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b(then)g(also)1882 4889 y(\000)p Fu(;)14 b(A)2033 4901 y Fv(x)2075 4889 y FA(\()p Fu(y)s FA(\))p 1882 4925 302 4 v 1910 4999 a(\000)p Fu(;)g Fx(8)p Fu(xA)2248 4944 y FA(is)28 b(a)f(correct)f Fx(8)p FA(-inference.)0 5185 y(Note)i(that)g(if)508 5131 y(\001)p 467 5166 152 4 v 467 5240 a(\000)p Fu(;)14 b(A)655 5185 y FA(is)28 b(an)f(\()p Fx(_)p FA(\)-,)h(\()p Fx(8)p FA(\)-,)g(or)e(\()p Fx(9)p FA(\)-inference)i(with)g(principal)f(form)n(ula)g Fu(A)h FA(and)f(minor)g(form)n(ula)g Fu(A)3528 5197 y Ft(0)3566 5185 y FA(,)g(then)h(w)n(e)0 5340 y(do)f(not)h(necessarily)e (ha)n(v)n(e)g(\001)e(=)e(\000)p Fu(;)14 b(A)1199 5352 y Ft(0)1236 5340 y FA(;)28 b(but)g(w)n(e)g(only)f(kno)n(w)g(\001)c(=)f (\000)2192 5310 y Fp(0)2216 5340 y Fu(;)14 b(A)2315 5352 y Ft(0)2380 5340 y FA(with)28 b(\000)2621 5310 y Fp(0)2644 5340 y Fu(;)14 b(A)23 b FA(=)g(\000)p Fu(;)14 b(A)p FA(.)37 b(Similarly)27 b(for)g(\()p Fx(^)p FA(\).)1929 5539 y(2)p eop %%Page: 3 4 3 3 bop 0 83 a Fz(De\014nition.)0 224 y FA(A)28 b Fs(derivation)h FA(is)f(a)f(tree)g(of)h(sequen)n(ts)f(generated)f(from)i(the)g(ab)r(o)n (v)n(e)e(axioms)g(and)i(rules.)0 390 y(The)g(sequen)n(t)f(at)g(the)h (ro)r(ot)f(of)h(a)f(deriv)-5 b(ation)27 b Fu(d)h FA(is)f(called)g(its)h Fs(endse)l(quent)p FA(.)0 531 y Fu(d)g FA(is)f(called)h(a)f Fs(derivation)k(of)g FA(\000)d(if)g(\000)f(is)h(its)f(endsequen)n(t.)0 722 y Fy(Examples:)379 842 y Fu(R)q(f)9 b(t;)14 b Fx(:)p Fu(R)q(f)9 b(t)p 195 884 720 4 v 933 900 a Ft(\()p Fp(_)p Ft(\))195 958 y Fx(:)p Fu(R)q(f)g(f)g(t)18 b Fx(_)h Fu(R)q(f)9 b(t;)14 b Fx(:)p Fu(R)q(f)9 b(t)p 88 1000 906 4 v 1012 1017 a Ft(\()p Fp(_)p Ft(\))88 1075 y Fx(:)p Fu(R)q(f)g(f)g(t)18 b Fx(_)h Fu(R)q(f)9 b(t;)14 b Fx(:)p Fu(R)q(f)9 b(t)17 b Fx(_)i Fu(R)q(t)p 28 1116 999 4 v 1046 1133 a Ft(\()p Fp(9)p Ft(\))28 1191 y Fx(9)p Fu(x)p FA(\()p Fx(:)p Fu(R)q(f)9 b(x)19 b Fx(_)g Fu(R)q(x)p FA(\))p Fu(;)14 b Fx(:)p Fu(R)q(f)9 b(t)18 b Fx(_)h Fu(R)q(t)p 0 1232 1027 4 v 1046 1249 a Ft(\()p Fp(9)p Ft(\))225 1307 y Fx(9)p Fu(x)p FA(\()p Fx(:)p Fu(R)q(f)9 b(x)19 b Fx(_)g Fu(R)q(x)p FA(\))1597 842 y Fx(:)p Fu(R)q(xy)s(;)14 b(R)q(xy)p 1539 884 492 4 v 2049 900 a Ft(\()p Fp(9)p Ft(\))1539 958 y Fx(9)p Fu(y)s Fx(:)p Fu(R)q(xy)s(;)g(R)q(xy)p 1478 1000 586 4 v 2082 1017 a Ft(\()p Fp(9)p Ft(\))1478 1075 y Fx(9)p Fu(y)s Fx(:)p Fu(R)q(xy)s(;)g Fx(9)p Fu(xR)q(xy)p 1417 1116 679 4 v 2115 1133 a Ft(\()p Fp(8)p Ft(\))1417 1191 y Fx(8)p Fu(x)p Fx(9)p Fu(y)s Fx(:)p Fu(R)q(xy)s(;)g Fx(9)p Fu(xR)q(xy)p 1359 1232 769 4 v 2146 1249 a Ft(\()p Fp(8)p Ft(\))1359 1307 y Fx(8)p Fu(x)p Fx(9)p Fu(y)s Fx(:)p Fu(R)q(xy)s(;)g Fx(8)p Fu(y)s Fx(9)p Fu(xR)q(xy)2598 842 y(G;)g Fx(:)p Fu(F)2808 854 y Ft(0)2846 842 y Fu(;)g(F)2936 854 y Ft(0)3057 842 y Fu(F)3110 854 y Ft(1)3147 842 y Fu(;)g Fx(:)p Fu(F)3292 854 y Ft(1)p 2570 884 760 4 v 3349 900 a(\()p Fp(^)p Ft(\))2579 958 y Fu(G;)g Fx(:)p Fu(F)2789 970 y Ft(0)2828 958 y Fu(;)g(F)2918 970 y Ft(1)2955 958 y Fu(;)g(F)3045 970 y Ft(0)3101 958 y Fx(^)19 b(:)p Fu(F)3283 970 y Ft(1)p 2552 1000 770 4 v 3340 1017 a(\()p Fp(9)p Ft(\))2748 1075 y Fu(G;)14 b Fx(:)p Fu(F)2958 1087 y Ft(0)2997 1075 y Fu(;)g(F)3087 1087 y Ft(1)3496 1075 y Fu(F)3549 1087 y Ft(2)3587 1075 y Fu(;)g Fx(:)p Fu(F)3732 1087 y Ft(2)p 2721 1116 1049 4 v 3788 1133 a(\()p Fp(^)p Ft(\))2874 1191 y Fu(G;)g Fx(:)p Fu(F)3084 1203 y Ft(0)3123 1191 y Fu(;)g(F)3213 1203 y Ft(2)3250 1191 y Fu(;)g(F)3340 1203 y Ft(1)3396 1191 y Fx(^)19 b(:)p Fu(F)3578 1203 y Ft(2)p 2847 1232 770 4 v 3635 1249 a(\()p Fp(9)p Ft(\))3043 1307 y Fu(G;)14 b Fx(:)p Fu(F)3253 1319 y Ft(0)3292 1307 y Fu(;)g(F)3382 1319 y Ft(2)0 1523 y FA(In)42 b(the)h(third)f(example)f(w)n(e)h(ha)n(v)n(e)f (used)h(the)g(follo)n(wing)f(abbreviations:)64 b Fu(G)48 b FA(:=)e Fx(9)p Fu(x)p FA(\()p Fu(F)12 b FA(\()p Fu(x)p FA(\))30 b Fx(^)f(:)p Fu(F)12 b FA(\()p Fr(S)i Fu(x)p FA(\)\),)47 b(i.e.)80 b Fu(G)47 b FA(=)0 1664 y Fx(:8)p Fu(x)p FA(\()p Fu(F)12 b FA(\()p Fu(x)p FA(\))24 b Fx(!)f Fu(F)12 b FA(\()p Fr(S)i Fu(x)p FA(\)\).)66 b Fu(F)897 1676 y Ft(0)958 1664 y FA(:=)23 b Fu(F)12 b FA(\(0\),)27 b Fu(F)1343 1676 y Ft(1)1404 1664 y FA(:=)c Fu(F)12 b FA(\()p Fr(S)i FA(0\),)27 b(etc.)0 1843 y(Actually)39 b(this)h(not)f(a)f(completely)h(correct)f(PL1)o(-deriv)-5 b(ation,)41 b(since)e(it)g(con)n(tains)f Fx(^)p FA(-inferences)3220 1789 y(\000)3272 1801 y Ft(0)3310 1789 y Fu(;)14 b(A)83 b FA(\000)3544 1801 y Ft(1)3581 1789 y Fu(;)14 b(B)p 3216 1824 474 4 v 3216 1898 a FA(\000)3268 1910 y Ft(0)3305 1898 y Fu(;)g FA(\000)3394 1910 y Ft(1)3431 1898 y Fu(;)g(A)19 b Fx(^)f Fu(B)3739 1843 y FA(with)0 1993 y(\000)52 2005 y Ft(0)112 1993 y Fx(6)p FA(=)23 b(\000)252 2005 y Ft(1)289 1993 y FA(.)37 b(But)28 b(in)g(an)f(ob)n(vious)f(w)n(a)n(y)h(it)h(can)f (b)r(e)h(tak)n(en)f(as)g(a)g(shorthand)g(for)g(the)h(follo)n(wing)e (PL1)o(-deriv)-5 b(ation:)111 2105 y Fu(G;)14 b Fx(:)p Fu(F)321 2117 y Ft(0)359 2105 y Fu(;)g(F)449 2117 y Ft(2)486 2105 y Fu(;)g(F)576 2117 y Ft(1)614 2105 y Fu(;)g(F)704 2117 y Ft(0)824 2105 y Fu(G;)g Fx(:)p Fu(F)1034 2117 y Ft(0)1073 2105 y Fu(;)g(F)1163 2117 y Ft(2)1200 2105 y Fu(;)g(F)1290 2117 y Ft(1)1328 2105 y Fx(:)p Fu(F)1436 2117 y Ft(1)p 83 2146 1391 4 v 1493 2163 a(\()p Fp(^)p Ft(\))344 2221 y Fu(G;)g Fx(:)p Fu(F)554 2233 y Ft(0)592 2221 y Fu(;)g(F)682 2233 y Ft(2)720 2221 y Fu(;)g(F)810 2233 y Ft(1)847 2221 y Fu(;)g(F)937 2233 y Ft(0)993 2221 y Fx(^)19 b(:)p Fu(F)1175 2233 y Ft(1)p 316 2263 897 4 v 1232 2279 a(\()p Fp(9)p Ft(\))513 2337 y Fu(G;)14 b Fx(:)p Fu(F)723 2349 y Ft(0)761 2337 y Fu(;)g(F)851 2349 y Ft(2)889 2337 y Fu(;)g(F)979 2349 y Ft(1)1640 2337 y Fu(G;)g Fx(:)p Fu(F)1850 2349 y Ft(0)1888 2337 y Fu(;)g(F)1978 2349 y Ft(2)2015 2337 y Fu(;)g(F)2105 2349 y Ft(1)2143 2337 y Fu(;)g Fx(:)p Fu(F)2288 2349 y Ft(2)p 485 2379 1841 4 v 2345 2395 a(\()p Fp(^)p Ft(\))1035 2454 y Fu(G;)g Fx(:)p Fu(F)1245 2466 y Ft(0)1283 2454 y Fu(;)g(F)1373 2466 y Ft(2)1410 2454 y Fu(;)g(F)1500 2466 y Ft(1)1556 2454 y Fx(^)19 b(:)p Fu(F)1738 2466 y Ft(2)p 1007 2495 770 4 v 1795 2512 a(\()p Fp(9)p Ft(\))1204 2570 y Fu(G;)14 b Fx(:)p Fu(F)1414 2582 y Ft(0)1452 2570 y Fu(;)g(F)1542 2582 y Ft(2)0 2786 y Fz(De\014nition.)0 2927 y FA(The)28 b Fs(cut-r)l(ank)e FA(of)i(a)f(deriv)-5 b(ation)27 b Fu(d)h FA(is)55 b Fr(crk)p FA(\()p Fu(d)p FA(\))24 b(:=)f(sup)p Fx(f)p FA(rk)n(\()p Fu(C)6 b FA(\)+)q(1)22 b(:)i Fu(C)33 b FA(cut-form)n(ula)27 b(of)h Fu(d)p Fx(g)p FA(.)0 3068 y Fu(d)g FA(is)f(called)h Fs(cutfr)l(e)l(e)f FA(if)h Fr(crk)p FA(\()p Fu(d)p FA(\))c(=)e(0.)0 3234 y(The)28 b Fs(height)h Fr(hgt)o FA(\()p Fu(d)p FA(\))f(of)g(a)f(deriv) -5 b(ation)27 b Fu(d)h FA(is)g(de\014ned)g(recursiv)n(ely)d(b)n(y)0 3375 y Fr(hgt)o FA(\()p Fu(d)p FA(\))f(:=)f(sup)481 3395 y Fv(i)d FA(0)g(and)g(\000)g(=)g(\000) 2410 669 y Ft(0)2447 657 y Fu(;)14 b Fx(8)p Fu(xA)34 b FA(with)h Fx(`)2921 626 y Fv(k)q Fp(\000)p Ft(1)2921 677 y Fv(m)3081 657 y FA(\000)3133 669 y Ft(0)3170 657 y Fu(;)14 b(A)35 b FA(and)f Fu(x)g Fx(62)h FA(FV)q(\(\000\):)0 795 y(Cho)r(ose)27 b Fu(y)e Fx(62)f FA(FV)r(\(\000)p Fu(\022)r FA(\))k(and)f(let)1025 773 y(~)1018 795 y Fu(\022)f FA(:=)c Fu(\022)1234 764 y Fv(y)1232 815 y(x)1274 795 y FA(.)0 933 y(I.H.)28 b Fx(\))23 b(`)323 902 y Fv(k)q Fp(\000)p Ft(1)323 953 y Fv(m)472 933 y FA(\000)524 945 y Ft(0)568 911 y FA(~)561 933 y Fu(\022)r(;)14 b(A)708 911 y FA(~)701 933 y Fu(\022)789 885 y Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(0)821 933 y Fx(\))54 b(`)1009 902 y Fv(k)1009 953 y(m)1095 933 y FA(\000)1154 911 y(~)1147 933 y Fu(\022)s FA(.)36 b(F)-7 b(rom)28 b Fu(x)23 b Fx(62)h FA(FV)q(\(\000\))k(it)g(follo)n(ws)f(that)h(\000)2472 911 y(~)2465 933 y Fu(\022)d FA(=)e(\000)p Fu(\022)r FA(.)0 1092 y(2.)36 b(In)28 b(all)g(other)f(cases)f(the)i(claim)f (follo)n(ws)g(immediately)h(from)f(the)h(I.H.)g(and)f(L.1.0.)0 1274 y Fz(Lemma)j(1.2.)77 b FA(\(W)-7 b(eak)n(ening\))0 1412 y(PL1)22 b Fx(`)224 1381 y Fv(k)224 1432 y(m)310 1412 y FA(\000)46 b(&)g(\000)22 b Fx(\022)h FA(\000)733 1381 y Fp(0)802 1412 y FA(=)-14 b Fx(\))46 b FA(PL1)22 b Fx(`)1206 1381 y Fv(k)1206 1432 y(m)1292 1412 y FA(\000)1344 1381 y Fp(0)1367 1412 y FA(.)0 1571 y(Pro)r(of)k(b)n(y)i(induction)g (on)f Fu(k)s FA(:)0 1709 y(1.)36 b(Assume)28 b Fu(k)e(>)d FA(0)k(and)g(\000)c(=)g(\000)1011 1721 y Ft(0)1048 1709 y Fu(;)14 b Fx(8)p Fu(xA)27 b FA(with)h Fx(`)1508 1679 y Fv(k)q Fp(\000)p Ft(1)1508 1730 y Fv(m)1657 1709 y FA(\000)1709 1721 y Ft(0)1746 1709 y Fu(;)14 b(A)28 b FA(and)g Fu(x)23 b Fx(62)g FA(FV)r(\(\000)2385 1721 y Ft(0)2422 1709 y FA(\):)0 1847 y(Cho)r(ose)k Fu(y)e Fx(62)f FA(FV)r(\(\000)636 1817 y Fp(0)659 1847 y FA(\).)65 b Fx(`)830 1817 y Fv(k)q Fp(\000)p Ft(1)830 1868 y Fv(m)979 1847 y FA(\000)1031 1859 y Ft(0)1068 1847 y Fu(;)14 b(A)1213 1800 y Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(1)1245 1847 y Fx(\))55 b(`)1434 1817 y Fv(k)q Fp(\000)p Ft(1)1434 1868 y Fv(m)1582 1847 y FA(\000)1634 1859 y Ft(0)1671 1847 y Fu(;)14 b(A)1770 1859 y Fv(x)1812 1847 y FA(\()p Fu(y)s FA(\))1967 1800 y Ft(I)p Fv(:)p Ft(H)p Fv(:)1981 1847 y Fx(\))38 b(`)2153 1817 y Fv(k)q Fp(\000)p Ft(1)2153 1868 y Fv(m)2302 1847 y FA(\000)2354 1817 y Fp(0)2377 1847 y Fu(;)14 b(A)2476 1859 y Fv(x)2518 1847 y FA(\()p Fu(y)s FA(\))2672 1793 y Ft(\()p Fp(8)p Ft(\))2676 1847 y Fx(\))26 b(`)2836 1817 y Fv(k)2836 1868 y(m)2922 1847 y FA(\000)2974 1817 y Fp(0)2997 1847 y FA(.)0 1985 y(F)-7 b(or)27 b(the)h(last)f(step)h (note)f(that)h Fx(8)p Fu(xA)23 b Fx(2)h FA(\000)1299 1955 y Fp(0)1350 1985 y FA(and)j Fu(y)f Fx(62)d FA(FV)r(\(\000)1858 1955 y Fp(0)1881 1985 y FA(\).)0 2161 y(2.)59 b(In)36 b(all)f(other)g(cases)f(the)h(claim)g(follo)n(ws)g(immediately)g(from)g (the)h(I.H.)f(\(Note)h(that)g(if)f(\000)h Fx(\022)f FA(\000)3220 2131 y Fp(0)3279 2161 y FA(and)3458 2107 y Fu(:::)p FA(\000)3579 2119 y Fv(i)3607 2107 y Fu(:::)p 3458 2142 218 4 v 3541 2216 a FA(\000)3721 2161 y(is)g(an)0 2379 y(inference)27 b(\(LogAx\),)h(\()p Fx(^)p FA(\),)g(\()p Fx(_)p FA(\),)h(\()p Fx(9)p FA(\))f(or)f(\(Cut\))i(then)1716 2324 y Fu(:::)p FA(\000)1837 2294 y Fp(0)1860 2324 y Fu(;)14 b FA(\000)1949 2336 y Fv(i)1977 2324 y Fu(:::)p 1716 2360 330 4 v 1843 2434 a FA(\000)1895 2404 y Fp(0)2083 2379 y FA(is)28 b(an)f(inference)h(of)f(the)h(same)f(kind.)37 b(\))0 2594 y Fz(Corollary)-8 b(.)0 2732 y FA(PL1)22 b Fx(`)224 2702 y Fv(k)224 2752 y(m)310 2732 y FA(\000)55 b(i\013)28 b(one)f(of)h(the)g(follo)n(wing)f(cases)f(holds)0 2870 y(\(LogAx\))83 b Fx(f)p Fu(A;)14 b Fx(:)p Fu(A)p Fx(g)22 b(\022)h FA(\000)28 b(for)f(some)g(atomic)g Fu(A)p FA(,)0 3008 y(\()p Fx(^)p FA(\))270 b Fu(A)451 3020 y Ft(0)488 3008 y Fx(^)p Fu(A)605 3020 y Ft(1)666 3008 y Fx(2)23 b FA(\000)28 b(and)f(PL1)22 b Fx(`)1209 2978 y Fv(k)q Fp(\000)p Ft(1)1209 3028 y Fv(m)1358 3008 y FA(\000)p Fu(;)14 b(A)1509 3020 y Fv(i)1564 3008 y FA(for)27 b(eac)n(h)g Fu(i)c Fx(2)g(f)p FA(0)p Fu(;)14 b FA(1)p Fx(g)p FA(,)0 3146 y(\()p Fx(_)p FA(\))270 b Fu(A)451 3158 y Ft(0)488 3146 y Fx(_)p Fu(A)605 3158 y Ft(1)666 3146 y Fx(2)23 b FA(\000)28 b(and)f(PL1)22 b Fx(`)1209 3116 y Fv(k)q Fp(\000)p Ft(1)1209 3166 y Fv(m)1358 3146 y FA(\000)p Fu(;)14 b(A)1509 3158 y Fv(i)1564 3146 y FA(for)27 b(some)g Fu(i)c Fx(2)h(f)p FA(0)p Fu(;)14 b FA(1)p Fx(g)p FA(,)0 3284 y(\()p Fx(8)p FA(\))278 b Fx(8)p Fu(xA)22 b Fx(2)i FA(\000)j(and)h(PL1)22 b Fx(`)1111 3254 y Fv(k)q Fp(\000)p Ft(1)1111 3304 y Fv(m)1259 3284 y FA(\000)p Fu(;)14 b(A)28 b FA(with)g Fu(x)c Fx(62)f FA(FV)r(\(\000\),)0 3422 y(\()p Fx(9)p FA(\))279 b Fx(9)p Fu(xA)23 b Fx(2)h FA(\000)j(and)h(PL1)22 b Fx(`)1111 3392 y Fv(k)q Fp(\000)p Ft(1)1111 3442 y Fv(m)1259 3422 y FA(\000)p Fu(;)14 b(A)1410 3434 y Fv(x)1452 3422 y FA(\()p Fu(t)p FA(\),)0 3560 y(\(Cut\))187 b(PL1)22 b Fx(`)613 3530 y Fv(k)q Fp(\000)p Ft(1)613 3580 y Fv(m)761 3560 y FA(\000)p Fu(;)14 b(C)29 b FA(&)23 b(PL1)f Fx(`)1250 3530 y Fv(k)q Fp(\000)p Ft(1)1250 3580 y Fv(m)1399 3560 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)34 b FA(with)28 b(rk)o(\()p Fu(C)6 b FA(\))24 b Fu(<)f(m)p FA(.)0 3719 y(In)28 b(all)f(cases)g (except)g(\(LogAx\))g(it)h(is)g(tacitly)g(assumed)f(that)g Fu(k)f(>)d FA(0.)0 3922 y Fz(Lemma)30 b(1.3.)77 b FA(\(In)n(v)n (ersion\))0 4060 y(\(a\))28 b(PL1)22 b Fx(`)358 4030 y Fv(k)358 4081 y(m)444 4060 y FA(\000)p Fu(;)14 b Fx(8)p Fu(xA)45 b FA(=)-14 b Fx(\))46 b FA(PL1)22 b Fx(`)1138 4030 y Fv(k)1138 4081 y(m)1224 4060 y FA(\000)p Fu(;)14 b(A)1375 4072 y Fv(x)1417 4060 y FA(\()p Fu(t)p FA(\);)0 4198 y(\(b\))28 b(PL1)22 b Fx(`)362 4168 y Fv(k)362 4219 y(m)448 4198 y FA(\000)p Fu(;)14 b(A)599 4210 y Ft(0)655 4198 y Fx(^)19 b Fu(A)791 4210 y Ft(1)874 4198 y FA(=)-14 b Fx(\))46 b FA(PL1)22 b Fx(`)1278 4168 y Fv(k)1278 4219 y(m)1364 4198 y FA(\000)p Fu(;)14 b(A)1515 4210 y Fv(i)1570 4198 y FA(for)28 b Fu(i)22 b FA(=)h(0)p Fu(;)14 b FA(1;)0 4336 y(\(c\))28 b(PL1)22 b Fx(`)353 4306 y Fv(k)353 4357 y(m)439 4336 y FA(\000)p Fu(;)14 b(A)k Fx(_)h Fu(B)51 b FA(=)-14 b Fx(\))45 b FA(PL1)22 b Fx(`)1199 4306 y Fv(k)1199 4357 y(m)1285 4336 y FA(\000)p Fu(;)14 b(A;)g(B)t FA(.)0 4496 y(Pro)r(of)26 b(of)i(\(a\))g(b)n(y)f(induction)h(on)f Fu(k)s FA(:)0 4634 y(1.)36 b(Assume)28 b(that)g Fx(8)p Fu(xA)g FA(is)f(principal)g(part)g(of)h(the)g(last)f(inference)h(of)f (the)h(giv)n(en)f(deriv)-5 b(ation.)0 4772 y(Then)28 b(this)g(has)f(to)g(b)r(e)h(a)f Fx(8)p FA(-inference,)g(and)g(w)n(e)g (ha)n(v)n(e)g Fx(`)1782 4742 y Fv(k)q Fp(\000)p Ft(1)1782 4793 y Fv(m)1930 4772 y FA(\000)p Fu(;)14 b Fx(8)p Fu(xA;)g(A)28 b FA(with)g Fu(x)23 b Fx(62)h FA(FV)r(\(\000\).)0 4910 y(By)j(L.1.1)g(w)n(e)g(obtain)h Fx(`)771 4880 y Fv(k)q Fp(\000)p Ft(1)771 4930 y Fv(m)919 4910 y FA(\000)p Fu(;)14 b Fx(8)p Fu(xA;)g(A)1263 4922 y Fv(x)1305 4910 y FA(\()p Fu(t)p FA(\),)28 b(and)f(then,)i(b)n(y)e(I.H.,)h Fx(`)2179 4880 y Fv(k)q Fp(\000)p Ft(1)2179 4930 y Fv(m)2328 4910 y FA(\000)p Fu(;)14 b(A)2479 4922 y Fv(x)2521 4910 y FA(\()p Fu(t)p FA(\).)0 5048 y(2.)36 b Fx(`)152 5018 y Fv(k)q Fp(\000)p Ft(1)152 5068 y Fv(m)301 5048 y FA(\000)p Fu(;)14 b Fx(8)p Fu(xA;)g(B)31 b FA(with)e Fx(8)p Fu(y)s(B)d Fx(2)d FA(\000)28 b(and)f Fu(y)f Fx(62)d FA(FV)r(\(\000)p Fu(;)14 b Fx(8)p Fu(xA)p FA(\):)37 b(Let)28 b Fu(z)e Fx(62)d FA(FV)r(\(\000)p Fu(;)14 b(A)2591 5060 y Fv(x)2633 5048 y FA(\()p Fu(t)p FA(\)\).)0 5186 y Fx(`)51 5156 y Fv(k)q Fp(\000)p Ft(1)51 5206 y Fv(m)200 5186 y FA(\000)p Fu(;)g Fx(8)p Fu(xA;)g(B)594 5139 y Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(1)601 5186 y FA(=)-14 b Fx(\))29 b(`)815 5156 y Fv(k)q Fp(\000)p Ft(1)815 5206 y Fv(m)964 5186 y FA(\000)p Fu(;)14 b Fx(8)p Fu(xA;)g(B)1309 5198 y Fv(y)1348 5186 y FA(\()p Fu(z)t FA(\))1512 5139 y Ft(I)p Fv(:)p Ft(H)p Fv(:)1501 5186 y FA(=)-14 b Fx(\))23 b(`)1709 5156 y Fv(k)q Fp(\000)p Ft(1)1709 5206 y Fv(m)1858 5186 y FA(\000)p Fu(;)14 b(A)2009 5198 y Fv(x)2050 5186 y FA(\()p Fu(t)p FA(\))p Fu(;)g(B)2244 5198 y Fv(y)2285 5186 y FA(\()p Fu(z)t FA(\))2460 5132 y Ft(\()p Fp(8)p Ft(\))2438 5186 y FA(=)-14 b Fx(\))23 b(`)2646 5156 y Fv(k)2646 5206 y(m)2731 5186 y FA(\000)p Fu(;)14 b(A)2882 5198 y Fv(x)2924 5186 y FA(\()p Fu(t)p FA(\).)0 5324 y(3.)36 b(In)28 b(all)g(other)f (cases)f(the)i(claim)f(is)h(trivial)f(or)g(follo)n(ws)f(immediately)i (from)f(the)h(I.H.)1929 5539 y(4)p eop %%Page: 5 6 5 5 bop 0 83 a Fz(Cutelimination)0 248 y(Lemma)30 b(1.4.)0 389 y FA(PL1)22 b Fx(`)224 358 y Fv(k)224 409 y(m)310 389 y FA(\000)p Fu(;)14 b(C)52 b FA(&)46 b(PL1)22 b Fx(`)845 358 y Fv(l)845 409 y(m)931 389 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)52 b FA(&)46 b(rk)o(\()p Fu(C)6 b FA(\))24 b Fx(\024)f Fu(m)46 b FA(=)-14 b Fx(\))46 b FA(PL1)21 b Fx(`)2136 358 y Fv(k)q Ft(+)p Fv(l)2136 409 y(m)2272 389 y FA(\000.)0 554 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f Fu(k)21 b FA(+)d Fu(l)r FA(:)0 694 y(Assume)28 b Fu(d)23 b Fx(`)425 664 y Fv(k)425 715 y(m)511 694 y FA(\000)p Fu(;)14 b(C)34 b FA(and)27 b Fu(e)c Fx(`)967 664 y Fv(l)967 715 y(m)1053 694 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)6 b FA(.)0 859 y(1.)36 b Fu(C)e FA(is)28 b(not)f(a)h(principal)f(form)n (ula)f(of)i Fr(last)p FA(\()p Fu(d)p FA(\):)0 1000 y(Let)g Fu(A)211 1012 y Ft(1)248 1000 y Fu(;)14 b(:::;)g(A)453 1012 y Fv(n)526 1000 y FA(\()p Fu(n)24 b Fx(\024)e FA(2\))28 b(b)r(e)g(the)g(minor)f(form)n(ulas)f(of)i Fr(last)o FA(\()p Fu(d)p FA(\),)h(so)e(that)h(w)n(e)f(ha)n(v)n(e)f Fx(`)2675 970 y Fv(k)q Fp(\000)p Ft(1)2675 1020 y Fv(m)2824 1000 y FA(\000)p Fu(;)14 b(C)q(;)g(A)3072 1012 y Fv(i)3128 1000 y FA(for)27 b Fu(i)22 b FA(=)h(1)p Fu(;)14 b(:::;)g(n)p FA(.)0 1294 y(Then)28 b(the)g(claim)f(is)h(obtained)f(as)g(sho)n(wn)g (in)h(the)g(follo)n(wing)e(diagram)2356 1240 y Fx(`)2407 1210 y Fv(k)q Fp(\000)p Ft(1)2407 1261 y Fv(m)2556 1240 y FA(\000)p Fu(;)14 b(C)q(;)g(A)2804 1252 y Fv(i)2990 1124 y Fx(`)3041 1094 y Fv(l)3041 1144 y(m)3127 1124 y FA(\000)p Fu(;)g Fx(:)p Fu(C)p 2915 1165 469 4 v 3402 1182 a Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(2)2915 1240 y Fx(`)2966 1210 y Fv(l)2966 1261 y(m)3052 1240 y FA(\000)p Fu(;)g Fx(:)p Fu(C)q(;)g(A)3355 1252 y Fv(i)p 2329 1282 1056 4 v 3402 1298 a Ft(IH)2488 1356 y Fu(:)g(:)g(:)50 b Fx(`)2686 1326 y Fv(k)q Fp(\000)p Ft(1+)p Fv(l)2686 1377 y(m)2907 1356 y FA(\000)p Fu(;)14 b(A)3058 1368 y Fv(i)3127 1356 y Fu(:)g(:)g(:)p 2460 1398 765 4 v 3243 1414 a Fl(last)p Ft(\()p Fv(d)p Ft(\))2723 1473 y Fx(`)2774 1442 y Fv(k)q Ft(+)p Fv(l)2774 1493 y(m)2910 1473 y FA(\000)0 1660 y(F)-7 b(or)27 b Fu(n)c FA(=)g(0)k(w)n(e)g(ha)n(v)n(e)f Fr(last)p FA(\()p Fu(d)p FA(\))e(=)e(LogAx)o(,)28 b(and)g(the)f (diagram)g(reduces)f(to)p 2359 1641 309 4 v 2405 1714 a Fx(`)2456 1684 y Fv(k)q Ft(+)p Fv(l)2456 1735 y(m)2592 1714 y FA(\000)2677 1655 y Ft(\(LogAx\))2920 1660 y FA(.)0 1838 y(1'.)37 b Fx(:)p Fu(C)d FA(is)27 b(not)h(a)f(principal)g(form)n (ula)g(of)g Fr(last)p FA(\()p Fu(e)p FA(\):)37 b(symmetric)27 b(to)h(1.)0 2003 y(2.)36 b Fu(C)e FA(is)28 b(principal)f(form)n(ula)g (of)g Fr(last)p FA(\()p Fu(d)p FA(\),)h(and)g Fx(:)p Fu(C)34 b FA(is)28 b(principal)f(form)n(ula)f(of)i Fr(last)o FA(\()p Fu(e)p FA(\):)0 2168 y(2.1.)36 b Fu(C)e FA(is)27 b(a)h(literal:)0 2309 y(Then)g Fx(f)p Fu(C)q(;)14 b Fx(:)p Fu(C)6 b Fx(g)23 b(\022)g FA(\000)18 b Fx([)h(f)p Fu(C)6 b Fx(g)27 b FA(and)g Fx(f)p Fu(C)q(;)14 b Fx(:)p Fu(C)6 b Fx(g)23 b(\022)g FA(\000)18 b Fx([)h(f:)p Fu(C)6 b Fx(g)p FA(.)37 b(Hence)28 b Fx(f)p Fu(C)q(;)14 b Fx(:)p Fu(C)6 b Fx(g)23 b(\022)g FA(\000,)k(and)h(therefore)e Fx(`)3252 2279 y Fv(k)q Ft(+)p Fv(l)3252 2330 y(m)3388 2309 y FA(\000.)0 2474 y(2.2.)36 b Fu(C)29 b FA(=)23 b Fx(9)p Fu(xA)p FA(:)38 b(Then)28 b Fx(:)p Fu(C)h FA(=)23 b Fx(8)p Fu(x)p Fx(:)p Fu(A)28 b FA(and)f(w)n(e)g(ha)n(v)n(e)g Fx(`)1771 2444 y Fv(k)q Fp(\000)p Ft(1)1771 2495 y Fv(m)1919 2474 y FA(\000)p Fu(;)14 b(C)q(;)g(A)2167 2486 y Fv(x)2210 2474 y FA(\()p Fu(t)p FA(\))28 b(and)g Fx(`)2545 2444 y Fv(l)p Fp(\000)p Ft(1)2545 2495 y Fv(m)2678 2474 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)q(;)g Fx(:)p Fu(A)29 b FA(with)f Fu(x)23 b Fx(62)h FA(FV)q(\(\000)p Fu(;)14 b Fx(:)p Fu(C)6 b FA(\).)0 2615 y(No)n(w)27 b(the)h(claim)f(is)h (obtained)f(as)g(sho)n(wn)g(in)h(the)g(follo)n(wing)e(diagram:)55 2842 y Fx(`)106 2812 y Fv(k)q Fp(\000)p Ft(1)106 2862 y Fv(m)255 2842 y FA(\000)p Fu(;)14 b(C)q(;)g(A)503 2854 y Fv(x)545 2842 y FA(\()p Fu(t)p FA(\))852 2726 y Fx(`)903 2696 y Fv(l)903 2746 y(m)989 2726 y FA(\000)p Fu(;)g Fx(:)p Fu(C)p 723 2767 578 4 v 1319 2784 a Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(2)723 2842 y Fx(`)774 2812 y Fv(l)774 2862 y(m)860 2842 y FA(\000)p Fu(;)g Fx(:)p Fu(C)q(;)g(A)1163 2854 y Fv(x)1206 2842 y FA(\()p Fu(t)p FA(\))p 28 2883 1273 4 v 1319 2900 a Ft(IH)384 2958 y Fx(`)435 2928 y Fv(k)q Fp(\000)p Ft(1+)p Fv(l)435 2979 y(m)656 2958 y FA(\000)p Fu(;)g(A)807 2970 y Fv(x)849 2958 y FA(\()p Fu(t)p FA(\))1734 2726 y Fx(`)1785 2696 y Fv(k)1785 2746 y(m)1871 2726 y FA(\000)p Fu(;)g(C)p 1577 2767 578 4 v 2173 2784 a Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(2)1577 2842 y Fx(`)1628 2812 y Fv(k)1628 2862 y(m)1714 2842 y FA(\000)p Fu(;)g(C)q(;)g Fx(:)p Fu(A)2017 2854 y Fv(x)2060 2842 y FA(\()p Fu(t)p FA(\))2485 2726 y Fx(`)2536 2696 y Fv(l)p Fp(\000)p Ft(1)2536 2746 y Fv(m)2669 2726 y FA(\000)p Fu(;)g Fx(:)p Fu(C)q(;)g Fx(:)p Fu(A)p 2403 2767 680 4 v 3102 2784 a Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(1)2403 2842 y Fx(`)2454 2812 y Fv(l)p Fp(\000)p Ft(1)2454 2862 y Fv(m)2587 2842 y FA(\000)p Fu(;)g Fx(:)p Fu(C)q(;)g Fx(:)p Fu(A)2945 2854 y Fv(x)2989 2842 y FA(\()p Fu(t)p FA(\))p 1549 2883 1534 4 v 3102 2900 a Ft(IH)2009 2958 y Fx(`)2060 2928 y Fv(k)q Ft(+)p Fv(l)p Fp(\000)p Ft(1)2060 2979 y Fv(m)2281 2958 y FA(\000)p Fu(;)g Fx(:)p Fu(A)2487 2970 y Fv(x)2529 2958 y FA(\()p Fu(t)p FA(\))p 357 3000 2267 4 v 2642 3016 a Ft(\(Cut\))1371 3074 y Fx(`)1422 3044 y Fv(k)q Ft(+)p Fv(l)1422 3095 y(m)1558 3074 y FA(\000)0 3239 y(F)-7 b(or)27 b(the)h(last)f(step)h(note)f(that)h(rk\()p Fu(A)1160 3251 y Fv(x)1202 3239 y FA(\()p Fu(t)p FA(\)\))c Fu(<)f FA(rk)o(\()p Fu(C)6 b FA(\))24 b Fx(\024)e Fu(m)p FA(.)0 3404 y(2.2'.)36 b Fu(C)29 b FA(=)23 b Fx(8)p Fu(xA)28 b FA(or)e Fu(A)712 3416 y Ft(0)768 3404 y Fx(^)19 b Fu(A)904 3416 y Ft(1)969 3404 y FA(or)27 b Fu(A)1133 3416 y Ft(0)1189 3404 y Fx(_)19 b Fu(A)1325 3416 y Ft(1)1362 3404 y FA(:)37 b(analogous)25 b(to)j(2.2.)0 3618 y Fz(Theorem)i(1.5)d FA(\(Cut-Elimination\).)0 3759 y(PL1)22 b Fx(`)224 3729 y Fv(k)224 3779 y(m)p Ft(+1)394 3759 y FA(\000)46 b(=)-14 b Fx(\))46 b FA(PL1)22 b Fx(`)896 3729 y Ft(2)929 3704 y Fn(k)896 3779 y Fv(m)992 3759 y FA(\000.)0 3924 y(Pro)r(of)k(b)n(y)i (induction)g(on)f Fu(k)s FA(:)0 4074 y(Let)h Fu(d)23 b Fx(`)266 4043 y Fv(k)266 4094 y(m)p Ft(+1)436 4074 y FA(\000)28 b(and)f(assume)g(that)h Fr(last)o FA(\()p Fu(d)p FA(\))c(=)1494 4019 y(\000)p Fu(;)14 b(C)89 b FA(\000)p Fu(;)14 b Fx(:)p Fu(C)p 1494 4054 447 4 v 1691 4128 a FA(\000)1978 4074 y(with)28 b(rk)o(\()p Fu(C)6 b FA(\))24 b(=)f Fu(m)p FA(.)0 4261 y Fx(`)51 4225 y Fv(k)q Fp(\000)p Ft(1)51 4283 y Fv(m)p Ft(+1)221 4261 y FA(\000)p Fu(;)14 b(C)34 b FA(and)27 b Fx(`)615 4225 y Fv(k)q Fp(\000)p Ft(1)615 4283 y Fv(m)p Ft(+1)785 4261 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)1041 4214 y Ft(I)p Fv(:)p Ft(H)p Fv(:)1055 4261 y Fx(\))61 b(`)1250 4231 y Ft(2)1283 4206 y Fn(k)q Fk(\000)p Fo(1)1250 4281 y Fv(m)1420 4261 y FA(\000)p Fu(;)14 b(C)33 b FA(and)28 b Fx(`)1814 4231 y Ft(2)1847 4206 y Fn(k)q Fk(\000)p Fo(1)1814 4281 y Fv(m)1983 4261 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)2239 4214 y Ft(L)p Fv(:)p Ft(1)p Fv(:)p Ft(4)2245 4261 y FA(=)-14 b Fx(\))76 b(`)2506 4231 y Ft(2)2539 4206 y Fn(k)q Fk(\000)p Fo(1)2648 4231 y Ft(+2)2732 4206 y Fn(k)q Fk(\000)p Fo(1)2506 4281 y Fv(m)2869 4261 y FA(\000.)0 4426 y(In)28 b(all)f(other)g(cases)g(the)h(claim)f(follo)n (ws)g(immediately)g(from)h(the)g(I.H.)0 4640 y Fz(De\014nition.)138 b Fu(B)41 b FA(is)36 b(a)h Fs(subformula)g FA(of)g Fu(A)g FA(if)h Fu(B)j FA(can)36 b(b)r(e)i(obtained)e(from)h Fu(A)g FA(b)n(y)g(\014nitely)g(man)n(y)f(steps)h(of)g(the)g(kind)0 4780 y Fr(Q)p Fu(xA)24 b Fx(7!)f Fu(A)362 4792 y Fv(x)404 4780 y FA(\()p Fu(t)p FA(\))28 b(or)f Fu(A)690 4792 y Ft(0)746 4780 y Fx(\005)17 b Fu(A)868 4792 y Ft(1)928 4780 y Fx(7!)24 b Fu(A)1097 4792 y Fv(i)1124 4780 y FA(.)37 b(Esp)r(ecially)27 b Fu(A)h FA(is)f(a)h(subform)n(ula)e(of)i(itself.)0 4994 y Fz(Remark.)78 b FA(Cutfree)28 b(deriv)-5 b(ations)26 b(are)h(distinguished)h(b)n(y)f(the)h(follo)n(wing)0 5159 y Fz(Subform)m(ula)j(prop)s(ert)m(y)0 5299 y FA(If)d Fu(d)g FA(is)f(a)h(cutfree)f(deriv)-5 b(ation)27 b(of)h(\000)f(then)h (ev)n(ery)f(form)n(ula)f(o)r(ccurring)h(in)g Fu(d)h FA(is)g(a)f (subform)n(ula)g(of)g(some)g Fu(A)c Fx(2)h FA(\000.)1929 5539 y(5)p eop %%Page: 6 7 6 6 bop 0 83 a Fz(P)m(artial)33 b(Cut)f(Elimination)0 244 y FA(Let)d Fm(S)219 214 y Ft(+)302 244 y FA(b)r(e)g(a)f(family)g (of)h(additional)f(inference)g(rules)g(of)g(the)h(form)2224 189 y(\000)p Fu(;)14 b FA(\001)2382 201 y Ft(0)2516 189 y Fu(:)g(:)g(:)97 b FA(\000)p Fu(;)14 b FA(\001)2868 201 y Fv(n)p Fp(\000)p Ft(1)p 2224 225 774 4 v 2532 299 a FA(\000)p Fu(;)g FA(\001)3008 244 y(,)29 b(and)f(b)r(eing)g(closed)g (under)0 398 y(substitution.)41 b(As)29 b(in)g(PL1)o(,)h(form)n(ulas)d Fu(A)f Fx(2)g FA(\001)j([)p Fu(A)c Fx(2)h FA(\001)1788 410 y Fv(i)1816 398 y FA(,)j(resp.])40 b(are)28 b(called)h(the)g (principal)f([minor,)h(resp.])40 b(form)n(ulas)28 b(of)0 537 y(the)h(resp)r(ectiv)n(e)e(inference.)38 b(Some)28 b(of)g(these)g(rules)g(ma)n(y)f(b)r(e)i(restricted)e(b)n(y)h(a)g(\\v)-5 b(ariable)27 b(condition")g(of)h(the)h(kind)f(that)g(a)0 675 y(certain)f(v)-5 b(ariable)27 b(\(called)g(the)h Fs(eigenvariable)j FA(of)c(the)h(resp.)f(inference\))h(m)n(ust)g(not)f (o)r(ccur)g(free)g(in)h(the)g(conclusion)f(\000)p Fu(;)14 b FA(\001.)0 836 y(Let)28 b Fm(S)23 b FA(:=)f(PL1)c(+)g Fm(S)672 806 y Ft(+)727 836 y FA(.)0 997 y(\010)23 b(:=)g(\010\()p Fm(S)p FA(\))g(:=)g Fx(f)p Fu(A)g FA(:)g Fu(A)28 b FA(is)f(a)g (principal)h(form)n(ula)e(of)i(an)f Fm(S)1866 966 y Ft(+)1921 997 y FA(-inference)o Fx(g)0 1157 y Fy(Remark.)73 b FA(Since)28 b(the)g(rules)f(of)g Fm(S)1104 1127 y Ft(+)1187 1157 y FA(are)f(closed)h(under)h(substitution,)g(w)n(e)f(ha)n(v)n(e)g Fx(8)p Fu(\022)r Fx(8)p Fu(A)22 b Fx(2)h FA(\010\()p Fu(A\022)j Fx(2)d FA(\010\).)0 1318 y(Let)p 149 1251 60 4 v 28 w(\010)g(:=)f(\010)d Fx([)g(f:)p Fu(A)k FA(:)g Fu(A)g Fx(2)h FA(\010)p Fx(g)p FA(.)0 1479 y Fz(De\014nition)31 b(of)h FA(rk)631 1491 y Ft(\010)682 1479 y FA(\()p Fu(A)p FA(\))0 1617 y(1.)k(rk)178 1629 y Ft(\010)229 1617 y FA(\()p Fu(A)p FA(\))24 b(:=)f Fx(\000)p FA(1,)k(if)h Fu(A)23 b Fx(2)p 887 1551 V 24 w FA(\010;)0 1756 y(2.)36 b(rk)178 1768 y Ft(\010)229 1756 y FA(\()p Fu(A)p FA(\))24 b(:=)f(0,)k(if)h Fu(A)g FA(is)g(a)f(literal)g(and)g Fu(A)d Fx(62)p 1466 1689 V 23 w FA(\010;)0 1894 y(3.)36 b(rk)178 1906 y Ft(\010)229 1894 y FA(\()p Fu(A)19 b Fx(\005)e Fu(B)t FA(\))24 b(:=)e(max)p Fx(f)p FA(rk)908 1906 y Ft(\010)959 1894 y FA(\()p Fu(A)p FA(\))p Fu(;)14 b FA(rk)1199 1906 y Ft(\010)1251 1894 y FA(\()p Fu(B)t FA(\))p Fx(g)19 b FA(+)f(1)27 b(,)h(if)g Fu(A)18 b Fx(\005)f Fu(B)28 b Fx(62)p 2031 1828 V 23 w FA(\010;)0 2033 y(4.)36 b(rk)178 2045 y Ft(\010)229 2033 y FA(\()p Fr(Q)q Fu(xA)p FA(\))24 b(:=)f(rk)675 2045 y Ft(\010)726 2033 y FA(\()p Fu(A)p FA(\))d(+)e(1)27 b(,)h(if)g Fr(Q)p Fu(xA)c Fx(62)p 1423 1966 V 23 w FA(\010.)0 2216 y Fy(Remark.)73 b FA(rk)458 2228 y Ft(\010)510 2216 y FA(\()p Fu(A\022)r FA(\))24 b Fx(\024)f FA(rk)865 2228 y Ft(\010)916 2216 y FA(\()p Fu(A)p FA(\))29 b(and)e(rk)1309 2228 y Ft(\010)1360 2216 y FA(\()p Fx(:)p Fu(A)p FA(\))e(=)d(rk)1730 2228 y Ft(\010)1781 2216 y FA(\()p Fu(A)p FA(\).)0 2376 y Fz(De\014nition.)83 b FA(The)28 b(relation)g Fm(S)c Fx(`)1146 2346 y Fv(k)1146 2399 y Ft(\010)p Fv(;m)1300 2376 y FA(\000)k(is)g(de\014ned)h(in)g(the) f(same)g(w)n(a)n(y)f(as)h(PL1)23 b Fx(`)2698 2346 y Fv(k)2698 2397 y(m)2785 2376 y FA(\000,)29 b(but)g(with)g(the)g(di\013erence)f (that)0 2515 y(no)n(w)f Fu(m)h FA(refers)e(to)i(rk)676 2527 y Ft(\010)755 2515 y FA(instead)g(of)f(rk)o(.)0 2676 y Fz(Lemma)j(1.6.)0 2814 y FA(\(a\))e Fm(S)23 b Fx(`)277 2826 y Ft(\010)p Fv(;m)430 2814 y FA(\000)46 b(=)-14 b Fx(\))45 b Fm(S)23 b Fx(`)850 2826 y Ft(\010)p Fv(;m)1003 2814 y FA(\000)p Fu(\022)r FA(;)0 2953 y(\(b\))28 b Fm(S)23 b Fx(`)281 2923 y Fv(k)281 2976 y Ft(\010)p Fv(;m)434 2953 y FA(\000)46 b(&)g(\000)23 b Fx(\022)f FA(\000)857 2923 y Fp(0)927 2953 y FA(=)-14 b Fx(\))46 b Fm(S)23 b Fx(`)1250 2923 y Fv(k)1250 2976 y Ft(\010)p Fv(;m)1402 2953 y FA(\000)1454 2923 y Fp(0)1478 2953 y FA(.)0 3158 y Fz(Lemma)30 b(1.7)0 3296 y Fm(S)23 b Fx(`)143 3266 y Fv(k)143 3319 y Ft(\010)p Fv(;m)296 3296 y FA(\000)p Fu(;)14 b(C)52 b FA(&)46 b Fm(S)23 b Fx(`)750 3266 y Fv(l)750 3319 y Ft(\010)p Fv(;m)903 3296 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)52 b FA(&)46 b(0)22 b Fx(\024)h FA(rk)1497 3308 y Ft(\010)1549 3296 y FA(\()p Fu(C)6 b FA(\))24 b Fx(\024)e Fu(m)46 b FA(=)-14 b Fx(\))46 b Fm(S)23 b Fx(`)2231 3261 y Fv(k)q Ft(+)p Fv(l)2231 3321 y Ft(\010)p Fv(;m)2384 3296 y FA(\000.)0 3457 y(Pro)r(of)j(b)n(y)i (induction)g(on)f Fu(k)21 b FA(+)d Fu(l)r FA(:)0 3596 y(The)29 b(pro)r(of)g(pro)r(ceeds)f(almost)g(literally)h(as)f(the)i (pro)r(of)e(of)h(Lemma)g(1.4.)40 b(In)30 b(case)e(2.2)g(one)h (concludes)f(from)h(0)c Fx(\024)g FA(rk)3719 3608 y Ft(\010)3770 3596 y FA(\()p Fu(C)6 b FA(\))0 3734 y(that)28 b Fu(C)q(;)14 b Fx(:)p Fu(C)30 b Fx(62)24 b FA(\010)j(and)h(therefore)e(\(b)n(y)i 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Fx(`)705 4713 y Fv(k)705 4766 y Ft(\010)p Fv(;m)858 4743 y FA(\000)p Fu(;)14 b Fx(8)p Fu(xA)45 b FA(=)-14 b Fx(\))46 b Fm(S)23 b Fx(`)1471 4713 y Fv(k)1471 4766 y Ft(\010)p Fv(;m)1624 4743 y FA(\000)p Fu(;)14 b(A)1775 4755 y Fv(x)1817 4743 y FA(\()p Fu(t)p FA(\);)0 4882 y(\(b\))28 b Fu(A)200 4894 y Ft(0)256 4882 y Fx(^)19 b Fu(A)392 4894 y Ft(1)453 4882 y Fx(62)k FA(\010)g(&)g Fm(S)g Fx(`)845 4851 y Fv(k)845 4904 y Ft(\010)p Fv(;m)998 4882 y FA(\000)p Fu(;)14 b(A)1149 4894 y Ft(0)1204 4882 y Fx(^)19 b Fu(A)1340 4894 y Ft(1)1424 4882 y FA(=)-14 b Fx(\))46 b Fm(S)23 b Fx(`)1747 4851 y Fv(k)1747 4904 y Ft(\010)p Fv(;m)1899 4882 y FA(\000)p Fu(;)14 b(A)2050 4894 y Fv(i)2106 4882 y FA(for)27 b Fu(i)c FA(=)f(0)p Fu(;)14 b FA(1;)0 5020 y(\(c\))28 b Fu(A)19 b Fx(_)g Fu(B)27 b Fx(62)c FA(\010)g(&)g Fm(S)g Fx(`)766 4990 y Fv(k)766 5043 y Ft(\010)p Fv(;m)919 5020 y FA(\000)p Fu(;)14 b(A)k Fx(_)h Fu(B)50 b FA(=)-14 b Fx(\))46 b Fm(S)23 b Fx(`)1598 4990 y Fv(k)1598 5043 y Ft(\010)p Fv(;m)1751 5020 y FA(\000)p Fu(;)14 b(A;)g(B)t FA(.)0 5181 y(Pro)r(of)25 b(as)g(for)h(L.1.3:)35 b(In)27 b(\(a\))f(the)h (condition)e Fx(8)p Fu(xA)e Fx(62)h FA(\010)i(guaran)n(tees)e(that)i Fx(8)p Fu(xA)h FA(cannot)e(b)r(e)i(the)f(principal)g(form)n(ula)f(of)h (an)0 5319 y(inference)h(other)g(than)h(\()p Fx(8)p FA(\).)37 b(Similarly)27 b(for)g(\(b\),\(c\).)1929 5539 y(6)p eop %%Page: 7 8 7 7 bop 0 83 a Fz(Completeness)28 b(of)k FA(PL1)0 249 y(Assuming)c(that)h Fx(L)g FA(is)f(coun)n(table)g(w)n(e)g(will)g(pro)n (v)n(e)f(the)i(completeness)f(of)g(PL1)g(without)h(cut)f(rule.)39 b(This)29 b(\(together)f(with)0 390 y(the)g(correctness)e(of)h(PL1)o (\))h(yields)g(a)f(so-called)f Fy(seman)n(tical)h(cut)g(elimination)h (pro)r(of)f FA(for)g(PL1)o(.)0 556 y Fz(De\014nition)0 697 y FA(\006)c Fx(j)-14 b FA(=)23 b Fu(C)75 b Fx(\()-14 b(\))70 b Fu(C)34 b FA(is)27 b(a)g(logical)g(consequence)f(from)i(\006) 82 b(\(\006)28 b(a)g(set)f(of)h(form)n(ulas\))0 839 y(\006)23 b Fx(j)-14 b FA(=)23 b Fx(f)p Fu(A)284 851 y Ft(1)321 839 y Fu(;)14 b(:::;)g(A)526 851 y Fv(n)571 839 y Fx(g)69 b(\()-14 b(\))69 b FA(\006)23 b Fx(j)-14 b FA(=)23 b Fu(A)1145 851 y Ft(1)1201 839 y Fx(_)c Fu(:::)g Fx(_)f Fu(A)1498 851 y Fv(n)0 980 y Fx(j)-14 b FA(=)23 b(\000)69 b Fx(\()-14 b(\))69 b(;)23 b(j)-14 b FA(=)23 b(\000.)0 1171 y Fz(Theorem)30 b(1.10)0 1312 y Fx(j)-14 b FA(=)23 b(\000)46 b(=)-14 b Fx(\))46 b FA(PL1)22 b Fx(`)599 1324 y Ft(0)659 1312 y FA(\000.)0 1478 y Fz(Corollary)0 1619 y FA(\006)h Fx(j)-14 b FA(=)23 b Fu(C)52 b FA(=)-14 b Fx(\))28 b FA(There)f(are)f Fu(A)893 1631 y Ft(1)931 1619 y Fu(;)14 b(:::;)g(A)1136 1631 y Fv(n)1204 1619 y Fx(2)24 b FA(\006)j(suc)n(h)h(that)g(PL1)22 b Fx(`)1962 1631 y Ft(0)2022 1619 y Fx(:)p Fu(A)2139 1631 y Ft(1)2176 1619 y Fu(;)14 b(:::;)g Fx(:)p Fu(A)2436 1631 y Fv(n)2482 1619 y Fu(;)g(C)6 b FA(.)0 1785 y(Pro)r(of:)36 b(\006)23 b Fx(j)-14 b FA(=)23 b Fu(C)52 b Fx(\))28 b FA(There)f(are)f Fu(A)1102 1797 y Ft(1)1140 1785 y Fu(;)14 b(:::;)g(A)1345 1797 y Fv(n)1413 1785 y Fx(2)24 b FA(\006)j(with)h Fx(f)p Fu(A)1872 1797 y Ft(1)1910 1785 y Fu(;)14 b(:::;)g(A)2115 1797 y Fv(n)2160 1785 y Fx(g)23 b(j)-14 b FA(=)22 b Fu(C)53 b Fx(\))23 b(j)-14 b FA(=)23 b Fx(:)p Fu(A)2753 1797 y Ft(1)2791 1785 y Fu(;)14 b(:::;)g Fx(:)p Fu(A)3051 1797 y Fv(n)3096 1785 y Fu(;)g(C)6 b FA(.)0 2100 y(Pro)r(of)26 b(of)i(Theorem)f(1.10:)0 2267 y(AX)d(:=)j(set)h(of)f(\014nite)h (sequences)f(\()p Fu(A)1170 2279 y Ft(0)1208 2267 y Fu(;)14 b(:::;)g(A)1413 2279 y Fv(l)1439 2267 y FA(\))28 b(suc)n(h)f(that)h (there)f(is)h(a)f(prime)g(form)n(ula)g Fu(A)h FA(with)g Fx(f)p Fu(A;)14 b Fx(:)p Fu(A)p Fx(g)23 b(\022)f(f)p Fu(A)3564 2279 y Ft(0)3601 2267 y Fu(;)14 b(:::;)g(A)3806 2279 y Fv(l)3832 2267 y Fx(g)p FA(.)0 2433 y(Let)28 b(\005)g(b)r(e)g(a) f(\014xed)h(\014nite)g(sequence)f(of)g(form)n(ulas)g(.)0 2599 y Fu(t)30 2611 y Ft(0)67 2599 y Fu(;)14 b(t)134 2611 y Ft(1)171 2599 y Fu(;)g(:::)p FA(:)37 b(en)n(umeration)27 b(of)h(T)-7 b(er.)0 2765 y Fu(\026;)14 b(\027)33 b FA(are)26 b(ranging)h(o)n(v)n(er)e(\014nite)k(0-1-sequences)c(\(elemen)n(ts)i(of) h Fx(f)p FA(0)p Fu(;)14 b FA(1)p Fx(g)2176 2735 y Fv()f FA(0)k(while)h(all)f(form)n (ulas)g(in)g(\005)2305 3847 y Fp(0)2357 3877 y FA(are)f(literals:)0 4018 y(3.1.)36 b Fu(A)23 b FA(=)g Fu(A)401 4030 y Ft(0)457 4018 y Fx(^)c Fu(A)593 4030 y Ft(1)630 4018 y FA(:)37 b(\005)752 4033 y Fv(\027)t Fp(\003)q(h)p Fv(i)p Fp(i)928 4018 y FA(:=)23 b(\005)1101 3988 y Fp(0)1124 4018 y Fu(;)14 b(A)1223 4030 y Fv(i)1251 4018 y Fu(;)g FA(\005)1350 3988 y Fp(00)1393 4018 y FA(,)0 4159 y(3.2.)36 b Fu(A)23 b FA(=)g Fu(A)401 4171 y Ft(0)457 4159 y Fx(_)c Fu(A)593 4171 y Ft(1)630 4159 y FA(:)37 b(\005)752 4174 y Fv(\027)t Fp(\003)q(h)p Fv(i)p Fp(i)928 4159 y FA(:=)23 b(\005)1101 4129 y Fp(0)1124 4159 y Fu(;)14 b(A)1223 4171 y Ft(0)1261 4159 y Fu(;)g(A)1360 4171 y Ft(1)1397 4159 y Fu(;)g FA(\005)1496 4129 y Fp(00)1539 4159 y FA(,)0 4301 y(3.3.)36 b Fu(A)23 b FA(=)g Fx(8)p Fu(xB)t FA(:)37 b(\005)622 4316 y Fv(\027)t Fp(\003h)p Fv(i)p 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y(\005)1780 4615 y Fv(\027)t Fp(\003)q(h)p Ft(0)p Fp(i)2026 4600 y FA(\005)2088 4615 y Fv(\027)t Fp(\003h)p Ft(1)p Fp(i)p 1718 4636 532 4 v 1933 4710 a FA(\005)1995 4722 y Fv(\027)2289 4655 y FA(\(in)i(case)e(3.1\),)2798 4600 y(\005)2860 4615 y Fv(\027)t Fp(\003h)p Ft(0)p Fp(i)p 2798 4636 225 4 v 2858 4710 a FA(\005)2920 4722 y Fv(\027)3061 4655 y FA(\(in)i(cases)e(3.3,)g(3.4\))h(is)f(an)0 4810 y(inference)f(of)h (PL1)o(.)37 b(In)28 b(case)e(3.2,)h(\005)1151 4822 y Fv(\027)1220 4810 y FA(is)h(obtained)f(from)h(\005)1904 4825 y Fv(\027)t Fp(\003h)p Ft(0)p Fp(i)2094 4810 y FA(b)n(y)f(t)n(w)n (o)g(\()p Fx(_)p FA(\)-inferences.)0 5025 y Fz(Assumption:)0 5167 y(\(A\))83 b FA(\()p Fu(i)290 5179 y Fv(n)336 5167 y FA(\))368 5179 y Fv(n)p Fp(2)p Ft(I)-11 b(N)547 5167 y FA(is)28 b(a)f(0-1-sequence)e(suc)n(h)i(that)56 b Fx(8)p Fu(n)22 b Fx(2)h FA(I)-14 b(N)q(\()14 b(\005)1961 5182 y Fp(h)p Fv(i)2011 5190 y Fo(0)2044 5182 y Fv(;:::;i)2167 5190 y Fn(n)p Fk(\000)p Fo(1)2280 5182 y Fp(i)2334 5167 y Fx(62)24 b FA(AX)14 b(\).)0 5333 y(Abbreviation:)36 b Fu(\027)5 b FA(\()p Fu(n)p FA(\))24 b(:=)f Fx(h)p Fu(i)891 5345 y Ft(0)928 5333 y Fu(;)14 b(:::;)g(i)1100 5345 y Fv(n)p Fp(\000)p Ft(1)1230 5333 y Fx(i)p FA(,)55 b Fx(F)31 b FA(:=)1542 5270 y Fq(S)1611 5358 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1777 5333 y FA(\005)1839 5348 y Fv(\027)t Ft(\()p Fv(n)p Ft(\))1929 5539 y FA(7)p eop %%Page: 8 9 8 8 bop 0 83 a FA(De\014nition:)38 b(If)28 b(not)f(all)g(form)n(ulas)g (in)h(\005)1261 95 y Fv(\027)1330 83 y FA(are)f(literals)f(let)i(df)7 b(\(\005)2033 95 y Fv(\027)2075 83 y FA(\))28 b(b)r(e)g(the)g(\014rst)f (form)n(ula)g(in)h(\005)3026 95 y Fv(\027)3095 83 y FA(whic)n(h)g(is)f (not)h(a)f(literal.)0 282 y(Prop)r(osition)f(1.)36 b(\005)610 297 y Fv(\027)t Ft(\()p Fv(n)p Ft(\))768 282 y FA(=)23 b(\005)918 252 y Fp(0)941 282 y Fu(;)14 b(A;)g FA(\005)1139 252 y Fp(00)1205 282 y FA(&)23 b(rk)o(\()p Fu(A)p FA(\))h Fu(>)e FA(0)46 b(=)-14 b Fx(\))46 b(9)p Fu(k)26 b Fx(\025)d Fu(n)p FA(\()14 b(df)7 b(\(\005)2345 297 y Fv(\027)t Ft(\()p Fv(k)q Ft(\))2475 282 y FA(\))23 b(=)g Fu(A)14 b FA(\).)0 439 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f(the)h(n)n(um)n (b)r(er)f(of)h(logical)e(sym)n(b)r(ols)h Fx(^)p Fu(;)14 b Fx(_)p Fu(;)g Fx(8)p Fu(;)g Fx(9)27 b FA(o)r(ccurring)f(in)i(\005) 2809 409 y Fp(0)2833 439 y FA(.)0 597 y(Prop)r(osition)e(2.)0 734 y(\(a\))i(rk)o(\()p Fu(A)p FA(\))c(=)f(0)45 b(=)-14 b Fx(\))46 b Fu(A)23 b Fx(62)h(F)35 b FA(or)27 b Fx(:)p Fu(A)d Fx(62)f(F)8 b FA(,)0 871 y(\(b\))28 b Fu(A)200 883 y Ft(0)238 871 y Fx(^)p Fu(A)355 883 y Ft(1)416 871 y Fx(2)23 b(F)54 b FA(=)-14 b Fx(\))46 b Fu(A)850 883 y Ft(0)911 871 y Fx(2)23 b(F)36 b FA(or)26 b Fu(A)1248 883 y Ft(1)1309 871 y Fx(2)d(F)8 b FA(,)0 1007 y(\(c\))28 b Fu(A)191 1019 y Ft(0)229 1007 y Fx(_)p Fu(A)346 1019 y Ft(1)407 1007 y Fx(2)23 b(F)54 b FA(=)-14 b Fx(\))46 b Fu(A)841 1019 y Ft(0)901 1007 y Fx(2)24 b(F)35 b FA(and)28 b Fu(A)1299 1019 y Ft(1)1359 1007 y Fx(2)c(F)8 b FA(,)0 1144 y(\(d\))28 b Fx(8)p Fu(xB)f Fx(2)d(F)54 b FA(=)-14 b Fx(\))45 b(9)p Fu(u)23 b Fx(2)h Fr(V)n(a)n(rs)o FA(\()14 b Fu(B)1149 1156 y Fv(x)1191 1144 y FA(\()p Fu(y)s FA(\))24 b Fx(2)f(F)f FA(\),)0 1281 y(\(e\))28 b Fx(9)p Fu(xB)g Fx(2)23 b(F)54 b FA(=)-14 b Fx(\))46 b(8)p Fu(t)22 b Fx(2)i FA(T)-7 b(er)o(\()14 b Fu(B)1094 1293 y Fv(x)1136 1281 y FA(\()p Fu(t)p FA(\))24 b Fx(2)f(F)f FA(\).)0 1439 y(Pro)r(of:)0 1576 y(\(a\))28 b(rk)o(\()p Fu(A)p FA(\))c(=)f(0)f(&)h Fu(A)g Fx(2)h FA(\005)826 1591 y Fv(\027)t Ft(\()p Fv(n)p Ft(\))983 1576 y FA(&)f Fx(:)p Fu(A)g Fx(2)h FA(\005)1352 1591 y Fv(\027)t Ft(\()p Fv(m)p Ft(\))1550 1576 y Fx(\))47 b(8)p Fu(k)25 b Fx(\025)d FA(max)p Fx(f)p Fu(m;)14 b(n)p Fx(g)p FA(\()p Fu(A;)g Fx(:)p Fu(A)22 b Fx(2)i FA(\005)2692 1591 y Fv(\027)t Ft(\()p Fv(k)q Ft(\))2822 1576 y FA(\).)0 1733 y(Assume)k(that)g Fu(A)23 b Fx(2)g(F)36 b FA(with)28 b(rk)o(\()p Fu(A)p FA(\))c Fu(>)f FA(0.)36 b(Then)28 b Fu(A)23 b FA(=)g(df)7 b(\(\005)1913 1748 y Fv(\027)t Ft(\()p Fv(k)q Ft(\))2043 1733 y FA(\))28 b(for)f(some)g Fu(k)s FA(.)0 1891 y(\(b\))h Fu(A)c FA(=)e Fu(A)373 1903 y Ft(0)411 1891 y Fx(^)p Fu(A)528 1903 y Ft(1)566 1891 y FA(:)37 b(Then)27 b Fu(A)904 1903 y Fv(i)955 1891 y Fx(2)d FA(\005)1096 1906 y Fv(\027)t Ft(\()p Fv(k)q Ft(\))p Fp(\003)q(h)p Fv(i)p Fp(i)1365 1891 y FA(for)j Fu(i)c FA(=)f(0)p Fu(;)14 b FA(1,)27 b(and)g(therefore)g Fu(A)2374 1903 y Ft(0)2435 1891 y Fx(2)c FA(\005)2575 1906 y Fv(\027)t Ft(\()p Fv(k)q Ft(+1\))2817 1891 y FA(or)k Fu(A)2981 1903 y Ft(1)3041 1891 y Fx(2)d FA(\005)3182 1906 y Fv(\027)t Ft(\()p Fv(k)q Ft(+1\))3396 1891 y FA(.)0 2027 y(\(c\))k Fu(A)23 b FA(=)g Fu(A)364 2039 y Ft(0)402 2027 y Fx(_)p Fu(A)519 2039 y Ft(1)556 2027 y FA(:)37 b(Then)28 b Fu(A)895 2039 y Ft(0)933 2027 y Fu(;)14 b(A)1032 2039 y Ft(1)1092 2027 y Fx(2)24 b FA(\005)1233 2042 y Fv(\027)t Ft(\()p Fv(k)q Ft(\))p Fp(\003)q(h)p Fv(i)p Fp(i)1502 2027 y FA(for)j Fu(i)22 b FA(=)h(0)p Fu(;)14 b FA(1,)27 b(and)g(therefore)g Fu(A)2511 2039 y Ft(0)2571 2027 y Fx(2)d FA(\005)2712 2042 y Fv(\027)t Ft(\()p Fv(k)q Ft(+1\))2954 2027 y FA(and)j Fu(A)3177 2039 y Ft(1)3238 2027 y Fx(2)c FA(\005)3378 2042 y Fv(\027)t Ft(\()p Fv(k)q Ft(+1\))3592 2027 y FA(.)0 2164 y(\(d\))28 b Fu(A)c FA(=)e Fx(8)p Fu(xB)t FA(:)37 b(Then)28 b Fu(B)812 2176 y Fv(x)854 2164 y FA(\()p Fu(y)s FA(\))23 b Fx(2)g FA(\005)1125 2179 y Fv(\027)t Ft(\()p Fv(k)q Ft(\))p Fp(\003)q(h)p Fv(i)p Fp(i)1394 2164 y FA(for)k(some)g Fu(y)f Fx(2)e Fr(V)n(a)n(rs)o FA(.)0 2301 y(\(e\))k Fu(A)23 b FA(=)g Fx(9)p Fu(xB)t FA(:)38 b(Then)28 b Fx(8)p Fu(n)22 b Fx(\025)g Fu(k)s FA(\()14 b Fx(9)p Fu(xB)28 b Fx(2)23 b FA(\005)1362 2316 y Fv(\027)t Ft(\()p Fv(n)p Ft(\))1511 2301 y FA(\))28 b(\()p Fx(\003)p FA(\))f(.)0 2438 y(By)g(induction)h (on)g Fu(m)f FA(w)n(e)g(pro)n(v)n(e)f Fu(B)1124 2450 y Fv(x)1166 2438 y FA(\()p Fu(t)1228 2450 y Fv(m)1291 2438 y FA(\))e Fx(2)f(F)8 b FA(:)0 2575 y(Assume)24 b(that)h Fu(B)t FA(\()p Fu(t)610 2587 y Fv(i)638 2575 y FA(\))e Fx(2)h(F)32 b FA(for)24 b(all)g Fu(i)f(<)f(m)p FA(.)36 b(By)24 b(Prop)r(osition)f(1)h(and)g(\()p Fx(\003)p FA(\))h(there)f(is) g(an)g Fu(n)f Fx(\025)g Fu(k)k FA(suc)n(h)d(that)h(df)7 b(\(\005)3462 2590 y Fv(\027)t Ft(\()p Fv(n)p Ft(\))3596 2575 y FA(\))24 b(=)e Fx(9)p Fu(xB)0 2712 y FA(and)27 b Fx(8)p Fu(i)22 b(<)h(m)p Fx(9)p Fu(j)28 b Fx(\024)23 b Fu(n)p FA(\()14 b Fu(B)t FA(\()p Fu(t)841 2724 y Fv(i)869 2712 y FA(\))23 b Fx(2)g FA(\005)1064 2727 y Fv(\027)t Ft(\()p Fv(j)s Ft(\))1202 2712 y FA(\))28 b(\()p Fx(\003\003)p FA(\).)0 2849 y(By)f(de\014nition)h(of)g(\005)656 2864 y Fv(\027)t Ft(\()p Fv(n)p Ft(+1\))902 2849 y FA(w)n(e)f(get)1163 2781 y Fq(\000)1215 2849 y Fx(8)p Fu(j)g Fx(\024)22 b Fu(n)p FA(\()14 b Fu(B)t FA(\()p Fu(t)1635 2861 y Fv(m)1699 2849 y FA(\))23 b Fx(62)h FA(\005)1895 2864 y Fv(\027)t Ft(\()p Fv(j)s Ft(\))2033 2849 y FA(\))f Fx(\))g Fu(B)t FA(\()p Fu(t)2323 2861 y Fv(m)2387 2849 y FA(\))g Fx(2)g FA(\005)2582 2864 y Fv(\027)t Ft(\()p Fv(n)p Ft(+1\))2815 2781 y Fq(\001)2880 2849 y FA(and)28 b(th)n(us)g Fu(B)t FA(\()p Fu(t)3354 2861 y Fv(m)3417 2849 y FA(\))23 b Fx(2)h(F)8 b FA(.)0 3027 y Fx(jMj)23 b FA(:=)g(T)-7 b(er)o(,)55 b Fu(f)530 2997 y Fp(M)612 3027 y FA(\()p Fu(s)683 3039 y Ft(1)721 3027 y Fu(;)14 b(:::;)g(s)903 3039 y Fv(n)948 3027 y FA(\))23 b(:=)g Fu(f)9 b(s)1203 3039 y Ft(1)1240 3027 y Fu(:::s)1348 3039 y Fv(n)1393 3027 y FA(,)55 b Fu(p)1513 2997 y Fp(M)1595 3027 y FA(\()p Fu(s)1666 3039 y Ft(1)1704 3027 y Fu(;)14 b(:::;)g(s)1886 3039 y Fv(n)1931 3027 y FA(\))23 b(:)p Fx(,)g Fu(ps)2196 3039 y Ft(1)2233 3027 y Fu(;)14 b(:::;)g(s)2415 3039 y Fv(n)2483 3027 y Fx(62)24 b(F)0 3184 y FA(Let)k Fu(\030)f FA(:)c Fr(V)n(a)n(r)g Fx(!)g FA(T)-7 b(er)27 b(suc)n(h)h(that)f Fu(\030)t FA(\()p Fu(x)p FA(\))e(=)d Fu(x)28 b FA(for)g Fu(x)23 b Fx(2)g Fr(V)n(a)n(rs)p FA(.)0 3362 y(Prop)r(osition)j(3.)0 3499 y(\(a\))i Fu(t)164 3469 y Fp(M)246 3499 y FA([)p Fu(\030)t FA(])23 b(=)g Fu(t)28 b FA(for)f(eac)n(h)f Fu(t)d Fx(2)h FA(T)-7 b(er)o(,)0 3636 y(\(b\))28 b Fu(A)c Fx(2)f(F)54 b(\))46 b(M)23 b(6j)-14 b FA(=)23 b Fu(A)p FA([)p Fu(\030)t FA(].)0 3794 y(Pro)r(of)j(of)i(\(b\))g(b)n(y)g(induction)g(on)f(rk)o (\()p Fu(A)p FA(\):)0 3931 y Fx(8)p Fu(xB)g Fx(2)c(F)54 b(\))46 b Fu(B)568 3943 y Fv(x)610 3931 y FA(\()p Fu(y)s FA(\))23 b Fx(62)h(F)35 b FA(for)28 b(some)f Fu(y)e Fx(2)f Fr(V)n(a)n(rs)1598 3884 y Ft(IH)1593 3931 y Fx(\))46 b(M)23 b(6j)-14 b FA(=)22 b Fu(B)2004 3943 y Fv(x)2046 3931 y FA(\()p Fu(y)s FA(\)[)p Fu(\030)t FA(])47 b Fx(\))f(M)23 b(6j)-14 b FA(=)23 b(\()p Fx(8)p Fu(xB)t FA(\)[)p Fu(\030)t FA(].)0 4088 y Fx(9)p Fu(xB)28 b Fx(2)23 b(F)54 b(\))46 b Fu(B)568 4100 y Fv(x)610 4088 y FA(\()p Fu(t)p FA(\))24 b Fx(2)f(F)36 b FA(for)27 b(all)g Fu(t)c Fx(2)h FA(T)-7 b(er)1449 4041 y Ft(IH)1444 4088 y Fx(\))46 b(M)23 b(6j)-14 b FA(=)23 b Fu(B)1856 4100 y Fv(x)1898 4088 y FA(\()p Fu(t)p FA(\)[)p Fu(\030)t FA(])28 b(for)f(all)h Fu(t)23 b Fx(2)g FA(T)-7 b(er)2649 4034 y Ft(\(a\))2650 4088 y Fx(\))0 4225 y(\))46 b(M)23 b(6j)-14 b FA(=)23 b Fu(B)t FA([)p Fu(\030)479 4195 y Fv(t)475 4246 y(x)517 4225 y FA(])28 b(for)f(all)g Fu(t)c Fx(2)h FA(T)-7 b(er)45 b Fx(\))h(M)23 b(6j)-14 b FA(=)23 b(\()p Fx(9)p Fu(xB)t FA(\)[)p Fu(\030)t FA(].)0 4424 y(No)n(w,)k(since)h(\005)23 b Fx(\022)g(F)8 b FA(,)27 b(Prop)r(osition)f(3b)h(yields)h Fx(6j)-14 b FA(=)23 b(\005.)0 4602 y(No)n(w)37 b(assume)h Fx(j)-14 b FA(=)40 b(\005.)68 b(Then)38 b(the)g(ab)r(o)n(v)n(e)e (assumption)i Fz(\(A\))g FA(is)g(false,)i(and)e(it)g(follo)n(ws)f(that) h(for)g(ev)n(ery)e(0-1-sequence)0 4739 y(\()p Fu(i)61 4751 y Fv(n)106 4739 y FA(\))138 4751 y Fv(n)p Fp(2)p Ft(I)-11 b(N)318 4739 y FA(there)27 b(exists)g(an)h Fu(n)f FA(with)h(\005)1203 4754 y Fp(h)p Fv(i)1253 4762 y Fo(0)1286 4754 y Fv(;:::;i)1409 4762 y Fn(n)p Fk(\000)p Fo(1)1522 4754 y Fp(i)1576 4739 y Fx(2)c FA(AX.)0 4896 y(By)j(K\177)-42 b(onigs)27 b(Lemma)g(\(and)h(since)f(\005)1163 4908 y Fv(\027)1228 4896 y Fx(2)c FA(AX)28 b(implies)g(\005)1802 4911 y Fv(\027)t Fp(\003h)p Fv(i)p Fp(i)1978 4896 y Fx(2)23 b FA(AX)q(\))28 b(the)g(set)f Fx(f)p Fu(\027)h FA(:)23 b(\005)2732 4908 y Fv(\027)2797 4896 y Fx(62)g FA(AX)q Fx(g)k FA(is)h(\014nite.)0 5033 y(Let)g Fu(m)23 b FA(:=)f(max)p Fx(f)p Fu(l)r(h)p FA(\()p Fu(\027)5 b FA(\))23 b(:)g(\005)868 5045 y Fv(\027)932 5033 y Fx(62)h FA(AX)p Fx(g)18 b FA(+)g(1.)0 5191 y(By)27 b(induction)h(on)g Fu(m)18 b Fx(\000)727 5174 y(\001)792 5191 y Fu(l)r(h)p FA(\()p Fu(\027)5 b FA(\))28 b(one)f(easily)f(pro)n(v)n(es)g(PL1)c Fx(`)1866 5203 y Ft(0)1926 5191 y FA(\005)1988 5203 y Fv(\027)2057 5191 y FA(\(cf.)29 b(the)f Fy(Remark)e FA(follo)n(wing)h(the)h (de\014nition)g(of)f(\005)3677 5203 y Fv(\027)3719 5191 y FA(\).)0 5328 y(Hence)h(PL1)22 b Fx(`)471 5340 y Ft(0)531 5328 y FA(\005.)1929 5539 y(8)p eop %%Page: 9 10 9 9 bop 0 83 a Fx(x)p Fz(2)63 b(An)32 b(application)g(of)g(partial)g (cut)g(elimination;)c(pro)m(v)-5 b(ably)33 b(recursiv)m(e)g(functions)e (of)h(PRA)g(and)g FA(I\006)3733 95 y Ft(1)0 274 y Fw(The)27 b(axiom)h(system)f(PRA)g(of)h(primitiv)n(e)f(recursiv)n(e)f(arithmetic) 0 440 y FA(Inductiv)n(e)i(De\014nition)g(of)g(sets)f(PR)1127 404 y Fv(n)1200 440 y FA(of)h Fu(n)p FA(-ary)e(function)i(sym)n(b)r (ols)0 606 y(\(PR)g(1\))f Fz(0)327 576 y Fv(n)395 606 y Fx(2)c FA(PR)591 570 y Fv(n)659 606 y FA(\()p Fu(n)g Fx(\025)g FA(0\),)55 b Fr(S)24 b Fx(2)f FA(PR)1270 570 y Ft(1)1307 606 y FA(,)55 b(I)1415 576 y Fv(n)1415 628 y(i)1483 606 y Fx(2)24 b FA(PR)1679 570 y Fv(n)1748 606 y FA(\(1)f Fx(\024)f Fu(i)h Fx(\024)f Fu(n)p FA(\).)0 747 y(\(PR)28 b(2\))f Fu(h)c Fx(2)g FA(PR)546 711 y Fv(m)632 747 y FA(&)g Fu(g)760 759 y Ft(1)797 747 y Fu(;)14 b(:::;)g(g)980 759 y Fv(m)1065 747 y Fx(2)24 b FA(PR)1261 711 y Fv(n)1330 747 y FA(&)e Fu(m;)14 b(n)23 b Fx(\025)g FA(1)45 b(=)-14 b Fx(\))46 b FA(\()p Fx(\016)p Fu(hg)2117 759 y Ft(1)2153 747 y Fu(:::g)2262 759 y Fv(m)2325 747 y FA(\))24 b Fx(2)f FA(PR)2577 711 y Fv(n)2622 747 y FA(.)0 888 y(\(PR)28 b(3\))f Fu(g)f Fx(2)d FA(PR)541 852 y Fv(n)609 888 y FA(&)g Fu(h)f Fx(2)i FA(PR)964 852 y Fv(n)p Ft(+2)1139 888 y FA(=)-14 b Fx(\))46 b FA(\(R)p Fu(g)s(h)p FA(\))23 b Fx(2)g FA(PR)1754 852 y Fv(n)p Ft(+1)1883 888 y FA(.)0 1079 y(Abbreviation:)36 b(PR)23 b(:=)787 1017 y Fq(S)856 1104 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1021 1079 y FA(PR)1139 1043 y Fv(n)1184 1079 y FA(,)55 b Fz(0)23 b FA(:=)g Fz(0)1492 1049 y Ft(0)1557 1079 y FA(.)0 1245 y Fx(L)57 1257 y Ft(0)118 1245 y FA(:=)f(PR)c Fx([)h(f)p FA(=)o Fx(g)27 b FA(,)h(where)f(=)g(is)h(a)f(binary)g(relation)g(sym)n(b)r(ol)g (\(equalit)n(y\).)0 1442 y(The)h Fx(L)228 1454 y Ft(0)265 1442 y FA(-terms)f Fz(0)p Fu(;)h Fr(S)14 b Fz(0)p Fu(;)27 b Fr(S)14 b(S)g Fz(0)p Fu(;)g(::::)28 b FA(are)e(called)h(n)n(umerals.) 36 b(F)-7 b(or)27 b Fu(n)c Fx(2)g FA(I)-14 b(N)29 b(let)e Fu(n)p 2391 1455 50 4 v 23 w FA(:=)2649 1308 y Fv(n)2575 1359 y Fq(z)p 2612 1359 20 10 v 20 w(}|)p 2706 1359 V 20 w({)2575 1442 y Fr(S)14 b Fu(:::)g Fr(S)g Fz(0)p FA(.)0 1608 y Fu(T)49 1620 y Ft(0)109 1608 y FA(:=)27 b(set)h(of)f(all)h (closed)f Fx(L)866 1620 y Ft(0)903 1608 y FA(-terms.)0 1749 y(If)h Fu(t)23 b Fx(2)g 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y FA(\(G2\))28 b(\000)p Fu(;)14 b Fx(:)p FA(\()p Fu(s)p FA(=)p Fu(t)p FA(\))p Fu(;)g Fx(:)p Fu(A)695 4127 y Fv(x)737 4115 y FA(\()p Fu(s)p FA(\))p Fu(;)g(A)939 4127 y Fv(x)982 4115 y FA(\()p Fu(t)p FA(\))28 b(,)g(for)f(eac)n(h)g(atomic)g Fx(L)1798 4127 y Ft(0)1835 4115 y FA(-form)n(ula)g Fu(A)0 4256 y FA(\(S0\))h(\000)p Fu(;)14 b Fx(:)p FA(\()p Fr(S)g Fu(t)p FA(=)p Fz(0)p FA(\))0 4397 y(\(S1\))28 b(\000)p Fu(;)14 b Fx(:)p FA(\()p Fr(S)g Fu(s)p FA(=)g Fr(S)g Fu(t)p FA(\))p Fu(;)g(s)p FA(=)o Fu(t)0 4538 y FA(\(PR0\))27 b(\000)p Fu(;)14 b Fz(0)388 4508 y Fv(n)433 4538 y Fu(t)463 4550 y Ft(1)500 4538 y Fu(:::t)599 4550 y Fv(n)645 4538 y FA(=)o Fz(0)0 4680 y FA(\(PR1\))27 b(\000)p Fu(;)14 b FA(I)370 4649 y Fv(n)370 4701 y(i)415 4680 y Fu(t)445 4692 y Ft(1)483 4680 y Fu(:::t)582 4692 y Fv(n)627 4680 y FA(=)o Fu(t)721 4692 y Fv(i)0 4821 y FA(\(PR2\))27 b(\000)p Fu(;)14 b FA(\()p Fx(\016)p Fu(hg)502 4833 y Ft(1)539 4821 y Fu(:::g)648 4833 y Fv(m)710 4821 y FA(\))p Fu(t)772 4833 y Ft(1)810 4821 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5337 a(\000)p Fu(;)g Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)1421 5349 y Fv(x)1463 5337 y FA(\()p Fu(t)p FA(\))1623 5282 y(\()p Fu(F)35 b Fx(2)24 b Fr(QF)k FA(and)f Fu(x)d Fx(62)f FA(FV)r(\(\000)p Fu(;)14 b(F)2560 5294 y Fv(x)2602 5282 y FA(\()p Fu(t)p FA(\)\)\).)1929 5539 y(9)p eop %%Page: 10 11 10 10 bop 0 83 a FA(One)27 b(easily)g(sees)g(that)h(the)g(rules)f(of)g Fr(PRA)h FA(are)f(closed)g(under)g(substitution)h(\(cf.)h Fx(x)o FA(3\).)0 224 y(Therefore)d(1.6-1.9)g(apply)h(to)h Fr(PRA)f FA(with)h(\010)23 b(=)g Fr(QF)p FA(.)0 390 y(The)28 b(follo)n(wing)e(Lemma)i(sho)n(ws)e(the)i(T)-7 b(ait)28 b(system)f Fr(PRA)h FA(pro)n(v)n(es)d(exactly)i(the)h(logical)f (consequences)f(of)h(PRA.)0 556 y Fz(Lemma)j(2.1.)0 697 y Fr(PRA)23 b Fx(`)g FA(\000)46 b Fx(\()-14 b(\))46 b FA(PL1)22 b Fx(`)h(:)p Fu(A)919 709 y Ft(1)957 697 y Fu(;)14 b(:::;)g Fx(:)p Fu(A)1217 709 y Fv(n)1263 697 y Fu(;)g FA(\000)27 b(for)g(some)g Fu(A)1776 709 y Ft(1)1814 697 y Fu(;)14 b(:::;)g(A)2019 709 y Fv(n)2087 697 y Fx(2)24 b FA(PRA)o(.)0 863 y(Pro)r(of:)36 b(Exercise.)0 1079 y Fy(Some)27 b(sp)r(ecial)h(function)g(sym)n(b)r(ols)0 1245 y FA(There)k(are)f(function)h(sym)n(b)r(ols)f(+)p Fu(;)14 b Fx(\001)p Fu(;)g Fx(\000)1221 1229 y(\001)1298 1245 y(2)31 b FA(PR)1501 1209 y Ft(2)1571 1245 y FA(and)g Fr(p)n(rd)g Fx(2)g FA(PR)2082 1209 y Ft(1)2151 1245 y FA(suc)n(h)h(that)h(the)f(follo)n(wing)f(equations)g(are)g(axioms)g(of) 0 1386 y Fr(PRA)p FA(:)k Fu(s)10 b FA(+)g(0)22 b(=)g Fu(s)p FA(,)i Fu(s)10 b FA(+)g Fr(S)j Fu(t)23 b FA(=)g Fr(S)p FA(\()p Fu(s)10 b FA(+)g Fu(t)p FA(\),)24 b Fu(s)10 b Fx(\001)g FA(0)22 b(=)h(0,)h Fu(s)10 b Fx(\001)g Fr(S)j Fu(t)23 b FA(=)g Fu(s)p Fx(\001)p Fu(t)10 b FA(+)g Fu(s)p FA(,)24 b Fr(p)n(rd)13 b FA(0)23 b(=)f(0,)i Fr(p)n(rd)14 b(S)g Fu(t)23 b FA(=)g Fu(t)p FA(,)h Fu(s)10 b Fx(\000)2924 1370 y(\001)2980 1386 y FA(0)22 b(=)h Fu(s)p FA(,)h Fu(s)10 b Fx(\000)3288 1370 y(\001)3344 1386 y Fr(S)k Fu(t)23 b FA(=)g Fr(p)n(rd)o FA(\()p Fu(s)10 b Fx(\000)3759 1370 y(\001)3815 1386 y Fu(t)p FA(\).)0 1577 y Fz(Lemma)30 b(2.2.)77 b FA(The)28 b(follo)n(wing)f(form)n(ulas)f(are)h(pro)n(v)-5 b(able)26 b(in)i Fr(PRA)p FA(.)0 1719 y(\(a\))g Fu(x)23 b Fx(6)p FA(=)g(0)f Fx(!)i Fu(x)f FA(=)g Fr(S)14 b(p)n(rd)f Fu(x)0 1860 y FA(\(b\))28 b Fu(x)19 b FA(+)f Fu(y)26 b FA(=)d Fu(y)e FA(+)d Fu(x)h Fx(^)g FA(\()p Fu(x)g FA(+)f Fu(y)s FA(\))g(+)g Fu(z)26 b FA(=)d Fu(x)c FA(+)f(\()p Fu(y)j FA(+)d Fu(z)t FA(\))0 2001 y(\(c\))28 b Fu(x)19 b FA(+)f Fu(y)26 b FA(=)c(0)h Fx($)g Fu(x)h FA(=)e(0)c Fx(^)h Fu(y)26 b FA(=)c(0)0 2142 y(\(d\))28 b Fu(x)19 b Fx(\001)g Fu(y)26 b FA(=)c Fu(y)f Fx(\001)e Fu(x)g Fx(^)f FA(\()p Fu(x)i Fx(\001)e Fu(y)s FA(\))h Fx(\001)f Fu(z)26 b FA(=)d Fu(x)c Fx(\001)f FA(\()p Fu(y)k Fx(\001)c Fu(z)t FA(\))g Fx(^)h Fu(x)g Fx(\001)g FA(\()p Fu(y)i FA(+)d Fu(z)t FA(\))23 b(=)f Fu(x)d Fx(\001)g Fu(y)i FA(+)d Fu(x)h Fx(\001)f Fu(z)0 2283 y FA(\(e\))28 b Fu(x)19 b Fx(\001)g Fu(y)25 b FA(=)e(0)g Fx($)g Fu(x)g FA(=)g(0)18 b Fx(_)g Fu(y)26 b FA(=)d(0)0 2424 y(\(f)6 b(\))29 b Fr(S)14 b Fu(x)19 b Fx(\000)271 2408 y(\001)336 2424 y Fr(S)14 b Fu(y)25 b FA(=)e Fu(x)c Fx(\000)637 2408 y(\001)702 2424 y Fu(y)0 2565 y FA(\(g\))28 b Fu(x)19 b FA(+)f(1)k(=)h Fr(S)14 b Fu(x)19 b Fx(^)g Fu(x)f Fx(\001)h FA(1)k(=)f Fu(x)d Fx(^)g Fr(p)n(rd)14 b FA(1)22 b(=)h(0)18 b Fx(^)g Fu(x)h Fx(\000)1533 2549 y(\001)1597 2565 y FA(1)k(=)f Fr(p)n(rd)14 b Fu(x)56 b FA(\(where)27 b(1)c(:=)g Fr(S)13 b FA(0\))0 2731 y(Pro)r(of)26 b(\(sk)n(etc)n(h\):)0 2873 y(\(a\))i Fr(QF)p FA(-Ind:)37 b(0)23 b(=)f(0)46 b Fx(\))g FA(0)23 b Fx(6)p FA(=)f(0)h Fx(!)g FA(0)f(=)h Fr(S)14 b(p)n(rd)g FA(0.)36 b Fu(x)24 b FA(=)e Fr(p)n(rd)14 b(S)g Fu(x)47 b Fx(\))f Fr(S)14 b Fu(x)23 b FA(=)g Fr(S)14 b(p)n(rd)f(S)h Fu(x)p FA(.)0 3039 y(\(b\),\(d\))29 b(Pro)r(of)e(b)n(y)g (A)n(t-Ind,)h(i.e.)37 b(induction)28 b(with)g(atomic)f(induction)h (form)n(ula)f Fu(F)12 b FA(.)0 3205 y(\(c\))28 b(\\)p Fx( )p FA(":)36 b(0)18 b(+)g(0)23 b(=)f(0.)37 b(\\)p Fx(!)p FA(":)f Fu(y)26 b Fx(6)p FA(=)c(0)h Fx(!)g Fu(x)c FA(+)f Fu(y)1519 3150 y Ft(\(a\))1529 3205 y FA(=)33 b Fu(x)18 b FA(+)g Fr(S)c(p)n(rd)g Fu(y)26 b FA(=)c Fr(S)q FA(\()p Fu(x)d FA(+)f Fr(p)n(rd)c Fu(y)s FA(\))23 b Fx(6)p FA(=)f(0.)0 3371 y(\(e\))28 b(\\)p Fx( )p FA(":)36 b Fu(y)26 b FA(=)d(0)f Fx(!)h Fu(x)c Fx(\001)g Fu(y)25 b FA(=)e(0,)k Fu(x)d FA(=)e(0)h Fx(!)g Fu(x)c Fx(\001)f Fu(y)26 b FA(=)d Fu(y)e Fx(\001)d Fu(x)24 b FA(=)f(0;)0 3512 y(\\)p Fx(!)p FA(":)36 b Fu(x)24 b Fx(6)p FA(=)e(0)c Fx(^)h Fu(y)26 b Fx(6)p FA(=)c(0)h Fx(!)g Fu(x)c Fx(\001)f Fu(y)26 b FA(=)d Fu(x)c Fx(\001)f Fr(S)c(p)n(rd)g Fu(y)26 b FA(=)c Fu(x)d Fx(\001)g Fr(p)n(rd)13 b Fu(y)21 b FA(+)d Fu(x)24 b FA(=)f Fu(x)18 b Fx(\001)h Fr(p)n(rd)14 b Fu(y)21 b FA(+)d Fr(S)c(p)n(rd)f Fu(x)24 b FA(=)f Fr(S)p FA(\()p Fu(:::)p FA(\))h Fx(6)p FA(=)e(0.)0 3678 y(\(f)6 b(\))26 b(A)n(t-Ind:)35 b Fr(S)14 b Fu(x)e Fx(\000)561 3661 y(\001)619 3678 y Fr(S)i FA(0)22 b(=)h Fr(p)n(rd)p FA(\()p Fr(S)14 b Fu(x)e Fx(\000)1115 3661 y(\001)1173 3678 y FA(0\))23 b(=)f Fr(p)n(rd)14 b(S)g Fu(x)24 b FA(=)e Fu(x)i FA(=)e Fu(x)12 b Fx(\000)1940 3661 y(\001)1997 3678 y FA(0.)36 b Fr(S)14 b Fu(x)e Fx(\000)2238 3661 y(\001)2295 3678 y Fr(S)i(S)g Fu(y)26 b FA(=)d Fr(p)n(rd)p FA(\()p Fr(S)14 b Fu(x)e Fx(\000)2854 3661 y(\001)2912 3678 y Fr(S)i Fu(y)s FA(\))3071 3631 y Ft(IH)3075 3678 y FA(=)27 b Fr(p)n(rd)p FA(\()p Fu(x)12 b Fx(\000)3391 3661 y(\001)3449 3678 y Fu(y)s FA(\))23 b(=)f Fu(x)12 b Fx(\000)3716 3661 y(\001)3773 3678 y Fr(S)i Fu(y)s FA(.)0 3844 y(\(g\))26 b Fu(x)16 b FA(+)f Fr(S)f FA(0)22 b(=)h Fr(S)p FA(\()p Fu(x)16 b FA(+)f(0\))22 b(=)h Fr(S)14 b Fu(x)p FA(,)27 b Fu(x)16 b Fx(\001)f Fr(S)f FA(0)22 b(=)h Fu(x)16 b Fx(\001)f FA(0)g(+)f Fu(x)24 b FA(=)f(0)14 b(+)h Fu(x)24 b FA(=)e Fu(x)16 b FA(+)f(0)22 b(=)h Fu(x)p FA(,)k Fr(p)n(rd)13 b(S)h FA(0)23 b(=)g(0,)j Fu(x)15 b Fx(\000)2956 3827 y(\001)3018 3844 y Fr(S)f FA(0)22 b(=)h Fr(p)n(rd)p FA(\()p Fu(x)16 b Fx(\000)3458 3827 y(\001)3519 3844 y FA(0\))23 b(=)g Fr(p)n(rd)13 b Fu(x)p FA(.)0 4035 y Fz(Lemma)30 b(2.3.)105 b FA(The)28 b(follo)n(wing)e(form)n(ulas)h(are)f(pro)n(v)-5 b(able)27 b(in)g Fr(PRA)p FA(.)0 4176 y(\(a\))h Fu(x)23 b Fx(6)p FA(=)g(0)f Fx($)i FA(1)18 b Fx(\000)544 4159 y(\001)608 4176 y Fu(x)24 b FA(=)e(0)0 4317 y(\(b\))28 b Fu(y)22 b Fx(\000)222 4301 y(\001)286 4317 y Fu(x)i Fx(6)p FA(=)e(0)h Fx(!)g Fu(y)j FA(=)c Fu(x)d FA(+)f(\()p Fu(y)k Fx(\000)1034 4301 y(\001)1099 4317 y Fu(x)p FA(\))0 4458 y(\(c\))28 b Fu(y)21 b Fx(\000)212 4442 y(\001)277 4458 y FA(\()p Fu(x)e FA(+)f Fu(y)s FA(\))23 b(=)g(0)0 4599 y(\(d\))28 b(\()p Fu(x)20 b FA(+)e Fu(y)s FA(\))g Fx(\000)435 4583 y(\001)500 4599 y Fu(y)26 b FA(=)c Fu(x)0 4741 y FA(\(e\))28 b Fu(x)19 b Fx(\000)216 4724 y(\001)281 4741 y Fu(y)26 b FA(=)c(0)h Fx(!)g Fu(x)g FA(=)g Fu(y)e Fx(_)e Fu(y)i Fx(\000)983 4724 y(\001)1048 4741 y Fu(x)i Fx(6)p FA(=)g(0)0 4882 y(\(f)6 b(\))29 b Fu(x)19 b Fx(\000)211 4865 y(\001)276 4882 y Fu(y)26 b FA(=)c(0)h Fx(!)g Fu(y)j FA(=)c Fu(x)d FA(+)f(\()p Fu(y)k Fx(\000)1020 4865 y(\001)1084 4882 y Fu(x)p FA(\))0 5048 y(Pro)r(of:)0 5189 y(\(a\))28 b(By)f(A)n(t-Ind)h(w)n(e)f(obtain)g(0)18 b Fx(\000)996 5172 y(\001)1060 5189 y Fu(x)24 b FA(=)e(0)27 b([0)18 b Fx(\000)1392 5172 y(\001)1456 5189 y FA(0)23 b(=)f(0,)28 b(0)18 b Fx(\000)1782 5172 y(\001)1846 5189 y Fr(S)c Fu(x)24 b FA(=)f Fr(p)n(rd)o FA(\(0)c Fx(\000)2290 5172 y(\001)2354 5189 y Fu(x)p FA(\))24 b(=)f Fr(p)n(rd)14 b FA(0)22 b(=)h(0].)0 5330 y Fu(x)h Fx(6)p FA(=)e(0)h Fx(!)g FA(1)18 b Fx(\000)410 5313 y(\001)475 5330 y Fu(x)23 b FA(=)g Fr(S)14 b FA(0)k Fx(\000)774 5313 y(\001)838 5330 y Fr(S)c(p)n(rd)g Fu(x)24 b FA(=)e(0)c Fx(\000)1263 5313 y(\001)1328 5330 y Fr(p)n(rd)c Fu(x)23 b FA(=)g(0;)55 b Fu(x)23 b FA(=)g(0)f Fx(!)h FA(1)18 b Fx(\000)2142 5313 y(\001)2206 5330 y Fu(x)24 b FA(=)e(1)h Fx(6)p FA(=)g(0.)1908 5539 y(10)p eop %%Page: 11 12 11 11 bop 0 83 a FA(\(b\))28 b Fr(QF)q FA(-Ind:)37 b(1.)f Fu(y)26 b FA(=)c(0)c(+)g(\()p Fu(y)k Fx(\000)971 66 y(\001)1036 83 y FA(0\).)0 224 y(2.)36 b Fu(y)22 b Fx(\000)185 207 y(\001)249 224 y Fr(S)14 b Fu(x)24 b Fx(6)p FA(=)e(0)h Fx(\))g Fr(p)n(rd)p FA(\()p Fu(y)e Fx(\000)866 207 y(\001)930 224 y Fu(x)p FA(\))j Fx(6)p FA(=)f(0)f Fx(\))h Fu(y)e Fx(\000)1375 207 y(\001)1439 224 y Fu(x)j Fx(6)p FA(=)e(0)h Fx(\))g Fu(y)e Fx(\000)1851 207 y(\001)1916 224 y Fu(x)i FA(=)g Fr(S)14 b(p)n(rd)p FA(\()p Fu(y)21 b Fx(\000)2362 207 y(\001)2426 224 y Fu(x)p FA(\))j(=)f Fr(S)p FA(\()p Fu(y)e Fx(\000)2778 207 y(\001)2843 224 y Fr(S)14 b Fu(x)p FA(\),)0 364 y Fu(y)21 b Fx(\000)83 347 y(\001)148 364 y Fr(S)14 b Fu(x)23 b Fx(6)p FA(=)g(0)g Fx(\))g Fu(y)603 317 y Ft(IH)608 364 y FA(=)j Fu(x)19 b FA(+)f(\()p Fu(y)k Fx(\000)964 347 y(\001)1029 364 y Fu(x)p FA(\))h(=)g Fu(x)c FA(+)f Fr(S)p FA(\()p Fu(y)j Fx(\000)1530 347 y(\001)1594 364 y Fr(S)14 b Fu(x)p FA(\))24 b(=)f Fr(S)14 b Fu(x)19 b FA(+)f(\()p Fu(y)j Fx(\000)2169 347 y(\001)2234 364 y Fr(S)14 b Fu(x)p FA(\).)0 529 y(\(c\))28 b(A)n(t-Ind:)37 b(1.)g(0)18 b Fx(\000)614 512 y(\001)678 529 y FA(\()p Fu(x)i FA(+)e(0\))23 b(=)f(0)c Fx(\000)1125 512 y(\001)1190 529 y Fu(x)23 b FA(=)g(0.)0 669 y(2.)36 b Fr(S)14 b Fu(y)22 b Fx(\000)245 653 y(\001)309 669 y FA(\()p Fu(x)e FA(+)e Fr(S)c Fu(y)s FA(\))23 b(=)f Fr(S)14 b Fu(y)21 b Fx(\000)880 653 y(\001)945 669 y Fr(S)p FA(\()p Fu(x)e FA(+)f Fu(y)s FA(\))23 b(=)g Fu(y)e Fx(\000)1442 653 y(\001)1507 669 y FA(\()p Fu(x)e FA(+)f Fu(y)s FA(\))1787 622 y Ft(IH)1791 669 y FA(=)27 b(0.)0 834 y(\(d\))h(A)n(t-Ind:)37 b(1.)g(\()p Fu(x)19 b FA(+)f(0\))g Fx(\000)837 817 y(\001)901 834 y FA(0)23 b(=)f Fu(x)p FA(.)38 b(2.)e(\()p Fu(x)19 b FA(+)f Fr(S)c Fu(y)s FA(\))19 b Fx(\000)1619 817 y(\001)1684 834 y Fr(S)14 b Fu(y)25 b FA(=)e Fr(S)p FA(\()p Fu(x)c FA(+)f Fu(y)s FA(\))h Fx(\000)2241 817 y(\001)2306 834 y Fr(S)13 b Fu(y)26 b FA(=)d(\()p Fu(x)c FA(+)f Fu(y)s FA(\))g Fx(\000)2817 817 y(\001)2881 834 y Fu(y)2948 787 y Ft(IH)2952 834 y FA(=)27 b Fu(x)p FA(.)0 999 y(\(e\))h Fr(QF)q FA(-Ind:)36 b(1.)h Fu(x)19 b Fx(\000)636 982 y(\001)701 999 y FA(0)j(=)h(0)f Fx(!)h Fu(x)h FA(=)f(0.)0 1139 y(2.)36 b Fu(x)19 b Fx(\000)189 1123 y(\001)253 1139 y Fr(S)14 b Fu(y)26 b FA(=)c(0)h Fx(\))g Fr(p)n(rd)p FA(\()p Fu(x)c Fx(\000)869 1123 y(\001)934 1139 y Fu(y)s FA(\))k(=)g(0)f Fx(\))h Fu(x)c Fx(\000)1378 1123 y(\001)1443 1139 y Fu(y)26 b FA(=)c(0)c Fx(_)h Fu(x)g Fx(\000)1818 1123 y(\001)1883 1139 y Fu(y)26 b FA(=)c Fr(S)14 b FA(0)27 b([)h(since)f Fu(x)19 b Fx(\000)2508 1123 y(\001)2572 1139 y Fu(y)26 b Fx(6)p FA(=)d(0)f Fx(!)h Fu(x)c Fx(\000)2984 1123 y(\001)3049 1139 y Fu(y)26 b FA(=)c Fr(S)14 b(p)n(rd)p FA(\()p Fu(x)19 b Fx(\000)3494 1123 y(\001)3559 1139 y Fu(y)s FA(\))28 b(])0 1233 y Ft(I)p Fv(:)p Ft(H)p Fv(:)15 1280 y Fx(\))61 b Fu(x)23 b FA(=)g Fu(y)e Fx(_)e Fu(y)i Fx(\000)536 1263 y(\001)601 1280 y Fu(x)i Fx(6)p FA(=)g(0)18 b Fx(_)g Fu(x)h Fx(\000)980 1263 y(\001)1044 1280 y Fu(y)26 b FA(=)d Fr(S)13 b FA(0.)0 1420 y Fu(x)24 b FA(=)e Fu(y)k Fx(\))d Fr(S)14 b Fu(y)21 b Fx(\000)474 1404 y(\001)539 1420 y Fu(x)i FA(=)g(\(1)18 b(+)g Fu(y)s FA(\))h Fx(\000)988 1404 y(\001)1052 1420 y Fu(y)1119 1366 y Ft(\(d\))1131 1420 y FA(=)35 b(1)22 b Fx(6)p FA(=)h(0.)0 1561 y Fu(y)e Fx(\000)83 1544 y(\001)148 1561 y Fu(x)i Fx(6)p FA(=)g(0)371 1507 y Ft(\(b\))373 1561 y Fx(\))j Fu(y)g FA(=)d Fu(x)c FA(+)f(\()p Fu(y)j Fx(\000)901 1544 y(\001)966 1561 y Fu(x)p FA(\))j Fx(\))f Fr(S)14 b Fu(y)21 b Fx(\000)1318 1544 y(\001)1383 1561 y Fu(x)i FA(=)g(\()p Fu(x)c FA(+)f Fr(S)p FA(\()p Fu(y)j Fx(\000)1884 1544 y(\001)1948 1561 y Fu(x)p FA(\)\))f Fx(\000)2100 1544 y(\001)2164 1561 y Fu(x)2235 1507 y Ft(\(d\))2247 1561 y FA(=)35 b Fr(S)p FA(\()p Fu(y)21 b Fx(\000)2508 1544 y(\001)2573 1561 y Fu(x)p FA(\))j Fx(6)p FA(=)e(0.)0 1701 y Fu(x)d Fx(\000)87 1685 y(\001)152 1701 y Fu(y)25 b FA(=)e Fr(S)14 b FA(0)23 b Fx(\))g Fu(x)c Fx(\000)624 1685 y(\001)688 1701 y Fu(y)26 b Fx(6)p FA(=)d(0)907 1647 y Ft(\(b\))910 1701 y Fx(\))j Fu(x)d FA(=)g Fu(y)e FA(+)d(\()p Fu(x)h Fx(\000)1442 1685 y(\001)1506 1701 y Fu(y)s FA(\))k(=)g Fu(y)e FA(+)d Fr(S)c FA(0)23 b(=)f Fr(S)14 b Fu(y)s FA(.)0 1866 y(\(f)6 b(\))29 b Fu(x)24 b FA(=)e Fu(y)k Fx(\))d Fu(y)j FA(=)c Fu(x)i FA(=)f Fu(x)18 b FA(+)g(\()p Fu(x)i Fx(\000)1036 1849 y(\001)1100 1866 y Fu(x)p FA(\))k(=)f Fu(x)c FA(+)f(\()p Fu(y)j Fx(\000)1555 1849 y(\001)1620 1866 y Fu(x)p FA(\);)56 b Fu(y)21 b Fx(\000)1862 1849 y(\001)1926 1866 y Fu(x)j Fx(6)p FA(=)e(0)2149 1812 y Ft(\(b\))2152 1866 y Fx(\))j Fu(y)h FA(=)d Fu(x)c FA(+)f(\()p Fu(y)j Fx(\000)2679 1849 y(\001)2744 1866 y Fu(x)p FA(\).)0 2079 y Fz(Lemma)30 b(2.4.)0 2220 y FA(\(a\))55 b(F)-7 b(or)27 b(eac)n(h)g Fu(A)d Fx(2)f Fr(QF)28 b FA(there)f(exists)h(a)f(term)g Fu(t)1536 2232 y Fv(A)1618 2220 y FA(suc)n(h)g(that)h Fr(PRA)23 b Fx(`)g Fu(A)g Fx($)h Fu(t)2466 2232 y Fv(A)2543 2220 y FA(=)e(0.)0 2360 y(\(b\))51 b(F)-7 b(or)27 b(eac)n(h)g Fu(A)d Fx(2)f Fr(QF)28 b FA(and)f(terms)h Fu(t)1220 2372 y Ft(0)1257 2360 y Fu(;)14 b(t)1324 2372 y Ft(1)1389 2360 y FA(there)27 b(exist)h(a)f(term)g Fu(d)2108 2372 y Fv(A)2163 2360 y FA(\()p Fu(t)2225 2372 y Ft(0)2262 2360 y Fu(;)14 b(t)2329 2372 y Ft(1)2366 2360 y FA(\))28 b(suc)n(h)g(that)161 2501 y Fr(PRA)c Fx(`)e Fu(A)i Fx(!)f Fu(d)655 2513 y Fv(A)709 2501 y FA(\()p Fu(t)771 2513 y Ft(0)809 2501 y Fu(;)14 b(t)876 2513 y Ft(1)913 2501 y FA(\))23 b(=)g Fu(t)1086 2513 y Ft(0)1151 2501 y FA(and)k Fr(PRA)c Fx(`)g(:)p Fu(A)g Fx(!)h Fu(d)1861 2513 y Fv(A)1915 2501 y FA(\()p Fu(t)1977 2513 y Ft(0)2014 2501 y Fu(;)14 b(t)2081 2513 y Ft(1)2119 2501 y FA(\))23 b(=)f Fu(t)2291 2513 y Ft(1)2329 2501 y FA(.)0 2641 y(\(c\))60 b(F)-7 b(or)27 b(eac)n(h)g(PR-term)g Fu(t)h FA(with)g(FV)q(\()p Fu(t)p FA(\))c Fx(\022)f(f)p Fu(x)1500 2653 y Ft(1)1537 2641 y Fu(;)14 b(:::;)g(x)1727 2653 y Fv(n)1773 2641 y Fx(g)27 b FA(there)g(is)h(a)f(function)h(sym)n(b)r(ol)f Fu(f)32 b Fx(2)23 b FA(PR)3085 2605 y Fv(n)3158 2641 y FA(suc)n(h)k(that)161 2782 y Fr(PRA)d Fx(`)e Fu(t)h FA(=)g Fu(f)9 b(x)658 2794 y Ft(1)695 2782 y Fu(:::x)811 2794 y Fv(n)857 2782 y FA(.)0 2947 y(Pro)r(of:)0 3087 y(\(a\))28 b Fu(t)164 3099 y Fv(r)r Ft(=)p Fv(s)306 3087 y FA(:=)23 b(\()p Fu(s)18 b Fx(\000)527 3070 y(\001)592 3087 y Fu(r)r FA(\))i(+)e(\()p Fu(r)j Fx(\000)877 3070 y(\001)942 3087 y Fu(s)p FA(\))55 b([)p Fu(s)23 b Fx(6)p FA(=)g Fu(r)1304 3040 y Ft(2)p Fv(:)p Ft(3e)1320 3087 y Fx(!)39 b Fu(s)19 b Fx(\000)1521 3070 y(\001)1585 3087 y Fu(r)26 b Fx(6)p FA(=)d(0)18 b Fx(_)h Fu(r)i Fx(\000)1949 3070 y(\001)2014 3087 y Fu(s)i Fx(6)p FA(=)f(0];)0 3228 y Fu(t)30 3243 y Fp(:)p Ft(\()p Fv(r)r Ft(=)p Fv(s)p Ft(\))269 3228 y FA(:=)g(1)c Fx(\000)461 3211 y(\001)525 3228 y Fu(t)555 3240 y Fv(r)r Ft(=)p Fv(s)674 3228 y FA(;)56 b Fu(t)783 3240 y Fv(A)p Fp(^)p Fv(B)957 3228 y FA(:=)23 b Fu(t)1098 3240 y Fv(A)1171 3228 y FA(+)18 b Fu(t)1284 3240 y Fv(B)1341 3228 y FA(;)55 b Fu(t)1449 3240 y Fv(A)p Fp(_)p Fv(B)1624 3228 y FA(:=)23 b Fu(t)1765 3240 y Fv(A)1837 3228 y Fx(\001)c Fu(t)1909 3240 y Fv(B)1966 3228 y FA(.)0 3392 y(\(b\))28 b(Let)g Fu(d)330 3404 y Fv(A)384 3392 y FA(\()p Fu(t)446 3404 y Ft(0)484 3392 y Fu(;)14 b(t)551 3404 y Ft(1)588 3392 y FA(\))23 b(:=)g Fu(t)784 3404 y Ft(0)821 3392 y Fx(\001)p FA(\(1)c Fx(\000)958 3376 y(\001)1022 3392 y Fu(t)1052 3404 y Fv(A)1106 3392 y FA(\))g(+)f Fu(t)1270 3404 y Ft(1)1307 3392 y Fx(\001)q FA(\(1)g Fx(\000)1444 3376 y(\001)1509 3392 y FA(\(1)g Fx(\000)1622 3376 y(\001)1687 3392 y Fu(t)1717 3404 y Fv(A)1771 3392 y FA(\)\).)0 3557 y(\(c\))28 b(cf.)g(Logic)f(I.)0 3746 y Fz(Abbreviations.)0 3887 y Fu(s)c Fx(\024)g Fu(t)g FA(:=)f(\()p Fu(s)d Fx(\000)424 3870 y(\001)489 3887 y Fu(t)k FA(=)f(0\);)56 b Fu(s)23 b(<)f(t)h FA(:=)g(\()p Fu(s)g Fx(\024)g Fu(t)18 b Fx(^)h Fu(s)k Fx(6)p FA(=)g Fu(t)p FA(\).)0 4027 y Fx(9)p Fu(x)h Fx(\024)e Fu(tA)i FA(:=)e Fx(9)p Fu(x)p FA(\()p Fu(x)j Fx(\024)d Fu(t)d Fx(^)g Fu(A)p FA(\))28 b(and)f Fx(8)p Fu(x)c Fx(\024)f Fu(tA)i FA(:=)e Fx(8)p Fu(x)p FA(\()p Fu(x)i Fx(\024)e Fu(t)h Fx(!)h Fu(A)p FA(\))k(if)g Fu(x)23 b Fx(62)h FA(FV)q(\()p Fu(t)p FA(\).)0 4216 y Fz(Lemma)30 b(2.5.)77 b FA(The)28 b(follo)n(wing)f(form)n(ulas)f(are)h(pro)n(v)-5 b(able)26 b(in)i Fr(PRA)p FA(.)0 4357 y(\(a\))g Fu(x)23 b Fx(\024)g Fu(y)j Fx($)d Fu(y)i FA(=)e Fu(x)c FA(+)f(\()p Fu(y)j Fx(\000)884 4340 y(\001)948 4357 y Fu(x)p FA(\))j Fx($)f(9)p Fu(z)t FA(\()p Fu(y)j FA(=)c Fu(x)d FA(+)f Fu(z)t FA(\);)0 4497 y(\(b\))28 b Fu(x)c Fx(\024)f Fu(x)18 b Fx(^)h FA(\()p Fu(x)24 b Fx(\024)f Fu(y)e Fx(^)e Fu(y)25 b Fx(\024)e Fu(x)g Fx(!)h Fu(x)f FA(=)g Fu(y)s FA(\);)0 4638 y(\(c\))28 b Fu(x)c Fx(\024)e Fu(y)f Fx(^)e Fu(y)26 b Fx(\024)d Fu(z)j Fx(!)d Fu(x)h Fx(\024)e Fu(z)t FA(;)0 4778 y(\(d\))28 b Fu(x)c Fx(\024)f Fu(y)e Fx(_)d Fu(y)26 b Fx(\024)d Fu(x)p FA(;)0 4919 y(\(e\))28 b(0)23 b Fx(\024)f Fu(x)p FA(;)0 5059 y(\(f)6 b(\))29 b Fu(x)24 b(<)e(y)k Fx($)d(:)p FA(\()p Fu(y)j Fx(\024)d Fu(x)p FA(\);)0 5200 y(\(g\))28 b Fu(x)23 b(<)g Fr(S)14 b Fu(y)26 b Fx($)d Fu(x)g Fx(\024)g Fu(y)s FA(;)0 5340 y(\(h\))28 b(\()p Fu(x)c(<)f(y)e Fx(^)e Fu(y)25 b Fx(\024)e Fu(z)t FA(\))18 b Fx(_)h FA(\()p Fu(x)24 b Fx(\024)e Fu(y)f Fx(^)e Fu(y)26 b(<)d(z)t FA(\))f Fx(!)h Fu(x)h(<)e(z)t FA(.)1908 5539 y(11)p eop %%Page: 12 13 12 12 bop 0 83 a FA(Pro)r(of:)36 b(\(a\))27 b Fu(x)19 b Fx(\000)481 66 y(\001)545 83 y Fu(y)26 b FA(=)c(0)h Fx(!)g Fu(y)j FA(=)d Fu(x)18 b FA(+)g(\()p Fu(y)k Fx(\000)1289 66 y(\001)1354 83 y Fu(x)p FA(\))h Fx(!)h(9)p Fu(z)t FA(\()p Fu(y)h FA(=)e Fu(x)18 b FA(+)h Fu(z)t FA(\).)36 b Fu(y)26 b FA(=)c Fu(x)d FA(+)f Fu(z)27 b Fx(!)c Fu(x)c Fx(\000)2683 66 y(\001)2747 83 y Fu(y)26 b FA(=)d Fu(x)c Fx(\000)2989 66 y(\001)3053 83 y FA(\()p Fu(z)j FA(+)c Fu(x)p FA(\))24 b(=)f(0.)0 224 y(\(b\))28 b Fu(x)19 b Fx(\000)225 208 y(\001)290 224 y Fu(x)24 b FA(=)e Fu(x)d Fx(\000)535 208 y(\001)600 224 y FA(\(0)f(+)g Fu(x)p FA(\))24 b(=)f(0.)36 b Fu(x)19 b Fx(\000)1154 208 y(\001)1219 224 y Fu(y)25 b FA(=)e(0)g Fx(!)g Fu(x)g FA(=)g Fu(y)e Fx(_)e Fu(y)i Fx(\000)1921 208 y(\001)1986 224 y Fu(x)i Fx(6)p FA(=)g(0,)k(hence)h Fu(x)19 b Fx(\000)2554 208 y(\001)2618 224 y Fu(y)26 b FA(=)d(0)f(=)h Fu(y)e Fx(\000)3008 208 y(\001)3073 224 y Fu(x)i Fx(!)h Fu(x)f FA(=)g Fu(y)s FA(.)0 365 y(\(c\))28 b Fu(x)c Fx(\024)e Fu(y)f Fx(^)e Fu(y)26 b Fx(\024)d Fu(z)j Fx(\))d(9)p Fu(u;)14 b(v)s FA(\()p Fu(x)19 b FA(+)f Fu(u)23 b FA(=)f Fu(y)f Fx(^)e Fu(y)i FA(+)d Fu(v)27 b FA(=)22 b Fu(z)t FA(\))h Fx(\))g(9)p Fu(u;)14 b(v)s FA(\()p Fu(z)26 b FA(=)d Fu(x)c FA(+)f Fu(u)g FA(+)g Fu(v)s FA(\))23 b Fx(\))g Fu(x)h Fx(\024)f Fu(z)t FA(.)0 506 y(\(d\))28 b Fx(:)p FA(\()p Fu(x)d Fx(\024)d Fu(y)s FA(\))h Fx(\))g Fu(x)c Fx(\000)676 490 y(\001)741 506 y Fu(y)26 b Fx(6)p FA(=)c(0)h Fx(\))g Fu(x)g FA(=)g Fu(y)e FA(+)d(\()p Fu(x)h Fx(\000)1489 490 y(\001)1553 506 y Fu(y)s FA(\))k Fx(\))g Fu(y)j Fx(\024)d Fu(x)p FA(.)0 648 y(\(e\))28 b Fu(x)c FA(=)e(0)c(+)g Fu(x)p FA(.)0 789 y(\(f)6 b(\))29 b Fu(x)24 b(<)e(y)k Fx($)d Fu(x)h Fx(\024)e Fu(y)f Fx(^)e Fu(x)24 b Fx(6)p FA(=)e Fu(y)k Fx($)d Fu(x)c Fx(\000)1167 772 y(\001)1232 789 y Fu(y)26 b FA(=)c(0)c Fx(^)h Fu(x)k Fx(6)p FA(=)g Fu(y)1769 734 y Ft(\(e\))1768 789 y Fx(!)46 b Fu(y)21 b Fx(\000)1980 772 y(\001)2045 789 y Fu(x)i Fx(6)p FA(=)g(0)g Fx($)g(:)p FA(\()p Fu(y)j Fx(\024)d Fu(x)p FA(\).)0 930 y Fu(y)e Fx(\000)83 913 y(\001)148 930 y Fu(x)i Fx(6)p FA(=)g(0)g Fx(!)g Fu(x)g Fx(6)p FA(=)g Fu(y)e Fx(^)e Fu(y)26 b FA(=)c Fu(x)d FA(+)f(\()p Fu(y)k Fx(\000)1190 913 y(\001)1254 930 y Fu(x)p FA(\);)29 b Fu(y)d FA(=)c Fu(x)d FA(+)f Fu(z)26 b Fx(!)d Fu(x)c Fx(\000)1947 913 y(\001)2011 930 y Fu(y)26 b FA(=)d Fu(x)18 b Fx(\000)2253 913 y(\001)2317 930 y FA(\()p Fu(x)h FA(+)f Fu(z)t FA(\))23 b(=)g(0.)0 1071 y(\(g\))e(\\)p Fx(!)p FA(":)32 b Fu(x)23 b(<)g Fr(S)14 b Fu(y)26 b Fx(\))d(:)p FA(\()p Fr(S)14 b Fu(y)26 b Fx(\024)d Fu(x)p FA(\))h Fx(\))f Fu(u)f FA(:=)h Fr(S)14 b Fu(y)7 b Fx(\000)1561 1054 y(\001)1611 1071 y Fu(x)23 b Fx(6)p FA(=)g(0)g Fx(\))g Fr(S)14 b Fu(y)25 b FA(=)e Fu(x)t FA(+)t Fu(u)g FA(=)f Fu(x)t FA(+)t Fr(S)14 b(p)n(rd)g Fu(u)23 b FA(=)f Fr(S)p FA(\()p Fu(x)t FA(+)t Fr(p)n(rd)14 b Fu(u)p FA(\))23 b Fx(\))g Fu(y)j FA(=)d Fu(x)t FA(+)t Fr(p)n(rd)13 b Fu(u)p FA(.)0 1212 y(\\)p Fx( )p FA(":)36 b Fu(x)19 b Fx(\000)313 1196 y(\001)378 1212 y Fu(y)25 b FA(=)e(0)g Fx(!)g Fu(x)c Fx(\000)790 1196 y(\001)854 1212 y Fr(S)14 b Fu(y)26 b FA(=)d Fr(p)n(rd)p FA(\()p Fu(x)c Fx(\000)1300 1196 y(\001)1365 1212 y Fu(y)s FA(\))k(=)f(0.)64 b Fu(x)24 b FA(=)f Fr(S)13 b Fu(y)22 b Fx(^)c Fu(x)24 b Fx(\024)f Fu(y)i Fx(!)e FA(1)g(=)g(\(1)18 b(+)g Fu(y)s FA(\))g Fx(\000)2809 1196 y(\001)2873 1212 y Fu(y)26 b FA(=)c Fr(S)14 b Fu(y)22 b Fx(\000)3171 1196 y(\001)3235 1212 y Fu(y)k FA(=)d(0.)0 1353 y(\(h\))28 b Fu(x)c(<)f(y)e Fx(^)d Fu(y)26 b Fx(\024)d Fu(z)j Fx(\))d Fu(x)h Fx(\024)e Fu(y)g Fx(^)c Fu(y)26 b Fx(\024)d Fu(z)j Fx(\))d Fu(x)h Fx(\024)e Fu(z)t FA(.)36 b Fu(x)24 b FA(=)f Fu(z)e Fx(^)e Fu(y)26 b Fx(\024)c Fu(z)27 b Fx(\))c Fu(y)j Fx(\024)c Fu(x)i Fx(\))f(:)p FA(\()p Fu(x)h(<)f(y)s FA(\).)0 1494 y Fu(x)h Fx(\024)e Fu(y)f Fx(^)e Fu(y)26 b(<)c(z)27 b Fx(\))c Fu(x)g Fx(\024)g Fu(y)e Fx(^)e Fu(y)26 b Fx(\024)c Fu(z)27 b Fx(\))c Fu(x)g Fx(\024)g Fu(z)t FA(.)36 b Fu(x)24 b FA(=)e Fu(z)g Fx(^)d Fu(x)k Fx(\024)g Fu(y)j Fx(\))d Fu(z)j Fx(\024)d Fu(y)i Fx(\))e(:)p FA(\()p Fu(y)k(<)22 b(z)t FA(\).)0 1710 y Fz(Lemma)30 b(2.6)c FA(\(P)n(airing\))0 1851 y(There)h(are)g(function)h(sym)n(b)r(ols)f Fu(\031)f Fx(2)d FA(PR)1290 1815 y Ft(2)1327 1851 y FA(,)28 b Fu(\031)1425 1863 y Ft(1)1463 1851 y Fu(;)14 b(\031)1547 1863 y Ft(2)1607 1851 y Fx(2)24 b FA(PR)1803 1815 y Ft(1)1868 1851 y FA(suc)n(h)j(that)h Fr(PRA)23 b Fx(`)g Fu(\031)2541 1863 y Fv(i)2569 1851 y Fu(\031)s(x)2666 1863 y Ft(1)2704 1851 y Fu(x)2751 1863 y Ft(2)2812 1851 y FA(=)f Fu(x)2946 1863 y Fv(i)2993 1851 y Fx(^)d Fu(\031)s(\031)3164 1863 y Ft(1)3202 1851 y Fu(x\031)3296 1863 y Ft(2)3334 1851 y Fu(x)k FA(=)g Fu(x)p FA(.)0 2017 y(Pro)r(of:)0 2159 y(W)-7 b(e)28 b(argue)e (informally)-7 b(,)27 b(but)h(so)f(that)h(all)g(steps)f(are)g(easily)f (formalizable)h(in)h Fr(PRA)p FA(.)0 2325 y Fu(\031)s FA(\()p Fu(a;)14 b(b)p FA(\))23 b(:=)g Fu(f)9 b FA(\()p Fu(a)18 b FA(+)g Fu(b)p FA(\))h(+)f Fu(b)27 b FA(with)h Fu(f)9 b FA(\()p Fu(n)p FA(\))23 b(:=)1312 2262 y Fq(P)1399 2350 y Fv(i)p Fp(\024)p Fv(n)1534 2325 y Fu(i)k FA(\(i.e.,)h Fu(f)9 b FA(\(0\))23 b(=)g(0,)k Fu(f)9 b FA(\()p Fu(n)p FA(+)o(1\))23 b(=)g Fu(f)9 b FA(\()p Fu(n)p FA(\))18 b(+)g Fu(n)g FA(+)g(1\).)0 2514 y Fu(h)p FA(\(0\))23 b(:=)g(0,)k Fu(h)p FA(\()p Fu(k)21 b FA(+)d(1\))23 b(:=)815 2397 y Fq(\032)891 2468 y Fu(h)p FA(\()p Fu(k)s FA(\))c(+)f(1)82 b(if)28 b Fu(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)s FA(\))19 b(+)f(1\))23 b Fx(\024)f Fu(k)s FA(+1)891 2560 y Fu(h)p FA(\()p Fu(k)s FA(\))226 b(otherwise)0 2688 y(\(1\))28 b Fu(f)9 b(h)p FA(\()p Fu(k)s FA(\))23 b Fx(\024)f Fu(k)k(<)d(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)s FA(\))18 b(+)g(1\))0 2854 y(Induction)28 b(step:)0 2995 y(Case)f(1.)36 b Fu(f)9 b(h)p FA(\()p Fu(k)21 b FA(+)d(1\))23 b(=)g Fu(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)s FA(\))18 b(+)g(1\))23 b Fx(\024)g Fu(k)e FA(+)d(1)1500 2926 y Ft(IH)1504 2995 y Fx(\024)27 b Fu(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)s FA(\))18 b(+)g(1\))23 b(=)g Fu(f)9 b(h)p FA(\()p Fu(k)21 b FA(+)d(1\))23 b Fu(<)f(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)22 b FA(+)c(1\))g(+)g(1\).)0 3137 y(Case)27 b(2.)36 b Fu(f)9 b(h)p FA(\()p Fu(k)21 b FA(+)d(1\))23 b(=)g Fu(f)9 b(h)p FA(\()p Fu(k)s FA(\))993 3067 y Ft(IH)997 3137 y Fx(\024)27 b Fu(k)f(<)c(k)g FA(+)c(1)k Fu(<)h(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)s FA(\))18 b(+)g(1\))23 b(=)g Fu(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)21 b FA(+)d(1\))g(+)g(1\).)0 3303 y Fy(De\014nition.)74 b Fu(\031)502 3315 y Ft(2)540 3303 y FA(\()p Fu(k)s FA(\))24 b(:=)e Fu(k)f Fx(\000)870 3286 y(\001)934 3303 y Fu(f)9 b(h)p FA(\()p Fu(k)s FA(\),)28 b Fu(\031)1240 3315 y Ft(1)1277 3303 y FA(\()p Fu(k)s FA(\))c(:=)f Fu(h)p FA(\()p Fu(k)s FA(\))18 b Fx(\000)1719 3286 y(\001)1784 3303 y Fu(\031)1831 3315 y Ft(2)1869 3303 y FA(\()p Fu(k)s FA(\).)0 3469 y Fu(k)26 b(<)d(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)s FA(\))18 b(+)g(1\))23 b(=)g Fu(f)9 b FA(\()p Fu(h)p FA(\()p Fu(k)s FA(\)\))18 b(+)g Fu(h)p FA(\()p Fu(k)s FA(\))h(+)f(1)23 b(&)f Fu(f)9 b(h)p FA(\()p Fu(k)s FA(\))23 b Fx(\024)g Fu(k)49 b Fx(\))d Fu(\031)2055 3481 y Ft(2)2093 3469 y FA(\()p Fu(k)s FA(\))23 b Fx(\024)g Fu(h)p FA(\()p Fu(k)s FA(\))46 b Fx(\))g Fu(\031)2694 3481 y Ft(1)2732 3469 y FA(\()p Fu(k)s FA(\))19 b(+)f Fu(\031)2991 3481 y Ft(2)3028 3469 y FA(\()p Fu(k)s FA(\))24 b(=)e Fu(h)p FA(\()p Fu(k)s FA(\).)0 3635 y Fu(\031)s FA(\()p Fu(\031)129 3647 y Ft(1)167 3635 y FA(\()p Fu(k)s FA(\))p Fu(;)14 b(\031)361 3647 y Ft(2)399 3635 y FA(\()p Fu(k)s FA(\)\))24 b(=)e Fu(f)9 b FA(\()p Fu(\031)781 3647 y Ft(1)819 3635 y FA(\()p Fu(k)s FA(\))19 b(+)f Fu(\031)1078 3647 y Ft(2)1115 3635 y FA(\()p Fu(k)s FA(\)\))h(+)f Fu(\031)1406 3647 y Ft(2)1444 3635 y FA(\()p Fu(k)s FA(\))24 b(=)e Fu(f)9 b(h)p FA(\()p Fu(k)s FA(\))18 b(+)g(\()p Fu(k)k Fx(\000)2092 3618 y(\001)2157 3635 y Fu(f)9 b(h)p FA(\()p Fu(k)s FA(\)\))23 b(=)g Fu(k)s FA(.)0 3801 y(Let)28 b Fu(k)e FA(:=)c Fu(\031)s FA(\()p Fu(a;)14 b(b)p FA(\).)38 b(Then)27 b Fu(f)9 b FA(\()p Fu(a)19 b FA(+)f Fu(b)p FA(\))23 b Fx(\024)f Fu(k)k(<)d(f)9 b FA(\()p Fu(a)18 b FA(+)g Fu(b)g FA(+)g(1\))28 b(and)f(therefore)g Fu(h)p FA(\()p Fu(k)s FA(\))c(=)g Fu(a)18 b FA(+)g Fu(b)p FA(.)0 3967 y Fu(\031)47 3979 y Ft(2)85 3967 y FA(\()p Fu(k)s FA(\))23 b(=)g(\()p Fu(f)9 b FA(\()p Fu(a)18 b FA(+)g Fu(b)p FA(\))h(+)f Fu(b)p FA(\))g Fx(\000)842 3950 y(\001)907 3967 y Fu(f)9 b FA(\()p Fu(a)18 b FA(+)g Fu(b)p FA(\))23 b(=)g Fu(b)k FA(and)g Fu(\031)1584 3979 y Ft(1)1622 3967 y FA(\()p Fu(k)s FA(\))c(=)g(\()p Fu(a)c FA(+)f Fu(b)p FA(\))g Fx(\000)2128 3950 y(\001)2193 3967 y Fu(b)k FA(=)h Fu(a)p FA(.)0 4183 y Fz(Lemma)30 b(2.7.)77 b FA(\(Bounded)28 b Fu(\026)p FA(-op)r(erator\))0 4324 y(Let)h Fu(A)24 b Fx(2)h Fr(QF)k FA(and)f(FV)r(\()p Fu(A)p FA(\))d Fx(\022)f(f)p Fu(x)1062 4336 y Ft(1)1100 4324 y Fu(;)14 b(:::;)g(x)1290 4336 y Fv(n)1335 4324 y Fu(;)g(y)s Fx(g)p FA(.)39 b(Then)28 b(there)h(exists)f(an)g Fu(f)33 b Fx(2)25 b FA(PR)2568 4288 y Fv(n)p Ft(+1)2726 4324 y FA(suc)n(h)j(that)h(the)g(follo)n(wing)e(form)n(ulas)0 4465 y(are)g(pro)n(v)-5 b(able)26 b(in)i Fr(PRA)f FA(\(where)h Fu(A)p FA(\()p Fu(t)p FA(\))c(:=)e Fu(A)1382 4477 y Fv(y)1423 4465 y FA(\()p Fu(t)p FA(\)\):)0 4606 y(\(a\))28 b Fx(9)p Fu(z)e Fx(\024)d Fu(y)s(A)p FA(\()p Fu(z)t FA(\))g Fx(!)g Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(xy)s FA(\))19 b Fx(^)g(8)p Fu(z)t(<)n(f)8 b(~)-41 b(xy)16 b Fx(:)p Fu(A)p FA(\()p Fu(z)t FA(\))56 b(\(i.e.)37 b Fx(9)p Fu(z)27 b Fx(\024)22 b Fu(y)s(A)p FA(\()p Fu(z)t FA(\))h Fx(!)g Fu(f)8 b(~)-41 b(xy)26 b FA(=)d(min)p Fx(f)p Fu(z)j FA(:)d Fu(A)p FA(\()p Fu(z)t FA(\))p Fx(g)p FA(\).)0 4747 y(\(b\))28 b Fx(9)p Fu(z)f Fx(\024)22 b Fu(y)s(A)p FA(\()p Fu(z)t FA(\))h Fx($)g Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(x)q(y)s FA(\))0 4888 y(\(c\))28 b Fx(:)p Fu(A)p FA(\(0\))19 b Fx(^)g Fu(A)p FA(\()p Fu(y)s FA(\))24 b Fx(!)f(:)p Fu(A)p FA(\()p Fu(p)p FA(\))c Fx(^)g Fu(A)p FA(\()p Fu(p)p FA(+1\),)27 b(where)g Fu(p)c FA(:=)g Fu(f)8 b(~)-41 b(xy)21 b Fx(\000)1982 4872 y(\001)2047 4888 y FA(1.)0 5054 y Fy(Notation.)p 417 5009 51 4 v 73 w Fu(\026)467 5075 y Fv(z)r Fp(\024)p Fv(y)593 5054 y Fu(A)p FA(\()p Fu(z)t FA(\))j(:=)e Fu(f)8 b(~)-41 b(xy)s FA(.)1908 5539 y(12)p eop %%Page: 13 14 13 13 bop 0 83 a FA(Pro)r(of:)0 224 y(By)27 b(Lemma)h(2.4)e(there)i(is) f(a)g(function)i(sym)n(b)r(ol)e Fu(h)c Fx(2)g FA(PR)1802 188 y Fv(n)p Ft(+2)1959 224 y FA(suc)n(h)k(that)0 365 y Fr(PRA)c Fx(`)g Fu(A)p FA(\()p Fu(z)t FA(\))18 b Fx(_)h(:)p Fu(A)p FA(\()p Fr(S)c Fu(y)s FA(\))23 b Fx(!)g Fu(hx)1030 377 y Ft(1)1068 365 y Fu(:::x)1184 377 y Fv(n)1230 365 y Fu(y)s(z)j FA(=)c Fu(z)31 b FA(and)d Fr(PRA)23 b Fx(`)f(:)p Fu(A)p FA(\()p Fu(z)t FA(\))d Fx(^)g Fu(A)p FA(\()p Fr(S)c Fu(y)s FA(\))23 b Fx(!)g Fu(hx)2688 377 y Ft(1)2725 365 y Fu(:::x)2841 377 y Fv(n)2887 365 y Fu(y)s(z)j FA(=)d Fr(S)14 b Fu(y)s FA(.)0 531 y(Let)28 b Fu(f)j FA(:=)23 b(\(R0)467 501 y Fv(n)512 531 y Fu(h)p FA(\).)37 b(Then)28 b(the)g(follo)n(wing)f(form)n(ulas)f(are)h(pro)n(v)-5 b(able)26 b(in)i Fr(PRA)0 697 y FA(\(1\))g Fu(f)8 b(~)-41 b(x)p FA(0)22 b(=)h(0)0 839 y(\(2\))28 b Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(xy)s FA(\))19 b Fx(_)g(:)p Fu(A)p FA(\()p Fr(S)14 b Fu(y)s FA(\))23 b Fx(!)h Fu(f)8 b(~)-41 b(x)13 b Fr(S)h Fu(y)26 b FA(=)d Fu(f)8 b(~)-41 b(xy)0 980 y FA(\(3\))28 b Fx(:)p Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(x)q(y)s FA(\))18 b Fx(^)h Fu(A)p FA(\()p Fr(S)14 b Fu(y)s FA(\))23 b Fx(!)h Fu(f)8 b(~)-41 b(x)13 b Fr(S)h Fu(y)26 b FA(=)d Fr(S)14 b Fu(y)0 1121 y FA(\(4\))28 b Fu(f)8 b(~)-41 b(xy)26 b Fx(\024)c Fu(y)0 1287 y FA(\(5\))28 b Fu(A)p FA(\()p Fu(y)s FA(\))23 b Fx(!)g Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(x)q(y)s FA(\))0 1453 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f Fu(y)s FA(:)37 b(1.)f Fu(y)26 b FA(=)c(0:)37 b(trivial.)0 1594 y(2.)f Fu(A)p FA(\()p Fr(S)15 b Fu(y)s FA(\))355 1540 y Ft(\(2\))p Fv(;)p Ft(\(3\))408 1594 y Fx(\))77 b FA(\()p Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(x)q(y)s FA(\))18 b Fx(^)h Fu(f)8 b(~)-41 b(x)14 b Fr(S)g Fu(y)25 b FA(=)e Fu(f)8 b(~)-41 b(xy)s FA(\))19 b Fx(_)f Fu(f)8 b(~)-41 b(x)14 b Fr(S)g Fu(y)26 b FA(=)d Fr(S)13 b Fu(y)26 b Fx(\))d Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(x)15 b Fr(S)f Fu(y)s FA(\).)0 1760 y(\(6\))28 b Fu(A)p FA(\()p Fu(y)269 1772 y Ft(0)306 1760 y FA(\))19 b Fx(^)g Fu(y)472 1772 y Ft(0)532 1760 y Fx(\024)j Fu(y)k Fx(!)d Fu(f)8 b(~)-41 b(xy)930 1772 y Ft(0)990 1760 y FA(=)23 b Fu(f)8 b(~)-41 b(xy)0 1926 y FA(Pro)r(of)26 b(b)n(y)i(induction)g(on)f Fu(y)s FA(:)37 b(1.)f Fu(y)26 b FA(=)c Fu(y)1227 1938 y Ft(0)1264 1926 y FA(:)37 b(trivial.)0 2067 y(2.)f Fu(y)26 b FA(=)d Fr(S)14 b Fu(z)22 b Fx(^)c Fu(y)491 2079 y Ft(0)551 2067 y Fx(\024)23 b Fu(z)t FA(:)36 b Fu(A)p FA(\()p Fu(y)876 2079 y Ft(0)914 2067 y FA(\))969 2013 y Ft(\(5\)+IH)1032 2067 y Fx(\))86 b Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(x)q(y)1434 2079 y Ft(0)1471 2067 y FA(\))18 b Fx(^)h Fu(f)8 b(~)-41 b(xy)1733 2079 y Ft(0)1793 2067 y FA(=)23 b Fu(f)8 b(~)-41 b(xz)26 b Fx(\))d Fu(f)8 b(~)-41 b(xy)26 b FA(=)d Fu(f)8 b(~)-41 b(x)13 b Fr(S)h Fu(z)2637 2013 y Ft(\(2\))2647 2067 y FA(=)33 b Fu(f)8 b(~)-41 b(xz)26 b FA(=)d Fu(f)8 b(~)-41 b(xy)3133 2079 y Ft(0)3170 2067 y FA(.)0 2233 y(\(7\))28 b Fu(z)e(<)d(f)8 b(~)-41 b(xy)25 b Fx(!)f(:)p Fu(A)p FA(\()p Fu(z)t FA(\))0 2399 y(Pro)r(of:)36 b Fu(A)p FA(\()p Fu(z)t FA(\))18 b Fx(^)h Fu(z)26 b(<)d(f)8 b(~)-41 b(xy)838 2345 y Ft(\(4\))839 2399 y Fx(\))24 b Fu(A)p FA(\()p Fu(z)t FA(\))18 b Fx(^)h Fu(z)26 b(<)d(y)1427 2345 y Ft(\(6\))p Fv(;)p Ft(\(4\))1480 2399 y Fx(\))77 b Fu(f)8 b(~)-41 b(xy)25 b FA(=)e Fu(f)8 b(~)-41 b(xz)26 b Fx(\024)d Fu(z)t FA(.)36 b(Con)n(tradiction.)0 2565 y(No)n(w)27 b(\(a\),\(b\))i(follo)n(w)d(from)i(\(4\)-\(7\).)0 2731 y(Pro)r(of)e(of)i(\(c\):)37 b Fx(:)p Fu(A)p FA(\(0\))20 b Fx(^)e Fu(A)p FA(\()p Fu(y)s FA(\))993 2677 y Ft(\(5\))994 2731 y Fx(\))24 b(:)p Fu(A)p FA(\(0\))19 b Fx(^)g Fu(A)p FA(\()p Fu(f)8 b(~)-41 b(xy)s FA(\))24 b Fx(\))f Fu(f)8 b(~)-41 b(xy)25 b Fx(6)p FA(=)e(0)g Fx(\))g Fu(f)8 b(~)-41 b(xy)25 b FA(=)e Fu(p)p FA(+)o(1)2658 2677 y Ft(\(7\))2659 2731 y Fx(\))h(:)p Fu(A)p FA(\()p Fu(p)p FA(\).)0 2947 y Fz(De\014nition.)0 3088 y FA(The)k(\001)240 3100 y Ft(0)277 3088 y FA(-)p Fs(formulas)g FA(are)f(generated)f(from)i Fx(L)1416 3100 y Ft(0)1453 3088 y FA(-literals)f(b)n(y)g(means)g(of)h Fx(^)p FA(,)g Fx(_)p FA(,)g Fx(8)p Fu(x)22 b Fx(\024)h Fu(t)p FA(,)28 b Fx(9)p Fu(x)23 b Fx(\024)g Fu(t)55 b FA(\()p Fu(x)24 b Fx(62)g FA(FV)q(\()p Fu(t)p FA(\)\).)0 3254 y(The)k(\006-)p Fs(formulas)g FA(are)f(generated)f(from)h Fx(L)1369 3266 y Ft(0)1407 3254 y FA(-literals)f(b)n(y)i(means)f(of)g Fx(^)p FA(,)h Fx(_)p FA(,)g Fx(8)p Fu(x)23 b Fx(\024)g Fu(t)k FA(\()p Fu(x)d Fx(62)g FA(FV)q(\()p Fu(t)p FA(\)\),)29 b Fx(9)p Fu(x)p FA(.)0 3421 y(A)f(form)n(ula)f(is)g(called)g(\006)773 3433 y Ft(1)811 3421 y FA(-)p Fs(formula)h FA(if)g(it)g(is)f(in)h Fr(QF)g FA(or)f(has)g(the)h(form)f Fx(9)p Fu(xA)i FA(with)f Fu(A)23 b Fx(2)h Fr(QF)p FA(.)0 3611 y Fz(Lemma)30 b(2.8.)0 3753 y FA(F)-7 b(or)27 b(eac)n(h)g(\001)405 3765 y Ft(0)442 3753 y FA(-form)n(ula)g Fu(A)g FA(there)h(exists)f(a)g(PR-term)g Fu(t)1749 3765 y Fv(A)1831 3753 y FA(suc)n(h)g(that)h Fr(PRA)23 b Fx(`)g Fu(A)g Fx($)g Fu(t)2678 3765 y Fv(A)2755 3753 y FA(=)g(0.)0 3919 y(Pro)r(of:)0 4060 y(By)k(Lemma)h(2.4a)e(it)i (su\016ces)f(to)h(pro)n(v)n(e)e(that)h(ev)n(ery)g(\001)1761 4072 y Ft(0)1798 4060 y FA(-form)n(ula)g(is)g(equiv)-5 b(alen)n(t)27 b(to)h(a)f(quan)n(ti\014erfree)g(form)n(ula.)0 4226 y(1.)64 b(Let)37 b Fu(A)h FA(=)g Fx(9)p Fu(y)j Fx(\024)d Fu(sB)t FA(\()p Fu(y)s FA(\))f(with)g(FV)r(\()p Fu(A)p FA(\))i(=)f Fx(f)o Fu(~)-41 b(x)p Fx(g)p FA(.)64 b(By)36 b(I.H.)h(there)g(is)g(a)f Fu(C)6 b FA(\()p Fu(y)s FA(\))39 b Fx(2)f Fr(QF)g FA(with)f(FV)q(\()p Fu(C)6 b FA(\))40 b Fx(\022)d(f)o Fu(~)-41 b(x;)14 b(y)s Fx(g)36 b FA(and)0 4367 y Fx(`)23 b Fu(B)t FA(\()p Fu(y)s FA(\))g Fx($)g Fu(C)6 b FA(\()p Fu(y)s FA(\).)38 b(By)27 b(Lemma)g(2.7)g(there)h(is)f (a)g(function)h(sym)n(b)r(ol)g Fu(f)36 b FA(suc)n(h)27 b(that)h Fx(`)23 b(9)p Fu(z)t Fx(\024)o Fu(y)s(C)6 b FA(\()p Fu(z)t FA(\))23 b Fx($)g Fu(C)6 b FA(\()p Fu(f)i(~)-41 b(xy)s FA(\).)0 4508 y(Hence)28 b Fx(`)23 b(9)p Fu(y)s Fx(\024)o Fu(sB)t FA(\()p Fu(y)s FA(\))g Fx($)g(9)p Fu(y)s Fx(\024)p Fu(sC)6 b FA(\()p Fu(y)s FA(\))23 b Fx($)g Fu(C)6 b FA(\()p Fu(f)i(~)-41 b(x)q(s)p FA(\).)0 4674 y(2.)35 b(If)25 b Fu(A)f FA(=)e Fx(8)p Fu(x)h Fx(\024)f Fu(sB)29 b FA(then)c(\(as)f(w)n(e)g(ha)n(v)n(e)f(just)i(sho)n(wn\))f Fx(`)f(9)p Fu(x)h Fx(\024)e Fu(s)p Fx(:)p Fu(B)28 b Fx($)23 b Fu(C)31 b FA(\(for)24 b(some)g Fu(C)29 b Fx(2)23 b Fr(QF)q FA(\))h(and)h(hence)f Fx(`)f Fu(A)g Fx($)g(:)p Fu(C)6 b FA(.)0 4890 y Fz(De\014nition.)80 b FA(A)29 b(recursiv)n(e)d(function)i Fu(f)33 b FA(:)23 b(I)-14 b(N)1484 4854 y Fv(n)1553 4890 y Fx(!)23 b FA(I)-14 b(N)29 b(called)f Fs(pr)l(ovably)k(r)l(e)l(cursive)c FA(\(or)f Fs(pr)l(ovably)32 b(total)p FA(\))d Fs(in)h Fm(S)e FA(if)g(there)g(is)0 5031 y(a)f(\006)129 5043 y Ft(1)166 5031 y FA(-form)n(ula)g Fu(A)p FA(\()o Fu(~)-41 b(x)q(;)14 b(y)s FA(\))27 b(suc)n(h)h(that)83 b(\(i\))56 b Fm(S)23 b Fx(`)f(8)o Fu(~)-41 b(x)p Fx(9)p Fu(y)17 b(A)p FA(\()o Fu(~)-41 b(x)q(;)14 b(y)s FA(\))55 b(and)g(\(ii\))28 b Fx(8)-5 b Fu(~)-37 b(a)o(;)14 b(b)23 b Fx(2)g FA(I)-14 b(N)q(\()14 b Fu(f)9 b FA(\()-5 b Fu(~)-37 b(a)p FA(\))23 b(=)g Fu(b)36 b Fx(,)h(N)e(j)-14 b FA(=)23 b Fu(A)p FA([)-5 b Fu(~)-37 b(a;)14 b(b)p FA(])g(\).)1908 5539 y(13)p eop %%Page: 14 15 14 14 bop 0 83 a Fz(Theorem)30 b(2.9)0 224 y FA(If)e Fr(PRA)23 b Fx(`)g FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A)27 b FA(where)g Fu(A)d Fx(2)f Fr(QF)28 b FA(and)g Fx(8)f FA(do)r(es)g(not)g(o)r(ccur)g(in)h(\000,)g(then)g Fr(PRA)23 b Fx(`)g FA(\000)p Fu(;)14 b(A)2743 236 y Fv(y)2783 224 y FA(\()p Fu(t)p FA(\))28 b(for)f(some)g(PR-term)g Fu(t)p FA(.)0 388 y(Pro)r(of:)0 529 y(By)j(Theorem)f(1.8)g(\(partial)h(cut)g (elimination\))h(w)n(e)f(ha)n(v)n(e)e(\(for)i(some)g Fu(k)s FA(\))g Fr(PRA)d Fx(`)2578 499 y Fv(k)2578 552 y Fl(QF)o Fv(;)p Ft(0)2747 529 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A)p FA(,)30 b(i.e.)45 b(there)30 b(exists)g(a)f Fr(PRA)p FA(-)0 669 y(deriv)-5 b(ation)23 b(of)h(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A)24 b FA(where)f(all)h(cut)g(form)n(ulas)f (are)g(quan)n(ti\014erfree.)34 b(W)-7 b(e)25 b(pro)r(ceed)e(b)n(y)h (induction)g(on)g(suc)n(h)f(a)h(deriv)-5 b(ation.)0 834 y(1.)36 b(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A)28 b FA(is)f(an)h(axiom:)36 b(Then)28 b(\000)f(is)h(an)f(axiom)g(to)r(o,)g(and)h(w)n(e)f(ma)n(y)g (set)g Fu(t)c FA(:=)g(0.)0 975 y(2.)36 b Fx(`)152 939 y Fv(k)q Fp(\000)p Ft(1)152 1000 y Fl(QF)o Fv(;)p Ft(0)316 975 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A;)g(B)657 987 y Ft(0)722 975 y FA(and)28 b Fx(`)935 939 y Fv(k)q Fp(\000)p Ft(1)935 1000 y Fl(QF)o Fv(;)p Ft(0)1099 975 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A;)g(B)1440 987 y Ft(1)1504 975 y FA(with)28 b Fu(B)1756 987 y Ft(0)1812 975 y Fx(^)19 b Fu(B)1949 987 y Ft(1)2009 975 y Fx(2)k FA(\000:)0 1115 y(Let)28 b Fu(t)23 b FA(:=)f Fu(d)355 1130 y Fv(A)p Ft(\()p Fv(t)456 1138 y Fo(0)489 1130 y Ft(\))519 1115 y FA(\()p Fu(t)581 1127 y Ft(0)619 1115 y Fu(;)14 b(t)686 1127 y Ft(1)723 1115 y FA(\))28 b(where)f(b)n(y)g(I.H.)h Fx(`)23 b FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)1591 1127 y Fv(i)1619 1115 y FA(\))p Fu(;)g(B)1751 1127 y Fv(i)1806 1115 y FA(\()p Fu(i)23 b FA(=)g(0)p Fu(;)14 b FA(1\))55 b(\(1\).)0 1280 y Fx(`)23 b Fu(A)p FA(\()p Fu(t)198 1292 y Ft(0)236 1280 y FA(\))g Fx(!)g Fu(t)g FA(=)g Fu(t)568 1292 y Ft(0)651 1280 y Fx(\))g(`)g Fu(A)p FA(\()p Fu(t)955 1292 y Ft(0)993 1280 y FA(\))g Fx(!)g Fu(A)p FA(\()p Fu(t)p FA(\))56 b(\(2\).)0 1420 y Fx(`)23 b(:)p Fu(A)p FA(\()p Fu(t)253 1432 y Ft(0)291 1420 y FA(\))g Fx(!)g Fu(t)g FA(=)g Fu(t)623 1432 y Ft(1)706 1420 y Fx(\))g(`)g(:)p Fu(A)p FA(\()p Fu(t)1065 1432 y Ft(0)1103 1420 y FA(\))h Fx(!)f FA(\()p Fu(A)p FA(\()p Fu(t)1421 1432 y Ft(1)1459 1420 y FA(\))g Fx(!)g Fu(A)p FA(\()p Fu(t)p FA(\)\))1856 1366 y Ft(\(2\))1857 1420 y Fx(\))h(`)e Fu(A)p FA(\()p Fu(t)2161 1432 y Ft(1)2199 1420 y FA(\))i Fx(!)f Fu(A)p FA(\()p Fu(t)p FA(\))56 b(\(3\).)0 1585 y(F)-7 b(rom)27 b(\(1\),)h(\(2\),)g(\(3\))g(w)n(e)f (obtain)g Fx(`)c FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\))p Fu(;)g(B)1465 1597 y Fv(i)1521 1585 y FA(\(for)27 b Fu(i)c FA(=)f(0)p Fu(;)14 b FA(1\))27 b(and)h(then)g Fx(`)23 b FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\).)0 1750 y(3.)36 b Fx(`)152 1714 y Fv(k)q Fp(\000)p Ft(1)152 1775 y Fl(QF)o Fv(;)p Ft(0)316 1750 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A;)g(B)657 1762 y Fv(k)726 1750 y FA(with)28 b Fu(B)978 1762 y Ft(0)1033 1750 y Fx(_)19 b Fu(B)1170 1762 y Ft(1)1231 1750 y Fx(2)k FA(\000:)37 b(Let)28 b Fu(t)23 b FA(:=)f Fu(t)1763 1762 y Ft(0)1801 1750 y FA(,)27 b(where)g(b)n(y)h(I.H.)g Fx(`)22 b FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)2659 1762 y Ft(0)2697 1750 y FA(\))p Fu(;)g(B)2829 1762 y Fv(k)2870 1750 y FA(.)0 1890 y(4.)36 b Fx(`)152 1855 y Fv(k)q Fp(\000)p Ft(1)152 1916 y Fl(QF)o Fv(;)p Ft(0)316 1890 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A;)g(B)657 1902 y Fv(x)699 1890 y FA(\()p Fu(s)p FA(\))28 b(with)g Fx(9)p Fu(xB)g Fx(2)23 b FA(\000:)37 b(As)28 b(3.)0 2031 y(5.)36 b Fx(`)152 1995 y Fv(k)q Fp(\000)p Ft(1)152 2056 y Fl(QF)o Fv(;)p Ft(0)316 2031 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A;)g(A)p FA(\()p Fu(s)p FA(\):)38 b(Let)28 b Fu(t)23 b FA(:=)f Fu(d)1175 2046 y Fv(A)p Ft(\()p Fv(t)1276 2054 y Fo(0)1309 2046 y Ft(\))1339 2031 y FA(\()p Fu(t)1401 2043 y Ft(0)1439 2031 y Fu(;)14 b(s)p FA(\))27 b(where)g(b)n(y)h(I.H.)g Fx(`)23 b FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)2383 2043 y Ft(0)2420 2031 y FA(\))p Fu(;)g(A)p FA(\()p Fu(s)p FA(\).)0 2171 y(As)28 b(in)g(2.)f(w)n(e)g(obtain)g Fx(`)c Fu(A)p FA(\()p Fu(t)890 2183 y Ft(0)928 2171 y FA(\))g Fx(!)g Fu(A)p FA(\()p Fu(t)p FA(\))29 b(and)e Fx(`)c Fu(A)p FA(\()p Fu(s)p FA(\))h Fx(!)f Fu(A)p FA(\()p Fu(t)p FA(\).)38 b(Hence)27 b Fx(`)c FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\).)0 2336 y(6.)36 b Fx(`)152 2301 y Fv(k)q Fp(\000)p Ft(1)152 2361 y Fl(QF)o Fv(;)p Ft(0)316 2336 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A;)g(B)32 b FA(and)27 b Fx(`)901 2301 y Fv(k)q Fp(\000)p Ft(1)901 2361 y Fl(QF)o Fv(;)p Ft(0)1065 2336 y FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s(A;)g Fx(:)p Fu(B)32 b FA(with)c Fu(B)g Fx(2)23 b Fr(QF)p FA(:)0 2477 y(Let)28 b Fu(t)23 b FA(:=)f Fu(d)355 2492 y Fv(A)p Ft(\()p Fv(t)456 2500 y Fo(0)489 2492 y Ft(\))519 2477 y FA(\()p Fu(t)581 2489 y Ft(0)619 2477 y Fu(;)14 b(t)686 2489 y Ft(1)723 2477 y FA(\),)28 b(where)f(b)n(y)g(I.H.)h Fx(`)23 b FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)1614 2489 y Ft(0)1652 2477 y FA(\))p Fu(;)g(B)32 b FA(and)27 b Fx(`)c FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)2264 2489 y Ft(1)2301 2477 y FA(\))p Fu(;)g Fx(:)p Fu(B)t FA(.)0 2617 y(As)28 b(in)g(2.)f(w)n(e)g(obtain)g Fx(`)c FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\))p Fu(;)g(B)32 b FA(und)c Fx(`)23 b FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\))p Fu(;)g Fx(:)p Fu(B)t FA(.)38 b(Hence)28 b Fx(`)23 b FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\).)0 2782 y(7.)36 b(\000)23 b(=)g(\000)316 2752 y Fp(0)339 2782 y Fu(;)14 b Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))28 b(and)g Fx(`)1048 2747 y Fv(k)q Fp(\000)p Ft(1)1048 2807 y Fl(QF)o Fv(;)p Ft(0)1212 2782 y FA(\000)1264 2752 y Fp(0)1287 2782 y Fu(;)14 b Fx(:)p Fu(F)e FA(\()p Fu(x)p FA(\))p Fu(;)i(F)e FA(\()p Fr(S)j Fu(x)p FA(\))p Fu(;)f Fx(9)p Fu(y)s(A)p FA(:)0 2947 y(W)-7 b(e)28 b(ha)n(v)n(e)e(to)i(pro)n(v)n(e)e Fx(`)d FA(\000)786 2917 y Fp(0)809 2947 y Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\))p Fu(;)g Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))29 b(for)e(some)g(term)g Fu(t)p FA(.)0 3112 y(W)-7 b(e)28 b(set)g Fu(r)310 3124 y Ft(0)370 3112 y FA(:=)p 481 3066 51 4 v 23 w Fu(\026)531 3132 y Fv(z)r Fp(\024)p Fv(s)653 3112 y Fx(:)p Fu(F)12 b FA(\()p Fu(z)t FA(\))32 b Fx(\000)933 3095 y(\001)998 3112 y FA(1.)0 3276 y(Then)c Fx(`)23 b(:)p Fu(F)12 b FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))p Fu(;)i(F)e FA(\()p Fu(r)893 3288 y Ft(0)931 3276 y FA(\))28 b(and)f Fx(`)c(:)p Fu(F)12 b FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))p Fu(;)i Fx(:)p Fu(F)e FA(\()p Fr(S)j Fu(r)1944 3288 y Ft(0)1982 3276 y FA(\).)0 3441 y([[)28 b(By)f(Lemma)g(2.7c,)g Fx(`)c Fu(F)12 b FA(\(0\))18 b Fx(^)h(:)p Fu(F)12 b FA(\()p Fu(y)s FA(\))24 b Fx(!)f Fu(F)12 b FA(\()p Fu(p)p FA(\()p Fu(y)s FA(\)\))19 b Fx(^)g(:)p Fu(F)12 b FA(\()p Fu(p)p FA(\()p Fu(y)s FA(\))19 b(+)f(1\))27 b(with)h Fu(p)p FA(\()p Fu(y)s FA(\))23 b(:=)p 2739 3396 V 23 w Fu(\026)2789 3462 y Fv(z)r Fp(\024)p Fv(y)2915 3441 y Fx(:)p Fu(F)12 b FA(\()p Fu(z)t FA(\))32 b Fx(\000)3196 3425 y(\001)3260 3441 y FA(1.)37 b(]])0 3655 y(No)n(w)27 b(w)n(e)g(conclude)h(as)f(follo)n (ws)110 3774 y(\000)162 3744 y Fp(0)185 3774 y Fu(;)14 b Fx(9)p Fu(y)s(A)p FA(\()p Fu(y)s FA(\))p Fu(;)g Fx(:)p Fu(F)e FA(\()p Fu(x)p FA(\))p Fu(;)i(F)e FA(\()p Fr(S)k Fu(x)p FA(\))-79 3832 y Ft(Subst)p 83 3817 13 5 v 108 3817 V 133 3817 V 158 3817 V 183 3817 V 208 3817 V 232 3817 V 257 3817 V 282 3817 V 307 3817 V 332 3817 V 357 3817 V 382 3817 V 407 3817 V 432 3817 V 457 3817 V 482 3817 V 506 3817 V 531 3817 V 556 3817 V 581 3817 V 606 3817 V 631 3817 V 656 3817 V 681 3817 V 706 3817 V 731 3817 V 756 3817 V 780 3817 V 805 3817 V 830 3817 V 855 3817 V 880 3817 V 905 3817 V 930 3817 V 955 3817 V 980 3817 V 1005 3817 V 83 3890 a FA(\000)135 3860 y Fp(0)158 3890 y Fu(;)e Fx(9)p Fu(y)s(A)p FA(\()p Fu(y)s FA(\))p Fu(;)g Fx(:)p Fu(F)e FA(\()p Fu(r)681 3902 y Ft(0)720 3890 y FA(\))p Fu(;)i(F)e FA(\()p Fr(S)i Fu(r)983 3902 y Ft(0)1021 3890 y FA(\))-19 3948 y Ft(IV)p 55 3933 V 80 3933 V 105 3933 V 130 3933 V 155 3933 V 180 3933 V 205 3933 V 230 3933 V 255 3933 V 280 3933 V 304 3933 V 329 3933 V 354 3933 V 379 3933 V 404 3933 V 429 3933 V 454 3933 V 479 3933 V 504 3933 V 529 3933 V 553 3933 V 578 3933 V 603 3933 V 628 3933 V 653 3933 V 678 3933 V 703 3933 V 728 3933 V 753 3933 V 778 3933 V 803 3933 V 827 3933 V 852 3933 V 877 3933 V 902 3933 V 927 3933 V 952 3933 V 977 3933 V 1002 3933 V 1027 3933 V 121 4006 a FA(\000)173 3976 y Fp(0)196 4006 y Fu(;)g(A)p FA(\()p Fu(t)p FA(\))p Fu(;)g Fx(:)p Fu(F)e FA(\()p Fu(r)615 4018 y Ft(0)654 4006 y FA(\))p Fu(;)i(F)e FA(\()p Fr(S)i Fu(r)917 4018 y Ft(0)955 4006 y FA(\))420 b Fx(:)p Fu(F)12 b FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))p Fu(;)i(F)e FA(\()p Fu(r)2009 4018 y Ft(0)2048 4006 y FA(\))p 66 4048 2015 4 v 2099 4064 a Ft(Cut)554 4122 y FA(\000)606 4092 y Fp(0)629 4122 y Fu(;)i(A)p FA(\()p Fu(t)p FA(\))p Fu(;)g Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))p Fu(;)i(F)e FA(\()p Fr(S)j Fu(r)1522 4134 y Ft(0)1560 4122 y FA(\))654 b Fx(:)p Fu(F)12 b FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))p Fu(;)i Fx(:)p Fu(F)e FA(\()p Fr(S)j Fu(r)2964 4134 y Ft(0)3002 4122 y FA(\))p 554 4164 2481 4 v 3053 4180 a Ft(Cut)1425 4239 y FA(\000)1477 4208 y Fp(0)1500 4239 y Fu(;)f(A)p FA(\()p Fu(t)p FA(\))p Fu(;)g Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(s)p FA(\))0 4452 y Fz(Corollary)32 b(2.9.)78 b FA(The)27 b(pro)n(v)-5 b(ably)26 b(recursiv)n(e)g(functions)i(of)g Fr(PRA)f FA(are)g(exactly)g(the)h(primitiv)n(e)f(recursiv)n(e)f (functions.)0 4617 y(Pro)r(of:)35 b(Ob)n(viously)26 b(ev)n(ery)f (primitiv)n(e)i(recursiv)n(e)e(function)i Fu(f)36 b FA(is)26 b(pro)n(v)-5 b(ably)26 b(recursiv)n(e)f(in)i Fr(PRA)p FA(:)36 b(let)27 b Fu(A)p FA(\()o Fu(~)-41 b(x)q(;)14 b(y)s FA(\))23 b(:=)g(\()p Fu(f)8 b(~)-41 b(x)23 b FA(=)g Fu(y)s FA(\).)0 4757 y(No)n(w)k(let)g Fu(A)h FA(b)r(e)f(a)g(\006)639 4769 y Ft(1)676 4757 y FA(-form)n(ula)f(with)i(\(i\))g Fr(PRA)23 b Fx(`)g(8)o Fu(~)-41 b(x)p Fx(9)p Fu(y)s(A)p FA(\()o Fu(~)g(x;)14 b(y)s FA(\))28 b(and)f(\(ii\))h Fx(8)-5 b Fu(~)-37 b(a)o(;)14 b(b)p FA(\()p Fu(f)9 b FA(\()-5 b Fu(~)-37 b(a)o FA(\))24 b(=)e Fu(b)h Fx(,)g(N)35 b(j)-14 b FA(=)23 b Fu(A)p FA([)-5 b Fu(~)-37 b(a;)14 b(b)p FA(]\).)37 b(First)27 b(notice)0 4898 y(that)j(the)h(statemen)n (t)f(of)g(Theorem)f(2.9)g(also)g(holds)h(for)f Fu(A)f Fx(2)f FA(\006)2052 4910 y Ft(1)2089 4898 y FA(:)42 b Fx(`)27 b FA(\000)p Fu(;)14 b Fx(9)p Fu(y)s Fx(9)p Fu(xB)t FA(\()p Fu(x;)g(y)s FA(\))55 b(=)-14 b Fx(\))27 b(`)g FA(\000)p Fu(;)14 b Fx(9)p Fu(z)t(B)t FA(\()p Fu(\031)3381 4910 y Ft(1)3418 4898 y Fu(z)t(;)g(\031)3545 4910 y Ft(2)3582 4898 y Fu(z)t FA(\))3710 4851 y Ft(Th)p Fv(:)p Ft(2)p Fv(:)p Ft(9)3738 4898 y FA(=)-14 b Fx(\))0 5038 y(`)40 b FA(\000)p Fu(;)14 b(B)t FA(\()p Fu(\031)326 5050 y Ft(1)364 5038 y Fu(t;)g(\031)478 5050 y Ft(2)515 5038 y Fu(t)p FA(\))80 b(=)-14 b Fx(\))40 b(`)g FA(\000)p Fu(;)14 b Fx(9)p Fu(xB)t FA(\()p Fu(x;)g(\031)1334 5050 y Ft(2)1372 5038 y Fu(t)p FA(\).)68 b(Then)38 b(w)n(e)f(conclude)h(as)f (follo)n(ws:)56 b Fr(PRA)40 b Fx(`)g(8)o Fu(~)-41 b(x)p Fx(9)p Fu(y)s(A)p FA(\()o Fu(~)g(x;)14 b(y)s FA(\))80 b Fx(\))g Fr(PRA)40 b Fx(`)0 5179 y(9)p Fu(y)s(A)p FA(\()o Fu(~)-41 b(x)q(;)14 b(y)s FA(\))87 b Fx(\))f Fr(PRA)44 b Fx(`)f Fu(A)p FA(\()o Fu(~)-41 b(x)q(;)14 b(t)p FA(\()o Fu(~)-41 b(x)q FA(\)\))40 b(for)f(some)h(PR)o(-term)g Fu(t)p FA(.)73 b(Then)40 b Fx(8)-5 b Fu(~)-37 b(a)o FA(\()p Fx(N)56 b(j)-14 b FA(=)44 b Fu(A)p FA(\()o Fu(~)-41 b(x)q(;)14 b(t)p FA(\()o Fu(~)-41 b(x)p FA(\)\)[)-5 b Fu(~)-37 b(a)q FA(]\))40 b(and)g(th)n(us,)j(b)n(y)c(\(ii\),)0 5319 y Fx(8)-5 b Fu(~)-37 b(a)o FA(\()p Fu(f)9 b FA(\()-5 b Fu(~)-37 b(a)p FA(\))23 b(=)g Fu(t)421 5289 y Fp(N)488 5319 y FA([)-5 b Fu(~)-37 b(a)p FA(]\).)1908 5539 y(14)p eop %%Page: 15 16 15 15 bop 0 83 a Fz(De\014nition)31 b(of)h FA(I\006)645 95 y Ft(1)0 224 y FA(The)c(\(T)-7 b(ait)28 b(st)n(yle\))f(system)g (I\006)972 236 y Ft(1)1037 224 y FA(is)h(the)g(same)f(as)g Fr(PRA)g FA(only)g(that)h(the)g Fr(QF)q FA(-induction)f(rule)h(is)f (replaced)g(b)n(y)g(the)0 403 y(\006)60 415 y Ft(1)97 403 y Fs(-induction)j(rule)700 349 y FA(\000)p Fu(;)14 b(F)842 361 y Fv(x)884 349 y FA(\(0\))83 b(\000)p Fu(;)14 b Fx(:)p Fu(F)r(;)g(F)1362 361 y Fv(x)1405 349 y FA(\()p Fr(S)h Fu(x)p FA(\))p 700 384 877 4 v 1000 458 a(\000)p Fu(;)f(F)1142 470 y Fv(x)1184 458 y FA(\()p Fu(t)p FA(\))1643 403 y(\()p Fu(F)35 b Fx(2)23 b FA(\006)1901 415 y Ft(1)1966 403 y FA(and)28 b Fu(x)23 b Fx(62)h FA(FV)q(\(\000)p Fu(;)14 b(F)2568 415 y Fv(x)2610 403 y FA(\()p Fu(t)p FA(\)\)\))0 628 y Fy(Remark.)73 b FA(As)28 b(for)f Fr(PRA)g FA(w)n(e)h(see)f(that)h(the)g(results)f(1.6-1.9)e(apply)j(to)f(I\006) 2352 640 y Ft(1)2389 628 y FA(,)h(no)n(w)f(with)h(\010)23 b(=)g(\006)3033 640 y Ft(1)3070 628 y FA(.)0 844 y Fz(Theorem)30 b(2.10)0 985 y FA(I\006)90 997 y Ft(1)150 985 y Fx(`)23 b FA(\000)28 b(with)g(\000)23 b Fx(\022)f FA(\006)715 997 y Ft(1)799 985 y FA(=)-14 b Fx(\))46 b Fr(PRA)23 b Fx(`)f FA(\000.)0 1151 y(Bew)n(eis:)0 1292 y(By)33 b(Theorem)g(1.8)g(w)n(e)h(ha)n(v)n(e)e(\(for)i(some)f Fu(k)s FA(\))h(I\006)1541 1304 y Ft(1)1611 1292 y Fx(`)1662 1262 y Fv(k)1662 1315 y Ft(\006)1709 1323 y Fo(1)1742 1315 y Fv(;)p Ft(0)1832 1292 y FA(\000,)i(i.e.)55 b(there)34 b(exists)f(an)h(I\006)2770 1304 y Ft(1)2807 1292 y FA(-deriv)-5 b(ation)33 b(of)g(\000)h(where)f(all)h(cut)0 1433 y(form)n(ulas)26 b(are)h(in)p 573 1367 98 4 v 28 w(\006)633 1445 y Ft(1)670 1433 y FA(.)37 b(W)-7 b(e)28 b(pro)r(ceed)f(b)n(y)g(induction)h(on)g (suc)n(h)f(a)g(deriv)-5 b(ation.)0 1599 y(1.)36 b(I\006)191 1611 y Ft(1)252 1599 y Fx(`)303 1564 y Fv(k)q Fp(\000)p Ft(1)303 1624 y(\006)350 1632 y Fo(1)382 1624 y Fv(;)p Ft(0)462 1599 y Fx(9)p Fu(xA;)14 b FA(\000)29 b(and)e(I\006)986 1611 y Ft(1)1046 1599 y Fx(`)1097 1564 y Fv(k)q Fp(\000)p Ft(1)1097 1624 y(\006)1144 1632 y Fo(1)1177 1624 y Fv(;)p Ft(0)1257 1599 y Fx(8)p Fu(x)p Fx(:)p Fu(A;)14 b FA(\000)27 b(with)i Fu(A)23 b Fx(2)g Fr(QF)q FA(:)0 1765 y(I\006)90 1777 y Ft(1)150 1765 y Fx(`)201 1730 y Fv(k)q Fp(\000)p Ft(1)201 1790 y(\006)248 1798 y Fo(1)281 1790 y Fv(;)p Ft(0)361 1765 y Fx(9)p Fu(xA;)14 b FA(\000)682 1718 y Ft(IH)652 1765 y FA(=)-14 b Fx(\))45 b Fr(PRA)23 b Fx(`)g FA(\000)p Fu(;)14 b Fx(9)p Fu(xA)56 b FA(\()p Fx(\003)p FA(\).)0 1916 y(I\006)90 1928 y Ft(1)150 1916 y Fx(`)201 1880 y Fv(k)q Fp(\000)p Ft(1)201 1940 y(\006)248 1948 y Fo(1)281 1940 y Fv(;)p Ft(0)361 1916 y Fx(8)p Fu(x)p Fx(:)p Fu(A;)14 b FA(\000)707 1868 y Ft(In)n(v)n(ersion)771 1916 y FA(=)-14 b Fx(\))110 b FA(I\006)1105 1928 y Ft(1)1166 1916 y Fx(`)1217 1880 y Fv(k)q Fp(\000)p Ft(1)1217 1940 y(\006)1264 1948 y Fo(1)1296 1940 y Fv(;)p Ft(0)1376 1916 y Fx(:)p Fu(A)1493 1928 y Fv(x)1536 1916 y FA(\()p Fu(y)s FA(\))p Fu(;)14 b FA(\000)1784 1868 y Ft(IH)1779 1916 y Fx(\))46 b Fr(PRA)23 b Fx(`)g(:)p Fu(A)2284 1928 y Fv(x)2326 1916 y FA(\()p Fu(y)s FA(\))p Fu(;)14 b FA(\000)46 b Fx(\))g Fr(PRA)23 b Fx(`)g(8)p Fu(x)p Fx(:)p Fu(A;)14 b FA(\000)3303 1861 y Ft(\()p Fp(\003)p Ft(\))3305 1916 y Fx(\))47 b Fr(PRA)23 b Fx(`)g FA(\000.)0 2082 y(2.)36 b(\000)23 b(=)g(\000)316 2051 y Fp(0)339 2082 y Fu(;)14 b Fx(9)p Fu(xA)p FA(\()p Fu(x;)g(t)p FA(\))30 b(and)d(I\006)990 2094 y Ft(1)1050 2082 y Fx(`)1101 2046 y Fv(k)q Fp(\000)p Ft(1)1101 2106 y(\006)1148 2114 y Fo(1)1181 2106 y Fv(;)p Ft(0)1261 2082 y FA(\000)1313 2051 y Fp(0)1336 2082 y Fu(;)14 b Fx(9)p Fu(xA)p FA(\()p Fu(x;)g FA(0\),)29 b(I\006)1860 2094 y Ft(1)1920 2082 y Fx(`)1971 2046 y Fv(k)q Fp(\000)p Ft(1)1971 2106 y(\006)2018 2114 y Fo(1)2051 2106 y Fv(;)p Ft(0)2131 2082 y FA(\000)2183 2051 y Fp(0)2206 2082 y Fu(;)14 b Fx(:9)p Fu(xA)p FA(\()p Fu(x;)g(y)s FA(\))p Fu(;)g Fx(9)p Fu(xA)p FA(\()p Fu(x;)g Fr(S)j Fu(y)s FA(\))28 b(with)g Fu(y)d Fx(62)f FA(FV)q(\(\000\):)0 2248 y(\(Due)k(to)g(our)f (general)f(con)n(v)n(en)n(tions)g(w)n(e)h(also)f(ha)n(v)n(e)h Fu(x)c Fx(62)h FA(FV)q(\()p Fu(t)p FA(\))29 b(and)e Fu(x;)14 b(y)26 b Fx(62)e FA(FV)q(\()p Fu(A)p FA(\(0)p Fu(;)14 b FA(0\)\).)38 b(\))0 2414 y(By)e(In)n(v)n(ersion)f(and)h(I.H.)h(w)n(e) f(obtain)g Fr(PRA)i Fx(`)f FA(\000)1587 2384 y Fp(0)1611 2414 y Fu(;)14 b Fx(9)p Fu(xA)p FA(\()p Fu(x;)g FA(0\))37 b(and)f Fr(PRA)i Fx(`)g FA(\000)2541 2384 y Fp(0)2564 2414 y Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(x;)g(y)s FA(\))p Fu(;)g Fx(9)p Fu(xA)p FA(\()p Fu(x;)g Fr(S)i Fu(y)s FA(\))37 b(where)f(w.l.o.g.)0 2555 y Fu(x)24 b Fx(62)f FA(FV)r(\(\000)351 2525 y Fp(0)374 2555 y FA(\).)0 2721 y(F)-7 b(rom)23 b(this)i(it)f(follo)n(ws)f(b)n(y)g(Theorem)g(2.9)g(that)i Fr(PRA)e Fx(`)g FA(\000)1797 2691 y Fp(0)1820 2721 y Fu(;)14 b(A)p FA(\()p Fu(q)s(;)g FA(0\))24 b(and)g Fr(PRA)f Fx(`)f FA(\000)2594 2691 y Fp(0)2618 2721 y Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(x;)g(y)s FA(\))p Fu(;)g(A)p FA(\()p Fu(p)p FA(\()p Fu(x;)g(y)s FA(\))p Fu(;)g Fr(S)h Fu(y)s FA(\))24 b(for)g(certain)0 2862 y(PR-terms)j Fu(p;)14 b(q)s FA(.)0 3028 y(There)27 b(exists)g(a)g(PR-term)g Fu(r)r FA(\()p Fu(y)s FA(\))i(suc)n(h)e(that)h Fr(PRA)23 b Fx(`)g Fu(r)r FA(\(0\))h(=)f Fu(q)e Fx(^)e Fu(r)r FA(\()p Fr(S)c Fu(y)s FA(\))23 b(=)g Fu(p)p FA(\()p Fu(r)r FA(\()p Fu(y)s FA(\))p Fu(;)14 b(y)s FA(\).)0 3194 y(No)n(w)27 b(w)n(e)g(obtain)h Fr(PRA)23 b Fx(`)f FA(\000)880 3164 y Fp(0)904 3194 y Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(r)r FA(\()p Fu(y)s FA(\))p Fu(;)g(y)s FA(\))p Fu(;)g(A)p FA(\()p Fu(r)r FA(\()p Fr(S)j Fu(y)s FA(\))p Fu(;)d Fr(S)g Fu(y)s FA(\),)27 b(and)h(then)g Fr(PRA)23 b Fx(`)g FA(\000)2576 3164 y Fp(0)2599 3194 y Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(r)r FA(\(0\))p Fu(;)g FA(0\))p Fu(;)g(A)p FA(\()p Fu(r)r FA(\()p Fu(t)p FA(\))p Fu(;)g(t)p FA(\))31 b(b)n(y)c(\(Ind\).)0 3360 y(T)-7 b(ogether)26 b(with)j Fr(PRA)23 b Fx(`)f Fu(r)r FA(\(0\))i(=)f Fu(q)31 b FA(and)c Fr(PRA)c Fx(`)g FA(\000)1593 3330 y Fp(0)1616 3360 y Fu(;)14 b(A)p FA(\()p Fu(q)s(;)g FA(0\))28 b(this)g(yields)f Fr(PRA)c Fx(`)g FA(\000)2632 3330 y Fp(0)2655 3360 y Fu(;)14 b(A)p FA(\()p Fu(r)r FA(\()p Fu(t)p FA(\))p Fu(;)g(t)p FA(\).)0 3501 y(F)-7 b(rom)27 b(this)h(w)n(e)f(conclude)h Fr(PRA)23 b Fx(`)f FA(\000)1154 3471 y Fp(0)1178 3501 y Fu(;)14 b Fx(9)p Fu(xA)p FA(\()p Fu(x;)g(t)p FA(\).)28 4008 y Fu(r)r FA(\()p Fr(S)h Fu(y)s FA(\))23 b(=)f Fu(p)p FA(\()p Fu(r)r FA(\()p Fu(y)s FA(\))p Fu(;)14 b(y)s FA(\))1036 3692 y(\000)1088 3662 y Fp(0)1111 3692 y Fu(;)g Fx(:)p Fu(A)p FA(\()p Fu(x;)g(y)s FA(\))p Fu(;)g Fx(9)p Fu(xA)p FA(\()p Fu(x;)g Fr(S)j Fu(y)s FA(\))745 3750 y Ft(\()p Fv(T)9 b(h:)p Ft(2)p Fv(:)p Ft(9\))p 989 3736 13 5 v 1014 3736 V 1039 3736 V 1064 3736 V 1089 3736 V 1114 3736 V 1139 3736 V 1164 3736 V 1189 3736 V 1214 3736 V 1238 3736 V 1263 3736 V 1288 3736 V 1313 3736 V 1338 3736 V 1363 3736 V 1388 3736 V 1413 3736 V 1438 3736 V 1463 3736 V 1488 3736 V 1512 3736 V 1537 3736 V 1562 3736 V 1587 3736 V 1612 3736 V 1637 3736 V 1662 3736 V 1687 3736 V 1712 3736 V 1737 3736 V 1762 3736 V 1786 3736 V 1811 3736 V 1836 3736 V 1861 3736 V 1886 3736 V 1911 3736 V 989 3850 a FA(\000)1041 3820 y Fp(0)1065 3850 y Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(x;)g(y)s FA(\))p Fu(;)g(A)p FA(\()p Fu(p)p FA(\()p Fu(x;)g(y)s FA(\))p Fu(;)g Fr(S)h Fu(y)s FA(\))654 3908 y Ft(\()p Fv(S)s(ubst)p Ft(\))p 875 3893 V 900 3893 V 925 3893 V 950 3893 V 975 3893 V 1000 3893 V 1024 3893 V 1049 3893 V 1074 3893 V 1099 3893 V 1124 3893 V 1149 3893 V 1174 3893 V 1199 3893 V 1224 3893 V 1249 3893 V 1274 3893 V 1298 3893 V 1323 3893 V 1348 3893 V 1373 3893 V 1398 3893 V 1423 3893 V 1448 3893 V 1473 3893 V 1498 3893 V 1523 3893 V 1547 3893 V 1572 3893 V 1597 3893 V 1622 3893 V 1647 3893 V 1672 3893 V 1697 3893 V 1722 3893 V 1747 3893 V 1772 3893 V 1797 3893 V 1821 3893 V 1846 3893 V 1871 3893 V 1896 3893 V 1921 3893 V 1946 3893 V 1971 3893 V 1996 3893 V 875 4008 a FA(\000)927 3978 y Fp(0)950 4008 y Fu(;)f Fx(:)p Fu(A)p FA(\()p Fu(r)r FA(\()p Fu(y)s FA(\))p Fu(;)g(y)s FA(\))p Fu(;)g(A)p FA(\()p Fu(p)p FA(\()p Fu(r)r FA(\()p Fu(y)s FA(\))p Fu(;)g(y)s FA(\))p Fu(;)g Fr(S)j Fu(y)s FA(\))p 28 4049 2010 4 v 515 4165 a(\000)567 4135 y Fp(0)590 4165 y Fu(;)d Fx(:)p Fu(A)p FA(\()p Fu(r)r FA(\()p Fu(y)s FA(\))p Fu(;)g(y)s FA(\))p Fu(;)g(A)p FA(\()p Fu(r)r FA(\()p Fr(S)j Fu(y)s FA(\))p Fu(;)d Fr(S)g Fu(y)s FA(\))p 515 4207 1036 4 v 1569 4224 a Ft(\()p Fv(I)5 b(nd)p Ft(\))591 4323 y FA(\000)643 4293 y Fp(0)666 4323 y Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(r)r FA(\(0\))p Fu(;)g FA(0\))p Fu(;)g(A)p FA(\()p Fu(r)r FA(\()p Fu(t)p FA(\))p Fu(;)g(t)p FA(\))472 b Fu(r)r FA(\(0\))24 b(=)e Fu(q)p 591 4365 1649 4 v 1026 4481 a FA(\000)1078 4451 y Fp(0)1101 4481 y Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(q)s(;)g FA(0\))p Fu(;)g(A)p FA(\()p Fu(r)r FA(\()p Fu(t)p FA(\))p Fu(;)g(t)p FA(\))466 b(\000)2319 4451 y Fp(0)2342 4481 y Fu(;)14 b(A)p FA(\()p Fu(q)s(;)g FA(0\))p 998 4522 1626 4 v 2643 4539 a Ft(Cut)1591 4639 y FA(\000)1643 4609 y Fp(0)1666 4639 y Fu(;)g(A)p FA(\()p Fu(r)r FA(\()p Fu(t)p FA(\))p Fu(;)g(t)p FA(\))p 1588 4680 447 4 v 2073 4697 a Ft(\()p Fp(9)p Ft(\))1588 4796 y FA(\000)1640 4766 y Fp(0)1663 4796 y Fu(;)g Fx(9)p Fu(xA)p FA(\()p Fu(x;)g(t)p FA(\))0 5012 y(3.)36 b(In)28 b(all)g(other)f(cases)f(the)i(claim)f(follo)n(ws)g(immediately)h(from)f (the)h(I.H.)0 5278 y Fz(De\014nition.)78 b FA(A)28 b(form)n(ula)e(of)i (the)g(form)f Fx(8)o Fu(~)-41 b(x)p Fx(9)o Fu(~)g(y)s(A)28 b FA(with)g Fu(A)23 b Fx(2)g Fr(QF)28 b FA(is)g(called)f(a)g(\005)2560 5248 y Ft(0)2560 5299 y(2)2598 5278 y FA(-form)n(ula.)1908 5539 y(15)p eop %%Page: 16 17 16 16 bop 0 83 a Fz(Corollary)32 b(2.10.)0 224 y FA(I\006)90 236 y Ft(1)161 224 y FA(is)i(\005)313 194 y Ft(0)313 245 y(2)350 224 y FA(-conserv)-5 b(ativ)n(e)32 b(o)n(v)n(er)g Fr(PRA)p FA(,)j(i.e.)55 b(I\006)1508 236 y Ft(1)1579 224 y FA(is)34 b(an)f(extension)g(of)h Fr(PRA)g FA(whic)n(h)f(pro)n(v)n (es)f(the)i(same)f(\005)3391 194 y Ft(0)3391 245 y(2)3429 224 y FA(-sen)n(tences)g(as)0 365 y Fr(PRA)p FA(.)k(Esp)r(ecially)-7 b(,)27 b(I\006)722 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Fx(9)p Fu(y)s(B)2932 1664 y Fv(y)2972 1694 y FA(.)38 b(No)n(w)27 b(b)n(y)h(the)g(Prop)r(osition)0 1835 y(b)r(elo)n(w)f(w)n(e)g(obtain)h(I\006)706 1847 y Ft(1)766 1835 y Fx(`)23 b Fu(C)29 b Fx(!)23 b(9)p Fu(z)t Fx(8)p Fu(x)f Fx(\024)h Fu(tB)1424 1805 y Fv(z)0 2001 y Fy(Prop)r(osition.)72 b FA(I\006)606 2013 y Ft(1)666 2001 y Fx(`)23 b(8)p Fu(x)g Fx(\024)f Fu(x)991 2013 y Ft(1)1029 2001 y Fx(9)p Fu(y)s(B)1186 1971 y Fv(y)1249 2001 y Fx(!)h(9)p Fu(z)t Fx(8)p Fu(x)f Fx(\024)h Fu(x)1695 2013 y Ft(1)1732 2001 y Fu(B)1799 1971 y Fv(z)1866 2001 y FA(for)k(eac)n(h)f(\006-form)n(ula)h Fu(B)0 2167 y FA(Pro)r(of)f(\(in)j(I\006)447 2179 y Ft(1)484 2167 y FA(\):)37 b(Assume)28 b Fx(8)p Fu(x)22 b Fx(\024)h Fu(x)1135 2179 y Ft(1)1173 2167 y Fx(9)p Fu(y)s(B)t FA(\()p Fu(x)p FA(\))1441 2137 y Fv(y)1537 2167 y FA(\()p Fx(\003)p FA(\).)0 2308 y(No)n(w)k(b)n(y)g(induction)h(on)g Fu(x)836 2320 y Ft(0)901 2308 y FA(w)n(e)f(pro)n(v)n(e)f Fx(9)p Fu(z)t Fx(8)p Fu(x)c Fx(\024)h Fu(x)1587 2320 y Ft(0)1625 2308 y FA(\()p Fu(x)g Fx(\024)g Fu(x)1862 2320 y Ft(1)1923 2308 y Fx(!)g Fu(B)t FA(\()p Fu(x)p FA(\))2207 2278 y Fv(z)2246 2308 y FA(\).)0 2449 y(Due)28 b(to)g(Lemma)f(2.8)g(this)h (induction)g(is)f(admissible)g(in)h(I\006)1909 2461 y Ft(1)1946 2449 y FA(.)0 2615 y(Start:)37 b Fx(`)22 b Fu(B)t FA(\(0\))491 2585 y Fv(y)555 2615 y Fx(!)h(8)p Fu(x)f Fx(\024)h FA(0\()p Fu(x)g Fx(\024)g Fu(x)1144 2627 y Ft(1)1205 2615 y Fx(!)g Fu(B)t FA(\()p Fu(x)p FA(\))1489 2585 y Fv(y)1530 2615 y FA(\))46 b Fx(\))23 b(`)g(9)p Fu(y)s(B)t FA(\(0\))2051 2585 y Fv(y)2114 2615 y Fx(!)g(9)p Fu(z)t Fx(8)p Fu(x)f Fx(\024)h FA(0\()p Fu(x)g Fx(\024)g Fu(x)2792 2627 y Ft(1)2853 2615 y Fx(!)g Fu(B)t FA(\()p Fu(x)p FA(\))3137 2585 y Fv(z)3176 2615 y FA(\)\).)0 2781 y(Step:)37 b(By)28 b(IH)g(w)n(e)f(ha)n(v)n(e)f(a)h Fu(z)k FA(with)d Fx(8)p Fu(x)23 b Fx(\024)g Fu(x)1365 2793 y Ft(0)1402 2781 y FA(\()p Fu(x)h Fx(\024)f Fu(x)1640 2793 y Ft(1)1701 2781 y Fx(!)g Fu(B)t FA(\()p Fu(x)p FA(\))1985 2751 y Fv(z)2024 2781 y FA(\).)0 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b(8)p Fu(z)26 b Fx(2)d FA(FV)r(\()p Fu(t)p FA(\)\()p Fu(z)t(\017)2037 3930 y Fv(y)2037 3981 y(x)2101 3960 y FA(=)g Fu(z)t FA(\))2287 3906 y Ft(\(S1\))p Fv(;)p Ft(\(S2\))2377 3960 y Fx(,)113 b Fu(t\017)2637 3930 y Fv(y)2637 3981 y(x)2701 3960 y FA(=)23 b Fu(t)p FA(.)0 4126 y(\(b\))28 b(Let)g Fu(x)c Fx(6)p FA(=)e Fu(y)s FA(.)37 b(Then:)g Fu(x)23 b Fx(62)h FA(FV)r(\()p Fu(t\022)r FA(\))1223 4072 y Ft(\(a\))1224 4126 y Fx(,)g Fu(t\022)r(\017)1436 4096 y Fv(y)1436 4147 y(x)1501 4126 y FA(=)f Fu(t\022)1683 4072 y Ft(\(S1\))p Fv(;)p Ft(\(S3\))1773 4126 y Fx(,)113 b(8)p Fu(z)26 b Fx(2)d FA(FV)r(\()p Fu(t)p FA(\)\()p Fu(z)t(\022)r(\017)2521 4096 y Fv(y)2521 4147 y(x)2586 4126 y FA(=)f Fu(z)t(\022)r FA(\))2812 4072 y Ft(\(a\))2813 4126 y Fx(,)i(8)p Fu(z)i Fx(2)d FA(FV)r(\()p Fu(t)p FA(\)\()p Fu(x)h Fx(62)g FA(FV)q(\()p Fu(z)t(\022)r FA(\)\).)0 4292 y(\(c\))k(1.)37 b Fu(x\022)319 4262 y Fv(y)317 4313 y(x)359 4292 y Fu(\017)393 4262 y Fv(s)393 4313 y(y)456 4292 y FA(=)23 b Fu(y)s(\017)622 4262 y Fv(s)622 4313 y(y)684 4292 y FA(=)g Fu(s)g FA(=)f Fu(x\022)1009 4262 y Fv(s)1007 4313 y(x)1049 4292 y FA(.)0 4433 y(2.)36 b Fu(x)24 b Fx(6)p FA(=)f Fu(z)j Fx(2)d FA(FV)r(\()p Fu(r)r FA(\))47 b Fx(\))g Fu(z)26 b Fx(2)d FA(FV)r(\()p Fu([)5 b(x:r)r FA(\))1320 4379 y Ft(\(b\))1323 4433 y Fx(\))48 b Fu(y)26 b Fx(62)e FA(FV)q(\()p Fu(z)t(\022)r FA(\))46 b Fx(\))h Fu(z)t(\022)2125 4403 y Fv(y)2123 4454 y(x)2164 4433 y Fu(\017)2198 4403 y Fv(s)2198 4454 y(y)2261 4433 y FA(=)22 b Fu(z)t(\022)r(\017)2466 4403 y Fv(s)2466 4454 y(y)2529 4379 y Ft(\(S1\))p Fv(;)p Ft(\(S2\))2628 4433 y FA(=)122 b Fu(z)t(\022)24 b FA(=)f Fu(z)t(\022)3093 4403 y Fv(s)3091 4454 y(x)3132 4433 y FA(.)0 4575 y(No)n(w)k(the)h (claim)f(follo)n(ws)g(b)n(y)g(\(S1\),\(S3\).)0 4741 y(\(d\))h(FV)r(\()p Fu(t)318 4753 y Fv(x)360 4741 y FA(\()p Fu(s)p FA(\)\))519 4686 y Ft(\(b\))531 4741 y FA(=)631 4678 y Fq(S)700 4765 y Fv(z)r Fp(2)p Ft(FV\()p Fv(t)p Ft(\))966 4741 y FA(FV)r(\()p Fu(z)1155 4753 y Fv(x)1197 4741 y FA(\()p Fu(s)p FA(\)\))23 b(=)1443 4678 y Fq(S)1512 4765 y Fv(z)r Fp(2)p Ft(FV)q(\()p Fv(t)p Ft(\))p Fp(nf)p Fv(x)p Fp(g)1905 4741 y Fx(f)p Fu(z)t Fx(g)16 b([)j FA(FV)r(\()p Fu(s)p FA(\))k(=)g(\(FV)r(\()p Fu(t)p FA(\))c Fx(n)f(f)p Fu(x)p Fx(g)p FA(\))g Fx([)h FA(FV)q(\()p Fu(s)p FA(\).)0 4907 y(\(e\))28 b(follo)n(ws)f(from)g (\(c\))h(with)g Fu(\022)e FA(=)c Fu(\017)p FA(.)0 5073 y(\(f)6 b(\))29 b(1.)37 b Fu(x)273 5085 y Fv(x)315 5073 y FA(\()p Fu(s)p FA(\))p Fu(\022)26 b FA(=)c Fu(s\022)k FA(=)c Fu(x\022)849 5043 y Fv(s\022)847 5093 y(x)919 5073 y FA(.)64 b(2.)37 b Fu(y)26 b Fx(6)p FA(=)c Fu(x)i Fx(\))f Fu(y)1480 5085 y Fv(x)1521 5073 y FA(\()p Fu(s)p FA(\))p Fu(\022)j FA(=)d Fu(y)s(\022)i FA(=)d Fu(y)s(\022)2057 5043 y Fv(s\022)2055 5093 y(x)2126 5073 y FA(.)65 b(No)n(w)27 b(the)h(claim)f(follo)n(ws)g(b)n(y)g(\(S1\),\(S3\).)0 5239 y(\(g\))h Fu(r)171 5251 y Fv(x)213 5239 y FA(\()p Fu(s)p FA(\))p Fu(\022)381 5184 y Ft(\(f)5 b(\))387 5239 y FA(=)29 b Fu(r)r(\022)561 5209 y Fv(s\022)559 5259 y(x)654 5184 y Ft(\(c\))662 5239 y FA(=)i Fu(r)r(\022)838 5209 y Fv(y)836 5259 y(x)879 5239 y Fu(\017)913 5209 y Fv(s\022)913 5259 y(y)982 5239 y FA(.)1908 5539 y(18)p eop %%Page: 19 20 19 19 bop 0 83 a Fz(Lemma)30 b(3.3.)105 b FA(F)-7 b(or)27 b Fu(r)n(;)14 b(r)854 53 y Fp(0)901 83 y Fx(2)24 b(T)49 b FA(the)28 b(follo)n(wing)e(holds)0 224 y(\(B1\))i Fu([)5 b(x:r)26 b FA(=)c Fu([)5 b(x:r)596 194 y Fp(0)667 224 y FA(=)-14 b Fx(\))46 b Fu(r)26 b FA(=)c Fu(r)1036 194 y Fp(0)1060 224 y FA(.)0 365 y(\(B2\))28 b Fu(y)d Fx(62)f FA(FV)q(\(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)r FA(\))49 b(=)-14 b Fx(\))46 b FA(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)27 b FA(=)c Fu([)5 b(y)s(:r)r(\022)1547 335 y Fv(y)1545 386 y(x)1587 365 y FA(.)0 531 y(The)28 b(pro)r(of)f(will)g(b)r(e)h(giv) n(en)f(b)r(elo)n(w.)0 722 y Fz(Lemma)j(3.4.)105 b FA(F)-7 b(or)27 b Fu(r)n(;)14 b(r)854 692 y Fp(0)901 722 y Fx(2)24 b(T)49 b FA(w)n(e)27 b(ha)n(v)n(e)0 863 y(\(a\))h Fu(y)e Fx(62)d FA(FV)r(\()p Fu([)5 b(x:r)r FA(\))47 b(=)-14 b Fx(\))46 b Fu([)5 b(x:r)27 b FA(=)22 b Fu([)5 b(y)s(:r)1233 875 y Fv(x)1275 863 y FA(\()p Fu(y)s FA(\).)0 1005 y(\(b\))28 b Fu([)5 b(x:r)27 b FA(=)22 b Fu([)5 b(y)s(:r)539 974 y Fp(0)609 1005 y FA(=)-14 b Fx(\))46 b Fu(r)828 974 y Fp(0)875 1005 y FA(=)23 b Fu(r)1000 1017 y Fv(x)1042 1005 y FA(\()p Fu(y)s FA(\).)0 1146 y(\(c\))28 b Fu([)5 b(x:r)26 b FA(=)d Fu([)5 b(y)s(:r)530 1116 y Fp(0)623 1146 y Fx(\()-14 b(\))69 b(8)p Fu(s)22 b Fx(2)i(T)d FA(\()p Fu(r)1166 1158 y Fv(x)1209 1146 y FA(\()p Fu(s)p FA(\))i(=)g Fu(r)1462 1116 y Fp(0)1460 1166 y Fv(y)1500 1146 y FA(\()p Fu(s)p FA(\)\).)0 1312 y(Pro)r(of:)0 1453 y(\(a\))28 b(follo)n(ws)e(from)i(\(B2\))f(with)h Fu(\022)e FA(:=)c Fu(\017)p FA(.)0 1594 y(\(b\))29 b Fu([)5 b(x:r)27 b FA(=)d Fu([)5 b(y)s(:r)542 1564 y Fp(0)614 1540 y Ft(\(B0\))638 1594 y Fx(\))72 b Fu(y)27 b Fx(62)d FA(FV)r(\()p Fu([)5 b(x:r)r FA(\))1318 1540 y Ft(\(B2\))1342 1594 y Fx(\))72 b Fu([)5 b(y)s(:r)1640 1564 y Fp(0)1688 1594 y FA(=)23 b Fu([)5 b(x:r)28 b FA(=)23 b Fu([)5 b(y)s(:r)2177 1606 y Fv(x)2219 1594 y FA(\()p Fu(y)s FA(\))2375 1540 y Ft(\(B1\))2400 1594 y Fx(\))72 b Fu(r)2594 1564 y Fp(0)2642 1594 y FA(=)24 b Fu(r)2768 1606 y Fv(x)2810 1594 y FA(\()p Fu(y)s FA(\).)39 b(\(c\))29 b(\\)p Fx(\))p FA(":)37 b Fu([)5 b(x:r)27 b FA(=)d Fu([)5 b(y)s(:r)3740 1564 y Fp(0)3811 1540 y Ft(\(b\))3814 1594 y Fx(\))0 1735 y Fu(y)26 b Fx(62)d FA(FV)r(\()p Fu([)5 b(x:r)r FA(\))25 b(&)d Fu(r)622 1747 y Fv(x)664 1735 y FA(\()p Fu(y)s FA(\))i(=)e Fu(r)922 1705 y Fp(0)993 1688 y Ft(L)p Fv(:)p Ft(3)p Fv(:)p Ft(2e)1039 1735 y Fx(\))93 b Fu(r)1252 1747 y Fv(x)1294 1735 y FA(\()p Fu(s)p FA(\))24 b(=)e Fu(r)1545 1747 y Fv(x)1588 1735 y FA(\()p Fu(y)s FA(\))1696 1747 y Fv(y)1736 1735 y FA(\()p Fu(s)p FA(\))h(=)g Fu(r)1989 1705 y Fp(0)1987 1756 y Fv(y)2027 1735 y FA(\()p Fu(s)p FA(\).)0 1876 y(\\)p Fx(\()p FA(":)36 b Fu(r)26 b FA(=)c Fu(r)413 1888 y Fv(x)456 1876 y FA(\()p Fu(x)p FA(\))i(=)e Fu(r)717 1846 y Fp(0)715 1897 y Fv(y)756 1876 y FA(\()p Fu(x)p FA(\))914 1829 y Ft(L)p Fv(:)p Ft(3)p Fv(:)p Ft(2d)964 1876 y Fx(\))96 b FA(FV)r(\()p Fu(r)r FA(\))24 b(=)f(FV)r(\()p Fu(r)1665 1846 y Fp(0)1663 1897 y Fv(y)1703 1876 y FA(\()p Fu(x)p FA(\)\))i Fx(\022)d FA(\(FV)r(\()p Fu(r)2179 1846 y Fp(0)2204 1876 y FA(\))c Fx(n)g(f)p Fu(y)s Fx(g)p FA(\))g Fx([)g(f)p Fu(x)p Fx(g)46 b(\))0 2017 y(\))g Fu(y)26 b Fx(62)d FA(FV)r(\()p Fu(r)r FA(\))d Fx(n)e(f)p Fu(x)p Fx(g)k FA(=)h(FV)q(\()p Fu([)5 b(x:r)r FA(\))1191 1963 y Ft(\(a\))1192 2017 y Fx(\))47 b Fu([)5 b(x:r)27 b FA(=)22 b Fu([)5 b(y)s(:r)1721 2029 y Fv(x)1763 2017 y FA(\()p Fu(y)s FA(\))23 b(=)g Fu([)5 b(y)s(:r)2125 1987 y Fp(0)2123 2038 y Fv(y)2163 2017 y FA(\()p Fu(y)s FA(\))23 b(=)g Fu([)5 b(y)s(:r)2525 1987 y Fp(0)2549 2017 y FA(.)0 2208 y Fz(De\014nition.)0 2350 y Fu([)p Fx(T)45 b FA(:=)22 b Fx(f)p Fu([)5 b(x:r)26 b FA(:)d Fu(x)h Fx(2)f Fr(V)n(a)n(rs)g FA(&)g Fu(r)i Fx(2)f(T)d(g)0 2491 y Fu(\014)27 b FA(:)d Fu([)p Fx(T)d(\002T)44 b(!)23 b(T)e FA(,)28 b Fu(\014)t FA(\()p Fu([)5 b(x:r)n(;)14 b(s)p FA(\))25 b(:=)d Fu(r)1101 2503 y Fv(x)1144 2491 y FA(\()p Fu(s)p FA(\))55 b(\(due)28 b(to)g(Lemma)f(3.4c,)g Fu(\014)32 b FA(is)27 b(w)n(ell)h(de\014ned\))0 2682 y Fz(Lemma)i(3.5.)0 2823 y FA(\(a\))e Fu(r)n(;)14 b(s)23 b Fx(2)g(T)68 b(\))46 b FA(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)27 b Fx(2)c Fu([)p Fx(T)45 b FA(&)23 b Fu(\014)t FA(\(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)r(;)14 b(s)p FA(\))25 b(=)e Fu(r)r(\022)1787 2793 y Fv(s)1785 2843 y(x)1828 2823 y FA(.)0 2964 y(\(b\))28 b Fu(t)23 b Fx(2)h Fu([)p Fx(T)44 b FA(&)23 b Fu(y)j Fx(62)d FA(FV)r(\()p Fu(t)p FA(\))47 b Fx(\))f Fu(t)23 b FA(=)f Fu([)5 b(y)s(:\014)t FA(\()p Fu(t;)14 b(y)s FA(\).)0 3105 y(\(c\))28 b Fu(t)23 b Fx(2)h Fu([)p Fx(T)44 b FA(&)23 b Fu(s)g Fx(2)g(T)68 b(\))46 b Fu(\014)t FA(\()p Fu(t;)14 b(s)p FA(\))p Fu(\022)26 b FA(=)d Fu(\014)t FA(\()p Fu(t\022)r(;)14 b(s\022)r FA(\).)0 3271 y(Pro)r(of:)0 3412 y(\(a\))28 b(Let)g Fu(y)d Fx(62)f FA(FV)q(\(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)r FA(\).)40 b(Then)28 b Fu(\014)t FA(\(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)r(;)14 b(s)p FA(\))25 b(=)e Fu(\014)t FA(\()p Fu([)5 b(y)s(:r)r(\022)1962 3382 y Fv(y)1960 3433 y(x)2003 3412 y Fu(;)14 b(s)p FA(\))23 b(=)g(\()p Fu(r)r(\022)2334 3382 y Fv(y)2332 3433 y(x)2375 3412 y FA(\))2407 3424 y Fv(y)2448 3412 y FA(\()p Fu(s)p FA(\))2574 3365 y Ft(L3)p Fv(:)p Ft(2c)2620 3412 y FA(=)69 b Fu(r)r(\022)2834 3382 y Fv(s)2832 3433 y(x)2874 3412 y FA(.)0 3578 y(\(b\))28 b(Let)g Fu(t)23 b FA(=)g Fu([)5 b(x:r)r FA(.)38 b(Then)28 b Fu(t)23 b FA(=)f Fu([)5 b(y)s(:r)1133 3590 y Fv(x)1175 3578 y FA(\()p Fu(y)s FA(\))24 b(=)e Fu([)5 b(y)s(:\014)t FA(\()p Fu(t;)14 b(y)s FA(\).)0 3744 y(\(c\))28 b(Let)g Fu(t)23 b FA(=)g Fu([)5 b(x:r)r FA(.)38 b Fu(\014)t FA(\(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)r(;)14 b(s\022)r FA(\))1135 3690 y Ft(\(a\))1145 3744 y FA(=)33 b Fu(r)r(\022)1323 3714 y Fv(s\022)1321 3765 y(x)1416 3697 y Ft(3)p Fv(:)p Ft(2f)1439 3744 y FA(=)46 b Fu(r)1587 3756 y Fv(x)1629 3744 y FA(\()p Fu(s)p FA(\))p Fu(\022)26 b FA(=)d Fu(\014)t FA(\()p Fu([)5 b(x:r)n(;)14 b(s)p FA(\))p Fu(\022)r FA(.)0 3960 y Fz(Remark.)114 b FA(Giv)n(en)33 b(a)g(term)g Fu([)5 b(x:r)36 b FA(and)e(a)e(substitution)i Fu(\022)i FA(with)d(dom\()p Fu(\022)r FA(\))g(=)f Fx(f)p Fu(y)j FA(:)d Fu(y)k Fx(6)p FA(=)c Fu(y)s(\022)r Fx(g)g FA(\014nite,)j(one)e(ma)n(y)g(assume)0 4101 y(w.l.o.g.)27 b(that)h Fu(x)23 b Fx(62)h FA(FV)q(\(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)r FA(\))21 b Fx([)e FA(dom\()p Fu(\022)r FA(\),)29 b(and)e(so)g(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)27 b FA(=)c Fu([)5 b(x:r)r(\022)2270 4071 y Fv(x)2268 4122 y(x)2336 4101 y FA(=)23 b Fu([)5 b(x:r)r(\022)r FA(.)p 0 4225 3900 4 v 0 4412 a Fz(Pro)s(of)32 b(of)f(\(S0\),)h(\(B1\),)f(\(B2\))0 4578 y Fw(Inductiv)n(e)d (de\014nition)g(of)f(sets)g Fx(T)1046 4590 y Fv(n)1115 4578 y FA(=)c Fx(T)1248 4590 y Fv(n)1293 4578 y FA(\()p Fr(V)n(a)n(rs)p FA(;)14 b Fx(F)8 b FA(;)14 b Fx(B)s FA(\))27 b Fw(of)g(quasiterms)0 4719 y FA(1.)36 b Fr(V)n(a)n(rs)19 b Fx([)g(f\016)429 4731 y Fv(k)491 4719 y FA(:)24 b Fu(k)h(<)e(n)p Fx(g)g(\022)f(T)941 4731 y Fv(n)987 4719 y FA(;)0 4860 y(2.)36 b Fu(f)c Fx(2)24 b(F)313 4872 y Fv(m)398 4860 y FA(&)f Fu(t)516 4872 y Ft(1)553 4860 y Fu(;)14 b(:::;)g(t)726 4872 y Fv(m)812 4860 y Fx(2)24 b(T)936 4872 y Fv(n)1027 4860 y FA(=)-14 b Fx(\))46 b Fu(f)9 b(t)1287 4872 y Ft(1)1324 4860 y Fu(:::t)1423 4872 y Fv(m)1509 4860 y Fx(2)24 b(T)1633 4872 y Fv(n)1678 4860 y FA(;)0 5001 y(3.)36 b Fu(r)26 b Fx(2)e(T)288 5013 y Fv(n)p Ft(+1)440 5001 y FA(&)f Fu([)28 b Fx(2)23 b(B)48 b(\))f Fu([)5 b(r)25 b Fx(2)f(T)1122 5013 y Fv(n)1167 5001 y Fu(:)0 5167 y Fy(Remark.)73 b Fu(n)23 b(<)f(m)46 b Fx(\))h(T)836 5179 y Fv(n)904 5167 y Fx(\022)23 b(T)1037 5179 y Fv(m)1100 5167 y FA(.)120 b Fx(T)1309 5137 y Fp(0)1356 5167 y FA(=)1443 5105 y Fq(S)1512 5192 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1678 5167 y Fx(T)1723 5179 y Fv(n)1768 5167 y FA(.)1908 5539 y(19)p eop %%Page: 20 21 20 20 bop 0 83 a Fz(Lemma)30 b(3.6.)0 224 y FA(\(a\))e Fu(t)23 b Fx(2)g(T)310 236 y Fv(n)402 224 y Fx(\))46 b Fu(t)561 236 y Fv(x)603 224 y FA([)p Fu(n)p FA(])23 b Fx(2)g(T)845 236 y Fv(n)p Ft(+1)975 224 y FA(.)0 365 y(\(b\))28 b Fx(T)45 b FA(=)22 b Fx(T)360 377 y Ft(0)398 365 y FA(.)0 506 y(\(c\))28 b Fu(t)23 b Fx(2)h(T)306 518 y Fv(n)397 506 y Fx(\))46 b Fu(t\022)26 b Fx(2)d(T)744 518 y Fv(n)789 506 y FA(.)0 648 y(\(d\))28 b Fu(t;)14 b(t)235 617 y Fp(0)282 648 y Fx(2)23 b(T)405 660 y Fv(n)473 648 y FA(&)g Fu(t)591 660 y Fv(x)633 648 y FA([)p Fu(n)p FA(])g(=)g Fu(t)870 617 y Fp(0)870 668 y Fv(x)911 648 y FA([)p Fu(n)p FA(])46 b Fx(\))h Fu(t)23 b FA(=)f Fu(t)1353 617 y Fp(0)1376 648 y FA(.)0 789 y(\(e\))28 b Fu(t)23 b Fx(2)h(T)306 801 y Fv(n)374 789 y FA(&)f Fu(y)i FA(=)e Fu(x\022)j Fx(62)806 726 y Fq(S)875 814 y Fv(z)r Fp(2)p Ft(FV)q(\()p Fv(t)p Ft(\))p Fp(nf)p Fv(x)p Fp(g)1282 789 y FA(FV)q(\()p Fu(z)t(\022)r FA(\))47 b Fx(\))f Fu(t)1753 801 y Fv(x)1794 789 y FA([)p Fu(n)p FA(])p Fu(\022)26 b FA(=)c(\()p Fu(t\022)r FA(\))2177 801 y Fv(y)2218 789 y FA([)p Fu(n)p FA(].)0 955 y(Pro)r(of:)0 1096 y(\(a\))28 b(ob)n(vious.)0 1237 y(\(b\))g(\\)p Fx(T)44 b(\022)23 b(T)402 1249 y Ft(0)439 1237 y FA(":)37 b Fu(r)26 b Fx(2)d(T)727 1249 y Ft(0)811 1183 y(\(a\))812 1237 y Fx(\))47 b Fu(r)979 1249 y Fv(x)1021 1237 y FA([0])23 b Fx(2)g(T)1255 1249 y Ft(1)1339 1237 y Fx(\))46 b Fu([)5 b(x:r)26 b FA(=)d Fu([)5 b(r)1800 1249 y Fv(x)1842 1237 y FA([0])23 b Fx(2)g(T)2076 1249 y Ft(0)2113 1237 y FA(.)0 1378 y(\\)p Fx(T)87 1390 y Ft(0)147 1378 y Fx(\022)g(T)e FA(":)36 b(One)28 b(easily)e(pro)n(v)n (es:)35 b(\(*\))28 b Fu(t)1259 1348 y Fp(0)1306 1378 y Fx(2)23 b(T)1429 1390 y Fv(n)p Ft(+1)1581 1378 y FA(&)g Fu(x)h Fx(62)f FA(FV)r(\()p Fu(t)1998 1348 y Fp(0)2021 1378 y FA(\))47 b Fx(\))f(9)p Fu(t)23 b Fx(2)g(T)2451 1390 y Fv(n)2497 1378 y FA(\()p Fu(t)2559 1348 y Fp(0)2605 1378 y FA(=)g Fu(t)2723 1390 y Fv(x)2765 1378 y FA([)p Fu(n)p FA(]\).)0 1519 y(No)n(w)k(b)n(y)g(induction)h(on)g(lh\()p Fu(t)p FA(\))g(on)f(pro)n(v)n(es)f(\()p Fu(t)d Fx(2)h(T)1561 1531 y Ft(0)1621 1519 y Fx(\))f Fu(t)g Fx(2)h(T)d FA(\):)0 1660 y(Let)28 b Fu(t)23 b FA(=)f Fu([)5 b(r)365 1630 y Fp(0)417 1660 y FA(with)28 b Fu(r)645 1630 y Fp(0)692 1660 y Fx(2)c(T)816 1672 y Ft(1)853 1660 y FA(.)37 b(T)-7 b(ak)n(e)27 b Fu(x)c Fx(62)h FA(FV)q(\()p Fu(r)1451 1630 y Fp(0)1476 1660 y FA(\).)37 b(Then)28 b(b)n(y)f(\(*\))h(there)f(is)h (an)f Fu(r)f Fx(2)d(T)2631 1672 y Ft(0)2696 1660 y FA(with)28 b Fu(r)2924 1630 y Fp(0)2971 1660 y FA(=)23 b Fu(r)3096 1672 y Fv(x)3138 1660 y FA([0].)0 1802 y(No)n(w)k(lh\()p Fu(r)r FA(\))e(=)d(lh)q(\()p Fu(r)614 1771 y Fp(0)638 1802 y FA(\))h Fu(<)g FA(lh\()p Fu(t)p FA(\))28 b(and)g(therefore)e(b)n (y)i(I.H.)g Fu(r)e Fx(2)d(T)49 b FA(and)27 b(th)n(us)h Fu(t)23 b FA(=)g Fu([)5 b(r)2558 1814 y Fv(x)2600 1802 y FA([0])23 b(=)f Fu([)5 b(x:r)26 b Fx(2)e(T)d FA(.)0 1968 y(\(c\))28 b(follo)n(ws)f(from)g(\(b\))h(b)n(y)g(induction)g(on)f Fx(T)1381 1980 y Fv(n)1426 1968 y FA(.)0 2134 y(\(d\))h(1.)37 b Fu(t)23 b Fx(2)g Fr(V)n(a)n(rs)c Fx([)f(f\016)698 2146 y Fv(k)761 2134 y FA(:)23 b Fu(k)j(<)d(n)p Fx(g)p FA(:)36 b(Then)28 b(also)f Fu(t)1529 2104 y Fp(0)1575 2134 y Fx(2)c Fr(V)n(a)n(rs)c Fx([)f(f\016)1980 2146 y Fv(k)2043 2134 y FA(:)23 b Fu(k)j(<)d(n)p Fx(g)p FA(.)0 2275 y(1.1.)36 b Fu(t)23 b FA(=)g Fu(x)p FA(:)37 b Fu(t)444 2245 y Fp(0)444 2295 y Fv(x)486 2275 y FA([)p Fu(n)p FA(])23 b(=)g Fu(t)723 2287 y Fv(x)764 2275 y FA([)p Fu(n)p FA(])g(=)g Fx(\016)1013 2287 y Fv(n)1104 2275 y Fx(\))46 b Fu(t)1263 2245 y Fp(0)1309 2275 y FA(=)23 b Fu(x)p FA(.)0 2416 y(1.2.)36 b Fu(t)23 b Fx(6)p FA(=)g Fu(x)p FA(:)37 b Fu(t)444 2386 y Fp(0)444 2436 y Fv(x)486 2416 y FA([)p Fu(n)p FA(])23 b(=)g Fu(t)723 2428 y Fv(x)764 2416 y FA([)p Fu(n)p FA(])g(=)g Fu(t)g Fx(6)p FA(=)g Fx(\016)1154 2428 y Fv(n)1244 2416 y Fx(\))47 b Fu(t)1404 2386 y Fp(0)1450 2416 y FA(=)23 b Fu(t)1568 2386 y Fp(0)1568 2436 y Fv(x)1609 2416 y FA([)p Fu(n)p FA(])g(=)g Fu(t)p FA(.)0 2557 y(2.)36 b Fu(t)24 b FA(=)e Fu([)5 b(r)30 b FA(with)f Fu(r)c Fx(2)f(T)722 2569 y Fv(n)p Ft(+1)851 2557 y FA(:)37 b(Then)28 b Fu(t)1158 2527 y Fp(0)1204 2557 y FA(=)23 b Fu([)5 b(r)1368 2527 y Fp(0)1420 2557 y FA(with)28 b Fu(r)1648 2527 y Fp(0)1695 2557 y Fx(2)23 b(T)1818 2569 y Fv(n)p Ft(+1)1975 2557 y FA(and)28 b Fu([)5 b(r)2211 2569 y Fv(x)2253 2557 y FA([)p Fu(n)p FA(+)o(1])23 b(=)g Fu(t)2596 2569 y Fv(x)2637 2557 y FA([)p Fu(n)p FA(])h(=)e Fu(t)2874 2527 y Fp(0)2874 2578 y Fv(x)2916 2557 y FA([)p Fu(n)p FA(])h(=)g Fu([)5 b(r)3199 2527 y Fp(0)3197 2578 y Fv(x)3239 2557 y FA([)p Fu(n)p FA(+)o(1].)0 2698 y(Hence)28 b Fu(r)284 2710 y Fv(x)326 2698 y FA([)p Fu(n)p FA(+)o(1])23 b(=)g Fu(r)678 2668 y Fp(0)676 2719 y Fv(x)718 2698 y FA([)p Fu(n)p FA(+)o(1])28 b(and)f(b)n(y)g(I.H.)i Fu(r)c FA(=)e Fu(r)1580 2668 y Fp(0)1632 2698 y FA(whic)n(h)k(yields)h Fu(t)23 b FA(=)f Fu(t)2273 2668 y Fp(0)2297 2698 y FA(.)0 2864 y(\(e\))28 b(1.)37 b Fu(t)23 b FA(=)f Fu(x)p FA(:)38 b Fu(t)509 2876 y Fv(x)550 2864 y FA([)p Fu(n)p FA(])p Fu(\022)26 b FA(=)c Fx(\016)840 2876 y Fv(n)908 2864 y FA(=)h Fu(y)1037 2876 y Fv(y)1076 2864 y FA([)p Fu(n)p FA(])g(=)g(\()p Fu(x\022)r FA(\))1435 2876 y Fv(y)1476 2864 y FA([)p Fu(n)p FA(].)0 3005 y(2.)36 b Fu(x)24 b Fx(6)p FA(=)f Fu(t)g Fx(2)g Fr(V)n(a)n(rs)18 b Fx([)h(f\016)718 3017 y Fv(k)781 3005 y FA(:)k Fu(k)j(<)d(n)p Fx(g)p FA(:)36 b Fu(t)1165 3017 y Fv(x)1207 3005 y FA([)p Fu(n)p FA(])p Fu(\022)25 b FA(=)e Fu(t\022)i FA(=)e(\()p Fu(t\022)r FA(\))1772 3017 y Fv(y)1812 3005 y FA([)p Fu(n)p FA(],)28 b(since)g Fu(y)d Fx(62)f FA(FV)q(\()p Fu(t\022)r FA(\).)0 3147 y(3.)36 b Fu(t)24 b FA(=)e Fu([)5 b(r)30 b FA(with)f Fu(r)c Fx(2)f(T)722 3159 y Fv(n)p Ft(+1)851 3147 y FA(:)37 b Fu(t)941 3159 y Fv(x)983 3147 y FA([)p Fu(n)p FA(])p Fu(\022)25 b FA(=)e Fu([)5 b(r)1305 3159 y Fv(x)1347 3147 y FA([)p Fu(n)p FA(+1])p Fu(\022)1613 3099 y Ft(IH)1618 3147 y FA(=)27 b Fu([)5 b FA(\()p Fu(r)r(\022)r FA(\))1891 3159 y Fv(y)1932 3147 y FA([)p Fu(n)p FA(+)o(1])23 b(=)g(\()p Fu([)5 b FA(\()p Fu(r)r(\022)r FA(\)\))2490 3159 y Fv(y)2532 3147 y FA([)p Fu(n)p FA(])23 b(=)f(\()p Fu(t\022)r FA(\))2873 3159 y Fv(y)2914 3147 y FA([)p Fu(n)p FA(].)0 3412 y(\(S0\))28 b Fu(t)23 b Fx(2)g(T)401 3365 y Ft(3)p Fv(:)p Ft(6b)420 3412 y Fx(\))43 b Fu(t)23 b Fx(2)h(T)723 3424 y Ft(0)783 3365 y(3)p Fv(:)p Ft(6c)800 3412 y Fx(\))39 b Fu(t\022)25 b Fx(2)f(T)1140 3424 y Ft(0)1200 3365 y(3)p Fv(:)p Ft(6b)1220 3412 y Fx(\))43 b Fu(t\022)25 b Fx(2)e(T)f FA(.)0 3578 y(\(B1\))28 b Fu(r)n(;)14 b(r)304 3548 y Fp(0)351 3578 y Fx(2)23 b(T)45 b FA(&)22 b Fu([)5 b(x:r)26 b FA(=)d Fu([)5 b(x:r)1010 3548 y Fp(0)1058 3531 y Ft(3)p Fv(:)p Ft(6b)1077 3578 y Fx(\))43 b Fu(r)n(;)14 b(r)1314 3548 y Fp(0)1361 3578 y Fx(2)24 b(T)1485 3590 y Ft(0)1545 3578 y FA(&)f Fu(r)1670 3590 y Fv(x)1712 3578 y FA([0])g(=)f Fu(r)1949 3548 y Fp(0)1947 3599 y Fv(x)1990 3578 y FA([0])2100 3531 y Ft(3)p Fv(:)p Ft(6d)2120 3578 y Fx(\))43 b Fu(r)26 b FA(=)d Fu(r)2436 3548 y Fp(0)2460 3578 y FA(.)0 3744 y(\(B2\))28 b Fu(y)d Fx(62)f FA(FV)q(\(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)r FA(\))796 3697 y Ft(L)p Fv(:)p Ft(3)p Fv(:)p Ft(2b)855 3744 y FA(=)1002 3682 y Fq(S)1072 3769 y Fv(z)r Fp(2)p Ft(FV\()p Fv([)t(x:r)r Ft(\))1434 3744 y FA(FV)q(\()p Fu(z)t(\022)r FA(\))1722 3690 y Ft(\(B0\))1756 3744 y FA(=)1877 3682 y Fq(S)1946 3769 y Fv(z)r Fp(2)p Ft(FV)q(\()p Fv(r)r Ft(\))p Fp(nf)p Fv(x)p Fp(g)2360 3744 y FA(FV)q(\()p Fu(z)t(\022)r FA(\))47 b Fx(\))0 3898 y(\))f FA(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)405 3844 y Ft(\(B0\))p Fv(;)p Ft(\(S1\))509 3898 y FA(=)127 b(\()p Fu([)5 b(x:r)r FA(\))p Fu(\022)952 3868 y Fv(y)950 3919 y(x)1017 3898 y FA(=)23 b Fu([)18 b(r)1192 3910 y Fv(x)1235 3898 y FA([0])p Fu(\022)1364 3868 y Fv(y)1362 3919 y(x)1427 3847 y Ft(3)p Fv(:)p Ft(6b)p Fv(;)p Ft(e)1480 3898 y FA(=)76 b Fu([)5 b FA(\()p Fu(r)r(\022)1770 3868 y Fv(y)1768 3919 y(x)1812 3898 y FA(\))1844 3910 y Fv(y)1884 3898 y FA([0])22 b(=)h Fu([)5 b(y)s(:r)r(\022)2266 3868 y Fv(y)2264 3919 y(x)2307 3898 y FA(.)p 0 4025 3900 4 v 0 4207 a(No)n(w)27 b(w)n(e)g(come)h(bac)n(k)e(to)i(the)g(language)e (of)h(1st)g(order)g(predicate)g(logic.)0 4373 y(Let)h Fr(V)n(a)n(rs)23 b FA(:=)f Fx(f)p Fu(v)515 4385 y Ft(0)552 4373 y Fu(;)14 b(v)629 4385 y Ft(1)667 4373 y Fu(;)g(:::)p Fx(g)p FA(,)27 b(and)g Fx(L)h FA(a)g(1st)f(order)f(language)g(as)h(in)n (tro)r(duced)g(in)h Fx(x)p FA(1.)0 4539 y Fx(L)p FA(-terms)c(and)g Fx(L)p FA(-form)n(ulas)f(are)g(in)n(tro)r(duced)h(literally)f(as)h(in)g Fx(x)p FA(1,)g(but)h(with)g(the)f(di\013erence)g(that)h(no)n(w)e Fr(Q)p Fu(xA)i FA(is)f(considered)0 4680 y(as)37 b(a)g(shorthand)g(for) g Fr(Q)q Fu(x:A)p FA(,)k(i.e.,)f Fx(L)p FA(-terms)e(and)f Fx(L)p FA(-form)n(ulas)g(are)f(considered)h(as)g(elemen)n(ts)h(of)f Fx(T)22 b FA(\()p Fr(V)n(a)n(rs)p FA(;)14 b Fx(F)8 b FA(;)14 b Fx(B)s FA(\))37 b(with)0 4821 y Fx(F)31 b FA(:=)23 b Fx(L)18 b([)h(f:)p Fu(;)14 b Fx(_)p Fu(;)g Fx(^g)28 b FA(and)f Fx(B)e FA(:=)e Fx(f8)p Fu(;)14 b Fx(9g)p FA(.)0 5012 y Fs(In)29 b(the)h(fol)t(lowing,)j Fu(\022)f Fs(r)l(anges)e(over)g (substitutions)f(having)i(the)f(pr)l(op)l(erty)h(that)e Fu(x\022)k Fs(is)d(an)g Fx(L)p Fs(-term)f(for)i(e)l(ach)f Fu(x)24 b Fx(2)f Fr(V)n(a)n(rs)p Fs(.)0 5153 y(Mor)l(e)l(over)31 b(we)f(assume)g(that)g FA(dom)o(\()p Fu(\022)r FA(\))24 b(:=)f Fx(f)p Fu(x)g Fx(2)g Fr(V)n(a)n(rs)g FA(:)g Fu(x\022)j Fx(6)p FA(=)d Fu(x)p Fx(g)30 b Fs(is)g(\014nite.)1908 5539 y FA(20)p eop %%Page: 21 22 21 21 bop 0 83 a Fz(Lemma)30 b(3.7)0 224 y FA(\(a\))e(If)g Fu(t)f FA(is)h(an)f Fx(L)p FA(-term)h(then)g Fu(t\022)i FA(is)d(an)h Fx(L)p FA(-term.)0 365 y(\(b\))g(If)g Fu(C)34 b FA(is)28 b(an)f Fx(L)p FA(-form)n(ula)g(then)h Fu(C)6 b(\022)30 b FA(is)e(an)f Fx(L)p FA(-form)n(ula.)0 531 y(Pro)r(of)f(b)n(y)i(induction)g(on)f(the)h(de\014nition)g(of)g Fx(L)p FA(-terms)f(and)g Fx(L)p FA(-form)n(ulas:)0 672 y(The)36 b(only)f(non)n(trivial)f(case)h(is)g Fu(C)43 b FA(=)36 b Fx(8)p Fu(xA)p FA(.)60 b(Cho)r(ose)35 b Fu(y)k Fx(62)d FA(FV)r(\()p Fu(C)6 b(\022)r FA(\).)62 b(Then)36 b Fu(C)6 b(\022)39 b FA(=)c Fx(8)p Fu(y)s(A\022)2975 642 y Fv(y)2973 693 y(x)3015 672 y FA(,)i(and)f(b)n(y)f(I.H.)h Fu(A\022)3645 642 y Fv(y)3643 693 y(x)3721 672 y FA(is)f(an)0 814 y Fx(L)p FA(-form)n(ula.)h(This)27 b(yields)h(the)g(claim.)0 1029 y Fz(De\014nition)f FA(of)g(the)h(truth)g(v)-5 b(alue)28 b([)-14 b([)9 b Fu(C)d FA(])-14 b(])1249 999 y Fp(M)1249 1053 y Fv(\030)1360 1029 y FA(of)27 b(a)g(form)n(ula)g Fu(C)34 b FA(in)28 b(an)f(in)n(terpretation)g(\()p Fx(M)p Fu(;)14 b(\030)t FA(\))0 1196 y(Let)28 b Fx(M)f FA(b)r(e)h(an)f Fx(L)p FA(-structure)g(with)i(univ)n(erse)d Fu(M)9 b FA(,)27 b(and)h(let)g Fu(\030)t(;)14 b(\021)31 b FA(range)26 b(o)n(v)n(er)g Fx(M)p FA(-assignmen)n(ts,)0 1337 y(i.e.,)i(functions)g Fu(\030)f FA(:)c Fr(V)n(a)n(rs)g Fx(!)g Fu(M)9 b FA(.)0 1503 y(F)-7 b(or)33 b(eac)n(h)g Fx(M)p FA(-assignmen)n(t)g Fu(\030)t FA(,)j(the)e Fs(value)h FA([)-14 b([)9 b Fu(t)p FA(])-14 b(])1480 1473 y Fp(M)1480 1526 y Fv(\030)1596 1503 y Fx(2)34 b Fu(M)43 b FA(of)34 b(an)f Fx(L)p FA(-term)h Fu(t)g FA(and)g(the)g Fs(truth)h(value)g FA([)-14 b([)9 b Fu(A)p FA(])-14 b(])3269 1473 y Fp(M)3269 1526 y Fv(\030)3385 1503 y Fx(2)34 b(f)p FA(0)p Fu(;)14 b FA(1)p Fx(g)32 b FA(of)i(an)0 1644 y Fx(L)p FA(-form)n(ula)29 b(are)g(de\014ne)h(as)f (usual.)43 b(Only)29 b(the)h(quan)n(ti\014er)f(case)g(requires)g(some)g (additional)g(care;)h(here)f(w)n(e)h(mak)n(e)f(use)g(of)0 1785 y(some)e(previously)f(\014xed)i(function)g Fr(v)g FA(whic)n(h)g(assigns)e(to)h(eac)n(h)g(form)n(ula)g Fu(C)34 b FA(a)27 b(v)-5 b(ariable)26 b Fr(v)q FA(\()p Fu(C)6 b FA(\))24 b Fx(62)f FA(FV)r(\()p Fu(C)6 b FA(\):)0 1951 y Fs(If)30 b Fu(C)g FA(=)273 1914 y Fp(8)273 1980 y(9)321 1951 y Fu(xA)g Fs(with)g Fu(x)24 b FA(=)f Fr(v)p FA(\()p Fu(C)6 b FA(\))30 b Fs(then)g FA([)-14 b([)9 b Fu(C)d FA(])-14 b(])1319 1921 y Fp(M)1319 1975 y Fv(\030)1425 1951 y FA(:=)1552 1914 y Ft(min)1546 1980 y(max)1678 1951 y Fx(f)p FA([)g([)9 b Fu(A)p FA(])-14 b(])1855 1921 y Fp(M)1855 1975 y Fv(\030)1887 1958 y Fn(a)1885 1991 y(x)1961 1951 y FA(:)23 b Fu(a)g Fx(2)h Fu(M)9 b Fx(g)p Fs(.)0 2117 y FA(Of)29 b(course,)f(this)h(de\014nition)g(is)f(only)g (reasonable)f(if)i([)-14 b([)9 b Fu(C)d FA(])-14 b(])1821 2087 y Fp(M)1821 2141 y Fv(\030)1933 2117 y FA(do)r(es)29 b(not)f(dep)r(end)i(on)e(the)h(c)n(hoice)f(of)h Fu(x)p FA(,)g(i.e.,)h(if)f([)-14 b([)9 b Fx(8)p Fu(xA)p FA(])-14 b(])3728 2087 y Fp(M)3728 2141 y Fv(\030)3835 2117 y FA(=)0 2258 y(min)p Fx(f)p FA([)g([)9 b Fu(A)p FA(])-14 b(])315 2228 y Fp(M)315 2282 y Fv(\030)347 2265 y Fn(a)345 2298 y(x)421 2258 y FA(:)23 b Fu(a)g Fx(2)h Fu(M)9 b Fx(g)26 b FA(also)h(in)h(case)f(that)g Fu(x)d Fx(6)p FA(=)f Fr(v)p FA(\()p Fx(8)p Fu(xA)p FA(\).)37 b(This)28 b(will)g(b)r(e)g(sho)n(wn)e(no)n(w.)0 2474 y Fz(Lemma)k(3.8.)105 b Fx(8)p Fu(z)26 b Fx(2)d FA(FV)r(\()p Fu(C)6 b FA(\)\()p Fu(\030)t FA(\()p Fu(z)t FA(\))24 b(=)e Fu(\021)s FA(\()p Fu(z)t FA(\)\))47 b(=)-14 b Fx(\))46 b FA([)-14 b([)9 b Fu(C)d FA(])-14 b(])1869 2444 y Fp(M)1869 2498 y Fv(\030)1975 2474 y FA(=)23 b([)-14 b([)9 b Fu(C)d FA(])-14 b(])2201 2444 y Fp(M)2201 2495 y Fv(\021)2284 2474 y FA(.)0 2640 y(Pro)r(of:)36 b(If)28 b Fu(C)h FA(=)23 b Fx(8)p Fu(xA)k FA(with)h Fu(x)c FA(=)e Fr(v)q FA(\()p Fu(C)6 b FA(\))28 b(then)g([)-14 b([)9 b Fu(C)d FA(])-14 b(])1572 2652 y Fv(\030)1633 2640 y FA(=)22 b(min)q Fx(f)p FA([)-14 b([)9 b Fu(A)p FA(])-14 b(])2036 2652 y Fv(\030)2068 2636 y Fn(a)2066 2669 y(x)2132 2640 y FA(:)23 b Fu(a)g Fx(2)h Fu(M)9 b Fx(g)2478 2593 y Ft(IH)2482 2640 y FA(=)27 b(min)p Fx(f)p FA([)-14 b([)9 b Fu(A)p FA(])-14 b(])2889 2652 y Fv(\021)2925 2636 y Fn(a)2923 2669 y(x)2990 2640 y FA(:)23 b Fu(a)g Fx(2)g Fu(M)9 b Fx(g)22 b FA(=)h([)-14 b([)9 b Fu(C)d FA(])-14 b(])3561 2652 y Fv(\021)3602 2640 y FA(.)0 2831 y Fz(Lemma)30 b(3.9.)105 b Fx(8)p Fu(z)26 b Fx(2)d FA(FV)r(\()p Fu(C)6 b FA(\)\([)-14 b([)9 b Fu(z)t(\022)r FA(])-14 b(])1220 2801 y Fp(M)1220 2855 y Fv(\030)1326 2831 y FA(=)23 b([)-14 b([)9 b Fu(z)t FA(])-14 b(])1530 2843 y Fv(\021)1570 2831 y FA(\))47 b(=)-14 b Fx(\))46 b FA([)-14 b([)9 b Fu(C)d(\022)r FA(])-14 b(])2008 2801 y Fp(M)2008 2855 y Fv(\030)2114 2831 y FA(=)23 b([)-14 b([)9 b Fu(C)d FA(])-14 b(])2340 2843 y Fv(\021)2381 2831 y FA(.)0 2997 y(Pro)r(of:)36 b(Let)27 b Fu(C)j FA(=)22 b Fx(8)p Fu(xA)28 b FA(with)g Fu(x)c FA(=)e Fr(v)p FA(\()p Fu(C)6 b FA(\);)29 b(then)f Fu(C)6 b(\022)26 b FA(=)c Fx(8)p Fu(y)s(A\022)1934 2967 y Fv(y)1932 3018 y(x)2001 2997 y FA(with)28 b Fu(y)e FA(=)d Fr(v)p FA(\()p Fu(C)6 b(\022)r FA(\).)0 3163 y([)-14 b([)9 b Fu(C)d(\022)r FA(])-14 b(])179 3175 y Fv(\030)240 3163 y FA(=)22 b([)-14 b([)9 b Fx(8)p Fu(y)s(A\022)562 3133 y Fv(y)560 3184 y(x)602 3163 y FA(])-14 b(])634 3175 y Fv(\030)694 3163 y FA(=)22 b(min)q Fx(f)p FA([)-14 b([)9 b Fu(A\022)1106 3133 y Fv(y)1104 3184 y(x)1146 3163 y FA(])-14 b(])1178 3175 y Fv(\030)1210 3159 y Fn(a)1208 3192 y(y)1274 3163 y FA(:)23 b Fu(a)g Fx(2)h Fu(M)9 b Fx(g)1620 3109 y Ft(IH+\()p Fp(\003)p Ft(\))1693 3163 y FA(=)95 b(min)p Fx(f)p FA([)-14 b([)9 b Fu(A)p FA(])-14 b(])2168 3175 y Fv(\021)2204 3159 y Fn(a)2202 3192 y(x)2269 3163 y FA(:)23 b Fu(a)g Fx(2)g Fu(M)9 b Fx(g)22 b FA(=)h([)-14 b([)9 b Fu(C)d FA(])-14 b(])2840 3175 y Fv(\021)2881 3163 y FA(.)0 3329 y(\()p Fx(\003)p FA(\))55 b(1.)37 b([)-14 b([)9 b Fu(x\022)392 3299 y Fv(y)390 3350 y(x)433 3329 y FA(])-14 b(])465 3341 y Fv(\030)497 3325 y Fn(a)495 3358 y(y)561 3329 y FA(=)23 b Fu(a)g FA(=)f([)-14 b([)9 b Fu(x)p FA(])-14 b(])923 3341 y Fv(\021)959 3325 y Fn(a)957 3358 y(x)1001 3329 y FA(.)0 3470 y(2.)36 b(If)28 b Fu(z)f Fx(2)c FA(FV)r(\()p Fu(A)p FA(\))c Fx(n)f(f)p Fu(x)p Fx(g)23 b FA(=)f(FV)r(\()p Fu(C)6 b FA(\))28 b(then)g Fu(y)e Fx(62)e FA(FV)q(\()p Fu(z)t(\022)r FA(\))k(\(since)g Fu(y)d Fx(62)f FA(FV)r(\()p Fu(C)6 b(\022)r FA(\)\))29 b(and)e(th)n(us)0 3611 y([)-14 b([)9 b Fu(z)t(\022)125 3581 y Fv(y)123 3632 y(x)165 3611 y FA(])-14 b(])197 3623 y Fv(\030)229 3607 y Fn(a)227 3641 y(y)293 3611 y FA(=)23 b([)-14 b([)9 b Fu(z)t(\022)r FA(])-14 b(])538 3623 y Fv(\030)570 3607 y Fn(a)568 3641 y(y)634 3564 y Ft(L)p Fv(:)p Ft(3)p Fv(:)p Ft(8)675 3611 y FA(=)63 b([)-14 b([)9 b Fu(z)t(\022)r FA(])-14 b(])960 3623 y Fv(\030)1020 3611 y FA(=)23 b([)-14 b([)9 b Fu(z)t FA(])-14 b(])1224 3623 y Fv(\021)1287 3611 y FA(=)23 b([)-14 b([)9 b Fu(z)t FA(])-14 b(])1491 3623 y Fv(\021)1527 3607 y Fn(a)1525 3641 y(x)1568 3611 y FA(.)0 3827 y Fz(Lemma)30 b(3.10.)105 b FA([)-14 b([)9 b Fx(8)p Fu(xA)p FA(])-14 b(])871 3839 y Fv(\030)931 3827 y FA(=)22 b(min)q Fx(f)p FA([)-14 b([)9 b Fu(A)p FA(])-14 b(])1334 3839 y Fv(\030)1366 3823 y Fn(a)1364 3856 y(x)1430 3827 y FA(:)23 b Fu(a)g Fx(2)h Fu(M)9 b Fx(g)0 3993 y FA(Pro)r(of:)36 b(Let)27 b Fu(y)f FA(:=)d Fr(v)p FA(\()p Fx(8)p Fu(xA)p FA(\).)38 b(Then)27 b Fx(8)p Fu(xA)c FA(=)g Fx(8)p Fu(y)s(A)1541 4005 y Fv(x)1582 3993 y FA(\()p Fu(y)s FA(\),)28 b(and)f(th)n(us)0 4134 y([)-14 b([)9 b Fx(8)p Fu(xA)p FA(])-14 b(])229 4146 y Fv(\030)289 4134 y FA(=)23 b(min)p Fx(f)p FA([)-14 b([)9 b Fu(A)660 4146 y Fv(x)702 4134 y FA(\()p Fu(y)s FA(\)])-14 b(])842 4146 y Fv(\030)874 4130 y Fn(a)872 4164 y(y)938 4134 y FA(:)24 b Fu(a)e Fx(2)i Fu(M)9 b Fx(g)1284 4080 y Ft(L)p Fv(:)p Ft(3)p Fv(:)p Ft(9+\()p Fp(\003)p Ft(\))1394 4134 y FA(=)132 b(min)p Fx(f)p FA([)-14 b([)9 b Fu(A)p FA(])-14 b(])1906 4146 y Fv(\030)1938 4130 y Fn(a)1936 4164 y(x)2002 4134 y FA(:)23 b Fu(a)g Fx(2)h Fu(M)9 b Fx(g)p FA(.)0 4301 y(\()p Fx(\003)p FA(\):)37 b(cf.)28 b(\()p Fx(\003)p FA(\))g(with)g Fu(\022)d FA(:=)e Fu(\017)k FA(in)h(the)g(pro)r(of)f(of)h(3.9.)0 4566 y Fz(Lemma)i(3.11.)0 4707 y FA(All)e(rules)f(of)h Fr(PRA)f FA(and)h(I\006)871 4719 y Ft(1)936 4707 y FA(are)e(closed)h(under)h (substitution.)0 4873 y(Pro)r(of:)0 5015 y(1.)158 4960 y(\000)p Fu(;)14 b(A)p 111 4995 245 4 v 111 5069 a FA(\000)p Fu(;)g Fx(8)p Fu(xA)394 5015 y FA(with)28 b Fu(x)23 b Fx(62)h FA(FV)q(\(\000\):)0 5169 y(In)k(this)g(case)e(w)n(e)i(only)f (ha)n(v)n(e)f(to)i(consider)e(substitutions)i Fu(\022)i FA(with)e Fu(y)e FA(:=)d Fu(x\022)i Fx(2)f Fr(V)n(a)n(rs)18 b Fx(n)g FA(FV)r(\(\000)p Fu(\022)r(;)c FA(\()p Fx(8)p Fu(xA)p FA(\))p Fu(\022)r FA(\).)0 5310 y Fu(y)26 b FA(=)c Fu(x\022)k Fx(62)e FA(FV)q(\(\()p Fx(8)p Fu(xA)p FA(\))p Fu(\022)r FA(\))48 b Fx(\))e FA(\()p Fx(8)p Fu(xA)p FA(\))p Fu(\022)25 b FA(=)e Fx(8)p Fu(y)16 b(A\022)1543 5280 y Fv(y)1541 5331 y(x)1606 5310 y FA(=)23 b Fx(8)p Fu(y)16 b(A\022)r FA(.)1908 5539 y(21)p eop %%Page: 22 23 22 22 bop 0 125 a FA(2.)111 71 y(\000)p Fu(;)14 b(A)262 83 y Fv(x)304 71 y FA(\()p Fu(t)p FA(\))p 111 106 288 4 v 133 180 a(\000)p Fu(;)g Fx(9)p Fu(xA)409 125 y FA(:)37 b(Let)28 b Fu(y)d Fx(62)f FA(FV)q(\(\()p Fx(9)p Fu(xA)p FA(\))p Fu(\022)r FA(\).)40 b(Then)27 b(\()p Fx(9)p Fu(xA)p FA(\))p Fu(\022)g FA(=)c Fx(9)p Fu(y)16 b(A\022)2062 95 y Fv(y)2060 146 y(x)2130 125 y FA(and)28 b Fu(A)2354 137 y Fv(x)2396 125 y FA(\()p Fu(t)p FA(\))p Fu(\022)e FA(=)c(\()p Fu(A\022)2777 95 y Fv(y)2775 146 y(x)2818 125 y FA(\))2850 137 y Fv(y)2890 125 y FA(\()p Fu(t\022)r FA(\).)0 297 y(3.)36 b(\()p Fx(:)p FA(\()p Fu(s)24 b FA(=)f Fu(t)p FA(\))p Fu(;)14 b Fx(:)p Fu(A)587 309 y Fv(x)630 297 y FA(\()p Fu(s)p FA(\))p Fu(;)g(A)832 309 y Fv(x)874 297 y FA(\()p Fu(t)p FA(\)\))p Fu(\022)26 b FA(=)d Fx(:)p FA(\()p Fu(s\022)j FA(=)c Fu(t\022)r FA(\))p Fu(;)14 b Fx(:)p FA(\()p Fu(A\022)1761 267 y Fv(y)1759 318 y(x)1803 297 y FA(\))1835 309 y Fv(y)1875 297 y FA(\()p Fu(s\022)r FA(\))p Fu(;)g FA(\()p Fu(A\022)2191 267 y Fv(y)2189 318 y(x)2233 297 y FA(\))2265 309 y Fv(y)2305 297 y FA(\()p Fu(t\022)r FA(\),)28 b(if)g Fu(y)e Fx(62)e FA(FV)q(\(\()p Fx(8)p Fu(xA)p FA(\))p Fu(\022)r FA(\).)0 472 y(4.)146 417 y(\000)p Fu(;)14 b Fx(:)p Fu(F)r(;)g(F)435 429 y Fv(x)477 417 y FA(\()p Fr(S)h Fu(x)p FA(\))p 111 453 573 4 v 111 527 a(\000)p Fu(;)f Fx(:)p Fu(F)308 539 y Fv(x)351 527 y FA(\(0\))p Fu(;)g(F)547 539 y Fv(x)589 527 y FA(\()p Fu(t)p FA(\))721 472 y(or)833 417 y(\000)p Fu(;)g(F)975 429 y Fv(x)1017 417 y FA(\(0\))83 b(\000)p Fu(;)14 b Fx(:)p Fu(F)r(;)g(F)1495 429 y Fv(x)1538 417 y FA(\()p Fr(S)g Fu(x)p FA(\))p 833 453 877 4 v 1132 527 a(\000)p Fu(;)g(F)1274 539 y Fv(x)1316 527 y FA(\()p Fu(t)p FA(\))1747 472 y(with)29 b Fu(x)23 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b(if)j Fu(F)35 b Fx(2)23 b Fr(QF)28 b FA([\006)826 940 y Ft(1)864 928 y FA(,)f(resp.])37 b(then)28 b Fu(F)12 b(\022)25 b Fx(2)e Fr(QF)29 b FA([\006)1762 940 y Ft(1)1799 928 y FA(,)f(resp.])0 1149 y Fx(x)p Fz(4)63 b(An)32 b(alternativ)m(e)h (presen)m(tation)f(of)f(the)h(T)-8 b(ait)32 b(st)m(yle)g(sequen)m(t)g (calculus)0 1317 y FA(W)-7 b(e)27 b(in)n(tro)r(duce)f(a)h(new)f(notion) h(of)f(deriv)-5 b(ation)26 b(for)g(the)h(T)-7 b(ait)27 b(st)n(yle)f(sequen)n(t)h(calculus)f(whic)n(h)g(di\013ers)h(from)f(the) h(usual)f(one)0 1451 y(\(in)n(tro)r(duced)e(in)g Fx(x)p FA(1\))f(in)h(so)f(far)h(as)f(the)h(new)g(deriv)-5 b(ations)22 b(ha)n(v)n(e)h(so-called)f(inference)i(sym)n(b)r(ols)f(\(denoting)h (inferences\))f(and)0 1584 y(not)30 b(sequen)n(ts)f(assigned)g(to)g (their)h(no)r(des.)43 b(The)30 b(sequen)n(t)f(\\b)r(elonging")g(to)g(a) h(certain)f(no)r(de)h Fu(\034)39 b FA(of)30 b(a)f(deriv)-5 b(ation)29 b Fu(d)h FA(is)g(not)0 1718 y(explicitely)f(displa)n(y)n (ed,)g(but)g(can)g(b)r(e)h(computed)f(b)n(y)g(tree)g(recursion)e(from)i Fu(d)g FA(\(similarly)g(as)f(the)i(free)f(assumptions)f(in)h(a)0 1851 y(natural)e(deduction)h(st)n(yle)f(deriv)-5 b(ation\).)36 b(This)28 b(approac)n(h)e(is)h(particularly)f(useful)i(for)f(our)g (further)h(purp)r(oses.)0 2037 y Fz(Pro)s(of)k(systems)0 2188 y FA(A)c(pro)r(of)f(system)g Fm(S)h FA(is)f(giv)n(en)g(b)n(y)0 2339 y({)55 b(a)27 b(set)h(of)f(formal)g(expressions)f(called)h Fs(infer)l(enc)l(e)j(symb)l(ols)f FA(\(syn)n(tactic)e(v)-5 b(ariable)27 b Fx(I)6 b FA(\))0 2490 y({)55 b(for)22 b(eac)n(h)g(inference)g(sym)n(b)r(ol)g Fx(I)29 b FA(a)23 b(set)f Fx(jI)6 b(j)23 b FA(\(the)h Fs(arity)f FA(of)g Fx(I)6 b FA(\),)24 b(a)e(sequen)n(t)h(\001\()p Fx(I)6 b FA(\))23 b(and)g(a)f(family)h(of)f(sequen)n(ts)g(\(\001)3545 2502 y Fv(\023)3575 2490 y FA(\()p Fx(I)6 b FA(\)\))3722 2505 y Fv(\023)p Fp(2jI)t(j)3877 2490 y FA(.)97 2624 y(The)28 b(elemen)n(ts)f(of)h(\001\()p Fx(I)6 b FA(\))28 b([)965 2561 y Fq(S)1034 2649 y 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Fx(^)h(:)p Fu(F)12 b FA(\()p Fu(S)5 b(S)g FA(0\))p Fu(;)14 b(F)e FA(\()p Fu(S)5 b(S)g FA(0\))p 2201 481 1370 4 v 3589 498 a Ft(\()p Fp(9)p Ft(\))2562 556 y Fx(:)p Fu(F)12 b FA(\(0\))p Fu(;)i(G;)g(F)e FA(\()p Fu(S)5 b(S)g FA(0\))0 693 y(where)27 b Fu(G)c FA(:=)g Fx(9)p Fu(x)p FA(\()p Fu(F)12 b FA(\()p Fu(x)p FA(\))20 b Fx(^)f(:)p Fu(F)12 b FA(\()p Fu(S)5 b(x)p FA(\)\),)0 877 y Fz(The)32 b(pro)s(of)g(system)e Fj(Z)i Fz(of)f(1st)h(order)g(arithmetic)0 1027 y FA(The)24 b Fs(language)h FA(of)f Fj(Z)g FA(is)g Fx(L)803 1039 y Ft(0)841 1027 y FA(\()p Fx(X)12 b FA(\))24 b(:=)f Fx(L)1168 1039 y Ft(0)1216 1027 y Fx([)11 b(f)p Fu(X)1393 1039 y Ft(0)1431 1027 y Fu(;)j(X)1537 1039 y Ft(1)1574 1027 y Fu(;)g(:::)p Fx(g)p FA(,)24 b(where)g Fu(X)2075 1039 y Ft(0)2112 1027 y Fu(;)14 b(X)2218 1039 y Ft(1)2255 1027 y Fu(;)g(:::)24 b FA(are)f(unary)g(predicate)h(sym)n (b)r(ols;)g(w)n(e)g(call)g(them)0 1160 y Fy(set)32 b(v)-5 b(ariables)p FA(.)47 b(But)32 b(note)f(that)h(they)g(are)f(not)g 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Ft(\001)185 2042 y FA(\))277 b(\001)83 b(for)27 b(\001)c Fx(2)h(AX)12 b FA(\()p Fj(Z)q FA(\))28 b(,)0 2231 y(\()p Fr(Ind)141 2191 y Fv(x;t)141 2255 y(F)228 2231 y FA(\))332 2176 y Fx(:)p Fu(F)r(;)14 b(F)532 2188 y Fv(x)575 2176 y FA(\()p Fr(S)g Fu(x)p FA(\))p 298 2211 484 4 v 298 2285 a Fx(:)p Fu(F)406 2297 y Fv(x)448 2285 y FA(\(0\))p Fu(;)g(F)644 2297 y Fv(x)686 2285 y FA(\()p Fu(t)p FA(\))874 2231 y(!)p Fu(x)p FA(!)37 b(.)0 2436 y(In)28 b(this)g(section)f Fx(I)34 b FA(is)27 b(alw)n(a)n(ys)f(an)h (inference)h(sym)n(b)r(ol)f(of)g Fj(Z)q FA(,)g(and)h Fu(d;)14 b(d)2197 2405 y Fp(0)2220 2436 y Fu(;)g(:::)28 b FA(denote)g Fj(Z)p FA(-deriv)-5 b(ations.)0 2586 y Fz(De\014nition)0 2719 y Fr(Ax)94 2731 y Ft(\001)152 2719 y Fu(\022)26 b FA(:=)c Fr(Ax)421 2731 y Ft(\001)p Fv(\022)541 2719 y FA(,)592 2656 y Fq(V)661 2744 y Fv(A)715 2719 y Fu(\022)j FA(:=)890 2656 y Fq(V)959 2744 y Fv(A\022)1074 2719 y FA(,)1125 2656 y Fq(W)1194 2677 y Fv(k)1194 2744 y(A)1248 2719 y Fu(\022)g FA(:=)1423 2656 y Fq(W)1492 2677 y Fv(k)1492 2744 y(A\022)1608 2719 y FA(,)i Fr(Cut)1784 2731 y Fv(C)1840 2719 y Fu(\022)e FA(:=)e Fr(Cut)2141 2731 y Fv(C)t(\022)2258 2719 y FA(,)2309 2656 y Fq(W)2378 2677 y Fv(s)2378 2744 y Fp(9)p Fv(xA)2507 2719 y Fu(\022)i FA(:=)2682 2656 y Fq(W)2751 2677 y Fv(s\022)2751 2744 y Ft(\()p Fp(9)p Fv(xA)p Ft(\))p Fv(\022)2994 2719 y FA(,)0 2789 y Fq(V)69 2810 y Fv(x)69 2877 y Fp(8)p Fv(xA)199 2852 y Fu(\022)g FA(:=)374 2789 y Fq(V)443 2810 y Fv(x\022)443 2877 y Ft(\()p Fp(8)p Fv(xA)p Ft(\))p Fv(\022)685 2852 y FA(,)j Fr(Ind)845 2812 y Fv(x;t)845 2876 y(F)931 2852 y Fu(\022)e FA(:=)c Fr(Ind)1215 2812 y Fv(x\022)r(;t\022)1215 2877 y(F)9 b(\022)1369 2852 y FA(.)0 3002 y(In)28 b(the)g(last)f(t)n(w) n(o)g(cases)f(it)i(is)g(required)e(that)i Fu(x\022)e Fx(2)e Fr(V)n(a)n(rs)18 b Fx(n)g FA(FV)q(\(\001\()p Fx(I)6 b FA(\))p Fu(\022)r FA(\).)0 3168 y(Then)28 b(the)g(follo)n(wing)e (holds)i(for)f(ev)n(ery)f Fj(Z)p FA(-inference)h Fx(I)6 b FA(:)0 3301 y({)27 b Fx(I)6 b Fu(\022)30 b FA(is)e(a)f Fj(Z)p FA(-inferences)g(b)r(elonging)g(to)h(the)g(same)f(rule)g(as)g Fx(I)6 b FA(,)28 b(\(cf.)g(pro)r(of)f(of)h(L....\))0 3434 y({)f Fx(jI)6 b Fu(\022)r Fx(j)24 b FA(=)f Fx(jI)6 b(j)28 b FA(and)f(Eig\()p Fx(I)6 b Fu(\022)r FA(\))25 b(=)d(Eig)q(\()p Fx(I)6 b FA(\))p Fu(\022)r FA(,)0 3568 y({)27 b(\001\()p Fx(I)6 b Fu(\022)r FA(\))25 b(=)d(\001\()p Fx(I)6 b FA(\))p Fu(\022)r FA(,)0 3701 y({)27 b(\()p Fx(8)p Fu(i)22 b Fx(2)i(jI)6 b(j)p FA(\))14 b(\001)490 3713 y Fv(i)518 3701 y FA(\()p Fx(I)6 b Fu(\022)r FA(\))24 b(=)f(\001)855 3713 y Fv(i)883 3701 y FA(\()p Fx(I)6 b FA(\))p Fu(\022)r FA(.)0 3918 y Fz(De\014nition)31 b(of)h Fu(d\022)0 4068 y FA(F)-7 b(or)27 b Fu(d)c FA(=)g Fx(I)6 b Fu(d)397 4080 y Ft(0)435 4068 y Fu(:::d)547 4080 y Fv(n)p Fp(\000)p Ft(1)705 4068 y FA(w)n(e)27 b(set)55 b Fu(d\022)26 b FA(:=)d(\()p Fx(I)1293 4046 y FA(~)1286 4068 y Fu(\022)s FA(\))p Fu(d)1403 4080 y Ft(0)1447 4046 y FA(~)1440 4068 y Fu(\022)16 b(:)e(:)g(:)g(d)1649 4080 y Fv(n)p Fp(\000)p Ft(1)1786 4046 y FA(~)1780 4068 y Fu(\022)29 b FA(with)7 4218 y(~)0 4240 y Fu(\022)c FA(:=)175 4123 y Fq(\032)251 4176 y Fu(\022)292 4146 y Fv(y)290 4197 y(x)415 4176 y FA(if)j(Eig)q(\()p Fx(I)6 b FA(\))24 b(=)e Fx(f)p Fu(x)p Fx(g)110 b FA(where)28 b Fu(y)d Fx(2)f Fr(V)n(a)n(r)18 b Fx(n)g FA(FV)r(\(\000\()p Fu(d)p FA(\))p Fu(\022)r FA(\))251 4305 y Fu(\022)125 b FA(if)28 b(Eig)q(\()p Fx(I)6 b FA(\))24 b(=)e Fx(;)0 4413 y FA(In)28 b(the)g(\014rst)f(case,) g(if)h Fu(x)c Fx(62)f FA(FV)r(\(\000\()p Fu(d)p FA(\))p Fu(\022)r FA(\))29 b(w)n(e)e(tak)n(e)g Fu(y)f FA(:=)c Fu(x)p FA(.)0 4580 y Fz(Remark.)78 b FA(Let)28 b Fu(d)23 b FA(=)g Fx(I)6 b Fu(d)834 4592 y Ft(0)871 4580 y Fu(:::d)983 4592 y Fv(n)p Fp(\000)p Ft(1)1114 4580 y FA(.)0 4713 y(\(a\))28 b Fu(d\017)23 b FA(=)f Fu(d)p FA(.)0 4862 y(\(b\))28 b(If)g Fu(\022)e FA(=)c Fu(\017)407 4832 y Fv(t)407 4883 y(x)477 4862 y FA(with)28 b(FV)q(\()p Fu(t)p FA(\))c(=)f Fx(;)k FA(then)h Fu(d\022)d FA(=)1442 4745 y Fq(\032)1518 4819 y Fu(d)538 b FA(if)28 b(Eig)q(\()p Fx(I)6 b FA(\))24 b(=)e Fx(f)p Fu(x)p Fx(g)1518 4906 y(I)6 b Fu(\022)r(d)1653 4918 y Ft(0)1691 4906 y Fu(\022)r(:::d)1844 4918 y Fv(n)p Fp(\000)p Ft(1)1975 4906 y Fu(\022)85 b FA(otherwise)0 5028 y(Pro)r(of)26 b(of)i(\(b\):)0 5211 y(W)-7 b(e)28 b(ha)n(v)n(e)e Fu(x)e Fx(62)f FA(FV)r(\(\000\()p Fu(d)p FA(\))p Fu(\022)r FA(\))29 b(and)f(therefore)1411 5189 y(~)1404 5211 y Fu(\022)e FA(=)1556 5069 y Fq(\()1637 5120 y Fu(\022)1678 5090 y Fv(x)1676 5140 y(x)1743 5120 y FA(=)d Fu(\017)86 b FA(if)29 b(Eig\()p Fx(I)6 b FA(\))24 b(=)e Fx(f)p Fu(x)p Fx(g)1637 5211 y Fu(\022)1678 5181 y Fv(z)1676 5232 y(z)1740 5211 y FA(=)g Fu(\022)85 b FA(if)29 b(Eig\()p Fx(I)6 b FA(\))24 b(=)e Fx(f)p Fu(z)t Fx(g)g(6)p FA(=)g Fx(f)p Fu(x)p Fx(g)1637 5302 y Fu(\022)275 b FA(if)29 b(Eig\()p Fx(I)6 b FA(\))24 b(=)e Fx(;)1908 5539 y FA(24)p eop %%Page: 25 26 25 25 bop 0 83 a Fz(Lemma)30 b(4.1.)105 b Fj(Z)23 b Fx(3)h Fu(d)f Fx(`)g FA(\000)46 b Fx(\))g Fj(Z)23 b Fx(3)h Fu(d\022)h Fx(`)e FA(\000)p Fu(\022)r FA(.)0 236 y(Pro)r(of:)36 b(Let)27 b Fu(d)d FA(=)e Fx(I)6 b Fu(d)656 248 y Ft(0)694 236 y Fu(:::d)806 248 y Fv(n)p Fp(\000)p Ft(1)937 236 y FA(.)37 b(W.l.o.g.)f(\000\()p Fu(d)p FA(\))24 b(=)f(\000.)0 371 y(\(1\))28 b(\000)193 349 y(~)186 371 y Fu(\022)d FA(=)e(\000)p Fu(\022)r FA(,)28 b(since)f(Eig\()p Fx(I)6 b FA(\))20 b Fx(\\)e FA(FV)r(\(\000\))23 b(=)g Fx(;)p FA(.)0 506 y(\(2\))28 b(Eig\()p Fx(I)346 484 y FA(~)339 506 y Fu(\022)s FA(\))19 b Fx(\\)f FA(FV)r(\(\000)p Fu(\022)r FA(\))24 b(=)e Fx(;)p FA(,)28 b(since)f(Eig\()p Fx(I)1399 484 y FA(~)1392 506 y Fu(\022)s FA(\))d(=)e(Eig\()p Fx(I)6 b FA(\))1821 484 y(~)1814 506 y Fu(\022)31 b FA(and)d(\(Eig\()p Fx(I)6 b FA(\))24 b(=)f Fx(f)p Fu(x)p Fx(g)f(\))h Fu(x)2740 484 y FA(~)2733 506 y Fu(\022)j Fx(62)e FA(FV)q(\(\000)p Fu(\022)r FA(\)\).)0 659 y Fu(d)i Fx(`)f FA(\000)52 b Fx(\))f FA(\001\()p Fx(I)6 b FA(\))27 b Fx(\022)e FA(\000)g(&)h Fx(8)p Fu(i)e(<)h(n)p FA(\()p Fu(d)1167 671 y Fv(i)1221 659 y Fx(`)g FA(\000)p Fu(;)14 b FA(\001)1455 671 y Fv(i)1483 659 y FA(\()p Fx(I)6 b FA(\)\))1683 605 y Ft(IH+\(1\))1746 659 y Fx(\))114 b FA(\001\()p Fx(I)6 b FA(\))2134 637 y(~)2127 659 y Fu(\022)29 b Fx(\022)c FA(\000)p Fu(\022)j FA(&)d Fx(8)p Fu(i)g(<)g(n)p FA(\()p Fu(d)2810 671 y Fv(i)2845 637 y FA(~)2838 659 y Fu(\022)j Fx(`)d FA(\000)p Fu(\022)r(;)14 b FA(\001)3180 671 y Fv(i)3208 659 y FA(\()p Fx(I)6 b FA(\))3330 637 y(~)3323 659 y Fu(\022)s FA(\))52 b Fx(\))f FA(\001\()p Fx(I)3742 637 y FA(~)3735 659 y Fu(\022)s FA(\))26 b Fx(\022)0 794 y FA(\000)p Fu(\022)f FA(&)e Fx(8)p Fu(i)f(<)g(n)p FA(\()p Fu(d)514 806 y Fv(i)549 772 y FA(~)542 794 y Fu(\022)k Fx(`)c FA(\000)p Fu(\022)r(;)14 b FA(\001)879 806 y Fv(i)907 794 y FA(\()p Fx(I)997 772 y FA(~)990 794 y Fu(\022)s FA(\)\))1143 740 y Ft(\(2\))1144 794 y Fx(\))47 b Fu(d\022)26 b Fx(`)c FA(\000)p Fu(\022)r FA(.)0 999 y Fz(De\014nition.)106 b FA(FV)q(\()p Fx(I)6 b FA(\))24 b(:=)915 882 y Fq(\032)991 959 y FA(FV)r(\(\001\()p Fx(I)6 b FA(\)\))20 b Fx([)e FA(FV)r(\()p Fu(t)p FA(\))84 b(if)28 b Fx(I)h FA(=)1984 896 y Fq(W)2053 917 y Fv(t)2053 984 y Fp(9)p Fv(xA)2210 959 y FA(or)e Fr(Ind)2421 919 y Fv(x;t)2421 983 y(F)991 1054 y FA(FV)r(\(\001\()p Fx(I)6 b FA(\)\))389 b(otherwise)0 1167 y Fz(Remark.)106 b FA(Eig\()p Fx(I)6 b FA(\))19 b Fx(\\)g FA(FV)r(\()p Fx(I)6 b FA(\))24 b(=)e Fx(;)p FA(.)0 1320 y Fz(De\014nition.)106 b FA(FV)q(\()p Fx(I)6 b Fu(d)791 1332 y Ft(0)829 1320 y Fu(:::d)941 1332 y Fv(n)p Fp(\000)p Ft(1)1072 1320 y FA(\))23 b(:=)g(FV)q(\()p Fx(I)6 b FA(\))20 b Fx([)1564 1258 y Fq(S)1633 1345 y Fv(i)154 4518 y(>)154 4543 y(<)154 4692 y(>)154 4717 y(>)154 4742 y(:)418 4499 y Fu(d)461 4511 y Fv(\023)p 452 4594 5 59 v 228 4670 a Fu(:)14 b(:)g(:)f FA(\000)390 4682 y Fv(\023)442 4670 y FA(:)23 b Fu(\013)541 4682 y Fv(\023)584 4670 y Fu(:)14 b(:)g(:)p 228 4711 454 4 v 700 4728 a Fp(I)367 4786 y FA(\000)23 b(:)g Fu(\013)811 4638 y FA(:)p Fx(,)69 b Fu(d)24 b FA(=)e Fx(I)6 b FA(\()p Fu(d)1266 4650 y Fv(\023)1296 4638 y FA(\))1328 4653 y Fv(\023)p Fp(2jI)t(j)1506 4638 y FA(&)22 b Fx(8)p Fu(\023)h Fx(2)g(jI)6 b(j)p FA(\(\000\()p Fu(d)2026 4650 y Fv(\023)2056 4638 y FA(\))23 b Fx(\022)g FA(\000)2251 4650 y Fv(\023)2303 4638 y FA(&)f Fr(hgt)p FA(\()p Fu(d)2580 4650 y Fv(\023)2609 4638 y FA(\))h Fx(\024)g Fu(\013)2805 4650 y Fv(\023)2857 4638 y Fu(<)f(\013)p FA(\))i(&)3151 4584 y Fu(:::)p FA(\000)3272 4596 y Fv(\023)3301 4584 y Fu(:::)p FA(\()p Fu(\023)p Fx(2)q(jI)6 b(j)p FA(\))p 3151 4619 466 4 v 3358 4693 a(\000)3626 4633 y Fp(I)3695 4638 y FA(,)0 4975 y(where)278 4921 y Fu(:::)p FA(\000)399 4933 y Fv(\023)428 4921 y Fu(:::)p FA(\()p Fu(\023)p Fx(2)q(jI)g(j)p FA(\))p 278 4956 V 485 5030 a(\000)753 4970 y Fp(I)841 4975 y FA(:)p Fx(,)46 b FA(\001\()p Fx(I)6 b FA(\))24 b Fx(\022)f FA(\000)g(&)f Fx(8)p Fu(\023)h Fx(2)g(jI)6 b(j)p FA(\(\000)1809 4987 y Fv(\023)1862 4975 y Fx(\022)22 b FA(\000)p Fu(;)14 b FA(\001)2107 4987 y Fv(\023)2136 4975 y FA(\()p Fx(I)6 b FA(\)\).)0 5243 y(Note)28 b(that)g Fu(d)23 b Fx(')534 5126 y Fq(\032)620 5188 y Fu(:)14 b(:)g(:)g(:)g(:)g(:)p 607 5224 222 4 v 630 5298 a FA(\000)23 b(:)g Fu(\013)838 5238 y Fp(I)917 5243 y FA(implies)28 b(\000\()p Fu(d)p FA(\))c Fx(\022)e FA(\000)28 b(and)f Fr(hgt)p FA(\()p Fu(d)p FA(\))c Fx(\024)g Fu(\013)p FA(.)1908 5539 y(27)p eop %%Page: 28 29 28 28 bop 0 83 a Fz(Theorem)30 b(and)j(De\014nition)e(5.1)0 222 y FA(F)-7 b(or)27 b(eac)n(h)g(form)n(ula)f Fu(C)34 b FA(w)n(e)28 b(de\014ne)f(an)h(op)r(erator)e Fx(R)1616 234 y Fv(C)1699 222 y FA(suc)n(h)i(that:)0 360 y Fu(d)23 b Fx(`)117 330 y Fv(\013)117 381 y(m)203 360 y FA(\000)p Fu(;)14 b(C)52 b FA(&)46 b Fu(e)23 b Fx(`)627 330 y Fv(\014)627 381 y(m)713 360 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)52 b FA(&)46 b(rk)o(\()p Fu(C)6 b FA(\))24 b Fx(\024)f Fu(m)46 b FA(=)-14 b Fx(\))46 b(R)1765 372 y Fv(C)1821 360 y FA(\()p Fu(d;)14 b(e)p FA(\))23 b Fx(`)2078 330 y Fv(\013)p Ft(#)p Fv(\014)2078 381 y(m)2243 360 y FA(\000.)0 521 y(Pro)r(of:)0 660 y Fx(R)70 672 y Fv(C)126 660 y FA(\()p Fu(d;)14 b(e)p FA(\))28 b(is)g(de\014ned)g(b)n(y)f(recursion)f(on)h Fu(\013)p FA(#)p Fu(\014)t FA(.)0 931 y Fz(1.)36 b Fu(C)29 b Fx(62)24 b FA(\001\()p Fr(last)p FA(\()p Fu(d)p FA(\)\):)38 b(Then)28 b Fu(d)23 b Fx(')1072 711 y Fq(8)1072 786 y(>)1072 811 y(>)1072 836 y(<)1072 985 y(>)1072 1010 y(>)1072 1035 y(:)1438 792 y Fu(d)1481 804 y Fv(\023)p 1472 887 5 59 v 1145 963 a Fu(:)14 b(:)g(:)g FA(\000)p Fu(;)g(C)q(;)g FA(\001)1511 975 y Fv(\023)1563 963 y FA(:)24 b Fu(\013)1663 975 y Fv(\023)1705 963 y Fu(:)14 b(:)g(:)p FA(\()p Fu(\023)p Fx(2)q Fu(I)7 b FA(\))p 1145 1004 849 4 v 2013 1021 a Fp(I)1335 1079 y FA(\000)p Fu(;)14 b(C)30 b FA(:)23 b Fu(\013)2139 931 y FA(where)k Fx(I)i FA(:=)23 b Fr(last)o FA(\()p Fu(d)p FA(\),)29 b Fu(I)h FA(:=)23 b Fx(jI)6 b(j)p FA(.)0 1211 y(By)27 b(IH)h(w)n(e)f(get)h Fx(R)581 1223 y Fv(C)637 1211 y FA(\()p Fu(d)712 1223 y Fv(\023)741 1211 y Fu(;)14 b(e)p FA(\))23 b Fx(`)923 1181 y Fv(\013)966 1189 y Fn(\023)993 1181 y Ft(#)p Fv(\014)923 1231 y(m)1115 1211 y FA(\000)p Fu(;)14 b FA(\001)1273 1223 y Fv(\023)1330 1211 y FA(for)27 b(all)g Fu(\023)d Fx(2)f Fu(I)7 b FA(.)37 b(F)-7 b(urther)27 b(w)n(e)h(ha)n(v)n(e)e Fu(\013)2474 1223 y Fv(\023)2503 1211 y FA(#)p Fu(\014)h(<)c(\013)p FA(#)p Fu(\014)33 b FA(for)27 b(all)g Fu(\023)d Fx(2)f Fu(I)7 b FA(.)0 1482 y(Hence)28 b Fx(R)317 1494 y Fv(C)373 1482 y FA(\()p Fu(d;)14 b(e)p FA(\))23 b(:=)g Fx(I)6 b FA(\()p Fx(R)843 1494 y Fv(C)900 1482 y FA(\()p Fu(d)975 1494 y Fv(\023)1004 1482 y Fu(;)14 b(e)p FA(\)\))1144 1494 y Fv(\023)p Fp(2)p Fv(I)1275 1482 y Fx(')1362 1262 y Fq(8)1362 1337 y(>)1362 1362 y(>)1362 1387 y(<)1362 1536 y(>)1362 1561 y(>)1362 1586 y(:)1607 1343 y Fx(R)1677 1355 y Fv(C)1733 1343 y FA(\()p Fu(d)1808 1355 y Fv(\023)1837 1343 y Fu(;)g(e)p FA(\))p 1774 1438 5 59 v 1436 1514 a Fu(:)g(:)g(:)g FA(\000)p Fu(;)g FA(\001)1705 1526 y Fv(\023)1757 1514 y FA(:)23 b Fu(\013)1856 1526 y Fv(\023)1885 1514 y FA(#)p Fu(\014)18 b(:)c(:)g(:)p FA(\()p Fu(\023)p Fx(2)q Fu(I)7 b FA(\))p 1436 1555 872 4 v 2327 1572 a Fp(I)1628 1630 y FA(\000)23 b(:)h Fu(\013)p FA(#)p Fu(\014)2397 1482 y FA(is)k(a)f(deriv)-5 b(ation)27 b(as)g(required.)0 1776 y Fz(1'.)36 b Fx(:)p Fu(C)29 b Fx(62)24 b FA(\001\()p Fr(last)p FA(\()p Fu(e)p FA(\)\):)37 b(symmetric)28 b(to)f(1.)0 1937 y Fz(2.)36 b Fu(C)29 b Fx(2)24 b FA(\001\()p Fr(last)p FA(\()p Fu(d)p FA(\)\))29 b(and)e Fx(:)p Fu(C)j Fx(2)23 b FA(\001\()p Fr(last)p FA(\()p Fu(e)p FA(\)\):)0 2098 y Fz(2.1.)35 b Fu(C)f FA(is)28 b(a)f(literal:)36 b(Then)28 b Fr(last)p FA(\()p Fu(d)p FA(\))c(=)e Fr(Ax)1354 2110 y Ft(\001)1409 2094 y Fk(0)1463 2098 y FA(and)27 b Fr(last)p FA(\()p Fu(e)p FA(\))c(=)g Fr(Ax)2053 2110 y Ft(\001)2108 2094 y Fk(00)2180 2098 y FA(with)28 b(\001)2438 2068 y Fp(0)2485 2098 y Fx(2)23 b(AX)13 b FA(\()p Fj(Z)2781 2061 y Fp(1)2852 2098 y FA(\))28 b(and)f(\001)3142 2068 y Fp(00)3208 2098 y Fx(2)c(AX)13 b FA(\()p Fj(Z)3504 2061 y Fp(1)3575 2098 y FA(\).)0 2237 y(Hence)28 b(\001)23 b(:=)g(\(\001)551 2207 y Fp(0)593 2237 y Fx(n)18 b(f)p Fu(C)6 b Fx(g)p FA(\))18 b Fx([)h FA(\(\001)1027 2207 y Fp(00)1088 2237 y Fx(n)f(f:)p Fu(C)6 b Fx(g)p FA(\))23 b Fx(\022)g FA(\000,)k(and)h(\001)23 b Fx(2)g(AX)13 b FA(\()p Fj(Z)2148 2200 y Fp(1)2218 2237 y FA(\))28 b(\(cf.)g(ab)r(o)n (v)n(e\).)36 b(W)-7 b(e)28 b(set)f Fx(R)3065 2249 y Fv(C)3122 2237 y FA(\()p Fu(d;)14 b(e)p FA(\))23 b(:=)g Fr(Ax)3532 2249 y Ft(\001)3591 2237 y FA(.)0 2398 y Fz(2.2.)35 b Fu(C)30 b Fx(')361 2336 y Fq(W)431 2423 y Fv(\023)p Fp(2)p Fv(J)546 2398 y Fu(C)605 2410 y Fv(\023)634 2398 y FA(:)0 2678 y(Then)e Fx(:)p Fu(C)h Fx(')448 2616 y Fq(V)517 2703 y Fv(\023)p Fp(2)p Fv(J)633 2678 y Fx(:)p Fu(C)747 2690 y Fv(\023)804 2678 y FA(and)e Fu(d)d Fx(')1119 2458 y Fq(8)1119 2533 y(>)1119 2557 y(>)1119 2582 y(<)1119 2732 y(>)1119 2757 y(>)1119 2782 y(:)1378 2538 y Fu(d)1421 2550 y Ft(0)p 1416 2634 5 59 v 1193 2709 a FA(\000)p Fu(;)14 b(C)q(;)g(C)1438 2721 y Fv(\026)1506 2709 y FA(:)24 b Fu(\013)1606 2721 y Ft(0)p 1193 2751 450 4 v 1662 2767 a Fx(_)1717 2740 y Fn(\026)1717 2789 y(C)1280 2826 y FA(\000)p Fu(;)14 b(C)29 b FA(:)23 b Fu(\013)1916 2678 y FA(,)83 b Fu(e)23 b Fx(')2172 2458 y Fq(8)2172 2533 y(>)2172 2557 y(>)2172 2582 y(<)2172 2732 y(>)2172 2757 y(>)2172 2782 y(:)2588 2538 y Fu(e)2627 2550 y Fv(\023)p 2619 2634 5 59 v 2246 2709 a Fu(:)14 b(:)g(:)f FA(\000)p Fu(;)h Fx(:)p Fu(C)q(;)g Fx(:)p Fu(C)2711 2721 y Fv(\023)2764 2709 y FA(:)23 b Fu(\014)2857 2721 y Fv(\023)2900 2709 y Fu(:)14 b(:)g(:)p FA(\()p Fu(\023)p Fx(2)q Fu(J)8 b FA(\))p 2246 2751 955 4 v 3219 2767 a Fx(^)3274 2776 y Fk(:)p Fn(C)2456 2826 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)30 b FA(:)23 b Fu(\014)3408 2678 y FA(.)0 2972 y(By)k(IH)h(w)n(e)f (get)55 b Fx(R)608 2984 y Fv(C)665 2972 y FA(\()p Fu(d)740 2984 y Ft(0)777 2972 y Fu(;)14 b(e)p FA(\))23 b Fx(`)959 2942 y Fv(\013)1002 2950 y Fo(0)1035 2942 y Ft(#)p Fv(\014)959 2993 y(m)1157 2972 y FA(\000)p Fu(;)14 b(C)1305 2984 y Fv(\026)1377 2972 y FA(and)28 b Fx(R)1609 2984 y Fv(C)1665 2972 y FA(\()p Fu(d;)14 b(e)1816 2984 y Fv(\026)1861 2972 y FA(\))23 b Fx(`)1967 2927 y Fv(\013)p Ft(#)p Fv(\014)2103 2935 y Fn(\026)1967 2982 y Fv(m)2169 2972 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)2372 2984 y Fv(\026)2417 2972 y FA(.)0 3133 y(F)-7 b(urther)27 b(rk\()p Fu(C)469 3145 y Fv(\026)514 3133 y FA(\))c Fu(<)g FA(rk)o(\()p Fu(C)6 b FA(\))24 b Fx(\024)f Fu(m)p FA(.)0 3408 y(Hence)83 b Fx(R)372 3420 y Fv(C)428 3408 y FA(\()p Fu(d;)14 b(e)p FA(\))24 b(:=)e Fr(Cut)871 3420 y Fv(C)919 3428 y Fn(\026)963 3408 y Fx(R)1033 3420 y Fv(C)1089 3408 y FA(\()p Fu(d)1164 3420 y Ft(0)1201 3408 y Fu(;)14 b(e)p FA(\))p Fx(R)1379 3420 y Fv(C)1436 3408 y FA(\()p Fu(d;)g(e)1587 3420 y Fv(\026)1631 3408 y FA(\))23 b Fx(')1774 3188 y Fq(8)1774 3262 y(>)1774 3287 y(>)1774 3312 y(<)1774 3462 y(>)1774 3487 y(>)1774 3512 y(:)1911 3268 y Fx(R)1981 3280 y Fv(C)2037 3268 y FA(\()p Fu(d)2112 3280 y Ft(0)2150 3268 y Fu(;)14 b(e)p FA(\))p 2082 3364 5 59 v 1848 3439 a(\000)p Fu(;)g(C)1996 3451 y Fv(\026)2064 3439 y FA(:)23 b Fu(\013)2163 3451 y Ft(0)2200 3439 y FA(#)p Fu(\014)2520 3268 y Fx(R)2590 3280 y Fv(C)2646 3268 y FA(\()p Fu(d;)14 b(e)2797 3280 y Fv(\026)2842 3268 y FA(\))p 2695 3364 V 2431 3439 a(\000)p Fu(;)g Fx(:)p Fu(C)2634 3451 y Fv(\026)2702 3439 y FA(:)23 b Fu(\013)p FA(#)p Fu(\014)2917 3451 y Fv(\026)p 1848 3481 1115 4 v 2981 3497 a Fl(Cut)3079 3506 y Fn(C)3120 3514 y(\026)2258 3556 y FA(\000)g(:)g Fu(\013)p FA(#)p Fu(\014)3245 3408 y FA(.)0 3679 y Fz(2.2'.)35 b Fu(C)29 b Fx(')388 3617 y Fq(V)457 3704 y Fv(\023)p Fp(2)p Fv(J)573 3679 y Fu(C)632 3691 y Fv(\023)661 3679 y FA(:)37 b(symmetric)27 b(to)h(\(Case)f(2.2\).)0 3930 y Fz(Theorem)j(and)j(De\014nition)e(5.2)0 4068 y FA(W)-7 b(e)28 b(de\014ne)g(an)f(op)r(erator)f Fx(E)35 b FA(suc)n(h)27 b(that)h(the)g(follo)n(wing)f(holds:)36 b Fu(d)23 b Fx(`)2139 4038 y Fv(\013)2139 4089 y(m)p Ft(+1)2309 4068 y FA(\000)46 b(=)-14 b Fx(\))46 b(E)7 b FA(\()p Fu(d)p FA(\))24 b Fx(`)2820 4038 y Ft(3)2853 4013 y Fn(\013)2820 4089 y Fv(m)2922 4068 y FA(\000.)0 4229 y(Pro)r(of:)0 4502 y Fz(1.)82 b Fu(d)24 b Fx(')311 4282 y Fq(8)311 4357 y(>)311 4382 y(>)311 4407 y(<)311 4556 y(>)311 4581 y(>)311 4606 y(:)502 4363 y Fu(d)545 4375 y Ft(0)p 540 4458 5 59 v 385 4534 a FA(\000)p Fu(;)14 b(C)29 b FA(:)23 b Fu(\013)661 4546 y Ft(0)954 4363 y Fu(d)997 4375 y Ft(1)p 992 4458 V 809 4534 a FA(\000)p Fu(;)14 b Fx(:)p Fu(C)30 b FA(:)23 b Fu(\013)1141 4546 y Ft(1)p 385 4575 794 4 v 1197 4592 a Fl(Cut)1295 4600 y Fn(C)694 4650 y FA(\000)g(:)g Fu(\013)1398 4502 y FA(:)34 b(Then)22 b(rk)o(\()p Fu(C)6 b FA(\))24 b Fx(\024)f Fu(m)e FA(and,)i(b)n(y)f(IH,)44 b Fx(E)7 b FA(\()p Fu(d)2652 4514 y Ft(0)2690 4502 y FA(\))23 b Fx(`)2796 4472 y Ft(3)2829 4447 y Fn(\013)2867 4459 y Fo(0)2796 4523 y Fv(m)2931 4502 y FA(\000)p Fu(;)14 b(C)28 b FA(and)22 b Fx(E)7 b FA(\()p Fu(d)3389 4514 y Ft(1)3427 4502 y FA(\))23 b Fx(`)3533 4472 y Ft(3)3566 4447 y Fn(\013)3604 4459 y Fo(1)3533 4523 y Fv(m)3668 4502 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)6 b FA(.)0 4785 y(Hence)28 b Fx(R)317 4797 y Fv(C)373 4785 y FA(\()p Fx(E)7 b FA(\()p Fu(d)531 4797 y Ft(0)569 4785 y FA(\))p Fu(;)14 b Fx(E)7 b FA(\()p Fu(d)764 4797 y Ft(1)802 4785 y FA(\)\))24 b Fx(`)941 4755 y Ft(3)974 4729 y Fn(\013)1012 4741 y Fo(0)1048 4755 y Ft(#3)1136 4729 y Fn(\013)1174 4741 y Fo(1)941 4805 y Fv(m)1238 4785 y FA(\000)55 b(b)n(y)27 b(Theorem)g(5.1.)36 b(So)27 b(w)n(e)h(could)f(de\014ne)h Fx(E)7 b FA(\()p Fu(d)p FA(\))28 b(to)g(b)r(e)g Fx(R)3147 4797 y Fv(C)3203 4785 y FA(\()p Fx(E)7 b FA(\()p Fu(d)3361 4797 y Ft(0)3399 4785 y FA(\))p Fu(;)14 b Fx(E)7 b FA(\()p Fu(d)3594 4797 y Ft(1)3633 4785 y FA(\)\).)0 4923 y(But)28 b(for)f(reasons)f(whic)n(h) h(b)r(ecome)h(clear)e(later)h(w)n(e)g(set)56 b Fx(E)7 b FA(\()p Fu(d)p FA(\))24 b(:=)f Fr(Rep)13 b Fx(R)2301 4935 y Fv(C)2357 4923 y FA(\()p Fx(E)7 b FA(\()p Fu(d)2515 4935 y Ft(0)2553 4923 y FA(\))p Fu(;)14 b Fx(E)7 b FA(\()p Fu(d)2748 4935 y Ft(1)2786 4923 y FA(\)\).)0 5195 y Fz(2.)36 b FA(otherwise:)g Fx(E)7 b FA(\()p Fu(d)p FA(\))24 b(:=)f Fx(I)6 b FA(\()p Fx(E)h FA(\()p Fu(d)1015 5207 y Fv(\023)1045 5195 y FA(\)\))1109 5207 y Fv(\023)p Fp(2)p Fv(I)1240 5195 y Fx(')1328 4975 y Fq(8)1328 5049 y(>)1328 5074 y(>)1328 5099 y(<)1328 5249 y(>)1328 5274 y(>)1328 5298 y(:)1604 5055 y Fx(E)g FA(\()p Fu(d)1730 5067 y Fv(\023)1760 5055 y FA(\))p 1696 5151 5 59 v 1401 5226 a Fu(:)14 b(:)g(:)g FA(\000)p Fu(;)g FA(\001)1670 5238 y Fv(\023)1722 5226 y FA(:)23 b(3)1810 5196 y Fv(\013)1853 5204 y Fn(\023)1898 5226 y Fu(:)14 b(:)g(:)p FA(\()p Fu(\023)p Fx(2)q Fu(I)7 b FA(\))p 1401 5268 786 4 v 2206 5284 a Fp(I)1593 5342 y FA(\000)23 b(:)g(3)1756 5312 y Fv(\013)2303 5195 y FA(if)56 b Fu(d)24 b FA(=)2561 4975 y Fq(8)2561 5049 y(>)2561 5074 y(>)2561 5099 y(<)2561 5249 y(>)2561 5274 y(>)2561 5298 y(:)2879 5055 y Fu(d)2922 5067 y Fv(\023)p 2912 5151 5 59 v 2635 5226 a Fu(:)14 b(:)g(:)g FA(\000)p Fu(;)g FA(\001)2904 5238 y Fv(\023)2955 5226 y FA(:)24 b Fu(\013)3055 5238 y Fv(\023)3097 5226 y Fu(:)14 b(:)g(:)p FA(\()p Fu(\023)p Fx(2)q Fu(I)7 b FA(\))p 2635 5268 752 4 v 3405 5284 a Fp(I)2827 5342 y FA(\000)23 b(:)g Fu(\013)1908 5539 y FA(28)p eop %%Page: 29 30 29 29 bop 0 83 a Fz(Theorem)30 b(5.3)d FA(\(In)n(v)n(ersion\))0 217 y(\(a\))h Fu(A)23 b Fx(')307 155 y Fq(V)376 242 y Fv(\023)p Fp(2)p Fv(J)492 217 y Fu(A)554 229 y Fv(\023)629 217 y FA(&)f Fj(Z)765 180 y Fp(1)858 217 y Fx(`)909 187 y Fv(\013)909 237 y(m)995 217 y FA(\000)p Fu(;)14 b(A)46 b FA(=)-14 b Fx(\))46 b Fj(Z)1420 180 y Fp(1)1513 217 y Fx(`)1564 187 y Fv(\013)1564 237 y(m)1650 217 y FA(\000)p Fu(;)14 b(A)1801 229 y Fv(\023)1858 217 y FA(for)27 b(eac)n(h)g Fu(\023)c Fx(2)g Fu(J)8 b FA(.)0 351 y(\(b\))28 b Fj(Z)187 314 y Fp(1)280 351 y Fx(`)331 321 y Fv(\013)331 371 y(m)417 351 y FA(\000)p Fu(;)14 b(A)568 363 y Ft(0)624 351 y Fx(_)k Fu(A)759 363 y Ft(1)843 351 y FA(=)-14 b Fx(\))46 b Fj(Z)1071 314 y Fp(1)1164 351 y Fx(`)1215 321 y Fv(\013)1215 371 y(m)1301 351 y FA(\000)p Fu(;)14 b(A)1452 363 y Ft(0)1490 351 y Fu(;)g(A)1589 363 y Ft(1)1626 351 y FA(.)0 573 y Fz(De\014nition.)106 b Fx(L)605 585 y Ft(0)642 573 y FA(\()p Fu(X)7 b FA(\))23 b(:=)g Fx(L)973 585 y Ft(0)1029 573 y Fx([)18 b(f)p Fu(X)7 b Fx(g)p FA(,)0 707 y(A)28 b(closed)f Fx(L)392 719 y Ft(0)429 707 y FA(\()p Fu(X)7 b FA(\)-form)n(ula)27 b(\(sequen)n(t\))g(is)h(called)f Fu(X)7 b FA(-p)r(ositiv)n(e)26 b(if)i(it)g(con)n(tains)f(no)g(subform)n (ula)g Fx(:)p Fu(X)7 b(t)p FA(.)0 859 y Fx(j)-14 b FA(=)74 829 y Fv(\013)144 859 y Fu(A)47 b FA(:)p Fx(,)f FA(\()p Fx(N)12 b Fu(;)i Fx(f)p Fu(n)23 b FA(:)g Fx(j)p Fu(n)p Fx(j)811 871 y Fp(\036)890 859 y Fu(<)g(\013)p Fx(g)p FA(\))g Fx(j)-14 b FA(=)23 b Fu(A)p FA(,)0 993 y Fx(j)-14 b FA(=)74 963 y Fv(\013)144 993 y Fx(f)p Fu(A)248 1005 y Ft(1)285 993 y Fu(;)14 b(:::;)g(A)490 1005 y Fv(k)531 993 y Fx(g)46 b FA(:)p Fx(,)23 b(j)-14 b FA(=)822 963 y Fv(\013)892 993 y Fu(A)954 1005 y Ft(1)1010 993 y Fx(_)19 b Fu(:::)g Fx(_)f Fu(A)1307 1005 y Fv(k)0 1180 y Fz(Theorem)30 b(5.4)d FA(\(Boundedness\).)65 b(Let)27 b(\000)h(b)r(e)g Fu(X)7 b FA(-p)r(ositiv)n(e.)0 1314 y Fj(Z)48 1277 y Fp(1)142 1314 y Fx(`)193 1274 y Fv(\014)193 1336 y Ft(1)260 1314 y Fx(:)p FA(Prog)489 1334 y Fp(\036)545 1314 y FA(\()p Fu(X)g FA(\))p Fu(;)14 b Fx(:)p Fu(X)7 b(s)892 1326 y Ft(1)928 1314 y Fu(;)14 b(:::;)g Fx(:)p Fu(X)7 b(s)1241 1326 y Fv(k)1282 1314 y Fu(;)14 b FA(\000)46 b(&)f Fx(j)p Fu(s)1589 1326 y Ft(1)1627 1314 y Fx(j)1650 1326 y Fp(\036)1706 1314 y Fu(;)14 b(:::;)g Fx(j)p Fu(s)1911 1326 y Fv(k)1951 1314 y Fx(j)1974 1326 y Fp(\036)2054 1314 y Fx(\024)22 b Fu(\013)47 b FA(=)-14 b Fx(\))46 b(j)-14 b FA(=)2495 1284 y Fv(\013)p Ft(+2)2622 1259 y Fn(\014)2688 1314 y FA(\000.)0 1501 y Fz(Corollary)-8 b(.)106 b Fj(Z)567 1464 y Fp(1)660 1501 y Fx(`)711 1461 y Fv(\014)711 1523 y Ft(1)779 1501 y FA(TI)869 1513 y Fp(\036)925 1501 y FA(\()p Fu(X)7 b FA(\))46 b(=)-14 b Fx(\))46 b(k\036)o(k)22 b(\024)h FA(2)1591 1471 y Fv(\014)1635 1501 y FA(.)0 1652 y(Pro)r(of)j(of)i(the)g(Corollary:)0 1786 y Fj(Z)48 1749 y Fp(1)142 1786 y Fx(`)193 1746 y Fv(\014)193 1808 y Ft(1)260 1786 y FA(TI)350 1798 y Fp(\036)406 1786 y FA(\()p Fu(X)7 b FA(\))592 1739 y Ft(5)p Fv(:)p Ft(3)594 1786 y Fx(\))47 b Fj(Z)772 1749 y Fp(1)866 1786 y Fx(`)917 1746 y Fv(\014)917 1808 y Ft(1)984 1786 y Fx(:)p FA(Prog)1213 1807 y Fp(\036)1269 1786 y FA(\()p Fu(X)7 b FA(\))p Fu(;)14 b(X)7 b(n)p 1522 1799 50 4 v 27 w FA(for)27 b(all)g Fu(n)1937 1739 y Ft(5)p Fv(:)p Ft(4)1938 1786 y Fx(\))48 b(j)p Fu(n)p Fx(j)2165 1798 y Fp(\036)2244 1786 y Fu(<)23 b FA(2)2374 1756 y Fv(\014)2446 1786 y FA(for)k(all)g Fu(n)46 b Fx(\))g(k\036)o(k)23 b(\024)f FA(2)3213 1756 y Fv(\014)3258 1786 y FA(.)0 1938 y(Pro)r(of)k(of)i(Lemma)f(5.4)g(b)n(y)g(induction)h (on)g Fu(\014)t FA(:)0 2072 y(Abbreviations:)36 b(\003)23 b(:=)f Fx(f:)p Fu(X)7 b(s)971 2084 y Ft(1)1008 2072 y Fu(;)14 b(:::;)g Fx(:)p Fu(X)7 b(s)1321 2084 y Fv(k)1361 2072 y Fx(g)p FA(.)0 2206 y(Let)28 b Fu(d)23 b Fx(`)266 2166 y Fv(\014)266 2228 y Ft(1)334 2206 y Fx(:)p FA(Prog)o(\()p Fu(X)7 b FA(\))p Fu(;)14 b FA(\003)p Fu(;)g FA(\000.)0 2357 y(1.1.)36 b Fr(last)p FA(\()p Fu(d)p FA(\))23 b(=)g Fr(Ax)600 2369 y Ft(\001)686 2357 y FA(and)k(\001)19 b Fx(\\)g FA(TR)n(UE)1246 2369 y Ft(0)1307 2357 y Fx(6)p FA(=)j Fx(;)p FA(:)37 b(Then)28 b(\000)18 b Fx(\\)h FA(TR)n(UE)2094 2369 y Ft(0)2155 2357 y Fx(6)p FA(=)j Fx(;)27 b FA(and)h(the)g(claim)f (is)h(trivial.)0 2509 y(1.2.)36 b Fr(last)p FA(\()p Fu(d)p FA(\))23 b(=)g Fr(Ax)600 2521 y Ft(\001)686 2509 y FA(and)k Fu(X)7 b(t;)14 b Fx(:)p Fu(X)7 b(s)22 b Fx(\022)h FA(\001)28 b(with)g Fu(t)1586 2479 y Fp(N)1676 2509 y FA(=)23 b Fu(s)1803 2479 y Fp(N)1870 2509 y FA(:)0 2643 y(Then)28 b Fu(X)7 b(t)22 b Fx(2)i FA(\000)j(and)h Fx(:)p Fu(X)7 b(s)22 b Fx(2)i FA(\003.)36 b(Hence)28 b Fx(j)p Fu(t)p Fx(j)1376 2655 y Fp(\036)1455 2643 y FA(=)23 b Fx(j)p Fu(s)p Fx(j)1628 2655 y Fp(\036)1707 2643 y Fx(\024)g Fu(\013)g(<)g(\013)c FA(+)f(2)2156 2613 y Fv(\014)2228 2643 y FA(and)27 b(th)n(us)h Fx(j)-14 b FA(=)2646 2613 y Fv(\013)p Ft(+2)2773 2588 y Fn(\014)2839 2643 y FA(\000.)0 2795 y(2.)36 b Fr(last)p FA(\()p Fu(d)p FA(\))24 b(=)441 2732 y Fq(W)511 2753 y Fv(s)542 2761 y Fo(0)511 2820 y Fp(:)p Ft(Prog\()p Fv(X)5 b Ft(\))808 2795 y FA(:)37 b(Then)28 b Fx(`)1136 2755 y Fv(\014)1174 2763 y Fo(0)1136 2817 y Ft(1)1233 2795 y Fx(:)p FA(Prog)o(\()p Fu(X)7 b FA(\))p Fu(;)14 b FA(\003)p Fu(;)g FA(\000)p Fu(;)g Fx(8)p Fu(y)s Fx(\036)n Fu(s)2015 2807 y Ft(0)2066 2795 y Fu(X)7 b(y)20 b Fx(^)f(:)p Fu(X)7 b(s)2447 2807 y Ft(0)2512 2795 y FA(with)28 b Fu(\014)2748 2807 y Ft(0)2808 2795 y Fu(<)23 b(\014)t FA(.)0 2929 y(By)k(5.3a)g(\(In)n(v)n(ersion\))f(w)n (e)h(get)55 b(\(1\))28 b Fx(`)1199 2889 y Fv(\014)1237 2897 y Fo(0)1199 2951 y Ft(1)1296 2929 y Fx(:)p FA(Prog\()p Fu(X)7 b FA(\))p Fu(;)14 b FA(\003)p Fu(;)g FA(\000)p Fu(;)g Fx(8)p Fu(y)s Fx(\036)m Fu(s)2078 2941 y Ft(0)2129 2929 y Fu(X)7 b(y)s FA(,)54 b(and)i(\(2\))27 b Fx(`)2700 2889 y Fv(\014)2738 2897 y Fo(0)2700 2951 y Ft(1)2797 2929 y Fx(:)p FA(Prog\()p Fu(X)7 b FA(\))p Fu(;)14 b Fx(:)p Fu(X)7 b(s)3373 2941 y Ft(0)3409 2929 y Fu(;)14 b FA(\003)p Fu(;)g FA(\000.)0 3080 y(By)27 b(IH)h(from)f(\(1\))h(w)n(e) f(get)56 b Fx(j)-14 b FA(=)943 3050 y Fv(\013)p Ft(+2)1070 3025 y Fn(\014)1103 3037 y Fo(0)1166 3080 y FA(\000)p Fu(;)14 b Fx(8)p Fu(y)s Fx(\036)n Fu(s)1448 3092 y Ft(0)1499 3080 y Fu(X)7 b(y)s FA(.)0 3232 y(\(Case)27 b(1\))h Fx(j)-14 b FA(=)407 3202 y Fv(\013)p Ft(+2)534 3177 y Fn(\014)567 3189 y Fo(0)630 3232 y FA(\000:)37 b(Then)28 b(also)e Fx(j)-14 b FA(=)1199 3202 y Fv(\013)p Ft(+2)1326 3177 y Fn(\014)1393 3232 y FA(\000,)27 b(since)h Fu(\014)1746 3244 y Ft(0)1806 3232 y Fx(\024)23 b Fu(\014)32 b FA(and)27 b(\000)h(is)f Fu(X)7 b FA(-p)r(ositiv)n(e.)0 3366 y(\(Case)27 b(2\))h Fx(j)-14 b FA(=)407 3336 y Fv(\013)p Ft(+2)534 3311 y Fn(\014)567 3323 y Fo(0)630 3366 y Fx(8)p Fu(y)s Fx(\036)o Fu(s)824 3378 y Ft(0)875 3366 y Fu(X)7 b(y)s FA(:)35 b(Then)28 b Fx(j)p Fu(m)p Fx(j)1389 3378 y Fp(\036)1468 3366 y Fu(<)23 b(\013)c FA(+)f(2)1753 3336 y Fv(\014)1791 3344 y Fo(0)1854 3366 y FA(for)27 b(all)h Fu(m)23 b Fx(\036)f Fu(s)2319 3336 y Fp(N)2319 3386 y Ft(0)2386 3366 y FA(,)28 b(i.e.)37 b Fx(j)p Fu(s)2642 3378 y Ft(0)2679 3366 y Fx(j)2702 3378 y Fp(\036)2782 3366 y Fx(\024)22 b Fu(\013)d FA(+)f(2)3066 3336 y Fv(\014)3104 3344 y Fo(0)3140 3366 y FA(.)0 3500 y(F)-7 b(rom)27 b(this)h(together)f(with)h Fx(j)p Fu(s)959 3512 y Ft(1)996 3500 y Fx(j)1019 3512 y Fp(\036)1075 3500 y Fu(;)14 b(:::;)g Fx(j)p Fu(s)1280 3512 y Fv(k)1321 3500 y Fx(j)1344 3512 y Fp(\036)1423 3500 y Fx(\024)23 b Fu(\013)28 b FA(and)f(\(2\))h(b)n(y)f(IH)h(w)n(e)g (obtain)f Fx(j)-14 b FA(=)2577 3470 y Fv(\013)p Ft(+2)2704 3445 y Fn(\014)2737 3457 y Fo(0)2773 3470 y Ft(+2)2857 3445 y Fn(\014)2890 3457 y Fo(0)2954 3500 y FA(\000,)27 b(and)h(th)n(us)g Fx(j)-14 b FA(=)3475 3470 y Fv(\013)p Ft(+2)3602 3445 y Fn(\014)3668 3500 y FA(\000.)0 3651 y(3.)44 b Fr(last)p FA(\()p Fu(d)p FA(\))28 b(=)458 3589 y Fq(V)527 3676 y Fv(C)613 3651 y FA(with)j Fu(C)i Fx(')989 3589 y Fq(V)1058 3676 y Fv(\023)p Fp(2)p Fv(I)1166 3651 y Fu(C)1225 3663 y Fv(\023)1281 3651 y Fx(2)28 b FA(\000:)42 b(Then,)31 b(for)e(all)h Fu(\023)e Fx(2)g Fu(I)7 b FA(,)31 b Fx(`)2259 3611 y Fv(\014)2297 3619 y Fn(\023)2259 3673 y Ft(1)2355 3651 y Fx(:)p FA(Prog)o(\()p Fu(X)7 b FA(\))p Fu(;)14 b FA(\003)p Fu(;)g FA(\000)p Fu(;)g(C)3003 3663 y Fv(\023)3061 3651 y FA(and)30 b Fu(\014)3272 3663 y Fv(\023)3328 3651 y Fu(<)d(\014)t FA(.)45 b(Hence,)31 b(b)n(y)0 3785 y(IH,)d Fx(j)-14 b FA(=)217 3755 y Fv(\013)p Ft(+2)344 3730 y Fn(\014)377 3738 y(\023)435 3785 y FA(\000)p Fu(;)14 b(C)583 3797 y Fv(\023)640 3785 y FA(for)27 b(all)g Fu(\023)p FA(.)37 b(Since)28 b(\000)g(is)f Fu(X)7 b FA(-p)r(ositiv)n (e,)26 b(this)i(implies)g Fx(j)-14 b FA(=)2302 3755 y Fv(\013)p Ft(+2)2429 3730 y Fn(\014)2495 3785 y FA(\000)p Fu(;)14 b(C)2643 3797 y Fv(\023)2700 3785 y FA(for)27 b(all)g Fu(\023)p FA(,)i(and)e(th)n(us)h Fx(j)-14 b FA(=)3441 3755 y Fv(\013)p Ft(+2)3568 3730 y Fn(\014)3634 3785 y FA(\000)p Fu(;)14 b(C)6 b FA(.)0 3937 y(4.)36 b Fr(last)p FA(\()p Fu(d)p FA(\))24 b(=)441 3875 y Fq(W)511 3895 y Fv(\026)511 3962 y(C)594 3937 y FA(with)k Fu(C)i Fx(2)23 b FA(\000:)37 b(as)27 b(3.)0 4089 y(5.)36 b Fr(last)p FA(\()p Fu(d)p FA(\))24 b(=)e Fr(Rep)p FA(:)37 b(immediate)28 b(b)n(y)f(IH.)0 4240 y(6.)36 b Fr(last)p FA(\()p Fu(d)p FA(\))24 b(=)e Fr(Cut)567 4252 y Fv(C)651 4240 y FA(with)28 b Fu(C)h Fx(2)24 b FA(TR)n(UE)1244 4252 y Ft(0)1282 4240 y FA(:)37 b(Then)27 b(b)n(y)h(I.H.)g Fx(j)-14 b FA(=)1914 4210 y Fv(\013)p Ft(+2)2041 4185 y Fn(\014)2074 4197 y Fo(0)2137 4240 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)34 b FA(and)28 b(th)n(us)f Fx(j)-14 b FA(=)2792 4210 y Fv(\013)p Ft(+2)2919 4185 y Fn(\014)2986 4240 y FA(\000.)0 4392 y(7.)35 b Fr(last)p FA(\()p Fu(d)p FA(\))24 b(=)e Fr(Cut)566 4404 y Fv(Y)15 b(s)680 4392 y FA(with)25 b Fu(Y)42 b Fx(6)p FA(=)22 b Fu(X)7 b FA(:)35 b(Then)25 b Fx(`)1442 4352 y Fv(\014)1480 4360 y Fo(0)1442 4414 y Ft(1)1539 4392 y Fx(:)p FA(Prog\()p Fu(X)7 b FA(\))p Fu(;)14 b FA(\003)p Fu(;)g FA(\000)p Fu(;)g Fx(:)p Fu(Y)k(s)24 b FA(whic)n(h)h(implies)g Fx(`)2878 4352 y Fv(\014)2916 4360 y Fo(0)2878 4414 y Ft(1)2975 4392 y Fx(:)p FA(Prog)o(\()p Fu(X)7 b FA(\))p Fu(;)14 b FA(\003)p Fu(;)g FA(\000)p Fu(;)g Fx(:)p FA(\(0)23 b(=)f(0\).)0 4526 y(No)n(w)27 b(the)h(claim)f(follo)n(ws)g(as)g(in)h(6.)0 4677 y(8.)36 b Fr(last)p FA(\()p Fu(d)p FA(\))24 b(=)e Fr(Cut)567 4689 y Fv(X)5 b(s)657 4697 y Fo(0)694 4677 y FA(:)37 b(Then)28 b(\(1\))g Fx(`)1156 4637 y Fv(\014)1194 4645 y Fo(0)1156 4699 y Ft(1)1253 4677 y Fx(:)p FA(Prog)o(\()p Fu(X)7 b FA(\))p Fu(;)14 b FA(\003)p Fu(;)g FA(\000)p Fu(;)g(X)7 b(s)1957 4689 y Ft(0)2048 4677 y FA(and)55 b(\(2\))28 b Fx(`)2422 4637 y Fv(\014)2460 4645 y Fo(0)2422 4699 y Ft(1)2519 4677 y Fx(:)p FA(Prog\()p Fu(X)7 b FA(\))p Fu(;)14 b Fx(:)p Fu(X)7 b(s)3095 4689 y Ft(0)3131 4677 y Fu(;)14 b FA(\003)p Fu(;)g FA(\000.)0 4829 y(By)27 b(IH)h(from)f(\(1\))h(w)n(e)f(get)56 b Fx(j)-14 b FA(=)943 4799 y Fv(\013)p Ft(+2)1070 4774 y Fn(\014)1103 4786 y Fo(0)1166 4829 y FA(\000)p Fu(;)14 b(X)7 b(s)1370 4841 y Ft(0)1406 4829 y FA(.)0 4981 y(\(Case)27 b(1\))h Fx(j)-14 b FA(=)407 4950 y Fv(\013)p Ft(+2)534 4925 y Fn(\014)567 4937 y Fo(0)630 4981 y FA(\000:)37 b(Then)28 b(also)e Fx(j)-14 b FA(=)1199 4950 y Fv(\013)p Ft(+2)1326 4925 y Fn(\014)1393 4981 y FA(\000.)0 5115 y(\(Case)27 b(2\))h Fx(j)-14 b FA(=)407 5084 y Fv(\013)p Ft(+2)534 5059 y Fn(\014)567 5071 y Fo(0)631 5115 y Fu(X)7 b(s)746 5127 y Ft(0)782 5115 y FA(:)38 b(Then)28 b Fx(j)p Fu(s)1122 5127 y Ft(0)1159 5115 y Fx(j)1182 5127 y Fp(\036)1261 5115 y Fu(<)23 b(\013)c FA(+)f(2)1546 5084 y Fv(\014)1584 5092 y Fo(0)1620 5115 y FA(.)38 b(F)-7 b(rom)27 b(this)h(together)f (with)h Fx(j)p Fu(s)2640 5127 y Ft(1)2678 5115 y Fx(j)2701 5127 y Fp(\036)2757 5115 y Fu(;)14 b(:::;)g Fx(j)p Fu(s)2962 5127 y Fv(k)3003 5115 y Fx(j)3026 5127 y Fp(\036)3105 5115 y Fx(\024)23 b Fu(\013)28 b FA(and)g(\(2\))g(b)n(y)f(IH)h(w)n(e)0 5248 y(obtain)f Fx(j)-14 b FA(=)332 5218 y Fv(\013)p Ft(+2)459 5193 y Fn(\014)492 5205 y Fo(0)529 5218 y Ft(+2)613 5193 y Fn(\014)646 5205 y Fo(0)709 5248 y FA(\000,)28 b(and)f(th)n(us)h Fx(j)-14 b FA(=)1230 5218 y Fv(\013)p Ft(+2)1357 5193 y Fn(\014)1423 5248 y FA(\000.)1908 5539 y(29)p eop %%Page: 30 31 30 30 bop 0 83 a Fz(Em)m(b)s(edding)29 b(of)j Fj(Z)g Fz(in)f Fj(Z)853 46 y Fp(1)0 247 y FA(F)-7 b(rom)27 b(no)n(w)g(on)g(w)n (e)h(denote)f Fj(Z)944 210 y Fp(1)1015 247 y FA(-deriv)-5 b(ations)26 b(b)n(y)h Fi(d)p Fu(;)14 b Fi(e)p Fu(;)g Fi(c)o FA(.)0 412 y(F)-7 b(or)27 b(eac)n(h)g(closed)g Fj(Z)p FA(-deriv)-5 b(ation)27 b Fu(d)g FA(w)n(e)h(de\014ne)f(its)h(in) n(terpretation)f Fu(d)2167 382 y Fp(1)2260 412 y Fx(2)d Fj(Z)2387 375 y Fp(1)2485 412 y FA(as)j(follo)n(ws:)0 576 y(1.)64 b(\()161 514 y Fq(V)231 534 y Fv(x)231 601 y Fp(8)p Fv(xA)360 576 y Fu(d)403 588 y Ft(0)440 576 y FA(\))472 546 y Fp(1)566 576 y FA(:=)677 514 y Fq(V)746 601 y Fp(8)p Fv(xA)875 509 y Fq(\000)913 576 y Fu(d)956 588 y Ft(0)994 576 y FA(\()p Fu(x=t)p FA(\))1177 546 y Fp(1)1248 509 y Fq(\001)1286 609 y Fv(t)p Fp(2)p Fv(T)1395 617 y Fo(0)1459 576 y FA(,)0 779 y(2.)g(\()p Fr(Ind)270 739 y Fv(x;t)270 803 y(F)357 779 y Fu(d)400 791 y Ft(0)437 779 y FA(\))469 748 y Fp(1)563 779 y FA(:=)674 662 y Fq(\032)750 726 y Fi(c)792 686 y Fv(x;t)792 750 y(F)1412 726 y FA(if)28 b Fu(n)23 b FA(=)g(0)750 834 y Fr(Cut)876 849 y Fv(F)9 b Ft(\()p Fv(n)p 953 862 42 4 v Ft(\))1024 834 y Fi(e)1070 846 y Fv(n)p Fp(\000)p Ft(1)1200 834 y Fi(c)1243 794 y Fv(x;t)1243 858 y(F)1412 834 y FA(if)28 b Fu(n)23 b(>)g FA(0)1894 779 y(where)129 1009 y Fu(n)g FA(:=)g Fu(t)343 979 y Fp(N)410 1009 y FA(,)55 b Fi(c)531 969 y Fv(x;t)531 1033 y(F)641 1009 y Fx(`)692 966 y Ft(2rk)o(\()p Fv(F)9 b Ft(\))692 1031 y(0)915 1009 y Fx(:)p Fu(F)1023 1021 y Fv(x)1066 1009 y FA(\()p Fu(n)p 1098 1022 50 4 v FA(\))p Fu(;)14 b(F)1270 1021 y Fv(x)1312 1009 y FA(\()p Fu(t)p FA(\),)56 b Fi(e)1531 1021 y Ft(0)1592 1009 y FA(:=)22 b Fu(d)1745 1021 y Ft(0)1783 1009 y FA(\()p Fu(x=)p FA(0\))1978 979 y Fp(1)2048 1009 y FA(,)56 b Fi(e)2173 1021 y Fv(i)2223 1009 y FA(:=)23 b Fr(Cut)2460 1024 y Fv(F)9 b Ft(\()p Fv(i)p 2537 1037 24 4 v Ft(\))2591 1009 y Fi(e)2636 1021 y Fv(i)p Fp(\000)p Ft(1)2749 1009 y Fu(d)2792 1021 y Ft(0)2830 1009 y FA(\()p Fu(x=i)p 2951 1022 29 4 v -1 w FA(\))3011 979 y Fp(1)3110 1009 y FA(for)27 b Fu(i)22 b(>)h FA(0)k(.)129 1448 y(\()p Fr(Ind)270 1408 y Fv(x;t)270 1472 y(F)357 1448 y Fu(d)400 1460 y Ft(0)437 1448 y FA(\))469 1418 y Fp(1)563 1448 y FA(=)651 1191 y Fu(d)694 1203 y Ft(0)731 1191 y FA(\()p Fu(x=)p FA(0\))926 1161 y Fp(1)1080 1191 y Fu(d)1123 1203 y Ft(0)1160 1191 y FA(\()p Fu(x=)p FA(1\))1355 1161 y Fp(1)p 651 1233 775 4 v 905 1308 a Fr(Cut)1031 1323 y Fv(F)9 b Ft(\(1\))1453 1308 y Fu(d)1496 1320 y Ft(0)1533 1308 y FA(\()p Fu(x=)p FA(2\))1728 1278 y Fp(1)p 905 1350 894 4 v 1340 1412 a FA(.)1340 1446 y(.)1340 1479 y(.)464 b Fu(d)1870 1491 y Ft(0)1907 1479 y FA(\()p Fu(x=n)p Fx(\000)p FA(1\))2217 1449 y Fp(1)p 1340 1520 947 4 v 1634 1595 a Fr(Cut)1760 1610 y Fv(F)9 b Ft(\()p Fv(n)p Fp(\000)p Ft(1\))2370 1595 y Fi(c)2412 1555 y Fv(x;t)2412 1619 y(F)p 1634 1637 865 4 v 1930 1712 a Fr(Cut)2055 1727 y Fv(F)g Ft(\()p Fv(n)p Ft(\))0 1884 y FA(3.)64 b(Otherwise:)36 b(\()p Fx(I)6 b Fu(d)680 1896 y Ft(0)718 1884 y Fu(:::d)830 1896 y Fv(n)p Fp(\000)p Ft(1)961 1884 y FA(\))993 1854 y Fp(1)1087 1884 y FA(:=)22 b Fx(I)6 b Fu(d)1291 1854 y Fp(1)1291 1905 y Ft(0)1376 1884 y Fu(:)14 b(:)g(:)g(d)1530 1854 y Fp(1)1530 1905 y Fv(n)p Fp(\000)p Ft(1)1688 1884 y FA(.)0 2097 y Fz(De\014nition)31 b(of)36 b FA(~)-46 b Fu(o)p FA(\()p Fu(d)p FA(\))32 b Fz(and)h Fr(dg)o FA(\()p Fu(d)p FA(\))g Fz(for)f(eac)m(h)h Fj(Z)p Fz(-deriv)-5 b(ation)31 b Fu(d)0 2261 y FA(Let)d Fu(d)23 b FA(=)g Fx(I)6 b Fu(d)397 2273 y Ft(0)434 2261 y Fu(:::d)546 2273 y Fv(n)p Fp(\000)p Ft(1)677 2261 y FA(.)0 2425 y(~)-42 b(o\()p Fu(d)p FA(\))23 b(:=)283 2308 y Fq(\032)359 2382 y FA(~)-42 b(o\()p Fu(d)476 2394 y Ft(0)513 2382 y FA(\))19 b(+)f Fu(!)336 b FA(if)28 b Fx(I)i FA(=)22 b Fr(Ind)1382 2343 y Fv(x;t)1382 2407 y(F)359 2477 y FA(sup)484 2498 y Fv(i)h FA(0:)53 b Fu(d)3460 4366 y Ft(0)3497 4354 y FA(\()p Fu(x=i)p 3618 4367 29 4 v FA(\))3679 4324 y Fp(1)3786 4354 y Fx(`)3837 4324 y Fv(\013)3837 4374 y(m)0 4494 y FA(\000)p Fu(;)14 b Fx(:)p Fu(F)e FA(\()p Fu(i)p 241 4507 V FA(\))p Fu(;)i(F)e FA(\()p Fu(i)p FA(+1)p 436 4514 135 4 v -1 w(\))28 b(and)f Fi(e)838 4506 y Fv(i)p Fp(\000)p Ft(1)974 4494 y Fx(`)1025 4464 y Fv(\013)p Ft(+)p Fv(i)p Fp(\000)p Ft(1)1025 4515 y Fv(m)1255 4494 y FA(\000)p Fu(;)14 b(F)e FA(\()p Fu(i)p 1441 4507 29 4 v -1 w FA(\))1548 4440 y Ft(rk)o(\(F\))p Fv(<)p Ft(m)1612 4494 y FA(=)-14 b Fx(\))110 b Fi(e)1902 4506 y Fv(i)1952 4494 y FA(=)23 b Fr(Cut)2166 4509 y Fv(F)9 b Ft(\()p Fv(i)p 2243 4522 24 4 v Ft(\))2296 4494 y Fi(e)2342 4506 y Fv(i)p Fp(\000)p Ft(1)2455 4494 y Fu(d)2498 4506 y Ft(0)2536 4494 y FA(\()p Fu(x=i)p 2657 4507 29 4 v -1 w FA(\))2717 4464 y Fp(1)2811 4494 y Fx(`)2862 4464 y Fv(\013)p Ft(+)p Fv(i)2862 4515 y(m)3007 4494 y FA(\000)p Fu(;)14 b(F)e FA(\()p Fu(i)p FA(+)o(1)p 3193 4514 135 4 v -1 w(\).)0 4658 y Fu(n)23 b FA(=)g(0:)36 b Fu(d)305 4628 y Fp(1)399 4658 y FA(=)22 b Fi(c)529 4619 y Fv(x;t)529 4683 y(F)639 4658 y Fx(`)690 4623 y Fv(\013)p Ft(+)p Fv(!)690 4681 y Ft(0)855 4658 y Fx(:)p Fu(F)12 b FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(t)p FA(\).)0 4823 y Fu(n)23 b(>)g FA(0:)36 b Fi(e)308 4835 y Fv(n)p Fp(\000)p Ft(1)461 4823 y Fx(`)512 4793 y Fv(\013)p Ft(+)p Fv(n)p Fp(\000)p Ft(1)512 4843 y Fv(m)759 4823 y FA(\000)p Fu(;)14 b(F)e FA(\()p Fu(n)p 945 4836 50 4 v FA(\))28 b(and)g Fi(c)1259 4783 y Fv(x;t)1259 4847 y(F)1369 4823 y Fx(`)1420 4793 y Fv(k)1420 4843 y Ft(0)1483 4823 y Fx(:)p Fu(F)12 b FA(\()p Fu(n)p 1635 4836 V 1 w FA(\))p Fu(;)i(F)e FA(\()p Fu(t)p FA(\))1961 4769 y Ft(rk)o(\()p Fv(F)d Ft(\))p Fv()g FA(0)0 3022 y(Pro)r(of:)0 3163 y(Let)d(\000)h(b)r(e)f(closed.)34 b(By)20 b(partial)g(cut-elimination)g(and)g(Lemma)g(4.3,)h(if)g Fj(Z)2258 3175 y Fv(m)2344 3163 y Fx(`)i FA(\000)d(then)h(there)f (exists)g(a)g(closed)g Fj(Z)3447 3175 y Fv(m)3510 3163 y FA(-deriv)-5 b(ation)0 3304 y Fu(d)28 b FA(of)f(\000)h(with)g(rk)510 3316 y Fl(QF)599 3304 y FA(\()p Fu(C)6 b FA(\))24 b Fu(<)e(m)28 b FA(for)f(all)g(cut)h(or)f(induction)h(form)n(ulas)e Fu(C)6 b FA(.)0 3470 y(Then)28 b(also)e(rk)460 3482 y Fl(QF)548 3470 y FA(\()p Fu(C)6 b FA(\))24 b Fu(<)f(m)k FA(for)g(all)g(cut-form)n(ulas)g Fu(C)34 b FA(of)27 b Fu(d)1842 3440 y Fp(1)1913 3470 y FA(.)0 3636 y(So)g(w)n(e)h(ha)n(v)n (e)e Fu(d)472 3606 y Fp(1)566 3636 y Fx(`)617 3606 y Fv(!)r Fp(\001)p Fv(k)617 3660 y Fl(QF)o Fv(;m)806 3636 y FA(\000)i(for)f(some)g Fu(k)f Fx(\025)d FA(1.)0 3802 y(By)k(insp)r(ecting)h(the)g(pro)r(ofs)f(of)g(5.1)g(and)h(5.2)e(one)i (sees)f(that)g(the)h(follo)n(wing)f(holds:)0 3944 y Fi(d)c Fx(`)125 3913 y Fv(\013)125 3967 y Fl(QF)o Fv(;n)p Ft(+1)381 3944 y FA(\000)46 b(=)-14 b Fx(\))46 b(E)7 b FA(\()p Fi(d)p FA(\))23 b Fx(`)899 3913 y Ft(3)932 3888 y Fn(\013)899 3967 y Fl(QF)o Fv(;n)1071 3944 y FA(\000.)0 4110 y(Hence)28 b Fx(E)298 4079 y Fv(m)361 4110 y FA(\()p Fu(d)436 4079 y Fp(1)507 4110 y FA(\))23 b Fx(`)613 4066 y Ft(3)646 4074 y Fn(m)701 4066 y Ft(\()p Fv(!)r Fp(\001)p Fv(k)q Ft(\))613 4135 y Fl(QF)o Fv(;)p Ft(0)881 4110 y FA(\000.)36 b(Moreo)n(v)n(er)25 b(there)j(is)f(an)h Fu(`)22 b Fx(2)i FA(I)-14 b(N)28 b(so)f(that)g(rk\()p Fu(C)6 b FA(\))24 b Fx(\024)e Fu(`)28 b FA(for)f(all)g(cut)h(form)n(ulas)e Fu(C)34 b FA(of)28 b Fx(E)3633 4079 y Fv(m)3696 4110 y FA(\()p Fu(d)3771 4079 y Fp(1)3842 4110 y FA(\).)0 4276 y Fy(Abbreviation.)71 b Fj(Z)619 4239 y Fp(1)712 4276 y Fx(`)763 4246 y Fv(\013)763 4296 y(n;)p Ft(0)884 4276 y FA(\000)46 b(:)p Fx(,)g Fi(d)23 b Fx(`)1259 4246 y Fv(\013)1259 4296 y(n)1329 4276 y FA(\000)i(for)g(some)f Fj(Z)1784 4239 y Fp(1)1855 4276 y FA(-deriv)-5 b(ation)24 b Fi(d)g FA(in)i(whic)n(h)f(all)g(cut)g(form)n(ulas)f(are)g(quan)n (ti\014erfree.)0 4442 y(The)k(follo)n(wing)e(prop)r(ositions)h(are)f (easily)h(pro)n(v)n(ed)f(\(where)h(\(1\))h(and)f(\(2\))h(are)f(needed)h (for)f(the)h(pro)r(of)f(of)g(\(3\)\):)0 4583 y(\(1\))h Fj(Z)182 4546 y Fp(1)275 4583 y Fx(`)326 4553 y Fv(\013)326 4603 y(n;)p Ft(0)447 4583 y FA(\000)p Fu(;)14 b(A)598 4595 y Ft(0)654 4583 y Fx(^)19 b Fu(A)790 4595 y Ft(1)873 4583 y Fx(\))46 b Fj(Z)1051 4546 y Fp(1)1144 4583 y Fx(`)1195 4553 y Fv(\013)1195 4603 y(n;)p Ft(0)1316 4583 y FA(\000)p Fu(;)14 b(A)1467 4595 y Fv(i)1495 4583 y FA(;)0 4724 y(\(2\))28 b Fj(Z)182 4687 y Fp(1)275 4724 y Fx(`)326 4694 y Fv(\013)326 4745 y(n;)p Ft(0)447 4724 y FA(\000)p Fu(;)14 b(A)598 4736 y Ft(0)654 4724 y Fx(_)19 b Fu(A)790 4736 y Ft(1)873 4724 y Fx(\))46 b Fj(Z)1051 4687 y Fp(1)1144 4724 y Fx(`)1195 4694 y Fv(\013)1195 4745 y(n;)p Ft(0)1316 4724 y FA(\000)p Fu(;)14 b(A)1467 4736 y Ft(0)1504 4724 y Fu(;)g(A)1603 4736 y Ft(1)1641 4724 y FA(;)0 4865 y(\(3\))28 b Fj(Z)182 4828 y Fp(1)275 4865 y Fx(`)326 4835 y Fv(\013)326 4886 y(n)p Ft(+1)p Fv(;)p Ft(0)531 4865 y FA(\000)23 b(&)g Fu(n)g Fx(\025)f FA(1)46 b Fx(\))g Fj(Z)1119 4828 y Fp(1)1213 4865 y Fx(`)1264 4835 y Ft(2)p Fp(\001)p Fv(\013)1264 4886 y(n;)p Ft(0)1387 4865 y FA(\000.)0 5031 y(By)27 b(the)h(ab)r(o)n(v)n(e)f(w)n(e)g(ha)n(v)n(e)f Fj(Z)870 4994 y Fp(1)964 5031 y Fx(`)1015 4988 y Ft(3)1048 4996 y Fn(m)1103 4988 y Ft(\()p Fv(!)r Fp(\001)p Fv(k)q Ft(\))1015 5056 y Fv(`)p Ft(+1)p Fv(;)p Ft(0)1282 5031 y FA(\000.)37 b(F)-7 b(rom)27 b(this)h(b)n(y)f(\(3\))h(w)n(e)f(get)h Fj(Z)2332 4994 y Fp(1)2425 5031 y Fx(`)2476 4988 y Ft(2)2509 4963 y Fn(`)2538 4988 y Fp(\001)p Ft(3)2591 4996 y Fn(m)2646 4988 y Ft(\()p Fv(!)r Fp(\001)p Fv(k)q Ft(\))2476 5053 y(1)2825 5031 y FA(\000.)0 5197 y(So)f(w)n(e)h(ha)n(v)n(e:)35 b Fj(Z)509 5209 y Fv(m)596 5197 y Fx(`)22 b FA(TI)759 5209 y Fp(\036)815 5197 y FA(\()p Fu(X)7 b FA(\))46 b Fx(\))g Fj(Z)1179 5160 y Fp(1)1272 5197 y Fx(`)1323 5167 y Fv(\013)1323 5218 y Ft(1)1393 5197 y FA(TI)1483 5209 y Fp(\036)1539 5197 y FA(\()p Fu(X)7 b FA(\))1725 5150 y Ft(5)p Fv(:)p Ft(8+5)p Fv(:)p Ft(3)1795 5197 y Fx(\))116 b(k\036)o(k)23 b(\024)f Fu(\013)i FA(=)e(2)2458 5167 y Fv(`)2508 5197 y Fx(\001)d FA(3)2592 5209 y Fv(m)2654 5197 y FA(\()p Fu(!)j Fx(\001)c Fu(k)s FA(\))24 b Fu(<)2990 5105 y Fq(n)3059 5168 y Fu(!)3114 5137 y Ft(2)3341 5168 y FA(if)k Fu(m)23 b FA(=)g(0)3059 5234 y Fu(!)3111 5246 y Fv(m)p Ft(+1)3341 5234 y FA(if)28 b Fu(m)23 b Fx(\025)g FA(1)1908 5539 y(31)p eop %%Page: 32 33 32 32 bop 0 83 a Fz(Arithmetization)30 b(of)i(ordinals)f Fu(<)23 b(")1298 95 y Ft(0)0 236 y FA(In)28 b(the)g(follo)n(wing)e Fu(a;)14 b(b;)g(c;)g(x;)g(y)s(;)g(z)30 b FA(denote)e(natural)f(n)n(um)n (b)r(ers.)0 389 y(W)-7 b(e)28 b(assume)f(that)h(I)-14 b(N)688 353 y Fv()288 1665 y(<)288 1815 y(>)288 1840 y(:)375 1618 y FA(o\()p Fu(h)497 1630 y Ft(0)534 1618 y FA(\)#o\()p Fu(h)757 1630 y Ft(1)795 1618 y FA(\))229 b(if)28 b Fx(I)i FA(=)22 b Fr(Cut)1420 1630 y Fv(C)375 1723 y FA(o\()p Fu(h)497 1735 y Ft(0)534 1723 y FA(\))d Fx(\001)f Fu(!)378 b FA(if)28 b Fx(I)i FA(=)22 b Fr(Ind)1403 1683 y Fv(x;t)1403 1747 y(C)375 1829 y FA(3)417 1799 y Ft(o\()p Fv(h)515 1807 y Fo(0)547 1799 y Ft(\))1056 1829 y FA(if)28 b Fx(I)i FA(=)22 b Fr(E)375 1900 y FA(sup)500 1920 y Fv(i)h FA(0)2085 4902 y(where)129 5123 y Fu(n)g FA(:=)g Fu(t)343 5093 y Fp(N)410 5123 y FA(,)55 b Fi(c)531 5083 y Fv(x;t)531 5148 y(F)641 5123 y Fx(`)692 5090 y Fv()h FA(0)k(.)0 5339 y(5.)64 b(Otherwise:)36 b(\()p Fx(I)6 b Fu(h)685 5351 y Ft(0)723 5339 y Fu(:::h)840 5351 y Fv(n)p Fp(\000)p Ft(1)970 5339 y FA(\))1002 5309 y Fi(!)1089 5339 y FA(:=)23 b Fx(I)6 b Fu(h)1299 5309 y Fi(!)1299 5360 y Ft(0)1377 5339 y Fu(:)14 b(:)g(:)f(h)1535 5309 y Fi(!)1535 5360 y Fv(n)p Fp(\000)p Ft(1)1693 5339 y FA(.)1908 5539 y(34)p eop %%Page: 35 36 35 35 bop 0 83 a Fz(Remark)0 222 y FA(With)28 b(the)g(help)g(of)g (Theorems)e(5.1,5.2)g(one)i(easily)e(v)n(eri\014es)h(that)h Fu(h)2165 192 y Fi(!)2256 222 y FA(is)f(a)h Fj(Z)2457 185 y Fp(1)2527 222 y FA(-deriv)-5 b(ation)27 b(with)h Fu(h)3182 192 y Fi(!)3268 222 y Fx(`)3319 179 y Ft(o\()p Fv(h)p Ft(\))3319 250 y Fl(deg)p Ft(\()p Fv(h)p Ft(\))3531 222 y FA(\000\()p Fu(h)p FA(\).)0 384 y(Let)g(us)f(lo)r(ok)g(at)h(the)g Fr(Ind)o FA(-case:)0 545 y(Let)g Fu(\013)202 557 y Ft(0)262 545 y FA(:=)23 b(o\()p Fu(h)495 557 y Ft(0)532 545 y FA(\),)28 b Fu(\013)668 557 y Fv(i)p Ft(+1)803 545 y FA(:=)22 b Fu(\013)966 557 y Fv(i)994 545 y FA(#)p Fu(\013)1116 557 y Ft(0)1181 545 y FA(and)28 b Fu(m)23 b FA(:=)f Fr(deg)p FA(\()p Fu(h)p FA(\))h(=)g(max)o Fx(f)p FA(rk)o(\()p Fu(F)12 b FA(\))p Fu(;)i Fr(deg)p FA(\()p Fu(h)2534 557 y Ft(0)2571 545 y FA(\))p Fx(g)p FA(.)0 707 y(F)-7 b(rom)27 b Fu(h)266 719 y Ft(0)303 707 y FA(\()p Fu(x=i)p FA(\))485 677 y Fi(!)572 707 y Fx(`)623 677 y Fv(\013)666 685 y Fo(0)623 728 y Fv(m)726 707 y FA(\000)p Fu(;)14 b Fx(:)p Fu(F)e FA(\()p Fu(i)p FA(\))p Fu(;)i(F)e FA(\()p Fu(i)p FA(+)o(1\))28 b(and)f Fi(e)1564 719 y Fv(i)p Fp(\000)p Ft(1)1700 707 y Fx(`)1751 663 y Fv(\013)1794 671 y Fn(i)p Fk(\000)p Fo(1)1751 717 y Fv(m)1920 707 y FA(\000)p Fu(;)14 b Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(i)p FA(\))28 b(w)n(e)f(obtain)0 846 y Fi(e)46 858 y Fv(i)97 846 y FA(=)22 b Fx(R)254 861 y Fv(F)9 b Ft(\()p Fv(i)p 331 874 24 4 v 1 w Ft(\))385 846 y FA(\()p Fi(e)463 858 y Fv(i)p Fp(\000)p Ft(1)576 846 y Fu(;)14 b(h)661 858 y Ft(0)698 846 y FA(\()p Fu(x=i)p 819 859 29 4 v FA(\))880 816 y Fi(!)944 846 y FA(\))23 b Fx(`)1050 816 y Fv(\013)1093 824 y Fn(i)1050 867 y Fv(m)1147 846 y FA(\000)p Fu(;)14 b Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(i)18 b FA(+)g(1\).)0 1008 y Fu(n)23 b(>)g FA(0:)36 b(F)-7 b(rom)27 b Fi(e)526 1020 y Fv(n)p Fp(\000)p Ft(1)679 1008 y Fx(`)730 964 y Fv(\013)773 972 y Fn(n)p Fk(\000)p Fo(1)730 1018 y Fv(m)914 1008 y FA(\000)p Fu(;)14 b Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(n)p FA(\))28 b(and)g Fi(c)1677 968 y Fv(x;t)1677 1032 y(F)1787 1008 y Fx(`)1838 978 y Fv(k)1838 1029 y Ft(0)1902 1008 y Fx(:)p Fu(F)12 b FA(\()p Fu(n)p 2054 1021 50 4 v FA(\))p Fu(;)i(F)e FA(\()p Fu(t)p FA(\))29 b(w)n(e)e(obtain)0 1147 y Fr(Rep)o Fx(R)203 1162 y Fv(F)9 b Ft(\()p Fv(n)p 280 1175 42 4 v 1 w Ft(\))352 1147 y FA(\()p Fi(e)430 1159 y Fv(n)p Fp(\000)p Ft(1)560 1147 y Fu(;)14 b Fi(c)640 1107 y Fv(x;t)640 1171 y(F)726 1147 y FA(\))24 b Fx(`)833 1117 y Fv(\013)876 1125 y Fo(0)908 1117 y Fp(\001)p Fv(!)833 1167 y(m)999 1147 y FA(\000)p Fu(;)14 b Fx(:)p Fu(F)e FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(t)p FA(\).)0 1309 y Fu(n)23 b FA(=)g(0:)36 b(Then)28 b Fi(c)521 1269 y Fv(x;t)521 1333 y(F)631 1309 y Fx(`)682 1272 y Fv(\013)725 1280 y Fo(0)757 1272 y Fp(\001)p Fv(!)682 1331 y Ft(0)848 1309 y Fx(:)p Fu(F)12 b FA(\(0\))p Fu(;)i(F)e FA(\()p Fu(t)p FA(\).)38 b(\(1)23 b Fx(\024)f Fu(\013)1568 1321 y Ft(0)1629 1309 y Fx(\))h Fu(!)j Fx(\024)c Fu(\013)1953 1321 y Ft(0)2009 1309 y Fx(\001)d Fu(!)s FA(\))0 1561 y Fz(De\014nition)31 b(of)h Fr(tp)o FA(\()p Fu(h)p FA(\))h Fz(and)f Fu(h)p FA([)p Fu(\023)p FA(])g Fz(for)g(closed)f Fj(Z)1590 1524 y Fp(\003)1629 1561 y Fz(-deriv)-5 b(ations)30 b Fu(h)i Fz(and)g Fu(\023)24 b Fx(2)f(j)p Fr(tp)p FA(\()p Fu(h)p FA(\))p Fx(j)0 1723 y FA(By)29 b(recursion)f(on)h(the)g(build-up)h(of)f Fu(h)g FA(w)n(e)g(de\014ne)h(a)f Fj(Z)1743 1686 y Fp(1)1813 1723 y FA(-inference)g Fr(tp)p FA(\()p Fu(h)p FA(\))h(and)f(closed)f Fj(Z)2864 1686 y Fp(\003)2902 1723 y FA(-deriv)-5 b(ation\(s\))28 b Fu(h)p FA([)p Fu(\023)p FA(])i(in)g(suc)n(h)e(a)0 1862 y(w)n(a)n(y)e(that)999 2019 y Fu(h)1047 1984 y Fi(!)1134 2019 y FA(=)d Fr(tp)o FA(\()p Fu(h)p FA(\))1406 1926 y Fq(\020)1456 2019 y Fu(h)p FA([)p Fu(\023)p FA(])1579 1984 y Fi(!)1644 1926 y Fq(\021)1693 2077 y Fv(\023)p Fp(2j)p Fl(tp)p Ft(\()p Fv(h)p Ft(\))p Fp(j)1976 2019 y FA(=)2088 1957 y Fu(:)14 b(:)g(:)f(h)p FA([)p Fu(\023)p FA(])2321 1927 y Fi(!)2399 1957 y Fu(:)h(:)g(:)g FA(\()p Fu(\023)p Fx(2)q(j)p Fr(tp)p FA(\()p Fu(h)p FA(\))p Fx(j)p FA(\))p 2074 1999 817 4 v 2307 2080 a Fr(tp)p FA(\()p Fu(h)p FA(\))0 2247 y(The)28 b(de\014nition)g(clauses)f(for)g Fu(h)d FA(=)f Fr(Cut)1230 2259 y Fv(C)1286 2247 y Fu(h)1334 2259 y Ft(0)1371 2247 y Fu(h)1419 2259 y Ft(1)1484 2247 y FA(and)28 b Fu(h)23 b FA(=)g Fr(E)p Fu(h)1903 2259 y Ft(0)1968 2247 y FA(can)k(b)r(e)h(read)f(o\013)h(from)g(the)g 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Fq(V)394 1775 y Fv(A)444 1783 y Fo(0)476 1775 y Fp(^)p Fv(A)571 1783 y Fo(1)607 1750 y Fu(h)655 1762 y Ft(0)692 1750 y Fu(h)740 1762 y Ft(1)805 1750 y FA(or)k Fu(h)c FA(=)1065 1688 y Fq(W)1135 1708 y Fv(k)1135 1775 y(A)1185 1783 y Fo(0)1217 1775 y Fp(_)p Fv(A)1312 1783 y Fo(1)1348 1750 y FA(:)37 b(analogous)25 b(to)j(2.1)f(and)g(2.2.)0 1915 y(4.1.)36 b Fu(h)23 b FA(=)g Fr(Cut)450 1927 y Fv(C)506 1915 y Fu(h)554 1927 y Ft(0)591 1915 y Fu(h)639 1927 y Ft(1)704 1915 y FA(with)28 b Fu(C)i FA(=)22 b Fx(9)p Fu(xA)p FA(,)29 b Fr(tp)p FA(\()p Fu(h)1429 1927 y Ft(0)1466 1915 y FA(\))23 b(=)1609 1853 y Fq(W)1678 1873 y Fv(t)1678 1940 y(C)1734 1915 y FA(,)28 b Fr(tp)p FA(\()p Fu(h)1938 1927 y Ft(1)1975 1915 y FA(\))23 b(=)2118 1853 y Fq(V)2187 1940 y Fp(:)p Fv(C)2288 1915 y FA(:)37 b(Then)28 b Fu(t)23 b Fx(2)g Fu(T)2745 1927 y Ft(0)2809 1915 y FA(and)28 b Fr(tp)p FA(\()p Fu(h)p FA(\))23 b(=)g Fr(Cut)3393 1930 y Fv(A)p Ft(\()p Fv(t)p Ft(\))3524 1915 y FA(.)0 2080 y(Let)28 b Fu(\015)f FA(:=)c(o\()p Fu(h)452 2092 y Ft(0)489 2080 y FA(\),)28 b Fu(\014)f FA(:=)c(o\()p Fu(h)879 2092 y Ft(1)916 2080 y FA(\).)0 2245 y(Then)28 b Fu(h)265 2257 y Ft(0)325 2245 y Fx(`)362 2226 y Fk(\016)374 2194 y Fv(\015)374 2278 y(m)437 2245 y FA(\000)p Fu(;)14 b(C)34 b FA(,)27 b Fu(h)717 2257 y Ft(1)778 2245 y Fx(`)815 2226 y Fk(\016)827 2194 y Fv(\014)827 2278 y(m)890 2245 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)34 b FA(and)27 b(rk\()p Fu(A)p FA(\()p Fu(t)p FA(\)\))d Fu(<)f FA(rk)o(\()p Fu(C)6 b FA(\))24 b Fx(\024)e Fu(m)p FA(.)0 2385 y(By)27 b(IH)h(w)n(e)f(obtain)h Fu(h)679 2397 y Ft(0)716 2385 y FA([0])23 b Fx(`)864 2367 y Fk(\016)876 2335 y Fv(\015)911 2343 y Fo(0)876 2418 y Fv(m)947 2385 y FA(\000)p Fu(;)14 b(C)q(;)g(A)p FA(\()p Fu(t)p FA(\))29 b(with)f Fu(\015)1550 2397 y Fv(t)1602 2385 y Fu(<)23 b(\015)5 b FA(,)27 b(and)h Fu(h)1998 2397 y Ft(1)2035 2385 y FA([)p Fu(t)p FA(])23 b Fx(`)2171 2367 y Fk(\016)2183 2335 y Fv(\014)2221 2343 y Fn(t)2183 2418 y Fv(m)2252 2385 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)q(;)g Fx(:)p Fu(A)p FA(\()p Fu(t)p FA(\))30 b(with)e Fu(\014)2970 2397 y Fv(t)3022 2385 y Fu(<)23 b(\014)t FA(.)0 2526 y(Hence)28 b Fu(h)p FA([0])22 b(=)h Fr(Cut)619 2538 y Fv(C)675 2526 y Fu(h)723 2538 y Ft(0)760 2526 y FA([0])p Fu(h)896 2538 y Ft(1)956 2526 y Fx(`)993 2507 y Fk(\016)1005 2476 y Fv(\015)1040 2484 y Fo(0)1072 2476 y Ft(#)p Fv(\014)1005 2559 y(m)1171 2526 y FA(\000)p Fu(;)14 b(A)p FA(\()p Fu(t)p FA(\))28 b(and)g Fu(h)p FA([1])22 b(=)h Fr(Cut)1978 2538 y Fv(C)2034 2526 y Fu(h)2082 2538 y Ft(0)2119 2526 y Fu(h)2167 2538 y Ft(1)2204 2526 y FA([)p Fu(t)p FA(])g Fx(`)2340 2507 y Fk(\016)2352 2476 y Fv(\015)t Ft(#)p Fv(\014)2484 2484 y Fn(t)2352 2559 y Fv(m)2514 2526 y FA(\000)p Fu(;)14 b Fx(:)p Fu(A)p FA(\()p Fu(t)p FA(\))0 2666 y(with)28 b Fu(\015)232 2678 y Ft(0)269 2666 y FA(#)p Fu(\014)19 b(;)27 b(\015)5 b FA(#)p Fu(\014)618 2678 y Fv(t)670 2666 y Fu(<)23 b(\015)5 b FA(#)p Fu(\014)27 b FA(=)c(o\()p Fu(h)p FA(\))g Fx(\024)g Fu(\013)p FA(.)0 2831 y(4.2.)36 b Fu(h)23 b FA(=)g Fr(Cut)450 2843 y Fv(C)506 2831 y Fu(h)554 2843 y Ft(0)591 2831 y Fu(h)639 2843 y Ft(1)704 2831 y FA(with)28 b Fr(tp)p FA(\()p Fu(h)1046 2843 y Ft(0)1084 2831 y FA(\))23 b(=)f Fx(I)34 b FA(and)28 b Fu(C)h Fx(62)24 b FA(\001\()p Fx(I)6 b FA(\):)38 b(Let)27 b Fu(\015)h FA(:=)23 b(o\()p Fu(h)2331 2843 y Ft(0)2368 2831 y FA(\),)28 b Fu(\014)f FA(:=)c(o\()p Fu(h)2758 2843 y Ft(1)2795 2831 y FA(\).)0 2971 y(Then)28 b Fu(h)265 2983 y Ft(0)325 2971 y Fx(`)362 2953 y Fk(\016)374 2921 y Fv(\015)374 3005 y(m)437 2971 y FA(\000)p Fu(;)14 b(C)34 b FA(,)27 b Fu(h)717 2983 y Ft(1)778 2971 y Fx(`)815 2953 y Fk(\016)827 2921 y Fv(\014)827 3005 y(m)890 2971 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)6 b FA(.)0 3112 y(By)27 b(IH)h(w)n(e)f(obtain)h Fu(h)679 3124 y Ft(0)716 3112 y FA([)p Fu(\023)p FA(])23 b Fx(`)851 3094 y Fk(\016)863 3062 y Fv(\015)898 3070 y Fn(\023)863 3145 y Fv(m)929 3112 y FA(\000)p Fu(;)14 b(C)q(;)g FA(\001)1184 3124 y Fv(\023)1214 3112 y FA(\()p Fx(I)6 b FA(\))28 b(with)g Fu(\015)1589 3124 y Fv(\023)1641 3112 y Fu(<)23 b(\015)5 b FA(,)27 b(for)g(all)g Fu(\023)d Fx(2)f(jI)6 b(j)p FA(.)0 3252 y(Hence)28 b Fu(h)p FA([)p Fu(\023)p FA(])23 b(=)g Fr(Cut)607 3264 y Fv(C)663 3252 y Fu(h)711 3264 y Ft(0)748 3252 y FA([)p Fu(\023)p FA(])p Fu(h)871 3264 y Ft(1)931 3252 y Fx(`)968 3234 y Fk(\016)980 3202 y Fv(\015)1015 3210 y Fn(\023)1042 3202 y Ft(#)p Fv(\014)980 3286 y(m)1141 3252 y FA(\000)p Fu(;)14 b FA(\001)1299 3264 y Fv(\023)1328 3252 y FA(\()p Fx(I)6 b FA(\))29 b(with)f Fu(\015)1704 3264 y Fv(\023)1733 3252 y FA(#)p Fu(\014)f(<)c(\015)5 b FA(#)p Fu(\014)27 b FA(=)c(o\()p Fu(h)p FA(\))g Fx(\024)f Fu(\013)p FA(.)0 3417 y Fu(h)48 3429 y Ft(0)108 3417 y Fx(`)145 3399 y Fk(\016)157 3367 y Fv(\015)157 3450 y(m)220 3417 y FA(\000)p Fu(;)14 b(C)29 b FA(&)23 b Fr(tp)p FA(\()p Fu(h)638 3429 y Ft(0)675 3417 y FA(\))h(=)e Fx(I)30 b FA(&)22 b Fu(C)30 b Fx(62)23 b FA(\001\()p Fx(I)6 b FA(\))1360 3370 y Ft(IH)1355 3417 y Fx(\))23 b FA(\001\()p Fx(I)6 b FA(\))24 b Fx(\022)f FA(\000)g(&)g(\()p Fx(I)29 b FA(=)23 b Fr(Cut)2240 3429 y Fv(A)2317 3417 y Fx(\))g FA(rk)o(\()p Fu(A)p FA(\))h Fu(<)f(m)p FA(\).)0 3582 y(4.3.)36 b Fu(h)23 b FA(=)g Fr(Cut)450 3594 y Fv(C)506 3582 y Fu(h)554 3594 y Ft(0)591 3582 y Fu(h)639 3594 y Ft(1)704 3582 y FA(with)28 b(rk\()p Fu(C)6 b FA(\))24 b(=)e(0)27 b(and)h Fu(C)h Fx(2)24 b FA(\001)1677 3552 y Ft(0)1737 3582 y FA(:=)f(\001\()p Fr(tp)p FA(\()p Fu(h)2102 3594 y Ft(0)2139 3582 y FA(\)\),)29 b Fx(:)p Fu(C)g Fx(2)24 b FA(\001)2546 3552 y Ft(1)2606 3582 y FA(:=)f(\001\()p Fr(tp)p FA(\()p Fu(h)2971 3594 y Ft(1)3009 3582 y FA(\)\):)0 3723 y(Then)28 b Fx(I)h FA(=)23 b Fr(Ax)472 3735 y Ft(\001)559 3723 y FA(with)28 b(\001)23 b(:=)g(\(\001)1052 3692 y Ft(0)1108 3723 y Fx(n)18 b(f)p Fu(C)6 b Fx(g)p FA(\))18 b Fx([)h FA(\(\001)1542 3692 y Ft(0)1598 3723 y Fx(n)f(f:)p Fu(C)6 b Fx(g)p FA(\),)28 b(and)f(b)n(y)g(IH)h(\001)2410 3692 y Fv(i)2461 3723 y Fx(\022)23 b FA(\000\()p Fu(h)2681 3735 y Fv(i)2708 3723 y FA(\).)38 b(Hence)27 b(\001\()p Fx(I)6 b FA(\))25 b(=)d(\001)h Fx(\022)g FA(\000\()p Fu(h)p FA(\))g Fx(\022)g FA(\000.)0 3887 y(5.)36 b Fu(h)23 b FA(=)g Fr(E)p Fu(h)358 3899 y Ft(0)422 3887 y FA(with)28 b Fr(tp)p FA(\()p Fu(h)764 3899 y Ft(0)802 3887 y FA(\))23 b(=)g Fr(Cut)1071 3899 y Fv(C)1126 3887 y FA(:)37 b(Then)28 b Fr(tp)p FA(\()p Fu(h)p FA(\))c(=)e Fr(Rep)27 b FA(,)h Fu(h)p FA([0])22 b(=)h Fr(Cut)2283 3899 y Fv(C)2339 3887 y Fr(E)o Fu(h)2436 3899 y Ft(0)2474 3887 y FA([0])p Fr(E)o Fu(h)2659 3899 y Ft(0)2696 3887 y FA([1])k(and)h Fr(deg)o FA(\()p Fu(h)3174 3899 y Ft(0)3211 3887 y FA(\))c Fx(\024)e Fu(m)p FA(+1.)0 4052 y(Let)28 b Fu(\015)f FA(:=)c(o\()p Fu(h)452 4064 y Ft(0)489 4052 y FA(\).)37 b(Then)28 b Fu(h)846 4064 y Ft(0)906 4052 y Fx(`)943 4034 y Fk(\016)955 4002 y Fv(\015)955 4085 y(m)p Ft(+1)1102 4052 y FA(\000.)0 4217 y(By)f(IH)h(w)n(e)f(ha)n(v)n(e)g(rk)o(\()p Fu(C)6 b FA(\))24 b Fu(<)f(m)p FA(+)o(1)k(and)h Fu(h)1297 4229 y Ft(0)1334 4217 y FA([0])23 b Fx(`)1482 4199 y Fk(\016)1494 4167 y Fv(\015)1529 4175 y Fo(0)1494 4250 y Fv(m)p Ft(+1)1641 4217 y FA(\000)p Fu(;)14 b(C)33 b FA(,)56 b Fu(h)1949 4229 y Ft(0)1986 4217 y FA([1])23 b Fx(`)2134 4199 y Fk(\016)2146 4167 y Fv(\015)2181 4175 y Fo(1)2146 4250 y Fv(m)p Ft(+1)2293 4217 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)34 b FA(with)28 b Fu(\015)2762 4229 y Ft(0)2799 4217 y Fu(;)14 b(\015)2879 4229 y Ft(1)2939 4217 y Fu(<)23 b(\015)5 b FA(.)0 4357 y(Hence)28 b Fr(E)o Fu(h)344 4369 y Ft(0)381 4357 y FA([0])23 b Fx(`)529 4339 y Fk(\016)541 4307 y Ft(3)574 4282 y Fn(\015)605 4294 y Fo(0)541 4391 y Fv(m)646 4357 y FA(\000)p Fu(;)14 b(C)34 b FA(and)27 b Fr(E)p Fu(h)1087 4369 y Ft(0)1124 4357 y FA([1])22 b Fx(`)1271 4339 y Fk(\016)1283 4307 y Ft(3)1316 4282 y Fn(\015)1347 4294 y Fo(1)1283 4391 y Fv(m)1388 4357 y FA(\000)p Fu(;)14 b Fx(:)p Fu(C)6 b FA(.)0 4522 y(F)-7 b(rom)27 b(this)h(together)f(with)h(rk)o(\()p Fu(C)6 b FA(\))p Fx(\024)q Fu(m)27 b FA(w)n(e)g(get)h Fu(h)p FA([0])22 b(=)h Fr(Cut)1901 4534 y Fv(C)1957 4522 y Fr(E)o Fu(h)2054 4534 y Ft(0)2091 4522 y FA([0])p Fr(E)p Fu(h)2277 4534 y Ft(0)2314 4522 y FA([1])f Fx(`)2461 4504 y Fk(\016)2473 4472 y Ft(3)2506 4447 y Fn(\015)2537 4459 y Fo(1)2574 4472 y Ft(#3)2662 4447 y Fn(\015)2693 4459 y Fo(1)2473 4555 y Fv(m)2733 4522 y FA(\000)28 b(and)f(3)3016 4492 y Fv(\015)3051 4500 y Fo(0)3087 4522 y FA(#3)3198 4492 y Fv(\015)3233 4500 y Fo(1)3269 4522 y Fu(<)p FA(3)3376 4492 y Fv(\015)3441 4522 y FA(=)c(o\()p Fu(h)p FA(\))g Fx(\024)f Fu(\013)p FA(.)0 4687 y(6.)36 b Fu(h)23 b FA(=)g Fr(Ind)369 4647 y Fv(x;t)369 4711 y(F)455 4687 y Fu(h)503 4699 y Ft(0)541 4687 y FA(:)36 b(Then)28 b Fx(I)i FA(=)22 b Fr(Rep)p FA(,)27 b(rk\()p Fu(F)12 b FA(\))23 b Fx(\024)g Fu(m)p FA(,)28 b(and)f Fu(h)1813 4699 y Ft(0)1873 4687 y Fx(`)1910 4669 y Fk(\016)1922 4637 y Fv(\015)1922 4720 y(m)1985 4687 y FA(\000)p Fu(;)14 b Fx(:)p Fu(F)e FA(\()p Fu(x)p FA(\))p Fu(;)i(F)e FA(\()p Fr(S)j Fu(x)p FA(\))29 b(with)f Fu(\015)g FA(:=)22 b(o\()p Fu(h)3100 4699 y Ft(0)3137 4687 y FA(\),)28 b Fu(\015)23 b FA(+)18 b Fu(!)26 b Fx(\024)d Fu(\013)p FA(.)0 5080 y Fu(h)p FA([0])g(=)246 4823 y Fu(h)294 4835 y Ft(0)331 4823 y FA(\()p Fu(y)s(=)p FA(0\))83 b Fu(h)654 4835 y Ft(0)691 4823 y FA(\()p Fu(y)s(=)p FA(1\))p 246 4865 636 4 v 431 4940 a Fr(Cut)557 4955 y Fv(F)9 b Ft(\(1\))910 4940 y Fu(h)958 4952 y Ft(0)995 4940 y FA(\()p Fu(y)s(=)p FA(2\))p 431 4982 755 4 v 797 5044 a(.)797 5078 y(.)797 5111 y(.)394 b Fu(h)1262 5123 y Ft(0)1299 5111 y FA(\()p Fu(y)s(=n)p Fx(\000)o FA(1\))p 797 5152 808 4 v 1021 5227 a Fr(Cut)1147 5242 y Fv(F)9 b Ft(\()p Fv(n)p Fp(\000)p Ft(1\))1687 5227 y Fu(c)1723 5187 y Fv(y)r(;t)1723 5251 y(F)p 1021 5269 787 4 v 1278 5344 a Fr(Cut)1403 5359 y Fv(F)g Ft(\()p Fv(n)p Ft(\))1891 5080 y FA(with)28 b Fj(Z)23 b Fx(3)h Fu(c)2266 5040 y Fv(y)r(;t)2266 5104 y(F)2374 5080 y Fx(`)2425 5092 y Ft(1)2485 5080 y Fx(:)p Fu(F)12 b FA(\()p Fu(n)p FA(\))p Fu(;)i(F)e FA(\()p Fu(t)p FA(\).)1908 5539 y(36)p eop %%Page: 37 38 37 37 bop 0 83 a Fz(Lemma)30 b(6.3)c FA(\(Consistency)h(of)h Fj(Z)p FA(\))0 224 y(Let)g Fj(Z)197 187 y Fp(\003)197 245 y(?)281 224 y FA(b)r(e)g(the)g(set)f(of)h(all)f(closed)g Fj(Z)1169 187 y Fp(\003)1207 224 y FA(-deriv)-5 b(ations)27 b Fu(h)g FA(with)h(\000\()p Fu(h)p FA(\))c(=)e Fx(;)h FA(&)g Fr(deg)o FA(\()p Fu(h)p FA(\))g(=)g(0.)0 365 y(\(a\))28 b Fu(h)23 b Fx(2)g Fj(Z)331 328 y Fp(\003)331 386 y(?)433 365 y FA(=)-14 b Fx(\))46 b Fu(h)p FA([0])23 b Fx(2)g Fj(Z)898 328 y Fp(\003)898 386 y(?)978 365 y FA(&)f(o\()p Fu(h)p FA([0]\))h Fu(<)f FA(o\()p Fu(h)p FA(\),)0 506 y(\(b\))28 b(There)f(is)h(no)f Fj(Z)p FA(-deriv)-5 b(ation)27 b Fu(d)h FA(with)g(\000\()p Fu(d)p FA(\))c(=)e Fx(;)p FA(.)0 697 y(Pro)r(of:)0 839 y(\(a\))28 b Fu(h)23 b Fx(2)g Fj(Z)331 801 y Fp(\003)331 859 y(?)433 791 y Ft(6)p Fv(:)p Ft(2a)451 839 y Fx(\))65 b FA(\001\()p Fr(tp)p FA(\()p Fu(h)p FA(\)\))24 b Fx(\022)e FA(\000\()p Fu(h)p FA(\))i(=)e Fx(;)h FA(&)f Fr(deg)p FA(\()p Fu(h)p FA(\))h(=)g(0)1888 791 y Ft(6)p Fv(:)p Ft(2b)1907 839 y Fx(\))66 b Fr(tp)p FA(\()p Fu(h)p FA(\))24 b(=)e Fr(Rep)p FA(.)0 980 y Fu(h)h Fx(2)g Fj(Z)198 943 y Fp(\003)198 1000 y(?)277 980 y FA(&)f Fr(tp)p FA(\()p Fu(h)p FA(\))i(=)e Fr(Rep)840 933 y Ft(6)p Fv(:)p Ft(2c)856 980 y Fx(\))62 b Fu(h)p FA([0])23 b Fx(2)g Fj(Z)1286 943 y Fp(\003)1286 1000 y(?)1366 980 y FA(&)f(o\()p Fu(h)p FA([0]\))h Fu(<)f FA(o\()p Fu(h)p FA(\).)0 1146 y(\(b\))28 b(By)g(trans\014nite)f (induction)h(up)g(to)f Fu(")1295 1158 y Ft(0)1360 1146 y FA(from)g(\(a\))h(w)n(e)f(get)h Fj(Z)1999 1109 y Fp(\003)1999 1166 y(?)2078 1146 y FA(=)23 b Fx(;)p FA(.)0 1287 y(No)n(w)k(assume)g (that)h Fu(d)g FA(is)f(a)g Fj(Z)q FA(-deriv)-5 b(ation)26 b(with)i(\000\()p Fu(d)p FA(\))c(=)f Fx(;)p FA(.)36 b(W.l.o.g.)28 b(w)n(e)f(ma)n(y)g(assume)g(that)g Fu(d)h FA(is)g(closed.)0 1428 y(Let)g Fu(m)23 b FA(:=)f Fr(deg)p FA(\()p Fu(d)p FA(\).)37 b(Then)28 b Fr(E)911 1391 y Fv(m)974 1428 y Fu(d)23 b FA(=)g Fr(E)o Fu(:::)p Fr(E)p Fu(d)g Fx(2)h Fj(Z)1489 1391 y Fp(\003)1489 1449 y(?)1545 1428 y FA(.)37 b Fs(Contr)l(adiction.)0 1644 y Fz(Theorem)30 b(6.4.)106 b FA(PRA)18 b(+)g Fr(QF)q FA(-TI)o(\()p Fu(")1239 1656 y Ft(0)1277 1644 y FA(\))23 b Fx(`)g FA(Con)o(\()p Fj(Z)q FA(\).)0 1810 y(Pro)r(of:)36 b(The)27 b(\(trans\014nite\))h(induction)g (form)n(ula)f(in)h(6.3b)f(is)g(the)h(\005)2131 1822 y Ft(1)2169 1810 y FA(-form)n(ula)e Fu(F)12 b FA(\()p Fu(\013)p FA(\))24 b(:=)f Fx(8)p Fu(h)p FA(\(o\()p Fu(h)p FA(\))f(=)g Fu(\013)i Fx(!)f Fu(h)g Fx(62)g Fj(Z)3589 1773 y Fp(\003)3589 1830 y(?)3645 1810 y FA(\):)0 1951 y Fx(8)p Fu(\014)k Fx(\036)22 b Fu(\013F)12 b FA(\()p Fu(\014)t FA(\))24 b(&)f(o\()p Fu(h)p FA(\))g(=)g Fu(\013)g FA(&)g Fu(h)g Fx(2)g Fj(Z)1179 1914 y Fp(\003)1179 1972 y(?)1281 1904 y Ft(6)p Fv(:)p Ft(3a)1299 1951 y Fx(\))65 b FA(\(o\()p Fu(h)p FA([0]\))23 b(=)f(o\()p Fu(h)p FA([0]\))h Fx(!)g Fu(h)p FA([0])f Fx(62)i Fj(Z)2487 1914 y Fp(\003)2487 1972 y(?)2543 1951 y FA(\))f(&)g Fu(h)p FA([0])g Fx(2)g Fj(Z)2971 1914 y Fp(\003)2971 1972 y(?)3073 1951 y Fx(\))46 b(?)p FA(.)0 2117 y(It)28 b(remains)f(to)g(pro)n(v)n(e)f(that)i(\005) 967 2129 y Ft(1)1004 2117 y FA(-TI\()p Fu(")1193 2129 y Ft(0)1230 2117 y FA(\))g(can)f(b)r(e)h(deriv)n(ed)f(from)h Fr(QF)p FA(-TI\()p Fu(")2340 2129 y Ft(0)2377 2117 y FA(\).)0 2258 y(Let)g Fu(F)12 b FA(\()p Fu(x)p FA(\))24 b(=)e Fx(8)p Fu(y)s(A)p FA(\()p Fu(x;)14 b(y)s FA(\))28 b(with)g Fu(A)23 b Fx(2)h Fr(QF)p FA(.)0 2424 y(There)j(are)g(primitiv) n(e)g(recursiv)n(e)f(functions)i Fu(q)s(;)14 b(r)n(;)g(p)27 b FA(suc)n(h)h(that)f Fu(p)p FA(\()2098 2393 y Fk(d)2126 2424 y Fu(\013)2191 2393 y Fk(e)2219 2424 y Fu(;)14 b(k)s FA(\))23 b(=)2444 2393 y Fk(d)2472 2424 y Fu(!)s Fx(\001)p Fu(\013)c FA(+)f Fu(k)2762 2393 y Fk(e)2817 2424 y FA(for)28 b(all)f Fu(\013)c(<)g(")3263 2436 y Ft(0)3300 2424 y FA(,)28 b Fu(k)e(<)c(!)s FA(,)28 b(and)0 2565 y Fr(PRA)23 b Fx(`)g Fu(x)g Fx(2)h FA(OT)36 b Fx(!)46 b Fu(p)p FA(\()p Fu(x;)14 b(y)s FA(\))24 b Fx(2)f FA(OT)18 b Fx(^)h Fu(q)s FA(\()p Fu(p)p FA(\()p Fu(x;)14 b(y)s FA(\)\))24 b(=)f Fu(x)c Fx(^)f Fu(r)r FA(\()p Fu(p)p FA(\()p Fu(x;)c(y)s FA(\)\))25 b(=)e Fu(y)s FA(;)0 2707 y Fr(PRA)g Fx(`)g(8)p Fu(x;)14 b(y)s FA(\()p Fu(q)s FA(\()p Fu(y)s FA(\))23 b Fx(\036)f Fu(q)s FA(\()p Fu(x)p FA(\))i Fx(!)g Fu(y)h Fx(\036)e Fu(x)p FA(\).)0 2873 y(Abbreviation:)36 b Fu(G)p FA(\()p Fu(z)t FA(\))23 b(:)p Fx(\021)37 b Fu(A)p FA(\()p Fu(q)s FA(\()p Fu(z)t FA(\))p Fu(;)14 b(r)r FA(\()p Fu(z)t FA(\)\))0 3039 y(\(*\))28 b(Prog)307 3059 y Fp(\036)363 3039 y FA(\()p Fu(F)12 b FA(\))24 b Fx(!)f FA(Prog)795 3059 y Fp(\036)851 3039 y FA(\()p Fu(G)p FA(\).)0 3205 y(Pro)r(of:)36 b(Assume)55 b(\(1\))28 b(Prog)902 3225 y Fp(\036)958 3205 y FA(\()p Fu(F)12 b FA(\),)56 b(\(2\))28 b Fx(8)p Fu(z)d Fx(\036)e Fu(cG)p FA(\()p Fu(z)t FA(\).)0 3346 y(T)-7 b(o)27 b(pro)n(v)n(e:)35 b Fu(G)p FA(\()p Fu(c)p FA(\),)29 b(i.e.)37 b Fu(A)p FA(\()p Fu(q)s FA(\()p Fu(c)p FA(\))p Fu(;)14 b(r)r FA(\()p Fu(c)p FA(\)\).)39 b(By)28 b(\(1\))f(this)h(can)g(b)r(e)g(obtained)f(from)g Fx(8)p Fu(x)c Fx(\036)f Fu(q)s FA(\()p Fu(c)p FA(\))p 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Fr(deg)260 5258 y Fl(QF)348 5238 y FA(\()p Fu(h)p FA([)p Fu(\023)p FA(]\))c Fx(\024)f Fr(deg)768 5258 y Fl(QF)856 5238 y FA(\()p Fu(h)p FA(\))28 b(for)g(eac)n(h)e Fu(\023)e Fx(2)f(j)p Fr(tp)p FA(\()p Fu(h)p FA(\))p Fx(j)p FA(.)1908 5539 y(37)p eop %%Page: 38 39 38 38 bop 0 83 a Fw(Characterization)28 b(of)g(the)f(pro)n(v)-5 b(ably)28 b(recursiv)n(e)d(functions)j(of)g Fj(Z)0 299 y FA(Let)g(\(OT)p Fu(;)14 b Fx(\036)p FA(\))27 b(b)r(e)h(the)g(w)n (ellordering)e(of)h(ordert)n(yp)r(e)g Fu(")1702 311 y Ft(0)1766 299 y FA(as)g(in)n(tro)r(duced)g(in)h Fx(x)p FA(5.)0 490 y Fz(Theorem)i(6.6)0 631 y FA(If)39 b(\005)156 643 y Fv(m)p Ft(+1)303 631 y FA(-IA)j Fx(`)f(8)p Fu(x)p Fx(9)p Fu(y)s(A)p FA(\()p Fu(x;)14 b(y)s FA(\))77 b(\()p Fu(A)40 b FA(a)e(quan)n(ti\014erfree)f Fx(L)1859 643 y Ft(0)1897 631 y FA(-form)n(ula\))g(then)j(there)e(are)g(primitiv)n(e) g(recursiv)n(e)f(functions)0 772 y Fu(g)26 b FA(:)d(I)-14 b(N)190 736 y Ft(2)250 772 y Fx(!)24 b FA(I)-14 b(N,)28 b Fu(\022)d FA(:)e(I)-14 b(N)675 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y Fv(\025)p Ft([)p Fv(n)p Ft(])1935 1094 y Fx(\024)23 b Fu(\014)2070 1106 y Ft(0)2130 1094 y FA(=)g Fu(!)2273 1064 y Fv(\015)2338 1094 y Fu(<)g(!)2481 1064 y Fv(\025)2524 1094 y FA(.)0 1235 y(If)28 b Fu(k)e FA(=)d(0)k(then)h Fu(\025)p FA([)p Fu(n)p FA(])23 b Fu(<)g(\015)28 b(<)22 b(\025)28 b FA(and)g(therefore)e(\(b)n(y)i(IH\))g Fu(\025)p FA([)p Fu(n)p FA(])c Fx(\024)e Fu(\015)5 b FA([0].)37 b(Hence)27 b Fu(!)2549 1205 y Fv(\025)p Ft([)p Fv(n)p Ft(])2694 1235 y Fx(\024)c Fu(!)2837 1205 y Fv(\015)t Ft([0])2973 1235 y FA(=)f Fu(!)3115 1205 y Fv(\015)3158 1235 y FA([0])g(=)h Fu(\014)t FA([0].)0 1376 y(If)28 b Fu(k)e(>)d FA(0)k(then)h Fu(!)553 1346 y Fv(\025)p Ft([)p Fv(n)p Ft(])698 1376 y Fx(\024)22 b Fu(\014)832 1388 y Ft(0)888 1376 y FA(+)c Fu(:::)h FA(+)f Fu(\014)1189 1388 y Fv(k)q Fp(\000)p Ft(1)1333 1376 y FA(+)g Fu(\014)1463 1388 y Fv(k)1504 1376 y FA([0])k(=)h Fu(\014)t FA([0].)0 1517 y(3.)36 b(Assume)26 b Fu(\013)e FA(=)549 1529 y Fv(N)-5 b(F)674 1517 y Fu(\013)727 1529 y Ft(0)780 1517 y FA(+)15 b Fu(:::)g FA(+)g Fu(\013)1077 1529 y Fv(m)1140 1517 y FA(,)27 b Fu(m)c(>)g FA(0)i(and)h Fu(\013)p FA([)p Fu(n)p FA(])e(=)e Fu(\013)1914 1529 y Ft(0)1967 1517 y FA(+)15 b Fu(:::)h FA(+)f Fu(\013)2265 1529 y Fv(m)p Fp(\000)p Ft(1)2428 1517 y FA(+)g Fu(\013)2561 1529 y Fv(m)2624 1517 y FA([)p Fu(n)p FA(])23 b Fu(<)g(\014)28 b(<)22 b(\013)p FA(.)37 b(Then)26 b Fu(m)d Fx(\024)g Fu(k)s FA(,)j Fu(\013)3653 1529 y Fv(m)3716 1517 y FA([)p Fu(n)p FA(])d Fu(<)0 1658 y(\014)47 1670 y Fv(m)124 1658 y FA(+)14 b Fu(:::)g FA(+)g Fu(\014)412 1670 y Fv(k)476 1658 y Fu(<)23 b(\013)617 1670 y Fv(m)705 1658 y FA(and)j Fu(\013)918 1670 y Fv(i)968 1658 y FA(=)d Fu(\014)1103 1670 y Fv(i)1156 1658 y FA(for)i Fu(i)e(<)g(m)p FA(.)36 b(By)25 b(IH)h(w)n(e)f(get)g Fu(\013)2108 1670 y Fv(m)2171 1658 y FA([)p Fu(n)p FA(])e Fx(\024)g FA(\()p Fu(\014)2457 1670 y Fv(m)2534 1658 y FA(+)14 b Fu(:::)g FA(+)g Fu(\014)2822 1670 y Fv(k)2863 1658 y FA(\)[0])23 b(=)g Fu(\014)3141 1670 y Fv(m)3218 1658 y FA(+)14 b Fu(:::)g FA(+)g Fu(\014)3506 1670 y Fv(k)q Fp(\000)p Ft(1)3646 1658 y FA(+)g Fu(\014)3772 1670 y Fv(k)3812 1658 y FA([0])0 1799 y(and)27 b(then)i Fu(\013)p FA([)p Fu(n)p FA(])23 b Fx(\024)g Fu(\014)658 1811 y Ft(0)713 1799 y FA(+)18 b Fu(:::)h FA(+)f Fu(\014)1014 1811 y Fv(k)q Fp(\000)p Ft(1)1158 1799 y FA(+)g Fu(\014)1288 1811 y Fv(k)1329 1799 y FA([0])23 b(=)f Fu(\014)t FA([0].)0 1965 y(\(d\))28 b(By)g(\(c\))g(w)n(e)f(ha)n(v)n(e)f Fu(\013)p FA([)p Fu(n)p FA(])e(=)f Fu(\014)t FA([0])p Fu(:::)p FA([0].)36 b(Hence)28 b Fu(N)9 b(\013)p FA([)p Fu(n)p FA(])23 b Fx(\024)g Fu(N)9 b(\014)t FA([0])23 b Fu(<)f(N)9 b(\014)t FA(.)0 2130 y(\(e\))29 b(Let)g Fu(\013)c Fx(2)g Fu(Lim)p FA(.)39 b(According)28 b(to)h(\(a\),\(d\))g(w)n(e)f(then)i(ha) n(v)n(e)d Fx(8)p Fu(n)p FA(\()p Fu(N)9 b(\013)p FA([)p Fu(n)p FA(])24 b Fu(<)h(N)9 b(\013)p FA([)p Fu(n)p FA(+)o(1]\),)29 b(and)g(therefore)e Fx(8)p Fu(n)p FA(\()p Fu(n)d Fx(\024)h Fu(N)9 b(\013)p FA([)p Fu(n)p FA(]\).)0 2280 y(No)n(w)27 b(the)h(claim)f(is)h(obtained)f(as)g(follo)n(ws:)36 b Fu(\013)p FA([)p Fu(N)9 b(\014)t FA(])24 b Fu(<)e(\014)28 b(<)22 b(\013)47 b Fx(\))f Fu(N)9 b(\013)p FA([)p Fu(N)g(\014)t FA(])2439 2211 y Ft(\(d\))2451 2280 y Fu(<)34 b(N)9 b(\014)27 b Fx(\024)c Fu(N)9 b(\013)p FA([)p Fu(N)g(\014)t FA(].)37 b(Con)n(tradiction.)0 2495 y Fz(De\014nition)27 b FA(of)g Fu(H)605 2507 y Fv(\013)676 2495 y FA(:)c(I)-14 b(N)23 b Fx(!)g FA(I)-14 b(N)28 b(for)f Fu(\013)d(<)f(")1366 2507 y Ft(0)0 2636 y Fu(H)69 2648 y Ft(0)106 2636 y FA(\()p Fu(n)p FA(\))h(:=)e Fu(n)p FA(,)111 b Fu(H)607 2648 y Fv(\013)654 2636 y FA(\()p Fu(n)p FA(\))24 b(:=)e Fu(H)971 2651 y Fv(\013)p Ft([)p Fv(n)p Ft(])1097 2636 y FA(\()p Fu(n)p FA(+1\))28 b(for)f Fu(\013)c(>)g FA(0.)0 2826 y Fz(Lemma)30 b(6.8)0 2967 y FA(\(a\))55 b Fu(H)230 2979 y Fv(\013)278 2967 y FA(\()p Fu(n)p FA(\))23 b Fu(<)g(H)572 2979 y Fv(\013)619 2967 y FA(\()p Fu(n)c FA(+)f(1\),)0 3108 y(\(b\))51 b Fu(\013)p FA([)p Fu(m)p FA(])24 b Fu(<)f(\014)k(<)c (\013)46 b Fx(\))g Fu(H)904 3123 y Fv(\013)p Ft([)p Fv(m)p Ft(])1048 3108 y FA(\()p Fu(n)p FA(+1\))23 b Fx(\024)f Fu(H)1448 3120 y Fv(\014)1493 3108 y FA(\()p Fu(n)p FA(\),)0 3249 y(\(c\))60 b Fu(\014)28 b(<)22 b(\013)i FA(&)f Fu(N)9 b(\014)27 b Fx(\024)22 b Fu(n)46 b Fx(\))g Fu(H)1019 3261 y Fv(\014)1064 3249 y FA(\()p Fu(n)p FA(\))24 b Fu(<)e(H)1358 3261 y Fv(\013)1406 3249 y FA(\()p Fu(n)p FA(\),)0 3390 y(\(d\))51 b Fu(\013)24 b(>)f FA(0)45 b Fx(\))h Fu(H)611 3402 y Fv(\013)659 3390 y FA(\()p Fu(n)p FA(\))23 b(=)g(min)p Fx(f)p Fu(k)j Fx(\025)c Fu(n)h FA(:)g Fu(\013)p FA([)p Fu(n)p FA(])p Fu(:::)p FA([)p Fu(k)s Fx(\000)p FA(1])g(=)f(0)p Fx(g)h FA(=)f Fu(n)c FA(+)g(min)q Fx(f)p Fu(l)24 b FA(:)37 b Fu(\013)p FA([)p Fu(n)p FA(][)p Fu(n)p FA(+1])14 b Fu(:)g(:)g(:)f FA([)p Fu(n)p FA(+)o Fu(l)r Fx(\000)o FA(1])23 b(=)f(0)p Fx(g)p FA(.)0 3556 y(Pro)r(of:)0 3697 y(\(a\),\(b\))29 b(sim)n(ultaneous)d(induction)i(on) g Fu(\013)p FA(:)37 b(Let)28 b Fu(\013)23 b(>)g FA(0.)0 3838 y(\(a\))28 b(1.)36 b Fu(\013)24 b Fx(2)f Fu(Lim)p FA(:)36 b Fu(H)677 3850 y Fv(\013)724 3838 y FA(\()p Fu(n)p FA(\))24 b(=)e Fu(H)1018 3853 y Fv(\013)p Ft([)p Fv(n)p Ft(])1144 3838 y FA(\()p Fu(n)p FA(+1\))1388 3776 y Ft(IHa)1409 3838 y Fu(<)43 b(H)1586 3853 y Fv(\013)p Ft([)p Fv(n)p Ft(])1712 3838 y FA(\()p Fu(n)p FA(+3\))1956 3768 y Ft(IHb)1978 3838 y Fx(\024)i Fu(H)2157 3853 y Fv(\013)p Ft([)p Fv(n)p Ft(+1])2367 3838 y FA(\()p Fu(n)p FA(+2\))23 b(=)g Fu(H)2768 3850 y Fv(\013)2815 3838 y FA(\()p Fu(n)p FA(+1\).)0 3978 y(2.)36 b Fu(\013)24 b FA(=)f Fu(\013)319 3990 y Ft(0)356 3978 y FA(+1:)36 b Fu(H)591 3990 y Fv(\013)638 3978 y FA(\()p Fu(n)p FA(\))24 b(=)e Fu(H)932 3990 y Fv(\013)975 3998 y Fo(0)1012 3978 y FA(\()p Fu(n)p FA(+1\))1256 3917 y Ft(IHa)1276 3978 y Fu(<)44 b(H)1454 3990 y Fv(\013)1497 3998 y Fo(0)1534 3978 y FA(\()p Fu(n)p FA(+)o(2\))23 b(=)g Fu(H)1934 3990 y Fv(\013)1981 3978 y FA(\()p Fu(n)p FA(+1\).)0 4119 y(\(b\))33 b(F)-7 b(rom)32 b Fu(\013)p FA([)p Fu(m)p FA(])f Fu(<)g(\014)k(<)c(\013)i FA(w)n(e)f(obtain)g Fu(\013)p FA([)p Fu(m)p FA(])f Fx(\024)g Fu(\014)t FA([)p Fu(n)p FA(])g Fu(<)f(\013)j FA(b)n(y)f(Lemma)g(6.7.)83 b(If)33 b Fu(\013)p FA([)p Fu(m)p FA(])e(=)g Fu(\014)t FA([)p Fu(n)p FA(])h(then)h Fu(H)3440 4134 y Fv(\013)p Ft([)p Fv(m)p Ft(])3584 4119 y FA(\()p Fu(n)p FA(+1\))d(=)0 4275 y Fu(H)69 4290 y Fv(\014)s Ft([)p Fv(n)p Ft(])192 4275 y FA(\()p Fu(n)p FA(+1\))23 b(=)g Fu(H)593 4287 y Fv(\014)637 4275 y FA(\()p Fu(n)p FA(\).)38 b(Otherwise)27 b Fu(H)1274 4290 y Fv(\013)p Ft([)p Fv(m)p Ft(])1417 4275 y FA(\()p Fu(n)p FA(+1\))1661 4206 y Ft(IHb)1683 4275 y Fx(\024)46 b Fu(H)1863 4290 y Fv(\014)s Ft([)p Fv(n)p Ft(])1986 4275 y FA(\()p Fu(n)p FA(\))2124 4214 y Ft(IHa)2144 4275 y Fu(<)e(H)2322 4290 y Fv(\014)s Ft([)p Fv(n)p Ft(])2445 4275 y FA(\()p Fu(n)p FA(+1\))23 b(=)f Fu(H)2845 4287 y Fv(\014)2890 4275 y FA(\()p Fu(n)p FA(\).)0 4441 y(\(c\))28 b(Induction)g(on)f Fu(\013)p FA(:)38 b Fu(\014)27 b(<)c(\013)972 4394 y Ft(L)p Fv(:)p Ft(6)p Fv(:)p Ft(7e)1019 4441 y Fx(\))93 b Fu(\014)27 b Fx(\024)c Fu(\013)p FA([)p Fu(N)9 b(\014)t FA(])23 b Fx(\024)g Fu(\013)p FA([)p Fu(n)p FA(])1889 4387 y Ft(\(a\)+IH)1952 4441 y Fx(\))109 b Fu(H)2213 4453 y Fv(\014)2258 4441 y FA(\()p Fu(n)p FA(\))24 b Fu(<)e(H)2552 4453 y Fv(\014)2597 4441 y FA(\()p Fu(n)p FA(+1\))h Fx(\024)f Fu(H)2997 4456 y Fv(\013)p Ft([)p Fv(n)p Ft(])3123 4441 y FA(\()p Fu(n)p FA(+1\))h(=)g Fu(H)3524 4453 y Fv(\013)3571 4441 y FA(\()p Fu(n)p FA(\).)0 4607 y(\(d\))28 b(Let)g Fu(k)e Fx(\025)d Fu(n)k FA(minimal)h(suc)n(h)f(that)h Fu(\013)p FA([)p Fu(n)p FA(])p Fu(:::)p FA([)p Fu(k)s Fx(\000)p FA(1])23 b(=)f(0.)0 4748 y(Then)28 b Fu(H)286 4760 y Fv(\013)333 4748 y FA(\()p Fu(n)p FA(\))c(=)e Fu(H)627 4763 y Fv(\013)p Ft([)p Fv(n)p Ft(])753 4748 y FA(\()p Fu(n)p FA(+1\))h(=)f Fu(::::)i FA(=)e Fu(H)1356 4763 y Fv(\013)p Ft([)p Fv(n)p Ft(])p Fv(:::)p Ft([)p Fv(k)q Fp(\000)p Ft(1])1701 4748 y FA(\()p Fu(k)s FA(\))h(=)g Fu(H)1991 4760 y Ft(0)2028 4748 y FA(\()p Fu(k)s FA(\))g(=)g Fu(k)s FA(.)0 4963 y Fy(Abbreviation.)0 5116 y Fu(N)-5 b(F)12 b FA(\()p Fu(\013;)i(\014)t FA(\))47 b(:)p Fx(,)508 4974 y Fq(\()575 5048 y Fu(\013)24 b FA(=)e(0)28 b(or)e Fu(\014)i FA(=)22 b(0)27 b(or)575 5189 y([)14 b Fu(\013)24 b FA(=)e Fu(!)831 5159 y Fv(\013)874 5167 y Fo(0)929 5189 y FA(+)c Fu(:::)h FA(+)f Fu(!)1238 5159 y Fv(\013)1281 5167 y Fn(n)1348 5189 y FA(&)23 b Fu(\014)28 b FA(=)22 b Fu(!)1653 5159 y Fv(\014)1691 5167 y Fo(0)1745 5189 y FA(+)c Fu(:::)h FA(+)f Fu(!)2054 5159 y Fv(\014)2092 5167 y Fn(m)2178 5189 y FA(with)28 b Fu(\013)2420 5201 y Ft(0)2481 5189 y Fx(\025)23 b Fu(:::)g Fx(\025)f Fu(\013)2801 5201 y Fv(n)2870 5189 y Fx(\025)g Fu(\014)3004 5201 y Ft(0)3065 5189 y Fx(\025)g Fu(:::)i Fx(\025)e Fu(\014)3379 5201 y Fv(m)3456 5189 y FA(])3517 5116 y(.)0 5340 y Fz(Prop)s(osition.)77 b Fu(N)-5 b(F)12 b FA(\()p Fu(\013;)i(\014)t FA(\))24 b(&)f Fu(\014)k(>)c FA(0)46 b Fx(\))g FA(\()p Fu(\013)19 b FA(+)f Fu(\014)t FA(\)[)p Fu(n)p FA(])24 b(=)e Fu(\013)d FA(+)f Fu(\014)t FA([)p Fu(n)p FA(])23 b(&)g Fu(N)-5 b(F)12 b FA(\()p Fu(\013;)i(\014)t FA([)p Fu(n)p FA(]\).)1908 5539 y(40)p eop %%Page: 41 42 41 41 bop 0 83 a Fz(Lemma)30 b(6.9)0 224 y FA(\(a\))e Fu(N)-5 b(F)12 b FA(\()p Fu(\013;)i(\014)t FA(\))47 b Fx(\))f Fu(H)711 236 y Fv(\013)p Ft(+)p Fv(\014)873 224 y FA(=)23 b Fu(H)1030 236 y Fv(\013)1095 224 y Fx(\016)18 b Fu(H)1224 236 y Fv(\014)1269 224 y FA(.)0 365 y(\(b\))28 b Fu(H)207 381 y Fv(!)251 364 y Fn(\013)p Fo(+1)368 365 y FA(\()p Fu(n)p FA(\))c(=)f Fu(H)670 322 y Ft(\()p Fv(n)p Ft(+1\))663 384 y Fv(!)707 368 y Fn(\013)850 365 y FA(\()p Fu(n)p FA(+1\))28 b(and)f Fu(H)1329 382 y Fv(!)1373 366 y Fn(\025)1416 365 y FA(\()p Fu(n)p FA(\))d(=)e Fu(H)1710 384 y Fv(!)1754 368 y Fn(\025)p Fo([)p Fn(n)p Fo(])1867 365 y FA(\()p Fu(n)p FA(+1\))27 b(for)h Fu(\025)23 b Fx(2)g Fu(Lim)p FA(.)0 506 y(\(c\))28 b(F)-7 b(or)27 b(eac)n(h)g(primitiv)n(e)g(recursiv)n(e)f(function)i Fu(f)37 b FA(there)27 b(exists)g(a)g Fu(k)f Fx(2)e FA(I)-14 b(N)28 b(suc)n(h)f(that)h Fx(8)o Fu(~)-41 b(x)p FA(\()14 b Fu(f)9 b FA(\()o Fu(~)-41 b(x)p FA(\))23 b Fu(<)g(H)3182 523 y Fv(!)3226 507 y Fn(k)3266 506 y FA(\(max)p Fx(f)o Fu(~)-41 b(x)p Fx(g)p FA(\))14 b(\).)0 672 y(Pro)r(of:)0 814 y(\(a\))28 b(Induction)g(on)f Fu(\014)t FA(:)37 b(1.)g Fu(H)907 826 y Fv(\013)p Ft(+0)1038 814 y FA(\()p Fu(n)p FA(\))24 b(=)e Fu(H)1332 826 y Fv(\013)1380 814 y FA(\()p Fu(n)p FA(\))h(=)g Fu(H)1674 826 y Fv(\013)1721 814 y FA(\()p Fu(H)1822 826 y Ft(0)1860 814 y FA(\()p Fu(n)p FA(\)\).)0 955 y(2.)36 b Fu(\014)28 b(>)23 b FA(0:)36 b Fu(H)434 967 y Fv(\013)p Ft(+)p Fv(\014)573 955 y FA(\()p Fu(n)p FA(\))23 b(=)g Fu(H)867 970 y Ft(\()p Fv(\013)p Ft(+)p Fv(\014)s Ft(\)[)p Fv(n)p Ft(])1136 955 y FA(\()p Fu(n)p FA(+1\))g(=)g Fu(H)1537 970 y Fv(\013)p Ft(+)p Fv(\014)s Ft([)p Fv(n)p Ft(])1754 955 y FA(\()p Fu(n)p FA(+1\))1998 908 y Ft(IH)2002 955 y FA(=)k Fu(H)2163 967 y Fv(\013)2210 955 y FA(\()p Fu(H)2311 970 y Fv(\014)s Ft([)p Fv(n)p Ft(])2435 955 y FA(\()p Fu(n)p FA(+1\)\))c(=)f Fu(H)2867 967 y Fv(\013)2915 955 y FA(\()p Fu(H)3016 967 y Fv(\014)3061 955 y FA(\()p Fu(n)p FA(\)\).)0 1096 y(\(b\))28 b Fu(H)207 1112 y Fv(!)251 1095 y Fn(\013)p Fo(+1)368 1096 y FA(\()p Fu(n)p FA(\))c(=)f Fu(H)663 1111 y Fv(!)707 1094 y Fn(\013)749 1111 y Fp(\001)p Ft(\()p Fv(n)p Ft(+1\))949 1096 y FA(\()p Fu(n)p FA(+1\))1193 1042 y Ft(\(a\))1203 1096 y FA(=)33 b Fu(H)1377 1053 y Ft(\()p Fv(n)p Ft(+1\))1370 1115 y Fv(!)1414 1098 y Fn(\013)1558 1096 y FA(\()p Fu(n)p FA(+1\).)0 1237 y(\(c\))28 b(F)-7 b(rom)27 b(\(b\))i(it)f(follo)n(ws)e(that)i(\()p Fu(k)s(;)14 b(n)p FA(\))23 b Fx(7!)g Fu(H)1416 1254 y Fv(!)1460 1237 y Fn(k)10 b FA(\()p Fu(n)p FA(\))28 b(is)f(\(a)h(v)-5 b(arian)n(t)26 b(of)6 b(\))29 b(the)f(Ac)n(k)n(ermann)e(function.)0 1453 y Fz(Theorem)k(6.10)0 1594 y FA(Let)e Fu(\022)d FA(:)e(I)-14 b(N)19 b Fx(\002)f FA(I)-14 b(N)24 b Fx(!)f FA(OT)k(b)r(e)h(primitiv)n(e)f(recursiv)n(e,)f(and)i Fu(\013)23 b(<)g(!)2015 1606 y Fv(m)p Ft(+2)2189 1594 y FA(suc)n(h)k(that)h Fx(8)p Fu(n)p FA(\()14 b Fu(\022)r FA(\()p Fu(n;)g FA(0\))23 b Fx(\026)3043 1562 y Fk(d)3071 1594 y Fu(\013)3136 1562 y Fk(e)3177 1594 y FA(\).)0 1735 y(Then)28 b(there)f(is)h(an)35 b(~)-50 b Fu(\013)23 b(<)g(!)844 1747 y Fv(m)p Ft(+2)1019 1735 y FA(suc)n(h)k(that)h(min)p Fx(f)p Fu(l)c FA(:)f Fu(\022)r FA(\()p Fu(n;)14 b(l)r FA(+)o(1\))23 b Fx(6\036)g Fu(\022)r FA(\()p Fu(n;)14 b(l)r FA(\))p Fx(g)23 b Fu(<)f(H)2544 1747 y Ft(~)-40 b Fv(\013)2584 1735 y FA(\()p Fu(n)p FA(\))29 b(\()p Fx(8)p Fu(n)p FA(\).)0 1901 y(Pro)r(of:)0 2042 y(Abbreviation:)542 2020 y(^)535 2042 y Fu(\022)s FA(\()p Fu(n;)14 b(l)r FA(\))22 b(:=)h Fu(o)p FA(\()p Fu(\022)r FA(\()p Fu(n;)14 b(l)r FA(\)\),)29 b(where)e Fu(o)p FA(\()p Fx(\001)p FA(\))h(is)g(the)g(isomorphism)e(from)h(\(OT)p Fu(;)14 b Fx(\036)p FA(\))28 b(on)n(to)e(\()p Fu(")3142 2054 y Ft(0)3180 2042 y Fu(;)14 b(<)p FA(\).)0 2183 y(W.l.o.g.)37 b Fx(8)p Fu(n)p FA(\()469 2162 y(^)464 2183 y Fu(\022)q FA(\()p Fu(n;)14 b FA(0\))23 b(=)f Fu(\013)15 b FA(\).)0 2325 y(Let)28 b Fu(w)r FA(\()p Fu(i;)14 b(n;)g(l)r FA(\))23 b(:=)f Fu(N)9 b FA(\()p Fu(!)750 2294 y Fv(i)796 2325 y Fx(\001)19 b FA(\()877 2303 y(^)870 2325 y Fu(\022)r FA(\()p Fu(n;)14 b(l)r FA(+)o(1\)+1\)\).)37 b(One)27 b(easily)g(sees)g(that)h Fu(w)i FA(is)d(primitiv)n(e)h(recursiv)n(e.)0 2491 y(Let)g Fu(g)s FA(\()p Fu(i;)14 b(n;)g(l)r FA(\))22 b(:=)g(max)p Fx(f)p Fu(w)r FA(\()p Fu(i;)14 b(n;)g(l)r FA(\))p Fu(;)g(i;)g(n;)g(l)r Fx(g)p FA(.)0 2657 y(There)27 b(exists)g(a)g Fu(k)f Fx(\025)d FA(1)k(suc)n(h)g(that)h Fu(g)s FA(\()p Fu(i;)14 b(n;)g(l)r FA(+)n(1\))23 b Fu(<)g(H)1703 2674 y Fv(!)1747 2657 y Fn(k)9 b FA(\(max)p Fx(f)p Fu(i;)14 b(n;)g(l)r Fx(g)p FA(\))26 b(and)h Fu(g)s FA(\()p Fu(i;)14 b(n;)g FA(0\))22 b Fu(<)h(H)2938 2674 y Fv(!)2982 2657 y Fn(k)9 b FA(\(max)p Fx(f)p Fu(i;)14 b(n)p Fx(g)p FA(\))26 b(\()p Fx(8)p Fu(i;)14 b(n;)g(l)r FA(\).)0 2823 y(Then)28 b(w)n(e)f(ha)n(v)n(e)0 2989 y(\(1\))55 b Fu(g)s FA(\()p Fu(k)s(;)14 b(n;)g(l)r FA(+)o(1\))23 b 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Fx(\001)19 b FA(\()1762 3789 y(^)1755 3810 y Fu(\022)r FA(\()p Fu(n;)14 b(l)r FA(+)o(1\)+1\)\))23 b(=)f Fu(w)r FA(\()p Fu(k)s(;)14 b(n;)g(l)r FA(\))24 b Fx(\024)e Fu(g)s FA(\()p Fu(k)s(;)14 b(n;)g(l)r FA(\).)0 4001 y(\(3\))55 b Fx(9)p Fu(l)25 b Fx(\024)e Fu(')p FA(\()p Fu(n;)14 b FA(0\)\()g Fu(')p FA(\()p Fu(n;)g(l)r FA(+)o(1\))23 b Fx(6)p Fu(<)g(')p FA(\()p Fu(n;)14 b(l)r FA(\))g(\).)0 4167 y(Pro)r(of:)36 b([)p Fx(8)p Fu(l)23 b Fx(\024)g Fu(j)5 b FA(\()14 b Fu(')p FA(\()p Fu(n;)g(l)r FA(+)o(1\))23 b Fu(<)g(')p FA(\()p Fu(n;)14 b(l)r FA(\))g(\))23 b Fx(\))g Fu(j)28 b(<)23 b(')p FA(\()p Fu(n;)14 b FA(0\)])28 b(and)f(therefore)g Fy(not)g Fx(8)p Fu(l)d Fx(\024)f Fu(')p FA(\()p Fu(n;)14 b FA(0\)\()g Fu(')p FA(\()p Fu(n;)g(l)r FA(+)o(1\))23 b Fu(<)g(')p FA(\()p Fu(n;)14 b(l)r FA(\))g(\).)0 4358 y(\(2\))23 b(&)g(\(3\))g(&)46 b Fu(\013)23 b FA(=)628 4337 y(^)621 4358 y Fu(\022)r FA(\()p Fu(n;)14 b FA(0\))37 b Fx(\))46 b(9)14 b Fu(l)24 b Fx(\024)f Fu(H)1287 4375 y Fv(!)1331 4359 y Fn(k)1367 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Fx(\001)18 b Fu(!)2158 4685 y Fv(!)2200 4693 y Fn(m)p Fo(+1)2353 4715 y FA(=)23 b Fu(!)2496 4685 y Fv(!)2538 4693 y Fn(m)p Fo(+1)2691 4715 y FA(=)f Fu(!)2830 4727 y Fv(m)p Ft(+2)2977 4715 y FA(.)0 4931 y Fz(Theorem)30 b(6.11)0 5072 y FA(If)e(\005)145 5084 y Fv(m)p Ft(+1)292 5072 y FA(-IA)23 b Fx(`)g(8)p Fu(x)p Fx(9)p Fu(y)s(A)p FA(\()p Fu(x;)14 b(y)s FA(\))56 b(\()p Fu(A)28 b FA(a)f(quan)n (ti\014erfree)g Fx(L)1757 5084 y Ft(0)1794 5072 y FA(-form)n(ula\))g (then)h(there)f(is)h(an)f Fu(\013)d(<)e(!)2975 5084 y Fv(m)p Ft(+2)0 5214 y FA(suc)n(h)27 b(that)h Fx(8)p Fu(n)p Fx(9)14 b Fu(l)24 b(<)e(H)729 5226 y Fv(\013)776 5214 y FA(\()p Fu(n)p FA(\))14 b(I)-14 b(N)24 b Fx(j)-14 b FA(=)23 b Fu(A)p FA(\()p Fu(n;)14 b(l)r FA(\).)1908 5539 y(41)p eop %%Page: 42 43 42 42 bop 0 83 a FA(Pro)r(of:)0 224 y(By)25 b(Theorem)f(6.6)g(there)h (are)f(primitiv)n(e)h(recursiv)n(e)e(functions)i Fu(g)s(;)14 b(\022)27 b FA(and)e(an)g Fu(\013)2476 236 y Ft(0)2536 224 y Fu(<)e(!)2676 236 y Fv(m)p Ft(+2)2848 224 y FA(suc)n(h)h(that)i Fx(8)p Fu(n)p FA(\()14 b Fu(\022)r FA(\()p Fu(n;)g FA(0\))22 b(=)3697 193 y Fk(d)3724 224 y Fu(\013)3777 236 y Ft(0)3826 193 y Fk(e)3868 224 y FA(\))0 365 y(and)27 b(I)-14 b(N)24 b Fx(j)-14 b FA(=)23 b Fx(8)p Fu(xA)p FA(\()p Fu(x;)14 b(f)9 b FA(\()p Fu(x)p FA(\)\),)29 b(where)e Fu(f)9 b FA(\()p Fu(n)p FA(\))23 b(:=)g Fu(g)s FA(\()p Fu(n;)14 b(f)1627 335 y Fp(\003)1664 365 y FA(\()p Fu(n)p FA(\)\))28 b(,)g Fu(f)1939 335 y Fp(\003)1977 365 y FA(\()p Fu(n)p FA(\))23 b(:=)g(min)p Fx(f)p Fu(l)h FA(:)g Fu(\022)r FA(\()p Fu(n;)14 b(l)r FA(+)o(1\))23 b Fx(6\036)f Fu(\022)r FA(\()p Fu(n;)14 b(l)r FA(\))p Fx(g)p FA(\).)0 506 y(By)27 b(6.10)g(there)g(exists)g Fu(\014)h(<)22 b(!)961 518 y Fv(m)p Ft(+2)1136 506 y FA(with)28 b Fx(8)p Fu(n)p FA(\()p Fu(f)1504 476 y Fp(\003)1541 506 y FA(\()p Fu(n)p FA(\))23 b Fu(<)g(H)1835 518 y Fv(\014)1880 506 y FA(\()p Fu(n)p FA(\)\).)37 b(F)-7 b(urther)28 b(there)f(exists)g Fu(k)f(<)d(!)30 b FA(with)0 648 y Fx(8)p Fu(n;)14 b(i)p FA(\()g Fu(g)s FA(\()p Fu(n;)g(i)p FA(\))21 b Fu(<)i(H)610 664 y Fv(!)654 648 y Fn(k)9 b FA(\(max)p Fx(f)p Fu(n;)14 b(i)p Fx(g)p FA(\))g(\),)26 b(hence)i Fu(f)9 b FA(\()p Fu(n)p FA(\))23 b Fu(<)g(H)1783 664 y Fv(!)1827 648 y Fn(k)9 b FA(\(max)p Fx(f)p Fu(n;)14 b(f)2233 617 y Fp(\003)2270 648 y FA(\()p Fu(n)p FA(\))p Fx(g)p FA(\))23 b Fx(\024)g Fu(H)2638 664 y Fv(!)2682 648 y Fn(k)9 b Fu(H)2791 660 y Fv(\014)2836 648 y FA(\()p Fu(n)p FA(\).)0 789 y(Since)28 b Fu(!)269 801 y Fv(m)p Ft(+2)439 789 y FA(=)d(sup)526 858 y Fv(i)p Fp(2)p Ft(I)-11 b(N)670 789 y Fu(!)722 801 y Fv(m)p Ft(+1)869 789 y FA(\()p Fu(i)p FA(\),)28 b(there)f(is)h Fu(\015)f FA(:=)c Fu(!)1542 801 y Fv(m)p Ft(+1)1689 789 y FA(\()p Fu(i)p FA(\))28 b(suc)n(h)f(that)h Fu(!)2232 759 y Fv(k)2272 789 y Fu(;)14 b(\014)28 b(<)22 b(\015)5 b FA(.)0 967 y(It)28 b(follo)n(ws)f(that)g(there)h(is)f(an)h Fu(n)1004 979 y Ft(0)1068 967 y FA(suc)n(h)g(that)f Fu(H)1504 983 y Fv(!)1548 967 y Fn(k)10 b Fu(H)1658 979 y Fv(\014)1703 967 y FA(\()p Fu(n)p FA(\))23 b Fx(\024)g Fu(H)1997 979 y Fv(\015)2039 967 y Fu(H)2108 979 y Fv(\015)2151 967 y FA(\()p Fu(n)p FA(\))g(=)g Fu(H)2445 979 y Fv(\015)t Ft(+)p Fv(\015)2577 967 y FA(\()p Fu(n)p FA(\))28 b(for)f(all)h Fu(n)22 b Fx(\025)h Fu(n)3172 979 y Ft(0)3209 967 y FA(.)0 1108 y(Hence)28 b Fu(f)9 b FA(\()p Fu(n)p FA(\))23 b Fu(<)f(H)590 1125 y Fv(!)634 1108 y Fn(k)10 b Fu(H)744 1120 y Fv(\014)789 1108 y FA(\()p Fu(n)871 1120 y Ft(0)908 1108 y FA(+)p Fu(n)p FA(\))23 b Fx(\024)f Fu(H)1234 1120 y Fv(\015)t Ft(+)p Fv(\015)1366 1108 y FA(\()p Fu(n)1448 1120 y Ft(0)1486 1108 y FA(+)o Fu(n)p FA(\))h(=)g Fu(H)1812 1120 y Fv(\015)t Ft(+)p Fv(\015)t Ft(+)p Fv(n)2033 1128 y Fo(0)2069 1108 y FA(\()p Fu(n)p FA(\))28 b(\(and)f Fu(\015)5 b FA(+)p Fu(\015)g FA(+)o Fu(n)2679 1120 y Ft(0)2739 1108 y Fu(<)22 b(!)2878 1120 y Fv(m)p Ft(+2)3025 1108 y FA(\).)0 1324 y Fz(Corollary)-8 b(.)106 b FA(\005)580 1336 y Fv(m)p Ft(+1)728 1324 y FA(-IA)23 b Fx(6`)k FA(`)c Fx(8)p Fu(n)p Fx(9)p Fu(l)r FA(\()14 b Fu(!)1263 1336 y Fv(m)p Ft(+2)1409 1324 y FA([)p Fu(n)p FA(][)p Fu(n)p FA(+)o(1])p Fu(:::)p FA([)p Fu(l)r FA(])23 b(=)f(0)14 b(\))23 b('.)0 1490 y(Pro)r(of:)36 b(Assume)27 b(\005)629 1502 y Fv(m)p Ft(+1)777 1490 y FA(-IA)c Fx(`)k FA(`)14 b Fx(8)p Fu(n)p Fx(9)p Fu(l)r FA(\()p Fu(::::)p FA(\))g('.)0 1631 y(Then)28 b(there)f(is)h(an)f Fu(\013)c(<)g(!)844 1643 y Fv(m)p Ft(+2)1019 1631 y FA(suc)n(h)k(that)h Fx(8)p Fu(n)p Fx(9)p Fu(l)23 b(<)g(H)1734 1643 y Fv(\013)1781 1631 y FA(\()p Fu(n)p FA(\)\()14 b Fu(!)1993 1643 y Fv(m)p Ft(+2)2141 1631 y FA([)p Fu(n)p FA(][)p Fu(n)p FA(+)o(1])p Fu(:::)p FA([)p Fu(l)r FA(])22 b(=)h(0)14 b(\),)27 b(i.e.)0 1772 y Fx(8)p Fu(n)p Fx(9)p Fu(l)d(<)e(H)348 1784 y Fv(\013)396 1772 y FA(\()p Fu(n)p FA(\)\()14 b Fu(H)625 1784 y Fv(!)667 1792 y Fn(m)p Fo(+2)797 1772 y FA(\()p Fu(n)p FA(\))24 b Fx(\024)e Fu(l)r FA(+)o(1)14 b(\).)37 b(This)27 b(implies)h Fx(8)p Fu(n)p FA(\()14 b Fu(H)1944 1784 y Fv(!)1986 1792 y Fn(m)p Fo(+2)2115 1772 y FA(\()p Fu(n)p FA(\))24 b Fx(\024)e Fu(H)2409 1784 y Fv(\013)2457 1772 y FA(\()p Fu(n)p FA(\))14 b(\).)0 1913 y(But)28 b(b)n(y)f(L.6.8c)g(w)n(e)g(ha)n (v)n(e)f Fx(8)p Fu(n)c Fx(\025)h Fu(N)9 b FA(\()p Fu(\013)p FA(\)\()14 b Fu(H)1354 1925 y Fv(\013)1402 1913 y FA(\()p Fu(n)p FA(\))24 b Fu(<)e(H)1696 1925 y Fv(!)1738 1933 y Fn(m)p Fo(+2)1868 1913 y FA(\()p Fu(n)p FA(\))14 b(\).)38 b(Con)n(tradiction.)0 2129 y Fs(Below)31 b(we)f(wil)t(l)h(show)61 b FA(\005)821 2141 y Fv(m)p Ft(+1)968 2129 y FA(-IA)23 b Fx(`)29 b Fs(`)24 b Fx(8)p Fu(n)p Fx(9)p Fu(l)r FA(\()14 b Fu(\013)p FA([)p Fu(n)p FA(][)p Fu(n)p FA(+)n(1])p Fu(:::)p FA([)p Fu(l)r FA(])23 b(=)f(0)14 b(\))23 b Fs(')60 b(for)31 b(e)l(ach)f Fu(\013)24 b(<)e(!)2789 2141 y Fv(m)p Ft(+2)2936 2129 y Fs(.)0 2395 y Fz(De\014nition.)106 b Fu(F)601 2407 y Fv(\013)671 2395 y FA(:=)23 b Fu(H)851 2407 y Fv(!)895 2390 y Fn(\013)996 2395 y Fs(\(F)-6 b(ast-Gr)l(owing)31 b(Hier)l(ar)l(chy\))0 2561 y Fz(Corollary)-8 b(.)106 b Fu(F)571 2573 y Ft(0)609 2561 y FA(\()p Fu(n)p FA(\))24 b(=)e 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Fx(S)1154 407 y Fv(n)1200 395 y FA(\()p Fu(x)p FA(\))h(b)r(e)g(the)f(n)n(um)n(b)r (er)g(obtained)g(b)n(y)g(writing)g Fu(x)h FA(in)f(complete)g(Can)n(tor) f(normal)g(form)0 532 y(at)g(base)g Fu(n)p FA(+1)g(and)g(then)i (replacing)d(the)i(base)f Fu(n)p FA(+)o(1)g(b)n(y)h Fu(n)p FA(+)o(2.)0 688 y(The)g Fs(Go)l(o)l(dstein)i(se)l(quenc)l(e)e FA(for)f Fu(a)c Fx(2)g FA(I)-14 b(N)56 b(\(GS\()p Fu(a;)14 b(n)p FA(\)\))1667 700 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1847 688 y FA(is)28 b(de\014ned)f(b)n(y)0 824 y(GS\()p Fu(a;)14 b FA(0\))23 b(:=)g(GS\()p Fu(a;)14 b FA(1\))23 b(:=)g Fu(a;)60 b FA(GS\()p Fu(a;)14 b(n)k FA(+)g(1\))23 b(:=)g Fx(S)1624 836 y Fv(n)1669 824 y FA(\(GS)q(\()p Fu(a;)14 b(n)p FA(\)\))19 b Fx(\000)2080 808 y(\001)2145 824 y FA(1)27 b(for)g Fu(n)c Fx(\025)f FA(1.)0 980 y(W)-7 b(e)28 b(will)g(pro)n(v)n(e:)0 1117 y(\(i\))56 b(Ev)n(ery)26 b(Go)r(o)r(dstein)i(sequence)f(terminates)g(\(i.e.)37 b Fx(8)p Fu(a)p Fx(9)p Fu(n:)p FA(GS)o(\()p Fu(a;)14 b(n)p FA(\))24 b(=)e(0\);)0 1253 y(\(ii\))28 b Fj(Z)c Fx(6`)e FA(`)14 b Fx(8)p Fu(a)p Fx(9)p Fu(n:)p FA(GS\()p Fu(a;)g(n)p FA(\))23 b(=)g(0)1003 1223 y Fp(0)1025 1253 y FA(.)0 1449 y Fz(De\014nition)k FA(v)n(on)g(\010)659 1419 y Fv(\013)659 1473 y(b)729 1449 y FA(:)c Fu(!)j Fx(!)d Fu(")998 1461 y Ft(0)1062 1449 y FA(f)r(\177)-44 b(ur)28 b(2)23 b Fx(\024)f Fu(b)h(<)g(!)30 b FA(und)e Fu(b)23 b(<)f(\013)i(<)e(")2090 1461 y Ft(0)0 1585 y FA(\010)60 1555 y Fv(\013)60 1609 y(b)107 1585 y FA(\()p Fu(x)p FA(\))i(:=)f Fu(\013)406 1555 y Ft(\010)453 1530 y Fn(\013)453 1572 y(b)496 1555 y Ft(\()p Fv(x)560 1563 y Fo(1)592 1555 y Ft(\))629 1585 y Fx(\001)7 b Fu(n)709 1597 y Ft(1)753 1585 y FA(+)g Fu(:)14 b(:)g(:)6 b FA(+)h Fu(\013)1053 1555 y Ft(\010)1100 1530 y Fn(\013)1100 1572 y(b)1143 1555 y Ft(\()p Fv(x)1207 1564 y Fn(k)1243 1555 y Ft(\))1280 1585 y Fx(\001)g Fu(n)1360 1597 y Fv(k)1400 1585 y FA(,)23 b(falls)f Fu(x)i FA(=)1772 1523 y Fq(P)1860 1544 y Fv(k)1860 1610 y(i)p Ft(=1)1985 1585 y Fu(b)2021 1555 y Fv(x)2059 1563 y Fn(i)2096 1585 y Fx(\001)7 b Fu(n)2176 1597 y Fv(i)2226 1585 y FA(mit)22 b Fu(x)2419 1597 y Ft(1)2480 1585 y Fu(>)g(:)14 b(:)g(:)23 b(>)g(x)2822 1597 y Fv(k)2885 1585 y FA(und)f Fu(n)3095 1597 y Ft(1)3133 1585 y Fu(;)14 b(:::;)g(n)3326 1597 y Fv(k)3389 1585 y Fx(2)24 b(f)p FA(1)p Fu(;)14 b(:::;)g(b)p Fx(\000)n FA(1)p Fx(g)p FA(.)0 1762 y Fx(S)50 1774 y Fv(n)95 1762 y FA(\()p Fu(x)p FA(\))25 b(:=)d(\010)401 1726 y Fv(n)p Ft(+2)401 1784 y Fv(n)p Ft(+1)530 1762 y FA(\()p Fu(x)p FA(\))57 b(\()p Fu(n)23 b Fx(\025)g FA(1,)k Fu(x)c Fx(2)h Fu(!)s FA(\).)120 b(Abk.:)37 b Fu(\022)1636 1774 y Fv(n)1681 1762 y FA(\()p Fu(x)p FA(\))24 b(:=)f(\010)1987 1731 y Fv(!)1987 1782 y(n)p Ft(+1)2116 1762 y FA(\()p Fu(x)p FA(\))112 b(\()p Fu(n)23 b Fx(\025)g FA(1,)k Fu(x)c Fx(2)h Fu(!)s FA(\).)0 1938 y Fz(Lemma)30 b(7.1)54 b FA(Sei)28 b Fu(n)23 b Fx(\025)f FA(1.)0 2074 y(\(a\))28 b Fu(x)23 b(<)g(y)j(<)c(!)49 b Fx(\))d(S)726 2086 y Fv(n)772 2074 y FA(\()p Fu(x)p FA(\))24 b Fu(<)e Fx(S)1044 2086 y Fv(n)1090 2074 y FA(\()p Fu(y)s FA(\))h(&)g Fu(\022)1348 2086 y Fv(n)1393 2074 y FA(\()p Fu(x)p FA(\))h Fu(<)f(\022)1655 2086 y Fv(n)1700 2074 y FA(\()p Fu(y)s FA(\),)0 2210 y(\(b\))28 b Fu(\022)177 2222 y Fv(n)223 2210 y FA(\()p Fu(x)p FA(\))c(=)e Fu(\022)484 2222 y Fv(n)p Ft(+1)613 2210 y Fx(S)663 2222 y Fv(n)709 2210 y FA(\()p Fu(x)p FA(\).)0 2407 y(Sei)28 b(n)n(un)f Fu(a)341 2419 y Fv(n)410 2407 y FA(:=)22 b(GS)q(\()p Fu(a;)14 b(n)p FA(\).)37 b(Dann)28 b(gilt:)0 2543 y Fu(a)44 2555 y Fv(n)112 2543 y Fu(>)23 b FA(0)45 b Fx(\))i(S)467 2555 y Fv(n)512 2543 y FA(\()p Fu(a)588 2555 y Fv(n)633 2543 y FA(\))24 b Fu(>)e FA(0)46 b Fx(\))g Fu(\022)1032 2555 y Fv(n)p Ft(+1)1161 2543 y FA(\()p Fu(a)1237 2555 y Fv(n)p Ft(+1)1367 2543 y FA(\))23 b(=)g Fu(\022)1549 2555 y Fv(n)p Ft(+1)1678 2543 y FA(\()p Fx(S)1760 2555 y Fv(n)1806 2543 y FA(\()p Fu(a)1882 2555 y Fv(n)1927 2543 y FA(\))c Fx(\000)f FA(1\))2158 2481 y Ft(7)p Fv(:)p Ft(1a)2185 2543 y Fu(<)50 b(\022)2339 2555 y Fv(n)p Ft(+1)2468 2543 y FA(\()p Fx(S)2550 2555 y Fv(n)2596 2543 y FA(\()p Fu(a)2672 2555 y Fv(n)2717 2543 y FA(\)\))2805 2496 y Ft(7)p Fv(:)p Ft(1b)2834 2543 y FA(=)h Fu(\022)2989 2555 y Fv(n)3035 2543 y FA(\()p Fu(a)3111 2555 y Fv(n)3156 2543 y FA(\).)0 2719 y Fz(De\014nition)j FA(P)526 2731 y Fv(n)571 2719 y FA(\(0\))23 b(:=)g(0)p Fu(;)36 b FA(P)969 2731 y Fv(n)1014 2719 y FA(\()p Fu(\013)19 b FA(+)f(1\))23 b(:=)g Fu(\013;)60 b FA(P)1602 2731 y Fv(n)1647 2719 y FA(\()p Fu(\025)p FA(\))24 b(:=)f(P)1951 2731 y Fv(n)1995 2719 y FA(\()p Fu(\025)p FA([)p Fu(n)p FA(]\).)0 2895 y Fz(Lemma)30 b(7.2)c FA(Sei)i Fu(n)23 b(>)f FA(0.)0 3031 y(\(a\))28 b Fu(\013)23 b(>)g FA(0)f Fx(\))i FA(P)526 3043 y Fv(n)570 3031 y FA(\()p Fu(\013)p FA(\))g Fu(<)f(\013)p FA(,)0 3168 y(\(b\))28 b(P)195 3180 y Fv(n)240 3168 y FA(\()p Fu(\013)19 b FA(+)f Fu(\014)t FA(\))24 b(=)f Fu(\013)18 b FA(+)g(P)833 3180 y Fv(n)878 3168 y FA(\()p Fu(\014)t FA(\),)29 b(falls)e Fu(N)-5 b(F)12 b FA(\()p Fu(\013;)i(\014)t FA(\),)0 3304 y(\(c\))28 b(P)186 3316 y Fv(n)231 3304 y FA(\()p Fu(!)318 3274 y Fv(\013)365 3304 y FA(\))23 b(=)g(P)565 3316 y Fv(n)610 3304 y FA(\()p Fu(!)697 3274 y Ft(P)742 3282 y Fn(n)782 3274 y Ft(\()p Fv(\013)p Ft(\))900 3304 y Fx(\001)18 b FA(\()p Fu(n)p FA(+1\)\),)28 b(falls)f Fu(\013)d(>)e FA(0,)0 3440 y(\(d\))28 b(P)195 3452 y Fv(n)240 3440 y FA(\()p Fu(\022)311 3452 y Fv(n)357 3440 y FA(\()p Fu(x)p FA(\)\))c(=)f Fu(\022)651 3452 y Fv(n)696 3440 y FA(\()p Fu(x)c Fx(\000)815 3424 y(\001)880 3440 y FA(1\),)0 3576 y(\(e\))28 b Fu(n)23 b Fx(\025)g FA(1)45 b Fx(\))h Fu(\022)545 3588 y Fv(n)591 3576 y FA(GS\()p Fu(a;)14 b(n)p FA(\))23 b(=)g(P)1065 3588 y Fv(n)1109 3576 y Fu(:::)p FA(P)1235 3588 y Ft(2)1272 3576 y Fu(\022)1311 3588 y Ft(1)1349 3576 y FA(\()p Fu(a)p FA(\).)0 3753 y Fz(Lemma)30 b(7.3)0 3889 y FA(\(a\))e Fu(\013)23 b(>)g FA(0)46 b Fx(\))g Fu(h)563 3901 y Fv(\013)610 3889 y FA(\()p Fu(n)p FA(\))23 b(=)g Fu(h)883 3904 y Ft(P)928 3912 y Fn(n)968 3904 y Ft(\()p Fv(\013)p Ft(\))1067 3889 y FA(\()p Fu(n)c FA(+)f(1\).)0 4025 y(\(b\))28 b Fu(h)186 4037 y Fv(\013)234 4025 y FA(\()p Fu(n)p FA(\))23 b(=)g(min)p Fx(f)p Fu(k)i Fx(\025)e Fu(n)g FA(:)37 b(P)985 4037 y Fv(k)q Fp(\000)p Ft(1)1110 4025 y Fu(:::)p FA(P)1236 4037 y Fv(n)1281 4025 y FA(\()p Fu(\013)p FA(\))24 b(=)f(0)p Fx(g)p FA(.)0 4181 y Fz(Satz)33 b(7.4)0 4318 y Fx(Z)d(6`)23 b(8)p Fu(x)p Fx(9)p Fu(y)s FA([)14 b(GS)o(\()p Fu(x;)g(y)s FA(\))24 b(=)e(0)14 b(].)0 4474 y(Pro)r(of)0 4610 y(Sei)28 b Fu(e)p FA(\(0\))23 b(:=)f(1)p Fu(;)37 b(e)p FA(\()p Fu(m)18 b FA(+)g(1\))23 b(:=)g(2)1009 4580 y Fv(e)p Ft(\()p Fv(m)p Ft(\))1155 4610 y FA(.)0 4746 y(Annahme:)37 b Fx(Z)30 b(`)23 b(8)p Fu(x)p Fx(9)p Fu(y)s FA([)14 b(GS)o(\()p Fu(x;)g(y)s FA(\))24 b(=)f(0)14 b(].)0 4882 y Fx(\))46 b(Z)30 b(`)23 b(8)p Fu(x)p Fx(9)p Fu(y)s FA([)14 b(GS)o(\()p Fu(e)p FA(\()p Fu(x)p FA(\))20 b(+)e Fu(x;)c(y)s FA(\))23 b(=)g(0)14 b(])1304 4835 y Ft(6)p Fv(:)p Ft(12)1322 4882 y Fx(\))64 b(9)p Fu(\013)24 b(<)f(")1719 4894 y Ft(0)1756 4882 y Fx(8)p Fu(m)p Fx(9)p Fu(n)f(<)g(H)2150 4894 y Fv(\013)2198 4882 y FA(\()p Fu(m)p FA(\)[)14 b(GS\()p Fu(e)p FA(\()p Fu(m)p FA(\))19 b(+)f Fu(m;)c(n)p FA(\))23 b(=)g(0)14 b(])45 b Fx(\))0 5019 y(\))h(9)p Fu(p;)14 b(n)p FA([)g Fu(n)23 b(<)f(H)570 5031 y Fv(!)612 5039 y Fn(p)651 5019 y FA(\()p Fu(p)p FA(\))h(&)g(GS\()p Fu(e)p FA(\()p Fu(p)p FA(\))c(+)f Fu(p;)c(n)p FA(\))23 b(=)g(0)14 b(].)36 b(\()p Fu(p)23 b Fx(\025)g Fu(N)9 b FA(\()p Fu(\013)p FA(\))28 b(mit)g Fu(\013)c(<)e(!)2442 5031 y Fv(p)2481 5019 y FA(;)27 b(v)n(ergl.)f(6.9c\))0 5175 y(GS\()p Fu(e)p FA(\()p Fu(p)p FA(\))s(+)s Fu(p;)14 b(n)p FA(\))22 b(=)h(0)46 b Fx(\))g Fu(n)23 b Fx(\025)f FA(2)h(&)g(P)1217 5187 y Fv(n)1261 5175 y Fu(:::)p FA(P)1387 5187 y Ft(2)1424 5175 y FA(\()p Fu(!)1508 5187 y Fv(p)1549 5175 y FA(+)s Fu(p)p FA(\))g(=)f Fu(\022)1840 5187 y Fv(n)1886 5175 y FA(GS\()p Fu(e)p FA(\()p Fu(p)p FA(\))s(+)s Fu(p;)14 b(n)p FA(\))22 b(=)h(0)45 b Fx(\))h Fu(H)2801 5187 y Fv(!)2843 5195 y Fn(p)2878 5187 y Ft(+)p Fv(p)2968 5175 y FA(\(1\))23 b Fu(<)f(h)3232 5187 y Fv(!)3274 5195 y Fn(p)3309 5187 y Ft(+)p Fv(p)3398 5175 y FA(\(2\))h Fx(\024)g Fu(n)p FA(+)o(1)46 b Fx(\))0 5311 y Fu(H)69 5323 y Fv(!)111 5331 y Fn(p)150 5311 y FA(\()p Fu(p)p FA(\))23 b(=)g Fu(H)436 5323 y Fv(!)478 5331 y Fn(p)513 5323 y Ft(+)p Fv(p)602 5311 y FA(\(0\))g Fu(<)g(H)888 5323 y Fv(!)930 5331 y Fn(p)964 5323 y Ft(+)p Fv(p)1054 5311 y FA(\(1\))g Fx(\024)g Fu(n)p FA(.)36 b Fy(Widerspruc)n(h)p FA(.)1908 5539 y(44)p eop %%Page: 45 46 45 45 bop 0 83 a Fw(The)27 b(P)n(aris-Harrington)f(Result)0 241 y Fz(Abbreviations:)0 378 y FA(Let)i Fu(k)s(;)14 b(m;)g(n;)g(r)25 b Fx(2)e FA(I)-14 b(N)14 b(\(=)23 b Fu(!)s FA(\),)28 b Fu(\024)g FA(a)f(cardinal,)f Fu(N)37 b FA(a)27 b(set.)120 b([)p Fu(N)9 b FA(])1945 348 y Fv(m)2031 378 y FA(:=)23 b Fx(f)p Fu(X)28 b Fx(\022)23 b Fu(N)32 b FA(:)23 b Fu(car)r(d)p FA(\()p Fu(X)7 b FA(\))24 b(=)e Fu(m)p Fx(g)0 536 y FA(Let)28 b Fu(f)36 b FA(b)r(e)28 b(a)f(function)h(with)g(dom\()p Fu(f)9 b FA(\))24 b(=)e([)p Fu(N)9 b FA(])1426 506 y Fv(m)1489 536 y FA(:)0 673 y Fu(X)34 b FA(is)27 b Fu(f)9 b FA(-homogeneous)49 b Fx(\()-14 b(\))69 b(;)23 b(6)p FA(=)f Fu(X)30 b Fx(\022)22 b Fu(N)32 b FA(&)23 b Fu(f)9 b Fh(\026)o FA([)p Fu(X)e FA(])1747 643 y Fv(m)1837 673 y FA(constan)n(t.)0 831 y Fu(N)32 b Fx(\000)-14 b(!)23 b FA(\()p Fu(\024)p FA(\))368 801 y Fv(m)368 852 y(r)477 831 y FA(:)p Fx(,)46 b(8)p Fu(f)31 b FA(:)23 b([)p Fu(N)9 b FA(])916 801 y Fv(m)1002 831 y Fx(!)23 b Fu(r)r Fx(9)p Fu(X)7 b FA(\()p Fu(X)35 b(f)9 b FA(-homogeneous)20 b(&)j Fu(car)r(d)p FA(\()p Fu(X)7 b FA(\))24 b Fx(\025)e Fu(\024)p FA(\))0 989 y Fu(N)149 942 y Fp(\003)99 989 y Fx(\000)-14 b(!)23 b FA(\()p Fu(\024)p FA(\))368 959 y Fv(m)368 1010 y(r)477 989 y FA(:)p Fx(,)46 b(8)p Fu(f)31 b FA(:)23 b([)p Fu(N)9 b FA(])916 959 y Fv(m)1002 989 y Fx(!)23 b Fu(r)r Fx(9)p Fu(X)7 b FA(\()p Fu(X)35 b(f)9 b FA(-homogeneous)20 b(&)j Fu(car)r(d)p FA(\()p Fu(X)7 b FA(\))24 b Fx(\025)e FA(max)p Fx(f)p Fu(\024;)14 b FA(min)o(\()p Fu(X)7 b FA(\))p Fx(g)p FA(\))166 b(\(for)27 b Fu(N)32 b Fx(\022)23 b FA(I)-14 b(N\))0 1147 y Fz(Ramsey)30 b(Theorem)54 b Fx(8)p Fu(m;)14 b(r)25 b Fx(2)e Fu(!)s FA(\()14 b Fu(!)26 b Fx(\000)-14 b(!)23 b FA(\()p Fu(!)s FA(\))1542 1117 y Fv(m)1542 1167 y(r)1619 1147 y FA(\))0 1305 y Fz(Finite)31 b(Ramsey)f(Theorem)54 b Fx(8)p Fu(m;)14 b(r)n(;)g(\024)22 b Fx(2)h Fu(!)s Fx(9)p Fu(N)32 b Fx(2)23 b Fu(!)s FA(\()14 b Fu(N)32 b Fx(\000)-14 b(!)23 b FA(\()p Fu(\024)p FA(\))2193 1275 y Fv(m)2193 1325 y(r)2270 1305 y FA(\))0 1463 y Fz(PH)55 b Fx(8)p Fu(m;)14 b(r)n(;)g(\024)22 b Fx(2)i Fu(!)s Fx(9)p Fu(N)31 b Fx(2)24 b Fu(!)s FA(\()14 b Fu(N)1100 1416 y Fp(\003)1050 1463 y Fx(\000)-14 b(!)23 b FA(\()p Fu(\024)p FA(\))1319 1433 y Fv(m)1319 1483 y(r)1396 1463 y FA(\))0 1621 y(\(Pro)r(of)k(of)g (the)h(Finite)h(Ramsey)d(Theorem)h(in)h(P)-7 b(A:)28 b(cf.)37 b(Ha)5 b(jek,)27 b(Pudlak)g(Ch.2,)g(Sec.1\))0 1779 y(Pro)r(of)f(of)i(PH:)f(Let)h Fu(m;)14 b(r)n(;)g(\024)23 b Fx(2)g Fu(!)31 b FA(b)r(e)d(\014xed.)37 b(T)-7 b(o)27 b(pro)n(v)n(e:)35 b Fx(9)p Fu(n)23 b Fx(2)g Fu(!)s Fx(8)p Fu(f)f Fx(:)p FA(\010\()p Fu(f)t(;)14 b(n)p FA(\))0 1916 y(where)55 b(\010\()p Fu(f)t(;)14 b(n)p FA(\))46 b(:)p Fx(,)g Fu(f)32 b FA(:)23 b([)p Fu(n)p FA(])937 1886 y Fv(m)1023 1916 y Fx(!)g Fu(r)j FA(&)d Fx(8)p Fu(X)7 b FA(\()p Fu(X)32 b(f)9 b FA(-hom)50 b Fx(\))c Fu(car)r(d)p FA(\()p Fu(X)7 b FA(\))24 b Fu(<)f FA(max)o Fx(f)p Fu(\024;)14 b FA(min\()p Fu(X)7 b FA(\))p Fx(g)p FA(\))27 b(.)0 2074 y(Assumption:)37 b Fx(8)p Fu(n)p Fx(9)p Fu(f)22 b FA(\010\()p Fu(f)t(;)14 b(n)p FA(\).)65 b(By)27 b(K\177)-42 b(onig's)26 b(Lemma)h(there)h(is)f(a)g(function)i Fu(f)2509 2044 y Fp(\003)2546 2074 y FA([)p Fu(!)s FA(])2647 2044 y Fv(m)2733 2074 y Fx(!)23 b Fu(r)31 b FA(suc)n(h)c(that)0 2211 y(\(+\))h Fx(8)p Fu(n)14 b FA(\010\()p Fu(f)410 2181 y Fp(\003)447 2211 y Fh(\026)o FA([)p Fu(n)p FA(])577 2181 y Fv(m)640 2211 y Fu(;)g(n)p FA(\))28 b(.)0 2369 y(By)34 b(Ramsey's)f(Theorem)g(there)h(exists)f(an)h(in\014nite)h Fu(f)1783 2339 y Fp(\003)1820 2369 y FA(-homogeneous)d(set)i Fu(X)40 b Fx(\022)33 b Fu(!)s FA(.)56 b(W)-7 b(e)34 b(c)n(ho)r(ose)f Fu(N)42 b(<)34 b(!)i FA(suc)n(h)e(that)0 2506 y Fu(car)r(d)p FA(\()p Fu(X)26 b Fx(\\)19 b Fu(N)9 b FA(\))24 b Fu(>)g FA(max)o Fx(f)p Fu(\024;)14 b FA(min\()p Fu(X)7 b FA(\))p Fx(g)p FA(.)38 b(F)-7 b(or)27 b Fu(f)33 b FA(:=)24 b Fu(f)1631 2476 y Fp(\003)1668 2506 y Fh(\026)p FA([)p Fu(N)9 b FA(])1825 2476 y Fv(m)1916 2506 y FA(w)n(e)28 b(then)g(ha)n(v)n(e)f Fu(f)9 b Fh(\026)p FA([)p Fu(X)25 b Fx(\\)19 b Fu(N)9 b FA(])2795 2476 y Fv(m)2882 2506 y FA(=)23 b Fu(f)3020 2476 y Fp(\003)3058 2506 y Fh(\026)p FA([)p Fu(X)i Fx(\\)19 b Fu(N)9 b FA(])3383 2476 y Fv(m)3470 2506 y FA(=)23 b Fu(constant)p FA(.)0 2643 y(Hence)h Fu(X)16 b Fx(\\)10 b Fu(N)33 b FA(is)23 b Fu(f)9 b FA(-homogeneous)21 b(and)j Fu(car)r(d)p FA(\()p Fu(X)17 b Fx(\\)10 b Fu(N)f FA(\))24 b Fu(>)e FA(max)p Fx(f)p Fu(\024;)14 b FA(min)o(\()p Fu(X)7 b FA(\))p Fx(g)23 b FA(=)g(max)o Fx(f)p Fu(\024;)14 b FA(min\()p Fu(X)i Fx(\\)10 b Fu(N)f FA(\))p Fx(g)p FA(,)25 b(i.e.,)f Fx(:)p FA(\010\()p Fu(f)t(;)14 b(N)9 b FA(\).)0 2780 y Fs(Contr)l(adiction)29 b FA(to)f(\(+\).)0 2938 y([[)21 b(Construction)f(of)h Fu(f)697 2908 y Fp(\003)735 2938 y FA(:)34 b(Let)21 b(\010)994 2950 y Fv(n)1062 2938 y FA(:=)i Fx(f)p Fu(f)31 b FA(:)23 b(\010\()p Fu(f)t(;)14 b(n)p FA(\))p Fx(g)21 b FA(and)g Fu(M)9 b FA(\()p Fu(f)g FA(\))23 b(:=)f Fx(f)p Fu(i)g FA(:)i Fx(9)p Fu(g)h Fx(2)f FA(\010)2534 2950 y Fv(i)2561 2938 y FA(\()p Fu(f)32 b Fx(\022)23 b Fu(g)s FA(\))p Fx(g)p FA(.)34 b(|)21 b(Starting)g(with)g Fu(f)3572 2950 y Ft(0)3632 2938 y FA(:=)i Fx(;)d FA(w)n(e)0 3075 y(de\014ne)28 b(a)f(sequence)g(\()p Fu(f)726 3087 y Fv(n)771 3075 y FA(\))803 3087 y Fv(n)p Fp(2)p Fv(!)965 3075 y FA(suc)n(h)g(that)h Fu(f)1373 3087 y Fv(n)1441 3075 y Fx(2)23 b FA(\010)1579 3087 y Fv(n)1648 3075 y FA(&)f Fu(f)1776 3087 y Fv(n)1844 3075 y Fx(\022)h Fu(f)1973 3087 y Fv(n)p Ft(+1)2125 3075 y FA(&)f Fu(car)r(d)p FA(\()p Fu(M)9 b FA(\()p Fu(f)2569 3087 y Fv(n)2615 3075 y FA(\)\))24 b Fx(\025)f Fu(!)s FA(,)k(and)g(w)n(e)h(set)f Fu(f)3359 3045 y Fp(\003)3420 3075 y FA(:=)3531 3013 y Fq(S)3600 3100 y Fv(n)p Fp(2)p Fv(!)3748 3075 y Fu(f)3789 3087 y Fv(n)3833 3075 y FA(.)0 3212 y(De\014nition)36 b(of)e Fu(f)536 3224 y Fv(n)p Ft(+1)665 3212 y FA(:)52 b(Let)35 b Fu(E)40 b FA(:=)35 b Fx(f)p Fu(f)43 b Fx(2)35 b FA(\010)1396 3224 y Fv(n)p Ft(+1)1561 3212 y FA(:)g Fu(f)1660 3224 y Fv(n)1740 3212 y Fx(\022)f Fu(f)9 b Fx(g)p FA(.)58 b(Then)35 b Fu(M)9 b FA(\()p Fu(f)2399 3224 y Fv(n)2444 3212 y FA(\))35 b(=)g Fx(f)p Fu(n)p Fx(g)22 b([)2846 3150 y Fq(S)2915 3237 y Fv(f)7 b Fp(2)p Fv(E)3069 3212 y Fu(M)i FA(\()p Fu(f)g FA(\),)36 b(and)f Fu(E)40 b FA(is)34 b(\014nite.)0 3349 y(This)j(together)e(with)j Fx(8)p Fu(i)f(>)g(n)p Fx(8)p Fu(f)46 b Fx(2)39 b FA(\010)1287 3361 y Fv(i)1314 3349 y FA(\()14 b Fu(f)9 b Fh(\026)p FA([)p Fu(n)p FA(+)o(1])1647 3319 y Fv(m)1748 3349 y Fx(2)38 b FA(\010)1901 3361 y Fv(n)p Ft(+1)2044 3349 y FA(\))g(implies)e(the)i(existence)e(of)h(an)f Fu(f)3188 3361 y Fv(n)p Ft(+1)3355 3349 y Fx(2)j Fu(E)i FA(suc)n(h)c(that)0 3486 y Fu(car)r(d)p FA(\()p Fu(M)9 b FA(\()p Fu(f)357 3498 y Fv(n)p Ft(+1)487 3486 y FA(\)\))23 b Fx(\025)g Fu(!)s FA(.]])0 3686 y Fz(Theorem)30 b(7.5)55 b Fx(8)p Fu(m)22 b Fx(\025)g FA(1)p Fx(8)p Fu(k)s FA(\()14 b Fu(H)1062 3701 y Fv(!)1104 3709 y Fn(m)1158 3701 y Ft(\()p Fv(k)q Ft(\))1251 3686 y FA(\()p Fu(k)s FA(+)o(1\))23 b Fu(<)g(R)1641 3698 y Fv(m)1704 3686 y FA(\()p Fu(k)s FA(\))14 b(\))28 b(with)g Fu(R)2140 3698 y Fv(m)2203 3686 y FA(\()p Fu(k)s FA(\))c(:=)f(min)p Fx(f)p Fu(N)31 b FA(:)23 b Fu(N)2921 3639 y Fp(\003)2871 3686 y Fx(\000)-14 b(!)23 b FA(\(2)p Fu(m)p FA(+)o Fu(k)s FA(+4\))3424 3651 y Fv(m)p Ft(+1)3424 3715 y Fv(k)q Ft(+\006)3558 3723 y Fn(i)22 b FA(0)1725 1180 y(;)56 b Fu(\014)1855 1150 y Ft(+)1933 1180 y FA(:=)23 b(min)p Fx(f)p FA(\012)2284 1192 y Fv(\033)r Ft(+1)2436 1180 y FA(:)g Fu(\014)k(<)c FA(\012)2704 1192 y Fv(\033)r Ft(+1)2833 1180 y Fx(g)p FA(.)0 1344 y Fx(<)g FA(:=)k(the)h(class)f(of)h(all)f(uncoun)n(table)g (regular)f(cardinals.)0 1554 y Fz(De\014nition.)0 1694 y FA(Giv)n(en)h(a)h(coun)n(table)f(set)g Fx(F)36 b FA(of)27 b(ordinal)g(functions)h(and)f(an)h(ordinal)e Fu(\014)32 b FA(let)0 1834 y(Cl\()p Fx(F)8 b FA(;)14 b Fu(\014)t FA(\))24 b(:=)j(the)h(closure)f(of)g Fu(\014)32 b FA(under)c(+)f(and)g (all)h(functions)f(in)h Fx(F)8 b FA(.)0 2021 y Fz(Lemma)30 b(8.1.)0 2161 y FA(If)e Fx(F)36 b FA(is)27 b(a)g(coun)n(table)g(set)h (of)f(ordinal)g(functions,)h(then)g(for)f(eac)n(h)g Fu(\024)c Fx(2)g(<)0 2301 y FA(the)28 b(set)g Fx(f)p Fu(\014)f(<)22 b(\024)h FA(:)g(Cl\()p Fx(F)8 b FA(;)14 b Fu(\014)t FA(\))19 b Fx(\\)g Fu(\024)k Fx(\022)g Fu(\014)t Fx(g)k FA(is)h(closed)f(un)n(b) r(ounded)h(in)f Fu(\024)p FA(.)0 2465 y(Pro)r(of:)0 2604 y Fy(un)n(b)r(ounded:)37 b FA(Let)28 b Fu(\014)657 2616 y Ft(0)717 2604 y Fu(<)23 b(\024)k FA(b)r(e)h(giv)n(en.)36 b(W)-7 b(e)28 b(set)g Fu(\014)1562 2616 y Fv(n)p Ft(+1)1714 2604 y FA(:=)23 b(min)p Fx(f)p Fu(\021)j FA(:)d(Cl\()p Fx(F)8 b FA(;)14 b Fu(\014)2385 2616 y Fv(n)2431 2604 y FA(\))k Fx(\\)h Fu(\024)k Fx(\022)g Fu(\021)s Fx(g)p FA(,)k Fu(\014)g FA(:=)c(sup)3160 2625 y Fv(n)f FA(0)0 1944 y(\(otherwise)26 b Fu( )455 1956 y Ft(0)493 1944 y FA(\()p Fu(\013)p FA(\))e(=)e(1)h Fx(\022)f Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\)\).)38 b(As)27 b(w)n(e)f(will)h(sho)n(w)f(b)r(elo)n(w,)h(the)g(IH)g(implies)g (that)h Fu(\014)f FA(:=)c Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\))j Fx(\\)h FA(\012)3407 1956 y Ft(1)3471 1944 y FA(is)26 b(in)h(fact)g(an)0 2083 y(ordinal.)36 b(Then)28 b Fu(C)6 b FA(\()p Fu(\013;)14 b(\014)t FA(\))20 b Fx(\\)e FA(\012)954 2095 y Ft(1)1015 2083 y Fx(\022)k Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\))19 b Fx(\\)g FA(\012)1516 2095 y Ft(1)1576 2083 y FA(=)k Fu(\014)32 b FA(and)c(th)n(us)f Fu( )2141 2095 y Ft(0)2179 2083 y FA(\()p Fu(\013)p FA(\))d Fx(\024)e Fu(\014)t FA(,)28 b(since)g Fu(\014)f Fx(\025)c FA(1)f(=)h(\012)3087 2048 y Fp(\000)3087 2106 y Ft(1)3143 2083 y FA(.)0 2246 y Fs(Claim:)39 b Fu(\015)27 b Fx(2)d Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\))19 b Fx(\\)g FA(\012)842 2258 y Ft(1)925 2246 y Fx(\))46 b Fu(\015)28 b Fx(\022)23 b Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\).)0 2386 y(Pro)r(of)26 b(b)n(y)i(side)f(induction)h(on)f(the)h(de\014nition)g(of)g Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\):)0 2525 y(1.)36 b Fu(\015)28 b FA(=)23 b Fu( )314 2537 y Ft(0)351 2525 y FA(\()p Fu(\030)t FA(\))28 b(with)h Fu(\030)e(<)22 b(\013)p FA(:)0 2665 y(By)27 b(the)h(ab)r(o)n(v)n(e)f(IH)g(w)n(e)h(ha)n (v)n(e)e Fu(C)1001 2677 y Ft(0)1039 2665 y FA(\()p Fu(\030)t FA(\))d(=)g Fu(C)6 b FA(\()p Fu(\030)t(;)14 b FA(1\).)37 b(Hence)28 b Fu(\015)g FA(=)22 b Fu( )2021 2677 y Ft(0)2059 2665 y FA(\()p Fu(\030)t FA(\))i Fx(\022)e Fu(C)6 b FA(\()p Fu(\030)t(;)14 b FA(1\))24 b Fx(\022)e Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\).)0 2804 y(2.)36 b Fu(\015)28 b FA(=)23 b Fu(\015)303 2816 y Ft(0)358 2804 y FA(+)18 b Fu(\015)484 2816 y Ft(1)549 2804 y FA(with)28 b Fu(\015)781 2816 y Ft(0)819 2804 y Fu(;)14 b(\015)899 2816 y Ft(1)959 2804 y Fx(2)23 b Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\))19 b Fx(\\)g FA(\012)1451 2816 y Ft(1)1488 2804 y FA(:)0 2944 y(Then)28 b(b)n(y)f(SIH)h Fu(\015)541 2956 y Ft(0)578 2944 y Fu(;)14 b(\015)658 2956 y Ft(1)719 2944 y Fx(\022)22 b Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\))29 b(whic)n(h)e(\(together)g (with)h Fu(\015)1925 2956 y Ft(0)1985 2944 y Fx(2)c Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\)\))28 b(yields)g Fu(\015)2662 2956 y Ft(0)2718 2944 y FA(+)18 b Fu(\015)2844 2956 y Ft(1)2904 2944 y Fx(\022)23 b Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\).)0 3107 y Fz(Corollary)-8 b(.)106 b Fu( )572 3119 y Ft(0)610 3107 y FA(\()p Fu(\013)p FA(\))24 b(=)e Fu(C)6 b FA(\()p Fu(\013;)14 b FA(1\))20 b Fx(\\)e FA(\012)1252 3119 y Ft(1)0 3293 y Fz(Lemma)30 b(8.4.)0 3432 y FA(If)c Fu(\015)h FA(=)c Fu(\015)282 3444 y Ft(0)333 3432 y FA(+)14 b Fu(:::)g FA(+)f Fu(\015)616 3444 y Fv(n)687 3432 y FA(with)26 b(additiv)n(e)f(principal)g(n)n(um)n(b)r(ers)g Fu(\015)1910 3444 y Ft(0)1970 3432 y Fx(\025)e Fu(:::)g Fx(\025)f Fu(\015)2280 3444 y Fv(n)2351 3432 y FA(then:)62 b Fu(\015)27 b Fx(2)d Fu(C)6 b FA(\()p Fu(\013;)14 b(\014)t FA(\))47 b Fx(,)g Fu(\015)3236 3444 y Ft(0)3273 3432 y Fu(;)14 b(:::;)g(\015)3459 3444 y Fv(n)3527 3432 y Fx(2)23 b Fu(C)6 b FA(\()p Fu(\013;)14 b(\014)t FA(\).)0 3595 y(Pro)r(of)26 b(of)i(\\)p Fx(\))p FA(")f(b)n(y)g(induction)h(on)f (the)h(de\014nition)g(of)g Fu(C)6 b FA(\()p Fu(\013;)14 b(\014)t FA(\):)0 3735 y(F)-7 b(or)27 b Fu(\015)h(<)22 b(\014)32 b FA(or)27 b Fu(n)c FA(=)g(0)k(the)h(claim)f(is)h(trivial.)36 b(Otherwise)27 b Fu(\015)g FA(=)c Fu(\030)f FA(+)c Fu(\021)31 b FA(with)d Fu(\030)t(;)14 b(\021)27 b Fx(2)c Fu(C)6 b FA(\()p Fu(\013;)14 b(\014)t FA(\).)0 3874 y(Then)28 b Fu(\030)f FA(=)c Fu(\015)411 3886 y Ft(0)466 3874 y FA(+)18 b Fu(:::)h FA(+)f Fu(\015)763 3886 y Fv(k)q Fp(\000)p Ft(1)907 3874 y FA(+)g Fu(\030)1026 3886 y Ft(1)1082 3874 y FA(+)g Fu(:::)h FA(+)f Fu(\030)1372 3886 y Fv(l)1425 3874 y FA(and)28 b Fu(\021)e FA(=)d Fu(\015)1785 3886 y Fv(k)1844 3874 y FA(+)18 b Fu(:::)h FA(+)f Fu(\015)2141 3886 y Fv(n)2186 3874 y FA(,)28 b(and)f(the)h(claim)f(follo)n(ws)g(b)n (y)g(I.H.)0 4154 y(In)h(order)e(to)h(a)n(v)n(oid)g(some)g(tec)n(hnical) g(complications)f(w)n(e)i(no)n(w)f(assume)g Fu(\027)2343 4166 y Ft(0)2403 4154 y FA(=)c Fu(!)s FA(.)0 4317 y Fs(F)-6 b(r)l(om)30 b(now)g(on)f(the)h(letters)g Fu(\033)n(;)14 b(\032;)g(\026;)g(\027)35 b Fs(r)l(ange)30 b(over)h(numb)l(ers)e Fu(<)22 b(!)s Fs(.)0 4503 y FA(Belo)n(w)i(w)n(e)h(will)g(in)n(tro)r (duce)f(a)h(system)g(of)g(ordinal)f(notations)g(based)g(on)h(the)g (ordinal)f(functions)i Fu( )3126 4515 y Fv(\033)3170 4503 y FA(.)37 b(The)25 b(canonical)e(w)n(a)n(y)0 4642 y(for)28 b(that)h(is)g(to)f(consider)g(the)h(set)g(T)g(of)f(all)h (terms)f(whic)n(h)h(are)f(generated)f(from)i(the)g(constan)n(t)f(0)g(b) n(y)h(means)f(of)h(function)0 4782 y(sym)n(b)r(ols)331 4774 y Fk(\010)376 4782 y Fu(;)14 b(D)482 4794 y Ft(0)518 4782 y Fu(;)g(D)624 4794 y Ft(1)661 4782 y Fu(;)g(:::)42 b FA(for)e(the)i(ordinal)e(functions)i(+)p Fu(;)14 b( )1929 4794 y Ft(0)1966 4782 y Fu(;)g( )2057 4794 y Ft(1)2094 4782 y Fu(;)g(:::)p FA(.)78 b(Then)41 b(one)g(lo)r(oks)g(for)f(a)h (\(primitiv)n(e\))h(recursiv)n(e)0 4921 y(c)n(haracterization)36 b(of)j(the)g(relation)e Fu(<)1258 4933 y Ft(o)1295 4921 y FA(:=)k Fx(f)p FA(\()p Fu(a;)14 b(b)p FA(\))41 b Fx(2)h FA(T)p Fx(\002)p FA(T)f(:)h(o\()p Fu(a)p FA(\))f Fu(<)g FA(o\()p Fu(b)p FA(\))p Fx(g)p FA(,)g(where)d(o\()p Fu(a)p FA(\))k Fx(2)g Fu(O)r(n)d FA(is)f(the)h(canonical)0 5061 y(in)n(terpretation)26 b(of)h Fu(a)c Fx(2)h FA(T.)37 b(It)27 b(turns)g(out)g(that)h(the)g(relation)e Fu(<)2033 5073 y Ft(o)2097 5061 y FA(has)h(a)f(particularly)g(simple)i(c)n (haracterization)c(when)k(it)0 5200 y(is)f(restricted)g(to)g(the)h (subset)f(OT)c Fx(\022)f FA(T)28 b(of)f(those)g(terms)g Fu(a)c Fx(2)g FA(T)28 b(whic)n(h)f(are)f(in)i(\\normalform")d(\(i.e.)j (o\()p Fu(b)p FA(\))22 b Fx(2)i Fu(C)3522 5212 y Fv(\033)3567 5200 y FA(\(o\()p Fu(b)p FA(\)\))k(for)0 5340 y(eac)n(h)f(subterm)g Fu(D)579 5352 y Fv(\033)624 5340 y Fu(b)g FA(of)h Fu(a)p FA(,)f(and)h(o\()p Fu(a)1156 5352 y Fv(n)1201 5340 y FA(\))23 b Fx(\024)g Fu(:)14 b(:)g(:)22 b Fx(\024)h FA(o\()p Fu(a)1669 5352 y Ft(0)1706 5340 y FA(\))28 b(for)f(eac)n(h)g(subterm)g Fu(a)2447 5352 y Ft(0)2485 5332 y Fk(\010)2543 5340 y Fu(:)14 b(:)g(:)2654 5332 y Fk(\010)2699 5340 y Fu(a)2743 5352 y Fv(n)2816 5340 y FA(of)27 b Fu(a)p FA(\).)1908 5539 y(47)p eop %%Page: 48 49 48 48 bop 0 83 a FA(No)n(w)26 b(w)n(e)g(de\014ne)g(the)h(set)f(T)g(of)g (terms,)g(a)g(linear)g(ordering)e Fx(\036)i FA(on)g(T,)g(for)g(an)n(y)f Fu(a)e Fx(2)h FA(T)i(and)g Fu(\033)h(<)22 b(!)29 b FA(a)d(set)g Fu(G)3363 95 y Fv(\033)3408 83 y Fu(a)g FA(of)g(subterms)0 224 y(of)i Fu(a)p FA(,)f(and)h(the)g(set)f(OT)g(of)h Fs(or)l(dinal)j(terms)c FA(\(i.e.)h(terms)g(in)g(normalform\))e(in)i (suc)n(h)f(a)g(w)n(a)n(y)g(that,)h(for)f(all)g Fu(a;)14 b(c)23 b Fx(2)g FA(OT,)0 365 y(\(a\))28 b Fu(c)23 b Fx(\036)f Fu(a)46 b Fx(,)h FA(o\()p Fu(c)p FA(\))23 b Fu(<)f FA(o\()p Fu(a)p FA(\))28 b(and)f(\(b\))i Fu(G)1295 377 y Fv(\033)1340 365 y Fu(c)23 b Fx(\036)f Fu(a)46 b Fx(,)g FA(o\()p Fu(c)p FA(\))24 b Fx(2)f Fu(C)2008 377 y Fv(\033)2053 365 y FA(\(o\()p Fu(a)p FA(\)\).)0 531 y(\(Here)28 b Fu(G)294 543 y Fv(\033)339 531 y Fu(a)22 b Fx(\036)h Fu(c)28 b FA(abbreviates)e Fx(8)p Fu(x)d Fx(2)g Fu(G)1258 543 y Fv(\033)1303 531 y Fu(a)p FA(\()p Fu(x)h Fx(\036)e Fu(c)p FA(\).\))0 747 y Fz(Inductiv)m(e)33 b(de\014nition)d(of)i FA(T)0 888 y(1.)k(0)23 b Fx(2)g FA(T.)0 1029 y(2.)36 b(If)28 b Fu(a)23 b Fx(2)h FA(T)j(and)h Fu(\033)e(<)d(!)s FA(,)k(then)i Fu(D)1104 1041 y Fv(\033)1148 1029 y Fu(a)23 b Fx(2)g FA(T;)28 b(w)n(e)f(call)g Fu(D)1747 1041 y Fv(\033)1792 1029 y Fu(a)g FA(a)h Fs(princip)l(al)j(term)p FA(.)0 1171 y(3.)36 b(If)28 b Fu(a)228 1183 y Ft(0)266 1171 y Fu(;)14 b(:::;)g(a)453 1183 y Fv(n)521 1171 y Fx(2)23 b FA(T)28 b(are)e(principal)i(terms)f(and)g Fu(n)c Fx(\025)g FA(1,)k(then)h(\()p Fu(a)2082 1183 y Ft(0)2120 1163 y Fk(\010)2165 1171 y Fu(:::)2234 1163 y Fk(\010)2279 1171 y Fu(a)2323 1183 y Fv(n)2368 1171 y FA(\))23 b Fx(2)h FA(T.)0 1337 y Fy(De\014nition.)74 b FA(F)-7 b(or)27 b Fu(a)c Fx(2)h FA(T)j(let)h Fu(`)p FA(\()p Fu(a)p FA(\))g(b)r(e)g(the) g Fs(length)g FA(\(n)n(um)n(b)r(er)g(of)f(sym)n(b)r(ols\))g(of)h Fu(a)p FA(.)0 1561 y Fy(Notation.)73 b FA(F)-7 b(or)27 b(principal)h(terms)f Fu(a)1188 1573 y Ft(0)1225 1561 y Fu(;)14 b(:::;)g(a)1412 1573 y Fv(n)p Fp(\000)p Ft(1)1570 1561 y FA(and)27 b Fu(n)c Fx(\025)g FA(0)k(w)n(e)g(set)h Fu(a)2257 1573 y Ft(0)2294 1553 y Fk(\010)2339 1561 y Fu(:::)2408 1553 y Fk(\010)2453 1561 y Fu(a)2497 1573 y Fv(n)p Fp(\000)p Ft(1)2650 1561 y FA(:=)2761 1419 y Fq(\()2841 1484 y FA(0)520 b(if)28 b Fu(n)23 b FA(=)g(0)2841 1550 y Fu(a)2885 1562 y Ft(0)3403 1550 y FA(if)28 b Fu(n)23 b FA(=)g(1)2841 1633 y(\()p Fu(a)2917 1645 y Ft(0)2955 1625 y Fk(\010)3000 1633 y Fu(:::)3069 1625 y Fk(\010)3114 1633 y Fu(a)3158 1645 y Fv(n)p Fp(\000)p Ft(1)3288 1633 y FA(\))83 b(if)28 b Fu(n)23 b(>)g FA(1)0 1805 y(So)k(ev)n(ery)g Fu(a)c Fx(2)g FA(T)28 b(can)f(b)r(e)h(uniquely)g(written)f(as)g Fu(a)c FA(=)g Fu(D)1786 1817 y Fv(\033)1824 1825 y Fo(0)1861 1805 y Fu(a)1905 1817 y Ft(0)1942 1797 y Fk(\010)2001 1805 y Fu(:)14 b(:)g(:)2111 1797 y Fk(\010)2156 1805 y Fu(D)2225 1817 y Fv(\033)2263 1825 y Fn(n)p Fk(\000)p Fo(1)2382 1805 y Fu(a)2426 1817 y Fv(n)p Fp(\000)p Ft(1)2583 1805 y FA(with)28 b Fu(n)23 b Fx(\025)g FA(0)k(and)g Fu(a)3207 1817 y Ft(0)3245 1805 y Fu(;)14 b(:::;)g(a)3432 1817 y Fv(n)p Fp(\000)p Ft(1)3585 1805 y Fx(2)23 b FA(T.)0 1971 y(F)-7 b(urther)27 b(w)n(e)h(de\014ne:)37 b(\()p Fu(a)772 1983 y Ft(0)809 1963 y Fk(\010)854 1971 y Fu(:::)923 1963 y Fk(\010)968 1971 y Fu(a)1012 1983 y Fv(n)p Fp(\000)p Ft(1)1142 1971 y FA(\))19 b Fx(\010)f FA(\()p Fu(b)1344 1983 y Ft(0)1381 1963 y Fk(\010)1426 1971 y Fu(:::)1495 1963 y Fk(\010)1540 1971 y Fu(b)1576 1983 y Fv(m)p Fp(\000)p Ft(1)1724 1971 y FA(\))23 b(:=)g Fu(a)1934 1983 y Ft(0)1971 1963 y Fk(\010)2016 1971 y Fu(:::)2085 1963 y Fk(\010)2130 1971 y Fu(a)2174 1983 y Fv(n)p Fp(\000)p Ft(1)2304 1963 y Fk(\010)2349 1971 y Fu(b)2385 1983 y Ft(0)2422 1963 y Fk(\010)2467 1971 y Fu(:::)2536 1963 y Fk(\010)2581 1971 y Fu(b)2617 1983 y Fv(m)p Fp(\000)p Ft(1)2765 1971 y FA(,)0 2112 y(and)k Fu(a)19 b Fx(\001)f Fu(n)23 b FA(:=)g Fu(a)493 2104 y Fk(\010)538 2112 y Fu(:::)607 2104 y Fk(\010)652 2112 y Fu(a)449 2147 y Fq(|)p 486 2147 49 10 v 49 w({z)p 609 2147 V 49 w(})552 2222 y Fv(n)723 2112 y FA(for)28 b(principal)f(terms)g Fu(a)1472 2124 y Fv(i)1499 2112 y Fu(;)14 b(b)1572 2124 y Fv(i)1600 2112 y Fu(;)g(a)p FA(.)0 2371 y Fz(De\014nition)31 b(of)h FA(o)22 b(:)h(T)g Fx(\000)-14 b(!)23 b Fu(O)r(n)0 2512 y FA(o\()p Fu(D)143 2524 y Fv(\033)181 2532 y Fo(0)218 2512 y Fu(a)262 2524 y Ft(0)299 2504 y Fk(\010)357 2512 y Fu(:)14 b(:)g(:)468 2504 y Fk(\010)513 2512 y Fu(D)582 2524 y Fv(\033)620 2532 y Fn(n)p Fk(\000)p Fo(1)738 2512 y Fu(a)782 2524 y Fv(n)p Fp(\000)p Ft(1)913 2512 y FA(\))23 b(:=)g Fu( )1133 2524 y Fv(\033)1171 2532 y Fo(0)1208 2512 y FA(o\()p Fu(a)1326 2524 y Ft(0)1363 2512 y FA(\))18 b(+)g Fu(:)c(:)g(:)19 b FA(+)f Fu( )1749 2524 y Fv(\033)1787 2532 y Fn(n)p Fk(\000)p Fo(1)1905 2512 y FA(o\()p Fu(a)2023 2524 y Fv(n)p Fp(\000)p Ft(1)2153 2512 y FA(\))0 2728 y Fz(De\014nition)31 b(of)h Fu(a)23 b Fx(\036)f Fu(b)32 b Fz(for)g Fu(a;)14 b(b)22 b Fx(2)i FA(T)0 2869 y(1.)64 b(0)23 b Fx(\036)f Fu(b)46 b FA(:)23 b Fx(\()-14 b(\))70 b Fu(b)22 b Fx(6)p FA(=)h(0)0 3010 y(2.)64 b Fu(D)198 3022 y Fv(\033)243 3010 y Fu(a)32 b Fx(\010)h FA(~)-43 b Fu(a)23 b Fx(\036)f Fu(D)639 3022 y Fv(\032)678 3010 y Fu(b)31 b Fx(\010)839 2988 y FA(~)842 3010 y Fu(b)46 b FA(:)23 b Fx(\()-14 b(\))70 b Fu(\033)26 b(<)d(\032)k FA(or)g(\()h Fu(\033)e FA(=)d Fu(\032)g FA(&)g Fu(a)g Fx(\036)f Fu(b)28 b FA(\))g(or)e(\()i Fu(\033)f FA(=)22 b Fu(\032)h FA(&)g Fu(a)g FA(=)f Fu(b)h FA(&)h(~)-43 b Fu(a)23 b Fx(\036)3106 2988 y FA(~)3109 3010 y Fu(b)28 b FA(\))0 3226 y Fz(Remark.)35 b Fx(\036)28 b FA(is)f(a)g(linear)g (ordering)f(on)h(T,)h(but)g(it's)g(not)g(a)f(w)n(ellordering)f(\(e.g.)h Fu(:)14 b(:)g(:)23 b Fx(\036)g Fu(D)2862 3238 y Ft(0)2899 3226 y Fu(D)2968 3238 y Ft(0)3005 3226 y Fu(D)3074 3238 y Ft(1)3111 3226 y FA(0)f Fx(\036)h Fu(D)3332 3238 y Ft(0)3369 3226 y Fu(D)3438 3238 y Ft(1)3475 3226 y FA(0)f Fx(\036)h Fu(D)3696 3238 y Ft(1)3733 3226 y FA(0\).)0 3417 y Fy(Abbreviations.)101 b FA(F)-7 b(or)27 b Fu(X)r(;)14 b(Y)41 b Fx(\022)22 b FA(T)28 b(and)g Fu(a)22 b Fx(2)i FA(T)k(let)0 3558 y Fu(X)h Fx(\026)23 b Fu(Y)65 b FA(:)p Fx(,)46 b(8)p Fu(x)23 b Fx(2)g Fu(X)7 b Fx(9)p Fu(y)25 b Fx(2)f Fu(Y)18 b FA(\()p Fu(x)24 b Fx(\026)f Fu(y)s FA(\);)0 3699 y Fu(X)29 b Fx(\036)23 b Fu(a)46 b FA(:)p Fx(,)g(8)p Fu(x)23 b Fx(2)g Fu(X)7 b FA(\()p Fu(x)23 b Fx(\036)g Fu(a)p FA(\);)0 3841 y Fu(a)g Fx(\026)g Fu(X)52 b FA(:)p Fx(,)46 b(:)p FA(\()p Fu(X)16 b Fx(\036)23 b Fu(a)p FA(\))46 b([)14 b Fx(,)46 b(9)p Fu(x)24 b Fx(2)f Fu(X)7 b FA(\()p Fu(a)23 b Fx(\026)g Fu(x)p FA(\)].)0 4031 y Fz(De\014nition)31 b(of)h Fu(G)620 4043 y Fv(\033)665 4031 y Fu(a)0 4198 y FA(1.)k Fu(G)166 4210 y Fv(\033)211 4198 y FA(\()p Fu(a)287 4210 y Ft(0)325 4190 y Fk(\010)370 4198 y Fu(:::)439 4190 y Fk(\010)484 4198 y Fu(a)528 4210 y Fv(n)p Fp(\000)p Ft(1)658 4198 y FA(\))23 b(:=)824 4135 y Fq(S)893 4222 y Fv(i)d FA(0.)1908 5539 y(48)p eop %%Page: 49 50 49 49 bop 0 83 a Fz(Theorem)30 b(8.5.)106 b FA(F)-7 b(or)27 b Fu(a;)14 b(c)23 b Fx(2)g FA(OT)k(w)n(e)h(ha)n(v)n(e)0 218 y(\(a\))g Fu(c)23 b Fx(\036)f Fu(a)46 b Fx(,)h FA(o\()p Fu(c)p FA(\))23 b Fu(<)f FA(o\()p Fu(a)p FA(\);)0 352 y(\(b\))28 b Fu(G)203 364 y Fv(\033)248 352 y Fu(c)23 b Fx(\036)g Fu(a)46 b Fx(,)g FA(o\()p Fu(c)p FA(\))23 b Fx(2)h Fu(C)917 364 y Fv(\033)962 352 y FA(\(o\()p Fu(a)p FA(\)\).)0 505 y(Pro)r(of)i(b)n(y)i(induction)g(on)f(the)h (length)g(of)f Fu(c)h FA(sim)n(ultaneous)f(for)g(\(a\),\(b\):)0 640 y(\(a\))h(W)-7 b(e)28 b(only)f(pro)n(v)n(e)f(\\)p Fx(\))p FA(".)36 b(The)28 b(rev)n(erse)d(implication)j(follo)n(ws)e (from)i(\\)p Fx(\))p FA(",)e(since)i Fx(\036)f FA(is)h(total.)0 774 y(Let)g Fu(c)23 b FA(=)f Fu(D)364 786 y Fv(\033)409 774 y Fu(c)445 786 y Ft(0)496 766 y Fk(\010)541 774 y Fu(c)577 786 y Ft(1)614 766 y Fk(\010)659 774 y Fu(:::)728 766 y Fk(\010)773 774 y Fu(c)809 786 y Fv(m)872 774 y FA(,)28 b Fu(a)23 b FA(=)f Fu(D)1146 786 y Fv(\032)1185 774 y Fu(a)1229 786 y Ft(0)1280 766 y Fk(\010)1325 774 y Fu(a)1369 786 y Ft(1)1406 766 y Fk(\010)1451 774 y Fu(:::)1520 766 y Fk(\010)1565 774 y Fu(a)1609 786 y Fv(n)1682 774 y FA(with)28 b(principal)f(terms)g Fu(c)2484 786 y Ft(1)2521 774 y Fu(;)14 b(:::;)g(c)2700 786 y Fv(m)2763 774 y Fu(;)g(a)2844 786 y Ft(1)2881 774 y Fu(;)g(:::;)g(a)3068 786 y Fv(n)3113 774 y FA(.)0 909 y(1.)40 b Fu(\033)28 b(<)d(\032)p FA(:)39 b(F)-7 b(rom)29 b Fu(c)631 921 y Fv(m)719 909 y Fx(\026)24 b Fu(:)14 b(:)g(:)25 b Fx(\026)g Fu(c)1056 921 y Ft(1)1118 909 y Fx(\026)g Fu(D)1277 921 y Fv(\033)1321 909 y Fu(c)1357 921 y Ft(0)1423 909 y FA(w)n(e)j(get)h(b)n(y)f(IH)i(o\()p Fu(c)2034 921 y Fv(m)2096 909 y FA(\))c Fx(\024)e Fu(:)14 b(:)g(:)25 b Fx(\024)g FA(o\()p Fu(c)2565 921 y Ft(1)2602 909 y FA(\))g Fx(\024)g FA(o\()p Fu(D)2892 921 y Fv(\033)2936 909 y Fu(c)2972 921 y Ft(0)3009 909 y FA(\))g(=)g Fu( )3210 921 y Fv(\033)3255 909 y FA(o\()p Fu(c)3365 921 y Ft(0)3402 909 y FA(\))g Fu(<)g FA(\012)3609 921 y Fv(\033)r Ft(+1)3766 909 y FA(and)0 1043 y(th)n(us)j(o\()p Fu(c)p FA(\))23 b Fu(<)f FA(\012)495 1055 y Fv(\033)r Ft(+1)647 1043 y Fx(\024)h FA(\012)795 1055 y Fv(\032)856 1043 y Fx(\024)g FA(o\()p Fu(D)1087 1055 y Fv(\032)1125 1043 y Fu(a)1169 1055 y Ft(0)1206 1043 y FA(\))h Fx(\024)e FA(o\()p Fu(a)p FA(\).)0 1178 y(2.)65 b Fu(\033)43 b FA(=)38 b Fu(\032)f FA(and)g Fu(c)610 1190 y Ft(0)686 1178 y Fx(\036)i Fu(a)834 1190 y Ft(0)871 1178 y FA(:)56 b(By)37 b(IH)h(o\()p Fu(c)1330 1190 y Ft(0)1367 1178 y FA(\))h Fu(<)f FA(o\()p Fu(a)1659 1190 y Ft(0)1696 1178 y FA(\).)66 b(Since)38 b Fu(D)2113 1190 y Fv(\033)2157 1178 y Fu(c)2193 1190 y Ft(0)2269 1178 y Fx(2)i FA(OT)o(,)g(w)n(e)d(ha)n(v)n(e)f Fu(G)2949 1190 y Fv(\033)2994 1178 y Fu(c)3030 1190 y Ft(0)3106 1178 y Fx(\036)i Fu(c)3245 1190 y Ft(0)3320 1178 y FA(and)f(th)n(us)g (b)n(y)g(IH)0 1313 y(o\()p Fu(c)110 1325 y Ft(0)147 1313 y FA(\))c Fx(2)h Fu(C)6 b FA(\(o\()p Fu(c)508 1325 y Ft(0)546 1313 y FA(\))p Fu(;)14 b( )669 1325 y Fv(\033)714 1313 y FA(o\()p Fu(c)824 1325 y Ft(0)861 1313 y FA(\)\).)55 b(Hence)34 b Fu( )1310 1325 y Fv(\033)1355 1313 y FA(o\()p Fu(c)1465 1325 y Ft(0)1502 1313 y FA(\))f Fu(<)g( )1719 1325 y Fv(\033)1764 1313 y FA(o\()p Fu(a)1882 1325 y Ft(0)1919 1313 y FA(\))h(b)n(y)f(Theorem)g(8.2d.)55 b(No)n(w)33 b(o\()p Fu(c)p FA(\))g Fx(\036)g FA(o\()p Fu(a)p FA(\))h(follo)n(ws)e (as)h(in)h(1.)0 1447 y(\(using)28 b(that)f Fu( )483 1459 y Fv(\033)528 1447 y FA(o\()p Fu(a)p FA(\))h(is)f(additiv)n(ely)g (closed\).)0 1582 y(3.)36 b Fu(\033)27 b FA(=)c Fu(\032)g FA(&)f Fu(c)452 1594 y Ft(0)512 1582 y FA(=)h Fu(a)644 1594 y Ft(0)704 1582 y FA(&)g Fu(c)828 1594 y Ft(1)865 1574 y Fk(\010)910 1582 y Fu(:::)979 1574 y Fk(\010)1024 1582 y Fu(c)1060 1594 y Fv(m)1146 1582 y Fx(\036)g Fu(a)1278 1594 y Ft(1)1315 1574 y Fk(\010)1360 1582 y Fu(:::)1429 1574 y Fk(\010)1474 1582 y Fu(a)1518 1594 y Fv(n)1563 1582 y FA(:)37 b(Immediate)28 b(b)n(y)f(IH.)0 1735 y(\(b\))h (Abbreviation:)37 b Fu(C)29 b FA(:=)23 b Fu(C)932 1747 y Fv(\033)977 1735 y FA(\(o\()p Fu(a)p FA(\)\).)0 1869 y(1.)53 b Fu(c)32 b FA(=)f Fu(c)318 1881 y Ft(0)356 1861 y Fk(\010)401 1869 y Fu(:::)470 1861 y Fk(\010)515 1869 y Fu(c)551 1881 y Fv(n)596 1869 y FA(:)48 b(Then)33 b(o\()p Fu(c)999 1881 y Ft(0)1036 1869 y FA(\))p Fu(;)14 b(:::;)g FA(o\()p Fu(c)1321 1881 y Fv(n)1366 1869 y FA(\))33 b(are)f(additiv)n (e)h(principal)g(n)n(um)n(b)r(ers)f(with)i(o\()p Fu(c)2896 1881 y Ft(0)2933 1869 y FA(\))e Fx(\025)g Fu(:::)g Fx(\025)f FA(o\()p Fu(c)3401 1881 y Fv(n)3446 1869 y FA(\))j(b)n(y)e(I.H.\(a\).)0 2004 y Fu(G)65 2016 y Fv(\033)110 2004 y Fu(c)23 b Fx(\036)g Fu(a)46 b Fx(,)476 1942 y Fq(V)545 2029 y Fv(i)p Fp(\024)p Fv(n)666 2004 y Fu(G)731 2016 y Fv(\033)775 2004 y Fu(c)811 2016 y Fv(i)862 2004 y Fx(\036)23 b Fu(a)1045 1957 y Ft(IH)1040 2004 y Fx(,)1169 1942 y Fq(V)1238 2029 y Fv(i)p Fp(\024)p Fv(n)1359 2004 y FA(o\()p Fu(c)1469 2016 y Fv(i)1496 2004 y FA(\))g Fx(2)h Fu(C)1741 1957 y Ft(L)p Fv(:)p Ft(8)p Fv(:)p Ft(4)1773 2004 y Fx(,)78 b FA(o\()p Fu(c)p FA(\))23 b Fx(2)g Fu(C)6 b FA(.)0 2138 y(2.)36 b Fu(c)23 b FA(=)g Fu(D)317 2150 y Fv(\026)361 2138 y Fu(c)397 2150 y Ft(0)462 2138 y FA(with)28 b Fu(\026)23 b(<)g(\033)s FA(:)37 b(Then)28 b Fu(G)1204 2150 y Fv(\033)1249 2138 y Fu(c)23 b FA(=)g Fx(;)k FA(and)g(o\()p Fu(c)p FA(\))c Fx(2)h FA(\012)1930 2150 y Fv(\026)p Ft(+1)2081 2138 y Fx(\022)f FA(\012)2229 2150 y Fv(\033)2297 2138 y Fx(\022)f Fu(C)6 b FA(.)0 2273 y(3.)36 b Fu(c)23 b FA(=)g Fu(D)317 2285 y Fv(\026)361 2273 y Fu(c)397 2285 y Ft(0)462 2273 y FA(with)28 b Fu(\033)f Fx(\024)22 b Fu(\026)p FA(:)37 b(Then)28 b Fu(G)1204 2285 y Fv(\026)1249 2273 y Fu(c)1285 2285 y Ft(0)1345 2273 y Fx(\036)23 b Fu(c)1469 2285 y Ft(0)1534 2273 y FA(and)k(therefore)g(b)n(y)g(I.H.)h (o\()p Fu(c)2435 2285 y Ft(0)2472 2273 y FA(\))23 b Fx(2)h Fu(C)2665 2285 y Fv(\026)2710 2273 y FA(\(o\()p Fu(c)2852 2285 y Ft(0)2889 2273 y FA(\)\))56 b(\()p Fx(\003)p FA(\).)0 2408 y(\\)p Fx(\))p FA(":)36 b Fx(f)p Fu(c)304 2420 y Ft(0)341 2408 y Fx(g)18 b([)g Fu(G)539 2420 y Fv(\033)584 2408 y Fu(c)620 2420 y Ft(0)681 2408 y FA(=)k Fu(G)833 2420 y Fv(\033)878 2408 y Fu(c)h Fx(\036)g Fu(a)1120 2360 y Ft(IH)1115 2408 y Fx(\))46 b FA(o\()p Fu(c)1354 2420 y Ft(0)1391 2408 y FA(\))23 b Fu(<)g FA(o\()p Fu(a)p FA(\))g(&)g(o\()p Fu(c)1905 2420 y Ft(0)1942 2408 y FA(\))g Fx(2)g Fu(C)2187 2353 y Ft(\()p Fp(\003)p Ft(\))2188 2408 y Fx(\))48 b FA(o\()p Fu(c)p FA(\))23 b(=)g Fu( )2626 2420 y Fv(\026)2670 2408 y FA(o\()p Fu(c)2780 2420 y Ft(0)2817 2408 y FA(\))g Fx(2)h Fu(C)6 b FA(.)0 2542 y(\\)p Fx(\()p FA(":)36 b(1.)g Fu(\033)27 b FA(=)c Fu(\026)p FA(:)37 b Fu( )653 2554 y 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5246 y FA(\(\012)1577 5258 y Fv(!)1626 5246 y FA(\).)37 b(This)27 b(implies)h Fx(k)p FA(OT)2355 5258 y Ft(0)2392 5246 y Fu(;)14 b Fx(\036)23 b(k)g FA(=)f Fu( )2723 5258 y Ft(0)2760 5246 y FA(\(\012)2852 5258 y Fv(!)2901 5246 y FA(\).)1908 5539 y(49)p eop %%Page: 50 51 50 50 bop 0 83 a Fz(F)-8 b(undamen)m(tal)31 b(sequences)0 249 y FA(In)20 b(order)e(to)i(get)f(a)h(b)r(etter)g(insigh)n(t)f(in)n (to)h(the)g(structure)f(\(T)p Fu(;)14 b Fx(\036)p FA(\))20 b(and)f(a)g(b)r(etter)i(understanding)e(of)g(the)h(collapsing)f (functions)0 390 y Fu( )54 402 y Fv(\033)125 390 y FA(w)n(e)25 b(no)n(w)h(presen)n(t)f(an)g(assignmen)n(t)g(of)h(\(fundamen)n(tal\))h (sequences)e(to)h(the)g(elemen)n(ts)g(of)g(T.)36 b(F)-7 b(or)25 b(eac)n(h)g(term)h Fu(a)d Fx(2)h FA(T)h(w)n(e)0 531 y(de\014ne)30 b(its)h Fs(\(c)l(o\014nality\))h(typ)l(e)e Fr(tp)p FA(\()p Fu(a)p FA(\))e Fx(2)f(f)p FA(0)p Fu(;)14 b FA(1)p Fu(;)g(!)s Fx(g)k([)i(f)p FA(\012)1744 543 y Fv(\026)p Ft(+1)1900 531 y FA(:)27 b Fu(\026)g(<)g(!)s Fx(g)i 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FA(in)g(the)g(prop)r(er)g(sense.)35 b(But)25 b(later)g(w)n(e)f (will)h(giv)n(e)f(a)h(natural)f(in)n(terpretation)g(of)h(the)g(terms)0 2067 y Fu(a)e Fx(2)g FA(T)28 b(as)f(w)n(ellfounded)h(trees)f (\(so-called)f Fs(tr)l(e)l(e)j(or)l(dinals)p FA(\))h(whic)n(h)d (harmonizes)f(with)i(the)g(assignmen)n(t)f(\()p Fu(a;)14 b(x)p FA(\))24 b Fx(7!)f Fu(a)p FA([)p Fu(x)p FA(].)0 2258 y Fz(De\014nition)31 b(of)h Fr(tp)o FA(\()p Fu(a)p FA(\))h Fz(and)f Fu(a)p FA([)p Fu(x)p FA(])g Fz(for)g Fu(a)23 b Fx(2)h FA(T)p Fz(,)31 b Fu(x)24 b Fx(2)f(j)p Fr(tp)p FA(\()p Fu(a)p FA(\))p Fx(j)0 2424 y FA(1.)36 b Fr(tp)p FA(\(0\))23 b(:=)g(0.)0 2565 y(2.)36 b Fr(tp)p FA(\()p Fu(D)275 2577 y Ft(0)313 2565 y FA(0\))22 b(:=)h(1,)55 b(\()p Fu(D)741 2577 y Ft(0)778 2565 y FA(0\)[0])23 b(:=)g(0.)0 2707 y(3.)36 b Fr(tp)p FA(\()p Fu(D)275 2719 y Fv(\026)p Ft(+1)404 2707 y FA(0\))23 b(:=)g(\012)672 2719 y Fv(\026)p Ft(+1)800 2707 y FA(,)55 b(\()p Fu(D)979 2719 y Fv(\026)p Ft(+1)1108 2707 y FA(0\)[)p Fu(x)p FA(])23 b(:=)g Fu(x)p FA(.)0 2848 y(4.)36 b Fr(tp)p FA(\()p Fu(a)p FA(\))24 b(=)e(1)46 b Fx(\))g Fr(tp)p FA(\()p Fu(D)784 2860 y Fv(\033)829 2848 y Fu(a)p FA(\))23 b(:=)g Fu(!)s FA(,)55 b(\()p Fu(D)1273 2860 y Fv(\033)1318 2848 y Fu(a)p FA(\)[)p Fu(i)p FA(])23 b(:=)f(\()p Fu(D)1703 2860 y Fv(\033)1748 2848 y Fu(a)p FA([0]\))c Fx(\001)h FA(\()p Fu(i)p FA(+)o(1\).)0 2989 y(5.)36 b Fr(tp)p FA(\()p Fu(a)p FA(\))24 b Fx(2)f(f)p Fu(!)s Fx(g)17 b([)i(f)p FA(\012)716 3001 y Fv(\026)p Ft(+1)867 2989 y FA(:)k Fu(\026)h(<)e(\033)s Fx(g)46 b(\))g Fr(tp)p FA(\()p Fu(D)1515 3001 y Fv(\033)1560 2989 y Fu(a)p FA(\))23 b(:=)g Fr(tp)p FA(\()p Fu(a)p FA(\),)56 b(\()p Fu(D)2131 3001 y Fv(\033)2175 2989 y Fu(a)p FA(\)[)p Fu(x)p FA(])24 b(:=)f Fu(D)2548 3001 y Fv(\033)2592 2989 y Fu(a)p FA([)p Fu(x)p FA(].)0 3130 y(6.)36 b Fr(tp)p FA(\()p Fu(a)p FA(\))24 b(=)e(\012)453 3142 y Fv(\026)p Ft(+1)605 3130 y FA(&)h Fu(\026)g Fx(\025)f Fu(\033)50 b Fx(\))c Fr(tp)p FA(\()p Fu(D)1253 3142 y Fv(\033)1298 3130 y Fu(a)p FA(\))23 b(:=)g Fu(!)s FA(,)55 b(\()p Fu(D)1742 3142 y Fv(\033)1786 3130 y Fu(a)p FA(\)[)p Fu(i)p FA(])23 b(:=)g Fu(D)2140 3142 y Fv(\033)2185 3130 y Fu(a)p FA([)p Fu(x)2299 3142 y Fv(i)2327 3130 y FA(])k(with)h Fu(x)2613 3142 y Ft(0)2674 3130 y FA(:=)23 b(\012)2845 3142 y Fv(\026)2889 3130 y FA(,)28 b Fu(x)2987 3142 y Fv(n)p Ft(+1)3140 3130 y FA(:=)23 b Fu(D)3320 3142 y Fv(\026)3364 3130 y Fu(a)p FA([)p Fu(x)3478 3142 y Fv(n)3523 3130 y FA(].)0 3271 y(7.)36 b Fr(tp)p FA(\()p Fu(a)250 3283 y Ft(0)288 3263 y Fk(\010)333 3271 y Fu(:::)402 3263 y Fk(\010)447 3271 y Fu(a)491 3283 y Fv(n)536 3271 y FA(\))23 b(:=)g Fr(tp)p FA(\()p Fu(a)851 3283 y Fv(n)896 3271 y FA(\),)56 b(\()p Fu(a)1083 3283 y Ft(0)1120 3263 y Fk(\010)1165 3271 y Fu(:::)1234 3263 y Fk(\010)1279 3271 y Fu(a)1323 3283 y Fv(n)1369 3271 y FA(\)[)p Fu(x)p FA(])23 b(:=)g(\()p Fu(a)1704 3283 y Ft(0)1742 3263 y Fk(\010)1787 3271 y Fu(:::)1856 3263 y Fk(\010)1901 3271 y Fu(a)1945 3283 y Fv(n)p Fp(\000)p Ft(1)2075 3271 y FA(\))c Fx(\010)f Fu(a)2253 3283 y Fv(n)2298 3271 y FA([)p Fu(x)p FA(])194 b(\()p Fu(n)23 b Fx(\025)g FA(1\).)0 3462 y(F)-7 b(or)27 b(tec)n(hnical)g(reasons)f(w)n(e)h(also)g(set)g Fu(a)p FA([)p Fu(n)p FA(])c(:=)g Fu(a)p FA([0],)k(if)h Fr(tp)p FA(\()p Fu(a)p FA(\))c(=)e(1.)0 3678 y(Pro)r(of)k(of)i(Theorem) f(8.7:)0 3819 y(\(a\),\(b\),\(c\))i(are)e(easily)f(v)n(eri\014ed)h(b)n (y)h(induction)g(on)f Fu(`)p FA(\()p Fu(a)p FA(\).)0 3985 y(\(d\))32 b(is)g(also)f(pro)n(v)n(ed)f(b)n(y)h(induction)h(on)f Fu(`)p FA(\()p Fu(a)p FA(\).)50 b(Here)31 b(all)g(cases)g(except)g(5.,) i(6.)e(are)g(straigh)n(tforw)n(ard.)46 b(So)31 b(let)h(us)g(assume)0 4126 y Fu(D)69 4138 y Fv(\033)113 4126 y Fu(c)19 b Fx(\010)h FA(~)-44 b Fu(c)23 b Fx(\036)f Fu(D)466 4138 y Fv(\033)511 4126 y Fu(a)p FA(.)37 b(Then)28 b Fu(c)23 b Fx(\036)f Fu(a)p FA(,)28 b(and)f(b)n(y)h(IH)f Fu(c)c Fx(\036)g Fu(a)p FA([)p Fu(x)p FA(])28 b(for)f(some)g Fu(x)d Fx(2)f FA(OT)18 b Fx(\\)h(j)p Fr(tp)p FA(\()p Fu(a)p FA(\))p Fx(j)p FA(.)37 b(Hence)28 b Fu(D)3085 4138 y Fv(\033)3130 4126 y Fu(c)18 b Fx(\010)i FA(~)-44 b Fu(c)23 b Fx(\036)g Fu(D)3483 4138 y Fv(\033)3527 4126 y Fu(a)p FA([)p Fu(x)p FA(].)0 4267 y(If)28 b Fr(tp)p FA(\()p Fu(a)p FA(\))23 b Fx(2)h(f)p Fu(!)s Fx(g)17 b([)i(f)p FA(\012)698 4279 y Fv(\026)p Ft(+1)849 4267 y FA(:)k Fu(\026)g(<)g(\033)s Fx(g)p FA(,)k(w)n(e)h(ha)n(v)n(e)e(\()p Fu(D)1613 4279 y Fv(\033)1658 4267 y Fu(a)p FA(\)[)p Fu(x)p FA(])e(=)e Fu(D)2007 4279 y Fv(\033)2052 4267 y Fu(a)p FA([)p Fu(x)p FA(])28 b(and)f(w)n(e)g(are)g(done.)0 4433 y(No)n(w)g(let)h(us)g (assume)e(that)i Fr(tp)p FA(\()p Fu(a)p FA(\))c(=)e(\012)1234 4445 y Fv(\026)p Ft(+1)1391 4433 y FA(with)28 b Fu(\026)23 b Fx(\025)f Fu(\033)s FA(.)0 4599 y(By)27 b(induction)h(on)g Fu(`)p FA(\()p Fu(a)p FA(\))f(one)h(can)f(pro)n(v)n(e)0 4741 y(\(1\))h Fr(tp)p FA(\()p Fu(a)p FA(\))23 b(=)g(\012)486 4753 y Fv(\026)p Ft(+1)637 4741 y FA(&)g Fu(c)g Fx(2)g FA(OT)g(&)g Fu(a)p FA([\012)1225 4753 y Fv(\026)1269 4741 y FA(])g Fx(\026)g Fu(c)g Fx(\036)f Fu(a)46 b FA(=)-14 b Fx(\))46 b(9)p Fu(x)24 b FA(=)f Fu(D)2093 4753 y Fv(\026)2137 4741 y FA(\()p Fu(b)p Fx(\010)o FA(1\))g Fx(2)h FA(OT)o(\()14 b Fu(b)23 b Fx(2)g Fu(G)2817 4753 y Fv(\026)2862 4741 y Fu(c)g FA(&)g Fu(c)g Fx(\036)f Fu(a)p FA([)p Fu(x)p FA(])14 b(\))0 4907 y(from)27 b(whic)n(h)h(w)n(e)f(conclude)0 5048 y(\(2\))h Fr(tp)p FA(\()p Fu(a)p FA(\))23 b(=)g(\012)486 5060 y Fv(\026)p Ft(+1)637 5048 y FA(&)g Fu(c)g Fx(2)g FA(OT)g(&)g Fu(a)p FA([\012)1225 5060 y Fv(\026)1269 5048 y FA(])g Fx(\026)g Fu(c)g FA(&)f Fx(f)p Fu(c)p Fx(g)c([)h Fu(G)1826 5060 y Fv(\026)1870 5048 y Fu(c)k Fx(\036)g Fu(a)46 b FA(=)-14 b Fx(\))1665 5189 y FA(=)g Fx(\))46 b(9)p Fu(b)23 b Fx(2)g FA(OT\()14 b Fu(`)p FA(\()p Fu(b)p FA(\))23 b Fu(<)f(`)p FA(\()p Fu(c)p FA(\))h(&)g Fx(f)p Fu(b)p Fx(g)17 b([)i Fu(G)2966 5201 y Fv(\026)3011 5189 y Fu(b)j Fx(\036)h Fu(a)g FA(&)g Fu(c)g Fx(\036)f Fu(a)p FA([)p Fu(D)3594 5201 y Fv(\026)3638 5189 y FA(\()p Fu(b)p Fx(\010)p FA(1\)]\))1908 5539 y(50)p eop %%Page: 51 52 51 51 bop 0 83 a FA(Ob)n(viously)26 b(\(2\))i(suggests)e(to)i(de\014ne) g Fu(x)1237 95 y Ft(0)1297 83 y FA(:=)23 b(\012)1468 95 y Fv(\026)1512 83 y FA(,)28 b Fu(x)1610 95 y Fv(n)p Ft(+1)1763 83 y FA(:=)23 b Fu(D)1943 95 y Fv(\026)1987 83 y Fu(a)p FA([)p Fu(x)2101 95 y Fv(n)2146 83 y FA(])28 b(in)g(order)e(to)i(obtain)f(b)n(y)g(induction)h(on)g Fu(`)p FA(\()p Fu(c)p FA(\))0 224 y(\(3\))g Fr(tp)p FA(\()p Fu(a)p FA(\))23 b(=)g(\012)486 236 y Fv(\026)p Ft(+1)637 224 y FA(&)g Fu(c)g Fx(2)g FA(OT)g(&)g Fx(f)p Fu(c)p Fx(g)17 b([)i Fu(G)1374 236 y Fv(\026)1419 224 y Fu(c)k Fx(\036)f Fu(a)46 b FA(=)-14 b Fx(\))46 b(9)p Fu(n)p FA(\()p Fu(c)23 b Fx(\036)g Fu(a)p FA([)p Fu(x)2224 236 y Fv(n)2270 224 y FA(]\).)0 415 y(\(Induction)28 b(step:)37 b(premise)23 b(&)f Fu(a)p FA([)p Fu(x)1118 427 y Ft(0)1156 415 y FA(])h Fx(\026)g Fu(c)1372 361 y Ft(\(2\))1373 415 y Fx(\))47 b(9)p Fu(b)23 b Fx(2)g FA(OT\()14 b Fu(`)p FA(\()p Fu(b)p FA(\))23 b Fu(<)f(`)p FA(\()p Fu(c)p FA(\))h(&)g Fx(f)p Fu(b)p Fx(g)17 b([)i Fu(G)2624 427 y Fv(\026)2669 415 y Fu(b)j Fx(\036)h Fu(a)g FA(&)g Fu(c)g Fx(\036)f Fu(a)p FA([)p Fu(D)3252 427 y Fv(\026)3296 415 y FA(\()p Fu(b)p Fx(\010)p FA(1\)]\))3609 368 y Ft(IH)3604 415 y Fx(\))0 556 y(\))46 b(9)p Fu(b)23 b Fx(2)g FA(OT)p Fx(9)p Fu(n)p FA(\()14 b Fu(b)23 b Fx(\036)f Fu(a)p FA([)p Fu(x)839 568 y Fv(n)885 556 y FA(])h(&)g Fu(c)g Fx(\036)f Fu(a)p FA([)p Fu(D)1301 568 y Fv(\026)1345 556 y FA(\()p Fu(b)p Fx(\010)p FA(1\)])14 b(\))46 b Fx(\))g(9)p Fu(b)23 b Fx(2)g FA(OT)p Fx(9)p Fu(n)p FA(\()p Fu(c)g Fx(\036)g Fu(a)p FA([)p Fu(D)2515 568 y Fv(\026)2559 556 y FA(\()p Fu(b)p Fx(\010)p FA(1\)])f Fx(\026)h Fu(a)p FA([)p Fu(D)3035 568 y Fv(\026)3079 556 y Fu(a)p FA([)p Fu(x)3193 568 y Fv(n)3239 556 y FA(]])g(=)g Fu(a)p FA([)p Fu(x)3510 568 y Fv(n)p Ft(+1)3639 556 y FA(]\))28 b(\))0 747 y(No)n(w)f Fx(9)p Fu(n)p FA(\()p Fu(D)386 759 y Fv(\033)431 747 y Fu(c)18 b Fx(\010)i FA(~)-44 b Fu(c)23 b Fx(\036)g FA(\()p Fu(D)816 759 y Fv(\033)861 747 y Fu(a)p FA(\)[)p Fu(n)p FA(]\))28 b(is)f(obtained)g(as)g(follo)n(ws:)0 888 y(OT)c Fx(3)g Fu(D)295 900 y Fv(\033)339 888 y Fu(c)c Fx(\010)h FA(~)-44 b Fu(c)23 b Fx(\036)f Fu(D)692 900 y Fv(\033)737 888 y Fu(a)827 837 y Fv(\033)r Fp(\024)p Fv(\026)827 888 y FA(=)-14 b Fx(\))46 b Fu(G)1072 900 y Fv(\026)1117 888 y Fu(c)23 b Fx(\022)f Fu(G)1328 900 y Fv(\033)1373 888 y Fu(c)h Fx(\036)g Fu(c)g Fx(\036)f Fu(a)1756 834 y Ft(\(3\))1757 888 y Fx(\))48 b(9)p Fu(n)p FA(\()p Fu(D)2085 900 y Fv(\033)2129 888 y Fu(c)19 b Fx(\010)g FA(~)-43 b Fu(c)23 b Fx(\036)f Fu(D)2482 900 y Fv(\033)2527 888 y Fu(a)p FA([)p Fu(x)2641 900 y Fv(n)2686 888 y FA(])h(=)g(\()p Fu(D)2921 900 y Fv(\033)2966 888 y Fu(a)p FA(\)[)p Fu(n)p FA(]\).)0 1104 y(Pro)r(of)j(of)i(\(1\):)0 1245 y(1.)36 b Fu(a)23 b FA(=)g(\012)316 1257 y Fv(\026)p Ft(+1)445 1245 y FA(:)37 b(Then)27 b Fu(c)c FA(=)g Fu(D)937 1257 y Fv(\026)981 1245 y Fu(c)1017 1257 y Ft(0)1073 1245 y Fx(\010)d FA(~)-44 b Fu(c)23 b Fx(\036)f Fu(D)1371 1257 y Fv(\026)1416 1245 y FA(\()p Fu(c)1484 1257 y Ft(0)1521 1245 y Fx(\010)p FA(1\).)37 b(Let)27 b Fu(b)c FA(:=)g Fu(c)2074 1257 y Ft(0)2111 1245 y FA(.)0 1386 y(2.)36 b Fu(a)23 b FA(=)g Fu(a)300 1398 y Ft(0)356 1386 y Fx(\010)18 b Fu(a)483 1398 y Ft(1)547 1386 y FA(with)29 b Fr(tp)o FA(\()p Fu(a)885 1398 y Ft(1)923 1386 y FA(\))23 b(=)g(\012)1126 1398 y Fv(\026)p Ft(+1)1254 1386 y FA(:)0 1528 y(Then)28 b Fu(c)23 b FA(=)f Fu(a)407 1540 y Ft(0)463 1528 y Fx(\010)c Fu(c)582 1540 y Ft(1)647 1528 y FA(with)28 b Fu(a)880 1540 y Ft(1)917 1528 y FA([\012)1000 1540 y Fv(\026)1045 1528 y FA(])23 b Fx(\026)f Fu(c)1214 1540 y Ft(1)1275 1528 y Fx(\036)g Fu(a)1406 1540 y Ft(1)1443 1528 y FA(,)28 b(and)g(the)g(claim)f(follo)n(ws)g(immediately)g(from)h(the)g(IH.)0 1669 y(3.)36 b Fu(a)23 b FA(=)g Fu(D)325 1681 y Fv(\032)363 1669 y Fu(a)407 1681 y Ft(0)472 1669 y FA(with)28 b Fu(\026)23 b(<)g(\032)p FA(:)37 b(Then)28 b Fr(tp)p FA(\()p Fu(a)1291 1681 y Ft(0)1328 1669 y FA(\))23 b(=)g(\012)1531 1681 y Fv(\026)p Ft(+1)1687 1669 y FA(and)28 b Fu(a)p FA([)p Fu(x)p FA(])23 b(=)g Fu(D)2166 1681 y Fv(\032)2204 1669 y Fu(a)2248 1681 y Ft(0)2285 1669 y FA([)p Fu(x)p FA(].)0 1810 y(F)-7 b(urther)27 b Fu(c)c FA(=)g Fu(D)517 1822 y Fv(\032)555 1810 y Fu(c)591 1822 y Ft(0)647 1810 y Fx(\010)d FA(~)-44 b Fu(c)28 b FA(with)g Fu(a)1027 1822 y Ft(0)1064 1810 y FA([\012)1147 1822 y Fv(\026)1191 1810 y FA(])c Fx(\026)e Fu(c)1361 1822 y Ft(0)1421 1810 y Fx(\036)h Fu(a)1553 1822 y Ft(0)1590 1810 y FA(.)0 1951 y(By)k(IH)h(w)n(e)f(get)h Fu(c)547 1963 y Ft(0)607 1951 y Fx(\036)23 b Fu(a)739 1963 y Ft(0)776 1951 y FA([)p Fu(x)p FA(])28 b(for)f(some)g Fu(x)d FA(=)e Fu(D)1459 1963 y Fv(\026)1504 1951 y FA(\()p Fu(b)p Fx(\010)o FA(1\))h Fx(2)g FA(OT)28 b(with)g Fu(b)22 b Fx(2)i Fu(G)2355 1963 y Fv(\026)2400 1951 y Fu(c)2436 1963 y Ft(0)2473 1951 y FA(.)0 2092 y(Since)k Fu(\026)23 b(<)f(\032)p FA(,)28 b Fu(G)536 2104 y Fv(\026)581 2092 y Fu(c)617 2104 y Ft(0)677 2092 y Fx(\022)23 b Fu(G)830 2104 y Fv(\026)875 2092 y Fu(c)p FA(.)37 b(F)-7 b(rom)27 b Fu(c)1225 2104 y Ft(0)1285 2092 y Fx(\036)c Fu(a)1417 2104 y Ft(0)1454 2092 y FA([)p Fu(x)p FA(])28 b(w)n(e)f(get)h Fu(c)23 b Fx(\036)f Fu(D)2051 2104 y Fv(\032)2089 2092 y Fu(a)2133 2104 y Ft(0)2171 2092 y FA([)p Fu(x)p FA(])h(=)g Fu(a)p FA([)p Fu(x)p FA(].)0 2308 y(F)-7 b(or)27 b(the)h(pro)r(of)f(of)h(8.7e) e(w)n(e)h(need)h(some)f(preparations.)0 2474 y Fz(De\014nition.)0 2615 y Fu(b)14 b Fx(\036)-24 b(\036)156 2627 y Fv(x)206 2615 y Fu(a)46 b FA(:)23 b Fx(\()-14 b(\))69 b Fu(b)23 b Fx(\036)f Fu(a)h FA(&)g Fx(8)p Fu(\033)s Fx(8)p Fu(c)p FA(\()p Fu(b)f Fx(\026)g Fu(c)h Fx(\026)g Fu(a)g Fx(\))g Fu(G)1606 2627 y Fv(\033)1651 2615 y Fu(b)g Fx(\026)f Fu(G)1862 2627 y Fv(\033)1907 2615 y Fu(c)d Fx([)f Fu(G)2100 2627 y Fv(\033)2145 2615 y Fu(x)p FA(\).)0 2806 y Fz(Lemma)30 b(8.8.)0 2947 y Fu(b)14 b Fx(\036)-24 b(\036)156 2959 y Fv(x)206 2947 y Fu(a)23 b FA(&)f Fu(G)425 2959 y Fv(\033)470 2947 y Fu(a)h Fx(\036)g Fu(a)g FA(&)f Fu(G)844 2959 y Fv(\033)889 2947 y Fu(x)i Fx(\036)f Fu(b)45 b FA(=)-14 b Fx(\))46 b Fu(G)1374 2959 y Fv(\033)1419 2947 y Fu(b)23 b Fx(\036)f Fu(b)p FA(.)0 3113 y(Pro)r(of:)36 b(W)-7 b(e)28 b(ha)n(v)n(e)e Fu(G)659 3125 y Fv(\033)704 3113 y Fu(b)d Fx(\026)f Fu(G)915 3125 y Fv(\033)960 3113 y Fu(a)d Fx([)f Fu(G)1161 3125 y Fv(\033)1206 3113 y Fu(x)24 b Fx(\036)f Fu(a)p FA(.)0 3254 y(Assumption:)49 b Fu(b)33 b Fx(\026)f Fu(G)735 3266 y Fv(\033)780 3254 y Fu(b)p FA(.)54 b(Then)34 b(there)g(exists)f(a)g(subterm)g Fu(d)h FA(of)g Fu(b)f FA(with)h(minimal)g(length)f(suc)n(h)g(that)h Fu(b)f Fx(\026)f Fu(G)3614 3266 y Fv(\033)3659 3254 y Fu(d)i Fx(\036)e Fu(a)p FA(.)0 3396 y(By)h(the)h(minimalit)n(y)g(of)f Fu(d)h FA(w)n(e)f(ha)n(v)n(e)f Fu(d)h FA(=)g Fu(D)1456 3408 y Fv(\026)1500 3396 y Fu(c)h FA(with)g Fu(G)1830 3408 y Fv(\033)1875 3396 y Fu(c)e Fx(\036)h Fu(b)f Fx(\026)h Fu(c)g Fx(\036)f Fu(a)p FA(.)55 b(Using)33 b Fu(b)14 b Fx(\036)-24 b(\036)2889 3408 y Fv(x)2940 3396 y Fu(a)33 b FA(and)g Fu(G)3249 3408 y Fv(\033)3294 3396 y Fu(x)h Fx(\036)e Fu(b)h FA(w)n(e)g(obtain)0 3537 y Fu(G)65 3549 y Fv(\033)110 3537 y Fu(b)23 b Fx(\026)f Fu(G)321 3549 y Fv(\033)366 3537 y Fu(c)d Fx([)f Fu(G)559 3549 y Fv(\033)604 3537 y Fu(x)24 b Fx(\036)f Fu(b)p FA(.)36 b(Con)n(tradiction.)0 3728 y Fz(Lemma)30 b(8.9.)0 3869 y Fu(b)36 3881 y Ft(0)87 3869 y Fx(\036)-24 b(\036)193 3881 y Fv(x)243 3869 y Fu(b)46 b FA(=)-14 b Fx(\))46 b Fu(a)18 b Fx(\010)g Fu(b)686 3881 y Ft(0)737 3869 y Fx(\036)-24 b(\036)843 3881 y Fv(x)893 3869 y Fu(a)18 b Fx(\010)g Fu(b)27 b FA(and)h Fu(D)1332 3881 y Fv(\026)1376 3869 y Fu(b)1412 3881 y Ft(0)1463 3869 y Fx(\036)-24 b(\036)1569 3881 y Fv(x)1619 3869 y Fu(D)1688 3881 y Fv(\026)1732 3869 y Fu(b)p FA(.)0 4035 y(Pro)r(of:)0 4176 y(1.)36 b(Supp)r(ose)28 b Fu(a)18 b Fx(\010)g Fu(b)608 4188 y Ft(0)668 4176 y Fx(\026)23 b Fu(c)g Fx(\026)g Fu(a)18 b Fx(\010)g Fu(b)p FA(.)37 b(Then)27 b Fu(c)c FA(=)g Fu(a)18 b Fx(\010)g Fu(c)1688 4188 y Ft(0)1753 4176 y FA(with)28 b Fu(b)1978 4188 y Ft(0)2038 4176 y Fx(\026)23 b Fu(c)2162 4188 y Ft(0)2222 4176 y Fx(\026)g Fu(b)p FA(.)0 4317 y(Hence)28 b Fu(G)312 4329 y Fv(\033)357 4317 y FA(\()p Fu(a)18 b Fx(\010)g Fu(b)570 4329 y Ft(0)607 4317 y FA(\))24 b(=)e Fu(G)815 4329 y Fv(\033)860 4317 y Fu(a)d Fx([)f Fu(G)1061 4329 y Fv(\033)1106 4317 y Fu(b)1142 4329 y Ft(0)1202 4317 y Fx(\026)23 b Fu(G)1355 4329 y Fv(\033)1400 4317 y Fu(a)18 b Fx([)h Fu(G)1601 4329 y Fv(\033)1646 4317 y Fu(c)f Fx([)h Fu(G)1839 4329 y Fv(\033)1884 4317 y Fu(x)24 b FA(=)e Fu(G)2107 4329 y Fv(\033)2152 4317 y Fu(c)d Fx([)f Fu(G)2345 4329 y Fv(\033)2390 4317 y Fu(x)p FA(.)0 4458 y(2.)36 b(Supp)r(ose)28 b Fu(D)496 4470 y Fv(\026)540 4458 y Fu(b)576 4470 y Ft(0)636 4458 y Fx(\026)23 b Fu(c)g Fx(\026)g Fu(D)940 4470 y Fv(\026)984 4458 y Fu(b)p FA(.)36 b(Then)28 b Fu(c)23 b FA(=)g(\()p Fu(D)1544 4470 y Fv(\026)1588 4458 y Fu(c)1624 4470 y Ft(0)1662 4458 y FA(\))18 b Fx(\010)g Fu(c)1831 4470 y Ft(1)1896 4458 y FA(with)28 b Fu(b)2121 4470 y Ft(0)2181 4458 y Fx(\026)23 b Fu(c)2305 4470 y Ft(0)2365 4458 y Fx(\026)g Fu(b)p FA(.)64 b(If)28 b Fu(\026)23 b(<)g(\033)s FA(,)28 b(then)g Fu(G)3175 4470 y Fv(\033)3220 4458 y FA(\()p Fu(D)3321 4470 y Fv(\026)3365 4458 y Fu(b)3401 4470 y Ft(0)3438 4458 y FA(\))c(=)e Fx(;)p FA(.)0 4599 y(No)n(w)27 b(let)h Fu(\026)23 b Fx(\025)g Fu(\033)s FA(.)65 b(Using)27 b(the)h(premise)f Fu(b)1325 4611 y Ft(0)1376 4599 y Fx(\036)-24 b(\036)1482 4611 y Fv(x)1532 4599 y Fu(b)27 b FA(w)n(e)h(obtain)f Fu(G)2041 4611 y Fv(\033)2086 4599 y Fu(b)2122 4611 y Ft(0)2182 4599 y Fx(\026)c Fu(G)2335 4611 y Fv(\033)2380 4599 y Fu(c)2416 4611 y Ft(0)2471 4599 y Fx([)c Fu(G)2610 4611 y Fv(\033)2655 4599 y Fu(x)p FA(,)28 b(and)g(then)0 4741 y Fu(G)65 4753 y Fv(\033)110 4741 y FA(\()p Fu(D)211 4753 y Fv(\026)256 4741 y Fu(b)292 4753 y Ft(0)328 4741 y FA(\))c(=)e Fx(f)p Fu(b)549 4753 y Ft(0)586 4741 y Fx(g)c([)h Fu(G)785 4753 y Fv(\033)830 4741 y Fu(b)866 4753 y Ft(0)925 4741 y Fx(\026)k(f)p Fu(c)1091 4753 y Ft(0)1128 4741 y Fx(g)18 b([)h Fu(G)1327 4753 y Fv(\033)1372 4741 y Fu(c)1408 4753 y Ft(0)1463 4741 y Fx([)g Fu(G)1602 4753 y Fv(\033)1647 4741 y Fu(x)k Fx(\022)g Fu(G)1870 4753 y Fv(\033)1915 4741 y Fu(c)18 b Fx([)h Fu(G)2108 4753 y Fv(\033)2153 4741 y Fu(x)p FA(.)0 4932 y Fz(Lemma)30 b(8.10.)0 5073 y FA(\(a\))e Fu(a)23 b Fx(2)g FA(T)g(&)g Fr(tp)p FA(\()p Fu(a)p FA(\))g(=)g(\012)802 5085 y Fv(\026)p Ft(+1)953 5073 y FA(&)g Fu(x)h Fx(2)f(j)p Fr(tp)p FA(\()p Fu(a)p FA(\))p Fx(j)47 b FA(=)-14 b Fx(\))45 b Fu(a)p FA([)p Fu(x)p FA(])14 b Fx(\036)-23 b(\036)1901 5085 y Fv(x)1951 5073 y Fu(a)p FA(.)0 5214 y(\(b\))28 b Fu(a)23 b Fx(2)h FA(T)f(&)f Fr(tp)p FA(\()p Fu(a)p FA(\))i Fx(2)f(f)p FA(1)p Fu(;)14 b(!)s Fx(g)44 b FA(=)-14 b Fx(\))46 b Fu(a)p FA([)p Fu(j)5 b FA(])14 b Fx(\036)-23 b(\036)1428 5226 y Ft(0)1479 5214 y Fu(a)1908 5539 y FA(51)p eop %%Page: 52 53 52 52 bop 0 83 a FA(Pro)r(of)26 b(b)n(y)i(induction)g(on)f Fu(`)p FA(\()p Fu(a)p FA(\):)0 210 y(\(a\))h(By)f(8.7a)f(w)n(e)i(ha)n (v)n(e)e Fu(a)p FA([)p Fu(x)p FA(])d Fx(\036)g Fu(a)p FA(.)37 b(|)28 b(Supp)r(ose)f Fu(a)p FA([)p Fu(x)p FA(])d Fx(\026)e Fu(c)h Fx(\026)g Fu(a)p FA(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(to)i(pro)n(v)n(e)e Fu(G)2765 222 y Fv(\032)2804 210 y Fu(a)p FA([)p Fu(x)p FA(])d Fx(\026)g Fu(G)3117 222 y Fv(\032)3156 210 y Fu(c)18 b Fx([)h Fu(G)3349 222 y Fv(\032)3388 210 y Fu(x)p FA(.)0 337 y(1.)36 b Fu(a)23 b FA(=)g Fu(D)325 349 y Ft(0)362 337 y FA(0)k(or)g Fu(a)c FA(=)f Fu(D)756 349 y Fv(\026)p Ft(+1)885 337 y FA(0:)36 b(trivial.)0 464 y(2.)g Fu(a)23 b FA(=)g Fu(D)325 476 y Fv(\033)369 464 y Fu(b)28 b FA(with)g Fr(tp)p FA(\()p Fu(b)p FA(\))23 b(=)g(\012)966 476 y Fv(\026)p Ft(+1)1117 464 y FA(&)g Fu(\026)g(<)g(\033)s FA(:)0 591 y(By)k(I.H.)h(w)n(e)g(get) f Fu(b)p FA([)p Fu(x)p FA(])14 b Fx(\036)-24 b(\036)806 603 y Fv(x)856 591 y Fu(b)27 b FA(and)h(then)g Fu(a)p FA([)p Fu(x)p FA(])c(=)e Fu(D)1587 603 y Fv(\033)1632 591 y Fu(b)p FA([)p Fu(x)p FA(])14 b Fx(\036)-24 b(\036)1881 603 y Fv(x)1931 591 y Fu(D)2000 603 y Fv(\033)2044 591 y Fu(b)23 b FA(=)g Fu(a)k FA(b)n(y)g(8.9.)0 718 y(3.)36 b Fu(a)23 b FA(=)g(\()p Fu(a)332 730 y Ft(0)369 710 y Fk(\010)414 718 y Fu(:::)483 710 y Fk(\010)529 718 y Fu(a)573 730 y Fv(n)618 718 y FA(\))55 b(\()p Fu(n)23 b Fx(\025)g FA(1\):)0 845 y(By)k(I.H.)h(w)n(e)g(get)f Fu(a)601 857 y Fv(n)646 845 y FA([)p Fu(x)p FA(])14 b Fx(\036)-23 b(\036)860 857 y Fv(x)910 845 y Fu(a)954 857 y Fv(n)1026 845 y FA(and)28 b(then)g Fu(a)p FA([)p Fu(x)p FA(])23 b(=)g(\()p Fu(a)1701 857 y Ft(0)1738 837 y Fk(\010)1783 845 y Fu(:::)1852 837 y Fk(\010)1898 845 y Fu(a)1942 857 y Fv(n)p Fp(\000)p Ft(1)2072 845 y FA(\))18 b Fx(\010)g Fu(a)2249 857 y Fv(n)2294 845 y FA([)p Fu(x)p FA(])c Fx(\036)-23 b(\036)2508 857 y Fv(x)2558 845 y FA(\()p Fu(a)2634 857 y Ft(0)2671 837 y Fk(\010)2716 845 y Fu(:::)2785 837 y Fk(\010)2831 845 y Fu(a)2875 857 y Fv(n)p Fp(\000)p Ft(1)3005 845 y FA(\))18 b Fx(\010)g Fu(a)3182 857 y Fv(n)3251 845 y FA(=)k Fu(a)28 b FA(b)n(y)f(8.9.)0 997 y(\(b\))h(By)g(8.7a)e(w)n(e)h(ha)n(v)n(e)g Fu(a)p FA([)p Fu(j)5 b FA(])23 b Fx(\036)f Fu(a)p FA(.)37 b(|)28 b(Supp)r(ose)f Fu(a)p FA([)p Fu(j)5 b FA(])23 b Fx(\026)g Fu(c)g Fx(\026)g Fu(a)p FA(.)37 b(W)-7 b(e)27 b(ha)n(v)n(e)g(to)g(pro)n (v)n(e)f Fu(G)2752 1009 y Fv(\032)2791 997 y Fu(a)p FA([)p Fu(j)5 b FA(])23 b Fx(\026)g Fu(G)3096 1009 y Fv(\032)3135 997 y Fu(c)p FA(.)0 1149 y(1.)36 b Fu(a)23 b FA(=)g Fu(D)325 1161 y Fv(\033)369 1149 y Fu(b)28 b FA(with)g Fr(tp)p FA(\()p Fu(b)p FA(\))23 b(=)g(1:)36 b(Then)28 b Fu(a)p FA([)p Fu(j)5 b FA(])23 b(=)g(\()p Fu(D)1565 1161 y Fv(\033)1609 1149 y Fu(b)p FA([0]\))18 b Fx(\001)h FA(\()p Fu(j)5 b FA(+1\))27 b(and)h Fu(G)2289 1161 y Fv(\032)2327 1149 y Fu(a)p FA([)p Fu(j)5 b FA(])23 b(=)g Fu(G)2632 1161 y Fv(\032)2671 1149 y FA(\()p Fu(D)2772 1161 y Fv(\033)2817 1149 y Fu(b)p FA([0]\).)0 1276 y(By)k(I.H.)h(and)g(3.5)f(w)n(e)g(get)g Fu(D)921 1288 y Fv(\033)966 1276 y Fu(b)p FA([0])14 b Fx(\036)-25 b(\036)1209 1288 y Ft(0)1246 1276 y Fu(D)1315 1288 y Fv(\033)1359 1276 y Fu(b)23 b FA(=)g Fu(a)p FA(.)0 1403 y(W)-7 b(e)28 b(also)e(ha)n(v)n(e)h Fu(D)570 1415 y Fv(\033)614 1403 y Fu(b)p FA([0])c Fx(\026)f Fu(c)h Fx(\026)g Fu(a)k FA(and)h(therefore)f Fu(G)1642 1415 y Fv(\032)1680 1403 y FA(\()p Fu(D)1781 1415 y Fv(\033)1826 1403 y Fu(b)p FA([0]\))c Fx(\026)f Fu(G)2157 1415 y Fv(\032)2196 1403 y Fu(c)p FA(.)0 1555 y(2.)36 b Fu(a)23 b FA(=)g Fu(D)325 1567 y Fv(\033)369 1555 y Fu(b)28 b FA(and)f Fr(tp)p FA(\()p Fu(b)p FA(\))c(=)g(\012)938 1567 y Fv(\026)p Ft(+1)1094 1555 y FA(with)28 b Fu(\033)f Fx(\024)22 b Fu(\026)p FA(:)37 b(Then)28 b Fu(a)p FA([)p Fu(j)5 b FA(])23 b(=)g Fu(D)2080 1567 y Fv(\033)2124 1555 y Fu(b)p FA([)p Fu(x)2230 1567 y Fv(j)2265 1555 y FA(])28 b(with)g Fu(x)2552 1567 y Ft(0)2613 1555 y FA(=)23 b(\012)2761 1567 y Fv(\026)2805 1555 y FA(,)28 b Fu(x)2903 1567 y Fv(i)p Ft(+1)3038 1555 y FA(=)23 b Fu(D)3195 1567 y Fv(\026)3239 1555 y Fu(b)p FA([)p Fu(x)3345 1567 y Fv(i)3373 1555 y FA(].)0 1682 y(Supp)r(ose)28 b(that)g Fu(\032)22 b Fx(\024)h Fu(\033)s FA(,)28 b(since)g(otherwise)e Fu(G)1398 1694 y Fv(\032)1437 1682 y Fu(a)p FA([)p Fu(j)5 b FA(])23 b(=)g Fx(;)p FA(.)0 1809 y(F)-7 b(rom)27 b Fu(a)p FA([)p Fu(j)5 b FA(])23 b Fx(\026)g Fu(c)g Fx(\026)f Fu(a)28 b FA(it)g(follo)n(ws)f(that)g Fu(c)c FA(=)g(\()p Fu(D)1459 1821 y Fv(\033)1504 1809 y Fu(c)1540 1821 y Ft(0)1577 1809 y FA(\))c Fx(\010)f Fu(c)1747 1821 y Ft(1)1812 1809 y FA(with)28 b Fu(b)p FA([)p Fu(x)2107 1821 y Fv(j)2142 1809 y FA(])23 b Fx(\026)g Fu(c)2312 1821 y Ft(0)2372 1809 y Fx(\026)f Fu(b)p FA(.)0 1936 y(By)27 b(\(a\))h(w)n(e)f(ha)n(v)n (e)g Fx(8)p Fu(i)p FA(\()p Fu(b)p FA([)p Fu(x)792 1948 y Fv(i)818 1936 y FA(])14 b Fx(\036)-23 b(\036)961 1948 y Fv(x)999 1956 y Fn(i)1030 1936 y Fu(b)p FA(\).)36 b(By)28 b(side)f(induction)h(on)g Fu(i)f FA(w)n(e)g(pro)n(v)n(e)f Fu(G)2406 1948 y Fv(\032)2445 1936 y Fu(b)p FA([)p Fu(x)2551 1948 y Fv(i)2579 1936 y FA(])d Fx(\026)f Fu(G)2777 1948 y Fv(\032)2816 1936 y Fu(c)2852 1948 y Ft(0)2908 1936 y Fx([)d(f)p FA(1)p Fx(g)26 b FA(for)h Fu(i)c Fx(\024)f Fu(j)5 b FA(:)0 2097 y Fu(b)p FA([)p Fu(x)106 2109 y Fv(i)134 2097 y FA(])14 b Fx(\036)-24 b(\036)277 2109 y Fv(x)315 2117 y Fn(i)345 2097 y Fu(b)23 b FA(&)f Fu(b)p FA([)p Fu(x)597 2109 y Fv(i)625 2097 y FA(])h Fx(\026)g Fu(c)795 2109 y Ft(0)855 2097 y Fx(\026)g Fu(b)45 b Fx(\))h Fu(G)1218 2109 y Fv(\032)1257 2097 y Fu(b)p FA([)p Fu(x)1363 2109 y Fv(i)1391 2097 y FA(])23 b Fx(\026)g Fu(G)1590 2109 y Fv(\032)1629 2097 y Fu(c)1665 2109 y Ft(0)1720 2097 y Fx([)c Fu(G)1859 2109 y Fv(\032)1898 2097 y Fu(x)1945 2109 y Fv(i)1996 2020 y Ft(\()p Fp(\003)p Ft(\))2007 2097 y Fx(\026)33 b(f)p Fu(c)2183 2109 y Ft(0)2220 2097 y Fx(g)18 b([)h Fu(G)2419 2109 y Fv(\032)2457 2097 y Fu(c)2493 2109 y Ft(0)2549 2097 y Fx([)g(f)p FA(1)p Fx(g)p FA(.)0 2274 y(\()p Fx(\003)p FA(\))55 b Fu(G)226 2286 y Fv(\032)265 2274 y Fu(x)312 2286 y Fv(i)354 2132 y Fq(\()435 2199 y Fx(\022)23 b(f)p FA(0)p Fu(;)14 b FA(1)p Fx(g)1438 b FA(if)29 b Fu(i)22 b FA(=)h(0)435 2346 y(=)g Fx(f)p Fu(b)p FA([)p Fu(x)671 2358 y Fv(i)p Fp(\000)p Ft(1)783 2346 y FA(])p Fx(g)18 b([)h Fu(G)1005 2358 y Fv(\032)1044 2346 y Fu(b)p FA([)p Fu(x)1150 2358 y Fv(i)p Fp(\000)p Ft(1)1262 2346 y FA(])1308 2276 y Ft(SIH)1331 2346 y Fx(\026)45 b(f)p Fu(c)1519 2358 y Ft(0)1556 2346 y Fx(g)18 b([)h Fu(G)1755 2358 y Fv(\032)1793 2346 y Fu(c)1829 2358 y Ft(0)1885 2346 y Fx([)g(f)p FA(1)p Fx(g)81 b FA(if)29 b Fu(i)22 b(>)h FA(0)0 2539 y(No)n(w)k(w)n(e)g(obtain)h Fu(G)635 2551 y Fv(\032)673 2539 y Fu(a)p FA([)p Fu(j)5 b FA(])24 b(=)e Fx(f)p Fu(b)p FA([)p Fu(x)1061 2551 y Fv(j)1096 2539 y FA(])p Fx(g)c([)h Fu(G)1318 2551 y Fv(\032)1356 2539 y Fu(b)p FA([)p Fu(x)1462 2551 y Fv(j)1498 2539 y FA(])k Fx(\026)f(f)p Fu(c)1709 2551 y Ft(0)1746 2539 y Fx(g)c([)h Fu(G)1945 2551 y Fv(\032)1984 2539 y Fu(c)2020 2551 y Ft(0)2075 2539 y Fx([)g(f)p FA(1)p Fx(g)2297 2463 y Ft(\(+\))2316 2539 y Fx(\026)42 b Fu(G)2488 2551 y Fv(\032)2526 2539 y Fu(c)p FA(.)37 b([\(+\))28 b Fu(b)p FA([0])23 b Fx(\036)f Fu(b)p FA([)p Fu(x)3142 2551 y Ft(0)3180 2539 y FA(])h Fx(\026)f Fu(c)3349 2551 y Ft(0)3410 2539 y Fx(\))h FA(1)f Fx(\026)h Fu(c)3704 2551 y Ft(0)3741 2539 y FA(.])0 2691 y(3.)36 b Fu(a)23 b FA(=)g Fu(D)325 2703 y Fv(\033)369 2691 y Fu(b)28 b FA(with)g Fr(tp)p FA(\()p Fu(b)p FA(\))23 b(=)g Fu(!)58 b FA(or)c Fu(a)23 b FA(=)g(\()p Fu(a)1376 2703 y Ft(0)1413 2683 y Fk(\010)1458 2691 y Fu(:::)1527 2683 y Fk(\010)1572 2691 y Fu(a)1616 2703 y Fv(n)1661 2691 y FA(\))56 b(\()p Fu(n)23 b Fx(\025)g FA(1\))k(with)h Fr(tp)p FA(\()p Fu(a)2381 2703 y Fv(n)2427 2691 y FA(\))23 b(=)g Fu(!)s FA(:)36 b(as)27 b(in)h(\(a\).)0 2868 y Fz(Pro)s(of)k(of)f(8.7e)c FA(b)n(y)g(induction)h(on)f Fu(`)p FA(\()p Fu(a)p FA(\):)0 2995 y(1.)36 b Fu(a)23 b FA(=)g(\()p Fu(a)332 3007 y Ft(0)369 2987 y Fk(\010)414 2995 y Fu(:::)483 2987 y Fk(\010)529 2995 y Fu(a)573 3007 y Fv(n)618 2995 y FA(\))g Fx(2)g FA(OT)55 b(\()p Fu(n)23 b Fx(\025)g FA(1\):)37 b(Then)28 b Fu(a)1519 3007 y Ft(0)1556 2995 y Fu(;)14 b(:::;)g(a)1743 3007 y Fv(n)1811 2995 y Fx(2)23 b FA(OT)28 b(and)f Fu(a)2247 3007 y Fv(n)2292 2995 y FA([)p Fu(x)p FA(])d Fx(\036)e Fu(a)2540 3007 y Fv(n)2608 2995 y Fx(\026)h Fu(:)14 b(:)g(:)23 b Fx(\026)g Fu(a)2948 3007 y Ft(0)2985 2995 y FA(.)0 3122 y(By)k(I.H.)h(w)n(e)g(ha)n(v)n(e)e Fu(a)654 3134 y Fv(n)699 3122 y FA([)p Fu(x)p FA(])e Fx(2)f FA(OT.)37 b(Hence)27 b Fu(a)p FA([)p Fu(x)p FA(])d(=)f(\()p Fu(a)1650 3134 y Ft(0)1687 3114 y Fk(\010)1732 3122 y Fu(:::)1801 3114 y Fk(\010)1846 3122 y Fu(a)1890 3134 y Fv(n)p Fp(\000)p Ft(1)2020 3122 y FA(\))c Fx(\010)f Fu(a)2198 3134 y Fv(n)2243 3122 y FA([)p Fu(x)p FA(])24 b Fx(2)f FA(OT.)0 3274 y(2.)36 b Fu(a)23 b FA(=)g Fu(D)325 3286 y Fv(\033)369 3274 y Fu(b)g Fx(2)h FA(OT)o(:)37 b(Then)28 b Fu(b)23 b Fx(2)g FA(OT)k(and)h Fu(G)1424 3286 y Fv(\033)1469 3274 y Fu(b)22 b Fx(\036)h Fu(b)p FA(.)0 3426 y(2.1.)36 b Fr(tp)p FA(\()p Fu(b)p FA(\))23 b(=)g(1:)36 b(By)28 b(I.H.)g(and)f(8.10)f(w)n(e)i (obtain)f Fu(b)p FA([0])22 b Fx(2)i FA(OT)j(and)g Fu(b)p FA([0])14 b Fx(\036)-24 b(\036)2346 3438 y Ft(0)2384 3426 y Fu(b)p FA(.)0 3553 y(F)-7 b(rom)27 b Fu(b)p FA([0])14 b Fx(\036)-24 b(\036)461 3565 y Ft(0)499 3553 y Fu(b)27 b FA(and)g Fu(G)788 3565 y Fv(\033)833 3553 y Fu(b)c Fx(\036)g Fu(b)k FA(w)n(e)g(get)h Fu(G)1369 3565 y Fv(\033)1414 3553 y Fu(b)p FA([0])22 b Fx(\036)g Fu(b)p FA([0])27 b(b)n(y)h(8.8.)36 b(Hence)28 b Fu(a)p FA([)p Fu(x)p FA(])23 b(=)g(\()p Fu(D)2676 3565 y Fv(\033)2720 3553 y Fu(b)p FA([0]\))18 b Fx(\001)h FA(\()p Fu(x)p FA(+1\))k Fx(2)h FA(OT)o(.)0 3705 y(2.2.)36 b Fu(a)23 b FA(=)g Fu(D)390 3717 y Fv(\033)434 3705 y Fu(b)k FA(with)h Fr(tp)p FA(\()p Fu(b)p FA(\))c Fx(2)f(f)p Fu(!)s Fx(g)17 b([)i(f)p FA(\012)1293 3717 y Fv(\026)p Ft(+1)1444 3705 y FA(:)k Fu(\026)g(<)g(\033)s Fx(g)p FA(:)37 b(By)27 b(I.H.)h(and)g(8.10)e(w)n(e)h(ha)n(v)n(e)f Fu(b)p FA([)p Fu(x)p FA(])e Fx(2)f FA(OT)k(and)h Fu(b)p FA([)p Fu(x)p FA(])14 b Fx(\036)-24 b(\036)3543 3717 y Fv(x)3593 3705 y Fu(b)p FA(.)0 3832 y(Since)28 b Fu(x)23 b Fx(2)h(j)p Fr(tp)p FA(\()p Fu(b)p FA(\))p Fx(j)k FA(with)g Fr(tp)p FA(\()p Fu(b)p FA(\))23 b Fx(2)g(f)p Fu(!)s Fx(g)17 b([)i(f)p FA(\012)1408 3844 y Fv(\026)p Ft(+1)1559 3832 y FA(:)k Fu(\026)g(<)g(\033)s Fx(g)p FA(,)28 b(w)n(e)f(ha)n(v)n(e)f Fu(G)2287 3844 y Fv(\033)2332 3832 y Fu(x)e Fx(\036)f Fu(b)p FA([)p Fu(x)p FA(].)0 3959 y(By)k(8.8)g(from)g Fu(b)p FA([)p Fu(x)p FA(])14 b Fx(\036)-23 b(\036)709 3971 y Fv(x)760 3959 y Fu(b)22 b FA(&)h Fu(G)971 3971 y Fv(\033)1016 3959 y Fu(b)g Fx(\036)f Fu(b)h FA(&)f Fu(G)1373 3971 y Fv(\033)1418 3959 y Fu(x)i Fx(\036)f Fu(b)p FA([)p Fu(x)p FA(])28 b(w)n(e)f(get)g Fu(G)2059 3971 y Fv(\033)2104 3959 y Fu(b)p FA([)p Fu(x)p FA(])c Fx(\036)g Fu(b)p FA([)p Fu(x)p FA(].)37 b(Hence)28 b Fu(a)p FA([)p Fu(x)p FA(])23 b(=)g Fu(D)3097 3971 y Fv(\033)3142 3959 y Fu(b)p FA([)p Fu(x)p FA(])g Fx(2)g FA(OT.)0 4111 y(2.3.)36 b Fr(tp)p FA(\()p Fu(b)p FA(\))23 b(=)g(\012)510 4123 y Fv(\026)p Ft(+1)666 4111 y FA(with)28 b Fu(\033)f Fx(\024)22 b Fu(\026)p FA(:)37 b(Then)28 b Fu(a)p FA([)p Fu(x)p FA(])c(=)e Fu(D)1660 4123 y Fv(\033)1705 4111 y Fu(b)p FA([)p Fu(x)1811 4123 y Fv(j)1846 4111 y FA(])27 b(with)i Fu(x)2133 4123 y Ft(0)2193 4111 y FA(=)23 b(\012)2341 4123 y Fv(\026)2385 4111 y FA(,)28 b Fu(x)2483 4123 y Fv(i)p Ft(+1)2618 4111 y FA(=)23 b Fu(D)2775 4123 y Fv(\026)2819 4111 y Fu(b)p FA([)p Fu(x)2925 4123 y Fv(i)2953 4111 y FA(].)0 4238 y(W)-7 b(e)28 b(ha)n(v)n(e)e(to)i(sho)n(w)f Fu(D)711 4250 y Fv(\033)755 4238 y Fu(b)p FA([)p Fu(x)861 4250 y Fv(j)896 4238 y FA(])c Fx(2)h FA(OT)o(.)0 4390 y(\(1\))k Fu(G)199 4402 y Fv(\026)244 4390 y Fu(b)p FA([)p Fu(x)350 4402 y Fv(i)377 4390 y FA(])23 b Fx(\022)g Fu(G)576 4402 y Fv(\033)621 4390 y Fu(b)p FA([)p Fu(x)727 4402 y Fv(i)755 4390 y FA(])2606 b([)28 b(since)f Fu(\033)g Fx(\024)22 b Fu(\026)28 b FA(])0 4542 y(\(2\))g Fx(8)p Fu(i)p FA(\()p Fu(x)289 4554 y Fv(i)339 4542 y Fx(2)23 b FA(OT)g Fx(\))g Fu(b)p FA([)p Fu(x)777 4554 y Fv(i)805 4542 y FA(])g Fx(2)g FA(OT\))2459 b([)28 b(b)n(y)f(I.H.)h(])0 4694 y(\(3\))g Fu(G)199 4706 y Fv(\033)244 4694 y Fu(x)291 4706 y Fv(i)342 4694 y Fx(\036)23 b Fu(b)p FA([)p Fu(x)536 4706 y Fv(i)563 4694 y FA(])46 b Fx(\))h Fu(G)827 4706 y Fv(\033)872 4694 y Fu(b)p FA([)p Fu(x)978 4706 y Fv(i)1005 4694 y FA(])24 b Fx(\036)e Fu(b)p FA([)p Fu(x)1245 4706 y Fv(i)1273 4694 y FA(])453 b([)28 b Fu(b)p FA([)p Fu(x)1906 4706 y Fv(i)1933 4694 y FA(])14 b Fx(\036)-23 b(\036)2076 4706 y Fv(x)2114 4714 y Fn(i)2144 4694 y Fu(b)23 b FA(&)g Fu(G)2356 4706 y Fv(\033)2401 4694 y Fu(b)f Fx(\036)h Fu(b)g FA(&)f Fu(G)2758 4706 y Fv(\033)2803 4694 y Fu(x)2850 4706 y Fv(i)2901 4694 y Fx(\036)h Fu(b)p FA([)p Fu(x)3095 4706 y Fv(i)3123 4694 y FA(])3192 4647 y Ft(8)p Fv(:)p Ft(8)3194 4694 y Fx(\))38 b Fu(G)3380 4706 y Fv(\033)3425 4694 y Fu(b)p FA([)p Fu(x)3531 4706 y Fv(i)3559 4694 y FA(])23 b Fx(\036)f Fu(b)p FA([)p Fu(x)3798 4706 y Fv(i)3826 4694 y FA(])28 b(])0 4846 y(\(4\))g Fu(G)199 4858 y Fv(\033)244 4846 y Fu(x)291 4858 y Fv(i)342 4846 y Fx(\036)23 b Fu(b)p FA([)p Fu(x)536 4858 y Fv(i)563 4846 y FA(])553 5003 y([Side)28 b(Ind.)g(on)f Fu(i)p FA(:)37 b Fu(G)1198 5015 y Fv(\033)1243 5003 y Fu(x)1290 5015 y Ft(0)1350 5003 y Fx(\022)23 b(f)p FA(0)p Fu(;)14 b FA(1)p Fx(g)21 b(\036)i Fu(b)p FA([)p Fu(x)1858 5015 y Ft(0)1895 5003 y FA(].)65 b Fu(G)2071 5015 y Fv(\033)2116 5003 y FA(\()p Fu(x)2195 5015 y Fv(i)p Ft(+1)2307 5003 y FA(\))24 b(=)e Fx(f)p Fu(b)p FA([)p Fu(x)2598 5015 y Fv(i)2625 5003 y FA(])p Fx(g)c([)h Fu(G)2847 5015 y Fv(\033)2892 5003 y Fu(b)p FA([)p Fu(x)2998 5015 y Fv(i)3026 5003 y FA(])3072 4927 y Ft(SIH+\(3\))3163 5003 y Fx(\026)113 b Fu(b)p FA([)p Fu(x)3447 5015 y Fv(i)3475 5003 y FA(])23 b Fx(\036)f Fu(b)p FA([)p Fu(x)3714 5015 y Fv(i)p Ft(+1)3826 5003 y FA(])28 b(])0 5155 y(\(5\))g Fu(x)181 5167 y Fv(i)232 5155 y Fx(2)23 b FA(OT)28 b(and)f Fu(a)p FA([)p Fu(i)p FA(])c Fx(2)g FA(OT.)1908 5539 y(52)p eop %%Page: 53 54 53 53 bop 0 83 a FA(Pro)r(of)26 b(b)n(y)i(induction)g(on)f Fu(i)p FA(:)37 b(1.)f Fu(x)1064 95 y Ft(0)1125 83 y FA(=)22 b(\012)1272 95 y Fv(\026)1340 83 y Fx(2)h FA(OT.)0 221 y(2.)36 b Fu(x)148 233 y Fv(i)200 221 y Fx(2)23 b FA(OT)449 167 y Ft(\(2\))p Fp(\000)p Ft(\(4\))518 221 y Fx(\))116 b Fu(b)p FA([)p Fu(x)823 233 y Fv(i)850 221 y FA(])24 b Fx(2)f FA(OT)g(&)f Fu(G)1275 233 y Fv(\033)1320 221 y Fu(b)p FA([)p Fu(x)1426 233 y Fv(i)1454 221 y FA(])h Fx(\036)g Fu(b)p FA([)p Fu(x)1694 233 y Fv(i)1722 221 y FA(])1791 167 y Ft(\(1\))1792 221 y Fx(\))47 b Fu(x)1969 233 y Fv(i)p Ft(+1)2104 221 y FA(=)23 b Fu(D)2261 233 y Fv(\026)2305 221 y Fu(b)p FA([)p Fu(x)2411 233 y Fv(i)2439 221 y FA(])g Fx(2)g FA(OT)k(and)h Fu(a)p FA([)p Fu(i)p FA(])23 b(=)f Fu(D)3175 233 y Fv(\033)3220 221 y Fu(b)p FA([)p Fu(x)3326 233 y Fv(i)3353 221 y FA(])i Fx(2)f FA(OT)o(.)0 390 y Fz(Lemma)30 b(8.11)0 524 y FA(0)23 b Fx(6)p FA(=)f Fu(a)h Fx(2)h FA(OT)422 536 y Ft(0)505 524 y FA(=)-14 b Fx(\))46 b Fr(tp)p FA(\()p Fu(a)p FA(\))24 b Fx(2)f(f)p FA(1)p Fu(;)14 b(!)s Fx(g)26 b FA(and)h(o\()p Fu(a)p FA(\))c(=)1634 407 y Fq(\032)1710 476 y FA(o\()p Fu(a)p FA([0]\))18 b(+)g(1)395 b(if)28 b Fr(tp)p FA(\()p Fu(a)p FA(\))c(=)e(1)1710 567 y(sup)1835 588 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1987 567 y FA(\(o\()p Fu(a)p FA([)p Fu(n)p FA(]\)+)o(1\))83 b(if)28 b Fr(tp)p FA(\()p Fu(a)p FA(\))c(=)e Fu(!)0 675 y FA(Pro)r(of:)0 809 y(W)-7 b(e)28 b(only)f(sho)n(w)g(that)h(o\()p Fu(a)p FA(\))23 b(=)f(sup)1096 829 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1248 809 y FA(\(o\()p Fu(a)p FA([)p Fu(n)p FA(]\)+1\))27 b(if)h Fr(tp)p FA(\()p Fu(a)p FA(\))c(=)e Fu(!)s FA(.)37 b(Let)28 b Fu(a)23 b Fx(2)g FA(OT)2594 821 y Ft(0)2659 809 y FA(with)28 b Fr(tp)p FA(\()p Fu(a)p FA(\))23 b(=)g Fu(!)s FA(.)0 943 y(Then)31 b Fu(a)p FA([)p Fu(n)p FA(])e Fx(\036)g Fu(a)g FA(&)g Fu(a)p FA([)p Fu(n)p FA(])g Fx(2)g FA(OT)1028 955 y Ft(0)1096 943 y FA(\(b)n(y)j(8.7a,e\))e(and)h(o\()p Fu(a)p FA([)p Fu(n)p FA(]\))e Fu(<)g FA(o\()p Fu(a)p FA(\))i(\(b)n(y)g(8.5a\).)47 b(No)n(w)31 b(let)g Fu(\015)j(<)29 b FA(o\()p Fu(a)p FA(\).)48 b(Then)31 b(\(b)n(y)h(8.6b\))0 1076 y Fu(\015)c FA(=)22 b(o\()p Fu(c)p FA(\))28 b(for)f(some)g Fu(c)c Fx(2)g FA(OT)925 1088 y Ft(0)990 1076 y FA(with)28 b Fu(c)23 b Fx(\036)g Fu(a)p FA(.)36 b(Theorem)27 b(8.7d)g(yields)h Fu(c)23 b Fx(\036)f Fu(a)p FA([)p Fu(n)p FA(])28 b(for)f(some)g Fu(n)c Fx(2)g FA(I)-14 b(N.)37 b(Hence)28 b Fu(\015)g(<)22 b FA(o\()p Fu(a)p FA([)p Fu(n)p FA(]\).)0 1245 y Fz(Lemma)30 b(8.12.)0 1379 y FA(\(a\))e Fu(\013)23 b(<)g(")337 1391 y Ft(0)420 1379 y Fx(\))46 b Fu(\013)24 b Fx(2)f Fu(C)763 1391 y Ft(0)801 1379 y FA(\()p Fu(\013)p FA(\))h(&)e Fu( )1083 1391 y Ft(0)1121 1379 y FA(\()p Fu(\013)p FA(\))i(=)e Fu(!)1404 1349 y Fv(\013)1451 1379 y FA(.)0 1513 y(\(b\))28 b(0)23 b Fu(<)g(\033)j FA(&)d Fu(\013)g(<)g(")655 1525 y Ft(\012)702 1533 y Fn(\033)741 1525 y Ft(+1)876 1513 y Fx(\))46 b Fu(\013)23 b Fx(2)h Fu(C)1219 1525 y Fv(\033)1264 1513 y FA(\()p Fu(\013)p FA(\))g(&)e Fu( )1546 1525 y Fv(\033)1591 1513 y FA(\()p Fu(\013)p FA(\))i(=)f Fu(!)1875 1483 y Ft(\012)1922 1491 y Fn(\033)1961 1483 y Ft(+)p Fv(\013)2083 1513 y FA(=)f(\012)2230 1525 y Fv(\033)2294 1513 y Fx(\001)c Fu(!)2390 1483 y Fv(\013)2437 1513 y FA(.)0 1664 y(Pro)r(of:)36 b(Exercise.)0 1833 y Fz(Lemma)30 b(8.13.)0 1967 y Fu(a)23 b Fx(2)g FA(OT)46 b(=)-14 b Fx(\))46 b FA(\()p Fu(a)23 b Fx(\036)g Fu(D)752 1979 y Ft(0)789 1967 y FA(\012)849 1979 y Fv(\027)t Ft(+1)1020 1967 y Fx(,)28 b FA(no)f Fu(D)1315 1979 y Fv(\033)1387 1967 y FA(with)h Fu(\033)f(>)c(\027)33 b FA(o)r(ccurs)26 b(in)i Fu(a)p FA(\).)0 2118 y(Pro)r(of:)0 2252 y(\\)p Fx(\))p FA(":)36 b Fu(a)23 b FA(=)g Fu(D)450 2264 y Ft(0)487 2252 y Fu(a)531 2264 y Ft(1)568 2244 y Fk(\010)613 2252 y Fu(:::)682 2244 y Fk(\010)727 2252 y Fu(D)796 2264 y Ft(0)833 2252 y Fu(a)877 2264 y Fv(n)950 2252 y FA(with)28 b Fu(G)1204 2264 y Ft(0)1241 2252 y Fu(a)1285 2264 y Fv(i)1336 2252 y Fx(\036)22 b Fu(a)1467 2264 y Fv(i)1518 2252 y Fx(\036)h FA(\012)1666 2264 y Fv(\027)t Ft(+1)1791 2252 y FA(.)0 2385 y(By)k(induction)h(on)g Fu(`)p FA(\()p Fu(a)p FA(\))f(one)h(obtains:)36 b Fu(a)23 b Fx(2)g FA(T)g(&)g Fx(f)p Fu(a)p Fx(g)17 b([)i Fu(G)1861 2397 y Ft(0)1899 2385 y Fu(a)j Fx(\036)h FA(\012)2113 2397 y Fv(\027)t Ft(+1)2285 2385 y Fx(\))46 b Fu(a)23 b Fx(2)g FA(T\()p Fu(\027)5 b FA(\))24 b(:=)f(T\(0)p Fu(;)3035 2377 y Fk(\010)3079 2385 y Fu(;)14 b(D)3185 2397 y Ft(0)3222 2385 y Fu(;)g(:::;)g(D)3434 2397 y Fv(\027)3475 2385 y FA(\).)0 2537 y([)p Fu(a)23 b FA(=)g Fu(D)247 2549 y Fv(\033)291 2537 y Fu(a)335 2549 y Ft(0)395 2537 y FA(&)g Fx(f)p Fu(a)p Fx(g)17 b([)i Fu(G)767 2549 y Ft(0)805 2537 y Fu(a)k Fx(\036)f FA(\012)1019 2549 y Fv(\027)t Ft(+1)1168 2537 y FA(=)g Fu(D)1324 2549 y Fv(\027)t Ft(+1)1449 2537 y FA(0)46 b Fx(\))g Fu(\033)27 b(<)22 b(\027)5 b FA(+1)23 b(&)f Fx(f)p Fu(a)2176 2549 y Ft(0)2213 2537 y Fx(g)c([)h Fu(G)2412 2549 y Ft(0)2449 2537 y Fu(a)2493 2549 y Ft(0)2553 2537 y FA(=)k Fu(G)2706 2549 y Ft(0)2744 2537 y Fu(a)f Fx(\036)h FA(\012)2958 2549 y Fv(\027)t Ft(+1)3083 2537 y FA(.)37 b(])0 2688 y(\\)p Fx(\()p FA(":)f(left)28 b(to)g(the)g(reader.)0 2980 y Fz(T)-8 b(ree)32 b(ordinals)0 3131 y(Inductiv)m(e)h (de\014nition)d(of)i(classes)f Fg(T)1319 3143 y Fv(\033)1395 3131 y Fz(of)h(tree)f(ordinals)0 3265 y FA(1.)36 b Fz(0)23 b FA(:=)g(\()14 b(\))23 b Fx(2)h Fg(T)519 3277 y Fv(\033)0 3398 y FA(2.)36 b Fi(\013)24 b Fx(2)f Fg(T)322 3410 y Fv(\033)412 3398 y Fx(\))46 b Fi(\013)q FA(+)o Fz(1)23 b FA(:=)g(\()p Fi(\013)p FA(\))g Fx(2)h Fg(T)1136 3410 y Fv(\033)0 3532 y FA(3.)36 b Fx(8)p Fu(n)22 b Fx(2)i FA(I)-14 b(N\()p Fi(\013)473 3544 y Fv(n)541 3532 y Fx(2)23 b Fg(T)675 3544 y Fv(\033)719 3532 y FA(\))47 b Fx(\))f FA(\()p Fi(\013)1022 3544 y Fv(n)1068 3532 y FA(\))1100 3544 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1275 3532 y Fx(2)23 b Fg(T)1409 3544 y Fv(\033)0 3666 y FA(4.)36 b Fu(\026)24 b(<)e(\033)27 b FA(&)22 b Fx(8)p Fi(\030)h Fx(2)g Fg(T)669 3678 y Fv(\026)713 3666 y FA(\()p Fi(\013)808 3678 y Ff(\030)865 3666 y Fx(2)h Fg(T)1000 3678 y Fv(\033)1044 3666 y FA(\))46 b Fx(\))g FA(\()p Fi(\013)1347 3678 y Ff(\030)1380 3666 y FA(\))1412 3678 y Ff(\030)p Fp(2)p Fe(T)1532 3686 y Fn(\026)1592 3666 y Fx(2)24 b Fg(T)1727 3678 y Fv(\033)0 3835 y Fg(T)56 3847 y Fv()577 732 y(<)577 881 y(>)577 906 y(:)664 630 y(\000)702 698 y Fg(D)756 710 y Fv(\033)807 698 y FA(\()p Fi(\013)903 710 y Ff(\030)936 698 y FA(\))968 630 y Fq(\001)1007 731 y Ff(\030)o Fp(2)p Fv(I)1935 698 y FA(if)28 b Fu(I)i Fx(2)24 b(f)p FA(I)-14 b(N)p Fx(g)17 b([)i(f)p Fg(T)2507 710 y Fv(\026)2574 698 y FA(:)k Fu(\026<)o(\033)s Fx(g)664 783 y Fq(\000)702 850 y Fg(D)756 862 y Fv(\033)807 850 y FA(\()p Fi(\013)903 862 y Ff(\030)932 878 y Fn(n)977 850 y FA(\))1009 783 y Fq(\001)1047 883 y Fv(n)p Fp(2)p Ft(I)-11 b(N)1935 850 y FA(if)28 b Fu(I)i FA(=)23 b Fg(T)2221 862 y Fv(\026)2292 850 y FA(with)28 b Fu(\026)c Fx(\025)e Fu(\033)664 958 y FA(with)29 b Fi(\030)896 979 y Ft(0)956 958 y FA(:=)23 b Fz(\012)1136 970 y Fv(\026)1180 958 y Fu(;)37 b Fi(\030)1282 979 y Fv(n)p Ft(+1)1435 958 y FA(:=)22 b Fg(D)1599 970 y Fv(\026)1650 958 y FA(\()p Fi(\013)1745 970 y Ff(\030)1775 986 y Fn(n)1820 958 y FA(\))0 1177 y Fz(Remark)0 1330 y FA(F)-7 b(or)27 b Fi(\013)c FA(=)g(\()p Fi(\013)418 1342 y Ff(\030)452 1330 y FA(\))484 1342 y Ff(\030)p Fp(2)p Fv(I)620 1330 y Fx(2)g Fg(T)754 1342 y Fv(\033)817 1330 y Fx(n)18 b(f)p FA(0)p Fx(g)26 b FA(w)n(e)h(ha)n(v)n(e)f Fg(D)1396 1342 y Fv(\033)1447 1330 y FA(\()p Fi(\013)q FA(\))d(=)1685 1188 y Fq(\()1766 1197 y(\000)1804 1264 y Fg(D)1858 1276 y Fv(\033)1909 1264 y FA(\()p Fi(\013)2004 1276 y Ft(0)2042 1264 y FA(\))18 b Fx(\001)h FA(\()p Fu(n)p FA(+1\))2355 1197 y Fq(\001)2392 1297 y Fv(n)p Fp(2)p Ft(I)-11 b(N)2627 1264 y FA(if)28 b Fu(I)i FA(=)23 b Fx(f)p FA(0)p Fx(g)1766 1313 y Fq(\000)1804 1381 y Fg(D)1858 1393 y Fv(\033)1909 1381 y FA(\()p Fi(\013)2004 1393 y Ff(\030)2038 1381 y FA(\))2070 1313 y Fq(\001)2108 1414 y Ff(\030)p Fp(2)p Fv(I)2627 1381 y FA(otherwise)2991 1330 y(.)0 1525 y(This)28 b(means)f(that)h(on)f Fg(T)795 1537 y Fv(\033)867 1525 y FA(the)h(function)g Fg(D)1389 1537 y Fv(\033)1467 1525 y FA(b)r(eha)n(v)n(es)f(lik)n(e)g(the)h (ordinal)e(function)0 1666 y Fu(\013)d Fx(7!)h Fu(!)238 1636 y Ft(\012)285 1644 y Fn(\033)324 1636 y Ft(+)p Fv(\013)450 1666 y FA(\(if)k Fu(\033)f(>)c FA(0\))k(or)g Fu(\013)c Fx(7!)g Fu(!)1160 1636 y Fv(\013)1235 1666 y FA(\(if)28 b Fu(\033)f FA(=)22 b(0\).)0 1832 y(No)n(w)j(w)n(e)g(are)g(going)f(to)i (pro)n(v)n(e)e(that)i Fx(k)p Fg(D)1258 1844 y Ft(0)1301 1832 y Fg(D)1355 1789 y Ft(\()p Fv(m)p Ft(\))1355 1842 y Fv(\027)1476 1832 y FA(\()p Fz(0)p FA(\))p Fx(k)g FA(equals)f Fu( )1960 1844 y Ft(0)1997 1832 y Fu( )2054 1789 y Ft(\()p Fv(m)p Ft(\))2051 1842 y Fv(\027)2169 1832 y FA(\(0\).)36 b(By)26 b(comparing)e(the)i(de\014nition)g(of)f Fg(D)3521 1844 y Fv(\033)3598 1832 y FA(with)h(the)0 1973 y(assignmen)n(t)h(of)h (fundamen)n(tal)g(sequences)g(ab)r(o)n(v)n(e)f(and)h(taking)f(Theorems) g(8.5,)h(8.7)f(in)n(to)h(consideration)e(this)j(should)f(b)r(e)0 2114 y(more)g(or)g(less)g(clear.)40 b(T)-7 b(o)29 b(obtain)f(a)h (rigorous)d(pro)r(of)i(w)n(e)h(in)n(tro)r(duce)f(the)i(canonical)d(in)n (terpretation)h Fz(t)e FA(:)f(T)g Fx(\000)-14 b(!)25 b Fg(T)3638 2126 y Fv()c(\027)5 b FA(,)28 b(then)g Fm(A)2493 2586 y Fv(\027)2534 2574 y FA(\()p Fu(X)7 b FA(\))22 b Fx(\022)h Fu(X)29 b Fx(!)24 b Fu(a)f Fx(2)g Fu(X)7 b FA(.)0 2765 y Fz(Lemma)30 b(11.5.)77 b FA(If)28 b Fu(a)23 b Fx(2)h FA(T)903 2777 y Ft(0)967 2765 y FA(con)n(tains)j(no)g(sym)n(b)r(ol)g Fu(D)1761 2777 y Fv(\033)1834 2765 y FA(with)h Fu(\033)e(>)d(\027)5 b FA(,)28 b(then)g Fu(a)23 b Fx(2)g Fu(W)2693 2777 y Ft(0)2731 2765 y FA(.)0 2980 y Fz(Theorem)30 b(11.6.)78 b Fx(j)p FA(ID)797 2992 y Fv(\027)839 2980 y Fx(j)23 b FA(=)g Fu( )1027 2992 y Ft(0)1064 2980 y FA(\()p Fu(")1135 2992 y Ft(\012)1182 3000 y Fn(\027)1219 2992 y Ft(+1)1307 2980 y FA(\).)0 3147 y(Pro)r(of:)36 b(1.)g(By)27 b(Corollary)f(10.5)g (w)n(e)h(ha)n(v)n(e)f Fx(j)p FA(ID)1464 3159 y Fv(\027)1505 3147 y Fx(j)d(\024)g Fu( )1693 3159 y Ft(0)1730 3147 y FA(\()p Fu(")1801 3159 y Ft(\012)1848 3167 y Fn(\027)1885 3159 y Ft(+1)1974 3147 y FA(\).)0 3313 y(2.)36 b(Let)28 b Fu(\013)c(<)e( )468 3325 y Ft(0)506 3313 y FA(\()p Fu(")577 3325 y Ft(\012)624 3333 y Fn(\027)661 3325 y Ft(+1)749 3313 y FA(\);)28 b(then)g Fu(\013)c(<)e Fx(k)p Fg(D)1281 3325 y Ft(0)1324 3313 y FA(\()p Fg(D)1410 3269 y Ft(\()p Fv(m)p Ft(\))1410 3322 y Fv(\027)1531 3313 y FA(\()p Fz(0)p FA(\)\))p Fx(k)28 b FA(for)f(some)g Fu(m)p FA(.)0 3479 y(Let)h Fu(a)23 b FA(:=)f Fu(D)395 3491 y Ft(0)432 3479 y Fu(D)503 3435 y Ft(\()p Fv(m)p Ft(\))501 3488 y Fv(\027)618 3479 y FA(0.)37 b(As)28 b(sho)n(wn)e(ab)r(o)n(v)n(e,)h(ID)1446 3491 y Fv(\027)1510 3479 y Fx(`)c Fu(a)g Fx(2)h Fu(W)1808 3491 y Ft(0)1845 3479 y FA(;)k(hence)g Fu(\013)23 b(<)g Fx(k)p Fg(D)2386 3491 y Ft(0)2429 3479 y FA(\()p Fg(D)2516 3435 y Ft(\()p Fv(m)p Ft(\))2516 3488 y Fv(\027)2637 3479 y FA(\()p Fz(0)p FA(\)\))p Fx(k)g FA(=)f Fx(k)p Fz(t)p FA(\()p Fu(a)p FA(\))p Fx(k)h FA(=)g Fx(j)p Fu(a)p Fx(j)3363 3491 y Fv(W)3425 3499 y Fo(0)3485 3479 y Fx(\024)f(j)p FA(ID)3689 3491 y Fv(\027)3730 3479 y Fx(j)p FA(.)1908 5539 y(55)p eop %%Page: 56 57 56 56 bop 0 83 a Fx(x)p Fz(9)63 b(Theories)31 b(for)h(Iterated)h (Inductiv)m(e)g(De\014nitions)0 274 y(De\014nition)0 415 y FA(Let)28 b Fu(M)36 b FA(b)r(e)28 b(a)f(set,)h(and)f(\010)c(:)g Fm(P)p FA(\()p Fu(M)9 b FA(\))23 b Fx(!)g Fm(P)p FA(\()p Fu(M)9 b FA(\))28 b Fs(monotone)p FA(,)g(i.e.)37 b Fx(8)p Fu(X)r(;)14 b(Y)40 b Fx(2)24 b Fm(P)p FA(\()p Fu(M)9 b FA(\)\()14 b Fu(X)29 b Fx(\022)23 b Fu(Y)55 b Fx(\))37 b FA(\010\()p Fu(X)7 b FA(\))23 b Fx(\022)g FA(\010\()p Fu(Y)c FA(\))14 b(\).)0 581 y(I)30 604 y Ft(\010)105 581 y FA(:=)215 519 y Fq(T)284 581 y Fx(f)p Fu(X)29 b Fx(2)24 b Fm(P)p FA(\()p Fu(M)9 b FA(\))23 b(:)g(\010\()p Fu(X)7 b FA(\))23 b Fx(\022)f Fu(X)7 b Fx(g)54 b Fs(\(the)30 b(interse)l(ction)g(of)h(al)t(l)g FA(\010)p Fs(-close)l(d)f(subsets)f (of)i Fu(M)9 b Fs(\))0 747 y FA(W)-7 b(e)28 b(sa)n(y)e(that)i(the)g (set)g(I)769 770 y Ft(\010)848 747 y FA(is)g Fs(inductively)j(de\014ne) l(d)f(by)g FA(\010.)0 888 y(De\014nitions)e(of)g(this)f(kind)h(are)f (called)g Fs(\(gener)l(alize)l(d\))k(inductive)g(de\014nitions)p FA(.)0 1079 y Fz(Lemma)f(9.1)0 1220 y FA(\(a\))e(\010\()p Fu(X)7 b FA(\))23 b Fx(\022)f Fu(X)43 b FA(=)-14 b Fx(\))37 b FA(I)757 1243 y Ft(\010)832 1220 y Fx(\022)22 b Fu(X)7 b FA(,)55 b(for)27 b(eac)n(h)g(set)h Fu(X)h Fx(\022)23 b Fu(M)9 b FA(.)0 1362 y(\(b\))28 b(\010\(I)260 1384 y Ft(\010)312 1362 y FA(\))23 b(=)g(I)485 1384 y Ft(\010)537 1362 y FA(.)64 b(So,)28 b(I)793 1384 y Ft(\010)872 1362 y FA(is)f(the)h(least)f(\010-closed)g(set)h(and)f(also)g(the)h(least)f (\014xp)r(oin)n(t)g(of)h(\010.)0 1503 y(\(c\))g(I)159 1526 y Ft(\010)229 1503 y Fx(\\)19 b FA(\010\()p Fu(X)7 b FA(\))23 b Fx(\022)f Fu(X)44 b FA(=)-14 b Fx(\))36 b FA(I)926 1526 y Ft(\010)1001 1503 y Fx(\022)23 b Fu(X)7 b FA(.)0 1669 y(Pro)r(of:)0 1810 y(\(a\))28 b(trivial.)0 1951 y(\(b\))i(HS:)f(\010\(I)422 1974 y Ft(\010)474 1951 y FA(\))c Fx(\022)g FA(I)651 1974 y Ft(\010)702 1951 y FA(.)70 b(Pro)r(of:)38 b Fx(8)p Fu(X)7 b FA(\(\010\()p Fu(X)g FA(\))23 b Fx(\022)i Fu(X)31 b Fx(\))25 b FA(I)1763 1974 y Ft(\010)1840 1951 y Fx(\022)f Fu(X)7 b FA(\))2087 1904 y Ft(\010)19 b(mon)p Fv(:)2151 1951 y Fx(\))113 b(8)p Fu(X)7 b FA(\(\010\()p Fu(X)g FA(\))24 b Fx(\022)h Fu(X)31 b Fx(\))25 b FA(\010\(I)3146 1974 y Ft(\010)3198 1951 y FA(\))g Fx(\022)g FA(\010\()p Fu(X)7 b FA(\))25 b Fx(\022)f Fu(X)7 b FA(\))50 b Fx(\))0 2092 y FA(\010\(I)122 2115 y Ft(\010)174 2092 y FA(\))23 b Fx(\022)317 2030 y Fq(T)386 2092 y Fx(f)p Fu(X)29 b FA(:)23 b(\010\()p Fu(X)7 b FA(\))23 b Fx(\022)f Fu(X)7 b Fx(g)22 b FA(=)h(I)1140 2115 y Ft(\010)1191 2092 y FA(.)37 b(|)28 b(HS)g Fx(\))46 b Fu(Y)c FA(:=)23 b(\010\(I)1950 2115 y Ft(\010)2002 2092 y FA(\))g Fx(\022)f FA(I)2174 2115 y Ft(\010)2272 2092 y Fx(\))46 b FA(\010\()p Fu(Y)19 b FA(\))24 b Fx(\022)e FA(\010\(I)2825 2115 y Ft(\010)2877 2092 y FA(\))h(=)g Fu(Y)3133 2038 y Ft(\(a\))3134 2092 y Fx(\))47 b FA(I)3294 2115 y Ft(\010)3368 2092 y Fx(\022)23 b Fu(Y)42 b FA(=)22 b(\010\(I)3755 2115 y Ft(\010)3807 2092 y FA(\).)0 2258 y(\(c\))28 b(I)159 2281 y Ft(\010)229 2258 y Fx(\\)19 b FA(\010\()p Fu(X)7 b FA(\))23 b Fx(\022)f Fu(X)53 b Fx(\))46 b FA(\010\(I)986 2281 y Ft(\010)1056 2258 y Fx(\\)19 b Fu(X)7 b FA(\))23 b Fx(\022)f FA(\010\(I)1470 2281 y Ft(\010)1522 2258 y FA(\))d Fx(\\)g FA(\010\()p Fu(X)7 b FA(\))22 b(=)h(I)1987 2281 y Ft(\010)2057 2258 y Fx(\\)c Fu(X)2252 2204 y Ft(\(a\))2253 2258 y Fx(\))47 b FA(I)2413 2281 y Ft(\010)2488 2258 y Fx(\022)23 b FA(I)2606 2281 y Ft(\010)2676 2258 y Fx(\\)18 b Fu(X)30 b Fx(\022)22 b Fu(X)7 b FA(.)0 2474 y Fz(De\014nition.)78 b FA(I)550 2444 y Fv(\013)550 2497 y Ft(\010)624 2474 y FA(:=)23 b(\010\(I)857 2441 y Fv(<\013)857 2498 y Ft(\010)957 2474 y FA(\))28 b(with)g(I)1236 2441 y Fv(<\013)1236 2498 y Ft(\010)1358 2474 y FA(:=)1469 2412 y Fq(S)1538 2499 y Fv(\030)r(<\013)1683 2474 y FA(I)1713 2434 y Fv(\030)1713 2498 y Ft(\010)1820 2474 y FA(\()p Fu(\013)c Fx(2)f Fu(O)r(n)p FA(\))0 2640 y Fz(Lemma)30 b(9.2.)0 2781 y FA(\(a\))e Fu(\013)23 b(<)g(\014)50 b Fx(\))c FA(I)554 2751 y Fv(\013)554 2804 y Ft(\010)629 2781 y Fx(\022)23 b FA(I)747 2741 y Fv(\014)747 2806 y Ft(\010)826 2781 y FA(;)544 b(\(b\))28 b(I)1561 2746 y Fv(\013)p Ft(+1)1561 2806 y(\010)1715 2781 y FA(=)23 b(\010\(I)1925 2751 y Fv(\013)1925 2804 y Ft(\010)1977 2781 y FA(\))28 b(;)0 2922 y(\(c\))g(I)159 2889 y Fv(<\013)159 2947 y Ft(\010)281 2922 y FA(=)23 b(I)399 2892 y Fv(\013)399 2945 y Ft(\010)478 2922 y FA(for)k(some)g Fu(\013)d Fx(2)f Fu(O)r(n)28 b FA(;)259 b(\(d\))28 b(If)g(I)1644 2889 y Fv(<\013)1644 2947 y Ft(\010)1766 2922 y FA(=)23 b(I)1884 2892 y Fv(\013)1884 2945 y Ft(\010)1935 2922 y FA(,)28 b(then)g Fx(8)p Fu(\014)f Fx(\025)22 b Fu(\013)p FA(\(I)2498 2882 y Fv(\014)2498 2947 y Ft(\010)2574 2922 y FA(=)g(I)2691 2945 y Ft(\010)2743 2922 y FA(\).)0 3088 y(Pro)r(of:)0 3230 y(\(a\))28 b(trivial.)0 3371 y(\(b\))g(I)168 3335 y Fv(\013)p Ft(+1)168 3395 y(\010)323 3371 y FA(=)22 b(\010\(I)532 3335 y Fv(<\013)p Ft(+1)532 3395 y(\010)716 3371 y FA(\))771 3316 y Ft(\(a\))781 3371 y FA(=)33 b(\010\(I)1001 3341 y Fv(\013)1001 3394 y Ft(\010)1053 3371 y FA(\).)0 3512 y(\(c\))28 b(Otherwise)f Fu(F)35 b FA(:)23 b Fu(O)r(n)h Fx(!)f Fm(P)p FA(\()p Fu(M)9 b FA(\))p Fu(;)36 b(\013)24 b Fx(7!)f FA(I)1396 3482 y Fv(\013)1396 3535 y Ft(\010)1475 3512 y FA(w)n(ould)k(b)r(e)h (injectiv)n(e.)37 b(But)28 b(then)g Fm(P)p FA(\()p Fu(M)9 b FA(\))28 b(w)n(ould)f(not)h(b)r(e)g(a)f(set.)0 3653 y(\(d\))h(1.)37 b(By)27 b(induction)h(on)f Fu(\014)32 b FA(w)n(e)c(get)f(I)1224 3613 y Fv(\014)1224 3677 y Ft(\010)1299 3653 y Fx(\022)22 b FA(I)1416 3676 y Ft(\010)1495 3653 y FA(for)28 b(all)f Fu(\014)t FA(:)37 b(I.H.)28 b Fx(\))23 b FA(I)2151 3613 y Fv(<\014)2151 3677 y Ft(\010)2271 3653 y Fx(\022)g FA(I)2389 3676 y Ft(\010)2463 3653 y Fx(\))g FA(I)2599 3613 y Fv(\014)2599 3677 y Ft(\010)2674 3653 y FA(=)g(\010\(I)2884 3613 y Fv(<\014)2884 3677 y Ft(\010)2980 3653 y FA(\))h Fx(\022)e FA(\010\(I)3245 3676 y Ft(\010)3297 3653 y FA(\))i(=)e(I)3470 3676 y Ft(\010)3522 3653 y FA(.)0 3794 y(2.)36 b(I)131 3761 y Fv(<\013)131 3819 y Ft(\010)254 3794 y FA(=)22 b(I)371 3764 y Fv(\013)371 3817 y Ft(\010)446 3794 y FA(=)h(\010\(I)656 3761 y Fv(<\013)656 3819 y Ft(\010)755 3794 y FA(\))g Fx(\))g FA(I)946 3817 y Ft(\010)1021 3794 y Fx(\022)g FA(I)1139 3761 y Fv(<\013)1139 3819 y Ft(\010)1261 3794 y Fx(\022)g FA(I)1379 3754 y Fv(\014)1379 3819 y Ft(\010)1458 3794 y FA(for)k Fu(\014)g Fx(\025)c Fu(\013)p FA(.)0 4010 y Fz(De\014nition.)0 4151 y FA(F)-7 b(or)27 b(eac)n(h)g(relation)f Fu(R)e Fx(\022)f Fu(M)9 b Fx(\002)o Fu(M)36 b FA(let)28 b(\010)1266 4163 y Fv(R)1344 4151 y FA(:)23 b Fm(P)p FA(\()p Fu(M)9 b FA(\))23 b Fx(!)g Fm(P)p FA(\()p Fu(M)9 b FA(\),)27 b(\010)2075 4163 y Fv(R)2130 4151 y FA(\()p Fu(X)7 b FA(\))23 b(:=)f Fx(f)p Fu(x)h Fx(2)h Fu(M)32 b FA(:)23 b Fx(8)p Fu(y)s(R)q(x)p FA(\()p Fu(y)i Fx(2)e Fu(X)7 b FA(\))p Fx(g)p FA(.)0 4317 y(Acc\()p Fu(M)t(;)14 b(R)q FA(\))23 b(:=)g(I)550 4329 y Ft(\010)597 4337 y Fn(R)759 4317 y Fs(\(the)30 b(ac)l(c)l(essible)h(p)l(art)f(of)h FA(\()p Fu(M)t(;)14 b(R)q FA(\))30 b Fs(\))0 4483 y FA(\(Acc\()p Fu(M)t(;)14 b(R)q FA(\))24 b(=)529 4421 y Fq(T)598 4483 y Fx(f)p Fu(X)7 b Fx(\022)o Fu(M)31 b FA(:)24 b Fx(8)p Fu(x)p Fx(2)o Fu(M)9 b FA(\()p Fx(8)p Fu(y)s(R)q(x)p FA(\()p Fu(y)25 b Fx(2)f Fu(X)7 b FA(\))22 b Fx(\))h Fu(x)h Fx(2)f Fu(X)7 b FA(\))p Fx(g)p FA(\))0 4674 y Fz(Lemma)30 b(9.4.)0 4815 y FA(Let)e Fu(R)g FA(b)r(e)g(a)f(binary)g (relation)g(on)g Fu(M)9 b FA(,)27 b(and)h(Acc)23 b(:=)g(Acc\()p Fu(M)t(;)14 b(R)q FA(\).)0 4956 y(\(a\))28 b Fx(8)p Fu(x)p FA([)p Fu(x)23 b Fx(2)g Fu(M)32 b FA(&)23 b Fx(8)p Fu(y)s(R)q(x)p FA(\()p Fu(y)i Fx(2)e FA(Acc\))46 b Fx(,)h Fu(x)23 b Fx(2)h FA(Acc)13 b(].)0 5098 y(\(b\))28 b Fx(8)p Fu(x)23 b Fx(2)g FA(Acc[)p Fx(8)p Fu(y)s(R)q(x)p FA(\()p Fu(y)j Fx(2)d Fu(X)7 b FA(\))23 b Fx(\))g Fu(x)g Fx(2)h Fu(X)7 b FA(])45 b(=)-14 b Fx(\))46 b FA(Acc)23 b Fx(\022)g Fu(X)7 b FA(,)27 b(for)g(ev)n(ery)f Fu(X)j Fx(\022)23 b Fu(M)9 b FA(.)37 b(\()p Fu(R)q Fh(\026)o FA(Acc)28 b(is)f(w)n(ellfounded\))0 5239 y(\(c\))h Fu(R)g FA(w)n(ellfounded)51 b Fx(\()-14 b(\))70 b Fu(M)31 b FA(=)23 b(Acc.)1908 5539 y(56)p eop %%Page: 57 58 57 57 bop 0 83 a FA(Pro)r(of:)0 224 y(\(a\))28 b(follo)n(ws)e(from)i (9.1b.)63 b(\(b\))29 b(follo)n(ws)d(from)i(9.1c.)0 365 y(\(c\))g(\\)p Fx(\))p FA(":)36 b(By)27 b(\(a\))h(w)n(e)f(ha)n(v)n(e)g Fx(8)p Fu(x)22 b Fx(2)i Fu(M)9 b FA(\()p Fx(8)p Fu(y)s(R)q(x)p FA(\()p Fu(y)25 b Fx(2)e FA(Acc\))h Fx(\))f Fu(x)g Fx(2)h FA(Acc\).)0 506 y(By)j Fu(R)q FA(-induction)h(from)f(this)h(w)n(e)f (get)g Fx(8)p Fu(x)c Fx(2)g Fu(M)9 b FA(\()p Fu(x)24 b Fx(2)f FA(Acc\).)0 648 y(\\)p Fx(\()p FA(":)36 b(follo)n(ws)27 b(from)g(\(b\).)0 863 y Fz(De\014nition.)78 b FA(F)-7 b(or)27 b Fu(x)c Fx(2)h FA(I)848 886 y Ft(\010)927 863 y FA(let)k Fx(j)p Fu(x)p Fx(j)1140 875 y Ft(\010)1215 863 y FA(:=)23 b(min)p Fx(f)p Fu(\013)g FA(:)g Fu(x)h Fx(2)f FA(I)1807 833 y Fv(\013)1807 886 y Ft(\010)1859 863 y Fx(g)0 1029 y Fz(Lemma)30 b(9.5.)77 b FA(If)28 b(\010)23 b(=)g(\010)880 1041 y Fv(R)962 1029 y FA(then)28 b Fx(j)p Fu(x)p Fx(j)1244 1041 y 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b(\()161 560 y Fq(V)231 580 y Fv(x)231 647 y Fp(8)p Fv(xA)360 622 y Fu(h)408 634 y Ft(0)445 622 y FA(\))477 592 y Fp(1)571 622 y FA(:=)681 560 y Fq(V)751 647 y Fp(8)p Fv(xA)880 555 y Fq(\000)918 622 y Fu(h)966 634 y Ft(0)1003 622 y FA(\()p Fu(x=t)p FA(\))1186 592 y Fp(1)1257 555 y Fq(\001)1295 655 y Fv(t)p Fp(2)p Fv(T)1404 663 y Fo(0)0 848 y FA(2.)g(\()p Fr(Ind)270 808 y Fv(x;t)270 872 y(F)357 848 y Fu(h)405 860 y Ft(0)442 848 y FA(\))474 818 y Fp(1)568 848 y FA(:=)678 731 y Fq(\032)754 795 y Fi(c)797 755 y Fv(x;t)797 820 y(F)1417 795 y FA(if)28 b Fu(n)23 b FA(=)f(0)754 904 y Fr(Cut)880 919 y Fv(F)9 b Ft(\()p Fv(n)p 957 932 42 4 v Ft(\))1028 904 y Fi(e)1074 916 y Fv(n)p Fp(\000)p Ft(1)1205 904 y Fi(c)1247 864 y Fv(x;t)1247 928 y(F)1417 904 y FA(if)28 b Fu(n)23 b(>)f FA(0)1899 848 y(where)129 1078 y Fu(n)h FA(:=)g Fu(t)343 1048 y Fp(N)410 1078 y FA(,)55 b Fi(c)531 1038 y Fv(x;t)531 1102 y(F)641 1078 y Fx(`)692 1035 y Ft(2rk)o(\()p Fv(F)9 b Ft(\))692 1100 y(0)915 1078 y Fx(:)p Fu(F)1023 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a(\012)1285 1380 y Fv(P)d(t)0 1515 y FA(4.)64 b(Otherwise:)36 b(\()p Fx(I)6 b Fu(h)685 1527 y Ft(0)723 1515 y Fu(:::h)840 1527 y Fv(n)p Fp(\000)p Ft(1)970 1515 y FA(\))1002 1485 y Fp(1)1096 1515 y FA(:=)22 b Fx(I)6 b Fu(h)1305 1485 y Fp(1)1305 1536 y Ft(0)1390 1515 y Fu(:)14 b(:)g(:)f(h)1548 1485 y Fp(1)1548 1536 y Fv(n)p Fp(\000)p Ft(1)0 1774 y Fz(Theorem)30 b(9.7)d FA(\(Em)n(b)r(edding\))0 1914 y(ID)93 1926 y Fv(\027)158 1914 y Fx(3)c Fu(h)g Fx(`)g FA(\000)46 b(&)g Fu(h)27 b FA(closed)50 b(=)-14 b Fx(\))46 b FA(ID)1206 1878 y Fp(1)1206 1935 y Fv(\027)1300 1914 y Fx(3)23 b Fu(h)1426 1884 y Fp(1)1519 1914 y Fx(`)1570 1926 y Fv(m)1656 1914 y FA(\000)28 b(for)f(some)g Fu(m)c Fx(2)g FA(I)-14 b(N)q(.)0 2102 y(Pro)r(of:)36 b(straigh)n(tforw)n(ard.) 0 2266 y(Esp)r(ecially)54 b(\()p Fr(Ind)562 2226 y Fv(P)r(;t)562 2290 y Fp(F)655 2266 y FA(\))687 2236 y Fp(1)781 2266 y Fx(`)832 2278 y Fv(m)918 2266 y Fx(:)p FA(\()p Fm(A)p FA(\()p Fx(F)8 b FA(\))p Fx(\022F)g FA(\))p Fu(;)14 b Fx(:)p 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Fq(\000)620 5331 y Fx(R)690 5343 y Fv(C)747 5331 y FA(\()p Fu(e;)14 b(d)898 5343 y Fv(\023)926 5331 y FA(\))958 5264 y Fq(\001)997 5365 y Fv(\023)p Fp(2)p Fv(I)1421 5331 y FA(otherwise)26 b(\(i.e.,)i(if)g Fx(:)p Fu(C)i Fx(62)23 b FA(\001\()p Fx(I)6 b FA(\)\))2699 5195 y(\()p Fu(C)29 b Fx(2)2898 5132 y Fq(V)2967 5153 y Ft(+)3022 5195 y FA(-F)-7 b(or)o(\))1908 5539 y(60)p eop %%Page: 61 62 61 61 bop 0 207 a Fx(E)7 b FA(\()p Fu(d)p FA(\))24 b(:=)293 12 y Fq(8)293 87 y(>)293 112 y(<)293 261 y(>)293 286 y(:)380 78 y Fx(R)450 90 y Fv(C)507 78 y FA(\()p Fx(E)7 b FA(\()p Fu(d)665 90 y Ft(0)703 78 y FA(\))p Fu(;)14 b Fx(E)7 b FA(\()p Fu(d)898 90 y Ft(1)936 78 y FA(\)\))128 b(if)29 b Fx(I)g FA(=)23 b Fr(Cut)1492 90 y Fv(C)1576 78 y FA(with)28 b Fu(C)h Fx(2)1932 15 y Fq(V)2001 36 y Ft(+)2056 78 y FA(-F)-7 b(or)380 206 y Fx(R)450 218 y Fp(:)p Fv(C)551 206 y FA(\()p Fx(E)7 b FA(\()p Fu(d)709 218 y Ft(1)748 206 y FA(\))p Fu(;)14 b Fx(E)7 b FA(\()p Fu(d)943 218 y Ft(0)981 206 y FA(\)\))83 b(if)29 b 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