%!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: Minlog7.dvi %%CreationDate: Tue Apr 28 12:51:00 1998 %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Symbol Times-Italic CourierNarrow %%+ Times-Bold Helvetica %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips Minlog7.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.04.28:1250 %%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 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D }B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginFont: Times-Roman % @psencodingfile{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "14 April 1995", % filename = "8r.enc", % email = "kb@cs.umb.edu", % address = "135 Center Hill Rd. // Plymouth, MA 02360", % codetable = "ISO/ASCII", % checksum = "xx", % docstring = "Encoding for TrueType or Type 1 fonts to be used with TeX." % } % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both ASCII and Windows. % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndFont %%BeginFont: CourierNarrow %!PS-AdobeFont-1.0: Courier 002.004 %%CreationDate: Tue Sep 17 07:47:14 1991 %%VMusage: 31037 39405 11 dict begin /FontInfo 10 dict dup begin /version (002.004) readonly def /Notice (Copyright (c) 1989, 1990, 1991 Adobe Systems Incorporated. All rights reserved.) readonly def /FullName (CourierNarrow) readonly def /FamilyName (CourierNarrow) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /CourierNarrow def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /grave put dup 1 /acute put dup 2 /circumflex put dup 3 /tilde put dup 4 /dieresis put dup 5 /hungarumlaut put dup 6 /ring put dup 7 /caron put dup 8 /breve put dup 9 /macron put dup 10 /dotaccent put dup 11 /cedilla put dup 12 /ogonek put dup 13 /quotesinglbase put dup 14 /guilsinglleft put dup 15 /guilsinglright put dup 16 /quotedblleft put dup 17 /quotedblright put dup 18 /quotedblbase put dup 19 /guillemotleft put dup 20 /guillemotright put dup 21 /endash put dup 22 /emdash put dup 23 /cwm put dup 24 /perthousand put dup 25 /dotlessi put dup 26 /dotlessj put dup 27 /ff put dup 28 /fi put dup 29 /fl put dup 30 /ffi put dup 31 /ffl put dup 32 /space put dup 33 /exclam put dup 34 /quotedbl put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /backslash put dup 93 /bracketright put dup 94 /asciicircum put dup 95 /underscore put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /braceleft put dup 124 /bar put dup 125 /braceright put dup 126 /asciitilde put dup 127 /sfthyphen put dup 128 /Abreve put dup 129 /Aogonek put dup 130 /Cacute put dup 131 /Ccaron put dup 132 /Dcaron put dup 133 /Ecaron put dup 134 /Eogonek put dup 135 /Gbreve put dup 136 /Lacute put dup 137 /Lcaron put dup 138 /Lslash put dup 139 /Nacute put dup 140 /Ncaron put dup 141 /Eng put dup 142 /Ohungarumlaut put dup 143 /Racute put dup 144 /Rcaron put dup 145 /Sacute put dup 146 /Scaron put dup 147 /Scedilla put dup 148 /Tcaron put dup 149 /Tcedilla put dup 150 /Uhungarumlaut put dup 151 /Uring put dup 152 /Ydieresis put dup 153 /Zacute put dup 154 /Zcaron put dup 155 /Zdotaccent put dup 156 /IJ put dup 157 /Idotaccent put dup 158 /dcroat put dup 159 /section put dup 160 /abreve put dup 161 /aogonek put dup 162 /cacute put dup 163 /ccaron put dup 164 /dcaron put dup 165 /ecaron put dup 166 /eogonek put dup 167 /gbreve put dup 168 /lacute put dup 169 /lcaron put dup 170 /lslash put dup 171 /nacute put dup 172 /ncaron put dup 173 /eng put dup 174 /ohungarumlaut put dup 175 /racute put dup 176 /rcaron put dup 177 /sacute put dup 178 /scaron put dup 179 /scedilla put dup 180 /tcaron put dup 181 /tcedilla put dup 182 /uhungarumlaut put dup 183 /uring put dup 184 /ydieresis put dup 185 /zacute put dup 186 /zcaron put dup 187 /zdotaccent put dup 188 /ij put dup 189 /exclamdown put dup 190 /questiondown put dup 191 /sterling put dup 192 /Agrave put dup 193 /Aacute put dup 194 /Acircumflex put dup 195 /Atilde put dup 196 /Adieresis put dup 197 /Aring put dup 198 /AE put dup 199 /Ccedilla put dup 200 /Egrave put dup 201 /Eacute put dup 202 /Ecircumflex put dup 203 /Edieresis put dup 204 /Igrave put dup 205 /Iacute put dup 206 /Icircumflex put dup 207 /Idieresis put dup 208 /Eth put dup 209 /Ntilde put dup 210 /Ograve put dup 211 /Oacute put dup 212 /Ocircumflex put dup 213 /Otilde put dup 214 /Odieresis put dup 215 /OE put dup 216 /Oslash put dup 217 /Ugrave put dup 218 /Uacute put dup 219 /Ucircumflex put dup 220 /Udieresis put dup 221 /Yacute put dup 222 /Thorn put dup 223 /SS put dup 224 /agrave put dup 225 /aacute put dup 226 /acircumflex put dup 227 /atilde put dup 228 /adieresis put dup 229 /aring put dup 230 /ae put dup 231 /ccedilla put dup 232 /egrave put dup 233 /eacute put dup 234 /ecircumflex put dup 235 /edieresis put dup 236 /igrave put dup 237 /iacute put dup 238 /icircumflex put dup 239 /idieresis put dup 240 /eth put dup 241 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b(28/04/1998;)g(12:50;)h(p.2)p eop %%Page: 3 3 3 2 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)161 b Fz(3)301 404 y(Finally)m(,)15 b(if)h(we)h(ha)o(v)o(e)f(some)g(additional)e(information)h(on)h(the)g (data)g(\(and)g(hence)243 457 y(enrich)h(the)h(speci\002cation\),)f(we) h(can)g(reuse)g(and)g(shorten)f(the)g(proof,)h(resulting)e(in)243 511 y(a)h(dif)o(ferent)g(\(possibly)f(more)h(ef)o(\002cient\))h (algorithm)e(which)h(in)g(general)g(cannot)g(be)243 565 y(obtained)10 b(by)g(optimizing)g(the)h(original)f(program.)520 730 y FC(2.)30 b Fz(L)r FA(O)r(G)r(I)r(C)15 b(F)r(O)r(R)g(C)r(O)r(M)r (P)r(U)r(T)o(A)r(B)s(L)r(E)i(F)r(U)r(N)r(C)r(T)r(I)r(O)r(N)r(A)r(L)r(S) 243 840 y Fz(The)h(theory)f(implemented)h(in)h(M)r FA(I)r(N)r(L)r(O)r (G)j Fz(is)17 b(He)o(yting)h(arithmetic)f(in)h(\002nite)g(types)243 894 y(\(T)n(roelstra)12 b(and)g(v)o(an)h(Dalen,)f(1988\),)g(arranged)h (around)f(a)h(kernel)f(of)g(minimal)g(logic.)243 947 y(Classical)d(arithmetic)g(is)g(contained)f(in)h(it)g(as)g(the)h (fragment)f(without)f(the)h(constructi)o(v)o(e)243 1001 y(quanti\002er)g Fv(9)452 985 y Fp(\003)471 1001 y Fz(.)h(From)g(the)f (classical)g(point)f(of)i(vie)o(w)f(this)f(fragment)i(is)f(suf)o (\002cient)g(since)243 1055 y(e)o(xistential)g(statements)i(can)g(be)g (e)o(xpressed)g(via)g Fv(:8:)p Fz(.)243 1165 y(2.1.)32 b Fx(The)11 b(term-calculus)243 1263 y Fz(As)h(objects)f(of)h (computational)e(interest)h(we)h(choose)g(the)g(terms)g(of)g(G\366del') n(s)f(System)243 1317 y Fx(T)6 b Fz(,)15 b(coming)f(with)f(types)h(\(e) o(v)o(en)g(if)h(the)f(type)g(information)f(is)h(mostly)f(suppressed)g (in)243 1371 y(this)h(te)o(xt\).)h(Besides)f(the)h(natural)f(numbers,)h (the)g(booleans)e(are)j(prede\002ned)f(ground)243 1425 y(types.)243 1505 y Fx(T)m(ypes.)91 b Fn(b)q(o)q(ole)10 b Fv(j)g Fn(nat)h Fv(j)f Fy(r)c Fv(\002)g Fy(s)11 b Fv(j)f Fy(r)g Fv(!)g Fy(s)p Fz(.)243 1584 y Fx(Constants.)89 b Fn(true)11 b Fv(j)e Fn(false)i Fv(j)e Fn(0)h Fv(j)g Fn(S)g Fv(j)g Fn(R)895 1593 y Fm(b)q(o)q(ole)p Fk(;)p Fo(r)1014 1584 y Fv(j)g Fn(R)1066 1591 y Fm(nat)p Fk(;)p Fo(r)1144 1584 y Fz(.)243 1664 y(The)f(successor)h Fn(S)g Fz(is)f(of)h(type)f Fn(nat)h Fv(!)f Fn(nat)p Fz(,)i(the)e(types)g(of)h (the)f(recursors)h(\(for)g(each)g(type)243 1718 y Fy(r)p Fz(\))k(will)f(be)h(made)h(precise)f(later)n(.)g(From)h(typed)e(v)o (ariables)g Fx(x)i Fz(and)f(constants)e Fx(c)j Fz(terms)243 1772 y(are)d(b)o(uilt)d(according)i(to)g(the)g(standard)f(of)h(simply)f (typed)h Fy(l)p Fz(-calculus.)243 1852 y Fx(T)l(erms.)90 b(x)11 b Fv(j)f Fx(c)g Fv(j)g(h)p Fx(r)l Fw(;)5 b Fx(s)p Fv(i)j(j)i Fy(p)703 1859 y Fq(0)721 1852 y Fx(r)h Fv(j)f Fy(p)798 1859 y Fq(1)816 1852 y Fx(r)h Fv(j)f Fy(l)p Fx(xr)h Fv(j)f Fx(r)q(s)p Fz(.)243 1932 y(General)j(bracketing)f(con)n (v)o(entions:)f(products)g(associate)i(to)f(the)h(left,)h(arro)o(ws)e (to)h(the)243 1986 y(right.)c(The)h(scope)g(of)g(a)g(binder)f(is)h(the) f(maximal)h(\(minimal\))g(possible)f(one,)h(whene)o(v)o(er)243 2040 y(the)h(bound)f(v)o(ariable)g(is)h(\(not\))g(follo)o(wed)f(by)g(a) i(dot.)301 2094 y(T)m(erms)f(are)h(computed)f(according)f(to)h(the)243 2179 y Fx(Con)n(versions.)250 b Fy(p)756 2186 y Fq(0)774 2179 y Fv(h)p Fx(r)l Fw(;)5 b Fx(s)p Fv(i)48 b(7!)j Fx(r)l Fw(;)731 2244 y Fy(p)756 2251 y Fq(1)774 2244 y Fv(h)p Fx(r)l Fw(;)5 b Fx(s)p Fv(i)48 b(7!)j Fx(s)p Fw(;)737 2309 y Fu(\()p Fy(l)p Fx(x)5 b(r)q Fu(\))p Fx(s)48 b Fv(7!)j Fx(r)1022 2316 y Fl(x)1038 2309 y Fu([)p Fx(s)p Fu(])p Fw(;)604 2373 y Fn(R)633 2382 y Fm(b)q(o)q(ole)o Fk(;)p Fo(r)742 2373 y Fx(r)q(s)5 b Fn(true)49 b Fv(7!)i Fx(r)l Fw(;)595 2438 y Fn(R)624 2446 y Fm(b)q(o)q(ole)p Fk(;)p Fo(r)733 2438 y Fx(r)q(s)5 b Fn(false)49 b Fv(7!)i Fx(s)p Fw(;)693 2503 y Fn(R)722 2510 y Fm(nat)p Fk(;)p Fo(r)800 2503 y Fx(r)q(s)p Fn(0)d Fv(7!)j Fx(r)l Fw(;)641 2568 y Fn(R)670 2575 y Fm(nat)p Fk(;)p Fo(r)748 2568 y Fx(r)q(s)p Fu(\()p Fn(S)o Fx(t)s Fu(\))c Fv(7!)k Fx(s)o(t)s Fu(\()p Fn(R)1084 2575 y Fm(nat)o Fk(;)p Fo(r)1160 2568 y Fx(r)q(s)o(t)s Fu(\))p Fw(;)911 3108 y Ft(Minlog7.tex;)18 b(28/04/1998;)g(12:50;)h(p.3)p eop %%Page: 4 4 4 3 bop 243 277 a Fz(4)85 b FC(BENL,)11 b(BERGER,)g(SCHWICHTENBERG,)f (SEISENBERGER,)h(ZUBER)243 404 y Fz(where)g Fx(r)383 411 y Fl(x)400 404 y Fu([)p Fx(s)p Fu(])f Fz(\(or)i Fx(r)q Fu([)p Fx(s)p Fu(])e Fz(for)h(short\))f(means)i Fx(s)f Fz(substituted)e(for)i Fx(x)h Fz(in)e Fx(r)q Fz(.)i(These)f(rules)g (sho)o(w)243 457 y(that)17 b Fn(R)358 466 y Fm(b)q(o)q(ole)p Fk(;)p Fo(r)467 457 y Fx(r)q(s)o(t)j Fz(\()p Fx(r)q Fz(,)e Fx(s)f Fz(of)h(type)f Fy(r)p Fz(\))h(corresponds)f(to)g Fi(if)f Fx(t)21 b Fi(then)e Fx(r)f Fi(else)g Fx(s)p Fz(,)g(whereas)243 511 y Fn(R)272 518 y Fm(nat)p Fk(;)p Fo(r)350 511 y Fx(r)q(s)o(t)g Fz(is)f(the)g(usual)f(recursor)h(with)f(base)i Fx(r)g Fz(\(of)f(type)f Fy(r)p Fz(\))i(and)f(loop)f Fx(s)h Fz(\(of)g(type)243 565 y Fn(nat)c Fv(!)f Fy(r)g Fv(!)h Fy(r)p Fz(\))i(to)f(be)h(passed)f (through)e Fx(t)s Fz(-times.)i(The)h(\002rst)g(three)f(rules)h(are)g (called)243 619 y Fy(b)p Fz(-con)n(v)o(ersions,)10 b(the)h(others)g(we) g(call)g Fn(R)p Fz(-con)n(v)o(ersions.)301 673 y(As)d(is)g(well)g(kno)o (wn)f(\(T)l(ait,)i(1967;)e(T)n(roelstra)i(and)f(v)o(an)g(Dalen,)h (1988\),)f(the)g(re)o(writing)243 727 y(procedure)j(stops)e(for)i(each) h(term)f(with)f(a)i(unique)d Fx(normal)h(form)p Fz(.)g(Hence)h(the)g (congru-)243 781 y(ence)e(gi)o(v)o(en)g(by)g(the)f(con)n(v)o(ersions)g (is)h(decidable.)g(In)g(our)g(setting,)f(congruent)g(terms)h(are)243 835 y(identi\002ed.)i(This)g Fx(P)l(oincar\351-equality)e Fz(mostly)i(sa)o(v)o(es)h(us)g(from)g(needing)f(an)h(equality-)243 889 y(calculus.)d(\(Ho)o(we)o(v)o(er)g(the)h(user)f(importing)g (self-de\002ned)g(con)n(v)o(ersions)f(is)h(responsible)243 943 y(for)i(their)g(termination)f(together)g(with)g Fy(b)i Fz(and)f Fn(R)p Fz(!\))243 1049 y(2.2.)32 b Fx(The)11 b(logical)e(fr)o(ame)o(work)243 1144 y Fz(No)o(w)j(the)g(basis)g(is)g (ready)h(to)f(introduce)g(the)g(objects)g(of)g(interest)g(for)h (theorem)g(pro)o(v-)243 1197 y(ing:)8 b(formulas.)i(It)g(is)f(crucial)h (b)o(ut)f(ne)o(v)o(ertheless)f(easy)i(to)f(distinguish)e(between)i(a)h (term)242 1251 y Fx(t)i Fz(of)d(type)f Fn(b)q(o)q(ole)i Fz(and)f(the)f(atomic)h(formula)g Fn(atom)o Fu(\()o Fx(t)s Fu(\))f Fz(saying)g(`)o Fx(t)14 b Fz(is)d Fn(true)q Fz('.)e(In)g (particular)243 1305 y(we)i(ha)o(v)o(e)h(the)f(atomic)g(formulas)734 1391 y Fv(>)50 b Fz(:)p Fu(=)g Fn(atom)o Fu(\()p Fn(true)p Fu(\))p Fw(;)734 1456 y Fv(?)g Fz(:)p Fu(=)g Fn(atom)o Fu(\()p Fn(false)p Fu(\))p Fw(:)243 1546 y Fz(From)20 b(the)g(atomic)g(formulas)f(\(which)h(are)g(decidable)f(since)h(their)f (characteristic)243 1600 y(terms)11 b(are)h(computable\))f(we)g(b)o (uild)f(the)243 1677 y Fx(F)-5 b(ormulas.)89 b Fn(atom)o Fu(\()o Fx(t)s Fu(\))9 b Fv(j)h Fx(A)c Fv(^)g Fx(B)k Fv(j)g Fx(A)g Fv(!)g Fx(B)g Fv(j)g(8)p Fx(x)5 b(A)10 b Fv(j)g(9)1118 1660 y Fp(\003)1137 1677 y Fx(x)5 b(A)p Fz(,)243 1753 y(where)12 b Fv(9)391 1737 y Fp(\003)423 1753 y Fz(is)g(the)g Fx(str)n(ong)f Fz(or)h Fx(constructive)f Fz(e)o(xistential)g(quanti\002er)h(\227)h(to)f(distinguish)243 1807 y(from)f(the)g(classical)g Fv(9)g Fz(which)g(can)g(be)h(vie)o(wed) e(as)h(de\002ned)g(by)772 1902 y Fv(:)p Fx(A)50 b Fz(:)p Fu(=)f Fx(A)10 b Fv(!)h(?)p Fw(;)751 1967 y Fv(9)p Fx(x)5 b(A)51 b Fz(:)p Fu(=)e Fv(:8)p Fx(x)5 b Fv(:)p Fx(A)p Fw(:)243 2058 y Fz(F)o(ormulas)k(without)f Fv(9)597 2041 y Fp(\003)626 2058 y Fz(are)i(called)f Fx(classical)p Fz(.)f(F)o(or)i(the)f(moment)h(we)g(will)e(concentrate)243 2112 y(on)j(the)f(classical)h(part.)g(W)l(e)h(will)e(meet)i Fv(9)903 2095 y Fp(\003)933 2112 y Fz(later)f(again.)301 2188 y(In)f(the)f(spirit)g(of)g(the)h(Curry-Ho)o(ward-correspondence)e (we)i(re)o(gard)g(proofs)g(in)f(\(the)243 2242 y Fv(^)p Fw(;)c Fv(!)p Fw(;)g Fv(8)p Fz(-fragment)12 b(of\))f(natural)g (deduction)e(as)j(terms)f(in)g(a)g(typed)g Fy(l)p Fz(-calculus,)f (where)243 2296 y(the)g(type)g(of)g(a)h(proof)f(term)h(corresponds)e (to)h(\(more)h(precisely:)e Fx(is)p Fz(\))h(its)g(end)g(formula)g(in) 243 2350 y(natural)g(deduction,)g(and)h(the)g(types)g(of)g(its)f(free)i (v)o(ariables)e Fx(ar)n(e)i Fz(its)e(free)i(assumptions.)301 2404 y(F)o(or)i(the)f(sake)g(of)g(clarity)f(we)i(annotate)e(types)h(as) g(superscripts)e(in)i(the)g(de\002nition)243 2458 y(of)j(proofs.)g(T)l (o)h(start)f(with,)f(an)i(assumption)e Fx(A)h Fz(is)g(introduced)f(by)h (an)g(assumption-)243 2512 y(v)o(ariable)11 b Fx(u)423 2496 y Fl(A)445 2512 y Fz(,)i(whereas)f(an)h(axiom)f Fx(A)g Fz(is)f(introduced)g(by)h(a)h(constant)d Fx(c)1375 2496 y Fl(A)1398 2512 y Fz(.)j(F)o(or)f(the)g(mo-)243 2566 y(ment)f(our)g(only)f(constant)g(is)h(the)g Fx(truth)e(axiom)i Fn(T)l(ruth)1109 2547 y Fp(>)1137 2566 y Fz(.)911 3108 y Ft(Minlog7.tex;)18 b(28/04/1998;)g(12:50;)h(p.4)p eop %%Page: 5 5 5 4 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)161 b Fz(5)243 407 y Fx(Pr)n(oofs.)114 b(u)512 391 y Fl(A)544 407 y Fv(j)10 b Fx(c)587 391 y Fl(A)620 407 y Fv(j)f(h)p Fx(M)700 391 y Fl(A)722 407 y Fw(;)c Fx(N)773 391 y Fl(B)795 407 y Fv(i)813 391 y Fl(A)p Fp(^)p Fl(B)888 407 y Fv(j)k Fu(\()p Fy(p)953 414 y Fq(0)972 407 y Fx(M)1012 391 y Fl(A)p Fp(^)p Fl(B)1076 407 y Fu(\))1094 391 y Fl(A)1126 407 y Fv(j)h Fu(\()p Fy(p)1192 414 y Fq(1)1210 407 y Fx(M)1250 391 y Fl(A)p Fp(^)p Fl(B)1315 407 y Fu(\))1333 391 y Fl(B)1365 407 y Fv(j)489 472 y Fu(\()p Fy(l)p Fx(u)555 456 y Fl(A)582 472 y Fx(M)622 456 y Fl(B)644 472 y Fu(\))662 456 y Fl(A)p Fp(!)p Fl(B)748 472 y Fv(j)f Fu(\()p Fx(M)828 456 y Fl(A)p Fp(!)p Fl(B)903 472 y Fx(N)936 456 y Fl(A)959 472 y Fu(\))977 456 y Fl(B)1009 472 y Fv(j)h Fu(\()p Fy(l)p Fx(x)1095 456 y Fo(r)1115 472 y Fx(M)1155 456 y Fl(A)1177 472 y Fu(\))1195 456 y Fp(8)p Fl(x)1229 444 y Fh(r)1244 456 y Fl(A)1276 472 y Fv(j)g Fu(\()p Fx(M)1357 456 y Fp(8)p Fl(x)1391 444 y Fh(r)1405 456 y Fl(A)1431 472 y Fx(t)1447 456 y Fo(r)1467 472 y Fu(\))1485 456 y Fl(A)1505 461 y Fg(x)1518 456 y Fj([)o Fl(t)r Fj(])1550 472 y Fw(:)243 561 y Fz(The)h(usual)f(restriction)g(on)h(the)g(object-v)o(ariable)e Fx(x)j Fz(applies)e(when)h(b)o(uilding)e Fy(l)p Fx(x)c(M)r Fz(.)243 679 y(2.3.)32 b Fx(Reasoning)9 b(in)i(classical)e(and)i (intuitioni)o(sti)o(c)e(arithmetic)243 784 y Fz(As)i(we)h(ha)o(v)o(e)f (said)g(nothing)f(special)h(about)f Fv(?)i Fz(so)f(far)n(,)h(we)g(ha)o (v)o(e)g(a)g(calculus)e(for)i(mini-)243 837 y(mal)f(logic.)g(T)l(o)g(e) o(xpand)g(it)g(to)f(arithmetic)h(and)g(incorporate)g(constructi)o(vity) d(we)j(add)243 929 y Fx(Induction)e(axioms.)115 b Fn(Ind)744 941 y Fp(8)p Fl(x)782 931 y Ff(b)q(o)q(ole)843 941 y Fl(A)915 929 y Fz(:)49 b Fx(A)1005 936 y Fl(x)1022 929 y Fu([)p Fn(true)p Fu(])10 b Fv(!)g Fx(A)1216 936 y Fl(x)1233 929 y Fu([)p Fn(false)p Fu(])g Fv(!)g(8)p Fx(x)5 b(A)p Fw(;)684 993 y Fn(Ind)744 1002 y Fp(8)p Fl(x)782 993 y Ff(nat)819 1002 y Fl(A)915 993 y Fz(:)49 b Fx(A)1005 1000 y Fl(x)1022 993 y Fu([)p Fn(0)p Fu(])9 b Fv(!)i(8)p Fx(x)p Fu(\()p Fx(A)f Fv(!)g Fx(A)1320 1000 y Fl(x)1337 993 y Fu([)p Fn(S)p Fx(x)p Fu(]\))f Fv(!)i(8)p Fx(x)5 b(A)p Fw(:)243 1094 y Fv(9)268 1078 y Fp(\003)298 1094 y Fx(axioms.)115 b Fv(9)579 1077 y Fp(\003)p Fj(+)579 1109 y Fl(x)p Fk(;)t Fl(A)712 1094 y Fz(:)50 b Fv(8)p Fx(x)p Fw(:)5 b Fx(A)10 b Fv(!)g(9)956 1078 y Fp(\003)975 1094 y Fx(x)5 b(A)p Fw(;)554 1161 y Fv(9)579 1143 y Fp(\003\000)579 1175 y Fl(x)p Fk(;)t Fl(A)p Fk(;)t Fl(B)712 1161 y Fz(:)50 b Fv(9)800 1144 y Fp(\003)819 1161 y Fx(x)5 b(A)10 b Fv(!)g(8)p Fx(x)p Fu(\()p Fx(A)h Fv(!)f Fx(B)p Fu(\))g Fv(!)g Fx(B)p Fw(;)243 1249 y Fz(where)k(in)f(the)g(latter)g Fx(x)h Fz(must)f(not)g(occur)h(free)g(in)f Fx(B)p Fz(.)h(The)f (notation)f Fx(M)r Fz(:)5 b Fx(A)13 b Fz(is)g(synon)o(y-)243 1302 y(mous)e(to)f Fx(M)439 1286 y Fl(A)461 1302 y Fz(.)301 1388 y(W)l(e)h(formulated)e(induction)f(and)i(e)o(xistence)g(as)g (axioms)f(since)h(we)g(re)o(gard)h(them)f(as)243 1442 y(e)o(xtensions)i(of)i(our)f(logical)g(kernel.)g(Y)-5 b(ou)13 b(might)g(miss)g(ne)o(gation)g(axioms.)g(But)h(the)o(y)243 1496 y(are)e(already)f(present:)301 1626 y Fx(Theor)n(em.)h Fz(\()p Fx(e)o(x-falso-quodlibet)p Fz(.\))c Fv(?)j(!)f Fx(A)h Fz(is)g(pro)o(v)o(able.)301 1757 y Fx(Pr)n(oof)o(.)17 b Fz(By)h(induction)d(on)i Fx(A)p Fz(.)h Fx(Base)o(.)g Fz(W)l(e)g(ha)o(v)o(e)g(to)f(sho)o(w)g Fv(?)d(!)g Fn(atom)o Fu(\()o Fx(t)s Fu(\))p Fz(.)j(This)243 1811 y(is)d(an)h(instance)f(of)h Fv(8)p Fx(x)p Fw(:)5 b Fv(?)13 b(!)g Fn(atom)o Fu(\()p Fx(x)p Fu(\))p Fz(.)i(T)l(o)g(pro)o(v)o(e)g(the)g(latter)n(,)g(we)g (use)g(the)g(boolean)243 1865 y(induction)i(axiom)i Fn(Ind)627 1875 y Fp(8)p Fl(x)p Fk(:)t Fp(?!)p Fm(atom)m Fj(\()p Fl(x)p Fj(\))850 1865 y Fz(.)h(It)f(forces)h(us)f(to)g(pro)o(v)o(e)g (both,)g Fv(?)c(!)g(>)20 b Fz(and)243 1919 y Fv(?)10 b(!)h(?)p Fz(.)j(Using)f(an)h(assumption)e Fx(u)h Fz(of)h(type)f Fv(?)p Fz(,)i(proofs)e(are)h(simply)f Fy(l)p Fx(u)p Fn(T)l(ruth)h Fz(resp.)243 1972 y Fy(l)p Fx(uu)p Fz(.)e(Summing)f(up,)h Fn(Ind)654 1983 y Fp(8)p Fl(x)p Fk(:)t Fp(?!)p Fm(atom)m Fj(\()p Fl(x)p Fj(\))882 1972 y Fy(l)p Fx(u)p Fn(T)l(ruth)6 b Fy(l)p Fx(uu)t(t)14 b Fz(is)d(a)h(proof)g(of)g Fv(?)e(!)h Fn(atom)o Fu(\()o Fx(t)s Fu(\))p Fz(.)243 2026 y Fx(Step.)16 b Fz(By)g(induction)e(hypothesis)g(we)j(ha)o(v)o(e)f(a)h(proof)f Fx(M)i Fz(of)e(type)g Fv(?)d(!)h Fx(A)p Fz(.)i(W)l(e)h(get)243 2080 y Fv(?)9 b(!)g(9)366 2064 y Fp(\003)385 2080 y Fx(x)c(A)11 b Fz(as)f(the)f(type)h(of)f Fy(l)p Fx(u)p Fw(:)c Fv(9)789 2063 y Fp(\003)p Fj(+)789 2095 y Fl(x)p Fk(;)t Fl(A)839 2080 y Fx(x)p Fu(\()p Fx(M)r(u)p Fu(\))p Fz(.)10 b(The)f(other)h(cases) g(of)g(the)f(induction)f(step)243 2134 y(are)k(easy)m(,)f(too.)301 2265 y(A)i(similar)g(consideration)e(sho)o(ws)i(that)f Fx(stability)f Fv(::)p Fx(A)h Fv(!)g Fx(A)h Fz(is)g(pro)o(v)o(able)g (for)g(all)243 2319 y(classical)c(formulas)g Fx(A)p Fz(.)h(Ne)o(v)o (ertheless)f(we)h(often)f(use)g(e)o(x-falso-quodlibet)e(and)j(stabil-) 243 2373 y(ity)g(as)h(axioms)g(in)g(order)g(to)g(a)o(v)o(oid)f(myriads) h(of)g(boolean)g(inductions.)301 2458 y(Summing)h(up,)h(with)e(the)i (appropriate)e(choice)i(of)f(a)h(term)g(calculus)e(we)i(get)f(He)o(y-) 243 2512 y(ting)i(arithmetic,)g(and)h(restriction)f(to)g(the)h (classical)f(fragment)h(yields)f(Peano)h(arith-)243 2566 y(metic.)911 3108 y Ft(Minlog7.tex;)j(28/04/1998;)g(12:50;)h(p.5)p eop %%Page: 6 6 6 5 bop 243 277 a Fz(6)85 b FC(BENL,)11 b(BERGER,)g(SCHWICHTENBERG,)f (SEISENBERGER,)h(ZUBER)243 404 y Fz(2.4.)32 b Fx(Semantics)243 513 y Fz(The)12 b(denotational)e(semantics)i(kept)f(in)h(the)g(design)f (of)j(M)r FA(I)r(N)r(L)r(O)r(G)i Fz(is)c(Scott')n(s)f(domain)243 567 y(theory)j(\(Scott,)i(1982\).)f(A)g(v)o(ariable)g(is)g(supposed)f (to)h(denote)g(a)g(partial)g(continuous)243 621 y(functional)e(of)i (its)f(type,)g(whereas)h(a)g(program)g(constant)f(is)g(interpreted)g (by)g(a)h(com-)243 675 y(putable)h(functional.)h(This)f(is)h (re\003ected)h(in)f(our)h(formalism)f(where)h(we)g(can)g(work)243 729 y(with)9 b(possibly)f(unde\002ned)i(objects)g(\(cf.)h(section)f (4\).)h(Since)f(in)g(some)h(applications)d(no)243 783 y(partial)g(objects)h(appear)n(,)h(we)g(in)f(addition)e(allo)o(w)i(a)g (second)g(kind)g(of)g(v)o(ariables)g(ranging)243 837 y(only)h(o)o(v)o(er)h(the)g(subclass)e(of)i(total)f(functionals.)f(F)o (or)j(the)e(theory)g(of)h(totality)e(in)h(higher)243 891 y(types)f(we)h(refer)g(to)g(\(Ber)o(ger)n(,)h(1993b\).)e (Computability)f(of)i(the)f(constants)g(implies)f(that)243 945 y(atomic)16 b(formulas)g(are)h(decidable)e(and)h(e)o(xtracted)g (programs)h(are)f(e)o(x)o(ecutable)h(\(See)243 998 y(also)10 b(section)g(5\).)695 1197 y FC(3.)30 b Fz(T)r FA(H)r(E)15 b Fz(M)r FA(I)r(N)r(L)r(O)r(G)h(S)r(Y)r(S)r(T)r(E)r(M)243 1323 y Fz(The)e(M)r FA(I)r(N)r(L)r(O)r(G)j Fz(system)c(is)f (implemented)h(in)h(S)r FA(C)r(H)r(E)r(M)r(E)q Fz(,)j(a)d(L)r FA(I)r(S)r(P)q Fz(-dialect.)g(Its)f(size)g(is)243 1377 y(about)g(1/2)h(MB,)h(which)f(is)g(rather)g(small)g(compared)h(with)e (other)h(theorem)g(pro)o(v)o(ers.)243 1431 y(It)d(is)f(an)h(interacti)o (v)o(e)f(system)h(which)f(ho)o(we)o(v)o(er)g(has)h(quite)f(elaborate)h (components)f(for)243 1485 y(automated)g(proof)h(search)h(and)f(term)g (re)o(writing.)243 1611 y(3.1.)32 b Fx(Inter)o(active)10 b(theor)n(em)h(pr)n(oving)243 1720 y Fz(F)o(or)g(pro)o(ving)e(a)i (theorem)f(in)h(M)r FA(I)r(N)r(L)r(O)r(G)k Fz(the)10 b(user)g(\002rst)h(has)f(to)g(specify)g(the)g(frame)o(work)243 1774 y(consisting)d(of)j(a)g(language)g(and)f(axioms.)h(This)f(can)h (be)g(done)f(by)h(declaring)f(ne)o(w)h(v)o(ari-)243 1828 y(ables)k(and)h(program)g(constants)f(of)h(possibly)e(user)o (-de\002ned)i(types.)f(W)l(e)i(allo)o(w)e(con-)243 1882 y(stants)g(parametric)h(in)g(types)f(and,)h(corresponding)f(to)g(that,) h(axioms)g(parametric)g(in)243 1936 y(formulas.)g(Canonical)h(e)o (xamples)f(are)i(the)e(recursion)g(operator)g(and)g(the)h(induction)243 1990 y(scheme.)f(Further)g(axioms)g(or)g(lemmas)g(to)f(be)h(pro)o(v)o (en)g(later)g(can)g(be)g(introduced)f(as)243 2044 y(so-called)f(global) g(assumptions.)f(The)i(user)g(may)h(take)e(an)o(ything)g(\(hopefully)g (v)o(alid\))243 2098 y(for)f(granted;)f(as)i(long)e(as)h(a)g(global)f (assumption)g(contains)g(no)h(computational)e(infor)o(-)243 2152 y(mation,)h(it)f(will)g(not)h(af)o(fect)h(program)f(e)o (xtraction\227as)g(we)g(will)f(see)i(in)e(section)g(5.)301 2243 y(In)17 b(the)h(sequel)e(we)i(want)f(to)g(gi)o(v)o(e)g(an)g(idea)h (ho)o(w)e(to)h(work)g(with)f(the)j(M)r FA(I)r(N)r(L)r(O)r(G)243 2297 y Fz(system.)11 b(Let)g(us)f(be)o(gin)h(with)f(a)i(small)f(e)o (xample:)594 2405 y Fu(\()p Fv(8)p Fx(x)p Fw(:)5 b Fx(A)p Fu([)p Fx(x)p Fu(])k Fv(!)i Fx(B)p Fu([)p Fx(x)p Fu(]\))e Fv(!)h(8)p Fx(xA)p Fu([)p Fx(x)p Fu(])g Fv(!)h(8)p Fx(xB)p Fu([)p Fx(x)p Fu(])p Fw(:)243 2512 y Fz(W)l(e)18 b(\002rst)g(present)f (a)h(natural)f(deduction)f(proof)i(of)f(this)g(formula)h(in)f(familiar) g(tree)243 2566 y(notation:)911 3108 y Ft(Minlog7.tex;)h(28/04/1998;)g (12:50;)h(p.6)p eop %%Page: 7 7 7 6 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)161 b Fz(7)529 442 y Fx(u)9 b Fz(:)h Fv(8)p Fx(x)p Fw(:)5 b Fx(A)p Fu([)p Fx(x)p Fu(])10 b Fv(!)g Fx(B)p Fu([)p Fx(x)p Fu(])93 b Fx(x)p 512 463 478 2 v 645 505 a(A)p Fu([)p Fx(x)p Fu(])9 b Fv(!)i Fx(B)p Fu([)p Fx(x)p Fu(])1066 442 y Fx(v)g Fz(:)e Fv(8)p Fx(xA)p Fu([)p Fx(x)p Fu(])93 b Fx(x)p 1050 463 319 2 v 1172 505 a(A)p Fu([)p Fx(x)p Fu(])p 628 527 635 2 v 909 569 a Fx(B)p Fu([)p Fx(x)p Fu(])811 606 y Fy(l)p Fx(x)p 869 591 153 2 v 886 633 a Fv(8)p Fx(xB)p Fu([)p Fx(x)p Fu(])719 670 y Fy(l)p Fx(v)p 777 654 337 2 v 794 697 a Fv(8)p Fx(xA)p Fu([)p Fx(x)p Fu(])9 b Fv(!)i(8)p Fx(xB)p Fu([)p Fx(x)p Fu(])529 734 y Fy(l)p Fx(u)p 589 718 713 2 v 605 760 a Fu(\()p Fv(8)p Fx(x)p Fw(:)5 b Fx(A)p Fu([)p Fx(x)p Fu(])10 b Fv(!)g Fx(B)p Fu([)p Fx(x)p Fu(]\))f Fv(!)i(8)p Fx(xA)p Fu([)p Fx(x)p Fu(])f Fv(!)g(8)p Fx(xB)p Fu([)p Fx(x)p Fu(])301 868 y Fz(Belo)o(w)j(the)g(protocol)f(of)i(the)f (interacti)o(v)o(e)f(proof)h(is)g(sho)o(wn.)g(The)g(lines)g(in)f (paren-)243 922 y(theses)i(are)i(the)g(commands)f(gi)o(v)o(en)g(by)g (the)h(M)r FA(I)r(N)r(L)r(O)r(G)k Fz(user;)15 b(those)f(starting)g (with)g(a)243 976 y(semicolon)f(are)j(the)e(responses)f(of)i(the)f (system,)g(e.g.)i(telling)d(us)h(current)g(goals)g(and)243 1030 y(actual)d(conte)o(xts)f(\(some)h(responses)f(are)i(dropped)f (here\).)301 1084 y(W)l(e)k(start)g(with)e(introducing)g(an)i (arbitrary)f(type)h Fe(arb)f Fz(and)h(declare)g(a)g(v)o(ariable)f Fx(x)243 1138 y Fz(of)g(type)f Fe(arb)h Fz(as)g(well)g(as)g(two)f (program)h(constants)e Fx(a)i Fz(and)g Fx(b)g Fz(of)g(type)g Fe(arb->boole)p Fz(,)243 1192 y(displayed)g(with)h(capital)g(letters,)g (representing)g(the)h(formulas)f Fx(A)p Fu([)p Fx(x)p Fu(])h Fz(and)f Fx(B)p Fu([)p Fx(x)p Fu(])p Fz(.)h(The)243 1246 y(numerical)11 b(ar)o(gument)g(0)f(in)h(the)f(declaration)g(of)h (the)g(program)g(constants)e(determines)243 1300 y(their)k(de)o(gree)h (of)g(totality)m(.)e(This)h(de)o(gree)h(has)g(impact)g(on)f(the)g(e)o (x)o(ecution)h(of)g(possible)243 1353 y(term)d(re)o(writing)f (discussed)g(belo)o(w)m(.)301 1407 y(After)j(stating)f(the)h(goal)g (the)g(proof)f(is)h(b)o(uilt)f(up)h(in)g(a)g(backward)g(reasoning)f (fash-)243 1461 y(ion.)f(F)o(or)i(conciseness,)e(the)h(system)g (displays)e(the)i(atomic)g(formula)g Fn(atom)o Fu(\()o Fx(t)s Fu(\))f Fz(by)h(the)243 1515 y(boolean)j(term)g Fx(t)j Fz(\(since)e(it)f(is)g(clear)i(from)f(the)f(conte)o(xt)h(that)f (a)h(formula)g(is)f(meant\).)243 1569 y(Hence)c(we)h(read)f Fe(\(A)22 b(x\))11 b Fz(for)g(the)g(internal)g Fe(\(atom)21 b(\(a)h(x\)\))o Fz(.)301 1623 y(The)d(symbols)e Fe(?)p Fz(,)i Fe(?-KERNEL)p Fz(,)e Fe(?1)i Fz(are)g(labels)g(to)f(identify)f (the)i(\(sub\)goals.)f(In)243 1677 y(I.3.15)13 b(the)g(role)f(of)h (these)g(`meta)o(v)o(ariables',)g(which)f(are)i(an)f(of)o(\002cial)g (part)g(of)g(the)f(for)o(-)243 1731 y(mal)h(language)f(of)h(T)l (ypelab,)g(is)g(thoroughly)d(in)n(v)o(estigated,)i(especially)g(with)g (respect)243 1785 y(to)i(the)g(problems)g(arising)g(in)g(conjunction)f (with)g Fy(b)p Fz(-reduction.)h(In)i(M)r FA(I)r(N)r(L)r(O)r(G)q Fz(,)i(ho)o(w-)243 1839 y(e)o(v)o(er)n(,)d(these)g(problems)f(do)g(not) g(occur)n(,)h(since)f(we)h(do)f(not)g(allo)o(w)f(normalization)h(for) 243 1893 y(incomplete)c(proofs.)301 1947 y(W)n(ith)h(the)h(command)h Fe(assume)e Fz(introduction)e(rules)j(for)g(implication)f(and)h(uni)o (v)o(er)o(-)243 2001 y(sal)e(quanti\002cation)g(can)h(be)g(treated)g (\(backwards\))f(in)h(one)g(step;)f(the)g(user)h(only)f(has)h(to)243 2055 y(choose)e(some)h(ne)o(w)f(labels)g(for)h(the)f(assumptions.)f (The)h(command)h Fe(use-with)e Fz(allo)o(ws)243 2109 y(to)i(pro)o(v)o(e)h(the)f(subgoal)f Fe(?-KERNEL:)f Fx(B)p Fu([)p Fx(x)p Fu(])i Fz(by)g(using)g(the)g(assumption)f Fx(u)h Fz(at)h Fx(x)g Fz(pro)o(vided)243 2163 y(one)i(can)g(\002nd)g(a) h(proof)f(for)g(the)g(premise)g Fx(A)p Fu([)p Fx(x)p Fu(])p Fz(.)g(This)g(is)f(the)h(ne)o(w)g(goal)g Fe(?1)p Fz(,)g(which)g(of)243 2217 y(course)e(can)g(be)g(pro)o(v)o(ed)h(with)e (a)h(further)g(application)f(of)h(an)g(elimination)f(rule.)243 2296 y Fe(\(add-ground-)o(type)19 b('arb)j('x\))243 2350 y(\(add-program)o(-con)o(sta)o(nt)e('a)i(\(cons-arrow)d('arb)j ('boole\))f(0)h("A"\))243 2404 y(\(add-program)o(-con)o(sta)o(nt)e('b)i (\(cons-arrow)d('arb)j('boole\))f(0)h("B"\))243 2512 y(\(set-goal)e('?)i(\(parse-formul)o(a)e("\(all)h(x)i(\(a)f(x)g(->)g(b) h(x\)\))44 b(->)924 2566 y(all)22 b(x)h(\(a)f(x\))45 b(->)g(all)21 b(x)i(\(b)f(x\)"\)\))911 3108 y Ft(Minlog7.tex;)c (28/04/1998;)g(12:50;)h(p.7)p eop %%Page: 8 8 8 7 bop 243 277 a Fz(8)85 b FC(BENL,)11 b(BERGER,)g(SCHWICHTENBERG,)f (SEISENBERGER,)h(ZUBER)243 404 y Fe(;?:\(all)21 b(x.A)g(x)i(->)f(B)g (x\))h(->)f(all)g(x)g(A)g(x)h(->)f(all)g(x)g(B)h(x)243 511 y(\(assume)e('u)h('v)g('x\))243 565 y(;ok,)f(under)h(these)f (assumptions)f(we)i(have)f(the)h(new)g(goal)243 619 y(;?-KERNEL:)e(B)i (x)h(from)243 673 y(;)45 b(u:all)21 b(x.A)h(x)h(->)f(B)g(x)243 727 y(;)45 b(v:all)21 b(x)i(A)f(x)243 781 y(;)45 b(x)243 889 y(\(use-with)20 b('u)i('x)g('?1\))243 943 y(;ok,)f(?-KERNEL)g(can)h (be)g(obtained)e(from)243 997 y(;?1:)h(A)i(x)f(from)243 1051 y(;)45 b(u:all)21 b(x.A)h(x)h(->)f(B)g(x)243 1105 y(;)45 b(v:all)21 b(x)i(A)f(x)243 1159 y(;)45 b(x)243 1267 y(\(use-with)20 b('v)i('x\))243 1321 y(;ok,)f(?1)i(is)f(proved.)e (Proof)i(finished.)301 1420 y Fz(No)o(w)8 b(let)h(us)f(ha)o(v)o(e)i(a)f (look)f(at)h(the)g(proof)f(term)i(corresponding)d(to)h(that)g(proof.)h (It)g(was)243 1474 y(generated)g(interacti)o(v)o(ely)g(by)g(editing)f (an)i(incomplete)f(deri)o(v)o(ation)f(term)i(with)e(holes)h(in)243 1528 y(it)i(which)g(are)i(the)e(open)h(goals.)f(In)h(our)g(e)o(xample)g (the)f(\(partial\))h(proof)f(term)i(de)o(v)o(elops)243 1582 y(as)e(follo)o(ws:)542 1661 y Fe(?)542 1727 y Fy(l)p Fx(u)p Fw(;)5 b Fx(v)m Fw(;)g Fx(x)p Fw(:)10 b Fe(?-KERNEL)542 1794 y Fy(l)p Fx(u)p Fw(;)5 b Fx(v)m Fw(;)g Fx(x)p Fw(:)10 b Fx(ux)5 b Fe(?1)542 1860 y Fy(l)p Fx(u)p Fw(;)g Fx(v)m Fw(;)g Fx(x)p Fw(:)10 b Fx(ux)p Fu(\()p Fx(vx)p Fu(\))p Fw(:)301 1939 y Fz(There)j(are)h(also)e(some)h(tools)e(for)i(forward)g (reasoning.)f(F)o(or)h(instance,)g(instead)f(of)243 1993 y(the)e(last)h(two)f(steps)g(in)g(the)h(interacti)o(v)o(e)f(proof,)h (it)f(is)h(also)f(possible)f(to)i(\002rst)g(instantiate)243 2047 y(the)g(assumption)f Fx(v)i Fz(with)e Fx(x)p Fz(.)j(Then)e(only)g (the)g(conte)o(xt)g(changes)g(since)g(we)h(ha)o(v)o(e)g(a)g(ne)o(w)243 2101 y(assumption)c Fx(v)480 2108 y Fl(i)491 2101 y Fz(:)d Fx(A)p Fu(\()p Fx(x)p Fu(\))p Fz(.)k(T)l(o)h(\002nish)e(the)h(proof)h (we)f(can)h(use)f Fx(u)h Fz(at)f Fx(x)h Fz(and)f Fx(v)1359 2108 y Fl(i)1371 2101 y Fz(.)h(This)e(dif)o(ferent)243 2155 y(proof)j(leads)f(to)h(dif)o(ferent)g(\(partial\))g(proof)f(terms) i(de)o(v)o(eloping)d(as)j(follo)o(ws:)542 2234 y Fe(?)542 2300 y Fy(l)p Fx(u)p Fw(;)5 b Fx(v)m Fw(;)g Fx(x)p Fw(:)10 b Fe(?-KERNEL)542 2367 y Fy(l)p Fx(u)p Fw(;)5 b Fx(v)m Fw(;)g Fx(x)p Fw(:)10 b Fu(\()p Fy(l)p Fx(v)749 2374 y Fl(i)760 2367 y Fw(:)h Fe(?-KERNEL-INS)o(T)m Fu(\)\()p Fx(vx)p Fu(\))542 2433 y Fy(l)p Fx(u)p Fw(;)5 b Fx(v)m Fw(;)g Fx(x)p Fw(:)10 b Fu(\()p Fy(l)p Fx(v)749 2440 y Fl(i)760 2433 y Fw(:)h Fx(uxv)847 2440 y Fl(i)858 2433 y Fu(\)\()p Fx(vx)p Fu(\))p Fw(:)243 2512 y Fz(Note)j(that)f(this)g (proof)h(term)h(can)f(be)h Fy(b)p Fz(-normalized)f(to)f(the)h(proof)g (term)h(abo)o(v)o(e)f(\(see)243 2566 y(subsection)9 b(3.4\).)911 3108 y Ft(Minlog7.tex;)18 b(28/04/1998;)g(12:50;)h(p.8)p eop %%Page: 9 9 9 8 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)161 b Fz(9)243 404 y(3.2.)32 b Fx(A)o(utomated)9 b(pr)n(oof-sear)n(c)o(h)243 498 y Fz(Of)16 b(course)g(our)f(\002rst)h(e)o(xample)h(was)e(an)h(easy)g(one) g(that)f(only)g(deals)g(with)g(logic.)h(A)243 552 y(useful)d(proof)h (system)f(should)g(be)h(able)g(to)g(pro)o(v)o(e)g(such)g(a)g(formula)h (without)d(human)243 606 y(interaction:)243 689 y Fe(\(set-goal)20 b('?)i(\(parse-formul)o(a)e("\(all)h(x)i(\(a)f(x)g(->)g(b)h(x\)\))44 b(->)924 743 y(all)22 b(x)h(\(a)f(x\))45 b(->)g(all)21 b(x)i(\(b)f(x\)"\)\))243 797 y(;?:\(all)f(x.A)g(x)i(->)f(B)g(x\))h(->)f (all)g(x)g(A)g(x)h(->)f(all)g(x)g(B)h(x)243 905 y(\(search\))243 959 y(;ok,)e(?)i(is)f(proved)f(by)h(minimal)f(quantifier)f(logic.)243 1013 y(;Proof)h(finished.)243 1097 y Fz(The)h(automatically)e (generated)i(proof)g(term,)h Fy(l)p Fx(u)1090 1104 y Fq(01)1125 1097 y Fw(;)5 b Fx(u)1166 1104 y Fq(02)1200 1097 y Fw(;)g Fx(x)p Fw(:)g Fx(u)1279 1104 y Fq(01)1314 1097 y Fx(x)g Fu(\()p Fx(u)1380 1104 y Fq(02)1415 1097 y Fx(x)p Fu(\))p Fz(,)23 b(is)e(equal)243 1151 y(modulo)c(bound)f (renaming)i(to)f(the)g(former)i(one.)f(It)f(is)h(obtained)e(by)h (breadth-\002rst)243 1205 y(proof-search)c(in)g(the)g(style)f(of)i (\(Miller)n(,)g(1991\).)e(Usually)m(,)h(this)f(de)o(vice)h(is)g (applied)f(to)243 1259 y(subgoals)j(pro)o(v)o(able)h(by)h(pure)f (predicate)h(logic)f(\(including)f(the)h(use)h(of)g(global)f(as-)243 1313 y(sumptions\).)243 1418 y(3.3.)32 b Fx(Pr)n(oof)10 b(pr)n(esentation)243 1512 y Fz(A)g(proof)h(term)g(can)g(be)f(used)g (to)g(automatically)g(produce)g(a)h(le)o(gible)f(presentation,)f(i.e.) 243 1566 y(a)i(T)294 1576 y(E)317 1566 y(X-script:)395 1658 y Fi(Pr)o(oposition.)22 b Fu(\()p Fv(8)p Fx(x)p Fw(:)p Fx(A)p Fu(\()p Fx(x)p Fu(\))10 b Fv(!)g Fx(B)p Fu(\()p Fx(x)p Fu(\)\))f Fv(!)i(8)p Fx(x)5 b(A)p Fu(\()p Fx(x)p Fu(\))10 b Fv(!)g(8)p Fx(x)5 b(B)p Fu(\()p Fx(x)p Fu(\))p Fz(.)395 1737 y Fx(Pr)n(oof)o(.)22 b Fz(F)o(or)g Fu(\()p Fv(8)p Fx(x)p Fw(:)p Fx(A)p Fu(\()p Fx(x)p Fu(\))16 b Fv(!)g Fx(B)p Fu(\()p Fx(x)p Fu(\)\))f Fv(!)i(8)p Fx(x)5 b(A)p Fu(\()p Fx(x)p Fu(\))16 b Fv(!)h(8)p Fx(x)5 b(B)p Fu(\()p Fx(x)p Fu(\))22 b Fz(assume)395 1791 y Fv(8)p Fx(x)p Fw(:)p Fx(A)p Fu(\()p Fx(x)p Fu(\))11 b Fv(!)h Fx(B)p Fu(\()p Fx(x)p Fu(\))h Fz([1],)g Fv(8)p Fx(x)5 b(A)p Fu(\()p Fx(x)p Fu(\))14 b Fz([2])f(and)g Fx(x)p Fz(.)h(Then)f(we)g(ha)o(v)o(e)h(to)f(sho)o(w)395 1845 y Fx(B)p Fu(\()p Fx(x)p Fu(\))p Fz(.)g(W)l(e)h(deri)o(v)o(e)f(this)e (from)j Fx(A)p Fu(\()p Fx(x)p Fu(\))c Fv(!)i Fx(B)p Fu(\()p Fx(x)p Fu(\))g Fz(\(as)i(an)f(instance)f(of)h([1]\))395 1899 y(and)e(ha)o(v)o(e)h(to)f(sho)o(w)f Fx(A)p Fu(\()p Fx(x)p Fu(\))p Fz(.)h(That)g(is)f(an)i(instance)e(of)h([2].)243 1989 y(This)i(purely)h(logical)f(e)o(xample)i(was)e(chosen)h(to)g (demonstrate)f(the)h(Curry-Ho)o(ward-)243 2043 y(correspondence)g(and)h (the)g(interacti)o(v)o(e)f(use)h(of)g(the)g(logical)f(rules.)i(M)r FA(I)r(N)r(L)r(O)r(G)q Fz(')n(s)i(em-)243 2097 y(phasis,)9 b(ho)o(we)o(v)o(er)n(,)h(lies)f(on)g(concrete)h(mathematical)g (problems;)f(data)g(types)g(and)h(con-)243 2151 y(stants)g(should)g(ha) o(v)o(e)j(a)f(\002x)o(ed)g(denotational)e(semantics)h(e)o(xpressed)g (operationally)f(by)243 2205 y(suitable)g(higher)g(type)h(re)o(writing) f(rules.)243 2310 y(3.4.)32 b Fx(Normalization)o(-by-e)o(valuati)o(on) 243 2404 y Fz(The)10 b(heart)h(of)h(M)r FA(I)r(N)r(L)r(O)r(G)j Fz(is)10 b(the)g(normalization-by-e)o(v)o(aluation)d(mechanism)k(pro)o (vid-)243 2458 y(ing)f(an)i(ef)o(\002cient)f(implementation)f(of)h(the) g(operational)f(semantics.)h(Since)h(terms)f(are)243 2512 y(represented)h(by)h(S)r FA(C)r(H)r(E)r(M)r(E)k Fz(e)o(xpressions)11 b(it)h(is)g(quite)f(natural)h(to)g(use)h(S)r FA(C)r(H)r(E)r(M)r(E)k Fz(e)o(v)o(al-)243 2566 y(uation)11 b(instead)h(of)h(e)o(xplicitly)e(programming)h(a)h(normalization)f (procedure)h(\(Ber)o(ger)911 3108 y Ft(Minlog7.tex;)18 b(28/04/1998;)g(12:50;)h(p.9)p eop %%Page: 10 10 10 9 bop 243 277 a Fz(10)62 b FC(BENL,)11 b(BERGER,)g(SCHWICHTENBERG,)f (SEISENBERGER,)h(ZUBER)243 404 y Fz(and)16 b(Schwichtenber)o(g,)h (1991;)e(Ber)o(ger)n(,)j(1993a;)e(Ber)o(ger)i(et)e(al.,)i(1998\).)e(T)m (erms)h(are)243 457 y(normalized)9 b(with)h(respect)f(to)h Fy(b)p Fn(R)p Fz(-con)n(v)o(ersion)g(and)f(additional)f(higher)i(order) g(re)o(write)243 511 y(rules)k(gi)o(v)o(en)h(by)g(the)f(user)n(.)h 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Fz(.)g(The)g(base)h(case)f Fn(even)q Fu(\()p Fz(2)p Fv(\003)o Fz(0)p Fu(\))p Fz(,)g(to)g(be)g(pro)o(v)o(en)h(\002rst,)f (normalizes)g(to)243 1237 y(the)j(truth)f(axiom.)i(In)g(the)f(step)g (case)h(we)f(also)g(use)g(term)h(re)o(writing)e(when)h(applying)243 1291 y(the)d(command)g Fe(use-with)p Fz(.)243 1379 y Fe(\(set-goal)20 b('?)i(\(parse-formul)o(a)e("all)h(n)i (even\(2*n\)"\)\))243 1487 y(\(ind\))243 1541 y(;ok,)e(?)i(can)f(be)g (obtained)e(from)243 1595 y(;?-STEP:)g(all)i(n.even\(2*n\))e(->)i (even\(2*\(n+1\))o(\))243 1649 y(;?-BASE:)e(even\(2*0\))243 1757 y(\(normalize-g)o(oal\))243 1811 y(;ok,)h(the)h(normalized)e(goal) i(is)243 1865 y(;?-BASE-NF:)d(T)243 1973 y(\(prop\))243 2027 y(;ok,)i(?-BASE-NF)g(is)h(proved)f(by)h(minimal)f(propositiona)o (l)f(logic.)243 2081 y(;The)h(active)g(goal)h(now)g(is)243 2135 y(;?-STEP:)e(all)i(n.even\(2*n\))e(->)i(even\(2*\(n+1\))o(\))243 2242 y(\(assume)f('n)h('ih\))243 2296 y(;ok,)f(under)h(these)f (assumptions)f(we)i(have)f(the)h(new)g(goal)243 2350 y(;?-STEP-KERN)o(EL:)d(even\(2*\(n+1\)\))g(from)243 2404 y(;)45 b(n)g(ih:even\(2*n\))243 2512 y(\(use-with)20 b('ih\))243 2566 y(;ok,)h(?-STEP-KERNEL)e(is)k(proved.)d(Proof)i (finished.)891 3108 y Ft(Minlog7.tex;)17 b(28/04/1998;)h(12:50;)i(p.10) p eop %%Page: 11 11 11 10 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)138 b Fz(11)243 404 y(Step)11 b(by)g(step)f(we)i(get)f(the)g(follo)o(wing)e(\(partial\))h(proof)h (terms:)606 505 y Fe(?)606 569 y Fz(nat-ind-at)782 579 y Fp(8)p Fl(n)t Fm(even)o Fj(\()p Fl(n)p Fj(\))942 569 y Fe(?-BASE)d(?-STEP)606 634 y Fz(nat-ind-at)782 644 y Fp(8)p Fl(n)t Fm(even)o Fj(\()p Fl(n)p Fj(\))942 634 y Fn(T)l(ruth)1048 615 y Fp(>)1086 634 y Fu(\()p Fy(l)p Fx(n)p Fw(;)d Fz(ih)n Fw(:)g Fz(ih)o Fu(\))243 730 y Fz(where)13 b(nat-ind-at)g(is)g(the)g(constant)f(representing)g(the)h (induction)f(axiom.)h(Note)g(that)243 784 y(although)g(the)h(re)o (write)g(steps)g(are)h(not)f(sho)o(wn)g(in)g(the)g(proof)g(term,)i(the) 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Fx(n)p Fw(;)5 b Fx(i)p Fu(\)\))j Fv(!)529 1773 y Fn(o)q(cc)q Fu(\()p Fx(n)e Fu(+)g Fz(1)p Fw(;)f Fx(i)p Fu(\))k(=)h Fn(o)q(cc)q Fu(\()p Fx(n)p Fw(;)5 b Fn(next)p Fu(\()p Fx(i)p Fu(\)\))p Fw(;)484 1838 y Fn(back)q Fu(\()p Fx(n)p Fu(\))k Fv(!)h Fn(act)q Fu(\()p Fx(n)p Fu(\))f(=)h Fx(i)g Fv(!)h Fn(def)s Fu(\()p Fn(o)q(cc)p Fu(\()p Fx(n)p Fw(;)5 b Fn(p)o(rev)p Fu(\()p Fx(i)p Fu(\)\)\))j Fv(!)529 1903 y Fn(o)q(cc)q Fu(\()p Fx(n)e Fu(+)g Fz(1)p Fw(;)f Fx(i)p Fu(\))k(=)h Fn(o)q(cc)q Fu(\()p Fx(n)p Fw(;)5 b Fx(i)p Fu(\))p Fw(;)484 1968 y Fn(back)q Fu(\()p Fx(n)p Fu(\))k Fv(!)h Fn(act)q Fu(\()p Fx(n)p Fu(\))f(=)h Fn(next)q Fu(\()p Fx(i)p Fu(\))f Fv(!)i Fn(def)t Fu(\()p Fn(o)q(cc)p Fu(\()p Fx(n)p Fw(;)5 b Fx(i)p Fu(\)\))j Fv(!)529 2032 y Fn(o)q(cc)q Fu(\()p Fx(n)e Fu(+)g Fz(1)p Fw(;)f Fx(i)p Fu(\))k(=)h Fn(o)q(cc)q Fu(\()p Fx(n)p Fw(;)5 b Fx(i)p Fu(\))p Fw(;)484 2097 y Fn(back)q Fu(\()p Fx(n)p Fu(\))k Fv(!)h Fn(act)q Fu(\()p Fx(n)p Fu(\))f Fv(6)p Fu(=)h Fn(next)q Fu(\()p Fx(i)p Fu(\))f Fv(!)i 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Fz(train)24 b Fx(j)18 b Fz(the)e(se)o(gment)i(where)f(the)g(train)g(is)g(located)f(at)i(this) e(time.)h(F)o(or)h Fn(lo)q(c)g Fz(we)f(re-)243 457 y(quire)9 b(the)h(follo)o(wing)e(property:)g Fx(The)j(tr)o(ain)e(being)g(in)g (the)h(active)f(se)n(gment)h(moves)f(one)243 511 y(se)n(gment)14 b(in)h(the)g(actual)g(dir)n(ection)f(only)g(if)h(this)f(se)n(gment)h (is)g(not)f Fd(occ)p Fx(upied\(!\),)i(and)243 565 y(nothing)9 b(happens)h(besides.)g Fz(In)i(clauses:)346 664 y Fn(lo)q(c)p Fu(\()p Fz(0)p Fw(;)g Fx(j)q Fu(\))c(=)j Fn(initlo)q(c)p Fu(\()c Fx(j)q Fu(\))p Fw(;)346 728 y Fn(act)p Fu(\()p Fx(n)p Fu(\))j(=)g Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\))c Fv(!)j Fn(back)q Fu(\()p Fx(n)p Fu(\))e Fv(!)h(:)p Fn(def)t Fu(\()p Fn(o)q(cc)p Fu(\()p Fx(n)p Fw(;)5 b Fn(p)o(rev)p Fu(\()p Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)12 b Fx(j)q Fu(\)\)\)\))7 b Fv(!)391 793 y Fn(lo)q(c)q Fu(\()p Fx(n)f Fu(+)g Fz(1)p Fw(;)12 b Fx(j)q Fu(\))d(=)h Fn(p)o(rev)p Fu(\()p Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)q 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y(whole)c(\002rst)h(perspecti)o(v)o (e)g(could)f(ha)o(v)o(e)i(been)f(a)o(v)o(oided)f(if)h(we)h(had)e(used)h (an)g(e)o(xistential)243 1298 y(quanti\002er)f(in)h(the)g(equations)f (abo)o(v)o(e,)h(as)h(follo)o(ws:)320 1397 y Fn(act)q Fu(\()p Fx(n)p Fu(\))d(=)h Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\))d Fv(!)h Fn(back)q Fu(\()p Fx(n)p Fu(\))f Fv(!)i(:9)c Fx(j)996 1404 y Fq(1)1014 1397 y Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)12 b Fx(j)1148 1404 y Fq(1)1165 1397 y Fu(\))e(=)g Fn(p)o(rev)q Fu(\()p Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\)\))c Fv(!)366 1461 y Fn(lo)q(c)p Fu(\()p Fx(n)e Fu(+)g Fz(1)p Fw(;)12 b Fx(j)q Fu(\))d(=)h Fn(p)o(rev)q Fu(\()p Fn(lo)q(c)o Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\)\))p Fw(;)320 1526 y Fn(act)q Fu(\()p Fx(n)p Fu(\))d(=)h Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\))d Fv(!)h Fn(back)q Fu(\()p Fx(n)p Fu(\))f Fv(!)i(9)c Fx(j)966 1533 y Fq(1)984 1526 y Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)12 b Fx(j)1118 1533 y Fq(1)1135 1526 y Fu(\))e(=)g Fn(p)o(rev)q Fu(\()p Fn(lo)q(c)o Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\)\))c Fv(!)366 1591 y Fn(lo)q(c)p Fu(\()p Fx(n)e Fu(+)g Fz(1)p Fw(;)12 b Fx(j)q Fu(\))d(=)h Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\))p Fw(:)243 1685 y Fz(Ho)o(we)o(v)o(er)n(,)f(since)f(we)g(want)g(to)g(shift)f(as)h (much)h(as)f(possible)f(of)h(the)g(b)o(urden)g(of)h(pro)o(ving)243 1739 y(something)j(to)i(our)g(re)o(write)f(apparatus,)h(we)g(make)h (this)d(e)o(xistential)h(quanti\002er)g(e)o(x-)243 1793 y(plicit)9 b(by)h(introducing)e(an)i(appropriate)g(function.)f(This)h (is)g(a)g(general)h(pattern:)e(when-)243 1847 y(e)o(v)o(er)k(in)g(a)g (gi)o(v)o(en)g(problem)g(there)g(is)f(a)i(functional)d(dependenc)o(y)i (of)g(some)g(data)g(from)243 1901 y(others,)g(introduce)f(a)h(function) f(to)g(describe)h(this)f(dependenc)o(y)h(in)g(an)g(e)o(xplicit)f(form.) 243 1955 y(This)j(function)f(can)i(then)f(be)h(used)f(in)h(a)g(re)o (writing)e(mechanism,)i(which)f(generally)243 2009 y(makes)c(pro)o (ving)f(much)h(simpler)n(.)243 2116 y(4.3.)32 b Fx(The)11 b(speci\002cation)e(and)i(its)f(pr)n(oof)243 2212 y Fz(No)o(w)h(we)i (can)f(formulate)g(the)g(safety)f(property)h(we)g(want)f(to)h(pro)o(v)o (e,)h(namely)f(that)f(no)243 2266 y(train)f(can)i(e)o(v)o(er)g(pass)e (another)h(one:)435 2369 y Fn(inito)q(cc)q Fu(\()p Fn(initlo)q(c)p Fu(\()c Fx(j)733 2376 y Fq(1)751 2369 y Fu(\)\))i(=)17 b Fx(j)861 2376 y Fq(1)890 2369 y Fv(!)10 b Fn(inito)q(cc)q Fu(\()p Fn(initlo)q(c)q Fu(\()d Fx(j)1244 2376 y Fq(2)1261 2369 y Fu(\)\))j(=)17 b Fx(j)1372 2376 y Fq(2)1400 2369 y Fv(!)435 2434 y Fn(initlo)q(c)p Fu(\()7 b Fx(j)589 2441 y Fq(1)607 2434 y Fu(\))j Fw(<)g Fn(initlo)q(c)q Fu(\()d Fx(j)835 2441 y Fq(2)853 2434 y Fu(\))i Fv(!)i(8)p Fx(n)5 b Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)12 b Fx(j)1123 2441 y Fq(1)1140 2434 y Fu(\))e Fw(<)g Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)1347 2441 y Fq(2)1364 2434 y Fu(\))p Fw(:)243 2512 y Fz(Rather)i(than)e(describing)g(the)i(interacti)o(v)o (e)e(construction)g(of)h(the)g(proof)g(in)g(the)i(M)r FA(I)r(N)r Fz(-)244 2566 y FA(L)r(O)r(G)i Fz(system,)d(we)h(restrict)f (ourselv)o(es)g(to)g(some)g(hints:)f(The)h(proof)h(is)f(by)g(induction) 891 3108 y Ft(Minlog7.tex;)j(28/04/1998;)h(12:50;)i(p.14)p eop %%Page: 15 15 15 14 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)138 b Fz(15)243 404 y(on)12 b Fx(n)p Fz(.)i(In)e(order)h(to)g(keep)f(it)h(comprehensible,)f(it)g(is)h (crucial)g(to)f(think)g(of)h(appropriate)243 457 y(lemmas)d(\002rst.)g (In)f(the)h(present)f(e)o(xample,)h(beside)f(some)h(basic)f (arithmetical)g(facts,)g(the)243 511 y(follo)o(wing)g(lemmas)j(are)f (pro)o(v)o(ed)h(separately)e(and)h(then)g(used:)512 605 y Fn(p)o(rev)q Fu(\()p Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)h Fx(j)q Fu(\)\))c Fv(\024)i Fn(lo)q(c)p Fu(\()p Fx(n)c Fu(+)g Fz(1)p Fw(;)12 b Fx(j)q Fu(\))p Fw(;)512 670 y Fn(lo)q(c)p Fu(\()p Fx(n)6 b Fu(+)g Fz(1)p Fw(;)12 b Fx(j)q Fu(\))d Fv(\024)i Fn(next)q Fu(\()p Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)h Fx(j)q Fu(\)\))p Fw(;)512 735 y Fn(back)q Fu(\()p Fx(n)p Fu(\))d Fv(!)i Fn(lo)q(c)p Fu(\()p Fx(n)6 b Fu(+)g Fz(1)p Fw(;)12 b Fx(j)q Fu(\))d Fv(\024)h Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)i Fx(j)q Fu(\))p Fw(;)512 799 y Fv(:)p Fn(back)q Fu(\()p Fx(n)p Fu(\))d Fv(!)i Fn(lo)q(c)p Fu(\()p Fx(n)p Fw(;)h Fx(j)q Fu(\))c Fv(\024)j Fn(lo)q(c)p Fu(\()p Fx(n)6 b Fu(+)g Fz(1)p Fw(;)12 b Fx(j)q Fu(\))p Fw(;)512 864 y Fn(inito)q(cc)q Fu(\()p Fn(initlo)q(c)q Fu(\()7 b Fx(j)q Fu(\)\))h(=)18 b Fx(j)10 b Fv(!)h Fn(o)q(cc)p Fu(\()p Fx(n)p Fw(;)5 b Fn(lo)q(c)o Fu(\()p Fx(n)p Fw(;)12 b Fx(j)q Fu(\)\))c(=)18 b Fx(j)q Fw(:)243 953 y Fz(The)12 b(size)g(of)g(the)f(proof)h(term)g(\002nally)g (obtained)f(is)g(about)g(two)g(pages,)h(and)g(the)g(ef)o(fort)243 1007 y(to)f(generate)h(it)f(\(without)g(using)f(an)o(y)i(automated)f (search\))i(was)e(42)g(interactions.)g(The)243 1061 y(amount)f(for)i (each)f(lemma)i(was)d(approximately)g(the)h(same.)243 1166 y(4.4.)32 b Fx(F)n(inal)10 b(r)n(emarks)g(on)h(the)g(sc)o (heduling)f(e)o(xample)265 1260 y Fz(1.)19 b(The)13 b(constants)f Fn(o)q(cc)i Fz(and)f Fn(lo)q(c)h Fz(are)g(de\002ned)f(by)g(primiti)o(v) o(e)g(recursion)f(with)h(func-)318 1314 y(tion)g(parameters,)h(i.e.)h (within)d(G\366del')n(s)g(system)i Fx(T)6 b Fz(.)14 b(Since)g(we)g (only)f(ha)o(v)o(e)h(used)318 1368 y(the)e(recursion)f(equations)g(as)h (re)o(write)g(rules,)g(termination)f(as)i(well)e(as)i(preserv)o(a-)318 1422 y(tion)d(of)h(the)g(v)o(alues)f(is)h(guaranteed.)265 1476 y(2.)19 b(The)c(possibility)e(to)i(use)h(e.g.)g Fx(i)g Fz(as)g(well)f(as)1052 1467 y(\210)1050 1476 y Fx(i)h Fz(pro)o(vides)e(additional)g(e)o(xpressi)o(v)o(e)318 1530 y(po)o(wer)n(,)k(which)f(can)i(be)f(quite)f(useful:)g(cf.)h(the)g (use)g(of)g Fn(def)t Fu(\()p Fn(o)q(cc)p Fu(\()p Fx(n)p Fw(;)5 b Fx(i)p Fu(\)\))16 b Fz(for)i(`is)318 1584 y(occupied)10 b(by)h(a)h(\(real\))f(train'.)265 1638 y(3.)19 b(The)8 b(ground)g(type)g Fn(segment)h Fz(was)f(described)g(axiomatically)m(,)f (by)i(means)f(of)h(prop-)318 1692 y(erties)j(of)h Fn(p)o(rev)h Fz(and)f Fn(next)h Fz(like)e Fn(p)o(rev)p Fu(\()p Fn(next)q Fu(\()p Fx(i)p Fu(\)\))e(=)h Fx(i)p Fz(.)i(If)h(instead)e(we)h(ha)o(v)o (e)g(a)g(\002x)o(ed)318 1746 y(ground)i(type)g(like)g(the)h(inte)o (gers)g(for)g(se)o(gments)g(and)g(constants)e(for)i(particular)318 1800 y(se)o(gments,)f(we)f(obtain)g(an)h(e)o(x)o(ecutable)g (speci\002cation.)f(It)h(can)g(be)f(e)o(x)o(ecuted)i(for)318 1854 y(completely)10 b(concrete)i(data,)g(b)o(ut)f(also)g(for)g (partially)f(gi)o(v)o(en)h(ones)g(\(partial)g(e)o(v)o(alu-)318 1908 y(ation\).)556 2083 y FC(5.)31 b Fz(P)r FA(R)q(O)r(G)r(R)r(A)r(M) 16 b(E)r(X)r(T)r(R)r(A)q(C)r(T)r(I)r(O)s(N)h Fz(Q)r FA(U)r(I)r(C)r(K)r (S)r(O)r(RT)243 2188 y Fz(In)c(the)f(pre)o(vious)g(section,)g(we)h(sho) o(wed)f(ho)o(w)h(M)r FA(I)r(N)r(L)r(O)r(G)k Fz(can)d(be)f(used)f(to)g (v)o(erify)h(that)243 2242 y(a)j(gi)o(v)o(en)f(program)g(\()p Fn(lo)q(c)q Fz(\))h(meets)f(its)g(speci\002cation.)g(No)o(w)g(we)h (will)e(e)o(xplain)h(ho)o(w)g(to)243 2296 y(obtain)d(correct)i (programs)f(in)h(M)r 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Fu(\()p Fx(s)545 2518 y Fk(<)p Fl(m)596 2511 y Fu(\))6 b(+)g Fz(#)684 2518 y Fl(n)703 2511 y Fu(\()p Fx(s)739 2518 y Fp(\025)p Fl(m)790 2511 y Fu(\))243 2566 y Fz(11.)18 b(#)341 2573 y Fl(n)359 2566 y Fu(\()p Fx(s)395 2573 y Fq(1)419 2566 y Fv(\003)6 b Fx(s)466 2573 y Fq(2)485 2566 y Fu(\))k(=)g Fz(#)581 2573 y Fl(n)599 2566 y Fu(\()p Fx(s)635 2573 y Fq(1)653 2566 y Fu(\))c(+)g Fz(#)741 2573 y Fl(n)760 2566 y Fu(\()p Fx(s)796 2573 y Fq(2)814 2566 y Fu(\))891 3108 y Ft(Minlog7.tex;)17 b(28/04/1998;)h(12:50;)i (p.20)p eop %%Page: 21 21 21 20 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)138 b Fz(21)301 404 y(No)o(w)12 b(we)h(formulate)g(the)g(speci\002cation)f(of)h(the)f(desired)h (algorithm:)e(we)i(want)f(to)243 457 y(sho)o(w)d(that)h(e)o(v)o(ery)g (list)f(of)h(natural)g(numbers)g(can)g(be)h(sorted,)f(i.e.)h(for)f(e)o (v)o(ery)h(list)e Fx(s)h Fz(there)243 511 y(is)h(a)h(list)e Fx(t)k Fz(such)d(that)f Fx(t)15 b Fz(is)c(sorted)g(and)h(e)o(v)o(ery)g (number)f Fx(n)h Fz(occurs)g(in)e Fx(t)k Fz(as)e(often)f(as)h(in)f Fx(s)p Fz(.)243 565 y(F)o(ormally:)621 619 y Fv(8)p Fx(s)5 b Fv(9)694 601 y Fp(\003)712 619 y Fx(t)s Fw(:)g Fn(so)o(rted)p Fu(\()o Fx(t)s Fu(\))h Fv(^)g(8)p Fx(n)f Fz(#)1029 626 y Fl(n)1047 619 y Fu(\()p Fx(s)p Fu(\))k(=)h Fz(#)1178 626 y Fl(n)1197 619 y Fu(\()o Fx(t)s Fu(\))o Fw(:)243 697 y Fz(The)j(idea)g(is)g(to)f(take,)h(gi)o(v)o(en)g(a)g(list)f Fx(s)f Fu(=)h Fx(n)922 680 y Fp(0)933 676 y Fk(_)968 697 y Fx(s)986 680 y Fp(0)997 697 y Fz(,)i(the)f(\002rst)g(element)g Fx(n)1355 680 y Fp(0)1379 697 y Fz(and)g(split)f Fx(s)1568 680 y Fp(0)1592 697 y Fz(up)243 751 y(into)g Fx(s)345 734 y Fp(0)355 759 y Fk(<)p Fl(n)398 750 y Fc(0)422 751 y Fz(and)h Fx(s)519 734 y Fp(0)530 759 y(\025)p Fl(n)573 750 y Fc(0)583 751 y Fz(,)h(sort)e(these)g(shorter)h(lists)e(and)i(put) f(the)h(results)e(together)i(in)f(the)243 805 y(right)e(order)n(.)h(In) g(other)g(words,)f(we)i(pro)o(v)o(e)f(the)g(goal)f(by)h(induction)e(o)o (v)o(er)j(the)f Fx(length)f Fz(of)243 859 y(the)h(gi)o(v)o(en)f(list,)h (so)f(we)i(\002rst)f(pro)o(v)o(e)465 952 y Fv(8)p Fx(m)p Fw(;)5 b Fx(s)p Fw(:)g Fn(len)p Fu(\()p Fx(s)p Fu(\))k Fv(\024)h Fx(m)g Fv(!)h(9)863 934 y Fp(\003)881 952 y Fx(t)s Fw(:)5 b Fn(so)o(rted)o Fu(\()o Fx(t)s Fu(\))h Fv(^)g(8)p Fx(n)f Fz(#)1197 959 y Fl(n)1215 952 y Fu(\()p Fx(s)p Fu(\))10 b(=)g Fz(#)1347 959 y Fl(n)1365 952 y Fu(\()o Fx(t)s Fu(\))243 1046 y Fz(by)e(induction)e(on)j Fx(m)f Fz(and)g(then)g(get)h(the)f(original)f(goal)h(by)g (instantiating)e Fx(m)i Fz(with)g Fn(len)p Fu(\()p Fx(s)p Fu(\))p Fz(.)243 1100 y(The)i(proof)g(is)h(simple;)e(we)i(sho)o(w)f(it) g(an)o(yway)g(to)g(demonstrate)g(that)g(this)f(e)o(xample)i(can)243 1154 y(be)g(carried)h(out)e(in)i(M)r FA(I)r(N)r(L)r(O)r(G)j Fz(without)9 b(running)h(into)g(technical)h(dif)o(\002culties.)243 1208 y Fx(Base)e(m)f Fu(=)h Fz(0.)g(Let)h Fx(s)f Fz(with)f Fn(len)q Fu(\()p Fx(s)p Fu(\))f Fv(\024)i Fz(0)g(be)h(gi)o(v)o(en.)f (Then)g Fx(s)f Fu(=)h Fy(e)h Fz(by)f(lemma)h(1.)g(Let)e Fx(t)j Fz(:)p Fu(=)d Fy(e)p Fz(,)243 1262 y(then)i Fn(so)o(rted)q Fu(\()o Fx(t)s Fu(\))p Fv(&)543 1245 y Fp(\003)556 1262 y Fv(.>)p Fz(,)i(we)f(ha)o(v)o(e)g(to)f(sho)o(w)g Fv(8)p Fx(m)5 b Fz(#)1056 1269 y Fl(m)1082 1262 y Fu(\()p Fx(s)p Fu(\))k(=)h Fz(#)1213 1269 y Fl(m)1239 1262 y Fu(\()o Fx(t)s Fu(\))o Fz(.)h(This)f(follo)o(ws)f(from)243 1316 y Fx(s)h Fu(=)g Fy(e)g Fu(=)f Fx(t)s Fz(.)243 1369 y Fx(Step)h(m)g Fv(!)h Fx(m)6 b Fu(+)g Fz(1.)12 b(Let)f(s)g(with)f Fn(len)q Fu(\()p Fx(s)p Fu(\))f Fv(\024)h Fx(m)c Fu(+)g Fz(1)12 b(be)f(gi)o(v)o(en.)243 1423 y Fx(Note)p Fz(:)h(we)i(make)f(a)g (case)h(distinction)c(here)j(\()p Fx(s)e Fu(=)h Fy(e)h Fz(or)g Fx(s)e Fu(=)g Fx(n)1226 1407 y Fp(0)1237 1403 y Fk(_)1272 1423 y Fx(s)1290 1407 y Fp(0)1301 1423 y Fz(\).)j(F)o(ormally)m(,)f(this)f(is)243 1477 y(achie)o(v)o(ed)j(by)g (using)f(the)h(structural)f(induction)f(scheme)j(for)g Fn(seq)q Fz(,)g(without)d(making)243 1531 y(use)e(of)g(the)g(induction) e(hypothesis)g(in)h(the)h(step)g(case.)243 1585 y Fx(Case)g(s)f Fu(=)g Fy(e)p Fz(.)i(See)g Fx(m)e Fu(=)g Fz(0.)243 1639 y Fx(Case)16 b(s)c Fu(=)h Fx(n)451 1623 y Fp(0)462 1619 y Fk(_)497 1639 y Fx(s)515 1623 y Fp(0)526 1639 y Fz(.)j(No)o(w)g Fn(len)p Fu(\()p Fx(s)747 1623 y Fp(0)758 1639 y Fu(\))c Fv(\024)h Fx(m)p Fv(&)914 1623 y Fp(\003)928 1639 y Fv(.)p Fn(len)q Fu(\()p Fx(s)p Fu(\))e Fv(\024)i Fx(m)8 b Fu(+)h Fz(1,)16 b(which)f(we)h(ha)o(v)o(e)g(just)243 1693 y(assumed.)c(Using)f (lemma)h(2)g(and)g(3)g(we)g(conclude)f Fn(len)q Fu(\()p Fx(s)1159 1677 y Fp(0)1169 1702 y Fk(<)p Fl(n)1212 1692 y Fc(0)1223 1693 y Fu(\))f Fv(\024)h Fx(m)h Fz(and)g Fn(len)p Fu(\()p Fx(s)1510 1677 y Fp(0)1521 1702 y(\025)p Fl(n)1564 1692 y Fc(0)1574 1693 y Fu(\))e Fv(\024)243 1747 y Fx(m)p Fz(.)16 b(Using)e(the)i(induction)d(hypothesis)g(for)j Fx(s)983 1731 y Fp(0)994 1756 y Fk(<)p Fl(n)1037 1746 y Fc(0)1063 1747 y Fz(and)g Fx(s)1163 1731 y Fp(0)1174 1756 y(\025)p Fl(n)1217 1746 y Fc(0)1243 1747 y Fz(we)g(get)e Fx(t)1395 1754 y Fq(1)1429 1747 y Fz(and)h Fx(t)1523 1754 y Fq(2)1557 1747 y Fz(with)243 1801 y(\(1\))d Fn(so)o(rted)q Fu(\()o Fx(t)453 1808 y Fq(1)471 1801 y Fu(\))p Fz(,)g(\(2\))h Fv(8)p Fx(n)5 b Fz(#)654 1808 y Fl(n)672 1801 y Fu(\()p Fx(s)708 1785 y Fp(0)719 1810 y Fk(<)p Fl(n)762 1800 y Fc(0)772 1801 y Fu(\))11 b(=)g Fz(#)870 1808 y Fl(n)888 1801 y Fu(\()o Fx(t)918 1808 y Fq(1)936 1801 y Fu(\))p Fz(,)i(\(3\))f Fn(so)o(rted)q Fu(\()o Fx(t)1188 1808 y Fq(2)1205 1801 y Fu(\))h Fz(and)f(\(4\))g Fv(8)p Fx(n)5 b Fz(#)1455 1808 y Fl(n)1474 1801 y Fu(\()p Fx(s)1510 1785 y Fp(0)1520 1810 y(\025)p Fl(n)1563 1800 y Fc(0)1574 1801 y Fu(\))10 b(=)243 1855 y Fz(#)266 1862 y Fl(n)284 1855 y Fu(\()o Fx(t)314 1862 y Fq(2)332 1855 y Fu(\))p Fz(.)h(Let)f Fx(t)j Fz(:)p Fu(=)c Fx(t)539 1862 y Fq(1)564 1855 y Fv(\003)d Fu(\()p Fx(n)634 1839 y Fp(0)644 1835 y Fk(_)678 1855 y Fx(t)691 1862 y Fq(2)710 1855 y Fu(\))o Fz(.)12 b(W)l(e)g(ha)o(v)o(e)f(to)g(sho)o(w)f Fn(so)o(rted)q Fu(\()o Fx(t)s Fu(\))g Fz(and)h Fv(8)p Fx(n)5 b Fz(#)1398 1862 y Fl(n)1416 1855 y Fu(\()p Fx(s)p Fu(\))10 b(=)g Fz(#)1548 1862 y Fl(n)1566 1855 y Fu(\()o Fx(t)s Fu(\))o Fz(.)243 1909 y Fn(so)o(rted)p Fu(\()o Fx(t)s Fu(\))j Fz(:)g(lemma)h(7)g(yields)e Fx(s)761 1893 y Fp(0)772 1918 y Fk(<)p Fl(n)815 1908 y Fc(0)837 1909 y Fw(<)f Fx(n)906 1893 y Fp(0)931 1909 y Fz(and)i(hence,)h(using)e(lemma)j(5)e (and)h(\(2\),)g(we)243 1963 y(get)e(\(5\))f Fx(t)388 1970 y Fq(1)417 1963 y Fw(<)g Fx(n)486 1947 y Fp(0)496 1963 y Fz(.)i(In)f(the)g(same)h(way)m(,)f(using)e(lemma)j(6)f(and)g(4)g (instead)f(of)h(7)g(and)g(5,)h(we)243 2017 y(get)d Fx(t)322 2024 y Fq(2)351 2017 y Fv(\025)h Fx(n)420 2001 y Fp(0)443 2017 y Fz(and)g(conclude)g(\(6\))h Fx(n)784 2001 y Fp(0)795 1996 y Fk(_)830 2017 y Fx(t)843 2024 y Fq(2)871 2017 y Fv(\025)f Fx(n)940 2001 y Fp(0)963 2017 y Fz(using)f(normalization)h (again,)h(as)f(well)h(as)243 2071 y(\(7\))g Fn(so)o(rted)p Fu(\()p Fx(n)463 2055 y Fp(0)474 2050 y Fk(_)508 2071 y Fx(t)521 2078 y Fq(2)539 2071 y Fu(\))g Fz(by)g(lemma)h(9)e(and)h (\(3\).)h(Finally)e(with)f(lemma)j(8,)8 b(\(7\),)g(\(6\),)g(\(1\))k (and)243 2125 y(\(5\))f(we)g(get)g Fn(so)o(rted)q Fu(\()o Fx(t)583 2132 y Fq(1)607 2125 y Fv(\003)6 b Fu(\()p Fx(n)677 2108 y Fp(0)688 2104 y Fk(_)722 2125 y Fx(t)735 2132 y Fq(2)753 2125 y Fu(\)\))p Fz(.)243 2179 y(T)l(o)11 b(sho)o(w)f Fv(8)p Fx(n)5 b Fz(#)483 2186 y Fl(n)502 2179 y Fu(\()p Fx(s)p Fu(\))k(=)h Fz(#)633 2186 y Fl(n)651 2179 y Fu(\()o Fx(t)s Fu(\))o Fz(,)i(let)f Fx(n)g Fz(be)g(gi)o(v)o(en.) 243 2247 y Fx(Case)g(n)f Fu(=)g Fx(n)446 2231 y Fp(0)469 2247 y Fz(:)h(#)516 2254 y Fl(n)534 2247 y Fu(\()p Fx(n)575 2231 y Fp(0)586 2227 y Fk(_)621 2247 y Fx(s)639 2231 y Fp(0)650 2247 y Fu(\))o Fv(&)712 2231 y Fp(\003)726 2247 y Fv(.)q Fz(#)795 2254 y Fl(n)813 2247 y Fu(\()p Fx(s)849 2231 y Fp(0)860 2247 y Fu(\))6 b(+)g Fz(1)963 2222 y Fq(L.10)977 2247 y Fu(=)28 b Fz(#)1063 2254 y Fl(n)1082 2247 y Fu(\()p Fx(s)1118 2231 y Fp(0)1128 2256 y Fk(<)p Fl(n)1171 2246 y Fc(0)1182 2247 y Fu(\))6 b(+)g Fz(#)1270 2254 y Fl(n)1289 2247 y Fu(\()p Fx(s)1325 2231 y Fp(0)1335 2256 y(\025)p Fl(n)1378 2246 y Fc(0)1389 2247 y Fu(\))f(+)h Fz(1)258 2291 y Fq(\(2\),\(4\))283 2319 y Fu(=)41 b Fz(#)382 2326 y Fl(n)400 2319 y Fu(\()o Fx(t)430 2326 y Fq(1)448 2319 y Fu(\))6 b(+)g Fz(#)536 2326 y Fl(n)555 2319 y Fu(\()o Fx(t)585 2326 y Fq(2)603 2319 y Fu(\))g(+)g Fz(1)p Fv(&)736 2302 y Fp(\003)750 2319 y Fv(.)q Fz(#)819 2326 y Fl(n)837 2319 y Fu(\()o Fx(t)867 2326 y Fq(1)885 2319 y Fu(\))g(+)g Fz(#)973 2326 y Fl(n)992 2319 y Fu(\()p Fx(n)1033 2302 y Fp(0)1043 2298 y Fk(_)1078 2319 y Fx(t)1091 2326 y Fq(2)1109 2319 y Fu(\))1142 2293 y Fq(L.11)1155 2319 y Fu(=)29 b Fz(#)1242 2326 y Fl(n)1260 2319 y Fu(\()o Fx(t)1290 2326 y Fq(1)1314 2319 y Fv(\003)6 b Fx(n)1366 2302 y Fp(0)1377 2298 y Fk(_)1411 2319 y Fx(t)1424 2326 y Fq(2)1443 2319 y Fu(\))o Fw(:)243 2387 y Fx(Case)11 b(n)f Fv(6)p Fu(=)g Fx(n)446 2370 y Fp(0)469 2387 y Fz(:)h(#)516 2394 y Fl(n)534 2387 y Fu(\()p Fx(n)575 2370 y Fp(0)586 2366 y Fk(_)621 2387 y Fx(s)639 2370 y Fp(0)650 2387 y Fu(\))o Fv(&)712 2370 y Fp(\003)726 2387 y Fv(.)q Fz(#)795 2394 y Fl(n)813 2387 y Fu(\()p Fx(s)849 2370 y Fp(0)860 2387 y Fu(\))893 2361 y Fq(L.10)906 2387 y Fu(=)29 b Fz(#)993 2394 y Fl(n)1011 2387 y Fu(\()p Fx(s)1047 2370 y Fp(0)1058 2395 y Fk(<)p Fl(n)1101 2386 y Fc(0)1111 2387 y Fu(\))6 b(+)g Fz(#)1199 2394 y Fl(n)1218 2387 y Fu(\()p Fx(s)1254 2370 y Fp(0)1264 2395 y(\025)p Fl(n)1307 2386 y Fc(0)1318 2387 y Fu(\))258 2431 y Fq(\(2\),\(4\))283 2458 y Fu(=)41 b Fz(#)382 2465 y Fl(n)400 2458 y Fu(\()o Fx(t)430 2465 y Fq(1)448 2458 y Fu(\))6 b(+)g Fz(#)536 2465 y Fl(n)555 2458 y Fu(\()o Fx(t)585 2465 y Fq(2)603 2458 y Fu(\))o Fv(&)665 2442 y Fp(\003)680 2458 y Fv(.)p Fz(#)748 2465 y Fl(n)766 2458 y Fu(\()o Fx(t)796 2465 y Fq(1)814 2458 y Fu(\))g(+)g Fz(#)902 2465 y Fl(n)921 2458 y Fu(\()p Fx(n)962 2442 y Fp(0)973 2438 y Fk(_)1007 2458 y Fx(t)1020 2465 y Fq(2)1038 2458 y Fu(\))1071 2433 y Fq(L.11)1084 2458 y Fu(=)29 b Fz(#)1171 2465 y Fl(n)1189 2458 y Fu(\()o Fx(t)1219 2465 y Fq(1)1244 2458 y Fv(\003)6 b Fx(n)1296 2442 y Fp(0)1307 2438 y Fk(_)1341 2458 y Fx(t)1354 2465 y Fq(2)1372 2458 y Fu(\))p Fw(:)243 2512 y Fz(Here)12 b(is)f(the)h(program)g(e)o (xtracted)g(from)g(this)f(proof)g Fx(M)r Fz(.)h(W)l(e)h(sho)o(w)d(an)i (edited)f(v)o(ersion)243 2566 y(of)j(the)g(original)g(S)r FA(C)r(H)r(E)r(M)r(E)k Fz(e)o(xpression)13 b(\(we)i(ha)o(v)o(e)g(split) d(the)i(e)o(xpression)g(into)f(three)891 3108 y Ft(Minlog7.tex;)k (28/04/1998;)h(12:50;)i(p.21)p eop %%Page: 22 22 22 21 bop 243 277 a Fz(22)62 b FC(BENL,)11 b(BERGER,)g(SCHWICHTENBERG,) f(SEISENBERGER,)h(ZUBER)243 404 y Fz(parts)f(and)h(renamed)h(the)f (bound)f(v)o(ariables\):)281 504 y Fu([)-5 b([)p Fx(M)r Fu(])g(])47 b(=)k Fy(l)p Fx(s)p Fw(:)5 b Fx(g)g Fn(len)o Fu(\()p Fx(s)p Fu(\))p Fx(s)g Fw(;)94 b Fz(where)338 569 y Fx(g)49 b Fu(=)i Fn(R)525 576 y Fm(nat)p Fk(;)p Fm(seq)p Fp(!)p 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y(and)i(ele)o(gant)f(\227)i(do)e(not)h(immediately)f (yield)g(a)h(program,)h(since)e(the)o(y)h(contain)f(non-)243 1689 y(constructi)o(v)o(e)e(ar)o(guments.)i(Ho)o(we)o(v)o(er)n(,)g(it)g (is)f(well-kno)o(wn)f(that)h(such)g(classical)g(proofs)243 1743 y(under)k(certain)h(circumstances)f(can)h(be)g(translated)f(into)f (constructi)o(v)o(e)h(proofs,)g(and)243 1797 y(hence)c(yield)g (algorithms)f(via)h(program)g(e)o(xtraction)g(\(cf.)h(section)e(5\).)i (There)g(is)f(a)g(sub-)243 1851 y(stantial)e(literature)h(on)g(that)h (subject,)f(and)g(the)i(M)r FA(I)r(N)r(L)r(O)r(G)j Fz(system)10 b(supports)f(a)i(v)o(ariant,)243 1905 y(kno)o(wn)h(as)i(`)p Fx(A)p Fz(-translation',)e(which)g(goes)h(back)h(to)f(work)g(of)g (Friedman)h(\(1978\))f(and)243 1959 y(Lei)o(v)o(ant)c(\(1985\).)h(W)l (e)h(will)f(e)o(xplain)g(this)f(translation)g(by)h(means)h(of)f(an)h(e) o(xample)g(con-)243 2013 y(cerning)g(minimization)g(on)g(\002nite)h (binary)f(trees.)h(W)l(e)h(will)e(also)g(use)h(this)f(e)o(xample)h(to) 243 2067 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y(sake)d(of)i(better)f(readability)m(.)f(Hence)794 1685 y Fx(A)822 1692 y Fq(0)841 1685 y Fu([)p Fx(s)p Fu(])f Fz(:)p Fv(\021)h Fx(A)980 1692 y Fq(0)998 1685 y Fu([)o Fx(t)s Fw(;)5 b Fx(s)p Fu(])p Fw(:)243 1782 y Fz(Roughly)m(,)13 b(the)g(classical)g(proof)g(goes)g(as)h(follo)o (ws:)e(assume)h Fx(P)p Fu([)o Fx(t)s Fu(])g Fz(and)g Fv(8)p Fx(s)p Fv(:)p Fx(A)1491 1789 y Fq(0)1510 1782 y Fu([)p Fx(s)p Fu(])p Fz(.)g(W)l(e)243 1836 y(ha)o(v)o(e)j(to)f(deri)o (v)o(e)g(a)h(contradiction.)e(Using)g(the)h(second)g(assumption)f(we)i (can)f(easily)243 1890 y(pro)o(v)o(e)d Fv(8)p Fx(s)5 b Fw(:)g Fx(s)10 b Fv(\022)g Fx(t)j Fv(!)e Fx(Q)p Fu([)p Fx(s)p Fu(])g Fz(by)h(induction)e(on)i Fx(s)p Fz(.)g(No)o(w)m(,)g (setting)f Fx(s)f Fz(:)p Fu(=)f Fx(t)s Fz(,)k(we)f(obtain)f Fx(Q)p Fu([)o Fx(t)s Fu(])243 1944 y Fz(and)f(hence)g Fx(A)463 1951 y Fq(0)481 1944 y Fu([)o Fx(t)s Fu(])f Fz(\()p Fv(\021)h Fx(A)619 1951 y Fq(0)637 1944 y Fu([)o Fx(t)s Fw(;)t Fx(t)s Fu(])p Fz(\))f(using)f(the)i(assumption)e Fx(P)p Fu([)o Fx(t)s Fu(])p Fz(.)h(But)h(this)f(contradicts)g(our)243 1998 y(assumption)g Fv(8)p Fx(s)p Fv(:)p Fx(A)562 2005 y Fq(0)581 1998 y Fu([)p Fx(s)p Fu(])p Fz(.)243 2109 y Fx(A-tr)o(anslation)243 2163 y Fz(In)14 b(order)h(to)f(apply)g(the)h Fx(A)p Fz(-translation)d(to)i(this)g(proof)g(we)h(\002rst)f(ha)o(v)o(e) i(to)e(look)f(at)i(the)243 2217 y(shape)d(of)h(the)f(proof)g(in)g(some) h(detail.)f(Note)g(that)g(we)h(only)f(used)g(induction)e(on)j(trees)243 2271 y(and)e(the)g(follo)o(wing)e(facts)h(about)h Fx(P)p Fz(,)g Fx(Q)g Fz(and)g Fv(\022)p Fz(:)372 2373 y(ax)415 2380 y Fq(1)434 2373 y Fz(:)49 b Fv(8)p Fx(n)5 b Fw(:)g Fx(Q)p Fu([)p Fn(leaf)r Fu(\()p Fx(n)p Fu(\)])372 2438 y Fz(ax)415 2445 y Fq(2)434 2438 y Fz(:)49 b Fv(8)p Fx(s)539 2445 y Fq(1)558 2438 y Fw(;)5 b Fx(s)594 2445 y Fq(2)617 2438 y Fw(:)g Fv(:)p Fx(P)p Fu([)p Fx(s)724 2445 y Fq(1)741 2438 y Fu(])10 b Fv(!)g(:)p Fx(P)p Fu([)p Fx(s)908 2445 y Fq(2)926 2438 y Fu(])g Fv(!)h Fx(Q)p Fu([)p Fx(s)1069 2445 y Fq(1)1086 2438 y Fu(])f Fv(!)h Fx(Q)p Fu([)p Fx(s)1229 2445 y Fq(2)1247 2438 y Fu(])e Fv(!)i Fx(Q)p Fu([)p Fv(h)p Fx(s)1407 2445 y Fq(1)1424 2438 y Fw(;)5 b Fx(s)1460 2445 y Fq(2)1478 2438 y Fv(i)p Fu(])372 2503 y Fz(ax)415 2510 y Fq(3)434 2503 y Fz(:)49 b Fv(8)p Fx(s)5 b Fw(:)g Fx(s)k Fv(\022)i Fx(s)372 2568 y Fz(ax)415 2575 y Fq(4)434 2568 y Fz(:)49 b Fv(8)p Fx(s)539 2575 y Fq(1)558 2568 y Fw(;)5 b Fx(s)594 2575 y Fq(2)612 2568 y Fw(;)g Fx(s)g Fw(:)g Fv(h)p Fx(s)707 2575 y Fq(1)724 2568 y Fw(;)g Fx(s)760 2575 y Fq(2)778 2568 y Fv(i)k(\022)i Fx(s)e Fv(!)i Fx(s)952 2575 y Fq(1)981 2568 y Fv(\022)f Fx(s)c Fv(^)g Fx(s)1104 2575 y Fq(2)1133 2568 y Fv(\022)k Fx(s)p Fw(:)891 3108 y Ft(Minlog7.tex;)17 b(28/04/1998;)h(12:50;)i(p.23)p eop %%Page: 24 24 24 23 bop 243 277 a Fz(24)62 b FC(BENL,)11 b(BERGER,)g(SCHWICHTENBERG,) f(SEISENBERGER,)h(ZUBER)243 404 y Fz(If)f(we)g(vie)o(w)g Fx(Q)g Fz(as)f(a)i(primiti)o(v)o(e)e(predicate)g(then)h(these)f(are)i Fy(P)p Fz(-formulas,)f(i.e.)h(uni)o(v)o(ersal)243 457 y(formulas)e(with)f(a)i(quanti\002er)o(-free)f(kernel.)g(Moreo)o(v)o (er)n(,)i(the)e(kernel)f(of)i(our)f(goal,)g Fx(A)1565 464 y Fq(0)1583 457 y Fu([)p Fx(s)p Fu(])p Fz(,)243 511 y(is)g(quanti\002er)o(-free.)i(No)o(w)m(,)f(the)f Fx(A)p Fz(-translation)f(works)h(for)h(a)h(situation)c(like)i(this.)g(Let)h (us)243 565 y(brie\003y)h(e)o(xplain)f(its)h(crucial)g(idea)g(at)g(our) g(classical)f(proof.)301 619 y(Recall)18 b(that)f(abo)o(v)o(e)g(we)h (ha)o(v)o(e)g(pro)o(v)o(ed)f Fv(?)h Fz(from)g(the)g(axioms)f(ax)1385 626 y Fq(1)1403 619 y Fw(;)5 b(:)g(:)g(:)t(;)g Fz(ax)1534 626 y Fq(4)1553 619 y Fz(,)18 b(the)243 673 y(assumptions)e Fx(P)p Fu([)o Fx(t)s Fu(])g Fz(and)i Fv(8)p Fx(s)5 b Fw(:)g Fx(A)749 680 y Fq(0)767 673 y Fu([)p Fx(s)p Fu(])13 b Fv(!)h(?)p Fz(.)19 b(Observ)o(e)f(also)f(that)g(the)h(symbol)f Fv(?)p Fz(,)i(for)243 727 y(falsity)m(,)12 b(didn')o(t)g(play)h(a)h (special)f(role)g(in)g(the)g(proof)g(\(e.g.)i(we)e(used)g(neither)g(e)o (x-falso-)243 781 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Fz(\))g(also)g(af)o (fects)g(the)243 2050 y(formula)16 b Fx(A)431 2057 y Fq(0)450 2050 y Fz(,)g(b)o(ut)g(still)f(transforms)g(the)h(formula)h Fv(8)p Fx(s)5 b Fw(:)g Fx(A)1173 2057 y Fq(0)1191 2050 y Fu([)p Fx(s)p Fu(])11 b Fv(!)j(?)j Fz(into)e(a)h(pro)o(v)o(able)243 2104 y(formula.)301 2188 y Fx(Remarks)p Fz(:)9 b(1.)h(Friedman')n(s)f (original)g(translation)f(\(Friedman,)j(1978\))e(replaces)h(e)o(v-)243 2242 y(ery)16 b(atomic)h(formula)f Fx(R)g Fz(by)g Fx(R)8 b Fv(_)h Fx(A)16 b Fz(and)g(not,)g(as)g(we)h(did,)f(by)g Fu(\()p Fx(R)c Fv(!)h Fx(A)p Fu(\))g Fv(!)g Fx(A)p Fz(.)k(But)243 2296 y(clearly)10 b(the)g(formulas)g Fx(R)5 b Fv(_)g Fx(A)11 b Fz(and)f Fu(\()p Fx(R)f Fv(!)g Fx(A)p Fu(\))g Fv(!)h Fx(A)g Fz(are)h(constructi)o(v)o(ely)d(equi)o(v)o(alent)h(as-) 243 2350 y(suming)f(decidability)g(of)i Fx(R)p Fz(.)g(W)l(e)g(ha)o(v)o (e)g(chosen)f(the)h(latter)f(v)o(ariant,)g(since)g(in)i(M)r FA(I)r(N)r(L)r(O)r(G)243 2404 y Fz(we)g(prefer)h(reasoning)e(with)g (implications)f(rather)j(with)e(disjunctions.)301 2458 y(2.)h(Another)f(way)g(to)g(see)h(the)g Fx(A)p Fz(-translation,)e(is)h (to)g(say)h(that)f(in)g(constructi)o(v)o(e)g(logic)243 2512 y(we)15 b(may)g(pass)f(from)h(a)g(proof)f(of)g Fv(:8)p Fx(x)p Fv(:)p Fx(D)p Fu([)p Fx(x)p Fu(])h Fz(\()p Fx(D)g Fz(quanti\002er)o(-free\),)g(or)g(equi)o(v)o(alently)243 2566 y Fv(::9)328 2550 y Fp(\003)352 2566 y Fx(D)p Fu([)p Fx(x)p Fu(])p Fz(,)c(to)g(a)h(proof)e(of)i Fv(9)716 2550 y Fp(\003)735 2566 y Fx(x)5 b(D)p Fu([)p Fx(x)p Fu(])p Fz(.)11 b(This)f(is)h(kno)o(wn)f(as)h(Marko)o(v')n(s)f(rule.)891 3108 y Ft(Minlog7.tex;)17 b(28/04/1998;)h(12:50;)i(p.24)p eop %%Page: 25 25 25 24 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)138 b Fz(25)243 404 y Fx(The)11 b(tr)o(anslated)e(pr)n(oof)243 457 y Fz(In)17 b(M)r FA(I)r(N)r(L)r(O)r (G)j Fz(we)c(ha)o(v)o(e)g(implemented)g(a)g(re\002nement)h(of)f(the)f Fx(A)p Fz(-translation)f(which)243 511 y(does)j(not)f(replace)i(all)f (atomic)g(formulas)g Fx(R)g Fz(by)g Fu(\()p Fx(R)c Fv(!)h Fx(A)p Fu(\))e Fv(!)i Fx(A)p Fz(.)k(In)f(our)g(e)o(xample)243 565 y(this)c(is)g(only)g(necessary)h(for)g(formulas)g Fx(Q)p Fu([)p Fv(\001)p Fu(])p Fz(;)e(in)i(general,)g(it)g(can)g(be)g (decided)f(easily)243 619 y(whether)g(an)h(atomic)g(formula)f(has)h(to) f(be)h(replaced)g(or)g(not.)f(F)o(or)i(more)f(information)243 673 y(on)d(this)f(re\002nement)h(we)h(refer)g(to)f(\(Ber)o(ger)h(and)f (Schwichtenber)o(g,)g(1995\).)301 727 y(No)o(w)m(,)h(the)h(M)r FA(I)r(N)r(L)r(O)r(G)j Fz(system)c(transforms)g(our)g(classical)g (proof)g(into)f(a)h(construc-)243 781 y(ti)o(v)o(e)d(one.)i(W)l(e)f (sho)o(w)f(this)g(automatically)g(generated)h(proof)g(in)f(tree)i(form) f(belo)o(w)m(.)g(Due)243 835 y(to)h(lack)h(of)f(space)h(we)g (graphically)f(contracted)g(consecuti)o(v)o(e)g(applications)e(of)j (elimi-)243 889 y(nation)7 b(rules)h(to)g(one)h(rule.)g(Similarly)f 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pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B 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1544 y Ff(:)g Fh(P)6 b Ff([)p Fh(s)1270 1550 y Fg(1)1288 1544 y Ff(])1129 1593 y Fh(u)1153 1599 y Fg(6)1183 1593 y Ff(:)11 b Fh(Q)p Ff([)p Fh(s)1270 1599 y Fg(1)1289 1593 y Ff(])1434 1544 y Fh(u)1458 1550 y Fg(7)1487 1544 y Ff(:)g Fh(P)6 b Ff([)p Fh(s)1574 1550 y Fg(2)1593 1544 y Ff(])1434 1593 y Fh(u)1458 1599 y Fg(8)1487 1593 y Ff(:)11 b Fh(Q)p Ff([)p Fh(s)1574 1599 y Fg(2)1593 1593 y Ff(])1738 1544 y Fh(u)1762 1550 y Fg(9)1792 1544 y Ff(:)g Fd(:)p Fh(P)6 b Ff([)p Fh(s)1907 1550 y Fg(1)1925 1544 y Ff(])1738 1593 y Fh(u)1762 1599 y Fg(10)1808 1593 y Ff(:)11 b Fd(:)p Fh(P)6 b Ff([)p Fh(s)1923 1599 y Fg(2)1941 1593 y Ff(])2042 1544 y Fh(u)2066 1550 y Fg(11)2113 1544 y Ff(:)11 b Fh(Q)p Ff([)p Fh(s)2200 1550 y Fg(1)2218 1544 y Ff(])2042 1593 y Fh(u)2066 1599 y Fg(12)2113 1593 y Ff(:)g Fh(Q)p Ff([)p Fh(s)2200 1599 y Fg(2)2218 1593 y Ff(])p eop end userdict /end-hook known{end-hook}if %%EndDocument @endspecial 844 3378 a Fr(F)n(igur)n(e)g(0.)891 3108 y Ft(Minlog7.tex;)17 b(28/04/1998;)h(12:50;)i(p.26)p eop %%Page: 27 27 27 26 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)138 b Fz(27)243 404 y Fx(The)11 b(e)o(xtr)o(acted)g(pr)n(ogr)o(am)243 457 y Fz(From)h(the)f(translated) f(proof)g(we)i(obtain)e(the)h(follo)o(wing)e(program:)597 553 y Fx(f)d Fu(\()o Fx(t)s Fu(\))48 b(=)i Fx(g)p Fu(\()o Fx(t)s Fw(;)t Fx(t)s Fu(\))447 618 y Fx(g)p Fu(\()p Fn(leaf)s Fu(\()p Fx(n)p Fu(\))p Fw(;)t Fx(t)s Fu(\))d(=)i Fx(t)450 683 y(g)p Fu(\()p Fv(h)p Fx(s)527 690 y Fq(1)544 683 y Fw(;)5 b Fx(s)580 690 y Fq(2)598 683 y Fv(i)p Fw(;)t Fx(t)s Fu(\))48 b(=)i Fi(if)108 b Fx(P)p Fu([)p Fx(s)995 690 y Fq(1)1012 683 y Fu(])800 747 y Fi(then)47 b Fx(g)p Fu(\()p Fx(s)991 754 y Fq(1)1009 747 y Fw(;)5 b Fx(s)1045 754 y Fq(1)1063 747 y Fu(\))800 812 y Fi(else)62 b(if)107 b Fx(P)p Fu([)p Fx(s)1127 819 y Fq(2)1145 812 y Fu(])948 877 y Fi(then)47 b Fx(g)p Fu(\()p Fx(s)1139 884 y Fq(2)1156 877 y Fw(;)5 b Fx(s)1192 884 y Fq(2)1210 877 y Fu(\))948 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b(e)o(xpect)h(from)g(this)f(hopefully)f(simpler)h(proof)g(a)h (hopefully)e(optimized)h(program)243 1973 y(adapted)k(to)g(the)h (restriction)c Fx(C)q Fu([)o Fx(t)s Fu(])p Fz(.)j(Such)h(a)g (simpli\002ed)f(proof)g(may)h(be)g(obtained)f(as)243 2027 y(follo)o(ws:)10 b(in)h(the)h(subproof)f Fx(M)732 2010 y Fp(0)743 2027 y Fu([)p Fx(u)779 2010 y Fl(B)801 2027 y Fu(])g Fz(considered)h(abo)o(v)o(e)g(we)g(replace)h(the)e (assumption)243 2081 y Fx(u)266 2064 y Fl(B)304 2081 y Fz(by)16 b(a)g(proof)g Fx(K)550 2064 y Fl(B)589 2081 y Fz(which)f(uses)h(the)f(additional)f(assumption)f Fx(C)q Fu([)o Fx(t)s Fu(])i Fz(\(and)h(possibly)243 2135 y(other)e (assumptions)f(v)o(alid)g(at)i(the)f(particular)g(occurrence)h(of)g Fx(u)p Fz(\),)g(and)f(then)g(replace)243 2188 y(the)9 b(modi\002ed)g(proof)g Fx(M)r Fu([)p Fy(l)p Fx(u)688 2172 y Fl(B)709 2188 y Fx(M)749 2172 y Fp(0)760 2188 y Fu([)p Fx(K)806 2172 y Fl(B)828 2188 y Fu(]])f Fz(by)h Fx(M)957 2172 y Fp(0)968 2188 y Fu([)p Fx(K)1014 2172 y Fl(B)1036 2188 y 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(sho)o(ws)g(that)g(it)g(is)g(essential)g(that)g(the)g(pruning)243 2512 y(operation)j(is)g(done)h(on)f Fx(pr)n(oofs)p Fz(,)g(since)h (optimized)f Fx(pr)n(ogr)o(ams)f Fz(will)g(always)h(compute)243 2566 y(the)f(same)g(result)g(on)g(the)g(restricted)f(inputs.)891 3108 y Ft(Minlog7.tex;)17 b(28/04/1998;)h(12:50;)i(p.27)p eop %%Page: 28 28 28 27 bop 243 277 a Fz(28)62 b FC(BENL,)11 b(BERGER,)g(SCHWICHTENBERG,) f(SEISENBERGER,)h(ZUBER)243 404 y Fx(F)n(irst)e(e)o(xample)243 457 y Fz(Assume)i(we)g(enrich)g(our)g(speci\002cation)f(by)h(the)g (additional)e(information)533 539 y Fx(C)562 546 y Fq(1)581 539 y Fu([)o Fx(t)s Fu(])g Fz(:)p Fv(\021)h(8)p Fx(s)732 546 y Fq(1)750 539 y Fw(;)5 b Fx(s)786 546 y Fq(2)809 539 y Fw(:)g Fv(h)p Fx(s)863 546 y Fq(1)881 539 y Fw(;)g Fx(s)917 546 y Fq(2)935 539 y Fv(i)10 b(\022)f Fx(t)k Fv(!)d Fx(P)p Fu([)p Fx(s)1147 546 y Fq(1)1165 539 y Fu(])g Fv(!)g Fx(P)p Fu([)p Fx(s)1302 546 y Fq(2)1320 539 y Fu(])p Fw(;)243 621 y Fz(i.e.)h(if)g Fx(P)g Fz(holds)f(for)h(a)g 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y Fz(and)k Fv(h)p Fx(s)359 952 y Fq(1)377 945 y Fw(;)5 b Fx(s)413 952 y Fq(2)431 945 y Fv(i)11 b(\022)g Fx(t)18 b Fz(which)13 b(are)i(both)e(v)o(alid)h(at)g(that)f(point.)h(This)f(means)i(that)e (the)h(case)243 999 y(analysis)h(according)h(to)f(whether)h Fx(P)p Fu([)p Fx(s)875 1006 y Fq(1)893 999 y Fu(])g Fz(holds)f(can)i (be)f(remo)o(v)o(ed)i(from)f(the)f(proof,)243 1053 y(since)10 b(we)g(ha)o(v)o(e)h(sho)o(wn)e(that)h(in)g(fact)g Fv(:)p Fx(P)p Fu([)p Fx(s)925 1060 y Fq(1)943 1053 y Fu(])h Fz(holds.)e(In)i(the)f(proof)g(tree)h(the)f(simpli\002ed)243 1107 y(proof)i(is)h(obtained)e(by)i(replacing)f(the)h(subproof)e Fx(M)k Fz(by)d Fx(M)1202 1090 y Fp(0)1213 1107 y Fz(,)i(where)f(the)f (assumption)243 1161 y Fx(u)266 1168 y Fq(9)296 1161 y Fz(is)e(replaced)i(by)e(a)i(proof)f(using)d Fx(C)838 1168 y Fq(1)856 1161 y Fu([)o Fx(t)s Fu(])p Fz(,)j Fx(u)942 1168 y Fq(3)971 1161 y Fz(and)g Fx(u)1071 1168 y Fq(10)1106 1161 y Fz(.)301 1215 y(The)g(program)g(e)o(xtracted)h(from)f(the)g 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Fz(This)f(has)h(an)h(e)o(xtreme)g(ef)o(fect)g(on)f(the)g(proof,)h (since)f(already)g(the)g(outer)g(case)h(analysis)243 1924 y(on)j Fx(P)p Fu([)p Fx(s)363 1931 y Fq(1)380 1924 y Fu(])g Fz(is)g(decided)g(positi)o(v)o(ely)m(.)e(Hence)i(we)h(get)f(a) g(proof)g(without)e(case)j(analysis.)243 1978 y(This)d(means)i(that)e (in)h(the)g(proof)g(tree)h(we)g(replace)f(the)g(subproof)f Fx(M)k Fz(by)d Fx(M)1481 1962 y Fp(00)1501 1978 y Fz(,)g(where)243 2032 y(the)9 b(assumption)e Fx(u)547 2040 y Fq(5)575 2032 y Fz(is)h(replaced)i(by)e(a)i(proof)f(using)d Fx(C)1105 2039 y Fq(2)1123 2032 y Fu([)o Fx(t)s Fu(])p Fz(,)j Fx(u)1207 2039 y Fq(3)1234 2032 y Fz(and)g(ax)1352 2039 y Fq(4)1371 2032 y Fz(.)h(The)f(e)o(xtracted)243 2086 y(program)i(is)g(simply)832 2174 y Fx(f)6 b Fu(\()o Fx(t)s Fu(\))49 b(=)h Fx(g)p Fu(\()o Fx(t)s Fw(;)t Fx(t)s Fu(\))683 2238 y Fx(g)p Fu(\()p Fn(leaf)r Fu(\()p Fx(n)p Fu(\))p Fw(;)t Fx(t)s Fu(\))e(=)h Fx(t)685 2303 y(g)p Fu(\()p Fv(h)p Fx(s)762 2310 y Fq(1)780 2303 y Fw(;)5 b Fx(s)816 2310 y Fq(2)834 2303 y Fv(i)p Fw(;)t Fx(t)s Fu(\))48 b(=)i Fx(g)p Fu(\()p Fx(s)1095 2310 y Fq(1)1113 2303 y Fw(;)5 b Fx(s)1149 2310 y Fq(1)1167 2303 y Fu(\))p Fw(:)891 3108 y Ft(Minlog7.tex;)17 b(28/04/1998;)h(12:50;)i(p.28)p eop %%Page: 29 29 29 28 bop 426 277 a FC(PR)n(OGRAM)10 b(DEVELOPMENT)i(IN)e(THE)i(M)r(I)r (N)r(L)r(O)r(G)g(SYSTEM)138 b Fz(29)243 404 y Fx(Thir)n(d)10 b(e)o(xample)243 457 y Fz(Our)h(last)f(pruning)g(condition)f(is)727 557 y Fx(C)756 564 y Fq(3)785 557 y Fz(:)p Fv(\021)h(8)p Fx(s)5 b Fw(:)g Fv(:)p Fx(P)p Fu([)p Fx(s)p Fu(])k Fv(!)h Fx(Q)p Fu([)p Fx(s)p Fu(])243 657 y Fz(which)f(is)h(in)g(fact)g(an)h(e) o(xtra)f(condition)f(on)h Fx(P)p Fz(.)g(W)l(e)h(look)e(again)h(at)g (the)h Fx(case)f Fv(:)p Fx(P)p Fu([)p Fx(s)1531 664 y Fq(1)1549 657 y Fu(])g Fz(and)243 711 y(therein)e(at)h(the)g Fx(sub-case)g Fv(:)p Fx(P)p Fu([)p Fx(s)743 718 y Fq(2)761 711 y Fu(])p Fz(.)g(T)l(o)g(pro)o(v)o(e)g Fx(A)g Fz(we)h(used)e(both)g (induction)f(hypotheses,)243 765 y Fx(B)p Fu([)p Fx(s)302 772 y Fq(1)319 765 y Fu(])14 b Fz(and)g Fx(B)p Fu([)p Fx(s)485 772 y Fq(2)503 765 y Fu(])p Fz(.)g(W)l(e)h(ended)f(up)g(in)g (a)g(situation)e(where)i(we)h(had)f(to)f(pro)o(v)o(e)i Fx(A)f Fz(under)243 819 y(the)9 b(e)o(xtra)h(assumptions)e Fx(Q)p Fu([)p Fx(s)707 826 y Fq(1)725 819 y Fu(])h Fz(\(from)h(ih)896 826 y Fq(1)914 819 y Fz(\))g(and)g Fx(Q)p Fu([)p Fx(s)1079 826 y Fq(2)1097 819 y Fu(])f Fz(\(from)h(ih)1268 826 y Fq(2)1286 819 y Fz(\).)h(No)o(w)e(from)h Fv(:)p Fx(P)p Fu([)p Fx(s)1607 826 y Fq(1)1625 819 y Fu(])243 873 y Fz(and)f Fv(:)p Fx(P)p Fu([)p Fx(s)407 880 y Fq(2)425 873 y Fu(])h Fz(and)g(the)f(pruning)g(condition)d Fx(C)950 880 y Fq(3)978 873 y Fz(we)k(can)h Fx(pr)n(ove)e(Q)p Fu([)p Fx(s)1290 880 y Fq(1)1307 873 y Fu(])h Fz(and)g Fx(Q)p Fu([)p Fx(s)1470 880 y Fq(2)1487 873 y Fu(])p Fz(.)g(Hence)243 927 y(we)k(do)h(not)e(need)i(to)f(use)g(the)g (induction)f(hypotheses)f(at)i(that)g(point.)g(F)o(or)h(the)f(proof)243 981 y(tree)d(this)g(means)g(that)g(the)g(subproof)f Fx(N)15 b Fz(is)c(replaced)g(by)g Fx(N)1189 964 y Fp(0)1200 981 y Fz(,)h(where)g(the)f(assumptions)243 1034 y Fx(u)266 1041 y Fq(11)315 1034 y Fz(and)k Fx(u)419 1041 y Fq(12)468 1034 y Fz(are)g(replaced)g(by)f(a)h(proof)f(using)d Fx(C)1055 1041 y Fq(3)1074 1034 y Fz(,)j Fx(u)1122 1041 y Fq(9)1156 1034 y Fz(and)g Fx(u)1259 1041 y Fq(10)1294 1034 y Fz(.)h(The)f(ef)o (fect)h(on)f(the)243 1088 y(e)o(xtracted)d(program)g(is)g(that)g(the)g (nested)f(recursi)o(v)o(e)h(call)g(disappears:)699 1194 y Fx(f)6 b Fu(\()o Fx(t)s Fu(\))49 b(=)h Fx(g)p Fu(\()o Fx(t)s Fw(;)t Fx(t)s Fu(\))550 1258 y Fx(g)p Fu(\()p Fn(leaf)r Fu(\()p Fx(n)p Fu(\))p Fw(;)t Fx(t)s Fu(\))e(=)h Fx(t)552 1323 y(g)p Fu(\()p Fv(h)p Fx(s)629 1330 y Fq(1)647 1323 y Fw(;)5 b Fx(s)683 1330 y Fq(2)701 1323 y Fv(i)p Fw(;)t Fx(t)s Fu(\))48 b(=)i Fi(if)107 b Fx(P)p Fu([)p Fx(s)1097 1330 y Fq(1)1115 1323 y Fu(])903 1388 y Fi(then)46 b Fx(g)p Fu(\()p Fx(s)1093 1395 y Fq(1)1111 1388 y Fw(;)5 b 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