(and)g(will)f(b)q(e)i(denoted)g(b)o(y)f Fo(l)13 b Fq(!)e Fo(r)i Fq(\()e Fo(c)19 b Fw(in)340 1691 y(the)h(sequel.)f(Ob)o(viously)m(,)e Fo(\033)789 1697 y Fm(1)827 1691 y Fw(=)k Fo(\033)904 1697 y Fm(2)934 1691 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))q(],)d(i.e.,)f(the)j (restrictions)g(of)e Fo(\033)1560 1697 y Fm(1)1598 1691 y Fw(and)g Fo(\033)1707 1697 y Fm(2)1744 1691 y Fw(to)340 1741 y Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))i(coincide.)f(It)f(remains)g (to)h(sho)o(w)f Fo(\033)1047 1747 y Fm(1)1085 1741 y Fw(=)i Fo(\033)1161 1747 y Fm(2)1192 1741 y Fw([)p Fq(E)t(V)s Fo(ar)q Fw(\()p Fo(l)12 b Fq(!)f Fo(r)i Fq(\()e Fo(c)p Fw(\)].)18 b(W)m(e)g(sho)o(w)h(b)o(y)340 1795 y(induction)j(on)g Fo(i)g Fw(that)g Fo(\033)757 1801 y Fm(1)800 1795 y Fw(=)k Fo(\033)882 1801 y Fm(2)912 1795 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))16 b Fq([)1097 1764 y Fp(S)1132 1774 y Fl(i)1132 1808 y(j)r Fm(=1)1198 1795 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)1299 1801 y Fl(j)1317 1795 y Fw(\))q(].)21 b(If)g Fo(i)k Fw(=)h(0,)21 b(then)i Fo(\033)1706 1801 y Fm(1)1749 1795 y Fw(=)340 1845 y Fo(\033)364 1851 y Fm(2)395 1845 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))q(].)13 b(Let)h Fo(i)e(>)g Fw(0.)h(According)h(to)f(the)i (inductiv)o(e)f(h)o(yp)q(othesis,)f Fo(\033)1488 1851 y Fm(1)1518 1845 y Fw(=)f Fo(\033)1586 1851 y Fm(2)1617 1845 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))e Fq([)340 1869 y Fp(S)375 1879 y Fl(i)p Fn(\000)p Fm(1)375 1912 y Fl(j)r Fm(=1)441 1900 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)542 1906 y Fl(j)561 1900 y Fw(\)].)17 b(Since)h Fq(V)s Fo(ar)q Fw(\()p Fo(s)835 1906 y Fl(i)850 1900 y Fw(\))g Fq(\022)h(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))13 b Fq([)1102 1869 y Fp(S)1137 1879 y Fl(i)p Fn(\000)p Fm(1)1137 1912 y Fl(j)r Fm(=1)1203 1900 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)1304 1906 y Fl(j)1323 1900 y Fw(\),)k(it)h(is)f(su\016cien)o(t)i(to)e(sho)o(w)340 1950 y Fo(\033)364 1956 y Fm(1)397 1950 y Fw(=)d Fo(\033)467 1956 y Fm(2)498 1950 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(t)611 1956 y Fl(i)625 1950 y Fw(\)].)g(No)o(w)h(\()p Fo(s)810 1956 y Fl(i)825 1950 y Fo(\033)849 1956 y Fm(1)867 1950 y Fw(==)q Fo(t)947 1956 y Fl(i)961 1950 y Fo(\033)985 1956 y Fm(1)1003 1950 y Fw(\))p Fq(!)1061 1934 y Fn(\003)1061 1966 y(R)1090 1956 y Fj(d)1090 1976 y(m)p Fd(\000)p Fe(1)1155 1950 y Fo(tr)q(ue)h Fw(and)f(\()p Fo(s)1366 1956 y Fl(i)1380 1950 y Fo(\033)1404 1956 y Fm(2)1422 1950 y Fw(==)q Fo(t)1502 1956 y Fl(i)1516 1950 y Fo(\033)1540 1956 y Fm(2)1558 1950 y Fw(\))p Fq(!)1616 1934 y Fn(\003)1616 1966 y(R)1645 1956 y Fj(d)1645 1976 y(n)p Fd(\000)p Fe(1)1703 1950 y Fo(tr)q(ue)340 2012 y Fw(imply)m(,)f(b)o(y)j(Prop)q(osition)g(5,)f (that)i(there)g(exist)g(ground)f(constructor)h(terms)f Fo(u)1619 2018 y Fm(1)1655 2012 y Fw(and)g Fo(u)1763 2018 y Fm(2)340 2062 y Fw(suc)o(h)f(that)e Fo(s)544 2068 y Fl(i)558 2062 y Fo(\033)582 2068 y Fm(1)601 2062 y Fq(!)643 2047 y Fn(\003)643 2078 y(R)672 2068 y Fj(d)672 2089 y(m)p Fd(\000)p Fe(1)737 2062 y Fo(u)761 2068 y Fm(1)805 2047 y Fn(\003)805 2078 y(R)834 2068 y Fj(d)834 2089 y(m)p Fd(\000)p Fe(1)870 2062 y Fq( )g Fo(t)941 2068 y Fl(i)954 2062 y Fo(\033)978 2068 y Fm(1)1011 2062 y Fw(and)g Fo(s)1111 2068 y Fl(i)1126 2062 y Fo(\033)1150 2068 y Fm(2)1168 2062 y Fq(!)1210 2047 y Fn(\003)1210 2078 y(R)1239 2068 y Fj(d)1239 2089 y(n)p Fd(\000)p Fe(1)1297 2062 y Fo(u)1321 2068 y Fm(2)1366 2047 y Fn(\003)1366 2078 y(R)1395 2068 y Fj(d)1395 2089 y(n)p Fd(\000)p Fe(1)1423 2062 y Fq( )g Fo(t)1494 2068 y Fl(i)1507 2062 y Fo(\033)1531 2068 y Fm(2)1550 2062 y Fw(.)g(It)g(is)g(an)g(im-)340 2127 y(mediate)g(consequence)k(of)c Fo(s)801 2133 y Fl(i)815 2127 y Fo(\033)839 2133 y Fm(1)871 2127 y Fw(=)f Fo(s)935 2133 y Fl(i)949 2127 y Fo(\033)973 2133 y Fm(2)992 2127 y Fw(,)h Fo(s)1037 2133 y Fl(i)1051 2127 y Fo(\033)1075 2133 y Fm(1)1106 2127 y Fq(k)-30 b(!)1139 2112 y Fn(\003)1139 2143 y(R)1168 2133 y Fj(d)1168 2154 y(m)p Fd(\000)p Fe(1)1233 2127 y Fo(u)1257 2133 y Fm(1)1276 2127 y Fw(,)14 b Fo(s)1321 2133 y Fl(i)1335 2127 y Fo(\033)1359 2133 y Fm(2)1390 2127 y Fq(k)-30 b(!)1423 2112 y Fn(\003)1423 2143 y(R)1452 2133 y Fj(d)1452 2154 y(n)p Fd(\000)p Fe(1)1510 2127 y Fo(u)1534 2133 y Fm(2)1552 2127 y Fw(,)15 b(and)f(the)i(in-)340 2193 y(ductiv)o(e)d(h)o(yp)q(othesis)g(\(on)f Fo(k)q Fw(\))g(that)h(the)f(t)o(w)o(o)g(ground)g(normal)e(forms)h Fo(u)1460 2199 y Fm(1)1491 2193 y Fw(and)h Fo(u)1594 2199 y Fm(2)1625 2193 y Fw(coincide.)340 2243 y(Hence)18 b Fo(t)481 2249 y Fl(i)495 2243 y Fo(\033)519 2249 y Fm(1)537 2243 y Fq(!)579 2227 y Fn(\003)579 2259 y(R)608 2248 y Fj(d)608 2269 y(m)p Fd(\000)p Fe(1)673 2243 y Fo(u)697 2249 y Fm(1)742 2227 y Fn(\003)742 2259 y(R)771 2248 y Fj(d)771 2269 y(n)p Fd(\000)p Fe(1)800 2243 y Fq( )13 b Fo(t)870 2249 y Fl(i)884 2243 y Fo(\033)908 2249 y Fm(2)926 2243 y Fw(.)i(Th)o(us,)h(for)g(all)f(v)n(ariables)g Fo(x)f Fq(2)h(V)s Fo(ar)q Fw(\()p Fo(t)1554 2249 y Fl(i)1569 2243 y Fw(\))c Fq(n)f(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))i Fq([)340 2286 y Fp(S)375 2296 y Fl(i)p Fn(\000)p Fm(1)375 2329 y Fl(j)r Fm(=1)441 2317 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)542 2323 y Fl(j)561 2317 y Fw(\),)k(it)f(follo)o(ws)g(from)g(the)h (fact)h(that)f Fo(t)1153 2323 y Fl(i)1183 2317 y Fw(is)g(a)g(linear)g (constructor)i(term)d(that)340 2366 y Fo(x\033)388 2372 y Fm(1)406 2366 y Fq(!)448 2351 y Fn(\003)448 2382 y(R)477 2372 y Fj(d)477 2393 y(m)p Fd(\000)p Fe(1)543 2366 y Fo(u)567 2372 y Fl(x)614 2351 y Fn(\003)614 2382 y(R)643 2372 y Fj(d)643 2393 y(n)p Fd(\000)p Fe(1)671 2366 y Fq( )f Fo(x\033)775 2372 y Fm(2)803 2366 y Fw(for)c(some)g(subterm)g Fo(u)1146 2372 y Fl(x)1177 2366 y Fw(of)g Fo(u)1245 2372 y Fm(1)1264 2366 y Fw(.)g(Finally)m(,)d(since)12 b Fo(u)1554 2372 y Fl(x)1585 2366 y Fw(is)e(a)g(ground)340 2432 y(constructor)i (term)e(and,)g(b)o(y)g(de\014nition)g(of)f Fq(!)1061 2440 y Fn(R)1090 2432 y Fj(d)18 b Fw(,)9 b(for)h(ev)o(ery)h Fo(x)h Fq(2)f(E)t(V)s Fo(ar)q Fw(\()p Fo(l)i Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(\))q(,)e Fo(x\033)1763 2438 y Fm(1)p eop %%Page: 7 7 7 6 bop 183 194 a Fw(and)13 b Fo(x\033)311 200 y Fm(2)343 194 y Fw(are)i(ground)e(constructor)j(terms,)d(w)o(e)h(infer)g Fo(x\033)1105 200 y Fm(1)1135 194 y Fw(=)d Fo(u)1202 200 y Fl(x)1235 194 y Fw(=)g Fo(x\033)1326 200 y Fm(2)1345 194 y Fw(.)183 244 y(\(2\))i(Since)g(parallel)f(reduction)i(con)o (tracts)h(pairwise)e(disjoin)o(t)f(redexes,)i(it)f(is)g(su\016cien)o(t) h(to)183 293 y(pro)o(v)o(e)d(the)g(lemma)d(for)j(the)g(case)h(where)g (b)q(oth)f Fo(t)i Fq(k)-30 b(!)978 302 y Fn(R)1007 293 y Fj(d)1007 310 y(m)1035 293 y Fo(t)1050 299 y Fm(1)1079 293 y Fw(and)11 b Fo(t)i Fq(k)-30 b(!)1218 302 y Fn(R)1247 293 y Fj(d)1247 310 y(n)1268 293 y Fo(t)1283 299 y Fm(2)1312 293 y Fw(consist)12 b(of)e(a)h(single)183 343 y Fq(!)225 352 y Fn(R)254 343 y Fj(d)30 b Fw(step.)14 b(In)f(other)g(w)o(ords,)g (w)o(e)h(ma)o(y)d(assume)h Fo(t)p Fq(!)1035 352 y Fn(R)1064 343 y Fj(d)1064 360 y(m)1093 343 y Fo(t)1108 349 y Fm(1)1140 343 y Fw(and)h Fo(t)p Fq(!)1277 352 y Fn(R)1306 343 y Fj(d)1306 360 y(n)1327 343 y Fo(t)1342 349 y Fm(2)1361 343 y Fw(.)f(F)m(urthermore,)183 393 y(b)q(ecause)k Fq(!)379 401 y Fn(R)408 393 y Fj(d)31 b Fw(is)15 b(deterministic,)e(the)i(only)f (in)o(teresting)h(case)g(is)f(that)h(where)g Fo(t)g Fw(is)f(a)g(re-)183 443 y(dex,)h(sa)o(y)g Fo(t)f Fw(=)g Fo(l)q(\033)q Fq(!)501 451 y Fn(R)530 443 y Fj(d)530 459 y(m)559 443 y Fo(r)q(\033)h Fw(=)f Fo(t)679 449 y Fm(1)713 443 y Fw(for)h(some)f(rule)i Fo(l)d Fq(!)e Fo(r)h Fq(\()f Fo(c)j Fq(2)f(R)q Fw(,)h(con)o(taining)h (a)g(prop)q(er)183 493 y(subredex)g Fo(s)f Fw(whic)o(h)g(is)g(con)o (tracted)h(to)f Fo(s)825 478 y Fn(0)851 493 y Fw(in)f(the)i(step)g Fo(t)p Fq(!)1116 501 y Fn(R)1145 493 y Fj(d)1145 509 y(n)1166 493 y Fo(t)1181 499 y Fm(2)1200 493 y Fw(.)245 543 y(Since)c Fq(R)385 549 y Fl(u)418 543 y Fw(is)f(orthogonal,)f (there)j(is)e(a)h(v)n(ariable)e Fo(x)j Fq(2)f(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))h(suc)o(h)f(that)g Fo(s)g Fw(is)f(a)h(subterm)183 593 y(of)j Fo(x\033)q Fw(.)g(So)h Fo(x\033)f Fw(=)f Fo(C)s Fw([)p Fo(s)p Fw(])h(for)h(some)f(con)o(text)h Fo(C)s Fw([)d(].)i(Let)h Fo(q)h Fw(b)q(e)f(the)h(p)q(osition)e(in)g Fo(t)h Fw(suc)o(h)g(that)183 642 y Fo(t)p Fq(j)210 648 y Fl(q)239 642 y Fw(=)d Fo(s)p Fw(.)f(Consequen)o(tly)m(,)f(for)h(ev)o (ery)h(descendan)o(t)h Fo(q)986 627 y Fn(0)1008 642 y Fw(of)e Fo(q)h Fw(in)e Fo(t)1144 648 y Fm(1)1163 642 y Fw(,)h(w)o(e)g(ha)o(v)o(e)g Fo(t)1352 648 y Fm(1)1370 642 y Fq(j)1382 648 y Fl(q)1398 640 y Fd(0)1423 642 y Fw(=)h Fo(s)p Fw(.)f(De\014ne)183 692 y Fo(t)198 698 y Fm(3)228 692 y Fw(=)h Fo(t)287 698 y Fm(1)305 692 y Fw([)p Fo(q)337 677 y Fn(0)360 692 y Fq( )f Fo(s)432 677 y Fn(0)457 692 y Fq(j)h Fo(q)501 677 y Fn(0)524 692 y Fq(2)f Fo(q)q Fq(n)p Fo(t)p Fq(!)661 701 y Fn(R)690 692 y Fj(d)690 709 y(m)718 692 y Fo(t)733 698 y Fm(1)752 692 y Fw(].)i(Clearly)m(,)f Fo(t)956 698 y Fm(1)987 692 y Fq(k)-30 b(!)1020 701 y Fn(R)1049 692 y Fj(d)1049 709 y(n)1070 692 y Fo(t)1085 698 y Fm(3)1104 692 y Fw(.)245 742 y(It)12 b(remains)f(to)h(sho)o(w)g Fo(t)606 748 y Fm(2)637 742 y Fq(k)-30 b(!)670 751 y Fn(R)699 743 y Fj(d)699 759 y(m)728 742 y Fo(t)743 748 y Fm(3)761 742 y Fw(.)12 b(T)m(o)f(this)h(end,)g(let)h(us)f(consider)h Fo(t)f Fw(=)f Fo(l)q(\033)q Fq(!)1431 751 y Fn(R)1460 743 y Fj(d)1460 759 y(m)1490 742 y Fo(r)q(\033)h Fw(=)g Fo(t)1605 748 y Fm(1)183 792 y Fw(again.)f(By)j(de\014nition)f(of)25 b Fq(k)-30 b(!)645 801 y Fn(R)674 792 y Fj(d)674 809 y(m)702 792 y Fw(,)13 b(there)h(exist)g(ground)f(constructor)i(terms)e Fo(u)1428 798 y Fl(i)1455 792 y Fw(suc)o(h)g(that)183 842 y Fo(s)202 848 y Fl(i)216 842 y Fo(\033)h Fq(k)-30 b(!)287 827 y Fn(\003)287 858 y(R)316 848 y Fj(d)316 869 y(m)p Fd(\000)p Fe(1)381 842 y Fo(u)405 848 y Fl(i)432 842 y Fw(and)13 b Fo(t)527 848 y Fl(i)541 842 y Fo(\033)g Fq(k)-30 b(!)611 827 y Fn(\003)611 858 y(R)640 848 y Fj(d)640 869 y(m)p Fd(\000)p Fe(1)705 842 y Fo(u)729 848 y Fl(i)756 842 y Fw(for)13 b(all)f Fo(s)895 848 y Fl(i)910 842 y Fw(==)p Fo(t)989 848 y Fl(i)1016 842 y Fw(in)h Fo(c)p Fw(.)g(De\014ne)h Fo(\033)1260 827 y Fn(0)1285 842 y Fw(b)o(y)f Fo(y)q(\033)h Fw(=)e Fo(y)q(\033)1491 827 y Fn(0)1517 842 y Fw(for)h(all)183 907 y Fo(y)g Fq(6)p Fw(=)f Fo(x)f Fw(and)f Fo(x\033)421 892 y Fn(0)444 907 y Fw(=)i Fo(C)s Fw([)p Fo(s)552 892 y Fn(0)564 907 y Fw(].)e(W)m(e)g(sho)o(w)h Fo(s)785 913 y Fl(i)799 907 y Fo(\033)824 892 y Fn(0)849 907 y Fq(k)-30 b(!)882 892 y Fn(\003)882 923 y(R)911 913 y Fj(d)911 934 y(m)p Fd(\000)p Fe(1)976 907 y Fo(u)1000 913 y Fl(i)1024 907 y Fw(and)11 b Fo(t)1117 913 y Fl(i)1131 907 y Fo(\033)1156 892 y Fn(0)1180 907 y Fq(k)-30 b(!)1213 892 y Fn(\003)1213 923 y(R)1242 913 y Fj(d)1242 934 y(m)p Fd(\000)p Fe(1)1307 907 y Fo(u)1331 913 y Fl(i)1345 907 y Fw(.)10 b(It)h(then)h(follo)o(ws) 183 973 y(that)k Fo(t)290 979 y Fm(2)324 973 y Fw(=)g Fo(t)p Fw([)p Fo(q)h Fq( )e Fo(s)511 958 y Fn(0)523 973 y Fw(])g(=)h Fo(l)q(\033)636 958 y Fn(0)648 973 y Fq(!)690 981 y Fn(R)719 973 y Fj(d)719 989 y(m)748 973 y Fo(r)q(\033)793 958 y Fn(0)821 973 y Fw(=)g Fo(t)884 979 y Fm(3)902 973 y Fw(.)g(Since)h Fo(s)1060 979 y Fl(i)1074 973 y Fo(\033)d Fq(k)-30 b(!)1145 958 y Fn(\003)1145 987 y(R)1174 979 y Fj(d)1174 995 y(n)1195 973 y Fo(s)1214 979 y Fl(i)1229 973 y Fo(\033)1254 958 y Fn(0)1265 973 y Fw(,)16 b Fo(s)1312 979 y Fl(i)1327 973 y Fo(\033)d Fq(k)-30 b(!)1397 958 y Fn(\003)1397 989 y(R)1426 979 y Fj(d)1426 999 y(m)p Fd(\000)p Fe(1)1491 973 y Fo(u)1515 979 y Fl(i)1529 973 y Fw(,)16 b(and)183 1038 y Fo(u)207 1044 y Fl(i)233 1038 y Fw(is)c(a)g(normal)e(form,)g(it)i(follo)o(ws)f(from)g(the)i(inductiv) o(e)f(h)o(yp)q(othesis)h(that)g Fo(s)1384 1044 y Fl(i)1398 1038 y Fo(\033)1423 1023 y Fn(0)1447 1038 y Fq(k)-30 b(!)1480 1023 y Fn(\003)1480 1054 y(R)1509 1044 y Fj(d)1509 1064 y(m)p Fd(\000)p Fe(1)1574 1038 y Fo(u)1598 1044 y Fl(i)1612 1038 y Fw(.)183 1103 y(Analogously)m(,)11 b(w)o(e)j(obtain)f Fo(t)631 1109 y Fl(i)645 1103 y Fo(\033)670 1088 y Fn(0)695 1103 y Fq(k)-30 b(!)728 1088 y Fn(\003)728 1119 y(R)757 1109 y Fj(d)757 1130 y(m)p Fd(\000)p Fe(1)822 1103 y Fo(u)846 1109 y Fl(i)859 1103 y Fw(.)183 1212 y Ff(Corollary)6 b(9.)21 b Fq(!)487 1220 y Fn(R)516 1212 y Fj(d)31 b Fr(is)12 b(level-c)n(on\015uent)i(\(i.e.,)d(for)h(every)h Fo(n)e Fq(2)g Fk(I)-6 b(N)q Fr(,)12 b Fq(!)1324 1220 y Fn(R)1353 1212 y Fj(d)1353 1228 y(n)1387 1212 y Fr(is)g(c)n (on\015uent\).)183 1305 y(Pr)n(o)n(of.)20 b Fw(Immediate)d(consequence) 22 b(of)d(Lemma)d(8.)j(\(Sho)o(w)g(that)h(for)f(ev)o(ery)h(div)o (ergence)183 1355 y Fo(t)198 1361 y Fm(1)229 1339 y Fn(\003)229 1369 y(R)258 1361 y Fj(d)258 1377 y(n)277 1355 y Fq( )14 b Fo(t)p Fq(!)390 1339 y Fn(\003)390 1369 y(R)419 1361 y Fj(d)419 1377 y(n)440 1355 y Fo(t)455 1361 y Fm(2)488 1355 y Fw(of)f(length)h Fo(k)q Fw(,)f(there)i(is)f(a)g(v)n(alley)e Fo(t)1025 1361 y Fm(1)1056 1355 y Fq(k)-30 b(!)1089 1339 y Fn(\003)1089 1369 y(R)1118 1361 y Fj(d)1118 1377 y(n)1139 1355 y Fo(t)1154 1361 y Fm(3)1187 1339 y Fn(\003)1187 1369 y(R)1216 1361 y Fj(d)1216 1377 y(n)1236 1355 y Fq( )g(k)13 b Fo(t)1297 1361 y Fm(2)1329 1355 y Fw(of)g(length)h Fo(k)q Fw(.\))183 1453 y Ff(Theorem)6 b(10.)21 b Fr(Every)15 b(functional)g(CTRS)g Fq(R)g Fr(is)g(level-c)n(on\015uent.)183 1546 y(Pr)n(o)n(of.)20 b Fw(It)13 b(follo)o(ws)g(from)e Fq(!)633 1555 y Fn(R)662 1546 y Fj(d)662 1563 y(n)695 1546 y Fq(\022)h(!)781 1552 y Fn(R)810 1556 y Fj(n)845 1546 y Fw(and)i(Prop)q(osition)f(11)g(that)h Fq(!)1336 1552 y Fn(R)1365 1556 y Fj(n)1400 1546 y Fw(is)g(a)f(compat-)183 1596 y(ible)g(re\014nemen)o(t)h(of)f Fq(!)554 1605 y Fn(R)583 1596 y Fj(d)583 1613 y(n)604 1596 y Fw(.)g(Hence,)i(b)o(y)e (the)h(aforemen)o(tioned)e(result)j(of)e(Staples)h([Sta75)o(],)183 1646 y Fq(!)225 1652 y Fn(R)254 1656 y Fj(n)289 1646 y Fw(is)g(con\015uen)o(t)g(if)f(and)h(only)f(if)g Fq(!)799 1654 y Fn(R)828 1646 y Fj(d)828 1663 y(n)864 1646 y Fw(is)g(con\015uen) o(t.)183 1739 y(As)f(a)g(matter)f(of)g(fact,)g(Theorem)h(10)f(is)h(a)g (sp)q(ecial)g(case)h(of)e(a)g(theorem)h(sho)o(wn)g(in)f([SMI95].)183 1789 y(The)i(pro)q(of)g(tec)o(hniques,)h(ho)o(w)o(ev)o(er,)f(are)h (di\013eren)o(t.)g(Suzuki)f Fr(et)h(al.)f Fw([SMI95)o(])g(sho)o(w)o(ed) g(their)183 1839 y(result)h(b)o(y)g(using)g(an)f(extended)j(parallel)c (rewriting)i(relation.)183 1923 y Ff(Prop)q(ositi)o(on)5 b(11.)21 b Fr(If)14 b Fo(s)p Fq(!)619 1908 y Fn(\003)619 1935 y(R)648 1939 y Fj(n)685 1923 y Fo(t)p Fr(,)g(then)i(ther)n(e)e(is) h(a)g(term)f Fo(u)h Fr(s.t.)f Fo(s)p Fq(!)1274 1908 y Fn(\003)1274 1938 y(R)1303 1929 y Fj(d)1303 1946 y(n)1325 1923 y Fo(u)g Fr(and)i Fo(t)p Fq(!)1501 1908 y Fn(\003)1501 1938 y(R)1530 1929 y Fj(d)1530 1946 y(n)1551 1923 y Fo(u)p Fr(.)183 2023 y(Pr)n(o)n(of.)k Fw(W)m(e)10 b(pro)q(ceed)j(b)o(y)e (induction)g(on)g(the)g(depth)h Fo(n)f Fw(of)g Fo(s)p Fq(!)1149 2008 y Fn(\003)1149 2034 y(R)1193 2023 y Fo(t)p Fw(.)g(The)g(prop)q(osition)g(holds)183 2072 y(v)n(acuously)g(for)g Fo(n)g Fw(=)h(0.)f(So)g(let)g Fo(n)h(>)g Fw(0.)e(W)m(e)i(further)g(pro) q(ceed)h(b)o(y)e(induction)g(on)h(the)g(length)183 2122 y Fo(k)19 b Fw(of)f(the)h(reduction)g(sequence)i Fo(s)p Fq(!)779 2107 y Fn(\003)779 2134 y(R)808 2138 y Fj(n)843 2122 y Fo(t)p Fw(.)d(Again,)g(the)h(case)g Fo(k)h Fw(=)g(0)e(holds)g(v) n(acuously)m(.)183 2172 y(Supp)q(ose)j(the)g(claim)d(is)j(true)g(for)f Fo(k)q Fw(.)g(In)g(order)h(to)g(sho)o(w)f(it)g(for)g Fo(k)14 b Fw(+)g(1,)20 b(w)o(e)h(consider)183 2222 y Fo(s)16 b Fw(=)h Fo(C)s Fw([)p Fo(l)q(\033)q Fw(])p Fq(!)404 2228 y Fn(R)433 2232 y Fj(n)453 2222 y Fo(C)s Fw([)p Fo(r)q(\033)q Fw(])f(=)g Fo(t)634 2207 y Fn(0)646 2222 y Fq(!)688 2207 y Fl(k)688 2233 y Fn(R)717 2237 y Fj(n)752 2222 y Fo(t)p Fw(.)g(It)h(follo)o(ws)e(from)g(the)i(inductiv)o(e)g(h)o (yp)q(othesis)g(that)183 2272 y(there)g(is)e(a)g(term)g Fo(u)495 2257 y Fn(0)522 2272 y Fw(suc)o(h)i(that)e Fo(t)724 2257 y Fn(0)736 2272 y Fq(!)778 2257 y Fn(\003)778 2286 y(R)807 2278 y Fj(d)807 2294 y(n)829 2272 y Fo(u)853 2257 y Fn(0)880 2272 y Fw(and)g Fo(t)p Fq(!)1019 2257 y Fn(\003)1019 2286 y(R)1048 2278 y Fj(d)1048 2294 y(n)1069 2272 y Fo(u)1093 2257 y Fn(0)1105 2272 y Fw(.)g(No)o(w)g(if)g Fq(E)t(V)t Fo(ar)q Fw(\()p Fo(l)d Fq(!)f Fo(r)i Fq(\()e Fo(c)p Fw(\))j(=)183 2330 y Fq(;)p Fw(,)i(then)j Fo(s)p Fq(!)392 2338 y Fn(R)421 2330 y Fj(d)421 2346 y(n)443 2330 y Fo(t)458 2315 y Fn(0)487 2330 y Fw(and)e(the)h(claim)e(follo)o (ws.)f(Supp)q(ose)k Fq(E)t(V)s Fo(ar)q Fw(\()p Fo(l)13 b Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(\))18 b Fq(6)p Fw(=)g Fq(;)f Fw(and)g(let)183 2379 y Fo(x)e Fq(2)f(E)t(V)t Fo(ar)q Fw(\()p Fo(l)f Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(\).)16 b(Then)g Fo(x)f Fq(2)g(V)s Fo(ar)q Fw(\()p Fo(t)894 2385 y Fl(j)913 2379 y Fw(\))h(for)f(some)h Fo(s)1136 2385 y Fl(j)1154 2379 y Fw(==)p Fo(t)1233 2385 y Fl(j)1267 2379 y Fw(in)f Fo(c)p Fw(.)h(Since)h Fo(s)p Fq(!)1535 2385 y Fn(R)1564 2389 y Fj(n)1585 2379 y Fo(t)1600 2364 y Fn(0)1612 2379 y Fw(,)183 2429 y(there)12 b(is)f(a)f(ground)h (constructor)i(term)d Fo(u)831 2435 y Fl(j)859 2429 y Fw(suc)o(h)i(that)f Fo(s)1056 2435 y Fl(j)1074 2429 y Fo(\033)q Fq(!)1141 2414 y Fn(\003)1141 2441 y(R)1170 2445 y Fj(n)p Fd(\000)p Fe(1)1228 2429 y Fo(u)1252 2435 y Fl(j)1283 2414 y Fn(\003)1283 2441 y(R)1312 2445 y Fj(n)p Fd(\000)p Fe(1)1368 2429 y Fq( )j Fo(t)1439 2435 y Fl(j)1456 2429 y Fo(\033)q Fw(.)d(By)g(the)p eop %%Page: 8 8 8 7 bop 340 194 a Fw(inductiv)o(e)17 b(h)o(yp)q(othesis)g(on)f Fo(n)h Fw(and)f(the)h(fact)f(that)h Fo(u)1190 200 y Fl(j)1224 194 y Fw(is)f(a)g(normal)f(form,)f(w)o(e)j(conclude)340 244 y Fo(s)359 250 y Fl(j)377 244 y Fo(\033)q Fq(!)444 228 y Fn(\003)444 260 y(R)473 249 y Fj(d)473 270 y(n)p Fd(\000)p Fe(1)532 244 y Fo(u)556 250 y Fl(j)599 228 y Fn(\003)599 260 y(R)628 249 y Fj(d)628 270 y(n)p Fd(\000)p Fe(1)657 244 y Fq( )c Fo(t)727 250 y Fl(j)745 244 y Fo(\033)q Fw(.)c(It)g(follo)o(ws)f(that)i Fo(x\033)q Fq(!)1140 228 y Fn(\003)1140 260 y(R)1169 249 y Fj(d)1169 270 y(n)p Fd(\000)p Fe(1)1227 244 y Fo(u)1251 250 y Fl(x)1281 244 y Fw(for)f(some)f(ground)i(constructor)340 306 y(subterm)i Fo(u)524 312 y Fl(x)556 306 y Fw(of)f Fo(u)625 312 y Fl(j)654 306 y Fw(b)q(ecause)i Fo(u)829 312 y Fl(j)858 306 y Fw(is)e(a)g(ground)h(constructor)h(term)e(and)g Fo(t)1475 312 y Fl(j)1504 306 y Fw(is)g(a)h(constructor)340 356 y(term.)d(Note)i(that)f Fo(u)654 362 y Fl(x)684 356 y Fw(is)g(unique)g(b)q(ecause)i Fq(!)1045 366 y Fn(R)1074 356 y Fj(d)1074 377 y(n)p Fd(\000)p Fe(1)1142 356 y Fw(is)e(con\015uen) o(t)h(\()p Fq(!)1413 365 y Fn(R)1442 356 y Fj(d)27 b Fw(is)10 b(lev)o(el-con\015uen)o(t)340 416 y(b)o(y)19 b(Corollary)f(9\).)h(De\014ne)h Fo(\033)819 401 y Fn(0)850 416 y Fw(b)o(y)f Fo(x\033)962 401 y Fn(0)994 416 y Fw(=)i Fo(u)1071 422 y Fl(x)1110 416 y Fw(for)e(ev)o(ery)h Fo(x)g Fq(2)h(E)t(V)s Fo(ar)q Fw(\()p Fo(l)13 b Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(\))19 b(and)340 465 y Fo(y)q(\033)386 450 y Fn(0)419 465 y Fw(=)g Fo(y)q(\033)i Fw(otherwise.)e(Observ)o(e)h (that)f Fo(z)r(\033)q Fq(!)1084 450 y Fn(\003)1084 481 y(R)1113 471 y Fj(d)1113 492 y(n)p Fd(\000)p Fe(1)1171 465 y Fo(z)r(\033)1217 450 y Fn(0)1248 465 y Fw(for)f(ev)o(ery)i(v)n (ariable)d Fo(z)22 b Fq(2)d(D)q Fw(\()p Fo(\033)q Fw(\))q(.)340 531 y(Let)c Fo(s)434 516 y Fn(0)459 531 y Fw(=)d Fo(C)s Fw([)p Fo(r)q(\033)593 516 y Fn(0)604 531 y Fw(].)h(According)i(to)f (the)h(ab)q(o)o(v)o(e,)f Fo(t)1107 516 y Fn(0)1118 531 y Fq(!)1160 516 y Fn(\003)1160 547 y(R)1189 537 y Fj(d)1189 557 y(n)p Fd(\000)p Fe(1)1247 531 y Fo(s)1266 516 y Fn(0)1279 531 y Fw(.)f(Observ)o(e)j(that)f(also)e Fo(s)p Fq(!)1699 539 y Fn(R)1728 531 y Fj(d)1728 547 y(n)1750 531 y Fo(s)1769 516 y Fn(0)340 596 y Fw(b)q(ecause)18 b Fo(s)515 602 y Fl(j)533 596 y Fo(\033)558 581 y Fn(0)569 596 y Fq(!)611 581 y Fn(\003)611 612 y(R)640 602 y Fj(d)640 622 y(n)p Fd(\000)p Fe(1)698 596 y Fo(u)722 602 y Fl(j)766 581 y Fn(\003)766 612 y(R)795 602 y Fj(d)795 622 y(n)p Fd(\000)p Fe(1)824 596 y Fq( )13 b Fo(t)894 602 y Fl(j)911 596 y Fo(\033)936 581 y Fn(0)964 596 y Fw(for)i(ev)o(ery)h Fo(s)1159 602 y Fl(j)1177 596 y Fw(==)q Fo(t)1257 602 y Fl(j)1290 596 y Fw(in)f Fo(c)g Fw(\(it)g(is)h(a)f(consequence)j(of) 340 661 y Fo(s)359 667 y Fl(j)377 661 y Fo(\033)q Fq(!)444 646 y Fn(\003)444 677 y(R)473 667 y Fj(d)473 688 y(n)p Fd(\000)p Fe(1)532 661 y Fo(u)556 667 y Fl(j)573 661 y Fw(,)e Fo(s)620 667 y Fl(j)638 661 y Fo(\033)q Fq(!)705 646 y Fn(\003)705 677 y(R)734 667 y Fj(d)734 688 y(n)p Fd(\000)p Fe(1)793 661 y Fo(s)812 667 y Fl(j)830 661 y Fo(\033)855 646 y Fn(0)884 661 y Fw(and)h(con\015uence)h(of)f Fq(!)1264 671 y Fn(R)1293 661 y Fj(d)1293 682 y(n)p Fd(\000)p Fe(1)1368 661 y Fw(that)g Fo(s)1480 667 y Fl(j)1498 661 y Fo(\033)1523 646 y Fn(0)1535 661 y Fq(!)1577 646 y Fn(\003)1577 