%!PS-Adobe-2.0 %%Creator: dvips 5.516 Copyright 1986, 1993 Radical Eye Software %%Title: conflu.dvi %%CreationDate: Fri Jan 14 14:34:56 1994 %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o conflu.ps conflu %DVIPSSource: TeX output 1994.01.13:1627 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 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(programs)e(in)h(a)g(mo)q(dular)f(w)o(a)o(y)h(in)g(order)h(to)g(handle) f(large)g(systems.)h(Th)o(us,)183 1585 y(from)f(a)i(practical)g(p)q (oin)o(t)g(of)g(view,)f(it)h(is)g(w)o(orth)h(kno)o(wing)e(under)i(what) f(conditions)g(the)183 1635 y(com)o(bined)10 b(program)f(inherits)j (prop)q(erties)g(from)e(its)h(constituen)o(t)h(mo)q(dules.)d(F)m(or)i (this)g(rea-)183 1685 y(son)g(it)g(is)h(not)f(astonishing)g(that)g(the) i(sub)r(ject)f(of)f(mo)q(dular)f(prop)q(erties)j(of)e(term)g(rewriting) 183 1734 y(systems)i(\(TRSs\))h(is)f(receiving)g(a)g(lot)g(of)g(atten)o (tion.)f(The)i(\014rst)g(imp)q(ortan)o(t)e(result)i(in)e(this)183 1784 y(area)19 b(states)h(that)f(con\015uence)i(is)d(mo)q(dular)g(for)g (disjoin)o(t)g(TRSs.)h(More)g(precisely)m(,)g(the)183 1834 y(com)o(bination)12 b(\()p Fp(F)t Fo(;)7 b Fp(R)p Ft(\))14 b(=)g(\()p Fp(F)646 1840 y Fn(1)674 1834 y Fp([)c(F)742 1840 y Fn(2)760 1834 y Fo(;)d Fp(R)814 1840 y Fn(1)843 1834 y Fp([)i(R)915 1840 y Fn(2)934 1834 y Ft(\))15 b(of)g(t)o(w)o(o)g (con\015uen)o(t)h(TRSs)f(\()p Fp(F)1435 1840 y Fn(1)1453 1834 y Fo(;)7 b Fp(R)1507 1840 y Fn(1)1525 1834 y Ft(\))16 b(and)183 1884 y(\()p Fp(F)229 1890 y Fn(2)247 1884 y Fo(;)7 b Fp(R)301 1890 y Fn(2)320 1884 y Ft(\))16 b(is)g(also)g (con\015uen)o(t)h(if)e(the)i(signatures)g Fp(F)1006 1890 y Fn(1)1041 1884 y Ft(and)f Fp(F)1154 1890 y Fn(2)1189 1884 y Ft(are)g(disjoin)o(t.)f(It)i(w)o(as)f(\014rst)183 1934 y(pro)o(v)o(ed)h(b)o(y)g(T)m(o)o(y)o(ama)e([T)m(o)o(y87b)n(])i (and)g(is)g(b)o(y)g(no)o(w)g(referred)j(to)d(as)g(T)m(o)o(y)o(ama's)e (Theorem.)183 1983 y(Not)g(long)f(ago)g(a)h(simpli\014ed)e(pro)q(of)i (of)f(T)m(o)o(y)o(ama')o(s)f(Theorem)h(w)o(as)h(giv)o(en)g(b)o(y)g (Klop)f(et)i(al.)183 2033 y([KMTV91)o(].)h(In)h(con)o(trast)g(to)g (this)g(encouraging)g(result,)g(termination)e(and)h(complete-)183 2083 y(ness)h(turned)h(out)e(to)h(lac)o(k)f(a)g(mo)q(dular)e(b)q(eha)o (vior)j(\(see)g([T)m(o)o(y87a)n(])f(and)h(also)e([Ohl93b)o(]\).)183 2133 y(Th)o(us)c(sev)o(eral)g(su\016cien)o(t)g(criteria)g(ensuring)h (their)f(mo)q(dularit)o(y)d(ha)o(v)o(e)j(b)q(een)h(giv)o(en)e(\(for)h (an)183 2183 y(o)o(v)o(erview)g(see)i(e.g.)e([Mid90)o(,)g(Gra93)o(,)g (Ohl93a)o(]\).)g(In)h(order)g(to)g(pro)o(v)o(e)g(mo)q(dularit)o(y)d(of) i(com-)183 2232 y(pleteness,)j(one)f(can)g(of)g(course)h(use)g(the)g (con\015uence)g(of)f(the)g(com)o(bined)f(system)h(to)g(sho)o(w)183 2282 y(its)e(termination.)e(F)m(or)i(example)f(the)i(deep)g(theorem)f (that)g(completeness)h(is)f(mo)q(dular)f(for)183 2332 y(left-linear)i(disjoin)o(t)g(TRSs)g([TKB89])g(crucially)h(dep)q(ends)h (on)f(T)m(o)o(y)o(ama')o(s)e(Theorem.)245 2382 y(In)21 b(recen)o(t)h(in)o(v)o(estigations)e(\(cf.)h([K)o(O92)o(,)g(MT93)o(,)f (Ohl93a,)g(Gra93)o(]\),)g(one)h(tries)h(to)183 2432 y(w)o(eak)o(en)16 b(the)h(disjoin)o(tness)g(requiremen)o(t.)e(One)i(w)o(a)o(y)f(to)g(do)g (it)g(is)g(to)h(allo)o(w)d(shared)j(con-)p eop %%Page: 2 2 2 1 bop 340 194 a Ft(structors)12 b(\(function)d(sym)o(b)q(ols)g(that)h (do)f(not)h(o)q(ccur)h(at)e(the)h(ro)q(ot)g(p)q(osition)f(of)g(the)i (left-hand)340 244 y(side)17 b(of)f(an)o(y)g(rewrite)h(rule\).)f (Unfortunately)m(,)g(con\015uence)i(is)e(not)g(preserv)o(ed)j(under)e (the)340 293 y(com)o(bination)10 b(of)h(constructor-sharing)i(TRSs.)e (In)h([K)o(O92)o(],)f(Kurihara)g(and)h(Oh)o(uc)o(hi)g(ga)o(v)o(e)340 343 y(the)j(follo)o(wing)c(coun)o(terexample:)340 440 y Fm(Example)16 b(1.1)k Ft(Consider)d(the)h(TRSs)f Fp(R)1022 446 y Fn(1)1057 440 y Ft(=)f Fp(f)p Fo(F)6 b Ft(\()p Fo(x;)h(x)p Ft(\))15 b Fp(!)h Fo(A;)7 b(F)f Ft(\()p Fo(x;)h Fl(C)p Ft(\()p Fo(x)p Ft(\)\))17 b Fp(!)e Fo(B)r Fp(g)i Ft(and)340 490 y Fp(R)375 496 y Fn(2)406 490 y Ft(=)12 b Fp(f)p Fo(a)f Fp(!)g Fl(C)r Ft(\()p Fo(a)p Ft(\))p Fp(g)p Ft(.)g(Both)i(systems)f(are)g(con\015uen)o(t)h(and)f(they)h (share)g(the)f(constructor)i Fl(C)r Ft(.)340 540 y(The)h(term)e Fo(F)6 b Ft(\()p Fo(a;)h(a)p Ft(\))14 b(rewrites)h(to)f Fo(A)g Ft(as)g(w)o(ell)g(as)g(to)g Fo(B)r Ft(,)g(i.e.,)f(it)g(has)i(t)o (w)o(o)e(di\013eren)o(t)i(normal)340 590 y(forms)e(w.r.t.)g(to)g Fp(!)658 596 y Fk(R)687 600 y Fj(1)703 596 y Fk([R)754 600 y Fj(2)772 590 y Ft(.)g(This)h(sho)o(ws)g(that)g Fp(R)1136 596 y Fn(1)1164 590 y Fp([)9 b(R)1236 596 y Fn(2)1269 590 y Ft(is)k(not)h(con\015uen)o(t.)403 686 y(In)19 b([K)o(O92)o(,)g(MT93)o(],)g(the)g(mo)q(dularit)o(y)e(of)h (completeness)i(for)f(constructor-sharing)340 736 y(TRSs)c(\(under)h (certain)g(assumptions\))e(is)h(pro)o(v)o(ed)h(as)f(follo)o(ws:)e(Let)i Fp(R)1474 742 y Fn(1)1508 736 y Ft(and)g Fp(R)1625 742 y Fn(2)1659 736 y Ft(b)q(e)h(t)o(w)o(o)340 786 y(complete)c(TRS)g(whic) o(h)g(share)h(at)g(most)e(constructors.)j(First)e(of)g(all)f(their)i (com)o(bined)e(sys-)340 835 y(tem)k Fp(R)h Ft(=)f Fp(R)558 841 y Fn(1)587 835 y Fp([)c(R)661 841 y Fn(2)695 835 y Ft(is)16 b(lo)q(cally)e(con\015uen)o(t)j(b)q(ecause)h(lo)q(cal)d (con\015uence)j(is)d(mo)q(dular)g(\(cf.)340 885 y([Mid90)o(]\).)g 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b(6)e(is)g(dedicated)i(to)f(the)g(mo)q (dularit)o(y)d(of)i(semi-completeness.)340 2432 y(Finally)m(,)c(the)i (last)g(section)g(con)o(tains)g(concluding)g(remarks.)p eop %%Page: 3 3 3 2 bop 183 194 a Fq(2)56 b(Preliminarie)o(s)183 290 y Ft(In)14 b(this)g(section,)h(w)o(e)f(brie\015y)h(recall)f(the)h (basic)f(notions)g(of)f(term)h(rewriting)g(as)g(surv)o(ey)o(ed)183 340 y(in)f(e.g.)g(Dersho)o(witz)i(and)e(Jouannaud)h([DJ90)o(])f(and)h (Klop)g([Klo92)n(].)245 389 y(A)20 b Fi(signatur)n(e)f Ft(is)h(a)f(coun)o(table)h(set)h Fp(F)j Ft(of)19 b Fi(function)h(symb)n (ols)g Ft(or)g Fi(op)n(er)n(ators)p Ft(,)e(where)183 439 y(ev)o(ery)h Fo(f)25 b Fp(2)19 b(F)k Ft(is)c(asso)q(ciated)g(with)g (a)f(natural)h(n)o(um)o(b)q(er)f(denoting)g(its)h(arit)o(y)m(.)e (Nullary)183 489 y(op)q(erators)e(are)g(called)f Fi(c)n(onstants)p Ft(.)g(The)h(set)g Fp(T)c Ft(\()p Fp(F)t Fo(;)c Fp(V)s Ft(\))14 b(of)g Fi(terms)g Ft(built)g(from)e(a)i(signature)183 539 y Fp(F)20 b Ft(and)d(a)f(coun)o(table)h(set)h(of)e 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Ft(is)183 2332 y(con\015uen)o(t)j(if)f(and)h(only)f(if)g(ev)o(ery)i (pair)e(of)g(con)o(v)o(ertible)h(terms)g(is)g(joinable.)e(The)i (relation)183 2382 y Fp(!)d Ft(is)g Fi(lo)n(c)n(al)r(ly)h(c)n (on\015uent)h Ft(if)e(for)g(all)f(terms)i Fo(s;)7 b(t)902 2388 y Fn(1)920 2382 y Fo(;)g(t)954 2388 y Fn(2)985 2382 y Ft(with)12 b Fo(t)1093 2388 y Fn(1)1123 2382 y Fp( )f Fo(s)h Fp(!)f Fo(t)1275 2388 y Fn(2)1307 2382 y Ft(w)o(e)h(ha)o(v)o(e)h Fo(t)1476 2388 y Fn(1)1506 2382 y Fp(#)e Fo(t)1553 2388 y Fn(2)1572 2382 y Ft(.)h(If)183 2432 y Fp(!)h Ft(is)g(con\015uen)o(t)h (and)g(terminating,)d(it)i(is)h(called)f Fi(c)n(omplete)g Ft(or)h Fi(c)n(onver)n(gent)p Ft(.)f(The)h(famous)p eop %%Page: 4 4 4 3 bop 340 194 a Ft(Newman's)9 b(Lemma)e(states)12 b(that)e (termination)e(and)i(lo)q(cal)f(con\015uence)i(imply)d(con\015uence.) 340 244 y(If)i Fp(!)g Ft(is)g(con\015uen)o(t)h(and)f(normalizing,)d (then)k(it)f(is)g(called)g Fi(semi-c)n(omplete)p Ft(.)f(Sometimes)f (this)340 293 y(prop)q(ert)o(y)15 b(is)f(called)g Fi(unique)i (normalization)e Ft(b)q(ecause)i(it)d(is)h(equiv)n(alen)o(t)g(to)g(the) g(prop)q(ert)o(y)340 343 y(that)h(ev)o(ery)g(term)e(has)i(a)f(unique)g (normal)e(form.)g(Some)h(of)h(these)h(notions)f(are)h(also)e(used)340 393 y(in)f(their)g(sp)q(ecialization)g(to)g(terms;)f(the)i(phrase)g(\\) p Fo(t)f Ft(is)g(terminating")e(means)h(for)h(instance)340 443 y(that)i(ev)o(ery)h(reduction)g(sequence)h(starting)d(from)g Fo(t)g Ft(is)h(\014nite.)403 493 y(A)19 b Fi(term)g(r)n(ewriting)f (system)h Ft(\(TRS)g(for)g(short\))h(is)f(a)g(pair)g(\()p Fp(F)t Fo(;)7 b Fp(R)p Ft(\))19 b(consisting)g(of)g(a)340 542 y(signature)12 b Fp(F)k Ft(and)11 b(a)h(set)g Fp(R)g(\032)g(T)e Ft(\()p Fp(F)t Fo(;)d Fp(V)s Ft(\))e Fp(\002)g(T)11 b Ft(\()p Fp(F)t Fo(;)c Fp(V)s Ft(\))12 b(of)f Fi(r)n(ewrite)g(rules)g Ft(or)h Fi(r)n(e)n(duction)g(rules)p Ft(.)340 592 y(Ev)o(ery)k(rewrite) h(rule)e(\()p Fo(l)q(;)7 b(r)q Ft(\))15 b(m)o(ust)g(satisfy)g(the)h 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Fp(V)s Ft(\))19 b(as)h(follo)o(ws:)d Fo(s)k Fp(!)1254 897 y Fk(R)1305 891 y Fo(t)e Ft(if)g(there)h(exists)g(a)f(rewrite)340 941 y(rule)e Fo(l)g Fp(!)f Fo(r)h Ft(in)f Fp(R)p Ft(,)g(a)h (substitution)f Fo(\033)i Ft(and)e(a)h(con)o(text)g Fo(C)s Ft([)11 b(])16 b(suc)o(h)i(that)e Fo(s)h Ft(=)f Fo(C)s Ft([)p Fo(l)q(\033)q Ft(])f(and)340 991 y Fo(t)e Ft(=)g Fo(C)s Ft([)p Fo(r)q(\033)q Ft(].)g(W)m(e)i(sa)o(y)f(that)h Fo(s)g Ft(rewrites)h(to)f Fo(t)f Ft(b)o(y)h Fi(c)n(ontr)n(acting)f Ft(redex)i Fo(l)q(\033)q Ft(.)f(W)m(e)f(call)g Fo(s)f Fp(!)1723 997 y Fk(R)1766 991 y Fo(t)340 1041 y Ft(a)i Fi(r)n(ewrite)e(step)i Ft(or)f Fi(r)n(e)n(duction)i(step)p Ft(.)e(A)h(TRS)f(\()p Fp(F)t Fo(;)7 b Fp(R)p Ft(\))14 b(has)h(one)f(of)g(the)i(ab)q(o)o(v)o(e)e(prop)q(erties)340 1090 y(\(e.g.)j(termination\))f(if)g(its)i(rewrite)g(relation)f(has)h (the)g(resp)q(ectiv)o(e)h(prop)q(ert)o(y)m(.)e(W)m(e)g(often)340 1140 y(simply)c(write)i Fp(R)f Ft(instead)h(of)f(\()p Fp(F)t Fo(;)7 b Fp(R)p Ft(\))14 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1898 y Fn(1)979 1892 y Fo(;)7 b Fp(R)1033 1898 y Fn(1)1051 1892 y Ft(\))17 b Fi(and)h Ft(\()p Fp(F)1213 1898 y Fn(2)1232 1892 y Fo(;)7 b Fp(R)1286 1898 y Fn(2)1304 1892 y Ft(\))18 b Fi(with)e(shar)n(e)n(d)i(c)n(onstructors)340 1942 y Fp(C)r Ft(.)c(In)g(this)g(case)h(w)o(e)f(de\014ne)h Fp(D)824 1948 y Fn(1)854 1942 y Ft(=)d Fp(F)928 1948 y Fn(1)955 1942 y Fp(n)d(C)r Ft(,)14 b Fp(D)1067 1948 y Fn(2)1097 1942 y Ft(=)e Fp(F)1171 1948 y Fn(2)1199 1942 y Fp(n)d(C)r Ft(,)k(and)h Fp(D)f Ft(=)f Fp(D)1480 1948 y Fn(1)1507 1942 y Fp(])d(D)1576 1948 y Fn(2)1595 1942 y Ft(.)340 1992 y(A)14 b(prop)q(ert)o(y)h Fp(P)i Ft(of)c(TRSs)g(is)h(called)f Fi(mo)n(dular)h Ft(if)f(for)g(all)g(TRSs)g Fp(R)1387 1998 y Fn(1)1420 1992 y Ft(and)g Fp(R)1535 1998 y Fn(2)1568 1992 y Ft(whic)o(h)g(share)340 2042 y(at)f(most)f(constructors,)j(their)e(union)f Fp(R)980 2048 y Fn(1)1004 2042 y Fp([)5 b(R)1072 2048 y Fn(2)1103 2042 y Ft(has)12 b(the)h(prop)q(ert)o(y)g Fp(P)i Ft(if)c(and)h(only)f (if)g(b)q(oth)340 2092 y 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b(:)g(:)g(:)e(;)i(s)1110 2427 y Fl(i)1122 2431 y Ff(m)1152 2421 y Fp(g)-14 b(g)p Ft(.)p eop %%Page: 7 7 7 6 bop 183 194 a Fm(Lemma)15 b(3.12)21 b Ft(Let)15 b Fo(s;)7 b(t)14 b Fp(2)f(T)d Ft(\()p Fp(F)t Fo(;)d Fp(V)s Ft(\))q(.)14 b(If)h Fo(s)f Fp(!)932 179 y Fl(i)959 194 y Fo(t)p Ft(,)g(then)i Fo(s)e Ft(=)g Fo(C)s Ft([)-7 b([)p Fo(s)1244 200 y Fn(1)1262 194 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1374 200 y Fl(j)1391 194 y Fo(;)g(:)g(:)g(:)e(;)i(s)1503 200 y Fl(n)1526 194 y Ft(])-7 b(])14 b(and)183 244 y Fo(t)d Ft(=)h Fo(C)s Ft([)p Fo(s)317 250 y Fn(1)335 244 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)447 228 y Fk(0)447 254 y Fl(j)464 244 y Fo(;)g(:)g(:)g(:)e(;)i(s)576 250 y Fl(n)599 244 y Ft(])k(for)g(some)f(con)o(text)i Fo(C)s Ft([)p Fo(;)7 b(:)g(:)g(:)t(;)g Ft(])j(whic)o(h)h(is)h(either)g(blac)o(k)f(or)g (white,)183 298 y(and)k(terms)h Fo(s)402 304 y Fn(1)421 298 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)533 304 y Fl(n)556 298 y Fo(;)g(s)594 283 y Fk(0)594 309 y Fl(j)627 298 y Ft(with)16 b Fo(s)743 304 y Fl(j)775 298 y 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Ft(if)h(all)f(its)i(blac)o(k)g(\(white\))g(principal)f(subterms)h (are)g(preserv)o(ed.)245 1266 y(Clearly)m(,)g(a)i(preserv)o(ed)h(term)f (is)f(b)q(oth)h(blac)o(k)g(and)f(white)h(preserv)o(ed.)h(If)f(a)f(top)h (blac)o(k)183 1316 y(\(top)d(white\))h(term)f(is)h(blac)o(k)f (\(white\))h(preserv)o(ed,)h(then)f(it)g(is)f(preserv)o(ed.)i(Note)f (also)f(that)183 1366 y(the)i(prop)q(erties)h(preserv)o(ed)h(and)e (blac)o(k)f(\(white\))h(preserv)o(ed)i(are)e(b)q(oth)g(conserv)o(ed)i (under)183 1416 y(reduction.)183 1493 y Fm(De\014niti)o(on)c(4.2)20 b Ft(W)m(e)15 b(write)h Fo(s)e Fp(!)743 1499 y Fl(c)773 1493 y Fo(t)h Ft(if)g(there)h(exists)g(a)f(con)o(text)h Fo(C)s Ft([)c(])i(and)h(terms)g Fo(s)1552 1499 y Fn(1)1572 1493 y Fo(;)7 b(t)1606 1499 y Fn(1)183 1542 y Ft(suc)o(h)12 b(that)g Fo(s)g Ft(=)g Fo(C)s Ft([)p Fo(s)501 1548 y Fn(1)519 1542 y Ft(])p Fo(;)7 b(t)k Ft(=)g Fo(C)s Ft([)p Fo(t)679 1548 y Fn(1)697 1542 y Ft(])p Fo(;)c(s)747 1548 y Fn(1)777 1542 y Ft(is)12 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Fn(1)1430 1496 y Fo(;)7 b(:)g(:)g(:)e(;)i(C)1553 1502 y Fl(n)1590 1496 y Ft(of)16 b(joinable)340 1546 y(terms.)h(Since)g (these)h(classes)g(are)g(\014nite,)e(for)h(ev)o(ery)g Fo(C)1228 1552 y Fl(j)1262 1546 y Ft(=)g Fp(f)p Fo(s)1351 1552 y Fn(1)1369 1546 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1481 1552 y Fl(m)1513 1546 y Fp(g)16 b Ft(there)i(exist)g(a)340 1596 y(common)8 b(reduct)j Fo(t)643 1602 y Fl(j)671 1596 y Ft(of)e Fo(s)733 1602 y Fn(1)752 1596 y Fo(;)e(:)g(:)g(:)e(;)i(s)864 1602 y Fl(m)895 1596 y Ft(,)j(i.e.,)f Fo(s)1011 1602 y Fl(l)1035 1596 y Fp(!)1077 1581 y Fk(\003)1107 1596 y Fo(t)1122 1602 y Fl(j)1150 1596 y Ft(for)h(ev)o(ery)h Fo(l)h Fp(2)f(f)p Ft(1)p Fo(;)c(:)g(:)g(:)e(;)i(m)p Fp(g)p Ft(.)i(Ob)o(viously)m(,)340 1646 y(the)15 b(set)g Fp(f)o Fo(t)512 1652 y Fn(1)531 1646 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)639 1652 y Fl(n)661 1646 y Fp(g)13 b Ft(represen)o(ts)k Fo(S)r Ft(.)d Fh(2)340 1728 y Fm(Lemma)i(5.4)k Ft(Preserv)o(ed)d(terms)c(are)i (con\015uen)o(t.)340 1778 y Fm(Pro)q(of:)9 b 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y(o)q(ccurring)f(in)f(the)g(con) o(v)o(ersion.)g(By)h(the)f(induction)g(h)o(yp)q(othesis,)g Fo(S)j Ft(consists)e(of)e(con\015uen)o(t)340 2127 y(terms)20 b(b)q(ecause)i(ev)o(ery)f(elemen)o(t)f(of)g Fo(S)j Ft(is)d(preserv)o (ed)j(and)d(has)g(rank)g(less)h(than)g Fo(k)q Ft(.)e(It)340 2177 y(follo)o(ws)c(from)f(Lemma)f(5.3)i(that)h Fo(S)i Ft(can)e(b)q(e)h(represen)o(ted)h(b)o(y)e(a)g(set)1462 2166 y(^)1455 2177 y Fo(S)s Ft(.)f(W)m(e)h(write)j(~)-24 b Fo(u)16 b Ft(for)340 2227 y(the)j(term)e(obtained)h(from)e Fo(u)i Ft(b)o(y)f(replacing)h(eac)o(h)h(white)f(principal)f(subterm)h (with)f(its)340 2276 y(represen)o(tativ)o(e.)f(Note)e(that)g Fo(u)d Fp(!)886 2261 y Fk(\003)919 2276 y Ft(~)-24 b Fo(u)p Ft(.)403 2326 y(W)m(e)17 b(claim)f(that)688 2319 y(~)687 2326 y Fo(t)702 2332 y Fn(1)f Fk(R)763 2336 y Fj(1)751 2303 y Fk(\003)e Fl(o)766 2326 y Fp( )823 2319 y Ft(~)822 2326 y Fo(t)862 2303 y Fl(o)g Fk(\003)852 2326 y Fp(!)891 2332 y Fk(R)920 2336 y Fj(1)952 2319 y Ft(~)951 2326 y Fo(t)966 2332 y Fn(2)985 2326 y Ft(.)k(Let)i Fo(u)1117 2332 y Fn(1)1154 2326 y Fp(!)e Fo(u)1237 2332 y Fn(2)1274 2326 y Ft(b)q(e)h(a)g(step)h(in)e(the)i(con)o(v)o(ersion) 340 2376 y Fo(t)355 2382 y Fn(1)395 2361 y Fk(\003)405 2376 y Fp( )13 b Fo(t)i Fp(!)532 2361 y Fk(\003)566 2376 y Fo(t)581 2382 y Fn(2)599 2376 y Ft(.)h(Since)g Fo(u)761 2382 y Fn(1)795 2376 y Ft(is)g(top)g(blac)o(k,)f(w)o(e)h(ma)o(y)e (write)j Fo(u)1325 2382 y Fn(1)1358 2376 y Ft(=)e Fo(C)1438 2361 y Fl(b)1435 2386 y Fn(1)1455 2376 y Fp(f)-14 b(f)p Fo(s)1502 2382 y Fn(1)1520 2376 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1632 2382 y Fl(n)1655 2376 y Fp(g)-14 b(g)p Ft(.)15 b(The)340 2432 y(claim)c(will)h(b)q(e)h(pro)o(v)o(ed)g(b)o(y)g(sho)o(wing)i(~)-24 b Fo(u)955 2438 y Fn(1)998 2408 y Fl(o)13 b Fk(\003)988 2432 y Fp(!)1027 2438 y Fk(R)1056 2442 y Fj(1)1091 2432 y Ft(~)-24 b Fo(u)1112 2438 y Fn(2)1130 2432 y Ft(.)13 b(W)m(e)f(distinguish)h(the)g(follo)o(wing)d(cases.)p eop %%Page: 9 9 9 8 bop 200 194 a Ft(1.)20 b(If)e Fo(u)323 200 y Fn(1)355 194 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582 y(~)183 589 y Fo(t)198 595 y Fn(2)230 589 y Ft(ha)o(v)o(e)i(a)f(common) e(reduct,)k(whic)o(h)f(is)g(also)f(a)h(common)d(reduct)k(of)e Fo(t)1303 595 y Fn(1)1336 589 y Ft(and)g Fo(t)1431 595 y Fn(2)1450 589 y Ft(.)183 639 y(Case)h(\(ii\):)f Fo(t)h Ft(is)f(top)h(white.)g(Analogous)f(to)g(case)i(\(i\).)183 689 y(Case)f(\(iii\):)e Fo(t)i Ft(is)g(top)g(transparen)o(t.)183 739 y(Let)19 b Fo(t)h Ft(=)g Fo(C)382 724 y Fl(t)397 739 y Ft([)-7 b([)p Fo(s)433 745 y Fn(1)451 739 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)563 745 y Fl(n)585 739 y Ft(])-7 b(])18 b(and)h(consider)h(a)f(con)o(v)o(ersion)g Fo(t)1135 745 y Fn(1)1174 724 y Fk(\003)1184 739 y Fp( )13 b Fo(t)20 b Fp(!)1316 724 y Fk(\003)1355 739 y Fo(t)1370 745 y Fn(2)1388 739 y Ft(.)f(Clearly)m(,)e(w)o(e)183 789 y(ma)o(y)d(write)k Fo(t)400 795 y Fn(1)435 789 y Ft(=)e Fo(C)516 773 y Fl(t)530 789 y Ft([)p Fo(u)566 795 y Fn(1)584 789 y Fo(;)7 b(:)g(:)g(:)e(;)i(u) 701 795 y Fl(n)723 789 y Ft(])16 b(and)h Fo(t)850 795 y Fn(2)885 789 y Ft(=)f Fo(C)966 773 y Fl(t)981 789 y Ft([)p Fo(v)1013 795 y Fn(1)1031 789 y Fo(;)7 b(:)g(:)g(:)e(;)i(v)1144 795 y Fl(n)1166 789 y Ft(])16 b(where)i Fo(u)1341 795 y Fl(j)1379 773 y Fk(\003)1389 789 y Fp( )13 b Fo(s)1463 795 y Fl(j)1498 789 y Fp(!)1540 773 y Fk(\003)1574 789 y Fo(v)1594 795 y Fl(j)1612 789 y Ft(.)183 838 y(By)18 b(cases)i(\(i\))e(and)g(\(ii\),)f(there)j(are)f(terms)f Fo(w)928 844 y Fn(1)946 838 y Fo(;)7 b(:)g(:)g(:)e(;)i(w)1069 844 y Fl(n)1109 838 y Ft(suc)o(h)19 b(that)g Fo(u)1326 844 y Fl(j)1362 838 y Fp(!)1404 823 y Fk(\003)1441 838 y Fo(w)1471 844 y Fl(j)1509 823 y Fk(\003)1519 838 y Fp( )13 b Fo(v)1594 844 y Fl(j)1612 838 y Ft(.)183 888 y(Ob)o(viously)m(,)f Fo(C)419 873 y Fl(t)433 888 y Ft([)p Fo(w)475 894 y Fn(1)493 888 y Fo(;)7 b(:)g(:)g(:)e(;)i(w)616 894 y Fl(n)638 888 y Ft(])13 b(is)h(a)f(common)e(reduct)16 b(of)d Fo(t)1097 894 y Fn(1)1129 888 y Ft(and)h Fo(t)1225 894 y Fn(2)1244 888 y Ft(.)f Fh(2)183 967 y Fm(Prop)q(ositi)o(on)g(5.5) 20 b Ft(White)14 b(\(blac)o(k\))g(preserv)o(ed)i(terms)d(are)i (con\015uen)o(t.)183 1017 y Fm(Pro)q(of:)g Ft(W)m(e)g(sho)o(w)h(that)f (ev)o(ery)i(white)f(preserv)o(ed)i(term)d(is)g(con\015uen)o(t.)h(So)g (supp)q(ose)h Fo(t)f Ft(is)183 1066 y(a)g(white)g(preserv)o(ed)j(term)d (and)g(consider)i(a)e(con)o(v)o(ersion)h Fo(t)1130 1072 y Fn(1)1169 1051 y Fk(\003)1179 1066 y Fp( )c Fo(t)j Fp(!)1307 1051 y Fk(\003)1342 1066 y Fo(t)1357 1072 y Fn(2)1375 1066 y Ft(.)g(It)h(has)g(to)f(b)q(e)183 1116 y(sho)o(wn)f(that)g(the)g(terms)g Fo(t)605 1122 y Fn(1)638 1116 y Ft(and)g Fo(t)735 1122 y Fn(2)768 1116 y Ft(are)h(joinable.)d (As)i(in)g(the)g(pro)q(of)g(of)f(Lemma)e(5.4,)h(let)183 1166 y Fo(S)k Ft(b)q(e)d(the)h(set)g(of)f(all)f(white)h(principal)f (subterms)i(o)q(ccurring)g(in)e(the)i(con)o(v)o(ersion.)f(Notice)183 1216 y(that)e(if)g Fo(u)g Ft(is)g(a)g(top)h(white)f(term)g(o)q (ccurring)h(in)f(the)h(con)o(v)o(ersion,)g(then)g Fo(u)f Ft(itself)g(b)q(elongs)h(to)183 1266 y Fo(S)r Ft(.)j(By)g(Lemma)d(5.4,) h Fo(S)19 b Ft(consists)d(of)g(con\015uen)o(t)g(terms)g(b)q(ecause)h (ev)o(ery)g(elemen)o(t)e(of)g Fo(S)k Ft(is)183 1315 y(preserv)o(ed.)12 b(By)g(Lemma)c(5.3,)i Fo(S)k Ft(can)d(b)q(e)h(represen)o(ted)i(b)o(y)d (a)f(set)1199 1305 y(^)1192 1315 y Fo(S)s Ft(.)h(Recall)f(that)k(~)-24 b Fo(u)11 b Ft(denotes)183 1365 y(the)h(result)g(of)e(replacing)h(ev)o (ery)h(white)g(principal)e(subterm)h(in)g Fo(u)g Ft(with)g(its)g (represen)o(tativ)o(e.)183 1415 y(Again)i(w)o(e)i(claim)d(that)569 1407 y(~)568 1415 y Fo(t)583 1421 y Fn(1)k Fk(R)645 1425 y Fj(1)632 1392 y Fk(\003)d Fl(o)647 1415 y Fp( )704 1407 y Ft(~)703 1415 y Fo(t)743 1392 y Fl(o)g Fk(\003)733 1415 y Fp(!)772 1421 y Fk(R)801 1425 y Fj(1)833 1407 y Ft(~)832 1415 y Fo(t)847 1421 y Fn(2)866 1415 y Ft(.)h(Let)h Fo(u)991 1421 y Fn(1)1022 1415 y Fp(!)d Fo(u)1100 1421 y Fn(2)1133 1415 y Ft(b)q(e)j(a)g(step)g(in)f(the)h(con)o(v)o(ersion) 183 1465 y Fo(t)198 1471 y Fn(1)237 1450 y Fk(\003)247 1465 y Fp( )e Fo(t)f Fp(!)371 1450 y Fk(\003)401 1465 y Fo(t)416 1471 y Fn(2)434 1465 y Ft(.)i(The)g(claim)e(will)g(b)q(e)j (pro)o(v)o(ed)f(b)o(y)f(distinguishing)g(the)i(follo)o(wing)c(cases.) 200 1538 y(1.)20 b(Supp)q(ose)12 b Fo(u)437 1544 y Fn(1)467 1538 y Ft(is)f(a)g(top)g(blac)o(k)f(or)h(top)g(transparen)o(t)i(term,)d (i.e.,)f Fo(u)1261 1544 y Fn(1)1291 1538 y Ft(=)j Fo(C)1368 1523 y Fl(b)1365 1548 y Fn(1)1384 1538 y Fp(f)-14 b(f)p Fo(s)1431 1544 y Fn(1)1450 1538 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1562 1544 y Fl(n)1584 1538 y Fp(g)-14 b(g)p Ft(.)253 1588 y(If)10 b Fo(u)315 1594 y Fn(1)348 1588 y Fp(!)390 1572 y Fl(o)390 1599 y Fk(R)419 1603 y Fj(1)450 1588 y Fo(u)474 1594 y Fn(2)492 1588 y Ft(,)g(then)i Fo(u)630 1594 y Fn(2)659 1588 y Ft(can)e(b)q(e)i(written)f(as)g Fo(u)999 1594 y Fn(2)1028 1588 y Ft(=)h Fo(C)1105 1572 y Fl(b)1102 1598 y Fn(2)1122 1588 y Fp(h)-7 b(h)p Fo(s)1166 1594 y Fl(i)1178 1598 y Fj(1)1196 1588 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1308 1594 y Fl(i)1320 1598 y Ff(m)1349 1588 y Fp(i)-7 b(i)11 b Ft(and)g(it)f(follo)o(ws)253 1642 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1752 y Fn(1)608 1742 y Ft([)-7 b([)o Fo(s)643 1748 y Fn(1)662 1742 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)774 1748 y Fl(j)791 1742 y Fo(;)g(:)g(:)g(:)e(;)i(s)903 1748 y Fl(n)926 1742 y Ft(])-7 b(])14 b Fp(!)g Fo(C)1046 1727 y Fl(b)1043 1752 y Fn(1)1063 1742 y Ft([)-7 b([)o Fo(s)1098 1748 y Fn(1)1117 1742 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1229 1727 y Fk(0)1229 1753 y Fl(j)1246 1742 y Fo(;)g(:)g(:)g(:)e(;)i(s)1358 1748 y Fl(n)1380 1742 y Ft(])-7 b(])15 b(=)g Fo(u)1483 1748 y Fn(2)1517 1742 y Ft(where)253 1798 y Fo(s)272 1804 y Fl(j)302 1798 y Fp(!)c Fo(s)374 1783 y Fk(0)374 1809 y Fl(j)392 1798 y Ft(.)i(Clearly)m(,)h(^)-23 b Fo(s)588 1804 y Fl(j)618 1798 y Ft(=)13 b(^)-22 b Fo(s)681 1783 y Fk(0)681 1809 y Fl(j)712 1798 y Ft(and)14 b(hence)k(~)-24 b Fo(u)932 1804 y Fn(1)962 1798 y Ft(=)12 b Fo(C)1039 1783 y Fl(b)1036 1808 y Fn(1)1056 1798 y Ft([)r(^)-23 b Fo(s)1087 1804 y Fn(1)1105 1798 y Fo(;)7 b(:)g(:)g(:)e(;)j Ft(^)-22 b Fo(s)1217 1804 y Fl(j)1234 1798 y Fo(;)7 b(:)g(:)g(:)e(;)j Ft(^)-22 b Fo(s)1346 1804 y Fl(n)1369 1798 y Ft(])11 b(=)j(~)-23 b Fo(u)1460 1804 y Fn(2)1478 1798 y Ft(.)200 1848 y(2.)20 b(Supp)q(ose)e Fo(u)443 1854 y Fn(1)477 1848 y Ft(is)f(top)f(white.)g(Since)h Fo(u)860 1854 y Fn(1)895 1848 y Ft(is)g(preserv)o(ed,)h Fo(u)1163 1854 y Fn(2)1198 1848 y Ft(m)o(ust)d(also)h(b)q(e)h(top)f(white)253 1898 y(and)g(preserv)o(ed.)i(Hence)g Fo(u)685 1904 y Fn(1)719 1898 y Ft(and)e Fo(u)826 1904 y Fn(2)861 1898 y Ft(are)g(b)q(oth)h(in)e Fo(S)r Ft(.)i(Of)f(course,)h(they)g(m)o(ust)e (ha)o(v)o(e)253 1947 y(the)g(same)e(represen)o(tativ)o(e.)i(So)h(~)-23 b Fo(u)789 1953 y Fn(1)819 1947 y Ft(=)14 b(^)-24 b Fo(u)886 1953 y Fn(1)916 1947 y Ft(=)15 b(^)-24 b Fo(u)984 1953 y Fn(2)1014 1947 y Ft(=)15 b(~)-24 b Fo(u)1082 1953 y Fn(2)1100 1947 y Ft(.)183 2020 y(This)10 b(sho)o(ws)g(the)h(claim.)d (Since)j Fp(!)723 2005 y Fl(o)723 2032 y Fk(R)752 2036 y Fj(1)779 2020 y Ft(is)f(con\015uen)o(t,)h(the)g(terms)1184 2013 y(~)1184 2020 y Fo(t)1199 2026 y Fn(1)1227 2020 y Ft(and)1305 2013 y(~)1305 2020 y Fo(t)1320 2026 y Fn(2)1348 2020 y Ft(ha)o(v)o(e)f(a)g(common)183 2075 y(reduct)312 2068 y(~)312 2075 y Fo(t)327 2081 y Fn(3)346 2075 y Ft(.)j(W)m(e)g (conclude)i(that)f Fo(t)718 2081 y Fn(1)748 2075 y Fp(!)790 2060 y Fk(\003)821 2068 y Ft(~)820 2075 y Fo(t)835 2081 y Fn(3)875 2060 y Fk(\003)884 2075 y Fp( )g Fo(t)955 2081 y Fn(2)973 2075 y Ft(.)183 2125 y Fh(2)245 2204 y Ft(F)m(or)i(the)i(rest)g(of)e(this)h(section,)g(w)o(e)g(additionally) e(assume)h Fp(!)1253 2210 y Fl(c)1287 2204 y Ft(to)g(b)q(e)i (normalizing.)183 2254 y(Under)c(this)g(condition)g(con\015uence)h (turns)g(out)f(to)f(b)q(e)i(mo)q(dular.)183 2332 y Fm(Lemma)g(5.6)21 b Ft(Ev)o(ery)14 b(term)f Fo(t)h Ft(has)g(a)g(preserv)o(ed)i(reduct.) 183 2382 y Fm(Pro)q(of:)c Ft(Since)h Fp(!)474 2388 y Fl(c)503 2382 y Ft(is)g(normalizing,)d Fo(t)h Fp(!)849 2367 y Fk(\003)849 2392 y Fl(c)880 2382 y Fo(t)895 2367 y Fk(0)919 2382 y Ft(for)i(some)f Fo(t)1100 2367 y Fk(0)1123 2382 y Fp(2)f Fo(N)5 b(F)h Ft(\()p Fp(!)1291 2388 y 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Fm(Prop)q(ositi)o(on)13 b(6.4)20 b Ft(If)15 b Fp(R)600 1304 y Fn(1)634 1298 y Ft(and)g Fp(R)751 1304 y Fn(2)784 1298 y Ft(are)h(semi-complete,)c(then)k(the)g (relation)e Fp(!)1501 1304 y Fl(c)1532 1298 y Ft(is)h(in-)183 1348 y(nermost)e(normalizing)e(on)j Fp(T)c Ft(\()p Fp(D)q Fo(;)d Fp(C)r Fo(;)g Fp(V)s Ft(\))q(.)183 1399 y Fm(Pro)q(of:)14 b Ft(It)h(is)h(sho)o(wn)f(that)g(ev)o(ery)h(term)f Fo(t)g Ft(is)g(innermost)g(normalizing)d(w.r.t.)i Fp(!)1497 1405 y Fl(c)1529 1399 y Ft(using)183 1448 y(induction)i(on)h Fo(r)q(ank)q Ft(\()p Fo(t)p Ft(\))g(=)g Fo(k)q Ft(.)g(The)g(case)h Fo(k)g Ft(=)f(0)g(is)g(trivially)e(true)j(and)f(for)g Fo(k)h Ft(=)f(1)g(the)183 1498 y(assertion)h(follo)o(ws)f(from)f(Lemma) f(6.2)i(in)g(conjunction)h(with)g(case)h(\(iii\))e(b)q(elo)o(w.)g(So)g (let)183 1548 y Fo(r)q(ank)q Ft(\()p Fo(t)p Ft(\))11 b(=)h Fo(k)g(>)g Ft(1.)h(The)i(follo)o(wing)c(case)k(analysis)e(yields) h(the)g(result.)183 1598 y(Case)g(\(i\):)f Fo(t)h Ft(is)g(top)g(blac)o (k.)183 1648 y(Then)25 b(w)o(e)f(ma)o(y)f(write)h Fo(t)29 b Ft(=)h Fo(C)729 1633 y Fl(b)745 1648 y Ft([)-7 b([)p Fo(t)777 1654 y Fn(1)795 1648 y Fo(;)7 b(:)g(:)g(:)t(;)g(t)902 1654 y Fl(n)924 1648 y Ft(])-7 b(].)23 b(Since)i Fo(r)q(ank)q Ft(\()p Fo(t)1216 1654 y Fl(j)1234 1648 y Ft(\))k Fo(<)g(r)q(ank)q Ft(\()p Fo(t)p Ft(\),)24 b(it)g(fol-)183 1697 y(lo)o(ws)19 b(from)f(the)j(induction)e(h)o(yp)q(othesis)i(that,)f(for)f(ev)o(ery)i Fo(j)j Fp(2)d(f)p Ft(1)p Fo(;)7 b(:)g(:)g(:)e(;)i(n)p Fp(g)p Ft(,)18 b(there)j(ex-)183 1747 y(ists)k(an)g(innermost)f(deriv)n (ation)g Fo(t)763 1753 y Fl(j)810 1747 y Fp(!)852 1732 y Fk(\003)852 1758 y Fl(c)901 1747 y Fo(t)916 1732 y Fk(0)916 1758 y Fl(j)958 1747 y Ft(suc)o(h)i(that)f Fo(t)1179 1732 y Fk(0)1179 1758 y Fl(j)1227 1747 y Fp(2)k Fo(N)5 b(F)h Ft(\()p Fp(!)1413 1753 y Fl(c)1429 1747 y Ft(\).)25 b(Accord-)183 1803 y(ing)c(to)i(Lemma)d(6.3,)h Fo(t)k Ft(=)i Fo(C)696 1788 y Fl(b)712 1803 y Ft([)-7 b([)p Fo(t)744 1809 y Fn(1)762 1803 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)870 1809 y Fl(n)892 1803 y Ft(])-7 b(])25 b Fp(!)976 1788 y Fk(\003)976 1813 y Fl(c)1020 1803 y Fo(C)1053 1788 y Fl(b)1069 1803 y Ft([)p Fo(t)1096 1788 y Fk(0)1096 1813 y Fn(1)1115 1803 y Fo(;)7 b(:)g(:)g(:)t(;)g(t)1222 1788 y Fk(0)1222 1813 y Fl(n)1244 1803 y Ft(].)22 b(Observ)o(e)i(that)e (this)183 1853 y(is)f(also)h(an)g(innermost)f(deriv)n(ation.)f(Moreo)o (v)o(er,)i(since)h Fo(t)1133 1838 y Fk(0)1133 1864 y Fl(j)1173 1853 y Ft(can)f(b)q(e)g(written)h(as)f Fo(t)1549 1838 y Fk(0)1549 1864 y Fl(j)1591 1853 y Ft(=)192 1902 y(^)183 1913 y Fo(C)216 1898 y Fl(b)213 1924 y(j)232 1913 y Fp(h)-7 b(h)p Fo(t)272 1893 y Fl(j)272 1924 y Fn(1)291 1913 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)399 1898 y Fl(j)399 1923 y(m)428 1927 y Ff(j)445 1913 y Fp(i)-7 b(i)p Ft(,)17 b(where)i(ev)o(ery)f(top)f(white)h Fo(t)945 1893 y Fl(j)945 1924 y(i)979 1913 y Ft(is)g(an)f(elemen)o(t)f(of)h Fo(N)5 b(F)h Ft(\()p Fp(!)1422 1919 y Fl(c)1438 1913 y Ft(\),)17 b(w)o(e)g(ma)o(y)183 1975 y(write)h Fo(C)326 1960 y Fl(b)342 1975 y Ft([)p Fo(t)369 1960 y Fk(0)369 1985 y Fn(1)387 1975 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)495 1960 y Fk(0)495 1985 y Fl(n)517 1975 y Ft(])18 b(=)607 1964 y(^)598 1975 y Fo(C)631 1960 y Fl(b)647 1975 y Fp(f)-14 b(f)p Fo(s)694 1981 y Fn(1)713 1975 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)825 1981 y Fl(m)856 1975 y Fp(g)-14 b(g)18 b Ft(for)f(some)h(blac)o(k)f (con)o(text)1351 1964 y(^)1342 1975 y Fo(C)1375 1960 y Fl(b)1391 1975 y Fp(f)-14 b(f)p Fo(;)7 b(:)g(:)g(:)e(;)i Fp(g)-14 b(g)17 b Ft(and)183 2025 y(top)e(white)h(terms)f Fo(s)509 2031 y Fn(1)528 2025 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)640 2031 y Fl(m)686 2025 y Fp(2)14 b Fo(N)5 b(F)h Ft(\()p Fp(!)857 2031 y Fl(c)873 2025 y Ft(\).)15 b(Cho)q(ose)h(fresh)h(v)n (ariables)d Fo(x)1362 2031 y Fn(1)1381 2025 y Fo(;)7 b(:)g(:)g(:)e(;)i(x)1498 2031 y Fl(m)1544 2025 y Ft(suc)o(h)183 2074 y(that)14 b Fo(<)6 b(s)330 2080 y Fn(1)350 2074 y Fo(;)h(:)g(:)g(:)e(;)i(s)462 2080 y Fl(m)499 2074 y Fo(>)14 b Fp(1)e Fo(<)6 b(x)661 2080 y Fn(1)680 2074 y Fo(;)h(:)g(:)g(:)e(;)i(x)797 2080 y Fl(m)834 2074 y Fo(>)p Ft(.)14 b(Since)1011 2064 y(^)1001 2074 y Fo(C)1034 2059 y Fl(b)1051 2074 y Fp(f)p Fo(x)1096 2080 y Fn(1)1114 2074 y Fo(;)7 b(:)g(:)g(:)e(;)i(x)1231 2080 y Fl(m)1261 2074 y Fp(g)13 b(2)g(T)d Ft(\()p Fp(D)1417 2080 y Fn(1)1436 2074 y Fo(;)d Fp(C)r Fo(;)g Fp(V)s Ft(\))15 b(and)183 2124 y Fp(!)225 2130 y Fl(c)260 2124 y Ft(is)20 b(terminating)d(on)j Fp(T)10 b Ft(\()p Fp(D)684 2130 y Fn(1)702 2124 y Fo(;)d Fp(C)r Fo(;)g Fp(V)s Ft(\))q(,)19 b(it)g(follo)o(ws)e(that)j(eac)o(h)g (innermost)e Fp(!)1464 2130 y Fl(c)1500 2124 y Ft(deriv)n(a-)183 2174 y(tion)c(starting)i(from)534 2164 y(^)525 2174 y Fo(C)558 2159 y Fl(b)574 2174 y Fp(f)p Fo(x)619 2180 y Fn(1)637 2174 y Fo(;)7 b(:)g(:)g(:)e(;)i(x)754 2180 y Fl(m)785 2174 y Fp(g)15 b Ft(ends)h(in)f(some)1082 2164 y(\026)1072 2174 y Fo(C)1105 2159 y Fl(b)1122 2174 y Fp(h)p Fo(x)1162 2180 y Fl(i)1174 2184 y Fj(1)1191 2174 y Fo(;)7 b(:)g(:)g(:)e(;)i(x)1308 2180 y Fl(i)1320 2184 y Ff(l)1333 2174 y Fp(i)14 b(2)g Fo(N)5 b(F)h Ft(\()p Fp(!)1534 2180 y Fl(c)1550 2174 y Ft(\).)14 b(It)183 2228 y(follo)o(ws)h(from)433 2218 y(^)423 2228 y Fo(C)456 2213 y Fl(b)473 2228 y Fp(f)p Fo(x)518 2234 y Fn(1)536 2228 y Fo(;)7 b(:)g(:)g(:)e(;)i(x)653 2234 y Fl(m)683 2228 y Fp(g)17 b(!)763 2213 y Fk(\003)763 2238 y Fl(c)808 2218 y Ft(\026)798 2228 y Fo(C)831 2213 y Fl(b)847 2228 y Fp(h)p Fo(x)887 2234 y Fl(i)899 2238 y Fj(1)917 2228 y Fo(;)7 b(:)g(:)g(:)e(;)i(x)1034 2234 y Fl(i)1046 2238 y Ff(l)1059 2228 y Fp(i)17 b Ft(and)g(the)h(sp)q(ecial)f(c)o(hoice)h (of)e(the)183 2282 y(v)n(ariables)f Fo(x)381 2288 y Fn(1)400 2282 y Fo(;)7 b(:)g(:)g(:)e(;)i(x)517 2288 y Fl(m)564 2282 y Ft(that)16 b Fo(t)g Fp(!)729 2267 y Fk(\003)729 2293 y Fl(c)773 2272 y Ft(^)764 2282 y Fo(C)797 2267 y Fl(b)813 2282 y Fp(f)-14 b(f)p Fo(s)860 2288 y Fn(1)879 2282 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)991 2288 y Fl(m)1022 2282 y Fp(g)-14 b(g)16 b(!)1108 2267 y Fk(\003)1108 2293 y Fl(c)1152 2272 y Ft(\026)1142 2282 y Fo(C)1175 2267 y Fl(b)1192 2282 y Fp(h)-7 b(h)p Fo(s)1236 2288 y Fl(i)1248 2292 y Fj(1)1266 2282 y Fo(;)7 b(:)g(:)g(:)e(;)i(s)1378 2288 y Fl(i)1390 2292 y Ff(l)1404 2282 y Fp(i)-7 b(i)16 b Ft(=)g Fo(t)1508 2267 y Fk(0)1520 2282 y Ft(.)g(It)h(is)183 2332 y(easy)h(to)f(v)o(erify)h(that)f Fo(t)560 2317 y Fk(0)590 2332 y Fp(2)g Fo(N)5 b(F)h Ft(\()p Fp(!)764 2338 y Fl(c)780 2332 y Ft(\))18 b(and)g(that)f Fo(t)h Fp(!)1067 2317 y Fk(\003)1067 2342 y Fl(c)1104 2332 y Fo(t)1119 2317 y Fk(0)1148 2332 y Ft(is)f(an)h(innermost)f(reduction) 183 2382 y(sequence.)183 2432 y(Case)d(\(ii\):)f Fo(t)h Ft(is)f(top)h(white.)g(Analogous)f(to)g(case)i(\(i\).)p eop %%Page: 14 14 14 13 bop 340 194 a Ft(Case)15 b(\(iii\):)d Fo(t)i Ft(is)g(top)f (transparen)o(t.)340 244 y(Let)k Fo(t)f Ft(=)f Fo(C)528 228 y Fl(t)543 244 y Ft([)-7 b([)p Fo(t)575 250 y Fn(1)592 244 y Fo(;)7 b(:)g(:)g(:)e(;)i(t)700 250 y Fl(n)722 244 y Ft(])-7 b(].)15 b(According)i(to)f(cases)h(\(i\))f(and)g(\(ii\),)g (ev)o(ery)h Fo(t)1479 250 y Fl(j)1512 244 y Ft(reduces)i(via)c(in-)340 293 y(nermost)d(rewriting)g(to)f(some)g Fo(t)839 278 y Fk(0)839 304 y Fl(j)868 293 y Fp(2)h Fo(N)5 b(F)h Ft(\()p Fp(!)1037 299 y Fl(c)1052 293 y Ft(\).)12 b(Clearly)m(,)e Fo(t)1257 278 y Fk(0)1280 293 y Ft(=)i Fo(C)1357 278 y Fl(t)1372 293 y Ft([)p Fo(t)1399 278 y Fk(0)1399 304 y Fn(1)1417 293 y Fo(;)7 b(:)g(:)g(:)t(;)g(t)1524 278 y Fk(0)1524 304 y Fl(n)1546 293 y Ft(])12 b Fp(2)f Fo(N)5 b(F)h Ft(\()p Fp(!)1738 299 y Fl(c)1754 293 y Ft(\),)340 343 y Fo(t)12 b Fp(!)409 328 y Fk(\003)409 353 y Fl(c)439 343 y Fo(t)454 328 y Fk(0)466 343 y Ft(,)h(and)h(the)g(deriv)n(ation)f (is)h(innermost.)f Fh(2)340 449 y Fm(Theorem)i(6.5)21 b Ft(Semi-completeness)13 b(is)h(mo)q(dular)e(for)h (constructor-sharing)i(TRSs.)340 503 y Fm(Pro)q(of:)k Ft(Let)g Fp(R)603 509 y Fn(1)641 503 y Ft(and)g Fp(R)762 509 y Fn(2)800 503 y Ft(b)q(e)g(constructor-sharing)i(TRSs)e(and)f(let) h Fp(R)i Ft(=)f Fp(R)1644 509 y Fn(1)1676 503 y Fp([)12 b(R)1751 509 y Fn(2)1770 503 y Ft(.)340 553 y(W)m(e)k(ha)o(v)o(e)h(to)f (sho)o(w)h(that)f Fp(R)h Ft(is)f(semi-complete)f(if)g(and)i(only)e(if)h Fp(R)1426 559 y Fn(1)1461 553 y Ft(and)g Fp(R)1580 559 y Fn(2)1615 553 y Ft(are)h(semi-)340 603 y(complete.)f(The)h(only-if)e (direction)h(is)h(trivial.)d(So)j(let)f Fp(R)1266 609 y Fn(1)1301 603 y Ft(and)g Fp(R)1419 609 y Fn(2)1454 603 y Ft(b)q(e)i(semi-complete.)340 652 y(According)f(to)e(Prop)q (osition)h(6.4,)e Fp(!)940 658 y Fl(c)973 652 y Ft(is)h(innermost)g (normalizing)f(and)h(hence)j(normal-)340 702 y(izing.)d(Consequen)o (tly)m(,)g Fp(R)h Ft(is)f(con\015uen)o(t)h(b)o(y)g(Theorem)f(5.10.)e (Finally)m(,)g(Prop)q(osition)j(3.14)340 752 y(states)f(that)f Fp(R)g Ft(is)g(also)f(normalizing.)e Fh(2)340 905 y Fq(7)56 b(Conclusions)340 1024 y Ft(Sev)o(eral)15 b(su\016cien)o(t)g (conditions)g(for)f(the)h(mo)q(dularit)o(y)e(of)h(con\015uence)i(in)e (the)i(presence)h(of)340 1073 y(shared)f(constructors)g(ha)o(v)o(e)f(b) q(een)h(stated)f(in)f(this)h(pap)q(er.)f(After)i(all,)d(the)i(whole)f (matter)340 1123 y(b)q(oils)e(do)o(wn)g(to)g(ensure)i(normalization)c (of)i(the)g(collapsing)g(reduction)h(relation)e Fp(!)1665 1129 y Fl(c)1682 1123 y Ft(.)h(P)o(er-)340 1173 y(haps)j(this)g (approac)o(h)f(leads)h(to)f(more)g(in)o(teresting)h(su\016cien)o(t)g (criteria.)g(The)g(mo)q(dularit)o(y)340 1223 y(of)f(con\015uence)i(for) f(left-linear)e(TRSs)i([R)-5 b(V80)o(])14 b(and)g(our)g(results)i(are)f (complemen)o(tary)m(.)c(It)340 1273 y(has)i(b)q(een)h(p)q(oin)o(ted)f (out)g(that)g(semi-completeness)g(of)f(the)h(constituen)o(t)h(systems)f (ensures)340 1322 y(normalization)h(of)i Fp(!)698 1328 y Fl(c)714 1322 y Ft(.)g(In)g(fact,)g(with)g(the)h(aid)e(of)h(Theorem)g (6.5,)f(one)h(can)h(sho)o(w)f(that)340 1372 y Fp(!)382 1378 y Fl(c)415 1372 y Ft(is)h(ev)o(en)g(terminating)e(under)j(these)g (circumstances.)f(It)g(is)f(p)q(ossible)h(that)g(a)f(direct)340 1422 y(pro)q(of)f(of)g(the)h(termination)d(of)i Fp(!)890 1428 y Fl(c)921 1422 y Ft(will)f(turn)i(out)f(to)g(b)q(e)h(simpler)e (than)h(the)h(giv)o(en)f(one.)340 1472 y(This)f(p)q(oin)o(t)g(needs)h (further)g(in)o(v)o(estigations.)403 1525 y(In)c([KK90],)f(Kurihara)h (and)g(Ka)r(ji)g(in)o(tro)q(duced)h(a)f(v)o(ery)g(in)o(teresting)h (relation)f Fh(;)g Ft(called)340 1575 y(\\mo)q(dular)21 b(reduction".)h(Roughly)g(sp)q(eaking,)g(if)f Fp(R)1207 1581 y Fn(1)1226 1575 y Fo(;)7 b(:)g(:)g(:)e(;)i Fp(R)1354 1581 y Fl(n)1399 1575 y Ft(are)23 b(pairwise)f(disjoin)o(t)340 1625 y(TRSs,)c(then)g(reduction)h(steps)g(ha)o(v)o(e)f(to)g(b)q(e)g(p)q (erformed)g(using)g(the)g(same)f(constituen)o(t)340 1675 y(TRS)f Fp(R)475 1681 y Fl(j)509 1675 y Ft(as)g(long)f(as)h(p)q (ossible.)g(More)h(precisely)m(,)f Fo(s)f Fh(;)g Fo(t)h Ft(if)f(and)h(only)g(if)f Fo(s)h Fp(!)1623 1657 y Fn(+)1623 1687 y Fk(R)1652 1691 y Ff(j)1683 1675 y Fo(t)g Ft(and)340 1731 y Fo(t)21 b Fp(2)g Fo(N)5 b(F)h Ft(\()p Fp(!)554 1737 y Fk(R)583 1741 y Ff(j)599 1731 y Ft(\))19 b(for)h(some)e Fo(j)24 b Fp(2)c(f)p Ft(1)p Fo(;)7 b(:)g(:)g(:)t(;)g(n)p Fp(g)p Ft(.)19 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y(X)g(macro)g(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