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Fq(d)p Fr(\()p Fq(e)p Fr(\))23 b(=)g(0)29 b Fo(and)h Fq(e)g Fo(is)g(close)l(d.)51 714 y(4.)39 b Fq(e)29 b Fo(is)h(c)l(al)t(le)l(d)h Fr(arithmetical)e Fo(if)i Fq(e)e Fo(c)l(ontains)h(no)g(1-variable.)51 814 y(5.)43 b Fg(TRUE)28 b(\(F)-7 b(ALSE\))32 b Fo(denotes)g(the)f(set)g (of)h(al)t(l)g(true)f(\(false\))h(simple)g(formulas.)44 b([Note)32 b(that)f(a)h(simple)g(formula)g(c)l(ontains)51 913 y(no)c(variables)i(and)e(is)g(c)l(onstructe)l(d)f(fr)l(om)i(c)l (omputable)g(atomic)f(formulas)h(by)g(b)l(o)l(ole)l(an)f(c)l(onne)l (ctives.)39 b(Every)29 b(simple)g(term)51 1013 y(is)h(a)g(numer)l(al].) 51 1112 y(6.)39 b Fr(I)-14 b(B)232 1124 y Fn(0)292 1112 y Fr(:=)23 b(I)-14 b(N)p Fo(;)30 b Fr(I)-14 b(B)611 1124 y Fn(1)648 1112 y Fo(:=)30 b(the)g(set)g(of)g(al)t(l)h(c)l(anonic)l(al) g(arithmetic)l(al)g Fq(\025)p Fo(-terms.)51 1256 y Fr(The)f(ob)5 b(jects)30 b(to)h(b)r(e)g(immediately)g(ev)-5 b(aluated)30 b(are)g(canonical)f Fq(\017)p Fr(-terms,)i(and)f(the)h(v)-5 b(alues)31 b(of)f Fq(\023)p 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b(y)s(;)g(z)t Fr(\))19 b Fp(^)g Fq(add)p Fr(\()p Fq(z)t(;)14 b(x;)g(z)1789 1864 y Fn(1)1826 1852 y Fr(\))23 b Fp(!)g Fq(pr)r(od)p Fr(\()p Fq(x;)14 b Fg(S)p Fq(y)s(;)g(z)2433 1864 y Fn(1)2471 1852 y Fr(\))51 1952 y(\(3\))27 b(Equalit)n(y)g(axioms:)36 b Fq(x)23 b Fr(=)g Fq(x)56 b Fr(and)f Fq(x)23 b Fr(=)g Fq(y)j Fp(!)d Fr(\()p Fq(F)12 b Fr([)p Fq(x)p Fr(])24 b Fp(!)f Fq(F)12 b Fr([)p Fq(y)s Fr(]\),)51 2051 y(\(4\))27 b(P)n(eano)f(axioms)h(for)g Fg(S)p Fr(:)37 b Fg(S)p Fq(x)24 b Fp(6)p Fr(=)e(0)55 b(and)g Fg(S)p Fq(x)24 b Fr(=)e Fg(S)p Fq(y)k Fp(!)d Fq(x)h Fr(=)f Fq(y)30 b Fr(,)51 2151 y(\(5\))d(Induction)h(axioms:)36 b Fq(F)12 b Fr([0])23 b Fp(!)g(8)p Fq(x)p Fr(\()p Fq(F)12 b Fr([)p Fq(x)p Fr(])23 b Fp(!)g Fq(F)12 b Fr([)p Fg(S)p Fq(x)p Fr(]\))24 b Fp(!)f(8)p Fq(xF)12 b Fr([)p Fq(x)p Fr(],)51 2250 y(\(6\))27 b(First)h(order)e (existen)n(tial)h(axioms:)36 b Fq(F)12 b Fr([)p Fq(t)p Fr(])23 b Fp(!)g(9)p Fq(xF)12 b Fr([)p Fq(x)p Fr(],)51 2350 y(\(7\))27 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Fr(=)g Fq(F)3495 4425 y Fe(\003)3533 4455 y Fr([)p Fq(\017\030)t(F)3695 4425 y Fe(\003)3733 4455 y Fr([)p Fq(\030)t Fr(]])g Fp(!)51 4555 y Fq(G)116 4525 y Fe(\003)154 4555 y Fr(,)28 b(and)f(therefore)g(closure)f(of)i Fq(E)5 b(A)1215 4525 y Fe(0)1266 4555 y Fr(under)28 b Fp(9)23 b(!)28 b Fr(follo)n(ws)e(from)h(the)h(fact)g(that)g Fq(E)5 b(A\017)28 b Fr(is)f(closed)g(under)h(substitution.\))51 4655 y(The)f(*-translation)f(of)h(an)n(y)g Fq(E)5 b(A)p Fr(-axiom)27 b(of)h(kind)f(\(1\){\(4\))h(is)f(an)g Fq(E)5 b(A\017)p Fr(-axiom)27 b(of)h(the)g(same)f(kind.)51 4754 y(The)g(*-translation)f(of)h(an)n(y)g(existen)n(tial)g(axiom)g(of)h Fq(E)5 b(A)28 b Fr(is)f(a)g(critical)g(form)n(ula.)51 4854 y(Belo)n(w)h(w)n(e)h(sho)n(w)f(that)i(the)g(*-translation)d(of)j (ev)n(ery)e(induction)h(axiom)g(of)g Fq(E)5 b(A)30 b Fr(is)f(deriv)-5 b(able)29 b(in)h Fq(E)5 b(A\017)p Fr(.)42 b(This)29 b(will)h(\014nish)51 4953 y(the)e(pro)r(of)f(of)g(a\).)51 5053 y(Observ)n(e)h(that)j(b\))g(is)f(an)h(immediate)f(consequence)g (of)g(a\):)43 b(Giv)n(en)30 b(an)g Fq(E)5 b(A\017)p Fr(-deriv)-5 b(ation)30 b Fq(d)g Fr(of)h(a)f(closed)g(form)n(ula)f Fq(F)43 b Fr(one)51 5153 y(simply)27 b(replaces)g(ev)n(ery)f(free)h Fq(\023)p Fr(-v)-5 b(ariable)27 b(in)h Fq(d)g Fr(b)n(y)f(0)1703 5123 y Fm(\023)1731 5153 y Fr(.)51 5272 y(No)n(w)g(w)n(e)g(consider)f (the)i(*-translation)e Fq(A)i Fr(of)g(an)f(induction)h(axiom.)51 5372 y(Ob)n(viously)e Fq(A)i Fr(is)f(of)h(the)g(form)f Fq(G)p Fr([0])c Fp(!)p 1330 5305 66 4 v 23 w Fq(G)p Fr([)p Fq(u)p Fr(])g Fp(!)g Fq(G)p Fr([)p Fq(e)p Fr(])28 b(with)p 1985 5305 V 28 w Fq(G)p Fr([)p Fq(x)p Fr(])c(:=)f Fq(G)p Fr([)p Fq(x)p Fr(])g Fp(!)h Fq(G)p Fr([)p Fg(S)p Fq(x)p Fr(],)k Fq(u)23 b Fr(:=)g Fq(\017x)p Fp(:)p 3139 5305 V Fq(G)p Fr([)p Fq(x)p Fr(],)29 b Fq(e)22 b Fr(:=)h Fq(\017x)p Fp(:)p Fq(G)p Fr([)p Fq(x)p Fr(].)51 5471 y(Let)k Fq(e)238 5441 y Fe(\000)317 5471 y Fr(:=)c Fq(\017y)s Fr(\()p Fq(e)f Fr(=)h Fg(S)p Fq(y)s 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Fp(!)g(:)p 3789 5704 V Fq(G)q Fr([)p Fq(u)p Fr(])51 5870 y(and)j(the)h(equalit)n(y)f (axiom)g Fq(e)22 b Fr(=)h Fg(S)p Fq(e)1152 5840 y Fe(\000)1231 5870 y Fp(!)g Fq(G)p Fr([)p Fg(S)p Fq(e)1510 5840 y Fe(\000)1566 5870 y Fr(])g Fp(!)g Fq(G)p Fr([)p Fq(e)p Fr(].)65 b Fd(2)1979 6119 y Fr(6)p eop %%Page: 7 7 7 6 bop 51 614 a Fk(3)137 b(Computations)47 b(with)g(the)g Fj(\017)p Fk(-Substitutions)51 796 y Fh(De\014nition)30 b(10)51 896 y Fo(A)n(n)f Fq(\017)p Fr(-substitution)g Fo(is)h(a)g(function)g Fq(S)35 b Fo(such)29 b(that)51 995 y Fq(dom)p Fr(\()p Fq(S)5 b Fr(\))30 b Fo(\(domain)h(of)f Fq(S)5 b Fo(\))30 b(is)g(a)g(set)g(of)g(c)l(anonic)l(al)h Fq(\017)p Fo(-terms,)51 1104 y(if)f Fq(e)23 b Fp(2)g Fq(dom)p Fr(\()p Fq(S)5 b Fr(\))31 b Fo(then)e Fq(S)5 b Fr(\()p Fq(e)p Fr(\))23 b Fp(2)h Fr(I)-14 b(B)1098 1068 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))1228 1104 y Fp([)19 b(f)p Fr(?)p Fp(g)p Fo(.)51 1204 y(A)n(n)29 b Fq(\017)p Fo(-substitution)f Fq(S)34 b Fo(is)c(c)l(al)t(le)l(d)h Fr(total)57 b Fo(if)31 b Fq(dom)p Fr(\()p Fq(S)5 b Fr(\))30 b Fo(is)g(the)g(set)g(of)g(al)t(l)h(c)l(anonic)l(al)g Fq(\017)p Fo(-terms.)p 51 1237 56 4 v 51 1304 a Fq(S)c Fr(:=)c Fq(S)g Fp([)c(f)p Fr(\()p Fq(e;)14 b Fr(?\))23 b(:)g Fq(e)29 b Fo(is)h(a)h(c)l(anonic)l(al)g Fq(\017)p Fo(-term)22 b Fp(62)h Fq(dom)p Fr(\()p Fq(S)5 b Fr(\))p Fp(g)30 b Fo(is)g(c)l(al)t(le)l(d)h(the)f Fr(standard)d(extension)g(of) i Fq(S)5 b Fo(.)51 1471 y Fh(Commen)m(t)p Fr(.)31 b(W)-7 b(e)22 b(consider)e(a)g(function)i(as)e(a)h(set)g(of)f(ordered)g (pairs.)34 b(So)20 b(an)h Fq(\017)p Fr(-substitution)g(is)g(a)f(set)h (of)g(pairs)f(\()p Fq(e;)14 b(u)p Fr(\))21 b(where)51 1580 y Fq(e)28 b Fr(is)g(a)g(canonical)f Fq(\017)p Fr(-term)h(and)g Fq(u)c Fp(2)g Fr(I)-14 b(B)1286 1544 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))1417 1580 y Fp([)20 b(f)p Fr(?)p Fp(g)p Fr(.)38 b(Hilb)r(ert's)29 b(\014nitist)g(p)r(osition)f(allo)n(w)n(ed)f (only)h(for)g(\014nite)h Fq(\017)p Fr(-substitutions)51 1680 y Fq(S)5 b Fr(.)44 b(All)31 b(\(canonical\))f Fq(\017)p 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b(de\014nition)g(of)g Fq(e)23 b(,)-14 b Fp(!)1650 2486 y Fn(1)1650 2539 y Fm(S)1721 2516 y Fq(e)1760 2486 y Fe(0)1783 2516 y Fr(\))51 2616 y Fo(1.1.)40 b(If)30 b Fr(\()p Fq(e;)14 b(u)p Fr(\))22 b Fp(2)i Fq(S)34 b Fo(and)d Fq(u)22 b Fp(6)p Fr(=)14 b(?)29 b Fo(then)h Fq(e)23 b(,)-14 b Fp(!)1404 2586 y Fn(1)1404 2639 y Fm(S)1475 2616 y Fq(u)51 2715 y Fo(1.2.)40 b(If)30 b Fr(\()p Fq(e;)d Fr(?\))d Fp(2)f Fq(S)35 b Fo(then)29 b Fq(e)23 b(,)-14 b Fp(!)1030 2685 y Fn(1)1030 2738 y Fm(S)1101 2715 y Fr(0)1143 2685 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))51 2815 y Fo(2.)39 b(If)30 b Fr(1)22 b Fp(\024)h Fq(i)g Fp(\024)f Fq(n)p Fo(,)30 b Fq(e)679 2827 y Fm(i)730 2815 y Fq(,)-14 b Fp(!)822 2785 y Fn(1)822 2838 y Fm(S)893 2815 y Fq(e)932 2785 y Fe(0)932 2837 y Fm(i)989 2815 y Fo(and)30 b Fq(e)1189 2785 y Fe(0)1189 2837 y Fm(i)1246 2815 y Fo(is)g(not)g(a)g Fq(\025)p Fo(-term)g(then)f Fq(pe)2096 2827 y Fn(1)2147 2815 y Fq(:)14 b(:)g(:)f(e)2296 2827 y Fm(n)2364 2815 y Fq(,)-14 b Fp(!)2456 2785 y Fn(1)2456 2838 y Fm(S)2528 2815 y Fq(pe)2609 2827 y Fn(1)2659 2815 y Fq(:)14 b(:)g(:)g(e)2809 2827 y Fm(i)p Fe(\000)p Fn(1)2921 2815 y Fq(e)2960 2785 y Fe(0)2960 2837 y Fm(i)2987 2815 y Fq(e)3026 2827 y Fm(i)p Fn(+1)3152 2815 y Fq(:)g(:)g(:)f(e)3301 2827 y Fm(n)51 2915 y Fo(3.)39 b(If)30 b Fr(\()p Fq(P)r(;)14 b(\025z)t(G)p Fr(\))24 b Fp(2)f Fq(S)35 b Fo(then)29 b Fg(App)p Fq(P)12 b(t)23 b(,)-14 b Fp(!)1279 2885 y Fn(1)1279 2938 y Fm(S)1351 2915 y Fq(G)p Fr([)p Fq(t)p Fr(])51 3014 y Fo(4.)39 b(If)30 b Fq(F)35 b(,)-14 b Fp(!)424 2984 y Fn(1)424 3037 y Fm(S)495 3014 y Fq(F)560 2984 y Fe(0)613 3014 y Fo(then)30 b Fq(\017\030)t(F)35 b(,)-14 b Fp(!)1052 2984 y Fn(1)1052 3037 y Fm(S)1123 3014 y Fq(\017\030)t(F)1262 2984 y Fe(0)51 3114 y Fo(5.)39 b(If)30 b Fq(G)23 b(,)-14 b Fp(!)424 3084 y Fn(1)424 3137 y Fm(S)496 3114 y Fq(G)561 3084 y Fe(0)614 3114 y Fo(then)29 b Fq(\025z)t(G)23 b(,)-14 b Fp(!)1069 3084 y Fn(1)1069 3137 y Fm(S)1141 3114 y Fq(\025z)t(G)1297 3084 y Fe(0)1320 3114 y Fo(.)51 3268 y Fh(Commen)m(t)p Fr(.)33 b(Since)27 b Fq(S)k Fr(is)26 b(de\014ned)h(only)f(for)g(canonical)f(terms)i(whic)n(h)f(do)g(not)h (con)n(tain)e(an)n(y)h(prop)r(er)f(closed)h Fq(\017)p Fr(-subterms,)51 3368 y(computations)36 b(pro)r(ceed)g(from)h(inside,)i (and)e(since)f(canonical)g(terms)g(are)g(closed,)j(no)d(term)h(con)n (taining)f(v)-5 b(ariables)35 b(is)51 3467 y(immediately)27 b(computed)h(according)e(to)h Fq(S)5 b Fr(:)37 b(only)27 b(its)h(closed)f(subterms)g(can)h(b)r(e)g(replaced.)51 3589 y Fh(Note)p Fr(.)36 b(If)28 b Fq(e)23 b(,)-14 b Fp(!)551 3559 y Fn(1)551 3612 y Fm(S)622 3589 y Fq(\025z)t(G)28 b Fr(and)f Fq(e)g Fr(is)h(not)g(a)f Fq(\025)p Fr(-term)g(then)h(either) g(\()p Fq(e;)14 b(\025z)t(G)p Fr(\))23 b Fp(2)h Fq(S)32 b Fr(or)27 b(\()p Fq(e;)14 b Fr(?\))23 b Fp(2)g Fq(S)33 b Fr(and)27 b Fq(\025z)t(G)c Fr(=)g(0)3450 3559 y Fn(1)3487 3589 y Fr(.)51 3757 y Fh(De\014nition)30 b(12)51 3857 y Fo(1.)37 b Fq(e)23 b Fo(is)h(c)l(al)t(le)l(d)h Fq(S)5 b Fo(-)p Fr(reducible)22 b Fo(if)j(ther)l(e)e(exists)h(an)f Fq(e)1607 3826 y Fe(0)1654 3857 y Fo(with)h Fq(e)f(,)-14 b Fp(!)1982 3826 y Fn(1)1982 3879 y Fm(S)2053 3857 y Fq(e)2092 3826 y Fe(0)2115 3857 y Fo(.)36 b(Otherwise)25 b Fq(e)e Fo(is)h(c)l(al)t(le)l(d)h Fq(S)5 b Fo(-)p Fr(irreducible)22 b Fo(or)i Fr(in)d Fq(S)5 b Fr(-normal)51 3956 y(form)p Fo(.)51 4056 y(2.)39 b Fq(,)-14 b Fp(!)249 4068 y Fm(S)327 4056 y Fo(denotes)30 b(the)g(tr)l(ansitive)g(and)g(r)l(e\015exive)g (closur)l(e)g(of)h Fq(,)-14 b Fp(!)2086 4026 y Fn(1)2086 4079 y Fm(S)2134 4056 y Fo(.)51 4224 y Fh(Note)p Fr(.)61 b Fq(e)35 b Fr(is)h Fq(S)5 b Fr(-reducible)35 b(i\013)i Fq(e)e Fr(con)n(tains)g(canonical)g Fq(\017)p Fr(-subterms)g(in)h(the)g (domain)f(of)h Fq(S)5 b Fr(.)62 b(In)36 b(particular)e(no)i(default)51 4323 y(computations)27 b(are)f(allo)n(w)n(ed)g(unless)i Fq(S)k Fr(con)n(tains)27 b(pairs)f(\()p Fq(e;)14 b Fr(?\).)217 4445 y(W)-7 b(e)24 b(are)e(going)h(to)g(pro)n(v)n(e)f(that)i Fq(,)-14 b Fp(!)1293 4457 y Fm(S)1365 4445 y Fr(is)24 b(w)n(ell-founded,)g(i.e.)35 b(ev)n(ery)23 b(sequence)g(of)g (computation)h(steps)f(is)h(terminating,)51 4545 y(b)n(y)j(sho)n(wing)f (that)i(eac)n(h)f(step)h(decreases)e(the)i(depth)g Fq(d)p Fr(\()p Fq(e)p Fr(\))g(\(De\014nition)h(4\).)51 4713 y Fh(Lemma)g(3.1)41 b Fo(If)30 b Fq(e)23 b(,)-14 b Fp(!)794 4682 y Fn(1)794 4735 y Fm(S)865 4713 y Fq(e)904 4682 y Fe(0)957 4713 y Fo(then)29 b Fq(F)12 b(V)19 b Fr(\()p Fq(e)1344 4682 y Fe(0)1367 4713 y Fr(\))24 b(=)e Fq(F)12 b(V)19 b Fr(\()p Fq(e)p Fr(\))p Fo(.)51 4880 y(Pr)l(o)l(of)p Fr(.)46 b(Consider)30 b(the)h(cases)e(in)i(the)f(de\014nition)h(of)g Fq(,)-14 b Fp(!)1826 4850 y Fn(1)1826 4903 y Fm(S)1874 4880 y Fr(.)45 b(If)31 b(the)g(computation)f(is)h(b)n(y)f(1.1)f(or)h (1.2,)g(then)h Fq(F)12 b(V)19 b Fr(\()p Fq(e)p Fr(\))28 b(=)f Fp(;)h Fr(=)51 4980 y Fq(F)12 b(V)18 b Fr(\()p Fq(e)253 4950 y Fe(0)277 4980 y Fr(\).)37 b(In)28 b(the)g(cases)f(4,5)g (use)g(the)h(induction)h(h)n(yp)r(othesis.)36 b(The)28 b(same)f(w)n(orks)f(in)i(case)f(2)h(when)f Fq(p)d Fp(6)p Fr(=)e Fg(App)28 b Fr(or)f Fq(p)c Fr(=)g Fg(App)51 5080 y Fr(but)28 b(the)g(second)f(argumen)n(t)f(is)i(c)n(hanged.)51 5179 y(Let)g(no)n(w)g Fq(e)23 b Fr(=)h Fg(App)p Fq(P)12 b(t)28 b Fr(and)g Fq(e)990 5149 y Fe(0)1037 5179 y Fr(=)c Fg(App)p Fr(\()p Fq(P)1364 5149 y Fe(0)1388 5179 y Fq(;)14 b(t)p Fr(\))28 b(with)h Fq(P)36 b(,)-14 b Fp(!)1886 5149 y Fn(1)1886 5202 y Fm(S)1958 5179 y Fq(P)2023 5149 y Fe(0)2075 5179 y Fr(\(recall)27 b(the)i(de\014nition)f(6\).)39 b(Then)29 b Fq(P)40 b Fr(is)28 b(an)g Fq(\017)p Fr(-term,)g(since)51 5279 y(1-v)-5 b(ariables)25 b(are)i(not)g(reducible.)51 5379 y(1.)36 b Fq(P)k Fr(is)27 b(canonical.)36 b(Then)28 b Fq(P)1007 5348 y Fe(0)1053 5379 y Fr(=)22 b Fq(\025z)t(G)h Fp(2)h Fr(I)-14 b(B)1473 5391 y Fn(1)1510 5379 y Fq(;)42 b(F)12 b(V)18 b Fr(\()p Fq(G)p Fr(\))24 b(=)f Fp(f)p Fq(z)t Fp(g)j Fr(and)h Fq(e)2300 5348 y Fe(0)2346 5379 y Fr(=)c Fq(G)p Fr([)p Fq(t)p Fr(].)37 b(Hence)28 b Fq(F)12 b(V)19 b Fr(\()p Fq(e)3085 5348 y Fe(0)3108 5379 y Fr(\))k(=)g Fq(F)12 b(V)19 b Fr(\()p Fq(t)p Fr(\))k(=)g Fq(F)12 b(V)19 b Fr(\()p Fq(e)p Fr(\).)51 5503 y(2.)35 b Fq(P)h Fr(is)24 b(not)g(canonical.)35 b(Then)24 b Fq(e)1112 5473 y Fe(0)1158 5503 y Fr(=)f Fg(App)p Fq(P)1452 5473 y Fe(0)1475 5503 y Fq(t)i Fr(and)f Fq(F)12 b(V)18 b Fr(\()p Fq(e)1890 5473 y Fe(0)1914 5503 y Fr(\))23 b(=)g Fq(F)12 b(V)18 b Fr(\()p Fq(P)2285 5473 y Fe(0)2309 5503 y Fr(\))12 b Fp([)g Fq(F)g(V)19 b Fr(\()p Fq(t)p Fr(\))2669 5456 y Fm(I)5 b(H)2683 5503 y Fr(=)37 b Fq(F)12 b(V)19 b Fr(\()p Fq(P)12 b Fr(\))g Fp([)g Fq(F)g(V)19 b Fr(\()p Fq(t)p Fr(\))k(=)g Fq(F)12 b(V)19 b Fr(\()p Fq(e)p Fr(\))p Fq(:)166 b Fd(2)51 5671 y Fh(Lemma)29 b(3.2)41 b Fo(F)-6 b(or)30 b(e)l(ach)g(arithmetic)l(al)i(expr)l(ession)e Fq(e)f Fo(one)h(has:)51 5770 y(a\))g Fq(d)p Fr(\()p Fq(e)p Fr(\))23 b Fq(<)g(!)s Fo(,)51 5870 y(b\))29 b Fq(d)p Fr(\()p Fq(e)p Fr([)p Fq(y)s(=t)p Fr(]\))23 b Fq(<)g Fr(\()p Fq(d)p Fr(\()p Fq(t)p Fr(\))c(+)f(1\))h Fp(\001)f Fq(!)s Fo(.)1979 6119 y Fr(7)p eop %%Page: 8 8 8 7 bop 51 614 a Fo(Pr)l(o)l(of)p Fr(.)37 b(a\))25 b(is)h(ob)n(vious,)e (since)h Fg(App)h Fr(do)r(es)f(not)h(o)r(ccur)e(in)i Fq(e)f Fr(and)g(all)h(clauses)e(of)h(the)h(de\014nition)g(4)f(except)g (3)g(add)h(only)f(\014nite)51 714 y(amoun)n(t.)51 814 y(b\))j(Induction)g(on)f Fq(e)p Fr(.)37 b(Let)27 b Fq(y)f Fp(2)d Fq(F)12 b(V)19 b Fr(\()p Fq(e)p Fr(\),)28 b(since)g(otherwise)e Fq(d)p Fr(\()p Fq(e)p Fr([)p Fq(y)s(=t)p Fr(]\))d(=)g Fq(d)p Fr(\()p Fq(e)p Fr(\))g Fq(<)g(!)s Fr(.)36 b(Set)28 b Fq(\013)c Fr(:=)f(\()p Fq(d)p Fr(\()p Fq(t)p Fr(\))c(+)f(1\))g Fp(\001)h Fq(!)s Fr(.)51 913 y(1.)36 b(If)28 b Fq(e)23 b Fr(=)f Fq(y)31 b Fr(then)d Fq(d)p Fr(\()p Fq(e)p Fr([)p Fq(y)s(=t)p Fr(]\))23 b(=)f Fq(d)p Fr(\()p Fq(t)p Fr(\))i Fq(<)f(d)p Fr(\()p Fq(t)p Fr(\))c(+)f(1)23 b Fq(<)f(\013)p Fr(.)51 1013 y(2.)41 b(If)29 b Fq(e)c Fr(=)g Fq(pe)476 1025 y Fn(1)527 1013 y Fq(:)14 b(:)g(:)f(e)676 1025 y Fm(n)750 1013 y Fr(with)30 b Fq(p)25 b Fp(6)p Fr(=)g Fg(App)30 b Fr(then)f Fq(d)p Fr(\()p Fq(e)p Fr([)p Fq(t)p Fr(]\))d(=)g Fq(d)p Fr(\()p Fq(pe)1954 1025 y Fn(1)1991 1013 y Fr([)p Fq(t)p Fr(])14 b Fq(:)g(:)g(:)f(e)2230 1025 y Fm(n)2275 1013 y Fr([)p Fq(t)p Fr(]\))26 b(=)f Fq(d)p Fr(\()p Fq(e)2613 1025 y Fn(1)2651 1013 y Fr([)p Fq(t)p Fr(]\)#)14 b Fq(:)g(:)g(:)g Fr(#)p Fq(d)p Fr(\()p Fq(e)3136 1025 y Fm(n)3181 1013 y Fr([)p Fq(t)p Fr(]\))26 b Fq(<)f(\013)p Fr(,)30 b(since)f(b)n(y)g(the)51 1112 y(induction)f(h)n(yp)r(othesis)f(\(IH\))h Fq(d)p Fr(\()p Fq(e)1127 1124 y Fm(i)1155 1112 y Fr([)p Fq(t)p Fr(]\))23 b Fq(<)g(\013)28 b Fr(and)g Fq(\013)g Fr(is)f(closed)g(under)h(#.)51 1251 y(3.)36 b(If)28 b Fq(e)23 b Fr(=)f Fq(\017xF)40 b Fr(then)28 b(w)n(e)g(can)f(assume)g Fq(x)c Fp(62)h Fq(F)12 b(V)18 b Fr(\()p Fq(t)p Fr(\))29 b(and)e Fq(d)p Fr(\()p Fq(e)p Fr([)p Fq(t)p Fr(]\))d(=)e Fq(d)p Fr(\()p Fq(\017xF)12 b Fr([)p Fq(t)p Fr(]\))24 b Fp(\024)f Fr(1)18 b(+)g Fq(d)p Fr(\()p Fq(F)12 b Fr([)p Fq(t)p Fr(]\))3062 1190 y Fm(I)5 b(H)3076 1251 y Fq(<)37 b Fr(1)18 b(+)g Fq(\013)23 b Fr(=)g Fq(\013)p Fr(.)51 1351 y(4.)36 b(If)28 b Fq(e)23 b Fr(=)f Fq(\025z)t(F)12 b Fr(,)28 b(apply)f(IH.)194 b Fd(2)51 1533 y Fh(Lemma)29 b(3.3)41 b Fo(If)30 b Fq(e)23 b(,)-14 b Fp(!)794 1503 y Fn(1)794 1556 y Fm(S)865 1533 y Fq(e)904 1503 y Fe(0)957 1533 y Fo(then)29 b Fq(d)p Fr(\()p Fq(e)1255 1503 y Fe(0)1279 1533 y Fr(\))23 b Fq(<)g(d)p Fr(\()p Fq(e)p Fr(\))p Fo(.)51 1716 y(Pr)l(o)l(of)28 b Fr(b)n(y)g(induction)g(on)f(the)h(de\014nition)g(of)g Fq(,)-14 b Fp(!)1570 1686 y Fn(1)1570 1739 y Fm(S)1618 1716 y Fr(.)51 1815 y(1.1,)26 b(1.2.)36 b(If)28 b Fq(e)g Fr(is)f(a)g(canonical)g Fq(\017)p Fr(-term)g(then)h Fq(d)p Fr(\()p Fq(e)1603 1785 y Fe(0)1626 1815 y Fr(\))c(=)e(0)h Fq(<)g Fr(1)f(=)h Fq(d)p Fr(\()p Fq(e)p Fr(\).)51 1915 y(2.)36 b(If)28 b Fq(e)23 b Fr(=)f Fq(pe)465 1927 y Fn(1)516 1915 y Fq(:)14 b(:)g(:)g(e)666 1927 y Fm(n)710 1915 y Fq(;)42 b(e)814 1885 y Fe(0)860 1915 y Fr(=)23 b Fq(pe)1029 1927 y Fn(1)1079 1915 y Fq(:)14 b(:)g(:)g(e)1229 1885 y Fe(0)1229 1937 y Fm(i)1270 1915 y Fq(:)g(:)g(:)g(e)1420 1927 y Fm(n)1492 1915 y Fr(and)28 b Fq(e)1693 1927 y Fm(i)1743 1915 y Fq(,)-14 b Fp(!)1835 1885 y Fn(1)1835 1938 y Fm(S)1906 1915 y Fq(e)1945 1885 y Fe(0)1945 1937 y Fm(i)2000 1915 y Fr(,)28 b(then:)51 2056 y(2.1.)36 b(if)28 b Fq(p)23 b Fp(6)p Fr(=)f Fg(App)28 b Fr(then)g Fq(d)p Fr(\()p Fq(e)917 2026 y Fe(0)941 2056 y Fr(\))23 b(=)g Fq(d)p Fr(\()p Fq(e)1198 2068 y Fn(1)1235 2056 y Fr(\)#)14 b Fq(:)g(:)g(:)g Fr(#)p Fq(d)p Fr(\()p Fq(e)1644 2026 y Fe(0)1644 2077 y Fm(i)1672 2056 y Fr(\)#)g Fq(:)g(:)g(:)g Fr(#)p Fq(d)p Fr(\()p Fq(e)2081 2068 y Fm(n)2126 2056 y Fr(\))2182 1994 y Fm(I)5 b(H)2196 2056 y Fq(<)37 b(d)p Fr(\()p Fq(e)2412 2068 y Fn(1)2449 2056 y Fr(\)#)14 b Fq(:)g(:)g(:)g Fr(#)p Fq(d)p Fr(\()p Fq(e)2858 2068 y Fm(n)2904 2056 y Fr(\))23 b(=)g Fq(d)p Fr(\()p Fq(e)p Fr(\))51 2195 y(2.2.)36 b(if)28 b Fq(e)22 b Fr(=)h Fg(App)p Fq(P)12 b(t)28 b Fr(and)f Fq(e)906 2165 y Fe(0)952 2195 y Fr(=)c Fg(App)p Fq(P)1246 2165 y Fe(0)1269 2195 y Fq(t)p Fr(,)28 b(then)g Fq(d)p Fr(\()p Fq(e)1653 2165 y Fe(0)1676 2195 y Fr(\))c(=)e Fq(!)f Fp(\001)e Fr(\()p Fq(d)p Fr(\()p Fq(P)2106 2165 y Fe(0)2130 2195 y Fr(\)#)p Fq(d)p Fr(\()p Fq(t)p Fr(\)\))2424 2134 y Fm(I)5 b(H)2439 2195 y Fq(<)36 b(!)21 b Fp(\001)e Fr(\()p Fq(d)p Fr(\()p Fq(P)12 b Fr(\)#)p Fq(d)p Fr(\()p Fq(t)p Fr(\)\))25 b(=)e Fq(d)p Fr(\()p Fq(e)p Fr(\).)51 2295 y(2.3.)36 b(if)28 b Fq(e)22 b Fr(=)h Fg(App)p Fq(P)12 b(t)28 b Fr(and)f Fq(e)906 2264 y Fe(0)952 2295 y Fr(=)c Fg(App)p Fq(P)12 b(t)1276 2264 y Fe(0)1299 2295 y Fr(,)28 b(one)f(pro)r(ceeds)g(as)g(in)g(the)h(case)f(2.2.)51 2394 y(3.)36 b(Let)28 b Fq(e)23 b Fr(=)f Fg(App)p Fq(P)12 b(t)28 b Fr(and)f Fq(e)914 2364 y Fe(0)960 2394 y Fr(=)c Fq(G)p Fr([)p Fq(t)p Fr(].)37 b(Note)28 b(that)g Fp(8)p Fq(\013)22 b(<)h(!)1895 2364 y Fm(!)1943 2394 y Fr(\()p Fq(\013)c Fp(\001)f Fq(!)26 b Fp(\024)d Fq(!)e Fp(\001)d Fq(\013)p Fr(\).)51 2494 y(Then)27 b(b)n(y)h(Lemma)f(3.2)g Fq(d)p Fr(\()p Fq(e)927 2464 y Fe(0)950 2494 y Fr(\))d Fq(<)e Fr(\()p Fq(d)p Fr(\()p Fq(t)p Fr(\))e(+)e(1\))g Fp(\001)g Fq(!)26 b Fp(\024)d Fq(!)e Fp(\001)d Fr(\()p Fq(d)p Fr(\()p Fq(t)p Fr(\))i(+)e(1\))23 b Fp(\024)f Fq(!)g Fp(\001)c Fr(\()p Fq(d)p Fr(\()p Fq(P)12 b Fr(\)#)p Fq(d)p Fr(\()p Fq(t)p Fr(\)\))25 b(=)e Fq(d)p Fr(\()p Fq(e)p Fr(\).)51 2593 y(4,)33 b(5.)50 b(If)33 b Fq(e)e Fr(=)f Fq(\017\030)t(F)r(;)14 b(e)722 2563 y Fe(0)776 2593 y Fr(=)31 b Fq(\017\030)t(F)1011 2563 y Fe(0)1066 2593 y Fr(or)h Fq(e)e Fr(=)h Fq(\025z)t(F)r(;)14 b(e)1560 2563 y Fe(0)1614 2593 y Fr(=)31 b Fq(\025z)t(F)1866 2563 y Fe(0)1921 2593 y Fr(with)i Fq(F)43 b(,)-14 b Fp(!)2303 2563 y Fn(1)2303 2616 y Fm(S)2382 2593 y Fq(F)2447 2563 y Fe(0)2502 2593 y Fr(then,)35 b(since)d Fq(F)12 b(V)18 b Fr(\()p Fq(F)12 b Fr(\))32 b(=)e Fq(F)12 b(V)19 b Fr(\()p Fq(F)3545 2563 y Fe(0)3569 2593 y Fr(\),)34 b(w)n(e)d(ha)n(v)n(e)51 2734 y Fq(d)p Fr(\()p Fq(e)165 2704 y Fe(0)188 2734 y Fr(\))23 b(=)g Fq(j)g Fr(+)18 b Fq(d)p Fr(\()p Fq(F)611 2704 y Fe(0)635 2734 y Fr(\))690 2673 y Fm(I)5 b(H)705 2734 y Fq(<)36 b(j)24 b Fr(+)18 b Fq(d)p Fr(\()p Fq(F)12 b Fr(\))23 b(=)g Fq(d)p Fr(\()p Fq(e)p Fr(\))28 b(for)f(some)g Fq(j)h Fr(=)23 b(0)p Fq(;)14 b Fr(1.)202 b Fd(2)51 2883 y Fr(W)-7 b(e)28 b(are)e(going)h(to)g(pro)n(v)n(e)f(that)i Fq(,)-14 b Fp(!)1151 2853 y Fn(1)1151 2906 y Fm(S)1227 2883 y Fr(is)27 b(lo)r(cally)g(con\015uen)n(t.)51 3066 y Fh(Lemma)i(3.4)41 b Fo(If)30 b Fq(e)f Fo(is)h(arithmetic)l(al)i(and)e Fq(t)23 b(,)-14 b Fp(!)1563 3036 y Fn(1)1563 3089 y Fm(S)1634 3066 y Fq(t)1664 3036 y Fe(0)1717 3066 y Fo(then)30 b Fq(e)p Fr([)p Fq(t)p Fr(])23 b Fq(,)-14 b Fp(!)2132 3078 y Fm(S)2203 3066 y Fq(e)p Fr([)p Fq(t)2295 3036 y Fe(0)2318 3066 y Fr(])p Fo(.)51 3249 y(Pr)l(o)l(of)28 b Fr(is)g(routine.)36 b Fd(2)51 3415 y Fh(Lemma)29 b(3.5)41 b Fo(If)30 b Fq(e)23 b(,)-14 b Fp(!)794 3385 y Fn(1)794 3438 y Fm(S)865 3415 y Fq(e)904 3385 y Fe(0)957 3415 y Fo(and)30 b Fq(e)23 b(,)-14 b Fp(!)1272 3385 y Fn(1)1272 3438 y Fm(S)1343 3415 y Fq(e)1382 3385 y Fe(00)1454 3415 y Fo(then)29 b(ther)l(e)h(is)g(an)g(expr)l(ession)g Fq(u)g Fo(such)f(that)h Fq(e)2928 3385 y Fe(0)2974 3415 y Fq(,)-14 b Fp(!)3066 3427 y Fm(S)3138 3415 y Fq(u)29 b Fo(and)h Fq(e)3415 3385 y Fe(00)3480 3415 y Fq(,)-14 b Fp(!)3572 3427 y Fm(S)3644 3415 y Fq(u)29 b Fo(.)51 3581 y(Pr)l(o)l(of)f Fr(b)n(y)g(induction)g(on)f Fq(e)p Fr(.)51 3680 y(1.)36 b(If)28 b Fq(e)23 b Fp(2)g Fq(dom)p Fr(\()p Fq(S)5 b Fr(\))29 b(then)f Fq(e)908 3650 y Fe(0)954 3680 y Fr(=)22 b Fq(e)1080 3650 y Fe(00)1122 3680 y Fr(.)51 3780 y(2.)36 b(Let)28 b Fq(e)23 b Fr(=)f Fq(pe)531 3792 y Fn(1)582 3780 y Fq(:)14 b(:)g(:)f(e)731 3792 y Fm(i)773 3780 y Fq(:)h(:)g(:)f(e)922 3792 y Fm(n)967 3780 y Fq(;)42 b(e)1071 3750 y Fe(0)1117 3780 y Fr(=)22 b Fq(pe)1285 3792 y Fn(1)1336 3780 y Fq(:)14 b(:)g(:)f(e)1485 3750 y Fe(0)1485 3802 y Fm(i)1527 3780 y Fq(:)h(:)g(:)f(e)1676 3792 y Fm(n)1749 3780 y Fr(and)27 b Fq(e)1949 3792 y Fm(i)2000 3780 y Fq(,)-14 b Fp(!)2092 3750 y Fn(1)2092 3803 y Fm(S)2163 3780 y 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b(e)28 b(pro)n(v)n(e)e(next)h(that)h Fq(,)-14 b Fp(!)876 5835 y Fm(S)952 5823 y Fr(is)28 b(preserv)n(ed)e(under)h(substitution.) 38 b(The)27 b(main)h(problem)f(will)h(b)r(e)g(with)g Fq(\025)p Fr(-terms.)1979 6119 y(8)p eop %%Page: 9 9 9 8 bop 51 614 a Fh(Lemma)29 b(3.7)51 714 y Fo(a\))h Fq(e)22 b(,)-14 b Fp(!)310 684 y Fn(1)310 737 y Fm(S)382 714 y Fq(e)421 684 y Fe(0)473 714 y Fo(implies)32 b Fq(e)p Fr([)p Fq(\021)s(=u)p Fr(])22 b Fq(,)-14 b Fp(!)1091 726 y Fm(S)1162 714 y Fq(e)1201 684 y Fe(0)1224 714 y Fr([)p Fq(\021)s(=u)p Fr(])p Fo(.)51 814 y(b\))29 b Fq(u)23 b(,)-14 b Fp(!)315 783 y Fn(1)315 836 y Fm(S)386 814 y Fq(u)434 783 y Fe(0)487 814 y Fo(implies)31 b Fq(e)p Fr([)p Fq(\021)s(=u)p Fr(])22 b Fq(,)-14 b Fp(!)1104 826 y Fm(S)1176 814 y Fq(e)p Fr([)p Fq(\021)s(=u)1372 783 y Fe(0)1394 814 y Fr(])p Fo(.)51 913 y(c\))29 b Fq(e)p Fr([)p Fq(\021)s(=u)p Fr(])23 b Fq(,)-14 b Fp(!)486 925 y Fm(S)557 913 y Fp(j)p Fq(e)p Fp(j)642 925 y Fm(S)690 913 y Fr([)p Fq(\021)s(=)p Fp(j)p Fq(u)p Fp(j)893 925 y Fm(S)940 913 y Fr(])p Fq(:)51 1082 y Fo(Pr)l(o)l(of)p Fr(.)38 b(W)-7 b(e)28 b(\014rst)f(pro)n(v)n(e)f(a\))h(and)h(b\))g (under)f(the)h(assumption)g(that)f Fq(u)h Fr(is)f(not)h(a)f Fq(\025)p Fr(-term.)51 1182 y(a\))g(W)-7 b(e)28 b(use)f(induction)h(on) g(the)g(de\014nition)g(of)f Fq(,)-14 b Fp(!)1621 1152 y Fn(1)1621 1205 y Fm(S)1670 1182 y Fr(.)51 1281 y(The)27 b(only)g(non-trivial)g(case)g(is)g Fq(e)c Fr(=)f Fg(App)q Fq(P)12 b(t;)41 b(e)1560 1251 y Fe(0)1606 1281 y Fr(=)23 b Fq(G)p Fr([)p Fq(z)t(=t)p Fr(])p Fq(;)40 b(P)35 b(,)-14 b Fp(!)2163 1251 y Fn(1)2163 1304 y Fm(S)2235 1281 y Fq(\025z)t(G)p Fr(.)51 1381 y(Then)21 b(since)g Fq(P)34 b Fr(is)21 b(closed)f(and)i Fq(F)12 b(V)18 b Fr(\()p Fq(G)p Fr(\))24 b(=)f Fp(f)p Fq(z)t Fp(g)p Fr(,)e(one)g(has)g Fq(e)p Fr([)p Fq(u)p Fr(])h(=)h Fg(App)p Fq(P)12 b(t)p Fr([)p Fq(u)p Fr(])p Fq(;)35 b(e)2517 1351 y Fe(0)2540 1381 y Fr([)p Fq(u)p Fr(])22 b(=)h Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)p Fr(]].)34 b(Hence)21 b(b)n(y)g(the)h(de\014nition)51 1481 y(of)27 b Fq(,)-14 b Fp(!)237 1451 y Fn(1)237 1504 y Fm(S)313 1481 y Fr(one)27 b(has)g Fq(e)p Fr([)p Fq(u)p Fr(])c Fq(,)-14 b Fp(!)861 1451 y Fn(1)861 1504 y Fm(S)932 1481 y Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)p Fr(]])22 b(=)h Fq(e)1401 1451 y Fe(0)1424 1481 y Fr([)p Fq(u)p Fr(].)51 1580 y(b\))k(W)-7 b(e)28 b(use)f(induction)h(on)f Fq(e)g Fr(assuming)g Fq(\021)f Fp(2)e Fq(F)12 b(V)18 b Fr(\()p Fq(e)p Fr(\).)37 b(Again)27 b(the)h(only)f(non-trivial)f (case)h(is)g Fq(e)c Fr(=)f Fg(App)q Fq(P)12 b(t;)41 b(P)12 b Fr([)p Fq(u)3613 1550 y Fe(0)3635 1580 y Fr(])24 b(=)e Fq(\025z)t(G)p Fr(.)51 1680 y(Then)37 b Fq(e)p Fr([)p Fq(u)p Fr(])i(=)g Fg(App)p Fq(P)12 b Fr([)p Fq(u)p Fr(])p Fq(t)p Fr([)p Fq(u)p Fr(])p Fq(;)51 b(e)p Fr([)p Fq(u)1161 1650 y Fe(0)1183 1680 y Fr(])40 b(=)f Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)1624 1650 y Fe(0)1646 1680 y Fr(]])f(and)f(b)n(y)g(IH)h Fq(P)12 b Fr([)p Fq(u)p Fr(])39 b Fq(,)-14 b Fp(!)2446 1692 y Fm(S)2534 1680 y Fq(\025z)t(G;)52 b(t)p Fr([)p Fq(u)p Fr(])39 b Fq(,)-14 b Fp(!)3020 1692 y Fm(S)3108 1680 y Fq(t)p Fr([)p Fq(u)3209 1650 y Fe(0)3231 1680 y Fr(].)67 b(Since)38 b Fq(u)f Fr(is)g(not)h(a)51 1780 y Fq(\025)p Fr(-term,)26 b(w)n(e)g(ha)n(v)n(e)f Fq(\025z)t(G)e Fp(2)g Fr(I)-14 b(B)990 1792 y Fn(1)1028 1780 y Fr(.)36 b(Hence)26 b(there)g(exists)g(a)g(canonical)f Fq(\017)p Fr(-term)g Fq(Q)h Fr(suc)n(h)g(that)h Fq(P)12 b Fr([)p Fq(u)p Fr(])22 b Fq(,)-14 b Fp(!)3190 1792 y Fm(S)3262 1780 y Fq(Q)22 b(,)-14 b Fp(!)3442 1750 y Fn(1)3442 1802 y Fm(S)3514 1780 y Fq(\025z)t(G)p Fr(.)36 b(Hence)51 1879 y Fq(e)p Fr([)p Fq(u)p Fr(])22 b Fq(,)-14 b Fp(!)298 1891 y Fm(S)369 1879 y Fg(App)q Fq(Qt)p Fr([)p Fq(u)678 1849 y Fe(0)700 1879 y Fr(])23 b Fq(,)-14 b Fp(!)838 1849 y Fn(1)838 1902 y Fm(S)910 1879 y Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)1184 1849 y Fe(0)1205 1879 y Fr(]])24 b(=)e Fq(e)p Fr([)p Fq(u)1472 1849 y Fe(0)1495 1879 y Fr(].)51 1979 y(No)n(w)27 b(w)n(e)g(assume)g(that)h Fq(u)f Fr(is)g(a)h Fq(\025)p Fr(-term.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(to)h(consider)g(t)n(w)n(o)g(additional)g(cases.)51 2079 y(a'\))f Fq(e)d Fr(=)g Fg(App)p Fq(P)12 b(t)27 b Fr(and)f Fq(P)12 b Fr([)p Fq(u)p Fr(])23 b(=)g Fq(\025z)t(G)p Fr(.)36 b(Then)27 b Fq(P)35 b Fr(=)23 b Fq(\021)30 b Fr(\(otherwise)c Fq(P)39 b Fr(con)n(tains)26 b Fq(\025)p Fr(\))i(and)f(th)n(us)g Fq(u)22 b Fr(=)h Fq(\025z)t(G)p Fr(,)k Fq(e)3367 2048 y Fe(0)3413 2079 y Fr(=)c Fg(App)p Fq(P)12 b(t)3737 2048 y Fe(0)3787 2079 y Fr(with)51 2178 y Fq(t)23 b(,)-14 b Fp(!)196 2148 y Fn(1)196 2201 y Fm(S)268 2178 y Fq(t)298 2148 y Fe(0)321 2178 y Fr(.)38 b(Hence)28 b Fq(e)p Fr([)p Fq(u)p Fr(])23 b(=)g Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)p Fr(]],)k Fq(e)1282 2148 y Fe(0)1305 2178 y Fr([)p Fq(u)p Fr(])c(=)g Fq(G)p Fr([)p Fq(z)t(=t)1713 2148 y Fe(0)1735 2178 y Fr([)p Fq(u)p Fr(]])28 b(and)g Fq(t)p Fr([)p Fq(u)p Fr(])23 b Fq(,)-14 b Fp(!)2281 2190 y Fm(S)2353 2178 y Fq(t)2383 2148 y Fe(0)2406 2178 y Fr([)p Fq(u)p Fr(])28 b(b)n(y)f(IH.)h(Since)h Fq(t)p Fr([)p Fq(u)p Fr(])e(is)h(not)g(a)f Fq(\025)p Fr(-term,)h(b)n(y)g(b\)) 51 2278 y(w)n(e)f(obtain)g Fq(e)p Fr([)p Fq(u)p Fr(])22 b Fq(,)-14 b Fp(!)678 2290 y Fm(S)750 2278 y Fq(e)789 2248 y Fe(0)812 2278 y Fr([)p Fq(u)p Fr(].)51 2377 y(b'\))30 b Fq(e)c Fr(=)h Fg(App)p Fq(P)12 b(t)30 b Fr(and)f Fq(P)12 b Fr([)p Fq(u)p Fr(])27 b(=)f Fq(\025z)t(G)p Fr(.)44 b(Then)30 b Fq(P)38 b Fr(=)27 b Fq(\021)33 b Fr(\(otherwise)c Fq(P)42 b Fr(con)n(tains)29 b Fq(\025)p Fr(\))h(and)g(th)n(us)g Fq(u)c Fr(=)h Fq(\025z)t(G;)14 b(u)3437 2347 y Fe(0)3486 2377 y Fr(=)27 b Fq(\025z)t(G)3734 2347 y Fe(0)3787 2377 y Fr(with)51 2477 y Fq(G)c(,)-14 b Fp(!)231 2489 y Fm(S)302 2477 y Fq(G)367 2447 y Fe(0)391 2477 y Fr(.)37 b(Hence)26 b Fq(e)p Fr([)p Fq(u)p Fr(])d(=)f Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)p Fr(]])k(and)g Fq(e)p Fr([)p Fq(u)1555 2447 y Fe(0)1577 2477 y Fr(])e(=)e Fq(G)1776 2447 y Fe(0)1800 2477 y Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)2009 2447 y Fe(0)2030 2477 y Fr(]].)37 b(By)26 b(IH)h Fq(t)p Fr([)p Fq(u)p Fr(])c Fq(,)-14 b Fp(!)2623 2489 y Fm(S)2694 2477 y Fq(t)p Fr([)p Fq(u)2795 2447 y Fe(0)2818 2477 y Fr(].)37 b(Since)27 b Fq(t)p Fr([)p Fq(u)p Fr(])f(is)g(not)h(a)f Fq(\025)p Fr(-term,)h(b)n(y)51 2577 y(a\),)g(b\))h(w)n(e)f(obtain)h Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)p Fr(]])22 b Fq(,)-14 b Fp(!)1096 2589 y Fm(S)1167 2577 y Fq(G)1232 2547 y Fe(0)1256 2577 y Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)p Fr(]])21 b Fq(,)-14 b Fp(!)1624 2589 y Fm(S)1696 2577 y Fq(G)1761 2547 y Fe(0)1784 2577 y Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)1993 2547 y Fe(0)2015 2577 y Fr(]].)51 2698 y(c\))25 b(By)f(induction)h(on)g (the)g(n)n(um)n(b)r(er)g(of)g(computation)f(steps,)i(w)n(e)e(obtain)h (v)n(ersions)e(of)h(a\),b\))i(with)f Fq(,)-14 b Fp(!)3251 2668 y Fn(1)3251 2721 y Fm(S)3324 2698 y Fr(replaced)24 b(b)n(y)h Fq(,)-14 b Fp(!)3854 2710 y Fm(S)3925 2698 y Fq(:)51 2798 y Fr(These)27 b(imply)h(c\):)37 b Fq(e)p Fr([)p Fq(u)p Fr(])22 b Fq(,)-14 b Fp(!)900 2810 y Fm(S)972 2798 y Fp(j)p Fq(e)p Fp(j)1057 2810 y Fm(S)1105 2798 y Fr([)p Fp(j)p Fq(u)p Fp(j)p Fr(])22 b Fq(,)-14 b Fp(!)1359 2810 y Fm(S)1431 2798 y Fp(j)p Fq(e)p Fp(j)1516 2810 y Fm(S)1564 2798 y Fr([)p Fp(j)p Fq(u)p Fp(j)1681 2810 y Fm(S)1729 2798 y Fr(])27 b(.)37 b Fd(2)217 2942 y Fr(The)e(next)h (prop)r(osition)f(clari\014es)g(the)h(structure)f(of)h(normal)f(forms)g (for)g(partial)g(substitutions)h Fq(S)5 b Fr(,)38 b(where)d(some)51 3042 y(canonical)26 b Fq(\017)p Fr(-terms)h(are)f(irreducible.)37 b(It)27 b(is)h(needed)g(in)g(section)f(6.3.)51 3211 y Fh(Lemma)i(3.8)51 3310 y Fo(a\))h(If)g Fq(v)j Fo(is)d(an)g Fq(S)5 b Fo(-irr)l(e)l(ducible)30 b(subterm)f(of)i Fq(e;)43 b(d)p Fr(\()p Fq(v)s Fr(\))24 b Fq(>)f Fr(0)29 b Fo(and)h Fq(e)23 b(,)-14 b Fp(!)2179 3280 y Fn(1)2179 3333 y Fm(S)2250 3310 y Fq(e)2289 3280 y Fe(0)2342 3310 y Fo(then)30 b Fq(v)j Fo(is)d(also)g(a)h(subterm)e(of)h Fq(e)3382 3280 y Fe(0)3405 3310 y Fo(.)51 3410 y(b\))f(If)i Fq(e)p Fr([)p Fq(\021)s(=u)p Fr(])d Fo(is)i Fq(S)5 b Fo(-c)l(omputable)30 b(and)h Fq(\021)26 b Fp(2)d Fq(F)12 b(V)19 b Fr(\()p Fq(e)p Fr(\))30 b Fo(then)g Fq(u)f Fo(is)h Fq(S)5 b Fo(-c)l(omputable.) 51 3510 y(c\))29 b(If)i Fq(e)e Fo(is)h Fq(S)5 b Fo(-c)l(omputable)30 b(and)g Fq(e)23 b(,)-14 b Fp(!)p 1226 3488 49 4 v 27 x Fm(S)1297 3510 y Fq(e)1336 3479 y Fe(0)1389 3510 y Fo(then)30 b Fq(e)22 b(,)-14 b Fp(!)1727 3522 y Fm(S)1799 3510 y Fq(e)1838 3479 y Fe(0)1861 3510 y Fo(.)51 3679 y(Pr)l(o)l(of)p Fr(.)38 b(a\))27 b(W)-7 b(e)28 b(use)f(induction)h(on)f (the)g(de\014nition)h(of)g Fq(,)-14 b Fp(!)1873 3648 y Fn(1)1873 3701 y Fm(S)1921 3679 y Fr(.)37 b(Since)27 b Fq(e)g Fr(is)h(reducible,)f Fq(v)k Fr(has)26 b(to)i(b)r(e)g(a)e(prop) r(er)h(subterm)g(of)h Fq(e)51 3778 y Fr(and)f(th)n(us)h Fq(e)f Fr(cannot)g(b)r(e)h(a)f(canonical)g Fq(\017)p Fr(-term.)51 3878 y(1.)36 b(Let)27 b Fq(e)c Fr(=)f Fq(pe)530 3890 y Fn(1)581 3878 y Fq(:)14 b(:)g(:)g(e)731 3890 y Fm(i)772 3878 y Fq(:)g(:)g(:)f(e)921 3890 y Fm(n)966 3878 y Fq(;)41 b(e)1069 3848 y Fe(0)1115 3878 y Fr(=)23 b Fq(pe)1284 3890 y Fn(1)1334 3878 y Fq(:)14 b(:)g(:)g(e)1484 3848 y Fe(0)1484 3899 y Fm(i)1525 3878 y Fq(:)g(:)g(:)g(e)1675 3890 y Fm(n)1747 3878 y Fr(and)27 b Fq(e)1947 3890 y Fm(i)1997 3878 y Fq(,)-14 b Fp(!)2089 3848 y Fn(1)2089 3901 y Fm(S)2160 3878 y Fq(e)2199 3848 y Fe(0)2199 3899 y Fm(i)2254 3878 y Fr(.)37 b(Then)27 b(the)g(claim)g(follo)n(ws)f (immediately)h(from)g(the)51 3977 y(IH.)51 4077 y(2.)49 b Fq(e)30 b Fr(=)g Fg(App)p Fq(P)12 b(t;)46 b(e)673 4047 y Fe(0)726 4077 y Fr(=)30 b Fq(G)p Fr([)p Fq(z)t(=t)p Fr(])h(with)i Fq(P)42 b(,)-14 b Fp(!)1459 4047 y Fn(1)1459 4100 y Fm(S)1537 4077 y Fq(\025z)t(G)p Fr(.)50 b(Then)32 b Fq(P)44 b Fr(is)32 b(a)g(canonical)e Fq(\017)p Fr(-term)i Fp(6)p Fr(=)e Fq(v)s Fr(,)j(and)f(therefore)f Fq(v)k Fr(is)d(not)g(a)51 4177 y(subterm)27 b(of)h Fq(P)12 b Fr(.)36 b(Hence)28 b Fq(v)j Fr(is)d(a)f(subterm)g(of)h Fq(t)g Fr(and)f(th)n(us)h(of)f Fq(e)2017 4147 y Fe(0)2068 4177 y Fr(since)g Fq(z)k Fr(o)r(ccurs)c(free)g(in)h Fq(G)p Fr(.)51 4276 y(3.)36 b(The)28 b(cases)e Fq(e)d Fr(=)g Fq(\017\030)t(F)r(;)14 b(e)886 4246 y Fe(0)932 4276 y Fr(=)23 b Fq(\017\030)t(F)1159 4246 y Fe(0)1209 4276 y Fr(or)k Fq(e)c Fr(=)f Fq(\025z)t(F)r(;)14 b(e)1682 4246 y Fe(0)1729 4276 y Fr(=)22 b Fq(\025z)t(F)1972 4246 y Fe(0)1995 4276 y Fr(with)28 b Fq(F)35 b(,)-14 b Fp(!)2364 4246 y Fn(1)2364 4299 y Fm(S)2436 4276 y Fq(F)2501 4246 y Fe(0)2552 4276 y Fr(follo)n(w)26 b(immediately)i(from)f(IH.)51 4376 y(b\))36 b(Assume)g(for)f(con)n(tradiction)g(that)h Fq(d)p Fr(\()p Fp(j)p Fq(u)p Fp(j)1487 4388 y Fm(S)1535 4376 y Fr(\))i Fq(>)e Fr(0.)62 b(Then)36 b Fp(j)p Fq(u)p Fp(j)2152 4388 y Fm(S)2235 4376 y Fr(and)g(hence)g Fp(j)p Fq(e)p Fp(j)2729 4388 y Fm(S)2777 4376 y Fr([)p Fp(j)p Fq(u)p Fp(j)2894 4388 y Fm(S)2942 4376 y Fr(])g(con)n(tains)f(an)h Fq(S)5 b Fr(-irreducible,)51 4476 y(closed)27 b Fq(\017)p Fr(-subterm)g Fq(v)s Fr(.)38 b(By)28 b(Lemma)g(3.7c)f(and)g(Theorem)g (3.6)g(w)n(e)h(ha)n(v)n(e)f Fp(j)p Fq(e)p Fp(j)2430 4488 y Fm(S)2478 4476 y Fr([)p Fp(j)p Fq(u)p Fp(j)2595 4488 y Fm(S)2642 4476 y Fr(])d Fq(,)-14 b Fp(!)2781 4488 y Fm(S)2853 4476 y Fp(j)p Fq(e)p Fr([)p Fq(u)p Fr(])p Fp(j)3032 4488 y Fm(S)3080 4476 y Fr(,)28 b(and)f(therefore)g([b)n(y)h(a\)])g Fq(v)51 4575 y Fr(is)f(a)g(subterm)h(of)f Fp(j)p Fq(e)p Fr([)p Fq(u)p Fr(])p Fp(j)800 4587 y Fm(S)848 4575 y Fr(.)37 b(This)27 b(con)n(tradicts)g(the)h Fq(S)5 b Fr(-computabilit)n (y)27 b(of)g Fq(e)p Fr([)p Fq(u)p Fr(].)51 4675 y(c\))c(It)g(su\016ces) f(to)h(pro)n(v)n(e)e(the)j(Lemma)e(for)g(one)h(step)g(reductions.)35 b(Assume)22 b(that)i Fq(e)e Fr(is)h Fq(S)5 b Fr(-computable)22 b(and)h Fq(e)f(,)-14 b Fp(!)3579 4645 y Fn(1)p 3579 4665 V 3579 4714 a Fm(S)3651 4675 y Fq(e)3690 4645 y Fe(0)3736 4675 y Fr(holds.)51 4790 y(By)27 b(b\))h(ev)n(ery)e(canonical)h Fq(\017)p Fr(-subterm)g Fq(v)j Fr(of)e Fq(e)f Fr(is)h Fq(S)5 b Fr(-computable)27 b(and)g(therefore)g(in)h Fq(dom)p Fr(\()p Fq(S)5 b Fr(\).)37 b(Hence)28 b Fq(e)23 b(,)-14 b Fp(!)3446 4760 y Fn(1)3446 4813 y Fm(S)3517 4790 y Fq(e)3556 4760 y Fe(0)3579 4790 y Fr(.)37 b Fd(2)217 4934 y Fr(Let)27 b(us)h(establish)f(some)g(prop)r(erties)g(of)g (regular)f Fq(\025)p Fr(-terms)i(\(De\014nition)g(8\))g(needed)f(in)h (section)f(5.)51 5103 y Fh(Lemma)i(3.9)51 5203 y Fo(a\))h(If)g Fq(T)41 b Fo(is)30 b(r)l(e)l(gular)g(and)g Fq(T)k(,)-14 b Fp(!)1035 5173 y Fn(1)1035 5226 y Fm(S)1106 5203 y Fq(T)1167 5173 y Fe(0)1219 5203 y Fo(then)30 b Fq(T)1465 5173 y Fe(0)1517 5203 y Fo(is)g(r)l(e)l(gular.)51 5302 y(b\))f(If)i Fq(T)40 b Fo(is)30 b(a)h(close)l(d)f(r)l(e)l(gular)g Fq(\025)p Fo(-term)g(with)g Fq(d)p Fr(\()p Fq(T)12 b Fr(\))23 b(=)g(0)29 b Fo(then)g Fq(T)35 b Fp(2)23 b Fr(I)-14 b(B)2236 5314 y Fn(1)2273 5302 y Fo(.)51 5402 y(c\))29 b(If)i Fq(T)40 b Fo(is)30 b(a)h(close)l(d)f(r)l(e)l(gular)g Fq(\025)p Fo(-term)g(then)f Fp(j)p Fq(T)12 b Fp(j)p 1576 5380 V 28 x Fm(S)1647 5402 y Fp(2)23 b Fr(I)-14 b(B)1800 5414 y Fn(1)1837 5402 y Fo(.)51 5571 y(Pr)l(o)l(of)p Fr(.)38 b(W)-7 b(e)28 b(ha)n(v)n(e)e Fq(T)34 b Fr(=)23 b Fq(e)p Fr([)p Fq(\021)912 5583 y Fn(1)949 5571 y Fq(=u)1039 5583 y Fn(1)1075 5571 y Fq(;)14 b(:::;)g(\021)1259 5583 y Fm(n)1305 5571 y Fq(=u)1395 5583 y Fm(n)1439 5571 y Fr(])28 b(where)f Fq(u)1778 5583 y Fn(1)1815 5571 y Fq(;)14 b(:::;)g(u)2006 5583 y Fm(n)2078 5571 y Fr(are)27 b Fq(\017)p Fr(-terms)f(and)i(no)f(b)r(ound)h(1-v)-5 b(ariable)26 b(o)r(ccurs)g(in)i Fq(e)p Fr(.)51 5671 y(a\))37 b(Without)i(loss)e(of)h (generalit)n(y)e(w)n(e)h(ma)n(y)g(assume)g(that)i(either)e Fq(T)2289 5641 y Fe(0)2352 5671 y Fr(=)i Fq(e)p Fr([)p Fq(u)2566 5641 y Fe(0)2566 5691 y Fn(1)2603 5671 y Fq(;)14 b(u)2688 5683 y Fn(2)2724 5671 y Fq(;)g(:::;)g(u)2915 5683 y Fm(n)2960 5671 y Fr(])38 b(with)g Fq(u)3268 5683 y Fn(1)3345 5671 y Fq(,)-14 b Fp(!)3437 5641 y Fn(1)3437 5694 y Fm(S)3526 5671 y Fq(u)3574 5641 y Fe(0)3574 5691 y Fn(1)3648 5671 y Fr(or)37 b Fq(T)3821 5641 y Fe(0)3884 5671 y Fr(=)51 5770 y Fq(e)90 5740 y Fe(0)113 5770 y Fr([)p Fq(u)184 5782 y Fn(1)220 5770 y Fq(;)14 b(:::;)g(u)411 5782 y Fm(n)456 5770 y Fr(])29 b(with)h Fq(e)25 b(,)-14 b Fp(!)855 5740 y Fn(1)855 5793 y Fm(S)929 5770 y Fq(e)968 5740 y Fe(0)991 5770 y Fr(.)42 b(In)29 b(b)r(oth)h(cases)e(one)h (easily)f(sees)h(that)g Fq(T)2364 5740 y Fe(0)2416 5770 y Fr(is)g(again)f(regular.)40 b(\(Only)29 b(if)g Fq(u)3407 5740 y Fe(0)3407 5791 y Fn(1)3470 5770 y Fp(2)d Fr(I)-14 b(B)3626 5782 y Fn(1)3692 5770 y Fr(a)29 b(short)51 5870 y(argumen)n(t)d(is)i(needed.