677 y(R)1606 667 y Fj(d)1606 688 y(n)p Fd(\000)p Fe(1)1664 661 y Fo(u)1688 667 y Fl(j)1705 661 y Fw(\).)g(It)340 727 y(no)o(w)10 b(follo)o(ws)f(from)g(con\015uence)j(of)e Fq(!)934 735 y Fn(R)963 727 y Fj(d)963 743 y(n)995 727 y Fw(in)g(conjunction)g(with)g Fo(t)1367 712 y Fn(0)1379 727 y Fq(!)1421 712 y Fn(\003)1421 743 y(R)1450 733 y Fj(d)1450 753 y(n)p Fd(\000)p Fe(1)1508 727 y Fo(s)1527 712 y Fn(0)1550 727 y Fw(and)g Fo(t)1642 712 y Fn(0)1653 727 y Fq(!)1695 712 y Fn(\003)1695 741 y(R)1724 733 y Fj(d)1724 749 y(n)1746 727 y Fo(u)1770 712 y Fn(0)340 792 y Fw(that)g Fo(s)445 777 y Fn(0)467 792 y Fw(and)g Fo(u)568 777 y Fn(0)589 792 y Fw(ha)o(v)o(e)g(a)g(common)d(reduct)k Fo(u)f Fw(w.r.t.)f Fq(!)1182 800 y Fn(R)1211 792 y Fj(d)1211 809 y(n)1232 792 y Fw(.)g(Clearly)m(,)g Fo(u)g Fw(is)h(a)g(common)d (reduct)340 842 y(of)14 b Fo(s)g Fw(and)g Fo(t)g Fw(w.r.t.)e Fq(!)683 850 y Fn(R)712 842 y Fj(d)712 858 y(n)748 842 y Fw(as)i(w)o(ell.)340 936 y Ff(Lemma)8 b(12.)20 b Fr(L)n(et)e Fq(R)g Fr(b)n(e)f(a)h(functional)g(CTRS.)g(Then,)f(for)g(every)h Fo(n)e Fq(\025)h Fw(0)p Fr(,)g(the)h(sets)f(of)340 986 y(normal)e(forms)f Fo(N)5 b(F)h Fw(\()p Fq(!)729 992 y Fn(R)758 996 y Fj(n)779 986 y Fw(\))15 b Fr(and)h Fo(N)5 b(F)h Fw(\()p Fq(!)1020 994 y Fn(R)1049 986 y Fj(d)1049 1002 y(n)1070 986 y Fw(\))15 b Fr(c)n(oincide.)340 1080 y(Pr)n(o)n(of.)20 b Fw(Ob)o(viously)m(,)e Fo(N)5 b(F)h Fw(\()p Fq(!)808 1086 y Fn(R)837 1090 y Fj(n)858 1080 y Fw(\))20 b Fq(\022)g Fo(N)5 b(F)h Fw(\()p Fq(!)1075 1088 y Fn(R)1104 1080 y Fj(d)1104 1096 y(n)1125 1080 y Fw(\))19 b(b)q(ecause)i Fq(!)1361 1088 y Fn(R)1390 1080 y Fj(d)1390 1096 y(n)1431 1080 y Fq(\022)g(!)1526 1086 y Fn(R)1555 1090 y Fj(n)1576 1080 y Fw(.)e(W)m(e)f(pro)o(v)o(e)340 1129 y Fo(N)5 b(F)h Fw(\()p Fq(!)469 1138 y Fn(R)498 1130 y Fj(d)498 1146 y(n)519 1129 y Fw(\))19 b Fq(\022)f Fo(N)5 b(F)h Fw(\()p Fq(!)733 1135 y Fn(R)762 1139 y Fj(n)783 1129 y Fw(\))18 b(indirectly)m(.)f(T)m(o)g(this)h(end,)f(supp) q(ose)j(there)f(is)e(a)h(term)f Fo(s)i Fq(2)340 1179 y Fo(N)5 b(F)h Fw(\()p Fq(!)469 1188 y Fn(R)498 1179 y Fj(d)498 1196 y(n)519 1179 y Fw(\))23 b(but)f Fo(s)k Fq(62)f Fo(N)5 b(F)h Fw(\()p Fq(!)869 1185 y Fn(R)898 1189 y Fj(n)919 1179 y Fw(\).)22 b(Since)h Fo(s)f Fw(is)g(not)h(a)f (normal)e(form)g(w.r.t.)h Fq(!)1719 1185 y Fn(R)1748 1189 y Fj(n)1770 1179 y Fw(,)340 1229 y(there)f(is)d(a)h(rule)g Fo(l)13 b Fq(!)e Fo(r)h Fq(\()f Fo(c)18 b Fq(2)g(R)p Fw(,)f(a)h(con)o(text)h Fo(C)s Fw([)11 b(])17 b(and)h(a)g(substitution) g Fo(\033)h Fw(suc)o(h)f(that)340 1279 y Fo(s)j Fw(=)g Fo(C)s Fw([)p Fo(l)q(\033)q Fw(])p Fq(!)570 1285 y Fn(R)599 1289 y Fj(n)619 1279 y Fo(C)s Fw([)p Fo(r)q(\033)q Fw(].)d(In)h (particular,)g(for)f(ev)o(ery)i Fo(s)1219 1285 y Fl(j)1237 1279 y Fw(==)q Fo(t)1317 1285 y Fl(j)1354 1279 y Fw(in)e Fo(c)p Fw(,)h(there)h(is)f(a)g(ground)340 1329 y(constructor)k(term)e Fo(u)698 1335 y Fl(j)737 1329 y Fw(suc)o(h)h(that)f Fo(s)954 1335 y Fl(j)972 1329 y Fo(\033)q Fq(!)1039 1314 y Fn(\003)1039 1340 y(R)1068 1344 y Fj(n)p Fd(\000)p Fe(1)1127 1329 y Fo(u)1151 1335 y Fl(j)1182 1314 y Fn(\003)1182 1340 y(R)1211 1344 y Fj(n)p Fd(\000)p Fe(1)1267 1329 y Fq( )13 b Fo(t)1337 1335 y Fl(j)1355 1329 y Fo(\033)q Fw(.)21 b(It)g(follo)o(ws)f(as)i(in)f(the)340 1383 y(pro)q(of)h(of)g(Prop)q (osition)g(11)g(that)h Fo(s)927 1389 y Fl(j)944 1383 y Fo(\033)969 1368 y Fn(0)981 1383 y Fq(!)1023 1368 y Fn(\003)1023 1399 y(R)1052 1389 y Fj(d)1052 1410 y(n)p Fd(\000)p Fe(1)1110 1383 y Fo(u)1134 1389 y Fl(j)1178 1368 y Fn(\003)1178 1399 y(R)1207 1389 y Fj(d)1207 1410 y(n)p Fd(\000)p Fe(1)1236 1383 y Fq( )13 b Fo(t)1306 1389 y Fl(j)1323 1383 y Fo(\033)1348 1368 y Fn(0)1360 1383 y Fw(.)22 b(Hence)i Fo(s)i Fw(=)g Fo(C)s Fw([)p Fo(l)q(\033)q Fw(])f(=)340 1448 y Fo(C)s Fw([)p Fo(l)q(\033)423 1433 y Fn(0)435 1448 y Fw(])p Fq(!)489 1457 y Fn(R)518 1449 y Fj(d)518 1465 y(n)539 1448 y Fo(C)s Fw([)p Fo(r)q(\033)629 1433 y Fn(0)640 1448 y Fw(].)12 b(This)i(is)g(a)g(con)o(tradiction)f (to)h Fo(s)e Fq(2)f Fo(N)5 b(F)h Fw(\()p Fq(!)1350 1457 y Fn(R)1379 1449 y Fj(d)1379 1465 y(n)1400 1448 y Fw(\).)340 1542 y(T)m(erm)14 b(rewriting)h(is)g(mainly)d(concerned)17 b(with)d(computing)g(normal)f(forms.)g(Our)i(results)340 1592 y(indicate)e(that,)f(from)f(a)h(computational)f(p)q(oin)o(t)h(of)g (view,)g(it)g(is)g(more)g(reasonable)h(to)g(w)o(ork)340 1642 y(with)e(the)g(deterministic)g(rewrite)h(relation)e(rather)i(than) f(the)g(ordinary)g(rewrite)g(relation.)340 1780 y Fs(4)56 b(Conditional)19 b(T)-5 b(erm)17 b(Graph)i(Rewriting)340 1883 y Fw(In)10 b(this)g(section,)f(w)o(e)h(use)g(the)h(term)e(based)h (approac)o(h)f(of)g([K)o(O95])g(to)g(term)g(graph)g(rewriting)340 1933 y(rather)19 b(than)e(those)h(of)e([BEG)833 1918 y Fm(+)861 1933 y Fw(87)o(])h(or)g([Plu93)o(])g(\(although)f(the)i (other)g(approac)o(hes)g(are)340 1983 y(more)11 b(graphic\).)g(Doing)g (so,)g(it)h(is)f(p)q(ossible)h(to)g(completely)e(argue)i(within)f(the)i (framew)o(ork)340 2033 y(of)g(term)g(rewriting)g(and)g(to)h(a)o(v)o (oid)e(concepts)j(from)d(di\013eren)o(t)i(\014elds.)f(Note)h(that)f (the)h(\014rst)340 2082 y(part)g(of)g(Section)g(4)g(is)f(a)h(mirror)f (image)f(of)h(Section)h(3,)f(where)i(graph)f(rewriting)g(replaces)340 2132 y(term)g(rewriting.)f(W)m(e)g(\014rst)i(recapitulate)g(some)e (basic)h(notions)f(from)f([K)o(O95].)403 2183 y(Let)f Fo(M)16 b Fw(b)q(e)c(a)e(coun)o(tably)h(in\014nite)g(set)h(of)e(ob)r (jects)i(called)f Fr(marks)p Fw(.)f(Let)i Fq(F)1539 2165 y Fn(\003)1569 2183 y Fw(=)g Fq(f)p Fo(f)1658 2168 y Fl(\026)1693 2183 y Fq(j)g Fo(f)17 b Fq(2)340 2232 y(F)t Fo(;)7 b(\026)26 b Fq(2)f Fo(M)5 b Fq(g)22 b Fw(b)q(e)h(the)g(set)g(of) f Fr(marke)n(d)h Fw(function)f(sym)o(b)q(ols.)e(F)m(or)i(all)f Fo(f)1534 2217 y Fl(\026)1583 2232 y Fq(2)26 b(F)1671 2214 y Fn(\003)1690 2232 y Fw(,)c(the)340 2282 y(arit)o(y)g(of)f Fo(f)527 2267 y Fl(\026)571 2282 y Fw(coincides)i(with)e(that)h(of)f Fo(f)t Fw(.)h(Moreo)o(v)o(er,)g(w)o(e)g(de\014ne)h Fo(sy)q(mbol)q Fw(\()p Fo(f)1634 2267 y Fl(\026)1659 2282 y Fw(\))i(=)g Fo(f)340 2332 y Fw(and)18 b Fo(mar)q(k)q Fw(\()p Fo(f)566 2317 y Fl(\026)590 2332 y Fw(\))h(=)g Fo(\026)p Fw(.)e(In)h(the)h (sequel)g(w)o(e)g(use)g(natural)e(n)o(um)o(b)q(ers)h(as)g(marks.)f(The) i(el-)340 2382 y(emen)o(ts)c(of)g Fq(T)563 2364 y Fn(\003)596 2382 y Fw(=)e Fq(T)e Fw(\()p Fq(F)725 2367 y Fn(\003)744 2382 y Fo(;)c Fq(V)s Fw(\))15 b(are)h(called)f Fr(marke)n(d)f Fw(terms.)h(Note)g(that)h(v)n(ariables)e(are)h(not)340 2432 y(mark)o(ed.)d(The)i(set)h Fq(T)682 2417 y Fn(\003)672 2442 y Fl(w)714 2432 y Fw(of)e Fr(wel)r(l-marke)n(d)g(terms)g Fw(o)o(v)o(er)h Fq(F)1228 2414 y Fn(\003)1260 2432 y Fw(is)f(the)i(subset)f(of)f Fq(T)1579 2414 y Fn(\003)1612 2432 y Fw(suc)o(h)h(that)p eop %%Page: 9 9 9 8 bop 183 194 a Fo(t)14 b Fq(2)g(T)287 179 y Fn(\003)277 204 y Fl(w)321 194 y Fw(if)h(and)g(only)g(if,)f(for)h(ev)o(ery)h(pair)f (\()p Fo(t)881 200 y Fm(1)900 194 y Fo(;)7 b(t)934 200 y Fm(2)952 194 y Fw(\))15 b(of)g(subterms)h(of)f Fo(t)p Fw(,)g Fo(mar)q(k)q Fw(\()p Fo(r)q(oot)p Fw(\()p Fo(t)1526 200 y Fm(1)1545 194 y Fw(\)\))f(=)183 244 y Fo(mar)q(k)q Fw(\()p Fo(r)q(oot)p Fw(\()p Fo(t)406 250 y Fm(2)425 244 y Fw(\)\))20 b(implies)f Fo(t)640 250 y Fm(1)681 244 y Fw(=)j Fo(t)750 250 y Fm(2)769 244 y Fw(.)d(F)m(or)h(example,)f (the)i(term)f Fo(pl)q(us)1323 228 y Fm(0)1342 244 y Fw(\(0)1379 228 y Fm(1)1397 244 y Fo(;)7 b Fw(0)1437 228 y Fm(1)1455 244 y Fw(\))20 b(is)g(w)o(ell-)183 293 y(mark)o(ed)c(but)i Fo(pl)q(us)491 278 y Fm(1)510 293 y Fw(\(0)547 278 y Fm(1)566 293 y Fo(;)7 b Fw(0)606 278 y Fm(1)624 293 y Fw(\))18 b(is)f(not.)g(W)m(ell-mark)o(ed)f(terms)h(exactly)h(corresp)q (ond)i(to)d(di-)183 343 y(rected)i(acyclic)e(graphs;)h(the)g(reader)h (is)e(referred)i(to)f([K)o(O95)o(])f(for)g(details.)g(In)h(con)o(trast) 183 393 y(to)c([K)o(O95)o(],)f(w)o(e)i(are)f(solely)g(in)o(terested)i 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Fw(b)o(y)h(erasing)g(all)e(marks.)g(Tw)o(o)h(mark)o (ed)g(terms)g Fo(s)h Fw(and)g Fo(t)f Fw(are)h(called)g Fr(e)n(quivalent)g Fw(\()p Fo(s)f Fq(\030)g Fo(t)p Fw(\))g(if)183 734 y(and)d(only)g(if)h Fo(s)g Fw(and)g Fo(t)f Fw(are)i(isomorphic)d (as)i(dags;)f(to)h(b)q(e)g(precise:)h(\(i\))f Fo(e)p Fw(\()p Fo(s)p Fw(\))j(=)f Fo(e)p Fw(\()p Fo(t)p Fw(\))e(and)g(\(ii\))f (for)183 784 y(ev)o(ery)16 b(pair)e(of)h(marks)f Fo(\026)h Fw(and)g Fo(\027)i Fw(for)e(whic)o(h)g(there)i(is)e(a)g(p)q(osition)f Fo(p)g Fq(2)f(P)s Fo(os)p Fw(\()p Fo(s)p Fw(\))i(=)f Fq(P)s Fo(os)p Fw(\()p Fo(t)p Fw(\))183 833 y(with)j Fo(mar)q(k)q Fw(\()p Fo(s)p Fq(j)429 839 y Fl(p)449 833 y Fw(\))h(=)g Fo(\026)g Fw(and)g Fo(mar)q(k)q Fw(\()p Fo(t)p Fq(j)805 839 y Fl(p)824 833 y Fw(\))g(=)g Fo(\027)s Fw(,)f(w)o(e)h(ha)o(v)o(e)g Fq(f)p Fo(q)h Fq(2)e(P)s Fo(os)p Fw(\()p Fo(s)p Fw(\))d Fq(j)e Fo(mar)q(k)q Fw(\()p Fo(s)p Fq(j)1538 839 y Fl(q)1557 833 y Fw(\))18 b(=)183 883 y Fo(\026)p Fq(g)e Fw(=)h Fq(f)p Fo(q)h Fq(2)e(P)s Fo(os)p Fw(\()p Fo(t)p Fw(\))e Fq(j)e Fo(mar)q(k)q Fw(\()p Fo(t)p Fq(j)696 889 y Fl(q)714 883 y Fw(\))17 b(=)g Fo(\027)s Fq(g)p Fw(.)e(Tw)o(o)h(mark)o(ed)g(substitutions)i Fo(\033)1390 889 y Fm(1)1425 883 y Fw(and)f Fo(\033)1533 889 y Fm(2)1568 883 y Fw(are)183 933 y Fr(e)n(quivalent)d Fw(if)f Fo(x\033)461 939 y Fm(1)490 933 y Fq(\030)f Fo(x\033)582 939 y Fm(2)614 933 y Fw(for)i(ev)o(ery)g(v)n(ariable)f Fo(x)p Fw(.)183 1023 y(F)m(or)h(instance,)h Fo(pl)q(us)509 1008 y Fm(0)528 1023 y Fw(\(0)565 1008 y Fm(1)584 1023 y Fo(;)7 b Fw(0)624 1008 y Fm(1)641 1023 y Fw(\))13 b Fq(\030)h Fo(pl)q(us)793 1008 y Fm(0)812 1023 y Fw(\(0)849 1008 y Fm(2)867 1023 y Fo(;)7 b Fw(0)907 1008 y Fm(2)925 1023 y Fw(\))15 b(but)g Fo(pl)q(us)1110 1008 y Fm(0)1129 1023 y Fw(\(0)1166 1008 y Fm(1)1184 1023 y Fo(;)7 b Fw(0)1224 1008 y Fm(1)1242 1023 y Fw(\))13 b Fq(6\030)h Fo(pl)q(us)1394 1008 y Fm(0)1413 1023 y Fw(\(0)1450 1008 y Fm(1)1468 1023 y Fo(;)7 b Fw(0)1508 1008 y Fm(2)1526 1023 y Fw(\).)14 b(On)183 1073 y(the)e(other)g(hand,)f Fo(e)p Fw(\()p Fo(pl)q(us)583 1057 y Fm(0)603 1073 y Fw(\(0)640 1057 y Fm(1)658 1073 y Fo(;)c Fw(0)698 1057 y Fm(1)716 1073 y Fw(\)\))12 b(=)g Fo(e)p Fw(\()p Fo(pl)q(us)916 1057 y Fm(0)935 1073 y Fw(\(0)972 1057 y Fm(1)991 1073 y Fo(;)7 b Fw(0)1031 1057 y Fm(2)1049 1073 y Fw(\)\).)k(Observ)o(e)i (that)f(the)g(equiv)n(alence)183 1122 y(relation)h Fq(\030)i Fw(de\014ned)g(ab)q(o)o(v)o(e)f(di\013ers)h(from)d(a)i(similar)e(one)i (in)g([K)o(O95])f(whic)o(h)h(requires)i(\(i\))183 1172 y(only)m(.)11 b(In)i(the)h(sequel,)g(the)g(marks)e(of)g(a)h(term)g Fo(s)h Fw(are)f(called)g Fr(fr)n(esh)g Fw(w.r.t.)f(another)i(mark)o(ed) 183 1222 y(term)f Fo(t)h Fw(if)f(no)g(marks)g(o)q(ccurring)i(in)e Fo(t)h Fw(do)g(also)f(o)q(ccur)i(in)e Fo(s)p Fw(.)183 1314 y Ff(De\014niti)o(on)5 b(14.)20 b Fw(Let)13 b Fq(R)g Fw(b)q(e)f(a)g(left-linear)g(3-CTRS.)f(A)h(rule)h Fo(l)1184 1299 y Fn(\003)1215 1314 y Fq(!)e Fo(r)1288 1299 y Fn(\003)1318 1314 y Fq(\()g Fo(c)1389 1299 y Fn(\003)1420 1314 y Fw(is)i(a)f Fr(marke)n(d)183 1364 y(version)h Fw(of)h(a)f(rule)h Fo(l)f Fq(!)e Fo(r)i Fq(\()e Fo(c)i Fw(in)h Fq(R)g Fw(if)f Fo(e)p Fw(\()p Fo(l)869 1349 y Fn(\003)889 1364 y Fw(\))e(=)h Fo(l)q Fw(,)i Fo(e)p Fw(\()p Fo(r)1054 1349 y Fn(\003)1073 1364 y Fw(\))e(=)g Fo(r)q Fw(,)h(and)h Fo(e)p Fw(\()p Fo(c)1324 1349 y Fn(\003)1343 1364 y Fw(\))e(=)g Fo(c)p Fw(.)183 1453 y(Since)k(w)o(e)f(are)h(solely)f(in)o(terested)j(in)d (left-linear)g(CTRSs,)g(w)o(e)g(extend)i(the)f(de\014nition)f(of)183 1503 y([BEG)285 1488 y Fm(+)312 1503 y Fw(87)o(])g(for)h(left-linear)e (unconditional)h(term)g(graph)g(rewriting)g(to)h(the)g(conditional)183 1553 y(case.)e(As)h(a)f(tec)o(hnical)g(consequence,)i(w)o(e)e(don't)g (ha)o(v)o(e)g(to)g(mark)e(v)n(ariables.)h(So)h(v)n(ariables)183 1603 y(are)g(maxim)o(all)o(y)d(shared.)k(On)f(the)h(other)f(hand,)g(b)o (y)f(using)h(fresh)h(and)f(m)o(utually)d(distinct)183 1652 y(marks)e(for)i(the)g(righ)o(t-hand)f(side)h(and)g(the)g (conditional)e(part)i(of)f(a)h(rewrite)g(rule,)g(w)o(e)g(adopt)183 1702 y(a)k("minima)o(l)e(structure)18 b(sharing)e(sc)o(heme")g (\(di\013eren)o(t)h(structure)i(sharing)c(sc)o(hemes)i(are)183 1752 y(discussed)e(in)f([K)o(O95)o(]\).)f(In)h(our)g(opinion,)e(these)j (c)o(hoices)g(are)f(v)o(ery)h(natural.)183 1844 y Ff(De\014niti)o(on)5 b(15.)20 b Fw(Let)e Fq(R)f Fw(b)q(e)h(a)f(left-linear)f(3-CTRS)g(with)h (strict)h(equalit)o(y)m(.)e(Let)h Fo(s)h Fw(and)183 1894 y Fo(t)d Fw(b)q(e)i(mark)o(ed)e(terms.)g(Let)h Fq(\))669 1900 y Fn(R)698 1904 y Fe(0)715 1894 y Fw(=)g Fq(;)f Fw(and)h(for)f Fo(n)g(>)g Fw(0,)g(de\014ne)i Fo(s)e Fq(\))1280 1900 y Fn(R)1309 1904 y Fj(n)1344 1894 y Fo(t)h Fw(if)f(there)i(exists) 183 1944 y(a)g(mark)o(ed)e(v)o(ersion)j Fo(l)529 1928 y Fn(\003)560 1944 y Fq(!)11 b Fo(r)633 1928 y Fn(\003)663 1944 y Fq(\()g Fo(c)734 1928 y Fn(\003)770 1944 y Fw(of)17 b(a)g(rewrite)h(rule)f Fo(l)c Fq(!)e Fo(r)i Fq(\()e Fo(c)17 b Fw(from)e Fq(R)p Fw(,)i(a)g(mark)o(ed)183 1993 y(substitution)d Fo(\033)h Fw(and)e(a)h(mark)o(ed)f(con)o(text)h Fo(C)s Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(])13 b(suc)o(h)i(that)209 2085 y Ff({)20 b Fo(s)12 b Fw(=)g Fo(C)s Fw([)p Fo(l)386 2070 y Fn(\003)404 2085 y Fo(\033)o(;)7 b(:)g(:)g(:)e(;)i(l)533 2070 y Fn(\003)552 2085 y Fo(\033)q Fw(])13 b(and)h Fo(t)e Fw(=)g Fo(C)s Fw([)p Fo(r)819 2070 y Fn(\003)837 2085 y Fo(\033)o(;)7 b(:)g(:)g(:)t(;)g(r)972 2070 y Fn(\003)991 2085 y Fo(\033)q Fw(],)209 2137 y Ff({)20 b Fo(l)266 2122 y Fn(\003)286 2137 y Fo(\033)15 b Fw(is)e(not)h(a)g(subterm)f(of)h Fo(C)s Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(],)209 2189 y Ff({)20 b Fw(for)14 b(ev)o(ery)h Fo(s)446 2195 y Fl(i)461 2189 y Fw(==)p Fo(t)540 2195 y Fl(i)569 2189 y Fw(in)e Fo(c)635 2174 y Fn(\003)655 2189 y Fw(,)g(w)o(e)i(ha)o(v)o(e)f(\()p Fo(s)873 2195 y Fl(i)888 2189 y Fo(\033)q Fw(==)q Fo(t)993 2195 y Fl(i)1006 2189 y Fo(\033)q Fw(\))f Fq(\))1102 2174 y Fn(\003)1102 2201 y(R)1131 2205 y Fj(n)p Fd(\000)p Fe(1)1187 2201 y Fn([R)1238 2205 y Fj(eq)1283 2189 y Fo(tr)q(ue)1361 2174 y Fn(\003)1380 2189 y Fw(,)h(where)i Fo(tr)q(ue)1605 2174 y Fn(\003)253 2239 y Fw(is)e(a)g(mark)o(ed)e(v)o (ersion)i(of)g Fo(tr)q(ue)p Fw(.)209 2291 y Ff({)20 b Fw(all)14 b(marks)g(on)h(function)f(sym)o(b)q(ols)g(in)g Fo(r)888 2276 y Fn(\003)907 2291 y Fw(,)h Fo(c)952 2276 y Fn(\003)971 2291 y Fw(,)f(and)h Fo(x\033)h Fw(\(for)f(ev)o(ery)h (extra)f(v)n(ariable)f Fo(x)253 2341 y Fw(in)g Fo(l)e Fq(!)f Fo(r)i Fq(\()e Fo(c)p Fw(\))j(are)g(m)o(utually)d(distinct)k (and)e(fresh)i(w.r.t.)e Fo(s)p Fw(.)183 2430 y(W)m(e)g(call)27 b Fq(\))386 2436 y Fn(R)442 2430 y Fw(=)485 2399 y Fp(S)520 2443 y Fl(n)p Fn(\025)p Fm(0)606 2430 y Fq(\))648 2436 y Fn(R)677 2440 y Fj(n)726 2430 y Fr(nonc)n(opying)15 b Fw(or)f Fr(gr)n(aph)g Fw(rewrite)h(relation)e(w.r.t.)g Fq(R)p Fw(.)p eop %%Page: 10 10 10 9 bop 340 194 a Fo(l)353 179 y Fn(\003)373 194 y Fo(\033)16 b Fw(is)e(called)h(the)g Fr(c)n(ontr)n(acte)n(d)h(marke)n(d)g(r)n(e)n (dex)f Fw(in)f Fo(s)p Fw(.)h(W)m(e)f(use)i(the)f(notation)f Fo(s)h Fq(\))1653 179 y Fl(l)1663 166 y Fd(\003)1680 179 y Fl(\033)1653 205 y Fn(R)1682 209 y Fj(n)1717 194 y Fo(t)g Fw(in)340 244 y(order)f(to)e(sp)q(ecify)h(the)g(con)o(tracted) g(mark)o(ed)f(redex.)h(Note)f(that)h(all)e(shared)i(subterms)g Fo(l)1737 228 y Fn(\003)1756 244 y Fo(\033)340 293 y Fw(are)j(replaced)h(sim)o(ultaneously)c(b)o(y)j Fo(r)940 278 y Fn(\003)959 293 y Fo(\033)q Fw(.)f(Observ)o(e)i(moreo)o(v)o(er)e (that)g(whenev)o(er)i Fo(s)e Fq(\))1704 299 y Fn(R)1748 293 y Fo(t)1763 299 y Fm(1)340 343 y Fw(and)e Fo(s)h Fq(\))495 349 y Fn(R)539 343 y Fo(t)554 349 y Fm(2)585 343 y Fw(b)o(y)e(reducing)h(the)h(same)d(mark)o(ed)h(redex,)h(then)g Fo(t)1358 349 y Fm(1)1388 343 y Fq(\030)f Fo(t)1447 349 y Fm(2)1478 343 y Fw(\()p Fo(t)1509 349 y Fm(1)1539 343 y Fw(=)g Fo(t)1598 349 y Fm(2)1629 343 y Fw(do)q(es)h(not)340 393 y(hold)h(in)f(general)h(b)q(ecause)i(the)e(in)o(tro)q(duced)h (fresh)g(marks)d(ma)o(y)g(di\013er\).)403 443 y(In)i(the)g(follo)o (wing,)d Fq(R)j Fw(denotes)h(a)f(functional)e(3-CTRS)h(unless)i(stated) g(otherwise.)340 529 y Ff(Prop)q(osition)5 b(16.)20 b Fr(The)15 b(fol)r(lowing)f(statements)h(ar)n(e)g(e)n(quivalent)g(for)f (al)r(l)g Fo(s;)7 b(t)12 b Fq(2)f(T)1666 514 y Fn(\003)1656 539 y Fl(w)1685 529 y Fr(:)356 615 y(1.)21 b Fw(\()p Fo(s)p Fw(==)q Fo(t)p Fw(\))12 b Fq(\))596 599 y Fn(\003)596 626 y(R[R)676 630 y Fj(eq)719 615 y Fo(tr)q(ue)797 599 y Fn(\003)356 665 y Fr(2.)21 b(Ther)n(e)d(exist)h(marke)n(d)g(gr)n (ound)g(c)n(onstructor)g(terms)f Fo(u)h Fr(and)h Fo(v)g Fr(such)g(that)f Fo(s)c Fq(\))1700 650 y Fn(\003)1700 676 y(R)1745 665 y Fo(u)p Fr(,)411 715 y Fo(t)g Fq(\))483 700 y Fn(\003)483 726 y(R)528 715 y Fo(v)q Fr(,)g(and)h Fo(e)p Fw(\()p Fo(u)p Fw(\))11 b(=)h Fo(e)p Fw(\()p Fo(v)q Fw(\))p Fr(.)873 700 y Fm(3)340 809 y Ff(De\014nition)t(17.)21 b Fw(The)d Fr(deterministic)e Fw(noncop)o(ying)h(reduction)h(relation) 31 b Fq(\))1602 818 y Fn(R)1631 809 y Fj(d)k Fw(is)17 b(de-)340 859 y(\014ned)12 b(analogous)e(to)25 b Fq(\))732 865 y Fn(R)775 859 y Fw(:)11 b(in)g(a)24 b Fq(\))931 868 y Fn(R)960 859 y Fj(d)960 876 y(n)1006 859 y Fw(rewrite)13 b(step,)e(it)g(is)g(additionally)d(required)k(that)340 909 y Fo(x\033)i Fw(is)e(a)g(mark)o(ed)f(ground)h(constructor)j(term)c (for)h(ev)o(ery)h(extra)g(v)n(ariable)f Fo(x)g Fw(in)g Fo(l)g Fq(!)f Fo(r)i Fq(\()e Fo(c)p Fw(.)340 1003 y(In)j(order)g(to)g (illustrate)f(ho)o(w)h(graph)f(rewriting)h(w)o(orks,)f(let)h Fq(R)g Fw(b)q(e)g(the)g(functional)f(CTRS)340 1053 y Fq(R)376 1059 y Fl(f)s(ib)437 1053 y Fw(from)e(Section)j(3)e(augmen)o (ted)h(b)o(y)g(the)g(rewrite)i(rules)e Fo(doubl)q(e)p Fw(\()p Fo(x)p Fw(\))f Fq(!)f Fo(pl)q(us)p Fw(\()p Fo(x;)c(x)p Fw(\))12 b(and)340 1102 y Fo(snd)p Fw(\()p Fo(pair)q Fw(\()p Fo(x;)7 b(y)q Fw(\)\))12 b Fq(!)g Fo(y)q Fw(.)i(There)h(is)f (the)28 b Fq(\))1013 1108 y Fn(R)1071 1102 y Fw(\(in)13 b(fact,)27 b Fq(\))1284 1111 y Fn(R)1313 1103 y Fj(d)1346 1102 y Fw(\))14 b(reduction)g(sequence:)346 1191 y Fo(doubl)q(e)462 1176 y Fm(0)481 1191 y Fw(\()p Fo(snd)563 1176 y Fm(1)581 1191 y Fw(\()p Fo(f)t(ib)653 1176 y Fm(2)673 1191 y Fw(\()p Fo(s)708 1176 y Fm(3)727 1191 y Fw(\(0)764 1176 y Fm(4)783 1191 y Fw(\)\)\)\))26 b Fq(\))915 1197 y Fn(R)970 1191 y Fo(pl)q(us)1047 1176 y Fm(5)1066 1191 y Fw(\()p Fo(snd)1148 1176 y Fm(1)1167 1191 y Fw(\()p Fo(f)t(ib)1239 1176 y Fm(2)1259 1191 y Fw(\()p Fo(s)1294 1176 y Fm(3)1313 1191 y Fw(\(0)1350 1176 y Fm(4)1369 1191 y Fw(\)\)\))p Fo(;)7 b(snd)1502 1176 y Fm(1)1520 1191 y Fw(\()p Fo(f)t(ib)1592 1176 y Fm(2)1612 1191 y Fw(\()p Fo(s)1647 1176 y Fm(3)1666 1191 y Fw(\(0)1703 1176 y Fm(4)1721 1191 y Fw(\)\)\)\))873 1241 y Fq(\))915 1247 y Fn(R)970 1241 y Fo(pl)q(us)1047 1226 y Fm(5)1066 1241 y Fw(\()p Fo(snd)1148 1226 y Fm(1)1167 1241 y Fw(\()p Fo(t)p Fw(\))p Fo(;)g(snd)1299 1226 y Fm(1)1317 1241 y Fw(\()p Fo(t)p Fw(\)\))873 1291 y Fq(\))915 1297 y Fn(R)970 1291 y Fo(pl)q(us)1047 1276 y Fm(5)1066 1291 y Fw(\()p Fo(pl)q(us)1159 1276 y Fm(12)1195 1291 y Fw(\(0)1232 1276 y Fm(8)1250 1291 y Fo(;)g(s)1288 1276 y Fm(9)1307 1291 y Fw(\(0)1344 1276 y Fm(10)1379 1291 y Fw(\)\))p Fo(;)g(pl)q(us)1507 1276 y Fm(12)1542 1291 y Fw(\(0)1579 1276 y Fm(8)1598 1291 y Fo(;)g(s)1636 1276 y Fm(9)1654 1291 y Fw(\(0)1691 1276 y Fm(10)1726 1291 y Fw(\)\)\))873 1340 y Fq(\))915 1346 y Fn(R)970 1340 y Fo(pl)q(us)1047 1325 y Fm(5)1066 1340 y Fw(\()p Fo(s)1101 1325 y Fm(9)1121 1340 y Fw(\(0)1158 1325 y Fm(10)1193 1340 y Fw(\))p Fo(;)g(s)1247 1325 y Fm(9)1265 1340 y Fw(\(0)1302 1325 y Fm(10)1337 1340 y Fw(\)\))340 1433 y(where)21 b Fo(t)f Fw(denotes)h(the)f(mark)o(ed)e(term)h Fo(pair)1069 1418 y Fm(11)1105 1433 y Fw(\()p Fo(s)1140 1418 y Fm(9)1159 1433 y Fw(\(0)1196 1418 y Fm(10)1231 1433 y Fw(\))p Fo(;)7 b(pl)q(us)1343 1418 y Fm(12)1378 1433 y Fw(\(0)1415 1418 y Fm(8)1433 1433 y Fo(;)g(s)1471 1418 y Fm(9)1490 1433 y Fw(\(0)1527 1418 y Fm(10)1562 1433 y Fw(\)\)\),)20 b(b)q(ecause)340 1482 y Fo(f)t(ib)396 1467 y Fm(6)416 1482 y Fw(\(0)453 1467 y Fm(4)471 1482 y Fw(\))14 b Fq(\))543 1488 y Fn(R)587 1482 y Fo(pair)664 1467 y Fm(7)683 1482 y Fw(\(0)720 1467 y Fm(8)739 1482 y Fo(;)7 b(s)777 1467 y Fm(9)795 1482 y Fw(\(0)832 1467 y Fm(10)867 1482 y Fw(\)\).)340 1568 y Ff(Lemma)h(18.)20 b Fr(L)n(et)13 b Fq(\))f Fr(denote)29 b Fq(\))895 1576 y Fn(R)924 1568 y Fj(d)924 1585 y(n)973 1568 y Fr(or)f Fq(\))1082 1574 y Fn(R)1111 1578 y Fj(n)1147 1568 y Fr(.)12 b(If)h Fo(s)f Fq(\030)g Fo(t)f Fq(\))1357 1553 y Fn(\003)1387 1568 y Fo(u)p Fr(,)i(wher)n(e)f(the)h(nonc)n(opy-)340 1618 y(ing)k(r)n(e)n(duction)g(of)f Fo(t)h Fr(to)g Fo(u)f Fr(c)n(onsists)h(of)f Fo(k)i Fr(steps,)e(then)h(ther)n(e)g(is)f(a)h (marke)n(d)g(term)f Fo(v)i Fr(such)340 1668 y(that)d Fo(s)d Fq(\))498 1653 y Fn(\003)528 1668 y Fo(v)i Fq(\030)d Fo(u)p Fr(,)k(wher)n(e)f Fo(s)h Fr(r)n(e)n(duc)n(es)g(to)g Fo(v)h Fr(in)f Fo(k)h Fr(steps.)340 1762 y(Pr)n(o)n(of.)k Fw(W)m(e)12 b(sho)o(w)f(the)i(lemma)8 b(for)j Fo(k)i Fw(=)f(1,)e(the)j(whole)e(claim)e(then)k(follo)o(ws)d(b)o(y)h (induction.)340 1812 y(Supp)q(ose)19 b Fo(s)f Fq(\030)g Fo(t)f Fq(\))668 1797 y Fl(l)678 1784 y Fd(\003)696 1797 y Fl(\033)736 1812 y Fo(u)p Fw(.)g(W)m(e)g(ma)o(y)e(write)j Fo(t)g Fw(=)g Fo(C)s Fw([)p Fo(l)1208 1797 y Fn(\003)1226 1812 y Fo(\033)o(;)7 b(:)g(:)g(:)e(;)i(l)1355 1797 y Fn(\003)1373 1812 y Fo(\033)q Fw(],)17 b(where)i Fo(l)1576 1797 y Fn(\003)1595 1812 y Fo(\033)g Fw(is)e(not)h(a)340 1862 y(subterm)c(of)e(the)i(mark)o(ed)f(con)o(text)h Fo(C)s Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(].)k(Then)j Fo(u)e Fw(=)f Fo(C)s Fw([)p Fo(r)1338 1847 y Fn(\003)1356 1862 y Fo(\033)o(;)c(:)g(:)g(:)e(;)i(r)1492 1847 y Fn(\003)1511 1862 y Fo(\033)q Fw(].)12 b(Since)i Fo(s)e Fq(\030)g Fo(t)p Fw(,)340 1912 y Fo(s)j Fw(ma)o(y)d(b)q(e)j(written)f(as)g Fo(s)f Fw(=)802 1901 y(\026)793 1912 y Fo(C)s Fw([)838 1901 y(\026)838 1912 y Fo(l)851 1896 y Fn(\003)872 1912 y Fw(\026)-24 b Fo(\033)r(;)7 b(:)g(:)g(:)t(;)987 1901 y Fw(\026)987 1912 y Fo(l)1000 1896 y Fn(\003)1021 1912 y Fw(\026)-23 b Fo(\033)q Fw(],)13 b(where)1201 1901 y(\026)1202 1912 y Fo(l)1215 1896 y Fn(\003)1236 1912 y Fw(\026)-23 b Fo(\033)15 b Fw(is)f(not)g(a)g(subterm)g(of)1643 1901 y(\026)1633 1912 y Fo(C)s Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(])340 1961 y(and)20 b Fo(C)s Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(])21 b Fq(\030)661 1951 y Fw(\026)652 1961 y Fo(C)r Fw([)p Fo(;)7 b(:)g(:)g(:)e(;)i Fw(].)18 b(Note)j(that)f Fo(C)s Fw([)p Fo(l)1092 1946 y Fn(\003)1111 1961 y Fo(\033)o(;)7 b(:)g(:)g(:)t(;)g(l)1239 1946 y Fn(\003)1258 1961 y Fo(\033)q Fw(])22 b Fq(\030)1381 1951 y Fw(\026)1371 1961 y Fo(C)s Fw([)1416 1950 y(\026)1416 1961 y Fo(l)1429 1946 y Fn(\003)1450 1961 y Fw(\026)-23 b Fo(\033)q(;)7 b(:)g(:)g(:)e(;)1566 1950 y Fw(\026)1566 1961 y Fo(l)1579 1946 y Fn(\003)1600 1961 y Fw(\026)-24 b Fo(\033)r Fw(])19 b(implies)340 2011 y Fo(e)p Fw(\()p Fo(x\033)q Fw(\))c Fq(\030)g Fo(e)p Fw(\()p Fo(x)r Fw(\026)-23 b Fo(\033)q Fw(\))16 b(as)g(w)o(ell)e(as)i Fo(x\033)f Fw(=)g Fo(y)q(\033)j Fw(if)c(and)i(only)f(if)f Fo(x)r Fw(\026)-23 b Fo(\033)16 b Fw(=)e Fo(y)t Fw(\026)-24 b Fo(\033)18 b Fw(for)d(all)f Fo(x;)7 b(y)16 b Fq(2)e(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))q(.)340 2061 y(Let)416 2050 y(\026)416 2061 y Fo(l)429 2046 y Fn(\003)463 2061 y Fq(!)i Fw(\026)-23 b Fo(r)539 2046 y Fn(\003)572 2061 y Fq(\()15 b Fw(\026)-22 b Fo(c)646 2046 y Fn(\003)680 2061 y Fw(b)q(e)17 b(a)e(mark)o(ed)f(v)o(ersion)i(of)f Fo(l)e Fq(!)e Fo(r)h Fq(\()f Fo(c)k Fw(suc)o(h)i(that)e(all)g(marks)f (on)j(\026)-23 b Fo(r)1762 2046 y Fn(\003)340 2111 y Fw(and)19 b(\026)-22 b Fo(c)443 2096 y Fn(\003)480 2111 y Fw(are)19 b(fresh)g(w.r.t.)e Fo(s)h Fw(and)g(m)o(utually)d(distinct.) j(F)m(or)g(ev)o(ery)h(extra)f(v)n(ariable)f Fo(x)h Fw(in)340 2161 y Fo(l)13 b Fq(!)e Fo(r)i Fq(\()e Fo(c)h Fw(let)i Fo(x)r Fw(\026)-23 b Fo(\033)13 b Fw(b)q(e)h(a)f(mark)o(ed)e(v)o (ersion)i(of)g Fo(e)p Fw(\()p Fo(x\033)q Fw(\))g(suc)o(h)h(that)f(all)f (marks)g(are)h(m)o(utually)340 2210 y(distinct)g(and)g(fresh)g(w.r.t.)f Fo(s)p Fw(,)j(\026)-23 b Fo(r)842 2195 y Fn(\003)873 2210 y Fw(and)14 b(\026)-22 b Fo(c)971 2195 y Fn(\003)990 2210 y Fw(.)12 b(It)h(is)g(fairly)e(simple)g(to)i(pro)o(v)o(e)g(that) 1583 2199 y(\026)1584 2210 y Fo(l)1597 2195 y Fn(\003)1618 2210 y Fw(\026)-23 b Fo(\033)13 b Fq(\))f Fw(\026)-22 b Fo(r)1726 2195 y Fn(\003)1747 2210 y Fw(\026)f Fo(\033)q Fw(.)340 2260 y(Th)o(us)15 b Fo(s)d Fq(\))g Fo(v)h Fw(=)618 2250 y(\026)608 2260 y Fo(C)s Fw([)r(\026)-23 b Fo(r)673 2245 y Fn(\003)694 2260 y Fw(\026)f Fo(\033)r(;)7 b(:)g(:)g(:)t(;)i Fw(\026)-23 b Fo(r)829 2245 y Fn(\003)850 2260 y Fw(\026)g Fo(\033)q Fw(].)13 b(It)h(follo)o(ws)f(from)f Fo(C)s Fw([)p Fo(l)1248 2245 y Fn(\003)1267 2260 y Fo(\033)o(;)7 b(:)g(:)g(:)t(;)g(l)1395 2245 y Fn(\003)1414 2260 y Fo(\033)q Fw(])12 b Fq(\030)1516 2250 y Fw(\026)1507 2260 y Fo(C)s Fw([)1552 2249 y(\026)1552 2260 y Fo(l)1565 2245 y Fn(\003)1586 2260 y Fw(\026)-24 b Fo(\033)r(;)7 b(:)g(:)g(:)t(;)1701 2249 y Fw(\026)1701 2260 y Fo(l)1714 2245 y Fn(\003)1735 2260 y Fw(\026)-23 b Fo(\033)q Fw(],)340 2310 y(the)16 b(prop)q(erties)g(of)e Fo(\033)h Fw(and)i(\026)-23 b Fo(\033)q Fw(,)14 b(and)g(the)h(fact)g(that)g(b)q(oth)f Fo(r)1274 2295 y Fn(\003)1308 2310 y Fw(and)i(\026)-23 b Fo(r)1409 2295 y Fn(\003)1443 2310 y Fw(are)15 b(freshly)f(mark)o(ed) 340 2360 y(that)g Fo(u)e Fq(\030)f Fo(v)q Fw(.)p 340 2389 237 2 v 349 2416 a Ft(3)379 2432 y Fv(Note)i(that)g Fg(u)e Fi(\030)f Fg(v)k Fv(is)f(not)h(required.)p eop %%Page: 11 11 11 10 bop 183 194 a Fw(The)13 b(next)h(lemma)d(sho)o(ws)i(that)h(the)g (deterministic)f(graph)g(rewrite)h(relation)f(is)g(sub)q(com-)183 244 y(m)o(utativ)o(e)f(\(cf.)h([Klo92)o(]\))h(mo)q(dulo)d Fq(\030)p Fw(.)183 330 y Ff(Lemma)c(19.)21 b Fr(F)m(or)14 b(al)r(l)h Fo(m;)7 b(n)k Fq(2)g Fk(I)-6 b(N)q Fr(,)14 b(the)h(fol)r(lowing)f(statements)h(hold:)199 416 y(1.)20 b(If)c Fo(s)d Fw(=)h Fo(l)389 401 y Fn(\003)388 427 y Fm(1)408 416 y Fo(\033)432 422 y Fm(1)465 416 y Fq(\))507 425 y Fn(R)536 416 y Fj(d)536 433 y(m)580 416 y Fo(r)600 401 y Fn(\003)599 427 y Fm(1)619 416 y Fo(\033)643 422 y Fm(1)677 416 y Fr(and)i Fo(s)e Fw(=)f Fo(l)849 401 y Fn(\003)848 427 y Fm(2)868 416 y Fo(\033)892 422 y Fm(2)926 416 y Fq(\))968 425 y Fn(R)997 416 y Fj(d)997 433 y(n)1033 416 y Fo(r)1053 401 y Fn(\003)1052 427 y Fm(2)1072 416 y Fo(\033)1096 422 y Fm(2)1114 416 y Fr(,)j(then)g(\(i\)) f Fo(l)1311 401 y Fn(\003)1310 427 y Fm(1)1344 416 y Fq(!)d Fo(r)1418 401 y Fn(\003)1417 427 y Fm(1)1450 416 y Fq(\()h Fo(c)1523 401 y Fn(\003)1523 427 y Fm(1)1558 416 y Fr(and)253 466 y Fo(l)266 451 y Fn(\003)265 476 y Fm(2)298 466 y Fq(!)g Fo(r)373 451 y Fn(\003)372 476 y Fm(2)404 466 y Fq(\()g Fo(c)477 451 y Fn(\003)477 476 y Fm(2)511 466 y Fr(ar)n(e)j(marke)n(d)f(versions)h(of)f(the)h(same)f (r)n(ewrite)f(rule)h Fo(l)e Fq(!)e Fo(r)h Fq(\()f Fo(c)i Fq(2)g(R)p Fr(,)253 516 y(\(ii\))h Fo(\033)351 522 y Fm(1)381 516 y Fq(\030)e Fo(\033)449 522 y Fm(2)468 516 y Fr(,)i(and)i(\(iii\))e Fo(r)683 501 y Fn(\003)682 526 y Fm(1)702 516 y Fo(\033)726 522 y Fm(1)756 516 y Fq(\030)d Fo(r)819 501 y Fn(\003)818 526 y Fm(2)838 516 y Fo(\033)862 522 y Fm(2)881 516 y Fr(.)199 576 y(2.)20 b(If)15 b Fo(s)e Fq(\))371 554 y Fl(l)381 541 y Fd(\003)381 562 y Fe(1)399 554 y Fl(\033)418 558 y Fe(1)371 591 y Fn(R)400 583 y Fj(d)400 600 y(m)449 576 y Fo(s)468 582 y Fm(1)487 576 y Fr(,)i Fo(t)e Fq(\))585 554 y Fl(l)595 541 y Fd(\003)595 562 y Fe(2)612 554 y Fl(\033)631 558 y Fe(2)585 591 y Fn(R)614 583 y Fj(d)614 600 y(n)662 576 y Fo(t)677 582 y Fm(1)696 576 y Fr(,)i(and)h Fo(s)d Fq(\030)g Fo(t)p Fr(,)i(then)h(ther)n(e)f(ar)n(e)g(marke)n(d)h(terms)e Fo(s)1474 582 y Fm(2)1509 576 y Fr(and)i Fo(t)1605 582 y Fm(2)253 648 y Fr(such)i(that)f(\(i\))f Fo(s)519 654 y Fm(1)554 648 y Fq(\))596 626 y Fl(l)606 613 y Fd(\003)606 634 y Fe(2)623 626 y Fl(\033)642 630 y Fe(2)596 663 y Fn(R)625 655 y Fj(d)625 672 y(n)676 648 y Fo(s)695 654 y Fm(2)731 648 y Fr(or)h Fo(s)806 654 y Fm(1)840 648 y Fw(=)f Fo(s)907 654 y Fm(2)926 648 y Fr(,)g(\(ii\))h Fo(t)1047 654 y Fm(1)1081 648 y Fq(\))1123 626 y Fl(l)1133 613 y Fd(\003)1133 634 y Fe(1)1150 626 y Fl(\033)1169 630 y Fe(1)1123 663 y Fn(R)1152 655 y Fj(d)1152 672 y(m)1203 648 y Fo(t)1218 654 y Fm(2)1253 648 y Fr(or)g Fo(t)1324 654 y Fm(1)1358 648 y Fw(=)e Fo(t)1420 654 y Fm(2)1439 648 y Fr(,)i(and)g(\(iii\))253 698 y Fo(s)272 704 y Fm(2)303 698 y Fq(\030)12 b Fo(t)362 704 y Fm(2)380 698 y Fr(.)183 794 y(Pr)n(o)n(of.)20 b Fw(The)13 b(pro)q(of)f(is)g(similar)e(to)i (that)h(of)e(Lemma)f(8.)i(Again,)f(w)o(e)i(pro)q(ceed)h(b)o(y)e (induction)183 844 y(on)i Fo(m)c Fw(+)g Fo(n)p Fw(.)k(The)g(base)h (case)h Fo(m)10 b Fw(+)g Fo(n)i Fw(=)h(0)h(holds)g(v)n(acuously)m(.)f (Supp)q(ose)i(the)h(lemma)11 b(holds)183 893 y(for)i(all)f Fo(m)339 878 y Fn(0)364 893 y Fw(and)h Fo(n)469 878 y Fn(0)494 893 y Fw(with)g Fo(m)624 878 y Fn(0)645 893 y Fw(+)8 b Fo(n)710 878 y Fn(0)733 893 y Fo(<)k(k)q Fw(.)h(In)g(the)h (induction)f(step,)h(w)o(e)g(ha)o(v)o(e)f(to)g(pro)o(v)o(e)g(that)183 943 y(the)18 b(lemma)c(holds)j(for)g(all)f Fo(m)i Fw(and)f Fo(n)g Fw(with)g Fo(m)12 b Fw(+)g Fo(n)17 b Fw(=)g Fo(k)q Fw(.)g(By)h(using)f(Lemma)d(18,)j(it)g(is)183 993 y(not)g(di\016cult)f (to)h(pro)o(v)o(e)h(that)f(the)h(inductiv)o(e)f(h)o(yp)q(othesis)h (implies)d(the)j(v)n(alidit)o(y)d(of)i(the)183 1043 y(diagrams)12 b(in)h(Figure)h(2,)f(where)i Fo(m)742 1028 y Fn(0)763 1043 y Fw(+)10 b Fo(n)830 1028 y Fn(0)853 1043 y Fo(<)i(k)j Fw(and)e Fq(!)h Fw(stands)g(for)g Fq(\))p Fw(.)271 1196 y Fg(:)c Fi(\030)h Fg(:)416 1167 y Fi(R)449 1152 y Fh(d)449 1177 y(n)468 1170 y Fd(0)552 1191 y Fg(:)287 1342 y Fi(R)320 1326 y Fh(d)320 1352 y(m)347 1345 y Fd(0)570 1342 y Fi(R)603 1326 y Fh(d)603 1352 y(m)630 1345 y Fd(0)269 1474 y Fg(:)346 1451 y Fi(R)379 1435 y Fh(d)379 1460 y(n)398 1453 y Fd(0)488 1479 y Fg(:)g Fi(\030)f Fg(:)p 369 1189 166 2 v 493 1188 a Fb(-)p 273 1449 2 237 v 274 1449 a(?)p 557 1222 2 10 v 557 1241 V 557 1260 V 557 1278 V 557 1297 V 557 1316 V 557 1335 V 557 1354 V 557 1373 V 557 1392 V 557 1411 V 557 1430 V 557 1449 V 557 1449 2 24 v 242 w(?)p 307 1473 10 2 v 326 1473 V 345 1473 V 364 1473 V 383 1473 V 402 1473 V 421 1473 V 440 1473 24 2 v 422 1472 a(-)735 1196 y Fg(:)g Fi(\030)g Fg(:)880 1167 y Fi(R)913 1152 y Fh(d)913 1177 y(n)932 1170 y Fd(0)1016 1191 y Fg(:)750 1342 y Fi(R)783 1326 y Fh(d)783 1352 y(m)810 1345 y Fd(0)750 1448 y Fi(\003)1034 1342 y(R)1067 1326 y Fh(d)1067 1352 y(m)1094 1345 y Fd(0)1034 1448 y Fi(\003)732 1474 y Fg(:)809 1451 y Fi(R)842 1435 y Fh(d)842 1460 y(n)861 1453 y Fd(0)952 1479 y Fg(:)g Fi(\030)h Fg(:)p 832 1189 166 2 v 956 1188 a Fb(-)p 737 1449 2 237 v 738 1449 a(?)p 1020 1222 2 10 v 1020 1241 V 1020 1260 V 1020 1278 V 1020 1297 V 1020 1316 V 1020 1335 V 1020 1354 V 1020 1373 V 1020 1392 V 1020 1411 V 1020 1430 V 1020 1449 V 1020 1449 2 24 v 241 w(?)p 771 1473 10 2 v 790 1473 V 809 1473 V 828 1473 V 846 1473 V 865 1473 V 884 1473 V 903 1473 24 2 v 885 1472 a(-)1198 1196 y Fg(:)f Fi(\030)h Fg(:)1344 1167 y Fi(R)1377 1152 y Fh(d)1377 1177 y(n)1396 1170 y Fd(0)1479 1191 y Fg(:)1437 1174 y Fi(\003)1214 1342 y(R)1247 1326 y Fh(d)1247 1352 y(m)1274 1345 y Fd(0)1214 1448 y Fi(\003)1497 1342 y(R)1530 1326 y Fh(d)1530 1352 y(m)1557 1345 y Fd(0)1497 1448 y Fi(\003)1196 1474 y Fg(:)1273 1451 y Fi(R)1306 1435 y Fh(d)1306 1460 y(n)1325 1453 y Fd(0)1367 1457 y Fi(\003)1415 1479 y Fg(:)g Fi(\030)f Fg(:)p 1296 1189 166 2 v 1420 1188 a Fb(-)p 1201 1449 2 237 v 1201 1449 a(?)p 1484 1222 2 10 v 1484 1241 V 1484 1260 V 1484 1278 V 1484 1297 V 1484 1316 V 1484 1335 V 1484 1354 V 1484 1373 V 1484 1392 V 1484 1411 V 1484 1430 V 1484 1449 V 1484 1449 2 24 v 242 w(?)p 1234 1473 10 2 v 1253 1473 V 1272 1473 V 1291 1473 V 1310 1473 V 1329 1473 V 1348 1473 V 1367 1473 24 2 v 1349 1472 a(-)839 1601 y Fu(Fig.)c(2.)183 1746 y Fw(\(1\))12 b(Let)h Fo(s)f Fw(=)g Fo(l)409 1731 y Fn(\003)408 1756 y Fm(1)428 1746 y Fo(\033)452 1752 y Fm(1)484 1746 y Fq(\))526 1754 y Fn(R)555 1746 y Fj(d)555 1763 y(m)597 1746 y Fo(r)617 1731 y Fn(\003)616 1756 y Fm(1)636 1746 y Fo(\033)660 1752 y Fm(1)691 1746 y Fw(and)g Fo(s)g Fw(=)g Fo(l)858 1731 y Fn(\003)857 1756 y Fm(2)877 1746 y Fo(\033)901 1752 y Fm(2)933 1746 y Fq(\))975 1754 y Fn(R)1004 1746 y Fj(d)1004 1763 y(n)1040 1746 y Fo(r)1060 1731 y Fn(\003)1059 1756 y Fm(2)1079 1746 y Fo(\033)1103 1752 y Fm(2)1121 1746 y Fw(.)g(Clearly)m(,)f(\(i\))h(follo)o(ws)e (directly)183 1796 y(from)i(orthogonalit)o(y)g(of)i Fq(R)620 1802 y Fl(u)656 1796 y Fw(and)g(\(iii\))f(is)h(a)g(consequence)j(of)c (\(i\))h(and)g(\(ii\).)f(Let)i(us)f(pro)o(v)o(e)183 1846 y(\(ii\).)19 b(Since)i Fo(\033)409 1852 y Fm(1)449 1846 y Fw(=)i Fo(\033)528 1852 y Fm(2)559 1846 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))q(],)c(it)h(remains)f(to)i(sho)o(w)f Fo(\033)1130 1852 y Fm(1)1171 1846 y Fq(\030)i Fo(\033)1249 1852 y Fm(2)1280 1846 y Fw([)p Fq(E)t(V)s Fo(ar)q Fw(\()p Fo(l)13 b Fq(!)e Fo(r)h Fq(\()f Fo(c)p Fw(\))q(].)183 1900 y(W)m(e)19 b(sho)o(w)h(b)o(y)g(induction)g(on)g Fo(i)g Fw(that)g Fo(\033)842 1906 y Fm(1)883 1900 y Fq(\030)i Fo(\033)961 1906 y Fm(2)992 1900 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))14 b Fq([)1174 1869 y Fp(S)1208 1879 y Fl(i)1208 1913 y(j)r Fm(=1)1275 1900 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)1376 1906 y Fl(j)1394 1900 y Fw(\)].)19 b(If)h Fo(i)i Fw(=)g(0,)183 1950 y(then)e Fo(\033)307 1956 y Fm(1)347 1950 y Fw(=)i Fo(\033)425 1956 y Fm(2)456 1950 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\)].)d(So)h(let)g Fo(i)i(>)g Fw(0.)d(According)h(to)g(the)g(inductiv)o(e)g(h)o(yp)q (othesis,)183 2005 y Fo(\033)207 2011 y Fm(1)242 2005 y Fq(\030)e Fo(\033)316 2011 y Fm(2)347 2005 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))12 b Fq([)526 1974 y Fp(S)560 1984 y Fl(i)p Fn(\000)p Fm(1)560 2017 y Fl(j)r Fm(=1)627 2005 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)728 2011 y Fl(j)746 2005 y Fw(\)].)17 b(Since)h Fq(V)s Fo(ar)q Fw(\()p Fo(s)1020 2011 y Fl(i)1035 2005 y Fw(\))f Fq(\022)h(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))13 b Fq([)1285 1974 y Fp(S)1320 1984 y Fl(i)p Fn(\000)p Fm(1)1320 2017 y Fl(j)r Fm(=1)1386 2005 y Fq(V)s Fo(ar)q Fw(\()p Fo(t)1487 2011 y Fl(j)1506 2005 y Fw(\),)k(it)g(is)183 2055 y(su\016cien)o(t)12 b(to)g(sho)o(w)f Fo(\033)528 2061 y Fm(1)558 2055 y Fq(\030)h Fo(\033)626 2061 y Fm(2)657 2055 y Fw([)p Fq(V)s Fo(ar)q Fw(\()p Fo(t)770 2061 y Fl(i)784 2055 y Fw(\)].)f(There)i(are)f(mark)o (ed)f(ground)g(constructor)j(terms)183 2104 y Fo(u)207 2110 y Fm(1)225 2104 y Fo(;)7 b(u)268 2110 y Fm(2)286 2104 y Fo(;)g(v)325 2110 y Fm(1)343 2104 y Fo(;)g(v)382 2110 y Fm(2)412 2104 y Fw(suc)o(h)13 b(that)f Fo(s)611 2110 y Fl(i)626 2104 y Fo(\033)650 2110 y Fm(1)682 2104 y Fq(\))724 2089 y Fn(\003)724 2120 y(R)753 2110 y Fj(d)753 2131 y(m)p Fd(\000)p Fe(1)818 2104 y Fo(u)842 2110 y Fm(1)861 2104 y Fw(,)f Fo(t)899 2110 y Fl(i)913 2104 y Fo(\033)937 2110 y Fm(1)969 2104 y Fq(\))1011 2089 y Fn(\003)1011 2120 y(R)1040 2110 y Fj(d)1040 2131 y(m)p Fd(\000)p Fe(1)1105 2104 y Fo(u)1129 2110 y Fm(2)1148 2104 y Fw(,)g(where)j Fo(e)p Fw(\()p Fo(u)1349 2110 y Fm(1)1368 2104 y Fw(\))d(=)h Fo(e)p Fw(\()p Fo(u)1498 2110 y Fm(2)1517 2104 y Fw(\),)g(and)183 2170 y Fo(s)202 2176 y Fl(i)216 2170 y Fo(\033)240 2176 y Fm(2)272 2170 y Fq(\))314 2154 y Fn(\003)314 2186 y(R)343 2176 y Fj(d)343 2196 y(n)p Fd(\000)p Fe(1)401 2170 y Fo(v)421 2176 y Fm(1)440 2170 y Fw(,)i Fo(t)481 2176 y Fl(i)495 2170 y Fo(\033)519 2176 y Fm(2)551 2170 y Fq(\))593 2154 y Fn(\003)593 2186 y(R)622 2176 y Fj(d)622 2196 y(n)p Fd(\000)p Fe(1)680 2170 y Fo(v)700 2176 y Fm(2)719 2170 y Fw(,)h(where)h Fo(e)p Fw(\()p Fo(v)922 2176 y Fm(1)941 2170 y Fw(\))d(=)h Fo(e)p Fw(\()p Fo(v)1071 2176 y Fm(2)1090 2170 y Fw(\).)h(It)g(no)o(w)f (follo)o(ws)f(from)h(the)h(in-)183 2232 y(ductiv)o(e)i(h)o(yp)q (othesis)h(on)f Fo(k)h Fw(in)e(com)o(bination)f(with)i Fo(s)1048 2238 y Fl(i)1062 2232 y Fo(\033)1086 2238 y Fm(1)1121 2232 y Fq(\030)h Fo(s)1190 2238 y Fl(i)1204 2232 y Fo(\033)1228 2238 y Fm(2)1263 2232 y Fw(that)f Fo(u)1380 2238 y Fm(1)1416 2232 y Fq(\030)g Fo(v)1485 2238 y Fm(1)1504 2232 y Fw(.)f(Th)o(us)183 2282 y Fo(e)p Fw(\()p Fo(u)242 2288 y Fm(2)261 2282 y Fw(\))i(=)h Fo(e)p Fw(\()p Fo(u)405 2288 y Fm(1)424 2282 y Fw(\))f(=)h Fo(e)p Fw(\()p Fo(v)564 2288 y Fm(1)583 2282 y Fw(\))g(=)f Fo(e)p Fw(\()p Fo(v)723 2288 y Fm(2)743 2282 y Fw(\).)f(As)i(in)e(the)i(pro)q (of)f(of)f(Lemma)e(8,)j(for)f(ev)o(ery)i(extra)183 2332 y(v)n(ariable)g Fo(x)p Fw(,)g(there)i(are)f(mark)o(ed)f(ground)h (constructor)i(terms)d Fo(u)1259 2338 y Fl(x)1300 2332 y Fw(and)h Fo(v)1407 2338 y Fl(x)1448 2332 y Fw(suc)o(h)g(that)183 2382 y Fo(x\033)231 2388 y Fm(1)265 2382 y Fw(=)c Fo(u)337 2388 y Fl(x)357 2382 y Fw(,)g Fo(x\033)433 2388 y Fm(2)467 2382 y Fw(=)g Fo(v)535 2388 y Fl(x)556 2382 y Fw(,)g(and)h Fo(e)p Fw(\()p Fo(u)727 2388 y Fl(x)748 2382 y Fw(\))f(=)g Fo(e)p Fw(\()p Fo(v)883 2388 y Fl(x)904 2382 y Fw(\).)g(Since)h(the)g (marks)f(on)g Fo(x\033)1368 2388 y Fm(1)1402 2382 y Fw(and)h Fo(x\033)1534 2388 y Fm(2)1568 2382 y Fw(are)183 2432 y(fresh)d(and)g(m)o(utually)d(distinct,)j(w)o(e)g(\014nally)f(deriv)o (e)h Fo(u)1034 2438 y Fl(x)1066 2432 y Fq(\030)e Fo(v)1130 2438 y Fl(x)1151 2432 y Fw(.)p eop %%Page: 12 12 12 11 bop 340 197 a Fw(\(2\))13 b(By)g(Lemma)d(18,)i(it)g(is)g (su\016cien)o(t)i(to)e(pro)o(v)o(e)h(that)g(if)e Fo(t)h Fq(\))1291 174 y Fl(l)1301 162 y Fd(\003)1301 183 y Fe(1)1318 174 y Fl(\033)1337 178 y Fe(1)1291 212 y Fn(R)1320 204 y Fj(d)1320 220 y(m)1368 197 y Fo(s)1387 203 y Fm(1)1418 197 y Fw(and)h Fo(t)e Fq(\))1566 174 y Fl(l)1576 162 y Fd(\003)1576 183 y Fe(2)1594 174 y Fl(\033)1613 178 y Fe(2)1566 212 y Fn(R)1595 204 y Fj(d)1595 220 y(n)1643 197 y Fo(t)1658 203 y Fm(1)1676 197 y Fw(,)i(then)340 269 y(\(i\))i Fo(s)418 275 y Fm(1)450 269 y Fq(\))492 247 y Fl(l)502 234 y Fd(\003)502 255 y Fe(2)519 247 y Fl(\033)538 251 y Fe(2)492 284 y Fn(R)521 276 y Fj(d)521 293 y(n)569 269 y Fo(s)588 275 y Fm(2)622 269 y Fw(or)f Fo(s)692 275 y Fm(1)724 269 y Fw(=)f Fo(s)788 275 y Fm(2)807 269 y Fw(,)h(\(ii\))g Fo(t)918 275 y Fm(1)950 269 y Fq(\))992 247 y Fl(l)1002 234 y Fd(\003)1002 255 y Fe(1)1019 247 y Fl(\033)1038 251 y Fe(1)992 284 y Fn(R)1021 276 y Fj(d)1021 293 y(m)1069 269 y Fo(t)1084 275 y Fm(2)1118 269 y Fw(or)g Fo(t)1184 275 y Fm(1)1215 269 y Fw(=)f Fo(t)1275 275 y Fm(2)1294 269 y Fw(,)h(and)h(\(iii\))e Fo(s)1502 275 y Fm(2)1534 269 y Fq(\030)g Fo(t)1594 275 y Fm(2)1627 269 y Fw(for)h(some)340 326 y(mark)o(ed)f(terms)h Fo(s)623 332 y Fm(2)656 326 y Fw(and)f Fo(t)751 332 y Fm(2)770 326 y Fw(.)g(W)m(e)h(distinguish)f(three)i(cases:)340 425 y(\(a\))f Fo(l)420 410 y Fn(\003)419 436 y Fm(1)440 425 y Fo(\033)464 431 y Fm(1)493 425 y Fw(=)e Fo(l)550 410 y Fn(\003)549 436 y Fm(2)570 425 y Fo(\033)594 431 y Fm(2)612 425 y Fw(,)340 475 y(\(b\))j Fo(l)423 460 y Fn(\003)422 485 y Fm(1)442 475 y Fo(\033)466 481 y Fm(1)498 475 y Fw(is)f(neither)h(a)e(subterm)h(of)f Fo(l)937 460 y Fn(\003)936 485 y Fm(2)957 475 y Fo(\033)981 481 y Fm(2)1013 475 y Fw(nor)h(con)o(v)o(ersely)m(,)340 525 y(\(c\))h Fo(l)418 510 y Fn(\003)417 535 y Fm(1)437 525 y Fo(\033)461 531 y Fm(1)493 525 y Fw(is)f(a)g(prop)q(er)h(subterm)e (of)h Fo(l)925 510 y Fn(\003)924 535 y Fm(2)944 525 y Fo(\033)968 531 y Fm(2)986 525 y Fw(.)340 624 y(\(a\))g(With)g(the)g (aid)f(of)g(\(1\),)h(this)g(follo)o(ws)e(easily)m(.)340 674 y(\(b\))j(The)f(pro)q(of)f(is)h(analogous)f(to)h(Prop)q(osition)f (3.19,)f(\(1\),)i(case)g(1)g(in)f([K)o(O95].)340 724 y(\(c\))i(W)m(e)e(pro)q(ceed)j(in)d(analogy)g(to)g(Prop)q(osition)h (3.19,)e(\(1\),)h(case)i(2)f(in)f([K)o(O95].)g(W)m(e)g(ha)o(v)o(e)584 806 y Fo(t)e Fw(=)h Fo(C)s Fw([)p Fo(l)712 789 y Fn(\003)711 817 y Fm(1)730 806 y Fo(\033)754 812 y Fm(1)773 806 y Fo(;)7 b(:)g(:)g(:)e(;)i(l)879 789 y Fn(\003)878 817 y Fm(2)897 806 y Fo(\033)921 812 y Fm(2)940 806 y Fw(])k(=)h Fo(C)s Fw([)p Fo(l)1065 789 y Fn(\003)1064 817 y Fm(1)1083 806 y Fo(\033)1107 812 y Fm(1)1125 806 y Fo(;)7 b(:)g(:)g(:)e(;)i(C) 1251 789 y Fn(0)1262 806 y Fw([)p Fo(l)1287 789 y Fn(\003)1286 817 y Fm(1)1306 806 y Fo(\033)1330 812 y Fm(1)1348 806 y Fo(;)g(:)g(:)g(:)e(;)i(l)1454 789 y Fn(\003)1453 817 y Fm(1)1473 806 y Fo(\033)1497 812 y Fm(1)1515 806 y Fw(]])340 889 y(where)15 b Fo(l)473 874 y Fn(\003)472 899 y Fm(2)492 889 y Fo(\033)516 895 y Fm(2)546 889 y Fw(=)d Fo(C)623 874 y Fn(0)634 889 y Fw([)p Fo(l)659 874 y Fn(\003)658 899 y Fm(1)678 889 y Fo(\033)702 895 y Fm(1)720 889 y Fo(;)7 b(:)g(:)g(:)e(;)i(l)826 874 y Fn(\003)825 899 y Fm(1)845 889 y Fo(\033)869 895 y Fm(1)887 889 y Fw(],)12 b(and)h Fo(l)1016 874 y Fn(\003)1015 899 y Fm(1)1035 889 y Fo(\033)1059 895 y Fm(1)1091 889 y Fw(is)g(neither)h(a)f(subterm)g(of)g Fo(C)s Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(])12 b(nor)h(of)340 939 y Fo(C)373 924 y Fn(0)385 939 y Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(].)12 b(Hence)738 987 y Fo(s)757 993 y Fm(1)788 987 y Fw(=)g Fo(C)s Fw([)p Fo(r)897 970 y Fn(\003)896 997 y Fm(1)915 987 y Fo(\033)939 993 y Fm(1)957 987 y Fo(;)7 b(:)g(:)g(:)e(;)i(C)1083 970 y Fn(0)1094 987 y Fw([)p Fo(r)1126 970 y Fn(\003)1125 997 y Fm(1)1144 987 y Fo(\033)1168 993 y Fm(1)1187 987 y Fo(;)g(:)g(:)g(:)t(;)g(r)1299 970 y Fn(\003)1298 997 y Fm(1)1318 987 y Fo(\033)1342 993 y Fm(1)1360 987 y Fw(]])572 1056 y Fo(t)587 1062 y Fm(1)617 1056 y Fw(=)12 b Fo(C)s Fw([)p Fo(l)719 1039 y Fn(\003)718 1066 y Fm(1)737 1056 y Fo(\033)761 1062 y Fm(1)780 1056 y Fo(;)7 b(:)g(:)g(:)e(;)i(r)893 1039 y Fn(\003)892 1066 y Fm(2)911 1056 y Fo(\033)935 1062 y Fm(2)953 1056 y Fw(])k(=)h Fo(C)s Fw([)p Fo(l)1078 1039 y Fn(\003)1077 1066 y Fm(1)1097 1056 y Fo(\033)1121 1062 y Fm(1)1139 1056 y Fo(;)7 b(:)g(:)g(:)e(;)i(C)1265 1039 y Fn(00)1285 1056 y Fw([)p Fo(l)1310 1039 y Fn(\003)1309 1066 y Fm(1)1329 1056 y Fo(\033)1353 1062 y Fm(1)1371 1056 y Fo(;)g(:)g(:)g(:)e(;)i(l) 