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1 %!PS-Adobe-2.0 |
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2 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software |
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3 %%Title: nomu-csl.dvi |
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4 %%Pages: 15 |
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5 %%PageOrder: Ascend |
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9 %%+ CMTI9 CMTT9 CMSY9 CMTI10 TeX-cmex9 CMSY6 CMSY8 CMR8 CMBXTI10 CMBX10 |
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10 %%+ CMMIB10 MSBM10 CMMI6 MSBM7 CMR5 CMMI5 EUSM10 CMMI8 CMTT8 CMBSY7 |
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11 %%+ Courier |
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12 %%EndComments |
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13 %DVIPSWebPage: (www.radicaleye.com) |
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14 %DVIPSCommandLine: dvips -o nomu-csl.ps nomu-csl.dvi |
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16 %DVIPSSource: TeX output 2003.06.14:1231 |
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17 %%BeginProcSet: texc.pro |
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22 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ |
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23 landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize |
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24 mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ |
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25 matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round |
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44 A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy |
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54 bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A |
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55 mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ |
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57 userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X |
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58 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 |
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59 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N |
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61 /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) |
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65 grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} |
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73 |
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74 %%EndProcSet |
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75 %%BeginProcSet: 8r.enc |
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76 % File 8r.enc as of 2002-03-12 for PSNFSS 9 |
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77 % |
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78 % This is the encoding vector for Type1 and TrueType fonts to be used |
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79 % with TeX. This file is part of the PSNFSS bundle, version 9 |
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80 % |
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81 % Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt |
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82 % |
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83 % Idea is to have all the characters normally included in Type 1 fonts |
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84 % available for typesetting. This is effectively the characters in Adobe |
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85 % Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro. |
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86 % |
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87 % Character code assignments were made as follows: |
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88 % |
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89 % (1) the Windows ANSI characters are almost all in their Windows ANSI |
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90 % positions, because some Windows users cannot easily reencode the |
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91 % fonts, and it makes no difference on other systems. The only Windows |
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92 % ANSI characters not available are those that make no sense for |
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93 % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen |
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94 % (173). quotesingle and grave are moved just because it's such an |
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95 % irritation not having them in TeX positions. |
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96 % |
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97 % (2) Remaining characters are assigned arbitrarily to the lower part |
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98 % of the range, avoiding 0, 10 and 13 in case we meet dumb software. |
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99 % |
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100 % (3) Y&Y Lucida Bright includes some extra text characters; in the |
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101 % hopes that other PostScript fonts, perhaps created for public |
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102 % consumption, will include them, they are included starting at 0x12. |
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103 % |
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104 % (4) Remaining positions left undefined are for use in (hopefully) |
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105 % upward-compatible revisions, if someday more characters are generally |
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106 % available. |
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107 % |
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108 % (5) hyphen appears twice for compatibility with both ASCII and Windows. |
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109 % |
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110 % (6) /Euro is assigned to 128, as in Windows ANSI |
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111 % |
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112 /TeXBase1Encoding [ |
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113 % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) |
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114 /.notdef /dotaccent /fi /fl |
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115 /fraction /hungarumlaut /Lslash /lslash |
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116 /ogonek /ring /.notdef |
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117 /breve /minus /.notdef |
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118 % These are the only two remaining unencoded characters, so may as |
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119 % well include them. |
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120 /Zcaron /zcaron |
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121 % 0x10 |
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122 /caron /dotlessi |
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123 % (unusual TeX characters available in, e.g., Lucida Bright) |
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124 /dotlessj /ff /ffi /ffl |
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125 /.notdef /.notdef /.notdef /.notdef |
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126 /.notdef /.notdef /.notdef /.notdef |
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127 % very contentious; it's so painful not having quoteleft and quoteright |
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128 % at 96 and 145 that we move the things normally found there down to here. |
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129 /grave /quotesingle |
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130 % 0x20 (ASCII begins) |
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131 /space /exclam /quotedbl /numbersign |
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132 /dollar /percent /ampersand /quoteright |
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133 /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash |
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134 % 0x30 |
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135 /zero /one /two /three /four /five /six /seven |
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136 /eight /nine /colon /semicolon /less /equal /greater /question |
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137 % 0x40 |
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138 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O |
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139 % 0x50 |
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140 /P /Q /R /S /T /U /V /W |
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141 /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore |
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142 % 0x60 |
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143 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o |
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144 % 0x70 |
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145 /p /q /r /s /t /u /v /w |
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146 /x /y /z /braceleft /bar /braceright /asciitilde |
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147 /.notdef % rubout; ASCII ends |
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148 % 0x80 |
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149 /Euro /.notdef /quotesinglbase /florin |
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150 /quotedblbase /ellipsis /dagger /daggerdbl |
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151 /circumflex /perthousand /Scaron /guilsinglleft |
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152 /OE /.notdef /.notdef /.notdef |
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153 % 0x90 |
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154 /.notdef /.notdef /.notdef /quotedblleft |
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155 /quotedblright /bullet /endash /emdash |
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156 /tilde /trademark /scaron /guilsinglright |
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157 /oe /.notdef /.notdef /Ydieresis |
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158 % 0xA0 |
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159 /.notdef % nobreakspace |
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160 /exclamdown /cent /sterling |
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161 /currency /yen /brokenbar /section |
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162 /dieresis /copyright /ordfeminine /guillemotleft |
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163 /logicalnot |
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164 /hyphen % Y&Y (also at 45); Windows' softhyphen |
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165 /registered |
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166 /macron |
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167 % 0xD0 |
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168 /degree /plusminus /twosuperior /threesuperior |
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169 /acute /mu /paragraph /periodcentered |
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170 /cedilla /onesuperior /ordmasculine /guillemotright |
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171 /onequarter /onehalf /threequarters /questiondown |
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172 % 0xC0 |
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173 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla |
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174 /Egrave /Eacute /Ecircumflex /Edieresis |
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175 /Igrave /Iacute /Icircumflex /Idieresis |
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176 % 0xD0 |
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177 /Eth /Ntilde /Ograve /Oacute |
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178 /Ocircumflex /Otilde /Odieresis /multiply |
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179 /Oslash /Ugrave /Uacute /Ucircumflex |
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180 /Udieresis /Yacute /Thorn /germandbls |
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181 % 0xE0 |
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182 /agrave /aacute /acircumflex /atilde |
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183 /adieresis /aring /ae /ccedilla |
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184 /egrave /eacute /ecircumflex /edieresis |
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185 /igrave /iacute /icircumflex /idieresis |
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186 % 0xF0 |
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187 /eth /ntilde /ograve /oacute |
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188 /ocircumflex /otilde /odieresis /divide |
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189 /oslash /ugrave /uacute /ucircumflex |
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190 /udieresis /yacute /thorn /ydieresis |
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191 ] def |
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192 |
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193 %%EndProcSet |
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194 %%BeginProcSet: aae443f0.enc |
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195 % Thomas Esser, Dec 2002. public domain |
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196 % |
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197 % Encoding for: |
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198 % cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10 |
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199 % |
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200 /TeXaae443f0Encoding [ |
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201 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega |
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202 /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa |
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203 /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi |
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204 /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf |
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205 /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft |
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206 /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle |
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207 /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle |
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208 /sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash |
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209 /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N |
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210 /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow |
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211 /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p |
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212 /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector |
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213 /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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214 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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215 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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216 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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217 /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi |
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218 /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta |
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219 /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon |
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220 /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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221 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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222 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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223 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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224 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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225 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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226 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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227 /.notdef /.notdef /.notdef /.notdef /.notdef |
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228 ] def |
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229 |
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230 %%EndProcSet |
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231 %%BeginProcSet: bbad153f.enc |
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232 % Thomas Esser, Dec 2002. public domain |
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233 % |
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234 % Encoding for: |
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235 % cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9 |
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236 % |
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237 /TeXbbad153fEncoding [ |
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238 /minus /periodcentered /multiply /asteriskmath /divide /diamondmath |
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239 /plusminus /minusplus /circleplus /circleminus /circlemultiply |
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240 /circledivide /circledot /circlecopyrt /openbullet /bullet |
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241 /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal |
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242 /greaterequal /precedesequal /followsequal /similar /approxequal |
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243 /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows |
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244 /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast |
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245 /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup |
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246 /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional |
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247 /prime /infinity /element /owner /triangle /triangleinv /negationslash |
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248 /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur |
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249 /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K |
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250 /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection |
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251 /unionmulti /logicaland /logicalor /turnstileleft /turnstileright |
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252 /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright |
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253 /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv |
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254 /backslash /wreathproduct /radical /coproduct /nabla /integral |
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255 /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section |
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256 /dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft |
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257 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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258 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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259 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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260 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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261 /minus /periodcentered /multiply /asteriskmath /divide /diamondmath |
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262 /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef |
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263 /circlemultiply /circledivide /circledot /circlecopyrt /openbullet |
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264 /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset |
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265 /lessequal /greaterequal /precedesequal /followsequal /similar |
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266 /approxequal /propersubset /propersuperset /lessmuch /greatermuch |
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267 /precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef |
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268 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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269 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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270 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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271 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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272 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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273 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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274 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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275 ] def |
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276 |
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277 %%EndProcSet |
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278 %%BeginProcSet: 09fbbfac.enc |
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279 % Thomas Esser, Dec 2002. public domain |
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280 % |
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281 % Encoding for: |
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282 % cmsltt10 cmtt10 cmtt12 cmtt8 cmtt9 |
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283 /TeX09fbbfacEncoding [ |
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284 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi |
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285 /Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown |
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286 /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla |
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287 /germandbls /ae /oe /oslash /AE /OE /Oslash /visiblespace /exclam |
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288 /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft |
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289 /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one |
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290 /two /three /four /five /six /seven /eight /nine /colon /semicolon /less |
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291 /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N |
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292 /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright |
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293 /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l |
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294 /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright |
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295 /asciitilde /dieresis /visiblespace /.notdef /.notdef /.notdef /.notdef |
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296 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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297 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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298 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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299 /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi |
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300 /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /arrowup /arrowdown |
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301 /quotesingle /exclamdown /questiondown /dotlessi /dotlessj /grave /acute |
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302 /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE |
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303 /OE /Oslash /visiblespace /dieresis /.notdef /.notdef /.notdef /.notdef |
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304 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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305 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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306 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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307 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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308 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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309 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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310 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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311 ] def |
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312 |
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313 %%EndProcSet |
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314 %%BeginProcSet: f7b6d320.enc |
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315 % Thomas Esser, Dec 2002. public domain |
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316 % |
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317 % Encoding for: |
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319 % cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8 |
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320 % cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10 |
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321 % cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10 |
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322 % |
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323 /TeXf7b6d320Encoding [ |
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324 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega |
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325 /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve |
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326 /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash |
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327 /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand |
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328 /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen |
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329 /period /slash /zero /one /two /three /four /five /six /seven /eight |
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330 /nine /colon /semicolon /exclamdown /equal /questiondown /question /at |
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331 /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X |
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332 /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent |
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333 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u |
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334 /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress |
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335 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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336 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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337 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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340 /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute |
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341 /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE |
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342 /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef |
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343 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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350 ] def |
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351 |
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352 %%EndProcSet |
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353 %%BeginProcSet: 10037936.enc |
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354 % Thomas Esser, Dec 2002. public domain |
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355 % |
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356 % Encoding for: |
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358 % |
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359 /TeX10037936Encoding [ |
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360 /minus /periodcentered /multiply /asteriskmath /divide /diamondmath |
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361 /plusminus /minusplus /circleplus /circleminus /circlemultiply |
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362 /circledivide /circledot /circlecopyrt /openbullet /bullet |
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363 /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal |
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364 /greaterequal /precedesequal /followsequal /similar /approxequal |
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365 /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows |
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366 /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast |
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367 /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup |
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368 /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional |
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369 /prime /infinity /element /owner /triangle /triangleinv /negationslash |
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370 /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur |
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371 /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K |
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372 /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection |
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373 /unionmulti /logicaland /logicalor /turnstileleft /turnstileright |
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374 /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright |
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375 /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv |
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376 /backslash /wreathproduct /radical /coproduct /nabla /integral |
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377 /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section |
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378 /dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft |
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384 /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef |
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385 /circlemultiply /circledivide /circledot /circlecopyrt /openbullet |
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386 /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset |
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387 /lessequal /greaterequal /precedesequal /followsequal /similar |
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388 /approxequal /propersubset /propersuperset /lessmuch /greatermuch |
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399 %%EndProcSet |
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400 %%BeginProcSet: 74afc74c.enc |
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401 % Thomas Esser, Dec 2002. public domain |
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402 % |
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403 % Encoding for: |
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405 % |
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406 /TeX74afc74cEncoding [ |
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407 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega |
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408 /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve |
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409 /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash |
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410 /suppress /exclam /quotedblright /numbersign /sterling /percent |
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411 /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma |
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412 /hyphen /period /slash /zero /one /two /three /four /five /six /seven |
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413 /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question |
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414 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W |
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415 /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent |
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416 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u |
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417 /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress |
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418 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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424 /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE |
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425 /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef |
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434 |
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435 %%EndProcSet |
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436 %%BeginProcSet: 0ef0afca.enc |
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437 % Thomas Esser, Dec 2002. public domain |
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438 % |
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439 % Encoding for: |
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440 % cmr5 |
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441 % |
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442 /TeX0ef0afcaEncoding [ |
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443 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega |
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444 /arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi |
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445 /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls |
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446 /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright |
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447 /numbersign /dollar /percent /ampersand /quoteright /parenleft |
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448 /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one |
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449 /two /three /four /five /six /seven /eight /nine /colon /semicolon |
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450 /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K |
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451 /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft |
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452 /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h |
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453 /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash |
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454 /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef |
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455 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef |
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458 /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda |
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459 /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /arrowup |
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460 /arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj |
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461 /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe |
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462 /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef |
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610 C94E41977843140A6FD0340C39C88E86CC0C058B87234810E1E900ADFB99C232 |
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611 12C88EF58715C1998419546382D0CFAE4297C48C2BD7 |
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612 0000000000000000000000000000000000000000000000000000000000000000 |
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613 0000000000000000000000000000000000000000000000000000000000000000 |
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614 0000000000000000000000000000000000000000000000000000000000000000 |
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615 0000000000000000000000000000000000000000000000000000000000000000 |
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616 0000000000000000000000000000000000000000000000000000000000000000 |
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617 0000000000000000000000000000000000000000000000000000000000000000 |
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618 0000000000000000000000000000000000000000000000000000000000000000 |
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619 0000000000000000000000000000000000000000000000000000000000000000 |
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620 cleartomark |
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621 %%EndFont |
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622 %%BeginFont: CMBSY7 |
|
623 %!PS-AdobeFont-1.1: CMBSY7 001.000 |
|
624 %%CreationDate: 1992 Oct 22 12:18:11 |
|
625 % Computer Modern fonts were designed by Donald E. Knuth. |
|
626 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
627 11 dict begin |
|
628 /FontInfo 7 dict dup begin |
|
629 /version (001.000) readonly def |
|
630 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
631 /FullName (CMBSY7) readonly def |
|
632 /FamilyName (Computer Modern) readonly def |
|
633 /Weight (Bold) readonly def |
|
634 /ItalicAngle -14.035 def |
|
635 /isFixedPitch false def |
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636 end readonly def |
|
637 /FontName /CMBSY7 def |
|
638 /PaintType 0 def |
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639 /FontType 1 def |
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640 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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641 /Encoding 256 array |
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642 0 1 255 {1 index exch /.notdef put} for |
|
643 dup 1 /periodcentered put |
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644 readonly def |
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645 /FontBBox{0 -927 1542 750}readonly def |
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646 /UniqueID 5032008 def |
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647 currentdict end |
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648 currentfile eexec |
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679 940655DE1405914E2D32B06EADFDFA08B9B7B2F9B88902FC1A6DC36F968F2D33 |
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680 5860D1D1D1768392BB9CFA |
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681 0000000000000000000000000000000000000000000000000000000000000000 |
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682 0000000000000000000000000000000000000000000000000000000000000000 |
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683 0000000000000000000000000000000000000000000000000000000000000000 |
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684 0000000000000000000000000000000000000000000000000000000000000000 |
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685 0000000000000000000000000000000000000000000000000000000000000000 |
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686 0000000000000000000000000000000000000000000000000000000000000000 |
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687 0000000000000000000000000000000000000000000000000000000000000000 |
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688 0000000000000000000000000000000000000000000000000000000000000000 |
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689 cleartomark |
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690 %%EndFont |
|
691 %%BeginFont: CMTT8 |
|
692 %!PS-AdobeFont-1.1: CMTT8 1.0 |
|
693 %%CreationDate: 1991 Aug 20 16:46:05 |
|
694 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
695 11 dict begin |
|
696 /FontInfo 7 dict dup begin |
|
697 /version (1.0) readonly def |
|
698 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
699 /FullName (CMTT8) readonly def |
|
700 /FamilyName (Computer Modern) readonly def |
|
701 /Weight (Medium) readonly def |
|
702 /ItalicAngle 0 def |
|
703 /isFixedPitch true def |
|
704 end readonly def |
|
705 /FontName /CMTT8 def |
|
706 /PaintType 0 def |
|
707 /FontType 1 def |
|
708 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
709 /Encoding 256 array |
|
710 0 1 255 {1 index exch /.notdef put} for |
|
711 dup 0 /.notdef put |
|
712 readonly def |
|
713 /FontBBox{-5 -232 545 699}readonly def |
|
714 /UniqueID 5000830 def |
|
715 currentdict end |
|
716 currentfile eexec |
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718 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 |
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768 07277A4E5B30B31A1DF870B571EC276A8EC80A11794DC5F1D444218690D89DCA |
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770 DB2773064DE8EB005CEF2D1F822E85B889104DB7DFB72D7E93068495531315EA |
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771 C4983BFA1186ACE72A45E709B846D2D5861AF4745D360B8E39F14E37575041C4 |
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772 F72697AF00EC75F46C4021E4B03A6E1FE26326667B08F6FB584F6B7E0133D5FC |
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773 DC9CB6C5B67BB5C7C52D115D189B796C |
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774 0000000000000000000000000000000000000000000000000000000000000000 |
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775 0000000000000000000000000000000000000000000000000000000000000000 |
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776 0000000000000000000000000000000000000000000000000000000000000000 |
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777 0000000000000000000000000000000000000000000000000000000000000000 |
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778 0000000000000000000000000000000000000000000000000000000000000000 |
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779 0000000000000000000000000000000000000000000000000000000000000000 |
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780 0000000000000000000000000000000000000000000000000000000000000000 |
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781 0000000000000000000000000000000000000000000000000000000000000000 |
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782 cleartomark |
|
783 %%EndFont |
|
784 %%BeginFont: CMMI8 |
|
785 %!PS-AdobeFont-1.1: CMMI8 1.100 |
|
786 %%CreationDate: 1996 Jul 23 07:53:54 |
|
787 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
788 11 dict begin |
|
789 /FontInfo 7 dict dup begin |
|
790 /version (1.100) readonly def |
|
791 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
792 /FullName (CMMI8) readonly def |
|
793 /FamilyName (Computer Modern) readonly def |
|
794 /Weight (Medium) readonly def |
|
795 /ItalicAngle -14.04 def |
|
796 /isFixedPitch false def |
|
797 end readonly def |
|
798 /FontName /CMMI8 def |
|
799 /PaintType 0 def |
|
800 /FontType 1 def |
|
801 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
802 /Encoding 256 array |
|
803 0 1 255 {1 index exch /.notdef put} for |
|
804 dup 0 /.notdef put |
|
805 readonly def |
|
806 /FontBBox{-24 -250 1110 750}readonly def |
|
807 /UniqueID 5087383 def |
|
808 currentdict end |
|
809 currentfile eexec |
|
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811 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B |
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813 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B |
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877 |
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879 0000000000000000000000000000000000000000000000000000000000000000 |
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880 0000000000000000000000000000000000000000000000000000000000000000 |
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881 0000000000000000000000000000000000000000000000000000000000000000 |
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882 0000000000000000000000000000000000000000000000000000000000000000 |
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883 0000000000000000000000000000000000000000000000000000000000000000 |
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884 0000000000000000000000000000000000000000000000000000000000000000 |
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885 0000000000000000000000000000000000000000000000000000000000000000 |
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886 cleartomark |
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887 %%EndFont |
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888 %%BeginFont: EUSM10 |
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889 %!PS-AdobeFont-1.1: EUSM10 2.1 |
|
890 %%CreationDate: 1992 Nov 20 17:36:44 |
|
891 % Euler fonts were designed by Hermann Zapf. |
|
892 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
893 11 dict begin |
|
894 /FontInfo 7 dict dup begin |
|
895 /version (2.1) readonly def |
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896 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
897 /FullName (EUSM10) readonly def |
|
898 /FamilyName (Euler) readonly def |
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899 /Weight (Medium) readonly def |
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900 /ItalicAngle 0 def |
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901 /isFixedPitch false def |
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902 end readonly def |
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903 /FontName /EUSM10 def |
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904 /PaintType 0 def |
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905 /FontType 1 def |
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906 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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907 /Encoding 256 array |
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908 0 1 255 {1 index exch /.notdef put} for |
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909 dup 85 /U put |
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910 readonly def |
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911 /FontBBox{-8 -194 963 738}readonly def |
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912 /UniqueID 5031988 def |
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913 currentdict end |
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914 currentfile eexec |
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958 0000000000000000000000000000000000000000000000000000000000000000 |
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961 0000000000000000000000000000000000000000000000000000000000000000 |
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963 cleartomark |
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964 %%EndFont |
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965 %%BeginFont: MSBM10 |
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966 %!PS-AdobeFont-1.1: MSBM10 2.1 |
|
967 %%CreationDate: 1993 Sep 17 11:10:37 |
|
968 % Math Symbol fonts were designed by the American Mathematical Society. |
|
969 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
970 11 dict begin |
|
971 /FontInfo 7 dict dup begin |
|
972 /version (2.1) readonly def |
|
973 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
974 /FullName (MSBM10) readonly def |
|
975 /FamilyName (Euler) readonly def |
|
976 /Weight (Medium) readonly def |
|
977 /ItalicAngle 0 def |
|
978 /isFixedPitch false def |
|
979 end readonly def |
|
980 /FontName /MSBM10 def |
|
981 /PaintType 0 def |
|
982 /FontType 1 def |
|
983 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
984 /Encoding 256 array |
|
985 0 1 255 {1 index exch /.notdef put} for |
|
986 dup 63 /emptyset put |
|
987 readonly def |
|
988 /FontBBox{-55 -420 2343 920}readonly def |
|
989 /UniqueID 5031982 def |
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990 currentdict end |
|
991 currentfile eexec |
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1016 90CCFAADF118304646955B0310C65F6CA51BEEEF87AFFE294D08C443636DF75B |
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1021 3EAA1D1B61597E6AAB0C0CE342E7332A947102F48F28D169AE3575F8EC625964 |
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1022 520ACB3FFF574B9D39D7BA6CEAD0ACE429F4F147B848FD26C3C2E187CF76E7A6 |
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1023 FE0CDD9A53C3E4586B1C252E056BF41347B2B8A28510B8E53183D24F5B758BF9 |
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1024 2A4B744E2DFE23911B0D40A6AA8559D89128A29EE51A806F533B599CA94C44EB |
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1025 3EEBB6304B96F2FCEBCE45929D139C3107DC10ED807E1D8B5229216787C20B3E |
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1027 67DE12C994F006556E99AB540962208FAB61 |
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1028 0000000000000000000000000000000000000000000000000000000000000000 |
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1029 0000000000000000000000000000000000000000000000000000000000000000 |
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1030 0000000000000000000000000000000000000000000000000000000000000000 |
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1032 0000000000000000000000000000000000000000000000000000000000000000 |
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1033 0000000000000000000000000000000000000000000000000000000000000000 |
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1034 0000000000000000000000000000000000000000000000000000000000000000 |
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1035 0000000000000000000000000000000000000000000000000000000000000000 |
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1036 cleartomark |
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1037 %%EndFont |
|
1038 %%BeginFont: CMMI5 |
|
1039 %!PS-AdobeFont-1.1: CMMI5 1.100 |
|
1040 %%CreationDate: 1996 Aug 02 08:21:10 |
|
1041 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1042 11 dict begin |
|
1043 /FontInfo 7 dict dup begin |
|
1044 /version (1.100) readonly def |
|
1045 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1046 /FullName (CMMI5) readonly def |
|
1047 /FamilyName (Computer Modern) readonly def |
|
1048 /Weight (Medium) readonly def |
|
1049 /ItalicAngle -14.04 def |
|
1050 /isFixedPitch false def |
|
1051 end readonly def |
|
1052 /FontName /CMMI5 def |
|
1053 /PaintType 0 def |
|
1054 /FontType 1 def |
|
1055 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1056 /Encoding 256 array |
|
1057 0 1 255 {1 index exch /.notdef put} for |
|
1058 dup 0 /.notdef put |
|
1059 readonly def |
|
1060 /FontBBox{37 -250 1349 750}readonly def |
|
1061 /UniqueID 5087380 def |
|
1062 currentdict end |
|
1063 currentfile eexec |
|
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1100 98F74525CD02C031B917B0F3B713F779C6F688A6DACCBA0A7CB924C1F4B6E52C |
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1101 D1A52A2216B92AAB440499E8EA04033624E9DA018543FD401BD38496FC59ABBA |
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1102 4C19614E0072FCDE2FD770EA52807278F203F0703AFFA233833AA63FF44812AB |
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1103 CAA3A659434A08E17123CFC2CEC39C706212FF8EB2FA4A1BD87D6FFB8EE2CA2E |
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1104 FA9192DAE432B288FAC7E3FB423B8D1BD5B2D81FAE1047EA843971F47BBE9E4D |
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1105 182FF0D3959C6328D632FE1C1DEE744A50D54C25AC7129FA1EFCC4A2A51BB055 |
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1106 4BA5DC6C5558A0DED3A9EB3B70A71EE14C762FC564A6E05613CF37CC4DE74B17 |
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1107 727DBB90B3EE742F91D5D9A62898970398DC953B5352DD8ECAFE2C7721D1FB1A |
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1108 8800A3E0D0025890BBFF0E36B04134226BC5698D146232A1247E128A8AB77E19 |
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1109 F4E35082B473A91F0B28336BFE89EF6DFCAEEB4ED013BDBFBD20763A448C71F6 |
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1110 7F8791B394432D797FD77058483F4AFB37BF8FED18CDABE950320B89557200C8 |
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1111 0504241C52F215F487F5A917DC641312170746806D4555EA5EFC5E6B1C3E48B9 |
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1112 0E93C9400B1334FB62E82F0C956EF81E6702A933193D66B49471181242 |
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1113 0000000000000000000000000000000000000000000000000000000000000000 |
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1114 0000000000000000000000000000000000000000000000000000000000000000 |
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1115 0000000000000000000000000000000000000000000000000000000000000000 |
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1116 0000000000000000000000000000000000000000000000000000000000000000 |
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1117 0000000000000000000000000000000000000000000000000000000000000000 |
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1118 0000000000000000000000000000000000000000000000000000000000000000 |
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1119 0000000000000000000000000000000000000000000000000000000000000000 |
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1120 0000000000000000000000000000000000000000000000000000000000000000 |
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1121 cleartomark |
|
1122 %%EndFont |
|
1123 %%BeginFont: CMR5 |
|
1124 %!PS-AdobeFont-1.1: CMR5 1.00B |
|
1125 %%CreationDate: 1992 Feb 19 19:55:02 |
|
1126 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1127 11 dict begin |
|
1128 /FontInfo 7 dict dup begin |
|
1129 /version (1.00B) readonly def |
|
1130 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1131 /FullName (CMR5) readonly def |
|
1132 /FamilyName (Computer Modern) readonly def |
|
1133 /Weight (Medium) readonly def |
|
1134 /ItalicAngle 0 def |
|
1135 /isFixedPitch false def |
|
1136 end readonly def |
|
1137 /FontName /CMR5 def |
|
1138 /PaintType 0 def |
|
1139 /FontType 1 def |
|
1140 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1141 /Encoding 256 array |
|
1142 0 1 255 {1 index exch /.notdef put} for |
|
1143 dup 0 /.notdef put |
|
1144 readonly def |
|
1145 /FontBBox{-341 -250 1304 965}readonly def |
|
1146 /UniqueID 5000788 def |
|
1147 currentdict end |
|
1148 currentfile eexec |
|
1149 D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 |
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1150 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 |
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1151 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F |
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1152 D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 |
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1155 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F |
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1156 D1F017CE45884D76EF2CB9BC5821FD25365DDEA1F9B0FF4CFF25B8E64D0747A3 |
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1166 FB3B7CA10E1BF26E44C24E1696CB089CB0055BD692C89B237CF269F77A31DC81 |
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1167 0F4B75C8400ABCFDCEC6443CD0E81871CD71AA3064ABDE882C4C52322C27FA8B |
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1168 41C689F827FB0F8AAF8022CF3C1F41C0B45601190C1328831857CBF9B1E7D1AA |
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1169 246117E56D6B7938488055F4E63E2A1C8D57C17D213729C68349FEC2C3466F41 |
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1170 171E00413D39DF1F67BC15912F30775AFDF7FB3312587E20A68CF77AD3906040 |
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1171 842D63C45E19278622DD228C18ABDD024DD9613CDC0B109095DB0ADC3A3C0CB5 |
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1172 AB597D490189EA81239E39202CBC7A829EB9B313A8F962F7879D374ADF529BD0 |
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1174 D961E1362D29A5F4C3393CEA88D53E01D0FDAE7050612947AD1B04F42B0F3B0C |
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1175 4ECD493606D6321E7773557228E0C71A0C5EEB809E9853FCD689BFE16A61E8BF |
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1176 D7E8683252EAE940B67546EF86DAD7CB9D786603060FDA494D3297F3F70864DD |
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1177 1C37DA22110220A988706B11409E2F475F486C749DDCF8199C25965925D091D1 |
|
1178 819EB866D2CE64D29464F7A6045778F1740A06610F1DD94F2A4EE817CAFAF919 |
|
1179 8D241C7C3A7A4BE1016C9A8BA70ED76B436646786E4CE058A6B056B9BF766110 |
|
1180 41EE4F93833911E1EAE9B21DF6925E77E2746E2FA82F76494E50D61E33B160DA |
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1181 780549CEF58DE177E36F5858137038EEBBE70A3A26B51508291B3049E2D392CF |
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1182 3E |
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1183 0000000000000000000000000000000000000000000000000000000000000000 |
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1184 0000000000000000000000000000000000000000000000000000000000000000 |
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1185 0000000000000000000000000000000000000000000000000000000000000000 |
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1186 0000000000000000000000000000000000000000000000000000000000000000 |
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1187 0000000000000000000000000000000000000000000000000000000000000000 |
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1188 0000000000000000000000000000000000000000000000000000000000000000 |
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1189 0000000000000000000000000000000000000000000000000000000000000000 |
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1190 0000000000000000000000000000000000000000000000000000000000000000 |
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1191 cleartomark |
|
1192 %%EndFont |
|
1193 %%BeginFont: CMMI6 |
|
1194 %!PS-AdobeFont-1.1: CMMI6 1.100 |
|
1195 %%CreationDate: 1996 Jul 23 07:53:52 |
|
1196 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1197 11 dict begin |
|
1198 /FontInfo 7 dict dup begin |
|
1199 /version (1.100) readonly def |
|
1200 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1201 /FullName (CMMI6) readonly def |
|
1202 /FamilyName (Computer Modern) readonly def |
|
1203 /Weight (Medium) readonly def |
|
1204 /ItalicAngle -14.04 def |
|
1205 /isFixedPitch false def |
|
1206 end readonly def |
|
1207 /FontName /CMMI6 def |
|
1208 /PaintType 0 def |
|
1209 /FontType 1 def |
|
1210 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1211 /Encoding 256 array |
|
1212 0 1 255 {1 index exch /.notdef put} for |
|
1213 dup 0 /.notdef put |
|
1214 readonly def |
|
1215 /FontBBox{11 -250 1241 750}readonly def |
|
1216 /UniqueID 5087381 def |
|
1217 currentdict end |
|
1218 currentfile eexec |
|
1219 D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE |
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1221 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 |
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1222 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B |
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1224 D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 |
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1225 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC |
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1226 4391C9DF440285B8FC159D0E98D4258FC57892DDF0342CA1080743A076089583 |
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1227 6AD6FB2DC4C13F077F17789476E48402796E685107AF60A63FB0DE0266D55CF1 |
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1228 8D0AD65B9342CB686E564758C96164FFA711B11C1CE8C726F3C7BB1044BBD283 |
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1229 9AA4675747DF61E130A55E297CA5F0182A3F12F9085AF2F503481071724077A9 |
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1230 387E27879A9649AD5F186F33500FAC8F7FA26634BDCE1221EC0ED0E359E5EA5E |
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1231 6166526FEB90C30D30099FBDC1BC2F9B62EFEEC48345160804AA98F8D0AA54B7 |
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1232 A480E715426651865C8E444EDB798C7E11040AF6E5A7ED1888653C6DBF5E6169 |
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1233 70BCD9C063B63B561EF165BF3AF11F8E519F37C6FDA2827685739DE2C48B5ADE |
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1234 EE84F067D704D4511DBFA49E166D543CFD9ECD7417055D8A827F51E087CD2927 |
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1235 BAFC7E6CFBD70B0FE969F890A11149D3D44D422C3370495DA9951AEE7253A49F |
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1236 3A9444C8CD9158D84117299F7F2332FEB0F94E6ED8BC7AA789A3219BC2F227D3 |
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1242 C06C71BFC0A2F3AC9951CFBB143C66FB84F9C4ED27DF70869352D61BD5E11508 |
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1243 0797B87C709E3C151EB44E478CA576D257DF226C00BEF712425856AF0377038F |
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1245 D383A2BB84CA8D0ACABE15ADE0A4E16A21C67B63701C22EAB3D4965FAD4B4013 |
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1247 ED93D7DC7E66828533B1E89492378286A329DD59517B9AA2AA50BA846F47514F |
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1250 F5E66146FEA53F33D7F98EFCAFA2B74E40BEA16A9956004AB554A50FED460D4D |
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1252 E18065C5FBEC0A5A22C6AD7764BAB978E5190881A388DF7BFBC664129B099686 |
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1253 A17652E7EB5DDBDF557CFC0A93F8B703EA7962D8F1343158359FB63C893F8E7C |
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1254 D32BFA93B63A9FF4CA03CFAA0CAD0F1A113A4626B5DA09148349C90F68DD0D3D |
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1255 749B258484F6E383A21C2DBA7278685C29026D43E9B76FBA07A1B448627A489D |
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1256 3500A4217A45F355845308382219E5C66F39518405354F4A32B3237878BB73B6 |
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1257 614CE0A2276709B17605BE8DADF01E261141E18C807F5426DA9857EBAD9F36C4 |
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1258 077C32E5495F7B004917FDC05135E1C066F1618621DD643B190CC07EFCE250DA |
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1259 6C48 |
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1260 0000000000000000000000000000000000000000000000000000000000000000 |
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1261 0000000000000000000000000000000000000000000000000000000000000000 |
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1262 0000000000000000000000000000000000000000000000000000000000000000 |
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1263 0000000000000000000000000000000000000000000000000000000000000000 |
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1264 0000000000000000000000000000000000000000000000000000000000000000 |
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1265 0000000000000000000000000000000000000000000000000000000000000000 |
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1266 0000000000000000000000000000000000000000000000000000000000000000 |
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1267 0000000000000000000000000000000000000000000000000000000000000000 |
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1268 cleartomark |
|
1269 %%EndFont |
|
1270 %%BeginFont: CMMIB10 |
|
1271 %!PS-AdobeFont-1.1: CMMIB10 1.100 |
|
1272 %%CreationDate: 1996 Jul 23 07:54:00 |
|
1273 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1274 11 dict begin |
|
1275 /FontInfo 7 dict dup begin |
|
1276 /version (1.100) readonly def |
|
1277 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1278 /FullName (CMMIB10) readonly def |
|
1279 /FamilyName (Computer Modern) readonly def |
|
1280 /Weight (Bold) readonly def |
|
1281 /ItalicAngle -14.04 def |
|
1282 /isFixedPitch false def |
|
1283 end readonly def |
|
1284 /FontName /CMMIB10 def |
|
1285 /PaintType 0 def |
|
1286 /FontType 1 def |
|
1287 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1288 /Encoding 256 array |
|
1289 0 1 255 {1 index exch /.notdef put} for |
|
1290 dup 0 /.notdef put |
|
1291 readonly def |
|
1292 /FontBBox{-15 -250 1216 750}readonly def |
|
1293 /UniqueID 5087392 def |
|
1294 currentdict end |
|
1295 currentfile eexec |
|
1296 D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE |
|
1297 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B |
|
1298 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 |
|
1299 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B |
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1300 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE |
|
1301 D919C2DDD26BDC0D99398B9F4D004B836D34E88C25F6CE738846C8E2E59A2BCF |
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1302 4ACF80A26D78872E9343A0537BC3BD7715F32ACD958D5AAED865BFE129278935 |
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1303 063A31C2634DE2F9077E0AAAAEB224466B779096D8E3FF0A12AD5157F6603DED |
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1304 1A82F3511359143311179080C510740B401C930C96270FD1AB3ECBCFEF5DE53F |
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1305 E846BAAE95828D5790922640EF8AB9D7CEBE7669FEA02B72F86872D3D8754A18 |
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1306 A1629C40A7C00C956F140BC63362478279C36EE353638CD3E249897207A94504 |
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1308 CEE3671644A75F1E696DA906B4C66870DBE87F5B4A176920C078ADBE24F55C09 |
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1309 3D18CDE21B5FBC1C6A8AB18E05EDBEF9D1C1C18B3E6377BA2A688579D4F708F9 |
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1310 A5CF4F56C5E39E2726106E9713E638775E606464CD674E5DC25CE9A696A65806 |
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1320 E993D5A48AF0019A459ECEEF20675D31E556D7BA13264C8FC790D9545A587CA2 |
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|
1324 A6E8865864634E7BB61C7B5DE2DCF13325A2517ADAEB75CA77A08EDA6BD7871B |
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|
1326 F0FD6DC1855D05370D147FE21BA2A766DE7D262DE8006047E457F31CBDDEDAFB |
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1327 1B32A386337EEA85547406B6BFFA4D1AABAA32858B7CEFAA5F931D3FDCF6025F |
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1328 E72ECFB5CB786E721AB28979D23B69020A7E7D3FDE3F5137EE53A36B77D37A4A |
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1330 CB11569735C43DBE0D45A874FB3171407B8206DD1BFF73901E67FF0B5D6870A3 |
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1331 B1BD280224C3E590B4A8F6E36095F1487CEBC180B9AE94496A6344FA5C996B0A |
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1332 A905513C12AA5D3815CB4057D33221CA48D28660A05AD6E6B0E118178595A51A |
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1333 37A3B0E57D87E18B446E4470AED95C7BDB924559C6D2096CA9572A790E13A301 |
|
1334 52A28D983E1A7CDE3AD9BDCB7D837CFCFE9D8848C3D64176397D57AF80D67779 |
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1335 EA1D62300D5D57FDDC470F36431DC140 |
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1336 0000000000000000000000000000000000000000000000000000000000000000 |
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1337 0000000000000000000000000000000000000000000000000000000000000000 |
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1338 0000000000000000000000000000000000000000000000000000000000000000 |
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1339 0000000000000000000000000000000000000000000000000000000000000000 |
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1340 0000000000000000000000000000000000000000000000000000000000000000 |
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1341 0000000000000000000000000000000000000000000000000000000000000000 |
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1342 0000000000000000000000000000000000000000000000000000000000000000 |
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1343 0000000000000000000000000000000000000000000000000000000000000000 |
|
1344 cleartomark |
|
1345 %%EndFont |
|
1346 %%BeginFont: CMBX10 |
|
1347 %!PS-AdobeFont-1.1: CMBX10 1.00B |
|
1348 %%CreationDate: 1992 Feb 19 19:54:06 |
|
1349 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1350 11 dict begin |
|
1351 /FontInfo 7 dict dup begin |
|
1352 /version (1.00B) readonly def |
|
1353 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1354 /FullName (CMBX10) readonly def |
|
1355 /FamilyName (Computer Modern) readonly def |
|
1356 /Weight (Bold) readonly def |
|
1357 /ItalicAngle 0 def |
|
1358 /isFixedPitch false def |
|
1359 end readonly def |
|
1360 /FontName /CMBX10 def |
|
1361 /PaintType 0 def |
|
1362 /FontType 1 def |
|
1363 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1364 /Encoding 256 array |
|
1365 0 1 255 {1 index exch /.notdef put} for |
|
1366 dup 0 /.notdef put |
|
1367 readonly def |
|
1368 /FontBBox{-301 -250 1164 946}readonly def |
|
1369 /UniqueID 5000768 def |
|
1370 currentdict end |
|
1371 currentfile eexec |
|
1372 D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 |
|
1373 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 |
|
1374 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F |
|
1375 D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 |
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1418 0000000000000000000000000000000000000000000000000000000000000000 |
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1419 cleartomark |
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1420 %%EndFont |
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1421 %%BeginFont: CMBXTI10 |
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1422 %!PS-AdobeFont-1.1: CMBXTI10 1.0 |
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1423 %%CreationDate: 1991 Aug 18 17:46:30 |
|
1424 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1425 11 dict begin |
|
1426 /FontInfo 7 dict dup begin |
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1427 /version (1.0) readonly def |
|
1428 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1429 /FullName (CMBXTI10) readonly def |
|
1430 /FamilyName (Computer Modern) readonly def |
|
1431 /Weight (Bold) readonly def |
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1432 /ItalicAngle -14.04 def |
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1433 /isFixedPitch false def |
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1434 end readonly def |
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1435 /FontName /CMBXTI10 def |
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1436 /PaintType 0 def |
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1437 /FontType 1 def |
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1438 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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1439 /Encoding 256 array |
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1440 0 1 255 {1 index exch /.notdef put} for |
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1441 dup 0 /.notdef put |
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1442 readonly def |
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1445 currentdict end |
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1502 0000000000000000000000000000000000000000000000000000000000000000 |
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1503 cleartomark |
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1504 %%EndFont |
|
1505 %%BeginFont: CMR8 |
|
1506 %!PS-AdobeFont-1.1: CMR8 1.0 |
|
1507 %%CreationDate: 1991 Aug 20 16:39:40 |
|
1508 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1509 11 dict begin |
|
1510 /FontInfo 7 dict dup begin |
|
1511 /version (1.0) readonly def |
|
1512 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1513 /FullName (CMR8) readonly def |
|
1514 /FamilyName (Computer Modern) readonly def |
|
1515 /Weight (Medium) readonly def |
|
1516 /ItalicAngle 0 def |
|
1517 /isFixedPitch false def |
|
1518 end readonly def |
|
1519 /FontName /CMR8 def |
|
1520 /PaintType 0 def |
|
1521 /FontType 1 def |
|
1522 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1523 /Encoding 256 array |
|
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|
1525 dup 0 /.notdef put |
|
1526 readonly def |
|
1527 /FontBBox{-36 -250 1070 750}readonly def |
|
1528 /UniqueID 5000791 def |
|
1529 currentdict end |
|
1530 currentfile eexec |
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1617 0000000000000000000000000000000000000000000000000000000000000000 |
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1618 cleartomark |
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1619 %%EndFont |
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1620 %%BeginFont: CMSY8 |
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1621 %!PS-AdobeFont-1.1: CMSY8 1.0 |
|
1622 %%CreationDate: 1991 Aug 15 07:22:10 |
|
1623 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1624 11 dict begin |
|
1625 /FontInfo 7 dict dup begin |
|
1626 /version (1.0) readonly def |
|
1627 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1628 /FullName (CMSY8) readonly def |
|
1629 /FamilyName (Computer Modern) readonly def |
|
1630 /Weight (Medium) readonly def |
|
1631 /ItalicAngle -14.035 def |
|
1632 /isFixedPitch false def |
|
1633 end readonly def |
|
1634 /FontName /CMSY8 def |
|
1635 /PaintType 0 def |
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1636 /FontType 1 def |
|
1637 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1638 /Encoding 256 array |
|
1639 0 1 255 {1 index exch /.notdef put} for |
|
1640 dup 0 /.notdef put |
|
1641 readonly def |
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1643 /UniqueID 5000818 def |
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1644 currentdict end |
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1645 currentfile eexec |
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1700 9A47B3B78B769CC9AE80E81EBAD96AC8182E1CEB0E |
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1701 0000000000000000000000000000000000000000000000000000000000000000 |
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1703 0000000000000000000000000000000000000000000000000000000000000000 |
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1704 0000000000000000000000000000000000000000000000000000000000000000 |
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1705 0000000000000000000000000000000000000000000000000000000000000000 |
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1706 0000000000000000000000000000000000000000000000000000000000000000 |
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1707 0000000000000000000000000000000000000000000000000000000000000000 |
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1708 0000000000000000000000000000000000000000000000000000000000000000 |
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1709 cleartomark |
|
1710 %%EndFont |
|
1711 %%BeginFont: CMSY6 |
|
1712 %!PS-AdobeFont-1.1: CMSY6 1.0 |
|
1713 %%CreationDate: 1991 Aug 15 07:21:34 |
|
1714 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1715 11 dict begin |
|
1716 /FontInfo 7 dict dup begin |
|
1717 /version (1.0) readonly def |
|
1718 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1719 /FullName (CMSY6) readonly def |
|
1720 /FamilyName (Computer Modern) readonly def |
|
1721 /Weight (Medium) readonly def |
|
1722 /ItalicAngle -14.035 def |
|
1723 /isFixedPitch false def |
|
1724 end readonly def |
|
1725 /FontName /CMSY6 def |
|
1726 /PaintType 0 def |
|
1727 /FontType 1 def |
|
1728 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1729 /Encoding 256 array |
|
1730 0 1 255 {1 index exch /.notdef put} for |
|
1731 dup 0 /.notdef put |
|
1732 readonly def |
|
1733 /FontBBox{-4 -948 1329 786}readonly def |
|
1734 /UniqueID 5000816 def |
|
1735 currentdict end |
|
1736 currentfile eexec |
|
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1774 cleartomark |
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1775 %%EndFont |
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1776 %%BeginFont: TeX-cmex9 |
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1777 %!PS-AdobeFont-1.0: TeX-cmex9 001.001 |
|
1778 % Filtered by type1fix.pl 0.05 |
|
1779 %%EndComments |
|
1780 13 dict dup begin |
|
1781 /FontInfo 16 dict dup begin |
|
1782 /Copyright (see\040copyright\040of\040original\040TeX\040font) def |
|
1783 /FamilyName (TeX\040cmex9) def |
|
1784 /FullName (TeX\040cmex9\040Regular) def |
|
1785 /ItalicAngle 0 def |
|
1786 /Notice (converted\040after\040April\0402001) def |
|
1787 /UnderlinePosition -100 def |
|
1788 /UnderlineThickness 50 def |
|
1789 /Weight (Regular) def |
|
1790 /isFixedPitch false def |
|
1791 /version (001.001) def |
|
1792 end readonly def |
|
1793 /FontName /TeX-cmex9 def |
|
1794 /PaintType 0 def |
|
1795 /FontType 1 def |
|
1796 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1797 /Encoding 256 array |
|
1798 0 1 255 {1 index exch /.notdef put} for |
|
1799 dup 27 /ff put |
|
1800 dup 56 /eight put |
|
1801 dup 58 /colon put |
|
1802 dup 60 /less put |
|
1803 dup 62 /greater put |
|
1804 readonly def |
|
1805 /FontBBox {-26 -2961 1503 773} readonly def |
|
1806 /UniqueID 4314405 def |
|
1807 currentdict end |
|
1808 currentfile eexec |
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1871 0000000000000000000000000000000000000000000000000000000000000000 |
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1872 cleartomark |
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1873 %%EndFont |
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1874 %%BeginFont: CMBSY10 |
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1875 %!PS-AdobeFont-1.1: CMBSY10 1.00 |
|
1876 %%CreationDate: 1992 Jul 23 21:21:18 |
|
1877 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1878 11 dict begin |
|
1879 /FontInfo 7 dict dup begin |
|
1880 /version (1.00) readonly def |
|
1881 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1882 /FullName (CMBSY10) readonly def |
|
1883 /FamilyName (Computer Modern) readonly def |
|
1884 /Weight (Bold) readonly def |
|
1885 /ItalicAngle -14.035 def |
|
1886 /isFixedPitch false def |
|
1887 end readonly def |
|
1888 /FontName /CMBSY10 def |
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1889 /PaintType 0 def |
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1890 /FontType 1 def |
|
1891 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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1892 /Encoding 256 array |
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1893 0 1 255 {1 index exch /.notdef put} for |
|
1894 dup 1 /periodcentered put |
|
1895 readonly def |
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1896 /FontBBox{-27 -940 1332 825}readonly def |
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1897 /UniqueID 5000762 def |
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1898 currentdict end |
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1899 currentfile eexec |
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1929 85DA381440E32FFF9546B9BFD2B14508D42F198C975076E2269C8B2BBF1AE20E |
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1930 C463B0EBE68BEF1F29F27E86E7600E0A7ECF879B5350A8B74101625D3DDDAC09 |
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1931 083BCA5E10DACF |
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1941 %%EndFont |
|
1942 %%BeginFont: CMTI10 |
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1943 %!PS-AdobeFont-1.1: CMTI10 1.00B |
|
1944 %%CreationDate: 1992 Feb 19 19:56:16 |
|
1945 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1946 11 dict begin |
|
1947 /FontInfo 7 dict dup begin |
|
1948 /version (1.00B) readonly def |
|
1949 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1950 /FullName (CMTI10) readonly def |
|
1951 /FamilyName (Computer Modern) readonly def |
|
1952 /Weight (Medium) readonly def |
|
1953 /ItalicAngle -14.04 def |
|
1954 /isFixedPitch false def |
|
1955 end readonly def |
|
1956 /FontName /CMTI10 def |
|
1957 /PaintType 0 def |
|
1958 /FontType 1 def |
|
1959 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1960 /Encoding 256 array |
|
1961 0 1 255 {1 index exch /.notdef put} for |
|
1962 dup 0 /.notdef put |
|
1963 readonly def |
|
1964 /FontBBox{-163 -250 1146 969}readonly def |
|
1965 /UniqueID 5000828 def |
|
1966 currentdict end |
|
1967 currentfile eexec |
|
1968 D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE |
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1969 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B |
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1971 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B |
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1973 D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 |
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1974 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F |
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1975 21EC3643079EECE0C9AB54B4772E5DCA82D0D4ACC7F42FB493AA04A3BF4A1BD6 |
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1976 06ECE186315DBE9CFDCB1A0303E8D3E83027CD3AFA8F0BD466A8E8CA0E7164CF |
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1980 1485831624D24F37C39BF9972D74E6EC4784727AC00B9C4A3AD3DA1C22BD6961 |
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1981 7E0ADAF55422F22ACA5E4DCD4DF9FCD187A566B7FB661D0530454D0DD6C6C50A |
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1982 7A3875C6CBF8EC7769F32A1F3F7FC1C072BADEC97794D4E90E0035282A170402 |
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1983 356E5A9CD9ABD80AC4342A5283E458A7269252F4541CBB6452B39ED54D336D0B |
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1984 19928E9CD1AB26AD83EB209E2EC75011A2643813053B5DBB0246097C4821B5F2 |
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1985 C92554E9140BE35B2DBFCD98809A8EC9FC910FDE9E0D86457C70ACB056EBF90F |
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1986 244DC0A5BBD455E15D6E3180311D52CF50B0BF7D0A7F64F3A1821E0AEDBC2E7B |
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1987 AEB549FE1D51088C153799C6E089B5D5D65E1C4E2D2B430CDF1FFA23CCB25D95 |
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1989 C2144CFE49A9AD892D4D31764F0AE3A10644AE3966B0A790684B14D11FA49785 |
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1990 EC5565D2B2E584CBFD85125F3FAC133338DE35361943DCE9AF05FCF2840CE512 |
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1998 E768F27D74A613F3ABF883222A8596B542EBF54E9DCE327B5682AEE5F1A7A225 |
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1999 BF26E2AEA0F352B9C950B47ADF650E1B2AE31E883EDD884EC90F94761C470EB1 |
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2001 5EE89221BFD28AE8117B69C318A5EF626F95048D62CEA5EC1E8EFD8955D3ABD8 |
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2002 0A98F103318B58ACC4D520AB607550EB6E430CF787310F2A00E75E6399514B04 |
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2004 95E12BCFB23A029E0245411CA7389B00CAC4E4EB98B35C5EABFDDFC75C04CCDF |
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2005 522B50526860CE41E607EA50F7A17396A67407302E624FEC49C59A30EA1F4843 |
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2007 282903CA6390DBA8DDE0025055CAA90D4A1170C7FD891FE699D55AA42FF93D2C |
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2009 A67C0B5AF6D103A67BD25EA16E4A5EA8C73FFB7188E67C7373AE2B6F045CD2AE |
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2012 87169D2824F8112DEFB5000FB6C5BE298156D057E2F75F123543C098B1E865A0 |
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2013 340B298CC6A373549E762EFE3495FFA4103130A20FBAEECBCB5A59B7E2D3799C |
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2014 3D6D0B692F14578E0E74CAD6BDD04297EF2E2BEDD821A81A60F1CC03F0985943 |
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2015 B520F940E71CBA0944800046BE533A0C55DF4D7FA9FF28A2404BDF50F5EFEEF8 |
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2016 B64E4EFBA2F41983B1AE564D9E192D61C5E2BDDF439DA6AF7FE4A2944D7A2633 |
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2017 C373D22C1776F84D9A74FA21A6CA78CE70C29BE2F3EF83C08FAFF81EBDB9BE24 |
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2018 2FE44526DE800CEC0E80A78021FA8D6578FAEAB3A588EAE6568E484E4BE7347D |
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2085 cleartomark |
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2086 %%EndFont |
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2087 %%BeginFont: CMSY9 |
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2088 %!PS-AdobeFont-1.1: CMSY9 1.0 |
|
2089 %%CreationDate: 1991 Aug 15 07:22:27 |
|
2090 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2091 11 dict begin |
|
2092 /FontInfo 7 dict dup begin |
|
2093 /version (1.0) readonly def |
|
2094 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2095 /FullName (CMSY9) readonly def |
|
2096 /FamilyName (Computer Modern) readonly def |
|
2097 /Weight (Medium) readonly def |
|
2098 /ItalicAngle -14.035 def |
|
2099 /isFixedPitch false def |
|
2100 end readonly def |
|
2101 /FontName /CMSY9 def |
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2102 /PaintType 0 def |
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2103 /FontType 1 def |
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2104 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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2105 /Encoding 256 array |
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2106 0 1 255 {1 index exch /.notdef put} for |
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2107 dup 0 /.notdef put |
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2108 readonly def |
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2111 currentdict end |
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2211 0000000000000000000000000000000000000000000000000000000000000000 |
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2212 0000000000000000000000000000000000000000000000000000000000000000 |
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2213 0000000000000000000000000000000000000000000000000000000000000000 |
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2214 0000000000000000000000000000000000000000000000000000000000000000 |
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2215 cleartomark |
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2216 %%EndFont |
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2217 %%BeginFont: CMTT9 |
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2218 %!PS-AdobeFont-1.1: CMTT9 1.0 |
|
2219 %%CreationDate: 1991 Aug 20 16:46:24 |
|
2220 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2221 11 dict begin |
|
2222 /FontInfo 7 dict dup begin |
|
2223 /version (1.0) readonly def |
|
2224 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2225 /FullName (CMTT9) readonly def |
|
2226 /FamilyName (Computer Modern) readonly def |
|
2227 /Weight (Medium) readonly def |
|
2228 /ItalicAngle 0 def |
|
2229 /isFixedPitch true def |
|
2230 end readonly def |
|
2231 /FontName /CMTT9 def |
|
2232 /PaintType 0 def |
|
2233 /FontType 1 def |
|
2234 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2235 /Encoding 256 array |
|
2236 0 1 255 {1 index exch /.notdef put} for |
|
2237 dup 0 /.notdef put |
|
2238 readonly def |
|
2239 /FontBBox{-6 -233 542 698}readonly def |
|
2240 /UniqueID 5000831 def |
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2241 currentdict end |
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2242 currentfile eexec |
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2353 2A4D |
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2356 0000000000000000000000000000000000000000000000000000000000000000 |
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2357 0000000000000000000000000000000000000000000000000000000000000000 |
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2358 0000000000000000000000000000000000000000000000000000000000000000 |
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2359 0000000000000000000000000000000000000000000000000000000000000000 |
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2360 0000000000000000000000000000000000000000000000000000000000000000 |
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2361 0000000000000000000000000000000000000000000000000000000000000000 |
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2362 cleartomark |
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2363 %%EndFont |
|
2364 %%BeginFont: CMTI9 |
|
2365 %!PS-AdobeFont-1.1: CMTI9 1.0 |
|
2366 %%CreationDate: 1991 Aug 18 21:08:07 |
|
2367 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2368 11 dict begin |
|
2369 /FontInfo 7 dict dup begin |
|
2370 /version (1.0) readonly def |
|
2371 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2372 /FullName (CMTI9) readonly def |
|
2373 /FamilyName (Computer Modern) readonly def |
|
2374 /Weight (Medium) readonly def |
|
2375 /ItalicAngle -14.04 def |
|
2376 /isFixedPitch false def |
|
2377 end readonly def |
|
2378 /FontName /CMTI9 def |
|
2379 /PaintType 0 def |
|
2380 /FontType 1 def |
|
2381 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2382 /Encoding 256 array |
|
2383 0 1 255 {1 index exch /.notdef put} for |
|
2384 dup 0 /.notdef put |
|
2385 readonly def |
|
2386 /FontBBox{-35 -250 1148 750}readonly def |
|
2387 /UniqueID 5000827 def |
|
2388 currentdict end |
|
2389 currentfile eexec |
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2467 0000000000000000000000000000000000000000000000000000000000000000 |
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2469 0000000000000000000000000000000000000000000000000000000000000000 |
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2470 0000000000000000000000000000000000000000000000000000000000000000 |
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2471 cleartomark |
|
2472 %%EndFont |
|
2473 %%BeginFont: CMR9 |
|
2474 %!PS-AdobeFont-1.1: CMR9 1.0 |
|
2475 %%CreationDate: 1991 Aug 20 16:39:59 |
|
2476 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2477 11 dict begin |
|
2478 /FontInfo 7 dict dup begin |
|
2479 /version (1.0) readonly def |
|
2480 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2481 /FullName (CMR9) readonly def |
|
2482 /FamilyName (Computer Modern) readonly def |
|
2483 /Weight (Medium) readonly def |
|
2484 /ItalicAngle 0 def |
|
2485 /isFixedPitch false def |
|
2486 end readonly def |
|
2487 /FontName /CMR9 def |
|
2488 /PaintType 0 def |
|
2489 /FontType 1 def |
|
2490 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2491 /Encoding 256 array |
|
2492 0 1 255 {1 index exch /.notdef put} for |
|
2493 dup 0 /.notdef put |
|
2494 readonly def |
|
2495 /FontBBox{-39 -250 1036 750}readonly def |
|
2496 /UniqueID 5000792 def |
|
2497 currentdict end |
|
2498 currentfile eexec |
|
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2569 3B710D801DFD60E1ED36 |
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2570 0000000000000000000000000000000000000000000000000000000000000000 |
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2571 0000000000000000000000000000000000000000000000000000000000000000 |
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2572 0000000000000000000000000000000000000000000000000000000000000000 |
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2573 0000000000000000000000000000000000000000000000000000000000000000 |
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2574 0000000000000000000000000000000000000000000000000000000000000000 |
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2575 0000000000000000000000000000000000000000000000000000000000000000 |
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2576 0000000000000000000000000000000000000000000000000000000000000000 |
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2577 0000000000000000000000000000000000000000000000000000000000000000 |
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2578 cleartomark |
|
2579 %%EndFont |
|
2580 %%BeginFont: CMR6 |
|
2581 %!PS-AdobeFont-1.1: CMR6 1.0 |
|
2582 %%CreationDate: 1991 Aug 20 16:39:02 |
|
2583 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2584 11 dict begin |
|
2585 /FontInfo 7 dict dup begin |
|
2586 /version (1.0) readonly def |
|
2587 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2588 /FullName (CMR6) readonly def |
|
2589 /FamilyName (Computer Modern) readonly def |
|
2590 /Weight (Medium) readonly def |
|
2591 /ItalicAngle 0 def |
|
2592 /isFixedPitch false def |
|
2593 end readonly def |
|
2594 /FontName /CMR6 def |
|
2595 /PaintType 0 def |
|
2596 /FontType 1 def |
|
2597 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2598 /Encoding 256 array |
|
2599 0 1 255 {1 index exch /.notdef put} for |
|
2600 dup 0 /.notdef put |
|
2601 readonly def |
|
2602 /FontBBox{-20 -250 1193 750}readonly def |
|
2603 /UniqueID 5000789 def |
|
2604 currentdict end |
|
2605 currentfile eexec |
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2676 0000000000000000000000000000000000000000000000000000000000000000 |
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2677 0000000000000000000000000000000000000000000000000000000000000000 |
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2678 0000000000000000000000000000000000000000000000000000000000000000 |
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2679 0000000000000000000000000000000000000000000000000000000000000000 |
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2680 0000000000000000000000000000000000000000000000000000000000000000 |
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2681 0000000000000000000000000000000000000000000000000000000000000000 |
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2682 0000000000000000000000000000000000000000000000000000000000000000 |
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2683 0000000000000000000000000000000000000000000000000000000000000000 |
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2684 cleartomark |
|
2685 %%EndFont |
|
2686 %%BeginFont: CMMI7 |
|
2687 %!PS-AdobeFont-1.1: CMMI7 1.100 |
|
2688 %%CreationDate: 1996 Jul 23 07:53:53 |
|
2689 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2690 11 dict begin |
|
2691 /FontInfo 7 dict dup begin |
|
2692 /version (1.100) readonly def |
|
2693 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2694 /FullName (CMMI7) readonly def |
|
2695 /FamilyName (Computer Modern) readonly def |
|
2696 /Weight (Medium) readonly def |
|
2697 /ItalicAngle -14.04 def |
|
2698 /isFixedPitch false def |
|
2699 end readonly def |
|
2700 /FontName /CMMI7 def |
|
2701 /PaintType 0 def |
|
2702 /FontType 1 def |
|
2703 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2704 /Encoding 256 array |
|
2705 0 1 255 {1 index exch /.notdef put} for |
|
2706 dup 0 /.notdef put |
|
2707 readonly def |
|
2708 /FontBBox{0 -250 1171 750}readonly def |
|
2709 /UniqueID 5087382 def |
|
2710 currentdict end |
|
2711 currentfile eexec |
|
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2781 0000000000000000000000000000000000000000000000000000000000000000 |
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2782 0000000000000000000000000000000000000000000000000000000000000000 |
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2783 0000000000000000000000000000000000000000000000000000000000000000 |
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2784 0000000000000000000000000000000000000000000000000000000000000000 |
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2785 0000000000000000000000000000000000000000000000000000000000000000 |
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2786 0000000000000000000000000000000000000000000000000000000000000000 |
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2787 0000000000000000000000000000000000000000000000000000000000000000 |
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2788 cleartomark |
|
2789 %%EndFont |
|
2790 %%BeginFont: CMR7 |
|
2791 %!PS-AdobeFont-1.1: CMR7 1.0 |
|
2792 %%CreationDate: 1991 Aug 20 16:39:21 |
|
2793 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2794 11 dict begin |
|
2795 /FontInfo 7 dict dup begin |
|
2796 /version (1.0) readonly def |
|
2797 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2798 /FullName (CMR7) readonly def |
|
2799 /FamilyName (Computer Modern) readonly def |
|
2800 /Weight (Medium) readonly def |
|
2801 /ItalicAngle 0 def |
|
2802 /isFixedPitch false def |
|
2803 end readonly def |
|
2804 /FontName /CMR7 def |
|
2805 /PaintType 0 def |
|
2806 /FontType 1 def |
|
2807 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2808 /Encoding 256 array |
|
2809 0 1 255 {1 index exch /.notdef put} for |
|
2810 dup 0 /.notdef put |
|
2811 readonly def |
|
2812 /FontBBox{-27 -250 1122 750}readonly def |
|
2813 /UniqueID 5000790 def |
|
2814 currentdict end |
|
2815 currentfile eexec |
|
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2865 95302D059AC126F57E432F369006A26DCF6675377685422F39E4D15B628EC9DC |
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2880 503D1260A24CFA7ED5F22D83B45F5435FD3D83B4F518980A4B8B012A |
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2881 0000000000000000000000000000000000000000000000000000000000000000 |
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2882 0000000000000000000000000000000000000000000000000000000000000000 |
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2883 0000000000000000000000000000000000000000000000000000000000000000 |
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2884 0000000000000000000000000000000000000000000000000000000000000000 |
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2885 0000000000000000000000000000000000000000000000000000000000000000 |
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2886 0000000000000000000000000000000000000000000000000000000000000000 |
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2887 0000000000000000000000000000000000000000000000000000000000000000 |
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2888 0000000000000000000000000000000000000000000000000000000000000000 |
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2889 cleartomark |
|
2890 %%EndFont |
|
2891 %%BeginFont: CMSY7 |
|
2892 %!PS-AdobeFont-1.1: CMSY7 1.0 |
|
2893 %%CreationDate: 1991 Aug 15 07:21:52 |
|
2894 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2895 11 dict begin |
|
2896 /FontInfo 7 dict dup begin |
|
2897 /version (1.0) readonly def |
|
2898 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2899 /FullName (CMSY7) readonly def |
|
2900 /FamilyName (Computer Modern) readonly def |
|
2901 /Weight (Medium) readonly def |
|
2902 /ItalicAngle -14.035 def |
|
2903 /isFixedPitch false def |
|
2904 end readonly def |
|
2905 /FontName /CMSY7 def |
|
2906 /PaintType 0 def |
|
2907 /FontType 1 def |
|
2908 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2909 /Encoding 256 array |
|
2910 0 1 255 {1 index exch /.notdef put} for |
|
2911 dup 0 /.notdef put |
|
2912 readonly def |
|
2913 /FontBBox{-15 -951 1252 782}readonly def |
|
2914 /UniqueID 5000817 def |
|
2915 currentdict end |
|
2916 currentfile eexec |
|
2917 D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 |
|
2918 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 |
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2919 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 |
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2920 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A |
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2921 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 |
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2922 8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D |
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2923 F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443 |
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2924 E611BE0A41A1F8C89C3BC16B352C3443AB6F665EAC5E0CC4229DECFC58E15765 |
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2925 424C919C273E7FA240BE7B2E951AB789D127625BBCB7033E005050EB2E12B1C8 |
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2926 E5F3AD1F44A71957AD2CC53D917BFD09235601155886EE36D0C3DD6E7AA2EF9C |
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2927 C402C77FF1549E609A711FC3C211E64E8F263D60A57E9F2B47E3480B978AAF63 |
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2930 97F78C821D430D69EC5C7E548AE3E7E75ABFC24370F314AA568E293EC4473C82 |
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2932 9129C602CC9BA1165DC528E4AD8B160B601E6A43103B0284E4B0EE1BFD7BCACE |
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2933 6AF68BD281AAC1218EF825B5247BEE54268C815113E5E5C8C9B87ACFB3D9A398 |
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2935 499F4A50B90096E345579FCE56BA57E55619387EF4288ED13E33BFB7B80CA66A |
|
2936 A705D1726C5C86FA3009BDB87737B26AD49A9718A709635B2FE39D06ADC17C49 |
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2937 336D4C49CEDA90C69E1CA649189B27EA425E8D580072E582317B4228213072C0 |
|
2938 A4CA5A70B500B2873D7050D13929D043D68A0202A0BD5C2B6D3E8B17EE986777 |
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|
2940 3F787F2E500B3EFB310DFB05EAE2CD8ECC11BB5315066A7EDD02A8F30C8495EB |
|
2941 D2CE5B2EC4EF67CE433FB119BF68310B7BEF5E3D851BFE1C570D12FE030055ED |
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2942 8FFB3B9158ED507ED51274E4D39AE9473DC08CB00A4BB3EE83A3C4AA67519BD5 |
|
2943 E052EFBE78E7D2F87838221CE7F1EB0F43589B98E4BF7E5AACAEA46CE2B09217 |
|
2944 D2D6D6D3C3444690265E573D63FD30EF32F2B21C43ADA3D9CB2BEC72523054F5 |
|
2945 425F3119F538A4D803E1C0518D2A04750DB8435CE0D0DF7110A82D0A189A5E05 |
|
2946 E4303B6DFD8FE673E64CCE9ADDD35A8029B641C41A461B7F661DCBC0BB76A364 |
|
2947 AF55619F114FAAD59CFB3806BA6A9907897DA0EF53C17C5099B4F09157CB841D |
|
2948 D31793 |
|
2949 0000000000000000000000000000000000000000000000000000000000000000 |
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2950 0000000000000000000000000000000000000000000000000000000000000000 |
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2951 0000000000000000000000000000000000000000000000000000000000000000 |
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2952 0000000000000000000000000000000000000000000000000000000000000000 |
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2953 0000000000000000000000000000000000000000000000000000000000000000 |
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2954 0000000000000000000000000000000000000000000000000000000000000000 |
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2955 0000000000000000000000000000000000000000000000000000000000000000 |
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2956 0000000000000000000000000000000000000000000000000000000000000000 |
|
2957 cleartomark |
|
2958 %%EndFont |
|
2959 %%BeginFont: CMR10 |
|
2960 %!PS-AdobeFont-1.1: CMR10 1.00B |
|
2961 %%CreationDate: 1992 Feb 19 19:54:52 |
|
2962 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2963 11 dict begin |
|
2964 /FontInfo 7 dict dup begin |
|
2965 /version (1.00B) readonly def |
|
2966 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2967 /FullName (CMR10) readonly def |
|
2968 /FamilyName (Computer Modern) readonly def |
|
2969 /Weight (Medium) readonly def |
|
2970 /ItalicAngle 0 def |
|
2971 /isFixedPitch false def |
|
2972 end readonly def |
|
2973 /FontName /CMR10 def |
|
2974 /PaintType 0 def |
|
2975 /FontType 1 def |
|
2976 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2977 /Encoding 256 array |
|
2978 0 1 255 {1 index exch /.notdef put} for |
|
2979 dup 0 /.notdef put |
|
2980 readonly def |
|
2981 /FontBBox{-251 -250 1009 969}readonly def |
|
2982 /UniqueID 5000793 def |
|
2983 currentdict end |
|
2984 currentfile eexec |
|
2985 D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 |
|
2986 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 |
|
2987 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F |
|
2988 D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 |
|
2989 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 |
|
2990 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 |
|
2991 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F |
|
2992 D1F017CE45884D76EF2CB9BC5821FD25365DDEA6E45F332B5F68A44AD8A530F0 |
|
2993 92A36FAC8D27F9087AFEEA2096F839A2BC4B937F24E080EF7C0F9374A18D565C |
|
2994 295A05210DB96A23175AC59A9BD0147A310EF49C551A417E0A22703F94FF7B75 |
|
2995 409A5D417DA6730A69E310FA6A4229FC7E4F620B0FC4C63C50E99E179EB51E4C |
|
2996 4BC45217722F1E8E40F1E1428E792EAFE05C5A50D38C52114DFCD24D54027CBF |
|
2997 2512DD116F0463DE4052A7AD53B641A27E81E481947884CE35661B49153FA19E |
|
2998 0A2A860C7B61558671303DE6AE06A80E4E450E17067676E6BBB42A9A24ACBC3E |
|
2999 B0CA7B7A3BFEA84FED39CCFB6D545BB2BCC49E5E16976407AB9D94556CD4F008 |
|
3000 24EF579B6800B6DC3AAF840B3FC6822872368E3B4274DD06CA36AF8F6346C11B |
|
3001 43C772CC242F3B212C4BD7018D71A1A74C9A94ED0093A5FB6557F4E0751047AF |
|
3002 D72098ECA301B8AE68110F983796E581F106144951DF5B750432A230FDA3B575 |
|
3003 5A38B5E7972AABC12306A01A99FCF8189D71B8DBF49550BAEA9CF1B97CBFC7CC |
|
3004 96498ECC938B1A1710B670657DE923A659DB8757147B140A48067328E7E3F9C3 |
|
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3071 5446ADCFA98C66409C0309 |
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3072 0000000000000000000000000000000000000000000000000000000000000000 |
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3073 0000000000000000000000000000000000000000000000000000000000000000 |
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3074 0000000000000000000000000000000000000000000000000000000000000000 |
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3075 0000000000000000000000000000000000000000000000000000000000000000 |
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3076 0000000000000000000000000000000000000000000000000000000000000000 |
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3077 0000000000000000000000000000000000000000000000000000000000000000 |
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3078 0000000000000000000000000000000000000000000000000000000000000000 |
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3079 0000000000000000000000000000000000000000000000000000000000000000 |
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3080 cleartomark |
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3081 %%EndFont |
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3082 %%BeginFont: CMTT10 |
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3083 %!PS-AdobeFont-1.1: CMTT10 1.00B |
|
3084 %%CreationDate: 1992 Apr 26 10:42:42 |
|
3085 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
3086 11 dict begin |
|
3087 /FontInfo 7 dict dup begin |
|
3088 /version (1.00B) readonly def |
|
3089 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
3090 /FullName (CMTT10) readonly def |
|
3091 /FamilyName (Computer Modern) readonly def |
|
3092 /Weight (Medium) readonly def |
|
3093 /ItalicAngle 0 def |
|
3094 /isFixedPitch true def |
|
3095 end readonly def |
|
3096 /FontName /CMTT10 def |
|
3097 /PaintType 0 def |
|
3098 /FontType 1 def |
|
3099 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
3100 /Encoding 256 array |
|
3101 0 1 255 {1 index exch /.notdef put} for |
|
3102 dup 0 /.notdef put |
|
3103 readonly def |
|
3104 /FontBBox{-4 -235 731 800}readonly def |
|
3105 /UniqueID 5000832 def |
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3106 currentdict end |
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3107 currentfile eexec |
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3261 0000000000000000000000000000000000000000000000000000000000000000 |
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3262 0000000000000000000000000000000000000000000000000000000000000000 |
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3263 0000000000000000000000000000000000000000000000000000000000000000 |
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3264 0000000000000000000000000000000000000000000000000000000000000000 |
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3265 cleartomark |
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3266 %%EndFont |
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3267 %%BeginFont: CMSY10 |
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3268 %!PS-AdobeFont-1.1: CMSY10 1.0 |
|
3269 %%CreationDate: 1991 Aug 15 07:20:57 |
|
3270 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
3271 11 dict begin |
|
3272 /FontInfo 7 dict dup begin |
|
3273 /version (1.0) readonly def |
|
3274 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
3275 /FullName (CMSY10) readonly def |
|
3276 /FamilyName (Computer Modern) readonly def |
|
3277 /Weight (Medium) readonly def |
|
3278 /ItalicAngle -14.035 def |
|
3279 /isFixedPitch false def |
|
3280 end readonly def |
|
3281 /FontName /CMSY10 def |
|
3282 /PaintType 0 def |
|
3283 /FontType 1 def |
|
3284 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
3285 /Encoding 256 array |
|
3286 0 1 255 {1 index exch /.notdef put} for |
|
3287 dup 0 /.notdef put |
|
3288 readonly def |
|
3289 /FontBBox{-29 -960 1116 775}readonly def |
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3290 /UniqueID 5000820 def |
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3291 currentdict end |
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3292 currentfile eexec |
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3397 F38106264F00E7842D4BA4CF999662 |
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3400 0000000000000000000000000000000000000000000000000000000000000000 |
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3401 0000000000000000000000000000000000000000000000000000000000000000 |
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3402 0000000000000000000000000000000000000000000000000000000000000000 |
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3403 0000000000000000000000000000000000000000000000000000000000000000 |
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3404 0000000000000000000000000000000000000000000000000000000000000000 |
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3405 0000000000000000000000000000000000000000000000000000000000000000 |
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3406 cleartomark |
|
3407 %%EndFont |
|
3408 %%BeginFont: CMMI10 |
|
3409 %!PS-AdobeFont-1.1: CMMI10 1.100 |
|
3410 %%CreationDate: 1996 Jul 23 07:53:57 |
|
3411 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
3412 11 dict begin |
|
3413 /FontInfo 7 dict dup begin |
|
3414 /version (1.100) readonly def |
|
3415 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
3416 /FullName (CMMI10) readonly def |
|
3417 /FamilyName (Computer Modern) readonly def |
|
3418 /Weight (Medium) readonly def |
|
3419 /ItalicAngle -14.04 def |
|
3420 /isFixedPitch false def |
|
3421 end readonly def |
|
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3640 0000000000000000000000000000000000000000000000000000000000000000 |
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3641 0000000000000000000000000000000000000000000000000000000000000000 |
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3642 0000000000000000000000000000000000000000000000000000000000000000 |
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3643 0000000000000000000000000000000000000000000000000000000000000000 |
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3644 0000000000000000000000000000000000000000000000000000000000000000 |
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3645 0000000000000000000000000000000000000000000000000000000000000000 |
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3646 0000000000000000000000000000000000000000000000000000000000000000 |
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3647 cleartomark |
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3648 %%EndFont |
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3649 %%BeginFont: CMMI9 |
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3650 %!PS-AdobeFont-1.1: CMMI9 1.100 |
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3651 %%CreationDate: 1996 Jul 23 07:53:55 |
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3652 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
3653 11 dict begin |
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3654 /FontInfo 7 dict dup begin |
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3655 /version (1.100) readonly def |
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3656 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
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3657 /FullName (CMMI9) readonly def |
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3658 /FamilyName (Computer Modern) readonly def |
|
3659 /Weight (Medium) readonly def |
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3660 /ItalicAngle -14.04 def |
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3661 /isFixedPitch false def |
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3662 end readonly def |
|
3663 /FontName /CMMI9 def |
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3664 /PaintType 0 def |
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3665 /FontType 1 def |
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3666 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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3667 /Encoding 256 array |
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3668 0 1 255 {1 index exch /.notdef put} for |
|
3669 dup 0 /.notdef put |
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3670 readonly def |
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3671 /FontBBox{-29 -250 1075 750}readonly def |
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3672 /UniqueID 5087384 def |
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3673 currentdict end |
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3674 currentfile eexec |
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3949 TeXDict begin 1 0 bop 1448 448 a FX(Nominal)30 b(Uni\002cation)1002 |
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3950 737 y FW(Christian)20 b(Urban)186 b(Andre)n(w)19 b(Pitts)187 |
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3951 b(Murdoch)18 b(Gabbay)1325 911 y FV(Uni)n(v)o(ersity)h(of)g(Cambridge,) |
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3952 h(Cambridge,)f(UK.)759 1221 y FU(Abstract.)41 b FV(W)-6 |
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3953 b(e)18 b(present)h(a)e(generalisation)i(of)f(\002rst-order)g |
|
3954 (uni\002cation)g(to)g(the)g(practically)759 1313 y(important)i(case)h |
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3955 (of)f(equations)h(between)f(terms)g(in)m(v)o(olving)h |
|
3956 FT(binding)g(oper)o(ations)p FV(.)f(A)g(sub-)759 1404 |
|
3957 y(stitution)26 b(of)f(terms)h(for)f(v)n(ariables)h(solv)o(es)g(such)h |
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3958 (an)f(equation)g(if)f(it)g(mak)o(es)i(the)e(equated)759 |
|
3959 1495 y(terms)20 b FS(\013)p FT(-equivalent)p FV(,)h(i.e.)f(equal)g(up)h |
|
3960 (to)f(renaming)h(bound)h(names.)f(F)o(or)e(the)i(applications)759 |
|
3961 1587 y(we)27 b(ha)o(v)o(e)f(in)h(mind,)f(we)h(must)g(consider)g(the)g |
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3962 (simple,)f(te)o(xtual)g(form)h(of)f(substitution)i(in)759 |
|
3963 1678 y(which)h(names)g(occurring)g(in)f(terms)g(may)g(be)g(captured)i |
|
3964 (within)d(the)i(scope)g(of)f(binders)759 1769 y(upon)22 |
|
3965 b(substitution.)f(W)-6 b(e)21 b(are)g(able)g(to)g(tak)o(e)g(a)g |
|
3966 (`nominal')g(approach)i(to)d(binding)i(in)f(which)759 |
|
3967 1861 y(bound)k(entities)d(are)h(e)o(xplicitly)f(named)i(\(rather)f |
|
3968 (than)g(using)h(nameless,)f(de)g(Bruijn-style)759 1952 |
|
3969 y(representations\))d(and)f(yet)f(get)h(a)f(v)o(ersion)h(of)f(this)h |
|
3970 (form)f(of)g(substitution)h(that)g(respects)g FS(\013)p |
|
3971 FV(-)759 2043 y(equi)n(v)n(alence)f(and)f(possesses)g(good)g |
|
3972 (algorithmic)g(properties.)f(W)-6 b(e)16 b(achie)n(v)o(e)h(this)e(by)i |
|
3973 (adapt-)759 2135 y(ing)25 b(an)f(e)o(xisting)g(idea)h(and)f |
|
3974 (introducing)i(a)e(k)o(e)o(y)h(ne)n(w)f(idea.)g(The)g(e)o(xisting)h |
|
3975 (idea)f(is)g(terms)759 2226 y(in)m(v)o(olving)c(e)o(xplicit)e |
|
3976 (substitutions)i(of)f(names)g(for)g(names,)g(e)o(xcept)g(that)g(here)g |
|
3977 (we)g(only)g(use)759 2317 y FT(e)o(xplicit)k(permutations)h |
|
3978 FV(\(bijecti)n(v)o(e)f(substitutions\).)h(The)f(k)o(e)o(y)h(ne)n(w)g |
|
3979 (idea)f(is)g(that)g(the)g(uni-)759 2409 y(\002cation)e(algorithm)h |
|
3980 (should)g(solv)o(e)g(not)f(only)h(equational)g(problems,)g(b)o(ut)f |
|
3981 (also)h(problems)759 2500 y(about)k(the)f FT(fr)m(eshness)h |
|
3982 FV(of)f(names)h(for)f(terms.)g(There)g(is)f(a)h(simple)g |
|
3983 (generalisation)h(of)f(the)759 2591 y(classical)c(\002rst-order)f |
|
3984 (uni\002cation)h(algorithm)h(to)e(this)h(setting)g(which)g(retains)f |
|
3985 (the)h(latter')l(s)759 2682 y(pleasant)i(properties:)f(uni\002cation)h |
|
3986 (problems)f(in)m(v)o(olving)h FS(\013)p FV(-equi)n(v)n(alence)h(and)f |
|
3987 (freshness)759 2774 y(are)c(decidable;)h(and)g(solv)n(able)f(problems)h |
|
3988 (possess)g(most)f(general)h(solutions.)523 3042 y FR(1)99 |
|
3989 b(Intr)n(oduction)523 3231 y FW(Decidability)25 b(of)h(uni\002cation)f |
|
3990 (for)g(equations)g(between)h(\002rst-order)e(terms)j(and)e(algorithms)g |
|
3991 (for)523 3330 y(computing)d(most)i(general)f(uni\002ers)h(form)f(a)h |
|
3992 (fundamental)e(tool)i(of)g(computational)e(logic)h(with)523 |
|
3993 3430 y(man)o(y)17 b(applications)g(to)h(programming)d(languages)i(and)g |
|
3994 (computer)n(-aided)e(reasoning.)i(Ho)n(we)n(v)o(er)m(,)523 |
|
3995 3530 y(v)o(ery)e(man)o(y)f(potential)h(applications)g(f)o(all)h |
|
3996 (outside)g(the)f(scope)h(of)f(\002rst-order)g(uni\002cation,)f(because) |
|
3997 523 3629 y(the)o(y)21 b(in)m(v)n(olv)o(e)g(term)h(languages)e(with)j |
|
3998 (binding)d(operations)h(where)g(at)i(the)f(v)o(ery)f(least)i(we)f(do)g |
|
3999 (not)523 3729 y(wish)f(to)f(distinguish)f(terms)h(dif)n(fering)f(up)g |
|
4000 (to)i(the)f(renaming)e(of)i(bound)e(names.)648 3828 y(There)23 |
|
4001 b(is)j(a)f(lar)o(ge)f(body)f(of)h(w)o(ork)g(studying)f(languages)g |
|
4002 (with)i(binders)f(through)e(the)j(use)g(of)523 3928 y(v)n(arious)d |
|
4003 FQ(\025)p FW(-calculi)g(as)i(term)e(representation)f(languages,)g |
|
4004 (leading)g(to)i FP(higher)n(-or)m(der)f(uni\002cation)523 |
|
4005 4028 y FW(algorithms)d(for)i(solving)e(equations)h(between)g |
|
4006 FQ(\025)p FW(-terms)h(modulo)e FQ(\013\014)t(\021)s FW(-equi)n(v)n |
|
4007 (alence.)g(Ho)n(we)n(v)o(er)m(,)523 4127 y(higher)n(-order)j |
|
4008 (uni\002cation)i(is)i(technically)e(complicated)f(without)h(being)h |
|
4009 (completely)e(satisf)o(ac-)523 4227 y(tory)g(from)g(a)i(pragmatic)d |
|
4010 (point)i(of)f(vie)n(w)-5 b(.)23 b(The)h(reason)f(lies)i(in)f(the)g(dif) |
|
4011 n(ference)e(between)i(substi-)523 4327 y(tution)j(for)g(\002rst-order)g |
|
4012 (terms)h(and)f(for)g FQ(\025)p FW(-terms.)h(The)f(former)g(is)h(a)h |
|
4013 (simple)f(operation)e(of)h(te)o(x-)523 4426 y(tual)19 |
|
4014 b(replacement)f(\(sometimes)g(called)h FP(gr)o(afting)f |
|
4015 FW([7],)h(or)f FP(conte)n(xt)h(substitution)g FW([12,)f(Sect.)h |
|
4016 (2.1]\),)523 4526 y(whereas)g(the)g(latter)h(also)g(in)m(v)n(olv)o(es)e |
|
4017 (renamings)g(to)h(a)n(v)n(oid)g(capture.)f(Capture-a)n(v)n(oidance)f |
|
4018 (ensures)523 4625 y(that)22 b(substitution)g(respects)g |
|
4019 FQ(\013)p FW(-equi)n(v)n(alence,)e(b)n(ut)i(it)h(complicates)e(higher)n |
|
4020 (-order)e(uni\002cation)i(al-)523 4725 y(gorithms.)e(Furthermore)f(it)k |
|
4021 (is)f(the)g(simple)f(te)o(xtual)g(form)g(of)g(substitution)g(rather)g |
|
4022 (than)g(the)g(more)523 4825 y(complicated)15 b(capture-a)n(v)n(oiding)e |
|
4023 (form)j(which)g(occurs)f(in)i(man)o(y)e(informal)g(applications)g(of)i |
|
4024 (`uni-)523 4924 y(\002cation)30 b(modulo)f FQ(\013)p |
|
4025 FW(-equi)n(v)n(alence'.)e(F)o(or)j(e)o(xample,)f(consider)g(the)h |
|
4026 (follo)n(wing)f(schematic)h(rule)p eop end |
|
4027 %%Page: 2 2 |
|
4028 TeXDict begin 2 1 bop 523 448 a FW(which)23 b(might)g(form)g(part)g(of) |
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4029 g(the)h(inducti)n(v)o(e)e(de\002nition)h(of)g(a)h(binary)e(e)n(v)n |
|
4030 (aluation)g(relation)h FO(+)h FW(for)523 548 y(the)c(e)o(xpressions)f |
|
4031 (of)h(an)g(imaginary)f(functional)f(programming)f(language:)1580 |
|
4032 671 y FN(app)o FM(\()p FN(fn)j FQ(a:Y)5 b(;)14 b(X)7 |
|
4033 b FM(\))45 b FO(+)h FQ(V)p 1559 712 787 4 v 1559 786 |
|
4034 a FN(let)20 b FQ(a)j FM(=)f FQ(X)27 b FN(in)20 b FQ(Y)65 |
|
4035 b FO(+)46 b FQ(V)2357 733 y(:)928 b FW(\(1\))523 897 |
|
4036 y(Here)22 b FQ(X)7 b FW(,)22 b FQ(Y)41 b FW(and)21 b |
|
4037 FQ(V)42 b FW(are)22 b(meta)n(v)n(ariables)f(standing)f(for)i(unkno)n |
|
4038 (wn)d(programming)g(language)h(e)o(x-)523 997 y(pressions.)27 |
|
4039 b(The)g(binders)g FN(fn)20 b FQ(a:)p FM(\()p FO(\000)p |
|
4040 FM(\))28 b FW(and)f FN(let)20 b FQ(a)37 b FM(=)f FQ(X)27 |
|
4041 b FN(in)20 b FM(\()p FO(\000)p FM(\))28 b FW(may)g(v)o(ery)e(well)i |
|
4042 (capture)f(free)523 1097 y(occurrences)16 b(of)i FQ(a)h |
|
4043 FW(when)f(we)h(instantiate)f(the)g(schematic)g(rule)g(by)g(replacing)f |
|
4044 FQ(Y)37 b FW(with)19 b(an)f(e)o(xpres-)523 1196 y(sion.)i(F)o(or)g |
|
4045 (instance,)f(using)h(the)g(rule)g(scheme)g(in)g(a)h(bottom-up)c(search) |
|
4046 j(for)g(a)g(proof)f(of)1612 1325 y FN(let)g FQ(a)k FM(=)g(1)d |
|
4047 FN(in)g FQ(a)46 b FO(+)g FM(1)991 b FW(\(2\))523 1464 |
|
4048 y(we)27 b(w)o(ould)g(use)g(a)g(substitution)f(that)h(in)m(v)n(olv)o(es) |
|
4049 f(capture,)f(namely)h FM([)p FQ(X)16 b FM(:=)9 b(1)o |
|
4050 FQ(;)14 b(Y)28 b FM(:=)9 b FQ(a;)14 b(V)28 b FM(:=)9 |
|
4051 b(1)o(])p FW(,)27 b(to)523 1564 y(unify)20 b(the)h(goal)g(with)g(the)g |
|
4052 (conclusion)f(of)h(\(1\)\227generating)d(the)j(ne)n(w)g(goal)f |
|
4053 FN(app)o FM(\()p FN(fn)g FQ(a:a;)14 b FM(1\))25 b FO(+)f |
|
4054 FM(1)523 1664 y FW(from)g(the)h(hypothesis)e(of)h(the)h(rule.)f(The)h |
|
4055 (problem)e(with)i(this)g(is)h(that)f(in)g(informal)e(practice)h(we)523 |
|
4056 1763 y(usually)k(identify)g(terms)h(up)f(to)h FQ(\013)p |
|
4057 FW(-equi)n(v)n(alence,)e(whereas)h(te)o(xtual)g(substitution)g(does)g |
|
4058 (not)h(re-)523 1863 y(spect)20 b FQ(\013)p FW(-equi)n(v)n(alence.)e(F)o |
|
4059 (or)i(e)o(xample,)f(up)g(to)i FQ(\013)p FW(-equi)n(v)n(alence,)d(the)i |
|
4060 (goal)1620 1991 y FN(let)f FQ(b)k FM(=)g(1)d FN(in)g |
|
4061 FQ(b)46 b FO(+)g FM(1)999 b FW(\(3\))523 2131 y(is)16 |
|
4062 b(the)g(same)f(as)i(\(2\))n(.)f(W)-7 b(e)17 b(might)d(think)h |
|
4063 (\(erroneously!\))d(that)j(the)h(conclusion)e(of)h(\(1\))f(is)j(the)e |
|
4064 (same)h(as)523 2231 y FN(let)k FQ(b)12 b FM(=)g FQ(X)h |
|
4065 FN(in)19 b FQ(Y)31 b FO(+)12 b FQ(V)41 b FW(without)21 |
|
4066 b(changing)e(the)j(rule')-5 b(s)22 b(hypothesis\227after)d(all,)j(if)g |
|
4067 (we)g(are)g(trying)523 2330 y(to)17 b(mak)o(e)f FQ(\013)p |
|
4068 FW(-equi)n(v)n(alence)f(disappear)h(into)g(the)h(infrastructure,)d |
|
4069 (then)j(we)g(must)g(be)g(able)f(to)h(replace)523 2430 |
|
4070 y(an)o(y)j FP(part)i FW(of)e(what)h(we)g(ha)n(v)o(e)f(with)g(an)h(equi) |
|
4071 n(v)n(alent)e(part.)h(So)g(we)h(might)f(be)h(tempted)e(to)i(unify)e |
|
4072 (the)523 2530 y(conclusion)i(with)h(\(3\))g(via)g(the)h(te)o(xtual)e |
|
4073 (substitution)h FM([)p FQ(X)15 b FM(:=)9 b(1)p FQ(;)14 |
|
4074 b(Y)27 b FM(:=)9 b FQ(b;)14 b(V)28 b FM(:=)9 b(1)o(])p |
|
4075 FW(,)23 b(and)f(then)f(apply)523 2629 y(this)27 b(substitution)f(to)i |
|
4076 (the)e(hypothesis)g(to)h(obtain)f(a)h(wrong)f(goal,)g |
|
4077 FN(app)o FM(\()p FN(fn)20 b FQ(a:b;)14 b FM(1\))35 b |
|
4078 FO(+)g FM(1)p FW(.)27 b(Using)523 2729 y FQ(\025)p FW(-calculus)d(and)g |
|
4079 (higher)n(-order)d(uni\002cation)i(sa)n(v)o(es)h(us)h(from)e(such)h |
|
4080 (slopp)o(y)f(thinking,)g(b)n(ut)h(at)g(the)523 2829 y(e)o(xpense)17 |
|
4081 b(of)i(ha)n(ving)e(to)i(mak)o(e)f(e)o(xplicit)g(the)h(dependence)d(of)i |
|
4082 (meta)n(v)n(ariables)g(on)g(bindable)f(names)523 2928 |
|
4083 y(via)h(the)f(use)h(of)f(function)f(v)n(ariables)h(and)g(application.)f |
|
4084 (F)o(or)h(e)o(xample,)f(\(1\))h(w)o(ould)g(be)h(replaced)e(by)593 |
|
4085 3075 y FN(app)d FM(\()p FN(fn)g FQ(\025a:F)26 b(a)p FM(\))14 |
|
4086 b FQ(X)53 b FO(+)46 b FQ(V)p 593 3115 848 4 v 644 3194 |
|
4087 a FN(let)12 b FQ(X)21 b FM(\()p FQ(\025a:F)26 b(a)p FM(\))46 |
|
4088 b FO(+)g FQ(V)1547 3147 y FW(or)m(,)19 b(modulo)g FQ(\021)s |
|
4089 FW(-equi)n(v)n(alence,)2520 3079 y FN(app)12 b FM(\()p |
|
4090 FN(fn)i FQ(F)e FM(\))i FQ(X)52 b FO(+)46 b FQ(V)p 2520 |
|
4091 3120 675 4 v 2603 3194 a FN(let)12 b FQ(X)20 b(F)58 b |
|
4092 FO(+)46 b FQ(V)3217 3147 y FW(.)3308 3139 y(\(4\))523 |
|
4093 3330 y(No)n(w)30 b(goal)f(\(3\))h(becomes)f FN(let)12 |
|
4094 b FM(1)i FQ(\025b:b)41 b FO(+)g FM(1)30 b FW(and)f(there)h(is)h(no)e |
|
4095 (problem)g(unifying)f(it)i(with)g(the)523 3430 y(conclusion)16 |
|
4096 b(of)i(\(4\))g(via)g(a)g(capture-a)n(v)n(oiding)d(substitution)i(of)h |
|
4097 FM(1)g FW(for)g FQ(X)7 b FW(,)17 b FQ(\025c:c)i FW(for)f |
|
4098 FQ(F)30 b FW(and)18 b FM(1)g FW(for)f FQ(V)i FW(.)648 |
|
4099 3530 y(This)31 b(is)i(all)f(v)o(ery)e(\002ne,)h(b)n(ut)g(the)h |
|
4100 (situation)f(is)h(not)f(as)h(pleasant)f(as)i(for)d(\002rst-order)g |
|
4101 (terms:)523 3629 y(higher)n(-order)19 b(uni\002cation)h(problems)g(can) |
|
4102 i(be)f(undecidable,)e(decidable)i(b)n(ut)g(lack)h(most)g(general)523 |
|
4103 3729 y(uni\002ers,)17 b(or)h(ha)n(v)o(e)g(such)f(uni\002ers)h(only)f |
|
4104 (by)h(imposing)e(some)i(restrictions)g([20];)f(see)h([6])g(for)f(a)h |
|
4105 (sur)n(-)523 3828 y(v)o(e)o(y)e(of)g(higher)n(-order)e(uni\002cation.)h |
|
4106 (W)-7 b(e)18 b(started)e(out)h(w)o(anting)f(to)g(compute)g(with)h |
|
4107 (binders)e(modulo)523 3928 y FQ(\013)p FW(-equi)n(v)n(alence,)j(and)h |
|
4108 (someho)n(w)g(the)h(process)g(of)f(making)g(possibly-capturing)e |
|
4109 (substitution)i(re-)523 4028 y(spectable)26 b(has)h(led)g(to)f |
|
4110 (function)f(v)n(ariables,)h(application,)f(capture-a)n(v)n(oiding)e |
|
4111 (substitution)j(and)523 4127 y FQ(\014)t(\021)s FW(-equi)n(v)n(alence.) |
|
4112 18 b(Does)i(it)h(ha)n(v)o(e)f(to)g(be)g(so?)h(No!)648 |
|
4113 4227 y(F)o(or)14 b(one)h(thing,)f(se)n(v)o(eral)h(authors)f(ha)n(v)o(e) |
|
4114 h(already)f(noted)g(that)h(one)g(can)g(mak)o(e)g(sense)g(of)g |
|
4115 (possibly-)523 4327 y(capturing)k(substitution)g(modulo)g |
|
4116 FQ(\013)p FW(-equi)n(v)n(alence)f(by)i(using)g FP(e)n(xplicit)h |
|
4117 (substitutions)f FW(in)g(the)h(term)523 4426 y(representation)28 |
|
4118 b(language:)h(see)i([7,)12 b(13,)h(15,)g(26].)29 b(Compared)g(with)h |
|
4119 (those)g(w)o(orks,)g(we)g(mak)o(e)g(a)523 4526 y(number)20 |
|
4120 b(of)h(simpli\002cations.)g(First,)i(we)f(\002nd)f(that)h(we)g(do)f |
|
4121 FP(not)i FW(need)e(to)h(use)g(function)e(v)n(ariables,)523 |
|
4122 4625 y(application)d(or)i FQ(\014)t(\021)s FW(-equi)n(v)n(alence)d(in)j |
|
4123 (our)f(representation)f(language\227lea)n(ving)e(just)k(binders)f(and) |
|
4124 523 4725 y FQ(\013)p FW(-equi)n(v)n(alence.)e(Secondly)-5 |
|
4125 b(,)16 b(instead)h(of)h(using)f(e)o(xplicit)h(substitutions)f(of)h |
|
4126 (names)f(for)h(names,)f(we)523 4825 y(use)g FP(e)n(xplicit)g |
|
4127 (permutations)f FW(of)g(names.)h(The)f(idea)h(of)f(using)h |
|
4128 (name-permutations,)c(and)j(in)h(partic-)523 4924 y(ular)23 |
|
4129 b(name-sw)o(appings,)f(when)h(dealing)f(with)i FQ(\013)p |
|
4130 FW(-con)m(v)o(ersion)d(dates)j(back)f(to)h([10])e(and)h(there)h(is)p |
|
4131 eop end |
|
4132 %%Page: 3 3 |
|
4133 TeXDict begin 3 2 bop 523 448 a FW(gro)n(wing)14 b(e)n(vidence)g(of)h |
|
4134 (its)i(usefulness)e(\(see)h([3,)d(4],)i(for)g(e)o(xample\).)e(When)j(a) |
|
4135 g(name)f(substitution)g(is)523 548 y(actually)i(a)h(permutation,)d(the) |
|
4136 i(function)f(it)i(induces)e(from)h(terms)g(to)h(terms)f(is)h(a)g |
|
4137 (bijection;)f(this)h(bi-)523 648 y(jecti)n(vity)g(gi)n(v)o(es)f(the)h |
|
4138 (operation)e(of)i(permuting)e(names)i(v)o(ery)f(good)f(logical)i |
|
4139 (properties)f(compared)523 747 y(with)j(name)f(substitution.)f |
|
4140 (Consider)h(for)g(e)o(xample)f(the)h FQ(\013)p FW(-equi)n(v)n(alent)f |
|
4141 (terms)i FN(fn)13 b FQ(a:b)20 b FW(and)f FN(fn)13 b FQ(c:b)p |
|
4142 FW(,)523 847 y(where)j FQ(a)p FW(,)h FQ(b)g FW(and)f |
|
4143 FQ(c)h FW(are)g(distinct.)f(If)h(we)g(apply)e(the)i(substitution)f |
|
4144 FM([)p FQ(b)p FL(7!)p FQ(a)p FM(])h FW(\(renaming)d(all)j(free)g(occur) |
|
4145 n(-)523 946 y(rences)h(of)g FQ(b)h FW(to)f(be)h FQ(a)p |
|
4146 FW(\))f(to)h(them)f(we)g(get)h FN(fn)13 b FQ(a:a)19 b |
|
4147 FW(and)f FN(fn)13 b FQ(c:a)p FW(,)18 b(which)g(are)g(no)g(longer)f |
|
4148 FQ(\013)p FW(-equi)n(v)n(alent.)523 1046 y(Thus)23 b(renaming)e |
|
4149 (substitutions)i(do)g(not)g(respect)g FQ(\013)p FW(-equi)n(v)n(alence)e |
|
4150 (in)j(general,)e(and)h(an)o(y)f(uni\002ca-)523 1146 y(tion)i(algorithm) |
|
4151 g(using)g(them)g(needs)g(to)h(tak)o(e)g(e)o(xtra)f(precautions)f(to)h |
|
4152 (not)h(inadv)o(ertently)d(change)523 1245 y(the)e(intended)f(meaning)f |
|
4153 (of)i(terms.)g(The)g(traditional)f(solution)g(for)h(this)h(problem)d |
|
4154 (is)j(to)f(introduce)523 1345 y(a)32 b(more)g(complicated)e(form)h(of)h |
|
4155 (renaming)e(substitution)h(that)h(a)n(v)n(oids)g(capture)f(of)g(names)h |
|
4156 (by)523 1445 y(binders.)19 b(In)h(contrast,)f(the)h(simple)g(operation) |
|
4157 e(of)i(name-permutation)c(respects)k FQ(\013)p FW(-equi)n(v)n(alence;) |
|
4158 523 1544 y(for)j(e)o(xample,)e(applying)h(the)h(name-permutation)d |
|
4159 FM(\()p FQ(a)14 b(b)p FM(\))24 b FW(that)f(sw)o(aps)h(all)g |
|
4160 (occurrences)d(of)i FQ(a)h FW(and)523 1644 y FQ(b)k FW(\(be)f(the)o(y)g |
|
4161 (free,)g(bound)e(or)i(binding\))f(to)i(the)f(terms)h(abo)o(v)o(e)e(gi)n |
|
4162 (v)o(es)h FN(fn)13 b FQ(b:a)27 b FW(and)g FN(fn)13 b |
|
4163 FQ(c:a)p FW(,)28 b(which)523 1743 y(are)22 b(still)i |
|
4164 FQ(\013)p FW(-equi)n(v)n(alent.)c(W)-7 b(e)23 b(e)o(xploit)e(such)h |
|
4165 (good)f(properties)g(of)h(name-permutations)c(to)23 b(gi)n(v)o(e)e(a) |
|
4166 523 1843 y(conceptually)d(simple)i(uni\002cation)f(algorithm.)648 |
|
4167 1944 y(In)f(addition)g(to)h(the)g(use)g(of)g(e)o(xplicit)f |
|
4168 (name-permutations,)e(we)j(also)g(compute)f(symbolically)523 |
|
4169 2044 y(with)i(predicates)g(e)o(xpressing)e FP(fr)m(eshness)j |
|
4170 FW(of)f(names)g(for)g(terms.)g(This)g(seems)h(to)f(be)g(the)g(k)o(e)o |
|
4171 (y)g(no)o(v-)523 2143 y(elty)i(of)f(our)g(approach.)f(Although)f(it)k |
|
4172 (arises)f(naturally)f(from)f(the)i(w)o(ork)f(reported)f(in)i([11,)12 |
|
4173 b(23],)21 b(it)523 2243 y(is)i(easy)g(to)f(see)h(directly)e(why)h |
|
4174 (there)g(is)h(a)g(need)e(for)h(computing)e(with)i(freshness,)g(gi)n(v)o |
|
4175 (en)f(that)h(we)523 2343 y(tak)o(e)i(a)g(`nominal')e(approach)g(to)i |
|
4176 (binders.)f(\(In)g(other)g(w)o(ords)h(we)g(stick)h(with)f(concrete)e(v) |
|
4177 o(ersions)523 2442 y(of)d(binding)f(and)h FQ(\013)p FW(-equi)n(v)n |
|
4178 (alence)e(in)i(which)g(the)h(bound)d(entity)i(is)h(named)f(e)o |
|
4179 (xplicitly)-5 b(,)17 b(rather)i(than)523 2542 y(using)f(de)g |
|
4180 (Bruijn-style)g(representations,)f(as)i(for)f(e)o(xample)f(in)h([7,)13 |
|
4181 b(26].\))k(A)i(basic)g(instance)f(of)g(our)523 2641 y(generalised)27 |
|
4182 b(form)g(of)h FQ(\013)p FW(-equi)n(v)n(alence)e(identi\002es)i |
|
4183 FN(fn)20 b FQ(a:X)35 b FW(with)28 b FN(fn)20 b FQ(b:)p |
|
4184 FM(\()p FQ(a)14 b(b)p FM(\))p FK(\001)o FQ(X)35 b FW(pro)o(vided)25 |
|
4185 b FQ(b)k FP(is)523 2741 y(fr)m(esh)22 b(for)i FQ(X)7 |
|
4186 b FW(,)21 b(where)f(the)i(subterm)e FM(\()p FQ(a)14 b(b)p |
|
4187 FM(\))p FK(\001)p FQ(X)28 b FW(indicates)21 b(an)g(e)o(xplicit)g |
|
4188 (permutation\227namely)c(the)523 2841 y(sw)o(apping)i(of)i |
|
4189 FQ(a)g FW(and)f FQ(b)p FW(\227w)o(aiting)g(to)g(be)h(applied)e(to)i |
|
4190 FQ(X)7 b FW(.)20 b(W)-7 b(e)21 b(write)g(`)p FQ(b)f FW(is)i(fresh)e |
|
4191 (for)g FQ(X)7 b FW(')20 b(symboli-)523 2940 y(cally)e(as)h |
|
4192 FQ(b)j FM(#)i FQ(X)7 b FW(;)18 b(the)g(intended)e(meaning)h(of)g(this)i |
|
4193 (relation)e(is)i(that)f FQ(b)g FW(does)g(not)g(occur)f(free)g(in)h(an)o |
|
4194 (y)523 3040 y(\(ground\))d(term)j(that)h(may)e(be)i(substituted)e(for)h |
|
4195 FQ(X)7 b FW(.)18 b(If)g(we)g(kno)n(w)f(more)h(about)f |
|
4196 FQ(X)25 b FW(we)19 b(may)e(be)i(able)523 3140 y(to)24 |
|
4197 b(eliminate)f(the)h(e)o(xplicit)f(permutation)e(in)j |
|
4198 FM(\()p FQ(a)14 b(b)p FM(\))p FK(\001)p FQ(X)7 b FW(;)23 |
|
4199 b(for)g(e)o(xample,)g(if)h(we)g(kne)n(w)f(that)g FQ(a)30 |
|
4200 b FM(#)g FQ(X)523 3239 y FW(holds)21 b(as)i(well)f(as)g |
|
4201 FQ(b)k FM(#)f FQ(X)7 b FW(,)22 b(then)f FM(\()p FQ(a)14 |
|
4202 b(b)p FM(\))p FK(\001)o FQ(X)29 b FW(can)21 b(be)h(replaced)e(by)i |
|
4203 FQ(X)7 b FW(.)21 b(It)h(should)e(already)h(be)h(clear)523 |
|
4204 3339 y(from)g(these)i(simple)f(e)o(xamples)f(that)i(in)f(our)g(setting) |
|
4205 g(the)g(appropriate)e(notion)i(of)g(term-equality)523 |
|
4206 3438 y(is)e(not)f(a)h(bare)e(equation,)g FQ(t)k FO(\031)f |
|
4207 FQ(t)1445 3408 y FL(0)1469 3438 y FW(,)e(b)n(ut)g(rather)g(a)g |
|
4208 (hypothetical)e(judgement)g(of)i(the)h(form)1784 3571 |
|
4209 y FO(r)i(`)g FQ(t)g FO(\031)g FQ(t)2121 3537 y FL(0)3308 |
|
4210 3571 y FW(\(5\))523 3701 y(where)d FO(r)h FW(is)g(a)f |
|
4211 FP(fr)m(eshness)h(en)m(vir)l(onment)q FW(\227a)e(\002nite)i(set)g |
|
4212 FO(f)p FQ(a)2238 3713 y FJ(1)2284 3701 y FM(#)9 b FQ(X)2431 |
|
4213 3713 y FJ(1)2468 3701 y FQ(;)14 b(:)g(:)g(:)g(;)g(a)2697 |
|
4214 3713 y FI(n)2751 3701 y FM(#)9 b FQ(X)2898 3713 y FI(n)2944 |
|
4215 3701 y FO(g)20 b FW(of)g(freshness)523 3801 y(assumptions.)h(F)o(or)h |
|
4216 (e)o(xample)f FO(f)p FQ(a)13 b FM(#)g FQ(X)r(;)h(b)f |
|
4217 FM(#)g FQ(X)7 b FO(g)13 b(`)g FN(fn)19 b FQ(a:X)h FO(\031)13 |
|
4218 b FN(fn)19 b FQ(b:X)29 b FW(is)24 b(a)e(v)n(alid)g(judgement)f(of)523 |
|
4219 3900 y(our)e FP(nominal)f(equational)g(lo)o(gic)p FW(.)h(Similarly)-5 |
|
4220 b(,)19 b(judgements)f(about)h(freshness)g(itself)h(will)h(tak)o(e)e |
|
4221 (the)523 4000 y(form)1763 4097 y FO(r)24 b(`)e FQ(a)h |
|
4222 FM(#)h FQ(t)f(:)1143 b FW(\(6\))523 4227 y FH(T)-8 b(o)18 |
|
4223 b(summarise:)f FW(W)-7 b(e)18 b(will)f(represent)f(languages)f(in)m(v)n |
|
4224 (olving)g(binders)h(using)g(the)h(usual)f(notion)g(of)523 |
|
4225 4327 y(\002rst-order)j(terms)i(o)o(v)o(er)f(a)h(man)o(y-sorted)d |
|
4226 (signature.)h(These)i(gi)n(v)o(e)f(us)h(terms)g(with:)g(distinguished) |
|
4227 523 4426 y(constants)j(naming)f(bindable)g(entities,)h(that)g(we)h |
|
4228 (call)g FP(atoms)p FW(;)f(terms)g FQ(a:t)h FW(e)o(xpressing)e(a)i |
|
4229 (generic)523 4526 y(form)16 b(of)h FP(binding)e FW(of)i(an)g(atom)g |
|
4230 FQ(a)h FW(in)f(a)g(term)g FQ(t)p FW(;)h(and)f(terms)g |
|
4231 FQ(\031)s FK(\001)p FQ(X)24 b FW(representing)15 b(an)i(e)o(xplicit)f |
|
4232 FP(permu-)523 4625 y(tation)21 b FW(of)h(atoms)g FQ(\031)k |
|
4233 FW(w)o(aiting)c(to)g(be)g(applied)f(to)h(whate)n(v)o(er)e(term)i(is)h |
|
4234 (substituted)e(for)h(the)g(v)n(ariable)523 4725 y FQ(X)7 |
|
4235 b FW(.)19 b(Section)f(2)i(presents)e(this)i(term-language)c(together)i |
|
4236 (with)h(a)g(syntax-directed)e(inducti)n(v)o(e)g(def-)523 |
|
4237 4825 y(inition)31 b(of)h(the)g(pro)o(v)n(able)e(judgements)g(of)i(the)g |
|
4238 (form)f(\(5\))h(and)f(\(6\))h(which)f(for)h FP(gr)l(ound)g |
|
4239 FW(terms)523 4924 y(\(i.e.)23 b(ones)g(with)g(no)f(v)n(ariables\))g |
|
4240 (agrees)h(with)g(the)g(usual)g(notions)f(of)h FQ(\013)p |
|
4241 FW(-equi)n(v)n(alence)e(and)i(`not)f(a)p eop end |
|
4242 %%Page: 4 4 |
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4243 TeXDict begin 4 3 bop 523 448 a FW(free)27 b(v)n(ariable)f(of)5 |
|
4244 b('.)26 b(Ho)n(we)n(v)o(er)m(,)f(on)i(open)f(terms)h(our)f(judgements)g |
|
4245 FP(dif)o(fer)j FW(from)d(these)i(standard)523 548 y(notions)h(and)g |
|
4246 (appear)f(to)i(be)g(an)f(e)o(xtension)f(that)i(has)g(not)f(yet)h(been)f |
|
4247 (studied)g(in)h(the)f(literature)523 648 y(\(including)22 |
|
4248 b([11,)12 b(24,)h(23]\).)22 b(Section)h(3)h(considers)f(uni\002cation)g |
|
4249 (in)h(this)g(setting.)g(Solving)e(equali-)523 747 y(ties)g(between)f |
|
4250 (abstractions)g FQ(a:t)k FO(\031)o FM(?)h FQ(a)1673 717 |
|
4251 y FL(0)1696 747 y FQ(:t)1749 717 y FL(0)1794 747 y FW(entails)c |
|
4252 (solving)f(both)f(equalities)h FQ(t)26 b FO(\031)o FM(?)g(\()p |
|
4253 FQ(a)14 b(a)3128 717 y FL(0)3151 747 y FM(\))p FK(\001)p |
|
4254 FQ(t)3240 717 y FL(0)3285 747 y FW(and)523 847 y(freshness)j(problems)f |
|
4255 FQ(a)23 b FM(#?)g FQ(t)1405 817 y FL(0)1428 847 y FW(.)18 |
|
4256 b(Therefore)d(our)i(general)f(form)g(of)h FP(nominal)f(uni\002cation)f |
|
4257 (pr)l(oblem)523 946 y FW(is)k(a)g(\002nite)f(collection)f(of)h(indi)n |
|
4258 (vidual)e(equality)i(and)f(freshness)h(problems.)e(Such)i(a)h(problem)d |
|
4259 FQ(P)31 b FW(is)523 1046 y(solv)o(ed)21 b(by)h(pro)o(viding)d(not)j |
|
4260 (only)f(a)i(substitution)e FQ(\033)26 b FW(\(of)c(terms)g(for)f(v)n |
|
4261 (ariables\),)g(b)n(ut)h(also)h(a)f(fresh-)523 1146 y(ness)d(en)m |
|
4262 (vironment)d FO(r)k FW(\(as)f(abo)o(v)o(e\),)e(which)h(together)g(ha)n |
|
4263 (v)o(e)g(the)h(property)e(that)i FO(r)9 b(`)g FQ(\033)s |
|
4264 FM(\()p FQ(t)p FM(\))24 b FO(\031)f FQ(\033)s FM(\()p |
|
4265 FQ(t)3349 1116 y FL(0)3373 1146 y FM(\))523 1245 y FW(and)16 |
|
4266 b FO(r)9 b(`)g FQ(a)g FM(#)g FQ(\033)s FM(\()p FQ(t)1041 |
|
4267 1215 y FL(00)1085 1245 y FM(\))18 b FW(hold)e(for)g(each)g(indi)n |
|
4268 (vidual)f(equality)h FQ(t)23 b FO(\031)o FM(?)h FQ(t)2430 |
|
4269 1215 y FL(0)2470 1245 y FW(and)16 b(freshness)h FQ(a)22 |
|
4270 b FM(#?)i FQ(t)3162 1215 y FL(00)3222 1245 y FW(in)16 |
|
4271 b(the)523 1345 y(problem)23 b FQ(P)12 b FW(.)26 b(Our)e(main)h(result)g |
|
4272 (with)g(respect)g(to)g(uni\002cation)f(is)i(that)g FP(solvability)e(is) |
|
4273 i(decidable)523 1445 y(and)g(that)h(solvable)g(pr)l(oblems)g(possess)h |
|
4274 (most)g(g)o(ener)o(al)e(solutions)h FW(\(for)f(a)i(reasonably)d(ob)o |
|
4275 (vious)523 1544 y(notion)j(of)g(`most)h(general'\).)e(The)i(proof)e(is) |
|
4276 j(via)f(a)g(uni\002cation)f(algorithm)f(that)j(is)f(v)o(ery)f(simi-)523 |
|
4277 1644 y(lar)f(to)g(the)g(\002rst-order)f(algorithm)f(gi)n(v)o(en)h(in)h |
|
4278 (the)g(no)n(w-common)c(transformational)i(style)i([18].)523 |
|
4279 1743 y(\(See)21 b([17,)f(Sect.)h(2.6])f(or)h([1,)f(Sect.)i(4.6])e(for)g |
|
4280 (e)o(xpositions)f(of)i(this.\))g(Section)g(4)g(considers)f(the)h(re-) |
|
4281 523 1843 y(lationship)e(of)h(our)f(v)o(ersion)f(of)i(`uni\002cation)e |
|
4282 (modulo)g FQ(\013)p FW(-equi)n(v)n(alence')g(to)i(e)o(xisting)f |
|
4283 (approaches.)523 1943 y(Section)26 b(5)h(assesses)h(what)f(has)g(been)f |
|
4284 (achie)n(v)o(ed)f(and)h(the)g(prospects)g(for)g(applications.)f(T)-7 |
|
4285 b(o)27 b(ap-)523 2042 y(preciate)c(the)i(kind)e(of)h(problem)e(that)i |
|
4286 (nominal)f(uni\002cation)g(solv)o(es,)g(you)h(might)f(lik)o(e)h(to)h |
|
4287 (try)e(the)523 2142 y(follo)n(wing)c(quiz)h(about)f(the)i |
|
4288 FQ(\025)p FW(-calculus)f([2])f(before)g(we)i(apply)f(our)f(algorithm)g |
|
4289 (to)i(solv)o(e)e(it)j(at)e(the)523 2242 y(end)g(of)g(Section)f(3.)p |
|
4290 523 2304 2882 4 v 523 2912 4 609 v 551 2381 a FU(Quiz)j |
|
4291 FT(Assuming)h(that)f FS(a)g FT(and)h FS(b)f FT(ar)m(e)h(distinct)f |
|
4292 (variables)p FV(,)h(is)f(it)f(possible)i(to)f(\002nd)h |
|
4293 FS(\025)p FV(-terms)e FS(M)3062 2389 y FG(1)3097 2381 |
|
4294 y FS(;)13 b(:)h(:)f(:)g(;)g(M)3342 2389 y FG(7)551 2472 |
|
4295 y FV(that)19 b(mak)o(e)h(the)f(follo)n(wing)g(pairs)g(of)g(terms)g |
|
4296 FS(\013)p FV(-equi)n(v)n(alent?)737 2581 y(1.)k FS(\025a:\025b:)p |
|
4297 FF(\()p FS(M)1126 2589 y FG(1)1173 2581 y FS(b)p FF(\))c |
|
4298 FV(and)g FS(\025b:\025a:)p FF(\()p FS(a)12 b(M)1744 2589 |
|
4299 y FG(1)1779 2581 y FF(\))283 b FV(3.)24 b FS(\025a:\025b:)p |
|
4300 FF(\()p FS(b)12 b(M)2527 2589 y FG(4)2562 2581 y FF(\))19 |
|
4301 b FV(and)g FS(\025b:\025a:)p FF(\()p FS(a)12 b(M)3100 |
|
4302 2589 y FG(5)3135 2581 y FF(\))737 2684 y FV(2.)23 b FS(\025a:\025b:)p |
|
4303 FF(\()p FS(M)1126 2692 y FG(2)1173 2684 y FS(b)p FF(\))c |
|
4304 FV(and)g FS(\025b:\025a:)p FF(\()p FS(a)12 b(M)1744 2692 |
|
4305 y FG(3)1779 2684 y FF(\))283 b FV(4.)24 b FS(\025a:\025b:)p |
|
4306 FF(\()p FS(b)12 b(M)2527 2692 y FG(6)2562 2684 y FF(\))19 |
|
4307 b FV(and)g FS(\025a:\025a:)p FF(\()p FS(a)12 b(M)3108 |
|
4308 2692 y FG(7)3143 2684 y FF(\))p FV(.)551 2778 y(If)26 |
|
4309 b(it)g(is)h(possible)g(to)f(\002nd)h(a)g(solution)g(for)f(an)o(y)i(of)e |
|
4310 (these)h(four)g(problems,)g(can)g(you)h(describe)f(what)g(all)551 |
|
4311 2869 y(possible)20 b(solutions)f(for)g(that)g(problem)h(are)f(lik)o(e?) |
|
4312 3377 2836 y |
|
4313 gsave currentpoint currentpoint translate 180 neg rotate neg exch |
|
4314 neg exch translate |
|
4315 3377 2836 a FU(Answers:)f FV(see)h(Example)h(2.)4149 |
|
4316 2836 y |
|
4317 currentpoint grestore moveto |
|
4318 4149 2836 a 3402 2912 4 609 v 523 2915 2882 4 |
|
4319 v 523 3137 a FR(2)99 b(Nominal)25 b(equational)g(logic)523 |
|
4320 3330 y FW(W)-7 b(e)27 b(tak)o(e)f(a)g(concrete)f(approach)e(to)j(the)g |
|
4321 (syntax)f(of)h(binders)f(in)h(which)f(bound)f(entities)i(are)g(e)o(x-) |
|
4322 523 3430 y(plicitly)18 b(named.)f(Furthermore)f(we)j(separate)f(the)g |
|
4323 (names)g(of)h(bound)d(entities)j(from)e(the)i(names)f(of)523 |
|
4324 3530 y(v)n(ariables,)h(which)g(is)h(inspired)f(for)g(e)o(xample)f(by)h |
|
4325 (the)h FQ(\031)s FW(-calculus)f([21],)f(in)i(which)f(the)g(restriction) |
|
4326 523 3629 y(operator)28 b(binds)h(channel)g(names)g(and)g(these)h(are)g |
|
4327 (quite)f(dif)n(ferent)f(from)h(names)h(of)f(unkno)n(wn)523 |
|
4328 3729 y(processes.)e(Names)h(of)g(bound)e(entities)i(will)g(be)g(called) |
|
4329 g FP(atoms)p FW(.)f(This)h(is)h(partly)e(for)g(historical)523 |
|
4330 3828 y(reasons)e(\(stemming)f(from)g(the)h(w)o(ork)g(by)g(the)g(second) |
|
4331 f(tw)o(o)i(authors)e([11]\))g(and)g(partly)h(to)g(indi-)523 |
|
4332 3928 y(cate)j(that)g(the)g(internal)f(structure)g(of)h(such)g(names)f |
|
4333 (is)i(irrele)n(v)n(ant)e(to)h(us:)g(all)h(we)f(care)g(about)f(is)523 |
|
4334 4028 y(their)c(identity)f(\(i.e.)h(whether)f(or)h(not)f(one)h(atom)g |
|
4335 (is)h(the)f(same)g(as)h(another\))d(and)h(that)h(the)h(supply)523 |
|
4336 4127 y(of)c(atoms)g(is)h(ine)o(xhaustible.)648 4227 y(Although)h(there) |
|
4337 h(are)h(se)n(v)o(eral)f(general)g(frame)n(w)o(orks)f(in)i(the)g |
|
4338 (literature)f(for)g(specifying)g(lan-)523 4327 y(guages)30 |
|
4339 b(with)h(binders,)e(not)h(all)h(of)g(them)f(meet)g(the)h(requirements)e |
|
4340 (mentioned)f(in)j(the)g(pre)n(vi-)523 4426 y(ous)23 b(paragraph.)e(Use) |
|
4341 j(of)f(the)g(simply)g(typed)f FQ(\025)p FW(-calculus)h(for)f(this)i |
|
4342 (purpose)e(is)i(common;)e(b)n(ut)h(as)523 4526 y(discussed)e(in)h(the)f |
|
4343 (Introduction,)d(it)23 b(leads)e(to)h(a)g(problematic)d(uni\002cation)h |
|
4344 (theory)-5 b(.)20 b(Among)g FP(\002r)o(st-)523 4625 y(or)m(der)e |
|
4345 FW(frame)n(w)o(orks,)c(Plotkin')-5 b(s)16 b(notion)f(of)h |
|
4346 FP(binding)f(signatur)m(e)g FW([25,)d(9],)k(being)f(unsorted,)g |
|
4347 (equates)523 4725 y(names)k(used)g(in)h(binding)d(with)j(names)f(of)g |
|
4348 (v)n(ariables)f(standing)h(for)f(unkno)n(wn)f(terms;)j(so)f(it)i(is)f |
|
4349 (not)523 4825 y(suf)n(\002ciently)i(general)g(for)h(us.)g(A)g |
|
4350 (\002rst-order)f(frame)n(w)o(ork)f(that)i(does)g(meet)g(our)f |
|
4351 (requirements)g(is)523 4924 y(the)17 b(notion)g(of)g |
|
4352 FP(nominal)f(alg)o(ebr)o(as)g FW(in)i([16].)e(The)h FP(nominal)f |
|
4353 (signatur)m(es)h FW(that)h(we)f(use)h(in)g(this)g(paper)p |
|
4354 eop end |
|
4355 %%Page: 5 5 |
|
4356 TeXDict begin 5 4 bop 523 448 a FW(are)26 b(a)h(mild)e(\(b)n(ut)h |
|
4357 (practically)f(useful\))g(generalisation)g(of)h(nominal)e(algebras)i |
|
4358 (in)g(which)f(name-)523 548 y(abstraction)h(and)g(pairing)g(can)g(be)h |
|
4359 (mix)o(ed)f(freely)g(in)h(arities)g(\(rather)f(than)g(insisting)h(as)h |
|
4360 (in)f([16])523 648 y(that)20 b(the)h(ar)o(gument)c(sort)k(of)f(a)g |
|
4361 (function)f(symbol)g(be)h(normalised)f(to)h(a)h(tuple)e(of)h |
|
4362 (abstractions\).)523 796 y FH(De\002nition)g(1.)41 b |
|
4363 FP(A)19 b FE(nominal)g(signature)g FP(is)h(speci\002ed)e(by:)g(a)h(set) |
|
4364 h(of)f FE(sorts)h(of)f(atoms)f FP(\(typical)g(sym-)523 |
|
4365 895 y(bol)j FQ(\027)5 b FP(\);)22 b(a)g(disjoint)f(set)i(of)f |
|
4366 FE(sorts)h(of)e(data)g FP(\(typical)g(symbol)g FQ(\016)s |
|
4367 FP(\);)h(and)f(a)g(set)i(of)f FE(function)g(symbols)523 |
|
4368 995 y FP(\(typical)g(symbol)h FQ(f)9 b FP(\),)22 b(eac)o(h)g(of)h(whic) |
|
4369 o(h)g(has)f(an)h FE(arity)g FP(of)g(the)g(form)g FQ(\034)38 |
|
4370 b FO(!)28 b FQ(\016)s FP(.)23 b(Her)m(e)h FQ(\034)33 |
|
4371 b FP(r)o(ang)o(es)22 b(o)o(ver)523 1094 y(\(compound\))d |
|
4372 FE(sorts)24 b FP(given)e(by)g(the)h(gr)o(ammar)e FQ(\034)37 |
|
4373 b FM(::=)27 b FQ(\027)33 b FO(j)27 b FQ(\016)k FO(j)c |
|
4374 FM(1)g FO(j)g FQ(\034)j FO(\002)20 b FQ(\034)37 b FO(j)27 |
|
4375 b(h)p FQ(\027)5 b FO(i)p FQ(\034)38 b FP(.)23 b(T)-8 |
|
4376 b(erms)24 b(of)f(sort)523 1194 y FO(h)p FQ(\027)5 b FO(i)p |
|
4377 FQ(\034)35 b FP(ar)m(e)25 b(binding)d(abstr)o(actions)h(of)h(atoms)g |
|
4378 (of)g(sort)h FQ(\027)30 b FP(o)o(ver)25 b(terms)g(of)f(sort)h |
|
4379 FQ(\034)9 b FP(.)25 b(W)-8 b(e)25 b(will)g(e)n(xplain)523 |
|
4380 1294 y(the)20 b(syntax)g(and)f(pr)l(operties)i(of)f(suc)o(h)g(terms)h |
|
4381 (in)f(a)h(moment.)523 1423 y(Example)e(1.)41 b FW(Here)20 |
|
4382 b(is)h(a)g(nominal)e(signature)g(for)g(e)o(xpressions)g(in)i(a)f(small) |
|
4383 h(fragment)d(of)i(ML:)706 1562 y FV(sort)f(of)g(atoms:)42 |
|
4384 b FD(vid)229 b FV(function)20 b(symbols:)h FC(vr)h FF(:)g |
|
4385 FD(vid)29 b FB(!)21 b FD(exp)706 1653 y FV(sort)e(of)g(data:)96 |
|
4386 b FD(exp)776 b FC(app)22 b FF(:)g FD(exp)h FB(\002)16 |
|
4387 b FD(exp)27 b FB(!)21 b FD(exp)2044 1744 y FC(fn)h FF(:)g |
|
4388 FB(h)p FD(vid)7 b FB(i)p FD(exp)27 b FB(!)21 b FD(exp)2044 |
|
4389 1836 y FC(lv)h FF(:)g FD(exp)g FB(\002)17 b(h)p FD(vid)8 |
|
4390 b FB(i)p FD(exp)26 b FB(!)c FD(exp)2044 1927 y FC(lf)g |
|
4391 FF(:)g FB(h)p FD(vid)7 b FB(i)p FF(\(\()p FB(h)p FD(vid)g |
|
4392 FB(i)p FD(exp)f FF(\))17 b FB(\002)f FD(exp)6 b FF(\))21 |
|
4393 b FB(!)g FD(exp)27 b FV(.)523 2068 y FW(The)34 b(function)f(symbol)g |
|
4394 FN(vr)i FW(constructs)e(terms)i(of)f(sort)g FA(exp)41 |
|
4395 b FW(representing)32 b(v)n(alue)i(identi\002ers)523 2167 |
|
4396 y(\(named)19 b(by)h(atoms)g(of)h(sort)f FA(vid)10 b FW(\);)21 |
|
4397 b FN(app)e FW(constructs)h(application)f(e)o(xpressions)g(from)h(pairs) |
|
4398 g(of)g(e)o(x-)523 2267 y(pressions;)d FN(fn)o FW(,)h |
|
4399 FN(lv)e FW(and)h FN(lf)g FW(construct)f(terms)h(representing)e |
|
4400 (respecti)n(v)o(ely)h(function)f(abstractions)523 2366 |
|
4401 y(\()p FN(fn)20 b(x)i FM(=)p FQ(>)g FN(e)p FW(\),)e(local)h(v)n(alue)e |
|
4402 (declarations)g(\()p FN(let)g(val)h(x)i FM(=)h FN(e1)d(in)g(e2)g(end)o |
|
4403 FW(\))g(and)g(local)g(recursi)n(v)o(e)523 2466 y(function)k |
|
4404 (declarations)g(\()p FN(let)19 b(fun)g(f)i(x)h FM(=)h |
|
4405 FN(e1)d(in)g(e2)g(end)n FW(\).)26 b(The)e(arities)i(of)f(the)g |
|
4406 (function)f(sym-)523 2566 y(bols)d(specify)g(which)g(are)h(binders)e |
|
4407 (and)h(in)h(which)f(w)o(ay)g(their)g(ar)o(guments)f(are)h(bound.)f |
|
4408 (This)h(kind)523 2665 y(of)k(speci\002cation)g(of)g(binding)f(scopes)h |
|
4409 (is)h(of)f(course)g(a)h(feature)e(of)i FP(higher)n(-or)m(der)e(abstr)o |
|
4410 (act)h(syn-)523 2765 y(tax)16 b FW([22],)f(using)h(function)e(types)i |
|
4411 FQ(\027)5 b FO(!)p FQ(\034)27 b FW(in)16 b(simply)g(typed)f |
|
4412 FQ(\025)p FW(-calculus)g(where)h(we)g(use)h(abstraction)523 |
|
4413 2865 y(sorts)k FO(h)p FQ(\027)5 b FO(i)p FQ(\034)k FW(.)22 |
|
4414 b(W)-7 b(e)22 b(shall)e(see)h(that)g(the)f(latter)h(ha)n(v)o(e)f(much)f |
|
4415 (more)h(elementary)f(\(indeed,)f(\002rst-order\))523 |
|
4416 2964 y(properties)27 b(compared)g(with)h(the)h(former)-5 |
|
4417 b(.)27 b(T)-7 b(o)29 b(mak)o(e)f(this)h(point)f(clear)g(we)h |
|
4418 (deliberately)e(use)i(a)523 3064 y(\002rst-order)g(syntax)g(for)g |
|
4419 (terms,)h(and)g FP(not)h FW(higher)n(-order)c(abstract)j(syntax,)f |
|
4420 (although)f(we)i(often)523 3163 y(refer)19 b(to)i(abstractions,)e |
|
4421 (binders)g(and)h(free)f(atoms)i(by)e(analogy)g(with)h(the)h |
|
4422 FQ(\025)p FW(-calculus.)523 3326 y FH(De\002nition)f(2.)41 |
|
4423 b FP(Given)26 b(a)g(nominal)f(signatur)m(e)o(,)g(we)h(assume)g(that)g |
|
4424 (ther)m(e)g(ar)m(e)g(countably)e(in\002nite)523 3425 |
|
4425 y(and)j(pairwise)i(disjoint)f(sets)i(of)e FE(atoms)g |
|
4426 FP(\(typical)g(symbol)g FQ(a)p FP(\))g(for)h(eac)o(h)f(sort)h(of)f |
|
4427 (atoms)g FQ(\027)5 b FP(,)29 b(and)523 3525 y FE(variables)22 |
|
4428 b FP(\(typical)g(symbol)g FQ(X)7 b FP(\))22 b(for)h(eac)o(h)f(sort)h |
|
4429 FQ(\034)9 b FP(.)24 b(The)f FE(terms)g FP(o)o(ver)g(a)f(nominal)g |
|
4430 (signatur)m(e)f(and)523 3624 y(their)27 b(sorts)h(ar)m(e)f(inductively) |
|
4431 f(de\002ned)f(as)i(follows,)g(wher)m(e)g(we)h(write)g |
|
4432 FQ(t)35 b FM(:)g FQ(\034)i FP(to)27 b(indicate)f(that)g(a)523 |
|
4433 3724 y(term)21 b FQ(t)g FP(has)f(sort)h FQ(\034)9 b FP(.)523 |
|
4434 3878 y FH(Unit)21 b(v)o(alue)40 b FO(hi)24 b FM(:)f(1)p |
|
4435 FP(.)523 3975 y FH(P)o(airs)41 b FO(h)p FQ(t)810 3987 |
|
4436 y FJ(1)848 3975 y FQ(;)14 b(t)915 3987 y FJ(2)952 3975 |
|
4437 y FO(i)23 b FM(:)g FQ(\034)1089 3987 y FJ(1)1145 3975 |
|
4438 y FO(\002)18 b FQ(\034)1264 3987 y FJ(2)1302 3975 y FP(,)i(if)h |
|
4439 FQ(t)1440 3987 y FJ(1)1501 3975 y FM(:)i FQ(\034)1583 |
|
4440 3987 y FJ(1)1641 3975 y FP(and)c FQ(t)1816 3987 y FJ(2)1877 |
|
4441 3975 y FM(:)k FQ(\034)1959 3987 y FJ(2)1996 3975 y FP(.)523 |
|
4442 4072 y FH(Data)40 b FQ(f)23 b(t)g FM(:)g FQ(\016)s FP(,)d(if)h |
|
4443 FQ(f)30 b FP(is)21 b(a)f(function)f(symbol)h(of)g(arity)h |
|
4444 FQ(\034)32 b FO(!)24 b FQ(\016)f FP(and)d FQ(t)j FM(:)g |
|
4445 FQ(\034)9 b FP(.)523 4169 y FH(Atoms)41 b FQ(a)23 b FM(:)g |
|
4446 FQ(\027)5 b FP(,)21 b(if)g FQ(a)g FP(is)g(an)e(atom)h(of)h(sort)g |
|
4447 FQ(\027)5 b FP(.)523 4267 y FH(Atom-abstraction)39 b |
|
4448 FQ(a:t)23 b FM(:)g FO(h)p FQ(\027)5 b FO(i)p FQ(\034)k |
|
4449 FP(,)22 b(if)f FQ(a)f FP(is)i(an)d(atom)h(of)g(sort)h |
|
4450 FQ(\027)26 b FP(and)20 b FQ(t)j FM(:)g FQ(\034)9 b FP(.)523 |
|
4451 4364 y FH(Suspension)42 b FQ(\031)s FK(\001)p FQ(X)d |
|
4452 FM(:)34 b FQ(\034)9 b FP(,)26 b(if)g FQ(\031)37 b FM(=)32 |
|
4453 b(\()p FQ(a)1624 4376 y FJ(1)1675 4364 y FQ(b)1711 4376 |
|
4454 y FJ(1)1748 4364 y FM(\)\()p FQ(a)1856 4376 y FJ(2)1908 |
|
4455 4364 y FQ(b)1944 4376 y FJ(2)1981 4364 y FM(\))14 b FO(\001)g(\001)g |
|
4456 (\001)f FM(\()p FQ(a)2213 4376 y FI(n)2273 4364 y FQ(b)2309 |
|
4457 4376 y FI(n)2354 4364 y FM(\))26 b FP(is)g(a)g(\002nite)f(list)i(whose) |
|
4458 e(elements)664 4463 y FM(\()p FQ(a)740 4475 y FI(i)782 |
|
4459 4463 y FQ(b)818 4475 y FI(i)845 4463 y FM(\))g FP(ar)m(e)f(pair)o(s)g |
|
4460 (of)g(atoms,)g(with)g FQ(a)1772 4475 y FI(i)1824 4463 |
|
4461 y FP(and)f FQ(b)2009 4475 y FI(i)2061 4463 y FP(of)h(the)f(same)i |
|
4462 (sort,)f(and)f FQ(X)31 b FP(is)24 b(a)g(variable)f(of)664 |
|
4463 4563 y(sort)e FQ(\034)9 b FP(.)21 b(In)f(the)g(case)h(that)f |
|
4464 FQ(\031)k FP(is)d(the)f(empty)g(list)h FM([])p FP(,)g(we)g(just)g |
|
4465 (write)g FQ(X)27 b FP(for)21 b FQ(\031)s FK(\001)p FQ(X)7 |
|
4466 b FP(.)523 4725 y FW(Recall)28 b(that)g(e)n(v)o(ery)e(\002nite)i |
|
4467 (permutation)d(can)i(be)h(e)o(xpressed)e(as)i(a)g(composition)e(of)h |
|
4468 (sw)o(appings)523 4825 y FM(\()p FQ(a)599 4837 y FI(i)641 |
|
4469 4825 y FQ(b)677 4837 y FI(i)704 4825 y FM(\))p FW(;)f(the)f(list)h |
|
4470 FQ(\031)j FW(of)c(pairs)g(of)f(atoms)h(occurring)e(in)i(a)h(suspension) |
|
4471 e(term)g FQ(\031)s FK(\001)p FQ(X)32 b FW(speci\002es)26 |
|
4472 b(a)f(\002-)523 4924 y(nite)c(permutation)d(of)i(atoms)h(w)o(aiting)f |
|
4473 (to)h(be)f(applied)g(once)f(we)i(kno)n(w)f(more)f(about)h(the)h(v)n |
|
4474 (ariable)p eop end |
|
4475 %%Page: 6 6 |
|
4476 TeXDict begin 6 5 bop 523 369 2882 4 v 523 975 4 607 |
|
4477 v 1156 532 a FF([])q Fz(\001)p FS(a)1288 489 y Fy(def)1295 |
|
4478 532 y FF(=)30 b FS(a)784 742 y FF(\(\()p FS(a)885 750 |
|
4479 y FG(1)932 742 y FS(a)973 750 y FG(2)1007 742 y FF(\))22 |
|
4480 b(::)g FS(\031)s FF(\))p Fz(\001)o FS(a)1288 699 y Fy(def)1295 |
|
4481 742 y FF(=)1385 566 y Fx(8)1385 634 y(>)1385 656 y(<)1385 |
|
4482 791 y(>)1385 813 y(:)1453 636 y FS(a)1494 644 y FG(1)1640 |
|
4483 636 y FV(if)c FS(\031)s Fz(\001)o FS(a)j FF(=)g FS(a)1959 |
|
4484 644 y FG(2)1453 745 y FS(a)1494 753 y FG(2)1640 745 y |
|
4485 FV(if)d FS(\031)s Fz(\001)o FS(a)j FF(=)g FS(a)1959 753 |
|
4486 y FG(1)1453 855 y FS(\031)s Fz(\001)o FS(a)75 b FV(otherwise)2520 |
|
4487 469 y FS(\031)s Fz(\001)p FB(hi)2675 426 y Fy(def)2682 |
|
4488 469 y FF(=)30 b FB(hi)2362 587 y FS(\031)s Fz(\001)o |
|
4489 FB(h)p FS(t)2491 595 y FG(1)2525 587 y FS(;)13 b(t)2587 |
|
4490 595 y FG(2)2622 587 y FB(i)2675 545 y Fy(def)2682 587 |
|
4491 y FF(=)30 b FB(h)p FS(\031)s Fz(\001)o FS(t)2901 595 |
|
4492 y FG(1)2935 587 y FS(;)14 b(\031)s Fz(\001)o FS(t)3069 |
|
4493 595 y FG(2)3103 587 y FB(i)2435 705 y FS(\031)s Fz(\001)o |
|
4494 FF(\()p FS(f)21 b(t)p FF(\))2675 663 y Fy(def)2682 705 |
|
4495 y FF(=)30 b FS(f)21 b FF(\()p FS(\031)s Fz(\001)o FS(t)p |
|
4496 FF(\))2431 824 y FS(\031)s Fz(\001)o FF(\()p FS(a:t)p |
|
4497 FF(\))2675 781 y Fy(def)2682 824 y FF(=)30 b(\()p FS(\031)s |
|
4498 Fz(\001)o FS(a)p FF(\))p FS(:)p FF(\()p FS(\031)s Fz(\001)o |
|
4499 FS(t)p FF(\))2358 942 y FS(\031)s Fz(\001)o FF(\()p FS(\031)2506 |
|
4500 910 y Fw(0)2528 942 y Fz(\001)o FS(X)6 b FF(\))2675 899 |
|
4501 y Fy(def)2682 942 y FF(=)30 b(\()p FS(\031)s FF(@)p FS(\031)2956 |
|
4502 910 y Fw(0)2977 942 y FF(\))p Fz(\001)p FS(X)d(:)p 3402 |
|
4503 975 V 523 978 2882 4 v 1360 1121 a FU(Fig)o(.)12 b(1.)18 |
|
4504 b FV(Permutation)h(action)h(on)f(terms,)f FS(\031)s Fz(\001)p |
|
4505 FS(t)p FV(.)523 1423 y FQ(X)36 b FW(\(by)28 b(substituting)f(for)h(it,) |
|
4506 i(for)e(e)o(xample\).)e(W)-7 b(e)30 b(represent)e(\002nite)h |
|
4507 (permutations)d(in)j(this)g(w)o(ay)523 1523 y(because)f(it)g(is)h |
|
4508 (really)f(the)g(operation)f(of)g(sw)o(apping)h(which)f(plays)h(a)h |
|
4509 (fundamental)c(role)j(in)g(the)523 1622 y(theory)-5 b(.)19 |
|
4510 b(Since,)i(semantically)g(speaking)f(\(see)h(Remark)g(1)g(belo)n(w)g |
|
4511 (about)f(semantics\),)h(sw)o(apping)523 1722 y(commutes)i(with)i(all)g |
|
4512 (term-forming)d(operations,)g(we)j(can)f(normalise)g(terms)g(in)m(v)n |
|
4513 (olving)f(an)h(e)o(x-)523 1821 y(plicit)j(sw)o(apping)f(operation)g(by) |
|
4514 g(pushing)g(the)h(sw)o(ap)g(in)h(as)f(f)o(ar)g(as)h(it)g(will)g(go,)e |
|
4515 (until)h(it)h(reaches)523 1921 y(a)d(v)n(ariable)e(\(cf.)g(Fig.)h(1\);) |
|
4516 g(the)g(terms)h(in)f(Def.)g(2)g(are)g(all)h(normalised)d(in)j(this)f(w) |
|
4517 o(ay)-5 b(,)24 b(with)g(e)o(xplicit)523 2021 y(sw)o(appings)19 |
|
4518 b(`piled)h(up')f(in)i(front)e(of)h(v)n(ariables)f(gi)n(ving)g(what)h |
|
4519 (we)h(ha)n(v)o(e)e(called)h FP(suspensions)p FW(.)648 |
|
4520 2128 y(W)-7 b(e)16 b(wish)g(to)g(gi)n(v)o(e)f(a)h(de\002nition)e(of)h |
|
4521 FQ(\013)p FW(-equi)n(v)n(alence)f(for)h(terms)g(o)o(v)o(er)g(a)h |
|
4522 (nominal)e(signature)g(that)523 2227 y(is)26 b(respected)e(by)g |
|
4523 (substitution)h(of)f(terms)h(for)g(v)n(ariables,)e(e)n(v)o(en)h(though) |
|
4524 g(the)g(latter)i(may)e(in)m(v)n(olv)o(e)523 2327 y(capture)19 |
|
4525 b(of)g(atoms)h(by)f(binders.)g(T)-7 b(o)20 b(do)f(so)h(we)h(will)f |
|
4526 (need)f(to)h(mak)o(e)g(use)g(of)f(an)h(auxiliary)e(relation)523 |
|
4527 2426 y(of)24 b FP(fr)m(eshness)h FW(between)e(atoms)i(and)e(terms,)h |
|
4528 (whose)g(intended)f(meaning)g(is)i(that)g(the)f(atom)g(does)523 |
|
4529 2526 y(not)j(occur)f(free)h(in)h(an)o(y)e(substitution)h(instance)g(of) |
|
4530 g(the)g(term.)g(As)h(discussed)f(in)h(the)f(Introduc-)523 |
|
4531 2626 y(tion,)21 b(our)f(judgements)f(about)i(term)f(equi)n(v)n(alence)f |
|
4532 (\()p FQ(t)25 b FO(\031)f FQ(t)2227 2596 y FL(0)2250 |
|
4533 2626 y FW(\))e(need)e(to)h(contain)f(hypotheses)f(about)523 |
|
4534 2725 y(the)26 b(freshness)g(of)f(atoms)h(with)h(respect)e(to)i(v)n |
|
4535 (ariables)e(\()p FQ(a)33 b FM(#)i FQ(X)7 b FW(\);)25 |
|
4536 b(and)h(the)g(same)g(goes)g(for)f(our)523 2825 y(judgements)g(about)g |
|
4537 (freshness)h(itself)g(\()p FQ(a)34 b FM(#)g FQ(t)p FW(\).)27 |
|
4538 b(Figure)e(2)h(gi)n(v)o(es)g(a)h(syntax-directed)c(inducti)n(v)o(e)523 |
|
4539 2925 y(de\002nition)c(of)h(equi)n(v)n(alence)e(and)i(freshness)f(using) |
|
4540 h(judgements)f(of)h(the)g(form)1463 3122 y FO(r)k(`)e |
|
4541 FQ(t)i FO(\031)e FQ(t)1800 3087 y FL(0)1906 3122 y FW(and)82 |
|
4542 b FO(r)24 b(`)e FQ(a)h FM(#)h FQ(t)523 3319 y FW(where)d |
|
4543 FQ(t)h FW(and)f FQ(t)972 3289 y FL(0)1018 3319 y FW(are)g(terms)h(of)f |
|
4544 (the)h(same)g(sort)g(o)o(v)o(er)e(a)i(gi)n(v)o(en)e(nominal)h |
|
4545 (signature,)f FQ(a)i FW(is)h(an)e(atom,)523 3419 y(and)31 |
|
4546 b(the)g FH(fr)o(eshness)g(en)m(vir)o(onment)f FO(r)i |
|
4547 FW(is)g(a)g(\002nite)f(set)h(of)f FH(fr)o(eshness)g(constraints)g |
|
4548 FQ(a)43 b FM(#)h FQ(X)7 b FW(,)523 3518 y(each)21 b(speci\002ed)g(by)g |
|
4549 (an)h(atom)f(and)f(a)i(v)n(ariable.)e(Rules)i(\()p FO(\031)p |
|
4550 FW(-abstraction-2\),)c(\()p FO(\031)p FW(-suspension\))h(and)523 |
|
4551 3618 y(\()p FM(#)p FW(-suspension\))f(in)j(Fig.)f(2)g(mak)o(e)g(use)g |
|
4552 (of)g(the)g(follo)n(wing)f(de\002nitions.)523 3828 y |
|
4553 FH(De\002nition)h(3.)41 b FP(Recall)21 b(fr)l(om)h(Def)o(.)g(2)g(that)f |
|
4554 (we)h(specify)g FE(\002nite)g(permutations)f(of)g(atoms)g |
|
4555 FP(by)h(\002nite)523 3928 y(lists)17 b FM(\()p FQ(a)749 |
|
4556 3940 y FJ(1)801 3928 y FQ(b)837 3940 y FJ(1)873 3928 |
|
4557 y FQ(;)d FM(\)\()p FQ(a)1018 3940 y FJ(2)1070 3928 y |
|
4558 FQ(b)1106 3940 y FJ(2)1143 3928 y FM(\))g FO(\001)g(\001)g(\001)g |
|
4559 FM(\()p FQ(a)1376 3940 y FI(n)1435 3928 y FQ(b)1471 3940 |
|
4560 y FI(n)1516 3928 y FM(\))i FP(r)m(epr)m(esenting)f(the)h(composition)f |
|
4561 (of)h(\002nitely)f(many)g(swappings)523 4028 y FM(\()p |
|
4562 FQ(a)599 4040 y FI(i)641 4028 y FQ(b)677 4040 y FI(i)704 |
|
4563 4028 y FM(\))p FP(,)25 b(with)h FQ(a)995 4040 y FI(i)1048 |
|
4564 4028 y FP(and)e FQ(b)1234 4040 y FI(i)1287 4028 y FP(of)h(the)g(same)g |
|
4565 (sort.)g(Since)g(we)g(will)h(apply)e(permutations)g(to)h(terms)h(on)523 |
|
4566 4127 y(the)g(left,)g(the)f(or)m(der)h(of)g(the)f(composition)f(is)j(fr) |
|
4567 l(om)f(right)g(to)g(left.)g(So)f(with)h(this)g(r)m(epr)m(esentation,) |
|
4568 523 4227 y(the)32 b(composition)f(of)i(a)f(permutation)f |
|
4569 FQ(\031)37 b FP(followed)32 b(by)g(a)h(swap)f FM(\()p |
|
4570 FQ(a)14 b(b)p FM(\))33 b FP(is)h(given)d(by)i(list-cons,)523 |
|
4571 4327 y(written)28 b FM(\()p FQ(a)14 b(b)p FM(\))36 b(::)g |
|
4572 FQ(\031)s FP(;)28 b(the)f(composition)f(of)h FQ(\031)k |
|
4573 FP(followed)c(by)g(another)f(permutation)g FQ(\031)3091 |
|
4574 4296 y FL(0)3142 4327 y FP(is)i(given)523 4426 y(by)22 |
|
4575 b(list-concatenation,)d(written)j(as)h FQ(\031)1671 4396 |
|
4576 y FL(0)1694 4426 y FM(@)p FQ(\031)s FP(;)f(the)g FE(identity)g |
|
4577 (permutation)g FP(is)h(given)e(by)h(the)g(empty)523 4526 |
|
4578 y(list)28 b FM([])p FP(;)e(and)g(the)g FE(in)m(v)o(erse)h |
|
4579 FP(of)g(a)f(permutation)f(is)j(given)d(by)i(list)g(r)m(e)o(ver)o(sal,)g |
|
4580 (written)g(as)g FQ(\031)3143 4496 y FL(\000)p FJ(1)3233 |
|
4581 4526 y FP(.)f(The)523 4625 y FE(permutation)21 b(action)p |
|
4582 FP(,)f FQ(\031)s FK(\001)p FQ(t)p FP(,)i(of)f(a)g(\002nite)g |
|
4583 (permutation)f(of)h(atoms)g FQ(\031)k FP(on)c(a)g(term)h |
|
4584 FQ(t)g FP(is)g(de\002ned)e(as)h(in)523 4725 y(F)l(ig)o(.)j(1;)g(it)h |
|
4585 (pushes)f(the)h(list)g FQ(\031)k FP(into)24 b(the)g(structur)m(e)h(of)g |
|
4586 (the)f(term)h FQ(t)g FP(until)g(it)g(`piles)f(up')g(in)h(fr)l(ont)f(of) |
|
4587 523 4825 y(suspensions)g(\(applying)f(the)i(actual)f(permutation)f |
|
4588 (that)i FQ(\031)k FP(r)m(epr)m(esents)c(to)g(atoms)g(that)f(it)i(meets) |
|
4589 523 4924 y(on)31 b(the)g(way\).)f(The)h FE(disagreement)f(set)i |
|
4590 FP(of)f(two)h(permutations)d FQ(\031)35 b FP(and)30 b |
|
4591 FQ(\031)2757 4894 y FL(0)2812 4924 y FP(\(used)h(in)g(rule)g(\()p |
|
4592 FO(\031)p FP(-)p eop end |
|
4593 %%Page: 7 7 |
|
4594 TeXDict begin 7 6 bop 523 369 2882 4 v 523 1702 4 1334 |
|
4595 v 556 483 376 4 v 556 555 a FB(r)21 b(`)g(hi)h(\031)f(hi)953 |
|
4596 499 y Fv(\()p Fu(\031)p Fv(-unit\))1220 446 y FB(r)h(`)f |
|
4597 FS(t)1402 454 y FG(1)1458 446 y FB(\031)g FS(t)1567 415 |
|
4598 y Fw(0)1567 459 y FG(1)1676 446 y FB(r)g(`)g FS(t)1857 |
|
4599 454 y FG(2)1913 446 y FB(\031)g FS(t)2022 415 y Fw(0)2022 |
|
4600 459 y FG(2)p 1220 479 836 4 v 1292 555 a FB(r)g(`)g(h)p |
|
4601 FS(t)1503 563 y FG(1)1538 555 y FS(;)13 b(t)1600 563 |
|
4602 y FG(2)1634 555 y FB(i)21 b(\031)g(h)p FS(t)1824 524 |
|
4603 y Fw(0)1824 569 y FG(1)1859 555 y FS(;)13 b(t)1921 524 |
|
4604 y Fw(0)1921 569 y FG(2)1955 555 y FB(i)2078 495 y Fv(\()p |
|
4605 Fu(\031)p Fv(-pair\))2403 459 y FB(r)22 b(`)f FS(t)g |
|
4606 FB(\031)g FS(t)2715 428 y Fw(0)p 2345 479 451 4 v 2345 |
|
4607 555 a FB(r)h(`)f FS(f)g(t)g FB(\031)g FS(f)g(t)2773 524 |
|
4608 y Fw(0)2817 495 y Fv(\()p Fu(\031)p Fv(-function)e(symbol\))613 |
|
4609 694 y FB(r)j(`)f FS(t)g FB(\031)g FS(t)925 662 y Fw(0)p |
|
4610 551 714 458 4 v 551 790 a FB(r)h(`)f FS(a:t)g FB(\031)g |
|
4611 FS(a:t)987 758 y Fw(0)1031 730 y Fv(\()p Fu(\031)p Fv(-abstraction-1\)) |
|
4612 1648 675 y FS(a)g FB(6)p FF(=)g FS(a)1832 643 y Fw(0)1929 |
|
4613 675 y FB(r)h(`)f FS(t)g FB(\031)g FF(\()p FS(a)12 b(a)2337 |
|
4614 643 y Fw(0)2359 675 y FF(\))p Fz(\001)p FS(t)2442 643 |
|
4615 y Fw(0)2539 675 y FB(r)21 b(`)g FS(a)g FF(#)g FS(t)2867 |
|
4616 643 y Fw(0)p 1648 714 1242 4 v 2029 790 a FB(r)g(`)g |
|
4617 FS(a:t)h FB(\031)f FS(a)2416 758 y Fw(0)2438 790 y FS(:t)2487 |
|
4618 758 y Fw(0)2911 730 y Fv(\()p Fu(\031)p Fv(-abstraction-2\))p |
|
4619 805 937 338 4 v 805 1005 a FB(r)g(`)h FS(a)f FB(\031)g |
|
4620 FS(a)1164 953 y Fv(\()p Fu(\031)p Fv(-atom\))1610 891 |
|
4621 y FF(\()p FS(a)g FF(#)h FS(X)6 b FF(\))21 b FB(2)h(r)f |
|
4622 FV(for)e(all)i FS(a)g FB(2)g FD(ds)7 b FF(\()p FS(\031)s(;)13 |
|
4623 b(\031)2637 859 y Fw(0)2659 891 y FF(\))p 1610 929 1079 |
|
4624 4 v 1870 1005 a FB(r)21 b(`)g FS(\031)s Fz(\001)p FS(X)27 |
|
4625 b FB(\031)21 b FS(\031)2313 973 y Fw(0)2335 1005 y Fz(\001)p |
|
4626 FS(X)2710 945 y Fv(\()p Fu(\031)p Fv(-suspension\))p |
|
4627 562 1133 361 4 v 562 1206 a FB(r)g(`)g FS(a)g FF(#)g |
|
4628 FB(hi)944 1150 y Fv(\()p Ft(#)p Fv(-unit\))1260 1099 |
|
4629 y FB(r)g(`)h FS(a)f FF(#)g FS(t)1589 1107 y FG(1)1698 |
|
4630 1099 y FB(r)g(`)h FS(a)f FF(#)g FS(t)2027 1107 y FG(2)p |
|
4631 1260 1133 802 4 v 1401 1206 a FB(r)g(`)h FS(a)f FF(#)g |
|
4632 FB(h)p FS(t)1760 1214 y FG(1)1794 1206 y FS(;)13 b(t)1856 |
|
4633 1214 y FG(2)1891 1206 y FB(i)2083 1149 y Fv(\()p Ft(#)p |
|
4634 Fv(-pair\))2428 1103 y FB(r)21 b(`)h FS(a)f FF(#)g FS(t)p |
|
4635 2399 1137 387 4 v 2399 1206 a FB(r)g(`)h FS(a)f FF(#)g |
|
4636 FS(f)g(t)2808 1153 y Fv(\()p Ft(#)p Fv(-function)e(symbol\))p |
|
4637 874 1367 391 4 v 874 1436 a FB(r)i(`)h FS(a)f FF(#)g |
|
4638 FS(a:t)1287 1384 y Fv(\()p Ft(#)p Fv(-abstraction-1\))1951 |
|
4639 1325 y FS(a)g FB(6)p FF(=)g FS(a)2135 1294 y Fw(0)2231 |
|
4640 1325 y FB(r)h(`)f FS(a)g FF(#)g FS(t)p 1951 1360 610 |
|
4641 4 v 2049 1436 a FB(r)g(`)g FS(a)g FF(#)h FS(a)2391 1404 |
|
4642 y Fw(0)2413 1436 y FS(:t)2582 1376 y Fv(\()p Ft(#)p Fv |
|
4643 (-abstraction-2\))1112 1552 y FS(a)f FB(6)p FF(=)g FS(a)1296 |
|
4644 1520 y Fw(0)p 1033 1587 364 4 v 1033 1663 a FB(r)g(`)g |
|
4645 FS(a)g FF(#)h FS(a)1375 1631 y Fw(0)1419 1603 y Fv(\()p |
|
4646 Ft(#)p Fv(-atom\))1869 1556 y FF(\()p FS(\031)1946 1524 |
|
4647 y Fw(\000)p FG(1)2028 1556 y Fz(\001)o FS(a)f FF(#)h |
|
4648 FS(X)6 b FF(\))21 b FB(2)h(r)p 1869 1594 588 4 v 1942 |
|
4649 1663 a(r)f(`)h FS(a)f FF(#)g FS(\031)s Fz(\001)o FS(X)2478 |
|
4650 1610 y Fv(\()p Ft(#)p Fv(-suspension\))p 3402 1702 4 |
|
4651 1334 v 523 1705 2882 4 v 1374 1849 a FU(Fig)o(.)12 b(2.)18 |
|
4652 b FV(Inducti)n(v)o(e)i(de\002nition)f(of)g FB(\031)g |
|
4653 FV(and)g FF(#)p FV(.)523 2139 y FP(suspension\))g(in)h(F)l(ig)o(.)g |
|
4654 (2\))g(is)h(de\002ned)e(by)1442 2351 y FA(ds)8 b FM(\()p |
|
4655 FQ(\031)s(;)14 b(\031)1695 2316 y FL(0)1719 2351 y FM(\))1774 |
|
4656 2299 y Fs(def)1777 2351 y FM(=)26 b FO(f)p FQ(a)c FO(j)i |
|
4657 FQ(\031)s FK(\001)o FQ(a)f FO(6)p FM(=)g FQ(\031)2304 |
|
4658 2316 y FL(0)2328 2351 y FK(\001)o FQ(a)p FO(g)g FQ(:)822 |
|
4659 b FW(\(7\))523 2538 y(Note)17 b(that)g(e)n(v)o(ery)f(disagreement)f |
|
4660 (set)j(of)f(the)g(lists)i FQ(\031)i FW(and)16 b FQ(\031)2226 |
|
4661 2508 y FL(0)2267 2538 y FW(is)i(a)g(subset)f(of)g(the)g |
|
4662 FP(\002nite)g FW(set)h(of)f(atoms)523 2638 y(occurring)e(in)i(either)f |
|
4663 (of)h(the)f(lists,)i(because)f(if)g FQ(a)g FW(does)f(not)h(occur)f(in)h |
|
4664 (those)f(lists,)i(then)f(from)f(Fig.)g(1)523 2737 y(we)21 |
|
4665 b(get)f FQ(\031)s FK(\001)p FQ(a)j FM(=)f FQ(a)h FM(=)g |
|
4666 FQ(\031)1199 2707 y FL(0)1222 2737 y FK(\001)p FQ(a)p |
|
4667 FW(.)d(T)-7 b(o)21 b(illustrate)f(the)g(use)h(of)f(disagreement)e |
|
4668 (sets,)j(consider)1235 2923 y FO(f)p FQ(a)h FM(#)i FQ(X)6 |
|
4669 b(;)14 b(c)23 b FM(#)g FQ(X)6 b FO(g)23 b(`)g FM(\()p |
|
4670 FQ(a)14 b(c)p FM(\)\()p FQ(a)g(b)p FM(\))p FK(\001)o |
|
4671 FQ(X)30 b FO(\031)22 b FM(\()p FQ(b)14 b(c)p FM(\))p |
|
4672 FK(\001)p FQ(X)523 3109 y FW(which)h(holds)f(by)h(\()p |
|
4673 FO(\031)p FW(-suspension\),)e(since)i(the)g(disagreement)f(set)i(of)8 |
|
4674 b FM(\()p FQ(a)14 b(c)p FM(\))-8 b(\()p FQ(a)14 b(b)p |
|
4675 FM(\))8 b FW(and)g FM(\()p FQ(b)14 b(c)p FM(\))8 b FW(is)g |
|
4676 FO(f)p FQ(a;c)p FO(g)p FW(.)523 3292 y FH(Lemma)21 b(1.)41 |
|
4677 b FO(r)30 b(`)f(\000)g(\031)h(\000)24 b FP(is)g(an)g(equivalence)e(r)m |
|
4678 (elation;)h(it)h(is)h(pr)m(eserved)f(by)f(all)h(of)g(the)g(term-)523 |
|
4679 3391 y(forming)g(oper)o(ations)f(in)h(Def)o(.)h(2;)f(and)f(it)i(r)m |
|
4680 (espects)g(the)g(fr)m(eshness)g(r)m(elation)f(\(i.e)o(.)f(if)j |
|
4681 FO(r)31 b(`)f FQ(a)h FM(#)g FQ(t)523 3491 y FP(and)24 |
|
4682 b FO(r)32 b(`)f FQ(t)h FO(\031)f FQ(t)1044 3461 y FL(0)1067 |
|
4683 3491 y FP(,)26 b(then)e FO(r)32 b(`)f FQ(a)h FM(#)g FQ(t)1672 |
|
4684 3461 y FL(0)1695 3491 y FP(\).)25 b(Both)f FO(\031)h |
|
4685 FP(and)f FM(#)i FP(ar)m(e)f(pr)m(eserved)f(by)h(the)g(permutation)523 |
|
4686 3591 y(action)c(given)h(in)g(F)l(ig)o(.)g(1)g(in)g(the)g(following)g |
|
4687 (sense:)g(if)35 b FO(r)26 b(`)h FQ(t)f FO(\031)g FQ(t)2470 |
|
4688 3560 y FL(0)2494 3591 y FP(,)c(then)g FO(r)k(`)h FQ(\031)s |
|
4689 FK(\001)o FQ(t)g FO(\031)f FQ(\031)s FK(\001)p FQ(t)3207 |
|
4690 3560 y FL(0)3230 3591 y FP(;)c(and)523 3690 y(if)33 b |
|
4691 FO(r)23 b(`)g FQ(a)g FM(#)g FQ(t)p FP(,)e(then)f FO(r)j(`)g |
|
4692 FQ(\031)s FK(\001)o FQ(a)g FM(#)h FQ(\031)s FK(\001)o |
|
4693 FQ(t)p FP(.)523 3790 y(Pr)l(oof)o(.)e FW(Although)i(reasoning)h(about)g |
|
4694 FO(\031)h FW(and)g FM(#)h FW(is)g(rather)e(pleasant)h(once)f(the)h(abo) |
|
4695 o(v)o(e)f(f)o(acts)i(are)523 3889 y(pro)o(v)o(ed,)g(establishing)i |
|
4696 (them)h(\002rst)g(is)h(rather)e(trick)o(y\227mainly)f(because)h(of)h |
|
4697 (the)f(lar)o(ge)g(number)523 3989 y(of)21 b(cases,)i(b)n(ut)e(also)h |
|
4698 (because)f(the)h(f)o(acts)g(in)g(the)f(lemma)g(are)h(inter)n |
|
4699 (-dependent;)d(in)i(addition)g(some)523 4089 y(further)14 |
|
4700 b(properties)g(of)h(the)h(permutation)d(action)i(and)g(disagreement)f |
|
4701 (sets)j(need)e(to)g(be)h(established)523 4188 y(\002rst)21 |
|
4702 b(\(statements)f(omitted\).)1376 4158 y Fy(1)3350 4188 |
|
4703 y FO(u)-55 b(t)648 4369 y FW(The)22 b(main)g(reason)f(for)h(using)g |
|
4704 (suspensions)g(in)h(the)f(syntax)g(of)g(terms)h(is)g(to)g(enable)e(a)i |
|
4705 (de\002ni-)523 4469 y(tion)e(of)g FP(substitution)f(of)i(terms)g(for)g |
|
4706 (variables)f FW(which)g(allo)n(ws)g(capture)f(of)h(free)g(atoms)g(by)g |
|
4707 (atom-)523 4568 y(abstractions)c(while)h(still)h(respecting)d |
|
4708 FQ(\013)p FW(-equi)n(v)n(alence.)f(The)j(follo)n(wing)e(lemma)h |
|
4709 (establishes)h(this.)523 4668 y(First)j(we)g(gi)n(v)o(e)e(some)h |
|
4710 (terminology)e(and)h(notation)g(for)h(term-substitution.)p |
|
4711 523 4746 473 4 v 558 4801 a Fr(1)606 4833 y FV(A)h(machine-check)o(ed)j |
|
4712 (proof)d(of)g(all)g(the)g(results)g(using)h(the)f(theorem)g(pro)o(v)o |
|
4713 (er)h(Isabelle)f(can)h(be)f(found)h(at)606 4924 y FC(http)g |
|
4714 FF(:)g FS(==)p FC(www)p FS(:)p FC(cl)p FS(:)p FC(cam)p |
|
4715 FS(:)p FC(ac)p FS(:)p FC(uk)p FS(=)1480 4916 y Fw(\030)1538 |
|
4716 4924 y FC(cu200)p FS(=)p FC(Unification)t FV(.)p eop |
|
4717 end |
|
4718 %%Page: 8 8 |
|
4719 TeXDict begin 8 7 bop 523 448 a FH(De\002nition)20 b(4.)41 |
|
4720 b FP(A)32 b FE(substitution)h FQ(\033)j FP(is)d(a)f(sort-r)m(especting) |
|
4721 f(function)f(fr)l(om)j(variables)e(to)h(terms)523 548 |
|
4722 y(with)22 b(the)g(pr)l(operty)g(that)f FQ(\033)s FM(\()p |
|
4723 FQ(X)7 b FM(\))27 b(=)e FQ(X)k FP(for)22 b(all)g(b)n(ut)g(\002nitely)g |
|
4724 (many)f(variables)g FQ(X)7 b FP(.)22 b(W)-8 b(e)22 b(shall)g(write)523 |
|
4725 648 y Fq(dom)8 b Fp(\()p Fo(\033)s Fp(\))23 b FP(for)f(the)f(\002nite)g |
|
4726 (set)h(of)g(variables)f FQ(X)28 b FP(satisfying)21 b |
|
4727 FQ(\033)s FM(\()p FQ(X)7 b FM(\))26 b FO(6)p FM(=)f FQ(X)7 |
|
4728 b FP(.)21 b(If)g FA(dom)7 b FM(\()p FQ(\033)s FM(\))23 |
|
4729 b FP(consists)f(of)523 747 y(variables)e FQ(X)922 759 |
|
4730 y FJ(1)959 747 y FQ(;)14 b(:)g(:)g(:)f(;)h(X)1212 759 |
|
4731 y FI(n)1278 747 y FP(and)19 b FQ(\033)s FM(\()p FQ(X)1574 |
|
4732 759 y FI(i)1602 747 y FM(\))24 b(=)e FQ(t)1775 759 y |
|
4733 FI(i)1824 747 y FP(for)e FQ(i)j FM(=)g(1)p FQ(::n)p FP(,)d(we)h(shall)f |
|
4734 (sometimes)h(write)g FQ(\033)j FP(as)1442 896 y FQ(\033)i |
|
4735 FM(=)d([)p FQ(X)1695 908 y FJ(1)1755 896 y FM(:=)g FQ(t)1896 |
|
4736 908 y FJ(1)1933 896 y FQ(;)14 b(:)g(:)g(:)f(;)h(X)2186 |
|
4737 908 y FI(n)2254 896 y FM(:=)23 b FQ(t)2395 908 y FI(n)2440 |
|
4738 896 y FM(])p FQ(:)822 b FW(\(8\))523 1044 y FP(W)-8 b(e)27 |
|
4739 b(write)h FQ(\033)s FM(\()p FQ(t)p FM(\))g FP(for)f(the)f(r)m(esult)h |
|
4740 (of)g FE(applying)e(a)h(substitution)i FQ(\033)i FP(to)d(a)f(term)h |
|
4741 FQ(t)p FP(;)g(this)g(is)g(the)g(term)523 1144 y(obtained)19 |
|
4742 b(fr)l(om)i FQ(t)h FP(by)f(r)m(eplacing)e(eac)o(h)h(suspension)g |
|
4743 FQ(\031)s FK(\001)p FQ(X)27 b FP(in)21 b FQ(t)h FP(\(as)e |
|
4744 FQ(X)28 b FP(r)o(ang)o(es)20 b(o)o(ver)h FA(dom)8 b FM(\()p |
|
4745 FQ(\033)s FM(\))p FP(\))22 b(by)523 1244 y(the)17 b(term)h |
|
4746 FQ(\031)s FK(\001)p FQ(\033)s FM(\()p FQ(X)7 b FM(\))18 |
|
4747 b FP(got)e(by)h(letting)g FQ(\031)k FP(act)c(on)g(the)g(term)h |
|
4748 FQ(\033)s FM(\()p FQ(X)7 b FM(\))17 b FP(using)g(the)g(de\002nition)e |
|
4749 (in)j(F)l(ig)o(.)e(1.)h(F)-9 b(or)523 1343 y(e)n(xample)o(,)25 |
|
4750 b(if)h FQ(\033)35 b FM(=)d([)p FQ(X)25 b FM(:=)18 b FO(h)p |
|
4751 FQ(b;)c(Y)19 b FO(i)p FM(])26 b FP(and)e FQ(t)33 b FM(=)f |
|
4752 FQ(a:)p FM(\()p FQ(a)14 b(b)p FM(\))p FK(\001)o FQ(X)7 |
|
4753 b FP(,)25 b(then)f FQ(\033)s FM(\()p FQ(t)p FM(\))34 |
|
4754 b(=)e FQ(a:)p FO(h)p FQ(a;)14 b FM(\()p FQ(a)g(b)p FM(\))p |
|
4755 FK(\001)o FQ(Y)19 b FO(i)p FP(.)26 b(Given)523 1443 y(substitutions)20 |
|
4756 b FQ(\033)k FP(and)19 b FQ(\033)1230 1413 y FL(0)1254 |
|
4757 1443 y FP(,)h(and)g(fr)m(eshness)g(en)m(vir)l(onments)f |
|
4758 FO(r)i FP(and)f FO(r)2533 1413 y FL(0)2556 1443 y FP(,)h(we)g(write) |
|
4759 1161 1592 y FM(\()p FQ(a)p FM(\))84 b FO(r)1422 1557 |
|
4760 y FL(0)1468 1592 y FO(`)23 b FQ(\033)s FM(\()p FO(r)p |
|
4761 FM(\))167 b FP(and)e FM(\()p FQ(b)p FM(\))83 b FO(r)23 |
|
4762 b(`)g FQ(\033)j FO(\031)d FQ(\033)2743 1557 y FL(0)3308 |
|
4763 1592 y FW(\(9\))523 1740 y FP(to)32 b(mean)f(that)g(\(for)h(a\))f |
|
4764 FO(r)1320 1710 y FL(0)1388 1740 y FO(`)44 b FQ(a)g FM(#)g |
|
4765 FQ(\033)s FM(\()p FQ(X)7 b FM(\))33 b FP(holds)e(for)h(eac)o(h)f |
|
4766 FM(\()p FQ(a)44 b FM(#)h FQ(X)7 b FM(\))43 b FO(2)i(r)32 |
|
4767 b FP(and)f(\(for)h(b\))523 1840 y FO(r)23 b(`)g FQ(\033)s |
|
4768 FM(\()p FQ(X)7 b FM(\))23 b FO(\031)g FQ(\033)1040 1810 |
|
4769 y FL(0)1064 1840 y FM(\()p FQ(X)7 b FM(\))20 b FP(holds)g(for)h(all)f |
|
4770 FQ(X)29 b FO(2)24 b FA(dom)7 b FM(\()p FQ(\033)s FM(\))19 |
|
4771 b FO([)g FA(dom)7 b FM(\()p FQ(\033)2435 1810 y FL(0)2459 |
|
4772 1840 y FM(\))p FP(.)523 1980 y FH(Lemma)21 b(2)41 b(\(Substitution\).) |
|
4773 14 b FP(Substitution)g(commutes)h(with)h(the)f(permutation)e(action:)h |
|
4774 FQ(\033)s FM(\()p FQ(\031)s FK(\001)q FQ(t)p FM(\))23 |
|
4775 b(=)523 2080 y FQ(\031)s FK(\001)p FM(\()p FQ(\033)s |
|
4776 FM(\()p FQ(t)p FM(\)\))p FP(.)f(Substitution)c(pr)m(eserves)j |
|
4777 FO(\031)g FP(and)e FM(#)i FP(in)f(the)g(following)g(sense:)581 |
|
4778 2196 y FO(\017)41 b FP(if)21 b FO(r)800 2166 y FL(0)847 |
|
4779 2196 y FO(`)h FQ(\033)s FM(\()p FO(r)p FM(\))g FP(and)e |
|
4780 FO(r)j(`)g FQ(t)g FO(\031)f FQ(t)1607 2166 y FL(0)1631 |
|
4781 2196 y FP(,)e(then)g FO(r)1905 2166 y FL(0)1951 2196 |
|
4782 y FO(`)j FQ(\033)s FM(\()p FQ(t)p FM(\))h FO(\031)f FQ(\033)s |
|
4783 FM(\()p FQ(t)2393 2166 y FL(0)2417 2196 y FM(\))p FP(;)581 |
|
4784 2289 y FO(\017)41 b FP(if)21 b FO(r)800 2259 y FL(0)847 |
|
4785 2289 y FO(`)h FQ(\033)s FM(\()p FO(r)p FM(\))g FP(and)e |
|
4786 FO(r)j(`)g FQ(a)g FM(#)g FQ(t)p FP(,)d(then)g FO(r)1900 |
|
4787 2259 y FL(0)1947 2289 y FO(`)i FQ(a)h FM(#)h FQ(\033)s |
|
4788 FM(\()p FQ(t)p FM(\))p FP(.)523 2411 y(Pr)l(oof)o(.)e |
|
4789 FW(The)d(\002rst)i(sentence)e(follo)n(ws)h(by)f(induction)f(on)h(the)h |
|
4790 (structure)f(of)g FQ(t)p FW(.)i(The)e(second)g(follo)n(ws)523 |
|
4791 2511 y(by)h(induction)f(on)h(the)h(proofs)e(of)h FO(r)k(`)f |
|
4792 FQ(t)h FO(\031)f FQ(t)1851 2481 y FL(0)1896 2511 y FW(and)d |
|
4793 FO(r)k(`)f FQ(a)h FM(#)g FQ(t)d FW(from)e(the)i(rules)f(in)h(Fig.)g(2,) |
|
4794 f(using)523 2611 y(the)g(\002rst)h(sentence)f(and)f(the)i(\(proof)d |
|
4795 (of\))h(Lemma)h(1.)1302 b FO(u)-55 b(t)648 2751 y FW(W)-7 |
|
4796 b(e)27 b(claim)f(that)g(the)h(relation)e FO(\031)h FW(de\002ned)f(in)i |
|
4797 (Fig.)f(2)g(gi)n(v)o(es)g(the)g(correct)f(notion)g(of)h |
|
4798 FQ(\013)p FW(-equi-)523 2851 y(v)n(alence)19 b(for)h(terms)h(o)o(v)o |
|
4799 (er)e(a)i(nominal)e(signature.)g(This)h(is)i(reasonable,)c(gi)n(v)o(en) |
|
4800 i(Lemma)f(1)i(and)e(the)523 2950 y(f)o(act)e(that,)g(by)f |
|
4801 (de\002nition,)f(it)j(satis\002es)g(rules)f(\()p FO(\031)p |
|
4802 FW(-abstraction-1\))c(and)j(\()p FO(\031)p FW(-abstraction-2\).)e |
|
4803 (Further)523 3050 y(e)n(vidence)23 b(is)j(pro)o(vided)c(by)i(the)h |
|
4804 (follo)n(wing)e(theorem,)g(which)h(sho)n(ws)h(that)f(for)g(ground)f |
|
4805 (terms)h FO(\031)523 3149 y FW(agrees)c(with)g(the)g(follo)n(wing)f |
|
4806 (more)g(traditional)g(de\002nition)g(of)h FQ(\013)p FW(-equi)n(v)n |
|
4807 (alence.)523 3290 y FH(De\002nition)g(5)41 b(\(Na)n(\250)-26 |
|
4808 b(\021v)o(e)20 b Fo(\013)p FH(-equi)o(v)o(alence\).)f |
|
4809 FP(De\002ne)h(the)g(binary)g(r)m(elation)f FQ(t)k FM(=)2797 |
|
4810 3302 y FI(\013)2854 3290 y FQ(t)2884 3260 y FL(0)2928 |
|
4811 3290 y FP(between)c(terms)523 3389 y(o)o(ver)j(a)g(nominal)e(signatur)m |
|
4812 (e)h(to)h(be)f(the)h(least)g(sort-r)m(especting)f(congruence)f(r)m |
|
4813 (elation)h(satisfying)523 3489 y FQ(a:t)j FM(=)709 3501 |
|
4814 y FI(\013)766 3489 y FQ(b:)p FM([)p FQ(a)p FL(7!)p FQ(b)p |
|
4815 FM(])p FQ(t)d FP(whene)o(ver)f FQ(b)h FP(is)g(an)g(atom)f(\(of)h(the)f |
|
4816 (same)h(sort)h(as)f FQ(a)p FP(\))f(not)h(occurring)f(at)g(all)h(in)g |
|
4817 FQ(t)p FP(.)523 3589 y(Her)m(e)g FM([)p FQ(a)p FL(7!)p |
|
4818 FQ(b)p FM(])p FQ(t)f FP(indicates)g(the)g(r)m(esult)h(of)f(r)m |
|
4819 (eplacing)f(all)h(fr)m(ee)h(occurr)m(ences)f(of)g FQ(a)h |
|
4820 FP(with)f FQ(b)h FP(in)f FQ(t)p FP(.)523 3729 y FH(Theor)o(em)g(1)41 |
|
4821 b(\(Adequacy\).)17 b FP(If)h FQ(t)g FP(and)f FQ(t)1688 |
|
4822 3699 y FL(0)1729 3729 y FP(ar)m(e)h FH(gr)o(ound)g(terms)g |
|
4823 FW(\(i.e.)f(terms)h(with)g(no)f(v)n(ariables)g(and)523 |
|
4824 3829 y(hence)e(no)h(suspensions\))f FP(o)o(ver)h(a)g(nominal)f |
|
4825 (signatur)m(e)o(,)g(then)h(the)g(r)m(elation)g FQ(t)23 |
|
4826 b FM(=)2804 3841 y FI(\013)2860 3829 y FQ(t)2890 3799 |
|
4827 y FL(0)2930 3829 y FP(of)16 b(Def)o(.)g(5)g(holds)523 |
|
4828 3928 y(if)23 b(and)e(only)h(if)h Fn(?)j FO(`)g FQ(t)h |
|
4829 FO(\031)f FQ(t)1320 3898 y FL(0)1366 3928 y FP(is)d(pr)l(o)o(vable)e |
|
4830 (fr)l(om)i(the)f(rules)h(in)f(F)l(ig)o(.)g(2.)g(Furthermor)m(e)o(,)f |
|
4831 Fn(?)26 b FO(`)h FQ(a)f FM(#)h FQ(t)523 4028 y FP(is)21 |
|
4832 b(pr)l(o)o(vable)e(if)i(and)f(only)f(if)i FQ(a)g FP(is)g(not)f(in)g |
|
4833 (the)h(set)g FA(F)-8 b(A)o FM(\()p FQ(t)p FM(\))21 b |
|
4834 FP(of)g(fr)m(ee)f(atoms)g(of)h FQ(t)p FP(.)523 4128 y(Pr)l(oof)o(.)h |
|
4835 FW(The)e(proof)e(is)k(similar)e(to)g(the)g(proof)f(of)h([11,)f |
|
4836 (Proposition)g(2.2].)713 b FO(u)-55 b(t)648 4268 y FW(F)o(or)28 |
|
4837 b(non-ground)d(terms,)j(the)h(relations)f FM(=)1953 4280 |
|
4838 y FI(\013)2029 4268 y FW(and)g FO(\031)h FP(dif)o(fer)p |
|
4839 FW(.)f(F)o(or)h(e)o(xample)e FQ(a:X)45 b FM(=)3199 4280 |
|
4840 y FI(\013)3271 4268 y FQ(b:X)523 4368 y FW(al)o(w)o(ays)26 |
|
4841 b(holds,)e(whereas)h Fn(?)32 b FO(`)g FQ(a:X)38 b FO(\031)32 |
|
4842 b FQ(b:X)g FW(is)26 b(not)f(pro)o(v)n(able)e(unless)i |
|
4843 FQ(a)32 b FM(=)g FQ(b)p FW(.)25 b(This)h(disagree-)523 |
|
4844 4467 y(ment)c(is)i(to)f(be)f(e)o(xpected,)f(since)i(we)g(noted)e(in)i |
|
4845 (the)g(Introduction)c(that)k FM(=)2698 4479 y FI(\013)2768 |
|
4846 4467 y FW(is)h FP(not)f FW(preserv)o(ed)e(by)523 4567 |
|
4847 y(substitution,)e(whereas)h(from)f(Lemma)g(2)i(we)f(kno)n(w)f(that)i |
|
4848 FO(\031)f FW(is.)523 4707 y FP(Remark)g(1.)41 b FW(Further)17 |
|
4849 b(e)n(vidence)g(for)h(the)g(status)i(of)e FO(\031)g FW(and)g |
|
4850 FM(#)h FW(is)h(pro)o(vided)c(by)i(a)h(natural)e(interpre-)523 |
|
4851 4807 y(tation)22 b(of)h(judgements)e(pro)o(v)n(able)f(from)h(the)i |
|
4852 (rules)g(in)f(Fig.)h(2)g(in)g(the)f(uni)n(v)o(erse)f(of)i(FM-sets)g |
|
4853 ([11].)523 4906 y(The)d(details)g(will)h(appear)e(in)i(the)f(full)g(v)o |
|
4854 (ersion)f(of)h(this)h(paper)-5 b(.)p eop end |
|
4855 %%Page: 9 9 |
|
4856 TeXDict begin 9 8 bop 523 448 a FR(3)99 b(Uni\002cation)523 |
|
4857 648 y FW(Gi)n(v)o(en)21 b(terms)g FQ(t)h FW(and)f FQ(t)1175 |
|
4858 618 y FL(0)1221 648 y FW(of)g(the)h(same)f(sort)h(o)o(v)o(er)e(a)i |
|
4859 (nominal)e(signature,)g(can)i(we)g(decide)f(whether)523 |
|
4860 748 y(or)31 b(not)g(there)g(is)h(a)g(substitution)e(of)h(terms)h(for)e |
|
4861 (the)i(v)n(ariables)e(in)i FQ(t)f FW(and)g FQ(t)2784 |
|
4862 717 y FL(0)2839 748 y FW(that)h(mak)o(es)f(them)523 847 |
|
4863 y(equal)20 b(in)i(the)f(sense)g(of)g(the)g(relation)f |
|
4864 FO(\031)h FW(introduced)d(in)k(the)f(pre)n(vious)e(section?)i(Since)g |
|
4865 (instances)523 947 y(of)26 b FO(\031)h FW(in)g(general)e(are)i |
|
4866 (established)f(modulo)f(freshness)h(constraints,)g(it)h(mak)o(es)g |
|
4867 (more)e(sense)i(to)523 1046 y(ask)i(whether)e(or)i(not)f(there)g(is)h |
|
4868 (both)f(a)h(substitution)f FQ(\033)k FW(and)c(a)h(freshness)f(en)m |
|
4869 (vironment)e FO(r)j FW(for)523 1146 y(which)21 b FO(r)26 |
|
4870 b(`)g FQ(\033)s FM(\()p FQ(t)p FM(\))h FO(\031)f FQ(\033)s |
|
4871 FM(\()p FQ(t)1294 1116 y FL(0)1318 1146 y FM(\))c FW(holds.)f(As)i(for) |
|
4872 e(ordinary)f(\002rst-order)g(uni\002cation,)g(solving)h(such)h(an)523 |
|
4873 1246 y(equational)f(problem)h(may)g(thro)n(w)h(up)f FP(se)o(ver)o(al)h |
|
4874 FW(equational)f(subproblems;)f(b)n(ut)i(an)f(added)g(com-)523 |
|
4875 1345 y(plication)j(here)g(is)h(that)g(because)f(of)g(rule)g(\()p |
|
4876 FO(\031)p FW(-abstraction-2\))d(in)k(Fig.)g(2,)f(equational)f(problems) |
|
4877 523 1445 y(may)c(generate)f FP(fr)m(eshness)i FW(problems,)e(i.e.)i |
|
4878 (ones)f(in)m(v)n(olving)f(the)h(relation)g FM(#)p FW(.)h(W)-7 |
|
4879 b(e)22 b(are)e(thus)h(led)f(to)523 1545 y(the)g(follo)n(wing)f |
|
4880 (de\002nition)g(of)h(uni\002cation)f(problems)f(for)i(nominal)f |
|
4881 (equational)g(logic.)523 1719 y FH(De\002nition)h(6.)41 |
|
4882 b FP(A)17 b FE(uni\002cation)f(problem)g FQ(P)28 b FP(o)o(ver)17 |
|
4883 b(a)f(nominal)f(signatur)m(e)h(is)h(a)f(\002nite)g(set)h(of)g(atomic) |
|
4884 523 1819 y(pr)l(oblems,)27 b(eac)o(h)f(of)h(whic)o(h)g(is)h(either)f |
|
4885 (an)g FE(equational)f(problem)g FQ(t)36 b FO(\031)o FM(?)g |
|
4886 FQ(t)2728 1789 y FL(0)2779 1819 y FP(wher)m(e)28 b FQ(t)f |
|
4887 FP(and)f FQ(t)3246 1789 y FL(0)3297 1819 y FP(ar)m(e)523 |
|
4888 1919 y(terms)21 b(of)e(the)h(same)g(sort)g(o)o(ver)g(the)g(signatur)m |
|
4889 (e)o(,)e(or)i(a)g FE(freshness)h(problem)e FQ(a)k FM(#?)g |
|
4890 FQ(t)d FP(wher)m(e)g FQ(a)g FP(is)h(an)523 2018 y(atom)g(and)f |
|
4891 FQ(t)i FP(a)g(term)g(o)o(ver)f(the)h(signatur)m(e)o(.)d(A)j |
|
4892 FE(solution)g FP(for)g FQ(P)33 b FP(consists)22 b(of)g(a)f(pair)g |
|
4893 FM(\()p FO(r)p FQ(;)14 b(\033)s FM(\))23 b FP(wher)m(e)523 |
|
4894 2118 y FO(r)e FP(is)g(a)f(fr)m(eshness)h(en)m(vir)l(onment)e(and)g |
|
4895 FQ(\033)24 b FP(is)d(a)g(substitution)e(satisfying)581 |
|
4896 2284 y FO(\017)41 b(r)23 b(`)g FQ(a)g FM(#)g FQ(\033)s |
|
4897 FM(\()p FQ(t)p FM(\))45 b FP(for)21 b(eac)o(h)42 b FM(\()p |
|
4898 FQ(a)23 b FM(#?)h FQ(t)p FM(\))f FO(2)h FQ(P)55 b FP(and)581 |
|
4899 2384 y FO(\017)41 b(r)23 b(`)g FQ(\033)s FM(\()p FQ(t)p |
|
4900 FM(\))h FO(\031)f FQ(\033)s FM(\()p FQ(t)1198 2354 y |
|
4901 FL(0)1222 2384 y FM(\))44 b FP(for)21 b(eac)o(h)42 b |
|
4902 FM(\()p FQ(t)23 b FO(\031)p FM(?)g FQ(t)1857 2354 y FL(0)1880 |
|
4903 2384 y FM(\))h FO(2)f FQ(P)12 b FP(.)523 2550 y(Suc)o(h)21 |
|
4904 b(a)i(pair)f(is)h(a)g FE(most)f(general)g(solution)h |
|
4905 FP(for)g FQ(P)34 b FP(if)23 b(given)f(any)g(other)g(solution)g |
|
4906 FM(\()p FO(r)3052 2520 y FL(0)3075 2550 y FQ(;)14 b(\033)3162 |
|
4907 2520 y FL(0)3186 2550 y FM(\))p FP(,)23 b(then)523 2650 |
|
4908 y(ther)m(e)d(is)h(a)f(substitution)f FQ(\033)1307 2620 |
|
4909 y FL(00)1370 2650 y FP(satisfying)h FO(r)1778 2620 y |
|
4910 FL(0)1824 2650 y FO(`)j FQ(\033)1948 2620 y FL(00)1991 |
|
4911 2650 y FM(\()p FO(r)p FM(\))e FP(and)e FO(r)2359 2620 |
|
4912 y FL(0)2406 2650 y FO(`)k FQ(\033)2530 2620 y FL(00)2590 |
|
4913 2650 y FO(\016)16 b FQ(\033)27 b FO(\031)c FQ(\033)2860 |
|
4914 2620 y FL(0)2883 2650 y FP(.)e(\(Her)m(e)f(we)g(have)523 |
|
4915 2749 y(used)e(the)g(notation)f(of)31 b FW(\(9\))n FP(;)19 |
|
4916 b(and)e FQ(\033)1542 2719 y FL(00)1597 2749 y FO(\016)11 |
|
4917 b FQ(\033)22 b FP(denotes)17 b(the)i FE(substitution)g(composition)e |
|
4918 FP(of)i FQ(\033)j FP(followed)523 2875 y(by)e FQ(\033)672 |
|
4919 2844 y FL(00)715 2875 y FP(,)h(given)e(by)h FM(\()p FQ(\033)1138 |
|
4920 2844 y FL(00)1200 2875 y FO(\016)e FQ(\033)s FM(\)\()p |
|
4921 FQ(X)7 b FM(\))1505 2823 y Fs(def)1509 2875 y FM(=)25 |
|
4922 b FQ(\033)1649 2844 y FL(00)1692 2875 y FM(\()p FQ(\033)s |
|
4923 FM(\()p FQ(X)7 b FM(\)\))p FP(.\))523 3116 y FH(Theor)o(em)20 |
|
4924 b(2)41 b(\(Nominal)27 b(uni\002cation\).)g FP(Ther)m(e)g(is)i(an)e |
|
4925 (algorithm)f(whic)o(h,)h(given)g(any)g(nominal)523 3207 |
|
4926 y(uni\002cation)d(pr)l(oblem,)h(decides)h(whether)g(or)g(not)g(it)g |
|
4927 (has)g(a)g(solution)f(and)g(if)i(it)g(does,)e(r)m(eturns)h(a)523 |
|
4928 3299 y(most)21 b(g)o(ener)o(al)e(solution.)523 3390 y(Pr)l(oof)o(.)j |
|
4929 FW(W)-7 b(e)16 b(describe)f(an)g(algorithm)f(using)h(labelled)f |
|
4930 (transformations)f(directly)i(generalising)e(the)523 |
|
4931 3481 y(presentation)19 b(of)h(\002rst-order)f(uni\002cation)g(in)i |
|
4932 ([17,)e(Sect.)i(2.6],)e(which)h(in)h(turn)f(is)h(based)f(upon)f(the)523 |
|
4933 3573 y(approach)e(in)i([18].)f(\(See)h(also)g([1,)f(Sect.)i(4.6])e(for) |
|
4934 g(a)h(detailed)g(e)o(xposition,)e(b)n(ut)i(not)f(using)h(labels.\))523 |
|
4935 3664 y(W)-7 b(e)21 b(use)g(tw)o(o)f(types)g(of)g(labelled)g |
|
4936 (transformation)d(between)j(uni\002cation)f(problems,)f(namely)1527 |
|
4937 3799 y FS(P)1651 3757 y Fm(\033)1608 3799 y FF(=)-13 |
|
4938 b FB(\))21 b FS(P)1813 3764 y Fw(0)1910 3799 y FV(and)76 |
|
4939 b FS(P)2210 3757 y Fw(r)2174 3799 y FF(=)-13 b FB(\))21 |
|
4940 b FS(P)2379 3764 y Fw(0)523 3932 y FW(where)i(the)g(substitution)f |
|
4941 FQ(\033)27 b FW(is)d(either)f(the)g(identity)f FQ(")p |
|
4942 FW(,)h(or)g(a)h(single)f(replacement)e FM([)p FQ(X)35 |
|
4943 b FM(:=)28 b FQ(t)p FM(])p FW(;)c(and)523 4032 y(where)h(the)i |
|
4944 (freshness)e(en)m(vironment)e FO(r)k FW(is)g(either)f(empty)f |
|
4945 Fn(?)p FW(,)h(or)g(a)g(singleton)f FO(f)p FQ(a)33 b FM(#)i |
|
4946 FQ(X)7 b FO(g)p FW(.)25 b(The)523 4132 y(le)o(gal)17 |
|
4947 b(transformations)e(are)i(gi)n(v)o(en)f(in)i(Fig.)f(3.)g(This)h |
|
4948 (\002gure)e(uses)i(the)f(notation)g FQ(P)i FO(])8 b FQ(P)3018 |
|
4949 4102 y FL(0)3060 4132 y FW(to)17 b(indicate)523 4231 |
|
4950 y FP(disjoint)26 b(union)e FW(of)i(problem)e(sets;)k(and)d(the)h |
|
4951 (notation)f FQ(\033)s(P)39 b FW(to)26 b(indicate)f(the)h(problem)e |
|
4952 (resulting)523 4331 y(from)19 b(applying)g(the)h(substitution)f |
|
4953 FQ(\033)24 b FW(to)d(all)f(the)h(terms)f(occurring)e(in)i(the)g |
|
4954 (problem)f FQ(P)12 b FW(.)648 4422 y(Gi)n(v)o(en)25 b(a)i |
|
4955 (uni\002cation)f(problem)f FQ(P)12 b FW(,)27 b(the)f(algorithm)f |
|
4956 (proceeds)h(in)g(tw)o(o)h(phases.)g(In)f(the)h(\002rst)523 |
|
4957 4529 y(phase)17 b(it)i(applies)e(as)i(man)o(y)1384 4482 |
|
4958 y FI(\033)1337 4529 y FM(=)-14 b FO(\))18 b FW(transformations)e(as)i |
|
4959 (possible)g(\(non-deterministically\).)13 b(If)18 b(this)523 |
|
4960 4620 y(results)23 b(in)f(a)h(problem)e(containing)g(no)h(equational)f |
|
4961 (subproblems)f(then)i(it)h(proceeds)e(to)i(the)f(sec-)523 |
|
4962 4742 y(ond)i(phase;)g(otherwise)g(it)h(halts)g(with)g(f)o(ailure.)f(In) |
|
4963 g(the)h(second)f(phase)g(it)h(applies)f(as)i(man)o(y)3310 |
|
4964 4695 y FL(r)3271 4742 y FM(=)-14 b FO(\))523 4833 y FW(transformations) |
|
4965 19 b(as)j(possible)f(\(non-deterministically\).)d(If)j(this)h(does)f |
|
4966 (not)g(result)g(in)h(the)f(empty)523 4924 y(problem,)c(then)i(it)h |
|
4967 (halts)f(with)g(f)o(ailure;)g(otherwise)f(o)o(v)o(erall)g(it)i(has)f |
|
4968 (constructed)f(a)h(transformation)p eop end |
|
4969 %%Page: 10 10 |
|
4970 TeXDict begin 10 9 bop 523 369 2882 4 v 523 2089 4 1721 |
|
4971 v 546 456 a Fv(\()p Fu(\031)p Ft(?)q Fv(-unit\))558 b |
|
4972 FB(fhi)22 b(\031)p FF(?)f FB(hig)c(])h FS(P)1912 413 |
|
4973 y Fm(")1865 456 y FF(=)-13 b FB(\))23 b FS(P)546 556 |
|
4974 y Fv(\()p Fu(\031)p Ft(?)q Fv(-pair\))241 b FB(fh)p FS(t)1140 |
|
4975 564 y FG(1)1174 556 y FS(;)14 b(t)1237 564 y FG(2)1271 |
|
4976 556 y FB(i)21 b(\031)p FF(?)g FB(h)p FS(t)1497 524 y |
|
4977 Fw(0)1497 569 y FG(1)1532 556 y FS(;)13 b(t)1594 524 |
|
4978 y Fw(0)1594 569 y FG(2)1628 556 y FB(ig)k(])h FS(P)1912 |
|
4979 513 y Fm(")1865 556 y FF(=)-13 b FB(\))23 b(f)p FS(t)2078 |
|
4980 564 y FG(1)2134 556 y FB(\031)o FF(?)f FS(t)2279 524 |
|
4981 y Fw(0)2279 569 y FG(1)2313 556 y FS(;)13 b(t)2375 564 |
|
4982 y FG(2)2431 556 y FB(\031)p FF(?)21 b FS(t)2576 524 y |
|
4983 Fw(0)2576 569 y FG(2)2611 556 y FB(g)c([)g FS(P)546 656 |
|
4984 y Fv(\()p Fu(\031)p Ft(?)q Fv(-function)i(symbol\))153 |
|
4985 b FB(f)p FS(f)21 b(t)g FB(\031)p FF(?)h FS(f)f(t)1636 |
|
4986 624 y Fw(0)1658 656 y FB(g)c(])h FS(P)1912 613 y Fm(")1865 |
|
4987 656 y FF(=)-13 b FB(\))23 b(f)p FS(t)e FB(\031)p FF(?)g |
|
4988 FS(t)2244 624 y Fw(0)2266 656 y FB(g)d([)f FS(P)546 756 |
|
4989 y Fv(\()p Fu(\031)p Ft(?)q Fv(-abstraction-1\))235 b |
|
4990 FB(f)p FS(a:t)22 b FB(\031)o FF(?)g FS(a:t)1636 724 y |
|
4991 Fw(0)1658 756 y FB(g)17 b(])h FS(P)1912 713 y Fm(")1865 |
|
4992 756 y FF(=)-13 b FB(\))23 b(f)p FS(t)e FB(\031)p FF(?)g |
|
4993 FS(t)2244 724 y Fw(0)2266 756 y FB(g)d([)f FS(P)546 855 |
|
4994 y Fv(\()p Fu(\031)p Ft(?)q Fv(-abstraction-2\))213 b |
|
4995 FB(f)p FS(a:t)21 b FB(\031)p FF(?)g FS(a)1564 824 y Fw(0)1587 |
|
4996 855 y FS(:t)1636 824 y Fw(0)1658 855 y FB(g)c(])h FS(P)1912 |
|
4997 813 y Fm(")1865 855 y FF(=)-13 b FB(\))23 b(f)p FS(t)e |
|
4998 FB(\031)p FF(?)g(\()p FS(a)13 b(a)2341 824 y Fw(0)2363 |
|
4999 855 y FF(\))p Fz(\001)o FS(t)2445 824 y Fw(0)2467 855 |
|
5000 y FS(;)g(a)21 b FF(#?)h FS(t)2713 824 y Fw(0)2735 855 |
|
5001 y FB(g)t([)t FS(P)33 b FV(pro)o(vided)20 b FS(a)i FB(6)p |
|
5002 FF(=)f FS(a)3382 824 y Fw(0)546 955 y Fv(\()p Fu(\031)p |
|
5003 Ft(?)q Fv(-atom\))567 b FB(f)p FS(a)21 b FB(\031)p FF(?)g |
|
5004 FS(a)p FB(g)c(])h FS(P)1912 913 y Fm(")1865 955 y FF(=)-13 |
|
5005 b FB(\))23 b FS(P)546 1055 y Fv(\()p Fu(\031)p Ft(?)q |
|
5006 Fv(-suspension\))187 b FB(f)p FS(\031)s Fz(\001)o FS(X)28 |
|
5007 b FB(\031)p FF(?)21 b FS(\031)1542 1023 y Fw(0)1564 1055 |
|
5008 y Fz(\001)p FS(X)6 b FB(g)17 b(])h FS(P)1912 1012 y Fm(")1865 |
|
5009 1055 y FF(=)-13 b FB(\))23 b(f)p FS(a)e FF(#?)h FS(X)27 |
|
5010 b FB(j)22 b FS(a)f FB(2)g FD(ds)7 b FF(\()p FS(\031)s(;)13 |
|
5011 b(\031)2736 1023 y Fw(0)2758 1055 y FF(\))p FB(g)k([)g |
|
5012 FS(P)546 1193 y Fv(\()p Fu(\031)p Ft(?)q Fv(-v)n(ariable\))1310 |
|
5013 1147 y FB(f)p FS(t)22 b FB(\031)o FF(?)g FS(\031)s Fz(\001)o |
|
5014 FS(X)6 b FB(g)18 b(])f FS(P)1310 1238 y FB(f)p FS(\031)s |
|
5015 Fz(\001)p FS(X)27 b FB(\031)p FF(?)22 b FS(t)p FB(g)17 |
|
5016 b(])g FS(P)1826 1087 y Fx(\033)1908 1150 y Fm(\033)1865 |
|
5017 1193 y FF(=)-13 b FB(\))23 b FS(\033)s(P)2204 1169 y |
|
5018 FV(with)c FS(\033)24 b FF(=)d([)p FS(X)28 b FF(:=)21 |
|
5019 b FS(\031)2766 1137 y Fw(\000)p FG(1)2848 1169 y Fz(\001)p |
|
5020 FS(t)p FF(])2204 1260 y FV(pro)o(vided)f FS(X)25 b FV(does)20 |
|
5021 b(not)f(occur)h(in)f FS(t)546 1371 y Fv(\()p Ft(#?)q |
|
5022 Fv(-unit\))569 b FB(f)p FS(a)22 b FF(#?)f FB(hig)c(])h |
|
5023 FS(P)1903 1328 y Fl(?)1865 1371 y FF(=)-13 b FB(\))23 |
|
5024 b FS(P)546 1482 y Fv(\()p Ft(#?)q Fv(-pair\))410 b FB(f)p |
|
5025 FS(a)22 b FF(#?)f FB(h)p FS(t)1497 1490 y FG(1)1532 1482 |
|
5026 y FS(;)13 b(t)1594 1490 y FG(2)1628 1482 y FB(ig)k(])h |
|
5027 FS(P)1903 1439 y Fl(?)1865 1482 y FF(=)-13 b FB(\))23 |
|
5028 b(f)p FS(a)e FF(#?)h FS(t)2262 1490 y FG(1)2296 1482 |
|
5029 y FS(;)13 b(a)21 b FF(#?)h FS(t)2542 1490 y FG(2)2576 |
|
5030 1482 y FB(g)17 b([)h FS(P)546 1592 y Fv(\()p Ft(#?)q |
|
5031 Fv(-function)h(symbol\))212 b FB(f)p FS(a)21 b FF(#?)h |
|
5032 FS(f)f(t)p FB(g)c(])h FS(P)1903 1549 y Fl(?)1865 1592 |
|
5033 y FF(=)-13 b FB(\))23 b(f)p FS(a)e FF(#?)h FS(t)p FB(g)17 |
|
5034 b([)g FS(P)546 1703 y Fv(\()p Ft(#?)q Fv(-abstraction-1\))298 |
|
5035 b FB(f)p FS(a)22 b FF(#?)f FS(a:t)p FB(g)c(])h FS(P)1903 |
|
5036 1660 y Fl(?)1865 1703 y FF(=)-13 b FB(\))23 b FS(P)546 |
|
5037 1814 y Fv(\()p Ft(#?)q Fv(-abstraction-2\))276 b FB(f)p |
|
5038 FS(a)21 b FF(#?)h FS(a)1587 1782 y Fw(0)1609 1814 y FS(:t)p |
|
5039 FB(g)17 b(])h FS(P)1903 1771 y Fl(?)1865 1814 y FF(=)-13 |
|
5040 b FB(\))23 b(f)p FS(a)e FF(#?)h FS(t)p FB(g)17 b([)g |
|
5041 FS(P)85 b FV(pro)o(vided)21 b FS(a)g FB(6)p FF(=)g FS(a)2987 |
|
5042 1782 y Fw(0)546 1924 y Fv(\()p Ft(#?)q Fv(-atom\))536 |
|
5043 b FB(f)p FS(a)21 b FF(#?)h FS(a)1636 1893 y Fw(0)1658 |
|
5044 1924 y FB(g)17 b(])h FS(P)1903 1881 y Fl(?)1865 1924 |
|
5045 y FF(=)-13 b FB(\))23 b FS(P)85 b FV(pro)o(vided)20 b |
|
5046 FS(a)h FB(6)p FF(=)g FS(a)2613 1893 y Fw(0)546 2037 y |
|
5047 Fv(\()p Ft(#?)q Fv(-suspension\))300 b FB(f)p FS(a)22 |
|
5048 b FF(#?)f FS(\031)s Fz(\001)p FS(X)6 b FB(g)17 b(])h |
|
5049 FS(P)1901 1994 y Fw(r)1865 2037 y FF(=)-13 b FB(\))23 |
|
5050 b FS(P)85 b FV(with)19 b FB(r)i FF(=)g FB(f)p FS(\031)2549 |
|
5051 2005 y Fw(\000)p FG(1)2632 2037 y Fz(\001)o FS(a)g FF(#)g |
|
5052 FS(X)6 b FB(g)p 3402 2089 V 523 2092 2882 4 v 1476 2235 |
|
5053 a FU(Fig)o(.)12 b(3.)18 b FV(Labelled)h(transformations.)523 |
|
5054 2500 y FW(sequence)g(of)h(the)g(form)1441 2636 y FS(P)1550 |
|
5055 2590 y Fm(\033)1586 2600 y Fk(1)1522 2636 y FF(=)-13 |
|
5056 b FB(\))21 b(\001)13 b(\001)g(\001)1802 2591 y Fm(\033)1838 |
|
5057 2599 y Fj(n)1778 2636 y FF(=)-13 b FB(\))21 b FS(P)1983 |
|
5058 2600 y Fw(0)2046 2590 y(r)2097 2600 y Fk(1)2026 2636 |
|
5059 y FF(=)-13 b FB(\))21 b(\001)14 b(\001)f(\001)2291 2591 |
|
5060 y Fw(r)2342 2599 y Fj(m)2282 2636 y FF(=)-13 b FB(\))21 |
|
5061 b Fi(?)793 b FV(\(10\))523 2758 y FW(\(where)21 b FQ(P)841 |
|
5062 2727 y FL(0)886 2758 y FW(does)h(not)f(contain)g(an)o(y)g(equational)f |
|
5063 (subproblems\))f(and)j(the)f(algorithm)g(returns)f(the)523 |
|
5064 2857 y(solution)f FM(\()p FO(r)912 2869 y FJ(1)969 2857 |
|
5065 y FO([)f(\001)c(\001)g(\001)19 b([)f(r)1300 2869 y FI(m)1364 |
|
5066 2857 y FQ(;)27 b(\033)1461 2869 y FI(n)1526 2857 y FO(\016)17 |
|
5067 b(\001)d(\001)g(\001)19 b(\016)f FQ(\033)1808 2869 y |
|
5068 FJ(1)1845 2857 y FM(\))p FW(.)648 2957 y(It)j(is)g(not)g(hard)f(to)h |
|
5069 (de)n(vise)f(a)i(well-founded)c(ordering)g(on)j(nominal)e |
|
5070 (uni\002cation)h(problems)f(to)523 3056 y(sho)n(w)h(that)g(each)g |
|
5071 (phase)g(of)g(the)g(algorithm)f(must)h(terminate.)f(So)h(one)g(just)h |
|
5072 (has)f(to)h(sho)n(w)e(that)531 3195 y(\(a\))40 b(if)21 |
|
5073 b(the)f(algorithm)e(f)o(ails)j(on)f FQ(P)12 b FW(,)20 |
|
5074 b(then)g FQ(P)33 b FW(has)20 b(no)g(solution;)f(and)526 |
|
5075 3289 y(\(b\))40 b(if)16 b(the)f(algorithm)f(succeeds)i(on)f |
|
5076 FQ(P)d FW(,)j(then)g(the)h(result)f(it)i(produces)c(is)k(a)f(most)f |
|
5077 (general)g(solution.)523 3425 y(When)22 b(f)o(ailure)g(happens)f(it)i |
|
5078 (is)h(because)d(of)h(certain)g(subproblems)f(that)h(manifestly)g(ha)n |
|
5079 (v)o(e)f(no)h(so-)523 3516 y(lution)d(\(e.g.)f(in)h(the)g(\002rst)h |
|
5080 (phase,)f FQ(a)k FO(\031)o FM(?)h FQ(a)1726 3486 y FL(0)1769 |
|
5081 3516 y FW(with)19 b FQ(a)k FO(6)p FM(=)g FQ(a)2135 3486 |
|
5082 y FL(0)2158 3516 y FW(,)c(and)g FQ(\031)s FK(\001)p FQ(X)29 |
|
5083 b FO(\031)p FM(?)23 b FQ(f)f(t)e FW(or)f FQ(f)k(t)g FO(\031)o |
|
5084 FM(?)g FQ(\031)s FK(\001)p FQ(X)j FW(with)523 3607 y |
|
5085 FQ(X)j FW(occurring)19 b(in)j FQ(t)p FW(;)g(in)g(the)g(second)f(phase,) |
|
5086 g FQ(a)k FM(#?)i FQ(a)p FW(\).)21 b(Therefore)f(part)h(\(a\))h(is)g(a)g |
|
5087 (consequence)e(of)523 3699 y(the)g(follo)n(wing)f(tw)o(o)h(properties)f |
|
5088 (of)h(transformations,)e(where)i(we)g(write)h Fh(U)p |
|
5089 FM(\()p FQ(P)12 b FM(\))21 b FW(for)f(the)g(set)h(of)f(all)523 |
|
5090 3790 y(solutions)g(for)f(a)i(problem)d FQ(P)12 b FW(:)683 |
|
5091 3930 y FV(if)18 b FF(\()p FB(r)841 3894 y Fw(0)863 3930 |
|
5092 y FS(;)13 b(\033)944 3894 y Fw(0)966 3930 y FF(\))21 |
|
5093 b FB(2)h Fg(U)p FF(\()p FS(P)11 b FF(\))18 b FV(and)h |
|
5094 FS(P)1534 3887 y Fm(\033)1491 3930 y FF(=)-13 b FB(\))21 |
|
5095 b FS(P)1696 3894 y Fw(0)1718 3930 y FS(;)13 b FV(then)20 |
|
5096 b FF(\()p FB(r)1994 3894 y Fw(0)2016 3930 y FS(;)13 b(\033)2097 |
|
5097 3894 y Fw(0)2119 3930 y FF(\))21 b FB(2)h Fg(U)p FF(\()p |
|
5098 FS(P)2389 3894 y Fw(0)2410 3930 y FF(\))d FV(and)h FB(r)2650 |
|
5099 3894 y Fw(0)2693 3930 y FB(`)i FS(\033)2809 3894 y Fw(0)2848 |
|
5100 3930 y FB(\016)17 b FS(\033)24 b FB(\031)d FS(\033)3099 |
|
5101 3894 y Fw(0)3280 3930 y FV(\(11\))682 4090 y(if)d FF(\()p |
|
5102 FB(r)840 4054 y Fw(0)863 4090 y FS(;)13 b(\033)944 4054 |
|
5103 y Fw(0)966 4090 y FF(\))21 b FB(2)g Fg(U)p FF(\()p FS(P)11 |
|
5104 b FF(\))18 b FV(and)i FS(P)1527 4048 y Fw(r)1491 4090 |
|
5105 y FF(=)-13 b FB(\))21 b FS(P)1696 4054 y Fw(0)1718 4090 |
|
5106 y FS(;)13 b FV(then)19 b FF(\()p FB(r)1993 4054 y Fw(0)2016 |
|
5107 4090 y FS(;)13 b(\033)2097 4054 y Fw(0)2119 4090 y FF(\))21 |
|
5108 b FB(2)g Fg(U)p FF(\()p FS(P)2388 4054 y Fw(0)2410 4090 |
|
5109 y FF(\))d FV(and)i FB(r)2649 4054 y Fw(0)2693 4090 y |
|
5110 FB(`)h FS(\033)2808 4054 y Fw(0)2830 4090 y FF(\()p FB(r)p |
|
5111 FF(\))p FS(:)305 b FV(\(12\))523 4204 y FW(F)o(or)20 |
|
5112 b(part)g(\(b\),)f(one)h(\002rst)h(sho)n(ws)680 4319 y |
|
5113 FV(if)d FF(\()p FB(r)838 4283 y Fw(0)860 4319 y FS(;)c(\033)942 |
|
5114 4283 y Fw(0)964 4319 y FF(\))21 b FB(2)g Fg(U)p FF(\()p |
|
5115 FS(P)1233 4283 y Fw(0)1255 4319 y FF(\))d FV(and)i FS(P)1554 |
|
5116 4276 y Fm(\033)1511 4319 y FF(=)-13 b FB(\))21 b FS(P)1716 |
|
5117 4283 y Fw(0)1738 4319 y FS(;)13 b FV(then)20 b FF(\()p |
|
5118 FB(r)2014 4283 y Fw(0)2036 4319 y FS(;)13 b(\033)2117 |
|
5119 4283 y Fw(0)2156 4319 y FB(\016)k FS(\033)s FF(\))k FB(2)h |
|
5120 Fg(U)p FF(\()p FS(P)11 b FF(\))722 b FV(\(13\))678 4479 |
|
5121 y(if)19 b FF(\()p FB(r)837 4443 y Fw(0)859 4479 y FS(;)13 |
|
5122 b(\033)940 4443 y Fw(0)962 4479 y FF(\))21 b FB(2)h Fg(U)p |
|
5123 FF(\()p FS(P)1232 4443 y Fw(0)1253 4479 y FF(\))p FS(;)35 |
|
5124 b(P)1456 4436 y Fw(r)1420 4479 y FF(=)-13 b FB(\))21 |
|
5125 b FS(P)1625 4443 y Fw(0)1665 4479 y FV(and)f FB(r)1856 |
|
5126 4443 y Fw(00)1918 4479 y FB(`)h FS(\033)2033 4443 y Fw(0)2055 |
|
5127 4479 y FF(\()p FB(r)p FF(\))p FS(;)13 b FV(then)19 b |
|
5128 FF(\()p FB(r)2454 4443 y Fw(0)2494 4479 y FB([)e(r)2626 |
|
5129 4443 y Fw(00)2667 4479 y FS(;)c(\033)2748 4443 y Fw(0)2770 |
|
5130 4479 y FF(\))21 b FB(2)g Fg(U)p FF(\()p FS(P)11 b FF(\))p |
|
5131 FS(:)190 b FV(\(14\))523 4601 y FW(From)19 b(these)h(and)g(the)f(f)o |
|
5132 (act)h(that)g FM(\()p Fn(?)p FQ(;)14 b(")p FM(\))23 b |
|
5133 FO(2)h Fh(U)p FM(\()p Fn(?)p FM(\))p FW(,)c(one)f(gets)i(that)e(if)i(a) |
|
5134 f(sequence)f(lik)o(e)h(\(10\))e(e)o(xists,)523 4725 y(then)28 |
|
5135 b FM(\()p FO(r)p FQ(;)14 b(\033)s FM(\))955 4678 y Fy(def)960 |
|
5136 4725 y FM(=)43 b(\()p FO(r)1169 4737 y FJ(1)1232 4725 |
|
5137 y FO([)25 b(\001)14 b(\001)g(\001)24 b([)i(r)1583 4737 |
|
5138 y FI(m)1646 4725 y FQ(;)i(\033)1744 4737 y FI(n)1814 |
|
5139 4725 y FO(\016)c(\001)14 b(\001)g(\001)25 b(\016)f FQ(\033)2115 |
|
5140 4737 y FJ(1)2153 4725 y FM(\))29 b FW(is)h(in)f Fh(U)p |
|
5141 FM(\()p FQ(P)12 b FM(\))p FW(.)30 b(Furthermore)c(from)i(\(11\))523 |
|
5142 4825 y(and)g(\(12\))o(,)h(we)g(get)g(that)f(an)o(y)g(other)g(solution)g |
|
5143 FM(\()p FO(r)2017 4795 y FL(0)2041 4825 y FQ(;)14 b(\033)2128 |
|
5144 4795 y FL(0)2152 4825 y FM(\))39 b FO(2)g Fh(U)p FM(\()p |
|
5145 FQ(P)12 b FM(\))30 b FW(satis\002es)g FO(r)2896 4795 |
|
5146 y FL(0)2959 4825 y FO(`)38 b FQ(\033)3098 4795 y FL(0)3122 |
|
5147 4825 y FM(\()p FO(r)p FM(\))30 b FW(and)523 4924 y FO(r)592 |
|
5148 4894 y FL(0)639 4924 y FO(`)22 b FQ(\033)762 4894 y FL(0)805 |
|
5149 4924 y FO(\016)17 b FQ(\033)27 b FO(\031)c FQ(\033)1076 |
|
5150 4894 y FL(0)1099 4924 y FW(,)e(so)f(that)h FM(\()p FO(r)p |
|
5151 FQ(;)14 b(\033)s FM(\))22 b FW(is)f(indeed)e(a)h(most)h(general)e |
|
5152 (solution.)614 b FO(u)-55 b(t)p eop end |
|
5153 %%Page: 11 11 |
|
5154 TeXDict begin 11 10 bop 523 369 2882 4 v 523 1962 4 1594 |
|
5155 v 533 453 a Ff(P)578 462 y FG(1)654 415 y Fm(")612 453 |
|
5156 y Ft(=)-12 b Fu(\))o(f)p Fe(fn)13 b Ff(b:)p Fe(app)q |
|
5157 Fu(h)p Ff(X)1084 462 y FG(1)1119 453 y Ff(;)e Fe(vr)h |
|
5158 Ff(b)p Fu(i)21 b(\031)p Ft(?)e Fe(fn)13 b Ff(b:)p Fe(app)p |
|
5159 Fu(h)p Fe(vr)g Ff(b;)f Ft(\()p Ff(a)g(b)p Ft(\))p Fd(\001)q |
|
5160 Ff(X)2042 462 y FG(1)2076 453 y Fu(i)p Ff(;)20 b(a)g |
|
5161 Ft(#?)g Fe(fn)12 b Ff(a:)p Fe(app)q Fu(h)p Fe(vr)h Ff(a;)f(X)2794 |
|
5162 462 y FG(1)2828 453 y Fu(ig)q Fv(\()p Fu(\031)p Ft(?)p |
|
5163 Fv(-abstraction-2\))654 508 y Fm(")612 546 y Ft(=)-12 |
|
5164 b Fu(\))o(f)p Fe(app)r Fu(h)p Ff(X)952 555 y FG(1)987 |
|
5165 546 y Ff(;)11 b Fe(vr)h Ff(b)p Fu(i)21 b(\031)o Ft(?)f |
|
5166 Fe(app)q Fu(h)p Fe(vr)13 b Ff(b;)f Ft(\()p Ff(a)g(b)p |
|
5167 Ft(\))p Fd(\001)p Ff(X)1777 555 y FG(1)1812 546 y Fu(i)p |
|
5168 Ff(;)20 b(a)g Ft(#?)g Fe(fn)12 b Ff(a:)p Fe(app)q Fu(h)p |
|
5169 Fe(vr)h Ff(a;)e(X)2529 555 y FG(1)2564 546 y Fu(ig)265 |
|
5170 b Fv(\()p Fu(\031)p Ft(?)p Fv(-abstraction-1\))622 615 |
|
5171 y Fu(\001)11 b(\001)g(\001)21 b(\001)12 b(\001)f(\001)2083 |
|
5172 b(\001)11 b(\001)h(\001)654 646 y Fm(")612 685 y Ft(=)-12 |
|
5173 b Fu(\))o(f)p Ff(X)818 694 y FG(1)873 685 y Fu(\031)p |
|
5174 Ft(?)20 b Fe(vr)12 b Ff(b;)31 b Fe(vr)13 b Ff(b)19 b |
|
5175 Fu(\031)p Ft(?)h(\()p Ff(a)13 b(b)p Ft(\))p Fd(\001)p |
|
5176 Ff(X)1599 694 y FG(1)1634 685 y Ff(;)19 b(a)h Ft(#?)g |
|
5177 Fe(fn)12 b Ff(a:)p Fe(app)q Fu(h)p Fe(vr)h Ff(a;)f(X)2324 |
|
5178 694 y FG(1)2358 685 y Fu(ig)471 b Fv(\()p Fu(\031)p Ft(?)p |
|
5179 Fv(-pair\))650 739 y Fm(\033)612 778 y Ft(=)-12 b Fu(\))o(f)p |
|
5180 Fe(vr)13 b Ff(b)20 b Fu(\031)p Ft(?)g Fe(vr)12 b Ff(a;)31 |
|
5181 b(a)20 b Ft(#?)g Fe(fn)13 b Ff(a:)p Fe(app)p Fu(h)p Fe(vr)g |
|
5182 Ff(a;)f Fe(vr)g Ff(b)p Fu(ig)68 b Fv(with)20 b Ff(\033)j |
|
5183 Ft(=)c([)p Ff(X)2358 787 y FG(1)2412 778 y Ft(:=)g Fe(vr)13 |
|
5184 b Ff(b)p Ft(])252 b Fv(\()p Fu(\031)p Ft(?)p Fv(-v)n(ariable\))654 |
|
5185 832 y Fm(")612 871 y Ft(=)-12 b Fu(\))o(f)p Ff(b)20 b |
|
5186 Fu(\031)p Ft(?)g Ff(a;)32 b(a)20 b Ft(#?)g Fe(fn)12 b |
|
5187 Ff(a:)p Fe(app)q Fu(h)p Fe(vr)h Ff(a;)e Fe(vr)i Ff(b)p |
|
5188 Fu(ig)1117 b Fv(\()p Fu(\031)p Ft(?)p Fv(-fnctn)19 b(symbol\))725 |
|
5189 954 y Ft(F)-8 b(AIL)533 1054 y Ff(P)578 1063 y FG(4)654 |
|
5190 1016 y Fm(")612 1054 y Ft(=)c Fu(\))o(f)p Fe(fn)13 b |
|
5191 Ff(b:)p Fe(app)q Fu(h)p Fe(vr)f Ff(b;)g(X)1228 1063 y |
|
5192 FG(6)1263 1054 y Fu(i)20 b(\031)p Ft(?)f Fe(fn)13 b Ff(a:)p |
|
5193 Fe(app)q Fu(h)p Fe(vr)g Ff(a;)e(X)1899 1063 y FG(7)1934 |
|
5194 1054 y Fu(ig)895 b Fv(\()p Fu(\031)p Ft(?)p Fv(-abstraction-1\))654 |
|
5195 1109 y Fm(")612 1147 y Ft(=)-12 b Fu(\))o(f)p Fe(app)r |
|
5196 Fu(h)p Fe(vr)12 b Ff(b;)g(X)1096 1156 y FG(6)1130 1147 |
|
5197 y Fu(i)21 b(\031)o Ft(?)f Fe(app)q Fu(h)p Fe(vr)13 b |
|
5198 Ff(b;)f Ft(\()p Ff(b)g(a)p Ft(\))p Fd(\001)p Ff(X)1777 |
|
5199 1156 y FG(7)1812 1147 y Fu(i)p Ff(;)32 b(b)19 b Ft(#?)h |
|
5200 Fe(app)q Fu(h)p Fe(vr)13 b Ff(a;)f(X)2395 1156 y FG(7)2429 |
|
5201 1147 y Fu(ig)400 b Fv(\()p Fu(\031)p Ft(?)p Fv(-abstraction-2\))622 |
|
5202 1216 y Fu(\001)11 b(\001)g(\001)21 b(\001)12 b(\001)f(\001)2083 |
|
5203 b(\001)11 b(\001)h(\001)654 1247 y Fm(")612 1286 y Ft(=)-12 |
|
5204 b Fu(\))o(f)p Ff(b)20 b Fu(\031)p Ft(?)g Ff(b;)31 b(X)1057 |
|
5205 1295 y FG(6)1112 1286 y Fu(\031)p Ft(?)19 b(\()p Ff(b)13 |
|
5206 b(a)p Ft(\))p Fd(\001)p Ff(X)1434 1295 y FG(7)1469 1286 |
|
5207 y Ff(;)31 b(b)20 b Ft(#?)g Fe(app)p Fu(h)p Fe(vr)13 b |
|
5208 Ff(a;)f(X)2024 1295 y FG(7)2059 1286 y Fu(ig)770 b Fv(\()p |
|
5209 Fu(\031)p Ft(?)p Fv(-fnctn)19 b(symbol\))654 1340 y Fm(")612 |
|
5210 1379 y Ft(=)-12 b Fu(\))o(f)p Ff(X)818 1388 y FG(6)873 |
|
5211 1379 y Fu(\031)p Ft(?)20 b(\()p Ff(b)12 b(a)p Ft(\))p |
|
5212 Fd(\001)q Ff(X)1196 1388 y FG(7)1230 1379 y Ff(;)31 b(b)20 |
|
5213 b Ft(#?)g Fe(app)q Fu(h)p Fe(vr)12 b Ff(a;)g(X)1785 1388 |
|
5214 y FG(7)1820 1379 y Fu(ig)1009 b Fv(\()p Fu(\031)p Ft(?)p |
|
5215 Fv(-atom\))650 1433 y Fm(\033)612 1472 y Ft(=)-12 b Fu(\))o(f)p |
|
5216 Ff(b)20 b Ft(#?)g Fe(app)q Fu(h)p Fe(vr)13 b Ff(a;)f(X)1265 |
|
5217 1481 y FG(7)1299 1472 y Fu(ig)67 b Fv(with)21 b Ff(\033)h |
|
5218 Ft(=)e([)p Ff(X)1781 1481 y FG(6)1835 1472 y Ft(:=)f(\()p |
|
5219 Ff(b)12 b(a)p Ft(\))p Fd(\001)q Ff(X)2144 1481 y FG(7)2178 |
|
5220 1472 y Ft(])693 b Fv(\()p Fu(\031)p Ft(?)p Fv(-v)n(ariable\))645 |
|
5221 1537 y Fl(?)612 1576 y Ft(=)-12 b Fu(\))o(f)p Ff(b)20 |
|
5222 b Ft(#?)g Fu(h)p Fe(vr)13 b Ff(a;)f(X)1159 1585 y FG(7)1193 |
|
5223 1576 y Fu(ig)1636 b Fv(\()p Ft(#?)p Fv(-fnctn)19 b(symbol\))622 |
|
5224 1645 y Fu(\001)11 b(\001)g(\001)21 b(\001)12 b(\001)f(\001)2083 |
|
5225 b(\001)11 b(\001)h(\001)645 1687 y Fl(?)612 1726 y Ft(=)-12 |
|
5226 b Fu(\))o(f)p Ff(b)20 b Ft(#?)g Ff(a;)32 b(b)19 b Ft(#?)h |
|
5227 Ff(X)1230 1735 y FG(7)1265 1726 y Fu(g)1591 b Fv(\()p |
|
5228 Ft(#?)p Fv(-fnctn)19 b(symbol\))645 1791 y Fl(?)612 1831 |
|
5229 y Ft(=)-12 b Fu(\))o(f)p Ff(b)20 b Ft(#?)g Ff(X)980 1840 |
|
5230 y FG(7)1015 1831 y Fu(g)1841 b Fv(\()p Ft(#?)p Fv(-atom\))643 |
|
5231 1898 y Fw(r)612 1936 y Ft(=)-12 b Fu(\))o Fc(?)67 b Fv(with)20 |
|
5232 b Fu(r)g Ft(=)f Fu(f)p Ff(b)h Ft(#)g Ff(X)1359 1945 y |
|
5233 FG(7)1393 1936 y Fu(g)1463 b Fv(\()p Ft(#?)p Fv(-suspension\))p |
|
5234 3402 1962 V 523 1965 2882 4 v 1553 2109 a FU(Fig)o(.)12 |
|
5235 b(4.)19 b FV(Example)g(deri)n(v)n(ations)523 2377 y FP(Example)g(2.)41 |
|
5236 b FW(Using)25 b(the)g(\002rst)h(three)e(function)f(symbols)i(of)f(the)h |
|
5237 (nominal)f(signature)g(of)h(Exam-)523 2468 y(ple)18 b(1)g(to)h |
|
5238 (represent)e FQ(\025)p FW(-terms,)h(the)g(Quiz)g(at)h(the)f(end)f(of)h |
|
5239 (the)h(Introduction)c(translates)j(into)g(the)g(fol-)523 |
|
5240 2559 y(lo)n(wing)k(four)h(uni\002cation)f(problems)f(o)o(v)o(er)h(that) |
|
5241 i(signature,)e(where)g FQ(a)i FW(and)f FQ(b)g FW(are)g(distinct)h |
|
5242 (atoms)523 2651 y(of)c(sort)g FA(vid)31 b FW(and)19 b |
|
5243 FQ(X)1103 2663 y FJ(1)1140 2651 y FQ(;)14 b(:)g(:)g(:)g(;)g(X)1394 |
|
5244 2663 y FJ(7)1452 2651 y FW(are)20 b(distinct)g(v)n(ariables)f(of)h |
|
5245 (sort)h FA(exp)5 b FW(:)1008 2804 y FS(P)1057 2812 y |
|
5246 FG(1)1113 2761 y Fy(def)1120 2804 y FF(=)26 b FB(f)p |
|
5247 FC(fn)13 b FS(a:)p FC(fn)g FS(b:)p FC(app)q FB(h)p FS(X)1753 |
|
5248 2812 y FG(1)1788 2804 y FS(;)g FC(vr)h FS(b)p FB(i)21 |
|
5249 b(\031)p FF(?)g FC(fn)13 b FS(b:)p FC(fn)h FS(a:)p FC(app)q |
|
5250 FB(h)p FC(vr)f FS(a;)g(X)2791 2812 y FG(1)2826 2804 y |
|
5251 FB(ig)p FV(,)1008 2922 y FS(P)1057 2930 y FG(2)1113 2879 |
|
5252 y Fy(def)1120 2922 y FF(=)26 b FB(f)p FC(fn)13 b FS(a:)p |
|
5253 FC(fn)g FS(b:)p FC(app)q FB(h)p FS(X)1753 2930 y FG(2)1788 |
|
5254 2922 y FS(;)g FC(vr)h FS(b)p FB(i)21 b(\031)p FF(?)g |
|
5255 FC(fn)13 b FS(b:)p FC(fn)h FS(a:)p FC(app)q FB(h)p FC(vr)f |
|
5256 FS(a;)g(X)2791 2930 y FG(3)2826 2922 y FB(ig)p FV(,)1008 |
|
5257 3040 y FS(P)1057 3048 y FG(3)1113 2997 y Fy(def)1120 |
|
5258 3040 y FF(=)26 b FB(f)p FC(fn)13 b FS(a:)p FC(fn)g FS(b:)p |
|
5259 FC(app)q FB(h)p FC(vr)h FS(b;)f(X)1912 3048 y FG(4)1947 |
|
5260 3040 y FB(i)21 b(\031)p FF(?)g FC(fn)13 b FS(b:)p FC(fn)h |
|
5261 FS(a:)p FC(app)q FB(h)p FC(vr)f FS(a;)g(X)2791 3048 y |
|
5262 FG(5)2826 3040 y FB(ig)p FV(,)1008 3158 y FS(P)1057 3166 |
|
5263 y FG(4)1113 3115 y Fy(def)1120 3158 y FF(=)26 b FB(f)p |
|
5264 FC(fn)13 b FS(a:)p FC(fn)g FS(b:)p FC(app)q FB(h)p FC(vr)h |
|
5265 FS(b;)f(X)1912 3166 y FG(6)1947 3158 y FB(i)21 b(\031)p |
|
5266 FF(?)g FC(fn)13 b FS(a:)p FC(fn)h FS(a:)p FC(app)p FB(h)p |
|
5267 FC(vr)f FS(a;)g(X)2798 3166 y FG(7)2833 3158 y FB(ig)p |
|
5268 FV(.)523 3288 y FW(Applying)18 b(the)j(nominal)e(uni\002cation)f |
|
5269 (algorithm)h(described)g(abo)o(v)o(e,)f(we)j(\002nd)f(that)581 |
|
5270 3429 y FO(\017)41 b FQ(P)717 3441 y FJ(1)775 3429 y FW(has)21 |
|
5271 b(no)f(solution;)581 3524 y FO(\017)41 b FQ(P)717 3536 |
|
5272 y FJ(2)775 3524 y FW(has)21 b(a)g(most)f(general)f(solution)g(gi)n(v)o |
|
5273 (en)g(by)h FO(r)2075 3536 y FJ(2)2122 3524 y FM(=)9 b |
|
5274 Fn(?)20 b FW(and)f FQ(\033)2468 3536 y FJ(2)2515 3524 |
|
5275 y FM(=)9 b([)p FQ(X)2681 3536 y FJ(2)2727 3524 y FM(:=)g |
|
5276 FN(vr)k FQ(b;)h(X)3067 3536 y FJ(3)3113 3524 y FM(:=)9 |
|
5277 b FN(vr)k FQ(a)p FM(])p FW(;)581 3618 y FO(\017)41 b |
|
5278 FQ(P)717 3630 y FJ(3)775 3618 y FW(has)21 b(a)g(most)f(general)f |
|
5279 (solution)g(gi)n(v)o(en)g(by)h FO(r)2075 3630 y FJ(3)2122 |
|
5280 3618 y FM(=)9 b Fn(?)20 b FW(and)f FQ(\033)2468 3630 |
|
5281 y FJ(3)2515 3618 y FM(=)9 b([)p FQ(X)2681 3630 y FJ(4)2727 |
|
5282 3618 y FM(:=)g(\()p FQ(a)14 b(b)p FM(\))p FK(\001)p FQ(X)3078 |
|
5283 3630 y FJ(5)3115 3618 y FM(])p FW(;)581 3713 y FO(\017)41 |
|
5284 b FQ(P)717 3725 y FJ(4)773 3713 y FW(has)17 b(a)h(most)f(general)f |
|
5285 (solution)h(gi)n(v)o(en)f(by)h FO(r)2052 3725 y FJ(4)2098 |
|
5286 3713 y FM(=)9 b FO(f)p FQ(b)22 b FM(#)i FQ(X)2434 3725 |
|
5287 y FJ(7)2471 3713 y FO(g)17 b FW(and)g FQ(\033)2715 3725 |
|
5288 y FJ(3)2762 3713 y FM(=)9 b([)p FQ(X)2928 3725 y FJ(6)2974 |
|
5289 3713 y FM(:=)g(\()p FQ(b)14 b(a)p FM(\))p FK(\001)o FQ(X)3324 |
|
5290 3725 y FJ(7)3361 3713 y FM(])p FW(.)523 3859 y(Deri)n(v)n(ations)21 |
|
5291 b(for)g FQ(P)1101 3871 y FJ(1)1161 3859 y FW(and)g FQ(P)1356 |
|
5292 3871 y FJ(4)1416 3859 y FW(are)g(sk)o(etched)g(in)h(Fig.)g(4.)f(Using)h |
|
5293 (the)g(Adequac)o(y)d(Theorem)h(1,)i(one)523 3958 y(can)30 |
|
5294 b(interpret)e(these)i(solutions)f(as)i(the)e(follo)n(wing)g(statements) |
|
5295 g(about)g(the)h FQ(\025)p FW(-terms)g(from)e(the)523 |
|
5296 4058 y(quiz.)p 523 4122 V 523 4963 4 841 v 551 4198 a |
|
5297 FU(Quiz)19 b(answers)595 4282 y FV(1.)41 b(There)19 b(is)g(no)g |
|
5298 FS(\025)p FV(-term)g FS(M)1347 4290 y FG(1)1400 4282 |
|
5299 y FV(making)h(the)f(\002rst)f(pair)h(of)g(terms)f FS(\013)p |
|
5300 FV(-equi)n(v)n(alent.)595 4374 y(2.)41 b(The)19 b(only)h(solution)f |
|
5301 (for)g(the)g(second)h(problem)g(is)f(to)f(tak)o(e)i FS(M)2310 |
|
5302 4382 y FG(2)2366 4374 y FF(=)h FS(b)d FV(and)i FS(M)2699 |
|
5303 4382 y FG(3)2755 4374 y FF(=)h FS(a)p FV(.)595 4465 y(3.)41 |
|
5304 b(F)o(or)21 b(the)g(third)g(problem)h(we)e(can)i(tak)o(e)f |
|
5305 FS(M)1810 4473 y FG(5)1866 4465 y FV(to)g(be)g(an)o(y)h |
|
5306 FS(\025)p FV(-term,)e(so)h(long)g(as)g(we)g(tak)o(e)h |
|
5307 FS(M)3060 4473 y FG(4)3115 4465 y FV(to)f(be)g(the)692 |
|
5308 4556 y(result)e(of)g(sw)o(apping)h(all)f(occurrences)h(of)f |
|
5309 FS(a)g FV(and)g FS(b)g FV(throughout)h FS(M)2470 4564 |
|
5310 y FG(5)2505 4556 y FV(.)595 4647 y(4.)41 b(F)o(or)22 |
|
5311 b(the)h(last)f(problem,)h(we)f(can)h(tak)o(e)g FS(M)1802 |
|
5312 4655 y FG(7)1859 4647 y FV(to)f(be)h(an)o(y)g FS(\025)p |
|
5313 FV(-term)f(that)g FT(does)h(not)g(contain)h(fr)m(ee)e(occur)o(-)692 |
|
5314 4739 y(r)m(ences)f(of)f FS(b)p FV(,)f(so)h(long)h(as)f(we)f(tak)o(e)i |
|
5315 FS(M)1698 4747 y FG(6)1752 4739 y FV(to)f(be)g(the)g(result)f(of)h(sw)o |
|
5316 (apping)h(all)f(occurrences)h(of)f FS(b)g FV(and)g FS(a)692 |
|
5317 4830 y FV(throughout)k FS(M)1116 4838 y FG(7)1151 4830 |
|
5318 y FV(,)e(or)g(equi)n(v)n(alently)i(\(since)e FS(b)g FV(is)g(not)g(free) |
|
5319 g(in)h FS(M)2401 4838 y FG(7)2435 4830 y FV(\),)f(taking)h |
|
5320 FS(M)2784 4838 y FG(6)2840 4830 y FV(to)g(be)f(the)g(result)g(of)692 |
|
5321 4921 y(replacing)e(all)f(free)f(occurrences)j(of)e FS(a)f |
|
5322 FV(in)h FS(M)1884 4929 y FG(7)1937 4921 y FV(with)g FS(b)p |
|
5323 FV(.)p 3402 4963 V 523 4966 2882 4 v eop end |
|
5324 %%Page: 12 12 |
|
5325 TeXDict begin 12 11 bop 523 448 a FP(Remark)20 b(2)41 |
|
5326 b(\(Atoms)22 b(ar)m(e)g(not)f(variables\).)g FW(Nominal)h |
|
5327 (uni\002cation)e(uni\002es)i(v)n(ariables,)f(b)n(ut)h(it)h(does)523 |
|
5328 548 y(not)i(unify)e(atoms.)i(Indeed)e(the)i(operation)e(of)h |
|
5329 (identifying)f(tw)o(o)i(atoms)g(by)f(renaming)f(does)i(not)523 |
|
5330 648 y(necessarily)e(preserv)o(e)e(the)j(v)n(alidity)e(of)h(the)g |
|
5331 (judgements)f(in)h(Fig.)g(2.)g(F)o(or)g(e)o(xample,)f |
|
5332 Fn(?)28 b FO(`)g FQ(a:b)g FO(\031)523 747 y FQ(c:b)23 |
|
5333 b FW(holds)g(if)h FQ(b)k FO(6)p FM(=)g FQ(a;)14 b(c)p |
|
5334 FW(;)24 b(b)n(ut)f(renaming)e FQ(b)j FW(to)f(be)g FQ(a)h |
|
5335 FW(in)f(this)h(judgement)d(we)j(get)f Fn(?)28 b FO(`)h |
|
5336 FQ(a:a)f FO(\031)g FQ(c:a)p FW(,)523 847 y(which)18 b(does)f(not)h |
|
5337 (hold)f(so)i(long)e(as)i FQ(a)j FO(6)p FM(=)h FQ(c)p |
|
5338 FW(.)18 b(Referring)f(to)h(Def.)g(2,)g(you)f(will)i(see)f(that)h(we)f |
|
5339 (do)g(allo)n(w)523 946 y(v)n(ariables)k(ranging)g(o)o(v)o(er)f(sorts)j |
|
5340 (of)f(atoms;)g(and)f(such)h(v)n(ariables)f(can)h(be)g(uni\002ed)g(lik)o |
|
5341 (e)g(an)o(y)f(other)523 1046 y(v)n(ariables.)17 b(Ho)n(we)n(v)o(er)m(,) |
|
5342 e(if)j FQ(A)h FW(is)f(such)f(a)i(v)n(ariable,)d(then)h(it)h(cannot)f |
|
5343 (appear)f(in)i(abstraction)f(position,)523 1146 y(i.e.)26 |
|
5344 b(as)h FQ(A:t)p FW(.)g(This)f(is)i(because)d(we)i(speci\002cally)f |
|
5345 (restricted)f(abstraction)g(to)i(range)e(o)o(v)o(er)g(atoms,)523 |
|
5346 1245 y(rather)31 b(than)g(o)o(v)o(er)g(arbitrary)f(terms)h(of)h(atom)f |
|
5347 (sort.)h(Such)f(a)h(restriction)f(seems)h(necessary)f(to)523 |
|
5348 1345 y(obtain)16 b(single,)g(most)g(general,)f(solutions)h(to)h |
|
5349 (nominal)e(uni\002cation)g(problems.)g(F)o(or)h(without)g(such)523 |
|
5350 1445 y(a)j(restriction,)e(because)h(of)g(rule)g(\()p |
|
5351 FO(\031)p FW(-abstraction-2\))d(we)j(w)o(ould)g(also)g(ha)n(v)o(e)g(to) |
|
5352 h(allo)n(w)f(v)n(ariables)f(to)523 1544 y(appear)24 b(on)g(the)h |
|
5353 (left-hand)f(side)h(of)g(freshness)f(relations)g(and)h(in)g(suspended)e |
|
5354 (permutations.)g(So)523 1644 y(then)e(we)g(w)o(ould)g(get)g |
|
5355 (uni\002cation)f(problems)g(lik)o(e)i FO(f)p FM(\()p |
|
5356 FQ(A)14 b(B)t FM(\))p FK(\001)p FQ(C)31 b FO(\031)p FM(?)25 |
|
5357 b FQ(C)6 b FO(g)p FW(,)21 b(where)g FQ(A)p FW(,)h FQ(B)k |
|
5358 FW(and)20 b FQ(C)29 b FW(are)523 1743 y(v)n(ariables)20 |
|
5359 b(of)g(atom)g(sort;)h(this)g(has)g(tw)o(o)g(incomparable)d(solutions,)i |
|
5360 (namely)g FM(\()p Fn(?)p FQ(;)14 b FM([)p FQ(A)24 b FM(:=)f |
|
5361 FQ(B)t FM(]\))f FW(and)523 1843 y FM(\()p FO(f)p FQ(A)h |
|
5362 FM(#)g FQ(C)q(;)14 b(B)28 b FM(#)23 b FQ(C)6 b FO(g)p |
|
5363 FQ(;)14 b(")p FM(\))p FW(.)523 2166 y FR(4)99 b(Related)26 |
|
5364 b(w)o(ork)523 2422 y FW(Most)31 b(pre)n(vious)e(w)o(ork)i(on)f |
|
5365 (uni\002cation)g(for)g(languages)f(with)i(binders)f(is)i(based)e(on)h |
|
5366 (forms)f(of)523 2522 y(higher)n(-order)j(uni\002cation,)h(i.e.)i |
|
5367 (solving)e(equations)h(between)g FQ(\025)p FW(-terms)h(modulo)e |
|
5368 FQ(\013\014)t(\021)s FW(-equi-)523 2622 y(v)n(alence)15 |
|
5369 b(by)h(capture-a)n(v)n(oiding)d(substitution)i(of)h(terms)g(for)f |
|
5370 (function)f(v)n(ariables.)h(Notable)h(among)523 2721 |
|
5371 y(that)i(w)o(ork)f(is)h(Miller')-5 b(s)18 b FP(higher)n(-or)m(der)f |
|
5372 (pattern)g(uni\002cation)e FW(used)j(in)f(his)h FQ(L)2707 |
|
5373 2733 y FI(\025)2769 2721 y FW(logic)f(programming)523 |
|
5374 2821 y(language)g([20].)g(This)h(kind)g(of)g(uni\002cation)f(retains)i |
|
5375 (the)f(good)f(properties)g(of)h(\002rst-order)f(uni\002ca-)523 |
|
5376 2920 y(tion:)22 b(a)h(linear)n(-time)e(decision)h(procedure)d(and)j(e)o |
|
5377 (xistence)f(of)h(most)g(general)f(uni\002ers.)h(Ho)n(we)n(v)o(er)523 |
|
5378 3020 y(it)f(imposes)g(a)g(restriction)f(on)g(the)g(form)g(of)g |
|
5379 FQ(\025)p FW(-terms)h(to)g(be)f(uni\002ed;)g(namely)g(that)g(function)f |
|
5380 (v)n(ari-)523 3120 y(ables)e(may)f(only)g(be)h(applied)f(to)h(distinct) |
|
5381 g(bound)e(v)n(ariables.)h(An)h(empirical)f(study)g(by)g(Michaylo)o(v) |
|
5382 523 3219 y(and)24 b(Pfenning)g([19])f(suggests)i(that)g(most)g |
|
5383 (uni\002cations)f(arising)h(dynamically)e(in)i(higher)n(-order)523 |
|
5384 3319 y(logic)16 b(programming)c(satisfy)k(Miller')-5 |
|
5385 b(s)16 b(restriction,)f(b)n(ut)h(that)g(it)h(rules)e(out)h(some)g |
|
5386 (useful)f(program-)523 3419 y(ming)22 b(idioms.)g(F)o(or)g(us,)g(the)h |
|
5387 (main)f(disadv)n(antage)e(of)i FQ(L)2158 3431 y FI(\025)2224 |
|
5388 3419 y FW(is)i(one)e(common)e(to)j(most)f(approaches)523 |
|
5389 3518 y(based)i(on)g(higher)n(-order)d(abstract)j(syntax:)g(one)f |
|
5390 (cannot)g FP(dir)m(ectly)i FW(e)o(xpress)e(the)i(common)d(idiom)523 |
|
5391 3618 y(of)c(possibly-capturing)c(substitution)j(of)h(terms)g(for)f |
|
5392 (meta)n(v)n(ariables.)g(Instead)g(one)h(has)g(to)g(replace)523 |
|
5393 3717 y(meta)n(v)n(ariables,)j FQ(X)7 b FW(,)21 b(with)h(function)f(v)n |
|
5394 (ariables)g(applied)g(to)h(distinct)g(lists)h(of)f(\(bound\))e(v)n |
|
5395 (ariables,)523 3817 y FQ(F)26 b(x)649 3829 y FJ(1)700 |
|
5396 3817 y FQ(:)14 b(:)g(:)g(x)858 3829 y FI(n)904 3817 y |
|
5397 FW(,)20 b(and)g(use)g(capture-a)n(v)n(oiding)d(substitution.)648 |
|
5398 3928 y(Do)n(wek)25 b FP(et)i(al)f FW([8])f(present)h(a)h(v)o(ersion)d |
|
5399 (of)i(higher)n(-order)d(pattern)j(uni\002cation)f(for)g |
|
5400 FQ(\025\033)31 b FW(\(a)26 b FQ(\025)p FW(-)523 4028 |
|
5401 y(calculus)i(with)h(de-Bruijn)e(indices)i(and)f(e)o(xplicit)g |
|
5402 (substitutions\))f(in)i(which)f(uni\002cation)f(prob-)523 |
|
5403 4127 y(lems)c(are)f(solv)o(ed,)g(lik)o(e)h(in)f(nominal)f |
|
5404 (uni\002cation,)h(by)g(te)o(xtual)f(replacements)h(of)g(terms)g(for)g |
|
5405 (v)n(ari-)523 4227 y(ables;)i(ho)n(we)n(v)o(er)e(a)j(`pre-cooking')20 |
|
5406 b(operation)i(ensures)h(that)h(the)g(te)o(xtual)f(replacements)g(can)h |
|
5407 (be)523 4327 y(\(f)o(aithfully\))e(related)i(to)g(capture-a)n(v)n |
|
5408 (oiding)d(substitutions.)j(It)g(seems)h(that)f(nominal)f(uni\002cation) |
|
5409 523 4426 y(problems)i(can)h(be)h(encoded)d(into)i(higher)n(-order)d |
|
5410 (pattern)j(uni\002cation)f(problems)g(using)h(a)h FP(non-)523 |
|
5411 4526 y(trivial)21 b FW(translation)f(\(the)g(details)g(of)h(this)g |
|
5412 (encoding)d(still)j(remain)f(to)h(be)f(in)m(v)o(estigated\).)e(But)j(e) |
|
5413 n(v)o(en)523 4625 y(if)h(it)h(turns)e(out)h(that)g(it)h(is)f(possible)g |
|
5414 (to)g(simulate)g(nominal)f(uni\002cation)f(by)i(higher)n(-order)c |
|
5415 (pattern)523 4725 y(uni\002cation,)k(the)h(calculations)g(in)m(v)n(olv) |
|
5416 o(ed)e(in)j(translating)f(our)f(terms)i(into)f(higher)n(-order)d |
|
5417 (pattern)523 4825 y(terms)30 b(and)g(then)f(using)g(higher)n(-order)e |
|
5418 (pattern)i(uni\002cation)g(are)h(f)o(ar)g(more)f(intricate)h(than)f |
|
5419 (our)523 4924 y(simple)20 b(algorithm)f(that)h(solv)o(es)g(nominal)f |
|
5420 (uni\002cation)g(problems)g(directly)-5 b(.)p eop end |
|
5421 %%Page: 13 13 |
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5422 TeXDict begin 13 12 bop 648 448 a FW(Hamana)20 b([13,)12 |
|
5423 b(14])21 b(manages)f(to)i(add)f(possibly-capturing)d(substitution)i(to) |
|
5424 i(a)g(language)d(lik)o(e)523 548 y(Miller')-5 b(s)21 |
|
5425 b FQ(L)864 560 y FI(\025)907 548 y FW(.)f(This)h(is)g(achie)n(v)o(ed)e |
|
5426 (by)g(adding)g(syntax)h(for)f(e)o(xplicit)h(renaming)f(operations)f |
|
5427 (and)i(by)523 648 y(recording)g(implicit)i(dependencies)e(of)i(v)n |
|
5428 (ariables)f(upon)g(bindable)f(names)i(in)h(a)f(typing)f(conte)o(xt.)523 |
|
5429 747 y(The)k(mathematical)g(foundation)d(for)j(Hamana')-5 |
|
5430 b(s)25 b(system)h(is)h(the)e(model)g(of)g(binding)f(syntax)h(of)523 |
|
5431 847 y(Fiore)f FP(et)g(al)g FW([9].)f(The)h(mathematical)f(foundation)e |
|
5432 (for)i(our)g(w)o(ork)g(appeared)g(concurrently)e([10])523 |
|
5433 946 y(and)k(is)g(in)h(a)f(sense)g(complementary)-5 b(.)22 |
|
5434 b(F)o(or)j(in)g(Hamana')-5 b(s)24 b(system)i(the)f(typing)e(conte)o(xt) |
|
5435 h(restricts)523 1046 y(which)d(terms)h(may)g(be)f(substituted)h(for)f |
|
5436 (a)h(v)n(ariable)f(by)g(gi)n(ving)g(a)h(\002nite)g(set)h(of)f(names)f |
|
5437 (that)h FP(must)523 1146 y(contain)27 b FW(the)i(free)f(names)g(of)g |
|
5438 (such)g(a)h(term;)f(whereas)g(we)h(gi)n(v)o(e)f(a)h(\002nite)f(set)i |
|
5439 (of)e(names)g(which)523 1245 y(the)c(term')-5 b(s)24 |
|
5440 b(free)g(v)n(ariables)f FP(must)h(avoid)p FW(.)f(Since)h |
|
5441 FQ(\013)p FW(-con)m(v)o(ersion)d(is)k(phrased)d(in)j(terms)e(of)h(a)n |
|
5442 (v)n(oid-)523 1345 y(ance,)18 b(i.e.)g(freshness)g(of)g(names,)f(our)h |
|
5443 (approach)e(seems)j(more)e(natural)h(if)g(one)g(w)o(ants)h(to)f |
|
5444 (compute)523 1445 y FQ(\013)p FW(-equi)n(v)n(alences)25 |
|
5445 b(concretely)-5 b(.)25 b(On)i(top)g(of)g(that,)g(our)f(use)h(of)g(name) |
|
5446 g(permutations,)e(rather)h(than)523 1544 y(arbitrary)18 |
|
5447 b(renaming)f(functions,)h(leads)i(to)f(technical)g(simpli\002cations.)g |
|
5448 (In)g(an)o(y)g(case,)g(the)h(bottom)523 1644 y(line)i(is)h(that)e |
|
5449 (Hamana')-5 b(s)22 b(system)g(seems)g(more)f(complicated)f(than)h(the)h |
|
5450 (one)f(presented)g(here)g(and)523 1743 y(does)f(not)g(possess)h(most)f |
|
5451 (general)f(uni\002ers.)523 2017 y FR(5)99 b(Conclusion)523 |
|
5452 2225 y FW(In)21 b(this)h(paper)e(we)i(ha)n(v)o(e)e(proposed)g(a)h |
|
5453 (solution)g(to)g(the)g(problem)f(of)h(\002nding)f(possibly-capturing) |
|
5454 523 2324 y(substitutions)k(that)g(unify)f(terms)i(in)m(v)n(olving)d |
|
5455 (binders)h(up)h(to)h FQ(\013)p FW(-con)m(v)o(ersion.)c(T)-7 |
|
5456 b(o)25 b(do)f(so)h(we)f(con-)523 2424 y(sidered)f(a)h(man)o(y-sorted)d |
|
5457 (\002rst-order)h(term)h(language)f(with)h(distinguished)f(collections)h |
|
5458 (of)g(con-)523 2523 y(stants)k(called)f FP(atoms)g FW(and)f(with)i |
|
5459 FP(atom-abstr)o(action)c FW(operations)i(for)g(binding)f(atoms)i(in)h |
|
5460 (terms.)523 2623 y(This)e(pro)o(vides)d(a)j(simple,)f(b)n(ut)g(\003e)o |
|
5461 (xible,)g(frame)n(w)o(ork)e(for)i(specifying)f(binding)f(operations)h |
|
5462 (and)523 2723 y(their)g(scopes,)g(in)g(which)f(the)h(bound)f(entities)h |
|
5463 (are)g(e)o(xplicitly)f(named.)g(By)h(using)g(v)n(ariables)f(pre-)523 |
|
5464 2822 y(\002x)o(ed)30 b(with)h(suspended)e(permutations,)g(one)h(can)h |
|
5465 (ha)n(v)o(e)f(substitution)g(of)g(terms)h(for)f(v)n(ariables)523 |
|
5466 2922 y(both)21 b(allo)n(w)g(capture)g(of)g(atoms)h(by)f(binders)g(and)g |
|
5467 (respect)g FQ(\013)p FW(-equi)n(v)n(alence)f(\(renaming)f(of)i(bound) |
|
5468 523 3022 y(atoms\).)c(The)f(de\002nition)g(of)h FQ(\013)p |
|
5469 FW(-equi)n(v)n(alence)e(for)i(the)g(term)g(language)e(mak)o(es)i(use)h |
|
5470 (of)f(an)g(auxiliary)523 3121 y FP(fr)m(eshness)23 b |
|
5471 FW(relation)f(between)g(atoms)h(and)f(terms)h(which)f(generalises)g |
|
5472 (the)h(`not)f(a)h(free)f(atom)h(of)5 b(')523 3221 y(relation)15 |
|
5473 b(from)f(ground)g(terms)h(to)h(terms)f(with)h(v)n(ariables;)f |
|
5474 (furthermore,)d(because)j(v)n(ariables)g(stand)523 3320 |
|
5475 y(for)i(unkno)n(wn)f(terms,)h(hence)h(with)g(unkno)n(wn)d(free)j |
|
5476 (atoms,)f(it)i(is)g(necessary)e(to)h(mak)o(e)f(hypotheses)523 |
|
5477 3420 y(about)26 b(the)h(freshness)f(of)h(atoms)f(for)h(v)n(ariables)f |
|
5478 (in)h(judgements)e(about)h(term)g(equi)n(v)n(alence)f(and)523 |
|
5479 3520 y(freshness.)20 b(This)i(reliance)e(on)h(`freshness')f(is)i(the)f |
|
5480 (main)g(no)o(v)o(elty\227it)e(arises)j(from)e(the)h(w)o(ork)g(re-)523 |
|
5481 3619 y(ported)e(in)i([11,)12 b(23].)19 b(It)i(leads)g(to)f(a)h(ne)n(w)g |
|
5482 (notion)e(of)h(uni\002cation)f(problem)g(in)h(which)g(instances)h(of) |
|
5483 523 3719 y(both)c(equi)n(v)n(alence)e(and)i(freshness)g(ha)n(v)o(e)g |
|
5484 (to)h(be)f(solv)o(ed)g(by)g(gi)n(ving)g(term-substitutions)e(and)i |
|
5485 (\(pos-)523 3819 y(sibly\))i(freshness)h(conditions)e(on)h(v)n |
|
5486 (ariables)g(in)h(the)g(solution.)e(W)-7 b(e)21 b(sho)n(wed)e(that)h |
|
5487 (this)g(uni\002cation)523 3918 y(problem)f(is)i(decidable)e(and)g |
|
5488 (unitary)-5 b(.)648 4011 y(Chene)o(y)30 b(and)i(Urban)f([5])g(are)h(in) |
|
5489 m(v)o(estigating)d(the)j(e)o(xtent)g(to)g(which)f(nominal)g |
|
5490 (uni\002cation)523 4102 y(can)c(be)g(used)g(in)g(resolution-based)e |
|
5491 (proof)g(search)i(for)f(a)i(form)e(of)h(\002rst-order)e(logic)i |
|
5492 (program-)523 4194 y(ming)f(for)h(languages)e(with)i(binders)f(\(with)h |
|
5493 (a)g(vie)n(w)g(to)g(pro)o(viding)d(better)j(machine-assistance)523 |
|
5494 4285 y(for)d(structural)f(operational)f(semantics\).)i(Such)g(a)h |
|
5495 (logic)e(programming)e(language)i(should)g(per)n(-)523 |
|
5496 4376 y(mit)j(a)g(concrete,)e(`nominal')g(approach)g(to)i(bound)e |
|
5497 (entities)i(in)g(programs)d(while)j(ensuring)e(that)523 |
|
5498 4468 y(computation)e(\(which)h(in)h(this)h(case)f(is)h(the)f |
|
5499 (computation)e(of)i(answers)g(to)g(queries\))f(respects)h |
|
5500 FQ(\013)p FW(-)523 4559 y(equi)n(v)n(alence)f(between)i(terms.)g(This)h |
|
5501 (is)g(illustrated)f(with)g(the)h(Prolog-lik)o(e)e(program)f(in)i(Fig.)h |
|
5502 (5,)523 4650 y(which)18 b(implements)f(a)h(simple)g(typing)f(algorithm) |
|
5503 f(for)i FQ(\025)p FW(-terms.)g(Interesting)e(is)j(the)f(third)g |
|
5504 (clause.)523 4742 y(First,)23 b(note)g(the)f(term)h Fb(\(lam)49 |
|
5505 b(x.M\))p FW(,)22 b(which)g(uni\002es)h(with)g(an)o(y)f |
|
5506 FQ(\025)p FW(-abstraction.)g(The)g(binder)f Fb(x)p FW(,)523 |
|
5507 4833 y(roughly)d(speaking,)g(has)i(in)g(the)f(`nominal')f(approach)f(a) |
|
5508 k(v)n(alue)e(which)g(can)g(be)h(used)f(in)h(the)g(body)523 |
|
5509 4924 y(of)e(the)g(clause,)f(for)g(e)o(xample)g(for)g(adding)g |
|
5510 Fb(\(pair)48 b(x)i(A\))18 b FW(to)g(the)g(conte)o(xt)f |
|
5511 Fb(Gamma)p FW(.)g(Second,)f(the)p eop end |
|
5512 %%Page: 14 14 |
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5513 TeXDict begin 14 13 bop 523 369 2882 4 v 523 946 4 578 |
|
5514 v 551 460 a Fa(type)40 b(Gamma)28 b(\(var)39 b(X\))g(A)29 |
|
5515 b(:-)f(mem)40 b(\(pair)f(X)h(A\))f(Gamma.)551 562 y(type)h(Gamma)28 |
|
5516 b(\(app)39 b(M)h(N\))f(B)29 b(:-)f(type)39 b(Gamma)h(M)f(\(arrow)g(A)h |
|
5517 (B\),)f(type)h(Gamma)f(N)g(A.)551 664 y(type)h(Gamma)28 |
|
5518 b(\(lam)39 b(x.M\))g(\(arrow)g(A)h(B\))17 b(/)h(x#Gamma)27 |
|
5519 b(:-)i(type)39 b(\(pair)g(x)h(A\))8 b Ft(:)19 b(:)k Fa(Gamma)39 |
|
5520 b(M)h(B.)551 794 y(mem)g(A)f(A)8 b Ft(:)19 b(:)k Fa(Tail.)551 |
|
5521 896 y(mem)40 b(A)f(B)8 b Ft(:)19 b(:)k Fa(Tail)29 b(:-)f(mem)39 |
|
5522 b(A)h(Tail.)p 3402 946 V 523 949 2882 4 v 571 1092 a |
|
5523 FU(Fig)o(.)11 b(5.)19 b FV(A)g(Prolog-lik)o(e)g(program)h(implementing) |
|
5524 g(the)f(typing)g(rules)g(for)g(the)g(simply-typed)h FS(\025)p |
|
5525 FV(-calculus.)523 1375 y FW(freshness)k(constraint)f |
|
5526 Fb(x#Gamma)g FW(ensures)h(that)g Fb(Gamma)g FW(cannot)f(be)h(replaced)f |
|
5527 (by)h(a)g(term)g(that)523 1466 y(contains)g Fb(x)i FW(freely)-5 |
|
5528 b(.)23 b(Since)i(this)h(clause)f(is)h(intended)e(to)h(implement)f(the)h |
|
5529 (usual)g(rule)f(for)h(typing)523 1557 y FQ(\025)p FW(-abstractions)1610 |
|
5530 1689 y FB(f)p FS(x)c FF(:)h FS(A)p FB(g)17 b([)g FS(\000)49 |
|
5531 b(.)39 b(M)30 b FF(:)21 b FS(B)p 1610 1728 709 4 v 1639 |
|
5532 1797 a(\000)49 b(.)39 b(\025x:M)29 b FF(:)21 b FS(A)g |
|
5533 FB(\033)g FS(B)523 1903 y FW(its)29 b(operational)c(beha)n(viour)h(is)i |
|
5534 (gi)n(v)o(en)e(by:)h(choose)g(fresh)g(names)g(for)g Fb(Gamma)p |
|
5535 FW(,)g Fb(x)p FW(,)h Fb(M)p FW(,)f Fb(A)h FW(and)f Fb(B)523 |
|
5536 2003 y FW(\(this)g(is)g(standard)e(in)h(Prolog-lik)o(e)f(languages\),)f |
|
5537 (unify)h(the)i(head)e(of)h(the)h(clause)f(with)h(the)f(goal)523 |
|
5538 2102 y(formula,)17 b(apply)g(the)h(resulting)g(uni\002er)g(to)g(the)h |
|
5539 (body)d(of)j(the)f(clause)g(and)g(mak)o(e)g(sure)g(that)h |
|
5540 Fb(Gamma)523 2202 y FW(is)30 b(not)f(replaced)f(by)h(a)h(term)e(that)i |
|
5541 (contains)e(freely)h(the)g(fresh)g(name)f(we)i(ha)n(v)o(e)e(chosen)h |
|
5542 (for)f Fb(x)p FW(.)523 2301 y(Similar)f(f)o(acilities)g(for)f |
|
5543 FP(functional)f(pr)l(o)o(gr)o(amming)g FW(already)h(e)o(xist)h(in)g |
|
5544 (the)f(FreshML)g(language,)523 2401 y(b)n(uilt)17 b(upon)f(the)i(same)f |
|
5545 (foundations:)e(see)j([27])e(and)h FN(www)p FQ(:)p FN(freshml)p |
|
5546 FQ(:)p FN(org)-6 b FW(.)18 b(W)-7 b(e)18 b(are)g(also)f(interested)523 |
|
5547 2501 y(in)22 b(the)g(special)g(case)g(of)g(`nominal)e(matching')g(and)h |
|
5548 (its)i(application)e(to)h(term-re)n(writing)d(modulo)523 |
|
5549 2600 y FQ(\013)p FW(-equi)n(v)n(alence.)648 2700 y(If)j(these)h |
|
5550 (applications)f(sho)n(w)g(that)h(nominal)f(uni\002cation)f(is)j |
|
5551 (practically)e(useful,)g(then)g(it)i(be-)523 2800 y(comes)31 |
|
5552 b(important)f(to)h(study)g(its)h(comple)o(xity)-5 b(.)29 |
|
5553 b(The)i(presentations)f(of)h(the)g(term)g(language)e(in)523 |
|
5554 2899 y(Section)f(2)g(and)f(of)h(the)g(algorithm)e(in)j(Section)e(3)h |
|
5555 (were)g(chosen)f(for)h(clarity)f(and)h(to)g(mak)o(e)f(the)523 |
|
5556 2999 y(proof)22 b(of)i(correctness)e(easier)1412 2969 |
|
5557 y Fy(2)1469 2999 y FW(rather)h(than)h(for)f(ef)n(\002cienc)o(y)-5 |
|
5558 b(.)21 b(In)j(an)o(y)f(case,)h(it)g(remains)f(to)h(be)g(in-)523 |
|
5559 3098 y(v)o(estigated)30 b(whether)f(the)i(sw)o(apping)f(and)h |
|
5560 (freshness)f(computations)f(that)i(we)g(ha)n(v)o(e)g(added)e(to)523 |
|
5561 3198 y(ordinary)-5 b(,)17 b(\002rst-order)i(uni\002cation)g(result)h |
|
5562 (in)h(greater)e(than)h(linear)n(-time)f(comple)o(xity)-5 |
|
5563 b(.)523 3326 y FU(Ackno)o(wledgements:)38 b FV(W)-6 b(e)19 |
|
5564 b(thank)h(Gilles)e(Do)n(wek,)i(Ro)o(y)g(Dyckhof)n(f,)g(Dale)g(Miller)m |
|
5565 (,)e(Francois)i(Pottier)e(and)523 3426 y(Helmut)g(Schwichtenber)o(g)g |
|
5566 (for)g(comments)g(on)g(this)g(w)o(ork.)g(This)f(research)h(w)o(as)g |
|
5567 (supported)h(by)f(UK)g(EPSRC)523 3526 y(grants)h(GR/R29697)h(\(Urban\)) |
|
5568 f(and)h(GR/R07615)g(\(Pitts)d(and)j(Gabbay\).)523 3786 |
|
5569 y FR(Refer)n(ences)560 3970 y FV(1.)42 b(F)-6 b(.)18 |
|
5570 b(Baader)h(and)h(T)-6 b(.)18 b(Nipk)o(o)n(w)-5 b(.)28 |
|
5571 b FT(T)-7 b(erm)19 b(Re)o(writing)f(and)i(All)e(That)p |
|
5572 FV(.)26 b(Cambridge)20 b(Uni)n(v)o(ersity)f(Press,)f(1998.)560 |
|
5573 4059 y(2.)42 b(H.)18 b(P)-8 b(.)18 b(Barendre)o(gt.)27 |
|
5574 b FT(The)19 b(Lambda)h(Calculus:)f(Its)f(Syntax)i(and)g(Semantics)p |
|
5575 FV(.)28 b(North-Holland,)19 b(1984.)560 4149 y(3.)42 |
|
5576 b(L.)23 b(Caires)g(and)h(L.)f(Cardelli.)41 b(A)23 b(spatial)g(logic)h |
|
5577 (for)f(concurrenc)o(y)j(II.)41 b(In)23 b FT(Pr)m(oc.)g(of)g(CONCUR)g |
|
5578 (2002)p FV(,)658 4241 y(v)o(olume)c(2421)h(of)f FT(LNCS)p |
|
5579 FV(,)f(pages)i(209\226225.)h(Springer)o(-V)-8 b(erlag,)18 |
|
5580 b(2002.)560 4331 y(4.)42 b(L.)23 b(Cardelli,)g(P)-8 b(.)23 |
|
5581 b(Gardner)m(,)h(and)h(G.)e(Ghelli.)43 b(Manipulating)25 |
|
5582 b(trees)f(with)f(hidden)i(labels.)44 b(In)23 b FT(Pr)m(oc.)h(of)658 |
|
5583 4422 y(FOSSA)n(CS)p FV(,)18 b(v)o(olume)i(2620)g(of)f |
|
5584 FT(LNCS)p FV(,)f(pages)h(216\226232.)i(Springer)o(-V)-8 |
|
5585 b(erlag,)18 b(2003.)560 4512 y(5.)42 b(J.)31 b(Chene)o(y)i(and)g(C.)e |
|
5586 (Urban.)69 b FS(\013)p FV(Prolog,)32 b(a)g(fresh)g(approach)i(to)e |
|
5587 (logic)g(programming)i(modulo)f FS(\013)p FV(-)658 4603 |
|
5588 y(equi)n(v)n(alence.)42 b(In)24 b FT(Pr)m(oc.)e(of)h(UNIF)f(2003)p |
|
5589 FV(,)i(number)h(DSIC-II/12/03)d(in)i(Departamento)g(de)f(Sistemas)658 |
|
5590 4694 y(Inform)t(\264)-29 b(aticos)22 b(y)f(Computaci)6 |
|
5591 b(\264)-31 b(on)23 b(T)-5 b(echnical)22 b(Report)f(Series.)g(Uni)n(v)o |
|
5592 (ersidad)h(Polit)t(\264)-29 b(ecnica)21 b(de)h(V)-8 b(alencia,)658 |
|
5593 4786 y(2003.)p 523 4837 473 4 v 558 4893 a Fr(2)606 4924 |
|
5594 y FV(See)19 b FC(http)j FF(:)g FS(==)p FC(www)p FS(:)p |
|
5595 FC(cl)p FS(:)p FC(cam)p FS(:)p FC(ac)p FS(:)p FC(uk)p |
|
5596 FS(=)1607 4916 y Fw(\030)1664 4924 y FC(cu200)p FS(=)p |
|
5597 FC(Unification)h FV(for)c(the)g(Isabelle)g(proof)g(scripts.)p |
|
5598 eop end |
|
5599 %%Page: 15 15 |
|
5600 TeXDict begin 15 14 bop 560 448 a FV(6.)42 b(G.)15 b(Do)n(wek.)22 |
|
5601 b(Higher)o(-order)16 b(uni\002cation)h(and)f(matching.)22 |
|
5602 b(In)16 b(A.)f(Robinson)j(and)e(A.)g(V)-10 b(oronk)o(o)o(v)-5 |
|
5603 b(,)18 b(editors,)658 540 y FT(Handbook)j(of)e(A)o(utomated)g |
|
5604 (Reasoning)p FV(,)g(chapter)h(16,)f(pages)h(1009\2261062.)i(Else)n |
|
5605 (vier)m(,)c(2001.)560 631 y(7.)42 b(G.)17 b(Do)n(wek,)h(T)-6 |
|
5606 b(.)16 b(Hardin,)i(and)g(C.)f(Kirchner)l(.)23 b(Higher)o(-order)18 |
|
5607 b(uni\002cation)g(via)f(e)o(xplicit)h(substitutions.)24 |
|
5608 b(In)658 722 y FT(10th)19 b(Symposium)i(of)d(LICS)p FV(,)g(pages)i |
|
5609 (366\226374.)h(IEEE)d(Computer)h(Society)g(Press,)f(1995.)560 |
|
5610 814 y(8.)42 b(G.)25 b(Do)n(wek,)i(T)-6 b(.)25 b(Hardin,)h(C.)g |
|
5611 (Kirchner)m(,)g(and)h(F)-6 b(.)25 b(Pfenning.)50 b(Higher)o(-order)26 |
|
5612 b(uni\002cation)h(via)f(e)o(xplicit)658 905 y(substitutions:)19 |
|
5613 b(the)g(case)h(of)e(higher)o(-order)i(patterns.)27 b(In)19 |
|
5614 b FT(Pr)m(oc.)f(of)h(JICSLP)p FV(,)f(pages)i(259\226273,)h(1996.)560 |
|
5615 996 y(9.)42 b(M.)29 b(P)-8 b(.)27 b(Fiore,)i(G.)f(D.)g(Plotkin,)h(and)g |
|
5616 (D.)g(T)m(uri.)58 b(Abstract)29 b(syntax)h(and)f(v)n(ariable)h |
|
5617 (binding.)60 b(In)29 b FT(14th)658 1088 y(Symposium)20 |
|
5618 b(of)f(LICS)p FV(,)f(pages)i(193\226202.)g(IEEE)e(Computer)i(Society)e |
|
5619 (Press,)g(1999.)523 1179 y(10.)42 b(M.)19 b(J.)f(Gabbay)i(and)f(A.)f |
|
5620 (M.)h(Pitts.)25 b(A)19 b(ne)n(w)g(approach)h(to)f(abstract)g(syntax)h |
|
5621 (in)m(v)o(olving)f(binders.)27 b(In)19 b FT(14th)658 |
|
5622 1270 y(Symposium)h(of)f(LICS)p FV(,)f(pages)i(214\226224.)g(IEEE)e |
|
5623 (Computer)i(Society)e(Press,)g(1999.)523 1362 y(11.)42 |
|
5624 b(M.)26 b(J.)f(Gabbay)i(and)g(A.)e(M.)h(Pitts.)48 b(A)26 |
|
5625 b(ne)n(w)g(approach)i(to)e(abstract)g(syntax)h(with)e(v)n(ariable)i |
|
5626 (binding.)658 1453 y FT(F)-8 b(ormal)19 b(Aspects)g(of)g(Computing)p |
|
5627 FV(,)g(13:341\226363,)j(2002.)523 1544 y(12.)42 b(C.)21 |
|
5628 b(A.)g(Gunter)l(.)35 b FT(Semantics)23 b(of)e(Pr)m(o)o(gr)o(amming)i |
|
5629 (Langua)o(g)o(es:)g(Structur)m(es)g(and)f(T)-7 b(ec)o(hniques)p |
|
5630 FV(.)37 b(F)o(ounda-)658 1636 y(tions)19 b(of)g(Computing.)h(MIT)e |
|
5631 (Press,)g(1992.)523 1727 y(13.)42 b(M.)22 b(Hamana.)40 |
|
5632 b(A)22 b(logic)h(programming)h(language)g(based)g(on)f(binding)h |
|
5633 (algebras.)39 b(In)23 b FT(Pr)m(oc.)f(of)h(T)l(A)n(CS)658 |
|
5634 1818 y(2001)p FV(,)d(v)o(olume)f(2215)h(of)f FT(LNCS)p |
|
5635 FV(,)f(pages)i(243\226262.)h(Springer)o(-V)-8 b(erlag,)18 |
|
5636 b(2001.)523 1910 y(14.)42 b(M.)24 b(Hamana.)45 b(Simple)24 |
|
5637 b FS(\014)1358 1918 y FG(0)1393 1910 y FV(-uni\002cation)h(for)f(terms) |
|
5638 h(with)f(conte)o(xt)h(holes.)45 b(In)24 b FT(Pr)m(oc.)g(of)g(UNIF)g |
|
5639 (2002)p FV(,)658 2001 y(2002.)k(Unpublished)21 b(proceedings.)523 |
|
5640 2092 y(15.)42 b(M.)19 b(Hashimoto)g(and)h(A.)e(Ohori.)27 |
|
5641 b(A)19 b(typed)g(conte)o(xt)h(calculus.)27 b FT(TCS)p |
|
5642 FV(,)19 b(266:249\226271,)i(2001.)523 2183 y(16.)42 b(F)-6 |
|
5643 b(.)17 b(Honsell,)h(M.)g(Miculan,)g(and)h(I.)e(Scagnetto.)25 |
|
5644 b(An)18 b(axiomatic)h(approach)g(to)f(metareasoning)i(on)e(nom-)658 |
|
5645 2275 y(inal)25 b(algebras)h(in)f(HO)m(AS.)46 b(In)25 |
|
5646 b FT(Pr)m(oc.)f(of)h(ICALP)f(2001)p FV(,)i(v)o(olume)f(2076)i(of)e |
|
5647 FT(LNCS)p FV(,)f(pages)i(963\226978.)658 2366 y(Springer)o(-V)-8 |
|
5648 b(erlag,)18 b(2001.)523 2457 y(17.)42 b(J.)19 b(W)-7 |
|
5649 b(.)18 b(Klop.)27 b(T)-5 b(erm)19 b(re)n(writing)f(systems.)28 |
|
5650 b(In)19 b(S.)f(Abramsk)o(y)-5 b(,)20 b(D.)e(M.)h(Gabbay)-5 |
|
5651 b(,)20 b(and)g(T)-6 b(.)19 b(S.)f(E.)g(Maibaum,)658 2549 |
|
5652 y(editors,)h FT(Handbook)i(of)d(Lo)o(gic)i(in)e(Computer)i(Science)o(,) |
|
5653 g(V)-8 b(olume)18 b(2)p FV(,)h(pages)h(1\226116.)g(OUP)-8 |
|
5654 b(,)18 b(1992.)523 2640 y(18.)42 b(A.)17 b(Martelli)g(and)h(U.)f |
|
5655 (Montanari.)24 b(An)18 b(ef)n(\002cient)f(uni\002cation)h(algorithm.)24 |
|
5656 b FT(A)n(CM)17 b(T)l(r)o(ans.)g(Pr)m(o)o(gr)o(amming)658 |
|
5657 2731 y(Langua)o(g)o(es)k(and)f(Systems)p FV(,)f(4\(2\):258\226282,)i |
|
5658 (1982.)523 2823 y(19.)42 b(S.)18 b(Michaylo)o(v)j(and)g(F)-6 |
|
5659 b(.)18 b(Pfenning.)30 b(An)19 b(empirical)h(study)g(of)f(the)h(runtime) |
|
5660 g(beha)o(viour)g(of)g(higher)o(-order)658 2914 y(logic)d(programs.)22 |
|
5661 b(In)16 b FT(Pr)m(oc.)g(W)-7 b(orkshop)18 b(on)f(the)g |
|
5662 FS(\025)p FT(Pr)m(olo)o(g)f(Pr)m(o)o(gr)o(amming)h(Langua)o(g)o(e)p |
|
5663 FV(,)h(pages)g(257\226271,)658 3005 y(1992.)28 b(CIS)18 |
|
5664 b(T)-5 b(echnical)19 b(Report)g(MS-CIS-92-86.)523 3097 |
|
5665 y(20.)42 b(D.)21 b(Miller)l(.)34 b(A)21 b(logic)h(programming)h |
|
5666 (language)g(with)e(lambda-abstraction,)h(function)g(v)n(ariables,)g |
|
5667 (and)658 3188 y(simple)d(uni\002cation.)27 b FT(J)n(ournal)20 |
|
5668 b(of)f(Lo)o(gic)g(and)h(Computation)p FV(,)g(1:497\226536,)h(1991.)523 |
|
5669 3279 y(21.)42 b(R.)20 b(Milner)m(,)h(J.)f(P)o(arro)n(w)-5 |
|
5670 b(,)20 b(and)h(D.)g(W)-6 b(alk)o(er)l(.)32 b(A)21 b(calculus)g(of)g |
|
5671 (mobile)g(processes)h(\(parts)e(I)h(and)g(II\).)32 b |
|
5672 FT(Infor)o(-)658 3371 y(mation)19 b(and)h(Computation)p |
|
5673 FV(,)g(100:1\22677,)g(1992.)523 3462 y(22.)42 b(F)-6 |
|
5674 b(.)13 b(Pfenning)i(and)h(C.)d(Elliott.)j(Higher)o(-order)f(abstract)f |
|
5675 (syntax.)19 b(In)14 b FT(Pr)m(oc.)g(A)n(CM-SIGPLAN)e(Confer)m(ence)658 |
|
5676 3553 y(on)19 b(Pr)m(o)o(gr)o(amming)h(Langua)o(g)o(e)h(Design)e(and)h |
|
5677 (Implementation)p FV(,)g(pages)f(199\226208.)i(A)m(CM)e(Press,)f(1988.) |
|
5678 523 3645 y(23.)42 b(A.)24 b(M.)g(Pitts.)44 b(Nominal)25 |
|
5679 b(logic:)f(A)g(\002rst)g(order)h(theory)g(of)f(names)i(and)f(binding.) |
|
5680 46 b(In)24 b FT(Pr)m(oc.)g(of)h(T)l(A)n(CS)658 3736 y(2001)p |
|
5681 FV(,)20 b(v)o(olume)f(2215)h(of)f FT(LNCS)p FV(,)f(pages)i(219\226242.) |
|
5682 h(Springer)o(-V)-8 b(erlag,)18 b(2001.)523 3827 y(24.)42 |
|
5683 b(A.)18 b(M.)h(Pitts)e(and)j(M.)e(J.)h(Gabbay)-5 b(.)27 |
|
5684 b(A)19 b(metalanguage)h(for)f(programming)h(with)f(bound)h(names)g |
|
5685 (modulo)658 3919 y(renaming.)40 b(In)22 b FT(Pr)m(oc.)g(of)h(MPC2000)p |
|
5686 FV(,)g(v)o(olume)g(1837)h(of)f FT(LNCS)p FV(,)e(pages)j(230\226255.)h |
|
5687 (Springer)o(-V)-8 b(erlag,)658 4010 y(2000.)523 4101 |
|
5688 y(25.)42 b(G.)27 b(D.)h(Plotkin.)56 b(An)28 b(illati)n(v)o(e)f(theory)i |
|
5689 (of)f(relations.)57 b(In)28 b FT(Situation)h(Theory)f(and)h(its)e |
|
5690 (Applications)p FV(,)658 4193 y(v)o(olume)19 b(22)h(of)f |
|
5691 FT(CSLI)f(Lecur)m(e)h(Notes)p FV(,)g(pages)h(133\226146.)h(Stanford)e |
|
5692 (Uni)n(v)o(ersity)-5 b(,)19 b(1990.)523 4284 y(26.)42 |
|
5693 b(M.)27 b(Sato,)f(T)-6 b(.)26 b(Sakurai,)g(and)i(Y)-10 |
|
5694 b(.)26 b(Kame)o(yama.)53 b(A)26 b(simply)h(typed)h(conte)o(xt)f |
|
5695 (calculus)g(with)g(\002rst-class)658 4375 y(en)m(vironments.)h |
|
5696 FT(J)n(ournal)21 b(of)d(Functional)i(and)g(Lo)o(gic)f(Pr)m(o)o(gr)o |
|
5697 (amming)p FV(,)g(2002\(4\),)h(2002.)523 4467 y(27.)42 |
|
5698 b(M.)20 b(R.)f(Shinwell,)g(A.)h(M.)g(Pitts,)e(and)j(M.)f(J.)f(Gabbay)-5 |
|
5699 b(.)32 b(FreshML:)20 b(Programming)h(with)e(binders)i(made)658 |
|
5700 4558 y(simple.)27 b(In)19 b FT(Pr)m(oc.)f(of)h(ICFP)e(2003)p |
|
5701 FV(.)j(A)m(CM)f(Press,)f(2003.)p eop end |
|
5702 %%Trailer |
|
5703 |
|
5704 userdict /end-hook known{end-hook}if |
|
5705 %%EOF |