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3424 y Fk(\026)1447 3460 y Fr(\))f(=)f Fo(\026)p Fr(.)61 b(In)39 b(the)g(sequel)h(w)m(e)f(use)h(natural)d(n)m(um)m(b)s(ers)j(as)120 3580 y(marks.)50 b(The)35 b(set)h(of)e Fn(marke)-5 b(d)33 b Fr(terms)i(o)m(v)m(er)h Fq(F)1830 3538 y Fl(\003)1904 3580 y Fr(and)f Fq(V)2165 3538 y Fl(\003)2239 3580 y Fr(is)f(denoted)i(b)m(y)g Fq(T)25 b Fr(\()p Fq(F)3047 3544 y Fl(\003)3086 3580 y Fo(;)17 b Fq(V)3199 3544 y Fl(\003)3238 3580 y Fr(\).)50 b(Tw)m(o)120 3700 y(subterms)30 b Fo(t)570 3715 y Fp(1)639 3700 y Fr(and)f Fo(t)860 3715 y Fp(2)928 3700 y Fr(of)g(a)g(mark)m(ed)g(term)g Fo(t)g Fr(are)g(shared)h(in)e Fo(t)h Fr(if)f Fo(t)2511 3715 y Fp(1)2579 3700 y Fr(=)f Fo(t)2717 3715 y Fp(2)2757 3700 y Fr(;)j(e.g.)f(0)3038 3664 y Fp(1)3107 3700 y Fr(and)g(0)3342 3664 y Fp(1)3410 3700 y Fr(are)120 3821 y(shared)34 b(in)f Fo(eq)637 3785 y Fp(0)676 3821 y Fr(\(0)763 3785 y Fp(1)803 3821 y Fo(;)17 b Fr(0)896 3785 y Fp(1)934 3821 y Fr(\).)46 b(If)34 b Fo(t)f Fr(is)g(a)g(mark)m(ed)h(term,)f(then)h Fo(e)p 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y(but)i Fo(eq)393 4748 y Fp(1)432 4784 y Fr(\(0)519 4748 y Fp(1)558 4784 y Fo(;)17 b Fr(0)651 4748 y Fp(1)690 4784 y Fr(\))34 b(is)g(not.)49 b(W)-8 b(ell-mark)m(ed)33 b(terms)h(exactly)h(corresp)s(ond)g(to)f (directed)g(acyclic)120 4904 y(graphs;)44 b(the)c(reader)h(is)e (referred)i(to)e([K)m(O95])h(for)f(details.)64 b(In)40 b(con)m(trast)h(to)e([K)m(O95],)j(w)m(e)120 5025 y(de\014ne)34 b(graph)e(rewriting)f(solely)h(on)h(w)m(ell-mark)m(ed)e(terms.)266 5145 y(Let)i Fq(R)h Fr(b)s(e)f(a)f(TRS)i(o)m(v)m(er)g(the)f(signature)f Fq(F)10 b Fr(.)44 b(A)33 b(rule)g Fo(l)2289 5109 y Fl(\003)2356 5145 y Fq(!)28 b Fo(r)2531 5109 y Fl(\003)2603 5145 y Fr(is)33 b(a)f Fn(marke)-5 b(d)34 b(version)e Fr(of)120 5265 y(a)f(rule)f Fo(l)g Fq(!)e Fo(r)33 b Fr(in)d Fq(R)i Fr(if)e Fo(e)p Fr(\()p Fo(l)1087 5229 y Fl(\003)1126 5265 y Fr(\))e(=)g Fo(l)r Fr(,)j Fo(e)p Fr(\()p Fo(r)1515 5229 y Fl(\003)1555 5265 y Fr(\))c(=)h Fo(r)s Fr(,)j(and)g(ev)m(ery)i (mark)m(ed)e(v)-5 b(ariable)29 b(in)i Fo(r)3142 5229 y Fl(\003)3212 5265 y Fr(app)s(ears)120 5386 y(also)44 b(in)h Fo(l)486 5350 y Fl(\003)525 5386 y Fr(.)82 b(The)46 b Fn(term)g(gr)-5 b(aph)46 b(r)-5 b(ewrite)46 b(r)-5 b(elation)78 b Fq(\))2209 5401 y Fl(R)2354 5386 y Fq(\022)50 b(T)2535 5401 y Fk(w)2592 5386 y Fr(\()p Fq(F)2712 5350 y Fl(\003)2751 5386 y Fo(;)17 b Fq(V)2864 5350 y Fl(\003)2903 5386 y Fr(\))31 b Fq(\002)g(T)3134 5401 y Fk(w)3191 5386 y Fr(\()p Fq(F)3311 5350 y Fl(\003)3350 5386 y Fo(;)17 b Fq(V)3463 5350 y Fl(\003)3502 5386 y Fr(\))p eop %%Page: 3 3 3 2 bop 120 407 a Fr(w.r.t.)39 b Fq(R)h Fr(is)e(de\014ned)i(as)f(follo) m(ws:)55 b Fo(s)32 b Fq(\))1631 422 y Fl(R)1728 407 y Fo(t)39 b Fr(if)f(there)h(exists)h(a)e(mark)m(ed)i(v)m(ersion)f Fo(l)3238 371 y Fl(\003)3316 407 y Fq(!)f Fo(r)3501 371 y Fl(\003)120 527 y Fr(of)e(a)g(rewrite)g(rule)g Fo(l)h Fq(!)d Fo(r)39 b Fr(from)c Fq(R)p Fr(,)i(a)g(mark)m(ed)f(substitution)g Fo(\033)k Fr(and)d(a)f(mark)m(ed)h(con)m(text)120 648 y Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])32 b(con)m(taining)f (at)i(least)f(one)h(hole)f Fj(2)g Fr(suc)m(h)j(that)265 840 y Fq(\017)49 b Fo(s)27 b Fr(=)h Fo(C)7 b Fr([)p Fo(l)676 804 y Fl(\003)716 840 y Fo(\033)n(;)17 b(:)g(:)g(:)f(;)h(l)1019 804 y Fl(\003)1059 840 y Fo(\033)t Fr(])32 b(and)h Fo(t)28 b Fr(=)f Fo(C)7 b Fr([)p Fo(r)1684 804 y Fl(\003)1723 840 y Fo(\033)n(;)17 b(:)g(:)g(:)g(;)g(r)2043 804 y Fl(\003)2082 840 y Fo(\033)t Fr(],)265 1040 y Fq(\017)49 b Fo(l)395 1004 y Fl(\003)435 1040 y Fo(\033)36 b Fr(is)c(not)h(a)f(subterm)h(of)f Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(].)120 1233 y Fo(l)151 1197 y Fl(\003)191 1233 y Fo(\033)42 b Fr(is)d(called)e(the) i Fn(c)-5 b(ontr)g(acte)g(d)40 b(marke)-5 b(d)40 b(r)-5 b(e)g(dex)38 b Fr(in)g Fo(s)p Fr(.)62 b(W)-8 b(e)39 b(use)h(the)f (notation)e Fo(s)c Fq(\))3272 1197 y Fk(l)3294 1173 y Fi(\003)3305 1197 y Fk(\033)3272 1258 y Fl(R)3385 1233 y Fo(t)39 b Fr(in)120 1353 y(order)32 b(to)f(sp)s(ecify)h(the)g(con)m (tracted)g(mark)m(ed)g(redex.)44 b(Note)32 b(that)f(all)f(shared)i (subterms)g Fo(l)3441 1317 y Fl(\003)3481 1353 y Fo(\033)120 1474 y Fr(are)44 b(replaced)g(sim)m(ultaneously)f(with)h Fo(r)1645 1438 y Fl(\003)1684 1474 y Fo(\033)t Fr(.)78 b(Observ)m(e)46 b(moreo)m(v)m(er)e(that)g(if)f Fo(s)33 b Fq(\))3168 1489 y Fl(R)3264 1474 y Fo(t)3299 1489 y Fp(1)3383 1474 y Fr(and)120 1594 y Fo(s)f Fq(\))298 1609 y Fl(R)395 1594 y Fo(t)430 1609 y Fp(2)502 1594 y Fr(b)m(y)i(con)m (tracting)e(the)h(same)f(mark)m(ed)h(redex,)h(then)f Fo(t)2456 1609 y Fp(1)2524 1594 y Fq(\030)28 b Fo(t)2664 1609 y Fp(2)2703 1594 y Fr(.)266 1715 y(The)40 b(purp)s(ose)f(of)f(a)h (term)f(graph)g(rewrite)h(relation)e(is)h(to)g(mo)s(del)f (non-deterministic)120 1835 y(computations.)63 b(F)-8 b(rom)37 b(this)i(p)s(oin)m(t)g(of)g(view,)i(the)f(ab)s(o)m(v)m(e)g (de\014nition)e(seems)i(to)f(b)s(e)h(a)f(bit)120 1955 y(unsatisfactory)23 b(b)s(ecause)i(it)d(do)s(esn't)i(sp)s(ecify)g(ho)m (w)g(the)g(righ)m(t-hand)e(side)h Fo(r)j Fr(of)d(a)g(rule)g(should)120 2076 y(b)s(e)44 b(mark)m(ed.)78 b(On)44 b(the)h(other)f(hand,)j(it)c (allo)m(ws)g(ev)m(ery)j(marking)d(sc)m(heme)i(that)f(ensures)120 2196 y(that)39 b(the)h(reduct)g(of)f(a)g(w)m(ell-mark)m(ed)g(term)g(is) g(again)e(w)m(ell-mark)m(ed.)64 b(So)39 b(it)f(includes)h(for)120 2316 y(instance)28 b(the)f(sc)m(heme)i(that)e(requires)h(that)f(the)h (marks)f(on)g(function)g(sym)m(b)s(ols)g(in)f Fo(r)k Fr(should)120 2437 y(b)s(e)45 b(\\fresh")h(and)f(m)m(utually)f (distinct.)81 b(More)45 b(details)f(on)i(di\013eren)m(t)f(structure)i (sharing)120 2557 y(sc)m(hemes)34 b(can)f(b)s(e)g(found)g(in)e([K)m (O95].)120 2771 y Fh(Example)36 b(2.1)49 b Fr(Let)40 b Fq(R)e Fr(=)h Fq(f)p Fo(eq)t Fr(\()p Fo(x;)17 b(x)p Fr(\))38 b Fq(!)g Fo(tr)s(ue)p Fq(g)g Fr(and)h Fo(s)g Fr(=)f Fo(eq)2499 2735 y Fp(0)2538 2771 y Fr(\()p Fo(eq)2668 2735 y Fp(1)2708 2771 y Fr(\(0)2795 2735 y Fp(2)2834 2771 y Fo(;)17 b Fr(0)2927 2735 y Fp(2)2966 2771 y Fr(\))p Fo(;)g(eq)3140 2735 y Fp(1)3179 2771 y Fr(\(0)3266 2735 y Fp(2)3305 2771 y Fo(;)g Fr(0)3398 2735 y Fp(2)3437 2771 y Fr(\)\).)120 2892 y(Then)45 b(w)m(e)h(ha)m(v)m(e)f Fo(s)33 b Fq(\))957 2907 y Fl(R)1053 2892 y Fo(eq)1145 2856 y Fp(0)1185 2892 y Fr(\()p Fo(tr)s(ue)1406 2856 y Fp(4)1445 2892 y Fo(;)17 b(tr)s(ue)1672 2856 y Fp(4)1711 2892 y Fr(\))44 b(b)m(y)h(using)f(the)h(mark)m(ed)f(rule)g Fo(eq)3044 2856 y Fp(1)3083 2892 y Fr(\()p Fo(x)3176 2856 y Fp(3)3216 2892 y Fo(;)17 b(x)3315 2856 y Fp(3)3355 2892 y Fr(\))47 b Fq(!)120 3012 y Fo(tr)s(ue)303 2976 y Fp(4)342 3012 y Fr(.)h(Since)35 b(the)g(same)f(v)-5 b(ariable)32 b(in)i(the)g(left-hand)f(side)i(of)e(a)h(rule)g(ma)m(y)g (ha)m(v)m(e)i(di\013eren)m(t)120 3132 y(marks,)25 b(w)m(e)g(also)d(ha)m (v)m(e)j(e.g.)f Fo(eq)1229 3096 y Fp(0)1269 3132 y Fr(\()p Fo(eq)1399 3096 y Fp(1)1438 3132 y Fr(\(0)1525 3096 y Fp(2)1564 3132 y Fo(;)17 b Fr(0)1657 3096 y Fp(3)1696 3132 y Fr(\))p Fo(;)g(eq)1870 3096 y Fp(1)1909 3132 y Fr(\(0)1996 3096 y Fp(2)2035 3132 y Fo(;)g Fr(0)2128 3096 y Fp(3)2167 3132 y Fr(\)\))33 b Fq(\))2376 3147 y Fl(R)2472 3132 y Fo(eq)2564 3096 y Fp(0)2604 3132 y Fr(\()p Fo(tr)s(ue)2825 3096 y Fp(6)2864 3132 y Fo(;)17 b(tr)s(ue)3091 3096 y Fp(6)3129 3132 y Fr(\))24 b(b)m(y)g(using)120 3253 y(the)33 b(rule)f Fo(eq)575 3217 y Fp(1)615 3253 y Fr(\()p Fo(x)708 3217 y Fp(4)747 3253 y Fo(;)17 b(x)846 3217 y Fp(5)886 3253 y Fr(\))28 b Fq(!)f Fo(tr)s(ue)1262 3217 y Fp(6)1301 3253 y Fr(.)120 3540 y Fm(2.2)135 b(Basic)45 b(Notions)h(of)f(Disjoin)l(t)i(Unions)120 3725 y Fr(Next)d(w)m(e)f(giv) m(e)g(a)f(brief)g(o)m(v)m(erview)j(of)d(the)h(basic)f(notions)h(of)f (disjoin)m(t)f(unions)i(of)f(graph)120 3845 y(rewrite)31 b(relations.)41 b(Let)31 b(\()p Fq(F)1165 3860 y Fp(1)1204 3845 y Fo(;)17 b Fq(R)1332 3860 y Fp(1)1372 3845 y Fr(\))30 b(and)h(\()p Fq(F)1738 3860 y Fp(2)1777 3845 y Fo(;)17 b Fq(R)1905 3860 y Fp(2)1945 3845 y Fr(\))30 b(b)s(e)h(t)m(w)m(o)g (disjoin)m(t)f(rewrite)h(systems,)h(i.e.,)120 3965 y Fq(F)192 3980 y Fp(1)253 3965 y Fq(\\)23 b(F)414 3980 y Fp(2)480 3965 y Fr(=)28 b Fq(;)p Fr(.)266 4086 y(In)d(the)f(sequel)h (let)e Fo(t)28 b Fq(2)g(T)e Fr(\()p Fq(F)1310 4050 y Fl(\003)1300 4110 y Fp(1)1353 4086 y Fq(])t(F)1505 4050 y Fl(\003)1495 4110 y Fp(2)1545 4086 y Fo(;)17 b Fq(V)1658 4050 y Fl(\003)1697 4086 y Fr(\).)41 b(Supp)s(ose)24 b Fo(t)k Fr(=)g Fo(C)7 b Fr([)p Fo(t)2482 4101 y Fp(1)2522 4086 y Fo(;)17 b(:)g(:)g(:)e(;)i(t)2775 4101 y Fk(n)2822 4086 y Fr(])24 b(with)g Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])27 b Fq(6)p Fr(=)120 4206 y Fj(2)p Fr(.)43 b(W)-8 b(e)29 b(write)g Fo(t)f Fr(=)g Fo(C)7 b Fr([)-17 b([)p Fo(t)990 4221 y Fp(1)1030 4206 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)1284 4221 y Fk(n)1330 4206 y Fr(])-17 b(])30 b(if)e Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])27 b Fq(2)i(T)c Fr(\()p Fq(F)2154 4170 y Fl(\003)2144 4232 y Fk(k)2193 4206 y Fo(;)17 b Fq(V)2306 4170 y Fl(\003)2345 4206 y Fr(\))29 b(and)h Fo(r)s(oot)p Fr(\()p Fo(t)2848 4221 y Fp(1)2887 4206 y Fr(\))p Fo(;)f(:)17 b(:)g(:)f(r)s(oot)p Fr(\()p Fo(t)3361 4221 y Fk(n)3408 4206 y Fr(\))28 b Fq(2)120 4327 y(F)202 4290 y Fl(\003)192 4352 y Fp(3)p Fl(\000)p Fk(k)368 4327 y Fr(for)43 b(some)h Fo(k)49 b Fq(2)e(f)p Fr(1)p Fo(;)17 b Fr(2)p Fq(g)p Fr(.)76 b(In)44 b(this)f(case)i(w)m(e)f(de\014ne)h(the)f Fn(set)g Fo(S)6 b Fr(\()p Fo(t)p Fr(\))43 b(of)h(all)d Fn(princip)-5 b(al)120 4447 y Fr(subterms)30 b(of)f Fo(t)h Fr(to)f(b)s(e)h Fo(S)6 b Fr(\()p Fo(t)p Fr(\))28 b(=)f Fq(f)p Fo(t)1347 4462 y Fp(1)1387 4447 y Fo(;)17 b(:)g(:)g(:)e(;)i(t)1640 4462 y Fk(n)1687 4447 y Fq(g)p Fr(.)43 b(The)30 b(motiv)-5 b(ation)26 b(for)j(this)h(de\014nition)e(of)h Fo(S)6 b Fr(\()p Fo(t)p Fr(\))120 4567 y(is)36 b(the)i(follo)m(wing:)49 b(If)37 b Fo(t)990 4582 y Fk(i)1056 4567 y Fr(and)g Fo(t)1285 4582 y Fk(j)1321 4567 y Fr(,)i Fo(i)c Fq(6)p Fr(=)g Fo(j)6 b Fr(,)38 b(o)s(ccur)f(in)f Fo(t)g Fr(=)f Fo(C)7 b Fr([)-17 b([)p Fo(t)2393 4582 y Fp(1)2433 4567 y Fo(;)17 b(:)g(:)g(:)f(;)h(t) 2687 4582 y Fk(n)2734 4567 y Fr(])-17 b(])37 b(and)g Fo(t)3037 4582 y Fk(i)3101 4567 y Fq(\030)e Fo(t)3248 4582 y Fk(j)3285 4567 y Fr(,)j(then)120 4688 y(these)e(mark)m(ed)f (terms)g(should)g(b)s(oth)g(b)s(e)g(represen)m(ted)i(only)d(if)g(they)i (are)f(not)g(shared)g(\(i.e.)120 4808 y Fo(t)155 4823 y Fk(i)211 4808 y Fq(6)p Fr(=)28 b Fo(t)350 4823 y Fk(j)386 4808 y Fr(\).)44 b(Moreo)m(v)m(er,)34 b(w)m(e)g(de\014ne)f(for)f(all)f Fo(t)234 5077 y(r)s(ank)s Fr(\()p Fo(t)p Fr(\))d(=)686 4936 y Fg(\032)803 5016 y Fr(1)1248 b(,)33 b(if)e Fo(t)d Fq(2)g(T)e Fr(\()p Fq(F)2606 4980 y Fl(\003)2596 5040 y Fp(1)2645 5016 y Fo(;)17 b Fq(V)2758 4980 y Fl(\003)2797 5016 y Fr(\))22 b Fq([)h(T)i Fr(\()p Fq(F)3145 4980 y Fl(\003)3135 5040 y Fp(2)3184 5016 y Fo(;)17 b Fq(V)3297 4980 y Fl(\003)3337 5016 y Fr(\))803 5136 y(1)k(+)i Fo(max)p Fq(f)p Fo(r)s(ank)s Fr(\()p Fo(t)1496 5151 y Fk(j)1533 5136 y Fr(\))17 b Fq(j)g Fr(1)26 b Fq(\024)j Fo(j)k Fq(\024)28 b Fo(n)p Fq(g)p Fr(,)33 b(if)e Fo(t)d Fr(=)g Fo(C)7 b Fr([)-17 b([)p Fo(t)2565 5151 y Fp(1)2605 5136 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)2859 5151 y Fk(n)2906 5136 y Fr(])-17 b(])120 5342 y(Note)33 b(that)f Fo(r)s(ank)s Fr(\()p Fo(t)p Fr(\))c(=)f Fo(r)s(ank)s Fr(\()p Fo(e)p Fr(\()p Fo(t)p Fr(\)\),)33 b(that)f(is,)g(the)h(rank)g(of)f(a)g(term)g(is)g (indep)s(enden)m(t)i(of)d(its)120 5463 y(marking.)p eop %%Page: 4 4 4 3 bop 266 407 a Fr(The)32 b(topmost)e(homogeneous)g(part)g(of)g Fo(t)p Fr(,)h(denoted)h(b)m(y)f Fo(top)p Fr(\()p Fo(t)p Fr(\),)g(is)f(obtained)g(from)f Fo(t)i Fr(b)m(y)120 527 y(replacing)g(all)g(principal)f(subterms)k(with)e Fj(2)p Fr(,)h(i.e.,)926 789 y Fo(top)p Fr(\()p Fo(t)p Fr(\))28 b(=)1300 648 y Fg(\032)1416 728 y Fo(t)397 b Fr(if)82 b Fo(r)s(ank)s Fr(\()p Fo(t)p Fr(\))28 b(=)g(1)1416 848 y Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])82 b(if)g Fo(t)28 b Fr(=)g Fo(C)7 b Fr([)-17 b([)p Fo(t)2304 863 y Fp(1)2344 848 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)2598 863 y Fk(n)2645 848 y Fr(])-17 b(])266 1050 y(F)-8 b(rom)31 b(no)m(w)j(on)e Fq(\))g Fr(denotes)i Fq(\))1446 1065 y Fl(R)1506 1074 y Ff(1)1540 1065 y Fl(]R)1647 1074 y Ff(2)1687 1050 y Fr(.)266 1170 y(A)h(subterm)g(of)g(a)f(mark)m(ed)h (term)g Fo(s)f Fr(is)h(an)g Fn(inner)f Fr(subterm)h(if)f(it)f(o)s (ccurs)j(in)e(one)h(of)g(the)120 1291 y(principal)23 b(subterms)i(of)g Fo(s)p Fr(,)h(otherwise)g(it)d(is)i(called)f Fn(outer)h Fr(subterm)g(of)f Fo(s)p Fr(.)41 b(A)25 b(reduction)g(step) 120 1411 y Fo(s)p Fq(\))p Fo(t)36 b Fr(is)g(called)g Fn(inner)g Fr(\(denoted)h(b)m(y)h Fo(s)p Fq(\))1677 1375 y Fk(i)1705 1411 y Fo(t)p Fr(\),)g(if)e(at)g(least)g(one)h(of)f(the)h (o)s(ccurrences)h(of)f(the)120 1531 y(con)m(tracted)31 b(redex)h(is)d(an)h(inner)g(subterm.)43 b(Otherwise,)31 b(w)m(e)h(sp)s(eak)f(of)e(an)h Fn(outer)h Fr(reduction)120 1652 y(step)i(and)f(write)f Fo(s)p Fq(\))909 1616 y Fk(o)947 1652 y Fo(t)p Fr(.)44 b(A)32 b(rewrite)f(step)i Fo(s)p Fq(\))p Fo(t)f Fr(is)f Fn(destructive)j(at)h(level)e(1)f Fr(if)f Fo(r)s(oot)p Fr(\()p Fo(s)p Fr(\))c Fq(2)h(F)3501 1616 y Fl(\003)3491 1678 y Fk(k)120 1772 y Fr(and)37 b Fo(r)s(oot)p Fr(\()p Fo(t)p Fr(\))f Fq(2)g(F)821 1736 y Fl(\003)811 1798 y Fp(3)p Fl(\000)p Fk(k)980 1772 y Fr(for)h(some)g Fo(k)h Fq(2)e(f)p Fr(1)p Fo(;)17 b Fr(2)p Fq(g)p Fr(.)57 b(Note)37 b(that)g(an)g(inner)g(graph)g(rewrite)g(step) 120 1893 y(cannot)g(b)s(e)h(destructiv)m(e)g(at)f(lev)m(el)g(1.)57 b(A)37 b(reduction)h(step)g Fo(s)p Fq(\))o Fo(t)g Fr(is)e Fn(destructive)j(at)h(level)e(2)120 2013 y Fr(if)j Fo(s)j Fr(=)g Fo(C)7 b Fr([)-17 b([)p Fo(s)589 2028 y Fp(1)629 2013 y Fo(;)17 b(:)g(:)g(:)f(;)h(s)894 2028 y Fk(j)930 2013 y Fo(;)g(:)g(:)g(:)f(;)h(s)1195 2028 y Fk(n)1241 2013 y Fr(])-17 b(])p Fq(\))q Fo(C)7 b Fr([)p Fo(t)1518 2028 y Fp(1)1557 2013 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)1811 2028 y Fk(j)1848 2013 y Fo(;)g(:)g(:)g(:)e(;)i(t)2101 2028 y Fk(n)2148 2013 y Fr(])44 b(=)g Fo(t)f Fr(with)f Fo(s)2695 2028 y Fk(j)2731 2013 y Fq(\))p Fo(t)2866 2028 y Fk(j)2945 2013 y Fr(destructiv)m(e)h(at)120 2133 y(lev)m(el)34 b(1)g(for)g(at)g(least)g(one)h Fo(j)i Fq(2)31 b(f)p Fr(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)p Fq(g)p Fr(.)49 b(Ob)m(viously)-8 b(,)35 b(if)e(a)h(graph)g(rewrite)h(step)g Fo(s)p Fq(\))p Fo(t)f Fr(is)120 2254 y(destructiv)m(e)g(at)e(lev)m(el)h(2,)f(then)h Fo(s)p Fq(\))1437 2218 y Fk(i)1465 2254 y Fo(t)p Fr(.)120 2585 y Fs(3)161 b(The)53 b(Simple)h(Pro)t(of)120 2804 y Fr(Let)38 b(\()p Fe(N)9 b Fo(;)17 b(>)p Fr(\))43 b(denote)c(the)f (usual)g(w)m(ell-founded)f(ordering)g(on)h(the)g(natural)f(n)m(um)m(b)s (ers)i(and)120 2924 y(\()p Fq(M)p Fr(\()p Fe(N)8 b Fr(\))p Fo(;)17 b(>)539 2888 y Fk(mul)q(t)700 2924 y Fr(\))32 b(its)g(w)m(ell-founded)g(m)m(ultiset)f(extension.)45 b(W)-8 b(e)33 b(de\014ne)1108 3133 y Fo(]s)27 b Fr(=)h([)p Fo(r)s(ank)s Fr(\()p Fo(t)p Fr(\))33 b Fq(j)f Fo(t)c Fq(2)g Fo(S)6 b Fr(\()p Fo(s)p Fr(\)])27 b Fq(2)i(M)p Fr(\()p Fe(N)8 b Fr(\))p Fo(;)120 3342 y Fr(i.e.,)32 b Fo(]s)h Fr(denotes)g(the)g(m)m(ultiset)f(of)g(the)h(ranks)g(of)f(the) h(terms)g(in)f(the)h(set)g Fo(S)6 b Fr(\()p Fo(s)p Fr(\).)120 3557 y Fh(Lemma)37 b(3.1)49 b Fr(If)33 b Fo(s)p Fq(\))952 3521 y Fl(\003)992 3557 y Fo(t)p Fr(,)f(then)i Fo(r)s(ank)s Fr(\()p Fo(s)p Fr(\))27 b Fq(\025)i Fo(r)s(ank)s Fr(\()p Fo(t)p Fr(\).)120 3773 y Fh(Lemma)37 b(3.2)49 b Fr(If)33 b Fo(s)p Fq(\))o Fo(t)g Fr(is)f(a)g(non-destructiv)m(e)i(graph)f (rewrite)f(step,)i(then)f Fo(]s)27 b Fq(\025)3139 3737 y Fk(mul)q(t)3322 3773 y Fo(]t)p Fr(.)120 3988 y Fh(Pro)s(of:)53 b Fr(Supp)s(ose)39 b Fo(r)s(oot)p Fr(\()p Fo(s)p Fr(\))d Fq(2)h(F)1382 3947 y Fl(\003)1382 4013 y Fp(1)1459 3988 y Fr(and)h Fo(s)e Fr(=)g Fo(C)7 b Fr([)-17 b([)p Fo(s)2008 4003 y Fp(1)2048 3988 y Fo(;)17 b(:)g(:)g(:)f(;)h(s)2313 4003 y Fk(n)2360 3988 y Fr(])-17 b(])33 b Fq(\))2530 3952 y Fk(l)2552 3929 y Fi(\003)2564 3952 y Fk(\033)2643 3988 y Fo(t)p Fr(.)59 b(If)38 b Fo(s)32 b Fq(\))3045 4003 y Fl(R)3105 4012 y Ff(2)3176 3988 y Fo(t)p Fr(,)39 b(then)g Fo(t)120 4109 y Fr(can)h(b)s(e)g(written)g(as)g Fo(C)7 b Fr([)-17 b([)p Fo(t)1071 4124 y Fp(1)1111 4109 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)1365 4124 y Fk(n)1412 4109 y Fr(])-17 b(],)42 b(where)f(either)f Fo(s)2137 4124 y Fk(j)2214 4109 y Fr(=)f Fo(t)2364 4124 y Fk(j)2441 4109 y Fr(or)h Fo(s)2614 4124 y Fk(j)2683 4109 y Fq(\))2783 4073 y Fk(l)2805 4049 y Fi(\003)2816 4073 y Fk(\033)2896 4109 y Fo(t)2931 4124 y Fk(j)2968 4109 y Fr(,)h(b)s(ecause)h(the)120 4229 y(reduction)36 b(step)g(is)g(non-destructiv)m(e)h(at)e(lev)m(el)h (2.)53 b(In)36 b(particular,)f Fo(r)s(ank)s Fr(\()p Fo(s)2937 4244 y Fk(j)2973 4229 y Fr(\))e Fq(\025)h Fo(r)s(ank)s Fr(\()p Fo(t)3438 4244 y Fk(j)3475 4229 y Fr(\).)120 4349 y(Since)45 b Fo(s)433 4364 y Fk(i)510 4349 y Fr(=)j Fo(s)680 4364 y Fk(j)761 4349 y Fr(implies)42 b Fo(t)1139 4364 y Fk(i)1216 4349 y Fr(=)49 b Fo(t)1376 4364 y Fk(j)1412 4349 y Fr(,)f(it)c(ob)m(viously)h(follo)m(ws)e(that)i Fo(]s)j Fq(\025)2806 4313 y Fk(mul)q(t)3010 4349 y Fo(]t)p Fr(.)80 b(Supp)s(ose)120 4470 y(otherwise)33 b Fo(s)f Fq(\))732 4485 y Fl(R)792 4494 y Ff(1)863 4470 y Fo(t)h Fr(is)f(non-destructiv)m(e)h(at)f(lev)m(el)g(1.)43 b(If)32 b Fo(s)h Fq(\))2453 4485 y Fl(R)2513 4494 y Ff(1)2584 4470 y Fo(t)g Fr(is)e(an)i(outer)f(reduction)120 4590 y(step,)50 b(then)c Fo(t)j Fr(=)h Fo(C)892 4554 y Fl(0)915 4590 y Fr([)-17 b([)p Fo(t)987 4605 y Fp(1)1027 4590 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)1281 4605 y Fk(m)1348 4590 y Fr(])-17 b(],)49 b(where)e Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)e(;)i Fr(])33 b Fq(\))2238 4554 y Fk(l)2260 4531 y Fi(\003)2272 4554 y Fk(\033)2351 4590 y Fo(C)2428 4554 y Fl(0)2451 4590 y Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])45 b(and)h Fo(S)6 b Fr(\()p Fo(t)p Fr(\))49 b Fq(\022)h Fo(S)6 b Fr(\()p Fo(s)p Fr(\).)120 4711 y(Hence)38 b Fo(]s)d Fq(\025)610 4674 y Fk(mul)q(t)800 4711 y Fo(]t)p Fr(.)57 b(If)36 b Fo(s)d Fq(\))1237 4726 y Fl(R)1297 4735 y Ff(1)1368 4711 y Fo(t)k Fr(is)f(an)h(inner)g(reduction)g(step,)h (then)g Fo(t)f Fr(can)g(b)s(e)g(written)120 4831 y(as)j Fo(C)324 4795 y Fl(0)347 4831 y Fr([)-17 b([)p Fo(t)419 4846 y Fp(1)460 4831 y Fo(;)17 b(:)g(:)g(:)e(;)i(t)713 4846 y Fk(m)780 4831 y Fr(])-17 b(],)42 b(where)g(either)d Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])40 b(=)h Fo(C)2043 4795 y Fl(0)2066 4831 y Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])39 b(or)h Fo(C)7 b Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(])32 b Fq(\))2987 4795 y Fk(l)3009 4771 y Fi(\003)3021 4795 y Fk(\033)3101 4831 y Fo(C)3178 4795 y Fl(0)3201 4831 y Fr([)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(].)3501 4795 y Fp(1)120 4951 y Fr(Moreo)m(v)m(er,)42 b(for)c(ev)m(ery)i Fo(k)h Fq(2)e(f)p Fr(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(m)p Fq(g)p Fr(,)40 b(there)f(is)f(a)h Fo(j)2204 4966 y Fk(k)2285 4951 y Fq(2)f(f)p Fr(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(n)p Fq(g)39 b Fr(suc)m(h)h(that)e(either)120 5072 y Fo(s)166 5087 y Fk(j)195 5099 y Fd(k)264 5072 y Fr(=)28 b Fo(t)403 5087 y Fk(k)478 5072 y Fr(or)33 b Fo(s)644 5087 y Fk(j)673 5099 y Fd(k)747 5072 y Fq(\))847 5036 y Fk(l)869 5012 y Fi(\003)881 5036 y Fk(\033)960 5072 y Fo(t)995 5087 y Fk(k)1038 5072 y Fr(.)44 b(Again,)31 b(it)h(follo)m(ws)f(that)h Fo(]s)c Fq(\025)2238 5036 y Fk(mul)q(t)2421 5072 y Fo(]t)p Fr(.)120 5287 y Fh(Lemma)37 b(3.3)49 b Fr(If)33 b Fo(s)p Fq(\))952 5251 y Fk(i)980 5287 y Fo(t)g Fr(is)f(destructiv)m(e)i(at)f (lev)m(el)f(2,)g(then)h Fo(]s)28 b(>)2505 5251 y Fk(mul)q(t)2687 5287 y Fo(]t)p Fr(.)p 120 5372 1368 4 v 232 5433 a Fc(1)269 5463 y Fb(Note)g(that)g(the)g(latter)f(cannot)g(happ)r(en)h(in)g(pure)f (term)h(rewriting.)p eop %%Page: 5 5 5 4 bop 120 407 a Fh(Pro)s(of:)45 b Fr(W)-8 b(e)33 b(ha)m(v)m(e)i Fo(s)29 b Fr(=)g Fo(C)7 b Fr([)-17 b([)p Fo(s)1202 422 y Fp(1)1242 407 y Fo(;)17 b(:)g(:)g(:)e(;)i(s)1506 422 y Fk(n)1553 407 y Fr(])-17 b(])33 b Fq(\))1723 371 y Fk(l)1745 347 y Fi(\003)1757 371 y Fk(\033)1836 407 y Fo(C)7 b Fr([)p Fo(t)1975 422 y Fp(1)2015 407 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)2269 422 y Fk(n)2316 407 y Fr(])29 b(=)g Fo(t)p Fr(,)k(where)i(either)e Fo(s)3178 422 y Fk(j)3244 407 y Fr(=)28 b Fo(t)3383 422 y Fk(j)3453 407 y Fr(or)120 527 y Fo(s)166 542 y Fk(j)235 527 y Fq(\))335 491 y Fk(l)357 468 y Fi(\003)369 491 y Fk(\033)448 527 y Fo(t)483 542 y Fk(j)558 527 y Fr(for)38 b(1)e Fq(\024)i Fo(j)43 b Fq(\024)38 b Fo(n)p Fr(.)60 b(Note)38 b(that)g(the)g(outer)h (con)m(text)g Fo(C)45 b Fr(is)38 b(not)g(a\013ected)h(b)m(y)g(the)120 648 y(reduction)i(step.)68 b(Moreo)m(v)m(er,)44 b(there)d(is)g(at)f (least)g(one)h(index)g Fo(k)i Fr(suc)m(h)f(that)f Fo(s)3068 663 y Fk(k)3143 648 y Fq(\))3243 611 y Fk(l)3265 588 y Fi(\003)3277 611 y Fk(\033)3356 648 y Fo(t)3391 663 y Fk(k)3474 648 y Fr(is)120 768 y(destructiv)m(e)c(at)e(lev)m(el)f(1.) 51 b(Apparen)m(tly)-8 b(,)37 b([)p Fo(r)s(ank)s Fr(\()p Fo(s)1951 783 y Fk(k)1994 768 y Fr(\)])32 b Fo(>)2167 732 y Fk(mul)q(t)2354 768 y Fo(]t)2427 783 y Fk(k)2469 768 y Fr(.)52 b(It)35 b(is)g(not)g(di\016cult)f(to)h(see)120 888 y(that)d Fo(]s)c(>)519 852 y Fk(mul)q(t)701 888 y Fo(]t)33 b Fr(follo)m(ws)e(as)i(a)f(consequence)k(b)s(ecause)e Fo(s)2286 903 y Fk(i)2342 888 y Fr(=)27 b Fo(s)2491 903 y Fk(j)2560 888 y Fr(implies)j Fo(t)2926 903 y Fk(i)2982 888 y Fr(=)e Fo(t)3121 903 y Fk(j)3158 888 y Fr(.)120 1085 y Fh(De\014nition)36 b(3.4)49 b Fr(F)-8 b(or)43 b(an)m(y)h(rewrite)f(deriv)-5 b(ation)42 b Fo(D)48 b Fr(:)e Fo(s)2274 1100 y Fp(1)2314 1085 y Fq(\))o Fo(s)2459 1100 y Fp(2)2499 1085 y Fq(\))o Fo(s)2644 1100 y Fp(3)2684 1085 y Fq(\))16 b Fo(:)h(:)g(:)43 b Fr(w)m(e)h(de\014ne)h(the)120 1205 y(rank)33 b(of)f Fo(D)j Fr(to)d(b)s(e)h Fo(r)s(ank)s Fr(\()p Fo(D)s Fr(\))28 b(=)f Fo(r)s(ank)s Fr(\()p Fo(s)1619 1220 y Fp(1)1659 1205 y Fr(\).)120 1401 y Fh(Theorem)37 b(3.5)49 b Fr(Let)34 b Fq(R)1047 1416 y Fp(1)1121 1401 y Fr(and)f Fq(R)1395 1416 y Fp(2)1469 1401 y Fr(b)s(e)h(disjoin)m(t)e (TRSs.)48 b(Their)34 b(com)m(bined)f(graph)h(rewrite)120 1522 y(relation)d Fq(\))h Fr(is)g(terminating)e(if)h(and)i(only)f(if)g (b)s(oth)64 b Fq(\))2184 1537 y Fl(R)2244 1546 y Ff(1)2348 1522 y Fr(and)h Fq(\))2670 1537 y Fl(R)2730 1546 y Ff(2)2834 1522 y Fr(are)33 b(terminating.)120 1718 y Fh(Pro)s(of:)60 b Fr(The)42 b(only)e(if)g(direction)g(is)h(eviden)m(tly)g(true.)69 b(F)-8 b(or)40 b(an)h(indirect)f(pro)s(of)g(of)h(the)g(if)120 1838 y(direction,)32 b(supp)s(ose)h(that)g(there)g(is)f(an)h (in\014nite)e Fq(\))p Fr(-deriv)-5 b(ation)1404 2030 y Fo(D)30 b Fr(:)e Fo(s)1616 2045 y Fp(1)1655 2030 y Fq(\))p Fo(s)1801 2045 y Fp(2)1840 2030 y Fq(\))p Fo(s)1986 2045 y Fp(3)2025 2030 y Fq(\))17 b Fo(:)g(:)g(:)120 2222 y Fr(W.l.o.g.)43 b(w)m(e)j(ma)m(y)e(assume)g(that)g Fo(D)j Fr(is)d(of)g(minimal)c(rank,)47 b(i.e.,)g(an)m(y)e Fq(\))o Fr(-deriv)-5 b(ation)42 b(of)120 2343 y(smaller)31 b(rank)h(is)h (\014nite.)266 2463 y(First)38 b(of)g(all)f(notice)h(that)h Fo(r)s(ank)s Fr(\()p Fo(s)1574 2478 y Fk(j)1611 2463 y Fr(\))f(=)g Fo(r)s(ank)s Fr(\()p Fo(D)s Fr(\))g(for)g(all)f(indices)i Fo(j)6 b Fr(.)62 b(In)39 b(particular,)120 2583 y(there)d(is)f(no)g (reduction)g(step)i(whic)m(h)e(is)g(destructiv)m(e)i(at)e(lev)m(el)g (1.)51 b(Moreo)m(v)m(er,)38 b(w.l.o.g.)51 b(w)m(e)120 2704 y(ma)m(y)32 b(assume)h(that)g Fo(r)s(oot)p Fr(\()p Fo(s)1140 2719 y Fp(1)1179 2704 y Fr(\))27 b Fq(2)i(F)1421 2668 y Fl(\003)1411 2728 y Fp(1)1492 2704 y Fr(and)k(hence)h Fo(r)s(oot)p Fr(\()p Fo(s)2213 2719 y Fk(j)2249 2704 y Fr(\))27 b Fq(2)h(F)2490 2668 y Fl(\003)2480 2728 y Fp(1)2562 2704 y Fr(for)k(all)e Fo(j)k Fq(\025)28 b Fr(1.)266 2824 y(F)-8 b(urthermore,)39 b(there)f(are)g(only)f(\014nitely)g(man)m (y)h(inner)f(reduction)h(steps)h(in)e Fo(D)j Fr(whic)m(h)120 2944 y(are)33 b(destructiv)m(e)h(at)e(lev)m(el)g(2.)43 b(This)33 b(is)f(b)s(ecause)i Fo(>)2000 2908 y Fk(mul)q(t)2187 2944 y Fr(is)e(w)m(ell-founded)g(and)265 3124 y Fq(\017)49 b Fo(s)410 3139 y Fk(j)446 3124 y Fq(\))546 3088 y Fk(i)574 3124 y Fo(s)620 3139 y Fk(j)t Fp(+1)779 3124 y Fr(destructiv)m(e)34 b(at)f(lev)m(el)f(2)g(implies)e Fo(]s)2116 3139 y Fk(j)2180 3124 y Fo(>)2256 3088 y Fk(mul)q(t)2439 3124 y Fo(]s)2523 3139 y Fk(j)t Fp(+1)2682 3124 y Fr(b)m(y)j(Lemma)e(3.3,)265 3319 y Fq(\017)49 b Fo(s)410 3334 y Fk(j)446 3319 y Fq(\))546 3283 y Fk(i)574 3319 y Fo(s)620 3334 y Fk(j)t Fp(+1)779 3319 y Fr(non-destructiv)m(e)34 b(at)e(lev)m(el)h(2)f(implies)e Fo(]s)2306 3334 y Fk(j)2370 3319 y Fq(\025)2447 3283 y Fk(mul)q(t)2630 3319 y Fo(]s)2714 3334 y Fk(j)t Fp(+1)2873 3319 y Fr(b)m(y)k(Lemma)d(3.2,)265 3515 y Fq(\017)49 b Fo(s)410 3530 y Fk(j)446 3515 y Fq(\))546 3478 y Fk(o)584 3515 y Fo(s)630 3530 y Fk(j)t Fp(+1)788 3515 y Fr(implies)30 b Fo(]s)1203 3530 y Fk(j)1267 3515 y Fq(\025)1344 3478 y Fk(mul)q(t)1527 3515 y Fo(]s)1611 3530 y Fk(j)t Fp(+1)1769 3515 y Fr(b)m(y)i(Lemma)e(3.2)i(b)s(ecause)g(the)g(reduction)g(step)364 3635 y(is)g(not)g(destructiv)m(e)j(at)d(lev)m(el)g(1.)120 3814 y(Th)m(us,)42 b(without)d(loss)f(of)h(generalit)m(y)-8 b(,)40 b(w)m(e)g(ma)m(y)f(further)g(assume)g(that)g(there)h(is)e(no)h (inner)120 3935 y(reduction)33 b(step)g(in)f Fo(D)j Fr(whic)m(h)e(is)f (destructiv)m(e)i(at)f(lev)m(el)f(2.)266 4055 y(It)46 b(follo)m(ws)e(for)h(an)m(y)h(outer)f(step)i Fo(s)1610 4070 y Fk(j)1679 4055 y Fq(\))1779 4019 y Fk(l)1801 3995 y Fi(\003)1812 4019 y Fk(\033)1779 4080 y Fl(R)1839 4089 y Ff(1)1910 4055 y Fo(s)1956 4070 y Fk(j)t Fp(+1)2128 4055 y Fr(in)d Fo(D)k Fr(that)d Fo(top)p Fr(\()p Fo(s)2822 4070 y Fk(j)2859 4055 y Fr(\))32 b Fq(\))3029 4019 y Fk(l)3051 3995 y Fi(\003)3063 4019 y Fk(\033)3029 4080 y Fl(R)3089 4089 y Ff(1)3160 4055 y Fo(top)p Fr(\()p Fo(s)3375 4070 y Fk(j)t Fp(+1)3502 4055 y Fr(\))120 4175 y(and)41 b(for)g(ev)m(ery)h(inner)f(reduction)g(step)h Fo(s)1702 4190 y Fk(j)1771 4175 y Fq(\))1871 4139 y Fk(l)1893 4116 y Fi(\003)1905 4139 y Fk(\033)1984 4175 y Fo(s)2030 4190 y Fk(j)t Fp(+1)2198 4175 y Fr(w)m(e)g(ha)m(v)m(e)g Fo(top)p Fr(\()p Fo(s)2798 4190 y Fk(j)2835 4175 y Fr(\))g(=)g Fo(top)p Fr(\()p Fo(s)3248 4190 y Fk(j)t Fp(+1)3374 4175 y Fr(\))f(or)120 4296 y Fo(top)p Fr(\()p Fo(s)335 4311 y Fk(j)371 4296 y Fr(\))33 b Fq(\))542 4260 y Fk(l)564 4236 y Fi(\003)576 4260 y Fk(\033)542 4321 y Fl(R)602 4330 y Ff(1)673 4296 y Fo(top)p Fr(\()p Fo(s)888 4311 y Fk(j)t Fp(+1)1015 4296 y Fr(\).)80 b(Since)e Fq(\))1560 4311 y Fl(R)1620 4320 y Ff(1)1736 4296 y Fr(is)44 b(terminating,)i (there)g(can)f(b)s(e)g(only)g(\014nitely)120 4416 y(man)m(y)h(graph)g (reduction)g(steps)h(with)f Fo(top)p Fr(\()p Fo(s)1845 4431 y Fk(j)1881 4416 y Fr(\))33 b Fq(\))2052 4380 y Fk(l)2074 4357 y Fi(\003)2086 4380 y Fk(\033)2052 4441 y Fl(R)2112 4450 y Ff(1)2183 4416 y Fo(top)p Fr(\()p Fo(s)2398 4431 y Fk(j)t Fp(+1)2525 4416 y Fr(\))46 b(in)f Fo(D)s Fr(.)83 b(So)46 b(w.l.o.g.)g(w)m(e)120 4537 y(ma)m(y)40 b(assume)h(that)e Fo(D)k Fr(con)m(tains)d(none)h(of)e(these)j (reduction)e(steps,)j(or)d(in)g(other)g(w)m(ords,)120 4657 y Fo(top)p Fr(\()p Fo(s)335 4672 y Fk(j)371 4657 y Fr(\))30 b(=)f Fo(top)p Fr(\()p Fo(s)759 4672 y Fk(j)t Fp(+1)886 4657 y Fr(\))k(for)g(all)f Fo(j)j Fq(\025)30 b Fr(1.)47 b(Th)m(us,)35 b(if)e Fo(s)1961 4672 y Fp(1)2030 4657 y Fr(=)c Fo(C)7 b Fr([)-17 b([)p Fo(t)2284 4672 y Fp(1)2324 4657 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)2578 4672 y Fk(n)2625 4657 y Fr(])-17 b(],)34 b(then)h(there)f(m)m(ust)g(b)s(e) 120 4777 y(an)f(in\014nite)f Fq(\))p Fr(-deriv)-5 b(ation)31 b(starting)h(from)f(some)i Fo(t)2052 4792 y Fk(k)2124 4777 y Fq(2)28 b Fo(S)6 b Fr(\()p Fo(s)2368 4792 y Fp(1)2408 4777 y Fr(\).)44 b(But)34 b(this)e(con)m(tradicts)i(the)120 4898 y(fact)e(that)h Fo(r)s(ank)s Fr(\()p Fo(D)s Fr(\))f(is)g(minimal)c (b)s(ecause)34 b Fo(r)s(ank)s Fr(\()p Fo(t)2047 4913 y Fk(k)2090 4898 y Fr(\))28 b Fo(<)g(r)s(ank)s Fr(\()p Fo(s)2554 4913 y Fp(1)2593 4898 y Fr(\).)266 5102 y(If)44 b(w)m(e)g(restrict)g(ourselv)m(es)h(to)e(the)h(mo)s(del)e(in)h(whic)m (h)h(v)-5 b(ariables)42 b(are)i Fn(not)f Fr(mark)m(ed)h(at)120 5222 y(all)f(\(whic)m(h)i(means)g(that)f(v)-5 b(ariables)44 b(are)g(maximally)d(shared\),)49 b(then)d(the)f(de\014nition)e(of)120 5342 y(graph)30 b(rewriting)e(used)k(here)e(b)s(ecomes)h(equiv)-5 b(alen)m(t)29 b(to)h(the)g(corresp)s(onding)g(de\014nitions)g(in)120 5463 y([BEG)359 5427 y Fp(+)418 5463 y Fr(87,)i(Plu93a)o(,)h(Plu93b)o (,)g(KKSV94)o(,)g(AK96,)f(AKP98].)p eop %%Page: 6 6 6 5 bop 120 407 a Fh(Example)36 b(3.6)49 b Fr(Let)40 b Fq(R)g Fr(=)g Fq(f)p Fo(eq)t Fr(\()p Fo(x;)17 b(x)p Fr(\))39 b Fq(!)h Fo(tr)s(ue)p Fq(g)f Fr(as)g(in)g(Example)g(2.1.)64 b(If)40 b(v)-5 b(ariables)38 b(are)120 527 y(unmark)m(ed,)44 b(then)d Fo(eq)935 491 y Fp(0)975 527 y Fr(\()p Fo(eq)1105 491 y Fp(1)1144 527 y Fr(\(0)1231 491 y Fp(2)1270 527 y Fo(;)17 b Fr(0)1363 491 y Fp(3)1402 527 y Fr(\))p Fo(;)g(eq)1576 491 y Fp(1)1615 527 y Fr(\(0)1702 491 y Fp(2)1741 527 y Fo(;)g Fr(0)1834 491 y Fp(3)1873 527 y Fr(\)\))41 b(do)s(es)g(not)g (reduce)h(to)f Fo(eq)2937 491 y Fp(0)2976 527 y Fr(\()p Fo(tr)s(ue)3197 491 y Fp(4)3236 527 y Fo(;)17 b(tr)s(ue)3463 491 y Fp(4)3502 527 y Fr(\))120 648 y(b)s(ecause)44 b(0)540 611 y Fp(2)622 648 y Fr(and)f(0)871 611 y Fp(3)953 648 y Fr(are)f(not)h(shared)h(and)e(th)m(us)i(the)f(rule)g Fo(eq)2529 611 y Fp(1)2568 648 y Fr(\()p Fo(x;)17 b(x)p Fr(\))45 b Fq(!)g Fo(tr)s(ue)3171 611 y Fp(4)3253 648 y Fr(cannot)120 768 y(b)s(e)e(applied.)74 b(In)44 b(order)f(to)g(o)m(v) m(ercome)h(this)f(phenomenon,)j(one)e(can)f(use)h(in)m(termediate)120 888 y Fn(folding)35 b Fr(steps)651 852 y Fp(2)761 888 y Fq(\))861 903 y Fl(F)990 888 y Fr(as)h(suggested)i(in)d([Plu93a],)i (De\014nition)d(2.9.)54 b(F)-8 b(or)36 b(example,)g(after)120 1009 y(the)d(folding)d(step)611 1229 y Fo(eq)703 1188 y Fp(0)743 1229 y Fr(\()p Fo(eq)873 1188 y Fp(1)912 1229 y Fr(\(0)999 1188 y Fp(2)1038 1229 y Fo(;)17 b Fr(0)1131 1188 y Fp(3)1170 1229 y Fr(\))p Fo(;)g(eq)1344 1188 y Fp(1)1383 1229 y Fr(\(0)1470 1188 y Fp(2)1509 1229 y Fo(;)g Fr(0)1602 1188 y Fp(3)1641 1229 y Fr(\)\))33 b Fq(\))1850 1244 y Fl(F)1943 1229 y Fo(eq)2035 1188 y Fp(0)2074 1229 y Fr(\()p Fo(eq)2204 1188 y Fp(1)2244 1229 y Fr(\(0)2331 1188 y Fp(3)2370 1229 y Fo(;)17 b Fr(0)2463 1188 y Fp(3)2502 1229 y Fr(\))p Fo(;)g(eq)2676 1188 y Fp(1)2715 1229 y Fr(\(0)2802 1188 y Fp(3)2841 1229 y Fo(;)g Fr(0)2934 1188 y Fp(3)2973 1229 y Fr(\)\))120 1449 y(the)33 b(rule)f Fo(eq)575 1413 y Fp(1)615 1449 y Fr(\()p Fo(x;)17 b(x)p Fr(\))28 b Fq(!)f Fo(tr)s(ue)1183 1413 y Fp(4)1255 1449 y Fr(is)32 b(applicable,)e(so)j(that)801 1669 y Fo(eq)893 1628 y Fp(0)933 1669 y Fr(\()p Fo(eq)1063 1628 y Fp(1)1102 1669 y Fr(\(0)1189 1628 y Fp(3)1228 1669 y Fo(;)17 b Fr(0)1321 1628 y Fp(3)1360 1669 y Fr(\))p Fo(;)g(eq)1534 1628 y Fp(1)1573 1669 y Fr(\(0)1660 1628 y Fp(3)1700 1669 y Fo(;)g Fr(0)1793 1628 y Fp(3)1831 1669 y Fr(\)\))33 b Fq(\))2040 1684 y Fl(R)2136 1669 y Fo(eq)2228 1628 y Fp(0)2268 1669 y Fr(\()p Fo(tr)s(ue)2489 1628 y Fp(4)2528 1669 y Fo(;)17 b(tr)s(ue)2755 1628 y Fp(4)2794 1669 y Fr(\))p Fo(:)266 1897 y Fr(It)34 b(is)e(fairly)f (simple)h(to)h(v)m(erify)g(that)g(the)h(ab)s(o)m(v)m(e)g(pro)s(of)e (also)g(holds)h(in)g(the)g(presence)i(of)120 2017 y(in)m(termediate)c (folding)g(steps)j(b)s(ecause)g Fo(s)e Fq(\))1793 2032 y Fl(F)1886 2017 y Fo(t)h Fr(implies)d Fo(S)6 b Fr(\()p Fo(t)p Fr(\))28 b Fq(\022)g Fo(S)6 b Fr(\()p Fo(s)p Fr(\).)120 2350 y Fs(4)161 b(Concluding)53 b(Remarks)120 2569 y Fr(It)25 b(should)g(b)s(e)f(p)s(oin)m(ted)h(out)g(that)f(in)g(the)h (con)m(text)h(of)f(sim)m(ulating)d(term)i(rewriting)f(b)m(y)j(graph)120 2690 y(rewriting)31 b(one)i(has)g(to)f(address)i(sev)m(eral)g(other)e (problems:)265 2893 y Fq(\017)49 b Fr(Is)33 b(the)g(c)m(hosen)i(mo)s (del)c(of)h(graph)g(rewriting)g(a)g(sound)i(and)f(complete)f(implemen)m (ta-)364 3013 y(tion)f(\(in)h(what)h(sense\))h(of)e(term)g(rewriting?) 265 3217 y Fq(\017)49 b Fr(Under)33 b(whic)m(h)g(conditions)f(is)65 b Fq(\))1636 3232 y Fl(R)1765 3217 y Fr(terminating?)265 3420 y Fq(\017)49 b Fr(Under)33 b(whic)m(h)g(conditions)f(is)65 b Fq(\))1636 3435 y Fl(R)1765 3420 y Fr(con\015uen)m(t?)266 3624 y(The)32 b(discussion)e(of)g(these)i(problems)d(is)h(b)s(ey)m(ond) i(the)e(scop)s(e)h(of)f(this)g(note,)h(w)m(e)h(refer)e(to)120 3744 y([BEG)359 3708 y Fp(+)418 3744 y Fr(87,)i(Plu93a)o(,)h(Plu93b)o (,)g(KKSV94)o(,)g(K)m(O95,)f(AK96,)g(Ohl97)o(,)h(AKP98])f(instead.)120 3985 y Fh(Ac)m(kno)m(wledgemen)m(ts:)60 b Fr(I)42 b(thank)g(Masahito)f (Kurihara)g(for)g(a)g(v)-5 b(aluable)40 b(commen)m(t)h(on)120 4105 y(a)32 b(previous)h(v)m(ersion)g(of)g(this)f(note.)120 4438 y Fs(References)120 4657 y Fr([AK96])171 b(Z.)44 b(M.)h(Ariola)d(and)j(J.)g(W.)g(Klop.)78 b(Equational)43 b(T)-8 b(erm)45 b(Graph)f(Rewriting.)592 4777 y Fn(F)-7 b(undamentae)33 b(Informatic)-5 b(ae)33 b Fh(26)p Fr(,)g(pages)g (207{240,)e(1996.)120 4981 y([AKP98])105 b(Z.)30 b(M.)g(Ariola,)f(J.)h (W.)h(Klop,)f(and)g(D.)g(Plump.)39 b(Bisimilarit)m(y)26 b(in)j(T)-8 b(erm)30 b(Graph)592 5101 y(Rewriting.)42 b(1998.)g(Submitted.)p 120 5189 1368 4 v 232 5250 a Fc(2)269 5280 y Fb([AKP98)o(])28 b(uses)f(the)h(phrase)e Fa(c)l(ol)t(lapsing)k Fb(instead.)p eop %%Page: 7 7 7 6 bop 120 407 a Fr([BEG)359 371 y Fp(+)418 407 y Fr(87])49 b(H.P)-8 b(.)34 b(Barendregt,)h(M.C.J.D.)g(v)-5 b(an)34 b(Eek)m(elen,)h(J.R.W.)g(Glauert,)e(J.R.)h(Kenn-)592 527 y(a)m(w)m(a)m(y)-8 b(,)41 b(M.J.)f(Plasmeijer,)f(and)g(M.R.)h (Sleep.)62 b(T)-8 b(erm)39 b(Graph)g(Rewriting.)61 b(In)592 648 y Fn(Pr)-5 b(o)g(c.)47 b(Par)-5 b(al)5 b(lel)46 b(A)n(r)-5 b(chite)g(ctur)g(es)48 b(and)f(L)-5 b(anguages)46 b(Eur)-5 b(op)g(e)p Fr(,)50 b(pages)c(141{158.)592 768 y(LNCS)33 b Fh(259)p Fr(,)g(1987.)120 966 y([BN98])178 b(F.)31 b(Baader)h(and)f(T.)i(Nipk)m(o)m(w.)42 b Fn(T)-7 b(erm)33 b(R)-5 b(ewriting)34 b(and)f(A)n(l)5 b(l)34 b(That)p Fr(.)42 b(Cam)m(bridge)592 1087 y(Univ)m(ersit)m(y)33 b(Press,)h(1998.)120 1285 y([Gra96])156 b(B.)42 b(Gramlic)m(h.)70 b Fn(T)-7 b(ermination)42 b(and)i(Con\015uenc)-5 b(e)43 b(Pr)-5 b(op)g(erties)44 b(of)f(Structur)-5 b(e)g(d)592 1405 y(R)g(ewrite)34 b(Systems)p Fr(.)43 b(PhD)33 b(thesis,)g(Univ)m (ersit\177)-49 b(at)32 b(Kaiserslautern,)g(1996.)120 1603 y([KKSV94])49 b(J.R.)38 b(Kenna)m(w)m(a)m(y)-8 b(,)41 b(J.W.)e(Klop,)f(M.R.)h(Sleep,)g(and)f(F.-J.)g(de)h(V)-8 b(ries.)59 b(On)39 b(the)592 1724 y(Adequacy)f(of)e(T)-8 b(erm)36 b(Graph)f(Rewriting)g(for)h(Sim)m(ulating)d(T)-8 b(erm)36 b(Rewriting.)592 1844 y Fn(A)n(CM)30 b(T)-7 b(r)i(ansaction)29 b(on)h(Pr)-5 b(o)g(gr)g(amming)29 b(L)-5 b(anguages)30 b(and)g(Systems)g Fh(16)p Fr(,)f(pages)592 1965 y(493{523,)i(1994.)120 2163 y([K)m(O95])171 b(M.)27 b(Kurihara)g(and)g(A.)h(Oh)m(uc)m(hi.)35 b(Mo)s(dularit)m(y)26 b(in)h(Noncop)m(ying)h(T)-8 b(erm)27 b(Rewrit-)592 2283 y(ing.)42 b Fn(The)-5 b(or)g(etic)g(al)34 b(Computer)g(Scienc)-5 b(e)34 b Fh(152)p Fr(,)f(pages)g(139{169,)e(1995.)120 2481 y([Mid90])149 b(A.)30 b(Middeldorp.)39 b Fn(Mo)-5 b(dular)33 b(Pr)-5 b(op)g(erties)33 b(of)f(T)-7 b(erm)32 b(R)-5 b(ewriting)32 b(Systems)p Fr(.)40 b(PhD)592 2602 y(thesis,)33 b(V)-8 b(rije)31 b(Univ)m(ersiteit)h(te)h(Amsterdam,)f (1990.)120 2800 y([Ohl93])162 b(E.)31 b(Ohlebusc)m(h.)43 b(A)31 b(Simple)e(Pro)s(of)i(of)g(Su\016cien)m(t)h(Conditions)e(for)h (the)g(T)-8 b(ermi-)592 2920 y(nation)33 b(of)h(the)g(Disjoin)m(t)f (Union)h(of)f(T)-8 b(erm)35 b(Rewriting)d(Systems.)50 b Fn(Bul)5 b(letin)36 b(of)592 3041 y(the)30 b(Eur)-5 b(op)g(e)g(an)31 b(Asso)-5 b(ciation)30 b(for)h(The)-5 b(or)g(etic)g(al)30 b(Computer)g(Scienc)-5 b(e)30 b Fh(49)p Fr(,)f(pages)592 3161 y(178{183,)i(1993.)120 3359 y([Ohl94])162 b(E.)24 b(Ohlebusc)m(h.)30 b Fn(Mo)-5 b(dular)27 b(Pr)-5 b(op)g(erties)27 b(of)g(Comp)-5 b(osable)25 b(T)-7 b(erm)26 b(R)-5 b(ewriting)26 b(Sys-)592 3480 y(tems)p Fr(.)43 b(PhD)32 b(thesis,)h(Univ)m(ersit\177)-49 b(at)33 b(Bielefeld,)e(1994.) 120 3678 y([Ohl97])162 b(E.)33 b(Ohlebusc)m(h.)46 b(Conditional)31 b(T)-8 b(erm)33 b(Graph)f(Rewriting.)44 b(In)33 b Fn(Pr)-5 b(o)g(c)g(e)g(e)g(dings)34 b(of)592 3798 y(the)25 b(6th)g (International)e(Confer)-5 b(enc)g(e)24 b(on)g(A)n(lgebr)-5 b(aic)24 b(and)h(L)-5 b(o)g(gic)24 b(Pr)-5 b(o)g(gr)g(amming)p Fr(,)592 3919 y(pages)33 b(144{158.)e(LNCS)i Fh(1298)p Fr(,)g(1997.)120 4117 y([Plu90])172 b(D.)33 b(Plump.)46 b(Implemen)m(ting)32 b(T)-8 b(erm)34 b(Rewriting)e(b)m(y)j(Graph)e (Reduction:)46 b(T)-8 b(er-)592 4237 y(mination)38 b(of)i(Com)m(bined)h (Systems.)69 b(In)41 b Fn(Pr)-5 b(o)g(c)g(e)g(e)g(dings)41 b(of)h(the)h(2nd)f(Interna-)592 4358 y(tional)34 b(Workshop)h(on)f (Conditional)g(and)g(T)-7 b(yp)i(e)g(d)34 b(R)-5 b(ewriting)35 b(Systems)p Fr(,)d(pages)592 4478 y(307{317.)f(LNCS)i Fh(516)p Fr(,)g(1990.)120 4676 y([Plu93a])123 b(D.)29 b(Plump.)39 b(Collapsed)29 b(T)-8 b(ree)32 b(Rewriting:)40 b(Completeness,)32 b(Con\015uence,)h(and)592 4797 y(Mo)s(dularit)m(y)-8 b(.)94 b(In)50 b Fn(Pr)-5 b(o)g(c)g(e)g(e)g(dings)50 b(of)h(the)g(3r)-5 b(d)50 b(International)g(Workshop)g(on)592 4917 y(Conditional)30 b(T)-7 b(erm)31 b(R)-5 b(ewriting)31 b(Systems)g(1992)p Fr(,)e(pages)h(97{112.)e(LNCS)i Fh(656)p Fr(,)592 5038 y(1993.)120 5236 y([Plu93b])118 b(D.)59 b(Plump.)126 b Fn(Evaluation)60 b(of)g(F)-7 b(unctional)59 b(Expr)-5 b(essions)59 b(by)i(Hyp)-5 b(er)g(gr)g(aph)592 5356 y(R)g(ewriting)p Fr(.)42 b(PhD)33 b(thesis,)g(Univ)m(ersit\177)-49 b(at)32 b(Bremen,)h(1993.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF