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% File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30
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%
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% @@psencodingfile@{
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% author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry,
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% W. Schmidt, P. Lehman",
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% version = "2.0",
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% date = "30 October 2002",
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% filename = "8r.enc",
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% email = "tex-fonts@@tug.org",
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% docstring = "This is the encoding vector for Type1 and TrueType
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% fonts to be used with TeX. This file is part of the
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% PSNFSS bundle, version 9"
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% @}
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%
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% The idea is to have all the characters normally included in Type 1 fonts
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% available for typesetting. This is effectively the characters in Adobe
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% Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol,
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% MacRoman, and some extra characters from Lucida.
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%
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% Character code assignments were made as follows:
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%
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% (1) the Windows ANSI characters are almost all in their Windows ANSI
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% positions, because some Windows users cannot easily reencode the
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% fonts, and it makes no difference on other systems. The only Windows
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% ANSI characters not available are those that make no sense for
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% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
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% (173). quotesingle and grave are moved just because it's such an
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% irritation not having them in TeX positions.
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%
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% (2) Remaining characters are assigned arbitrarily to the lower part
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% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
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%
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% (3) Y&Y Lucida Bright includes some extra text characters; in the
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% hopes that other PostScript fonts, perhaps created for public
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% consumption, will include them, they are included starting at 0x12.
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% These are /dotlessj /ff /ffi /ffl.
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%
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% (4) hyphen appears twice for compatibility with both ASCII and Windows.
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%
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% (5) /Euro was assigned to 128, as in Windows ANSI
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%
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% (6) Missing characters from MacRoman encoding incorporated as follows:
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%
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% PostScript MacRoman TeXBase1
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% -------------- -------------- --------------
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% /notequal 173 0x16
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% /infinity 176 0x17
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% /lessequal 178 0x18
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% /greaterequal 179 0x19
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% /partialdiff 182 0x1A
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% /summation 183 0x1B
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% /product 184 0x1C
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% /pi 185 0x1D
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% /integral 186 0x81
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% /Omega 189 0x8D
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% /radical 195 0x8E
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% /approxequal 197 0x8F
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% /Delta 198 0x9D
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% /lozenge 215 0x9E
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%
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/TeXBase1Encoding [
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% 0x00
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/.notdef /dotaccent /fi /fl
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/fraction /hungarumlaut /Lslash /lslash
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/ogonek /ring /.notdef /breve
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/minus /.notdef /Zcaron /zcaron
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% 0x10
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/caron /dotlessi /dotlessj /ff
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/ffi /ffl /notequal /infinity
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/lessequal /greaterequal /partialdiff /summation
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/product /pi /grave /quotesingle
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% 0x20
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/space /exclam /quotedbl /numbersign
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/dollar /percent /ampersand /quoteright
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/parenleft /parenright /asterisk /plus
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/comma /hyphen /period /slash
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% 0x30
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/zero /one /two /three
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/four /five /six /seven
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/eight /nine /colon /semicolon
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/less /equal /greater /question
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% 0x40
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/at /A /B /C
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/D /E /F /G
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/H /I /J /K
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/L /M /N /O
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% 0x50
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/P /Q /R /S
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/T /U /V /W
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/X /Y /Z /bracketleft
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/backslash /bracketright /asciicircum /underscore
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% 0x60
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/quoteleft /a /b /c
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/d /e /f /g
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155 |
/h /i /j /k
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/l /m /n /o
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% 0x70
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/p /q /r /s
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159 |
/t /u /v /w
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160 |
/x /y /z /braceleft
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161 |
/bar /braceright /asciitilde /.notdef
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162 |
% 0x80
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163 |
/Euro /integral /quotesinglbase /florin
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164 |
/quotedblbase /ellipsis /dagger /daggerdbl
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165 |
/circumflex /perthousand /Scaron /guilsinglleft
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166 |
/OE /Omega /radical /approxequal
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167 |
% 0x90
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168 |
/.notdef /.notdef /.notdef /quotedblleft
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169 |
/quotedblright /bullet /endash /emdash
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170 |
/tilde /trademark /scaron /guilsinglright
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171 |
/oe /Delta /lozenge /Ydieresis
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172 |
% 0xA0
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173 |
/.notdef /exclamdown /cent /sterling
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174 |
/currency /yen /brokenbar /section
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175 |
/dieresis /copyright /ordfeminine /guillemotleft
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176 |
/logicalnot /hyphen /registered /macron
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177 |
% 0xD0
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178 |
/degree /plusminus /twosuperior /threesuperior
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179 |
/acute /mu /paragraph /periodcentered
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180 |
/cedilla /onesuperior /ordmasculine /guillemotright
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181 |
/onequarter /onehalf /threequarters /questiondown
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182 |
% 0xC0
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183 |
/Agrave /Aacute /Acircumflex /Atilde
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184 |
/Adieresis /Aring /AE /Ccedilla
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185 |
/Egrave /Eacute /Ecircumflex /Edieresis
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186 |
/Igrave /Iacute /Icircumflex /Idieresis
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187 |
% 0xD0
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188 |
/Eth /Ntilde /Ograve /Oacute
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189 |
/Ocircumflex /Otilde /Odieresis /multiply
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190 |
/Oslash /Ugrave /Uacute /Ucircumflex
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191 |
/Udieresis /Yacute /Thorn /germandbls
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% 0xE0
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/agrave /aacute /acircumflex /atilde
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194 |
/adieresis /aring /ae /ccedilla
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195 |
/egrave /eacute /ecircumflex /edieresis
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196 |
/igrave /iacute /icircumflex /idieresis
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197 |
% 0xF0
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198 |
/eth /ntilde /ograve /oacute
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199 |
/ocircumflex /otilde /odieresis /divide
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200 |
/oslash /ugrave /uacute /ucircumflex
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201 |
/udieresis /yacute /thorn /ydieresis
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] def
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index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll
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exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0
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ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{
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pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get
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div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type
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/nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end
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definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup
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sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll
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mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[
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exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if}
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forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def
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/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
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/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
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/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
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/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
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X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
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/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
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/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
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{userdict/md get type/dicttype eq{userdict begin md length 10 add md
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maxlength ge{/md md dup length 20 add dict copy def}if end md begin
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235 |
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
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236 |
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
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itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
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transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
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239 |
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
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240 |
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
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241 |
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
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242 |
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
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get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
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yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
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neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
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246 |
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
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90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
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neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
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1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
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2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
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-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
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TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
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Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
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}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
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save N userdict maxlength dict begin/magscale true def normalscale
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currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
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/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
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psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
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psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
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TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def
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@MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll
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newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto
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closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N
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TeXDict begin 1 0 bop 705 448 a FB(Classical)30 b(Logic)f(is)h(better)g
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861 |
(than)h(Intuitionistic)f(Logic:)681 598 y(A)g(Conjectur)n(e)h(about)f
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862 |
(Double-Negation)h(T)-9 b(ranslations)1417 886 y FA(Christian)21
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863 |
b(Urban)e(and)h(Diana)g(Ratiu)1410 1060 y Fz(f)p Fy(urban,ratiu)p
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864 |
Fz(g)p Fy(@math.lmu.de)1642 1152 y Fx(Uni)n(v)o(ersity)f(of)g(Munich)
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865 |
759 1462 y Fw(Abstract.)41 b Fx(It)20 b(is)h(well-kno)n(wn)g(that)f(in)
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866 |
h(terms)f(of)g(consistenc)o(y)i(classical)f(logic)f(and)i(intu-)759
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867 |
1553 y(itionistic)f(logic)h(ha)o(v)o(e)g(equal)g(strength:)g(e)n(v)o
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868 |
(ery)h(intuitionistic)e(proof)h(is)g(a)f(classical)h(proof)759
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869 |
1645 y(and)29 b(e)n(v)o(ery)f(classical)g(proof)h(can)f(be)g(embedded)i
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870 |
(into)e(intuitionistic)f(logic)h(via)g(double)759 1736
|
|
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871 |
y(ne)o(gation)21 b(translations.)f(It)f(is)h(also)g(well-kno)n(wn)h
|
|
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872 |
(that)f(intuitionistic)g(proofs)h(contain)f(wit-)759
|
|
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873 |
1827 y(nesses)33 b(for)f(e)o(xistential)g(statements,)g(which)h(is)f
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874 |
(not)g(al)o(w)o(ays)h(the)f(case)h(with)f(classical)759
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875 |
1919 y(proofs.)f(Ho)n(we)n(v)o(er)m(,)g(here)h(we)e(study)i(classical)e
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876 |
(and)i(intuitionistic)e(logic)h(as)f(reduction)759 2010
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877 |
y(systems.)24 b(From)g(this)f(perspecti)n(v)o(e,)i(we)f(conjecture)h
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878 |
(that)e(classical)h(logic)g(is)g(more)g(po)n(w-)759 2101
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879 |
y(erful)g(than)h(intuitionistic)f(logic.)g(The)g(conjecture)i(links)e
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880 |
(double-ne)o(gation)i(translations)759 2193 y(to)21 b(the)g(colour)o
|
|
|
881 |
(-protocol)i(introduced)f(by)g(Danos)g(et)e(al.)h(for)g
|
|
|
882 |
(cut-elimination)h(in)f(classical)759 2284 y(logic.)27
|
|
|
883 |
b(If)g(the)g(conjecture)i(turns)e(out)h(to)f(be)g(true,)g(then)h(we)f
|
|
|
884 |
(can)h(conclude)g(that)f Fv(not)h(all)759 2375 y Fx(cut-reductions)20
|
|
|
885 |
b(can)g(be)f(simulated)g(by)h(double-ne)o(gation)g(translations.)523
|
|
|
886 |
2644 y Fu(1)99 b(Intr)n(oduction)523 2832 y FA(Since)25
|
|
|
887 |
b(the)f(w)o(orks)g(on)h(double-ne)o(gation)20 b(translations)k(by)g
|
|
|
888 |
(Gentzen,)f(G)7 b(\250)-35 b(odel)24 b(and)g(K)m(olmogoro)o(v)-5
|
|
|
889 |
b(,)523 2932 y(one)26 b(kno)n(ws)g(that)g(classical)h(logic)f(and)g
|
|
|
890 |
(intuitionistic)g(logic)g(ha)n(v)o(e)g(equal)f(strength)h(in)g(terms)h
|
|
|
891 |
(of)523 3031 y(consistenc)o(y:)33 b(intuitionistic)h(sequent-proofs)e
|
|
|
892 |
(can)i(be)h(seen)g(as)g(classical)g(proofs)f(where)f(the)523
|
|
|
893 |
3131 y(right-hand)27 b(side)i(of)g(the)g(sequents)g(is)h(restricted)e
|
|
|
894 |
(to)i(maximal)e(one)g(formula)g(and)g(e)n(v)o(ery)g(clas-)523
|
|
|
895 |
3231 y(sical)19 b(sequent-proof)c(can)k(be)f(embedded)e(into)j
|
|
|
896 |
(intuitionistic)e(logic)h(via)h(double-ne)o(gation)14
|
|
|
897 |
b(trans-)523 3330 y(lations.)27 b(As)h(a)f(result,)g(consistenc)o(y)f
|
|
|
898 |
(of)h(one)f(logic)h(implies)g(consistenc)o(y)f(of)h(the)g(other)-5
|
|
|
899 |
b(.)26 b(In)h(this)523 3430 y(paper)m(,)i(ho)n(we)n(v)o(er)m(,)f(we)j
|
|
|
900 |
(focus)f(on)h(the)f(correspondence)e(of)i(intuitionistic)g(and)g
|
|
|
901 |
(classical)i(logic)523 3530 y(with)20 b(respect)g(to)h(term-re)n
|
|
|
902 |
(writing,)c(proof-normalisation)f(and)k(cut-elimination.)648
|
|
|
903 |
3629 y(According)c(to)i(the)g(Curry-Ho)n(w)o(ard)d(correspondence,)f
|
|
|
904 |
(the)k(simply-typed)e(lambda-calculus)523 3729 y(can)28
|
|
|
905 |
b(be)f(vie)n(wed)g(as)h(a)h(term-assignment)c(for)i(intuitionistic)g
|
|
|
906 |
(proofs)g(formalised)f(in)i(Gentzen')-5 b(s)523 3828
|
|
|
907 |
y(natural)24 b(deduction)f(calculus)i(NJ.)h(T)-6 b(erm-re)n(writing)22
|
|
|
908 |
b(in)k(the)f(simply-typed)e(lambda-calculus)g(is)523
|
|
|
909 |
3928 y(a)e(form)f(of)h(computation)e(that)i(con)m(v)o(erts)e(a)i(term)g
|
|
|
910 |
(to)g(its)h(simplest)g(form,)d(analogous)g(to)i(symbolic)523
|
|
|
911 |
4028 y(e)n(v)n(aluation.)29 b(On)h(the)h(other)e(hand,)g(normalisation)
|
|
|
912 |
g(is)j(a)f(method)e(for)h(eliminating)f(certain)h(re-)523
|
|
|
913 |
4127 y(dundancies)23 b(in)j(proofs.)e(Applied)g(iterati)n(v)o(ely)-5
|
|
|
914 |
b(,)24 b(it)i(transforms)d(a)j(proof)e(to)h(one)g(in)h(normalform.)523
|
|
|
915 |
4227 y(Using)19 b(the)f(Curry-Ho)n(w)o(ard)e(correspondence)f(we)k(see)
|
|
|
916 |
g(that)g(the)g(tw)o(o)f(notions)g(coincide)f(and)h(con-)523
|
|
|
917 |
4327 y(sequently)h(we)i(can)f(talk)g(of)g(a)g(computational)e
|
|
|
918 |
(interpretation)g(of)i(intuitionistic)g(proofs.)648 4426
|
|
|
919 |
y(T)-7 b(w)o(o)20 b(properties)e(hold)h(for)g(term-re)n(writing)e(in)j
|
|
|
920 |
(the)g(simply-typed)d(lambda-calculus)h(and)h(by)523
|
|
|
921 |
4526 y(the)k(Curry-Ho)n(w)o(ard)d(correspondence)f(also)k(for)f
|
|
|
922 |
(normalisation)e(in)j(NJ:)h(the)o(y)d(are)i(strongly)e(nor)n(-)523
|
|
|
923 |
4625 y(malising)k(and)h(Church-Rosser)-5 b(.)24 b(W)m(ith)i(some)g
|
|
|
924 |
(limitations)g([8,)12 b(15,)h(18,)g(23],)24 b(the)i(Curry-Ho)n(w)o(ard)
|
|
|
925 |
523 4725 y(correspondence)18 b(applies)i(also)h(to)h(the)f
|
|
|
926 |
(sequent-calculus)d(LJ)k(and)e(to)h(the)g(process)g(of)f(cut-elimi-)523
|
|
|
927 |
4825 y(nation.)28 b(F)o(or)g(sak)o(e)h(of)g(simplicity)-5
|
|
|
928 |
b(,)28 b(we)h(shall)g(ignore)f(these)h(limitations)f(here)h(and)f(re)o
|
|
|
929 |
(gard)f(cut-)523 4924 y(elimination)k(in)h(intuitionistic)f(logic)g(as)
|
|
|
930 |
i(strongly)d(normalising)g(and)h(as)i(ha)n(ving)e(\(morally)f(at)p
|
|
|
931 |
eop end
|
|
|
932 |
%%Page: 2 2
|
|
|
933 |
TeXDict begin 2 1 bop 523 448 a FA(least\))24 b(the)f(Church-Rosser)e
|
|
|
934 |
(property)-5 b(.)21 b(This)i(ignorance)e(can)i(be)g(partially)f
|
|
|
935 |
(justi\002ed)i(if)f(one)g(sees)523 548 y(behind)c(e)n(v)o(ery)f
|
|
|
936 |
(LJ-proof)g(an)i(NJ-proof)f(where)g(the)h(inessential)g(dif)n(ferences)
|
|
|
937 |
f(present)g(in)h(sequent)523 648 y(proofs)31 b(\223disappear\224,)g
|
|
|
938 |
(and)h(sees)h(cut-elimination)d(as)k(an)e(approximation)d(of)k
|
|
|
939 |
(proof-normali-)523 747 y(sation.)648 850 y(Although)19
|
|
|
940 |
b(it)i(has)h(been)e(sho)n(wn)g(that)h(some)g(cut-elimination)e
|
|
|
941 |
(procdures)g(for)h(classical)i(logic)523 950 y(are)i(strongly)f
|
|
|
942 |
(normalising)g(as)i(well,)f(cut-elimination)e(in)j(classical)g(logic)f
|
|
|
943 |
(is,)h(gi)n(v)o(en)e(a)h(sensible)523 1049 y(notion)i(of)h
|
|
|
944 |
(cut-reductions,)d Ft(not)k FA(Church-Rosser)n(\227not)d(e)n(v)o(en)h
|
|
|
945 |
(morally)-5 b(.)25 b(The)i(lack)g(of)g(Church-)523 1149
|
|
|
946 |
y(Rosser)18 b(in)f(classical)h(logic)f(is)h(a)g(main)f(theme)f(running)
|
|
|
947 |
f(through)g(the)j(w)o(orks)e([2,)d(3,)g(19,)g(22],)j(which)523
|
|
|
948 |
1249 y(analyse)j(what)g(this)h(means)f(from)f(a)i(computational)d
|
|
|
949 |
(point)h(of)h(vie)n(w)-5 b(.)19 b(F)o(or)g(e)o(xample)f(in)h([19])f(it)
|
|
|
950 |
i(has)523 1348 y(been)31 b(sho)n(wn)g(that)g(a)h(lambda-calculus)d
|
|
|
951 |
(with)j(a)g(non-deterministic)c(choice-operator)g(can)k(be)523
|
|
|
952 |
1448 y(embedded)19 b(into)i(a)g(fragment)f(of)h(classical)h(logic.)e(A)
|
|
|
953 |
i(simple)f(classical)h(proof)d(tak)o(en)i(from)f([6,)13
|
|
|
954 |
b(9])523 1547 y(shall)21 b(illustrate)f(the)g(lack)g(of)g
|
|
|
955 |
(Church-Rosser:)1158 1725 y Fs(A)p 1238 1713 10 38 v
|
|
|
956 |
1248 1696 42 4 v 88 w(A)83 b(A)p 1534 1713 10 38 v 1543
|
|
|
957 |
1696 42 4 v 88 w(A)p 1158 1745 509 4 v 1178 1818 a(A)19
|
|
|
958 |
b Fr(_)g Fs(A)p 1414 1806 10 38 v 1423 1790 42 4 v 88
|
|
|
959 |
w(A;)14 b(A)1707 1762 y Fr(_)1762 1774 y Fq(L)p 1178
|
|
|
960 |
1854 467 4 v 1228 1928 a Fs(A)19 b Fr(_)g Fs(A)p 1463
|
|
|
961 |
1916 10 38 v 1473 1899 42 4 v 88 w(A)1686 1874 y(contr)1879
|
|
|
962 |
1886 y Fq(R)2017 1725 y Fs(A)p 2098 1713 10 38 v 2107
|
|
|
963 |
1696 42 4 v 88 w(A)84 b(A)p 2393 1713 10 38 v 2403 1696
|
|
|
964 |
42 4 v 88 w(A)p 2017 1745 509 4 v 2056 1818 a(A;)14 b(A)p
|
|
|
965 |
2236 1806 10 38 v 2246 1790 42 4 v 89 w(A)p Fr(^)p Fs(A)2567
|
|
|
966 |
1762 y Fr(^)2622 1774 y Fq(R)p 2056 1854 430 4 v 2106
|
|
|
967 |
1928 a Fs(A)p 2187 1916 10 38 v 2196 1899 42 4 v 88 w(A)p
|
|
|
968 |
Fr(^)q Fs(A)2527 1874 y(contr)2720 1886 y Fq(L)p 1228
|
|
|
969 |
1948 1209 4 v 1608 2021 a Fs(A)p Fr(_)q Fs(A)p 1807 2009
|
|
|
970 |
10 38 v 1816 1992 42 4 v 88 w(A)p Fr(^)q Fs(A)2478 1973
|
|
|
971 |
y(cut)3308 2021 y FA(\(1\))523 2183 y(The)32 b(cut)g(in)g(this)h(proof)
|
|
|
972 |
e(can)h(be)g(eliminated)f(by)h(reducing)e(it)j(to)f(one)g(of)g(the)g
|
|
|
973 |
(follo)n(wing)e(tw)o(o)523 2283 y(normalforms)1169 2460
|
|
|
974 |
y Fs(A)p 1250 2448 10 38 v 1259 2432 42 4 v 88 w(A)84
|
|
|
975 |
b(A)p 1545 2448 10 38 v 1555 2432 42 4 v 88 w(A)p 1169
|
|
|
976 |
2480 509 4 v 1209 2554 a(A;)14 b(A)p 1388 2542 10 38
|
|
|
977 |
v 1398 2525 42 4 v 88 w(A)p Fr(^)q Fs(A)1719 2497 y Fr(^)1774
|
|
|
978 |
2510 y Fq(R)p 1209 2590 430 4 v 1258 2663 a Fs(A)p 1339
|
|
|
979 |
2651 10 38 v 1348 2635 42 4 v 88 w(A)p Fr(^)q Fs(A)1679
|
|
|
980 |
2609 y(contr)1872 2621 y Fq(L)2005 2460 y Fs(A)p 2086
|
|
|
981 |
2448 10 38 v 2096 2432 42 4 v 89 w(A)83 b(A)p 2382 2448
|
|
|
982 |
10 38 v 2391 2432 42 4 v 88 w(A)p 2005 2480 509 4 v 2045
|
|
|
983 |
2554 a(A;)14 b(A)p 2225 2542 10 38 v 2234 2525 42 4 v
|
|
|
984 |
88 w(A)p Fr(^)q Fs(A)2555 2497 y Fr(^)2610 2510 y Fq(R)p
|
|
|
985 |
2045 2590 430 4 v 2094 2663 a Fs(A)p 2175 2651 10 38
|
|
|
986 |
v 2185 2635 42 4 v 89 w(A)p Fr(^)p Fs(A)2516 2609 y(contr)2709
|
|
|
987 |
2621 y Fq(L)p 1258 2683 1167 4 v 1509 2756 a Fs(A)p Fr(_)q
|
|
|
988 |
Fs(A)p 1707 2744 10 38 v 1717 2728 42 4 v 88 w(A)p Fr(^)q
|
|
|
989 |
Fs(A;)g(A)p Fr(^)p Fs(A)2466 2700 y Fr(_)2521 2712 y
|
|
|
990 |
Fq(L)p 1509 2792 665 4 v 1617 2866 a Fs(A)p Fr(_)q Fs(A)p
|
|
|
991 |
1816 2854 10 38 v 1825 2837 42 4 v 88 w(A)p Fr(^)q Fs(A)2215
|
|
|
992 |
2812 y(contr)2408 2824 y Fq(R)3308 2866 y FA(\(2\))1146
|
|
|
993 |
3103 y Fs(A)p 1227 3091 10 38 v 1236 3074 42 4 v 88 w(A)83
|
|
|
994 |
b(A)p 1522 3091 10 38 v 1532 3074 42 4 v 89 w(A)p 1146
|
|
|
995 |
3123 509 4 v 1167 3196 a(A)19 b Fr(_)f Fs(A)p 1402 3184
|
|
|
996 |
10 38 v 1412 3167 42 4 v 89 w(A;)c(A)1695 3140 y Fr(_)1751
|
|
|
997 |
3152 y Fq(L)p 1167 3232 467 4 v 1216 3306 a Fs(A)19 b
|
|
|
998 |
Fr(_)g Fs(A)p 1452 3294 10 38 v 1461 3277 42 4 v 88 w(A)1675
|
|
|
999 |
3251 y(contr)1868 3263 y Fq(R)2005 3103 y Fs(A)p 2086
|
|
|
1000 |
3091 10 38 v 2096 3074 42 4 v 89 w(A)83 b(A)p 2382 3091
|
|
|
1001 |
10 38 v 2391 3074 42 4 v 88 w(A)p 2005 3123 509 4 v 2026
|
|
|
1002 |
3196 a(A)19 b Fr(_)g Fs(A)p 2262 3184 10 38 v 2271 3167
|
|
|
1003 |
42 4 v 88 w(A;)14 b(A)2555 3140 y Fr(_)2610 3152 y Fq(L)p
|
|
|
1004 |
2026 3232 467 4 v 2076 3306 a Fs(A)19 b Fr(_)g Fs(A)p
|
|
|
1005 |
2311 3294 10 38 v 2321 3277 42 4 v 88 w(A)2534 3251 y(contr)2727
|
|
|
1006 |
3263 y Fq(R)p 1216 3325 1227 4 v 1497 3399 a Fs(A)p Fr(_)q
|
|
|
1007 |
Fs(A;)14 b(A)p Fr(_)q Fs(A)p 1913 3387 10 38 v 1922 3370
|
|
|
1008 |
42 4 v 88 w(A)p Fr(^)q Fs(A)2485 3342 y Fr(^)2540 3355
|
|
|
1009 |
y Fq(R)p 1497 3435 665 4 v 1606 3508 a Fs(A)p Fr(_)p
|
|
|
1010 |
Fs(A)p 1804 3496 10 38 v 1814 3480 42 4 v 89 w(A)p Fr(^)p
|
|
|
1011 |
Fs(A)2203 3454 y(contr)2396 3466 y Fq(L)3308 3508 y FA(\(3\))523
|
|
|
1012 |
3671 y(which)i(are)h(obtained,)e(respecti)n(v)o(ely)-5
|
|
|
1013 |
b(,)15 b(by)h(either)g(permuting)f(the)i(cut)f(to)h(the)g(left)g(o)o(v)
|
|
|
1014 |
o(er)f(the)g Fs(contr)3322 3683 y Fq(R)3377 3671 y FA(-)523
|
|
|
1015 |
3770 y(rule)f(or)h(to)f(the)h(right)f(o)o(v)o(er)f(the)i
|
|
|
1016 |
Fs(contr)1597 3782 y Fq(L)1647 3770 y FA(-rule.)e(Another)h(e)o(xample)
|
|
|
1017 |
f(sho)n(wing)g(that)i(in)f(classical)i(logic)523 3870
|
|
|
1018 |
y(one)j(can,)f(in)i(general,)e(reach)g(more)g(than)h(one)g(normalform)d
|
|
|
1019 |
(is)k(Lafont')-5 b(s)20 b(proof)e([10,)i(P)o(age)f(151].)648
|
|
|
1020 |
3973 y(In)26 b(light)h(of)f(the)h(absence)f(of)g(the)h(Church-Rosser)e
|
|
|
1021 |
(property)g(for)h(cut-elimination)e(in)j(clas-)523 4072
|
|
|
1022 |
y(sical)h(logic)e(and)g(in)h(light)g(of)f(the)h(w)o(ork)f(on)g
|
|
|
1023 |
(double-ne)o(gation)d(translations,)i(there)i(seem)g(to)g(be)523
|
|
|
1024 |
4172 y(ob)o(vious)k(questions:)g(What)i(is)g(the)g(correspondence)28
|
|
|
1025 |
b(between)k(cut-elimination)e(in)j(classical)523 4272
|
|
|
1026 |
y(logic)15 b(and)g(the)h(embeddings)d(of)i(classical)h(proofs)e(into)i
|
|
|
1027 |
(intuitionistic)e(logic)i(via)f(double-ne)o(gation)523
|
|
|
1028 |
4371 y(translations?)23 b(Since)h(cut-elimination)e(in)i
|
|
|
1029 |
(intuitionistic)g(logic)f(is)i(Church-Rosser)m(,)3030
|
|
|
1030 |
4341 y Fp(1)3085 4371 y FA(which)f(re-)523 4471 y(striction)e(is)h
|
|
|
1031 |
(tacitly)f(enforced)f(by)g(a)i(double-ne)o(gation)18
|
|
|
1032 |
b(translation)j(so)i(that)f(eliminating)f(cuts)h(in)523
|
|
|
1033 |
4570 y(the)i(double-ne)o(gated)c(v)o(ersion)j(of)h(a)g(classical)h
|
|
|
1034 |
(proof)e(leads)h(to)g(only)g(a)g(single)g(normalform?)d(Or)p
|
|
|
1035 |
523 4654 473 4 v 558 4710 a Fo(1)606 4742 y Fx(As)j(mentioned)i
|
|
|
1036 |
(earlier)f(we)f(ignore)h(what)g(we)g(belie)n(v)o(e)g(to)f(be)h
|
|
|
1037 |
(super\002cial)g(v)n(ariations)g(between)g(dif)n(fer)o(-)606
|
|
|
1038 |
4833 y(ent)c(normalforms)g(reachable)h(from)f(an)g(intuitionistic)f
|
|
|
1039 |
(sequent-proof.)i(Ho)n(we)n(v)o(er)f(see)g([18])g(for)g(a)f(more)606
|
|
|
1040 |
4924 y(thorough)h(analysis)e(of)g(this)g(aspect.)p eop
|
|
|
1041 |
end
|
|
|
1042 |
%%Page: 3 3
|
|
|
1043 |
TeXDict begin 3 2 bop 523 448 a FA(more)24 b(concisely)f(ask)o(ed,)i
|
|
|
1044 |
(do)f(double-ne)o(gation)c(translations)k(correspond)e(to)i(particular)
|
|
|
1045 |
f(strate-)523 548 y(gies)e(of)g(ho)n(w)f(to)h(eliminate)g(cuts?)g(Does)
|
|
|
1046 |
g(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(lead)g(to)h(the)g
|
|
|
1047 |
(same)523 648 y(normalform)i(\(by)j Ft(same)g FA(we)h(mean)e
|
|
|
1048 |
(corresponding)e(to)j(one)g(particular)f(normalform)e(obtained)523
|
|
|
1049 |
747 y(by)j(cut-elimination)e(in)j(classical)g(logic\)?)f(If)g(not,)g
|
|
|
1050 |
(then)g(can)g(one)g(\002nd)g(for)g(e)n(v)o(ery)f(normalform)523
|
|
|
1051 |
847 y(of)h(a)g(classical)h(proof)d(a)j(corresponding)22
|
|
|
1052 |
b(double-ne)o(gation)g(translation)j(that)h(will)h(produce)c(the)523
|
|
|
1053 |
946 y(double-ne)o(gated)g(v)o(ersion)j(of)i(this)g(normalform\227that)c
|
|
|
1054 |
(is,)k(can)f(e)n(v)o(ery)f(reduction)g(sequence)g(in)523
|
|
|
1055 |
1046 y(classical)16 b(logic)f(be)h(simulated)f(by)g(a)h(\(probably)d
|
|
|
1056 |
(carefully)h(chosen\))g(double)g(ne)o(gation)f(translation)523
|
|
|
1057 |
1146 y(and)26 b(performing)e(cut-elimination)h(in)i(intuitionistic)f
|
|
|
1058 |
(logic?)h(Are)g(there)f(an)o(y)g(double-ne)o(gation)523
|
|
|
1059 |
1245 y(translations)20 b(that)g(lead)g(to)g(normalforms)e(that)i(ha)n
|
|
|
1060 |
(v)o(e)g(no)f(equi)n(v)n(alent)g(amongst)g(the)h(normalforms)523
|
|
|
1061 |
1345 y(reachable)i(by)h(cut-elimination)f(in)h(classical)h(logic?)f
|
|
|
1062 |
(Can)h(one)f(characterise)f(someho)n(w)-5 b(,)22 b(which)523
|
|
|
1063 |
1445 y(normalforms)16 b(can)i(be)g(reached)f(by)h(double-ne)o(gation)c
|
|
|
1064 |
(translations)k(and)g(which)g(can)g(not?)g(In)g(this)523
|
|
|
1065 |
1544 y(paper)h(we)i(conjecture)d(answers)i(for)g(all)h(these)f
|
|
|
1066 |
(questions.)648 1654 y(Although)k(some)i(special)h(cases)g(seem)g(to)f
|
|
|
1067 |
(be)h(answered)e(by)h(e)o(xisting)g(w)o(ork,)f(for)h(e)o(xample)523
|
|
|
1068 |
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
|
|
|
1069 |
(w)o(ork)g(that)g(treat)h(these)g(questions)e(in)i(full)f(generality)-5
|
|
|
1070 |
b(.)21 b(The)523 1853 y(answers)27 b(we)g(shall)g(gi)n(v)o(e)f(to)h
|
|
|
1071 |
(these)f(questions)g(are)h(a)g(lot)g(inspired)f(by)g(the)h(comments)e
|
|
|
1072 |
(made)h(in)523 1952 y([6,)d(Sec.)g(7].)g(Ho)n(we)n(v)o(er)m(,)e(there)i
|
|
|
1073 |
(only)g(one)g(half)g(of)g(the)g(correspondence)d(is)k(considered,)e
|
|
|
1074 |
(namely)523 2052 y(ho)n(w)31 b(their)h(v)o(ersion)f(of)h(classical)g
|
|
|
1075 |
(logic)g(and)f(cut-elimination)f(can)i(be)g(embedded)e(via)i(some)523
|
|
|
1076 |
2152 y(speci\002c)21 b(double-ne)o(gation)c(translations)j(into)g
|
|
|
1077 |
(intuitionistic)h(logic.)f(W)-7 b(e)22 b(conjecture)d(also)j(a)f(cor)n
|
|
|
1078 |
(-)523 2251 y(respondence)i(in)i(the)g(other)f(direction,)g(namely)g
|
|
|
1079 |
(that)h(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(and)523
|
|
|
1080 |
2351 y(corresponding)12 b(reduction)i(sequences)h(can)g(be)g(simulated)
|
|
|
1081 |
g(by)g(their)h(cut-elimination)d(procedure.)523 2451
|
|
|
1082 |
y(Since)19 b(we)g(shall)h(use)f(as)h(\223point)e(of)g(reference\224)f
|
|
|
1083 |
(a)j(more)e(general)g(cut-elimination)f(procedure)f(for)523
|
|
|
1084 |
2550 y(cut-elimination)h(in)j(classical)g(logic)f(than)g(the)g(one)g
|
|
|
1085 |
(described)f(in)h([6],)g(we)h(are)f(also)h(able)f(to)g(dra)o(w)523
|
|
|
1086 |
2650 y(the)26 b(conclusion)e(that)h(gi)n(v)o(en)g(our)g(conjecture)f
|
|
|
1087 |
(is)i(true,)f(then)g(double-ne)o(gation)d(translations)j(are)523
|
|
|
1088 |
2749 y(not)20 b(enough)e(to)i(describe)g(the)g Ft(full)h
|
|
|
1089 |
FA(computational)c(meaning)i(of)h(a)h(classical)g(proof.)648
|
|
|
1090 |
2859 y(As)h(can)f(be)h(seen)g(to)f(answer)h(the)f(correspondence)d
|
|
|
1091 |
(questions,)j(we)h(\002rst)g(ha)n(v)o(e)f(to)h(mak)o(e)f(pre-)523
|
|
|
1092 |
2959 y(cise)28 b(what)e(we)i(mean)e(by)g(cut-elimination)f(in)i
|
|
|
1093 |
(classical)h(logic.)e(Most)h(cut-elimination)e(proce-)523
|
|
|
1094 |
3058 y(dures,)d(including)f(Gentzen')-5 b(s)23 b(original)f(one,)g
|
|
|
1095 |
(only)g(terminate)g(if)h(a)h(particular)d(strate)o(gy)h(for)h(cut-)523
|
|
|
1096 |
3158 y(elimination)e(is)i(emplo)o(yed.)c(Common)i(e)o(xamples)g(being)g
|
|
|
1097 |
(an)h(innermost)e(reduction)g(strate)o(gy)-5 b(,)21 b(or)523
|
|
|
1098 |
3257 y(the)d(elimination)g(of)g(the)g(cut)h(with)f(the)h(highest)f
|
|
|
1099 |
(rank.)f(Using)h(those)g(cut-elimination)f(procedures)523
|
|
|
1100 |
3357 y(we)29 b(cannot)f(characterise)f(what)i(the)f(set)i(of)e
|
|
|
1101 |
Ft(all)h FA(normalforms)d(of)i(a)h(classical)h(proof)d(is\227the)o(y)
|
|
|
1102 |
523 3457 y(w)o(ould)c(produce)e(only)i(one)g(or)h(a)g(limited)f(number)
|
|
|
1103 |
f(of)h(normalforms.)e(W)-7 b(e)25 b(shall)f(therefore)d(base)523
|
|
|
1104 |
3556 y(our)f(ar)o(guments)f(on)i(the)g(cut-elimination)e(procedure)f
|
|
|
1105 |
(de)n(v)o(eloped)h(by)h(Urban)h(and)f(Bierman)h([20,)523
|
|
|
1106 |
3656 y(21],)h(which)g(is)i(lik)o(e)g(Gentzen')-5 b(s)22
|
|
|
1107 |
b(procedure)f(e)o(xcept)g(it)j(imposes)f(one)f(slight)h(restriction)f
|
|
|
1108 |
(on)h(ho)n(w)523 3756 y(commuting)g(cuts)j(need)f(to)h(be)g(analysed.)e
|
|
|
1109 |
(Since)i(this)g(cut-elimination)d(procedure)g(is)k(strongly)523
|
|
|
1110 |
3855 y(normalising,)g(we)i(can)g(calculate)f(all)h(cut-free)f
|
|
|
1111 |
(normalforms)e(of)j(a)g(classical)g(proof.)e(Because)523
|
|
|
1112 |
3955 y(this)19 b(procedure)d(is)k(not)f(Church-Rosser)m(,)d(the)j
|
|
|
1113 |
(collection)f(of)g(normalforms)e(for)i(a)i(classical)f(proof)523
|
|
|
1114 |
4054 y(contains)f(in)h(general)e(more)h(than)g(one)g(element\227as)g
|
|
|
1115 |
(can)h(be)f(seen)h(for)f(e)o(xample)f(with)i(the)g(proofs)523
|
|
|
1116 |
4154 y(\(2\))24 b(and)g(\(3\).)f(As)j(this)f(cut-elimination)d
|
|
|
1117 |
(procedure)g(puts)j(only)e(v)o(ery)h(slight)g(restrictions)g(on)g(the)
|
|
|
1118 |
523 4254 y(process)i(of)h(cut-elimination)d(we)j(belie)n(v)o(e)f(a)h
|
|
|
1119 |
(good)e(case)i(can)g(be)g(made)f(that)g(the)h(collection)f(of)523
|
|
|
1120 |
4353 y(normalforms)18 b(calculated)i(by)h(this)g(procedure)d(includes)i
|
|
|
1121 |
(all)i(\223essential\224)f(normalforms.)3167 4323 y Fp(2)3218
|
|
|
1122 |
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
|
|
|
1123 |
(shall)f(not)g(be)g(concerned)e(with)j(in)f(this)h(paper)-5
|
|
|
1124 |
b(.)p 523 4563 473 4 v 558 4619 a Fo(2)606 4650 y Fx(Making)31
|
|
|
1125 |
b(such)e(a)h(case)f(is)g(hopeless)h(for)f(other)h(strongly-normalising)
|
|
|
1126 |
h(cut-elimination)e(procedures,)606 4742 y(lik)o(e)f(the)g(one)h(by)f
|
|
|
1127 |
(Dragalin)g([7],)g(because)h(although)g(the)o(y)f(are)g
|
|
|
1128 |
(strongly-normalising,)i(the)o(y)e(enforce)606 4833 y(quite)21
|
|
|
1129 |
b(strong)g(restrictions)g(on)g(ho)n(w)h(cuts)f(can)g(be)g(eliminated.)g
|
|
|
1130 |
(F)o(or)f(e)o(xample)i(Dragalin)e(does)i(not)f(allo)n(w)606
|
|
|
1131 |
4924 y(\(multi\)cuts)e(to)f(permute)i(o)o(v)o(er)f(other)g
|
|
|
1132 |
(\(multi\)cuts,)g(see)g([19].)p eop end
|
|
|
1133 |
%%Page: 4 4
|
|
|
1134 |
TeXDict begin 4 3 bop 648 448 a FA(The)30 b(cut-elimination)f
|
|
|
1135 |
(procedure)f(of)j(Urban)f(and)g(Bierman)g(will)i(be)f(described)e(in)i
|
|
|
1136 |
(more)523 548 y(detail)24 b(in)g(Sec.)h(3,)f(together)e(with)i(a)h(v)n
|
|
|
1137 |
(ariant\227the)e(colour)f(protocol\227de)n(v)o(eloped)e(by)j(Danos)h
|
|
|
1138 |
(et)523 648 y(al.)d([6,)12 b(12].)20 b(Beforehand,)e(ho)n(we)n(v)o(er)m
|
|
|
1139 |
(,)f(we)k(present)f(some)g(preliminaries)f(about)g(double-ne)o(gation)
|
|
|
1140 |
523 747 y(translations)28 b(in)h(Sec.)f(2.)h(W)-7 b(e)29
|
|
|
1141 |
b(will)h(state)f(the)f(conjecture)f(in)i(Sec.)f(4,)h(gi)n(v)o(e)f(some)
|
|
|
1142 |
g(e)n(vidence)f(on)523 847 y(why)e(this)h(conjecture)e(is)j(plausible)e
|
|
|
1143 |
(and)g(present)g(some)g(ideas)h(on)f(ho)n(w)g(to)h(pro)o(v)o(e)e(it.)i
|
|
|
1144 |
(In)g(Sec.)f(5)523 946 y(we)32 b(shall)g(dra)o(w)f(some)h(conclusions)e
|
|
|
1145 |
(with)i(respect)g(to)g(the)g(computational)d(interpretation)h(of)523
|
|
|
1146 |
1046 y(classical)21 b(proofs.)523 1300 y Fu(2)99 b(Pr)n(eliminaries)25
|
|
|
1147 |
b(on)g(Double-Negation)h(T)-7 b(ranslations)523 1487
|
|
|
1148 |
y FA(W)g(e)21 b(assume)e(the)h(reader)e(has)i(acquaintance)d(with)j
|
|
|
1149 |
(sequent-calculus)d(formulations)g(of)j(classical)523
|
|
|
1150 |
1586 y(and)e(intuitionistic)f(logic.)h(Because)g(there)g(e)o(xist)g
|
|
|
1151 |
(sequents)g(that)h(are)f(pro)o(v)n(able)e(in)i(classical)h(logic,)523
|
|
|
1152 |
1686 y(b)n(ut)25 b(unpro)o(v)n(able)d(in)j(intuitionistic)g(logic,)g
|
|
|
1153 |
(the)g(interesting)f(point)g(of)h(double-ne)o(gation)c(transla-)523
|
|
|
1154 |
1786 y(tions)g(is)i(that)e(one)g(can)g(embed)f(classical)j(logic)e
|
|
|
1155 |
(into)g(intuitionistic)f(logic)h(so)h(that)g(pro)o(v)n(ability)d(is)523
|
|
|
1156 |
1885 y(preserv)o(ed.)f(F)o(or)i(e)o(xample)f(the)h(follo)n(wing)e
|
|
|
1157 |
(translation)i(de\002ned)f(o)o(v)o(er)g(formulae)1515
|
|
|
1158 |
2075 y Fs(A)1577 2044 y Fn(\003)1639 2028 y Fp(def)1643
|
|
|
1159 |
2075 y Fm(=)28 b Fr(::)p Fs(A)22 b FA(with)e Fs(A)h FA(being)e(atomic)
|
|
|
1160 |
1390 2199 y Fm(\()p Fr(:)p Fs(B)t Fm(\))1576 2169 y Fn(\003)1639
|
|
|
1161 |
2152 y Fp(def)1643 2199 y Fm(=)28 b Fr(:)p Fm(\()p Fs(B)1890
|
|
|
1162 |
2169 y Fn(\003)1929 2199 y Fm(\))1325 2324 y(\()p Fs(B)t
|
|
|
1163 |
Fr(^)q Fs(C)6 b Fm(\))1577 2294 y Fn(\003)1639 2277 y
|
|
|
1164 |
Fp(def)1643 2324 y Fm(=)28 b Fs(B)1803 2294 y Fn(\003)1841
|
|
|
1165 |
2324 y Fr(^)q Fs(C)1962 2294 y Fn(\003)1316 2449 y Fm(\()p
|
|
|
1166 |
Fs(B)t Fr(\033)p Fs(C)6 b Fm(\))1577 2419 y Fn(\003)1639
|
|
|
1167 |
2402 y Fp(def)1643 2449 y Fm(=)28 b Fs(B)1803 2419 y
|
|
|
1168 |
Fn(\003)1841 2449 y Fr(\033)p Fs(C)1971 2419 y Fn(\003)1325
|
|
|
1169 |
2574 y Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))1577 2544
|
|
|
1170 |
y Fn(\003)1639 2527 y Fp(def)1643 2574 y Fm(=)28 b Fr(:)p
|
|
|
1171 |
Fm(\()p Fr(:)p Fm(\()p Fs(B)1977 2544 y Fn(\003)2017
|
|
|
1172 |
2574 y Fm(\))p Fr(^:)p Fm(\()p Fs(C)2256 2544 y Fn(\003)2295
|
|
|
1173 |
2574 y Fm(\)\))3308 2313 y FA(\(4\))523 2737 y(can)20
|
|
|
1174 |
b(be)g(used)g(to)g(sho)n(w)g(that)h(e)n(v)o(ery)d(classical)k(proof)c
|
|
|
1175 |
(with)i(the)h(end-sequent)1854 2904 y Fs(\000)p 1935
|
|
|
1176 |
2892 10 38 v 1945 2876 42 4 v 100 w(\001)523 3072 y FA(can)f(be)g
|
|
|
1177 |
(translated)g(to)g(an)g(intuitionistic)g(proof)e(with)j(the)f
|
|
|
1178 |
(end-sequent)1747 3240 y Fs(\000)1810 3209 y Fn(\003)1848
|
|
|
1179 |
3240 y Fs(;)14 b Fr(:)p Fs(\001)2009 3209 y Fn(\003)p
|
|
|
1180 |
2066 3228 10 38 v 2075 3211 42 4 v 2158 3240 a Fs(:)1127
|
|
|
1181 |
b FA(\(5\))523 3407 y(W)-7 b(e)28 b(use)e(the)h(con)m(v)o(ention)c
|
|
|
1182 |
(that)j(if)h Fs(\000)39 b FA(is)27 b(the)f(sequent-conte)o(xt)e
|
|
|
1183 |
Fr(f)p Fs(B)2502 3419 y Fl(1)2539 3407 y Fs(;)14 b(:)g(:)g(:)f(;)h(B)
|
|
|
1184 |
2786 3419 y Fq(n)2831 3407 y Fr(g)27 b FA(then)f Fs(\000)3133
|
|
|
1185 |
3377 y Fn(\003)3197 3407 y FA(stands)523 3507 y(for)f(the)g
|
|
|
1186 |
(sequent-conte)o(xt)d Fr(f)p Fs(B)1432 3477 y Fn(\003)1428
|
|
|
1187 |
3527 y Fl(1)1470 3507 y Fs(;)14 b(:)g(:)g(:)f(;)h(B)1721
|
|
|
1188 |
3477 y Fn(\003)1717 3527 y Fq(n)1763 3507 y Fr(g)p FA(.)25
|
|
|
1189 |
b(Similarly)f(for)h Fr(:)p Fs(\001)2432 3477 y Fn(\003)2471
|
|
|
1190 |
3507 y FA(.)g(W)-7 b(e)27 b(shall)e(also)h(use)f(the)g(con-)523
|
|
|
1191 |
3606 y(v)o(ention)17 b(that)h Fs(A)h FA(stands)f(for)f(an)h(atomic)g
|
|
|
1192 |
(formula)f(and)g Fs(B)t(;)d(:)g(:)g(:)19 b FA(for)e(arbitrary)g
|
|
|
1193 |
(formulae.)f(A)i(similar)523 3706 y(embedding)g(can)i(be)g(obtained)f
|
|
|
1194 |
(with)h(the)g(translation:)1729 3895 y Fs(A)1791 3865
|
|
|
1195 |
y Fn(\016)1852 3848 y Fp(def)1857 3895 y Fm(=)28 b Fr(::)p
|
|
|
1196 |
Fs(A)1604 4020 y Fm(\()p Fr(:)p Fs(B)t Fm(\))1790 3990
|
|
|
1197 |
y Fn(\016)1852 3973 y Fp(def)1857 4020 y Fm(=)g Fr(:)p
|
|
|
1198 |
Fm(\()p Fs(B)2104 3990 y Fn(\016)2143 4020 y Fm(\))1539
|
|
|
1199 |
4145 y(\()p Fs(B)t Fr(^)p Fs(C)6 b Fm(\))1790 4115 y
|
|
|
1200 |
Fn(\016)1852 4098 y Fp(def)1857 4145 y Fm(=)28 b Fr(::)p
|
|
|
1201 |
Fm(\()p Fs(B)2159 4115 y Fn(\016)2198 4145 y Fr(^)q Fs(C)2319
|
|
|
1202 |
4115 y Fn(\016)2357 4145 y Fm(\))1529 4270 y(\()p Fs(B)t
|
|
|
1203 |
Fr(\033)p Fs(C)6 b Fm(\))1790 4240 y Fn(\016)1852 4223
|
|
|
1204 |
y Fp(def)1857 4270 y Fm(=)28 b Fr(::)p Fm(\()p Fs(B)2159
|
|
|
1205 |
4240 y Fn(\016)2198 4270 y Fr(\033)p Fs(C)2328 4240 y
|
|
|
1206 |
Fn(\016)2366 4270 y Fm(\))1539 4395 y(\()p Fs(B)t Fr(_)p
|
|
|
1207 |
Fs(C)6 b Fm(\))1790 4365 y Fn(\016)1852 4348 y Fp(def)1857
|
|
|
1208 |
4395 y Fm(=)28 b Fr(::)p Fm(\()p Fs(B)2159 4365 y Fn(\016)2198
|
|
|
1209 |
4395 y Fr(_)q Fs(C)2319 4365 y Fn(\016)2357 4395 y Fm(\))3308
|
|
|
1210 |
4133 y FA(\(6\))648 4558 y(The)i(usual)h(proof)e(\(see)i(for)g(e)o
|
|
|
1211 |
(xample)e([4]\))h(for)h(establishing)f(that)h(e)n(v)o(ery)e(classical)j
|
|
|
1212 |
(proof)523 4657 y(can)21 b(be)g(translated)f(to)i(an)f(intuitionistic)f
|
|
|
1213 |
(proof)g(proceeds)f(inducti)n(v)o(ely)g(by)i(translating)f(stepwise)523
|
|
|
1214 |
4757 y(e)n(v)o(ery)f(inference)f(rule)i(in)h(a)f(proof.)f(F)o(or)g
|
|
|
1215 |
(instance)h(an)g(axiom)g(of)g(the)g(form)1858 4924 y
|
|
|
1216 |
Fs(A)p 1938 4912 10 38 v 1948 4896 42 4 v 88 w(A)p eop
|
|
|
1217 |
end
|
|
|
1218 |
%%Page: 5 5
|
|
|
1219 |
TeXDict begin 5 4 bop 523 448 a FA(is)21 b(translated)f(by)f
|
|
|
1220 |
Fm(\()p Fr(\000)p Fm(\))1175 418 y Fn(\003)1235 448 y
|
|
|
1221 |
FA(to)h(the)g(proof)1674 603 y Fr(::)p Fs(A)p 1865 591
|
|
|
1222 |
10 38 v 1875 574 42 4 v 89 w Fr(::)p Fs(A)p 1628 623
|
|
|
1223 |
527 4 v 1628 696 a Fr(::)p Fs(A;)14 b Fr(:::)p Fs(A)p
|
|
|
1224 |
2084 684 10 38 v 2094 668 42 4 v 2195 635 a Fr(:)2250
|
|
|
1225 |
647 y Fq(L)523 865 y FA(which)i(conforms)g(with)h(the)g(desired)f
|
|
|
1226 |
(property)f(stated)i(in)g(\(5\).)f(When)h(translating)f(a)i(proof)d
|
|
|
1227 |
(ending)523 964 y(with)20 b(an)h Fr(_)846 976 y Fq(L)896
|
|
|
1228 |
964 y FA(-rule)1648 1021 y Fm(:)1466 1094 y Fs(B)t(;)14
|
|
|
1229 |
b(\000)1621 1106 y Fl(1)p 1676 1082 10 38 v 1686 1066
|
|
|
1230 |
42 4 v 1746 1094 a Fs(\001)1815 1106 y Fl(1)2114 1021
|
|
|
1231 |
y Fm(:)1935 1094 y Fs(C)q(;)g(\000)2083 1106 y Fl(2)p
|
|
|
1232 |
2140 1082 10 38 v 2149 1066 42 4 v 2209 1094 a Fs(\001)2278
|
|
|
1233 |
1106 y Fl(2)p 1466 1130 851 4 v 1505 1204 a Fs(B)t Fr(_)q
|
|
|
1234 |
Fs(C)q(;)g(\000)1776 1216 y Fl(1)1813 1204 y Fs(;)g(\000)1901
|
|
|
1235 |
1216 y Fl(2)p 1957 1192 10 38 v 1967 1175 42 4 v 2027
|
|
|
1236 |
1204 a Fs(\001)2096 1216 y Fl(1)2133 1204 y Fs(;)g(\001)2239
|
|
|
1237 |
1216 y Fl(2)2357 1147 y Fr(_)2413 1160 y Fq(L)3308 1204
|
|
|
1238 |
y FA(\(7\))523 1345 y(then)20 b(by)g(induction)e(hypothesis)g(we)j(ha)n
|
|
|
1239 |
(v)o(e)f(tw)o(o)g(intuitionistic)g(proofs)f(ending)f(with)1522
|
|
|
1240 |
1476 y Fm(:)1268 1549 y Fs(B)1335 1519 y Fn(\003)1374
|
|
|
1241 |
1549 y Fs(;)c(\000)1474 1519 y Fn(\003)1462 1570 y Fl(1)1511
|
|
|
1242 |
1549 y Fs(;)g Fr(:)p Fs(\001)1672 1519 y Fn(\003)1672
|
|
|
1243 |
1570 y Fl(1)p 1729 1537 10 38 v 1739 1521 42 4 v 1882
|
|
|
1244 |
1549 a FA(and)2338 1476 y Fm(:)2085 1549 y Fs(C)2150
|
|
|
1245 |
1519 y Fn(\003)2188 1549 y Fs(;)g(\000)2288 1519 y Fn(\003)2276
|
|
|
1246 |
1570 y Fl(2)2326 1549 y Fs(;)g Fr(:)p Fs(\001)2487 1519
|
|
|
1247 |
y Fn(\003)2487 1570 y Fl(2)p 2544 1537 10 38 v 2554 1521
|
|
|
1248 |
42 4 v 2637 1549 a Fs(:)523 1718 y FA(W)-7 b(e)21 b(can)f(then)g(form)f
|
|
|
1249 |
(the)h(intuitionistic)g(proof)1486 1849 y Fm(:)1232 1923
|
|
|
1250 |
y Fs(B)1299 1892 y Fn(\003)1338 1923 y Fs(;)14 b(\000)1438
|
|
|
1251 |
1892 y Fn(\003)1426 1943 y Fl(1)1475 1923 y Fs(;)g Fr(:)p
|
|
|
1252 |
Fs(\001)1636 1892 y Fn(\003)1636 1943 y Fl(1)p 1693 1911
|
|
|
1253 |
10 38 v 1703 1894 42 4 v 1223 1963 549 4 v 1223 2037
|
|
|
1254 |
a Fs(\000)1286 2007 y Fn(\003)1274 2058 y Fl(1)1324 2037
|
|
|
1255 |
y Fs(;)g Fr(:)p Fs(\001)1485 2007 y Fn(\003)1485 2058
|
|
|
1256 |
y Fl(1)p 1542 2025 10 38 v 1551 2008 42 4 v 1611 2037
|
|
|
1257 |
a Fr(:)p Fs(B)1733 2007 y Fn(\003)1814 1975 y Fr(:)1869
|
|
|
1258 |
1987 y Fq(R)2268 1849 y Fm(:)2016 1923 y Fs(C)2081 1892
|
|
|
1259 |
y Fn(\003)2119 1923 y Fs(;)g(\000)2219 1892 y Fn(\003)2207
|
|
|
1260 |
1943 y Fl(2)2257 1923 y Fs(;)g Fr(:)p Fs(\001)2418 1892
|
|
|
1261 |
y Fn(\003)2418 1943 y Fl(2)p 2475 1911 10 38 v 2484 1894
|
|
|
1262 |
42 4 v 2006 1963 548 4 v 2006 2037 a Fs(\000)2069 2007
|
|
|
1263 |
y Fn(\003)2057 2058 y Fl(2)2107 2037 y Fs(;)g Fr(:)p
|
|
|
1264 |
Fs(\001)2268 2007 y Fn(\003)2268 2058 y Fl(2)p 2325 2025
|
|
|
1265 |
10 38 v 2335 2008 42 4 v 2395 2037 a Fr(:)p Fs(C)2515
|
|
|
1266 |
2007 y Fn(\003)2595 1975 y Fr(:)2650 1987 y Fq(R)p 1223
|
|
|
1267 |
2077 1331 4 v 1338 2151 a Fs(\000)1401 2121 y Fn(\003)1389
|
|
|
1268 |
2172 y Fl(1)1439 2151 y Fs(;)g(\000)1539 2121 y Fn(\003)1527
|
|
|
1269 |
2172 y Fl(2)1577 2151 y Fs(;)g Fr(:)p Fs(\001)1738 2121
|
|
|
1270 |
y Fn(\003)1738 2172 y Fl(1)1776 2151 y Fs(;)g Fr(:)p
|
|
|
1271 |
Fs(\001)1937 2121 y Fn(\003)1937 2172 y Fl(2)p 1994 2139
|
|
|
1272 |
10 38 v 2004 2123 42 4 v 2064 2151 a Fr(:)p Fs(B)2186
|
|
|
1273 |
2121 y Fn(\003)2224 2151 y Fr(^)q(:)p Fs(C)2400 2121
|
|
|
1274 |
y Fn(\003)2595 2094 y Fr(^)2650 2107 y Fq(R)p 1260 2192
|
|
|
1275 |
1258 4 v 1260 2271 a Fr(:)p Fm(\()p Fr(:)p Fs(B)1469
|
|
|
1276 |
2240 y Fn(\003)1508 2271 y Fr(^:)p Fs(C)1683 2240 y Fn(\003)1722
|
|
|
1277 |
2271 y Fm(\))p Fs(;)g(\000)1854 2240 y Fn(\003)1842 2291
|
|
|
1278 |
y Fl(1)1892 2271 y Fs(;)g(\000)1992 2240 y Fn(\003)1980
|
|
|
1279 |
2291 y Fl(2)2030 2271 y Fs(;)g Fr(:)p Fs(\001)2191 2240
|
|
|
1280 |
y Fn(\003)2191 2291 y Fl(1)2229 2271 y Fs(;)g Fr(:)p
|
|
|
1281 |
Fs(\001)2390 2240 y Fn(\003)2390 2291 y Fl(2)p 2447 2259
|
|
|
1282 |
10 38 v 2457 2242 42 4 v 2558 2203 a Fr(:)2613 2215 y
|
|
|
1283 |
Fq(L)3308 2271 y FA(\(8\))523 2439 y(as)32 b(the)f(translation)g(of)g
|
|
|
1284 |
(\(7\).)f(F)o(or)h(the)g(sak)o(e)g(of)g(more)g(clarity)g(we)g(will)h
|
|
|
1285 |
(omit)f(in)h(what)f(follo)n(ws)523 2539 y(the)22 b(sequent-conte)o(xts)
|
|
|
1286 |
d(whene)n(v)o(er)h(the)o(y)h(are)h(unimportant.)d(Thus)j(we)g(shall)g
|
|
|
1287 |
(gi)n(v)o(e)f(for)h(the)f(proof-)523 2638 y(fragment)e(sho)n(wn)g(in)h
|
|
|
1288 |
(\(8\))g(only)f(the)h(follo)n(wing)f(simpli\002ed)h(inference)e(rules:)
|
|
|
1289 |
1636 2769 y Fm(:)1551 2843 y Fs(B)1618 2813 y Fn(\003)p
|
|
|
1290 |
1675 2831 10 38 v 1684 2814 42 4 v 1523 2863 249 4 v
|
|
|
1291 |
1542 2925 10 38 v 1551 2908 42 4 v 1611 2937 a Fr(:)p
|
|
|
1292 |
Fs(B)1733 2907 y Fn(\003)1814 2875 y Fr(:)1869 2887 y
|
|
|
1293 |
Fq(R)2118 2769 y Fm(:)2034 2843 y Fs(C)2099 2813 y Fn(\003)p
|
|
|
1294 |
2156 2831 10 38 v 2166 2814 42 4 v 2006 2863 247 4 v
|
|
|
1295 |
2025 2925 10 38 v 2034 2908 42 4 v 2094 2937 a Fr(:)p
|
|
|
1296 |
Fs(C)2214 2907 y Fn(\003)2295 2875 y Fr(:)2350 2887 y
|
|
|
1297 |
Fq(R)p 1523 2957 730 4 v 1675 3018 10 38 v 1685 3002
|
|
|
1298 |
42 4 v 1745 3030 a Fr(:)p Fs(B)1867 3000 y Fn(\003)1906
|
|
|
1299 |
3030 y Fr(^:)p Fs(C)2081 3000 y Fn(\003)2295 2974 y Fr(^)2350
|
|
|
1300 |
2986 y Fq(R)p 1597 3050 583 4 v 1597 3129 a Fr(:)p Fm(\()p
|
|
|
1301 |
Fr(:)p Fs(B)1806 3099 y Fn(\003)1845 3129 y Fr(^)q(:)p
|
|
|
1302 |
Fs(C)2021 3099 y Fn(\003)2059 3129 y Fm(\))p 2110 3117
|
|
|
1303 |
10 38 v 2120 3101 42 4 v 2221 3062 a Fr(:)2276 3074 y
|
|
|
1304 |
Fq(L)523 3298 y FA(When)i(translating)f(a)i(classical)g(proof)e(ending)
|
|
|
1305 |
f(with)j(an)f Fr(^)2243 3310 y Fq(R)2298 3298 y FA(-rule)1759
|
|
|
1306 |
3429 y Fm(:)p 1711 3490 10 38 v 1721 3473 42 4 v 1781
|
|
|
1307 |
3502 a Fs(B)1996 3429 y Fm(:)p 1949 3490 10 38 v 1959
|
|
|
1308 |
3473 42 4 v 73 x Fs(C)p 1693 3522 392 4 v 1769 3583 10
|
|
|
1309 |
38 v 1778 3567 42 4 v 1838 3595 a(B)t Fr(^)q Fs(C)2126
|
|
|
1310 |
3539 y Fr(^)2181 3551 y Fq(R)3308 3595 y FA(\(9\))523
|
|
|
1311 |
3764 y(we)h(ha)n(v)o(e)e(by)h(induction)e(hypothesis)h(tw)o(o)h
|
|
|
1312 |
(intuitionistic)g(proofs)f(ending)g(in)1663 3895 y Fm(:)1550
|
|
|
1313 |
3969 y Fr(:)p Fs(B)1672 3938 y Fn(\003)p 1729 3957 10
|
|
|
1314 |
38 v 1739 3940 42 4 v 1882 3969 a FA(and)2197 3895 y
|
|
|
1315 |
Fm(:)2085 3969 y Fr(:)p Fs(C)2205 3938 y Fn(\003)p 2262
|
|
|
1316 |
3957 10 38 v 2272 3940 42 4 v 2355 3969 a Fs(:)523 4137
|
|
|
1317 |
y FA(In)g(order)f(to)i(form)e(an)h(intuitionistic)g(proof)f(ending)g
|
|
|
1318 |
(with)h(the)g(sequent)g Fr(:)p Fm(\()p Fs(B)2773 4107
|
|
|
1319 |
y Fn(\003)2812 4137 y Fr(^)p Fs(C)2932 4107 y Fn(\003)2971
|
|
|
1320 |
4137 y Fm(\))p 3021 4125 10 38 v 3031 4109 42 4 v 88
|
|
|
1321 |
w FA(,)g(we)h(need)523 4237 y(to)i(e)o(xploit)f(the)h(property)e(of)i
|
|
|
1322 |
Fm(\()p Fr(\000)p Fm(\))1511 4207 y Fn(\003)1572 4237
|
|
|
1323 |
y FA(that)g(one)g(can)g(al)o(w)o(ays)g(pro)o(v)o(e)e
|
|
|
1324 |
(intuitionistically)h(the)h(sequent)523 4336 y Fr(::)p
|
|
|
1325 |
Fm(\()p Fr(\000)p Fm(\))762 4306 y Fn(\003)p 819 4324
|
|
|
1326 |
10 38 v 829 4308 42 4 v 889 4336 a Fm(\()p Fr(\000)p
|
|
|
1327 |
Fm(\))1018 4306 y Fn(\003)1056 4336 y FA(.)g(F)o(or)f(e)o(xample)f(in)h
|
|
|
1328 |
(the)g(atomic)g(case)h(one)e(has)i(for)e Fr(::)p Fs(A)2723
|
|
|
1329 |
4306 y Fn(\003)p 2781 4324 10 38 v 2790 4308 42 4 v 2850
|
|
|
1330 |
4336 a Fs(A)2912 4306 y Fn(\003)2973 4336 y FA(the)h(intuition-)523
|
|
|
1331 |
4436 y(istic)g(proof:)1727 4519 y Fr(:)p Fs(A)p 1863
|
|
|
1332 |
4507 10 38 v 1872 4490 42 4 v 88 w Fr(:)p Fs(A)p 1681
|
|
|
1333 |
4539 416 4 v 1681 4612 a Fr(:)p Fs(A;)14 b Fr(::)p Fs(A)p
|
|
|
1334 |
2027 4600 10 38 v 2036 4584 42 4 v 2138 4551 a Fr(:)2193
|
|
|
1335 |
4563 y Fq(L)p 1671 4648 434 4 v 1671 4722 a Fr(:)p Fs(A)p
|
|
|
1336 |
1807 4710 10 38 v 1817 4693 42 4 v 89 w Fr(:::)p Fs(A)2147
|
|
|
1337 |
4660 y Fr(:)2202 4672 y Fq(R)p 1625 4742 527 4 v 1625
|
|
|
1338 |
4815 a Fr(::::)p Fs(A;)g Fr(:)p Fs(A)p 2082 4803 10 38
|
|
|
1339 |
v 2091 4786 42 4 v 2193 4753 a Fr(:)2248 4765 y Fq(L)p
|
|
|
1340 |
1616 4851 545 4 v 1616 4924 a Fr(::::)p Fs(A)p 1918 4912
|
|
|
1341 |
10 38 v 1928 4896 42 4 v 90 w Fr(::)p Fs(A)2202 4863
|
|
|
1342 |
y Fr(:)2257 4875 y Fq(R)p eop end
|
|
|
1343 |
%%Page: 6 6
|
|
|
1344 |
TeXDict begin 6 5 bop 523 448 a FA(Using)26 b(proofs)e(for)i
|
|
|
1345 |
Fr(::)p Fs(B)1287 418 y Fn(\003)p 1344 436 10 38 v 1354
|
|
|
1346 |
420 42 4 v 1414 448 a Fs(B)1481 418 y Fn(\003)1546 448
|
|
|
1347 |
y FA(and)f Fr(::)p Fs(C)1867 418 y Fn(\003)p 1925 436
|
|
|
1348 |
10 38 v 1934 420 42 4 v 1994 448 a Fs(C)2059 418 y Fn(\003)2097
|
|
|
1349 |
448 y FA(,)i(we)f(can)g(construct)f(the)h(follo)n(wing)e(trans-)523
|
|
|
1350 |
548 y(lated)c(proof)f(for)g(\(9\):)963 694 y Fm(:)850
|
|
|
1351 |
768 y Fr(:)p Fs(B)972 738 y Fn(\003)p 1030 756 10 38
|
|
|
1352 |
v 1039 739 42 4 v 823 788 304 4 v 841 850 10 38 v 851
|
|
|
1353 |
833 42 4 v 911 862 a Fr(::)p Fs(B)1088 831 y Fn(\003)1168
|
|
|
1354 |
799 y Fr(:)1223 811 y Fq(R)1554 788 y Fm(:)1361 862 y
|
|
|
1355 |
Fr(::)p Fs(B)1538 831 y Fn(\003)p 1596 850 10 38 v 1605
|
|
|
1356 |
833 42 4 v 1665 862 a Fs(B)1732 831 y Fn(\003)p 823 881
|
|
|
1357 |
948 4 v 1218 943 10 38 v 1228 927 42 4 v 1288 955 a Fs(B)1355
|
|
|
1358 |
925 y Fn(\003)1812 907 y Fs(cut)2148 694 y Fm(:)2036
|
|
|
1359 |
768 y Fr(:)p Fs(C)2156 738 y Fn(\003)p 2213 756 10 38
|
|
|
1360 |
v 2223 739 42 4 v 2008 788 303 4 v 2027 850 10 38 v 2036
|
|
|
1361 |
833 42 4 v 2096 862 a Fr(::)p Fs(C)2271 831 y Fn(\003)2352
|
|
|
1362 |
799 y Fr(:)2407 811 y Fq(R)2736 788 y Fm(:)2545 862 y
|
|
|
1363 |
Fr(::)p Fs(C)2720 831 y Fn(\003)p 2777 850 10 38 v 2787
|
|
|
1364 |
833 42 4 v 2847 862 a Fs(C)2912 831 y Fn(\003)p 2008
|
|
|
1365 |
881 942 4 v 2402 943 10 38 v 2412 927 42 4 v 2472 955
|
|
|
1366 |
a Fs(C)2537 925 y Fn(\003)2992 907 y Fs(cut)p 1200 975
|
|
|
1367 |
1376 4 v 1730 1037 10 38 v 1739 1020 42 4 v 1799 1049
|
|
|
1368 |
a(B)1866 1019 y Fn(\003)1905 1049 y Fr(^)p Fs(C)2025
|
|
|
1369 |
1019 y Fn(\003)2616 992 y Fr(^)2672 1004 y Fq(R)p 1651
|
|
|
1370 |
1069 472 4 v 1651 1148 a Fr(:)p Fm(\()p Fs(B)1805 1118
|
|
|
1371 |
y Fn(\003)1844 1148 y Fr(^)q Fs(C)1965 1118 y Fn(\003)2003
|
|
|
1372 |
1148 y Fm(\))p 2054 1136 10 38 v 2063 1119 42 4 v 2165
|
|
|
1373 |
1081 a Fr(:)2220 1093 y Fq(L)523 1332 y FA(W)-7 b(e)31
|
|
|
1374 |
b(shall)f(refer)f(to)h(the)f(cuts)h(introduced)e(by)h(the)h(double-ne)o
|
|
|
1375 |
(gation)25 b(translation)k(as)h Ft(auxiliary)523 1432
|
|
|
1376 |
y(cuts)p FA(.)25 b(F)o(or)f(a)h(number)e(of)h(reasons)g(\(one)g(of)g
|
|
|
1377 |
(them)g(being)g(to)g(minimise)g(the)h(amount)e(of)h(writing\))523
|
|
|
1378 |
1531 y(we)d(shall)f(use)h(a)f(ne)n(w)g(inference)f(rule,)g(namely)p
|
|
|
1379 |
1746 1686 10 38 v 1756 1670 42 4 v 1816 1698 a Fr(::)p
|
|
|
1380 |
Fs(B)p 1728 1718 266 4 v 1802 1780 10 38 v 1811 1763
|
|
|
1381 |
42 4 v 1871 1792 a(B)2035 1730 y Fr(::)2145 1742 y Fq(R)523
|
|
|
1382 |
1976 y FA(to)h(stand)g(for)g(auxiliary)f(cuts,)h(which)g(ha)n(v)o(e)f
|
|
|
1383 |
(al)o(w)o(ays)i(the)f(form:)1644 2122 y Fm(:)1491 2195
|
|
|
1384 |
y Fs(\000)p 1572 2183 10 38 v 1582 2167 42 4 v 100 w
|
|
|
1385 |
Fr(::)p Fs(B)2058 2122 y Fm(:)1903 2195 y Fr(::)p Fs(B)p
|
|
|
1386 |
2099 2183 10 38 v 2109 2167 42 4 v 92 w(B)p 1491 2215
|
|
|
1387 |
745 4 v 1755 2289 a(\000)p 1836 2277 10 38 v 1845 2260
|
|
|
1388 |
42 4 v 99 w(B)2277 2241 y(cut)j(:)523 2473 y FA(Clearly)-5
|
|
|
1389 |
b(,)28 b(this)h(ne)n(w)g(rule)g(does)f(not)g(af)n(fect)h(the)g(pro)o(v)
|
|
|
1390 |
n(ability)d(of)i(sequents.)g(Issues)i(whether)d(the)523
|
|
|
1391 |
2573 y Fr(::)633 2585 y Fq(R)688 2573 y FA(-rule)22 b(\(or)g(an)g
|
|
|
1392 |
(auxiliary)f(cut\))i(af)n(fects)f(the)g(beha)n(viour)f(under)g
|
|
|
1393 |
(cut-elimination)f(are)j(delayed)523 2672 y(until)f(Sec.)g(4.)g(W)m
|
|
|
1394 |
(ith)g(this)g(ne)n(w)g(inference)e(rule)i(we)g(can)g(gi)n(v)o(e)f(the)g
|
|
|
1395 |
(translation)g(of)h(\(9\))f(more)g(com-)523 2772 y(pactly)f(as:)1553
|
|
|
1396 |
2836 y Fm(:)1440 2909 y Fr(:)p Fs(B)1562 2879 y Fn(\003)p
|
|
|
1397 |
1620 2897 10 38 v 1629 2881 42 4 v 1413 2929 304 4 v
|
|
|
1398 |
1431 2991 10 38 v 1441 2974 42 4 v 1501 3003 a Fr(::)p
|
|
|
1399 |
Fs(B)1678 2973 y Fn(\003)1758 2941 y Fr(:)1813 2953 y
|
|
|
1400 |
Fq(R)p 1413 3023 304 4 v 1487 3085 10 38 v 1496 3068
|
|
|
1401 |
42 4 v 1556 3097 a Fs(B)1623 3067 y Fn(\003)1758 3035
|
|
|
1402 |
y Fr(::)1868 3047 y Fq(R)2146 2836 y Fm(:)2034 2909 y
|
|
|
1403 |
Fr(:)p Fs(C)2154 2879 y Fn(\003)p 2211 2897 10 38 v 2221
|
|
|
1404 |
2881 42 4 v 2006 2929 303 4 v 2025 2991 10 38 v 2034
|
|
|
1405 |
2974 42 4 v 2094 3003 a Fr(::)p Fs(C)2269 2973 y Fn(\003)2350
|
|
|
1406 |
2941 y Fr(:)2405 2953 y Fq(R)p 2006 3023 303 4 v 2080
|
|
|
1407 |
3085 10 38 v 2090 3068 42 4 v 2150 3097 a Fs(C)2215 3067
|
|
|
1408 |
y Fn(\003)2350 3035 y Fr(::)2460 3047 y Fq(R)p 1468 3117
|
|
|
1409 |
786 4 v 1703 3178 10 38 v 1713 3162 42 4 v 1773 3190
|
|
|
1410 |
a Fs(B)1840 3160 y Fn(\003)1878 3190 y Fr(^)p Fs(C)1998
|
|
|
1411 |
3160 y Fn(\003)2295 3134 y Fr(^)2350 3146 y Fq(R)p 1625
|
|
|
1412 |
3210 472 4 v 1625 3289 a Fr(:)p Fm(\()p Fs(B)1779 3259
|
|
|
1413 |
y Fn(\003)1818 3289 y Fr(^)p Fs(C)1938 3259 y Fn(\003)1976
|
|
|
1414 |
3289 y Fm(\))p 2027 3277 10 38 v 2037 3261 42 4 v 2138
|
|
|
1415 |
3222 a Fr(:)2193 3234 y Fq(L)523 3441 y FA(The)h(translations)f(for)g
|
|
|
1416 |
(the)h(rules)f Fs(contr)1694 3453 y Fq(L)1745 3441 y
|
|
|
1417 |
FA(,)h Fs(contr)1980 3453 y Fq(R)2035 3441 y FA(,)g Fs(w)r(eak)2264
|
|
|
1418 |
3453 y Fq(L)2314 3441 y FA(,)g Fs(w)r(eak)2543 3453 y
|
|
|
1419 |
Fq(R)2598 3441 y FA(,)g Fr(:)2695 3453 y Fq(L)2745 3441
|
|
|
1420 |
y FA(,)g Fr(:)2842 3453 y Fq(R)2897 3441 y FA(,)g Fr(_)2995
|
|
|
1421 |
3453 y Fq(R)3045 3461 y Fk(i)3075 3441 y FA(,)g Fr(^)3173
|
|
|
1422 |
3453 y Fq(L)3219 3461 y Fk(i)3249 3441 y FA(,)g Fr(\033)3355
|
|
|
1423 |
3453 y Fq(L)523 3541 y FA(and)f Fr(\033)728 3553 y Fq(R)803
|
|
|
1424 |
3541 y FA(are)h(left)f(as)h(e)o(x)o(ercises)e(to)i(the)f(reader)-5
|
|
|
1425 |
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
|
|
|
1426 |
(gation)c(translations)j(of)h(the)g(tw)o(o)h(subproofs)1033
|
|
|
1427 |
3808 y Fs(A)p 1114 3796 10 38 v 1123 3779 42 4 v 88 w(A)84
|
|
|
1428 |
b(A)p 1409 3796 10 38 v 1419 3779 42 4 v 88 w(A)p 1033
|
|
|
1429 |
3828 509 4 v 1054 3901 a(A)19 b Fr(_)f Fs(A)p 1289 3889
|
|
|
1430 |
10 38 v 1299 3872 42 4 v 89 w(A;)c(A)1583 3845 y Fr(_)1638
|
|
|
1431 |
3857 y Fq(L)p 1054 3937 467 4 v 1104 4011 a Fs(A)k Fr(_)h
|
|
|
1432 |
Fs(A)p 1339 3999 10 38 v 1348 3982 42 4 v 88 w(A)1562
|
|
|
1433 |
3956 y(contr)1755 3968 y Fq(R)2142 3808 y Fs(A)p 2222
|
|
|
1434 |
3796 10 38 v 2232 3779 42 4 v 88 w(A)83 b(A)p 2518 3796
|
|
|
1435 |
10 38 v 2527 3779 42 4 v 88 w(A)p 2142 3828 509 4 v 2181
|
|
|
1436 |
3901 a(A;)14 b(A)p 2361 3889 10 38 v 2370 3872 42 4 v
|
|
|
1437 |
88 w(A)p Fr(^)q Fs(A)2691 3845 y Fr(^)2746 3857 y Fq(R)p
|
|
|
1438 |
2181 3937 430 4 v 2231 4011 a Fs(A)p 2311 3999 10 38
|
|
|
1439 |
v 2321 3982 42 4 v 88 w(A)p Fr(^)p Fs(A)2652 3956 y(contr)2845
|
|
|
1440 |
3968 y Fq(L)523 4195 y FA(sho)n(wn)19 b(in)i(\(1\).)e(The)h
|
|
|
1441 |
(translations)g(are:)574 4429 y Fz(::)p Fj(A)p 751 4417
|
|
|
1442 |
9 34 v 760 4402 38 4 v 80 w Fz(::)p Fj(A)p 532 4449 486
|
|
|
1443 |
4 v 532 4516 a Fz(::)p Fj(A;)13 b Fz(:::)p Fj(A)p 954
|
|
|
1444 |
4504 9 34 v 963 4489 38 4 v 1058 4461 a Fz(:)1109 4469
|
|
|
1445 |
y Fi(L)p 523 4551 503 4 v 523 4618 a Fz(:::)p Fj(A)p
|
|
|
1446 |
751 4606 9 34 v 760 4591 38 4 v 80 w Fz(:::)p Fj(A)1067
|
|
|
1447 |
4563 y Fz(:)1118 4571 y Fi(R)1294 4429 y Fz(::)p Fj(A)p
|
|
|
1448 |
1471 4417 9 34 v 1480 4402 38 4 v 80 w Fz(::)p Fj(A)p
|
|
|
1449 |
1252 4449 486 4 v 1252 4516 a Fz(::)p Fj(A;)g Fz(:::)p
|
|
|
1450 |
Fj(A)p 1674 4504 9 34 v 1683 4489 38 4 v 1779 4461 a
|
|
|
1451 |
Fz(:)1830 4469 y Fi(L)p 1243 4551 503 4 v 1243 4618 a
|
|
|
1452 |
Fz(:::)p Fj(A)p 1471 4606 9 34 v 1480 4591 38 4 v 80
|
|
|
1453 |
w Fz(:::)p Fj(A)1787 4563 y Fz(:)1838 4571 y Fi(R)p 523
|
|
|
1454 |
4638 1223 4 v 629 4706 a Fz(:::)p Fj(A;)h Fz(:::)p Fj(A)p
|
|
|
1455 |
1103 4694 9 34 v 1111 4679 38 4 v 80 w Fz(:::)p Fj(A)p
|
|
|
1456 |
Fz(^:::)p Fj(A)1787 4655 y Fz(^)1838 4663 y Fi(R)p 557
|
|
|
1457 |
4740 1156 4 v 557 4813 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p
|
|
|
1458 |
Fz(^:::)p Fj(A)p Fh(\))p Fj(;)f Fz(:::)p Fj(A;)h Fz(:::)p
|
|
|
1459 |
Fj(A)p 1649 4801 9 34 v 1657 4786 38 4 v 1753 4752 a
|
|
|
1460 |
Fz(:)1804 4760 y Fi(L)p 557 4852 1156 4 v 679 4924 a
|
|
|
1461 |
Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p
|
|
|
1462 |
Fj(;)f Fz(:::)p Fj(A)p 1526 4912 9 34 v 1535 4897 38
|
|
|
1463 |
4 v 1753 4871 a(contr)1932 4879 y Fi(L)2030 4341 y Fz(::)p
|
|
|
1464 |
Fj(A)p 2207 4329 9 34 v 2216 4314 38 4 v 80 w Fz(::)p
|
|
|
1465 |
Fj(A)p 1987 4361 486 4 v 1987 4429 a Fz(::)p Fj(A;)g
|
|
|
1466 |
Fz(:::)p Fj(A)p 2410 4417 9 34 v 2418 4402 38 4 v 2514
|
|
|
1467 |
4373 a Fz(:)2565 4381 y Fi(L)p 1979 4463 503 4 v 1979
|
|
|
1468 |
4531 a Fz(::)p Fj(A)p 2156 4519 9 34 v 2164 4504 38 4
|
|
|
1469 |
v 80 w Fz(::::)p Fj(A)2523 4475 y Fz(:)2574 4483 y Fi(R)p
|
|
|
1470 |
1979 4551 503 4 v 2030 4618 a Fz(::)p Fj(A)p 2207 4606
|
|
|
1471 |
9 34 v 2216 4591 38 4 v 80 w Fz(::)p Fj(A)2523 4563 y
|
|
|
1472 |
Fz(::)2625 4571 y Fi(R)2801 4341 y Fz(::)p Fj(A)p 2978
|
|
|
1473 |
4329 9 34 v 2987 4314 38 4 v 80 w Fz(::)p Fj(A)p 2759
|
|
|
1474 |
4361 486 4 v 2759 4429 a Fz(::)p Fj(A;)g Fz(:::)p Fj(A)p
|
|
|
1475 |
3181 4417 9 34 v 3190 4402 38 4 v 3286 4373 a Fz(:)3337
|
|
|
1476 |
4381 y Fi(L)p 2750 4463 503 4 v 2750 4531 a Fz(::)p Fj(A)p
|
|
|
1477 |
2927 4519 9 34 v 2936 4504 38 4 v 80 w Fz(::::)p Fj(A)3294
|
|
|
1478 |
4475 y Fz(:)3345 4483 y Fi(R)p 2750 4551 503 4 v 2801
|
|
|
1479 |
4618 a Fz(::)p Fj(A)p 2978 4606 9 34 v 2987 4591 38 4
|
|
|
1480 |
v 80 w Fz(::)p Fj(A)3294 4563 y Fz(::)3396 4571 y Fi(R)p
|
|
|
1481 |
2030 4638 1172 4 v 2213 4706 a Fz(::)p Fj(A;)g Fz(::)p
|
|
|
1482 |
Fj(A)p 2584 4694 9 34 v 2593 4679 38 4 v 80 w Fz(::)p
|
|
|
1483 |
Fj(A)p Fz(^::)p Fj(A)3243 4655 y Fz(^)3294 4663 y Fi(R)p
|
|
|
1484 |
2141 4740 951 4 v 2141 4813 a Fz(::)p Fj(A;)g Fz(::)p
|
|
|
1485 |
Fj(A;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p Fj(A)p
|
|
|
1486 |
Fh(\))p 3028 4801 9 34 v 3036 4786 38 4 v 3132 4752 a
|
|
|
1487 |
Fz(:)3183 4760 y Fi(L)p 2141 4852 951 4 v 2238 4924 a
|
|
|
1488 |
Fz(::)p Fj(A;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p
|
|
|
1489 |
Fj(A)p Fh(\))p 2931 4912 9 34 v 2939 4897 38 4 v 3132
|
|
|
1490 |
4871 a Fj(contr)3311 4879 y Fi(L)p eop end
|
|
|
1491 |
%%Page: 7 7
|
|
|
1492 |
TeXDict begin 7 6 bop 523 448 a FA(T)-7 b(o)16 b(gi)n(v)o(e)f(a)i
|
|
|
1493 |
(double-ne)o(gation)11 b(translation)16 b(for)f(the)h(whole)f(proof,)g
|
|
|
1494 |
(we)h(need)f(to)i(be)f(able)f(to)i(translate)523 548
|
|
|
1495 |
y(instances)k(of)f(the)h(cut-rule.)e(While)i(the)g(logical)g(inference)
|
|
|
1496 |
e(rules)h(and)h(the)f(structural)g(rules)h(ha)n(v)o(e)523
|
|
|
1497 |
648 y(relati)n(v)o(ely)g(canonical)h(double-ne)o(gation)17
|
|
|
1498 |
b(translations,)22 b(there)g(is)i(a)e(choice)g(for)g(ho)n(w)g(to)h
|
|
|
1499 |
(translate)523 747 y(cut-rules.)c(Consider)h(the)g(follo)n(wing)e
|
|
|
1500 |
(cut-instance:)1725 886 y Fs(\031)1772 898 y Fl(1)1756
|
|
|
1501 |
951 y Fm(:)p 1708 1012 10 38 v 1718 996 42 4 v 1778 1024
|
|
|
1502 |
a Fs(B)1963 886 y(\031)2010 898 y Fl(2)1994 951 y Fm(:)1928
|
|
|
1503 |
1024 y Fs(B)p 2014 1012 10 38 v 2023 996 42 4 v 1690
|
|
|
1504 |
1044 394 4 v 1861 1098 10 38 v 1871 1081 42 4 v 2125
|
|
|
1505 |
1070 a(cut)3267 1110 y FA(\(10\))523 1287 y(By)f(induction)e
|
|
|
1506 |
(hypothesis)h(we)h(ha)n(v)o(e)f(tw)o(o)h(intuitionistic)g(proofs)e
|
|
|
1507 |
Fs(\031)2484 1257 y Fn(\003)2481 1307 y Fl(1)2540 1287
|
|
|
1508 |
y FA(and)h Fs(\031)2727 1257 y Fn(\003)2724 1307 y Fl(2)2783
|
|
|
1509 |
1287 y FA(with)h(end-sequents:)1574 1450 y Fs(\031)1624
|
|
|
1510 |
1420 y Fn(\003)1621 1470 y Fl(1)1607 1523 y Fm(:)1494
|
|
|
1511 |
1597 y Fr(:)p Fs(B)1616 1566 y Fn(\003)p 1673 1585 10
|
|
|
1512 |
38 v 1683 1568 42 4 v 1909 1597 a FA(and)2247 1450 y
|
|
|
1513 |
Fs(\031)2297 1420 y Fn(\003)2294 1470 y Fl(2)2280 1523
|
|
|
1514 |
y Fm(:)2195 1597 y Fs(B)2262 1566 y Fn(\003)p 2318 1585
|
|
|
1515 |
10 38 v 2328 1568 42 4 v 2411 1597 a Fs(:)523 1773 y
|
|
|
1516 |
FA(T)-7 b(o)17 b(form)e(an)i(intuitionistic)f(proof)f(with)h(the)h
|
|
|
1517 |
(end-sequent)p 2254 1761 10 38 v 2263 1745 42 4 v 119
|
|
|
1518 |
w(\(remember)e(we)i(omit)f(the)h(sequent-)523 1873 y(conte)o(xts\),)i
|
|
|
1519 |
(we)h(can)g(translate)g(\(10\))f(either)h(as:)1005 2217
|
|
|
1520 |
y Fs(\031)1055 2187 y Fn(\003)1052 2238 y Fl(1)1038 2290
|
|
|
1521 |
y Fm(:)925 2364 y Fr(:)p Fs(B)1047 2334 y Fn(\003)p 1104
|
|
|
1522 |
2352 10 38 v 1113 2336 42 4 v 1337 2124 a Fs(\031)1387
|
|
|
1523 |
2094 y Fn(\003)1384 2144 y Fl(2)1369 2197 y Fm(:)1284
|
|
|
1524 |
2270 y Fs(B)1351 2240 y Fn(\003)p 1408 2258 10 38 v 1417
|
|
|
1525 |
2242 42 4 v 1256 2290 249 4 v 1275 2352 10 38 v 1284
|
|
|
1526 |
2336 42 4 v 1344 2364 a Fr(:)p Fs(B)1466 2334 y Fn(\003)1547
|
|
|
1527 |
2302 y Fr(:)1602 2314 y Fq(R)p 925 2384 581 4 v 1189
|
|
|
1528 |
2438 10 38 v 1199 2421 42 4 v 1547 2410 a Fs(cut)1826
|
|
|
1529 |
2332 y FA(or)2169 2030 y Fs(\031)2219 2000 y Fn(\003)2216
|
|
|
1530 |
2051 y Fl(1)2202 2103 y Fm(:)2089 2177 y Fr(:)p Fs(B)2211
|
|
|
1531 |
2147 y Fn(\003)p 2268 2165 10 38 v 2278 2148 42 4 v 2061
|
|
|
1532 |
2197 304 4 v 2080 2258 10 38 v 2089 2242 42 4 v 2149
|
|
|
1533 |
2270 a Fr(::)p Fs(B)2326 2240 y Fn(\003)2407 2208 y Fr(:)2462
|
|
|
1534 |
2220 y Fq(R)p 2061 2290 304 4 v 2135 2352 10 38 v 2145
|
|
|
1535 |
2336 42 4 v 2205 2364 a Fs(B)2272 2334 y Fn(\003)2407
|
|
|
1536 |
2302 y Fr(::)2517 2314 y Fq(R)2707 2217 y Fs(\031)2757
|
|
|
1537 |
2187 y Fn(\003)2754 2238 y Fl(2)2740 2290 y Fm(:)2655
|
|
|
1538 |
2364 y Fs(B)2722 2334 y Fn(\003)p 2779 2352 10 38 v 2788
|
|
|
1539 |
2336 42 4 v 2117 2384 732 4 v 2457 2438 10 38 v 2466
|
|
|
1540 |
2421 42 4 v 2890 2410 a Fs(cut)523 2627 y FA(W)-7 b(e)21
|
|
|
1541 |
b(refer)f(to)g(these)h(choices)e(as)i Ft(left-)g FA(and)e
|
|
|
1542 |
Ft(right-tr)o(anslation)f FA(of)i(a)h(cut,)f(respecti)n(v)o(ely)-5
|
|
|
1543 |
b(.)648 2726 y(Coming)15 b(back)g(to)h(the)g(proof)e(gi)n(v)o(en)h(in)h
|
|
|
1544 |
(\(1\),)f(let)h(us)h(call)f(the)g(left-)g(and)f(right-translation)e(of)
|
|
|
1545 |
j(this)523 2826 y(proof)22 b Fs(\031)777 2796 y Fn(\003)774
|
|
|
1546 |
2849 y Fq(L)847 2826 y FA(and)h Fs(\031)1041 2796 y Fn(\003)1038
|
|
|
1547 |
2849 y Fq(R)1093 2826 y FA(,)g(respecti)n(v)o(ely)-5
|
|
|
1548 |
b(.)21 b(Eliminating)h(all)h(cuts)h(\(including)d(auxiliary)h(cuts\))h
|
|
|
1549 |
(from)f Fs(\031)3358 2796 y Fn(\003)3355 2849 y Fq(L)523
|
|
|
1550 |
2925 y FA(and)e Fs(\031)714 2895 y Fn(\003)711 2948 y
|
|
|
1551 |
Fq(R)766 2925 y FA(,)h(we)g(obtain)f(the)h(tw)o(o)g(cut-free)f(proofs)f
|
|
|
1552 |
Fs(\031)1986 2895 y Fn(0\003)1983 2948 y Fq(L)2065 2925
|
|
|
1553 |
y FA(and)h Fs(\031)2256 2895 y Fn(0\003)2253 2948 y Fq(R)2335
|
|
|
1554 |
2925 y FA(sho)n(wn)g(in)h(Fig.)g(1.)g(It)g(turns)f(out)g(\(we)523
|
|
|
1555 |
3025 y(ho)n(we)n(v)o(er)d(lea)n(v)o(e)i(out)f(the)h(calculations\))f
|
|
|
1556 |
(that)h(had)f(we)i(double-ne)o(gation)14 b(translated)k(the)h(tw)o(o)g
|
|
|
1557 |
(nor)n(-)523 3125 y(malforms)h(of)i(\(1\))f(and)g(then)g(eliminated)g
|
|
|
1558 |
(all)h(auxiliary)e(cuts)i(from)f(the)h(double-ne)o(gated)17
|
|
|
1559 |
b(proofs,)523 3224 y(we)i(w)o(ould)e(ha)n(v)o(e)h(also)g(obtained)f
|
|
|
1560 |
(the)h(proofs)f Fs(\031)1895 3194 y Fn(0\003)1892 3247
|
|
|
1561 |
y Fq(L)1972 3224 y FA(and)g Fs(\031)2160 3194 y Fn(0\003)2157
|
|
|
1562 |
3247 y Fq(R)2218 3224 y FA(:)i(The)f(normalform)d(\(2\))j(obtained)f
|
|
|
1563 |
(from)523 3324 y(\(1\))g(by)h(commuting)e(the)i(cut)g(to)g(the)h(left)f
|
|
|
1564 |
(leads)g(to)g Fs(\031)2032 3294 y Fn(0\003)2029 3347
|
|
|
1565 |
y Fq(L)2090 3324 y FA(\227the)g(normalform)d(of)j(the)g
|
|
|
1566 |
(left-translation)523 3424 y(of)23 b(\(1\),)f(while)g(\(3\))h(obtained)
|
|
|
1567 |
e(by)h(commuting)f(the)i(cut)g(to)g(the)f(right)h(leads)g(to)g
|
|
|
1568 |
Fs(\031)2877 3393 y Fn(0\003)2874 3446 y Fq(R)2934 3424
|
|
|
1569 |
y FA(\227the)g(normal-)523 3523 y(form)30 b(of)g(the)g
|
|
|
1570 |
(right-translation)e(of)j(\(1\).)e(W)-7 b(e)32 b(tak)o(e)f(this)g(as)g
|
|
|
1571 |
(a)g(\002rst)g(hint)f(that)h(double-ne)o(gation)523 3623
|
|
|
1572 |
y(translations)d(seem)g(to)h(be)f(able)g(to)h(simulate)f(the)h(beha)n
|
|
|
1573 |
(viour)d(of)i(cut-elimination)e(in)j(classical)523 3722
|
|
|
1574 |
y(logic.)648 3822 y(T)-7 b(o)16 b(sum)g(up)f(this)i(section,)e(let)i
|
|
|
1575 |
(us)f(remark)f(that)h(a)g(similar)g(\223story\224)f(can)h(be)g(told)g
|
|
|
1576 |
(for)f(the)h(transla-)523 3922 y(tion)k Fm(\()p Fr(\000)p
|
|
|
1577 |
Fm(\))802 3892 y Fn(\016)862 3922 y FA(gi)n(v)o(en)f(in)i(\(6\).)e(In)i
|
|
|
1578 |
(f)o(act,)f(it)h(can)g(be)f(told)g(for)g(an)o(y)g(sensible)g(notion)g
|
|
|
1579 |
(of)g(double-ne)o(gation)523 4021 y(translation.)27 b(F)o(or)h(e)o
|
|
|
1580 |
(xample)f(it)i(w)o(ould)e(be)h(completely)f(immaterial)h(to)g(our)f
|
|
|
1581 |
(\223story\224)h(if)g(we)h(had)523 4121 y(translated)g(atomic)f
|
|
|
1582 |
(formulae)g Fs(A)i FA(as)g Fr(::)p Fs(A)p FA(,)g Fr(::::)p
|
|
|
1583 |
Fs(A)h FA(or)e Fr(::::::)p Fs(A)p FA(.)j(The)d(most)g(important)523
|
|
|
1584 |
4221 y(property)17 b(we)j(distill)g(from)f(the)g(ar)o(guments)f(abo)o
|
|
|
1585 |
(v)o(e)g(is)i(that)g(we)g(re)o(gard)d(a)j(double-ne)o(gation)15
|
|
|
1586 |
b(trans-)523 4320 y(lation,)20 b(say)g Fm(\()p Fr(\000)p
|
|
|
1587 |
Fm(\))1014 4290 y Fq(x)1056 4320 y FA(,)h(as)f(a)h(translation)e(of)h
|
|
|
1588 |
(a)h(classical)g(proof)d(ha)n(ving)i(an)g(end-sequent)1952
|
|
|
1589 |
4462 y Fm(:)1854 4535 y Fs(\000)p 1935 4523 10 38 v 1945
|
|
|
1590 |
4507 42 4 v 100 w(\001)523 4712 y FA(to)g(an)h(intuitionistic)e(proof)g
|
|
|
1591 |
(with)h(the)g(end-sequent)1952 4851 y Fm(:)1766 4924
|
|
|
1592 |
y Fs(\000)1829 4894 y Fq(x)1871 4924 y Fs(;)14 b Fr(:)p
|
|
|
1593 |
Fs(\001)2032 4894 y Fq(x)p 2092 4912 10 38 v 2102 4896
|
|
|
1594 |
42 4 v eop end
|
|
|
1595 |
%%Page: 8 8
|
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|
1596 |
TeXDict begin 8 7 bop 523 369 2882 4 v 523 2639 4 2271
|
|
|
1597 |
v 593 457 a Fg(\031)649 426 y Ff(0\003)646 473 y Fe(L)734
|
|
|
1598 |
457 y Fd(=)727 536 y Fz(:)p Fj(A)p 853 524 9 34 v 861
|
|
|
1599 |
509 38 4 v 80 w Fz(:)p Fj(A)p 685 556 383 4 v 685 623
|
|
|
1600 |
a Fz(:)p Fj(A;)12 b Fz(::)p Fj(A)p 1004 611 9 34 v 1013
|
|
|
1601 |
597 38 4 v 1109 568 a Fz(:)1160 576 y Fi(L)p 676 658
|
|
|
1602 |
401 4 v 676 726 a Fz(:)p Fj(A)p 801 714 9 34 v 810 699
|
|
|
1603 |
38 4 v 79 w Fz(:::)p Fj(A)1117 670 y Fz(:)1168 678 y
|
|
|
1604 |
Fi(R)1345 536 y Fz(:)p Fj(A)p 1470 524 9 34 v 1479 509
|
|
|
1605 |
38 4 v 79 w Fz(:)p Fj(A)p 1302 556 383 4 v 1302 623 a
|
|
|
1606 |
Fz(:)p Fj(A;)h Fz(::)p Fj(A)p 1622 611 9 34 v 1630 597
|
|
|
1607 |
38 4 v 1726 568 a Fz(:)1777 576 y Fi(L)p 1293 658 401
|
|
|
1608 |
4 v 1293 726 a Fz(:)p Fj(A)p 1419 714 9 34 v 1428 699
|
|
|
1609 |
38 4 v 80 w Fz(:::)p Fj(A)1735 670 y Fz(:)1786 678 y
|
|
|
1610 |
Fi(R)p 676 746 1018 4 v 782 813 a Fz(:)p Fj(A;)g Fz(:)p
|
|
|
1611 |
Fj(A)p 1051 801 9 34 v 1059 786 38 4 v 80 w Fz(:::)p
|
|
|
1612 |
Fj(A)p Fz(^:::)p Fj(A)1735 762 y Fz(^)1786 770 y Fi(R)p
|
|
|
1613 |
709 848 951 4 v 709 920 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p
|
|
|
1614 |
Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fj(A;)g Fz(:)p
|
|
|
1615 |
Fj(A)p 1597 908 9 34 v 1605 893 38 4 v 1701 860 a Fz(:)1752
|
|
|
1616 |
868 y Fi(L)p 709 959 951 4 v 781 1032 a Fz(:)p Fh(\()p
|
|
|
1617 |
Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p
|
|
|
1618 |
Fj(A)p 1525 1020 9 34 v 1534 1005 38 4 v 1701 978 a(contr)1880
|
|
|
1619 |
986 y Fi(L)p 772 1070 825 4 v 772 1143 a Fz(:)p Fh(\()p
|
|
|
1620 |
Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p 1374 1131
|
|
|
1621 |
9 34 v 1382 1116 38 4 v 80 w Fz(::)p Fj(A)1638 1082 y
|
|
|
1622 |
Fz(:)1689 1090 y Fi(R)2053 536 y Fz(:)p Fj(A)p 2178 524
|
|
|
1623 |
9 34 v 2187 509 38 4 v 79 w Fz(:)p Fj(A)p 2010 556 383
|
|
|
1624 |
4 v 2010 623 a Fz(:)p Fj(A;)g Fz(::)p Fj(A)p 2330 611
|
|
|
1625 |
9 34 v 2338 597 38 4 v 2434 568 a Fz(:)2485 576 y Fi(L)p
|
|
|
1626 |
2001 658 401 4 v 2001 726 a Fz(:)p Fj(A)p 2127 714 9
|
|
|
1627 |
34 v 2136 699 38 4 v 80 w Fz(:::)p Fj(A)2443 670 y Fz(:)2494
|
|
|
1628 |
678 y Fi(R)2670 536 y Fz(:)p Fj(A)p 2796 524 9 34 v 2805
|
|
|
1629 |
509 38 4 v 80 w Fz(:)p Fj(A)p 2628 556 383 4 v 2628 623
|
|
|
1630 |
a Fz(:)p Fj(A;)g Fz(::)p Fj(A)p 2948 611 9 34 v 2956
|
|
|
1631 |
597 38 4 v 3052 568 a Fz(:)3103 576 y Fi(L)p 2619 658
|
|
|
1632 |
401 4 v 2619 726 a Fz(:)p Fj(A)p 2745 714 9 34 v 2754
|
|
|
1633 |
699 38 4 v 80 w Fz(:::)p Fj(A)3061 670 y Fz(:)3112 678
|
|
|
1634 |
y Fi(R)p 2001 746 1018 4 v 2108 813 a Fz(:)p Fj(A;)f
|
|
|
1635 |
Fz(:)p Fj(A)p 2376 801 9 34 v 2385 786 38 4 v 80 w Fz(:::)p
|
|
|
1636 |
Fj(A)p Fz(^)q(:::)p Fj(A)3061 762 y Fz(^)3112 770 y Fi(R)p
|
|
|
1637 |
2035 848 951 4 v 2035 920 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p
|
|
|
1638 |
Fz(^:::)p Fj(A)p Fh(\))p Fj(;)i Fz(:)p Fj(A;)f Fz(:)p
|
|
|
1639 |
Fj(A)p 2922 908 9 34 v 2931 893 38 4 v 3027 860 a Fz(:)3078
|
|
|
1640 |
868 y Fi(L)p 2035 959 951 4 v 2106 1032 a Fz(:)p Fh(\()p
|
|
|
1641 |
Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p
|
|
|
1642 |
Fj(A)p 2851 1020 9 34 v 2860 1005 38 4 v 3027 978 a(contr)3206
|
|
|
1643 |
986 y Fi(L)p 2098 1070 825 4 v 2098 1143 a Fz(:)p Fh(\()p
|
|
|
1644 |
Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p Fh(\))p 2700 1131 9
|
|
|
1645 |
34 v 2708 1116 38 4 v 81 w Fz(::)p Fj(A)2964 1082 y Fz(:)3015
|
|
|
1646 |
1090 y Fi(R)p 772 1181 2151 4 v 1020 1254 a Fz(:)p Fh(\()p
|
|
|
1647 |
Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p
|
|
|
1648 |
Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p Fh(\))p 2240
|
|
|
1649 |
1242 9 34 v 2249 1227 38 4 v 80 w Fz(::)p Fj(A)p Fz(^)q(::)p
|
|
|
1650 |
Fj(A)2964 1198 y Fz(^)3015 1206 y Fi(R)p 948 1293 1800
|
|
|
1651 |
4 v 948 1365 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p
|
|
|
1652 |
Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(:::)p Fj(A)p
|
|
|
1653 |
Fz(^)q(:::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(::)p
|
|
|
1654 |
Fj(A)p Fz(^::)p Fj(A)p Fh(\))p 2684 1353 9 34 v 2693
|
|
|
1655 |
1338 38 4 v 2789 1305 a Fz(:)2840 1313 y Fi(L)p 948 1404
|
|
|
1656 |
1800 4 v 1257 1477 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p
|
|
|
1657 |
Fj(A)p Fh(\))p Fj(;)h Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p
|
|
|
1658 |
Fj(A)p Fh(\))p 2375 1465 9 34 v 2383 1450 38 4 v 2789
|
|
|
1659 |
1423 a Fj(contr)2968 1431 y Fi(L)593 1568 y Fg(\031)649
|
|
|
1660 |
1536 y Ff(0\003)646 1583 y Fe(R)734 1568 y Fd(=)727 1646
|
|
|
1661 |
y Fz(:)p Fj(A)p 853 1634 9 34 v 861 1619 38 4 v 80 w
|
|
|
1662 |
Fz(:)p Fj(A)p 685 1666 383 4 v 685 1734 a Fz(::)p Fj(A;)e
|
|
|
1663 |
Fz(:)p Fj(A)p 1004 1722 9 34 v 1013 1707 38 4 v 1109
|
|
|
1664 |
1678 a Fz(:)1160 1686 y Fi(L)p 676 1768 401 4 v 676 1836
|
|
|
1665 |
a Fz(::)p Fj(A)p 853 1824 9 34 v 861 1809 38 4 v 80 w
|
|
|
1666 |
Fz(::)p Fj(A)1117 1780 y Fz(:)1168 1788 y Fi(R)1345 1646
|
|
|
1667 |
y Fz(:)p Fj(A)p 1470 1634 9 34 v 1479 1619 38 4 v 79
|
|
|
1668 |
w Fz(:)p Fj(A)p 1302 1666 383 4 v 1302 1734 a Fz(::)p
|
|
|
1669 |
Fj(A;)h Fz(:)p Fj(A)p 1622 1722 9 34 v 1630 1707 38 4
|
|
|
1670 |
v 1726 1678 a Fz(:)1777 1686 y Fi(L)p 1293 1768 401 4
|
|
|
1671 |
v 1293 1836 a Fz(::)p Fj(A)p 1470 1824 9 34 v 1479 1809
|
|
|
1672 |
38 4 v 80 w Fz(::)p Fj(A)1735 1780 y Fz(:)1786 1788 y
|
|
|
1673 |
Fi(R)p 676 1856 1018 4 v 782 1924 a Fz(::)p Fj(A;)g Fz(::)p
|
|
|
1674 |
Fj(A)p 1153 1912 9 34 v 1162 1897 38 4 v 80 w Fz(::)p
|
|
|
1675 |
Fj(A)p Fz(^::)p Fj(A)1735 1873 y Fz(^)1786 1881 y Fi(R)p
|
|
|
1676 |
709 1958 951 4 v 709 2031 a Fz(:)p Fh(\()p Fz(::)p Fj(A)p
|
|
|
1677 |
Fz(^)q(::)p Fj(A)p Fh(\))p Fj(;)g Fz(::)p Fj(A;)g Fz(::)p
|
|
|
1678 |
Fj(A)p 1597 2019 9 34 v 1605 2004 38 4 v 1701 1970 a
|
|
|
1679 |
Fz(:)1752 1978 y Fi(L)p 709 2069 951 4 v 806 2142 a Fz(:)p
|
|
|
1680 |
Fh(\()p Fz(::)p Fj(A)p Fz(^)q(::)p Fj(A)p Fh(\))p Fj(;)g
|
|
|
1681 |
Fz(::)p Fj(A)p 1500 2130 9 34 v 1508 2115 38 4 v 1701
|
|
|
1682 |
2088 a(contr)1880 2096 y Fi(L)p 798 2181 774 4 v 798
|
|
|
1683 |
2253 a Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p Fj(A)p
|
|
|
1684 |
Fh(\))p 1297 2241 9 34 v 1306 2226 38 4 v 80 w Fz(:::)p
|
|
|
1685 |
Fj(A)1613 2193 y Fz(:)1664 2201 y Fi(R)2053 1646 y Fz(:)p
|
|
|
1686 |
Fj(A)p 2178 1634 9 34 v 2187 1619 38 4 v 79 w Fz(:)p
|
|
|
1687 |
Fj(A)p 2010 1666 383 4 v 2010 1734 a Fz(::)p Fj(A;)g
|
|
|
1688 |
Fz(:)p Fj(A)p 2330 1722 9 34 v 2338 1707 38 4 v 2434
|
|
|
1689 |
1678 a Fz(:)2485 1686 y Fi(L)p 2001 1768 401 4 v 2001
|
|
|
1690 |
1836 a Fz(::)p Fj(A)p 2178 1824 9 34 v 2187 1809 38 4
|
|
|
1691 |
v 80 w Fz(::)p Fj(A)2443 1780 y Fz(:)2494 1788 y Fi(R)2670
|
|
|
1692 |
1646 y Fz(:)p Fj(A)p 2796 1634 9 34 v 2805 1619 38 4
|
|
|
1693 |
v 80 w Fz(:)p Fj(A)p 2628 1666 383 4 v 2628 1734 a Fz(::)p
|
|
|
1694 |
Fj(A;)g Fz(:)p Fj(A)p 2948 1722 9 34 v 2956 1707 38 4
|
|
|
1695 |
v 3052 1678 a Fz(:)3103 1686 y Fi(L)p 2619 1768 401 4
|
|
|
1696 |
v 2619 1836 a Fz(::)p Fj(A)p 2796 1824 9 34 v 2805 1809
|
|
|
1697 |
38 4 v 80 w Fz(::)p Fj(A)3061 1780 y Fz(:)3112 1788 y
|
|
|
1698 |
Fi(R)p 2001 1856 1018 4 v 2108 1924 a Fz(::)p Fj(A;)g
|
|
|
1699 |
Fz(::)p Fj(A)p 2479 1912 9 34 v 2487 1897 38 4 v 80 w
|
|
|
1700 |
Fz(::)p Fj(A)p Fz(^::)p Fj(A)3061 1873 y Fz(^)3112 1881
|
|
|
1701 |
y Fi(R)p 2035 1958 951 4 v 2035 2031 a Fz(:)p Fh(\()p
|
|
|
1702 |
Fz(::)p Fj(A)p Fz(^::)p Fj(A)p Fh(\))p Fj(;)g Fz(::)p
|
|
|
1703 |
Fj(A;)g Fz(::)p Fj(A)p 2922 2019 9 34 v 2931 2004 38
|
|
|
1704 |
4 v 3027 1970 a Fz(:)3078 1978 y Fi(L)p 2035 2069 951
|
|
|
1705 |
4 v 2132 2142 a Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p
|
|
|
1706 |
Fj(A)p Fh(\))p Fj(;)g Fz(::)p Fj(A)p 2825 2130 9 34 v
|
|
|
1707 |
2834 2115 38 4 v 3027 2088 a(contr)3206 2096 y Fi(L)p
|
|
|
1708 |
2124 2181 774 4 v 2124 2253 a Fz(:)p Fh(\()p Fz(::)p
|
|
|
1709 |
Fj(A)p Fz(^::)p Fj(A)p Fh(\))p 2623 2241 9 34 v 2631
|
|
|
1710 |
2226 38 4 v 80 w Fz(:::)p Fj(A)2938 2193 y Fz(:)2989
|
|
|
1711 |
2201 y Fi(R)p 798 2292 2100 4 v 1071 2364 a Fz(:)p Fh(\()p
|
|
|
1712 |
Fz(::)p Fj(A)p Fz(^)q(::)p Fj(A)p Fh(\))p Fj(;)g Fz(:)p
|
|
|
1713 |
Fh(\()p Fz(::)p Fj(A)p Fz(^::)p Fj(A)p Fh(\))p 2087 2352
|
|
|
1714 |
9 34 v 2095 2338 38 4 v 80 w Fz(:::)p Fj(A)p Fz(^:::)p
|
|
|
1715 |
Fj(A)2938 2308 y Fz(^)2990 2317 y Fi(R)p 999 2403 1697
|
|
|
1716 |
4 v 999 2476 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p
|
|
|
1717 |
Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^)q(::)p
|
|
|
1718 |
Fj(A)p Fh(\))p Fj(;)g Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p
|
|
|
1719 |
Fj(A)p Fh(\))p 2633 2464 9 34 v 2641 2449 38 4 v 2737
|
|
|
1720 |
2415 a Fz(:)2788 2423 y Fi(L)p 999 2514 1697 4 v 1257
|
|
|
1721 |
2587 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p
|
|
|
1722 |
Fh(\))p Fj(;)h Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p
|
|
|
1723 |
Fj(A)p Fh(\))p 2375 2575 9 34 v 2383 2560 38 4 v 2737
|
|
|
1724 |
2533 a Fj(contr)2916 2541 y Fi(L)p 3402 2639 4 2271 v
|
|
|
1725 |
523 2642 2882 4 v 523 2731 a Fw(Fig)o(.)e(1.)36 b Fx(One)h(can)g
|
|
|
1726 |
(obtain)g(the)f(\002rst)f(proof)i(by)g(double-ne)o(gation)i
|
|
|
1727 |
(translation)d(of)h(\(1\))f(using)h(the)f(left-)523 2822
|
|
|
1728 |
y(translation)27 b(for)g(the)g(cut,)g(and)g(then)h(eliminating)f(all)f
|
|
|
1729 |
(cuts)h(including)h(the)f(auxiliary)g(cuts.)g(Equally)-5
|
|
|
1730 |
b(,)27 b(one)523 2913 y(can)21 b(\002rst)f(reduce)i(\(1\))f(to)g
|
|
|
1731 |
(\(2\),)f(double-ne)o(gate)j(translate)e(\(2\))f(and)i(then)f
|
|
|
1732 |
(eliminate)g(all)f(cuts.)h(Similarly)f(with)523 3005
|
|
|
1733 |
y(the)f(second)h(proof:)f(it)f(can)h(be)g(obtained)g(by)g
|
|
|
1734 |
(right-translating)g(the)g(cut)g(in)f(\(1\))h(and)g(then)g(eliminate)g
|
|
|
1735 |
(all)f(cuts;)523 3096 y(or)h(by)g(double-ne)o(gate)i(translate)e(\(3\))
|
|
|
1736 |
f(and)i(then)f(eliminate)g(all)g(cuts.)523 3364 y FA(preserving)30
|
|
|
1737 |
b(the)i(\223structure\224)f(of)h(the)g(classical)h(proof.)e(In)g(order)
|
|
|
1738 |
g(to)i(achie)n(v)o(e)e(this)h(one)g(needs)523 3464 y(the)19
|
|
|
1739 |
b(property)e(that)i Fr(::)p Fm(\()p Fr(\000)p Fm(\))1328
|
|
|
1740 |
3434 y Fq(x)p 1389 3452 10 38 v 1398 3435 42 4 v 1458
|
|
|
1741 |
3464 a Fm(\()p Fr(\000)p Fm(\))1587 3434 y Fq(x)1649
|
|
|
1742 |
3464 y FA(is)g(intuitionistically)f(deri)n(v)n(able.)f(Ho)n(we)n(v)o
|
|
|
1743 |
(er)m(,)g(this)i(lea)n(v)o(es)g(us)523 3564 y(with)26
|
|
|
1744 |
b(man)o(y)e(possible)i(double)e(ne)o(gation)f(translations\227clearly)h
|
|
|
1745 |
(the)i(ones)f(gi)n(v)o(en)f(by)i(Gentzen,)523 3663 y(G)7
|
|
|
1746 |
b(\250)-35 b(odel)20 b(and)f(K)m(olmogoro)o(v)e(are)j(not)g(the)g(only)
|
|
|
1747 |
g(ones)g(that)g(satisfy)h(these)f(constraints.)523 3909
|
|
|
1748 |
y Fu(3)99 b(Cut-Elimination)26 b(and)f(Its)g(Colour)n(ed)h(V)-9
|
|
|
1749 |
b(ariant)523 4089 y FA(Urban)19 b(and)h(Bierman)f(ha)n(v)o(e)h(sho)n
|
|
|
1750 |
(wn)f(in)h([19,)12 b(21])19 b(that)i(only)e(a)h(small)h(restriction)e
|
|
|
1751 |
(on)h(the)g(standard)523 4188 y(cut-elimination)30 b(procedure)f(for)i
|
|
|
1752 |
(classical)i(and)e(intuitionistic)h(logic)f(is)i(suf)n(\002cient)e(to)h
|
|
|
1753 |
(obtain)523 4288 y(strongly)f(normalising)g(proof-transformations.)c
|
|
|
1754 |
(The)32 b Ft(lo)o(gical)g(cuts)p FA(,)g(also)h(sometimes)f(called)523
|
|
|
1755 |
4387 y Ft(k)o(e)n(y-cuts)p FA(,)c(are)h(transformed)e(by)h(this)i
|
|
|
1756 |
(cut-elimination)d(procedure)f(in)k(a)f(completely)f(standard)523
|
|
|
1757 |
4487 y(f)o(ashion)19 b([10].)g(F)o(or)h(e)o(xample)f(the)h(logical)g
|
|
|
1758 |
(cut)1459 4607 y Fs(\031)1506 4619 y Fl(1)1490 4672 y
|
|
|
1759 |
Fm(:)p 1442 4733 10 38 v 1452 4717 42 4 v 1512 4745 a
|
|
|
1760 |
Fs(B)1696 4607 y(\031)1743 4619 y Fl(2)1727 4672 y Fm(:)p
|
|
|
1761 |
1680 4733 10 38 v 1690 4717 42 4 v 73 x Fs(C)p 1424 4765
|
|
|
1762 |
392 4 v 1500 4827 10 38 v 1510 4810 42 4 v 1570 4839
|
|
|
1763 |
a(B)t Fr(^)p Fs(C)1857 4782 y Fr(^)1912 4795 y Fq(R)2145
|
|
|
1764 |
4607 y Fs(\031)2192 4619 y Fl(3)2176 4672 y Fm(:)2110
|
|
|
1765 |
4745 y Fs(B)p 2195 4733 10 38 v 2205 4717 42 4 v 2049
|
|
|
1766 |
4765 276 4 v 2049 4839 a(B)t Fr(^)q Fs(C)p 2256 4827
|
|
|
1767 |
10 38 v 2265 4810 42 4 v 2367 4778 a Fr(^)2422 4790 y
|
|
|
1768 |
Fq(L)2468 4798 y Fc(1)p 1482 4859 844 4 v 1878 4912 10
|
|
|
1769 |
38 v 1887 4896 42 4 v 2367 4884 a Fs(cut)3267 4924 y
|
|
|
1770 |
FA(\(11\))p eop end
|
|
|
1771 |
%%Page: 9 9
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1772 |
TeXDict begin 9 8 bop 523 448 a FA(is)21 b(transformed)d(to)1725
|
|
|
1773 |
496 y Fs(\031)1772 508 y Fl(1)1756 560 y Fm(:)p 1708
|
|
|
1774 |
622 10 38 v 1718 605 42 4 v 1778 634 a Fs(B)1963 496
|
|
|
1775 |
y(\031)2010 508 y Fl(3)1994 560 y Fm(:)1928 634 y Fs(B)p
|
|
|
1776 |
2014 622 10 38 v 2023 605 42 4 v 1690 654 394 4 v 1861
|
|
|
1777 |
707 10 38 v 1871 691 42 4 v 2125 679 a(cut)523 872 y
|
|
|
1778 |
FA(and)i(so)h(on)f(for)f(the)i(other)e(connecti)n(v)o(es.)g(As)i
|
|
|
1779 |
(before)e(we)i(con)m(v)o(eniently)c(ignore)i(all)i(matters)f(to)h(do)
|
|
|
1780 |
523 972 y(with)e(ho)n(w)f(the)h(sequent-conte)o(xts)d(should)h(be)i
|
|
|
1781 |
(adjusted.)f(A)h(logical)f(cut)h(can)f(be)h(characterised)e(as)523
|
|
|
1782 |
1071 y(a)25 b(cut)f(where)g(in)g(both)f(subproofs)g(the)h(cut-formulas)
|
|
|
1783 |
e(are)i Ft(fr)m(eshly)h(intr)l(oduced)g FA(by)f(logical)g(rules)523
|
|
|
1784 |
1171 y(directly)19 b(abo)o(v)o(e)g(the)h(cut.)g(F)o(or)g(e)o(xample)f
|
|
|
1785 |
(in)h(the)g(proof)1898 1317 y Fs(\031)1912 1370 y Fm(:)p
|
|
|
1786 |
1846 1390 156 4 v 1864 1451 10 38 v 1874 1434 42 4 v
|
|
|
1787 |
1934 1463 a Fs(B)2042 1408 y(r)523 1648 y FA(we)33 b(say)f(the)h
|
|
|
1788 |
(formula)e Fs(B)37 b FA(is)c(freshly)f(introduced)e(if)i(it)h(is)h
|
|
|
1789 |
(what)e(usually)g(is)h(called)f(the)h(main)523 1748 y(formula)24
|
|
|
1790 |
b(of)h(the)g(logical)g(inference)f(rule)h Fs(r)r FA(.)i(Consequently)-5
|
|
|
1791 |
b(,)23 b(a)j(logical)f(cut)g(is)h(a)g(cut)g(where)e(the)523
|
|
|
1792 |
1847 y(cut-formula)16 b(is)j(freshly)f(introduced)e(in)i(the)h(tw)o(o)g
|
|
|
1793 |
(immediate)e(subproofs)f(of)j(the)f(cut.)g(In)g(all)h(other)523
|
|
|
1794 |
1947 y(cases)f(we)f(ha)n(v)o(e)g(a)g Ft(commuting)f(cut)p
|
|
|
1795 |
FA(.)h(The)g(cut)g(in)g(\(1\),)f(for)h(e)o(xample,)e(is)k(a)e
|
|
|
1796 |
(commuting)e(cut)i(because)523 2047 y(the)26 b(cut-formula)d
|
|
|
1797 |
Fs(A)j FA(is)h(in)f(both)f(subproofs)e(introduced)h(by)h(a)h
|
|
|
1798 |
(contraction-rule,)c(which)j(is)i(not)523 2146 y(considered)18
|
|
|
1799 |
b(to)j(be)f(a)h(logical)e(inference)g(rule.)648 2247
|
|
|
1800 |
y(Gentzen)d(introduced)f(proof-transformations)e(that)k(permute)f
|
|
|
1801 |
(commuting)f(cuts)j(upw)o(ards)e(in)523 2346 y(a)25 b(stepwise)h(f)o
|
|
|
1802 |
(ashion)e(only)g(by)h(re)n(writing)f(neighboring)e(inference)h(rules.)i
|
|
|
1803 |
(In)g(contrast,)f(the)h(cut-)523 2446 y(elimination)c(procedure)e(of)j
|
|
|
1804 |
(Urban)f(and)g(Bierman)h(contains)f(proof-transformations)c(that)22
|
|
|
1805 |
b(push)523 2545 y(commuting)15 b(cuts)i(upw)o(ards)f(in)i(a)f(single)g
|
|
|
1806 |
(\223big\224)g(step)g(to)n(w)o(ards)g(all)g(places)g(where)g(the)g
|
|
|
1807 |
(cut-formula)523 2645 y(w)o(as)k(introduced.)d(Consider)h(the)h(follo)n
|
|
|
1808 |
(wing)f(picture)1106 3842 y @beginspecial 134 @llx 377
|
|
|
1809 |
@lly 592 @urx 666 @ury 2061 @rwi @clip @setspecial
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1810 |
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1811 |
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|
1955 |
load } if ] cvx newpath 3 1 roll T exec } def
|
|
|
1956 |
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
|
|
|
1957 |
cvi def } def
|
|
|
1958 |
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
|
|
|
1959 |
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
|
|
|
1960 |
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
|
|
|
1961 |
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
|
|
|
1962 |
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
|
|
|
1963 |
ifelse { pop } repeat a } def
|
|
|
1964 |
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
|
|
|
1965 |
and { pop pop /n n 1 sub def } if } def
|
|
|
1966 |
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
|
|
|
1967 |
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
|
|
|
1968 |
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
|
|
|
1969 |
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
|
|
|
1970 |
Lineto pop pop closepath } ifelse } def
|
|
|
1971 |
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
|
|
|
1972 |
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
|
|
|
1973 |
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
|
|
|
1974 |
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
|
|
|
1975 |
setmatrix } def
|
|
|
1976 |
% DG modification begin - Jan. 15, 1997
|
|
|
1977 |
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
|
|
|
1978 |
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
|
|
|
1979 |
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
|
|
|
1980 |
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
|
|
|
1981 |
%setmatrix } def
|
|
|
1982 |
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
|
|
|
1983 |
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
|
|
|
1984 |
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
|
|
|
1985 |
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
|
|
|
1986 |
% DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis)
|
|
|
1987 |
% setmatrix } def
|
|
|
1988 |
setmatrix pop } def
|
|
|
1989 |
% DG/SR modification end
|
|
|
1990 |
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
|
|
|
1991 |
def } def
|
|
|
1992 |
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
|
|
|
1993 |
def } def
|
|
|
1994 |
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
|
|
|
1995 |
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
|
|
|
1996 |
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
|
|
|
1997 |
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
|
|
|
1998 |
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
|
|
|
1999 |
m mul neg def } def
|
|
|
2000 |
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
|
|
|
2001 |
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
|
|
|
2002 |
def
|
|
|
2003 |
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
|
|
|
2004 |
/NC { CC x1 y1 x2 y2 x y curveto } def
|
|
|
2005 |
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
|
|
|
2006 |
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
|
|
|
2007 |
/NAC { x2 y2 x y curveto CC x1 y1 } def
|
|
|
2008 |
/EAC { x2 y2 x y ArrowB curveto pop pop } def
|
|
|
2009 |
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
|
|
|
2010 |
n { NC } repeat EOC } ifelse } def
|
|
|
2011 |
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
|
|
|
2012 |
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
|
|
|
2013 |
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
|
|
|
2014 |
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
|
|
|
2015 |
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
|
|
|
2016 |
repeat closepath pop pop } ifelse } def
|
|
|
2017 |
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
|
|
|
2018 |
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
|
|
|
2019 |
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
|
|
|
2020 |
def
|
|
|
2021 |
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
|
|
|
2022 |
rotate matrix concatmatrix exch findfont exch makefont setfont } def
|
|
|
2023 |
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
|
|
|
2024 |
x1 y1 lineto closepath } def
|
|
|
2025 |
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
|
|
|
2026 |
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
|
|
|
2027 |
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
|
|
|
2028 |
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
|
|
|
2029 |
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
|
|
|
2030 |
ifelse } def
|
|
|
2031 |
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
|
|
|
2032 |
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
|
|
|
2033 |
pop pop Rect } { OvalFrame } ifelse } def
|
|
|
2034 |
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
|
|
|
2035 |
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
|
|
|
2036 |
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
|
|
|
2037 |
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
|
|
|
2038 |
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
|
|
|
2039 |
ifelse } def
|
|
|
2040 |
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
|
|
|
2041 |
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
|
|
|
2042 |
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
|
|
|
2043 |
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
|
|
|
2044 |
def
|
|
|
2045 |
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
|
|
|
2046 |
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
|
|
|
2047 |
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
|
|
|
2048 |
y0 x0 2 mul x1 sub y1 ] def } def
|
|
|
2049 |
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
|
|
|
2050 |
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
|
|
|
2051 |
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
|
|
|
2052 |
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
|
|
|
2053 |
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
|
|
|
2054 |
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
|
|
|
2055 |
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
|
|
|
2056 |
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
|
|
|
2057 |
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
|
|
|
2058 |
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
|
|
|
2059 |
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
|
|
|
2060 |
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
|
|
|
2061 |
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
|
|
|
2062 |
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
|
|
|
2063 |
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
|
|
|
2064 |
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
|
|
|
2065 |
w add def } repeat grestore gsave n 0 gt
|
|
|
2066 |
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
|
|
|
2067 |
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
|
|
|
2068 |
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
|
|
|
2069 |
% DG/SR modification end
|
|
|
2070 |
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
|
|
|
2071 |
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
|
|
|
2072 |
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
|
|
|
2073 |
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
|
|
|
2074 |
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
|
|
|
2075 |
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
|
|
|
2076 |
exch t L stroke /i i h add def } repeat grestore } def
|
|
|
2077 |
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
|
|
|
2078 |
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
|
|
|
2079 |
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
|
|
|
2080 |
mul neg d } def
|
|
|
2081 |
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
|
|
|
2082 |
/Rot { CP CP translate 3 -1 roll neg rotate NET } def
|
|
|
2083 |
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
|
|
|
2084 |
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
|
|
|
2085 |
dup a add ] cvx def } def
|
|
|
2086 |
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
|
|
|
2087 |
cvx def } def
|
|
|
2088 |
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
|
|
|
2089 |
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
|
|
|
2090 |
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
|
|
|
2091 |
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
|
|
|
2092 |
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
|
|
|
2093 |
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
|
|
|
2094 |
h1 abs h sub dup s mul abs } ifelse } def
|
|
|
2095 |
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
|
|
|
2096 |
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
|
|
|
2097 |
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
|
|
|
2098 |
exch } def
|
|
|
2099 |
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
|
|
|
2100 |
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
|
|
|
2101 |
} if } ifelse } def
|
|
|
2102 |
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
|
|
|
2103 |
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
|
|
|
2104 |
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
|
|
|
2105 |
/IfVisible true def } def
|
|
|
2106 |
end
|
|
|
2107 |
% END pstricks.pro
|
|
|
2108 |
|
|
|
2109 |
%%EndProcSet
|
|
|
2110 |
%%BeginProcSet: pst-dots.pro 0 0
|
|
|
2111 |
%!PS-Adobe-2.0
|
|
|
2112 |
%%Title: Dot Font for PSTricks
|
|
|
2113 |
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
|
|
|
2114 |
%%Creation Date: May 7, 1993
|
|
|
2115 |
%% Version 97 patch 1, 99/12/16
|
|
|
2116 |
%% Modified by Etienne Riga <etienne.riga@skynet.be> - Dec. 16, 1999
|
|
|
2117 |
%% to add /Diamond, /SolidDiamond and /BoldDiamond
|
|
|
2118 |
10 dict dup begin
|
|
|
2119 |
/FontType 3 def
|
|
|
2120 |
/FontMatrix [ .001 0 0 .001 0 0 ] def
|
|
|
2121 |
/FontBBox [ 0 0 0 0 ] def
|
|
|
2122 |
/Encoding 256 array def
|
|
|
2123 |
0 1 255 { Encoding exch /.notdef put } for
|
|
|
2124 |
Encoding
|
|
|
2125 |
dup (b) 0 get /Bullet put
|
|
|
2126 |
dup (c) 0 get /Circle put
|
|
|
2127 |
dup (C) 0 get /BoldCircle put
|
|
|
2128 |
dup (u) 0 get /SolidTriangle put
|
|
|
2129 |
dup (t) 0 get /Triangle put
|
|
|
2130 |
dup (T) 0 get /BoldTriangle put
|
|
|
2131 |
dup (r) 0 get /SolidSquare put
|
|
|
2132 |
dup (s) 0 get /Square put
|
|
|
2133 |
dup (S) 0 get /BoldSquare put
|
|
|
2134 |
dup (q) 0 get /SolidPentagon put
|
|
|
2135 |
dup (p) 0 get /Pentagon put
|
|
|
2136 |
dup (P) 0 get /BoldPentagon put
|
|
|
2137 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
|
2138 |
dup (l) 0 get /SolidDiamond put
|
|
|
2139 |
dup (d) 0 get /Diamond put
|
|
|
2140 |
(D) 0 get /BoldDiamond put
|
|
|
2141 |
% DG/SR modification end
|
|
|
2142 |
/Metrics 13 dict def
|
|
|
2143 |
Metrics begin
|
|
|
2144 |
/Bullet 1000 def
|
|
|
2145 |
/Circle 1000 def
|
|
|
2146 |
/BoldCircle 1000 def
|
|
|
2147 |
/SolidTriangle 1344 def
|
|
|
2148 |
/Triangle 1344 def
|
|
|
2149 |
/BoldTriangle 1344 def
|
|
|
2150 |
/SolidSquare 886 def
|
|
|
2151 |
/Square 886 def
|
|
|
2152 |
/BoldSquare 886 def
|
|
|
2153 |
/SolidPentagon 1093.2 def
|
|
|
2154 |
/Pentagon 1093.2 def
|
|
|
2155 |
/BoldPentagon 1093.2 def
|
|
|
2156 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
|
2157 |
/SolidDiamond 1008 def
|
|
|
2158 |
/Diamond 1008 def
|
|
|
2159 |
/BoldDiamond 1008 def
|
|
|
2160 |
% DG/SR modification end
|
|
|
2161 |
/.notdef 0 def
|
|
|
2162 |
end
|
|
|
2163 |
/BBoxes 13 dict def
|
|
|
2164 |
BBoxes begin
|
|
|
2165 |
/Circle { -550 -550 550 550 } def
|
|
|
2166 |
/BoldCircle /Circle load def
|
|
|
2167 |
/Bullet /Circle load def
|
|
|
2168 |
/Triangle { -571.5 -330 571.5 660 } def
|
|
|
2169 |
/BoldTriangle /Triangle load def
|
|
|
2170 |
/SolidTriangle /Triangle load def
|
|
|
2171 |
/Square { -450 -450 450 450 } def
|
|
|
2172 |
/BoldSquare /Square load def
|
|
|
2173 |
/SolidSquare /Square load def
|
|
|
2174 |
/Pentagon { -546.6 -465 546.6 574.7 } def
|
|
|
2175 |
/BoldPentagon /Pentagon load def
|
|
|
2176 |
/SolidPentagon /Pentagon load def
|
|
|
2177 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
|
2178 |
/Diamond { -428.5 -742.5 428.5 742.5 } def
|
|
|
2179 |
/BoldDiamond /Diamond load def
|
|
|
2180 |
/SolidDiamond /Diamond load def
|
|
|
2181 |
% DG/SR modification end
|
|
|
2182 |
/.notdef { 0 0 0 0 } def
|
|
|
2183 |
end
|
|
|
2184 |
/CharProcs 20 dict def
|
|
|
2185 |
CharProcs begin
|
|
|
2186 |
/Adjust {
|
|
|
2187 |
2 copy dtransform floor .5 add exch floor .5 add exch idtransform
|
|
|
2188 |
3 -1 roll div 3 1 roll exch div exch scale
|
|
|
2189 |
} def
|
|
|
2190 |
/CirclePath { 0 0 500 0 360 arc closepath } def
|
|
|
2191 |
/Bullet { 500 500 Adjust CirclePath fill } def
|
|
|
2192 |
/Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath
|
|
|
2193 |
eofill } def
|
|
|
2194 |
/BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath
|
|
|
2195 |
eofill } def
|
|
|
2196 |
/BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
|
|
|
2197 |
/TrianglePath { 0 660 moveto -571.5 -330 lineto 571.5 -330 lineto
|
|
|
2198 |
closepath } def
|
|
|
2199 |
/SolidTriangle { TrianglePath fill } def
|
|
|
2200 |
/Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
|
|
|
2201 |
/BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
|
|
|
2202 |
/SquarePath { -450 450 moveto 450 450 lineto 450 -450 lineto
|
|
|
2203 |
-450 -450 lineto closepath } def
|
|
|
2204 |
/SolidSquare { SquarePath fill } def
|
|
|
2205 |
/Square { SquarePath .89 .89 scale SquarePath eofill } def
|
|
|
2206 |
/BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
|
|
|
2207 |
/PentagonPath {
|
|
|
2208 |
-337.8 -465 moveto
|
|
|
2209 |
337.8 -465 lineto
|
|
|
2210 |
546.6 177.6 lineto
|
|
|
2211 |
0 574.7 lineto
|
|
|
2212 |
-546.6 177.6 lineto
|
|
|
2213 |
closepath
|
|
|
2214 |
} def
|
|
|
2215 |
/SolidPentagon { PentagonPath fill } def
|
|
|
2216 |
/Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
|
|
|
2217 |
/BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
|
|
|
2218 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
|
2219 |
/DiamondPath { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto
|
|
|
2220 |
428.5 0 lineto closepath } def
|
|
|
2221 |
/SolidDiamond { DiamondPath fill } def
|
|
|
2222 |
/Diamond { DiamondPath .85 .85 scale DiamondPath eofill } def
|
|
|
2223 |
/BoldDiamond { DiamondPath .7 .7 scale DiamondPath eofill } def
|
|
|
2224 |
% DG/SR modification end
|
|
|
2225 |
/.notdef { } def
|
|
|
2226 |
end
|
|
|
2227 |
/BuildGlyph {
|
|
|
2228 |
exch
|
|
|
2229 |
begin
|
|
|
2230 |
Metrics 1 index get exec 0
|
|
|
2231 |
BBoxes 3 index get exec
|
|
|
2232 |
setcachedevice
|
|
|
2233 |
CharProcs begin load exec end
|
|
|
2234 |
end
|
|
|
2235 |
} def
|
|
|
2236 |
/BuildChar {
|
|
|
2237 |
1 index /Encoding get exch get
|
|
|
2238 |
1 index /BuildGlyph get exec
|
|
|
2239 |
} bind def
|
|
|
2240 |
end
|
|
|
2241 |
/PSTricksDotFont exch definefont pop
|
|
|
2242 |
%END pst-dots.pro
|
|
|
2243 |
|
|
|
2244 |
%%EndProcSet
|
|
|
2245 |
%%BeginProcSet: pst-node.pro 0 0
|
|
|
2246 |
%!
|
|
|
2247 |
% PostScript prologue for pst-node.tex.
|
|
|
2248 |
% Version 97 patch 1, 97/05/09.
|
|
|
2249 |
% For distribution, see pstricks.tex.
|
|
|
2250 |
%
|
|
|
2251 |
/tx@NodeDict 400 dict def tx@NodeDict begin
|
|
|
2252 |
tx@Dict begin /T /translate load def end
|
|
|
2253 |
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
|
|
|
2254 |
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
|
|
|
2255 |
grestore } def
|
|
|
2256 |
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
|
|
|
2257 |
} def
|
|
|
2258 |
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
|
|
|
2259 |
Sin mul } def } def
|
|
|
2260 |
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
|
|
|
2261 |
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
|
|
|
2262 |
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
|
|
|
2263 |
dup Sin mul Cos Div } ifelse } def
|
|
|
2264 |
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
|
|
|
2265 |
/u ED /NodePos { GetRnodePos } def } def
|
|
|
2266 |
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
|
|
|
2267 |
Cos mul exch Sin mul } def
|
|
|
2268 |
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
|
|
|
2269 |
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
|
|
|
2270 |
} ifelse } def
|
|
|
2271 |
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
|
|
|
2272 |
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
|
|
|
2273 |
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
|
|
|
2274 |
sin def /NodePos { TriNodePos } def } def
|
|
|
2275 |
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
|
|
|
2276 |
hh mul Atan dup cos ww mul exch sin hh mul } def
|
|
|
2277 |
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
|
|
|
2278 |
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
|
|
|
2279 |
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
|
|
|
2280 |
{ 0 Dist Sin mul } ifelse } ifelse Do } def
|
|
|
2281 |
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
|
|
|
2282 |
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
|
|
|
2283 |
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
|
|
|
2284 |
XYPos } ifelse } def
|
|
|
2285 |
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
|
|
|
2286 |
roll sin mul sub exch } ifelse } def
|
|
|
2287 |
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
|
|
|
2288 |
AddOffset yA add /yA1 ED xA add /xA1 ED } def
|
|
|
2289 |
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
|
|
|
2290 |
AddOffset yB add /yB1 ED xB add /xB1 ED } def
|
|
|
2291 |
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
|
|
|
2292 |
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
|
|
|
2293 |
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
|
|
|
2294 |
} ifelse } def
|
|
|
2295 |
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
|
|
|
2296 |
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
|
|
|
2297 |
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
|
|
|
2298 |
} ifelse } def
|
|
|
2299 |
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
|
|
|
2300 |
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
|
|
|
2301 |
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
|
|
|
2302 |
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
|
|
|
2303 |
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
|
|
|
2304 |
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
|
|
|
2305 |
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
|
|
|
2306 |
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
|
|
|
2307 |
} repeat LPutLine cleartomark } def
|
|
|
2308 |
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
|
|
|
2309 |
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
|
|
|
2310 |
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
|
|
|
2311 |
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
|
|
|
2312 |
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
|
|
|
2313 |
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
|
|
|
2314 |
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
|
|
|
2315 |
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
|
|
|
2316 |
} def
|
|
|
2317 |
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
|
|
|
2318 |
/NAngle yyB yyA sub xxB xxA sub Atan def } def
|
|
|
2319 |
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def
|
|
|
2320 |
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
|
|
|
2321 |
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
|
|
|
2322 |
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd
|
|
|
2323 |
} def
|
|
|
2324 |
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
|
|
|
2325 |
} def
|
|
|
2326 |
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
|
|
|
2327 |
/NAngle yyB yyA sub xxB xxA sub Atan def } def
|
|
|
2328 |
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def
|
|
|
2329 |
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
|
|
|
2330 |
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
|
|
|
2331 |
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
|
|
|
2332 |
VPosEnd } def
|
|
|
2333 |
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
|
|
|
2334 |
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
|
|
|
2335 |
def grestore exec /LPutVar /SaveLPutVar load def } def
|
|
|
2336 |
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
|
|
|
2337 |
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
|
|
|
2338 |
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
|
|
|
2339 |
LPutVar VPutLine } def LPutVar } def
|
|
|
2340 |
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
|
|
|
2341 |
end } def
|
|
|
2342 |
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
|
|
|
2343 |
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
|
|
|
2344 |
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
|
|
|
2345 |
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
|
|
|
2346 |
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
|
|
|
2347 |
ifelse } def
|
|
|
2348 |
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
|
|
|
2349 |
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
|
|
|
2350 |
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
|
|
|
2351 |
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
|
|
|
2352 |
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
|
|
|
2353 |
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
|
|
|
2354 |
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
|
|
|
2355 |
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
|
|
|
2356 |
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
|
|
|
2357 |
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
|
|
|
2358 |
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
|
|
|
2359 |
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
|
|
|
2360 |
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
|
|
|
2361 |
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
|
|
|
2362 |
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
|
|
|
2363 |
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
|
|
|
2364 |
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
|
|
|
2365 |
VPutLines } def } def
|
|
|
2366 |
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
|
|
|
2367 |
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
|
|
|
2368 |
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
|
|
|
2369 |
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
|
|
|
2370 |
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
|
|
|
2371 |
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
|
|
|
2372 |
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
|
|
|
2373 |
VPutLines } def } def
|
|
|
2374 |
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
|
|
|
2375 |
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
|
|
|
2376 |
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
|
|
|
2377 |
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
|
|
|
2378 |
def
|
|
|
2379 |
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
|
|
|
2380 |
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
|
|
|
2381 |
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
|
|
|
2382 |
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
|
|
|
2383 |
def
|
|
|
2384 |
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
|
|
|
2385 |
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
|
|
|
2386 |
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
|
|
|
2387 |
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
|
|
|
2388 |
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
|
|
|
2389 |
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
|
|
|
2390 |
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
|
|
|
2391 |
VPutLines } def } def
|
|
|
2392 |
% DG/SR modification begin - May 9, 1997 - Patch 1
|
|
|
2393 |
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
|
|
|
2394 |
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
|
|
|
2395 |
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
|
|
|
2396 |
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
|
|
|
2397 |
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
|
|
|
2398 |
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
|
|
|
2399 |
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
|
|
|
2400 |
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
|
|
|
2401 |
% DG/SR modification end
|
|
|
2402 |
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
|
|
|
2403 |
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
|
|
|
2404 |
Div def /y ED /x ED } def
|
|
|
2405 |
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
|
|
|
2406 |
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
|
|
|
2407 |
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
|
|
|
2408 |
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
|
|
|
2409 |
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
|
|
|
2410 |
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
|
|
|
2411 |
tx@Dict begin false Polygon end } def
|
|
|
2412 |
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
|
|
|
2413 |
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
|
|
|
2414 |
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
|
|
|
2415 |
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
|
|
|
2416 |
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
|
|
|
2417 |
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
|
|
|
2418 |
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
|
|
|
2419 |
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
|
|
|
2420 |
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
|
|
|
2421 |
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
|
|
|
2422 |
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
|
|
|
2423 |
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
|
|
|
2424 |
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
|
|
|
2425 |
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
|
|
|
2426 |
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
|
|
|
2427 |
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
|
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|
2428 |
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
|
|
|
2429 |
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
|
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2430 |
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
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|
2431 |
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
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|
2432 |
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
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2433 |
LPutPos } def } def
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/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
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2435 |
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
|
|
|
2436 |
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
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2437 |
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
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|
2438 |
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
|
|
|
2439 |
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
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|
2440 |
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
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|
2441 |
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
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2442 |
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
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|
2443 |
exch X sub neg exch moveto setmatrix CP grestore } def
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2444 |
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
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|
2445 |
def } ifelse LPutCoor } def
|
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|
2446 |
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
|
|
|
2447 |
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
|
|
|
2448 |
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
|
|
|
2449 |
def
|
|
|
2450 |
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
|
|
|
2451 |
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
|
|
|
2452 |
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
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2453 |
def
|
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end
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% END pst-node.pro
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%%EndProcSet
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%!
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% PostScript header file pst-text.pro
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2461 |
% Version 97, 94/04/20; patched MV 10-09-99 00:36
|
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|
2462 |
% For distribution, see pstricks.tex.
|
|
|
2463 |
|
|
|
2464 |
/tx@TextPathDict 40 dict def
|
|
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2465 |
tx@TextPathDict begin
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2466 |
|
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2467 |
% Syntax: <dist> PathPosition -
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2468 |
% Function: Searches for position of currentpath distance <dist> from
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% beginning. Sets (X,Y)=position, and Angle=tangent.
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/PathPosition
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{ /targetdist exch def
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/pathdist 0 def
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/continue true def
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gsave
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flattenpath
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{ movetoproc } { linetoproc } { } { firstx firsty linetoproc }
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2478 |
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grestore
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/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
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2483 |
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2484 |
/@movetoproc
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{ /newy exch def /newx exch def
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2486 |
/firstx newx def /firsty newy def
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2487 |
} def
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|
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2489 |
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
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2490 |
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2491 |
/@linetoproc
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{
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2493 |
/oldx newx def /oldy newy def
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2496 |
/dy newy oldy sub def
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2498 |
/pathdist pathdist dist add def
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pathdist targetdist ge
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2500 |
{ pathdist targetdist sub dist div dup
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2501 |
dy mul neg newy add /Y exch def
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2502 |
dx mul neg newx add /X exch def
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2503 |
/Angle dy dx atan def
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2504 |
/continue false def
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} if
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2506 |
} def
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2507 |
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/TextPathShow
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{ /String exch def
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2510 |
/CharCount 0 def
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2511 |
String length
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{ String CharCount 1 getinterval ShowChar
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2513 |
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2514 |
} repeat
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2515 |
} def
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2516 |
|
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2517 |
% Syntax: <pathlength> <position> InitTextPath -
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2518 |
/InitTextPath
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2519 |
{ gsave
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|
2520 |
currentpoint /Y exch def /X exch def
|
|
|
2521 |
exch X Hoffset sub sub mul
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2522 |
Voffset Hoffset sub add
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2523 |
neg X add /Hoffset exch def
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2524 |
/Voffset Y def
|
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|
2525 |
grestore
|
|
|
2526 |
} def
|
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2527 |
|
|
|
2528 |
/Transform
|
|
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2529 |
{ PathPosition
|
|
|
2530 |
dup
|
|
|
2531 |
Angle cos mul Y add exch
|
|
|
2532 |
Angle sin mul neg X add exch
|
|
|
2533 |
translate
|
|
|
2534 |
Angle rotate
|
|
|
2535 |
} def
|
|
|
2536 |
|
|
|
2537 |
/ShowChar
|
|
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2538 |
{ /Char exch def
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2539 |
gsave
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|
|
2540 |
Char end stringwidth
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2541 |
tx@TextPathDict begin
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2542 |
2 div /Sy exch def 2 div /Sx exch def
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|
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2543 |
|
|
|
2544 |
%%% MV 10-09-99 00:36
|
|
|
2545 |
/sc?currentpoint where {pop sc?currentpoint} {currentpoint} ifelse
|
|
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2546 |
% currentpoint
|
|
|
2547 |
|
|
|
2548 |
Voffset sub Sy add exch
|
|
|
2549 |
Hoffset sub Sx add
|
|
|
2550 |
Transform
|
|
|
2551 |
Sx neg Sy neg moveto
|
|
|
2552 |
Char end tx@TextPathSavedShow
|
|
|
2553 |
tx@TextPathDict begin
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|
2554 |
grestore
|
|
|
2555 |
Sx 2 mul Sy 2 mul rmoveto
|
|
|
2556 |
} def
|
|
|
2557 |
|
|
|
2558 |
end
|
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2559 |
% END pst-text.pro
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2560 |
|
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2561 |
%%EndProcSet
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2562 |
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2563 |
%!
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2565 |
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2566 |
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/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
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2569 |
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
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2570 |
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2571 |
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2573 |
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2574 |
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2576 |
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2577 |
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
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2578 |
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
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2580 |
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
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2581 |
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
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2582 |
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2587 |
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
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2588 |
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
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-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
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currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
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/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
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2597 |
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2609 |
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3032 |
@endspecial 523 4028 a(where)f(the)h(cut-formula)e(on)h(the)h
|
|
|
3033 |
(right-hand)d(side)k(is)f Ft(not)i FA(freshly)d(introduced,)e(rather)i
|
|
|
3034 |
(it)i(is)f(intro-)523 4127 y(duced)25 b(some)n(where)f(deeper)h(inside)
|
|
|
3035 |
h(the)g(subproof)e(and)h(because)h(of)g(contractions)e(possibly)h(in)
|
|
|
3036 |
523 4227 y(se)n(v)o(eral)d(places.)h(W)-7 b(e)24 b(ha)n(v)o(e)f
|
|
|
3037 |
(indicated)e(three)i(cases)h(for)e(a)h(cut-formula)e(being)h
|
|
|
3038 |
(introduced:)f(by)h(a)523 4327 y(logical)h(inference)f(rule,)h(by)h(an)
|
|
|
3039 |
f(axiom)g(and)g(by)g(a)i(weak)o(ening-rule\227in)20 b(general)j(we)h
|
|
|
3040 |
(can)f(ha)n(v)o(e)523 4426 y(an)o(y)f(mixture)f(of)h(these)h(cases.)g
|
|
|
3041 |
(T)-7 b(o)23 b(eliminate)f(such)g(commuting)f(cuts)i(the)f(procedure)e
|
|
|
3042 |
(of)i(Urban)523 4526 y(and)d(Bierman)h(pushes)f(up,)h(roughly)e
|
|
|
3043 |
(speaking,)g(the)i(cut-rule)f(until)h(it)g(reaches)g
|
|
|
3044 |
Ft(all)g FA(places)g(where)523 4625 y(the)29 b(cut-formula)d(is)k
|
|
|
3045 |
(introduced)d(in)i Ft(one)f FA(step.)h(In)g(case)g(it)h(reaches)e(a)h
|
|
|
3046 |
(logical)g(inference-rule,)523 4725 y(then)e(the)h(proof)e(on)i(the)f
|
|
|
3047 |
(left-hand)f(side)j(will)f(be)g(cut)g(against)f(this)h(logical)f
|
|
|
3048 |
(inference-rule;)e(in)523 4825 y(case)h(of)g(an)g(axiom,)f(the)g(proof)
|
|
|
3049 |
g(on)g(the)h(left-hand)e(side)i(replaces)g(the)g(axiom)f(and)g(in)h
|
|
|
3050 |
(case)g(of)g(a)523 4924 y(weak)o(ening-rule,)16 b(the)j(proof)f(on)h
|
|
|
3051 |
(the)g(left-hand)f(side)h(is)i(deleted.)d(\(Again)g(all)i(matters)f(to)
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3052 |
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3054 |
TeXDict begin 10 9 bop 523 448 a FA(adjusting)22 b(the)h(sequent-conte)
|
|
|
3055 |
o(xts)d(are)j(omitted)g(in)g(this)g(paper)-5 b(.)23 b(The)f(w)o(ork)h
|
|
|
3056 |
(reported)e(in)i([19,)12 b(21])523 548 y(formulates)23
|
|
|
3057 |
b(these)h(proof-transformation)18 b(as)25 b(term-re)n(writing)d(rules)h
|
|
|
3058 |
(where)h(such)f(adjustments)523 648 y(are)31 b(b)n(uilt)g(into)g(the)g
|
|
|
3059 |
(inference)f(rules,)h(v)o(ery)f(similar)h(to)g(term-re)n(writing)e(in)j
|
|
|
3060 |
(the)f(simply-typed)523 747 y(lambda-calculus.\))648
|
|
|
3061 |
850 y(The)24 b(important)f(property)f(of)i(the)h(cut-elimination)e
|
|
|
3062 |
(procedure)f(of)i(Urban)f(and)h(Bierman)h(is)523 950
|
|
|
3063 |
y(the)d(f)o(act)g(that)f(it)i(is)f(strongly-normalising.)c(This)k
|
|
|
3064 |
(property)d(is)k(not)e(ob)o(vious:)f(the)i(reduction-rule)523
|
|
|
3065 |
1050 y(for)16 b(commuting)f(cuts)j(allo)n(ws)f(a)h(cut-rule)d(to)j
|
|
|
3066 |
(\223jump\224)e(o)o(v)o(er)f(other)i(cut-rules\227a)f(highly)g
|
|
|
3067 |
(problem-)523 1149 y(atic)26 b(reduction)e(if)i(one)f(tries)i(to)f
|
|
|
3068 |
(construct)e(a)i(decreasing)f(measure)g(for)g(cut-elimination.)e(Also)
|
|
|
3069 |
523 1249 y(this)g(rule)e(might)h(generate)e(se)n(v)o(eral)i(copies)g
|
|
|
3070 |
(of)f(a)i(subproof)c(when)j(a)g(cut-formula)e(is)j(introduced)523
|
|
|
3071 |
1349 y(in)k(se)n(v)o(eral)f(places.)h(Urban)f(and)g(Bierman,)g
|
|
|
3072 |
(therefore,)f(had)h(to)h(resort)f(to)h(a)h(quite)e(complicated)523
|
|
|
3073 |
1448 y(logical)20 b(relations)f(ar)o(gument)f(to)j(sho)n(w)f(strong)f
|
|
|
3074 |
(normalisation.)648 1551 y(Recall)33 b(that)f(the)h(cut-elimination)d
|
|
|
3075 |
(procedure)g(of)i(Urban)f(and)h(Bierman)g(is)h Ft(not)h
|
|
|
3076 |
FA(Church-)523 1651 y(Rosser:)29 b(when)e(on)h(both)f(sides)i(of)e(a)i
|
|
|
3077 |
(commuting)d(cut)i(the)g(cut-formula)e(is)j(not)e(freshly)g(intro-)523
|
|
|
3078 |
1751 y(duced,)19 b(then)i(this)g(cut)g(can)g(be)f(mo)o(v)o(ed)f(either)
|
|
|
3079 |
i(to)g(the)f(left)i(or)e(to)h(the)g(right,)f(leading)g(in)h(general)e
|
|
|
3080 |
(to)523 1850 y(tw)o(o)k(\223non-joinable\224)c(proofs.)h(F)o(or)i(e)o
|
|
|
3081 |
(xample,)f(the)h(proof)f(sho)n(wn)g(in)h(\(1\))g(can)g(be)g(reduced)f
|
|
|
3082 |
(in)h(one)523 1950 y(step)h(to)g(the)g(normalform)d(\(2\))j(or)f(in)h
|
|
|
3083 |
(one)g(step)g(to)g(\(3\).)f(Which)h(choice)f(is)i(tak)o(en)f(is)g(left)
|
|
|
3084 |
h(unspeci-)523 2050 y(\002ed)f(by)g(the)g(procedure.)e(Ne)n(v)o
|
|
|
3085 |
(ertheless)h(with)i(this)f(cut-elimination)f(procedure)e(we)k(ha)n(v)o
|
|
|
3086 |
(e)f(some)523 2149 y(ef)n(fecti)n(v)o(e)d(means)g(to)h(calculate)g(for)
|
|
|
3087 |
f(a)i(classical)g(proof)d(its)j(collection)e(of)h Ft(all)g
|
|
|
3088 |
FA(normalforms\227for)523 2249 y(e)o(xample)e(by)h(na)n(\250)-26
|
|
|
3089 |
b(\021v)o(ely)19 b(trying)g(out)g(all)i(possible)f(reductions.)2303
|
|
|
3090 |
2219 y Fp(3)648 2352 y FA(The)26 b(cut-elimination)f(procedure)g(of)i
|
|
|
3091 |
(Urban)f(and)h(Bierman)f(w)o(as)i(in)g(part)f(inspired)f(by)g(the)523
|
|
|
3092 |
2452 y(w)o(ork)h(of)h(Danos)g(et)g(al.)h([6].)e(The)h(main)f(dif)n
|
|
|
3093 |
(ference)f(is)j(that)f(their)g(cut-elimination)e(procedure)523
|
|
|
3094 |
2551 y Ft(is)c FA(Church-Rosser)-5 b(.)20 b(The)o(y)g(achie)n(v)o(e)g
|
|
|
3095 |
(this)h(by)g(pre-determining)c(e)n(v)o(ery)j(choice)g(that)h(can)g(be)g
|
|
|
3096 |
(made)523 2651 y(during)16 b(cut-elimination.)f(This)j
|
|
|
3097 |
(pre-determination)13 b(is)19 b(done)d(via)i Ft(colour)o(s)p
|
|
|
3098 |
FA(,)f(which)g(are)g(annotated)523 2751 y(to)24 b(e)n(v)o(ery)f
|
|
|
3099 |
(formula)g(and)g(subformula)f(in)i(a)h(proof.)d(T)-7
|
|
|
3100 |
b(o)24 b(see)h(ho)n(w)e(colours)g(w)o(ork,)h(consider)f(again)523
|
|
|
3101 |
2850 y(the)17 b(proof)e(sho)n(wn)i(in)g(\(1\).)f(The)h(choice)f(about)g
|
|
|
3102 |
(which)g(direction)g(is)i(tak)o(en)e(for)h(the)g(commuting)d(cut)523
|
|
|
3103 |
2950 y(is)26 b(determined)c(by)i(annotating)f(the)i(colour)e(`)p
|
|
|
3104 |
Fs(\()p FA(')h(or)h(`)p Fs(*)p FA(')f(to)h(the)f(cut-formula)e
|
|
|
3105 |
Fs(A)p FA(.)k(The)e(colour)n(-)523 3050 y(protocol)14
|
|
|
3106 |
b(of)i(Danos)g(et)g(al.)g(pre-scribes)f(that)h(in)g(the)g(former)f
|
|
|
3107 |
(case)h(it)h(is)g(\002rst)g(attempted)e(to)h(permute)523
|
|
|
3108 |
3149 y(the)i(cut)f(to)h(the)f(left)h(and)f(in)h(the)f(second)g(case)h
|
|
|
3109 |
(to)g(the)f(right)g(\(hence)f(the)i(use)g(of)f(an)g(arro)n(w)g(to)h
|
|
|
3110 |
(denote)523 3249 y(a)26 b(colour!\).)d(F)o(or)j(e)o(xample,)e(if)h(we)h
|
|
|
3111 |
(w)o(ant)g(to)g(reach)f(from)f(\(1\))h(the)h(normalform)d(\(2\))h(we)i
|
|
|
3112 |
(need)f(to)523 3348 y(orient)19 b(the)i(colour)e(of)g(the)i
|
|
|
3113 |
(cut-formula)c Fs(A)k FA(to)g(the)f(left,)g(as)h(sho)n(wn)e(belo)n(w)
|
|
|
3114 |
1087 3489 y Fq(\()1089 3563 y Fs(A)p 1194 3543 10 38
|
|
|
3115 |
v 1204 3526 42 4 v 1259 3489 a Fq(\()1261 3563 y Fs(A)1408
|
|
|
3116 |
3489 y Fq(\()1410 3563 y Fs(A)p 1516 3543 10 38 v 1526
|
|
|
3117 |
3526 42 4 v 1581 3489 a Fq(\()1583 3563 y Fs(A)p 1087
|
|
|
3118 |
3583 561 4 v 1101 3643 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g(\000)1098
|
|
|
3119 |
3691 y Fq(\()1100 3764 y Fs(A)1187 3756 y Fr(_)1266 3691
|
|
|
3120 |
y Fq(\()1268 3764 y Fs(A)p 1374 3744 10 38 v 1383 3728
|
|
|
3121 |
42 4 v 1466 3691 a Fq(\()1468 3764 y Fs(A)1532 3756 y(;)1569
|
|
|
3122 |
3691 y Fq(\()1571 3764 y Fs(A)1689 3599 y Fr(_)1744 3612
|
|
|
3123 |
y Fq(L)p 1098 3792 538 4 v 1153 3853 a(\()g Fn(\000)g(\000)g(\000)g
|
|
|
3124 |
(\000)1150 3901 y Fq(\()1152 3974 y Fs(A)1239 3966 y
|
|
|
3125 |
Fr(_)1317 3901 y Fq(\()1319 3974 y Fs(A)p 1425 3954 10
|
|
|
3126 |
38 v 1435 3938 42 4 v 1518 3901 a Fq(\()1520 3974 y Fs(A)1677
|
|
|
3127 |
3812 y(contr)1870 3824 y Fq(R)2008 3489 y(\()2010 3563
|
|
|
3128 |
y Fs(A)p 2116 3543 10 38 v 2125 3526 42 4 v 2180 3489
|
|
|
3129 |
a Fq(\()2182 3563 y Fs(A)2330 3489 y Fq(\()2332 3563
|
|
|
3130 |
y Fs(A)p 2437 3543 10 38 v 2447 3526 42 4 v 2502 3489
|
|
|
3131 |
a Fq(\()2504 3563 y Fs(A)p 2008 3583 561 4 v 2020 3691
|
|
|
3132 |
a Fq(\()2022 3764 y Fs(A)2086 3756 y(;)2123 3691 y Fq(\()2125
|
|
|
3133 |
3764 y Fs(A)p 2230 3744 10 38 v 2240 3728 42 4 v 2326
|
|
|
3134 |
3643 a Fn(\000)g(\000)g(\000)g(\000)g Fq(*)2323 3691
|
|
|
3135 |
y(\()2325 3764 y Fs(A)2412 3756 y Fr(^)2491 3691 y Fq(\()2493
|
|
|
3136 |
3764 y Fs(A)2610 3599 y Fr(^)2665 3612 y Fq(R)p 2020
|
|
|
3137 |
3792 538 4 v 2071 3901 a(\()2073 3974 y Fs(A)p 2179 3954
|
|
|
3138 |
10 38 v 2188 3938 42 4 v 2274 3853 a Fn(\000)g(\000)g(\000)g(\000)g
|
|
|
3139 |
Fq(*)2271 3901 y(\()2273 3974 y Fs(A)2361 3966 y Fr(^)2439
|
|
|
3140 |
3901 y Fq(\()2441 3974 y Fs(A)2598 3812 y(contr)2791
|
|
|
3141 |
3824 y Fq(L)p 1150 3994 1356 4 v 1529 4055 a(\()g Fn(\000)g(\000)g
|
|
|
3142 |
(\000)g(\000)1527 4102 y Fq(\()1529 4176 y Fs(A)1616
|
|
|
3143 |
4168 y Fr(_)1694 4102 y Fq(\()1696 4176 y Fs(A)p 1802
|
|
|
3144 |
4156 10 38 v 1812 4139 42 4 v 1897 4055 a Fn(\000)g(\000)g(\000)g(\000)
|
|
|
3145 |
g Fq(*)1895 4102 y(\()1897 4176 y Fs(A)1984 4168 y Fr(^)2062
|
|
|
3146 |
4102 y Fq(\()2064 4176 y Fs(A)2547 4019 y(cut)523 4361
|
|
|
3147 |
y FA(If)26 b(the)f(normalform)e(\(3\))i(is)i(to)f(be)f(reached,)g(then)
|
|
|
3148 |
g(accordingly)e(we)j(need)f(to)h(orient)f(the)h(colour)523
|
|
|
3149 |
4461 y(of)e(the)h(cut-formula)d(to)j(the)g(right.)f(Note)g(ho)n(we)n(v)
|
|
|
3150 |
o(er)f(that)i(choosing)e(colours)g(has)i(nothing)e(to)i(do)523
|
|
|
3151 |
4560 y(with)j(imposing)e(a)i(strate)o(gy)f(for)g(cut-elimination:)f(it)
|
|
|
3152 |
i(is)h(not)e(cuts)h(that)g(are)f(selected)h(by)f(them,)523
|
|
|
3153 |
4660 y(b)n(ut)f(rather)f(the)h(w)o(ay)h(ho)n(w)e(cuts)h(are)g(reduced.)
|
|
|
3154 |
f(Important)f(for)h(our)g(discussion)h(is)h(the)f(f)o(act)g(that)p
|
|
|
3155 |
523 4746 473 4 v 558 4801 a Fo(3)606 4833 y Fx(A)15 b(less)g(na)n(\250)
|
|
|
3156 |
-23 b(\021v)o(e)16 b(method,)g(which)g(only)g(tries)f(out)g(all)g
|
|
|
3157 |
(possible)h(reduction)g(for)g(outermost)g(cuts,)f(is)g(described)606
|
|
|
3158 |
4924 y(in)k([19].)p eop end
|
|
|
3159 |
%%Page: 11 11
|
|
|
3160 |
TeXDict begin 11 10 bop 523 448 a FA(one)21 b(is,)i(ho)n(we)n(v)o(er)m
|
|
|
3161 |
(,)c(not)i(completely)g(free)g(about)g(ho)n(w)g(to)h(annotate)f
|
|
|
3162 |
(colours)f(to)i(a)g(sequent)f(proof.)523 548 y(In)k(f)o(act)g(once)g
|
|
|
3163 |
(the)g(colour)f(`)p Fs(\()p FA(')h(is)h(chosen)e(for)h(the)g
|
|
|
3164 |
(cut-formula)d Fs(A)k FA(in)g(\(1\),)e(all)i(occurrences)d(of)523
|
|
|
3165 |
648 y Fs(A)j FA(must)g(ha)n(v)o(e)f(this)h(colour)-5
|
|
|
3166 |
b(.)24 b(The)i(only)e(\223free\224)h(choices)g(in)h(this)g(proof)e(are)
|
|
|
3167 |
h(the)h(colours)f(for)f(the)523 747 y(formulae)16 b Fs(A)p
|
|
|
3168 |
Fr(_)q Fs(A)i FA(and)f Fs(A)p Fr(^)q Fs(A)p FA(\227for)g(them)g(we)h
|
|
|
3169 |
(can)g(mak)o(e)f(an)o(y)g(choice,)g(b)n(ut)g(it)i(has)e(to)h(be)g
|
|
|
3170 |
(consistent)523 847 y(throughout)k(the)j(proof.)f(Danos)h(et)h(al.)f
|
|
|
3171 |
(state)h(this)g(consistenc)o(y)e(requirement)f(using)h(the)i(notion)523
|
|
|
3172 |
946 y(of)j(an)g Ft(identity)f(class)i FA(in)f(a)h(proof)d(\(the)i
|
|
|
3173 |
(follo)n(wing)e(de\002nition)h(is)i(slightly)f(adapted)f(from)g([16,)
|
|
|
3174 |
523 1046 y(P)o(age)20 b(107]\):)523 1205 y Fb(De\002nition)g(1.)41
|
|
|
3175 |
b Ft(Occurr)m(ences)18 b(of)g(\(sub\)formulae)e(in)j(a)f(pr)l(oof)g(ar)
|
|
|
3176 |
m(e)g FA(identi\002ed)f Ft(whene)o(ver)h(the)n(y)f(ar)m(e)523
|
|
|
3177 |
1304 y(the)j(corr)m(esponding)e(occurr)m(ences)i(of)g(the)g(same)h
|
|
|
3178 |
(\(sub\)formula)d(in)581 1455 y Fr(\017)41 b Ft(the)20
|
|
|
3179 |
b(two)h(formulae)f(in)g(an)g(axiom,)581 1551 y Fr(\017)41
|
|
|
3180 |
b Ft(the)20 b(cut-formulae)f(in)h(a)h(cut)f(and)581 1648
|
|
|
3181 |
y Fr(\017)41 b Ft(the)26 b(up)f(and)g(down)g(occurr)m(ences)g(of)h(a)f
|
|
|
3182 |
(formula)h(in)f(an)h(infer)m(ence)f(rule)h(\(this)g(includes)e(the)664
|
|
|
3183 |
1747 y(contr)o(acted)19 b(occurr)m(ences)g(in)i(contr)o(actions)d
|
|
|
3184 |
(rules\).)523 1901 y(An)32 b FA(identity)f(class)i Ft(in)f(a)f(pr)l
|
|
|
3185 |
(oof)h(is)h(the)f(r)m(e\003e)n(xive)o(,)f(symmetric)i(and)e(tr)o
|
|
|
3186 |
(ansitive)h(closur)m(e)f(of)h(the)523 2000 y(identi\002cation)18
|
|
|
3187 |
b(r)m(elation.)2086 b Fr(u)-55 b(t)523 2159 y FA(The)25
|
|
|
3188 |
b(consistenc)o(y)e(requirement)g(can)i(then)f(be)h(stated)g(as)h(follo)
|
|
|
3189 |
n(ws:)f(Whene)n(v)o(er)e(colours)h(are)h(an-)523 2259
|
|
|
3190 |
y(notated)18 b(to)h(a)g(proof,)e(then)i(e)n(v)o(ery)e(formula)h(in)h
|
|
|
3191 |
(an)g(identity)f(class)i(must)f(recei)n(v)o(e)f(the)h(same)g(colour)-5
|
|
|
3192 |
b(.)648 2358 y(The)21 b(interesting)g(point)g(of)h(colour)n
|
|
|
3193 |
(-annotations)d(is)j(the)g(f)o(act)h(that)f(the)o(y)f(determine)f
|
|
|
3194 |
(uniquely)523 2458 y(a)27 b(normalform.)d(In)j(light)g(of)g(this,)g(it)
|
|
|
3195 |
h(seems)f(reasonable)f(to)h(re)o(gard)e(as)j(the)f(collection)f(of)g
|
|
|
3196 |
(nor)n(-)523 2557 y(malforms)17 b(reachable)f(from)h(a)i(classical)f
|
|
|
3197 |
(proof)f(all)h(those)g(for)g(which)f(a)h(colour)f(annotation)f(e)o
|
|
|
3198 |
(xists)523 2657 y(that)28 b(mak)o(es)h(them)e(reachable.)g(Then)h(the)g
|
|
|
3199 |
(question)f(arises:)i(Can)g(we)f(\002nd)g(for)g(e)n(v)o(ery)f(normal-)
|
|
|
3200 |
523 2757 y(form)17 b(reachable)f(by)i(the)g(\(un-coloured\))c
|
|
|
3201 |
(cut-elimination)h(procedure)h(of)h(Urban)g(and)h(Bierman)f(a)523
|
|
|
3202 |
2856 y(colour)n(-annotation)12 b(that)17 b(mak)o(es)f(them)g(reachable)
|
|
|
3203 |
f(by)h(the)g(cut-elimination)e(procedure)g(of)i(Danos)523
|
|
|
3204 |
2956 y(et)21 b(al.?)f(The)g(answer)g(is)h(no)f(and)f(for)h(deep)f
|
|
|
3205 |
(reasons!)h(Consider)f(the)i(follo)n(wing)d(classical)j(proof)1213
|
|
|
3206 |
3178 y Fm(\(1\))1254 3251 y(:)1042 3325 y Fs(A)p Fr(_)p
|
|
|
3207 |
Fs(A)p 1240 3313 10 38 v 1250 3296 42 4 v 89 w(A)p Fr(^)p
|
|
|
3208 |
Fs(A)1598 3106 y(A)p Fr(^)p Fs(A)p 1796 3094 10 38 v
|
|
|
3209 |
1806 3078 42 4 v 89 w(A)p Fr(^)p Fs(A)84 b(A)p Fr(^)p
|
|
|
3210 |
Fs(A)p 2327 3094 10 38 v 2336 3078 42 4 v 88 w(A)p Fr(^)q
|
|
|
3211 |
Fs(A)p 1573 3126 1029 4 v 1573 3205 a(A)p Fr(^)p Fs(A;)14
|
|
|
3212 |
b(A)p Fr(^)q Fs(A)p 1988 3193 10 38 v 1997 3176 42 4
|
|
|
3213 |
v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p
|
|
|
3214 |
Fs(A)p Fr(^)q Fs(A)p Fm(\))2643 3143 y Fr(^)2698 3156
|
|
|
3215 |
y Fq(R)p 1573 3246 1029 4 v 1681 3325 a Fs(A)p Fr(^)q
|
|
|
3216 |
Fs(A)p 1879 3313 10 38 v 1889 3296 42 4 v 88 w Fm(\()p
|
|
|
3217 |
Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q
|
|
|
3218 |
Fs(A)p Fm(\))2643 3265 y Fs(contr)2836 3277 y Fq(L)p
|
|
|
3219 |
1042 3365 1452 4 v 1361 3444 a Fs(A)p Fr(_)q Fs(A)p 1560
|
|
|
3220 |
3432 10 38 v 1569 3416 42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q
|
|
|
3221 |
Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2535
|
|
|
3222 |
3391 y Fs(cut)3267 3444 y FA(\(12\))523 3611 y(where)k(we)g(cut)h(the)f
|
|
|
3223 |
(proof)e(from)i(\(1\))f(against)h(a)g(proof)f(whose)h(cut-formula,)e
|
|
|
3224 |
Fs(A)p Fr(^)p Fs(A)p FA(,)j(is)g(contracted)523 3711
|
|
|
3225 |
y(in)k(the)g(right-subproof.)18 b(W)-7 b(e)24 b(can)e(reduce)g(the)g
|
|
|
3226 |
(lo)n(wer)h(cut)f(so)h(that)g(we)g(obtain)f(tw)o(o)h(copies)f(of)g(the)
|
|
|
3227 |
523 3810 y(proof)d(\(1\):)1503 3899 y Fm(\(1\))1545 3972
|
|
|
3228 |
y(:)1332 4045 y Fs(A)p Fr(_)q Fs(A)p 1531 4033 10 38
|
|
|
3229 |
v 1540 4017 42 4 v 88 w(A)p Fr(^)q Fs(A)2034 3899 y Fm(\(1\))2075
|
|
|
3230 |
3972 y(:)1863 4045 y Fs(A)p Fr(_)q Fs(A)p 2061 4033 10
|
|
|
3231 |
38 v 2071 4017 42 4 v 88 w(A)p Fr(^)q Fs(A)p 1307 4065
|
|
|
3232 |
1029 4 v 1307 4144 a(A)p Fr(_)q Fs(A;)14 b(A)p Fr(_)q
|
|
|
3233 |
Fs(A)p 1722 4132 10 38 v 1732 4116 42 4 v 88 w Fm(\()p
|
|
|
3234 |
Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q
|
|
|
3235 |
Fs(A)p Fm(\))2378 4082 y Fr(^)2433 4095 y Fq(R)p 1307
|
|
|
3236 |
4185 1029 4 v 1416 4264 a Fs(A)p Fr(_)p Fs(A)p 1614 4252
|
|
|
3237 |
10 38 v 1623 4235 42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p
|
|
|
3238 |
Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2378
|
|
|
3239 |
4204 y Fs(contr)2571 4216 y Fq(L)3267 4264 y FA(\(13\))523
|
|
|
3240 |
4404 y(W)m(ithout)20 b(colours,)f(we)h(can)g(then)g(reduce)f(each)h
|
|
|
3241 |
(cop)o(y)f(completely)g(independently)e(as)k(follo)n(ws:)1503
|
|
|
3242 |
4560 y Fm(\(2\))1545 4633 y(:)1332 4706 y Fs(A)p Fr(_)q
|
|
|
3243 |
Fs(A)p 1531 4694 10 38 v 1540 4677 42 4 v 88 w(A)p Fr(^)q
|
|
|
3244 |
Fs(A)2034 4560 y Fm(\(3\))2075 4633 y(:)1863 4706 y Fs(A)p
|
|
|
3245 |
Fr(_)q Fs(A)p 2061 4694 10 38 v 2071 4677 42 4 v 88 w(A)p
|
|
|
3246 |
Fr(^)q Fs(A)p 1307 4726 1029 4 v 1307 4805 a(A)p Fr(_)q
|
|
|
3247 |
Fs(A;)14 b(A)p Fr(_)q Fs(A)p 1722 4793 10 38 v 1732 4776
|
|
|
3248 |
42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q
|
|
|
3249 |
Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2378 4743 y Fr(^)2433
|
|
|
3250 |
4755 y Fq(R)p 1307 4845 1029 4 v 1416 4924 a Fs(A)p Fr(_)p
|
|
|
3251 |
Fs(A)p 1614 4912 10 38 v 1623 4896 42 4 v 88 w Fm(\()p
|
|
|
3252 |
Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q
|
|
|
3253 |
Fs(A)p Fm(\))2378 4865 y Fs(contr)2571 4877 y Fq(L)3267
|
|
|
3254 |
4924 y FA(\(14\))p eop end
|
|
|
3255 |
%%Page: 12 12
|
|
|
3256 |
TeXDict begin 12 11 bop 523 448 a FA(Such)24 b(a)g(beha)n(viour)e
|
|
|
3257 |
(cannot)h(be)g(achie)n(v)o(ed)g(by)g(using)h(colours:)e(the)i(colours)f
|
|
|
3258 |
(must)h(be)g(annotated)523 548 y(before)17 b(cut-elimination)f
|
|
|
3259 |
(commences)g(and)i(is)h(in)m(v)n(ariant)d(under)h(cut-reductions.)e
|
|
|
3260 |
(Consequently)-5 b(,)523 648 y(whene)n(v)o(er)22 b(a)i(cut)g(is)h
|
|
|
3261 |
(duplicated)e(in)h(a)g(reduction)e(sequence)h(\(as)i(in)f(the)g
|
|
|
3262 |
(reduction)e(\(12\))p Fr(!)p FA(\(13\)\),)523 747 y(the)29
|
|
|
3263 |
b(colour)n(-annotation)d(pre)n(v)o(ents)h(both)i(instances)g(from)f
|
|
|
3264 |
(reducing)g(dif)n(ferently)-5 b(.)26 b(\(The)j(deeper)523
|
|
|
3265 |
847 y(reason)e(mentioned)g(earlier)g(is)j(that)e(one)f(just)i(cannot)e
|
|
|
3266 |
(pre-determine)f(the)i(choices)f(in)i(a)f(com-)523 946
|
|
|
3267 |
y(pletely)20 b(non-deterministic)d(reduction)i(system.\))648
|
|
|
3268 |
1061 y(Comparing)26 b(the)j(cut-elimination)e(procedure)f(of)j(Urban)e
|
|
|
3269 |
(and)i(Bierman)f(with)h(the)g(one)f(of)523 1161 y(Danos)16
|
|
|
3270 |
b(et)i(al.,)f(tw)o(o)g(points)f(stand)g(out:)h(Both)g(cut-elimination)d
|
|
|
3271 |
(procedures)h(are)i(strongly)e(normal-)523 1261 y(ising)685
|
|
|
3272 |
1230 y Fp(4)742 1261 y FA(and)22 b(also)i(determine)e(a)h(collection)f
|
|
|
3273 |
(of)h(normalforms)e(reachable)h(from)g(a)i(sequent-proof)523
|
|
|
3274 |
1360 y(in)c(classical)g(logic.)f(As)h(sho)n(wn)f(by)g(e)o(xample,)f
|
|
|
3275 |
(these)i(collections)f(contain)f(in)i(general)e(more)h(than)523
|
|
|
3276 |
1460 y(one)g(element.)g(Also)i(as)f(sho)n(wn)f(by)h(e)o(xample,)e(the)i
|
|
|
3277 |
(collection)f(determined)f(by)h(the)h(procedure)e(of)523
|
|
|
3278 |
1559 y(Danos)k(et)g(al.)g(is)h(generally)e(a)h(proper)f(subset)h(of)f
|
|
|
3279 |
(the)h(collection)f(determined)f(by)i(the)g(procedure)523
|
|
|
3280 |
1659 y(of)f(Urban)g(and)g(Bierman.)g(The)g(colour)n(-annotations)d(in)k
|
|
|
3281 |
(the)f(procedure)e(of)i(Danos)h(et)g(al.)g(cannot)523
|
|
|
3282 |
1759 y(fully)e(account)f(for)g(the)h(non-determinism)d(present)j(in)g
|
|
|
3283 |
(classical)h(logic.)648 1873 y(Both)f(cut-elimination)d(procedures)h
|
|
|
3284 |
(can)i(also)g(be)g(used)f(for)h(reducing)e(intuitionistic)h(proofs.)523
|
|
|
3285 |
1973 y(Because)k(of)g(the)h(restrictions)e(imposed)g(upon)g
|
|
|
3286 |
(intuitionistic)h(sequents,)f(non-deterministic)f(re-)523
|
|
|
3287 |
2073 y(duction)e(sequences)h(such)g(as)i(\(12\))p Fr(!)p
|
|
|
3288 |
FA(\(13\))p Fr(!)p FA(\(14\))16 b(cannot)j(be)i(constructed.)d(But)k
|
|
|
3289 |
(still)f(the)g(proce-)523 2172 y(dure)d(of)g(Urban)g(and)g(Bierman)g
|
|
|
3290 |
(is)h Ft(not)h FA(Church-Rosser)d(in)i(the)f(intuitionistic)g(case,)h
|
|
|
3291 |
(and)f(also)h(dif-)523 2272 y(ferent)f(colour)n(-annotations)d(of)j(an)
|
|
|
3292 |
g(intuitionistic)g(proof)f(might)h(lead)g(to)h(dif)n(ferent)d
|
|
|
3293 |
(normalforms.)523 2372 y(Ho)n(we)n(v)o(er)m(,)31 b(as)i(mentioned)e
|
|
|
3294 |
(earlier)m(,)h(we)h(re)o(gard)e(the)i(dif)n(ferences)e(between)h(the)h
|
|
|
3295 |
(normalforms)523 2471 y(reachable)19 b(from)g(an)h(intuitionistic)f
|
|
|
3296 |
(sequent-proof)e(as)k(inessential)f(and)g(re)o(gard)e(cut-elimination)
|
|
|
3297 |
523 2571 y(as)23 b(morally)e(Church-Rosser)-5 b(.)21
|
|
|
3298 |
b(That)h(in)h(turn)e(means)h(that)h(in)f(the)g(intuitionistic)g(case)h
|
|
|
3299 |
(there)f(is)h(no)523 2670 y(dif)n(ference)18 b(between)i(coloured)e
|
|
|
3300 |
(and)h(un-coloured)e(cut-elimination\227at)h(least)j(morally)-5
|
|
|
3301 |
b(.)523 3012 y Fu(4)99 b(Conjectur)n(e)523 3287 y FA(W)-7
|
|
|
3302 |
b(e)19 b(ha)n(v)o(e)e(already)g(seen)h(that)f(by)h(translating)f(the)g
|
|
|
3303 |
(cut)h(in)g(\(1\))f(using)g(a)h(left-)g(and)f(right-translation,)523
|
|
|
3304 |
3387 y(we)k(can)f(simulate)g(the)h(reductions)e(\(1\))p
|
|
|
3305 |
Fr(!)p FA(\(2\))f(and)i(\(1\))p Fr(!)p FA(\(3\))f(by)h(double-ne)o
|
|
|
3306 |
(gations.)15 b(Ho)n(we)n(v)o(er)k(in)523 3486 y(general,)g(a)h(left-)g
|
|
|
3307 |
(or)g(right-translation)e(of)i(a)g(cut)h(is)g(not)f(suf)n(\002cient)f
|
|
|
3308 |
(to)h(simulate)h(all)f(cut-reduction)523 3586 y(sequences)f(in)i
|
|
|
3309 |
(classical)g(logic.)e(Consider)h(the)g(follo)n(wing)f(instance)g(of)h
|
|
|
3310 |
(a)h(logical)f(cut:)1520 3790 y Fs(\031)1567 3802 y Fl(1)1550
|
|
|
3311 |
3855 y Fm(:)p 1503 3916 10 38 v 1512 3900 42 4 v 1572
|
|
|
3312 |
3928 a Fs(B)p 1424 3948 276 4 v 1442 4010 10 38 v 1452
|
|
|
3313 |
3993 42 4 v 1512 4022 a(B)t Fr(_)p Fs(C)1741 3961 y Fr(_)1797
|
|
|
3314 |
3974 y Fq(R)1847 3982 y Fc(1)2001 3790 y Fs(\031)2048
|
|
|
3315 |
3802 y Fl(2)2032 3855 y Fm(:)1966 3928 y Fs(B)p 2052
|
|
|
3316 |
3916 10 38 v 2061 3900 42 4 v 2239 3790 a(\031)2286 3802
|
|
|
3317 |
y Fl(3)2269 3855 y Fm(:)2204 3928 y Fs(C)p 2288 3916
|
|
|
3318 |
10 38 v 2298 3900 42 4 v 1966 3948 392 4 v 2024 4022
|
|
|
3319 |
a(B)t Fr(_)p Fs(C)p 2230 4010 10 38 v 2240 3993 42 4
|
|
|
3320 |
v 2399 3965 a Fr(_)2454 3978 y Fq(L)p 1424 4042 876 4
|
|
|
3321 |
v 1836 4095 10 38 v 1846 4079 42 4 v 2341 4067 a Fs(cut)3267
|
|
|
3322 |
4107 y FA(\(15\))523 4318 y(W)-7 b(e)21 b(can)f(reduce)f(this)i(cut)f
|
|
|
3323 |
(to)1725 4408 y Fs(\031)1772 4420 y Fl(1)1756 4473 y
|
|
|
3324 |
Fm(:)p 1708 4534 10 38 v 1718 4518 42 4 v 1778 4546 a
|
|
|
3325 |
Fs(B)1963 4408 y(\031)2010 4420 y Fl(2)1994 4473 y Fm(:)1928
|
|
|
3326 |
4546 y Fs(B)p 2014 4534 10 38 v 2023 4518 42 4 v 1690
|
|
|
3327 |
4566 394 4 v 1861 4620 10 38 v 1871 4603 42 4 v 2125
|
|
|
3328 |
4592 a(cut)p 523 4746 473 4 v 558 4801 a Fo(4)606 4833
|
|
|
3329 |
y Fx(Danos)29 b(et)g(al.)f(sho)n(wed)i(strong)f(normalisation)g(of)g
|
|
|
3330 |
(their)f(cut-elimination)h(procedure)h(by)f(translating)606
|
|
|
3331 |
4924 y(reduction)20 b(sequences)h(in)d(classical)h(logic)g(to)g
|
|
|
3332 |
(reduction)h(sequences)h(of)e(proof-nets)g(in)g(linear)g(logic.)p
|
|
|
3333 |
eop end
|
|
|
3334 |
%%Page: 13 13
|
|
|
3335 |
TeXDict begin 13 12 bop 523 448 a FA(and)28 b(assuming)g(that)h(the)f
|
|
|
3336 |
(cut-formula)e Fs(B)34 b FA(is)29 b(not)f(freshly)g(introduced)e(in)j
|
|
|
3337 |
Fs(\031)2821 460 y Fl(1)2859 448 y FA(,)g(we)g(can)f(further)523
|
|
|
3338 |
548 y(permute)19 b Fs(\031)863 560 y Fl(2)921 548 y FA(inside)h
|
|
|
3339 |
Fs(\031)1187 560 y Fl(1)1225 548 y FA(.)g(This)h(beha)n(viour)d
|
|
|
3340 |
(correspond)f(to)k(the)f(colour)n(-annotation)1508 786
|
|
|
3341 |
y Fs(\031)1555 798 y Fl(1)1539 851 y Fm(:)p 1480 952
|
|
|
3342 |
10 38 v 1489 935 42 4 v 1575 898 a Fq(\()1572 972 y Fs(B)p
|
|
|
3343 |
1389 991 322 4 v 1408 1092 10 38 v 1417 1075 42 4 v 1503
|
|
|
3344 |
1038 a Fq(\()1500 1112 y Fs(B)1591 1104 y Fr(_)p Fs(C)1753
|
|
|
3345 |
1004 y Fr(_)1808 1016 y Fq(R)1858 1024 y Fc(1)2024 786
|
|
|
3346 |
y Fs(\031)2071 798 y Fl(2)2055 851 y Fm(:)1980 898 y
|
|
|
3347 |
Fq(\()1978 972 y Fs(B)p 2086 952 10 38 v 2096 935 42
|
|
|
3348 |
4 v 2273 825 a(\031)2320 837 y Fl(3)2304 890 y Fm(:)2239
|
|
|
3349 |
964 y Fs(C)p 2323 952 10 38 v 2332 935 42 4 v 1978 991
|
|
|
3350 |
415 4 v 2038 1038 a Fq(\()2035 1112 y Fs(B)2126 1104
|
|
|
3351 |
y Fr(_)p Fs(C)p 2265 1092 10 38 v 2274 1075 42 4 v 2434
|
|
|
3352 |
1008 a Fr(_)2489 1020 y Fq(L)p 1389 1132 945 4 v 1836
|
|
|
3353 |
1185 10 38 v 1846 1169 42 4 v 2376 1157 a Fs(cut)3267
|
|
|
3354 |
1197 y FA(\(16\))523 1519 y(where)d(we)h(lea)n(v)o(e)g(the)g(colour)e
|
|
|
3355 |
(annotation)g(for)h Fs(C)25 b FA(and)17 b Fs(B)t Fr(_)p
|
|
|
3356 |
Fs(C)25 b FA(unspeci\002ed,)16 b(since)i(it)h(is)f(not)g(impor)n(-)523
|
|
|
3357 |
1619 y(tant)25 b(for)f(the)i(ar)o(gument)c(at)k(hand.)d(The)i(beha)n
|
|
|
3358 |
(viour)e(of)i(\(16\))f(can)g(be)h(simulated)g(by)f(the)h(double-)523
|
|
|
3359 |
1719 y(ne)o(gation)18 b(translation)h Fm(\()p Fr(\000)p
|
|
|
3360 |
Fm(\))1328 1688 y Fn(\003)1367 1719 y FA(.)h(The)g(double-ne)o(gated)c
|
|
|
3361 |
(v)o(ersion)j(of)h(\(15\))f(is)i(as)g(follo)n(ws:)1364
|
|
|
3362 |
2163 y Fs(\031)1414 2133 y Fn(\003)1411 2184 y Fl(1)1397
|
|
|
3363 |
2236 y Fm(:)1284 2310 y Fr(:)p Fs(B)1406 2280 y Fn(\003)p
|
|
|
3364 |
1463 2298 10 38 v 1473 2281 42 4 v 1177 2330 463 4 v
|
|
|
3365 |
1177 2404 a Fr(:)p Fs(B)1299 2373 y Fn(\003)1338 2404
|
|
|
3366 |
y Fr(^:)p Fs(C)1513 2373 y Fn(\003)p 1571 2392 10 38
|
|
|
3367 |
v 1580 2375 42 4 v 1682 2342 a Fr(^)1737 2355 y Fq(L)1783
|
|
|
3368 |
2363 y Fc(1)p 1117 2423 583 4 v 1136 2490 10 38 v 1145
|
|
|
3369 |
2474 42 4 v 1205 2502 a Fr(:)p Fm(\()p Fr(:)p Fs(B)1414
|
|
|
3370 |
2472 y Fn(\003)1454 2502 y Fr(^:)p Fs(C)1629 2472 y Fn(\003)1668
|
|
|
3371 |
2502 y Fm(\))1742 2435 y Fr(:)1797 2447 y Fq(R)p 1090
|
|
|
3372 |
2543 639 4 v 1090 2622 a Fr(::)p Fm(\()p Fr(:)p Fs(B)1354
|
|
|
3373 |
2592 y Fn(\003)1393 2622 y Fr(^)q(:)p Fs(C)1569 2592
|
|
|
3374 |
y Fn(\003)1607 2622 y Fm(\))p 1658 2610 10 38 v 1668
|
|
|
3375 |
2593 42 4 v 1769 2555 a Fr(:)1824 2567 y Fq(L)2037 2069
|
|
|
3376 |
y Fs(\031)2087 2039 y Fn(\003)2084 2090 y Fl(2)2070 2142
|
|
|
3377 |
y Fm(:)1985 2216 y Fs(B)2052 2186 y Fn(\003)p 2109 2204
|
|
|
3378 |
10 38 v 2118 2188 42 4 v 1957 2236 249 4 v 1976 2298
|
|
|
3379 |
10 38 v 1985 2281 42 4 v 2045 2310 a Fr(:)p Fs(B)2167
|
|
|
3380 |
2280 y Fn(\003)2247 2248 y Fr(:)2302 2260 y Fq(R)2519
|
|
|
3381 |
2069 y Fs(\031)2569 2039 y Fn(\003)2566 2090 y Fl(3)2552
|
|
|
3382 |
2142 y Fm(:)2468 2216 y Fs(C)2533 2186 y Fn(\003)p 2590
|
|
|
3383 |
2204 10 38 v 2599 2188 42 4 v 2440 2236 247 4 v 2459
|
|
|
3384 |
2298 10 38 v 2468 2281 42 4 v 2528 2310 a Fr(:)p Fs(C)2648
|
|
|
3385 |
2280 y Fn(\003)2728 2248 y Fr(:)2783 2260 y Fq(R)p 1957
|
|
|
3386 |
2330 730 4 v 2109 2392 10 38 v 2119 2375 42 4 v 2179
|
|
|
3387 |
2404 a Fr(:)p Fs(B)2301 2373 y Fn(\003)2339 2404 y Fr(^)q(:)p
|
|
|
3388 |
Fs(C)2515 2373 y Fn(\003)2728 2347 y Fr(^)2784 2359 y
|
|
|
3389 |
Fq(R)p 2031 2423 583 4 v 2031 2502 a Fr(:)p Fm(\()p Fr(:)p
|
|
|
3390 |
Fs(B)2240 2472 y Fn(\003)2279 2502 y Fr(^:)p Fs(C)2454
|
|
|
3391 |
2472 y Fn(\003)2493 2502 y Fm(\))p 2544 2490 10 38 v
|
|
|
3392 |
2553 2474 42 4 v 2655 2435 a Fr(:)2710 2447 y Fq(L)p
|
|
|
3393 |
2003 2543 639 4 v 2022 2610 10 38 v 2031 2593 42 4 v
|
|
|
3394 |
2091 2622 a Fr(::)p Fm(\()p Fr(:)p Fs(B)2355 2592 y Fn(\003)2395
|
|
|
3395 |
2622 y Fr(^:)p Fs(C)2570 2592 y Fn(\003)2609 2622 y Fm(\))2683
|
|
|
3396 |
2555 y Fr(:)2738 2567 y Fq(R)p 1090 2663 1552 4 v 1840
|
|
|
3397 |
2716 10 38 v 1849 2700 42 4 v 2683 2688 a Fs(cut)523
|
|
|
3398 |
3064 y FA(which)f(reduces)f(in)h(three)g(steps)h(to)f(the)g(proof)1676
|
|
|
3399 |
3508 y Fs(\031)1726 3478 y Fn(\003)1723 3529 y Fl(1)1709
|
|
|
3400 |
3581 y Fm(:)1596 3655 y Fr(:)p Fs(B)1718 3625 y Fn(\003)p
|
|
|
3401 |
1775 3643 10 38 v 1785 3626 42 4 v 2008 3414 a Fs(\031)2058
|
|
|
3402 |
3384 y Fn(\003)2055 3435 y Fl(2)2041 3487 y Fm(:)1956
|
|
|
3403 |
3561 y Fs(B)2023 3531 y Fn(\003)p 2079 3549 10 38 v 2089
|
|
|
3404 |
3532 42 4 v 1928 3581 249 4 v 1946 3643 10 38 v 1956
|
|
|
3405 |
3626 42 4 v 2016 3655 a Fr(:)p Fs(B)2138 3625 y Fn(\003)2218
|
|
|
3406 |
3593 y Fr(:)2273 3605 y Fq(R)p 1596 3675 581 4 v 1861
|
|
|
3407 |
3728 10 38 v 1871 3712 42 4 v 2218 3700 a Fs(cut)523
|
|
|
3408 |
4076 y FA(Because)f(the)g Fr(:)995 4088 y Fq(R)1049 4076
|
|
|
3409 |
y FA(-rule)f(introduces)f(freshly)h(the)h(cut-formula)d
|
|
|
3410 |
Fr(:)p Fs(B)2494 4046 y Fn(\003)2533 4076 y FA(,)j(the)f(cut)h(is)h
|
|
|
3411 |
(\223block)o(ed\224)d(from)523 4175 y(reducing)h(to)j(the)f(right.)f
|
|
|
3412 |
(It)i(must)f(\002rst)h(reduce)e(to)h(the)g(left)g(just)h(as)g(the)f
|
|
|
3413 |
(colour)f(annotation)f(in)j(\(16\))523 4275 y(prescribed.)d(If)i(we)h
|
|
|
3414 |
(w)o(anted)f(to)g(simulated)g(the)g(opposite)f(colouring)f(for)h
|
|
|
3415 |
Fs(B)t FA(,)i(namely)1508 4513 y Fs(\031)1555 4525 y
|
|
|
3416 |
Fl(1)1539 4578 y Fm(:)p 1480 4679 10 38 v 1489 4662 42
|
|
|
3417 |
4 v 1575 4625 a Fq(*)1572 4699 y Fs(B)p 1389 4718 322
|
|
|
3418 |
4 v 1408 4819 10 38 v 1417 4802 42 4 v 1503 4765 a Fq(*)1500
|
|
|
3419 |
4839 y Fs(B)1591 4831 y Fr(_)p Fs(C)1753 4731 y Fr(_)1808
|
|
|
3420 |
4743 y Fq(R)1858 4751 y Fc(1)2024 4513 y Fs(\031)2071
|
|
|
3421 |
4525 y Fl(2)2055 4578 y Fm(:)1980 4625 y Fq(*)1978 4699
|
|
|
3422 |
y Fs(B)p 2086 4679 10 38 v 2096 4662 42 4 v 2273 4552
|
|
|
3423 |
a(\031)2320 4564 y Fl(3)2304 4617 y Fm(:)2239 4691 y
|
|
|
3424 |
Fs(C)p 2323 4679 10 38 v 2332 4662 42 4 v 1978 4718 415
|
|
|
3425 |
4 v 2038 4765 a Fq(*)2035 4839 y Fs(B)2126 4831 y Fr(_)p
|
|
|
3426 |
Fs(C)p 2265 4819 10 38 v 2274 4802 42 4 v 2434 4735 a
|
|
|
3427 |
Fr(_)2489 4747 y Fq(L)p 1389 4859 945 4 v 1836 4912 10
|
|
|
3428 |
38 v 1846 4896 42 4 v 2376 4884 a Fs(cut)3267 4924 y
|
|
|
3429 |
FA(\(17\))p eop end
|
|
|
3430 |
%%Page: 14 14
|
|
|
3431 |
TeXDict begin 14 13 bop 523 448 a FA(it)29 b(turns)g(out)f(we)h(ha)n(v)
|
|
|
3432 |
o(e)f(to)h(double-ne)o(gate)c(translate)j(\(15\))g(using)g(the)h
|
|
|
3433 |
(translation)e Fm(\()p Fr(\000)p Fm(\))3156 418 y Fn(\016)3224
|
|
|
3434 |
448 y FA(gi)n(v)o(en)523 548 y(in)20 b(\(6\).)g(The)g(resulting)f
|
|
|
3435 |
(intuitionistic)g(proof)g(is:)1436 721 y Fs(\031)1486
|
|
|
3436 |
691 y Fn(\016)1483 741 y Fl(1)1469 794 y Fm(:)1356 867
|
|
|
3437 |
y Fr(:)p Fs(B)1478 837 y Fn(\016)p 1535 855 10 38 v 1545
|
|
|
3438 |
839 42 4 v 1328 887 304 4 v 1347 949 10 38 v 1356 932
|
|
|
3439 |
42 4 v 1416 961 a Fr(::)p Fs(B)1593 931 y Fn(\016)1674
|
|
|
3440 |
899 y Fr(:)1729 911 y Fq(R)p 1328 981 304 4 v 1402 1043
|
|
|
3441 |
10 38 v 1412 1026 42 4 v 1472 1055 a Fs(B)1539 1025 y
|
|
|
3442 |
Fn(\016)1674 993 y Fr(::)1784 1005 y Fq(R)p 1304 1075
|
|
|
3443 |
353 4 v 1323 1136 10 38 v 1332 1120 42 4 v 1392 1148
|
|
|
3444 |
a Fs(B)1459 1118 y Fn(\016)1498 1148 y Fr(_)p Fs(C)1618
|
|
|
3445 |
1118 y Fn(\016)1698 1087 y Fr(_)1753 1100 y Fq(R)1803
|
|
|
3446 |
1108 y Fc(1)p 1244 1168 472 4 v 1244 1247 a Fr(:)p Fm(\()p
|
|
|
3447 |
Fs(B)1398 1217 y Fn(\016)1437 1247 y Fr(_)q Fs(C)1558
|
|
|
3448 |
1217 y Fn(\016)1596 1247 y Fm(\))p 1647 1235 10 38 v
|
|
|
3449 |
1656 1219 42 4 v 1758 1180 a Fr(:)1813 1192 y Fq(L)p
|
|
|
3450 |
1217 1288 528 4 v 1235 1355 10 38 v 1245 1338 42 4 v
|
|
|
3451 |
1305 1367 a Fr(::)p Fm(\()p Fs(B)1514 1337 y Fn(\016)1553
|
|
|
3452 |
1367 y Fr(_)p Fs(C)1673 1337 y Fn(\016)1712 1367 y Fm(\))1786
|
|
|
3453 |
1300 y Fr(:)1841 1312 y Fq(R)p 1189 1407 583 4 v 1189
|
|
|
3454 |
1486 a Fr(:::)p Fm(\()p Fs(B)1453 1456 y Fn(\016)1493
|
|
|
3455 |
1486 y Fr(_)p Fs(C)1613 1456 y Fn(\016)1651 1486 y Fm(\))p
|
|
|
3456 |
1702 1474 10 38 v 1712 1458 42 4 v 1813 1419 a Fr(:)1868
|
|
|
3457 |
1431 y Fq(L)2111 908 y Fs(\031)2161 878 y Fn(\016)2158
|
|
|
3458 |
929 y Fl(2)2144 981 y Fm(:)2059 1055 y Fs(B)2126 1025
|
|
|
3459 |
y Fn(\016)p 2183 1043 10 38 v 2192 1026 42 4 v 2387 908
|
|
|
3460 |
a Fs(\031)2437 878 y Fn(\016)2434 929 y Fl(3)2419 981
|
|
|
3461 |
y Fm(:)2335 1055 y Fs(C)2400 1025 y Fn(\016)p 2457 1043
|
|
|
3462 |
10 38 v 2467 1026 42 4 v 2059 1075 468 4 v 2117 1148
|
|
|
3463 |
a Fs(B)2184 1118 y Fn(\016)2222 1148 y Fr(_)p Fs(C)2342
|
|
|
3464 |
1118 y Fn(\016)p 2399 1136 10 38 v 2409 1120 42 4 v 2568
|
|
|
3465 |
1092 a Fr(_)2623 1104 y Fq(L)p 2057 1168 472 4 v 2075
|
|
|
3466 |
1235 10 38 v 2085 1219 42 4 v 2145 1247 a Fr(:)p Fm(\()p
|
|
|
3467 |
Fs(B)2299 1217 y Fn(\016)2338 1247 y Fr(_)p Fs(C)2458
|
|
|
3468 |
1217 y Fn(\016)2496 1247 y Fm(\))2570 1180 y Fr(:)2625
|
|
|
3469 |
1192 y Fq(R)p 2029 1288 528 4 v 2029 1367 a Fr(::)p Fm(\()p
|
|
|
3470 |
Fs(B)2238 1337 y Fn(\016)2277 1367 y Fr(_)q Fs(C)2398
|
|
|
3471 |
1337 y Fn(\016)2436 1367 y Fm(\))p 2487 1355 10 38 v
|
|
|
3472 |
2496 1338 42 4 v 2598 1300 a Fr(:)2653 1312 y Fq(L)p
|
|
|
3473 |
2001 1407 583 4 v 2020 1474 10 38 v 2029 1458 42 4 v
|
|
|
3474 |
2089 1486 a Fr(:::)p Fm(\()p Fs(B)2353 1456 y Fn(\016)2393
|
|
|
3475 |
1486 y Fr(_)p Fs(C)2513 1456 y Fn(\016)2552 1486 y Fm(\))2625
|
|
|
3476 |
1419 y Fr(:)2680 1431 y Fq(R)p 1189 1527 1395 4 v 1861
|
|
|
3477 |
1581 10 38 v 1871 1564 42 4 v 2625 1553 a Fs(cut)523
|
|
|
3478 |
1785 y FA(which)h(after)g(four)f(steps)h(reduces)g(to)g(the)g(proof)
|
|
|
3479 |
1601 1958 y Fs(\031)1651 1927 y Fn(\016)1648 1978 y Fl(1)1634
|
|
|
3480 |
2030 y Fm(:)1521 2104 y Fr(:)p Fs(B)1643 2074 y Fn(\016)p
|
|
|
3481 |
1700 2092 10 38 v 1709 2076 42 4 v 1493 2124 304 4 v
|
|
|
3482 |
1512 2186 10 38 v 1521 2169 42 4 v 1581 2198 a Fr(::)p
|
|
|
3483 |
Fs(B)1758 2168 y Fn(\016)1839 2136 y Fr(:)1894 2148 y
|
|
|
3484 |
Fq(R)p 1493 2218 304 4 v 1567 2280 10 38 v 1576 2263
|
|
|
3485 |
42 4 v 1636 2292 a Fs(B)1703 2262 y Fn(\016)1839 2230
|
|
|
3486 |
y Fr(::)1949 2242 y Fq(R)2139 2145 y Fs(\031)2189 2115
|
|
|
3487 |
y Fn(\016)2186 2166 y Fl(2)2172 2218 y Fm(:)2087 2292
|
|
|
3488 |
y Fs(B)2154 2262 y Fn(\016)p 2210 2280 10 38 v 2220 2263
|
|
|
3489 |
42 4 v 1548 2312 732 4 v 1889 2365 10 38 v 1898 2349
|
|
|
3490 |
42 4 v 2321 2337 a Fs(cut)3267 2377 y FA(\(18\))523 2569
|
|
|
3491 |
y(What)g(happens)e(ne)o(xt,)g(ho)n(we)n(v)o(er)m(,)f(is)j(not)f(clear)g
|
|
|
3492 |
(at)h(\002rst)g(sight.)g(If)f(we)g(e)o(xpand)f(the)h
|
|
|
3493 |
Fr(::)3010 2581 y Fq(R)3065 2569 y FA(-rule)g(to)h(an)523
|
|
|
3494 |
2669 y(auxiliary)e(cut,)i(then)f(the)g(cut)h(can)f(reduce)g(into)g
|
|
|
3495 |
(both)g(directions.)f(If)i(we)g(re)o(gard)d(the)j Fr(::)3104
|
|
|
3496 |
2681 y Fq(R)3159 2669 y FA(-rule)f(as)523 2769 y(an)k(inference)f(rule)
|
|
|
3497 |
g(in)i(its)g(o)n(wn)e(right,)h(then)f(the)h(cut-formula)e
|
|
|
3498 |
Fs(B)2441 2738 y Fn(\016)2503 2769 y FA(is)j(freshly)e(introduced)f(in)
|
|
|
3499 |
i(the)523 2868 y(subproof)h(on)i(the)g(left-hand)f(side)h(and)g
|
|
|
3500 |
(therefore)e(the)j(cut)f(can)g(only)g(mo)o(v)o(e)e(to)j(the)f
|
|
|
3501 |
(right\227just)523 2968 y(as)21 b(prescribed)d(by)i(the)g(colour)n
|
|
|
3502 |
(-annotation)d(in)j(\(17\).)f(Although)f(we)j(do)f(not)f(ha)n(v)o(e)h
|
|
|
3503 |
(a)h(proof)d(of)i(this)523 3067 y(f)o(act,)i(e)o(xperiments)f(with)h
|
|
|
3504 |
([17])g(ha)n(v)o(e)g(con)m(vinced)d(us)k(that)g(when)e(cut-elimination)
|
|
|
3505 |
g(is)i(concerned,)523 3167 y(we)18 b(can)g(indeed)e(re)o(gard)g(the)i
|
|
|
3506 |
Fr(::)1470 3179 y Fq(R)1525 3167 y FA(-rule)f(as)h(a)h(proper)d
|
|
|
3507 |
(inference)g(rule)h(with)h(the)g(consequence)d(that)523
|
|
|
3508 |
3267 y(in)j(the)g(proof)e(abo)o(v)o(e)g Fs(B)1205 3237
|
|
|
3509 |
y Fn(\016)1262 3267 y FA(is)j(freshly)e(introduced.)e(\(Roughly)h
|
|
|
3510 |
(speaking,)g(if)j(we)f(had)f(e)o(xpanded)e(the)523 3366
|
|
|
3511 |
y Fr(::)633 3378 y Fq(R)688 3366 y FA(-rule)26 b(to)g(an)g(auxiliary)f
|
|
|
3512 |
(cut)h(in)h(the)f(proof)f(\(18\),)f(then)i(mo)o(ving)e(the)j(cut)f(to)g
|
|
|
3513 |
(the)g(left)h(means)523 3466 y(it)d(cannot)e(mo)o(v)o(e)g(v)o(ery)g(f)o
|
|
|
3514 |
(ar)m(,)g(namely)g(only)h(to)g(the)g(place)g(where)g
|
|
|
3515 |
Fs(B)2492 3436 y Fn(\016)2554 3466 y FA(is)h(introduced)d(in)i(the)g
|
|
|
3516 |
(proof)523 3566 y(of)k(the)h(sequent)f Fr(::)p Fs(B)1209
|
|
|
3517 |
3535 y Fn(\016)p 1266 3554 10 38 v 1276 3537 42 4 v 1336
|
|
|
3518 |
3566 a Fs(B)1403 3535 y Fn(\016)1469 3566 y FA(and)g(then)g(the)h(cut)g
|
|
|
3519 |
(has)f(to)h(mo)o(v)o(e)e(right.)h(In)g(ef)n(fect)g(we)h(obtain)f(a)523
|
|
|
3520 |
3665 y(beha)n(viour)18 b(which)i(is)h(almost)f(identical)g(to)g(mo)o
|
|
|
3521 |
(ving)e(the)j(cut)f(to)g(the)g(right)g(in)g(the)g(\002rst)h(place.\))
|
|
|
3522 |
648 3768 y(Since)29 b(the)h(colour)n(-protocol)c(of)k(Danos)f(et)i(al.)
|
|
|
3523 |
f(allo)n(ws)g(us)g(to)g(annotate)e(in)i(man)o(y)f(circum-)523
|
|
|
3524 |
3868 y(stances)f(either)f(colour)g(`)p Fs(\()p FA(')g(or)h(`)p
|
|
|
3525 |
Fs(*)p FA(')f(to)h(the)g(formulae)e(in)i(a)g(classical)g(proof,)e(we)i
|
|
|
3526 |
(need)f(ho)n(w-)523 3967 y(e)n(v)o(er)21 b(to)h(depart)g(from)f(the)h
|
|
|
3527 |
(traditional)f(double-ne)o(gation)c(technique)k(that)h(translates)g(a)h
|
|
|
3528 |
(classical)523 4067 y(proof)k(uniformly)e(using)j(a)g(single)g
|
|
|
3529 |
(double-ne)o(gation)c(translation.)j(T)-7 b(o)28 b(simulate)g(the)g
|
|
|
3530 |
(coloured)523 4166 y(cut-elimination)14 b(procedure)h(in)h(a)h
|
|
|
3531 |
(meaningful)e(w)o(ay)-5 b(,)15 b(we)i(need)f(to)h(allo)n(w)f(more)g
|
|
|
3532 |
(than)g(one)g(double-)523 4266 y(ne)o(gation)f(translation.)h(Let)h(us)
|
|
|
3533 |
h(e)o(xplain)e(this)h(f)o(act)h(with)f(a)g(classical)h(proof)e
|
|
|
3534 |
Fs(\031)21 b FA(ending)15 b(in)j(a)f(cut)g(with)523 4366
|
|
|
3535 |
y(the)j(cut-formula)1815 4429 y Fq(\()-11 b Fn(\000)g(\000)g(\000)g
|
|
|
3536 |
(\000)1815 4476 y Fq(*)1812 4550 y Fs(B)1902 4542 y Fr(_)1984
|
|
|
3537 |
4476 y Fq(\()1981 4550 y Fs(C)2093 4542 y(:)523 4703
|
|
|
3538 |
y FA(W)k(e)27 b(will)f(sho)n(w)f(that)g(the)g(beha)n(viour)f(of)h(this)
|
|
|
3539 |
h(\(coloured\))c(cut)k(can)f(be)g(simulated)g(by)g(a)g(double-)523
|
|
|
3540 |
4825 y(ne)o(gation)30 b(translation)g(with)i(the)g(clause)g
|
|
|
3541 |
Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))2028 4795 y Fn(\017)2111
|
|
|
3542 |
4778 y Fp(def)2116 4825 y Fm(=)48 b Fr(::)p Fm(\()p Fs(B)2438
|
|
|
3543 |
4795 y Fn(\017)2478 4825 y Fr(_::)p Fs(C)2708 4795 y
|
|
|
3544 |
Fn(\017)2747 4825 y Fm(\))p FA(.)33 b(T)-7 b(o)31 b(sho)n(w)h(this)g
|
|
|
3545 |
(we)523 4924 y(analyse)26 b(all)h(cases)g(ho)n(w)f(the)h(cut)f(in)h
|
|
|
3546 |
Fs(\031)j FA(could)c(ha)n(v)o(e)f(arisen.)h(Consider)g(\002rst)h(the)g
|
|
|
3547 |
(case)g(where)f Fs(\031)p eop end
|
|
|
3548 |
%%Page: 15 15
|
|
|
3549 |
TeXDict begin 15 14 bop 523 448 a FA(ends)20 b(with)g(the)h(follo)n
|
|
|
3550 |
(wing)d(logical)i(cut)1496 602 y Fs(\031)1543 614 y Fl(1)1527
|
|
|
3551 |
666 y Fm(:)p 1468 767 10 38 v 1477 751 42 4 v 1563 714
|
|
|
3552 |
a Fq(*)1560 787 y Fs(B)p 1365 807 346 4 v 1384 969 10
|
|
|
3553 |
38 v 1393 952 42 4 v 1480 868 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g
|
|
|
3554 |
(\000)1479 915 y Fq(*)1476 989 y Fs(B)1567 981 y Fr(_)1648
|
|
|
3555 |
915 y Fq(\()1645 989 y Fs(C)1753 820 y Fr(_)1808 832
|
|
|
3556 |
y Fq(R)1858 840 y Fc(1)2024 602 y Fs(\031)2071 614 y
|
|
|
3557 |
Fl(2)2055 666 y Fm(:)1980 714 y Fq(*)1978 787 y Fs(B)p
|
|
|
3558 |
2086 767 10 38 v 2096 751 42 4 v 2285 602 a(\031)2332
|
|
|
3559 |
614 y Fl(3)2316 666 y Fm(:)2242 714 y Fq(\()2239 787
|
|
|
3560 |
y Fs(C)p 2347 767 10 38 v 2356 751 42 4 v 1978 807 439
|
|
|
3561 |
4 v 2027 868 a Fq(\()g Fn(\000)g(\000)g(\000)g(\000)2026
|
|
|
3562 |
915 y Fq(*)2024 989 y Fs(B)2114 981 y Fr(_)2196 915 y
|
|
|
3563 |
Fq(\()2193 989 y Fs(C)p 2300 969 10 38 v 2310 952 42
|
|
|
3564 |
4 v 2458 824 a Fr(_)2513 836 y Fq(L)p 1365 1008 1005
|
|
|
3565 |
4 v 1842 1062 10 38 v 1852 1045 42 4 v 2411 1034 a Fs(cut)3267
|
|
|
3566 |
1074 y FA(\(19\))523 1269 y(which)20 b(can)g(reduce)f(to)1714
|
|
|
3567 |
1327 y Fs(\031)1761 1339 y Fl(1)1744 1391 y Fm(:)p 1685
|
|
|
3568 |
1492 10 38 v 1695 1476 42 4 v 1780 1439 a Fq(*)1778 1512
|
|
|
3569 |
y Fs(B)1975 1327 y(\031)2022 1339 y Fl(2)2006 1391 y
|
|
|
3570 |
Fm(:)1931 1439 y Fq(*)1928 1512 y Fs(B)p 2037 1492 10
|
|
|
3571 |
38 v 2046 1476 42 4 v 1667 1532 440 4 v 1861 1586 10
|
|
|
3572 |
38 v 1871 1569 42 4 v 2148 1557 a(cut)3267 1598 y FA(\(20\))523
|
|
|
3573 |
1764 y(The)h Fm(\()p Fr(\000)p Fm(\))802 1733 y Fn(\017)840
|
|
|
3574 |
1764 y FA(-translated)f(v)o(ersion)g(of)h Fs(\031)1381
|
|
|
3575 |
1941 y(\031)1431 1911 y Fn(\017)1428 1962 y Fl(1)1414
|
|
|
3576 |
2014 y Fm(:)1301 2088 y Fr(:)p Fs(B)1423 2058 y Fn(\017)p
|
|
|
3577 |
1480 2076 10 38 v 1489 2059 42 4 v 1273 2108 304 4 v
|
|
|
3578 |
1292 2169 10 38 v 1301 2153 42 4 v 1361 2181 a Fr(::)p
|
|
|
3579 |
Fs(B)1538 2151 y Fn(\017)1619 2119 y Fr(:)1674 2131 y
|
|
|
3580 |
Fq(R)p 1273 2201 304 4 v 1347 2263 10 38 v 1356 2247
|
|
|
3581 |
42 4 v 1416 2275 a Fs(B)1483 2245 y Fn(\017)1619 2213
|
|
|
3582 |
y Fr(::)1729 2225 y Fq(R)p 1194 2295 463 4 v 1212 2357
|
|
|
3583 |
10 38 v 1222 2340 42 4 v 1282 2369 a Fs(B)1349 2339 y
|
|
|
3584 |
Fn(\017)1387 2369 y Fr(_::)p Fs(C)1617 2339 y Fn(\017)1698
|
|
|
3585 |
2308 y Fr(_)1753 2320 y Fq(R)1803 2328 y Fc(1)p 1134
|
|
|
3586 |
2389 583 4 v 1134 2468 a Fr(:)p Fm(\()p Fs(B)1288 2438
|
|
|
3587 |
y Fn(\017)1327 2468 y Fr(_::)p Fs(C)1557 2438 y Fn(\017)1596
|
|
|
3588 |
2468 y Fm(\))p 1647 2456 10 38 v 1656 2439 42 4 v 1758
|
|
|
3589 |
2400 a Fr(:)1813 2412 y Fq(L)p 1106 2508 639 4 v 1124
|
|
|
3590 |
2575 10 38 v 1134 2559 42 4 v 1194 2587 a Fr(::)p Fm(\()p
|
|
|
3591 |
Fs(B)1403 2557 y Fn(\017)1442 2587 y Fr(_)q(::)p Fs(C)1673
|
|
|
3592 |
2557 y Fn(\017)1712 2587 y Fm(\))1786 2520 y Fr(:)1841
|
|
|
3593 |
2532 y Fq(R)p 1078 2628 694 4 v 1078 2707 a Fr(:::)p
|
|
|
3594 |
Fm(\()p Fs(B)1342 2677 y Fn(\017)1382 2707 y Fr(_::)p
|
|
|
3595 |
Fs(C)1612 2677 y Fn(\017)1651 2707 y Fm(\))p 1702 2695
|
|
|
3596 |
10 38 v 1712 2678 42 4 v 1813 2640 a Fr(:)1868 2652 y
|
|
|
3597 |
Fq(L)2111 2128 y Fs(\031)2161 2098 y Fn(\017)2158 2149
|
|
|
3598 |
y Fl(2)2144 2201 y Fm(:)2059 2275 y Fs(B)2126 2245 y
|
|
|
3599 |
Fn(\017)p 2183 2263 10 38 v 2192 2247 42 4 v 2442 1941
|
|
|
3600 |
a Fs(\031)2492 1911 y Fn(\017)2489 1962 y Fl(3)2475 2014
|
|
|
3601 |
y Fm(:)2390 2088 y Fs(C)2455 2058 y Fn(\017)p 2512 2076
|
|
|
3602 |
10 38 v 2522 2059 42 4 v 2363 2108 247 4 v 2381 2169
|
|
|
3603 |
10 38 v 2391 2153 42 4 v 2451 2181 a Fr(:)p Fs(C)2571
|
|
|
3604 |
2151 y Fn(\017)2651 2119 y Fr(:)2706 2131 y Fq(R)p 2335
|
|
|
3605 |
2201 303 4 v 2335 2275 a Fr(::)p Fs(C)2510 2245 y Fn(\017)p
|
|
|
3606 |
2568 2263 10 38 v 2577 2247 42 4 v 2679 2213 a Fr(:)2734
|
|
|
3607 |
2225 y Fq(L)p 2059 2295 579 4 v 2117 2369 a Fs(B)2184
|
|
|
3608 |
2339 y Fn(\017)2222 2369 y Fr(_::)p Fs(C)2452 2339 y
|
|
|
3609 |
Fn(\017)p 2510 2357 10 38 v 2519 2340 42 4 v 2679 2312
|
|
|
3610 |
a Fr(_)2734 2324 y Fq(L)p 2057 2389 583 4 v 2075 2456
|
|
|
3611 |
10 38 v 2085 2439 42 4 v 2145 2468 a Fr(:)p Fm(\()p Fs(B)2299
|
|
|
3612 |
2438 y Fn(\017)2338 2468 y Fr(_::)p Fs(C)2568 2438 y
|
|
|
3613 |
Fn(\017)2607 2468 y Fm(\))2681 2400 y Fr(:)2736 2412
|
|
|
3614 |
y Fq(R)p 2029 2508 639 4 v 2029 2587 a Fr(::)p Fm(\()p
|
|
|
3615 |
Fs(B)2238 2557 y Fn(\017)2277 2587 y Fr(_)q(::)p Fs(C)2508
|
|
|
3616 |
2557 y Fn(\017)2547 2587 y Fm(\))p 2597 2575 10 38 v
|
|
|
3617 |
2607 2559 42 4 v 2709 2520 a Fr(:)2764 2532 y Fq(L)p
|
|
|
3618 |
2001 2628 694 4 v 2020 2695 10 38 v 2029 2678 42 4 v
|
|
|
3619 |
2089 2707 a Fr(:::)p Fm(\()p Fs(B)2353 2677 y Fn(\017)2393
|
|
|
3620 |
2707 y Fr(_::)p Fs(C)2623 2677 y Fn(\017)2662 2707 y
|
|
|
3621 |
Fm(\))2736 2640 y Fr(:)2791 2652 y Fq(R)p 1078 2747 1617
|
|
|
3622 |
4 v 1861 2801 10 38 v 1871 2785 42 4 v 2736 2773 a Fs(cut)523
|
|
|
3623 |
3008 y FA(reduces)f(to)1601 3087 y Fs(\031)1651 3057
|
|
|
3624 |
y Fn(\017)1648 3108 y Fl(1)1634 3160 y Fm(:)1521 3234
|
|
|
3625 |
y Fr(:)p Fs(B)1643 3204 y Fn(\017)p 1700 3222 10 38 v
|
|
|
3626 |
1709 3205 42 4 v 1493 3254 304 4 v 1512 3315 10 38 v
|
|
|
3627 |
1521 3299 42 4 v 1581 3327 a Fr(::)p Fs(B)1758 3297 y
|
|
|
3628 |
Fn(\017)1839 3265 y Fr(:)1894 3277 y Fq(R)p 1493 3347
|
|
|
3629 |
304 4 v 1567 3409 10 38 v 1576 3393 42 4 v 1636 3421
|
|
|
3630 |
a Fs(B)1703 3391 y Fn(\017)1839 3359 y Fr(::)1949 3371
|
|
|
3631 |
y Fq(R)2139 3274 y Fs(\031)2189 3244 y Fn(\017)2186 3295
|
|
|
3632 |
y Fl(2)2172 3347 y Fm(:)2087 3421 y Fs(B)2154 3391 y
|
|
|
3633 |
Fn(\017)p 2210 3409 10 38 v 2220 3393 42 4 v 1548 3441
|
|
|
3634 |
732 4 v 1889 3495 10 38 v 1898 3478 42 4 v 2321 3467
|
|
|
3635 |
a Fs(cut)523 3673 y FA(where)g(\(remember)f(we)j(re)o(gard)d(the)i
|
|
|
3636 |
Fr(::)1712 3685 y Fq(R)1767 3673 y FA(-rule)f(as)i(proper)d(inference)g
|
|
|
3637 |
(rule\))i(the)g(proof)e Fs(\031)3150 3643 y Fn(\017)3147
|
|
|
3638 |
3693 y Fl(1)3209 3673 y FA(has)i(to)523 3772 y(mo)o(v)o(e)f(inside)h
|
|
|
3639 |
Fs(\031)995 3742 y Fn(\017)992 3793 y Fl(2)1054 3772
|
|
|
3640 |
y FA(just)h(lik)o(e)f(the)g(beha)n(viour)f(of)g(\(20\).)g(If)h
|
|
|
3641 |
Fs(\031)k FA(ends)c(with)h(the)f(logical)g(cut)1496 3928
|
|
|
3642 |
y Fs(\031)1543 3940 y Fl(1)1527 3993 y Fm(:)p 1468 4094
|
|
|
3643 |
10 38 v 1478 4077 42 4 v 1564 4040 a Fq(\()1561 4114
|
|
|
3644 |
y Fs(C)p 1365 4134 346 4 v 1384 4295 10 38 v 1393 4279
|
|
|
3645 |
42 4 v 1480 4194 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g(\000)1479
|
|
|
3646 |
4242 y Fq(*)1476 4315 y Fs(B)1567 4307 y Fr(_)1648 4242
|
|
|
3647 |
y Fq(\()1645 4315 y Fs(C)1753 4146 y Fr(_)1808 4159 y
|
|
|
3648 |
Fq(R)1858 4167 y Fc(2)2024 3928 y Fs(\031)2071 3940 y
|
|
|
3649 |
Fl(2)2055 3993 y Fm(:)1980 4040 y Fq(*)1978 4114 y Fs(B)p
|
|
|
3650 |
2086 4094 10 38 v 2096 4077 42 4 v 2285 3928 a(\031)2332
|
|
|
3651 |
3940 y Fl(3)2316 3993 y Fm(:)2242 4040 y Fq(\()2239 4114
|
|
|
3652 |
y Fs(C)p 2347 4094 10 38 v 2356 4077 42 4 v 1978 4134
|
|
|
3653 |
439 4 v 2027 4194 a Fq(\()g Fn(\000)g(\000)g(\000)g(\000)2026
|
|
|
3654 |
4242 y Fq(*)2024 4315 y Fs(B)2114 4307 y Fr(_)2196 4242
|
|
|
3655 |
y Fq(\()2193 4315 y Fs(C)p 2300 4295 10 38 v 2310 4279
|
|
|
3656 |
42 4 v 2458 4150 a Fr(_)2513 4163 y Fq(L)p 1365 4335
|
|
|
3657 |
1005 4 v 1842 4389 10 38 v 1852 4372 42 4 v 2411 4360
|
|
|
3658 |
a Fs(cut)3267 4401 y FA(\(21\))523 4596 y(which)20 b(can)g(reduce)f(to)
|
|
|
3659 |
1771 4653 y Fs(\031)1818 4665 y Fl(1)1802 4718 y Fm(:)p
|
|
|
3660 |
1743 4819 10 38 v 1753 4802 42 4 v 1838 4765 a Fq(\()1836
|
|
|
3661 |
4839 y Fs(C)2031 4653 y(\031)2078 4665 y Fl(3)2062 4718
|
|
|
3662 |
y Fm(:)1988 4765 y Fq(\()1985 4839 y Fs(C)p 2092 4819
|
|
|
3663 |
10 38 v 2102 4802 42 4 v 1725 4859 438 4 v 1918 4912
|
|
|
3664 |
10 38 v 1927 4896 42 4 v 3267 4924 a FA(\(22\))p eop
|
|
|
3665 |
end
|
|
|
3666 |
%%Page: 16 16
|
|
|
3667 |
TeXDict begin 16 15 bop 523 448 a FA(then,)20 b(the)g
|
|
|
3668 |
Fm(\()p Fr(\000)p Fm(\))959 418 y Fn(\017)997 448 y FA(-translated)f(v)
|
|
|
3669 |
o(ersion)g(of)h Fs(\031)1381 699 y(\031)1431 669 y Fn(\017)1428
|
|
|
3670 |
720 y Fl(1)1414 772 y Fm(:)1302 846 y Fr(:)p Fs(C)1422
|
|
|
3671 |
816 y Fn(\017)p 1479 834 10 38 v 1488 817 42 4 v 1274
|
|
|
3672 |
866 303 4 v 1292 927 10 38 v 1302 911 42 4 v 1362 939
|
|
|
3673 |
a Fr(::)p Fs(C)1537 909 y Fn(\017)1618 877 y Fr(:)1673
|
|
|
3674 |
889 y Fq(R)p 1194 959 463 4 v 1212 1021 10 38 v 1222
|
|
|
3675 |
1005 42 4 v 1282 1033 a Fs(B)1349 1003 y Fn(\017)1387
|
|
|
3676 |
1033 y Fr(_::)p Fs(C)1617 1003 y Fn(\017)1698 972 y Fr(_)1753
|
|
|
3677 |
985 y Fq(R)1803 993 y Fc(1)p 1134 1053 583 4 v 1134 1132
|
|
|
3678 |
a Fr(:)p Fm(\()p Fs(B)1288 1102 y Fn(\017)1327 1132 y
|
|
|
3679 |
Fr(_::)p Fs(C)1557 1102 y Fn(\017)1596 1132 y Fm(\))p
|
|
|
3680 |
1647 1120 10 38 v 1656 1103 42 4 v 1758 1065 a Fr(:)1813
|
|
|
3681 |
1077 y Fq(L)p 1106 1173 639 4 v 1124 1239 10 38 v 1134
|
|
|
3682 |
1223 42 4 v 1194 1251 a Fr(::)p Fm(\()p Fs(B)1403 1221
|
|
|
3683 |
y Fn(\017)1442 1251 y Fr(_)q(::)p Fs(C)1673 1221 y Fn(\017)1712
|
|
|
3684 |
1251 y Fm(\))1786 1184 y Fr(:)1841 1196 y Fq(R)p 1078
|
|
|
3685 |
1292 694 4 v 1078 1371 a Fr(:::)p Fm(\()p Fs(B)1342 1341
|
|
|
3686 |
y Fn(\017)1382 1371 y Fr(_::)p Fs(C)1612 1341 y Fn(\017)1651
|
|
|
3687 |
1371 y Fm(\))p 1702 1359 10 38 v 1712 1342 42 4 v 1813
|
|
|
3688 |
1304 a Fr(:)1868 1316 y Fq(L)2111 793 y Fs(\031)2161
|
|
|
3689 |
763 y Fn(\017)2158 813 y Fl(2)2144 866 y Fm(:)2059 939
|
|
|
3690 |
y Fs(B)2126 909 y Fn(\017)p 2183 927 10 38 v 2192 911
|
|
|
3691 |
42 4 v 2442 605 a Fs(\031)2492 575 y Fn(\017)2489 626
|
|
|
3692 |
y Fl(3)2475 678 y Fm(:)2390 752 y Fs(C)2455 722 y Fn(\017)p
|
|
|
3693 |
2512 740 10 38 v 2522 723 42 4 v 2363 772 247 4 v 2381
|
|
|
3694 |
834 10 38 v 2391 817 42 4 v 2451 846 a Fr(:)p Fs(C)2571
|
|
|
3695 |
816 y Fn(\017)2651 784 y Fr(:)2706 796 y Fq(R)p 2335
|
|
|
3696 |
866 303 4 v 2335 939 a Fr(::)p Fs(C)2510 909 y Fn(\017)p
|
|
|
3697 |
2568 927 10 38 v 2577 911 42 4 v 2679 877 a Fr(:)2734
|
|
|
3698 |
889 y Fq(L)p 2059 959 579 4 v 2117 1033 a Fs(B)2184 1003
|
|
|
3699 |
y Fn(\017)2222 1033 y Fr(_::)p Fs(C)2452 1003 y Fn(\017)p
|
|
|
3700 |
2510 1021 10 38 v 2519 1005 42 4 v 2679 976 a Fr(_)2734
|
|
|
3701 |
989 y Fq(L)p 2057 1053 583 4 v 2075 1120 10 38 v 2085
|
|
|
3702 |
1103 42 4 v 2145 1132 a Fr(:)p Fm(\()p Fs(B)2299 1102
|
|
|
3703 |
y Fn(\017)2338 1132 y Fr(_::)p Fs(C)2568 1102 y Fn(\017)2607
|
|
|
3704 |
1132 y Fm(\))2681 1065 y Fr(:)2736 1077 y Fq(R)p 2029
|
|
|
3705 |
1173 639 4 v 2029 1251 a Fr(::)p Fm(\()p Fs(B)2238 1221
|
|
|
3706 |
y Fn(\017)2277 1251 y Fr(_)q(::)p Fs(C)2508 1221 y Fn(\017)2547
|
|
|
3707 |
1251 y Fm(\))p 2597 1239 10 38 v 2607 1223 42 4 v 2709
|
|
|
3708 |
1184 a Fr(:)2764 1196 y Fq(L)p 2001 1292 694 4 v 2020
|
|
|
3709 |
1359 10 38 v 2029 1342 42 4 v 2089 1371 a Fr(:::)p Fm(\()p
|
|
|
3710 |
Fs(B)2353 1341 y Fn(\017)2393 1371 y Fr(_::)p Fs(C)2623
|
|
|
3711 |
1341 y Fn(\017)2662 1371 y Fm(\))2736 1304 y Fr(:)2791
|
|
|
3712 |
1316 y Fq(R)p 1078 1412 1617 4 v 1861 1466 10 38 v 1871
|
|
|
3713 |
1449 42 4 v 2736 1437 a Fs(cut)523 1648 y FA(reduces)f(to)1677
|
|
|
3714 |
1808 y Fs(\031)1727 1778 y Fn(\017)1724 1829 y Fl(1)1710
|
|
|
3715 |
1881 y Fm(:)1598 1955 y Fr(:)p Fs(C)1718 1925 y Fn(\017)p
|
|
|
3716 |
1775 1943 10 38 v 1785 1926 42 4 v 2007 1714 a Fs(\031)2057
|
|
|
3717 |
1684 y Fn(\017)2054 1735 y Fl(3)2040 1787 y Fm(:)1956
|
|
|
3718 |
1861 y Fs(C)2021 1831 y Fn(\017)p 2078 1849 10 38 v 2087
|
|
|
3719 |
1832 42 4 v 1928 1881 247 4 v 1946 1943 10 38 v 1956
|
|
|
3720 |
1926 42 4 v 2016 1955 a Fr(:)p Fs(C)2136 1925 y Fn(\017)2216
|
|
|
3721 |
1893 y Fr(:)2271 1905 y Fq(R)p 1598 1975 577 4 v 1861
|
|
|
3722 |
2028 10 38 v 1871 2012 42 4 v 2216 2000 a Fs(cut)523
|
|
|
3723 |
2183 y FA(where)h(proof)f Fs(\031)998 2152 y Fn(\017)995
|
|
|
3724 |
2203 y Fl(3)1057 2183 y FA(has)i(to)f(mo)o(v)o(e)f(inside)h
|
|
|
3725 |
Fs(\031)1746 2152 y Fn(\017)1743 2203 y Fl(1)1806 2183
|
|
|
3726 |
y FA(just)h(lik)o(e)f(in)h(the)f(proof)f(\(22\).)g(The)h(only)g(case)h
|
|
|
3727 |
(we)f(still)523 2282 y(need)g(to)g(consider)f(is)i(when)f
|
|
|
3728 |
Fs(\031)k FA(ends)c(in)g(a)h(commuting)d(cut)i(of)g(the)g(form)1630
|
|
|
3729 |
2415 y Fs(\031)1677 2427 y Fl(1)1661 2480 y Fm(:)p 1518
|
|
|
3730 |
2642 10 38 v 1527 2625 42 4 v 1613 2541 a Fq(\()-11 b
|
|
|
3731 |
Fn(\000)g(\000)g(\000)g(\000)1613 2588 y Fq(*)1610 2662
|
|
|
3732 |
y Fs(B)1700 2654 y Fr(_)1782 2588 y Fq(\()1779 2662 y
|
|
|
3733 |
Fs(C)2059 2415 y(\031)2106 2427 y Fl(2)2089 2480 y Fm(:)1931
|
|
|
3734 |
2541 y Fq(\()g Fn(\000)g(\000)g(\000)g(\000)1931 2588
|
|
|
3735 |
y Fq(*)1928 2662 y Fs(B)2018 2654 y Fr(_)2100 2588 y
|
|
|
3736 |
Fq(\()2097 2662 y Fs(C)p 2204 2642 10 38 v 2214 2625
|
|
|
3737 |
42 4 v 1499 2681 775 4 v 1861 2735 10 38 v 1871 2718
|
|
|
3738 |
42 4 v 2315 2707 a(cut)3267 2747 y FA(\(23\))523 2917
|
|
|
3739 |
y(The)25 b(beha)n(viour)e(of)i(this)h(cut)g(is)g(determined)e(by)h(the)
|
|
|
3740 |
g(outermost)f(colour)g(`)p Fs(\()p FA('.)h(This)h(beha)n(viour)523
|
|
|
3741 |
3017 y(can)21 b(be)h(simulated)f(by)g(the)h Fm(\()p Fr(\000)p
|
|
|
3742 |
Fm(\))1464 2987 y Fn(\017)1502 3017 y FA(-translation,)e(pro)o(vided)f
|
|
|
3743 |
(we)j(use)g(a)g(left-translation)e(for)h(the)g(cut)523
|
|
|
3744 |
3117 y(in)f(\(23\).)f(The)h(translated)g(proof)e(is)j(then)f(as)h
|
|
|
3745 |
(follo)n(ws:)1454 3390 y Fs(\031)1504 3360 y Fn(\017)1501
|
|
|
3746 |
3411 y Fl(1)1487 3463 y Fm(:)1152 3542 y Fr(:::)p Fm(\()p
|
|
|
3747 |
Fs(B)1416 3512 y Fn(\017)1455 3542 y Fr(_)q(::)p Fs(C)1686
|
|
|
3748 |
3512 y Fn(\017)1725 3542 y Fm(\))p 1775 3530 10 38 v
|
|
|
3749 |
1785 3514 42 4 v 2230 3271 a Fs(\031)2280 3241 y Fn(\017)2277
|
|
|
3750 |
3291 y Fl(2)2263 3344 y Fm(:)1956 3423 y Fr(::)p Fm(\()p
|
|
|
3751 |
Fs(B)2165 3393 y Fn(\017)2204 3423 y Fr(_::)p Fs(C)2434
|
|
|
3752 |
3393 y Fn(\017)2473 3423 y Fm(\))p 2524 3411 10 38 v
|
|
|
3753 |
2534 3394 42 4 v 1928 3463 694 4 v 1946 3530 10 38 v
|
|
|
3754 |
1956 3514 42 4 v 2016 3542 a Fr(:::)p Fm(\()p Fs(B)2280
|
|
|
3755 |
3512 y Fn(\017)2320 3542 y Fr(_::)p Fs(C)2550 3512 y
|
|
|
3756 |
Fn(\017)2589 3542 y Fm(\))2663 3475 y Fr(:)2718 3487
|
|
|
3757 |
y Fq(R)p 1152 3583 1470 4 v 1861 3637 10 38 v 1871 3620
|
|
|
3758 |
42 4 v 2663 3608 a Fs(cut)523 3819 y FA(No)n(w)31 b Fs(\031)764
|
|
|
3759 |
3789 y Fn(\017)761 3840 y Fl(1)834 3819 y FA(will)h(freshly)e
|
|
|
3760 |
(introduce)g(the)h(cut-formula)d Fr(:::)p Fm(\()p Fs(B)2423
|
|
|
3761 |
3789 y Fn(\017)2463 3819 y Fr(_::)p Fs(C)2693 3789 y
|
|
|
3762 |
Fn(\017)2732 3819 y Fm(\))k FA(only)f(if)g Fs(\031)3105
|
|
|
3763 |
3831 y Fl(1)3174 3819 y FA(freshly)523 3919 y(introduces)20
|
|
|
3764 |
b(the)i(formula)f Fs(B)t Fr(_)p Fs(C)29 b FA(\(recall)22
|
|
|
3765 |
b(that)g(double-ne)o(gation)17 b(translations)22 b(need)f(to)h(preserv)
|
|
|
3766 |
o(e)523 4018 y(the)g(structure)f(of)g(a)i(classical)f(proof\).)e
|
|
|
3767 |
(Consequently)-5 b(,)19 b(the)j(translated)f(proof)f(simulates)i(e)o
|
|
|
3768 |
(xactly)523 4118 y(the)e(beha)n(viour)e(of)i(\(23\).)648
|
|
|
3769 |
4218 y(What)32 b(this)g(e)o(xample)f(sho)n(ws)h(is)g(that)g(the)g
|
|
|
3770 |
Fm(\()p Fr(\000)p Fm(\))2070 4187 y Fn(\017)2109 4218
|
|
|
3771 |
y FA(-translation)e(of)i Fs(\031)k FA(can)31 b(simulate)h(the)g(be-)523
|
|
|
3772 |
4317 y(ha)n(viour)19 b(where)g(the)i(cut-formula)c(is)k(annotated)e
|
|
|
3773 |
(with)h(the)g(colours)1815 4442 y Fq(\()-11 b Fn(\000)g(\000)g(\000)g
|
|
|
3774 |
(\000)1815 4489 y Fq(*)1812 4563 y Fs(B)1902 4555 y Fr(_)1984
|
|
|
3775 |
4489 y Fq(\()1981 4563 y Fs(C)2093 4555 y(:)523 4725
|
|
|
3776 |
y FA(Note)27 b(that)g(there)f(are)h(other)f(double)g(ne)o(gation)e
|
|
|
3777 |
(translation)i(which)h(can)f(be)h(used)g(for)f(a)h(similar)523
|
|
|
3778 |
4825 y(simulation)g(of)h(this)g(particular)f(colour)n(-annotation.)d
|
|
|
3779 |
(From)j(our)g(discussion)h(it)g(seems)h(reason-)523 4924
|
|
|
3780 |
y(able)e(to)g(e)o(xpect)f(that)g(one)h(can)f(\002nd)h(corresponding)c
|
|
|
3781 |
(double-ne)o(gation)f(translations)27 b(for)f(e)n(v)o(ery)p
|
|
|
3782 |
eop end
|
|
|
3783 |
%%Page: 17 17
|
|
|
3784 |
TeXDict begin 17 16 bop 523 448 a FA(possible)20 b(colour)n
|
|
|
3785 |
(-annotation.)15 b(Since)20 b(the)g(formulae)e Fs(B)25
|
|
|
3786 |
b FA(and)19 b Fs(C)27 b FA(can)19 b(be)h(compound)d(with)j(further)523
|
|
|
3787 |
548 y(colour)n(-annotations)j(inside,)i(we)i(need)e(some)h(\003e)o
|
|
|
3788 |
(xibility)g(of)f(ho)n(w)h(to)g(double-ne)o(gate)d(translate)523
|
|
|
3789 |
669 y(formulae.)30 b(One)h(has)h(to)g(be)g(able)g(to)g(b)n(uild)f(into)
|
|
|
3790 |
h(the)f(clause)h Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))2653
|
|
|
3791 |
639 y Fn(\017)2736 622 y Fp(def)2741 669 y Fm(=)49 b
|
|
|
3792 |
Fr(::)p Fm(\()p Fs(B)3064 639 y Fn(\017)3103 669 y Fr(_)q(::)p
|
|
|
3793 |
Fs(C)3334 639 y Fn(\017)3373 669 y Fm(\))523 769 y FA(that)26
|
|
|
3794 |
b(the)g(translations)f(of)g Fs(B)31 b FA(and)25 b Fs(C)32
|
|
|
3795 |
b FA(might)26 b(follo)n(w)f(a)h(completely)e(dif)n(ferent)g(double-ne)o
|
|
|
3796 |
(gation)523 868 y(scheme.)k(Ho)n(w)g(to)h(do)f(this)h(ele)o(gantly)d
|
|
|
3797 |
(is)k(not)e(kno)n(wn)f(to)h(us.)h(On)g(the)f(other)g(hand,)f(we)i
|
|
|
3798 |
(cannot)523 968 y(e)o(xpect)23 b(complete)h(\223freedom\224)e(in)j(a)f
|
|
|
3799 |
(double-ne)o(gation)c(translation)k(as)h(we)g(ha)n(v)o(e)e(to)i(mak)o
|
|
|
3800 |
(e)f(sure,)523 1068 y(roughly)15 b(speaking,)g(that)i(dif)n(ferent)e
|
|
|
3801 |
(double-ne)o(gation)d(translation)k(still)h(\002t)h(together)d(in)i
|
|
|
3802 |
(the)g(trans-)523 1167 y(lated)j(proof.)e(This)i(means)g(we)h(ha)n(v)o
|
|
|
3803 |
(e)e(to)h(mak)o(e)g(sure)g(that)g(double-ne)o(gation)15
|
|
|
3804 |
b(translations)20 b(respect)523 1267 y(the)j(identity-class)g
|
|
|
3805 |
(constraint)g(from)f(the)h(colour)n(-annotations.)d(F)o(or)j(e)o
|
|
|
3806 |
(xample)f(we)i(cannot)e(ha)n(v)o(e)523 1367 y(an)e(axiom)f(in)i(a)f
|
|
|
3807 |
(translated)g(proof)f(where)g(the)h(double-ne)o(gation)c(translations)k
|
|
|
3808 |
(disagree)f(as)i(in)1815 1550 y Fs(B)1882 1520 y Fn(\003)p
|
|
|
3809 |
1938 1538 10 38 v 1948 1522 42 4 v 2008 1550 a Fs(B)2075
|
|
|
3810 |
1520 y Fn(\016)523 1735 y FA(This)c(point)g(about)f(\002tting)h
|
|
|
3811 |
(double-ne)o(gation)12 b(translations)17 b(together)e(and)i(the)g
|
|
|
3812 |
(identity-class)f(con-)523 1834 y(straint)22 b(of)g(colours)e(we)j(tak)
|
|
|
3813 |
o(e)e(as)i(a)f(further)e(e)n(vidence)h(that)h(colours)f(and)g(double)f
|
|
|
3814 |
(ne)o(gation)g(trans-)523 1934 y(lation)g(must)g(ha)n(v)o(e)g
|
|
|
3815 |
(something)e(to)j(do)f(with)g(each)g(other)-5 b(.)648
|
|
|
3816 |
2034 y(From)22 b(the)i(observ)n(ations)e(made)g(abo)o(v)o(e)g(we)i
|
|
|
3817 |
(conjecture)e(that)h(e)n(v)o(ery)f(colour)n(-annotation)e(de-)523
|
|
|
3818 |
2134 y(termining)31 b(a)i(single)f(normalform)e(of)i(a)h(classical)g
|
|
|
3819 |
(proof)e(can)h(be)g(equally)g(determined)e(by)i(a)523
|
|
|
3820 |
2233 y(double)24 b(ne)o(gation)g(translation,)g(and)h(e)n(v)o(ery)g
|
|
|
3821 |
(double-ne)o(gation)c(translation)k(determining)e(a)j(nor)n(-)523
|
|
|
3822 |
2333 y(malform)17 b(can)i(be)g(equally)e(determined)g(by)i(a)g(colour)n
|
|
|
3823 |
(-annotation.)14 b(In)19 b(ef)n(fect)f(we)h(conjecture)e(that)523
|
|
|
3824 |
2433 y(double-ne)o(gation)f(translations)j(can)h(be)g(simulated)g(by)g
|
|
|
3825 |
(colours)f(and)h(vice)g(v)o(ersa.)648 2533 y(While)h(establishing)f
|
|
|
3826 |
(the)h(simulation)f(properties)g(is)h(a)h(\002rst)f(step)h(for)e
|
|
|
3827 |
(understanding)e(the)j(re-)523 2632 y(lation)d(between)f(double-ne)o
|
|
|
3828 |
(gation)c(translations)18 b(and)f(colour)n(-annotations,)d(we)k
|
|
|
3829 |
(consider)f(this)i(as)523 2732 y(not)k(yet)h(gi)n(ving)f(the)g
|
|
|
3830 |
(complete)g(\223picture\224.)f(F)o(or)h(this)i(consider)d(the)i
|
|
|
3831 |
(collection)f(of)g(normalforms)523 2832 y(of)30 b(a)g(classical)g
|
|
|
3832 |
(proof)e(determined)g(by)i(double-ne)o(gation)25 b(translations)k(and)g
|
|
|
3833 |
(by)g(colour)g(anno-)523 2931 y(tations.)e(W)-7 b(e)28
|
|
|
3834 |
b(conjecture)d(that)i(both)f(are)h(the)g(\223same\224)g(collection,)e
|
|
|
3835 |
(whereby)h(one)g(needs)g(a)i(\(yet)523 3031 y(unkno)n(wn\))c(v)o(ery)i
|
|
|
3836 |
(cle)n(v)o(er)g(notion)g(of)h(\223sameness\224.)g(Clearly)-5
|
|
|
3837 |
b(,)26 b(there)h(are)g(more)f(double-ne)o(gation)523
|
|
|
3838 |
3131 y(translations)17 b(of)h(a)h(classical)f(proof)f(than)g(there)h
|
|
|
3839 |
(are)g(colour)n(-annotations)c(\(there)k(are)f(only)h(\002nitely)523
|
|
|
3840 |
3230 y(man)o(y)k(colour)n(-annotations,)e(b)n(ut)j(there)g(are)h
|
|
|
3841 |
(in\002nitely)f(man)o(y)f(double)g(ne)o(gation)f(translations)i(as)523
|
|
|
3842 |
3330 y(already)j(the)h(translations)g(of)g(atomic)f(formulae)g(as)i
|
|
|
3843 |
Fr(::)p Fs(A)p FA(,)g Fr(::::)p Fs(A;)14 b(:)g(:)g(:)29
|
|
|
3844 |
b FA(indicate\).)d(F)o(or)h(us)h(it)523 3429 y(is,)19
|
|
|
3845 |
b(ho)n(we)n(v)o(er)m(,)d(clear)i(that)h(one)e(can)i(group)d(double-ne)o
|
|
|
3846 |
(gation)e(translations)k(into)g(dif)n(ferent)f(classes,)523
|
|
|
3847 |
3529 y(where)j(each)f(class)j(corresponds)c(to)i(a)g(colour)n
|
|
|
3848 |
(-annotation.)648 3629 y(Let)32 b(us)h(consider)f(what)h(is)g
|
|
|
3849 |
(necessary)f(to)h(turn)f(these)h(conjectures)e(into)h(theorems.)g
|
|
|
3850 |
(First)523 3729 y(we)e(ha)n(v)o(e)f(to)h(mak)o(e)g(precise)f(what)h(we)
|
|
|
3851 |
g(mean)f(by)h Ft(all)g FA(double-ne)o(gation)25 b(translation.)k(As)h
|
|
|
3852 |
(seen)523 3828 y(abo)o(v)o(e,)22 b(the)h(notion)f(of)h(double-ne)o
|
|
|
3853 |
(gation)c(translation)k(has)g(to)h(be)f(a)h(generalised)e(v)o(ersion)g
|
|
|
3854 |
(of)h(the)523 3928 y(traditional)g(notion\227lik)o(e)f(the)h(ones)h(gi)
|
|
|
3855 |
n(v)o(en)e(by)h(Gentzen,)g(G)7 b(\250)-35 b(odel)23 b(and)g(K)m
|
|
|
3856 |
(olmogoro)o(v\227because)523 4028 y(one)g(needs)g(to)g(tak)o(e)h(into)f
|
|
|
3857 |
(account)f(the)i(dif)n(ferent)d(colour)n(-annotations)f(by)j(v)n
|
|
|
3858 |
(arying)f(the)h(double-)523 4127 y(ne)o(gations)g(translation)h(when)g
|
|
|
3859 |
(inducti)n(v)o(ely)e(descending)h(a)i(formula.)e(Further)m(,)h(one)g(w)
|
|
|
3860 |
o(ould)g(ide-)523 4227 y(ally)g(lik)o(e)f(to)h(ha)n(v)o(e)f(a)h(rather)
|
|
|
3861 |
f(general)f(notion)g(of)i(double-ne)o(gation)19 b(translation)j(so)i
|
|
|
3862 |
(that)g(one)f(can)523 4327 y(meaningful)h(state)k(properties)d(for)h
|
|
|
3863 |
(all)h Ft(possible)g FA(double-ne)o(gation)22 b(translation.)k(Ho)n(we)
|
|
|
3864 |
n(v)o(er)m(,)e(we)523 4426 y(ha)n(v)o(e)f(been)f(unable)h(to)g
|
|
|
3865 |
(formulate)f(such)h(a)g(general)g(notion.)f(Then)g(we)i(ha)n(v)o(e)e
|
|
|
3866 |
(to)i(cate)o(gorise)e(ho)n(w)523 4526 y(the)k(double-ne)o(gation)c
|
|
|
3867 |
(translations)k(beha)n(v)o(e)f(under)f(cut-elimination.)g(Finally)-5
|
|
|
3868 |
b(,)26 b(one)f(has)i(to)f(es-)523 4625 y(tablish)e(the)h
|
|
|
3869 |
(\223simulation-property\224\227which)19 b(unfortunately)i(f)o(ails)k
|
|
|
3870 |
(if)f(one)g(tak)o(es)h(a)f(v)o(ery)f(na)n(\250)-26 b(\021v)o(e)523
|
|
|
3871 |
4725 y(vie)n(w)17 b(on)g(sequent-proofs)d(and)i(cut-elimination.)f(One)
|
|
|
3872 |
i(problem)e(is)j(that)f(double-ne)o(gation)c(trans-)523
|
|
|
3873 |
4825 y(lations)29 b(might)g(introduce)e(auxiliary)h(cuts.)h(Such)g
|
|
|
3874 |
(auxiliary)f(cuts)h(only)g(occur)f(in)h(the)g(double-)523
|
|
|
3875 |
4924 y(ne)o(gated)22 b(proofs)g(and)i(are)f(thus)h(not)g(tak)o(en)f
|
|
|
3876 |
(account)g(of)g(by)g(colour)n(-annotations.)d(Therefore)i(we)p
|
|
|
3877 |
eop end
|
|
|
3878 |
%%Page: 18 18
|
|
|
3879 |
TeXDict begin 18 17 bop 523 448 a FA(ha)n(v)o(e)25 b(to)h(mak)o(e)f
|
|
|
3880 |
(sure)g(that)h(such)g(auxiliary)e(cuts)i(cannot)e(\223mess)i(up\224)f
|
|
|
3881 |
(the)h(normalform)d(reached)523 548 y(by)28 b(eliminating)g(cuts)h(in)f
|
|
|
3882 |
(the)h(double-ne)o(gated)24 b(proof.)j(This)i(can)f(be)h(relati)n(v)o
|
|
|
3883 |
(ely)e(easily)i(sho)n(wn)523 648 y(pro)o(vided)23 b(the)i(auxiliary)f
|
|
|
3884 |
(cut)h(occurs)g(as)h(the)f(lo)n(wermost)f(inference)g(in)h(a)h
|
|
|
3885 |
(proof\227then)c(a)k(tech-)523 747 y(nique)h(introduced)f(in)j([19])e
|
|
|
3886 |
(sho)n(ws)h(that)h(one)e(can)h(restrict)h(attention)e(to)i(only)e
|
|
|
3887 |
(outermost)g(cuts)523 847 y(when)c(calculating)f(the)h(collection)g(of)
|
|
|
3888 |
g(normalforms.)d(Ho)n(we)n(v)o(er)i(in)i(the)f(general)g(case)g(we)h
|
|
|
3889 |
(ha)n(v)o(e)523 946 y(for)17 b(this)h(not)g(\223messing)f(up\224)g
|
|
|
3890 |
(only)g(empirical)g(e)n(vidence)f(obtained)h(from)f(man)o(y)h
|
|
|
3891 |
(calculations)g(\(the)523 1046 y(tools)22 b(with)g(which)f(we)h(do)g
|
|
|
3892 |
(such)f(calculations)g(are)h(gi)n(v)o(en)f(in)h([17]\).)e(These)i
|
|
|
3893 |
(calculations)f(indeed)523 1146 y(v)n(alidate)f(the)g(assumption)f
|
|
|
3894 |
(that)h(the)g Fr(::)1703 1158 y Fq(R)1758 1146 y FA(-rule)p
|
|
|
3895 |
1746 1309 10 38 v 1756 1293 42 4 v 1816 1321 a Fr(::)p
|
|
|
3896 |
Fs(B)p 1728 1341 266 4 v 1802 1403 10 38 v 1811 1386
|
|
|
3897 |
42 4 v 1871 1415 a(B)2035 1353 y Fr(::)2145 1365 y Fq(R)523
|
|
|
3898 |
1612 y FA(can)27 b(be)h(re)o(garded)d(as)j(introducing)d(the)j(formula)
|
|
|
3899 |
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
|
|
|
3900 |
(act,)e(ho)n(we)n(v)o(er)m(,)523 1711 y(we)d(guess)g(one)f(needs)g(a)h
|
|
|
3901 |
(\223full-blo)n(wn\224)d(conte)o(xt-lemma)g(using)i(Ho)n(we')-5
|
|
|
3902 |
b(s)24 b(method.)e(The)h(biggest)523 1811 y(problem)17
|
|
|
3903 |
b(we)j(see)g(in)f(establishing)f(a)i(one-to-one)c(correspondence)g
|
|
|
3904 |
(between)i(the)h(collections)g(of)523 1911 y(normalforms)e(determined)i
|
|
|
3905 |
(by)g(double-ne)o(gations)d(translations)j(and)h(by)f(colour)n
|
|
|
3906 |
(-annotations)e(is)523 2010 y(to)j(\002nd)g(a)h(meaningful)d(equi)n(v)n
|
|
|
3907 |
(alence)g(relation)h(on)h(double-ne)o(gation)c(translations.)648
|
|
|
3908 |
2115 y(Further)m(,)25 b(the)j(colour)n(-annotations)c(are)j(gi)n(v)o
|
|
|
3909 |
(en)f(for)h(sequent-proofs,)d(for)j(which)g(it)h(is)g(well-)523
|
|
|
3910 |
2214 y(kno)n(wn)f(that)i(the)o(y)f(mak)o(e)h(\223inessential\224)f(dif)
|
|
|
3911 |
n(ferences)f(that)i(w)o(ould)g(not)f(materialise)h(if)g(we)g(had)523
|
|
|
3912 |
2314 y(a)23 b(natural-deduction)18 b(formulation)i(or)j(proof-net)c
|
|
|
3913 |
(formulation.)h(Ho)n(we)n(v)o(er)m(,)g(such)j(formulations)523
|
|
|
3914 |
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
|
|
|
3915 |
(enough)d(to)j(be)f(useful)g(in)h(getting)e(rid)i(of)f(the)g
|
|
|
3916 |
(inessential)523 2513 y(dif)n(ferences)f(between)h(classical)h
|
|
|
3917 |
(sequent-proofs.)d(This)i(problem)f(also)i(sho)n(ws)g(up)f(with)h
|
|
|
3918 |
(intu-)523 2613 y(itionistic)21 b(sequent-proofs)c(where)j
|
|
|
3919 |
(cut-elimination,)f(despite)h(our)g(w)o(orking)f(hypothesis)g(stating)
|
|
|
3920 |
523 2713 y(the)j(contrary)-5 b(,)19 b(is)k Ft(not)g FA(Church-Rosser)-5
|
|
|
3921 |
b(.)20 b(Ho)n(we)n(v)o(er)m(,)g(for)h(fragments)f(of)i(intuitionistic)f
|
|
|
3922 |
(logic)g(there)523 2812 y(are)28 b(already)f(good)g(tools\227for)g(e)o
|
|
|
3923 |
(xample)g(natural)h(deduction)e(and)i(contraction-free)d(sequent-)523
|
|
|
3924 |
2912 y(calculi\227which)g(pro)o(vide)f(a)j(canonical)d(notion)h(of)h
|
|
|
3925 |
(what)g(an)g(intuitionistic)g(proof)e(is.)j(Because)523
|
|
|
3926 |
3011 y(of)k(all)g(these)h(dif)n(\002culties,)e(we)h(ha)n(v)o(e,)g(at)g
|
|
|
3927 |
(the)g(moment,)f(to)h(content)f(ourselv)o(es)g(with)h(e)o(xample)523
|
|
|
3928 |
3111 y(calculations)19 b(that,)h(ho)n(we)n(v)o(er)m(,)e(all)j(seem)f
|
|
|
3929 |
(to)h(v)n(alidate)e(the)h(conjecture.)523 3401 y Fu(5)99
|
|
|
3930 |
b(Conclusion)523 3624 y FA(Although)14 b(there)h(is)i(plenty)e(of)g
|
|
|
3931 |
(literature)g(on)g(classical)i(logic)e(and)g(double-ne)o(gation)c
|
|
|
3932 |
(translations,)523 3724 y(there)34 b(is)i(surprising)d(little)i
|
|
|
3933 |
(literature)f(that)h(studies)g(the)f(relation)g(between)g(double-ne)o
|
|
|
3934 |
(gation)523 3824 y(translations)27 b(and)h(the)g(process)f(of)h
|
|
|
3935 |
(normalising)e(a)i(classical)h(proof.)d(W)-7 b(e)29 b(\002nd)f(notable)
|
|
|
3936 |
f(e)o(xcep-)523 3923 y(tions)20 b(are)g([1,)12 b(5,)h(6,)g(11,)g(13],)
|
|
|
3937 |
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)
|
|
|
3938 |
i(about)f(this)h(relation)f(or)523 4023 y(consider)k(only)g(special)h
|
|
|
3939 |
(cases.)h(F)o(or)f(e)o(xample,)e(the)o(y)h(relate)h(one)g(kind)f(of)h
|
|
|
3940 |
(colour)n(-annotation)c(to)523 4123 y(one)f(double-ne)o(gation)c
|
|
|
3941 |
(translation,)j(or)h(consider)f(only)h(speci\002c)h(double-ne)o(gation)
|
|
|
3942 |
14 b(translations.)648 4227 y(If)19 b(our)h(conjecture)e(turns)h(out)h
|
|
|
3943 |
(to)g(be)g(true,)f(then)h(one)f(has)i(a)f(v)o(ery)f(simple)h
|
|
|
3944 |
(characterisation)e(of)523 4327 y(all)24 b(double-ne)o(gation)18
|
|
|
3945 |
b(translations)k(in)i(terms)f(of)f(colours.)g(This)h(is)h(desirable,)e
|
|
|
3946 |
(because)g(we)i(\002nd)523 4426 y(it)19 b(is)f(rather)f(mysterious)g
|
|
|
3947 |
(ho)n(w)g(double-ne)o(gation)c(translation)k(can)h(turn)f(a)h
|
|
|
3948 |
(classical)h(proof)d(into)i(an)523 4526 y(intuitionistic)f(proof,)f
|
|
|
3949 |
(whose)h(computational)f(interpretation)f(as)k(e)o(xplained)c(in)j(the)
|
|
|
3950 |
g(introduction)523 4625 y(is)32 b(by)e(the)h(Curry-Ho)n(w)o(ard)d
|
|
|
3951 |
(correspondence)g(well-understood,)f(while)k(it)h(is)f(not)g
|
|
|
3952 |
(understood)523 4725 y(at)c(all)h(for)e(the)h(classical)g(proof)e(we)i
|
|
|
3953 |
(started)g(with)g(\(see)g([2,)13 b(19])26 b(for)g(tw)o(o)h(interesting)
|
|
|
3954 |
f(e)o(xamples)523 4825 y(e)o(xtracting)e(computational)g(meaning)h
|
|
|
3955 |
(from)g(a)h(classical)h(proof)d(that)j(because)e(of)h(the)g(the)g(non-)
|
|
|
3956 |
523 4924 y(determinism)19 b(cannot)g(be)h(e)o(xtracted)f(by)h(double)e
|
|
|
3957 |
(ne)o(gation)g(translations\).)p eop end
|
|
|
3958 |
%%Page: 19 19
|
|
|
3959 |
TeXDict begin 19 18 bop 648 448 a FA(The)24 b(main)h(point)g(we)g(tak)o
|
|
|
3960 |
(e)g(a)o(w)o(ay)g(from)f(the)h(conjecture)f(is)i(that)f(double-ne)o
|
|
|
3961 |
(gation)c(transla-)523 548 y(tions)j(are)f(not)g(enough)e(to)j
|
|
|
3962 |
(characterise)e(the)i(full)f(computational)e(content)i(of)g(classical)h
|
|
|
3963 |
(proofs,)523 648 y(where)31 b(by)g Ft(full)h FA(computational)e
|
|
|
3964 |
(content)g(we)j(mean)e(the)g(collection)g(of)g(normalforms)f(reach-)523
|
|
|
3965 |
747 y(able)h(by)h(the)f(cut-elimination)e(procedure)g(of)j(Urban)e(and)
|
|
|
3966 |
h(Bierman.)g(This)h(is)g(because)f(there)523 847 y(are)h(some)f
|
|
|
3967 |
(normalforms)e(that)j(can)g(be)f(reached)g(by)g(this)h(cut-elimination)
|
|
|
3968 |
e(procedure)f(which)523 946 y(cannot)i(be)h(reached)e(by)i(an)o(y)f
|
|
|
3969 |
(colour)n(-annotation,)c(and)32 b(by)f(the)h(conjecture)e(also)i(not)g
|
|
|
3970 |
(by)f(an)o(y)523 1046 y(double-ne)o(gation)15 b(translation.)k(These)h
|
|
|
3971 |
(\223unreachable\224)d(normalforms)g(embody)h(a)i(form)f(of)h(non-)523
|
|
|
3972 |
1146 y(determinism)g(present)i(in)f(classical)i(logic,)e(b)n(ut)h(not)f
|
|
|
3973 |
(present)g(in)h(intuitionistic)g(logic.)f(Therefore)523
|
|
|
3974 |
1245 y(the)h(conclusion)d(we)j(dra)o(w)f(from)g(this)h(is)g(that)g
|
|
|
3975 |
(intuitionistic)f(logic)g(and)g(double-ne)o(gation)c(trans-)523
|
|
|
3976 |
1345 y(lations)k(can)f(only)g(gi)n(v)o(e)f(some)i(hints)f(for)g
|
|
|
3977 |
(understanding)e(the)j Ft(full)g FA(computational)d(meaning)h(of)h(a)
|
|
|
3978 |
523 1445 y(classical)h(proof.)648 1544 y(Completely)15
|
|
|
3979 |
b(untouched)e(by)j(our)f(treatment)g(here)h(is)h(the)f(correspondence)c
|
|
|
3980 |
(between)j(double-)523 1644 y(ne)o(gation)29 b(translations)i(and)f
|
|
|
3981 |
(linear)h(logic.)g(The)g(colour)e(annotation)h(also)h(connects)g(to)g
|
|
|
3982 |
(linear)523 1743 y(logic,)25 b(where)g(cut-elimination)e(formulated)h
|
|
|
3983 |
(with)h(proof-nets)f(is)i(also)g(Church-Rosser)m(,)e(to)h(the)523
|
|
|
3984 |
1843 y(cut-elimination)18 b(procedure)g(with)i(colours)f(\(see)i([6,)13
|
|
|
3985 |
b(12]\).)523 2104 y Fu(Refer)n(ences)560 2290 y Fx(1.)42
|
|
|
3986 |
b(F)-6 b(.)14 b(Barbanera)j(and)f(S.)e(Berardi.)19 b(A)c(Symmetric)g
|
|
|
3987 |
(Lambda)h(Calculus)g(for)f(\223Classical\224)h(Program)f(Extrac-)658
|
|
|
3988 |
2381 y(tion.)41 b(In)24 b Fv(Theor)m(etical)g(Aspects)f(of)g(Computer)i
|
|
|
3989 |
(Softwar)m(e)p Fx(,)e(v)o(olume)h(789)g(of)g Fv(LNCS)p
|
|
|
3990 |
Fx(,)e(pages)j(495\226515.)658 2472 y(Springer)19 b(V)-8
|
|
|
3991 |
b(erlag,)18 b(1994.)560 2562 y(2.)42 b(F)-6 b(.)28 b(Barbanera,)h(S.)f
|
|
|
3992 |
(Berardi,)g(and)i(M.)e(Schi)n(v)n(alocchi.)60 b(\223Classical\224)29
|
|
|
3993 |
b(Programming-with-Proofs)g(in)658 2654 y Fj(\025)703
|
|
|
3994 |
2622 y Fi(sy)r(m)824 2654 y Fx(:)20 b(An)i(Analysis)f(of)g
|
|
|
3995 |
(Non-Con\003uence.)36 b(In)21 b Fv(Theor)m(etical)h(Aspects)f(of)g
|
|
|
3996 |
(Computer)h(Softwar)m(e)p Fx(,)f(v)o(ol-)658 2745 y(ume)e(1281)h(of)f
|
|
|
3997 |
Fv(LNCS)p Fx(,)f(pages)i(365\226390.)h(Springer)e(V)-8
|
|
|
3998 |
b(erlag,)18 b(1997.)560 2835 y(3.)42 b(G.)21 b(Bellin,)f(M.)h(Hyland,)g
|
|
|
3999 |
(E.)g(Robinson,)h(and)g(C.)e(Urban.)35 b(Proof)21 b(Theory)g(of)h
|
|
|
4000 |
(Classical)e(Propositional)658 2927 y(Calculus.)27 b(\(Accepted)20
|
|
|
4001 |
b(for)f(publication)g(in)g(Theoretical)g(Computer)h(Science\),)e(2005.)
|
|
|
4002 |
560 3017 y(4.)42 b(S.)18 b(R.)h(Buss.)29 b(An)20 b(Introduction)g(to)g
|
|
|
4003 |
(Proof-Theory)-5 b(.)29 b(In)20 b Fv(Handbook)h(of)f(Pr)m(oof)f(Theory)
|
|
|
4004 |
p Fx(,)g(v)o(olume)h(137)h(of)658 3108 y Fv(Studies)e(in)g(Lo)o(gic)g
|
|
|
4005 |
(and)h(the)f(F)-8 b(oundations)21 b(of)e(Mathematics)p
|
|
|
4006 |
Fx(,)g(pages)h(1\22678.)g(North-Holland,)f(1998.)560
|
|
|
4007 |
3198 y(5.)42 b(P)-8 b(.-L.)22 b(Curien)i(and)h(H.)e(Herbelin.)43
|
|
|
4008 |
b(The)24 b(Duality)g(of)g(Computation.)44 b(In)24 b Fv(Confer)m(ence)i
|
|
|
4009 |
(on)e(Functional)658 3290 y(Pr)m(o)o(gr)o(amming)p Fx(,)19
|
|
|
4010 |
b(pages)h(233\226243.)h(A)m(CM)e(Press,)f(2000.)560 3380
|
|
|
4011 |
y(6.)42 b(V)-10 b(.)15 b(Danos,)g(J.-B.)f(Joinet,)h(and)g(H.)g
|
|
|
4012 |
(Schellinx.)j(A)d(Ne)n(w)f(Deconstructi)n(v)o(e)j(Logic:)d(Linear)h
|
|
|
4013 |
(Logic.)j Fv(J)n(ournal)658 3471 y(of)h(Symbolic)g(Lo)o(gic)p
|
|
|
4014 |
Fx(,)g(62\(3\):755\226807,)j(1997.)560 3561 y(7.)42 b(A.)27
|
|
|
4015 |
b(G.)h(Dragalin.)56 b Fv(Mathematical)29 b(Intuitionism:)f(Intr)m
|
|
|
4016 |
(oduction)h(to)f(Pr)m(oof)g(Theory)p Fx(,)g(v)o(olume)h(67)g(of)658
|
|
|
4017 |
3653 y Fv(T)l(r)o(anslations)19 b(of)g(Mathematical)h(Mono)o(gr)o(aphs)
|
|
|
4018 |
p Fx(.)29 b(American)20 b(Mathematical)f(Society)-5 b(,)19
|
|
|
4019 |
b(1988.)560 3743 y(8.)42 b(R.)21 b(Dyckhof)n(f)i(and)g(L.)e(Pinto.)37
|
|
|
4020 |
b(Permutability)21 b(of)h(Proofs)g(in)g(Intuitionistic)g(Sequent)g
|
|
|
4021 |
(Calculi.)37 b Fv(Theo-)658 3834 y(r)m(etical)19 b(Computer)h(Scince)p
|
|
|
4022 |
Fx(,)f(212\(1-2\):141\226155,)j(1999.)560 3924 y(9.)42
|
|
|
4023 |
b(J.)24 b(Gallier)l(.)45 b(Constructi)n(v)o(e)25 b(Logics.)g(Part)f(I:)
|
|
|
4024 |
g(A)g(Tutorial)g(on)h(Proof)g(Systems)g(and)g(Typed)g
|
|
|
4025 |
Fj(\025)p Fx(-calculi.)658 4016 y Fv(Theor)m(etical)19
|
|
|
4026 |
b(Computer)h(Science)p Fx(,)g(110\(2\):249\226239,)h(1993.)523
|
|
|
4027 |
4106 y(10.)42 b(J.-Y)-10 b(.)22 b(Girard,)f(Y)-10 b(.)22
|
|
|
4028 |
b(Lafont,)g(and)h(P)-8 b(.)21 b(T)-6 b(aylor)l(.)38 b
|
|
|
4029 |
Fv(Pr)m(oofs)22 b(and)h(Types)p Fx(,)f(v)o(olume)h(7)f(of)g
|
|
|
4030 |
Fv(Cambridg)o(e)i(T)l(r)o(acts)e(in)658 4197 y(Theor)m(etical)d
|
|
|
4031 |
(Computer)h(Science)p Fx(.)28 b(Cambridge)19 b(Uni)n(v)o(ersity)h
|
|
|
4032 |
(Press,)e(1989.)523 4287 y(11.)42 b(T)-6 b(.)26 b(Grif)n(\002n.)49
|
|
|
4033 |
b(A)26 b(Formulae-as-Types)h(Notion)f(of)g(Control.)51
|
|
|
4034 |
b(In)26 b Fv(Principles)g(of)g(Pr)m(o)o(gr)o(amming)h(Lan-)658
|
|
|
4035 |
4379 y(gua)o(g)o(es)p Fx(,)20 b(pages)g(47\22658.)g(A)m(CM)f(Press,)f
|
|
|
4036 |
(1990.)523 4469 y(12.)42 b(J.-B.)19 b(Joinet,)i(H.)f(Schellinx,)g(and)h
|
|
|
4037 |
(L.)f(T)-6 b(ortora)20 b(de)h(F)o(alco.)31 b(SN)20 b(and)h(CR)f(for)h
|
|
|
4038 |
(Free-Style)e(LK)3111 4437 y Fi(tq)3169 4469 y Fx(:)h(Linear)658
|
|
|
4039 |
4560 y(Decorations)k(and)g(Simulation)f(of)g(Normalization.)41
|
|
|
4040 |
b Fv(J)n(ournal)24 b(of)f(Symbolic)h(Lo)o(gic)p Fx(,)f
|
|
|
4041 |
(67\(1\):162\226196,)658 4652 y(2002.)523 4742 y(13.)42
|
|
|
4042 |
b(J.)25 b(Laird.)48 b(A)25 b(Deconstruction)i(of)e(Non-Deterministic)h
|
|
|
4043 |
(Cut)f(Elimination.)48 b(In)26 b Fv(Pr)m(oceedings)g(of)g(the)658
|
|
|
4044 |
4833 y(5th)e(International)g(Confer)m(ence)h(on)f(T)-6
|
|
|
4045 |
b(yped)25 b(Lambda)f(Calculi)f(and)h(Applications)p Fx(,)g(v)o(olume)g
|
|
|
4046 |
(2044)h(of)658 4924 y Fv(LNCS)p Fx(,)18 b(pages)i(268\226282.)h
|
|
|
4047 |
(Springer)e(V)-8 b(erlag,)18 b(2001.)p eop end
|
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|
4048 |
%%Page: 20 20
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|
4049 |
TeXDict begin 20 19 bop 523 448 a Fx(14.)42 b(M.)23 b(P)o(arigot.)42
|
|
|
4050 |
b Fj(\025\026)p Fx(-calculus:)23 b(An)h(Algorithmic)g(Interpretation)g
|
|
|
4051 |
(of)f(Classical)g(Logic.)42 b(In)24 b Fv(Lo)o(gic)f(Pr)m(o-)658
|
|
|
4052 |
540 y(gr)o(amming)16 b(and)f(A)o(utomated)g(Deduction)p
|
|
|
4053 |
Fx(,)h(v)o(olume)f(624)h(of)f Fv(LNCS)p Fx(,)f(pages)h(190\226201.)i
|
|
|
4054 |
(Springer)e(V)-8 b(erlag,)658 631 y(1992.)523 722 y(15.)42
|
|
|
4055 |
b(G.)20 b(Pottinger)l(.)31 b(Normalisation)21 b(as)f(Homomorphic)i
|
|
|
4056 |
(Image)f(of)g(Cut-Elimination.)31 b Fv(Annals)21 b(of)f(Mathe-)658
|
|
|
4057 |
814 y(matical)f(Lo)o(gic)p Fx(,)g(12:323\226357,)i(1977.)523
|
|
|
4058 |
905 y(16.)42 b(H.)16 b(Schellinx.)23 b Fv(The)17 b(Noble)g(Art)g(of)g
|
|
|
4059 |
(Linear)g(Decor)o(ating)p Fx(.)23 b(PhD)17 b(thesis,)g(Institute)g(for)
|
|
|
4060 |
f(Logic,)h(Language)658 996 y(and)j(Computation,)f(Uni)n(v)o(ersity)g
|
|
|
4061 |
(of)g(Amsterdam,)g(1994.)28 b(ILLC)18 b(dissertation)h(series.)523
|
|
|
4062 |
1088 y(17.)42 b(C.)18 b(Urban.)27 b(http://www4.in.tum.de/)1677
|
|
|
4063 |
1071 y Fa(\030)1726 1088 y Fx(urbanc/cut/cutapplet.html.)523
|
|
|
4064 |
1179 y(18.)42 b(C.)22 b(Urban.)40 b(Re)n(visiting)23
|
|
|
4065 |
b(Zuck)o(er')l(s)g(Work)g(on)g(the)g(Correspondence)i(Between)e
|
|
|
4066 |
(Cut-elimination)g(and)658 1270 y(Normalisation.)k(\(T)-6
|
|
|
4067 |
b(o)19 b(apear)g(in)g(Adv)n(ances)h(in)f(Natural)g(Deduction\).)523
|
|
|
4068 |
1362 y(19.)42 b(C.)24 b(Urban.)45 b Fv(Classical)24 b(Lo)o(gic)h(and)g
|
|
|
4069 |
(Computation)p Fx(.)46 b(PhD)24 b(thesis,)g(Cambridge)i(Uni)n(v)o
|
|
|
4070 |
(ersity)-5 b(,)24 b(October)658 1453 y(2000.)523 1544
|
|
|
4071 |
y(20.)42 b(C.)18 b(Urban.)27 b(Strong)19 b(Normalisation)g(for)f(a)h
|
|
|
4072 |
(Gentzen-lik)o(e)g(Cut-Elimination)g(Procedure.)27 b(In)18
|
|
|
4073 |
b Fv(Pr)m(oceed-)658 1636 y(ings)f(of)f(the)g(5th)h(International)g
|
|
|
4074 |
(Confer)m(ence)h(on)f(T)-6 b(yped)17 b(Lambda)g(Calculi)f(and)i
|
|
|
4075 |
(Applications)p Fx(,)e(v)o(olume)658 1727 y(2044)k(of)f
|
|
|
4076 |
Fv(LNCS)p Fx(,)f(pages)i(415\226429.)h(Springer)e(V)-8
|
|
|
4077 |
b(erlag,)18 b(2001.)523 1818 y(21.)42 b(C.)20 b(Urban)h(and)g(G.)f(M.)g
|
|
|
4078 |
(Bierman.)32 b(Strong)20 b(Normalisation)h(of)g(Cut-Elimination)f(in)g
|
|
|
4079 |
(Classical)g(Logic.)658 1910 y Fv(Fundamenta)g(Informaticae)p
|
|
|
4080 |
Fx(,)g(45\(1\2262\):123\226155,)i(2001.)523 2001 y(22.)42
|
|
|
4081 |
b(S.)30 b(v)n(an)i(Bak)o(el,)e(S.)g(Lengrand,)i(and)g(P)-8
|
|
|
4082 |
b(.)29 b(Lescanne.)66 b(The)31 b(Language)h(X:)f(Circuits,)f
|
|
|
4083 |
(Computations)658 2092 y(and)25 b(Classical)g(Logic.)45
|
|
|
4084 |
b(In)25 b Fv(Pr)m(oc.)f(of)h(9th)g(Italian)g(Confer)m(ence)h(on)f
|
|
|
4085 |
(Theor)m(etical)g(Computer)h(Science)658 2183 y(\(ICTCS\))p
|
|
|
4086 |
Fx(,)18 b(v)o(olume)h(3701)h(of)f Fv(LNCS)p Fx(,)f(pages)i(81\22696,)g
|
|
|
4087 |
(2005.)523 2275 y(23.)42 b(J.)27 b(Zuck)o(er)l(.)56 b(The)28
|
|
|
4088 |
b(Correspondence)j(Between)d(Cut-Elimination)f(and)i(Normalisation.)56
|
|
|
4089 |
b Fv(Annals)28 b(of)658 2366 y(Mathematical)20 b(Lo)o(gic)p
|
|
|
4090 |
Fx(,)f(7:1\226112,)h(1974.)p eop end
|
|
|
4091 |
%%Trailer
|
|
|
4092 |
|
|
|
4093 |
userdict /end-hook known{end-hook}if
|
|
|
4094 |
%%EOF
|