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b(b)m(y)g(an)f(argumen)m(t)g(whic)m(h)i(also)e(applies)g (to)g(t)m(yp)s(e{free)h(terms,)f(i.e.)43 b(terms)32 b(without)g(form)m (ula)0 578 y(sup)s(erscripts.)44 b(The)30 b(main)f(idea)g(of)h(the)g (pro)s(of)f(\(due)i(to)e(J.)g(B.)g(Rosser)h(and)g(W.)g(W.)f(T)-8 b(ait\))28 b(is)i(to)f(use)0 698 y(the)34 b(relation)e Fk(r)f Fl(!)c Fk(r)788 661 y Fg(0)849 698 y Fn(de\014ned)35 b(in)e(Section)h(1.2.)43 b(Its)33 b(crucial)h(prop)s(ert)m(y)f(is)g (giv)m(en)g(b)m(y)0 877 y Fd(Lemma)38 b(1.3.1.)58 b Fc(If)33 b Fk(r)e Fl(!)c Fk(r)1081 841 y Fg(0)1142 877 y Fc(and)34 b Fk(t)27 b Fl(!)h Fk(t)1563 841 y Fg(0)1624 877 y Fc(then)34 b Fk(r)1896 892 y Fj(v)1943 877 y Fn([)p Fk(t)p Fn(])27 b Fl(!)g Fk(r)2237 841 y Fg(0)2234 902 y Fj(v)2282 877 y Fn([)p Fk(t)2346 841 y Fg(0)2373 877 y Fn(])p Fc(.)0 1056 y Fn(The)44 b(pro)s(of)g(is)f(b)m(y)g(induction)h(on)g(the)g (de\014nition)g(of)f Fk(r)48 b Fl(!)c Fk(r)2389 1020 y Fg(0)2417 1056 y Fn(.)74 b(All)43 b(cases)h(are)g(ob)m(vious)f (except)0 1176 y(p)s(ossibly)33 b(Rule)h(4.)44 b(So)33 b(assume)g Fk(r)e Fl(!)c Fk(r)1472 1140 y Fg(0)1500 1176 y Fn(,)i Fk(~)-47 b(s)28 b Fl(!)1767 1151 y Fk(~)1762 1176 y(s)1809 1147 y Fg(0)1870 1176 y Fn(and)34 b Fk(s)27 b Fl(!)g Fk(s)2312 1140 y Fg(0)2340 1176 y Fn(.)44 b(Then)795 1415 y Fk(r)840 1430 y Fj(v)888 1415 y Fn([)p Fk(t)p Fn(])27 b Fl(!)g Fk(r)1182 1374 y Fg(0)1179 1440 y Fj(v)1226 1415 y Fn([)p Fk(t)1290 1374 y Fg(0)1318 1415 y Fn(])p Fk(;)112 b(~)-47 b(s)1536 1430 y Fj(v)1583 1415 y Fn([)p Fk(t)p Fn(])27 b Fl(!)1834 1391 y Fk(~)1830 1415 y(s)1877 1386 y Fg(0)1904 1430 y Fj(v)1952 1415 y Fn([)p Fk(t)2016 1374 y Fg(0)2043 1415 y Fn(])100 b(and)g Fk(s)2478 1430 y Fj(v)2525 1415 y Fn([)p Fk(t)p Fn(])27 b Fl(!)g Fk(s)2818 1374 y Fg(0)2818 1440 y Fj(v)2865 1415 y Fn([)p Fk(t)2929 1374 y Fg(0)2957 1415 y Fn(])0 1654 y(b)m(y)33 b(induction)h(h)m(yp)s (othesis,)g(and)f(hence)839 1894 y(\()p Fk(\025)o(~)-49 b(uu:r)1123 1909 y Fj(v)1170 1894 y Fn([)p Fk(t)p Fn(]\))l Fk(~)j(s)1348 1909 y Fj(v)1394 1894 y Fn([)p Fk(t)p Fn(])p Fk(s)1533 1909 y Fj(v)1579 1894 y Fn([)p Fk(t)p Fn(])839 1960 y Fb(|)p 884 1960 327 12 v 327 w({z)p 1301 1960 V 327 w(})993 2059 y Fh(\(\()p Fj(\025~)-41 b(u)q(u:r)r Fh(\))m Fj(~)j(ss)p Fh(\))1398 2069 y Fe(v)1440 2059 y Fh([)p Fj(t)p Fh(])1698 1894 y Fl(!)27 b Fn(\()p Fk(\025)o(~)-49 b(u:)p Fn(\()p Fk(r)2094 1852 y Fg(0)2091 1918 y Fj(v)2138 1894 y Fn([)p Fk(t)2202 1852 y Fg(0)2229 1894 y Fn(]\))2296 1909 y Fj(u)2348 1894 y Fn([)p Fk(s)2423 1852 y Fg(0)2423 1918 y Fj(v)2470 1894 y Fn([)p Fk(t)2534 1852 y Fg(0)2561 1894 y Fn(]]\))2661 1869 y Fk(~)2656 1894 y(s)2703 1865 y Fg(0)2730 1909 y Fj(v)2777 1894 y Fn([)p Fk(t)2841 1852 y Fg(0)2869 1894 y Fn(])1825 1960 y Fb(|)p 1870 1960 447 12 v 447 w({z)p 2407 1960 V 447 w(})2038 2075 y Fh(\(\()p Fj(\025~)-41 b(u)q(:r)2258 2055 y Fa(0)2256 2094 y Fe(u)2302 2075 y Fh([)p Fj(s)2363 2055 y Fa(0)2390 2075 y Fh(]\))2451 2057 y Fj(~)2444 2075 y(s)2482 2055 y Fa(0)2508 2075 y Fh(\))2539 2085 y Fe(v)2582 2075 y Fh([)p Fj(t)2635 2055 y Fa(0)2661 2075 y Fh(])2913 1894 y Fk(:)0 2310 y Fn(b)m(y)33 b(de\014nition)h(of)g Fl(!)p Fn(.)p 866 2310 4 77 v 870 2238 92 4 v 870 2310 V 962 2310 4 77 v 0 2489 a Fd(Lemma)45 b(1.3.2.)58 b Fc(Assume)39 b Fk(r)h Fl(!)d Fk(r)1383 2453 y Fg(0)1450 2489 y Fc(and)i Fk(r)i Fl(!)c Fk(r)1920 2453 y Fg(00)1970 2489 y Fc(.)62 b(Then)40 b(w)m(e)g(can)f(\014nd)h(a)f(term)g Fk(r)3254 2453 y Fg(000)3366 2489 y Fc(suc)m(h)i(that)0 2609 y Fk(r)48 2573 y Fg(0)103 2609 y Fl(!)28 b Fk(r)279 2573 y Fg(000)385 2609 y Fc(and)34 b Fk(r)627 2573 y Fg(00)705 2609 y Fl(!)28 b Fk(r)881 2573 y Fg(000)954 2609 y Fc(.)0 2788 y Fn(The)46 b(pro)s(of)g(is)g(b)m(y)g(induction)h(on)f(the)g (de\014nition)g(of)g Fk(r)52 b Fl(!)c Fk(r)2419 2752 y Fg(0)2447 2788 y Fn(.)82 b(Again)45 b(all)g(cases)h(are)g(ob)m(vious) 0 2908 y(except)34 b(p)s(ossibly)f(the)h(situation)f(where)h(either)g Fk(r)d Fl(!)d Fk(r)2107 2872 y Fg(0)2168 2908 y Fn(or)33 b Fk(r)e Fl(!)d Fk(r)2542 2872 y Fg(00)2626 2908 y Fn(is)33 b(obtained)h(via)f(Rule)h(4.)44 b(By)0 3028 y(symmetry)27 b(w)m(e)j(ma)m(y)f(assume)g(the)h(former.)43 b(But)29 b(then)h(the)g(claim)f(follo)m(ws)g(from)g(Lemma)f(1.3.1:)41 b(If)1342 3267 y(\()p Fk(\025)o(~)-49 b(uu:r)s Fn(\))l Fk(~)j(s)o(s)27 b Fl(!)g Fn(\()p Fk(\025)o(~)-49 b(u:r)2145 3226 y Fg(0)2142 3291 y Fj(u)2194 3267 y Fn([)p Fk(s)2269 3226 y Fg(0)2296 3267 y Fn(]\))2368 3242 y Fk(~)2363 3267 y(s)2410 3238 y Fg(0)0 3506 y Fn(and)1304 3626 y(\()p Fk(\025)o(~)g(u)o(u:r)s Fn(\))l Fk(~)j(s)o(s)28 b Fl(!)f Fn(\()p Fk(\025)o(~)-49 b(uu:r)2164 3585 y Fg(00)2214 3626 y Fn(\))2269 3601 y Fk(~)2253 3626 y(s)2300 3597 y Fg(00)2351 3626 y Fk(s)2398 3585 y Fg(00)2448 3626 y Fk(;)0 3815 y Fn(then)945 3935 y(\()p Fk(\025)o(~)g(uu:r)s Fn(\))l Fk(~)j(s)o(s)27 b Fl(!)h Fn(\()p Fk(\025)o(~)-49 b(u:r)1749 3894 y Fg(0)1746 3959 y Fj(u)1797 3935 y Fn([)p Fk(s)1872 3894 y Fg(0)1900 3935 y Fn(]\))1972 3910 y Fk(~)1967 3935 y(s)2014 3906 y Fg(0)2068 3935 y Fl(!)28 b Fn(\()p Fk(\025)o(~)-49 b(u:r)2426 3894 y Fg(000)2423 3959 y Fj(u)2499 3935 y Fn([)p Fk(s)2574 3894 y Fg(000)2647 3935 y Fn(]\))2741 3910 y Fk(~)2714 3935 y(s)2761 3906 y Fg(000)0 4124 y Fn(and)907 4244 y(\()p Fk(\025)o(~)g(uu:r)s Fn(\))l Fk(~)j(s)o(s)27 b Fl(!)g Fn(\()p Fk(\025)o(~)-49 b(uu:r)1767 4203 y Fg(00)1818 4244 y Fn(\))1873 4219 y Fk(~)1857 4244 y(s)1904 4215 y Fg(00)1954 4244 y Fk(s)2001 4203 y Fg(00)2079 4244 y Fl(!)28 b Fn(\()p Fk(\025)o(~)-49 b(u)o(:r)2436 4203 y Fg(000)2433 4268 y Fj(u)2510 4244 y Fn([)p Fk(s)2585 4203 y Fg(000)2658 4244 y Fn(]\))2752 4219 y Fk(~)2725 4244 y(s)2772 4215 y Fg(000)2845 4244 y Fk(;)0 4433 y Fn(and)34 b(if)1138 4553 y(\()p Fk(\025)o(~)-49 b(uu)l(~)j(v)t(v)t(:r)s Fn(\))l Fk(~)g(s)o(s)1655 4534 y(~)1661 4553 y(tt)27 b Fl(!)h Fn(\()p Fk(\025)o(~)-49 b(u)l(~)i(v)t(v)t(:r)2221 4512 y Fg(0)2218 4577 y Fj(u)2270 4553 y Fn([)p Fk(s)2345 4512 y Fg(0)2372 4553 y Fn(]\))2444 4528 y Fk(~)2439 4553 y(s)2486 4524 y Fg(0)2513 4528 y Fk(~)2513 4553 y(t)2549 4524 y Fg(0)2577 4553 y Fk(t)2613 4512 y Fg(0)0 4742 y Fn(and)1089 4862 y(\()p Fk(\025)o(~)e(uu)l(~)j(v)t (v)t(:r)s Fn(\))l Fk(~)g(s)o(s)1606 4843 y(~)1612 4862 y(t)o(t)28 b Fl(!)f Fn(\()p Fk(\025)o(~)-49 b(uu)l(~)j(v)t(:r)2177 4821 y Fg(00)2174 4886 y Fj(v)2227 4862 y Fn([)p Fk(t)2291 4821 y Fg(00)2342 4862 y Fn(]\))2425 4837 y Fk(~)2409 4862 y(s)2456 4833 y Fg(00)2506 4862 y Fk(s)2553 4821 y Fg(00)2614 4837 y Fk(~)2604 4862 y(t)2640 4833 y Fg(00)0 5051 y Fn(then)563 5171 y(\()p Fk(\025)o(~)d(uu)l(~)j(v)t(v)t(:r)s Fn(\))l Fk(~)g(s)o(s)1080 5152 y(~)1086 5171 y(t)o(t)28 b Fl(!)f Fn(\()p Fk(\025)o(~)-49 b(u)l(~)j(v)t(v)t(:r)1646 5130 y Fg(0)1643 5195 y Fj(u)1694 5171 y Fn([)p Fk(s)1769 5130 y Fg(0)1797 5171 y Fn(]\))1869 5146 y Fk(~)1864 5171 y(s)1911 5142 y Fg(0)1937 5146 y Fk(~)1938 5171 y(t)1974 5142 y Fg(0)2002 5171 y Fk(t)2038 5130 y Fg(0)2093 5171 y Fl(!)28 b Fn(\()p Fk(\025)o(~)-49 b(u)l(~)j(v)s(:r)2502 5130 y Fg(000)2499 5195 y Fj(u;v)2617 5171 y Fn([)p Fk(s)2692 5130 y Fg(000)2765 5171 y Fk(;)17 b(t)2846 5130 y Fg(000)2920 5171 y Fn(]\))3014 5146 y Fk(~)2987 5171 y(s)3034 5142 y Fg(000)3129 5146 y Fk(~)3107 5171 y(t)3143 5142 y Fg(000)0 5360 y Fn(and)500 5480 y(\()p Fk(\025)o(~)-49 b(uu)l(~)j(v)t(v)t(:r)s Fn(\))l Fk(~)g(s)n(s)1016 5462 y(~)1022 5480 y(tt)28 b Fl(!)f Fn(\()p Fk(\025)o(~)-49 b(uu)l(~)j(v)t(:r)1588 5439 y Fg(00)1585 5505 y Fj(v)1638 5480 y Fn([)p Fk(t)1702 5439 y Fg(00)1753 5480 y Fn(]\))1836 5455 y Fk(~)1820 5480 y(s)1867 5451 y Fg(00)1917 5480 y Fk(s)1964 5439 y Fg(00)2025 5455 y Fk(~)2014 5480 y(t)2050 5451 y Fg(00)2129 5480 y Fl(!)27 b Fn(\()p Fk(\025)o(~)-49 b(u)l(~)j(v)t(:r)2538 5439 y Fg(000)2535 5505 y Fj(u;v)2653 5480 y Fn([)p Fk(s)2728 5439 y Fg(000)2801 5480 y Fk(;)17 b(t)2882 5439 y Fg(000)2955 5480 y Fn(]\))3049 5455 y Fk(~)3022 5480 y(s)3069 5451 y Fg(000)3164 5455 y Fk(~)3142 5480 y(t)3178 5451 y Fg(000)3252 5480 y Fk(:)p 0 5669 V 4 5597 92 4 v 4 5669 V 95 5669 4 77 v eop %%Page: 7 7 7 6 bop 0 100 a Fd(Theorem)38 b(1.3.3.)58 b(\(Ch)m(urc)m(h{Rosser\))31 b Fc(Assume)j Fk(r)d Fl(!)2236 63 y Fg(\003)2310 100 y Fk(r)2358 63 y Fg(0)2419 100 y Fc(and)j Fk(r)d Fl(!)2789 63 y Fg(\003)2863 100 y Fk(r)2911 63 y Fg(00)2962 100 y Fk(:)i Fc(Then)h(w)m(e)g(can)g(\014nd)0 219 y(a)f(term)f Fk(r)369 183 y Fg(000)476 219 y Fc(suc)m(h)j(that)e Fk(r)965 183 y Fg(0)1020 219 y Fl(!)1120 183 y Fg(\003)1193 219 y Fk(r)1241 183 y Fg(000)1348 219 y Fc(and)h Fk(r)1590 183 y Fg(00)1668 219 y Fl(!)1768 183 y Fg(\003)1841 219 y Fk(r)1889 183 y Fg(000)1963 219 y Fc(.)0 392 y Fn(The)g(pro)s(of)f (is)g(immediate)e(from)i(Lemma)f(1.3.2.)p 1920 392 4 77 v 1924 320 92 4 v 1924 392 V 2016 392 4 77 v 0 564 a Fd(Corollary)j(1.3.4.)58 b(\(Uniqueness)33 b(of)h(the)g(normal)g (form\))d Fc(Assume)f Fk(r)g Fl(!)3035 528 y Fg(\003)3108 564 y Fk(r)3156 528 y Fg(0)3214 564 y Fc(and)h Fk(r)f Fl(!)3580 528 y Fg(\003)3653 564 y Fk(r)3701 528 y Fg(00)3752 564 y Fc(,)0 684 y(where)k(b)s(oth)f Fk(r)571 648 y Fg(0)632 684 y Fc(and)h Fk(r)874 648 y Fg(00)958 684 y Fc(are)f(in)g(normal)f 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b(precisely)-8 b(,)34 b(w)m(e)h(will)e(construct)i(terms)e Fk(r)2313 1906 y Fj(n)2402 1891 y Fn(of)h(length)g(3)p Fk(n)g Fn(and)h(sho)m(w)g(that)f(an)m(y)0 2011 y(normalization)28 b(algorithm)g(needs)j(at)f(least)f(2)1745 2026 y Fj(n)p Fg(\000)p Fh(2)1917 2011 y Fl(\000)15 b Fk(n)30 b Fn(con)m(v)m(ersions) i(\(with)d(2)2933 2026 y Fh(0)3005 2011 y Fn(:=)f(1)p Fk(;)17 b Fn(2)3283 2026 y Fj(n)p Fh(+1)3465 2011 y Fn(:=)27 b(2)3647 1975 y Fh(2)3687 1985 y Fe(n)3741 2011 y Fn(\))0 2131 y(to)33 b(reduce)i Fk(r)483 2146 y Fj(n)570 2131 y Fn(to)e(its)g(normal)f(form.)199 2250 y(The)41 b(fact)f(that)f(there) i(is)f(no)g(elemen)m(tary)f(algorithm)f(\(i.e.)63 b(whose)41 b(time)e(is)g(exp)s(onen)m(tially)0 2370 y(b)s(ounded\))28 b(to)e(compute)g(the)h(normal)e(form)h(of)g(terms)g(also)g(follo)m(ws)g (from)g(\(Statman)f(1979\),)i(where)0 2489 y(it)h(is)g(sho)m(wn)i(more) d(generally)h(that)h(the)f(problem)g(whether)i(t)m(w)m(o)e(terms)g Fk(r)2791 2504 y Fh(1)2864 2489 y Fn(and)h Fk(r)3098 2504 y Fh(2)3171 2489 y Fn(ha)m(v)m(e)g(the)g(same)0 2609 y(normal)d(form)g(is)h(not)h(elemen)m(tary)e(recursiv)m(e.)43 b(The)28 b(simple)e(example)g(treated)i(here)g(is)f(tak)m(en)g(from)0 2728 y(\(Sc)m(h)m(wic)m(h)m(ten)m(b)s(erg)36 b(1982,)c(p.)h(455\).)199 2848 y(The)49 b(pure)h(t)m(yp)s(es)e Fk(k)k Fn(are)c(de\014ned)j (inductiv)m(ely)d(b)m(y)h(0)j(:=)i Fk(P)62 b Fn(\(some)47 b(\014xed)i(prop)s(ositional)0 2967 y(v)-6 b(ariable\))33 b(and)g Fk(k)25 b Fn(+)e(1)28 b(=)g Fk(k)i Fl(!)e Fk(k)s Fn(.)44 b(W)-8 b(e)33 b(de\014ne)i(iteration)d(terms)h Fk(I)2488 2982 y Fj(n)2575 2967 y Fn(of)h(pure)g(t)m(yp)s(e)f Fk(k)25 b Fn(+)e(2)33 b(b)m(y)1195 3176 y Fk(I)1239 3191 y Fj(n)1321 3176 y Fn(:)p Fl(\021)27 b Fk(\025f)11 b(\025u)p Fn(\()p Fk(f)g Fn(\()p Fk(f)g Fn(\()p Fk(:)17 b(:)g(:)c(f)e Fn(\()p Fk(f)g(u)p Fn(\))17 b Fk(:)g(:)g(:)m Fn(\)\)\))p Fk(;)0 3384 y Fn(with)30 b Fk(n)g Fn(o)s(ccurrences)j(of)d Fk(f)41 b Fn(after)31 b Fk(\025f)11 b(\025u)p Fn(;)31 b(here)g Fk(f)5 b(;)17 b(u)30 b 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Fj(m)1533 4548 y Fk(f)11 b Fn(\))21 b Fl(\016)h Fn(\()p Fk(I)1808 4563 y Fj(n)1862 4548 y Fk(f)11 b Fn(\))27 b(=)h Fk(I)2137 4563 y Fj(m)p Fh(+)p Fj(n)2323 4548 y Fk(f)5 b(;)1649 4697 y(I)1693 4712 y Fj(m)1790 4697 y Fl(\016)22 b Fk(I)1906 4712 y Fj(n)1988 4697 y Fn(=)28 b Fk(I)2137 4712 y Fj(m)p Fg(\001)p Fj(n)2286 4697 y Fk(;)1743 4847 y(I)1787 4862 y Fj(m)1862 4847 y Fk(I)1906 4862 y Fj(n)1988 4847 y Fn(=)g Fk(I)2137 4862 y Fj(n)2186 4842 y Fe(m)2257 4847 y Fk(:)p 2285 4847 4 77 v 2289 4775 92 4 v 2289 4847 V 2380 4847 4 77 v 0 5017 a Fn(As)33 b(an)h(immediate)d(consequence)k(w)m(e)f(ha)m(v) m(e)1404 5225 y Fk(r)1449 5240 y Fj(n)1530 5225 y Fn(:)p Fl(\021)28 b Fk(I)1707 5240 y Fh(2)1752 5225 y Fk(I)1796 5240 y Fh(2)1857 5225 y Fk(:)17 b(:)g(:)e(I)2034 5240 y Fh(2)1663 5282 y Fb(|)p 1708 5282 118 12 v 118 w({z)p 1916 5282 V 118 w(})1846 5372 y Fj(n)2106 5225 y Fn(=)28 b Fk(I)2255 5240 y Fh(2)2295 5250 y Fe(n)2348 5225 y Fk(:)0 5550 y Fn(No)m(w)g(consider)g(an)m(y)g(sequence)i(of)e (reduction)g(steps)h(transforming)e Fk(r)2615 5565 y Fj(n)2697 5550 y Fn(in)m(to)h(its)g(normal)e(form,)i(and)0 5669 y(let)33 b Fk(s)191 5684 y Fj(n)278 5669 y Fn(denote)h(the)g (total)e(n)m(um)m(b)s(er)h(of)h(reduction)g(steps)g(in)f(this)h (sequence.)p eop %%Page: 8 8 8 7 bop 0 100 a Fd(Theorem)38 b(1.4.2.)57 b Fk(s)858 115 y Fj(n)940 100 y Fl(\025)28 b Fn(2)1095 115 y Fj(n)p Fg(\000)p Fh(2)1273 100 y Fl(\000)23 b Fk(n)p Fc(.)0 278 y Fn(Pro)s(of.)57 b(The)38 b(length)g(of)g Fk(r)1006 293 y Fj(n)1098 278 y Fn(is)g(3)p Fk(n)p Fn(.)57 b(Note)37 b(that)g(an)m(y)h(con)m(v)m(ersion)h(step)f(can)g(at)g(most)e(square)i (the)0 397 y(length)33 b(of)h(the)f(original)f(term.)44 b(Hence)34 b(w)m(e)f(ha)m(v)m(e)911 626 y(2)961 641 y Fj(n)1043 626 y Fk(<)28 b Fn(length\()p Fk(I)1502 641 y Fh(2)1542 651 y Fe(n)1595 626 y Fn(\))199 b(\(the)34 b(normal)e(form)g(of)h Fk(r)2775 641 y Fj(n)2830 626 y Fn(\))1043 794 y Fl(\024)28 b Fn(length\()p Fk(r)1503 809 y Fj(n)1557 794 y Fn(\))1596 753 y Fh(2)1636 722 y Fe(s)1669 732 y(n)1043 961 y Fn(=)g(\(3)p Fk(n)p Fn(\))1336 920 y Fh(2)1376 890 y Fe(s)1409 900 y(n)1043 1140 y Fl(\024)g Fn(2)1198 1099 y Fh(2)1238 1069 y Fe(n)p Ff(+)p Fe(s)1366 1079 y(n)1624 1140 y Fn(\(since)34 b(3)p Fk(n)27 b Fl(\024)h Fn(2)2199 1099 y Fh(2)2239 1069 y Fe(n)2293 1140 y Fn(\))p Fk(;)0 1376 y Fn(and)34 b(the)f(theorem)g(is)g(pro)m(v)m(ed.)p 1216 1376 4 77 v 1220 1304 92 4 v 1220 1376 V 1311 1376 4 77 v 0 1730 a Fo(1.5.)71 b(Strong)52 b(normalization)0 1967 y Fn(In)35 b(Section)h(1.2)e(w)m(e)i(ha)m(v)m(e)f(pro)m(v)m(ed)h (that)f(an)m(y)g(term)f(can)h(b)s(e)h(reduced)g(to)f(a)g(normal)f (form,)g(and)i(in)0 2087 y(Section)i(1.3)f(w)m(e)h(ha)m(v)m(e)f(seen)i (that)e(this)h(normal)e(form)h(is)g(uniquely)g(determined.)57 b(But)37 b(it)g(is)g(still)0 2206 y(conceiv)-6 b(able)40 b(that)e(there)h(migh)m(t)f(b)s(e)h(an)g(o)s(dd)g(reduction)h(sequence) g(whic)m(h)g(do)s(es)f(not)g(terminate)0 2326 y(at)34 b(all.)48 b(It)34 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Fn(of)f(lev)m(el)g(0)g(b)s(oth)g (claims)e(are)0 4237 y(ob)m(vious.)58 b(No)m(w)37 b(consider)i Fk(')c Fl(!)g Fk( )t Fn(.)57 b(F)-8 b(or)37 b(1,)i(assume)f(that)f Fk(r)2343 4201 y Fj(')p Fg(!)p Fj( )2572 4237 y Fn(is)h(strongly)f (computable.)57 b(By)0 4356 y(induction)29 b(h)m(yp)s(othesis)g(2)f (and)g(the)h(de\014nition)f(of)h(strong)f(computabilit)m(y)e(w)m(e)j (kno)m(w)f(that)g(\()p Fk(r)s(u)p Fn(\))3625 4320 y Fj( )3713 4356 y Fn(is)0 4476 y(strongly)e(computable)h(and)g(hence)h(that)f(an)m (y)g(reduction)g(sequence)i(starting)d(with)h Fk(r)s(u)f Fn(terminates)0 4595 y(\(b)m(y)31 b(induction)h(h)m(yp)s(othesis)g (1\).)42 b(But)31 b(this)g(ob)m(viously)g(implies)f(that)h(the)g(same)g (is)g(true)h(for)f Fk(r)s Fn(.)43 b(F)-8 b(or)0 4715 y(2,)41 b(assume)g(that)c Fk(~)-47 b(r)43 b Fn(are)d(strongly)g (computable.)64 b(W)-8 b(e)41 b(ha)m(v)m(e)g(to)e(sho)m(w)i(that)f Fk(u)m(~)-47 b(r)43 b Fn(\(whic)m(h)e(is)f(to)g(b)s(e)0 4835 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Fk(r)2191 5514 y Fg(0)2215 5550 y Fk(~)-47 b(s)40 b Fn(terminates.)65 b(But)40 b(this)g(is)g(ob)m (viously)0 5669 y(true,)33 b(b)s(ecause)i(otherwise)f(w)m(e)f(w)m(ould) h(also)f(ha)m(v)m(e)g(an)h(in\014nite)f(reduction)h(sequence)h(for)f Fk(r)o(~)-46 b(s)o(:)p 3557 5669 V 3561 5597 92 4 v 3561 5669 V 3652 5669 4 77 v eop %%Page: 9 9 9 8 bop 0 100 a Fd(Lemma)38 b(1.5.3.)58 b Fc(An)m(y)33 b(term)f Fk(r)k Fc(is)d(strongly)g(computable)g(under)h(substitution.)0 290 y Fn(The)g(pro)s(of)f(is)g(b)m(y)h(induction)g(on)f(the)g(heigh)m (t)h(of)g Fk(r)s Fn(.)199 414 y Fm(Case)f Fk(u)p Fn(.)44 b(Ob)m(vious.)199 537 y Fm(Case)38 b Fk(r)s(s)p Fn(.)56 b(Let)802 519 y Fk(~)808 537 y(t)38 b Fn(b)s(e)g(strongly)f (computable.)56 b(W)-8 b(e)38 b(ha)m(v)m(e)g(to)g(sho)m(w)g(that)f Fk(r)s Fn([)3078 519 y Fk(~)3084 537 y(t)o Fn(])p Fk(s)p Fn([)3216 519 y Fk(~)3222 537 y(t)o Fn(])h(is)f(strongly)0 657 y(computable.)44 b(But)32 b(this)i(holds,)f(since)h(b)m(y)f 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b Fn(deg)r(\()p Fk(')p Fn(\))27 b(=)i(2)k(for)g Fk(')g Fn(atomic,)100 3688 y Fl(\017)49 b Fn(deg)r(\()p Fk(')28 b Fl(!)f Fk( )t Fn(\))g(=)h(1)22 b(+)h(deg)q(\()p Fk(')p Fn(\))f Fl(\001)g Fn(deg)r(\()p Fk( )t Fn(\).)0 3853 y(F)-8 b(or)33 b(then)h(w)m(e)g(ha)m(v)m(e,)g(writing)f Fk(a)28 b Fn(:=)g(deg)r(\()p Fk(')p Fn(\),)33 b Fk(b)28 b Fn(:=)h(deg)r(\()p Fk( )t Fn(\))j(and)i Fk(c)29 b Fn(:=)f(deg)r(\()p Fk(C)7 b Fn(\),)33 b(1)22 b(+)h Fk(ab)f Fn(+)h(1)f(+)h Fk(bc)28 b Fn(=)0 3973 y(2)22 b(+)h(\()p Fk(a)e Fn(+)i Fk(c)p Fn(\))p Fk(b)k Fl(\024)h Fn(2)22 b(+)h Fk(abc)k(<)h Fn(1)22 b(+)h Fk(c)f Fn(+)h Fk(abc)k Fn(=)h(1)22 b(+)h(\(1)e(+)i Fk(ab)p Fn(\))p Fk(c)32 b Fn(and)i(hence)h(1.1.)199 4092 y(More)j(formally)-8 b(,)36 b(w)m(e)j(de\014ne)g(the)f(relation)f Fl(`)1930 4056 y Fj(\013)1930 4117 y(m)2040 4092 y Fn(\000)f Fl(\))f Fk(')i Fn(\(to)g(b)s(e)h(read:)54 b(\000)35 b Fl(\))g Fk(')i Fn(is)h(deriv)-6 b(able)0 4212 y(with)44 b(heigh)m(t)i Fl(\024)h Fk(\013)e Fn(and)g(cut{rank)g Fl(\024)i Fk(m)p Fn(\))e(with)f Fk(\013;)17 b(m)45 b Fn(natural)f(n)m(um)m(b)s(ers)i(inductiv)m(ely)e(b)m(y)h(the)0 4332 y(follo)m(wing)33 b(rules.)199 4497 y Fm(Axiom.)44 b Fl(`)618 4461 y Fj(\013)618 4521 y(m)722 4497 y Fn(\000)p Fk(;)17 b(')27 b Fl(\))g Fk(')34 b Fn(for)f Fk(')g Fn(atomic.)199 4639 y Fl(!)p Fm({right.)43 b Fn(If)34 b Fl(`)788 4603 y Fj(\013)788 4664 y(m)891 4639 y Fn(\000)p Fk(;)17 b(')27 b Fl(\))h Fk( )t Fn(,)k(then)i Fl(`)1635 4603 y Fj(\013)p Fh(+1)1635 4664 y Fj(m)1820 4639 y Fn(\000)28 b Fl(\))f Fk(')h Fl(!)f Fk( )t Fn(.)199 4781 y Fl(!)p Fm({left.)44 b Fn(If)33 b Fl(`)720 4745 y Fj(\013)720 4806 y(m)823 4781 y Fn(\000)p Fk(;)17 b(')28 b Fl(!)f Fk( )k Fl(\))d Fk(')33 b Fn(and)h Fl(`)1727 4745 y Fj(\013)1727 4806 y(m)1830 4781 y Fn(\000)p Fk(;)17 b(')27 b Fl(!)h Fk( )t(;)17 b( )29 b Fl(\))f Fk(\037)p Fn(,)33 b(then)h Fl(`)2904 4745 y Fj(\013)p Fh(+1)2904 4806 y Fj(m)3089 4781 y Fn(\000)p Fk(;)17 b(')28 b Fl(!)f Fk( )k Fl(\))c Fk(\037)p Fn(.)199 4924 y Fm(Cut.)45 b Fn(If)34 b Fl(`)593 4888 y Fj(\013)593 4948 y(m)696 4924 y Fn(\000)28 b Fl(\))f Fk(\037)34 b Fn(and)f Fl(`)1263 4888 y Fj(\013)1263 4948 y(m)1367 4924 y Fn(\000)p Fk(;)17 b(\037)27 b Fl(\))g Fk(')p Fn(,)33 b(then)h Fl(`)2104 4881 y Fj(\013)p Fh(+1)2104 4958 y Fj(m)p Fh(+deg)q(\()p Fj(\037)p Fh(\))2501 4924 y Fn(\000)28 b Fl(\))f Fk(')p Fn(.)0 5100 y(Then)34 b(the)g(b)s(ound)g(men)m(tioned) f(ab)s(o)m(v)m(e)g(is)h(a)f(consequence)i(of)e(the)h(follo)m(wing)0 5265 y Fd(Theorem)k(1.7.1.)57 b(\(Cut)37 b(Elimination)g(Theorem\))32 b Fc(If)i Fl(`)2393 5229 y Fj(\013)2393 5290 y(m)p Fh(+1)2597 5265 y Fn(\000)28 b Fl(\))f Fk(')p Fc(,)33 b(then)h Fl(`)3228 5229 y Fh(2)p Fj(\013)3228 5290 y(m)3352 5265 y Fn(\000)28 b Fl(\))f Fk(')p Fc(.)0 5430 y Fn(This)e(theorem)g(is)g(due)h(to)f (\(Hudelmaier)f(1989\);)i(its)f(presen)m(t)i(form)m(ulation)d(and)h (pro)s(of)h(is)f(the)g(result)0 5550 y(of)33 b(Buc)m(hholz')h(analysis) f(\(Buc)m(hholz)h(1989\))e(of)h(Hudelmaier's)g(argumen)m(ts.)199 5669 y(W)-8 b(e)34 b(need)g(some)f(Lemmata)e(b)s(efore)j(w)m(e)g(can)g (giv)m(e)e(the)i(pro)s(of.)p eop %%Page: 11 11 11 10 bop 0 100 a Fd(Lemma)38 b(1.7.2.)58 b(\(W)-10 b(eak)m(ening)37 b(Lemma\))c Fc(If)g Fl(`)1963 63 y Fj(\013)1963 124 y(m)2067 100 y Fn(\000)28 b Fl(\))f Fk(')p Fc(,)33 b(then)h Fl(`)2698 63 y Fj(\013)2698 124 y(m)2801 100 y Fn(\000)p Fk(;)17 b Fn(\001)27 b Fl(\))h Fk(')p Fc(.)p 3283 100 4 77 v 3287 28 92 4 v 3287 100 V 3378 100 4 77 v 0 277 a Fd(Lemma)38 b(1.7.3.)58 b(\(In)m(v)m(ersion\))94 397 y Fc(i.)49 b(If)34 b Fl(`)360 361 y Fj(\013)360 422 y(m)463 397 y Fn(\000)28 b Fl(\))f Fk(')h Fl(!)f Fk( )t Fc(,)33 b(then)h Fl(`)1318 361 y Fj(\013)1318 422 y(m)1421 397 y Fn(\000)p Fk(;)17 b(')27 b Fl(\))h Fk( )t Fc(.)66 516 y(ii.)49 b(If)34 b Fl(`)360 480 y Fj(\013)360 541 y(m)463 516 y Fn(\000)p Fk(;)17 b(')28 b Fl(!)f Fk( )k Fl(\))c Fk(\037)p Fc(,)33 b(then)h Fl(`)1424 480 y Fj(\013)1424 541 y(m)1528 516 y Fn(\000)p Fk(;)17 b( )30 b Fl(\))e Fk(\037)p Fc(.)39 636 y(iii.)48 b(If)34 b Fl(`)360 600 y Fj(\013)360 661 y(m)463 636 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')27 b Fl(!)h Fk( )t Fn(\))e Fl(!)i Fk(\037)g Fl(\))f Fk(#)p Fc(,)33 b(then)h Fl(`)1716 600 y Fj(\013)1716 661 y(m)1819 636 y Fn(\000)p Fk(;)17 b(';)g( )30 b Fl(!)d Fk(\037)h Fl(\))g Fk(#)p Fc(.)0 814 y Fn(Pro)s(of.)44 b(By)32 b(induction)i(on)f Fk(\013)p Fn(.)44 b(W)-8 b(e)34 b(only)e(treat)h(one)h(case)g(of)f (iii.)43 b(Assume)33 b(that)1356 1046 y Fl(`)1417 1005 y Fj(\013)1417 1071 y(m)1520 1046 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')27 b Fl(!)g Fk( )t Fn(\))g Fl(!)h Fk(\037)g Fl(\))f Fk(#)0 1278 y Fn(w)m(as)34 b(inferred)g(from)363 1510 y Fl(`)424 1469 y Fj(\013)p Fg(\000)p Fh(1)424 1535 y Fj(m)610 1510 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')27 b Fl(!)h Fk( )t Fn(\))f Fl(!)g Fk(\037)h Fl(\))f Fk(')h Fl(!)f Fk( )103 b Fn(and)128 b Fl(`)2192 1469 y Fj(\013)p Fg(\000)p Fh(1)2192 1535 y Fj(m)2378 1510 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')28 b Fl(!)f Fk( )t Fn(\))g Fl(!)g Fk(\037;)17 b(\037)28 b Fl(\))f Fk(#:)0 1742 y Fn(By)32 b(induction)i(h)m(yp)s(othesis)g(w)m(e)g(get)551 1974 y Fl(`)612 1933 y Fj(\013)p Fg(\000)p Fh(1)612 1999 y Fj(m)798 1974 y Fn(\000)p Fk(;)17 b(';)g( )31 b Fl(!)c Fk(\037)h Fl(\))f Fk(')h Fl(!)f Fk( )103 b Fn(and)128 b Fl(`)2192 1933 y Fj(\013)p Fg(\000)p Fh(1)2192 1999 y Fj(m)2378 1974 y Fn(\000)p Fk(;)17 b(';)g( )31 b Fl(!)c Fk(\037;)17 b(\037)28 b Fl(\))f Fk(#:)0 2206 y Fn(Hence)34 b(b)m(y)f(i)1390 2326 y Fl(`)1451 2285 y Fj(\013)p Fg(\000)p Fh(1)1451 2351 y Fj(m)1637 2326 y Fn(\000)p Fk(;)17 b(';)g( )30 b Fl(!)d Fk(\037)h Fl(\))g Fk( )t(:)0 2512 y Fn(No)m(w)33 b Fl(!)p Fn({left)g(yields)g Fl(`)892 2476 y Fj(\013)892 2537 y(m)995 2512 y Fn(\000)p Fk(;)17 b(';)g( )30 b Fl(!)d Fk(\037)h Fl(\))g Fk(#)p Fn(.)p 1783 2512 V 1787 2440 92 4 v 1787 2512 V 1878 2512 4 77 v 0 2690 a Fd(Lemma)38 b(1.7.4.)58 b(\(Cut)37 b(Elimination)g(Lemma\))94 2821 y Fc(i.)49 b(If)34 b Fl(`)360 2785 y Fj(\013)360 2846 y Fh(0)444 2821 y Fn(\000)28 b Fl(\))f Fk(')34 b Fc(and)g Fl(`)1015 2773 y Fj(\014)1015 2848 y Fh(0)1096 2821 y Fn(\000)p Fk(;)17 b(')27 b Fl(\))h Fk( )36 b Fc(and)e Fk(')f Fc(is)g(atomic,)f(then)i Fl(`)2559 2773 y Fj(\013)p Fh(+)p Fj(\014)2559 2848 y Fh(0)2753 2821 y Fn(\000)28 b Fl(\))f Fk( )t Fc(.)66 2954 y(ii.)49 b(If)e Fl(`)373 2918 y Fj(\013)373 2979 y Fh(0)478 2954 y Fn(\000)j Fl(\))f Fk(')g Fl(!)g Fk( )g Fc(and)e Fl(`)1384 2906 y Fj(\014)1384 2980 y Fh(0)1486 2954 y Fn(\000)p Fk(;)17 b(')49 b Fl(!)g Fk( )k Fl(\))48 b Fk(\037)f Fc(and)f Fk(')g Fc(is)g(atomic)f(and)h Fk(\014)55 b Fl(\024)50 b Fk(\013)p Fc(,)f(then)199 3073 y Fl(`)260 3037 y Fj(\013)p Fh(+)p Fj(\014)260 3098 y(m)455 3073 y Fn(\000)27 b Fl(\))h Fk(\037)33 b Fc(with)g Fk(m)28 b Fn(=)h(deg)q(\()p Fk( )t Fn(\))p Fc(.)39 3205 y(iii.)48 b(If)36 b Fl(`)362 3168 y Fj(\013)362 3229 y Fh(0)451 3205 y Fn(\000)c Fl(\))f Fn(\()p Fk(')h Fl(!)g Fk( )t Fn(\))f Fl(!)g Fk(\037)36 b Fc(and)g Fl(`)1569 3157 y Fj(\014)1569 3231 y Fh(0)1655 3205 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')31 b Fl(!)h Fk( )t Fn(\))f Fl(!)g Fk(\037)h Fl(\))g Fk(#)p Fc(,)k(then)g Fl(`)2938 3168 y Fj(\013)p Fh(+)p Fj(\014)t Fh(+2)2938 3229 y Fj(m)3237 3205 y Fn(\000)c Fl(\))g Fk(#)p Fc(,)k(with)199 3324 y Fk(m)28 b Fn(=)h(deg)q(\()p Fk(')f Fl(!)f Fk( )t Fn(\))22 b(+)g(deg)r(\()p Fk( )31 b Fl(!)c Fk(\037)p Fn(\))p Fc(.)0 3502 y Fn(Pro)s(of.)44 b(i.)f(By)33 b(induction)g(on)h Fk(\014)5 b Fn(.)44 b(ii.)g(Consider)34 b(also)39 3680 y(ii'.)48 b(If)39 b Fl(`)365 3644 y Fj(\013)365 3704 y Fh(0)458 3680 y Fn(\000)e Fl(\))f Fk(')h Fl(!)f Fk( )42 b Fn(and)d Fl(`)1298 3644 y Fj(\013)1298 3704 y Fh(0)1391 3680 y Fn(\000)p Fk(;)17 b( )40 b Fl(\))c Fk(\037)j Fn(and)g Fl(`)2100 3632 y Fj(\014)2100 3706 y Fh(0)2190 3680 y Fn(\000)p Fk(;)17 b(')36 b Fl(!)g Fk( )k Fl(\))c Fk(')j Fn(and)g Fk(')g Fn(is)f(atomic)f(and)199 3799 y Fk(\014)d Fl(\024)28 b Fk(\013)p Fn(,)33 b(then)h Fl(`)807 3763 y Fj(\013)p Fh(+)p Fj(\014)t Fh(+1)807 3824 y Fj(m)1102 3799 y Fn(\000)28 b Fl(\))f Fk(\037)p Fn(,)33 b(with)g Fk(m)28 b Fn(=)g(deg)r(\()p Fk( )t Fn(\).)0 3977 y(W)-8 b(e)34 b(pro)m(v)m(e)f(ii)g(and)h(ii')e(sim)m(ultaneously)g (b)m(y)i(induction)f(on)h Fk(\014)5 b Fn(.)199 4097 y(ii.)44 b(Assume)834 4329 y Fl(`)895 4281 y Fj(\014)t Fg(\000)p Fh(1)895 4355 y(0)1078 4329 y Fn(\000)p Fk(;)17 b(')28 b Fl(!)f Fk( )k Fl(\))c Fk(')34 b Fn(and)61 b Fl(`)2009 4281 y Fj(\014)t Fg(\000)p Fh(1)2009 4355 y(0)2193 4329 y Fn(\000)p Fk(;)17 b(')27 b Fl(!)g Fk( )t(;)17 b( )30 b Fl(\))d Fk(\037:)0 4561 y Fn(Then)34 b(w)m(e)g(ha)m(v)m(e)g Fl(`)698 4525 y Fj(\013)698 4585 y Fh(0)782 4561 y Fn(\000)p Fk(;)17 b( )31 b Fl(\))c Fk(\037)33 b Fn(\(since)h Fk(\014)f Fl(\024)c Fk(\013)p Fn(\).)43 b(Hence)35 b(induction)e(h)m(yp)s (othesis)h(ii')e(yields)1571 4804 y Fl(`)1632 4757 y Fj(\013)p Fh(+)p Fj(\014)1632 4838 y Fh(deg)q(\()p Fj( )s Fh(\))1902 4804 y Fn(\000)c Fl(\))f Fk(\037:)199 5047 y Fn(ii'.)43 b Fm(Case)36 b Fk(\014)d Fn(=)28 b(0)p Fm(.)44 b Fn(Then)34 b Fk(')28 b Fl(2)g Fn(\000,)33 b(hence)1571 5280 y Fl(`)1632 5237 y Fj(\013)p Fh(+1)1632 5313 y(deg)q(\()p Fj( )s Fh(\))1902 5280 y Fn(\000)28 b Fl(\))f Fk(\037:)199 5550 y Fm(Case)43 b Fl(`)512 5502 y Fj(\014)t Fg(\000)p Fh(1)512 5576 y(0)709 5550 y Fn(\000)p Fk(;)17 b(')41 b Fl(!)g Fk( )k Fl(\))c Fk(')i Fm(and)g Fl(`)1684 5502 y Fj(\014)t Fg(\000)p Fh(1)1684 5576 y(0)1881 5550 y Fn(\000)p Fk(;)17 b(')41 b Fl(!)g Fk( )t(;)17 b( )44 b Fl(\))d Fk(')p Fm(.)69 b Fn(Then)42 b(the)g(claim)e(follo)m(ws)0 5669 y(immediately)30 b(from)j(the)g(induction)h(h)m(yp)s(othesis.)p eop %%Page: 12 12 12 11 bop 199 100 a Fm(Case)48 b Fl(`)517 52 y Fj(\014)t Fg(\000)p Fh(1)517 126 y(0)724 100 y Fn(\000)p Fk(;)17 b(')50 b Fl(!)h Fk( )j Fl(\))c Fk(#)f Fm(and)f Fl(`)1741 52 y Fj(\014)t Fg(\000)p Fh(1)1741 126 y(0)1947 100 y Fn(\000)p Fk(;)17 b(')51 b Fl(!)f Fk( )t(;)17 b(\021)54 b Fl(\))c Fk(')f Fm(with)f Fk(#)j Fl(!)f Fk(\021)55 b Fl(2)c Fn(\000)e Fm(and)0 219 y Fk(#)28 b Fl(!)f Fk(\021)32 b Fl(6)p Fn(=)c Fk(')g Fl(!)f Fk( )t Fm(.)43 b Fn(Then)34 b(induction)g(h)m(yp)s(othesis)g(ii)f(yields)1559 461 y Fl(`)1620 413 y Fj(\013)p Fh(+)p Fj(\014)t Fg(\000)p Fh(1)1620 495 y(deg)q(\()p Fj( )s Fh(\))1916 461 y Fn(\000)28 b Fl(\))g Fk(#;)0 713 y Fn(and)34 b(induction)g(h)m(yp)s(othesis)f(ii') g(yields)1522 955 y Fl(`)1583 907 y Fj(\013)p Fh(+)p Fj(\014)1583 989 y Fh(deg)q(\()p Fj( )s Fh(\))1853 955 y Fn(\000)p Fk(;)17 b(\021)31 b Fl(\))d Fk(\037:)0 1208 y Fn(No)m(w)33 b Fl(!)p Fn({left)g(giv)m(es)1558 1330 y Fl(`)1619 1282 y Fj(\013)p Fh(+)p Fj(\014)t Fh(+1)1619 1364 y(deg)q(\()p Fj( )s Fh(\))1914 1330 y Fn(\000)28 b Fl(\))f Fk(\037:)199 1533 y Fn(iii.)43 b(By)33 b(induction)g(on)h Fk(\014)5 b Fn(.)44 b(Assume)432 1775 y Fl(`)493 1727 y Fj(\014)t Fg(\000)p Fh(1)493 1801 y(0)677 1775 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')27 b Fl(!)g Fk( )t Fn(\))g Fl(!)g Fk(\037)h Fl(\))g Fk(')f Fl(!)h Fk( )36 b Fn(and)62 b Fl(`)2126 1727 y Fj(\014)t Fg(\000)p Fh(1)2126 1801 y(0)2309 1775 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')27 b Fl(!)g Fk( )t Fn(\))g Fl(!)h Fk(\037;)17 b(\037)27 b Fl(\))h Fk(#:)0 2016 y Fn(F)-8 b(rom)41 b Fl(`)332 1980 y Fj(\013)332 2041 y Fh(0)431 2016 y Fn(\000)i Fl(\))f Fn(\()p Fk(')g Fl(!)g Fk( )t Fn(\))g Fl(!)g Fk(\037)g Fn(w)m(e)h(get)f Fl(`)1754 1980 y Fj(\013)1754 2041 y Fh(0)1853 2016 y Fn(\000)p Fk(;)17 b(')42 b Fl(!)g Fk( )k Fl(\))c Fk(\037)p Fn(,)i(hence)g Fl(`)2943 1980 y Fj(\013)2943 2041 y Fh(0)3042 2016 y Fn(\000)p Fk(;)17 b( )46 b Fl(\))c Fk(\037)p Fn(,)i(hence)0 2136 y Fl(`)61 2093 y Fj(\013)p Fh(+1)61 2163 y(0)260 2136 y Fn(\000)e Fl(\))f Fk( )k Fl(!)c Fk(\037)p Fn(.)69 b(On)42 b(the)g(other)g(hand,)i(the)e(In)m(v)m(ersion)g(Lemma,)g(P)m (arts)f(i)g(and)h(iii,)h(yields)0 2269 y Fl(`)61 2221 y Fj(\014)t Fg(\000)p Fh(1)61 2295 y(0)244 2269 y Fn(\000)p Fk(;)17 b(';)g( )30 b Fl(!)e Fk(\037)g Fl(\))f Fk( )t Fn(,)32 b(hence)j Fl(`)1368 2221 y Fj(\014)1368 2295 y Fh(0)1449 2269 y Fn(\000)p Fk(;)17 b( )31 b Fl(!)c Fk(\037)h Fl(\))g Fk(')g Fl(!)f Fk( )t Fn(.)43 b(So)1393 2524 y Fl(`)1454 2476 y Fj(\013)p Fh(+)p Fj(\014)t Fh(+1)1454 2558 y(deg)q(\()p Fj( )s Fg(!)p Fj(\037)p Fh(\))1853 2524 y Fn(\000)28 b Fl(\))f Fk(')h Fl(!)g Fk( )t(:)0 2803 y Fn(F)-8 b(urthermore,)53 b(from)c Fl(`)939 2755 y Fj(\014)t Fg(\000)p Fh(1)939 2830 y(0)1150 2803 y Fn(\000)p Fk(;)17 b Fn(\()p Fk(')55 b Fl(!)g Fk( )t Fn(\))f Fl(!)h Fk(\037;)17 b(\037)55 b Fl(\))g Fk(#)50 b Fn(w)m(e)g(get)f Fl(`)2782 2755 y Fj(\014)t Fg(\000)p Fh(1)2782 2830 y(0)2993 2803 y Fn(\000)p Fk(;)17 b(\037)55 b Fl(\))g Fk(#)p Fn(.)94 b(Since)0 2923 y Fl(`)61 2887 y Fj(\013)61 2948 y Fh(0)145 2923 y Fn(\000)p Fk(;)17 b(')28 b Fl(!)f Fk( )k Fl(\))c Fk(\037)p Fn(,)34 b(a)f(cut)g(yields)1409 3176 y Fl(`)1470 3128 y Fj(\013)p Fh(+)p Fj(\014)1470 3210 y Fh(deg)q(\()p Fj(\037)p Fh(\))1734 3176 y Fn(\000)p Fk(;)17 b(')28 b Fl(!)f Fk( )k Fl(\))c Fk(#:)0 3444 y Fn(No)m(w)33 b(one)h(further)g (cut)g(giv)m(es)f Fl(`)1222 3408 y Fj(\013)p Fh(+)p Fj(\014)t Fh(+2)1222 3469 y Fj(m)1517 3444 y Fn(\000)28 b Fl(\))f Fk(#)p Fn(.)p 1865 3444 4 77 v 1869 3372 92 4 v 1869 3444 V 1960 3444 4 77 v 199 3564 a(W)-8 b(e)34 b(no)m(w)f(pro)m(v)m(e)h (the)g(Cut)f(Elimination)e(Theorem,)h(b)m(y)i(induction)f(on)h Fk(\013)p Fn(.)44 b(Assume)1181 3806 y Fl(`)1242 3763 y Fj(\013)p Fg(\000)p Fh(1)1242 3836 y Fj(k)1428 3806 y Fn(\000)28 b Fl(\))f Fk(\037)34 b Fn(and)61 b Fl(`)2023 3763 y Fj(\013)p Fg(\000)p Fh(1)2023 3836 y Fj(k)2210 3806 y Fn(\000)p Fk(;)17 b(\037)27 b Fl(\))h Fk(')0 4048 y Fn(with)33 b Fk(m)23 b Fn(+)f(1)28 b(=)g Fk(k)d Fn(+)e(deg)q(\()p Fk(\037)p Fn(\).)44 b(If)34 b Fk(k)c Fl(6)p Fn(=)f(0,)j(then)i(b)m(y)g (induction)f(h)m(yp)s(othesis)1065 4309 y Fl(`)1126 4257 y Fh(2\()p Fj(\013)p Fg(\000)p Fh(1\))1126 4339 y Fj(k)r Fg(\000)p Fh(1)1414 4309 y Fn(\000)28 b Fl(\))g Fk(\037)33 b Fn(and)62 b Fl(`)2010 4257 y Fh(2\()p Fj(\013)p Fg(\000)p Fh(1\))2010 4339 y Fj(k)r Fg(\000)p Fh(1)2298 4309 y Fn(\000)p Fk(;)17 b(\037)27 b Fl(\))h Fk(';)0 4550 y Fn(hence)40 b(b)m(y)e(cut)g Fl(`)662 4514 y Fh(2)p Fj(\013)662 4575 y(m)794 4550 y Fn(\000)e Fl(\))f Fk(')p Fn(,)k(since)g Fk(k)28 b Fl(\000)e 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