\))1979 6119 y(9)p eop %%Page: 10 10 10 9 bop 51 614 a Fr(b\))28 b(F)-7 b(or)28 b(con)n(tradiction)f(let)i (us)f(assume)g(that)h Fq(\021)1551 626 y Fm(i)1603 614 y Fp(2)c Fq(F)12 b(V)18 b Fr(\()p Fq(e)p Fr(\).)40 b(Then,)29 b(since)f Fq(T)40 b Fr(is)28 b(closed,)g Fq(u)2915 626 y Fm(i)2970 614 y Fr(is)h(a)f(closed)f Fq(\017)p Fr(-term)h(and)g(th)n (us)51 714 y(0)22 b Fq(<)h(d)p Fr(\()p Fq(u)326 726 y Fm(i)353 714 y Fr(\))h Fp(\024)e Fq(d)p Fr(\()p Fq(T)12 b Fr(\).)36 b(Hence)24 b(none)f(of)h Fq(\021)1292 726 y Fn(1)1329 714 y Fq(;)14 b(:::;)g(\021)1513 726 y Fm(n)1582 714 y Fr(o)r(ccurs)23 b(free)h(in)g Fq(e)f Fr(and)h(w)n(e)f(ha)n(v)n(e) f Fq(T)34 b Fr(=)23 b Fq(e)h Fr(whic)n(h)f(means)g(that)h Fq(T)35 b Fr(is)24 b(a)f(closed)51 814 y Fq(\025)p Fr(-term)k(con)n (taining)g(no)g(1-v)-5 b(ariables.)35 b(Hence)28 b Fq(T)34 b Fp(2)23 b Fr(I)-14 b(B)1773 826 y Fn(1)1810 814 y Fr(,)28 b(since)g Fq(d)p Fr(\()p Fq(T)12 b Fr(\))22 b(=)h(0.)51 913 y(c\))k(follo)n(ws)g(from)g(a\))h(and)f(b\).)65 b Fd(2)51 1188 y Fk(4)137 b(The)46 b(rank)f(function)51 1370 y Fr(The)37 b(rank)f(will)i(measure)e(nesting)i(of)f(b)r(ound)h(v) -5 b(ariables.)65 b(W)-7 b(e)38 b(extend)f(to)g(the)h Fq(\017)p Fr(-language)d(a)i(de\014nition)h(kno)n(wn)f(for)51 1469 y(Rami\014ed)27 b(Analysis)g([16)o(].)51 1569 y(Set)h Fq(o)p Fr(\()p Fq(x)p Fr(\))c(:=)f(0,)k Fq(o)p Fr(\()p Fq(X)7 b Fr(\))23 b(:=)g Fq(!)s Fr(,)k Fq(o)p Fr(\()p Fg(App)q Fr(\))c(:=)g Fq(!)s Fr(,)55 b Fq(o)p Fr(\()p Fq(p)p Fr(\))24 b(:=)e Fq(o)p Fr(\(0\))i(:=)e(0,)28 b(for)f Fq(p)22 b Fp(6)p Fr(=)h Fg(App)p Fr(.)51 1693 y(In)k(the)h(follo)n (wing)f Fq(\033)k Fr(denotes)c(elemen)n(ts)h(from)f(V)-7 b(ar)18 b Fp([)h(f\003g)p Fr(.)51 1876 y Fh(De\014nition)30 b(15)41 b Fr(De\014nition)29 b(of)e(rk)1191 1888 y Fm(\033)1235 1876 y Fr(\()p Fq(e)p Fr(\))51 1976 y Fo(1.)39 b(If)30 b Fq(\033)c Fp(62)e Fr(FV)q(\()p Fq(e)p Fr(\))19 b Fp([)g(f\003g)28 b Fo(then)i Fr(rk)1124 1988 y Fm(\033)1169 1976 y Fr(\()p Fq(e)p Fr(\))23 b(:=)g(0)p Fo(.)51 2075 y(2.)39 b(F)-6 b(or)30 b Fq(\033)c Fp(2)e Fr(FV)q(\()p Fq(e)p Fr(\))19 b Fp([)g(f\003g)28 b Fo(we)i(de\014ne:)51 2316 y Fr(rk)127 2328 y Fm(\033)172 2316 y Fr(\()p Fq(e)p Fr(\))23 b(:=)409 2121 y Ff(8)409 2196 y(>)409 2221 y(<)409 2370 y(>)409 2395 y(:)496 2167 y Fq(o)p Fr(\()p Fq(e)p Fr(\))1037 b Fo(if)31 b Fq(e)22 b Fp(2)i Fr(V)-7 b(ar)18 b Fp([)g(f)p Fr(0)p Fp(g)496 2266 y Fr(max)p Fp(f)p Fq(o)p Fr(\()p Fq(p)p Fr(\))p Fq(;)c Fr(rk)952 2278 y Fm(\033)997 2266 y Fr(\()p Fq(e)1068 2278 y Fn(1)1105 2266 y Fr(\))p Fq(;)g(:::;)g Fr(rk)1357 2278 y Fm(\033)1401 2266 y Fr(\()p Fq(e)1472 2278 y Fm(n)1517 2266 y Fr(\))p Fp(g)85 b Fo(if)31 b Fq(e)22 b Fr(=)h Fq(pe)1987 2278 y Fn(1)2024 2266 y Fq(:::e)2132 2278 y Fm(n)496 2366 y Fr(max)p Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)950 2378 y Fm(\033)995 2366 y Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)1238 2378 y Fm(\030)1274 2366 y Fr(\()p Fq(F)e Fr(\))19 b(+)f(1)p Fp(g)87 b Fo(if)31 b Fq(e)22 b Fr(=)h Fq(\017\030)t(F)496 2466 y Fr(rk)573 2478 y Fm(\033)617 2466 y Fr(\()p Fq(G)p Fr(\))930 b Fo(if)31 b Fq(e)22 b Fr(=)h Fq(\025z)t(G)2253 2316 y Fo(.)51 2640 y Fh(Note)31 b(1)p Fr(.)36 b(rk)470 2652 y Fm(\033)515 2640 y Fr(\()p Fq(e)p Fr(\))23 b Fq(<)f(!)g Fp(\001)c Fr(2.)51 2739 y Fh(Note)31 b(2)p Fr(.)36 b(rk)470 2751 y Fm(\033)514 2739 y Fr(\()p Fq(e)p Fr(\))28 b(is)g(a)f(measure)g (of)g(nesting)g(of)h(b)r(ound)g(v)-5 b(ariables)26 b(in)i(subterms)f (of)h Fq(e)f Fr(con)n(taining)g(free)g(v)-5 b(ariable)27 b Fq(\033)k Fr(,)c(and)51 2839 y(rk)127 2851 y Fe(\003)193 2839 y Fr(tak)n(es)f(accoun)n(t)h(of)h(all)f Fq(\017)p Fr(-subterms.)36 b(More)27 b(precisely)-7 b(,)51 3005 y Fh(Lemma)29 b(4.1)1101 3105 y Fr(rk)1177 3117 y Fe(\003)1215 3105 y Fr(\()p Fq(e)p Fr(\))23 b(=)g(sup)p Fp(f)p Fq(o)p Fr(\()p Fq(\021)s Fr(\))p Fq(;)14 b Fr(rk)1858 3117 y Fm(\033)1902 3105 y Fr(\()p Fq(F)e Fr(\)+1)53 b(:)23 b Fq(\021)s(;)14 b(\017\033)s(F)42 b Fo(o)l(c)l(cur)29 b(in)h Fq(e)p Fp(g)943 b Fr(\(4\))51 3271 y Fo(Pr)l(o)l(of)28 b Fr(b)n(y)g(induction)g(on)f Fq(e)p Fr(.)37 b(De\014ne)510 3453 y Fq(o)p Fr(\()p Fq(e)p Fr(\))23 b(:=)g(sup)p Fp(f)p Fq(o)p Fr(\()p Fq(\021)s Fr(\))h(:)50 b Fq(\021)31 b Fr(o)r(ccurs)c(in)h Fq(e)p Fp(g)p Fq(;)179 b Fr(rk)1984 3465 y Fn(1)2022 3453 y Fr(\()p Fq(e)p Fr(\))23 b(:=)g(sup)p Fp(f)p Fr(rk)2501 3465 y Fm(\033)2546 3453 y Fr(\()p Fq(F)12 b Fr(\)+1)50 b(:)24 b Fq(\017\033)s(F)39 b Fr(o)r(ccurs)27 b(in)h Fq(e)p Fp(g)51 3636 y Fr(Note)f(that)h Fq(o)p Fr(\()p Fq(pe)584 3648 y Fn(1)621 3636 y Fq(:::e)729 3648 y Fm(n)774 3636 y Fr(\))c(=)e(max)p Fp(f)p Fq(o)p Fr(\()p Fq(e)1225 3648 y Fn(1)1262 3636 y Fr(\))p Fq(;)14 b(:::;)g(o)p Fr(\()p Fq(e)1548 3648 y Fm(n)1593 3636 y Fr(\))p Fp(g)p Fq(:)51 3736 y Fr(1.)36 b Fq(e)23 b Fp(2)g Fr(V)-7 b(ar)18 b Fp([)h(f)p Fr(0)p Fp(g)p Fr(.)35 b(Then)28 b(rk)991 3748 y Fe(\003)1030 3736 y Fr(\()p Fq(e)p Fr(\))23 b(=)g Fq(o)p Fr(\()p Fq(e)p Fr(\))p Fq(;)42 b Fr(rk)1528 3748 y Fn(1)1565 3736 y Fr(\()p Fq(e)p Fr(\))23 b(=)g(0)51 3835 y(2.)36 b Fq(e)23 b Fr(=)f Fq(pe)382 3847 y Fn(1)419 3835 y Fq(:::e)527 3847 y Fm(n)572 3835 y Fr(.)37 b(Then)51 3955 y(rk)127 3967 y Fe(\003)165 3955 y Fr(\()p Fq(e)p Fr(\))291 3908 y Fm(I)5 b(H)305 3955 y Fr(=)37 b(max)p Fp(f)p Fq(o)p Fr(\()p Fq(p)p Fr(\))p Fq(;)14 b Fr(max)941 3967 y Fm(i)983 3955 y Fr(max)o Fp(f)p Fq(o)p Fr(\()p Fq(e)1290 3967 y Fm(i)1318 3955 y Fr(\))p Fq(;)g Fr(rk)1463 3967 y Fn(1)1500 3955 y Fr(\()p Fq(e)1571 3967 y Fm(i)1599 3955 y Fr(\))p Fp(gg)22 b Fr(=)51 4055 y(=)g(max)p Fp(f)p Fq(o)p Fr(\()p Fq(p)p Fr(\))p Fq(;)14 b Fr(max)672 4067 y Fm(i)714 4055 y Fq(o)p Fr(\()p Fq(e)825 4067 y Fm(i)853 4055 y Fr(\))p Fq(;)g Fr(max)1076 4067 y Fm(i)1118 4055 y Fr(rk)1194 4067 y Fn(1)1231 4055 y Fr(\()p Fq(e)1302 4067 y Fm(i)1330 4055 y Fr(\))p Fp(g)23 b Fr(=)f(max)p Fp(f)p Fq(o)p Fr(\()p Fq(e)p Fr(\))p Fq(;)14 b Fr(rk)1967 4067 y Fn(1)2004 4055 y Fr(\()p Fq(e)p Fr(\))p Fp(g)p Fr(.)51 4154 y(3.)36 b Fq(e)23 b Fr(=)f Fq(\017\030)t(F)12 b Fr(.)37 b(Then)51 4274 y(rk)127 4286 y Fe(\003)165 4274 y Fr(\()p Fq(e)p Fr(\))23 b(=)g(max)p Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)833 4286 y Fe(\003)871 4274 y Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)1114 4286 y Fm(\030)1150 4274 y Fr(\()p Fq(F)e Fr(\)+)q(1)p Fp(g)1451 4227 y Fm(I)5 b(H)1465 4274 y Fr(=)36 b(max)p Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b(o)p Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)2227 4286 y Fn(1)2265 4274 y Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)2507 4286 y Fm(\030)2544 4274 y Fr(\()p Fq(F)e Fr(\)+1)p Fp(g)22 b Fr(=)g(max)p Fp(f)p Fq(o)p 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Fo(.)51 5007 y(c\))29 b Fq(!)d Fp(\024)d Fr(rk)394 5019 y Fe(\003)432 5007 y Fr(\()p Fq(P)12 b Fr(\))p Fo(,)31 b(for)f(e)l(ach)h(1-term)f Fq(P)12 b Fo(.)51 5190 y(Pr)l(o)l(of)p Fr(.)38 b(Easy)26 b(induction)i(on)f Fq(e;)14 b(P)e Fr(.)51 5372 y Fh(Lemma)29 b(4.3)51 5472 y Fo(If)h Fq(d)p Fr(\()p Fq(e)p Fr(\))23 b(=)g(0)29 b Fo(and)h Fq(e)g Fo(is)g(not)f(a)h Fq(\025)p Fo(-term)g(then)59 b Fr(rk)1570 5484 y Fe(\003)1609 5472 y Fr(\()p Fq(e)p Fr(\))23 b Fp(\024)g Fr(sup)p Fp(f)p Fr(rk)2065 5484 y Fm(\033)2110 5472 y Fr(\()p Fq(e)p Fr(\))g(:)h Fq(\033)i Fp(2)d Fr(FV)r(\()p Fq(e)p Fr(\))p Fp(g)p Fo(.)51 5654 y(Pr)l(o)l(of:)51 5754 y Fr(1.)36 b Fq(e)23 b Fr(=)f(0:)37 b(Then)28 b(rk)696 5766 y Fe(\003)734 5754 y Fr(\()p Fq(e)p Fr(\))23 b(=)g(0.)37 b Fq(e)22 b Fr(=)h Fq(\030)k Fp(2)d Fr(V)-7 b(ar)o(:)37 b(Then)28 b(rk)1823 5766 y Fe(\003)1861 5754 y Fr(\()p Fq(e)p Fr(\))23 b(=)g Fq(o)p Fr(\()p Fq(\030)t Fr(\))h(=)f(rk)2407 5766 y Fm(\030)2443 5754 y Fr(\()p Fq(e)p Fr(\).)51 5854 y(2.)36 b Fq(e)23 b Fr(=)f Fq(\017\030)t(F)12 b Fr(:)37 b(Then,)28 b(since)f Fq(d)p Fr(\()p Fq(e)p Fr(\))d(=)f(0,)k(w)n(e)g(ha)n(v)n(e)f Fq(d)p Fr(\()p Fq(F)12 b Fr(\))24 b(=)f(0)k(and)g(FV)r(\()p Fq(e)p Fr(\))c Fp(6)p Fr(=)g Fp(;)p Fr(.)36 b(Therefore)1958 6119 y(10)p eop %%Page: 11 11 11 10 bop 51 614 a Fr(max)o Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)505 626 y Fm(\030)541 614 y Fr(\()p Fq(F)e Fr(\)+1)p Fp(g)22 b(\024)h Fr(sup)p Fp(f)p Fr(rk)1172 626 y Fm(\033)1216 614 y Fr(\()p Fq(e)p Fr(\))h(:)f Fq(\033)j Fp(2)e Fr(FV)q(\()p Fq(e)p Fr(\))p Fp(g)k Fr(and,)f(b)n(y)h(I.H.,)g(rk) 2396 626 y Fe(\003)2434 614 y Fr(\()p Fq(F)12 b Fr(\))24 b Fp(\024)e Fr(sup)p Fp(f)p Fr(rk)2917 626 y Fm(\033)2962 614 y Fr(\()p Fq(F)12 b Fr(\))24 b(:)f Fq(\033)j Fp(2)e Fr(FV)q(\()p Fq(F)12 b Fr(\))p Fp(g)p Fr(.)51 714 y(Hence)27 b(rk)374 726 y Fe(\003)412 714 y Fr(\()p Fq(e)p Fr(\))c(=)g(max)o Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)1080 726 y Fe(\003)1118 714 y Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)1361 726 y Fm(\030)1397 714 y Fr(\()p Fq(F)e Fr(\)+1)p Fp(g)22 b(\024)h Fr(sup)p Fp(f)p Fr(rk)2028 726 y Fm(\033)2073 714 y Fr(\()p Fq(e)p Fr(\))g(:)g Fq(\033)j Fp(2)e Fr(FV)q(\()p Fq(e)p Fr(\))p Fp(g)p Fr(.)51 814 y(3.)36 b Fq(e)23 b Fr(=)f Fq(pe)382 826 y Fn(1)419 814 y Fq(:::e)527 826 y Fm(n)572 814 y Fr(.)37 b(Since)28 b Fq(!)e Fp(\024)c Fr(rk)1091 826 y Fe(\003)1129 814 y Fr(\()p Fq(P)12 b Fr(\))28 b(for)f(eac)n(h)g(1-term)f Fq(P)12 b Fr(,)28 b(w)n(e)f(ha)n(v)n(e)51 960 y(rk)127 972 y Fe(\003)165 960 y Fr(\()p Fq(e)p Fr(\))c(=)g(max)p Fp(f)p Fr(rk)651 972 y Fe(\003)690 960 y Fr(\()p Fq(e)761 972 y Fn(1)798 960 y Fr(\))p Fq(;)14 b(:::;)g Fr(rk)1049 972 y Fe(\003)1088 960 y Fr(\()p Fq(e)1159 972 y Fm(n)1204 960 y Fr(\))p Fp(g)1301 891 y Fn(IH)1305 960 y Fp(\024)27 b Fr(sup)p Fp(f)p Fr(rk)1639 972 y Fm(\033)1684 960 y Fr(\()p Fq(e)1755 972 y Fm(i)1783 960 y Fr(\))c(:)g Fq(\033)k Fp(2)c Fr(FV)r(\()p Fq(e)2225 972 y Fm(i)2252 960 y Fr(\))14 b(&)g(1)p Fp(\024)o Fq(i)p Fp(\024)o Fq(n)p Fp(g)22 b(\024)h Fr(sup)p Fp(f)p Fr(rk)3021 972 y Fm(\033)3066 960 y Fr(\()p Fq(e)p Fr(\))g(:)g Fq(\033)k Fp(2)c Fr(FV)r(\()p Fq(e)p Fr(\))p Fp(g)p Fr(.)k Fd(2)51 1126 y Fh(Lemma)i(4.4)51 1226 y Fo(If)h Fq(\017\030)t(F)41 b Fo(is)30 b(c)l(anonic)l(al)h(then)f Fr(rk)1021 1238 y Fe(\003)1059 1226 y Fr(\()p Fq(F)12 b Fr(\))24 b Fp(\024)f Fr(rk)1376 1238 y Fm(\030)1412 1226 y Fr(\()p Fq(F)12 b Fr(\))30 b Fo(and)h Fr(rk)1809 1238 y Fe(\003)1847 1226 y Fr(\()p Fq(\017\030)t(F)12 b Fr(\))24 b(=)e(max)p Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)2615 1238 y Fm(\030)2652 1226 y Fr(\()p Fq(F)e Fr(\)+1)p Fp(g)p Fo(.)51 1392 y(Pr)l(o)l(of:)51 1492 y Fr(Since)27 b Fq(\017\030)t(F)40 b Fr(is)27 b(canonical,)g(w)n(e)g(ha)n(v)n(e)f Fq(d)p Fr(\()p Fq(F)12 b Fr(\))24 b(=)f(0.)36 b(By)28 b(4.3)e(this)i(yields)51 1591 y(rk)127 1603 y Fe(\003)165 1591 y Fr(\()p Fq(F)12 b Fr(\))24 b Fp(\024)e Fr(sup)p Fp(f)p Fr(rk)648 1603 y Fm(\033)693 1591 y Fr(\()p Fq(F)12 b Fr(\))23 b(:)37 b Fq(\033)27 b Fp(2)c Fr(FV)r(\()p Fq(F)12 b Fr(\))p Fp(g)28 b Fr(and)f(th)n(us)h(\(since)f(FV)r(\()p Fq(F)12 b Fr(\))23 b Fp(\022)g(f)p Fq(\030)t Fp(g)k Fr(\))h(rk)2598 1603 y Fe(\003)2636 1591 y Fr(\()p Fq(F)12 b Fr(\))24 b Fp(\024)f Fr(rk)2953 1603 y Fm(\030)2989 1591 y Fr(\()p Fq(F)12 b Fr(\).)51 1691 y(Hence)27 b(rk)374 1703 y Fe(\003)412 1691 y Fr(\()p Fq(\017\030)t(F)12 b Fr(\))23 b(=)g(max)o Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)1180 1703 y Fe(\003)1218 1691 y Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)1461 1703 y Fm(\030)1497 1691 y Fr(\()p Fq(F)e Fr(\))19 b(+)f(1)p Fp(g)k Fr(=)h(max)p Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)2376 1703 y Fm(\030)2413 1691 y Fr(\()p Fq(F)e Fr(\))19 b(+)f(1)p Fp(g)p Fr(.)26 b Fd(2)217 1834 y Fr(The)h(next)h(t)n(w)n(o)f(lemmas)g (sho)n(w)g(that)h(the)g(rank)e(do)r(es)i(not)f(increase)g(during)g (computation.)51 2000 y Fh(Lemma)i(4.5)51 2100 y Fr(rk)127 2112 y Fm(\033)172 2100 y Fr(\()p Fq(e)p Fr([)p Fq(y)s(=t)p Fr(]\))22 b Fq(<)h Fr(max)o Fp(f)p Fq(!)s(;)14 b Fr(rk)911 2112 y Fm(\033)955 2100 y Fr(\()p Fq(t)p Fr(\)+)q(1)p Fp(g)p Fo(,)29 b(if)h Fq(e)g Fo(is)g(arithmetic)l(al.)51 2266 y(Pr)l(o)l(of:)38 b Fr(Let)28 b Fq(y)e Fp(2)d Fr(FV)r(\()p Fq(e)p Fr(\))28 b(and)f Fq(\033)f Fp(2)e Fr(FV)q(\()p Fq(e)p Fr([)p Fq(t)p Fr(]\))19 b Fp([)g(f\003g)p Fr(.)36 b(\(Otherwise)27 b(rk)2237 2278 y Fm(\033)2282 2266 y Fr(\()p Fq(e)p Fr([)p Fq(t)p Fr(]\))c(=)g(rk)2648 2278 y Fm(\033)2693 2266 y Fr(\()p Fq(e)p Fr(\))g Fq(<)g(!)30 b Fr(or)d(rk)3167 2278 y Fm(\033)3212 2266 y Fr(\()p Fq(e)p Fr([)p Fq(t)p Fr(]\))c(=)g(0.\))51 2365 y(1.)36 b Fq(e)23 b Fr(=)f Fq(y)s Fr(:)37 b(rk)481 2377 y Fm(\033)526 2365 y Fr(\()p Fq(e)p Fr([)p Fq(t)p Fr(]\))24 b(=)e(rk)892 2377 y Fm(\033)937 2365 y Fr(\()p Fq(t)p Fr(\).)51 2465 y(2.)36 b Fq(e)23 b Fr(=)f Fq(pe)382 2477 y Fn(1)419 2465 y Fq(:::e)527 2477 y Fm(n)572 2465 y Fr(:)37 b(Then)28 b(rk)925 2477 y Fm(\033)970 2465 y Fr(\()p Fq(e)p Fr([)p Fq(t)p Fr(]\))23 b(=)g(max)o Fp(f)p Fq(o)p 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Fq(t)p Fr(]\))23 b Fq(<)g Fr(max)p Fp(f)p Fq(!)s(;)14 b Fr(rk)2163 2676 y Fm(x)2205 2664 y Fr(\()p Fq(t)p Fr(\))19 b(+)f(1)p Fp(g)k Fr(=)h Fq(!)s Fr(.)51 2764 y(The)k(last)h(equation)f(holds)g(b)n (y)g(clause)g(1)g(in)h(the)g(de\014nition)g(of)g(rank,)e(since)i Fq(x)23 b Fp(62)h Fr(FV)q(\()p Fq(t)p Fr(\))19 b Fp([)g(f\003g)p Fr(.)51 2864 y(4.)36 b Fq(e)23 b Fr(=)f Fq(\025z)t(G)p Fr(.)37 b(The)28 b(assertion)e(follo)n(ws)h(b)n(y)g(I.H.)h Fd(2)51 3030 y Fh(Lemma)h(4.6)51 3129 y Fo(If)h Fq(e)22 b(,)-14 b Fp(!)291 3099 y Fn(1)291 3152 y Fm(S)363 3129 y Fq(e)402 3099 y Fe(0)455 3129 y Fo(then)29 b Fr(rk)715 3141 y Fm(\033)760 3129 y Fr(\()p Fq(e)831 3099 y Fe(0)854 3129 y Fr(\))24 b Fp(\024)e Fr(rk)1074 3141 y Fm(\033)1118 3129 y Fr(\()p Fq(e)p Fr(\))p Fo(.)51 3295 y(Pr)l(o)l(of)30 b(by)h(induction)f(on)g(the)g(de\014nition)g(of)g Fq(,)-14 b Fp(!)1568 3265 y Fn(1)1568 3318 y Fm(S)1617 3295 y Fo(:)51 3395 y Fr(Let)27 b Fq(\033)g Fp(2)c Fr(FV)r(\()p Fq(e)540 3365 y Fe(0)563 3395 y Fr(\))c Fp([)g(f\003g)p Fr(.)35 b(\(Otherwise)27 b(rk)1373 3407 y Fm(\033)1418 3395 y Fr(\()p Fq(e)1489 3365 y Fe(0)1512 3395 y Fr(\))d(=)e(0.\))51 3495 y(1.1.)36 b(If)28 b Fq(e)f Fr(is)g(a)h(canonical)e(0-)p Fq(\017)p Fr(-term)g(then)i(rk)1478 3507 y Fm(\033)1522 3495 y Fr(\()p Fq(e)1593 3465 y Fe(0)1617 3495 y Fr(\))23 b(=)g(0,)k(since)g Fq(e)2094 3465 y Fe(0)2140 3495 y Fp(2)d Fr(I)-14 b(N.)51 3594 y(1.2.)36 b(If)28 b Fq(e)f Fr(is)g(a)h(canonical)e(1-)p Fq(\017)p Fr(-term)g(then)i(rk)1478 3606 y Fm(\033)1522 3594 y Fr(\()p Fq(e)1593 3564 y Fe(0)1617 3594 y Fr(\))23 b Fq(<)g(!)i Fp(\024)e Fr(rk)2001 3606 y Fm(\033)2046 3594 y Fr(\()p Fq(e)p Fr(\),)28 b(since)f Fq(e)2442 3564 y Fe(0)2493 3594 y Fr(is)g(arithmetical)g(and)h Fq(\033)e Fr(=)d Fp(\003)p Fr(.)51 3694 y(2.)36 b Fq(e)23 b Fr(=)f Fq(pe)382 3706 y Fn(1)419 3694 y Fq(:::e)527 3706 y Fm(n)600 3694 y Fr(or)27 b Fq(e)22 b Fr(=)h Fq(\025y)s(F)12 b Fr(:)51 3794 y(2.1.)36 b Fq(e)22 b Fr(=)h Fg(App)p Fq(P)12 b(t)28 b Fr(and)f Fq(e)830 3763 y Fe(0)876 3794 y Fr(=)c Fq(G)p Fr([)p Fq(z)t(=t)p Fr(])k(with)h Fq(P)35 b(,)-14 b Fp(!)1586 3763 y Fn(1)1586 3816 y Fm(S)1657 3794 y Fq(\025z)t(G)p Fr(.)37 b(Then)51 3940 y(rk)127 3952 y Fm(\033)172 3940 y Fr(\()p Fq(e)243 3910 y Fe(0)266 3940 y Fr(\))321 3871 y Fn(4)p Fm(:)p Fn(5)332 3940 y Fp(\024)c Fr(max)p Fp(f)p Fq(!)s(;)14 b Fr(rk)794 3952 y Fm(\033)839 3940 y Fr(\()p Fq(t)p Fr(\))p Fp(g)23 b(\024)g Fr(rk)1162 3952 y Fm(\033)1207 3940 y Fr(\()p Fq(e)p Fr(\))g(=)g(max)o Fp(f)p Fq(!)s(;)14 b Fr(rk)1784 3952 y Fm(\033)1829 3940 y Fr(\()p Fq(P)e Fr(\))p Fq(;)i Fr(rk)2072 3952 y Fm(\033)2117 3940 y Fr(\()p Fq(t)p Fr(\))p Fp(g)p Fr(.)51 4040 y(2.2.)36 b(otherwise:)g(immediate)27 b(from)h(I.H.)51 4139 y(3.)36 b Fq(e)23 b Fr(=)f Fq(\017\030)t(F)40 b Fr(and)27 b Fq(e)668 4109 y Fe(0)714 4139 y Fr(=)c Fq(\017\030)t(F)941 4109 y Fe(0)992 4139 y Fr(with)28 b Fq(F)35 b(,)-14 b Fp(!)1361 4109 y Fn(1)1361 4162 y Fm(S)1432 4139 y Fq(F)1497 4109 y Fe(0)1520 4139 y Fr(:)51 4288 y(rk)127 4300 y Fm(\033)172 4288 y Fr(\()p Fq(e)243 4258 y Fe(0)266 4288 y Fr(\))23 b(=)g(max)o Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)863 4300 y Fm(\033)908 4288 y Fr(\()p Fq(F)1005 4258 y Fe(0)1028 4288 y Fr(\))p Fq(;)g Fr(rk)1174 4300 y Fm(\030)1210 4288 y Fr(\()p Fq(F)1307 4258 y Fe(0)1331 4288 y Fr(\)+)o(1)p Fp(g)1534 4218 y Fn(IH)1538 4288 y Fp(\024)27 b Fr(max)o Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)2084 4300 y Fm(\033)2129 4288 y Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)2371 4300 y Fm(\030)2408 4288 y Fr(\()p Fq(F)e Fr(\)+1)p Fp(g)22 b Fr(=)g(rk)2872 4300 y Fm(\033)2916 4288 y Fr(\()p Fq(e)p Fr(\).)28 b Fd(2)51 4409 y Fr(Substitution)g(of)g(a)f(v)-5 b(ariable)26 b(b)n(y)i(an)f(appropriate)f(canonical)g(v)-5 b(alue)28 b(also)e(do)r(es)i(not)f(increase)f(the)i(rank.)51 4575 y Fh(Lemma)h(4.7)51 4675 y Fr(rk)127 4687 y Fm(\033)172 4675 y Fr(\()p Fq(e)p Fr([)p Fq(\021)s(=u)p Fr(]\))22 b Fp(\024)h Fr(rk)641 4687 y Fm(\033)686 4675 y Fr(\()p Fq(e)p Fr(\))p Fo(,)30 b(for)h(e)l(ach)g Fq(u)22 b Fp(2)i Fr(I)-14 b(B)1388 4690 y Fm(\023)p Fn(\()p Fm(\021)r Fn(\))1505 4675 y Fo(.)51 4841 y(Pr)l(o)l(of)30 b(by)h(induction)f(on)g Fq(e)p Fo(:)37 b Fr(Let)27 b Fq(e)1158 4811 y Fe(0)1204 4841 y Fr(:=)c Fq(e)p Fr([)p Fq(\021)s(=u)p Fr(])k(and)g(assume)g(that) h Fq(\021)e Fp(2)e Fr(FV)q(\()p Fq(e)p Fr(\))k(and)g Fq(\033)e Fp(2)e Fr(FV)q(\()p Fq(e)3085 4811 y Fe(0)3108 4841 y Fr(\))19 b Fp([)g(f\003g)p Fr(.)51 4941 y(Note)27 b(that)h(FV)r(\()p Fq(e)620 4910 y Fe(0)643 4941 y Fr(\))23 b(=)g(FV)r(\()p Fq(e)p Fr(\))18 b Fp(n)g(f)p Fq(\021)s Fp(g)p Fr(.)51 5040 y(1.)36 b Fq(e)23 b Fr(=)f Fq(\021)s Fr(:)38 b(Then)27 b Fq(\033)g Fr(=)22 b Fp(\003)28 b Fr(and)f(rk)1090 5052 y Fe(\003)1129 5040 y Fr(\()p Fq(e)1200 5010 y Fe(0)1223 5040 y Fr(\))c(=)g(0)k(\(if)h Fq(\023)p Fr(\()p Fq(\021)s Fr(\))d(=)d(0\))28 b(or)e(rk)2072 5052 y Fe(\003)2110 5040 y Fr(\()p Fq(e)2181 5010 y Fe(0)2204 5040 y Fr(\))e Fq(<)e(!)k Fr(=)c(rk)2589 5052 y Fe(\003)2627 5040 y Fr(\()p Fq(e)p Fr(\))28 b(\(if)g Fq(\023)p Fr(\()p Fq(\021)s Fr(\))d(=)d(1\).)51 5140 y(2.)36 b Fq(e)23 b Fr(=)f Fg(App)q Fq(\021)s(t)27 b Fr(and)h Fq(u)22 b Fr(=)h Fq(\025z)t(G)p Fr(:)37 b(Then)28 b Fq(e)1336 5110 y Fe(0)1382 5140 y Fr(=)22 b Fq(G)p Fr([)p Fq(z)t(=t)p Fr([)p Fq(u)p Fr(]])27 b(and)g(th)n(us)51 5286 y(rk)127 5298 y Fm(\033)172 5286 y Fr(\()p Fq(e)243 5256 y Fe(0)266 5286 y Fr(\))321 5217 y Fn(4)p Fm(:)p Fn(5)332 5286 y Fp(\024)33 b Fr(max)p Fp(f)p Fq(!)s(;)14 b Fr(rk)794 5298 y Fm(\033)839 5286 y Fr(\()p Fq(t)p Fr([)p Fq(u)p Fr(]\))p Fp(g)1092 5217 y Fn(IH)1096 5286 y Fp(\024)27 b Fr(max)o Fp(f)p Fq(!)s(;)14 b Fr(rk)1552 5298 y Fm(\033)1596 5286 y Fr(\()p Fq(t)p Fr(\))p Fp(g)23 b Fr(=)g(rk)1919 5298 y Fm(\033)1964 5286 y Fr(\()p Fq(e)p Fr(\).)51 5386 y(3.)36 b Fq(e)23 b Fr(=)f Fq(\017\030)t(F)12 b Fr(.)37 b(Then)51 5528 y(rk)127 5540 y Fm(\033)172 5528 y Fr(\()p Fq(e)243 5498 y Fe(0)266 5528 y Fr(\))23 b(=)g(rk)485 5540 y Fm(\033)530 5528 y Fr(\()p Fq(\017\030)t(F)701 5498 y Fe(0)724 5528 y Fr(\))h(=)e(max)p Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)1321 5540 y Fm(\033)1366 5528 y Fr(\()p Fq(F)1463 5498 y Fe(0)1487 5528 y Fr(\))p Fq(;)g Fr(rk)1632 5540 y Fm(\030)1669 5528 y Fr(\()p Fq(F)1766 5498 y Fe(0)1789 5528 y Fr(\))19 b(+)f(1)p Fp(g)2029 5458 y Fn(IH)2033 5528 y Fp(\024)27 b Fr(max)o Fp(f)p Fq(o)p Fr(\()p Fq(\030)t Fr(\))p Fq(;)14 b Fr(rk)2579 5540 y Fm(\033)2624 5528 y Fr(\()p Fq(F)e Fr(\))p Fq(;)i Fr(rk)2867 5540 y Fm(\030)2903 5528 y Fr(\()p Fq(F)e Fr(\))19 b(+)f(1)p Fp(g)k Fr(=)h(rk)3404 5540 y Fm(\033)3449 5528 y Fr(\()p Fq(e)p Fr(\).)51 5627 y(4.)36 b(In)28 b(all)f(other)g(cases)g(the)h(assertion)e(follo)n(ws)g (immediately)i(from)f(IH.)56 b Fd(2)217 5770 y Fr(The)25 b(next)h(statemen)n(t)f(sho)n(ws)f(that)i(our)f(de\014nition)g(of)h (rank)e(is)i(suitable:)35 b(the)26 b(rank)e(decreases)g(when)i(the)g ('b)r(o)r(dy')f(of)51 5870 y(a)i(canonical)f Fq(\017)p Fr(-term)h(is)h(substituted)g(b)n(y)f(a)h(canonical)e(v)-5 b(alue.)1958 6119 y(11)p eop %%Page: 12 12 12 11 bop 51 614 a Fh(Lemma)29 b(4.8)51 714 y Fo(If)h Fq(\017\030)t(F)41 b Fo(is)30 b(c)l(anonic)l(al)h(then)f Fr(rk)1021 726 y Fe(\003)1059 714 y Fr(\()p Fq(F)12 b Fr([)p Fq(u)p Fr(]\))24 b Fq(<)e Fr(rk)1470 726 y Fe(\003)1508 714 y Fr(\()p Fq(\017\030)t(F)12 b Fr(\))p Fo(,)30 b(for)h(e)l(ach)f Fq(u)23 b Fp(2)g Fr(I)-14 b(B)2309 729 y Fm(\023)p Fn(\()p Fm(\030)r Fn(\))2422 714 y Fo(.)51 946 y(Pr)l(o)l(of:)38 b Fr(rk)382 958 y Fe(\003)421 946 y Fr(\()p Fq(F)12 b Fr([)p Fq(u)p Fr(]\))667 877 y Fn(4)p Fm(:)p Fn(7)677 946 y Fp(\024)34 b Fr(rk)852 958 y Fe(\003)890 946 y Fr(\()p Fq(F)12 b Fr(\))1043 877 y Fn(4)p Fm(:)p Fn(4)1054 946 y Fp(\024)33 b Fr(rk)1228 958 y Fm(\030)1264 946 y Fr(\()p Fq(F)12 b Fr(\))24 b Fq(<)f Fr(rk)1581 958 y Fe(\003)1619 946 y Fr(\()p Fq(\017\030)t(F)12 b Fr(\).)28 b Fd(2)51 1125 y Fh(De\014nition)i(16)41 b Fr(rk\()p Fq(e)p Fr(\))23 b(:=)g(rk)1023 1137 y Fe(\003)1061 1125 y Fr(\()p Fq(e)p Fr(\))30 b Fo(is)g(c)l(al)t(le)l(d)h(the)f Fr(rank)f 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Fo(of)g(r)l(ank)g Fp(\024)23 b Fq(r)r Fo(.)51 1961 y(Pr)l(o)l(of:)51 2060 y Fr(Since)k(all)h(subterms)f(of)h(an)f (expression)f Fq(e)h Fr(ha)n(v)n(e)g(ranks)f Fp(\024)d Fr(rk)o(\()p Fq(e)p Fr(\),)28 b(w)n(e)f(ha)n(v)n(e:)51 2160 y(rk)o(\()p Fq(e)p Fr(\))c Fp(\024)g Fq(r)40 b Fp(\))c(8)p Fq(e)623 2130 y Fe(0)646 2160 y Fr(\()14 b Fq(e)22 b(,)-14 b Fp(!)845 2172 y Fm(S)917 2160 y Fq(e)956 2130 y Fe(0)1016 2160 y Fp(,)37 b Fq(e)22 b(,)-14 b Fp(!)1289 2172 y Fm(S)1333 2156 y Fc(0)1383 2160 y Fq(e)1422 2130 y Fe(0)1459 2160 y Fr(\).)51 2259 y(T)-7 b(ogether)26 b(with)i(Lemma)g(4.6)e(this)i (yields)g(the)g(assertion)e(b)n(y)h(induction)h(on)f Fq(d)p Fr(\()p Fq(e)p Fr(\).)h Fd(2)51 2533 y Fk(5)137 b(The)46 b(H-pro)t(cess)51 2715 y Fr(Let)27 b(us)h(recall)f(that)g (critical)g(form)n(ulas)g(are)f(form)n(ulas)h(of)g(three)h(t)n(yp)r (es:)51 2815 y Fq(F)12 b Fr([)p Fq(t)p Fr(])23 b Fp(!)g Fq(F)12 b Fr([)p Fq(\017xF)g Fr([)p Fq(x)p Fr(]],)51 2914 y Fq(s)23 b Fp(6)p Fr(=)f(0)h Fp(!)g Fq(F)12 b Fr([)p Fq(\017xF)g Fr([)p Fq(x)p Fr(]])28 b(with)g Fq(F)35 b Fr(:=)23 b(\()p Fq(s)g Fr(=)g Fg(S)p Fq(x)p Fr(\),)51 3014 y Fq(F)12 b Fr([)p Fq(T)g Fr(])22 b Fp(!)h Fq(F)12 b Fr([)p Fq(\017X)7 b(F)12 b Fr([)p Fq(X)7 b Fr(]])26 b(with)i Fq(T)39 b Fr(a)27 b(1-)p Fq(\017)p Fr(-term)f(or)h(regular)f (lam)n(b)r(da-term.)51 3138 y Fh(W)-8 b(e)31 b(assume)f(that)i Fq(C)6 b(r)854 3150 y Fn(0)892 3138 y Fq(;)14 b(:::;)g(C)6 b(r)1137 3150 y Fm(N)1233 3138 y Fh(\()p Fr(with)32 b Fq(N)g Fp(2)23 b Fr(I)-14 b(N)q Fh(\))32 b(is)f(a)h(\014xed)g(sequence) g(of)g(closed)f(critical)h(form)m(ulas)p Fr(.)217 3262 y(In)19 b(this)g(section)g(w)n(e)g(de\014ne)g(a)g(successiv)n(e)f (appro)n(ximation)f(pro)r(cess)h(for)h(\014nding)g(a)g(solution)g(of)g (this)g(system)g(according)51 3361 y(to)25 b(Hilb)r(ert's)i(approac)n (h.)34 b(It)27 b(is)e(useful)i(to)f(recall)f(here)g(a)h(Commen)n(t)g (from)f(Section)h(3.)36 b(The)26 b(H-pro)r(cess)f(will)h(b)r(e)h (arranged)51 3461 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Fr(is)h(the)g(H-term)f(and)h Fq(v)j Fr(the)d(H-v)-5 b(alue)27 b(of)h Fq(S)5 b Fr(,)27 b(then)h Fq(v)e Fp(2)e Fr(I)-14 b(B)2399 1298 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))2511 1283 y Fr(.)37 b(\(F)-7 b(or)27 b Fq(\023)p Fr(\()p Fq(e)p Fr(\))d(=)e(1)28 b(cf.)g(Lemma)f(3.9c.\))51 1405 y(The)g(next)h(de\014nition)g(is)g(cen)n(tral)e(for)h(the)h (substitution)g(metho)r(d.)51 1576 y Fh(De\014nition)i(21)41 b Fo(If)p 721 1509 56 4 v 31 w Fq(S)34 b Fo(is)c(nonsolving)h(then)51 1676 y Fr(H\()p Fq(S)5 b Fr(\))23 b(:=)g(\()p Fq(S)g Fp(n)18 b(f)p Fr(\()p Fq(e;)c Fr(?\))p Fp(g)p Fr(\))828 1691 y Fe(\024)p Fn(rk\()p Fm(e)p Fn(\))1047 1676 y Fp([)k(f)p Fr(\()p Fq(e;)c(v)s Fr(\))p Fp(g)p Fo(,)30 b(wher)l(e)g Fq(e)g Fo(is)g(the)g(H-term)f(and)h Fq(v)j Fo(the)d(H-value)f(of)i Fq(S)5 b Fo(.)51 1864 y Fr(Let)27 b(us)h(pro)n(v)n(e)e(that)p 710 1791 183 4 v 28 w(H\()p Fq(S)5 b Fr(\))28 b(is)f(indeed)h(a)f (correct)g Fq(\017)p Fr(-substitution)g(if)p 2212 1797 56 4 v 28 w Fq(S)33 b Fr(is)27 b(correct)f(and)i(nonsolving.)51 2034 y Fh(Lemma)h(5.1)51 2134 y Fo(L)l(et)g Fq(S)34 b Fo(b)l(e)c(an)g Fq(\017)p Fo(-substitution)e(such)i(that)p 1370 2067 V 30 w Fq(S)k Fo(is)c(c)l(orr)l(e)l(ct)g(and)g(nonsolving,)51 2234 y(and)g(let)g Fq(e)f Fo(b)l(e)h(the)g(H-term,)f Fq(v)k Fo(the)d(H-value)f(of)i Fq(S)5 b Fo(.)38 b(Then)31 b(the)f(fol)t(lowing)i(holds:)51 2333 y(a\))59 b Fr(\()p Fq(e;)14 b Fr(?\))24 b Fp(2)p 467 2267 V 23 w Fq(S)5 b Fo(,)51 2433 y(b\))63 b Fp(j)p Fq(e)p Fp(j)271 2448 y Fn(H\()p Fm(S)s Fn(\))443 2433 y Fr(=)23 b Fq(v)j Fp(6)p Fr(=)d(0)727 2403 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))838 2433 y Fo(,)51 2550 y(c\))p 186 2477 183 4 v 63 w Fr(H\()p Fq(S)5 b Fr(\))31 b Fo(is)f(c)l(orr)l(e)l(ct.)51 2720 y(Pr)l(o)l(of:)51 2820 y Fr(Let)d Fq(C)6 b(r)r Fr(\()p Fq(S)f Fr(\))25 b(=)d Fq(F)588 2832 y Fn(0)649 2820 y Fp(!)h Fq(F)12 b Fr([)p Fq(\017\030)t(F)g Fr(].)37 b(Then)28 b Fq(e)22 b Fr(=)h Fq(\017\030)t Fp(j)p Fq(F)12 b Fp(j)p 1616 2798 49 4 v 28 x Fm(S)1664 2820 y Fr(.)51 2929 y(a\))32 b Fo(Assumption:)48 b Fr(\()p Fq(e;)14 b(w)r Fr(\))32 b Fp(2)g Fq(S)38 b Fr(and)32 b Fq(w)j Fp(6)p Fr(=)14 b(?.)52 b(Then,)34 b(since)p 1974 2863 56 4 v 33 w Fq(S)j Fr(is)c(correct,)g(b)n(y)f(Lemma)h(3.7c)f Fq(F)12 b Fr([)p Fq(\017\030)t(F)g Fr(])31 b Fq(,)-14 b Fp(!)p 3429 2907 49 4 v 28 x Fm(S)3509 2929 y Fp(j)p Fq(F)12 b Fp(j)p 3620 2907 V 28 x Fm(S)3668 2929 y Fr([)p Fq(w)r Fr(])33 b Fq(,)-14 b Fp(!)p 3900 2907 V 28 x Fm(S)51 3029 y Fg(TRUE)n Fr(.)51 3129 y(On)27 b(the)h(other)f(hand,)h(since)f(Cr)o(\()p Fq(S)5 b Fr(\))24 b Fq(,)-14 b Fp(!)p 1311 3107 V 27 x Fm(S)1382 3129 y Fg(F)-7 b(ALSE)q Fr(,)27 b(w)n(e)h(ha)n(v)n(e)e Fq(F)12 b Fr([)p Fq(\017\030)t(F)g Fr(])23 b Fq(,)-14 b Fp(!)p 2348 3107 V 27 x Fm(S)2419 3129 y Fg(F)-7 b(ALSE)q Fr(.)37 b Fo(Contr)l(adiction.)51 3238 y Fr(b\))28 b(By)f(Lemma)g(3.7b) g(w)n(e)g(ha)n(v)n(e)g Fq(F)12 b Fr([)p Fq(e)p Fr(])23 b Fq(,)-14 b Fp(!)p 1342 3216 V 28 x Fm(S)1413 3238 y Fg(F)-7 b(ALSE)28 b Fr(and)f Fq(F)12 b Fr([)p Fq(v)s Fr(])24 b Fq(,)-14 b Fp(!)p 2108 3216 V 28 x Fm(S)2179 3238 y Fg(TRUE)o Fr(.)37 b(Hence,)28 b(b)n(y)f(3.7b,)g Fq(v)f Fp(6)p Fr(=)d Fp(j)p Fq(e)p Fp(j)p 3283 3216 V 28 x Fm(S)3354 3238 y Fr(=)g(0)3484 3208 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))3595 3238 y Fr(.)51 3338 y(The)k(equation)g Fp(j)p Fq(e)p Fp(j)645 3353 y Fn(H\()p Fm(S)s Fn(\))817 3338 y Fr(=)c Fq(v)31 b Fr(holds,)c(since)g(\()p Fq(e;)14 b(v)s Fr(\))24 b Fp(2)f Fr(H\()p Fq(S)5 b Fr(\).)51 3438 y(c\))26 b(Let)f(\()p Fq(\017\030)t(A;)14 b(w)r Fr(\))25 b Fp(2)e Fr(H\()p Fq(S)5 b Fr(\))26 b(with)h Fq(w)e Fp(6)p Fr(=)e(?.)36 b(Then)26 b(\()p Fq(\017\030)t(A;)14 b(w)r Fr(\))25 b Fp(2)e Fq(S)30 b Fr(or)25 b Fq(\017\030)t(A)h Fr(is)g Fq(e)f Fr(and)g Fq(w)k Fr(is)c Fq(v)s Fr(.)37 b(One)25 b(has)g Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)p 3451 3416 V 27 x Fm(S)3525 3438 y Fp(2)23 b Fg(TRUE)o Fr(:)36 b(in)51 3547 y(the)30 b(\014rst)f(case)g(since)p 752 3480 56 4 v 29 w Fq(S)34 b Fr(is)c(correct,)e(and)i(in)f(the)h (second)f(case)g(b)n(y)g(de\014nition)h(of)g(H\()p Fq(S)5 b Fr(\))30 b(and)f(Lemma)h(3.7c.)41 b(By)30 b(Lemma)51 3646 y(4.8)c(one)i(has)f(rk)o(\()p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(]\))26 b Fq(<)c Fr(rk\()p Fq(e)p Fr(\),)28 b(and)f(b)n(y)g(Lemma)h(4.9)e Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)p 2103 3625 149 4 v 32 x Fn(H\()p Fm(S)s Fn(\))2278 3646 y Fr(=)22 b Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)p 2602 3625 49 4 v 28 x Fm(S)2653 3646 y Fr(.)27 b Fd(2)51 3830 y Fh(De\014nition)j (22)51 3929 y Fo(The)g Fr(H-pro)r(cess)f Fo(\(for)h Fq(C)6 b(r)870 3941 y Fn(0)908 3929 y Fq(;)14 b(:::;)g(C)6 b(r)1153 3941 y Fm(N)1217 3929 y Fo(\))30 b(is)g(de\014ne)l(d)g(as)g(fol)t (lows:)51 4079 y Fq(S)102 4091 y Fn(0)162 4079 y Fr(:=)23 b Fp(;)p Fo(,)84 b Fq(S)475 4091 y Fm(n)p Fn(+1)628 4079 y Fr(:=)738 3962 y Ff(\032)814 4043 y Fr(H\()p Fq(S)959 4055 y Fm(n)1005 4043 y Fr(\))h Fo(if)p 1203 3976 97 4 v 31 w Fq(S)1254 4055 y Fm(n)1329 4043 y Fo(is)30 b(nonsolving)814 4142 y Fp(;)266 b Fo(otherwise)1810 4079 y(.)51 4239 y(The)30 b(H-pr)l(o)l(c)l(ess)g Fr(terminates)f Fo(i\013)h(ther)l(e)g (exists)f(an)h Fq(n)23 b Fp(2)g Fr(I)-14 b(N)31 b Fo(such)e(that)p 2283 4173 V 30 w Fq(S)2334 4251 y Fm(n)2409 4239 y Fo(is)h(solving.)51 4410 y Fr(The)25 b(next)h(de\014nition)f(determines)g(when)h(the)g (substitution)f Fq(S)31 b Fr(itself,)26 b(\(and)f(not)p 2644 4343 56 4 v 26 w Fq(S)t Fr(\))h(con)n(tains)e(su\016cien)n(t)i (information)e(to)51 4510 y(compute)j(all)h(necessary)d(v)-5 b(alues.)37 b(An)28 b(imp)r(ortan)n(t)f(instance)h(is)f(the)h(pro)n(v) -5 b(able)26 b(incorrectness)g(of)i Fq(S)5 b Fr(.)51 4680 y Fh(De\014nition)30 b(23)51 4780 y Fo(L)l(et)f Fq(S)34 b Fo(b)l(e)c(an)g Fq(\017)p Fo(-substitution.)51 4880 y Fq(S)k Fo(is)c Fr(computationally)d(inconsisten)n(t)g(\(ci\))j Fo(i\013)g Fq(A)24 b(,)-14 b Fp(!)1734 4892 y Fm(S)1805 4880 y Fg(F)-7 b(ALSE)30 b Fo(for)h(some)f Fq(A)23 b Fp(2)h(F)8 b Fr(\()p Fq(S)d Fr(\))p Fo(.)51 4979 y(Otherwise)30 b Fq(S)k Fo(is)c Fr(computationally)d(consisten)n(t)g(\(cc\))p Fo(.)51 5079 y(A)n(n)i(expr)l(ession)h Fq(e)f Fo(is)h Fq(S)5 b Fr(-computable)29 b Fo(i\013)h Fq(d)p Fr(\()p Fp(j)p Fq(e)p Fp(j)1543 5091 y Fm(S)1592 5079 y Fr(\))23 b(=)g(0)p Fo(.)51 5179 y Fq(S)34 b Fo(is)c Fr(computing)g Fo(i\013)g(al)t(l)h(formulas)f Fq(A)24 b Fp(2)f(F)8 b Fr(\()p Fq(S)d Fr(\))30 b Fo(ar)l(e)g Fq(S)5 b Fo(-c)l(omputable.)51 5278 y Fq(S)34 b Fo(is)c Fr(deciding)g Fo(i\013)g Fq(S)k Fo(is)c(c)l(omputing)g(and)g(the)g(critic)l(al)h(formulas)g Fq(C)6 b(r)2257 5290 y Fn(0)2295 5278 y Fq(;)14 b(:::;)g(C)6 b(r)2540 5290 y Fm(N)2633 5278 y Fo(ar)l(e)30 b Fq(S)5 b Fo(-c)l(omputable.)51 5449 y Fh(Remark)51 5549 y Fr(1.)36 b(A)28 b(closed)f(form)n(ula)g Fq(A)h Fr(is)f Fq(S)5 b Fr(-computable)27 b(i\013)h Fq(A)23 b(,)-14 b Fp(!)1772 5561 y Fm(S)1844 5549 y Fg(TRUE)26 b Fr(or)h Fq(A)c(,)-14 b Fp(!)2367 5561 y Fm(S)2439 5549 y Fg(F)-7 b(ALSE)p Fr(.)51 5648 y(2.)36 b Fq(e)27 b Fr(is)h Fq(S)5 b Fr(-computable)27 b(i\013)h Fp(j)p Fq(e)p Fp(j)1017 5660 y Fm(S)1088 5648 y Fr(=)22 b Fp(j)p Fq(e)p Fp(j)p 1260 5626 49 4 v 28 x Fm(S)1308 5648 y Fr(.)51 5748 y(3.)36 b(If)28 b Fq(S)k Fr(is)c(computing)f(and)h(cc)f(then)h Fq(S)33 b Fr(is)27 b(correct.)51 5870 y(The)g(next)h(de\014nition)g(requires)e(that)i(all) f(steps)h(for)f(computing)h(H\()p Fq(S)5 b Fr(\))28 b(are)e(p)r (ossible)i(in)f(terms)h(of)f Fq(S)33 b Fr(itself.)1958 6119 y(13)p eop %%Page: 14 14 14 13 bop 51 614 a Fh(De\014nition)30 b(24)51 714 y Fo(L)l(et)f Fq(S)34 b Fo(b)l(e)c(an)g Fq(\017)f Fo(substitution.)38 b(We)30 b(say)g(that)g Fr(the)e(H-rule)f(applies)g(to)j Fq(S)k Fo(i\013)51 814 y(\(1\))c Fq(S)k Fo(is)c(c)l(c,)g(de)l(ciding,)i (nonsolving,)f(and)51 913 y(\(2\))f(if)g Fr(Cr\()p Fq(S)5 b Fr(\))23 b(=)g Fq(F)12 b Fr([)p Fq(t)p Fr(])23 b Fp(!)g Fq(F)12 b Fr([)p Fq(\017xF)g Fr(])30 b Fo(then)f(ther)l(e)h(exists)f (an)h Fq(n)23 b Fp(2)h Fr(I)-14 b(N)30 b Fo(with)g Fp(j)p Fq(F)12 b Fp(j)2440 925 y Fm(S)2488 913 y Fr([)-12 b([)p Fq(n)p Fr(])g(])24 b Fq(,)-14 b Fp(!)2722 925 y Fm(S)2794 913 y Fg(TRUE)n Fo(.)51 1096 y Fh(Lemma)29 b(5.2)51 1196 y Fo(L)l(et)g Fq(S)34 b Fo(b)l(e)c(c)l(c,)g(de)l(ciding,)i(nonsolving,) f(and)g Fr(Cr)o(\()p Fq(S)5 b Fr(\))24 b(=)e Fq(F)12 b Fr([)p Fq(t)p Fr(])23 b Fp(!)g Fq(F)12 b Fr([)p Fq(\017xF)g Fr(])p Fo(.)39 b(Then)51 1295 y(a\))30 b Fp(j)p Fq(t)p Fp(j)233 1307 y Fm(S)304 1295 y Fp(2)23 b Fr(I)-14 b(N)30 b Fo(and)h Fq(F)12 b Fr([)p Fp(j)p Fq(t)p Fp(j)816 1307 y Fm(S)864 1295 y Fr(])23 b Fq(,)-14 b Fp(!)1002 1307 y Fm(S)1073 1295 y Fg(TRUE)o Fo(.)51 1395 y(b\))29 b(If)i(the)f (formulas)g Fq(F)12 b Fr([)p Fq(i)p Fr(])23 b(\()p Fq(i)g(<)f Fp(j)p Fq(t)p Fp(j)1130 1407 y Fm(S)1179 1395 y Fr(\))30 b Fo(ar)l(e)g Fq(S)5 b Fo(-c)l(omputable)30 b(then)f(the)h(H-rule)f (applies)j(to)e Fq(S)5 b Fo(.)51 1577 y(Pr)l(o)l(of:)51 1677 y Fr(a\))29 b(Since)g(Cr\()p Fq(S)5 b Fr(\))26 b Fq(,)-14 b Fp(!)p 703 1655 49 4 v 28 x Fm(S)777 1677 y Fg(F)-7 b(ALSE)29 b Fr(and)g(Cr\()p Fq(S)5 b Fr(\))29 b(is)g Fq(S)5 b Fr(-computable,)29 b(w)n(e)g(ha)n(v)n(e)f Fq(F)12 b Fr([)p Fq(t)p Fr(])26 b Fq(,)-14 b Fp(!)2664 1689 y Fm(S)2738 1677 y Fg(TRUE)o Fr(.)41 b(No)n(w)29 b(the)h(assertion)d(follo)n(ws)51 1777 y(b)n(y)g(the)h(Lemmas)f(3.7b)g (and)g(3.8b.)51 1876 y(b\))h(Let)f Fq(m)c Fr(:=)g Fp(j)p Fq(t)p Fp(j)588 1888 y Fm(S)636 1876 y Fr(.)37 b(Since)28 b Fq(F)12 b Fr([)p Fq(m)p Fr(])23 b Fq(,)-14 b Fp(!)1212 1888 y Fm(S)1283 1876 y Fg(TRUE)27 b Fr(and)g(since)h Fq(F)12 b Fr([0])p Fq(;)i(:::;)g(F)e Fr([)p Fq(m)p Fp(\000)o Fr(1])27 b(are)g Fq(S)5 b Fr(-computable,)26 b(for)51 1976 y Fq(n)c Fr(:=)h(min)q Fp(f)p Fq(k)i Fr(:)37 b Fq(F)12 b Fr([)p Fq(k)s Fr(])23 b Fq(,)-14 b Fp(!)815 1988 y Fm(S)886 1976 y Fg(TRUE)o Fp(g)27 b Fr(w)n(e)g(ha)n(v)n(e)g Fq(n)c Fp(\024)f Fq(m)28 b Fr(and)f Fp(j)p Fq(F)12 b Fp(j)2019 1988 y Fm(S)2068 1976 y Fr([)-12 b([)p Fq(n)p Fr(])g(])24 b Fq(,)-14 b Fp(!)2302 1988 y Fm(S)2373 1976 y Fg(TRUE)o Fr(.)28 b Fd(2)51 2159 y Fh(Lemma)h(5.3)51 2258 y Fo(If)h(the)g(H-rule)f(applies)j(to)d Fq(S)35 b Fo(then)29 b Fr(H\()p Fq(S)5 b Fr(\))31 b Fo(is)f(c)l(orr)l(e)l(ct.) 51 2441 y(Pr)l(o)l(of:)38 b Fr(\(cf.)28 b(pro)r(of)f(of)h(5.1\))51 2540 y(Let)33 b(\()p Fq(\017\030)t(A;)14 b(w)r Fr(\))34 b Fp(2)g Fr(H\()p Fq(S)5 b Fr(\))34 b(with)g Fq(w)i Fp(6)p Fr(=)c(?.)55 b(W)-7 b(e)34 b(ha)n(v)n(e)e(to)i(pro)n(v)n(e)e Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)2267 2555 y Fn(H\()p Fm(S)s Fn(\))2451 2540 y Fp(2)33 b Fg(TRUE)o Fr(.)55 b(But)34 b(rk)o(\()p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(]\))36 b Fq(<)c Fr(rk)o(\()p Fq(e)p Fr(\))i(\(with)h Fq(e)51 2640 y Fr(the)e(H-term)h(of)f Fq(S)5 b Fr(\))33 b(and)g(therefore)g Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)1474 2655 y Fn(H\()p Fm(S)s Fn(\))1658 2640 y Fr(=)32 b Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)1992 2652 y Fm(S)2042 2640 y Fr(.)54 b(If)34 b(\()p Fq(\017\030)t(A;)14 b(w)r Fr(\))34 b Fp(2)f Fq(S)38 b Fr(then)c Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)3149 2652 y Fm(S)3232 2640 y Fp(2)33 b Fg(TRUE)n Fr(,)i(since)e Fq(S)38 b Fr(is)51 2740 y(correct.)46 b(Otherwise)30 b Fq(\017\030)t(A)f Fr(=)g Fq(e)i Fr(and)g Fq(w)h Fr(=)d(H-v)-5 b(alue)27 b(of)h Fq(S)t Fr(.)48 b(W)-7 b(e)32 b(treat)e(the)i(case)e(where)h(Cr\()p Fq(S)5 b Fr(\))29 b(=)g Fq(F)12 b Fr([)p Fq(T)g Fr(])28 b Fp(!)h Fq(F)12 b Fr([)p Fq(\017X)7 b(F)12 b Fr([)p Fq(X)7 b Fr(]])51 2839 y(and)32 b(lea)n(v)n(e)f(the)i(t)n(w)n(o)f(other)g(cases) f(to)h(the)h(reader.)50 b(Then)33 b Fq(\017\030)t(A)e Fr(=)g Fq(e)g Fr(=)g Fq(\017\030)t Fp(j)p Fq(F)12 b Fp(j)p 2563 2817 V 28 x Fm(S)2643 2839 y Fr(and)33 b Fq(w)g Fr(=)e Fp(j)p Fq(T)12 b Fp(j)p 3105 2817 V 28 x Fm(S)3152 2839 y Fr(.)52 b(Since)33 b Fq(S)k Fr(is)32 b(deciding,)51 2939 y Fq(F)12 b Fr([)p Fq(T)g Fr(])28 b(is)h Fq(S)5 b Fr(-computable)28 b(and)h(therefore)g(\(b)n(y)g(Lemmas)f(3.7c)h(and)g (3.8c\))f Fp(j)p Fq(F)12 b Fr([)p Fq(T)g Fr(])p Fp(j)2619 2951 y Fm(S)2692 2939 y Fr(=)25 b Fp(j)14 b(j)p Fq(F)e Fp(j)p 2930 2917 V 28 x Fm(S)2978 2939 y Fr([)i Fp(j)p Fq(T)e Fp(j)p 3122 2917 V 28 x Fm(S)3183 2939 y Fr(])i Fp(j)3243 2951 y Fm(S)3317 2939 y Fr(=)25 b Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(w)r Fr(])g(])p Fp(j)3644 2951 y Fm(S)3695 2939 y Fr(.)41 b(Since)51 3039 y Fp(j)p Fr(Cr)o(\()p Fq(S)5 b Fr(\))p Fp(j)p 309 3017 V 27 x Fm(S)381 3039 y Fp(2)23 b Fg(F)-7 b(ALSE)28 b Fr(and)g Fq(S)k Fr(is)27 b(deciding,)h(w)n(e)f(ha)n(v)n(e)g Fp(j)p Fq(F)12 b Fr([)p Fq(T)g Fr(])p Fp(j)1934 3051 y Fm(S)2004 3039 y Fr(=)23 b Fp(j)p Fq(F)12 b Fr([)p Fq(T)g Fr(])p Fp(j)p 2310 3017 V 27 x Fm(S)2380 3039 y Fp(2)23 b Fg(TRUE)o Fr(.)65 b Fd(2)51 3313 y Fk(6)137 b(T)-11 b(ermination)45 b(pro)t(of)g(via)h (Cut-elimination)217 3495 y Fr(In)36 b(this)h(section)f(w)n(e)g(apply)h (the)f(general)g(sc)n(hema)f(from)h([12)o(])h(for)f(reducing)g(the)h (termination)f(problem)g(for)g(the)51 3595 y(H-pro)r(cess)29 b(to)h(the)h(cut-elimination)f(problem)g(in)g(some)g(sp)r(ecially)g (devised)g(sequen)n(t)g(calculus)g(with)h(a)e(kind)i(of)f Fq(!)s Fr(-rule.)51 3694 y(The)36 b(termination)f(pro)r(of)h(is)g (constructiv)n(e)f(and)h(uses)f(induction)i(up)f(to)g Fq(\017)2487 3706 y Fm(\017)2515 3714 y Fl(0)2551 3694 y Fr(.)63 b(In)36 b(the)g(sequel)g(w)n(e)g(use)g Fq(r)j Fr(as)c(syn)n(tactic)51 3794 y(v)-5 b(ariable)26 b(for)h(ordinals)g(\() p Fq(<)22 b(\017)957 3806 y Fm(\017)985 3814 y Fl(0)1021 3794 y Fr(\).)51 4026 y Fs(6.1)112 b(The)38 b(system)f Fi(\017E)6 b(A)217 4180 y Fr(The)26 b(original)e(in\014nite)j(deriv)-5 b(ation)26 b(constructed)f(for)h(a)g(giv)n(en)f(system)h Fq(E)31 b Fr(of)26 b(critical)g(form)n(ulas)e(represen)n(ts)h(the)i (tree)51 4279 y(of)k(all)f(\014nite)i Fq(\017)p Fr(-substitutions:)43 b(empt)n(y)31 b(substitution)h(is)f(put)g(at)g(the)g(b)r(ottom,)i(and)d (all)h(one-comp)r(onen)n(t)f(extensions)g(of)51 4379 y Fq(S)j Fr(are)27 b(placed)h(ab)r(o)n(v)n(e)f Fq(S)33 b Fr(\(the)c(rule)f Fg(Cut)g Fr(b)r(elo)n(w\).)39 b(The)29 b(branc)n(h)e(of)h(this)h(tree)f(is)g(terminated)g(\(cf.)40 b(Axioms)28 b(b)r(elo)n(w\))g(when)51 4478 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Fr(traceable)g(to)h(the)g(main)g(form)n(ula)f(\()p Fq(e;)14 b Fr(?\))24 b(of)g Fg(CutF)n(r)p Fr(.)36 b(Then)24 b(the)g Fg(AxS)h Fr(is)e(replaced)g(b)n(y)h(the)g(rule)g Fg(H)o Fr(,)h(and)f(the)g(deriv)-5 b(ation)23 b(of)51 4932 y(the)k(corresp)r(onding)e(righ)n(t)h(premise)g(of)h(the)g(cut)g (is)g(placed)g(o)n(v)n(er)e(the)i(rule)f Fg(H)p Fr(.)37 b(After)27 b(all)g(cuts)g(of)g(rank)e Fq(r)30 b Fr(are)c(eliminated,)51 5032 y(these)j Fg(CutF)n(r)g Fr(will)g(b)r(e)h(pruned)f(to)g Fg(F)n(r)p Fr(.)41 b(So)29 b(\014nally)g(cuts)g(of)g(rank)f Fq(r)k Fr(will)d(b)r(e)h(replaced)e(b)n(y)h Fg(F)n(r)g Fr(of)g(rank)f Fq(r)r Fr(.)43 b(This)29 b(motiv)-5 b(ates)51 5131 y(the)28 b(follo)n(wing)51 5293 y Fh(De\014nition)i(28)41 b Fo(L)l(et)30 b Fq(d)g Fo(b)l(e)f(a)i(de)l(duction.)51 5393 y Fq(d)f Fo(is)g(an)f Fq(r)r Fr(-deduction)90 b Fo(i\013)60 b Fg(Cut)p Fr(\()p Fq(d)p Fr(\))24 b Fq(<)e(r)k Fr(&)d Fg(CutF)n(r)p Fr(\()p Fq(d)p Fr(\))h Fq(<)f Fr(0)f(&)h Fg(F)n(r)p Fr(\()p Fq(d)p Fr(\))h Fp(\025)f Fq(r)i Fr(&)e Fg(H)p Fr(\()p Fq(d)p Fr(\))h Fp(\025)e Fq(r)r Fo(.)51 5493 y Fq(d)30 b Fo(is)g(an)f Fq(r)370 5463 y Fn(+)426 5493 y Fr(-deduction)h Fo(i\013)60 b Fg(Cut)p Fr(\()p Fq(d)p Fr(\))23 b Fq(<)g(r)j Fr(&)c Fg(CutF)n(r)q Fr(\()p Fq(d)p Fr(\))i(=)e Fq(r)k Fr(&)d Fg(F)n(r)p Fr(\()p Fq(d)p Fr(\))h Fq(>)e(r)k Fr(&)d Fg(H)p Fr(\()p Fq(d)p Fr(\))h Fp(\025)e Fq(r)r Fo(.)51 5671 y Fh(Lemma)29 b(6.1)51 5770 y Fo(Every)34 b Fq(r)331 5740 y Fn(+)387 5770 y Fo(-derivation)i Fq(d)d Fo(of)i Fr(\002)e Fo(c)l(an)h(b)l(e)g(tr)l (ansforme)l(d)g(into)g(an)g Fq(r)r Fo(-derivation)i Fq(d)2625 5740 y Fe(0)2682 5770 y Fo(of)f Fr(\002)e Fo(with)i Fr(h\()p Fq(d)3188 5740 y Fe(0)3212 5770 y Fr(\))30 b Fp(\024)g Fr(h\()p Fq(d)p Fr(\))35 b Fo(by)f(pruning)51 5870 y(e)l(ach)c Fg(CutF)n(r)g Fo(to)g Fg(F)n(r)p Fo(.)1958 6119 y Fr(15)p eop %%Page: 16 16 16 15 bop 51 614 a Fo(Pr)l(o)l(of)28 b Fr(is)g(ob)n(vious.)35 b Fd(2)51 783 y Fh(Lemma)29 b(6.2)41 b Fr(\(Prop)r(erties)26 b(of)i(\014xed)f(and)h(temp)r(orary)e(parts)h(of)g(a)h(sequen)n(t\))51 883 y Fo(a\))59 b(If)31 b Fr(\002)e Fo(is)h(a)g(se)l(quent)f(in)h(an)g Fq(r)r Fr(+1)p Fo(-de)l(duction)f(of)i Fp(;)e Fo(then)h Fr(\002)p Fq(t)22 b(>)h(r)33 b Fo(and)d Fr(\002)p Fq(f)h Fp(\024)23 b Fq(r)r Fo(.)51 982 y(b\))63 b(If)31 b Fr(\006)e Fo(is)h(a)g(se)l(quent)f(in)h(an)g Fq(r)1077 952 y Fn(+)1133 982 y Fo(-de)l(duction)g(of)g(a)h(se)l(quent)d Fr(\002)i Fo(then)186 1082 y(\(1\))60 b Fr(\002)421 1094 y Fe(\024)p Fm(r)528 1082 y Fp(n)18 b Fr(\002)p Fq(t)k Fp(\022)h Fr(\006)p Fo(,)186 1182 y(\(2\))60 b Fr(\(\006)p Fq(f)9 b Fr(\))530 1194 y Fe(\025)p Fm(r)642 1182 y Fp(\022)23 b Fr(\002)p Fo(,)186 1281 y(\(3\))60 b Fr(\002)p Fq(t)23 b Fp(\025)f Fq(r)40 b Fp(\))d Fr(\006)p Fq(t)23 b Fp(\025)g Fq(r)r Fo(.)51 1450 y(Pr)l(o)l(of.)51 1550 y Fr(a\))i(is)g(pro)n(v)n (ed)e(b)n(y)i(the)h(b)r(ottom-up)f(induction)h(on)e(the)i(giv)n(en)e (deduction.)37 b(This)25 b(statemen)n(t)g(is)g(ob)n(viously)f(true)h (for)f Fp(;)p Fr(,)h(and)51 1649 y(is)k(inherited)h(from)g(conclusion)f (to)g(the)i(premise)e(of)h(a)f(rule:)41 b(temp)r(orary)29 b(part)g(\002)p Fq(t)h Fr(is)f(increased)g(only)g(b)n(y)h Fg(F)n(r)g Fr(whic)n(h)g(has)51 1749 y(rank)c Fp(\025)d Fq(r)r Fr(+1)g Fq(>)f(r)r Fr(,)29 b(and)e(\002)p Fq(f)36 b Fr(is)28 b(increased)e(b)n(y)h Fg(Cut)h Fr(whic)n(h)g(has)f(rank)f Fq(<)d(r)e Fr(+)d(1)27 b(\(i.e.)38 b Fp(\024)22 b Fq(r)r Fr(\).)51 1849 y(b\)\(1\))28 b(The)g(only)f(non)n(trivial)f(case)h(is)h (\()p Fg(H)p Fr(\).)37 b(Let)28 b(\002)23 b(=)f(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\007)28 b(and)f(\006)c(=)g(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\007)2687 1864 y Fe(\024)p Fn(rk\()p Fm(e)p Fn(\))2887 1849 y Fr(.)37 b(Then)28 b(\002)3229 1861 y Fe(\024)p Fm(r)3336 1849 y Fp(n)18 b Fr(\002)p Fq(t)k Fp(\022)h Fr(\007)3666 1861 y Fe(\024)p Fm(r)3778 1849 y Fp(\022)f Fr(\006,)51 1948 y(since)27 b Fq(r)f Fp(\024)c Fr(rk\()p Fq(e)p Fr(\).)51 2048 y(\(2\))30 b(Going)f(from)h(\006)g(do)n(wn)f(to)h(\002)f(the)i(only)e(p)r(oin)n (ts)h(where)g(some)f(\()p Fq(e;)14 b Fr(?)2324 2018 y Fe(\016)2362 2048 y Fr(\))30 b(could)g(v)-5 b(anish)30 b(are)f Fg(Cut)p Fr(-inferences.)43 b(But)30 b(eac)n(h)51 2148 y Fg(Cut)d Fr(in)h(an)f Fq(r)455 2117 y Fn(+)511 2148 y Fr(-deriv)-5 b(ation)27 b(has)g(rank)g Fq(<)22 b(r)r Fr(.)51 2247 y(\(3\))27 b(holds,)h(since)f Fg(F)n(r)p Fr(\()p Fq(d)p Fr(\))d Fq(>)f(r)30 b Fr(and)e Fg(CutF)n(r)p Fr(\()p Fq(d)p Fr(\))c(=)e Fq(r)r Fr(,)29 b(so)e(that)h(form)n(ulas)e (added)i(to)f(\002)p Fq(t)g Fr(are)g(of)g(the)h(rank)f Fp(\025)c Fq(r)r Fr(.)204 b Fd(2)217 2391 y Fr(Applicabilit)n(y)26 b(of)h(the)g(standard)e(cut-elimination)i(transformations)d(in)j(our)f (case)f(is)i(seriously)e(restricted)h(b)n(y)g(non-)51 2491 y(admissibilit)n(y)37 b(\(in)i(general\))e(of)h(the)g(w)n(eak)n (ening)f(rule:)57 b(adding)38 b(comp)r(onen)n(ts)f(to)h(a)g(sequen)n(t) f(can)h(fail)g(to)g(pro)r(duce)g(a)51 2590 y(sequen)n(t.)51 2759 y Fh(De\014nition)30 b(29)41 b Fo(Two)31 b(se)l(quents)f Fr(\002)p Fq(;)14 b Fr(\006)29 b Fo(ar)l(e)h Fr(m)n(ultiplicable)h Fo(if)g Fr(\002)18 b Fp([)h Fr(\006)30 b Fo(is)g(a)h(function.)39 b(In)30 b(this)h(c)l(ase)f(we)g(write)h Fr(\002)18 b Fp(\003)g Fr(\006)30 b Fo(for)51 2859 y Fr(\002)18 b Fp([)g Fr(\006)p Fo(,)31 b(and)f(say)g(that)g Fr(\002)18 b Fp(\003)g Fr(\006)30 b Fo(is)g Fr(de\014ned)p Fo(.)51 3028 y Fh(Lemma)f(6.3)51 3127 y Fo(L)l(et)g Fq(d)h Fo(b)l(e)g(an)f Fq(r)526 3097 y Fn(+)582 3127 y Fo(-derivation)j(of)e Fr(\002)p Fo(.)38 b(L)l(et)30 b Fr(\006)23 b Fp(\024)f Fq(r)33 b Fo(b)l(e)d(a)g(c)l(orr)l(e)l(ct)f(se)l(quent)g(such)h(that)g Fr(\002)18 b Fp(\003)g Fr(\006)29 b Fo(is)i(de\014ne)l(d)f(and)51 3227 y Fr(\(\006)p Fq(f)9 b Fr(\))225 3239 y Fe(\025)p Fm(r)336 3227 y Fp(\022)23 b Fr(\002)p Fq(;)37 b Fr(\006)p Fq(t)23 b Fp(\025)f Fq(r)r Fo(.)40 b(Then)30 b(ther)l(e)g(exists)f(an)h Fq(r)1661 3197 y Fn(+)1717 3227 y Fo(-derivation)h Fq(d)19 b Fp(\003)f Fr(\006)30 b Fo(of)g Fr(\002)18 b Fp(\003)g Fr(\006)30 b Fo(with)g Fr(h\()p Fq(d)19 b Fp(\003)f Fr(\006\))23 b Fp(\024)g Fr(h\()p Fq(d)p Fr(\))p Fo(.)51 3396 y(Pr)l(o)l(of)30 b(by)h(induction)f(on)g Fq(d)p Fo(:)51 3495 y Fr(W)-7 b(e)28 b(distinguish)f(cases)g(according)f(to)h(the)h(last)f(inference) h(of)f Fq(d)p Fr(.)51 3595 y(1.)36 b Fg(Cut)28 b Fr(with)g(main)f(term) h Fq(e)p Fr(:)37 b(Then)27 b(rk\()p Fq(e)p Fr(\))c Fq(<)g(r)r Fr(.)51 3695 y(\(a\))k Fq(e)c Fp(62)g Fr(dom\(\006\):)38 b Fq(d)18 b Fp(\003)g Fr(\006)28 b(is)g(obtained)f(from)g(\()p Fq(d)1571 3707 y Fm(u)1633 3695 y Fp(\003)18 b Fr(\006\))1785 3710 y Fm(u)p Fe(2)p Fn(I)-11 b(B)1929 3721 y Fa(\023)p Fl(\()p Fa(e)p Fl(\))2029 3710 y Fe([f)p Fn(?)2139 3693 y Fc(\016)2174 3710 y Fe(g)2239 3695 y Fr(b)n(y)28 b(the)g(same)f(Cut.) 51 3794 y(\(b\))h(\()p Fq(e;)14 b Fr(?\))23 b Fp(2)h Fr(\006:)37 b(This)27 b(cannot)g(happ)r(en,)h(since)g(\006)p Fq(t)23 b Fp(\025)f Fq(r)31 b Fr(and)c(rk\()p Fq(e)p Fr(\))c Fq(<)g(r)r Fr(.)51 3894 y(\(c\))28 b(\()p Fq(e;)14 b(u)p Fr(\))22 b Fp(2)i Fr(\006)k(with)g Fq(u)22 b Fp(2)i Fr(I)-14 b(B)970 3909 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))1100 3894 y Fp([)19 b(f)p Fr(?)1255 3864 y Fe(\016)1293 3894 y Fp(g)p Fr(:)51 3994 y(Then)26 b(\(\()p Fq(e;)14 b(u)p Fr(\))p Fq(;)g Fr(\002\))i Fp(\003)g Fr(\006)23 b(=)g(\002)15 b Fp(\003)h Fr(\006,)27 b(and)f(therefore)g(the)h Fg(Cut)f Fr(is)h(pruned:)36 b(the)27 b(deriv)-5 b(ation)26 b Fq(d)16 b Fp(\003)g Fr(\006)23 b(:=)f Fq(d)3195 4006 y Fm(u)3255 3994 y Fp(\003)16 b Fr(\006)27 b(has)f(endsequen)n(t)51 4093 y(\002)18 b Fp(\003)g Fr(\006.)51 4193 y(2.)36 b Fg(CutF)n(r)28 b Fr(with)g(main)g(term)f Fq(e)p Fr(.)37 b(Then)28 b(rk)o(\()p Fq(e)p Fr(\))23 b(=)g Fq(r)30 b Fr(and)e(\()p Fq(e;)14 b Fr(?)1956 4163 y Fe(\016)1994 4193 y Fr(\))23 b Fp(62)h Fr(\002.)36 b(Hence)28 b(\()p Fq(e;)14 b Fr(?)2646 4163 y Fe(\016)2684 4193 y Fr(\))24 b Fp(62)f Fr(\006,)28 b(since)f(\(\006)p Fq(f)9 b Fr(\))3306 4205 y Fe(\025)p Fm(r)3418 4193 y Fp(\022)23 b Fr(\002.)51 4292 y(\(a\))k Fq(e)c Fp(62)g Fr(dom\(\006\):)38 b(As)27 b(\(a\))h(ab)r(o)n(v)n(e.)51 4392 y(\(b\))g(\()p Fq(e;)14 b(u)p Fr(\))23 b Fp(2)g Fr(\006)28 b(with)g Fq(u)22 b Fp(2)i Fr(I)-14 b(B)979 4407 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))1109 4392 y Fp([)19 b(f)p Fr(?)p Fp(g)p Fr(:)36 b(as)27 b(\(c\))h(ab)r(o)n(v)n(e.)51 4492 y(3.)36 b Fg(F)n(r)28 b Fr(with)g(main)g(term)f Fq(e)p Fr(:)37 b(Then)28 b(\(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\002\))19 b Fp(\003)e Fr(\006)28 b(is)g(de\014ned,)g(since)f(\006)c Fp(\024)g Fq(r)j(<)c Fr(rk)o(\()p Fq(e)p Fr(\).)51 4591 y(4.)36 b Fg(H)28 b Fr(with)g(main)f(term)h Fq(e)p Fr(:)36 b(Then)28 b(rk\()p Fq(e)p Fr(\))23 b Fp(\025)f Fq(r)r Fr(,)29 b(and)e(\002)c(=)g(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\007)27 b(is)h(deriv)n(ed)e(from)i(\002) 2811 4561 y Fe(0)2857 4591 y Fr(:=)22 b(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\007)3252 4606 y Fe(\024)p Fn(rk\()p Fm(e)p Fn(\))3452 4591 y Fr(.)51 4691 y(>F)-7 b(rom)27 b(\006)c Fp(\024)g Fq(r)i Fp(\024)e Fr(rk)o(\()p Fq(e)p Fr(\))28 b(it)g(follo)n(ws)f(that)h(\006)1432 4661 y Fe(0)1478 4691 y Fr(:=)23 b(\006)18 b Fp(n)g(f)p Fr(\()p Fq(e;)c(:::)p Fr(\))p Fp(g)27 b Fr(is)h(still)g(correct,)e(and)h(that)h (\(\006)3026 4661 y Fe(0)3050 4691 y Fq(f)9 b Fr(\))3132 4703 y Fe(\025)p Fm(r)3243 4691 y Fp(\022)23 b Fr(\002)3396 4661 y Fe(0)3419 4691 y Fr(.)51 4791 y(>F)-7 b(rom)27 b Fq(e)c Fp(62)g Fr(dom\(\006)697 4760 y Fe(0)720 4791 y Fr(\))28 b(and)g(\002)1007 4760 y Fe(0)1053 4791 y Fp(\022)22 b Fr(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\007)28 b(it)g(follo)n(ws)f(that)g(\002)2053 4760 y Fe(0)2095 4791 y Fp(\003)18 b Fr(\006)2215 4760 y Fe(0)2266 4791 y Fr(is)27 b(de\014ned.)51 4890 y(Since)c(\006)323 4860 y Fe(0)369 4890 y Fp(\024)g Fq(r)j Fp(\024)c Fr(rk\()p Fq(e)p Fr(\),)i(w)n(e)f(also)f(get)h(\002)1313 4860 y Fe(0)1346 4890 y Fp(\003)10 b Fr(\006)1458 4860 y Fe(0)1503 4890 y Fr(=)23 b(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\007)1876 4905 y Fe(\024)p Fn(rk)o(\()p Fm(e)p Fn(\))2085 4890 y Fp(\003)c Fr(\006)2197 4860 y Fe(0)2243 4890 y Fr(=)22 b(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\(\007)c Fp(\003)g Fr(\006)2769 4860 y Fe(0)2792 4890 y Fr(\))2824 4905 y Fe(\024)p Fn(rk\()p Fm(e)p Fn(\))3047 4890 y Fr(and)23 b(\002)10 b Fp(\003)g Fr(\006)22 b(=)g(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\007)c Fp(\003)g Fr(\006)3903 4860 y Fe(0)3925 4890 y Fr(.)51 4990 y(So)27 b Fq(d)19 b Fp(\003)f Fr(\006)27 b(is)h(obtained)f(from)g Fq(d)1039 5002 y Fn(0)1095 4990 y Fp(\003)18 b Fr(\006)1215 4960 y Fe(0)1266 4990 y Fr(b)n(y)27 b(an)h Fg(H)o Fr(-inference)g(of)f(the)h(same)f (kind.)51 5089 y(5.)36 b(Axioms:)h(If)28 b(\002)f(is)g(an)h(axiom)e (then)j(\002)18 b Fp(\003)g Fr(\006)27 b(is)h(an)f(axiom)g(of)g(the)h (same)f(kind.)203 b Fd(2)51 5233 y Fr(W)-7 b(e)28 b(con)n(tin)n(ue)f (to)g(in)n(v)n(estigate)f(admissibilit)n(y)i(of)f(w)n(eak)n(ening)f(.) 51 5402 y Fh(De\014nition)k(30)41 b Fr(\(\002)730 5414 y Fn(0)768 5402 y Fq(;)14 b(:::;)g Fr(\002)976 5414 y Fm(n)1020 5402 y Fr(\))30 b Fo(is)g(an)g Fq(r)r Fr(-path)e(\(for)g (\002)1776 5414 y Fm(n)1820 5402 y Fr(\))i Fo(if)h(it)f(is)g(a)g(p)l (ath)g(in)g(some)g Fq(r)r Fo(-de)l(duction)h(of)g Fr(\002)3307 5414 y Fn(0)3367 5402 y Fr(=)22 b Fp(;)p Fo(.)51 5571 y Fh(Lemma)29 b(6.4)51 5671 y Fo(L)l(et)g Fr(\(\002)291 5683 y Fn(0)328 5671 y Fq(;)14 b(:::;)g Fr(\002)536 5683 y Fm(n)581 5671 y Fr(\))30 b Fo(b)l(e)g(an)f Fq(r)r Fr(+)q(1)p Fo(-p)l(ath)g(for)i Fr(\002)23 b(:=)f(\002)1615 5683 y Fm(n)1660 5671 y Fo(.)39 b(L)l(et)29 b Fr(\006)23 b Fp(\024)g Fq(r)32 b Fo(b)l(e)e(a)g(c)l(orr)l(e)l(ct)g(se)l(quent)e (such)i(that)g Fr(\002)3269 5683 y Fe(\024)p Fm(r)3380 5671 y Fp(\022)23 b Fr(\006)p Fo(.)51 5770 y(Then)30 b Fr(\002)17 b Fp(\003)h Fr(\006)29 b Fo(is)h(de\014ne)l(d,)h(and)f (ther)l(e)f(exists)h(a)g(de)l(duction)g(of)g Fr(\006)g Fo(fr)l(om)g Fr(\002)17 b Fp(\003)h Fr(\006)29 b Fo(c)l(onsisting)h (only)g(of)h Fg(F)n(r)p Fo(-)e(and)h Fg(H)p Fo(-infer)l(enc)l(es)51 5870 y(of)g(r)l(anks)g Fq(>)23 b(r)r Fo(.)1958 6119 y Fr(16)p eop %%Page: 17 17 17 16 bop 51 614 a Fo(Pr)l(o)l(of)30 b(by)h(induction)f(on)g Fq(n)p Fo(:)37 b Fr(Let)28 b Fq(n)22 b(>)h Fr(0)k(and)h(\002)1587 584 y Fe(0)1633 614 y Fr(:=)22 b(\002)1808 626 y Fm(n)p Fe(\000)p Fn(1)1938 614 y Fr(.)51 714 y(\002)15 b Fp(\003)g Fr(\006)26 b(is)g(de\014ned,)h(since)f(\002)931 726 y Fe(\024)p Fm(r)1042 714 y Fp(\022)d Fr(\006)g(=)g(\006)1361 726 y Fe(\024)p Fm(r)1449 714 y Fr(.)37 b(Note)26 b(that)h(\002)1952 684 y Fe(0)1952 737 y(\024)p Fm(r)2063 714 y Fp(\022)22 b Fr(\002)2215 726 y Fe(\024)p Fm(r)2304 714 y Fr(,)k(and)g(therefore)g (b)n(y)f(I.H.)i(there)f(is)g(a)g(deduction)g(of)51 814 y(\006)d(from)g(\002)391 783 y Fe(0)425 814 y Fp(\003)10 b Fr(\006.)35 b(W)-7 b(e)24 b(no)n(w)f(sho)n(w)g(that)h(either)f(\002) 1577 783 y Fe(0)1611 814 y Fp(\003)10 b Fr(\006)23 b(=)f(\002)10 b Fp(\003)g Fr(\006)24 b(or)e(\002)2206 783 y Fe(0)2240 814 y Fp(\003)10 b Fr(\006)23 b(deriv)n(es)g(from)g(\002)10 b Fp(\003)g Fr(\006)24 b(b)n(y)f(a)g Fg(F)n(r)p Fr(-)h(or)e Fg(H)p Fr(-inference)h(of)51 913 y(rank)j Fq(>)c(r)r Fr(.)38 b(F)-7 b(or)26 b(this)i(w)n(e)f(distinguish)g(cases)f (according)f(to)i(the)g(topmost)g(inference)g(in)h(\(\002)2970 925 y Fn(0)3007 913 y Fq(;)14 b(:::;)g Fr(\002)3215 925 y Fm(n)3260 913 y Fr(\),)27 b(i.e.)37 b(the)27 b(inference)51 1013 y(from)g(\002)g(to)h(\002)506 983 y Fe(0)528 1013 y Fr(.)51 1112 y(1.)62 b Fg(Cut)36 b Fr(:)54 b(W)-7 b(e)37 b(ha)n(v)n(e)e(\002)h(=)h(\()p Fq(e;)14 b(u)p Fr(\))p Fq(;)g Fr(\002)1262 1082 y Fe(0)1321 1112 y Fr(and)36 b(rk)o(\()p Fq(e)p Fr(\))i Fp(\024)f Fq(r)r Fr(.)63 b(Then)37 b(the)f Fg(Cut)g Fr(is)g(pruned.)63 b(Indeed,)38 b(\()p Fq(e;)14 b(u)p Fr(\))37 b Fp(2)h Fr(\006)e(and)g(th)n(us)51 1212 y(\002)18 b Fp(\003)g Fr(\006)23 b(=)f(\002)429 1182 y Fe(0)471 1212 y Fp(\003)c Fr(\006.)51 1312 y(2.)36 b Fg(CutF)n(r)28 b Fr(:)37 b(This)28 b(cannot)f(happ)r(en.)51 1411 y(3.)36 b Fg(F)n(r)28 b Fr(with)g(main)g(term)f Fq(e)p Fr(,)h(and)f(\002)c(=)f(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\002)1555 1381 y Fe(0)1578 1411 y Fr(:)37 b(Then)28 b(rk\()p Fq(e)p Fr(\))23 b Fq(>)f(r)31 b Fr(and)c(\002)18 b Fp(\003)g Fr(\006)23 b(=)g(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\002)2969 1381 y Fe(0)3011 1411 y Fp(\003)k Fr(\006.)51 1511 y(4.)36 b Fg(H)28 b Fr(with)g(main)f(term)h Fq(e)p Fr(,)f(and)h(\002)22 b(=)h(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\007)1545 1526 y Fe(\024)p Fn(rk)o(\()p Fm(e)p Fn(\))1745 1511 y Fq(;)37 b Fr(\002)1870 1481 y Fe(0)1916 1511 y Fr(=)22 b(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\007:)51 1611 y(Then)20 b(\002)325 1580 y Fe(0)350 1611 y Fp(\003)s Fr(\006)g(is)f(deriv)n(ed)g(from)h(\002)s Fp(\003)s Fr(\006)f(b)n(y)g Fg(H)p Fr(.)34 b(Indeed)20 b(rk\()p Fq(e)p Fr(\))j Fq(>)g(r)r Fr(,)42 b(\002)s Fp(\003)s Fr(\006)22 b(=)g(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\(\007)s Fp(\003)s Fr(\006\))2833 1626 y Fe(\024)p Fn(rk\()p Fm(e)p Fn(\))3033 1611 y Fr(,)22 b(and)d(\002)3296 1580 y Fe(0)3322 1611 y Fp(\003)s Fr(\006)j(=)h(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\007)s Fp(\003)s Fr(\006.)51 1710 y(The)27 b(H-rule)g(applies)h(to)f(\002)921 1680 y Fe(0)962 1710 y Fp(\003)18 b Fr(\006)28 b(\(cf.)g(De\014nition)h(24\),)e(since)g(the) h(H-rule)f(applies)g(to)h(\002)2854 1680 y Fe(0)2877 1710 y Fr(,)g(and)f(\006)h(is)f(correct.)202 b Fd(2)217 1857 y Fr(Applicabilit)n(y)30 b(of)h(the)f(cut-reduction)g (transformation)f(is)i(restricted)e(more)h(or)f(less)h(to)h(sub)r (deriv)-5 b(ations)29 b(of)i(`go)r(o)r(d')51 1957 y(deriv)-5 b(ations)26 b(of)i(an)f(empt)n(y)h(sequen)n(t)f Fp(;)p Fr(.)51 2134 y Fh(Lemma)i(6.5)51 2234 y Fo(L)l(et)36 b Fq(d)h Fo(b)l(e)g(a)g(derivation)h(ending)g(with)f(a)g(cut)f Fp(C)42 b Fo(of)37 b(r)l(ank)g Fq(r)i Fo(such)e(that)g(the)g(imme)l (diate)h(sub)l(derivations)g(of)g Fq(d)f Fo(ar)l(e)g Fq(r)3863 2204 y Fn(+)3919 2234 y Fo(-)51 2333 y(derivations,)32 b(and)e(ther)l(e)g(exists)f(an)h Fr(\()p Fq(r)r Fr(+1\))p Fo(-p)l(ath)g(for)h(the)f(endse)l(quent)f Fr(\002)g Fo(of)i Fq(d)p Fo(.)51 2433 y(Then)f(ther)l(e)g(exists)f(an)h Fq(r)859 2403 y Fn(+)915 2433 y Fo(-derivation)h Fq(d)1381 2403 y Fe(0)1434 2433 y Fo(of)g Fr(\002)e Fo(with)i Fr(h\()p Fq(d)1928 2403 y Fe(0)1952 2433 y Fr(\))23 b Fp(\024)f Fr(h\()p Fq(d)p Fr(\))e(+)e Fq(!)j Fr(+)d(h\()p Fq(d)p Fr(\))p Fo(.)51 2610 y(Pr)l(o)l(of.)38 b Fr(Let)28 b Fq(e)f Fr(b)r(e)h(the)g(main)g(term)f(of)h Fp(C)5 b Fr(,)27 b(and)h Fq(d)1582 2622 y Fm(u)1653 2610 y Fr(the)g(immediate)g(sub)r (deriv)-5 b(ations)27 b(of)g Fq(d)p Fr(.)51 2710 y(W)-7 b(e)28 b(transform)e Fq(d)i Fr(as)f(follo)n(ws:)51 2809 y(\(1\))60 b(The)31 b(cut)g Fp(C)k Fr(is)30 b(turned)g(in)n(to)g(an)g (inference)h Fg(CutF)n(r)f Fr(b)n(y)g(c)n(hanging)f(ev)n(ery)g(sequen)n (t)h(\()p Fq(e;)14 b Fr(?)2983 2779 y Fe(\016)3021 2809 y Fr(\))p Fq(;)g Fr(\002)3155 2779 y Fe(0)3209 2809 y Fr(of)30 b Fq(d)3349 2821 y Fn(?)3380 2805 y Fc(\016)3450 2809 y Fr(in)n(to)g(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\002)3902 2779 y Fe(0)3925 2809 y Fr(.)51 2909 y(The)31 b(only)h(rules)f(that)h (are)f(damaged)g(b)n(y)g(this)i(transformation)d(are)h(axioms)f Fg(AxH)j Fr(of)f(the)g(form)f(\()p Fq(e;)14 b Fr(?)3357 2879 y Fe(\016)3395 2909 y Fr(\))p Fq(;)g Fr(\007)32 b(whic)n(h)g(no)n(w)51 3009 y(b)r(ecome)19 b(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\007.)34 b(A)n(t)20 b(eac)n(h)f(of)g(these)h(p)r (oin)n(ts)g(w)n(e)f(extend)h(the)g(deduction)f(b)n(y)h(the)g(corresp)r (onding)d Fg(H)p Fr(-inference,)k(obtaining)51 3108 y(thereb)n(y)27 b(the)h(new)f(top)h(sequen)n(t)f(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\007)1397 3120 y Fe(\024)p Fm(r)1486 3108 y Fr(.)51 3208 y(\(2\))55 b(W)-7 b(e)28 b(consider)f(no)n(w)g(one) g(suc)n(h)g(top)h(sequen)n(t)f(\006)c(:=)g(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\007)2118 3220 y Fe(\024)p Fm(r)2206 3208 y Fr(.)51 3307 y(By)27 b(Lemma)g(5.3)41 b(\006)23 b(=)f(\006)855 3319 y Fe(\024)p Fm(r)972 3307 y Fr(is)27 b(correct,)g(and)g(b)n(y)g(6.2a,b\(1\))g(w)n(e)g(get)g(\002)2308 3319 y Fe(\024)p Fm(r)2420 3307 y Fr(=)22 b(\002)2572 3319 y Fe(\024)p Fm(r)2679 3307 y Fp(n)c Fr(\002)p Fq(t)23 b Fp(\022)f Fr(\006.)51 3407 y(\(3\))52 b(By)25 b(Lemma)h(6.4)39 b(\002)15 b Fp(\003)f Fr(\006)26 b(is)g(de\014ned,)h(and)e(there)h (exists)g(a)f(deduction)h(of)g(\006)g(from)g(\002)14 b Fp(\003)h Fr(\006)26 b(consisting)f(only)h(of)f Fg(F)n(r)q Fr(-)g(and)51 3507 y Fg(H)o Fr(-inferences)i(of)h(ranks)e Fq(>)d(r)r Fr(.)51 3606 y(\(4\))h(T)-7 b(o)24 b(deriv)n(e)f(top)h (sequen)n(ts)g(\002)12 b Fp(\003)g Fr(\006)22 b(in)j(\(3\))f(w)n(e)g (consider)f(no)n(w)h(the)g(sub)r(deriv)-5 b(ation)24 b Fq(d)2757 3618 y Fm(v)2821 3606 y Fr(of)g(\002)2977 3576 y Fe(0)3023 3606 y Fr(:=)f(\()p Fq(e;)14 b(v)s Fr(\))p Fq(;)g Fr(\002)24 b(in)g(the)h(original)51 3706 y(deriv)-5 b(ation)27 b Fq(d)p Fr(.)51 3806 y(\002)116 3775 y Fe(0)161 3806 y Fp(\003)22 b Fr(\006)35 b(is)e(de\014ned,)k(since)c(\002)23 b Fp(\003)f Fr(\006)34 b(is)g(de\014ned)g(and)g(\()p Fq(e;)14 b(v)s Fr(\))34 b Fp(2)h Fr(\006.)56 b(By)33 b(Lemma)h(6.2b\(2\),\(3\))g(\(applied)g(to)g Fq(d)3519 3818 y Fn(?)3550 3801 y Fc(\016)3589 3806 y Fr(\))h(w)n(e)e(ha)n(v)n(e) 51 3905 y(\(\()p Fq(e;)14 b Fr(?)230 3875 y Fe(\016)268 3905 y Fr(\))p Fq(;)g Fr(\007\))p Fq(f)475 3917 y Fe(\025)p Fm(r)592 3905 y Fp(\022)29 b Fr(\()p Fq(e;)14 b Fr(?)833 3875 y Fe(\016)871 3905 y Fr(\))p Fq(;)g Fr(\002)31 b(and)f(\()14 b(\(\()p Fq(e;)g Fr(?)1425 3875 y Fe(\016)1464 3905 y Fr(\))p Fq(;)g Fr(\002\))p Fq(t)29 b Fp(\025)f Fq(r)k Fp(\))c Fr(\(\()p Fq(e;)14 b Fr(?)2141 3875 y Fe(\016)2180 3905 y Fr(\))p Fq(;)g Fr(\007\))p Fq(t)29 b Fp(\025)f Fq(r)17 b Fr(\).)47 b(By)31 b(6.2a)f(\(applied)h(to)g(the)g(\()p Fq(r)r Fr(+)q(1\)-path)51 4005 y(for)24 b(\002\))h(w)n(e)g(ha)n(v)n(e)f (\002)p Fq(t)e(>)h(r)r Fr(.)37 b(Hence)25 b(\(\006)p Fq(f)9 b Fr(\))1328 4017 y Fe(\025)p Fm(r)1440 4005 y Fp(\022)23 b Fr(\002)1593 3975 y Fe(0)1641 4005 y Fr(and)i(\006)p Fq(t)e Fp(\025)f Fq(r)r Fr(.)37 b(No)n(w)25 b(b)n(y)g(Lemma)f(6.3)h (there)f(is)h(an)g Fq(r)3266 3975 y Fn(+)3322 4005 y Fr(-deriv)-5 b(ation)24 b Fq(d)3780 4017 y Fm(v)3833 4005 y Fp(\003)13 b Fr(\006)51 4104 y(of)27 b(\002)210 4074 y Fe(0)251 4104 y Fp(\003)18 b Fr(\006,)28 b(i.e.)37 b(of)28 b(\002)18 b Fp(\003)g Fr(\006.)51 4204 y(\(5\))70 b(The)35 b(structure)g(of)g Fq(d)914 4174 y Fe(0)973 4204 y Fr(is)g(the)g(follo)n(wing:)51 b(to)35 b(some)g(tops)g(of)g(the) g(deduction)h Fq(d)2811 4174 y Fe(0)2811 4228 y Fn(?)2842 4211 y Fc(\016)2916 4204 y Fr(of)f(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\002)35 b(\(whic)n(h)h(is)f(an)g Fq(r)3865 4174 y Fn(+)3921 4204 y Fr(-)51 4304 y(deduction\))d(one-branc)n(h)e (deductions)i(from)f(Lemmma)h(6.4)f(are)f(added,)j(and)f(the)g(tops)f (of)h(the)g(latter)g(deductions)f(are)51 4403 y Fq(r)90 4373 y Fn(+)145 4403 y Fr(-deriv)n(ed)c(b)n(y)g(Lemma)h(6.3.)36 b(Hence)27 b(the)h(en)n(tire)g(deriv)-5 b(ation)27 b(is)g(an)g Fq(r)2292 4373 y Fn(+)2348 4403 y Fr(-deriv)-5 b(ation.)202 b Fd(2)51 4551 y Fr(No)n(w)27 b(cut)h(reduction)f(is)g(iterated)h(in)g (the)f(standard)g(w)n(a)n(y)-7 b(.)51 4712 y Fh(Lemma)29 b(6.6)51 4812 y Fo(If)38 b Fq(d)g Fo(is)h(an)f Fq(r)r Fr(+)q(1)p Fo(-derivation)h(of)g Fr(\002)p Fo(,)h(and)f Fr(\002)e Fo(has)i(an)f Fq(r)r Fr(+)q(1)p Fo(-p)l(ath)g(then)g(ther)l (e)g(exists)g(an)g Fq(r)2997 4782 y Fn(+)3053 4812 y Fo(-derivation)h Fq(d)3527 4782 y Fe(0)3589 4812 y Fo(of)g Fr(\002)f Fo(with)51 4911 y Fr(h\()p Fq(d)172 4881 y Fe(0)196 4911 y Fr(\))23 b Fp(\024)f Fq(!)393 4881 y Fn(h\()p Fm(d)p Fn(\)+1)604 4911 y Fo(.)51 5073 y(Pr)l(o)l(of)30 b(by)h(induction)f(on)g Fr(h\()p Fq(d)p Fr(\))p Fo(:)51 5173 y Fr(If)e Fq(h)p Fr(\()p Fq(d)p Fr(\))23 b(=)g(0,)k(i.e.)37 b Fq(d)28 b Fr(consists)f(of)g(an)h(axiom,)f(the)h(assertion)e(is)h(ob) n(vious.)36 b(Assume)27 b Fq(h)p Fr(\()p Fq(d)p Fr(\))d Fq(>)f Fr(0.)51 5272 y(Let)32 b Fp(I)38 b Fr(b)r(e)33 b(the)f(last)g(inference)g(of)g Fq(d)p Fr(.)51 b(Let)32 b(\()p Fq(d)1511 5284 y Fm(u)1555 5272 y Fr(\))1587 5284 y Fm(u)p Fe(2)p Fm(I)1741 5272 y Fr(b)r(e)h(the)f(family)g(of)g (immediate)h(sub)r(deriv)-5 b(ations)31 b(of)h Fq(d)p Fr(,)i(and)e(\002)3758 5284 y Fm(u)3833 5272 y Fr(the)51 5372 y(endsequen)n(t)c(of)g Fq(d)619 5384 y Fm(u)663 5372 y Fr(.)39 b(Then)28 b(b)n(y)g(I.H.)h(for)f(eac)n(h)f Fq(u)d Fp(2)h Fq(I)35 b Fr(w)n(e)28 b(ha)n(v)n(e)f(an)h Fq(r)2233 5342 y Fn(+)2289 5372 y Fr(-deriv)-5 b(ation)27 b Fq(d)2750 5342 y Fe(0)2750 5392 y Fm(u)2822 5372 y Fr(of)i(\002)2983 5384 y Fm(u)3054 5372 y Fr(with)g(h\()p Fq(d)3365 5342 y Fe(0)3365 5392 y Fm(u)3409 5372 y Fr(\))24 b Fp(\024)g Fq(!)3609 5342 y Fn(h\()p Fm(d)3707 5350 y Fa(u)3745 5342 y Fn(\)+1)3884 5372 y Fp(\024)51 5471 y Fq(!)106 5441 y Fn(h\()p Fm(d)p Fn(\))233 5471 y Fr(.)36 b(Let)28 b Fq(d)484 5441 y Fn(+)567 5471 y Fr(b)r(e)g(the)g(deriv)-5 b(ation)27 b(of)h(\002)f(whic)n(h)h(is)f(obtained)g(from)h(the)g (family)f(\()p Fq(d)2730 5441 y Fe(0)2730 5492 y Fm(u)2774 5471 y Fr(\))2806 5483 y Fm(u)p Fe(2)p Fm(I)2956 5471 y Fr(b)n(y)g Fp(I)6 b Fr(.)51 5571 y(Ob)n(viously)26 b(h\()p Fq(d)562 5541 y Fn(+)618 5571 y Fr(\))d Fp(\024)g Fq(!)816 5541 y Fn(h\()p Fm(d)p Fn(\))961 5571 y Fr(+)18 b(1.)51 5671 y(1.)36 b Fp(I)e Fr(is)28 b(not)f(a)g(cut)h(of)g(rank)f Fq(r)r Fr(:)37 b(Then)28 b Fq(d)1320 5641 y Fn(+)1403 5671 y Fr(is)g(an)f Fq(r)1641 5641 y Fn(+)1697 5671 y Fr(-deriv)-5 b(ation)26 b(of)i(\002.)51 5770 y(2.)42 b Fp(I)36 b Fr(is)30 b(a)f(cut)h(of)f(rank)g Fq(r)r Fr(:)42 b(In)30 b(that)g(case)e(w)n(e)i(apply)f(Lemma)g(6.5)g(to)h Fq(d)2336 5740 y Fn(+)2391 5770 y Fr(,)g(and)g(obtain)f(an)g Fq(r)3024 5740 y Fn(+)3080 5770 y Fr(-deriv)-5 b(ation)29 b Fq(d)3543 5740 y Fe(0)3596 5770 y Fr(of)h(\002)f(with)51 5870 y(h\()p Fq(d)172 5840 y Fe(0)196 5870 y Fr(\))23 b Fp(\024)f Fr(h\()p Fq(d)459 5840 y Fn(+)515 5870 y Fr(\))d(+)f Fq(!)j Fr(+)d(h\()p Fq(d)926 5840 y Fn(+)982 5870 y Fr(\))23 b Fq(<)g(!)1180 5840 y Fn(h\()p Fm(d)p Fn(\)+1)1391 5870 y Fr(.)203 b Fd(2)1958 6119 y Fr(17)p eop %%Page: 18 18 18 17 bop 51 614 a Fh(Lemma)29 b(6.7)51 714 y Fo(If)34 b Fq(d)g Fo(is)g(an)g Fq(!)s Fo(-derivation)h(of)g Fr(\002)p Fo(,)g(and)f Fr(\002)f Fo(has)i(an)f Fq(!)s Fo(-p)l(ath)g Fq(f)42 b Fo(in)34 b(which)i(al)t(l)f(cuts)e(ar)l(e)h(of)h(r)l(ank)e Fp(\024)e Fq(r)36 b Fo(\(with)f Fq(r)e(<)d(!)s Fo(\))j(then)51 814 y(ther)l(e)c(exists)h(an)g Fq(r)643 783 y Fn(+)698 814 y Fo(-derivation)i Fq(d)1165 783 y Fe(0)1218 814 y Fo(of)f Fr(\002)e Fo(with)h Fr(h\()p Fq(d)1711 783 y Fe(0)1735 814 y Fr(\))23 b Fp(\024)g Fq(\017)1912 829 y Fn(h\()p Fm(d)p Fn(\)+1)2123 814 y Fo(.)51 996 y(Pr)l(o)l(of)30 b(by)h(induction)f(on)g Fr(h\()p Fq(d)p Fr(\))p Fo(:)51 1096 y Fr(1.)36 b(Supp)r(ose)28 b(that)f Fq(d)h Fr(ends)f(in)h(a)f(cut) h(of)f(rank)g Fq(r)21 b Fr(+)d Fq(n)p Fr(.)36 b(Let)28 b Fq(d)1954 1108 y Fm(u)2025 1096 y Fr(b)r(e)g(the)g Fq(!)s Fr(-sub)r(deriv)-5 b(ations)26 b(of)h(its)h(premises.)36 b(By)27 b(I.H.)h(eac)n(h)51 1196 y Fq(d)94 1208 y Fm(u)163 1196 y Fr(can)e(b)r(e)h(transformed)e(in)n(to)h(an)g(\()p Fq(r)19 b Fr(+)c Fq(n)p Fr(\))1420 1165 y Fn(+)1475 1196 y Fr(-deriv)-5 b(ation)25 b Fq(d)1934 1165 y Fe(0)1934 1216 y Fm(u)1978 1196 y Fr(.)36 b(By)26 b(replacemen)n(t)g(of)g Fq(d)2765 1208 y Fm(u)2835 1196 y Fr(b)n(y)g Fq(d)2992 1165 y Fe(0)2992 1216 y Fm(u)3061 1196 y Fr(for)g(all)g Fq(u)g Fr(the)g(deriv)-5 b(ation)26 b Fq(d)51 1295 y Fr(is)h(turned)h(in)n(to)f(a)g(deriv)-5 b(ation)27 b Fq(c)h Fr(with)g(h\()p Fq(c)p Fr(\))c(=)e(sup)1664 1315 y Fm(u)1708 1295 y Fr(\(h\()p Fq(d)1861 1265 y Fe(0)1861 1316 y Fm(u)1905 1295 y Fr(\))d(+)f(1\))23 b Fp(\024)f Fr(sup)2348 1315 y Fm(u)2392 1295 y Fr(\()p Fq(\017)2458 1310 y Fn(h\()p Fm(d)2556 1318 y Fa(u)2594 1310 y Fn(\)+1)2727 1295 y Fr(+)c(1\))23 b Fp(\024)f Fq(\017)3028 1310 y Fn(h\()p Fm(d)p Fn(\))3174 1295 y Fr(+)c(1.)51 1395 y(T)-7 b(o)18 b(the)h(deriv)-5 b(ation)18 b Fq(c)h Fr(w)n(e)f(apply)h(Lemma)f (6.5)g(and)h(obtain)f(an)h(\()p Fq(r)s Fr(+)q Fq(n)p Fr(\))2208 1365 y Fn(+)2263 1395 y Fr(-deriv)-5 b(ation)18 b Fq(c)2708 1365 y Fe(0)2750 1395 y Fr(of)h(\002)f(with)i(h\()p Fq(c)3214 1365 y Fe(0)3237 1395 y Fr(\))k Fp(\024)e Fr(h\()p Fq(c)p Fr(\))q(+)q Fq(!)t Fr(+)q(h\()p Fq(c)p Fr(\))h Fq(<)51 1494 y(\017)85 1509 y Fn(h\()p Fm(d)p Fn(\)+1)296 1494 y Fr(.)51 1594 y(No)n(w)k Fq(n)g Fr(applications)g(of)g(Lemmata)h (6.1,6.6)e(yield)h(the)h(claim.)51 1694 y(Note)f(that)h(for)f(eac)n(h)g Fq(i)c Fp(2)g Fr(I)-14 b(N)28 b Fq(f)36 b Fr(is)28 b(an)f(\()p Fq(r)r Fr(+)q Fq(i)p Fr(+)o(1\)-path,)g(and)h(\()p Fq(\013)23 b(<)g(\017)2197 1709 y Fn(h\()p Fm(d)p Fn(\)+1)2454 1694 y Fp(\))46 b Fq(!)2638 1664 y Fm(\013)p Fn(+1)2792 1694 y Fq(<)23 b(\017)2914 1709 y Fn(h\()p Fm(d)p Fn(\)+1)3125 1694 y Fr(\).)51 1793 y(2.)34 b(Supp)r(ose)22 b(that)h Fq(d)f Fr(ends)g(in)h(some)e(other)h(inference)g Fp(I)6 b Fr(.)35 b(Again)22 b(b)n(y)g(I.H.)g(eac)n(h)g(immediate)g(sub)r (deriv)-5 b(ation)22 b Fq(d)3496 1805 y Fm(u)3562 1793 y Fr(transforms)51 1893 y(in)n(to)f(an)g Fq(r)361 1863 y Fn(+)417 1893 y Fr(-deriv)-5 b(ation)20 b Fq(d)871 1863 y Fe(0)871 1913 y Fm(u)915 1893 y Fr(.)35 b(By)21 b(replacemen)n(t)g(of)g Fq(d)1686 1905 y Fm(u)1751 1893 y Fr(b)n(y)g Fq(d)1903 1863 y Fe(0)1903 1913 y Fm(u)1969 1893 y Fr(for)f(all)i Fq(u)f Fr(the)h(deriv)-5 b(ation)20 b Fq(d)i Fr(is)f(turned)h(in)n(to)f(an)g Fq(r)3502 1863 y Fn(+)3558 1893 y Fr(-deriv)-5 b(ation)51 1993 y Fq(d)94 1962 y Fe(0)145 1993 y Fr(with)28 b(h\()p Fq(d)455 1962 y Fe(0)479 1993 y Fr(\))23 b(=)g(sup)747 2013 y Fm(u)790 1993 y Fr(\(h\()p Fq(d)943 1962 y Fe(0)943 2013 y Fm(u)987 1993 y Fr(\))c(+)f(1\))23 b Fp(\024)g Fr(sup)1431 2013 y Fm(u)1474 1993 y Fr(\()p Fq(\017)1540 2008 y Fn(h\()p Fm(d)1638 2016 y Fa(u)1677 2008 y Fn(\)+1)1809 1993 y Fr(+)18 b(1\))23 b Fp(\024)g Fq(\017)2111 2008 y Fn(h\()p Fm(d)p Fn(\)+1)2322 1993 y Fr(.)203 b Fd(2)51 2175 y Fh(Lemma)29 b(6.8)41 b Fr(\(Cut-elimination\))51 2275 y Fo(L)l(et)29 b Fq(d)h Fo(b)l(e)g(an)f Fq(r)r Fo(-derivation)k(of)d Fp(;)f Fo(with)i Fq(r)25 b(<)e(!)e Fr(+)d Fq(!)33 b Fo(and)d Fr(h\()p Fq(d)p Fr(\))24 b Fq(<)e(\017)2150 2287 y Fn(0)2187 2275 y Fo(.)51 2374 y(Then)30 b(ther)l(e)g(exists)f(a)h Fr(0)p Fo(-derivation)h Fq(d)1281 2344 y Fe(0)1334 2374 y Fo(of)g Fp(;)e Fo(with)h Fr(h\()p Fq(d)1804 2344 y Fe(0)1828 2374 y Fr(\))24 b Fq(<)e(\017)2005 2386 y Fm(\017)2033 2394 y Fl(0)2069 2374 y Fo(.)51 2557 y(Pr)l(o)l(of:)67 b Fr(w.l.o.g.)37 b Fq(r)26 b Fr(=)e Fq(!)d Fr(+)d Fq(n)p Fr(.)38 b(Then)28 b Fq(n)g Fr(applications)f(of)h(6.6,6.1)e(yield)i(an) g Fq(!)s Fr(-deriv)-5 b(ation)27 b Fq(d)2998 2527 y Fe(\016)3064 2557 y Fr(of)h Fp(;)f Fr(with)i(h\()p Fq(d)3539 2527 y Fe(\016)3578 2557 y Fr(\))24 b Fq(<)f(\017)3756 2569 y Fn(0)3793 2557 y Fr(.)38 b(T)-7 b(o)51 2657 y Fq(d)94 2627 y Fe(\016)160 2657 y Fr(w)n(e)27 b(apply)g(6.7,)g(6.1)g(and)g (obtain)g(a)h(0-deriv)-5 b(ation)26 b Fq(d)1792 2627 y Fe(0)1843 2657 y Fr(of)i Fp(;)f Fr(with)h(h\()p Fq(d)2317 2627 y Fe(0)2341 2657 y Fr(\))23 b Fp(\024)g Fq(\017)2518 2672 y Fn(h\()p Fm(d)2616 2655 y Fc(\016)2649 2672 y Fn(\)+1)2787 2657 y Fq(<)f(\017)2908 2669 y Fm(\017)2936 2677 y Fl(0)2972 2657 y Fr(.)203 b Fd(2)217 2806 y Fr(The)19 b(next)g(Lemma)g(sa)n(ys)f(in)i(fact)f(that)h(a)f(cut-free)g(deriv)-5 b(ation)18 b(of)i(the)f(empt)n(y)h(sequen)n(t)e(is)i(a)f(proto)r(col)f (of)h(a)g(terminating)51 2906 y(H-pro)r(cess.)51 3088 y Fh(Lemma)29 b(6.9)51 3188 y Fo(A)g(0-derivation)j Fq(d)e Fo(of)g Fp(;)f Fo(c)l(onsists)h(of)h(exactly)f(one)g(br)l(anch)g(and)g (the)g(fol)t(lowing)j(holds:)51 3288 y(\(a\))d(al)t(l)g(se)l(quents)f (in)h Fq(d)g Fo(ar)l(e)g(c)l(orr)l(e)l(ct;)51 3387 y(\(b\))f(the)h(top) g(se)l(quent)f(of)i Fq(d)f Fo(is)g(an)g(axiom)g Fg(AxS)p Fo(,)h(and)f(al)t(l)h(other)f(infer)l(enc)l(es)h(in)e Fq(d)h Fo(ar)l(e)g(of)h(the)f(kind)g Fg(F)n(r)g Fo(or)h Fg(H)o Fo(.)51 3570 y(Pr)l(o)l(of.)51 3670 y Fr(\(a\))26 b(Pro)r(of)g(b)n(y)g(b)r(ottom-up)h(induction:)37 b Fg(H)26 b Fr(:)37 b(cf.)g(Lemma)26 b(5.3.)63 b Fg(F)n(r)27 b Fr(:)36 b(If)27 b(\002)g(is)f(correct)f(and)i Fq(e)c Fp(62)g Fr(dom\(\002\))k(then)g(\()p Fq(e;)14 b Fr(?\))p Fq(;)g Fr(\002)26 b(is)51 3769 y(correct.)51 3869 y(\(b\))32 b(Since)g Fq(d)g Fr(is)g(a)g(0-deriv)-5 b(ation,)31 b(there)h(are)f(no) g Fg(Cut)p Fr(-)h(or)f Fg(CutF)n(r)p Fr(-inferences)g(in)i Fq(d)p Fr(.)50 b(Hence)32 b Fq(d)g Fr(is)g(linear.)49 b(By)31 b(b)r(ottom-up)51 3968 y(induction)f(w)n(e)f(obtain)g(\002)p Fq(f)35 b Fr(=)26 b Fp(;)j Fr(for)g(eac)n(h)g(sequen)n(t)h(\002)f(in)h Fq(d)p Fr(.)43 b(Since)30 b Fq(d)g Fr(is)f(w)n(ellfounded,)i(there)e (exists)g(a)h(top)f(sequen)n(t)h(\007.)51 4068 y(This)d(has)g(to)h(b)r (e)g(an)f(axiom.)36 b(By)27 b(\(a\))h(\007)f(is)h(not)f(ci.)37 b(Since)28 b(\002)p Fq(f)j Fr(=)23 b Fp(;)p Fr(,)55 b(\007)27 b(cannot)g(b)r(e)h(an)g(axiom)e Fg(AxH)p Fr(.)195 b Fd(2)51 4300 y Fs(6.3)112 b(Construction)36 b(of)i(the)f(original)f(deriv)-6 b(ation)217 4454 y Fr(Here)24 b(w)n(e)f(construct)h(the)h(tree)f(of)g (\014nite)h Fq(\017)p Fr(-substitutions)f(men)n(tioned)g(at)g(the)h(b)r (eginning)f(of)g(the)h(section)f(6.)35 b(General)51 4553 y(idea)25 b(here)g(is)h(the)g(same)f(as)g(in)h([12)o(].)36 b(A)n(t)26 b(eac)n(h)f(stage)g(lea)n(v)n(es)f(of)h(the)h(tree)g(are)e (extended)i(\(b)n(y)g(the)g(b)r(ottom-up)g(application)51 4653 y(of)32 b Fg(Cut)p Fr(\))g(to)g(mak)n(e)f(them)i(\\more)d (computed")i(till)h(the)f(axioms)f(are)g(reac)n(hed.)49 b(Subterms)32 b(of)g(the)h(non-computed)e(\(but)51 4753 y(needed\))c Fq(\017)p Fr(-terms)g(of)g(maxim)n(um)h(rank)e(are)g (computed)i(till)g(these)g(maxim)n(um)f Fq(\017)p Fr(-terms)f(can)h(b)r (e)h(reduced)f(to)h(a)f(canonical)51 4852 y(form)g(and)h(then)h (computed.)38 b(Note)28 b(that)h(the)f(follo)n(wing)f(de\014nitions)h (are)f(stated)h(for)g(a)g(giv)n(en)f(substitution)h Fq(S)5 b Fr(,)28 b(and)g(not)51 4952 y(for)f(its)g(completion)p 718 4885 56 4 v 28 w Fq(S)t Fr(.)51 5135 y Fh(De\014nition)j(31)41 b Fo(L)l(et)30 b Fq(S)k Fo(b)l(e)c(an)g Fq(\017)p Fo(-substitution)e (and)i Fr(\010)g Fo(a)g(\014nite)f(set)h(of)g(close)l(d)h(formulas.)51 5234 y Fq(\032)94 5246 y Fm(S)142 5234 y Fr(\(\010\))23 b(:=)g(max)o Fp(f)p Fr(rk)o(\()p Fp(j)p Fq(A)p Fp(j)812 5246 y Fm(S)861 5234 y Fr(\))g(:)37 b Fq(A)24 b Fp(2)f Fr(\010)p Fq(;)37 b(d)p Fr(\()p Fp(j)p Fq(A)p Fp(j)1443 5246 y Fm(S)1492 5234 y Fr(\))23 b Fq(>)g Fr(0)p Fp(g)17 b([)i(f)p Fr(0)p Fp(g)51 5334 y Fq(\027)92 5346 y Fm(S)140 5334 y Fr(\(\010\))k(:=)g Fq(!)453 5304 y Fm(!)519 5334 y Fp(\001)c Fq(r)i Fr(+)d(#)771 5346 y Fm(S)819 5334 y Fr(\(\010)p Fq(;)c(r)r Fr(\))31 b Fo(wher)l(e)g Fq(r)25 b Fr(:=)e Fq(\032)1501 5346 y Fm(S)1549 5334 y Fr(\(\010\))p Fo(,)51 5433 y Fr(#)120 5445 y Fm(S)168 5433 y Fr(\(\010)p Fq(;)14 b(r)r Fr(\))24 b(:=)f Fq(d)546 5445 y Fm(r)583 5433 y Fr(\()p Fp(j)p Fq(A)700 5445 y Fn(1)738 5433 y Fp(j)761 5445 y Fm(S)809 5433 y Fr(\)#)14 b Fq(:)g(:)g(:)g Fr(#)p Fq(d)1147 5445 y Fm(r)1184 5433 y Fr(\()p Fp(j)p Fq(A)1301 5445 y Fm(n)1347 5433 y Fp(j)1370 5445 y Fm(S)1418 5433 y Fr(\))p Fo(,)30 b(wher)l(e)h Fr(\010)23 b(=)f Fp(f)p Fq(A)2014 5445 y Fn(1)2052 5433 y Fq(;)14 b(:::;)g(A)2257 5445 y Fm(n)2302 5433 y Fp(g)29 b Fo(without)h(r)l(ep)l(etitions,)51 5583 y Fq(d)94 5595 y Fm(r)131 5583 y Fr(\()p Fq(F)12 b Fr(\))23 b(:=)394 5466 y Ff(\032)470 5533 y Fr(0)215 b Fo(if)31 b Fr(rk)o(\()p Fq(F)12 b Fr(\))24 b Fq(<)f(r)470 5633 y(d)p Fr(\()p Fq(F)12 b Fr(\))85 b Fo(otherwise)1174 5583 y(.)51 5821 y Fr(Note)27 b(that)h Fq(\032)474 5833 y Fm(S)522 5821 y Fr(\(\010\))c Fq(<)e(!)f Fr(+)d Fq(!)s(;)37 b Fr(#)1097 5833 y Fm(S)1145 5821 y Fr(\(\010)p Fq(;)14 b(r)r Fr(\))25 b Fq(<)d(!)1512 5790 y Fm(!)1560 5821 y Fq(;)37 b(\027)1661 5833 y Fm(S)1709 5821 y Fr(\(\010\))24 b Fq(<)e(!)1999 5790 y Fm(!)r Fn(+1)2150 5821 y Fp(\001)c Fr(3.)1958 6119 y(18)p eop %%Page: 19 19 19 18 bop 51 614 a Fh(Lemma)29 b(6.10)40 b Fr(\(One)28 b(step)g(of)f(extension\))51 714 y Fo(L)l(et)i Fq(S)34 b Fo(b)l(e)c(an)g Fq(\017)p Fo(-substitution)e(and)i Fr(\010)g Fo(a)g(\014nite)f(set)h(of)g(close)l(d)h(formulas.)51 814 y(L)l(et)e Fq(e)23 b Fr(=)f Fq(\017\030)t(F)49 b Fp(62)23 b Fr(dom\()p Fq(S)5 b Fr(\))30 b Fo(b)l(e)g(a)g(c)l(anonic)l (al)h Fq(\017)p Fo(-subterm)d(of)j(a)f(formula)h Fp(j)p Fq(A)2385 826 y Fn(0)2422 814 y Fp(j)2445 826 y Fm(S)2523 814 y Fo(with)g Fq(A)2766 826 y Fn(0)2826 814 y Fp(2)24 b Fr(\010)p Fq(;)37 b Fr(rk)o(\()p Fp(j)p Fq(A)3218 826 y Fn(0)3256 814 y Fp(j)3279 826 y Fm(S)3327 814 y Fr(\))23 b(=)g Fq(\032)3513 826 y Fm(S)3561 814 y Fr(\(\010\))p Fo(.)51 913 y(L)l(et)29 b Fq(u)22 b Fp(2)i Fr(I)-14 b(B)418 928 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))548 913 y Fp([)19 b(f)p Fr(?)p Fp(g)p Fo(,)30 b Fq(S)856 883 y Fe(0)902 913 y Fr(:=)22 b Fq(S)h Fp([)c(f)p Fr(\()p Fq(e;)14 b(u)p Fr(\))p Fp(g)p Fo(,)29 b(and)h Fr(\010)1707 883 y Fe(0)1754 913 y Fr(:=)45 b Fg(if)28 b Fq(u)23 b Fr(=)14 b(?)27 b Fg(then)h Fr(\010)f Fg(else)g Fr(\010)19 b Fp([)f(f)p Fq(F)12 b Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(g)p Fo(.)51 1013 y(Then)30 b Fq(\032)310 1025 y Fm(S)354 1009 y Fc(0)380 1013 y Fr(\(\010)472 983 y Fe(0)496 1013 y Fr(\))23 b Fp(\024)g Fq(\032)682 1025 y Fm(S)730 1013 y Fr(\(\010\))30 b Fo(and)h Fq(\027)1087 1025 y Fm(S)1131 1009 y Fc(0)1157 1013 y Fr(\(\010)1249 983 y Fe(0)1273 1013 y Fr(\))23 b Fq(<)g(\027)1457 1025 y Fm(S)1505 1013 y Fr(\(\010\))p Fo(.)51 1189 y(Pr)l(o)l(of:)51 1288 y Fr(\(a\))k(Since)h Fq(S)g Fp(\022)23 b Fq(S)624 1258 y Fe(0)647 1288 y Fr(,)k(w)n(e)g(ha)n (v)n(e)g Fp(jj)p Fq(w)r Fp(j)1141 1300 y Fm(S)1190 1288 y Fp(j)1213 1300 y Fm(S)1257 1284 y Fc(0)1307 1288 y Fr(=)22 b Fp(j)p Fq(w)r Fp(j)1501 1300 y Fm(S)1545 1284 y Fc(0)1600 1288 y Fr(and)28 b(th)n(us)f(rk\()p Fp(j)p Fq(w)r Fp(j)2160 1300 y Fm(S)2204 1284 y Fc(0)2231 1288 y Fr(\))d Fp(\024)e Fr(rk\()p Fp(j)p Fq(w)r Fp(j)2590 1300 y Fm(S)2639 1288 y Fr(\),)28 b Fq(d)p Fr(\()p Fp(j)p Fq(w)r Fp(j)2904 1300 y Fm(S)2948 1284 y Fc(0)2976 1288 y Fr(\))23 b Fp(\024)g Fq(d)p Fr(\()p Fp(j)p Fq(w)r Fp(j)3301 1300 y Fm(S)3350 1288 y Fr(\))28 b(for)f(eac)n(h)g Fq(w)r Fr(.)51 1388 y(Let)g Fq(r)f Fr(:=)d Fq(\032)416 1400 y Fm(S)464 1388 y Fr(\(\010\))28 b(and)g Fq(r)817 1358 y Fe(0)864 1388 y Fr(:=)22 b Fq(\032)1017 1400 y Fm(S)1061 1384 y Fc(0)1088 1388 y Fr(\(\010)1180 1358 y Fe(0)1203 1388 y Fr(\).)51 1488 y(One)36 b(easily)g(sees)g(that)h Fq(r)877 1457 y Fe(0)940 1488 y Fp(\024)h Fq(r)r Fr(.)65 b(Indeed,)40 b(rk)o(\()p Fp(j)p Fq(F)12 b Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(j)1810 1500 y Fm(S)1854 1483 y Fc(0)1882 1488 y Fr(\))38 b Fp(\024)g Fr(rk\()p Fq(F)12 b Fr([)-12 b([)p Fq(u)p Fr(])g(]\))39 b Fq(<)f Fr(rk\()p Fq(e)p Fr(\))g Fp(\024)g Fq(r)r Fr(,)i(and)d(for)f(remaining)g(terms)h Fq(w)51 1587 y Fr(o)r(ccurring)26 b(in)i(\010)f(cf)h(\(a\))g(.)51 1687 y(Let)f Fq(r)238 1657 y Fe(0)285 1687 y Fr(=)c Fq(r)31 b Fr(\(for)c Fq(r)639 1657 y Fe(0)686 1687 y Fq(<)c(r)30 b Fr(the)e(claim)f(is)h(trivial:)36 b Fq(\027)1609 1699 y Fm(S)1653 1683 y Fc(0)1680 1687 y Fr(\(\010)1772 1657 y Fe(0)1795 1687 y Fr(\))24 b Fq(<)e(!)1993 1657 y Fm(!)2059 1687 y Fp(\001)d Fr(\()p Fq(r)2172 1657 y Fe(0)2215 1687 y Fr(+)f(1\))23 b Fp(\024)f Fq(!)2537 1657 y Fm(!)2603 1687 y Fp(\001)d Fq(r)r Fr(\).)51 1786 y(F)-7 b(or)27 b(eac)n(h)f Fq(A)e Fp(2)f Fr(\010)28 b(w)n(e)f(ha)n(v)n(e)f(rk\()p Fp(j)p Fq(A)p Fp(j)1168 1798 y Fm(S)1212 1782 y Fc(0)1239 1786 y Fr(\))d Fp(\024)g Fr(rk)o(\()p Fp(j)p Fq(A)p Fp(j)1598 1798 y Fm(S)1647 1786 y Fr(\))p Fq(;)37 b(d)p Fr(\()p Fp(j)p Fq(A)p Fp(j)1922 1798 y Fm(S)1966 1782 y Fc(0)1994 1786 y Fr(\))23 b Fp(\024)g Fq(d)p Fr(\()p Fp(j)p Fq(A)p Fp(j)2320 1798 y Fm(S)2369 1786 y Fr(\))28 b(and)f(th)n(us)h Fq(d)2816 1798 y Fm(r)2853 1786 y Fr(\()p Fp(j)p Fq(A)p Fp(j)2993 1798 y Fm(S)3037 1782 y Fc(0)3064 1786 y Fr(\))23 b Fp(\024)g Fq(d)3250 1798 y Fm(r)3287 1786 y Fr(\()p Fp(j)p Fq(A)p Fp(j)3427 1798 y Fm(S)3476 1786 y Fr(\).)51 1886 y(Moreo)n(v)n(er)h Fp(j)p Fq(A)500 1898 y Fn(0)538 1886 y Fp(j)561 1898 y Fm(S)637 1886 y Fr(is)j Fq(S)776 1856 y Fe(0)799 1886 y Fr(-reducible)g(and)h(rk)o(\()p Fp(j)p Fq(A)1537 1898 y Fn(0)1575 1886 y Fp(j)1598 1898 y Fm(S)1646 1886 y Fr(\))23 b(=)g Fq(r)r Fr(.)38 b(Hence)28 b Fq(d)2179 1898 y Fm(r)2216 1886 y Fr(\()p Fp(j)p Fq(A)2333 1898 y Fn(0)2370 1886 y Fp(j)2393 1898 y Fm(S)2437 1882 y Fc(0)2464 1886 y Fr(\))23 b Fq(<)g(d)2650 1898 y Fm(r)2687 1886 y Fr(\()p Fp(j)p Fq(A)2804 1898 y Fn(0)2842 1886 y Fp(j)2865 1898 y Fm(S)2913 1886 y Fr(\))28 b(\(Lemma)g(3.3\).)51 1986 y(Finally)f Fq(d)376 1998 y Fm(r)413 1986 y Fr(\()p Fp(j)p Fq(F)12 b Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(j)672 1998 y Fm(S)716 1982 y Fc(0)744 1986 y Fr(\))23 b(=)g(0,)k(since)g (rk\()p Fp(j)p Fq(F)12 b Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(j)1518 1998 y Fm(S)1562 1982 y Fc(0)1589 1986 y Fr(\))24 b Fq(<)e(r)r Fr(.)204 b Fd(2)51 2162 y Fh(Lemma)29 b(6.11)40 b Fr(\(Rank)28 b(reduction\))51 2261 y Fo(L)l(et)h Fr(\002)g Fo(b)l(e)h(a)g(se)l(quent,)f Fq(L)h Fo(a)g(\014nite)f(set)g(of)i(close) l(d)g(formulas,)g(and)f Fq(r)c Fr(:=)d Fq(\032)2368 2273 y Fn(\002)2423 2261 y Fr(\()p Fp(F)8 b Fr(\(\002\))19 b Fp([)f Fq(L)p Fr(\))30 b Fo(\(cf.)39 b(De\014nition)30 b(19\).)51 2361 y(Then)24 b(ther)l(e)f(is)h(a)f(de)l(duction)h Fq(d)g Fo(of)g Fr(\002)e Fo(by)i(cuts)f(of)h(r)l(anks)f Fp(\024)g Fq(r)j Fo(fr)l(om)d(c)l(omputing)h(se)l(quents)e Fr(\007)h Fo(c)l(ontaining)h Fr(\002)e Fo(and)i(c)l(omputing)51 2460 y(al)t(l)30 b(formulas)h(in)f Fq(L)p Fo(.)38 b(Mor)l(e)l(over)31 b Fr(h\()p Fq(d)p Fr(\))24 b Fp(\024)f Fq(\027)1405 2472 y Fn(\002)1460 2460 y Fr(\()p Fp(F)8 b Fr(\(\002\))19 b Fp([)f Fq(L)p Fr(\))p Fo(.)51 2636 y(Pr)l(o)l(of)30 b(by)h(induction)f(on)g Fq(\027)911 2648 y Fn(\002)966 2636 y Fr(\()p Fp(F)8 b Fr(\(\002\))19 b Fp([)f Fq(L)p Fr(\))p Fo(:)51 2736 y Fr(Let)27 b(\010)c(:=)g Fp(F)8 b Fr(\(\002\))19 b Fp([)f Fq(L)p Fr(.)37 b(If)28 b(\002)f(computes)h (all)f(form)n(ulas)f(in)i(\010)g(w)n(e)f(are)g(done.)51 2836 y(Otherwise)f(there)i(exists)f(a)g(canonical)g Fq(\017)p Fr(-subterm)f Fq(e)d Fr(=)g Fq(\017\030)t(F)39 b Fr(of)28 b(a)f(form)n(ula)g Fp(j)p Fq(A)2573 2848 y Fn(0)2610 2836 y Fp(j)2633 2848 y Fn(\002)2716 2836 y Fr(with)h Fq(A)2967 2848 y Fn(0)3028 2836 y Fp(2)23 b Fr(\010)p Fq(;)37 b Fr(rk)o(\()p Fp(j)p Fq(A)3419 2848 y Fn(0)3457 2836 y Fp(j)3480 2848 y Fn(\002)3535 2836 y Fr(\))24 b(=)e Fq(r)r Fr(.)51 2935 y(Let)27 b Fq(u)c Fp(2)g Fr(I)-14 b(B)423 2950 y Fm(\023)p Fn(\()p Fm(e)p Fn(\))554 2935 y Fp([)19 b(f)p Fr(?)709 2905 y Fe(\016)747 2935 y Fp(g)27 b Fr(and)g(\002)1042 2905 y Fe(0)1088 2935 y Fr(:=)c(\()p Fq(e;)14 b(u)p Fr(\))p Fq(;)g Fr(\002.)36 b(Then)28 b Fp(F)8 b Fr(\(\002)1930 2905 y Fe(0)1953 2935 y Fr(\))23 b(=)46 b Fg(if)28 b Fq(u)22 b Fr(=)14 b(?)2380 2905 y Fe(\016)2446 2935 y Fg(then)27 b Fp(F)8 b Fr(\(\002\))28 b Fg(else)f Fp(F)8 b Fr(\(\002\))18 b Fp([)h(f)p Fq(F)12 b Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(g)p Fr(.)51 3035 y(By)27 b(the)h(Lemma)f(6.10)45 b Fq(r)853 3005 y Fe(0)900 3035 y Fr(:=)23 b Fq(\032)1054 3047 y Fn(\002)1105 3031 y Fc(0)1131 3035 y Fr(\()p Fp(F)8 b Fr(\(\002)1328 3005 y Fe(0)1352 3035 y Fr(\))18 b Fp([)h Fq(L)p Fr(\))k Fp(\024)g Fq(r)30 b Fr(and)d Fq(\027)1945 3047 y Fn(\002)1996 3031 y Fc(0)2023 3035 y Fr(\()p Fp(F)8 b Fr(\(\002)2220 3005 y Fe(0)2243 3035 y Fr(\))19 b Fp([)g Fq(L)p Fr(\))k Fq(<)f(\027)2608 3047 y Fn(\002)2664 3035 y Fr(\()p Fp(F)8 b Fr(\(\002\))18 b Fp([)h Fq(L)p Fr(\).)51 3134 y(Hence)j(\(b)n(y)f(I.H.\))i(there)f (exists)f(a)h(deduction)g Fq(d)1540 3146 y Fm(u)1605 3134 y Fr(of)g(\002)1759 3104 y Fe(0)1804 3134 y Fr(b)n(y)f(cuts)h(of)g (ranks)f Fp(\024)h Fq(r)2517 3104 y Fe(0)2563 3134 y Fr(from)g(computing)g(sequen)n(ts)f(\007)g(con)n(taining)51 3234 y(\002)116 3204 y Fe(0)166 3234 y Fr(and)28 b(computing)f(all)g (form)n(ulas)g(in)h Fq(L)p Fr(,)f(and)g(h\()p Fq(d)1677 3246 y Fm(u)1721 3234 y Fr(\))d Fp(\024)e Fq(\027)1905 3246 y Fn(\002)1956 3230 y Fc(0)1983 3234 y Fr(\()p Fp(F)8 b Fr(\(\002)2180 3204 y Fe(0)2203 3234 y Fr(\))19 b Fp([)g Fq(L)p Fr(\).)51 3334 y(A)28 b(cut)g(with)g(main)f(term)h Fq(e)f Fr(yields)g(the)h(desired)f(deduction)h Fq(d)p Fr(.)204 b Fd(2)51 3510 y Fh(Lemma)29 b(6.12)51 3609 y Fo(Ther)l(e)h(exists)g(an)f Fq(r)d(<)d(!)e Fr(+)d Fq(!)32 b Fo(and)f(an)e Fq(r)r Fr(+)q(1)p Fo(-derivation)i Fq(d)e Fo(of)i(the)f(empty)g(se)l(quent)f(c)l(ontaining)h(only)g(axioms)h(and) f(cuts.)51 3709 y(In)f(addition)j(we)e(have)h Fr(h\()p Fq(d)p Fr(\))24 b Fq(<)e(!)1117 3679 y Fm(!)r Fn(+2)1249 3709 y Fo(.)51 3885 y(Pr)l(o)l(of:)51 3984 y Fr(First)h(apply)h(Lemma)g (6.11)e(to)i(the)h(empt)n(y)f(sequen)n(t)f(and)h(the)g(set)g Fq(L)f Fr(:=)g Fp(f)p Fq(C)6 b(r)2480 3996 y Fn(0)2517 3984 y Fq(;)14 b(:::;)g(C)6 b(r)2762 3996 y Fm(N)2826 3984 y Fp(g)p Fr(.)35 b(Let)24 b Fq(r)i Fr(:=)d Fq(\032)3288 3999 y Fe(;)3326 3984 y Fr(\()p Fq(L)p Fr(\),)i(and)e(consider)51 4084 y(an)i(arbitrary)e(top)j(sequen)n(t)f(\002)h(of)f(the)h(resulting) f Fq(r)r Fr(+)q(1-deduction)g Fq(d)2221 4054 y Fe(\016)2285 4084 y Fr(whic)n(h)g(is)h(not)f(an)h(axiom.)35 b(Then)26 b(\002)f(is)g(cc,)h(deciding)51 4184 y(and)g(nonsolving.)36 b(Since)27 b(the)g(only)g(inferences)g(in)g Fq(d)1717 4153 y Fe(\016)1782 4184 y Fr(are)f(cuts)i(of)f(rank)f Fp(\024)c Fq(r)r Fr(,)28 b(w)n(e)f(ha)n(v)n(e)f(\002)p Fq(t)c Fr(=)h Fp(;)k Fr(and)f(rk\()p Fq(e)p Fr(\))d Fp(\024)g Fq(r)30 b Fr(for)c(eac)n(h)51 4283 y Fq(e)c Fp(2)i Fr(dom\(\002\).)36 b(Let)27 b Fq(e)f Fr(b)r(e)g(the)h(H-term)f(of)g(\002.)36 b(Since)27 b(\002)f(is)g(deciding,)g Fq(e)g Fr(is)g(\002-computable)g (\(cf.)37 b(Lemmas)25 b(3.7c)h(and)g(3.8b\).)51 4383 y(T)-7 b(ogether)31 b(with)i(Lemma)g(5.1a)e(and)i(\002)p Fq(t)e Fr(=)g Fp(;)i Fr(this)g(implies)g(\()p Fq(e;)14 b Fr(?)2145 4353 y Fe(\016)2183 4383 y Fr(\))32 b Fp(2)g Fr(\002.)52 b(Since)33 b(\002)f(is)h(not)f(an)h(axiom)f Fg(AxH)p Fr(,)h(it)g(follo)n(ws)51 4482 y(that)c(Cr\(\002\))g(is)g(of)g (the)h(form)e Fq(F)12 b Fr([)p Fq(t)p Fr(])26 b Fp(!)g Fq(F)12 b Fr([)p Fq(\017xF)g Fr(])29 b(and)g Fq(e)c Fr(=)h Fq(\017xA)j Fr(with)h Fq(A)c Fr(:=)f Fp(j)p Fq(F)12 b Fp(j)2561 4494 y Fn(\002)2616 4482 y Fr(.)42 b(By)29 b(Lemma)g(5.2)f Fq(A)p Fr([)p Fq(n)p Fr(])e Fq(,)-14 b Fp(!)3522 4494 y Fn(\002)3603 4482 y Fg(TRUE)28 b Fr(for)51 4582 y Fq(n)22 b Fr(:=)h Fp(j)p Fq(t)p Fp(j)310 4594 y Fn(\002)365 4582 y Fr(.)51 4682 y(No)n(w)34 b(let)h Fq(L)431 4652 y Fe(0)489 4682 y Fr(:=)f Fp(f)p Fq(A)p Fr([)p Fq(n)p Fp(\000)o Fr(1])p Fq(;)14 b(:)g(:)g(:)f(;)h(A)p Fr([0])p Fp(g)p Fr(.)58 b(Then)35 b Fq(\032)1641 4694 y Fn(\002)1696 4682 y Fr(\()p Fp(F)8 b Fr(\(\002\))23 b Fp([)h Fq(L)2084 4652 y Fe(0)2107 4682 y Fr(\))35 b Fp(\024)f Fq(r)r Fr(.)60 b(Apply)35 b(Lemma)f(6.11)f(to)i(\002,)h Fq(L)3426 4652 y Fe(0)3484 4682 y Fr(and)e(consider)51 4781 y(an)n(y)28 b(cc)h(top)h(sequen)n(t)f(\007)g(of)g(the)h(resulting) f Fq(r)r Fr(+1-deduction.)42 b(\007)29 b(con)n(tains)f(\002)h(and)h (computes)f(all)g(form)n(ulas)f(in)i Fq(L)3699 4751 y Fe(0)3721 4781 y Fr(.)43 b(No)n(w)51 4881 y(Lemma)27 b(5.2b)g(yields)g(that)h(\007)f(is)h(an)f(axiom)g Fg(AxH)h Fr(with)g(main)g(term)f Fq(e)p Fr(.)203 b Fd(2)51 5057 y Fh(Theorem)30 b(6.13)40 b Fo(.)f(The)31 b(H-pr)l(o)l(c)l(ess)e (terminates.)51 5233 y(Pr)l(o)l(of)p Fr(.)38 b(Com)n(bine)27 b(Lemmata)g(6.12,)f(6.8,)h(6.9.)36 b(Cf.)28 b([12)o(].)51 5464 y Fs(6.4)112 b(Pro)s(ducing)37 b(a)g(substitution)f(in)h(terms)g (of)g(\014nite)g(predicates)51 5617 y Fh(6.4.1)93 b(Soundness)51 5770 y(Lemma)29 b(6.14)40 b Fo(L)l(et)f Fq(S)44 b Fo(b)l(e)39 b(a)g(c)l(orr)l(e)l(ct)g(and)h(total)f Fq(\017)p Fo(-substitution.)65 b(Then)40 b(al)t(l)g(close)l(d)g(axioms)g(of)g Fq(E)5 b(A\017)39 b Fo(exc)l(ept)f(mayb)l(e)51 5870 y(critic)l(al)31 b(formulas)f(ar)l(e)g(satis\014e)l(d)g(by)h Fq(S)5 b Fo(.)38 b(Mo)l(dus)30 b(p)l(onens)g(rule)g(pr)l(eserves)g(truth)f (under)h Fq(S)5 b Fo(.)1958 6119 y Fr(19)p eop %%Page: 20 20 20 19 bop 51 614 a Fo(Pr)l(o)l(of.)57 b Fr(All)34 b(instances)g(of)g (prop)r(ositional)e(tautologies)h(and)g(de\014ning)h(axioms)f(for)h (predicate)f(constan)n(ts)g(are)g(satis\014ed)51 714 y(b)n(y)h Fq(S)k Fr(b)n(y)c(the)h(Lemma)f(3.7.)56 b(Mo)r(dus)34 b(p)r(onens)g(preserv)n(es)e(truth)j(under)f Fq(S)5 b Fr(,)36 b(since)e(v)-5 b(alues)33 b(of)i(comp)r(osite)e(form)n(ulas)g (are)51 814 y(calculated)g(accordingly)g(to)h(standard)f(b)r(o)r(olean) h(rules.)56 b(Equalit)n(y)33 b(axioms)g(are)g(satis\014ed,)j(since)e(b) n(y)g(Lemma)g(3.7c:)49 b(if)51 913 y Fp(j)p Fq(t)p Fp(j)127 925 y Fm(S)198 913 y Fr(=)22 b Fp(j)p Fq(u)p Fp(j)379 925 y Fm(S)455 913 y Fr(for)27 b(0-terms)f Fq(t)p Fr(,)i Fq(u)f Fr(then)h Fp(j)p Fq(e)p Fr([)p Fq(t)p Fr(])p Fp(j)1388 925 y Fm(S)1460 913 y Fr(=)22 b Fp(j)p Fq(e)p Fr([)p Fq(u)p Fr(])p Fp(j)1726 925 y Fm(S)1801 913 y Fr(for)27 b(an)n(y)g(expression)f Fq(e)p Fr(.)51 1013 y(Consider)33 b(a)h(minimalit)n(y)h(axiom)e Fq(\017xF)12 b Fr([)p Fq(x)p Fr(])35 b(=)f Fg(S)p Fq(t)h Fp(!)f(:)p Fq(F)12 b Fr([)p Fq(t)p Fr(]:)86 b(Assume)34 b Fp(j)p Fq(\017xF)12 b Fr([)p Fq(x)p Fr(])36 b(=)e Fg(S)p Fq(t)p Fp(j)2876 1025 y Fm(S)2958 1013 y Fp(2)h Fg(TRUE)f Fr(and)g(let)h Fq(n)f Fr(:=)g Fp(j)p Fq(t)p Fp(j)3877 1025 y Fm(S)3925 1013 y Fr(.)51 1112 y(Then)c(\()p Fq(\017x)p Fp(j)p Fq(F)12 b Fp(j)494 1124 y Fm(S)542 1112 y Fq(;)i(n)p Fr(+1\))27 b Fp(2)g Fq(S)35 b Fr(and)30 b(b)n(y)f(correctness)f(and)i(Lemma)g(3.7)f(w)n(e)h (ha)n(v)n(e)f Fp(j)p Fq(F)12 b Fr([)-12 b([)p Fq(n)p Fr(+1])g(])p Fp(j)2926 1124 y Fm(S)3001 1112 y Fr(=)27 b Fp(j)14 b(j)p Fq(F)e Fp(j)3241 1124 y Fm(S)3289 1112 y Fr([)-12 b([)p Fq(n)p Fr(+1])g(])14 b Fp(j)3551 1124 y Fm(S)3626 1112 y Fp(2)27 b Fg(TRUE)o Fr(,)51 1212 y(in)g(particular)g Fp(j:)p Fq(F)12 b Fr([)p Fq(t)p Fr(])p Fp(j)773 1224 y Fm(S)844 1212 y Fr(=)23 b Fp(j:)p Fq(F)12 b Fr([)p Fq(n)p Fr(])p Fp(j)1194 1224 y Fm(S)1266 1212 y Fp(2)23 b Fg(TRUE)o Fr(.)249 b Fd(2)51 1428 y Fh(6.4.2)93 b Fq(\017)p Fh(-free)32 b(deriv)-5 b(ation)217 1581 y Fr(Let)31 b Fg(d*)f Fr(b)r(e)i(a)f(closed)f(deriv)-5 b(ation)30 b(in)i Fq(E)5 b(A\017)p Fr(,)32 b(and)f(let)g Fq(S)36 b Fr(b)r(e)31 b(a)g(correct,)g(total,)g(and)g(solving)f Fq(\017)p Fr(-substitution)h (for)g(the)51 1681 y(system)24 b Fq(C)6 b(r)425 1693 y Fn(0)463 1681 y Fq(;)14 b(:::;)g(C)6 b(r)708 1693 y Fm(N)797 1681 y Fr(of)25 b(critical)f(form)n(ulas)g(of)g Fg(d*)p Fr(.)36 b(\(F)-7 b(or)24 b(example)h Fq(S)j Fr(=)p 2400 1614 97 4 v 22 w Fq(S)2451 1693 y Fm(n)2496 1681 y Fr(,)e(where)e Fq(S)2833 1693 y Fm(n)2903 1681 y Fr(is)h(pro)r(duced) g(b)n(y)f(the)i(H-pro)r(cess)51 1780 y(for)32 b Fq(C)6 b(r)285 1792 y Fn(0)323 1780 y Fq(;)14 b(:::;)g(C)6 b(r)568 1792 y Fm(N)632 1780 y Fr(,)34 b(cf.)f(De\014nition)h(22.\))52 b(Since)33 b Fq(S)38 b Fr(is)33 b(correct)f(and)g(total,)i(all)f (axioms)f(of)h Fg(d*)f Fr(except)h(critical)g(form)n(ulas)51 1880 y(are)28 b(satis\014ed)h(b)n(y)f Fq(S)34 b Fr(and)29 b(mo)r(dus)h(p)r(onens)f(rule)g(preserv)n(es)e(truth)i(under)g Fq(S)5 b Fr(.)42 b(Since)29 b Fq(S)34 b Fr(is)29 b(solving,)g(critical) f(form)n(ulas)g(are)51 1980 y(satis\014ed,)f(to)r(o.)36 b(Hence)28 b(all)f(form)n(ulas)g(in)h Fg(d*)f Fr(are)f(true)i(under)f Fq(S)5 b Fr(.)217 2079 y(Closed)17 b(form)n(ulas)h(are)f(constructed)h (b)n(y)g(prop)r(ositional)f(connectiv)n(es)g(>from)h(atomic)g(form)n (ulas)g(of)g(the)h(form)f Fq(pe)3704 2091 y Fn(1)3754 2079 y Fq(:)c(:)g(:)g(e)3904 2091 y Fm(n)51 2179 y Fr(and)32 b Fg(App)p Fr(\()p Fq(\017X)7 b(F)12 b Fr(\))p Fq(e)636 2191 y Fn(1)705 2179 y Fr(where)31 b Fq(e)988 2191 y Fm(i)1048 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b(W)-7 b(e)23 b(obtain)f(an)g Fq(\017)p Fr(-free)g(deriv)-5 b(ation)21 b(from)i(form)n(ulas)51 2959 y(whic)n(h)31 b(are)g(true)h(under)g(the)g(standard)f(in)n (terpretation)g(of)h(predicate)f(constan)n(ts)g(and)h(b)r(o)r(olean)f (connectiv)n(es,)h(and)g(the)51 3058 y(rule)806 3249 y(\()14 b Fg(App)q Fp(f)p Fq(n)1086 3261 y Fn(1)1122 3249 y Fq(;)g(:)g(:)g(:)g(;)g(n)1357 3261 y Fm(k)1397 3249 y Fp(g)p Fr(\()p Fq(n)p Fr(\))28 b(is)f(true)h(\))1947 3198 y Fm(def)1923 3249 y Fp(\()-14 b(\))51 b Fr(\()p Fq(n)28 b Fr(o)r(ccurs)e(among)i Fq(n)2811 3261 y Fn(1)2848 3249 y Fq(;)14 b(:)g(:)g(:)g(;)g(n)3083 3261 y Fm(k)3123 3249 y Fr(\))g Fq(:)217 3432 y Fr(Note)22 b(that)g(thereb)n(y)g(an)g Fq(E)5 b(A\017)22 b Fr(deriv)-5 b(ation)21 b Fg(d*)h Fr(of)g(a)g(form)n(ula)f Fq(F)12 b Fr([)p Fq(\017xF)g Fr(])22 b(\(=)h(\()p Fp(9)p Fq(xF)12 b Fr(\))2739 3402 y Fe(\003)2778 3432 y Fr(\))23 b(or)e Fq(F)12 b Fr([)p Fq(\017X)7 b(F)12 b Fr(])21 b(\(=)i(\()p Fp(9)p Fq(X)7 b(F)12 b Fr(\))3607 3402 y Fe(\003)3645 3432 y Fr(\))23 b(with)f Fq(F)51 3532 y Fr(quan)n(ti\014er-)i(and)i Fq(\017)p Fr(-free)f(is)h(transformed)f(in)n(to)h(a)f(deriv)-5 b(ation)26 b(of)g Fq(F)12 b Fr([)p Fq(n)p Fr(])25 b(or)h Fq(F)12 b Fr([)p Fq(P)g Fr(])25 b(for)h(some)f(n)n(umeral)g Fq(n)h Fr(or)f(\014nite)i(predicate)51 3631 y Fq(P)12 b Fr(.)51 3906 y Fk(7)137 b(Non-constructiv)l(e)47 b(pro)t(of)e(of)g (termination)51 4088 y Fr(In)23 b(this)h(section)f Fq(S)5 b Fr(,)24 b Fq(S)735 4100 y Fm(n)781 4088 y Fq(;)14 b(:::)23 b Fr(alw)n(a)n(ys)f(denote)h Fq(\017)p Fr(-substitutions)g(with)h Fp(f)p Fq(e)f Fp(2)g Fr(dom\()p Fq(S)5 b Fr(\))23 b(:)37 b Fq(S)5 b Fr(\()p Fq(e)p Fr(\))23 b(=)14 b(?)p Fp(g)22 b Fr(=)h Fp(;)p Fr(.)35 b(F)-7 b(or)23 b(eac)n(h)g(pair)g(\()p Fq(e;)14 b(u)p Fr(\))51 4187 y(w)n(e)27 b(set)g(rk\(\()p Fq(e;)14 b(u)p Fr(\)\))23 b(:=)g(rk)o(\()p Fq(e)p Fr(\).)51 4370 y Fh(De\014nition)30 b(32)41 b Fo(L)l(et)30 b Fq(S)k Fo(b)l(e)c(an)g Fq(\017)p Fo(-substitution)e(such)i(that)p 1952 4303 56 4 v 29 w Fq(S)35 b 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Fr(\).)51 3138 y(Pro)r(of)28 b(b)n(y)h(induction)h(on)f Fq(d)p Fr(\()p Fq(e)p Fr(\).)44 b(If)30 b Fq(d)p Fr(\()p Fq(e)p Fr(\))c(=)g(0)j(then)h Fp(j)p Fq(e)p Fp(j)1793 3150 y Fm(i)1847 3138 y Fr(=)c Fq(e)g Fr(=)g Fp(j)p Fq(e)p Fp(j)2179 3150 y Fm(n)2224 3138 y Fr(.)43 b(Assume)29 b(no)n(w)g(that)h Fq(d)p Fr(\()p Fq(e)p Fr(\))d Fq(>)e Fr(0)30 b(and)f(let)h Fq(u)f Fr(b)r(e)h(some)51 3237 y(canonical)e Fq(\017)p Fr(-subterm)h(of)g Fq(e)p Fr(.)43 b(By)29 b(\(1\))h(there)f(is)g(an)h Fq(m)f Fr(suc)n(h)g(that)h Fp(j)p Fq(u)p Fp(j)2257 3249 y Fm(i)2311 3237 y Fr(=)c Fp(j)p Fq(u)p Fp(j)2496 3249 y Fm(m)2588 3237 y Fr(for)j(all)g Fq(i)d Fp(\025)g Fq(m)p Fr(.)43 b(Let)29 b Fq(e)3308 3207 y Fe(0)3361 3237 y Fr(result)g(from)g Fq(e)h Fr(b)n(y)51 3337 y(`con)n(traction')d(of)h Fq(u)p Fr(.)39 b(Then)29 b Fq(e)24 b(,)-14 b Fp(!)1112 3307 y Fn(1)p 1112 3327 72 4 v 1112 3376 a Fm(S)1153 3384 y Fa(i)1208 3337 y Fq(e)1247 3307 y Fe(0)1298 3337 y Fr(for)28 b(all)g Fq(i)c Fp(\025)g Fq(m)p Fr(,)29 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Fm(n)2650 3555 y Fr(for)22 b(eac)n(h)f(critical)h(form)n(ula)g Fq(F)3578 3567 y Fn(0)3615 3555 y Fp(!)p Fq(F)12 b Fr([)p Fq(\017\030)t(F)g Fr(])51 3655 y(from)28 b(the)h(list)f Fq(C)6 b(r)633 3667 y Fn(0)671 3655 y Fq(;)14 b(:::;)g(C)6 b(r)916 3667 y Fm(N)980 3655 y Fr(.)40 b(So)28 b(esp)r(ecially)g(for)g (the)h(H-term)f Fq(e)c Fr(=)h Fq(\017\030)t Fp(j)p Fq(F)12 b Fp(j)2433 3667 y Fm(n)2506 3655 y Fr(of)29 b Fq(S)2653 3667 y Fm(n)2726 3655 y Fr(w)n(e)g(ha)n(v)n(e)e Fp(j)p Fq(e)p Fp(j)3127 3667 y Fm(n)p Fn(+1)3280 3655 y Fr(=)e Fp(j)p Fq(\017\030)t Fp(j)p Fq(F)12 b Fp(j)3578 3667 y Fm(n)p Fn(+1)3707 3655 y Fp(j)3730 3667 y Fm(n)p Fn(+1)3884 3655 y Fr(=)51 3754 y Fp(j)p Fq(\017\030)t(F)g Fp(j)236 3766 y Fm(n)p Fn(+1)388 3754 y Fr(=)22 b Fp(j)p Fq(\017\030)t(F)12 b Fp(j)660 3766 y Fm(n)729 3754 y Fr(=)22 b Fp(j)p Fq(e)p Fp(j)901 3766 y Fm(n)946 3754 y Fr(.)37 b(\(Note)27 b(that)f Fp(j)p Fq(\017\030)t(F)12 b Fp(j)1601 3766 y Fm(S)1672 3754 y Fr(=)23 b Fp(j)14 b Fq(\017\030)t Fp(j)p Fq(F)e Fp(j)1982 3766 y Fm(S)2044 3754 y Fp(j)2067 3766 y Fm(S)2115 3754 y Fr(\).)37 b(But,)27 b(since)f Fq(S)2647 3766 y Fm(n)p Fn(+1)2799 3754 y Fr(=)d(H\()p Fq(S)3032 3766 y Fm(n)3077 3754 y Fr(\),)28 b(this)e(is)h(a)f(con)n(tradiction)51 3854 y(to)h(Lemma)g(5.1b.)203 b Fd(2)51 4001 y Fr(Eac)n(h)26 b(in\014nite)i Fq(r)r Fr(-pro)r(cess)f(can)g(b)r(e)h(extended)g(to)g (an)f(in\014nite)h Fq(r)22 b Fr(+)c(1-pro)r(cess.)51 4179 y Fh(Theorem)30 b(7.3)51 4278 y Fo(L)l(et)i Fr(\()p Fq(S)280 4290 y Fm(i)307 4278 y Fr(\))339 4290 y Fm(i)p Fe(2)p Fm(!)488 4278 y Fo(b)l(e)h(an)f Fq(r)r Fo(-pr)l(o)l(c)l(ess.)48 b(Then)33 b(\(by)g(adding)h(p)l(airs)f(of)h(r)l(ank)e Fq(r)r Fo(\))h Fq(S)2396 4290 y Fn(0)2466 4278 y Fo(c)l(an)f(b)l(e)h (extende)l(d)f(to)h(an)f Fr(\()p Fq(r)r Fr(+)q(1\))p Fo(-substitution)51 4378 y Fq(S)5 b Fr(+)29 b Fo(such)g(that)h(the)g Fr(\()p Fq(r)r Fr(+)q(1\))p Fo(-pr)l(o)l(c)l(ess)f(b)l(e)l(ginning)h (with)h Fq(S)5 b Fr(+)29 b Fo(is)h(in\014nite.)51 4556 y(Pr)l(o)l(of:)38 b Fr(Again)27 b(w)n(e)g(set)h Fp(j)p Fq(e)p Fp(j)885 4568 y Fm(n)953 4556 y Fr(:=)23 b Fp(j)p Fq(e)p Fp(j)p 1149 4534 86 4 v 27 x Fm(S)1190 4591 y Fa(n)1234 4556 y Fr(.)51 4655 y Fq(S)5 b Fr(+)22 b(:=)305 4593 y Ff(S)374 4680 y Fm(i)p Fe(2)p Fm(!)490 4655 y Fr(\()p Fq(S)573 4667 y Fm(i)601 4655 y Fr(\))633 4667 y Fe(\024)p Fm(r)722 4655 y Fr(.)37 b(By)26 b(Lemma)h(7.1)40 b Fq(S)5 b Fr(+)27 b(is)f(a)h(function.)37 b(W)-7 b(e)28 b(sho)n(w)e(that)h(the)h Fq(r)20 b Fr(+)d(1-pro)r(cess)25 b(b)r(eginning)i(with)g Fq(S)5 b Fr(+)26 b(is)51 4755 y(an)h(accelerated)f(v)n(ersion)g(of)i(the)g(giv)n(en)e(in\014nite)j Fq(r)r Fr(-pro)r(cess)e(\()p Fq(S)2053 4767 y Fm(i)2080 4755 y Fr(\))2112 4767 y Fm(i)p Fe(2)p Fm(!)2229 4755 y Fr(.)37 b(\(cf.)h(\(2\))27 b(b)r(elo)n(w\).)51 4855 y(\(1\))p 212 4788 121 4 v 55 w Fq(S)5 b Fr(+)27 b(is)g(correct.)51 4954 y(Pro)r(of:)51 b(Let)35 b(\()p Fq(\017\030)t(A;)14 b(u)p Fr(\))36 b Fp(2)g Fr(\()p Fq(S)977 4966 y Fm(i)1005 4954 y Fr(\))1037 4966 y Fe(\024)p Fm(r)1126 4954 y Fr(.)60 b(Then)35 b(for)g(eac)n(h)f Fq(v)39 b Fp(2)d Fr(I)-14 b(B)2007 4969 y Fm(\023)p Fn(\()p Fm(\030)r Fn(\))2120 4954 y Fr(,)37 b(rk\()p Fq(A)p Fr([)p Fq(v)s Fr(]\))g Fq(<)e Fr(rk)o(\()p Fq(\017\030)t(A)p Fr(\))i Fp(\024)e Fq(r)j Fr(and)d(therefore)g Fp(j)p Fq(A)p Fr([)p Fq(v)s Fr(])p Fp(j)3820 4966 y Fm(i)3884 4954 y Fr(=)51 5054 y Fp(j)p Fq(A)p Fr([)p Fq(v)s Fr(])p Fp(j)248 5066 y Fn(0)309 5054 y Fr(=)22 b Fp(j)p Fq(A)p Fr([)p Fq(v)s Fr(])p Fp(j)p 593 5032 100 4 v 27 x Fm(S)s Fn(+)693 5054 y Fr(.)37 b(Hence)28 b Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(j)1224 5066 y Fm(i)1276 5054 y Fr(=)22 b Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(j)p 1587 5032 V 27 x Fm(S)s Fn(+)1687 5054 y Fr(.)37 b(Since)28 b Fq(S)2015 5066 y Fm(i)2070 5054 y Fr(is)g(correct,)e(w)n(e)h(ha)n(v)n (e)g Fp(j)p Fq(A)p Fr([)-12 b([)p Fq(u)p Fr(])g(])p Fp(j)2992 5066 y Fm(i)3044 5054 y Fp(2)23 b Fg(TRUE)o Fr(.)51 5163 y(Let)k(\()p Fq(S)5 b Fr(+)352 5175 y Fm(i)380 5163 y Fr(\))412 5175 y Fm(i<\027)556 5163 y Fr(b)r(e)28 b(the)g(\()p Fq(r)r Fr(+)q(1\)-pro)r(cess)e(starting)g(with)j Fq(S)5 b Fr(+)o(.)120 b(Abbreviation:)36 b Fp(j)p Fq(e)p Fp(j)2722 5133 y Fn(+)2722 5183 y Fm(n)2800 5163 y Fr(:=)23 b Fp(j)p Fq(e)p Fp(j)p 2996 5141 140 4 v 28 x Fm(S)s Fn(+)3090 5202 y Fa(n)3135 5163 y Fr(.)51 5272 y(The)k(follo)n(wing)g(prop)r (osition)f(\(2a\))i(together)e(with)i(\(1\))g(yields)g(the)g(theorem.) 51 5372 y(\(2\))55 b(F)-7 b(or)27 b(ev)n(ery)f Fq(i)d Fp(2)g Fq(!)31 b Fr(holds)212 5471 y(\(a\))55 b Fq(i)23 b(<)g(\027)5 b Fr(,)212 5571 y(\(b\))51 b(there)28 b(exists)f Fq(j)h Fp(2)c Fq(!)30 b Fr(suc)n(h)d(that)h Fq(S)5 b Fr(+)1525 5583 y Fm(i)1581 5571 y Fr(is)27 b(an)g(extension)g(of)h Fq(S)2292 5583 y Fm(j)2355 5571 y Fr(b)n(y)f(some)g(pairs)g(of)g(rank)g Fq(r)r Fr(.)51 5671 y(Pro)r(of)f(b)n(y)h(induction)h(on)g Fq(i)p Fr(:)51 5770 y(I.)f Fq(i)c Fr(=)g(0:)36 b(\(a\))28 b(trivial.)64 b(\(b\))28 b(T)-7 b(ak)n(e)27 b Fq(j)h Fr(:=)22 b(0.)51 5870 y(I)r(I.)28 b Fq(i)22 b Fp(!)h Fq(i)p Fr(+1:)1958 6119 y(21)p eop %%Page: 22 22 22 21 bop 51 614 a Fr(By)27 b(I.H.)h Fq(i)23 b(<)f(\027)5 b Fr(,)28 b(and)g(there)f(is)h(a)f Fq(j)32 b Fr(suc)n(h)c(that)g Fq(S)5 b Fr(+)1664 626 y Fm(i)1714 614 y Fr(=)23 b Fq(S)1853 626 y Fm(j)1906 614 y Fp(])c Fr(\011)28 b(with)g Fp(8)p Fq(\031)d Fp(2)f Fr(\011\()14 b(rk)o(\()p Fq(\031)s Fr(\))23 b(=)g Fq(r)r Fr(\).)51 714 y(\(The)28 b(sym)n(b)r(ol)f Fp(])h Fr(indicates)f(disjoin)n(t)h(union\).)51 814 y(Let)e(\001)h(b)r (e)g(the)f(\014nite)h(set)g(of)f(all)g(pairs)g Fq(\031)g Fp(2)d Fq(S)5 b Fr(+)26 b(of)g(rank)g Fq(r)j Fr(whic)n(h)d(are)g(used)g (in)h(the)g(computation)f(of)g Fp(j)p Fq(C)6 b(r)3446 826 y Fm(I)3485 814 y Fp(j)3508 778 y Fn(+)3508 836 y(0)3563 814 y Fq(;)14 b(:::;)g Fp(j)p Fq(C)6 b(r)3831 826 y Fm(I)3870 814 y Fp(j)3893 778 y Fn(+)3893 837 y Fm(i)51 913 y Fr(\()p Fq(I)30 b Fr(=)23 b(0)p Fq(;)14 b(:::;)g(N)9 b Fr(\))27 b(as)g(w)n(ell)g(as)g Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)1132 925 y Fn(0)1169 913 y Fr(\))p Fq(;)14 b(:)g(:)g(:)g(;)g(\031)s Fr(\()p Fq(S)5 b Fr(+)1589 925 y Fm(i)1616 913 y Fr(\).)51 1013 y(Let)27 b Fq(k)f Fr(:=)d(min)p Fp(f)p Fq(l)h Fp(\025)f Fq(j)28 b Fr(:)37 b(\001)23 b Fp(\022)g Fq(S)1049 1025 y Fm(l)1088 1013 y Fr(&)14 b(rk)o(\()p Fq(S)1326 1025 y Fm(l)1351 1013 y Fr(\))24 b Fq(>)e(r)r Fp(g)28 b Fr(\(cf.)38 b(Lemma)27 b(7.2\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e Fq(k)g Fp(\025)c Fq(j)d Fr(&)14 b(\001)23 b Fp(\022)g Fq(S)3106 1025 y Fm(k)3160 1013 y Fr(&)14 b(rk)o(\()p Fq(S)3398 1025 y Fm(k)3439 1013 y Fr(\))23 b Fq(>)g(r)51 1112 y Fr(CASE)k(A:)h Fq(k)e Fr(=)d Fq(j)5 b Fr(.)51 1212 y(\(a\))24 b(Then)h Fq(S)5 b Fr(+)515 1224 y Fm(i)566 1212 y Fr(=)23 b Fq(S)705 1224 y Fm(k)758 1212 y Fp(])12 b Fr(\011,)26 b(and)e(\011)g(is)h(not)f(used)h(in)g(the)g(computation)f(of)h Fp(j)p Fq(C)6 b(r)2530 1224 y Fm(I)2569 1212 y Fp(j)2592 1177 y Fn(+)2592 1235 y Fm(i)2671 1212 y Fr(since)25 b Fq(S)2923 1224 y Fm(k)2988 1212 y Fr(already)e(con)n(tains)h(\001.)36 b(Hence)51 1312 y Fp(j)p Fq(C)6 b(r)176 1324 y Fm(I)214 1312 y Fp(j)237 1276 y Fn(+)237 1335 y Fm(i)319 1312 y Fr(=)27 b Fp(j)p Fq(C)6 b(r)536 1324 y Fm(I)574 1312 y Fp(j)597 1324 y Fm(k)668 1312 y Fr(for)29 b Fq(I)34 b Fr(=)26 b(0)p Fq(;)14 b(:::;)g(N)9 b Fr(.)43 b(Since)p 1504 1245 92 4 v 30 w Fq(S)1555 1324 y Fm(k)1626 1312 y Fr(is)29 b(nonsolving)g(and)g(rk\()p Fq(S)2450 1324 y Fm(k)2491 1312 y Fr(\))e Fq(>)f(r)r Fr(,)31 b(it)f(follo)n(ws)f(that) p 3276 1245 148 4 v 30 w Fq(S)5 b Fr(+)3396 1324 y Fm(i)3454 1312 y Fr(is)30 b(nonsolving,)51 1411 y Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)253 1423 y Fm(i)281 1411 y Fr(\))23 b(=)g Fq(\031)s Fr(\()p Fq(S)557 1423 y Fm(k)598 1411 y Fr(\),)28 b(and)g(rk)o(\()p Fq(S)5 b Fr(+)1072 1423 y Fm(i)1099 1411 y Fr(\))24 b Fp(\025)e Fq(r)g Fr(+)c(1.)36 b(Hence)28 b Fq(i)18 b Fr(+)g(1)k Fq(<)h(\027)5 b Fr(.)51 1511 y(\(b\))24 b Fq(S)5 b Fr(+)305 1523 y Fm(i)p Fn(+1)440 1511 y Fr(=)23 b(\()p Fq(S)5 b Fr(+)680 1523 y Fm(i)708 1511 y Fr(\))740 1526 y Fe(\024)p Fn(rk\()p Fm(S)s Fn(+)974 1537 y Fa(i)1000 1526 y Fn(\))1041 1511 y Fp(])11 b(f)p Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)1352 1523 y Fm(i)1380 1511 y Fr(\))p Fp(g)23 b Fr(=)f(\()p Fq(S)1647 1523 y Fm(k)1699 1511 y Fp(])11 b Fr(\011\))1862 1526 y Fe(\024)p Fn(rk\()p Fm(S)2042 1535 y Fa(k)2078 1526 y Fn(\))2119 1511 y Fp(])g(f)p Fq(\031)s Fr(\()p Fq(S)2360 1523 y Fm(k)2401 1511 y Fr(\))p Fp(g)23 b Fr(=)g(\()p Fq(S)2669 1523 y Fm(k)2710 1511 y Fr(\))2742 1526 y Fe(\024)p Fn(rk\()p Fm(S)2922 1535 y Fa(k)2958 1526 y Fn(\))2999 1511 y Fp(])11 b(f)p Fq(\031)s Fr(\()p Fq(S)3240 1523 y Fm(k)3281 1511 y Fr(\))p Fp(g)g(])g Fr(\011)23 b(=)f Fq(S)3658 1523 y Fm(k)q Fn(+1)3794 1511 y Fp(])11 b Fr(\011.)51 1611 y(CASE)33 b(B:)h Fq(j)k(<)33 b(k)s Fr(.)56 b(Then)34 b(rk)o(\()p Fq(S)1102 1623 y Fm(k)q Fe(\000)p Fn(1)1228 1611 y Fr(\))g(=)f Fq(r)r Fr(.)56 b(Otherwise)33 b(\()p Fq(S)1992 1623 y Fm(k)2033 1611 y Fr(\))2065 1623 y Fe(\024)p Fm(r)2188 1611 y Fr(=)g(\()p Fq(S)2369 1623 y Fm(k)q Fe(\000)p Fn(1)2495 1611 y Fr(\))2527 1623 y Fe(\024)p Fm(r)2650 1611 y Fr(and)g(rk\()p Fq(S)2977 1623 y Fm(k)q Fe(\000)p Fn(1)3103 1611 y Fr(\))g Fq(>)g(r)r Fr(,)k(so)c(that)h Fq(k)j Fr(is)c(not)51 1710 y(minimal.)51 1810 y(Hence)26 b(in)h(the)f(step)h(from)f Fq(S)954 1822 y Fm(k)q Fe(\000)p Fn(1)1106 1810 y Fr(to)g Fq(S)1257 1822 y Fm(k)1324 1810 y Fr(all)g(pairs)g(of)g(rank)f Fq(>)e(r)29 b Fr(are)d(remo)n(v)n(ed.)34 b(It)27 b(follo)n(ws)e(that)i Fq(S)5 b Fr(+)3236 1822 y Fn(0)3297 1810 y Fr(=)22 b Fq(S)3435 1822 y Fm(k)3492 1810 y Fp(])16 b Fr(\010)26 b(where)g(\010)51 1910 y(consists)g(of)i(pairs)f(of)g(rank)g Fq(r)j Fr(whic)n(h)e(are)e(not)i(used)g(in)f(the)h(computation)g(of)f Fp(j)p Fq(C)6 b(r)2665 1922 y Fm(I)2704 1910 y Fp(j)2727 1874 y Fn(+)2727 1932 y(0)2782 1910 y Fq(;)14 b(:::;)g Fp(j)p Fq(C)6 b(r)3050 1922 y Fm(I)3089 1910 y Fp(j)3112 1874 y Fn(+)3112 1933 y Fm(i)3195 1910 y Fr(and)27 b Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)3559 1922 y Fm(i)3587 1910 y Fr(\).)51 2009 y Fo(Pr)l(op)l(osition)p Fr(.)38 b Fq(S)5 b Fr(+)643 2021 y Fm(l)691 2009 y Fr(=)23 b Fq(S)830 2021 y Fm(k)q Fn(+)p Fm(l)961 2009 y Fp(])c Fr(\010,)28 b(for)f Fq(l)d Fr(=)f(0)p Fq(;)14 b(:::;)g Fr(min)p Fp(f)p Fq(i)j Fr(+)h(1)p Fq(;)c(\027)23 b Fp(\000)18 b Fr(1)p Fp(g)83 b Fr(\()p Fp(\003)p Fr(\).)51 2109 y(Pro)r(of)26 b(b)n(y)h(induction)h(on)g Fq(l)r Fr(:)36 b(Let)28 b Fq(l)c(<)f Fr(min)p Fp(f)p Fq(i)18 b Fr(+)g(1)p Fq(;)c(\027)23 b Fp(\000)18 b Fr(1)p Fp(g)27 b Fr(and)g Fq(S)5 b Fr(+)2178 2121 y Fm(l)2227 2109 y Fr(=)23 b Fq(S)2366 2121 y Fm(k)q Fn(+)p Fm(l)2497 2109 y Fp(])c Fr(\010.)51 2208 y(By)j(the)i (de\014nition)f(of)g(\010,)h(all)f(pairs)f(of)h(rank)f Fq(r)k Fr(whic)n(h)d(are)f(used)h(in)g(the)h(computation)e(of)h Fp(j)p Fq(C)6 b(r)3003 2220 y Fm(I)3042 2208 y Fp(j)3065 2173 y Fn(+)3065 2231 y(0)3120 2208 y Fq(;)14 b(:::;)g Fp(j)p Fq(C)6 b(r)3388 2220 y Fm(I)3427 2208 y Fp(j)3450 2173 y Fn(+)3450 2232 y Fm(i)3529 2208 y Fr(and)22 b Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)3888 2220 y Fm(i)3916 2208 y Fr(\))51 2308 y(are)19 b(con)n(tained)i(in)g(\001)i Fp(\022)g Fq(S)873 2320 y Fm(k)913 2308 y Fr(.)35 b(Since)21 b Fq(l)k Fp(\024)d Fq(i)p Fr(,)g(w)n(e)f(ha)n(v)n(e)e Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)1895 2320 y Fm(l)1921 2308 y Fr(\))23 b(=)g Fq(\031)s Fr(\()p Fq(S)2197 2320 y Fm(k)q Fn(+)p Fm(l)2310 2308 y Fr(\))e(and)g(rk)o(\()p Fq(S)5 b Fr(+)2747 2320 y Fm(l)2773 2308 y Fr(\))23 b(=)f(rk\()p Fq(S)3075 2320 y Fm(k)q Fn(+)p Fm(l)3188 2308 y Fr(\).)35 b(Since)21 b Fq(l)7 b Fr(+)e(1)22 b Fq(<)g(\027)5 b Fr(,)23 b(the)51 2408 y(substitution)k Fq(S)5 b Fr(+)633 2420 y Fm(l)684 2408 y Fr(is)27 b(not)g(terminal,)f(and)h(w)n(e)f(ha)n(v)n (e)g(rk)o(\()p Fq(S)5 b Fr(+)1970 2420 y Fm(l)1995 2408 y Fr(\))24 b Fp(\025)e Fq(r)r Fr(+)q(1.)36 b(Hence)27 b Fq(S)5 b Fr(+)2710 2420 y Fm(l)p Fn(+1)2842 2408 y Fr(=)23 b(\()p Fq(S)5 b Fr(+)3083 2420 y Fm(l)3108 2408 y Fr(\))3140 2423 y Fe(\024)p Fn(rk\()p Fm(S)s Fn(+)3374 2434 y Fa(l)3398 2423 y Fn(\))3445 2408 y Fp([)17 b(f)p Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)3761 2420 y Fm(l)3787 2408 y Fr(\))p Fp(g)23 b Fr(=)51 2507 y(\()p Fq(S)134 2519 y Fm(k)q Fn(+)p Fm(l)265 2507 y Fp(])c Fr(\010\))431 2522 y Fe(\024)p Fn(rk\()p Fm(S)611 2531 y Fa(k)q Fl(+)p Fa(l)710 2522 y Fn(\))758 2507 y Fp(])g(f)p Fq(\031)s Fr(\()p Fq(S)1007 2519 y Fm(k)q Fn(+)p Fm(l)1120 2507 y Fr(\))p Fp(g)k Fr(=)g(\()p Fq(S)1388 2519 y Fm(k)q Fn(+)p Fm(l)1501 2507 y Fr(\))1533 2522 y Fe(\024)p Fn(rk\()p Fm(S)1713 2531 y Fa(k)q Fl(+)p Fa(l)1812 2522 y Fn(\))1860 2507 y Fp(])c(f)p Fq(\031)s Fr(\()p Fq(S)2109 2519 y Fm(k)q Fn(+)p Fm(l)2222 2507 y Fr(\))p Fp(g)f(])h Fr(\010)k(=)g Fq(S)2610 2519 y Fm(k)q Fn(+)p Fm(l)p Fn(+1)2825 2507 y Fp(])c Fr(\010.)51 2618 y(\(a\))32 b(The)h(ab)r(o)n(v)n(e)e(prop)r (osition)g(\()p Fp(\003)p Fr(\))i(yields)f Fp(j)p Fq(C)6 b(r)1548 2630 y Fm(I)1587 2618 y Fp(j)1610 2582 y Fn(+)1610 2641 y Fm(i)1697 2618 y Fr(=)30 b Fp(j)p Fq(C)6 b(r)1917 2630 y Fm(I)1956 2618 y Fp(j)1979 2630 y Fm(k)q Fn(+)p Fm(i)2127 2618 y Fr(since)32 b(\011)h(is)f(not)g(used)h(in)g(the)g (computation)f(of)g Fp(j)p Fq(C)6 b(r)3831 2630 y Fm(I)3870 2618 y Fp(j)3893 2582 y Fn(+)3893 2641 y Fm(i)51 2717 y Fr(.Hence)p 334 2651 V 41 w Fq(S)f Fr(+)454 2729 y Fm(i)522 2717 y Fr(is)41 b(nonsolving,)i(since)p 1289 2651 167 4 v 41 w Fq(S)1340 2729 y Fm(k)q Fn(+)p Fm(i)1496 2717 y Fr(is)d(nonsolving.)75 b(No)n(w)41 b(b)n(y)f(\()p Fp(\003)p Fr(\))h(w)n(e)g(also)e(ha)n(v)n(e)h Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)3278 2729 y Fm(i)3306 2717 y Fr(\))45 b(=)g Fq(\031)s Fr(\()p Fq(S)3626 2729 y Fm(k)q Fn(+)p Fm(i)3741 2717 y Fr(\))c(and)51 2817 y(rk)o(\()p Fq(S)5 b Fr(+)279 2829 y Fm(i)307 2817 y Fr(\))30 b(=)g(rk)o(\()p Fq(S)623 2829 y Fm(k)q Fn(+)p Fm(i)739 2817 y Fr(\).)49 b(Assume)32 b(rk)o(\()p Fq(S)1314 2829 y Fm(k)q Fn(+)p Fm(i)1430 2817 y Fr(\))e(=)f Fq(r)r Fr(;)35 b(then)d Fq(\031)s Fr(\()p Fq(S)2009 2829 y Fm(k)q Fn(+)p Fm(i)2125 2817 y Fr(\))e Fp(2)g Fq(S)5 b Fr(+)29 b Fp(\022)h Fq(S)5 b Fr(+)2637 2829 y Fm(i)2696 2817 y Fr(and)32 b(th)n(us)g Fq(\031)s Fr(\()p Fq(S)3182 2829 y Fm(k)q Fn(+)p Fm(i)3297 2817 y Fr(\))e Fp(6)p Fr(=)g Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)3657 2829 y Fm(i)3684 2817 y Fr(\),)34 b(since)51 2917 y Fq(\031)s Fr(\()p Fq(S)5 b Fr(+)253 2929 y Fm(i)281 2917 y Fr(\))23 b Fp(62)h Fq(S)5 b Fr(+)535 2929 y Fm(i)563 2917 y Fr(.)37 b(Con)n(tradiction.)51 3016 y(Hence)27 b(rk\()p Fq(S)5 b Fr(+)526 3028 y Fm(i)554 3016 y Fr(\))23 b(=)g(rk)o(\()p Fq(S)856 3028 y Fm(k)q Fn(+)p Fm(i)972 3016 y Fr(\))g Fp(\025)g Fq(r)r Fr(+1)k(and)h(th)n(us)f Fq(i)18 b Fr(+)g(1)23 b Fq(<)g(\027)5 b Fr(.)51 3116 y(\(b\))28 b(As)g(in)g(the)f(pro)r(of)h(of)f(\()p Fp(\003)p Fr(\))h(w)n(e)f(get)g Fq(S)5 b Fr(+)1378 3128 y Fm(i)p Fn(+1)1512 3116 y Fr(=)23 b Fq(S)1651 3128 y Fm(k)q Fn(+)p Fm(i)p Fn(+1)1869 3116 y Fp(])c Fr(\010.)203 b Fd(2)51 3299 y Fh(Theorem)30 b(7.4)51 3398 y Fo(The)g(0-pr)l(o)l(c)l(ess)h Fr(\005)f Fo(b)l(e)l(ginning)f(with)i(the)f(empty)g(substitution)f Fp(;)g Fo(terminates)g(in)h(a)g(solving)h(substitution.)51 3581 y(Pr)l(o)l(of:)38 b Fr(Ob)n(viously)26 b(it)i(su\016ces)g(to)f (pro)n(v)n(e)f(that)i(\005)g(is)f(\014nite)h(\(terminates\).)51 3680 y(F)-7 b(or)27 b(con)n(tradiction)f(w)n(e)h(assume)g(that)h(\005)g (is)f(in\014nite.)51 3780 y(Belo)n(w)f(w)n(e)h(de\014ne)h (substitutions)g Fq(S)1209 3750 y Fm(r)1273 3780 y Fr(for)f(all)h Fq(r)e Fp(2)d Fq(O)r(n)28 b Fr(suc)n(h)f(that:)51 3880 y(\(1\))55 b Fq(S)268 3850 y Fn(0)328 3880 y Fr(=)23 b Fp(;)p Fr(,)51 3979 y(\(2\))55 b Fq(S)268 3949 y Fm(r)332 3979 y Fr(is)28 b(an)f Fq(r)r Fr(-substitution,)51 4079 y(\(3\))55 b(the)28 b Fq(r)r Fr(-pro)r(cess)f(starting)g(with)h Fq(S)1267 4049 y Fm(r)1331 4079 y Fr(is)f(in\014nite,)51 4179 y(\(4\))55 b Fq(S)268 4148 y Fm(q)327 4179 y Fp(\022)23 b Fq(S)471 4148 y Fm(r)508 4179 y Fr(,)k(for)g(all)h Fq(q)e(<)c(r)r Fr(.)51 4303 y(Let)34 b Fq(R)h Fr(:=)f(max)o Fp(f)p Fr(rk)o(\()p Fq(C)6 b(r)832 4315 y Fm(I)871 4303 y Fr(\))34 b(:)h Fq(I)41 b Fp(\024)33 b Fq(N)9 b Fp(g)23 b Fr(+)f(1.)56 b(Then)35 b(rk)o(\()p Fq(S)5 b Fr(\))34 b Fq(<)g(R)h Fr(for)f(eac)n(h)f(substitution)i Fq(S)5 b Fr(.)57 b(But)34 b(on)g(the)h(other)e(hand)51 4403 y(rk)o(\()p Fq(S)215 4373 y Fm(R)269 4403 y Fr(\))24 b Fp(\025)e Fq(R)q Fr(,)28 b(since)f(b)n(y)g(\(3\))h(the)g Fq(R)q Fr(-pro)r(cess)e(starting)h(with)h Fq(S)2058 4373 y Fm(R)2140 4403 y Fr(is)f(in\014nite.)38 b Fo(Contr)l(adiction.)51 4527 y Fr(De\014nition)28 b(of)f Fq(S)587 4497 y Fm(r)652 4527 y Fr(b)n(y)g(trans\014nite)g(recursion)f(on)i Fq(r)r Fr(:)51 4627 y Fq(S)107 4597 y Fn(0)167 4627 y Fr(:=)22 b Fp(;)p Fr(,)51 4727 y Fq(S)107 4696 y Fm(r)r Fn(+1)250 4727 y Fr(:=)h(\()p Fq(S)449 4696 y Fm(r)486 4727 y Fr(\)+)k(\(cf.)38 b(Theorem)26 b(7.3\),)51 4826 y Fq(S)107 4796 y Fm(r)166 4826 y Fr(:=)277 4764 y Ff(S)346 4851 y Fm(q)r(from)h(I.H.\(2\),\(4\),)i(since)e(eac)n(h) g(particular)e(v)-5 b(alue)32 b(\()p Fq(e;)14 b(u)p Fr(\))32 b(in)h Fq(S)3409 5120 y Fm(r)3477 5150 y Fr(is)f(v)n(eri\014ed)g(b)n(y) 51 5250 y(computation)27 b(in)h(some)f Fq(S)896 5219 y Fm(q)932 5250 y Fq(;)h(q)e(<)d(r)r Fr(.)51 5349 y(F)-7 b(or)29 b(\(3\))h(assume)f(that)h(the)g Fq(r)r Fr(-pro)r(cess)f(for)g Fq(S)1498 5319 y Fm(r)1564 5349 y Fr(is)h(\014nite.)43 b(Then)30 b(it)g(uses)g(information)f(only)g(from)g Fq(S)3273 5319 y Fm(q)3340 5349 y Fr(for)g(\014nitely)h(man)n(y)51 5449 y Fq(q)c(<)c(r)r Fr(,)29 b(hence)e(there)h(is)f(a)h Fq(q)e(<)c(r)31 b Fr(suc)n(h)c(that)h Fq(S)1529 5419 y Fm(q)1593 5449 y Fr(is)f(\014nite)i(whic)n(h)e(con)n(tradicts)f (I.H.\(3\).)167 b Fd(2)1958 6119 y Fr(22)p eop %%Page: 23 23 23 22 bop 51 614 a Fk(References)92 796 y Fr([1])41 b(W.Ac)n(k)n (ermann,)24 b(Begr)r(\177)-44 b(undung)23 b(des)h(T)-7 b(ertium)25 b(non)e(datur)h(mittels)h(der)f(Hilb)r(ertsc)n(hen)g (Theorie)f(der)h(Widerspruc)n(hs-)221 896 y(freiheit,)k(Math.Ann.)h (1925,)c(93,)i(1-36)92 1062 y([2])41 b(W.Ac)n(k)n(ermann,)27 b(Zur)g(Widerspruc)n(hsfreiheit)g(der)g(Zahlen)n(theorie,)f(Math.Ann.)j (1940,)c(117,)i(162-194)92 1228 y([3])41 b(N.)28 b(Bourbaki,)e(Theorie) h(des)h(ensem)n(bles,)f(Hermann,)g(1958)92 1394 y([4])41 b(G.)28 b(Gen)n(tzen,)g(Die)g(Widerspruc)n(hsfreiheit)f(der)g(reinen)g (Zahlen)n(theorie,)f(Math)i(Ann.,)g(1936,)e(112,)h(N4,)g(493-565)92 1560 y([5])41 b(J.-Y.Girard,)26 b(Une)h(extension)f(de)h(l'in)n (terpretation)f(de)h(G\177)-42 b(odel)27 b(a)f(l'analyse)g(et)h(la)f (application)g(a)h(l'elimination)f(des)221 1660 y(coupures)33 b(dans)f(l'analyse)g(et)i(la)f(theorie)g(des)g(t)n(yp)r(es,)h(Pro)r(c.) e(2-nd)h(Scand.Logic)f(Symp.,)j(North-Holland,)f(1972,)221 1759 y(63-92)92 1925 y([6])41 b(D.Hilb)r(ert,)29 b(Probleme)d(der)i (Grundlegung)f(der)g(Mathematik,)g(Math.Ann.)i(1929,)c(102,)i(1-9)92 2091 y([7])41 b(D.Hilb)r(ert,)29 b(P)-7 b(.Berna)n(ys,)25 b(Grundlagen)i(der)h(Mathematik,)f(Bd.2,)g(Springer,)g(1970)92 2257 y([8])41 b(G.Kreisel,)27 b(On)g(the)h(In)n(terpretation)f(of)g (Non-\014nitist)h(pro)r(ofs)f(I,)h(J.)f(Sym)n(b)r(olic)h(Logic)e(1951,) g(16,)h(241-267)92 2423 y([9])41 b(G.Kreisel)27 b(On)g(the)h(In)n (terpretation)f(of)g(Non-\014nitist)h(pro)r(ofs)f(I)r(I,)h(J.)f(Sym)n (b)r(olic)h(Logic)e(1952,)g(17,)h(43-58)51 2589 y([10])40 b(G.Min)n(ts,)i(Simpli\014ed)d(Consistency)f(Pro)r(of)f(for)h (Arithmetic)h(\(Russian\),)i(Pro)r(c.)d(Estonian)f(Acad.)h(of)h(Sci.)f (Fiz.-)221 2689 y(Math.1982,)26 b(31)h(N4,)g(376-382)51 2855 y([11])40 b(G.)f(Min)n(ts,)i(Epsilon)d(Substitution)h(Metho)r(d)g (for)f(the)h(Theory)e(of)h(Hereditarily)g(Finite)h(Sets)f(\(Russian\),) j(Pro)r(c.)221 2955 y(Eston.)27 b(Acad.)h(of)f(Sci.)h(Fiz.-Math.)g (1989)d(N2,)j(154-164)51 3121 y([12])40 b(G.Min)n(ts,)29 b(Gen)n(tzen-t)n(yp)r(e)f(Systems)h(and)f(Hilb)r(ert's)h(Epsilon)e (Substitution)i(Metho)r(d.)g(I.)g(In:)38 b(Logic,)28 b(Metho)r(d.)g(and)221 3220 y(Philos.)f(of)h(Sci.)f(IX,)h(Elsevier,)f (1994,)e(91-122)51 3386 y([13])40 b(G.)29 b(Min)n(ts,)g(S.)g(T)-7 b(upailo,)28 b(Epsilon)g(Substitution)i(Metho)r(d)f(for)f(Elemen)n (tary)f(Analysis,)h(Rep)r(ort)h(No.)f(CSLI-93-175,)221 3486 y(1993,)e(CSLI,)i(Stanford)f(Univ)n(ersit)n(y)-7 b(,)27 b(F)-7 b(ebruary)26 b(1993)51 3652 y([14])40 b(Kleene)27 b(S.C.,)h(In)n(tro)r(duction)g(to)f(Methamathematics,)g(V)-7 b(an)28 b(Nostrand,)f(1952)51 3818 y([15])40 b(J.v)n(on)27 b(Neumann,)h(Zur)f(Hilb)r(ertsc)n(hen)g(Bew)n(eistheorie,)g(Math.)g (Zeitsc)n(hrift)h(26,)e(1927,)g(1-46)51 3984 y([16])40 b(K.)28 b(Sc)n(h)n(utte,)f(Pro)r(of)g(Theory)-7 b(,)26 b(Springer,)h(1977)51 4150 y([17])40 b(W.T)-7 b(ait,)28 b(F)-7 b(unctionals)28 b(De\014ned)g(b)n(y)f(T)-7 b(rans\014nite)27 b(Recursion,)g(J.)g(Sym)n(b)r(olic)h(Logic)e(1965,)g(30)h(N2,)g (155-174)51 4316 y([18])40 b(W.T)-7 b(ait,)28 b(The)g(Substitution)g (Metho)r(d,)h(J.)e(Sym)n(b)r(olic)g(Logic)g(1965,)f(30,)g(N2,)i (175-192)51 4482 y([19])40 b(H.W)-7 b(eyl,)29 b(Da)n(vid)e(Hilb)r(ert)h (and)f(His)h(Mathem)n(tical)f(W)-7 b(ork,)27 b(Bull.)h(Amer.)g(Math.)g (So)r(ciet)n(y)-7 b(,)27 b(1944,)f(50,)h(612-654)1958 6119 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF