%!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: mod93.dvi %%CreationDate: Tue Jan 31 15:15:09 1995 %%Pages: 41 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips mod93 %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1995.01.31:1514 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 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Fo(\).)-31 1604 y(\()p Fn(^)21 1586 y Fm(\000)55 1604 y Fo(\))26 b(If)16 b Fl(d)175 1586 y Fj(A)205 1591 y Fh(0)225 1586 y Fm(^)p Fj(A)282 1591 y Fh(1)320 1604 y Fo(is)g(a)h(deriv)m(ation)f(term,)g(then)882 1701 y Fl(\031)910 1708 y Fj(i)927 1701 y Fo(\()p Fl(d)972 1681 y Fj(A)1002 1686 y Fh(0)1021 1681 y Fm(^)p Fj(A)1078 1686 y Fh(1)1100 1701 y Fo(\))1119 1681 y Fj(A)1149 1686 y Fg(i)100 1798 y Fo(is)g(a)g(deriv)m(ation)g(term)g(with)h(F)-6 b(A\()p Fl(\031)769 1805 y Fj(i)787 1798 y Fo(\()p Fl(d)832 1780 y Fj(A)862 1785 y Fh(0)881 1780 y Fm(^)p Fj(A)938 1785 y Fh(1)960 1798 y Fo(\))979 1780 y Fj(A)1009 1785 y Fg(i)1028 1798 y Fo(\))14 b(:=)f(F)-6 b(A)q(\()p Fl(d)1237 1780 y Fj(A)1267 1785 y Fh(0)1286 1780 y Fm(^)p Fj(A)1343 1785 y Fh(1)1365 1798 y Fo(\))17 b(\(for)g Fl(i)d Fn(2)g(f)p Fo(0)p Fl(;)8 b Fo(1)p Fn(g)p Fo(\).)-25 1870 y(\()p Fn(8)23 1852 y Fi(+)55 1870 y Fo(\))26 b(If)16 b Fl(d)175 1852 y Fj(A)224 1870 y Fo(is)h(a)f(deriv)m(ation)g(term)g(and)g Fl(x)794 1852 y Fj(\032)831 1870 y Fn(62)879 1833 y Ff(S)920 1870 y Fn(f)p Fo(FV\()p Fl(B)r Fo(\))p Fn(j)p Fl(u)1136 1852 y Fj(B)1185 1870 y Fn(2)e Fo(F)-6 b(A\()p Fl(d)1341 1852 y Fj(A)1374 1870 y Fo(\))p Fn(g)p Fo(,)17 b(then)879 1968 y(\()p Fl(\025x)955 1947 y Fj(\032)988 1968 y Fl(d)1014 1947 y Fj(A)1046 1968 y Fo(\))1065 1947 y Fm(8)p Fj(x)1112 1932 y Fg(\032)1138 1947 y Fj(A)100 2065 y Fo(is)f(a)g(deriv)m(ation)g (term)g(with)h(F)-6 b(A\()p Fl(\025x)798 2047 y Fj(\032)831 2065 y Fl(d)857 2047 y Fj(A)889 2065 y Fo(\))14 b(=)g(F)-6 b(A\()p Fl(d)1084 2047 y Fj(A)1117 2065 y Fo(\).)-25 2137 y(\()p Fn(8)23 2119 y Fm(\000)55 2137 y Fo(\))26 b(If)16 b Fl(d)175 2119 y Fm(8)p Fj(x)222 2104 y Fg(\032)248 2119 y Fj(A)297 2137 y Fo(is)g(a)h(deriv)m(ation)f(term)g(and)g Fl(t)857 2119 y Fj(\032)896 2137 y Fo(is)g(a)h(term,)f(then)853 2234 y(\()p Fl(d)898 2214 y Fm(8)p Fj(x)945 2199 y Fg(\032)970 2214 y Fj(A)1003 2234 y Fl(t)1021 2214 y Fj(\032)1044 2234 y Fo(\))1063 2214 y Fj(A)p Fi([)p Fj(t=x)1162 2199 y Fg(\032)1183 2214 y Fi(])100 2331 y Fo(is)g(a)g(deriv)m(ation)g(term) g(with)h(F)-6 b(A\()p Fl(d)767 2313 y Fm(8)p Fj(x)814 2298 y Fg(\032)840 2313 y Fj(A)872 2331 y Fl(t)890 2313 y Fj(\032)914 2331 y Fo(\))14 b(=)f(F)-6 b(A)q(\()p Fl(d)1109 2313 y Fm(8)p Fj(x)1156 2298 y Fg(\032)1182 2313 y Fj(A)1214 2331 y Fo(\).)0 2416 y(Recall)17 b(that)i(w)o(e)e(iden)o(tify)h(terms)f (with)g(the)h(same)f Fl(\014)s(\021)r Fo(-normal)e(form.)25 b(Hence)19 b(e.g.)e Fl(d)1638 2398 y Fj(R)p Fi(\()p Fj(r)q Fi(\))p Fm(!)p Fj(B)1793 2416 y Fl(e)1816 2398 y Fj(R)p Fi(\()p Fj(s)p Fi(\))1917 2416 y Fo(is)0 2476 y(a)g(correct)f(deriv)m (ation)g(pro)o(vided)f Fl(r)j Fo(and)e Fl(s)h Fo(ha)o(v)o(e)f(the)h (same)e Fl(\014)s(\021)r Fo(-normal)f(forms.)100 2535 y(A)j(deriv)m(ation)f(term)g Fl(d)533 2517 y Fj(A)582 2535 y Fo(is)g(called)g Fk(close)m(d)5 b Fo(,)18 b(if)e(F)-6 b(A)q(\()p Fl(d)1091 2517 y Fj(A)1124 2535 y Fo(\))14 b(=)f Fn(;)p Fo(.)22 b(W)l(e)17 b(write)793 2633 y Fl(d)819 2612 y Fj(B)853 2633 y Fo([)p Fl(u)896 2610 y Fj(A)926 2615 y Fh(1)896 2646 y Fi(1)947 2633 y Fl(;)8 b(:)g(:)g(:)h(;)f(u)1087 2612 y Fj(A)1117 2617 y Fg(n)1087 2645 y Fj(n)1143 2633 y Fo(])0 2730 y(to)18 b(indicate)f(that)h(the)f(assumption)f(v)m (ariables)g(free)h(in)g Fl(d)1099 2712 y Fj(B)1151 2730 y Fo(are)g(in)g(the)h(list)f Fl(u)1494 2708 y Fj(A)1524 2713 y Fh(1)1494 2743 y Fi(1)1545 2730 y Fl(;)8 b(:)g(:)g(:)h(;)f(u) 1685 2712 y Fj(A)1715 2717 y Fg(n)1685 2742 y Fj(n)1741 2730 y Fo(.)25 b(W)l(e)18 b(also)0 2790 y(use)e(the)h(notation)f Fl(d)p Fo(:)8 b Fl(A)17 b Fo(instead)f(of)h Fl(d)728 2771 y Fj(A)760 2790 y Fo(.)p eop %%Page: 4 4 4 3 bop 522 50 a Fo(1.)22 b(Pro)q(ofs)16 b(in)g(minimal)f(logic)h(and)g (arithmetic)p 0 76 1950 2 v 0 169 a Fe(De\014nition.)30 b Fd(A)21 b(form)o(ula)d Fl(A)j Fd(is)e(called)h Fk(derivable)25 b Fd(from)19 b(assumptions)f Fl(A)1454 176 y Fi(1)1476 169 y Fl(;)8 b(:)g(:)g(:)h(;)f(A)1624 176 y Fj(n)1652 169 y Fd(,)21 b(is)e(there)h(is)g(a)0 229 y(deriv)m(ation)c(term)g Fl(d)379 211 y Fj(B)413 229 y Fo([)p Fl(u)456 207 y Fj(A)486 212 y Fh(1)456 242 y Fi(1)508 229 y Fl(;)8 b(:)g(:)g(:)g(;)g(u)647 211 y Fj(A)677 216 y Fg(n)647 241 y Fj(n)704 229 y Fo(])16 b Fd(with)h(di\013eren)o(t)e(assumption)g(v)m(ariables)g Fl(u)1543 207 y Fj(A)1573 212 y Fh(1)1543 242 y Fi(1)1595 229 y Fl(;)8 b(:)g(:)g(:)g(;)g(u)1734 211 y Fj(A)1764 216 y Fg(n)1734 241 y Fj(n)1791 229 y Fd(.)100 320 y Fo(Let)20 b Fl(S)i Fo(b)q(e)e(a)g(\(\014nite)f(or)g(in\014nite\))h(set) f(of)h(form)o(ulas.)29 b(W)l(e)20 b(write)g Fl(S)h Fn(`)f Fl(B)r Fo(,)g(if)g(the)g(form)o(ula)e Fl(B)k Fo(is)0 380 y(deriv)m(able)16 b(from)g(\014nitely)g(man)o(y)g(assumptions)e Fl(A)953 387 y Fi(1)975 380 y Fl(;)8 b(:)g(:)g(:)h(;)f(A)1123 387 y Fj(n)1164 380 y Fn(2)14 b Fl(S)s Fo(.)100 440 y Fk(Examples)21 b Fo(for)16 b(deriv)m(able)g(form)o(ulas)e(are)520 561 y Fl(P)21 b Fn(!)14 b Fo(\()p Fl(Q)g Fn(!)g Fl(P)7 b Fo(\))p Fl(;)520 635 y Fo(\()p Fl(P)21 b Fn(!)14 b Fl(Q)g Fn(!)g Fl(R)p Fo(\))g Fn(!)g Fo(\()p Fl(P)21 b Fn(!)14 b Fl(Q)p Fo(\))g Fn(!)g Fl(P)21 b Fn(!)13 b Fl(R;)520 710 y Fo(\()p Fn(8)p Fl(x:P)7 b Fo(\()p Fl(x)p Fo(\))15 b Fn(!)e Fl(Q)p Fo(\()p Fl(x)p Fo(\)\))j Fn(!)e(8)p Fl(x)8 b(P)f Fo(\()p Fl(x)p Fo(\))14 b Fn(!)g(8)p Fl(x)8 b(Q)p Fo(\()p Fl(x)p Fo(\))p Fl(:)0 831 y Fe(De\014nition.)30 b Fd(F)l(or)15 b(an)o(y)h(deriv)m(ation)g Fl(d)h Fd(w)o(e)f(de\014ne)g (its)h(set)f Fo(FV)q(\()p Fl(d)p Fo(\))h Fd(of)g Fk(fr)m(e)m(e)k Fd(\(ob)s(ject\))d Fk(variables)k Fd(b)o(y)683 952 y Fo(FV)q(\()p Fl(u)802 931 y Fj(A)834 952 y Fo(\))14 b(:=)f(FV)q(\()p Fl(A)p Fo(\))p Fl(:)594 1027 y Fo(FV\()p Fl(\025u)741 1006 y Fj(A)774 1027 y Fl(d)800 1006 y Fj(B)834 1027 y Fo(\))h(:=)f(FV)q(\()p Fl(A)p Fo(\))f Fn([)g Fo(FV\()p Fl(d)1251 1006 y Fj(B)1286 1027 y Fo(\))p Fl(:)558 1101 y Fo(FV)q(\()p Fl(d)674 1081 y Fj(A)p Fm(!)p Fj(B)778 1101 y Fl(e)801 1081 y Fj(A)834 1101 y Fo(\))i(:=)f(FV)q(\()p Fl(d)1049 1081 y Fj(A)p Fm(!)p Fj(B)1153 1101 y Fo(\))f Fn([)f Fo(FV)q(\()p Fl(e)1341 1081 y Fj(A)1374 1101 y Fo(\))p Fl(:)567 1176 y Fo(FV)q(\()p Fn(h)p Fl(d)702 1155 y Fj(A)735 1176 y Fl(;)d(e)780 1155 y Fj(B)814 1176 y Fn(i)p Fo(\))15 b(:=)e(FV)q(\()p Fl(d)1049 1155 y Fj(A)1082 1176 y Fo(\))e Fn([)h Fo(FV\()p Fl(e)1269 1155 y Fj(B)1304 1176 y Fo(\))p Fl(:)543 1251 y Fo(FV)q(\()p Fl(\031)661 1258 y Fj(i)678 1251 y Fo(\()p Fl(d)723 1230 y Fj(A)p Fm(^)p Fj(B)814 1251 y Fo(\)\))j(:=)e(FV)q(\()p Fl(d)1049 1230 y Fj(A)p Fm(^)p Fj(B)1140 1251 y Fo(\))p Fl(:)620 1325 y Fo(FV\()p Fl(\025x)8 b(d)800 1305 y Fj(A)834 1325 y Fo(\))14 b(=)g(FV\()p Fl(d)1035 1305 y Fj(A)1068 1325 y Fo(\))d Fn(n)g(f)p Fl(x)p Fn(g)p Fl(:)616 1400 y Fo(FV\()p Fl(d)731 1380 y Fm(8)p Fj(x)5 b(A)816 1400 y Fl(t)p Fo(\))14 b(:=)f(FV)q(\()p Fl(d)1049 1380 y Fm(8)p Fj(x)5 b(A)1134 1400 y Fo(\))11 b Fn([)g Fo(FV)q(\()p Fl(t)p Fo(\))p Fl(:)100 1521 y Fk(Example)t Fo(:)22 b(Let)17 b Fl(d)f Fo(b)q(e)h(the)g(deriv)m(ation)f(giv)o(en)g(b)o(y)827 1639 y Fl(u)p Fo(:)8 b Fl(R)p Fo(\()p Fl(x)p Fo(\))p 753 1665 305 2 v 753 1713 a Fl(R)p Fo(\()p Fl(x)p Fo(\))15 b Fn(!)31 b Fl(R)p Fo(\()p Fl(x)p Fo(\))1078 1676 y Fn(!)1128 1656 y Fi(+)1174 1676 y Fl(u)0 1831 y Fo(Then)16 b(w)o(e)g(ha)o(v)o(e) 843 1884 y Fl(d)e Fo(=)f Fl(\025u)993 1863 y Fj(R)p Fi(\()p Fj(x)p Fi(\))1088 1884 y Fl(u)1117 1863 y Fj(R)p Fi(\()p Fj(x)p Fi(\))1203 1884 y Fl(;)740 1958 y Fo(F)-6 b(A\()p Fl(d)p Fo(\))15 b(=)e Fn(;)p Fl(;)734 2033 y Fo(FV\()p Fl(d)p Fo(\))i(=)e Fn(f)p Fl(x)p Fn(g)p Fl(:)100 2127 y Fo(F)l(or)h(deriv)m(ation)i(terms)f Fl(d)h Fo(w)o(e)f(ha)o(v)o(e)g(t) o(w)o(o)h(kinds)f(of)h(substitution:)k(w)o(e)c(can)g(substitute)f(a)h (deriv)m(a-)0 2186 y(tion)g(term)f Fl(f)249 2168 y Fj(A)299 2186 y Fo(for)g(a)h(free)g(assumption)e(v)m(ariable)i Fl(u)989 2168 y Fj(A)1021 2186 y Fo(,)g(denoted)f Fl(d)p Fo([)p Fl(f)5 b(=u)p Fo(],)16 b(and)f(w)o(e)h(can)g(substitute)f(an)0 2246 y(ob)s(ject)i(term)f Fl(t)h Fo(for)f(a)g(free)h(ob)s(ject)g(v)m (ariable)f Fl(x)p Fo(,)h(denoted)f Fl(d)p Fo([)p Fl(t=x)p Fo(].)0 2337 y Fe(Lemma.)27 b Fd(\(i\))20 b(If)g Fl(d)p Fo(:)8 b 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Fo(\))14 b(=)g Fl(\025)o(~)-24 b(x)9 b(r)346 1352 y Fi(1)369 1345 y Fl(;)f(:)g(:)g(:)g(;)g(\025)o(~) -24 b(x)9 b(r)567 1352 y Fj(k)592 1345 y Fo(.)22 b(In)16 b(the)h(last)g(clause)f(the)g(\(omitted\))i(t)o(yp)q(es)e(are)614 1478 y Fl(x)642 1458 y Fj(\032)666 1478 y Fl(;)58 b Fo(\()p Fl(~)-25 b(y)r Fo(\))802 1458 y Fj(\034)t Fi(\()p Fj(A)p Fi(\))939 1478 y Fo(and)49 b Fl(z)1094 1451 y Fj(\032)p Fm(!)p Fj(\034)t Fi(\()p Fj(A)p Fi(\))p Fm(!)p Fj(\033)1302 1456 y Fg(j)1092 1492 y Fj(j)1322 1478 y Fl(;)0 1600 y Fo(where)16 b Fl(\034)6 b Fo(\()p Fl(B)r Fo(\))14 b(=)g Fl(\033)345 1607 y Fi(1)368 1600 y Fl(;)8 b(:)g(:)g(:)g(;)g(\033)506 1607 y Fj(n)534 1600 y Fo(.)100 1661 y(The)16 b(follo)o(wing)f(can)i(b) q(e)f(pro)o(v)o(ed)g(easily)l(.)0 1760 y Fe(Lemma.)27 b Fo(FV\(ets)q(\()p Fl(d)p Fo(\)\))15 b Fn(\022)e Fo(FV)q(\()p Fl(d)p Fo(\))e Fn([)h(f)o Fl(~)-24 b(x)774 1735 y Fj(\034)t Fi(\()p Fj(A)p Fi(\))774 1766 y Fj(u)874 1760 y Fo(:)14 b Fl(u)931 1742 y Fj(A)976 1760 y Fn(2)g Fo(F)-6 b(A)q(\()p Fl(d)p Fo(\))p Fn(g)p Fd(.)0 1853 y Fe(Lemma.)27 b Fd(W)l(e)16 b(ha)o(v)o(e)g Fo(ets\()p Fl(d)p Fo([)p Fl(t=x)p Fo(]\))f(=)f(ets\()p Fl(d)p Fo(\)[)p Fl(t=x)p Fo(])j Fd(and)f Fo(ets\()p Fl(d)p Fo([)p Fl(e=u)p Fo(]\))e(=)f(ets)q(\()p Fl(d)p Fo(\)[ets\()p Fl(e)p Fo(\))p Fl(=)o(~)-24 b(x)1670 1860 y Fj(u)1698 1853 y Fo(])p Fd(,)16 b(hence)607 1975 y Fl(d)e Fo(=)686 1982 y Fj(\014)r Fm(9)734 1973 y Fb(\003)770 1975 y Fl(e)64 b Fo(=)-8 b Fn(\))63 b Fo(ets\()p Fl(d)p Fo(\))14 b(=)1179 1982 y Fj(\014)1220 1975 y Fo(ets\()p Fl(e)p Fo(\))653 2050 y Fl(d)g Fo(=)732 2057 y Fj(\021)770 2050 y Fl(e)64 b Fo(=)-8 b Fn(\))63 b Fo(ets\()p Fl(d)p Fo(\))14 b(=)1179 2057 y Fj(\021)1217 2050 y Fo(ets\()p Fl(e)p Fo(\))572 b Fc(\003)100 2173 y Fo(Hence)17 b(w)o(e)f(can)g(safely)h(iden)o(tify)f (deriv)m(ations)g(with)g(the)h(same)f Fl(\014)s(\021)r Fn(9)1396 2155 y Fm(\003)1418 2173 y Fo({normal)f(forms.)0 2265 y Fe(Soundness)k(Theorem.)27 b Fd(Assume)17 b(that)h(to)g(an)o(y)g 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Fn(!)f Fl(A)0 462 y Fo(and)i(w)o(e)g(ha)o (v)o(e)443 569 y Fl(e)466 576 y Fj(ij)501 569 y Fo([)p Fl(u)544 548 y Fm(\000)544 583 y Fj(i)577 569 y Fo(]:)8 b Fn(\021)14 b Fo(Stab)765 576 y Fj(C)794 581 y Fg(ij)828 569 y Fl(\025v)883 548 y Fm(:)p Fj(C)939 553 y Fg(ij)973 569 y Fl(:u)1016 548 y Fm(\000)1016 583 y Fj(i)1049 569 y Fl(\025)o(~)-24 b(u)1116 540 y Fj(~)1107 548 y(C)1136 553 y Fg(i)1153 569 y Fl(:)p Fo(Efq)1243 581 y Fj(P)1268 586 y Fg(i)1286 569 y Fo(\()p Fl(v)r(u)1360 576 y Fj(j)1381 569 y Fo(\):)8 b Fl(C)1458 576 y Fj(ij)1493 569 y Fl(;)443 661 y(e)466 668 y Fj(i)483 661 y Fo([)p Fl(u)526 639 y Fm(\000)526 674 y Fj(i)559 661 y Fo(]:)g Fn(\021)13 b Fl(\025w)713 638 y Fj(P)738 643 y Fg(i)712 674 y Fj(i)757 661 y Fl(:u)800 639 y Fm(\000)800 674 y Fj(i)833 661 y Fl(\025)o(~)-24 b(u)900 631 y Fj(~)891 640 y(C)920 645 y Fg(i)937 661 y Fl(w)973 668 y Fj(i)989 661 y Fo(:)8 b Fn(:)p Fl(P)1076 668 y Fj(i)1093 661 y Fl(:)0 768 y Fo(By)17 b(IH)g(w)o(e)f(ha)o(v)o(e)g Fl(d)364 775 y Fj(C)393 780 y Fg(ij)427 768 y Fo(:)8 b Fl(C)485 775 y Fj(ij)534 768 y Fn(!)14 b Fl(C)638 749 y Fj(A)634 781 y(ij)669 768 y Fo(.)22 b(Hence)473 885 y(~)-27 b Fl(v)495 892 y Fj(i)512 885 y Fo(\()p Fl(d)557 892 y Fj(C)586 897 y Fg(i)p Fh(1)621 885 y Fl(e)644 892 y Fj(i)p Fi(1)681 885 y Fo(\))8 b Fl(:)g(:)g(:)h Fo(\()p Fl(d)820 892 y Fj(C)849 897 y Fg(in)884 906 y(i)905 885 y Fl(e)928 892 y Fj(in)967 897 y Fg(i)985 885 y Fo(\)\()p Fl(\025w)1089 863 y Fj(P)1114 868 y Fg(i)1088 899 y Fj(i)1133 885 y Fl(:)p Fo(Efq)1223 898 y Fj(A)1255 885 y Fo(\()p Fl(e)1297 892 y Fj(i)1314 885 y Fl(w)1350 892 y Fj(i)1367 885 y Fo(\)\):)f Fl(A:)100 1001 y Fk(Case)20 b Fl(u)253 980 y Fi(+)253 1015 y Fj(i)286 1001 y Fo(:)8 b Fl(B)346 1008 y Fj(i)379 1001 y Fo(for)16 b(all)g Fl(i)p Fo(.)23 b(Then)792 1061 y Fl(w)q Fo(\()s(~)-28 b Fl(uu)906 1040 y Fi(+)906 1074 y(1)947 1061 y Fl(:)8 b(:)g(:)g(u)1042 1040 y Fi(+)1042 1073 y Fj(m)1079 1061 y Fo(\):)g Fl(A)100 1150 y Fo(The)16 b(extracted)h(terms)f(for)g(this)g(deriv)m(ation)g(are)254 1257 y Fl(d)280 1236 y Fi(ets)346 1257 y Fn(\021)d Fl(\025)o(~)-24 b(x)455 1264 y Fi(1)478 1257 y Fl(;)8 b(:)g(:)g(:)h(;)f(~)-25 b(x)617 1264 y Fj(m)655 1257 y Fl(;)7 b(~)-24 b(z)s(:)16 b Fe(if)50 b Fn(:)p Fl(B)887 1264 y Fi(1)959 1257 y Fe(then)g Fl(~)-25 b(x)1149 1264 y Fi(1)1172 1257 y Fl(d)1198 1236 y Fi(ets)1198 1269 y Fj(C)1227 1274 y Fh(11)1274 1257 y Fl(:)8 b(:)g(:)g(d)1366 1236 y Fi(ets)1366 1269 y Fj(C)1395 1274 y Fh(1)p Fg(n)1434 1281 y Fh(1)1454 1246 y Fl(~)1458 1257 y Fo(0)50 b Fe(else)741 1332 y Fl(:)8 b(:)g(:)733 1406 y Fe(if)50 b Fn(:)p Fl(B)887 1413 y Fj(m)975 1406 y Fe(then)f Fl(~)-24 b(x)1165 1413 y Fj(m)1203 1406 y Fl(d)1229 1386 y Fi(ets)1229 1419 y Fj(C)1258 1424 y Fg(m)p Fh(1)1318 1406 y Fl(:)8 b(:)g(:)h(d)1411 1386 y Fi(ets)1411 1419 y Fj(C)1440 1424 y Fg(mn)1493 1429 y(m)1526 1396 y Fl(~)1530 1406 y Fo(0)50 b Fe(else)732 1481 y Fl(~)-24 b(z)52 b Fe(\014)8 b Fl(:)g(:)g(:)h Fe(\014)p Fl(;)0 1588 y Fo(where)15 b Fl(~)-24 b(x)172 1595 y Fj(i)189 1588 y Fl(;)7 b(~)-24 b(z)19 b Fo(are)d(the)h(lists)f(of)h(v)m (ariables)e(asso)q(ciated)i(with)h(~)-27 b Fl(v)1163 1595 y Fj(i)1180 1588 y Fo(:)8 b Fl(B)1242 1570 y Fj(A)1240 1601 y(i)1275 1588 y Fl(;)g(w)q Fo(:)g Fl(R)14 b Fn(!)g Fl(A)p Fo(.)388 b Fc(\003)100 1648 y Fo(Here)16 b(w)o(e)f(ha)o(v)o(e)g (used)g(case)h(splitting)e(according)h(to)h(the)g(quan)o(ti\014er)e (free)i(form)o(ulas)e Fl(B)1743 1655 y Fj(i)1775 1648 y Fo(whic)o(h)h(is)0 1708 y(admissible)f(b)o(y)j(a)f(remark)f(ab)q(o)o (v)o(e.)100 1767 y(If)24 b(w)o(e)f(w)o(an)o(t)g(to)h(use)f(the)h Fl(A)p Fo({translation)e(to)i(extract)h(the)f(computational)e(con)o (ten)o(t)h(from)g(a)0 1827 y(classical)14 b(pro)q(of)h(w)o(e)g(ha)o(v)o (e)g(to)h(c)o(ho)q(ose)f(a)g(particular)f Fl(A)h Fo(in)o(v)o(olving)f (the)i(strong)e(existen)o(tial)h(quan)o(ti\014er.)0 1914 y Fe(Lemma)h(2.)29 b Fd(Let)16 b Fl(B)402 1921 y Fj(i)419 1914 y Fo([)o Fl(~)-24 b(x;)8 b(~)-25 b(y)r Fo(])16 b Fd(b)q(e)g(quan)o(ti\014er{free)f(form)o(ulas)f(and)h Fl(A)p Fo([)o Fl(~)-24 b(x)q Fo(])14 b(:=)f Fn(9)1434 1896 y Fm(\003)1456 1914 y Fl(~)-24 b(y)1491 1877 y Ff(V)-30 b(V)1544 1929 y Fj(i)1569 1914 y Fl(B)1607 1921 y Fj(i)1623 1914 y Fo([)o Fl(~)-24 b(x)q(;)8 b(~)-25 b(y)r Fo(])p Fd(.)22 b(Then)15 b(w)o(e)0 1974 y(can)h(\014nd)g(a)g(deriv)m(ation)g (of)h Fo(\()p Fn(8)p Fl(~)-25 b(y)q(:B)648 1981 y Fi(1)670 1974 y Fo([)o Fl(~)h(x)q(;)8 b(~)-25 b(y)r Fo(])14 b Fn(!)f Fl(:)8 b(:)g(:)14 b Fn(!)g Fl(B)1026 1981 y Fj(m)1064 1974 y Fo([)o Fl(~)-24 b(x;)8 b(~)-25 b(y)r Fo(])14 b Fn(!)g(?)p Fo(\))1304 1956 y Fj(A)p Fi([)o Fj(~)-19 b(x)p Fi(])1381 1974 y Fd(.)0 2067 y Fk(Pr)m(o)m(of.)24 b Fo(Let)17 b Fl(~)-24 b(y)18 b Fo(b)q(e)f(giv)o(en)f(and)g(assume)h(~)-27 b Fl(v)776 2074 y Fj(i)793 2067 y Fo(:)823 2054 y Fl(~)815 2067 y(B)855 2048 y Fj(A)853 2079 y(i)904 2067 y Fo(and)16 b Fl(w)q Fo(:)8 b Fn(?)14 b(!)g Fl(A)p Fo(.)22 b(W)l(e)17 b(m)o(ust)e(sho)o(w)h Fl(A)p Fo(.)100 2133 y Fk(Case)k Fl(u)253 2111 y Fm(\000)253 2147 y Fj(i)286 2133 y Fo(:)8 b Fn(:)p Fl(B)379 2140 y Fj(i)413 2133 y Fo(for)16 b(some)g Fl(i)p Fo(.)22 b(Let)17 b Fl(B)794 2140 y Fj(i)824 2133 y Fn(\021)887 2120 y Fl(~)877 2133 y(C)913 2140 y Fj(i)943 2133 y Fn(!)d Fl(P)1039 2140 y Fj(i)1072 2133 y Fo(with)i(atoms)g Fl(P)1364 2140 y Fj(i)1381 2133 y Fo(.)22 b(Then)706 2258 y(~)-27 b Fl(v)728 2265 y Fj(i)745 2258 y Fo(:)777 2246 y Fl(~)767 2258 y(C)803 2265 y Fj(i)819 2223 y(A)865 2258 y Fn(!)14 b Fo(\()p Fl(P)980 2265 y Fj(i)1011 2258 y Fn(!)f Fl(A)p Fo(\))i Fn(!)f Fl(A)0 2365 y Fo(and)i(w)o(e)g(ha)o(v)o (e)443 2472 y Fl(e)466 2479 y Fj(ij)501 2472 y Fo([)p Fl(u)544 2451 y Fm(\000)544 2486 y Fj(i)577 2472 y Fo(]:)8 b Fn(\021)14 b Fo(Stab)765 2479 y Fj(C)794 2484 y Fg(ij)828 2472 y Fl(\025v)883 2452 y Fm(:)p Fj(C)939 2457 y Fg(ij)973 2472 y Fl(:u)1016 2451 y Fm(\000)1016 2486 y Fj(i)1049 2472 y Fl(\025)o(~)-24 b(u)1116 2443 y Fj(~)1107 2452 y(C)1136 2457 y Fg(i)1153 2472 y Fl(:)p Fo(Efq)1243 2485 y Fj(P)1268 2490 y Fg(i)1286 2472 y Fo(\()p Fl(v)r(u)1360 2479 y Fj(j)1381 2472 y Fo(\):)8 b Fl(C)1458 2479 y Fj(ij)1493 2472 y Fl(;)443 2564 y(e)466 2571 y Fj(i)483 2564 y Fo([)p Fl(u)526 2543 y Fm(\000)526 2578 y Fj(i)559 2564 y Fo(]:)g Fn(\021)13 b Fl(\025w)713 2542 y Fj(P)738 2547 y Fg(i)712 2578 y Fj(i)757 2564 y Fl(:u)800 2543 y Fm(\000)800 2578 y Fj(i)833 2564 y Fl(\025)o(~)-24 b(u)900 2535 y Fj(~)891 2544 y(C)920 2549 y Fg(i)937 2564 y Fl(w)973 2571 y Fj(i)989 2564 y Fo(:)8 b Fn(:)p Fl(P)1076 2571 y Fj(i)1093 2564 y Fl(:)0 2671 y Fo(Using)16 b Fl(d)166 2678 y Fj(C)195 2683 y Fg(ij)229 2671 y Fo(:)8 b Fl(C)287 2678 y Fj(ij)336 2671 y Fn(!)13 b Fl(C)439 2653 y Fj(A)435 2684 y(ij)487 2671 y Fo(from)j(Lemma)g(1)h(w)o(e)f(obtain)473 2789 y(~)-27 b Fl(v)495 2796 y Fj(i)512 2789 y Fo(\()p Fl(d)557 2796 y Fj(C)586 2801 y Fg(i)p Fh(1)621 2789 y Fl(e)644 2796 y Fj(i)p Fi(1)681 2789 y Fo(\))8 b Fl(:)g(:)g(:)h Fo(\()p Fl(d)820 2796 y Fj(C)849 2801 y Fg(in)884 2809 y(i)905 2789 y Fl(e)928 2796 y Fj(in)967 2801 y Fg(i)985 2789 y Fo(\)\()p Fl(\025w)1089 2767 y Fj(P)1114 2772 y Fg(i)1088 2803 y Fj(i)1133 2789 y Fl(:)p Fo(Efq)1223 2801 y Fj(A)1255 2789 y Fo(\()p Fl(e)1297 2796 y Fj(i)1314 2789 y Fl(w)1350 2796 y Fj(i)1367 2789 y Fo(\)\):)f Fl(A:)p eop %%Page: 17 17 17 16 bop 627 50 a Fo(2.)22 b(Classical)15 b(pro)q(ofs)h(as)g(programs) p 0 76 1950 2 v 100 169 a Fk(Case)k Fl(u)253 148 y Fi(+)253 183 y Fj(i)286 169 y Fo(:)8 b Fl(B)346 176 y Fj(i)379 169 y Fo(for)16 b(all)g Fl(i)p Fo(.)23 b(Then)763 281 y Fn(9)791 260 y Fi(+)823 281 y Fl(~)-24 b(y)r Fn(h)p Fl(u)898 259 y Fi(+)898 294 y(1)931 281 y Fl(;)8 b(:)g(:)g(:)h(;)f(u) 1071 260 y Fi(+)1071 293 y Fj(m)1108 281 y Fn(i)p Fo(:)g Fl(A)100 392 y Fo(The)16 b(extracted)h(terms)f(for)g(this)g(deriv)m (ation)g(are)230 503 y Fl(d)256 482 y Fi(ets)321 503 y Fn(\021)e Fl(\025~)-25 b(y)r(;)8 b(~)-25 b(x)479 510 y Fi(1)502 503 y Fl(;)8 b(:)g(:)g(:)h(;)f(~)-25 b(x)641 510 y Fj(m)679 503 y Fl(;)7 b(~)-24 b(z)s(:)16 b Fe(if)50 b Fn(:)p 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b(the)h(form)f(\(3\))h(of)g Fl(d)436 2771 y Fi(ets)436 2802 y Fj(u)487 2790 y Fo(,)g(since)f(all)g Fl(B)747 2797 y Fj(i)763 2790 y Fo([)n Fl(~)-23 b(r)r(;)6 b(~)-23 b(s)p Fo(])14 b Fn(\021)g Fo(\()957 2777 y Fl(~)946 2790 y(B)984 2797 y Fj(i)1014 2790 y Fn(!)g Fl(P)1110 2797 y Fj(i)1126 2790 y Fo(\)[)n Fl(~)-23 b(r)s(;)6 b(~)-23 b(s)p Fo(])16 b(are)h(true.)p eop %%Page: 20 20 20 19 bop 635 50 a Fo(2.)22 b(Classical)15 b(pro)q(ofs)h(as)g(programs) p 0 76 1950 2 v 100 169 a Fk(Case)24 b Fo(2.)33 b(There)20 b(is)g(a)g(minimal)f Fl(i)h Fo(suc)o(h)f(that)i Fl(e)1033 176 y Fj(i)1070 169 y Fo(deriv)o(es)e(a)i(false)f(form)o(ula,)f Fl(D)1644 176 y Fj(i)p Fi(1)1681 169 y Fo([)n Fl(~)-23 b(s)p Fo(])20 b Fn(!)g(\001)8 b(\001)g(\001)20 b(!)0 229 y Fl(D)41 236 y Fj(in)80 241 y Fg(i)98 229 y Fo([)n Fl(~)-23 b(s)p Fo(])21 b Fn(!)g(?)f Fo(sa)o(y)l(.)34 b(Then)20 b Fl(D)589 236 y Fj(i)p Fi(1)626 229 y Fo([)n Fl(~)-23 b(s)p Fo(])p Fl(;)8 b(:)g(:)g(:)h(;)f(D)829 236 y Fj(in)868 241 y Fg(i)886 229 y 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b(Then)20 b(w)o(e)h(are)f(to)h(pac)o(k)0 2610 y(the)d(blo)q(c)o(ks)f Fa(i0::i1::nil)j Fo(in)o(to)d(the)h(bins)e Fa(j0::j1::nil)k Fo(with)d(the)h(side)f(condition)g(that)h(the)g (\014rst)0 2670 y(blo)q(c)o(k)d Fa(i0)g Fo(has)f(to)h(go)f(in)o(to)g (the)h(\014rst)f(bin)g Fa(j0)p Fo(.)22 b(If)14 b(w)o(e)h(sp)q(ecialize) f(the)h(general)f(pro)q(of)g(to)h(this)g(particular)0 2730 y(situation)h(and)f(normalize)g(it,)i(then)f(b)q(oth)h(inductions) e(unfold)g(completely)h(and)g(w)o(e)g(obtain)g(a)g(pro)q(of)0 2790 y(as)g(sho)o(wn)f(in)h(App)q(endix)h(2.)p eop %%Page: 30 30 30 29 bop 414 50 a Fo(4.)22 b(An)17 b(example)f(of)h(program)d(dev)o (elopmen)o(t)h(b)o(y)i(pruning)p 0 76 1950 2 v 100 169 a(Note)g(that)g(this)f(pro)q(of)g(has)g(the)h(follo)o(wing)e (structure.)0 229 y(Case)h Fa(u26)q Fo(:)8 b Fl(i)237 236 y Fi(0)273 229 y Fn(\024)14 b Fl(j)347 236 y Fi(0)369 229 y Fo(.)100 303 y(Sub)q(case)i Fa(u27)q Fo(:)8 b Fl(i)408 310 y Fi(0)442 303 y Fo(+)j Fl(i)509 310 y Fi(1)547 303 y Fn(\024)k 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y(second)f(instance\))g(and)f (then)i(carry)e(out)i(a)f(somewhat)f(elab)q(orate)i(argumen)o(t)d(to)j (arriv)o(e)e(at)i(some)0 1196 y(conclusion)e(\(as)h(it)g(turns)f(out,)j (atomic)d(form)o(ulas)g(in)g(b)q(oth)i(instances\).)35 b(W)l(e)21 b(could)g(use)f(the)i(\014rst)0 1256 y(pruning)17 b(step)j(to)g(remo)o(v)o(e)e(these)h(and)g(other)g(unneccessary)g (parts)f(of)i(the)g(pro)q(of.)30 b(Ho)o(w)o(ev)o(er,)19 b(suc)o(h)0 1315 y(an)e(application)f(of)h(pruning)e(do)q(es)i(not)h(y) o(et)f(sho)o(w)f(the)i(e\013ect)g(w)o(e)f(w)o(an)o(t)f(to)i (demonstrate,)e(since)h(the)0 1375 y(algorithm)e(represen)o(ted)g(b)o (y)h(the)h(pro)q(of)f(has)g(not)g(c)o(hanged.)100 1435 y(T)l(o)j(ac)o(hiev)o(e)g(this)g(desired)f(e\013ect,)j(w)o(e)f(\014rst) e(ha)o(v)o(e)h(to)h(pro)o(vide)e(some)h(additional)f(assumptions)0 1495 y(on)d(the)h(input)f(data.)21 b(So)16 b(let)g(us)f(assume)f(that)i (the)g(second)f(blo)q(c)o(k)g(is)g(less)g(than)g(or)h(equal)f(to)h(the) g(\014rst)0 1555 y(one,)g(and)g(that)h(the)g(\014rst)f(bin)f(is)h(less) g(than)h(or)f(equal)g(to)h(the)g(second)e(one,)i(i.e.)k(that)612 1666 y Fa(v1)p Fo(:)8 b Fl(i)703 1673 y Fi(1)740 1666 y Fn(\024)13 b Fl(i)809 1673 y Fi(0)932 1666 y Fo(and)98 b Fa(v2)q Fo(:)8 b Fl(j)1207 1673 y Fi(0)1243 1666 y Fn(\024)13 b Fl(j)1316 1673 y Fi(1)0 1777 y Fo(holds.)28 b(Then)19 b(w)o(e)g(can)g(replace)g(quite)g(a)g(n)o(um)o(b)q(er)e(of)i (subpro)q(ofs)f(b)o(y)h(di\013eren)o(t)f(ones)g(|)h(this)g(is)g(the)0 1837 y(\014rst)c(pruning)e(step)i(of)h(Section)f(2)g(|,)g(where)g(w)o (e)g(alw)o(a)o(ys)f(aim)h(at)g(replacing)f(the)i(use)f(of)g(assumption) 0 1897 y(v)m(ariables)21 b(b)q(ound)g(later)h(in)f(the)h(pro)q(of)g (\(e.g.)g(case)g(assumptions\))e(b)o(y)h(the)i(use)e(of)h(our)f (additional)0 1956 y(assumptions)d Fa(v1)i Fo(and)f Fa(v2)i Fo(and)e(|)h(if)g(necessary)f(|)h(some)f(other)h(assumption)e(v)m (ariables.)31 b(Let)21 b(us)0 2016 y(lo)q(ok)c(at)g(these)f(replacemen) o(ts)f(individually)l(.)20 b(The)d(result)f(is)g(sho)o(wn)f(in)h(App)q (endix)g(3.)11 2104 y(\(1\))26 b(Here)14 b(the)g(mac)o(hine)f(replaced) g(the)h(deriv)m(ation)g(using)f(Lemmata)g(3)h(and)g(5)g(of)g(an)g(atom) g(from)f(the)100 2164 y(false)j(assumption)f Fa(u29:delta-seq)28 b(0::0::nil)g(->)e(F)18 b Fo(|)f(whic)o(h)f(rewrites)g(in)o(to)g Fa(T)27 b(->)f(F)17 b Fo(|)100 2224 y(b)o(y)f(a)g(trivial)h(deriv)m (ation)f(using)f(ex{falso{quo)q(dlib)q(et)h(and)g(the)h(truth{axiom)f Fa(T)p Fo(.)11 2297 y(\(2\))26 b(The)e(case)g(assumption)e Fa(u32)p Fo(:)8 b Fl(i)711 2304 y Fi(1)760 2297 y Fn(\024)27 b Fl(j)847 2304 y Fi(1)893 2297 y Fo(is)c(replaced)h(b)o(y)f(a)i (simple)d(deriv)m(ation)i(\(in)o(v)o(olving)f(a)100 2357 y(global)17 b(assumption)e Fa(trans)k Fo(expressing)d(transitivit)o(y)i (of)g Fn(\024)p Fo(\))g(of)g Fl(i)1372 2364 y Fi(1)1410 2357 y Fn(\024)e Fl(j)1486 2364 y Fi(1)1525 2357 y Fo(from)h(the)h (additional)100 2417 y(assumptions)i Fa(v1)p Fo(:)8 b Fl(i)480 2424 y Fi(1)527 2417 y Fn(\024)24 b Fl(i)607 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y(\(\(add)g(0\))g(\(\(add)g(0\))f(nil\)\)\))209 2379 y(0\))183 2439 y(truth-axiom-symbol\))157 2499 y(u44\)\)\)\)\)\))p eop %%Page: 40 40 40 39 bop 0 179 a Fp(References)25 434 y Fo([1])24 b(Ulric)o(h)15 b(Berger.)21 b(Program)14 b(extraction)i(from)f(normalization)f(pro)q (ofs.)21 b(In)15 b(M.)h(Bezem)g(and)f(J.F.)102 494 y(Gro)q(ote,)g (editors,)f Fk(T)l(yp)m(e)m(d)k(L)m(amb)m(da)h(Calculi)c(and)i(Applic)m (ations)p Fo(,)g(v)o(olume)d(664)h(of)g Fk(L)m(e)m(ctur)m(e)i(Notes)102 553 y(in)h(Computer)g(Scienc)m(e)p Fo(,)f(pages)f(91{106,)f(Berlin,)h (1993.)g(Springer.)25 641 y([2])24 b(Ro)o(y)e(Dyc)o(kho\013.)39 b(Con)o(traction{free)21 b(sequen)o(t)h(calculi)f(for)h(in)o (tuitionistic)f(logic.)39 b Fk(Journal)23 b(of)102 700 y(Symb)m(olic)c(L)m(o)m(gic)p Fo(,)f(57:793{807,)d(1992.)25 787 y([3])24 b(Harv)o(ey)c(F)l(riedman.)26 b(Classically)18 b(and)h(in)o(tuitionistically)e(pro)o(v)m(ably)h(recursiv)o(e)g (functions.)28 b(In)102 847 y(Dana)16 b(S.)f(Scott)i(and)e(Gert)h(H.)g 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b(Dale)18 b(Miller.)25 b(A)19 b(logic)e(programming)e (language)i(with)h(lam)o(b)q(da{abstraction,)d(function)j(v)m(ari-)102 1554 y(ables)e(and)g(simple)f(uni\014cation.)21 b Fk(Journal)d(of)h(L)m (o)m(gic)g(and)g(Computation)p Fo(,)e(2\(4\):497{536,)f(1991.)25 1642 y([8])24 b(G.E.)17 b(Min)o(ts.)j(On)c(e{theorems)g(\(in)g (russian\).)21 b Fk(Zapiski)p Fo(,)c(40:110{118,)f(1974.)25 1729 y([9])24 b(Chetan)g(Murth)o(y)l(.)43 b(Extracting)23 b(constructiv)o(e)h(con)o(ten)o(t)f(from)g(classical)g(pro)q(ofs.)43 b(T)l(ec)o(hnical)102 1788 y(Rep)q(ort)18 b(90{1151,)e(Dep.of)i (Comp.Science,)e(Cornell)g(Univ.,)h(Ithaca,)h(New)g(Y)l(ork,)f(1990.)24 b(PhD)102 1848 y(thesis.)0 1935 y([10])g(T)l(obias)18 b(Nipk)o(o)o(w.)28 b(F)l(unctional)18 b(uni\014cation)f(of)i (higher{order)e(patterns.)28 b(In)19 b Fk(Pr)m(o)m(c.)i(8th)f(IEEE)102 1995 y(Symp.)f(L)m(o)m(gic)g(in)g(Computer)e(Scienc)m(e)p Fo(,)g(pages)f(64{74,)g(1993.)0 2082 y([11])24 b(Dag)g(Pra)o(witz.)41 b Fk(Natur)m(al)25 b(De)m(duction)p 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b(Springer,)17 b(Berlin,)102 441 y(1973.)0 533 y([18])24 b(Anne)d(S.)g(T)l(ro)q(elstra)f(and)g(Dirk)g(v)m(an)h(Dalen.) 35 b Fk(Constructivism)21 b(in)i(Mathematics.)f(A)o(n)f(Intr)m(o-)102 592 y(duction)p Fo(,)h(v)o(olume)f(121,)g(123)g(of)g Fk(Studies)i(in)g(L)m(o)m(gic)h(and)f(the)f(F)l(oundations)i(of)f (Mathematics)p Fo(.)102 652 y(North{Holland,)16 b(Amsterdam,)f(1988.)0 744 y([19])24 b(An)o(ton)19 b(W)l(allner.)27 b(Komplexe)19 b(Existenzb)q(ew)o(eise)f(in)g(der)h(Arithmetik.)28 b(Master's)18 b(thesis,)g(Ma-)102 804 y(thematisc)o(hes)e(Institut)g(der)g(Univ)o (ersit\177)-25 b(at)16 b(M)q(\177)-26 b(unc)o(hen,)15 b(1993.)0 896 y([20])24 b(Hermann)e(W)l(eyl.)497 883 y(\177)491 896 y(Ub)q(er)h(die)g(neue)g(Grundlagenkrise)e(der)h (Mathematik.)41 b Fk(Mathematische)102 956 y(Zeitschrift)p Fo(,)17 b(10,)f(1921.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF