diff -r 0b4a5595cbe4 -r 680070975206 Publications/clc-06.ps --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Publications/clc-06.ps Sat Oct 22 12:11:38 2011 +0100 @@ -0,0 +1,4094 @@ +%!PS-Adobe-2.0 +%%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software +%%Title: double.dvi +%%Pages: 20 +%%PageOrder: Ascend +%%BoundingBox: 0 0 595 842 +%%DocumentFonts: Times-Bold Times-Roman Courier Times-Italic +%%DocumentPaperSizes: a4 +%%EndComments +%DVIPSWebPage: (www.radicaleye.com) +%DVIPSCommandLine: dvips double.dvi -o double.ps +%DVIPSParameters: dpi=600 +%DVIPSSource: TeX output 2006.06.26:1209 +%%BeginProcSet: tex.pro 0 0 +%! +/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S +N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 +mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 +0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ +landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize +mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ +matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round +exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ +statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] +N/FBB[0 0 0 0]N/nn 0 N/IEn 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 IEn N end A{/foo setfont}2 +array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/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 A +definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get +}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} +B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr +1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S +/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy +setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask +restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn +/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 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 A 1 get A +mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ +SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ +userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X +1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 +index cvrs cvn put}for pop 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/Rx 0 N/Ry 0 N/V{}B/RV/v{ +/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) +(LaserWriter 16/600)]{A length product length le{A length product exch 0 +exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse +end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask +grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} +imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round +exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto +fill grestore}B/a{moveto}B/delta 0 N/tail{A/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 +%%BeginProcSet: 8r.enc 0 0 +% File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 +% +% @@psencodingfile@{ +% author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, +% W. Schmidt, P. Lehman", +% version = "2.0", +% date = "30 October 2002", +% filename = "8r.enc", +% email = "tex-fonts@@tug.org", +% docstring = "This is the encoding vector for Type1 and TrueType +% fonts to be used with TeX. This file is part of the +% PSNFSS bundle, version 9" +% @} +% +% The 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, Windows ANSI including the euro symbol, +% MacRoman, and some 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. +% These are /dotlessj /ff /ffi /ffl. +% +% (4) hyphen appears twice for compatibility with both ASCII and Windows. +% +% (5) /Euro was assigned to 128, as in Windows ANSI +% +% (6) Missing characters from MacRoman encoding incorporated as follows: +% +% PostScript MacRoman TeXBase1 +% -------------- -------------- -------------- +% /notequal 173 0x16 +% /infinity 176 0x17 +% /lessequal 178 0x18 +% /greaterequal 179 0x19 +% /partialdiff 182 0x1A +% /summation 183 0x1B +% /product 184 0x1C +% /pi 185 0x1D +% /integral 186 0x81 +% /Omega 189 0x8D +% /radical 195 0x8E +% /approxequal 197 0x8F +% /Delta 198 0x9D +% /lozenge 215 0x9E +% +/TeXBase1Encoding [ +% 0x00 + /.notdef /dotaccent /fi /fl + /fraction /hungarumlaut /Lslash /lslash + /ogonek /ring /.notdef /breve + /minus /.notdef /Zcaron /zcaron +% 0x10 + /caron /dotlessi /dotlessj /ff + /ffi /ffl /notequal /infinity + /lessequal /greaterequal /partialdiff /summation + /product /pi /grave /quotesingle +% 0x20 + /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 +% 0x80 + /Euro /integral /quotesinglbase /florin + /quotedblbase /ellipsis /dagger /daggerdbl + /circumflex /perthousand /Scaron /guilsinglleft + /OE /Omega /radical /approxequal +% 0x90 + /.notdef /.notdef /.notdef /quotedblleft + /quotedblright /bullet /endash /emdash + /tilde /trademark /scaron /guilsinglright + /oe /Delta /lozenge /Ydieresis +% 0xA0 + /.notdef /exclamdown /cent /sterling + /currency /yen /brokenbar /section + /dieresis /copyright /ordfeminine /guillemotleft + /logicalnot /hyphen /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 + + +%%EndProcSet +%%BeginProcSet: texps.pro 0 0 +%! +TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 +index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll +exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 +ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ +pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get +div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type +/nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end +definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup +sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll +mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ +exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} +forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def +end + +%%EndProcSet +%%BeginProcSet: special.pro 0 0 +%! +TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N +/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N +/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N +/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ +/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho +X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B +/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ +/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known +{userdict/md get type/dicttype eq{userdict begin md length 10 add md +maxlength ge{/md md dup length 20 add dict copy def}if end md begin +/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S +atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ +itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll +transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll +curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf +pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} +if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 +-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 +get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip +yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub +neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ +noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop +90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get +neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr +1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr +2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 +-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S +TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ +Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale +}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState +save N userdict maxlength dict begin/magscale true def normalscale +currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts +/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x +psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx +psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub +TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def +@MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll +newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto +closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N +/@beginspecial{SDict begin/SpecialSave save N gsave normalscale +currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} +N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs +neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate +rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse +scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg +lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx +ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N +/setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ +pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave +restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B +/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 +setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY +moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix +matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc +savematrix setmatrix}N end + +%%EndProcSet +TeXDict begin 39139632 55387786 1000 600 600 (double.dvi) +@start +%DVIPSBitmapFont: Fa cmsy5 5 1 +/Fa 1 25 df<07E00001801FFC0001803FFE0001803FFF0001807C3FC003807007E00380 +E003F00700E001FE1F00C0007FFE00C0003FFE00C0001FFC00C00003F000210C7B8F2D> +24 D E +%EndDVIPSBitmapFont +/Fb 139[28 4[42 46 4[23 3[37 32[60 18[42 2[21 43[46 2[{ +TeXBase1Encoding ReEncodeFont}9 83.022 /Times-Bold rf +%DVIPSBitmapFont: Fc cmr5 5 2 +/Fc 2 51 df<00600001E0000FE000FFE000F1E00001E00001E00001E00001E00001E000 +01E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E000 +01E00001E00001E00001E0007FFF807FFF80111C7B9B1C>49 D<03FC000FFF003C0FC070 +03E07801F0FC00F0FC00F8FC00F8FC00787800780000F80000F00000F00001E00003C000 +0780000F00001C0000380000E00001C0180380180600180C00383FFFF07FFFF0FFFFF0FF +FFF0151C7D9B1C>I E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fd cmbx9 9 1 +/Fd 1 62 df<7FFFFFFFFFFFFF80FFFFFFFFFFFFFFC0FFFFFFFFFFFFFFC0FFFFFFFFFFFF +FFC07FFFFFFFFFFFFF800000000000000000000000000000000000000000000000000000 +000000000000000000000000000000000000000000000000000000000000000000000000 +000000000000000000000000000000000000000000000000000000000000000000000000 +0000000000007FFFFFFFFFFFFF80FFFFFFFFFFFFFFC0FFFFFFFFFFFFFFC0FFFFFFFFFFFF +FFC07FFFFFFFFFFFFF803A177B9D45>61 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fe cmmib6 6 2 +/Fe 2 83 df<00FFFFF00000FFFFF00000FFFFF000000FF80000000FF80000000FF00000 +000FF00000001FF00000001FF00000001FE00000001FE00000003FE00000003FE0000000 +3FC00000003FC00000007FC00000007FC00000007F800000007F80000000FF80000000FF +80000000FF00006000FF0000F001FF0001F001FF0001E001FE0003E001FE0007C003FE00 +0FC003FE001F8003FC003F8003FC01FF80FFFFFFFF00FFFFFFFF00FFFFFFFE0024227CA1 +2E>76 D<00FFFFFFE00000FFFFFFFE0000FFFFFFFF80000FF8007FC0000FF8001FE0000F +F0000FF0000FF0000FF0001FF0000FF0001FF0000FF0001FE0000FF0001FE0000FF0003F +E0001FE0003FE0003FC0003FC0007F80003FC003FF00007FFFFFFC00007FFFFFE000007F +FFFFF800007F800FFC0000FF8003FE0000FF8003FE0000FF0001FE0000FF0001FE0001FF +0003FE0001FF0003FE0001FE0003FC0001FE0003FC0003FE0003FC0003FE0007FC1E03FC +0007F81E03FC0007FC3EFFFFE003FC3CFFFFE001FFF8FFFFE000FFF0000000001FC02F23 +7CA134>82 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Ff cmbsy6 6 2 +/Ff 2 49 df<003C00003C00003E00003C00703C0EFC3C3FFE3C7FFFBDFF7FDBFE0FFFF0 +01FF8001FF800FFFF07FDBFEFFBDFFFE3C7FFC3C3F703C0E003C00003E00003C00003C00 +18167B9723>3 D<00F801FC03FE03FE03FE07FE07FE07FC07FC0FF80FF80FF01FF01FE0 +1FE01FC03FC03F803F807F007F007E007E00FC00FC0078000F1A7D9B15>48 +D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fg cmmib9 9 1 +/Fg 1 26 df<001FFFFFFFFE007FFFFFFFFF01FFFFFFFFFF07FFFFFFFFFF0FFFFFFFFFFF +1FFFFFFFFFFF3FFFFFFFFFFE3FFFFFFFFFF87E0078078000F800F8078000F000F80F8000 +6001F00F80000001F00F80000001F00F80000003F00F80000003E01F80000007E01F8000 +0007E01F8000000FE01FC000000FC01FC000001FC01FC000001FC01FE000003FC01FE000 +003F801FE000007F801FF000007F801FF00000FF801FF80001FF001FF80001FF001FFC00 +01FF001FFC0001FF000FF80001FE000FF80001FC0007F00000700003C00030227DA035> +25 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fh cmr9 9 2 +/Fh 2 42 df<0000C00001C0000380000F00000E00001C00003C0000780000F00000F000 +01E00003C00003C00007C0000780000F80000F00001F00001F00001E00003E00003E0000 +3E00003C00007C00007C00007C00007C00007C0000F80000F80000F80000F80000F80000 +F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F800007C0000 +7C00007C00007C00007C00003C00003E00003E00003E00001E00001F00001F00000F0000 +0F800007800007C00003C00003C00001E00000F00000F000007800003C00001C00000E00 +000F000003800001C00000C0124A79B71E>40 DI +E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fi cmmi6 6 7 +/Fi 7 122 df<00FFFF800000FFFF00000007C000000007C00000000F800000000F8000 +00000F800000000F800000001F000000001F000000001F000000001F000000003E000000 +003E000000003E000000003E000000007C000000007C000000007C000000007C00000000 +F800000000F800018000F800018000F800030001F000030001F000060001F000060001F0 +000E0003E0001C0003E0003C0003E000F80007E003F800FFFFFFF800FFFFFFF00021227C +A12A>76 D<00FFFFFC0000FFFFFF800007C00FC00007C003F0000F8001F0000F8001F800 +0F8000F8000F8000F8001F0001F8001F0001F8001F0001F8001F0003F0003E0003E0003E +0007C0003E001F80003E007E00007FFFF800007FFFE000007C01F000007C00780000F800 +7C0000F8003C0000F8003E0000F8003E0001F0007E0001F0007E0001F0007E0001F0007E +0003E000FC0003E000FE0603E000FE0607E0007E0CFFFE007E18FFFE003FF000000007E0 +27237CA12E>82 D<0F007E00FC001F81FF83FF0031C383C7078061EE03CC038061EC01F8 +03C0C1F801F003C0C1F001F003C0C1E001E003C003E003C0078003C003C0078003C003C0 +078003C003C00F00078007800F00078007800F00078007801E04078007801E060F000F00 +1E0C0F000F003C0C0F000F003C180F000F003C181E001E001C701E001E001FE00C000C00 +07802F177D9536>109 D<001F0200FF8601E0CE03807E07007C0F003C1E003C3E003C3C +00787C00787C00787C0078F800F0F800F0F800F0F800F0F801E07801E07803E03807E01C +1FC00FFBC007E3C00003C0000780000780000780000780000F00000F0000FFF001FFF017 +207E951C>113 D<003F8000FFE001E0F00380300300700700F00700F007804007C00007 +FE0003FF8001FFC0003FE00003F03000F07800F0F800E0F800E0F000C06001C0780F803F +FE0007F80014177D951D>115 D<00300000780000F00000F00000F00000F00001E00001 +E00001E00001E00003C000FFFF80FFFF8003C0000780000780000780000780000F00000F +00000F00000F00001E00001E00001E01001E01803C03003C06003C06003C0C001C38000F +F00007C00011217D9F18>I<07C000000FE0030018F0078030F0078060F00F00C0F00F00 +C0F00F00C1E00F0001E01E0003C01E0003C01E0003C01E0007803C0007803C0007803C00 +07803C0007807800078078000780F8000781F80003C3F00001FFF00000FCF0000000F000 +0001E0001E01E0003E03C0003E0380003C070000300E0000383C00001FF8000007C00000 +19217D9520>121 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fj cmmi9 9 9 +/Fj 9 117 df<007C000000007F800000001FE00000000FE000000007F000000007F000 +000003F800000003F800000003F800000001FC00000001FC00000001FC00000000FE0000 +0000FE00000000FF000000007F000000007F000000003F800000003F800000003F800000 +001FC00000001FC00000001FE00000000FE00000000FE000000007F000000007F0000000 +07F000000007F80000000FF80000001FF80000003DFC00000079FC000000F8FE000001F0 +FE000003E0FE000007C07F00000F807F00001F007F00003E003F80007E003F8000FC003F +C001F8001FC003F0001FC007E0000FE01FC0000FE03FC0000FE07F800007F0FF000007F0 +FE000007F8FC000003F8F8000001FCF0000000FC26357CB32D>21 +D<000700000000000FC0003800001FC0007C00001FC000FC00001F8000FC00001F8000FC +00003F8001FC00003F8001FC00003F0001F800003F0001F800007F0003F800007F0003F8 +00007E0003F000007E0003F00000FE0007F00000FE0007F00000FC0007E00000FC0007E0 +0001FC000FE00001FC000FE00001F8000FC00001F8000FC08003F8001FC0C003F8001FC1 +C003F0001F818003F0001F818007F0003F838007F8007F830007F8007F030007F800FF07 +000FFC03CF86000FFE070F8E000FDFFE07FC000FC3F801F0001FC0000000001FC0000000 +001F80000000001F80000000003F80000000003F80000000003F00000000003F00000000 +007F00000000007F00000000007E00000000007E0000000000FE0000000000FE00000000 +00FC00000000003800000000002A327FA02E>I<3C007E00FF00FF00FF80FF807F803D80 +0180018001800180038003000300070006000E000C001C0038007000600009177A8715> +59 D<0000000003000000000000070000000000000F0000000000000F0000000000001F +8000000000001F8000000000003F8000000000007F8000000000007F800000000000FF80 +0000000000FF800000000001BF8000000000033F8000000000033FC000000000063FC000 +000000061FC0000000000C1FC000000000181FC000000000181FC000000000301FC00000 +0000701FC000000000601FC000000000C01FC000000000C01FE000000001801FE0000000 +03800FE000000003000FE000000006000FE000000006000FE00000000C000FE00000001C +000FE000000018000FE000000030000FF000000030000FF00000007FFFFFF0000000FFFF +FFF0000000FFFFFFF0000001800007F0000001800007F0000003000007F0000006000007 +F0000006000007F800000C000007F800000C000003F8000018000003F8000030000003F8 +000070000003F8000060000003F80000E0000003F80001E0000003F80007F0000007FC00 +FFFF0001FFFFF0FFFF0001FFFFF0FFFE0001FFFFF034367DB53A>65 +D<00007F000003FFC0000FC0F0003F0038007C003800F800F801F001F803E003F807E003 +F80FC003F80F8001F01F8000003F8000003F0000003F0000007F0000007E0000007E0000 +007E0000007E000000FC000000FC000000FC0000007C0000007C00000C7C00001C7C0000 +383E0000703E0000E01F0003C00F800F0007C07E0001FFF000007F80001E227EA021>99 +D<01E000FE000007F803FF80000E3E0F07E0001C3E3C03F000181F7001F000381FE001F0 +00303FC001F800703FC001F800603F8001F800603F0001F800603F0003F800E07F0003F0 +00407E0003F000007E0003F000007E0007F00000FE0007E00000FC0007E00000FC000FE0 +0000FC000FC00001FC000FC00001F8001FC00001F8001F808001F8001F81C003F8003F81 +8003F0003F018003F0003F038003F0007F030007F0007E030007E0007E070007E0003E0E +0007E0003E1C000FE0001E38000FC0000FF00003800003C0002A227EA02E>110 +D<00007F00000003FFC000000FC1F000003F00F800007C007C0000F8003E0001F0003E00 +03E0001F0007E0001F000FC0001F000F80001F001F80001F803F80001F803F00001F803F +00003F007F00003F007E00003F007E00003F007E00007F007E00007E00FC00007E00FC00 +00FE00FC0000FC007C0000F8007C0001F8007C0003F0007C0003E0003E0007C0003E000F +80001F001F00000F807E000007C1F8000001FFE00000007F00000021227EA025>I<03E0 +03E00FF81FF81C7C3C1C187C703E383EE0FE303FC0FE307F80FE707F00FC607E00FC607E +0070E07E0000C0FE000040FC000000FC000000FC000001FC000001F8000001F8000001F8 +000003F8000003F0000003F0000003F0000007F0000007E0000007E0000007E000000FE0 +00000FC000000FC000000FC000001FC000001F800000070000001F227EA023>114 +D<000380000FC0000FC0000FC0001FC0001FC0001F80001F80003F80003F80003F00003F +00007F00007F00007E007FFFFE7FFFFEFFFFFE00FC0000FC0001FC0001FC0001F80001F8 +0003F80003F80003F00003F00007F00007F00007E00007E0000FE0000FE0000FC0000FC0 +081FC01C1FC0181F80181F80381F80701F80601F00E01F01C00F83800F870007FE0001F8 +0017307FAE1C>116 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fk cmmi5 5 1 +/Fk 1 106 df<007000F800F800F000E00000000000000000000000000F801FC031E061 +E061E0C3C003C00780078007800F000F081E181E181E301E700FE007800D1D7D9C16> +105 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fl cmr7 7 3 +/Fl 3 52 df<00380000780001F8001FF800FEF800E0F80000F80000F80000F80000F800 +00F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F800 +00F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F800 +00F80001FC00FFFFF8FFFFF815267BA521>49 D<00FF000003FFE0000E03F0001800F800 +30007C0060007E0078003F00FC003F00FE001F80FE001F80FE001F80FE001F807C001F80 +00001F8000001F0000003F0000003E0000007E0000007C000000F8000001F0000003E000 +0003C00000078000000E0000001C0000003800000070018000E001800180018003000300 +060003000C0003001FFFFF003FFFFF007FFFFE00FFFFFE00FFFFFE0019267DA521>I<00 +FF000003FFE0000F01F8001C007C0030007E003C003E007E003F007E003F007E003F007E +003F003C003F0000003E0000007E0000007C000000F8000001F0000007E00001FF800001 +FF00000001E0000000F00000007C0000003E0000003F0000001F0000001F8000001F8038 +001F807C001F80FE001F80FE001F80FE001F00FC003F0078003E0070007C003800F8001F +01F00007FFC00000FF000019277DA521>I E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fm cmr10 10 7 +/Fm 7 62 df<0000600000E00001C0000380000700000E00001E00003C00007800007800 +00F00001E00001E00003C00003C00007C0000780000F80000F00000F00001F00001E0000 +1E00003E00003E00003E00007C00007C00007C00007C00007C00007C0000F80000F80000 +F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000 +F80000F80000F80000F800007C00007C00007C00007C00007C00007C00003E00003E0000 +3E00001E00001E00001F00000F00000F00000F800007800007C00003C00003C00001E000 +01E00000F000007800007800003C00001E00000E000007000003800001C00000E0000060 +135278BD20>40 DI<0001C0000003C0000007C000001FC00000FFC000FFFFC000FFFFC0 +00FF1FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC0 +00001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC0 +00001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC0 +00001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC0 +00001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC0 +00003FE0007FFFFFF07FFFFFF07FFFFFF01C3879B72A>49 D<000FF00000007FFE000001 +FFFF800003E03FE0000F000FF0000E0007F8001C0003FC00380001FE00300001FE007000 +00FF00600000FF00FC0000FF00FF00007F80FF80007F80FF80007F80FF80007F80FF8000 +7F80FF80007F807F00007F801C00007F800000007F80000000FF00000000FF00000000FF +00000001FE00000001FC00000003FC00000003F800000007F000000007E00000000FE000 +00001FC00000003F800000003F000000007C00000000F800000001F000000003E0000000 +07C00000000F800000000F000000001E000180003C000180007800018000F000038001E0 +00030003C000030007800003000E000007000FFFFFFF001FFFFFFF003FFFFFFF007FFFFF +FE00FFFFFFFE00FFFFFFFE00FFFFFFFE0021387CB72A>I<0007F80000003FFF0000007F +FFC00001F80FF00003C007F800078003FC000E0001FC000F0001FE001FE000FE001FF000 +FF001FF000FF001FF000FF001FF000FF001FF000FF000FE000FF0007C000FF00000000FE +00000001FE00000001FE00000001FC00000003F800000003F800000007F000000007E000 +00000F800000007E0000001FFC0000001FFF800000000FE000000007F000000001FC0000 +0001FE00000000FF000000007F800000007F800000007FC00000007FC00000003FC00000 +003FE00000003FE01E00003FE07F80003FE0FFC0003FE0FFC0003FE0FFC0003FE0FFC000 +3FE0FFC0003FC0FF80007FC07F80007F807E00007F80700000FF00380001FE001E0001FE +000F8003F80007F00FF00001FFFFC000007FFF0000000FF80000233A7DB72A>I<1C007F +00FF80FF80FF80FF80FF807F001C00000000000000000000000000000000000000000000 +0000000000000000000000000000001C007F00FF80FF80FF80FF80FF807F001C00092479 +A317>58 D<7FFFFFFFFFFFF8FFFFFFFFFFFFFCFFFFFFFFFFFFFC7FFFFFFFFFFFF8000000 +000000000000000000000000000000000000000000000000000000000000000000000000 +000000000000000000000000000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000007FFFFFFFFFFFF8FFFFFFFFFFFF +FCFFFFFFFFFFFFFC7FFFFFFFFFFFF836167B9F41>61 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fn cmsy7 7 5 +/Fn 5 49 df0 +D<00380000380000380000380000380060380CF8383EFC387EFE38FE3FBBF807FFC001FF +00007C0001FF0007FFC03FBBF8FE38FEFC387EF8383E60380C0038000038000038000038 +0000380017197B9A22>3 D<007F000001FFC00007FFF0000FC1F8001F007C003C001E00 +38000E0078000F0070000700F0000780E0000380E0000380E0000380E0000380E0000380 +F00007807000070078000F0038000E003C001E001F007C000FC1F80007FFF00001FFC000 +007F000019197C9A22>14 D<007F000001FFC00007FFF0000FFFF8001FFFFC003FFFFE00 +3FFFFE007FFFFF007FFFFF00FFFFFF80FFFFFF80FFFFFF80FFFFFF80FFFFFF80FFFFFF80 +FFFFFF807FFFFF007FFFFF003FFFFE003FFFFE001FFFFC000FFFF80007FFF00001FFC000 +007F000019197C9A22>I<00E001F003F803F803F807F007F007F007E007E00FE00FC00F +C00FC01F801F801F001F003F003E003E003E007C007C007C007800F800F800F00010000D +1E7D9F13>48 D E +%EndDVIPSBitmapFont +/Fo 203[25 25 25 25 49[{TeXBase1Encoding ReEncodeFont}4 +49.8132 /Times-Roman rf /Fp 153[19 26 29 47[29 29 29 +29 49[{TeXBase1Encoding ReEncodeFont}7 58.1154 /Times-Roman +rf +%DVIPSBitmapFont: Fq cmmi7 7 6 +/Fq 6 121 df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ndDVIPSBitmapFont +%DVIPSBitmapFont: Fr cmsy10 10 11 +/Fr 11 118 df<7FFFFFFFFFFF80FFFFFFFFFFFFC0FFFFFFFFFFFFC07FFFFFFFFFFF8032 +04799641>0 D<001FF00000FFFE0001FFFF0007FFFFC00FFFFFE01FFFFFF03FFFFFF83F +FFFFF87FFFFFFC7FFFFFFC7FFFFFFCFFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFEFF +FFFFFEFFFFFFFEFFFFFFFEFFFFFFFE7FFFFFFC7FFFFFFC7FFFFFFC3FFFFFF83FFFFFF81F +FFFFF00FFFFFE007FFFFC001FFFF0000FFFE00001FF0001F1F7BA42A>15 +D<7FFFFFFC000000FFFFFFFFC00000FFFFFFFFF000007FFFFFFFFC000000000007FE0000 +000000007F8000000000001FC0000000000007E0000000000003F0000000000001F80000 +00000000FC0000000000007C0000000000003E0000000000003E0000000000001F000000 +0000001F0000000000000F8000000000000F8000000000000780000000000007C0000000 +000007C0000000000003C0000000000003C0000000000003C0000000000003C000000000 +0003C0000000000003C0000000000003C0000000000003C0000000000007C00000000000 +07C00000000000078000000000000F8000000000000F8000000000001F0000000000001F +0000000000003E0000000000003E0000000000007C000000000000FC000000000001F800 +0000000003F0000000000007E000000000001FC000000000007F800000000007FE00007F +FFFFFFFC0000FFFFFFFFF00000FFFFFFFFC000007FFFFFFC000000323279AD41>27 +D<0000000000001E00000000000000001E00000000000000001E00000000000000001E00 +000000000000001F00000000000000000F00000000000000000F00000000000000000F80 +0000000000000007800000000000000007C00000000000000003E00000000000000003E0 +0000000000000001F00000000000000000F80000000000000000FC00000000000000007E +00000000000000003F00000000000000001F80000000000000000FC00000000000000007 +F07FFFFFFFFFFFFFFFFCFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFFFFFFFFF +FC0000000000000007F0000000000000000FC0000000000000001F80000000000000003F +00000000000000007E0000000000000000FC0000000000000000F80000000000000001F0 +0000000000000003E00000000000000003E00000000000000007C0000000000000000780 +000000000000000F80000000000000000F00000000000000000F00000000000000001F00 +000000000000001E00000000000000001E00000000000000001E00000000000000001E00 +00482C7BAA53>33 D<7FFFFFFFFFF8FFFFFFFFFFFCFFFFFFFFFFFC7FFFFFFFFFFC000000 +00003C00000000003C00000000003C00000000003C00000000003C00000000003C000000 +00003C00000000003C00000000003C00000000003C00000000003C00000000003C000000 +00003C00000000003C00000000003C00000000003C00000000003C00000000003C000000 +0000182E177C9D37>58 D<00000300000000000780000000000FC0000000000FC0000000 +001FE0000000001FE0000000001FE0000000003FF0000000003FF0000000007CF8000000 +007CF800000000F87C00000000F87C00000000F03C00000001F03E00000001F03E000000 +03E01F00000003E01F00000007C00F80000007C00F8000000F8007C000000F8007C00000 +0F0003C000001F0003E000001F0003E000003E0001F000003E0001F000007C0000F80000 +7C0000F80000780000780000F800007C0000F800007C0001F000003E0001F000003E0003 +E000001F0003E000001F0007C000000F8007C000000F800780000007800F80000007C00F +80000007C01F00000003E01F00000003E03E00000001F03E00000001F03C00000000F07C +00000000F87C00000000F8F8000000007CF8000000007CF0000000003C6000000000182E +347CB137>94 D<600000000018F0000000003CF8000000007CF8000000007C7C00000000 +F87C00000000F83C00000000F03E00000001F03E00000001F01F00000003E01F00000003 +E00F80000007C00F80000007C007800000078007C000000F8007C000000F8003E000001F +0003E000001F0001F000003E0001F000003E0000F800007C0000F800007C000078000078 +00007C0000F800007C0000F800003E0001F000003E0001F000001F0003E000001F0003E0 +00000F0003C000000F8007C000000F8007C0000007C00F80000007C00F80000003E01F00 +000003E01F00000001F03E00000001F03E00000000F03C00000000F87C00000000F87C00 +0000007CF8000000007CF8000000003FF0000000003FF0000000001FE0000000001FE000 +0000001FE0000000000FC0000000000FC000000000078000000000030000002E347CB137 +>I<000001F800000FF800003F800000FC000001F8000003F0000007E0000007E000000F +E000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000F +C000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000F +C000000FC000000FC000000FC000000FC000000FC000000FC000000FC000001FC000001F +8000003F8000007F000000FE000003F800007FE00000FF0000007FE0000003F8000000FE +0000007F0000003F8000001F8000001FC000000FC000000FC000000FC000000FC000000F +C000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000F +C000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000F +C000000FC000000FC000000FE0000007E0000007E0000003F0000001F8000000FC000000 +3F8000000FF8000001F81D537ABD2A>102 DI<6000 +00000060F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000 +000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000 +000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000 +000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000 +000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000 +000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000 +000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000 +000000F0F000000000F0F000000000F0F000000000F0FFFFFFFFFFF0FFFFFFFFFFF0FFFF +FFFFFFF07FFFFFFFFFE02C327BB137>116 D<7FFFFFFFFFE0FFFFFFFFFFF0FFFFFFFFFF +F0FFFFFFFFFFF0F000000000F0F000000000F0F000000000F0F000000000F0F000000000 +F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000 +F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000 +F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000 +F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000 +F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000 +F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000F0F000000000 +F0F000000000F0F000000000F0F000000000F0F000000000F06000000000602C327BB137 +>I E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fs cmmi10 10 20 +/Fs 20 120 df<0003FFFFFFFFFF800007FFFFFFFFFF800007FFFFFFFFFF80000007F800 +00FF80000007F000001F80000007F000000F0000000FF000000F0000000FF00000070000 +000FE00000070000000FE00000070000001FE00000070000001FE00000070000001FC000 +00070000001FC00000060000003FC00000060000003FC00000060000003F800000060000 +003F800000060000007F800000060000007F800000000000007F000000000000007F0000 +0000000000FF00000000000000FF00000000000000FE00000000000000FE000000000000 +01FE00000000000001FE00000000000001FC00000000000001FC00000000000003FC0000 +0000000003FC00000000000003F800000000000003F800000000000007F8000000000000 +07F800000000000007F000000000000007F00000000000000FF00000000000000FF00000 +000000000FE00000000000000FE00000000000001FE00000000000001FE0000000000000 +1FC00000000000001FC00000000000003FC00000000000003FC00000000000003F800000 +000000003F800000000000007F800000000000007F800000000000007F00000000000000 +FF800000000000FFFFFFC000000000FFFFFFC000000000FFFFFFC00000000039397DB833 +>0 D<00000000000C000000000000001C000000000000003E000000000000007E000000 +000000007E00000000000000FE00000000000001FF00000000000003FF00000000000003 +FF000000000000067F0000000000000C7F8000000000001C7F800000000000187F800000 +000000303F800000000000603FC00000000000E03FC00000000000C03FC0000000000180 +1FC00000000003001FE00000000007001FE00000000006001FE0000000000C000FE00000 +000018000FF00000000038000FF00000000030000FF00000000060000FF000000000C000 +07F800000001C00007F800000001800007F800000003000007F800000007000003F80000 +0006000003FC0000000C000003FC00000018000003FC00000038000001FC000000300000 +01FE00000060000001FE000000C0000001FE000001C0000000FE00000180000000FF0000 +0300000000FF00000600000000FF00000E000000007F00000C000000007F800018000000 +007F800030000000007F800070000000003F800060000000003FC000C0000000003FC001 +80000000003FC00380000000001FC00300000000001FE00600000000001FE00C00000000 +001FE01FFFFFFFFFFFFFE01FFFFFFFFFFFFFF03FFFFFFFFFFFFFF07FFFFFFFFFFFFFF07F +FFFFFFFFFFFFF0FFFFFFFFFFFFFFF03C3C7CBB45>I<003FFFFFFFE000FFFFFFFFF001FF +FFFFFFF007FFFFFFFFF007FFFFFFFFE00F80700600001E00600E00003C00600C00003800 +E00C00007000C00C0000E000C01C0000C001C01C00000001C01C00000001801C00000003 +80380000000380380000000780380000000700380000000700380000000F00380000000F +00780000001E007C0000001E007C0000001E007C0000003E007C0000003C007C0000007C +007C0000007C007E000000FC007E000000F8007E000001F8007E000001F8007F000003F8 +007F000003F0003F000003F0003F000003F0003F000001C0001C00002C257EA32F>25 +D<0000780000000000000000780000000000000000780000000000000000F80000000000 +000000F00000000000000001F00000000000000001E00000000000000003E00000000000 +000007C00000000000000007C0000000000000000F80000000000000001F800000000000 +00001F00000000000000003E00000000000000007C0000000000000000FC000000000000 +0001F80000000000000003F0000000000000000FE0000000000000001FC0000000000000 +003FFFFFFFFFFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF +FE48187BAA53>40 D<0000000000001E00000000000000001E00000000000000001E0000 +0000000000001F00000000000000000F00000000000000000F8000000000000000078000 +00000000000007C00000000000000003E00000000000000003E00000000000000001F000 +00000000000001F80000000000000000F800000000000000007C00000000000000003E00 +000000000000003F00000000000000001F80000000000000000FC00000000000000007F0 +0000000000000003F87FFFFFFFFFFFFFFFFCFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF +7FFFFFFFFFFFFFFFFF48187BAA53>42 D<1C007F00FF80FF80FF80FF80FF807F001C0009 +09798817>58 D<1C007F00FF80FF80FFC0FFC0FFC07FC01CC000C000C000C000C001C001 +80018003800300070006000E001C003800700060000A19798817>I<0000000000600000 +0000000070000000000000F0000000000001F0000000000001F0000000000003F0000000 +000003F0000000000007F000000000000FF000000000000FF000000000001FF800000000 +001FF8000000000033F8000000000073F8000000000063F80000000000C3F80000000000 +C3F8000000000183F8000000000183F8000000000303F8000000000603F8000000000603 +FC000000000C03FC000000000C01FC000000001801FC000000003001FC000000003001FC +000000006001FC000000006001FC00000000C001FC00000001C001FC000000018001FC00 +0000030001FE000000030001FE000000060000FE0000000E0000FE0000000C0000FE0000 +00180000FE0000001FFFFFFE0000003FFFFFFE0000003FFFFFFE000000600000FE000000 +C00000FE000000C00000FF000001800000FF0000018000007F0000030000007F00000600 +00007F0000060000007F00000C0000007F00000C0000007F0000180000007F0000380000 +007F0000700000007F0000F00000007F8001F80000007F8007F8000000FF80FFFF80003F +FFFFFFFF80007FFFFFFFFF80007FFFFF383C7DBB3E>65 D<0003FFFFFFFF80000007FFFF +FFFFF0000007FFFFFFFFFC00000007F80003FE00000007F00000FF00000007F000007F80 +00000FF000003FC000000FF000001FC000000FE000001FE000000FE000001FE000001FE0 +00001FE000001FE000001FE000001FC000001FE000001FC000001FE000003FC000001FE0 +00003FC000001FC000003F8000003FC000003F8000003F8000007F8000007F8000007F80 +00007F0000007F000000FE0000007F000001FC000000FF000003F8000000FF00000FF000 +0000FE00001FC0000000FE0000FF00000001FFFFFFFC00000001FFFFFFF800000001FC00 +00FF00000001FC00003FC0000003FC00000FE0000003FC000007F0000003F8000007F000 +0003F8000003F8000007F8000003F8000007F8000003FC000007F0000001FC000007F000 +0001FC00000FF0000001FC00000FF0000003FC00000FE0000003FC00000FE0000003FC00 +001FE0000003FC00001FE0000007F800001FC0000007F800001FC000000FF000003FC000 +000FF000003FC000001FE000003F8000003FC000003F8000007F8000007F800000FF0000 +007F800001FE0000007F000007FC000000FF00003FF00000FFFFFFFFFFC00000FFFFFFFF +FF000000FFFFFFFFF80000003B397DB83F>I<00000000FF8001C00000000FFFE001C000 +00007FFFF80380000001FF807E0780000007F8000F0F8000001FE000079F8000003F8000 +03BF000000FF000001FF000001FC000000FF000003F8000000FF000007F00000007E0000 +0FE00000007E00001FC00000007E00003F800000003E00007F800000003C0000FF000000 +003C0000FE000000003C0001FE000000003C0003FC00000000380003F800000000380007 +F80000000038000FF00000000038000FF00000000030001FF00000000030001FE0000000 +0000001FE00000000000003FC00000000000003FC00000000000003FC00000000000007F +C00000000000007F800000000000007F800000000000007F80000000000000FF80000000 +000000FF00000000000000FF00000000000000FF00000000000000FF00000000000000FF +00000000030000FF00000000030000FF00000000070000FF00000000060000FF00000000 +0600007F000000000E00007F000000000C00007F000000001C00007F000000003800003F +800000003800003F800000007000001F80000000E000001FC0000001C000000FE0000003 +8000000FE000000780000007F000000E00000003F800003C00000001FC00007800000000 +FF0001F0000000003FE00FC0000000000FFFFF000000000003FFFC0000000000007FC000 +0000003A3D7CBA3B>I<00007E00000003FF8000000FC1C380001F00EFC0007E007FC000 +FC003FC001F8003FC003F0001F8007F0001F8007E0001F800FE0003F801FC0003F001FC0 +003F003F80003F003F80007F007F80007E007F00007E007F00007E007F0000FE00FF0000 +FC00FE0000FC00FE0000FC00FE0001FC00FE0001F800FC0001F80CFC0001F80CFC0003F8 +0CFC0003F01CFC0003F018FC0007F0187C0007F0387E000FF0303E001FF0303E007BF070 +1F00E1F0E00F83C0F9C003FF007F8000FC001F0026267DA42C>97 +D<00003FC00001FFF00007E03C000F800E003F0007007E001F00FC007F01F800FF03F000 +FF07E000FF0FE000FF0FC000FE1FC000383F8000003F8000007F8000007F0000007F0000 +007F000000FF000000FE000000FE000000FE000000FE000000FC000000FC000000FC0000 +00FC000003FC0000077E0000067E00000E3E00003C3F0000701F0000E00F8007C007C03F +0001FFF800003FC00020267DA424>99 D<00003FC00001FFF00007E078001F801C007E00 +1E00FC000E01F8000E03F0000E07F0000E0FE0000E0FC0001E1FC0001C1FC0003C3F8000 +F83F8003E07F803FC07FFFFE007FFFE0007F000000FF000000FE000000FE000000FE0000 +00FE000000FE000000FE000000FE0000007E0000037E0000077E0000063E00000E3E0000 +3C1F0000700F8000E00F8007C003E03F0001FFF800003FC00020267DA427>101 +D<0003F0000001FFF0000001FFF0000001FFF000000007F000000007E000000007E00000 +0007E00000000FE00000000FC00000000FC00000000FC00000001FC00000001F80000000 +1F800000001F800000003F800000003F000000003F000000003F000000007F000000007E +0007C0007E001FF0007E00783800FE00E0F800FC01C1FC00FC0383FC00FC0707FC01FC0E +07FC01F81C07F801F83803F001F87001E003F8E0000003F1C0000003F380000003F70000 +0007FE00000007FE00000007FFE0000007E7F800000FE0FE00000FC07F00000FC03F8000 +0FC01F80001FC01FC0001F800FC0301F800FC0301F800FC0703F801FC0603F001F80603F +001F80603F001F80E07F001F80C07E001F81C07E000F81807E000F8380FE00078700FC00 +03FE00380000F800263B7CB92B>107 D<03E0007F000007F801FFE0000E3C0781F0001C +3E1E00F800383F3800FC00303F7000FC00303FE0007C00703FC0007C00603F80007C0060 +3F80007C00E03F0000FC00C07F0000FC00C07E0000FC00C07E0000FC00007E0001FC0000 +FE0001F80000FC0001F80000FC0001F80000FC0003F80001FC0003F00001F80003F00001 +F80007F00001F80007E00003F80007E00003F0000FE03003F0000FC03003F0001FC07007 +F0001F806007E0001F806007E0001F80E007E0001F00C00FE0001F01C00FC0001F01800F +C0001F03800FC0001F07001FC0000F0E001F800007FC0007000001F0002C267EA432> +110 D<00001FC0000000FFF8000007E07E00000F801F00003F000F80007E000FC000FC00 +07C001F80007E003F00007E007E00003F00FE00003F00FC00003F01FC00003F03F800007 +F03F800007F07F800007F07F000007F07F000007F07F00000FF0FF00000FF0FE00000FE0 +FE00000FE0FE00001FE0FE00001FC0FE00001FC0FC00003F80FC00003F00FC00007F00FC +00007E007E0000FC007E0001F8003E0003F0003F0007E0001F000FC0000F801F000007E0 +7E000001FFF00000003F80000024267DA428>I<03E001F80007F807FE000E3C1E07001C +3E381F00183F703F80383FE07F80303FC0FF80703F80FF80603F80FF00603F007E00603F +003C00E07F000000C07E000000C07E000000007E00000000FE00000000FC00000000FC00 +000000FC00000001FC00000001F800000001F800000001F800000003F800000003F00000 +0003F000000003F000000007F000000007E000000007E000000007E00000000FE0000000 +0FC00000000FC00000000FC00000001FC00000001F80000000070000000021267EA425> +114 D<0001C0000003E0000007E0000007E0000007E0000007E000000FE000000FC00000 +0FC000000FC000001FC000001F8000001F8000001F8000003F8000003F00007FFFFF807F +FFFF80FFFFFF80007E0000007E0000007E000000FE000000FC000000FC000000FC000001 +FC000001F8000001F8000001F8000003F8000003F0000003F0000003F0000007F0000007 +E0000007E0000007E000000FE000000FC006000FC006000FC00E001FC00C001F801C001F +8018001F8038001F8070001F8060001F80E0000F81C0000787800003FE000000F8000019 +357EB31E>116 D<00F80000000003FE00003800070F00007C000E0F8000FC001C0F8000 +FC00180F8000FC00380F8001FC00300F8001FC00701F8001F800601F8001F800601F8003 +F800E03F8003F800C03F0003F000C07F0003F000007E0007F000007E0007F00000FE0007 +E00000FC0007E00000FC000FE00001FC000FE00001F8000FC00001F8000FC00001F8001F +C00003F8001FC00003F0001F80C003F0001F80C003F0003F80C003F0003F81C003F0003F +018003F0003F018003F0007F038003F000FF030001F000FF030001F001FF070000F8079F +0E00007C0E0F1C00003FFC07F8000007F001F0002A267EA430>I<00F800000000F003FE +0000E001F8070F0001F003F80E0F8003F003FC1C0F8003F003FC180F8003F003FC380F80 +07F001FC300F8007E000FC701F8007E0007C601F8007E0007C601F800FE0003CE03F800F +E00038C03F000FC00038C07F000FC00038007E001FC00038007E001FC0003000FE001F80 +003000FC001F80003000FC003F80007001FC003F80006001F8003F00006001F8003F0000 +6001F8003F0000E003F8007F0000C003F0007E0000C003F0007E0001C003F0007E000180 +03F0007E00038003F0007E00030003F0007E00070003F000FE00060003F000FE000E0001 +F001FE001C0001F801BF00380000FC039F807000007E0F0FC0E000001FFC03FFC0000003 +F0007F000036267EA43B>119 D E +%EndDVIPSBitmapFont +/Ft 134[37 37 55 37 42 23 32 32 1[42 42 42 60 23 37 1[23 +42 42 23 37 42 37 42 42 17[60 13[51 18[21 28 21 2[28 +28 36[42 42 2[{TeXBase1Encoding ReEncodeFont}32 83.022 +/Times-Italic rf /Fu 138[55 33 39 44 2[50 55 83 28 1[33 +28 1[50 33 44 55 44 55 50 10[72 1[66 1[72 1[61 1[72 4[39 +3[66 72 72 13[50 50 50 50 50 3[33 45[{TeXBase1Encoding ReEncodeFont}32 +99.6264 /Times-Bold rf /Fv 134[33 1[50 1[37 21 29 29 +1[37 37 37 54 21 33 1[21 37 37 21 33 37 33 37 37 12[42 +37 2[46 1[50 62 42 1[33 25 54 1[46 1[54 50 1[46 6[25 +37 3[37 6[19 25 3[25 25 40[{TeXBase1Encoding ReEncodeFont}41 +74.7198 /Times-Italic rf /Fw 139[25 29 33 8[21 1[37 3[33 +42 37 26[46 4[54 15[37 2[19 46[{TeXBase1Encoding ReEncodeFont}12 +74.7198 /Times-Bold rf /Fx 87[25 17[37 1[33 33 24[33 +37 37 54 37 37 21 29 25 37 37 37 37 58 21 37 21 21 37 +37 25 33 37 33 37 33 3[25 1[25 46 54 54 71 54 54 46 42 +50 1[42 54 54 66 46 54 29 25 54 54 42 46 54 50 50 54 +5[21 21 37 37 37 37 37 37 37 37 37 37 21 19 25 19 2[25 +25 25 21[21 13[42 42 2[{TeXBase1Encoding ReEncodeFont}79 +74.7198 /Times-Roman rf /Fy 138[45 45 1[45 3[45 45 45 +2[45 45 2[45 45 1[45 45 32[45 17[45 1[45 44[{ +TeXBase1Encoding ReEncodeFont}15 74.7198 /Courier rf +%DVIPSBitmapFont: Fz cmsy9 9 4 +/Fz 4 104 df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ndDVIPSBitmapFont +/FA 87[28 16[83 2[37 37 24[37 42 42 60 42 42 23 32 28 +42 42 42 42 65 23 42 23 23 42 42 28 37 42 37 42 37 28 +2[28 1[28 3[78 1[60 51 46 55 1[46 60 60 74 51 60 32 28 +60 60 46 51 60 55 55 60 1[37 3[23 23 42 42 42 42 42 42 +42 42 42 42 1[21 28 21 2[28 28 28 5[28 15[23 13[46 46 +2[{TeXBase1Encoding ReEncodeFont}77 83.022 /Times-Roman +rf /FB 138[66 40 47 53 2[60 66 1[33 1[40 33 66 60 1[53 +1[53 66 60 12[80 5[86 1[80 2[47 4[86 86 1[86 6[40 12[40 +45[{TeXBase1Encoding ReEncodeFont}24 119.552 /Times-Bold +rf end +%%EndProlog +%%BeginSetup +%%Feature: *Resolution 600dpi +TeXDict begin +%%PaperSize: A4 + end +%%EndSetup +%%Page: 1 1 +TeXDict begin 1 0 bop 705 448 a FB(Classical)30 b(Logic)f(is)h(better)g +(than)h(Intuitionistic)f(Logic:)681 598 y(A)g(Conjectur)n(e)h(about)f +(Double-Negation)h(T)-9 b(ranslations)1417 886 y FA(Christian)21 +b(Urban)e(and)h(Diana)g(Ratiu)1410 1060 y Fz(f)p Fy(urban,ratiu)p +Fz(g)p Fy(@math.lmu.de)1642 1152 y Fx(Uni)n(v)o(ersity)f(of)g(Munich) +759 1462 y Fw(Abstract.)41 b Fx(It)20 b(is)h(well-kno)n(wn)g(that)f(in) +h(terms)f(of)g(consistenc)o(y)i(classical)f(logic)f(and)i(intu-)759 +1553 y(itionistic)f(logic)h(ha)o(v)o(e)g(equal)g(strength:)g(e)n(v)o +(ery)h(intuitionistic)e(proof)h(is)g(a)f(classical)h(proof)759 +1645 y(and)29 b(e)n(v)o(ery)f(classical)g(proof)h(can)f(be)g(embedded)i +(into)e(intuitionistic)f(logic)h(via)g(double)759 1736 +y(ne)o(gation)21 b(translations.)f(It)f(is)h(also)g(well-kno)n(wn)h +(that)f(intuitionistic)g(proofs)h(contain)f(wit-)759 +1827 y(nesses)33 b(for)f(e)o(xistential)g(statements,)g(which)h(is)f +(not)g(al)o(w)o(ays)h(the)f(case)h(with)f(classical)759 +1919 y(proofs.)f(Ho)n(we)n(v)o(er)m(,)g(here)h(we)e(study)i(classical)e +(and)i(intuitionistic)e(logic)h(as)f(reduction)759 2010 +y(systems.)24 b(From)g(this)f(perspecti)n(v)o(e,)i(we)f(conjecture)h +(that)e(classical)h(logic)g(is)g(more)g(po)n(w-)759 2101 +y(erful)g(than)h(intuitionistic)f(logic.)g(The)g(conjecture)i(links)e +(double-ne)o(gation)i(translations)759 2193 y(to)21 b(the)g(colour)o +(-protocol)i(introduced)f(by)g(Danos)g(et)e(al.)h(for)g +(cut-elimination)h(in)f(classical)759 2284 y(logic.)27 +b(If)g(the)g(conjecture)i(turns)e(out)h(to)f(be)g(true,)g(then)h(we)f +(can)h(conclude)g(that)f Fv(not)h(all)759 2375 y Fx(cut-reductions)20 +b(can)g(be)f(simulated)g(by)h(double-ne)o(gation)g(translations.)523 +2644 y Fu(1)99 b(Intr)n(oduction)523 2832 y FA(Since)25 +b(the)f(w)o(orks)g(on)h(double-ne)o(gation)20 b(translations)k(by)g +(Gentzen,)f(G)7 b(\250)-35 b(odel)24 b(and)g(K)m(olmogoro)o(v)-5 +b(,)523 2932 y(one)26 b(kno)n(ws)g(that)g(classical)h(logic)f(and)g +(intuitionistic)g(logic)g(ha)n(v)o(e)g(equal)f(strength)h(in)g(terms)h +(of)523 3031 y(consistenc)o(y:)33 b(intuitionistic)h(sequent-proofs)e +(can)i(be)h(seen)g(as)g(classical)g(proofs)f(where)f(the)523 +3131 y(right-hand)27 b(side)i(of)g(the)g(sequents)g(is)h(restricted)e +(to)i(maximal)e(one)g(formula)g(and)g(e)n(v)o(ery)g(clas-)523 +3231 y(sical)19 b(sequent-proof)c(can)k(be)f(embedded)e(into)j +(intuitionistic)e(logic)h(via)h(double-ne)o(gation)14 +b(trans-)523 3330 y(lations.)27 b(As)h(a)f(result,)g(consistenc)o(y)f +(of)h(one)f(logic)h(implies)g(consistenc)o(y)f(of)h(the)g(other)-5 +b(.)26 b(In)h(this)523 3430 y(paper)m(,)i(ho)n(we)n(v)o(er)m(,)f(we)j +(focus)f(on)h(the)f(correspondence)e(of)i(intuitionistic)g(and)g +(classical)i(logic)523 3530 y(with)20 b(respect)g(to)h(term-re)n +(writing,)c(proof-normalisation)f(and)k(cut-elimination.)648 +3629 y(According)c(to)i(the)g(Curry-Ho)n(w)o(ard)d(correspondence,)f +(the)k(simply-typed)e(lambda-calculus)523 3729 y(can)28 +b(be)f(vie)n(wed)g(as)h(a)h(term-assignment)c(for)i(intuitionistic)g +(proofs)g(formalised)f(in)i(Gentzen')-5 b(s)523 3828 +y(natural)24 b(deduction)f(calculus)i(NJ.)h(T)-6 b(erm-re)n(writing)22 +b(in)k(the)f(simply-typed)e(lambda-calculus)g(is)523 +3928 y(a)e(form)f(of)h(computation)e(that)i(con)m(v)o(erts)e(a)i(term)g +(to)g(its)h(simplest)g(form,)d(analogous)g(to)i(symbolic)523 +4028 y(e)n(v)n(aluation.)29 b(On)h(the)h(other)e(hand,)g(normalisation) +g(is)j(a)f(method)e(for)h(eliminating)f(certain)h(re-)523 +4127 y(dundancies)23 b(in)j(proofs.)e(Applied)g(iterati)n(v)o(ely)-5 +b(,)24 b(it)i(transforms)d(a)j(proof)e(to)h(one)g(in)h(normalform.)523 +4227 y(Using)19 b(the)f(Curry-Ho)n(w)o(ard)e(correspondence)f(we)k(see) +g(that)g(the)g(tw)o(o)f(notions)g(coincide)f(and)h(con-)523 +4327 y(sequently)h(we)i(can)f(talk)g(of)g(a)g(computational)e +(interpretation)g(of)i(intuitionistic)g(proofs.)648 4426 +y(T)-7 b(w)o(o)20 b(properties)e(hold)h(for)g(term-re)n(writing)e(in)j +(the)g(simply-typed)d(lambda-calculus)h(and)h(by)523 +4526 y(the)k(Curry-Ho)n(w)o(ard)d(correspondence)f(also)k(for)f +(normalisation)e(in)j(NJ:)h(the)o(y)d(are)i(strongly)e(nor)n(-)523 +4625 y(malising)k(and)h(Church-Rosser)-5 b(.)24 b(W)m(ith)i(some)g +(limitations)g([8,)12 b(15,)h(18,)g(23],)24 b(the)i(Curry-Ho)n(w)o(ard) +523 4725 y(correspondence)18 b(applies)i(also)h(to)h(the)f +(sequent-calculus)d(LJ)k(and)e(to)h(the)g(process)g(of)f(cut-elimi-)523 +4825 y(nation.)28 b(F)o(or)g(sak)o(e)h(of)g(simplicity)-5 +b(,)28 b(we)h(shall)g(ignore)f(these)h(limitations)f(here)h(and)f(re)o +(gard)f(cut-)523 4924 y(elimination)k(in)h(intuitionistic)f(logic)g(as) +i(strongly)d(normalising)g(and)h(as)i(ha)n(ving)e(\(morally)f(at)p +eop end +%%Page: 2 2 +TeXDict begin 2 1 bop 523 448 a FA(least\))24 b(the)f(Church-Rosser)e +(property)-5 b(.)21 b(This)i(ignorance)e(can)i(be)g(partially)f +(justi\002ed)i(if)f(one)g(sees)523 548 y(behind)c(e)n(v)o(ery)f +(LJ-proof)g(an)i(NJ-proof)f(where)g(the)h(inessential)g(dif)n(ferences) +f(present)g(in)h(sequent)523 648 y(proofs)31 b(\223disappear\224,)g +(and)h(sees)h(cut-elimination)d(as)k(an)e(approximation)d(of)k +(proof-normali-)523 747 y(sation.)648 850 y(Although)19 +b(it)i(has)h(been)e(sho)n(wn)g(that)h(some)g(cut-elimination)e +(procdures)g(for)h(classical)i(logic)523 950 y(are)i(strongly)f +(normalising)g(as)i(well,)f(cut-elimination)e(in)j(classical)g(logic)f +(is,)h(gi)n(v)o(en)e(a)h(sensible)523 1049 y(notion)i(of)h +(cut-reductions,)d Ft(not)k FA(Church-Rosser)n(\227not)d(e)n(v)o(en)h +(morally)-5 b(.)25 b(The)i(lack)g(of)g(Church-)523 1149 +y(Rosser)18 b(in)f(classical)h(logic)f(is)h(a)g(main)f(theme)f(running) +f(through)g(the)j(w)o(orks)e([2,)d(3,)g(19,)g(22],)j(which)523 +1249 y(analyse)j(what)g(this)h(means)f(from)f(a)i(computational)d +(point)h(of)h(vie)n(w)-5 b(.)19 b(F)o(or)g(e)o(xample)f(in)h([19])f(it) +i(has)523 1348 y(been)31 b(sho)n(wn)g(that)g(a)h(lambda-calculus)d +(with)j(a)g(non-deterministic)c(choice-operator)g(can)k(be)523 +1448 y(embedded)19 b(into)i(a)g(fragment)f(of)h(classical)h(logic.)e(A) +i(simple)f(classical)h(proof)d(tak)o(en)i(from)f([6,)13 +b(9])523 1547 y(shall)21 b(illustrate)f(the)g(lack)g(of)g +(Church-Rosser:)1158 1725 y Fs(A)p 1238 1713 10 38 v +1248 1696 42 4 v 88 w(A)83 b(A)p 1534 1713 10 38 v 1543 +1696 42 4 v 88 w(A)p 1158 1745 509 4 v 1178 1818 a(A)19 +b Fr(_)g Fs(A)p 1414 1806 10 38 v 1423 1790 42 4 v 88 +w(A;)14 b(A)1707 1762 y Fr(_)1762 1774 y Fq(L)p 1178 +1854 467 4 v 1228 1928 a Fs(A)19 b Fr(_)g Fs(A)p 1463 +1916 10 38 v 1473 1899 42 4 v 88 w(A)1686 1874 y(contr)1879 +1886 y Fq(R)2017 1725 y Fs(A)p 2098 1713 10 38 v 2107 +1696 42 4 v 88 w(A)84 b(A)p 2393 1713 10 38 v 2403 1696 +42 4 v 88 w(A)p 2017 1745 509 4 v 2056 1818 a(A;)14 b(A)p +2236 1806 10 38 v 2246 1790 42 4 v 89 w(A)p Fr(^)p Fs(A)2567 +1762 y Fr(^)2622 1774 y Fq(R)p 2056 1854 430 4 v 2106 +1928 a Fs(A)p 2187 1916 10 38 v 2196 1899 42 4 v 88 w(A)p +Fr(^)q Fs(A)2527 1874 y(contr)2720 1886 y Fq(L)p 1228 +1948 1209 4 v 1608 2021 a Fs(A)p Fr(_)q Fs(A)p 1807 2009 +10 38 v 1816 1992 42 4 v 88 w(A)p Fr(^)q Fs(A)2478 1973 +y(cut)3308 2021 y FA(\(1\))523 2183 y(The)32 b(cut)g(in)g(this)h(proof) +e(can)h(be)g(eliminated)f(by)h(reducing)e(it)j(to)f(one)g(of)g(the)g +(follo)n(wing)e(tw)o(o)523 2283 y(normalforms)1169 2460 +y Fs(A)p 1250 2448 10 38 v 1259 2432 42 4 v 88 w(A)84 +b(A)p 1545 2448 10 38 v 1555 2432 42 4 v 88 w(A)p 1169 +2480 509 4 v 1209 2554 a(A;)14 b(A)p 1388 2542 10 38 +v 1398 2525 42 4 v 88 w(A)p Fr(^)q Fs(A)1719 2497 y Fr(^)1774 +2510 y Fq(R)p 1209 2590 430 4 v 1258 2663 a Fs(A)p 1339 +2651 10 38 v 1348 2635 42 4 v 88 w(A)p Fr(^)q Fs(A)1679 +2609 y(contr)1872 2621 y Fq(L)2005 2460 y Fs(A)p 2086 +2448 10 38 v 2096 2432 42 4 v 89 w(A)83 b(A)p 2382 2448 +10 38 v 2391 2432 42 4 v 88 w(A)p 2005 2480 509 4 v 2045 +2554 a(A;)14 b(A)p 2225 2542 10 38 v 2234 2525 42 4 v +88 w(A)p Fr(^)q Fs(A)2555 2497 y Fr(^)2610 2510 y Fq(R)p +2045 2590 430 4 v 2094 2663 a Fs(A)p 2175 2651 10 38 +v 2185 2635 42 4 v 89 w(A)p Fr(^)p Fs(A)2516 2609 y(contr)2709 +2621 y Fq(L)p 1258 2683 1167 4 v 1509 2756 a Fs(A)p Fr(_)q +Fs(A)p 1707 2744 10 38 v 1717 2728 42 4 v 88 w(A)p Fr(^)q +Fs(A;)g(A)p Fr(^)p Fs(A)2466 2700 y Fr(_)2521 2712 y +Fq(L)p 1509 2792 665 4 v 1617 2866 a Fs(A)p Fr(_)q Fs(A)p +1816 2854 10 38 v 1825 2837 42 4 v 88 w(A)p Fr(^)q Fs(A)2215 +2812 y(contr)2408 2824 y Fq(R)3308 2866 y FA(\(2\))1146 +3103 y Fs(A)p 1227 3091 10 38 v 1236 3074 42 4 v 88 w(A)83 +b(A)p 1522 3091 10 38 v 1532 3074 42 4 v 89 w(A)p 1146 +3123 509 4 v 1167 3196 a(A)19 b Fr(_)f Fs(A)p 1402 3184 +10 38 v 1412 3167 42 4 v 89 w(A;)c(A)1695 3140 y Fr(_)1751 +3152 y Fq(L)p 1167 3232 467 4 v 1216 3306 a Fs(A)19 b +Fr(_)g Fs(A)p 1452 3294 10 38 v 1461 3277 42 4 v 88 w(A)1675 +3251 y(contr)1868 3263 y Fq(R)2005 3103 y Fs(A)p 2086 +3091 10 38 v 2096 3074 42 4 v 89 w(A)83 b(A)p 2382 3091 +10 38 v 2391 3074 42 4 v 88 w(A)p 2005 3123 509 4 v 2026 +3196 a(A)19 b Fr(_)g Fs(A)p 2262 3184 10 38 v 2271 3167 +42 4 v 88 w(A;)14 b(A)2555 3140 y Fr(_)2610 3152 y Fq(L)p +2026 3232 467 4 v 2076 3306 a Fs(A)19 b Fr(_)g Fs(A)p +2311 3294 10 38 v 2321 3277 42 4 v 88 w(A)2534 3251 y(contr)2727 +3263 y Fq(R)p 1216 3325 1227 4 v 1497 3399 a Fs(A)p Fr(_)q +Fs(A;)14 b(A)p Fr(_)q Fs(A)p 1913 3387 10 38 v 1922 3370 +42 4 v 88 w(A)p Fr(^)q Fs(A)2485 3342 y Fr(^)2540 3355 +y Fq(R)p 1497 3435 665 4 v 1606 3508 a Fs(A)p Fr(_)p +Fs(A)p 1804 3496 10 38 v 1814 3480 42 4 v 89 w(A)p Fr(^)p +Fs(A)2203 3454 y(contr)2396 3466 y Fq(L)3308 3508 y FA(\(3\))523 +3671 y(which)i(are)h(obtained,)e(respecti)n(v)o(ely)-5 +b(,)15 b(by)h(either)g(permuting)f(the)i(cut)f(to)h(the)g(left)g(o)o(v) +o(er)f(the)g Fs(contr)3322 3683 y Fq(R)3377 3671 y FA(-)523 +3770 y(rule)f(or)h(to)f(the)h(right)f(o)o(v)o(er)f(the)i +Fs(contr)1597 3782 y Fq(L)1647 3770 y FA(-rule.)e(Another)h(e)o(xample) +f(sho)n(wing)g(that)i(in)f(classical)i(logic)523 3870 +y(one)j(can,)f(in)i(general,)e(reach)g(more)g(than)h(one)g(normalform)d +(is)k(Lafont')-5 b(s)20 b(proof)e([10,)i(P)o(age)f(151].)648 +3973 y(In)26 b(light)h(of)f(the)h(absence)f(of)g(the)h(Church-Rosser)e +(property)g(for)h(cut-elimination)e(in)j(clas-)523 4072 +y(sical)h(logic)e(and)g(in)h(light)g(of)f(the)h(w)o(ork)f(on)g +(double-ne)o(gation)d(translations,)i(there)i(seem)g(to)g(be)523 +4172 y(ob)o(vious)k(questions:)g(What)i(is)g(the)g(correspondence)28 +b(between)k(cut-elimination)e(in)j(classical)523 4272 +y(logic)15 b(and)g(the)h(embeddings)d(of)i(classical)h(proofs)e(into)i +(intuitionistic)e(logic)i(via)f(double-ne)o(gation)523 +4371 y(translations?)23 b(Since)h(cut-elimination)e(in)i +(intuitionistic)g(logic)f(is)i(Church-Rosser)m(,)3030 +4341 y Fp(1)3085 4371 y FA(which)f(re-)523 4471 y(striction)e(is)h +(tacitly)f(enforced)f(by)g(a)i(double-ne)o(gation)18 +b(translation)j(so)i(that)f(eliminating)f(cuts)h(in)523 +4570 y(the)i(double-ne)o(gated)c(v)o(ersion)j(of)h(a)g(classical)h +(proof)e(leads)h(to)g(only)g(a)g(single)g(normalform?)d(Or)p +523 4654 473 4 v 558 4710 a Fo(1)606 4742 y Fx(As)j(mentioned)i +(earlier)f(we)f(ignore)h(what)g(we)g(belie)n(v)o(e)g(to)f(be)h +(super\002cial)g(v)n(ariations)g(between)g(dif)n(fer)o(-)606 +4833 y(ent)c(normalforms)g(reachable)h(from)f(an)g(intuitionistic)f +(sequent-proof.)i(Ho)n(we)n(v)o(er)f(see)g([18])g(for)g(a)f(more)606 +4924 y(thorough)h(analysis)e(of)g(this)g(aspect.)p eop +end +%%Page: 3 3 +TeXDict begin 3 2 bop 523 448 a FA(more)24 b(concisely)f(ask)o(ed,)i +(do)f(double-ne)o(gation)c(translations)k(correspond)e(to)i(particular) +f(strate-)523 548 y(gies)e(of)g(ho)n(w)f(to)h(eliminate)g(cuts?)g(Does) +g(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(lead)g(to)h(the)g +(same)523 648 y(normalform)i(\(by)j Ft(same)g FA(we)h(mean)e +(corresponding)e(to)j(one)g(particular)f(normalform)e(obtained)523 +747 y(by)j(cut-elimination)e(in)j(classical)g(logic\)?)f(If)g(not,)g +(then)g(can)g(one)g(\002nd)g(for)g(e)n(v)o(ery)f(normalform)523 +847 y(of)h(a)g(classical)h(proof)d(a)j(corresponding)22 +b(double-ne)o(gation)g(translation)j(that)h(will)h(produce)c(the)523 +946 y(double-ne)o(gated)g(v)o(ersion)j(of)i(this)g(normalform\227that)c +(is,)k(can)f(e)n(v)o(ery)f(reduction)g(sequence)g(in)523 +1046 y(classical)16 b(logic)f(be)h(simulated)f(by)g(a)h(\(probably)d +(carefully)h(chosen\))g(double)g(ne)o(gation)f(translation)523 +1146 y(and)26 b(performing)e(cut-elimination)h(in)i(intuitionistic)f +(logic?)h(Are)g(there)f(an)o(y)g(double-ne)o(gation)523 +1245 y(translations)20 b(that)g(lead)g(to)g(normalforms)e(that)i(ha)n +(v)o(e)g(no)f(equi)n(v)n(alent)g(amongst)g(the)h(normalforms)523 +1345 y(reachable)i(by)h(cut-elimination)f(in)h(classical)h(logic?)f +(Can)h(one)f(characterise)f(someho)n(w)-5 b(,)22 b(which)523 +1445 y(normalforms)16 b(can)i(be)g(reached)f(by)h(double-ne)o(gation)c +(translations)k(and)g(which)g(can)g(not?)g(In)g(this)523 +1544 y(paper)h(we)i(conjecture)d(answers)i(for)g(all)h(these)f +(questions.)648 1654 y(Although)k(some)i(special)h(cases)g(seem)g(to)f +(be)h(answered)e(by)h(e)o(xisting)g(w)o(ork,)f(for)h(e)o(xample)523 +1753 y([5,)13 b(6,)g(14],)21 b(we)i(are)g(una)o(w)o(are)e(of)h(an)o(y)g +(w)o(ork)g(that)g(treat)h(these)g(questions)e(in)i(full)f(generality)-5 +b(.)21 b(The)523 1853 y(answers)27 b(we)g(shall)g(gi)n(v)o(e)f(to)h +(these)f(questions)g(are)h(a)g(lot)g(inspired)f(by)g(the)h(comments)e +(made)h(in)523 1952 y([6,)d(Sec.)g(7].)g(Ho)n(we)n(v)o(er)m(,)e(there)i +(only)g(one)g(half)g(of)g(the)g(correspondence)d(is)k(considered,)e +(namely)523 2052 y(ho)n(w)31 b(their)h(v)o(ersion)f(of)h(classical)g +(logic)g(and)f(cut-elimination)f(can)i(be)g(embedded)e(via)i(some)523 +2152 y(speci\002c)21 b(double-ne)o(gation)c(translations)j(into)g +(intuitionistic)h(logic.)f(W)-7 b(e)22 b(conjecture)d(also)j(a)f(cor)n +(-)523 2251 y(respondence)i(in)i(the)g(other)f(direction,)g(namely)g +(that)h(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(and)523 +2351 y(corresponding)12 b(reduction)i(sequences)h(can)g(be)g(simulated) +g(by)g(their)h(cut-elimination)d(procedure.)523 2451 +y(Since)19 b(we)g(shall)h(use)f(as)h(\223point)e(of)g(reference\224)f +(a)j(more)e(general)g(cut-elimination)f(procedure)f(for)523 +2550 y(cut-elimination)h(in)j(classical)g(logic)f(than)g(the)g(one)g +(described)f(in)h([6],)g(we)h(are)f(also)h(able)f(to)g(dra)o(w)523 +2650 y(the)26 b(conclusion)e(that)h(gi)n(v)o(en)g(our)g(conjecture)f +(is)i(true,)f(then)g(double-ne)o(gation)d(translations)j(are)523 +2749 y(not)20 b(enough)e(to)i(describe)g(the)g Ft(full)h +FA(computational)c(meaning)i(of)h(a)h(classical)g(proof.)648 +2859 y(As)h(can)f(be)h(seen)g(to)f(answer)h(the)f(correspondence)d +(questions,)j(we)h(\002rst)g(ha)n(v)o(e)f(to)h(mak)o(e)f(pre-)523 +2959 y(cise)28 b(what)e(we)i(mean)e(by)g(cut-elimination)f(in)i +(classical)h(logic.)e(Most)h(cut-elimination)e(proce-)523 +3058 y(dures,)d(including)f(Gentzen')-5 b(s)23 b(original)f(one,)g +(only)g(terminate)g(if)h(a)h(particular)d(strate)o(gy)h(for)h(cut-)523 +3158 y(elimination)e(is)i(emplo)o(yed.)c(Common)i(e)o(xamples)g(being)g +(an)h(innermost)e(reduction)g(strate)o(gy)-5 b(,)21 b(or)523 +3257 y(the)d(elimination)g(of)g(the)g(cut)h(with)f(the)h(highest)f +(rank.)f(Using)h(those)g(cut-elimination)f(procedures)523 +3357 y(we)29 b(cannot)f(characterise)f(what)i(the)f(set)i(of)e +Ft(all)h FA(normalforms)d(of)i(a)h(classical)h(proof)d(is\227the)o(y) +523 3457 y(w)o(ould)c(produce)e(only)i(one)g(or)h(a)g(limited)f(number) +f(of)h(normalforms.)e(W)-7 b(e)25 b(shall)f(therefore)d(base)523 +3556 y(our)f(ar)o(guments)f(on)i(the)g(cut-elimination)e(procedure)f +(de)n(v)o(eloped)h(by)h(Urban)h(and)f(Bierman)h([20,)523 +3656 y(21],)h(which)g(is)i(lik)o(e)g(Gentzen')-5 b(s)22 +b(procedure)f(e)o(xcept)g(it)j(imposes)f(one)f(slight)h(restriction)f +(on)h(ho)n(w)523 3756 y(commuting)g(cuts)j(need)f(to)h(be)g(analysed.)e +(Since)i(this)g(cut-elimination)d(procedure)g(is)k(strongly)523 +3855 y(normalising,)g(we)i(can)g(calculate)f(all)h(cut-free)f +(normalforms)e(of)j(a)g(classical)g(proof.)e(Because)523 +3955 y(this)19 b(procedure)d(is)k(not)f(Church-Rosser)m(,)d(the)j +(collection)f(of)g(normalforms)e(for)i(a)i(classical)f(proof)523 +4054 y(contains)f(in)h(general)e(more)h(than)g(one)g(element\227as)g +(can)h(be)f(seen)h(for)f(e)o(xample)f(with)i(the)g(proofs)523 +4154 y(\(2\))24 b(and)g(\(3\).)f(As)j(this)f(cut-elimination)d +(procedure)g(puts)j(only)e(v)o(ery)h(slight)g(restrictions)g(on)g(the) +523 4254 y(process)i(of)h(cut-elimination)d(we)j(belie)n(v)o(e)f(a)h +(good)e(case)i(can)g(be)g(made)f(that)g(the)h(collection)f(of)523 +4353 y(normalforms)18 b(calculated)i(by)h(this)g(procedure)d(includes)i +(all)i(\223essential\224)f(normalforms.)3167 4323 y Fp(2)3218 +4353 y FA(Ho)n(w-)523 4453 y(e)n(v)o(er)e(this)i(is)g(a)g(point)e(we)i +(shall)f(not)g(be)g(concerned)e(with)j(in)f(this)h(paper)-5 +b(.)p 523 4563 473 4 v 558 4619 a Fo(2)606 4650 y Fx(Making)31 +b(such)e(a)h(case)f(is)g(hopeless)h(for)f(other)h(strongly-normalising) +h(cut-elimination)e(procedures,)606 4742 y(lik)o(e)f(the)g(one)h(by)f +(Dragalin)g([7],)g(because)h(although)g(the)o(y)f(are)g +(strongly-normalising,)i(the)o(y)e(enforce)606 4833 y(quite)21 +b(strong)g(restrictions)g(on)g(ho)n(w)h(cuts)f(can)g(be)g(eliminated.)g +(F)o(or)f(e)o(xample)i(Dragalin)e(does)i(not)f(allo)n(w)606 +4924 y(\(multi\)cuts)e(to)f(permute)i(o)o(v)o(er)f(other)g +(\(multi\)cuts,)g(see)g([19].)p eop end +%%Page: 4 4 +TeXDict begin 4 3 bop 648 448 a FA(The)30 b(cut-elimination)f +(procedure)f(of)j(Urban)f(and)g(Bierman)g(will)i(be)f(described)e(in)i +(more)523 548 y(detail)24 b(in)g(Sec.)h(3,)f(together)e(with)i(a)h(v)n +(ariant\227the)e(colour)f(protocol\227de)n(v)o(eloped)e(by)j(Danos)h +(et)523 648 y(al.)d([6,)12 b(12].)20 b(Beforehand,)e(ho)n(we)n(v)o(er)m +(,)f(we)k(present)f(some)g(preliminaries)f(about)g(double-ne)o(gation) +523 747 y(translations)28 b(in)h(Sec.)f(2.)h(W)-7 b(e)29 +b(will)h(state)f(the)f(conjecture)f(in)i(Sec.)f(4,)h(gi)n(v)o(e)f(some) +g(e)n(vidence)f(on)523 847 y(why)e(this)h(conjecture)e(is)j(plausible)e +(and)g(present)g(some)g(ideas)h(on)f(ho)n(w)g(to)h(pro)o(v)o(e)e(it.)i +(In)g(Sec.)f(5)523 946 y(we)32 b(shall)g(dra)o(w)f(some)h(conclusions)e +(with)i(respect)g(to)g(the)g(computational)d(interpretation)h(of)523 +1046 y(classical)21 b(proofs.)523 1300 y Fu(2)99 b(Pr)n(eliminaries)25 +b(on)g(Double-Negation)h(T)-7 b(ranslations)523 1487 +y FA(W)g(e)21 b(assume)e(the)h(reader)e(has)i(acquaintance)d(with)j +(sequent-calculus)d(formulations)g(of)j(classical)523 +1586 y(and)e(intuitionistic)f(logic.)h(Because)g(there)g(e)o(xist)g +(sequents)g(that)h(are)f(pro)o(v)n(able)e(in)i(classical)h(logic,)523 +1686 y(b)n(ut)25 b(unpro)o(v)n(able)d(in)j(intuitionistic)g(logic,)g +(the)g(interesting)f(point)g(of)h(double-ne)o(gation)c(transla-)523 +1786 y(tions)g(is)i(that)e(one)g(can)g(embed)f(classical)j(logic)e +(into)g(intuitionistic)f(logic)h(so)h(that)g(pro)o(v)n(ability)d(is)523 +1885 y(preserv)o(ed.)f(F)o(or)i(e)o(xample)f(the)h(follo)n(wing)e +(translation)i(de\002ned)f(o)o(v)o(er)g(formulae)1515 +2075 y Fs(A)1577 2044 y Fn(\003)1639 2028 y Fp(def)1643 +2075 y Fm(=)28 b Fr(::)p Fs(A)22 b FA(with)e Fs(A)h FA(being)e(atomic) +1390 2199 y Fm(\()p Fr(:)p Fs(B)t Fm(\))1576 2169 y Fn(\003)1639 +2152 y Fp(def)1643 2199 y Fm(=)28 b Fr(:)p Fm(\()p Fs(B)1890 +2169 y Fn(\003)1929 2199 y Fm(\))1325 2324 y(\()p Fs(B)t +Fr(^)q Fs(C)6 b Fm(\))1577 2294 y Fn(\003)1639 2277 y +Fp(def)1643 2324 y Fm(=)28 b Fs(B)1803 2294 y Fn(\003)1841 +2324 y Fr(^)q Fs(C)1962 2294 y Fn(\003)1316 2449 y Fm(\()p +Fs(B)t Fr(\033)p Fs(C)6 b Fm(\))1577 2419 y Fn(\003)1639 +2402 y Fp(def)1643 2449 y Fm(=)28 b Fs(B)1803 2419 y +Fn(\003)1841 2449 y Fr(\033)p Fs(C)1971 2419 y Fn(\003)1325 +2574 y Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))1577 2544 +y Fn(\003)1639 2527 y Fp(def)1643 2574 y Fm(=)28 b Fr(:)p +Fm(\()p Fr(:)p Fm(\()p Fs(B)1977 2544 y Fn(\003)2017 +2574 y Fm(\))p Fr(^:)p Fm(\()p Fs(C)2256 2544 y Fn(\003)2295 +2574 y Fm(\)\))3308 2313 y FA(\(4\))523 2737 y(can)20 +b(be)g(used)g(to)g(sho)n(w)g(that)h(e)n(v)o(ery)d(classical)k(proof)c +(with)i(the)h(end-sequent)1854 2904 y Fs(\000)p 1935 +2892 10 38 v 1945 2876 42 4 v 100 w(\001)523 3072 y FA(can)f(be)g +(translated)g(to)g(an)g(intuitionistic)g(proof)e(with)j(the)f +(end-sequent)1747 3240 y Fs(\000)1810 3209 y Fn(\003)1848 +3240 y Fs(;)14 b Fr(:)p Fs(\001)2009 3209 y Fn(\003)p +2066 3228 10 38 v 2075 3211 42 4 v 2158 3240 a Fs(:)1127 +b FA(\(5\))523 3407 y(W)-7 b(e)28 b(use)e(the)h(con)m(v)o(ention)c +(that)j(if)h Fs(\000)39 b FA(is)27 b(the)f(sequent-conte)o(xt)e +Fr(f)p Fs(B)2502 3419 y Fl(1)2539 3407 y Fs(;)14 b(:)g(:)g(:)f(;)h(B) +2786 3419 y Fq(n)2831 3407 y Fr(g)27 b FA(then)f Fs(\000)3133 +3377 y Fn(\003)3197 3407 y FA(stands)523 3507 y(for)f(the)g +(sequent-conte)o(xt)d Fr(f)p Fs(B)1432 3477 y Fn(\003)1428 +3527 y Fl(1)1470 3507 y Fs(;)14 b(:)g(:)g(:)f(;)h(B)1721 +3477 y Fn(\003)1717 3527 y Fq(n)1763 3507 y Fr(g)p FA(.)25 +b(Similarly)f(for)h Fr(:)p Fs(\001)2432 3477 y Fn(\003)2471 +3507 y FA(.)g(W)-7 b(e)27 b(shall)e(also)h(use)f(the)g(con-)523 +3606 y(v)o(ention)17 b(that)h Fs(A)h FA(stands)f(for)f(an)h(atomic)g +(formula)f(and)g Fs(B)t(;)d(:)g(:)g(:)19 b FA(for)e(arbitrary)g +(formulae.)f(A)i(similar)523 3706 y(embedding)g(can)i(be)g(obtained)f +(with)h(the)g(translation:)1729 3895 y Fs(A)1791 3865 +y Fn(\016)1852 3848 y Fp(def)1857 3895 y Fm(=)28 b Fr(::)p +Fs(A)1604 4020 y Fm(\()p Fr(:)p Fs(B)t Fm(\))1790 3990 +y Fn(\016)1852 3973 y Fp(def)1857 4020 y Fm(=)g Fr(:)p +Fm(\()p Fs(B)2104 3990 y Fn(\016)2143 4020 y Fm(\))1539 +4145 y(\()p Fs(B)t Fr(^)p Fs(C)6 b Fm(\))1790 4115 y +Fn(\016)1852 4098 y Fp(def)1857 4145 y Fm(=)28 b Fr(::)p +Fm(\()p Fs(B)2159 4115 y Fn(\016)2198 4145 y Fr(^)q Fs(C)2319 +4115 y Fn(\016)2357 4145 y Fm(\))1529 4270 y(\()p Fs(B)t +Fr(\033)p Fs(C)6 b Fm(\))1790 4240 y Fn(\016)1852 4223 +y Fp(def)1857 4270 y Fm(=)28 b Fr(::)p Fm(\()p Fs(B)2159 +4240 y Fn(\016)2198 4270 y Fr(\033)p Fs(C)2328 4240 y +Fn(\016)2366 4270 y Fm(\))1539 4395 y(\()p Fs(B)t Fr(_)p +Fs(C)6 b Fm(\))1790 4365 y Fn(\016)1852 4348 y Fp(def)1857 +4395 y Fm(=)28 b Fr(::)p Fm(\()p Fs(B)2159 4365 y Fn(\016)2198 +4395 y Fr(_)q Fs(C)2319 4365 y Fn(\016)2357 4395 y Fm(\))3308 +4133 y FA(\(6\))648 4558 y(The)i(usual)h(proof)e(\(see)i(for)g(e)o +(xample)e([4]\))h(for)h(establishing)f(that)h(e)n(v)o(ery)e(classical)j +(proof)523 4657 y(can)21 b(be)g(translated)f(to)i(an)f(intuitionistic)f +(proof)g(proceeds)f(inducti)n(v)o(ely)g(by)i(translating)f(stepwise)523 +4757 y(e)n(v)o(ery)f(inference)f(rule)i(in)h(a)f(proof.)f(F)o(or)g +(instance)h(an)g(axiom)g(of)g(the)g(form)1858 4924 y +Fs(A)p 1938 4912 10 38 v 1948 4896 42 4 v 88 w(A)p eop +end +%%Page: 5 5 +TeXDict begin 5 4 bop 523 448 a FA(is)21 b(translated)f(by)f +Fm(\()p Fr(\000)p Fm(\))1175 418 y Fn(\003)1235 448 y +FA(to)h(the)g(proof)1674 603 y Fr(::)p Fs(A)p 1865 591 +10 38 v 1875 574 42 4 v 89 w Fr(::)p Fs(A)p 1628 623 +527 4 v 1628 696 a Fr(::)p Fs(A;)14 b Fr(:::)p Fs(A)p +2084 684 10 38 v 2094 668 42 4 v 2195 635 a Fr(:)2250 +647 y Fq(L)523 865 y FA(which)i(conforms)g(with)h(the)g(desired)f +(property)f(stated)i(in)g(\(5\).)f(When)h(translating)f(a)i(proof)d +(ending)523 964 y(with)20 b(an)h Fr(_)846 976 y Fq(L)896 +964 y FA(-rule)1648 1021 y Fm(:)1466 1094 y Fs(B)t(;)14 +b(\000)1621 1106 y Fl(1)p 1676 1082 10 38 v 1686 1066 +42 4 v 1746 1094 a Fs(\001)1815 1106 y Fl(1)2114 1021 +y Fm(:)1935 1094 y Fs(C)q(;)g(\000)2083 1106 y Fl(2)p +2140 1082 10 38 v 2149 1066 42 4 v 2209 1094 a Fs(\001)2278 +1106 y Fl(2)p 1466 1130 851 4 v 1505 1204 a Fs(B)t Fr(_)q +Fs(C)q(;)g(\000)1776 1216 y Fl(1)1813 1204 y Fs(;)g(\000)1901 +1216 y Fl(2)p 1957 1192 10 38 v 1967 1175 42 4 v 2027 +1204 a Fs(\001)2096 1216 y Fl(1)2133 1204 y Fs(;)g(\001)2239 +1216 y Fl(2)2357 1147 y Fr(_)2413 1160 y Fq(L)3308 1204 +y FA(\(7\))523 1345 y(then)20 b(by)g(induction)e(hypothesis)g(we)j(ha)n +(v)o(e)f(tw)o(o)g(intuitionistic)g(proofs)f(ending)f(with)1522 +1476 y Fm(:)1268 1549 y Fs(B)1335 1519 y Fn(\003)1374 +1549 y Fs(;)c(\000)1474 1519 y Fn(\003)1462 1570 y Fl(1)1511 +1549 y Fs(;)g Fr(:)p Fs(\001)1672 1519 y Fn(\003)1672 +1570 y Fl(1)p 1729 1537 10 38 v 1739 1521 42 4 v 1882 +1549 a FA(and)2338 1476 y Fm(:)2085 1549 y Fs(C)2150 +1519 y Fn(\003)2188 1549 y Fs(;)g(\000)2288 1519 y Fn(\003)2276 +1570 y Fl(2)2326 1549 y Fs(;)g Fr(:)p Fs(\001)2487 1519 +y Fn(\003)2487 1570 y Fl(2)p 2544 1537 10 38 v 2554 1521 +42 4 v 2637 1549 a Fs(:)523 1718 y FA(W)-7 b(e)21 b(can)f(then)g(form)f +(the)h(intuitionistic)g(proof)1486 1849 y Fm(:)1232 1923 +y Fs(B)1299 1892 y Fn(\003)1338 1923 y Fs(;)14 b(\000)1438 +1892 y Fn(\003)1426 1943 y Fl(1)1475 1923 y Fs(;)g Fr(:)p +Fs(\001)1636 1892 y Fn(\003)1636 1943 y Fl(1)p 1693 1911 +10 38 v 1703 1894 42 4 v 1223 1963 549 4 v 1223 2037 +a Fs(\000)1286 2007 y Fn(\003)1274 2058 y Fl(1)1324 2037 +y Fs(;)g Fr(:)p Fs(\001)1485 2007 y Fn(\003)1485 2058 +y Fl(1)p 1542 2025 10 38 v 1551 2008 42 4 v 1611 2037 +a Fr(:)p Fs(B)1733 2007 y Fn(\003)1814 1975 y Fr(:)1869 +1987 y Fq(R)2268 1849 y Fm(:)2016 1923 y Fs(C)2081 1892 +y Fn(\003)2119 1923 y Fs(;)g(\000)2219 1892 y Fn(\003)2207 +1943 y Fl(2)2257 1923 y Fs(;)g Fr(:)p Fs(\001)2418 1892 +y Fn(\003)2418 1943 y Fl(2)p 2475 1911 10 38 v 2484 1894 +42 4 v 2006 1963 548 4 v 2006 2037 a Fs(\000)2069 2007 +y Fn(\003)2057 2058 y Fl(2)2107 2037 y Fs(;)g Fr(:)p +Fs(\001)2268 2007 y Fn(\003)2268 2058 y Fl(2)p 2325 2025 +10 38 v 2335 2008 42 4 v 2395 2037 a Fr(:)p Fs(C)2515 +2007 y Fn(\003)2595 1975 y Fr(:)2650 1987 y Fq(R)p 1223 +2077 1331 4 v 1338 2151 a Fs(\000)1401 2121 y Fn(\003)1389 +2172 y Fl(1)1439 2151 y Fs(;)g(\000)1539 2121 y Fn(\003)1527 +2172 y Fl(2)1577 2151 y Fs(;)g Fr(:)p Fs(\001)1738 2121 +y Fn(\003)1738 2172 y Fl(1)1776 2151 y Fs(;)g Fr(:)p +Fs(\001)1937 2121 y Fn(\003)1937 2172 y Fl(2)p 1994 2139 +10 38 v 2004 2123 42 4 v 2064 2151 a Fr(:)p Fs(B)2186 +2121 y Fn(\003)2224 2151 y Fr(^)q(:)p Fs(C)2400 2121 +y Fn(\003)2595 2094 y Fr(^)2650 2107 y Fq(R)p 1260 2192 +1258 4 v 1260 2271 a Fr(:)p Fm(\()p Fr(:)p Fs(B)1469 +2240 y Fn(\003)1508 2271 y Fr(^:)p Fs(C)1683 2240 y Fn(\003)1722 +2271 y Fm(\))p Fs(;)g(\000)1854 2240 y Fn(\003)1842 2291 +y Fl(1)1892 2271 y Fs(;)g(\000)1992 2240 y Fn(\003)1980 +2291 y Fl(2)2030 2271 y Fs(;)g Fr(:)p Fs(\001)2191 2240 +y Fn(\003)2191 2291 y Fl(1)2229 2271 y Fs(;)g Fr(:)p +Fs(\001)2390 2240 y Fn(\003)2390 2291 y Fl(2)p 2447 2259 +10 38 v 2457 2242 42 4 v 2558 2203 a Fr(:)2613 2215 y +Fq(L)3308 2271 y FA(\(8\))523 2439 y(as)32 b(the)f(translation)g(of)g +(\(7\).)f(F)o(or)h(the)g(sak)o(e)g(of)g(more)g(clarity)g(we)g(will)h +(omit)f(in)h(what)f(follo)n(ws)523 2539 y(the)22 b(sequent-conte)o(xts) +d(whene)n(v)o(er)h(the)o(y)h(are)h(unimportant.)d(Thus)j(we)g(shall)g +(gi)n(v)o(e)f(for)h(the)f(proof-)523 2638 y(fragment)e(sho)n(wn)g(in)h +(\(8\))g(only)f(the)h(follo)n(wing)f(simpli\002ed)h(inference)e(rules:) +1636 2769 y Fm(:)1551 2843 y Fs(B)1618 2813 y Fn(\003)p +1675 2831 10 38 v 1684 2814 42 4 v 1523 2863 249 4 v +1542 2925 10 38 v 1551 2908 42 4 v 1611 2937 a Fr(:)p +Fs(B)1733 2907 y Fn(\003)1814 2875 y Fr(:)1869 2887 y +Fq(R)2118 2769 y Fm(:)2034 2843 y Fs(C)2099 2813 y Fn(\003)p +2156 2831 10 38 v 2166 2814 42 4 v 2006 2863 247 4 v +2025 2925 10 38 v 2034 2908 42 4 v 2094 2937 a Fr(:)p +Fs(C)2214 2907 y Fn(\003)2295 2875 y Fr(:)2350 2887 y +Fq(R)p 1523 2957 730 4 v 1675 3018 10 38 v 1685 3002 +42 4 v 1745 3030 a Fr(:)p Fs(B)1867 3000 y Fn(\003)1906 +3030 y Fr(^:)p Fs(C)2081 3000 y Fn(\003)2295 2974 y Fr(^)2350 +2986 y Fq(R)p 1597 3050 583 4 v 1597 3129 a Fr(:)p Fm(\()p +Fr(:)p Fs(B)1806 3099 y Fn(\003)1845 3129 y Fr(^)q(:)p +Fs(C)2021 3099 y Fn(\003)2059 3129 y Fm(\))p 2110 3117 +10 38 v 2120 3101 42 4 v 2221 3062 a Fr(:)2276 3074 y +Fq(L)523 3298 y FA(When)i(translating)f(a)i(classical)g(proof)e(ending) +f(with)j(an)f Fr(^)2243 3310 y Fq(R)2298 3298 y FA(-rule)1759 +3429 y Fm(:)p 1711 3490 10 38 v 1721 3473 42 4 v 1781 +3502 a Fs(B)1996 3429 y Fm(:)p 1949 3490 10 38 v 1959 +3473 42 4 v 73 x Fs(C)p 1693 3522 392 4 v 1769 3583 10 +38 v 1778 3567 42 4 v 1838 3595 a(B)t Fr(^)q Fs(C)2126 +3539 y Fr(^)2181 3551 y Fq(R)3308 3595 y FA(\(9\))523 +3764 y(we)h(ha)n(v)o(e)e(by)h(induction)e(hypothesis)h(tw)o(o)h +(intuitionistic)g(proofs)f(ending)g(in)1663 3895 y Fm(:)1550 +3969 y Fr(:)p Fs(B)1672 3938 y Fn(\003)p 1729 3957 10 +38 v 1739 3940 42 4 v 1882 3969 a FA(and)2197 3895 y +Fm(:)2085 3969 y Fr(:)p Fs(C)2205 3938 y Fn(\003)p 2262 +3957 10 38 v 2272 3940 42 4 v 2355 3969 a Fs(:)523 4137 +y FA(In)g(order)f(to)i(form)e(an)h(intuitionistic)g(proof)f(ending)g +(with)h(the)g(sequent)g Fr(:)p Fm(\()p Fs(B)2773 4107 +y Fn(\003)2812 4137 y Fr(^)p Fs(C)2932 4107 y Fn(\003)2971 +4137 y Fm(\))p 3021 4125 10 38 v 3031 4109 42 4 v 88 +w FA(,)g(we)h(need)523 4237 y(to)i(e)o(xploit)f(the)h(property)e(of)i +Fm(\()p Fr(\000)p Fm(\))1511 4207 y Fn(\003)1572 4237 +y FA(that)g(one)g(can)g(al)o(w)o(ays)g(pro)o(v)o(e)e +(intuitionistically)h(the)h(sequent)523 4336 y Fr(::)p +Fm(\()p Fr(\000)p Fm(\))762 4306 y Fn(\003)p 819 4324 +10 38 v 829 4308 42 4 v 889 4336 a Fm(\()p Fr(\000)p +Fm(\))1018 4306 y Fn(\003)1056 4336 y FA(.)g(F)o(or)f(e)o(xample)f(in)h +(the)g(atomic)g(case)h(one)e(has)i(for)e Fr(::)p Fs(A)2723 +4306 y Fn(\003)p 2781 4324 10 38 v 2790 4308 42 4 v 2850 +4336 a Fs(A)2912 4306 y Fn(\003)2973 4336 y FA(the)h(intuition-)523 +4436 y(istic)g(proof:)1727 4519 y Fr(:)p Fs(A)p 1863 +4507 10 38 v 1872 4490 42 4 v 88 w Fr(:)p Fs(A)p 1681 +4539 416 4 v 1681 4612 a Fr(:)p Fs(A;)14 b Fr(::)p Fs(A)p +2027 4600 10 38 v 2036 4584 42 4 v 2138 4551 a Fr(:)2193 +4563 y Fq(L)p 1671 4648 434 4 v 1671 4722 a Fr(:)p Fs(A)p +1807 4710 10 38 v 1817 4693 42 4 v 89 w Fr(:::)p Fs(A)2147 +4660 y Fr(:)2202 4672 y Fq(R)p 1625 4742 527 4 v 1625 +4815 a Fr(::::)p Fs(A;)g Fr(:)p Fs(A)p 2082 4803 10 38 +v 2091 4786 42 4 v 2193 4753 a Fr(:)2248 4765 y Fq(L)p +1616 4851 545 4 v 1616 4924 a Fr(::::)p Fs(A)p 1918 4912 +10 38 v 1928 4896 42 4 v 90 w Fr(::)p Fs(A)2202 4863 +y Fr(:)2257 4875 y Fq(R)p eop end +%%Page: 6 6 +TeXDict begin 6 5 bop 523 448 a FA(Using)26 b(proofs)e(for)i +Fr(::)p Fs(B)1287 418 y Fn(\003)p 1344 436 10 38 v 1354 +420 42 4 v 1414 448 a Fs(B)1481 418 y Fn(\003)1546 448 +y FA(and)f Fr(::)p Fs(C)1867 418 y Fn(\003)p 1925 436 +10 38 v 1934 420 42 4 v 1994 448 a Fs(C)2059 418 y Fn(\003)2097 +448 y FA(,)i(we)f(can)g(construct)f(the)h(follo)n(wing)e(trans-)523 +548 y(lated)c(proof)f(for)g(\(9\):)963 694 y Fm(:)850 +768 y Fr(:)p Fs(B)972 738 y Fn(\003)p 1030 756 10 38 +v 1039 739 42 4 v 823 788 304 4 v 841 850 10 38 v 851 +833 42 4 v 911 862 a Fr(::)p Fs(B)1088 831 y Fn(\003)1168 +799 y Fr(:)1223 811 y Fq(R)1554 788 y Fm(:)1361 862 y +Fr(::)p Fs(B)1538 831 y Fn(\003)p 1596 850 10 38 v 1605 +833 42 4 v 1665 862 a Fs(B)1732 831 y Fn(\003)p 823 881 +948 4 v 1218 943 10 38 v 1228 927 42 4 v 1288 955 a Fs(B)1355 +925 y Fn(\003)1812 907 y Fs(cut)2148 694 y Fm(:)2036 +768 y Fr(:)p Fs(C)2156 738 y Fn(\003)p 2213 756 10 38 +v 2223 739 42 4 v 2008 788 303 4 v 2027 850 10 38 v 2036 +833 42 4 v 2096 862 a Fr(::)p Fs(C)2271 831 y Fn(\003)2352 +799 y Fr(:)2407 811 y Fq(R)2736 788 y Fm(:)2545 862 y +Fr(::)p Fs(C)2720 831 y Fn(\003)p 2777 850 10 38 v 2787 +833 42 4 v 2847 862 a Fs(C)2912 831 y Fn(\003)p 2008 +881 942 4 v 2402 943 10 38 v 2412 927 42 4 v 2472 955 +a Fs(C)2537 925 y Fn(\003)2992 907 y Fs(cut)p 1200 975 +1376 4 v 1730 1037 10 38 v 1739 1020 42 4 v 1799 1049 +a(B)1866 1019 y Fn(\003)1905 1049 y Fr(^)p Fs(C)2025 +1019 y Fn(\003)2616 992 y Fr(^)2672 1004 y Fq(R)p 1651 +1069 472 4 v 1651 1148 a Fr(:)p Fm(\()p Fs(B)1805 1118 +y Fn(\003)1844 1148 y Fr(^)q Fs(C)1965 1118 y Fn(\003)2003 +1148 y Fm(\))p 2054 1136 10 38 v 2063 1119 42 4 v 2165 +1081 a Fr(:)2220 1093 y Fq(L)523 1332 y FA(W)-7 b(e)31 +b(shall)f(refer)f(to)h(the)f(cuts)h(introduced)e(by)h(the)h(double-ne)o +(gation)25 b(translation)k(as)h Ft(auxiliary)523 1432 +y(cuts)p FA(.)25 b(F)o(or)f(a)h(number)e(of)h(reasons)g(\(one)g(of)g +(them)g(being)g(to)g(minimise)g(the)h(amount)e(of)h(writing\))523 +1531 y(we)d(shall)f(use)h(a)f(ne)n(w)g(inference)f(rule,)g(namely)p +1746 1686 10 38 v 1756 1670 42 4 v 1816 1698 a Fr(::)p +Fs(B)p 1728 1718 266 4 v 1802 1780 10 38 v 1811 1763 +42 4 v 1871 1792 a(B)2035 1730 y Fr(::)2145 1742 y Fq(R)523 +1976 y FA(to)h(stand)g(for)g(auxiliary)f(cuts,)h(which)g(ha)n(v)o(e)f +(al)o(w)o(ays)i(the)f(form:)1644 2122 y Fm(:)1491 2195 +y Fs(\000)p 1572 2183 10 38 v 1582 2167 42 4 v 100 w +Fr(::)p Fs(B)2058 2122 y Fm(:)1903 2195 y Fr(::)p Fs(B)p +2099 2183 10 38 v 2109 2167 42 4 v 92 w(B)p 1491 2215 +745 4 v 1755 2289 a(\000)p 1836 2277 10 38 v 1845 2260 +42 4 v 99 w(B)2277 2241 y(cut)j(:)523 2473 y FA(Clearly)-5 +b(,)28 b(this)h(ne)n(w)g(rule)g(does)f(not)g(af)n(fect)h(the)g(pro)o(v) +n(ability)d(of)i(sequents.)g(Issues)i(whether)d(the)523 +2573 y Fr(::)633 2585 y Fq(R)688 2573 y FA(-rule)22 b(\(or)g(an)g +(auxiliary)f(cut\))i(af)n(fects)f(the)g(beha)n(viour)f(under)g +(cut-elimination)f(are)j(delayed)523 2672 y(until)f(Sec.)g(4.)g(W)m +(ith)g(this)g(ne)n(w)g(inference)e(rule)i(we)g(can)g(gi)n(v)o(e)f(the)g +(translation)g(of)h(\(9\))f(more)g(com-)523 2772 y(pactly)f(as:)1553 +2836 y Fm(:)1440 2909 y Fr(:)p Fs(B)1562 2879 y Fn(\003)p +1620 2897 10 38 v 1629 2881 42 4 v 1413 2929 304 4 v +1431 2991 10 38 v 1441 2974 42 4 v 1501 3003 a Fr(::)p +Fs(B)1678 2973 y Fn(\003)1758 2941 y Fr(:)1813 2953 y +Fq(R)p 1413 3023 304 4 v 1487 3085 10 38 v 1496 3068 +42 4 v 1556 3097 a Fs(B)1623 3067 y Fn(\003)1758 3035 +y Fr(::)1868 3047 y Fq(R)2146 2836 y Fm(:)2034 2909 y +Fr(:)p Fs(C)2154 2879 y Fn(\003)p 2211 2897 10 38 v 2221 +2881 42 4 v 2006 2929 303 4 v 2025 2991 10 38 v 2034 +2974 42 4 v 2094 3003 a Fr(::)p Fs(C)2269 2973 y Fn(\003)2350 +2941 y Fr(:)2405 2953 y Fq(R)p 2006 3023 303 4 v 2080 +3085 10 38 v 2090 3068 42 4 v 2150 3097 a Fs(C)2215 3067 +y Fn(\003)2350 3035 y Fr(::)2460 3047 y Fq(R)p 1468 3117 +786 4 v 1703 3178 10 38 v 1713 3162 42 4 v 1773 3190 +a Fs(B)1840 3160 y Fn(\003)1878 3190 y Fr(^)p Fs(C)1998 +3160 y Fn(\003)2295 3134 y Fr(^)2350 3146 y Fq(R)p 1625 +3210 472 4 v 1625 3289 a Fr(:)p Fm(\()p Fs(B)1779 3259 +y Fn(\003)1818 3289 y Fr(^)p Fs(C)1938 3259 y Fn(\003)1976 +3289 y Fm(\))p 2027 3277 10 38 v 2037 3261 42 4 v 2138 +3222 a Fr(:)2193 3234 y Fq(L)523 3441 y FA(The)h(translations)f(for)g +(the)h(rules)f Fs(contr)1694 3453 y Fq(L)1745 3441 y +FA(,)h Fs(contr)1980 3453 y Fq(R)2035 3441 y FA(,)g Fs(w)r(eak)2264 +3453 y Fq(L)2314 3441 y FA(,)g Fs(w)r(eak)2543 3453 y +Fq(R)2598 3441 y FA(,)g Fr(:)2695 3453 y Fq(L)2745 3441 +y FA(,)g Fr(:)2842 3453 y Fq(R)2897 3441 y FA(,)g Fr(_)2995 +3453 y Fq(R)3045 3461 y Fk(i)3075 3441 y FA(,)g Fr(^)3173 +3453 y Fq(L)3219 3461 y Fk(i)3249 3441 y FA(,)g Fr(\033)3355 +3453 y Fq(L)523 3541 y FA(and)f Fr(\033)728 3553 y Fq(R)803 +3541 y FA(are)h(left)f(as)h(e)o(x)o(ercises)e(to)i(the)f(reader)-5 +b(.)648 3641 y(W)e(e)21 b(can)f(no)n(w)g(gi)n(v)o(e)f(the)h(double-ne)o +(gation)c(translations)j(of)h(the)g(tw)o(o)h(subproofs)1033 +3808 y Fs(A)p 1114 3796 10 38 v 1123 3779 42 4 v 88 w(A)84 +b(A)p 1409 3796 10 38 v 1419 3779 42 4 v 88 w(A)p 1033 +3828 509 4 v 1054 3901 a(A)19 b Fr(_)f Fs(A)p 1289 3889 +10 38 v 1299 3872 42 4 v 89 w(A;)c(A)1583 3845 y Fr(_)1638 +3857 y Fq(L)p 1054 3937 467 4 v 1104 4011 a Fs(A)k Fr(_)h +Fs(A)p 1339 3999 10 38 v 1348 3982 42 4 v 88 w(A)1562 +3956 y(contr)1755 3968 y Fq(R)2142 3808 y Fs(A)p 2222 +3796 10 38 v 2232 3779 42 4 v 88 w(A)83 b(A)p 2518 3796 +10 38 v 2527 3779 42 4 v 88 w(A)p 2142 3828 509 4 v 2181 +3901 a(A;)14 b(A)p 2361 3889 10 38 v 2370 3872 42 4 v +88 w(A)p Fr(^)q Fs(A)2691 3845 y Fr(^)2746 3857 y Fq(R)p +2181 3937 430 4 v 2231 4011 a Fs(A)p 2311 3999 10 38 +v 2321 3982 42 4 v 88 w(A)p Fr(^)p Fs(A)2652 3956 y(contr)2845 +3968 y Fq(L)523 4195 y FA(sho)n(wn)19 b(in)i(\(1\).)e(The)h +(translations)g(are:)574 4429 y Fz(::)p Fj(A)p 751 4417 +9 34 v 760 4402 38 4 v 80 w Fz(::)p Fj(A)p 532 4449 486 +4 v 532 4516 a Fz(::)p Fj(A;)13 b Fz(:::)p Fj(A)p 954 +4504 9 34 v 963 4489 38 4 v 1058 4461 a Fz(:)1109 4469 +y Fi(L)p 523 4551 503 4 v 523 4618 a Fz(:::)p Fj(A)p +751 4606 9 34 v 760 4591 38 4 v 80 w Fz(:::)p Fj(A)1067 +4563 y Fz(:)1118 4571 y Fi(R)1294 4429 y Fz(::)p Fj(A)p +1471 4417 9 34 v 1480 4402 38 4 v 80 w Fz(::)p Fj(A)p +1252 4449 486 4 v 1252 4516 a Fz(::)p Fj(A;)g Fz(:::)p +Fj(A)p 1674 4504 9 34 v 1683 4489 38 4 v 1779 4461 a +Fz(:)1830 4469 y Fi(L)p 1243 4551 503 4 v 1243 4618 a +Fz(:::)p Fj(A)p 1471 4606 9 34 v 1480 4591 38 4 v 80 +w Fz(:::)p Fj(A)1787 4563 y Fz(:)1838 4571 y Fi(R)p 523 +4638 1223 4 v 629 4706 a Fz(:::)p Fj(A;)h Fz(:::)p Fj(A)p +1103 4694 9 34 v 1111 4679 38 4 v 80 w Fz(:::)p Fj(A)p +Fz(^:::)p Fj(A)1787 4655 y Fz(^)1838 4663 y Fi(R)p 557 +4740 1156 4 v 557 4813 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p +Fz(^:::)p Fj(A)p Fh(\))p Fj(;)f Fz(:::)p Fj(A;)h Fz(:::)p +Fj(A)p 1649 4801 9 34 v 1657 4786 38 4 v 1753 4752 a +Fz(:)1804 4760 y Fi(L)p 557 4852 1156 4 v 679 4924 a +Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p +Fj(;)f Fz(:::)p Fj(A)p 1526 4912 9 34 v 1535 4897 38 +4 v 1753 4871 a(contr)1932 4879 y Fi(L)2030 4341 y Fz(::)p +Fj(A)p 2207 4329 9 34 v 2216 4314 38 4 v 80 w Fz(::)p +Fj(A)p 1987 4361 486 4 v 1987 4429 a Fz(::)p Fj(A;)g +Fz(:::)p Fj(A)p 2410 4417 9 34 v 2418 4402 38 4 v 2514 +4373 a Fz(:)2565 4381 y Fi(L)p 1979 4463 503 4 v 1979 +4531 a Fz(::)p Fj(A)p 2156 4519 9 34 v 2164 4504 38 4 +v 80 w Fz(::::)p Fj(A)2523 4475 y Fz(:)2574 4483 y Fi(R)p +1979 4551 503 4 v 2030 4618 a Fz(::)p Fj(A)p 2207 4606 +9 34 v 2216 4591 38 4 v 80 w Fz(::)p Fj(A)2523 4563 y +Fz(::)2625 4571 y Fi(R)2801 4341 y Fz(::)p Fj(A)p 2978 +4329 9 34 v 2987 4314 38 4 v 80 w Fz(::)p Fj(A)p 2759 +4361 486 4 v 2759 4429 a Fz(::)p Fj(A;)g Fz(:::)p Fj(A)p +3181 4417 9 34 v 3190 4402 38 4 v 3286 4373 a Fz(:)3337 +4381 y Fi(L)p 2750 4463 503 4 v 2750 4531 a Fz(::)p Fj(A)p +2927 4519 9 34 v 2936 4504 38 4 v 80 w Fz(::::)p Fj(A)3294 +4475 y Fz(:)3345 4483 y Fi(R)p 2750 4551 503 4 v 2801 +4618 a Fz(::)p Fj(A)p 2978 4606 9 34 v 2987 4591 38 4 +v 80 w Fz(::)p Fj(A)3294 4563 y Fz(::)3396 4571 y Fi(R)p +2030 4638 1172 4 v 2213 4706 a Fz(::)p Fj(A;)g Fz(::)p +Fj(A)p 2584 4694 9 34 v 2593 4679 38 4 v 80 w Fz(::)p +Fj(A)p Fz(^::)p Fj(A)3243 4655 y Fz(^)3294 4663 y Fi(R)p +2141 4740 951 4 v 2141 4813 a Fz(::)p Fj(A;)g Fz(::)p +Fj(A;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p Fj(A)p +Fh(\))p 3028 4801 9 34 v 3036 4786 38 4 v 3132 4752 a +Fz(:)3183 4760 y Fi(L)p 2141 4852 951 4 v 2238 4924 a +Fz(::)p Fj(A;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p +Fj(A)p Fh(\))p 2931 4912 9 34 v 2939 4897 38 4 v 3132 +4871 a Fj(contr)3311 4879 y Fi(L)p eop end +%%Page: 7 7 +TeXDict begin 7 6 bop 523 448 a FA(T)-7 b(o)16 b(gi)n(v)o(e)f(a)i +(double-ne)o(gation)11 b(translation)16 b(for)f(the)h(whole)f(proof,)g +(we)h(need)f(to)i(be)f(able)f(to)i(translate)523 548 +y(instances)k(of)f(the)h(cut-rule.)e(While)i(the)g(logical)g(inference) +e(rules)h(and)h(the)f(structural)g(rules)h(ha)n(v)o(e)523 +648 y(relati)n(v)o(ely)g(canonical)h(double-ne)o(gation)17 +b(translations,)22 b(there)g(is)i(a)e(choice)g(for)g(ho)n(w)g(to)h +(translate)523 747 y(cut-rules.)c(Consider)h(the)g(follo)n(wing)e +(cut-instance:)1725 886 y Fs(\031)1772 898 y Fl(1)1756 +951 y Fm(:)p 1708 1012 10 38 v 1718 996 42 4 v 1778 1024 +a Fs(B)1963 886 y(\031)2010 898 y Fl(2)1994 951 y Fm(:)1928 +1024 y Fs(B)p 2014 1012 10 38 v 2023 996 42 4 v 1690 +1044 394 4 v 1861 1098 10 38 v 1871 1081 42 4 v 2125 +1070 a(cut)3267 1110 y FA(\(10\))523 1287 y(By)f(induction)e +(hypothesis)h(we)h(ha)n(v)o(e)f(tw)o(o)h(intuitionistic)g(proofs)e +Fs(\031)2484 1257 y Fn(\003)2481 1307 y Fl(1)2540 1287 +y FA(and)h Fs(\031)2727 1257 y Fn(\003)2724 1307 y Fl(2)2783 +1287 y FA(with)h(end-sequents:)1574 1450 y Fs(\031)1624 +1420 y Fn(\003)1621 1470 y Fl(1)1607 1523 y Fm(:)1494 +1597 y Fr(:)p Fs(B)1616 1566 y Fn(\003)p 1673 1585 10 +38 v 1683 1568 42 4 v 1909 1597 a FA(and)2247 1450 y +Fs(\031)2297 1420 y Fn(\003)2294 1470 y Fl(2)2280 1523 +y Fm(:)2195 1597 y Fs(B)2262 1566 y Fn(\003)p 2318 1585 +10 38 v 2328 1568 42 4 v 2411 1597 a Fs(:)523 1773 y +FA(T)-7 b(o)17 b(form)e(an)i(intuitionistic)f(proof)f(with)h(the)h +(end-sequent)p 2254 1761 10 38 v 2263 1745 42 4 v 119 +w(\(remember)e(we)i(omit)f(the)h(sequent-)523 1873 y(conte)o(xts\),)i +(we)h(can)g(translate)g(\(10\))f(either)h(as:)1005 2217 +y Fs(\031)1055 2187 y Fn(\003)1052 2238 y Fl(1)1038 2290 +y Fm(:)925 2364 y Fr(:)p Fs(B)1047 2334 y Fn(\003)p 1104 +2352 10 38 v 1113 2336 42 4 v 1337 2124 a Fs(\031)1387 +2094 y Fn(\003)1384 2144 y Fl(2)1369 2197 y Fm(:)1284 +2270 y Fs(B)1351 2240 y Fn(\003)p 1408 2258 10 38 v 1417 +2242 42 4 v 1256 2290 249 4 v 1275 2352 10 38 v 1284 +2336 42 4 v 1344 2364 a Fr(:)p Fs(B)1466 2334 y Fn(\003)1547 +2302 y Fr(:)1602 2314 y Fq(R)p 925 2384 581 4 v 1189 +2438 10 38 v 1199 2421 42 4 v 1547 2410 a Fs(cut)1826 +2332 y FA(or)2169 2030 y Fs(\031)2219 2000 y Fn(\003)2216 +2051 y Fl(1)2202 2103 y Fm(:)2089 2177 y Fr(:)p Fs(B)2211 +2147 y Fn(\003)p 2268 2165 10 38 v 2278 2148 42 4 v 2061 +2197 304 4 v 2080 2258 10 38 v 2089 2242 42 4 v 2149 +2270 a Fr(::)p Fs(B)2326 2240 y Fn(\003)2407 2208 y Fr(:)2462 +2220 y Fq(R)p 2061 2290 304 4 v 2135 2352 10 38 v 2145 +2336 42 4 v 2205 2364 a Fs(B)2272 2334 y Fn(\003)2407 +2302 y Fr(::)2517 2314 y Fq(R)2707 2217 y Fs(\031)2757 +2187 y Fn(\003)2754 2238 y Fl(2)2740 2290 y Fm(:)2655 +2364 y Fs(B)2722 2334 y Fn(\003)p 2779 2352 10 38 v 2788 +2336 42 4 v 2117 2384 732 4 v 2457 2438 10 38 v 2466 +2421 42 4 v 2890 2410 a Fs(cut)523 2627 y FA(W)-7 b(e)21 +b(refer)f(to)g(these)h(choices)e(as)i Ft(left-)g FA(and)e +Ft(right-tr)o(anslation)f FA(of)i(a)h(cut,)f(respecti)n(v)o(ely)-5 +b(.)648 2726 y(Coming)15 b(back)g(to)h(the)g(proof)e(gi)n(v)o(en)h(in)h +(\(1\),)f(let)h(us)h(call)f(the)g(left-)g(and)f(right-translation)e(of) +j(this)523 2826 y(proof)22 b Fs(\031)777 2796 y Fn(\003)774 +2849 y Fq(L)847 2826 y FA(and)h Fs(\031)1041 2796 y Fn(\003)1038 +2849 y Fq(R)1093 2826 y FA(,)g(respecti)n(v)o(ely)-5 +b(.)21 b(Eliminating)h(all)h(cuts)h(\(including)d(auxiliary)h(cuts\))h +(from)f Fs(\031)3358 2796 y Fn(\003)3355 2849 y Fq(L)523 +2925 y FA(and)e Fs(\031)714 2895 y Fn(\003)711 2948 y +Fq(R)766 2925 y FA(,)h(we)g(obtain)f(the)h(tw)o(o)g(cut-free)f(proofs)f +Fs(\031)1986 2895 y Fn(0\003)1983 2948 y Fq(L)2065 2925 +y FA(and)h Fs(\031)2256 2895 y Fn(0\003)2253 2948 y Fq(R)2335 +2925 y FA(sho)n(wn)g(in)h(Fig.)g(1.)g(It)g(turns)f(out)g(\(we)523 +3025 y(ho)n(we)n(v)o(er)d(lea)n(v)o(e)i(out)f(the)h(calculations\))f +(that)h(had)f(we)i(double-ne)o(gation)14 b(translated)k(the)h(tw)o(o)g +(nor)n(-)523 3125 y(malforms)h(of)i(\(1\))f(and)g(then)g(eliminated)g +(all)h(auxiliary)e(cuts)i(from)f(the)h(double-ne)o(gated)17 +b(proofs,)523 3224 y(we)i(w)o(ould)e(ha)n(v)o(e)h(also)g(obtained)f +(the)h(proofs)f Fs(\031)1895 3194 y Fn(0\003)1892 3247 +y Fq(L)1972 3224 y FA(and)g Fs(\031)2160 3194 y Fn(0\003)2157 +3247 y Fq(R)2218 3224 y FA(:)i(The)f(normalform)d(\(2\))j(obtained)f +(from)523 3324 y(\(1\))g(by)h(commuting)e(the)i(cut)g(to)g(the)h(left)f +(leads)g(to)g Fs(\031)2032 3294 y Fn(0\003)2029 3347 +y Fq(L)2090 3324 y FA(\227the)g(normalform)d(of)j(the)g +(left-translation)523 3424 y(of)23 b(\(1\),)f(while)g(\(3\))h(obtained) +e(by)h(commuting)f(the)i(cut)g(to)g(the)f(right)h(leads)g(to)g +Fs(\031)2877 3393 y Fn(0\003)2874 3446 y Fq(R)2934 3424 +y FA(\227the)g(normal-)523 3523 y(form)30 b(of)g(the)g +(right-translation)e(of)j(\(1\).)e(W)-7 b(e)32 b(tak)o(e)f(this)g(as)g +(a)g(\002rst)g(hint)f(that)h(double-ne)o(gation)523 3623 +y(translations)d(seem)g(to)h(be)f(able)g(to)h(simulate)f(the)h(beha)n +(viour)d(of)i(cut-elimination)e(in)j(classical)523 3722 +y(logic.)648 3822 y(T)-7 b(o)16 b(sum)g(up)f(this)i(section,)e(let)i +(us)f(remark)f(that)h(a)g(similar)g(\223story\224)f(can)h(be)g(told)g +(for)f(the)h(transla-)523 3922 y(tion)k Fm(\()p Fr(\000)p +Fm(\))802 3892 y Fn(\016)862 3922 y FA(gi)n(v)o(en)f(in)i(\(6\).)e(In)i +(f)o(act,)f(it)h(can)g(be)f(told)g(for)g(an)o(y)g(sensible)g(notion)g +(of)g(double-ne)o(gation)523 4021 y(translation.)27 b(F)o(or)h(e)o +(xample)f(it)i(w)o(ould)e(be)h(completely)f(immaterial)h(to)g(our)f +(\223story\224)h(if)g(we)h(had)523 4121 y(translated)g(atomic)f +(formulae)g Fs(A)i FA(as)g Fr(::)p Fs(A)p FA(,)g Fr(::::)p +Fs(A)h FA(or)e Fr(::::::)p Fs(A)p FA(.)j(The)d(most)g(important)523 +4221 y(property)17 b(we)j(distill)g(from)f(the)g(ar)o(guments)f(abo)o +(v)o(e)g(is)i(that)g(we)g(re)o(gard)d(a)j(double-ne)o(gation)15 +b(trans-)523 4320 y(lation,)20 b(say)g Fm(\()p Fr(\000)p +Fm(\))1014 4290 y Fq(x)1056 4320 y FA(,)h(as)f(a)h(translation)e(of)h +(a)h(classical)g(proof)d(ha)n(ving)i(an)g(end-sequent)1952 +4462 y Fm(:)1854 4535 y Fs(\000)p 1935 4523 10 38 v 1945 +4507 42 4 v 100 w(\001)523 4712 y FA(to)g(an)h(intuitionistic)e(proof)g +(with)h(the)g(end-sequent)1952 4851 y Fm(:)1766 4924 +y Fs(\000)1829 4894 y Fq(x)1871 4924 y Fs(;)14 b Fr(:)p +Fs(\001)2032 4894 y Fq(x)p 2092 4912 10 38 v 2102 4896 +42 4 v eop end +%%Page: 8 8 +TeXDict begin 8 7 bop 523 369 2882 4 v 523 2639 4 2271 +v 593 457 a Fg(\031)649 426 y Ff(0\003)646 473 y Fe(L)734 +457 y Fd(=)727 536 y Fz(:)p Fj(A)p 853 524 9 34 v 861 +509 38 4 v 80 w Fz(:)p Fj(A)p 685 556 383 4 v 685 623 +a Fz(:)p Fj(A;)12 b Fz(::)p Fj(A)p 1004 611 9 34 v 1013 +597 38 4 v 1109 568 a Fz(:)1160 576 y Fi(L)p 676 658 +401 4 v 676 726 a Fz(:)p Fj(A)p 801 714 9 34 v 810 699 +38 4 v 79 w Fz(:::)p Fj(A)1117 670 y Fz(:)1168 678 y +Fi(R)1345 536 y Fz(:)p Fj(A)p 1470 524 9 34 v 1479 509 +38 4 v 79 w Fz(:)p Fj(A)p 1302 556 383 4 v 1302 623 a +Fz(:)p Fj(A;)h Fz(::)p Fj(A)p 1622 611 9 34 v 1630 597 +38 4 v 1726 568 a Fz(:)1777 576 y Fi(L)p 1293 658 401 +4 v 1293 726 a Fz(:)p Fj(A)p 1419 714 9 34 v 1428 699 +38 4 v 80 w Fz(:::)p Fj(A)1735 670 y Fz(:)1786 678 y +Fi(R)p 676 746 1018 4 v 782 813 a Fz(:)p Fj(A;)g Fz(:)p +Fj(A)p 1051 801 9 34 v 1059 786 38 4 v 80 w Fz(:::)p +Fj(A)p Fz(^:::)p Fj(A)1735 762 y Fz(^)1786 770 y Fi(R)p +709 848 951 4 v 709 920 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p +Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fj(A;)g Fz(:)p +Fj(A)p 1597 908 9 34 v 1605 893 38 4 v 1701 860 a Fz(:)1752 +868 y Fi(L)p 709 959 951 4 v 781 1032 a Fz(:)p Fh(\()p +Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p +Fj(A)p 1525 1020 9 34 v 1534 1005 38 4 v 1701 978 a(contr)1880 +986 y Fi(L)p 772 1070 825 4 v 772 1143 a Fz(:)p Fh(\()p +Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p 1374 1131 +9 34 v 1382 1116 38 4 v 80 w Fz(::)p Fj(A)1638 1082 y +Fz(:)1689 1090 y Fi(R)2053 536 y Fz(:)p Fj(A)p 2178 524 +9 34 v 2187 509 38 4 v 79 w Fz(:)p Fj(A)p 2010 556 383 +4 v 2010 623 a Fz(:)p Fj(A;)g Fz(::)p Fj(A)p 2330 611 +9 34 v 2338 597 38 4 v 2434 568 a Fz(:)2485 576 y Fi(L)p +2001 658 401 4 v 2001 726 a Fz(:)p Fj(A)p 2127 714 9 +34 v 2136 699 38 4 v 80 w Fz(:::)p Fj(A)2443 670 y Fz(:)2494 +678 y Fi(R)2670 536 y Fz(:)p Fj(A)p 2796 524 9 34 v 2805 +509 38 4 v 80 w Fz(:)p Fj(A)p 2628 556 383 4 v 2628 623 +a Fz(:)p Fj(A;)g Fz(::)p Fj(A)p 2948 611 9 34 v 2956 +597 38 4 v 3052 568 a Fz(:)3103 576 y Fi(L)p 2619 658 +401 4 v 2619 726 a Fz(:)p Fj(A)p 2745 714 9 34 v 2754 +699 38 4 v 80 w Fz(:::)p Fj(A)3061 670 y Fz(:)3112 678 +y Fi(R)p 2001 746 1018 4 v 2108 813 a Fz(:)p Fj(A;)f +Fz(:)p Fj(A)p 2376 801 9 34 v 2385 786 38 4 v 80 w Fz(:::)p +Fj(A)p Fz(^)q(:::)p Fj(A)3061 762 y Fz(^)3112 770 y Fi(R)p +2035 848 951 4 v 2035 920 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p +Fz(^:::)p Fj(A)p Fh(\))p Fj(;)i Fz(:)p Fj(A;)f Fz(:)p +Fj(A)p 2922 908 9 34 v 2931 893 38 4 v 3027 860 a Fz(:)3078 +868 y Fi(L)p 2035 959 951 4 v 2106 1032 a Fz(:)p Fh(\()p +Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p +Fj(A)p 2851 1020 9 34 v 2860 1005 38 4 v 3027 978 a(contr)3206 +986 y Fi(L)p 2098 1070 825 4 v 2098 1143 a Fz(:)p Fh(\()p +Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p Fh(\))p 2700 1131 9 +34 v 2708 1116 38 4 v 81 w Fz(::)p Fj(A)2964 1082 y Fz(:)3015 +1090 y Fi(R)p 772 1181 2151 4 v 1020 1254 a Fz(:)p Fh(\()p +Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p +Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p Fh(\))p 2240 +1242 9 34 v 2249 1227 38 4 v 80 w Fz(::)p Fj(A)p Fz(^)q(::)p +Fj(A)2964 1198 y Fz(^)3015 1206 y Fi(R)p 948 1293 1800 +4 v 948 1365 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p +Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(:::)p Fj(A)p +Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(::)p +Fj(A)p Fz(^::)p Fj(A)p Fh(\))p 2684 1353 9 34 v 2693 +1338 38 4 v 2789 1305 a Fz(:)2840 1313 y Fi(L)p 948 1404 +1800 4 v 1257 1477 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p +Fj(A)p Fh(\))p Fj(;)h Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p +Fj(A)p Fh(\))p 2375 1465 9 34 v 2383 1450 38 4 v 2789 +1423 a Fj(contr)2968 1431 y Fi(L)593 1568 y Fg(\031)649 +1536 y Ff(0\003)646 1583 y Fe(R)734 1568 y Fd(=)727 1646 +y Fz(:)p Fj(A)p 853 1634 9 34 v 861 1619 38 4 v 80 w +Fz(:)p Fj(A)p 685 1666 383 4 v 685 1734 a Fz(::)p Fj(A;)e +Fz(:)p Fj(A)p 1004 1722 9 34 v 1013 1707 38 4 v 1109 +1678 a Fz(:)1160 1686 y Fi(L)p 676 1768 401 4 v 676 1836 +a Fz(::)p Fj(A)p 853 1824 9 34 v 861 1809 38 4 v 80 w +Fz(::)p Fj(A)1117 1780 y Fz(:)1168 1788 y Fi(R)1345 1646 +y Fz(:)p Fj(A)p 1470 1634 9 34 v 1479 1619 38 4 v 79 +w Fz(:)p Fj(A)p 1302 1666 383 4 v 1302 1734 a Fz(::)p +Fj(A;)h Fz(:)p Fj(A)p 1622 1722 9 34 v 1630 1707 38 4 +v 1726 1678 a Fz(:)1777 1686 y Fi(L)p 1293 1768 401 4 +v 1293 1836 a Fz(::)p Fj(A)p 1470 1824 9 34 v 1479 1809 +38 4 v 80 w Fz(::)p Fj(A)1735 1780 y Fz(:)1786 1788 y +Fi(R)p 676 1856 1018 4 v 782 1924 a Fz(::)p Fj(A;)g Fz(::)p +Fj(A)p 1153 1912 9 34 v 1162 1897 38 4 v 80 w Fz(::)p +Fj(A)p Fz(^::)p Fj(A)1735 1873 y Fz(^)1786 1881 y Fi(R)p +709 1958 951 4 v 709 2031 a Fz(:)p Fh(\()p Fz(::)p Fj(A)p +Fz(^)q(::)p Fj(A)p Fh(\))p Fj(;)g Fz(::)p Fj(A;)g Fz(::)p +Fj(A)p 1597 2019 9 34 v 1605 2004 38 4 v 1701 1970 a +Fz(:)1752 1978 y Fi(L)p 709 2069 951 4 v 806 2142 a Fz(:)p +Fh(\()p Fz(::)p Fj(A)p Fz(^)q(::)p Fj(A)p Fh(\))p Fj(;)g +Fz(::)p Fj(A)p 1500 2130 9 34 v 1508 2115 38 4 v 1701 +2088 a(contr)1880 2096 y Fi(L)p 798 2181 774 4 v 798 +2253 a Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p Fj(A)p +Fh(\))p 1297 2241 9 34 v 1306 2226 38 4 v 80 w Fz(:::)p +Fj(A)1613 2193 y Fz(:)1664 2201 y Fi(R)2053 1646 y Fz(:)p +Fj(A)p 2178 1634 9 34 v 2187 1619 38 4 v 79 w Fz(:)p +Fj(A)p 2010 1666 383 4 v 2010 1734 a Fz(::)p Fj(A;)g +Fz(:)p Fj(A)p 2330 1722 9 34 v 2338 1707 38 4 v 2434 +1678 a Fz(:)2485 1686 y Fi(L)p 2001 1768 401 4 v 2001 +1836 a Fz(::)p Fj(A)p 2178 1824 9 34 v 2187 1809 38 4 +v 80 w Fz(::)p Fj(A)2443 1780 y Fz(:)2494 1788 y Fi(R)2670 +1646 y Fz(:)p Fj(A)p 2796 1634 9 34 v 2805 1619 38 4 +v 80 w Fz(:)p Fj(A)p 2628 1666 383 4 v 2628 1734 a Fz(::)p +Fj(A;)g Fz(:)p Fj(A)p 2948 1722 9 34 v 2956 1707 38 4 +v 3052 1678 a Fz(:)3103 1686 y Fi(L)p 2619 1768 401 4 +v 2619 1836 a Fz(::)p Fj(A)p 2796 1824 9 34 v 2805 1809 +38 4 v 80 w Fz(::)p Fj(A)3061 1780 y Fz(:)3112 1788 y +Fi(R)p 2001 1856 1018 4 v 2108 1924 a Fz(::)p Fj(A;)g +Fz(::)p Fj(A)p 2479 1912 9 34 v 2487 1897 38 4 v 80 w +Fz(::)p Fj(A)p Fz(^::)p Fj(A)3061 1873 y Fz(^)3112 1881 +y Fi(R)p 2035 1958 951 4 v 2035 2031 a Fz(:)p Fh(\()p +Fz(::)p Fj(A)p Fz(^::)p Fj(A)p Fh(\))p Fj(;)g Fz(::)p +Fj(A;)g Fz(::)p Fj(A)p 2922 2019 9 34 v 2931 2004 38 +4 v 3027 1970 a Fz(:)3078 1978 y Fi(L)p 2035 2069 951 +4 v 2132 2142 a Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p +Fj(A)p Fh(\))p Fj(;)g Fz(::)p Fj(A)p 2825 2130 9 34 v +2834 2115 38 4 v 3027 2088 a(contr)3206 2096 y Fi(L)p +2124 2181 774 4 v 2124 2253 a Fz(:)p Fh(\()p Fz(::)p +Fj(A)p Fz(^::)p Fj(A)p Fh(\))p 2623 2241 9 34 v 2631 +2226 38 4 v 80 w Fz(:::)p Fj(A)2938 2193 y Fz(:)2989 +2201 y Fi(R)p 798 2292 2100 4 v 1071 2364 a Fz(:)p Fh(\()p +Fz(::)p Fj(A)p Fz(^)q(::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p +Fh(\()p Fz(::)p Fj(A)p Fz(^::)p Fj(A)p Fh(\))p 2087 2352 +9 34 v 2095 2338 38 4 v 80 w Fz(:::)p Fj(A)p Fz(^:::)p +Fj(A)2938 2308 y Fz(^)2990 2317 y Fi(R)p 999 2403 1697 +4 v 999 2476 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p +Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^)q(::)p +Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p +Fj(A)p Fh(\))p 2633 2464 9 34 v 2641 2449 38 4 v 2737 +2415 a Fz(:)2788 2423 y Fi(L)p 999 2514 1697 4 v 1257 +2587 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p +Fh(\))p Fj(;)h Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p +Fj(A)p Fh(\))p 2375 2575 9 34 v 2383 2560 38 4 v 2737 +2533 a Fj(contr)2916 2541 y Fi(L)p 3402 2639 4 2271 v +523 2642 2882 4 v 523 2731 a Fw(Fig)o(.)e(1.)36 b Fx(One)h(can)g +(obtain)g(the)f(\002rst)f(proof)i(by)g(double-ne)o(gation)i +(translation)d(of)h(\(1\))f(using)h(the)f(left-)523 2822 +y(translation)27 b(for)g(the)g(cut,)g(and)g(then)h(eliminating)f(all)f +(cuts)h(including)h(the)f(auxiliary)g(cuts.)g(Equally)-5 +b(,)27 b(one)523 2913 y(can)21 b(\002rst)f(reduce)i(\(1\))f(to)g +(\(2\),)f(double-ne)o(gate)j(translate)e(\(2\))f(and)i(then)f +(eliminate)g(all)f(cuts.)h(Similarly)f(with)523 3005 +y(the)f(second)h(proof:)f(it)f(can)h(be)g(obtained)g(by)g +(right-translating)g(the)g(cut)g(in)f(\(1\))h(and)g(then)g(eliminate)g +(all)f(cuts;)523 3096 y(or)h(by)g(double-ne)o(gate)i(translate)e(\(3\)) +f(and)i(then)f(eliminate)g(all)g(cuts.)523 3364 y FA(preserving)30 +b(the)i(\223structure\224)f(of)h(the)g(classical)h(proof.)e(In)g(order) +g(to)i(achie)n(v)o(e)e(this)h(one)g(needs)523 3464 y(the)19 +b(property)e(that)i Fr(::)p Fm(\()p Fr(\000)p Fm(\))1328 +3434 y Fq(x)p 1389 3452 10 38 v 1398 3435 42 4 v 1458 +3464 a Fm(\()p Fr(\000)p Fm(\))1587 3434 y Fq(x)1649 +3464 y FA(is)g(intuitionistically)f(deri)n(v)n(able.)f(Ho)n(we)n(v)o +(er)m(,)g(this)i(lea)n(v)o(es)g(us)523 3564 y(with)26 +b(man)o(y)e(possible)i(double)e(ne)o(gation)f(translations\227clearly)h +(the)i(ones)f(gi)n(v)o(en)f(by)i(Gentzen,)523 3663 y(G)7 +b(\250)-35 b(odel)20 b(and)f(K)m(olmogoro)o(v)e(are)j(not)g(the)g(only) +g(ones)g(that)g(satisfy)h(these)f(constraints.)523 3909 +y Fu(3)99 b(Cut-Elimination)26 b(and)f(Its)g(Colour)n(ed)h(V)-9 +b(ariant)523 4089 y FA(Urban)19 b(and)h(Bierman)f(ha)n(v)o(e)h(sho)n +(wn)f(in)h([19,)12 b(21])19 b(that)i(only)e(a)h(small)h(restriction)e +(on)h(the)g(standard)523 4188 y(cut-elimination)30 b(procedure)f(for)i +(classical)i(and)e(intuitionistic)h(logic)f(is)i(suf)n(\002cient)e(to)h +(obtain)523 4288 y(strongly)f(normalising)g(proof-transformations.)c +(The)32 b Ft(lo)o(gical)g(cuts)p FA(,)g(also)h(sometimes)f(called)523 +4387 y Ft(k)o(e)n(y-cuts)p FA(,)c(are)h(transformed)e(by)h(this)i +(cut-elimination)d(procedure)f(in)k(a)f(completely)f(standard)523 +4487 y(f)o(ashion)19 b([10].)g(F)o(or)h(e)o(xample)f(the)h(logical)g +(cut)1459 4607 y Fs(\031)1506 4619 y Fl(1)1490 4672 y +Fm(:)p 1442 4733 10 38 v 1452 4717 42 4 v 1512 4745 a +Fs(B)1696 4607 y(\031)1743 4619 y Fl(2)1727 4672 y Fm(:)p +1680 4733 10 38 v 1690 4717 42 4 v 73 x Fs(C)p 1424 4765 +392 4 v 1500 4827 10 38 v 1510 4810 42 4 v 1570 4839 +a(B)t Fr(^)p Fs(C)1857 4782 y Fr(^)1912 4795 y Fq(R)2145 +4607 y Fs(\031)2192 4619 y Fl(3)2176 4672 y Fm(:)2110 +4745 y Fs(B)p 2195 4733 10 38 v 2205 4717 42 4 v 2049 +4765 276 4 v 2049 4839 a(B)t Fr(^)q Fs(C)p 2256 4827 +10 38 v 2265 4810 42 4 v 2367 4778 a Fr(^)2422 4790 y +Fq(L)2468 4798 y Fc(1)p 1482 4859 844 4 v 1878 4912 10 +38 v 1887 4896 42 4 v 2367 4884 a Fs(cut)3267 4924 y +FA(\(11\))p eop end +%%Page: 9 9 +TeXDict begin 9 8 bop 523 448 a FA(is)21 b(transformed)d(to)1725 +496 y Fs(\031)1772 508 y Fl(1)1756 560 y Fm(:)p 1708 +622 10 38 v 1718 605 42 4 v 1778 634 a Fs(B)1963 496 +y(\031)2010 508 y Fl(3)1994 560 y Fm(:)1928 634 y Fs(B)p +2014 622 10 38 v 2023 605 42 4 v 1690 654 394 4 v 1861 +707 10 38 v 1871 691 42 4 v 2125 679 a(cut)523 872 y +FA(and)i(so)h(on)f(for)f(the)i(other)e(connecti)n(v)o(es.)g(As)i +(before)e(we)i(con)m(v)o(eniently)c(ignore)i(all)i(matters)f(to)h(do) +523 972 y(with)e(ho)n(w)f(the)h(sequent-conte)o(xts)d(should)h(be)i +(adjusted.)f(A)h(logical)f(cut)h(can)f(be)h(characterised)e(as)523 +1071 y(a)25 b(cut)f(where)g(in)g(both)f(subproofs)g(the)h(cut-formulas) +e(are)i Ft(fr)m(eshly)h(intr)l(oduced)g FA(by)f(logical)g(rules)523 +1171 y(directly)19 b(abo)o(v)o(e)g(the)h(cut.)g(F)o(or)g(e)o(xample)f +(in)h(the)g(proof)1898 1317 y Fs(\031)1912 1370 y Fm(:)p +1846 1390 156 4 v 1864 1451 10 38 v 1874 1434 42 4 v +1934 1463 a Fs(B)2042 1408 y(r)523 1648 y FA(we)33 b(say)f(the)h +(formula)e Fs(B)37 b FA(is)c(freshly)f(introduced)e(if)i(it)h(is)h +(what)e(usually)g(is)h(called)f(the)h(main)523 1748 y(formula)24 +b(of)h(the)g(logical)g(inference)f(rule)h Fs(r)r FA(.)i(Consequently)-5 +b(,)23 b(a)j(logical)f(cut)g(is)h(a)g(cut)g(where)e(the)523 +1847 y(cut-formula)16 b(is)j(freshly)f(introduced)e(in)i(the)h(tw)o(o)g +(immediate)e(subproofs)f(of)j(the)f(cut.)g(In)g(all)h(other)523 +1947 y(cases)f(we)f(ha)n(v)o(e)g(a)g Ft(commuting)f(cut)p +FA(.)h(The)g(cut)g(in)g(\(1\),)f(for)h(e)o(xample,)e(is)k(a)e +(commuting)e(cut)i(because)523 2047 y(the)26 b(cut-formula)d +Fs(A)j FA(is)h(in)f(both)f(subproofs)e(introduced)h(by)h(a)h +(contraction-rule,)c(which)j(is)i(not)523 2146 y(considered)18 +b(to)j(be)f(a)h(logical)e(inference)g(rule.)648 2247 +y(Gentzen)d(introduced)f(proof-transformations)e(that)k(permute)f +(commuting)f(cuts)j(upw)o(ards)e(in)523 2346 y(a)25 b(stepwise)h(f)o +(ashion)e(only)g(by)h(re)n(writing)f(neighboring)e(inference)h(rules.)i +(In)g(contrast,)f(the)h(cut-)523 2446 y(elimination)c(procedure)e(of)j +(Urban)f(and)g(Bierman)h(contains)f(proof-transformations)c(that)22 +b(push)523 2545 y(commuting)15 b(cuts)i(upw)o(ards)f(in)i(a)f(single)g +(\223big\224)g(step)g(to)n(w)o(ards)g(all)g(places)g(where)g(the)g +(cut-formula)523 2645 y(w)o(as)k(introduced.)d(Consider)h(the)h(follo)n +(wing)f(picture)1106 3842 y @beginspecial 134 @llx 377 +@lly 592 @urx 666 @ury 2061 @rwi @clip @setspecial +%%BeginDocument: transport.ps +%!PS-Adobe-2.0 +%%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software +%%Title: transport.dvi +%%Pages: 1 +%%PageOrder: Ascend +%%BoundingBox: 0 0 595 842 +%%DocumentPaperSizes: a4 +%%EndComments +%DVIPSWebPage: (www.radicaleye.com) +%DVIPSCommandLine: dvips transport.dvi -o transport.ps +%DVIPSParameters: dpi=600 +%DVIPSSource: TeX output 2006.03.30:1633 +%%BeginProcSet: tex.pro 0 0 +%! +/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S +N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 +mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 +0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ +landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize +mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ +matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round +exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ +statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] +N/FBB[0 0 0 0]N/nn 0 N/IEn 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 IEn N end A{/foo setfont}2 +array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/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 A +definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get +}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} +B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr +1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S +/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy +setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask +restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn +/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 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 A 1 get A +mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ +SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ +userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X +1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 +index cvrs cvn put}for pop 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/Rx 0 N/Ry 0 N/V{}B/RV/v{ +/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) +(LaserWriter 16/600)]{A length product length le{A length product exch 0 +exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse +end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask +grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} +imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round +exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto +fill grestore}B/a{moveto}B/delta 0 N/tail{A/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 +%%BeginProcSet: pstricks.pro 0 0 +%! +% PostScript prologue for pstricks.tex. +% Version 97 patch 4, 04/05/10 +% For distribution, see pstricks.tex. +% +/tx@Dict 200 dict def tx@Dict begin +/ADict 25 dict def +/CM { matrix currentmatrix } bind def +/SLW /setlinewidth load def +/CLW /currentlinewidth load def +/CP /currentpoint load def +/ED { exch def } bind def +/L /lineto load def +/T /translate load def +/TMatrix { } def +/RAngle { 0 } def +/Atan { /atan load stopped { pop pop 0 } if } def +/Div { dup 0 eq { pop } { div } ifelse } def +/NET { neg exch neg exch T } def +/Pyth { dup mul exch dup mul add sqrt } def +/PtoC { 2 copy cos mul 3 1 roll sin mul } def +/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def } +def +/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def +} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ } +/pathforall load stopped { pop pop pop pop } if z } def +/STP { .996264 dup scale } def +/STV { SDict begin normalscale end STP } def +% +%%-------------- DG begin patch 15 ---------------%% +%/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def +%PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y +%mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul +%/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ] +%0 } ifelse setdash stroke } def +/DashLine { + dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def PathLength } ifelse + /b ED /x1 ED /y1 ED /x ED /y ED + /z y x add y1 add x1 add def + /Coef b a .5 sub 2 mul y mul sub z Div round + z mul a .5 sub 2 mul y mul add b exch Div def + /y y Coef mul def /x x Coef mul def /y1 y1 Coef mul def /x1 x1 Coef mul def + x1 0 gt y1 0 gt x 0 gt y 0 gt and { [ y x y1 x1 ] 1 a sub y mul} + { [ 1 0] 0 } ifelse setdash stroke +} def +%%-------------- DG end patch 15 ---------------%% +/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt +{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add +def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if +] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1 +setlinecap stroke } def +/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch +2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25 +% DG/SR modification begin - Dec. 12, 1997 - Patch 2 +%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a +itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a +% DG/SR modification end +Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict +/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1 +% DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) +% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore } +% def +a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore +pop pop } def +% DG/SR modification end +/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg +exch 3 index sub exch Atan rotate newpath } def +/EndArrow { @mtrx setmatrix CP grestore end } def +/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1 +scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill +grestore } def +/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0 +CLW moveto } def +/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2 +div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0 +CLW moveto } def +/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW +2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0 +curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto } +def +/SD { 0 360 arc fill } def +/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS +CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def +/Shadow { [ { /moveto load } { /lineto load } { /curveto load } { +/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto +load } if ] cvx newpath 3 1 roll T exec } def +/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch +cvi def } def +/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if +f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def +/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub +def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def +/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 } +ifelse { pop } repeat a } def +/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq +and { pop pop /n n 1 sub def } if } def +/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop } +repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1 +ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1 +moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto +Lineto pop pop closepath } ifelse } def +/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul +neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add +def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2 +div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx +setmatrix } def +% DG modification begin - Jan. 15, 1997 +%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq { +%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 +%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0 +%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx +%setmatrix } def +/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup +CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 +div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0 +d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx +% DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) +% setmatrix } def +setmatrix pop } def +% DG/SR modification end +/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth +def } def +/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth +def } def +/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1 +def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp +mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos +abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def +/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul +m mul neg def } def +/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if } +ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def } +def +/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def +/NC { CC x1 y1 x2 y2 x y curveto } def +/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def +/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def +/NAC { x2 y2 x y curveto CC x1 y1 } def +/EAC { x2 y2 x y ArrowB curveto pop pop } def +/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def + n { NC } repeat EOC } ifelse } def +/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload +/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop +pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def +/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt { +CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC } +repeat closepath pop pop } ifelse } def +/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def +/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def +/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore } +def +/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix +rotate matrix concatmatrix exch findfont exch makefont setfont } def +/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto +x1 y1 lineto closepath } def +/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1 +y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div +exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse +/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto +x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath } +ifelse } def +/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add +/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq { +pop pop Rect } { OvalFrame } ifelse } def +/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop +} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ] +aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def +/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6 +2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto } +ifelse } def +/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv { +6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def +/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1 +sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore } +def +/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1 +sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA +x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0 +y0 x0 2 mul x1 sub y1 ] def } def +/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 } +if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if +/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div +round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div +round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub +0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def +/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def } +if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def } +{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add +exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n +div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1 +dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt { +dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt { +pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup +stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg} +{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i +w add def } repeat grestore gsave n 0 gt +% DG/SR modification begin - Nov. 7, 1997 - Patch 1 +%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash } +{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash } +% DG/SR modification end +{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul +n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n +div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop +1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs +dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt +{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g +exch t L stroke /i i h add def } repeat grestore } def +/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0 +1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add +exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c +mul neg d } def +/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def +/Rot { CP CP translate 3 -1 roll neg rotate NET } def +/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 } +def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle +dup a add ] cvx def } def +/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ] +cvx def } def +/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def +/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def +/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def +/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s +abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b +div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } { +h1 abs h sub dup s mul abs } ifelse } def +/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s +mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q +{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add +exch } def +/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible +/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def +} if } ifelse } def +/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def +/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit +neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def +/IfVisible true def } def +end +% END pstricks.pro + +%%EndProcSet +%%BeginProcSet: pst-dots.pro 0 0 +%!PS-Adobe-2.0 +%%Title: Dot Font for PSTricks +%%Creator: Timothy Van Zandt +%%Creation Date: May 7, 1993 +%% Version 97 patch 1, 99/12/16 +%% Modified by Etienne Riga - Dec. 16, 1999 +%% to add /Diamond, /SolidDiamond and /BoldDiamond +10 dict dup begin + /FontType 3 def + /FontMatrix [ .001 0 0 .001 0 0 ] def + /FontBBox [ 0 0 0 0 ] def + /Encoding 256 array def + 0 1 255 { Encoding exch /.notdef put } for + Encoding + dup (b) 0 get /Bullet put + dup (c) 0 get /Circle put + dup (C) 0 get /BoldCircle put + dup (u) 0 get /SolidTriangle put + dup (t) 0 get /Triangle put + dup (T) 0 get /BoldTriangle put + dup (r) 0 get /SolidSquare put + dup (s) 0 get /Square put + dup (S) 0 get /BoldSquare put + dup (q) 0 get /SolidPentagon put + dup (p) 0 get /Pentagon put + dup (P) 0 get /BoldPentagon put +% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga + dup (l) 0 get /SolidDiamond put + dup (d) 0 get /Diamond put + (D) 0 get /BoldDiamond put +% DG/SR modification end + /Metrics 13 dict def + Metrics begin + /Bullet 1000 def + /Circle 1000 def + /BoldCircle 1000 def + /SolidTriangle 1344 def + /Triangle 1344 def + /BoldTriangle 1344 def + /SolidSquare 886 def + /Square 886 def + /BoldSquare 886 def + /SolidPentagon 1093.2 def + /Pentagon 1093.2 def + /BoldPentagon 1093.2 def +% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga + /SolidDiamond 1008 def + /Diamond 1008 def + /BoldDiamond 1008 def +% DG/SR modification end + /.notdef 0 def + end + /BBoxes 13 dict def + BBoxes begin + /Circle { -550 -550 550 550 } def + /BoldCircle /Circle load def + /Bullet /Circle load def + /Triangle { -571.5 -330 571.5 660 } def + /BoldTriangle /Triangle load def + /SolidTriangle /Triangle load def + /Square { -450 -450 450 450 } def + /BoldSquare /Square load def + /SolidSquare /Square load def + /Pentagon { -546.6 -465 546.6 574.7 } def + /BoldPentagon /Pentagon load def + /SolidPentagon /Pentagon load def +% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga + /Diamond { -428.5 -742.5 428.5 742.5 } def + /BoldDiamond /Diamond load def + /SolidDiamond /Diamond load def +% DG/SR modification end + /.notdef { 0 0 0 0 } def + end + /CharProcs 20 dict def + CharProcs begin + /Adjust { + 2 copy dtransform floor .5 add exch floor .5 add exch idtransform + 3 -1 roll div 3 1 roll exch div exch scale + } def + /CirclePath { 0 0 500 0 360 arc closepath } def + /Bullet { 500 500 Adjust CirclePath fill } def + /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath + eofill } def + /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath + eofill } def + /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def + /TrianglePath { 0 660 moveto -571.5 -330 lineto 571.5 -330 lineto + closepath } def + /SolidTriangle { TrianglePath fill } def + /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def + /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def + /SquarePath { -450 450 moveto 450 450 lineto 450 -450 lineto + -450 -450 lineto closepath } def + /SolidSquare { SquarePath fill } def + /Square { SquarePath .89 .89 scale SquarePath eofill } def + /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def + /PentagonPath { + -337.8 -465 moveto + 337.8 -465 lineto + 546.6 177.6 lineto + 0 574.7 lineto + -546.6 177.6 lineto + closepath + } def + /SolidPentagon { PentagonPath fill } def + /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def + /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def +% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga + /DiamondPath { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto + 428.5 0 lineto closepath } def + /SolidDiamond { DiamondPath fill } def + /Diamond { DiamondPath .85 .85 scale DiamondPath eofill } def + /BoldDiamond { DiamondPath .7 .7 scale DiamondPath eofill } def +% DG/SR modification end + /.notdef { } def + end + /BuildGlyph { + exch + begin + Metrics 1 index get exec 0 + BBoxes 3 index get exec + setcachedevice + CharProcs begin load exec end + end + } def + /BuildChar { + 1 index /Encoding get exch get + 1 index /BuildGlyph get exec + } bind def +end +/PSTricksDotFont exch definefont pop +%END pst-dots.pro + +%%EndProcSet +%%BeginProcSet: pst-node.pro 0 0 +%! +% PostScript prologue for pst-node.tex. +% Version 97 patch 1, 97/05/09. +% For distribution, see pstricks.tex. +% +/tx@NodeDict 400 dict def tx@NodeDict begin +tx@Dict begin /T /translate load def end +/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def } +if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end +grestore } def +/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def +} def +/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch +Sin mul } def } def +/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def +} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def } +ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx +dup Sin mul Cos Div } ifelse } def +/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub +/u ED /NodePos { GetRnodePos } def } def +/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup +Cos mul exch Sin mul } def +/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h +mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul +} ifelse } def +/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d +ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d +sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan +sin def /NodePos { TriNodePos } def } def +/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos +hh mul Atan dup cos ww mul exch sin hh mul } def +/GetCenter { begin X Y NodeMtrx transform CM itransform end } def +/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist +Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg } +{ 0 Dist Sin mul } ifelse } ifelse Do } def +/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform +exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx +dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop +XYPos } ifelse } def +/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1 +roll sin mul sub exch } ifelse } def +/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA +AddOffset yA add /yA1 ED xA add /xA1 ED } def +/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB +AddOffset yB add /yB1 ED xB add /xB1 ED } def +/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA +AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED +ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED +} ifelse } def +/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB +AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED +ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED +} ifelse } def +/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED +/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b +known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA +ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def +/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub +mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def +/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt +{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop +} repeat LPutLine cleartomark } def +/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED +/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3 +mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx +sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul +add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add +y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul +mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def +/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def +} def +/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def +/NAngle yyB yyA sub xxB xxA sub Atan def } def +/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def +/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def +} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit +} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd +} def +/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def +} def +/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def +/NAngle yyB yyA sub xxB xxA sub Atan def } def +/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def +/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def +} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check { +exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark +VPosEnd } def +/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2 +roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx +def grestore exec /LPutVar /SaveLPutVar load def } def +/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def +GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos { +LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos { +LPutVar VPutLine } def LPutVar } def +/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop +end } def +/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub +Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan +/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1 +yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos { +LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } +ifelse } def +/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1 +roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA +GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict +begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ] +cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def +/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate +def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx +itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2 +yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 +yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def +/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def +/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2 +yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform +/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1 +tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ] +cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { +VPutLines } def } def +/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def +xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx +transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def } +{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB +0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict +begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx +def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { +VPutLines } def } def +/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if +xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end +/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { +LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } +def +/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED +GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin +false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { +LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } +def +/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate +def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2 +mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3 +mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 +xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false +Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ] +cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { +VPutLines } def } def +% DG/SR modification begin - May 9, 1997 - Patch 1 +%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r +%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add +%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 +%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED +/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r +r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add +exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 +mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED +% DG/SR modification end +} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add +AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r +Div def /y ED /x ED } def +/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB +180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos +mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def +/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy +add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos +{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar +tx@Dict begin false Polygon end } def +/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan +def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2 +mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def +/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div +def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def +GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul +sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0 +eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn } +{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def +0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2 +roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0 +0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2 +roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat } +ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos +{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le { +r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC } +{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add +add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub +mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t +sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse +y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos { +LPutPos } def } def +/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB +sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1 +ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB +w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos +{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop } +{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine } +def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin +/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def +/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg +exch X sub neg exch moveto setmatrix CP grestore } def +/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0 +def } ifelse LPutCoor } def +/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg } +if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop } +ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED } +def +/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if +r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop } +ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED } +def +end +% END pst-node.pro + +%%EndProcSet +%%BeginProcSet: pst-text.pro 0 0 +%! +% PostScript header file pst-text.pro +% Version 97, 94/04/20; patched MV 10-09-99 00:36 +% For distribution, see pstricks.tex. + +/tx@TextPathDict 40 dict def +tx@TextPathDict begin + +% Syntax: PathPosition - +% Function: Searches for position of currentpath distance from +% beginning. Sets (X,Y)=position, and Angle=tangent. +/PathPosition +{ /targetdist exch def + /pathdist 0 def + /continue true def + /X { newx } def /Y { newy } def /Angle 0 def + gsave + flattenpath + { movetoproc } { linetoproc } { } { firstx firsty linetoproc } + /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if + grestore +} def + +/movetoproc { continue { @movetoproc } { pop pop } ifelse } def + +/@movetoproc +{ /newy exch def /newx exch def + /firstx newx def /firsty newy def +} def + +/linetoproc { continue { @linetoproc } { pop pop } ifelse } def + +/@linetoproc +{ + /oldx newx def /oldy newy def + /newy exch def /newx exch def + /dx newx oldx sub def + /dy newy oldy sub def + /dist dx dup mul dy dup mul add sqrt def + /pathdist pathdist dist add def + pathdist targetdist ge + { pathdist targetdist sub dist div dup + dy mul neg newy add /Y exch def + dx mul neg newx add /X exch def + /Angle dy dx atan def + /continue false def + } if +} def + +/TextPathShow +{ /String exch def + /CharCount 0 def + String length + { String CharCount 1 getinterval ShowChar + /CharCount CharCount 1 add def + } repeat +} def + +% Syntax: InitTextPath - +/InitTextPath +{ gsave + currentpoint /Y exch def /X exch def + exch X Hoffset sub sub mul + Voffset Hoffset sub add + neg X add /Hoffset exch def + /Voffset Y def + grestore +} def + +/Transform +{ PathPosition + dup + Angle cos mul Y add exch + Angle sin mul neg X add exch + translate + Angle rotate +} def + +/ShowChar +{ /Char exch def + gsave + Char end stringwidth + tx@TextPathDict begin + 2 div /Sy exch def 2 div /Sx exch def + +%%% MV 10-09-99 00:36 + /sc?currentpoint where {pop sc?currentpoint} {currentpoint} ifelse +% currentpoint + + Voffset sub Sy add exch + Hoffset sub Sx add + Transform + Sx neg Sy neg moveto + Char end tx@TextPathSavedShow + tx@TextPathDict begin + grestore + Sx 2 mul Sy 2 mul rmoveto +} def + +end +% END pst-text.pro + +%%EndProcSet +%%BeginProcSet: special.pro 0 0 +%! +TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N +/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N +/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N +/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ +/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho +X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B +/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ +/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known +{userdict/md get type/dicttype eq{userdict begin md length 10 add md +maxlength ge{/md md dup length 20 add dict copy def}if end md begin +/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S +atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ +itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll +transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll +curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf +pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} +if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 +-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 +get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip +yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub +neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ +noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop +90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get +neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr +1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr +2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 +-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S +TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ +Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale +}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState +save N userdict maxlength dict begin/magscale true def normalscale +currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts +/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x +psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx +psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub +TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def +@MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll +newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto +closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N +/@beginspecial{SDict begin/SpecialSave save N gsave normalscale +currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} +N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs +neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate +rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse +scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg +lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx +ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N +/setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ +pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave +restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B +/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 +setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY +moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix +matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc +savematrix setmatrix}N end + +%%EndProcSet +%%BeginProcSet: color.pro 0 0 +%! +TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop +setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll +}repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def +/TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{ +setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{ +/currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch +known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC +/Yellow{0 0 1 0 setcmykcolor}DC/Goldenrod{0 0.10 0.84 0 setcmykcolor}DC +/Dandelion{0 0.29 0.84 0 setcmykcolor}DC/Apricot{0 0.32 0.52 0 +setcmykcolor}DC/Peach{0 0.50 0.70 0 setcmykcolor}DC/Melon{0 0.46 0.50 0 +setcmykcolor}DC/YellowOrange{0 0.42 1 0 setcmykcolor}DC/Orange{0 0.61 +0.87 0 setcmykcolor}DC/BurntOrange{0 0.51 1 0 setcmykcolor}DC +/Bittersweet{0 0.75 1 0.24 setcmykcolor}DC/RedOrange{0 0.77 0.87 0 +setcmykcolor}DC/Mahogany{0 0.85 0.87 0.35 setcmykcolor}DC/Maroon{0 0.87 +0.68 0.32 setcmykcolor}DC/BrickRed{0 0.89 0.94 0.28 setcmykcolor}DC/Red{ +0 1 1 0 setcmykcolor}DC/OrangeRed{0 1 0.50 0 setcmykcolor}DC/RubineRed{ +0 1 0.13 0 setcmykcolor}DC/WildStrawberry{0 0.96 0.39 0 setcmykcolor}DC +/Salmon{0 0.53 0.38 0 setcmykcolor}DC/CarnationPink{0 0.63 0 0 +setcmykcolor}DC/Magenta{0 1 0 0 setcmykcolor}DC/VioletRed{0 0.81 0 0 +setcmykcolor}DC/Rhodamine{0 0.82 0 0 setcmykcolor}DC/Mulberry{0.34 0.90 +0 0.02 setcmykcolor}DC/RedViolet{0.07 0.90 0 0.34 setcmykcolor}DC +/Fuchsia{0.47 0.91 0 0.08 setcmykcolor}DC/Lavender{0 0.48 0 0 +setcmykcolor}DC/Thistle{0.12 0.59 0 0 setcmykcolor}DC/Orchid{0.32 0.64 0 +0 setcmykcolor}DC/DarkOrchid{0.40 0.80 0.20 0 setcmykcolor}DC/Purple{ +0.45 0.86 0 0 setcmykcolor}DC/Plum{0.50 1 0 0 setcmykcolor}DC/Violet{ +0.79 0.88 0 0 setcmykcolor}DC/RoyalPurple{0.75 0.90 0 0 setcmykcolor}DC +/BlueViolet{0.86 0.91 0 0.04 setcmykcolor}DC/Periwinkle{0.57 0.55 0 0 +setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC +/CornflowerBlue{0.65 0.13 0 0 setcmykcolor}DC/MidnightBlue{0.98 0.13 0 +0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1 +0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11 +0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0 +setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0 +0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC +/Aquamarine{0.82 0 0.30 0 setcmykcolor}DC/BlueGreen{0.85 0 0.33 0 +setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0 +0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0 +1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC +/PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0 +setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{ +0.26 0 0.76 0 setcmykcolor}DC/OliveGreen{0.64 0 0.95 0.40 setcmykcolor} +DC/RawSienna{0 0.72 1 0.45 setcmykcolor}DC/Sepia{0 0.83 1 0.70 +setcmykcolor}DC/Brown{0 0.81 1 0.60 setcmykcolor}DC/Tan{0.14 0.42 0.56 0 +setcmykcolor}DC/Gray{0 0 0 0.50 setcmykcolor}DC/Black{0 0 0 1 +setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end + +%%EndProcSet +TeXDict begin 39139632 55387786 1000 600 600 (transport.dvi) +@start +%DVIPSBitmapFont: Fa cmr12 12 1 +/Fa 1 50 df<000030000000F0000001F0000003F000001FF00000FFF000FFFFF000FFE7 +F000FF07F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007 +F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007 +F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007 +F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007 +F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007 +F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F000000F +F800001FFC007FFFFFFF7FFFFFFF7FFFFFFF204278C131>49 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fb cmr12 14.4 1 +/Fb 1 47 df<0F003FC07FE0FFF0FFF0FFF0FFF0FFF0FFF07FE03FC00F000C0C768B21> +46 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fc cmmib7 7 1 +/Fc 1 106 df<0001C00007F0000FF0000FF0000FF0000FE0000FE00007800000000000 +0000000000000000000000000000000000FC0003FF000FFF801F1FC03E1FC07C1FC0783F +C0783F80F87F80707F00007F0000FF0000FE0001FE0001FC0001FC0003FC1C03F83C07F8 +3C07F07C07F0F807F0F007F3F003FFC001FF80007E0016297EA81B>105 +D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fd cmmib10 10 1 +/Fd 1 77 df<0003FFFFFFF0000007FFFFFFF8000007FFFFFFF8000007FFFFFFF0000000 +07FF800000000007FF800000000007FF80000000000FFF80000000000FFF00000000000F +FF00000000000FFF00000000001FFF00000000001FFE00000000001FFE00000000001FFE +00000000003FFE00000000003FFC00000000003FFC00000000003FFC00000000007FFC00 +000000007FF800000000007FF800000000007FF80000000000FFF80000000000FFF00000 +000000FFF00000000000FFF00000000001FFF00000000001FFE00000000001FFE0000000 +0001FFE00000000003FFE00000000003FFC00000000003FFC00000000003FFC000003800 +07FFC00000780007FF800000780007FF800000F80007FF800000F8000FFF800000F0000F +FF000001F0000FFF000001E0000FFF000003E0001FFF000003E0001FFE000007C0001FFE +00000FC0001FFE00000FC0003FFE00001F80003FFC00003F80003FFC0000FF80003FFC00 +01FF00007FFC0007FF00007FF8007FFF00FFFFFFFFFFFE00FFFFFFFFFFFE00FFFFFFFFFF +FE00FFFFFFFFFFFC0035397CB83F>76 D E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Fe cmti12 14.4 3 +/Fe 3 118 df<0000003FC000000001FFF80000000FFFFE0000003FE07F0000007F001F +800001FE000F800003F80007C00007F00003C0000FE0001FC0001FC0003FC0003F80007F +C0007F80007FC000FF00007FC001FE00007FC001FE00007F8003FC00003F0007FC000008 +0007F8000000000FF8000000000FF8000000001FF0000000001FF0000000003FF0000000 +003FE0000000003FE0000000003FE0000000007FE0000000007FC0000000007FC0000000 +007FC0000000007FC000000000FF8000000000FF8000000000FF8000000000FF80000000 +00FF80000000007F80000000007F80000000007F80000001807F80000003C07F80000007 +C03F8000000F803F8000001F803F8000003F001FC000007E001FC00000FC000FE00003F0 +0007F0000FE00003F8003F800001FE03FE0000007FFFF80000001FFFE000000007FE0000 +002A3574B336>99 D<00000F000000001F800000003F800000003F800000003F80000000 +7F800000007F000000007F000000007F00000000FF00000000FE00000000FE00000000FE +00000001FE00000001FC00000001FC00000001FC00000003FC00000003F800000003F800 +000003F800000007F800000007F000000007F000007FFFFFFF80FFFFFFFF80FFFFFFFF00 +FFFFFFFF00000FE00000001FE00000001FC00000001FC00000001FC00000003FC0000000 +3F800000003F800000003F800000007F800000007F000000007F000000007F00000000FF +00000000FE00000000FE00000000FE00000001FE00000001FC00000001FC00000001FC00 +000003FC00000003F800000003F800000003F800000007F800000007F000000007F00000 +0007F00000000FF0003C000FE0003C000FE0007C000FE00078001FE00078001FC000F800 +1FC000F0001FC001F0001FC003E0001F8003C0001F8007C0001F800F80001F800F00000F +C03E00000FC07C000007E1F8000003FFF0000001FFC00000007E000000214C75CA27> +116 D<000FC000000000003FF0000003C0007FFC000007E000F07C00000FE001E07E0000 +0FE003C03E00001FE007803F00001FC00F803F00001FC00F007F00001FC01F007F00003F +C01E007F00003F803E007F00003F803C00FF00003F803C00FE00007F807C00FE00007F00 +7801FE00007F007801FC00007F00F803FC0000FF00F803F80000FE000003F80000FE0000 +07F80000FE000007F00001FE000007F00001FC00000FF00001FC00000FE00001FC00000F +E00003FC00001FE00003F800001FC00003F800001FC00003F800001FC00007F800003FC0 +0007F000003F800007F000003F800007F000003F80000FF000007F80000FE00F007F0000 +0FE00F007F00000FE00F007F00001FE00F007F00001FC01F007F00001FC01E007F00001F +C01E007F00003FC03E007F00007FC03C007F00007F803C003F0000FF807C003F0001FF80 +78003F8003FF80F8001F8007CFC0F0001FC00F8FC1F0000FF03F07E3E00003FFFC03FFC0 +0001FFF001FF8000003FC0007E00383577B33F>I E +%EndDVIPSBitmapFont +%DVIPSBitmapFont: Ff cmbsy10 14.4 1 +/Ff 1 95 df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ndDVIPSBitmapFont +%DVIPSBitmapFont: Fg cmmib10 14.4 6 +/Fg 6 120 df<00000FFFFFFFFFFFFFFE00000000001FFFFFFFFFFFFFFFE0000000001F +FFFFFFFFFFFFFFFC000000001FFFFFFFFFFFFFFFFF000000001FFFFFFFFFFFFFFFFFC000 +00001FFFFFFFFFFFFFFFFFE00000000001FFFE0000007FFFF80000000003FFFE0000000F +FFFC0000000003FFFE00000007FFFC0000000003FFFC00000003FFFE0000000003FFFC00 +000001FFFE0000000007FFFC00000001FFFF0000000007FFFC00000000FFFF0000000007 +FFF800000000FFFF0000000007FFF800000000FFFF800000000FFFF800000000FFFF8000 +00000FFFF800000000FFFF800000000FFFF000000000FFFF800000000FFFF000000000FF +FF800000001FFFF000000000FFFF800000001FFFF000000000FFFF000000001FFFE00000 +0001FFFF000000001FFFE000000001FFFF000000003FFFE000000001FFFE000000003FFF +E000000003FFFE000000003FFFC000000003FFFC000000003FFFC000000007FFFC000000 +007FFFC000000007FFF8000000007FFFC00000000FFFF0000000007FFF800000001FFFE0 +000000007FFF800000003FFFC000000000FFFF800000007FFF8000000000FFFF80000000 +FFFF0000000000FFFF00000003FFFE0000000000FFFF0000000FFFF80000000001FFFF00 +00003FFFE00000000001FFFF000003FFFF800000000001FFFFFFFFFFFFFE000000000001 +FFFFFFFFFFFFF0000000000003FFFFFFFFFFFFF0000000000003FFFFFFFFFFFFFE000000 +000003FFFFFFFFFFFFFF800000000003FFFC0000003FFFE00000000007FFFC0000000FFF +F80000000007FFFC00000003FFFC0000000007FFF800000001FFFE0000000007FFF80000 +0001FFFE000000000FFFF800000000FFFF000000000FFFF800000000FFFF800000000FFF +F0000000007FFF800000000FFFF0000000007FFF800000001FFFF0000000007FFFC00000 +001FFFF0000000007FFFC00000001FFFE0000000007FFFC00000001FFFE0000000007FFF +C00000003FFFE0000000007FFFC00000003FFFE0000000007FFFC00000003FFFC0000000 +007FFFC00000003FFFC0000000007FFFC00000007FFFC000000000FFFFC00000007FFFC0 +00000000FFFF800000007FFF8000000000FFFF800000007FFF8000000001FFFF80000000 +FFFF8000000001FFFF00000000FFFF8000000003FFFF00000000FFFF0000000003FFFE00 +000000FFFF0000000007FFFE00000001FFFF0000000007FFFC00000001FFFF000000000F +FFF800000001FFFE000000001FFFF800000001FFFE000000003FFFF000000003FFFE0000 +00007FFFE000000003FFFE00000001FFFFC000000003FFFC00000003FFFF8000000003FF +FC0000000FFFFE0000000007FFFC000000FFFFFC0000007FFFFFFFFFFFFFFFFFF0000000 +FFFFFFFFFFFFFFFFFFE0000000FFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFFFC00 +000000FFFFFFFFFFFFFFFFE000000000FFFFFFFFFFFFFFFC000000000061527BD168>66 +D<0000000000003FFF8000003800000000000FFFFFF80000FC0000000000FFFFFFFE0001 +F80000000007FFFFFFFF8003F8000000003FFFFFFFFFE007F800000000FFFFFFFFFFF01F +F800000003FFFFFE007FF83FF00000000FFFFFC0000FFE7FF00000003FFFFC000003FFFF +F00000007FFFF0000000FFFFF0000001FFFFC00000007FFFE0000003FFFF000000003FFF +E0000007FFFC000000001FFFE000000FFFF0000000000FFFE000003FFFE0000000000FFF +C000007FFFC00000000007FFC00000FFFF800000000007FFC00001FFFF000000000003FF +C00003FFFE000000000003FF800007FFFC000000000003FF800007FFF8000000000001FF +80000FFFF0000000000001FF80001FFFF0000000000001FF00003FFFE0000000000001FF +00007FFFC0000000000001FF00007FFFC0000000000001FF0000FFFF80000000000001FE +0000FFFF80000000000001FE0001FFFF00000000000001FE0003FFFF00000000000001FE +0003FFFE00000000000001FC0007FFFE00000000000001FC0007FFFE00000000000000FC +000FFFFC0000000000000000000FFFFC0000000000000000000FFFFC0000000000000000 +001FFFF80000000000000000001FFFF80000000000000000003FFFF80000000000000000 +003FFFF00000000000000000003FFFF00000000000000000003FFFF00000000000000000 +007FFFF00000000000000000007FFFE00000000000000000007FFFE00000000000000000 +007FFFE0000000000000000000FFFFE0000000000000000000FFFFC00000000000000000 +00FFFFC0000000000000000000FFFFC0000000000000000000FFFFC00000000000000000 +00FFFFC0000000000000000000FFFF80000000000000000000FFFF800000000000000000 +00FFFF800000000000001F0000FFFF800000000000003F0000FFFF800000000000003F00 +00FFFF800000000000007F0000FFFF800000000000007E0000FFFF800000000000007E00 +00FFFF80000000000000FE0000FFFF80000000000000FC00007FFF80000000000001FC00 +007FFF80000000000001F800007FFF80000000000003F800003FFFC0000000000007F000 +003FFFC000000000000FE000003FFFC000000000000FE000001FFFE000000000001FC000 +001FFFE000000000003F8000000FFFF000000000007F00000007FFF00000000000FE0000 +0007FFF80000000003FC00000003FFFC0000000007F800000001FFFE000000000FF00000 +0000FFFF800000003FE0000000007FFFC0000000FFC0000000003FFFF0000003FF000000 +00001FFFFE00001FFE000000000007FFFFF001FFF8000000000003FFFFFFFFFFF0000000 +000000FFFFFFFFFFC00000000000003FFFFFFFFE0000000000000007FFFFFFF800000000 +00000000FFFFFFC0000000000000000003FFF80000000000005E5679D362>I<0000001F +FC00000000000001FFFF80000000000007FFFFE07C000000001FFFFFF1FF000000007FFF +FFFBFF80000001FFFC0FFFFF80000003FFE001FFFF80000007FFC000FFFF8000000FFF00 +007FFF8000001FFE00003FFF8000003FFC00003FFF8000007FFC00003FFF800000FFF800 +003FFF000001FFF000003FFF000003FFF000003FFF000003FFE000007FFF000007FFE000 +007FFE00000FFFE000007FFE00000FFFC000007FFE00001FFFC00000FFFE00001FFFC000 +00FFFC00003FFF800000FFFC00003FFF800000FFFC00003FFF800001FFFC00007FFF8000 +01FFF800007FFF000001FFF800007FFF000001FFF800007FFF000003FFF80000FFFF0000 +03FFF00000FFFE000003FFF00000FFFE000003FFF00000FFFE000007FFF00000FFFE0000 +07FFE00000FFFC000007FFE00000FFFC000007FFE00000FFFC00000FFFE00F80FFFC0000 +0FFFC01F80FFFC00000FFFC01F80FFF800000FFFC01F80FFF800001FFFC03F80FFF80000 +1FFF803F00FFF800001FFF803F00FFF800001FFF803F007FF800003FFF807E007FFC0000 +7FFF807E003FFC0000FFFF80FC003FFC0001FFFF80FC001FFE0007FFFF81F8000FFF001F +FFFF83F80007FFC0FFF7FF87F00003FFFFFFC3FFFFE00001FFFFFF81FFFFC000007FFFFE +00FFFF8000001FFFF8003FFF00000001FF800007FC000041377AB54C>97 +D<00000003FFC0000000007FFFFC00000003FFFFFF0000000FFFFFFF8000003FFFFFFFC0 +0000FFFF00FFE00003FFF8003FF00007FFE0000FF0000FFF80000FF8001FFF000007F800 +3FFE000007F8007FFC000007F800FFF8000007F801FFF8000007F803FFF0000007F807FF +F000000FF807FFE000000FF00FFFE000001FF00FFFC000003FE01FFFC000007FE01FFFC0 +0001FFC03FFF80000FFF803FFF8000FFFF003FFFFFFFFFFC007FFFFFFFFFF8007FFFFFFF +FFC0007FFFFFFFFE00007FFFFFFFC00000FFFF0000000000FFFE0000000000FFFE000000 +0000FFFE0000000000FFFE0000000000FFFC0000000000FFFC0000000000FFFC00000000 +00FFFC0000000000FFFC0000000000FFFC00000000007FFC00000000007FFC0000000060 +7FFC00000000F07FFC00000001F83FFC00000003FC3FFE00000007F81FFE0000001FF00F +FF0000007FE007FF800001FFC003FFC0000FFF8001FFF801FFFE0000FFFFFFFFFC00007F +FFFFFFF000001FFFFFFF80000003FFFFFC000000007FFF80000036377AB542>101 +D<000007FF80000000001FFFFF80000000003FFFFF80000000003FFFFF80000000003FFF +FF80000000003FFFFF80000000003FFFFF0000000000003FFF0000000000007FFF000000 +0000007FFF0000000000007FFE0000000000007FFE000000000000FFFE000000000000FF +FE000000000000FFFC000000000000FFFC000000000001FFFC000000000001FFFC000000 +000001FFF8000000000001FFF8000000000003FFF8000000000003FFF8000000000003FF +F0000000000003FFF0000000000007FFF0000000000007FFF0000000000007FFE0000000 +000007FFE000000000000FFFE000000000000FFFE000007F00000FFFC00003FFC0000FFF +C0000FFFE0001FFFC0003FFFF0001FFFC0007FFFF0001FFF8000FF83F8001FFF8003FC01 +F8003FFF8007F80FF8003FFF800FE01FF8003FFF001F803FFC003FFF003F007FFC007FFF +007E00FFF8007FFF00FC01FFF8007FFE01F801FFF8007FFE03F001FFF800FFFE07C001FF +F800FFFE1F8001FFF000FFFC3F0001FFE000FFFC7E0000FFC001FFFDFC00007F8001FFFF +F800003F0001FFFFF00000000001FFFFE00000000003FFFF800000000003FFFFFE000000 +0003FFFFFFE000000003FFFFFFFC00000007FFFFFFFF00000007FFFFFFFF80000007FFE1 +FFFFC0000007FFE07FFFE000000FFFE01FFFF000000FFFE007FFF000000FFFC003FFF800 +000FFFC003FFF800001FFFC001FFF8007C1FFFC001FFF800FC1FFF8001FFF800FC1FFF80 +01FFF800FC3FFF8001FFF801FC3FFF8003FFF801F83FFF0003FFF001F83FFF0003FFF003 +F87FFF0003FFF003F07FFF0003FFF007F07FFE0003FFF007E07FFE0001FFF00FC0FFFE00 +01FFF01FC0FFFE0000FFF03F80FFFC00007FF87F00FFFC00003FFFFE00FFF800001FFFFC +007FF000000FFFF8003FE0000003FFF0000F800000007FC0003E547BD248>107 +D<0001FE00000000000000FC00000FFFC0000001F80003FE00003FFFF0000007FC0007FF +00007FFFF800000FFE000FFF8000FFFFFC00001FFE001FFFC001FF1FFE00003FFF001FFF +C003FC1FFF00003FFF003FFFE007F01FFF00007FFF003FFFE00FE01FFF00007FFF003FFF +E00FC01FFF80007FFE003FFFE01FC01FFF80007FFE003FFFE03F801FFF8000FFFE003FFF +E03F003FFF0000FFFC001FFFE07F003FFF0000FFFC000FFFE07E003FFF0000FFFC0007FF +E07E007FFF0001FFFC0001FFC0FE007FFE0001FFF80000FFC0FC00FFFE0001FFF800007F +C0FC00FFFC0001FFF800003FC0FC00FFFC0003FFF800003FC00001FFFC0003FFF000001F +800001FFF80003FFF000001F800003FFF80003FFF000001F800003FFF00007FFF000003F +800003FFF00007FFE000003F000007FFF00007FFE000003F000007FFE00007FFE000003F +000007FFE0000FFFE000007F00000FFFE0000FFFC000007E00000FFFC0000FFFC000007E +00000FFFC0000FFFC000007E00000FFFC0001FFFC00000FC00001FFFC0001FFF800000FC +00001FFF80001FFF800000FC00001FFF80001FFF800001F800001FFF80001FFF800001F8 +00001FFF80001FFF800001F800001FFF00001FFF000003F000001FFF00001FFF000003F0 +00001FFF00001FFF000007E000001FFF00001FFF000007E000001FFF00001FFF00000FC0 +00001FFF00001FFF00000FC000001FFF80001FFF00001F8000001FFF80003FFF80003F00 +00000FFF80007FFF80003F0000000FFFC0007FFF80007E00000007FFC000FFFFC000FC00 +000003FFF003FFFFE003F800000001FFFC0FFBFFFC0FF000000000FFFFFFF1FFFFFFE000 +0000007FFFFFC07FFFFFC0000000001FFFFF803FFFFF800000000007FFFF0007FFFE0000 +000000007FF800007FF80000005B377CB563>119 D E +%EndDVIPSBitmapFont +end +%%EndProlog +%%BeginSetup +%%Feature: *Resolution 600dpi +TeXDict begin +%%PaperSize: A4 + end +%%EndSetup +%%Page: 1 1 +TeXDict begin 1 0 bop 0 TeXcolorgray Black 0 TeXcolorgray +1 TeXcolorgray 0 TeXcolorgray 1 TeXcolorgray 0 TeXcolorgray +0 TeXcolorgray 0 TeXcolorgray 1 TeXcolorgray 0 TeXcolorgray +0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray +0 TeXcolorgray 0 TeXcolorgray 0.25 TeXcolorgray 0 TeXcolorgray +0.5 TeXcolorgray 0 TeXcolorgray -947 3755 a @beginspecial +@setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 0.5 TeXcolorgray 0 TeXcolorgray +@beginspecial @setspecial + tx@Dict begin STP newpath 2.0 SLW 0 setgray 2.0 SLW 0 setgray +/ArrowA { /lineto load stopped { moveto } if } def /ArrowB { } def +[ 284.52744 106.69778 327.20654 170.71646 284.52744 184.94283 256.07469 +142.26372 284.52744 106.69778 /currentpoint load stopped pop 1. 0.1 +0. /c ED /b ED /a ED false OpenCurve gsave 0.5 setgray fill grestore +gsave 2.0 SLW 0 setgray 0 setlinecap stroke grestore end + +@endspecial 0 TeXcolorgray +0 TeXcolorgray 1417 w @beginspecial @setspecial + tx@Dict begin STP newpath 2.0 SLW 0 setgray /ArrowA { moveto } def +/ArrowB { } def [ 338.58746 85.35823 85.35823 85.35823 /Lineto /lineto +load def false Line gsave 2.0 SLW 0 setgray 0 setlinecap stroke +grestore end + +@endspecial +-1417 w @beginspecial @setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 1301 2981 +14 52 v 1314 2958 59 6 v 1442 2993 a Fg(B)6 b Ff(^)o +Fg(C)1828 3755 y @beginspecial @setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial +@beginspecial @setspecial + tx@Dict begin STP newpath 2.0 SLW 0 setgray 2.0 SLW 0 setgray +/ArrowA { /lineto load stopped { moveto } if } def /ArrowB { } def +[ 170.71646 106.69778 227.62195 199.1692 227.62195 256.07469 199.1692 +312.98018 142.26372 334.31973 79.66776 321.51608 28.45274 270.30106 +28.45274 213.39557 99.5846 106.69778 /currentpoint load stopped pop +1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 2.0 SLW 0 setgray +0 setlinecap stroke grestore end + +@endspecial 59 w @beginspecial +@setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 2735 2993 a(B)g Ff(^)o Fg(C)p +3111 2981 14 52 v 3124 2958 59 6 v 0 TeXcolorgray 3369 +3084 a Fe(cut)p 0 TeXcolorgray 305 3755 a @beginspecial +@setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 2135 3161 14 52 v 2148 3138 +59 6 v 1523 w @beginspecial @setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial +1 TeXcolorgray 0 TeXcolorgray 1 TeXcolorgray 0 TeXcolorgray +@beginspecial @setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 14.22636 266.03333 +42.67911 274.56879 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 0 TeXcolorgray +0 TeXcolorgray @beginspecial @setspecial + tx@Dict begin STP newpath 2.0 SLW 0 setgray /ArrowA { moveto } def +/ArrowB { } def [ 91.04869 270.30106 21.33955 270.30106 /Lineto /lineto +load def false Line gsave 2.0 SLW 0 setgray 0 setlinecap stroke +grestore end + +@endspecial +2607 1529 a Ff(^)2699 1547 y Fd(L)2762 1559 y Fc(i)2109 +1647 y Fg(B)g Ff(^)o Fg(C)p 2485 1635 14 52 v 2498 1612 +59 6 v 2332 1367 a Fb(.)2332 1400 y(.)2332 1434 y(.)1828 +3755 y @beginspecial @setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 1 TeXcolorgray +0 TeXcolorgray 1 TeXcolorgray 0 TeXcolorgray @beginspecial +@setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 193.47873 310.13472 +216.24101 294.48607 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 2872 1292 a Fg(B)g Ff(^)o Fg(C)p +3210 1280 14 52 v 3223 1258 59 6 v 141 w(B)g Ff(^)o Fg(C)1828 +3755 y @beginspecial @setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 1 TeXcolorgray +0 TeXcolorgray 1 TeXcolorgray 0 TeXcolorgray @beginspecial +@setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 231.88968 217.6633 +223.3542 194.90146 .5 Frame gsave 1 setgray fill grestore end + +@endspecial 0 TeXcolorgray 0 TeXcolorgray +@beginspecial @setspecial + tx@Dict begin STP newpath 2.0 SLW 0 setgray /ArrowA { moveto } def +/ArrowB { } def [ 241.84831 213.39557 170.71646 213.39557 /Lineto +/lineto load def false Line gsave 2.0 SLW 0 setgray 0 setlinecap +stroke grestore end + +@endspecial 0 TeXcolorgray +3881 2016 a(w)s(eak)4197 2034 y Fd(L)p 0 TeXcolorgray +3314 2119 a Fg(B)g Ff(^)o Fg(C)p 3689 2107 14 52 v 3703 +2084 59 6 v 3584 1863 a Fb(.)3584 1896 y(.)3584 1930 +y(.)234 3755 y @beginspecial @setspecial + tx@Dict begin STP newpath 0.0 SLW 1 setgray 0. true 3.0 neg 3.0 +neg 3.0 3.0 .5 Frame gsave 1 setgray fill grestore end + +@endspecial +0 TeXcolorgray 0 TeXcolorgray @beginspecial @setspecial + tx@Dict begin STP newpath 3.0 SLW 0 setgray /ArrowA { moveto } def +/ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow + } def [ 244.69376 113.81097 227.62195 102.43004 210.55013 113.81097 + 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 3.0 SLW 0 setgray +0 setlinecap stroke grestore end + + +@endspecial 0 TeXcolorgray 1918 5251 a Fa(1)p 0 TeXcolorgray +eop end +%%Trailer + +userdict /end-hook known{end-hook}if +%%EOF + +%%EndDocument + @endspecial 523 4028 a(where)f(the)h(cut-formula)e(on)h(the)h +(right-hand)d(side)k(is)f Ft(not)i FA(freshly)d(introduced,)e(rather)i +(it)i(is)f(intro-)523 4127 y(duced)25 b(some)n(where)f(deeper)h(inside) +h(the)g(subproof)e(and)h(because)h(of)g(contractions)e(possibly)h(in) +523 4227 y(se)n(v)o(eral)d(places.)h(W)-7 b(e)24 b(ha)n(v)o(e)f +(indicated)e(three)i(cases)h(for)e(a)h(cut-formula)e(being)h +(introduced:)f(by)h(a)523 4327 y(logical)h(inference)f(rule,)h(by)h(an) +f(axiom)g(and)g(by)g(a)i(weak)o(ening-rule\227in)20 b(general)j(we)h +(can)f(ha)n(v)o(e)523 4426 y(an)o(y)f(mixture)f(of)h(these)h(cases.)g +(T)-7 b(o)23 b(eliminate)f(such)g(commuting)f(cuts)i(the)f(procedure)e +(of)i(Urban)523 4526 y(and)d(Bierman)h(pushes)f(up,)h(roughly)e +(speaking,)g(the)i(cut-rule)f(until)h(it)g(reaches)g +Ft(all)g FA(places)g(where)523 4625 y(the)29 b(cut-formula)d(is)k +(introduced)d(in)i Ft(one)f FA(step.)h(In)g(case)g(it)h(reaches)e(a)h +(logical)g(inference-rule,)523 4725 y(then)e(the)h(proof)e(on)i(the)f +(left-hand)f(side)j(will)f(be)g(cut)g(against)f(this)h(logical)f +(inference-rule;)e(in)523 4825 y(case)h(of)g(an)g(axiom,)f(the)g(proof) +g(on)g(the)h(left-hand)e(side)i(replaces)g(the)g(axiom)f(and)g(in)h +(case)g(of)g(a)523 4924 y(weak)o(ening-rule,)16 b(the)j(proof)f(on)h +(the)g(left-hand)f(side)h(is)i(deleted.)d(\(Again)g(all)i(matters)f(to) +g(do)g(with)p eop end +%%Page: 10 10 +TeXDict begin 10 9 bop 523 448 a FA(adjusting)22 b(the)h(sequent-conte) +o(xts)d(are)j(omitted)g(in)g(this)g(paper)-5 b(.)23 b(The)f(w)o(ork)h +(reported)e(in)i([19,)12 b(21])523 548 y(formulates)23 +b(these)h(proof-transformation)18 b(as)25 b(term-re)n(writing)d(rules)h +(where)h(such)f(adjustments)523 648 y(are)31 b(b)n(uilt)g(into)g(the)g +(inference)f(rules,)h(v)o(ery)f(similar)h(to)g(term-re)n(writing)e(in)j +(the)f(simply-typed)523 747 y(lambda-calculus.\))648 +850 y(The)24 b(important)f(property)f(of)i(the)h(cut-elimination)e +(procedure)f(of)i(Urban)f(and)h(Bierman)h(is)523 950 +y(the)d(f)o(act)g(that)f(it)i(is)f(strongly-normalising.)c(This)k +(property)d(is)k(not)e(ob)o(vious:)f(the)i(reduction-rule)523 +1050 y(for)16 b(commuting)f(cuts)j(allo)n(ws)f(a)h(cut-rule)d(to)j +(\223jump\224)e(o)o(v)o(er)f(other)i(cut-rules\227a)f(highly)g +(problem-)523 1149 y(atic)26 b(reduction)e(if)i(one)f(tries)i(to)f +(construct)e(a)i(decreasing)f(measure)g(for)g(cut-elimination.)e(Also) +523 1249 y(this)g(rule)e(might)h(generate)e(se)n(v)o(eral)i(copies)g +(of)f(a)i(subproof)c(when)j(a)g(cut-formula)e(is)j(introduced)523 +1349 y(in)k(se)n(v)o(eral)f(places.)h(Urban)f(and)g(Bierman,)g +(therefore,)f(had)h(to)h(resort)f(to)h(a)h(quite)e(complicated)523 +1448 y(logical)20 b(relations)f(ar)o(gument)f(to)j(sho)n(w)f(strong)f +(normalisation.)648 1551 y(Recall)33 b(that)f(the)h(cut-elimination)d +(procedure)g(of)i(Urban)f(and)h(Bierman)g(is)h Ft(not)h +FA(Church-)523 1651 y(Rosser:)29 b(when)e(on)h(both)f(sides)i(of)e(a)i +(commuting)d(cut)i(the)g(cut-formula)e(is)j(not)e(freshly)g(intro-)523 +1751 y(duced,)19 b(then)i(this)g(cut)g(can)g(be)f(mo)o(v)o(ed)f(either) +i(to)g(the)f(left)i(or)e(to)h(the)g(right,)f(leading)g(in)h(general)e +(to)523 1850 y(tw)o(o)k(\223non-joinable\224)c(proofs.)h(F)o(or)i(e)o +(xample,)f(the)h(proof)f(sho)n(wn)g(in)h(\(1\))g(can)g(be)g(reduced)f +(in)h(one)523 1950 y(step)h(to)g(the)g(normalform)d(\(2\))j(or)f(in)h +(one)g(step)g(to)g(\(3\).)f(Which)h(choice)f(is)i(tak)o(en)f(is)g(left) +h(unspeci-)523 2050 y(\002ed)f(by)g(the)g(procedure.)e(Ne)n(v)o +(ertheless)h(with)i(this)f(cut-elimination)f(procedure)e(we)k(ha)n(v)o +(e)f(some)523 2149 y(ef)n(fecti)n(v)o(e)d(means)g(to)h(calculate)g(for) +f(a)i(classical)g(proof)d(its)j(collection)e(of)h Ft(all)g +FA(normalforms\227for)523 2249 y(e)o(xample)e(by)h(na)n(\250)-26 +b(\021v)o(ely)19 b(trying)g(out)g(all)i(possible)f(reductions.)2303 +2219 y Fp(3)648 2352 y FA(The)26 b(cut-elimination)f(procedure)g(of)i +(Urban)f(and)h(Bierman)f(w)o(as)i(in)g(part)f(inspired)f(by)g(the)523 +2452 y(w)o(ork)h(of)h(Danos)g(et)g(al.)h([6].)e(The)h(main)f(dif)n +(ference)f(is)j(that)f(their)g(cut-elimination)e(procedure)523 +2551 y Ft(is)c FA(Church-Rosser)-5 b(.)20 b(The)o(y)g(achie)n(v)o(e)g +(this)h(by)g(pre-determining)c(e)n(v)o(ery)j(choice)g(that)h(can)g(be)g +(made)523 2651 y(during)16 b(cut-elimination.)f(This)j +(pre-determination)13 b(is)19 b(done)d(via)i Ft(colour)o(s)p +FA(,)f(which)g(are)g(annotated)523 2751 y(to)24 b(e)n(v)o(ery)f +(formula)g(and)g(subformula)f(in)i(a)h(proof.)d(T)-7 +b(o)24 b(see)h(ho)n(w)e(colours)g(w)o(ork,)h(consider)f(again)523 +2850 y(the)17 b(proof)e(sho)n(wn)i(in)g(\(1\).)f(The)h(choice)f(about)g +(which)g(direction)g(is)i(tak)o(en)e(for)h(the)g(commuting)d(cut)523 +2950 y(is)26 b(determined)c(by)i(annotating)f(the)i(colour)e(`)p +Fs(\()p FA(')h(or)h(`)p Fs(*)p FA(')f(to)h(the)f(cut-formula)e +Fs(A)p FA(.)k(The)e(colour)n(-)523 3050 y(protocol)14 +b(of)i(Danos)g(et)g(al.)g(pre-scribes)f(that)h(in)g(the)g(former)f +(case)h(it)h(is)g(\002rst)g(attempted)e(to)h(permute)523 +3149 y(the)i(cut)f(to)h(the)f(left)h(and)f(in)h(the)f(second)g(case)h +(to)g(the)f(right)g(\(hence)f(the)i(use)g(of)f(an)g(arro)n(w)g(to)h +(denote)523 3249 y(a)26 b(colour!\).)d(F)o(or)j(e)o(xample,)e(if)h(we)h +(w)o(ant)g(to)g(reach)f(from)f(\(1\))h(the)h(normalform)d(\(2\))h(we)i +(need)f(to)523 3348 y(orient)19 b(the)i(colour)e(of)g(the)i +(cut-formula)c Fs(A)k FA(to)g(the)f(left,)g(as)h(sho)n(wn)e(belo)n(w) +1087 3489 y Fq(\()1089 3563 y Fs(A)p 1194 3543 10 38 +v 1204 3526 42 4 v 1259 3489 a Fq(\()1261 3563 y Fs(A)1408 +3489 y Fq(\()1410 3563 y Fs(A)p 1516 3543 10 38 v 1526 +3526 42 4 v 1581 3489 a Fq(\()1583 3563 y Fs(A)p 1087 +3583 561 4 v 1101 3643 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g(\000)1098 +3691 y Fq(\()1100 3764 y Fs(A)1187 3756 y Fr(_)1266 3691 +y Fq(\()1268 3764 y Fs(A)p 1374 3744 10 38 v 1383 3728 +42 4 v 1466 3691 a Fq(\()1468 3764 y Fs(A)1532 3756 y(;)1569 +3691 y Fq(\()1571 3764 y Fs(A)1689 3599 y Fr(_)1744 3612 +y Fq(L)p 1098 3792 538 4 v 1153 3853 a(\()g Fn(\000)g(\000)g(\000)g +(\000)1150 3901 y Fq(\()1152 3974 y Fs(A)1239 3966 y +Fr(_)1317 3901 y Fq(\()1319 3974 y Fs(A)p 1425 3954 10 +38 v 1435 3938 42 4 v 1518 3901 a Fq(\()1520 3974 y Fs(A)1677 +3812 y(contr)1870 3824 y Fq(R)2008 3489 y(\()2010 3563 +y Fs(A)p 2116 3543 10 38 v 2125 3526 42 4 v 2180 3489 +a Fq(\()2182 3563 y Fs(A)2330 3489 y Fq(\()2332 3563 +y Fs(A)p 2437 3543 10 38 v 2447 3526 42 4 v 2502 3489 +a Fq(\()2504 3563 y Fs(A)p 2008 3583 561 4 v 2020 3691 +a Fq(\()2022 3764 y Fs(A)2086 3756 y(;)2123 3691 y Fq(\()2125 +3764 y Fs(A)p 2230 3744 10 38 v 2240 3728 42 4 v 2326 +3643 a Fn(\000)g(\000)g(\000)g(\000)g Fq(*)2323 3691 +y(\()2325 3764 y Fs(A)2412 3756 y Fr(^)2491 3691 y Fq(\()2493 +3764 y Fs(A)2610 3599 y Fr(^)2665 3612 y Fq(R)p 2020 +3792 538 4 v 2071 3901 a(\()2073 3974 y Fs(A)p 2179 3954 +10 38 v 2188 3938 42 4 v 2274 3853 a Fn(\000)g(\000)g(\000)g(\000)g +Fq(*)2271 3901 y(\()2273 3974 y Fs(A)2361 3966 y Fr(^)2439 +3901 y Fq(\()2441 3974 y Fs(A)2598 3812 y(contr)2791 +3824 y Fq(L)p 1150 3994 1356 4 v 1529 4055 a(\()g Fn(\000)g(\000)g +(\000)g(\000)1527 4102 y Fq(\()1529 4176 y Fs(A)1616 +4168 y Fr(_)1694 4102 y Fq(\()1696 4176 y Fs(A)p 1802 +4156 10 38 v 1812 4139 42 4 v 1897 4055 a Fn(\000)g(\000)g(\000)g(\000) +g Fq(*)1895 4102 y(\()1897 4176 y Fs(A)1984 4168 y Fr(^)2062 +4102 y Fq(\()2064 4176 y Fs(A)2547 4019 y(cut)523 4361 +y FA(If)26 b(the)f(normalform)e(\(3\))i(is)i(to)f(be)f(reached,)g(then) +g(accordingly)e(we)j(need)f(to)h(orient)f(the)h(colour)523 +4461 y(of)e(the)h(cut-formula)d(to)j(the)g(right.)f(Note)g(ho)n(we)n(v) +o(er)f(that)i(choosing)e(colours)g(has)i(nothing)e(to)i(do)523 +4560 y(with)j(imposing)e(a)i(strate)o(gy)f(for)g(cut-elimination:)f(it) +i(is)h(not)e(cuts)h(that)g(are)f(selected)h(by)f(them,)523 +4660 y(b)n(ut)f(rather)f(the)h(w)o(ay)h(ho)n(w)e(cuts)h(are)g(reduced.) +f(Important)f(for)h(our)g(discussion)h(is)h(the)f(f)o(act)g(that)p +523 4746 473 4 v 558 4801 a Fo(3)606 4833 y Fx(A)15 b(less)g(na)n(\250) +-23 b(\021v)o(e)16 b(method,)g(which)g(only)g(tries)f(out)g(all)g +(possible)h(reduction)g(for)g(outermost)g(cuts,)f(is)g(described)606 +4924 y(in)k([19].)p eop end +%%Page: 11 11 +TeXDict begin 11 10 bop 523 448 a FA(one)21 b(is,)i(ho)n(we)n(v)o(er)m +(,)c(not)i(completely)g(free)g(about)g(ho)n(w)g(to)h(annotate)f +(colours)f(to)i(a)g(sequent)f(proof.)523 548 y(In)k(f)o(act)g(once)g +(the)g(colour)f(`)p Fs(\()p FA(')h(is)h(chosen)e(for)h(the)g +(cut-formula)d Fs(A)k FA(in)g(\(1\),)e(all)i(occurrences)d(of)523 +648 y Fs(A)j FA(must)g(ha)n(v)o(e)f(this)h(colour)-5 +b(.)24 b(The)i(only)e(\223free\224)h(choices)g(in)h(this)g(proof)e(are) +h(the)h(colours)f(for)f(the)523 747 y(formulae)16 b Fs(A)p +Fr(_)q Fs(A)i FA(and)f Fs(A)p Fr(^)q Fs(A)p FA(\227for)g(them)g(we)h +(can)g(mak)o(e)f(an)o(y)g(choice,)g(b)n(ut)g(it)i(has)e(to)h(be)g +(consistent)523 847 y(throughout)k(the)j(proof.)f(Danos)h(et)h(al.)f +(state)h(this)g(consistenc)o(y)e(requirement)f(using)h(the)i(notion)523 +946 y(of)j(an)g Ft(identity)f(class)i FA(in)f(a)h(proof)d(\(the)i +(follo)n(wing)e(de\002nition)h(is)i(slightly)f(adapted)f(from)g([16,) +523 1046 y(P)o(age)20 b(107]\):)523 1205 y Fb(De\002nition)g(1.)41 +b Ft(Occurr)m(ences)18 b(of)g(\(sub\)formulae)e(in)j(a)f(pr)l(oof)g(ar) +m(e)g FA(identi\002ed)f Ft(whene)o(ver)h(the)n(y)f(ar)m(e)523 +1304 y(the)j(corr)m(esponding)e(occurr)m(ences)i(of)g(the)g(same)h +(\(sub\)formula)d(in)581 1455 y Fr(\017)41 b Ft(the)20 +b(two)h(formulae)f(in)g(an)g(axiom,)581 1551 y Fr(\017)41 +b Ft(the)20 b(cut-formulae)f(in)h(a)h(cut)f(and)581 1648 +y Fr(\017)41 b Ft(the)26 b(up)f(and)g(down)g(occurr)m(ences)g(of)h(a)f +(formula)h(in)f(an)h(infer)m(ence)f(rule)h(\(this)g(includes)e(the)664 +1747 y(contr)o(acted)19 b(occurr)m(ences)g(in)i(contr)o(actions)d +(rules\).)523 1901 y(An)32 b FA(identity)f(class)i Ft(in)f(a)f(pr)l +(oof)h(is)h(the)f(r)m(e\003e)n(xive)o(,)f(symmetric)i(and)e(tr)o +(ansitive)h(closur)m(e)f(of)h(the)523 2000 y(identi\002cation)18 +b(r)m(elation.)2086 b Fr(u)-55 b(t)523 2159 y FA(The)25 +b(consistenc)o(y)e(requirement)g(can)i(then)f(be)h(stated)g(as)h(follo) +n(ws:)f(Whene)n(v)o(er)e(colours)h(are)h(an-)523 2259 +y(notated)18 b(to)h(a)g(proof,)e(then)i(e)n(v)o(ery)e(formula)h(in)h +(an)g(identity)f(class)i(must)f(recei)n(v)o(e)f(the)h(same)g(colour)-5 +b(.)648 2358 y(The)21 b(interesting)g(point)g(of)h(colour)n +(-annotations)d(is)j(the)g(f)o(act)h(that)f(the)o(y)f(determine)f +(uniquely)523 2458 y(a)27 b(normalform.)d(In)j(light)g(of)g(this,)g(it) +h(seems)f(reasonable)f(to)h(re)o(gard)e(as)j(the)f(collection)f(of)g +(nor)n(-)523 2557 y(malforms)17 b(reachable)f(from)h(a)i(classical)f +(proof)f(all)h(those)g(for)g(which)f(a)h(colour)f(annotation)f(e)o +(xists)523 2657 y(that)28 b(mak)o(es)h(them)e(reachable.)g(Then)h(the)g +(question)f(arises:)i(Can)g(we)f(\002nd)g(for)g(e)n(v)o(ery)f(normal-) +523 2757 y(form)17 b(reachable)f(by)i(the)g(\(un-coloured\))c +(cut-elimination)h(procedure)h(of)h(Urban)g(and)h(Bierman)f(a)523 +2856 y(colour)n(-annotation)12 b(that)17 b(mak)o(es)f(them)g(reachable) +f(by)h(the)g(cut-elimination)e(procedure)g(of)i(Danos)523 +2956 y(et)21 b(al.?)f(The)g(answer)g(is)h(no)f(and)f(for)h(deep)f +(reasons!)h(Consider)f(the)i(follo)n(wing)d(classical)j(proof)1213 +3178 y Fm(\(1\))1254 3251 y(:)1042 3325 y Fs(A)p Fr(_)p +Fs(A)p 1240 3313 10 38 v 1250 3296 42 4 v 89 w(A)p Fr(^)p +Fs(A)1598 3106 y(A)p Fr(^)p Fs(A)p 1796 3094 10 38 v +1806 3078 42 4 v 89 w(A)p Fr(^)p Fs(A)84 b(A)p Fr(^)p +Fs(A)p 2327 3094 10 38 v 2336 3078 42 4 v 88 w(A)p Fr(^)q +Fs(A)p 1573 3126 1029 4 v 1573 3205 a(A)p Fr(^)p Fs(A;)14 +b(A)p Fr(^)q Fs(A)p 1988 3193 10 38 v 1997 3176 42 4 +v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p +Fs(A)p Fr(^)q Fs(A)p Fm(\))2643 3143 y Fr(^)2698 3156 +y Fq(R)p 1573 3246 1029 4 v 1681 3325 a Fs(A)p Fr(^)q +Fs(A)p 1879 3313 10 38 v 1889 3296 42 4 v 88 w Fm(\()p +Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q +Fs(A)p Fm(\))2643 3265 y Fs(contr)2836 3277 y Fq(L)p +1042 3365 1452 4 v 1361 3444 a Fs(A)p Fr(_)q Fs(A)p 1560 +3432 10 38 v 1569 3416 42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q +Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2535 +3391 y Fs(cut)3267 3444 y FA(\(12\))523 3611 y(where)k(we)g(cut)h(the)f +(proof)e(from)i(\(1\))f(against)h(a)g(proof)f(whose)h(cut-formula,)e +Fs(A)p Fr(^)p Fs(A)p FA(,)j(is)g(contracted)523 3711 +y(in)k(the)g(right-subproof.)18 b(W)-7 b(e)24 b(can)e(reduce)g(the)g +(lo)n(wer)h(cut)f(so)h(that)g(we)g(obtain)f(tw)o(o)h(copies)f(of)g(the) +523 3810 y(proof)d(\(1\):)1503 3899 y Fm(\(1\))1545 3972 +y(:)1332 4045 y Fs(A)p Fr(_)q Fs(A)p 1531 4033 10 38 +v 1540 4017 42 4 v 88 w(A)p Fr(^)q Fs(A)2034 3899 y Fm(\(1\))2075 +3972 y(:)1863 4045 y Fs(A)p Fr(_)q Fs(A)p 2061 4033 10 +38 v 2071 4017 42 4 v 88 w(A)p Fr(^)q Fs(A)p 1307 4065 +1029 4 v 1307 4144 a(A)p Fr(_)q Fs(A;)14 b(A)p Fr(_)q +Fs(A)p 1722 4132 10 38 v 1732 4116 42 4 v 88 w Fm(\()p +Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q +Fs(A)p Fm(\))2378 4082 y Fr(^)2433 4095 y Fq(R)p 1307 +4185 1029 4 v 1416 4264 a Fs(A)p Fr(_)p Fs(A)p 1614 4252 +10 38 v 1623 4235 42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p +Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2378 +4204 y Fs(contr)2571 4216 y Fq(L)3267 4264 y FA(\(13\))523 +4404 y(W)m(ithout)20 b(colours,)f(we)h(can)g(then)g(reduce)f(each)h +(cop)o(y)f(completely)g(independently)e(as)k(follo)n(ws:)1503 +4560 y Fm(\(2\))1545 4633 y(:)1332 4706 y Fs(A)p Fr(_)q +Fs(A)p 1531 4694 10 38 v 1540 4677 42 4 v 88 w(A)p Fr(^)q +Fs(A)2034 4560 y Fm(\(3\))2075 4633 y(:)1863 4706 y Fs(A)p +Fr(_)q Fs(A)p 2061 4694 10 38 v 2071 4677 42 4 v 88 w(A)p +Fr(^)q Fs(A)p 1307 4726 1029 4 v 1307 4805 a(A)p Fr(_)q +Fs(A;)14 b(A)p Fr(_)q Fs(A)p 1722 4793 10 38 v 1732 4776 +42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q +Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2378 4743 y Fr(^)2433 +4755 y Fq(R)p 1307 4845 1029 4 v 1416 4924 a Fs(A)p Fr(_)p +Fs(A)p 1614 4912 10 38 v 1623 4896 42 4 v 88 w Fm(\()p +Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q +Fs(A)p Fm(\))2378 4865 y Fs(contr)2571 4877 y Fq(L)3267 +4924 y FA(\(14\))p eop end +%%Page: 12 12 +TeXDict begin 12 11 bop 523 448 a FA(Such)24 b(a)g(beha)n(viour)e +(cannot)h(be)g(achie)n(v)o(ed)g(by)g(using)h(colours:)e(the)i(colours)f +(must)h(be)g(annotated)523 548 y(before)17 b(cut-elimination)f +(commences)g(and)i(is)h(in)m(v)n(ariant)d(under)h(cut-reductions.)e +(Consequently)-5 b(,)523 648 y(whene)n(v)o(er)22 b(a)i(cut)g(is)h +(duplicated)e(in)h(a)g(reduction)e(sequence)h(\(as)i(in)f(the)g +(reduction)e(\(12\))p Fr(!)p FA(\(13\)\),)523 747 y(the)29 +b(colour)n(-annotation)d(pre)n(v)o(ents)h(both)i(instances)g(from)f +(reducing)g(dif)n(ferently)-5 b(.)26 b(\(The)j(deeper)523 +847 y(reason)e(mentioned)g(earlier)g(is)j(that)e(one)f(just)i(cannot)e +(pre-determine)f(the)i(choices)f(in)i(a)f(com-)523 946 +y(pletely)20 b(non-deterministic)d(reduction)i(system.\))648 +1061 y(Comparing)26 b(the)j(cut-elimination)e(procedure)f(of)j(Urban)e +(and)i(Bierman)f(with)h(the)g(one)f(of)523 1161 y(Danos)16 +b(et)i(al.,)f(tw)o(o)g(points)f(stand)g(out:)h(Both)g(cut-elimination)d +(procedures)h(are)i(strongly)e(normal-)523 1261 y(ising)685 +1230 y Fp(4)742 1261 y FA(and)22 b(also)i(determine)e(a)h(collection)f +(of)h(normalforms)e(reachable)h(from)g(a)i(sequent-proof)523 +1360 y(in)c(classical)g(logic.)f(As)h(sho)n(wn)f(by)g(e)o(xample,)f +(these)i(collections)f(contain)f(in)i(general)e(more)h(than)523 +1460 y(one)g(element.)g(Also)i(as)f(sho)n(wn)f(by)h(e)o(xample,)e(the)i +(collection)f(determined)f(by)h(the)h(procedure)e(of)523 +1559 y(Danos)k(et)g(al.)g(is)h(generally)e(a)h(proper)f(subset)h(of)f +(the)h(collection)f(determined)f(by)i(the)g(procedure)523 +1659 y(of)f(Urban)g(and)g(Bierman.)g(The)g(colour)n(-annotations)d(in)k +(the)f(procedure)e(of)i(Danos)h(et)g(al.)g(cannot)523 +1759 y(fully)e(account)f(for)g(the)h(non-determinism)d(present)j(in)g +(classical)h(logic.)648 1873 y(Both)f(cut-elimination)d(procedures)h +(can)i(also)g(be)g(used)f(for)h(reducing)e(intuitionistic)h(proofs.)523 +1973 y(Because)k(of)g(the)h(restrictions)e(imposed)g(upon)g +(intuitionistic)h(sequents,)f(non-deterministic)f(re-)523 +2073 y(duction)e(sequences)h(such)g(as)i(\(12\))p Fr(!)p +FA(\(13\))p Fr(!)p FA(\(14\))16 b(cannot)j(be)i(constructed.)d(But)k +(still)f(the)g(proce-)523 2172 y(dure)d(of)g(Urban)g(and)g(Bierman)g +(is)h Ft(not)h FA(Church-Rosser)d(in)i(the)f(intuitionistic)g(case,)h +(and)f(also)h(dif-)523 2272 y(ferent)f(colour)n(-annotations)d(of)j(an) +g(intuitionistic)g(proof)f(might)h(lead)g(to)h(dif)n(ferent)d +(normalforms.)523 2372 y(Ho)n(we)n(v)o(er)m(,)31 b(as)i(mentioned)e +(earlier)m(,)h(we)h(re)o(gard)e(the)i(dif)n(ferences)e(between)h(the)h +(normalforms)523 2471 y(reachable)19 b(from)g(an)h(intuitionistic)f +(sequent-proof)e(as)k(inessential)f(and)g(re)o(gard)e(cut-elimination) +523 2571 y(as)23 b(morally)e(Church-Rosser)-5 b(.)21 +b(That)h(in)h(turn)e(means)h(that)h(in)f(the)g(intuitionistic)g(case)h +(there)f(is)h(no)523 2670 y(dif)n(ference)18 b(between)i(coloured)e +(and)h(un-coloured)e(cut-elimination\227at)h(least)j(morally)-5 +b(.)523 3012 y Fu(4)99 b(Conjectur)n(e)523 3287 y FA(W)-7 +b(e)19 b(ha)n(v)o(e)e(already)g(seen)h(that)f(by)h(translating)f(the)g +(cut)h(in)g(\(1\))f(using)g(a)h(left-)g(and)f(right-translation,)523 +3387 y(we)k(can)f(simulate)g(the)h(reductions)e(\(1\))p +Fr(!)p FA(\(2\))f(and)i(\(1\))p Fr(!)p FA(\(3\))f(by)h(double-ne)o +(gations.)15 b(Ho)n(we)n(v)o(er)k(in)523 3486 y(general,)g(a)h(left-)g +(or)g(right-translation)e(of)i(a)g(cut)h(is)g(not)f(suf)n(\002cient)f +(to)h(simulate)h(all)f(cut-reduction)523 3586 y(sequences)f(in)i +(classical)g(logic.)e(Consider)h(the)g(follo)n(wing)f(instance)g(of)h +(a)h(logical)f(cut:)1520 3790 y Fs(\031)1567 3802 y Fl(1)1550 +3855 y Fm(:)p 1503 3916 10 38 v 1512 3900 42 4 v 1572 +3928 a Fs(B)p 1424 3948 276 4 v 1442 4010 10 38 v 1452 +3993 42 4 v 1512 4022 a(B)t Fr(_)p Fs(C)1741 3961 y Fr(_)1797 +3974 y Fq(R)1847 3982 y Fc(1)2001 3790 y Fs(\031)2048 +3802 y Fl(2)2032 3855 y Fm(:)1966 3928 y Fs(B)p 2052 +3916 10 38 v 2061 3900 42 4 v 2239 3790 a(\031)2286 3802 +y Fl(3)2269 3855 y Fm(:)2204 3928 y Fs(C)p 2288 3916 +10 38 v 2298 3900 42 4 v 1966 3948 392 4 v 2024 4022 +a(B)t Fr(_)p Fs(C)p 2230 4010 10 38 v 2240 3993 42 4 +v 2399 3965 a Fr(_)2454 3978 y Fq(L)p 1424 4042 876 4 +v 1836 4095 10 38 v 1846 4079 42 4 v 2341 4067 a Fs(cut)3267 +4107 y FA(\(15\))523 4318 y(W)-7 b(e)21 b(can)f(reduce)f(this)i(cut)f +(to)1725 4408 y Fs(\031)1772 4420 y Fl(1)1756 4473 y +Fm(:)p 1708 4534 10 38 v 1718 4518 42 4 v 1778 4546 a +Fs(B)1963 4408 y(\031)2010 4420 y Fl(2)1994 4473 y Fm(:)1928 +4546 y Fs(B)p 2014 4534 10 38 v 2023 4518 42 4 v 1690 +4566 394 4 v 1861 4620 10 38 v 1871 4603 42 4 v 2125 +4592 a(cut)p 523 4746 473 4 v 558 4801 a Fo(4)606 4833 +y Fx(Danos)29 b(et)g(al.)f(sho)n(wed)i(strong)f(normalisation)g(of)g +(their)f(cut-elimination)h(procedure)h(by)f(translating)606 +4924 y(reduction)20 b(sequences)h(in)d(classical)h(logic)g(to)g +(reduction)h(sequences)h(of)e(proof-nets)g(in)g(linear)g(logic.)p +eop end +%%Page: 13 13 +TeXDict begin 13 12 bop 523 448 a FA(and)28 b(assuming)g(that)h(the)f +(cut-formula)e Fs(B)34 b FA(is)29 b(not)f(freshly)g(introduced)e(in)j +Fs(\031)2821 460 y Fl(1)2859 448 y FA(,)g(we)g(can)f(further)523 +548 y(permute)19 b Fs(\031)863 560 y Fl(2)921 548 y FA(inside)h +Fs(\031)1187 560 y Fl(1)1225 548 y FA(.)g(This)h(beha)n(viour)d +(correspond)f(to)k(the)f(colour)n(-annotation)1508 786 +y Fs(\031)1555 798 y Fl(1)1539 851 y Fm(:)p 1480 952 +10 38 v 1489 935 42 4 v 1575 898 a Fq(\()1572 972 y Fs(B)p +1389 991 322 4 v 1408 1092 10 38 v 1417 1075 42 4 v 1503 +1038 a Fq(\()1500 1112 y Fs(B)1591 1104 y Fr(_)p Fs(C)1753 +1004 y Fr(_)1808 1016 y Fq(R)1858 1024 y Fc(1)2024 786 +y Fs(\031)2071 798 y Fl(2)2055 851 y Fm(:)1980 898 y +Fq(\()1978 972 y Fs(B)p 2086 952 10 38 v 2096 935 42 +4 v 2273 825 a(\031)2320 837 y Fl(3)2304 890 y Fm(:)2239 +964 y Fs(C)p 2323 952 10 38 v 2332 935 42 4 v 1978 991 +415 4 v 2038 1038 a Fq(\()2035 1112 y Fs(B)2126 1104 +y Fr(_)p Fs(C)p 2265 1092 10 38 v 2274 1075 42 4 v 2434 +1008 a Fr(_)2489 1020 y Fq(L)p 1389 1132 945 4 v 1836 +1185 10 38 v 1846 1169 42 4 v 2376 1157 a Fs(cut)3267 +1197 y FA(\(16\))523 1519 y(where)d(we)h(lea)n(v)o(e)g(the)g(colour)e +(annotation)g(for)h Fs(C)25 b FA(and)17 b Fs(B)t Fr(_)p +Fs(C)25 b FA(unspeci\002ed,)16 b(since)i(it)h(is)f(not)g(impor)n(-)523 +1619 y(tant)25 b(for)f(the)i(ar)o(gument)c(at)k(hand.)d(The)i(beha)n +(viour)e(of)i(\(16\))f(can)g(be)h(simulated)g(by)f(the)h(double-)523 +1719 y(ne)o(gation)18 b(translation)h Fm(\()p Fr(\000)p +Fm(\))1328 1688 y Fn(\003)1367 1719 y FA(.)h(The)g(double-ne)o(gated)c +(v)o(ersion)j(of)h(\(15\))f(is)i(as)g(follo)n(ws:)1364 +2163 y Fs(\031)1414 2133 y Fn(\003)1411 2184 y Fl(1)1397 +2236 y Fm(:)1284 2310 y Fr(:)p Fs(B)1406 2280 y Fn(\003)p +1463 2298 10 38 v 1473 2281 42 4 v 1177 2330 463 4 v +1177 2404 a Fr(:)p Fs(B)1299 2373 y Fn(\003)1338 2404 +y Fr(^:)p Fs(C)1513 2373 y Fn(\003)p 1571 2392 10 38 +v 1580 2375 42 4 v 1682 2342 a Fr(^)1737 2355 y Fq(L)1783 +2363 y Fc(1)p 1117 2423 583 4 v 1136 2490 10 38 v 1145 +2474 42 4 v 1205 2502 a Fr(:)p Fm(\()p Fr(:)p Fs(B)1414 +2472 y Fn(\003)1454 2502 y Fr(^:)p Fs(C)1629 2472 y Fn(\003)1668 +2502 y Fm(\))1742 2435 y Fr(:)1797 2447 y Fq(R)p 1090 +2543 639 4 v 1090 2622 a Fr(::)p Fm(\()p Fr(:)p Fs(B)1354 +2592 y Fn(\003)1393 2622 y Fr(^)q(:)p Fs(C)1569 2592 +y Fn(\003)1607 2622 y Fm(\))p 1658 2610 10 38 v 1668 +2593 42 4 v 1769 2555 a Fr(:)1824 2567 y Fq(L)2037 2069 +y Fs(\031)2087 2039 y Fn(\003)2084 2090 y Fl(2)2070 2142 +y Fm(:)1985 2216 y Fs(B)2052 2186 y Fn(\003)p 2109 2204 +10 38 v 2118 2188 42 4 v 1957 2236 249 4 v 1976 2298 +10 38 v 1985 2281 42 4 v 2045 2310 a Fr(:)p Fs(B)2167 +2280 y Fn(\003)2247 2248 y Fr(:)2302 2260 y Fq(R)2519 +2069 y Fs(\031)2569 2039 y Fn(\003)2566 2090 y Fl(3)2552 +2142 y Fm(:)2468 2216 y Fs(C)2533 2186 y Fn(\003)p 2590 +2204 10 38 v 2599 2188 42 4 v 2440 2236 247 4 v 2459 +2298 10 38 v 2468 2281 42 4 v 2528 2310 a Fr(:)p Fs(C)2648 +2280 y Fn(\003)2728 2248 y Fr(:)2783 2260 y Fq(R)p 1957 +2330 730 4 v 2109 2392 10 38 v 2119 2375 42 4 v 2179 +2404 a Fr(:)p Fs(B)2301 2373 y Fn(\003)2339 2404 y Fr(^)q(:)p +Fs(C)2515 2373 y Fn(\003)2728 2347 y Fr(^)2784 2359 y +Fq(R)p 2031 2423 583 4 v 2031 2502 a Fr(:)p Fm(\()p Fr(:)p +Fs(B)2240 2472 y Fn(\003)2279 2502 y Fr(^:)p Fs(C)2454 +2472 y Fn(\003)2493 2502 y Fm(\))p 2544 2490 10 38 v +2553 2474 42 4 v 2655 2435 a Fr(:)2710 2447 y Fq(L)p +2003 2543 639 4 v 2022 2610 10 38 v 2031 2593 42 4 v +2091 2622 a Fr(::)p Fm(\()p Fr(:)p Fs(B)2355 2592 y Fn(\003)2395 +2622 y Fr(^:)p Fs(C)2570 2592 y Fn(\003)2609 2622 y Fm(\))2683 +2555 y Fr(:)2738 2567 y Fq(R)p 1090 2663 1552 4 v 1840 +2716 10 38 v 1849 2700 42 4 v 2683 2688 a Fs(cut)523 +3064 y FA(which)f(reduces)f(in)h(three)g(steps)h(to)f(the)g(proof)1676 +3508 y Fs(\031)1726 3478 y Fn(\003)1723 3529 y Fl(1)1709 +3581 y Fm(:)1596 3655 y Fr(:)p Fs(B)1718 3625 y Fn(\003)p +1775 3643 10 38 v 1785 3626 42 4 v 2008 3414 a Fs(\031)2058 +3384 y Fn(\003)2055 3435 y Fl(2)2041 3487 y Fm(:)1956 +3561 y Fs(B)2023 3531 y Fn(\003)p 2079 3549 10 38 v 2089 +3532 42 4 v 1928 3581 249 4 v 1946 3643 10 38 v 1956 +3626 42 4 v 2016 3655 a Fr(:)p Fs(B)2138 3625 y Fn(\003)2218 +3593 y Fr(:)2273 3605 y Fq(R)p 1596 3675 581 4 v 1861 +3728 10 38 v 1871 3712 42 4 v 2218 3700 a Fs(cut)523 +4076 y FA(Because)f(the)g Fr(:)995 4088 y Fq(R)1049 4076 +y FA(-rule)f(introduces)f(freshly)h(the)h(cut-formula)d +Fr(:)p Fs(B)2494 4046 y Fn(\003)2533 4076 y FA(,)j(the)f(cut)h(is)h +(\223block)o(ed\224)d(from)523 4175 y(reducing)h(to)j(the)f(right.)f +(It)i(must)f(\002rst)h(reduce)e(to)h(the)g(left)g(just)h(as)g(the)f +(colour)f(annotation)f(in)j(\(16\))523 4275 y(prescribed.)d(If)i(we)h +(w)o(anted)f(to)g(simulated)g(the)g(opposite)f(colouring)f(for)h +Fs(B)t FA(,)i(namely)1508 4513 y Fs(\031)1555 4525 y +Fl(1)1539 4578 y Fm(:)p 1480 4679 10 38 v 1489 4662 42 +4 v 1575 4625 a Fq(*)1572 4699 y Fs(B)p 1389 4718 322 +4 v 1408 4819 10 38 v 1417 4802 42 4 v 1503 4765 a Fq(*)1500 +4839 y Fs(B)1591 4831 y Fr(_)p Fs(C)1753 4731 y Fr(_)1808 +4743 y Fq(R)1858 4751 y Fc(1)2024 4513 y Fs(\031)2071 +4525 y Fl(2)2055 4578 y Fm(:)1980 4625 y Fq(*)1978 4699 +y Fs(B)p 2086 4679 10 38 v 2096 4662 42 4 v 2273 4552 +a(\031)2320 4564 y Fl(3)2304 4617 y Fm(:)2239 4691 y +Fs(C)p 2323 4679 10 38 v 2332 4662 42 4 v 1978 4718 415 +4 v 2038 4765 a Fq(*)2035 4839 y Fs(B)2126 4831 y Fr(_)p +Fs(C)p 2265 4819 10 38 v 2274 4802 42 4 v 2434 4735 a +Fr(_)2489 4747 y Fq(L)p 1389 4859 945 4 v 1836 4912 10 +38 v 1846 4896 42 4 v 2376 4884 a Fs(cut)3267 4924 y +FA(\(17\))p eop end +%%Page: 14 14 +TeXDict begin 14 13 bop 523 448 a FA(it)29 b(turns)g(out)f(we)h(ha)n(v) +o(e)f(to)h(double-ne)o(gate)c(translate)j(\(15\))g(using)g(the)h +(translation)e Fm(\()p Fr(\000)p Fm(\))3156 418 y Fn(\016)3224 +448 y FA(gi)n(v)o(en)523 548 y(in)20 b(\(6\).)g(The)g(resulting)f +(intuitionistic)g(proof)g(is:)1436 721 y Fs(\031)1486 +691 y Fn(\016)1483 741 y Fl(1)1469 794 y Fm(:)1356 867 +y Fr(:)p Fs(B)1478 837 y Fn(\016)p 1535 855 10 38 v 1545 +839 42 4 v 1328 887 304 4 v 1347 949 10 38 v 1356 932 +42 4 v 1416 961 a Fr(::)p Fs(B)1593 931 y Fn(\016)1674 +899 y Fr(:)1729 911 y Fq(R)p 1328 981 304 4 v 1402 1043 +10 38 v 1412 1026 42 4 v 1472 1055 a Fs(B)1539 1025 y +Fn(\016)1674 993 y Fr(::)1784 1005 y Fq(R)p 1304 1075 +353 4 v 1323 1136 10 38 v 1332 1120 42 4 v 1392 1148 +a Fs(B)1459 1118 y Fn(\016)1498 1148 y Fr(_)p Fs(C)1618 +1118 y Fn(\016)1698 1087 y Fr(_)1753 1100 y Fq(R)1803 +1108 y Fc(1)p 1244 1168 472 4 v 1244 1247 a Fr(:)p Fm(\()p +Fs(B)1398 1217 y Fn(\016)1437 1247 y Fr(_)q Fs(C)1558 +1217 y Fn(\016)1596 1247 y Fm(\))p 1647 1235 10 38 v +1656 1219 42 4 v 1758 1180 a Fr(:)1813 1192 y Fq(L)p +1217 1288 528 4 v 1235 1355 10 38 v 1245 1338 42 4 v +1305 1367 a Fr(::)p Fm(\()p Fs(B)1514 1337 y Fn(\016)1553 +1367 y Fr(_)p Fs(C)1673 1337 y Fn(\016)1712 1367 y Fm(\))1786 +1300 y Fr(:)1841 1312 y Fq(R)p 1189 1407 583 4 v 1189 +1486 a Fr(:::)p Fm(\()p Fs(B)1453 1456 y Fn(\016)1493 +1486 y Fr(_)p Fs(C)1613 1456 y Fn(\016)1651 1486 y Fm(\))p +1702 1474 10 38 v 1712 1458 42 4 v 1813 1419 a Fr(:)1868 +1431 y Fq(L)2111 908 y Fs(\031)2161 878 y Fn(\016)2158 +929 y Fl(2)2144 981 y Fm(:)2059 1055 y Fs(B)2126 1025 +y Fn(\016)p 2183 1043 10 38 v 2192 1026 42 4 v 2387 908 +a Fs(\031)2437 878 y Fn(\016)2434 929 y Fl(3)2419 981 +y Fm(:)2335 1055 y Fs(C)2400 1025 y Fn(\016)p 2457 1043 +10 38 v 2467 1026 42 4 v 2059 1075 468 4 v 2117 1148 +a Fs(B)2184 1118 y Fn(\016)2222 1148 y Fr(_)p Fs(C)2342 +1118 y Fn(\016)p 2399 1136 10 38 v 2409 1120 42 4 v 2568 +1092 a Fr(_)2623 1104 y Fq(L)p 2057 1168 472 4 v 2075 +1235 10 38 v 2085 1219 42 4 v 2145 1247 a Fr(:)p Fm(\()p +Fs(B)2299 1217 y Fn(\016)2338 1247 y Fr(_)p Fs(C)2458 +1217 y Fn(\016)2496 1247 y Fm(\))2570 1180 y Fr(:)2625 +1192 y Fq(R)p 2029 1288 528 4 v 2029 1367 a Fr(::)p Fm(\()p +Fs(B)2238 1337 y Fn(\016)2277 1367 y Fr(_)q Fs(C)2398 +1337 y Fn(\016)2436 1367 y Fm(\))p 2487 1355 10 38 v +2496 1338 42 4 v 2598 1300 a Fr(:)2653 1312 y Fq(L)p +2001 1407 583 4 v 2020 1474 10 38 v 2029 1458 42 4 v +2089 1486 a Fr(:::)p Fm(\()p Fs(B)2353 1456 y Fn(\016)2393 +1486 y Fr(_)p Fs(C)2513 1456 y Fn(\016)2552 1486 y Fm(\))2625 +1419 y Fr(:)2680 1431 y Fq(R)p 1189 1527 1395 4 v 1861 +1581 10 38 v 1871 1564 42 4 v 2625 1553 a Fs(cut)523 +1785 y FA(which)h(after)g(four)f(steps)h(reduces)g(to)g(the)g(proof) +1601 1958 y Fs(\031)1651 1927 y Fn(\016)1648 1978 y Fl(1)1634 +2030 y Fm(:)1521 2104 y Fr(:)p Fs(B)1643 2074 y Fn(\016)p +1700 2092 10 38 v 1709 2076 42 4 v 1493 2124 304 4 v +1512 2186 10 38 v 1521 2169 42 4 v 1581 2198 a Fr(::)p +Fs(B)1758 2168 y Fn(\016)1839 2136 y Fr(:)1894 2148 y +Fq(R)p 1493 2218 304 4 v 1567 2280 10 38 v 1576 2263 +42 4 v 1636 2292 a Fs(B)1703 2262 y Fn(\016)1839 2230 +y Fr(::)1949 2242 y Fq(R)2139 2145 y Fs(\031)2189 2115 +y Fn(\016)2186 2166 y Fl(2)2172 2218 y Fm(:)2087 2292 +y Fs(B)2154 2262 y Fn(\016)p 2210 2280 10 38 v 2220 2263 +42 4 v 1548 2312 732 4 v 1889 2365 10 38 v 1898 2349 +42 4 v 2321 2337 a Fs(cut)3267 2377 y FA(\(18\))523 2569 +y(What)g(happens)e(ne)o(xt,)g(ho)n(we)n(v)o(er)m(,)f(is)j(not)f(clear)g +(at)h(\002rst)g(sight.)g(If)f(we)g(e)o(xpand)f(the)h +Fr(::)3010 2581 y Fq(R)3065 2569 y FA(-rule)g(to)h(an)523 +2669 y(auxiliary)e(cut,)i(then)f(the)g(cut)h(can)f(reduce)g(into)g +(both)g(directions.)f(If)i(we)g(re)o(gard)d(the)j Fr(::)3104 +2681 y Fq(R)3159 2669 y FA(-rule)f(as)523 2769 y(an)k(inference)f(rule) +g(in)i(its)g(o)n(wn)e(right,)h(then)f(the)h(cut-formula)e +Fs(B)2441 2738 y Fn(\016)2503 2769 y FA(is)j(freshly)e(introduced)f(in) +i(the)523 2868 y(subproof)h(on)i(the)g(left-hand)f(side)h(and)g +(therefore)e(the)j(cut)f(can)g(only)g(mo)o(v)o(e)e(to)j(the)f +(right\227just)523 2968 y(as)21 b(prescribed)d(by)i(the)g(colour)n +(-annotation)d(in)j(\(17\).)f(Although)f(we)j(do)f(not)f(ha)n(v)o(e)h +(a)h(proof)d(of)i(this)523 3067 y(f)o(act,)i(e)o(xperiments)f(with)h +([17])g(ha)n(v)o(e)g(con)m(vinced)d(us)k(that)g(when)e(cut-elimination) +g(is)i(concerned,)523 3167 y(we)18 b(can)g(indeed)e(re)o(gard)g(the)i +Fr(::)1470 3179 y Fq(R)1525 3167 y FA(-rule)f(as)h(a)h(proper)d +(inference)g(rule)h(with)h(the)g(consequence)d(that)523 +3267 y(in)j(the)g(proof)e(abo)o(v)o(e)g Fs(B)1205 3237 +y Fn(\016)1262 3267 y FA(is)j(freshly)e(introduced.)e(\(Roughly)h +(speaking,)g(if)j(we)f(had)f(e)o(xpanded)e(the)523 3366 +y Fr(::)633 3378 y Fq(R)688 3366 y FA(-rule)26 b(to)g(an)g(auxiliary)f +(cut)h(in)h(the)f(proof)f(\(18\),)f(then)i(mo)o(ving)e(the)j(cut)f(to)g +(the)g(left)h(means)523 3466 y(it)d(cannot)e(mo)o(v)o(e)g(v)o(ery)g(f)o +(ar)m(,)g(namely)g(only)h(to)g(the)g(place)g(where)g +Fs(B)2492 3436 y Fn(\016)2554 3466 y FA(is)h(introduced)d(in)i(the)g +(proof)523 3566 y(of)k(the)h(sequent)f Fr(::)p Fs(B)1209 +3535 y Fn(\016)p 1266 3554 10 38 v 1276 3537 42 4 v 1336 +3566 a Fs(B)1403 3535 y Fn(\016)1469 3566 y FA(and)g(then)g(the)h(cut)g +(has)f(to)h(mo)o(v)o(e)e(right.)h(In)g(ef)n(fect)g(we)h(obtain)f(a)523 +3665 y(beha)n(viour)18 b(which)i(is)h(almost)f(identical)g(to)g(mo)o +(ving)e(the)j(cut)f(to)g(the)g(right)g(in)g(the)g(\002rst)h(place.\)) +648 3768 y(Since)29 b(the)h(colour)n(-protocol)c(of)k(Danos)f(et)i(al.) +f(allo)n(ws)g(us)g(to)g(annotate)e(in)i(man)o(y)f(circum-)523 +3868 y(stances)f(either)f(colour)g(`)p Fs(\()p FA(')g(or)h(`)p +Fs(*)p FA(')f(to)h(the)g(formulae)e(in)i(a)g(classical)g(proof,)e(we)i +(need)f(ho)n(w-)523 3967 y(e)n(v)o(er)21 b(to)h(depart)g(from)f(the)h +(traditional)f(double-ne)o(gation)c(technique)k(that)h(translates)g(a)h +(classical)523 4067 y(proof)k(uniformly)e(using)j(a)g(single)g +(double-ne)o(gation)c(translation.)j(T)-7 b(o)28 b(simulate)g(the)g +(coloured)523 4166 y(cut-elimination)14 b(procedure)h(in)h(a)h +(meaningful)e(w)o(ay)-5 b(,)15 b(we)i(need)f(to)h(allo)n(w)f(more)g +(than)g(one)g(double-)523 4266 y(ne)o(gation)f(translation.)h(Let)h(us) +h(e)o(xplain)e(this)h(f)o(act)h(with)f(a)g(classical)h(proof)e +Fs(\031)21 b FA(ending)15 b(in)j(a)f(cut)g(with)523 4366 +y(the)j(cut-formula)1815 4429 y Fq(\()-11 b Fn(\000)g(\000)g(\000)g +(\000)1815 4476 y Fq(*)1812 4550 y Fs(B)1902 4542 y Fr(_)1984 +4476 y Fq(\()1981 4550 y Fs(C)2093 4542 y(:)523 4703 +y FA(W)k(e)27 b(will)f(sho)n(w)f(that)g(the)g(beha)n(viour)f(of)h(this) +h(\(coloured\))c(cut)k(can)f(be)g(simulated)g(by)g(a)g(double-)523 +4825 y(ne)o(gation)30 b(translation)g(with)i(the)g(clause)g +Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))2028 4795 y Fn(\017)2111 +4778 y Fp(def)2116 4825 y Fm(=)48 b Fr(::)p Fm(\()p Fs(B)2438 +4795 y Fn(\017)2478 4825 y Fr(_::)p Fs(C)2708 4795 y +Fn(\017)2747 4825 y Fm(\))p FA(.)33 b(T)-7 b(o)31 b(sho)n(w)h(this)g +(we)523 4924 y(analyse)26 b(all)h(cases)g(ho)n(w)f(the)h(cut)f(in)h +Fs(\031)j FA(could)c(ha)n(v)o(e)f(arisen.)h(Consider)g(\002rst)h(the)g +(case)g(where)f Fs(\031)p eop end +%%Page: 15 15 +TeXDict begin 15 14 bop 523 448 a FA(ends)20 b(with)g(the)h(follo)n +(wing)d(logical)i(cut)1496 602 y Fs(\031)1543 614 y Fl(1)1527 +666 y Fm(:)p 1468 767 10 38 v 1477 751 42 4 v 1563 714 +a Fq(*)1560 787 y Fs(B)p 1365 807 346 4 v 1384 969 10 +38 v 1393 952 42 4 v 1480 868 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g +(\000)1479 915 y Fq(*)1476 989 y Fs(B)1567 981 y Fr(_)1648 +915 y Fq(\()1645 989 y Fs(C)1753 820 y Fr(_)1808 832 +y Fq(R)1858 840 y Fc(1)2024 602 y Fs(\031)2071 614 y +Fl(2)2055 666 y Fm(:)1980 714 y Fq(*)1978 787 y Fs(B)p +2086 767 10 38 v 2096 751 42 4 v 2285 602 a(\031)2332 +614 y Fl(3)2316 666 y Fm(:)2242 714 y Fq(\()2239 787 +y Fs(C)p 2347 767 10 38 v 2356 751 42 4 v 1978 807 439 +4 v 2027 868 a Fq(\()g Fn(\000)g(\000)g(\000)g(\000)2026 +915 y Fq(*)2024 989 y Fs(B)2114 981 y Fr(_)2196 915 y +Fq(\()2193 989 y Fs(C)p 2300 969 10 38 v 2310 952 42 +4 v 2458 824 a Fr(_)2513 836 y Fq(L)p 1365 1008 1005 +4 v 1842 1062 10 38 v 1852 1045 42 4 v 2411 1034 a Fs(cut)3267 +1074 y FA(\(19\))523 1269 y(which)20 b(can)g(reduce)f(to)1714 +1327 y Fs(\031)1761 1339 y Fl(1)1744 1391 y Fm(:)p 1685 +1492 10 38 v 1695 1476 42 4 v 1780 1439 a Fq(*)1778 1512 +y Fs(B)1975 1327 y(\031)2022 1339 y Fl(2)2006 1391 y +Fm(:)1931 1439 y Fq(*)1928 1512 y Fs(B)p 2037 1492 10 +38 v 2046 1476 42 4 v 1667 1532 440 4 v 1861 1586 10 +38 v 1871 1569 42 4 v 2148 1557 a(cut)3267 1598 y FA(\(20\))523 +1764 y(The)h Fm(\()p Fr(\000)p Fm(\))802 1733 y Fn(\017)840 +1764 y FA(-translated)f(v)o(ersion)g(of)h Fs(\031)1381 +1941 y(\031)1431 1911 y Fn(\017)1428 1962 y Fl(1)1414 +2014 y Fm(:)1301 2088 y Fr(:)p Fs(B)1423 2058 y Fn(\017)p +1480 2076 10 38 v 1489 2059 42 4 v 1273 2108 304 4 v +1292 2169 10 38 v 1301 2153 42 4 v 1361 2181 a Fr(::)p +Fs(B)1538 2151 y Fn(\017)1619 2119 y Fr(:)1674 2131 y +Fq(R)p 1273 2201 304 4 v 1347 2263 10 38 v 1356 2247 +42 4 v 1416 2275 a Fs(B)1483 2245 y Fn(\017)1619 2213 +y Fr(::)1729 2225 y Fq(R)p 1194 2295 463 4 v 1212 2357 +10 38 v 1222 2340 42 4 v 1282 2369 a Fs(B)1349 2339 y +Fn(\017)1387 2369 y Fr(_::)p Fs(C)1617 2339 y Fn(\017)1698 +2308 y Fr(_)1753 2320 y Fq(R)1803 2328 y Fc(1)p 1134 +2389 583 4 v 1134 2468 a Fr(:)p Fm(\()p Fs(B)1288 2438 +y Fn(\017)1327 2468 y Fr(_::)p Fs(C)1557 2438 y Fn(\017)1596 +2468 y Fm(\))p 1647 2456 10 38 v 1656 2439 42 4 v 1758 +2400 a Fr(:)1813 2412 y Fq(L)p 1106 2508 639 4 v 1124 +2575 10 38 v 1134 2559 42 4 v 1194 2587 a Fr(::)p Fm(\()p +Fs(B)1403 2557 y Fn(\017)1442 2587 y Fr(_)q(::)p Fs(C)1673 +2557 y Fn(\017)1712 2587 y Fm(\))1786 2520 y Fr(:)1841 +2532 y Fq(R)p 1078 2628 694 4 v 1078 2707 a Fr(:::)p +Fm(\()p Fs(B)1342 2677 y Fn(\017)1382 2707 y Fr(_::)p +Fs(C)1612 2677 y Fn(\017)1651 2707 y Fm(\))p 1702 2695 +10 38 v 1712 2678 42 4 v 1813 2640 a Fr(:)1868 2652 y +Fq(L)2111 2128 y Fs(\031)2161 2098 y Fn(\017)2158 2149 +y Fl(2)2144 2201 y Fm(:)2059 2275 y Fs(B)2126 2245 y +Fn(\017)p 2183 2263 10 38 v 2192 2247 42 4 v 2442 1941 +a Fs(\031)2492 1911 y Fn(\017)2489 1962 y Fl(3)2475 2014 +y Fm(:)2390 2088 y Fs(C)2455 2058 y Fn(\017)p 2512 2076 +10 38 v 2522 2059 42 4 v 2363 2108 247 4 v 2381 2169 +10 38 v 2391 2153 42 4 v 2451 2181 a Fr(:)p Fs(C)2571 +2151 y Fn(\017)2651 2119 y Fr(:)2706 2131 y Fq(R)p 2335 +2201 303 4 v 2335 2275 a Fr(::)p Fs(C)2510 2245 y Fn(\017)p +2568 2263 10 38 v 2577 2247 42 4 v 2679 2213 a Fr(:)2734 +2225 y Fq(L)p 2059 2295 579 4 v 2117 2369 a Fs(B)2184 +2339 y Fn(\017)2222 2369 y Fr(_::)p Fs(C)2452 2339 y +Fn(\017)p 2510 2357 10 38 v 2519 2340 42 4 v 2679 2312 +a Fr(_)2734 2324 y Fq(L)p 2057 2389 583 4 v 2075 2456 +10 38 v 2085 2439 42 4 v 2145 2468 a Fr(:)p Fm(\()p Fs(B)2299 +2438 y Fn(\017)2338 2468 y Fr(_::)p Fs(C)2568 2438 y +Fn(\017)2607 2468 y Fm(\))2681 2400 y Fr(:)2736 2412 +y Fq(R)p 2029 2508 639 4 v 2029 2587 a Fr(::)p Fm(\()p +Fs(B)2238 2557 y Fn(\017)2277 2587 y Fr(_)q(::)p Fs(C)2508 +2557 y Fn(\017)2547 2587 y Fm(\))p 2597 2575 10 38 v +2607 2559 42 4 v 2709 2520 a Fr(:)2764 2532 y Fq(L)p +2001 2628 694 4 v 2020 2695 10 38 v 2029 2678 42 4 v +2089 2707 a Fr(:::)p Fm(\()p Fs(B)2353 2677 y Fn(\017)2393 +2707 y Fr(_::)p Fs(C)2623 2677 y Fn(\017)2662 2707 y +Fm(\))2736 2640 y Fr(:)2791 2652 y Fq(R)p 1078 2747 1617 +4 v 1861 2801 10 38 v 1871 2785 42 4 v 2736 2773 a Fs(cut)523 +3008 y FA(reduces)f(to)1601 3087 y Fs(\031)1651 3057 +y Fn(\017)1648 3108 y Fl(1)1634 3160 y Fm(:)1521 3234 +y Fr(:)p Fs(B)1643 3204 y Fn(\017)p 1700 3222 10 38 v +1709 3205 42 4 v 1493 3254 304 4 v 1512 3315 10 38 v +1521 3299 42 4 v 1581 3327 a Fr(::)p Fs(B)1758 3297 y +Fn(\017)1839 3265 y Fr(:)1894 3277 y Fq(R)p 1493 3347 +304 4 v 1567 3409 10 38 v 1576 3393 42 4 v 1636 3421 +a Fs(B)1703 3391 y Fn(\017)1839 3359 y Fr(::)1949 3371 +y Fq(R)2139 3274 y Fs(\031)2189 3244 y Fn(\017)2186 3295 +y Fl(2)2172 3347 y Fm(:)2087 3421 y Fs(B)2154 3391 y +Fn(\017)p 2210 3409 10 38 v 2220 3393 42 4 v 1548 3441 +732 4 v 1889 3495 10 38 v 1898 3478 42 4 v 2321 3467 +a Fs(cut)523 3673 y FA(where)g(\(remember)f(we)j(re)o(gard)d(the)i +Fr(::)1712 3685 y Fq(R)1767 3673 y FA(-rule)f(as)i(proper)d(inference)g +(rule\))i(the)g(proof)e Fs(\031)3150 3643 y Fn(\017)3147 +3693 y Fl(1)3209 3673 y FA(has)i(to)523 3772 y(mo)o(v)o(e)f(inside)h +Fs(\031)995 3742 y Fn(\017)992 3793 y Fl(2)1054 3772 +y FA(just)h(lik)o(e)f(the)g(beha)n(viour)f(of)g(\(20\).)g(If)h +Fs(\031)k FA(ends)c(with)h(the)f(logical)g(cut)1496 3928 +y Fs(\031)1543 3940 y Fl(1)1527 3993 y Fm(:)p 1468 4094 +10 38 v 1478 4077 42 4 v 1564 4040 a Fq(\()1561 4114 +y Fs(C)p 1365 4134 346 4 v 1384 4295 10 38 v 1393 4279 +42 4 v 1480 4194 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g(\000)1479 +4242 y Fq(*)1476 4315 y Fs(B)1567 4307 y Fr(_)1648 4242 +y Fq(\()1645 4315 y Fs(C)1753 4146 y Fr(_)1808 4159 y +Fq(R)1858 4167 y Fc(2)2024 3928 y Fs(\031)2071 3940 y +Fl(2)2055 3993 y Fm(:)1980 4040 y Fq(*)1978 4114 y Fs(B)p +2086 4094 10 38 v 2096 4077 42 4 v 2285 3928 a(\031)2332 +3940 y Fl(3)2316 3993 y Fm(:)2242 4040 y Fq(\()2239 4114 +y Fs(C)p 2347 4094 10 38 v 2356 4077 42 4 v 1978 4134 +439 4 v 2027 4194 a Fq(\()g Fn(\000)g(\000)g(\000)g(\000)2026 +4242 y Fq(*)2024 4315 y Fs(B)2114 4307 y Fr(_)2196 4242 +y Fq(\()2193 4315 y Fs(C)p 2300 4295 10 38 v 2310 4279 +42 4 v 2458 4150 a Fr(_)2513 4163 y Fq(L)p 1365 4335 +1005 4 v 1842 4389 10 38 v 1852 4372 42 4 v 2411 4360 +a Fs(cut)3267 4401 y FA(\(21\))523 4596 y(which)20 b(can)g(reduce)f(to) +1771 4653 y Fs(\031)1818 4665 y Fl(1)1802 4718 y Fm(:)p +1743 4819 10 38 v 1753 4802 42 4 v 1838 4765 a Fq(\()1836 +4839 y Fs(C)2031 4653 y(\031)2078 4665 y Fl(3)2062 4718 +y Fm(:)1988 4765 y Fq(\()1985 4839 y Fs(C)p 2092 4819 +10 38 v 2102 4802 42 4 v 1725 4859 438 4 v 1918 4912 +10 38 v 1927 4896 42 4 v 3267 4924 a FA(\(22\))p eop +end +%%Page: 16 16 +TeXDict begin 16 15 bop 523 448 a FA(then,)20 b(the)g +Fm(\()p Fr(\000)p Fm(\))959 418 y Fn(\017)997 448 y FA(-translated)f(v) +o(ersion)g(of)h Fs(\031)1381 699 y(\031)1431 669 y Fn(\017)1428 +720 y Fl(1)1414 772 y Fm(:)1302 846 y Fr(:)p Fs(C)1422 +816 y Fn(\017)p 1479 834 10 38 v 1488 817 42 4 v 1274 +866 303 4 v 1292 927 10 38 v 1302 911 42 4 v 1362 939 +a Fr(::)p Fs(C)1537 909 y Fn(\017)1618 877 y Fr(:)1673 +889 y Fq(R)p 1194 959 463 4 v 1212 1021 10 38 v 1222 +1005 42 4 v 1282 1033 a Fs(B)1349 1003 y Fn(\017)1387 +1033 y Fr(_::)p Fs(C)1617 1003 y Fn(\017)1698 972 y Fr(_)1753 +985 y Fq(R)1803 993 y Fc(1)p 1134 1053 583 4 v 1134 1132 +a Fr(:)p Fm(\()p Fs(B)1288 1102 y Fn(\017)1327 1132 y +Fr(_::)p Fs(C)1557 1102 y Fn(\017)1596 1132 y Fm(\))p +1647 1120 10 38 v 1656 1103 42 4 v 1758 1065 a Fr(:)1813 +1077 y Fq(L)p 1106 1173 639 4 v 1124 1239 10 38 v 1134 +1223 42 4 v 1194 1251 a Fr(::)p Fm(\()p Fs(B)1403 1221 +y Fn(\017)1442 1251 y Fr(_)q(::)p Fs(C)1673 1221 y Fn(\017)1712 +1251 y Fm(\))1786 1184 y Fr(:)1841 1196 y Fq(R)p 1078 +1292 694 4 v 1078 1371 a Fr(:::)p Fm(\()p Fs(B)1342 1341 +y Fn(\017)1382 1371 y Fr(_::)p Fs(C)1612 1341 y Fn(\017)1651 +1371 y Fm(\))p 1702 1359 10 38 v 1712 1342 42 4 v 1813 +1304 a Fr(:)1868 1316 y Fq(L)2111 793 y Fs(\031)2161 +763 y Fn(\017)2158 813 y Fl(2)2144 866 y Fm(:)2059 939 +y Fs(B)2126 909 y Fn(\017)p 2183 927 10 38 v 2192 911 +42 4 v 2442 605 a Fs(\031)2492 575 y Fn(\017)2489 626 +y Fl(3)2475 678 y Fm(:)2390 752 y Fs(C)2455 722 y Fn(\017)p +2512 740 10 38 v 2522 723 42 4 v 2363 772 247 4 v 2381 +834 10 38 v 2391 817 42 4 v 2451 846 a Fr(:)p Fs(C)2571 +816 y Fn(\017)2651 784 y Fr(:)2706 796 y Fq(R)p 2335 +866 303 4 v 2335 939 a Fr(::)p Fs(C)2510 909 y Fn(\017)p +2568 927 10 38 v 2577 911 42 4 v 2679 877 a Fr(:)2734 +889 y Fq(L)p 2059 959 579 4 v 2117 1033 a Fs(B)2184 1003 +y Fn(\017)2222 1033 y Fr(_::)p Fs(C)2452 1003 y Fn(\017)p +2510 1021 10 38 v 2519 1005 42 4 v 2679 976 a Fr(_)2734 +989 y Fq(L)p 2057 1053 583 4 v 2075 1120 10 38 v 2085 +1103 42 4 v 2145 1132 a Fr(:)p Fm(\()p Fs(B)2299 1102 +y Fn(\017)2338 1132 y Fr(_::)p Fs(C)2568 1102 y Fn(\017)2607 +1132 y Fm(\))2681 1065 y Fr(:)2736 1077 y Fq(R)p 2029 +1173 639 4 v 2029 1251 a Fr(::)p Fm(\()p Fs(B)2238 1221 +y Fn(\017)2277 1251 y Fr(_)q(::)p Fs(C)2508 1221 y Fn(\017)2547 +1251 y Fm(\))p 2597 1239 10 38 v 2607 1223 42 4 v 2709 +1184 a Fr(:)2764 1196 y Fq(L)p 2001 1292 694 4 v 2020 +1359 10 38 v 2029 1342 42 4 v 2089 1371 a Fr(:::)p Fm(\()p +Fs(B)2353 1341 y Fn(\017)2393 1371 y Fr(_::)p Fs(C)2623 +1341 y Fn(\017)2662 1371 y Fm(\))2736 1304 y Fr(:)2791 +1316 y Fq(R)p 1078 1412 1617 4 v 1861 1466 10 38 v 1871 +1449 42 4 v 2736 1437 a Fs(cut)523 1648 y FA(reduces)f(to)1677 +1808 y Fs(\031)1727 1778 y Fn(\017)1724 1829 y Fl(1)1710 +1881 y Fm(:)1598 1955 y Fr(:)p Fs(C)1718 1925 y Fn(\017)p +1775 1943 10 38 v 1785 1926 42 4 v 2007 1714 a Fs(\031)2057 +1684 y Fn(\017)2054 1735 y Fl(3)2040 1787 y Fm(:)1956 +1861 y Fs(C)2021 1831 y Fn(\017)p 2078 1849 10 38 v 2087 +1832 42 4 v 1928 1881 247 4 v 1946 1943 10 38 v 1956 +1926 42 4 v 2016 1955 a Fr(:)p Fs(C)2136 1925 y Fn(\017)2216 +1893 y Fr(:)2271 1905 y Fq(R)p 1598 1975 577 4 v 1861 +2028 10 38 v 1871 2012 42 4 v 2216 2000 a Fs(cut)523 +2183 y FA(where)h(proof)f Fs(\031)998 2152 y Fn(\017)995 +2203 y Fl(3)1057 2183 y FA(has)i(to)f(mo)o(v)o(e)f(inside)h +Fs(\031)1746 2152 y Fn(\017)1743 2203 y Fl(1)1806 2183 +y FA(just)h(lik)o(e)f(in)h(the)f(proof)f(\(22\).)g(The)h(only)g(case)h +(we)f(still)523 2282 y(need)g(to)g(consider)f(is)i(when)f +Fs(\031)k FA(ends)c(in)g(a)h(commuting)d(cut)i(of)g(the)g(form)1630 +2415 y Fs(\031)1677 2427 y Fl(1)1661 2480 y Fm(:)p 1518 +2642 10 38 v 1527 2625 42 4 v 1613 2541 a Fq(\()-11 b +Fn(\000)g(\000)g(\000)g(\000)1613 2588 y Fq(*)1610 2662 +y Fs(B)1700 2654 y Fr(_)1782 2588 y Fq(\()1779 2662 y +Fs(C)2059 2415 y(\031)2106 2427 y Fl(2)2089 2480 y Fm(:)1931 +2541 y Fq(\()g Fn(\000)g(\000)g(\000)g(\000)1931 2588 +y Fq(*)1928 2662 y Fs(B)2018 2654 y Fr(_)2100 2588 y +Fq(\()2097 2662 y Fs(C)p 2204 2642 10 38 v 2214 2625 +42 4 v 1499 2681 775 4 v 1861 2735 10 38 v 1871 2718 +42 4 v 2315 2707 a(cut)3267 2747 y FA(\(23\))523 2917 +y(The)25 b(beha)n(viour)e(of)i(this)h(cut)g(is)g(determined)e(by)h(the) +g(outermost)f(colour)g(`)p Fs(\()p FA('.)h(This)h(beha)n(viour)523 +3017 y(can)21 b(be)h(simulated)f(by)g(the)h Fm(\()p Fr(\000)p +Fm(\))1464 2987 y Fn(\017)1502 3017 y FA(-translation,)e(pro)o(vided)f +(we)j(use)g(a)g(left-translation)e(for)h(the)g(cut)523 +3117 y(in)f(\(23\).)f(The)h(translated)g(proof)e(is)j(then)f(as)h +(follo)n(ws:)1454 3390 y Fs(\031)1504 3360 y Fn(\017)1501 +3411 y Fl(1)1487 3463 y Fm(:)1152 3542 y Fr(:::)p Fm(\()p +Fs(B)1416 3512 y Fn(\017)1455 3542 y Fr(_)q(::)p Fs(C)1686 +3512 y Fn(\017)1725 3542 y Fm(\))p 1775 3530 10 38 v +1785 3514 42 4 v 2230 3271 a Fs(\031)2280 3241 y Fn(\017)2277 +3291 y Fl(2)2263 3344 y Fm(:)1956 3423 y Fr(::)p Fm(\()p +Fs(B)2165 3393 y Fn(\017)2204 3423 y Fr(_::)p Fs(C)2434 +3393 y Fn(\017)2473 3423 y Fm(\))p 2524 3411 10 38 v +2534 3394 42 4 v 1928 3463 694 4 v 1946 3530 10 38 v +1956 3514 42 4 v 2016 3542 a Fr(:::)p Fm(\()p Fs(B)2280 +3512 y Fn(\017)2320 3542 y Fr(_::)p Fs(C)2550 3512 y +Fn(\017)2589 3542 y Fm(\))2663 3475 y Fr(:)2718 3487 +y Fq(R)p 1152 3583 1470 4 v 1861 3637 10 38 v 1871 3620 +42 4 v 2663 3608 a Fs(cut)523 3819 y FA(No)n(w)31 b Fs(\031)764 +3789 y Fn(\017)761 3840 y Fl(1)834 3819 y FA(will)h(freshly)e +(introduce)g(the)h(cut-formula)d Fr(:::)p Fm(\()p Fs(B)2423 +3789 y Fn(\017)2463 3819 y Fr(_::)p Fs(C)2693 3789 y +Fn(\017)2732 3819 y Fm(\))k FA(only)f(if)g Fs(\031)3105 +3831 y Fl(1)3174 3819 y FA(freshly)523 3919 y(introduces)20 +b(the)i(formula)f Fs(B)t Fr(_)p Fs(C)29 b FA(\(recall)22 +b(that)g(double-ne)o(gation)17 b(translations)22 b(need)f(to)h(preserv) +o(e)523 4018 y(the)g(structure)f(of)g(a)i(classical)f(proof\).)e +(Consequently)-5 b(,)19 b(the)j(translated)f(proof)f(simulates)i(e)o +(xactly)523 4118 y(the)e(beha)n(viour)e(of)i(\(23\).)648 +4218 y(What)32 b(this)g(e)o(xample)f(sho)n(ws)h(is)g(that)g(the)g +Fm(\()p Fr(\000)p Fm(\))2070 4187 y Fn(\017)2109 4218 +y FA(-translation)e(of)i Fs(\031)k FA(can)31 b(simulate)h(the)g(be-)523 +4317 y(ha)n(viour)19 b(where)g(the)i(cut-formula)c(is)k(annotated)e +(with)h(the)g(colours)1815 4442 y Fq(\()-11 b Fn(\000)g(\000)g(\000)g +(\000)1815 4489 y Fq(*)1812 4563 y Fs(B)1902 4555 y Fr(_)1984 +4489 y Fq(\()1981 4563 y Fs(C)2093 4555 y(:)523 4725 +y FA(Note)27 b(that)g(there)f(are)h(other)f(double)g(ne)o(gation)e +(translation)i(which)h(can)f(be)h(used)g(for)f(a)h(similar)523 +4825 y(simulation)g(of)h(this)g(particular)f(colour)n(-annotation.)d +(From)j(our)g(discussion)h(it)g(seems)h(reason-)523 4924 +y(able)e(to)g(e)o(xpect)f(that)g(one)h(can)f(\002nd)h(corresponding)c +(double-ne)o(gation)f(translations)27 b(for)f(e)n(v)o(ery)p +eop end +%%Page: 17 17 +TeXDict begin 17 16 bop 523 448 a FA(possible)20 b(colour)n +(-annotation.)15 b(Since)20 b(the)g(formulae)e Fs(B)25 +b FA(and)19 b Fs(C)27 b FA(can)19 b(be)h(compound)d(with)j(further)523 +548 y(colour)n(-annotations)j(inside,)i(we)i(need)e(some)h(\003e)o +(xibility)g(of)f(ho)n(w)h(to)g(double-ne)o(gate)d(translate)523 +669 y(formulae.)30 b(One)h(has)h(to)g(be)g(able)g(to)g(b)n(uild)f(into) +h(the)f(clause)h Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))2653 +639 y Fn(\017)2736 622 y Fp(def)2741 669 y Fm(=)49 b +Fr(::)p Fm(\()p Fs(B)3064 639 y Fn(\017)3103 669 y Fr(_)q(::)p +Fs(C)3334 639 y Fn(\017)3373 669 y Fm(\))523 769 y FA(that)26 +b(the)g(translations)f(of)g Fs(B)31 b FA(and)25 b Fs(C)32 +b FA(might)26 b(follo)n(w)f(a)h(completely)e(dif)n(ferent)g(double-ne)o +(gation)523 868 y(scheme.)k(Ho)n(w)g(to)h(do)f(this)h(ele)o(gantly)d +(is)k(not)e(kno)n(wn)f(to)h(us.)h(On)g(the)f(other)g(hand,)f(we)i +(cannot)523 968 y(e)o(xpect)23 b(complete)h(\223freedom\224)e(in)j(a)f +(double-ne)o(gation)c(translation)k(as)h(we)g(ha)n(v)o(e)e(to)i(mak)o +(e)f(sure,)523 1068 y(roughly)15 b(speaking,)g(that)i(dif)n(ferent)e +(double-ne)o(gation)d(translation)k(still)h(\002t)h(together)d(in)i +(the)g(trans-)523 1167 y(lated)j(proof.)e(This)i(means)g(we)h(ha)n(v)o +(e)e(to)h(mak)o(e)g(sure)g(that)g(double-ne)o(gation)15 +b(translations)20 b(respect)523 1267 y(the)j(identity-class)g +(constraint)g(from)f(the)h(colour)n(-annotations.)d(F)o(or)j(e)o +(xample)f(we)i(cannot)e(ha)n(v)o(e)523 1367 y(an)e(axiom)f(in)i(a)f +(translated)g(proof)f(where)g(the)h(double-ne)o(gation)c(translations)k +(disagree)f(as)i(in)1815 1550 y Fs(B)1882 1520 y Fn(\003)p +1938 1538 10 38 v 1948 1522 42 4 v 2008 1550 a Fs(B)2075 +1520 y Fn(\016)523 1735 y FA(This)c(point)g(about)f(\002tting)h +(double-ne)o(gation)12 b(translations)17 b(together)e(and)i(the)g +(identity-class)f(con-)523 1834 y(straint)22 b(of)g(colours)e(we)j(tak) +o(e)e(as)i(a)f(further)e(e)n(vidence)h(that)h(colours)f(and)g(double)f +(ne)o(gation)g(trans-)523 1934 y(lation)g(must)g(ha)n(v)o(e)g +(something)e(to)j(do)f(with)g(each)g(other)-5 b(.)648 +2034 y(From)22 b(the)i(observ)n(ations)e(made)g(abo)o(v)o(e)g(we)i +(conjecture)e(that)h(e)n(v)o(ery)f(colour)n(-annotation)e(de-)523 +2134 y(termining)31 b(a)i(single)f(normalform)e(of)i(a)h(classical)g +(proof)e(can)h(be)g(equally)g(determined)e(by)i(a)523 +2233 y(double)24 b(ne)o(gation)g(translation,)g(and)h(e)n(v)o(ery)g +(double-ne)o(gation)c(translation)k(determining)e(a)j(nor)n(-)523 +2333 y(malform)17 b(can)i(be)g(equally)e(determined)g(by)i(a)g(colour)n +(-annotation.)14 b(In)19 b(ef)n(fect)f(we)h(conjecture)e(that)523 +2433 y(double-ne)o(gation)f(translations)j(can)h(be)g(simulated)g(by)g +(colours)f(and)h(vice)g(v)o(ersa.)648 2533 y(While)h(establishing)f +(the)h(simulation)f(properties)g(is)h(a)h(\002rst)f(step)h(for)e +(understanding)e(the)j(re-)523 2632 y(lation)d(between)f(double-ne)o +(gation)c(translations)18 b(and)f(colour)n(-annotations,)d(we)k +(consider)f(this)i(as)523 2732 y(not)k(yet)h(gi)n(ving)f(the)g +(complete)g(\223picture\224.)f(F)o(or)h(this)i(consider)d(the)i +(collection)f(of)g(normalforms)523 2832 y(of)30 b(a)g(classical)g +(proof)e(determined)g(by)i(double-ne)o(gation)25 b(translations)k(and)g +(by)g(colour)g(anno-)523 2931 y(tations.)e(W)-7 b(e)28 +b(conjecture)d(that)i(both)f(are)h(the)g(\223same\224)g(collection,)e +(whereby)h(one)g(needs)g(a)i(\(yet)523 3031 y(unkno)n(wn\))c(v)o(ery)i +(cle)n(v)o(er)g(notion)g(of)h(\223sameness\224.)g(Clearly)-5 +b(,)26 b(there)h(are)g(more)f(double-ne)o(gation)523 +3131 y(translations)17 b(of)h(a)h(classical)f(proof)f(than)g(there)h +(are)g(colour)n(-annotations)c(\(there)k(are)f(only)h(\002nitely)523 +3230 y(man)o(y)k(colour)n(-annotations,)e(b)n(ut)j(there)g(are)h +(in\002nitely)f(man)o(y)f(double)g(ne)o(gation)f(translations)i(as)523 +3330 y(already)j(the)h(translations)g(of)g(atomic)f(formulae)g(as)i +Fr(::)p Fs(A)p FA(,)g Fr(::::)p Fs(A;)14 b(:)g(:)g(:)29 +b FA(indicate\).)d(F)o(or)h(us)h(it)523 3429 y(is,)19 +b(ho)n(we)n(v)o(er)m(,)d(clear)i(that)h(one)e(can)i(group)d(double-ne)o +(gation)e(translations)k(into)g(dif)n(ferent)f(classes,)523 +3529 y(where)j(each)f(class)j(corresponds)c(to)i(a)g(colour)n +(-annotation.)648 3629 y(Let)32 b(us)h(consider)f(what)h(is)g +(necessary)f(to)h(turn)f(these)h(conjectures)e(into)h(theorems.)g +(First)523 3729 y(we)e(ha)n(v)o(e)f(to)h(mak)o(e)g(precise)f(what)h(we) +g(mean)f(by)h Ft(all)g FA(double-ne)o(gation)25 b(translation.)k(As)h +(seen)523 3828 y(abo)o(v)o(e,)22 b(the)h(notion)f(of)h(double-ne)o +(gation)c(translation)k(has)g(to)h(be)f(a)h(generalised)e(v)o(ersion)g +(of)h(the)523 3928 y(traditional)g(notion\227lik)o(e)f(the)h(ones)h(gi) +n(v)o(en)e(by)h(Gentzen,)g(G)7 b(\250)-35 b(odel)23 b(and)g(K)m +(olmogoro)o(v\227because)523 4028 y(one)g(needs)g(to)g(tak)o(e)h(into)f +(account)f(the)i(dif)n(ferent)d(colour)n(-annotations)f(by)j(v)n +(arying)f(the)h(double-)523 4127 y(ne)o(gations)g(translation)h(when)g +(inducti)n(v)o(ely)e(descending)h(a)i(formula.)e(Further)m(,)h(one)g(w) +o(ould)g(ide-)523 4227 y(ally)g(lik)o(e)f(to)h(ha)n(v)o(e)f(a)h(rather) +f(general)f(notion)g(of)i(double-ne)o(gation)19 b(translation)j(so)i +(that)g(one)f(can)523 4327 y(meaningful)h(state)k(properties)d(for)h +(all)h Ft(possible)g FA(double-ne)o(gation)22 b(translation.)k(Ho)n(we) +n(v)o(er)m(,)e(we)523 4426 y(ha)n(v)o(e)f(been)f(unable)h(to)g +(formulate)f(such)h(a)g(general)g(notion.)f(Then)g(we)i(ha)n(v)o(e)e +(to)i(cate)o(gorise)e(ho)n(w)523 4526 y(the)k(double-ne)o(gation)c +(translations)k(beha)n(v)o(e)f(under)f(cut-elimination.)g(Finally)-5 +b(,)26 b(one)f(has)i(to)f(es-)523 4625 y(tablish)e(the)h +(\223simulation-property\224\227which)19 b(unfortunately)i(f)o(ails)k +(if)f(one)g(tak)o(es)h(a)f(v)o(ery)f(na)n(\250)-26 b(\021v)o(e)523 +4725 y(vie)n(w)17 b(on)g(sequent-proofs)d(and)i(cut-elimination.)f(One) +i(problem)e(is)j(that)f(double-ne)o(gation)c(trans-)523 +4825 y(lations)29 b(might)g(introduce)e(auxiliary)h(cuts.)h(Such)g +(auxiliary)f(cuts)h(only)g(occur)f(in)h(the)g(double-)523 +4924 y(ne)o(gated)22 b(proofs)g(and)i(are)f(thus)h(not)g(tak)o(en)f +(account)g(of)g(by)g(colour)n(-annotations.)d(Therefore)i(we)p +eop end +%%Page: 18 18 +TeXDict begin 18 17 bop 523 448 a FA(ha)n(v)o(e)25 b(to)h(mak)o(e)f +(sure)g(that)h(such)g(auxiliary)e(cuts)i(cannot)e(\223mess)i(up\224)f +(the)h(normalform)d(reached)523 548 y(by)28 b(eliminating)g(cuts)h(in)f +(the)h(double-ne)o(gated)24 b(proof.)j(This)i(can)f(be)h(relati)n(v)o +(ely)e(easily)i(sho)n(wn)523 648 y(pro)o(vided)23 b(the)i(auxiliary)f +(cut)h(occurs)g(as)h(the)f(lo)n(wermost)f(inference)g(in)h(a)h +(proof\227then)c(a)k(tech-)523 747 y(nique)h(introduced)f(in)j([19])e +(sho)n(ws)h(that)h(one)e(can)h(restrict)h(attention)e(to)i(only)e +(outermost)g(cuts)523 847 y(when)c(calculating)f(the)h(collection)g(of) +g(normalforms.)d(Ho)n(we)n(v)o(er)i(in)i(the)f(general)g(case)g(we)h +(ha)n(v)o(e)523 946 y(for)17 b(this)h(not)g(\223messing)f(up\224)g +(only)g(empirical)g(e)n(vidence)f(obtained)h(from)f(man)o(y)h +(calculations)g(\(the)523 1046 y(tools)22 b(with)g(which)f(we)h(do)g +(such)f(calculations)g(are)h(gi)n(v)o(en)f(in)h([17]\).)e(These)i +(calculations)f(indeed)523 1146 y(v)n(alidate)f(the)g(assumption)f +(that)h(the)g Fr(::)1703 1158 y Fq(R)1758 1146 y FA(-rule)p +1746 1309 10 38 v 1756 1293 42 4 v 1816 1321 a Fr(::)p +Fs(B)p 1728 1341 266 4 v 1802 1403 10 38 v 1811 1386 +42 4 v 1871 1415 a(B)2035 1353 y Fr(::)2145 1365 y Fq(R)523 +1612 y FA(can)27 b(be)h(re)o(garded)d(as)j(introducing)d(the)j(formula) +e Fs(B)t FA(.)i(T)-7 b(o)27 b(gi)n(v)o(e)g(a)h(proof)e(of)h(this)i(f)o +(act,)e(ho)n(we)n(v)o(er)m(,)523 1711 y(we)d(guess)g(one)f(needs)g(a)h +(\223full-blo)n(wn\224)d(conte)o(xt-lemma)g(using)i(Ho)n(we')-5 +b(s)24 b(method.)e(The)h(biggest)523 1811 y(problem)17 +b(we)j(see)g(in)f(establishing)f(a)i(one-to-one)c(correspondence)g +(between)i(the)h(collections)g(of)523 1911 y(normalforms)e(determined)i +(by)g(double-ne)o(gations)d(translations)j(and)h(by)f(colour)n +(-annotations)e(is)523 2010 y(to)j(\002nd)g(a)h(meaningful)d(equi)n(v)n +(alence)g(relation)h(on)h(double-ne)o(gation)c(translations.)648 +2115 y(Further)m(,)25 b(the)j(colour)n(-annotations)c(are)j(gi)n(v)o +(en)f(for)h(sequent-proofs,)d(for)j(which)g(it)h(is)g(well-)523 +2214 y(kno)n(wn)f(that)i(the)o(y)f(mak)o(e)h(\223inessential\224)f(dif) +n(ferences)f(that)i(w)o(ould)g(not)f(materialise)h(if)g(we)g(had)523 +2314 y(a)23 b(natural-deduction)18 b(formulation)i(or)j(proof-net)c +(formulation.)h(Ho)n(we)n(v)o(er)m(,)g(such)j(formulations)523 +2414 y(ha)n(v)o(e)28 b(not)g(yet)h(been)e(de)n(v)o(eloped)f(f)o(ar)j +(enough)d(to)j(be)f(useful)g(in)h(getting)e(rid)i(of)f(the)g +(inessential)523 2513 y(dif)n(ferences)f(between)h(classical)h +(sequent-proofs.)d(This)i(problem)f(also)i(sho)n(ws)g(up)f(with)h +(intu-)523 2613 y(itionistic)21 b(sequent-proofs)c(where)j +(cut-elimination,)f(despite)h(our)g(w)o(orking)f(hypothesis)g(stating) +523 2713 y(the)j(contrary)-5 b(,)19 b(is)k Ft(not)g FA(Church-Rosser)-5 +b(.)20 b(Ho)n(we)n(v)o(er)m(,)g(for)h(fragments)f(of)i(intuitionistic)f +(logic)g(there)523 2812 y(are)28 b(already)f(good)g(tools\227for)g(e)o +(xample)g(natural)h(deduction)e(and)i(contraction-free)d(sequent-)523 +2912 y(calculi\227which)g(pro)o(vide)f(a)j(canonical)d(notion)h(of)h +(what)g(an)g(intuitionistic)g(proof)e(is.)j(Because)523 +3011 y(of)k(all)g(these)h(dif)n(\002culties,)e(we)h(ha)n(v)o(e,)g(at)g +(the)g(moment,)f(to)h(content)f(ourselv)o(es)g(with)h(e)o(xample)523 +3111 y(calculations)19 b(that,)h(ho)n(we)n(v)o(er)m(,)e(all)j(seem)f +(to)h(v)n(alidate)e(the)h(conjecture.)523 3401 y Fu(5)99 +b(Conclusion)523 3624 y FA(Although)14 b(there)h(is)i(plenty)e(of)g +(literature)g(on)g(classical)i(logic)e(and)g(double-ne)o(gation)c +(translations,)523 3724 y(there)34 b(is)i(surprising)d(little)i +(literature)f(that)h(studies)g(the)f(relation)g(between)g(double-ne)o +(gation)523 3824 y(translations)27 b(and)h(the)g(process)f(of)h +(normalising)e(a)i(classical)h(proof.)d(W)-7 b(e)29 b(\002nd)f(notable) +f(e)o(xcep-)523 3923 y(tions)20 b(are)g([1,)12 b(5,)h(6,)g(11,)g(13],) +19 b(which)g(ho)n(we)n(v)o(er)f(do)h(not)h(gi)n(v)o(e)f(much)f(insight) +i(about)f(this)h(relation)f(or)523 4023 y(consider)k(only)g(special)h +(cases.)h(F)o(or)f(e)o(xample,)e(the)o(y)h(relate)h(one)g(kind)f(of)h +(colour)n(-annotation)c(to)523 4123 y(one)f(double-ne)o(gation)c +(translation,)j(or)h(consider)f(only)h(speci\002c)h(double-ne)o(gation) +14 b(translations.)648 4227 y(If)19 b(our)h(conjecture)e(turns)h(out)h +(to)g(be)g(true,)f(then)h(one)f(has)i(a)f(v)o(ery)f(simple)h +(characterisation)e(of)523 4327 y(all)24 b(double-ne)o(gation)18 +b(translations)k(in)i(terms)f(of)f(colours.)g(This)h(is)h(desirable,)e +(because)g(we)i(\002nd)523 4426 y(it)19 b(is)f(rather)f(mysterious)g +(ho)n(w)g(double-ne)o(gation)c(translation)k(can)h(turn)f(a)h +(classical)h(proof)d(into)i(an)523 4526 y(intuitionistic)f(proof,)f +(whose)h(computational)f(interpretation)f(as)k(e)o(xplained)c(in)j(the) +g(introduction)523 4625 y(is)32 b(by)e(the)h(Curry-Ho)n(w)o(ard)d +(correspondence)g(well-understood,)f(while)k(it)h(is)f(not)g +(understood)523 4725 y(at)c(all)h(for)e(the)h(classical)g(proof)e(we)i +(started)g(with)g(\(see)g([2,)13 b(19])26 b(for)g(tw)o(o)h(interesting) +f(e)o(xamples)523 4825 y(e)o(xtracting)e(computational)g(meaning)h +(from)g(a)h(classical)h(proof)d(that)j(because)e(of)h(the)g(the)g(non-) +523 4924 y(determinism)19 b(cannot)g(be)h(e)o(xtracted)f(by)h(double)e +(ne)o(gation)g(translations\).)p eop end +%%Page: 19 19 +TeXDict begin 19 18 bop 648 448 a FA(The)24 b(main)h(point)g(we)g(tak)o +(e)g(a)o(w)o(ay)g(from)f(the)h(conjecture)f(is)i(that)f(double-ne)o +(gation)c(transla-)523 548 y(tions)j(are)f(not)g(enough)e(to)j +(characterise)e(the)i(full)f(computational)e(content)i(of)g(classical)h +(proofs,)523 648 y(where)31 b(by)g Ft(full)h FA(computational)e +(content)g(we)j(mean)e(the)g(collection)g(of)g(normalforms)f(reach-)523 +747 y(able)h(by)h(the)f(cut-elimination)e(procedure)g(of)j(Urban)e(and) +h(Bierman.)g(This)h(is)g(because)f(there)523 847 y(are)h(some)f +(normalforms)e(that)j(can)g(be)f(reached)g(by)g(this)h(cut-elimination) +e(procedure)f(which)523 946 y(cannot)i(be)h(reached)e(by)i(an)o(y)f +(colour)n(-annotation,)c(and)32 b(by)f(the)h(conjecture)e(also)i(not)g +(by)f(an)o(y)523 1046 y(double-ne)o(gation)15 b(translation.)k(These)h +(\223unreachable\224)d(normalforms)g(embody)h(a)i(form)f(of)h(non-)523 +1146 y(determinism)g(present)i(in)f(classical)i(logic,)e(b)n(ut)h(not)f +(present)g(in)h(intuitionistic)g(logic.)f(Therefore)523 +1245 y(the)h(conclusion)d(we)j(dra)o(w)f(from)g(this)h(is)g(that)g +(intuitionistic)f(logic)g(and)g(double-ne)o(gation)c(trans-)523 +1345 y(lations)k(can)f(only)g(gi)n(v)o(e)f(some)i(hints)f(for)g +(understanding)e(the)j Ft(full)g FA(computational)d(meaning)h(of)h(a) +523 1445 y(classical)h(proof.)648 1544 y(Completely)15 +b(untouched)e(by)j(our)f(treatment)g(here)h(is)h(the)f(correspondence)c +(between)j(double-)523 1644 y(ne)o(gation)29 b(translations)i(and)f +(linear)h(logic.)g(The)g(colour)e(annotation)h(also)h(connects)g(to)g +(linear)523 1743 y(logic,)25 b(where)g(cut-elimination)e(formulated)h +(with)h(proof-nets)f(is)i(also)g(Church-Rosser)m(,)e(to)h(the)523 +1843 y(cut-elimination)18 b(procedure)g(with)i(colours)f(\(see)i([6,)13 +b(12]\).)523 2104 y Fu(Refer)n(ences)560 2290 y Fx(1.)42 +b(F)-6 b(.)14 b(Barbanera)j(and)f(S.)e(Berardi.)19 b(A)c(Symmetric)g +(Lambda)h(Calculus)g(for)f(\223Classical\224)h(Program)f(Extrac-)658 +2381 y(tion.)41 b(In)24 b Fv(Theor)m(etical)g(Aspects)f(of)g(Computer)i +(Softwar)m(e)p Fx(,)e(v)o(olume)h(789)g(of)g Fv(LNCS)p +Fx(,)e(pages)j(495\226515.)658 2472 y(Springer)19 b(V)-8 +b(erlag,)18 b(1994.)560 2562 y(2.)42 b(F)-6 b(.)28 b(Barbanera,)h(S.)f +(Berardi,)g(and)i(M.)e(Schi)n(v)n(alocchi.)60 b(\223Classical\224)29 +b(Programming-with-Proofs)g(in)658 2654 y Fj(\025)703 +2622 y Fi(sy)r(m)824 2654 y Fx(:)20 b(An)i(Analysis)f(of)g +(Non-Con\003uence.)36 b(In)21 b Fv(Theor)m(etical)h(Aspects)f(of)g +(Computer)h(Softwar)m(e)p Fx(,)f(v)o(ol-)658 2745 y(ume)e(1281)h(of)f +Fv(LNCS)p Fx(,)f(pages)i(365\226390.)h(Springer)e(V)-8 +b(erlag,)18 b(1997.)560 2835 y(3.)42 b(G.)21 b(Bellin,)f(M.)h(Hyland,)g +(E.)g(Robinson,)h(and)g(C.)e(Urban.)35 b(Proof)21 b(Theory)g(of)h +(Classical)e(Propositional)658 2927 y(Calculus.)27 b(\(Accepted)20 +b(for)f(publication)g(in)g(Theoretical)g(Computer)h(Science\),)e(2005.) +560 3017 y(4.)42 b(S.)18 b(R.)h(Buss.)29 b(An)20 b(Introduction)g(to)g +(Proof-Theory)-5 b(.)29 b(In)20 b Fv(Handbook)h(of)f(Pr)m(oof)f(Theory) +p Fx(,)g(v)o(olume)h(137)h(of)658 3108 y Fv(Studies)e(in)g(Lo)o(gic)g +(and)h(the)f(F)-8 b(oundations)21 b(of)e(Mathematics)p +Fx(,)g(pages)h(1\22678.)g(North-Holland,)f(1998.)560 +3198 y(5.)42 b(P)-8 b(.-L.)22 b(Curien)i(and)h(H.)e(Herbelin.)43 +b(The)24 b(Duality)g(of)g(Computation.)44 b(In)24 b Fv(Confer)m(ence)i +(on)e(Functional)658 3290 y(Pr)m(o)o(gr)o(amming)p Fx(,)19 +b(pages)h(233\226243.)h(A)m(CM)e(Press,)f(2000.)560 3380 +y(6.)42 b(V)-10 b(.)15 b(Danos,)g(J.-B.)f(Joinet,)h(and)g(H.)g +(Schellinx.)j(A)d(Ne)n(w)f(Deconstructi)n(v)o(e)j(Logic:)d(Linear)h +(Logic.)j Fv(J)n(ournal)658 3471 y(of)h(Symbolic)g(Lo)o(gic)p +Fx(,)g(62\(3\):755\226807,)j(1997.)560 3561 y(7.)42 b(A.)27 +b(G.)h(Dragalin.)56 b Fv(Mathematical)29 b(Intuitionism:)f(Intr)m +(oduction)h(to)f(Pr)m(oof)g(Theory)p Fx(,)g(v)o(olume)h(67)g(of)658 +3653 y Fv(T)l(r)o(anslations)19 b(of)g(Mathematical)h(Mono)o(gr)o(aphs) +p Fx(.)29 b(American)20 b(Mathematical)f(Society)-5 b(,)19 +b(1988.)560 3743 y(8.)42 b(R.)21 b(Dyckhof)n(f)i(and)g(L.)e(Pinto.)37 +b(Permutability)21 b(of)h(Proofs)g(in)g(Intuitionistic)g(Sequent)g +(Calculi.)37 b Fv(Theo-)658 3834 y(r)m(etical)19 b(Computer)h(Scince)p +Fx(,)f(212\(1-2\):141\226155,)j(1999.)560 3924 y(9.)42 +b(J.)24 b(Gallier)l(.)45 b(Constructi)n(v)o(e)25 b(Logics.)g(Part)f(I:) +g(A)g(Tutorial)g(on)h(Proof)g(Systems)g(and)g(Typed)g +Fj(\025)p Fx(-calculi.)658 4016 y Fv(Theor)m(etical)19 +b(Computer)h(Science)p Fx(,)g(110\(2\):249\226239,)h(1993.)523 +4106 y(10.)42 b(J.-Y)-10 b(.)22 b(Girard,)f(Y)-10 b(.)22 +b(Lafont,)g(and)h(P)-8 b(.)21 b(T)-6 b(aylor)l(.)38 b +Fv(Pr)m(oofs)22 b(and)h(Types)p Fx(,)f(v)o(olume)h(7)f(of)g +Fv(Cambridg)o(e)i(T)l(r)o(acts)e(in)658 4197 y(Theor)m(etical)d +(Computer)h(Science)p Fx(.)28 b(Cambridge)19 b(Uni)n(v)o(ersity)h +(Press,)e(1989.)523 4287 y(11.)42 b(T)-6 b(.)26 b(Grif)n(\002n.)49 +b(A)26 b(Formulae-as-Types)h(Notion)f(of)g(Control.)51 +b(In)26 b Fv(Principles)g(of)g(Pr)m(o)o(gr)o(amming)h(Lan-)658 +4379 y(gua)o(g)o(es)p Fx(,)20 b(pages)g(47\22658.)g(A)m(CM)f(Press,)f +(1990.)523 4469 y(12.)42 b(J.-B.)19 b(Joinet,)i(H.)f(Schellinx,)g(and)h +(L.)f(T)-6 b(ortora)20 b(de)h(F)o(alco.)31 b(SN)20 b(and)h(CR)f(for)h +(Free-Style)e(LK)3111 4437 y Fi(tq)3169 4469 y Fx(:)h(Linear)658 +4560 y(Decorations)k(and)g(Simulation)f(of)g(Normalization.)41 +b Fv(J)n(ournal)24 b(of)f(Symbolic)h(Lo)o(gic)p Fx(,)f +(67\(1\):162\226196,)658 4652 y(2002.)523 4742 y(13.)42 +b(J.)25 b(Laird.)48 b(A)25 b(Deconstruction)i(of)e(Non-Deterministic)h +(Cut)f(Elimination.)48 b(In)26 b Fv(Pr)m(oceedings)g(of)g(the)658 +4833 y(5th)e(International)g(Confer)m(ence)h(on)f(T)-6 +b(yped)25 b(Lambda)f(Calculi)f(and)h(Applications)p Fx(,)g(v)o(olume)g +(2044)h(of)658 4924 y Fv(LNCS)p Fx(,)18 b(pages)i(268\226282.)h +(Springer)e(V)-8 b(erlag,)18 b(2001.)p eop end +%%Page: 20 20 +TeXDict begin 20 19 bop 523 448 a Fx(14.)42 b(M.)23 b(P)o(arigot.)42 +b Fj(\025\026)p Fx(-calculus:)23 b(An)h(Algorithmic)g(Interpretation)g +(of)f(Classical)g(Logic.)42 b(In)24 b Fv(Lo)o(gic)f(Pr)m(o-)658 +540 y(gr)o(amming)16 b(and)f(A)o(utomated)g(Deduction)p +Fx(,)h(v)o(olume)f(624)h(of)f Fv(LNCS)p Fx(,)f(pages)h(190\226201.)i +(Springer)e(V)-8 b(erlag,)658 631 y(1992.)523 722 y(15.)42 +b(G.)20 b(Pottinger)l(.)31 b(Normalisation)21 b(as)f(Homomorphic)i +(Image)f(of)g(Cut-Elimination.)31 b Fv(Annals)21 b(of)f(Mathe-)658 +814 y(matical)f(Lo)o(gic)p Fx(,)g(12:323\226357,)i(1977.)523 +905 y(16.)42 b(H.)16 b(Schellinx.)23 b Fv(The)17 b(Noble)g(Art)g(of)g +(Linear)g(Decor)o(ating)p Fx(.)23 b(PhD)17 b(thesis,)g(Institute)g(for) +f(Logic,)h(Language)658 996 y(and)j(Computation,)f(Uni)n(v)o(ersity)g +(of)g(Amsterdam,)g(1994.)28 b(ILLC)18 b(dissertation)h(series.)523 +1088 y(17.)42 b(C.)18 b(Urban.)27 b(http://www4.in.tum.de/)1677 +1071 y Fa(\030)1726 1088 y Fx(urbanc/cut/cutapplet.html.)523 +1179 y(18.)42 b(C.)22 b(Urban.)40 b(Re)n(visiting)23 +b(Zuck)o(er')l(s)g(Work)g(on)g(the)g(Correspondence)i(Between)e +(Cut-elimination)g(and)658 1270 y(Normalisation.)k(\(T)-6 +b(o)19 b(apear)g(in)g(Adv)n(ances)h(in)f(Natural)g(Deduction\).)523 +1362 y(19.)42 b(C.)24 b(Urban.)45 b Fv(Classical)24 b(Lo)o(gic)h(and)g +(Computation)p Fx(.)46 b(PhD)24 b(thesis,)g(Cambridge)i(Uni)n(v)o +(ersity)-5 b(,)24 b(October)658 1453 y(2000.)523 1544 +y(20.)42 b(C.)18 b(Urban.)27 b(Strong)19 b(Normalisation)g(for)f(a)h +(Gentzen-lik)o(e)g(Cut-Elimination)g(Procedure.)27 b(In)18 +b Fv(Pr)m(oceed-)658 1636 y(ings)f(of)f(the)g(5th)h(International)g +(Confer)m(ence)h(on)f(T)-6 b(yped)17 b(Lambda)g(Calculi)f(and)i +(Applications)p Fx(,)e(v)o(olume)658 1727 y(2044)k(of)f +Fv(LNCS)p Fx(,)f(pages)i(415\226429.)h(Springer)e(V)-8 +b(erlag,)18 b(2001.)523 1818 y(21.)42 b(C.)20 b(Urban)h(and)g(G.)f(M.)g +(Bierman.)32 b(Strong)20 b(Normalisation)h(of)g(Cut-Elimination)f(in)g +(Classical)g(Logic.)658 1910 y Fv(Fundamenta)g(Informaticae)p +Fx(,)g(45\(1\2262\):123\226155,)i(2001.)523 2001 y(22.)42 +b(S.)30 b(v)n(an)i(Bak)o(el,)e(S.)g(Lengrand,)i(and)g(P)-8 +b(.)29 b(Lescanne.)66 b(The)31 b(Language)h(X:)f(Circuits,)f +(Computations)658 2092 y(and)25 b(Classical)g(Logic.)45 +b(In)25 b Fv(Pr)m(oc.)f(of)h(9th)g(Italian)g(Confer)m(ence)h(on)f +(Theor)m(etical)g(Computer)h(Science)658 2183 y(\(ICTCS\))p +Fx(,)18 b(v)o(olume)h(3701)h(of)f Fv(LNCS)p Fx(,)f(pages)i(81\22696,)g +(2005.)523 2275 y(23.)42 b(J.)27 b(Zuck)o(er)l(.)56 b(The)28 +b(Correspondence)j(Between)d(Cut-Elimination)f(and)i(Normalisation.)56 +b Fv(Annals)28 b(of)658 2366 y(Mathematical)20 b(Lo)o(gic)p +Fx(,)f(7:1\226112,)h(1974.)p eop end +%%Trailer + +userdict /end-hook known{end-hook}if +%%EOF