1477 1039 y Fn(\003)1476 1066 y Fm(1)1496 1056 y Fo(\033)1520 1062 y Fm(1)1538 1056 y Fw(])340 1126 y(for)13 b(some)f(con)o(text)i Fo(C)686 1111 y Fn(00)707 1126 y Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g Fw(])13 b(whic)o(h)g(do)q(es)g(not)h(con)o(tain)e Fo(l)1278 1111 y Fn(\003)1277 1137 y Fm(1)1298 1126 y Fo(\033)1322 1132 y Fm(1)1340 1126 y Fw(.)h(Observ)o(e)h(that)f(no)g(o)q(ccur-)340 1176 y(rence)18 b(of)d Fo(C)533 1161 y Fn(0)544 1176 y Fw([)p Fo(r)576 1161 y Fn(\003)575 1187 y Fm(1)594 1176 y Fo(\033)618 1182 y Fm(1)637 1176 y Fo(;)7 b(:)g(:)g(:)t(;)g(r) 749 1161 y Fn(\003)748 1187 y Fm(1)768 1176 y Fo(\033)792 1182 y Fm(1)810 1176 y Fw(])15 b(o)q(ccurs)i(in)e Fo(s)1036 1182 y Fm(1)1071 1176 y Fw(aside)h(from)e(those)i(obtained)f(b)o(y)h (con)o(tract-)340 1226 y(ing)i(the)i(mark)o(ed)d(redex)j Fo(l)773 1211 y Fn(\003)772 1236 y Fm(1)792 1226 y Fo(\033)816 1232 y Fm(1)853 1226 y Fw(b)q(ecause)g(w)o(e)f(use)h(fresh)f(marks.)e (By)i(the)g(same)f(reason,)340 1276 y Fo(C)s Fw([)p Fo(;)7 b(:)g(:)g(:)t(;)g(C)510 1261 y Fn(00)531 1276 y Fw([)p Fo(;)g(:)g(:)g(:)t(;)g Fw(]])k(do)q(es)i(not)f(con)o(tain)f Fo(l)991 1261 y Fn(\003)990 1286 y Fm(1)1011 1276 y Fo(\033)1035 1282 y Fm(1)1053 1276 y Fw(.)h(No)o(w)g(if)f Fo(l)1219 1261 y Fn(\003)1218 1286 y Fm(1)1238 1276 y Fo(\033)1262 1282 y Fm(1)1293 1276 y Fw(is)h(not)g(a)g(subterm)g(of)f Fo(t)1658 1282 y Fm(1)1677 1276 y Fw(,)h(then)340 1326 y(let)17 b Fo(t)418 1332 y Fm(2)454 1326 y Fw(=)f Fo(t)517 1332 y Fm(1)536 1326 y Fw(.)g(Otherwise)j(de\014ne)f Fo(t)902 1332 y Fm(2)937 1326 y Fw(=)f Fo(C)s Fw([)p Fo(r)1051 1311 y Fn(\003)1050 1336 y Fm(1)1069 1326 y Fo(\033)1093 1332 y Fm(1)1111 1326 y Fo(;)7 b(:)g(:)g(:)e(;)i(C)1237 1311 y Fn(00)1257 1326 y Fw([)p Fo(r)1289 1311 y Fn(\003)1288 1336 y Fm(1)1308 1326 y Fo(\033)1332 1332 y Fm(1)1350 1326 y Fo(;)g(:)g(:)g(:)e(;)i(r)1463 1311 y Fn(\003)1462 1336 y Fm(1)1481 1326 y Fo(\033)1505 1332 y Fm(1)1524 1326 y Fw(]])16 b(and)g(observ)o(e)340 1385 y(that)e Fo(t)445 1391 y Fm(1)475 1385 y Fq(\))517 1362 y Fl(l)527 1350 y Fd(\003)527 1371 y Fe(1)545 1362 y Fl(\033)564 1366 y Fe(1)517 1400 y Fn(R)546 1392 y Fj(d)546 1408 y(m)594 1385 y Fo(t)609 1391 y Fm(2)628 1385 y Fw(.)403 1444 y(W)m(e)g(next)h(sho)o(w)g(that)f Fo(C)796 1429 y Fn(0)808 1444 y Fw([)p Fo(r)840 1429 y Fn(\003)839 1454 y Fm(1)858 1444 y Fo(\033)882 1450 y Fm(1)901 1444 y Fo(;)7 b(:)g(:)g(:)t(;)g(r)1013 1429 y Fn(\003)1012 1454 y Fm(1)1032 1444 y Fo(\033)1056 1450 y Fm(1)1074 1444 y Fw(])12 b Fq(\))1140 1452 y Fn(R)1169 1444 y Fj(d)1169 1460 y(n)1204 1444 y Fo(C)1237 1429 y Fn(00)1258 1444 y Fw([)p Fo(r)1290 1429 y Fn(\003)1289 1454 y Fm(1)1308 1444 y Fo(\033)1332 1450 y Fm(1)1350 1444 y Fo(;)7 b(:)g(:)g(:)e(;)i(r) 1463 1429 y Fn(\003)1462 1454 y Fm(1)1482 1444 y Fo(\033)1506 1450 y Fm(1)1524 1444 y Fw(].)14 b(T)m(o)g(this)g(end,)340 1494 y(recall)k(that)g Fo(l)562 1479 y Fn(\003)561 1504 y Fm(2)582 1494 y Fo(\033)606 1500 y Fm(2)643 1494 y Fw(=)g Fo(C)726 1479 y Fn(0)738 1494 y Fw([)p Fo(l)763 1479 y Fn(\003)762 1504 y Fm(1)781 1494 y Fo(\033)805 1500 y Fm(1)824 1494 y Fo(;)7 b(:)g(:)g(:)e(;)i(l)930 1479 y Fn(\003)929 1504 y Fm(1)948 1494 y Fo(\033)972 1500 y Fm(1)991 1494 y Fw(])13 b Fq(\))1058 1502 y Fn(R)1087 1494 y Fj(d)1087 1510 y(n)1122 1494 y Fo(r)1142 1479 y Fn(\003)1141 1504 y Fm(2)1161 1494 y Fo(\033)1185 1500 y Fm(2)1204 1494 y Fw(.)k(Th)o(us,)h(for)g(ev)o(ery)h Fo(s)1555 1500 y Fl(i)1569 1494 y Fw(==)p Fo(t)1648 1500 y Fl(i)1680 1494 y Fw(in)f Fo(c)1751 1479 y Fn(\003)1751 1504 y Fm(2)1770 1494 y Fw(,)340 1543 y(there)c(exist)f(mark)o(ed)e (ground)h(constructor)i(terms)e Fo(u)1183 1549 y Fl(i)1208 1543 y Fw(and)g Fo(v)1307 1549 y Fl(i)1333 1543 y Fw(suc)o(h)h(that)g Fo(s)1533 1549 y Fl(i)1547 1543 y Fo(\033)1571 1549 y Fm(2)1603 1543 y Fq(\))1645 1528 y Fn(\003)1645 1559 y(R)1674 1549 y Fj(d)1674 1570 y(n)p Fd(\000)p Fe(1)1732 1543 y Fo(u)1756 1549 y Fl(i)1770 1543 y Fw(,)340 1609 y Fo(t)355 1615 y Fl(i)369 1609 y Fo(\033)393 1615 y Fm(2)425 1609 y Fq(\))467 1594 y Fn(\003)467 1625 y(R)496 1615 y Fj(d)496 1635 y(n)p Fd(\000)p Fe(1)554 1609 y Fo(v)574 1615 y Fl(i)588 1609 y Fw(,)f(and)f Fo(e)p Fw(\()p Fo(u)749 1615 y Fl(i)763 1609 y Fw(\))h(=)g Fo(e)p Fw(\()p Fo(v)890 1615 y Fl(i)904 1609 y Fw(\).)f(Since)i Fq(R)1085 1615 y Fl(u)1118 1609 y Fw(is)f(orthogonal,)e(for)h(ev)o(ery)h(o)q (ccurrence)j(of)340 1674 y Fo(l)353 1659 y Fn(\003)352 1684 y Fm(1)373 1674 y Fo(\033)397 1680 y Fm(1)415 1674 y Fw(,)f(there)j(is)d(a)h(v)n(ariable)f Fo(x)e Fq(2)h(V)s Fo(ar)q Fw(\()p Fo(l)q Fw(\))j(suc)o(h)g(that)f Fo(x\033)1226 1680 y Fm(2)1257 1674 y Fw(=)e Fo(C)1332 1680 y Fl(x)1353 1674 y Fw([)p Fo(l)1378 1659 y Fn(\003)1377 1684 y Fm(1)1397 1674 y Fo(\033)1421 1680 y Fm(1)1439 1674 y Fo(;)7 b(:)g(:)g(:)e(;)i(l) 1545 1659 y Fn(\003)1544 1684 y Fm(1)1564 1674 y Fo(\033)1588 1680 y Fm(1)1606 1674 y Fw(])14 b(con)o(tains)340 1724 y(this)j(particular)f(o)q(ccurrence.)i(De\014ne)f Fo(\033)994 1709 y Fn(0)993 1734 y Fm(2)1027 1724 y Fw(b)o(y)f Fo(x\033)1136 1709 y Fn(0)1135 1734 y Fm(2)1169 1724 y Fw(=)f Fo(C)1246 1730 y Fl(x)1267 1724 y Fw([)p Fo(r)1299 1709 y Fn(\003)1298 1734 y Fm(1)1317 1724 y Fo(\033)1341 1730 y Fm(1)1360 1724 y Fo(;)7 b(:)g(:)g(:)t(;)g(r)1472 1709 y Fn(\003)1471 1734 y Fm(1)1491 1724 y Fo(\033)1515 1730 y Fm(1)1533 1724 y Fw(])16 b(for)g(all)f(those)340 1773 y(v)n(ariables)d Fo(x)g Fw(and)h Fo(y)q(\033)673 1758 y Fn(0)672 1784 y Fm(2)703 1773 y Fw(=)e Fo(y)q(\033)791 1779 y Fm(2)823 1773 y Fw(otherwise.)i(No)o(w)f Fo(l)1124 1758 y Fn(\003)1123 1784 y Fm(2)1144 1773 y Fo(\033)1169 1758 y Fn(0)1168 1784 y Fm(2)1197 1773 y Fq(\))1239 1782 y Fn(R)1268 1774 y Fj(d)1268 1790 y(n)1301 1773 y Fo(r)1321 1758 y Fn(\003)1320 1784 y Fm(2)1340 1773 y Fo(\033)1365 1758 y Fn(0)1364 1784 y Fm(2)1394 1773 y Fw(=)g Fo(C)1471 1758 y Fn(00)1492 1773 y Fw([)p Fo(r)1524 1758 y Fn(\003)1523 1784 y Fm(1)1542 1773 y Fo(\033)1566 1779 y Fm(1)1585 1773 y Fo(;)7 b(:)g(:)g(:)e(;)i(r) 1698 1758 y Fn(\003)1697 1784 y Fm(1)1716 1773 y Fo(\033)1740 1779 y Fm(1)1758 1773 y Fw(].)340 1823 y(In)12 b(order)h(to)f(see)h (this,)e(infer)h(from)e(the)j(inductiv)o(e)e(h)o(yp)q(othesis)i(on)f Fo(k)g Fw(in)g(conjunction)g(with)340 1873 y Fo(s)359 1879 y Fl(i)374 1873 y Fo(\033)398 1879 y Fm(2)430 1873 y Fq(\))472 1858 y Fn(\003)472 1889 y(R)501 1879 y Fj(d)501 1900 y(n)p Fd(\000)p Fe(1)559 1873 y Fo(u)583 1879 y Fl(i)614 1873 y Fw(and)17 b Fo(s)717 1879 y Fl(i)731 1873 y Fo(\033)755 1879 y Fm(2)787 1873 y Fq(\))829 1882 y Fn(R)858 1873 y Fj(d)858 1890 y(m)901 1873 y Fo(s)920 1879 y Fl(i)934 1873 y Fo(\033)959 1858 y Fn(0)958 1883 y Fm(2)994 1873 y Fw(that)g(there)i(is)e(a)g(mark)o(ed)f(ground)i (constructor)340 1938 y(term)e Fo(u)466 1923 y Fn(0)466 1949 y Fl(i)497 1938 y Fw(suc)o(h)h(that)g Fo(s)705 1944 y Fl(i)719 1938 y Fo(\033)744 1923 y Fn(0)743 1949 y Fm(2)775 1938 y Fq(\))817 1923 y Fn(\003)817 1954 y(R)846 1944 y Fj(d)846 1965 y(n)p Fd(\000)p Fe(1)904 1938 y Fo(u)928 1923 y Fn(0)928 1949 y Fl(i)959 1938 y Fw(and)f Fo(u)1066 1944 y Fl(i)1096 1938 y Fq(\030)g Fo(u)1168 1923 y Fn(0)1168 1949 y Fl(i)1182 1938 y Fw(.)g(Analogously)m(,)f (there)j(is)e(a)g(mark)o(ed)340 2004 y(ground)j(constructor)i(term)d Fo(v)836 1989 y Fn(0)835 2014 y Fl(i)868 2004 y Fw(suc)o(h)h(that)g Fo(t)1076 2010 y Fl(i)1090 2004 y Fo(\033)1115 1989 y Fn(0)1114 2014 y Fm(2)1146 2004 y Fq(\))1188 1989 y Fn(\003)1188 2020 y(R)1217 2010 y Fj(d)1217 2030 y(n)p Fd(\000)p Fe(1)1275 2004 y Fo(v)1296 1989 y 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Fn(R)757 2332 y Fj(d)757 2349 y(n)804 2332 y Fq(\022)g(\))903 2338 y Fn(R)932 2342 y Fj(n)968 2332 y Fw(,)11 b(Prop)q(osition)g(22,)g(and)g(Lemma)e(18,)i (w)o(e)h(conclude)340 2382 y(b)o(y)k(the)g(generalization)f(of)f (Staples')i(result)g(\(see)h(Section)f(2\))f(that)29 b Fq(\))1493 2388 y Fn(R)1522 2392 y Fj(n)1573 2382 y Fw(is)15 b(con\015uen)o(t)340 2432 y(if)e(and)h(only)f(if)27 b Fq(\))644 2440 y Fn(R)673 2432 y Fj(d)673 2448 y(n)722 2432 y Fw(is)14 b(con\015uen)o(t.)p eop %%Page: 13 13 13 12 bop 183 194 a Ff(Prop)q(ositi)o(on)5 b(22.)21 b Fr(If)c Fo(s)f Fq(\))638 179 y Fn(\003)638 205 y(R)667 209 y Fj(n)703 194 y Fo(t)p Fr(,)h(then)i(ther)n(e)e(ar)n(e)h(marke)n (d)g(terms)f Fo(u)h Fr(and)h Fo(v)g Fr(such)g(that)183 244 y Fo(s)c Fq(\))259 228 y Fn(\003)259 258 y(R)288 249 y Fj(d)288 266 y(n)324 244 y Fo(u)p Fr(,)g Fo(t)f Fq(\))447 228 y Fn(\003)447 258 y(R)476 249 y Fj(d)476 266 y(n)513 244 y Fo(v)q Fr(,)h(and)h Fo(u)11 b Fq(\030)h Fo(v)q Fr(.)183 340 y Ff(Lemma)7 b(23.)21 b Fr(F)m(or)14 b(every)h Fo(n)c Fq(2)h Fk(I)-6 b(N)p Fr(,)15 b(the)g(sets)f Fo(N)5 b(F)h Fw(\()15 b Fq(\))1046 346 y Fn(R)1075 350 y Fj(n)1111 340 y Fw(\))g Fr(and)h Fo(N)5 b(F)h Fw(\()14 b Fq(\))1366 348 y Fn(R)1395 340 y Fj(d)1395 356 y(n)1431 340 y Fw(\))h Fr(c)n(oincide.)183 428 y Fw(It)9 b(is)h(our)g(next)g (goal)f(to)g(sho)o(w)h(that,)f(for)h(ev)o(ery)g(functional)f(CTRS,)g (\()p Fq(T)1286 413 y Fn(\003)1276 438 y Fl(w)1305 428 y Fo(;)20 b Fq(\))1379 434 y Fn(R)1423 428 y Fw(\))10 b(is)g(a)f(sound)183 477 y(and)i(complete)g(implemen)o(tation)d(of)j (\()p Fq(T)g Fw(\()p Fq(F)t Fo(;)c Fq(V)s Fw(\))p Fo(;)g Fq(!)997 483 y Fn(R)1027 477 y Fw(\))k(in)h(the)g(sense)h(of)e (Barendregt)i Fr(et)g(al.)183 527 y Fw([BEG)285 512 y Fm(+)312 527 y Fw(87)o(].)18 b(T)m(o)f(this)h(end,)h(w)o(e)f(pro)o(v)o (e)g(that)g(the)h(Gross-Kn)o(uth)g(reduction)g(strategy)g(is)183 577 y(normalizing)11 b(\(for)i(unconditional)g(orthogonal)g(TRSs,)g (this)h(w)o(as)g(sho)o(wn)g(in)f([O'D77)o(]\).)183 658 y Ff(De\014niti)o(on)5 b(24.)20 b Fw(F)m(or)g(ev)o(ery)h Fo(n)g Fq(2)h Fk(I)-6 b(N)p Fw(,)20 b(the)g(Gross-Kn)o(uth)h(reduction) g(strategy)g(w.r.t.)183 707 y Fq(!)225 716 y Fn(R)254 707 y Fj(d)254 724 y(n)295 707 y Fw(is)e(de\014ned)i(as)f(follo)o(ws:)e (T)m(ak)o(e)h(all)f(the)i(redexes)i(in)d(a)h(term)f Fo(s)i Fq(2)g(T)11 b Fw(\()p Fq(F)t Fo(;)c Fq(V)s Fw(\))20 b(and)183 763 y(reduce)g(them)d(from)f(innermost)i(to)g(outermost,)f(pro)q (ducing)h Fo(t)g Fw(\(notation)g Fo(s)c Fq(!)1491 748 y Fl(GK)1491 777 y Fn(R)1520 769 y Fj(d)1520 786 y(n)1562 763 y Fo(t)p Fw(\).)1605 748 y Fm(4)183 819 y Fw(Rep)q(eat)g(the)h(pro) q(cess)g(with)f Fo(t)g Fw(and)f(so)h(on.)183 907 y Ff(Theorem)6 b(25.)21 b Fr(The)15 b(Gr)n(oss-Knuth)f(r)n(e)n(duction)h(str)n(ate)n (gy)f(is)g(normalizing,)g(i.e.,)g(if)g Fo(s)h Fr(has)183 957 y(a)g(normal)f(form)h(w.r.t.)e Fq(!)616 965 y Fn(R)645 957 y Fj(d)645 973 y(n)666 957 y Fr(,)i(then)g(every)30 b Fq(!)953 941 y Fl(GK)953 971 y Fn(R)982 963 y Fj(d)982 979 y(n)1040 957 y Fr(r)n(e)n(duction)14 b(se)n(quenc)n(e)i(is)f (\014nite.)183 1050 y(Pr)n(o)n(of.)20 b Fw(The)e(pro)q(of)f(is)g Fr(verb)n(atim)g Fw(the)h(same)f(as)g(O'Donnell's)g(pro)q(of)g(for)g (unconditional)183 1100 y(orthogonal)12 b(TRSs)i(and)g(not)g(rep)q (eated)h(here;)g(for)e(details)h(see)h([O'D77)o(],)e(Chapter)h(V.)183 1188 y(In)19 b(con)o(trast)i(to)e(the)i(unconditional)d(case,)j(the)f (Gross-Kn)o(uth)g(strategy)h(is)e(not)h(com-)183 1238 y(putable)11 b(for)h(CTRSs.)f(The)i(Gross-Kn)o(uth)f(reduction)h (strategy)f(w.r.t.)25 b Fq(\))1379 1246 y Fn(R)1408 1238 y Fj(d)1408 1255 y(n)1455 1238 y Fw(is)11 b(de\014ned)183 1288 y(in)i(analogy)e(to)j(De\014nition)e(24.)h(It)g(is)g(fairly)f (simple)g(to)i(see,)g(that)f(the)h(reduct)h Fo(t)e Fw(is)h(unique)183 1338 y(mo)q(dulo)d Fq(\030)p Fw(.)j(It)g(can)g(b)q(e)h(sho)o(wn)f(as)g (ab)q(o)o(v)o(e)f(that)28 b Fq(\))1007 1323 y Fl(GK)1007 1352 y Fn(R)1036 1344 y Fj(d)1036 1360 y(n)1092 1338 y Fw(is)14 b(normalizing)d(as)j(w)o(ell.)183 1426 y Ff(De\014niti)o(on) 5 b(26.)20 b Fw(Let)d Fq(\))e Fw(and)g Fq(!)h Fw(b)q(e)g(binary)f (relations)h(on)f Fq(T)1206 1411 y Fn(\003)1196 1436 y Fl(w)1241 1426 y Fw(and)h Fq(T)10 b Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\),)16 b(resp)q(ec-)183 1476 y(tiv)o(ely)m(.)g(\()p Fq(T)358 1461 y Fn(\003)348 1486 y Fl(w)377 1476 y Fo(;)7 b Fq(\))p Fw(\))17 b(is)g(called)h(a)f Fr(sound)j(implementation)d Fw(of)h(\()p Fq(T)10 b Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\))p Fo(;)g Fq(!)p Fw(\))17 b(if)g Fo(s)h Fq(\))1526 1461 y Fn(\003)1563 1476 y Fo(t)g Fq(2)183 1526 y Fo(N)5 b(F)h Fw(\()p Fq(\))p Fw(\))12 b(implies)g Fo(e)p Fw(\()p Fo(s)p Fw(\))h Fq(!)606 1511 y Fn(\003)636 1526 y Fo(e)p Fw(\()p Fo(t)p Fw(\))f Fq(2)f Fo(N)5 b(F)h Fw(\()p Fq(!)p Fw(\).)12 b(On)i(the)h(other)f(hand,)f(\()p Fq(T)1335 1511 y Fn(\003)1325 1536 y Fl(w)1354 1526 y Fo(;)7 b Fq(\))p Fw(\))13 b(is)g(called)h(a)183 1576 y Fr(c)n(omplete)f(implementation)e Fw(of)h(\()p Fq(T)e Fw(\()p Fq(F)t Fo(;)d Fq(V)s Fw(\))p Fo(;)g Fq(!)p 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(immediately)j(clear)d(b)o(y)221 2432 y(Lemma)g(8)g(\(2\).)p eop %%Page: 14 14 14 13 bop 340 194 a Fw(\(1\))16 b(W)m(e)f(pro)q(ceed)i(b)o(y)e (induction)g(on)g(the)h(length)f Fo(k)h Fw(of)f Fo(s)f Fq(\))1304 179 y Fn(\003)1304 208 y(R)1333 200 y Fj(d)1366 194 y Fo(t)p Fw(.)g(The)i(base)g(case)g Fo(k)f Fw(=)g(0)340 249 y(clearly)j(holds.)g(Th)o(us)g(consider)h Fo(s)14 b Fq(\))952 234 y Fl(l)962 222 y Fd(\003)980 234 y Fl(\033)952 264 y Fn(R)981 255 y Fj(d)981 272 y(n)1016 249 y Fo(t)1031 234 y Fn(0)1057 249 y Fq(\))1099 234 y Fl(k)1099 264 y Fn(R)1128 255 y Fj(d)1128 272 y(n)1163 249 y Fo(t)p Fw(.)k(According)g(to)g(the)h(inductiv)o(e)f(h)o(y-)340 313 y(p)q(othesis,)23 b Fo(e)p Fw(\()p Fo(t)571 298 y Fn(0)583 313 y Fw(\))p Fq(!)641 298 y Fn(\003)641 328 y(R)670 319 y Fj(d)670 336 y(n)691 313 y Fo(e)p Fw(\()p Fo(t)p Fw(\).)f(Since)g Fo(s)15 b Fq(\))983 298 y Fl(l)993 286 y Fd(\003)1010 298 y Fl(\033)983 328 y Fn(R)1012 319 y Fj(d)1012 336 y(n)1047 313 y Fo(t)1062 298 y Fn(0)1074 313 y 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y(con\015uen)o(t.)d(Th)o(us)f Fo(t)651 1683 y Fl(j)682 1677 y Fw(is)g(the)g(mark)o(ed)f(term)g(w)o(e)h(are)h(lo)q(oking)d (for.)340 1763 y Ff(Theorem)7 b(28.)36 b Fq(\))674 1769 y Fn(R)703 1773 y Fj(n)754 1763 y Fr(is)14 b(a)i(sound)f(and)h(c)n (omplete)f(implementation)g(of)g Fq(!)1592 1769 y Fn(R)1621 1773 y Fj(n)1642 1763 y Fr(.)340 1848 y(Pr)n(o)n(of.)20 b Fw(Again,)13 b(w)o(e)h(use)h(induction)e(on)h Fo(n)g Fw(to)g(sho)o(w:)358 1927 y(1.)20 b Fq(8)p Fo(s;)7 b(t)k Fq(2)h(T)571 1912 y Fn(\003)561 1937 y Fl(w)590 1927 y Fw(:)h(if)g Fo(s)i Fq(\))729 1912 y Fn(\003)729 1938 y(R)758 1942 y Fj(n)793 1927 y Fo(t)p Fw(,)e(then)i Fo(e)p Fw(\()p Fo(s)p Fw(\))p Fq(!)1040 1912 y Fn(\003)1040 1938 y(R)1069 1942 y Fj(n)1105 1927 y Fo(e)p Fw(\()p Fo(t)p Fw(\).)358 1977 y(2.)20 b Fq(8)p Fo(t)12 b Fq(2)f(T)533 1962 y Fn(\003)523 1987 y Fl(w)552 1977 y Fw(:)i Fo(t)f Fq(2)f Fo(N)5 b(F)h Fw(\()13 b Fq(\))785 1983 y Fn(R)814 1987 y Fj(n)850 1977 y Fw(\))h(if)f(and)g(only)h(if)f Fo(e)p Fw(\()p Fo(t)p Fw(\))f Fq(2)f Fo(N)5 b(F)h Fw(\()p Fq(!)1374 1983 y Fn(R)1403 1987 y Fj(n)1424 1977 y Fw(\).)358 2026 y(3.)20 b Fq(8)p Fo(s)13 b Fq(2)f(T)539 2011 y Fn(\003)529 2037 y Fl(w)558 2026 y Fw(:)i(if)g Fo(e)p Fw(\()p Fo(s)p Fw(\))p Fq(!)735 2011 y Fn(\003)735 2038 y(R)764 2042 y Fj(n)800 2026 y Fo(u)e Fq(2)h Fo(N)5 b(F)h Fw(\()p Fq(!)1006 2032 y Fn(R)1035 2036 y Fj(n)1056 2026 y Fw(\),)14 b(then)h(there)g(is)g(a)f(term)g Fo(t)e Fq(2)g(T)1578 2011 y Fn(\003)1567 2037 y Fl(w)1611 2026 y Fw(suc)o(h)j(that)411 2076 y Fo(s)f Fq(\))486 2061 y Fn(\003)486 2088 y(R)515 2092 y Fj(n)551 2076 y Fo(t)f Fw(and)h Fo(e)p Fw(\()p Fo(t)p Fw(\))e(=)g Fo(u)p Fw(.)340 2155 y(\(1\))i(The)h(pro)q(of)e(is)h (the)g(same)f(as)h(in)g(Prop)q(osition)f(27.)340 2205 y(\(2\))h(F)m(ollo)o(ws)f(from)f(Prop)q(osition)h(27)g(in)h (conjunction)g(with)f(Lemmata)e(12)j(and)f(23.)340 2254 y(\(3\))g(By)g(Prop)q(osition)f(11,)f(there)j(is)e(a)g(term)g Fo(v)i Fw(suc)o(h)g(that)e Fo(e)p Fw(\()p Fo(s)p Fw(\))p Fq(!)1360 2239 y 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Fo(u)p Fw(.)g(No)o(w)g(b)o(y)g(Theorem)g(28,)g(there) h(is)g(a)f(term)g Fo(t)g Fq(2)h(T)1426 413 y Fn(\003)1416 438 y Fl(w)1456 428 y Fw(suc)o(h)g(that)183 477 y Fo(s)i Fq(\))258 462 y Fn(\003)258 489 y(R)287 493 y Fj(n)322 477 y Fo(t)e Fq(2)f Fo(N)5 b(F)h Fw(\()13 b Fq(\))530 483 y Fn(R)559 487 y Fj(n)595 477 y Fw(\))h(and)f Fo(e)p Fw(\()p Fo(t)p Fw(\))f(=)g Fo(u)p Fw(.)h(It)h(is)g(easy)g(to)g(pro)o(v) o(e)g(that)g Fo(t)e Fq(2)f Fo(N)5 b(F)h Fw(\()13 b Fq(\))1515 483 y Fn(R)1559 477 y Fw(\).)183 607 y Fs(References)183 697 y Fv([AEH94])34 b(S.)13 b(An)o(to)o(y)m(,)j(R.)c(Ec)o(hahed,)18 b(and)f(M.)12 b(Han)o(us.)28 b(A)17 b(Needed)g(Narro)o(wing)g (Strategy.)28 b(In)361 742 y Fa(Pr)n(o)n(c.)10 b(21st)f(A)o(CM)i(Symp)n (osium)d(on)i(Principles)e(of)h(Pr)n(o)n(gr)n(amming)h(L)n(anguages)p Fv(,)c(pages)361 788 y(268{279,)14 b(1994.)183 834 y([BEG)277 818 y Ft(+)302 834 y Fv(87])21 b(H.P)m(.)14 b(Barendregt,)j(M.C.J.D.)11 b(v)n(an)16 b(Eek)o(elen,)g(J.R.W.)f(Glauert,)i(J.R.)d(Kenna)o(w)o(a)o (y)m(,)361 879 y(M.J.)g(Plasmeijer,)h(and)h(M.R.)e(Sleep.)22 b(T)m(erm)14 b(graph)i(rewriting.)22 b(In)15 b Fa(Pr)n(o)n(c.)f(Par)n (al)r(lel)361 925 y(A)o(r)n(chite)n(ctur)n(es)d(and)h(L)n(anguages)f (Eur)n(op)n(e)p Fv(,)g(pages)j(141{158.)g(LNCS)e Fu(259)p Fv(,)h(1987.)183 971 y([BK86])61 b(J.A.)14 b(Bergstra)h(and)h(J.W.)e (Klop.)23 b(Conditional)18 b(Rewrite)e(Rules:)g(Con\015uence)g(and)361 1016 y(T)m(ermination.)k Fa(Journal)13 b(of)g(Computer)h(and)e(System)h (Scienc)n(es)f Fu(32)p Fa(\(3\))p Fv(,)g(pages)i(323{)361 1062 y(362,)f(1986.)183 1108 y([DJ90])69 b(N.)12 b(Dersho)o(witz)j(and) g(J.-P)m(.)e(Jouannaud.)21 b(Rewrite)15 b(Systems.)20 b(In)14 b(J.)e(v)n(an)i(Leeu)o(w)o(en,)361 1153 y(editor,)e Fa(Handb)n(o)n(ok)d(of)j(The)n(or)n(etic)n(al)d(Computer)j(Scienc)n(e)p Fv(,)c(v)o(olume)14 b(B,)d(c)o(hapter)i(6.)e(El-)361 1199 y(sevier)j({)f(The)g(MIT)g(Press,)g(1990.)183 1244 y([Han94a])30 b(M.)13 b(Han)o(us.)20 b(On)14 b(Extra)g(V)m(ariables)h (in)g(\(Equational\))i(Logic)e(Programming.)21 b(T)m(ec)o(h-)361 1290 y(nical)15 b(Rep)q(ort)f(MPI-I-94-246,)f(Max-Plank-Institut)i(f)q (\177)-20 b(ur)13 b(Informatik,)h(1994.)183 1336 y([Han94b])28 b(M.)13 b(Han)o(us.)33 b(The)18 b(In)o(tegration)i(of)e(F)m(unctions)i (in)o(to)f(Logic)h(Programming:)g(F)m(rom)361 1381 y(Theory)15 b(to)e(Practice.)21 b Fa(The)14 b(Journal)f(of)g(L)n(o)n(gic)h(Pr)n(o)n (gr)n(amming)f Fu(19,20)p Fv(,)h(pages)h(583{)361 1427 y(628,)e(1994.)183 1473 y([Han95])49 b(M.)13 b(Han)o(us.)34 b(On)18 b(Extra)h(V)m(ariables)i(in)e(\(Equational\))i(Logic)f (Programming.)35 b(In)361 1518 y Fa(Pr)n(o)n(c)n(e)n(e)n(dings)17 b(of)h(the)g(12th)g(Internation)o(al)e(Confer)n(enc)n(e)h(on)h(L)n(o)n (gic)g(Pr)n(o)n(gr)n(amming)p Fv(,)361 1564 y(pages)c(665{679.)g(MIT)e (Press,)h(1995.)183 1610 y([Klo92])59 b(J.W.)18 b(Klop.)35 b(T)m(erm)18 b(Rewriting)i(Systems.)35 b(In)19 b(S.)12 b(Abramsky)m(,)19 b(D.)12 b(Gabba)o(y)m(,)20 b(and)361 1655 y(T.)12 b(Maibaum,)20 b(editors,)e Fa(Handb)n(o)n(ok)e(of)h(L)n(o) n(gic)h(in)f(Computer)h(Scienc)n(e)p Fv(,)d(v)o(olume)f(2,)361 1701 y(pages)g(1{116.)g(Oxford)f(Univ)o(ersit)o(y)i(Press,)e(1992.)183 1747 y([K)o(O95])59 b(M.)13 b(Kurihara)18 b(and)f(A.)12 b(Oh)o(uc)o(hi.)29 b(Mo)q(dularit)o(y)20 b(in)d(Noncop)o(ying)i(T)m (erm)d(Rewriting.)361 1792 y Fa(The)n(or)n(etic)n(al)c(Computer)h (Scienc)n(e)e Fu(152)p Fv(,)i(pages)g(139{169,)h(1995.)183 1838 y([MH94])54 b(A.)12 b(Middeldorp)j(and)d(E.)g(Hamo)q(en.)17 b(Completeness)c(Results)g(for)e(Basic)i(Narro)o(wing.)361 1884 y Fa(Applic)n(able)c(A)o(lgebr)n(a)g(in)i(Engine)n(ering,)e (Communic)n(ation)g(and)h(Computing)g Fu(5)p Fv(,)h(pages)361 1929 y(213{253,)j(1994.)183 1975 y([O'D77])48 b(M.J.)12 b(O'Donnell.)19 b Fa(Computing)12 b(in)h(Systems)f(Describ)n(e)n(d)g (by)h(Equations)p Fv(.)h(LNCS)f Fu(58)p Fv(,)361 2021 y(1977.)183 2066 y([Plu93])61 b(D.)13 b(Plump.)22 b Fa(Evaluation)12 b(of)i(F)m(unctional)e(Expr)n(essions)g(by)i(Hyp)n(er)n(gr)n(aph)e(R)n (ewriting)p Fv(.)361 2112 y(PhD)i(thesis,)g(Univ)o(ersit\177)-19 b(at)14 b(Bremen,)g(1993.)183 2158 y([SMI95])48 b(T.)12 b(Suzuki,)22 b(A.)12 b(Middeldorp)q(,)23 b(and)e(T.)12 b(Ida.)41 b(Lev)o(el-Con\015uence)23 b(of)d(Conditional)361 2203 y(Rewrite)14 b(Systems)g(with)f(Extra)h(V)m(ariables)g(in)g(Righ)o (t-Hand)h(Sides.)j(In)13 b Fa(Pr)n(o)n(c)n(e)n(e)n(dings)361 2249 y(of)i(the)g(6th)g(Internationa)o(l)e(Confer)n(enc)n(e)g(on)i(R)n (ewriting)g(T)m(e)n(chniques)e(and)h(Applic)n(a-)361 2295 y(tions)p Fv(,)d(pages)j(179{193.)g(LNCS)e Fu(914)p Fv(,)h(1995.)183 2340 y([Sta75])63 b(J.)12 b(Staples.)64 b(Ch)o(urc)o(h-Rosser)29 b(Theorems)g(for)f(Replacemen)o(t)i(Systems.) 63 b(In)361 2386 y(J.)12 b(Crosley)m(,)f(editor,)f Fa(A)o(lgebr)n(a)g (and)f(L)n(o)n(gic)p Fv(,)g(pages)i(291{307.)g(Lecture)f(Notes)g(in)h (Math-)361 2432 y(ematics)j Fu(450)p Fv(,)f(1975.)p eop %%Page: 16 16 16 15 bop 340 194 a Fv(This)14 b(article)g(w)o(as)f(pro)q(cessed)i (using)f(the)f(L)968 185 y Ft(A)985 194 y Fv(T)1007 202 y(E)1028 194 y(X)f(macro)h(pac)o(k)n(age)h(with)g(LLNCS)e(st)o(yle)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF