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b(satis\014ed)g(\(where)h(\000)2831 5177 y Fp(\027)2905 5165 y Fu([)g Fr(I)3007 5177 y Fp(\027)3049 5165 y Fu(,)h(resp.)e(])54 b(is)34 b(the)f(sequen)n(t)0 5319 y([inference)28 b(sym)n(b)r(ol])f(at) g(no)r(de)h Fo(\027)5 b Fu(\).)37 b(Consequen)n(tly)27 b(w)n(e)g(ha)n(v)n(e)g(\000\()p Fo(d)p Fr(j)2098 5331 y Fp(\027)2139 5319 y Fu(\))d Fr(\022)e Fu(\000)2334 5331 y Fp(\027)2376 5319 y Fu(,)27 b(if)i Fo(d)p Fr(j)2569 5331 y Fp(\027)2638 5319 y Fu(is)e(the)h(sub)r(deriv)-5 b(ation)28 b(with)g(ro)r(ot)e Fo(\027)5 b Fu(.)1929 5539 y(3)p eop %%Page: 4 5 4 4 bop 0 83 a Fu(Tw)n(o)27 b(examples:)36 b(Abbreviation:)g Fk(Ax)1206 95 y Fp(F)1284 83 y Fu(:=)23 b(LogAx)1636 103 y Fp(F)0 224 y Fu(I.)28 b Fo(d)55 b Fu(=)267 162 y Fl(W)336 182 y Fp(t)336 249 y Fi(9)p Fp(x)p Fn(\()p Fi(:)p Fp(Rf)7 b(x)p Fi(_)p Fp(Rx)p Fn(\))772 162 y Fl(W)841 182 y Fp(f)g(t)841 251 y Fi(9)p Fp(x)p Fn(\()p Fi(:)p Fp(Rf)g(x)p Fi(_)p Fp(Rx)p Fn(\))1276 162 y Fl(W)1346 182 y Fn(0)1346 249 y Fi(:)p Fp(Rf)g(t)p Fi(_)p Fp(Rt)1629 162 y Fl(W)1698 182 y Fn(1)1698 249 y Fi(:)p Fp(Rf)g(f)g(t)p Fi(_)p Fp(Rf)g(t)2059 224 y Fk(Ax)2153 236 y Fp(Rf)g(t)2294 224 y Fu(=)179 658 y(=)506 394 y Fk(Ax)599 406 y Fp(Rf)g(t)p 383 435 431 4 v 383 456 a Fl(W)452 477 y Fn(1)452 544 y Fi(:)p Fp(Rf)g(f)g(t)p Fi(_)p Fp(Rf)g(t)p 355 567 458 4 v 408 588 a Fl(W)477 609 y Fn(0)477 676 y Fi(:)p Fp(Rf)g(t)p Fi(_)p Fp(Rt)p 318 699 505 4 v 318 723 a Fl(W)387 744 y Fp(f)g(t)387 812 y Fi(9)p Fp(x)p Fn(\()p Fi(:)p Fp(Rf)g(x)p Fi(_)p Fp(Rx)p Fn(\))p 290 839 533 4 v 304 858 a Fl(W)373 879 y Fp(t)373 946 y Fi(9)p Fp(x)p Fn(\()p Fi(:)p Fp(Rf)g(x)p Fi(_)p Fp(Rx)p Fn(\))928 658 y Fu(=)1478 430 y Fo(R)q(f)i(t;)14 b Fr(:)p Fo(R)q(f)9 b(t)p 1294 471 720 4 v 2032 488 a Fn(\()p Fi(_)p Fn(\))1294 546 y Fr(:)p Fo(R)q(f)g(f)g(t)18 b Fr(_)h Fo(R)q(f)9 b(t;)14 b Fr(:)p Fo(R)q(f)9 b(t)p 1188 588 906 4 v 2112 604 a Fn(\()p Fi(_)p Fn(\))1188 662 y Fr(:)p Fo(R)q(f)g(f)g(t)17 b Fr(_)i Fo(R)q(f)9 b(t;)14 b Fr(:)p Fo(R)q(f)9 b(t)17 b Fr(_)i Fo(R)q(t)p 1127 704 999 4 v 2145 721 a Fn(\()p Fi(9)p Fn(\))1127 779 y Fr(9)p Fo(x)p Fu(\()p Fr(:)p Fo(R)q(f)9 b(x)19 b Fr(_)g Fo(R)q(x)p Fu(\))p Fo(;)14 b Fr(:)p Fo(R)q(f)9 b(t)18 b Fr(_)h Fo(R)q(t)p 1099 820 1027 4 v 2145 837 a Fn(\()p Fi(9)p Fn(\))1324 895 y Fr(9)p Fo(x)p Fu(\()p Fr(:)p Fo(R)q(f)9 b(x)19 b Fr(_)g Fo(R)q(x)p Fu(\))0 1105 y(I)r(I.)28 b(Let)g Fo(G)23 b Fu(:=)g Fr(9)p Fo(x)p Fu(\()p Fo(F)12 b Fu(\()p Fo(x)p Fu(\))20 b Fr(^)f(:)p Fo(F)12 b Fu(\()p Fo(S)5 b(x)p Fu(\)\),)0 1246 y Fo(d)56 b Fu(=)186 1184 y Fl(W)255 1205 y Fp(S)s Fn(0)255 1271 y Fp(G)337 1184 y Fl(V)406 1271 y Fp(F)9 b Fn(\()p Fp(S)s Fn(0\))p Fi(^:)p Fp(F)g Fn(\()p Fp(S)s(S)s Fn(0\))903 1184 y Fl(W)972 1205 y Fn(0)972 1271 y Fp(G)1028 1184 y Fl(V)1097 1271 y Fp(F)g Fn(\(0\))p Fi(^:)p Fp(F)g Fn(\()p Fp(S)s Fn(0\))1507 1246 y Fk(Ax)1601 1261 y Fp(F)g Fn(\(0\))1741 1246 y Fk(Ax)1835 1261 y Fp(F)g Fn(\()p Fp(S)s Fn(0\))2042 1246 y Fk(Ax)2135 1261 y Fp(F)g Fn(\()p Fp(S)s(S)s Fn(0\))2386 1246 y Fu(=)99 1664 y(=)320 1415 y Fk(Ax)414 1430 y Fp(F)g Fn(\(0\))637 1415 y Fk(Ax)730 1430 y Fp(F)g Fn(\()p Fp(S)s Fn(0\))p 292 1457 623 4 v 364 1469 a Fl(V)433 1556 y Fp(F)g Fn(\(0\))p Fi(^:)p Fp(F)g Fn(\()p Fp(S)s Fn(0\))p 336 1583 507 4 v 527 1604 a Fl(W)596 1625 y Fn(0)596 1691 y Fp(G)1081 1667 y Fk(Ax)1174 1682 y Fp(F)g Fn(\()p Fp(S)s(S)s Fn(0\))p 499 1709 903 4 v 667 1721 a Fl(V)737 1808 y Fp(F)g Fn(\()p Fp(S)s Fn(0\))p Fi(^:)p Fp(F)g Fn(\()p Fp(S)s(S)s Fn(0\))p 640 1835 595 4 v 862 1858 a Fl(W)931 1879 y Fp(S)s Fn(0)931 1946 y Fp(G)1508 1664 y Fu(=)1790 1435 y Fr(:)p Fo(F)j Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\(0\))83 b Fr(:)p Fo(F)12 b Fu(\()p Fo(S)5 b Fu(0\))p Fo(;)14 b(F)e Fu(\()p Fo(S)5 b Fu(0\))p 1762 1477 1193 4 v 2974 1493 a Fn(\()p Fi(^)p Fn(\))1771 1551 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\(0\))19 b Fr(^)g(:)p Fo(F)12 b Fu(\()p Fo(S)5 b Fu(0\))p Fo(;)14 b(F)e Fu(\()p Fo(S)5 b Fu(0\))p 1744 1593 1202 4 v 2964 1610 a Fn(\()p Fi(9)p Fn(\))2049 1668 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\()p Fo(S)5 b Fu(0\))480 b Fr(:)p Fo(F)12 b Fu(\()p Fo(S)5 b(S)g Fu(0\))p Fo(;)14 b(F)e Fu(\()p Fo(S)5 b(S)g Fu(0\))p 2021 1709 1758 4 v 3797 1726 a Fn(\()p Fi(^)p Fn(\))2229 1784 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\()p Fo(S)5 b Fu(0\))18 b Fr(^)h(:)p Fo(F)12 b Fu(\()p Fo(S)5 b(S)g Fu(0\))p Fo(;)14 b(F)e Fu(\()p Fo(S)5 b(S)g Fu(0\))p 2201 1825 1370 4 v 3589 1842 a Fn(\()p Fi(9)p Fn(\))2562 1900 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\()p Fo(S)5 b(S)g Fu(0\))0 2140 y Ft(De\014nition)0 2281 y Fu(\006)23 b Fr(j)-14 b Fu(=)23 b Fo(C)75 b Fr(\()-14 b(\))70 b Fo(C)34 b Fu(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 2423 y(\006)23 b Fr(j)-14 b Fu(=)23 b Fr(f)p Fo(A)284 2435 y Fn(1)321 2423 y Fo(;)14 b(:::;)g(A)526 2435 y Fp(n)571 2423 y Fr(g)69 b(\()-14 b(\))69 b Fu(\006)23 b Fr(j)-14 b Fu(=)23 b Fo(A)1145 2435 y Fn(1)1201 2423 y Fr(_)c Fo(:::)g Fr(_)f Fo(A)1498 2435 y Fp(n)0 2564 y Fr(j)-14 b Fu(=)23 b(\000)69 b Fr(\()-14 b(\))69 b(;)23 b(j)-14 b Fu(=)23 b(\000.)0 2730 y(An)28 b Fm(axiom)i(system)e Fu(is)g(a)f(set)g(of)h Fs(closed)f Fu(form)n(ulas.)0 2921 y Ft(Theorem)j(1.1)0 3062 y Fr(j)-14 b Fu(=)23 b(\000)46 b(=)-14 b Fr(\))46 b Fu(PL1)22 b Fr(`)599 3074 y Fn(0)659 3062 y Fu(\000.)0 3228 y Ft(Corollary)0 3369 y Fu(If)28 b(\006)g(is)f(an)g(axiom)g (system)h(and)f Fo(C)34 b Fu(a)27 b(decen)n(t)h(form)n(ula)e(then)i (the)g(follo)n(wing)f(holds:)0 3510 y(\006)c Fr(j)-14 b Fu(=)23 b Fo(C)52 b Fu(=)-14 b Fr(\))28 b Fu(There)f(are)f Fo(A)893 3522 y Fn(1)931 3510 y Fo(;)14 b(:::;)g(A)1136 3522 y Fp(n)1204 3510 y Fr(2)24 b Fu(\006)j(suc)n(h)h(that)g(PL1)22 b Fr(`)1962 3522 y Fn(0)2022 3510 y Fr(:)p Fo(A)2139 3522 y Fn(1)2176 3510 y Fo(;)14 b(:::;)g Fr(:)p Fo(A)2436 3522 y Fp(n)2482 3510 y Fo(;)g(C)6 b Fu(.)0 3826 y(Pro)r(of)26 b(of)i(Theorem)f(1.2:)0 3992 y(AX)d(:=)j(set)h(of)f(\014nite)h (sequences)f(\()p Fo(A)1170 4004 y Fn(0)1208 3992 y Fo(;)14 b(:::;)g(A)1413 4004 y Fp(l)1439 3992 y Fu(\))28 b(suc)n(h)f(that)h (there)f(is)h(a)f(prime)g(form)n(ula)g Fo(A)h Fu(with)g Fr(f)p Fo(A;)14 b Fr(:)p Fo(A)p Fr(g)23 b(\022)f(f)p Fo(A)3564 4004 y Fn(1)3601 3992 y Fo(;)14 b(:::;)g(A)3806 4004 y Fp(l)3832 3992 y Fr(g)p Fu(.)0 4158 y(Let)28 b(\005)g(b)r(e)g(a) f(\014nite)h(sequence)f(of)h(decen)n(t)f(form)n(ulas)g(.)0 4324 y Fo(t)30 4336 y Fn(0)67 4324 y Fo(;)14 b(t)134 4336 y Fn(1)171 4324 y Fo(;)g(:::)p Fu(:)37 b(en)n(umeration)27 b(of)h(T)-7 b(er.)0 4490 y Fo(\026;)14 b(\027)33 b Fu(are)26 b(ranging)h(o)n(v)n(er)e(\014nite)k(0-1-sequences)c(\(elemen)n(ts)i(of) h Fr(f)p Fu(0)p Fo(;)14 b Fu(1)p Fr(g)2176 4460 y Fp()f Fu(0)k(while)h(all)f(form)n(ulas)g(in)g(\005)2305 194 y Fi(0)2357 224 y Fu(are)f(literals:)0 365 y(3.1.)36 b Fo(A)23 b Fu(=)g Fo(A)401 377 y Fn(0)457 365 y Fr(^)c Fo(A)593 377 y Fn(1)630 365 y Fu(:)37 b(\005)752 380 y Fp(\027)t Fi(\003)q(h)p Fp(i)p Fi(i)928 365 y Fu(:=)23 b(\005)1101 335 y Fi(0)1124 365 y Fo(;)14 b(A)1223 377 y Fp(i)1251 365 y Fo(;)g Fu(\005)1350 335 y Fi(00)1393 365 y Fu(,)0 506 y(3.2.)36 b Fo(A)23 b Fu(=)g Fo(A)401 518 y Fn(0)457 506 y Fr(_)c Fo(A)593 518 y Fn(1)630 506 y Fu(:)37 b(\005)752 521 y Fp(\027)t Fi(\003)q(h)p Fp(i)p Fi(i)928 506 y Fu(:=)23 b(\005)1101 476 y Fi(0)1124 506 y Fo(;)14 b(A)1223 518 y Fn(0)1261 506 y Fo(;)g(A)1360 518 y Fn(1)1397 506 y Fo(;)g Fu(\005)1496 476 y Fi(00)1539 506 y Fu(,)0 648 y(3.3.)36 b Fo(A)23 b Fu(=)g Fr(8)p Fo(xB)t Fu(:)37 b(\005)622 663 y Fp(\027)t Fi(\003h)p Fp(i)p Fi(i)798 648 y Fu(:=)22 b(\005)970 617 y Fi(0)994 648 y Fo(;)14 b(B)1094 660 y Fp(x)1135 648 y Fu(\()p Fo(u)p Fu(\))p Fo(;)g Fu(\005)1346 617 y Fi(00)1389 648 y Fu(,)28 b(where)f Fo(u)g Fu(is)h(the)g(\014rst)f(v)-5 b(ariable)27 b(in)g Fk(FV)19 b Fr(n)f Fu(FV)r(\(\005)2954 660 y Fp(\027)2996 648 y Fu(\),)0 789 y(3.4.)36 b Fo(A)23 b Fu(=)g Fr(9)p Fo(xB)t Fu(:)38 b(\005)622 804 y Fp(\027)t Fi(\003h)p Fp(i)p Fi(i)798 789 y Fu(:=)22 b(\005)970 759 y Fi(0)994 789 y Fo(;)14 b(B)1094 801 y Fp(x)1135 789 y Fu(\()p Fo(t)1197 801 y Fp(k)1239 789 y Fu(\))p Fo(;)g Fu(\005)1370 759 y Fi(00)1412 789 y Fo(;)g(A)p Fu(,)28 b(where)f Fo(k)k Fu(is)c(minimal)h(s.t.)g(\()14 b Fo(x)24 b Fr(2)f Fu(FV)r(\()p Fo(B)t Fu(\))h Fr(\))f(8)p Fo(\026)f Fr(v)g Fo(\027)e(B)3325 801 y Fp(x)3366 789 y Fu(\()p Fo(t)3428 801 y Fp(k)3470 789 y Fu(\))j Fr(62)g Fu(\005)3665 801 y Fp(\026)3724 789 y Fu(\).)0 1005 y Ft(Assumption:)0 1146 y Fu(\()p Fo(i)61 1158 y Fp(n)106 1146 y Fu(\))138 1158 y Fp(n)p Fi(2)p Fn(I)-11 b(N)318 1146 y Fu(is)27 b(a)g(0-1-sequence)f(suc)n(h)h(that)56 b Fr(8)p Fo(n)22 b Fr(2)h Fu(I)-14 b(N\()14 b(\005)1731 1161 y Fi(h)p Fp(i)1781 1169 y Fg(0)1815 1161 y Fp(;:::;i)1938 1169 y Fh(n)p Ff(\000)p Fg(1)2051 1161 y Fi(i)2105 1146 y Fr(62)23 b Fu(AX)14 b(\).)0 1312 y(Abbreviation:)36 b Fo(\027)5 b Fu(\()p Fo(n)p Fu(\))24 b(:=)f Fr(h)p Fo(i)891 1324 y Fn(0)928 1312 y Fo(;)14 b(:::;)g(i)1100 1324 y Fp(n)p Fi(\000)p Fn(1)1230 1312 y Fr(i)p Fu(,)55 b(\010)23 b(:=)1534 1249 y Fl(S)1603 1337 y Fp(n)p Fi(2)p Fn(I)-11 b(N)1769 1312 y Fu(\005)1831 1327 y Fp(\027)t Fn(\()p Fp(n)p Fn(\))0 1528 y Fu(De\014nition:)38 b(If)28 b(not)f(all)g(form)n (ulas)g(in)h(\005)1261 1540 y Fp(\027)1330 1528 y Fu(are)f(literals)f (let)i(dp)q(\(\005)2048 1540 y Fp(\027)2089 1528 y Fu(\))g(b)r(e)g(the) g(\014rst)g(form)n(ula)e(in)i(\005)3040 1540 y Fp(\027)3109 1528 y Fu(whic)n(h)g(is)g(not)f(a)g(literal.)0 1743 y(Prop)r(osition)f (1)0 1885 y Fo(A)d Fr(2)h Fu(\005)226 1900 y Fp(\027)t Fn(\()p Fp(n)p Fn(\))383 1885 y Fu(&)f(rk)o(\()p Fo(A)p Fu(\))h Fo(>)f Fu(0)45 b(=)-14 b Fr(\))46 b(9)p Fo(k)26 b Fr(\025)d Fo(n)p Fu(\()14 b(dp\(\005)1537 1900 y Fp(\027)t Fn(\()p Fp(k)q Fn(\))1668 1885 y Fu(\))23 b(=)g Fo(A)14 b Fu(\))0 2051 y(Prop)r(osition)26 b(2)0 2192 y(a\))h(rk\()p Fo(A)p Fu(\))d(=)e(0)46 b(=)-14 b Fr(\))46 b Fo(A)23 b Fr(62)h Fu(\010)18 b Fr(_)h(:)p Fo(A)k Fr(62)h Fu(\010,)0 2333 y(b\))k Fo(A)168 2345 y Fn(0)206 2333 y Fr(^)p Fo(A)323 2345 y Fn(1)384 2333 y Fr(2)23 b Fu(\010)46 b(=)-14 b Fr(\))46 b Fo(A)810 2345 y Fn(0)870 2333 y Fr(2)24 b Fu(\010)k(or)e Fo(A)1200 2345 y Fn(1)1261 2333 y Fr(2)d Fu(\010,)0 2474 y(c\))28 b Fo(A)159 2486 y Fn(0)196 2474 y Fr(_)q Fo(A)314 2486 y Fn(1)374 2474 y Fr(2)c Fu(\010)46 b(=)-14 b Fr(\))46 b Fo(A)801 2486 y Fn(0)861 2474 y Fr(2)24 b Fu(\010)j(and)h Fo(A)1251 2486 y Fn(1)1311 2474 y Fr(2)c Fu(\010,)0 2615 y(d\))k Fr(8)p Fo(xB)f Fr(2)c Fu(\010)46 b(=)-14 b Fr(\))46 b(9)p Fo(u)23 b Fr(2)g Fk(FV)q Fu(\()14 b Fo(B)1061 2627 y Fp(x)1103 2615 y Fu(\()p Fo(u)p Fu(\))23 b Fr(2)g Fu(\010)14 b(\),)0 2756 y(e\))28 b Fr(9)p Fo(xB)g Fr(2)23 b Fu(\010)46 b(=)-14 b Fr(\))46 b(8)p Fo(t)22 b Fr(2)i Fu(T)-7 b(er)o(\()14 b Fo(B)1054 2768 y Fp(x)1096 2756 y Fu(\()p Fo(t)p Fu(\))24 b Fr(2)f Fu(\010)14 b(\).)0 2922 y(Pro)r(of:)0 3064 y(a\))27 b(rk\()p Fo(A)p Fu(\))d(=)e(0)h(&)g Fo(A)g Fr(2)g Fu(\005)793 3079 y Fp(\027)t Fn(\()p Fp(n)p Fn(\))951 3064 y Fu(&)f Fr(:)p Fo(A)i Fr(2)g Fu(\005)1320 3079 y Fp(\027)t Fn(\()p Fp(m)p Fn(\))1518 3064 y Fr(\))46 b(8)p Fo(k)25 b Fr(\025)e Fu(max)o Fr(f)p Fo(m;)14 b(n)p Fr(g)p Fu(\()p Fo(A;)g Fr(:)p Fo(A)23 b Fr(2)h Fu(\005)2660 3079 y Fp(\027)t Fn(\()p Fp(k)q Fn(\))2790 3064 y Fu(\).)0 3230 y(Assume)k(that)g Fo(A)23 b Fr(2)g Fu(\010)28 b(with)g(rk)o(\()p Fo(A)p Fu(\))c Fo(>)f Fu(0.)36 b(Then)28 b Fo(A)23 b Fu(=)g(dp\(\005)1919 3245 y Fp(\027)t Fn(\()p Fp(k)q Fn(\))2050 3230 y Fu(\))28 b(for)f(some)g Fo(k)s Fu(.)0 3396 y(b\))h Fo(A)23 b Fu(=)g Fo(A)341 3408 y Fn(0)379 3396 y Fr(^)p Fo(A)496 3408 y Fn(1)533 3396 y Fu(:)37 b(Then)28 b Fo(A)872 3408 y Fp(i)923 3396 y Fr(2)c Fu(\005)1064 3411 y Fp(\027)t Fn(\()p Fp(k)q Fn(\))p Fi(\003)q(h)p Fp(i)p Fi(i)1333 3396 y Fu(for)j Fo(i)22 b Fu(=)h(0)p Fo(;)14 b Fu(1,)27 b(and)g(therefore)g Fo(A)2342 3408 y Fn(0)2402 3396 y Fr(2)d Fu(\005)2543 3411 y Fp(\027)t Fn(\()p Fp(k)q Fn(+1\))2785 3396 y Fu(or)i Fo(A)2948 3408 y Fn(1)3009 3396 y Fr(2)d Fu(\005)3149 3411 y Fp(\027)t Fn(\()p Fp(k)q Fn(+1\))3364 3396 y Fu(.)0 3537 y(c\))28 b Fo(A)23 b Fu(=)g Fo(A)332 3549 y Fn(0)369 3537 y Fr(_)q Fo(A)487 3549 y Fn(1)524 3537 y Fu(:)37 b(Then)28 b Fo(A)863 3549 y Fn(0)900 3537 y Fo(;)14 b(A)999 3549 y Fn(1)1060 3537 y Fr(2)23 b Fu(\005)1200 3552 y Fp(\027)t Fn(\()p Fp(k)q Fn(\))p Fi(\003)q(h)p Fp(i)p Fi(i)1469 3537 y Fu(for)k Fo(i)c Fu(=)g(0)p Fo(;)14 b Fu(1,)26 b(and)i(therefore)f Fo(A)2479 3549 y Fn(0)2539 3537 y Fr(2)d Fu(\005)2680 3552 y Fp(\027)t Fn(\()p Fp(k)q Fn(+1\))2921 3537 y Fu(and)k Fo(A)3145 3549 y Fn(1)3205 3537 y Fr(2)c Fu(\005)3346 3552 y Fp(\027)t Fn(\()p Fp(k)q Fn(+1\))3560 3537 y Fu(.)0 3678 y(d\))k Fo(A)23 b Fu(=)g Fr(8)p Fo(xB)t Fu(:)37 b(Then)27 b Fo(B)779 3690 y Fp(x)821 3678 y Fu(\()p Fo(u)p Fu(\))c Fr(2)h Fu(\005)1097 3693 y Fp(\027)t Fn(\()p Fp(k)q Fn(\))p Fi(\003)q(h)p Fp(i)p Fi(i)1366 3678 y Fu(for)j(some)g Fo(u)c Fr(2)g Fk(V)n(a)n(r)1969 3690 y Fn(0)2006 3678 y Fu(.)0 3819 y(e\))28 b Fo(A)23 b Fu(=)g Fr(9)p Fo(xB)t Fu(:)37 b(Then)28 b Fr(8)p Fo(n)22 b Fr(\025)h Fo(k)957 3831 y Fn(0)994 3819 y Fu(\()14 b Fr(9)p Fo(xB)28 b Fr(2)c Fu(\005)1365 3834 y Fp(\027)t Fn(\()p Fp(n)p Fn(\))1513 3819 y Fu(\))k(\()p Fr(\003)p Fu(\))g(.)0 3960 y(By)f(induction)h(on)g Fo(m)f Fu(w)n(e)g(pro)n(v)n(e) f Fo(B)1124 3972 y Fp(x)1166 3960 y Fu(\()p Fo(t)1228 3972 y Fp(m)1291 3960 y Fu(\))e Fr(2)f Fu(\010:)0 4101 y(Assume)h(that)g Fo(B)t Fu(\()p Fo(t)609 4113 y Fp(i)637 4101 y Fu(\))g Fr(2)f Fu(\010)h(for)g(all)f Fo(i)g(<)g(m)p Fu(.)35 b(By)24 b(Prop)r(osition)e(1)i(and)g(\()p Fr(\003)p Fu(\))g(there)f(is)h(an)g Fo(n)f Fr(\025)f Fo(k)27 b Fu(suc)n(h)d(that)g(dp)q(\(\005)3462 4116 y Fp(\027)t Fn(\()p Fp(n)p Fn(\))3596 4101 y Fu(\))g(=)e Fr(9)p Fo(xB)0 4242 y Fu(and)27 b Fr(8)p Fo(i)22 b(<)h(m)p Fr(9)p Fo(j)28 b Fr(\024)23 b Fo(n)p Fu(\()14 b Fo(B)t Fu(\()p Fo(t)841 4254 y Fp(i)869 4242 y Fu(\))23 b Fr(2)g Fu(\005)1064 4257 y Fp(\027)t Fn(\()p Fp(j)s Fn(\))1202 4242 y Fu(\))28 b(\()p Fr(\003\003)p Fu(\).)0 4384 y(By)f(de\014nition)h(of)g(\005)656 4399 y Fp(\027)t Fn(\()p Fp(n)p Fn(+1\))902 4384 y Fu(w)n(e)f(get)1163 4316 y Fl(\000)1215 4384 y Fr(8)p Fo(j)g Fr(\024)22 b Fo(n)p Fu(\()14 b Fo(B)t Fu(\()p Fo(t)1635 4396 y Fp(m)1699 4384 y Fu(\))23 b Fr(62)h Fu(\005)1895 4399 y Fp(\027)t Fn(\()p Fp(j)s Fn(\))2033 4384 y Fu(\))f Fr(\))g Fo(B)t Fu(\()p Fo(t)2323 4396 y Fp(m)2387 4384 y Fu(\))g Fr(2)g Fu(\005)2582 4399 y Fp(\027)t Fn(\()p Fp(n)p Fn(+1\))2815 4316 y Fl(\001)2880 4384 y Fu(and)28 b(th)n(us)g Fo(B)t Fu(\()p Fo(t)3354 4396 y Fp(m)3417 4384 y Fu(\))23 b Fr(2)h Fu(\010.)0 4575 y Fr(jMj)f Fu(:=)g(T)-7 b(er)o(,)55 b Fo(f)530 4544 y Fi(M)612 4575 y Fu(\()p Fo(s)683 4587 y Fn(1)721 4575 y Fo(;)14 b(:::;)g(s)903 4587 y Fp(n)948 4575 y Fu(\))23 b(:=)g Fo(f)9 b(s)1203 4587 y Fn(1)1240 4575 y Fo(:::s)1348 4587 y Fp(n)1393 4575 y Fu(,)55 b Fo(p)1513 4544 y Fi(M)1595 4575 y Fu(\()p Fo(s)1666 4587 y Fn(1)1704 4575 y Fo(;)14 b(:::;)g(s)1886 4587 y Fp(n)1931 4575 y Fu(\))23 b(:)p Fr(,)g Fo(ps)2196 4587 y Fn(1)2233 4575 y Fo(;)14 b(:::;)g(s)2415 4587 y Fp(n)2483 4575 y Fr(62)24 b Fu(\010)0 4741 y(Let)k Fo(\030)f Fu(:)c Fk(V)n(a)n(r)g Fr(!)g Fu(T)-7 b(er)27 b(suc)n(h)h(that)f Fo(\030)t Fu(\()p Fo(x)p Fu(\))e(=)d Fo(x)28 b Fu(for)g Fo(x)23 b Fr(2)g Fk(FV)q Fu(.)0 4932 y(Prop)r(osition)0 5073 y(a\))k Fo(t)131 5043 y Fi(M)214 5073 y Fu([)p Fo(\030)t Fu(])c(=)g Fo(t)k Fu(for)g(eac)n(h)g Fo(t)c Fr(2)h Fu(T)-7 b(er)o(,)0 5214 y(b\))28 b Fo(A)23 b Fr(2)h Fu(\010)46 b Fr(\))g(M)23 b(6j)-14 b Fu(=)23 b Fo(A)p Fu([)p Fo(\030)t Fu(].)1929 5539 y(5)p eop %%Page: 6 7 6 6 bop 0 83 a Fu(Pro)r(of)26 b(of)i(b\))g(b)n(y)f(induction)h(on)g(rk) o(\()p Fo(A)p Fu(\):)0 224 y Fr(8)p Fo(xB)f Fr(2)c Fu(\010)46 b Fr(\))g Fo(B)560 236 y Fp(x)602 224 y Fu(\()p Fo(u)p Fu(\))23 b Fr(62)h Fu(\010)j(for)g(some)g Fo(u)c Fr(2)g Fk(FV)1541 177 y Fn(IH)1536 224 y Fr(\))46 b(M)23 b(6j)-14 b Fu(=)23 b Fo(B)1948 236 y Fp(x)1990 224 y Fu(\()p Fo(u)p Fu(\)[)p Fo(\030)t Fu(])46 b Fr(\))g(M)23 b(6j)-14 b Fu(=)23 b(\()p Fr(8)p Fo(xB)t Fu(\)[)p Fo(\030)t Fu(].)0 390 y Fr(9)p Fo(xB)45 b Fr(2)40 b Fu(\010)80 b Fr(\))f Fo(B)661 402 y Fp(x)703 390 y Fu(\()p Fo(t)p Fu(\))41 b Fr(2)f Fu(\010)e(for)f(all)g Fo(t)j Fr(2)g Fu(T)-7 b(er)1666 343 y Fn(IH)1661 390 y Fr(\))79 b(M)40 b(6j)-14 b Fu(=)40 b Fo(B)2140 402 y Fp(x)2181 390 y Fu(\()p Fo(t)p Fu(\)[)p Fo(\030)t Fu(])f(for)e(all)h Fo(t)h Fr(2)i Fu(T)-7 b(er)3042 336 y Fn(a\))3030 390 y Fr(\))80 b(M)39 b(6j)-14 b Fu(=)40 b Fo(B)t Fu([)p Fo(\030)3576 360 y Fp(t)3572 411 y(x)3614 390 y Fu(])e(for)f(all)0 531 y Fo(t)23 b Fr(2)g Fu(T)-7 b(er)46 b Fr(\))g(M)23 b(6j)-14 b Fu(=)23 b(\()p Fr(9)p Fo(xB)t Fu(\)[)p Fo(\030)t Fu(].)0 747 y(\005)g Fr(\022)g Fu(\010)g(&)g Fr(M)f(6j)-14 b Fu(=)23 b Fo(A)p Fu([)p Fo(\030)t Fu(])28 b(for)g(eac)n(h)e Fo(A)e Fr(2)f Fu(\010)46 b(=)-14 b Fr(\))23 b(6j)-14 b Fu(=)23 b(\005.)0 913 y(Hence)28 b(the)g(assumption)f(w)n(as)f(false.)0 1079 y(By)h(K\177)-42 b(onigs)27 b(Lemma)g(\(and)h(since)f(\005)1163 1091 y Fp(\027)1228 1079 y Fr(2)c Fu(AX)28 b(implies)g(\005)1802 1094 y Fp(\027)t Fi(\003h)p Fp(i)p Fi(i)1978 1079 y Fr(2)23 b Fu(AX)q(\))28 b(the)g(set)f Fr(f)p Fo(\027)h Fu(:)23 b(\005)2732 1091 y Fp(\027)2797 1079 y Fr(62)g Fu(AX)q Fr(g)k Fu(is)h(\014nite.)0 1220 y(Let)g Fo(m)23 b Fu(:=)f(max)p Fr(f)p Fo(l)r(h)p Fu(\()p Fo(\027)5 b Fu(\))23 b(:)g(\005)868 1232 y Fp(\027)932 1220 y Fr(62)h Fu(AX)p Fr(g)18 b Fu(+)g(1.)0 1386 y(By)27 b(recursion)f(on)i Fo(m)p Fr(\000)694 1370 y(\001)744 1386 y Fo(l)r(h)p Fu(\()p Fo(\027)5 b Fu(\))28 b(w)n(e)f(de\014ne)h(a)f(PL1)o(-deriv)-5 b(ation)26 b Fo(d)1998 1398 y Fp(\027)2068 1386 y Fu(with)i Fo(d)2300 1398 y Fp(\027)2364 1386 y Fr(`)2415 1398 y Fn(0)2475 1386 y Fu(\005)2537 1398 y Fp(\027)2579 1386 y Fu(:)0 1553 y(1.)36 b(\005)163 1565 y Fp(\027)228 1553 y Fr(2)24 b Fu(AX:)37 b(Then)28 b Fo(d)751 1565 y Fp(\027)815 1553 y Fu(:=)23 b(LogAx)1167 1573 y Fp(A)1249 1553 y Fu(with)28 b(suitable)f Fo(A)p Fu(.)0 1719 y(2.)36 b(\005)163 1731 y Fp(\027)228 1719 y Fr(62)24 b Fu(AX:)37 b(Then)28 b Fo(l)r(h)p Fu(\()p Fo(\027)5 b Fu(\))23 b Fo(<)g(m)k Fu(and)h(th)n(us)f Fo(m)p Fr(\000)1540 1702 y(\001)1590 1719 y Fo(l)r(h)p Fu(\()p Fo(\027)5 b Fr(\003h)p Fo(i)p Fr(i)p Fu(\))23 b Fo(<)g(m)p Fr(\000)2113 1702 y(\001)2163 1719 y Fo(l)r(h)p Fu(\()p Fo(\027)5 b Fu(\).)0 1860 y(Hence)28 b(b)n(y)f(IH)56 b(\(*\))27 b Fo(d)686 1875 y Fp(\027)t Fi(\003)q(h)p Fp(i)p Fi(i)862 1860 y Fr(`)913 1872 y Fn(0)973 1860 y Fu(\005)1035 1875 y Fp(\027)t Fi(\003)q(h)p Fp(i)p Fi(i)1216 1860 y Fu(for)g Fo(i)c Fu(=)f(0)p Fo(;)14 b Fu(1.)0 2026 y(No)n(w)27 b(not)h(all)f(form)n(ulas)f Fo(A)e Fr(2)f Fu(\005)1015 2038 y Fp(\027)1084 2026 y Fu(are)k(literals,)g(since)g(otherwise)g(\005)2154 2038 y Fp(\027)t Fi(\003)p Fp(\026)2293 2026 y Fu(=)c(\005)2443 2038 y Fp(\027)2507 2026 y Fr(62)h Fu(AX)k(for)f(all)g Fo(\026)p Fu(.)0 2167 y(Hence)h(one)f(of)h(the)g(follo)n(wing)e(cases)h (holds:)0 2333 y(2.1.)64 b(\005)256 2345 y Fp(\027)320 2333 y Fu(=)23 b(\005)470 2303 y Fi(0)494 2333 y Fo(;)14 b(A)593 2345 y Fn(0)630 2333 y Fr(^)p Fo(A)747 2345 y Fn(1)785 2333 y Fo(;)g Fu(\005)884 2303 y Fi(00)954 2333 y Fu(and)28 b(\005)1178 2348 y Fp(\027)t Fi(\003h)p Fp(i)p Fi(i)1353 2333 y Fu(=)23 b(\005)1503 2303 y Fi(0)1527 2333 y Fo(;)14 b(A)1626 2345 y Fp(i)1653 2333 y Fo(;)g Fu(\005)1752 2303 y Fi(00)1795 2333 y Fu(:)37 b Fo(d)1898 2345 y Fp(\027)1963 2333 y Fu(:=)2073 2271 y Fl(V)2142 2358 y Fp(A)2192 2366 y Fg(0)2225 2358 y Fi(^)p Fp(A)2320 2366 y Fg(1)2356 2333 y Fo(d)2399 2348 y Fp(\027)t Fi(\003h)p Fn(0)p Fi(i)2561 2333 y Fo(d)2604 2348 y Fp(\027)t Fi(\003)q(h)p Fn(1)p Fi(i)2767 2333 y Fu(.)0 2474 y(2.2.)64 b(\005)256 2486 y Fp(\027)320 2474 y Fu(=)23 b(\005)470 2444 y Fi(0)494 2474 y Fo(;)14 b(A)593 2486 y Fn(0)630 2474 y Fr(_)p Fo(A)747 2486 y Fn(1)785 2474 y Fo(;)g Fu(\005)884 2444 y Fi(00)954 2474 y Fu(and)28 b(\005)1178 2489 y Fp(\027)t Fi(\003h)p Fn(0)p Fi(i)1363 2474 y Fu(=)23 b(\005)1513 2444 y Fi(0)1536 2474 y Fo(;)14 b(A)1635 2486 y Fn(0)1673 2474 y Fo(;)g(A)1772 2486 y Fn(1)1809 2474 y Fo(;)g Fu(\005)1908 2444 y Fi(00)1951 2474 y Fu(:)37 b Fo(d)2054 2486 y Fp(\027)2118 2474 y Fu(:=)2229 2412 y Fl(W)2298 2432 y Fn(1)2298 2499 y Fp(A)2348 2507 y Fg(0)2380 2499 y Fi(_)p Fp(A)2475 2507 y Fg(1)2512 2412 y Fl(W)2581 2432 y Fn(0)2581 2499 y Fp(A)2631 2507 y Fg(0)2663 2499 y Fi(_)p Fp(A)2758 2507 y Fg(1)2794 2474 y Fo(d)2837 2489 y Fp(\027)t Fi(\003)q(h)p Fn(0)p Fi(i)3000 2474 y Fu(.)0 2615 y(2.3.)64 b(\005)256 2627 y Fp(\027)320 2615 y Fu(=)23 b(\005)470 2585 y Fi(0)494 2615 y Fo(;)14 b Fr(8)p Fo(xB)t(;)g Fu(\005)791 2585 y Fi(00)860 2615 y Fu(and)28 b(\005)1084 2630 y Fp(\027)t Fi(\003h)p Fn(0)p Fi(i)1269 2615 y Fu(=)23 b(\005)1419 2585 y Fi(0)1443 2615 y Fo(;)14 b(B)1543 2627 y Fp(x)1584 2615 y Fu(\()p Fo(u)p Fu(\))p Fo(;)g Fu(\005)1795 2585 y Fi(00)1866 2615 y Fu(with)28 b Fo(u)22 b Fr(2)i Fk(FV)19 b Fr(n)f Fu(FV)q(\(\005)2596 2627 y Fp(\027)2638 2615 y Fu(\):)37 b Fo(d)2773 2627 y Fp(\027)2838 2615 y Fu(:=)2949 2553 y Fl(V)3018 2573 y Fp(u)3018 2640 y Fi(8)p Fp(xB)3150 2615 y Fo(d)3193 2630 y Fp(\027)t Fi(\003)q(h)p Fn(0)p Fi(i)3356 2615 y Fu(.)0 2756 y(2.4.)64 b(\005)256 2768 y Fp(\027)320 2756 y Fu(=)23 b(\005)470 2726 y Fi(0)494 2756 y Fo(;)14 b Fr(9)p Fo(xB)t(;)g Fu(\005)790 2726 y Fi(00)860 2756 y Fu(and)28 b(\005)1084 2771 y Fp(\027)t Fi(\003h)p Fn(0)p Fi(i)1269 2756 y Fu(=)23 b(\005)1419 2726 y Fi(0)1443 2756 y Fo(;)14 b(B)1543 2768 y Fp(x)1584 2756 y Fu(\()p Fo(t)p Fu(\))p Fo(;)g Fu(\005)1777 2726 y Fi(00)1848 2756 y Fu(with)28 b Fo(t)23 b Fr(2)h Fu(T)-7 b(er)o(:)37 b Fo(d)2394 2768 y Fp(\027)2459 2756 y Fu(:=)2569 2694 y Fl(W)2638 2715 y Fp(t)2638 2781 y Fi(9)p Fp(xB)2771 2756 y Fo(d)2814 2771 y Fp(\027)t Fi(\003h)p Fn(0)p Fi(i)2976 2756 y Fu(.)1929 5539 y(6)p eop %%Page: 7 8 7 7 bop 0 83 a Fr(x)p Ft(2)63 b(Cut-Elimination)0 294 y(Substitution)0 458 y(De\014nition)0 598 y Fu(A)28 b(pro)r(of)f (system)g Fj(S)h Fu(is)f Fm(close)l(d)k(under)f(substitution)c Fu(i\013)i(it)h(satis\014es)d(the)i(follo)n(wing)f(conditions:)0 738 y(\(I\))88 b(F)-7 b(or)28 b(eac)n(h)g Fo(x)e Fr(2)f Fk(FV)q Fu(,)k Fo(t)c Fr(2)g Fu(T)-7 b(er)29 b(and)f Fr(I)j(2)26 b Fj(S)i Fu(with)i(Eig\()p Fr(I)6 b Fu(\))20 b Fr(\\)g Fu(\(FV)r(\()p Fo(t)p Fu(\))f Fr([)h(f)p Fo(x)p Fr(g)p Fu(\))25 b(=)f Fr(;)29 b Fu(an)f(inference)h(sym)n(b)r(ol)f Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\))26 b Fr(2)f Fj(S)182 878 y Fu(is)j(de\014ned)g(suc)n(h)f(that)182 1018 y(\(a\))h Fr(jI)6 b Fu(\()p Fo(x=t)p Fu(\))p Fr(j)24 b Fu(=)f Fr(jI)6 b(j)28 b Fu(and)f(Eig\()p Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\)\))25 b(=)e(Eig\()p Fr(I)6 b Fu(\),)182 1158 y(\(b\))29 b(\001\()p Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\)\))24 b(=)f(\001\()p Fr(I)6 b Fu(\))984 1170 y Fp(x)1027 1158 y Fu(\()p Fo(t)p Fu(\),)182 1298 y(\(c\))28 b(\()p Fr(8)p Fo(i)22 b Fr(2)i(jI)6 b(j)p Fu(\))14 b(\001)732 1310 y Fp(i)760 1298 y Fu(\()p Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\)\))25 b(=)d(\001)1239 1310 y Fp(i)1267 1298 y Fu(\()p Fr(I)6 b Fu(\))1382 1310 y Fp(x)1425 1298 y Fu(\()p Fo(t)p Fu(\),)182 1438 y(\(d\))29 b(deg\()p Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\)\))25 b(=)d(deg)q(\()p Fr(I)6 b Fu(\).)0 1602 y(\(I)r(I\))56 b(F)-7 b(or)27 b Fr(I)j(2)23 b Fj(S)k Fu(with)i(Eig\()p Fr(I)6 b Fu(\))24 b Fr(6)p Fu(=)e Fr(;)28 b Fu(and)f Fo(u)c Fr(2)g Fk(FV)c Fr(n)f Fu(FV)r(\(\001\()p Fr(I)6 b Fu(\)\))29 b(an)e(inference)h(sym)n (b)r(ol)f Fr(I)2873 1572 y Fp(u)2940 1602 y Fr(2)c Fj(S)28 b Fu(is)f(de\014ned)h(suc)n(h)f(that)182 1742 y(\(a\))h Fr(jI)390 1712 y Fp(u)434 1742 y Fr(j)23 b Fu(=)g Fr(jI)6 b(j)28 b Fu(and)f(Eig\()p Fr(I)1059 1712 y Fp(u)1103 1742 y Fu(\))d(=)e Fr(f)p Fo(u)p Fr(g)p Fu(,)182 1882 y(\(b\))29 b(\001\()p Fr(I)473 1852 y Fp(u)517 1882 y Fu(\))23 b(=)g(\001\()p Fr(I)6 b Fu(\),)182 2022 y(\(c\))28 b(\()p Fr(8)p Fo(i)22 b Fr(2)i(jI)6 b(j)p Fu(\))14 b(\001)732 2034 y Fp(i)760 2022 y Fu(\()p Fr(I)843 1992 y Fp(u)887 2022 y Fu(\))24 b(=)e(\001)1099 2034 y Fp(i)1127 2022 y Fu(\()p Fr(I)6 b Fu(\))1242 2034 y Fp(y)1283 2022 y Fu(\()p Fo(u)p Fu(\),)28 b(where)f(Eig\()p Fr(I)6 b Fu(\))24 b(=)f Fr(f)p Fo(y)s Fr(g)p Fu(,)182 2162 y(\(d\))29 b(deg\()p Fr(I)529 2132 y Fp(u)573 2162 y Fu(\))24 b(=)e(deg)q(\()p Fr(I)6 b Fu(\).)0 2373 y Ft(De\014nition)0 2513 y Fu(LogAx)241 2534 y Fp(A)295 2513 y Fu(\()p Fo(x=t)p Fu(\))24 b(:=)e(LogAx)853 2534 y Fp(A)903 2542 y Fh(x)940 2534 y Fn(\()p Fp(t)p Fn(\))1049 2513 y Fu(,)1100 2451 y Fl(V)1169 2538 y Fp(A)1223 2513 y Fu(\()p Fo(x=t)p Fu(\))i(:=)1540 2451 y Fl(V)1610 2538 y Fp(A)1660 2546 y Fh(x)1696 2538 y Fn(\()p Fp(t)p Fn(\))1805 2513 y Fu(,)1856 2451 y Fl(W)1925 2472 y Fp(k)1925 2538 y(A)1979 2513 y Fu(\()p Fo(x=t)p Fu(\))g(:=)2297 2451 y Fl(W)2366 2472 y Fp(k)2366 2538 y(A)2416 2546 y Fh(x)2453 2538 y Fn(\()p Fp(t)p Fn(\))2561 2513 y Fu(,)k Fk(Cut)2738 2525 y Fp(C)2794 2513 y Fu(\()p Fo(x=t)p Fu(\))c(:=)e Fk(Cut)3237 2528 y Fp(C)3285 2536 y Fh(x)3322 2528 y Fn(\()p Fp(t)p Fn(\))3431 2513 y Fu(,)0 2615 y Fl(W)69 2635 y Fp(s)69 2702 y Fi(9)p Fp(z)r(A)195 2677 y Fu(\()p Fo(x=t)p Fu(\))i(:=)512 2615 y Fl(W)581 2634 y Fp(s)612 2642 y Fh(x)650 2634 y Fn(\()p Fp(t)p Fn(\))581 2705 y(\()p Fi(9)p Fp(z)r(A)p Fn(\))755 2713 y Fh(x)792 2705 y Fn(\()p Fp(t)p Fn(\))901 2677 y Fu(,)951 2615 y Fl(V)1021 2635 y Fp(y)1021 2702 y Fi(8)p Fp(z)r(A)1146 2677 y Fu(\()p Fo(x=t)p Fu(\))g(:=)1464 2615 y Fl(V)1533 2635 y Fp(y)1533 2704 y Fn(\()p Fi(8)p Fp(z)r(A)p Fn(\))1707 2712 y Fh(x)1743 2704 y Fn(\()p Fp(t)p Fn(\))1852 2677 y Fu(.)120 b(\()2027 2615 y Fl(V)2097 2635 y Fp(y)2097 2702 y Fi(8)p Fp(z)r(A)2222 2677 y Fu(\))2254 2647 y Fp(u)2321 2677 y Fu(:=)2432 2615 y Fl(V)2501 2635 y Fp(u)2501 2702 y Fi(8)p Fp(z)r(A)0 2865 y Ft(Lemma)30 b(2.1)0 3005 y Fu(With)e(the)g(ab)r(o)n(v)n(e)f(de\014nitions)g(PL1)g(is)g(closed)g (under)h(substitution.)0 3169 y(Pro)r(of:)0 3309 y(ad)g(\(I)r(I\):)i (If)f(Eig\()p Fr(I)6 b Fu(\))26 b(=)e Fr(f)p Fo(y)s Fr(g)p Fu(,)k(then)h Fr(I)i Fu(=)1266 3246 y Fl(V)1335 3267 y Fp(y)1335 3333 y Fi(8)p Fp(z)r(A)1489 3309 y Fu(and)e Fr(I)1703 3278 y Fp(u)1771 3309 y Fu(=)1861 3246 y Fl(V)1930 3267 y Fp(u)1930 3333 y Fi(8)p Fp(z)r(A)2056 3309 y Fu(.)40 b(Hence)28 b Fr(jI)2440 3278 y Fp(u)2484 3309 y Fr(j)d Fu(=)f Fr(jI)6 b(j)p Fu(,)30 b(Eig\()p Fr(I)2976 3278 y Fp(u)3020 3309 y Fu(\))25 b(=)g Fr(f)p Fo(u)p Fr(g)p Fu(,)i(\001\()p Fr(I)3501 3278 y Fp(u)3545 3309 y Fu(\))e(=)g(\001\()p Fr(I)6 b Fu(\),)0 3449 y(and)27 b(\001)230 3461 y Fn(0)268 3449 y Fu(\()p Fr(I)351 3418 y Fp(u)395 3449 y Fu(\))c(=)g Fr(f)p Fo(A)642 3461 y Fp(z)680 3449 y Fu(\()p Fo(u)p Fu(\))p Fr(g)857 3394 y Fn(\()p Fi(\003)p Fn(\))867 3449 y Fu(=)34 b Fr(f)p Fo(A)1070 3461 y Fp(z)1108 3449 y Fu(\()p Fo(y)s Fu(\))1216 3461 y Fp(y)1256 3449 y Fu(\()p Fo(u)p Fu(\))p Fr(g)22 b Fu(=)h(\001)1589 3461 y Fn(0)1626 3449 y Fu(\()p Fr(I)6 b Fu(\))1741 3461 y Fp(y)1782 3449 y Fu(\()p Fo(u)p Fu(\).)65 b(\(*\))28 b Fo(y)e Fr(62)d Fu(FV)r(\()p Fr(8)p Fo(z)t(A)p Fu(\).)0 3612 y(ad)k(\(I\):)0 3752 y(1.)36 b Fr(I)30 b Fu(=)264 3690 y Fl(V)333 3711 y Fp(y)333 3777 y Fi(8)p Fp(z)r(A)458 3752 y Fu(:)37 b(Then)28 b Fo(y)e Fr(62)d Fu(FV)r(\()p Fr(8)p Fo(z)t(A)p Fu(\))18 b Fr([)g(f)p Fo(x)p Fr(g)p Fu(.)0 3892 y(One)27 b(easily)g(v)n(eri\014es)f(\(*\))i(\()p Fr(8)p Fo(z)t(A)p Fu(\))1031 3904 y Fp(x)1072 3892 y Fu(\()p Fo(t)p Fu(\))c(=)f Fr(8)p Fo(z)t(A)1430 3862 y Fi(0)1479 3892 y Fu(with)28 b Fo(A)1730 3862 y Fi(0)1730 3913 y Fp(z)1769 3892 y Fu(\()p Fo(y)s Fu(\))23 b(=)g Fo(A)2050 3904 y Fp(z)2088 3892 y Fu(\()p Fo(y)s Fu(\))2196 3904 y Fp(x)2238 3892 y Fu(\()p Fo(t)p Fu(\).)0 4032 y(By)34 b(\(*\))g(w)n(e)f(ha)n(v)n(e)g Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\))35 b(=)970 3970 y Fl(V)1039 3991 y Fp(y)1039 4057 y Fi(8)p Fp(z)r(A)1161 4041 y Ff(0)1188 4032 y Fu(,)g(and)f Fo(y)j Fr(62)d Fu(FV)q(\()p Fr(8)p Fo(z)t(A)1882 4002 y Fi(0)1904 4032 y Fu(\),)i(since)e Fr(8)p Fo(z)t(A)2357 4002 y Fi(0)2413 4032 y Fu(=)f(\()p Fr(8)p Fo(z)t(A)p Fu(\))2727 4044 y Fp(x)2768 4032 y Fu(\()p Fo(t)p Fu(\))i(and)f Fo(y)i Fr(62)e Fu(FV)r(\()p Fr(8)p Fo(z)t(A)p Fu(\))22 b Fr([)h Fu(FV)q(\()p Fo(t)p Fu(\).)0 4172 y(Hence)28 b Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\))28 b(is)g(an)f(inference)h(sym)n(b)r(ol)f(of)g(PL1)o(.)0 4312 y(\(a\),\(b\),\(d\))i(are)e(trivial.)0 4452 y(\(c\))h(\001)198 4464 y Fn(0)236 4452 y Fu(\()p Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\)\))24 b(=)f Fr(f)p Fo(A)750 4422 y Fi(0)750 4473 y Fp(z)788 4452 y Fu(\()p Fo(y)s Fu(\))p Fr(g)961 4398 y Fn(\()p Fi(\003)p Fn(\))971 4452 y Fu(=)34 b Fr(f)p Fo(A)1174 4464 y Fp(z)1212 4452 y Fu(\()p Fo(y)s Fu(\))1320 4464 y Fp(x)1362 4452 y Fu(\()p Fo(t)p Fu(\))p Fr(g)23 b Fu(=)f(\001)1677 4464 y Fn(0)1715 4452 y Fu(\()p Fr(I)6 b Fu(\))1830 4464 y Fp(x)1873 4452 y Fu(\()p Fo(t)p Fu(\))0 4616 y([Pro)r(of)26 b(of)i(\(*\):)37 b(If)28 b Fo(x)c Fr(62)f Fu(FV)r(\()p Fr(8)p Fo(z)t(A)p Fu(\))k(then)h(\()p Fr(8)p Fo(z)t(A)p Fu(\))1509 4628 y Fp(x)1550 4616 y Fu(\()p Fo(t)p Fu(\))23 b(=)g Fr(8)p Fo(z)t(A)j Fu(and)i Fo(A)2157 4628 y Fp(z)2195 4616 y Fu(\()p Fo(y)s Fu(\))23 b(=)g Fo(A)2476 4628 y Fp(z)2515 4616 y Fu(\()p Fo(y)s Fu(\))2623 4628 y Fp(x)2665 4616 y Fu(\()p Fo(t)p Fu(\))28 b(\(note)g(that)g Fo(x)23 b Fr(6)p Fu(=)g Fo(y)s Fu(\).)0 4756 y(If)28 b Fo(x)c Fr(2)f Fu(FV)r(\()p Fr(8)p Fo(z)t(A)p Fu(\))j(then)j(\()p Fr(8)p Fo(z)t(A)p Fu(\))998 4768 y Fp(x)1039 4756 y Fu(\()p Fo(t)p Fu(\))23 b(=)g Fr(8)p Fo(z)t(A)1396 4768 y Fp(x)1436 4756 y Fu(\()p Fo(t)p Fu(\))29 b(and)e Fo(A)1782 4768 y Fp(x)1824 4756 y Fu(\()p Fo(t)p Fu(\))1918 4768 y Fp(z)1957 4756 y Fu(\()p Fo(y)s Fu(\))c(=)g Fo(A)2238 4768 y Fp(z)2276 4756 y Fu(\()p Fo(y)s Fu(\))2384 4768 y Fp(x)2426 4756 y Fu(\()p Fo(t)p Fu(\),)29 b(since)e Fo(z)f Fr(62)e Fu(FV)q(\()p Fo(t)p Fu(\))29 b(and)e Fo(x)d Fr(6)p Fu(=)e Fo(y)s Fu(.])0 4920 y(2.)36 b Fr(I)30 b Fu(=)264 4858 y Fl(W)333 4878 y Fp(s)333 4945 y Fi(9)p Fp(z)r(A)458 4920 y Fu(:)37 b(One)28 b(easily)e(v)n(eri\014es)h(\(*\))h(\()p Fr(9)p Fo(z)t(A)p Fu(\))1549 4932 y Fp(x)1591 4920 y Fu(\()p Fo(t)p Fu(\))23 b(=)g Fr(9)p Fo(z)t(A)1947 4890 y Fi(0)1998 4920 y Fu(with)28 b Fo(A)2249 4890 y Fi(0)2249 4940 y Fp(z)2287 4920 y Fu(\()p Fo(s)2358 4932 y Fp(x)2400 4920 y Fu(\()p Fo(t)p Fu(\)\))c(=)f Fo(A)2700 4932 y Fp(z)2738 4920 y Fu(\()p Fo(s)p Fu(\))2841 4932 y Fp(x)2884 4920 y Fu(\()p Fo(t)p Fu(\).)0 5060 y(Hence)28 b Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\))24 b(=)592 4998 y Fl(W)661 5017 y Fp(s)692 5025 y Fh(x)730 5017 y Fn(\()p Fp(t)p Fn(\))661 5085 y Fi(9)p Fp(z)r(A)783 5068 y Ff(0)839 5060 y Fu(is)j(an)g (inference)h(sym)n(b)r(ol)f(of)h(PL1)n(.)0 5200 y(\(a\),\(b\),\(d\))h (are)e(trivial.)0 5340 y(\(c\))h(\001)198 5352 y Fn(0)236 5340 y Fu(\()p Fr(I)6 b Fu(\()p Fo(x=t)p Fu(\)\))24 b(=)f Fr(f)p Fo(A)750 5310 y Fi(0)750 5361 y Fp(z)788 5340 y Fu(\()p Fo(s)859 5352 y Fp(x)901 5340 y Fu(\()p Fo(t)p Fu(\)\))p Fr(g)1092 5286 y Fn(\()p Fi(\003)p Fn(\))1103 5340 y Fu(=)33 b Fr(f)p Fo(A)1305 5352 y Fp(z)1343 5340 y Fu(\()p Fo(s)p Fu(\))1446 5352 y Fp(x)1488 5340 y Fu(\()p Fo(t)p Fu(\))p Fr(g)24 b Fu(=)e(\001)1804 5352 y Fn(0)1842 5340 y Fu(\()p Fr(I)6 b Fu(\))1957 5352 y Fp(x)1999 5340 y Fu(\()p Fo(t)p Fu(\).)1929 5539 y(7)p eop %%Page: 8 9 8 8 bop 0 83 a Fu([Pro)r(of)34 b(of)h(\(*\):)51 b(If)36 b Fo(x)f Fr(62)h Fu(FV)r(\()p Fr(9)p Fo(z)t(A)p Fu(\))f(then)g(\()p Fr(9)p Fo(z)t(A)p Fu(\))1583 95 y Fp(x)1625 83 y Fu(\()p Fo(t)p Fu(\))h(=)f Fr(9)p Fo(z)t(A)g Fu(and)f Fo(A)2271 95 y Fp(z)2310 83 y Fu(\()p Fo(s)2381 95 y Fp(x)2423 83 y Fu(\()p Fo(t)p Fu(\)\))i(=)f Fo(A)2747 95 y Fp(z)2785 83 y Fu(\()p Fo(s)p Fu(\))2888 95 y Fp(x)2931 83 y Fu(\()p Fo(t)p Fu(\).)59 b(If)35 b Fo(x)h Fr(2)g Fu(FV)q(\()p Fr(9)p Fo(z)t(A)p Fu(\))g(then)0 224 y(\()p Fr(9)p Fo(z)t(A)p Fu(\))215 236 y Fp(x)257 224 y Fu(\()p Fo(t)p Fu(\))24 b(=)e Fr(9)p Fo(z)t(A)613 236 y Fp(x)655 224 y Fu(\()p Fo(t)p Fu(\))28 b(and)f Fo(A)1000 236 y Fp(x)1042 224 y Fu(\()p Fo(t)p Fu(\))1136 236 y Fp(z)1175 224 y Fu(\()p Fo(s)1246 236 y Fp(x)1288 224 y Fu(\()p Fo(t)p Fu(\)\))d(=)f Fo(A)1588 236 y Fp(z)1626 224 y Fu(\()p Fo(s)p Fu(\))1729 236 y Fp(x)1772 224 y Fu(\()p Fo(t)p Fu(\),)28 b(since)f Fo(z)g Fr(62)c Fu(FV)r(\()p Fo(t)p Fu(\))c Fr([)f(f)p Fo(x)p Fr(g)p Fu(.])0 389 y(3.)36 b(Otherwise:)g(trivial.)0 651 y Fq(No)n(w)27 b(let)h Fj(S)f Fq(b)r(e)h(a)g(\014nitary)g(pro)r(of) f(system)h(whic)n(h)f(is)h(closed)e(under)i(substitution)g(and)g (extends)g Fu(PL1)o Fq(.)0 792 y Fu(Let)g Fo(m)222 804 y Fn(0)282 792 y Fr(\024)23 b Fo(!)30 b Fu(suc)n(h)d(that)h(deg)q(\()p Fr(I)6 b Fu(\))24 b Fr(\024)e Fo(m)1244 804 y Fn(0)1309 792 y Fu(and)28 b(rk)o(\()p Fo(A)p Fu(\))c Fo(<)e(m)1857 804 y Fn(0)1922 792 y Fu(for)27 b(all)h Fr(I)h(2)23 b Fj(S)c Fr(n)f Fu(PL1)26 b(and)i Fo(A)23 b Fr(2)g Fu(\001\()p Fr(I)6 b Fu(\).)0 957 y(T)-7 b(o)27 b(simplify)h(the)g(writing)g(w)n(e) f(assume)g(in)g(addition)h(that)g Fr(jI)6 b(j)23 b Fu(=)g Fr(f)p Fu(0)p Fr(g)j Fu(whenev)n(er)h(Eig\()p Fr(I)6 b Fu(\))24 b Fr(6)p Fu(=)f Fr(;)p Fu(.)0 1146 y(F)-7 b(or)27 b(the)h(rest)f(of)h(this)g(section)f Fo(d;)14 b(d)1111 1158 y Fp(i)1166 1146 y Fu(denote)28 b Fj(S)p Fu(-deriv)-5 b(ations.)63 b(o\()p Fo(d)p Fu(\))23 b(:=)g Fo(l)r(g)s Fu(\()p Fo(d)p Fu(\))28 b(.)0 1360 y Ft(De\014nition)j(of)h Fo(d)p Fu(\()p Fo(x=t)p Fu(\))g Ft(for)g Fo(x)24 b Fr(2)f Fk(FV)q Ft(,)31 b Fo(t)23 b Fr(2)h Fu(T)-7 b(er)0 1621 y(F)g(or)27 b Fo(d)c Fu(=)g Fr(I)6 b Fo(d)397 1633 y Fn(0)435 1621 y Fo(:::d)547 1633 y Fp(n)p Fi(\000)p Fn(1)705 1621 y Fu(w)n(e)27 b(set)55 b Fo(d)p Fu(\()p Fo(x=t)p Fu(\))24 b(:=)1345 1425 y Fl(8)1345 1500 y(>)1345 1525 y(<)1345 1674 y(>)1345 1699 y(:)1433 1475 y Fo(d)1020 b Fu(if)28 b(Eig\()p Fr(I)6 b Fu(\))24 b(=)f Fr(f)p Fo(x)p Fr(g)1433 1621 y(I)1484 1590 y Fp(u)1527 1621 y Fu(\()p Fo(x=t)p Fu(\))14 b Fo(d)1767 1633 y Fn(0)1805 1621 y Fu(\()p Fo(y)s(=u)p Fu(\)\()p Fo(x=t)p Fu(\))310 b(if)28 b(Eig\()p Fr(I)6 b Fu(\))24 b(=)f Fr(f)p Fo(y)s Fr(g)j Fu(with)i Fo(y)e Fr(2)d Fu(FV)r(\()p Fo(t)p Fu(\))c Fr(n)f(f)p Fo(x)p Fr(g)1433 1766 y(I)6 b Fu(\()p Fo(x=t)p Fu(\))p Fo(d)1710 1778 y Fn(0)1748 1766 y Fu(\()p Fo(x=t)p Fu(\))14 b Fo(:)g(:)g(:)g(d)2099 1778 y Fp(n)p Fi(\000)p Fn(1)2229 1766 y Fu(\()p Fo(x=t)p Fu(\))84 b(otherwise)0 1906 y(where)27 b Fo(u)g Fu(is)h(the)g(\014rst)f(v)-5 b(ariable)27 b(in)h Fk(FV)19 b Fr(n)f Fu(\(FV)q(\(\000\()p Fo(d)p Fu(\)\))i Fr([)f Fu(FV)r(\()p Fo(t)p Fu(\))g Fr([)f(f)p Fo(x)p Fr(g)p Fu(\),)0 2095 y Ft(Lemma)30 b(2.2)54 b Fo(d)23 b Fr(`)633 2065 y Fp(k)633 2116 y(m)719 2095 y Fu(\000)46 b Fr(\))g Fo(d)p Fu(\()p Fo(x=t)p Fu(\))24 b Fr(`)1247 2065 y Fp(k)1247 2116 y(m)1333 2095 y Fu(\000)1385 2107 y Fp(x)1427 2095 y Fu(\()p Fo(t)p Fu(\).)0 2260 y(Pro)r(of:)0 2401 y(Let)k Fo(x;)14 b(t)28 b Fu(b)r(e)g(\014xed,)f(and)h Fo(d)23 b Fu(=)g Fr(I)6 b Fo(d)1037 2413 y Fn(0)1075 2401 y Fo(:::d)1187 2413 y Fp(n)p Fi(\000)p Fn(1)1317 2401 y Fu(.)37 b(W.l.o.g.)g(\000\()p Fo(d)p Fu(\))23 b(=)g(\000.)37 b({)27 b(W)-7 b(e)28 b(write)f Fo(U)2570 2371 y Fi(0)2621 2401 y Fu(for)g Fo(U)9 b Fu(\()p Fo(x=t)p Fu(\).)0 2566 y(1.)36 b(Eig)q(\()p Fr(I)6 b Fu(\))24 b(=)e Fr(f)p Fo(x)p Fr(g)p Fu(:)37 b Fo(d)684 2536 y Fi(0)730 2566 y Fu(=)23 b Fo(d)28 b Fu(and)f(\000)c(=)g(\000)1265 2578 y Fp(x)1307 2566 y Fu(\()p Fo(t)p Fu(\),)28 b(since)f Fo(x)d Fr(62)f Fu(FV)r(\(\000\).)0 2731 y(2.)36 b(Eig)q(\()p Fr(I)6 b Fu(\))24 b(=)e Fr(f)p Fo(y)s Fr(g)27 b Fu(with)h Fo(y)e Fr(2)d Fu(FV)r(\()p Fo(t)p Fu(\))c Fr(n)f(f)p Fo(x)p Fr(g)p Fu(:)36 b(Let)28 b Fo(e)22 b Fu(:=)h Fr(I)1792 2701 y Fp(u)1836 2731 y Fo(d)1879 2743 y Fn(0)1916 2731 y Fu(\()p Fo(y)s(=u)p Fu(\).)0 2871 y(W)-7 b(e)28 b(ha)n(v)n(e)e Fo(d)377 2883 y Fn(0)438 2871 y Fr(`)489 2841 y Fp(k)q Fi(\000)p Fn(1)489 2892 y Fp(m)638 2871 y Fu(\000)p Fo(;)14 b Fu(\001)796 2883 y Fn(0)833 2871 y Fu(\()p Fr(I)6 b Fu(\),)28 b(and)g(therefore)f(b)n(y)g(IH)55 b Fo(d)1815 2883 y Fn(0)1853 2871 y Fu(\()p Fo(y)s(=u)p Fu(\))22 b Fr(`)2124 2841 y Fp(k)q Fi(\000)p Fn(1)2124 2892 y Fp(m)2273 2871 y Fu(\000)p Fo(;)14 b Fu(\001)2431 2883 y Fn(0)2468 2871 y Fu(\()p Fr(I)6 b Fu(\))2583 2883 y Fp(y)2624 2871 y Fu(\()p Fo(u)p Fu(\).)0 3012 y(Since)28 b(\001)286 3024 y Fn(0)323 3012 y Fu(\()p Fr(I)6 b Fu(\))438 3024 y Fp(y)479 3012 y Fu(\()p Fo(u)p Fu(\))23 b(=)g(\001)771 3024 y Fn(0)808 3012 y Fu(\()p Fr(I)891 2982 y Fp(u)935 3012 y Fu(\))28 b(and)g Fo(u)22 b Fr(62)i Fu(FV)q(\(\000\),)k(w)n(e)g (obtain)f Fo(e)c Fr(`)2084 2982 y Fp(k)2084 3032 y(m)2170 3012 y Fu(\000.)0 3177 y(Since)28 b(Eig\()p Fr(I)422 3147 y Fp(u)466 3177 y Fu(\))23 b(=)g Fr(f)p Fo(u)p Fr(g)j Fu(and)i Fo(u)22 b Fr(62)i Fu(FV)q(\()p Fo(t)p Fu(\))19 b Fr([)g(f)p Fo(x)p Fr(g)p Fu(,)27 b(w)n(e)h(ha)n(v)n(e)e Fo(e)1916 3147 y Fi(0)1962 3177 y Fu(=)d(\()p Fr(I)2133 3147 y Fp(u)2177 3177 y Fu(\))2209 3147 y Fi(0)2232 3177 y Fo(d)2275 3189 y Fn(0)2313 3177 y Fu(\()p Fo(y)s(=u)p Fu(\))2511 3147 y Fi(0)2556 3177 y Fu(=)g Fo(d)2687 3147 y Fi(0)2710 3177 y Fu(,)28 b(and)g(b)n(y)f(case)g(3.)g(b)r(elo)n(w)55 b Fo(e)3608 3147 y Fi(0)3654 3177 y Fr(`)3705 3147 y Fp(k)3705 3197 y(m)3791 3177 y Fu(\000)3843 3147 y Fi(0)3866 3177 y Fu(.)0 3342 y(3.)36 b(Otherwise:)0 3482 y(F)-7 b(or)27 b(eac)n(h)g Fo(i)22 b(<)h(n)28 b Fu(w)n(e)f(ha)n(v)n(e)f Fo(d)909 3494 y Fp(i)960 3482 y Fr(`)1011 3452 y Fp(k)q Fi(\000)p Fn(1)1011 3503 y Fp(m)1160 3482 y Fu(\000)p Fo(;)14 b Fu(\001)1318 3494 y Fp(i)1345 3482 y Fu(\()p Fr(I)6 b Fu(\))29 b(and)e(therefore)g(b)n(y)g(IH)h Fo(d)2277 3452 y Fi(0)2277 3504 y Fp(i)2328 3482 y Fr(`)2379 3452 y Fp(k)q Fi(\000)p 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Fu(\))h(=)e(Eig\()p Fr(I)6 b Fu(\)\).)39 b(No)n(w)27 b(w)n(e)g(get)55 b Fo(d)1698 3758 y Fi(0)1745 3788 y Fu(=)22 b Fr(I)1883 3758 y Fi(0)1907 3788 y Fo(d)1950 3758 y Fi(0)1950 3808 y Fn(0)1987 3788 y Fo(:::d)2099 3758 y Fi(0)2099 3808 y Fp(n)p Fi(\000)p Fn(1)2253 3788 y Fr(`)2304 3758 y Fp(k)q Fi(\000)p Fn(1)2304 3808 y Fp(m)2452 3788 y Fu(\000)2504 3758 y Fi(0)2528 3788 y Fu(.)0 4002 y Ft(Remark)0 4142 y Fu(Giv)n(en)d(a)g(\014nite)h (set)f(of)g(v)-5 b(ariables)18 b Fo(V)39 b Fu(and)19 b(a)g(deriv)-5 b(ation)18 b Fo(d)24 b Fu(=)e Fr(I)6 b Fo(d)1970 4154 y Fn(0)2008 4142 y Fo(:::d)2120 4154 y Fp(n)p Fi(\000)p Fn(1)2270 4142 y Fu(with)20 b Fj(S)j Fr(3)g Fo(d)g Fr(`)2738 4112 y Fp(k)2738 4163 y(m)2824 4142 y Fu(\000)d(w)n(e)f(ma)n(y)f(from)h(no)n(w)g(on)g(assume)0 4283 y(without)30 b(loss)e(of)h(generalit)n(y)f(\(w.l.o.g.\))41 b(that)30 b(Eig\()p Fr(I)6 b Fu(\))20 b Fr(\\)g Fu(\(FV)r(\(\000\))g Fr([)g Fo(V)f Fu(\))26 b(=)f Fr(;)p Fu(.)41 b(Because)28 b(if)i(Eig\()p Fr(I)6 b Fu(\))27 b(=)e Fr(f)p Fo(y)s Fr(g)f(\022)i Fu(FV)q(\(\000\))20 b Fr([)g Fo(V)0 4423 y Fu(w)n(e)25 b(ma)n(y)f(c)n(ho)r(ose)g(some)h Fo(u)d Fr(2)i Fk(FV)14 b Fr(n)f Fu(\()p Fo(V)32 b Fr([)13 b Fu(FV)s(\(\000\)\))26 b(and)f(consider)f(the)h(mo)r(di\014ed)h(deriv)-5 b(ation)24 b Fo(d)2944 4393 y Fi(0)2991 4423 y Fu(:=)e Fr(I)3152 4393 y Fp(u)3196 4423 y Fo(d)3239 4435 y Fn(0)3277 4423 y Fu(\()p Fo(y)s(=u)p Fu(\).)35 b(According)0 4564 y(to)27 b(Lemma)h(2.2)f Fj(S)22 b Fr(3)i Fo(d)745 4534 y Fi(0)791 4564 y Fr(`)842 4534 y Fp(k)842 4584 y(m)928 4564 y Fu(\000.)0 4778 y Ft(Lemma)30 b(2.3)0 4918 y Fu(Assume)e Fj(S)22 b Fr(3)i Fo(d)f Fr(`)595 4888 y Fp(k)595 4939 y(m)681 4918 y Fu(\000)p Fo(;)14 b(C)34 b Fu(with)28 b(rk)o(\()p Fo(C)6 b Fu(\))24 b Fr(\025)f Fo(m)1442 4930 y Fn(0)1479 4918 y Fu(.)0 5059 y(a\))k(If)h Fo(C)i Fu(=)22 b Fr(8)p Fo(xA)p Fu(,)28 b(then)g Fj(S)23 b Fr(`)899 5029 y Fp(k)899 5079 y(m)985 5059 y Fu(\000)p Fo(;)14 b(A)1136 5071 y Fp(x)1178 5059 y Fu(\()p Fo(t)p Fu(\))28 b(for)f(eac)n(h)g Fo(t)c Fr(2)g Fu(T)-7 b(er.)0 5199 y(b\))28 b(If)g Fo(C)h Fu(=)23 b Fo(A)427 5211 y Fn(0)483 5199 y Fr(^)c Fo(A)619 5211 y Fn(1)656 5199 y Fu(,)28 b(then)g Fj(S)23 b Fr(`)1039 5169 y Fp(k)1039 5220 y(m)1125 5199 y Fu(\000)p Fo(;)14 b(A)1276 5211 y Fp(j)1339 5199 y Fu(for)27 b Fo(j)h Fr(2)23 b(f)p Fu(0)p Fo(;)14 b Fu(1)p Fr(g)p Fu(.)0 5340 y(c\))28 b(If)g Fo(C)h Fu(=)23 b Fo(A)418 5352 y Fn(0)474 5340 y Fr(_)c Fo(A)610 5352 y Fn(1)647 5340 y Fu(,)28 b(then)g Fj(S)23 b Fr(`)1030 5310 y Fp(k)1030 5361 y(m)1116 5340 y Fu(\000)p Fo(;)14 b(A)1267 5352 y Fn(0)1304 5340 y Fo(;)g(A)1403 5352 y Fn(1)1440 5340 y Fu(.)1929 5539 y(8)p eop %%Page: 9 10 9 9 bop 0 83 a Fu(Pro)r(of)26 b(of)i(a\):)37 b(Let)27 b Fo(d)d Fu(=)e Fr(I)6 b Fo(d)852 95 y Fn(0)890 83 y Fo(:::d)1002 95 y Fp(n)p Fi(\000)p Fn(1)1133 83 y Fu(,)27 b(and)h(\(w.l.o.g.\))37 b(\000\()p Fo(d)p Fu(\))23 b(=)g(\000)p Fo(;)14 b(C)6 b Fu(.)0 221 y(1.)41 b Fr(8)p Fo(xA)26 b Fr(2)g Fu(\001\()p Fr(I)6 b Fu(\):)41 b(Since)29 b(rk\()p Fr(8)p Fo(xA)p Fu(\))d Fr(\025)f Fo(m)1321 233 y Fn(0)1358 221 y Fu(,)30 b Fr(I)36 b Fu(has)28 b(to)h(b)r(e)h(a)f(PL1)o (-inference.)41 b(Hence)29 b Fr(I)j Fu(=)2908 159 y Fl(V)2978 179 y Fp(y)2978 246 y Fi(8)p Fp(xA)3107 221 y Fu(,)e Fo(y)e Fr(62)e Fu(FV)r(\(\000)p Fo(;)14 b Fr(8)p Fo(xA)p Fu(\))29 b(and)0 359 y Fo(d)43 371 y Fn(0)104 359 y Fr(`)155 329 y Fp(k)q Fi(\000)p Fn(1)155 380 y Fp(m)304 359 y Fu(\000)p Fo(;)14 b Fr(8)p Fo(xA;)g(A)648 371 y Fp(x)689 359 y Fu(\()p Fo(y)s Fu(\).)38 b(By)28 b(Lemma)f(2.2)g(w)n(e)h(get)f Fo(d)1723 371 y Fn(0)1761 359 y Fu(\()p Fo(y)s(=t)p Fu(\))c Fr(`)2015 329 y Fp(k)q Fi(\000)p Fn(1)2015 380 y Fp(m)2164 359 y Fu(\000)p Fo(;)14 b Fr(8)p Fo(xA;)g(A)2508 371 y Fp(x)2549 359 y Fu(\()p Fo(t)p Fu(\).)38 b(No)n(w)28 b(w)n(e)f(apply)g(the)i(IH)f(and)f(obtain)0 497 y Fj(S)c Fr(`)143 467 y Fp(k)q Fi(\000)p Fn(1)143 518 y Fp(m)291 497 y Fu(\000)p Fo(;)14 b(A)442 509 y Fp(x)484 497 y Fu(\()p Fo(t)p Fu(\).)0 657 y(2.)36 b Fr(8)p Fo(xA)23 b Fr(62)h 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Fp(x)3541 795 y Fu(\()p Fo(t)p Fu(\).)0 1041 y Ft(Cut-elimination)0 1201 y(Theorem)30 b(2.4)0 1339 y Fj(S)23 b Fr(`)143 1309 y Fp(k)143 1360 y(m)229 1339 y Fu(\000)p Fo(;)14 b(C)52 b Fu(&)46 b Fj(S)22 b Fr(`)682 1309 y Fp(l)682 1360 y(m)768 1339 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)53 b Fu(&)46 b Fo(m)1208 1351 y Fn(0)1268 1339 y Fr(\024)22 b Fu(rk\()p Fo(C)6 b Fu(\))24 b Fr(\024)e Fo(m)46 b Fu(=)-14 b Fr(\))46 b Fj(S)23 b Fr(`)2114 1309 y Fp(k)q Fn(+)p Fp(l)2114 1360 y(m)2250 1339 y Fu(\000.)0 1499 y(Pro)r(of)j(b)n(y)i(induction)g(on)f Fo(k)21 b Fu(+)d Fo(l)r Fu(:)0 1637 y(Assume)28 b Fo(d)23 b Fr(`)425 1607 y Fp(k)425 1657 y(m)511 1637 y Fu(\000)p Fo(;)14 b(C)34 b Fu(and)27 b Fo(e)c Fr(`)967 1607 y Fp(l)967 1657 y(m)1053 1637 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)6 b Fu(.)0 1797 y(1.)36 b Fo(C)e Fu(is)28 b(not)f(a)h(main)f(form)n(ula)g (of)g Fr(I)j Fu(:=)23 b Fk(last)o Fu(\()p Fo(d)p Fu(\):)0 1935 y(Then)28 b(\001\()p Fr(I)6 b Fu(\))24 b Fr(\022)f 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b(and)f(therefore)g Fr(`)1076 3023 y Fp(k)q Fn(+)p Fp(l)1076 3073 y(m)1212 3053 y Fu(\000.)0 3213 y(2.2.)36 b Fo(C)29 b Fu(=)23 b Fr(9)p Fo(xA)p Fu(:)38 b(Then)28 b Fr(:)p Fo(C)h Fu(=)23 b Fr(8)p Fo(xA)p Fu(,)k Fk(last)p Fu(\()p Fo(d)p Fu(\))d(=)1553 3150 y Fl(W)1622 3171 y Fp(t)1622 3237 y(C)1678 3213 y Fu(,)j(and)h Fk(last)o Fu(\()p Fo(e)p Fu(\))c(=)2225 3150 y Fl(V)2294 3171 y Fp(y)2294 3237 y Fi(:)p Fp(C)2395 3213 y Fu(.)0 3351 y(Hence)k Fo(d)290 3363 y Fn(0)350 3351 y Fr(`)401 3320 y Fp(k)436 3328 y Fg(0)401 3371 y Fp(m)496 3351 y Fu(\000)p Fo(;)14 b(C)q(;)g(A)744 3363 y Fp(x)786 3351 y Fu(\()p Fo(t)p Fu(\))29 b(,)55 b Fo(e)1026 3363 y Fn(0)1086 3351 y Fr(`)1137 3320 y Fp(l)1158 3328 y Fg(0)1137 3371 y Fp(m)1223 3351 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)q(;)g Fr(:)p Fo(A)1581 3363 y Fp(x)1624 3351 y Fu(\()p Fo(y)s Fu(\))28 b(with)g Fo(k)1992 3363 y Fn(0)2052 3351 y Fo(<)23 b(k)s Fu(,)28 b Fo(l)2262 3363 y Fn(0)2322 3351 y Fo(<)22 b(l)r Fu(.)0 3510 y(W.l.o.g.)27 b Fo(y)f Fr(62)d 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b(diesem)f(Absc)n(hnitt)i(b)r(ezeic)n(hnen)e(wir)h(mit)g Fo(A;)14 b(B)t(;)g(C)34 b Fu(stets)28 b(Klassen.)0 565 y(Die)35 b(Struktur)f(\()p Fo(O)r(n;)14 b(<)p Fu(\))35 b(der)f(Ordinalzahlen)f(ist)i(bis)g(auf)f(Isomorphie)f(eindeutig)i(b)r (estimm)n(t)g(durc)n(h)f(folgende)g(Eigen-)0 706 y(sc)n(haften:)0 847 y(\(O1\))55 b Fo(<)27 b Fu(ist)h(eine)g(lineare)e(Ordn)n(ung)h(auf) h Fo(O)r(n)p Fu(.)0 988 y(\(O2\))55 b(Jede)27 b(nic)n(h)n(tleere)g (Klasse)f Fo(C)k Fr(\022)22 b Fo(O)r(n)28 b Fu(b)r(esitzt)h(ein)e (kleinstes)h(Elemen)n(t.)0 1129 y(\(O3\))55 b(F)r(\177)-44 b(ur)28 b(jedes)f Fo(\013)d Fr(2)f Fo(O)r(n)28 b Fu(ist)g Fr(f)p Fo(\030)f Fr(2)c Fo(O)r(n)h Fu(:)37 b Fo(\030)27 b(<)22 b(\013)p Fr(g)28 b Fu(eine)g(Menge.)0 1270 y(\(O4\))55 b(Zu)28 b(jeder)f(Menge)g Fo(A)d Fr(\022)e Fo(O)r(n)28 b Fu(gibt)g(es)f Fo(\015)h Fr(2)23 b Fo(O)r(n)29 b Fu(mit)f Fr(8)p Fo(\013)p Fr(2)p Fo(A)p Fu(\()p Fo(\013)c(<)e(\015)5 b Fu(\).)226 1411 y(\()p Fo(O)r(n)28 b Fu(ist)g(also)f(k)n(eine)g (Menge,)g(sondern)g(eine)h(ec)n(h)n(te)f(Klasse\).)10 1581 y(\177)0 1602 y(Ublic)n(herw)n(eise)k(de\014niert)h(man)g(die)g (Ordinalzahlen)f(derart,)h(da\031)f(stets)h Fo(\013)f Fu(=)e Fr(f)p Fo(\030)34 b Fr(2)d Fo(O)r(n)g Fu(:)f Fo(\030)35 b Fr(2)30 b Fo(\013)p Fr(g)p Fu(.)50 b(Da)n(v)n(on)31 b(w)n(ollen)g(wir)0 1743 y(hier)c(auc)n(h)g(ausgehen,)g(da)g(es)h(an)f (manc)n(hen)g(Stellen)h(die)g(Notation)f(v)n(ereinfac)n(h)n(t.)0 1910 y Fo(\013;)14 b(\014)t(;)g(\015)5 b(;)14 b(\016)o(;)g(\030)t(;)g (\021)s(;)g(\020)34 b Fu(b)r(ezeic)n(hnen)27 b(im)h(folgenden)g(stets)f (Ordinalzahlen.)0 2051 y(Die)h(kleinste)g(Ordinalzahl)e(wird)h(mit)h(0) g(b)r(ezeic)n(hnet.)37 b(En)n(tsprec)n(hend)26 b(obiger)g(Annahme)i (ist)g(0)23 b(=)f Fr(;)p Fu(.)0 2217 y(V)28 b(sei)f(die)h(Allklasse)f (\(Klasse)f(aller)h(Mengen\).)0 2383 y Ft(Satz)33 b(3.1)p Fu(\(T)-7 b(rans\014nite)26 b(Induktion\))0 2524 y Fr(8)p Fo(\013)p Fu([)14 b Fr(8)p Fo(\030)t(<)n(\013)h(\036)p Fu(\()p Fo(\030)t Fu(\))38 b Fr(!)23 b Fo(\036)p Fu(\()p Fo(\013)p Fu(\))14 b(])24 b Fr(!)f(8)p Fo(\013)14 b(\036)p Fu(\()p Fo(\013)p Fu(\).)0 2690 y Fm(Beweis:)0 2831 y Fu(Wir)36 b(b)r(ew)n(eisen)g(die)h(Kon)n(trap)r(osition)d(der)i (Behauptung.)62 b(Gelte)37 b(also)e Fr(:8)p Fo(\013\036)p Fu(\()p Fo(\013)p Fu(\).)64 b(Dann)36 b(ist)h(die)f(Klasse)f Fo(C)44 b Fu(:=)37 b Fr(f)p Fo(\013)g Fu(:)0 2972 y Fr(:)p Fo(\036)p Fu(\()p Fo(\013)p Fu(\))p Fr(g)29 b Fu(nic)n(h)n(tleer)e(und) h(b)r(esitzt)g(daher)f(ein)h(kleinstes)f(Elemen)n(t)g Fo(\013)2144 2984 y Fn(0)2182 2972 y Fu(.)37 b(Somit)28 b(gilt)f Fr(8)p Fo(\030)t(<)o(\013)2833 2984 y Fn(0)2870 2972 y Fo(\036)p Fu(\()p Fo(\030)t Fu(\))20 b Fr(^)f(:)p Fo(\036)p Fu(\()p Fo(\013)3306 2984 y Fn(0)3344 2972 y Fu(\).)0 3138 y Ft(Satz)33 b(3.2)27 b Fu(\(T)-7 b(rans\014nite)27 b(Rekursion\))0 3279 y(Zu)h Fo(G)23 b Fu(:)g Fo(A)c Fr(\002)f Fo(O)r(n)h Fr(\002)f Fo(V)42 b Fr(!)23 b Fo(B)32 b Fu(gibt)27 b(es)g(genau)g(ein)h Fo(F)35 b Fu(:)23 b Fo(A)c Fr(\002)f Fo(O)r(n)23 b Fr(!)h Fo(B)31 b Fu(mit)0 3421 y Fo(F)12 b Fu(\()p Fo(x;)i(\013)p Fu(\))24 b(=)f Fo(G)p Fu(\()p Fo(x;)14 b(\013;)g(F)702 3433 y Fp(x)745 3421 y Fr(j)p Fo(\013)p Fu(\))29 b(f)r(\177)-44 b(ur)27 b(alle)h(\()p Fo(x;)14 b(\013)p Fu(\))24 b Fr(2)f Fo(A)c Fr(\002)f Fo(O)r(n)p Fu(.)38 b(\(Dab)r(ei)28 b(sei)f Fo(F)2255 3433 y Fp(x)2321 3421 y Fu(:=)22 b Fo(\025y)s(:F)12 b Fu(\()p Fo(x;)i(y)s Fu(\).)38 b(\))0 3587 y Ft(De\014nition)0 3728 y Fo(\013)p Fu(+1)23 b(:=)f(min)q Fr(f)p Fo(\030)k Fu(:)d Fo(\013)h(<)f(\030)t Fr(g)p Fu(.)0 3869 y Fo(\013)28 b Fu(hei\031t)g(Limeszahl,)f(falls)g Fo(\013)d Fr(6)p Fu(=)e(0)27 b(und)i Fr(:9)p Fo(\030)t Fu(\()p Fo(\013)24 b Fu(=)f Fo(\030)t Fu(+)o(1\).)0 4010 y Fo(Lim)k Fu(b)r(ezeic)n(hnet)h (die)f(Klasse)g(aller)f(Limeszahlen)h(und)h Fo(!)j Fu(die)c(kleinste)h (Limeszahl.)0 4151 y(\(Die)g(Existenz)f(v)n(on)g(Limeszahlen)g(folgt)h (aus)f(dem)h(sog.)e(Unendlic)n(hk)n(eitsaxiom)h(der)g(Mengenlehre.\))0 4292 y(Die)h(Ordinalzahlen)e Fo(<)d(!)30 b Fu(nenn)n(t)e(man)g Fm(end)t(liche)j(Or)l(dinalzahlen)f Fu(o)r(der)d Fm(nat)q(\177)-43 b(urliche)30 b(Zahlen)p Fu(.)0 4433 y(Die)e(Buc)n(hstab)r(en)f Fo(i;)14 b(j;)g(k)s(;)g(l)r(;)g(m;)g(n)26 b Fu(b)r(ezeic)n(hnen)i(im)g (folgenden)f(stets)h(nat)r(\177)-44 b(urlic)n(he)27 b(Zahlen.)0 4599 y Ft(De\014nition)0 4741 y Fu(F)r(\177)-44 b(ur)28 b(jede)f(Menge)h Fo(A)23 b Fr(\022)g Fo(O)r(n)28 b Fu(sei)f(sup\()p Fo(A)p Fu(\))d(:=)f(min)p Fr(f)p Fo(\015)k Fu(:)c Fr(8)p Fo(\030)k Fr(2)c Fo(A)p Fu(\()p Fo(\030)28 b Fr(\024)23 b Fo(\015)5 b Fu(\))p Fr(g)27 b Fu(und)0 4882 y(sup)125 4852 y Fn(+)180 4882 y Fu(\()p Fo(A)p Fu(\))d(:=)f(min)p Fr(f)p Fo(\015)k Fu(:)c Fr(8)p Fo(\030)k Fr(2)c Fo(A)p Fu(\()p Fo(\030)28 b(<)22 b(\015)5 b Fu(\))p Fr(g)23 b Fu(\(=)g(sup)p Fr(f)p Fo(\030)t Fu(+)o(1)g(:)g Fo(\030)k Fr(2)c Fo(A)p Fr(g)p Fu(\).)0 5048 y Ft(Lemma)30 b(3.3)0 5189 y Fu(F)r(\177)-44 b(ur)28 b(jede)f(nic)n(h)n(tleere)g(Menge)h Fo(A)23 b Fr(\022)f Fo(O)r(n)29 b Fu(gilt:)36 b(sup\()p Fo(A)p Fu(\))24 b Fr(62)g Fo(A)46 b Fr(\))g Fu(sup\()p Fo(A)p Fu(\))24 b Fr(2)g Fo(Lim)p Fu(.)1908 5539 y(10)p eop %%Page: 11 12 11 11 bop 0 83 a Fm(Beweis:)0 224 y Fu(Sei)28 b Fo(\013)23 b Fu(:=)g(sup\()p Fo(A)p Fu(\))h Fr(62)f Fo(A)p Fu(.)37 b(O\013en)n(bar)27 b(ist)h(dann)f Fo(\013)d Fr(6)p Fu(=)e(0.)37 b(Bleibt)28 b(zu)g(zeigen)f Fr(8)p Fo(\014)f(<)d(\013)p Fu(\()p Fo(\014)g Fu(+)18 b(1)23 b Fo(<)f(\013)p Fu(\).)0 365 y(Sei)28 b(also)e Fo(\014)i(<)22 b(\013)p Fu(.)38 b(Dann)28 b(existiert)f(ein)h Fo(\030)f Fr(2)c Fo(A)28 b Fu(mit)g Fo(\014)g(<)22 b(\030)t Fu(,)28 b(und)g(w)n(egen)f Fo(\013)c Fr(62)h Fo(A)k Fu(gilt)f Fo(\014)c Fu(+)18 b(1)23 b Fr(\024)f Fo(\030)28 b(<)22 b(\013)p Fu(.)0 531 y Ft(De\014nition)0 672 y Fu(Eine)27 b(bin\177)-42 b(are)27 b(Relation)g Fr(\036)h Fu(hei\031t)f Fm(fundiert)p Fu(,)h(w)n(enn)g(jede)g(nic)n(h)n(tleere)f(Menge)g(ein)h(bzgl.)f Fr(\036)g Fu(minimales)h(Elemen)n(t)f(b)r(esitzt.)0 839 y Ft(Lemma)j(3.4)0 980 y Fu(a\))d(Ist)h Fr(\036)f Fu(fundiert,)i(so)e (gibt)g(es)h(k)n(eine)f(unendlic)n(he)h(F)-7 b(olge)26 b(\()p Fo(a)1944 992 y Fp(n)1990 980 y Fu(\))2022 992 y Fp(n)g Fu(0.)0 2192 y(b\))28 b Fo(\025)c Fr(2)f Fo(Lim)36 b Fr(\))46 b Fo(F)12 b Fu(\()p Fo(\025)p Fu(\))24 b Fr(2)f Fo(Lim)p Fu(.)0 2333 y(c\))28 b Fr(8)p Fo(\015)f Fr(\025)22 b Fo(F)12 b Fu(\(0\))p Fr(9)p Fu(!)p Fo(\013)p Fu(\()i Fo(F)e Fu(\()p Fo(\013)p Fu(\))25 b Fr(\024)e Fo(\015)k(<)c(F)12 b Fu(\()p Fo(\013)19 b Fu(+)f(1\)\).)0 2474 y(d\))28 b Fo(G)g Fu(Normalfunktion)f Fr(\))46 b Fo(F)31 b Fr(\016)18 b Fo(G)28 b Fu(Normalfunktion.)0 2615 y(e\))g Fo(F)12 b Fu(\(sup\()p Fo(A)p Fu(\)\))24 b(=)f(sup\()p Fo(F)12 b Fu([)p Fo(A)p Fu(]\))47 b(\(=)23 b(sup)1243 2635 y Fp(\030)r Fi(2)p Fp(A)1388 2615 y Fo(F)12 b Fu(\()p Fo(\030)t Fu(\))i(\))28 b(f)r(\177)-44 b(ur)28 b(jede)g(nic)n(h)n(tleere)f(Menge)g Fo(A)c Fr(\022)g Fo(O)r(n)p Fu(.)0 2781 y Fm(Beweis:)0 2922 y Fu(a\))29 b(W)-7 b(egen)28 b Fr(8)p Fo(\030)g(<)d(\013)p Fu(\()p Fo(\030)f Fu(+)18 b(1)25 b Fr(\024)f Fo(\013)p Fu(\))29 b(gilt)g Fo(\015)g Fu(:=)c(sup)p Fr(f)p Fo(F)12 b Fu(\()p Fo(\030)23 b Fu(+)c(1\))24 b(:)h Fo(\030)k(<)24 b(\013)p Fr(g)h(\024)g Fo(F)12 b Fu(\()p Fo(\013)p Fu(\).)41 b(Ist)28 b Fo(\013)e Fu(=)e Fo(\014)f Fu(+)c(1,)29 b(so)f Fo(F)12 b Fu(\()p Fo(\013)p Fu(\))25 b Fr(2)h(f)p Fo(F)12 b Fu(\()p Fo(\030)23 b Fu(+)18 b(1\))25 b(:)0 3064 y Fo(\030)i(<)c(\013)p Fr(g)k Fu(und)h(deshalb)g Fo(F)12 b Fu(\()p Fo(\013)p Fu(\))24 b Fr(\024)e Fo(\015)5 b Fu(.)37 b(Ist)28 b Fo(\013)23 b Fr(2)g Fo(Lim)p Fu(,)k(so)g Fo(F)12 b Fu(\()p Fo(\013)p Fu(\))24 b(=)f(sup)2148 3084 y Fp(\030)r(<\013)2293 3064 y Fo(F)12 b Fu(\()p Fo(\030)t Fu(\))24 b Fr(\024)e Fu(sup)2698 3084 y Fp(\030)r(<\013)2844 3064 y Fo(F)12 b Fu(\()p Fo(\030)22 b Fu(+)c(1\))23 b(=)g Fo(\015)5 b Fu(.)0 3205 y(b\))28 b(Es)f(ist)h(0)22 b Fr(\024)h Fo(F)12 b Fu(\(0\))23 b Fo(<)g(F)12 b Fu(\()p Fo(\025)p Fu(\).)38 b(Aus)27 b Fo(\015)h(<)23 b(F)12 b Fu(\()p Fo(\025)p Fu(\))28 b(folgt)55 b Fr(9)p Fo(\030)28 b(<)22 b(\025)p Fu(\()p 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Fu(\()p Fo(\030)17 b Fu(+)11 b(1\))22 b Fr(\024)h Fo(\015)5 b Fu(\),)25 b(also)e Fo(F)12 b Fu(\()p Fo(\013)p Fu(\))24 b Fr(\024)e Fo(\015)5 b Fu(.)0 3628 y(d\))28 b(\()p Fo(F)j Fr(\016)18 b Fo(G)p Fu(\)\()p Fo(\025)p Fu(\))24 b(=)f Fo(F)12 b Fu(\()p Fo(G)p Fu(\()p Fo(\025)p Fu(\)\))25 b(=)d Fo(F)12 b Fu(\(sup)q(\()p Fr(f)p Fo(G)p Fu(\()p Fo(\030)t Fu(\))23 b(:)h Fo(\030)j(<)22 b(\025)p Fr(g)p Fu(\)\))i(=)f(sup)2098 3648 y Fp(\030)r(<\025)2239 3628 y Fo(F)12 b Fu(\()p Fo(G)p Fu(\()p Fo(\030)t Fu(\)\).)0 3769 y(e\))34 b(Sei)g Fr(;)e(6)p Fu(=)h Fo(A)h Fr(\022)e Fo(O)r(n)j Fu(und)f Fo(\013)f Fu(:=)g(sup\()p Fo(A)p Fu(\).)56 b(Ist)34 b Fo(\013)f Fr(2)h Fo(A)p Fu(,)h(so)e Fo(F)12 b Fu(\()p Fo(\013)p Fu(\))34 b(=)f(sup\()p Fo(F)12 b Fu([)p Fo(A)p Fu(]\),)36 b(da)e Fo(F)45 b Fu(ordn)n(ungstreu.)54 b(Ist)34 b Fo(\013)f Fr(62)h Fo(A)p Fu(,)0 3910 y(so)e Fo(\013)g Fr(2)g Fo(Lim)f Fu(und)i(deshalb)g Fo(F)12 b Fu(\()p Fo(\013)p Fu(\))32 b(=)f(sup)1381 3931 y Fp(\030)r(<\013)1526 3910 y Fo(F)12 b Fu(\()p Fo(\030)t Fu(\).)53 b(F)-7 b(erner)32 b(gilt)g Fr(8)p Fo(\030)j Fr(2)d Fo(A)p Fu(\()p Fo(\030)k(<)31 b(\013)p Fu(\))22 b Fr(^)h(8)p Fo(\030)34 b(<)d(\013)p Fr(9)p Fo(\021)s Fr(2)q Fo(A)p Fu(\()p Fo(\030)36 b(<)31 b(\021)s Fu(\),)k(w)n(oraus)0 4051 y(sup)125 4072 y Fp(\030)r(<\013)270 4051 y Fo(F)12 b Fu(\()p Fo(\030)t Fu(\))24 b(=)f(sup\()p Fo(F)12 b Fu([)p Fo(A)p Fu(]\))28 b(folgt.)0 4218 y Ft(De\014nition)j Fu(v)n(on)c Fo(\013)19 b Fu(+)f Fo(\014)32 b Fu(durc)n(h)27 b(trans\014nite)g(Rekursion)g(nac)n(h)g Fo(\014)0 4359 y(\013)19 b Fu(+)f(0)k(:=)h Fo(\013;)97 b(\013)19 b Fu(+)f(\()p Fo(\014)23 b Fu(+)18 b(1\))23 b(:=)g(\()p Fo(\013)c Fu(+)f Fo(\014)t Fu(\))h(+)f(1)p Fo(;)97 b(\013)19 b Fu(+)f Fo(\025)23 b Fu(:=)g(sup)2047 4379 y Fp(\021)r(<\025)2179 4359 y Fu(\()p Fo(\013)c Fu(+)f Fo(\021)s Fu(\).)0 4525 y(Abk)r(\177)-44 b(urzung:)36 b(1)23 b(:=)g(0)18 b(+)g(1.)0 4691 y(Bemerkung:)36 b Fo(\013)19 b Fu(+)f(1)27 b(hat)h(n)n(un)f(zw)n (ei)g(Bedeutungen,)h(die)g(ab)r(er)h(\177)-44 b(ub)r(ereinstimmen,)28 b(n\177)-42 b(amlic)n(h)0 4832 y(1.)36 b Fo(\013)19 b Fu(+)f(1)23 b(=)g(min)p Fr(f)p Fo(\030)k Fu(:)c Fo(\013)g(<)g(\030)t Fr(g)p Fu(,)0 4973 y(2.)36 b Fo(\013)19 b Fu(+)f(1)23 b(=)k(Summe)h(v)n(on)f Fo(\013)h Fu(und)g(1)g(gem\177)-42 b(a\031)26 b(obiger)g(De\014nition.)1908 5539 y(13)p eop %%Page: 14 15 14 14 bop 0 83 a Ft(Lemma)30 b(3.11)0 224 y Fu(a\))d(F)r(\177)-44 b(ur)28 b(jedes)g Fo(\013)g Fu(ist)g Fo(\014)f Fr(7!)c Fo(\013)c Fu(+)f Fo(\014)32 b Fu(eine)c(Normalfunktion.)0 365 y(b\))g Fo(\014)153 377 y Fn(0)213 365 y Fo(<)23 b(\014)348 377 y Fn(1)431 365 y Fr(\))46 b Fo(\013)19 b Fu(+)f Fo(\014)762 377 y Fn(0)823 365 y Fo(<)k(\013)d Fu(+)f Fo(\014)1112 377 y Fn(1)0 506 y Fu(c\))28 b Fo(\013;)14 b(\014)28 b Fr(\024)22 b Fo(\013)d Fu(+)f Fo(\014)0 648 y Fu(d\))28 b Fr(8)p Fo(\015)f Fr(\025)22 b Fo(\013)p Fr(9)p Fu(!)p Fo(\014)t Fu(\()p Fo(\013)f Fu(+)d Fo(\014)27 b Fu(=)c Fo(\015)5 b Fu(\))0 789 y(e\))28 b Fo(\013)150 801 y Fn(0)210 789 y Fr(\024)23 b Fo(\013)351 801 y Fn(1)434 789 y Fr(\))47 b Fo(\013)617 801 y Fn(0)672 789 y Fu(+)18 b Fo(\014)28 b Fr(\024)22 b Fo(\013)970 801 y Fn(1)1026 789 y Fu(+)c Fo(\014)0 930 y Fu(f)6 b(\))29 b(\()p Fo(\013)19 b Fu(+)f Fo(\014)t Fu(\))h(+)f Fo(\015)28 b Fu(=)22 b Fo(\013)d Fu(+)f(\()p Fo(\014)23 b Fu(+)18 b Fo(\015)5 b Fu(\))0 1071 y(g\))27 b Fo(\013;)14 b(\014)28 b(<)23 b(!)49 b Fr(\))d Fo(\013)19 b Fu(+)f Fo(\014)27 b Fu(=)c Fo(\014)f Fu(+)c Fo(\013)24 b(<)f(!)0 1212 y Fu(h\))28 b(0)23 b Fo(<)f(k)k(<)d(!)49 b Fr(\))d Fo(k)21 b Fu(+)d Fo(!)26 b Fu(=)c Fo(!)k(<)d(!)e Fu(+)d Fo(k)0 1378 y Fm(Beweis:)0 1519 y Fu(a\))27 b(folgt)h(aus)f(3.9.)36 b(b\),d\))29 b(folgen)e(aus)g(a\).)37 b(c\))27 b(folgt)h(aus)f(a\))g (und)h(3.7.)36 b(e\))28 b(Induktion)g(nac)n(h)f Fo(\014)t Fu(.)37 b(f)6 b(\))29 b(Induktion)f(nac)n(h)f Fo(\015)5 b Fu(.)0 1660 y(g\))27 b(Induktion)h(nac)n(h)f Fo(\014)t Fu(.)38 b(h\))28 b Fo(!)d Fr(\024)e Fo(k)e Fu(+)d Fo(!)26 b Fu(=)d(sup)1501 1681 y Fp(n)c(\014)306 4387 y Fn(0)366 4375 y Fo(>)g(:::)g(>)g(\014)681 4387 y Fp(n)740 4375 y Fu(&)14 b Fo(\016)856 4387 y Fn(0)892 4375 y Fo(;)g(:::;)g(\016)1072 4387 y Fp(n)1140 4375 y Fo(<)23 b(\013)47 b Fr(\))f Fo(\013)1510 4345 y Fp(\014)1578 4375 y Fo(>)23 b(\013)1719 4345 y Fp(\014)1757 4353 y Fg(0)1812 4375 y Fr(\001)18 b Fo(\016)1890 4387 y Fn(0)1946 4375 y Fu(+)g Fo(:::)g Fu(+)g Fo(\013)2252 4345 y Fp(\014)2290 4353 y Fh(n)2354 4375 y Fr(\001)g Fo(\016)2432 4387 y Fp(n)2477 4375 y Fu(.)0 4541 y Fm(Beweis:)0 4682 y Fu(e\))24 b(Ind.)g(nac)n(h)f Fo(n)p Fu(:)35 b(I.V.)25 b Fr(\))46 b Fo(\013)907 4652 y Fp(\014)945 4660 y Fg(0)1004 4682 y Fo(>)23 b(\013)1145 4652 y Fp(\014)1183 4660 y Fg(1)1231 4682 y Fr(\001)11 b Fo(\016)1302 4694 y Fn(1)1350 4682 y Fu(+)g Fo(:::)g Fu(+)g Fo(\013)1635 4652 y Fp(\014)1673 4660 y Fh(n)1727 4682 y Fr(\001)g Fo(\016)1798 4694 y Fp(n)1889 4682 y Fr(\))47 b Fo(a)2063 4652 y Fp(\014)2130 4682 y Fr(\025)23 b Fo(\013)2271 4652 y Fp(\014)2309 4660 y Fg(0)2356 4682 y Fr(\001)11 b Fo(\013)24 b Fr(\025)f Fo(\013)2608 4652 y Fp(\014)2646 4660 y Fg(0)2693 4682 y Fr(\001)11 b Fo(\016)2764 4694 y Fn(0)2812 4682 y Fu(+)g Fo(\013)2941 4652 y Fp(\014)2979 4660 y Fg(0)3038 4682 y Fo(>)23 b(\013)3179 4652 y Fp(\014)3217 4660 y Fg(0)3264 4682 y Fr(\001)11 b Fo(\016)3335 4694 y Fn(0)3383 4682 y Fu(+)g Fo(:::)g Fu(+)g Fo(\013)3668 4652 y Fp(\014)3706 4660 y Fh(n)3761 4682 y Fr(\001)g Fo(\016)3832 4694 y Fp(n)3877 4682 y Fu(.)0 4848 y Ft(Satz)33 b(3.14)0 4990 y Fu(a\))55 b(Zu)28 b Fo(\013)23 b Fr(\025)g Fu(2)k(und)h Fo(\015)g Fr(\025)22 b Fu(1)27 b(existieren)g(eindeutig)h Fo(\014)t(;)14 b(\016)o(;)g(\015)1816 5002 y Fn(0)1881 4990 y Fu(mit)28 b(0)22 b Fo(<)h(\016)j(<)d(\013)14 b Fu(&)g Fo(\015)2525 5002 y Fn(0)2585 4990 y Fo(<)22 b(\013)2725 4959 y Fp(\014)2784 4990 y Fu(&)14 b Fo(\015)28 b Fu(=)22 b Fo(\013)3074 4959 y Fp(\014)3138 4990 y Fr(\001)c Fo(\016)k Fu(+)c Fo(\015)3364 5002 y Fn(0)3401 4990 y Fu(.)0 5131 y(b\))51 b(Zu)28 b Fo(\013)23 b Fr(\025)g Fu(2)k(und)h Fo(\015)g Fr(\025)22 b Fu(1)27 b(existieren)g(eindeutig)h Fo(\014)1659 5143 y Fn(0)1720 5131 y Fo(>)22 b(:::)h(>)g(\014)2034 5143 y Fp(n)2107 5131 y Fu(und)28 b(0)22 b Fo(<)h(\016)2462 5143 y Fn(0)2499 5131 y Fo(;)14 b(:::;)g(\016)2679 5143 y Fp(n)2747 5131 y Fo(<)23 b(\013)28 b Fu(mit)129 5272 y Fo(\015)g Fu(=)22 b Fo(\013)340 5242 y Fp(\014)378 5250 y Fg(0)434 5272 y Fr(\001)c Fo(\016)512 5284 y Fn(0)568 5272 y Fu(+)g Fo(:::)g Fu(+)g Fo(\013)874 5242 y Fp(\014)912 5250 y Fh(n)976 5272 y Fr(\001)g Fo(\016)1054 5284 y Fp(n)1099 5272 y Fu(.)37 b Fm(\(Cantorsche)31 b(Normalform)g(von)f Fo(\015)35 b Fm(zur)29 b(Basis)i Fo(\013)f Fm(\))1908 5539 y Fu(14)p eop %%Page: 15 16 15 15 bop 0 83 a Fm(Beweis:)0 224 y Fu(a\))37 b Fm(Eindeutigkeit:)59 b Fu(Sei)38 b Fo(\015)44 b Fu(=)c Fo(\013)1057 194 y Fp(\014)1127 224 y Fr(\001)25 b Fo(\016)j Fu(+)d Fo(\015)1373 236 y Fn(0)1450 224 y Fu(=)39 b Fo(\013)1607 194 y Fp(\014)1645 202 y Fg(1)1707 224 y Fr(\001)25 b Fo(\016)1792 236 y Fn(1)1854 224 y Fu(+)g Fo(\015)1987 236 y Fn(1)2062 224 y Fu(mit)38 b(0)i Fo(<)f(\016)o(;)14 b(\016)2520 236 y Fn(1)2596 224 y Fo(<)39 b(\013)15 b Fu(&)f Fo(\015)2890 236 y Fn(0)2966 224 y Fo(<)39 b(\013)3123 194 y Fp(\014)3182 224 y Fu(&)14 b Fo(\015)3304 236 y Fn(1)3381 224 y Fo(<)39 b(\013)3538 194 y Fp(\014)3576 202 y Fg(1)3613 224 y Fu(.)67 b(Dann)0 365 y Fo(\013)53 335 y Fp(\014)142 365 y Fr(\024)44 b Fo(\015)49 b(<)44 b(\013)505 335 y Fp(\014)s Fn(+1)648 365 y Fu(&)14 b Fo(\013)780 335 y Fp(\014)818 343 y Fg(1)898 365 y Fr(\024)44 b Fo(\015)49 b(<)44 b(\013)1261 335 y Fp(\014)1299 343 y Fg(1)1331 335 y Fn(+1)1419 365 y Fu(,)g(also)39 b Fo(\014)49 b Fu(=)44 b Fo(\014)1917 377 y Fn(1)1954 365 y Fu(.)75 b(W)-7 b(eiter)40 b(folgt)g(n)n(un)h Fo(\013)2766 335 y Fp(\014)2838 365 y Fr(\001)27 b Fo(\016)47 b Fr(\024)d Fo(\015)49 b(<)44 b(\013)3335 335 y Fp(\014)3407 365 y Fr(\001)27 b Fu(\()p Fo(\016)j Fu(+)c(1\))41 b(und)0 506 y Fo(\013)53 476 y Fp(\014)117 506 y Fr(\001)18 b Fo(\016)195 518 y Fn(1)255 506 y Fr(\024)23 b Fo(\015)28 b(<)22 b(\013)554 476 y Fp(\014)618 506 y Fr(\001)c Fu(\()p Fo(\016)728 518 y Fn(1)784 506 y Fu(+)g(1\),)28 b(also)e(auc)n(h)h Fo(\016)g Fu(=)22 b Fo(\016)1542 518 y Fn(1)1579 506 y Fu(.)37 b(Aus)28 b Fo(\013)1861 476 y Fp(\014)1925 506 y Fr(\001)18 b Fo(\016)k Fu(+)c Fo(\015)2151 518 y Fn(0)2211 506 y Fu(=)23 b Fo(\013)2352 476 y Fp(\014)2415 506 y Fr(\001)c Fo(\016)i Fu(+)d Fo(\015)2641 518 y Fn(1)2706 506 y Fu(folgt)28 b(sc)n(hlie\031lic)n(h)e Fo(\015)3333 518 y Fn(0)3393 506 y Fu(=)d Fo(\015)3524 518 y Fn(1)3561 506 y Fu(.)0 648 y Fm(Existenz:)40 b Fu(Nac)n(h)29 b(3.10c)f(existiert) h(ein)g Fo(\014)34 b Fu(mit)29 b Fo(\013)1539 617 y Fp(\014)1610 648 y Fr(\024)d Fo(\015)k(<)25 b(\013)1917 617 y Fp(\014)s Fn(+1)2047 648 y Fu(,)k(d.h.)42 b Fo(\013)2332 617 y Fp(\014)2397 648 y Fr(\001)20 b Fu(1)25 b Fr(\024)g Fo(\015)31 b(<)25 b(\013)2814 617 y Fp(\014)2879 648 y Fr(\001)19 b Fo(\013)p Fu(.)42 b(Wiederum)30 b(mit)g(3.10)d(\(im)0 789 y(w)n(esen)n(tlic)n(hen\))i(folgt)g(daraus)f Fo(\013)1028 759 y Fp(\014)1093 789 y Fr(\001)20 b Fo(\016)29 b Fr(\024)d Fo(\015)k(<)c(\013)1510 759 y Fp(\014)1575 789 y Fr(\001)20 b Fu(\()p Fo(\016)j Fu(+)c(1\))26 b(=)g Fo(\013)2038 759 y Fp(\014)2102 789 y Fr(\001)20 b Fo(\016)j Fu(+)c Fo(\013)2342 759 y Fp(\014)2387 789 y Fu(.)43 b(Nac)n(h)29 b(3.11)f(existiert)h(deshalb)g(ein)h Fo(\015)3648 801 y Fn(0)3711 789 y Fo(<)c(\013)3855 759 y Fp(\014)0 930 y Fu(mit)i Fo(\015)g Fu(=)22 b Fo(\013)363 900 y Fp(\014)427 930 y Fr(\001)d Fo(\016)i Fu(+)d Fo(\015)653 942 y Fn(0)690 930 y Fu(.)0 1096 y(b\))28 b(folgt)f(aus)h(a\))f(und)h(3.13e)e(mittels) i(Induktion)g(nac)n(h)f Fo(\015)5 b Fu(.)0 1262 y Ft(Abk)s(\177)-50 b(urzung)0 1403 y Fo(\013)23 b Fu(=)141 1415 y Fp(C)-7 b(N)i(F)307 1403 y Fo(!)362 1373 y Fp(\013)405 1381 y Fg(0)460 1403 y Fr(\001)18 b Fo(k)544 1415 y Fn(0)600 1403 y Fu(+)g Fo(:)c(:)g(:)19 b Fu(+)f Fo(!)937 1373 y Fp(\013)980 1382 y Fh(k)1038 1403 y Fr(\001)h Fo(k)1123 1415 y Fp(n)1214 1403 y Fu(:)24 b Fr(\()-14 b(\))46 b Fo(\013)23 b Fu(=)g Fo(!)1678 1373 y Fp(\013)1721 1381 y Fg(0)1776 1403 y Fr(\001)18 b Fo(k)1860 1415 y Fn(0)1916 1403 y Fu(+)g Fo(:)c(:)g(:)19 b Fu(+)f Fo(!)2253 1373 y Fp(\013)2296 1382 y Fh(k)2354 1403 y Fr(\001)h Fo(k)2439 1415 y Fp(n)2498 1403 y Fu(&)14 b Fo(\013)2630 1415 y Fn(0)2690 1403 y Fo(>)23 b(:)14 b(:)g(:)22 b(>)h(\013)3038 1415 y Fp(n)0 1569 y Ft(De\014nition)59 b Fu(\(Additiv)n(e)28 b(Hauptzahlen\))0 1710 y Fo(\015)g Fr(2)23 b Fo(O)r(n)28 b Fu(hei\031t)g Fs(additiv)n(e)f(Hauptzahl)p Fu(,)h(falls)f Fo(\015)h(>)22 b Fu(0)c Fr(^)h(8)p Fo(\030)t(;)14 b(\021)26 b(<)c(\015)5 b Fu(\()p Fo(\030)23 b Fu(+)18 b Fo(\021)26 b(<)d(\015)5 b Fu(\).)0 1876 y Fo(H)30 b Fu(:=)d(Klasse)f(aller)h (additiv)n(en)g(Hauptzahlen.)0 2042 y Ft(Lemma)j(3.15)0 2183 y Fu(a\))d Fo(\013)d Fr(7!)f Fo(!)339 2153 y Fp(\013)414 2183 y Fu(ist)k(die)h(Ordn)n(ungsfunktion)f(der)g(Klasse)f Fo(H)35 b Fu(aller)26 b(additiv)n(en)i(Hauptzahlen.)0 2325 y(b\))g Fo(\015)g Fr(2)23 b Fo(H)53 b Fr(,)46 b Fo(\015)28 b(>)22 b Fu(0)c Fr(^)h(8)p Fo(\030)27 b(<)22 b(\015)5 b Fu(\()p Fo(\030)23 b Fu(+)18 b Fo(\015)27 b Fu(=)c Fo(\015)5 b Fu(\).)0 2491 y Fm(Beweis:)0 2632 y Fu(a\))27 b(1.)37 b(Durc)n(h)27 b(Induktion)h(nac)n(h)f Fo(\013)h Fu(zeigen)f(wir)h Fo(!)1561 2602 y Fp(\013)1631 2632 y Fr(2)23 b Fo(H)7 b Fu(:)0 2773 y(1.1.)36 b Fo(!)221 2743 y Fn(0)281 2773 y Fr(2)23 b Fo(H)35 b Fu(ist)28 b(trivial.)0 2914 y(1.2.)36 b Fo(\030)t(;)14 b(\021)26 b(<)d(!)453 2884 y Fp(\013)p Fn(+1)621 2914 y Fr(\))37 b Fo(\030)t(;)14 b(\021)26 b(<)d(!)1028 2884 y Fp(\013)1093 2914 y Fr(\001)c Fo(n)27 b Fu(f)r(\177)-44 b(ur)28 b(ein)g Fo(n)23 b(<)f(!)40 b Fr(\))d Fo(\030)22 b Fu(+)c Fo(\021)27 b(<)22 b(!)2201 2884 y Fp(\013)2267 2914 y Fr(\001)c Fo(n)g Fu(+)g Fo(!)2514 2884 y Fp(\013)2580 2914 y Fr(\001)g Fo(n)23 b Fu(=)g Fo(!)2837 2884 y Fp(\013)2902 2914 y Fr(\001)c Fu(\()p Fo(n)f Fu(+)g Fo(n)p Fu(\))24 b Fo(<)e(!)3375 2884 y Fp(\013)p Fn(+1)3506 2914 y Fu(.)0 3055 y(1.3.)36 b Fo(\030)t(;)14 b(\021)26 b(<)d(!)453 3025 y Fp(\025)533 3055 y Fr(\))37 b Fo(\030)t(;)14 b(\021)26 b(<)d(!)940 3025 y Fp(\013)1015 3055 y Fu(f)r(\177)-44 b(ur)27 b(ein)h Fo(\013)23 b(<)g(\025)1529 3008 y Fn(I)p Fp(:)p Fn(V)q Fp(:)1545 3055 y Fr(\))38 b Fo(\030)22 b Fu(+)c Fo(\021)27 b(<)22 b(!)2017 3025 y Fp(\013)2087 3055 y Fo(<)h(!)2230 3025 y Fp(\025)2273 3055 y Fu(.)0 3196 y(2.)36 b Fo(\015)28 b Fr(62)c(f)p Fo(!)348 3166 y Fp(\013)417 3196 y Fu(:)37 b Fo(\013)24 b Fr(2)f Fo(O)r(n)p Fr(g)46 b(\))g Fo(\015)28 b Fr(62)23 b Fo(H)7 b Fu(.)0 3337 y(Bew)n(eis:)41 b(Sei)30 b(1)c Fr(\024)h Fo(\015)k Fr(62)d(f)p Fo(!)861 3307 y Fp(\013)934 3337 y Fu(:)41 b Fo(\013)27 b Fr(2)h Fo(O)r(n)p Fr(g)p Fu(.)44 b(Dann)30 b Fo(\015)h Fu(=)c Fo(!)1833 3307 y Fp(\014)1897 3337 y Fr(\001)20 b Fo(n)g Fu(+)g Fo(\015)2138 3349 y Fn(0)2205 3337 y Fu(mit)31 b(0)26 b Fo(<)h(n)f(<)h(!)17 b Fu(&)d Fo(\015)2879 3349 y Fn(0)2942 3337 y Fo(<)27 b(!)3089 3307 y Fp(\014)3163 3337 y Fu(und)j(1)d Fo(<)f(n)20 b Fr(_)h Fu(0)26 b Fo(<)h(\015)3840 3349 y Fn(0)3877 3337 y Fu(.)0 3479 y(F)r(\177)-44 b(ur)28 b Fo(\021)e Fu(:=)c Fo(!)393 3448 y Fp(\014)456 3479 y Fr(\001)d Fu(\()p Fo(n)f Fr(\000)g Fu(1\))h(+)f Fo(\015)900 3491 y Fn(0)965 3479 y Fu(gilt)27 b(n)n(un)h(0)23 b Fo(<)f(\021)k(<)d (!)1638 3448 y Fp(\014)1701 3479 y Fr(\001)18 b Fo(n)23 b Fr(\024)g Fo(\015)32 b Fu(und)c Fo(!)2199 3448 y Fp(\014)2267 3479 y Fo(<)22 b(!)2409 3448 y Fp(\014)2472 3479 y Fu(+)c Fo(\021)27 b Fu(=)22 b Fo(\015)5 b Fu(,)28 b(d.h.)37 b Fo(\015)27 b Fr(62)d Fo(H)7 b Fu(.)0 3645 y(b\))34 b(1.)53 b(Sei)34 b Fo(\015)j Fr(2)c Fo(H)40 b Fu(und)34 b Fo(\030)j(<)32 b(\015)5 b Fu(.)54 b(Dann)33 b Fo(\030)27 b Fu(+)22 b Fo(\015)37 b Fu(=)32 b(sup)p Fr(f)p Fo(\030)26 b Fu(+)c Fo(\021)j Fu(+)d(1)32 b(:)h Fo(\021)j(<)c(\015)5 b Fr(g)31 b(\024)i Fo(\015)5 b Fu(,)34 b(da)f(\()p Fo(\030)t(;)14 b(\021)36 b(<)c(\014)51 b Fr(\))c Fo(\030)26 b Fu(+)c Fo(\021)36 b(<)c(\015)51 b Fr(\))0 3786 y Fo(\030)23 b Fu(+)18 b Fo(\021)j Fu(+)d(1)23 b Fr(\024)f Fo(\015)5 b Fu(\).)37 b(2.)g(Gelte)28 b Fo(\015)f(>)c Fu(0)14 b(&)g Fr(8)p Fo(\030)25 b(<)e(\015)5 b Fu(\()p Fo(\030)22 b Fu(+)c Fo(\015)28 b Fu(=)23 b Fo(\015)5 b Fu(\),)27 b(und)h(sei)g Fo(\030)t(;)14 b(\021)26 b(<)d(\015)5 b Fu(.)36 b(Dann)28 b Fo(\030)23 b Fu(+)18 b Fo(\021)26 b(<)c(\030)h Fu(+)18 b Fo(\015)28 b Fu(=)22 b Fo(\015)5 b Fu(.)0 4002 y Ft(Abk)s(\177)-50 b(urzung)56 b Fo(\013)23 b Fu(=)661 4014 y Fp(N)-5 b(F)786 4002 y Fo(\013)839 4014 y Fn(0)895 4002 y Fu(+)18 b Fo(:)c(:)g(:)k Fu(+)h Fo(\013)1230 4014 y Fp(n)1321 4002 y Fu(:)k Fr(\()-14 b(\))69 b Fo(\013)24 b Fu(=)f Fo(\013)1806 4014 y Fn(0)1861 4002 y Fu(+)18 b Fo(:)c(:)g(:)19 b Fu(+)f Fo(\013)2196 4014 y Fp(n)2264 4002 y Fu(&)23 b Fo(\013)2405 4014 y Fn(0)2442 4002 y Fo(;)14 b(:::;)g(\013)2638 4014 y Fp(n)2706 4002 y Fr(2)24 b Fo(H)30 b Fu(&)22 b Fo(\013)3024 4014 y Fn(0)3085 4002 y Fr(\025)g Fo(:)14 b(:)g(:)23 b Fr(\025)g Fo(\013)3433 4014 y Fp(n)3478 4002 y Fu(.)0 4218 y Ft(Lemma)30 b(3.16)0 4359 y Fu(a\))55 b(Zu)28 b(jedem)g Fo(\013)23 b(>)g Fu(0)k(existiert)g(genau)g(ein)h(T)-7 b(up)r(el)28 b Fo(\013)1712 4371 y Fn(0)1750 4359 y Fo(;)14 b(:::;)g(\013)1946 4371 y Fp(n)2014 4359 y Fr(2)23 b Fo(H)35 b Fu(mit)28 b Fo(\013)23 b Fu(=)2489 4371 y Fp(N)-5 b(F)2615 4359 y Fo(\013)2668 4371 y Fn(0)2724 4359 y Fu(+)18 b Fo(:::)g Fu(+)g Fo(\013)3030 4371 y Fp(n)3076 4359 y Fu(.)0 4500 y(b\))51 b(Ist)28 b Fo(\013)23 b Fu(=)g Fo(\013)469 4512 y Fn(0)525 4500 y Fu(+)18 b Fo(:::)g Fu(+)g Fo(\013)831 4512 y Fp(n)904 4500 y Fu(mit)29 b(additiv)n(en)e(Hauptzahlen)h Fo(\013)1961 4512 y Fn(0)2021 4500 y Fr(\025)23 b Fo(:)14 b(:)g(:)23 b Fr(\025)f Fo(\013)2369 4512 y Fp(n)2414 4500 y Fu(,)28 b(so)f(gilt)h(f)r(\177)-44 b(ur)27 b(jedes)h Fo(k)e(<)d(n)p Fu(:)129 4641 y Fo(\013)182 4653 y Fn(0)238 4641 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)545 4653 y Fp(k)609 4641 y Fo(<)k(\013)28 b Fu(und)g Fo(\013)996 4653 y Fp(k)q Fn(+1)1140 4641 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)1447 4653 y Fp(n)1515 4641 y Fo(<)23 b(\013)p Fu(.)0 4807 y Fm(Beweis:)0 4948 y Fu(a\))k(folgt)h(aus)f(dem)g(Satz)j(\177)-44 b(ub)r(er)27 b(die)h(Can)n(torsc)n(he-Normalform)c(\(zur)j(Basis)g Fo(!)s Fu(\))g(un)n(ter)g(Ber)r(\177)-44 b(uc)n(ksic)n(h)n(tigung)26 b(der)h(Gleic)n(h)n(ung)0 5089 y Fo(!)55 5059 y Fp(\014)118 5089 y Fr(\001)18 b Fo(n)23 b Fu(=)g Fo(!)375 5059 y Fp(\014)438 5089 y Fu(+)18 b Fo(:::)g Fu(+)g Fo(!)746 5059 y Fp(\014)791 5089 y Fu(.)0 5230 y(b\))28 b(Die)g(erste)f(Ungleic) n(h)n(ung)g(ist)h(trivial.)36 b(Nun)28 b(zur)f(zw)n(eiten)h(Ungleic)n (h)n(ung.)1908 5539 y(15)p eop %%Page: 16 17 16 16 bop 0 83 a Fu(F)-7 b(all)28 b(1:)36 b Fo(\013)317 95 y Fp(k)q Fn(+1)465 83 y Fo(<)23 b(\013)606 95 y Fp(k)647 83 y Fu(.)37 b(Dann)28 b Fo(\013)985 95 y Fp(k)q Fn(+1)1128 83 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)1435 95 y Fp(n)1503 83 y Fo(<)23 b(\013)1644 95 y Fp(k)1708 83 y Fo(<)f(\013)p Fu(.)0 224 y(F)-7 b(all)28 b(2:)36 b Fo(\013)317 236 y Fp(k)q Fn(+1)465 224 y Fu(=)23 b Fo(\013)606 236 y Fp(k)647 224 y Fu(.)37 b(Dann)28 b Fo(\013)985 236 y Fp(k)q Fn(+1)1128 224 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)1435 236 y Fp(n)1503 224 y Fr(\024)23 b Fo(\013)1644 236 y Fp(k)1703 224 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)2010 236 y Fp(n)p Fi(\000)p Fn(1)2163 224 y Fo(<)23 b(\013)2304 236 y Fp(k)2363 224 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)2670 236 y Fp(n)2738 224 y Fr(\024)23 b Fo(\013)p Fu(.)0 390 y Ft(De\014nition)31 b Fu(\(Nat)r(\177)-44 b(urlic)n(he)27 b(Summe)h(o)r(der)f(Hessen)n(b)r(erg-Summe\))0 531 y Fo(\013)p Fu(#0)c(:=)g(0#)p Fo(\013)g Fu(:=)g Fo(\013)p Fu(.)0 672 y(F)r(\177)-44 b(ur)28 b Fo(\013)23 b Fu(=)302 684 y Fp(N)-5 b(F)427 672 y Fo(\013)480 684 y Fn(0)536 672 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)843 684 y Fp(n)916 672 y Fu(und)28 b Fo(\014)f Fu(=)1221 684 y Fp(N)-5 b(F)1347 672 y Fo(\013)1400 684 y Fp(n)p Fn(+1)1547 672 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)1854 684 y Fp(n)p Fn(+)p Fp(m)2037 672 y Fu(sei)27 b Fo(\013)p Fu(#)p Fo(\014)h Fu(:=)23 b Fo(\013)2518 687 y Fp(p)p Fn(\(0\))2660 672 y Fu(+)18 b Fo(:::)g Fu(+)h Fo(\013)2967 687 y Fp(p)p Fn(\()p Fp(m)p Fn(+)p Fp(n)p Fn(\))3208 672 y Fu(,)0 814 y(w)n(ob)r(ei)27 b Fo(p)h Fu(eine)f(P)n(erm)n(utation)f(v)n(on)h Fo(m)19 b Fu(+)f Fo(n)g Fu(+)g(1)27 b(mit)h Fo(\013)1713 829 y Fp(p)p Fn(\(0\))1860 814 y Fr(\025)22 b Fo(:::)i Fr(\025)e Fo(\013)2180 829 y Fp(p)p Fn(\()p Fp(m)p Fn(+)p Fp(n)p Fn(\))2422 814 y Fu(.)0 980 y Ft(Bemerkung)0 1121 y Fu(Ist)28 b Fo(\013)23 b Fu(=)264 1133 y Fp(C)-7 b(N)i(F)430 1121 y Fo(!)485 1091 y Fp(\015)520 1099 y Fg(0)574 1121 y Fr(\001)19 b Fo(k)659 1133 y Fn(0)715 1121 y Fu(+)f Fo(:)c(:)g(:)k Fu(+)g Fo(!)1051 1091 y Fp(\015)1086 1099 y Fh(n)1149 1121 y Fr(\001)g Fo(k)1233 1133 y Fp(n)1307 1121 y Fu(und)28 b Fo(\014)f Fu(=)1612 1133 y Fp(C)-7 b(N)i(F)1778 1121 y Fo(!)1833 1091 y Fp(\015)1868 1099 y Fg(0)1922 1121 y Fr(\001)19 b Fo(l)1989 1133 y Fn(0)2044 1121 y Fu(+)f Fo(:)c(:)g(:)19 b Fu(+)f Fo(!)2381 1091 y Fp(\015)2416 1099 y Fh(n)2478 1121 y Fr(\001)h Fo(l)2545 1133 y Fp(n)2590 1121 y Fu(,)28 b(so)f(gilt)0 1262 y Fo(\013)p Fu(#)p Fo(\014)h Fu(=)23 b Fo(!)340 1232 y Fp(\015)375 1240 y Fg(0)429 1262 y Fr(\001)c Fu(\()p Fo(k)546 1274 y Fn(0)583 1262 y Fu(+)p Fo(l)673 1274 y Fn(0)710 1262 y Fu(\))g(+)f Fo(:)c(:)g(:)k Fu(+)g Fo(!)1097 1232 y Fp(\015)1132 1240 y Fh(n)1195 1262 y Fr(\001)g Fu(\()p Fo(k)1311 1274 y Fp(n)1357 1262 y Fu(+)p Fo(l)1447 1274 y Fp(n)1492 1262 y Fu(\))p Fo(:)0 1428 y Ft(Lemma)30 b(3.17)0 1569 y Fu(a\))d Fo(\013)p Fu(#)p Fo(\014)h Fu(=)23 b Fo(\014)t Fu(#)p Fo(\013)p Fu(,)0 1710 y(b\))28 b(\()p Fo(\013)p Fu(#)p Fo(\014)t Fu(\)#)p Fo(\015)h Fu(=)23 b Fo(\013)p Fu(#\()p Fo(\014)t Fu(#)p Fo(\015)5 b Fu(\),)0 1851 y(c\))28 b Fo(\013)150 1863 y Fn(0)210 1851 y Fr(\025)23 b Fo(:::)g Fr(\025)g Fo(\013)531 1863 y Fp(n)604 1851 y Fu(additiv)n(e)k (Hauptzahlen)h Fr(\))46 b Fo(\013)1591 1863 y Fn(0)1647 1851 y Fu(+)18 b Fo(:::)g Fu(+)g Fo(\013)1953 1863 y Fp(n)2022 1851 y Fu(=)k Fo(\013)2162 1863 y Fn(0)2200 1851 y Fu(#)p Fo(:::)p Fu(#)p Fo(\013)2460 1863 y Fp(n)2505 1851 y Fu(,)0 1993 y(d\))28 b Fo(\014)f(<)c(\015)51 b Fr(\))46 b Fo(\013)p Fu(#)p Fo(\014)28 b(<)23 b(\013)p Fu(#)p Fo(\015)5 b Fu(,)0 2134 y(e\))28 b Fo(\013;)14 b(\014)28 b(<)22 b(!)404 2104 y Fp(\015)493 2134 y Fr(\))46 b Fo(\013)p Fu(#)p Fo(\014)28 b(<)22 b(!)961 2104 y Fp(\015)1004 2134 y Fu(,)0 2275 y(f)6 b(\))29 b Fo(\013)19 b Fu(+)f Fo(\014)27 b Fr(\024)c Fo(\013)p Fu(#)p Fo(\014)t Fu(.)0 2441 y Fm(Beweis:)0 2582 y Fu(a\),b\),e\))28 b(klar.)0 2723 y(d\))g(W)-7 b(egen)27 b(b\))h(und)f(c\))h(reic)n(h)n(t)f(es,)g (die)g(Behauptung)g(f)r(\177)-44 b(ur)28 b Fo(\013)23 b Fr(2)g Fo(H)35 b Fu(zu)27 b(b)r(ew)n(eisen.)36 b(Sei)28 b Fo(\013)23 b Fu(=)g Fo(!)2925 2693 y Fp(\013)2968 2701 y Fg(0)3004 2723 y Fu(,)28 b Fo(\014)f Fu(=)3194 2735 y Fp(C)-7 b(N)i(F)3360 2723 y Fo(!)3415 2693 y Fp(\016)3445 2701 y Fg(0)3499 2723 y Fr(\001)18 b Fo(k)3583 2735 y Fn(0)3638 2723 y Fu(+)g Fo(:)c(:)g(:)j Fu(+)0 2864 y Fo(!)55 2834 y Fp(\016)85 2842 y Fh(n)144 2864 y Fr(\001)d Fo(k)224 2876 y Fp(n)295 2864 y Fu(und)26 b Fo(\015)h Fu(=)594 2876 y Fp(C)-7 b(N)i(F)760 2864 y Fo(!)815 2834 y Fp(\016)845 2842 y Fg(0)895 2864 y Fr(\001)15 b Fo(l)958 2876 y Fn(0)1009 2864 y Fu(+)e Fo(:)h(:)g(:)g Fu(+)g Fo(!)1332 2834 y Fp(\016)1362 2842 y Fh(n)1421 2864 y Fr(\001)g Fo(l)1483 2876 y Fp(n)1553 2864 y Fu(mit)26 b Fo(\013)1756 2876 y Fn(0)1817 2864 y Fu(=)c Fo(\016)1941 2876 y Fp(m)2030 2864 y Fu(f)r(\177)-44 b(ur)25 b(ein)h Fo(m)d Fr(\024)f Fo(n)p Fu(.)36 b(Dann)26 b Fo(\013)p Fu(#)p Fo(\014)i Fu(=)23 b Fo(!)3146 2834 y Fp(\016)3176 2842 y Fg(0)3226 2864 y Fr(\001)14 b Fo(k)3309 2834 y Fi(0)3306 2885 y Fn(0)3358 2864 y Fu(+)f Fo(:)h(:)g(:)g Fu(+)g Fo(!)3681 2834 y Fp(\015)3716 2842 y Fh(n)3774 2864 y Fr(\001)h Fo(k)3858 2834 y Fi(0)3855 2885 y Fp(n)0 3005 y Fu(und)28 b Fo(\013)p Fu(#)p Fo(\015)g Fu(=)23 b Fo(!)502 2975 y Fp(\016)532 2983 y Fg(0)587 3005 y Fr(\001)18 b Fo(l)655 2975 y Fi(0)653 3026 y Fn(0)709 3005 y Fu(+)g Fo(:)c(:)g(:)k Fu(+)g Fo(!)1045 2975 y Fp(\015)1080 2983 y Fh(n)1143 3005 y Fr(\001)g Fo(l)1211 2975 y Fi(0)1209 3026 y Fp(n)1254 3005 y Fu(,)28 b(w)n(ob)r(ei)f Fo(k)1586 2975 y Fi(0)1583 3026 y Fp(m)1670 3005 y Fu(=)22 b Fo(k)1800 3017 y Fp(m)1864 3005 y Fu(+)o(1)p Fo(;)37 b(l)2057 2975 y Fi(0)2055 3026 y Fp(m)2141 3005 y Fu(=)22 b Fo(l)2253 3017 y Fp(m)2316 3005 y Fu(+1)27 b(und)h Fo(k)2662 2975 y Fi(0)2659 3027 y Fp(i)2710 3005 y Fu(=)22 b Fo(k)2840 3017 y Fp(i)2868 3005 y Fo(;)37 b(l)2955 2975 y Fi(0)2953 3027 y Fp(i)3004 3005 y Fu(=)22 b Fo(l)e Fr(\000)e Fo(i)28 b Fu(f)r(\177)-44 b(ur)27 b Fo(i)c Fr(6)p Fu(=)f Fo(m)p Fu(.)0 3147 y(f)6 b(\))29 b(1.)36 b(Ist)28 b Fo(\013)23 b Fr(2)h Fo(H)7 b Fu(,)27 b(so)g Fo(\013)19 b Fu(+)f Fo(\014)28 b Fu(=)22 b Fo(\014)28 b Fr(\024)22 b Fo(\013)p Fu(#)p Fo(\014)33 b Fu(o)r(der)27 b Fo(\013)19 b Fu(+)f Fo(\014)27 b Fu(=)c Fo(\013)p Fu(#)p Fo(\014)t Fu(.)0 3288 y(2.)36 b(F)r(\177)-44 b(ur)28 b(b)r(eliebiges)f Fo(\013)h Fu(folgt)g(die)g(Behauptung)f(aus)g(1.)36 b(und)28 b(der)g(Assoziativit\177)-42 b(at)26 b(v)n(on)h(#.)1908 5539 y(16)p eop %%Page: 17 18 17 17 bop 0 83 a Fr(x)p Ft(4)63 b(Pro)s(of)32 b(theoretic)g(analysis)f (of)h Fr(Z)38 b Ft(via)33 b(the)e(in\014nitary)i(system)d Fe(Z)2483 46 y Fi(1)0 299 y Ft(The)i(axiom)e(system)g Fr(Z)39 b Ft(of)32 b(pure)g(n)m(um)m(b)s(er)f(theory)0 465 y Fu(Inductiv)n(e)d(De\014nition)g(of)g(sets)f(PR)1127 429 y Fp(n)1200 465 y Fu(of)h Fo(n)p Fu(-ary)e(function)i(sym)n(b)r (ols)0 631 y(\(PR)g(1\))f Ft(0)327 601 y Fp(n)395 631 y Fr(2)c Fu(PR)591 595 y Fp(n)659 631 y Fu(\()p Fo(n)g Fr(\025)g Fu(0\),)55 b Fk(S)24 b Fr(2)f Fu(PR)1270 595 y Fn(1)1307 631 y Fu(,)55 b(I)1415 601 y Fp(n)1415 653 y(i)1483 631 y Fr(2)24 b Fu(PR)1679 595 y Fp(n)1748 631 y Fu(\(1)f Fr(\024)f Fo(i)h Fr(\024)f Fo(n)p Fu(\).)0 772 y(\(PR)28 b(2\))f Fo(h)c Fr(2)g Fu(PR)546 736 y Fp(m)632 772 y Fu(&)g Fo(g)760 784 y Fn(1)797 772 y Fo(;)14 b(:::;)g(g)980 784 y Fp(m)1065 772 y Fr(2)24 b Fu(PR)1261 736 y Fp(n)1330 772 y Fu(&)e Fo(m;)14 b(n)23 b Fr(\025)g Fu(1)45 b(=)-14 b Fr(\))46 b Fu(\()p Fr(\016)p Fo(hg)2117 784 y Fn(1)2153 772 y Fo(:::g)2262 784 y Fp(m)2325 772 y Fu(\))24 b Fr(2)f Fu(PR)2577 736 y Fp(n)2622 772 y Fu(.)0 913 y(\(PR)28 b(3\))f Fo(g)f Fr(2)d Fu(PR)541 877 y Fp(n)609 913 y Fu(&)g Fo(h)f Fr(2)i Fu(PR)964 877 y Fp(n)p Fn(+2)1139 913 y Fu(=)-14 b Fr(\))46 b Fu(\(R)p Fo(g)s(h)p Fu(\))23 b Fr(2)g Fu(PR)1754 877 y Fp(n)p Fn(+1)1883 913 y Fu(.)0 1104 y(Abbreviation:)36 b(PR)23 b(:=)787 1042 y Fl(S)856 1129 y Fp(n)p Fi(2)p Fn(I)-11 b(N)1021 1104 y Fu(PR)1139 1068 y Fp(n)1184 1104 y Fu(,)55 b Ft(0)23 b Fu(:=)g Ft(0)1492 1074 y Fn(0)1557 1104 y Fu(.)0 1270 y Fr(L)57 1282 y Fn(0)118 1270 y Fu(:=)f(PR)c Fr([)h(f)p 488 1244 50 8 v 488 1262 V 66 w(g)27 b Fu(,)h(where)p 914 1244 V 914 1262 V 121 w(is)g(a)f(binary)g(relation)f(sym)n(b)r(ol)h(\(equalit)n (y\).)0 1467 y(The)h Fr(L)228 1479 y Fn(0)265 1467 y Fu(-terms)f Ft(0)p Fo(;)h Fk(S)p Ft(0)p Fo(;)f Fk(SS)p Ft(0)p Fo(;)14 b(::::)28 b Fu(are)f(called)g(n)n(umerals.)36 b(F)-7 b(or)27 b Fo(n)c Fr(2)g Fu(I)-14 b(N)28 b(let)g Fo(n)p 2350 1480 50 4 v 23 w Fu(:=)2594 1333 y Fp(n)2534 1384 y Fl(z)p 2571 1384 6 10 v 6 w(}|)p 2651 1384 V 6 w({)2534 1467 y Fk(S)p Fo(:::)p Fk(S)14 b Ft(0)p Fu(.)0 1633 y Fo(T)34 b Fu(:=)27 b(set)h(of)g(all)f(closed)g Fr(L)840 1645 y Fn(0)877 1633 y Fu(-terms.)0 1774 y(If)h Fo(t)23 b Fr(2)g Fo(T)39 b Fu(then)28 b Fk(val)p Fu(\()p Fo(t)p Fu(\))h(denotes)e(its)h(canonical)e Fm(value)p Fu(.)37 b(Hence)28 b Fk(val)p Fu(\()p Fo(n)p 2113 1787 50 4 v Fu(\))c(=)e Fo(n)p Fu(.)0 1940 y(TR)n(UE)238 1952 y Fn(0)298 1940 y Fu(:=)g(set)g(of)g(all)g(true)g(closed)f(literals)h (of)g Fr(L)1554 1952 y Fn(0)1613 1940 y Fu([=)h Fr(f)p Fo(s)p 1813 1914 50 8 v 1813 1932 V 66 w(t)g Fu(:)g Fo(s;)14 b(t)23 b Fr(2)g Fo(T)34 b Fu(&)23 b Fk(val)p Fu(\()p Fo(s)p Fu(\))h(=)e Fk(val)p Fu(\()p Fo(t)p Fu(\))p Fr(g)8 b([)g(f:)p Fu(\()p Fo(s)p 3141 1914 V 3141 1932 V 66 w(t)p Fu(\))24 b(:)f Fk(val)p Fu(\()p Fo(s)p Fu(\))g Fr(6)p Fu(=)g Fk(val)p Fu(\()p Fo(t)p Fu(\))p Fr(g)p Fu(])0 2131 y(The)31 b Ft(language)k(of)h Fr(Z)h Fu(is)31 b Fr(L)928 2143 y Fn(0)965 2131 y Fu(\()p Fr(X)12 b Fu(\))30 b(:=)e Fr(L)1303 2143 y Fn(0)1361 2131 y Fr([)21 b(f)p Fo(X)1548 2143 y Fn(0)1585 2131 y Fo(;)14 b(X)1691 2143 y Fn(1)1728 2131 y Fo(;)g(:::)p Fr(g)p Fu(,)31 b(where)f Fo(X)2242 2143 y Fn(0)2279 2131 y Fo(;)14 b(X)2385 2143 y Fn(1)2422 2131 y Fo(;)g(:::)31 b Fu(are)f(unary)g(predicate)g(sym)n (b)r(ols;)i(w)n(e)e(call)0 2272 y(them)21 b Fs(set)e(v)-5 b(ariables)p Fu(.)34 b(But)20 b(note)g(that)g(they)g(are)f(not)h (considered)f(as)g(v)-5 b(ariables)19 b(in)h(the)g(prop)r(er)f(sense)h (\(e.g.)34 b(FV)r(\()p Fo(X)3585 2284 y Fp(i)3612 2272 y Ft(0)p Fu(\))24 b(=)e Fr(;)p Fu(\).)0 2413 y(W)-7 b(e)28 b(use)f Fo(X)34 b Fu(as)27 b(syn)n(tactic)g(v)-5 b(ariable)27 b(for)g Fo(X)1350 2425 y Fn(0)1387 2413 y Fo(;)14 b(X)1493 2425 y Fn(1)1530 2413 y Fo(;)g(:::)p Fu(.)0 2579 y(The)28 b Ft(axioms)i(of)h Fr(Z)k Fu(are)27 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Fk(S)p Fo(x)p Fu(\)\))h Fr(!)f Fo(F)1118 4102 y Fp(x)1160 4090 y Fu(\()p Fo(z)t Fu(\),)28 b(for)f(eac)n(h)g Fr(L)1689 4102 y Fn(0)1726 4090 y Fu(\()p Fr(X)12 b Fu(\)-form)n(ula)28 b Fo(F)1908 5539 y Fu(17)p eop %%Page: 18 19 18 18 bop 0 83 a Ft(De\014nition)0 224 y Fu(Let)28 b Fo(R)g Fu(b)r(e)g(an)f Fr(L)525 236 y Fn(0)563 224 y Fu(-form)n(ula)f(with)i(FV)r(\()p Fo(R)q Fu(\))23 b(=)g Fr(f)p Fo(x;)14 b(y)s Fr(g)27 b Fu(suc)n(h)g(that)h(the)g(relation)0 365 y Fr(\036)p Fu(:=)22 b Fr(f)p Fu(\()p Fo(m;)14 b(n)p Fu(\))23 b Fr(2)g Fu(I)-14 b(N)621 329 y Fn(2)681 365 y Fu(:)23 b(I)-14 b(N)24 b Fr(j)-14 b Fu(=)23 b Fo(R)989 377 y Fp(y)r(;x)1086 365 y Fu(\()p Fo(m;)14 b(n)p Fu(\))p Fr(g)27 b Fu(is)h(w)n(ellfounded.)0 506 y(By)f(recursion)f(o)n(v)n(er)g Fr(\036)h Fu(one)h(de\014nes)f(the)h Fr(\036)p Fm(-norm)h Fr(j)p Fo(n)p Fr(j)1736 518 y Fi(\036)1820 506 y Fu(of)e Fo(n)c Fr(2)h Fu(I)-14 b(N:)0 648 y Fr(j)p Fo(n)p Fr(j)96 660 y Fi(\036)175 648 y Fu(:=)23 b(sup)p Fr(fj)p Fo(m)p Fr(j)572 660 y Fi(\036)646 648 y Fu(+)18 b(1)k(:)h Fo(m)g Fr(\036)g Fo(n)p Fr(g)p Fu(.)0 814 y(Abbreviations:)0 955 y Fo(s)g Fr(\036)g Fo(t)g Fu(:=)f Fo(R)376 967 y Fp(y)r(;x)474 955 y Fu(\()p Fo(s;)14 b(t)p Fu(\))28 b(,)83 b Fr(8)p Fo(y)s Fr(\036)n Fo(t)14 b(F)e Fu(\()p Fo(y)s Fu(\))46 b(:=)37 b Fr(8)p Fo(y)s Fu(\()p Fo(y)s Fr(\036)n Fo(t)23 b Fr(!)g Fo(F)12 b Fu(\()p Fo(y)s Fu(\)\),)0 1096 y Fr(j)p Fo(t)p Fr(j)76 1108 y Fi(\036)155 1096 y Fu(:=)23 b Fr(j)p Fk(val)p Fu(\()p Fo(t)p Fu(\))p Fr(j)504 1108 y Fi(\036)588 1096 y Fu(for)k Fo(t)c Fr(2)h Fo(T)12 b Fu(,)0 1237 y(Prog)173 1257 y Fi(\036)229 1237 y Fu(\()p Fo(F)g Fu(\))24 b(:=)e Fr(8)p Fo(x)p Fu(\()p Fr(8)p Fo(y)s Fr(\036)o Fo(xF)12 b Fu(\()p Fo(y)s Fu(\))37 b Fr(!)23 b Fo(F)12 b Fu(\()p Fo(x)p Fu(\)\),)0 1378 y(TI)90 1390 y Fi(\036)146 1378 y Fu(\()p Fo(F)r(;)i(t)p Fu(\))24 b(:=)f(Prog)640 1398 y Fi(\036)696 1378 y Fu(\()p Fo(F)12 b Fu(\))24 b Fr(!)f(8)p Fo(x)p Fr(\036)o Fo(tF)12 b Fu(\()p Fo(x)p Fu(\).)0 1544 y(In)29 b(this)h(section)f(w)n(e)f(will)i(sho)n(w) e(that)h Fm(tr)l(ans\014nite)i(induction)g(up)g(to)g Fo(")2244 1556 y Fn(0)2311 1544 y Fu(is)e(not)g(pro)n(v)-5 b(able)28 b(in)h Fr(Z)7 b Fu(,)29 b(more)g(precisely)f(w)n(e)h(will)0 1685 y(establish)e(the)h(follo)n(wing)0 1851 y Ft(Theorem)54 b Fr(Z)30 b(`)23 b Fu(TI)683 1863 y Fi(\036)739 1851 y Fu(\()p Fo(X)r(;)14 b(n)p 879 1864 50 4 v -1 w Fu(\))47 b(=)-14 b Fr(\))46 b(j)p Fo(n)p Fr(j)1283 1863 y Fi(\036)1362 1851 y Fo(<)22 b(")1488 1863 y Fn(0)1525 1851 y Fu(.)0 2017 y(Sk)n(etc)n(h)27 b(of)h(the)g(pro)r(of:)0 2183 y(W)-7 b(e)28 b(de\014ne)g(an)f(in\014nitary)g(pro)r(of)g(system)h Fe(Z)1401 2146 y Fi(1)1471 2183 y Fu(,)g(whic)n(h)g(\(essen)n(tially\)) f(results)g(from)g(PL1)g(b)n(y)0 2325 y(\(i\))107 b(replacing)26 b(eac)n(h)h(inference)g(sym)n(b)r(ol)1368 2262 y Fl(V)1438 2283 y Fp(y)1438 2350 y Fi(8)p Fp(xA)1595 2325 y Fu(b)n(y)g(its)h (in\014nitary)f(v)n(ersion)194 2483 y(\()226 2421 y Fl(V)295 2508 y Fi(8)p Fp(xA)425 2483 y Fu(\))527 2428 y Fo(:)14 b(:)g(:)f(A)699 2440 y Fp(x)741 2428 y Fu(\()p Fo(t)p Fu(\))h Fo(:)g(:)g(:)h Fu(\()p Fo(t)p Fr(2)p Fo(T)d Fu(\))p 513 2464 658 4 v 764 2538 a Fr(8)p Fo(xA)1291 2483 y Fu(\()p Fo(!)s Fu(-rule\))0 2663 y(\(ii\))84 b(adding)27 b(the)h(axioms)e Fo(A)i Fu(\(for)g Fo(A)23 b Fr(2)g Fu(TR)n(UE)1539 2675 y Fn(0)1576 2663 y Fu(\))28 b(and)g Fo(X)7 b(s;)14 b Fr(:)p Fo(X)7 b(t)26 b Fu(\(for)h Fo(s;)14 b(t)23 b Fr(2)h Fo(T)38 b Fu(with)28 b Fk(val)p Fu(\()p Fo(s)p Fu(\))c(=)f Fk(val)p Fu(\()p Fo(t)p Fu(\)\))0 2829 y(Then)28 b(w)n(e)f(pro)n(v)n(e:)0 2970 y(Lemma)g(4.5)55 b(PL1)22 b Fr(`)682 2940 y Fp(k)682 2990 y Fn(0)745 2970 y Fu(\000)h(&)g(FV)r (\(\000\))g(=)g Fr(;)45 b Fu(=)-14 b Fr(\))46 b Fe(Z)1568 2933 y Fi(1)1662 2970 y Fr(`)1713 2940 y Fp(k)1713 2990 y Fn(0)1776 2970 y Fu(\000.)0 3111 y(Lemma)27 b(4.8)55 b Fo(A)23 b Fr(2)h(Z)52 b Fu(=)-14 b Fr(\))46 b Fe(Z)962 3074 y Fi(1)1056 3111 y Fr(`)1107 3081 y Fp(\013)1107 3131 y Fn(0)1177 3111 y Fo(A)28 b Fu(for)f(some)g 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Fo(;)14 b(:::;)g Fr(:)p Fo(A)1590 3712 y Fp(l)1616 3700 y Fo(;)g Fu(TI)1743 3712 y Fi(\036)1799 3700 y Fu(\()p Fo(X)r(;)g(n)p 1939 3713 V Fu(\))28 b(for)f(some)g Fo(k)f(<)c(!)31 b Fu(and)c Fo(A)2846 3712 y Fn(1)2884 3700 y Fo(;)14 b(:::;)g(A)3089 3712 y Fp(l)3138 3700 y Fr(2)23 b(Z)3353 3653 y Fn(4)p Fp(:)p Fn(5)3329 3700 y Fu(=)-14 b Fr(\))0 3841 y Fe(Z)48 3804 y Fi(1)142 3841 y Fr(`)193 3811 y Fp(k)193 3862 y Fn(0)256 3841 y Fr(:)p Fo(A)373 3853 y Fn(1)411 3841 y Fo(;)14 b(:::;)g Fr(:)p Fo(A)671 3853 y Fp(l)697 3841 y Fo(;)g Fu(TI)824 3853 y Fi(\036)880 3841 y Fu(\()p Fo(X)r(;)g(n)p 1020 3854 V Fu(\))1172 3794 y Fn(4)p Fp(:)p Fn(8)1148 3841 y Fu(=)-14 b Fr(\))46 b Fe(Z)1376 3804 y Fi(1)1470 3841 y Fr(`)1521 3811 y Fp(\013)1521 3862 y(m)1607 3841 y Fu(TI)1696 3853 y Fi(\036)1752 3841 y Fu(\()p Fo(X)r(;)14 b(n)p 1892 3854 V Fu(\))28 b(with)g Fo(\013)c(<)e(!)f Fu(+)d Fo(!)31 b Fu(and)c Fo(m)c Fu(:=)g(max)3116 3853 y Fn(1)p Fi(\024)p Fp(i)p Fi(\024)p Fp(l)3316 3841 y Fu(rk)o(\()p Fo(A)3486 3853 y Fp(i)3515 3841 y Fu(\)+)o(1)3723 3794 y Fn(4)p Fp(:)p Fn(2)3699 3841 y Fu(=)-14 b Fr(\))0 3983 y Fe(Z)48 3945 y Fi(1)142 3983 y Fr(`)193 3952 y Fp(\014)193 4003 y(m)278 3983 y Fu(TI)368 3995 y Fi(\036)424 3983 y Fu(\()p Fo(X)r(;)14 b(n)p 564 3996 V Fu(\))83 b(with)29 b Fo(\014)e(<)c(")1120 3995 y Fn(0)1227 3936 y(4)p Fp(:)p Fn(4)1203 3983 y Fu(=)-14 b Fr(\))22 b(j)p Fo(n)p Fr(j)1455 3995 y Fi(\036)1535 3983 y Fr(\024)g Fu(2)1664 3952 y Fp(\014)1732 3983 y Fo(<)g(")1858 3995 y Fn(0)1895 3983 y Fu(.)1908 5539 y(18)p eop %%Page: 19 20 19 19 bop 0 83 a Ft(The)32 b(in\014nitary)g(pro)s(of)g(system)e Fe(Z)1230 46 y Fi(1)0 274 y Fu(The)e(language)e(of)h Fe(Z)659 237 y Fi(1)757 274 y Fu(is)h Fr(L)898 286 y Fn(0)935 274 y Fu(\()p Fr(X)12 b Fu(\).)0 440 y(W)-7 b(e)38 b(in)n(tro)r(duce)f(the)h(follo)n(wing)e(relation)g Fr(')h Fu(b)r(et)n(w)n(een)h Fr(L)1849 452 y Fn(0)1886 440 y Fu(\()p Fr(X)12 b Fu(\)-form)n(ulas)37 b(and)g(\(p)r(ossibly)g (in\014nitary\))h(conjunctions)f(or)0 581 y(disjunctions)28 b(of)f Fr(L)611 593 y Fn(0)649 581 y Fu(\()p Fr(X)12 b Fu(\)-form)n(ulas:)0 747 y Fo(A)62 759 y Fn(0)118 747 y Fr(^)19 b Fo(A)254 759 y Fn(1)314 747 y Fr(')402 685 y Fl(V)471 772 y Fp(i)p Fi(2f)p Fn(0)p Fp(;)p Fn(1)p Fi(g)697 747 y Fo(A)759 759 y Fp(i)787 747 y Fu(,)257 b Fr(8)p Fo(xA)23 b Fr(')1334 685 y Fl(V)1403 772 y Fp(t)p Fi(2)p Fp(T)1525 747 y Fo(A)1587 759 y Fp(x)1629 747 y Fu(\()p Fo(t)p Fu(\),)229 b Fo(A)2037 759 y Fn(0)2093 747 y Fr(_)19 b Fo(A)2229 759 y Fn(1)2290 747 y Fr(')2377 685 y Fl(W)2447 772 y Fp(i)p Fi(2f)p Fn(0)p Fp(;)p Fn(1)p Fi(g)2673 747 y Fo(A)2735 759 y Fp(i)2763 747 y Fu(,)229 b Fr(9)p Fo(xA)24 b Fr(')3281 685 y Fl(W)3350 772 y Fp(t)p Fi(2)p Fp(T)3473 747 y Fo(A)3535 759 y Fp(x)3577 747 y Fu(\()p Fo(t)p Fu(\))0 913 y(Then)k(w)n(e)f(ha)n(v)n(e)0 1054 y({)55 b Fo(A)23 b Fr(')g(\003)312 1066 y Fp(\023)p Fi(2)p Fp(J)427 1054 y Fo(A)489 1066 y Fp(\023)532 1054 y Fu(&)14 b Fo(\023)23 b Fr(2)g Fo(J)54 b Fu(=)-14 b Fr(\))46 b Fu(rk)o(\()p Fo(A)1191 1066 y Fp(\023)1221 1054 y Fu(\))23 b Fo(<)g Fu(rk)o(\()p Fo(A)p Fu(\),)0 1196 y({)55 b Fo(A)23 b Fr(')g(\003)312 1208 y Fp(\023)p Fi(2)p Fp(J)427 1196 y Fo(A)489 1208 y Fp(\023)564 1196 y Fu(=)-14 b Fr(\))46 b(:)p Fu(\()p Fo(A)p Fu(\))24 b Fr(')p 1037 1147 42 4 v 23 w(\003)1078 1208 y Fp(\023)p Fi(2)p Fp(J)1194 1196 y Fr(:)p Fu(\()p Fo(A)1343 1208 y Fp(\023)1373 1196 y Fu(\),)k(where)p 1696 1123 70 4 v 1696 1133 a Fl(W)1788 1196 y Fu(:=)1899 1133 y Fl(V)1968 1196 y Fu(,)p 2019 1123 V 2019 1133 a Fl(V)2111 1196 y Fu(:=)2222 1133 y Fl(W)2291 1196 y Fu(,)0 1386 y Ft(De\014nition)0 1528 y Fr(AX)13 b Fu(\()p Fe(Z)218 1490 y Fi(1)288 1528 y Fu(\))24 b(:=)j(set)h(of)f(all)h(sequen)n(ts)f(\001)g(suc)n(h)h(that) 0 1669 y({)f(all)g(elemen)n(ts)h(of)g(\001)f(are)g(literals,)0 1810 y({)g(\001)19 b Fr(\\)g Fu(TR)n(UE)468 1822 y Fn(0)528 1810 y Fr(6)p Fu(=)k Fr(;)55 b Fu(or)f(\001)28 b(con)n(tains)f(a)g (subset)h Fr(f)p Fo(X)7 b(s;)14 b Fr(:)p Fo(X)7 b(t)p Fr(g)25 b Fu(with)j Fk(val)p Fu(\()p Fo(s)p Fu(\))c(=)e Fk(val)p Fu(\()p Fo(t)p Fu(\).)0 1976 y Ft(Remark:)63 b Fu(\001)491 1946 y Fi(0)515 1976 y Fo(;)14 b Fu(\001)621 1946 y Fi(00)686 1976 y Fr(2)24 b(AX)12 b Fu(\()p Fe(Z)983 1939 y Fi(1)1053 1976 y Fu(\))47 b(=)-14 b Fr(\))45 b Fu(\(\001)1412 1946 y Fi(0)1455 1976 y Fr(n)18 b(f)p Fo(C)6 b Fr(g)p Fu(\))18 b Fr([)g Fu(\(\001)1888 1946 y Fi(00)1950 1976 y Fr(n)g(f:)p Fo(C)6 b Fr(g)p Fu(\))23 b Fr(2)g(AX)13 b Fu(\()p Fe(Z)2565 1939 y Fi(1)2636 1976 y Fu(\))0 2192 y Fe(Z)48 2155 y Fi(1)119 2192 y Ft(-inferences)0 2358 y Fu(\()p Fk(Ax)126 2321 y Fi(1)126 2378 y Fn(\001)196 2358 y Fu(\))495 b(\001)320 b(if)28 b(\001)23 b Fr(2)g(AX)13 b Fu(\()p Fe(Z)1576 2321 y Fi(1)1647 2358 y Fu(\))0 2562 y(\()32 2500 y Fl(V)101 2587 y Fp(A)156 2562 y Fu(\))499 2507 y Fo(:)h(:)g(:)g(A)672 2519 y Fp(\023)715 2507 y Fo(:)g(:)g(:)g Fu(\()p Fo(\023)p Fr(2)q Fo(J)8 b Fu(\))p 486 2543 544 4 v 643 2617 a Fo(A)1112 2562 y Fu(if)28 b Fo(A)23 b Fr(')1361 2500 y Fl(V)1430 2587 y Fp(\023)p Fi(2)p Fp(J)1545 2562 y Fo(A)1607 2574 y Fp(\023)0 2804 y Fu(\()32 2742 y Fl(W)101 2763 y Fp(\026)101 2829 y(A)156 2804 y Fu(\))704 2750 y Fo(A)766 2762 y Fp(\026)p 690 2785 135 4 v 726 2859 a Fo(A)1112 2804 y Fu(if)28 b Fo(A)23 b Fr(')1361 2742 y Fl(W)1430 2829 y Fp(\023)p Fi(2)p Fp(J)1545 2804 y Fo(A)1607 2816 y Fp(\023)1664 2804 y Fu(and)28 b Fo(\026)23 b Fr(2)g Fo(J)0 3047 y Fu(\()p Fk(Cut)158 3059 y Fp(C)214 3047 y Fu(\))623 2992 y Fo(C)89 b Fr(:)p Fo(C)p 623 3028 269 4 v 736 3102 a Fr(;)0 3289 y Fu(\()p Fk(Rep)p Fu(\))736 3235 y Fr(;)p 723 3270 70 4 v 737 3344 a(;)0 3518 y Fu(o\()p Fo(d)p Fu(\))23 b(:=)g Fo(l)r(g)s Fu(\()p Fo(d)p Fu(\),)28 b(deg\()p Fk(Cut)794 3530 y Fp(C)850 3518 y Fu(\))c(:=)e(rk\()p Fo(C)6 b Fu(\))19 b(+)f(1)27 b(and)h(deg\()p Fr(I)6 b Fu(\))24 b(:=)f(0)k(for)g(all)h (other)f(inferences.)0 3709 y(Remark:)36 b(A)n(t)28 b(momen)n(t)f(w)n (e)h(could)f(do)g(without)h Fk(Rep)g Fu(inferences.)36 b(They)27 b(will)h(b)r(ecome)g(imp)r(ortan)n(t)f(later.)0 3900 y(NOT)-7 b(A)g(TION)0 4042 y(W)g(e)28 b(use)f Fo(d;)14 b(d)409 4054 y Fn(0)447 4042 y Fo(;)g(d)527 4054 y Fn(1)564 4042 y Fo(;)g(e;)g(:::)28 b Fu(as)f(syn)n(tactic)g(v)-5 b(ariables)26 b(for)h Fe(Z)1746 4004 y Fi(1)1817 4042 y Fu(-deriv)-5 b(ations.)0 4336 y Fo(d)23 b Fr(')154 4116 y Fl(8)154 4190 y(>)154 4215 y(>)154 4240 y(<)154 4390 y(>)154 4415 y(>)154 4440 y(:)418 4196 y Fo(d)461 4208 y Fp(\023)p 452 4292 5 59 v 228 4367 a Fo(:)14 b(:)g(:)f Fu(\000)390 4379 y Fp(\023)442 4367 y Fu(:)23 b Fo(\013)541 4379 y Fp(\023)584 4367 y Fo(:)14 b(:)g(:)p 228 4409 454 4 v 700 4425 a Fi(I)367 4484 y Fu(\000)23 b(:)g Fo(\013)811 4336 y Fu(:)p Fr(,)69 b Fo(d)24 b Fu(=)e Fr(I)6 b Fu(\()p Fo(d)1266 4348 y Fp(\023)1296 4336 y Fu(\))1328 4351 y Fp(\023)p Fi(2jI)t(j)1506 4336 y Fu(&)22 b Fr(8)p Fo(\023)h Fr(2)g(jI)6 b(j)p Fu(\(\000\()p Fo(d)2026 4348 y Fp(\023)2056 4336 y Fu(\))23 b Fr(\022)g Fu(\000)2251 4348 y Fp(\023)2303 4336 y Fu(&)f(o\()p Fo(d)2507 4348 y Fp(\023)2536 4336 y Fu(\))h Fr(\024)g Fo(\013)2732 4348 y Fp(\023)2784 4336 y Fo(<)g(\013)p Fu(\))g(&)3078 4281 y Fo(:::)p Fu(\000)3199 4293 y Fp(\023)3228 4281 y Fo(:::)p Fu(\()p Fo(\023)p Fr(2)q(jI)6 b(j)p Fu(\))p 3078 4317 466 4 v 3285 4391 a(\000)3554 4331 y Fi(I)3623 4336 y Fu(,)0 4673 y(where)278 4618 y Fo(:::)p Fu(\000)399 4630 y Fp(\023)428 4618 y Fo(:::)p Fu(\()p Fo(\023)p Fr(2)q(jI)g(j)p Fu(\))p 278 4654 V 485 4728 a(\000)753 4668 y Fi(I)841 4673 y Fu(:)p Fr(,)46 b Fu(\001\()p Fr(I)6 b Fu(\))24 b Fr(\022)f Fu(\000)g(&)f Fr(8)p Fo(\023)h Fr(2)g(jI)6 b(j)p Fu(\(\000)1809 4685 y Fp(\023)1862 4673 y Fr(\022)22 b Fu(\000)p Fo(;)14 b Fu(\001)2107 4685 y Fp(\023)2136 4673 y Fu(\()p Fr(I)6 b Fu(\)\).)0 4940 y(Note)28 b(that)g Fo(d)23 b Fr(')534 4823 y Fl(\032)620 4885 y Fo(:)14 b(:)g(:)g(:)g(:)g(:)p 607 4921 222 4 v 630 4995 a Fu(\000)23 b(:)g Fo(\013)838 4935 y Fi(I)917 4940 y Fu(implies)28 b(\000\()p Fo(d)p Fu(\))c Fr(\022)e Fu(\000)28 b(and)f(o\()p Fo(d)p Fu(\))d Fr(\024)e Fo(\013)p Fu(.)1908 5539 y(19)p eop %%Page: 20 21 20 20 bop 0 83 a Ft(Theorem)30 b(and)j(De\014nition)e(4.1)0 224 y Fu(F)-7 b(or)27 b(eac)n(h)g(form)n(ula)f Fo(C)34 b Fu(w)n(e)28 b(de\014ne)f(an)h(op)r(erator)e Fr(R)1616 236 y Fp(C)1699 224 y Fu(suc)n(h)i(that:)0 365 y Fo(d)23 b Fr(`)117 335 y Fp(\013)117 386 y(m)203 365 y Fu(\000)p Fo(;)14 b(C)52 b Fu(&)46 b Fo(e)23 b Fr(`)627 335 y Fp(\014)627 386 y(m)713 365 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)52 b Fu(&)46 b(rk)o(\()p Fo(C)6 b Fu(\))24 b Fr(\024)f Fo(m)46 b Fu(=)-14 b Fr(\))46 b(R)1765 377 y Fp(C)1821 365 y Fu(\()p Fo(d;)14 b(e)p Fu(\))23 b Fr(`)2078 335 y Fp(\013)p Fn(#)p Fp(\014)2078 386 y(m)2243 365 y Fu(\000.)0 531 y(Pro)r(of:)36 b Fr(R)330 543 y Fp(C)386 531 y Fu(\()p Fo(d;)14 b(e)p Fu(\))28 b(is)f(de\014ned)h(b)n(y)g(recursion)e(on)h Fo(\013)p Fu(#)p Fo(\014)t Fu(.)0 805 y Ft(1.)36 b Fo(C)e Fu(is)28 b(not)f(a)g(main)h(form)n(ula)f(of)g Fr(I)j Fu(:=)22 b Fk(last)p Fu(\()p Fo(d)1493 817 y Fn(0)1530 805 y Fu(\):)38 b(Then)28 b Fo(d)23 b Fr(')1993 585 y Fl(8)1993 660 y(>)1993 685 y(>)1993 710 y(<)1993 859 y(>)1993 884 y(>)1993 909 y(:)2360 666 y Fo(d)2403 678 y Fp(\023)p 2393 761 5 59 v 2067 837 a Fo(:)14 b(:)g(:)g Fu(\000)p Fo(;)g(C)q(;)g Fu(\001)2433 849 y Fp(\023)2485 837 y Fu(:)23 b Fo(\013)2584 849 y Fp(\023)2627 837 y Fo(:)14 b(:)g(:)p Fu(\()p Fo(\023)p Fr(2)q Fo(I)7 b Fu(\))p 2067 878 849 4 v 2935 895 a Fi(I)2257 953 y Fu(\000)p Fo(;)14 b(C)29 b Fu(:)23 b Fo(\013)3005 805 y Fu(.)0 1088 y(By)k(IH)h(w)n(e)f(get)h Fr(R)581 1100 y Fp(C)637 1088 y Fu(\()p Fo(d)712 1100 y Fp(\023)741 1088 y Fo(;)14 b(e)p Fu(\))23 b Fr(`)923 1057 y Fp(\013)966 1065 y Fh(\023)993 1057 y Fn(#)p Fp(\014)923 1108 y(m)1115 1088 y Fu(\000)1167 1100 y Fp(\023)1224 1088 y Fu(for)k(all)g Fo(\023)d Fr(2)f Fo(I)7 b Fu(.)37 b(F)-7 b(urther)27 b(w)n(e)h(ha)n(v)n(e)e Fo(\013)2368 1100 y Fp(\023)2397 1088 y Fu(#)p Fo(\014)h(<)c(\013)p Fu(#)p Fo(\014)33 b Fu(for)27 b(all)g Fo(\023)c Fr(2)h Fo(I)7 b Fu(.)0 1361 y(Hence)28 b Fr(R)317 1373 y Fp(C)373 1361 y Fu(\()p Fo(d;)14 b(e)p Fu(\))23 b(:=)g Fr(I)6 b Fu(\()p Fr(R)843 1373 y Fp(C)900 1361 y Fu(\()p Fo(d)975 1373 y Fp(\023)1004 1361 y Fo(;)14 b(e)p Fu(\)\))1144 1373 y Fp(\023)p Fi(2)p Fp(I)1275 1361 y Fr(')1362 1141 y Fl(8)1362 1216 y(>)1362 1241 y(>)1362 1266 y(<)1362 1415 y(>)1362 1440 y(>)1362 1465 y(:)1607 1222 y Fr(R)1677 1234 y Fp(C)1733 1222 y Fu(\()p Fo(d)1808 1234 y Fp(\023)1837 1222 y Fo(;)g(e)p Fu(\))p 1774 1317 5 59 v 1436 1393 a Fo(:)g(:)g(:)g Fu(\000)p Fo(;)g Fu(\001)1705 1405 y Fp(\023)1757 1393 y Fu(:)23 b Fo(\013)1856 1405 y Fp(\023)1885 1393 y Fu(#)p Fo(\014)18 b(:)c(:)g(:)p Fu(\()p Fo(\023)p Fr(2)q Fo(I)7 b Fu(\))p 1436 1435 872 4 v 2327 1451 a Fi(I)1628 1509 y Fu(\000)23 b(:)h Fo(\013)p Fu(#)p Fo(\014)2397 1361 y Fu(is)k(a)f(deriv)-5 b(ation)27 b(as)g(required.)0 1660 y Ft(1'.)36 b Fr(:)p Fo(C)e Fu(is)27 b(not)h(main)g(form)n(ula)e (of)i Fk(last)o Fu(\()p Fo(e)p Fu(\):)38 b(symmetric)27 b(to)g(1.)0 1826 y Ft(2.)36 b Fo(C)e Fu(is)28 b(main)f(form)n(ula)g(of) g Fk(last)p Fu(\()p Fo(d)p Fu(\),)h(and)g Fr(:)p Fo(C)34 b Fu(is)28 b(main)f(form)n(ula)g(of)g Fk(last)p Fu(\()p Fo(e)p Fu(\):)0 1992 y Ft(2.1.)35 b Fo(C)f Fu(is)28 b(a)f(literal:)36 b(Then)28 b Fk(last)p Fu(\()p Fo(d)p Fu(\))c(=)e Fk(Ax)1354 1955 y Fi(1)1354 2013 y Fn(\001)1409 1996 y Ff(0)1463 1992 y Fu(and)27 b Fk(last)p Fu(\()p Fo(e)p Fu(\))c(=)g Fk(Ax)2053 1955 y Fi(1)2053 2013 y Fn(\001)2108 1996 y Ff(00)2180 1992 y Fu(with)28 b(\001)2438 1962 y Fi(0)2485 1992 y Fr(2)23 b(AX)13 b Fu(\()p Fe(Z)2781 1955 y Fi(1)2852 1992 y Fu(\))28 b(and)f(\001)3142 1962 y Fi(00)3208 1992 y Fr(2)c(AX)13 b Fu(\()p Fe(Z)3504 1955 y Fi(1)3575 1992 y Fu(\).)0 2134 y(Hence)28 b(\001)23 b(:=)g(\(\001)551 2103 y Fi(0)593 2134 y Fr(n)18 b(f)p Fo(C)6 b Fr(g)p Fu(\))18 b Fr([)h Fu(\(\001)1027 2103 y Fi(00)1088 2134 y Fr(n)f(f:)p Fo(C)6 b Fr(g)p Fu(\))23 b Fr(\022)g Fu(\000,)k(and)h (\001)23 b Fr(2)g(AX)13 b Fu(\()p Fe(Z)2148 2096 y Fi(1)2218 2134 y Fu(\))28 b(\(b)n(y)f(....\).)38 b(W)-7 b(e)28 b(set)f Fr(R)2952 2146 y Fp(C)3009 2134 y Fu(\()p Fo(d;)14 b(e)p Fu(\))23 b(:=)g Fk(Ax)3419 2096 y Fi(1)3419 2154 y Fn(\001)3490 2134 y Fu(.)0 2300 y Ft(2.2.)35 b Fo(C)30 b Fr(')361 2237 y Fl(W)431 2325 y Fp(\023)p Fi(2)p Fp(J)546 2300 y Fo(C)605 2312 y Fp(\023)634 2300 y Fu(:)0 2582 y(Then)e Fr(:)p Fo(C)h Fr(')448 2520 y Fl(V)517 2607 y Fp(\023)p Fi(2)p Fp(J)633 2582 y Fr(:)p Fo(C)747 2594 y Fp(\023)804 2582 y Fu(and)e Fo(d)d Fr(')1119 2362 y Fl(8)1119 2437 y(>)1119 2462 y(>)1119 2487 y(<)1119 2636 y(>)1119 2661 y(>)1119 2686 y(:)1378 2443 y Fo(d)1421 2455 y Fn(0)p 1416 2538 5 59 v 1193 2614 a Fu(\000)p Fo(;)14 b(C)q(;)g(C)1438 2626 y Fp(\026)1506 2614 y Fu(:)24 b Fo(\013)1606 2626 y Fn(0)p 1193 2655 450 4 v 1662 2672 a Fr(_)1717 2644 y Fh(\026)1717 2693 y(C)1280 2730 y Fu(\000)p Fo(;)14 b(C)29 b Fu(:)23 b Fo(\013)1999 2582 y Fu(,)56 b Fo(e)22 b Fr(')2227 2362 y Fl(8)2227 2437 y(>)2227 2462 y(>)2227 2487 y(<)2227 2636 y(>)2227 2661 y(>)2227 2686 y(:)2643 2443 y Fo(e)2682 2455 y Fp(\023)p 2674 2538 5 59 v 2301 2614 a Fo(:)14 b(:)g(:)g Fu(\000)p Fo(;)g Fr(:)p Fo(C)q(;)g Fr(:)p Fo(C)2767 2626 y Fp(\023)2820 2614 y Fu(:)23 b Fo(\014)2913 2626 y Fp(\023)2955 2614 y Fo(:)14 b(:)g(:)p Fu(\()p Fo(\023)p Fr(2)q Fo(J)8 b Fu(\))p 2301 2655 955 4 v 3274 2672 a Fr(^)3330 2680 y Ff(:)p Fh(C)2512 2730 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)29 b Fu(:)23 b Fo(\014)3463 2582 y Fu(.)0 2879 y(By)k(IH)h(w)n(e)f (get)55 b Fr(R)608 2891 y Fp(C)665 2879 y Fu(\()p Fo(d)740 2891 y Fn(0)777 2879 y Fo(;)14 b(e)p Fu(\))23 b Fr(`)959 2849 y Fp(\013)1002 2857 y Fg(0)1035 2849 y Fn(#)p Fp(\014)959 2900 y(m)1157 2879 y Fu(\000)p Fo(;)14 b(C)1305 2891 y Fp(\026)1377 2879 y Fu(and)28 b Fr(R)1609 2891 y Fp(C)1665 2879 y Fu(\()p Fo(d;)14 b(e)1816 2891 y Fp(\026)1861 2879 y Fu(\))23 b Fr(`)1967 2834 y Fp(\013)p Fn(#)p Fp(\014)2103 2842 y Fh(\026)1967 2889 y Fp(m)2169 2879 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)2372 2891 y Fp(\026)2417 2879 y Fu(.)0 3045 y(F)-7 b(urther)27 b(rk\()p Fo(C)469 3057 y Fp(\026)514 3045 y Fu(\))c Fo(<)g Fu(rk)o(\()p Fo(C)6 b Fu(\))24 b Fr(\024)f Fo(m)p Fu(.)0 3322 y(Hence)83 b Fr(R)372 3334 y Fp(C)428 3322 y Fu(\()p Fo(d;)14 b(e)p Fu(\))24 b(:=)e Fk(Cut)871 3334 y Fp(C)919 3342 y Fh(\026)963 3322 y Fr(R)1033 3334 y Fp(C)1089 3322 y Fu(\()p Fo(d)1164 3334 y Fn(0)1201 3322 y Fo(;)14 b(e)p Fu(\))p Fr(R)1379 3334 y Fp(C)1436 3322 y Fu(\()p Fo(d;)g(e)1587 3334 y Fp(\026)1631 3322 y Fu(\))23 b Fr(')1774 3102 y Fl(8)1774 3177 y(>)1774 3202 y(>)1774 3227 y(<)1774 3376 y(>)1774 3401 y(>)1774 3426 y(:)1911 3183 y Fr(R)1981 3195 y Fp(C)2037 3183 y Fu(\()p Fo(d)2112 3195 y Fn(0)2150 3183 y Fo(;)14 b(e)p Fu(\))p 2082 3278 5 59 v 1848 3354 a(\000)p Fo(;)g(C)1996 3366 y Fp(\026)2064 3354 y Fu(:)23 b Fo(\013)2163 3366 y Fn(0)2200 3354 y Fu(#)p Fo(\014)2520 3183 y Fr(R)2590 3195 y Fp(C)2646 3183 y Fu(\()p Fo(d;)14 b(e)2797 3195 y Fp(\026)2842 3183 y Fu(\))p 2695 3278 V 2431 3354 a(\000)p Fo(;)g Fr(:)p Fo(C)2634 3366 y Fp(\026)2702 3354 y Fu(:)23 b Fo(\013)p Fu(#)p Fo(\014)2917 3366 y Fp(\026)p 1848 3395 1115 4 v 2981 3412 a Fd(Cut)3088 3420 y Fh(C)3129 3428 y(\026)2258 3470 y Fu(\000)g(:)g Fo(\013)p Fu(#)p Fo(\014)3245 3322 y Fu(.)0 3596 y Ft(2.2'.)35 b Fo(C)29 b Fr(')388 3534 y Fl(V)457 3621 y Fp(\023)p Fi(2)p Fp(J)573 3596 y Fo(C)632 3608 y Fp(\023)661 3596 y Fu(:)37 b(symmetric)27 b(to)h(\(Case)f(2.2\).)0 3862 y Ft(Theorem)j(and)j(De\014nition)e(4.2)0 4003 y Fu(W)-7 b(e)28 b(de\014ne)g(an)f(op)r(erator)f Fr(E)35 b Fu(suc)n(h)27 b(that)h(the)g(follo)n(wing)f(holds:)36 b Fo(d)23 b Fr(`)2139 3973 y Fp(\013)2139 4024 y(m)p Fn(+1)2309 4003 y Fu(\000)46 b(=)-14 b Fr(\))46 b(E)7 b Fu(\()p Fo(d)p Fu(\))24 b Fr(`)2820 3973 y Fn(3)2853 3948 y Fh(\013)2820 4024 y Fp(m)2922 4003 y Fu(\000.)0 4169 y(Pro)r(of:)0 4447 y Ft(1.)82 b Fo(d)24 b Fr(')311 4227 y Fl(8)311 4302 y(>)311 4327 y(>)311 4352 y(<)311 4501 y(>)311 4526 y(>)311 4551 y(:)502 4308 y Fo(d)545 4320 y Fn(0)p 540 4403 5 59 v 385 4479 a Fu(\000)p Fo(;)14 b(C)29 b Fu(:)23 b Fo(\013)661 4491 y Fn(0)954 4308 y Fo(d)997 4320 y Fn(1)p 992 4403 V 809 4479 a Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)30 b Fu(:)23 b Fo(\013)1141 4491 y Fn(1)p 385 4520 794 4 v 1197 4537 a Fd(Cut)1304 4545 y Fh(C)694 4595 y Fu(\000)g(:)g Fo(\013)1398 4447 y Fu(:)34 b(Then)22 b(rk)o(\()p Fo(C)6 b Fu(\))24 b Fr(\024)f Fo(m)e Fu(and,)i(b)n(y)f(IH,)44 b Fr(E)7 b Fu(\()p Fo(d)2652 4459 y Fn(0)2690 4447 y Fu(\))23 b Fr(`)2796 4417 y Fn(3)2829 4392 y Fh(\013)2867 4404 y Fg(0)2796 4468 y Fp(m)2931 4447 y Fu(\000)p Fo(;)14 b(C)28 b Fu(and)22 b Fr(E)7 b Fu(\()p Fo(d)3389 4459 y Fn(1)3427 4447 y Fu(\))23 b Fr(`)3533 4417 y Fn(3)3566 4392 y Fh(\013)3604 4404 y Fg(1)3533 4468 y Fp(m)3668 4447 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)6 b Fu(.)0 4732 y(Hence)28 b Fr(R)317 4744 y Fp(C)373 4732 y Fu(\()p Fr(E)7 b Fu(\()p Fo(d)531 4744 y Fn(0)569 4732 y Fu(\))p Fo(;)14 b Fr(E)7 b Fu(\()p Fo(d)764 4744 y Fn(1)803 4732 y Fu(\)\))24 b Fr(`)942 4702 y Fn(3)975 4677 y Fh(\013)1013 4689 y Fg(0)1049 4702 y Fn(#3)1137 4677 y Fh(\013)1175 4689 y Fg(1)942 4752 y Fp(m)1239 4732 y Fu(\000)56 b(b)n(y)27 b(Theorem)g(4.1.)37 b(So)28 b(w)n(e)f(could)h(de\014ne)g Fr(E)7 b Fu(\()p Fo(d)p Fu(\))29 b(to)f(b)r(e)g Fr(R)3152 4744 y Fp(C)3208 4732 y Fu(\()p Fr(E)7 b Fu(\()p Fo(d)3366 4744 y Fn(0)3404 4732 y Fu(\))p Fo(;)14 b Fr(E)7 b Fu(\()p Fo(d)3599 4744 y Fn(1)3637 4732 y Fu(\)\).)39 b(But)0 4873 y(for)27 b(reasons)f(whic)n(h)h(b)r(ecome)h(clear)e(later)h(w)n(e)h(set)55 b Fr(E)7 b Fu(\()p Fo(d)p Fu(\))24 b(:=)f Fk(Rep)13 b Fr(R)2136 4885 y Fp(C)2192 4873 y Fu(\()p Fr(E)7 b Fu(\()p Fo(d)2350 4885 y Fn(0)2388 4873 y Fu(\))p Fo(;)14 b Fr(E)7 b Fu(\()p Fo(d)2583 4885 y Fn(1)2622 4873 y Fu(\)\).)0 5147 y Ft(2.)36 b Fu(otherwise:)g Fr(E)7 b Fu(\()p Fo(d)p Fu(\))24 b(:=)f Fr(I)6 b Fu(\()p Fr(E)h Fu(\()p Fo(d)1015 5159 y Fp(\023)1045 5147 y Fu(\)\))1109 5159 y Fp(\023)p Fi(2)p Fp(I)1240 5147 y Fr(')1328 4927 y Fl(8)1328 5002 y(>)1328 5027 y(>)1328 5052 y(<)1328 5201 y(>)1328 5226 y(>)1328 5251 y(:)1604 5008 y Fr(E)g Fu(\()p Fo(d)1730 5020 y Fp(\023)1760 5008 y Fu(\))p 1696 5103 5 59 v 1401 5179 a Fo(:)14 b(:)g(:)g Fu(\000)p Fo(;)g Fu(\001)1670 5191 y Fp(\023)1722 5179 y Fu(:)23 b(3)1810 5148 y Fp(\013)1853 5156 y Fh(\023)1898 5179 y Fo(:)14 b(:)g(:)p Fu(\()p Fo(\023)p Fr(2)q Fo(I)7 b Fu(\))p 1401 5220 786 4 v 2206 5237 a Fi(I)1593 5295 y Fu(\000)23 b(:)g(3)1756 5265 y Fp(\013)2303 5147 y Fu(if)56 b Fo(d)24 b Fu(=)2561 4927 y Fl(8)2561 5002 y(>)2561 5027 y(>)2561 5052 y(<)2561 5201 y(>)2561 5226 y(>)2561 5251 y(:)2879 5008 y Fo(d)2922 5020 y Fp(\023)p 2912 5103 5 59 v 2635 5179 a Fo(:)14 b(:)g(:)g Fu(\000)p Fo(;)g Fu(\001)2904 5191 y Fp(\023)2955 5179 y Fu(:)24 b Fo(\013)3055 5191 y Fp(\023)3097 5179 y Fo(:)14 b(:)g(:)p Fu(\()p Fo(\023)p Fr(2)q Fo(I)7 b Fu(\))p 2635 5220 752 4 v 3405 5237 a Fi(I)2827 5295 y Fu(\000)23 b(:)g Fo(\013)1908 5539 y Fu(20)p eop %%Page: 21 22 21 21 bop 0 83 a Ft(Theorem)30 b(4.3)d Fu(\(In)n(v)n(ersion\))0 224 y(a\))g Fo(A)d Fr(')274 162 y Fl(V)344 249 y Fp(\023)p Fi(2)p Fp(J)459 224 y Fo(A)521 236 y Fp(\023)596 224 y Fu(&)f Fe(Z)732 187 y Fi(1)826 224 y Fr(`)877 194 y Fp(\013)877 245 y(m)962 224 y Fu(\000)p Fo(;)14 b(A)47 b Fu(=)-14 b Fr(\))46 b Fe(Z)1388 187 y Fi(1)1481 224 y Fr(`)1532 194 y Fp(\013)1532 245 y(m)1618 224 y Fu(\000)p Fo(;)14 b(A)1769 236 y Fp(\023)1825 224 y Fu(for)28 b(eac)n(h)e Fo(\023)e Fr(2)f Fo(J)8 b Fu(.)0 365 y(b\))28 b Fe(Z)154 328 y Fi(1)248 365 y Fr(`)299 335 y Fp(\013)299 386 y(m)385 365 y Fu(\000)p Fo(;)14 b(A)536 377 y Fn(0)573 365 y Fo(;)g(A)672 377 y Fn(1)755 365 y Fu(=)-14 b Fr(\))46 b Fe(Z)983 328 y Fi(1)1077 365 y Fr(`)1128 335 y Fp(\013)1128 386 y(m)1214 365 y Fu(\000)p Fo(;)14 b(A)1365 377 y Fn(0)1402 365 y Fo(;)g(A)1501 377 y Fn(1)1538 365 y Fu(.)0 631 y Ft(De\014nition)0 772 y Fr(L)57 784 y Fn(0)95 772 y Fu([)p Fo(X)7 b Fu(])22 b(:=)h Fr(L)407 784 y Fn(0)463 772 y Fr([)18 b(f)p Fo(X)7 b Fr(g)p Fu(,)0 913 y(A)28 b(closed)f Fr(L)392 925 y Fn(0)429 913 y Fu([)p Fo(X)7 b Fu(]-form)n(ula)26 b(\(sequen)n(t\))i(is)f(called)h Fo(X)7 b Fu(-p)r(ositiv)n(e)26 b(if)i(it)g(con)n(tains)f(no)g(subform)n (ula)g Fr(:)p Fo(X)7 b(t)p Fu(.)0 1079 y Fr(j)-14 b Fu(=)74 1049 y Fp(\013)144 1079 y Fo(A)47 b Fu(:)p Fr(,)f Fu(\(I)-14 b(N)p Fo(;)14 b Fr(f)p Fo(n)23 b Fu(:)g Fr(j)p Fo(n)p Fr(j)809 1091 y Fi(\036)888 1079 y Fo(<)g(\013)p Fr(g)p Fu(\))g Fr(j)-14 b Fu(=)23 b Fo(A)p Fu(,)0 1220 y Fr(j)-14 b Fu(=)74 1190 y Fp(\013)144 1220 y Fr(f)p Fo(A)248 1232 y Fn(1)285 1220 y Fo(;)14 b(:::;)g(A)490 1232 y Fp(k)531 1220 y Fr(g)46 b Fu(:)p Fr(,)23 b(j)-14 b Fu(=)822 1190 y Fp(\013)892 1220 y Fo(A)954 1232 y Fn(1)1010 1220 y Fr(_)19 b Fo(:::)g Fr(_)f Fo(A)1307 1232 y Fp(k)0 1436 y Ft(Lemma)30 b(4.4)c Fu(\(Boundedness\))0 1577 y(Let)i(\000)f(b)r(e)h Fo(X)7 b Fu(-p)r(ositiv)n(e.)0 1719 y Fe(Z)48 1681 y Fi(1)142 1719 y Fr(`)193 1679 y Fp(\014)193 1741 y Fn(0)260 1719 y Fr(:)p Fu(Prog)489 1739 y Fi(\036)545 1719 y Fu(\()p Fo(X)g Fu(\))p Fo(;)14 b Fr(:)p Fo(X)7 b(s)892 1731 y Fn(1)928 1719 y Fo(;)14 b(:::;)g Fr(:)p Fo(X)7 b(s)1241 1731 y Fp(k)1282 1719 y Fo(;)14 b Fu(\000)46 b(&)f Fr(j)p Fo(s)1589 1731 y Fn(1)1627 1719 y Fr(j)1650 1731 y Fi(\036)1706 1719 y Fo(;)14 b(:::;)g Fr(j)p Fo(s)1911 1731 y Fp(k)1951 1719 y Fr(j)1974 1731 y Fi(\036)2054 1719 y Fr(\024)22 b Fo(\013)47 b Fu(=)-14 b Fr(\))46 b(j)-14 b Fu(=)2495 1688 y Fp(\013)p Fn(+2)2622 1663 y Fh(\014)2688 1719 y Fu(\000.)0 1885 y Ft(Corollary)55 b Fe(Z)496 1847 y Fi(1)590 1885 y Fr(`)641 1845 y Fp(\014)641 1907 y Fn(0)708 1885 y Fu(TI)798 1897 y Fi(\036)854 1885 y Fu(\()p Fo(X)r(;)14 b(n)p 994 1898 50 4 v Fu(\))47 b(=)-14 b Fr(\))45 b(j)p Fo(n)p Fr(j)1398 1897 y Fi(\036)1477 1885 y Fr(\024)23 b Fu(2)1607 1854 y Fp(\014)1651 1885 y Fu(.)0 2051 y(Pro)r(of)j(of)i (Corollary:)0 2192 y Fe(Z)48 2155 y Fi(1)142 2192 y Fr(`)193 2152 y Fp(\014)193 2214 y Fn(0)260 2192 y Fu(TI)350 2204 y Fi(\036)406 2192 y Fu(\()p Fo(X)r(;)14 b(n)p 546 2205 V Fu(\))674 2145 y Fn(4)p Fp(:)p Fn(3a)692 2192 y Fr(\))64 b Fe(Z)888 2155 y Fi(1)981 2192 y Fr(`)1032 2152 y Fp(\014)1032 2214 y Fn(0)1100 2192 y Fr(:)p Fu(Prog)1328 2212 y Fi(\036)1384 2192 y Fu(\()p Fo(X)7 b Fu(\))p Fo(;)14 b Fr(8)p Fo(y)25 b Fr(\036)d Fo(n)p 1761 2205 V(X)7 b(y)1953 2145 y Fn(4)p Fp(:)p Fn(4)1955 2192 y Fr(\))24 b(j)-14 b Fu(=)2136 2162 y Fn(2)2169 2137 y Fh(\014)2235 2192 y Fr(8)p Fo(y)25 b Fr(\036)e Fo(n)p 2436 2205 V(X)7 b(y)48 b Fr(\))e(j)p Fo(n)p Fr(j)2876 2204 y Fi(\036)2955 2192 y Fr(\024)23 b Fu(2)3085 2162 y Fp(\014)3129 2192 y Fu(.)0 2358 y(Pro)r(of)j(of)i (Lemma)f(b)n(y)h(induction)g(on)f Fo(\014)t Fu(:)0 2499 y(Abbreviations:)36 b(\003)23 b(:=)f Fr(f:)p Fo(X)7 b(s)971 2511 y Fn(1)1008 2499 y Fo(;)14 b(:::;)g Fr(:)p Fo(X)7 b(s)1321 2511 y Fp(k)1361 2499 y Fr(g)p Fu(.)0 2640 y(Let)28 b Fo(d)23 b Fr(`)266 2600 y Fp(\014)266 2662 y Fn(0)334 2640 y Fr(:)p Fu(Prog)562 2660 y Fi(\036)618 2640 y Fu(\()p Fo(X)7 b Fu(\))p Fo(;)14 b Fu(\003)p Fo(;)g Fu(\000.)0 2806 y(1.1.)36 b Fk(last)p Fu(\()p Fo(d)p Fu(\))23 b(=)g Fk(Ax)600 2769 y Fi(1)600 2827 y Fn(\001)698 2806 y Fu(and)k(\001)19 b Fr(\\)g Fu(TR)n(UE)1258 2818 y Fn(0)1318 2806 y Fr(6)p Fu(=)k Fr(;)p Fu(:)37 b(\000)18 b Fr(\\)h Fu(TR)n(UE)1889 2818 y Fn(0)1949 2806 y Fr(6)p Fu(=)k Fr(;)k Fu(and)h(the)g(claim)f(is) h(trivial.)0 2972 y(1.2.)36 b Fk(last)p Fu(\()p Fo(d)p Fu(\))23 b(=)g Fk(Ax)600 2935 y Fi(1)600 2993 y Fn(\001)698 2972 y Fu(and)k Fo(X)7 b(t;)14 b Fr(:)p Fo(X)7 b(s)22 b Fr(\022)h Fu(\001)k(with)i Fk(val)p Fu(\()p Fo(t)p Fu(\))23 b(=)g Fk(val)p Fu(\()p Fo(s)p Fu(\):)0 3113 y(Then)28 b Fo(X)7 b(t)22 b Fr(2)i Fu(\000)j(and)h Fr(:)p Fo(X)7 b(s)22 b Fr(2)i Fu(\003.)36 b(Hence)28 b Fr(j)p Fo(t)p Fr(j)1376 3125 y Fi(\036)1455 3113 y Fu(=)23 b Fr(j)p Fo(s)p Fr(j)1628 3125 y Fi(\036)1707 3113 y Fr(\024)g Fo(\013)g(<)g(\013)c Fu(+)f(2)2156 3083 y Fp(\014)2228 3113 y Fu(and)27 b(th)n(us)h Fr(j)-14 b Fu(=)2646 3083 y Fp(\013)p Fn(+2)2773 3058 y Fh(\014)2839 3113 y Fu(\000.)0 3279 y(2.)36 b Fk(last)p Fu(\()p Fo(d)p Fu(\))24 b(=)441 3217 y Fl(W)511 3238 y Fp(s)542 3246 y Fg(0)511 3304 y Fi(:)p Fn(Prog)693 3320 y Ff(\036)742 3304 y Fn(\()p Fp(X)5 b Fn(\))857 3279 y Fu(:)37 b(Then)28 b Fr(`)1185 3239 y Fp(\014)1223 3247 y Fg(0)1185 3302 y Fn(0)1282 3279 y Fr(:)p Fu(Prog)1510 3300 y Fi(\036)1567 3279 y Fu(\()p Fo(X)7 b Fu(\))p Fo(;)14 b Fu(\003)p Fo(;)g Fu(\000)p Fo(;)g Fr(8)p Fo(y)s Fr(\036)m Fo(s)2120 3291 y Fn(0)2171 3279 y Fo(X)7 b(y)20 b Fr(^)f(:)p Fo(X)7 b(s)2552 3291 y Fn(0)2617 3279 y Fu(with)28 b Fo(\014)2853 3291 y Fn(0)2913 3279 y Fo(<)23 b(\014)t Fu(.)0 3421 y(By)k(4.3a)g(\(In)n(v)n(ersion\))f (w)n(e)h(get)55 b(\(1\))28 b Fr(`)1199 3381 y Fp(\014)1237 3389 y Fg(0)1199 3443 y Fn(0)1296 3421 y Fr(:)p Fu(Prog)1525 3441 y Fi(\036)1581 3421 y Fu(\()p Fo(X)7 b Fu(\))p Fo(;)14 b Fu(\003)p Fo(;)g Fu(\000)p Fo(;)g Fr(8)p Fo(y)s Fr(\036)m Fo(s)2134 3433 y Fn(0)2185 3421 y Fo(X)7 b(y)s Fu(,)54 b(and)i(\(2\))27 b Fr(`)2756 3381 y Fp(\014)2794 3389 y Fg(0)2756 3443 y Fn(0)2853 3421 y Fr(:)p Fu(Prog)3082 3441 y Fi(\036)3138 3421 y Fu(\()p Fo(X)7 b Fu(\))p Fo(;)14 b Fr(:)p Fo(X)7 b(s)3485 3433 y Fn(0)3521 3421 y Fo(;)14 b Fu(\003)p Fo(;)g Fu(\000.)0 3587 y(By)27 b(IH)h(from)f(\(1\))h(w)n(e) f(get)56 b Fr(j)-14 b Fu(=)943 3556 y Fp(\013)p Fn(+2)1070 3531 y Fh(\014)1103 3543 y Fg(0)1166 3587 y Fu(\000)p Fo(;)14 b Fr(8)p Fo(y)s Fr(\036)n Fo(s)1448 3599 y Fn(0)1499 3587 y Fo(X)7 b(y)s Fu(.)0 3753 y(\(Case)27 b(1\))h Fr(j)-14 b Fu(=)407 3722 y Fp(\013)p Fn(+2)534 3697 y Fh(\014)567 3709 y Fg(0)630 3753 y Fu(\000:)37 b(Then)28 b(also)e Fr(j)-14 b Fu(=)1199 3722 y Fp(\013)p Fn(+2)1326 3697 y Fh(\014)1393 3753 y Fu(\000,)27 b(since)h Fo(\014)1746 3765 y Fn(0)1806 3753 y Fr(\024)23 b Fo(\014)32 b Fu(and)27 b(\000)h(is)f Fo(X)7 b Fu(-p)r(ositiv)n(e.)0 3894 y(\(Case)27 b(2\))h Fr(j)-14 b Fu(=)407 3864 y Fp(\013)p Fn(+2)534 3839 y Fh(\014)567 3851 y Fg(0)630 3894 y Fr(8)p Fo(y)s Fr(\036)o Fo(s)824 3906 y Fn(0)875 3894 y Fo(X)7 b(y)s Fu(:)35 b(Then)28 b Fr(j)p Fo(m)p Fr(j)1389 3906 y Fi(\036)1468 3894 y Fo(<)23 b(\013)c Fu(+)f(2)1753 3864 y Fp(\014)1791 3872 y Fg(0)1854 3894 y Fu(for)27 b(all)h Fo(m)23 b Fr(\036)f Fk(val)p Fu(\()p Fo(s)2449 3906 y Fn(0)2487 3894 y Fu(\),)28 b(i.e.)37 b Fr(j)p Fo(s)2775 3906 y Fn(0)2812 3894 y Fr(j)2835 3906 y Fi(\036)2914 3894 y Fr(\024)23 b Fo(\013)c Fu(+)f(2)3199 3864 y Fp(\014)3237 3872 y Fg(0)3272 3894 y Fu(.)0 4035 y(F)-7 b(rom)27 b(this)h(together)f(with)h Fr(j)p Fo(s)959 4047 y Fn(1)996 4035 y Fr(j)1019 4047 y Fi(\036)1075 4035 y Fo(;)14 b(:::;)g Fr(j)p Fo(s)1280 4047 y Fp(k)1321 4035 y Fr(j)1344 4047 y Fi(\036)1423 4035 y Fr(\024)23 b Fo(\013)28 b Fu(and)f(\(2\))h(b)n(y)f(IH)h(w)n(e)g (obtain)f Fr(j)-14 b Fu(=)2577 4005 y Fp(\013)p Fn(+2)2704 3980 y Fh(\014)2737 3992 y Fg(0)2773 4005 y Fn(+2)2857 3980 y Fh(\014)2890 3992 y Fg(0)2954 4035 y Fu(\000,)27 b(and)h(th)n(us)g Fr(j)-14 b Fu(=)3475 4005 y Fp(\013)p Fn(+2)3602 3980 y Fh(\014)3668 4035 y Fu(\000.)0 4201 y(3.)39 b Fk(last)p Fu(\()p Fo(d)p Fu(\))26 b(=)448 4139 y Fl(V)517 4226 y Fp(C)602 4201 y Fu(with)j Fo(C)i Fr(')971 4139 y Fl(V)1040 4226 y Fp(\023)p Fi(2)p Fp(I)1148 4201 y Fo(C)1207 4213 y Fp(\023)1260 4201 y Fr(2)26 b Fu(\000:)38 b(Then,)29 b(for)g(all)f Fo(\023)d Fr(2)g Fo(I)7 b Fu(,)29 b Fr(`)2220 4161 y Fp(\014)2258 4169 y Fh(\023)2220 4223 y Fn(0)2313 4201 y Fr(:)p Fu(Prog)2542 4221 y Fi(\036)2598 4201 y Fu(\()p Fo(X)7 b Fu(\))p Fo(;)14 b Fu(\003)p Fo(;)g Fu(\000)p Fo(;)g(C)3018 4213 y Fp(\023)3075 4201 y Fu(and)28 b Fo(\014)3284 4213 y Fp(\023)3338 4201 y Fo(<)c(\014)t Fu(.)40 b(Hence,)29 b(b)n(y)0 4342 y(IH,)f Fr(j)-14 b Fu(=)217 4312 y Fp(\013)p Fn(+2)344 4287 y Fh(\014)377 4295 y(\023)435 4342 y Fu(\000)p Fo(;)14 b(C)583 4354 y Fp(\023)640 4342 y Fu(for)27 b(all)g Fo(\023)p Fu(.)37 b(Since)28 b(\000)g(is)f Fo(X)7 b Fu(-p)r(ositiv)n(e,)26 b(this)i(implies)g Fr(j)-14 b Fu(=)2302 4312 y Fp(\013)p Fn(+2)2429 4287 y Fh(\014)2495 4342 y Fu(\000)p Fo(;)14 b(C)2643 4354 y Fp(\023)2700 4342 y Fu(for)27 b(all)g Fo(\023)p Fu(,)i(and)e(th)n(us)h Fr(j)-14 b Fu(=)3441 4312 y Fp(\013)p Fn(+2)3568 4287 y Fh(\014)3634 4342 y Fu(\000)p Fo(;)14 b(C)6 b Fu(.)0 4508 y(4.)101 4446 y Fl(W)171 4466 y Fp(\026)171 4533 y(C)254 4508 y Fu(with)28 b Fo(C)i Fr(2)23 b Fu(\000:)37 b(as)27 b(3.)0 4674 y(5.)36 b Fk(last)p Fu(\()p Fo(d)p Fu(\))24 b(=)e Fk(Rep)p Fu(:)37 b(immediate)28 b(b)n(y)f(IH.)1908 5539 y(21)p eop %%Page: 22 23 22 22 bop 0 83 a Ft(Em)m(b)s(edding)29 b(of)j Fu(PL1)17 b(+)h Fr(Z)39 b Ft(in)m(to)32 b Fe(Z)1205 46 y Fi(1)1307 83 y Ft(;)f(unpro)m(v)-5 b(abilit)m(y)33 b(of)e Fu(TI\()p Fo(")2219 95 y Fn(0)2257 83 y Fu(\))0 274 y Ft(Lemma)f(4.5)0 415 y Fu(PL1)22 b Fr(`)224 385 y Fp(k)224 436 y Fn(0)288 415 y Fu(\000)14 b(&)g(FV\(\000\))24 b(=)f Fr(;)36 b Fu(=)-14 b Fr(\))46 b Fe(Z)1083 378 y Fi(1)1176 415 y Fr(`)1227 385 y Fp(k)1227 436 y Fn(0)1291 415 y Fu(\000)0 581 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f Fo(k)s Fu(:)0 722 y(1.)36 b Fr(f:)p Fo(A;)14 b(A)p Fr(g)23 b(\022)g Fu(\000)28 b(where)f Fo(A)h Fu(is)f(atomic:)36 b(Then)28 b(either)g Fo(A)23 b Fr(2)g Fu(TR)n(UE)2163 734 y Fn(0)2228 722 y Fu(or)k Fr(:)p Fo(A)c Fr(2)h Fu(TR)n(UE)2786 734 y Fn(0)2851 722 y Fu(or)j Fo(A)c Fu(=)g Fo(X)7 b(s)p Fu(.)36 b(Hence)28 b Fe(Z)3595 685 y Fi(1)3688 722 y Fr(`)3739 692 y Fp(k)3739 743 y Fn(0)3803 722 y Fu(\000.)0 863 y(2.)36 b Fr(8)p Fo(xA)23 b Fr(2)h Fu(\000)j(und)h(PL1)22 b Fr(`)828 828 y Fp(k)q Fi(\000)p Fn(1)828 886 y(0)977 863 y Fu(\000)p Fo(;)14 b(A)1128 875 y Fp(x)1170 863 y Fu(\()p Fo(y)s Fu(\):)37 b(Then)28 b(b)n(y)f(2.2)55 b(PL1)22 b Fr(`)2056 828 y Fp(k)q Fi(\000)p Fn(1)2056 886 y(0)2204 863 y Fu(\000)p Fo(;)14 b(A)2355 875 y Fp(x)2397 863 y Fu(\()p Fo(t)p Fu(\))28 b(for)f(all)h Fo(t)23 b Fr(2)g Fo(T)12 b Fu(.)0 1005 y(Hence)28 b(b)n(y)f(IH)56 b Fe(Z)558 967 y Fi(1)651 1005 y Fr(`)702 969 y Fp(k)q Fi(\000)p Fn(1)702 1027 y(0)851 1005 y Fu(\000)p Fo(;)14 b(A)1002 1017 y Fp(x)1044 1005 y Fu(\()p Fo(t)p Fu(\))28 b(for)f(all)g Fo(t)c Fr(2)h Fo(T)39 b Fu(and)27 b(th)n(us)h Fe(Z)2020 967 y Fi(1)2113 1005 y Fr(`)2164 974 y Fp(k)2164 1025 y Fn(0)2228 1005 y Fu(\000,)f(since)h Fr(8)p Fo(xA)23 b Fr(')2800 942 y Fl(V)2870 1029 y Fp(t)p Fi(2)p Fp(T)2992 1005 y Fo(A)3054 1017 y Fp(x)3096 1005 y Fu(\()p Fo(t)p Fu(\).)0 1146 y(3.)52 b Fr(9)p Fo(xA)32 b Fr(2)g Fu(\000)h(and)f(PL1)f Fr(`)875 1110 y Fp(k)q Fi(\000)p Fn(1)875 1168 y(0)1032 1146 y Fu(\000)p Fo(;)14 b(A)1183 1158 y Fp(x)1225 1146 y Fu(\()p Fo(t)p Fu(\):)48 b(w.l.o.g.)j Fo(t)32 b Fr(2)g Fo(T)43 b Fu(\(cf.)34 b(Lemma)e(2.2\).)52 b(By)32 b(IH)66 b Fe(Z)2948 1109 y Fi(1)3050 1146 y Fr(`)3101 1110 y Fp(k)q Fi(\000)p Fn(1)3101 1168 y(0)3258 1146 y Fu(\000)p Fo(;)14 b(A)3409 1158 y Fp(x)3451 1146 y Fu(\()p Fo(t)p Fu(\))33 b(and)g(th)n(us)0 1287 y Fe(Z)48 1250 y Fi(1)142 1287 y Fr(`)193 1257 y Fp(k)193 1307 y Fn(0)256 1287 y Fu(\000.)0 1428 y(4.)j(\()p Fr(^)p Fu(\),\()p Fr(_)p Fu(\):)j(as)27 b(2.)37 b(and)27 b(3.)0 1594 y Ft(Lemma)j(4.6)0 1735 y Fo(s;)14 b(t)23 b Fr(2)g Fo(T)34 b Fu(&)23 b Fk(val)p Fu(\()p Fo(s)p Fu(\))h(=)e Fk(val)p Fu(\()p Fo(t)p Fu(\))i(&)e(FV)r(\() p Fo(F)12 b Fu(\))24 b Fr(\022)e(f)p Fo(x)p Fr(g)46 b Fu(=)-14 b Fr(\))46 b Fe(Z)1756 1698 y Fi(1)1849 1735 y Fr(`)1900 1692 y Fn(2)p Fi(\001)p Fn(rk\()p Fp(F)9 b Fn(\))1900 1757 y(0)2144 1735 y Fr(:)p Fo(F)2252 1747 y Fp(x)2294 1735 y Fu(\()p Fo(s)p Fu(\))p Fo(;)14 b(F)2487 1747 y Fp(x)2530 1735 y Fu(\()p Fo(t)p Fu(\).)0 1901 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f(rk)o(\()p Fo(F)12 b Fu(\).)0 2067 y Ft(Lemma)30 b(4.7)c Fu(\(Deriv)-5 b(abilit)n(y)28 b(of)f(Complete)h(Induction\))0 2208 y(Let)g Fr(8)p Fo(xF)39 b Fu(b)r(e)28 b(closed)f(and)g Fo(r)f Fu(:=)d(2)18 b Fr(\001)g Fu(rk\()p Fo(F)12 b Fu(\).)120 b(Then)28 b(for)f(eac)n(h)g Fo(t)c Fr(2)g Fo(T)39 b Fu(w)n(e)27 b(ha)n(v)n(e)0 2350 y Fe(Z)48 2312 y Fi(1)142 2350 y Fr(`)193 2314 y Fp(r)r Fn(+2)p Fp(n)193 2372 y Fn(0)377 2350 y Fr(:)p Fo(F)485 2362 y Fp(x)528 2350 y Fu(\(0\))p Fo(;)14 b Fr(:8)p Fo(x)p Fu(\()p Fo(F)35 b Fr(!)23 b Fo(F)1099 2362 y Fp(x)1141 2350 y Fu(\()p Fk(S)q Fo(x)p Fu(\)\))p Fo(;)28 b(F)1435 2362 y Fp(x)1478 2350 y Fu(\()p Fo(t)p Fu(\))g(where)f Fo(n)c Fu(:=)f Fk(val)p Fu(\()p Fo(t)p Fu(\).)0 2516 y(Pro)r(of)k(b)n(y)i(induction)g(on)f Fo(n)p Fu(:)0 2657 y(Let)h(\000)23 b(:=)f Fr(f:)p Fo(F)12 b Fu(\(0\))p Fo(;)i Fr(:8)p Fo(x)p Fu(\()p Fo(F)35 b Fr(!)23 b Fo(F)1067 2669 y Fp(x)1110 2657 y Fu(\()p Fk(S)p Fo(x)p Fu(\)\))p Fr(g)p Fu(,)28 b(i.e.)37 b(\000)23 b(=)g Fr(f:)p Fo(F)12 b Fu(\(0\))p Fo(;)i Fr(9)p Fo(x)p Fu(\()p Fo(F)31 b Fr(^)19 b(:)p Fo(F)2394 2669 y Fp(x)2436 2657 y Fu(\()p Fk(S)q Fo(x)p Fu(\)\))p Fr(g)p Fu(.)0 2823 y(1.)36 b Fk(val)p Fu(\()p Fo(t)p Fu(\))24 b(=)f(0:)36 b(Then)28 b Fr(`)774 2793 y Fp(r)774 2843 y Fn(0)834 2823 y Fu(\000)p Fo(;)14 b(F)e Fu(\(0\))27 b(b)n(y)h(Lemma)f(4.6.)0 2964 y(2.)36 b Fk(val)p Fu(\()p Fo(t)p Fu(\))24 b(=)f Fo(n)p Fu(+)o(1:)0 3105 y(\(1\))28 b Fr(`)185 3070 y Fp(r)r Fn(+2)p Fp(n)185 3127 y Fn(0)369 3105 y Fu(\000)p Fo(;)14 b(F)e Fu(\()p Fo(n)p 555 3118 50 4 v Fu(\))1509 b([)28 b(IH)g(])0 3246 y(\(2\))g Fr(`)185 3216 y Fp(r)185 3267 y Fn(0)245 3246 y Fr(:)p Fo(F)12 b Fu(\()p Fk(S)p Fo(n)p 443 3259 V Fu(\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))1337 b([)27 b(L.4.6)g(])0 3387 y(\(3\))h Fr(`)185 3352 y Fp(r)r Fn(+2)p Fp(n)p Fn(+1)185 3409 y(0)453 3387 y Fu(\000)p Fo(;)14 b(F)e Fu(\()p Fo(n)p 639 3400 V Fu(\))19 b Fr(^)g(:)p Fo(F)12 b Fu(\()p Fk(S)q Fo(n)p 1013 3400 V -1 w Fu(\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))598 b([\(1\),\(2\),\()p Fr(^)p Fu(\))29 b(])0 3528 y(\(4\))f Fr(`)185 3493 y Fp(r)r Fn(+2)p Fp(n)p Fn(+2)185 3551 y(0)453 3528 y Fu(\000)p Fo(;)14 b(F)e Fu(\()p Fo(t)p Fu(\))832 b([)28 b(\(3\),)1740 3466 y Fl(W)1810 3483 y Fp(n)p 1810 3496 46 4 v 1810 3559 a Fi(9)p Fp(x)p Fn(\()p Fp(F)9 b Fi(^:)p Fp(F)2095 3567 y Fh(x)2130 3559 y Fn(\()p Fd(S)q Fp(x)p Fn(\)\))2317 3528 y Fu(])0 3794 y Ft(Lemma)30 b(4.8)0 3935 y Fu(F)-7 b(or)27 b(eac)n(h)g Fo(C)i Fr(2)24 b(Z)34 b Fu(there)27 b(is)h(an)f Fo(\013)d(<)e(!)f Fu(+)d Fo(!)31 b Fu(suc)n(h)c(that)h Fe(Z)1826 3898 y Fi(1)1919 3935 y Fr(`)1970 3905 y Fp(\013)1970 3956 y Fn(0)2041 3935 y Fo(C)6 b Fu(.)0 4101 y(Pro)r(of:)0 4242 y(1.)36 b Fo(C)30 b Fu(=)22 b Fr(8)o Fo(~)-41 b(y)16 b Fr(8)p Fo(z)t Fu([)e Fo(F)e Fu(\(0)p Fo(;)h(~)-41 b(y)q Fu(\))23 b Fr(!)g(8)p Fo(x)p Fu(\()p Fo(F)12 b Fu(\()p Fo(x;)h(~)-41 b(y)s Fu(\))24 b Fr(!)f Fo(F)12 b Fu(\()p Fk(S)p Fo(x;)h(~)-41 b(y)s Fu(\)\))24 b Fr(!)f Fo(F)12 b Fu(\()p Fo(z)t(;)h(~)-41 b(y)r Fu(\)\)]:)0 4408 y(By)27 b(Lemma)h(4.7)e(w)n(e)i(ha)n(v)n(e)e Fr(`)925 4378 y Fp(!)925 4429 y Fn(0)996 4408 y Fr(:)p Fo(F)12 b Fu(\(0)p Fo(;)f(~)-39 b(s)p Fu(\))p Fo(;)14 b Fr(:8)p Fo(x)p Fu(\()p Fo(F)e Fu(\()p Fo(x;)f(~)-39 b(s)q Fu(\))23 b Fr(!)g Fo(F)12 b Fu(\()p Fk(S)p Fo(x;)f(~)-39 b(s)q Fu(\)\))p Fo(;)14 b(F)e Fu(\()p Fo(t;)f(~)-39 b(s)p Fu(\))28 b(for)f(all)d Fo(~)-39 b(s;)14 b(t)23 b Fr(2)h Fo(T)12 b Fu(.)36 b(This)27 b(yields)h(the)g(claim.)0 4575 y(2.)36 b Fo(C)30 b Fu(=)22 b Fr(8)o Fo(~)-41 b(y)r Fr(8)p Fo(x)461 4587 y Fn(1)498 4575 y Fr(8)p Fo(x)592 4587 y Fn(2)629 4575 y Fu(\()p Fo(x)708 4587 y Fn(1)p 754 4549 50 8 v 754 4566 V 812 4575 a Fo(x)859 4587 y Fn(2)920 4575 y Fr(!)23 b Fo(A)p Fu(\()p Fo(x)1167 4587 y Fn(1)1205 4575 y Fo(;)13 b(~)-41 b(y)s Fu(\))23 b Fr(!)g Fo(A)p Fu(\()p Fo(x)1588 4587 y Fn(2)1627 4575 y Fo(;)13 b(~)-41 b(y)r Fu(\)\))28 b(where)g Fo(A)f Fu(is)h(atomic:)0 4716 y(Then)36 b(for)g(an)n(y)f(closed)g Fr(L)836 4728 y Fn(0)873 4716 y Fu(-terms)e Fo(~)-39 b(s;)14 b(t)1247 4728 y Fn(1)1284 4716 y Fo(;)g(t)1351 4728 y Fn(2)1424 4716 y Fu(w)n(e)36 b(ha)n(v)n(e)e Fr(`)1805 4686 y Fn(0)1805 4736 y(0)1879 4716 y Fr(:)p Fu(\()p Fo(t)1996 4728 y Fn(1)p 2042 4690 V 2042 4707 V 2101 4716 a Fo(t)2131 4728 y Fn(2)2168 4716 y Fu(\))p Fo(;)14 b Fr(:)p Fo(A)p Fu(\()p Fo(t)2416 4728 y Fn(1)2454 4716 y Fo(;)d(~)-39 b(s)p Fu(\))p Fo(;)14 b(A)p Fu(\()p Fo(t)2723 4728 y Fn(2)2761 4716 y Fo(;)d(~)-39 b(s)p Fu(\).)62 b([)36 b(If)g Fk(val)p Fu(\()p Fo(t)3264 4728 y Fn(1)3302 4716 y Fu(\))h(=)g Fk(val)p Fu(\()p Fo(t)3633 4728 y Fn(2)3670 4716 y Fu(\))g(then)0 4857 y(either)31 b Fr(:)p Fo(A)p Fu(\()p Fo(t)418 4869 y Fn(1)456 4857 y Fo(;)11 b(~)-39 b(s)p Fu(\))28 b Fr(2)h Fu(TR)n(UE)913 4869 y Fn(0)981 4857 y Fu(or)h Fo(A)p Fu(\()p Fo(t)1210 4869 y Fn(2)1248 4857 y Fo(;)11 b(~)-39 b(s)p Fu(\))29 b Fr(2)f Fu(TR)n(UE)1706 4869 y Fn(0)1774 4857 y Fu(or)i Fr(f:)p Fo(A)p Fu(\()p Fo(t)2100 4869 y Fn(1)2137 4857 y Fo(;)11 b(~)-39 b(s)p Fu(\))p Fo(;)14 b(A)p Fu(\()p Fo(t)2406 4869 y Fn(2)2444 4857 y Fo(;)d(~)-39 b(s)p Fu(\))p Fr(g)28 b Fu(=)g Fr(f:)p Fo(X)7 b(t;)14 b(X)7 b(t)3061 4827 y Fi(0)3083 4857 y Fr(g)30 b Fu(with)h Fk(val)p Fu(\()p Fo(t)p Fu(\))e(=)f Fk(val)p Fu(\()p Fo(t)3821 4827 y Fi(0)3845 4857 y Fu(\).)0 4998 y(Otherwise)f Fr(:)p Fu(\()p Fo(t)510 5010 y Fn(1)p 556 4972 V 556 4990 V 614 4998 a Fo(t)644 5010 y Fn(2)681 4998 y Fu(\))d Fr(2)f Fu(TR)n(UE)1053 5010 y Fn(0)1090 4998 y Fu(.)37 b(])g(Hence)27 b Fr(`)1507 4968 y Fp(!)1507 5019 y Fn(0)1578 4998 y Fo(C)6 b Fu(.)0 5164 y(3.)36 b Fo(C)30 b Fu(=)22 b Fr(8)p Fo(x)p Fr(8)p Fo(y)s Fu(\()p Fk(S)o Fo(x)p 594 5138 V 594 5156 V 67 w Fk(S)p Fo(y)k Fr(!)d Fo(x)p 927 5138 V 927 5156 V 67 w(y)s Fu(\):)37 b(The)28 b(claim)f(holds,)g(since)h Fr(`)2009 5134 y Fn(0)2009 5185 y(0)2069 5164 y Fr(:)p Fk(S)p Fo(s)p 2217 5138 V 2217 5156 V 66 w Fk(S)q Fo(t;)14 b(s)p 2436 5138 V 2436 5156 V 66 w(t)27 b Fu(for)h(all)f Fo(s;)14 b(t)23 b Fr(2)g Fo(T)12 b Fu(.)0 5330 y(4.)36 b(All)29 b(other)e(axioms)f(are)h(of)g(the)h(form)f Fr(8)o Fo(~)-41 b(xA)28 b Fu(with)g Fo(A)1740 5343 y Fp(~)-33 b(x)1783 5330 y Fu(\()m Fo(~)-39 b(s)p Fu(\))24 b Fr(2)f Fu(TR)n(UE)2226 5342 y Fn(0)2291 5330 y Fu(for)k(all)d Fo(~)-39 b(s)23 b Fr(2)g Fo(T)12 b Fu(.)1908 5539 y(22)p eop %%Page: 23 24 23 23 bop 0 83 a Ft(Theorem)30 b(4.9)0 224 y Fu(If)e Fr(Z)i(`)23 b Fo(C)34 b Fu(and)27 b(FV)r(\()p Fo(C)6 b Fu(\))24 b(=)e Fr(;)27 b Fu(then)h Fe(Z)1166 187 y Fi(1)1259 224 y Fr(`)1310 194 y Fp(\013)1310 245 y Fn(0)1380 224 y Fo(C)34 b Fu(for)27 b(some)g Fo(\013)d(<)e(")2011 236 y Fn(0)2048 224 y Fu(.)0 390 y(Pro)r(of:)0 531 y(Assume)32 b Fr(Z)38 b(`)31 b Fo(C)6 b Fu(.)51 b(Then)32 b(there)g(are)g(axioms)f Fo(A)1559 543 y Fn(1)1596 531 y Fo(;)14 b(:::;)g(A)1801 543 y Fp(l)1858 531 y Fr(2)31 b(Z)39 b Fu(suc)n(h)32 b(that)h(PL1)c Fr(`)i(:)p Fo(A)2799 543 y Fn(1)2837 531 y Fo(;)14 b(:::;)g Fr(:)p Fo(A)3097 543 y Fp(l)3123 531 y Fo(;)g(C)6 b Fu(.)51 b(No)n(w)32 b(Lemma)g(4.5)0 672 y(yields)27 b Fr(`)284 642 y Fp(!)284 693 y Fn(0)355 672 y Fr(:)p Fo(A)472 684 y Fn(1)510 672 y Fo(;)14 b(:::;)g Fr(:)p Fo(A)770 684 y Fp(n)816 672 y Fo(;)g(C)6 b Fu(,)28 b(and)f(b)n(y)h(Lemma)f(4.8)g(w)n(e)g(ha)n(v)n(e)f Fr(`)2040 633 y Fp(\014)2040 695 y Fn(0)2108 672 y Fo(A)2170 684 y Fp(i)2225 672 y Fu(\()p Fo(i)d Fu(=)g(1)p Fo(;)14 b(:::;)g(l)r Fu(\))27 b(for)g(some)g Fo(\014)g(<)c(!)e Fu(+)d Fo(!)s Fu(.)0 839 y(Hence)27 b Fr(`)297 808 y Fp(!)r Fn(+)p Fp(!)297 859 y(m)463 839 y Fo(C)6 b Fu(,)28 b(with)f Fo(m)c Fu(:=)g(max)1129 851 y Fn(1)p Fi(\024)p Fp(i)p Fi(\024)p Fp(l)1328 839 y Fu(rk\()p Fo(A)1499 851 y Fp(i)1527 839 y Fu(\)+1.)36 b(No)n(w)27 b(w)n(e)f(apply)h(the)h(Cut-Elimination)f (Theorem)f(4.2)g(and)h(obtain)0 1011 y Fr(`)51 981 y Fp(\013)51 1032 y Fn(0)121 1011 y Fo(C)34 b Fu(with)28 b Fo(\013)c Fu(:=)e(3)632 981 y Fi(\001)652 956 y Ff(\001)671 940 y(\001)690 923 y Fn(3)723 898 y Fh(!)q Fg(+)p Fh(!)886 1011 y Fo(<)g(")1012 1023 y Fn(0)1049 1011 y Fu(.)0 1227 y Ft(Theorem)30 b(4.10)55 b Fr(Z)29 b(`)23 b Fu(TI)884 1239 y Fi(\036)940 1227 y Fu(\()p Fo(X)r(;)14 b(n)p 1080 1240 50 4 v Fu(\))46 b(=)-14 b Fr(\))46 b(j)p Fo(n)p Fr(j)1484 1239 y Fi(\036)1563 1227 y Fo(<)23 b(")1690 1239 y Fn(0)0 1393 y Fu(Pro)r(of:)0 1535 y Fr(Z)30 b(`)23 b Fu(TI)254 1547 y Fi(\036)310 1535 y Fu(\()p Fo(X)r(;)14 b(n)p 450 1548 V -1 w Fu(\))579 1487 y Fn(4)p Fp(:)p Fn(9)555 1535 y Fu(=)-14 b Fr(\))22 b Fe(Z)760 1497 y Fi(1)853 1535 y Fr(`)904 1504 y Fp(\013)904 1555 y Fn(0)974 1535 y Fu(TI)1064 1547 y Fi(\036)1120 1535 y Fu(\()p Fo(X)r(;)14 b(n)p 1260 1548 V Fu(\))1388 1480 y Fn(4)p Fp(:)p Fn(4\(Cor)p Fp(:)p Fn(\))1453 1535 y Fu(=)-14 b Fr(\))111 b(j)p Fo(n)p Fr(j)1794 1547 y Fi(\036)1873 1535 y Fr(\024)23 b Fu(2)2003 1504 y Fp(\013)2073 1535 y Fo(<)g(")2200 1547 y Fn(0)2237 1535 y Fu(.)0 1725 y Ft(Corollary)-8 b(.)64 b Fu(If)28 b Fo(")598 1737 y Fn(0)658 1725 y Fr(\024)23 b(k\036)o(k)f Fu(:=)h(sup)p Fr(fj)p Fo(n)p Fr(j)1290 1737 y Fi(\036)1364 1725 y Fu(+)18 b(1)23 b(:)g Fo(n)g Fr(2)g Fu(I)-14 b(N)p Fr(g)55 b Fu(then)h Fr(Z)30 b(6`)23 b Fu(Prog)2438 1746 y Fi(\036)2494 1725 y Fu(\()p Fo(X)7 b Fu(\))23 b Fr(!)g(8)p Fo(y)s(X)7 b(y)s Fu(.)0 1892 y(Pro)r(of:)0 2033 y Fr(Z)30 b(`)23 b Fu(Prog)337 2053 y Fi(\036)393 2033 y Fu(\()p Fo(X)7 b Fu(\))23 b Fr(!)g(8)p Fo(y)s(X)7 b(y)47 b Fr(\))f(Z)30 b(`)23 b Fu(TI)1299 2045 y Fi(\036)1356 2033 y Fu(\()p Fo(X)r(;)14 b(n)p 1496 2046 V -1 w Fu(\))28 b(\(for)g(all)f Fo(n)c Fr(2)g Fu(I)-14 b(N)q(\))2188 1978 y Fn(4)p Fp(:)p Fn(4\(Cor)p Fp(:)p Fn(\))2278 2033 y Fr(\))137 b(j)p Fo(n)p Fr(j)2594 2045 y Fi(\036)2673 2033 y Fr(\024)23 b Fu(2)2803 2003 y Fp(\013)2877 2033 y Fu(for)k(all)h Fo(n)50 b Fr(\))c(k\036)o(k)23 b Fo(<)g(")3647 2045 y Fn(0)3684 2033 y Fu(.)1908 5539 y(23)p eop %%Page: 24 25 24 24 bop 0 83 a Fq(Pro)n(v)-5 b(abilit)n(y)27 b(of)g(trans\014nite)i (induction)f(in)g Fr(Z)0 248 y Fu(W)-7 b(e)20 b(assume)g(a)f(bijectiv)n (e)i(co)r(ding)e(of)h(\014nite)h(sequences)e(of)h(natural)f(n)n(um)n(b) r(ers)h(I)-14 b(N)2493 211 y Fp()e(m)p Fu(.)0 4942 y Ft(Theorem)30 b(5.2)0 5083 y Fu(If)g Fr(Z)145 5095 y Fp(m)235 5083 y Fr(`)d Fo(A)j Fu(then)g Fe(Z)d Fr(`)722 5095 y Fp(m)p Fn(+1)896 5083 y Fo(A)p Fu(,)k(i.e.,)g(there)e(is)h(a)f Fe(Z)q Fu(-deriv)-5 b(ation)28 b Fo(d)j Fu(of)e Fo(A)i Fu(suc)n(h)e(that)h(rk\()p Fo(C)6 b Fu(\))27 b Fr(\024)g Fo(m)i Fu(for)h(eac)n(h)f Fk(Cut)3517 5095 y Fp(C)3602 5083 y Fu(or)g Fk(Ind)3815 5043 y Fp(y)r(;t)3815 5107 y(C)0 5224 y Fu(o)r(ccurring)d(in)i Fo(d)p Fu(.)1908 5539 y(27)p eop %%Page: 28 29 28 28 bop 0 83 a Fu(Pro)r(of:)0 224 y(Let)29 b Fe(Z)198 236 y Fp(m)289 224 y Fu(b)r(e)g(the)g(subsystem)f(of)h Fe(Z)f Fu(corresp)r(onding)f(to)h Fr(Z)1817 236 y Fp(m)1880 224 y Fu(,)h(i.e.,)g Fe(Z)2138 236 y Fp(m)2230 224 y Fu(arises)e(from)h Fe(Z)g Fu(b)n(y)g(remo)n(ving)f(all)h(inferences)g Fk(Ind)3815 184 y Fp(y)r(;t)3815 249 y(F)0 365 y Fu(with)g(rk)o(\()p Fo(F)12 b Fu(\))24 b Fo(>)f(m)p Fu(.)36 b(W)-7 b(e)28 b(\014rst)g(sho)n(w)f Fe(Z)1207 377 y Fp(m)1293 365 y Fr(`)1344 377 y Fn(0)1404 365 y Fo(A)h Fu(,)g(for)f(eac)n(h)f Fo(A)e Fr(2)f(Z)2082 377 y Fp(m)2145 365 y Fu(:)0 506 y(1.)36 b(If)28 b Fo(A)g Fu(is)g(not)g(an)f(induction)h(axiom)f(then)h Fe(Z)1474 518 y Fp(m)1560 506 y Fr(`)1611 518 y Fn(0)1671 506 y Fo(A)g Fu(is)f(easy)g(to)h(see.)0 648 y(2.)36 b(No)n(w)28 b(let)f Fo(A)d Fr(\021)e(8)m Fo(~)-39 b(v)s Fr(8)p Fo(z)t Fu(\()p Fo(F)848 660 y Fp(x)889 648 y Fu(\(0\))23 b Fr(!)g(8)p Fo(x)p Fu(\()p Fo(F)35 b Fr(!)23 b Fo(F)1497 660 y Fp(x)1539 648 y Fu(\()p Fk(S)p Fo(x)p Fu(\)\))h Fr(!)g Fo(F)1912 660 y Fp(x)1954 648 y Fu(\()p Fo(z)t Fu(\)\))k(with)g(rk)o(\()p Fo(F)12 b Fu(\))24 b Fr(\024)e Fo(m)p Fu(.)0 789 y(Let)28 b Fo(G)23 b Fu(:=)g Fr(8)p Fo(x)p Fu(\()p Fo(F)35 b Fr(!)23 b Fo(F)721 801 y Fp(x)763 789 y Fu(\()p Fk(S)p Fo(x)p Fu(\)\))29 b(and)e Fo(y)f Fr(2)e Fk(FV)19 b Fr(n)f Fu(FV)q(\()p Fo(G)p Fu(\):)0 930 y(PL1)24 b Fr(`)226 942 y Fn(0)288 930 y Fo(F)341 942 y Fp(x)384 930 y Fu(\()p Fo(y)s Fu(\))19 b Fr(^)h(:)p Fo(F)694 942 y Fp(x)736 930 y Fu(\()p Fk(S)q Fo(y)s Fu(\))p Fo(;)14 b Fr(:)p Fo(F)1036 942 y Fp(x)1078 930 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(F)1276 942 y Fp(x)1319 930 y Fu(\()p Fk(S)p Fo(y)s Fu(\))51 b Fr(\))f Fu(PL1)24 b Fr(`)1883 942 y Fn(0)1946 930 y Fr(:)p Fo(G;)14 b Fr(:)p Fo(F)2211 942 y Fp(x)2254 930 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(F)2452 942 y Fp(x)2494 930 y Fu(\()p Fk(S)p Fo(y)s Fu(\))2699 883 y Fk(Ind)2712 930 y Fr(\))63 b Fe(Z)2906 942 y Fp(m)2995 930 y Fr(`)3046 942 y Fn(0)3108 930 y Fr(:)p Fo(G;)14 b Fr(:)p Fo(F)3373 942 y Fp(x)3416 930 y Fu(\(0\))p Fo(;)g(F)3612 942 y Fp(x)3654 930 y Fu(\()p Fo(u)p Fu(\))51 b Fr(\))0 1071 y Fe(Z)48 1083 y Fp(m)134 1071 y Fr(`)185 1083 y Fn(0)245 1071 y Fr(8)p Fo(z)t Fu(\()p Fo(F)420 1083 y Fp(x)461 1071 y Fu(\(0\))23 b Fr(!)g Fo(G)h Fr(!)f Fo(F)944 1083 y Fp(x)986 1071 y Fu(\()p Fo(z)t Fu(\)\))46 b Fr(\))g Fe(Z)1349 1083 y Fp(m)1435 1071 y Fr(`)1486 1083 y Fn(0)1546 1071 y Fo(A)p Fu(.)0 1237 y(No)n(w)33 b(assume)h Fr(Z)549 1249 y Fp(m)646 1237 y Fr(`)f Fo(A)p Fu(.)56 b(Then)35 b(there)e(are)g Fo(A)1520 1249 y Fn(1)1558 1237 y Fo(;)14 b(:::;)g(A)1763 1249 y Fp(n)1842 1237 y Fr(2)34 b(Z)1991 1249 y Fp(m)2088 1237 y Fu(suc)n(h)g(that)g(PL1)f Fr(`)2703 1249 y Fn(0)2774 1237 y Fr(:)p Fo(A)2891 1249 y Fn(1)2928 1237 y Fo(;)14 b(:::;)g Fr(:)p Fo(A)3188 1249 y Fp(n)3234 1237 y Fo(;)g(A)34 b Fu(and)g(\(as)g(sho)n(wn)0 1378 y(ab)r(o)n(v)n(e\))27 b Fe(Z)316 1390 y Fp(m)402 1378 y Fr(`)453 1390 y Fn(0)513 1378 y Fo(A)575 1390 y Fp(i)630 1378 y Fu(for)h Fo(i)22 b Fu(=)h(1)p Fo(;)14 b(:::;)g(n)p Fu(.)36 b(This)28 b(yields)f Fe(Z)1662 1390 y Fp(m)1748 1378 y Fr(`)1799 1390 y Fp(r)1859 1378 y Fo(A)h Fu(with)g Fo(r)e Fu(:=)d(max)o Fr(f)p Fo(m;)14 b Fu(rk)o(\()p Fo(A)2788 1390 y Fn(1)2825 1378 y Fu(\))p Fo(;)g(:::;)g Fu(rk\()p Fo(A)3171 1390 y Fp(n)3217 1378 y Fu(\))p Fr(g)p Fu(+)o(1.)0 1544 y(No)n(w)27 b(w)n(e)g(apply)h (Theorem)e(2.5)h(\(with)i Fj(S)22 b Fu(:=)h Fe(Z)1496 1556 y Fp(m)1587 1544 y Fu(and)k Fo(m)1821 1556 y Fn(0)1882 1544 y Fu(:=)22 b Fo(m)p Fu(+1\))27 b(and)h(obtain)f Fe(Z)2699 1556 y Fp(m)2785 1544 y Fr(`)2836 1556 y Fp(m)p Fn(+1)3006 1544 y Fo(A)h Fu(,)g(i.e.,)g Fe(Z)23 b Fr(`)3426 1556 y Fp(m)p Fn(+1)3596 1544 y Fo(A)p Fu(.)0 1810 y Ft(De\014nition)31 b(of)h Fu(hgt)674 1830 y Fp(d)713 1810 y Fu(\()p Fo(\027)5 b Fu(\))33 b Ft(and)f Fo(O)1103 1822 y Fp(d)1142 1810 y Fu(\()p Fo(\027)5 b Fu(\))61 b(\(Gen)n(tzen\))0 1976 y(Let)28 b Fo(d)g Fu(b)r(e)g(a)f Fe(Z)p Fu(-deriv)-5 b(ation)27 b(and)g Fo(\027)33 b Fu(a)27 b(p)r(osition)h(in)g Fo(d)p Fu(.)0 2167 y Fr(I)45 2179 y Fp(d)84 2167 y Fu(\()p Fo(\027)5 b Fu(\))24 b(:=)j(the)h(inference)g (at)f(p)r(osition)g Fo(\027)33 b Fu(in)28 b Fo(d)0 2341 y Fu(rk)76 2353 y Fp(d)115 2341 y Fu(\()p Fo(\027)5 b Fu(\))24 b(:=)360 2224 y Fl(\032)436 2313 y Fu(rk)o(\()p Fo(C)6 b Fu(\))84 b(if)28 b Fr(I)846 2325 y Fp(d)885 2313 y Fu(\()p Fo(\027)5 b Fu(\))24 b(=)f Fk(Cut)1232 2325 y Fp(C)1316 2313 y Fu(or)k Fk(Ind)1527 2273 y Fp(y)r(;t)1527 2338 y(C)436 2403 y Fu(0)247 b(otherwise)0 2565 y(hgt)120 2586 y Fp(d)159 2565 y Fu(\()p Fo(\027)5 b Fu(\))24 b(:=)e(sup)p Fr(f)p Fu(rk)646 2577 y Fp(d)685 2565 y Fu(\()p Fo(\033)s Fu(\))i(:)f Fo(\033)31 b Fu(is)d(strictly)f(b)r(elo)n(w)g Fo(\027)5 b Fr(g)0 2939 y Fo(O)63 2951 y Fp(d)102 2939 y Fu(\()p Fo(\027)g Fu(\))24 b(:=)347 2719 y Fl(8)347 2794 y(>)347 2819 y(>)347 2844 y(<)347 2993 y(>)347 3018 y(>)347 3043 y(:)434 2782 y Fo(!)486 2794 y Fp(m)549 2782 y Fu(\()p Fo(O)644 2794 y Fp(d)683 2782 y Fu(\()p Fo(\027)5 b Fu(0\)#)p Fo(O)967 2794 y Fp(d)1007 2782 y Fu(\()p Fo(\027)g Fu(1\)\))84 b(if)28 b Fr(I)1396 2794 y Fp(d)1435 2782 y Fu(\()p Fo(\027)5 b Fu(\))24 b(=)e Fk(Cut)1782 2794 y Fp(C)434 2935 y Fo(!)486 2947 y Fp(m)549 2935 y Fu(\()p Fo(O)644 2947 y Fp(d)683 2935 y Fu(\()p Fo(\027)5 b Fu(0\))19 b Fr(\001)g Fo(!)s Fu(\))292 b(if)28 b Fr(I)1396 2947 y Fp(d)1435 2935 y Fu(\()p Fo(\027)5 b Fu(\))24 b(=)e Fk(Ind)1765 2895 y Fp(y)r(;t)1765 2960 y(C)434 3092 y Fu(\(sup)592 3113 y Fp(i)f(!)2546 5111 y Fp(m)2609 5099 y Fu(\()p Fo(O)2704 5111 y Fp(d)2743 5099 y Fu(\()p Fo(\027)5 b Fu(0\))23 b Fr(\001)g Fo(n)p Fu(\))35 b(=)f Fo(O)3243 5115 y Fp(d)3278 5098 y Ff(\000)3331 5099 y Fu(\()p Fo(\027)5 b Fu(\))36 b(\(with)f Fo(m)f Fu(:=)0 5240 y(rk)o(\()p Fo(F)12 b Fu(\))p Fr(\000)224 5223 y(\001)275 5240 y Fu(hgt)395 5260 y Fp(d)434 5240 y Fu(\()p Fo(\027)5 b Fu(\)\).)38 b(This)27 b(yields)h Fo(O)r Fu(\()p Fo(d)p Fu(\))c Fo(>)f(O)r Fu(\()p Fo(d)1484 5210 y Fi(\000)1541 5240 y Fu(\).)1908 5539 y(29)p eop %%Page: 30 31 30 30 bop 0 281 a Fu(Case)27 b(2:)36 b Fo(d)24 b Fu(=)552 55 y(~)537 77 y Fo(d)580 89 y Fn(0)507 151 y Fr(\001)36 b(\001)g(\001)531 189 y(\001)46 b(\001)543 230 y Fj(c)575 242 y Fn(0)881 55 y Fu(~)867 77 y Fo(d)910 89 y Fn(1)836 151 y Fr(\001)36 b(\001)h(\001)861 189 y(\001)46 b(\001)872 230 y Fj(c)904 242 y Fn(1)p 543 272 427 4 v 665 346 a Fk(Cut)791 358 y Fp(C)685 420 y Fr(\001)36 b(\001)h(\001)710 457 y(\001)46 b(\001)735 499 y Fj(a)1141 281 y Fu(with)28 b Fo(C)h Fr(2)24 b Fu(\001\()p Fk(last)p Fu(\()1767 259 y(~)1752 281 y Fo(d)1795 293 y Fn(0)1832 281 y Fu(\)\))29 b(and)e Fr(:)p Fo(C)j Fr(2)23 b Fu(\001\()p Fk(last)p Fu(\()2578 259 y(~)2563 281 y Fo(d)2606 293 y Fn(1)2644 281 y Fu(\)\).)0 619 y(Here)k(w)n(e)g(ha)n(v)n(e)g(four)g(sub)r(cases)g (according)f(to)h(the)h(shap)r(e)g(of)f Fo(C)6 b Fu(.)37 b(Let)28 b(us)g(consider)e(the)i(case)f Fo(C)i Fu(=)23 b Fr(8)p Fo(xA)p Fu(.)37 b(Then)0 1142 y Fo(d)23 b Fu(=)213 804 y Fo(d)256 816 y Fn(0)p 154 846 199 4 v 154 858 a Fl(V)223 879 y Fp(y)223 945 y Fi(8)p Fp(xA)182 994 y Fr(\001)37 b(\001)f(\001)207 1031 y(\001)46 b(\001)218 1073 y Fj(c)250 1085 y Fn(0)600 797 y Fo(d)643 809 y Fn(1)p 518 839 244 4 v 518 858 a Fl(W)588 879 y Fp(t)588 945 y Fi(9)p Fp(x)p Fi(:)p Fp(A)569 994 y Fr(\001)37 b(\001)f(\001)594 1031 y(\001)46 b(\001)605 1073 y Fj(c)637 1085 y Fn(1)p 218 1114 485 4 v 333 1189 a Fk(Cut)459 1201 y Fi(8)p Fp(xA)390 1263 y Fr(\001)36 b(\001)g(\001)414 1300 y(\001)47 b(\001)439 1342 y Fj(b)390 1416 y Fr(\001)36 b(\001)g(\001)414 1453 y(\001)47 b(\001)421 1494 y Fj(a)463 1506 y Fn(0)983 1142 y Fu(and)194 b Fo(d)1354 1112 y Fi(\000)1433 1142 y Fu(:=)1544 862 y Fo(d)1587 874 y Fn(0)1624 862 y Fu(\()p Fo(y)s(=t)p Fu(\))1603 936 y Fr(\001)36 b(\001)g(\001)1628 973 y(\001)46 b(\001)1639 1015 y Fj(c)1671 1027 y Fn(0)1968 739 y Fo(d)2011 751 y Fn(1)p 1887 780 244 4 v 1887 800 a Fl(W)1956 820 y Fp(t)1956 887 y Fi(9)p Fp(x)p Fi(:)p Fp(A)1938 936 y Fr(\001)36 b(\001)g(\001)1962 973 y(\001)46 b(\001)1974 1015 y Fj(c)2006 1027 y Fn(1)p 1639 1056 405 4 v 1713 1131 a Fk(Cut)1839 1143 y Fi(8)p Fp(xA)1770 1205 y Fr(\001)37 b(\001)f(\001)1795 1242 y(\001)46 b(\001)1820 1283 y Fj(b)2355 746 y Fo(d)2398 758 y Fn(0)p 2296 787 199 4 v 2296 800 a Fl(V)2365 820 y Fp(y)2365 887 y Fi(8)p Fp(xA)2324 936 y Fr(\001)37 b(\001)f(\001)2349 973 y(\001)46 b(\001)2361 1015 y Fj(c)2393 1027 y Fn(0)2734 862 y Fo(d)2777 874 y Fn(1)2704 936 y Fr(\001)36 b(\001)g(\001)2729 973 y(\001)46 b(\001)2740 1015 y Fj(c)2772 1027 y Fn(1)p 2361 1056 449 4 v 2457 1131 a Fk(Cut)2583 1143 y Fi(8)p Fp(xA)2514 1205 y Fr(\001)36 b(\001)h(\001)2539 1242 y(\001)46 b(\001)2564 1283 y Fj(b)p 1820 1325 787 4 v 2085 1400 a Fk(Cut)2210 1415 y Fp(A)p Fn(\()p Fp(t)p Fn(\))2142 1474 y Fr(\001)36 b(\001)h(\001)2167 1511 y(\001)46 b(\001)2174 1553 y Fj(a)2216 1565 y Fn(0)0 1727 y Fu(where)25 b Fj(a)d Fu(=)h Fj(a)432 1739 y Fn(0)469 1727 y Fr(f)p Fj(b)p Fr(g)g Fu(suc)n(h)i(that)g(the)h(ro)r(ot)e(of)h Fj(b)g Fu(\(i.e.)36 b(the)26 b(p)r(osition)f(of)g Fr(\003)f Fu(in)i Fj(a)2381 1739 y Fn(0)2417 1727 y Fu(\))g(is)f(the)g(upp)r(ermost)g(no)r(de)g Fo(\027)31 b Fu(b)r(elo)n(w)24 b(the)i(ro)r(ot)0 1868 y(of)i Fj(c)127 1880 y Fn(0)192 1868 y Fu(\(or)f Fj(c)358 1880 y Fn(1)395 1868 y Fu(\))h(suc)n(h)g(that)f(hgt)942 1889 y Fp(d)981 1868 y Fu(\()p Fo(\027)5 b Fu(\))24 b Fo(<)f Fu(hgt)1323 1889 y Fp(d)1361 1868 y Fu(\()p Fo(\027)5 b Fu(0\).)0 2059 y(Instead)27 b(of)h(pro)n(ving)e Fo(O)r Fu(\()p Fo(d)828 2029 y Fi(\000)885 2059 y Fu(\))d Fo(<)g(O)r Fu(\()p Fo(d)p Fu(\))29 b(in)f(general)e(w)n(e)h(treat)h(a)f (su\016cien)n(tly)g(informativ)n(e)g(example.)0 2325 y Ft(Example)0 2466 y Fu(By)g(:)c Fo(m)28 b Fu(w)n(e)f(indicate)h(the)g (heigh)n(t)f(hgt)1227 2486 y Fp(d)1266 2466 y Fu(\()p Fo(\027)5 b Fu(\))28 b(of)g(the)g(resp)r(ectiv)n(e)f(p)r(osition)g Fo(\027)5 b Fu(.)0 2930 y Fo(d)23 b Fu(=)213 2647 y Fo(d)256 2659 y Fn(0)p 154 2689 199 4 v 154 2701 a Fl(V)223 2722 y Fp(y)223 2788 y Fi(8)p Fp(xA)435 2764 y Fo(c)471 2776 y Fn(0)p 154 2805 355 4 v 176 2880 a Fk(Cut)302 2892 y Fp(G)354 2900 y Fg(0)390 2880 y Fu(:)p Fo(m)756 2640 y(d)799 2652 y Fn(1)p 675 2682 244 4 v 675 2701 a Fl(W)744 2722 y Fp(t)744 2788 y Fi(9)p Fp(x)p Fi(:)p Fp(A)1001 2764 y Fo(c)1037 2776 y Fn(1)p 675 2805 400 4 v 719 2880 a Fk(Cut)845 2892 y Fp(G)897 2900 y Fg(1)933 2880 y Fu(:)q Fo(m)p 176 2921 854 4 v 427 2996 a Fk(Cut)553 3008 y Fi(8)p Fp(xA)682 2996 y Fu(:)q Fo(m)323 b(b)p 427 3038 711 4 v 655 3112 a Fk(Cut)781 3124 y Fp(F)836 3112 y Fu(:)p Fo(n)311 b(a)p 655 3154 609 4 v 834 3229 a Fk(Cut)960 3241 y Fp(D)1021 3229 y Fu(:0)1375 2930 y(with)28 b(rk)o(\()p Fo(D)r Fu(\))c(=)f Fo(n)g(<)f Fu(rk\()p Fr(8)p Fo(xA)p Fu(\))h Fr(\024)g Fu(rk)o(\()p Fo(F)12 b Fu(\))24 b(=)e Fo(m)p Fu(.)0 3466 y(Then)55 b Fj(b)23 b Fu(=)426 3411 y Fr(\003)184 b Fo(b)p 408 3447 281 4 v 457 3521 a Fk(Cut)583 3533 y Fp(F)725 3466 y Fu(,)56 b Fj(a)846 3478 y Fn(0)906 3466 y Fu(=)1022 3411 y Fr(\003)184 b Fo(a)p 1003 3447 289 4 v 1054 3521 a Fk(Cut)1180 3533 y Fp(D)1357 3466 y Fu(and)0 4104 y Fo(d)43 4074 y Fi(\000)122 4104 y Fu(=)210 3881 y Fo(d)253 3893 y Fn(0)290 3881 y Fu(\()p Fo(y)s(=t)p Fu(\))83 b Fo(c)589 3893 y Fn(0)p 210 3922 417 4 v 263 3997 a Fk(Cut)389 4009 y Fp(G)441 4017 y Fg(0)477 3997 y Fu(:)p Fo(m)791 3753 y(d)834 3765 y Fn(1)p 709 3795 244 4 v 709 3819 a Fl(W)779 3839 y Fp(k)779 3906 y Fi(9)p Fp(x)p Fi(:)p Fp(A)1036 3881 y Fo(c)1072 3893 y Fn(1)p 709 3922 400 4 v 754 3997 a Fk(Cut)880 4009 y Fp(G)932 4017 y Fg(1)968 3997 y Fu(:)p Fo(m)p 263 4039 802 4 v 488 4113 a Fk(Cut)614 4125 y Fi(8)p Fp(xA)743 4113 y Fu(:)p Fo(m)297 b(b)p 488 4155 685 4 v 705 4230 a Fk(Cut)831 4242 y Fp(F)886 4230 y Fu(:)p Fo(k)1314 3765 y(d)1357 3777 y Fn(0)p 1255 3806 199 4 v 1255 3819 a Fl(V)1324 3839 y Fp(y)1324 3906 y Fi(8)p Fp(xA)1537 3881 y Fo(c)1573 3893 y Fn(0)p 1255 3922 355 4 v 1277 3997 a Fk(Cut)1403 4009 y Fp(G)1455 4017 y Fg(0)1492 3997 y Fu(:)p Fo(m)1693 3881 y(d)1736 3893 y Fn(1)1939 3881 y Fo(c)1975 3893 y Fn(1)p 1693 3922 320 4 v 1698 3997 a Fk(Cut)1824 4009 y Fp(G)1876 4017 y Fg(1)1912 3997 y Fu(:)p Fo(m)p 1277 4039 731 4 v 1467 4113 a Fk(Cut)1593 4125 y Fi(8)p Fp(xA)1722 4113 y Fu(:)p Fo(m)222 b(b)p 1467 4155 609 4 v 1646 4230 a Fk(Cut)1772 4242 y Fp(F)1828 4230 y Fu(:)p Fo(k)p 705 4271 1192 4 v 1136 4346 a Fk(Cut)1262 4361 y Fp(A)p Fn(\()p Fp(t)p Fn(\))1393 4346 y Fu(:)p Fo(n)638 b(a)p 1136 4388 1012 4 v 1516 4463 a Fk(Cut)1642 4475 y Fp(D)1702 4463 y Fu(:0)2341 4104 y(with)28 b Fo(k)e Fu(:=)d(max)o Fr(f)p Fo(n;)14 b Fu(rk)o(\()p Fo(A)p Fu(\()p Fo(t)p Fu(\)\))p Fr(g)24 b Fo(<)e(m)p Fu(.)0 4720 y Fo(O)r Fu(\()p Fo(d)140 4690 y Fi(\000)197 4720 y Fu(\))i(=)e Fo(!)392 4732 y Fp(n)437 4628 y Fl(\020)487 4720 y Fo(!)539 4732 y Fp(k)q Fi(\000)p Fp(n)672 4653 y Fl(\000)710 4720 y Fo(!)762 4732 y Fp(m)p Fi(\000)p Fp(k)913 4720 y Fu(\()p Fo(\015)993 4684 y Fi(\000)988 4742 y Fn(0)1049 4720 y Fu(#)p Fo(\015)1161 4732 y Fn(1)1199 4720 y Fu(#)p Fo(\014)t Fu(\))14 b(#)g Fo(!)1500 4732 y Fp(m)p Fi(\000)p Fp(k)1652 4720 y Fu(\()p Fo(\015)1727 4732 y Fn(0)1764 4720 y Fu(#)p Fo(\015)1881 4684 y Fi(\000)1876 4742 y Fn(1)1937 4720 y Fu(#)p Fo(\014)t Fu(\))2089 4653 y Fl(\001)2142 4720 y Fu(#)g Fo(\013)2278 4628 y Fl(\021)2351 4720 y Fo(<)0 4878 y(!)52 4890 y Fp(n)97 4786 y Fl(\020)146 4878 y Fo(!)198 4890 y Fp(k)q Fi(\000)p Fp(n)332 4810 y Fl(\000)370 4878 y Fo(!)422 4890 y Fp(m)p Fi(\000)p Fp(k)573 4878 y Fu(\()p Fo(\015)648 4890 y Fn(0)686 4878 y Fu(#)p Fo(\015)798 4890 y Fn(1)835 4878 y Fu(#)p Fo(\014)t Fu(\))987 4810 y Fl(\001)1040 4878 y Fu(#)g Fo(\013)1176 4786 y Fl(\021)1249 4878 y Fu(=)23 b Fo(!)1389 4890 y Fp(n)1433 4810 y Fl(\000)1472 4878 y Fo(!)1524 4890 y Fp(m)p Fi(\000)p Fp(n)1679 4878 y Fu(\()p Fo(\015)1754 4890 y Fn(0)1792 4878 y Fu(#)p Fo(\015)1904 4890 y Fn(1)1941 4878 y Fu(#)p Fo(\014)t Fu(\))14 b(#)g Fo(\013)2243 4810 y Fl(\001)2305 4878 y Fu(=)23 b Fo(O)r Fu(\()p Fo(d)p Fu(\))29 b(.)1908 5539 y(30)p eop %%Page: 31 32 31 31 bop 0 83 a Ft(Theorem)54 b Fu(PRA)18 b(+)g(TI)800 53 y Fn(qf)800 104 y Fp(")831 112 y Fg(0)891 83 y Fr(`)23 b Fu(Con\()p Fe(Z)p Fu(\).)0 249 y(Pro)r(of:)0 390 y(W)-7 b(e)28 b(assume)f(that)h Fe(Z)p Fu(-deriv)-5 b(ations)26 b(and)i(ordinals)e(are)h(co)r(ded)g(b)n(y)h(natural)e(n)n(um)n(b)r(ers) h(suc)n(h)h(that)0 531 y({)f Fo(O)r Fu(\()p Fo(d)p Fu(\))e Fr(2)e Fu(OT)k(for)g(eac)n(h)g Fo(d)c Fr(2)h Fo(Z)33 b Fu(\()p Fo(Z)c Fu(:=)e(set)h(of)g(all)f(\(co)r(des)g(of)6 b(\))29 b Fe(Z)p Fu(-deriv)-5 b(ations\),)0 672 y({)27 b(the)h(function)g Fo(d)c Fr(7!)f Fo(O)r Fu(\()p Fo(d)p Fu(\))29 b(is)e(primitiv)n(e)h(recursiv)n(e,)0 814 y({)f(there)h(is)f (a)g(primitiv)n(e)h(recursiv)n(e)d(function)k Fo(g)h Fu(with)e Fr(8)p Fo(d)22 b Fr(2)i Fo(Z)6 b Fu(\()p Fo(g)s Fu(\()p Fo(d)p Fu(\))23 b(=)g(0)f Fr($)h Fu(\000\()p Fo(d)p Fu(\))h(=)f Fr(;)p Fu(\).)0 1005 y(Abbreviation:)36 b Fo(F)12 b Fu(\()p Fo(y)s(;)i(z)t Fu(\))23 b(:)p Fr(\021)f Fo(z)27 b Fr(2)c Fo(Z)h Fr(^)19 b Fo(O)r Fu(\()p Fo(z)t Fu(\))k(=)g Fo(y)40 b Fr(!)d Fo(g)s Fu(\()p Fo(z)t Fu(\))22 b Fr(6)p Fu(=)h(0,)2241 984 y(~)2222 1005 y Fo(F)12 b Fu(\()p Fo(y)s Fu(\))23 b(:)p Fr(\021)g(8)p Fo(z)t(F)12 b Fu(\()p Fo(y)s(;)i(z)t Fu(\))0 1171 y(By)27 b(TI)220 1183 y Fi(\036)276 1171 y Fu(\()327 1150 y(~)308 1171 y Fo(F)12 b Fu(\))28 b(w)n(e)g(pro)n(v)n(e)e Fr(8)p Fo(y)888 1150 y Fu(~)871 1171 y Fo(F)10 b Fu(\()p Fo(y)s Fu(\),)28 b(i.e.)g Fr(8)p Fo(a)p Fr(8)p Fo(d)p Fu(\()p Fo(d)22 b Fr(2)h Fo(Z)h Fr(^)19 b Fo(O)r Fu(\()p Fo(d)p Fu(\))25 b(=)d Fo(a)h Fr(!)g Fo(g)s Fu(\()p Fo(d)p Fu(\))h Fr(6)p Fu(=)e(0\):)0 1337 y(Assume)28 b Fo(d)c Fr(2)f Fo(Z)i Fr(^)18 b Fo(O)r Fu(\()p Fo(d)p Fu(\))25 b(=)e Fo(a)p Fu(.)38 b(Then)28 b(there)f(is)h(a)g Fo(d)1624 1307 y Fi(\000)1703 1337 y Fr(2)c Fo(Z)34 b Fu(with)28 b Fo(g)s Fu(\()p Fo(d)p Fu(\))c(=)f(0)g Fr(!)g Fo(O)r Fu(\()p Fo(d)2635 1307 y Fi(\000)2692 1337 y Fu(\))h Fr(\036)f Fo(O)r Fu(\()p Fo(d)p Fu(\))i(=)e Fo(a)18 b Fr(^)h Fo(g)s Fu(\()p Fo(d)3375 1307 y Fi(\000)3431 1337 y Fu(\))24 b(=)f(0.)37 b(By)28 b(IH)0 1478 y(w)n(e)f(ha)n(v)n(e)g Fo(O)r Fu(\()p Fo(d)454 1448 y Fi(\000)511 1478 y Fu(\))c Fr(\036)g Fo(a)g Fr(!)g Fo(g)s Fu(\()p Fo(d)945 1448 y Fi(\000)1001 1478 y Fu(\))g Fr(6)p Fu(=)g(0.)36 b(Hence)28 b Fo(g)s Fu(\()p Fo(d)p Fu(\))c(=)e(0)h Fr(!)g(?)p Fu(,)k(i.e.)37 b Fo(g)s Fu(\()p Fo(d)p Fu(\))24 b Fr(6)p Fu(=)e(0.)0 1694 y(It)28 b(remains)f(to)g(deriv)n(e)g(TI)836 1706 y Fi(\036)892 1694 y Fu(\()943 1673 y(~)924 1694 y Fo(F)12 b Fu(\))28 b(from)f(TI)1335 1654 y Fn(qf)1335 1712 y Fi(\036)1399 1694 y Fu(.)0 1860 y(There)g(are)g(primitiv)n(e)g (recursiv)n(e)f(functions)i Fo(q)s(;)14 b(r)n(;)g(p)27 b Fu(suc)n(h)h(that)0 2001 y({)f Fo(a)c Fr(2)h Fu(OT)e Fr(!)46 b Fo(q)s Fu(\()p Fo(a)p Fu(\))24 b Fr(2)f Fu(OT)18 b Fr(^)h Fo(o)p Fu(\()p Fo(a)p Fu(\))24 b(=)e Fo(!)g Fr(\001)c Fo(o)p Fu(\()p Fo(q)s Fu(\()p Fo(a)p Fu(\)\))i(+)e Fo(r)r Fu(\()p Fo(a)p Fu(\),)0 2142 y({)27 b Fo(b)c Fr(2)g Fu(OT)18 b Fr(^)h Fo(k)26 b Fr(2)d Fu(I)-14 b(N)38 b Fr(!)46 b Fo(p)p Fu(\()p Fo(b;)14 b(k)s Fu(\))23 b Fr(2)g Fu(OT)18 b Fr(^)h Fo(q)s Fu(\()p Fo(p)p Fu(\()p Fo(b;)14 b(k)s Fu(\)\))23 b(=)g Fo(b)18 b Fr(^)h Fo(r)r Fu(\()p Fo(p)p Fu(\()p Fo(b;)14 b(k)s Fu(\)\))24 b(=)e Fo(k)s Fu(.)0 2308 y(Abb.:)38 b Fo(G)p Fu(\()p Fo(x)p Fu(\))24 b(:)p Fr(\021)36 b Fo(x)24 b Fr(2)f Fu(OT)g Fr(!)g Fo(F)12 b Fu(\()p Fo(q)s Fu(\()p Fo(x)p Fu(\))p Fo(;)i(r)r Fu(\()p Fo(x)p Fu(\)\))0 2474 y(\(*\))28 b(Prog)307 2494 y Fi(\036)363 2474 y Fu(\()414 2453 y(~)395 2474 y Fo(F)12 b Fu(\))24 b Fr(!)f Fu(Prog)795 2494 y Fi(\036)851 2474 y Fu(\()p Fo(G)p Fu(\).)0 2640 y(Pro)r(of:)0 2781 y(Assume)55 b(\(1\))28 b(Prog)642 2802 y Fi(\036)698 2781 y Fu(\()749 2760 y(~)730 2781 y Fo(F)13 b Fu(\),)55 b(\(2\))28 b Fr(8)p Fo(y)d Fr(\036)d Fo(xG)p Fu(\()p Fo(y)s Fu(\),)57 b(\(3\))28 b Fo(x)23 b Fr(2)h Fu(OT)o(.)0 2922 y(T)-7 b(o)27 b(pro)n(v)n(e:)35 b Fr(8)p Fo(y)25 b Fr(\036)e Fo(q)s Fu(\()p Fo(x)p Fu(\))p Fr(8)p Fo(z)t(F)12 b Fu(\()p Fo(y)s(;)i(z)t Fu(\))54 b([)28 b(from)f(this)h(using)f(Prog)1926 2943 y Fi(\036)1982 2922 y Fu(\()2033 2901 y(~)2014 2922 y Fo(F)13 b Fu(\))28 b(w)n(e)f(get)2419 2901 y(~)2400 2922 y Fo(F)12 b Fu(\()p Fo(q)s Fu(\()p Fo(x)p Fu(\)\))29 b(and)f(then)g Fo(F)12 b Fu(\()p Fo(q)s Fu(\()p Fo(x)p Fu(\))p Fo(;)i(r)r Fu(\()p Fo(x)p Fu(\)\))31 b(].)0 3088 y(Let)d Fo(b)22 b Fr(\036)h Fo(q)s Fu(\()p Fo(x)p Fu(\))29 b(and)e Fo(k)f Fr(2)e Fu(I)-14 b(N)28 b(b)r(e)g(giv)n(en.)36 b(W)-7 b(e)28 b(set)f Fo(a)c Fu(:=)g Fo(p)p Fu(\()p Fo(b;)14 b(k)s Fu(\).)37 b(Then)27 b Fo(a)c Fr(2)h Fu(OT)18 b Fr(^)h Fo(q)s Fu(\()p Fo(a)p Fu(\))k(=)g Fo(b)f Fr(\036)h Fo(q)s Fu(\()p Fo(x)p Fu(\))d Fr(^)e Fo(r)r Fu(\()p Fo(a)p Fu(\))25 b(=)d Fo(k)s Fu(.)0 3254 y Fo(a;)14 b(x)23 b Fr(2)h Fu(OT)18 b Fr(^)g Fo(q)s Fu(\()p Fo(a)p Fu(\))24 b Fr(\036)f Fo(q)s Fu(\()p Fo(x)p Fu(\))47 b Fr(\))f Fo(a)23 b Fr(\036)g Fo(x)1281 3200 y Fn(\(2\))1282 3254 y Fr(\))47 b Fo(G)p Fu(\()p Fo(a)p Fu(\))g Fr(\))f Fo(F)12 b Fu(\()p Fo(q)s Fu(\()p Fo(a)p Fu(\))p Fo(;)i(r)r Fu(\()p Fo(a)p Fu(\)\))48 b Fr(\))f Fo(F)12 b Fu(\()p Fo(b;)i(k)s Fu(\).)0 3470 y(Using)27 b(TI)323 3430 y Fn(qf)323 3488 y Fi(\036)388 3470 y Fu(\()p Fo(G)p Fu(\))h(from)f(\(*\))h(w)n(e)f(get:)0 3611 y(Prog)173 3632 y Fi(\036)229 3611 y Fu(\()280 3590 y(~)261 3611 y Fo(F)12 b Fu(\))24 b Fr(\))f(8)p Fo(xG)p Fu(\()p Fo(x)p Fu(\))h Fr(\))f(8)p Fo(x)g Fr(2)g Fu(OT)13 b Fo(F)f Fu(\()p Fo(q)s Fu(\()p Fo(x)p Fu(\))p Fo(;)i(r)r Fu(\()p Fo(x)p Fu(\)\))27 b Fr(\))c(8)p Fo(y)i Fr(2)e Fu(OT)o Fr(8)p Fo(z)t(F)12 b Fu(\()p Fo(y)s(;)i(z)t Fu(\))21 b Fr(\))i(8)p Fo(y)i Fr(2)f Fu(OT)2941 3590 y(~)2922 3611 y Fo(F)12 b Fu(\()p Fo(y)s Fu(\).)0 3778 y(T)-7 b(rivially)27 b(holds)g(Prog)725 3798 y Fi(\036)781 3778 y Fu(\()832 3757 y(~)813 3778 y Fo(F)12 b Fu(\))24 b Fr(!)f(8)p Fo(y)s Fr(62)o Fu(OT)1342 3757 y(~)1323 3778 y Fo(F)12 b Fu(\()p Fo(y)s Fu(\).)1908 5539 y(31)p eop %%Page: 32 33 32 32 bop 0 83 a Fr(x)p Ft(6)63 b(Notations)31 b(for)h(in\014nitary)g (deriv)-5 b(ations;)31 b(the)h(pro)s(of)f(system)f Fe(Z)2451 46 y Fi(\003)0 249 y Ft(The)i(pro)s(of-system)e Fe(Z)813 212 y Fi(\003)0 415 y Fu(The)e(system)f Fe(Z)494 378 y Fi(\003)560 415 y Fu(results)g(from)g Fe(Z)h Fu(b)n(y)g(adding)f(the) h(follo)n(wing)e(inference)i(sym)n(b)r(ols:)0 590 y({)f(\()p Fk(R)155 602 y Fp(C)211 590 y Fu(\))299 535 y Fo(C)90 b Fr(:)p Fo(C)p 299 571 269 4 v 413 645 a Fr(;)689 590 y Fu(with)28 b(deg)q(\()p Fk(R)1090 602 y Fp(C)1145 590 y Fu(\))c(:=)e(rk\()p Fo(C)6 b Fu(\),)0 807 y({)27 b(\()p Fk(E)p Fu(\))253 753 y Fr(;)p 239 788 70 4 v 253 862 a(;)429 807 y Fu(with)h(deg)q(\()p Fk(E)p Fu(\))g(unde\014ned,)0 1013 y(and)f(de\014ning)h(o\()p Fo(h)p Fu(\))g(and)f(deg)q(\()p Fo(h)p Fu(\))h(for)f Fe(Z)1259 976 y Fi(\003)1297 1013 y Fu(-deriv)-5 b(ations)27 b Fo(h)c Fu(=)f Fr(I)6 b Fo(h)2005 1025 y Fn(0)2043 1013 y Fo(:::h)2160 1025 y Fp(n)p Fi(\000)p Fn(1)2318 1013 y Fu(as)26 b(follo)n(ws)0 1297 y(o\()p Fo(h)p Fu(\))d(:=)288 1102 y Fl(8)288 1177 y(>)288 1202 y(<)288 1351 y(>)288 1376 y(:)375 1155 y Fu(o\()p Fo(h)497 1167 y Fn(0)534 1155 y Fu(\)#o\()p Fo(h)757 1167 y Fn(1)795 1155 y Fu(\))229 b(if)28 b Fr(I)i Fu(=)22 b Fk(R)1348 1167 y Fp(C)375 1259 y Fu(o\()p Fo(h)497 1271 y Fn(0)534 1259 y Fu(\))d(+)f Fo(!)336 b Fu(if)28 b Fr(I)i Fu(=)22 b Fk(Ind)1403 1219 y Fp(y)r(;t)1403 1284 y(C)375 1366 y Fu(3)417 1336 y Fn(o\()p Fp(h)515 1344 y Fg(0)547 1336 y Fn(\))1056 1366 y Fu(if)28 b Fr(I)i Fu(=)22 b Fk(E)375 1436 y Fu(sup)500 1457 y Fp(i)g Fu(0)2041 4134 y(where)129 4352 y Fo(n)g Fu(:=)g Fk(val)p Fu(\()p Fo(t)p Fu(\),)56 b Fc(c)628 4312 y Fp(y)r(;t)628 4376 y(F)735 4352 y Fr(`)786 4319 y Fp()f Fu(0)28 b(.)129 4847 y(\()p Fk(Ind)270 4807 y Fp(y)r(;t)270 4872 y(F)355 4847 y Fo(h)403 4859 y Fn(0)440 4847 y Fu(\))472 4817 y Fi(1)566 4847 y Fu(=)653 4533 y Fo(h)701 4545 y Fn(0)738 4533 y Fu(\()p Fo(y)s(=)p Fu(0\))930 4503 y Fi(1)1083 4533 y Fo(h)1131 4545 y Fn(0)1168 4533 y Fu(\()p Fo(y)s(=)p Fu(1\))1360 4503 y Fi(1)p 653 4574 777 4 v 909 4649 a Fk(Cut)1034 4664 y Fp(F)9 b Fn(\(1\))1457 4649 y Fo(h)1505 4661 y Fn(0)1543 4649 y Fu(\()p Fo(y)s(=)p Fu(2\))1735 4619 y Fi(1)p 909 4691 896 4 v 1345 4754 a Fu(.)1345 4787 y(.)1345 4820 y(.)464 b Fo(h)1880 4832 y Fn(0)1917 4820 y Fu(\()p Fo(y)s(=n)p Fr(\000)o Fu(1\))2223 4790 y Fi(1)p 1345 4862 949 4 v 1639 4936 a Fk(Cut)1765 4951 y Fp(F)9 b Fn(\()p Fp(n)p Fi(\000)p Fn(1\))2376 4936 y Fc(c)2420 4897 y Fp(y)r(;t)2420 4961 y(F)p 1639 4978 865 4 v 1935 5053 a Fk(Cut)2061 5068 y Fp(F)g Fn(\()p Fp(n)p Fn(\))p 1935 5095 275 4 v 2005 5170 a Fk(Rep)0 5340 y Fu(5.)64 b(Otherwise:)36 b(\()p Fr(I)6 b Fo(h)685 5352 y Fn(0)723 5340 y Fo(:::h)840 5352 y Fp(n)p Fi(\000)p Fn(1)970 5340 y Fu(\))1002 5310 y Fi(1)1096 5340 y Fu(:=)22 b Fr(I)6 b Fo(h)1305 5310 y Fi(1)1305 5361 y Fn(0)1390 5340 y Fo(:)14 b(:)g(:)f(h)1548 5310 y Fi(1)1548 5361 y Fp(n)p Fi(\000)p Fn(1)1706 5340 y Fu(.)1908 5539 y(33)p eop %%Page: 34 35 34 34 bop 0 83 a Ft(Remark)0 224 y Fu(With)28 b(the)g(help)g(of)g (Theorems)e(4.1,4.2)g(one)i(easily)e(v)n(eri\014es)h(that)h Fo(h)2165 194 y Fi(1)2262 224 y Fu(is)g(a)f Fe(Z)2463 187 y Fi(1)2534 224 y Fu(-deriv)-5 b(ation)26 b(with)i Fo(h)3188 194 y Fi(1)3282 224 y Fr(`)3333 181 y Fn(o\()p Fp(h)p Fn(\))3333 252 y(deg\()p Fp(h)p Fn(\))3551 224 y Fu(\000\()p Fo(h)p Fu(\).)0 390 y(Let)g(us)f(lo)r(ok)g(at)h(the)g Fk(Ind)o Fu(-case:)0 893 y(\()p Fk(Ind)141 853 y Fp(y)r(;t)141 918 y(F)226 893 y Fo(h)274 905 y Fn(0)311 893 y Fu(\))343 863 y Fi(1)437 893 y Fr(')673 494 y Fo(h)721 506 y Fn(0)758 494 y Fu(\()p Fo(y)s(=)p Fu(0\))950 464 y Fi(1)p 844 589 5 59 v 570 665 a Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(F)e Fu(\(1\))24 b(:)f Fo(\015)1307 494 y(h)1355 506 y Fn(0)1392 494 y Fu(\()p Fo(y)s(=)p Fu(1\))1584 464 y Fi(1)p 1478 589 V 1204 665 a Fr(:)p Fo(F)12 b Fu(\(1\))p Fo(;)i(F)e Fu(\(2\))24 b(:)f Fo(\015)p 570 707 1186 4 v 834 781 a Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(F)e Fu(\(2\))24 b(:)f Fo(\015)5 b Fu(+)o(1)1885 610 y Fo(h)1933 622 y Fn(0)1970 610 y Fu(\()p Fo(y)s(=)p Fu(2\))2162 580 y Fi(1)p 2057 706 5 59 v 1783 781 a Fr(:)p Fo(F)12 b Fu(\(2\))p Fo(;)i(F)e Fu(\(3\))23 b(:)h Fo(\015)p 834 823 1500 4 v 1573 885 a Fu(.)1573 919 y(.)1573 952 y(.)2468 781 y Fo(h)2516 793 y Fn(0)2553 781 y Fu(\()p Fo(y)s(=n)p Fr(\000)o Fu(1\))2859 751 y Fi(1)p 2697 876 5 59 v 2362 952 a Fr(:)p Fo(F)12 b Fu(\()p Fo(n)p Fr(\000)p Fu(1\))p Fo(;)i(F)e Fu(\()p Fo(n)p Fu(\))23 b(:)g Fo(\015)p 1573 993 1463 4 v 1914 1068 a Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(F)e Fu(\()p Fo(n)p Fu(\))24 b(:)f Fo(\015)5 b Fu(+)o Fo(n)p Fr(\000)p Fu(1)3327 897 y Fc(c)3371 857 y Fp(y)r(;t)3371 921 y(F)p 3390 993 5 59 v 3119 1068 a Fr(:)p Fo(F)12 b Fu(\()p Fo(n)p Fu(\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))24 b(:)f Fo(k)p 1914 1110 1751 4 v 2462 1184 a Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))24 b(:)f Fo(\015)5 b Fu(+)t(~)-46 b Fo(n)p 2460 1226 659 4 v 2460 1301 a Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))24 b(:)f Fo(\015)5 b Fu(+)o Fo(!)0 1527 y Ft(6.4.)41 b(De\014nition)31 b(of)h Fk(tp)o Fu(\()p Fo(h)p Fu(\))h Ft(and)f Fo(h)p Fu([)p Fo(\023)p Fu(])g Ft(for)g(closed)f Fe(Z)1781 1490 y Fi(\003)1820 1527 y Ft(-deriv)-5 b(ations)30 b Fo(h)i Ft(and)g Fo(\023)24 b Fr(2)f(j)p Fk(tp)p Fu(\()p Fo(h)p Fu(\))p Fr(j)0 1693 y Fu(By)29 b(recursion)f(on)h(the)g(build-up)h(of)f Fo(h)g Fu(w)n(e)g(de\014ne)h(a)f Fe(Z)1743 1656 y Fi(1)1813 1693 y Fu(-inference)g Fk(tp)p Fu(\()p Fo(h)p Fu(\))h(and)f(closed)f Fe(Z)2864 1656 y Fi(\003)2902 1693 y Fu(-deriv)-5 b(ation\(s\))28 b Fo(h)p Fu([)p Fo(\023)p Fu(])i(in)g(suc)n(h)e(a)0 1834 y(w)n(a)n(y)e(that)987 1990 y Fo(h)1035 1956 y Fi(1)1128 1990 y Fu(=)d Fk(tp)o Fu(\()p Fo(h)p Fu(\))1400 1898 y Fl(\020)1450 1990 y Fo(h)p Fu([)p Fo(\023)p Fu(])1573 1956 y Fi(1)1644 1898 y Fl(\021)1694 2048 y Fp(\023)p Fi(2j)p Fd(tp)p Fn(\()p Fp(h)p Fn(\))p Fi(j)1982 1990 y Fu(=)2094 1929 y Fo(:)14 b(:)g(:)f(h)p Fu([)p Fo(\023)p Fu(])2327 1899 y Fi(1)2412 1929 y Fo(:)h(:)g(:)g Fu(\()p Fo(\023)p Fr(2)q(j)p Fk(tp)p Fu(\()p Fo(h)p Fu(\))p Fr(j)p Fu(\))p 2080 1971 824 4 v 2316 2052 a Fk(tp)p Fu(\()p Fo(h)p Fu(\))0 2230 y(The)31 b(de\014nition)g(clauses)g(for)f Fo(h)f Fu(=)f Fk(R)1182 2242 y Fp(C)1237 2230 y Fo(h)1285 2242 y Fn(0)1322 2230 y Fo(h)1370 2242 y Fn(1)1439 2230 y Fu(and)i Fo(h)f Fu(=)f Fk(E)p Fo(h)1871 2242 y Fn(0)1939 2230 y Fu(can)j(b)r(e)g(read)f(o\013)h(from)g(the)g(corresp)r(onding)e (clauses)h(in)i(the)0 2371 y(de\014nitions)c(of)f Fr(R)566 2383 y Fp(C)650 2371 y Fu(and)h Fr(E)7 b Fu(.)0 2537 y(1.1.)54 b Fo(h)23 b Fu(=)g Fk(Ax)437 2549 y Fn(\001)495 2537 y Fu(:)65 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)g Fk(Ax)996 2500 y Fi(1)996 2557 y Fn(\001)1066 2537 y Fu(.)0 2703 y(1.2.)54 b Fo(h)23 b Fu(=)343 2641 y Fl(V)412 2728 y Fp(C)468 2703 y Fo(h)516 2715 y Fn(0)553 2703 y Fo(h)601 2715 y Fn(1)638 2703 y Fu(:)65 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)1045 2641 y Fl(V)1114 2728 y Fp(C)1170 2703 y Fu(,)28 b Fo(h)p Fu([)p Fo(i)p Fu(])22 b(:=)h Fo(h)1525 2715 y Fp(i)1552 2703 y Fu(.)0 2869 y(1.3.)54 b Fo(h)23 b Fu(=)343 2807 y Fl(V)412 2827 y Fp(y)412 2894 y(C)468 2869 y Fo(h)516 2881 y Fn(0)553 2869 y Fu(:)65 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)960 2807 y Fl(V)1029 2894 y Fp(C)1085 2869 y Fu(,)28 b Fo(h)p Fu([)p Fo(t)p Fu(])23 b(:=)f Fo(h)1441 2881 y Fn(0)1478 2869 y Fu(\()p Fo(y)s(=t)p Fu(\).)0 3035 y(1.4.)54 b Fo(h)23 b Fu(=)343 2973 y Fl(W)412 2993 y Fp(\023)412 3060 y(C)468 3035 y Fo(h)516 3047 y Fn(0)553 3035 y Fu(:)65 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)960 2973 y Fl(W)1029 2993 y Fp(\023)1029 3060 y(C)1085 3035 y Fu(,)28 b Fo(h)p Fu([0])22 b(:=)h Fo(h)1453 3047 y Fn(0)1490 3035 y Fu(.)0 3222 y(2.)119 b Fo(h)23 b Fu(=)g Fk(Ind)452 3182 y Fp(y)r(;t)452 3246 y(F)537 3222 y Fo(h)585 3234 y Fn(0)622 3222 y Fu(:)64 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))24 b(:=)e Fk(Rep)p Fu(,)27 b Fo(h)p Fu([0])c(:=)1482 3105 y Fl(\032)1558 3169 y Fo(c)1594 3129 y Fp(y)r(;t)1594 3193 y(F)2204 3169 y Fu(if)28 b Fo(n)23 b Fu(=)g(0)1558 3277 y Fk(Cut)1684 3292 y Fp(F)9 b Fn(\()p Fp(n)p 1761 3305 42 4 v Fn(\))1832 3277 y Fo(e)1871 3289 y Fp(n)p Fi(\000)p Fn(1)2001 3277 y Fo(c)2037 3237 y Fp(y)r(;t)2037 3302 y(F)2204 3277 y Fu(if)28 b Fo(n)23 b(>)g Fu(0)2686 3222 y(where)184 3443 y Fo(n)g Fu(:=)g Fk(val)p Fu(\()p Fo(t)p Fu(\),)56 b Fe(Z)687 3405 y Fi(\003)748 3443 y Fr(3)24 b Fo(c)863 3403 y Fp(y)r(;t)863 3467 y(F)971 3443 y Fr(`)1022 3410 y Fp()g Fu(0)k(.)0 3609 y(3.)119 b Fo(h)23 b Fu(=)g Fk(E)p Fo(h)441 3621 y Fn(0)478 3609 y Fu(:)0 3750 y(3.1.)54 b Fk(tp)p Fu(\()p Fo(h)337 3762 y Fn(0)375 3750 y Fu(\))23 b(=)g Fk(Cut)644 3762 y Fp(C)699 3750 y Fu(:)65 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)g Fk(Rep)p Fu(,)k Fo(h)p Fu([0])c(:=)g Fk(R)1613 3762 y Fp(C)1669 3750 y Fk(E)p Fo(h)1767 3762 y Fn(0)1804 3750 y Fu([0])p Fk(E)o Fo(h)1989 3762 y Fn(0)2026 3750 y Fu([1],)0 3891 y(3.2.)54 b(otherwise:)64 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)g Fk(tp)p Fu(\()p Fo(h)1086 3903 y Fn(0)1123 3891 y Fu(\),)28 b Fo(h)p Fu([)p Fo(\023)p Fu(])c(:=)f Fk(E)o Fo(h)1561 3903 y Fn(0)1598 3891 y Fu([)p Fo(\023)p Fu(].)0 4057 y(4.)119 b Fo(h)23 b Fu(=)g Fk(R)397 4069 y Fp(C)452 4057 y Fo(h)500 4069 y Fn(0)538 4057 y Fo(h)586 4069 y Fn(1)623 4057 y Fu(:)0 4198 y(4.1.)54 b Fo(C)30 b Fr(62)23 b Fu(\001\()p Fk(tp)q Fu(\()p Fo(h)606 4210 y Fn(0)643 4198 y Fu(\)\):)65 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))24 b(:=)e Fk(tp)p Fu(\()p Fo(h)1267 4210 y Fn(0)1304 4198 y Fu(\),)28 b Fo(h)p Fu([)p Fo(\023)p Fu(])c(:=)f Fk(R)1698 4210 y Fp(C)1754 4198 y Fo(h)1802 4210 y Fn(0)1839 4198 y Fu([)p Fo(\023)p Fu(])p Fo(h)1962 4210 y Fn(1)2000 4198 y Fu(.)0 4339 y(4.2.)54 b Fr(:)p Fo(C)30 b Fr(62)24 b Fu(\001\()p Fk(tp)p Fu(\()p Fo(h)661 4351 y Fn(1)698 4339 y Fu(\)\):)66 b Fk(tp)o Fu(\()p Fo(h)p Fu(\))24 b(:=)f Fk(tp)o Fu(\()p Fo(h)1322 4351 y Fn(1)1360 4339 y Fu(\),)28 b Fo(h)p Fu([)p Fo(\023)p Fu(])23 b(:=)g Fk(R)1754 4351 y Fp(C)1809 4339 y Fo(h)1857 4351 y Fn(0)1894 4339 y Fo(h)1942 4351 y Fn(1)1980 4339 y Fu([)p Fo(\023)p Fu(].)0 4480 y(4.3.)54 b Fo(C)30 b Fr(2)23 b Fu(\001\()p Fk(tp)q Fu(\()p Fo(h)606 4492 y Fn(0)643 4480 y Fu(\)\))28 b(and)g Fr(:)p Fo(C)h Fr(2)24 b Fu(\001\()p Fk(tp)p Fu(\()p Fo(h)1373 4492 y Fn(1)1410 4480 y Fu(\)\):)0 4621 y(4.3.0.)54 b(rk)o(\()p Fo(C)6 b Fu(\))24 b(=)f(0:)64 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)g Fk(Ax)1108 4584 y Fi(1)1108 4642 y Fn(\001)1206 4621 y Fu(with)28 b(\001)23 b(:=)g(\(\001\()p Fk(tp)p Fu(\()p Fo(h)1884 4633 y Fn(0)1922 4621 y Fu(\)\))c Fr(n)f(f)p Fo(C)6 b Fr(g)p Fu(\))18 b Fr([)h Fu(\(\001\()p Fk(tp)p Fu(\()p Fo(h)2624 4633 y Fn(1)2662 4621 y Fu(\)\))g Fr(n)f(f:)p Fo(C)6 b Fr(g)p Fu(\).)0 4763 y(4.3.1.)54 b Fo(C)29 b Fu(=)23 b Fr(8)p Fo(xA)p Fu(:)65 b(Then)27 b Fk(tp)p Fu(\()p Fo(h)1038 4775 y Fn(1)1076 4763 y Fu(\))c(=)1219 4700 y Fl(W)1288 4721 y Fp(t)1288 4787 y Fi(:)p Fp(C)1416 4763 y Fu(for)k(some)g Fo(t)c Fr(2)g Fo(T)39 b Fu(.)249 4904 y Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(:=)g Fk(Cut)694 4919 y Fp(A)744 4927 y Fh(x)781 4919 y Fn(\()p Fp(t)p Fn(\))862 4904 y Fu(,)28 b Fo(h)p Fu([0])22 b(:=)h Fk(R)1236 4916 y Fp(C)1292 4904 y Fo(h)1340 4916 y Fn(0)1377 4904 y Fu([)p Fo(t)p Fu(])p Fo(h)1501 4916 y Fn(1)1538 4904 y Fu(,)28 b Fo(h)p Fu([1])22 b(:=)h Fk(R)1911 4916 y Fp(C)1967 4904 y Fo(h)2015 4916 y Fn(0)2052 4904 y Fo(h)2100 4916 y Fn(1)2137 4904 y Fu([0].)0 5045 y(4.3.2.)54 b Fo(C)29 b Fu(=)23 b Fr(9)p Fo(xA)29 b Fu(or)d Fo(A)772 5057 y Fn(0)810 5045 y Fr(^)p Fo(A)927 5057 y Fn(1)992 5045 y Fu(or)h Fo(A)1156 5057 y Fn(0)1194 5045 y Fr(_)p Fo(A)1311 5057 y Fn(1)1348 5045 y Fu(:)37 b(analogous)26 b(to)h(4.3.1.)1908 5539 y(34)p eop %%Page: 35 36 35 35 bop 0 83 a Ft(Theorem)30 b(6.5)0 220 y Fu(If)e Fo(h)g Fu(is)f(a)g(closed)g Fe(Z)604 183 y Fi(\003)642 220 y Fu(-deriv)-5 b(ation)27 b(with)h Fr(I)i Fu(=)22 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))28 b(and)g Fo(h)23 b Fr(`)1908 190 y Fp(\013)1908 241 y(m)1993 220 y Fu(\000,)28 b(then)g(the)g(follo) n(wing)f(holds:)0 358 y(a\))g(\001\()p Fr(I)6 b Fu(\))25 b Fr(\022)d Fu(\000,)0 495 y(b\))28 b(deg)q(\()p Fr(I)6 b Fu(\))24 b Fr(\024)e Fo(m)28 b Fu(,)g(i.e.)37 b(\()p Fr(I)29 b Fu(=)23 b Fk(Cut)1073 507 y Fp(C)1152 495 y Fr(\))g Fu(rk)o(\()p Fo(C)6 b Fu(\))24 b Fo(<)f(m)p Fu(\),)0 632 y(c\))28 b(F)-7 b(or)27 b(eac)n(h)g Fo(\023)c Fr(2)g(jI)6 b(j)p Fu(:)38 b Fo(h)p Fu([)p Fo(\023)p Fu(])23 b Fr(`)918 602 y Fp(\013)961 610 y Fh(\023)918 653 y Fp(m)1015 632 y Fu(\000)p Fo(;)14 b Fu(\001)1173 644 y Fp(\023)1202 632 y Fu(\()p Fr(I)6 b Fu(\))29 b(with)f Fo(\013)1588 644 y Fp(\023)1640 632 y Fo(<)22 b(\013)p Fu(.)0 791 y Fm(Pr)l(o)l(of)31 b(by)f(str)l(aightforwar)l(d)i(induction)e(on)g (the)g(build-up)g(of)h Fo(h)p Fm(:)0 928 y Fu(W.l.o.g.)37 b(FV)q(\(\000\))24 b(=)e Fr(;)p Fu(.)0 1087 y(1.)36 b Fo(h)23 b Fu(=)g Fk(Ax)354 1099 y Fn(\001)412 1087 y Fu(:)37 b(Then)28 b Fr(I)h Fu(=)23 b Fk(Ax)945 1049 y Fi(1)945 1107 y Fn(\001)1043 1087 y Fu(and)k(th)n(us)h(\001\()p Fr(I)6 b Fu(\))24 b(=)f(\001)g(=)f(\000\()p Fo(h)p Fu(\))i Fr(\022)e Fu(\000)28 b(and)f Fr(jI)6 b(j)24 b Fu(=)f Fr(;)p Fu(.)0 1245 y(2.1.)36 b Fo(h)23 b Fu(=)325 1183 y Fl(V)394 1203 y Fp(y)394 1270 y Fi(8)p Fp(xA)523 1245 y Fo(h)571 1257 y Fn(0)608 1245 y Fu(:)37 b(Then)28 b Fr(I)h Fu(=)1047 1183 y Fl(V)1116 1270 y Fi(8)p Fp(xA)1245 1245 y Fu(,)f(\001\()p Fr(I)6 b Fu(\))24 b(=)f Fr(f8)p Fo(xA)p Fr(g)f(\022)h Fu(\000)k(and)h Fo(h)2231 1257 y Fn(0)2291 1245 y Fr(`)2342 1215 y Fp(\013)2385 1223 y Fg(0)2342 1266 y Fp(m)2444 1245 y Fu(\000)p Fo(;)14 b(A)2595 1257 y Fp(x)2637 1245 y Fu(\()p Fo(y)s Fu(\))28 b(with)g Fo(\013)3015 1257 y Fn(0)3075 1245 y Fo(<)23 b(\013)p Fu(.)0 1382 y(By)k(...)37 b(w)n(e)28 b(get)f Fo(h)p Fu([)p Fo(t)p Fu(])c(=)g Fo(h)780 1394 y Fn(0)817 1382 y Fu(\()p Fo(y)s(=t)p Fu(\))f Fr(`)1070 1352 y Fp(\015)1070 1403 y(m)1156 1382 y Fu(\000)p Fo(;)14 b(A)1307 1394 y Fp(x)1349 1382 y Fu(\()p Fo(t)p Fu(\))28 b(for)g(eac)n(h)e Fo(t)d Fr(2)h Fo(T)12 b Fu(.)0 1541 y(2.2.)36 b Fo(h)23 b Fu(=)325 1478 y Fl(W)394 1499 y Fp(t)394 1566 y Fi(9)p Fp(xA)523 1541 y Fo(h)571 1553 y Fn(0)608 1541 y Fu(:)37 b(Then)28 b Fo(t)23 b Fr(2)g Fo(T)12 b Fu(,)27 b Fr(I)j Fu(=)1289 1478 y Fl(W)1358 1499 y Fp(t)1358 1566 y Fi(9)p Fp(xA)1488 1541 y Fu(,)d(\001\()p Fr(I)6 b Fu(\))25 b(=)d Fr(f9)p Fo(xA)p Fr(g)h(\022)g Fu(\000)k(and)h Fo(h)2473 1553 y Fn(0)2533 1541 y Fr(`)2584 1511 y Fp(\013)2627 1519 y Fg(0)2584 1561 y Fp(m)2686 1541 y Fu(\000)p Fo(;)14 b(A)2837 1553 y Fp(x)2879 1541 y Fu(\()p Fo(t)p Fu(\))28 b(with)h Fo(\013)3244 1553 y Fn(0)3304 1541 y Fo(<)23 b(\013)p Fu(.)0 1699 y(2.3.)36 b Fo(h)23 b Fu(=)325 1637 y Fl(V)394 1724 y Fp(A)444 1732 y Fg(0)476 1724 y Fi(^)p Fp(A)571 1732 y Fg(1)607 1699 y Fo(h)655 1711 y Fn(0)692 1699 y Fo(h)740 1711 y Fn(1)805 1699 y Fu(or)k Fo(h)c Fu(=)1065 1637 y Fl(W)1135 1657 y Fp(k)1135 1724 y(A)1185 1732 y Fg(0)1217 1724 y Fi(_)p Fp(A)1312 1732 y Fg(1)1348 1699 y Fu(:)37 b(analogous)25 b(to)j(2.1)f(and)g(2.2.)0 1858 y(3.)35 b Fo(h)23 b Fu(=)f Fk(Cut)384 1870 y Fp(C)440 1858 y Fo(h)488 1870 y Fn(0)525 1858 y Fo(h)573 1870 y Fn(1)610 1858 y Fu(:)35 b(Then)22 b Fr(I)30 b Fu(=)22 b Fk(Cut)1167 1870 y Fp(C)1223 1858 y Fu(,)i(rk)o(\()p Fo(C)6 b Fu(\))24 b Fo(<)f(m)p Fu(,)g(and)g Fo(h)p Fu([0])f(=)h Fo(h)2157 1870 y Fn(0)2217 1858 y Fr(`)2268 1827 y Fp(\013)2311 1835 y Fg(0)2268 1878 y Fp(m)2371 1858 y Fu(\000)p Fo(;)14 b(C)6 b Fu(,)23 b Fo(h)p Fu([1])g(=)f Fo(h)2865 1870 y Fn(1)2926 1858 y Fr(`)2977 1827 y Fp(\013)3020 1835 y Fg(1)2977 1878 y Fp(m)3079 1858 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)29 b Fu(with)23 b Fo(\013)3548 1870 y Fn(0)3586 1858 y Fo(;)14 b(\013)3676 1870 y Fn(1)3736 1858 y Fo(<)23 b(\013)p Fu(.)0 2016 y(4.1.)36 b Fo(h)23 b Fu(=)g Fk(R)378 2028 y Fp(C)434 2016 y Fo(h)482 2028 y Fn(0)519 2016 y Fo(h)567 2028 y Fn(1)632 2016 y Fu(with)28 b Fo(C)h Fu(=)23 b Fr(8)p Fo(xA)p Fu(,)28 b Fk(tp)o Fu(\()p Fo(h)1356 2028 y Fn(0)1394 2016 y Fu(\))23 b(=)1537 1954 y Fl(V)1606 2041 y Fp(C)1662 2016 y Fu(,)k Fk(tp)p Fu(\()p Fo(h)1865 2028 y Fn(1)1903 2016 y Fu(\))c(=)2046 1954 y Fl(W)2115 1974 y Fp(t)2115 2041 y Fi(:)p Fp(C)2215 2016 y Fu(:)37 b(Then)28 b Fo(t)23 b Fr(2)h Fo(T)38 b Fu(and)28 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))23 b(=)g Fk(Cut)3295 2031 y Fp(A)p Fn(\()p Fp(t)p Fn(\))3426 2016 y Fu(.)0 2174 y(Let)28 b Fo(\015)f Fu(:=)c(o\()p Fo(h)452 2186 y Fn(0)489 2174 y Fu(\),)28 b Fo(\014)f Fu(:=)c(o\()p Fo(h)879 2186 y Fn(1)916 2174 y Fu(\).)0 2333 y(Then)28 b Fo(h)265 2345 y Fn(0)325 2333 y Fr(`)376 2303 y Fp(\015)376 2353 y(m)462 2333 y Fu(\000)p Fo(;)14 b(C)34 b Fu(,)27 b Fo(h)742 2345 y Fn(1)802 2333 y Fr(`)853 2303 y Fp(\014)853 2353 y(m)939 2333 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)34 b Fu(and)28 b(rk)o(\()p Fo(A)p Fu(\()p Fo(t)p Fu(\)\))d Fo(<)d Fu(rk\()p Fo(C)6 b Fu(\))23 b Fr(\024)g Fo(m)p Fu(.)0 2470 y(By)k(IH)h(w)n(e)f (obtain)h Fo(h)679 2482 y Fn(0)716 2470 y Fu([)p Fo(t)p Fu(])23 b Fr(`)866 2440 y Fp(\015)901 2448 y Fh(t)866 2491 y Fp(m)955 2470 y Fu(\000)p Fo(;)14 b(C)q(;)g(A)p Fu(\()p Fo(t)p Fu(\))29 b(with)f Fo(\015)1558 2482 y Fp(t)1611 2470 y Fo(<)22 b(\015)5 b Fu(,)28 b(and)f Fo(h)2006 2482 y Fn(1)2043 2470 y Fu([0])c Fr(`)2205 2440 y Fp(\014)2243 2448 y Fg(0)2302 2470 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)q(;)g Fr(:)p Fo(A)p Fu(\()p Fo(t)p Fu(\))29 b(with)f Fo(\014)3019 2482 y Fn(0)3080 2470 y Fo(<)22 b(\014)t Fu(.)0 2608 y(Hence)28 b Fo(h)p Fu([0])22 b(=)h Fk(R)547 2620 y Fp(C)602 2608 y Fo(h)650 2620 y Fn(0)687 2608 y Fu([)p Fo(t)p Fu(])p Fo(h)811 2620 y Fn(1)872 2608 y Fr(`)923 2577 y Fp(\015)958 2585 y Fh(t)985 2577 y Fn(#)p Fp(\014)923 2628 y(m)1107 2608 y Fu(\000)p Fo(;)14 b(A)p Fu(\()p Fo(t)p Fu(\))29 b(and)e Fo(h)p Fu([1])22 b(=)h Fk(R)1842 2620 y Fp(C)1898 2608 y Fo(h)1946 2620 y Fn(0)1983 2608 y Fo(h)2031 2620 y Fn(1)2068 2608 y Fu([0])f Fr(`)2229 2577 y Fp(\015)t Fn(#)p Fp(\014)2361 2585 y Fg(0)2229 2628 y Fp(m)2419 2608 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(A)p Fu(\()p Fo(t)p Fu(\))0 2745 y(with)28 b Fo(\015)232 2757 y Fp(t)261 2745 y Fu(#)p Fo(\014)19 b(;)27 b(\015)5 b Fu(#)p Fo(\014)610 2757 y Fn(0)670 2745 y Fo(<)23 b(\015)5 b Fu(#)p Fo(\014)27 b Fu(=)c(o\()p Fo(h)p Fu(\))g Fr(\024)g Fo(\013)p Fu(.)0 2903 y(4.2.)36 b Fo(h)23 b Fu(=)g Fk(R)378 2915 y Fp(C)434 2903 y Fo(h)482 2915 y Fn(0)519 2903 y Fo(h)567 2915 y Fn(1)632 2903 y Fu(with)28 b Fk(tp)p Fu(\()p Fo(h)974 2915 y Fn(0)1011 2903 y Fu(\))c(=)e Fr(I)34 b Fu(and)28 b Fo(C)h Fr(62)23 b Fu(\001\()p Fr(I)6 b Fu(\):)38 b(Let)28 b Fo(\015)g Fu(:=)23 b(o\()p Fo(h)2259 2915 y Fn(0)2295 2903 y Fu(\),)28 b Fo(\014)g Fu(:=)23 b(o\()p Fo(h)2686 2915 y Fn(1)2722 2903 y Fu(\).)0 3041 y(Then)28 b Fo(h)265 3053 y Fn(0)325 3041 y Fr(`)376 3011 y Fp(\015)376 3061 y(m)462 3041 y Fu(\000)p Fo(;)14 b(C)34 b Fu(,)27 b Fo(h)742 3053 y Fn(1)802 3041 y Fr(`)853 3011 y Fp(\014)853 3061 y(m)939 3041 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)6 b Fu(.)0 3178 y(By)27 b(IH)h(w)n(e)f(obtain)h Fo(h)679 3190 y Fn(0)716 3178 y Fu([)p Fo(\023)p Fu(])23 b Fr(`)865 3148 y Fp(\015)900 3156 y Fh(\023)865 3199 y Fp(m)954 3178 y Fu(\000)p Fo(;)14 b(C)q(;)g Fu(\001)1209 3190 y Fp(\023)1238 3178 y Fu(\()p Fr(I)6 b Fu(\))29 b(with)f Fo(\015)1614 3190 y Fp(\023)1666 3178 y Fo(<)23 b(\015)5 b Fu(,)27 b(for)g(all)g Fo(\023)d Fr(2)f(jI)6 b(j)p Fu(.)0 3315 y(Hence)28 b Fo(h)p Fu([)p Fo(\023)p Fu(])23 b(=)g Fk(R)534 3327 y Fp(C)590 3315 y Fo(h)638 3327 y Fn(0)675 3315 y Fu([)p Fo(\023)p Fu(])p Fo(h)798 3327 y Fn(1)859 3315 y Fr(`)910 3285 y Fp(\015)945 3293 y Fh(\023)972 3285 y Fn(#)p Fp(\014)910 3336 y(m)1094 3315 y Fu(\000)p Fo(;)14 b Fu(\001)1252 3327 y Fp(\023)1281 3315 y Fu(\()p Fr(I)6 b Fu(\))28 b(with)g Fo(\015)1656 3327 y Fp(\023)1685 3315 y Fu(#)p Fo(\014)g(<)22 b(\015)5 b Fu(#)p Fo(\014)27 b Fu(=)c(o\()p Fo(h)p Fu(\))g Fr(\024)g Fo(\013)p Fu(.)0 3474 y Fo(h)48 3486 y Fn(0)108 3474 y Fr(`)159 3444 y Fp(\015)159 3494 y(m)245 3474 y Fu(\000)p Fo(;)14 b(C)29 b Fu(&)23 b Fk(tp)p Fu(\()p Fo(h)663 3486 y Fn(0)700 3474 y Fu(\))g(=)g Fr(I)29 b Fu(&)23 b Fo(C)29 b Fr(62)24 b Fu(\001\()p Fr(I)6 b Fu(\))1385 3427 y Fn(IH)1380 3474 y Fr(\))23 b Fu(\001\()p Fr(I)6 b Fu(\))24 b Fr(\022)f Fu(\000)g(&)f(\()p Fr(I)30 b Fu(=)23 b Fk(Cut)2265 3486 y Fp(A)2342 3474 y Fr(\))g Fu(rk)o(\()p Fo(A)p Fu(\))h Fo(<)f(m)p Fu(\).)0 3632 y(4.3.)36 b Fo(h)23 b Fu(=)g Fk(R)378 3644 y Fp(C)434 3632 y Fo(h)482 3644 y Fn(0)519 3632 y Fo(h)567 3644 y Fn(1)632 3632 y Fu(with)28 b(rk)o(\()p Fo(C)6 b Fu(\))24 b(=)f(0)k(and)g Fo(C)j Fr(2)23 b Fu(\001)1604 3602 y Fn(0)1665 3632 y Fu(:=)f(\001\()p Fk(tp)q Fu(\()p Fo(h)2030 3644 y Fn(0)2067 3632 y Fu(\)\),)28 b Fr(:)p Fo(C)i Fr(2)24 b Fu(\001)2474 3602 y Fn(1)2534 3632 y Fu(:=)f(\001\()p Fk(tp)p Fu(\()p Fo(h)2899 3644 y Fn(1)2936 3632 y Fu(\)\):)0 3770 y(Then)28 b Fr(I)h Fu(=)23 b Fk(Ax)472 3782 y Fn(\001)559 3770 y Fu(with)28 b(\001)23 b(:=)g(\(\001)1052 3739 y Fn(0)1108 3770 y Fr(n)18 b(f)p Fo(C)6 b Fr(g)p Fu(\))18 b Fr([)h Fu(\(\001)1542 3739 y Fn(0)1598 3770 y Fr(n)f(f:)p Fo(C)6 b Fr(g)p Fu(\),)28 b(and)f(b)n(y)g(IH)h(\001)2410 3739 y Fp(i)2461 3770 y Fr(\022)23 b Fu(\000\()p Fo(h)2681 3782 y Fp(i)2708 3770 y Fu(\).)38 b(Hence)27 b(\001\()p Fr(I)6 b Fu(\))25 b(=)d(\001)h Fr(\022)g Fu(\000\()p Fo(h)p Fu(\))g Fr(\022)g Fu(\000.)0 3928 y(5.)36 b Fo(h)23 b Fu(=)g Fk(E)p Fo(h)358 3940 y Fn(0)422 3928 y Fu(with)28 b Fk(tp)p Fu(\()p Fo(h)764 3940 y Fn(0)802 3928 y Fu(\))23 b(=)g Fk(Cut)1071 3940 y Fp(C)1126 3928 y Fu(:)37 b(Then)28 b Fk(tp)p Fu(\()p Fo(h)p Fu(\))c(=)e Fk(Rep)27 b Fu(,)h Fo(h)p Fu([0])22 b(=)h Fk(R)2211 3940 y Fp(C)2267 3928 y Fk(E)o Fo(h)2364 3940 y Fn(0)2401 3928 y Fu([0])p Fk(E)o Fo(h)2586 3940 y Fn(0)2624 3928 y Fu([1])k(and)g(deg)q(\()p Fo(h)3106 3940 y Fn(0)3143 3928 y Fu(\))d Fr(\024)e Fo(m)p Fu(+1.)0 4086 y(Let)28 b Fo(\015)f Fu(:=)c(o\()p Fo(h)452 4098 y Fn(0)489 4086 y Fu(\).)37 b(Then)28 b Fo(h)846 4098 y Fn(0)906 4086 y Fr(`)957 4047 y Fp(\015)957 4109 y(m)p Fn(+1)1127 4086 y Fu(\000.)0 4245 y(By)f(IH)h(w)n(e)f(ha)n(v)n(e) g(rk)o(\()p Fo(C)6 b Fu(\))24 b Fo(<)f(m)p Fu(+)o(1)k(and)h Fo(h)1297 4257 y Fn(0)1334 4245 y Fu([0])23 b Fr(`)1496 4205 y Fp(\015)1531 4213 y Fg(0)1496 4267 y Fp(m)p Fn(+1)1666 4245 y Fu(\000)p Fo(;)14 b(C)33 b Fu(,)56 b Fo(h)1974 4257 y Fn(0)2011 4245 y Fu([1])22 b Fr(`)2172 4205 y Fp(\015)2207 4213 y Fg(1)2172 4267 y Fp(m)p Fn(+1)2342 4245 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)34 b Fu(with)29 b Fo(\015)2812 4257 y Fn(0)2849 4245 y Fo(;)14 b(\015)2929 4257 y Fn(1)2989 4245 y Fo(<)23 b(\015)5 b Fu(.)0 4382 y(Hence)28 b Fk(E)o Fo(h)344 4394 y Fn(0)381 4382 y Fu([0])23 b Fr(`)543 4352 y Fn(3)576 4327 y Fh(\015)607 4339 y Fg(0)543 4403 y Fp(m)671 4382 y Fu(\000)p Fo(;)14 b(C)33 b Fu(and)28 b Fk(E)o Fo(h)1111 4394 y Fn(0)1149 4382 y Fu([1])22 b Fr(`)1310 4352 y Fn(3)1343 4327 y Fh(\015)1374 4339 y Fg(1)1310 4403 y Fp(m)1438 4382 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(C)6 b Fu(.)0 4541 y(F)-7 b(rom)26 b(this)g(together)f(with) h(rk\()p Fo(C)6 b Fu(\))24 b Fr(\024)e Fo(m)k Fu(w)n(e)g(get)g Fo(h)p Fu([0])c(=)h Fk(R)1863 4553 y Fp(C)1919 4541 y Fk(E)p Fo(h)2017 4553 y Fn(0)2054 4541 y Fu([0])p Fk(E)o Fo(h)2239 4553 y Fn(0)2276 4541 y Fu([1])g Fr(`)2438 4510 y Fn(3)2471 4485 y Fh(\015)2502 4497 y Fg(1)2538 4510 y Fn(#3)2626 4485 y Fh(\015)2657 4497 y Fg(1)2438 4561 y Fp(m)2721 4541 y Fu(\000)j(and)f(3)3000 4510 y Fp(\015)3035 4518 y Fg(0)3071 4541 y Fu(#3)3182 4510 y Fp(\015)3217 4518 y Fg(1)3276 4541 y Fo(<)e Fu(3)3406 4510 y Fp(\015)3471 4541 y Fu(=)g(o\()p Fo(h)p Fu(\))g Fr(\024)g Fo(\013)p Fu(.)0 4699 y(6.)36 b Fo(h)23 b Fu(=)g Fk(Ind)369 4659 y Fp(y)r(;t)369 4723 y(F)454 4699 y Fo(h)502 4711 y Fn(0)539 4699 y Fu(:)37 b(Then)27 b Fr(I)j Fu(=)22 b Fk(Rep)p Fu(,)28 b(rk)o(\()p Fo(F)12 b Fu(\))23 b Fo(<)g(m)p Fu(,)28 b(and)f Fo(h)1811 4711 y Fn(0)1871 4699 y Fr(`)1922 4669 y Fp(\015)1922 4720 y(m)2008 4699 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(F)e Fu(\()p Fo(y)s Fu(\))p Fo(;)i(F)e Fu(\()p Fk(S)q Fo(y)s Fu(\))27 b(with)h Fo(\015)g Fu(:=)23 b(o\()p Fo(h)3102 4711 y Fn(0)3139 4699 y Fu(\),)28 b Fo(\015)23 b Fu(+)18 b Fo(!)25 b Fr(\024)e Fo(\013)p Fu(.)0 5080 y Fo(h)p Fu([0])g(=)246 4823 y Fo(h)294 4835 y Fn(0)331 4823 y Fu(\()p Fo(y)s(=)p Fu(0\))83 b Fo(h)654 4835 y Fn(0)691 4823 y Fu(\()p Fo(y)s(=)p Fu(1\))p 246 4865 636 4 v 431 4940 a Fk(Cut)557 4955 y Fp(F)9 b Fn(\(1\))910 4940 y Fo(h)958 4952 y Fn(0)995 4940 y Fu(\()p Fo(y)s(=)p Fu(2\))p 431 4982 755 4 v 797 5044 a(.)797 5078 y(.)797 5111 y(.)394 b Fo(h)1262 5123 y Fn(0)1299 5111 y Fu(\()p Fo(y)s(=n)p Fr(\000)o Fu(1\))p 797 5152 808 4 v 1021 5227 a Fk(Cut)1147 5242 y Fp(F)9 b Fn(\()p Fp(n)p Fi(\000)p Fn(1\))1687 5227 y Fo(c)1723 5187 y Fp(y)r(;t)1723 5251 y(F)p 1021 5269 787 4 v 1278 5344 a Fk(Cut)1403 5359 y Fp(F)g Fn(\()p Fp(n)p Fn(\))1891 5080 y Fu(with)28 b Fo(c)2116 5040 y Fp(y)r(;t)2116 5104 y(F)2224 5080 y Fr(`)2275 5047 y Fp()g Fu(0)f Fr(\))37 b Fu(3)516 2104 y Fp(m)579 2092 y Fu(\()p Fo(!)666 2062 y Fn(2)703 2092 y Fu(\))23 b(=)g Fo(!)898 2104 y Fp(m)p Fn(+1)1045 2092 y Fu(.)0 2258 y(Pro)r(of:)36 b(a\))27 b(3)403 2270 y Fp(m)p Fn(+1)550 2258 y Fu(\()p Fo(!)637 2228 y Fp(\013)684 2258 y Fu(\))c(=)g(3)869 2270 y Fp(m)932 2258 y Fu(\(3)1006 2228 y Fp(!)1050 2203 y Fh(\013)1096 2258 y Fu(\))g(=)f(3)1280 2270 y Fp(m)1343 2258 y Fu(\(3)1417 2228 y Fp(!)r Fi(\001)p Fp(!)1525 2203 y Fh(\013)1571 2258 y Fu(\))h(=)g(3)1756 2270 y Fp(m)1818 2258 y Fu(\(\(3)1924 2228 y Fp(!)1972 2258 y Fu(\))2004 2228 y Fp(!)2048 2203 y Fh(\013)2095 2258 y Fu(\))g(=)g(3)2280 2270 y Fp(m)2342 2258 y Fu(\()p Fo(!)2429 2228 y Fp(!)2473 2203 y Fh(\013)2519 2258 y Fu(\))2575 2211 y Fn(IH)2579 2258 y Fu(=)k Fo(!)2723 2270 y Fp(m)2786 2258 y Fu(\()p Fo(!)2873 2228 y Fp(!)2917 2203 y Fh(\013)2963 2258 y Fu(\))c(=)g Fo(!)3158 2270 y Fp(m)p Fn(+1)3304 2258 y Fu(\()p Fo(!)3391 2228 y Fp(\013)3439 2258 y Fu(\).)0 2399 y(b\))28 b(3)148 2411 y Fp(m)211 2399 y Fu(\()p Fo(!)298 2369 y Fn(2)335 2399 y Fu(\))23 b(=)g(3)520 2411 y Fp(m)p Fi(\000)p Fn(1)667 2399 y Fu(\(3)741 2369 y Fp(!)r Fi(\001)p Fp(!)853 2399 y Fu(\))g(=)g(3)1038 2411 y Fp(m)p Fi(\000)p Fn(1)1185 2399 y Fu(\()p Fo(!)1272 2369 y Fp(!)1320 2399 y Fu(\))1379 2345 y Fn(a\))1376 2399 y Fu(=)f Fo(!)1515 2411 y Fp(m)p Fi(\000)p Fn(1)1663 2399 y Fu(\()p Fo(!)1750 2369 y Fp(!)1798 2399 y Fu(\))h(=)g Fo(!)1993 2411 y Fp(m)p Fn(+1)2140 2399 y Fu(.)0 2615 y(Let)28 b(OT)22 b Fr(\022)h Fu(I)-14 b(N)28 b(and)f Fr(\036\022)c Fu(I)-14 b(N)19 b Fr(\002)f Fu(I)-14 b(N)28 b(b)r(e)g(the)g(primitiv)n(e)f(recursiv)n(e)f(relations)g(de\014ned)i (in)g Fr(x)p Fu(4.)0 2756 y(\(OT)p Fo(;)14 b Fr(\036)p Fu(\))27 b(has)g(ordert)n(yp)r(e)g Fo(")882 2768 y Fn(0)919 2756 y Fu(.)37 b(0)27 b(is)g(the)h Fr(\036)p Fu(-least)f(elemen)n(t)g (of)h(OT)o(.)0 2922 y(F)-7 b(or)27 b Fo(\013)c(<)g(")352 2934 y Fn(0)417 2922 y Fu(let)537 2891 y Ff(d)564 2922 y Fo(\013)629 2891 y Ff(e)685 2922 y Fu(b)r(e)28 b(the)g(corresp)r (onding)d(elemen)n(t)j(of)g(OT)o(.)0 3138 y Ft(Theorem)i(6.7)0 3279 y Fu(If)e Fr(Z)143 3291 y Fp(m)229 3279 y Fr(`)23 b(8)p Fo(x)p Fr(9)p Fo(y)s(A)p Fu(\()p Fo(x;)14 b(y)s Fu(\))55 b(\()p Fo(A)28 b Fu(atomic\))f(then)g(there)g(are)g(primitiv)n (e)g(recursiv)n(e)e(functions)j Fo(g)d Fu(:)f(I)-14 b(N)3010 3243 y Fn(2)3070 3279 y Fr(!)23 b Fu(I)-14 b(N)q(,)27 b Fo(\022)f Fu(:)d(I)-14 b(N)3494 3243 y Fn(2)3554 3279 y Fr(!)24 b Fu(I)-14 b(N)27 b(and)0 3421 y(an)g Fo(\013)d(<)e Fu(max)p Fr(f)p Fo(!)531 3390 y Fn(2)567 3421 y Fo(;)14 b(!)656 3433 y Fp(m)p Fn(+1)803 3421 y Fr(g)27 b Fu(suc)n(h)g(that)h Fr(8)p Fo(n)p Fu(\()p Fo(\022)r Fu(\()p Fo(n;)14 b Fu(0\))23 b(=)1712 3389 y Ff(d)1740 3421 y Fo(\013)1805 3389 y Ff(e)1833 3421 y Fu(\))28 b(and)f(I)-14 b(N)24 b Fr(j)-14 b Fu(=)22 b Fr(8)p Fo(xA)p Fu(\()p Fo(x;)14 b(f)9 b Fu(\()p Fo(x)p Fu(\)\))29 b(,)0 3562 y(where)e Fo(f)9 b Fu(\()p Fo(n)p Fu(\))23 b(:=)g Fo(g)s Fu(\()p Fo(n;)14 b Fu(min)p Fr(f)p Fo(k)25 b Fu(:)e Fo(\022)r Fu(\()p Fo(n;)14 b(k)s Fu(+1\))23 b Fr(6\036)g Fo(\022)r Fu(\()p Fo(n;)14 b(k)s Fu(\))p Fr(g)p Fu(\).)0 3728 y(Pro)r(of:)0 3869 y(W)-7 b(e)45 b(assume)e(a)h(canonical)f(arithmetization)h(\(co)r(ding\))h Fo(q)54 b Fr(7!)2105 3837 y Ff(d)2132 3869 y Fo(q)2184 3837 y Ff(e)2256 3869 y Fu(of)44 b(syn)n(tax)g(\(terms,)k(form)n(ulas,) g(sequen)n(ts,)g(\014nite)0 4010 y(deriv)-5 b(ations,)46 b(deriv)-5 b(ation)43 b(terms)g(etc.\).)84 b(A)43 b(set)g Fo(M)52 b Fu(of)43 b(syn)n(tactical)f(ob)5 b(jects)43 b(is)g(called)g(primitiv)n(e)g(recursiv)n(e)e(if)j(the)0 4151 y(set)g Fr(f)188 4120 y Ff(d)215 4151 y Fo(q)267 4120 y Ff(e)346 4151 y Fu(:)51 b Fo(q)j Fr(2)d Fo(M)9 b Fr(g)43 b Fu(is)h(primitiv)n(e)h(recursiv)n(e.)85 b(An)45 b(op)r(eration)e(\(function\))i(\010)g(on)f(syn)n(tactical)f(ob)5 b(jects)44 b(or)f(ordi-)0 4292 y(nals)37 b Fo(<)i(")324 4304 y Fn(0)399 4292 y Fu(is)e(called)g(primitiv)n(e)g(recursiv)n(e)f (if)i(there)g(is)f(a)g(primitiv)n(e)g(recursiv)n(e)f(function)i Fo(f)48 b Fu(:)40 b(I)-14 b(N)3227 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Fu(.)38 b(By)28 b(Prop)r(osition)e(c\))i(and)g(Theorem)f(6.5a)f(from)i (this)g(w)n(e)f(get)h Fk(tp)p Fu(\()p Fo(h)p Fu(\()p Fo(n;)14 b(k)s Fr(\000)o Fu(1\)\))24 b(=)3366 5031 y Fl(W)3435 5051 y Fp(t)3435 5118 y Fi(9)p Fp(y)r(A)p Fn(\()p Fp(n;y)r Fn(\))3739 5093 y Fu(with)0 5231 y Fo(A)p Fu(\()p Fo(n;)14 b(t)p Fu(\))24 b Fr(2)f Fk(TRUE)562 5194 y Fi(\000)562 5252 y Fn(0)618 5231 y Fu(.)37 b(Hence)28 b Fo(f)9 b Fu(\()p Fo(n)p Fu(\))23 b(=)f Fo(g)s Fu(\()p Fo(n;)14 b(k)s Fu(\))23 b(=)g Fk(val)p Fu(\()p Fo(t)p Fu(\))28 b(and)f(I)-14 b(N)24 b Fr(j)-14 b Fu(=)23 b Fo(A)p Fu(\()p Fo(n;)14 b(f)9 b Fu(\()p Fo(n)p Fu(\)\).)37 b(qed)1908 5539 y(37)p eop %%Page: 38 39 38 38 bop 0 83 a Fu(DIE)28 b(HARD)n(Y-HIERAR)n(CHIE)0 249 y Ft(De\014nition)f Fu(\(F)-7 b(undamen)n(tal)28 b(Sequences)f(for)g(ordinals)f Fo(<)d(")1934 261 y Fn(0)1971 249 y Fu(\))0 390 y(1.)36 b(0[)p Fo(n)p Fu(])23 b(:=)g(1[)p Fo(n)p Fu(])f(:=)h(0.)0 531 y(2.)36 b Fo(!)156 501 y Fp(\013)p Fn(+1)287 531 y Fu([)p Fo(n)p Fu(])24 b(:=)e Fo(!)572 501 y Fp(\013)638 531 y Fr(\001)c Fu(\()p Fo(n)h Fu(+)f(1\).)0 672 y(3.)36 b Fo(!)156 642 y Fp(\025)200 672 y Fu([)p Fo(n)p Fu(])23 b(:=)f Fo(!)484 642 y Fp(\025)p Fn([)p Fp(n)p Fn(])606 672 y Fu(,)28 b(f)r(\177)-44 b(ur)28 b Fo(\025)23 b Fr(2)h Fo(Lim)p Fu(.)0 814 y(4.)36 b Fo(\013)p Fu([)p Fo(n)p Fu(])24 b(:=)f Fo(\013)438 826 y Fn(0)493 814 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)800 826 y Fp(k)q Fi(\000)p Fn(1)944 814 y Fu(+)g Fo(\013)1080 826 y Fp(k)1121 814 y Fu([)p Fo(n)p Fu(],)28 b(falls)g Fo(\013)23 b Fu(=)1583 826 y Fp(N)-5 b(F)1708 814 y Fo(\013)1761 826 y Fn(0)1817 814 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\013)2124 826 y Fp(k)2165 814 y Fu(.)0 980 y(Prop)r(osition:)35 b(\()p Fo(\013)19 b Fu(+)f(1\)[)p Fo(n)p Fu(])23 b(=)g Fo(\013)p Fu(.)0 1146 y Ft(De\014nition)54 b Fo(N)9 b(\013)24 b Fu(:=)e Fo(N)9 b(\013)861 1158 y Fn(1)917 1146 y Fu(+)18 b Fo(:)c(:)g(:)k Fu(+)g Fo(N)9 b(\013)1327 1158 y Fp(k)1386 1146 y Fu(+)18 b Fo(k)59 b Fu(if)28 b Fo(\013)23 b Fu(=)g Fo(!)1866 1116 y Fp(\013)1909 1124 y Fg(1)1964 1146 y Fu(+)18 b Fo(:)c(:)g(:)f(!)2212 1116 y Fp(\013)2255 1125 y Fh(k)2323 1146 y Fu(with)28 b Fo(k)e Fr(\025)d Fu(0)k(and)h Fo(\013)2953 1158 y Fp(k)3017 1146 y Fr(\024)22 b Fo(:)14 b(:)g(:)23 b Fr(\024)g Fo(\013)3365 1158 y Fn(1)3425 1146 y Fo(<)g(")3552 1158 y Fn(0)0 1337 y Ft(Lemma)30 b(6.8)0 1478 y Fu(a\))d Fo(\013)d Fr(2)f Fo(Lim)45 b Fr(\))i(8)p Fo(n)p Fu(\()p Fo(\013)p Fu([)p Fo(n)p Fu(])22 b Fo(<)h(\013)p Fu([)p Fo(n)c Fu(+)f(1]\))23 b(&)f Fo(\013)i Fu(=)e(sup)p Fr(f)p Fo(\013)p Fu([)p Fo(n)p Fu(])h(:)37 b Fo(n)23 b Fr(2)h Fu(I)-14 b(N)p Fr(g)0 1619 y Fu(b\))28 b Fo(\013)c(>)e Fu(0)46 b Fr(\))g Fo(N)9 b(\013)p Fu([0])23 b Fo(<)f(N)9 b(\013)0 1760 y Fu(c\))28 b Fo(\013)p Fu([)p Fo(n)p Fu(])23 b Fo(<)g(\014)k(<)c(\013)47 b Fr(\))f Fo(\013)p Fu([)p Fo(n)p Fu(])23 b Fr(\024)g Fo(\014)t Fu([0])0 1901 y(d\))28 b Fo(\013)p Fu([)p Fo(n)p Fu(])c Fo(<)e(\014)28 b(<)22 b(\013)47 b Fr(\))f Fo(N)9 b(\013)p Fu([)p Fo(n)p Fu(])23 b Fo(<)g(N)9 b(\014)0 2042 y Fu(e\))28 b Fo(\014)f(<)c(\013)46 b Fr(\))h Fo(\014)27 b Fr(\024)c Fo(\013)p Fu([)p Fo(N)9 b(\014)t Fu(])0 2208 y(Pro)r(of:)0 2350 y(a\),b\))28 b(ob)n(vious.)0 2491 y(c\))g(Let)g Fo(\014)f Fu(=)385 2503 y Fp(N)-5 b(F)510 2491 y Fo(\014)557 2503 y Fn(0)613 2491 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\014)914 2503 y Fp(k)954 2491 y Fu(.)0 2632 y(1.)36 b(Assume)28 b Fo(!)464 2602 y Fp(\013)530 2632 y Fr(\001)18 b Fu(\()p Fo(n)h Fu(+)f(1\))k Fo(<)h(\014)28 b(<)22 b(!)1156 2602 y Fp(\013)p Fn(+1)1287 2632 y Fu(.)37 b(Then)28 b Fo(k)e(>)c(n)28 b Fu(and)f Fo(\014)2006 2644 y Fn(0)2067 2632 y Fu(=)22 b Fo(:::)i Fu(=)e Fo(\014)2381 2644 y Fp(n)2449 2632 y Fu(=)h Fo(!)2592 2602 y Fp(\013)2639 2632 y Fu(.)0 2773 y(F)-7 b(rom)27 b(this)h(w)n(e)f(get)h Fo(!)696 2743 y Fp(\013)761 2773 y Fr(\001)19 b Fu(\()p Fo(n)f Fu(+)g(1\))23 b Fr(\024)g Fo(\014)1218 2785 y Fn(0)1273 2773 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(\014)1574 2785 y Fp(k)q Fi(\000)p Fn(1)1718 2773 y Fu(+)g Fo(\014)1848 2785 y Fp(k)1889 2773 y Fu([0])23 b(=)f Fo(\014)t Fu([0].)0 2914 y(2.)36 b(Assume)28 b Fo(!)464 2884 y Fp(\025)p Fn([)p Fp(n)p Fn(])609 2914 y Fo(<)23 b(\014)k(<)c(!)914 2884 y Fp(\025)985 2914 y Fu(und)28 b Fo(\025)23 b Fr(2)g Fo(Lim)p Fu(.)36 b(Then)28 b Fo(!)1790 2884 y Fp(\025)p Fn([)p Fp(n)p Fn(])1935 2914 y Fr(\024)23 b Fo(\014)2070 2926 y Fn(0)2130 2914 y Fu(=)g Fo(!)2273 2884 y Fp(\015)2338 2914 y Fo(<)g(!)2481 2884 y Fp(\025)2524 2914 y Fu(.)0 3055 y(If)28 b Fo(k)e Fu(=)d(0)k(then)h Fo(\025)p Fu([)p Fo(n)p Fu(])23 b Fo(<)g(\015)28 b(<)22 b(\025)28 b Fu(and)g(therefore)e(\(b)n(y)i(IH\))g Fo(\025)p Fu([)p Fo(n)p Fu(])c Fr(\024)e Fo(\015)5 b Fu([0].)37 b(Hence)27 b Fo(!)2549 3025 y Fp(\025)p Fn([)p Fp(n)p Fn(])2694 3055 y Fr(\024)c Fo(!)2837 3025 y Fp(\015)t Fn([0])2973 3055 y Fu(=)f Fo(!)3115 3025 y Fp(\015)3158 3055 y Fu([0])g(=)h Fo(\014)t Fu([0].)0 3196 y(If)28 b Fo(k)e(>)d Fu(0)k(then)h Fo(!)553 3166 y Fp(\025)p Fn([)p Fp(n)p Fn(])698 3196 y Fr(\024)22 b Fo(\014)832 3208 y Fn(0)888 3196 y Fu(+)c Fo(:::)h Fu(+)f Fo(\014)1189 3208 y Fp(k)q Fi(\000)p Fn(1)1333 3196 y Fu(+)g Fo(\014)1463 3208 y Fp(k)1504 3196 y Fu([0])k(=)h Fo(\014)t Fu([0].)0 3337 y(3.)59 b(Assume)35 b Fo(\013)i Fu(=)594 3349 y Fp(N)-5 b(F)732 3337 y Fo(\013)785 3349 y Fn(0)845 3337 y Fu(+)24 b Fo(:::)f Fu(+)g Fo(\013)1167 3349 y Fp(m)1230 3337 y Fu(,)38 b Fo(m)d(>)h Fu(0)e(and)h Fo(\013)p Fu([)p Fo(n)p Fu(])h(=)g Fo(\013)2084 3349 y Fn(0)2145 3337 y Fu(+)23 b Fo(:::)g Fu(+)g Fo(\013)2466 3349 y Fp(m)p Fi(\000)p Fn(1)2638 3337 y Fu(+)g Fo(\013)2779 3349 y Fp(m)2842 3337 y Fu([)p Fo(n)p Fu(])36 b Fo(<)f(\014)40 b(<)c(\013)p Fu(.)60 b(Then)35 b Fo(m)h Fr(\024)f Fo(k)s Fu(,)0 3479 y Fo(\013)53 3491 y Fp(m)116 3479 y Fu([)p Fo(n)p Fu(])26 b Fo(<)f(\014)375 3491 y Fp(m)463 3479 y Fo(<)g(\013)606 3491 y Fp(m)699 3479 y Fu(and)j Fo(\013)914 3491 y Fp(i)968 3479 y Fu(=)d Fo(\014)1105 3491 y Fp(i)1161 3479 y Fu(for)k Fo(i)c(<)g(m)p Fu(.)41 b(By)29 b(IH)g(w)n(e)g(get)g Fo(\013)2141 3491 y Fp(m)2204 3479 y Fu([)p Fo(n)p Fu(])d Fr(\024)f Fo(\014)2463 3491 y Fp(m)2526 3479 y Fu([0])j(and)h(then)h Fo(\013)p Fu([)p Fo(n)p Fu(])c Fr(\024)f Fo(\014)3308 3491 y Fn(0)3364 3479 y Fu(+)19 b Fo(:::)h Fu(+)f Fo(\014)3668 3491 y Fp(m)p Fi(\000)p Fn(1)3835 3479 y Fu(+)0 3620 y Fo(\014)47 3632 y Fp(m)110 3620 y Fu([0])k Fr(\024)f Fo(\014)355 3632 y Fn(0)411 3620 y Fu(+)c Fo(:::)h Fu(+)f Fo(\014)712 3632 y Fp(k)q Fi(\000)p Fn(1)856 3620 y Fu(+)g Fo(\014)986 3632 y Fp(k)1027 3620 y Fu([0])k(=)h Fo(\014)t Fu([0].)0 3786 y(d\))28 b(By)f(c\))h(w)n(e)f(ha)n(v)n(e)g Fo(\013)p Fu([)p Fo(n)p Fu(])c(=)g Fo(\014)t Fu([0])p Fo(:::)p Fu([0].)37 b(Hence)27 b Fo(N)9 b(\013)p Fu([)p Fo(n)p Fu(])24 b Fr(\024)e Fo(N)9 b(\014)t Fu([0])23 b Fo(<)g(N)9 b(\014)t Fu(.)0 3927 y(e\))33 b(Let)g Fo(\013)f Fr(2)h Fo(Lim)p Fu(.)51 b(According)32 b(to)h(a\),d\))g(w)n(e)g(then)g (ha)n(v)n(e)f Fr(8)p Fo(n)p Fu(\()p Fo(N)9 b(\013)p Fu([)p Fo(n)p Fu(])31 b Fo(<)h(N)9 b(\013)p Fu([)p Fo(n)p Fu(+)o(1]\),)34 b(and)f(therefore)f Fo(N)9 b(\014)36 b Fr(\024)c Fo(N)9 b(\013)p Fu([)p Fo(N)g(\014)t Fu(].)0 4068 y(T)-7 b(ogether)26 b(with)j(d\))f(this)g(yields)f(the)h(assertion.)0 4284 y Ft(De\014nition)f Fu(of)g Fo(H)605 4296 y Fp(\013)676 4284 y Fu(:)c(I)-14 b(N)23 b Fr(!)g Fu(I)-14 b(N)28 b(for)f Fo(\013)d(<)f(")1366 4296 y Fn(0)0 4425 y Fu(1.)36 b Fo(H)170 4437 y Fn(0)208 4425 y Fu(\()p Fo(n)p Fu(\))23 b(:=)g Fo(n)p Fu(,)0 4566 y(2.)36 b Fo(H)170 4578 y Fp(\013)218 4566 y Fu(\()p Fo(n)p Fu(\))23 b(:=)g Fo(H)535 4581 y Fp(\013)p Fn([)p Fp(n)p Fn(])661 4566 y Fu(\()p Fo(n)18 b Fu(+)h(1\),)27 b(for)g Fo(\013)d(>)e Fu(0.)0 4732 y Ft(Lemma)30 b(6.9)0 4873 y Fu(a\))d Fo(H)170 4885 y Fp(\013)218 4873 y Fu(\()p Fo(n)p Fu(\))c Fo(<)g(H)512 4885 y Fp(\013)559 4873 y Fu(\()p Fo(n)c Fu(+)f(1\),)0 5015 y(b\))28 b Fo(\014)t Fu([)p Fo(m)p Fu(])c Fo(<)e(\013)i(<)e(\014)51 b Fr(\))46 b Fo(H)847 5030 y Fp(\014)s Fn([)p Fp(m)p Fn(])988 5015 y Fu(\()p Fo(n)19 b Fu(+)f(1\))23 b Fr(\024)f Fo(H)1425 5027 y Fp(\013)1473 5015 y Fu(\()p Fo(n)p Fu(\),)0 5156 y(c\))28 b Fo(\014)f(<)c(\013)g Fu(&)g Fo(N)9 b(\014)27 b Fr(\024)c Fo(n)46 b Fr(\))g Fo(H)955 5168 y Fp(\014)1000 5156 y Fu(\()p Fo(n)p Fu(\))23 b Fo(<)g(H)1294 5168 y Fp(\013)1341 5156 y Fu(\()p Fo(n)p Fu(\),)0 5297 y(d\))28 b Fo(\013)c(>)e Fu(0)46 b Fr(\))g Fo(H)556 5309 y Fp(\013)603 5297 y Fu(\()p Fo(n)p Fu(\))24 b(=)e(min)q Fr(f)p Fo(k)j Fr(\025)e Fo(n)g Fu(:)g Fo(\013)p Fu([)p Fo(n)p Fu(])p Fo(:::)p Fu([)p Fo(k)s Fr(\000)p Fu(1])f(=)h(0)p Fr(g)f Fu(=)h Fo(n)18 b Fu(+)g(min)q Fr(f)p Fo(l)23 b Fu(:)37 b Fo(\013)p Fu([)p Fo(n)p Fu(][)p Fo(n)19 b Fu(+)f(1])c Fo(:)g(:)g(:)f Fu([)p Fo(n)19 b Fu(+)f Fo(l)i Fr(\000)e Fu(1])k(=)h(0)p Fr(g)p Fu(.)1908 5539 y(38)p eop %%Page: 39 40 39 39 bop 0 83 a Fu(Pro)r(of:)0 224 y(a\),b\))28 b(sim)n(ultaneous)f (induction)h(on)f Fo(\013)p Fu(:)37 b(Let)28 b Fo(\013)c(>)e Fu(0.)0 365 y(a\))27 b(1.)37 b Fo(\013)23 b Fr(2)h Fo(Lim)p Fu(:)36 b Fo(H)645 377 y Fp(\013)692 365 y Fu(\()p Fo(n)p Fu(\))24 b(=)e Fo(H)986 380 y Fp(\013)p Fn([)p Fp(n)p Fn(])1112 365 y Fu(\()p Fo(n)p Fu(+1\))1356 304 y Fn(IHa)1376 365 y Fo(<)44 b(H)1554 380 y Fp(\013)p Fn([)p Fp(n)p Fn(])1680 365 y Fu(\()p Fo(n)p Fu(+3\))1923 296 y Fn(IHb)1946 365 y Fr(\024)h Fo(H)2125 380 y Fp(\013)p Fn([)p Fp(n)p Fn(+1])2335 365 y Fu(\()p Fo(n)p Fu(+2\))23 b(=)f Fo(H)2735 377 y Fp(\013)2783 365 y Fu(\()p Fo(n)p Fu(+)o(1\).)0 506 y(2.)36 b Fo(\013)24 b Fu(=)f Fo(\013)319 518 y Fn(0)356 506 y Fu(+1:)36 b Fo(H)591 518 y Fp(\013)638 506 y Fu(\()p Fo(n)p Fu(\))24 b(=)e Fo(H)932 518 y Fp(\013)975 526 y Fg(0)1012 506 y Fu(\()p Fo(n)p Fu(+1\))1256 445 y Fn(IHa)1276 506 y Fo(<)44 b(H)1454 518 y Fp(\013)1497 526 y Fg(0)1534 506 y Fu(\()p Fo(n)p Fu(+)o(2\))23 b(=)g Fo(H)1934 518 y Fp(\013)1981 506 y Fu(\()p Fo(n)p Fu(+1\).)0 648 y(b\))28 b(F)-7 b(rom)27 b Fo(\014)t Fu([)p Fo(m)p Fu(])d Fo(<)e(\013)i(<)e (\014)32 b Fu(w)n(e)c(obtain)f Fo(\014)t Fu([)p Fo(m)p Fu(])c Fr(\024)g Fo(\013)p Fu([)p Fo(n)p Fu(])g Fo(<)g(\014)32 b Fu(b)n(y)27 b(Lemma)h(6.8.)0 789 y(Hence)g Fo(H)316 804 y Fp(\014)s Fn([)p Fp(m)p Fn(])457 789 y Fu(\()p Fo(n)p Fu(\))595 719 y Fn(IHb)617 789 y Fr(\024)45 b Fo(H)796 804 y Fp(\013)p Fn([)p Fp(n)p Fn(])922 789 y Fu(\()p Fo(n)p Fu(\))1060 727 y Fn(IHa)1080 789 y Fo(<)f(H)1258 804 y Fp(\013)p Fn([)p Fp(n)p Fn(])1384 789 y Fu(\()p Fo(n)p Fu(+)o(1\))23 b(=)g Fo(H)1784 801 y Fp(\013)1831 789 y Fu(\()p Fo(n)p Fu(\).)0 955 y(c\))28 b(Induction)g(on)f Fo(\013)p Fu(:)37 b Fo(\014)28 b(<)22 b(\013)940 908 y Fn(L)p Fp(:)p Fn(6)p Fp(:)p Fn(8e)987 955 y Fr(\))92 b Fo(\014)28 b Fr(\024)22 b Fo(\013)p Fu([)p Fo(N)9 b(\014)t Fu(])24 b Fr(\024)f Fo(\013)p Fu([)p Fo(n)p Fu(])1857 901 y Fn(a\)+IH)1907 955 y Fr(\))96 b Fo(H)2155 967 y Fp(\014)2200 955 y Fu(\()p Fo(n)p Fu(\))24 b Fo(<)e(H)2494 967 y Fp(\014)2539 955 y Fu(\()p Fo(n)p Fu(+1\))h Fr(\024)f Fo(H)2939 970 y Fp(\013)p Fn([)p Fp(n)p Fn(])3065 955 y Fu(\()p Fo(n)p Fu(+1\))h(=)f Fo(H)3465 967 y Fp(\013)3513 955 y Fu(\()p Fo(n)p Fu(\).)0 1121 y(d\))28 b(Let)g Fo(k)e Fr(\025)c Fo(n)28 b Fu(minimal)g(suc)n(h)f(that)h Fo(\013)p Fu([)p Fo(n)p Fu(])p Fo(:::)p Fu([)p Fo(k)s Fr(\000)p Fu(1])23 b(=)f(0.)0 1262 y(Then)28 b Fo(H)286 1274 y Fp(\013)333 1262 y Fu(\()p Fo(n)p Fu(\))c(=)e Fo(H)627 1277 y Fp(\013)p Fn([)p Fp(n)p Fn(])753 1262 y Fu(\()p Fo(n)p Fu(+1\))h(=)f Fo(::::)i Fu(=)e Fo(H)1356 1277 y Fp(\013)p Fn([)p Fp(n)p Fn(])p Fp(:::)p Fn([)p Fp(k)q Fi(\000)p Fn(1])1701 1262 y Fu(\()p Fo(k)s Fu(\))h(=)g Fo(H)1991 1274 y Fn(0)2028 1262 y Fu(\()p Fo(k)s Fu(\))g(=)g Fo(k)s Fu(.)0 1478 y Ft(Abbreviation:)55 b Fo(N)-5 b(F)12 b Fu(\()p Fo(\013;)i(\014)t Fu(\))77 b(:)p Fr(,)g Fo(\013)38 b Fu(=)g(0)e(or)g Fo(\014)42 b Fu(=)c(0)e(or)g([)14 b Fo(\013)39 b Fu(=)e Fo(!)2273 1448 y Fp(\013)2316 1456 y Fg(0)2377 1478 y Fu(+)24 b Fo(:::)h Fu(+)f Fo(!)2704 1448 y Fp(\013)2747 1456 y Fh(n)2830 1478 y Fu(&)38 b Fo(\014)k Fu(=)c Fo(!)3180 1448 y Fp(\014)3218 1456 y Fg(0)3278 1478 y Fu(+)24 b Fo(:::)h Fu(+)f Fo(!)3605 1448 y Fp(\014)3643 1456 y Fh(m)3739 1478 y Fu(with)0 1619 y Fo(\013)53 1631 y Fn(0)113 1619 y Fr(\025)f Fo(:::)g Fr(\025)g Fo(\013)434 1631 y Fp(n)502 1619 y Fr(\025)g Fo(\014)637 1631 y Fn(0)697 1619 y Fr(\025)g Fo(:::)g Fr(\025)g Fo(\014)1012 1631 y Fp(m)1089 1619 y Fu(])k(.)0 1785 y Ft(Prop)s(osition:)35 b Fo(N)-5 b(F)12 b Fu(\()p Fo(\013;)i(\014)t Fu(\))24 b(&)f Fo(\014)k(>)c Fu(0)45 b Fr(\))h Fu(\()p Fo(\013)20 b Fu(+)e Fo(\014)t Fu(\)[)p Fo(n)p Fu(])23 b(=)g Fo(\013)c Fu(+)f Fo(\014)t Fu([)p Fo(n)p Fu(])23 b(&)g Fo(N)-5 b(F)12 b Fu(\()p Fo(\013;)i(\014)t Fu([)p Fo(n)p Fu(]\).)0 1951 y Ft(Lemma)30 b(6.10)0 2092 y Fu(a\))d Fo(N)-5 b(F)12 b Fu(\()p Fo(\013;)i(\014)t Fu(\))48 b Fr(\))e Fo(H)679 2104 y Fp(\013)p Fn(+)p Fp(\014)841 2092 y Fu(=)22 b Fo(H)997 2104 y Fp(\013)1063 2092 y Fr(\016)c Fo(H)1192 2104 y Fp(\014)1237 2092 y Fu(.)0 2233 y(b\))28 b Fo(H)175 2249 y Fp(!)219 2232 y Fh(\013)p Fg(+1)336 2233 y Fu(\()p Fo(n)p Fu(\))c(=)e Fo(H)637 2190 y Fn(\()p Fp(n)p Fn(+1\))630 2252 y Fp(!)674 2236 y Fh(\013)818 2233 y Fu(\()p Fo(n)p Fu(+1\))27 b(and)h Fo(H)1297 2250 y Fp(!)1341 2234 y Fh(\025)1384 2233 y Fu(\()p Fo(n)p Fu(\))23 b(=)g Fo(H)1678 2252 y Fp(!)1722 2236 y Fh(\025)p Fg([)p Fh(n)p Fg(])1835 2233 y Fu(\()p Fo(n)p Fu(+1\))k(for)g Fo(\025)d Fr(2)f Fo(Lim)p Fu(.)0 2374 y(c\))28 b(F)-7 b(or)27 b(eac)n(h)g(primitiv)n(e)g(recursiv)n(e)f (function)i Fo(f)36 b Fu(there)28 b(exists)f(a)g Fo(k)f Fr(2)d Fu(I)-14 b(N)28 b(suc)n(h)g(that)g Fr(8)o Fo(~)-41 b(x)o Fu(\()14 b Fo(f)9 b Fu(\()o Fo(~)-41 b(x)q Fu(\))23 b Fo(<)g(H)3150 2391 y Fp(!)3194 2375 y Fh(k)3234 2374 y Fu(\(max)p Fr(f)o Fo(~)-41 b(x)p Fr(g)p Fu(\))14 b(\).)0 2540 y(Pro)r(of:)0 2682 y(a\))27 b(Induction)h(on)g Fo(\014)t Fu(:)37 b(1.)f Fo(H)874 2694 y Fp(\013)p Fn(+0)1006 2682 y Fu(\()p Fo(n)p Fu(\))23 b(=)g Fo(H)1300 2694 y Fp(\013)1347 2682 y Fu(\()p Fo(n)p Fu(\))h(=)e Fo(H)1641 2694 y Fp(\013)1689 2682 y Fu(\()p Fo(H)1790 2694 y Fn(0)1827 2682 y Fu(\()p Fo(n)p Fu(\)\).)0 2823 y(2.)36 b Fo(\014)28 b(>)23 b Fu(0:)36 b Fo(H)434 2835 y Fp(\013)p Fn(+)p Fp(\014)573 2823 y Fu(\()p Fo(n)p Fu(\))23 b(=)g Fo(H)867 2838 y Fn(\()p Fp(\013)p Fn(+)p Fp(\014)s Fn(\)[)p Fp(n)p Fn(])1136 2823 y Fu(\()p Fo(n)p Fu(+1\))g(=)g Fo(H)1537 2838 y Fp(\013)p Fn(+)p Fp(\014)s Fn([)p Fp(n)p Fn(])1754 2823 y Fu(\()p Fo(n)c Fu(+)f(1\))2035 2776 y Fn(IH)2039 2823 y Fu(=)27 b Fo(H)2200 2835 y Fp(\013)2247 2823 y Fu(\()p Fo(H)2348 2838 y Fp(\014)s Fn([)p Fp(n)p Fn(])2472 2823 y Fu(\()p Fo(n)18 b Fu(+)g(1\)\))23 b(=)g Fo(H)2941 2835 y Fp(\013)2988 2823 y Fu(\()p Fo(H)3089 2835 y Fp(\014)3134 2823 y Fu(\()p Fo(n)p Fu(\)\).)0 2964 y(b\))28 b Fo(H)175 2980 y Fp(!)219 2963 y Fh(\013)p Fg(+1)336 2964 y Fu(\()p Fo(n)p Fu(\))c(=)e Fo(H)630 2979 y Fp(!)674 2962 y Fh(\013)716 2979 y Fi(\001)p Fn(\()p Fp(n)p Fn(+1\))917 2964 y Fu(\()p Fo(n)p Fu(+1\))1163 2910 y Fn(a\))1161 2964 y Fu(=)g Fo(H)1324 2921 y Fn(\()p Fp(n)p Fn(+1\))1317 2983 y Fp(!)1361 2966 y Fh(\013)1505 2964 y Fu(\()p Fo(n)p Fu(+1\).)0 3105 y(c\))28 b(F)-7 b(rom)27 b(b\))h(it)g(follo)n(ws)f(that)h(\()p Fo(k)s(;)14 b(n)p Fu(\))23 b Fr(7!)g Fo(H)1352 3122 y Fp(!)1396 3105 y Fh(k)9 b Fu(\()p Fo(n)p Fu(\))28 b(is)g(\(a)f(v)-5 b(arian)n(t)27 b(of)6 b(\))28 b(the)g(Ac)n(k)n(ermann)f(function.)0 3321 y Ft(Theorem)j(6.11)0 3462 y Fu(Let)e Fo(\022)d Fu(:)e(I)-14 b(N)19 b Fr(\002)f Fu(I)-14 b(N)24 b Fr(!)f Fu(OT)k(b)r(e)h(primitiv)n(e)f(recursiv)n(e,)f Fo(m)d Fr(\025)g Fu(1,)k(and)g Fo(\013)d(<)f(!)2291 3474 y Fp(m)p Fn(+1)2465 3462 y Fu(suc)n(h)k(that)h Fr(8)p Fo(n)p Fu(\()14 b Fo(\022)r Fu(\()p Fo(n;)g Fu(0\))22 b Fr(\026)3319 3430 y Ff(d)3347 3462 y Fo(\013)3412 3430 y Ff(e)3453 3462 y Fu(\).)0 3603 y(Then)28 b(there)f(is)h(an)35 b(~)-50 b Fo(\013)23 b(<)g(!)844 3615 y Fp(m)p Fn(+1)1019 3603 y Fu(suc)n(h)k(that)h(min)p Fr(f)p Fo(l)c Fu(:)f Fo(\022)r Fu(\()p Fo(n;)14 b(l)r Fu(+)o(1\))23 b Fr(6\036)g Fo(\022)r Fu(\()p Fo(n;)14 b(l)r Fu(\))p Fr(g)23 b Fo(<)f(H)2544 3615 y Fn(~)-40 b Fp(\013)2584 3603 y Fu(\()p Fo(n)p Fu(\))29 b(\()p Fr(8)p Fo(n)p Fu(\).)0 3769 y(Pro)r(of:)0 3910 y(W.l.o.g.)37 b Fr(8)p Fo(n)p 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b(&)g(\(3\))g(&)46 b Fo(\013)23 b Fu(=)g Fr(k)p Fo(\022)r Fu(\()p Fo(n;)14 b Fu(0\))p Fr(k)36 b(\))46 b(9)14 b Fo(l)24 b Fr(\024)f Fo(H)1370 1000 y Fp(!)1414 984 y Fh(k)1450 1000 y Fi(\001)p Fp(\013)1517 984 y Fu(\()p Fo(g)s Fu(\()p Fo(k)s(;)14 b(n;)g Fu(0\)\)[)g Fo(\022)r Fu(\()p Fo(n;)g(l)r Fu(+)o(1\))23 b Fr(6\036)g Fo(\022)r Fu(\()p Fo(n;)14 b(l)r Fu(\))g(].)0 1175 y Fo(H)69 1191 y Fp(!)113 1175 y Fh(k)149 1191 y Fi(\001)p Fp(\013)216 1175 y Fu(\()p Fo(g)s Fu(\()p Fo(k)s(;)g(n;)g Fu(0\)\))23 b Fo(<)g(H)779 1191 y Fp(!)823 1175 y Fh(k)859 1191 y Fi(\001)p Fp(\013)926 1175 y Fo(H)995 1191 y Fp(!)1039 1175 y Fh(k)9 b Fu(\(max)p Fr(f)p Fo(k)s(;)14 b(n)p Fr(g)p Fu(\))22 b Fr(\024)h Fo(H)1694 1191 y Fp(!)1738 1175 y Fh(k)1774 1191 y Fi(\001)p Fn(\()p Fp(\013)p Fn(+1\)+)p Fp(k)2064 1175 y Fu(\()p Fo(n)p Fu(\).)0 1341 y Fo(m)g Fr(\025)g Fu(1)f Fr(\))h Fo(!)j Fr(\024)d Fo(!)572 1353 y Fp(m)662 1341 y Fu(,)83 b Fo(\013)24 b(<)e(!)984 1353 y Fp(m)p Fn(+1)1154 1341 y Fu(=)h Fo(!)1297 1310 y Fp(!)1339 1318 y Fh(m)1434 1341 y Fr(\))37 b Fo(!)1609 1310 y Fp(k)1668 1341 y Fr(\001)19 b Fu(\()p Fo(\013)p Fu(+1\))f(+)g Fo(k)26 b(<)d(!)2247 1310 y Fp(k)2306 1341 y Fr(\001)18 b Fu(\()p Fo(\013)p Fu(+)q(2\))k Fo(<)h(!)2737 1310 y Fp(k)2796 1341 y Fr(\001)c Fo(!)2893 1310 y Fp(!)2935 1318 y Fh(m)3016 1341 y Fu(=)k Fo(!)3159 1310 y Fp(!)3201 1318 y Fh(m)3283 1341 y Fu(=)f Fo(!)3422 1353 y Fp(m)p Fn(+1)3569 1341 y Fu(.)0 1606 y Ft(Theorem)30 b(6.12)0 1747 y Fu(If)e Fr(Z)143 1759 y Fp(m)229 1747 y Fr(`)23 b(8)p Fo(x)p Fr(9)p Fo(y)s(A)p Fu(\()p Fo(x;)14 b(y)s Fu(\))56 b(\()p Fo(A)28 b Fu(atomic,)f Fo(m)c Fr(\025)g Fu(1\))k(then)h(there)g(is)f (an)g Fo(\013)d(<)f(!)2316 1759 y Fp(m)p Fn(+1)2490 1747 y Fu(suc)n(h)k(that)h Fr(8)p Fo(n)p Fr(9)p Fo(l)c(<)e(H)3205 1759 y Fp(\013)3253 1747 y Fu(\()p Fo(n)p Fu(\))14 b(I)-14 b(N)23 b Fr(j)-14 b Fu(=)23 b Fo(A)p Fu(\()p Fo(n;)14 b(l)r Fu(\).)0 1913 y(Pro)r(of:)0 2055 y(By)24 b(Theorem)f(6.7)g(there) 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Fo(n)1032 2817 y Fn(0)1102 2805 y Fu(suc)n(h)g(that)h Fo(H)1548 2822 y Fp(!)1592 2805 y Fh(k)9 b Fo(H)1701 2817 y Fp(\014)1746 2805 y Fu(\()p Fo(n)p Fu(\))31 b Fr(\024)g Fo(H)2056 2817 y Fp(\015)2098 2805 y Fo(H)2167 2817 y Fp(\015)2210 2805 y Fu(\()p Fo(n)p Fu(\))g(=)g Fo(H)2520 2817 y Fp(\015)t Fn(+)p Fp(\015)2652 2805 y Fu(\()p Fo(n)p Fu(\))i(for)f(all)g Fo(n)f Fr(\025)f Fo(n)3277 2817 y Fn(0)3315 2805 y Fu(.)51 b(Hence)32 b Fo(f)9 b Fu(\()p Fo(n)p Fu(\))31 b Fo(<)0 2946 y(H)69 2963 y Fp(!)113 2947 y Fh(k)153 2946 y Fo(H)222 2958 y Fp(\014)267 2946 y Fu(\()p Fo(n)349 2958 y Fn(0)387 2946 y Fu(+)o Fo(n)p Fu(\))23 b Fr(\024)g Fo(H)713 2958 y Fp(\015)t Fn(+)p Fp(\015)845 2946 y Fu(\()p Fo(n)927 2958 y Fn(0)964 2946 y Fu(+)p Fo(n)p Fu(\))g(=)g Fo(H)1291 2958 y Fp(\015)t Fn(+)p Fp(\015)t Fn(+)p Fp(n)1512 2966 y Fg(0)1547 2946 y Fu(\()p Fo(n)p Fu(\))28 b(\(and)g Fo(\015)5 b Fu(+)o Fo(\015)g Fu(+)o Fo(n)2157 2958 y Fn(0)2217 2946 y Fo(<)23 b(!)2357 2958 y Fp(m)p Fn(+1)2504 2946 y Fu(\).)0 3162 y Ft(Corollary:)64 b Fr(Z)544 3174 y Fp(m)630 3162 y Fr(6`)28 b Fu(`)23 b Fr(8)p Fo(n)p Fr(9)p Fo(l)r Fu(\()14 b Fo(!)1023 3174 y Fp(m)p Fn(+1)1168 3162 y Fu([)p Fo(n)p Fu(][)p Fo(n)p Fu(+1])p Fo(:::)p Fu([)p Fo(l)r Fu(])22 b(=)h(0)14 b(\))22 b('.)0 3328 y(Pro)r(of:)36 b(Assumption:)h Fr(Z)812 3340 y Fp(m)898 3328 y Fr(`)28 b Fu(`)14 b Fr(8)p Fo(n)p Fr(9)p Fo(l)r Fu(\()p Fo(::::)p Fu(\))g('.)0 3469 y(Then)19 b(there)g(is)f(an)h Fo(\013)k(<)g(!)809 3481 y Fp(m)p Fn(+1)975 3469 y Fu(suc)n(h)18 b(that)h Fr(8)p Fo(n)p Fr(9)p Fo(l)24 b(<)e(H)1672 3481 y Fp(\013)1720 3469 y Fu(\()p Fo(n)p Fu(\)\()14 b Fo(!)1932 3481 y Fp(m)p Fn(+1)2079 3469 y Fu([)p Fo(n)p Fu(][)p Fo(n)p Fu(+1])p Fo(:::)p Fu([)p Fo(l)r Fu(])22 b(=)g(0)14 b(\),)21 b(i.e.)34 b Fr(8)p Fo(n)p Fr(9)p Fo(l)23 b(<)g(H)3249 3481 y Fp(\013)3296 3469 y Fu(\()p Fo(n)p Fu(\)\()14 b Fo(H)3525 3481 y Fp(!)3567 3489 y Fh(m)p Fg(+1)3698 3469 y Fu(\()p Fo(n)p Fu(\))23 b Fr(\024)0 3611 y Fo(l)r Fu(+)o(1)14 b(\).)62 b(This)36 b(implies)g Fr(8)p Fo(n)p Fu(\()14 b Fo(H)964 3623 y Fp(!)1006 3631 y Fh(m)p Fg(+1)1135 3611 y Fu(\()p Fo(n)p Fu(\))37 b Fr(\024)g Fo(H)1457 3623 y Fp(\013)1504 3611 y Fu(\()p Fo(n)p Fu(\))14 b(\).)63 b(But)36 b(b)n(y)g(L.6.9c)f(w)n(e)h (ha)n(v)n(e)e Fr(8)p Fo(n)i Fr(\025)h Fo(N)9 b Fu(\()p Fo(\013)p Fu(\)\()14 b Fo(H)3174 3623 y Fp(\013)3222 3611 y Fu(\()p Fo(n)p Fu(\))38 b Fo(<)e(H)3544 3623 y Fp(!)3586 3631 y Fh(m)p Fg(+1)3716 3611 y Fu(\()p Fo(n)p Fu(\))14 b(\).)0 3752 y(Con)n(tradiction.)0 4017 y(Disgression)0 4183 y(Let)28 b Fo(h)197 4195 y Fp(\013)244 4183 y Fu(\()p Fo(n)p Fu(\))23 b(:=)g Fo(n)p Fu(,)28 b Fo(h)641 4195 y Fp(\013)p Fn(+1)772 4183 y Fu(\()p Fo(n)p Fu(\))23 b(:=)g Fo(h)1068 4195 y Fp(\013)1115 4183 y Fu(\()p Fo(n)c Fu(+)f(1\),)27 b(and)h Fo(h)1633 4195 y Fp(\025)1676 4183 y Fu(\()p Fo(n)p Fu(\))c(=)e Fo(h)1949 4198 y Fp(\025)p Fn([)p Fp(n)p Fn(])2071 4183 y Fu(\()p Fo(n)p Fu(\))28 b(for)f Fo(\025)d Fr(2)f Fo(Lim)p Fu(.)0 4349 y(Then)28 b Fo(h)265 4361 y Fp(\013)312 4349 y Fu(\()p Fo(n)p Fu(\))23 b Fr(\024)g Fo(H)606 4361 y Fp(\013)653 4349 y Fu(\()p Fo(n)p Fu(\))h Fo(<)e(h)926 4361 y Fp(\013)974 4349 y Fu(\()p Fo(n)c Fu(+)g(1\).)0 4515 y(Pro)r(of)26 b(b)n(y)i(induction)g (on)f Fo(\013)p Fu(:)0 4657 y(1.)36 b Fo(h)149 4669 y Fp(\013)p Fn(+1)281 4657 y Fu(\()p Fo(n)p Fu(\))23 b(=)g Fo(h)554 4669 y Fp(\013)601 4657 y Fu(\()p Fo(n)18 b Fu(+)h(1\))881 4587 y Fn(IH)886 4657 y Fr(\024)26 b Fo(H)1046 4669 y Fp(\013)1094 4657 y Fu(\()p Fo(n)18 b Fu(+)g(1\))23 b(=)g Fo(H)1531 4669 y Fp(\013)p Fn(+1)1662 4657 y Fu(\()p Fo(n)p Fu(\))28 b(and)g Fo(H)2035 4669 y Fp(\013)p Fn(+1)2166 4657 y Fu(\()p Fo(n)p Fu(\))c(=)e Fo(H)2460 4669 y Fp(\013)2508 4657 y Fu(\()p Fo(n)c Fu(+)g(1\))2788 4595 y Fn(IH)2792 4657 y Fo(<)27 b(h)2932 4669 y Fp(\013)2979 4657 y Fu(\()p Fo(n)19 b Fu(+)f(2\))23 b(=)f Fo(h)3395 4669 y Fp(\013)p Fn(+1)3526 4657 y Fu(\()p Fo(n)d Fu(+)f(1\).)0 4798 y(2.)65 b Fo(\013)39 b Fr(2)g Fo(Lim)p Fu(:)55 b Fo(h)601 4810 y Fp(\013)648 4798 y Fu(\()p Fo(n)p Fu(\))39 b(=)f Fo(h)952 4813 y Fp(\013)p Fn([)p Fp(n)p Fn(])1078 4798 y Fu(\()p Fo(n)p Fu(\))1231 4728 y Fn(IH)1236 4798 y Fr(\024)k Fo(H)1412 4813 y Fp(\013)p Fn([)p Fp(n)p Fn(])1538 4798 y Fu(\()p Fo(n)p Fu(\))d Fo(<)g(H)1864 4813 y Fp(\013)p Fn([)p Fp(n)p Fn(])1990 4798 y Fu(\()p Fo(n)24 b Fu(+)h(1\))38 b(=)h Fo(H)2471 4810 y Fp(\013)2518 4798 y Fu(\()p Fo(n)p Fu(\))e(and)g Fo(H)2909 4810 y Fp(\013)2957 4798 y Fu(\()p Fo(n)p Fu(\))i(=)f Fo(H)3282 4813 y Fp(\013)p Fn([)p Fp(n)p Fn(])3408 4798 y Fu(\()p Fo(n)25 b Fu(+)f(1\))3717 4728 y Fn(L)p Fp(:)p Fn(6)p Fp(:)p Fn(9b)3776 4798 y Fr(\024)0 4939 y Fo(H)69 4954 y Fp(\013)p Fn([)p Fp(n)p Fn(+1])279 4939 y Fu(\()p Fo(n)p Fu(\))416 4877 y Fn(IH)421 4939 y Fo(<)j(h)561 4954 y Fp(\013)p Fn([)p Fp(n)p Fn(+1])770 4939 y Fu(\()p Fo(n)19 b Fu(+)f(1\))23 b(=)f Fo(h)1186 4951 y Fp(\013)1234 4939 y Fu(\()p Fo(n)c Fu(+)g(1\).)1908 5539 y(40)p eop %%Page: 41 42 41 41 bop 0 83 a Ft(App)s(endix)31 b(to)h Fr(x)p Ft(6:)k Fu(Comparison)26 b(with)i(Gen)n(tzen's)g(consistency)f(pro)r(of)0 244 y(Mo)r(di\014cations)g(of)h Fe(Z)660 207 y Fi(\003)698 244 y Fu(:)0 405 y({)f(all)g Fk(Cut)310 417 y Fp(C)394 405 y Fu(are)g(deleted,)0 543 y({)g(in)h(the)g(de\014nition)g(of)g(\()p Fk(Ind)914 503 y Fp(y)r(;t)914 568 y(F)998 543 y Fo(h)1046 555 y Fn(0)1083 543 y Fu(\)[0])g(w)n(e)f(use)h Fk(R)1550 558 y Fp(F)9 b Fn(\()p Fp(i)p 1627 571 24 4 v 1 w Fn(\))1709 543 y Fu(instead)27 b(of)h Fk(Cut)2216 558 y Fp(F)9 b Fn(\()p Fp(i)p 2293 571 V Fn(\))2346 543 y Fu(,)0 682 y({)27 b(deg)q(\()p Fk(Ind)336 642 y Fp(y)r(;t)336 706 y(F)421 682 y Fu(\))c(:=)g(rk)o(\()p Fo(F)12 b Fu(\),)0 942 y({)27 b(o\()p Fo(h)p Fu(\))c(:=)357 747 y Fl(8)357 822 y(>)357 846 y(<)357 996 y(>)357 1021 y(:)444 799 y Fu(o\()p Fo(h)566 811 y Fn(0)603 799 y Fu(\)#o\()p Fo(h)826 811 y Fn(1)864 799 y Fu(\))280 b(if)28 b Fr(I)i Fu(=)22 b Fk(R)1468 811 y Fp(C)444 904 y Fu(o\()p Fo(h)566 916 y Fn(0)603 904 y Fu(\))d Fr(\001)g Fo(!)428 b Fu(if)28 b Fr(I)i Fu(=)22 b Fk(Ind)1523 864 y Fp(y)r(;t)1523 928 y(C)444 1010 y Fo(!)499 980 y Fn(o\()p Fp(h)597 988 y Fg(0)629 980 y Fn(\))1176 1010 y Fu(if)28 b Fr(I)i Fu(=)22 b Fk(E)444 1081 y Fu(\(sup)602 1101 y Fp(i)g(:)14 b(:)g(:)23 b(>)g(x)2822 618 y Fp(k)2885 606 y Fu(und)f Fo(n)3095 618 y Fn(1)3133 606 y Fo(;)14 b(:::;)g(n)3326 618 y Fp(k)3389 606 y Fr(2)24 b(f)p Fu(1)p Fo(;)14 b(:::;)g(b)p Fr(\000)n Fu(1)p Fr(g)p Fu(.)0 797 y Fr(S)50 809 y Fp(n)95 797 y Fu(\()p Fo(x)p Fu(\))25 b(:=)d(\010)401 761 y Fp(n)p Fn(+2)401 819 y Fp(n)p Fn(+1)530 797 y Fu(\()p Fo(x)p Fu(\))57 b(\()p Fo(n)23 b Fr(\025)g Fu(1,)k Fo(x)c Fr(2)h Fo(!)s Fu(\).)0 963 y(GS\()p Fo(a;)14 b Fu(0\))23 b(:=)g(GS\()p Fo(a;)14 b Fu(1\))23 b(:=)g Fo(a;)60 b Fu(GS\()p Fo(a;)14 b(n)k Fu(+)g(1\))23 b(:=)g Fr(S)1624 975 y Fp(n)1669 963 y Fu(\(GS)q(\()p Fo(a;)14 b(n)p Fu(\)\))p Fr(\000)2059 946 y(\001)2110 963 y Fu(1)27 b(f)r(\177)-44 b(ur)27 b Fo(n)c Fr(\025)g Fu(1.)0 1129 y(Die)28 b(F)-7 b(olge)27 b(\(GS)q(\()p Fo(a;)14 b(n)p Fu(\)\))740 1141 y Fp(n)p Fi(2)p Fn(I)-11 b(N)919 1129 y Fu(\(w)n(ob)r(ei)28 b Fo(a)23 b Fr(2)g Fo(!)s Fu(\))28 b(hei\031t)f Fs(Go)r(o)r(dstein-F)-7 b(olge)27 b Fu(zu)h Fo(a)p Fu(.)0 1295 y(Wir)g(w)n(erden)g(zeigen,)g(da\031)f(jede)i(Go)r (o)r(dstein-F)-7 b(olge)27 b(terminiert)h(\(d.h.)39 b Fr(8)p Fo(a)p Fr(9)p Fo(n:)p Fu(GS\()p Fo(a;)14 b(n)p Fu(\))24 b(=)g(0\),)k(und,)h(da\031)e(diese)h(T)-7 b(atsac)n(he)0 1436 y(nic)n(h)n(t)28 b(in)f(Z)h(b)r(ew)n(eisbar)e(ist.)0 1652 y(Abk.:)37 b Fo(\022)274 1664 y Fp(n)319 1652 y Fu(\()p Fo(x)p Fu(\))24 b(:=)f(\010)625 1622 y Fp(!)625 1673 y(n)p Fn(+1)754 1652 y Fu(\()p Fo(x)p Fu(\))112 b(\()p Fo(n)23 b Fr(\025)g Fu(1,)k Fo(x)d Fr(2)f Fo(!)s Fu(\).)0 1843 y Ft(Lemma)30 b(7.1)54 b Fu(Sei)28 b Fo(n)23 b Fr(\025)f Fu(1.)0 1984 y(a\))27 b Fo(x)d(<)f(y)i(<)e(!)49 b Fr(\))d(S)694 1996 y Fp(n)739 1984 y Fu(\()p Fo(x)p Fu(\))24 b Fo(<)f Fr(S)1012 1996 y Fp(n)1058 1984 y Fu(\()p Fo(y)s Fu(\))g(&)g Fo(\022)1316 1996 y Fp(n)1361 1984 y Fu(\()p Fo(x)p Fu(\))h Fo(<)e(\022)1622 1996 y Fp(n)1668 1984 y Fu(\()p Fo(y)s Fu(\),)0 2125 y(b\))28 b Fo(\022)145 2137 y Fp(n)190 2125 y Fu(\()p Fo(x)p Fu(\))c(=)f Fo(\022)452 2137 y Fp(n)p Fn(+1)581 2125 y Fr(S)631 2137 y Fp(n)677 2125 y Fu(\()p Fo(x)p Fu(\),)0 2267 y(c\))28 b Fo(x)23 b(>)g Fu(0)46 b Fr(\))g Fo(\022)511 2279 y Fp(n)556 2267 y Fu(\()p Fo(x)19 b Fr(\000)f Fu(1\))23 b Fo(<)g(\022)961 2279 y Fp(n)1006 2267 y Fu(\()p Fo(x)p Fu(\).)0 2433 y(Bew)n(eis)k(durc)n(h)g(Induktion)h(nac)n(h)f Fo(x)p Fu(:)0 2574 y(a\))g(Man)h(zeige:)36 b(2)23 b Fr(\024)f Fo(b)h(<)g(\013)g Fu(&)g Fo(x)g(<)g(y)49 b Fr(\))d Fu(\010)1428 2544 y Fp(\013)1428 2597 y(b)1475 2574 y Fu(\()p Fo(x)p Fu(\))24 b Fo(<)f Fu(\010)1758 2544 y Fp(\013)1758 2597 y(b)1805 2574 y Fu(\()p Fo(y)s Fu(\).)37 b(Dab)r(ei)28 b(b)r(en)n(utze)g(man)g(folgenden)f(Hilfsatz:)0 2715 y(Ist)h Fo(\030)f Fu(=)c Fo(\014)325 2685 y Fp(\030)355 2693 y Fg(0)410 2715 y Fr(\001)18 b Fo(\016)488 2727 y Fn(0)544 2715 y Fu(+)g Fo(:)c(:)g(:)k Fu(+)g Fo(\014)876 2685 y Fp(\030)906 2694 y Fh(k)966 2715 y Fr(\001)g Fo(\016)1044 2727 y Fp(k)1113 2715 y Fu(und)28 b Fo(\021)e Fu(=)d Fo(\014)1485 2685 y Fp(\021)1519 2693 y Fg(0)1574 2715 y Fr(\001)18 b Fo(\015)1658 2727 y Fn(0)1714 2715 y Fu(+)g Fo(:)c(:)g(:)k Fu(+)g Fo(\014)2046 2685 y Fp(\021)2080 2694 y Fh(l)2128 2715 y Fr(\001)g Fo(\015)2212 2727 y Fp(l)2265 2715 y Fu(mit)28 b Fo(\015)g Fr(\025)23 b Fu(2)k(,)h Fo(\030)2732 2727 y Fn(0)2792 2715 y Fo(>)23 b(:)14 b(:)g(:)23 b(>)f(\030)3123 2727 y Fp(k)3192 2715 y Fu(,)28 b Fo(\021)3284 2727 y Fn(0)3344 2715 y Fo(>)23 b(:)14 b(:)g(:)23 b(>)g(\021)3681 2727 y Fp(l)3734 2715 y Fu(und)0 2856 y(0)g Fo(<)f(\016)189 2868 y Fn(0)226 2856 y Fo(;)14 b(:::;)g(\016)406 2868 y Fp(k)447 2856 y Fo(;)g(\015)527 2868 y Fn(0)564 2856 y Fo(;)g(:::;)g(\015)750 2868 y Fp(l)799 2856 y Fo(<)22 b(\014)t Fu(,)28 b(so)f(gilt:)0 3076 y Fo(\030)g(<)c(\021)49 b Fr(\()-14 b(\))439 2934 y Fl(\()506 3008 y Fo(k)26 b(<)d(l)h Fu(&)f Fr(8)p Fo(i)f Fr(\024)g Fo(k)s Fu(\()p Fo(\030)1099 3020 y Fp(i)1151 3008 y Fu(=)g Fo(\021)1279 3020 y Fp(i)1330 3008 y Fu(&)h Fo(\016)1455 3020 y Fp(i)1505 3008 y Fu(=)g Fo(\015)1636 3020 y Fp(i)1664 3008 y Fu(\))28 b(o)r(der)506 3149 y Fr(9)p Fo(p)23 b Fr(\024)g Fu(min\()p Fo(k)s(;)14 b(l)r Fu(\))28 b(mit)g Fr(8)p Fo(i)22 b(<)g(p)p Fu(\()p Fo(\030)1492 3161 y Fp(i)1543 3149 y Fu(=)h Fo(\021)1672 3161 y Fp(i)1723 3149 y Fu(&)g Fo(\016)1848 3161 y Fp(i)1898 3149 y Fu(=)g Fo(\015)2029 3161 y Fp(i)2056 3149 y Fu(\))h(&)e([)14 b Fo(\030)2272 3161 y Fp(p)2334 3149 y Fo(<)23 b(\021)2463 3161 y Fp(p)2529 3149 y Fu(o)r(der)k(\()p Fo(\030)2784 3161 y Fp(p)2846 3149 y Fu(=)c Fo(\021)2975 3161 y Fp(p)3036 3149 y Fu(&)g Fo(\016)3161 3161 y Fp(p)3222 3149 y Fo(<)g(\021)3351 3161 y Fp(p)3390 3149 y Fu(\))14 b(])0 3295 y(b\))28 b(Sei)g Fo(x)23 b Fu(=)398 3232 y Fl(P)486 3253 y Fp(k)486 3319 y(i)p Fn(=1)597 3295 y Fu(\()p Fo(n)p Fu(+1\))818 3264 y Fp(x)856 3272 y Fh(i)904 3295 y Fr(\001)c Fo(n)996 3307 y Fp(i)1051 3295 y Fu(mit)28 b Fo(x)1250 3307 y Fn(1)1311 3295 y Fo(>)23 b(:)14 b(:)g(:)23 b(>)f(x)1653 3307 y Fp(k)1722 3295 y Fu(und)28 b Fo(n)1938 3307 y Fn(1)1975 3295 y Fo(;)14 b(:::;)g(n)2168 3307 y Fp(k)2232 3295 y Fr(2)23 b(f)p Fu(1)p Fo(;)14 b(:::;)g(n)p Fr(g)p Fu(.)0 3436 y(Es)27 b(gilt)h Fr(S)315 3448 y Fp(n)360 3436 y Fu(\()p Fo(x)p Fu(\))c(=)583 3373 y Fl(P)670 3394 y Fp(k)670 3461 y(i)p Fn(=1)782 3436 y Fu(\()p Fo(n)p Fu(+2\))1003 3406 y Fi(S)1044 3414 y Fh(n)1084 3406 y Fn(\()p Fp(x)1148 3414 y Fh(i)1174 3406 y Fn(\))1222 3436 y Fr(\001)19 b Fo(n)1314 3448 y Fp(i)1341 3436 y Fu(,)28 b(und)g(nac)n(h)f(a\))h Fr(S)1906 3448 y Fp(n)1951 3436 y Fu(\()p Fo(x)2030 3448 y Fn(1)2068 3436 y Fu(\))c Fo(>)e(:)14 b(:)g(:)23 b(>)g Fr(S)2469 3448 y Fp(n)2514 3436 y Fu(\()p Fo(x)2593 3448 y Fp(k)2635 3436 y Fu(\).)0 3577 y(F)-7 b(olglic)n(h)27 b Fo(\022)347 3589 y Fp(n)p Fn(+1)476 3577 y Fr(S)526 3589 y Fp(n)572 3577 y Fu(\()p Fo(x)p Fu(\))d(=)e Fo(!)849 3547 y Fp(\022)881 3555 y Fh(n)p Fg(+1)992 3547 y Fi(S)1033 3555 y Fh(n)1074 3547 y Fn(\()p Fp(x)1138 3555 y Fg(1)1170 3547 y Fn(\))1218 3577 y Fr(\001)d Fo(n)1310 3589 y Fn(1)1365 3577 y Fu(+)f Fo(:)c(:)g(:)19 b Fu(+)f Fo(!)1702 3547 y Fp(\022)1734 3555 y Fh(n)p Fg(+1)1845 3547 y Fi(S)1886 3555 y Fh(n)1926 3547 y Fn(\()p Fp(x)1990 3556 y Fh(k)2026 3547 y Fn(\))2075 3577 y Fr(\001)g Fo(n)2166 3589 y Fp(k)2230 3530 y Fn(IH)2234 3577 y Fu(=)27 b Fo(!)2381 3547 y Fp(\022)2413 3555 y Fh(n)2453 3547 y Fn(\()p Fp(x)2517 3555 y Fg(1)2549 3547 y Fn(\))2597 3577 y Fr(\001)19 b Fo(n)2689 3589 y Fn(1)2744 3577 y Fu(+)f Fo(:)c(:)g(:)19 b Fu(+)f Fo(!)3081 3547 y Fp(\022)3113 3555 y Fh(n)3153 3547 y Fn(\()p Fp(x)3217 3556 y Fh(k)3253 3547 y Fn(\))3301 3577 y Fr(\001)h Fo(n)3393 3589 y Fp(k)3456 3577 y Fu(=)k Fo(\022)3583 3589 y Fp(n)3628 3577 y Fu(\()p Fo(x)p Fu(\).)0 3743 y(c\))28 b(folgt)f(aus)g(7.2a,d.)0 3959 y(Sei)h(n)n(un)f Fo(a)341 3971 y Fp(n)410 3959 y Fu(:=)22 b(GS)q(\()p Fo(a;)14 b(n)p Fu(\).)37 b(Dann)28 b(gilt:)0 4100 y Fo(a)44 4112 y Fp(n)112 4100 y Fo(>)23 b Fu(0)45 b Fr(\))i(S)467 4112 y Fp(n)512 4100 y Fu(\()p Fo(a)588 4112 y Fp(n)633 4100 y Fu(\))24 b Fo(>)e Fu(0)46 b Fr(\))g Fo(\022)1032 4112 y Fp(n)p Fn(+1)1161 4100 y Fu(\()p Fo(a)1237 4112 y Fp(n)p Fn(+1)1367 4100 y Fu(\))23 b(=)g Fo(\022)1549 4112 y Fp(n)p Fn(+1)1678 4100 y Fu(\()p Fr(S)1760 4112 y Fp(n)1806 4100 y Fu(\()p Fo(a)1882 4112 y Fp(n)1927 4100 y Fu(\))c Fr(\000)f Fu(1\))2158 4039 y Fn(7)p Fp(:)p Fn(1c)2183 4100 y Fo(<)48 b(\022)2335 4112 y Fp(n)p Fn(+1)2464 4100 y Fu(\()p Fr(S)2546 4112 y Fp(n)2592 4100 y Fu(\()p Fo(a)2668 4112 y Fp(n)2714 4100 y Fu(\)\))2801 4053 y Fn(7)p Fp(:)p Fn(1b)2830 4100 y Fu(=)k Fo(\022)2986 4112 y Fp(n)3031 4100 y Fu(\()p Fo(a)3107 4112 y Fp(n)3152 4100 y Fu(\).)0 4366 y Ft(De\014nition)i Fu(P)526 4378 y Fp(n)571 4366 y Fu(\(0\))23 b(:=)g(0)p Fo(;)36 b Fu(P)969 4378 y Fp(n)1014 4366 y Fu(\()p Fo(\013)19 b Fu(+)f(1\))23 b(:=)g Fo(\013;)60 b Fu(P)1602 4378 y Fp(n)1647 4366 y Fu(\()p Fo(\025)p Fu(\))24 b(:=)f(P)1951 4378 y Fp(n)1995 4366 y Fu(\()p Fo(\025)p Fu([)p Fo(n)p Fu(]\).)0 4557 y Ft(Lemma)30 b(7.2)c Fu(Sei)i Fo(n)23 b(>)f Fu(0.)0 4698 y(a\))27 b Fo(\013)d(>)f Fu(0)f Fr(\))h Fu(P)493 4710 y Fp(n)538 4698 y Fu(\()p Fo(\013)p Fu(\))h Fo(<)f(\013)p Fu(,)0 4839 y(b\))28 b(P)163 4851 y Fp(n)208 4839 y Fu(\()p Fo(\013)19 b Fu(+)f Fo(\014)t Fu(\))24 b(=)e Fo(\013)d Fu(+)f(P)801 4851 y Fp(n)846 4839 y Fu(\()p Fo(\014)t Fu(\),)29 b(falls)e Fo(N)-5 b(F)12 b Fu(\()p Fo(\013;)i(\014)t Fu(\),)0 4980 y(c\))28 b(P)154 4992 y Fp(n)199 4980 y Fu(\()p Fo(!)286 4950 y Fp(\013)333 4980 y Fu(\))23 b(=)g(P)533 4992 y Fp(n)577 4980 y Fu(\()p Fo(!)664 4950 y Fn(P)709 4958 y Fh(n)750 4950 y Fn(\()p Fp(\013)p Fn(\))867 4980 y Fr(\001)c Fu(\()p Fo(n)p Fu(+1\)\),)27 b(falls)h Fo(\013)23 b(>)g Fu(0,)0 5121 y(d\))28 b(P)163 5133 y Fp(n)208 5121 y Fu(\()p Fo(\022)279 5133 y Fp(n)324 5121 y Fu(\()p Fo(x)p Fu(\)\))d(=)d Fo(\022)618 5133 y Fp(n)663 5121 y Fu(\()p Fo(x)p Fr(\000)761 5104 y(\001)812 5121 y Fu(1\),)0 5262 y(e\))28 b Fo(n)23 b Fr(\025)f Fu(1)46 b Fr(\))g Fo(\022)513 5274 y Fp(n)558 5262 y Fu(GS)q(\()p Fo(a;)14 b(n)p Fu(\))23 b(=)f(P)1032 5274 y Fp(n)1077 5262 y Fo(:::)p Fu(P)1203 5274 y Fn(2)1240 5262 y Fo(\022)1279 5274 y Fn(1)1316 5262 y Fu(\()p Fo(a)p Fu(\).)1908 5539 y(44)p eop %%Page: 45 46 45 45 bop 0 83 a Fu(Bew)n(eis:)0 217 y(a\),b\))28 b(klar.)0 352 y(c\))g(Induktion)g(nac)n(h)f Fo(\013)p Fu(:)0 486 y(1.)36 b(P)158 498 y Fp(n)203 486 y Fu(\()p Fo(!)290 456 y Fp(\013)p Fn(+1)421 486 y Fu(\))24 b(=)e(P)621 498 y Fp(n)666 486 y Fu(\()p Fo(!)753 456 y Fp(\013)819 486 y Fr(\001)c Fu(\()p Fo(n)p Fu(+1\)\))23 b(=)g(P)1281 498 y Fp(n)1326 486 y Fu(\()p Fo(!)1413 456 y Fn(P)1458 464 y Fh(n)1498 456 y Fn(\()p Fp(\013)p Fn(+1\))1700 486 y Fr(\001)18 b Fu(\()p Fo(n)p Fu(+1\)\).)0 621 y(2.)36 b(P)158 633 y Fp(n)203 621 y Fu(\()p Fo(!)290 591 y Fp(\025)334 621 y Fu(\))23 b(=)g(P)534 633 y Fp(n)578 621 y Fu(\()p Fo(!)665 591 y Fp(\025)p Fn([)p Fp(n)p Fn(])787 621 y Fu(\))843 574 y Fn(IH)847 621 y Fu(=)k(P)996 633 y Fp(n)1041 621 y Fu(\()p Fo(!)1128 591 y Fn(P)1173 599 y Fh(n)1213 591 y Fn(\()p Fp(\025)p Fn([)p Fp(n)p Fn(]\))1405 621 y Fr(\001)19 b Fu(\()p Fo(n)p Fu(+)o(1\)\))24 b(=)e(P)1867 633 y Fp(n)1912 621 y Fu(\()p Fo(!)1999 591 y Fn(P)2044 599 y Fh(n)2084 591 y Fn(\()p Fp(\025)p Fn(\))2198 621 y Fr(\001)d Fu(\()p Fo(n)f Fu(+)g(1\)\).)0 773 y(d\))28 b(Induktion)g(nac)n(h)f Fo(x)p Fu(:)37 b(F)r(\177)-44 b(ur)28 b Fo(x)23 b Fu(=)g(0)k(ist)h(die)g(Behauptung)f(trivial.)0 908 y(Sei)h(also)e(0)d Fo(<)f(x)i Fu(=)611 845 y Fl(P)698 866 y Fp(k)698 933 y(i)p Fn(=1)810 908 y Fu(\()p Fo(n)p Fu(+1\))1031 878 y Fp(x)1069 886 y Fh(i)1117 908 y Fr(\001)18 b Fo(n)1208 920 y Fp(i)1264 908 y Fu(mit)28 b Fo(x)1463 920 y Fn(1)1524 908 y Fo(>)22 b(:)14 b(:)g(:)23 b(>)g(x)1866 920 y Fp(k)1935 908 y Fu(und)28 b Fo(n)2151 920 y Fn(0)2188 908 y Fo(;)14 b(:::;)g(n)2381 920 y Fp(k)2445 908 y Fr(2)23 b(f)p Fu(1)p Fo(;)14 b(:::;)g(n)p Fr(g)p Fu(.)0 1060 y(Dann)28 b Fo(x)19 b Fr(\000)f Fu(1)k(=)526 998 y Fl(P)614 1019 y Fp(k)q Fi(\000)p Fn(1)614 1085 y Fp(i)p Fn(=1)739 1060 y Fu(\()p Fo(n)p Fu(+1\))960 1030 y Fp(x)998 1038 y Fh(i)1046 1060 y Fr(\001)d Fo(n)1138 1072 y Fp(i)1184 1060 y Fu(+)f(\()p Fo(n)p Fu(+1\))1488 1030 y Fp(x)1526 1039 y Fh(k)1584 1060 y Fr(\001)g Fu(\()p Fo(n)1707 1072 y Fp(k)1748 1060 y Fr(\000)p Fu(1\))g(+)g(\()p Fo(n)p Fu(+1\))2209 1030 y Fp(x)2247 1039 y Fh(k)2283 1030 y Fi(\000)p Fn(1)2390 1060 y Fr(\001)h Fo(n)f Fu(+)g Fo(:)c(:)g(:)k Fu(+)g(\()p Fo(n)p Fu(+1\))3002 1030 y Fn(0)3058 1060 y Fr(\001)g Fo(n)28 b Fu(und)g(folglic)n(h)0 1195 y Fo(\022)39 1207 y Fp(n)84 1195 y Fu(\()p Fo(x)19 b Fr(\000)f Fu(1\))23 b(=)g Fo(!)505 1165 y Fp(\022)537 1173 y Fh(n)577 1165 y Fn(\()p Fp(x)641 1173 y Fg(1)673 1165 y Fn(\))721 1195 y Fr(\001)c Fo(n)813 1207 y Fn(1)868 1195 y Fu(+)f Fo(:)c(:)g(:)19 b Fu(+)f Fo(!)1205 1165 y Fp(\022)1237 1173 y Fh(n)1277 1165 y Fn(\()p Fp(x)1341 1174 y Fh(k)1377 1165 y Fn(\))1425 1195 y Fr(\001)h Fu(\()p Fo(n)1549 1207 y Fp(k)1590 1195 y Fr(\000)o Fu(1\))g(+)f Fo(!)1885 1165 y Fp(\022)1917 1173 y Fh(n)1957 1165 y Fn(\()p Fp(x)2021 1174 y Fh(k)2056 1165 y Fi(\000)p Fn(1\))2190 1195 y Fr(\001)g Fo(n)h Fu(+)f Fo(:)c(:)g(:)k Fu(+)g Fo(!)2636 1165 y Fp(\022)2668 1173 y Fh(n)2708 1165 y Fn(\(0\))2816 1195 y Fr(\001)g Fo(n)2930 1148 y Fn(HS)2941 1195 y Fu(=)0 1329 y Fo(!)55 1299 y Fp(\022)87 1307 y Fh(n)127 1299 y Fn(\()p Fp(x)191 1307 y Fg(1)223 1299 y Fn(\))272 1329 y Fr(\001)g Fo(n)363 1341 y Fn(1)419 1329 y Fu(+)g Fo(:)c(:)g(:)k Fu(+)g Fo(!)755 1299 y Fp(\022)787 1307 y Fh(n)827 1299 y Fn(\()p Fp(x)891 1308 y Fh(k)927 1299 y Fn(\))975 1329 y Fr(\001)h Fu(\()p Fo(n)1099 1341 y Fp(k)1140 1329 y Fr(\000)o Fu(1\))g(+)f(P)1437 1341 y Fp(n)1481 1329 y Fu(\()p Fo(!)1568 1299 y Fp(\022)1600 1307 y Fh(n)1641 1299 y Fn(\()p Fp(x)1705 1308 y Fh(k)1741 1299 y Fn(\))1771 1329 y Fu(\))1826 1275 y Fn(b\))p Fp(;)p Fn(7)p Fp(:)p Fn(1a)1894 1329 y Fu(=)91 b(P)2107 1341 y Fp(n)2152 1329 y Fu(\()p Fo(!)2239 1299 y Fp(\022)2271 1307 y Fh(n)2311 1299 y Fn(\()p Fp(x)2375 1307 y Fg(1)2407 1299 y Fn(\))2456 1329 y Fr(\001)18 b Fo(n)2547 1341 y Fn(1)2603 1329 y Fu(+)g Fo(:)c(:)g(:)k Fu(+)g Fo(!)2939 1299 y Fp(\022)2971 1307 y Fh(n)3011 1299 y Fn(\()p Fp(x)3075 1308 y Fh(k)3111 1299 y Fn(\))3160 1329 y Fr(\001)g Fo(n)3251 1341 y Fp(k)3292 1329 y Fu(\))23 b(=)g(P)3492 1341 y Fp(n)3537 1329 y Fu(\()p Fo(\022)3608 1341 y Fp(n)3653 1329 y Fu(\()p Fo(x)p Fu(\)\).)0 1500 y(HS:)28 b Fo(z)e(<)d(x)47 b Fr(\))f Fu(P)592 1512 y Fp(n)636 1500 y Fu(\()p Fo(!)723 1470 y Fp(\022)755 1478 y Fh(n)796 1470 y Fn(\()p Fp(z)r Fn(\))886 1500 y Fu(\))23 b(=)g Fo(!)1084 1470 y Fp(\022)1116 1478 y Fh(n)1156 1470 y Fn(\()p Fp(z)r Fi(\000)p Fn(1\))1349 1500 y Fr(\001)c Fo(n)f Fu(+)g Fo(:)c(:)g(:)19 b Fu(+)f Fo(!)1796 1470 y Fp(\022)1828 1478 y Fh(n)1868 1470 y Fn(\(0\))1975 1500 y Fr(\001)h Fo(n)p Fu(.)0 1653 y(Bew)n(eis)27 b(durc)n(h)g(Neb)r(eninduktion)h(nac)n(h)f Fo(z)t Fu(:)0 1787 y(1.)36 b(P)158 1799 y Fp(n)203 1787 y Fu(\()p Fo(!)290 1757 y Fp(\022)322 1765 y Fh(n)362 1757 y Fn(\(0\))452 1787 y Fu(\))23 b(=)g(P)652 1799 y Fp(n)696 1787 y Fu(\(1\))g(=)g(0.)0 1921 y(2.)57 b Fo(z)t Fu(+)n(1)34 b Fo(<)g(x)p Fu(:)51 b(P)581 1933 y Fp(n)626 1921 y Fu(\()p Fo(!)713 1891 y Fp(\022)745 1899 y Fh(n)785 1891 y Fn(\()p Fp(z)r Fn(+1\))959 1921 y Fu(\))1030 1867 y Fn(c\))1026 1921 y Fu(=)33 b(P)1181 1933 y Fp(n)1226 1921 y Fu(\()p Fo(!)1313 1891 y Fn(P)1358 1899 y Fh(n)1398 1891 y Fp(\022)1430 1899 y Fh(n)1471 1891 y Fn(\()p Fp(z)r Fn(+1\))1668 1921 y Fr(\001)23 b Fu(\()p Fo(n)p Fu(+1\)\))2001 1874 y Fn(IH)2005 1921 y Fu(=)38 b(P)2165 1933 y Fp(n)2210 1921 y Fu(\()p Fo(!)2297 1891 y Fp(\022)2329 1899 y Fh(n)2369 1891 y Fn(\()p Fp(z)r Fn(\))2482 1921 y Fr(\001)23 b Fu(\()p Fo(n)p Fu(+1\)\))2816 1867 y Fn(b\))2815 1921 y Fu(=)34 b Fo(!)2969 1891 y Fp(\022)3001 1899 y Fh(n)3041 1891 y Fn(\()p Fp(z)r Fn(\))3154 1921 y Fr(\001)23 b Fo(n)g Fu(+)f(P)3417 1933 y Fp(n)3462 1921 y Fu(\()p Fo(!)3549 1891 y Fp(\022)3581 1899 y Fh(n)3621 1891 y Fn(\()p Fp(z)r Fn(\))3711 1921 y Fu(\))3778 1874 y Fn(NIH)3807 1921 y Fu(=)0 2056 y Fo(!)55 2026 y Fp(\022)87 2034 y Fh(n)127 2026 y Fn(\()p Fp(z)r Fn(\))236 2056 y Fr(\001)c Fo(n)g Fu(+)g Fo(!)483 2026 y Fp(\022)515 2034 y Fh(n)555 2026 y Fn(\()p Fp(z)r Fi(\000)p Fn(1\))749 2056 y Fr(\001)g Fo(n)h Fu(+)f Fo(:)c(:)g(:)k Fu(+)g Fo(!)1195 2026 y Fp(\022)1227 2034 y Fh(n)1267 2026 y Fn(\(0\))1375 2056 y Fr(\001)g Fo(n)p Fu(.)0 2208 y(e\))41 b(Sei)h Fo(a)302 2220 y Fp(i)375 2208 y Fu(:=)j(GS\()p Fo(a;)14 b(i)p Fu(\).)78 b(Dann)41 b Fo(\022)1171 2220 y Fp(n)p Fn(+1)1301 2208 y Fu(\()p Fo(a)1377 2220 y Fp(n)p Fn(+1)1506 2208 y Fu(\))1584 2161 y Fn(Def)1604 2208 y Fu(=)65 b Fo(\022)1773 2220 y Fp(n)p Fn(+1)1903 2208 y Fu(\()p Fr(S)1985 2220 y Fp(n)2030 2208 y Fu(\()p Fo(a)2106 2220 y Fp(n)2152 2208 y Fu(\))p Fr(\000)2203 2192 y(\001)2253 2208 y Fu(1\))2373 2161 y Fn(7)p Fp(:)p Fn(2d)2402 2208 y Fu(=)74 b(P)2598 2220 y Fp(n)p Fn(+1)2727 2208 y Fo(\022)2766 2220 y Fp(n)p Fn(+1)2895 2208 y Fr(S)2945 2220 y Fp(n)2990 2208 y Fu(\()p Fo(a)3066 2220 y Fp(n)3112 2208 y Fu(\))3190 2161 y Fn(7)p Fp(:)p Fn(1b)3219 2208 y Fu(=)g(P)3415 2220 y Fp(n)p Fn(+1)3544 2208 y Fo(\022)3583 2220 y Fp(n)3628 2208 y Fu(\()p Fo(a)3704 2220 y Fp(n)3749 2208 y Fu(\))3827 2161 y Fn(IH)3831 2208 y Fu(=)0 2343 y(P)57 2355 y Fp(n)p Fn(+1)186 2343 y Fu(P)243 2355 y Fp(n)287 2343 y Fo(:::)p Fu(P)413 2355 y Fn(2)450 2343 y Fo(\022)489 2355 y Fn(1)527 2343 y Fu(\()p Fo(a)p Fu(\).)0 2532 y Ft(De\014nition)54 b Fo(h)517 2544 y Fn(0)554 2532 y Fu(\()p Fo(n)p Fu(\))24 b(:=)f Fo(n;)96 b(h)1020 2544 y Fp(\013)p Fn(+1)1152 2532 y Fu(\()p Fo(n)p Fu(\))23 b(:=)g Fo(h)1448 2544 y Fp(\013)1495 2532 y Fu(\()p Fo(n)18 b Fu(+)g(1\))p Fo(;)97 b(h)1920 2544 y Fp(\025)1964 2532 y Fu(\()p Fo(n)p Fu(\))23 b(:=)g Fo(h)2260 2547 y Fp(\025)p Fn([)p Fp(n)p Fn(])2382 2532 y Fu(\()p Fo(n)p Fu(\).)0 2684 y Ft(Lemma)30 b(7.3)0 2819 y Fu(a\))d Fo(H)170 2831 y Fp(\013)218 2819 y Fu(\()p Fo(n)p Fu(\))c Fo(<)g(h)491 2831 y Fp(\013)538 2819 y Fu(\()p Fo(n)p Fu(+1\),)0 2953 y(b\))28 b Fo(\013)c(>)e Fu(0)46 b Fr(\))g Fo(h)535 2965 y Fp(\013)582 2953 y Fu(\()p Fo(n)p Fu(\))24 b(=)e Fo(h)855 2968 y Fn(P)900 2976 y Fh(n)940 2968 y Fn(\()p Fp(\013)p Fn(\))1040 2953 y Fu(\()p Fo(n)c Fu(+)g(1\).)0 3088 y(c\))28 b Fo(h)145 3100 y Fp(\013)192 3088 y Fu(\()p Fo(n)p Fu(\))23 b(=)g(min)p Fr(f)p Fo(k)j Fr(\025)d Fo(n)f Fu(:)37 b(P)943 3100 y Fp(k)q Fi(\000)p Fn(1)1069 3088 y Fo(:::)p Fu(P)1195 3100 y Fp(n)1240 3088 y Fu(\()p Fo(\013)p Fu(\))24 b(=)e(0)p Fr(g)p Fu(.)0 3240 y(Pro)r(of:)0 3375 y(a\))27 b(siehe)h Fr(x)p Fu(6.)0 3509 y(b\))g(Induktion)g(nac)n(h)f Fo(\013)p Fu(:)0 3644 y(1.)36 b Fo(\013)24 b Fu(=)f Fo(\014)f Fu(+)c(1:)37 b Fo(h)568 3656 y Fp(\013)615 3644 y Fu(\()p Fo(n)p Fu(\))23 b(=)g Fo(h)888 3656 y Fp(\014)933 3644 y Fu(\()p Fo(n)18 b Fu(+)g(1\))23 b(=)g Fo(h)1349 3659 y Fn(P)1394 3667 y Fh(n)1434 3659 y Fn(\()p Fp(\013)p Fn(\))1533 3644 y Fu(\()p Fo(n)c Fu(+)f(1\).)0 3778 y(2.)36 b Fo(\013)24 b Fr(2)f Fo(Lim)p Fu(:)36 b Fo(h)522 3790 y Fp(\013)569 3778 y Fu(\()p Fo(n)p Fu(\))24 b(=)f Fo(h)843 3793 y Fp(\013)p Fn([)p Fp(n)p Fn(])968 3778 y Fu(\()p Fo(n)p Fu(\))h(=)e Fo(h)1241 3793 y Fn(P)1286 3801 y Fh(n)1327 3793 y Fn(\()p Fp(\013)p Fn([)p Fp(n)p Fn(]\))1504 3778 y Fu(\()p Fo(n)d Fu(+)f(1\))23 b(=)g Fo(h)1921 3793 y Fn(P)1966 3801 y Fh(n)2006 3793 y Fn(\()p Fp(\013)p Fn(\))2105 3778 y Fu(\()p Fo(n)18 b Fu(+)g(1\).)0 3912 y(c\))28 b(F)r(\177)-44 b(ur)27 b Fo(k)f Fu(=)d(min)p Fr(f)p Fo(i)f Fr(\025)h Fo(n)g Fu(:)37 b(P)923 3924 y Fp(i)p Fi(\000)p Fn(1)1035 3912 y Fo(:::)p Fu(P)1161 3924 y Fp(n)1206 3912 y Fu(\()p Fo(\013)p Fu(\))24 b(=)f(0)p Fr(g)j Fu(gilt)i(o\013en)n (bar)0 4047 y Fo(h)48 4059 y Fp(\013)95 4047 y Fu(\()p Fo(n)p Fu(\))c(=)e Fo(h)368 4062 y Fn(P)413 4070 y Fh(n)453 4062 y Fn(\()p Fp(\013)p Fn(\))553 4047 y Fu(\()p Fo(n)c Fu(+)g(1\))23 b(=)g Fo(h)969 4062 y Fn(P)1014 4070 y Fh(n)p Fg(+1)1125 4062 y Fn(P)1170 4070 y Fh(n)1210 4062 y Fn(\()p Fp(\013)p Fn(\))1309 4047 y Fu(\()p Fo(n)c Fu(+)f(2\))23 b(=)f Fo(:)14 b(:)g(:)23 b Fu(=)g Fo(h)1933 4062 y Fn(P)1978 4071 y Fh(k)q Ff(\000)p Fg(1)2087 4062 y Fp(:::)p Fn(P)2192 4070 y Fh(n)2231 4062 y Fn(\()p Fp(\013)p Fn(\))2331 4047 y Fu(\()p Fo(k)s Fu(\))g(=)g Fo(h)2600 4059 y Fn(0)2637 4047 y Fu(\()p Fo(k)s Fu(\))g(=)g Fo(k)s Fu(.)0 4199 y Ft(Satz)33 b(7.4)0 4334 y Fr(Z)d(6`)23 b(8)p Fo(x)p Fr(9)p Fo(y)s Fu([)14 b(GS)o(\()p Fo(x;)g(y)s Fu(\))24 b(=)e(0)14 b(].)0 4486 y(Pro)r(of)0 4621 y(Sei)28 b Fo(e)p Fu(\(0\))23 b(:=)f(1)p Fo(;)37 b(e)p Fu(\()p Fo(m)18 b Fu(+)g(1\))23 b(:=)g(2)1009 4591 y Fp(e)p Fn(\()p Fp(m)p Fn(\))1155 4621 y Fu(.)0 4755 y(Annahme:)37 b Fr(Z)30 b(`)23 b(8)p Fo(x)p Fr(9)p Fo(y)s Fu([)14 b(GS)o(\()p Fo(x;)g(y)s Fu(\))24 b(=)f(0)14 b(].)0 4890 y Fr(\))46 b(Z)30 b(`)23 b(8)p Fo(x)p Fr(9)p Fo(y)s Fu([)14 b(GS)o(\()p Fo(e)p Fu(\()p Fo(x)p Fu(\))20 b(+)e Fo(x;)c(y)s Fu(\))23 b(=)g(0)14 b(])1304 4843 y Fn(6)p Fp(:)p Fn(12)1322 4890 y Fr(\))64 b(9)p Fo(\013)24 b(<)f(")1719 4902 y Fn(0)1756 4890 y Fr(8)p Fo(m)p Fr(9)p Fo(n)f(<)g(H)2150 4902 y Fp(\013)2198 4890 y Fu(\()p Fo(m)p Fu(\)[)14 b(GS\()p Fo(e)p Fu(\()p Fo(m)p Fu(\))19 b(+)f Fo(m;)c(n)p Fu(\))23 b(=)g(0)14 b(])45 b Fr(\))0 5024 y(\))h(9)p Fo(p;)14 b(n)p Fu([)g Fo(n)23 b(<)f(H)570 5036 y Fp(!)612 5044 y Fh(p)651 5024 y Fu(\()p Fo(p)p Fu(\))h(&)g(GS\()p Fo(e)p Fu(\()p Fo(p)p Fu(\))c(+)f Fo(p;)c(n)p Fu(\))23 b(=)g(0)14 b(].)36 b(\()p Fo(p)23 b Fr(\025)g Fo(N)9 b Fu(\()p Fo(\013)p Fu(\))28 b(mit)g Fo(\013)c(<)e(!)2442 5036 y Fp(p)2481 5024 y Fu(;)27 b(v)n(ergl.)f(6.9c\))0 5177 y(GS\()p Fo(e)p Fu(\()p Fo(p)p Fu(\))s(+)s Fo(p;)14 b(n)p Fu(\))22 b(=)h(0)46 b Fr(\))g Fo(n)23 b Fr(\025)f Fu(2)h(&)g(P)1217 5189 y Fp(n)1261 5177 y Fo(:::)p Fu(P)1387 5189 y Fn(2)1424 5177 y Fu(\()p Fo(!)1508 5189 y Fp(p)1549 5177 y Fu(+)s Fo(p)p Fu(\))g(=)f Fo(\022)1840 5189 y Fp(n)1886 5177 y Fu(GS\()p Fo(e)p Fu(\()p Fo(p)p Fu(\))s(+)s Fo(p;)14 b(n)p Fu(\))22 b(=)h(0)45 b Fr(\))h Fo(H)2801 5189 y Fp(!)2843 5197 y Fh(p)2878 5189 y Fn(+)p Fp(p)2968 5177 y Fu(\(1\))23 b Fo(<)f(h)3232 5189 y Fp(!)3274 5197 y Fh(p)3309 5189 y Fn(+)p Fp(p)3398 5177 y Fu(\(2\))h Fr(\024)g Fo(n)p Fu(+)o(1)46 b Fr(\))0 5311 y Fo(H)69 5323 y Fp(!)111 5331 y Fh(p)150 5311 y Fu(\()p Fo(p)p Fu(\))23 b(=)g Fo(H)436 5323 y Fp(!)478 5331 y Fh(p)513 5323 y Fn(+)p Fp(p)602 5311 y Fu(\(0\))g Fo(<)g(H)888 5323 y Fp(!)930 5331 y Fh(p)964 5323 y Fn(+)p Fp(p)1054 5311 y Fu(\(1\))g Fr(\024)g Fo(n)p Fu(.)36 b Fs(Widerspruc)n(h)p Fu(.)1908 5539 y(45)p eop %%Page: 46 47 46 46 bop 0 83 a Fq(The)27 b(P)n(aris-Harrington)f(Result)0 249 y Fu(\(The)32 b(pro)r(of)e(giv)n(en)h(here)g(is)g(based)f(on)h(M.)h (Lo)r(ebl)f(and)g(J.)g(Ne)l(\024)-37 b(set)l(\024)g(ril,)30 b Fm(A)n(n)j(unpr)l(ovable)h(R)l(amsey-typ)l(e)f(The)l(or)l(em)p Fu(,)g(Rep)r(ort)0 390 y(No)27 b(89598-OR,)e(Septem)n(b)r(er)j(1989.\)) 0 556 y Ft(Abbreviations:)0 697 y Fu(Let)g Fo(k)s(;)14 b(m;)g(n;)g(r)25 b Fr(2)e Fu(I)-14 b(N)14 b(\(=)23 b Fo(!)s Fu(\),)28 b Fo(\024)g Fu(a)f(cardinal,)f Fo(N)37 b Fu(a)27 b(set.)0 863 y([)p Fo(N)9 b Fu(])122 833 y Fp(m)208 863 y Fu(:=)23 b Fr(f)p Fo(X)29 b Fr(\022)22 b Fo(N)32 b Fu(:)23 b Fo(car)r(d)p Fu(\()p Fo(X)7 b Fu(\))24 b(=)e Fo(m)p Fr(g)0 1029 y Fu(Let)28 b Fo(f)36 b Fu(b)r(e)28 b(a)f(function)h(with)g(dom\()p Fo(f)9 b Fu(\))24 b(=)e([)p Fo(N)9 b Fu(])1426 999 y Fp(m)1489 1029 y Fu(:)0 1171 y Fo(X)34 b Fu(is)27 b Fo(f)9 b Fu(-homogeneous)49 b Fr(\()-14 b(\))69 b(;)23 b(6)p Fu(=)f Fo(X)30 b Fr(\022)22 b Fo(N)32 b Fu(&)23 b Fo(f)9 b Fb(\026)o Fu([)p Fo(X)e Fu(])1747 1140 y Fp(m)1837 1171 y Fu(constan)n(t.)0 1337 y Fo(N)32 b Fr(\000)-14 b(!)23 b Fu(\()p Fo(\024)p Fu(\))368 1307 y Fp(m)368 1357 y(r)477 1337 y Fu(:)p Fr(,)46 b(8)p Fo(f)31 b Fu(:)23 b([)p Fo(N)9 b Fu(])916 1307 y Fp(m)1002 1337 y Fr(!)23 b Fo(r)r Fr(9)p Fo(X)7 b Fu(\()p Fo(X)35 b(f)9 b Fu(-homogeneous)20 b(&)j Fo(car)r(d)p Fu(\()p Fo(X)7 b Fu(\))24 b Fr(\025)e Fo(\024)p Fu(\))0 1503 y Fo(N)149 1456 y Fi(\003)99 1503 y Fr(\000)-14 b(!)23 b Fu(\()p Fo(\024)p Fu(\))368 1473 y Fp(m)368 1523 y(r)477 1503 y Fu(:)p Fr(,)46 b(8)p Fo(f)31 b Fu(:)23 b([)p Fo(N)9 b Fu(])916 1473 y Fp(m)1002 1503 y Fr(!)23 b Fo(r)r Fr(9)p Fo(X)7 b Fu(\()p Fo(X)35 b(f)9 b Fu(-homogeneous)20 b(&)j Fo(car)r(d)p Fu(\()p Fo(X)7 b Fu(\))24 b Fr(\025)e Fu(max)p Fr(f)p Fo(\024;)14 b Fu(min)o(\()p Fo(X)7 b Fu(\))p Fr(g)p Fu(\))166 b(\(for)27 b Fo(N)32 b Fr(\022)23 b Fu(I)-14 b(N\))0 1669 y Ft(Ramsey)30 b(Theorem)54 b Fr(8)p Fo(m;)14 b(r)25 b Fr(2)e Fo(!)s Fu(\()14 b Fo(!)26 b Fr(\000)-14 b(!)23 b Fu(\()p Fo(!)s Fu(\))1542 1639 y Fp(m)1542 1689 y(r)1619 1669 y Fu(\))0 1835 y Ft(Finite)31 b(Ramsey)f(Theorem)54 b Fr(8)p Fo(m;)14 b(r)n(;)g(\024)22 b Fr(2)h Fo(!)s Fr(9)p Fo(N)32 b Fr(2)23 b Fo(!)s Fu(\()14 b Fo(N)32 b Fr(\000)-14 b(!)23 b Fu(\()p Fo(\024)p Fu(\))2193 1805 y Fp(m)2193 1855 y(r)2270 1835 y Fu(\))0 2001 y Ft(PH)55 b Fr(8)p Fo(m;)14 b(r)n(;)g(\024)22 b Fr(2)i Fo(!)s Fr(9)p Fo(N)31 b Fr(2)24 b Fo(!)s Fu(\()14 b Fo(N)1100 1954 y Fi(\003)1050 2001 y Fr(\000)-14 b(!)23 b Fu(\()p Fo(\024)p Fu(\))1319 1971 y Fp(m)1319 2021 y(r)1396 2001 y Fu(\))0 2167 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 2333 y(Pro)r(of)f(of)i(PH:)0 2474 y(Let)g Fo(m;)14 b(r)n(;)g(\024)22 b Fr(2)i Fo(!)30 b Fu(b)r(e)e(\014xed.)37 b(T)-7 b(o)27 b(pro)n(v)n(e:)36 b Fr(9)p Fo(n)23 b Fr(2)g Fo(!)s Fr(8)p Fo(f)e Fr(:)p Fu(\010\()p Fo(f)t(;)14 b(n)p Fu(\))0 2615 y(where)55 b(\010\()p Fo(f)t(;)14 b(n)p Fu(\))46 b(:)p Fr(,)g Fo(f)32 b Fu(:)23 b([)p Fo(n)p Fu(])937 2585 y Fp(m)1023 2615 y Fr(!)g Fo(r)j Fu(&)d Fr(8)p Fo(X)7 b Fu(\()p Fo(X)32 b(f)9 b Fu(-hom)50 b Fr(\))c Fo(car)r(d)p Fu(\()p Fo(X)7 b Fu(\))24 b Fo(<)f Fu(max)o Fr(f)p Fo(\024;)14 b Fu(min\()p Fo(X)7 b Fu(\))p Fr(g)p Fu(\))27 b(.)0 2781 y(Assumption:)37 b Fr(8)p Fo(n)p Fr(9)p Fo(f)22 b Fu(\010\()p Fo(f)t(;)14 b(n)p Fu(\).)0 2922 y(By)27 b(K\177)-42 b(onig's)27 b(Lemma)g(there)g(is)h(a)f(function)h Fo(f)1467 2892 y Fi(\003)1505 2922 y Fu([)p Fo(!)s Fu(])1606 2892 y Fp(m)1692 2922 y Fr(!)23 b Fo(r)30 b Fu(suc)n(h)e(that)0 3064 y(\(+\))g Fr(8)p Fo(n)14 b Fu(\010\()p Fo(f)410 3033 y Fi(\003)447 3064 y Fb(\026)o Fu([)p Fo(n)p Fu(])577 3033 y Fp(m)640 3064 y Fo(;)g(n)p Fu(\))28 b(.)0 3230 y(By)34 b(Ramsey's)f(Theorem)g(there)h(exists)f(an)h(in\014nite)h Fo(f)1783 3199 y Fi(\003)1820 3230 y Fu(-homogeneous)d(set)i Fo(X)40 b Fr(\022)33 b Fo(!)s Fu(.)56 b(W)-7 b(e)34 b(c)n(ho)r(ose)f Fo(N)42 b(<)34 b(!)i Fu(suc)n(h)e(that)0 3371 y Fo(car)r(d)p Fu(\()p Fo(X)26 b Fr(\\)19 b Fo(N)9 b Fu(\))24 b Fo(>)g Fu(max)o Fr(f)p Fo(\024;)14 b Fu(min\()p Fo(X)7 b Fu(\))p Fr(g)p Fu(.)38 b(F)-7 b(or)27 b Fo(f)33 b Fu(:=)24 b Fo(f)1631 3341 y Fi(\003)1668 3371 y Fb(\026)p Fu([)p Fo(N)9 b Fu(])1825 3341 y Fp(m)1916 3371 y Fu(w)n(e)28 b(then)g(ha)n(v)n(e)f Fo(f)9 b Fb(\026)p Fu([)p Fo(X)25 b Fr(\\)19 b Fo(N)9 b Fu(])2795 3341 y Fp(m)2882 3371 y Fu(=)23 b Fo(f)3020 3341 y Fi(\003)3058 3371 y Fb(\026)p Fu([)p Fo(X)i Fr(\\)19 b Fo(N)9 b Fu(])3383 3341 y Fp(m)3470 3371 y Fu(=)23 b Fo(constant)p Fu(.)0 3512 y(Hence)h Fo(X)16 b Fr(\\)10 b Fo(N)33 b Fu(is)23 b Fo(f)9 b Fu(-homogeneous)21 b(and)j Fo(car)r(d)p Fu(\()p Fo(X)17 b Fr(\\)10 b Fo(N)f Fu(\))24 b Fo(>)e Fu(max)p Fr(f)p Fo(\024;)14 b Fu(min)o(\()p Fo(X)7 b Fu(\))p Fr(g)23 b Fu(=)g(max)o Fr(f)p Fo(\024;)14 b Fu(min\()p Fo(X)i Fr(\\)10 b Fo(N)f Fu(\))p Fr(g)p Fu(,)25 b(i.e.,)f Fr(:)p Fu(\010\()p Fo(f)t(;)14 b(N)9 b Fu(\).)0 3653 y Fm(Contr)l(adiction)29 b Fu(to)f(\(+\).)0 3819 y([[)21 b(Construction)f(of)h Fo(f)697 3789 y Fi(\003)735 3819 y Fu(:)34 b(Let)21 b(\010)994 3831 y Fp(n)1062 3819 y Fu(:=)i Fr(f)p Fo(f)31 b Fu(:)23 b(\010\()p Fo(f)t(;)14 b(n)p Fu(\))p Fr(g)21 b Fu(and)g Fo(M)9 b Fu(\()p Fo(f)g Fu(\))23 b(:=)f Fr(f)p Fo(i)g Fu(:)i Fr(9)p Fo(g)h Fr(2)f Fu(\010)2534 3831 y Fp(i)2561 3819 y Fu(\()p Fo(f)32 b Fr(\022)23 b Fo(g)s Fu(\))p Fr(g)p Fu(.)34 b(|)21 b(Starting)g(with)g Fo(f)3572 3831 y Fn(0)3632 3819 y Fu(:=)i Fr(;)d Fu(w)n(e)0 3960 y(de\014ne)28 b(a)f(sequence)g(\()p Fo(f)726 3972 y Fp(n)771 3960 y Fu(\))803 3972 y Fp(n)p Fi(2)p Fp(!)965 3960 y Fu(suc)n(h)g(that)h Fo(f)1373 3972 y Fp(n)1441 3960 y Fr(2)23 b Fu(\010)1579 3972 y Fp(n)1648 3960 y Fu(&)f Fo(f)1776 3972 y Fp(n)1844 3960 y Fr(\022)h Fo(f)1973 3972 y Fp(n)p Fn(+1)2125 3960 y Fu(&)f Fo(car)r(d)p Fu(\()p Fo(M)9 b Fu(\()p Fo(f)2569 3972 y Fp(n)2615 3960 y Fu(\)\))24 b Fr(\025)f Fo(!)s Fu(,)k(and)g(w)n(e)h(set)f Fo(f)3359 3930 y Fi(\003)3420 3960 y Fu(:=)3531 3898 y Fl(S)3600 3985 y Fp(n)p Fi(2)p Fp(!)3748 3960 y Fo(f)3789 3972 y Fp(n)3833 3960 y Fu(.)0 4101 y(De\014nition)36 b(of)e Fo(f)536 4113 y Fp(n)p Fn(+1)665 4101 y Fu(:)52 b(Let)35 b Fo(E)40 b Fu(:=)35 b Fr(f)p Fo(f)43 b Fr(2)35 b Fu(\010)1396 4113 y Fp(n)p Fn(+1)1561 4101 y Fu(:)g Fo(f)1660 4113 y Fp(n)1740 4101 y Fr(\022)f Fo(f)9 b Fr(g)p Fu(.)58 b(Then)35 b Fo(M)9 b Fu(\()p Fo(f)2399 4113 y Fp(n)2444 4101 y Fu(\))35 b(=)g Fr(f)p Fo(n)p Fr(g)22 b([)2846 4039 y Fl(S)2915 4126 y Fp(f)7 b Fi(2)p Fp(E)3069 4101 y Fo(M)i Fu(\()p Fo(f)g Fu(\),)36 b(and)f Fo(E)40 b Fu(is)34 b(\014nite.)0 4242 y(This)j(together)e(with)j Fr(8)p Fo(i)f(>)g(n)p Fr(8)p Fo(f)46 b Fr(2)39 b Fu(\010)1287 4254 y Fp(i)1314 4242 y Fu(\()14 b Fo(f)9 b Fb(\026)p Fu([)p Fo(n)p Fu(+)o(1])1647 4212 y Fp(m)1748 4242 y Fr(2)38 b Fu(\010)1901 4254 y Fp(n)p Fn(+1)2044 4242 y Fu(\))g(implies)e(the)i(existence)e(of)h(an)f Fo(f)3188 4254 y Fp(n)p Fn(+1)3355 4242 y Fr(2)j Fo(E)i Fu(suc)n(h)c(that)0 4384 y Fo(car)r(d)p Fu(\()p Fo(M)9 b Fu(\()p Fo(f)357 4396 y Fp(n)p Fn(+1)487 4384 y Fu(\)\))23 b Fr(\025)g Fo(!)s Fu(.]])0 4649 y Ft(Theorem)30 b(7.5)55 b Fr(8)p Fo(m)22 b Fr(\025)g Fu(1)p Fr(8)p Fo(k)s Fu(\()14 b Fo(H)1062 4664 y Fp(!)1104 4672 y Fh(m)1158 4664 y Fn(\()p Fp(k)q Fn(\))1251 4649 y Fu(\()p Fo(k)s Fu(+)o(1\))23 b Fo(<)g(R)1641 4661 y Fp(m)1704 4649 y Fu(\()p Fo(k)s Fu(\))14 b(\))28 b(with)g Fo(R)2140 4661 y Fp(m)2203 4649 y Fu(\()p Fo(k)s Fu(\))c(:=)f(min)p Fr(f)p Fo(N)31 b Fu(:)23 b Fo(N)2921 4602 y Fi(\003)2871 4649 y Fr(\000)-14 b(!)23 b Fu(\(2)p Fo(m)p Fu(+)o Fo(k)s Fu(+4\))3424 4614 y Fp(m)p Fn(+1)3424 4678 y Fp(k)q Fn(+\006)3558 4686 y Fh(i)23 b Fu(0)14 b(&)g Fo(\013)22 b(>)g(\013)2326 1869 y Fn(1)2387 1857 y Fo(>)h(:::)g(>)f(\013)2707 1869 y Fp(t)2737 1857 y Fu(:)0 1990 y Fe(r)p Fu(\()p Fo(\013)p Fu(\))i(:=)e(max)p Fr(f)p Fo(t;)14 b(n)592 2002 y Fn(1)628 1990 y Fo(;)g(:::;)g(n)821 2002 y Fp(t)850 1990 y Fo(;)g Fe(r)p Fu(\()p Fo(\013)999 2002 y Fn(1)1037 1990 y Fu(\))p Fo(;)g(:::;)g Fe(r)o Fu(\()p Fo(\013)1323 2002 y Fp(t)1353 1990 y Fu(\))p Fr(g)p Fu(.)0 2141 y Ft(Lemma)30 b(7.6)0 2274 y Fu(a\))d Fe(r)p Fu(\()p Fo(\013)p Fu([)p Fo(k)s Fu(]\))d Fr(\024)f Fu(max)o Fr(f)p Fe(r)o Fu(\()p Fo(\013)p Fu(\))p Fo(;)14 b(k)s Fr(g)19 b Fu(+)f(1)0 2408 y(b\))28 b Fe(r)p Fu(\()p Fo(\013)p Fu(\))c Fo(<)e(n)h Fu(=)-14 b Fr(\))23 b Fe(r)p Fu(\()p Fo(\013)p Fu([)p Fo(n)p Fr(\000)p Fu(1]\))g Fo(<)f(n)p Fu(+1)0 2541 y(c\))28 b Fe(r)o Fu(\()p Fo(!)207 2553 y Fp(m)271 2541 y Fu(\()p Fo(k)s Fu(\))18 b(+)g Fo(n)p Fu(\))24 b Fr(\024)e Fo(n)c Fu(+)h(1,)27 b(falls)g(1)c Fr(\024)f Fo(m)28 b Fu(und)g Fo(k)e(<)d(n)p Fu(.)0 2692 y(Bew)n(eis)k(v)n(on)g(c\):)37 b(1.)f Fo(k)26 b Fu(=)d(0)k(und)h Fo(m)23 b Fu(=)g(1:)36 b Fo(!)1390 2704 y Fp(m)1453 2692 y Fu(\()p Fo(k)s Fu(\))19 b(+)f Fo(n)23 b Fu(=)f Fo(!)1880 2662 y Fn(0)1917 2692 y Fu(\()p Fo(n)d Fu(+)f(1\).)0 2825 y(2.)36 b Fo(k)26 b Fr(6)p Fu(=)d(0)k(und)h Fo(m)23 b Fu(=)g(1:)36 b Fe(r)p Fu(\()p Fo(!)889 2837 y Fp(m)952 2825 y Fu(\()p Fo(k)s Fu(\))19 b(+)f Fo(n)p Fu(\))23 b(=)g(max)o Fr(f)p Fu(2)p Fo(;)14 b Fu(1)p Fo(;)g(n;)g Fe(r)n Fu(\()p Fo(k)s Fu(\))p Fr(g)23 b(\024)f Fo(n)p Fu(+1.)0 2959 y(3.)36 b Fo(m)23 b(>)g Fu(1:)37 b Fe(r)o Fu(\()p Fo(!)497 2971 y Fp(m)560 2959 y Fu(\()p Fo(k)s Fu(\))19 b(+)f Fo(n)p Fu(\))23 b(=)g(max)o Fr(f)p Fu(2)p Fo(;)14 b Fu(1)p Fo(;)g(n;)g Fe(r)n Fu(\()p Fo(!)1515 2971 y Fp(m)p Fi(\000)p Fn(1)1663 2959 y Fu(\()p Fo(k)s Fu(\)\))p Fr(g)23 b Fu(=)g(max)p Fr(f)p Fu(2)p Fo(;)14 b(n;)g(k)s Fr(g)21 b(\024)h Fo(n)p Fu(+1.)0 3144 y Ft(De\014nition)0 3277 y Fu(Sei)28 b Fo(\013)23 b Fu(=)g Fo(!)353 3247 y Fp(\013)396 3255 y Fg(1)432 3277 y Fo(n)482 3289 y Fn(1)538 3277 y Fu(+)18 b Fo(:::)g Fu(+)g Fo(!)846 3247 y Fp(\013)889 3255 y Fh(t)921 3277 y Fo(n)971 3289 y Fp(t)1028 3277 y Fu(mit)28 b Fo(t)23 b Fr(\025)g Fu(0)14 b(&)g Fo(n)1506 3289 y Fn(1)1541 3277 y Fo(;)g(:::;)g(n)1734 3289 y Fp(t)1787 3277 y Fo(>)22 b Fu(0)14 b(&)g Fo(\013)22 b(>)h(\013)2225 3289 y Fn(1)2285 3277 y Fo(>)g(:::)g(>)g(\013)2606 3289 y Fp(t)2635 3277 y Fu(:)0 3411 y Fk(S)46 3423 y Fp(i)74 3411 y Fu(\()p Fo(\013)p Fu(\))h(:=)f Fo(!)381 3381 y Fp(\013)424 3389 y Fh(i)454 3411 y Fo(n)504 3423 y Fp(i)559 3411 y Fu(,)28 b Fk(E)659 3423 y Fp(i)687 3411 y Fu(\()p Fo(\013)p Fu(\))c(:=)f Fo(\013)992 3423 y Fp(i)1047 3411 y Fu(,)28 b Fk(K)1155 3423 y Fp(i)1183 3411 y Fu(\()p Fo(\013)p Fu(\))c(:=)f Fo(n)1485 3423 y Fp(i)1540 3411 y Fu(f)r(\177)-44 b(ur)28 b Fo(i)22 b Fr(2)i(f)p Fu(1)p Fo(;)14 b(:::;)g(t)p Fr(g)0 3544 y Fk(S)46 3556 y Fp(i)74 3544 y Fu(\()p Fo(\013)p Fu(\))24 b(:=)f Fk(E)375 3556 y Fp(i)403 3544 y Fu(\()p Fo(\013)p Fu(\))h(:=)e Fk(K)712 3556 y Fp(i)740 3544 y Fu(\()p Fo(\013)p Fu(\))i(:=)f(0)k(f)r(\177)-44 b(ur)27 b Fo(i)c Fr(62)g(f)p Fu(1)p Fo(;)14 b(:::;)g(t)p Fr(g)p Fu(.)0 3695 y Ft(De\014nition)0 3828 y Fu(F)r(\177)-44 b(ur)28 b Fo(\013)23 b(>)g(\014)32 b Fu(sei)27 b Fk(d)p Fu(\()p Fo(\013;)14 b(\014)t Fu(\))24 b(:=)f(min)p Fr(f)p Fo(i)f Fu(:)i Fk(S)1231 3840 y Fp(i)1258 3828 y Fu(\()p Fo(\013)p Fu(\))g Fo(>)f Fk(S)1533 3840 y Fp(i)1561 3828 y Fu(\()p Fo(\014)t Fu(\))p Fr(g)28 b Fu(,)g Fk(K)o Fu(\()p Fo(\013;)14 b(\014)t Fu(\))25 b(:=)d Fk(K)2252 3858 y(d)p Fn(\()p Fp(\013;\014)s Fn(\))2454 3828 y Fu(\()p Fo(\013)p Fu(\))29 b(,)f Fk(E)o Fu(\()p Fo(\013;)14 b(\014)t Fu(\))25 b(:=)d Fk(E)3090 3858 y(d)p Fn(\()p Fp(\013;\014)s Fn(\))3292 3828 y Fu(\()p Fo(\013)p Fu(\).)0 3979 y Ft(Lemma)30 b(7.7)0 4113 y Fo(\013)23 b(>)g(\014)28 b(>)22 b(\015)d Fu(&)14 b Fk(d)o Fu(\()p Fo(\013;)g(\014)t Fu(\))24 b Fr(\024)f Fk(d)p Fu(\()p Fo(\014)t(;)14 b(\015)5 b Fu(\))14 b(&)g Fk(K)o Fu(\()p Fo(\013;)g(\014)t Fu(\))24 b Fr(\024)f Fk(K)p Fu(\()p Fo(\014)t(;)14 b(\015)5 b Fu(\))23 b(=)-14 b Fr(\))23 b Fk(E)o Fu(\()p Fo(\013;)14 b(\014)t Fu(\))25 b Fo(>)d Fk(E)p Fu(\()p Fo(\014)t(;)14 b(\015)5 b Fu(\).)0 4263 y(Bew)n(eis:)0 4397 y(F)-7 b(all)28 b(1:)38 b Fk(d)p Fu(\()p Fo(\013;)14 b(\014)t Fu(\))25 b(=)f Fk(d)p Fu(\()p Fo(\014)t(;)14 b(\015)5 b Fu(\))25 b(=)f Fo(i)p Fu(.)38 b(Dann)29 b Fk(E)o Fu(\()p Fo(\013;)14 b(\014)t Fu(\))26 b(=)e Fk(E)1719 4409 y Fp(i)1747 4397 y Fu(\()p Fo(\013)p Fu(\),)29 b Fk(E)p Fu(\()p Fo(\014)t(;)14 b(\015)5 b Fu(\))25 b(=)e Fk(E)2329 4409 y Fp(i)2357 4397 y Fu(\()p Fo(\014)t Fu(\))29 b(und)g Fk(K)2725 4409 y Fp(i)2753 4397 y Fu(\()p Fo(\013)p Fu(\))c Fr(\024)f Fk(K)3042 4409 y Fp(i)3069 4397 y Fu(\()p Fo(\014)t Fu(\).)40 b(W)-7 b(egen)28 b Fo(\013)d(<)f(\014)33 b Fu(und)0 4530 y Fk(K)58 4542 y Fp(i)85 4530 y Fu(\()p Fo(\013)p Fu(\))24 b Fr(\024)f Fk(K)372 4542 y Fp(i)399 4530 y Fu(\()p Fo(\014)t Fu(\))29 b(m)n(u\031)e Fk(E)775 4542 y Fp(i)802 4530 y Fu(\()p Fo(\013)p Fu(\))d Fo(>)f Fk(E)1081 4542 y Fp(i)1108 4530 y Fu(\()p Fo(\014)t Fu(\))29 b(sein.)0 4681 y(F)-7 b(all)28 b(2:)36 b Fo(i)23 b Fu(=)f Fk(d)p Fu(\()p Fo(\013;)14 b(\014)t Fu(\))25 b Fo(<)d Fk(d)p Fu(\()p Fo(\014)t(;)14 b(\015)5 b Fu(\))24 b(=)e Fo(j)5 b Fu(.)37 b(Dann)28 b Fk(E)p Fu(\()p Fo(\013;)14 b(\014)t Fu(\))24 b(=)e Fk(E)1857 4693 y Fp(i)1885 4681 y Fu(\()p Fo(\013)p Fu(\))i Fr(\025)e Fk(E)2163 4693 y Fp(i)2191 4681 y Fu(\()p Fo(\014)t Fu(\))i Fo(>)e Fk(E)2467 4693 y Fp(j)2502 4681 y Fu(\()p Fo(\014)t Fu(\))i(=)e Fk(E)p Fu(\()p Fo(\014)t(;)14 b(\015)5 b Fu(\).)0 4866 y Ft(De\014nition)27 b Fu(\()p Fo(\037)c Fu(:)g([)p Fo(")657 4878 y Fn(0)694 4866 y Fu(])717 4836 y Fn(3)777 4866 y Fr(!)h(f)926 4848 y Fu(^)926 4866 y(0)o Fo(;)1004 4848 y Fu(^)1004 4866 y(1)o Fo(;)1082 4848 y Fu(^)1082 4866 y(2)o Fr(g)p Fu(\))0 4999 y(F)r(\177)-44 b(ur)28 b Fo(\014)208 5011 y Fn(0)268 4999 y Fo(>)22 b(\014)402 5011 y Fn(1)463 4999 y Fo(>)g(\014)597 5011 y Fn(2)662 4999 y Fu(sei)0 5211 y Fo(\037)p Fu(\()p Fo(\014)131 5223 y Fn(0)168 5211 y Fo(;)14 b(\014)252 5223 y Fn(1)290 5211 y Fo(;)g(\014)374 5223 y Fn(2)411 5211 y Fu(\))23 b(:=)577 5041 y Fl(8)577 5116 y(<)577 5265 y(:)664 5106 y Fu(^)664 5124 y(0)83 b(falls)27 b Fk(d)p Fu(\()p Fo(\014)1084 5136 y Fn(1)1122 5124 y Fo(;)14 b(\014)1206 5136 y Fn(2)1243 5124 y Fu(\))23 b Fo(<)g Fk(d)p Fu(\()p Fo(\014)1508 5136 y Fn(0)1545 5124 y Fo(;)14 b(\014)1629 5136 y Fn(1)1666 5124 y Fu(\))664 5211 y(^)664 5228 y(1)83 b(falls)27 b Fk(d)p Fu(\()p Fo(\014)1084 5240 y Fn(0)1122 5228 y Fo(;)14 b(\014)1206 5240 y Fn(1)1243 5228 y Fu(\))23 b Fr(\024)g Fk(d)p Fu(\()p Fo(\014)1508 5240 y Fn(1)1545 5228 y Fo(;)14 b(\014)1629 5240 y Fn(2)1666 5228 y Fu(\))g(&)g Fk(K)o Fu(\()p Fo(\014)1927 5240 y Fn(1)1965 5228 y Fo(;)g(\014)2049 5240 y Fn(2)2086 5228 y Fu(\))23 b Fo(<)g Fk(K)p Fu(\()p Fo(\014)2366 5240 y Fn(0)2403 5228 y Fo(;)14 b(\014)2487 5240 y Fn(1)2524 5228 y Fu(\))664 5315 y(^)664 5333 y(2)83 b(sonst)1908 5539 y(47)p eop %%Page: 48 49 48 48 bop 0 83 a Ft(Lemma)30 b(7.8)0 224 y Fu(Sei)e Fo(\014)181 236 y Fn(0)241 224 y Fo(>)23 b(:)14 b(:)g(:)23 b(>)f(\014)583 236 y Fp(`)643 224 y Fu(mit)28 b Fo(`)23 b Fr(\025)f Fu(2,)28 b(und)g(sei)f Fo(c)c Fr(2)g(f)1498 206 y Fu(^)1498 224 y(0)o Fo(;)1576 206 y Fu(^)1576 224 y(1)p Fo(;)1655 206 y Fu(^)1655 224 y(2)o Fr(g)k Fu(mit)i Fr(f)p Fo(\037)p Fu(\()p Fo(\014)2091 236 y Fp(i)2118 224 y Fo(;)14 b(\014)2202 236 y Fp(i)p Fn(+1)2313 224 y Fo(;)g(\014)2397 236 y Fp(i)p Fn(+2)2509 224 y Fu(\))23 b(:)h Fo(i)e Fr(\024)h Fo(`)18 b Fr(\000)g Fu(2)p Fr(g)k Fu(=)h Fr(f)p Fo(c)p Fr(g)p Fu(.)0 365 y(a\))k Fo(c)c Fr(2)h(f)281 348 y Fu(^)281 365 y(0)o Fo(;)359 348 y Fu(^)359 365 y(1)o Fr(g)46 b Fu(=)-14 b Fr(\))46 b Fo(`)22 b Fr(\024)h Fe(r)p Fu(\()p Fo(\014)919 377 y Fn(0)956 365 y Fu(\))0 506 y(b\))28 b Fo(c)23 b Fu(=)253 489 y(^)253 506 y(2)45 b(=)-14 b Fr(\))46 b Fk(E)p Fu(\()p Fo(\014)649 518 y Fn(0)686 506 y Fo(;)14 b(\014)770 518 y Fn(1)807 506 y Fu(\))24 b Fo(>)e(:)14 b(:)g(:)23 b(>)g Fk(E)p Fu(\()p Fo(\014)1287 518 y Fp(`)p Fi(\000)p Fn(1)1404 506 y Fo(;)14 b(\014)1488 518 y Fp(`)1519 506 y Fu(\))0 672 y(Bew)n(eis:)36 b(a\))28 b(Sei)f Fo(d)584 684 y Fp(i)635 672 y Fu(:=)c Fk(d)p Fu(\()p Fo(\014)868 684 y Fp(i)896 672 y Fo(;)14 b(\014)980 684 y Fp(i)p Fn(+1)1091 672 y Fu(\))28 b(und)g Fo(k)1360 684 y Fp(i)1411 672 y Fu(:=)23 b Fk(K)o Fu(\()p Fo(\014)1658 684 y Fp(i)1686 672 y Fo(;)14 b(\014)1770 684 y Fp(i)p Fn(+1)1882 672 y Fu(\))83 b(\()p Fo(i)23 b(<)g(`)p Fu(\).)0 839 y(O\013en)n(bar)k(ist)g Fo(d)500 851 y Fn(0)538 839 y Fo(;)14 b(k)618 851 y Fn(0)678 839 y Fr(\024)23 b Fe(r)o Fu(\()p Fo(\014)871 851 y Fn(0)909 839 y Fu(\).)37 b(Ist)28 b Fo(c)23 b Fu(=)1271 821 y(^)1271 839 y(0)o(,)28 b(so)f Fo(d)1508 851 y Fn(0)1568 839 y Fo(>)c(d)1699 851 y Fn(1)1759 839 y Fo(>)g(:)14 b(:)g(:)23 b(>)g(d)2098 851 y Fp(`)p Fi(\000)p Fn(1)2238 839 y Fr(\025)f Fu(1,)28 b(w)n(oraus)d Fo(`)e Fr(\024)g Fo(d)2886 851 y Fn(0)2951 839 y Fu(folgt.)0 980 y(Ist)28 b Fo(c)23 b Fu(=)269 962 y(^)269 980 y(1,)28 b(so)e Fo(k)506 992 y Fn(0)567 980 y Fo(>)d(k)698 992 y Fn(1)758 980 y Fo(>)g(:)14 b(:)g(:)23 b(>)f(k)1096 992 y Fp(`)p Fi(\000)p Fn(1)1237 980 y Fr(\025)g Fu(1,)27 b(w)n(oraus)f Fo(`)d Fr(\024)f Fo(k)1884 992 y Fn(0)1950 980 y Fu(folgt.)36 b(|)28 b(b\))g(folgt)g(aus)f(Lemma)g(7.7.)0 1171 y Ft(De\014nition)g Fu(\()p Fo(\037)526 1140 y Fp(k)526 1191 y(m)612 1171 y Fu(:)c([)p Fo(!)733 1183 y Fp(m)796 1171 y Fu(\()p Fo(k)s Fu(+1\)])1036 1140 y Fp(m)p Fn(+1)1206 1171 y Fr(!)g(C)1361 1140 y Fp(k)1356 1191 y(m)1446 1171 y Fu(f)r(\177)-44 b(ur)28 b Fo(m)23 b Fr(\025)g Fu(1)p Fo(;)k(k)f Fr(\025)c Fu(0\))0 1337 y(1.)36 b(F)r(\177)-44 b(ur)28 b Fo(\013)23 b Fu(=)g Fo(!)481 1307 y Fp(k)522 1337 y Fo(n)572 1349 y Fn(0)627 1337 y Fu(+)18 b Fo(:)c(:)g(:)k Fu(+)g Fo(!)963 1307 y Fn(0)1000 1337 y Fo(n)1050 1349 y Fp(k)1114 1337 y Fo(>)23 b(!)1257 1307 y Fp(k)1297 1337 y Fo(n)1347 1307 y Fi(0)1347 1357 y Fn(0)1403 1337 y Fu(+)18 b Fo(:)c(:)g(:)k Fu(+)g Fo(!)1739 1307 y Fn(0)1776 1337 y Fo(n)1826 1307 y Fi(0)1826 1360 y Fp(k)1890 1337 y Fu(=)k Fo(\014)32 b Fu(setzen)c(wir)f Fo(\037)2501 1307 y Fp(k)2501 1357 y Fn(1)2542 1337 y Fu(\()p Fo(\013;)14 b(\014)t Fu(\))24 b(:=)f(min)p Fr(f)p Fo(i)f Fu(:)i Fo(n)3210 1307 y Fi(0)3210 1358 y Fp(i)3260 1337 y Fo(<)f(n)3398 1349 y Fp(i)3425 1337 y Fr(g)p Fu(.)0 1503 y(2.)36 b(Sei)28 b Fo(m)23 b Fr(\025)g Fu(1)k(,)h Fo(!)591 1515 y Fp(m)p Fn(+1)738 1503 y Fu(\()p Fo(k)s Fu(+)o(1\))23 b Fo(>)g(\014)1112 1515 y Fn(0)1172 1503 y Fo(>)g(:)14 b(:)g(:)23 b(>)f(\014)1514 1515 y Fp(m)p Fn(+1)1689 1503 y Fu(,)28 b Fo(\016)1777 1515 y Fp(i)1827 1503 y Fu(:=)23 b Fk(E)p Fu(\()p Fo(\014)2067 1515 y Fp(i)2094 1503 y Fo(;)14 b(\014)2178 1515 y Fp(i)p Fn(+1)2290 1503 y Fu(\))28 b(,)g(und)g Fo(c)2603 1515 y Fp(i)2653 1503 y Fu(:=)23 b Fo(\037)p Fu(\()p Fo(\014)2895 1515 y Fp(i)2923 1503 y Fo(;)14 b(\014)3007 1515 y Fp(i)p Fn(+1)3118 1503 y Fo(;)g(\014)3202 1515 y Fp(i)p Fn(+2)3314 1503 y Fu(\).)0 1677 y Fo(\037)52 1647 y Fp(k)52 1698 y(m)p Fn(+1)199 1677 y Fu(\()p Fo(\014)278 1689 y Fn(0)316 1677 y Fo(;)g(:::;)g(\014)506 1689 y Fp(m)p Fn(+1)653 1677 y Fu(\))23 b(:=)819 1560 y Fl(\032)895 1638 y Fo(\037)947 1608 y Fp(k)947 1658 y(m)1010 1638 y Fu(\()p Fo(\016)1079 1650 y Fn(0)1116 1638 y Fo(;)14 b(:::;)g(\016)1296 1650 y Fp(m)1359 1638 y Fu(\))93 b(falls)27 b Fo(c)1693 1650 y Fn(0)1754 1638 y Fu(=)22 b Fo(:)14 b(:)g(:)23 b Fu(=)g Fo(c)2085 1650 y Fp(m)p Fi(\000)p Fn(1)2256 1638 y Fu(=)2344 1620 y(^)2344 1638 y(2)895 1729 y(\()p Fo(c)963 1741 y Fn(0)1000 1729 y Fo(;)14 b(:)g(:)g(:)g(;)g(c)1221 1741 y Fp(m)p Fi(\000)p Fn(1)1369 1729 y Fu(\))83 b(sonst)0 1865 y(\(Man)28 b(b)r(eac)n(h)n(te,)f(da\031)g(nac)n(h)g(L.7.8)f Fo(!)1162 1877 y Fp(m)1225 1865 y Fu(\()p Fo(k)s Fu(+1\))d Fo(>)f(\016)1589 1877 y Fn(0)1650 1865 y Fo(>)g(:)14 b(:)g(:)23 b(>)g(\016)1982 1877 y Fp(m)2045 1865 y Fu(,)k(falls)h Fo(c)2305 1877 y Fn(0)2365 1865 y Fu(=)23 b Fo(:)14 b(:)g(:)23 b Fu(=)f Fo(c)2696 1877 y Fp(m)p Fi(\000)p Fn(1)2867 1865 y Fu(=)2955 1847 y(^)2955 1865 y(2)o(.)0 2031 y(3.)36 b Fr(C)150 2000 y Fp(k)145 2051 y Fn(1)214 2031 y Fu(:=)23 b Fr(f)p Fu(0)p Fo(;)14 b(:)g(:)g(:)e(;)i(k)s Fr(g)p Fu(,)27 b Fr(C)779 2000 y Fp(k)774 2051 y(m)p Fn(+1)944 2031 y Fu(:=)c Fr(C)1104 2000 y Fp(k)1099 2051 y(m)1180 2031 y Fr([)c(f)1296 2013 y Fu(^)1296 2031 y(0)o Fo(;)1374 2013 y Fu(^)1374 2031 y(1)o Fo(;)1452 2013 y Fu(^)1452 2031 y(2)p Fr(g)1536 2000 y Fp(m)0 2246 y Ft(Lemma)30 b(7.9)0 2388 y Fu(Aus)e(1)22 b Fr(\024)h Fo(m)g(<)g(`)14 b Fu(&)g Fo(!)685 2400 y Fp(m)746 2388 y Fu(\()p Fo(k)22 b Fu(+)c(1\))23 b Fo(>)f(\014)1157 2400 y Fn(0)1218 2388 y Fo(>)g(:)14 b(:)g(:)23 b(>)g(\014)1560 2400 y Fp(`)1605 2388 y Fu(&)14 b Fo(c)23 b Fr(2)g(C)1870 2357 y Fp(k)1865 2408 y(m)1956 2388 y Fu(folgt:)0 2529 y Fr(8)p Fo(i)f Fr(\024)g Fo(`)p Fr(\000)p Fo(m)p Fu(\()14 b Fo(\037)456 2499 y Fp(k)456 2549 y(m)519 2529 y Fu(\()p Fo(\014)598 2541 y Fp(i)626 2529 y Fo(;)g(:::;)g(\014)816 2541 y Fp(i)p Fn(+)p Fp(m)953 2529 y Fu(\))23 b(=)g Fo(c)14 b Fu(\))46 b(=)-14 b Fr(\))46 b Fo(`)23 b(<)f Fe(r)p Fu(\()p Fo(\014)1655 2541 y Fn(0)1692 2529 y Fu(\))d(+)f Fo(m)p Fu(.)0 2695 y(Bew)n(eis)27 b(durc)n(h)g(Induktion)h(nac)n(h)f Fo(m)p Fu(:)0 2836 y(I.)h Fo(m)23 b Fu(=)f(1:)37 b(Sei)28 b Fo(\014)547 2848 y Fp(i)597 2836 y Fu(=)23 b Fo(!)740 2806 y Fp(k)780 2836 y Fo(n)830 2848 y Fp(i;)p Fn(0)929 2836 y Fu(+)18 b Fo(:)c(:)g(:)k Fu(+)g Fo(!)1265 2806 y Fn(0)1302 2836 y Fo(n)1352 2848 y Fp(i;k)1436 2836 y Fu(.)37 b(Dann)28 b Fo(n)1771 2848 y Fn(0)p Fp(;c)1881 2836 y Fo(>)22 b(:)14 b(:)g(:)23 b(>)g(n)2226 2848 y Fp(`;c)2335 2836 y Fu(und)28 b(somit)f Fo(`)c Fr(\024)f Fo(n)2922 2848 y Fn(0)p Fp(;c)3032 2836 y Fr(\024)h Fe(r)o Fu(\()p Fo(\014)3225 2848 y Fn(0)3263 2836 y Fu(\))g Fo(<)g Fe(r)o Fu(\()p Fo(\014)3511 2848 y Fn(0)3549 2836 y Fu(\))18 b(+)h Fo(m)p Fu(.)0 3002 y(I)r(I.)g Fo(m)k Fr(!)g Fo(m)p Fu(+1:)32 b(Sei)19 b(1)j Fr(\024)h Fo(m)14 b Fu(&)g Fo(m)p Fu(+)o(1)22 b Fo(<)h(`)14 b Fu(&)g Fo(c)22 b Fr(2)h(C)1586 2972 y Fp(k)1581 3023 y(m)p Fn(+1)1742 3002 y Fu(&)14 b Fo(!)1873 3014 y Fp(m)p Fn(+1)2019 3002 y Fu(\()p Fo(k)s Fu(+1\))22 b Fo(>)h(\014)2393 3014 y Fn(0)2453 3002 y Fo(>)g(:)14 b(:)g(:)23 b(>)g(\014)2796 3014 y Fp(`)2846 3002 y Fu(und)d Fo(\037)3056 2972 y Fp(k)3056 3023 y(m)p Fn(+1)3203 3002 y Fu(\()p Fo(\014)3282 3014 y Fp(i)3310 3002 y Fo(;)14 b(:::;)g(\014)3500 3014 y Fp(i)p Fn(+)p Fp(m)p Fn(+1)3721 3002 y Fu(\))23 b(=)g Fo(c)0 3143 y Fu(f)r(\177)-44 b(ur)28 b(alle)f Fo(i)c Fr(\024)f Fo(`)p Fr(\000)o Fo(m)p Fr(\000)p Fu(1.)0 3309 y(F)-9 b(ALL)28 b(1:)36 b Fo(c)23 b Fr(2)h(f)520 3291 y Fu(^)520 3309 y(0)o Fo(;)598 3291 y Fu(^)598 3309 y(1)o Fo(;)676 3291 y Fu(^)676 3309 y(2)o Fr(g)759 3279 y Fp(m)822 3309 y Fu(.)0 3450 y(Sei)k Fo(c)170 3462 y Fp(i)220 3450 y Fu(:=)23 b Fo(\037)p Fu(\()p Fo(\014)462 3462 y Fp(i)490 3450 y Fo(;)14 b(\014)574 3462 y Fp(i)p Fn(+1)685 3450 y Fo(;)g(\014)769 3462 y Fp(i)p Fn(+2)881 3450 y Fu(\))23 b(\()p Fo(i)g Fr(\024)g Fo(`)p Fr(\000)o Fu(2\).)37 b(F)r(\177)-44 b(ur)27 b(alle)h Fo(i)22 b Fr(\024)h Fo(`)p Fr(\000)o Fo(m)p Fr(\000)p Fu(2)k(hab)r(en)g(wir)h(dann)0 3591 y(\()p Fo(c)68 3603 y Fp(i)96 3591 y Fo(;)14 b(:::;)g(c)275 3603 y Fp(i)p Fn(+)p Fp(m)p Fi(\000)p Fn(1)497 3591 y Fu(\))24 b(=)e Fo(\037)692 3561 y Fp(k)692 3612 y(m)p Fn(+1)839 3591 y Fu(\()p Fo(\014)918 3603 y Fp(i)946 3591 y Fo(;)14 b(:::;)g(\014)1136 3603 y Fp(i)p Fn(+)p Fp(m)p Fn(+1)1358 3591 y Fu(\))23 b(=)g Fo(\037)1553 3561 y Fp(k)1553 3612 y(m)p Fn(+1)1700 3591 y Fu(\()p Fo(\014)1779 3603 y Fp(i)p Fn(+1)1891 3591 y Fo(;)14 b(:::;)g(\014)2081 3603 y Fp(i)p Fn(+)p Fp(m)p Fn(+2)2302 3591 y Fu(\))24 b(=)e(\()p Fo(c)2513 3603 y Fp(i)p Fn(+1)2625 3591 y Fo(;)14 b(:::;)g(c)2804 3603 y Fp(i)p Fn(+)p Fp(m)2942 3591 y Fu(\).)37 b(Das)27 b(impliziert)0 3733 y Fr(8)p Fo(i)k Fr(\024)i Fo(`)p Fr(\000)o Fo(m)p Fr(\000)o Fu(2\()14 b Fo(c)565 3745 y Fp(i)625 3733 y Fu(=)32 b Fo(c)758 3745 y Fp(i)p Fn(+1)903 3733 y Fu(=)g Fo(:)14 b(:)g(:)32 b Fu(=)g Fo(c)1262 3745 y Fp(i)p Fn(+)p Fp(m)1414 3733 y Fu(\))h(und)h(somit)f Fo(c)1919 3745 y Fn(0)1989 3733 y Fu(=)f Fo(:)14 b(:)g(:)33 b Fu(=)f Fo(c)2349 3745 y Fp(`)p Fi(\000)p Fn(2)2466 3733 y Fu(.)54 b(Zusammen)33 b(mit)h Fo(\037)3179 3702 y Fp(k)3179 3753 y(m)p Fn(+1)3326 3733 y Fu(\()p Fo(\014)3405 3745 y Fn(0)3443 3733 y Fo(;)14 b(:::;)g(\014)3633 3745 y Fp(m)p Fn(+1)3780 3733 y Fu(\))33 b Fr(2)0 3874 y(f)42 3856 y Fu(^)42 3874 y(0)o Fo(;)120 3856 y Fu(^)120 3874 y(1)o Fo(;)198 3856 y Fu(^)198 3874 y(2)p Fr(g)282 3844 y Fp(m)372 3874 y Fu(folgt)28 b(daraus)e Fo(c)868 3886 y Fn(0)928 3874 y Fr(2)d(f)1048 3856 y Fu(^)1048 3874 y(0)o Fo(;)1126 3856 y Fu(^)1126 3874 y(1)p Fr(g)k Fu(und)h(dann)g Fo(`)22 b(<)h Fe(r)o Fu(\()p Fo(\014)1861 3886 y Fn(0)1899 3874 y Fu(\))c(+)f Fo(m)g Fu(+)g(1)27 b(nac)n(h)g(Lemma)h(7.8a.)0 4040 y(F)-9 b(ALL)28 b(2:)36 b Fo(c)23 b Fr(2)h(C)527 4010 y Fp(k)522 4060 y(m)584 4040 y Fu(.)0 4181 y(Dann)32 b Fo(c)265 4193 y Fn(0)332 4181 y Fu(=)e Fo(:)14 b(:)g(:)30 b Fu(=)g Fo(c)685 4193 y Fp(`)p Fi(\000)p Fn(2)832 4181 y Fu(=)926 4163 y(^)926 4181 y(2.)49 b(Nac)n(h)32 b(L.7.8b)f(folgt)g(daraus)g Fo(\016)2021 4193 y Fn(0)2088 4181 y Fo(>)f(:)14 b(:)g(:)30 b(>)f(\016)2441 4193 y Fp(`)p Fi(\000)p Fn(1)2558 4181 y Fu(.)50 b(F)-7 b(erner)31 b Fo(c)f Fu(=)g Fo(\037)3108 4151 y Fp(k)3108 4201 y(m)p Fn(+1)3255 4181 y Fu(\()p Fo(\014)3334 4193 y Fp(i)3362 4181 y Fo(;)14 b(:::;)g(\014)3552 4193 y Fp(i)p Fn(+)p Fp(m)p Fn(+1)3773 4181 y Fu(\))30 b(=)0 4322 y Fo(\037)52 4292 y Fp(k)52 4343 y(m)115 4322 y Fu(\()p Fo(\016)184 4334 y Fp(i)212 4322 y Fo(;)14 b(:::;)g(\016)392 4334 y Fp(i)p Fn(+)p Fp(m)529 4322 y Fu(\))22 b(f)r(\177)-44 b(ur)21 b Fo(i)h Fr(\024)h Fo(`)p Fr(\000)o Fu(1)p Fr(\000)o Fo(m)p Fu(.)35 b(Nac)n(h)20 b(IV)i(folgt)f(daraus)e Fo(`)p Fr(\000)o Fu(1)k Fo(<)g Fe(r)o Fu(\()p Fo(\016)2295 4334 y Fn(0)2333 4322 y Fu(\))5 b(+)g Fo(m)p Fu(.)35 b(Also)20 b Fo(`)j(<)f Fe(r)p Fu(\()p Fo(\016)2992 4334 y Fn(0)3030 4322 y Fu(\))5 b(+)g Fo(m)g Fu(+)g(1)22 b Fr(\024)h Fe(r)p Fu(\()p Fo(\014)3543 4334 y Fn(0)3580 4322 y Fu(\))5 b(+)g Fo(m)g Fu(+)g(1.)0 4538 y Ft(Pro)s(of)32 b(of)f(Theorem)g(7.5:)0 4679 y Fu(Let)d Fo(m)23 b Fr(\025)f Fu(1,)28 b Fo(k)d Fr(\025)e Fu(0,)k Fo(n)c Fu(:=)g Fo(m)p Fu(+)o Fo(k)s Fu(+2,)k Fo(\024)c Fu(:=)g Fo(m)p Fu(+)o Fo(n)p Fu(+2)f(=)h(2)p Fo(m)p Fu(+)o Fo(k)s Fu(+)o(4,)k Fo(r)f Fu(:=)d Fo(k)e Fu(+)d(\006)2545 4691 y Fp(i)23 b(\013)3187 568 y Fp(i)p Fn(+1)3299 556 y Fu(\).)0 722 y(HS)28 b(1:)37 b Fr(8)p Fo(i)21 b Fr(\024)i Fo(n)p Fu(\()p Fe(r)p Fu(\()p Fo(\013)617 734 y Fp(i)645 722 y Fu(\))18 b(+)h Fo(m)j(<)h(\024)p Fu(\))28 b(und)g Fr(8)p Fo(i)22 b Fr(\025)g Fo(n)p Fu(\()p Fe(r)p Fu(\()p Fo(\013)1615 734 y Fp(i)1643 722 y Fu(\))d(+)f Fo(m)23 b(<)f(i)p Fu(\).)0 888 y(Bew)n(eis:)36 b Fo(i)23 b Fr(\024)f Fo(n)h Fr(\))g Fe(r)p Fu(\()p Fo(\013)736 900 y Fp(i)764 888 y Fu(\))c(+)f Fo(m)23 b Fr(\024)f Fe(r)p Fu(\()p Fo(\013)1193 900 y Fn(0)1231 888 y Fu(\))c(+)g Fo(m)23 b Fr(\024)g Fo(n)p Fu(+)o(1+)o Fo(m)g(<)g(\024)p Fu(.)0 1029 y Fe(r)p Fu(\()p Fo(\013)112 1041 y Fp(n)157 1029 y Fu(\))f(+)e Fo(m)30 b Fr(\024)f Fo(k)s Fu(+1)21 b(+)f Fo(m)30 b(<)f(n)p Fu(;)34 b(und)e(nac)n(h)f(L.7.6b)g(gilt:)45 b Fe(r)p Fu(\()p Fo(\013)1985 1041 y Fp(i)2013 1029 y Fu(\))21 b(+)g Fo(m)30 b(<)f(i)g Fr(\))h Fe(r)p Fu(\()p Fo(\013)2632 1041 y Fp(i)p Fn(+1)2744 1029 y Fu(\))g(=)g Fe(r)o Fu(\()p Fo(\013)3012 1041 y Fp(i)3040 1029 y Fu([)p Fo(i)p Fr(\000)o Fo(m)p Fr(\000)p Fu(1]\))g Fo(<)f(i)p Fr(\000)o Fo(m)p Fu(+1)g Fr(\))0 1171 y Fe(r)p Fu(\()p Fo(\013)112 1183 y Fp(i)p Fn(+1)224 1171 y Fu(\))18 b(+)g Fo(m)23 b(<)g(i)18 b Fu(+)g(1.)0 1337 y(HS2:)37 b Fo(H)30 b Fr(\024)22 b Fo(N)9 b Fu(.)0 1503 y(Bew)n(eis:)64 b Fo(n)23 b Fr(\024)f Fo(i)h(<)g(N)31 b Fr(\))23 b Fo(H)907 1515 y Fp(\013)950 1523 y Fh(i)981 1503 y Fu(\()p Fo(i)18 b Fr(\000)g Fo(m)h Fr(\000)f Fu(1\))23 b(=)f Fo(H)1571 1515 y Fp(\013)1614 1523 y Fh(i)p Fg(+1)1716 1503 y Fu(\()p Fo(i)c Fu(+)g(1)g Fr(\000)g Fo(m)g Fr(\000)h Fu(1\).)0 1669 y(Also)27 b Fo(H)j Fu(=)23 b Fo(H)443 1684 y Fp(!)485 1692 y Fh(m)540 1684 y Fn(\()p Fp(k)q Fn(\))632 1669 y Fu(\()p Fo(k)f Fu(+)c(1\))23 b(=)f Fo(H)1065 1681 y Fp(\013)1108 1689 y Fh(n)1154 1669 y Fu(\()p Fo(n)c Fr(\000)g Fo(m)g Fr(\000)g Fu(1\))23 b(=)g Fo(H)1765 1681 y Fp(\013)1808 1689 y Fh(N)1866 1669 y Fu(\()p Fo(N)k Fr(\000)18 b Fo(m)g Fr(\000)h Fu(1\))j(=)h Fo(N)k Fr(\000)18 b Fo(m)h Fr(\000)f Fu(1)k Fr(\024)h Fo(N)9 b Fu(.)0 1835 y(De\014nition:)38 b Fo(f)31 b Fu(:)23 b([)p Fo(N)9 b Fu(])659 1805 y Fp(m)p Fn(+1)829 1835 y Fr(!)23 b Fo(r)r Fu(,)29 b Fo(f)9 b Fu(\()p Fr(f)p Fo(i)1179 1847 y Fn(0)1215 1835 y Fo(;)14 b(:::;)g(i)1387 1847 y Fp(m)1450 1835 y Fr(g)1492 1847 y Fp(<)1547 1835 y Fu(\))24 b(:=)e Fo(\037)1765 1805 y Fp(k)1765 1855 y(m)1828 1835 y Fu(\()p Fr(f)p Fo(\013)1955 1847 y Fp(i)1978 1855 y Fg(0)2015 1835 y Fo(;)14 b(:::;)g(\013)2211 1847 y Fp(i)2234 1855 y Fh(m)2294 1835 y Fr(g)2336 1847 y Fp(>)2392 1835 y Fu(\).)0 2001 y(Sei)36 b Fo(X)43 b Fu(=)37 b Fr(f)p Fo(i)427 2013 y Fn(0)463 2001 y Fo(;)14 b(:::;)g(i)635 2013 y Fp(`)667 2001 y Fr(g)709 2013 y Fp(<)800 2001 y Fo(f)9 b Fu(-homogeneous.)60 b(Dann)36 b(ist)g Fr(f)p Fo(\013)1895 2013 y Fp(i)1918 2021 y Fg(0)1955 2001 y Fo(;)14 b(:::;)g(\013)2151 2013 y Fp(i)2174 2022 y Fh(`)2207 2001 y Fr(g)2249 2013 y Fp(>)2377 2001 y Fo(\037)2429 1971 y Fp(k)2429 2021 y(m)2492 2001 y Fu(-homogen,)37 b(und)f(w)n(egen)f(Lemma)h(7.9)f(gilt)0 2142 y Fo(car)r(d)p Fu(\()p Fo(X)7 b Fu(\))24 b(=)e Fo(`)c Fu(+)g(1)23 b Fr(\024)g Fe(r)o Fu(\()p Fo(\013)813 2154 y Fp(i)836 2162 y Fg(0)874 2142 y Fu(\))18 b(+)g Fo(m)1103 2081 y Fn(HS1)1131 2142 y Fo(<)49 b Fu(max)p Fr(f)p Fo(\024;)14 b(i)1556 2154 y Fn(0)1592 2142 y Fr(g)22 b Fu(=)h(max)p Fr(f)p Fo(\024;)14 b Fu(min)o(\()p Fo(X)7 b Fu(\))p Fr(g)p Fu(.)0 2507 y Fr(x)p Ft(8)63 b(The)32 b(collapsing)f(function)h Fo( )0 2723 y Fu(Abbreviation:)k(\012)23 b(:=)g Fr(@)780 2735 y Fn(1)817 2723 y Fu(.)0 2939 y Fq(De\014nition)k(of)h Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))29 b Fq(and)f Fo( )s Fu(\()p Fo(\013)p Fu(\))h Fq(b)n(y)e(recursion)f(on)h Fo(\013)0 3105 y(C)6 b Fu(\()p Fo(\013)p Fu(\))24 b(:=)k(closure)e(of)i Fr(f)p Fu(0)p Fr(g)e Fu(under)h(+,)h Fo(\025x:!)1370 3075 y Fn(\012+)p Fp(x)1538 3105 y Fu(and)f Fo(\030)h Fr(7!)23 b Fo( )s Fu(\()p Fo(\030)t Fu(\))28 b(\()p Fo(\030)g(<)22 b(\013)i Fu(&)e Fo(\030)28 b Fr(2)23 b Fo(C)6 b Fu(\()p Fo(\030)t Fu(\)\).)0 3271 y Fo( )s Fu(\()p Fo(\013)p Fu(\))24 b(:=)f(min)p Fr(f)p Fo(\030)k Fu(:)c Fo(\030)k Fr(62)d Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))p Fr(g)p Fu(.)0 3462 y Ft(Lemma)30 b(8.1)0 3603 y Fu(a\))55 b Fo(\013)24 b Fr(\024)e Fo(\014)28 b Fr(\))23 b Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))24 b Fr(\022)f Fo(C)6 b Fu(\()p Fo(\014)t Fu(\))24 b(&)f Fo( )s Fu(\()p Fo(\013)p Fu(\))h Fr(\024)e Fo( )s Fu(\()p Fo(\014)t Fu(\),)129 3744 y Fo(\013)i(<)e(\014)28 b Fu(&)22 b Fo(\013)i Fr(2)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))25 b Fr(\))e Fo( )s Fu(\()p Fo(\013)p Fu(\))h Fo(<)e( )s Fu(\()p Fo(\014)t Fu(\),)0 3885 y(b\))51 b Fo( )s Fu(\()p Fo(\013)p Fu(\))24 b Fo(<)f Fu(\012,)0 4027 y(c\))60 b Fo( )s Fu(\()p Fo(\013)p Fu(\))29 b(is)e(an)h(additiv)n(e)f (principal)g(n)n(um)n(b)r(er,)0 4168 y(d\))51 b Fo( )s Fu(\()p Fo(\013)p Fu(\))24 b(=)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))20 b Fr(\\)e Fu(\012,)0 4309 y(e\))60 b Fo(\013)24 b Fr(2)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))24 b Fr(,)f Fo( )s Fu(\()p Fo(\013)p Fu(\))i Fo(<)d( )s Fu(\()p Fo(\013)p Fu(+)q(1\).)0 4475 y(Pro)r(of:)0 4616 y(a\))31 b(The)g(\014rst)g(part)f (is)h(trivial.)47 b({)30 b(Second)h(part:)43 b Fo(\013)30 b(<)e(\014)33 b Fu(&)28 b Fo(\013)i Fr(2)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))30 b Fr(\))f Fo( )s Fu(\()p Fo(\013)p Fu(\))g Fr(\024)g Fo( )s Fu(\()p Fo(\014)t Fu(\))h(&)e Fo( )s Fu(\()p Fo(\013)p Fu(\))i Fr(2)f Fo(C)6 b Fu(\()p Fo(\014)t Fu(\))30 b Fr(\))f Fo( )s Fu(\()p Fo(\013)p Fu(\))g Fo(<)0 4757 y( )s Fu(\()p Fo(\014)t Fu(\).)38 b(The)28 b(last)f(implication)g(holds,)h(since)f Fo( )s Fu(\()p Fo(\014)t Fu(\))d Fr(62)g Fo(C)6 b Fu(\()p Fo(\014)t Fu(\).)0 4923 y(b\))28 b Fo(car)r(d)p Fu(\()p Fo(C)6 b Fu(\()p Fo(\013)p Fu(\)\))26 b Fo(<)c Fu(\012)h Fr(\))h(9)p Fo(\030)j(<)22 b Fu(\012\()p Fo(\030)28 b Fr(62)23 b Fo(C)6 b Fu(\()p Fo(\013)p Fu(\)\))25 b Fr(\))e Fo( )s Fu(\()p Fo(\013)p Fu(\))h Fo(<)f Fu(\012.)0 5089 y(c\))28 b(Assume)g(not.)36 b(Then)28 b Fo( )s Fu(\()p Fo(\013)p Fu(\))c(=)f Fo(\030)f Fu(+)d Fo(\021)30 b Fu(with)f Fo(\030)t(;)14 b(\021)26 b(<)c( )s Fu(\()p Fo(\013)p Fu(\),)29 b(and)f(th)n(us)f Fo( )s Fu(\()p Fo(\013)p Fu(\))d(=)f Fo(\030)g Fu(+)18 b Fo(\021)26 b Fr(2)d Fo(C)6 b Fu(\()p Fo(\013)p Fu(\).)39 b(Con)n(tradiction.)1908 5539 y(49)p eop %%Page: 50 51 50 50 bop 0 83 a Fu(d\))28 b(\\)p Fr(\022)p Fu(")e(follo)n(ws)h(from)g (b\).)0 222 y(Pro)r(of)f(of)i(\\)p Fr(\023)p Fu(")e(b)n(y)i Fo(C)6 b Fu(-induction,)28 b(i.e.)37 b(induction)27 b(on)h(the)g (de\014nition)g(of)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\):)0 361 y(Let)28 b Fo(\015)f Fr(2)d Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))20 b Fr(\\)e Fu(\012.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)f(to)g(pro)n (v)n(e)f Fo(\015)i(<)22 b( )s Fu(\()p Fo(\013)p Fu(\).)0 523 y(1.)36 b Fo(\015)28 b Fu(=)23 b(0:)36 b(0)23 b Fo(<)f( )s Fu(\()p Fo(\013)p Fu(\),)29 b(since)f(0)22 b Fr(2)i Fo(C)6 b Fu(\()p Fo(\013)p Fu(\).)0 662 y(2.)34 b Fo(\015)28 b Fu(=)23 b Fo(\030)9 b Fu(+)c Fo(\021)23 b Fu(with)f Fo(\030)t(;)14 b(\021)26 b Fr(2)e Fo(C)6 b Fu(\()p Fo(\013)p Fu(\):)34 b(Then)22 b Fo(\030)t(;)14 b(\021)26 b(<)c Fu(\012,)h(and)e(therefore)e(b)n(y)i(IH)g Fo(\030)t(;)14 b(\021)27 b(<)22 b( )s Fu(\()p Fo(\013)p Fu(\).)36 b(By)21 b(c\))g(w)n(e)f(get)h Fo(\015)28 b Fu(=)22 b Fo(\030)9 b Fu(+)c Fo(\021)26 b(<)d( )s Fu(\()p Fo(\013)p Fu(\).)0 801 y(3.)36 b Fo(\015)28 b Fu(=)23 b Fo(!)315 771 y Fn(\012+)p Fp(\030)449 801 y Fu(:)37 b(This)28 b(cannot)f(b)r(e.)0 940 y(4.)36 b Fo(\015)28 b Fu(=)23 b Fo( )s Fu(\()p Fo(\030)t Fu(\))28 b(with)g Fo(\030)g(<)22 b(\013)28 b Fu(and)g Fo(\030)f Fr(2)c Fo(C)6 b Fu(\()p Fo(\030)t Fu(\):)38 b(In)28 b(this)g(case)f(the)h(claim)f(follo)n(ws)g(b)n(y)g(a\).)0 1102 y(e\))h(\\)p Fr(\))p Fu(")f(follo)n(ws)f(from)h(a\).)37 b(\\)p Fr(\()p Fu(":)f Fo(\013)24 b Fr(62)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))24 b Fr(\))f Fo(C)6 b Fu(\()p Fo(\013)p Fu(+)q(1\))23 b(=)g Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))24 b Fr(\))f Fo( )s Fu(\()p Fo(\013)p Fu(+1\))g(=)g Fo( )s Fu(\()p Fo(\013)p Fu(\).)0 1310 y Ft(Remark)54 b Fo( )s Fu(\()p Fo(\013)528 1322 y Fn(0)566 1310 y Fu(\))23 b(=)g Fo( )s Fu(\()p Fo(\013)851 1322 y Fn(1)889 1310 y Fu(\))g(&)g Fo(\013)1085 1322 y Fp(i)1136 1310 y Fr(2)g Fo(C)6 b Fu(\()p Fo(\013)1364 1322 y Fp(i)1392 1310 y Fu(\))37 b(\()p Fo(i)23 b Fu(=)g(0)p Fo(;)14 b Fu(1\))45 b Fr(\))i Fo(\013)2014 1322 y Fn(0)2074 1310 y Fu(=)23 b Fo(\013)2215 1322 y Fn(1)2252 1310 y Fu(.)0 1472 y Ft(Lemma)30 b(8.2)0 1611 y Fu(a\))d Fo(!)156 1581 y Fp(\016)186 1589 y Fg(1)223 1611 y Fu(#)p Fo(:::)p Fu(#)p Fo(!)485 1581 y Fp(\016)515 1589 y Fh(n)583 1611 y Fr(2)d Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))47 b Fr(,)f Fo(!)1075 1581 y Fp(\016)1105 1589 y Fg(1)1141 1611 y Fo(;)14 b(:::;)g(!)1339 1581 y Fp(\016)1369 1589 y Fh(n)1437 1611 y Fr(2)24 b Fo(C)6 b Fu(\()p Fo(\013)p Fu(\),)0 1750 y(b\))28 b Fo(\030)23 b Fu(+)18 b Fo(\021)26 b Fr(2)d Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))25 b Fr(\))e Fo(\021)j Fr(2)d Fo(C)6 b Fu(\()p Fo(\013)p Fu(\).)0 1912 y(Pro)r(of:)0 2051 y(a\))27 b(Pro)r(of)g(of)g (\\)p Fr(\))p Fu(")g(b)n(y)g(C-induction.)37 b(W.l.o.g.)g Fo(\016)1579 2063 y Fn(0)1639 2051 y Fr(\025)23 b Fo(:)14 b(:)g(:)23 b Fr(\025)f Fo(\016)1971 2063 y Fp(n)2016 2051 y Fu(.)0 2190 y(1.)36 b Fo(!)156 2160 y Fp(\016)186 2168 y Fg(1)241 2190 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(!)550 2160 y Fp(\016)580 2168 y Fh(n)648 2190 y Fu(=)k Fo(\030)h Fu(+)18 b Fo(\021)31 b Fu(with)d Fo(\030)t(;)14 b(\021)26 b Fr(2)e Fo(C)6 b Fu(\()p Fo(\013)p Fu(\):)0 2329 y(Then)26 b Fo(\030)i Fu(=)344 2341 y Fp(N)6 b(F)480 2329 y Fo(!)535 2299 y Fp(\016)565 2307 y Fg(1)617 2329 y Fu(+)15 b Fo(:::)h Fu(+)g Fo(!)918 2299 y Fp(\016)948 2308 y Fh(k)1003 2329 y Fu(+)g Fo(!)1139 2299 y Fp(\030)1169 2307 y Fg(1)1221 2329 y Fu(+)f Fo(:::)h Fu(+)f Fo(!)1521 2299 y Fp(\030)1551 2308 y Fh(k)1618 2329 y Fu(and)26 b Fo(\021)g Fu(=)1910 2341 y Fp(N)6 b(F)2047 2329 y Fo(!)2102 2299 y Fp(\016)2132 2308 y Fh(k)q Fg(+1)2258 2329 y Fu(+)16 b Fo(:::)g Fu(+)f Fo(!)2559 2299 y Fp(\016)2589 2307 y Fh(n)2633 2329 y Fu(.)37 b(By)26 b(IH)g(w)n(e)g(get)g Fo(!)3253 2299 y Fp(\016)3283 2307 y Fg(1)3320 2329 y Fo(;)14 b(:::;)g(!)3518 2299 y Fp(\016)3548 2307 y Fh(n)3615 2329 y Fr(2)24 b Fo(C)6 b Fu(\()p Fo(\013)p Fu(\).)0 2469 y(2.)36 b Fo(!)156 2438 y Fp(\016)186 2446 y Fg(1)241 2469 y Fu(+)18 b Fo(:::)h Fu(+)f Fo(!)550 2438 y Fp(\016)580 2446 y Fh(n)648 2469 y Fu(=)k Fo( )s Fu(\()p Fo(\030)t Fu(\))29 b(or)e Fo(!)1082 2438 y Fn(\012+)p Fp(\030)1216 2469 y Fu(:)37 b(Then)28 b Fo(n)23 b Fu(=)f(1)27 b(and)h(the)g(claim)f(is)h(trivial.)0 2631 y(b\))g(follo)n(ws)f(from)g(a\).)0 2838 y Ft(Lemma)j(8.3)0 2977 y Fu(a\))d Fo( )s Fu(\()p Fo(\013)p Fu(\))e(=)d(min)q Fr(f)p Fo(\016)j Fr(2)f Fo(AP)35 b Fu(:)23 b Fr(8)p Fo(\030)j(<)d(\013) p Fu(\()p Fo(\030)28 b Fr(2)23 b Fo(C)6 b Fu(\()p Fo(\030)t Fu(\))24 b Fr(\))f Fo( )s Fu(\()p Fo(\030)t Fu(\))h Fo(<)f(\016)s Fu(\))p Fr(g)p Fu(.)0 3116 y(b\))28 b Fo(\013)c Fr(\024)e Fo(")309 3128 y Fn(0)392 3116 y Fr(\))46 b Fo( )s Fu(\()p Fo(\013)p Fu(\))25 b(=)d Fo(!)862 3086 y Fp(\013)909 3116 y Fu(.)0 3278 y(Pro)r(of:)0 3417 y(a\))27 b(Let)h Fo(\016)287 3429 y Fn(0)347 3417 y Fu(:=)23 b(min)p Fr(f)p Fo(\016)j Fr(2)e Fo(AP)35 b Fu(:)23 b Fr(8)p Fo(\030)j(<)d(\013)p Fu(\()p Fo(\030)28 b Fr(2)23 b Fo(C)6 b Fu(\()p Fo(\030)t Fu(\))24 b Fr(\))f Fo( )s Fu(\()p Fo(\030)t Fu(\))h Fo(<)f(\016)s Fu(\))p Fr(g)p Fu(.)0 3557 y(By)k(8.1a,c,d)g(w)n(e)g(ha)n(v)n(e)f Fo(\016)785 3569 y Fn(0)846 3557 y Fr(\024)c Fo( )s Fu(\()p Fo(\013)p Fu(\))i(=)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))20 b Fr(\\)e Fu(\012.)37 b(By)28 b(C-induction)f(w)n(e)h(get)f Fo(C)6 b Fu(\()p Fo(\013)p Fu(\))20 b Fr(\\)e Fu(\012)23 b Fr(\022)g Fo(\016)2945 3569 y Fn(0)2982 3557 y Fu(.)0 3696 y(b\))28 b(Induction)g(on)f Fo(\013)p Fu(:)38 b(Let)27 b Fo(\013)d Fr(\024)f Fo(")1063 3708 y Fn(0)1099 3696 y Fu(.)37 b(Then)28 b(b)n(y)g(IH)f Fr(8)p Fo(\030)g(<)22 b(\013)p Fu(\()p Fo(\030)28 b(<)23 b(!)2100 3666 y Fp(\030)2159 3696 y Fu(=)g Fo( )s Fu(\()p Fo(\030)t Fu(\)\).)0 3835 y(Hence)28 b Fo( )s Fu(\()p Fo(\013)p Fu(\))c(=)f(min)p Fr(f)p Fo(\016)i Fr(2)f Fo(AP)35 b Fu(:)23 b Fr(8)p Fo(\030)k(<)22 b(\013)p Fu(\()p Fo(!)1387 3805 y Fp(\030)1447 3835 y Fo(<)h(\016)s Fu(\))p Fr(g)f Fu(=)h Fo(!)1814 3805 y Fp(\013)1861 3835 y Fu(.)0 4042 y Fq(Inductiv)n(e)28 b(de\014nition)g(of)f(the)h(set)g Fu(T)f Fq(of)h(terms)0 4182 y Fu(\(T1\))55 b(0)23 b Fr(2)g Fu(T.)0 4321 y(\(T2\))55 b Fo(a)23 b Fr(2)h Fu(T)f(&)f Fo(\033)27 b Fr(2)c(f)p Fu(0)p Fo(;)14 b Fu(1)p Fr(g)44 b(\))j Fo(D)1137 4333 y Fp(\033)1181 4321 y Fo(a)23 b Fr(2)g Fu(T)56 b(\(terms)27 b(of)h(this)g(kind)f(are)g(called)g Fm(princip)l(al)32 b(terms)p Fu(\))0 4460 y(\(T3\))55 b Fo(a)265 4472 y Fn(0)303 4460 y Fo(;)14 b(:::;)g(a)490 4472 y Fp(n)558 4460 y Fr(2)23 b Fu(T)g(\()p Fo(n)g Fr(\025)g Fu(1\))k(principal)h (terms)50 b Fr(\))c Fu(\()p Fo(a)1819 4472 y Fn(0)1856 4460 y Fo(;)14 b(::;)g(a)2020 4472 y Fp(n)2065 4460 y Fu(\))24 b Fr(2)f Fu(T)0 4622 y(Notation:)36 b(W)-7 b(e)28 b(use)g Fo(a;)14 b(b;)g(c)27 b Fu(as)g(syn)n(tactic)f(v)-5 b(ariables)27 b(for)g(elemen)n(ts)g(of)h(T.)37 b(F)-7 b(or)26 b(principal)i(terms)f Fo(a)g Fu(w)n(e)g(set)h(\()p Fo(a)p Fu(\))23 b(:=)g Fo(a)p Fu(.)37 b(The)0 4761 y(empt)n(y)28 b(sequence)f(\()14 b(\))28 b(is)f(iden)n(ti\014ed)h(with)h(0.)0 4923 y Fq(De\014nition)e(of)h Fo(a)23 b Fr(\036)g Fo(b)k Fq(for)g Fo(a;)14 b(b)22 b Fr(2)i Fu(T)0 5062 y Fo(a)f Fr(\036)g Fo(b)k Fu(if)h(and)f(only)h(if)g(one)f(of)g(the)h(follo)n (wing)f(cases)f(holds)0 5201 y(\()p Fr(\036)d Fu(1\))55 b Fo(a)23 b Fu(=)g(0)k(and)g Fo(b)c Fr(6)p Fu(=)g(0)0 5340 y(\()p Fr(\036)g Fu(2\))55 b(\()p Fo(a)23 b Fu(=)g Fo(D)505 5352 y Fn(0)542 5340 y Fo(a)586 5352 y Fn(0)646 5340 y Fu(&)g Fo(b)f Fu(=)h Fo(D)949 5352 y Fn(1)986 5340 y Fo(b)1022 5352 y Fn(0)1059 5340 y Fu(\))28 b(or)f(\()p Fo(a)c Fu(=)f Fo(D)1476 5352 y Fp(\033)1521 5340 y Fo(a)1565 5352 y Fn(0)1625 5340 y Fu(&)h Fo(b)f Fu(=)h Fo(D)1928 5352 y Fp(\033)1972 5340 y Fo(b)2008 5352 y Fn(0)2068 5340 y Fu(&)g Fo(a)2200 5352 y Fn(0)2260 5340 y Fr(\036)g Fo(b)2384 5352 y Fn(0)2421 5340 y Fu(\))1908 5539 y(50)p eop %%Page: 51 52 51 51 bop 0 83 a Fu(\()p Fr(\036)23 b Fu(3\))55 b Fo(a)23 b Fu(=)g(\()p Fo(a)480 95 y Fn(0)517 83 y Fo(;)14 b(:::;)g(a)704 95 y Fp(n)749 83 y Fu(\))23 b(&)g Fo(b)g Fu(=)f(\()p Fo(b)1106 95 y Fn(0)1144 83 y Fo(;)14 b(:::;)g(b)1323 95 y Fp(m)1385 83 y Fu(\))24 b(&)e(1)h Fr(\024)f Fo(m)p Fu(+)p Fo(n)h Fu(&)249 224 y([\()p Fo(n)g(<)g(m)g Fu(&)g Fr(8)p Fo(i<)n(n)p Fu(\()p Fo(a)914 236 y Fp(i)965 224 y Fu(=)f Fo(b)1088 236 y Fp(i)1115 224 y Fu(\))28 b(or)f Fr(9)p Fo(k)f Fr(\024)d Fu(min)p Fr(f)p Fo(m;)14 b(n)p Fr(g)p Fu(\()p Fo(a)1938 236 y Fp(k)2001 224 y Fr(\036)22 b Fo(b)2124 236 y Fp(k)2188 224 y Fu(&)h Fr(8)p Fo(i<)n(k)s Fu(\()p Fo(a)2537 236 y Fp(i)2588 224 y Fu(=)f Fo(b)2711 236 y Fp(i)2739 224 y Fu(\)])0 390 y Ft(Lemma)30 b(8.4)54 b Fo(a)23 b Fr(6)p Fu(=)f Fo(b)46 b Fr(\))g Fo(a)23 b Fr(\036)g Fo(b)k Fu(or)g Fo(b)c Fr(\036)f Fo(a)p Fu(.)0 606 y Fq(De\014nition)27 b(of)h Fu(K)548 576 y Fi(\003)586 606 y Fo(a)f Fq(for)g Fo(a)c Fr(2)h Fu(T)0 747 y(1.)36 b(K)166 717 y Fi(\003)204 747 y Fu(0)23 b(:=)f Fr(;)0 888 y Fu(2.)36 b(K)166 858 y Fi(\003)204 888 y Fu(\()p Fo(a)280 900 y Fn(0)318 888 y Fo(;)14 b(:::;)g(a)505 900 y Fp(n)550 888 y Fu(\))23 b(:=)716 826 y Fl(S)785 913 y Fp(i)p Fi(\024)p Fp(n)919 888 y Fu(K)984 858 y Fi(\003)1022 888 y Fo(a)1066 900 y Fp(i)0 1029 y Fu(3.)36 b(K)166 999 y Fi(\003)204 1029 y Fo(D)273 1041 y Fn(1)310 1029 y Fo(a)23 b Fu(:=)g(K)553 999 y Fi(\003)591 1029 y Fo(a)0 1171 y Fu(4.)36 b(K)166 1140 y Fi(\003)204 1171 y Fo(D)273 1183 y Fn(0)310 1171 y Fo(a)23 b Fu(:=)g Fr(f)p Fo(a)p Fr(g)0 1337 y Fu(Abbreviation:)64 b(K)628 1307 y Fi(\003)666 1337 y Fo(a)23 b Fr(\036)f Fo(b)46 b Fu(:)p Fr(,)g(8)p Fo(x)p 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y Fu(\012)g Fr(\036)f Fu(\012.)46 b(Hence)31 b(the)g(mapping)f Fo(a)e Fr(7!)g Fo(o)p Fu(\()p Fo(a)p Fu(\))j(cannot)f(b)r(e)h(order)0 2374 y(preserving.)0 2590 y Fq(Inductiv)n(e)d(de\014nition)g(of)f(the)h (set)g Fu(OT)22 b Fr(\022)h Fu(T)0 2731 y(\(OT1\))55 b(0)23 b Fr(2)g Fu(OT)o(.)0 2873 y(\(OT2\))55 b Fo(a)330 2885 y Fn(0)367 2873 y Fo(;)14 b(:::;)g(a)554 2885 y Fp(n)622 2873 y Fr(2)24 b Fu(OT)e(\()p Fo(n)h Fr(\025)g Fu(1\))28 b(principal)f(terms)50 b(&)23 b Fo(\013)1884 2885 y Fp(n)1952 2873 y Fr(\026)g Fo(:::)g Fr(\026)g Fo(a)2264 2885 y Fn(0)2347 2873 y Fr(\))46 b Fu(\()p Fo(a)2552 2885 y Fn(0)2589 2873 y Fo(;)14 b(::;)g(a)2753 2885 y Fp(n)2798 2873 y Fu(\))24 b Fr(2)f Fu(OT)0 3014 y(\(OT3\))55 b Fo(a)23 b Fr(2)g Fu(OT)46 b Fr(\))g Fo(D)800 3026 y Fn(1)837 3014 y Fo(a)23 b Fr(2)g Fu(OT)0 3155 y(\(OT4\))55 b Fo(a)23 b Fr(2)g Fu(OT)g(&)g(K)732 3125 y Fi(\003)769 3155 y Fo(a)g Fr(\036)g Fo(a)46 b Fr(\))g Fo(D)1212 3167 y Fn(0)1249 3155 y Fo(a)23 b Fr(2)g Fu(OT)0 3346 y Ft(Theorem)30 b(8.5)55 b Fu(F)-7 b(or)27 b Fo(a;)14 b(c)23 b Fr(2)g Fu(OT)k(the)h(follo)n(wing)f(holds:)0 3487 y(a\))g Fo(c)c Fr(\036)g Fo(a)46 b Fr(,)g Fo(o)p Fu(\()p Fo(c)p Fu(\))24 b Fo(<)f(o)p Fu(\()p Fo(a)p Fu(\),)0 3628 y(b\))28 b(K)171 3598 y Fi(\003)209 3628 y Fo(c)23 b Fr(\036)f Fo(a)46 b Fr(,)h Fo(o)p Fu(\()p Fo(c)p Fu(\))23 b Fr(2)h Fo(C)6 b Fu(\()p Fo(o)p Fu(\()p Fo(a)p Fu(\)\).)0 3794 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f(the)h(length)g(of)f Fo(c)p Fu(:)0 3935 y(Due)h(to)g(8.4)e(for)h(a\))h(it)g(su\016ces)f(to)h (pro)n(v)n(e)e(\\)p Fr(\))p Fu(".)0 4101 y(1.)36 b Fo(c)23 b Fu(=)g(0:)37 b(a\))27 b Fo(c)c Fr(\036)g Fo(a)g Fr(\))g Fo(a)g Fr(6)p Fu(=)f(0)h Fr(\))g Fo(o)p Fu(\()p Fo(a)p Fu(\))h Fr(6)p Fu(=)e(0)h Fr(\))g Fo(o)p Fu(\()p Fo(c)p Fu(\))h Fo(<)e(o)p Fu(\()p Fo(a)p Fu(\).)0 4242 y(b\))28 b(K)171 4212 y Fi(\003)209 4242 y Fo(c)23 b Fu(=)f Fr(;)28 b Fu(aand)f Fo(o)p Fu(\()p Fo(c)p Fu(\))c(=)g(0)g Fr(2)g Fo(C)6 b Fu(\()p Fo(o)p Fu(\()p Fo(a)p Fu(\)\).)0 4408 y(2.)36 b Fo(c)23 b 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b Fo(\033)h Fu(=)c Fo(\034)37 b Fu(=)26 b(1)j(this)h(yields)0 4973 y Fo(o)p Fu(\()p Fo(c)p Fu(\))e Fo(<)f(o)p Fu(\()p Fo(a)p Fu(\).)46 b(F)-7 b(or)30 b Fo(\033)h Fu(=)c Fo(\034)37 b Fu(=)27 b(0,)k(since)f Fo(c)d Fr(2)h Fu(OT,)j(w)n(e)f(ha)n(v)n(e)f(K) 1975 4943 y Fi(\003)2013 4973 y Fo(b)e Fr(\036)g Fo(b)j Fu(and)g(therefore)f(b)n(y)h(IH)h Fo(o)p Fu(\()p Fo(b)p Fu(\))d Fr(2)g Fo(C)6 b Fu(\()p Fo(o)p Fu(\()p Fo(b)p Fu(\)\).)46 b(T)-7 b(ogether)0 5114 y(with)28 b Fo(o)p Fu(\()p Fo(b)p Fu(\))c Fo(<)e(o)p Fu(\()509 5092 y(~)512 5114 y Fo(b)p Fu(\))28 b(from)g(this)g(w)n(e)f(get)g Fo(o)p Fu(\()p Fo(c)p Fu(\))d(=)f Fo( )s Fu(\()p Fo(o)p Fu(\()p Fo(b)p Fu(\)\))h Fo(<)e( )s Fu(\()p Fo(o)p Fu(\()2009 5092 y(~)2012 5114 y Fo(b)q Fu(\)\))h(=)g Fo(o)p Fu(\()p Fo(D)2365 5126 y Fn(0)2400 5092 y Fu(~)2403 5114 y Fo(b)o Fu(\))h Fr(\024)e Fo(o)p Fu(\()p Fo(a)p Fu(\).)1908 5539 y(51)p eop %%Page: 52 53 52 52 bop 0 83 a Fu(b\))28 b(\(1\))g Fo(c)23 b Fu(=)f Fo(D)455 95 y Fn(1)492 83 y Fo(b)p Fu(:)37 b(K)653 53 y Fi(\003)691 83 y Fo(c)23 b Fu(=)f(K)902 53 y Fi(\003)940 83 y Fo(b)55 b Fu(and)g Fo(o)p Fu(\()p Fo(c)p Fu(\))24 b Fr(2)f Fo(C)6 b Fu(\()p Fo(o)p Fu(\()p Fo(a)p Fu(\)\))25 b Fr(,)e Fo(o)p Fu(\()p Fo(b)p Fu(\))h Fr(2)f Fo(C)6 b Fu(\()p Fo(o)p Fu(\()p Fo(a)p Fu(\)\).)0 224 y(\(2\))28 b Fo(c)23 b Fu(=)f Fo(D)349 236 y Fn(0)386 224 y Fo(b)p Fu(:)37 b(Then)28 b(K)764 194 y Fi(\003)801 224 y Fo(c)23 b Fu(=)g Fr(f)p Fo(b)p Fr(g)j Fu(and)i(K)1321 194 y Fi(\003)1359 224 y Fo(b)22 b Fr(\036)h Fo(b)p Fu(.)37 b(By)27 b(IH)h(the)g(latter)f (yields)g Fo(o)p Fu(\()p Fo(b)p Fu(\))d Fr(2)f Fo(C)6 b Fu(\()p Fo(o)p Fu(\()p Fo(b)p Fu(\)\))29 b(and)f(then:)0 365 y(K)65 335 y Fi(\003)103 365 y Fo(c)23 b Fr(\036)f Fo(a)h Fr(,)g Fo(b)g Fr(\036)g Fo(a)641 318 y Fn(IH)636 365 y Fr(,)g Fo(o)p Fu(\()p Fo(b)p Fu(\))g Fo(<)g(o)p Fu(\()p Fo(a)p Fu(\))h Fr(,)f Fo( )s Fu(\()p Fo(o)p Fu(\()p Fo(b)p Fu(\)\))h Fo(<)f( )s Fu(\()p Fo(o)p Fu(\()p Fo(a)p Fu(\)\))h Fr(,)f Fo(o)p Fu(\()p Fo(c)p Fu(\))h(=)e Fo( )s 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Fo(a)g Fu(=)g Fo(D)224 2220 y Fn(0)261 2208 y Fo(b)f Fr(2)i Fu(OT)e Fr(\))h Fo(b)g Fr(2)g Fu(OT)g(&)g(K)1089 2178 y Fi(\003)1126 2208 y Fo(b)g Fr(\036)g Fo(b)1336 2161 y Fn(IH)1331 2208 y Fr(\))h Fo(o)p Fu(\()p Fo(b)p Fu(\))f Fr(2)g Fo(C)6 b Fu(\()p Fo(")1815 2220 y Fn(\012+1)1951 2208 y Fu(\))24 b(&)e Fo(o)p Fu(\()p Fo(b)p Fu(\))i Fr(2)f Fo(C)6 b Fu(\(o\()p Fo(b)p Fu(\)\))2631 2154 y Fn(\(+\))2641 2208 y Fr(\))33 b Fo(o)p Fu(\()p Fo(a)p Fu(\))24 b(=)e Fo( )s Fu(\()p Fo(o)p Fu(\()p Fo(b)p Fu(\)\))i Fr(2)g Fo(C)6 b Fu(\()p Fo(")3516 2220 y Fn(\012+1)3652 2208 y Fu(\).)0 2374 y(2.)36 b(F)-7 b(rom)28 b(8.4)e(and)i(8.5a)e(it)i (follo)n(ws)f(that)h Fo(o)p Fr(j)p Fu(OT)f(is)h(injectiv)n(e.)0 2540 y(3.)36 b(By)28 b(C-induction)f(w)n(e)h(pro)n(v)n(e:)35 b Fo(\015)27 b Fr(2)d Fo(C)6 b Fu(\()p Fo(")1352 2552 y Fn(\012+1)1488 2540 y Fu(\))23 b Fr(\))g(9)p Fo(a)g Fr(2)h Fu(OT)o(\()p Fo(\015)k Fu(=)23 b Fo(o)p Fu(\()p Fo(a)p Fu(\)\).)0 2682 y(3.1.)36 b Fo(\015)28 b Fu(=)22 b(0:)37 b(trivial.)0 2823 y(3.2.)f Fo(\015)28 b Fu(=)22 b Fo(\030)h Fu(+)18 b Fo(\021)s Fu(:)37 b(.....)g Fo(\030)27 b Fu(=)c Fo(o)p Fu(\(\()p Fo(a)1021 2835 y Fn(0)1059 2823 y Fo(;)14 b(:::;)g(a)1246 2835 y Fp(n)1291 2823 y Fu(\)\),)28 b Fo(\021)e Fu(=)d Fo(o)p Fu(\(\()p Fo(b)1701 2835 y Fn(0)1739 2823 y Fo(;)14 b(:::;)g(b)1918 2835 y Fp(m)1980 2823 y Fu(\)\))28 b(with)g Fo(a)2305 2835 y Fp(i)2333 2823 y Fo(;)14 b(b)2406 2835 y Fp(i)2456 2823 y Fr(2)24 b Fu(OT)j(and)g Fo(a)2892 2835 y Fp(n)2960 2823 y Fr(\026)c Fo(:)14 b(:)g(:)23 b Fr(\026)f Fo(a)3299 2835 y Fn(0)3336 2823 y Fu(,)28 b Fo(b)3423 2835 y Fp(m)3509 2823 y Fr(\026)23 b Fo(:)14 b(:)g(:)22 b Fr(\026)h Fo(b)3840 2835 y Fn(0)3877 2823 y Fu(.)0 2964 y(Let)28 b Fo(b)185 2976 y Fp(m)271 2964 y Fr(\026)22 b Fo(a)402 2976 y Fp(k)471 2964 y Fu(and)27 b Fo(a)676 2976 y Fp(k)q Fn(+1)801 2964 y Fo(;)14 b(:::;)g(a)988 2976 y Fp(n)1056 2964 y Fr(\036)23 b Fo(b)1180 2976 y Fp(m)1242 2964 y Fu(.)37 b(Then)28 b Fo(a)23 b Fu(:=)g(\()p Fo(a)1773 2976 y Fn(0)1810 2964 y Fo(;)14 b(:::;)g(a)1997 2976 y Fp(k)2038 2964 y Fo(;)g(b)2111 2976 y Fn(0)2148 2964 y Fo(;)g(:::;)g(b)2327 2976 y Fp(m)2389 2964 y Fu(\))24 b Fr(2)f Fu(OT)k(and)h Fo(\015)f Fu(=)c Fo(o)p Fu(\()p Fo(a)p Fu(\).)0 3105 y(3.3.)37 b Fo(\015)28 b Fu(=)23 b Fo(!)381 3075 y Fn(\012+)p Fp(\015)514 3083 y Fg(0)578 3105 y Fu(with)28 b Fo(\015)810 3117 y Fn(0)871 3105 y Fr(2)c Fo(C)6 b Fu(\()p Fo(")1086 3117 y Fn(\012+1)1222 3105 y Fu(\):)37 b(By)28 b(IH)g(there)g(is)f Fo(a)1905 3117 y Fn(0)1966 3105 y Fr(2)d Fu(OT)j(with)i Fo(\015)2430 3117 y Fn(0)2490 3105 y Fu(=)23 b Fo(o)p Fu(\()p Fo(a)2694 3117 y Fn(0)2732 3105 y Fu(\).)38 b(Hence)28 b Fo(\015)g Fu(=)23 b Fo(o)p Fu(\()p Fo(D)3372 3117 y Fn(1)3409 3105 y Fo(a)3453 3117 y Fn(0)3490 3105 y Fu(\))h(&)f Fo(D)3703 3117 y Fn(1)3740 3105 y Fo(a)3784 3117 y Fn(0)3845 3105 y Fr(2)0 3246 y Fu(OT.)0 3387 y(3.4.)36 b Fo(\015)28 b Fu(=)22 b Fo( )s Fu(\()p Fo(\030)t Fu(\))29 b(with)f Fo(\030)f(<)c(\013)g Fu(&)g Fo(\030)k Fr(2)c Fo(C)6 b Fu(\()p Fo(")1295 3399 y Fn(\012+1)1431 3387 y Fu(\))24 b(&)e Fo(\030)28 b Fr(2)23 b Fo(C)6 b Fu(\()p Fo(\030)t Fu(\):)0 3528 y(By)27 b(IH)h(w)n(e)f(ha)n(v)n(e)g Fo(b)c Fr(2)g Fu(OT)k(with)h Fo(\030)f Fu(=)c Fo(o)p Fu(\()p Fo(b)p Fu(\).)65 b Fo(o)p Fu(\()p Fo(b)p Fu(\))24 b Fr(2)f Fo(C)6 b Fu(\(o\()p Fo(b)p Fu(\)\))1958 3481 y Fn(L)p Fp(:)p Fn(8)p Fp(:)p Fn(5)1990 3528 y Fr(\))78 b Fu(K)2216 3498 y Fi(\003)2253 3528 y Fo(b)23 b Fr(\036)g Fo(b)f Fr(\))h Fo(D)2633 3540 y Fn(0)2670 3528 y Fo(b)g Fr(2)g Fu(OT)28 b(and)f Fo(\015)h Fu(=)22 b Fo(o)p Fu(\()p Fo(D)3420 3540 y Fn(0)3458 3528 y Fo(b)p Fu(\).)0 3744 y Ft(De\014nition)54 b Fu(OT)594 3756 y Fn(0)654 3744 y Fu(:=)23 b Fr(f)p Fo(a)f Fr(2)i Fu(OT)e(:)h Fo(a)g Fr(\036)g Fo(D)1369 3756 y Fn(1)1406 3744 y Fu(0)p Fr(g)0 3910 y Ft(Lemma)30 b(8.7)0 4051 y Fu(a\))d Fo( )s Fu(\()p Fo(")229 4063 y Fn(\012+1)365 4051 y Fu(\))c(=)g Fr(k)p Fu(\(OT)706 4063 y Fn(0)744 4051 y Fo(;)14 b Fr(\036)p Fu(\))p Fr(k)27 b Fu(\(order)f(t)n(yp)r(e)i(of)g(the)f(w)n(ellordering)f(\(OT)2246 4063 y Fn(0)2283 4051 y Fo(;)14 b Fr(\036)p Fu(\)\))0 4193 y(b\))28 b(F)-7 b(or)27 b(eac)n(h)g Fo(a)c Fr(2)g Fu(OT)k(suc)n(h)h(that)g Fo(a)23 b Fr(\036)f Fo(D)1330 4205 y Fn(1)1367 4193 y Fo(a)28 b Fu(w)n(e)f(ha)n(v)n(e)54 b Fo(o)p Fu(\()p Fo(a)p Fu(\))24 b(=)e(sup)p Fr(f)p Fo(o)p Fu(\()p Fo(b)p Fu(\)+1)g(:)i Fo(b)e Fr(2)i Fu(OT)e(&)h Fo(b)g Fr(\036)f Fo(a)p Fr(g)p Fu(.)0 4359 y(Pro)r(of:)0 4500 y Fo(a)29 b Fr(7!)g Fo(o)p Fu(\()p Fo(a)p Fu(\))j(maps)f Fr(f)p Fo(a)e Fr(2)g Fu(OT)g(:)g Fo(a)h Fr(\036)e Fo(D)1227 4512 y Fn(1)1264 4500 y Fu(0)p Fr(g)j Fu(order)f(preserving)f(on)n(to)i Fo(C)6 b Fu(\()p Fo(")2330 4512 y Fn(\012+1)2466 4500 y Fu(\))21 b Fr(\\)g Fu(\012)29 b(=)g Fo( )s Fu(\()p Fo(")2906 4512 y Fn(\012+1)3042 4500 y Fu(\).)48 b(Hence)31 b Fr(k)p Fu(\(OT)3594 4512 y Fn(0)3631 4500 y Fo(;)14 b Fr(\036)p Fu(\))p Fr(k)28 b Fu(=)0 4641 y Fo( )s Fu(\()p Fo(")128 4653 y Fn(\012+1)264 4641 y Fu(\))g(and)f(further)h(\(for)g Fo(a)23 b Fr(2)g Fu(OT)1195 4653 y Fn(0)1232 4641 y Fu(\))28 b Fo(o)p Fu(\()p Fo(a)p Fu(\))c(=)f(sup)p Fr(f)p Fo(\014)t Fu(+1)g(:)g Fo(\014)28 b(<)22 b(o)p Fu(\()p Fo(a)p Fu(\))p Fr(g)i Fu(=)f(sup)p Fr(f)p Fo(o)p Fu(\()p Fo(b)p Fu(\)+)o(1)g(:)h Fo(b)f Fr(2)g Fu(OT)g(&)g Fo(b)g Fr(\036)g Fo(D)3482 4653 y Fn(1)3519 4641 y Fu(0)g(&)g Fo(o)p Fu(\()p Fo(b)p Fu(\))g Fo(<)0 4782 y(o)p Fu(\()p Fo(a)p Fu(\))p Fr(g)g Fu(=)g(sup)p Fr(f)p Fo(o)p Fu(\()p Fo(b)p Fu(\)+)o(1)g(:)g Fo(b)g Fr(2)g Fu(OT)g(&)f Fo(b)h Fr(\036)g Fo(a)p Fr(g)p Fu(.)1908 5539 y(52)p eop %%Page: 53 54 53 53 bop 0 83 a Fr(x)p Ft(9)63 b(Ordinal)32 b(analysis)f(of)h Fu(ID)1055 95 y Fn(1)0 294 y Ft(Inductiv)m(e)h(De\014nitions)0 482 y Fu(Let)28 b Fo(M)36 b Fu(b)r(e)28 b(a)f(set)h(and)f(\010)c(:)g Fr(P)7 b Fu(\()p Fo(M)i Fu(\))23 b Fr(!)g(P)7 b Fu(\()p Fo(M)i Fu(\))28 b Fm(monotone)p Fu(,)g(i.e.)37 b Fr(8)p Fo(X)r(;)14 b(Y)40 b Fr(2)24 b(P)7 b Fu(\()p Fo(M)i Fu(\)\()p Fo(X)29 b Fr(\022)23 b Fo(Y)41 b Fr(\))23 b Fu(\010\()p Fo(X)7 b Fu(\))23 b Fr(\022)g Fu(\010\()p Fo(Y)c Fu(\)\).)0 645 y(I)30 657 y Fn(\010)105 645 y Fu(:=)215 583 y Fl(T)284 645 y Fr(f)p Fo(X)29 b Fr(2)24 b(P)7 b Fu(\()p Fo(M)i Fu(\))23 b(:)g(\010\()p Fo(X)7 b Fu(\))23 b Fr(\022)f Fo(X)7 b Fr(g)54 b Fu(is)28 b(called)f Fm(the)j(inductively)h(de\014ne) l(d)f(set)f(given)i(by)f Fu(\010)g(.)0 809 y Ft(Lemma)g(9.1)0 949 y Fu(\(1\))e(\010\(I)256 961 y Fn(\010)308 949 y Fu(\))23 b(=)g(I)481 961 y Fn(\010)532 949 y Fu(.)0 1089 y(\(2\))28 b(\010\()p Fo(X)7 b Fu(\))23 b Fr(\022)f Fo(X)53 b Fu(=)-14 b Fr(\))46 b Fu(I)776 1101 y Fn(\010)850 1089 y Fr(\022)23 b Fo(X)7 b Fu(,)55 b(for)27 b(eac)n(h)f(set)i Fo(X)h Fr(\022)23 b Fo(M)9 b Fu(.)0 1229 y(\(I)62 1241 y Fn(\010)141 1229 y Fu(is)28 b(the)g(least)f(\010-closed)g(set)g(and)h (the)g(least)f(\014xed)h(p)r(oin)n(t)f(of)h(\010.\))0 1393 y(Pro)r(of:)0 1533 y(\(2\))g(trivial.)0 1672 y(\(1\))g(\(i\))g (\010\()p Fo(X)7 b Fu(\))23 b Fr(\022)g Fo(X)681 1618 y Fn(\(2\))682 1672 y Fr(\))47 b Fu(I)842 1684 y Fn(\010)917 1672 y Fr(\022)22 b Fo(X)1126 1625 y Fn(\010)d(mon)p Fp(:)1164 1672 y Fu(=)-14 b Fr(\))84 b Fu(\010\(I)1504 1684 y Fn(\010)1556 1672 y Fu(\))23 b Fr(\022)g Fu(\010\()p Fo(X)7 b Fu(\))23 b Fr(\022)f Fo(X)7 b Fu(.)0 1812 y(Hence)28 b(\010\(I)369 1824 y Fn(\010)421 1812 y Fu(\))23 b Fr(\022)564 1750 y Fl(T)633 1812 y Fr(f)p Fo(X)29 b Fr(2)23 b(P)7 b Fu(\()p Fo(M)i Fu(\))23 b(:)g(\010\()p Fo(X)7 b Fu(\))23 b Fr(\022)f Fo(X)7 b Fr(g)22 b Fu(=)h(I)1707 1824 y Fn(\010)1758 1812 y Fu(.)0 1952 y(\(ii\))28 b(Let)g Fo(Y)42 b Fu(:=)23 b(\010\(I)610 1964 y Fn(\010)661 1952 y Fu(\).)38 b(\(i\))855 1905 y Fn(\010)19 b(mon)p Fp(:)893 1952 y Fu(=)-14 b Fr(\))84 b Fu(\010\()p Fo(Y)19 b Fu(\))24 b Fr(\022)e Fu(\010\(I)1535 1964 y Fn(\010)1587 1952 y Fu(\))h(=)g Fo(Y)1843 1898 y Fn(\(2\))1844 1952 y Fr(\))47 b Fu(I)2004 1964 y Fn(\010)2079 1952 y Fr(\022)22 b Fo(Y)42 b Fu(=)23 b(\010\(I)2466 1964 y Fn(\010)2517 1952 y Fu(\).)0 2140 y Ft(De\014nition)54 b Fu(I)499 2110 y Fp(\013)499 2163 y Fn(\010)574 2140 y Fu(:=)23 b(\010\(I)807 2107 y Fp(<\013)807 2164 y Fn(\010)906 2140 y Fu(\),)28 b(where)f(I)1259 2107 y Fp(<\013)1259 2164 y Fn(\010)1381 2140 y Fu(:=)1492 2078 y Fl(S)1561 2165 y Fp(\030)r(<\013)1707 2140 y Fu(I)1737 2100 y Fp(\030)1737 2164 y Fn(\010)1899 2140 y Fu(\()p Fo(\013)d Fr(2)f Fo(O)r(n)p Fu(\))28 b(.)0 2303 y Ft(Lemma)i(9.2)0 2443 y Fu(a\))d Fo(\014)h(<)23 b(\013)46 b Fr(\))g Fu(I)522 2404 y Fp(\014)522 2468 y Fn(\010)597 2443 y Fr(\022)22 b Fu(I)714 2413 y Fp(\013)714 2466 y Fn(\010)766 2443 y Fu(.)0 2583 y(b\))28 b(I)136 2550 y Fp(<\016)136 2608 y Fn(\010)247 2583 y Fu(=)23 b(I)365 2553 y Fp(\016)365 2606 y Fn(\010)463 2583 y Fr(\))46 b Fu(I)622 2553 y Fp(\016)622 2606 y Fn(\010)696 2583 y Fu(=)23 b(I)814 2595 y Fn(\010)889 2583 y Fu(=)976 2521 y Fl(S)1045 2608 y Fp(\013)p Fi(2)p Fp(O)r(n)1245 2583 y Fu(I)1275 2553 y Fp(\013)1275 2606 y Fn(\010)1326 2583 y Fu(.)0 2723 y(c\))28 b Fr(9)p Fo(\016)e(<)d(car)r(d)p Fu(\()p Fo(M)9 b Fu(\))610 2693 y Fn(+)666 2723 y Fu(\()23 b(I)751 2693 y Fp(\016)751 2746 y Fn(\010)826 2723 y Fu(=)f(I)943 2690 y Fp(<\016)943 2748 y Fn(\010)1055 2723 y Fu(\).)0 2887 y(Bew)n(eis:)0 3027 y(b\))28 b(1.)37 b(By)27 b(induction)h(on)f Fo(\013)h Fu(w)n(e)f(pro)n(v)n(e)f(I)1279 2997 y Fp(\013)1279 3050 y Fn(\010)1354 3027 y Fr(\022)d Fu(I)1472 3039 y Fn(\010)1523 3027 y Fu(:)37 b(IH)28 b Fr(\))23 b Fu(I)1839 2994 y Fp(<\013)1839 3051 y Fn(\010)1961 3027 y Fr(\022)g Fu(I)2079 3039 y Fn(\010)2154 3027 y Fr(\))g Fu(I)2290 2997 y Fp(\013)2290 3050 y Fn(\010)2364 3027 y Fu(=)g(\010\(I)2574 2994 y Fp(<\013)2574 3051 y Fn(\010)2673 3027 y Fu(\))h Fr(\022)e Fu(\010\(I)2938 3039 y Fn(\010)2990 3027 y Fu(\))h(=)g(I)3163 3039 y Fn(\010)3215 3027 y Fu(.)0 3167 y(2.)36 b(F)-7 b(rom)28 b(\010\(I)442 3137 y Fp(\016)442 3190 y Fn(\010)493 3167 y Fu(\))c(=)e(\010\(I)758 3134 y Fp(<\016)758 3191 y Fn(\010)847 3167 y Fu(\))h(=)g(I)1020 3137 y Fp(\016)1020 3190 y Fn(\010)1099 3167 y Fu(it)28 b(follo)n(ws)f(that)h(I)1665 3179 y Fn(\010)1739 3167 y Fr(\022)23 b Fu(I)1857 3137 y Fp(\016)1857 3190 y Fn(\010)1908 3167 y Fu(.)37 b(Hence)28 b(I)2245 3137 y Fp(\016)2245 3190 y Fn(\010)2320 3167 y Fr(\022)2407 3105 y Fl(S)2476 3192 y Fp(\013)p Fi(2)p Fp(O)r(n)2676 3167 y Fu(I)2706 3137 y Fp(\013)2706 3190 y Fn(\010)2780 3167 y Fr(\022)23 b Fu(I)2898 3179 y Fn(\010)2972 3167 y Fr(\022)g Fu(I)3090 3137 y Fp(\016)3090 3190 y Fn(\010)3142 3167 y Fu(.)0 3331 y(c\))34 b(By)f(a\))h(w)n(e)f(ha)n(v)n(e)g(I)703 3298 y Fp(<\013)703 3355 y Fn(\010)835 3331 y Fr(\022)g Fu(I)963 3301 y Fp(\013)963 3354 y Fn(\010)1015 3331 y Fu(.)55 b(|)34 b(Assumption:)50 b(I)1745 3298 y Fp(<\013)1745 3355 y Fn(\010)1877 3331 y Fa($)33 b Fu(I)2005 3301 y Fp(\013)2005 3354 y Fn(\010)2090 3331 y Fu(for)g(all)h Fo(\013)g(<)e(car)r(d)p Fu(\()p Fo(M)9 b Fu(\))2845 3301 y Fn(+)2902 3331 y Fu(.)55 b(Then)34 b(there)f(is)h(a)f(mapping)0 3471 y Fo(f)f Fu(:)23 b Fo(car)r(d)p Fu(\()p Fo(M)9 b Fu(\))435 3441 y Fn(+)514 3471 y Fr(!)23 b Fo(M)36 b Fu(with)28 b Fo(f)9 b Fu(\()p Fo(\013)p Fu(\))24 b Fr(2)f Fu(I)1225 3441 y Fp(\013)1225 3494 y Fn(\010)1295 3471 y Fr(n)18 b Fu(I)1385 3438 y Fp(<\013)1385 3495 y Fn(\010)1512 3471 y Fu(for)27 b(all)h Fo(\013)23 b Fr(2)g Fo(car)r(d)p Fu(\()p Fo(M)9 b Fu(\))2225 3441 y Fn(+)2282 3471 y Fu(,)27 b(i.e.)37 b Fo(f)g Fu(injectiv)n(e.)g(Con)n(tradiction.)0 3658 y Ft(De\014nition)54 b Fu(F)-7 b(or)27 b Fo(b)c Fr(2)g Fu(I)785 3670 y Fn(\010)865 3658 y Fu(let)55 b Fr(j)p Fo(b)p Fr(j)1094 3670 y Fn(\010)1169 3658 y Fu(:=)22 b(min)q Fr(f)p Fo(\013)h Fu(:)g Fo(b)f Fr(2)i Fu(I)1749 3628 y Fp(\013)1749 3681 y Fn(\010)1800 3658 y Fr(g)p Fu(.)0 3822 y Ft(Corollary)-8 b(.)36 b Fu(F)-7 b(or)27 b Fo(b)c Fr(2)g Fu(I)764 3834 y Fn(\010)844 3822 y Fu(the)28 b(follo)n(wing)e(holds:)0 3962 y(a\))h Fo(b)c Fr(2)h Fu(I)269 3932 y Fp(\013)269 3985 y Fn(\010)343 3962 y Fr(,)f(j)p Fo(b)p Fr(j)531 3974 y Fn(\010)606 3962 y Fr(\024)f Fo(\013)28 b Fu(,)83 b Fo(b)23 b Fr(2)h Fu(I)1048 3929 y Fp(<\013)1048 3986 y Fn(\010)1170 3962 y Fr(,)f(j)p Fo(b)p Fr(j)1358 3974 y Fn(\010)1432 3962 y Fo(<)g(\013)0 4102 y Fu(b\))28 b Fr(j)p Fo(b)p Fr(j)188 4114 y Fn(\010)262 4102 y Fu(=)23 b(sup)p Fr(fj)p Fo(c)p Fr(j)599 4114 y Fn(\010)669 4102 y Fu(+)18 b(1)k(:)h Fo(c)g Fr(2)h Fu(I)1030 4059 y Fp(<)p Fi(j)p Fp(b)p Fi(j)1151 4067 y Fg(\010)1030 4126 y Fn(\010)1198 4102 y Fr(g)0 4265 y Fu(Pro)r(of)g(of)i(b\)\\)p Fr(\024)p Fu(":)35 b(Let)25 b Fo(\013)f Fu(:=)f Fo(r)r(ig)s(hthandside) n Fu(.)37 b(Then)25 b(I)1763 4222 y Fp(<)p Fi(j)p Fp(b)p Fi(j)1884 4230 y Fg(\010)1763 4290 y Fn(\010)1955 4265 y Fr(\022)e Fu(I)2073 4232 y Fp(<\013)2073 4290 y Fn(\010)2197 4265 y Fu(and)j(th)n(us)f(I)2567 4222 y Fi(j)p Fp(b)p Fi(j)2636 4230 y Fg(\010)2567 4290 y Fn(\010)2707 4265 y Fr(\022)e Fu(I)2825 4235 y Fp(\013)2825 4288 y Fn(\010)2876 4265 y Fu(.)37 b(Hence)25 b Fo(b)e Fr(2)g Fu(I)3347 4235 y Fp(\013)3347 4288 y Fn(\010)3399 4265 y Fu(,)j(i.e.)g Fr(j)p Fo(b)p Fr(j)3662 4277 y Fn(\010)3736 4265 y Fr(\024)d Fo(\013)p Fu(.)0 4453 y Ft(De\014nition)0 4593 y Fu(If)28 b Fr(\036)f Fu(is)h(a)f(binary)g(relation)f(on)i Fo(M)9 b Fu(,)27 b(and)g(\010\()p Fo(X)7 b Fu(\))23 b(=)g Fr(f)p Fo(a)f Fr(2)i Fo(M)32 b Fu(:)23 b Fr(8)p Fo(b)p Fr(\036)n Fo(a)p Fu(\()p Fo(b)g Fr(2)g Fo(X)7 b Fu(\))p Fr(g)27 b Fu(\()p Fo(X)j Fr(2)23 b(P)7 b Fu(\()p Fo(M)i Fu(\)\),)28 b(then)0 4733 y(Acc\()p Fr(\036)p Fu(\))23 b(:=)g(Acc\()p Fo(M)t(;)14 b Fr(\036)p Fu(\))23 b(:=)g(I)950 4745 y Fn(\010)1029 4733 y Fu(\(=)1149 4670 y Fl(S)1218 4733 y Fr(f)p Fo(X)29 b Fr(\022)23 b Fo(M)31 b Fu(:)23 b Fr(8)p Fo(a)f Fr(2)i Fo(M)9 b Fu(\()p Fr(8)p Fo(b)p Fr(\036)n Fo(a)p Fu(\()p Fo(b)23 b Fr(2)g Fo(X)7 b Fu(\))23 b Fr(\))g Fo(a)g Fr(2)g Fo(X)7 b Fu(\))p Fr(g)p Fu(\).)0 4920 y Ft(Lemma)30 b(9.3)0 5060 y Fu(a\))d Fo(a)c Fr(2)h Fu(Acc\()p Fr(\036)p Fu(\))69 b Fr(\()-14 b(\))70 b(8)p Fo(b)p Fr(\036)n Fo(a)p Fu(\()p Fo(b)23 b Fr(2)g Fu(Acc\()p Fr(\036)p Fu(\)\))84 b(\()p Fo(a)23 b Fr(2)g Fo(M)9 b Fu(\))0 5200 y(b\))28 b Fr(j)p Fo(a)p Fr(j)196 5212 y Fn(\010)271 5200 y Fu(=)22 b(sup)p Fr(fj)p Fo(b)p Fr(j)607 5212 y Fn(\010)677 5200 y Fu(+)c(1)k(:)h Fo(b)g Fr(\036)g Fo(a)p Fr(g)82 b Fu(\()p Fo(a)23 b Fr(2)h Fu(Acc\()p Fr(\036)p Fu(\)\))0 5340 y(c\))k Fr(\036)o(j)184 5355 y Fn(Acc\()p Fi(\036)p Fn(\))428 5340 y Fu(is)g(w)n(ellfounded)1908 5539 y(53)p eop %%Page: 54 55 54 54 bop 0 83 a Fu(d\))28 b Fo(S)g Fr(\022)22 b Fo(M)32 b Fu(&)23 b Fr(8)p Fo(a)p Fr(2)o Fo(S)5 b Fr(8)p Fo(b)p Fr(\036)n Fo(a)p Fu(\()p Fo(b)23 b Fr(2)g Fo(S)5 b Fu(\))23 b(&)46 b Fr(\036j)1343 95 y Fp(S)1419 83 y Fu(w)n(ellfounded)27 b(=)-14 b Fr(\))46 b Fo(S)28 b Fr(\022)22 b Fu(Acc)q(\()p Fr(\036)p Fu(\))0 222 y(e\))28 b Fr(\036)f Fu(w)n(ellfounded)h(=)-14 b Fr(\))46 b Fo(M)31 b Fu(=)23 b(Acc\()p Fr(\036)p Fu(\))0 383 y(Pro)r(of:)0 521 y(a\))k(follo)n(ws)g(from)g(9.1\(1\).)0 660 y(b\))h Fr(j)p Fo(a)p Fr(j)196 672 y Fn(\010)271 660 y Fr(\024)22 b Fo(\013)47 b Fr(,)f Fo(a)23 b Fr(2)g Fu(\010\(I)854 627 y Fp(<\013)854 685 y Fn(\010)954 660 y Fu(\))g(=)g Fr(f)p Fo(c)f Fr(2)i Fo(M)31 b Fu(:)23 b Fr(8)p Fo(b)p Fr(\036)o Fo(c)p Fu(\()p Fo(b)g Fr(2)g Fu(I)1816 627 y Fp(<\013)1816 685 y Fn(\010)1915 660 y Fu(\))p Fr(g)46 b(,)g(8)p Fo(b)p Fr(\036)o Fo(a)p Fu(\()p Fo(b)23 b Fr(2)g Fu(I)2554 627 y Fp(<\013)2554 685 y Fn(\010)2653 660 y Fu(\))47 b Fr(,)f(8)p Fo(b)p Fr(\036)n Fo(a)p Fu(\()p Fr(j)p Fo(b)p Fr(j)3165 672 y Fn(\010)3240 660 y Fo(<)22 b(\013)p Fu(\))p Fr(g)p Fu(.)0 799 y Fr(j)p Fo(a)p Fr(j)90 811 y Fn(\010)165 799 y Fu(=)g(min)q Fr(f)p Fo(\013)h Fu(:)g Fr(j)p Fo(a)p Fr(j)645 811 y Fn(\010)719 799 y Fr(\024)g Fo(\013)p Fr(g)g Fu(=)g(min)p Fr(f)p Fo(\013)g Fu(:)g Fr(8)p Fo(b)p Fr(\036)n Fo(a)p Fu(\()p Fr(j)p Fo(b)p Fr(j)1619 811 y Fn(\010)1694 799 y Fo(<)g(\013)p Fu(\))p Fr(g)g Fu(=)f(sup)p Fr(fj)p Fo(b)p Fr(j)2268 811 y Fn(\010)2338 799 y Fu(+)c(1)k(:)h Fo(b)g Fr(\036)g Fo(a)p Fr(g)p Fu(.)0 937 y(c\))28 b Fo(b)23 b Fr(\036)f Fo(a)h Fr(2)h Fu(Acc\()p Fr(\036)p Fu(\))46 b Fr(\))g(j)p Fo(b)p Fr(j)911 949 y Fn(\010)985 937 y Fo(<)23 b Fr(j)p Fo(a)p Fr(j)1163 949 y Fn(\010)1215 937 y Fu(.)0 1076 y(d\))28 b(By)f(induction)h(o)n(v)n(er)e Fr(\036j)p Fo(S)32 b Fu(w)n(e)27 b(pro)n(v)n(e:)36 b Fo(a)23 b Fr(2)g Fo(S)28 b Fr(\))23 b Fo(a)g Fr(2)g Fu(Acc\()p Fr(\036)p Fu(\).)0 1215 y Fo(a)g Fr(2)g Fo(S)28 b Fr(\))23 b(8)p Fo(b)p Fr(\036)o Fo(a)p Fu(\()p Fo(b)f Fr(2)i Fo(S)5 b Fu(\))806 1168 y Fn(IH)801 1215 y Fr(\))23 b(8)p Fo(b)p Fr(\036)n Fo(a)p Fu(\()p Fo(b)g Fr(2)h Fu(Acc\()p Fr(\036)p Fu(\)\))f Fr(\))g Fo(a)g Fr(2)h Fu(Acc\()p Fr(\036)p Fu(\).)0 1353 y(e\))k(trivial.)0 1559 y Ft(Example)0 1720 y Fr(O)r Fu(\()p Fo(X)r(;)14 b(x)p Fu(\))24 b(:=)f Fo(x)p 477 1695 50 8 v 477 1712 V 67 w Fu(0)18 b Fr(_)g Fu(\()p Fo(\031)748 1732 y Fn(1)786 1720 y Fo(x)p 841 1695 V 841 1712 V 67 w Fu(1)g Fr(^)h Fo(X)7 b(\031)1157 1732 y Fn(2)1194 1720 y Fo(x)p Fu(\))19 b Fr(_)g Fu(\()p Fo(\031)1445 1732 y Fn(1)1483 1720 y Fo(x)p 1538 1695 V 1538 1712 V 66 w Fu(2)f Fr(^)h(8)p Fo(y)s Fr(9)p Fo(z)t Fu(\()p Fr(f)p Fo(\031)2031 1732 y Fn(2)2067 1720 y Fo(x)p Fr(g)p Fu(\()p Fo(y)s Fu(\))k Fr(')g Fo(z)e Fr(^)e Fo(X)7 b(z)t Fu(\)\))0 1882 y Ft(Lemma)30 b(9.4)0 2020 y Fr(j)p Fu(0)p Fr(j)88 2032 y Fi(O)169 2020 y Fu(=)23 b(0)k(,)56 b Fr(j)p Fo(\031)s Fu(\(1)p Fo(;)14 b(a)p Fu(\))p Fr(j)688 2032 y Fi(O)769 2020 y Fu(=)23 b Fr(j)p Fo(a)p Fr(j)947 2032 y Fi(O)1024 2020 y Fu(+)18 b(1)27 b(,)56 b Fr(j)p Fo(\031)s Fu(\(2)p Fo(;)14 b(e)p Fu(\))p Fr(j)1533 2032 y Fi(O)1614 2020 y Fu(=)23 b(sup)p Fr(fjf)p Fo(e)p Fr(g)p Fu(\()p Fo(n)p Fu(\))p Fr(j)2152 2032 y Fi(O)2228 2020 y Fu(+)18 b(1)k(:)i Fo(n)e Fr(2)i Fu(I)-14 b(N)p Fr(g)0 2181 y Ft(Lemma)30 b(9.5)0 2320 y Fu(Let)55 b Fo(b)23 b Fr(\036)g Fo(a)46 b Fu(:)p Fr(,)g Fo(a)23 b Fu(=)g Fo(\031)s Fu(\(1)p Fo(;)14 b(b)p Fu(\))k Fr(_)h Fu(\()p Fo(\031)1120 2332 y Fn(1)1158 2320 y Fo(a)k Fu(=)f(2)c Fr(^)h(9)p Fo(y)s Fu(\()p Fr(f)p Fo(\031)1657 2332 y Fn(2)1694 2320 y Fo(a)p Fr(g)p Fu(\()p Fo(y)s Fu(\))k Fr(')f Fo(b)p Fu(\)\).)0 2459 y(Then)55 b Fr(O)26 b(\022)d Fu(Acc\()p Fr(\036)p Fu(\))55 b(and)g Fr(8)p Fo(a)22 b Fr(2)i(O)r Fu(\()14 b Fr(j)p Fo(a)p Fr(j)1329 2471 y Fi(O)1411 2459 y Fu(=)23 b Fr(j)p Fo(a)p Fr(j)1589 2471 y Fi(\036)1658 2459 y Fu(\).)0 2620 y(Pro)r(of:)0 2758 y(1.)36 b Fo(a)23 b Fu(=)g(0:)37 b(0)22 b Fr(2)i Fu(Acc,)j(since)h Fr(f)p Fo(c)22 b Fu(:)h Fo(c)g Fr(\036)g Fu(0)p Fr(g)p Fu(.)36 b Fr(j)p Fu(0)p Fr(j)1415 2770 y Fi(\036)1494 2758 y Fu(=)22 b(0)h(=)g Fr(j)p Fu(0)p Fr(j)1822 2770 y Fi(O)1880 2758 y Fu(.)0 2897 y(2.)36 b Fo(a)23 b Fu(=)g Fo(\031)s Fu(\(1)p Fo(;)14 b(a)461 2909 y Fn(0)498 2897 y Fu(\))28 b(and)g Fo(a)764 2909 y Fn(0)824 2897 y Fr(2)23 b(O)r Fu(:)38 b(IH)27 b Fr(\))47 b Fo(a)1324 2909 y Fn(0)1384 2897 y Fr(2)23 b Fu(Acc)g(&)g Fr(j)p Fo(a)1776 2909 y Fn(0)1813 2897 y Fr(j)1836 2909 y Fi(O)1918 2897 y Fu(=)f Fr(j)p Fo(a)2072 2909 y Fn(0)2110 2897 y Fr(j)2133 2909 y Fi(\036)2235 2897 y Fr(\))h Fu([since)k Fr(f)p Fo(c)c Fu(:)g Fo(c)g Fr(\036)g Fo(a)p Fr(g)f Fu(=)h Fr(f)p Fo(a)3143 2909 y Fn(0)3179 2897 y Fr(g)p Fu(])0 3036 y Fo(a)g Fr(2)g Fu(Acc)47 b(&)e Fr(j)p Fo(a)p Fr(j)528 3048 y Fi(\036)607 3036 y Fu(=)23 b Fr(j)p Fo(a)762 3048 y Fn(0)799 3036 y Fr(j)822 3048 y Fi(\036)897 3036 y Fu(+)18 b(1)k(=)h Fr(j)p Fo(a)1199 3048 y Fn(0)1236 3036 y Fr(j)1259 3048 y Fi(O)1336 3036 y Fu(+)18 b(1)23 b(=)f Fr(j)p Fo(a)p Fr(j)1661 3048 y Fi(O)1720 3036 y Fu(.)0 3174 y(3.)36 b Fo(a)23 b Fu(=)g Fo(\031)s Fu(\(2)p Fo(;)14 b(e)p Fu(\))23 b(&)g Fr(8)p Fo(y)s Fu(\()p Fr(f)p Fo(e)p Fr(g)p Fu(\()p Fo(y)s Fu(\))e Fr(2)i(O)r Fu(\):)38 b(IH)28 b Fr(\))37 b(8)p Fo(y)s Fu(\()p Fr(f)p Fo(e)p Fr(g)p Fu(\()p Fo(y)s Fu(\))21 b Fr(2)i Fu(Acc\))h(&)e Fr(8)p Fo(y)s Fu(\()p Fr(jf)p Fo(e)p Fr(g)p Fu(\()p Fo(y)s Fu(\))p Fr(j)2585 3186 y Fi(O)2665 3174 y Fu(=)h Fr(jf)p Fo(e)p Fr(g)p Fu(\()p Fo(y)s Fu(\))p Fr(j)3030 3186 y Fi(\036)3085 3174 y Fu(\))46 b Fr(\))0 3313 y Fu([since)27 b Fr(f)p Fo(c)c Fu(:)g Fo(c)g Fr(\036)g Fo(a)p Fr(g)f Fu(=)h Fr(ff)p Fo(e)p Fr(g)p Fu(\()p Fo(y)s Fu(\))e(:)i Fo(y)j Fr(2)e Fu(I)-14 b(N)p Fr(g)p Fu(])69 b Fr(j)p Fo(a)p Fr(j)1504 3325 y Fi(\036)1583 3313 y Fu(=)22 b(sup)p Fr(fjf)p Fo(e)p Fr(gj)2006 3325 y Fi(\036)2079 3313 y Fu(+)c(1)23 b(:)g Fo(n)g Fr(2)g Fu(I)-14 b(N)q Fr(g)22 b Fu(=)h(sup)p Fr(fjf)p Fo(e)p Fr(gj)2991 3325 y Fi(O)3066 3313 y Fu(+)18 b(1)23 b(:)g Fo(n)g Fr(2)g Fu(I)-14 b(N)q Fr(g)22 b Fu(=)h Fr(j)p Fo(a)p Fr(j)3732 3325 y Fi(O)3790 3313 y Fu(.)0 3519 y Ft(The)32 b(formal)f(theory)h Fu(ID)892 3531 y Fn(1)0 3680 y Fu(A)c Fm(p)l(ositive)j(op)l(er)l(ator)g(form)d Fu(is)g(an)f Fr(L)1170 3692 y Fn(0)1208 3680 y Fu(\()p Fr(X)12 b Fu(\)-form)n(ula)27 b Fr(A)p Fu(,)h(suc)n(h)f(that)0 3819 y({)g(FV)r(\()p Fr(A)p Fu(\))d Fr(\022)e(f)p Fo(v)510 3831 y Fn(0)547 3819 y Fr(g)p Fu(,)0 3957 y({)27 b Fr(A)h Fu(con)n(tains)f(no)g(set)h(v)-5 b(ariable)26 b(except)i Fo(X)1375 3969 y Fn(0)1412 3957 y Fu(,)0 4096 y({)f Fo(X)138 4108 y Fn(0)203 4096 y Fu(o)r(ccurs)g(only)g(p)r(ositiv)n(ely)g(in)h Fr(A)p Fu(.)0 4257 y(Notation:)37 b Fr(A)p Fu(\()p Fo(F)r(;)14 b(t)p Fu(\))29 b(denotes)e(the)h(result)f(of)h(replacing)e(in)i Fr(A)1949 4269 y Fp(v)1982 4277 y Fg(0)2019 4257 y Fu(\()p Fo(t)p Fu(\))g(ev)n(ery)f(subform)n(ula)f Fo(X)2859 4269 y Fn(0)2896 4257 y Fo(s)i Fu(b)n(y)f Fo(F)12 b Fu(\()p Fo(s)p Fu(\).)0 4418 y(\010)60 4430 y Fi(A)117 4418 y Fu(\()p Fo(X)7 b Fu(\))23 b(:=)g Fr(f)p Fo(n)f Fr(2)i Fu(I)-14 b(N)23 b(:)g(I)-14 b(N)24 b Fr(j)-14 b Fu(=)23 b Fr(A)p Fu(\()p Fo(X)r(;)14 b(n)p Fu(\))p Fr(g)p Fu(,)55 b(\()p Fo(X)30 b Fr(\022)22 b Fu(I)-14 b(N)q(\).)83 b(\010)1833 4430 y Fi(A)1918 4418 y Fu(is)28 b(monotone.)0 4579 y(Abbreviation:)36 b(W)-7 b(e)28 b(write)g(I)921 4591 y Fi(A)978 4579 y Fo(;)g Fu(I)1059 4549 y Fp(\013)1059 4602 y Fi(A)1116 4579 y Fo(;)g Fr(j)p Fo(n)p Fr(j)1263 4591 y Fi(A)1320 4579 y Fo(;)14 b(::::)28 b Fu(for)f(I)1634 4591 y Fn(\010)1681 4599 y Ff(A)1736 4579 y Fo(;)h Fu(I)1817 4549 y Fp(\013)1817 4602 y Fn(\010)1864 4610 y Ff(A)1919 4579 y Fo(;)g Fr(j)p Fo(n)p Fr(j)2066 4591 y Fn(\010)2113 4599 y Ff(A)2167 4579 y Fo(;)14 b(::::)0 4763 y Fr(L)p Fu(\(ID)183 4775 y Fn(1)220 4763 y Fu(\))24 b(:=)e Fr(L)443 4775 y Fn(0)481 4763 y Fu(\()p Fr(X)12 b Fu(\))19 b Fr([)g(f)p Fo(P)804 4775 y Fi(A)884 4763 y Fu(:)k Fr(A)28 b Fu(p)r(ositiv)n(e)g(op)r (erator)d(form)j Fr(g)82 b Fu(where)27 b(the)h Fo(P)2423 4775 y Fi(A)2481 4763 y Fu('s)g(are)e(new)i(unary)f(predicate)g(sym)n (b)r(ols.)0 4924 y(The)h(axioms)e(of)i(ID)638 4936 y Fn(1)703 4924 y Fu(are)f(the)h(axioms)e(of)i Fr(Z)34 b Fu(in)28 b(the)g(language)e Fr(L)p Fu(\(ID)2222 4936 y Fn(1)2259 4924 y Fu(\),)i(and)0 5063 y(\(ID1\))56 b Fr(8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(P)532 5075 y Fi(A)590 5063 y Fo(;)14 b(x)p Fu(\))24 b Fr(!)f Fo(P)889 5075 y Fi(A)947 5063 y Fo(x)p Fu(\),)0 5201 y(\(ID2\))56 b(for)27 b(eac)n(h)g Fr(L)p Fu(\(ID)752 5213 y Fn(1)789 5201 y Fu(\)-form)n(ula)g Fo(F)39 b Fu(the)28 b(univ)n(ersal)e(closure) h(of)255 5340 y Fr(8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)r(;)14 b(x)p Fu(\))25 b Fr(!)e Fo(F)12 b Fu(\()p Fo(x)p Fu(\)\))24 b Fr(!)f(8)p Fo(x)p Fu(\()p Fo(P)1298 5352 y Fi(A)1356 5340 y Fo(x)g Fr(!)g Fo(F)12 b Fu(\()p Fo(x)p Fu(\)\).)1908 5539 y(54)p eop %%Page: 55 56 55 55 bop 0 83 a Ft(Lemma)30 b(9.6)0 224 y Fu(ID)93 236 y Fn(1)154 224 y Fr(`)23 b(8)p Fo(x)p Fu(\()p Fo(P)407 236 y Fi(A)464 224 y Fo(x)h Fr(!)f(A)p Fu(\()p Fo(P)792 236 y Fi(A)850 224 y Fo(;)14 b(x)p Fu(\)\).)0 390 y(Pro)r(of:)36 b(Let)27 b Fo(F)12 b Fu(\()p Fo(x)p Fu(\))24 b(:=)f Fr(A)p Fu(\()p Fo(P)870 402 y Fi(A)928 390 y Fo(;)14 b(x)p Fu(\).)0 531 y(\(ID1\))47 b Fr(\))f(8)p Fo(x)p Fu(\()p Fo(F)12 b Fu(\()p Fo(x)p Fu(\))24 b Fr(!)f Fo(P)860 543 y Fi(A)917 531 y Fo(x)p Fu(\))47 b Fr(\))f(8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)r(;)14 b(x)p Fu(\))25 b Fr(!)e(A)p Fu(\()p Fo(P)1849 543 y Fi(A)1908 531 y Fo(;)14 b(x)p Fu(\))1698 587 y Fl(|)p 1735 587 89 10 v 89 w({z)p 1898 587 V 89 w(})1791 669 y Fp(F)9 b Fn(\()p Fp(x)p Fn(\))2024 531 y Fu(\))47 b Fr(\))f(8)p Fo(x)p Fu(\()p Fo(P)2411 543 y Fi(A)2469 531 y Fo(x)23 b Fr(!)g Fo(F)12 b Fu(\()p Fo(x)p Fu(\)\).)0 788 y(Question:)36 b(ID)477 800 y Fn(1)537 788 y Fr(`)23 b Fo(P)664 800 y Fi(A)722 788 y Fo(n)p 722 801 50 4 v 22 w Fu(=)-14 b Fr(\))23 b(j)p Fo(n)p 974 801 V Fr(j)1047 800 y Fi(A)1128 788 y Fo(<)p Fu(???)0 954 y(Remark:)36 b(Let)28 b Fr(A)p Fu(\()p Fo(X)r(;)14 b(x)p Fu(\))24 b(=)e Fr(8)p Fo(y)s Fr(\036)o Fo(xX)7 b(y)s Fu(.)0 1095 y(Then)28 b(Prog)o(\()p Fo(X)7 b Fu(\))23 b(=)g Fr(8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(X)r(;)14 b(x)p Fu(\))23 b Fr(!)g Fo(X)7 b(x)p Fu(\).)37 b(Hence)28 b(ID)1737 1107 y Fn(1)1797 1095 y Fr(`)23 b Fu(Prog)o(\()p Fo(P)2129 1107 y Fi(A)2187 1095 y Fu(\).)0 1261 y(ID)93 1273 y Fn(1)154 1261 y Fr(`)g Fu(TI)317 1273 y Fi(\036)373 1261 y Fu(\()p Fo(X)r(;)14 b(n)p 513 1274 V Fu(\))47 b(=)-14 b Fr(\))45 b Fu(ID)915 1273 y Fn(1)975 1261 y Fr(`)23 b Fo(P)1102 1273 y Fi(A)1160 1261 y Fo(n)p 1160 1274 V Fu(.)0 1427 y(Pro)r(of:)36 b(ID)353 1439 y Fn(1)413 1427 y Fr(`)23 b Fu(TI)577 1439 y Fi(\036)633 1427 y Fu(\()p Fo(X)r(;)14 b(n)p 773 1440 V Fu(\))46 b(=)-14 b Fr(\))46 b Fu(TI)1171 1439 y Fi(\036)1227 1427 y Fu(\()p Fo(P)1312 1439 y Fi(A)1370 1427 y Fo(;)14 b(n)p 1407 1440 V Fu(\))46 b Fr(\))g Fu(ID)1758 1439 y Fn(1)1818 1427 y Fr(`)23 b(8)p Fo(y)s Fr(\036)n Fo(n)p 2046 1440 V(P)2149 1439 y Fi(A)2253 1427 y Fr(\))46 b Fu(ID)2476 1439 y Fn(1)2536 1427 y Fr(`)23 b Fo(P)2663 1439 y Fi(A)2721 1427 y Fo(n)p 2721 1440 V -1 w Fu(.)0 1693 y Ft(The)32 b(language)g Fr(L)649 1662 y Fi(\003)0 1859 y Fu(F)-7 b(or)27 b(eac)n(h)g(p)r(ositiv)n(e)g(op)r(erator)f(form)h Fr(A)h Fu(and)f(eac)n(h)g(ordinal)g Fo(\013)c Fr(\024)g Fu(\012)28 b(let)g Fo(P)2335 1826 y Fp(<\013)2323 1883 y Fi(A)2461 1859 y Fu(b)r(e)g(a)f(new)h(unary)f(predicate)g(sym)n(b)r (ol.)0 2025 y Fr(L)57 1995 y Fi(\003)118 2025 y Fu(:=)c Fr(L)286 2037 y Fn(0)324 2025 y Fu(\()p Fr(X)12 b Fu(\))19 b Fr([)g(f)p Fo(P)659 1992 y Fp(<\013)647 2049 y Fi(A)780 2025 y Fu(:)24 b Fo(\013)f Fr(\024)g Fu(\012)14 b Fo(;)36 b Fr(A)28 b Fu(p)r(ositiv)n(e)g(op)r(erator)d(form)j Fr(g)0 2191 y Fu(W)-7 b(e)28 b(assume)f(that)h Fo(P)675 2158 y Fp(<)p Fn(\012)663 2215 y Fi(A)806 2191 y Fu(equals)f Fo(P)1111 2203 y Fi(A)1168 2191 y Fu(.)37 b(Hence)28 b Fr(L)p Fu(\(ID)1658 2203 y Fn(1)1695 2191 y Fu(\))c Fr(\022)e(L)1895 2161 y Fi(\003)1934 2191 y Fu(.)0 2357 y(De\014nition)28 b(of)g(rk)o(\()p Fo(A)p Fu(\))h(for)e(eac)n(h)g Fr(L)1083 2327 y Fi(\003)1121 2357 y Fu(-form)n(ula)f Fo(A)0 2498 y Fu(1.)36 b(rk\()p Fo(P)275 2468 y Fp(<\013)374 2498 y Fo(t)p Fu(\))23 b(:=)g(rk)o(\()p Fr(:)p Fo(P)798 2468 y Fp(<\013)898 2498 y Fo(t)p Fu(\))g(:=)g Fo(!)e Fr(\001)e Fo(\013)0 2639 y Fu(2.)36 b(rk\()p Fo(A)p Fu(\))24 b(:=)e(0,)28 b(if)g Fo(A)g Fu(is)f(an)g Fr(L)952 2651 y Fn(0)990 2639 y Fu(\()p Fr(X)12 b Fu(\)-literal)0 2780 y(3.)36 b(rk\()p Fo(A)19 b Fr(^)g Fo(B)t Fu(\))k(:=)g(rk)o(\()p Fo(A)c Fr(_)g Fo(B)t Fu(\))k(=)g(max)p Fr(f)p Fu(rk)n(\()p Fo(A)p Fu(\))p Fo(;)14 b Fu(rk\()p Fo(B)t Fu(\))p Fr(g)19 b Fu(+)f(1)0 2921 y(4.)36 b(rk\()p Fr(8)p Fo(xA)p Fu(\))23 b(:=)g(rk)o(\()p Fr(9)p Fo(xA)p Fu(\))i(:=)d(rk\()p Fo(A)p Fu(\)+1)0 3087 y(De\014nition)28 b(of)g Fk(k)521 3099 y Fn(\005)575 3087 y Fu(\()p Fo(A)p Fu(\))g(for)f(eac)n(h)g Fr(L)1100 3057 y Fi(\003)1138 3087 y Fu(-form)n(ula)g Fo(A)0 3228 y Fu(1.)36 b Fk(k)142 3240 y Fn(\005)195 3228 y Fu(\()p Fo(P)292 3198 y Fp(<\013)392 3228 y Fo(t)p Fu(\))23 b(:=)g Fr(f)p Fu(0)p Fr(g)p Fu(,)j Fk(k)803 3240 y Fn(\005)857 3228 y Fu(\()p Fr(:)p Fo(P)1009 3198 y Fp(<\013)1108 3228 y Fo(t)p Fu(\))e(:=)e Fr(f)p Fu(0)p Fo(;)14 b(\013)p Fr(g)p Fu(.)0 3370 y(2.)36 b Fk(k)142 3382 y Fn(\005)195 3370 y Fu(\()p Fo(A)p Fu(\))24 b(:=)f Fr(f)p Fu(0)p Fr(g)p Fu(,)j(if)i Fo(A)g Fu(is)g(an)f Fr(L)1053 3382 y Fn(0)1090 3370 y Fu(\()p Fr(X)12 b Fu(\)-literal.)0 3511 y(3.)36 b Fk(k)142 3523 y Fn(\005)195 3511 y Fu(\()p Fo(A)19 b Fr(^)g Fo(B)t Fu(\))24 b(:=)e Fk(k)656 3523 y Fn(\005)709 3511 y Fu(\()p Fo(A)d Fr(_)g Fo(B)t Fu(\))k(:=)g Fk(k)1170 3523 y Fn(\005)1223 3511 y Fu(\()p Fo(A)p Fu(\))c Fr([)g Fk(k)1482 3523 y Fn(\005)1536 3511 y Fu(\()p Fo(B)t Fu(\).)0 3652 y(4.)36 b Fk(k)142 3664 y Fn(\005)195 3652 y Fu(\()p Fr(8)p Fo(xA)p Fu(\))24 b(:=)e Fk(k)590 3664 y Fn(\005)643 3652 y Fu(\()p Fr(9)p Fo(xA)p Fu(\))j(:=)d Fk(k)1038 3664 y Fn(\005)1091 3652 y Fu(\()p Fo(A)p Fu(\).)0 3818 y(De\014nition)0 3959 y Fk(k)41 3971 y Fn(\006)92 3959 y Fu(\()p Fo(A)p Fu(\))i(:=)f Fk(k)393 3971 y Fn(\005)446 3959 y Fu(\()p Fr(:)p Fo(A)p Fu(\))29 b(and)f Fk(k)o Fu(\()p Fo(A)p Fu(\))c(:=)f Fk(k)1160 3971 y Fn(\005)1213 3959 y Fu(\()p Fo(A)p Fu(\))c Fr([)g Fk(k)1472 3971 y Fn(\006)1524 3959 y Fu(\()p Fo(A)p Fu(\).)38 b(\(Hence)28 b Fk(k)o Fu(\()p Fo(A)p Fu(\))c(=)f Fk(k)o Fu(\()p Fr(:)p Fo(A)p Fu(\).\))0 4100 y Fk(k)p Fu(\()p Fo(t)p Fu(\))g(:=)g Fr(;)k Fu(for)g Fo(t)c Fr(2)h Fo(T)12 b Fu(,)109 b Fk(k)p Fu(\()p Fo(\030)t Fu(\))24 b(:=)e Fr(f)p Fo(\030)t Fr(g)27 b Fu(for)g Fo(\030)h Fr(2)23 b Fo(O)r(n)p Fu(.)0 4266 y(No)n(w)k(w)n(e)g(`em)n(b)r(ed')h Fr(L)677 4236 y Fi(\003)743 4266 y Fu(in)n(to)g(in\014nitary)f(prop)r(ositional)f(logic:)0 4432 y(De\014nition)0 4573 y Fo(P)65 4540 y Fp(<\013)53 4598 y Fi(A)164 4573 y Fo(t)d Fr(')305 4511 y Fl(W)374 4598 y Fp(\030)r(<\013)505 4573 y Fr(A)p Fu(\()p Fo(P)668 4534 y Fp(<\030)656 4598 y Fi(A)757 4573 y Fo(;)14 b(t)p Fu(\))28 b(,)83 b Fr(:)p Fo(P)1110 4540 y Fp(<\013)1098 4598 y Fi(A)1209 4573 y Fo(t)23 b Fr(')1350 4511 y Fl(V)1419 4598 y Fp(\030)r(<\013)1550 4573 y Fr(:A)p Fu(\()p Fo(P)1768 4534 y Fp(<\030)1756 4598 y Fi(A)1857 4573 y Fo(;)14 b(t)p Fu(\))28 b(,)0 4715 y Fo(A)19 b Fr(^)g Fo(B)27 b Fr(')332 4652 y Fl(V)402 4739 y Fp(\023)p Fi(2f)p Fn(0)p 506 4752 34 4 v -1 w Fp(;)p Fn(1)p 558 4752 V Fi(g)629 4715 y Fo(A)691 4727 y Fp(\023)748 4715 y Fu(,)83 b Fo(A)18 b Fr(_)h Fo(B)27 b Fr(')1186 4652 y Fl(W)1255 4739 y Fp(\023)p Fi(2f)p Fn(0)p 1359 4752 V Fp(;)p Fn(1)p 1412 4752 V -1 w Fi(g)1483 4715 y Fo(A)1545 4727 y Fp(\023)1574 4715 y Fu(,)83 b(where)27 b Fo(A)1982 4727 y Fn(0)p 1982 4740 38 4 v 2042 4715 a Fu(:=)c Fo(A)p Fu(,)28 b Fo(A)2328 4727 y Fn(1)p 2328 4740 V 2389 4715 a Fu(:=)22 b Fo(B)t Fu(,)0 4856 y Fr(8)p Fo(xA)h Fr(')267 4793 y Fl(V)336 4881 y Fp(t)p Fi(2)p Fp(T)458 4856 y Fo(A)520 4868 y Fp(x)562 4856 y Fu(\()p Fo(t)p Fu(\),)84 b Fr(9)p Fo(xA)23 b Fr(')1029 4793 y Fl(W)1098 4881 y Fp(t)p Fi(2)p Fp(T)1220 4856 y Fo(A)1282 4868 y Fp(x)1324 4856 y Fu(\()p Fo(t)p Fu(\))0 5022 y Ft(Lemma)30 b(9.7)0 5163 y Fu(a\))d Fo(A)d Fr(')e(\003)316 5175 y Fp(\023)p Fi(2)p Fp(J)432 5163 y Fo(A)494 5175 y Fp(\023)569 5163 y Fu(=)-14 b Fr(\))46 b Fk(k)789 5175 y Fn(\005)842 5163 y Fu(\()p Fo(A)936 5175 y Fp(\023)966 5163 y Fu(\))23 b Fr(\022)g Fk(k)1149 5175 y Fn(\005)1202 5163 y Fu(\()p Fo(A)p Fu(\))d Fr([)e Fk(k)p Fu(\()p Fo(\023)p Fu(\),)0 5304 y(b\))28 b Fo(A)23 b Fr(')279 5242 y Fl(V)348 5329 y Fp(\023)p Fi(2)p Fp(J)464 5304 y Fo(A)526 5316 y Fp(\023)601 5304 y Fu(=)-14 b Fr(\))46 b(9)p Fo(\033)27 b Fr(2)c Fk(k)1019 5316 y Fn(\005)1073 5304 y Fu(\()p Fo(A)p Fu(\))p Fr(8)p Fo(\023)g Fr(2)g Fo(J)8 b Fu([)14 b Fk(k)p Fu(\()p Fo(\023)p Fu(\))24 b Fr(\022)f Fo(\033)17 b Fu(].)1908 5539 y(55)p eop %%Page: 56 57 56 56 bop 0 83 a Fu(c\))28 b(rk)o(\()p Fo(A)p Fu(\))c(=)f Fo(!)e Fr(\001)d Fo(\013)h Fu(+)f Fo(n)28 b Fu(with)g Fo(\013)23 b Fu(=)g(max)o(\()p Fk(k)p Fu(\()p Fo(A)p Fu(\)\))29 b(and)f Fo(n)22 b(<)h(!)s Fu(.)0 224 y(d\))28 b(rk)o(\()p Fo(A)p Fu(\))c(=)f(\012)46 b(=)-14 b Fr(\))46 b Fo(A)23 b Fu(=)g Fo(P)932 236 y Fi(A)990 224 y Fo(t)k Fu(or)g Fo(A)c Fu(=)g Fr(:)p Fo(P)1430 236 y Fi(A)1488 224 y Fo(t)p Fu(.)0 489 y Ft(The)32 b(in\014nitary)g(pro)s(of)g(system) e Fu(ID)1275 453 y Fi(1)1275 510 y Fn(1)0 680 y Fu(The)e(language)e(of) h(ID)705 644 y Fi(1)705 701 y Fn(1)803 680 y Fu(is)g Fr(L)943 650 y Fi(\003)981 680 y Fu(.)0 846 y(The)h(inference)f(sym)n (b)r(ols)g(of)h(ID)1024 810 y Fi(1)1024 866 y Fn(1)1122 846 y Fu(are)0 1062 y(\()p Fk(Ax)126 1024 y Fi(1)126 1082 y Fn(\001)196 1062 y Fu(\))495 b(\001)320 b(if)28 b(\001)23 b Fr(2)g(AX)13 b Fu(\()p Fe(Z)1576 1024 y Fi(1)1647 1062 y Fu(\))0 1266 y(\()32 1203 y Fl(V)101 1291 y Fp(A)156 1266 y Fu(\))499 1211 y Fo(:)h(:)g(:)g(A)672 1223 y Fp(\023)715 1211 y Fo(:)g(:)g(:)g Fu(\()p Fo(\023)p Fr(2)q Fo(J)8 b Fu(\))p 486 1247 544 4 v 643 1320 a Fo(A)1112 1266 y Fu(if)28 b Fo(A)23 b Fr(')1361 1203 y Fl(V)1430 1291 y Fp(\023)p Fi(2)p Fp(J)1545 1266 y Fo(A)1607 1278 y Fp(\023)0 1508 y Fu(\()32 1446 y Fl(W)101 1466 y Fp(\026)101 1533 y(A)156 1508 y Fu(\))704 1453 y Fo(A)766 1465 y Fp(\026)p 690 1489 135 4 v 726 1563 a Fo(A)1112 1508 y Fu(if)28 b Fo(A)23 b Fr(')1361 1446 y Fl(W)1430 1533 y Fp(\023)p Fi(2)p Fp(J)1545 1508 y Fo(A)1607 1520 y Fp(\023)1664 1508 y Fu(and)28 b Fo(\026)23 b Fr(2)g Fo(J)0 1750 y Fu(\()p Fk(Cut)158 1762 y Fp(C)214 1750 y Fu(\))623 1696 y Fo(C)89 b Fr(:)p Fo(C)p 623 1731 269 4 v 736 1805 a Fr(;)0 1993 y Fu(\()p Fk(Rep)p Fu(\))736 1938 y Fr(;)p 723 1974 70 4 v 737 2047 a(;)0 2235 y Fu(\()p Fk(Cl)105 2247 y Fp(P)147 2255 y Ff(A)198 2247 y Fp(t)227 2235 y Fu(\))603 2180 y Fr(A)p Fu(\()p Fo(P)754 2192 y Fi(A)812 2180 y Fo(;)14 b(t)p Fu(\))p 603 2216 309 4 v 687 2290 a Fo(P)740 2302 y Fi(A)798 2290 y Fo(t)0 2414 y Ft(De\014nition)31 b(of)h Fk(k)o Fu(\()p Fr(I)6 b Fu(\))29 b(for)e(eac)n(h)g(ID)1146 2378 y Fi(1)1146 2435 y Fn(1)1216 2414 y Fu(-inference)g Fr(I)0 2580 y Fk(k)p Fu(\()p Fk(Ax)167 2543 y Fi(1)167 2601 y Fn(\001)237 2580 y Fu(\))c(:=)g Fk(k)o Fu(\()p Fk(Rep)p Fu(\))h(:=)e Fr(;)27 b Fu(,)h Fk(k)p Fu(\()968 2518 y Fl(V)1037 2605 y Fp(A)1091 2580 y Fu(\))c(:=)e Fk(k)p Fu(\()p Fk(Cut)1456 2592 y Fp(A)1510 2580 y Fu(\))i(:=)e Fk(k)p Fu(\()p Fk(Cl)1822 2592 y Fp(A)1876 2580 y Fu(\))h(:=)g Fk(k)p Fu(\()p Fo(A)p Fu(\))28 b(,)g Fk(k)p Fu(\()2361 2518 y Fl(W)2430 2538 y Fp(\026)2430 2605 y(A)2484 2580 y Fu(\))23 b(:=)g Fk(k)p Fu(\()p Fo(\026)p Fu(\).)0 2796 y Ft(Inductiv)m(e)33 b(De\014nition)e(of)g Fu(ID)1073 2760 y Fi(1)1073 2816 y Fn(1)1143 2796 y Ft(-deriv)-5 b(ations)0 2962 y Fu(If)32 b Fr(I)37 b Fu(is)31 b(an)g(inference)g(sym) n(b)r(ol)g(of)g(ID)1207 2925 y Fi(1)1207 2982 y Fn(1)1277 2962 y Fu(,)h Fo(\013)g Fu(is)f(an)g(ordinal,)g(and)g(\()p Fo(d)2172 2974 y Fp(\023)2201 2962 y Fu(\))2233 2977 y Fp(\023)p Fi(2jI)t(j)2419 2962 y Fu(is)g(a)g(family)g(of)g(ID)3028 2925 y Fi(1)3028 2982 y Fn(1)3098 2962 y Fm(-derivations)i Fu(suc)n(h)e(that)0 3125 y Fo(o)p Fu(\()p Fo(d)115 3137 y Fp(\023)144 3125 y Fu(\))24 b Fo(<)e(\013)29 b Fu(for)e(all)g Fo(\023)c Fr(2)h(jI)6 b(j)p Fu(,)28 b(then)56 b Fo(d)23 b Fu(:=)g Fo(\013)p Fr(I)6 b Fu(\()p Fo(d)1463 3137 y Fp(\023)1493 3125 y Fu(\))1525 3140 y Fp(\023)p Fi(2jI)t(j)1702 3125 y Fu(:=)1837 3070 y Fo(:)14 b(:)g(:)f(d)1990 3082 y Fp(\023)2033 3070 y Fo(:)h(:)g(:)p 1823 3106 308 4 v 1890 3180 a(\013)23 b Fu(:)g Fr(I)2168 3125 y Fu(is)k(an)h(ID)2460 3089 y Fi(1)2460 3145 y Fn(1)2530 3125 y Fm(-derivation)i Fu(with)55 b Fo(o)p Fu(\()p Fo(d)p Fu(\))24 b(:=)f Fo(\013)p Fu(.)0 3324 y(The)32 b(cut-elimination)g(op)r(erators)e Fr(R)1198 3336 y Fp(C)1286 3324 y Fu(and)i Fr(E)39 b Fu(are)31 b(assumed)g(to)h(b)r(e)g(adapted)g(to)g(this)g(mo)r(di\014ed) g(notion)g(of)g(deriv)-5 b(ation)0 3465 y(where)27 b(for)g Fr(E)35 b Fu(w)n(e)27 b(use)h Fo(!)767 3435 y Fn(\012+)p Fp(\013)940 3465 y Fu(instead)f(of)h(3)1363 3435 y Fp(\013)1437 3465 y Fu(\(cf.)38 b(pro)r(ofs)26 b(of)i(9.8)f(and)g(9.9\).)0 3731 y Ft(De\014nition)0 3872 y Fu(A)h(set)g Fo(H)h Fr(\022)23 b Fo(O)r(n)28 b Fu(is)g(called)f Fm(nic)l(e)55 b Fu(i\013)h Fr(f)p Fu(0)p Fo(;)14 b Fu(1)p Fr(g)21 b(\022)i Fo(H)34 b Fu(and)28 b Fo(H)34 b Fu(is)28 b(closed)f(under)g(#)14 b Fo(;)28 b(\025\030)t(:)14 b(!)s Fr(\001)p Fo(\030)k(;)28 b(\025\030)t(:)14 b(!)3062 3842 y Fn(\012+)p Fp(\030)3197 3872 y Fu(.)22 4017 y(~)0 4038 y Fo(H)30 b Fu(:=)23 b Fr(f)p Fo(\023)f Fu(:)i Fk(k)o Fu(\()p Fo(\023)p Fu(\))g Fr(\022)f Fo(H)7 b Fr(g)p Fu(.)0 4204 y Ft(Con)m(v)m(en)m(tion:)37 b Fu(In)28 b(the)g(follo)n(wing)e Fo(H)35 b Fu(is)27 b(used)h(as)f(syn)n(tactical)f(v)-5 b(ariable)27 b(for)g(nice)g(sets.)0 4369 y Ft(Remark:)35 b Fu(Ev)n(ery)26 b Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))29 b(with)f Fo(\015)f Fr(\025)c Fo(!)30 b Fu(is)e(nice.)0 4535 y Ft(De\014nition)0 4676 y Fu(Let)g Fo(d)23 b Fu(=)g Fo(\013)p Fr(I)6 b Fu(\()p Fo(d)482 4688 y Fp(\023)512 4676 y Fu(\))544 4688 y Fp(\023)p Fi(2)p Fp(I)651 4676 y Fu(:)65 b Fo(d)19 b(/)f(H)34 b Fu(\()p Fo(d)28 b Fu(is)g Fo(H)7 b Fm(-c)l(ontr)l(ol)t(le)l(d)p Fu(\))28 b(i\013)g Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(\013)p Fr(g)j(\022)h Fo(H)34 b Fu(and)28 b Fr(8)p Fo(\023)22 b Fr(2)i Fo(I)h Fr(\\)2857 4655 y Fu(~)2835 4676 y Fo(H)7 b Fu(\()p Fo(d)2986 4688 y Fp(\023)3034 4676 y Fo(/)17 b(H)7 b Fu(\).)0 4892 y Ft(Theorem)30 b(9.8)0 5033 y Fo(d)43 5045 y Fn(0)99 5033 y Fo(/)18 b(H)53 b Fu(&)45 b Fo(d)434 5045 y Fn(1)490 5033 y Fo(/)18 b(H)53 b Fu(&)46 b(rk)o(\()p Fo(C)6 b Fu(\))24 b Fr(6)p Fu(=)f(\012)46 b(=)-14 b Fr(\))46 b(R)1456 5045 y Fp(C)1512 5033 y Fu(\()p Fo(d)1587 5045 y Fn(0)1625 5033 y Fo(;)14 b(d)1705 5045 y Fn(1)1742 5033 y Fu(\))19 b Fo(/)f(H)7 b Fu(.)0 5199 y(Pro)r(of:)0 5340 y(Abbr.:)37 b Fo(\013)24 b Fu(:=)e(o\()p Fo(d)574 5352 y Fn(0)611 5340 y Fu(\),)29 b Fo(\014)e Fu(:=)c(o\()p Fo(d)997 5352 y Fn(1)1034 5340 y Fu(\).)1908 5539 y(56)p eop %%Page: 57 58 57 57 bop 0 83 a Fu(\(Case)27 b(1\))h Fo(C)33 b Fu(is)28 b(not)g(main)f(form)n(ula)g(of)g Fr(I)j Fu(:=)22 b Fk(last)p Fu(\()p Fo(d)1645 95 y Fn(0)1683 83 y Fu(\):)0 249 y(W)-7 b(e)28 b(ha)n(v)n(e)e Fr(R)404 261 y Fp(C)461 249 y Fu(\()p Fo(d)536 261 y Fn(0)573 249 y Fo(;)14 b(d)653 261 y Fn(1)691 249 y Fu(\))23 b(=)g(\()p Fo(\013)p Fu(#)p Fo(\014)t Fu(\))p Fr(I)6 b Fu(\()p Fr(R)1224 261 y Fp(C)1282 249 y Fu(\()p Fo(d)1357 261 y Fn(0)p Fp(\023)1419 249 y Fo(;)14 b(d)1499 261 y Fn(1)1537 249 y Fu(\)\))1601 261 y Fp(\023)p Fi(2)p Fp(I)1736 249 y Fu(where)27 b Fo(d)2019 261 y Fn(0)2080 249 y Fu(=)c Fo(\013)p Fr(I)6 b Fu(\()p Fo(d)2347 261 y Fn(0)p Fp(\023)2410 249 y Fu(\))2442 261 y Fp(\023)p Fi(2)p Fp(I)2549 249 y Fu(.)0 440 y(Since)28 b Fo(d)260 452 y Fn(0)316 440 y Fo(/)18 b(H)7 b Fu(,)27 b(w)n(e)g(also)g(ha)n(v)n (e)f Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(\013)p Fr(g)k(\022)f Fo(H)35 b Fu(and)27 b Fo(d)1786 452 y Fn(0)p Fp(\023)1866 440 y Fo(/)18 b(H)35 b Fu(for)27 b(all)g Fo(\023)d Fr(2)f Fo(I)j Fr(\\)2560 419 y Fu(~)2538 440 y Fo(H)7 b Fu(.)0 606 y(F)-7 b(rom)27 b(this)h(\(together)f(with)h Fo(\014)g Fr(2)23 b Fo(H)e Fu(\))28 b(w)n(e)f(get)h Fk(k)o Fu(\()p Fr(I)6 b Fu(\))20 b Fr([)e(f)p Fo(\013)p Fu(#)p Fo(\014)t Fr(g)24 b(\022)e Fo(H)7 b Fu(,)28 b(and)f(\(b)n(y)h(IH\))g Fr(R)2767 618 y Fp(C)2823 606 y Fu(\()p Fo(d)2898 618 y Fn(0)p Fp(\023)2961 606 y Fo(;)14 b(d)3041 618 y Fn(1)3078 606 y Fu(\))19 b Fo(/)f(H)34 b Fu(for)27 b(all)h Fo(\023)23 b Fr(2)h Fo(I)h Fr(\\)3823 585 y Fu(~)3801 606 y Fo(H)7 b Fu(,)0 747 y(i.e.)37 b Fr(R)213 759 y Fp(C)269 747 y Fu(\()p Fo(d)344 759 y Fn(0)382 747 y Fo(;)14 b(d)462 759 y Fn(1)499 747 y Fu(\))19 b Fo(/)f(H)7 b Fu(.)0 913 y(\(Case)27 b(2\))h Fo(C)h Fr(')509 851 y Fl(W)578 938 y Fp(\023)p Fi(2)p Fp(J)693 913 y Fo(C)752 925 y Fp(\023)809 913 y Fu(is)f(main)f(form)n(ula)g(of)g Fk(last)p Fu(\()p Fo(d)1696 925 y Fn(0)1734 913 y Fu(\),)h(and)f Fr(:)p Fo(C)j Fr(')2209 851 y Fl(V)2279 938 y Fp(\023)p Fi(2)p Fp(J)2394 913 y Fr(:)p Fo(C)2508 925 y Fp(\023)2565 913 y Fu(is)e(main)f(form)n(ula)g(of)h Fk(last)o Fu(\()p Fo(d)3452 925 y Fn(1)3490 913 y Fu(\):)0 1054 y(Then)g Fk(last)o Fu(\()p Fo(d)413 1066 y Fn(1)451 1054 y Fu(\))23 b(=)594 992 y Fl(V)663 1079 y Fi(:)p Fp(C)764 1054 y Fu(,)28 b(and,)f(since)g(rk\()p Fo(C)6 b Fu(\))24 b Fr(6)p Fu(=)e(\012,)28 b Fk(last)p Fu(\()p Fo(d)1827 1066 y Fn(0)1864 1054 y Fu(\))c(=)2007 992 y Fl(W)2076 1013 y Fp(\026)2076 1079 y(C)2132 1054 y Fu(.)0 1196 y(Hence)55 b Fr(R)344 1208 y Fp(C)401 1196 y Fu(\()p Fo(d)476 1208 y Fn(0)513 1196 y Fo(;)14 b(d)593 1208 y Fn(1)631 1196 y Fu(\))23 b(=)g(\()p Fo(\013)p Fu(#)p Fo(\014)t Fu(\))p Fk(Cut)1138 1208 y Fp(C)1186 1216 y Fh(\026)1229 1196 y Fr(R)1299 1208 y Fp(C)1356 1196 y Fu(\()p Fo(d)1431 1208 y Fn(00)1501 1196 y Fo(;)14 b(d)1581 1208 y Fn(1)1619 1196 y Fu(\))p Fr(R)1721 1208 y Fp(C)1777 1196 y Fu(\()p Fo(d)1852 1208 y Fn(0)1890 1196 y Fo(;)g(d)1970 1208 y Fn(1)p Fp(\026)2048 1196 y Fu(\),)28 b(where)f Fo(d)2414 1208 y Fn(0)2474 1196 y Fu(=)c Fo(\013)2615 1133 y Fl(W)2685 1154 y Fp(\026)2685 1220 y(C)2740 1196 y Fo(d)2783 1208 y Fn(00)2882 1196 y Fu(and)k Fo(d)3086 1208 y Fn(1)3147 1196 y Fu(=)22 b Fo(\014)3285 1133 y Fl(V)3355 1220 y Fi(:)p Fp(C)3455 1196 y Fu(\()p Fo(d)3530 1208 y Fn(1)p Fp(\023)3593 1196 y Fu(\))3625 1208 y Fp(\023)p Fi(2)p Fp(J)3741 1196 y Fu(.)0 1362 y(W)-7 b(e)28 b(ha)n(v)n(e)e Fk(k)p Fu(\()p Fo(C)466 1374 y Fp(\026)511 1362 y Fu(\))19 b Fr([)g(f)p Fo(\013;)14 b(\014)t Fr(g)22 b(\022)h Fk(k)p Fu(\()p Fo(\026)p Fu(\))c Fr([)f Fk(k)p Fu(\()p Fo(C)6 b Fu(\))19 b Fr([)g(f)p Fo(\013;)14 b(\014)t Fr(g)23 b(\022)g Fo(H)7 b Fu(,)27 b Fo(d)1986 1374 y Fn(00)2075 1362 y Fo(/)18 b(H)7 b Fu(,)27 b(and)h Fo(d)2466 1374 y Fn(1)p Fp(\026)2562 1362 y Fo(/)18 b(H)35 b Fu(\(since)27 b Fk(k)p Fu(\()p Fo(\026)p Fu(\))d Fr(\022)e Fo(H)7 b Fu(\).)0 1528 y(F)-7 b(rom)36 b Fo(\013;)14 b(\014)43 b Fr(2)38 b Fo(H)43 b Fu(w)n(e)36 b(get)h Fo(\013)p Fu(#)p Fo(\014)43 b Fr(2)38 b Fo(H)7 b Fu(.)63 b(By)37 b(IH)f(w)n(e)h(get)f Fr(R)1975 1540 y Fp(C)2031 1528 y Fu(\()p Fo(d)2106 1540 y Fn(00)2177 1528 y Fo(;)14 b(d)2257 1540 y Fn(1)2294 1528 y Fu(\))25 b Fo(/)f(H)43 b Fu(and)36 b Fr(R)2769 1540 y Fp(C)2826 1528 y Fu(\()p Fo(d)2901 1540 y Fn(0)2938 1528 y Fo(;)14 b(d)3018 1540 y Fn(1)p Fp(\026)3096 1528 y Fu(\))25 b Fo(/)f(H)7 b Fu(.)63 b(T)-7 b(ogether)36 b(with)0 1669 y Fk(k)p Fu(\()p Fo(C)132 1681 y Fp(\026)177 1669 y Fu(\))18 b Fr([)h(f)p Fo(\013)p Fu(#)p Fo(\014)t Fr(g)k(\022)g Fo(H)34 b Fu(this)28 b(yields)g Fr(R)1238 1681 y Fp(C)1294 1669 y Fu(\()p Fo(d)1369 1681 y Fn(0)1407 1669 y Fo(;)14 b(d)1487 1681 y Fn(1)1524 1669 y Fu(\))19 b Fo(/)f(H)7 b Fu(.)0 1885 y Ft(Theorem)30 b(9.9)0 2026 y Fu(a\))d Fo(d)d Fr(`)219 1996 y Fp(\013)219 2046 y(\032)p Fn(+1)364 2026 y Fu(\000)46 b(&)g Fo(\032)23 b Fr(6)p Fu(=)g(\012)46 b(=)-14 b Fr(\))45 b(E)7 b Fu(\()p Fo(d)p Fu(\))25 b Fr(`)1246 1996 y Fp(!)1290 1971 y Fg(\012+)p Fh(\013)1246 2046 y Fp(\032)1441 2026 y Fu(\000,)0 2167 y(b\))j Fo(d)19 b(/)f(H)53 b Fu(=)-14 b Fr(\))45 b(E)7 b Fu(\()p Fo(d)p Fu(\))20 b Fo(/)e(H)7 b Fu(.)0 2333 y(Pro)r(of)26 b(of)i(b\):)0 2474 y(\(Case)f(1\))h Fo(d)23 b Fu(=)f Fo(\013)p Fk(Cut)666 2486 y Fp(C)722 2474 y Fo(d)765 2486 y Fn(0)802 2474 y Fo(d)845 2486 y Fn(1)910 2474 y Fu(with)28 b(rk\()p Fo(C)6 b Fu(\))24 b Fr(6)p Fu(=)e(\012.)0 2640 y(Then)28 b Fo(\013)23 b Fr(2)h Fo(H)34 b Fu(,)28 b Fo(d)569 2652 y Fn(0)606 2640 y Fo(;)14 b(d)686 2652 y Fn(1)742 2640 y Fo(/)k(H)34 b Fu(and)28 b Fr(E)7 b Fu(\()p Fo(d)p Fu(\))24 b(=)f Fo(!)1392 2610 y Fn(\012+)p Fp(\013)1537 2640 y Fk(Rep)o Fr(R)1740 2652 y Fp(C)1797 2640 y Fu(\()p Fr(E)7 b Fu(\()p Fo(d)1955 2652 y Fn(0)1993 2640 y Fu(\))p Fo(;)14 b Fr(E)7 b Fu(\()p Fo(d)2188 2652 y Fn(1)2226 2640 y Fu(\)\).)0 2806 y(By)27 b(IH)h Fr(E)7 b Fu(\()p Fo(d)376 2818 y Fn(0)414 2806 y Fu(\))p Fo(;)14 b Fr(E)7 b Fu(\()p Fo(d)609 2818 y Fn(1)647 2806 y Fu(\))19 b Fo(/)f(H)7 b Fu(,)28 b(and)f(therefore)g(b)n(y)g(9.8)g Fr(R)1714 2818 y Fp(C)1770 2806 y Fu(\()p Fr(E)7 b Fu(\()p Fo(d)1928 2818 y Fn(0)1966 2806 y Fu(\))p Fo(;)14 b Fr(E)7 b Fu(\()p Fo(d)2161 2818 y Fn(1)2199 2806 y Fu(\)\))19 b Fo(/)f(H)7 b Fu(.)0 2972 y(F)-7 b(rom)27 b Fo(\013)d Fr(2)f Fo(H)34 b Fu(w)n(e)28 b(get)f Fo(!)792 2942 y Fn(\012+)p Fp(\013)960 2972 y Fr(2)d Fo(H)34 b Fu(and)28 b(th)n(us)f Fr(E)7 b Fu(\()p Fo(d)p Fu(\))20 b Fo(/)e(H)7 b Fu(.)0 3138 y(\(Case)27 b(2\))h(otherwise:)36 b(Then)27 b Fo(d)d Fu(=)e Fo(\013)p Fr(I)6 b Fu(\()p Fo(d)1284 3150 y Fp(\023)1314 3138 y Fu(\))1346 3150 y Fp(\023)p Fi(2)p Fp(I)1482 3138 y Fu(and)27 b Fr(E)7 b Fu(\()p Fo(d)p Fu(\))24 b(=)f Fo(!)1968 3108 y Fn(\012+)p Fp(\013)2113 3138 y Fr(I)6 b Fu(\()p Fr(E)h Fu(\()p Fo(d)2322 3150 y Fp(\023)2352 3138 y Fu(\)\))2416 3150 y Fp(\023)p Fi(2)p Fp(I)2524 3138 y Fu(.)0 3304 y Fo(d)19 b(/)f(H)53 b Fr(\))73 b Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(\013)p Fr(g)k(\022)f Fo(H)30 b Fu(&)23 b Fo(d)1126 3316 y Fp(\023)1173 3304 y Fo(/)18 b(H)34 b Fu(for)27 b Fo(\023)d Fr(2)f Fo(I)j Fr(\\)1751 3283 y Fu(~)1730 3304 y Fo(H)1856 3257 y Fn(IH)1851 3304 y Fr(\))0 3445 y Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(!)346 3415 y Fn(\012+)p Fp(\013)491 3445 y Fr(g)j(\022)h Fo(H)30 b Fu(&)22 b Fr(E)7 b Fu(\()p Fo(d)955 3457 y Fp(\023)985 3445 y Fu(\))18 b Fo(/)g(H)35 b Fu(for)27 b Fo(\023)c Fr(2)h Fo(I)h Fr(\\)1614 3424 y Fu(~)1592 3445 y Fo(H)53 b Fr(\))46 b(E)7 b Fu(\()p Fo(d)p Fu(\))20 b Fo(/)d(H)7 b Fu(.)0 3661 y Ft(De\014nition)0 3802 y Fu(F)-7 b(or)29 b(eac)n(h)g Fr(L)p Fu(\(ID)523 3814 y Fn(1)560 3802 y Fu(\)-form)n(ula)f Fo(B)34 b Fu(let)c Fo(B)t Fu([)p Fo(\013;)14 b(\014)t Fu(])31 b(denote)e(the)h(result)f (of)h(replacing)e(in)i Fo(B)k Fu(ev)n(ery)28 b(negativ)n(e)g(o)r (ccurrence)h(of)g Fo(P)3842 3814 y Fi(A)0 3944 y Fu(b)n(y)e Fo(P)180 3911 y Fp(<\013)168 3968 y Fi(A)279 3944 y Fu(,)h(and)g(ev)n (ery)e(p)r(ositiv)n(e)h(o)r(ccurrence)f(of)i Fo(P)1579 3956 y Fi(A)1664 3944 y Fu(b)n(y)g Fo(P)1845 3904 y Fp(<\014)1833 3968 y Fi(A)1941 3944 y Fu(.)0 4110 y(\000[)p Fo(\013;)14 b(\014)t Fu(])24 b(:=)e Fr(f)p Fo(B)t Fu([)p Fo(\013;)14 b(\014)t Fu(])24 b(:)f Fo(B)k Fr(2)d Fu(\000)p Fr(g)0 4276 y(j)-14 b Fu(=)26 b Fr(f)p Fo(A)204 4288 y Fn(1)241 4276 y Fo(;)14 b(:::;)g(A)446 4288 y Fp(n)491 4276 y Fr(g)52 b Fu(:)26 b Fr(\()-14 b(\))104 b Fo(A)952 4288 y Fn(1)1009 4276 y Fr(_)20 b Fo(:::)g Fr(_)g Fo(A)1310 4288 y Fp(n)1385 4276 y Fu(is)29 b(true)g(in)h(the)f(standard)g(in)n (terpretation)f(where)h(the)h(set)f(v)-5 b(ariables)28 b(ma)n(y)h(b)r(e)0 4417 y(in)n(terpreted)e(arbitrarily)-7 b(.)1908 5539 y(57)p eop %%Page: 58 59 58 58 bop 0 83 a Ft(De\014nition)54 b Fu(Let)28 b Fr(H)h Fu(b)r(e)f(a)f(collection)g(of)g(nice)h(sets:)0 223 y Fo(d)19 b(/)104 193 y Fi(\003)160 223 y Fr(H)47 b Fu(:)p Fr(,)f(8)p Fo(H)29 b Fr(2)23 b(H)q Fu(\()p Fo(d)c(/)f(H)7 b Fu(\))28 b(,)166 b Fr(H)q Fu([)p Fo(\023)p Fu(])24 b(:=)e Fr(f)p Fo(H)30 b Fr(2)23 b(H)h Fu(:)f Fk(k)p Fu(\()p Fo(\023)p Fu(\))h Fr(\022)e Fo(H)7 b Fr(g)p Fu(.)0 412 y Ft(Lemma)30 b(9.10)0 553 y Fu(Let)e Fo(d)23 b Fu(=)g Fo(\013)p Fr(I)6 b Fu(\()p Fo(d)482 565 y Fp(\023)512 553 y Fu(\))544 565 y Fp(\023)p Fi(2)p Fp(I)651 553 y Fu(:)65 b Fo(d)19 b(/)843 523 y Fi(\003)899 553 y Fr(H)47 b(,)f Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(\013)p Fr(g)j(\022)1641 491 y Fl(T)1724 553 y Fr(H)47 b Fu(&)f Fr(8)p Fo(\023)22 b Fr(2)i Fo(I)7 b Fu(\()p Fo(d)2247 565 y Fp(\023)2294 553 y Fo(/)2336 523 y Fi(\003)2392 553 y Fr(H)q Fu([)p Fo(\023)p Fu(]\).)0 718 y(Pro)r(of:)0 858 y Fo(d)19 b(/)104 828 y Fi(\003)160 858 y Fr(H)47 b(,)0 999 y(8)p Fo(H)29 b Fr(2)23 b(H)q Fu(\()p Fo(d)c(/)f(H)7 b Fu(\))46 b Fr(,)0 1139 y(8)p Fo(H)29 b Fr(2)23 b(H)q Fu(\()p Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(\013)p Fr(g)k(\022)f Fo(H)30 b Fu(&)23 b Fr(8)p Fo(\023)f Fr(2)i Fo(I)h Fr(\\)1343 1118 y Fu(~)1321 1139 y Fo(H)7 b Fu(\()p Fo(d)1472 1151 y Fp(\023)1520 1139 y Fo(/)18 b(H)7 b Fu(\)\))46 b Fr(,)0 1280 y Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(\013)p Fr(g)j(\022)496 1217 y Fl(T)579 1280 y Fr(H)i Fu(&)f Fr(8)p Fo(\023)f Fr(2)h Fo(I)7 b Fr(8)p Fo(H)29 b Fr(2)24 b(H)q Fu(\()p Fk(k)p Fu(\()p Fo(\023)p Fu(\))g Fr(\022)e Fo(H)30 b Fr(\))23 b Fo(d)1800 1292 y Fp(\023)1848 1280 y Fo(/)18 b(H)7 b Fu(\))46 b Fr(,)0 1420 y Fk(k)p Fu(\()p Fr(I)6 b Fu(\))19 b Fr([)g(f)p Fo(\013)p Fr(g)j(\022)496 1358 y Fl(T)579 1420 y Fr(H)i Fu(&)f Fr(8)p Fo(\023)f Fr(2)h Fo(I)7 b Fu(\()p Fo(d)1055 1432 y Fp(\023)1103 1420 y Fo(/)1145 1390 y Fi(\003)1201 1420 y Fr(f)p Fo(H)30 b Fr(2)23 b(H)h Fu(:)f Fk(k)p Fu(\()p Fo(\023)p Fu(\))h Fr(\022)e Fo(H)7 b Fr(g)p Fu(\).)0 1624 y Ft(Abbreviation:)37 b Fr(H)683 1636 y Fp(\015)748 1624 y Fu(:=)23 b Fr(f)p Fo(C)6 b Fu(\()p Fo(\030)t Fu(\))24 b(:)f Fo(\030)k Fr(\025)c Fo(\015)5 b Fr(g)p Fu(.)36 b(|)27 b(Note)h(that)1931 1561 y Fl(T)2014 1624 y Fr(H)2084 1636 y Fp(\015)2150 1624 y Fu(=)22 b Fo(C)6 b Fu(\()p Fo(\015)f Fu(\),)28 b(and)g Fr(H)2697 1636 y Fp(\015)2740 1624 y Fu([)p Fo(\023)p Fu(])23 b(=)g Fr(H)2996 1636 y Fp(\015)3066 1624 y Fu(if)28 b Fk(k)p Fu(\()p Fo(\023)p Fu(\))c Fr(\022)e Fo(C)6 b Fu(\()p Fo(\015)f Fu(\).)0 1803 y Ft(Theorem)30 b(9.11)0 1959 y Fo(d)23 b Fr(`)117 1928 y Fp(\013)117 1981 y Fn(\012+1)276 1959 y Fu(\000)46 b(&)f Fo(!)26 b Fr(\024)d Fo(\015)k Fr(2)d Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))23 b(&)g Fo(d)c(/)1191 1928 y Fi(\003)1247 1959 y Fr(H)1317 1971 y Fp(\015)1406 1959 y Fu(=)-14 b Fr(\))22 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )s Fu(\()p Fo(\015)23 b Fu(+)18 b Fo(!)2169 1928 y Fn(\012+)p Fp(\013)2314 1959 y Fu(\)])0 2138 y(Pro)r(of:)0 2397 y(1.)36 b Fo(d)24 b Fu(=)255 2177 y Fl(8)255 2252 y(>)255 2277 y(>)255 2302 y(<)255 2451 y(>)255 2476 y(>)255 2501 y(:)461 2241 y Fo(d)504 2253 y Fn(0)p 499 2337 5 59 v 329 2429 a Fu(\000)p Fo(;)14 b(P)e(t)23 b Fu(:)g Fo(\013)635 2441 y Fn(0)942 2241 y Fo(d)985 2253 y Fn(1)p 980 2337 V 783 2429 a Fu(\000)p Fo(;)14 b Fr(:)p Fo(P)e(t)23 b Fu(:)g Fo(\013)1144 2441 y Fn(1)p 329 2470 853 4 v 1200 2487 a(\()p Fd(Cut)1333 2495 y Fh(P)8 b(t)1404 2487 y Fn(\))668 2562 y Fu(\000)23 b(:)g Fo(\013)1490 2397 y Fu(:)166 b(Let)28 b Fo(\015)1871 2409 y Fn(0)1931 2397 y Fu(:=)23 b Fo(\015)g Fu(+)18 b Fo(!)2246 2367 y Fn(\012+)p Fp(\013)2387 2375 y Fg(0)2424 2397 y Fu(,)27 b Fo(\015)2517 2409 y Fn(1)2578 2397 y Fu(:=)22 b Fo(\015)2731 2409 y Fn(0)2787 2397 y Fu(+)c Fo(!)2925 2367 y Fn(\012+)p Fp(\013)3066 2375 y Fg(1)3103 2397 y Fu(,)31 b(~)-45 b Fo(\015)27 b Fu(:=)c Fo(\015)g Fu(+)18 b Fo(!)3539 2367 y Fn(\012+)p Fp(\013)3684 2397 y Fu(.)0 2685 y Fo(d)h(/)104 2655 y Fi(\003)160 2685 y Fr(H)230 2697 y Fp(\015)319 2685 y Fr(\))46 b Fo(d)491 2697 y Fn(0)528 2685 y Fo(;)14 b(d)608 2697 y Fn(1)664 2685 y Fo(/)706 2655 y Fi(\003)762 2685 y Fr(H)832 2697 y Fp(\015)921 2685 y Fr(\))46 b Fo(d)1093 2697 y Fn(0)1149 2685 y Fo(/)1191 2655 y Fi(\003)1247 2685 y Fr(H)1317 2697 y Fp(\015)1383 2685 y Fu(&)23 b Fo(d)1514 2697 y Fn(1)1569 2685 y Fo(/)1611 2655 y Fi(\003)1668 2685 y Fr(H)1738 2697 y Fp(\015)1773 2705 y Fg(0)1809 2685 y Fu(.)65 b Fo(d)1940 2697 y Fn(0)1996 2685 y Fo(/)2038 2655 y Fi(\003)2094 2685 y Fr(H)2164 2697 y Fp(\015)2253 2685 y Fr(\))46 b Fo(\013)2435 2697 y Fn(0)2495 2685 y Fr(2)24 b Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))46 b Fr(\))g Fo(\015)2969 2697 y Fn(0)3030 2685 y Fr(2)23 b Fo(C)6 b Fu(\()p Fo(\015)3248 2697 y Fn(0)3286 2685 y Fu(\).)0 2841 y(IH)56 b(=)-14 b Fr(\))22 b(j)-14 b Fu(=)23 b(\000)p Fo(;)14 b(P)e(t)i Fu([)p Fo( )s(\015)5 b(;)14 b( )s(\015)864 2853 y Fn(0)901 2841 y Fu(])27 b(and)h Fr(j)-14 b Fu(=)23 b(\000)p Fo(;)14 b Fr(:)p Fo(P)e(t)i Fu([)p Fo( )s(\015)1586 2853 y Fn(0)1623 2841 y Fo(;)g( )s(\015)1760 2853 y Fn(1)1797 2841 y Fu(])55 b(\(and)28 b Fo(\015)g(<)22 b(\015)2270 2853 y Fn(0)2331 2841 y Fo(<)g(\015)2461 2853 y Fn(1)2522 2841 y Fo(<)j Fu(~)-45 b Fo(\015)5 b Fu(\))148 3020 y(=)-14 b Fr(\))22 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])p Fo(;)14 b(P)847 2990 y Fp(< )r(\015)980 2998 y Fg(0)1017 3020 y Fo(t)27 b Fu(and)h Fr(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])p Fo(;)14 b Fr(:)p Fo(P)1834 2990 y Fp(< )r(\015)1967 2998 y Fg(0)2003 3020 y Fo(t)51 b Fu(=)-14 b Fr(\))23 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])0 3224 y(F)-7 b(or)27 b(the)h(rest)f(of)h(this)g(pro)r(of)f(w)n(e) g(set)g Fo(\015)1222 3236 y Fp(\023)1274 3224 y Fu(:=)c Fo(\015)g Fu(+)18 b Fo(!)1589 3194 y Fn(\012+)p Fp(\013)1730 3202 y Fh(\023)1789 3224 y Fu(and)30 b(~)-45 b Fo(\015)28 b Fu(:=)23 b Fo(\015)g Fu(+)18 b Fo(!)2336 3194 y Fn(\012+)p Fp(\013)2481 3224 y Fu(.)0 3524 y(2.)36 b Fo(d)24 b Fu(=)255 3304 y Fl(8)255 3378 y(>)255 3403 y(>)255 3428 y(<)255 3577 y(>)255 3602 y(>)255 3627 y(:)444 3367 y Fo(d)487 3379 y Fn(0)p 482 3463 5 59 v 329 3555 a Fu(\000)p Fo(;)14 b(A)23 b Fu(:)g Fo(\013)602 3567 y Fn(0)893 3367 y Fo(d)936 3379 y Fn(1)p 931 3463 V 750 3555 a Fu(\000)p Fo(;)14 b Fr(:)p Fo(A)24 b Fu(:)f Fo(\013)1079 3567 y Fn(1)p 329 3597 788 4 v 1135 3613 a(\()p Fd(Cut)1268 3621 y Fh(A)1315 3613 y Fn(\))636 3688 y Fu(\000)f(:)i Fo(\013)1426 3524 y Fu(with)k(rk\()p Fo(A)p Fu(\))c Fo(<)e Fu(\012:)0 3812 y(IH)28 b(=)-14 b Fr(\))23 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )s(\015)691 3824 y Fn(0)727 3812 y Fu(])p Fo(;)g(A)28 b Fu(and)g Fr(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )s(\015)1453 3824 y Fn(1)1489 3812 y Fu(])p Fo(;)g Fr(:)p Fo(A)47 b Fu(=)-14 b Fr(\))23 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(].)0 4116 y(3.)36 b Fo(d)24 b Fu(=)255 3896 y Fl(8)255 3971 y(>)255 3996 y(>)255 4021 y(<)255 4170 y(>)255 4195 y(>)255 4220 y(:)540 3960 y Fo(d)583 3972 y Fn(0)p 578 4055 5 59 v 329 4148 a Fu(\000)p Fo(;)14 b Fr(A)p Fu(\()p Fo(P)r(;)g(t)p Fu(\))24 b(:)f Fo(\013)793 4160 y Fn(0)p 329 4189 502 4 v 850 4206 a(\()p Fd(Cl)938 4214 y Fh(P)7 b(t)1009 4206 y Fn(\))493 4280 y Fu(\000)23 b(:)g Fo(\013)1086 4116 y Fu(:)0 4404 y Fo(d)43 4416 y Fn(0)99 4404 y Fo(/)141 4374 y Fi(\003)197 4404 y Fr(H)267 4416 y Fp(\015)356 4404 y Fr(\))46 b Fo(\013)538 4416 y Fn(0)598 4404 y Fr(2)24 b Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))46 b Fr(\))h Fo(\015)1073 4416 y Fn(0)1133 4404 y Fr(2)23 b Fo(C)6 b Fu(\()p Fo(\015)1351 4416 y Fn(0)1389 4404 y Fu(\))46 b Fr(\))h Fo( )s(\015)1697 4416 y Fn(0)1757 4404 y Fo(<)23 b( )6 b Fu(~)-45 b Fo(\015)32 b Fu(\()p Fr(\003)p Fu(\))c(.)0 4584 y(IH)g(=)-14 b Fr(\)j)g Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )s(\015)668 4596 y Fn(0)704 4584 y Fu(])p Fo(;)g Fr(A)p Fu(\()p Fo(P)927 4554 y Fp(< )r(\015)1060 4562 y Fg(0)1097 4584 y Fo(;)g(t)p Fu(\))1266 4530 y Fn(\()p Fi(\003)p Fn(\))1242 4584 y Fu(=)-14 b Fr(\))23 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])p Fo(;)14 b(P)1942 4554 y Fp(< )5 b Fn(~)-36 b Fp(\015)2082 4584 y Fo(t)83 b Fu(\(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)5 b Fu(])27 b(,)h(since)f Fo(P)12 b(t)23 b Fr(2)h Fu(\000\).)0 4888 y(4.)36 b Fo(d)24 b Fu(=)255 4668 y Fl(8)255 4743 y(>)255 4768 y(>)255 4792 y(<)255 4942 y(>)255 4967 y(>)255 4992 y(:)685 4732 y Fo(d)728 4744 y Fp(\030)p 723 4827 5 59 v 329 4919 a Fo(:::)p Fu(\000)p Fo(;)14 b Fr(:A)p Fu(\()p Fo(P)705 4889 y Fp(<\030)794 4919 y Fo(;)g(t)p Fu(\))23 b(:)g Fo(\013)1015 4931 y Fp(\030)1052 4919 y Fo(:::)p Fu(\()p Fo(\030)28 b(<)22 b Fu(\012\))p 329 4961 1068 4 v 1415 4978 a Fn(\()p Fr(^)1497 4986 y Ff(:)p Fh(P)7 b(t)1607 4978 y Fn(\))638 5052 y Fu(\000)23 b(:)g Fo(\013)1665 4888 y Fu(:)37 b(Note)28 b(that)g Fr(f)p Fo(\030)f Fr(2)c Fu(\012)g(:)g Fk(k)p Fu(\()p Fo(\030)t Fu(\))h Fr(\022)e Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))p Fr(g)23 b Fu(=)g Fo( )s Fu(\()p Fo(\015)5 b Fu(\).)0 5185 y Fo(d)19 b(/)104 5155 y Fi(\003)160 5185 y Fr(H)230 5197 y Fp(\015)319 5185 y Fr(\))46 b(8)p Fo(\030)26 b(<)d( )s(\015)5 b Fu(\()p Fo(d)825 5197 y Fp(\030)880 5185 y Fo(/)922 5155 y Fi(\003)978 5185 y Fr(H)1048 5197 y Fp(\015)1091 5185 y Fu([)p Fo(\030)t Fu(])23 b(=)f Fr(H)1357 5197 y Fp(\015)1400 5185 y Fu(\))1484 5138 y Fn(IH)1479 5185 y Fr(\))46 b(8)p Fo(\030)26 b(<)d( )s(\015)41 b Fr(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])p Fo(;)14 b Fr(:A)p Fu(\()p Fo(P)2642 5155 y Fp(<\030)2731 5185 y Fo(;)g(t)p Fu(\))47 b Fr(\))0 5340 y(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])p Fo(;)14 b Fr(:)p Fo(P)598 5310 y Fp(< )r(\015)822 5340 y Fu(\(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])27 b(,)h(since)g Fr(:)p Fo(P)12 b(t)23 b Fr(2)g Fu(\000\).)1908 5539 y(58)p eop %%Page: 59 60 59 59 bop 0 227 a Fu(5.)36 b Fo(d)24 b Fu(=)255 7 y Fl(8)255 81 y(>)255 106 y(>)255 131 y(<)255 281 y(>)255 306 y(>)255 330 y(:)685 71 y Fo(d)728 83 y Fp(\030)p 723 166 5 59 v 329 258 a Fo(:::)p Fu(\000)p Fo(;)14 b Fr(:A)p Fu(\()p Fo(P)705 228 y Fp(<\030)794 258 y Fo(;)g(t)p Fu(\))23 b(:)g Fo(\013)1015 270 y Fp(\030)1052 258 y Fo(:::)p Fu(\()p Fo(\030)28 b(<)22 b(\016)s Fu(\))p 329 300 1048 4 v 1395 316 a Fn(\()p Fr(^)1477 337 y Ff(:)p Fh(P)1560 325 y(<\016)1637 337 y(t)1665 316 y Fn(\))638 391 y Fu(\000)h(:)g Fo(\013)1751 227 y Fu(with)28 b Fo(\016)e(<)d Fu(\012:)0 548 y Fo(d)16 b(/)101 518 y Fi(\003)154 548 y Fr(H)224 560 y Fp(\015)289 548 y Fr(\))24 b(8)p Fo(\030)i Fr(2)d Fo(\016)c Fr(\\)c Fo( )s Fu(\()p Fo(\015)5 b Fu(\)\()p Fo(d)953 560 y Fp(\030)1006 548 y Fo(/)1048 518 y Fi(\003)1101 548 y Fr(H)1171 560 y Fp(\015)1214 548 y Fu([)p Fo(\030)t Fu(])23 b(=)g Fr(H)1481 560 y Fp(\015)1523 548 y Fu(\))h(&)e Fr(f)p Fo(\016)s Fr(g)g Fu(=)h Fk(k)p Fu(\()1973 486 y Fl(V)2042 573 y Fi(:)p Fp(P)2138 557 y Fh(<\016)2216 573 y Fp(t)2245 548 y Fu(\))g Fr(\022)g Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))16 b Fr(\\)f Fu(\012)23 b(=)g Fo( )s Fu(\()p Fo(\015)5 b Fu(\))46 b Fr(\))h Fu(\()p Fr(8)p Fo(\030)26 b(<)d(\016)s Fu(\))p Fo(d)3511 560 y Fp(\030)3563 548 y Fo(/)3605 518 y Fi(\003)3658 548 y Fr(H)3728 560 y Fp(\015)3822 501 y Fn(IH)3817 548 y Fr(\))0 704 y Fu(\()p Fr(8)p Fo(\030)k(<)22 b(\016)s Fu(\))i Fr(j)-14 b Fu(=)22 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)5 b Fu(])p Fo(;)14 b Fr(:A)p Fu(\()p Fo(P)1021 674 y Fp(<\030)1110 704 y Fo(;)g(t)p Fu(\))46 b Fr(\))23 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])p Fo(;)14 b Fr(:)p Fo(P)1959 674 y Fp(<\016)2047 704 y Fo(t)p Fu(.)0 1011 y(6.)36 b Fo(d)24 b Fu(=)255 791 y Fl(8)255 866 y(>)255 891 y(>)255 916 y(<)255 1065 y(>)255 1090 y(>)255 1115 y(:)593 853 y Fo(d)636 865 y Fn(0)p 631 949 5 59 v 329 1041 a Fu(\000)p Fo(;)14 b Fr(A)p Fu(\()p Fo(P)581 1011 y Fp(<\026)678 1041 y Fo(;)g(t)p Fu(\))23 b(:)g Fo(\013)899 1053 y Fn(0)p 329 1082 608 4 v 955 1099 a(\()p Fr(_)1037 1049 y Fh(\026)1037 1132 y(P)1080 1120 y(<\016)1158 1132 y(t)1185 1099 y Fn(\))546 1178 y Fu(\000)g(:)g Fo(\013)1294 1011 y Fu(with)28 b Fo(\026)23 b(<)f(\016)27 b Fr(\024)22 b Fu(\012:)0 1335 y Fo(d)d(/)104 1305 y Fi(\003)160 1335 y Fr(H)230 1347 y Fp(\015)319 1335 y Fr(\))46 b Fo(d)491 1347 y Fn(0)547 1335 y Fo(/)589 1305 y Fi(\003)645 1335 y Fr(H)715 1347 y Fp(\015)809 1288 y Fn(IH)804 1335 y Fr(\))23 b(j)-14 b Fu(=)23 b(\000[)p Fo( )s(\015)5 b(;)14 b( )6 b Fu(~)-45 b Fo(\015)t Fu(])p Fo(;)14 b Fr(A)p Fu(\()p Fo(P)1551 1305 y Fp(<\026)1648 1335 y Fo(;)g(t)p Fu(\).)0 1515 y(It)28 b(remains)f(to)g(pro)n(v)n(e)f(\()p Fo(\016)g Fu(=)d(\012)g Fr(\))g Fo(\026)g(<)g( )6 b Fu(~)-45 b Fo(\015)5 b Fu(\):)64 b Fo(d)19 b(/)1586 1485 y Fi(\003)1642 1515 y Fr(H)1712 1527 y Fp(\015)1778 1515 y Fr(\))46 b(f)p Fo(\026)p Fr(g)22 b(\022)h Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))46 b Fr(\))h Fo(\026)23 b Fr(2)g Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))19 b Fr(\\)g Fu(\012)k(=)f Fo( )s Fu(\()p Fo(\015)5 b Fu(\))24 b Fr(\024)e Fo( )6 b Fu(~)-45 b Fo(\015)5 b Fu(.)0 1781 y Fq(Em)n(b)r(edding)28 b(of)g Fu(ID)627 1793 y Fn(1)692 1781 y Fq(in)n(to)f Fu(ID)954 1745 y Fi(1)954 1801 y Fn(1)0 1947 y Ft(De\014nition)0 2088 y Fr(H)70 2100 y Fn(\000)138 2088 y Fu(:=)c Fr(f)p Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))23 b(:)g Fk(k)p Fu(\(\000\))g Fr(\022)g Fo(C)6 b Fu(\()p Fo(\015)f Fu(\))23 b(&)g Fo(\015)28 b Fr(\025)22 b Fo(!)s Fr(g)p Fu(,)0 2229 y Fo(d)h Fb(\015)126 2199 y Fp(\013)126 2250 y(\032)197 2229 y Fu(\000)46 b(:)23 b Fr(\()-14 b(\))69 b Fo(d)23 b Fr(`)679 2199 y Fp(\013)679 2250 y(\032)750 2229 y Fu(\000)46 b(&)f Fo(d)19 b(/)1062 2199 y Fi(\003)1118 2229 y Fr(H)1188 2241 y Fn(\000)1233 2229 y Fu(,)0 2370 y(ID)93 2334 y Fi(1)93 2391 y Fn(1)187 2370 y Fb(\015)247 2340 y Fp(\013)247 2391 y(\032)317 2370 y Fu(\000)46 b(:)23 b Fr(\()-14 b(\))70 b Fo(d)23 b Fb(\015)809 2340 y Fp(\013)809 2391 y(\032)879 2370 y Fu(\000)56 b(for)27 b(some)g(ID)1415 2334 y Fi(1)1415 2391 y Fn(1)1485 2370 y Fu(-deriv)-5 b(ation)27 b Fo(d)0 2561 y Ft(Remark:)35 b Fo(A)24 b Fr(2)f Fu(\000)46 b Fr(\))g Fu(rk\()p Fo(A)p Fu(\))24 b Fr(2)1090 2499 y Fl(T)1173 2561 y Fr(H)1243 2573 y Fn(\000)0 2752 y Ft(Lemma)30 b(9.12)54 b Fu(F)-7 b(or)27 b Fo(\013)c Fr(2)867 2690 y Fl(T)951 2752 y Fr(H)1021 2764 y Fn(\000)1094 2752 y Fu(:)0 2893 y(a\))k Fo(A)d Fr(')274 2831 y Fl(V)344 2918 y Fp(\023)p Fi(2)p Fp(J)459 2893 y Fo(A)521 2905 y Fp(\023)573 2893 y Fr(2)g Fu(\000)f(&)f Fr(8)p Fo(\023)h Fr(2)g Fo(J)8 b Fu(\(ID)1171 2857 y Fi(1)1171 2914 y Fn(1)1264 2893 y Fb(\015)1324 2863 y Fp(\013)1367 2871 y Fh(\023)1324 2914 y Fp(\032)1422 2893 y Fu(\000)p Fo(;)14 b(A)1573 2905 y Fp(\023)1625 2893 y Fu(&)22 b Fo(\013)1765 2905 y Fp(\023)1817 2893 y Fo(<)h(\013)p Fu(\))47 b Fr(\))f Fu(ID)2259 2857 y Fi(1)2259 2914 y Fn(1)2352 2893 y Fb(\015)2412 2863 y Fp(\013)2412 2914 y(\032)2483 2893 y Fu(\000.)0 3034 y(b\))28 b Fo(A)23 b Fr(')279 2972 y Fl(W)348 3059 y Fp(\023)p Fi(2)p Fp(J)464 3034 y Fo(A)526 3046 y Fp(\023)578 3034 y Fr(2)g Fu(\000)g(&)g(ID)912 2998 y Fi(1)912 3055 y Fn(1)1006 3034 y Fb(\015)1066 3004 y Fp(\013)1109 3012 y Fg(0)1066 3055 y Fp(\032)1168 3034 y Fu(\000)p Fo(;)14 b(A)1319 3046 y Fp(\026)1387 3034 y Fu(&)23 b Fo(\013)1528 3046 y Fn(0)1588 3034 y Fo(<)g(\013)g Fu(&)g Fo(\026)g Fr(2)g Fo(J)31 b Fu(&)23 b Fk(k)p Fu(\()p Fo(\026)p Fu(\))g Fr(\022)2422 2972 y Fl(T)2505 3034 y Fr(H)2575 3046 y Fn(\000)2666 3034 y Fr(\))47 b Fu(ID)2889 2998 y Fi(1)2889 3055 y Fn(1)2982 3034 y Fb(\015)3042 3004 y Fp(\013)3042 3055 y(\032)3113 3034 y Fu(\000.)0 3176 y(c\))28 b(ID)190 3139 y Fi(1)190 3196 y Fn(1)284 3176 y Fb(\015)344 3145 y Fp(\013)387 3153 y Fg(0)344 3196 y Fp(\032)446 3176 y Fu(\000)p Fo(;)14 b(A)24 b Fu(&)e(ID)802 3139 y Fi(1)802 3196 y Fn(1)895 3176 y Fb(\015)955 3145 y Fp(\013)998 3153 y Fg(1)955 3196 y Fp(\032)1058 3176 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(A)23 b Fu(&)g Fo(\013)1428 3188 y Fn(0)1465 3176 y Fo(;)14 b(\013)1555 3188 y Fn(1)1615 3176 y Fo(<)23 b(\013)h Fu(&)e Fk(k)p Fu(\()p Fo(A)p Fu(\))i Fr(\022)2145 3113 y Fl(T)2228 3176 y Fr(H)2298 3188 y Fn(\000)2390 3176 y Fr(\))46 b Fu(ID)2612 3139 y Fi(1)2612 3196 y Fn(1)2706 3176 y Fb(\015)2766 3145 y Fp(\013)2766 3196 y(\032)2836 3176 y Fu(\000.)0 3342 y(Pro)r(of:)0 3483 y(W)-7 b(e)28 b(ha)n(v)n(e)e Fk(k)p Fu(\(\000\))19 b Fr([)g(f)p Fo(\013)p Fr(g)j(\022)831 3421 y Fl(T)914 3483 y Fr(H)984 3495 y Fn(\000)1029 3483 y Fu(.)0 3649 y(a\))27 b(Assume)h Fr(8)p Fo(\023)23 b Fr(2)g Fo(J)8 b Fu(\()p Fo(d)715 3661 y Fp(\023)767 3649 y Fb(\015)827 3619 y Fp(\013)870 3627 y Fh(\023)827 3669 y Fp(\032)925 3649 y Fu(\000)p Fo(;)14 b(A)1076 3661 y Fp(\023)1104 3649 y Fu(\),)29 b(and)e(let)h Fo(d)23 b Fu(:=)g Fo(\013)1699 3587 y Fl(V)1768 3674 y Fp(A)1823 3649 y Fu(\()p Fo(d)1898 3661 y Fp(\023)1927 3649 y Fu(\))1959 3661 y Fp(\023)p Fi(2)p Fp(J)2075 3649 y Fu(.)0 3790 y(Then)28 b Fk(k)o Fu(\()289 3728 y Fl(V)359 3815 y Fp(A)413 3790 y Fu(\))23 b(=)g Fk(k)o Fu(\()p Fo(A)p Fu(\))h Fr(\022)f Fk(k)p Fu(\(\000\),)28 b(and)f(it)h(remains)f(to)g(pro)n(v)n(e)f Fo(d)1964 3802 y Fp(\023)2012 3790 y Fo(/)2054 3760 y Fi(\003)2110 3790 y Fr(H)2180 3802 y Fn(\000)2225 3790 y Fu([)p Fo(\023)p Fu(]:)0 3931 y Fo(d)43 3943 y Fp(\023)95 3931 y Fb(\015)155 3901 y Fp(\013)198 3909 y Fh(\023)155 3952 y Fp(\032)252 3931 y Fu(\000)p Fo(;)14 b(A)403 3943 y Fp(\023)478 3931 y Fr(\))46 b Fo(d)650 3943 y Fp(\023)698 3931 y Fo(/)740 3901 y Fi(\003)796 3931 y Fr(H)866 3943 y Fn(\000)p Fp(;A)977 3951 y Fh(\023)1031 3877 y Fn(\()p Fi(\003)p Fn(\))1032 3931 y Fr(\))i Fo(d)1206 3943 y Fp(\023)1253 3931 y Fo(/)1295 3901 y Fi(\003)1351 3931 y Fr(H)1421 3943 y Fn(\000)1467 3931 y Fu([)p Fo(\023)p Fu(].)120 b(\()p Fr(\003)p Fu(\):)37 b Fk(k)p Fu(\(\000)p Fo(;)14 b(A)2075 3943 y Fp(\023)2104 3931 y Fu(\))23 b Fr(\022)g Fk(k)o Fu(\(\000\))c Fr([)g Fk(k)p Fu(\()p Fo(\023)p Fu(\).)0 4097 y(b\))k(Assume)f Fo(d)446 4109 y Fn(0)506 4097 y Fb(\015)566 4067 y Fp(\013)609 4075 y Fg(0)566 4118 y Fp(\032)669 4097 y Fu(\000)p Fo(;)14 b(A)820 4109 y Fp(\026)865 4097 y Fu(,)23 b(and)f(let)g Fo(d)i Fu(:=)e Fo(\013)1411 4035 y Fl(W)1481 4055 y Fp(\026)1481 4122 y(A)1535 4097 y Fo(d)1578 4109 y Fn(0)1615 4097 y Fu(.)36 b(Then)22 b Fk(k)o Fu(\()1957 4035 y Fl(W)2027 4055 y Fp(\026)2027 4122 y(A)2081 4097 y Fu(\))h(=)g Fk(k)o Fu(\()p Fo(\026)p Fu(\))h Fr(\022)2490 4035 y Fl(T)2573 4097 y Fr(H)2643 4109 y Fn(\000)2688 4097 y Fu(,)f(and)f(it)h(remains)e(to)h(pro)n(v)n(e)f Fo(d)3630 4109 y Fn(0)3674 4097 y Fo(/)3716 4067 y Fi(\003)3762 4097 y Fr(H)3832 4109 y Fn(\000)3877 4097 y Fu(:)0 4238 y Fk(k)p Fu(\()p Fo(A)135 4250 y Fp(\026)180 4238 y Fu(\))i Fr(\022)g Fk(k)o Fu(\(\000\))c Fr([)g Fk(k)p Fu(\()p Fo(\026)p Fu(\))k Fr(\022)838 4176 y Fl(T)921 4238 y Fr(H)991 4250 y Fn(\000)1082 4238 y Fr(\))46 b(H)1281 4250 y Fn(\000)1350 4238 y Fu(=)22 b Fr(H)1507 4250 y Fn(\000)p Fp(;A)1618 4258 y Fh(\026)1662 4238 y Fu(.)120 b Fo(d)1848 4250 y Fn(0)1908 4238 y Fb(\015)1968 4208 y Fp(\013)2011 4216 y Fg(0)1968 4259 y Fp(\032)2071 4238 y Fu(\000)p Fo(;)14 b(A)2222 4250 y Fp(\026)2313 4238 y Fr(\))46 b Fo(d)2485 4250 y Fn(0)2541 4238 y Fo(/)2583 4208 y Fi(\003)2639 4238 y Fr(H)2709 4250 y Fn(\000)p Fp(;A)2820 4258 y Fh(\026)2863 4238 y Fu(.)0 4404 y(c\))28 b(Assume)g Fo(d)448 4416 y Fn(0)508 4404 y Fb(\015)568 4374 y Fp(\013)611 4382 y Fg(0)568 4425 y Fp(\032)671 4404 y Fu(\000)p Fo(;)14 b(A)p Fu(,)28 b Fo(d)916 4416 y Fn(1)976 4404 y Fb(\015)1036 4374 y Fp(\013)1079 4382 y Fg(1)1036 4425 y Fp(\032)1139 4404 y Fu(\000)p Fo(;)14 b Fr(:)p Fo(A)p Fu(,)28 b(and)f(let)h Fo(d)c Fu(:=)e Fo(\013)p Fk(Cut)2034 4416 y Fp(A)2088 4404 y Fo(d)2131 4416 y Fn(0)2168 4404 y Fo(d)2211 4416 y Fn(1)2249 4404 y Fu(.)37 b(It)27 b(remains)g(to)h(pro)n(v)n(e)e Fo(d)3077 4416 y Fp(i)3123 4404 y Fo(/)3165 4374 y Fi(\003)3221 4404 y Fr(H)3291 4416 y Fn(\000)3336 4404 y Fu(:)0 4545 y Fk(k)p Fu(\()p Fo(A)p Fu(\))d Fr(\022)278 4483 y Fl(T)361 4545 y Fr(H)431 4557 y Fn(\000)523 4545 y Fr(\))46 b(H)722 4557 y Fn(\000)790 4545 y Fu(=)23 b Fr(H)948 4560 y Fn(\000)p Fp(;)p Fn(\()p Fi(:)p Fn(\))p Fp(A)1159 4545 y Fu(.)120 b Fo(d)1345 4557 y Fp(i)1396 4545 y Fb(\015)1456 4515 y Fp(\013)1499 4523 y Fh(i)1456 4566 y Fp(\032)1553 4545 y Fu(\000)p Fo(;)14 b Fu(\()p Fr(:)p Fu(\))p Fo(A)47 b Fr(\))f Fo(d)2042 4557 y Fp(i)2088 4545 y Fo(/)2130 4515 y Fi(\003)2187 4545 y Fr(H)2257 4560 y Fn(\000)p Fp(;)p Fn(\()p Fi(:)p Fn(\))p Fp(A)2468 4545 y Fu(.)0 4736 y Ft(Lemma)30 b(9.13)0 4878 y Fo(t)30 4890 y Fn(0)67 4878 y Fo(;)14 b(t)134 4890 y Fn(1)194 4878 y Fr(2)24 b Fo(T)34 b Fu(&)23 b Fk(val)p Fu(\()p Fo(t)604 4890 y Fn(0)641 4878 y Fu(\))g(=)g Fk(val)p Fu(\()p Fo(t)944 4890 y Fn(1)982 4878 y Fu(\))g(&)g(FV)q(\()p Fo(A)p Fu(\))h Fr(\022)f(f)p Fo(x)p Fr(g)45 b(\))h Fu(ID)1879 4841 y Fi(1)1879 4898 y Fn(1)1972 4878 y Fb(\015)2032 4834 y Fn(2)p Fi(\001)p Fn(rk\()p Fp(A)p Fn(\))2032 4900 y(0)2275 4878 y Fr(:)p Fo(A)p Fu(\()p Fo(x=t)2543 4890 y Fn(0)2581 4878 y Fu(\))p Fo(;)14 b(A)p Fu(\()p Fo(x=t)2863 4890 y Fn(1)2901 4878 y Fu(\).)1908 5539 y(59)p eop %%Page: 60 61 60 60 bop 0 83 a Fu(Pro)r(of)26 b(b)n(y)i(induction)g(on)f(rk)o(\()p Fo(A)p Fu(\):)0 224 y(1.)36 b Fo(A)24 b Fu(=)e(\()p Fr(:)p Fu(\))p Fo(X)7 b(t)28 b Fu(or)f(\()p Fr(:)p Fu(\)\()p Fo(r)p 828 198 50 8 v 828 216 V 70 w(s)p Fu(\):)38 b(trivial.)0 365 y(2.)e Fo(A)24 b Fr(')274 303 y Fl(W)344 390 y Fp(\023)p Fi(2)p Fp(J)459 365 y Fo(A)521 377 y Fp(\023)550 365 y Fu(:)37 b(Then)28 b Fo(A)p Fu(\()p Fo(x=t)1040 377 y Fp(i)1068 365 y Fu(\))c Fr(')1211 303 y Fl(W)1280 390 y Fp(\023)p Fi(2)p Fp(J)1396 365 y Fo(A)1458 377 y Fp(\023)1487 365 y Fu(\()p Fo(x=t)1638 377 y Fp(i)1666 365 y Fu(\).)0 506 y(IH)56 b Fr(\))46 b Fu(ID)370 470 y Fi(1)370 527 y Fn(1)464 506 y Fb(\015)524 463 y Fn(2)p Fi(\001)p Fn(rk)o(\()p Fp(A)713 471 y Fh(\023)740 463 y Fn(\))524 529 y(0)793 506 y Fr(:)p Fo(A)910 518 y Fp(\023)939 506 y Fu(\()p Fo(t)1001 518 y Fn(0)1039 506 y Fu(\))p Fo(;)14 b(A)1170 518 y Fp(\023)1199 506 y Fu(\()p Fo(t)1261 518 y Fn(1)1299 506 y Fu(\))83 b(\()p Fr(8)p Fo(\023)23 b Fr(2)g Fo(J)8 b Fu(\))148 593 y Fn(9)p Fp(:)p Fn(12)p Fp(;)p Fn(\()p Fi(\003)347 601 y Fg(1)379 593 y Fn(\))235 648 y Fr(\))133 b Fu(ID)544 611 y Fi(1)544 668 y Fn(1)638 648 y Fb(\015)698 604 y Fn(2)p Fi(\001)p Fn(rk)o(\()p Fp(A)887 612 y Fh(\023)914 604 y Fn(\)+1)698 670 y(0)1051 648 y Fr(:)p Fo(A)1168 660 y Fp(\023)1198 648 y Fu(\()p Fo(t)1260 660 y Fn(0)1297 648 y Fu(\))p Fo(;)14 b(A)p Fu(\()p Fo(t)1490 660 y Fn(1)1528 648 y Fu(\))83 b(\()p Fr(8)p Fo(\023)23 b Fr(2)h Fo(J)8 b Fu(\))148 734 y Fn(9)p Fp(:)p Fn(12)p Fp(;)p Fn(\()p Fi(\003)347 742 y Fg(2)379 734 y Fn(\))235 789 y Fr(\))133 b Fu(ID)544 753 y Fi(1)544 809 y Fn(1)638 789 y Fb(\015)698 746 y Fn(2)p Fi(\001)p Fn(rk)o(\()p Fp(A)p Fn(\))698 811 y(0)940 789 y Fr(:)p Fo(A)p Fu(\()p Fo(t)1119 801 y Fn(0)1157 789 y Fu(\))p Fo(;)14 b(A)p Fu(\()p Fo(t)1350 801 y Fn(1)1388 789 y Fu(\))0 955 y(\()p Fr(\003)74 967 y Fn(1)111 955 y Fu(\))28 b Fk(k)p Fu(\()p Fo(\023)p Fu(\))19 b Fr([)g(f)p Fu(2)p Fr(\001)o Fu(rk)o(\()p Fo(A)674 967 y Fp(\023)703 955 y Fu(\)+1)p Fr(g)j(\022)994 892 y Fl(T)1077 955 y Fr(H)1147 970 y Fi(:)p Fp(A)1242 978 y Fh(\023)1269 970 y Fn(\()p Fp(t)1320 978 y Fg(0)1352 970 y Fn(\))p Fp(;A)p Fn(\()p Fp(t)1499 978 y Fg(1)1531 970 y Fn(\))1589 955 y Fu(,)55 b(\()p Fr(\003)1741 967 y Fn(2)1779 955 y Fu(\))27 b(2)p Fr(\001)p Fu(rk)o(\()p Fo(A)p Fu(\))d Fr(2)2208 892 y Fl(T)2291 955 y Fr(H)2361 970 y Fi(:)p Fp(A)p Fn(\()p Fp(t)2507 978 y Fg(0)2539 970 y Fn(\))p Fp(;A)p Fn(\()p Fp(t)2686 978 y Fg(1)2718 970 y Fn(\))2748 955 y Fu(.)0 1146 y Ft(Lemma)30 b(9.14)0 1287 y Fu(PL1)22 b Fr(`)224 1257 y Fp(k)224 1307 y Fn(0)301 1287 y Fu(\000)h(&)g(FV)r(\(\000\))g(=)g Fr(;)45 b Fu(=)-14 b Fr(\))46 b Fu(ID)1170 1251 y Fi(1)1170 1307 y Fn(1)1263 1287 y Fb(\015)1323 1251 y Fn(\012+)p Fp(k)1323 1309 y Fn(0)1499 1287 y Fu(\000)0 1453 y(Pro)r(of:)0 1594 y(1.1.)36 b Fr(f:)p Fo(A;)14 b(A)p Fr(g)23 b(\022)g Fu(\000)k(with)h Fo(A)c Fu(=)e Fo(X)7 b(t)27 b Fu(or)g Fo(A)c Fu(=)g(\()p Fo(s)p 1505 1568 V 1505 1586 V 66 w(t)p Fu(\):)38 b(Then)27 b(\000)c Fr(2)h(AX)12 b Fu(\()p Fe(Z)2274 1557 y Fi(1)2344 1594 y Fu(\).)0 1735 y(1.2.)36 b Fr(f:)p Fo(P)12 b(t;)i(P)e(t)p Fr(g)22 b(\022)h Fu(\000:)36 b(The)28 b(claim)g(follo)n(ws)e(from)h (9.13.)0 1876 y(2.)36 b Fr(8)p Fo(xA)23 b Fr(2)h Fu(\000,)j Fo(k)f(>)d Fu(0)k(and)h(PL1)22 b Fr(`)1073 1841 y Fp(k)q Fi(\000)p Fn(1)1073 1898 y(0)1221 1876 y Fu(\000)p Fo(;)14 b(A)1372 1888 y Fp(x)1414 1876 y Fu(\()p Fo(y)s Fu(\):)0 2017 y(Then)28 b(PL1)22 b Fr(`)441 1982 y Fp(k)q Fi(\000)p Fn(1)441 2040 y(0)589 2017 y Fu(\000)p Fo(;)14 b(A)740 2029 y Fp(x)782 2017 y Fu(\()p Fo(t)p Fu(\))28 b(for)g(all)f Fo(t)c Fr(2)g Fo(T)12 b Fu(,)27 b(and)g(b)n(y)h(IH)55 b(ID)1907 1981 y Fi(1)1907 2038 y Fn(1)2000 2017 y Fb(\015)2060 1982 y Fn(\012+)p Fp(k)q Fi(\000)p Fn(1)2060 2040 y(0)2321 2017 y Fu(\000)p Fo(;)14 b(A)2472 2029 y Fp(x)2514 2017 y Fu(\()p Fo(t)p Fu(\))28 b(for)f(all)h Fo(t)23 b Fr(2)g Fo(T)12 b Fu(.)0 2159 y(By)27 b(9.12)g(this)g(implies)h(ID)843 2122 y Fi(1)843 2179 y Fn(1)936 2159 y Fb(\015)996 2123 y Fn(\012+)p Fp(k)996 2181 y Fn(0)1172 2159 y Fu(\000,)g(since)f Fr(8)p Fo(xA)c Fr(')1745 2096 y Fl(V)1814 2183 y Fp(t)p Fi(2)p Fp(T)1936 2159 y Fo(A)1998 2171 y Fp(x)2040 2159 y Fu(\()p Fo(t)p Fu(\).)0 2300 y(3.)36 b Fr(9)p Fo(xA)24 b Fr(2)g Fu(\000,)j Fo(k)f(>)d Fu(0)k(and)h(PL1)22 b Fr(`)1073 2264 y Fp(k)q Fi(\000)p Fn(1)1073 2322 y(0)1221 2300 y Fu(\000)p Fo(;)14 b(A)1372 2312 y Fp(x)1414 2300 y Fu(\()p Fo(t)p Fu(\):)38 b(w.l.o.g.)e Fo(t)23 b Fr(2)g Fo(T)12 b Fu(.)0 2441 y(IH)p Fr(\))46 b Fu(ID)315 2405 y Fi(1)315 2461 y Fn(1)408 2441 y Fb(\015)468 2405 y Fn(\012+)p Fp(k)q Fi(\000)p Fn(1)468 2463 y(0)729 2441 y Fu(\000)p Fo(;)14 b(A)880 2453 y Fp(x)922 2441 y Fu(\()p Fo(t)p Fu(\))1063 2387 y Fn(9)p Fp(:)p Fn(12)p Fp(;)d Fd(k)p Fn(\(t\)=)p Fi(;)1170 2441 y Fu(=)-14 b Fr(\))152 b Fu(ID)1550 2405 y Fi(1)1550 2461 y Fn(1)1643 2441 y Fb(\015)1703 2405 y Fn(\012+)p Fp(k)1703 2463 y Fn(0)1879 2441 y Fu(\000.)0 2657 y Ft(Lemma)30 b(9.15)0 2798 y Fu(If)e(FV)r(\()p Fr(8)p Fo(xF)12 b Fu(\))23 b(=)f Fr(;)p Fu(,)28 b Fo(t)23 b Fr(2)g Fo(T)12 b Fu(,)27 b(and)g Fo(\016)f Fu(=)d(2)p Fr(\001)p Fu(rk)o(\()p Fo(F)12 b Fu(\))28 b(then)0 2939 y(ID)93 2903 y Fi(1)93 2960 y Fn(1)187 2939 y Fb(\015)247 2896 y Fp(\016)r Fn(+2)p Fd(val)o Fn(\()p Fp(t)p Fn(\))247 2961 y(0)551 2939 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i Fr(:8)p Fo(x)p Fu(\()p Fo(F)e Fu(\()p Fo(x)p Fu(\))24 b Fr(!)f Fo(F)12 b Fu(\()p Fk(S)p Fo(x)p Fu(\)\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\).)0 3105 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f Fo(n)p Fu(:)0 3246 y(1.)36 b Fk(val)p Fu(\()p Fo(t)p Fu(\))24 b(=)f(0:)36 b(By)28 b(9.13)e(ID)905 3210 y Fi(1)905 3267 y Fn(1)999 3246 y Fb(\015)1059 3216 y Fp(\016)1059 3267 y Fn(0)1142 3246 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\).)0 3387 y(2.)36 b Fk(val)p Fu(\()p Fo(t)p Fu(\))24 b(=)f Fo(n)p Fu(+)o(1:)37 b(Let)27 b Fo(G)d Fu(:=)e Fr(:8)p Fo(x)p Fu(\()p Fo(F)12 b Fu(\()p Fo(x)p Fu(\))25 b Fr(!)e Fo(F)12 b Fu(\()p Fk(S)p Fo(x)p Fu(\)\).)10 3502 y(IH)28 b Fr(\))23 b Fb(\015)296 3467 y Fp(\016)r Fn(+2)p Fp(n)296 3524 y Fn(0)481 3502 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\()p Fo(n)p 943 3515 50 4 v Fu(\))10 3643 y(9.13)26 b Fr(\))d Fb(\015)351 3613 y Fp(\016)351 3664 y Fn(0)435 3643 y Fr(:)p Fo(F)12 b Fu(\()p Fk(S)p Fo(n)p 633 3656 V Fu(\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))1025 3420 y Fl(\))1115 3562 y Fr(\))23 b Fb(\015)1281 3526 y Fp(\016)r Fn(+2)p Fp(n)p Fn(+1)1281 3584 y(0)1550 3562 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\()p Fo(n)p 2012 3575 V Fu(\))19 b Fr(^)g(:)p Fo(F)12 b Fu(\()p Fk(S)q Fo(n)p 2386 3575 V -1 w Fu(\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))2710 3505 y Fd(k)q Fn(\(n)p 2771 3518 37 4 v -1 w(\)=)p Fi(;)2747 3562 y Fu(=)-14 b Fr(\))60 b Fb(\015)3001 3519 y Fp(\016)r Fn(+2\()p Fp(n)p Fn(+1\))3001 3584 y(0)3322 3562 y Fr(:)p Fo(F)12 b Fu(\(0\))p Fo(;)i(G;)g(F)e Fu(\()p Fo(t)p Fu(\).)0 3855 y Ft(Lemma)30 b(9.16)0 3996 y Fu(If)e Fo(B)t Fu(\()p Fo(X)7 b Fu(\))28 b(is)f(an)g Fo(X)7 b Fu(-p)r(ositiv)n(e)27 b Fr(L)984 4008 y Fn(1)1021 3996 y Fu(-form)n(ula)f(then)j(for)e(all)g Fo(\013)c Fr(\024)g Fu(\012:)0 4162 y(ID)93 4126 y Fi(1)93 4183 y Fn(1)187 4162 y Fb(\015)247 4127 y Fp(\016)r Fn(+)p Fp(\013)373 4102 y Ff(0)247 4185 y Fn(0)423 4162 y Fr(:8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)r(;)14 b(x)p Fu(\))25 b Fr(!)e Fo(F)12 b Fu(\()p Fo(x)p Fu(\)\))p Fo(;)29 b Fr(:)p Fo(B)t Fu(\()p Fo(P)1483 4132 y Fp(<\013)1583 4162 y Fu(\))p Fo(;)14 b(B)t Fu(\()p Fo(F)e Fu(\))28 b(,)56 b(where)27 b Fo(\016)f Fu(:=)d(2)p Fr(\001)o Fu(rk\()p Fo(F)12 b Fu(\))28 b(and)f Fo(\013)2881 4132 y Fi(0)2928 4162 y Fu(:=)c(2)p Fr(\001)o Fu(rk\()p Fo(B)t Fu(\()p Fo(P)3376 4132 y Fp(<\013)3475 4162 y Fu(\)\))c(+)f(2.)0 4328 y(Pro)r(of)26 b(b)n(y)i(induction)g(on)f(rk)o(\()p Fo(B)t Fu(\()p Fo(P)1099 4298 y Fp(<\013)1199 4328 y Fu(\)\):)38 b(Let)27 b Fo(G)d Fu(:=)e Fr(:8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)r(;)14 b(x)p Fu(\))26 b Fr(!)d Fo(F)12 b Fu(\()p Fo(x)p Fu(\)\))24 b([)14 b(=)22 b Fr(9)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)r(;)14 b(x)p Fu(\))22 b Fr(^)c(:)p Fo(F)12 b Fu(\()p Fo(x)p Fu(\)\))i(].)0 4470 y(1.)36 b Fo(B)t Fu(\()p Fo(X)7 b Fu(\))24 b Fr(\021)e Fo(X)7 b(t)p Fu(:)36 b(Then)28 b Fo(\013)854 4439 y Fi(0)901 4470 y Fu(=)23 b(2)p Fo(!)s(\013)18 b Fu(+)g(2)k(=)h Fo(!)s(\013)18 b Fu(+)g(2.)0 4611 y(IH)28 b Fr(\))23 b Fb(\015)286 4571 y Fp(\016)r Fn(+)p Fp(!)r(\030)r Fn(+)p Fp(m)286 4633 y Fn(0)582 4611 y Fo(G;)14 b Fr(:A)p Fu(\()p Fo(P)902 4581 y Fp(<\030)992 4611 y Fo(;)g(t)p Fu(\))p Fo(;)g Fr(A)p Fu(\()p Fo(F)r(;)g(t)p Fu(\))56 b(for)27 b(all)h Fo(\030)f(<)c(\013)0 4752 y Fr(\))g Fb(\015)166 4716 y Fp(\016)r Fn(+)p Fp(!)r(\013)166 4774 y Fn(0)364 4752 y Fo(G;)14 b Fr(:)p Fo(P)586 4722 y Fp(<\013)685 4752 y Fo(t;)g Fr(A)p Fu(\()p Fo(F)r(;)g(t)p Fu(\))0 4893 y Fr(\))23 b Fb(\015)166 4857 y Fp(\016)r Fn(+)p Fp(!)r(\013)p Fn(+1)166 4915 y(0)448 4893 y Fo(G;)14 b Fr(:)p Fo(P)670 4863 y Fp(<\013)769 4893 y Fo(t;)g Fr(A)p Fu(\()p Fo(F)r(;)g(t)p Fu(\))20 b Fr(^)f(:)p Fo(F)12 b Fu(\()p Fo(t)p Fu(\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))84 b([)28 b(since)f Fb(\015)1990 4863 y Fp(\016)1990 4914 y Fn(0)2050 4893 y Fr(:)p Fo(F)12 b Fu(\()p Fo(t)p Fu(\))p Fo(;)i(F)e Fu(\()p Fo(t)p Fu(\))29 b(])0 5034 y Fr(\))23 b Fb(\015)166 4999 y Fp(\016)r Fn(+)p Fp(!)r(\013)p Fn(+2)166 5056 y(0)448 5034 y Fo(G;)14 b Fr(:)p Fo(P)670 5004 y Fp(<\013)769 5034 y Fo(t;)g(F)e Fu(\()p Fo(t)p Fu(\).)0 5200 y(2.)36 b Fo(B)c Fr(L)253 5212 y Fn(0)291 5200 y Fu(-literal:)k(9.13.)1908 5539 y(60)p eop %%Page: 61 62 61 61 bop 0 83 a Fu(3.)36 b Fo(B)28 b Fr(\021)22 b(8)p Fo(y)s(B)433 95 y Fn(0)469 83 y Fu(\()p Fo(X)r(;)14 b(y)s Fu(\):)37 b(Then)28 b(rk)o(\()p Fo(B)t Fu(\()p Fo(P)1234 53 y Fp(<\013)1334 83 y Fu(\)\))c(=)e Fo(\014)1556 95 y Fn(0)1612 83 y Fu(+)c(1)27 b(mit)h Fo(\014)1963 95 y Fn(0)2024 83 y Fu(:=)22 b(rk\()p Fo(B)2306 95 y Fn(0)2343 83 y Fu(\()p Fo(P)2440 53 y Fp(<\013)2539 83 y Fo(;)14 b(y)s Fu(\)\).)0 224 y(IH)28 b Fr(\))23 b Fb(\015)286 184 y Fp(\016)r Fn(+2)p Fp(\014)440 192 y Fg(0)472 184 y Fn(+2)286 246 y(0)584 224 y Fo(G;)14 b Fr(:)p Fo(B)804 236 y Fn(0)841 224 y Fu(\()p Fo(P)938 194 y Fp(<\013)1038 224 y Fo(;)g(t)p Fu(\))p Fo(;)g(B)1237 236 y Fn(0)1274 224 y Fu(\()p Fo(F)r(;)g(t)p Fu(\))56 b(for)28 b(all)f Fo(t)c Fr(2)g Fo(T)0 365 y Fu(=)-14 b Fr(\))23 b Fb(\015)217 325 y Fp(\016)r Fn(+2)p Fp(\014)371 333 y Fg(0)403 325 y Fn(+3)217 387 y(0)514 365 y Fo(G;)14 b Fr(9)p Fo(y)s Fr(:)p Fo(B)824 377 y Fn(0)862 365 y Fu(\()p Fo(P)959 335 y Fp(<\013)1058 365 y Fo(;)g(y)s Fu(\))p Fo(;)g(B)1271 377 y Fn(0)1308 365 y Fu(\()p Fo(F)r(;)g(t)p Fu(\))57 b(for)27 b(all)g Fo(t)c Fr(2)h Fo(T)0 506 y Fu(=)-14 b Fr(\))23 b Fb(\015)217 463 y Fp(\016)r Fn(+2\()p Fp(\014)397 471 y Fg(0)429 463 y Fn(+1\)+2)217 529 y(0)650 506 y Fo(G;)14 b Fr(:8)p Fo(y)s(B)961 518 y Fn(0)998 506 y Fu(\()p Fo(P)1095 476 y Fp(<\013)1194 506 y Fo(;)g(y)s Fu(\))p Fo(;)g Fr(8)p Fo(y)s(B)1498 518 y Fn(0)1534 506 y Fu(\()p Fo(F)r(;)g(y)s Fu(\).)0 722 y Ft(Lemma)30 b(9.17)0 863 y Fu(a\))d(ID)195 827 y Fi(1)195 884 y Fn(1)288 863 y Fb(\015)348 828 y Fn(\012+)p Fp(!)348 886 y Fn(0)518 863 y Fr(8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(P)795 875 y Fi(A)853 863 y Fo(;)14 b(x)p Fu(\))24 b Fr(!)f Fo(P)1152 875 y Fi(A)1209 863 y Fo(x)p Fu(\),)0 1005 y(b\))28 b(ID)199 968 y Fi(1)199 1025 y Fn(1)293 1005 y Fb(\015)353 969 y Fn(\012)p Fi(\001)p Fn(2+)p Fp(!)353 1027 y Fn(0)575 1005 y Fr(8)o Fo(~)-41 b(y)r Fu([)p Fr(8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)12 b Fu(\()p Fr(\001)p Fo(;)h(~)-41 b(y)s Fu(\))p Fo(;)14 b(x)p Fu(\))24 b Fr(!)f Fo(F)12 b Fu(\()p Fo(x;)h(~)-41 b(y)t Fu(\)\))23 b Fr(!)g(8)p Fo(x)p Fu(\()p Fo(P)1989 1017 y Fi(A)2047 1005 y Fo(x)g Fr(!)h Fo(F)12 b Fu(\()p Fo(x;)h(~)-41 b(y)s Fu(\)\)].)0 1171 y(Pro)r(of:)0 1312 y(a\))25 b(9.13)f Fr(\))f Fb(\015)438 1275 y Fn(\012+)p Fp(k)571 1283 y Fg(0)438 1334 y Fn(0)631 1312 y Fr(:A)p Fu(\()p Fo(P)r(;)14 b(t)p Fu(\))p Fo(;)g Fr(A)p Fu(\()p Fo(P)r(;)g(t)p Fu(\))86 b([with)28 b(\012)18 b(+)g Fo(k)1729 1324 y Fn(0)1790 1312 y Fu(=)23 b(2rk)n(\()p Fr(A)p Fu(\()p Fo(P)r(;)14 b(x)p Fu(\)\)])2400 1258 y Fn(\()p Fd(Cl)2488 1266 y Fh(P)7 b(t)2559 1258 y Fn(\))2426 1312 y Fu(=)-14 b Fr(\))48 b Fb(\015)2668 1275 y Fn(\012+)p Fp(k)2801 1283 y Fg(0)2834 1275 y Fn(+1)2668 1334 y(0)2945 1312 y Fr(:A)p Fu(\()p Fo(P)r(;)14 b(t)p Fu(\))p Fo(;)g(P)e(t)85 b Fu(\()p Fr(8)p Fo(t)22 b Fr(2)i Fo(T)12 b Fu(\))45 b Fr(\))0 1453 y Fb(\015)60 1416 y Fn(\012+)p Fp(k)193 1424 y Fg(0)226 1416 y Fn(+4)60 1475 y(0)337 1453 y Fr(8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(P)r(;)14 b(x)p Fu(\))25 b Fr(!)e Fo(P)12 b(x)p Fu(\).)0 1594 y(b\))36 b(9.16)d Fr(\))j Fb(\015)475 1559 y Fp(\016)r Fn(+\012+2)475 1616 y(0)729 1594 y Fr(:8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)12 b Fu(\()p Fr(\001)p Fo(;)f(~)-39 b(s)q Fu(\))p Fo(;)14 b(x)p Fu(\))37 b Fr(!)e Fo(F)12 b Fu(\()p Fo(x;)f(~)-39 b(s)q Fu(\)\))p Fo(;)14 b Fr(:)p Fo(P)e(t;)i(F)e Fu(\()p Fo(t;)f(~)-39 b(s)p Fu(\)\))71 b(for)35 b(all)g Fo(t;)11 b(~)-39 b(s)35 b Fr(2)h Fo(T)81 b Fu([)36 b Fo(\016)i Fu(:=)e(2rk)n(\()p Fo(F)12 b Fu(\))36 b Fo(<)g Fu(\012)23 b(+)g Fo(!)38 b Fu(])0 1735 y Fr(\))23 b Fb(\015)166 1700 y Fn(\012)p Fi(\001)p Fn(2+5)166 1757 y(0)378 1735 y Fr(:8)p Fo(x)p Fu(\()p Fr(A)p Fu(\()p Fo(F)12 b Fu(\()p Fr(\001)p Fo(;)f(~)-39 b(s)p Fu(\))p Fo(;)14 b(x)p Fu(\))24 b Fr(!)f Fo(F)12 b Fu(\()p Fo(x;)f(~)-39 b(s)q Fu(\)\))p Fo(;)14 b Fr(8)p Fo(x)p Fu(\()p Fo(P)e(x)24 b Fr(!)f Fo(F)12 b Fu(\()p Fo(x;)f(~)-39 b(s)p Fu(\)\))28 b(for)f(all)e Fo(~)-39 b(s)23 b Fr(2)g Fo(T)12 b Fu(.)0 1951 y Ft(Lemma)30 b(9.18)54 b Fu(If)28 b Fo(A)g Fu(is)f(an)g(axiom)g(of)h(ID)1370 1963 y Fn(1)1435 1951 y Fu(then)56 b(ID)1745 1915 y Fi(1)1745 1972 y Fn(1)1839 1951 y Fb(\015)1899 1916 y Fn(\012)p Fi(\001)p Fn(2+)p Fp(!)1899 1973 y Fn(0)2135 1951 y Fo(A)p Fu(.)0 2117 y(Pro)r(of)26 b(b)n(y)i(means)f(of)g(9.13,)g(9.15,)f(9.17.) 0 2333 y Ft(Theorem)k(9.19)55 b Fu(ID)724 2345 y Fn(1)784 2333 y Fr(`)23 b Fo(P)911 2345 y Fi(A)969 2333 y Fo(n)p 969 2346 50 4 v 46 w Fr(\))46 b(j)p Fo(n)p Fr(j)1290 2345 y Fi(A)1370 2333 y Fo(<)23 b( )s Fu(\()p Fo(")1586 2345 y Fn(\012+1)1722 2333 y Fu(\).)0 2499 y(Pro)r(of:)0 2640 y(Assume)28 b(ID)401 2652 y Fn(1)461 2640 y Fr(`)23 b Fo(P)12 b(n)p 600 2653 V Fu(.)37 b(Then)28 b(there)f(are)g(ID)1371 2652 y Fn(1)1408 2640 y Fu(-axioms)f Fo(A)1777 2652 y Fn(1)1815 2640 y Fo(;)14 b(:::;)g(A)2020 2652 y Fp(l)2073 2640 y Fu(suc)n(h)28 b(that)f(PL1)22 b Fr(`)2664 2652 y Fn(0)2724 2640 y Fr(:)p Fo(A)2841 2652 y Fn(1)2879 2640 y Fo(;)14 b(:::;)g Fr(:)p Fo(A)3139 2652 y Fp(l)3165 2640 y Fo(;)g(P)e(n)p 3267 2653 V Fu(.)0 2781 y Fb(\015)60 2751 y Fn(\012)p Fi(\001)p Fn(2)60 2802 y(0)187 2781 y Fr(:)p Fo(A)304 2793 y Fn(1)342 2781 y Fo(;)i(:::;)g Fr(:)p Fo(A)602 2793 y Fp(l)628 2781 y Fo(;)g(P)e(n)p 730 2794 V 55 w Fu([)28 b(b)n(y)f(9.14)g(])55 b(and)g Fb(\015)1504 2746 y Fn(\012)p Fi(\001)p Fn(2+)p Fp(!)1504 2803 y Fn(0)1726 2781 y Fo(A)1788 2793 y Fp(i)1899 2781 y Fu(\()p Fo(i)23 b Fu(=)g(1)p Fo(;)14 b(:::;)g(l)r Fu(\))27 b([)g(b)n(y)h(9.18])0 2875 y Fn(9)p Fp(:)p Fn(12c)7 2922 y Fu(=)-14 b Fr(\))31 b Fb(\015)232 2892 y Fn(\012)p Fi(\001)p Fn(3)232 2945 y(\012+)p Fp(m)p Fn(+1)514 2922 y Fo(P)12 b(n)p 579 2935 V 83 w Fu(with)28 b(\012+)o Fo(m)23 b Fu(:=)g(max)o Fr(f)p Fu(\012)p Fo(;)14 b Fu(rk)o(\()p Fo(A)1695 2934 y Fn(1)1733 2922 y Fu(\))p Fo(;)g(:::;)g Fu(rk\()p Fo(A)2079 2934 y Fp(l)2105 2922 y Fu(\))p Fr(g)82 b Fu([note)28 b(that)g Fk(k)o Fu(\()p Fo(A)2783 2934 y Fp(i)2812 2922 y Fu(\))23 b Fr(\022)g(f)p Fu(0)p Fo(;)14 b Fu(\012)p Fr(g)p Fu(])24 3016 y Fn(9)p Fp(:)p Fn(9)0 3064 y Fu(=)-14 b Fr(\))23 b Fb(\015)217 3033 y Fp(\013)217 3086 y Fn(\012+1)375 3064 y Fo(P)12 b(n)p 440 3077 V 83 w Fu(with)28 b Fo(\013)c Fr(2)f Fo(C)6 b Fu(\()p Fo(")1053 3076 y Fn(\012+1)1189 3064 y Fu(\))19 b Fr(\\)g Fo(")1353 3076 y Fn(\012+1)7 3158 y(9)p Fp(:)p Fn(11)0 3205 y Fu(=)-14 b Fr(\))23 b(j)-14 b Fu(=)23 b Fo(P)319 3175 y Fp(<\014)415 3205 y Fo(n)p 415 3218 V 83 w Fu(with)28 b Fo(\014)g Fu(:=)22 b Fo( )s Fu(\()p Fo(!)g Fu(+)c Fo(!)1223 3175 y Fn(\012+)p Fp(\013)1368 3205 y Fu(\))23 b Fo(<)g( )s Fu(\()p Fo(")1639 3217 y Fn(\012+1)1775 3205 y Fu(\).)1908 5539 y(61)p eop %%Page: 62 63 62 62 bop 0 83 a Ft(APPENDIX)33 b(to)e Fr(x)p Ft(4)0 339 y(Remark:)k Fu(Theorem)27 b(4.16)g(follo)n(ws)f(from)h(4.19)g(b\).) 0 502 y(Zu)h(zeigen:)36 b Fr(Z)30 b(`)23 b(8)p Fo(h)p Fr(8)p Fu(\()p Fo(n)794 514 y Fp(i)819 502 y Fu(\))851 514 y Fp(i)g Fu(\012)28 b(is)f(closed)g (under)g(+)14 b Fo(;)28 b(\025x:!)1396 991 y Fn(\012+)p Fp(x)1536 1021 y Fu(,)g(and)f([)h(b)n(y)g(d\))g(])f(also)g(under)g Fo( )s Fu(.)37 b(This)28 b(yields)f(the)h(claim.)1908 5539 y(64)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF