%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: leeds90.dvi %%Pages: 33 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips leeds90.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1995.04.25:1051 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 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Fo(!)26 b Fn(=)c(\()p Fo(!)g Fn(+)c Fo(n)p Fn(\))1974 4404 y Fm(n)p Fi(2)p Ff(N)2122 4392 y Fo(;)1189 4575 y(!)1244 4540 y Fl(2)1303 4575 y Fn(=)23 b Fo(!)e Fp(\001)e Fo(!)25 b Fn(=)e(\()p Fo(!)e Fp(\001)e Fo(n)p Fn(\))1900 4587 y Fm(n)p Fi(2)p Ff(N)2048 4575 y Fo(;)1367 4757 y(!)1422 4723 y Fm(!)1493 4757 y Fn(=)j(\()p Fo(!)1667 4723 y Fm(n)1712 4757 y Fn(\))1744 4769 y Fm(n)p Fi(2)p Ff(N)0 4940 y Fq(and)33 b(similarly)-8 b(,)29 b(for)j Fn(\012)23 b(:=)f(\012)1010 4952 y Fl(1)1047 4940 y Fq(,)1106 5164 y Fn(\012)c Fp(\001)h Fn(2)j(=)h(\012)18 b(+)g(\012)23 b(=)g(\(\012)c(+)f Fo(\020)6 b Fn(\))1978 5176 y Fm(\020)t Fi(2T)2095 5184 y Fd(1)2131 5164 y Fo(;)p eop %%Page: 1 4 1 3 bop 1179 125 a Fn(\012)1239 90 y Fl(2)1299 125 y Fn(=)23 b(\012)c Fp(\001)f Fn(\012)23 b(=)g(\(\012)18 b Fp(\001)h Fo(\020)6 b Fn(\))1904 137 y Fm(\020)t Fi(2T)2021 145 y Fd(1)2058 125 y Fo(;)1349 307 y Fn(\012)1409 273 y Fl(\012)1484 307 y Fn(=)23 b(\(\012)1664 273 y Fm(\020)1702 307 y Fn(\))1734 319 y Fm(\020)t 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b(al\()p Fo(D)205 5076 y Fm(\033)250 5064 y Fo(a)p Fn(\))23 b(=)f Fp(D)500 5076 y Fm(\033)545 5064 y Fn(\(v)-5 b(al\()p Fo(a)p Fn(\)\))p Fq(,)p eop %%Page: 1 5 1 4 bop -157 125 a Fq(iii.)46 b Fn(v)-5 b(al\()p Fo(a)180 137 y Fl(1)217 125 y Fo(;)14 b(:)g(:)g(:)g(;)g(a)446 137 y Fm(k)486 125 y Fn(\))24 b(=)e(v)-5 b(al\()p Fo(a)809 137 y Fl(1)847 125 y Fn(\))18 b(+)g Fp(\001)c(\001)g(\001)19 b Fn(+)f(v)-5 b(al)o(\()p Fo(a)1358 137 y Fm(k)1399 125 y Fn(\))p Fq(.)0 299 y(F)d(or)33 b Fo(a;)14 b(b)23 b Fp(2)i Fn(T)33 b Fq(w)m(e)i(de\014ne)f Fo(a)19 b Fn(+)g Fo(b)33 b Fq(to)g(b)s(e)g(the)h(concatenation)f(of)g(the)h(lists)e Fo(a)i Fq(and)f Fo(b)p Fq(.)46 b(Then)0 423 y(clearly)28 b Fo(a)13 b Fn(+)g Fo(b)22 b Fp(2)h Fn(T)p Fq(,)30 b(and)g(also)e Fo(a)13 b Fn(+)g(0)21 b(=)i(0)13 b(+)g Fo(a)21 b Fn(=)i Fo(a)p Fq(,)30 b Fo(a)13 b Fn(+)g(\()p Fo(b)g Fn(+)g Fo(c)p Fn(\))21 b(=)i(\()p Fo(a)13 b Fn(+)g Fo(b)p Fn(\))g(+)g Fo(c)p Fq(.)40 b(W)-8 b(e)30 b(abbreviate)0 548 y Fo(a)18 b Fn(+)g 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y(iii.)46 b(If)33 b Fo(a)142 1531 y Fl(1)179 1519 y Fo(;)14 b(:)g(:)g(:)f(;)h(a)407 1531 y Fm(k)471 1519 y Fp(2)23 b Fn(HT)672 1531 y Fm(\033)749 1519 y Fq(with)32 b Fo(k)26 b Fp(\025)d Fn(0)p Fq(,)32 b(then)h Fn(\()p Fo(a)1527 1531 y Fl(1)1564 1519 y Fo(;)14 b(:)g(:)g(:)g(;)g(a)1793 1531 y Fm(k)1833 1519 y Fn(\))24 b Fp(2)f Fn(T)2027 1531 y Fm(\033)2072 1519 y Fq(.)0 1694 y(Clearly)32 b Fn(v)-5 b(al)o(\()p Fo(a)p Fn(\))24 b Fp(2)f(T)697 1706 y Fm(\033)775 1694 y Fq(for)32 b Fo(a)23 b Fp(2)g Fn(T)1129 1706 y Fm(\033)1174 1694 y Fq(.)0 1943 y Fn(T)60 1955 y Fl(0)135 1943 y Fq(consists)38 b(of)f Fn(0)p Fo(;)14 b Fn(1)p Fo(;)g Fn(1)19 b(+)i(1)p Fo(;)14 b Fn(1)20 b(+)h(1)g(+)g(1)p Fo(;)14 b(:)g(:)g(:)36 b Fq(and)i(hence)h(is)e(iden)m(ti\014ed)g(with)g (the)h(set)g Fg(N)f Fq(of)0 2067 y(natural)32 b(n)m(um)m(b)s(ers.)45 b(Let)33 b Fo(!)26 b Fn(:=)e Fo(D)1208 2079 y Fl(0)1245 2067 y Fn(0)p Fo(;)14 b Fn(\012)22 b(:=)i Fo(D)1587 2079 y Fl(1)1624 2067 y Fn(0)32 b Fq(and)h(generally)f Fn(\012)2363 2079 y Fm(\033)2431 2067 y Fn(:=)23 b Fo(D)2611 2079 y Fm(\033)2656 2067 y Fn(0)p Fq(.)44 b(The)33 b Fn(\012)3029 2079 y Fm(\033)3074 2067 y Fq('s)h(as)0 2192 y(w)m(ell)e(as)h(1)f(are)g (called)g Fj(r)-5 b(e)g(gular)43 b Fq(tree)33 b(notations.)0 2441 y(F)-8 b(or)38 b(an)m(y)h Fo(a)32 b Fp(2)g Fn(T)39 b Fq(w)m(e)g(ha)m(v)m(e)h Fn(v)-5 b(al\()p Fo(a)p Fn(\))32 b Fp(2)g(T)1389 2453 y Fm(\027)1431 2441 y Fq(,)40 b(i.e.)62 b Fn(v)-5 b(al)o(\()p Fo(a)p Fn(\):)14 b Fo(I)39 b Fp(!)32 b(T)2168 2453 y Fm(\027)2248 2441 y Fq(with)39 b Fo(I)g Fn(=)31 b Fp(;)p Fo(;)14 b Fp(f)p Fn(0)p Fp(g)36 b Fq(or)j Fp(T)3060 2453 y Fm(\033)3143 2441 y Fq(for)0 2565 y(some)d Fo(\033)31 b(<)c(\027)5 b Fq(.)53 b(W)-8 b(e)36 b(no)m(w)h(w)m(an)m(t)g (to)e(reco)m(v)m(er)j(from)c Fo(a)i Fq(its)f Fj(typ)-5 b(e)36 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fq(,)38 b(whic)m(h)e(is)f (to)h(b)s(e)g Fn(0)p Fo(;)14 b Fn(1)0 2690 y Fq(or)36 b Fn(\012)183 2702 y Fm(\033)264 2690 y Fq(if)g Fo(I)43 b Fq(is)36 b Fp(;)p Fo(;)14 b Fp(f)p Fn(0)p Fp(g)35 b Fq(or)h Fp(T)947 2702 y Fm(\033)992 2690 y Fq(,)h(resp)s(ectiv)m(ely)-8 b(.)57 b(In)37 b(the)g(case)g Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))30 b(=)e(\012)2474 2702 y Fm(\033)2519 2690 y Fq(,)37 b(w)m(e)h(also)e(w)m(an)m(t)h(to)0 2814 y(reco)m(v)m(er)44 b(from)c Fo(a)i Fq(its)g Fj(fundamental)g(se)-5 b(quenc)g(e)p Fq(,)44 b(i.e.)71 b(notations)41 b Fo(a)p Fn([)p Fo(z)t Fn(])35 b Fp(2)i Fn(T)42 b Fq(with)f(v)-5 b(alue)0 2939 y Fn(v)g(al\()p Fo(a)p Fn(\)\(v)g(al\()p Fo(z)t Fn(\)\))p Fq(,)33 b(for)f(all)f Fo(z)26 b Fp(2)d Fn(T)1035 2951 y Fm(\033)1080 2939 y Fq(.)0 3188 y(Let)h Fp(j)p Fn(0)p Fp(j)f Fn(:=)f Fp(;)p Fo(;)14 b Fp(j)p Fn(1)p Fp(j)22 b Fn(:=)h Fp(f)p Fn(0)p Fp(g)f Fq(and)i Fp(j)p Fn(\012)1099 3200 y Fm(\033)1144 3188 y Fp(j)f Fn(:=)g(T)1361 3200 y Fm(\033)1405 3188 y Fq(.)41 b(F)-8 b(or)23 b Fo(a)f Fp(2)i Fn(T)f Fq(w)m(e)i(de\014ne)g Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b Fp(2)g(f)p Fn(0)p Fo(;)14 b Fn(1)p Fp(g)s([)s(f)p Fn(\012)2899 3200 y Fm(\033)2965 3188 y Fn(:)23 b Fo(\033)j(<)d(\027)5 b Fp(g)0 3313 y Fq(and)33 b Fo(a)p Fn([)p Fo(z)t Fn(])22 b Fp(2)h Fn(T)33 b Fq(for)f Fo(z)26 b Fp(2)e(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)33 b Fq(b)m(y)h(induction)d(on)i Fo(a)p Fq(,)f(as)h(follo)m(ws.) -103 3487 y(i.)48 b(F)-8 b(or)32 b Fo(a)23 b Fp(2)g(f)p Fn(0)p Fo(;)14 b Fn(1)p Fp(g)j([)h(f)p Fn(\012)717 3499 y Fm(\033)785 3487 y Fn(:)23 b Fo(\033)j(<)d(\027)5 b Fp(g)32 b Fq(let)g Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(:=)f Fo(a)32 b Fq(and)h Fo(a)p Fn([)p Fo(z)t Fn(])22 b(:=)h Fo(z)t Fq(.)-130 3636 y(ii.)47 b(F)-8 b(or)32 b Fo(D)244 3648 y Fm(\033)288 3636 y Fo(a)h Fq(with)f Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(1)32 b Fq(let)g Fo(\034)9 b Fn(\()p Fo(D)1213 3648 y Fm(\033)1258 3636 y Fo(a)p Fn(\))23 b(:=)g Fo(!)35 b Fq(and)e Fn(\()p Fo(D)1846 3648 y Fm(\033)1891 3636 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])23 b(=)f(\()p Fo(D)2274 3648 y Fm(\033)2319 3636 y Fo(a)p Fn([0]\))c Fp(\001)h Fn(\()p Fo(n)f Fn(+)g(1\))p Fq(.)-157 3786 y(iii.)46 b(F)-8 b(or)32 b Fo(D)244 3798 y Fm(\033)288 3786 y Fo(a)h Fq(with)f Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)912 3798 y Fm(\045)983 3786 y Fq(with)32 b Fo(\045)23 b Fp(\024)g Fo(\033)35 b Fq(let)d Fo(\034)9 b Fn(\()p Fo(D)1728 3798 y Fm(\033)1774 3786 y Fo(a)p Fn(\))23 b(:=)g(\012)2044 3798 y Fm(\045)2115 3786 y Fq(and)32 b Fn(\()p Fo(D)2405 3798 y Fm(\033)2450 3786 y Fo(a)p Fn(\)[)p Fo(z)t Fn(])23 b(:=)g Fo(D)2818 3798 y Fm(\033)2862 3786 y Fo(a)p Fn([)p Fo(z)t Fn(])p Fq(.)-154 3935 y(iv.)48 b(F)-8 b(or)33 b Fo(D)245 3947 y Fm(\033)289 3935 y Fo(a)g Fq(with)h Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))25 b(=)f(\012)917 3947 y Fm(\026)p Fl(+1)1079 3935 y Fq(with)33 b Fo(\033)28 b(<)c(\026)19 b Fn(+)g(1)33 b Fq(let)g Fo(\034)9 b Fn(\()p Fo(D)1982 3947 y Fm(\033)2027 3935 y Fo(a)p Fn(\))25 b(:=)f Fo(!)36 b Fq(and)d Fn(\()p Fo(D)2619 3947 y Fm(\033)2664 3935 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])25 b(:=)f Fo(D)3042 3947 y Fm(\033)3086 3935 y Fo(a)p Fn([)p Fo(z)3192 3947 y Fm(n)3237 3935 y Fn(])0 4060 y Fq(with)32 b Fo(z)261 4072 y Fl(0)321 4060 y Fn(:=)23 b(\012)492 4072 y Fm(\026)536 4060 y Fq(,)33 b Fo(z)635 4072 y Fm(n)p Fl(+1)787 4060 y Fn(:=)23 b Fo(D)967 4072 y Fm(\026)1011 4060 y Fo(a)p Fn([)p Fo(z)1117 4072 y Fm(n)1162 4060 y Fn(])p Fq(.)-127 4209 y(v.)49 b Fo(\034)9 b Fn(\()p Fo(a)121 4221 y Fl(1)159 4209 y Fo(;)14 b(:)g(:)g(:)g(;)g(a)388 4221 y Fm(k)428 4209 y Fn(\))24 b(:=)e Fo(\034)9 b Fn(\()p Fo(a)715 4221 y Fm(k)757 4209 y Fn(\))33 b Fq(and)g Fn(\()p Fo(a)1088 4221 y Fl(1)1125 4209 y Fo(;)14 b(:)g(:)g(:)g(;)g(a)1354 4221 y Fm(k)1394 4209 y Fn(\)[)p Fo(z)t Fn(])23 b(:=)g(\()p Fo(a)1725 4221 y Fl(1)1762 4209 y Fo(;)14 b(:)g(:)g(:)g(;)g(a)1991 4221 y Fm(k)q Fi(\000)p Fl(1)2116 4209 y Fo(;)g(a)2197 4221 y Fm(k)2238 4209 y Fn([)p Fo(z)t Fn(]\))p Fq(.)0 4384 y(Then)36 b(clearly)e Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))28 b(=)e(0)53 b Fp(\()-14 b(\))53 b Fo(a)26 b Fn(=)g(0)35 b Fq(and)g Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))27 b(=)g(1)52 b Fp(\()-14 b(\))53 b Fo(a)27 b Fn(=)f Fo(a)p Fn([0])19 b(+)h(1)p Fq(.)50 b(Also,)35 b(if)f Fo(a)26 b Fp(2)h Fn(T)3215 4396 y Fm(\033)0 4508 y Fq(and)h Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b Fp(6)p Fn(=)f(0)p Fo(;)14 b Fn(1)p Fq(,)28 b(then)h Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)1169 4520 y Fm(\045)1235 4508 y Fq(for)28 b(some)g Fo(\045)23 b(<)f(\033)s Fq(,)30 b(and)e(in)f(this)h(case)h(w)m (e)g(ha)m(v)m(e)g Fo(a)p Fn([)p Fo(z)t Fn(])23 b Fp(2)g Fn(T)3215 4520 y Fm(\033)0 4633 y Fq(for)32 b(all)f Fo(z)26 b Fp(2)d Fn(T)488 4645 y Fm(\045)550 4633 y Fn(=)g Fp(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)p Fq(.)0 4807 y Fk(Lemma)45 b(2.1.)56 b Fc(If)40 b Fo(a)32 b Fp(2)h Fn(T)40 b Fc(and)f Fo(z)d Fp(2)d(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)p Fc(,)43 b(then)d Fn(v)-5 b(al\()p Fo(z)t Fn(\))32 b Fp(2)h Fn(dom\(v)-5 b(al\()p Fo(a)p Fn(\)\))40 b Fc(and)g Fn(v)-5 b(al\()p Fo(a)p Fn([)p Fo(z)t Fn(]\))32 b(=)0 4932 y(v)-5 b(al\()p Fo(a)p Fn(\)\(v)g(al\()p Fo(z)t Fn(\)\))p Fc(.)0 5106 y Fq(Pro)s(of.)54 b(First)35 b(note)i(that)f(if)f Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))30 b(=)e(0)p Fo(;)14 b Fn(1)35 b Fq(or)h Fn(\012)1701 5118 y Fm(\033)1745 5106 y Fq(,)i(then)f Fn(dom\(v)-5 b(al\()p Fo(a)p Fn(\)\))29 b(=)f Fp(;)p Fo(;)14 b Fp(f)p Fn(0)p Fp(g)34 b Fq(or)i Fp(T)2998 5118 y Fm(\033)3043 5106 y Fq(.)54 b(W)-8 b(e)0 5230 y(pro)m(v)m(e)41 b(the)e(Lemma)f(b)m(y)i (induction)f(on)g Fo(a)p Fq(,)i(and)e(treat)h(only)e(Case)j(iv,)f(i.e.) 64 b Fo(D)2943 5242 y Fm(\033)2987 5230 y Fo(a)39 b Fq(with)p eop %%Page: 1 6 1 5 bop 0 125 a Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)325 137 y Fm(\026)p Fl(+1)479 125 y Fq(and)j Fo(\033)h(<)22 b(\026)7 b Fn(+)g(1)p Fq(.)41 b(Let)26 b Fn(v)-5 b(al\()p Fo(a)p Fn(\))23 b(=)g(\()p Fo(\013)1638 137 y Fm(\020)1676 125 y Fn(\))1708 137 y Fm(\020)t Fi(2T)1825 145 y Fh(\026)p Fd(+1)1940 125 y Fq(.)41 b(Then)27 b(b)m(y)g(induction)e(h)m(yp)s (othesis)548 349 y Fn(v)-5 b(al\(\()p Fo(D)785 361 y Fm(\033)830 349 y Fo(a)p Fn(\)[)p Fo(n)p Fn(]\))24 b(=)e(v)-5 b(al\()p Fo(D)1350 361 y Fm(\033)1395 349 y Fo(a)p Fn([)p Fo(z)1501 361 y Fm(n)1546 349 y Fn(]\))23 b(=)g Fp(D)1776 361 y Fm(\033)1820 349 y Fn(\(v)-5 b(al\()p Fo(a)p Fn([)p Fo(z)2094 361 y Fm(n)2140 349 y Fn(]\)\))23 b(=)g Fp(D)2402 361 y Fm(\033)2447 349 y Fo(\013)2500 364 y Fl(v)l(al\()p Fm(z)2641 372 y Fh(n)2681 364 y Fl(\))0 573 y Fq(with)32 b Fo(z)261 585 y Fl(0)321 573 y Fn(=)23 b(\012)469 585 y Fm(\026)513 573 y Fo(;)14 b(z)589 585 y Fm(n)p Fl(+1)741 573 y Fn(=)23 b Fo(D)898 585 y Fm(\026)942 573 y Fo(a)p Fn([)p Fo(z)1048 585 y Fm(n)1093 573 y Fn(])p Fq(,)32 b(and)931 797 y Fn(v)-5 b(al\()p Fo(D)1136 809 y Fm(\033)1180 797 y Fo(a)p Fn(\)\()p Fo(n)p Fn(\))24 b(=)f Fp(D)1546 809 y Fm(\033)1590 797 y Fn(\(v)-5 b(al\()p Fo(a)p Fn(\)\)\()p Fo(n)p Fn(\))25 b(=)d Fp(D)2156 809 y Fm(\033)2201 797 y Fo(\013)2254 809 y Fm(\020)2284 817 y Fh(n)0 1021 y Fq(with)32 b Fo(\020)258 1033 y Fl(0)319 1021 y Fn(=)23 b(\012)467 1033 y Fm(\026)511 1021 y Fo(;)14 b(\020)584 1033 y Fm(n)p Fl(+1)737 1021 y Fn(=)23 b Fp(D)889 1033 y Fm(\026)933 1021 y Fo(\013)986 1033 y Fm(\020)1016 1041 y Fh(n)1061 1021 y Fq(.)44 b(Hence)34 b(it)e(su\016ces)j(to)d(pro) m(v)m(e)i Fn(v)-5 b(al\()p Fo(z)2409 1033 y Fm(n)2454 1021 y Fn(\))23 b(=)g Fo(\020)2633 1033 y Fm(n)2679 1021 y Fq(.)43 b(This)33 b(follo)m(ws)0 1146 y(b)m(y)g(induction)f(on)h Fo(n)f Fq(from)f Fn(v)-5 b(al\(\012)1213 1158 y Fm(\026)1258 1146 y Fn(\))23 b(=)g(\012)1461 1158 y Fm(\026)1538 1146 y Fq(and)568 1370 y Fn(v)-5 b(al)o(\()p Fo(z)742 1382 y Fm(n)p Fl(+1)872 1370 y Fn(\))23 b(=)g Fp(D)1079 1382 y Fm(\026)1123 1370 y Fn(v)-5 b(al\()p Fo(a)p Fn([)p Fo(z)1365 1382 y Fm(n)1410 1370 y Fn(]\))23 b(=)g Fp(D)1640 1382 y Fm(\026)1685 1370 y Fo(\013)1738 1385 y Fl(v)l(al\()p Fm(z)1879 1393 y Fh(n)1919 1385 y Fl(\))1972 1370 y Fn(=)g Fp(D)2124 1382 y Fm(\026)2169 1370 y Fo(\013)2222 1382 y Fm(\020)2252 1390 y Fh(n)2320 1370 y Fn(=)f Fo(\020)2443 1382 y Fm(n)p Fl(+1)2573 1370 y Fo(:)p 2596 1370 4 75 v 2600 1298 90 4 v 2600 1370 V 2689 1370 4 75 v 0 1619 a Fq(As)38 b(a)f(consequence,)42 b(w)m(e)d(can)f(infer)f(the)h (principle)e(of)h Fj(tr)-5 b(ans\014nite)39 b(induction)g(on)g Fn(T)3188 1631 y Fm(\033)3233 1619 y Fq(,)0 1743 y(i.e.)597 1868 y Fn(\()p Fp(8)p Fo(a)22 b Fp(2)h Fn(T)880 1880 y Fm(\033)925 1868 y Fo(:)p Fp(8)p Fo(z)j Fp(2)d(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)24 b Fn(:)f Fo(')p Fn(\()p Fo(a)p Fn([)p Fo(z)t Fn(]\))h Fp(!)f Fo(')p Fn(\()p Fo(a)p Fn(\)\))h Fp(!)f(8)p Fo(a)f Fp(2)h Fn(T)2363 1880 y Fm(\033)2431 1868 y Fn(:)g Fo(')p Fn(\()p Fo(a)p Fn(\))p Fo(;)0 2051 y Fq(from)40 b(the)i(principle)e(of)g(trans\014nite)i (induction)e(on)h Fp(T)2052 2063 y Fm(\033)2138 2051 y Fq(in)g(Section)g(1.)69 b(T)-8 b(o)42 b(see)g(this,)0 2175 y(assume)c(the)g(premise)f(and)g(let)g Fo(a)30 b Fp(2)g Fn(T)1437 2187 y Fm(\033)1481 2175 y Fq(.)58 b(W)-8 b(e)38 b(use)g(trans\014nite)g(induction)e(on)h Fn(v)-5 b(al\()p Fo(a)p Fn(\))30 b Fp(2)0 2300 y(T)45 2312 y Fm(\033)90 2300 y Fq(.)74 b(It)42 b(su\016ces)j(to)d(pro)m(v)m(e)i Fp(8)p Fo(z)c Fp(2)e(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)p Fo(:')p Fn(\()p Fo(a)p Fn([)p Fo(z)t Fn(]\))p Fq(.)75 b(So)43 b(let)e Fo(z)g Fp(2)d(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)p Fq(.)75 b(By)43 b(Lemma)e(2.1)0 2424 y Fn(v)-5 b(al\()p Fo(z)t Fn(\))38 b Fp(2)i Fn(dom)o(\(v)-5 b(al\()p Fo(a)p Fn(\)\))45 b Fq(and)f Fn(v)-5 b(al)o(\()p Fo(a)p Fn([)p Fo(z)t Fn(]\))39 b(=)f(v)-5 b(al\()p Fo(a)p Fn(\)\(v)g(al)q(\()p Fo(z)t Fn(\)\))p Fq(.)77 b(Hence)45 b Fn(v)-5 b(al)o(\()p Fo(a)p Fn([)p Fo(z)t Fn(]\))44 b Fq(comes)g(b)s(efore)0 2549 y Fn(v)-5 b(al\()p Fo(a)p Fn(\))38 b Fq(in)f(the)h(sense)h(of)f(the)g(inductiv)m(e)g(generation)f (of)g Fp(T)2158 2561 y Fm(\033)2203 2549 y Fq(.)59 b(So)37 b Fo(')p Fn(\()p Fo(a)p Fn([)p Fo(z)t Fn(]\))h Fq(b)m(y)h(induction)0 2673 y(h)m(yp)s(othesis.)0 2922 y Fk(3.)49 b(THE)32 b(SLO)m(W)h(GR)m(O) m(WING)e(HIERAR)m(CHY)0 3047 y Fq(Giv)m(en)e(a)g(tree)h(notation)e Fo(a)23 b Fp(2)h Fn(T)1142 3059 y Fl(1)1208 3047 y Fq(and)30 b(a)f(natural)f(n)m(um)m(b)s(er)i Fo(n)p Fq(,)g(w)m(e)g(ma)m(y)g (decide)f(to)h(clim)m(b)0 3171 y(do)m(wn)d(the)f(tree)g(\(whic)m(h)g (gro)m(ws)h(do)m(wn)m(w)m(ards\),)i(using)c Fo(n)h Fq(as)g(a)f (parameter.)41 b(This)26 b(is)f(done)0 3296 y(as)36 b(follo)m(ws.)52 b(If)36 b(the)g(no)s(de)g(w)m(e)h(are)f(at)g(is)f(formed)g(b)m(y)i(the) f(successor)j(op)s(eration,)c(then)0 3421 y(w)m(e)i(ha)m(v)m(e)g(to)e (do)h(some)f(w)m(ork)i(to)e(clim)m(b)f(do)m(wn)j(one)f(step.)53 b(If)36 b(on)f(the)i(other)e(hand)h(the)0 3545 y(no)s(de)d(is)g(formed) f(as)i(a)e(sequence)k(\(whic)m(h)e(m)m(ust)f(b)s(e)g(of)g(length)f Fo(!)s Fq(,)h(since)h Fo(a)24 b Fp(2)g Fn(T)2935 3557 y Fl(1)2972 3545 y Fq(\),)33 b(then)0 3670 y(w)m(e)28 b(don't)g(ha)m(v)m(e)h(to)e(w)m(ork)h(but)f(just)h(slip)e(do)m(wn)i(to) f(the)h Fo(n)p Fq({th)f(elemen)m(t)g(of)g(the)h(sequence.)0 3919 y(If)37 b(w)m(e)g(coun)m(t)h(the)f(pieces)g(of)g(w)m(ork)g(w)m(e)h (ha)m(v)m(e)g(done)f(un)m(til)f(w)m(e)h(reac)m(h)h(a)e(b)s(ottom)g(no)s (de,)0 4043 y(w)m(e)j(get)g(a)f(natural)f(n)m(um)m(b)s(er)i Fo(G)1178 4055 y Fm(a)1218 4043 y Fn(\()p Fo(n)p Fn(\))p Fq(.)61 b(These)40 b(functions)f Fo(G)2200 4055 y Fm(a)2240 4043 y Fn(:)14 b Fg(N)31 b Fp(!)g Fg(N)38 b Fq(for)g Fo(a)31 b Fp(2)h Fn(T)2987 4055 y Fl(1)3062 4043 y Fq(form)0 4168 y(the)40 b(so{called)f Fj(slow)h(gr)-5 b(owing)41 b(hier)-5 b(ar)g(chy)9 b Fq(;)42 b(the)e(formal)d(de\014nition)i(is)g (b)m(y)i(trans\014nite)0 4292 y(induction)32 b(on)g Fo(a)23 b Fp(2)g Fn(T)774 4304 y Fl(1)812 4292 y Fq(,)32 b(as)h(follo)m(ws)-103 4467 y(i.)48 b Fo(G)65 4479 y Fl(0)103 4467 y Fn(\()p Fo(n)p Fn(\))23 b(=)g(0)p Fq(,)-130 4616 y(ii.)47 b Fo(G)65 4628 y Fm(a)p Fl(+1)189 4616 y Fn(\()p Fo(n)p Fn(\))24 b(=)f Fo(G)480 4628 y Fm(a)520 4616 y Fn(\()p Fo(n)p Fn(\))c(+)f(1)p Fq(,)-157 4765 y(iii.)46 b Fo(G)65 4777 y Fm(a)105 4765 y Fn(\()p Fo(n)p Fn(\))24 b(=)f Fo(G)396 4780 y Fm(a)p Fl([)p Fm(n)p Fl(])515 4765 y Fn(\()p Fo(n)p Fn(\))33 b Fq(if)e Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f Fo(!)s Fq(.)0 4940 y(Note)44 b(that)f Fo(G)534 4952 y Fm(a)p Fl(+)p Fm(b)655 4940 y Fn(\()p Fo(n)p Fn(\))c(=)g Fo(G)977 4952 y Fm(a)1017 4940 y Fn(\()p Fo(n)p Fn(\))26 b(+)e Fo(G)1311 4952 y Fm(b)1345 4940 y Fn(\()p Fo(n)p Fn(\))p Fq(;)49 b(this)44 b(can)g(b)s(e)g(pro)m(v)m(ed)h(easily)e(b)m (y)i(trans\014nite)0 5064 y(induction)32 b(on)g Fo(b)23 b Fp(2)g Fn(T)766 5076 y Fl(1)803 5064 y Fq(.)p eop %%Page: 1 7 1 6 bop 0 125 a Fq(F)-8 b(or)32 b(example,)600 322 y Fo(G)665 334 y Fm(k)706 322 y Fn(\()p Fo(n)p Fn(\))24 b(=)e Fo(k)s(;)593 472 y(G)658 484 y Fm(!)706 472 y Fn(\()p Fo(n)p Fn(\))i(=)e Fo(G)996 487 y Fm(!)r Fl([)p Fm(n)p Fl(])1123 472 y Fn(\()p Fo(n)p Fn(\))i(=)f Fo(G)1414 484 y Fm(n)1459 472 y Fn(\()p Fo(n)p Fn(\))g(=)g Fo(n;)517 621 y(G)582 633 y Fm(D)636 641 y Fd(0)669 633 y Fl(1)706 621 y Fn(\()p Fo(n)p Fn(\))h(=)e Fo(G)996 636 y Fm(!)r Fi(\001)p Fl(\()p Fm(n)p Fl(+1\))1242 621 y Fn(\()p Fo(n)p Fn(\))h(=)g(\()p Fo(n)18 b Fn(+)g(1\))g Fp(\001)h Fo(G)1849 633 y Fm(!)1897 621 y Fn(\()p Fo(n)p Fn(\))24 b(=)e(\()p Fo(n)d Fn(+)f(1\))g Fp(\001)h Fo(n;)517 771 y(G)582 783 y Fm(D)636 791 y Fd(0)669 783 y Fl(2)706 771 y Fn(\()p Fo(n)p Fn(\))24 b(=)e Fo(G)996 786 y Fl(\()p Fm(D)1076 794 y Fd(0)1109 786 y Fl(1\))p Fi(\001)p Fl(\()p Fm(n)p Fl(+1\))1369 771 y Fn(\()p Fo(n)p Fn(\))i(=)e(\()p Fo(n)d Fn(+)f(1\))g Fp(\001)h Fo(G)1977 783 y Fm(D)2031 791 y Fd(0)2064 783 y Fl(1)2101 771 y Fn(\()p Fo(n)p Fn(\))k(=)g(\()p Fo(n)c Fn(+)f(1\))2584 736 y Fl(2)2639 771 y Fp(\001)h Fo(n;)506 920 y(G)571 932 y Fm(D)625 940 y Fd(0)658 932 y Fm(!)706 920 y Fn(\()p Fo(n)p Fn(\))24 b(=)e Fo(G)996 932 y Fm(D)1050 940 y Fd(0)1083 932 y Fm(n)1129 920 y Fn(\()p Fo(n)p Fn(\))h(=)g(\()p Fo(n)18 b Fn(+)g(1\))1611 886 y Fm(n)1675 920 y Fp(\001)g Fo(n:)0 1126 y Fq(Hence,)41 b(the)e(functions)g Fo(G)990 1138 y Fm(a)1069 1126 y Fq(with)f Fo(a)g Fq(built)f(up)i(from)e Fn(0)p Fo(;)14 b Fn(1)38 b Fq(b)m(y)h Fn(+)f Fq(and)h Fo(D)2669 1138 y Fl(0)2744 1126 y Fq(but)g(without)0 1251 y(nesting)33 b(of)f Fo(D)517 1263 y Fl(0)586 1251 y Fq(are)h(all)d(p)s(olynomials.)0 1500 y(The)48 b(functions)f Fo(G)715 1512 y Fm(a)802 1500 y Fq(with)g Fo(a)g Fq(of)f(the)h(form)f Fo(D)1751 1512 y Fl(0)1788 1500 y Fo(D)1859 1470 y Fm(m)1857 1521 y Fl(1)1922 1500 y Fn(0)h Fq(will)d(b)s(e)j(used)h(in)f(Section)f(4)h (to)0 1624 y(estimate)35 b(the)i(instances)g(pro)m(vided)g(b)m(y)g (existen)m(tial)e(pro)s(ofs)h(in)f(arithmetic.)53 b(W)-8 b(e)36 b(will)0 1749 y(also)43 b(sho)m(w)i(in)e(Section)h(5)g(that)g (this)f(result)h(is)g(b)s(est)h(p)s(ossible,)h(since)e(an)m(y)h(suc)m (h)g Fo(G)3219 1761 y Fm(a)0 1874 y Fq(actually)31 b(is)h(pro)m(v)-5 b(ably)33 b(total)e(in)g(arithmetic.)0 2123 y(In)e(order)g(to)f(ac)m (hiev)m(e)i(these)g(results)f(w)m(e)h(need)g(some)f(monotonicit)m(y)d (prop)s(erties)j(of)f(the)0 2247 y Fo(G)65 2259 y Fm(a)105 2247 y Fq(.)66 b(Since)40 b Fo(G)525 2259 y Fm(k)566 2247 y Fo(n)34 b Fn(=)f Fo(k)43 b Fq(and)d Fo(G)1096 2259 y Fm(!)1144 2247 y Fo(n)33 b Fn(=)h Fo(n)p Fq(,)41 b(w)m(e)g(cannot)f(ha)m(v)m(e)i(that)d Fn(v)-5 b(al\()p Fo(a)p Fn(\))34 b Fo(<)f Fn(v)-5 b(al\()p Fo(b)p Fn(\))40 b Fq(implies)0 2372 y Fo(G)65 2384 y Fm(a)105 2372 y Fo(n)28 b Fp(\024)g Fo(G)341 2384 y Fm(b)374 2372 y Fo(n)p Fq(,)37 b(for)f(all)d Fo(n)p Fq(.)54 b(Hence)37 b(w)m(e)g(in)m(tro)s (duce)f(appropriate)g(relations)e Fo(<)2781 2384 y Fm(k)2858 2372 y Fq(suc)m(h)j(that)0 2496 y Fo(a)23 b(<)132 2508 y Fm(k)195 2496 y Fo(b)33 b Fq(implies)d Fo(G)660 2508 y Fm(a)700 2496 y Fo(n)23 b Fp(\024)g Fo(G)926 2508 y Fm(b)959 2496 y Fo(n)33 b Fq(for)f(all)e Fo(n)23 b Fp(\025)g Fo(k)s Fq(.)0 2745 y(F)-8 b(or)30 b Fo(a)23 b Fp(6)p Fn(=)g(0)31 b Fq(let)f Fo(a)584 2715 y Fi(\000)663 2745 y Fn(:=)23 b Fo(a)p Fn([0])31 b Fq(if)f Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(1)30 b Fq(or)h Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))25 b(=)d Fo(!)s Fq(,)31 b(and)h Fo(a)2091 2715 y Fi(\000)2170 2745 y Fn(:=)22 b Fo(a)p Fn([\012)2407 2757 y Fm(\026)2452 2745 y Fn(])31 b Fq(if)f Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)2919 2757 y Fm(\026)p Fl(+1)3048 2745 y Fq(.)43 b(Let)0 2870 y Fo(a)23 b(<)132 2882 y Fm(k)195 2870 y Fo(b)30 b Fq(i\013)e(w)m(e)j(ha)m(v)m(e)g(a)f (\014nite)f(sequence)j Fo(a)23 b Fn(=)g Fo(a)1662 2882 y Fl(0)1699 2870 y Fo(;)14 b(a)1780 2882 y Fl(1)1817 2870 y Fo(;)g(:)g(:)g(:)g(;)g(a)2046 2882 y Fm(n)2114 2870 y Fn(=)22 b Fo(b)30 b Fq(with)f Fo(n)23 b(>)g Fn(0)29 b Fq(suc)m(h)i(that)e(for)0 2994 y(all)f Fo(i)22 b(<)h(n)29 b Fq(either)h Fo(a)669 3006 y Fm(i)720 2994 y Fn(=)22 b Fo(a)851 2959 y Fi(\000)851 3017 y Fm(i)p Fl(+1)993 2994 y Fq(or)29 b Fo(\034)9 b Fn(\()p Fo(a)1230 3006 y Fm(i)p Fl(+1)1343 2994 y Fn(\))23 b(=)f Fo(!)33 b Fq(and)d Fo(a)1801 3006 y Fm(i)1851 2994 y Fn(=)23 b Fo(a)1983 3006 y Fm(i)p Fl(+1)2095 2994 y Fn([)p Fo(j)5 b Fn(])29 b Fq(for)g(some)h Fo(j)35 b Fq(with)29 b Fn(1)23 b Fp(\024)f Fo(j)28 b Fp(\024)23 b Fo(k)s Fq(.)0 3119 y(Note)38 b(that)g(from)f Fo(a)31 b(<)834 3131 y Fm(k)905 3119 y Fo(b)38 b Fq(and)g Fo(k)c Fp(\024)d Fo(l)39 b Fq(w)m(e)h(can)e(ob)m(viously)g(conclude)h (that)f Fo(a)30 b(<)2945 3131 y Fm(l)3002 3119 y Fo(b)p Fq(.)59 b(W)-8 b(e)0 3243 y(write)32 b Fo(a)23 b Fp(\024)381 3255 y Fm(k)445 3243 y Fo(b)32 b Fq(for)g Fo(a)23 b(<)794 3255 y Fm(k)857 3243 y Fo(b)32 b Fq(or)h Fo(a)23 b Fn(=)f Fo(b)p Fq(.)0 3492 y(Some)k(of)f(our)h(later)f(argumen)m(ts)h(will)e(b) s(e)i(b)m(y)h(induction)e(on)h Fn(length\()p Fo(a)p Fn(\))p Fq(,)i(whic)m(h)f(is)e(de\014ned)0 3617 y(b)m(y)1028 3715 y Fn(length\(0\))e(=)g(0)p Fo(;)1028 3865 y Fn(length\(1\))g(=)g (1)p Fo(;)912 4014 y Fn(length\()p Fo(D)1239 4026 y Fm(\033)1284 4014 y Fo(a)p Fn(\))g(=)g(length\()p Fo(a)p Fn(\))c(+)f Fo(\033)k Fn(+)c(1)p Fo(;)719 4164 y Fn(length\()p Fo(a)1021 4176 y Fl(1)1059 4164 y Fo(;)c(:)g(:)g(:)f(;)h(a)1287 4176 y Fm(k)1328 4164 y Fn(\))23 b(=)g(length\()p Fo(a)1773 4176 y Fl(1)1810 4164 y Fn(\))c(+)f Fp(\001)c(\001)g(\001)k Fn(+)g(length\()p Fo(a)2444 4176 y Fm(k)2485 4164 y Fn(\))p Fo(:)0 4328 y Fq(Note)33 b(that)f Fn(length\()p Fo(a)19 b Fn(+)f Fo(b)p Fn(\))23 b(=)f(length\()p Fo(a)p Fn(\))d(+)f(length\()p Fo(b)p Fn(\))p Fq(.)0 4503 y Fk(Lemma)37 b(3.1.)-103 4627 y Fc(i.)48 b(If)33 b Fo(a)22 b Fp(6)p Fn(=)h(0)p Fc(,)32 b(then)h Fn(\()p Fo(D)676 4639 y Fm(\033)721 4627 y Fo(a)p Fn(\))797 4597 y Fi(\000)876 4627 y Fn(=)23 b Fo(D)1033 4639 y Fm(\033)1077 4627 y Fo(a)1121 4597 y Fi(\000)1177 4627 y Fc(.)-130 4752 y(ii.)47 b(If)33 b Fo(b)22 b Fp(6)p Fn(=)h(0)p Fc(,)32 b(then)h Fn(\()p Fo(a)19 b Fn(+)f Fo(b)p Fn(\))813 4721 y Fi(\000)892 4752 y Fn(=)k Fo(a)d Fn(+)f Fo(b)1161 4721 y Fi(\000)1216 4752 y Fc(.)-157 4876 y(iii.)46 b(If)33 b Fo(a)22 b Fp(6)p Fn(=)h(0)p Fc(,)32 b(then)h Fn(length\()p Fo(a)877 4846 y Fi(\000)934 4876 y Fn(\))23 b Fo(<)g Fn(length\()p Fo(a)p Fn(\))p Fc(.)0 5051 y Fq(Pro)s(of.)43 b(i.)f(If)33 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(1)32 b Fq(or)g Fo(!)s Fq(,)g(then)h Fo(\034)9 b Fn(\()p Fo(D)1440 5063 y Fm(\033)1486 5051 y Fo(a)p Fn(\))23 b(=)g Fo(!)35 b Fq(and)846 5275 y Fn(\()p Fo(D)947 5287 y Fm(\033)992 5275 y Fo(a)p Fn(\))1068 5240 y Fi(\000)1147 5275 y Fn(=)23 b(\()p Fo(D)1336 5287 y Fm(\033)1381 5275 y Fo(a)p Fn(\)[0])g(=)f(\()p Fo(D)1756 5287 y Fm(\033)1801 5275 y Fo(a)p Fn([0]\))c Fp(\001)h Fn(1)j(=)h Fo(D)2246 5287 y Fm(\033)2290 5275 y Fo(a)2334 5240 y Fi(\000)2390 5275 y Fo(:)p eop %%Page: 1 8 1 7 bop 0 125 a Fq(If)33 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)e(\012)422 137 y Fm(\026)p Fl(+1)583 125 y Fq(with)33 b Fo(\026)18 b Fn(+)g(1)23 b Fp(\024)f Fo(\033)s Fq(,)33 b(then)h Fo(\034)9 b Fn(\()p Fo(D)1588 137 y Fm(\033)1633 125 y Fo(a)p Fn(\))23 b(=)g(\012)1880 137 y Fm(\026)p Fl(+1)2041 125 y Fq(and)866 349 y Fn(\()p Fo(D)967 361 y Fm(\033)1012 349 y Fo(a)p Fn(\))1088 314 y Fi(\000)1167 349 y Fn(=)g(\()p Fo(D)1356 361 y Fm(\033)1401 349 y Fo(a)p Fn(\)[\012)1560 361 y Fm(\026)1605 349 y Fn(])g(=)f Fo(D)1807 361 y Fm(\033)1852 349 y Fo(a)p Fn([\012)1979 361 y Fm(\026)2023 349 y Fn(])h(=)g Fo(D)2226 361 y Fm(\033)2270 349 y Fo(a)2314 314 y Fi(\000)2370 349 y Fo(:)0 573 y Fq(If)33 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)e(\012)422 585 y Fm(\026)p Fl(+1)583 573 y Fq(with)33 b Fo(\033)26 b(<)d(\026)18 b Fn(+)g(1)p Fq(,)32 b(then)i Fo(\034)9 b Fn(\()p Fo(D)1588 585 y Fm(\033)1633 573 y Fo(a)p Fn(\))23 b(=)g Fo(!)35 b Fq(and)898 797 y Fn(\()p Fo(D)999 809 y Fm(\033)1044 797 y Fo(a)p Fn(\))1120 763 y Fi(\000)1199 797 y Fn(=)23 b(\()p Fo(D)1388 809 y Fm(\033)1432 797 y Fo(a)p Fn(\)[0])g(=)g Fo(D)1776 809 y Fm(\033)1820 797 y Fo(a)p Fn([\012)1947 809 y Fm(\026)1992 797 y Fn(])g(=)f Fo(D)2194 809 y Fm(\033)2239 797 y Fo(a)2283 763 y Fi(\000)2339 797 y Fo(:)0 1021 y Fq(ii.)39 b(The)26 b(claim)d(follo)m(ws)h(from)f Fo(\034)9 b Fn(\()p Fo(a)d Fn(+)g Fo(b)p Fn(\))23 b(=)g Fo(\034)9 b Fn(\()p Fo(b)p Fn(\))26 b Fq(and)f Fn(\()p Fo(a)6 b Fn(+)g Fo(b)p Fn(\)[)p Fo(z)t Fn(])22 b(=)g Fo(a)6 b Fn(+)g Fo(b)p Fn([)p Fo(z)t Fn(])p Fq(.)39 b(iii.)f(By)26 b(induction)0 1146 y(on)38 b(a.)61 b(Case)39 b Fn(1)p Fo(;)14 b(!)s Fq(.)60 b(Then)40 b Fo(a)1043 1116 y Fi(\000)1130 1146 y Fn(=)31 b Fo(a)p Fn([0])g(=)g(0)38 b Fq(and)g(the)h(claim)d(is)i(immediate.)58 b(Case)39 b Fo(D)3144 1158 y Fm(\033)3189 1146 y Fo(a)p Fq(.)0 1270 y(F)-8 b(or)32 b Fo(a)23 b Fn(=)f(0)32 b Fq(this)h(is)f(clear.)43 b(F)-8 b(or)31 b Fo(a)23 b Fp(6)p Fn(=)g(0)32 b Fq(w)m(e)i(ha)m(v)m(e)g(b)m(y)f(i)948 1468 y Fn(length\(\()p Fo(D)1307 1480 y Fm(\033)1352 1468 y Fo(a)p Fn(\))1428 1434 y Fi(\000)1484 1468 y Fn(\))23 b(=)g(length\()p Fo(D)1954 1480 y Fm(\033)1999 1468 y Fo(a)2043 1434 y Fi(\000)2099 1468 y Fn(\))1539 1618 y(=)g(length\()p Fo(a)1929 1583 y Fi(\000)1985 1618 y Fn(\))c(+)f Fo(\033)k Fn(+)c(1)1539 1767 y Fo(<)23 b Fn(length\()p Fo(a)p Fn(\))c(+)f Fo(\033)k Fn(+)c(1)1539 1916 y(=)23 b(length\()p Fo(D)1954 1928 y Fm(\033)1999 1916 y Fo(a)p Fn(\))p Fo(:)0 2128 y Fq(Case)34 b(a+b.)43 b(Similarly)-8 b(,)28 b(using)33 b(ii.)p 1296 2128 4 75 v 1300 2057 90 4 v 1300 2128 V 1389 2128 4 75 v 0 2302 a Fk(Lemma)k(3.2.)-103 2427 y Fc(i.)48 b(If)33 b Fo(b)134 2439 y Fl(0)193 2427 y Fo(<)258 2439 y Fm(k)322 2427 y Fo(b)p Fc(,)f(then)h Fo(a)19 b Fn(+)f Fo(b)821 2439 y Fl(0)881 2427 y Fo(<)946 2439 y Fm(k)1009 2427 y Fo(a)g Fn(+)h Fo(b)p Fc(.)-130 2552 y(ii.)47 b(If)33 b Fo(c)23 b Fp(6)p Fn(=)f(0)p Fc(,)32 b(then)h Fn(1)23 b Fp(\024)697 2564 y Fl(1)757 2552 y Fo(c)p Fc(.)-157 2676 y(iii.)46 b(If)33 b Fo(b)22 b(<)221 2688 y Fm(k)285 2676 y Fo(a)p Fc(,)32 b(then)h Fo(D)679 2688 y Fm(\033)724 2676 y Fo(b)23 b(<)848 2688 y Fm(k)911 2676 y Fo(D)980 2688 y Fm(\033)1024 2676 y Fo(a)p Fc(.)-154 2801 y(iv.)48 b Fn(\()p Fo(D)103 2770 y Fm(m)101 2821 y(\033)166 2801 y Fo(a)p Fn(\))19 b(+)f(1)23 b Fp(\024)474 2813 y Fl(1)533 2801 y Fo(D)604 2770 y Fm(m)602 2821 y(\033)668 2801 y Fn(\()p Fo(a)18 b Fn(+)g(1\))p Fc(.)0 2975 y Fq(Pro)s(of.)59 b(i.)f(The)39 b(claim)c(follo)m(ws)h(from)h(Lemma)f(3.1)i(ii)e (together)i(with)f Fo(\034)9 b Fn(\()p Fo(a)22 b Fn(+)f Fo(b)p Fn(\))31 b(=)f Fo(\034)9 b Fn(\()p Fo(b)p Fn(\))0 3099 y Fq(and)33 b Fn(\()p Fo(a)18 b Fn(+)g Fo(b)p Fn(\)[)p Fo(z)t Fn(])23 b(=)f Fo(a)d Fn(+)f Fo(b)p Fn([)p Fo(z)t Fn(])p Fq(.)0 3224 y(ii.)64 b(By)41 b(induction)e(on)g Fn(length\()p Fo(c)p Fn(\))p Fq(.)67 b(Case)41 b(1.)65 b Fn(1)33 b Fp(\024)1833 3236 y Fl(1)1904 3224 y Fn(1)p Fq(.)65 b(Case)41 b Fo(!)s Fq(.)66 b Fn(1)33 b(=)g Fo(!)s Fn([1])g Fo(<)2841 3236 y Fl(1)2911 3224 y Fo(!)s Fq(.)66 b(Case)0 3349 y Fn(\012)60 3361 y Fm(\026)p Fl(+1)189 3349 y Fq(.)53 b Fn(1)28 b Fp(\024)404 3361 y Fl(1)469 3349 y Fn(\012)529 3361 y Fm(\026)601 3349 y Fo(<)666 3361 y Fl(1)731 3349 y Fn(\012)791 3361 y Fm(\026)p Fl(+1)920 3349 y Fq(;)37 b(here)g Fn(1)28 b Fp(\024)f Fn(\012)1421 3361 y Fm(\026)1502 3349 y Fq(holds)35 b(b)m(y)i(induction)e(h)m(yp)s (othesis.)55 b(Case)38 b Fo(D)3172 3361 y Fm(\033)3216 3349 y Fo(a)0 3473 y Fq(with)h Fo(a)34 b Fp(6)p Fn(=)f(0)p Fq(.)64 b Fn(1)33 b Fp(\024)678 3485 y Fl(1)748 3473 y Fo(D)817 3485 y Fm(\033)862 3473 y Fo(a)906 3443 y Fi(\000)995 3473 y Fn(=)g(\()p Fo(D)1194 3485 y Fm(\033)1239 3473 y Fo(a)p Fn(\))1315 3443 y Fi(\000)1404 3473 y Fo(<)1469 3485 y Fl(1)1540 3473 y Fo(D)1609 3485 y Fm(\033)1653 3473 y Fo(a)p Fq(;)43 b(here)e(the)f(\014rst)h(inequalit)m(y)d(follo)m (ws)h(b)m(y)0 3598 y(induction)h(h)m(yp)s(othesis,)45 b(since)c Fn(length\()p Fo(a)1510 3567 y Fi(\000)1567 3598 y Fn(\))35 b Fo(<)g Fn(length\()p Fo(a)p Fn(\))p Fq(.)70 b(Case)42 b(a)f(+)g(b)g(with)g Fo(a;)14 b(b)34 b Fp(6)p Fn(=)h(0)p Fq(.)0 3722 y Fn(1)23 b Fp(\024)130 3734 y Fl(1)189 3722 y Fo(a)g(<)321 3734 y Fl(1)381 3722 y Fo(a)16 b Fn(+)g(1)22 b Fp(\024)651 3734 y Fl(1)711 3722 y Fo(a)16 b Fn(+)f Fo(b)p Fq(;)32 b(in)e(the)i(last)e(inequalit)m (y)g(w)m(e)i(ha)m(v)m(e)h(used)f(i)e(and)h(the)h(induction)0 3847 y(h)m(yp)s(othesis.)0 3971 y(iii.)80 b(This)46 b(follo)m(ws)f (from)f Fn(\()p Fo(D)1105 3983 y Fm(\033)1150 3971 y Fo(a)p Fn(\))1226 3941 y Fi(\000)1324 3971 y Fn(=)e Fo(D)1500 3983 y Fm(\033)1544 3971 y Fo(a)1588 3941 y Fi(\000)1690 3971 y Fq(and)k(the)g(fact)f(that,)k(if)c Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))43 b(=)e Fo(!)s Fq(,)49 b(then)0 4096 y Fo(\034)9 b Fn(\()p Fo(D)146 4108 y Fm(\033)191 4096 y Fo(a)p Fn(\))24 b(=)e Fo(!)35 b Fq(and)e Fn(\()p Fo(D)756 4108 y Fm(\033)801 4096 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])23 b(=)g Fo(D)1153 4108 y Fm(\033)1197 4096 y Fo(a)p Fn([)p Fo(n)p Fn(])p Fq(.)0 4220 y(iv.)48 b(By)35 b(induction)e(on)h Fo(m)p Fq(.)48 b(F)-8 b(or)33 b Fn(0)h Fq(there)h(is)f(nothing)f(to)h (sho)m(w,)h(and)g(in)e(the)i(induction)0 4345 y(step)d(w)m(e)g(ha)m(v)m (e)h Fn(\()p Fo(D)674 4315 y Fm(m)p Fl(+1)672 4365 y Fm(\033)821 4345 y Fo(a)p Fn(\))16 b(+)g(1)23 b Fp(\024)1124 4357 y Fl(1)1183 4345 y Fn(\()p Fo(D)1286 4315 y Fm(m)p Fl(+1)1284 4365 y Fm(\033)1434 4345 y Fo(a)p Fn(\))16 b Fp(\001)g Fn(2)23 b(=)f(\()p Fo(D)1818 4357 y Fm(\033)1863 4345 y Fn(\(\()p Fo(D)1998 4315 y Fm(m)1996 4365 y(\033)2062 4345 y Fo(a)p Fn(\))16 b(+)g(1\)\)[1])22 b Fo(<)2516 4357 y Fl(1)2576 4345 y Fo(D)2645 4357 y Fm(\033)2690 4345 y Fn(\(\()p Fo(D)2825 4315 y Fm(m)2823 4365 y(\033)2888 4345 y Fo(a)p Fn(\))17 b(+)e(1\))23 b Fp(\024)3223 4357 y Fl(1)0 4469 y Fo(D)71 4439 y Fm(m)p Fl(+1)69 4490 y Fm(\033)218 4469 y Fn(\()p Fo(a)c Fn(+)f(1\))p Fq(,)33 b(where)h(in)e(the)i(\014rst)f(inequalit)m(y)f(w)m(e)i(ha)m(v)m(e)g (used)g(ii,)e(and)h(in)f(the)h(last)g(one)0 4594 y(w)m(e)h(ha)m(v)m(e)g (used)f(the)g(induction)f(h)m(yp)s(othesis)i(and)e(iii.)p 2015 4594 V 2019 4522 90 4 v 2019 4594 V 2108 4594 4 75 v 0 4768 a Fk(Lemma)37 b(3.3.)-103 4893 y Fc(i.)48 b(If)33 b Fo(\034)9 b Fn(\()p Fo(c)p Fn(\))24 b(=)e(\012)414 4905 y Fm(\026)p Fl(+1)575 4893 y Fc(and)33 b Fo(x;)14 b(y)26 b Fp(2)e(j)p Fo(\034)9 b Fn(\()p Fo(c)p Fn(\))p Fp(j)33 b Fc(and)g Fo(x)24 b(<)1545 4905 y Fm(k)1608 4893 y Fo(y)s Fc(,)32 b(then)i Fo(c)p Fn([)p Fo(x)p Fn(])23 b Fo(<)2151 4905 y Fm(k)2215 4893 y Fo(c)p Fn([)p Fo(y)s Fn(])p Fc(.)-130 5017 y(ii.)47 b(If)33 b Fo(\034)9 b Fn(\()p Fo(c)p Fn(\))24 b(=)e(\012)414 5029 y Fm(\026)p Fl(+1)575 5017 y Fc(and)33 b Fo(x)24 b Fp(2)f(j)p Fo(\034)9 b Fn(\()p Fo(c)p Fn(\))p Fp(j)34 b Fc(then)f Fo(c)p Fn([)p Fo(x)p Fn(])19 b(+)f(1)23 b Fp(\024)1722 5029 y Fl(1)1782 5017 y Fo(c)p Fn([)p Fo(x)18 b Fn(+)g(1])p Fc(.)p eop %%Page: 1 9 1 8 bop -157 125 a Fc(iii.)46 b(If)33 b Fo(\034)9 b Fn(\()p Fo(c)p Fn(\))24 b(=)e Fo(!)s Fc(,)33 b(then)g Fo(c)p Fn([)p Fo(n)p Fn(])18 b(+)g(1)23 b Fp(\024)1054 137 y Fl(1)1114 125 y Fo(c)p Fn([)p Fo(n)18 b Fn(+)g(1])p Fc(.)0 299 y Fq(Pro)s(of.)63 b(i.)g(By)40 b(induction)e(on)h Fn(length\()p Fo(c)p Fn(\))p Fq(.)65 b(Case)40 b Fn(\012)1903 311 y Fm(\026)p Fl(+1)2032 299 y Fq(.)63 b(Ob)m(vious,)42 b(since)e Fn(\012)2844 311 y Fm(\026)p Fl(+1)2972 299 y Fn([)p Fo(z)t Fn(])32 b(=)g Fo(z)t Fq(.)0 423 y(Case)i Fo(D)303 435 y Fm(\033)347 423 y Fo(a)p Fq(.)43 b(Then)34 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)1041 435 y Fm(\026)p Fl(+1)1202 423 y Fq(and)33 b Fo(\026)18 b Fn(+)g(1)23 b Fp(\024)f Fo(\033)s Fq(,)33 b(hence)869 648 y Fn(\()p Fo(D)970 660 y Fm(\033)1015 648 y Fo(a)p Fn(\)[)p Fo(x)p Fn(])24 b(=)e Fo(D)1364 660 y Fm(\033)1409 648 y Fo(a)p Fn([)p Fo(x)p Fn(])h Fo(<)1634 660 y Fm(k)1698 648 y Fo(D)1767 660 y Fm(\033)1811 648 y Fo(a)p Fn([)p Fo(y)s Fn(])g(=)g(\()p Fo(D)2157 660 y Fm(\033)2201 648 y Fo(a)p Fn(\)[)p Fo(y)s Fn(])p Fo(;)0 872 y Fq(b)m(y)40 b(induction)d(h)m(yp)s(othesis)j(and)f(Lemma)e(3.2)i(iii.)59 b(Case)39 b(a)g(+)f(b.)62 b(Then)40 b Fo(\034)9 b Fn(\()p Fo(b)p Fn(\))33 b(=)e(\012)3131 884 y Fm(\026)p Fl(+1)0 996 y Fq(and)822 1121 y Fn(\()p Fo(a)19 b Fn(+)f Fo(b)p Fn(\)[)p Fo(x)p Fn(])23 b(=)g Fo(a)18 b Fn(+)g Fo(b)p Fn([)p Fo(x)p Fn(])23 b Fo(<)1634 1133 y Fm(k)1698 1121 y Fo(a)18 b Fn(+)g Fo(b)p Fn([)p Fo(y)s Fn(])23 b(=)f(\()p Fo(a)d Fn(+)f Fo(b)p Fn(\)[)p Fo(y)s Fn(])p Fo(:)0 1303 y Fq(ii.)40 b(By)28 b(induction)f(on)g Fn(length\()p Fo(c)p Fn(\))p Fq(.)42 b(Case)29 b Fn(\012)1514 1315 y Fm(\026)p Fl(+1)1642 1303 y Fq(.)42 b(Ob)m(vious,)29 b(since)f Fn(\012)2408 1315 y Fm(\026)p Fl(+1)2537 1303 y Fn([)p Fo(z)t Fn(])22 b(=)h Fo(z)t Fq(.)41 b(Case)28 b Fo(D)3144 1315 y Fm(\033)3189 1303 y Fo(a)p Fq(.)0 1428 y(Then)34 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)580 1440 y Fm(\026)p Fl(+1)741 1428 y Fq(and)32 b Fo(\026)19 b Fn(+)f(1)k Fp(\024)h Fo(\033)s Fq(,)33 b(hence)290 1652 y Fn(\()p Fo(D)391 1664 y Fm(\033)435 1652 y Fo(a)p Fn(\)[)p Fo(x)p Fn(])20 b(+)e(1)k(=)h Fo(D)928 1664 y Fm(\033)972 1652 y Fo(a)p Fn([)p Fo(x)p Fn(])c(+)f(1)23 b Fp(\024)1341 1664 y Fl(1)1401 1652 y Fo(D)1470 1664 y Fm(\033)1514 1652 y Fn(\()p Fo(a)p Fn([)p Fo(x)p Fn(])c(+)f(1\))23 b Fp(\024)1947 1664 y Fl(1)2007 1652 y Fo(D)2076 1664 y Fm(\033)2121 1652 y Fo(a)p Fn([)p Fo(x)18 b Fn(+)g(1])23 b(=)g(\()p Fo(D)2613 1664 y Fm(\033)2657 1652 y Fo(a)p Fn(\)[)p Fo(x)d Fn(+)e(1])0 1876 y Fq(where)38 b(in)d(the)i(\014rst)g (inequalit)m(y)e(w)m(e)j(ha)m(v)m(e)f(used)h(Lemma)d(3.2)h(iv,)h(and)f (in)g(the)g(second)0 2001 y(one)25 b(the)f(induction)g(h)m(yp)s (othesis)h(and)g(Lemma)e(3.2)h(iii.)38 b(Case)25 b(a)f(+)g(b.)41 b(Then)26 b Fo(\034)9 b Fn(\()p Fo(b)p Fn(\))24 b(=)e(\012)3131 2013 y Fm(\026)p Fl(+1)0 2125 y Fq(and)534 2250 y Fn(\()p Fo(a)d Fn(+)f Fo(b)p Fn(\)[)p Fo(x)p Fn(])h(+)f(1)k(=)h Fo(a)18 b Fn(+)g Fo(b)p Fn([)p Fo(x)p Fn(])h(+)f(1)k Fp(\024)1632 2262 y Fl(1)1692 2250 y Fo(a)d Fn(+)f Fo(b)p Fn([)p Fo(x)g Fn(+)g(1])23 b(=)g(\()p Fo(a)18 b Fn(+)g Fo(b)p Fn(\)[)p Fo(x)h Fn(+)f(1])p Fo(:)0 2433 y Fq(iii.)42 b(By)34 b(induction)e(on)h(length)f(\(c\).)45 b(Case)34 b Fo(!)s Fq(.)44 b(Then)35 b Fo(!)s Fn([)p Fo(n)p Fn(])18 b(+)g(1)23 b(=)h Fo(n)18 b Fn(+)g(1)24 b(=)f Fo(!)s Fn([)p Fo(n)18 b Fn(+)g(1])p Fq(.)45 b(Case)0 2557 y Fo(D)69 2569 y Fm(\033)113 2557 y Fo(a)33 b Fq(with)f Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(1)p Fq(.)43 b(Then)275 2781 y Fn(\()p Fo(D)376 2793 y Fm(\033)420 2781 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])19 b(+)f(1)k(=)h(\()p Fo(D)947 2793 y Fm(\033)992 2781 y Fo(a)p Fn([0]\))18 b Fp(\001)h Fn(\()p Fo(n)f Fn(+)g(1\))g(+)g(1)23 b Fp(\024)1704 2793 y Fl(1)1764 2781 y Fn(\()p Fo(D)1865 2793 y Fm(\033)1910 2781 y Fo(a)p Fn([0]\))18 b Fp(\001)g Fn(\()p Fo(n)h Fn(+)f(2\))23 b(=)f(\()p Fo(D)2602 2793 y Fm(\033)2647 2781 y Fo(a)p Fn(\)[)p Fo(n)d Fn(+)f(1])p Fo(:)0 3005 y Fq(Case)34 b Fo(D)303 3017 y Fm(\033)347 3005 y Fo(a)e Fq(with)h Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)e Fo(!)s Fq(.)44 b(Then)240 3230 y Fn(\()p Fo(D)341 3242 y Fm(\033)386 3230 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])18 b(+)g(1)23 b(=)f(\()p Fo(D)912 3242 y Fm(\033)957 3230 y Fo(a)p Fn([)p Fo(n)p Fn(]\))d(+)f(1)k Fp(\024)1360 3242 y Fl(1)1420 3230 y Fo(D)1489 3242 y Fm(\033)1534 3230 y Fn(\()p Fo(a)p Fn([)p Fo(n)p Fn(])c(+)g(1\))23 b Fp(\024)1969 3242 y Fl(1)2029 3230 y Fo(D)2098 3242 y Fm(\033)2142 3230 y Fo(a)p Fn([)p Fo(n)c Fn(+)f(1])k(=)h(\()p Fo(D)2637 3242 y Fm(\033)2682 3230 y Fo(a)p Fn(\)[)p Fo(n)18 b Fn(+)g(1])p Fo(:)0 3454 y Fq(Case)41 b Fo(D)310 3466 y Fm(\033)354 3454 y Fo(a)f Fq(with)f Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))35 b(=)e(\012)1013 3466 y Fm(\026)p Fl(+1)1141 3454 y Fo(;)14 b(\033)37 b(<)c(\026)23 b Fn(+)f(1)p Fq(.)64 b(Then)41 b Fn(\()p Fo(D)2016 3466 y Fm(\033)2061 3454 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])34 b(=)f Fo(D)2434 3466 y Fm(\033)2478 3454 y Fo(a)p Fn([)p Fo(z)2584 3466 y Fm(n)2629 3454 y Fn(])40 b Fq(with)f Fo(z)2960 3466 y Fl(0)3030 3454 y Fn(=)33 b(\012)3188 3466 y Fm(\026)3233 3454 y Fq(,)0 3578 y Fo(z)39 3590 y Fm(n)p Fl(+1)191 3578 y Fn(=)23 b Fo(D)348 3590 y Fm(\026)392 3578 y Fo(a)p Fn([)p Fo(z)498 3590 y Fm(n)543 3578 y Fn(])p Fq(.)43 b(It)33 b(su\016ces)h(to)f(pro)m(v)m (e)1347 3802 y Fo(z)1386 3814 y Fm(n)1449 3802 y Fn(+)18 b(1)23 b Fp(\024)1662 3814 y Fl(1)1722 3802 y Fo(z)1761 3814 y Fm(n)p Fl(+1)1890 3802 y Fo(;)1241 b Fn(\(1\))0 4027 y Fq(for)32 b(then)h(w)m(e)h(obtain)501 4251 y Fn(\()p Fo(D)602 4263 y Fm(\033)647 4251 y Fo(a)p Fn([)p Fo(z)753 4263 y Fm(n)797 4251 y Fn(]\))19 b(+)f(1)23 b Fp(\024)1084 4263 y Fl(1)1144 4251 y Fo(D)1213 4263 y Fm(\033)1257 4251 y Fn(\()p Fo(a)p Fn([)p Fo(z)1395 4263 y Fm(n)1440 4251 y Fn(])c(+)f(1\))23 b Fp(\024)1727 4263 y Fl(1)1786 4251 y Fo(D)1855 4263 y Fm(\033)1900 4251 y Fo(a)p Fn([)p Fo(z)2006 4263 y Fm(n)2069 4251 y Fn(+)18 b(1])23 b Fp(\024)2305 4263 y Fl(1)2365 4251 y Fo(D)2434 4263 y Fm(\033)2478 4251 y Fo(a)p Fn([)p Fo(z)2584 4263 y Fm(n)p Fl(+1)2713 4251 y Fn(])p Fo(;)0 4475 y Fq(using)39 b(ii,)f(\(1\))h(and)g(i.)61 b(W)-8 b(e)40 b(pro)m(v)m(e)g(\(1\))e(b)m(y)i(induction)e(on)h Fo(n)p Fq(.)62 b(The)40 b(base)f(case)h(follo)m(ws)0 4599 y(from)672 4724 y Fn(\012)732 4736 y Fm(\026)795 4724 y Fn(+)18 b(1)23 b Fp(\024)1008 4736 y Fl(1)1068 4724 y Fn(\()p Fo(D)1169 4736 y Fm(\026)1213 4724 y Fn(0\))18 b Fp(\001)h Fn(2)k(=)f(\()p Fo(D)1600 4736 y Fm(\026)1645 4724 y Fn(1\)[1])g Fo(<)1894 4736 y Fl(1)1954 4724 y Fo(D)2023 4736 y Fm(\026)2067 4724 y Fn(1)h Fp(\024)2197 4736 y Fl(1)2257 4724 y Fo(D)2326 4736 y Fm(\026)2370 4724 y Fo(a)p Fn([\012)2497 4736 y Fm(\026)2541 4724 y Fn(])p Fo(;)0 4907 y Fq(and)33 b(the)g(induction)e(step)j(follo)m(ws) d(from)501 5131 y Fn(\()p Fo(D)602 5143 y Fm(\026)647 5131 y Fo(a)p Fn([)p Fo(z)753 5143 y Fm(n)798 5131 y Fn(]\))18 b(+)g(1)23 b Fp(\024)1084 5143 y Fl(1)1144 5131 y Fo(D)1213 5143 y Fm(\026)1257 5131 y Fn(\()p Fo(a)p Fn([)p Fo(z)1395 5143 y Fm(n)1440 5131 y Fn(])c(+)f(1\))23 b Fp(\024)1727 5143 y Fl(1)1786 5131 y Fo(D)1855 5143 y Fm(\026)1900 5131 y Fo(a)p Fn([)p Fo(z)2006 5143 y Fm(n)2069 5131 y Fn(+)18 b(1])23 b Fp(\024)2305 5143 y Fl(1)2364 5131 y Fo(D)2433 5143 y Fm(\026)2478 5131 y Fo(a)p Fn([)p Fo(z)2584 5143 y Fm(n)p Fl(+1)2712 5131 y Fn(])p Fo(;)0 5355 y Fq(where)46 b(w)m(e)h(ha)m(v)m(e)f(used)g(ii,)h (the)e(induction)f(h)m(yp)s(othesis)j(and)e(i.)80 b(Case)46 b(a)f(+)g(b)g(with)0 5479 y Fo(\034)9 b Fn(\()p Fo(b)p Fn(\))24 b(=)f Fo(!)s Fq(.)43 b(Then)481 5704 y Fn(\()p Fo(a)19 b Fn(+)f Fo(b)p Fn(\)[)p Fo(n)p Fn(])g(+)g(1)23 b(=)f Fo(a)d Fn(+)f Fo(b)p Fn([)p Fo(n)p Fn(])g(+)g(1)k Fp(\024)1584 5716 y Fl(1)1644 5704 y Fo(a)d Fn(+)f Fo(b)p Fn([)p Fo(n)g Fn(+)g(1])k(=)h(\()p Fo(a)c Fn(+)f Fo(b)p Fn(\)[)p Fo(n)g Fn(+)g(1])p Fo(:)p 2683 5704 4 75 v 2687 5632 90 4 v 2687 5704 V 2775 5704 4 75 v 0 5953 a Fq(No)m(w)38 b(w)m(e)h(can)f(pro)m(v)m(e)h(the)f(monotonicit)m(y)e(prop)s(erties)i (of)f(the)h(functions)g Fo(G)2841 5965 y Fm(a)2919 5953 y Fq(w)m(e)g(w)m(ere)0 6077 y(lo)s(oking)30 b(for.)p eop %%Page: 1 10 1 9 bop 0 125 a Fk(Lemma)37 b(3.4.)57 b(\(Monotonicit)m(y)35 b(Prop)s(erties)h(of)h(the)h Fo(G)2298 137 y Fm(a)2338 125 y Fk(\))-103 249 y Fc(i.)48 b(If)33 b Fo(b)22 b(<)221 261 y Fm(k)285 249 y Fo(a)32 b Fc(and)h Fo(k)26 b Fp(\024)c Fo(n)p Fc(,)33 b(then)g Fo(G)1104 261 y Fm(b)1138 249 y Fn(\()p Fo(n)p Fn(\))23 b Fp(\024)g Fo(G)1428 261 y Fm(a)1468 249 y Fn(\()p Fo(n)p Fn(\))-130 374 y Fc(ii.)47 b Fo(G)65 386 y Fm(a)105 374 y Fn(\()p Fo(n)p Fn(\))24 b Fp(\024)f Fo(G)396 386 y Fm(a)436 374 y Fn(\()p Fo(n)c Fn(+)f(1\))p Fc(.)0 548 y Fq(Pro)s(of.)43 b(i.)f(By)33 b(trans\014nite)g(induction)f(on)g Fo(a)23 b Fp(2)g Fn(T)1796 560 y Fl(1)1834 548 y Fq(.)43 b(Case)34 b Fo(a)2182 518 y Fi(\000)2238 548 y Fq(.)43 b(If)33 b Fo(\034)9 b Fn(\()p Fo(a)p 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Fn([)p Fo(j)5 b Fn(])39 b Fq(with)f Fn(1)32 b Fp(\024)f Fo(j)37 b Fp(\024)32 b Fo(k)s Fq(.)62 b(Then)40 b(again)e(b)m(y)i (Lemma)d(3.3)i(iii)d(w)m(e)k(ha)m(v)m(e)g Fo(a)p Fn([)p Fo(j)5 b Fn(])32 b Fp(\024)3024 1581 y Fl(1)3093 1569 y Fo(a)p Fn([)p Fo(n)p Fn(])p Fq(,)0 1694 y(hence)i(b)m(y)f(induction)f (h)m(yp)s(othesis)1104 1918 y Fo(G)1169 1933 y Fm(a)p Fl([)p Fm(j)s Fl(])1278 1918 y Fn(\()p Fo(n)p Fn(\))24 b Fp(\024)e Fo(G)1568 1933 y Fm(a)p Fl([)p Fm(n)p Fl(])1687 1918 y Fn(\()p Fo(n)p Fn(\))i(=)f Fo(G)1978 1930 y Fm(a)2018 1918 y Fn(\()p Fo(n)p Fn(\))p Fo(:)0 2142 y Fq(ii.)41 b(By)31 b(trans\014nite)g(induction)f(on)h Fo(a)23 b Fp(2)g Fn(T)1511 2154 y Fl(1)1548 2142 y Fq(.)43 b(Case)32 b(0.)43 b Fo(G)2034 2154 y Fl(0)2071 2142 y Fn(\()p Fo(n)p Fn(\))24 b(=)e(0)h(=)g Fo(G)2514 2154 y Fl(0)2551 2142 y Fn(\()p Fo(n)16 b Fn(+)f(1\))p Fq(.)43 b(Case)31 b(a)g(+)0 2267 y(1.)643 2391 y Fo(G)708 2403 y Fm(a)p Fl(+1)832 2391 y Fn(\()p Fo(n)p Fn(\))24 b(=)e Fo(G)1122 2403 y Fm(a)1163 2391 y 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3271 y Fq(whic)m(h)h(enables)f(us)h(to)f(shift)g(a)0 3396 y(dep)s(ence)26 b(on)d Fo(n)h Fq(from)f(the)h(index)g(in)m(to)f (the)i(argumen)m(t.)40 b(This)24 b(will)e(b)s(e)i(used)h(in)e(Section)h (4.)0 3570 y Fk(Lemma)43 b(3.5.)57 b Fc(Let)38 b Fo(a)30 b Fn(=)g Fo(D)1052 3582 y Fl(0)1089 3570 y Fn(\()p Fo(c)22 b Fp(\001)f Fn(\()p Fo(n)h Fn(+)f(1\)\))38 b Fc(with)f Fo(c)31 b Fn(=)f Fo(D)2017 3540 y Fm(m)2015 3590 y(\033)2080 3570 y Fn(\(\012)2172 3582 y Fm(\033)2238 3570 y Fp(\001)22 b Fo(m)p Fn(\))38 b Fc(and)g Fn(1)30 b Fp(\024)g Fo(m)h Fp(\024)f Fo(n)p Fc(,)39 b(and)0 3694 y(furthermore)32 b Fo(d)24 b Fn(=)e Fo(D)762 3706 y Fl(0)799 3694 y Fo(D)870 3664 y Fm(m)p Fl(+2)868 3715 y Fm(\033)1017 3694 y Fn(0)p Fc(.)43 b(Then)34 b(w)m(e)g(ha)m(v)m(e)f Fo(G)1817 3706 y Fm(a)1858 3694 y Fn(\(1\))23 b Fp(\024)g Fo(G)2140 3706 y Fm(d)2179 3694 y Fn(\()p Fo(n)p Fn(\))p Fc(.)0 3869 y Fq(Pro)s(of.)43 b(First)31 b(note)i(that)897 4047 y Fo(c)18 b Fp(\001)g Fn(\()p Fo(n)h Fn(+)f(1\))23 b(=)g(\()p 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Fo(')23 b Fp(!)g Fo( )s Fq(.)41 b(Similarly)-8 b(,)24 b(if)h Fo(r)j Fq(has)f(t)m(yp)s(e)g Fo(\033)s Fq(,)h(then)e Fo(\025x)3013 3299 y Fm(\045)3053 3329 y Fo(r)i Fq(has)0 3454 y(t)m(yp)s(e)j Fo(\045)23 b Fp(!)g Fo(\033)s Fq(.)43 b(If)29 b Fo(r)k Fq(deriv)m(es)e Fo(')p Fn(\()p Fo(x)p Fn(\))g Fq(and)e(if)g(no)h(assumption)f(v)-5 b(ariable)28 b(free)i(in)f Fo(r)j Fq(assumes)f(a)0 3578 y(form)m(ula)e(with)i Fo(x)g Fq(among)f(its)h(free)g(v)-5 b(ariables)30 b(\(this)h(is)f(kno)m(wn)j(as)e Fj(variable)i(c)-5 b(ondition)p Fq(\),)0 3703 y(then)33 b Fo(\025xr)j Fq(deriv)m(es)e Fp(8)p Fo(x')p Fn(\()p Fo(x)p Fn(\))p Fq(.)-125 3852 y(5.)49 b(If)35 b Fo(r)i Fq(deriv)m(es)f Fo(')27 b Fp(!)f Fo( )38 b Fq(and)d Fo(s)f Fq(deriv)m(es)i Fo(')p Fq(,)g(then)g Fo(r)r(s)f Fq(deriv)m(es)h Fo( )s Fq(.)50 b(Similarly)-8 b(,)31 b(if)j Fo(r)k Fq(has)d(t)m(yp)s(e)0 3977 y Fo(\045)23 b Fp(!)g Fo(\033)35 b Fq(and)e Fo(s)e Fq(has)i(t)m(yp)s(e)g Fo(\045)p Fq(,)f(then)g Fo(r)r(s)h Fq(has)f(t)m(yp)s(e)h Fo(\033)s Fq(.)44 b(If)32 b Fo(r)j Fq(deriv)m(es)e Fp(8)p Fo(x')p Fn(\()p Fo(x)p Fn(\))g Fq(and)f Fo(s)f Fq(has)i(t)m(yp)s(e)0 4101 y Fn(nat)p Fq(,)g(then)g Fo(r)r(s)g Fq(deriv)m(es)h Fo(')p Fn(\()p Fo(s)p Fn(\))p Fq(.)-125 4251 y(6.)49 b(An)m(y)32 b(term/deriv)-5 b(ation)29 b(has)j(a)e(uniquely)i (determined)f(long)f(normal)f(form,)h(where)i(for)0 4375 y Fo(R)42 b Fq(w)m(e)g(ha)m(v)m(e)g(the)f(usual)g(con)m(v)m(ersion)h (rules)f Fo(R)q(r)r(s)p Fn(0)35 b Fp(7!)g Fo(r)44 b Fq(and)d Fo(R)q(r)r(s)p Fn(\()p Fo(S)5 b(t)p Fn(\))35 b Fp(7!)g Fo(st)p Fn(\()p Fo(R)q(r)r(st)p Fn(\))p Fq(.)70 b(F)-8 b(or)0 4500 y(example,)38 b(if)f Fo(F)49 b Fq(is)37 b(of)g(t)m(yp)s(e)i Fn(\(nat)30 b Fp(!)g Fn(nat\))g Fp(!)g Fn(\(nat)h Fp(!)f Fn(nat\))38 b Fq(and)f Fo(g)j Fq(is)d(of)g(t)m(yp)s(e)i Fn(nat)30 b Fp(!)g Fn(nat)p Fq(,)0 4624 y(then)j(the)g(long)f(normal)e (form)h(of)162 4848 y Fo(g)101 b Fq(is)c Fo(\025x:g)s(x)140 4973 y(F)110 b Fq(is)97 b Fo(\025z)t(x:F)12 b Fn(\()p Fo(\025y)s(:z)t(y)s Fn(\))p Fo(x)98 5098 y(F)g(g)100 b Fq(is)d Fo(\025x:F)12 b Fn(\()p Fo(\025y)s(:g)s(y)s Fn(\))p Fo(x)p eop %%Page: 1 14 1 13 bop 0 125 a Fq(W)-8 b(e)33 b(iden)m(tify)f(terms/deriv)-5 b(ations)31 b(with)h(the)h(same)g(long)e(normal)g(form.)0 374 y(F)-8 b(or)32 b(an)m(y)h(term/deriv)-5 b(ation)30 b Fo(r)36 b Fq(let)31 b Fn(lev)q(\()p Fo(r)r Fn(\))j Fq(denote)g(the)f(lev)m(el)f(of)g(its)g(t)m(yp)s(e/form)m(ula.)0 623 y(F)-8 b(or)39 b(terms/deriv)-5 b(ations)38 b Fo(r)k Fq(with)d Fn(lev)q(\()p Fo(r)r Fn(\))34 b(=)f(0)p Fq(,)41 b(tree)f(notations)e Fo(a)33 b Fp(2)h Fn(T)39 b Fq(\(see)i(Section)e (2,)0 747 y(tak)m(en)i(with)e Fo(\027)f Fn(=)33 b(2)p Fq(\))39 b(and)h Fo(m)33 b Fp(2)h Fg(N)39 b Fq(w)m(e)i(de\014ne)g (inductiv)m(ely)e(the)h(relation)e Fp(`)2843 717 y Fm(a)2843 768 y(m)2925 747 y Fo(r)r Fq(,)k(to)e(b)s(e)0 872 y(read)33 b Fo(r)i Fq(is)d(SDH{)p Fj(gener)-5 b(ate)g(d)42 b Fq(with)32 b(size)h Fo(a)f Fq(and)h(rank)g Fo(m)p Fq(,)f(b)m(y)i(the)f(follo)m (wing)d(rules.)-90 1046 y Fp(\017)48 b Fj(V)-7 b(ariable)34 b(R)n(ule.)43 b Fq(If)32 b Fp(`)797 1016 y Fm(a)797 1067 y(m)869 1046 y Fo(t)899 1058 y Fm(i)934 1046 y Fo(~)-49 b(y)968 1058 y Fm(i)1028 1046 y Fq(for)32 b Fo(i)23 b Fn(=)f(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(n)33 b Fq(with)f Fo(n)23 b Fp(\025)f Fn(0)p Fq(,)32 b(then)i Fp(`)2382 1016 y Fm(a)p Fl(+1)2382 1067 y Fm(m)2515 1046 y Fo(xt)2592 1058 y Fl(1)2643 1046 y Fo(:)14 b(:)g(:)g(t)2784 1058 y Fm(n)2829 1046 y Fq(.)-90 1196 y Fp(\017)48 b Fj(Closur)-5 b(e)34 b(R)n(ule)h(0.)43 b Fp(`)752 1165 y Fl(1)752 1216 y Fm(m)824 1196 y Fn(0)p Fq(.)-90 1345 y Fp(\017)48 b Fj(Closur)-5 b(e)34 b(R)n(ule)h Fo(S)5 b Fj(.)43 b Fq(If)32 b Fp(`)855 1315 y Fm(a)855 1365 y(m)927 1345 y Fo(r)r Fq(,)i(then)f Fp(`)1300 1315 y Fm(a)p Fl(+1)1300 1365 y Fm(m)1433 1345 y Fo(S)5 b(r)r Fq(.)-90 1494 y Fp(\017)48 b Fj(L)-5 b(emma{R)n(ule.)50 b Fq(Let)36 b Fo(L)f Fq(b)s(e)h(a)f(lemma) e(asserting)i(a)g(true)h Fn(\005)p Fq({form)m(ula)e Fo(')p Fq(.)52 b(If)35 b Fp(`)2898 1464 y Fm(a)2898 1515 y(m)2975 1494 y Fo(r)3012 1506 y Fm(i)3047 1494 y Fo(~)-49 b(y)3081 1506 y Fm(i)3143 1494 y Fq(for)0 1619 y Fo(i)23 b Fn(=)f(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(n)33 b Fq(with)f Fo(n)23 b Fp(\025)f Fn(0)p Fq(,)32 b(then)i Fp(`)1205 1589 y Fm(a)p Fl(+1)1205 1639 y Fm(m)1338 1619 y Fo(L)m(~)-39 b(r)r Fq(.)-90 1768 y Fp(\017)48 b Fo(!)55 1738 y Fl(+)110 1768 y Fj({R)n(ule.)42 b Fq(If)33 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)p Fo(;)14 b Fp(`)934 1738 y Fm(a)970 1713 y Fe(\000)934 1789 y Fm(m)1032 1768 y Fo(t)p Fq(,)33 b(and)f Fp(8)p Fo(z)26 b Fp(2)d Fn(T)1561 1780 y Fl(1)1598 1768 y Fp(8)p Fo(n:)f Fp(`)1791 1738 y Fm(z)1791 1789 y Fl(0)1839 1768 y Fo(n)g Fp(!`)2045 1725 y Fm(a)p Fl([)p Fm(z)r Fl(])2045 1778 y Fm(m)2166 1768 y Fo(R)q(r)r(sn)2353 1753 y(~)2358 1768 y(t)14 b Fq(,)33 b(then)g Fp(`)2735 1738 y Fm(a)2735 1789 y(m)2807 1768 y Fo(R)q(r)r(st)2974 1753 y(~)2979 1768 y(t)p Fq(.)-90 1918 y Fp(\017)48 b Fo(<)65 1930 y Fl(1)102 1918 y Fj({R)n(ule.)43 b Fq(If)32 b Fp(`)564 1888 y Fm(b)564 1938 y(m)636 1918 y Fo(r)j Fq(and)e Fo(b)23 b(<)1022 1930 y Fl(1)1082 1918 y Fo(a)p Fq(,)32 b(then)h Fp(`)1458 1888 y Fm(a)1458 1938 y(m)1530 1918 y Fo(r)r Fq(.)-90 2067 y Fp(\017)48 b Fj(Cut)38 b(R)n(ule.)53 b Fq(If)35 b Fp(`)617 2037 y Fm(a)617 2088 y(m)694 2067 y Fo(r)r(~)-42 b(y)39 b Fq(with)c Fn(lev)q(\()p Fo(r)r Fn(\))29 b Fp(\024)e Fo(m)36 b Fq(and)g Fp(`)1720 2037 y Fm(a)1720 2088 y(m)1796 2067 y Fo(t)1826 2079 y Fm(i)1861 2067 y Fo(~)-49 b(y)1895 2079 y Fm(i)1958 2067 y Fq(for)35 b Fo(i)27 b Fn(=)h(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(n)35 b Fq(with)h Fo(n)27 b Fp(\025)g Fn(1)p Fq(,)36 b(then)0 2192 y Fp(`)51 2162 y Fm(a)p Fl(+1)51 2212 y Fm(m)184 2192 y Fo(r)r(t)253 2204 y Fl(1)305 2192 y Fo(:)14 b(:)g(:)g(t)446 2204 y Fm(n)491 2192 y Fq(.)0 2441 y(More)38 b(precisely)-8 b(,)39 b(w)m(e)g(\014rst)f(inductiv)m(ely)f(de\014ne)i Fp(`)1880 2411 y Fm(a)1880 2461 y(m)1959 2441 y Fo(r)i Fq(for)c Fo(a)30 b Fp(2)h Fn(T)2411 2453 y Fl(1)2485 2441 y Fq(b)m(y)39 b(the)f(rules)g(giv)m(en)0 2565 y(excluding)k(the)h Fo(!)679 2535 y Fl(+)734 2565 y Fq({Rule,)h(and)f(based)h(on)e(this)g (relation)f(w)m(e)j(then)f(de\014ne)h Fp(`)2975 2535 y Fm(a)2975 2586 y(m)3061 2565 y Fo(r)h Fq(for)0 2690 y Fo(a)23 b Fp(2)g Fn(T)33 b Fq(b)m(y)g(all)e(the)i(rules)f(giv)m(en.)0 2864 y Fk(V)-9 b(ariable)36 b(Lemma)h(4.1.)57 b Fc(If)32 b Fo(c)23 b Fp(6)p Fn(=)g(0)32 b Fc(and)g Fn(lev)q(\()p Fo(x)p Fn(\))24 b Fo(<)f(k)s Fc(,)33 b(then)g Fp(`)2285 2834 y Fm(c)p Fi(\001)p Fm(k)2285 2885 y(m)2384 2864 y Fo(x)o(~)-41 b(y)s Fc(.)0 3039 y Fq(Pro)s(of.)75 b(By)44 b(induction)e(on)i Fn(lev)q(\()p Fo(x)p Fn(\))p Fq(.)77 b(By)43 b(induction)g(h)m(yp)s(othesis)h Fp(`)2562 2995 y Fm(c)p Fi(\001)p Fl(\()p Fm(k)q Fi(\000)p Fl(1\))2562 3048 y Fm(m)2814 3039 y Fo(y)2855 3051 y Fm(i)2888 3039 y Fo(~)-48 b(z)2921 3051 y Fm(i)2948 3039 y Fq(,)46 b(hence)0 3163 y Fp(`)51 3120 y Fm(c)p Fi(\001)p Fl(\()p Fm(k)q Fi(\000)p Fl(1\)+1)51 3173 y Fm(m)371 3163 y Fo(x)o(~)-41 b(y)36 b Fq(b)m(y)d(the)g(V)-8 b(ariable)31 b(Rule,)h(hence)i Fp(`)1761 3133 y Fm(c)p Fi(\001)p Fm(k)1761 3184 y(m)1860 3163 y Fo(x)o(~)-41 b(y)36 b Fq(b)m(y)d(the)g Fo(<)2352 3175 y Fl(1)2389 3163 y Fq({Rule.)p 2705 3163 4 75 v 2709 3092 90 4 v 2709 3163 V 2798 3163 4 75 v 0 3337 a Fk(Substitution)26 b(Lemma)h(4.2.)57 b Fc(If)23 b Fp(`)1389 3307 y Fm(a)1389 3358 y(m)1461 3337 y Fo(r)k Fc(and)d Fp(`)1757 3307 y Fm(b)1757 3358 y(m)1829 3337 y Fo(s)1868 3349 y Fm(j)1914 3337 y Fo(~)-53 b(y)1944 3349 y Fm(j)2003 3337 y Fc(with)23 b Fn(lev)q(\()p Fo(s)2392 3349 y Fm(j)2427 3337 y Fn(\))h Fp(\024)e Fo(m)i Fc(for)f Fo(j)28 b Fn(=)23 b(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(n)p Fc(,)0 3462 y(then)33 b Fp(`)273 3432 y Fm(b)p Fl(+)p Fm(a)273 3483 y(m)402 3462 y Fo(r)438 3475 y Fm(~)-33 b(x)482 3462 y Fn([)m Fo(~)-39 b(s)p Fn(])p Fc(.)0 3636 y Fq(Pro)s(of.)40 b(By)24 b(induction)e(on)h Fp(`)1047 3606 y Fm(a)1047 3657 y(m)1119 3636 y Fo(r)r Fq(.)41 b(W)-8 b(e)24 b(write)f Fo(t)1655 3606 y Fi(\003)1716 3636 y Fq(for)g Fo(t)1885 3649 y Fm(~)-33 b(x)1928 3636 y Fn([)m Fo(~)-39 b(s)p Fn(])p Fq(.)40 b Fj(V)-7 b(ariable)26 b(R)n(ule)p Fq(.)40 b(By)24 b(induction)0 3761 y(h)m(yp)s(othesis)40 b Fp(`)538 3731 y Fm(b)p Fl(+)p Fm(a)538 3781 y(m)677 3761 y Fo(t)707 3731 y Fi(\003)707 3783 y Fm(i)752 3761 y Fo(~)-49 b(y)786 3773 y Fm(i)814 3761 y Fq(,)41 b(hence)f Fp(`)1210 3731 y Fm(b)p Fl(+)p Fm(a)p Fl(+1)1210 3781 y Fm(m)1433 3761 y Fo(xt)1510 3731 y Fi(\003)1510 3782 y Fl(1)1562 3761 y Fo(:)14 b(:)g(:)g(t)1703 3731 y Fi(\003)1703 3781 y Fm(n)1788 3761 y Fq(b)m(y)40 b(the)f(V)-8 b(ariable)38 b(Rule.)63 b(No)m(w)40 b(if)e Fo(x)h Fq(is)0 3885 y(one)j(of)f(the)h(v) -5 b(ariables)40 b Fo(x)945 3897 y Fm(j)1022 3885 y Fq(to)i(b)s(e)f (substituted)i(b)m(y)f Fo(s)1995 3897 y Fm(j)2030 3885 y Fq(,)i(w)m(e)f(m)m(ust)f(use)g(the)g(Cut)g(Rule)0 4010 y(instead)34 b(of)g(the)g(V)-8 b(ariable)32 b(Rule.)47 b(This)34 b(is)g(p)s(ossible)f(since)i Fn(lev)q(\()p Fo(s)2386 4022 y Fm(j)2421 4010 y Fn(\))25 b Fp(\024)g Fo(m)34 b Fq(b)m(y)h(h)m(yp)s(othesis)0 4134 y(and)j Fp(`)246 4104 y Fm(b)p Fl(+)p Fm(a)246 4155 y(m)384 4134 y Fo(s)423 4146 y Fm(j)468 4134 y Fo(~)-53 b(y)498 4146 y Fm(j)571 4134 y Fq(b)m(y)39 b(h)m(yp)s(othesis)h(and)e(the)h Fo(<)1633 4146 y Fl(1)1669 4134 y Fq({Rule.)60 b(Then)39 b(\(if)e Fo(n)31 b(>)g Fn(0)p Fq(\))37 b(the)i(Cut)f(Rule)0 4259 y(yields)44 b Fp(`)337 4229 y Fm(b)p Fl(+)p Fm(a)p Fl(+1)337 4280 y Fm(m)566 4259 y Fo(s)605 4271 y Fm(j)640 4259 y Fo(t)670 4229 y Fi(\003)670 4280 y Fl(1)722 4259 y Fo(:)14 b(:)g(:)g(t)863 4229 y Fi(\003)863 4280 y Fm(n)908 4259 y Fq(,)47 b(as)e(required.)78 b(In)45 b(case)g Fo(n)39 b Fn(=)g(0)k Fq(there)i(are)f(no)g Fo(t)2810 4271 y Fm(i)2838 4259 y Fq('s)h(and)f(w)m(e)0 4384 y(ha)m(v)m(e)36 b(used)g(the)g(V)-8 b(ariable)33 b(Rule)h(to)h(generate)g Fp(`)1806 4353 y Fm(a)p Fl(+1)1806 4404 y Fm(m)1943 4384 y Fo(x)1990 4396 y Fm(j)2025 4384 y Fq(.)50 b(But)35 b(then)h Fp(`)2574 4353 y Fm(b)p Fl(+)p Fm(a)p Fl(+1)2574 4404 y Fm(m)2791 4384 y Fo(s)2830 4396 y Fm(j)2899 4384 y Fq(holds)f(b)m(y)0 4508 y(h)m(yp)s(othesis)e(and)f(the)g Fo(<)901 4520 y Fl(1)938 4508 y Fq({Rule.)42 b(F)-8 b(or)31 b(all)f(other)h(rules)h (the)g(claim)e(follo)m(ws)g(easily)h(from)0 4633 y(the)i(induction)f(h) m(yp)s(othesis)h(and)g(the)g(same)f(rule.)p 1917 4633 V 1921 4561 90 4 v 1921 4633 V 2010 4633 4 75 v 0 4807 a Fk(Cut)37 b(Elimination)d(Lemma)j(4.3.)56 b Fc(If)33 b Fp(`)1608 4777 y Fm(a)1608 4828 y(m)p Fl(+1)1764 4807 y Fo(r)r Fc(,)g(then)g Fp(`)2136 4777 y Fm(D)2190 4785 y Fd(1)2223 4777 y Fm(a)2136 4827 y(m)2272 4807 y Fo(r)r Fc(.)0 4981 y Fq(Pro)s(of.)86 b(By)47 b(induction)f(on)h Fp(`)1164 4951 y Fm(a)1164 5002 y(m)p Fl(+1)1341 4981 y Fo(r)r Fq(.)87 b Fj(V)-7 b(ariable)47 b(R)n(ule)p Fq(.)86 b(By)48 b(induction)e(h)m(yp)s(othesis)0 5106 y Fp(`)51 5076 y Fm(D)105 5084 y Fd(1)137 5076 y Fm(a)51 5126 y(m)197 5106 y Fo(t)227 5118 y Fm(i)262 5106 y Fo(~)-49 b(y)296 5118 y Fm(i)323 5106 y Fq(,)42 b(hence)g Fp(`)722 5063 y Fl(\()p Fm(D)802 5071 y Fd(1)834 5063 y Fm(a)p Fl(\)+1)722 5116 y Fm(m)1004 5106 y Fo(xt)1081 5118 y Fl(1)1132 5106 y Fo(:)14 b(:)g(:)g(t)1273 5118 y Fm(n)1358 5106 y Fq(b)m(y)41 b(the)f(V)-8 b(ariable)39 b(Rule,)i(hence)g Fp(`)2663 5063 y Fm(D)2717 5071 y Fd(1)2750 5063 y Fl(\()p Fm(a)p Fl(+1\))2663 5116 y Fm(m)2945 5106 y Fo(xt)3022 5118 y Fl(1)3074 5106 y Fo(:)14 b(:)g(:)g(t)3215 5118 y Fm(n)p eop %%Page: 1 15 1 14 bop 0 125 a Fq(b)m(y)41 b(the)g Fo(<)384 137 y Fl(1)420 125 y Fq({Rule.)66 b Fj(Closur)-5 b(e)41 b(R)n(ule)h(0)p Fq(.)66 b(Note)40 b(that)g Fn(1)33 b Fo(<)2096 137 y Fl(1)2167 125 y Fo(D)2236 137 y Fl(1)2273 125 y Fn(1)p Fq(.)66 b(Hence)41 b Fp(`)2756 94 y Fm(D)2810 102 y Fd(1)2842 94 y Fl(1)2756 145 y Fm(m)2900 125 y Fn(0)e Fq(b)m(y)i(the)0 249 y Fo(<)65 261 y Fl(1)102 249 y Fq({Rule.)60 b Fj(Closur)-5 b(e)40 b(R)n(ule)g Fo(S)5 b Fq(.)62 b(By)39 b(induction)e(h)m(yp)s (othesis)j Fp(`)2305 219 y Fm(D)2359 227 y Fd(1)2391 219 y Fm(a)2305 270 y(m)2449 249 y Fo(r)r Fq(,)h(hence)f Fp(`)2884 206 y Fl(\()p Fm(D)2964 214 y Fd(1)2996 206 y Fm(a)p Fl(\)+1)2884 259 y Fm(m)3164 249 y Fo(S)5 b(r)0 374 y Fq(b)m(y)39 b(the)f(Closure)g(Rule)f Fo(S)5 b Fq(,)38 b(hence)h Fp(`)1354 330 y Fm(D)1408 338 y Fd(1)1441 330 y Fl(\()p Fm(a)p Fl(+1\))1354 383 y Fm(m)1633 374 y Fo(S)5 b(r)41 b Fq(b)m(y)d(the)g Fo(<)2145 386 y Fl(1)2182 374 y Fq({Rule.)58 b Fj(L)-5 b(emma)39 b(R)n(ule)p Fq(.)58 b(By)0 498 y(induction)39 b(h)m(yp)s(othesis)j Fp(`)981 468 y Fm(D)1035 476 y Fd(1)1067 468 y Fm(a)981 519 y(m)1128 498 y Fo(r)1165 510 y Fm(i)1200 498 y Fo(~)-49 b(y)1234 510 y Fm(i)1261 498 y Fq(,)43 b(hence)f Fp(`)1661 455 y Fl(\()p Fm(D)1741 463 y Fd(1)1773 455 y Fm(a)p Fl(\)+1)1661 508 y Fm(m)1944 498 y Fo(L)m(~)-39 b(r)42 b Fq(b)m(y)g(the)e(Lemma)g (Rule,)h(hence)0 623 y Fp(`)51 580 y Fm(D)105 588 y Fd(1)137 580 y Fl(\()p Fm(a)p Fl(+1\))51 632 y Fm(m)323 623 y Fo(L)m(~)-39 b(r)36 b Fq(b)m(y)e(the)f Fo(<)822 635 y Fl(1)859 623 y Fq({Rule.)44 b Fo(!)1231 593 y Fl(+)1286 623 y Fj({R)n(ule)p Fq(.)g(Then)34 b Fp(`)1904 593 y Fm(a)1904 643 y(m)p Fl(+1)2061 623 y Fo(R)q(r)r(st)2228 607 y(~)2233 623 y(t)f Fq(has)g(b)s(een)h(inferred)f(from)0 747 y Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)p Fq(,)32 b Fp(`)435 717 y Fm(a)471 692 y Fe(\000)435 768 y Fm(m)p Fl(+1)592 747 y Fo(t)p Fq(,)g(and)1055 872 y Fp(8)p Fo(z)26 b Fp(2)d Fn(T)1305 884 y Fl(1)1342 872 y Fp(8)p Fo(n:)f Fp(`)1535 837 y Fm(z)1535 892 y Fl(0)1583 872 y Fo(n)g Fp(!`)1789 829 y Fm(a)p Fl([)p Fm(z)r Fl(])1789 894 y Fm(m)p Fl(+1)1946 872 y Fo(R)q(r)r(sn)2133 856 y(~)2138 872 y(t)14 b(:)0 1054 y Fq(By)33 b(induction)f(h)m(yp)s (othesis)h Fp(`)1118 1024 y Fm(D)1172 1032 y Fd(1)1205 1024 y Fm(a)1241 999 y Fe(\000)1118 1075 y Fm(m)1303 1054 y Fo(t)p Fq(,)g(and)1030 1279 y Fp(8)p Fo(z)25 b Fp(2)e Fn(T)1279 1291 y Fl(1)1317 1279 y Fp(8)p Fo(n:)f Fp(`)1510 1244 y Fm(z)1510 1299 y Fl(0)1557 1279 y Fo(n)h Fp(!`)1764 1244 y Fm(D)1818 1252 y Fd(1)1850 1244 y Fm(a)p Fl([)p Fm(z)r Fl(])1764 1299 y Fm(m)1971 1279 y Fo(R)q(r)r(sn)2158 1263 y(~)2163 1279 y(t)14 b(:)0 1503 y Fq(No)m(w)47 b Fo(D)305 1515 y Fl(1)342 1503 y Fo(a)386 1473 y Fi(\000)484 1503 y Fn(=)42 b(\()p Fo(D)692 1515 y Fl(1)730 1503 y Fo(a)p Fn(\))806 1473 y Fi(\000)862 1503 y Fq(,)50 b(and)c(since)h Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))43 b(=)f(\012)k Fq(w)m(e)i(ha)m(v)m(e) f Fo(D)2269 1515 y Fl(1)2306 1503 y Fo(a)p Fn([)p Fo(z)t Fn(])42 b(=)g(\()p Fo(D)2689 1515 y Fl(1)2726 1503 y Fo(a)p Fn(\)[)p Fo(z)t Fn(])p Fq(.)84 b(Hence)0 1627 y Fp(`)51 1597 y Fm(D)105 1605 y Fd(1)137 1597 y Fm(a)51 1648 y(m)196 1627 y Fo(R)q(r)r(st)363 1612 y(~)368 1627 y(t)39 b Fq(b)m(y)i(the)e Fo(!)809 1597 y Fl(+)864 1627 y Fq({Rule.)62 b Fo(<)1264 1639 y Fl(1)1301 1627 y Fj({R)n(ule)p Fq(.)h(By)39 b(induction)f(h)m(yp)s(othesis)j Fp(`)2770 1597 y Fm(D)2824 1605 y Fd(1)2856 1597 y Fm(b)2770 1648 y(m)2908 1627 y Fo(r)r Fq(.)64 b(Since)0 1752 y(from)45 b Fo(b)c(<)386 1764 y Fl(1)465 1752 y Fo(a)k Fq(w)m(e)i(can)f(infer)f Fo(D)1210 1764 y Fl(1)1247 1752 y Fo(b)d(<)1390 1764 y Fl(1)1468 1752 y Fo(D)1537 1764 y Fl(1)1574 1752 y Fo(a)p Fq(,)50 b(w)m(e)c(get)g Fp(`)2078 1722 y Fm(D)2132 1730 y Fd(1)2165 1722 y Fm(a)2078 1772 y(m)2233 1752 y Fo(r)i Fq(b)m(y)f(the)f Fo(<)2713 1764 y Fl(1)2750 1752 y Fq({Rule.)82 b Fj(Cut)0 1876 y(R)n(ule)p Fq(.)57 b(By)38 b(induction)e(h)m(yp)s(othesis)j Fp(`)1408 1846 y Fm(D)1462 1854 y Fd(1)1494 1846 y Fm(a)1408 1897 y(m)1550 1876 y Fo(r)r(~)-42 b(y)41 b Fq(and)c Fp(`)1916 1846 y Fm(D)1970 1854 y Fd(1)2003 1846 y Fm(a)1916 1897 y(m)2059 1876 y Fo(t)2089 1888 y Fm(i)2123 1876 y Fo(~)-49 b(y)2157 1888 y Fm(i)2185 1876 y Fq(.)57 b(Since)38 b Fn(lev)q(\()p Fo(r)r Fn(\))31 b Fp(\024)e Fo(m)21 b Fn(+)g(1)p Fq(,)38 b(w)m(e)0 2001 y(ha)m(v)m(e)d Fn(lev)q(\()p Fo(t)393 2013 y Fm(i)421 2001 y Fn(\))26 b Fp(\024)e Fo(m)34 b Fq(and)g(hence)i Fp(`)1190 1971 y Fm(D)1244 1979 y Fd(1)1276 1971 y Fm(a)p Fl(+)p Fm(D)1417 1979 y Fd(1)1450 1971 y Fm(a)1190 2021 y(m)1501 2001 y Fo(r)r(t)1570 2013 y Fl(1)1622 2001 y Fo(:)14 b(:)g(:)g(t)1763 2013 y Fm(n)1842 2001 y Fq(b)m(y)35 b(the)f(Substitution)f(Lemma.)47 b(But)0 2125 y Fn(\()p Fo(D)101 2137 y Fl(1)138 2125 y Fn(\()p Fo(a)19 b Fn(+)f(1\)\)[1])23 b(=)f Fo(D)689 2137 y Fl(1)726 2125 y Fo(a)d Fn(+)f Fo(D)941 2137 y Fl(1)978 2125 y Fo(a)p Fq(,)32 b(so)h Fp(`)1252 2082 y Fm(D)1306 2090 y Fd(1)1338 2082 y Fl(\()p Fm(a)p Fl(+1\))1252 2135 y Fm(m)1524 2125 y Fo(r)r(t)1593 2137 y Fl(1)1644 2125 y Fo(:)14 b(:)g(:)g(t)1785 2137 y Fm(n)1863 2125 y Fq(b)m(y)33 b(the)g Fo(<)2231 2137 y Fl(1)2268 2125 y Fq({Rule.)p 2584 2125 4 75 v 2588 2054 90 4 v 2588 2125 V 2677 2125 4 75 v 0 2374 a(W)-8 b(e)49 b(no)m(w)g(w)m(an)m(t)h(to)e(pro)m(v)m(e)i (the)f(Collapsing)d(Lemma)h(men)m(tioned)h(ab)s(o)m(v)m(e.)93 b(F)-8 b(or)47 b(its)0 2499 y(form)m(ulation)22 b(w)m(e)j(need)h(the)f (notion)e(of)h(the)h Fj(\014rst)i(instanc)-5 b(e)27 b Fp(j)p Fo(r)r Fp(j)h Fj(pr)-5 b(ovide)g(d)27 b(by)g(a)h(r)-5 b(efutation)0 2623 y Fo(r)33 b Fj(of)51 b Fn(\005)p Fj({assumptions)p Fq(.)41 b(So)27 b(let)g Fo(r)j 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Fm(i)1568 3495 y Fq(among)d Fo(~)-39 b(r)50 b Fq(actually)d(deriv)m(e)i(true)f(form)m(ulas)0 3620 y(\(whic)m(h)33 b(can)f(b)s(e)h(decided,)g(since)f(the)h(form)m(ulas)e(are)h(quan)m (ti\014er{free)h(and)f(can)h(clearly)0 3744 y(b)s(e)h(assumed)g(to)f(b) s(e)h(closed\),)g(let)f Fp(j)p Fo(r)r Fp(j)i Fq(b)s(e)e(the)h(list)f (of)g(all)e Fp(j)p Fo(r)2171 3756 y Fm(j)2207 3744 y Fp(j)p Fq(,)j Fo(r)2328 3756 y Fm(j)2397 3744 y Fq(term)f(among)c Fo(~)-39 b(r)s Fq(.)46 b(Oth-)0 3869 y(erwise,)33 b(let)e Fp(j)p Fo(r)r Fp(j)i Fq(b)s(e)f Fp(j)p Fo(r)770 3881 y Fm(i)798 3869 y Fo(~)-41 b(u)o Fp(j)p Fq(,)32 b(where)i Fo(r)1247 3881 y Fm(i)1307 3869 y Fq(is)d(the)i(\014rst)f(deriv)-5 b(ation)31 b(among)d Fo(~)-39 b(r)34 b Fq(deriving)e(a)f(false)0 3993 y(quan)m(ti\014er{free)i(form)m(ula)e(and)g Fo(~)-40 b(u)32 b Fq(are)g(lemmas)f(or)h(assumptions)h(of)f(true)h(form)m(ulas.) 0 4242 y Fj(Case)45 b Fo(r)d Fp(\021)d Fo(L)m(~)-39 b(r)46 b Fq(with)e Fo(L)f Fq(a)h(lemma.)76 b(Then)45 b(some)f Fo(r)1946 4254 y Fm(i)2018 4242 y Fq(among)c Fo(~)-39 b(r)47 b Fq(m)m(ust)d(deriv)m(e)h(a)e(false)0 4367 y(quan)m (ti\014er{free)f(form)m(ula,)g(since)g Fo(r)j Fq(deriv)m(es)d(a)g (false)f(form)m(ula)e(and)j(the)g(lemma)d Fo(L)i Fq(is)0 4491 y(assumed)h(to)e(b)s(e)h(true.)69 b(Let)41 b Fp(j)p Fo(r)r Fp(j)g Fq(b)s(e)h Fp(j)p Fo(r)1448 4503 y Fm(i)1475 4491 y Fo(~)-41 b(u)o Fp(j)41 b Fq(for)f(the)i(\014rst)f(suc)m(h)h Fo(r)2395 4503 y Fm(i)2423 4491 y Fq(,)h(with)d Fo(~)-41 b(u)40 b Fq(lemmas)f(or)0 4616 y(assumptions)32 b(of)h(true)f(form)m (ulas.)0 4790 y Fk(First)42 b(Collapsing)g(Lemma)h(4.4.)56 b Fc(Supp)s(ose)38 b Fp(`)1902 4760 y Fm(a)1902 4811 y Fl(0)1959 4790 y Fo(r)i Fc(with)d Fo(r)j Fc(a)d(closed)h(term)f(of)g (t)m(yp)s(e)0 4915 y Fn(nat)c Fc(or)g(else)h(a)f(refutation)f(of)h Fn(\005)p Fc({assumptions.)45 b(Let)34 b Fp(j)p Fo(r)r Fp(j)g Fc(b)s(e)f(the)h(n)m(umerical)e(v)-5 b(alue)32 b(of)h Fo(r)0 5039 y Fc(\(in)e(case)i Fo(r)i Fc(is)c(a)h(term\),)f(or)h (the)g(maxim)m(um)e(v)-5 b(alue)31 b(in)g(the)i(\014rst)f(instance)g (pro)m(vided)h(b)m(y)0 5164 y Fo(r)i Fc(\(in)d(case)i Fo(r)h Fc(is)d(a)g(deriv)-5 b(ation\).)42 b(Then)34 b Fp(`)1522 5127 y Fm(D)1576 5135 y Fd(0)1608 5127 y Fm(a)1522 5186 y Fl(0)1658 5164 y Fp(j)p Fo(r)r Fp(j)p Fc(.)p eop %%Page: 1 16 1 15 bop 0 125 a Fq(Pro)s(of.)60 b(By)39 b(induction)f(on)g Fp(`)1113 94 y Fm(a)1113 145 y Fl(0)1171 125 y Fo(r)r Fq(.)61 b Fj(V)-7 b(ariable)40 b(R)n(ule)p Fq(.)60 b(Then)40 b Fo(r)34 b Fp(\021)d Fo(u)2443 137 y Fm(i)2468 125 y Fo(~)-40 b(r)s Fq(,)40 b(and)f Fp(`)2824 89 y Fm(a)p Fl(+1)2824 147 y(0)2965 125 y Fo(u)3013 137 y Fm(i)3038 125 y Fo(~)-40 b(r)41 b Fq(has)0 249 y(b)s(een)c(inferred)g(from)e Fp(`)884 219 y Fm(a)884 270 y Fl(0)939 249 y Fo(r)976 261 y Fm(i)1003 249 y Fo(~)-41 b(y)s Fq(.)55 b(If)37 b(all)d(deriv)-5 b(ations)36 b Fo(r)1909 261 y Fm(i)1973 249 y Fq(among)d Fo(~)-39 b(r)39 b Fq(deriv)m(e)e(true)g(form)m(ulas,)0 374 y(then)c(b)m(y)g(de\014nition)e Fp(j)p Fo(r)r Fp(j)i 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835 y Fm(D)105 843 y Fd(0)137 835 y Fm(a)51 894 y Fl(0)197 872 y Fp(j)p Fo(r)257 884 y Fm(i)284 872 y Fo(~)-41 b(u)p Fp(j)39 b Fq(holds)h(b)m(y)h(induction)d(h)m(yp)s(othesis,)43 b(hence)e Fp(`)2087 829 y Fm(D)2141 837 y Fd(0)2174 829 y Fl(\()p Fm(a)p Fl(+1\))2087 894 y(0)2369 872 y Fp(j)p Fo(r)2429 884 y Fm(i)2456 872 y Fo(~)-40 b(u)o Fp(j)40 b Fq(b)m(y)h(the)f Fo(<)2951 884 y Fl(1)2988 872 y Fq({Rule.)0 996 y Fj(Closur)-5 b(e)45 b(R)n(ule)h(0)p Fq(.)79 b(Clear,)47 b(since)e Fp(`)1360 959 y Fm(D)1414 967 y Fd(0)1447 959 y Fl(1)1360 1018 y(0)1510 996 y Fn(0)f Fq(b)m(y)i(the)f Fo(<)1989 1008 y Fl(1)2026 996 y Fq({Rule.)78 b Fj(Closur)-5 b(e)46 b(R)n(ule)f Fo(S)5 b Fq(.)79 b(By)0 1121 y(induction)36 b(h)m(yp)s(othesis)i Fp(`)974 1084 y Fm(D)1028 1092 y Fd(0)1060 1084 y Fm(a)974 1143 y Fl(0)1116 1121 y Fp(j)p Fo(r)r Fp(j)p Fq(,)h(hence)f Fp(`)1593 1084 y Fm(D)1647 1092 y Fd(0)1680 1084 y Fm(a)p Fl(+1)1593 1143 y(0)1819 1121 y Fo(S)5 b Fp(j)p Fo(r)r Fp(j)38 b Fq(b)m(y)g(the)f(Closure)g (Rule)g Fo(S)5 b Fq(,)37 b(hence)0 1245 y Fp(`)51 1202 y Fm(D)105 1210 y Fd(0)137 1202 y Fl(\()p Fm(a)p Fl(+1\))51 1267 y(0)341 1245 y Fo(S)5 b Fp(j)p Fo(r)r Fp(j)46 b Fq(b)m(y)g(the)g Fo(<)922 1257 y Fl(1)958 1245 y Fq({Rule,)i(and)e Fp(j)p Fo(S)5 b(r)r Fp(j)41 b Fn(=)g Fo(S)5 b Fp(j)p Fo(r)r Fp(j)p Fq(.)82 b Fj(L)-5 b(emma{R)n(ule.)80 b Fq(Then)47 b Fo(r)d Fp(\021)c Fo(L)m(~)-39 b(r)s Fq(,)0 1370 y(and)35 b Fp(`)243 1334 y Fm(a)p Fl(+1)243 1392 y(0)380 1370 y Fo(L)m(~)-39 b(r)37 b Fq(has)f(b)s(een)g(inferred)f (from)e Fp(`)1567 1340 y Fm(a)1567 1390 y Fl(0)1620 1370 y Fo(r)1657 1382 y Fm(i)1685 1370 y Fo(~)-42 b(y)s Fq(.)51 b(By)36 b(de\014nition)e Fp(j)p Fo(r)r Fp(j)27 b Fn(=)f Fp(j)p Fo(r)2662 1382 y Fm(i)2690 1370 y Fo(~)-41 b(u)o Fp(j)35 b Fq(for)g(some)g Fo(r)3232 1382 y Fm(i)0 1494 y Fq(among)20 b Fo(~)-40 b(r)26 b Fq(deriving)d(a)g(false)g(quan)m (ti\014er{free)h(form)m(ula)d(with)h Fo(~)-41 b(u)23 b Fq(lemmas)f(or)h(assumptions)0 1619 y(of)33 b(true)h(form)m(ulas.)45 b(By)34 b(induction)e(h)m(yp)s(othesis)p Fp(`)1842 1582 y Fm(D)1896 1590 y Fd(0)1930 1582 y Fm(a)1842 1641 y Fl(0)1980 1619 y Fp(j)p Fo(r)2040 1631 y Fm(i)2067 1619 y Fo(~)-40 b(u)o Fp(j)p Fq(,)34 b(hence)h Fp(`)2523 1576 y Fm(D)2577 1584 y Fd(0)2609 1576 y Fl(\()p Fm(a)p Fl(+1\))2523 1641 y(0)2796 1619 y Fp(j)p Fo(r)2856 1631 y Fm(i)2883 1619 y Fo(~)-41 b(u)o Fp(j)34 b Fq(b)m(y)g(the)0 1743 y Fo(<)65 1755 y Fl(1)102 1743 y Fq({Rule.)42 b Fo(!)472 1713 y Fl(+)527 1743 y Fj({R)n(ule)p Fq(.)h(Then)34 b Fp(`)1144 1713 y Fm(a)1144 1764 y Fl(0)1193 1743 y Fo(R)q(r)1294 1755 y Fl(0)1331 1743 y Fo(st)1395 1728 y(~)1400 1743 y(t)f Fq(has)g(b)s(een)g(inferred)f(from)g Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)p Fq(,)32 b Fp(`)2893 1713 y Fm(a)2929 1688 y Fe(\000)2893 1764 y Fl(0)2991 1743 y Fo(t)p Fq(,)h(and)1055 1968 y Fp(8)p Fo(z)26 b Fp(2)d Fn(T)1305 1980 y Fl(1)1342 1968 y Fp(8)p Fo(n:)g Fp(`)1536 1933 y Fm(z)1536 1988 y Fl(0)1583 1968 y Fo(n)g Fp(!`)1790 1924 y Fm(a)p Fl([)p Fm(z)r Fl(])1790 1990 y(0)1910 1968 y Fo(R)q(r)2011 1980 y Fl(0)2049 1968 y Fo(sn)2133 1952 y(~)2138 1968 y(t)13 b(:)0 2192 y Fq(W)-8 b(e)23 b(ha)m(v)m(e)i(to)d(sho)m(w)i Fp(`)766 2155 y Fm(D)820 2163 y Fd(0)853 2155 y Fm(a)766 2214 y Fl(0)902 2192 y Fp(j)p Fo(R)q(r)1026 2204 y Fl(0)1063 2192 y Fo(sk)1142 2176 y(~)1148 2192 y(t)14 b Fp(j)23 b Fq(with)f Fo(k)k Fn(:=)d Fp(j)p Fo(t)p Fp(j)p Fq(.)40 b(F)-8 b(rom)22 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))24 b(=)f(\012)f Fq(w)m(e)i(get)f Fo(\034)9 b Fn(\()p Fo(D)2800 2204 y Fl(0)2838 2192 y Fo(a)p Fn(\))24 b(=)e Fo(!)k Fq(and)0 2316 y Fn(\()p Fo(D)101 2328 y Fl(0)138 2316 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])i(=)f Fo(D)499 2328 y Fl(0)536 2316 y Fo(a)p Fn([)p Fo(z)642 2328 y Fm(n)686 2316 y Fn(])36 b Fq(with)f Fo(z)1009 2328 y Fl(0)1073 2316 y Fn(=)27 b Fo(!)s Fq(,)36 b Fo(z)1322 2328 y Fm(n)p Fl(+1)1478 2316 y Fn(=)27 b Fo(D)1639 2328 y Fl(0)1676 2316 y Fo(a)p Fn([)p Fo(z)1782 2328 y Fm(n)1827 2316 y Fn(])p Fq(,)36 b(hence)h Fo(z)2226 2328 y Fl(1)2291 2316 y Fn(=)26 b Fo(D)2451 2328 y Fl(0)2488 2316 y Fo(a)p Fn([)p Fo(!)s Fn(])h(=)g Fo(D)2821 2328 y Fl(0)2858 2316 y Fo(a)2902 2286 y Fi(\000)2958 2316 y Fq(.)53 b(Since)0 2441 y Fo(D)69 2453 y Fl(0)106 2441 y Fo(a)150 2411 y Fi(\000)229 2441 y Fn(=)23 b(\()p Fo(D)418 2453 y Fl(0)455 2441 y Fo(a)p Fn(\))531 2411 y Fi(\000)587 2441 y Fq(,)29 b(the)g(induction)e(h)m(yp)s(othesis)i(yields)f Fp(`)2033 2398 y Fl(\()p Fm(D)2113 2406 y Fd(0)2145 2398 y Fm(a)p Fl(\))2207 2373 y Fe(\000)2033 2451 y Fm(a)2270 2441 y Fo(k)s Fq(.)42 b(Since)28 b Fn(\()p Fo(D)2736 2453 y Fl(0)2773 2441 y Fo(a)p Fn(\))2849 2411 y Fi(\000)2928 2441 y Fp(2)c Fo(T)3056 2453 y Fl(1)3093 2441 y Fq(,)29 b(w)m(e)0 2565 y(get)34 b Fp(`)215 2522 y Fm(a)p Fl([\()p Fm(D)350 2530 y Fd(0)382 2522 y Fm(a)p Fl(\))444 2497 y Fe(\000)493 2522 y Fl(])215 2588 y(0)527 2565 y Fp(j)p Fo(R)q(r)651 2577 y Fl(0)689 2565 y Fo(sk)768 2550 y(~)774 2565 y(t)13 b Fp(j)34 b Fq(from)f(our)g(assumption,)h(so)g(again)f(the) h(induction)f(h)m(yp)s(othesis)0 2690 y(yields)41 b Fp(`)334 2647 y Fm(D)388 2655 y Fd(0)421 2647 y Fm(a)p Fl([\()p Fm(D)556 2655 y Fd(0)588 2647 y Fm(a)p Fl(\))650 2622 y Fe(\000)699 2647 y Fl(])334 2712 y(0)744 2690 y Fp(j)p Fo(R)q(r)868 2702 y Fl(0)906 2690 y Fo(sk)985 2675 y(~)991 2690 y(t)13 b Fp(j)p Fq(.)71 b(But)42 b Fo(D)1427 2702 y Fl(0)1464 2690 y Fo(a)p Fn([)p Fo(z)1570 2702 y Fl(1)1607 2690 y Fn(])36 b(=)f(\()p Fo(D)1867 2702 y Fl(0)1905 2690 y Fo(a)p Fn(\)[1])p Fq(,)44 b(so)d(the)h Fo(<)2510 2702 y Fl(1)2547 2690 y Fq({Rule)f(giv)m(es)h Fp(`)3133 2653 y Fm(D)3187 2661 y Fd(0)3220 2653 y Fm(a)3133 2712 y Fl(0)0 2814 y Fp(j)p Fo(R)q(r)124 2826 y Fl(0)161 2814 y Fo(sk)240 2799 y(~)246 2814 y(t)14 b Fp(j)p Fq(.)85 b Fo(<)490 2826 y Fl(1)527 2814 y Fj({R)n(ule)p Fq(.)g(By)47 b(induction)e(h)m(yp)s(othesis)j Fp(`)2040 2778 y Fm(D)2094 2786 y Fd(0)2126 2778 y Fm(b)2040 2837 y Fl(0)2188 2814 y Fp(j)p Fo(r)r Fp(j)p Fq(,)j(hence)d Fp(`)2687 2778 y Fm(D)2741 2786 y Fd(0)2774 2778 y Fm(a)2687 2837 y Fl(0)2843 2814 y Fp(j)p Fo(r)r Fp(j)f Fq(b)m(y)g(the)0 2939 y Fo(<)65 2951 y Fl(1)102 2939 y Fq({Rule.)p 417 2939 4 75 v 421 2868 90 4 v 421 2939 V 510 2939 4 75 v 0 3113 a Fk(Second)38 b(Collapsing)e(Lemma)h(4.5.)57 b Fc(If)32 b Fp(`)1704 3083 y Fm(a)1704 3134 y Fl(0)1753 3113 y Fo(n)h Fc(with)f Fo(a)23 b Fp(2)g Fn(T)2263 3125 y Fl(1)2300 3113 y Fc(,)33 b(then)g Fp(`)2633 3070 y Fm(G)2685 3078 y Fh(a)2721 3070 y Fl(\(1\))2633 3135 y(0)2819 3113 y Fo(n)p Fc(.)0 3288 y Fq(Pro)s(of.)45 b(By)34 b(induction)e(on)i Fp(`)1083 3258 y Fm(a)1083 3308 y Fl(0)1133 3288 y Fo(n)p Fq(.)46 b Fj(Closur)-5 b(e)35 b(R)n(ule)g(0)p Fq(.)46 b Fp(`)2009 3245 y Fm(G)2061 3253 y Fd(1)2092 3245 y Fl(\(1\))2009 3310 y(0)2192 3288 y Fn(0)33 b Fq(since)g Fo(G)2571 3300 y Fl(1)2609 3288 y Fn(\(1\))24 b(=)g(1)p Fq(.)45 b Fj(Closur)-5 b(e)0 3412 y(R)n(ule)44 b Fo(S)5 b Fq(.)74 b(By)43 b(induction)f(h)m(yp)s (othesis)i Fp(`)1541 3369 y Fm(G)1593 3377 y Fh(a)1629 3369 y Fl(\(1\))1541 3434 y(0)1741 3412 y Fo(n)p Fq(,)i(hence)e Fp(`)2196 3368 y Fm(G)2248 3376 y Fh(a)p Fd(+1)2354 3368 y Fl(\(1\))2196 3434 y(0)2467 3412 y Fo(S)5 b(n)43 b Fq(b)m(y)g(the)h(Closure)0 3537 y(Rule)35 b Fo(S)5 b Fq(,)37 b(since)f Fo(G)659 3549 y Fm(a)p Fl(+1)783 3537 y Fn(\(1\))28 b(=)f Fo(G)1074 3549 y Fm(a)1115 3537 y Fn(\(1\))20 b(+)g(1)p Fq(.)53 b Fo(<)1513 3549 y Fl(1)1549 3537 y Fj({R)n(ule)p Fq(.)g(By)36 b(induction)f(h)m(yp)s(othesis)i Fp(`)2998 3494 y Fm(G)3050 3503 y Fh(b)3080 3494 y Fl(\(1\))2998 3559 y(0)3183 3537 y Fo(n)p Fq(,)0 3661 y(hence)30 b Fp(`)318 3618 y Fm(G)370 3626 y Fh(a)405 3618 y Fl(\(1\))318 3683 y(0)504 3661 y Fo(n)e Fq(b)m(y)h(the)f Fo(<)941 3673 y Fl(1)978 3661 y Fq({Rule,)g(since)h(from)e Fo(b)c(<)1864 3673 y Fl(1)1924 3661 y Fo(a)28 b Fq(w)m(e)h(can)f(conclude)h Fo(G)2771 3673 y Fm(b)2804 3661 y Fn(\(1\))24 b Fp(\024)e Fo(G)3086 3673 y Fm(a)3127 3661 y Fn(\(1\))p Fq(.)p 0 3786 V 4 3714 90 4 v 4 3786 V 93 3786 4 75 v 0 3960 a Fk(V)-9 b(alue)37 b(Lemma)g(4.6.)56 b Fc(If)33 b Fp(`)1088 3930 y Fm(k)1088 3981 y Fl(0)1138 3960 y Fo(n)p Fc(,)f(then)h Fo(n)23 b(<)g(k)s Fc(.)0 4134 y Fq(Pro)s(of.)63 b(By)40 b(induction)f(on)g Fp(`)1119 4104 y Fm(k)1119 4155 y Fl(0)1178 4134 y Fo(n)p Fq(.)64 b Fj(Closur)-5 b(e)41 b(R)n(ule)g(0)p Fq(.)63 b(Clear,)41 b(since)f Fp(j)p Fn(0)p Fp(j)32 b Fn(=)h(0)p Fq(.)63 b Fj(Closur)-5 b(e)0 4259 y(R)n(ule)35 b Fo(S)5 b Fq(.)45 b(By)33 b(induction)f(h)m(yp)s (othesis)i Fo(n)24 b(<)f(k)s Fq(,)33 b(hence)i Fo(S)5 b(n)23 b(<)h(k)d Fn(+)d(1)p Fq(.)45 b Fo(<)2506 4271 y Fl(1)2543 4259 y Fj({R)n(ule)p Fq(.)f(Note)33 b(\014rst)0 4384 y(that)h(from)f Fo(b)25 b(<)571 4396 y Fl(1)633 4384 y Fo(k)37 b Fq(w)m(e)e(can)f(conclude)h Fo(b)25 b Fp(2)h Fo(T)1632 4396 y Fl(0)1702 4384 y Fq(and)35 b(hence)g(that)f Fo(b)g Fq(is)g(a)f(n)m(umeral)h Fo(l)h Fq(with)0 4508 y Fo(l)24 b(<)f(k)s Fq(.)43 b(By)33 b(induction)f(h)m (yp)s(othesis)i Fo(n)23 b(<)f(l)r Fq(,)32 b(hence)i Fo(n)23 b(<)g(k)s Fq(.)p 2115 4508 V 2119 4437 90 4 v 2119 4508 V 2208 4508 4 75 v 0 4757 a(W)-8 b(e)35 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y Fo(~)-49 b(y)1368 3175 y Fm(i)1395 3163 y Fq(.)55 b(Note)36 b(that)g(b)m(y)h(the)g Fo(<)2307 3175 y Fl(1)2344 3163 y Fq({Rule)e(w)m(e)j(can)e(assume)0 3288 y(that)d(w)m(e)h(ha)m(v)m(e)h(the)e(same)h Fo(k)i Fq(in)c(all)f(cases.)47 b(Then)34 b Fp(`)1916 3258 y Fl(\012)p Fi(\001)p Fm(k)q Fl(+1)1916 3308 y Fm(m)2118 3288 y Fo(ts)o(~)-41 b(y)36 b Fq(b)m(y)e(the)g(Cut)g(Rule,)e(hence)0 3412 y Fp(`)51 3369 y Fl(\012)p Fi(\001)p Fl(\()p Fm(k)q Fl(+1\))51 3422 y Fm(m)304 3412 y Fo(ts)o(~)-41 b(y)35 b Fq(b)m(y)e(the)g Fo(<)817 3424 y Fl(1)854 3412 y Fq({Rule.)p 1170 3412 4 75 v 1174 3341 90 4 v 1174 3412 V 1263 3412 4 75 v 0 3661 a(No)m(w)26 b(w)m(e)h(obtain)d(the)i(desired)h(estimate)d (of)h(instances)i(in)e(existen)m(tial)f(pro)s(ofs,)j(in)e(terms)0 3786 y(of)32 b(the)h(slo)m(w)g(gro)m(wing)f(hierarc)m(h)m(y)-8 b(.)0 3960 y Fk(Theorem)36 b(4.8.)57 b Fc(Let)32 b Fo(r)j Fc(b)s(e)d(a)f(closed)i(term)e(of)h(t)m(yp)s(e)h Fn(nat)22 b Fp(!)i Fn(nat)p Fc(.)43 b(Then)33 b(there)g(is)e(an)h Fo(m)0 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b(only)f(treat)g(the)h(second)h(part,)h(since)e(the)g (pro)s(of)f(of)g(the)h(\014rst)g(part)f(is)0 5214 y(iden)m(tical)32 b(\(just)j(lea)m(v)m(e)g Fo(u)e Fq(out\).)48 b(Consider)35 b Fo(r)r Fn(\()p Fo(S)5 b(x)p Fn(\))p Fo(u)p Fq(.)49 b(By)34 b(the)h(Em)m(b)s(edding)e(Lemma)g(4.7)p eop %%Page: 1 18 1 17 bop 0 125 a Fq(w)m(e)41 b(\014nd)f(an)f Fo(m)h Fq(suc)m(h)h(that)e Fp(`)1104 94 y Fl(\012)p Fi(\001)p Fm(m)1104 145 y(m)1253 125 y Fo(r)r Fn(\()p Fo(S)5 b(x)p Fn(\))p Fo(u)p Fq(.)66 b(Hence)41 b Fp(`)1948 94 y Fm(c)1948 145 y Fl(0)2004 125 y Fo(r)r Fn(\()p Fo(S)5 b(x)p Fn(\))p Fo(u)41 b Fq(with)e Fo(c)33 b Fn(:=)g Fo(D)2789 94 y Fm(m)2787 145 y Fl(1)2852 125 y Fn(\(\012)23 b Fp(\001)f Fo(m)p Fn(\))40 b Fq(b)m(y)0 249 y(the)33 b(Cut)g(Elimination)c(Lemma)j(4.3.)43 b(Since)33 b Fp(`)1746 219 y Fm(c)p Fi(\001)p Fm(n)1746 270 y Fl(0)1850 249 y Fn(\()p Fo(n)19 b Fp(\000)f Fn(1\))32 b Fq(\(b)m(y)i(the)f (Closure)g(Rules)g(and)0 374 y(the)44 b Fo(<)244 386 y Fl(1)280 374 y 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4990 y Fo(a)g Fp(2)g Fo(W)2386 5002 y Fm(n)0 5172 y Fq(Pro)s(of.)43 b(Let)33 b Fo(a)f Fq(b)s(e)h(\014xed.)44 b(Assume)1125 5396 y Fp(8)p Fo(z)26 b Fp(2)d(j)-14 b(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)-14 b(j)1532 5408 y Fm(n)1602 5396 y Fn(:)23 b Fo(D)1717 5408 y Fl(0)1754 5396 y Fo(a)p Fn([)p Fo(z)t Fn(])g Fp(2)g Fo(W)2066 5408 y Fm(n)2112 5396 y Fo(:)1019 b Fn(\(1\))p eop %%Page: 1 19 1 18 bop 0 125 a Fq(W)-8 b(e)38 b(ha)m(v)m(e)i(to)e(sho)m(w)h(that)f Fo(D)1062 137 y Fl(0)1099 125 y Fo(a)30 b Fp(2)i Fo(W)1338 137 y Fm(n)1383 125 y Fq(,)40 b(i.e.)59 b(that)38 b Fp(9)p Fo(k)s Fn(\()p Fo(D)2044 137 y Fl(0)2081 125 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])2253 94 y Fm(k)2325 125 y Fn(=)31 b(0)p Fq(.)59 b Fj(Case)39 b Fo(a)31 b Fn(=)f(0)p Fq(.)60 b(The)0 249 y(claim)33 b(follo)m(ws)i(from)f Fo(!)s Fn([)p Fo(n)p Fn(])27 b(=)g Fo(n)36 b Fq(and)f Fo(m)p Fn([)p Fo(n)p Fn(])1537 219 y Fm(k)1606 249 y Fn(=)27 b Fo(m)20 b Fp(\000)g Fo(k)s Fq(.)52 b Fj(Case)37 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))29 b(=)e(1)p Fq(.)52 b(Note)36 b(\014rst)g(that)0 374 y(the)d(set)g Fo(W)398 386 y Fm(n)476 374 y Fq(clearly)e(is)h (closed)g(against)g(addition.)41 b(Since)33 b Fn(\()p Fo(D)2287 386 y Fl(0)2324 374 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])23 b(=)g(\()p Fo(D)2708 386 y Fl(0)2745 374 y Fo(a)p Fn([)p Fo(n)p Fn(]\))18 b Fp(\001)g Fn(\()p Fo(n)g Fn(+)g(1\))p Fq(,)0 498 y(the)38 b(claim)d(follo)m(ws)h(from)h(\(1\),)h(whic)m(h)g (in)f(our)g(case)i(is)e Fo(D)2154 510 y Fl(0)2191 498 y Fo(a)p Fn([)p Fo(n)p Fn(])30 b Fp(2)g Fo(W)2524 510 y Fm(n)2570 498 y Fq(.)58 b Fj(Case)39 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))31 b(=)f Fo(!)s Fq(.)0 623 y(Then)g Fn(\()p Fo(D)352 635 y Fl(0)390 623 y Fo(a)p Fn(\)[)p Fo(n)p Fn(])23 b(=)g Fo(D)742 635 y Fl(0)779 623 y Fo(a)p Fn([)p Fo(n)p Fn(])p Fq(,)29 b(and)h Fo(D)1231 635 y Fl(0)1268 623 y Fo(a)p Fn([)p Fo(n)p Fn(])23 b Fp(2)g Fo(W)1587 635 y Fm(n)1662 623 y Fq(b)m(y)30 b(\(1\).)42 b Fj(Case)31 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))25 b(=)d(\012)p Fq(.)43 b(Then)30 b Fn(\()p Fo(D)2971 635 y Fl(0)3008 623 y Fo(a)p Fn(\)[0])23 b(=)0 747 y Fo(D)69 759 y Fl(0)106 747 y Fo(a)p Fn([)p Fo(!)s Fn(])p Fq(,)39 b Fn(\()p Fo(D)418 759 y Fl(0)456 747 y Fo(a)p Fn(\)[)p Fo(m)21 b Fn(+)g(1])31 b(=)g Fo(D)996 759 y Fl(0)1033 747 y Fo(a)p Fn([\()p Fo(D)1201 759 y Fl(0)1238 747 y Fo(a)p Fn(\)[)p Fo(m)p Fn(]])p Fq(.)60 b(W)-8 b(e)39 b(sho)m(w)g Fn(\()p Fo(D)2065 759 y Fl(0)2102 747 y Fo(a)p Fn(\)[)p Fo(m)p Fn(])32 b Fp(2)f Fo(W)2493 759 y Fm(n)2577 747 y Fq(b)m(y)39 b(induction)e(on)0 872 y Fo(m)p Fq(.)56 b(F)-8 b(or)36 b Fo(m)29 b Fn(=)g(0)36 b Fq(w)m(e)i(get)f Fo(D)993 884 y Fl(0)1030 872 y Fo(a)p Fn([)p Fo(!)s Fn(])29 b Fp(2)g Fo(W)1366 884 y Fm(n)1449 872 y Fq(b)m(y)38 b(\(1\),)f(since)h Fo(!)31 b Fp(2)f Fo(W)2268 884 y Fm(n)2350 872 y Fq(\(see)38 b(Case)g Fo(a)29 b Fn(=)g(0)p Fq(\).)56 b(F)-8 b(or)0 996 y(the)32 b(induction)f(step)h(w)m(e)h(can)f(assume)g Fn(\()p Fo(D)1563 1008 y Fl(0)1601 996 y Fo(a)p Fn(\)[)p Fo(m)p Fn(])23 b Fp(2)g Fo(W)1975 1008 y Fm(n)2021 996 y Fq(.)43 b(But)32 b(then)g Fo(D)2574 1008 y Fl(0)2611 996 y Fo(a)p Fn([\()p Fo(D)2779 1008 y Fl(0)2817 996 y Fo(a)p Fn(\)[)p Fo(m)p Fn(]])23 b Fp(2)g Fo(W)3214 1008 y Fm(n)0 1121 y Fq(b)m(y)33 b(\(1\).)p 331 1121 4 75 v 335 1049 90 4 v 335 1121 V 423 1121 4 75 v 0 1295 a Fk(Lemma)k(5.2.)57 b Fc(If)32 b(the)h(form)m(ula)e Fo( )s Fn(\()p Fo(a)p Fn(\))i Fc(is)f Fo(n)p Fc({progressiv)m(e,)h (then)g(so)g(is)1049 1519 y Fo( )1106 1485 y Fi(\003)1145 1519 y Fn(\()p Fo(a)p Fn(\))23 b(:)p Fp(\021)g(8)p Fo(c: )s Fn(\()p Fo(c)p Fn(\))f Fp(!)i Fo( )s Fn(\()p Fo(c)18 b Fn(+)g Fo(D)2074 1531 y Fl(1)2111 1519 y Fo(a)p Fn(\))p Fo(:)0 1743 y Fq(Pro)s(of.)43 b(Let)33 b Fo( )s Fn(\()p Fo(a)p Fn(\))g Fq(b)s(e)f Fo(n)p Fq({progressiv)m(e,)i(i.e.)1026 1968 y Fp(8)p Fo(a:)p Fp(8)p Fo(z)25 b Fp(2)e(j)-14 b(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)-14 b(j)1546 1980 y Fm(n)1616 1968 y Fn(:)23 b Fo( )s Fn(\()p Fo(a)p Fn([)p Fo(z)t Fn(]\))g Fp(!)g Fo( )s Fn(\()p Fo(a)p Fn(\))p Fo(:)921 b Fn(\(2\))0 2192 y Fq(W)-8 b(e)39 b(ha)m(v)m(e)g(to)f(sho)m (w)h(that)f Fo( )1050 2162 y Fi(\003)1088 2192 y Fn(\()p Fo(a)p Fn(\))h Fq(is)f Fo(n)p Fq({progressiv)m(e.)61 b(So)38 b(let)f Fo(a)h Fq(b)s(e)h(giv)m(en)f(and)g(assume)0 2316 y(that)947 2441 y Fp(8)p Fo(z)25 b Fp(2)f(j)-14 b(j)p Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))p Fp(j)-14 b(j)1354 2453 y Fm(n)1401 2441 y Fp(8)p Fo(c: )s Fn(\()p Fo(c)p Fn(\))22 b Fp(!)i Fo( )s Fn(\()p Fo(c)18 b Fn(+)g Fo(D)2088 2453 y Fl(1)2125 2441 y Fo(a)p Fn([)p Fo(z)t Fn(]\))p Fo(:)841 b Fn(\(3\))0 2623 y Fq(W)-8 b(e)28 b(m)m(ust)g(sho)m(w)h Fo( )694 2593 y Fi(\003)732 2623 y Fn(\()p Fo(a)p Fn(\))p Fq(.)43 b(So)27 b(let)h(also)e Fo(c)i Fq(b)s(e)g(giv)m(en)g(and)g (assume)g Fo( )s Fn(\()p Fo(c)p Fn(\))p Fq(.)43 b(W)-8 b(e)28 b(ha)m(v)m(e)h(to)f(sho)m(w)0 2748 y Fo( )s Fn(\()p Fo(c)19 b Fn(+)f Fo(D)296 2760 y Fl(1)333 2748 y Fo(a)p Fn(\))p Fq(.)43 b(By)33 b(\(2\))g(it)e(su\016ces)k(to)d(pro)m(v)m(e) 1023 2972 y Fp(8)p Fo(z)25 b Fp(2)f(j)-14 b(j)p Fo(\034)9 b Fn(\()p Fo(D)1391 2984 y Fl(1)1429 2972 y Fo(a)p Fn(\))p Fp(j)-14 b(j)1537 2984 y Fm(n)1606 2972 y Fn(:)23 b Fo( )s Fn(\()p Fo(c)c Fn(+)f(\()p Fo(D)1980 2984 y Fl(1)2017 2972 y Fo(a)p Fn(\)[)p Fo(z)t Fn(]\))p Fo(:)917 b Fn(\(4\))0 3196 y Fj(Case)46 b Fo(a)40 b Fn(=)h(0)p Fq(.)80 b(W)-8 b(e)45 b(m)m(ust)h(sho)m(w)g Fp(8)p Fo(z)d Fp(2)e Fo(W)1582 3208 y Fm(n)1668 3196 y Fn(:)g Fo( )s Fn(\()p Fo(c)26 b Fn(+)f Fo(z)t Fn(\))p Fq(,)48 b(i.e.)80 b Fo(z)44 b Fp(2)e Fn(T)2568 3208 y Fl(1)2646 3196 y Fp(!)e Fo(z)t Fn([)p Fo(n)p Fn(])2908 3166 y Fm(k)2989 3196 y Fn(=)h(0)f Fp(!)0 3321 y Fo( )s Fn(\()p Fo(c)17 b Fn(+)g Fo(z)t Fn(\))p Fq(.)43 b(This)32 b(is)f(done)i(b)m(y)g(induction)d(on)i Fo(k)s Fq(.)43 b(F)-8 b(or)31 b Fo(k)26 b Fn(=)d(0)31 b Fq(the)h(claim)e(follo)m(ws)g(from)h(our)0 3445 y(assumption)37 b Fo( )s Fn(\()p Fo(c)p Fn(\))p Fq(.)59 b(F)-8 b(or)37 b(the)h(induction)f(step,)j(assume)e Fo(z)c Fp(2)d Fn(T)2357 3457 y Fl(1)2394 3445 y Fq(.)59 b(Then)38 b Fp(j)-14 b(j)p Fo(\034)9 b Fn(\()p Fo(z)t Fn(\))p Fp(j)-14 b(j)2955 3457 y Fm(n)3032 3445 y Fn(=)30 b Fp(f)p Fo(n)p Fp(g)0 3570 y Fq(\(if)42 b Fo(z)g Fp(6)p Fn(=)c(0)p Fq(,)46 b(but)e(the)g(case)g Fo(z)e Fn(=)d(0)k Fq(is)g(ob)m(vious\).)76 b(Hence)45 b(b)m(y)g(\(2\))e(with)g Fo(c)24 b Fn(+)h Fo(z)46 b Fq(for)d Fo(a)h Fq(it)0 3694 y(su\016ces)38 b(to)e(sho)m(w)h Fo( )s Fn(\()p Fo(c)20 b Fn(+)g Fo(z)t Fn([)p Fo(n)p Fn(]\))p Fq(.)53 b(But)37 b(since)f Fn(\()p Fo(z)t Fn([)p Fo(n)p Fn(]\)[)p Fo(n)p Fn(])1923 3664 y Fm(k)1991 3694 y Fn(=)28 b Fo(z)t Fn([)p Fo(n)p Fn(])2223 3664 y Fm(k)q Fl(+1)2375 3694 y Fn(=)f(0)36 b Fq(this)f(follo)m(ws)g (from)0 3819 y(the)40 b(induction)e(h)m(yp)s(othesis.)66 b Fj(Case)40 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))34 b(=)f(1)p Fq(.)63 b(W)-8 b(e)40 b(m)m(ust)g(sho)m(w)h Fo( )s Fn(\()p Fo(c)22 b Fn(+)g(\()p Fo(D)2825 3831 y Fl(1)2863 3819 y Fo(a)p Fn(\)[)p Fo(n)p Fn(]\))p Fq(,)41 b(i.e.)0 3944 y Fo( )s Fn(\()p Fo(c)r Fn(+)r(\()p Fo(D)295 3956 y Fl(1)331 3944 y Fo(a)p Fn([)p Fo(n)p Fn(]\))r Fp(\001)r Fn(\()p Fo(n)r Fn(+)r(1\)\))p Fq(.)e(W)-8 b(e)23 b(pro)m(v)m(e)h Fo( )s Fn(\()p Fo(c)r Fn(+)r(\()p Fo(D)1559 3956 y Fl(1)1595 3944 y Fo(a)p Fn([)p Fo(n)p Fn(]\))r Fp(\001)r Fo(m)p Fn(\))e Fq(b)m(y)i(induction)e(on)g Fo(m)p Fq(.)40 b(F)-8 b(or)22 b Fo(m)h Fn(=)g(0)f Fq(w)m(e)0 4068 y(ha)m(v)m(e)k Fo( )s Fn(\()p Fo(c)p Fn(\))g Fq(b)m(y)g(our)e(assumption.)41 b(F)-8 b(or)24 b(the)h(induction)f(step,)k(\(3\))c(with)h Fo(c)5 b Fn(+)g(\()p Fo(D)2802 4080 y Fl(1)2839 4068 y Fo(a)p Fn([)p Fo(n)p Fn(]\))g Fp(\001)g Fo(m)26 b Fq(for)0 4193 y Fo(c)g Fq(and)g(the)g(induction)e(h)m(yp)s(othesis)j(yield)e Fo( )s Fn(\()p Fo(c)7 b Fn(+)g(\()p Fo(D)1840 4205 y Fl(1)1877 4193 y Fo(a)p Fn([)p Fo(n)p Fn(]\))g Fp(\001)g Fo(m)g Fn(+)g Fo(D)2307 4205 y Fl(1)2343 4193 y Fo(a)p Fn([)p Fo(n)p Fn(]\))p Fq(.)42 b Fj(Case)27 b Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))25 b(=)d Fo(!)s(;)14 b Fn(\012)p Fq(.)0 4317 y(Since)31 b(in)e(this)h(case)i Fo(\034)9 b Fn(\()p Fo(D)903 4329 y Fl(1)940 4317 y Fo(a)p Fn(\))24 b(=)e Fo(\034)9 b Fn(\()p Fo(a)p Fn(\))32 b Fq(and)e Fn(\()p Fo(D)1600 4329 y Fl(1)1638 4317 y Fo(a)p Fn(\)[)p Fo(z)t Fn(])22 b(=)h Fo(D)1982 4329 y Fl(1)2019 4317 y Fo(a)p Fn([)p Fo(z)t Fn(])29 b Fq(w)m(e)j(get)e(the)h(claim)d(\(4\))i (from)0 4442 y(\(3\))i(and)h(our)f(assumption)g Fo( )s Fn(\()p Fo(c)p Fn(\))p Fq(.)p 1264 4442 V 1268 4370 90 4 v 1268 4442 V 1357 4442 4 75 v 0 4616 a Fk(Theorem)37 b(5.3.)56 b Fc(F)-8 b(or)32 b(an)m(y)h Fo(m)p Fc(,)g(w)m(e)h(can)e (formally)e(pro)m(v)m(e)k(in)e(arithmetic)1211 4840 y Fp(8)p Fo(x)p Fp(9)p Fo(y)s Fn(\()p Fo(D)1496 4852 y Fl(0)1532 4840 y Fo(D)1603 4806 y Fm(m)1601 4861 y Fl(1)1667 4840 y Fn(0\)[)p Fo(x)p Fn(])1834 4806 y Fm(y)1897 4840 y Fn(=)22 b(0)p Fo(:)0 5064 y Fq(Pro)s(of.)57 b(W)-8 b(e)38 b(giv)m(e)f(an)g(informal)d(pro)s(of)j(whic)m(h)g(can)h(easily)e (b)s(e)i(formalized.)55 b(Let)38 b Fo(n)f Fq(b)s(e)0 5189 y(\014xed.)79 b(Since)45 b(the)f(form)m(ula)f Fo(')p Fn(\()p Fo(a)p Fn(\))d Fp(\021)f Fo(D)1499 5201 y Fl(0)1536 5189 y Fo(a)g Fp(2)h Fo(W)1792 5201 y Fm(n)1882 5189 y Fq(is)j Fo(n)p Fq({progressiv)m(e)i(b)m(y)g(Lemma)e(5.1,)p eop %%Page: 1 20 1 19 bop 0 125 a Fq(w)m(e)36 b(kno)m(w)g(from)e(Lemma)g(5.2)g(that)h (also)f Fo(')1612 94 y Fi(\003)1651 125 y Fn(\()p Fo(a)p Fn(\))p Fq(,)i Fo(')1876 94 y Fi(\003\003)1948 125 y Fn(\()p Fo(a)p Fn(\))p Fo(;)14 b(:)g(:)g(:)g(;)g(')2295 94 y Fm(m)2359 125 y Fn(\()p Fo(a)p Fn(\))35 b Fq(are)g Fo(n)p Fq({progressiv)m(e.)0 249 y(Hence)k(w)m(e)f(ha)m(v)m(e)h Fo(')727 219 y Fm(m)790 249 y Fn(\(0\))p Fq(,)g(hence)g Fo(')1292 219 y Fm(m)p Fi(\000)p Fl(1)1440 249 y Fn(\()p Fo(D)1541 261 y Fl(1)1579 249 y Fn(0\))e Fq(b)m(y)h(the)g(de\014nition) f(of)g Fo( )2615 219 y Fi(\003)2653 249 y Fq(,)h(hence)h(induc-)0 374 y(tiv)m(ely)32 b Fo(')322 343 y Fi(\003)361 374 y Fn(\()p Fo(D)464 338 y Fm(m)p Fi(\000)p Fl(1)462 396 y(1)612 374 y Fn(0\))p Fq(,)h(hence)g Fo(')p Fn(\()p Fo(D)1173 343 y Fm(m)1171 394 y Fl(1)1237 374 y Fn(0\))p Fq(,)g(hence)h Fp(9)p Fo(y)s Fn(\()p Fo(D)1833 386 y Fl(0)1870 374 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