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b(Ob)m(viously)-8 b(,)40 b(application)e(is)h(monotone)f(in)h(the)0 5550 y(second)c(argumen)m(t,)d(i.e.)1416 5669 y Fn(u)27 b Fl(w)i Fn(u)1663 5628 y Ff(0)1718 5669 y Fl(\))e Fn(W)14 b(u)28 b Fl(w)g Fn(W)14 b(u)2308 5628 y Ff(0)2336 5669 y Fn(:)1211 b Fr(\(1)p Fn(:)p Fr(6\))1865 5908 y(2)p eop %%Page: 3 3 3 2 bop 0 -137 a Fr(No)m(w)33 b(de\014ne)i Fn(W)41 b Fl(w)28 b Fn(W)863 -173 y Ff(0)925 -137 y Fr(b)m(y)883 82 y Fn(W)42 b Fl(w)28 b(f)p Fr(\()p Fn(u)1270 41 y Ff(0)1270 107 y Fm(j)1312 82 y Fn(;)17 b(v)1409 41 y Ff(0)1405 107 y Fm(j)1446 82 y Fr(\):)g Fn(j)33 b Fl(2)28 b Fn(J)10 b Fl(g)27 b Fr(:)g Fl(\()-17 b(\))55 b(8)p Fn(j)34 b Fl(2)28 b Fn(J)o(:W)14 b(u)2604 41 y Ff(0)2604 107 y Fm(j)2673 82 y Fl(w)28 b Fn(v)2830 41 y Ff(0)2826 107 y Fm(j)2869 82 y Fn(:)678 b Fr(\(1)p Fn(:)p Fr(7\))0 301 y Fj(Lemma)38 b(1.2.)58 b Fi(If)33 b Fn(D)k Fi(and)d Fn(E)k Fi(are)33 b(information)f(systems,)h(then)h(so)f(is)g Fn(D)e Fl(!)c Fn(E)6 b Fi(.)0 476 y Fr(Pro)s(of.)44 b(W)-8 b(e)33 b(\014rst)h(sho)m(w)g(the)f(transitivit)m(y)f(of)h Fl(w)p Fr(.)44 b(So)34 b(let)973 695 y Fn(W)41 b Fl(w)28 b(f)p Fr(\()p Fn(u)1359 654 y Ff(0)1359 720 y Fm(j)1401 695 y Fn(;)17 b(v)1498 654 y Ff(0)1494 720 y Fm(j)1535 695 y Fr(\):)g Fn(j)33 b Fl(2)28 b Fn(J)10 b Fl(g)27 b(w)h(f)p Fr(\()p Fn(u)2180 654 y Ff(00)2180 720 y Fm(k)2231 695 y Fn(;)17 b(v)2328 654 y Ff(00)2324 720 y Fm(k)2378 695 y Fr(\):)g Fn(k)30 b Fl(2)e Fn(K)7 b Fl(g)p Fn(:)0 914 y Fr(Then)34 b(w)m(e)g(ha)m(v)m(e)g(for)f(all)f Fn(k)f Fl(2)d Fn(K)40 b Fr(b)m(y)33 b(\(1.6\))f(and)i(\(1.7\))774 1153 y Fn(W)14 b(u)939 1112 y Ff(00)939 1178 y Fm(k)1018 1153 y Fl(w)1123 1058 y Fe([)1234 1153 y Fl(f)p Fn(W)g(u)1449 1112 y Ff(0)1449 1178 y Fm(j)1491 1153 y Fr(:)j Fn(u)1593 1112 y Ff(00)1593 1178 y Fm(k)1670 1153 y Fl(w)29 b Fn(u)1833 1112 y Ff(0)1833 1178 y Fm(j)1875 1153 y Fl(g)e(w)2057 1058 y Fe([)2168 1153 y Fl(f)p Fn(v)2270 1112 y Ff(0)2266 1178 y Fm(j)2308 1153 y Fr(:)17 b Fn(u)2410 1112 y Ff(00)2410 1178 y Fm(k)2488 1153 y 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Fe(S)1649 1661 y Fm(j)t Ff(2)p Fm(J)1791 1641 y Fd(0)1839 1631 y Fn(u)1896 1595 y Ff(0)1896 1657 y Fm(j)1976 1631 y Fl(2)38 b Fr(Con)2258 1646 y Fm(D)2330 1631 y Fr(.)62 b(By)39 b(\(1.4\))f(w)m(e)i(ha)m(v)m(e) g(to)f(sho)m(w)h(that)0 1676 y Fe(S)83 1781 y Fm(j)t Ff(2)p Fm(J)225 1761 y Fd(0)273 1751 y Fn(v)325 1714 y Ff(0)321 1777 y Fm(j)391 1751 y Fl(2)28 b Fr(Con)662 1766 y Fm(E)729 1751 y Fr(.)44 b(But)33 b(this)g(follo)m(ws)h(from)947 1913 y Fe([)1058 2008 y Fl(f)p Fn(v)1156 2023 y Fm(i)1189 2008 y Fr(:)1262 1913 y Fe([)1234 2127 y Fm(j)t Ff(2)p Fm(J)1376 2107 y Fd(0)1419 2008 y Fn(u)1476 1967 y Ff(0)1476 2033 y Fm(j)1545 2008 y Fl(w)28 b Fn(u)1707 2023 y Fm(i)1741 2008 y Fl(g)f(w)1923 1913 y Fe([)2034 2008 y Fl(f)p Fn(v)2132 2023 y Fm(i)2165 2008 y Fr(:)17 b Fn(u)2267 1967 y Ff(0)2267 2033 y Fm(j)2336 2008 y Fl(w)28 b Fn(u)2498 2023 y Fm(i)2532 2008 y Fl(g)f(w)h Fn(v)2766 1967 y Ff(0)2762 2033 y Fm(j)2804 2008 y Fn(:)0 2336 y Fr(This)33 b(concludes)i(the)f(pro)s(of.)p 1147 2336 4 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(ely)g(deal)h(with)f(the)h(information)f(systems)g(built)h(up)g(from)f Fn(D)3368 3481 y Fm(\023)3448 3466 y Fr(and)h Fn(D)3735 3481 y Fm(o)0 3585 y Fr(b)m(y)d(the)g Fl(!)p Fr({op)s(eration.)69 b(More)42 b(formally)-8 b(,)41 b(de\014ne)j(the)e(notion)f(of)h(a)g Fh(typ)-5 b(e)44 b(symb)-5 b(ol)52 b Fr(and)43 b(its)e Fh(level)0 3705 y Fr(inductiv)m(ely)33 b(b)m(y)g(the)h(clauses)38 3879 y(1.)49 b Fn(\023)34 b Fr(and)g Fn(o)f Fr(are)g(t)m(yp)s(e)h(sym)m (b)s(ols,)d(and)j(lev)q(\()p Fn(\023)p Fr(\))27 b(=)i(lev)q(\()p Fn(o)p Fr(\))e(=)h(0.)38 3999 y(2.)49 b(If)36 b Fn(\032)e Fr(and)h Fn(\033)j Fr(are)d(t)m(yp)s(e)g(sym)m(b)s(ols,)f(then)h(so)g (is)g(\()p Fn(\032)29 b Fl(!)h Fn(\033)t Fr(\),)k(and)i(lev)q(\(\()p Fn(\032)29 b Fl(!)h Fn(\033)t Fr(\)\))f(=)i(max)n(\(lev)q(\()p Fn(\032)p Fr(\))22 b(+)165 4119 y(1)p Fn(;)17 b Fr(lev\()p Fn(\033)t Fr(\)\).)0 4293 y(As)37 b(usual)h(w)m(e)g(write)e Fn(\032)877 4308 y Fk(1)922 4293 y Fn(;)17 b(:)g(:)g(:)d(;)j(\032)1196 4308 y Fm(m)1305 4293 y Fl(!)34 b Fn(\033)40 b Fr(for)e(\()p Fn(\032)1784 4308 y Fk(1)1862 4293 y Fl(!)c Fr(\()p Fn(\032)2087 4308 y Fk(2)2165 4293 y Fl(!)g Fn(:)17 b(:)g(:)e Fr(\()p Fn(\032)2523 4308 y Fm(m)2632 4293 y Fl(!)34 b Fn(\033)t Fr(\))17 b Fn(:)g(:)g(:)m Fr(\)\).)56 b(Note)37 b(that)g(an)m(y)0 4413 y(t)m(yp)s(e)42 b(sym)m(b)s(ol)f(can)h(b)s(e)g(written)g(uniquely) g(in)g(the)g(form)f Fn(\032)2290 4428 y Fk(1)2334 4413 y Fn(;)17 b(:)g(:)g(:)d(;)j(\032)2608 4428 y Fm(m)2725 4413 y Fl(!)42 b Fn(\023)g Fr(or)g Fn(\032)3127 4428 y Fk(1)3171 4413 y Fn(;)17 b(:)g(:)g(:)d(;)j(\032)3445 4428 y Fm(m)3562 4413 y Fl(!)42 b Fn(o)p Fr(.)0 4533 y(F)-8 b(or)38 b(an)m(y)g(t)m(yp)s(e)g(sym)m(b)s(ol)e Fn(\032)i Fr(de\014ne)i(the)e(information)f(system)g Fn(D)2472 4548 y Fm(\032)2557 4533 y Fr(as)h(follo)m(ws.)59 b Fn(D)3147 4548 y Fm(\023)3220 4533 y Fr(and)38 b Fn(D)3500 4548 y Fm(o)3583 4533 y Fr(ha)m(v)m(e)0 4652 y(already)31 b(b)s(een)h(de\014ned,)i(and)e Fn(D)1227 4667 y Fm(\032)p Ff(!)p Fm(\033)1429 4652 y Fr(:=)c 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b Fl(2)h Fn(I)8 b Fl(g)27 b(2)h Fr(Con)17 b Fn(:)670 b Fr(\(1)p Fn(:)p Fr(9\))0 5550 y(Ob)m(viously)46 b Fn(D)563 5565 y Fm(\023)643 5550 y Fr(and)g Fn(D)931 5565 y Fm(o)1022 5550 y Fr(are)f(coheren)m(t.)83 b(No)m(w)45 b(the)h(coherence)i(of)e (all)e(standard)j(information)0 5669 y(systems)33 b Fn(D)452 5684 y Fm(\032)532 5669 y Fr(follo)m(ws)g(from)1865 5908 y(3)p eop %%Page: 4 4 4 3 bop 0 -137 a Fj(Lemma)31 b(1.3.)57 b Fi(If)28 b Fn(D)i Fi(and)d Fn(E)32 b Fi(are)27 b(information)f(systems)g(and)i Fn(E)k Fi(is)27 b(coheren)m(t,)i(then)f(so)e(is)h Fn(D)k Fl(!)d Fn(E)6 b Fi(.)0 24 y Fr(Pro)s(of.)44 b(Let)33 b Fl(f)p Fr(\()p Fn(u)636 39 y Fm(i)669 24 y Fn(;)17 b(v)762 39 y Fm(i)795 24 y Fr(\):)g Fn(i)27 b Fl(2)h Fn(I)8 b Fl(g)32 b Fr(b)s(e)i(\014nite)g(and)f(assume)1054 177 y Fl(8)p Fn(i;)17 b(j)34 b Fl(2)28 b Fn(I)8 b(:)p Fl(f)p Fr(\()p Fn(u)1583 192 y Fm(i)1615 177 y Fn(;)17 b(v)1708 192 y Fm(i)1741 177 y Fr(\))p Fn(;)g Fr(\()p Fn(u)1921 192 y Fm(j)1961 177 y Fn(;)g(v)2054 192 y 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Fm(\032)2145 1581 y Fc(m)2216 1556 y Fl(j)27 b(!)g(j)p Fn(D)2508 1571 y Fm(\033)2562 1556 y Fl(j)0 1800 y Fr(b)m(y)33 b(induction)h(on)g Fn(r)s Fr(,)e(as)h(follo)m(ws.)38 1983 y(1.)49 b Fl(j)o Fn(~)-49 b(x)28 b Fl(7!)f Fn(x)462 1998 y Fm(i)495 1983 y Fl(j)33 b Fr(is)g(the)h Fn(i)p Fr({th)f(pro)6 b(jection)34 b(function,)g(whic)m (h)g(is)f(clearly)f(con)m(tin)m(uous.)38 2104 y(2.)49 b Fl(j)o Fn(~)-49 b(x)28 b Fl(7!)f Fn(\025y)t(:r)s Fl(j)f Fr(:=)i Fl(j)o Fn(~)-49 b(x;)17 b(y)30 b Fl(7!)d Fn(r)s Fl(j)1191 2119 y Ff(\000)1291 2104 y Fr(\(cf.)45 b(Lemma)31 b(2.5\).)38 2225 y(3.)49 b Fl(j)o Fn(~)-49 b(x)49 b Fl(7!)f Fn(ts)p Fl(j)e Fr(is)g(the)g(result)g(of)h(substituting)f(the)g (functions)i Fl(j)o Fn(~)-49 b(x)48 b Fl(7!)h Fn(t)p Fl(j)c Fr(and)i Fl(j)o Fn(~)-49 b(x)48 b Fl(7!)h Fn(s)p Fl(j)c Fr(in)h(the)165 2345 y(con)m(tin)m(uous)29 b(binary)d (application)h(function.)43 b(Clearly)25 b(the)j(resulting)f(function)h (is)e(con)m(tin)m(uous.)0 2529 y(No)m(w)e(w)m(e)g(can)g(de\014ne)i(the) e 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y Fl(j)p Fn(;)g Fl(j)p Fn(D)1855 1403 y Fm(\023;\032)p Ff(!)p Fm(\032)2074 1388 y Fl(j)28 b(!)f(j)p Fn(D)2367 1403 y Fm(\032)2414 1388 y Fl(j)0 1636 y Fr(b)m(y)1634 1764 y Fn(h)1691 1779 y Fk(0)1736 1764 y Fr(\()p Fn(y)t(;)17 b(z)t Fr(\))26 b(=)i Fn(y)1243 1970 y(h)1300 1985 y Fm(m)p Fk(+1)1477 1970 y Fr(\()p Fn(y)t(;)17 b(z)t Fr(\))27 b(=)h Fn(z)t Fl(f)p Fn(m)p Fl(g)p Fr(\()p Fn(h)2168 1985 y Fm(m)2245 1970 y Fr(\()p Fn(y)t(;)17 b(z)t Fr(\)\))p Fn(:)0 2168 y Fr(Clearly)32 b(eac)m(h)i Fn(h)627 2183 y Fm(m)737 2168 y Fr(is)f(con)m(tin)m(uous)i (,)d(b)m(y)i(induction)g(on)f Fn(m)p Fr(.)44 b(No)m(w)33 b(de\014ne)i(a)e(function)1377 2415 y Fn(f)11 b Fr(:)17 b Fl(j)p Fn(D)1592 2430 y Fm(\023)1626 2415 y Fl(j)27 b(!)g(j)p Fn(D)1918 2433 y Fm(\032;)p Fk(\()p Fm(\023;\032)p Ff(!)p Fm(\023)p Fk(\))p Ff(!)p Fm(\032)2375 2415 y Fl(j)0 2663 y Fr(b)m(y)1705 2790 y Fn(f)11 b Fr(\()p Fl(;)p Fr(\))27 b(=)h Fl(;)1477 2996 y Fn(f)11 b Fr(\()p Fl(f)p Fn(m)p Fl(g)p Fr(\))27 b(=)h(\()p Fn(h)2030 3011 y 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5669 y Fl(j)637 5628 y Fm(pr)750 5669 y Fr(:=)28 b Fl(fj)p Fn(r)s Fl(j)p Fr(:)17 b Fn(r)34 b Fr(closed)g(primitiv)m(e)d(recursiv)m(e)j(term)e (of)42 b(t)m(yp)s(e)33 b Fn(\032)p Fl(g)27 b(\022)h(j)p Fn(D)3225 5684 y Fm(\032)3271 5669 y Fl(j)p Fn(:)1865 5908 y Fr(8)p eop %%Page: 9 9 9 8 bop 0 -137 a Fr(The)48 b(elemen)m(ts)f(of)g Fl(j)p Fn(D)877 -122 y Fm(\032)923 -137 y Fl(j)951 -173 y Fm(pr)1083 -137 y Fr(are)g(called)g Fh(p)-5 b(artial)48 b(primitive)f(r)-5 b(e)g(cursive)48 b(functionals)p Fr(.)84 b(Note)47 b(that)0 -17 y(an)m(y)35 b(elemen)m(t)g(of)g Fl(j)p Fn(D)785 -2 y Fm(\032)p Ff(!)p Fm(\033)960 -17 y Fl(j)988 -53 y Fm(pr)1108 -17 y Fr(can)g(b)s(e)h(view)m(ed)f(|)g(via)f(Lemma)g(2.5)g(|)h(as)g(a)g (con)m(tin)m(uous)i(function)0 103 y Fl(j)p Fn(D)110 118 y Fm(\032)156 103 y Fl(j)28 b(!)f(j)p Fn(D)449 118 y Fm(\033)503 103 y Fl(j)p Fr(.)165 225 y(It)j(seems)h(w)m(orth)m(wile) f(to)f(also)h(note)g(that)g(an)m(y)g(partial)f(primitiv)m(e)g(recursiv) m(e)i(functional)g(when)0 344 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b(\014rst)g(sho)m(w)h(that,)e(for)0 5669 y(su\016cien)m(tly)f(big)f Fn(k)s Fr(,)g(it)f(is)h(easy)h(to)e (estimate)g(the)i(heigh)m(t)g(of)f(a)g(tree)h(of)f(degree)h Fn(k)j Fr(for)c Fn(r)r(~)-49 b(y)s Fr(.)1840 5908 y(11)p eop %%Page: 12 12 12 11 bop 0 -137 a Fj(Lemma)38 b(4.1.)58 b Fi(F)-8 b(or)33 b(an)m(y)g(v)-6 b(ariable)33 b Fn(x)g Fi(w)m(e)g(ha)m(v)m(e,)h(with)f (an)g(arbitrary)f Fn(k)s Fi(,)1572 20 y Fl(j)p Fn(x)o(~)-49 b(y)t Fl(j)1738 35 y Fm(k)1814 20 y Fl(\024)28 b Fr(lev)q(\()p Fn(x)p Fr(\))p Fn(:)1367 b Fr(\(4)p Fn(:)p Fr(2\))0 176 y Fi(No)m(w)33 b(let)g Fn(r)j Fi(b)s(e)d(an)m(y)h(primitiv)m(e)d (recursiv)m(e)j(term.)43 b(If)34 b(all)e(subterms)i(of)f Fn(r)j Fi(ha)m(v)m(e)e(lev)m(els)f Fl(\024)28 b Fn(k)s Fi(,)33 b(then)1614 333 y Fl(j)p Fn(r)r(~)-49 b(y)r Fl(j)1769 348 y Fm(k)1846 333 y Fn(<)28 b(!)e Fl(\001)21 b Fr(2)p Fn(:)1409 b Fr(\(4)p Fn(:)p Fr(3\))0 489 y(Pro)s(of.)44 b(\(4.2\))32 b(can)h(b)s(e)h(seen)g(easily)f(b)m(y)g(induction)h(on)f (the)h(lev)m(el)f(of)g Fn(x)p 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Fr(,)i(b)m(y)h(induction)f(on)g(the)h(length)f(of) g Fn(u)p Fr(.)56 b(This)37 b(is)g(ob)m(vious)0 1474 y(\(b)m(y)f(Rule)g (3\))g(if)g Fn(u)g Fr(is)f(of)i(lev)m(el)e(0.)52 b(Otherwise,)37 b Fn(u)32 b Fr(=)h Fl(f)p Fr(\()p Fn(u)2153 1489 y Fm(i)2186 1474 y Fn(;)17 b(v)2279 1489 y Fm(i)2312 1474 y Fr(\):)g Fn(i)31 b Fl(2)i Fn(I)8 b Fl(g)35 b Fr(and)h(w)m(e)h(ha)m(v)m(e)f Fl(j)p Fr([)p Fn(w)3402 1489 y Fm(i)3435 1474 y Fr(])8 b Fn(~)-58 b(y)3512 1489 y Fm(i)3545 1474 y Fl(j)31 b Fn(<)i(!)0 1594 y Fr(for)g Fn(i)28 b Fr(=)g(1)p Fn(;)17 b(:)g(:)g(:)d(;)j(m)33 b Fr(b)m(y)g(induction)g(h)m(yp)s(othesis)h(and) f(hence)i Fl(j)p Fn(y)t Fr([)p Fn(w)2432 1609 y Fk(1)2476 1594 y Fr(])17 b Fn(:)g(:)g(:)d Fr([)p Fn(w)2752 1609 y Fm(m)2827 1594 y Fr(])p Fl(j)2883 1609 y Fm(k)2959 1594 y Fn(<)29 b(!)36 b Fr(b)m(y)d(Rule)h(2.)43 b(By)0 1713 y(Rule)34 b(3)f(w)m(e)h(can)f(conclude)i Fl(j)p Fr([)p Fl(f)p Fr(\()p Fn(u)1258 1728 y Fm(i)1290 1713 y Fn(;)17 b(v)1383 1728 y Fm(i)1416 1713 y Fr(\):)g Fn(i)27 b Fl(2)h Fn(I)8 b Fl(g)p Fr(])p Fn(y)s(~)-49 b(y)r Fl(j)1917 1728 y Fm(k)1993 1713 y Fn(<)28 b(!)t Fr(.)165 1833 y Fh(Case)43 b Fn(N)11 b Fh(.)44 b Fr(W)-8 b(e)34 b(ha)m(v)m(e)f Fl(j)p Fn(N)11 b(y)t Fl(j)1184 1848 y Fm(k)1260 1833 y Fl(\024)28 b Fr(1,)k(since)j Fl(j)p Fn(y)t Fl(j)1829 1848 y Fm(k)1904 1833 y Fl(\024)28 b Fr(0.)165 1952 y Fh(Case)44 b Fn(R)q Fh(.)j Fr(W)-8 b(e)35 b(\014rst)g(sho)m(w)g Fl(j)p Fn(R)q(my)t(z)t(~)-50 b(y)s Fl(j)1576 1967 y Fm(k)1655 1952 y Fn(<)30 b(!)t Fr(,)k(b)m(y)h(induction)g(on)f Fn(m)p Fr(.)48 b(Since)36 b Fl(j)p Fn(y)s(~)-49 b(y)r Fl(j)3198 1967 y Fm(k)3277 1952 y Fl(\024)30 b Fr(lev)q(\()p Fn(y)t Fr(\))j(b)m(y)0 2072 y(\(4.2\))39 b(w)m(e)i(ha)m(v)m(e)g Fl(j)p Fn(R)q Fr(0)p Fn(y)r(~)-48 b(z)r(~)f(y)s Fl(j)973 2087 y Fm(k)1061 2072 y Fl(\024)40 b Fr(lev)q(\()p Fn(y)t Fr(\))26 b(+)h(1)39 b Fn(<)h(!)k Fr(b)m(y)d(Rule)g(6.)65 b(F)-8 b(or)40 b(the)g(induction)h(step,)h(assume)0 2191 y Fl(j)p Fn(R)q(my)t(z)t(~)-50 b(y)s Fl(j)375 2206 y Fm(k)473 2191 y Fn(<)50 b(!)t Fr(.)82 b(Then)47 b(\(b)m(y)f(Rule)g(2)g (and)h(\(4.2\)\))d(w)m(e)j(also)f(ha)m(v)m(e)g Fl(j)p Fn(z)t(m)p Fr(\()p Fn(R)q(my)t(z)t Fr(\))o Fn(~)-49 b(y)t Fl(j)3283 2206 y Fm(k)3381 2191 y Fn(<)49 b(!)h Fr(and)0 2311 y(hence)37 b Fl(j)p Fn(R)q Fr(\()p Fn(m)23 b Fr(+)i(1\))p Fn(y)t(z)t(~)-50 b(y)s Fl(j)907 2326 y Fm(k)987 2311 y Fn(<)33 b(!)39 b Fr(b)m(y)d(Rule)g(7.)52 b(W)-8 b(e)36 b(no)m(w)g(sho)m(w)g Fl(j)p Fn(R)q(xy)t(z)t(~)-50 b(y)s Fl(j)2683 2326 y Fm(k)2763 2311 y Fl(\024)33 b Fn(!)t Fr(.)51 b(This)36 b(follo)m(ws)f(from)0 2430 y Fl(j)p Fn(R)q(my)t(z)t(~)-50 b(y)s Fl(j)375 2445 y Fm(k)451 2430 y Fn(<)29 b(!)36 b Fr(for)e(all)e Fn(m)i Fr(and)g Fl(j)p Fn(x)p Fl(j)1374 2445 y Fm(k)1450 2430 y Fl(\024)28 b Fr(0)33 b(b)m(y)g(Rule)h(5.)165 2550 y Fh(Case)h Fn(\025x:r)s Fh(.)40 b Fr(By)24 b(induction)h(h)m(yp)s(othesis)g(w)m(e)g(ha)m(v)m(e) g Fl(j)p Fn(r)s Fr([)p Fn(y)t Fr(])o Fn(~)-49 b(y)q Fl(j)2357 2565 y Fm(k)2434 2550 y Fn(<)28 b(!)8 b Fl(\001)t Fr(2,)25 b(and)g(hence)h Fl(j)p Fr(\()p Fn(\025x:r)s Fr(\))p Fn(y)s(~)-49 b(y)r Fl(j)3626 2565 y Fm(k)3702 2550 y Fn(<)0 2670 y(!)26 b Fl(\001)c Fr(2)33 b(b)m(y)g(Rule)h(1.)165 2789 y Fh(Case)46 b Fn(r)s(s)p Fh(.)52 b Fr(By)35 b(induction)i(h)m(yp)s(othesis)f(w)m(e) h(ha)m(v)m(e)f Fl(j)p Fn(r)s(y)t(y)2256 2804 y Fk(1)2316 2789 y Fn(:)17 b(:)g(:)e(y)2498 2804 y Fm(m)2574 2789 y Fl(j)2602 2804 y Fm(k)2683 2789 y Fn(<)33 b(!)27 b Fl(\001)d Fr(2)36 b(and)h Fl(j)p Fn(s)n(~)-48 b(z)t Fl(j)3370 2804 y Fm(k)3451 2789 y Fn(<)33 b(!)27 b Fl(\001)d Fr(2,)0 2909 y(and)29 b(b)m(y)h(\(4.2\))d Fl(j)p Fn(y)634 2924 y Fm(i)674 2909 y Fn(~)-57 b(z)713 2924 y Fm(i)747 2909 y Fl(j)775 2924 y Fm(k)851 2909 y Fl(\024)28 b Fr(lev)q(\()p Fn(y)1170 2924 y Fm(i)1203 2909 y Fr(\))f Fn(<)h(!)33 b Fr(for)c Fn(i)f Fr(=)g(1)p Fn(;)17 b(:)g(:)g(:)d(;)j(m)p Fr(.)43 b(By)28 b(the)h Fn(k)s Fr({Abbreviation{Rule)g(w)m(e)g(can)0 3028 y(conclude)35 b Fl(j)p Fn(r)s(sy)582 3043 y Fk(1)642 3028 y Fn(:)17 b(:)g(:)e(y)824 3043 y Fm(m)899 3028 y Fl(j)927 3043 y Fm(k)1004 3028 y Fn(<)28 b(!)d Fl(\001)d Fr(2.)p 1368 3028 4 77 v 1372 2956 92 4 v 1372 3028 V 1463 3028 4 77 v 165 3148 a(W)-8 b(e)34 b(no)m(w)f(w)m(an)m(t)g(to)g (sho)m(w)g(ho)m(w)h(from)e(an)h(estimate)f(for)h(the)g(heigh)m(t)h(of)f (a)g(tree)g(of)g(degree)h Fn(k)i Fr(w)m(e)0 3267 y(can)e(obtain)f(an)g (estimate)f(for)i(the)f(heigh)m(t)h(of)f(the)h(tree)f(of)h(degree)g(0.) 0 3428 y Fj(Lemma)g(4.2.)58 b Fi(Let)30 b Fn(r)m(;)17 b(s)947 3443 y Fk(1)991 3428 y Fn(;)g(:)g(:)g(:)d(;)j(s)1260 3443 y Fm(m)1364 3428 y Fi(b)s(e)30 b(primitiv)m(e)e(recursiv)m(e)j (terms)e(with)h Fr(lev)q(\()p Fn(r)s Fr(\))d(=)h(0)p Fi(.)42 b(If)31 b Fl(j)p Fn(r)s Fl(j)3535 3443 y Fm(k)3610 3428 y Fl(\024)d Fn(\013)0 3548 y Fi(and)34 b Fl(j)p Fn(s)269 3563 y Fm(j)323 3548 y Fn(~)-63 b(y)359 3563 y Fm(j)401 3548 y Fl(j)429 3563 y Fm(k)505 3548 y Fl(\024)28 b Fn(\014)39 b Fi(for)34 b Fn(j)f Fr(=)28 b(1)p Fn(;)17 b(:)g(:)g(:)d(;)j(m)p Fi(,)33 b(and)h(if)f Fn(s)1789 3563 y Fk(1)1833 3548 y Fn(;)17 b(:)g(:)g(:)d(;)j(s)2102 3563 y Fm(m)2210 3548 y Fi(ha)m(v)m(e)34 b(lev)m(els)f Fl(\024)28 b Fn(k)s Fi(,)33 b(then)1207 3704 y Fl(j)p Fn(r)1280 3719 y Fm(x)1325 3729 y Fb(1)1363 3719 y Fm(;:::)o(;x)1527 3729 y Fc(m)1598 3704 y Fr([)p Fn(s)1673 3719 y Fk(1)1717 3704 y Fn(;)17 b(:)g(:)g(:)d(;)j(s)1986 3719 y Fm(m)2061 3704 y Fr(])p Fl(j)2117 3719 y Fm(k)2193 3704 y Fl(\024)28 b Fn(\014)g Fr(+)22 b Fn(\013:)0 3861 y Fr(Pro)s(of.)48 b(W)-8 b(e)35 b(use)h(induction)f(on)g(the)g(generation)g(of)g Fl(j)p Fn(r)s Fl(j)2128 3876 y Fm(k)2206 3861 y Fl(\024)30 b Fn(\013)p Fr(,)35 b(and)g(|)g(for)g(readabilit)m(y)f(|)g(write)0 3980 y Fn(t)36 3944 y Ff(\003)115 3980 y Fr(for)f Fl(j)p Fn(t)331 3995 y Fm(x)376 4005 y Fb(1)415 3995 y Fm(;:::)o(;x)579 4005 y Fc(m)650 3980 y Fr([)p Fn(s)725 3995 y Fk(1)769 3980 y Fn(;)17 b(:)g(:)g(:)d(;)j(s)1038 3995 y Fm(m)1113 3980 y Fr(])p Fl(j)p Fr(.)165 4100 y Fh(R)m(ule)31 b(1)14 b Fr(.)41 b Fl(j)p Fn(r)600 4064 y Ff(\003)645 4100 y Fr([)p Fn(s)720 4064 y Ff(\003)765 4100 y Fr(])787 4082 y Fn(~)793 4100 y(t)829 4064 y Ff(\003)875 4100 y Fl(j)903 4115 y Fm(k)979 4100 y Fl(\024)28 b Fn(\014)14 b Fr(+)9 b Fn(\013)1304 4115 y Fk(0)1378 4100 y Fn(<)28 b(\014)14 b Fr(+)9 b Fn(\013)27 b Fr(b)m(y)g(induction)g(h)m(yp)s(othesis,)i (hence)f Fl(j)p Fr(\()p Fn(\025x:r)3340 4064 y Ff(\003)3385 4100 y Fr(\))p Fn(s)3471 4064 y Ff(\003)3510 4082 y Fn(~)3516 4100 y(t)3552 4064 y Ff(\003)3598 4100 y Fl(j)3626 4115 y Fm(k)3702 4100 y Fl(\024)0 4220 y Fn(\014)g Fr(+)22 b Fn(\013)34 b Fr(b)m(y)f(Rule)h(1.)165 4339 y Fh(R)m(ule)40 b(2)14 b Fr(.)54 b Fl(j)p Fn(t)610 4303 y Ff(\003)610 4365 y Fm(i)655 4339 y Fn(~)-49 b(y)709 4303 y Ff(\003)705 4365 y Fm(i)754 4339 y Fl(j)782 4354 y Fm(k)864 4339 y Fl(\024)34 b Fn(\014)c Fr(+)25 b Fn(\013)1227 4354 y Fm(i)1294 4339 y Fn(<)34 b(\014)c Fr(+)25 b Fn(\013)37 b Fr(for)g Fn(i)d Fr(=)g(1)p Fn(;)17 b(:)g(:)g(:)d(;)j(m)37 b Fr(b)m(y)g(induction)g(h)m(yp)s(othesis,)h(hence)0 4459 y Fl(j)p Fn(xt)121 4422 y Ff(\003)121 4483 y Fk(1)183 4459 y Fn(:)17 b(:)g(:)e(t)352 4422 y Ff(\003)352 4483 y Fm(m)427 4459 y Fl(j)455 4474 y Fm(k)532 4459 y Fl(\024)28 b Fn(\014)14 b Fr(+)9 b Fn(\013)p Fr(.)43 b(No)m(w)26 b(if)h Fn(x)g Fr(is)f(one)h(of)g(the)g(v)-6 b(ariables)27 b Fn(x)2322 4474 y Fm(j)2391 4459 y Fr(to)f(b)s(e)h(substituted)h(b)m (y)e Fn(s)3328 4474 y Fm(j)3370 4459 y Fr(,)i(w)m(e)f(m)m(ust)0 4578 y(use)34 b(the)g Fn(k)s Fr({Abbreviation{Rule)e(instead)i(of)g (Rule)f(2.)44 b(This)34 b(is)f(p)s(ossible)h(since)g(b)m(y)f(h)m(yp)s (othesis)h Fn(s)3738 4593 y Fm(j)0 4698 y Fr(has)48 b(lev)m(el)f Fl(\024)k Fn(k)s Fr(,)g(and)c(since)h(also)f(b)m(y)h(h)m(yp)s(othesis)g Fl(j)p Fn(s)2107 4713 y Fm(j)2159 4698 y Fn(~)-61 b(z)2194 4713 y Fm(j)2236 4698 y Fl(j)2264 4713 y Fm(k)2364 4698 y Fl(\024)51 b Fn(\014)5 b Fr(.)87 b(Then)48 b(\(if)f Fn(m)52 b(>)f Fr(0\))c(the)h Fn(k)s Fr({)0 4817 y(Abbreviation{Rule)38 b(yields)f Fl(j)p Fn(s)1218 4832 y Fm(j)1260 4817 y Fn(t)1296 4781 y Ff(\003)1296 4842 y Fk(1)1358 4817 y Fn(:)17 b(:)g(:)e(t)1527 4781 y Ff(\003)1527 4842 y Fm(m)1603 4817 y Fl(j)1631 4832 y Fm(k)1715 4817 y Fl(\024)36 b Fn(\014)31 b Fr(+)25 b Fn(\013)p Fr(,)39 b(as)f(required.)59 b(In)38 b(case)g Fn(m)e Fr(=)g(0)i(there)g(are)0 4937 y(no)32 b Fn(t)173 4952 y Fm(i)206 4937 y Fr('s)f(and)h(w)m(e)g(ha)m(v)m(e)g(used)h(Rule)f (2)f(to)g(generate)h Fl(j)p Fn(x)2010 4952 y Fm(j)2052 4937 y Fl(j)2080 4952 y Fm(k)2156 4937 y Fl(\024)c Fn(\013)p Fr(.)44 b(But)31 b(then)h Fl(j)p Fn(s)2893 4952 y Fm(j)2934 4937 y Fl(j)2962 4952 y Fm(k)3038 4937 y Fl(\024)d Fn(\014)24 b Fr(+)19 b Fn(\013)31 b Fr(holds)h(b)m(y)0 5056 y(h)m(yp)s(othesis.) 165 5176 y Fh(R)m(ule)c(3)14 b Fr(.)40 b Fl(j)p Fn(r)596 5140 y Ff(\003)641 5176 y Fr([)p Fn(w)740 5191 y Fk(1)785 5176 y Fr(])17 b Fn(:)g(:)g(:)d Fr([)p Fn(w)1061 5191 y Fm(m)1136 5176 y Fr(])p Fl(j)1192 5191 y Fm(k)1268 5176 y Fl(\024)29 b Fn(\014)7 b Fr(+)r Fn(\013)1580 5191 y Fm(w)1638 5201 y Fb(1)1677 5191 y Fm(:::)o(w)1806 5201 y Fc(m)1905 5176 y Fn(<)28 b(\014)7 b Fr(+)r Fn(\013)24 b Fr(for)g(all)e Fn(w)2582 5191 y Fk(1)2627 5176 y Fn(;)17 b(:)g(:)g(:)d(;)j(w)2920 5191 y Fm(m)3019 5176 y Fr(and)23 b Fl(j)p Fr([)3258 5101 y Fe(S)3341 5206 y Fm(i)p Ff(2)p Fm(I)3463 5186 y Fd(0)3512 5176 y Fn(v)3560 5191 y Fm(i)3593 5176 y Fr(])3615 5158 y Fn(~)3621 5176 y(t)3657 5140 y Ff(\003)3703 5176 y Fl(j)3731 5191 y Fm(k)0 5295 y Fl(\024)28 b Fn(\014)19 b Fr(+)13 b Fn(\013)334 5310 y Fm(I)374 5291 y Fd(0)434 5295 y Fn(<)29 b(\014)18 b Fr(+)13 b Fn(\013)30 b Fr(for)e(all)g Fn(I)1131 5259 y Ff(0)1187 5295 y Fl(\022)g Fn(I)36 b Fr(suc)m(h)30 b(that)1803 5221 y Fe(S)1886 5325 y Fm(i)p Ff(2)p Fm(I)2008 5305 y Fd(0)2058 5295 y Fn(v)2106 5310 y Fm(i)2168 5295 y Fr(is)e(consisten)m(t)i(b)m(y)f(induction)g(h)m(yp)s(othesis,)0 5430 y(hence)35 b Fl(j)p Fr([)p Fl(f)p Fr(\()p Fn(u)479 5445 y Fm(i)511 5430 y Fn(;)17 b(v)604 5445 y Fm(i)637 5430 y Fr(\):)g Fn(i)27 b Fl(2)h Fn(I)8 b Fl(g)p Fr(])p Fn(r)1047 5412 y(~)1054 5430 y(t)15 b Fl(j)1133 5445 y Fm(k)1209 5430 y Fl(\024)29 b Fn(\014)e Fr(+)c Fn(\013)33 b Fr(b)m(y)h(Rule)g(3.)165 5550 y Fh(R)m(ules)42 b(4{8)52 b Fr(and)39 b(the)g Fn(k)s Fh({A)m(bbr)-5 b(eviation{R)m(ule)46 b Fr(can)39 b(b)s(e)h(treated)f(similarly)d(\(i.)i(e.)61 b(as)39 b(Rules)h(1)0 5669 y(and)34 b(3\);)e(the)i(claim)d(alw)m(a)m (ys)i(follo)m(ws)g(b)m(y)h(induction)f(h)m(yp)s(othesis)h(and)g(the)g (same)e(rule.)p 3352 5669 V 3356 5597 92 4 v 3356 5669 V 3447 5669 4 77 v 1840 5908 a(12)p eop %%Page: 13 13 13 12 bop 0 -137 a Fj(Lemma)26 b(4.3.)58 b Fi(Let)23 b Fn(r)i Fi(b)s(e)e(a)g(primitiv)m(e)d(recursiv)m(e)k(term)e(of)h(lev)m (el)f(0.)41 b(If)23 b Fl(j)p Fn(r)s Fl(j)2746 -122 y Fm(k)r Fk(+1)2922 -137 y Fl(\024)28 b Fn(\013)p Fi(,)d(then)e Fl(j)p Fn(r)s Fl(j)3464 -122 y Fm(k)3540 -137 y Fl(\024)28 b Fr(2)3695 -173 y Fm(\013)3752 -137 y Fi(.)0 44 y Fr(Pro)s(of.)103 b(W)-8 b(e)53 b(again)g(use)h(induction)f(on)g(the)h(generation)f(of)g Fl(j)p Fn(r)s Fl(j)2596 59 y Fm(k)r Fk(+1)2805 44 y Fl(\024)62 b Fn(\013)p Fr(.)103 b(The)53 b(only)g(case)0 163 y(where)43 b(the)f(claim)e(do)s(es)i(not)f(follo)m(w)h(immediately)c(from)j(the)h (induction)h(h)m(yp)s(othesis)f(is)g(where)0 283 y Fl(j)p Fn(r)s(t)112 298 y Fk(1)173 283 y Fn(:)17 b(:)g(:)d(t)341 298 y Fm(m)417 283 y Fl(j)445 298 y Fm(k)r Fk(+1)634 283 y Fl(\024)40 b Fn(\013)h Fr(w)m(as)f(generated)h(from)f Fl(j)p Fn(r)r(~)-49 b(y)s Fl(j)1914 298 y Fm(k)r Fk(+1)2103 283 y Fl(\024)40 b Fn(\013)2284 298 y Fk(0)2368 283 y Fn(<)g(\013)g Fr(and)h Fl(j)p Fn(t)2854 298 y Fm(i)2895 283 y Fn(~)-58 b(y)2936 298 y Fm(i)2969 283 y Fl(j)2997 298 y Fm(k)r Fk(+1)3186 283 y Fl(\024)40 b Fn(\013)3367 298 y Fm(i)3439 283 y Fn(<)g(\013)g Fr(for)0 402 y Fn(i)28 b Fr(=)g(1)p Fn(;)17 b(:)g(:)g(:)d(;)j(m)32 b Fr(b)m(y)h(the)g Fn(k)s Fr({Abbreviation{Rule,)e(and)i(lev)q(\()p Fn(r)s Fr(\))27 b Fl(\024)h Fn(k)c Fr(+)d(1.)43 b(By)32 b(induction)h(h)m(yp)s (othesis)0 522 y(w)m(e)j(then)h(ha)m(v)m(e)f Fl(j)p Fn(r)r(~)-49 b(y)s Fl(j)767 537 y Fm(k)848 522 y Fl(\024)33 b Fr(2)1008 486 y Fm(\013)1060 496 y Fb(0)1139 522 y Fr(and)j Fl(j)p Fn(t)1399 537 y Fm(i)1441 522 y Fn(~)-59 b(y)1481 537 y Fm(i)1514 522 y Fl(j)1542 537 y Fm(k)1623 522 y Fl(\024)33 b Fr(2)1783 486 y Fm(\013)1835 496 y Fc(i)1907 522 y Fr(for)j Fn(i)c Fr(=)h(1)p Fn(;)17 b(:)g(:)g(:)d(;)j(m)p Fr(.)52 b(Using)35 b(Lemma)g(4.2)g(w)m(e)h(can)0 642 y(conclude)f Fl(j)p Fn(r)s(t)522 657 y Fk(1)582 642 y Fn(:)17 b(:)g(:)e(t)751 657 y Fm(m)827 642 y Fl(j)855 657 y Fm(k)931 642 y Fl(\024)28 b Fr(2)1086 605 y Fk(max)o(\()p Fm(\013)1316 615 y Fb(1)1355 605 y Fm(;:::)o(;\013)1526 615 y Fc(m)1591 605 y Fk(\))1649 642 y Fr(+)23 b(2)1799 605 y Fm(\013)1851 615 y Fb(0)1922 642 y Fl(\024)28 b Fr(2)2077 605 y Fm(\013)2134 642 y Fr(.)p 2206 642 4 77 v 2210 570 92 4 v 2210 642 V 2301 642 4 77 v 165 761 a(W)-8 b(e)39 b(no)m(w)g(sho)m(w)g(that,)g(from)e(an)i(estimate)e Fn(\013)i Fr(for)f(the)h(heigh)m(t)g(of)f(the)h(tree)g(of)g(degree)g(0) f(of)h(a)0 881 y(primitiv)m(e)22 b(recursiv)m(e)i(term)f Fn(r)r(~)-49 b(y)s Fr(,)25 b(w)m(e)f(can)f(conclude)i(|)e(b)m(y)h(an)f (application)g(of)h(trans\014nite)g(induction)0 1001 y(of)33 b(the)h(same)f(length)g Fn(\013)g Fr(|)g(that)g(the)h(leftmost) e(reduction)i(sequence)h(for)e Fn(r)j Fr(terminates.)0 1181 y Fj(Lemma)k(4.4.)57 b Fi(Let)35 b Fn(r)i Fi(b)s(e)e(a)f(primitiv) m(e)f(recursiv)m(e)j(term)d(of)i(arbitrary)e(lev)m(el.)48 b(If)35 b Fl(j)p Fn(r)r(~)-49 b(y)s Fl(j)3278 1196 y Fk(0)3352 1181 y Fl(\024)30 b Fn(\013)p Fi(,)35 b(then)0 1301 y(the)f(leftmost)e(\(or)h(standard\))g(reduction)h(sequence)h(for) f Fn(r)i Fi(terminates.)0 1481 y Fr(Pro)s(of.)45 b(W)-8 b(e)34 b(use)g(induction)h(on)e(the)h(generation)g(of)g Fl(j)p Fn(r)r(~)-49 b(y)s Fl(j)2169 1496 y Fk(0)2241 1481 y Fl(\024)29 b Fn(\013)p Fr(,)34 b(and)g(write)f Fn(t)-5 b Fl(#)34 b Fr(to)f(mean)g(that)g(the)0 1601 y(leftmost)f(reduction)i(sequence)h(for)f Fn(t)f Fr(terminates.)165 1720 y(If)h Fn(r)i Fr(is)d(a)g(v)-6 b(ariable)33 b(or)g(a)g(constan)m (t,)h(then)f(the)h(claim)e(is)h(trivial.)165 1840 y Fh(Case)38 b Fn(\025x:r)s Fr(.)51 b(Then)37 b Fl(j)p Fr([)p Fn(\025x:r)s Fr(])p Fn(y)s(~)-49 b(y)q Fl(j)1351 1855 y Fk(0)1428 1840 y Fl(\024)32 b Fn(\013)k Fr(w)m(as)g(generated)h(from)e Fl(j)p Fn(r)s Fr([)p Fn(y)t Fr(])o Fn(~)-49 b(y)q Fl(j)2789 1855 y Fk(0)2866 1840 y Fl(\024)32 b Fn(\013)3039 1855 y Fk(0)3115 1840 y Fn(<)h(\013)j Fr(b)m(y)g(Rule)g(1.)0 1960 y(By)c(induction)i(h)m(yp)s(othesis)g(w)m(e)g(kno)m(w)f Fn(r)s Fr([)p Fn(y)t Fr(])-6 b Fl(#)p Fr(,)32 b(and)h(this)h(ob)m (viously)e(implies)g(\()p Fn(\025x:r)s Fr(\))-6 b Fl(#)p Fr(.)165 2080 y Fh(Case)41 b Fn(xt)508 2095 y Fk(1)569 2080 y Fn(:)17 b(:)g(:)e(t)738 2095 y Fm(m)813 2080 y Fr(.)60 b(Then)39 b Fl(j)p Fn(xt)1287 2095 y Fk(1)1348 2080 y Fn(:)17 b(:)g(:)e(t)1517 2095 y Fm(m)1592 2080 y Fn(~)-49 b(y)s Fl(j)1673 2095 y Fk(0)1754 2080 y Fl(\024)37 b Fn(\013)i Fr(w)m(as)f(generated)i(from)e Fl(j)p Fn(t)2932 2095 y Fm(i)2973 2080 y Fn(~)-59 b(y)3013 2095 y Fm(i)3046 2080 y Fl(j)3074 2095 y Fk(0)3155 2080 y Fl(\024)37 b Fn(\013)3333 2095 y Fm(i)3402 2080 y Fn(<)g(\013)i Fr(and)0 2199 y Fl(j)p Fn(y)77 2214 y Fm(j)130 2199 y Fn(~)-62 b(z)164 2214 y Fm(j)207 2199 y Fl(j)235 2214 y Fk(0)313 2199 y Fl(\024)34 b Fn(\014)480 2214 y Fm(j)555 2199 y Fn(<)g(\013)j Fr(b)m(y)g(Rule)g(2.)55 b(By)35 b(induction)j(h)m(yp)s (othesis)f(w)m(e)g(kno)m(w)g Fn(t)2829 2214 y Fm(i)2862 2199 y Fl(#)p Fr(,)g(and)g(this)g(ob)m(viously)0 2319 y(implies)32 b(\()p Fn(xt)470 2334 y Fk(1)531 2319 y Fn(:)17 b(:)g(:)e(t)700 2334 y Fm(m)776 2319 y Fr(\))-6 b Fl(#)p Fr(.)165 2439 y Fh(Case)37 b Fr([)p Fl(f)p Fr(\()p Fn(u)585 2454 y Fm(i)617 2439 y Fn(;)17 b(v)710 2454 y Fm(i)743 2439 y Fr(\):)g Fn(i)28 b Fl(2)i Fn(I)8 b Fl(g)p Fr(])p Fn(r)1156 2420 y(~)1163 2439 y(t)o Fr(.)47 b(Then)35 b Fl(j)p Fr([)p Fl(f)p Fr(\()p Fn(u)1736 2454 y Fm(i)1768 2439 y Fn(;)17 b(v)1861 2454 y Fm(i)1894 2439 y Fr(\):)g Fn(i)28 b Fl(2)i Fn(I)8 b Fl(g)p Fr(])p Fn(r)2307 2420 y(~)2314 2439 y(t)n(~)-49 b(y)s Fl(j)2429 2454 y Fk(0)2503 2439 y Fl(\024)30 b Fn(\013)k Fr(w)m(as)g(generated)i (b)m(y)e(Rule)h(3)0 2558 y(from)d Fl(j)p Fn(r)s Fr([)p Fn(w)410 2573 y Fk(1)454 2558 y Fr(])17 b Fn(:)g(:)g(:)d Fr([)p Fn(w)730 2573 y Fm(m)806 2558 y Fr(])p Fl(j)862 2573 y Fk(0)934 2558 y Fl(\024)28 b Fn(\013)1103 2573 y Fm(w)1161 2583 y Fb(1)1199 2573 y Fm(:::)o(w)1328 2583 y Fc(m)1427 2558 y Fn(<)g(\013)33 b Fr(for)g(all)f(full)i(argumen)m(t)e (sequences)j Fn(w)3062 2573 y Fk(1)3107 2558 y Fn(;)17 b(:)g(:)g(:)d(;)j(w)3400 2573 y Fm(m)3509 2558 y Fr(in)33 b(an)m(y)0 2678 y(of)39 b(the)h Fn(u)354 2693 y Fm(i)426 2678 y Fr(\()p Fn(i)e Fl(2)g Fn(I)8 b Fr(\),)40 b(and)f(from)g Fl(j)p Fr([)1297 2603 y Fe(S)1379 2708 y Fm(i)p Ff(2)p Fm(I)1501 2688 y Fd(0)1550 2678 y Fn(v)1598 2693 y Fm(i)1632 2678 y Fr(])1654 2660 y Fn(~)1660 2678 y(t)o(~)-49 b(y)s Fl(j)1776 2693 y Fk(0)1858 2678 y Fl(\024)38 b Fn(\013)2037 2693 y Fm(I)2077 2673 y Fd(0)2147 2678 y Fn(<)g(\013)i Fr(for)f(all)f Fn(I)2720 2642 y Ff(0)2785 2678 y Fl(\022)h Fn(I)46 b Fr(suc)m(h)41 b(that)3444 2603 y Fe(S)3527 2708 y Fm(i)p Ff(2)p Fm(I)3649 2688 y Fd(0)3698 2678 y Fn(v)3746 2693 y Fm(i)0 2798 y Fr(is)j(consisten)m(t.)79 b(By)43 b(induction)i(h)m(yp)s(othesis)g(w)m(e)g(kno)m(w)f(\()p Fn(r)s Fr([)p Fn(w)2385 2813 y Fk(1)2429 2798 y Fr(])17 b Fn(:)g(:)g(:)d Fr([)p Fn(w)2705 2813 y Fm(m)2781 2798 y Fr(]\))f Fl(#)43 b Fr(for)i(all)e Fn(w)3338 2813 y Fk(1)3383 2798 y Fn(;)17 b(:)g(:)g(:)e(;)i(w)3677 2813 y Fm(m)3752 2798 y Fr(,)0 2917 y(and)43 b(also)f(\([)479 2842 y Fe(S)562 2947 y Fm(i)p Ff(2)p Fm(I)684 2927 y Fd(0)733 2917 y Fn(v)781 2932 y Fm(i)814 2917 y Fr(])836 2899 y Fn(~)842 2917 y(t)17 b Fr(\))10 b Fl(#)42 b Fr(for)h(all)e Fn(I)1397 2881 y Ff(0)1468 2917 y Fl(\022)j Fn(I)50 b Fr(suc)m(h)44 b(that)2142 2842 y Fe(S)2225 2947 y Fm(i)p Ff(2)p Fm(I)2347 2927 y Fd(0)2396 2917 y Fn(v)2444 2932 y Fm(i)2520 2917 y Fr(is)f(consisten)m(t.)73 b(No)m(w)42 b(from)g(the)0 3037 y(de\014nition)i(of)g(the)g(leftmost)f(reduction)i (pro)s(cedure)g(it)e(can)h(b)s(e)g(seen)h(easily)e(that)h(this)g (implies)0 3156 y(\([)p Fl(f)p Fr(\()p Fn(u)213 3171 y Fm(i)245 3156 y Fn(;)17 b(v)338 3171 y Fm(i)371 3156 y Fr(\):)g Fn(i)27 b Fl(2)h Fn(I)8 b Fl(g)p Fr(])p Fn(r)781 3138 y(~)788 3156 y(t)15 b Fr(\)\))-6 b Fl(#)p Fr(.)165 3276 y Fh(Case)47 b Fn(N)11 b(r)s Fr(.)82 b(Then)46 b Fl(j)p Fn(N)11 b(r)s Fl(j)1137 3291 y Fk(0)1229 3276 y Fl(\024)50 b Fn(\013)45 b Fr(w)m(as)h(generated)h(from)e Fl(j)p Fn(r)s Fl(j)2488 3291 y Fk(0)2581 3276 y Fl(\024)k Fn(\013)2771 3291 y Fk(0)2864 3276 y Fn(<)g(\013)d Fr(b)m(y)g(Rule)g (4.)81 b(By)0 3396 y(induction)34 b(h)m(yp)s(othesis)g(w)m(e)f(kno)m(w) h Fn(r)m Fl(#)p Fr(,)e(and)i(this)f(ob)m(viously)g(implies)f(\()p Fn(N)11 b(r)s Fr(\))-6 b Fl(#)p Fr(.)165 3516 y Fh(Case)29 b Fn(R)q(r)s(st)605 3497 y(~)611 3516 y(t)24 b Fr(with)h Fn(r)j Fr(not)d(a)h(constan)m(t.)41 b(Then)27 b Fl(j)p Fn(R)q(r)s(st)2128 3497 y(~)2134 3516 y(t)m(~)-49 b(y)t Fl(j)2249 3531 y Fk(0)2321 3516 y Fl(\024)28 b Fn(\013)d Fr(w)m(as)h(generated)g(b)m(y)g(Rule)g(5)f(from)0 3635 y Fl(j)p Fn(r)s Fl(j)104 3650 y Fk(0)183 3635 y Fl(\024)35 b Fn(\013)359 3599 y Ff(0)422 3635 y Fn(<)h(\013)i Fr(and)g Fl(j)p Fn(R)q(mst)1104 3617 y(~)1110 3635 y(t)n(~)-49 b(y)s Fl(j)1225 3650 y Fk(0)1305 3635 y Fl(\024)35 b Fn(\013)1481 3650 y Fm(m)1592 3635 y Fn(<)g(\013)j Fr(for)g(all)e (nonnegativ)m(e)i(in)m(tegers)h Fn(m)p Fr(.)57 b(By)37 b(induction)0 3755 y(h)m(yp)s(othesis)d(w)m(e)g(kno)m(w)f Fn(r)m Fl(#)g Fr(and)h(\()p Fn(R)q(mst)1497 3736 y(~)1503 3755 y(t)16 b Fr(\))-6 b Fl(#)33 b Fr(for)g(all)g Fn(m)p Fr(,)g(and)h(this)f(ob)m(viously)g(implies)f(\()p Fn(R)q(r)s(st)3518 3736 y(~)3524 3755 y(t)15 b Fr(\))-6 b Fl(#)p Fr(.)165 3875 y Fh(Case)35 b Fn(R)q(r)s(s)c Fr(with)h Fn(r)k Fr(not)c(a)g (constan)m(t.)45 b(Then)33 b Fl(j)p Fn(R)q(r)s(sz)t(~)-50 b(y)s Fl(j)2207 3890 y Fk(0)2279 3875 y Fl(\024)28 b Fn(\013)k Fr(w)m(as)h(generated)g(b)m(y)g(Rule)g(5)f(from)0 3994 y Fl(j)p Fn(r)s Fl(j)104 4009 y Fk(0)182 3994 y Fl(\024)i Fn(\013)357 3958 y Ff(0)419 3994 y Fn(<)h(\013)i Fr(and)g Fl(j)p Fn(R)q(msz)t(~)-50 b(y)t Fl(j)1199 4009 y Fk(0)1277 3994 y Fl(\024)35 b Fn(\013)1453 4009 y Fm(m)1562 3994 y Fn(<)f(\013)j Fr(for)h(all)e(nonnegativ)m(e)h(in)m(tegers)h Fn(m)p Fr(.)56 b(W)-8 b(e)37 b(also)g(kno)m(w)0 4114 y(that)45 b Fl(j)p Fn(R)q Fr(0)p Fn(sz)t(~)-50 b(y)r Fl(j)559 4129 y Fk(0)651 4114 y Fl(\024)47 b Fn(\013)839 4129 y Fk(0)929 4114 y Fr(w)m(as)e(generated)h(b)m(y)f(Rule)g(6)g(from) f Fl(j)p Fn(s)o(~)-49 b(y)s Fl(j)2490 4129 y Fk(0)2582 4114 y Fl(\024)47 b Fn(\013)2770 4078 y Ff(0)2770 4138 y Fk(0)2862 4114 y Fn(<)g(\013)3050 4129 y Fk(0)3095 4114 y Fr(.)79 b(By)44 b(induction)0 4233 y(h)m(yp)s(othesis)34 b(w)m(e)g(kno)m(w)f Fn(r)m Fl(#)g Fr(and)h Fn(s)-6 b Fl(#)p Fr(,)33 b(and)g(this)h(clearly)e(implies)g(\()p Fn(R)q(r)s(s)p Fr(\))-6 b Fl(#)p Fr(.)165 4353 y Fh(Case)37 b Fn(R)q(r)g Fr(with)e Fn(r)i Fr(not)e(a)g(constan)m(t.)49 b(Then)36 b Fl(j)p Fn(R)q(r)s(y)t(z)t(~)-50 b(y)r Fl(j)2187 4368 y Fk(0)2262 4353 y Fl(\024)30 b Fn(\013)35 b Fr(w)m(as)g (generated)h(b)m(y)f(Rule)h(5)e(from)0 4473 y Fl(j)p Fn(r)s Fl(j)104 4488 y Fk(0)184 4473 y Fl(\024)i Fn(\013)361 4437 y Ff(0)425 4473 y Fn(<)h(\013)h Fr(and)h Fl(j)p Fn(R)q(my)t(z)t(~)-50 b(y)s Fl(j)1215 4488 y Fk(0)1296 4473 y Fl(\024)36 b Fn(\013)1473 4488 y Fm(m)1585 4473 y Fn(<)g(\013)j Fr(for)f(all)g(nonnegativ)m(e)g(in)m(tegers)h Fn(m)p Fr(.)60 b(By)37 b(induction)0 4592 y(h)m(yp)s(othesis)d(w)m(e)g (kno)m(w)f Fn(r)m Fl(#)p Fr(,)g(and)g(this)h(clearly)e(implies)g(\()p Fn(R)q(r)s Fr(\))-6 b Fl(#)p Fr(.)165 4712 y Fh(Case)32 b Fn(R)q Fr(0)p Fn(st)610 4694 y(~)616 4712 y(t)o Fr(.)42 b(Then)30 b Fl(j)p Fn(R)q Fr(0)p Fn(st)1209 4694 y(~)1215 4712 y(t)n(~)-49 b(y)s Fl(j)1330 4727 y Fk(0)1402 4712 y Fl(\024)29 b Fn(\013)g Fr(w)m(as)g(generated)h(from)f Fl(j)p Fn(s)2539 4694 y(~)2545 4712 y(t)n(~)-49 b(y)s Fl(j)2660 4727 y Fk(0)2732 4712 y Fl(\024)29 b Fn(\013)2902 4727 y Fk(0)2974 4712 y Fn(<)f(\013)h Fr(b)m(y)g(Rule)h(6.)42 b(By)0 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y(conclude)j(\(\()p Fn(\025x:r)s Fr(\))p Fn(s)759 1049 y(~)765 1067 y(t)15 b Fr(\))-6 b Fl(#)p Fr(.)p 971 1067 4 77 v 975 995 92 4 v 975 1067 V 1066 1067 4 77 v 165 1189 a(T)e(o)33 b(summarize,)f(w)m (e)h(ha)m(v)m(e)h(pro)m(v)m(ed)g(the)g(follo)m(wing)e(result.)0 1377 y Fj(Theorem)25 b(4.5.)58 b Fi(Let)23 b Fn(r)i Fi(b)s(e)e(a)f (primitiv)m(e)f(recursiv)m(e)i(term)f(of)h(arbitrary)e(lev)m(el,)j(p)s (ossibly)e(con)m(taining)0 1496 y(free)i(v)-6 b(ariables)22 b(and)h(constan)m(ts)g(for)g(\014nite)g(functionals.)41 b(Then)24 b(w)m(e)f(can)g(\014nd)g(an)g(ordinal)f Fn(\013)28 b(<)g(")3568 1511 y Fk(0)3636 1496 y Fi(\(b)m(y)0 1616 y(the)k(constructions)h(in)f(Lemmas)e(4.1)h(and)i(4.3\))e(suc)m(h)i (that)e(the)i(termination)d(of)i(the)g(leftmost)f(\(or)0 1735 y(standard\))d(reduction)h(sequence)g(for)f Fn(r)i Fi(is)e(pro)m(v)-6 b(able)28 b(in)g(elemen)m(tary)f(arithmetic)f (\(more)g(precisely:)0 1855 y(primitiv)m(e)31 b(recursiv)m(e)k 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b(to)f(Mathematical)f(Logic.)65 b(North{Holland,)40 b(Amster-)165 3588 y(dam)33 b(1968,)f(pp.)h (109{120)38 3776 y(3.)49 b(Ersho)m(v,)42 b(Y)-8 b(u.)40 b(L.:)58 b(Mo)s(del)41 b Fn(C)48 b Fr(of)40 b(partial)g(con)m(tin)m (uous)i(functionals.)66 b(In)41 b(R.)f(Gandy)h(and)g(M.)165 3895 y(Hyland)34 b(\(eds.\):)45 b(Logic)34 b(Collo)s(quium)e(1976.)46 b(North)33 b(Holland,)g(Amsterdam)g(1977,)g(pp.)h(455{)165 4015 y(467)38 4202 y(4.)49 b(F)-8 b(riedman,)30 b(H.:)42 b(Equalit)m(y)29 b(b)s(et)m(w)m(een)j(functionals.)44 b(In)31 b(R.)f(P)m(arikh)g(\(ed.\):)43 b(Logic)30 b(Collo)s(quium,)165 4322 y(Lecture)35 b(Notes)e(in)g(Math)h(453.)43 b(Springer,)34 b(Berlin)e(1975,)h(pp.)g(22{37)38 4509 y(5.)49 b(G\177)-50 b(odel,)48 b(K.:)708 4484 y(\177)696 4509 y(Ub)s(er)c(eine)i(bisher)f (no)s(c)m(h)h(nic)m(h)m(t)f(b)s(en)s(\177)-53 b(utzte)46 b(Erw)m(eiterung)f(des)g(\014niten)h(Stand-)165 4629 y(punktes.)f(Dialectica)33 b Fh(12)p Fr(,)g(280{287)e(\(1958\))38 4816 y(6.)49 b(Ho)m(w)m(ard,)37 b(W.)f(A.:)48 b(Ordinal)36 b(analysis)g(of)g(terms)f(of)h(\014nite)g(t)m(yp)s(e.)52 b(The)37 b(Journal)f(of)g(Sym)m(b)s(olic)165 4936 y(Logic,)d Fh(45)g Fr(\(3\),)f(493{504)g(\(1980\))38 5123 y(7.)49 b(Kreisel,)d(G.:)63 b(In)m(terpretation)44 b(of)f(analysis)f(b)m(y)i (means)e(of)h(constructiv)m(e)h(functionals)g(of)f(\014-)165 5243 y(nite)f(t)m(yp)s(es.)70 b(In)42 b(A.)f(Heyting)g(\(ed.\):)61 b(Constructivit)m(y)40 b(in)i(Mathematics.)69 b(North)41 b(Holland,)165 5362 y(Amsterdam)32 b(1959,)g(pp.)i(101{128)38 5550 y(8.)49 b(Plotkin,)28 b(G.)h(D.:)41 b(LCF)29 b(considered)h(as)f (a)g(programming)d(language.)42 b(Theoretical)29 b(Computer)165 5669 y(Science.)46 b Fh(5)p Fr(,)33 b(223{255)e(\(1977\))1840 5908 y(14)p eop %%Page: 15 15 15 14 bop 38 -137 a Fr(9.)49 b(Plotkin,)24 b(G.)g(D.:)38 b Fj(T)918 -173 y Fm(!)999 -137 y Fr(as)24 b(a)g(univ)m(ersal)f (domain.)40 b(Journal)24 b(of)g(Computer)f(and)h(System)f(Sciences)165 -17 y Fh(17)p Fr(,)33 b(209{236)f(\(1978\))-12 162 y(10.)49 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b(D.)e(S.:)59 b(Domains)39 b(for)i(denotational)f(seman)m(tics.)66 b(In)40 b(M.)h(Nielsen,)h(E.)e (M.)h(Sc)m(hmidt)165 1418 y(\(eds.\):)54 b(Automata,)37 b(Languages)i(and)f(Programming.)56 b(Lecture)39 b(Notes)f(in)g (Computer)g(Sci-)165 1537 y(ence)d(150,)d(Springer,)i(Berlin)f(1982,)f (pp.)h(577{613)-12 1716 y(14.)49 b(Statman,)d(R.:)66 b(Equalit)m(y)43 b(b)s(et)m(w)m(een)i(functionals)g(revisited.)77 b(In)45 b(L.)f(A.)g(Harrington)f(et)h(al.)165 1836 y(\(eds.\):)68 b(Harv)m(ey)44 b(F)-8 b(riedman's)44 b(Researc)m(h)j(on)e(the)h(F)-8 b(oundations)45 b(of)g(Mathematics.)79 b(North)165 1956 y(Holland,)33 b(Amsterdam)e(1985,)h(pp.)i(331{338)165 2075 y(\032)1840 5908 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF