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1 %!PS-Adobe-2.0 |
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60 %%BeginProcSet: 8r.enc 0 0 |
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61 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 |
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62 % |
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63 % @@psencodingfile@{ |
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64 % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, |
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65 % W. Schmidt, P. Lehman", |
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66 % version = "2.0", |
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67 % date = "30 October 2002", |
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68 % filename = "8r.enc", |
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69 % email = "tex-fonts@@tug.org", |
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70 % docstring = "This is the encoding vector for Type1 and TrueType |
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71 % fonts to be used with TeX. This file is part of the |
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72 % PSNFSS bundle, version 9" |
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73 % @} |
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74 % |
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75 % The idea is to have all the characters normally included in Type 1 fonts |
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76 % available for typesetting. This is effectively the characters in Adobe |
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77 % Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol, |
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78 % MacRoman, and some extra characters from Lucida. |
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79 % |
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80 % Character code assignments were made as follows: |
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81 % |
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82 % (1) the Windows ANSI characters are almost all in their Windows ANSI |
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83 % positions, because some Windows users cannot easily reencode the |
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84 % fonts, and it makes no difference on other systems. The only Windows |
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85 % ANSI characters not available are those that make no sense for |
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86 % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen |
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87 % (173). quotesingle and grave are moved just because it's such an |
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88 % irritation not having them in TeX positions. |
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89 % |
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90 % (2) Remaining characters are assigned arbitrarily to the lower part |
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91 % of the range, avoiding 0, 10 and 13 in case we meet dumb software. |
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92 % |
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93 % (3) Y&Y Lucida Bright includes some extra text characters; in the |
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94 % hopes that other PostScript fonts, perhaps created for public |
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95 % consumption, will include them, they are included starting at 0x12. |
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96 % These are /dotlessj /ff /ffi /ffl. |
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97 % |
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98 % (4) hyphen appears twice for compatibility with both ASCII and Windows. |
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99 % |
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100 % (5) /Euro was assigned to 128, as in Windows ANSI |
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101 % |
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102 % (6) Missing characters from MacRoman encoding incorporated as follows: |
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103 % |
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104 % PostScript MacRoman TeXBase1 |
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105 % -------------- -------------- -------------- |
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106 % /notequal 173 0x16 |
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107 % /infinity 176 0x17 |
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108 % /lessequal 178 0x18 |
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109 % /greaterequal 179 0x19 |
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110 % /partialdiff 182 0x1A |
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111 % /summation 183 0x1B |
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112 % /product 184 0x1C |
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113 % /pi 185 0x1D |
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114 % /integral 186 0x81 |
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115 % /Omega 189 0x8D |
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116 % /radical 195 0x8E |
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117 % /approxequal 197 0x8F |
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118 % /Delta 198 0x9D |
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119 % /lozenge 215 0x9E |
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120 % |
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121 /TeXBase1Encoding [ |
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122 % 0x00 |
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123 /.notdef /dotaccent /fi /fl |
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124 /fraction /hungarumlaut /Lslash /lslash |
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125 /ogonek /ring /.notdef /breve |
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126 /minus /.notdef /Zcaron /zcaron |
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127 % 0x10 |
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128 /caron /dotlessi /dotlessj /ff |
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129 /ffi /ffl /notequal /infinity |
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130 /lessequal /greaterequal /partialdiff /summation |
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131 /product /pi /grave /quotesingle |
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132 % 0x20 |
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133 /space /exclam /quotedbl /numbersign |
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134 /dollar /percent /ampersand /quoteright |
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135 /parenleft /parenright /asterisk /plus |
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136 /comma /hyphen /period /slash |
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137 % 0x30 |
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138 /zero /one /two /three |
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139 /four /five /six /seven |
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140 /eight /nine /colon /semicolon |
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141 /less /equal /greater /question |
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142 % 0x40 |
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143 /at /A /B /C |
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144 /D /E /F /G |
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145 /H /I /J /K |
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146 /L /M /N /O |
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147 % 0x50 |
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148 /P /Q /R /S |
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149 /T /U /V /W |
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150 /X /Y /Z /bracketleft |
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151 /backslash /bracketright /asciicircum /underscore |
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152 % 0x60 |
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153 /quoteleft /a /b /c |
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154 /d /e /f /g |
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155 /h /i /j /k |
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156 /l /m /n /o |
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157 % 0x70 |
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158 /p /q /r /s |
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159 /t /u /v /w |
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160 /x /y /z /braceleft |
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161 /bar /braceright /asciitilde /.notdef |
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162 % 0x80 |
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163 /Euro /integral /quotesinglbase /florin |
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164 /quotedblbase /ellipsis /dagger /daggerdbl |
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165 /circumflex /perthousand /Scaron /guilsinglleft |
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166 /OE /Omega /radical /approxequal |
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167 % 0x90 |
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168 /.notdef /.notdef /.notdef /quotedblleft |
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169 /quotedblright /bullet /endash /emdash |
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170 /tilde /trademark /scaron /guilsinglright |
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171 /oe /Delta /lozenge /Ydieresis |
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172 % 0xA0 |
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173 /.notdef /exclamdown /cent /sterling |
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174 /currency /yen /brokenbar /section |
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175 /dieresis /copyright /ordfeminine /guillemotleft |
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176 /logicalnot /hyphen /registered /macron |
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177 % 0xD0 |
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178 /degree /plusminus /twosuperior /threesuperior |
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179 /acute /mu /paragraph /periodcentered |
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180 /cedilla /onesuperior /ordmasculine /guillemotright |
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181 /onequarter /onehalf /threequarters /questiondown |
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182 % 0xC0 |
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183 /Agrave /Aacute /Acircumflex /Atilde |
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184 /Adieresis /Aring /AE /Ccedilla |
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185 /Egrave /Eacute /Ecircumflex /Edieresis |
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186 /Igrave /Iacute /Icircumflex /Idieresis |
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187 % 0xD0 |
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188 /Eth /Ntilde /Ograve /Oacute |
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189 /Ocircumflex /Otilde /Odieresis /multiply |
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190 /Oslash /Ugrave /Uacute /Ucircumflex |
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191 /Udieresis /Yacute /Thorn /germandbls |
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192 % 0xE0 |
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193 /agrave /aacute /acircumflex /atilde |
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194 /adieresis /aring /ae /ccedilla |
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195 /egrave /eacute /ecircumflex /edieresis |
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196 /igrave /iacute /icircumflex /idieresis |
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197 % 0xF0 |
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198 /eth /ntilde /ograve /oacute |
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199 /ocircumflex /otilde /odieresis /divide |
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200 /oslash /ugrave /uacute /ucircumflex |
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201 /udieresis /yacute /thorn /ydieresis |
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202 ] def |
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203 |
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204 |
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227 /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N |
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228 /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ |
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232 /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known |
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233 {userdict/md get type/dicttype eq{userdict begin md length 10 add md |
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234 maxlength ge{/md md dup length 20 add dict copy def}if end md begin |
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235 /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S |
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236 atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ |
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2380 Fu(If)26 b Fm(x)i Fl(6\021)g Fm(y)g Fu(and)d Fm(x)j Fl(62)g |
|
2381 Fm(F)14 b(V)21 b Fk(\()p Fm(L)p Fk(\))p Fu(,)k(then)1013 |
|
2382 1811 y Fm(M)10 b Fk([)p Fm(x)29 b Fk(:=)f Fm(N)10 b Fk(][)p |
|
2383 Fm(y)31 b Fk(:=)d Fm(L)p Fk(])g Fl(\021)g Fm(M)10 b Fk([)p |
|
2384 Fm(y)32 b Fk(:=)c Fm(L)p Fk(][)p Fm(x)g Fk(:=)g Fm(N)10 |
|
2385 b Fk([)p Fm(y)31 b Fk(:=)d Fm(L)p Fk(]])p Fu(.)291 2043 |
|
2386 y(one)c(might)g(start)g(with)g(the)h(follo)n(wing)e(informal)h(proof)g |
|
2387 (gi)n(v)o(en)g(by)g(Barendre)o(gt)h([)p 0 0 1 TeXcolorrgb |
|
2388 3146 2043 a |
|
2389 SDict begin H.S end |
|
2390 3146 2043 a 0 0 1 TeXcolorrgb Fu(4)p 0 0 1 |
|
2391 TeXcolorrgb 3196 1975 a |
|
2392 SDict begin H.R end |
|
2393 3196 1975 a 3196 2043 a |
|
2394 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2395 (cite.Barendregt81) cvn H.B /ANN pdfmark end |
|
2396 3196 |
|
2397 2043 a Black Fu(]:)p Black Black 541 2243 a Fn(Pr)n(oof:)31 |
|
2398 b Fu(By)25 b(induction)e(on)i(the)g(structure)f(of)h |
|
2399 Fm(M)10 b Fu(.)541 2423 y Fn(Case)24 b(1:)31 b Fm(M)36 |
|
2400 b Fu(is)24 b(a)h(v)n(ariable.)659 2604 y(Case)g(1.1.)e |
|
2401 Fm(M)39 b Fl(\021)28 b Fm(x)p Fu(.)i(Then)19 b(both)g(sides)g(equal)g |
|
2402 Fm(N)10 b Fk([)p Fm(y)32 b Fk(:=)27 b Fm(L)p Fk(])20 |
|
2403 b Fu(since)g Fm(x)28 b Fl(6\021)g Fm(y)t Fu(.)659 2784 |
|
2404 y(Case)d(1.2.)e Fm(M)59 b Fl(\021)47 b Fm(y)t Fu(.)62 |
|
2405 b(Then)35 b(both)g(sides)g(equal)g Fm(L)p Fu(,)j(for)e |
|
2406 Fm(x)48 b Fl(62)g Fm(F)14 b(V)21 b Fk(\()p Fm(L)p Fk(\))1050 |
|
2407 2905 y Fu(implies)j Fm(L)p Fk([)p Fm(x)k Fk(:=)g Fm(:)17 |
|
2408 b(:)g(:)o Fk(])28 b Fl(\021)g Fm(L)p Fu(.)659 3085 y(Case)d(1.3.)e |
|
2409 Fm(M)39 b Fl(\021)28 b Fm(z)33 b Fl(6\021)28 b Fm(x;)17 |
|
2410 b(y)t Fu(.)30 b(Then)24 b(both)g(sides)h(equal)f Fm(z)t |
|
2411 Fu(.)541 3266 y Fn(Case)f(2:)30 b Fm(M)38 b Fl(\021)28 |
|
2412 b Fm(\025z)t(:M)1346 3281 y Fj(1)1387 3266 y Fu(.)i(By)23 |
|
2413 b(the)g(v)n(ariable)g(con)l(v)o(ention)e(we)j(may)e(assume)h(that)541 |
|
2414 3386 y Fm(z)32 b Fl(6\021)c Fm(x;)17 b(y)28 b Fu(and)d |
|
2415 Fm(z)30 b Fu(is)24 b(not)g(free)i(in)f Fm(N)5 b(;)17 |
|
2416 b(L)p Fu(.)31 b(Then)24 b(by)h(induction)e(hypothesis)777 |
|
2417 3590 y Fk(\()p Fm(\025z)t(:M)1042 3605 y Fj(1)1082 3590 |
|
2418 y Fk(\)[)p Fm(x)29 b Fk(:=)e Fm(N)10 b Fk(][)p Fm(y)32 |
|
2419 b Fk(:=)27 b Fm(L)p Fk(])48 b Fl(\021)g Fm(\025z)t(:)p |
|
2420 Fk(\()p Fm(M)2244 3605 y Fj(1)2284 3590 y Fk([)p Fm(x)29 |
|
2421 b Fk(:=)e Fm(N)10 b Fk(][)p Fm(y)32 b Fk(:=)27 b Fm(L)p |
|
2422 Fk(]\))1854 3771 y Fl(\021)48 b Fm(\025z)t(:)p Fk(\()p |
|
2423 Fm(M)2244 3786 y Fj(1)2284 3771 y Fk([)p Fm(y)31 b Fk(:=)d |
|
2424 Fm(L)p Fk(][)p Fm(x)h Fk(:=)e Fm(N)10 b Fk([)p Fm(y)32 |
|
2425 b Fk(:=)27 b Fm(L)p Fk(]]\))1854 3952 y Fl(\021)48 b |
|
2426 Fk(\()p Fm(\025z)t(:M)2244 3967 y Fj(1)2284 3952 y Fk(\)[)p |
|
2427 Fm(y)31 b Fk(:=)d Fm(L)p Fk(][)p Fm(x)g Fk(:=)g Fm(N)10 |
|
2428 b Fk([)p Fm(y)32 b Fk(:=)27 b Fm(L)p Fk(]])p Fu(.)541 |
|
2429 4156 y Fn(Case)c(3:)31 b Fm(M)38 b Fl(\021)29 b Fm(M)1215 |
|
2430 4171 y Fj(1)1254 4156 y Fm(M)1348 4171 y Fj(2)1388 4156 |
|
2431 y Fu(.)i(The)24 b(statement)e(follo)n(ws)h(again)g(from)h(the)g |
|
2432 (induction)541 4276 y(hypothesis.)2260 b Fi(2)437 4450 |
|
2433 y Fu(In)39 b(order)h(to)f(translate)g(this)f(informal)g(proof)h(to)g |
|
2434 (proof)g(steps)g(in)g(a)g(theorem)g(pro)o(v)o(er)l(,)291 |
|
2435 4570 y(one)28 b(has)h(to)g(decide)g(ho)n(w)f(to)g(encode)h |
|
2436 (lambda-terms)f(and)h(ho)n(w)f(to)g(de\002ne)i(the)e(substitution)291 |
|
2437 4691 y(operation.)i(A)24 b(na)m(\250)-30 b(\021v)o(e)25 |
|
2438 b(choice)g(w)o(ould)f(be)h(to)f(represent)h(the)g(lambda-terms)e(as)i |
|
2439 (the)g(datatype)291 4712 y |
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2440 SDict begin H.S end |
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2441 291 4712 a 291 4712 a |
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2442 SDict begin 14.5 H.A end |
|
2443 291 |
|
2444 4712 a 291 4712 a |
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2445 SDict begin [ /View [/XYZ H.V] /Dest (equation.1) cvn H.B /DEST pdfmark |
|
2446 end |
|
2447 291 4712 a Black Black 674 4954 a Fn(datatype)56 |
|
2448 b Fm(l)r(am)g Fk(=)f Fh(V)-7 b(ar)38 b Fm(name)57 b Fl(j)e |
|
2449 Fh(App)33 b Fm(l)r(am)c(l)r(am)56 b Fl(j)f Fh(L)-5 b(am)35 |
|
2450 b Fm(name)29 b(l)r(am)p Black -3099 w Fu(\(1\))p Black |
|
2451 291 5226 a(where)j(the)g(type)f Fm(name)i Fu(can,)h(for)f(e)o(xample,)f |
|
2452 (be)g(strings)f(or)h(natural)g(numbers.)51 b(Since)33 |
|
2453 b(the)291 5347 y(term-constructor)18 b Fh(L)-5 b(am)27 |
|
2454 b Fu(has)20 b(a)g(concrete)g(name,)h(one)f(has)f(to)h(pro)o(v)o(e)f |
|
2455 (the)g(substitution)e(lemma)p Black 1897 5513 a(2)p Black |
|
2456 eop end |
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2457 %%Page: 3 3 |
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2460 version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto |
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2462 grestore}if}{pop}ifelse}{pop}ifelse}if end |
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2463 0 0 a Black 0 TeXcolorgray |
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2464 291 -166 a |
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2465 SDict begin H.S end |
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2466 291 -166 a 0 TeXcolorgray 0 TeXcolorgray 291 |
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2467 -166 a |
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2468 SDict begin H.R end |
|
2469 291 -166 a 291 -166 a |
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2470 SDict begin [ /View [/XYZ H.V] /Dest (page.3) cvn H.B /DEST pdfmark |
|
2471 end |
|
2472 291 -166 a Black 1173 w |
|
2473 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(modulo)23 |
|
2474 b(an)i(e)o(xplicit)e(notion)h(of)h(alpha-equi)n(v)n(alence,)e(that)i |
|
2475 (is)f(one)h(has)f(to)h(pro)o(v)o(e)p Black Black 957 |
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2476 235 a Fm(M)10 b Fk([)p Fm(x)29 b Fk(:=)f Fm(N)10 b Fk(][)p |
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2477 Fm(y)31 b Fk(:=)d Fm(L)p Fk(])g Fl(\031)1853 250 y Fg(\013)1931 |
|
2478 235 y Fm(M)10 b Fk([)p Fm(y)31 b Fk(:=)d Fm(L)p Fk(][)p |
|
2479 Fm(x)h Fk(:=)e Fm(N)10 b Fk([)p Fm(y)32 b Fk(:=)27 b |
|
2480 Fm(L)p Fk(]])i Fm(:)291 396 y Fu(F)o(or)24 b(the)h(substitution)d |
|
2481 (operation)i(one)h(might)e(follo)n(w)h(Church)h([)p 0 0 1 |
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2482 TeXcolorrgb 2550 397 a |
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2483 SDict begin H.S end |
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2484 2550 397 a 0 0 1 TeXcolorrgb -1 |
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2485 x Fu(5)p 0 0 1 TeXcolorrgb 2600 328 a |
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2486 SDict begin H.R end |
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2487 2600 328 a 2600 |
|
2488 396 a |
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2489 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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2490 (cite.CurryFeys58) cvn H.B /ANN pdfmark end |
|
2491 2600 396 a Black Fu(])g(and)g(de\002ne)291 418 |
|
2492 y |
|
2493 SDict begin H.S end |
|
2494 291 418 a 291 418 a |
|
2495 SDict begin 14.5 H.A end |
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2496 291 418 a 291 418 a |
|
2497 SDict begin [ /View [/XYZ H.V] /Dest (equation.2) cvn H.B /DEST pdfmark |
|
2498 end |
|
2499 291 418 a Black |
|
2500 Black 1062 662 a Fk(\()p Fh(V)-7 b(ar)39 b Fm(y)t Fk(\)[)p |
|
2501 Fm(x)27 b Fk(:=)h Fm(N)10 b Fk(])1773 606 y Ff(def)1778 |
|
2502 662 y Fk(=)1891 458 y Fe(8)1891 547 y(<)1891 727 y(:)2027 |
|
2503 595 y Fm(N)264 b Fu(if)25 b Fm(x)j Fl(\021)g Fm(y)2027 |
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2504 752 y Fh(V)-7 b(ar)36 b Fm(y)102 b Fu(otherwise)807 943 |
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2505 y Fk(\()p Fh(App)33 b Fm(M)1144 958 y Fj(1)1212 943 y |
|
2506 Fm(M)1306 958 y Fj(2)1346 943 y Fk(\)[)p Fm(x)28 b Fk(:=)g |
|
2507 Fm(N)10 b Fk(])1773 887 y Ff(def)1778 943 y Fk(=)37 b |
|
2508 Fh(App)d Fk(\()p Fm(M)2229 958 y Fj(1)2269 943 y Fk([)p |
|
2509 Fm(x)28 b Fk(:=)g Fm(N)10 b Fk(]\))28 b(\()p Fm(M)2823 |
|
2510 958 y Fj(2)2863 943 y Fk([)p Fm(x)g Fk(:=)g Fm(N)10 b |
|
2511 Fk(]\))870 1123 y(\()p Fh(L)-5 b(am)35 b Fm(x)28 b(M)1306 |
|
2512 1138 y Fj(1)1346 1123 y Fk(\)[)p Fm(x)g Fk(:=)g Fm(N)10 |
|
2513 b Fk(])1773 1068 y Ff(def)1778 1123 y Fk(=)37 b Fh(L)-5 |
|
2514 b(am)35 b Fm(x)28 b(M)2289 1138 y Fj(1)874 1304 y Fk(\()p |
|
2515 Fh(L)-5 b(am)35 b Fm(y)c(M)1306 1319 y Fj(1)1346 1304 |
|
2516 y Fk(\)[)p Fm(x)d Fk(:=)g Fm(N)10 b Fk(])1773 1248 y |
|
2517 Ff(def)1778 1304 y Fk(=)37 b Fh(L)-5 b(am)35 b Fm(z)e |
|
2518 Fk(\()p Fm(M)2322 1319 y Fj(1)2361 1304 y Fk([)p Fm(y)e |
|
2519 Fk(:=)d Fm(z)t Fk(][)p Fm(x)h Fk(:=)f Fm(N)10 b Fk(]\))p |
|
2520 Black 291 933 a Fu(\(2\))p Black 291 1481 a(where)36 |
|
2521 b(in)f(the)h(last)f(clause)g(it)h(is)f(assumed)g(that)g |
|
2522 Fm(y)51 b Fl(6\021)d Fm(x)p Fu(,)39 b(and)c(if)h Fm(x)48 |
|
2523 b Fl(62)g Fm(F)14 b(V)22 b Fk(\()p Fm(M)3181 1496 y Fj(1)3220 |
|
2524 1481 y Fk(\))36 b Fu(or)g Fm(y)51 b Fl(62)291 1602 y |
|
2525 Fm(F)14 b(V)21 b Fk(\()p Fm(N)10 b Fk(\))30 b Fu(then)e |
|
2526 Fm(z)41 b Fl(\021)36 b Fm(y)t Fu(,)30 b(otherwise)f Fm(z)34 |
|
2527 b Fu(is)28 b(the)h(\002rst)h(v)n(ariable)f(in)f(the)h(sequence)h |
|
2528 Fm(v)3120 1617 y Fj(0)3159 1602 y Fm(;)17 b(v)3250 1617 |
|
2529 y Fj(1)3290 1602 y Fm(;)g(v)3381 1617 y Fj(2)3420 1602 |
|
2530 y Fm(;)g(:)g(:)g(:)291 1722 y Fu(not)24 b(in)g Fm(M)639 |
|
2531 1737 y Fj(1)704 1722 y Fu(or)h Fm(N)10 b Fu(.)437 1851 |
|
2532 y(Unfortunately)-6 b(,)37 b(with)e(these)h(na)m(\250)-30 |
|
2533 b(\021v)o(e)35 b(choices)h(the)g(translation)f(of)h(the)f(informal)g |
|
2534 (proof)291 1971 y(into)23 b(actual)i(reasoning)g(steps)f(is)g(a)h |
|
2535 (nightmare:)30 b(Already)25 b(the)f(simple)g(property)g(stating)g(that) |
|
2536 291 2092 y Fm(L)p Fk([)p Fm(x)33 b Fk(:=)g Fm(:)17 b(:)g(:)p |
|
2537 Fk(])33 b Fl(\031)860 2107 y Fg(\013)943 2092 y Fm(L)28 |
|
2538 b Fu(pro)o(vided)f Fm(x)33 b Fl(62)g Fm(F)14 b(V)22 b |
|
2539 Fk(\()p Fm(L)p Fk(\))28 b Fu(is)f(a)h(tour)f(de)h(force.)40 |
|
2540 b(In)28 b(nearly)g(all)f(reasoning)291 2212 y(steps)d(in)l(v)n(olving)f |
|
2541 Fh(L)-5 b(am)32 b Fu(one)24 b(needs)h(the)g(property)p |
|
2542 Black Black 807 2364 a(if)g Fm(M)38 b Fl(\031)1102 2379 |
|
2543 y Fg(\013)1180 2364 y Fm(M)1284 2323 y Fd(0)1333 2364 |
|
2544 y Fu(and)25 b Fm(N)38 b Fl(\031)1695 2379 y Fg(\013)1772 |
|
2545 2364 y Fm(N)1860 2323 y Fd(0)1909 2364 y Fu(then)25 b |
|
2546 Fm(M)10 b Fk([)p Fm(x)29 b Fk(:=)e Fm(N)10 b Fk(])29 |
|
2547 b Fl(\031)2672 2379 y Fg(\013)2749 2364 y Fm(M)2853 2323 |
|
2548 y Fd(0)2877 2364 y Fk([)p Fm(x)f Fk(:=)g Fm(N)3206 2323 |
|
2549 y Fd(0)3230 2364 y Fk(])291 2525 y Fu(in)35 b(order)i(to)e(manually)g |
|
2550 (massage)h(the)g(lambda-terms)f(to)h(a)g(suitable)f(form.)64 |
|
2551 b(The)37 b(\223rough)291 2645 y(sk)o(etches\224)22 b(Curry)h(gi)n(v)o |
|
2552 (es)d(for)j(this)e(property)h(e)o(xtend)g(o)o(v)o(er)f(10)h(pages)g([)p |
|
2553 0 0 1 TeXcolorrgb 2729 2646 a |
|
2554 SDict begin H.S end |
|
2555 2729 2646 a 0 0 1 TeXcolorrgb |
|
2556 -1 x Fu(5)p 0 0 1 TeXcolorrgb 2779 2578 a |
|
2557 SDict begin H.R end |
|
2558 2779 2578 a |
|
2559 2779 2645 a |
|
2560 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2561 (cite.CurryFeys58) cvn H.B /ANN pdfmark end |
|
2562 2779 2645 a Black Fu(,)h(P)o(ages)g(94\226104].)29 |
|
2563 b(As)291 2766 y(can)e(be)h(easily)f(imagined,)g(implementing)e(these)i |
|
2564 (sk)o(etches)g(results)g(into)f(a)i(rather)g(unpleas-)291 |
|
2565 2886 y(ant)j(e)o(xperience)g(with)g(theorem)g(pro)o(v)o(ers\227nothing) |
|
2566 e(of)i(the)h(sort)f(that)g(mak)o(es)g(formalising)291 |
|
2567 3007 y(proofs)h(\223addicti)n(v)o(e)f(in)h(a)g(videogame)g(kind)f(of)i |
|
2568 (w)o(ay\224)f([)p 0 0 1 TeXcolorrgb 2271 3008 a |
|
2569 SDict begin H.S end |
|
2570 2271 |
|
2571 3008 a 0 0 1 TeXcolorrgb -1 x Fu(8)p 0 0 1 TeXcolorrgb |
|
2572 2321 2939 a |
|
2573 SDict begin H.R end |
|
2574 2321 2939 a 2321 3007 a |
|
2575 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2576 (cite.Leroy06) cvn H.B /ANN pdfmark end |
|
2577 2321 3007 a Black |
|
2578 Fu(,)j(P)o(age)d(53].)54 b(One)32 b(reason)h(for)g(the)291 |
|
2579 3127 y(dif)n(\002culties)g(is)i(the)g(f)o(act)h(that)e(Curry')-5 |
|
2580 b(s)35 b(substitution)d(operation)j(is)f(not)h(equi)n(v)n |
|
2581 (ariant\227that)291 3247 y(means)24 b(is)g(not)h(independent)e(under)i |
|
2582 (renamings)f([)p 0 0 1 TeXcolorrgb 2050 3248 a |
|
2583 SDict begin H.S end |
|
2584 2050 3248 |
|
2585 a 0 0 1 TeXcolorrgb -1 x Fu(10)p 0 0 1 TeXcolorrgb 2150 |
|
2586 3179 a |
|
2587 SDict begin H.R end |
|
2588 2150 3179 a 2150 3247 a |
|
2589 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2590 (cite.Pitts03) cvn H.B /ANN pdfmark end |
|
2591 2150 3247 a Black Fu(].)437 |
|
2592 3376 y(The)33 b(main)e(point)h(of)h(de)f(Bruijn)h(indices)f(and)g |
|
2593 (names)g(from)h(the)f(nominal)g(logic)g(w)o(ork)291 3496 |
|
2594 y(is)j(to)h(allo)n(w)f(for)h(more)g(cle)n(v)o(er)f(methods)g(of)h |
|
2595 (representing)g(binders)f(and)h(to)g(substantially)291 |
|
2596 3617 y(reduce)29 b(the)f(amount)g(of)h(ef)n(fort)f(needed)h(to)g |
|
2597 (formalise)f(proofs.)42 b(In)28 b(Section)p 0 0 1 TeXcolorrgb |
|
2598 3006 3617 a |
|
2599 SDict begin H.S end |
|
2600 3006 3617 a 0 0 1 TeXcolorrgb Fu(2)p 0 0 1 |
|
2601 TeXcolorrgb 3056 3549 a |
|
2602 SDict begin H.R end |
|
2603 3056 3549 a 3056 3617 a |
|
2604 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2605 (section.0.2) cvn H.B /ANN pdfmark end |
|
2606 3056 |
|
2607 3617 a Black 29 w Fu(we)h(illustrate)291 3737 y(this)g(in)g(the)h |
|
2608 (conte)o(xt)f(of)h(the)g(substitution)d(lemma.)45 b(Section)p |
|
2609 0 0 1 TeXcolorrgb 2477 3738 a |
|
2610 SDict begin H.S end |
|
2611 2477 3738 a 0 0 1 TeXcolorrgb |
|
2612 -1 x Fu(3)p 0 0 1 TeXcolorrgb 2527 3669 a |
|
2613 SDict begin H.R end |
|
2614 2527 3669 a |
|
2615 2527 3737 a |
|
2616 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2617 (section.0.3) cvn H.B /ANN pdfmark end |
|
2618 2527 3737 a Black 30 w Fu(contains)29 b(a)h(brief)h(sk)o |
|
2619 (etch)e(of)291 3858 y(the)k(formalisations)f(for)i(the)f(narro)n(wing)g |
|
2620 (and)g(transiti)n(vity)e(proof)j(of)f(subtyping)f(from)i(the)291 |
|
2621 3978 y(POPLmark-Challenge)25 b([)p 0 0 1 TeXcolorrgb |
|
2622 1234 3979 a |
|
2623 SDict begin H.S end |
|
2624 1234 3979 a 0 0 1 TeXcolorrgb -1 x Fu(3)p |
|
2625 0 0 1 TeXcolorrgb 1284 3910 a |
|
2626 SDict begin H.R end |
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2627 1284 3910 a 1284 3978 a |
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2883 (equation.3) cvn H.B /ANN pdfmark end |
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2884 1877 |
|
2885 1600 a Black Fu(\))g(is)g(therefore)g(equal)g(to)g Fk(\()p |
|
2886 Fl(")2863 1564 y Fg(i)2863 1625 y Fj(0)2938 1600 y Fm(N)10 |
|
2887 b Fk(\)[)p Fm(j)42 b Fk(:=)36 b Fm(L)p Fk(])29 b Fu(and)390 |
|
2888 1721 y(because)h(we)g(kno)n(w)f(by)h(the)f(assumption)f |
|
2889 Fm(i)37 b Fl(\024)h Fm(j)e Fu(that)29 b Fm(n)37 b(<)g(j)32 |
|
2890 b Fk(+)25 b(1)p Fu(,)31 b(the)f(right-hand)f(side)390 |
|
2891 1841 y(is)d(equal)g(to)f Fl(")878 1805 y Fg(i)878 1866 |
|
2892 y Fj(0)947 1841 y Fk(\()p Fm(N)10 b Fk([)p Fm(j)30 b |
|
2893 Fl(\000)23 b Fm(i)30 b Fk(:=)g Fm(L)p Fk(]\))p Fu(.)k(No)n(w)26 |
|
2894 b(we)g(ha)n(v)o(e)f(to)h(sho)n(w)f(that)g(both)g(terms)h(are)h(equal.) |
|
2895 390 1961 y(F)o(or)e(this)f(we)h(pro)o(v)o(e)f(\002rst)h(the)f(lemma) |
|
2896 1733 1961 y |
|
2897 SDict begin H.S end |
|
2898 1733 1961 a 1733 1961 a |
|
2899 SDict begin 14.5 H.A end |
|
2900 1733 1961 a 1733 |
|
2901 1961 a |
|
2902 SDict begin [ /View [/XYZ H.V] /Dest (equation.4) cvn H.B /DEST pdfmark |
|
2903 end |
|
2904 1733 1961 a Black Black 955 2123 a Fl(8)p Fm(i;)17 |
|
2905 b(j:)29 b Fu(if)c Fm(i)j Fl(\024)g Fm(j)j Fu(and)25 b |
|
2906 Fm(j)33 b Fl(\024)28 b Fm(i)23 b Fk(+)f Fm(m)j Fu(then)52 |
|
2907 b Fl(")2391 2081 y Fg(n)2391 2147 y(j)2466 2123 y Fk(\()p |
|
2908 Fl(")2554 2081 y Fg(m)2554 2147 y(i)2648 2123 y Fm(N)10 |
|
2909 b Fk(\))28 b(=)p Fl(")2928 2081 y Fg(m)p Fj(+)p Fg(n)2928 |
|
2910 2148 y(i)3119 2123 y Fm(N)p Black -2807 w Fu(\(4\))p |
|
2911 Black 390 2284 a(which)37 b(can)g(be)g(pro)o(v)o(ed)f(by)h(induction)e |
|
2912 (on)i Fm(N)10 b Fu(.)68 b(\(The)37 b(quanti\002cation)f(o)o(v)o(er)g |
|
2913 Fm(i)i Fu(and)f Fm(j)43 b Fu(is)390 2404 y(necessary)26 |
|
2914 b(in)g(order)g(to)f(get)h(the)g Fh(L)-5 b(am)7 b Fu(-case)26 |
|
2915 b(through.\))33 b(This)25 b(lemma)g(helps)g(to)h(pro)o(v)o(e)f(the)390 |
|
2916 2525 y(ne)o(xt)f(lemma)881 2525 y |
|
2917 SDict begin H.S end |
|
2918 881 2525 a 881 2525 |
|
2919 a |
|
2920 SDict begin 14.5 H.A end |
|
2921 881 2525 a 881 2525 a |
|
2922 SDict begin [ /View [/XYZ H.V] /Dest (equation.5) cvn H.B /DEST pdfmark |
|
2923 end |
|
2924 881 2525 a Black Black 922 2686 |
|
2925 a Fl(8)p Fm(k)s(;)17 b(j:)28 b Fu(if)d Fm(k)31 b Fl(\024)d |
|
2926 Fm(j)j Fu(then)52 b Fl(")1788 2645 y Fg(i)1788 2710 y(k)1858 |
|
2927 2686 y Fk(\()p Fm(N)10 b Fk([)p Fm(j)34 b Fk(:=)28 b |
|
2928 Fm(L)p Fk(]\))g(=)g(\()p Fl(")2567 2645 y Fg(i)2567 2710 |
|
2929 y(k)2637 2686 y Fm(N)10 b Fk(\)[)p Fm(j)28 b Fk(+)22 |
|
2930 b Fm(i)28 b Fk(:=)g Fm(L)p Fk(])p Black -2851 w Fu(\(5\))p |
|
2931 Black 390 2847 a(which)21 b(too)f(can)i(be)f(pro)o(v)o(ed)f(by)h |
|
2932 (induction)e(on)i Fm(N)10 b Fu(.)30 b(\(Again)20 b(the)h |
|
2933 (quanti\002cation)f(are)i(crucial)390 2967 y(to)31 b(get)h(the)f |
|
2934 (induction)f(through.\))51 b(W)-8 b(e)32 b(can)g(no)n(w)f(instantiate)f |
|
2935 (this)g(lemma)h(with)g Fm(k)44 b Fl(7!)39 b Fk(0)390 |
|
2936 3088 y Fu(and)30 b Fm(j)42 b Fl(7!)36 b Fm(j)c Fl(\000)26 |
|
2937 b Fm(i)p Fu(,)31 b(which)e(mak)o(es)g(the)h(precondition)e(tri)n |
|
2938 (vially)g(true)h(and)h(thus)e(we)i(obtain)390 3208 y(the)25 |
|
2939 b(equation)p Black Black 1151 3369 a Fl(")1201 3328 y |
|
2940 Fg(i)1201 3394 y Fj(0)1268 3369 y Fk(\()p Fm(N)10 b Fk([)p |
|
2941 Fm(j)28 b Fl(\000)23 b Fm(i)28 b Fk(:=)g Fm(L)p Fk(]\))g(=)f(\()p |
|
2942 Fl(")2131 3328 y Fg(i)2131 3394 y Fj(0)2198 3369 y Fm(N)10 |
|
2943 b Fk(\)[)p Fm(j)29 b Fl(\000)22 b Fm(i)h Fk(+)f Fm(i)28 |
|
2944 b Fk(:=)f Fm(L)p Fk(])i Fm(:)390 3530 y Fu(The)36 b(term)g |
|
2945 Fk(\()p Fl(")888 3494 y Fg(i)888 3555 y Fj(0)976 3530 |
|
2946 y Fm(N)10 b Fk(\)[)p Fm(j)37 b Fl(\000)31 b Fm(i)g Fk(+)f |
|
2947 Fm(i)49 b Fk(:=)g Fm(L)p Fk(])37 b Fu(is)e(equal)h(to)g |
|
2948 Fk(\()p Fl(")2404 3494 y Fg(i)2404 3555 y Fj(0)2492 3530 |
|
2949 y Fm(N)10 b Fk(\)[)p Fm(j)55 b Fk(:=)49 b Fm(L)p Fk(])p |
|
2950 Fu(,)39 b(as)d(we)h(had)f(to)390 3651 y(sho)n(w)-6 b(.)55 |
|
2951 b(Ho)n(we)n(v)o(er)32 b(this)g(last)h(step)f(is)h(surprisingly)f |
|
2952 Fc(not)i Fu(immediate:)46 b(it)33 b(depends)g(on)g(the)390 |
|
2953 3771 y(assumption)f(that)i Fm(i)44 b Fl(\024)i Fm(j)6 |
|
2954 b Fu(.)58 b(This)33 b(is)g(because)i(in)e(theorem)h(pro)o(v)o(ers)f |
|
2955 (lik)o(e)g(Isabelle/HOL)390 3892 y(and)38 b(Coq)h(subtraction)e(o)o(v)o |
|
2956 (er)g(natural)h(numbers)g(is)f(de\002ned)i(so)f(that)g |
|
2957 Fk(0)32 b Fl(\000)g Fm(n)53 b Fk(=)f(0)38 b Fu(and)390 |
|
2958 4012 y(consequently)24 b(the)g(equation)g Fm(j)29 b Fl(\000)22 |
|
2959 b Fm(i)h Fk(+)f Fm(i)28 b Fk(=)f Fm(j)k Fu(does)25 b(not)f(hold)g(in)g |
|
2960 (general!)p Black 291 4162 a Fl(\017)p Black 49 w Fu(Case)31 |
|
2961 b Fm(n)22 b(>)37 b(i)p Fu(:)42 b(Since)31 b(the)f(right-hand)g(side)f |
|
2962 (of)i(\()p 0 0 1 TeXcolorrgb 2097 4163 a |
|
2963 SDict begin H.S end |
|
2964 2097 4163 a |
|
2965 0 0 1 TeXcolorrgb -1 x Fu(3)p 0 0 1 TeXcolorrgb 2147 |
|
2966 4094 a |
|
2967 SDict begin H.R end |
|
2968 2147 4094 a 2147 4162 a |
|
2969 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2970 (equation.3) cvn H.B /ANN pdfmark end |
|
2971 2147 4162 a Black Fu(\))f(equals)g |
|
2972 Fk(\()p Fm(V)22 b(ar)s Fk(\()p Fm(n)k Fl(\000)h Fk(1\)\)[)p |
|
2973 Fm(j)43 b Fk(:=)38 b Fm(L)p Fk(])p Fu(,)32 b(we)390 4283 |
|
2974 y(distinguish)22 b(further)j(three)f(subcases)h(\(namely)f |
|
2975 Fm(n)e Fl(\000)h Fk(1)k Fm(<)11 b(j)6 b Fu(,)24 b Fm(n)d |
|
2976 Fl(\000)h Fk(1)27 b(=)h Fm(j)i Fu(and)25 b Fm(n)c Fl(\000)g |
|
2977 Fk(1)11 b Fm(>)27 b(j)6 b Fu(\):)p Black 368 4463 a Fl(\017)p |
|
2978 Black 50 w Fu(Subcase)25 b Fm(n)c Fl(\000)h Fk(1)11 b |
|
2979 Fm(<)28 b(j)6 b Fu(:)30 b(W)-8 b(e)25 b(therefore)g(kno)n(w)f(also)g |
|
2980 (that)g Fm(n)11 b(<)28 b(j)f Fk(+)21 b(1)k Fu(and)f(thus)g(both)g |
|
2981 (sides)468 4584 y(of)h(\()p 0 0 1 TeXcolorrgb 609 4585 |
|
2982 a |
|
2983 SDict begin H.S end |
|
2984 609 4585 a 0 0 1 TeXcolorrgb -1 x Fu(3)p 0 0 1 TeXcolorrgb |
|
2985 659 4516 a |
|
2986 SDict begin H.R end |
|
2987 659 4516 a 659 4584 a |
|
2988 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
2989 (equation.3) cvn H.B /ANN pdfmark end |
|
2990 659 4584 a Black Fu(\))g(are)g(equal)g |
|
2991 (to)f Fh(V)-7 b(ar)36 b Fk(\()p Fm(n)22 b Fl(\000)h Fk(1\))p |
|
2992 Fu(.)p Black 368 4704 a Fl(\017)p Black 50 w Fu(Subcase)31 |
|
2993 b Fm(n)c Fl(\000)g Fk(1)39 b(=)f Fm(j)6 b Fu(:)43 b(T)-8 |
|
2994 b(aking)30 b(into)g(account)h(that)f Fm(n)23 b(>)38 b(i)31 |
|
2995 b Fu(implies)f Fk(0)38 b Fm(<)h(n)p Fu(,)33 b(we)e(ha)n(v)o(e)468 |
|
2996 4824 y(also)25 b Fm(n)k Fk(=)g Fm(j)f Fk(+)23 b(1)i Fu(\(remember)h |
|
2997 (that)f(because)h(of)f(the)h(\223quirk\224)f(with)g(subtraction,)f |
|
2998 (this)h(is)468 4945 y(not)e(ob)o(vious\).)29 b(Hence)24 |
|
2999 b(we)h(can)f(calculate)g(that)g(the)g(left-hand)f(side)h(of)g(\()p |
|
3000 0 0 1 TeXcolorrgb 3009 4946 a |
|
3001 SDict begin H.S end |
|
3002 3009 4946 a 0 0 1 TeXcolorrgb |
|
3003 -1 x Fu(3)p 0 0 1 TeXcolorrgb 3059 4877 a |
|
3004 SDict begin H.R end |
|
3005 3059 4877 a |
|
3006 3059 4945 a |
|
3007 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3008 (equation.3) cvn H.B /ANN pdfmark end |
|
3009 3059 4945 a Black Fu(\))g(equals)g Fl(")3445 |
|
3010 4897 y Fg(j)3445 4969 y Fj(0)3512 4945 y Fm(L)468 5065 |
|
3011 y Fu(and)30 b(the)f(right-hand)g(side)h(to)g Fk(\()p |
|
3012 Fl(")1625 5018 y Fg(j)t Fj(+1)1625 5089 y(0)1788 5065 |
|
3013 y Fm(L)p Fk(\)[)p Fm(i)38 b Fk(:=)f Fm(N)10 b Fk([)p |
|
3014 Fm(j)33 b Fl(\000)26 b Fm(i)38 b Fk(:=)f Fm(L)p Fk(]])p |
|
3015 Fu(.)46 b(T)-8 b(o)30 b(sho)n(w)f(that)h(these)468 5185 |
|
3016 y(terms)24 b(are)i(equal)e(we)h(need)g(the)g(lemma)1898 |
|
3017 5185 y |
|
3018 SDict begin H.S end |
|
3019 1898 5185 a 1898 5185 a |
|
3020 SDict begin 14.5 H.A end |
|
3021 1898 5185 a 1898 5185 |
|
3022 a |
|
3023 SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark |
|
3024 end |
|
3025 1898 5185 a Black Black 845 5347 a Fl(8)p Fm(k)s(;)17 |
|
3026 b(i:)28 b Fu(if)d Fm(k)31 b Fl(\024)d Fm(i)d Fu(and)g |
|
3027 Fm(i)j(<)g(k)d Fk(+)d(\()p Fm(j)28 b Fk(+)22 b(1\))j |
|
3028 Fu(then)f Fk(\()p Fl(")2525 5299 y Fg(j)t Fj(+1)2525 |
|
3029 5374 y Fg(k)2679 5347 y Fm(L)p Fk(\)[)p Fm(i)k Fk(:=)g |
|
3030 Fm(P)14 b Fk(])27 b(=)p Fl(")3259 5299 y Fg(j)3259 5374 |
|
3031 y(k)3329 5347 y Fm(L)p Black -2927 w Fu(\(6\))p Black |
|
3032 Black 1897 5513 a(5)p Black eop end |
|
3033 %%Page: 6 6 |
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3039 0 0 a Black 0 TeXcolorgray |
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3040 291 -166 a |
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3041 SDict begin H.S end |
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3042 291 -166 a 0 TeXcolorgray 0 TeXcolorgray 291 |
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3043 -166 a |
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3044 SDict begin H.R end |
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3045 291 -166 a 291 -166 a |
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3046 SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark |
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3047 end |
|
3048 291 -166 a Black 1173 w |
|
3049 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 468 83 a Fu(pro)o(v)o(ed)g(by) |
|
3050 g(induction)g(on)h Fm(L)p Fu(.)53 b(Instantiating)30 |
|
3051 b(this)i(lemma)f(with)g Fm(k)45 b Fl(7!)c Fk(0)p Fu(,)33 |
|
3052 b Fm(i)42 b Fl(7!)f Fm(i)32 b Fu(and)468 203 y(using)25 |
|
3053 b(the)i(assumption)d Fm(i)32 b Fl(\024)f Fm(j)6 b Fu(,)27 |
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3054 b(we)g(can)g(infer)g(that)f(the)h(preconditions)e(of)h(this)g(lemma)468 |
|
3055 324 y(hold)e(and)g(thus)g(can)i(conclude)e(that)h Fk(\()p |
|
3056 Fl(")1840 277 y Fg(j)t Fj(+1)1840 348 y(0)1994 324 y |
|
3057 Fm(L)p Fk(\)[)p Fm(i)j Fk(:=)g Fm(N)10 b Fk([)p Fm(j)28 |
|
3058 b Fl(\000)23 b Fm(i)28 b Fk(:=)g Fm(L)p Fk(]])g(=)p Fl(")3066 |
|
3059 277 y Fg(j)3066 348 y Fj(0)3133 324 y Fm(L)p Fu(.)p Black |
|
3060 368 444 a Fl(\017)p Black 50 w Fu(Subcase)d Fm(n)e Fl(\000)f |
|
3061 Fk(1)11 b Fm(>)28 b(j)6 b Fu(:)31 b(W)-8 b(e)25 b(therefore)h(also)e |
|
3062 (kno)n(w)g(that)h Fm(n)11 b(>)28 b(j)g Fk(+)22 b(1)p |
|
3063 Fu(.)31 b(These)25 b(inequalities)468 565 y(in)f(turn)h(imply)e(that)h |
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3064 (both)g(sides)g(of)h(\()p 0 0 1 TeXcolorrgb 1754 566 |
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3065 a |
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3066 SDict begin H.S end |
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3068 1804 497 a |
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3069 SDict begin H.R end |
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3070 1804 497 a 1804 565 a |
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3071 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3072 (equation.3) cvn H.B /ANN pdfmark end |
|
3073 1804 565 a Black Fu(\))g(are)h(equal)f |
|
3074 (to)f Fh(V)-7 b(ar)36 b Fk(\()p Fm(n)22 b Fl(\000)h Fk(2\))p |
|
3075 Fu(.)291 728 y(This)h(concludes)g(the)h(proof)f(of)h(the)g |
|
3076 (substitution)d(lemma.)1159 b Fi(2)291 917 y Fu(In)28 |
|
3077 b(this)f(formalisation)g(considerable)g(ingenuity)g(is)h(needed)g(when) |
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3078 g(in)l(v)o(enting)e(the)i(lemmas)291 1037 y(\()p 0 0 1 |
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3079 TeXcolorrgb 324 1037 a |
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3080 SDict begin H.S end |
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3081 324 1037 a 0 0 1 TeXcolorrgb Fu(4)p |
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3084 374 969 a 374 1037 a |
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3085 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3086 (equation.4) cvn H.B /ANN pdfmark end |
|
3087 374 |
|
3088 1037 a Black Fu(\),)35 b(\()p 0 0 1 TeXcolorrgb 500 1038 |
|
3089 a |
|
3090 SDict begin H.S end |
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3091 500 1038 a 0 0 1 TeXcolorrgb -1 x Fu(5)p 0 0 1 TeXcolorrgb |
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3092 550 969 a |
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3093 SDict begin H.R end |
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3094 550 969 a 550 1037 a |
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3095 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3096 (equation.5) cvn H.B /ANN pdfmark end |
|
3097 550 1037 a Black Fu(\))e(and)h(\()p |
|
3098 0 0 1 TeXcolorrgb 827 1038 a |
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3099 SDict begin H.S end |
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3100 827 1038 a 0 0 1 TeXcolorrgb |
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3101 -1 x Fu(6)p 0 0 1 TeXcolorrgb 877 969 a |
|
3102 SDict begin H.R end |
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3103 877 969 a 877 |
|
3104 1037 a |
|
3105 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3106 (equation.6) cvn H.B /ANN pdfmark end |
|
3107 877 1037 a Black Fu(\).)56 b(Also)32 b(the)o(y)g(are)i(quite)f |
|
3108 (\223brittle\224\227in)f(the)h(sense)g(that)g(the)o(y)f(seem)h(to)g(go) |
|
3109 291 1158 y(through)d(just)h(in)g(the)h(form)f(stated.)51 |
|
3110 b(T)-8 b(o)31 b(\002nd)h(them)f(can)h(be)g(a)g(daunting)e(task)h(for)h |
|
3111 (an)g(ine)o(x-)291 1278 y(perienced)27 b(user)g(of)h(theorem)f(pro)o(v) |
|
3112 o(ers)f(\(the)o(y)g(are)i(only)f(in)g(little)f(part)h(inspired)f(by)h |
|
3113 (the)g(f)o(acts)291 1399 y(needed)h(in)g(the)g(main)g(proof\).)41 |
|
3114 b(In)28 b(practice)h(ho)n(we)n(v)o(er)e(the)o(y)h(seem)g(to)g(cause)g |
|
3115 (fe)n(w)h(problems,)291 1519 y(because)23 b(the)o(y)f(\223carry)i(o)o |
|
3116 (v)o(er\224)f(from)g(language)g(to)f(language,)h(and)g(hence)h(one)f |
|
3117 (does)g(not)f(need)291 1639 y(to)g(\223in)l(v)o(ent)g(the)g(wheel\224)h |
|
3118 (again)f(for)h(a)g(ne)n(w)f(language.)30 b(Theorem)23 |
|
3119 b(pro)o(ving)e(e)o(xperts)h(just)g(cop)o(y)291 1760 y(these)32 |
|
3120 b(lemmas)g(from)h(e)o(xisting)e(formalisations.)53 b(Indeed)34 |
|
3121 b(when)e(submitting)f(his)h(solution)291 1880 y(of)37 |
|
3122 b(the)f(POPLmark-Challenge,)41 b(the)c(\002rst)g(author)f(only)h |
|
3123 (minimally)d(adapted)j(to)g(System)291 2000 y(F)346 2015 |
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3124 y Fg(<)p Fj(:)462 2000 y Fu(the)g(proofs)g(Nipk)o(o)n(w)f([)p |
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3125 0 0 1 TeXcolorrgb 1301 2002 a |
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3126 SDict begin H.S end |
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3128 -2 x Fu(9)p 0 0 1 TeXcolorrgb 1351 1932 a |
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3130 1351 1932 a |
|
3131 1351 2000 a |
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3132 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3133 (cite.Nipkow-JAR01) cvn H.B /ANN pdfmark end |
|
3134 1351 2000 a Black Fu(])h(ga)n(v)o(e)g(in)f(Isabelle/HOL)h |
|
3135 (for)h(the)f(lambda-calculus.)67 b(Nip-)291 2121 y(k)o(o)n(w)29 |
|
3136 b(in)h(turn)h(got)f(his)g(collection)f(of)i(lemmas)e(from)i(Rasmussen)f |
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3137 ([)p 0 0 1 TeXcolorrgb 2708 2121 a |
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3156 (cite.Nipkow-JAR01) cvn H.B /ANN pdfmark end |
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3157 1463 2241 a Black Fu(,)h(P)o(age)g(57]:)p |
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3158 Black Black 390 2441 a Fc(\223)g(Initially)30 b(I)g(tried)g(to)h |
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3159 (\002nd)f(and)g(pr)l(o)o(ve)g(these)h(lemmas)f(fr)l(om)g(scr)o(atc)o(h) |
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3160 f(b)n(ut)h(soon)g(de-)470 2561 y(cided)20 b(to)g(steal)g(them)g(fr)l |
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3161 (om)f(Rasmussen')l(s)g(ZF)i(pr)l(oofs)d(instead,)i(whic)o(h)g(has)g |
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3162 (obvious)470 2681 y(advanta)o(g)o(es:)517 2862 y(\226)25 |
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3163 b(I)g(did)f(not)h(have)g(to)f(\002nd)g(this)g(collection)g(of)h |
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3164 (non-obvious)e(lemmas)h(myself.)15 b(.)g(.)g(\224)291 |
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3165 3036 y Fu(Rasmussen)30 b(seems)g(to)h(ha)n(v)o(e)f(gotten)g(his)h |
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3166 (lemmas)e(from)i(a)g(formalisation)e(by)i(Huet)g([)p |
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3173 3437 3036 a |
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3174 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3175 (cite.Huet94) cvn H.B /ANN pdfmark end |
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3176 3437 3036 a Black Fu(])g(in)291 3156 y(Coq.)437 |
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3177 3285 y(In)f(light)e(of)i(the)g(subtleties)e(and)i(quirks)f(in)g(the)g |
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3178 (proof)h(based)g(on)f(de)h(Bruijn)g(indices,)g(it)291 |
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3179 3405 y(might)c(be)i(surprising)f(that)g(one)h(does)g(not)f(end)h(up)g |
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3180 (with)f(a)h(proof)g(script)f(of)h(more)g(than)g(100)291 |
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3181 3525 y(lines)21 b(of)h(code.)30 b(In)21 b(f)o(act)i(the)e(formalised)g |
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3182 (proof)h(by)g(Nipk)o(o)n(w)e(consists)h(of)g(only)g(a)i(fe)n(w)e |
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3190 2334 3578 a 2334 3646 a |
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3191 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3192 (equation.4) cvn H.B /ANN pdfmark end |
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3193 2334 3646 a Black |
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3194 Fu(\),)g(\()p 0 0 1 TeXcolorrgb 2446 3647 a |
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3195 SDict begin H.S end |
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3197 a 0 0 1 TeXcolorrgb -1 x Fu(5)p 0 0 1 TeXcolorrgb 2496 |
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3201 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3202 (equation.5) cvn H.B /ANN pdfmark end |
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3203 2496 3646 a Black Fu(\))g(and)g(\()p |
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3210 2798 3646 a |
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3211 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3212 (equation.6) cvn H.B /ANN pdfmark end |
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3213 2798 3646 a Black Fu(\).)29 b(The)21 b(reason)g(is)f(that) |
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3214 291 3766 y(one)28 b(can)h(\223optimise\224)f(proof)g(scripts)g(by)h |
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3215 (emplo)o(ying)e(automatic)h(proof)g(tools.)41 b(Such)29 |
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3216 b(proof)291 3887 y(tools)21 b(can)i(mak)o(e)f(case)h(distinctions)d |
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3217 (and)j(apply)e(de\002nitions)h(without)f(manual)h(interference.)291 |
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3218 4007 y(Ho)n(we)n(v)o(er)36 b(such)h(optimisations)d(are)k(done)g |
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3219 Fc(after)h Fu(one)e(has)g(a)h(formal)f(proof)g(lik)o(e)g(the)g(one)291 |
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3220 4127 y(described)32 b(abo)o(v)o(e.)52 b(As)32 b(we)g(mentioned)f |
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3221 (earlier)l(,)k(just)c(blindly)g(attacking)g(a)i(problem)e(with)291 |
|
3222 4248 y(automatic)25 b(proof)i(tools)e(leads)i(to)f(dead)h(ends,)g(e)o |
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3223 (xcept)f(in)g(the)h(most)e(tri)n(vial)g(proofs,)i(and)g(the)291 |
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3224 4368 y(substitution)g(lemma)i(is)h(already)g(too)g(complicated.)45 |
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3225 b(This)30 b(is)f(not)h(surprising)e(considering)291 4488 |
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3226 y(ho)n(w)33 b(much)h(ingenuity)f(one)h(needs)g(to)g(in)l(v)o(ent)g(the) |
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3233 2574 4420 a 2574 4488 a |
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3234 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3235 (equation.4) cvn H.B /ANN pdfmark end |
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3236 2574 4488 a Black Fu(\),)k(\()p |
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3237 0 0 1 TeXcolorrgb 2702 4489 a |
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3238 SDict begin H.S end |
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3246 2752 4488 a Black Fu(\))e(and)f(\()p 0 0 1 |
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3247 TeXcolorrgb 3031 4489 a |
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3248 SDict begin H.S end |
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3255 (equation.6) cvn H.B /ANN pdfmark end |
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3256 3081 4488 a Black Fu(\).)60 b(Ho)n(we)n(v)o(er)l(,)291 |
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3257 4609 y(once)29 b(one)h(kno)n(ws)e(ho)n(w)g(the)i(proof)f(proceeds,)h |
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3258 (one)g(can)g(guide)e(the)i(automatic)e(proof)h(tools)291 |
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3259 4729 y(by)j(pro)o(viding)e(e)o(xplicitly)h(the)h(lemmas)g(that)g(lead)g |
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3260 (to)g(a)h(proof.)54 b(In)32 b(case)h(of)g(the)f(de)h(Bruijn)291 |
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3261 4850 y(indices)h(v)o(ersion)h(of)g(the)h(substitution)c(lemma,)37 |
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3262 b(ho)n(we)n(v)o(er)l(,)g(this)e(kind)f(of)i(post-processing)291 |
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3263 4970 y(is)31 b(not)h(without)f(pitf)o(alls.)52 b(F)o(or)32 |
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3264 b(e)o(xample)f(it)h(helps)g(if)g(the)g(lemma)g(is)f(stated)h(the)g |
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3265 (other)g(w)o(ay)291 5090 y(around,)24 b(namely)g(as)p |
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3266 Black Black 878 5347 a Fm(M)10 b Fk([)p Fm(j)29 b Fk(+)22 |
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3267 b(1)27 b(:=)h Fm(L)p Fk(][)p Fm(i)h Fk(:=)e Fm(N)10 b |
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3268 Fk([)p Fm(j)29 b Fl(\000)23 b Fm(i)28 b Fk(:=)f Fm(L)p |
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3269 Fk(]])h(=)g Fm(M)10 b Fk([)p Fm(i)29 b Fk(:=)e Fm(N)10 |
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3270 b Fk(][)p Fm(j)35 b Fk(:=)27 b Fm(L)p Fk(])p Black 1897 |
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3287 291 -166 a Black 1173 w |
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3288 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(otherwise)d(the)h |
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3300 a 107 x Fc(2.2)99 b(V)-11 b(er)o(sion)24 b(using)f(the)i(Nominal)f |
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3301 (Datatype)g(P)-8 b(ac)n(ka)o(g)o(e)291 669 y Fu(The)27 |
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3320 (cite.UrbanTasson05) cvn H.B /ANN pdfmark end |
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3321 1802 669 a Black Fu(])f(allo)n(ws)f |
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3322 (to)h(reason)g(with)f(\223named\224)i(alpha-equi-)291 |
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3333 v 36 w(datatype)56 b Fh(lam)62 b Fk(=)55 b Fh(V)-7 b(ar)39 |
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3334 b Fm(name)56 b Fl(j)27 b Fh(App)34 b(lam)h(lam)62 b Fl(j)27 |
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3335 b Fh(L)-5 b(am)3089 933 y Fb(h)-11 b(h)3129 946 y Fh(name)3366 |
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3336 933 y Fb(i)g(i)3406 946 y Fh(lam)p Black -3270 w Fu(\(7\))p |
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3345 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3346 (cite.Pitts03) cvn H.B /ANN pdfmark end |
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3347 2106 1110 a Black Fu(]\227in)g(informal) |
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3348 g(proofs)g(atoms)f(are)j(usu-)291 1230 y(ally)33 b(referred)h(to)g(as)f |
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3349 (v)n(ariables;)1473 1217 y Fb(h)-11 b(h)1530 1230 y Fm(:)17 |
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3350 b(:)g(:)1661 1217 y Fb(i)-11 b(i)1734 1230 y Fu(indicates)33 |
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3351 b(that)g(a)h(name)f(is)g(bound)g(in)g Fh(L)-5 b(am)7 |
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3352 b Fu(.)57 b(This)291 1351 y(de\002nition)30 b(allo)n(ws)f(one)i(to)g |
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3353 (write)g(lambda-terms)e(as)j Fh(L)-5 b(am)38 b Fm(a)31 |
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3354 b Fk(\()p Fh(V)-7 b(ar)42 b Fm(a)p Fk(\))p Fu(.)49 b(Unlik)o(e)30 |
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3355 b(the)h(na)m(\250)-30 b(\021v)o(e)291 1471 y(representation)30 |
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3356 b(mentioned)f(in)i(the)f(Introduction,)h(ho)n(we)n(v)o(er)l(,)g(the)g |
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3357 (nominal)e(datatype)h Fh(lam)291 1591 y Fu(stands)24 |
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3358 b(for)h(alpha-equi)n(v)n(alence)e(classes,)i(that)f(means)h(one)f(has)h |
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3359 (equations)f(such)g(as)p Black Black 1322 1747 a Fh(L)-5 |
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3360 b(am)32 b Fm(x)25 b Fk(\()p Fh(V)-7 b(ar)36 b Fm(x)p |
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3361 Fk(\))28 b(=)f Fh(L)-5 b(am)32 b Fm(y)d Fk(\()p Fh(V)-7 |
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3362 b(ar)35 b Fm(y)t Fk(\))27 b Fm(:)291 1912 y Fu(When)e(formalising)g |
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3363 (the)g(substitution)e(lemma,)j(this)f(will)g(allo)n(w)f(us)i(to)f(reap) |
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3364 i(the)e(bene\002ts)h(of)291 2032 y(equational)j(reasoning.)48 |
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3365 b(Ho)n(we)n(v)o(er)l(,)31 b(it)f(raises)g(a)h(small)f(obstacle)g(for)h |
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3366 (the)f(de\002nition)g(of)h(the)291 2153 y(substitution)g(operation.)57 |
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3367 b(Using)33 b(the)h(infrastructure)f(of)h(the)g(nominal)e(datatype)i |
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3368 (package)291 2273 y(one)24 b(can)i(de\002ne)f(this)f(operation)g(as)p |
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3369 Black Black 798 2543 a Fk(\()p Fh(V)-7 b(ar)39 b Fm(y)t |
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3370 Fk(\)[)p Fm(x)27 b Fk(:=)h Fm(N)10 b Fk(])1509 2487 y |
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3371 Ff(def)1514 2543 y Fk(=)1627 2338 y Fe(8)1627 2428 y(<)1627 |
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3372 2607 y(:)1763 2475 y Fm(N)264 b Fu(if)25 b Fm(x)j Fl(\021)g |
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3373 Fm(y)1763 2632 y Fh(V)-7 b(ar)36 b Fm(y)102 b Fu(otherwise)543 |
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3374 2824 y Fk(\()p Fh(App)33 b Fm(M)880 2839 y Fj(1)948 2824 |
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3375 y Fm(M)1042 2839 y Fj(2)1082 2824 y Fk(\)[)p Fm(x)28 |
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3376 b Fk(:=)g Fm(N)10 b Fk(])1509 2768 y Ff(def)1514 2824 |
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3377 y Fk(=)37 b Fh(App)d Fk(\()p Fm(M)1965 2839 y Fj(1)2005 |
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3378 2824 y Fk([)p Fm(x)28 b Fk(:=)g Fm(N)10 b Fk(]\))28 b(\()p |
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3379 Fm(M)2559 2839 y Fj(2)2599 2824 y Fk([)p Fm(x)g Fk(:=)g |
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3380 Fm(N)10 b Fk(]\))610 3004 y(\()p Fh(L)-5 b(am)35 b Fm(y)c(M)1042 |
|
3381 3019 y Fj(1)1082 3004 y Fk(\)[)p Fm(x)d Fk(:=)g Fm(N)10 |
|
3382 b Fk(])1509 2948 y Ff(def)1514 3004 y Fk(=)37 b Fh(L)-5 |
|
3383 b(am)35 b Fm(y)c Fk(\()p Fm(M)2059 3019 y Fj(1)2099 3004 |
|
3384 y Fk([)p Fm(x)d Fk(:=)g Fm(N)10 b Fk(]\))199 b Fu(pro)o(vided)24 |
|
3385 b Fm(y)31 b Fk(#)d(\()p Fm(x;)17 b(N)10 b Fk(\))291 3185 |
|
3386 y Fu(where)26 b(the)f(side-constraint)g Fm(y)32 b Fk(#)e(\()p |
|
3387 Fm(x;)17 b(N)10 b Fk(\))26 b Fu(means)g(that)f Fm(y)32 |
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3388 b Fl(6)p Fk(=)e Fm(x)c Fu(and)g Fm(y)j Fu(not)c(free)h(in)g |
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3389 Fm(N)10 b Fu(.)34 b(Ho)n(w-)291 3306 y(e)n(v)o(er)h(to)g(ensure)g(that) |
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3390 h(one)f(has)g(indeed)h(de\002ned)g(a)g(function,)h(one)e(needs)h(to)f |
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3391 (v)o(erify)g(some)291 3426 y(properties)c(of)g(the)h(clauses)f(by)h |
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3392 (which)f(substitution)e(is)i(de\002ned)h(\(see)g([)p |
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3394 SDict begin H.S end |
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3397 SDict begin H.R end |
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3398 2960 3358 a 2960 |
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3399 3426 a |
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3400 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3401 (cite.Pitts05) cvn H.B /ANN pdfmark end |
|
3402 2960 3426 a Black Fu(,)p 0 0 1 TeXcolorrgb 2985 |
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3403 3427 a |
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3404 SDict begin H.S end |
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3405 2985 3427 a 0 0 1 TeXcolorrgb -1 x Fu(13)p 0 0 1 |
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3406 TeXcolorrgb 3085 3358 a |
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3407 SDict begin H.R end |
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3408 3085 3358 a 3085 3426 a |
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3409 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
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3410 (cite.UrbanBerghofer06) cvn H.B /ANN pdfmark end |
|
3411 3085 |
|
3412 3426 a Black Fu(])g(for)g(the)f(de-)291 3546 y(tails\).)62 |
|
3413 b(This)35 b(requires)g(some)g(small)g(proofs)g(that)g(ha)n(v)o(e)g(no)g |
|
3414 (counterpart)h(in)f(the)g(informal)291 3667 y(proof)29 |
|
3415 b(and)h(in)f(the)h(formalisation)e(based)h(on)h(de)g(Bruijn)f(indices.) |
|
3416 45 b(This)29 b(need)h(of)g(v)o(erifying)291 3787 y(some)d(properties)h |
|
3417 (arises)g(whene)n(v)o(er)g(a)g(function)g(is)f(de\002ned)i(by)f |
|
3418 (recursion)g(o)o(v)o(er)f(the)h(struc-)291 3908 y(ture)c(of)h |
|
3419 (alpha-equated)g(lambda-terms.)437 4036 y(W)l(ith)36 |
|
3420 b(the)g(de\002nition)f(of)h(the)h(nominal)e(datatype)g |
|
3421 Fh(lam)44 b Fu(comes)36 b(the)g(follo)n(wing)e Fc(str)l(ong)291 |
|
3422 4157 y Fu(structural)24 b(induction)f(principle)h([)p |
|
3423 0 0 1 TeXcolorrgb 1505 4157 a |
|
3424 SDict begin H.S end |
|
3425 1505 4157 a 0 0 1 TeXcolorrgb |
|
3426 Fu(14)p 0 0 1 TeXcolorrgb 1605 4089 a |
|
3427 SDict begin H.R end |
|
3428 1605 4089 a 1605 |
|
3429 4157 a |
|
3430 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3431 (cite.UrbanNorrish05) cvn H.B /ANN pdfmark end |
|
3432 1605 4157 a Black Fu(,)p 0 0 1 TeXcolorrgb 1630 |
|
3433 4158 a |
|
3434 SDict begin H.S end |
|
3435 1630 4158 a 0 0 1 TeXcolorrgb -1 x Fu(15)p 0 0 1 |
|
3436 TeXcolorrgb 1729 4089 a |
|
3437 SDict begin H.R end |
|
3438 1729 4089 a 1729 4157 a |
|
3439 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3440 (cite.UrbanTasson05) cvn H.B /ANN pdfmark end |
|
3441 1729 |
|
3442 4157 a Black Fu(]:)p Black Black 690 4348 a Fl(8)p Fm(c)17 |
|
3443 b(x:)56 b(P)41 b Fk(\()p Fh(V)-7 b(ar)36 b Fm(x)p Fk(\))28 |
|
3444 b Fm(c)690 4505 y Fl(8)p Fm(c)17 b(M)898 4520 y Fj(1)955 |
|
3445 4505 y Fm(M)1049 4520 y Fj(2)1089 4505 y Fm(:)55 b Fk(\()p |
|
3446 Fl(8)p Fm(d:)28 b(P)41 b(M)1568 4520 y Fj(1)1635 4505 |
|
3447 y Fm(d)p Fk(\))50 b Fl(^)g Fk(\()p Fl(8)p Fm(d:)28 b(P)41 |
|
3448 b(M)2287 4520 y Fj(2)2354 4505 y Fm(d)p Fk(\))55 b Fl(\))28 |
|
3449 b Fm(P)41 b Fk(\()p Fh(App)31 b Fm(M)3065 4520 y Fj(1)3132 |
|
3450 4505 y Fm(M)3226 4520 y Fj(2)3266 4505 y Fk(\))d Fm(c)690 |
|
3451 4662 y Fl(8)p Fm(c)17 b(z)22 b(M)5 b(:)55 b(z)33 b Fk(#)28 |
|
3452 b Fm(c)50 b Fl(^)g Fk(\()p Fl(8)p Fm(d:)28 b(P)41 b(M)d(d)p |
|
3453 Fk(\))55 b Fl(\))g Fm(P)30 b Fk(\()p Fh(L)-5 b(am)32 |
|
3454 b Fm(z)e(M)10 b Fk(\))17 b Fm(c)p 690 4736 2684 4 v 1904 |
|
3455 4821 a(P)30 b(M)d(c)291 4985 y Fu(This)32 b(induction)h(principle)g |
|
3456 (states)g(that)g(if)h(one)f(w)o(ants)g(to)h(establish)e(a)i(property)f |
|
3457 Fm(P)48 b Fu(for)34 b(all)291 5106 y(lambda-terms)29 |
|
3458 b Fm(M)10 b Fu(,)33 b(then,)f(as)e(e)o(xpected,)i(one)e(has)h(to)f(pro) |
|
3459 o(v)o(e)g(it)g(for)h(the)f(constructors)g Fh(V)-7 b(ar)10 |
|
3460 b Fu(,)291 5226 y Fh(App)37 b Fu(and)31 b Fh(L)-5 b(am)7 |
|
3461 b Fu(.)49 b(It)31 b(is)f(called)h Fc(str)l(ong)e Fu(induction)h |
|
3462 (principle)g(because)h(it)g(has)g(Barendre)o(gt')-5 b(s)291 |
|
3463 5347 y(v)n(ariable)38 b(con)l(v)o(ention)g(already)i(b)n(uilt)e(in.)74 |
|
3464 b(Barendre)o(gt)40 b(assumes)e(in)h(his)g(informal)f(proof)p |
|
3465 Black 1897 5513 a(7)p Black eop end |
|
3466 %%Page: 8 8 |
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3474 SDict begin H.S end |
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3477 SDict begin H.R end |
|
3478 291 -166 a 291 -166 a |
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3479 SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn H.B /DEST pdfmark |
|
3480 end |
|
3481 291 -166 a Black 1173 w |
|
3482 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(that)f(in)h(the)g |
|
3483 (lambda-case)g(the)g(binder)g Fm(z)36 b Fu(is)30 b(not)h(equal)g(to)f |
|
3484 Fm(x)i Fu(and)f Fm(y)t Fu(,)h(and)f(is)g(not)f(free)i(in)f |
|
3485 Fm(N)291 203 y Fu(and)j Fm(L)p Fu(.)62 b(Using)34 b(the)g(strong)g |
|
3486 (induction)g(principle,)i(we)f(will)f(be)h(able)g(to)f(mimic)g(the)h(v) |
|
3487 n(ari-)291 324 y(able)d(con)l(v)o(ention)f(by)i(instantiating)d |
|
3488 Fm(c)p Fu(,)k(we)f(call)g(this)e(the)i Fc(conte)n(xt)h |
|
3489 Fu(of)f(the)f(induction,)h(with)291 444 y Fm(c)h Fl(7!)g |
|
3490 Fk(\()p Fm(x;)17 b(y)t(;)g(N)5 b(;)17 b(L)p Fk(\))p Fu(.)990 |
|
3491 444 y |
|
3492 SDict begin H.S end |
|
3493 990 444 a 990 444 a |
|
3494 SDict begin 14.5 H.A end |
|
3495 990 444 a 990 444 a |
|
3496 SDict begin [ /View [/XYZ H.V] /Dest (footnote.1) cvn H.B /DEST pdfmark |
|
3497 end |
|
3498 990 444 |
|
3499 a 0 0 1 TeXcolorrgb 990 444 a |
|
3500 SDict begin H.S end |
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|
3502 444 y |
|
3503 SDict begin 14.5 H.L end |
|
3504 1062 444 a 1062 444 a |
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3505 SDict begin [ /Subtype /Link /Dest (Hfootnote.2) cvn /H /I /Border |
|
3506 [0 0 0] /Color [1 0 0] H.B /ANN pdfmark end |
|
3507 1062 444 a Black 42 w Fu(When)28 |
|
3508 b(it)g(then)g(comes)g(to)g(establishing)f(the)h Fh(L)-5 |
|
3509 b(am)7 b Fu(-case,)30 b(we)f(can)g(as-)291 565 y(sume)d(that)h(the)g |
|
3510 (binder)g Fm(z)32 b Fu(is)27 b(fresh)h(for)f Fk(\()p |
|
3511 Fm(x;)17 b(y)t(;)g(N)5 b(;)17 b(L)p Fk(\))p Fu(,)27 b(that)g(means)g |
|
3512 (is)f(not)h(equal)g(to)g Fm(x)h Fu(and)f Fm(y)t Fu(,)291 |
|
3513 685 y(and)33 b(is)g(not)f(free)j(in)d Fm(N)44 b Fu(and)34 |
|
3514 b Fm(L)p Fu(.)56 b(As)33 b(a)h(result,)h(the)e(induction)f(in)h(the)g |
|
3515 (substitution)d(lemma)291 805 y(will)j(go)i(through)f(smoothly)-6 |
|
3516 b(,)34 b(just)g(lik)o(e)g(in)h(Barendre)o(gt')-5 b(s)34 |
|
3517 b(informal)g(proof.)60 b(If)36 b(the)e(nomi-)291 926 |
|
3518 y(nal)26 b(datatype)f(package)i(had)f Fc(not)i Fu(pro)o(vided)d(such)g |
|
3519 (strong)h(induction)e(principles,)i(reasoning)291 1046 |
|
3520 y(w)o(ould)e(be)i(quite)f(incon)l(v)o(enient:)31 b(one)25 |
|
3521 b(w)o(ould)g(ha)n(v)o(e)h(to)f(rename)h(binders)f(so)g(that,)h(for)g(e) |
|
3522 o(xam-)291 1166 y(ple,)e(substitutions)e(can)j(be)g(mo)o(v)o(ed)e |
|
3523 (under)i(lambdas.)437 1295 y(Despite)20 b(the)h(e)o(xcellent)e(notes)i |
|
3524 (from)f(Barendre)o(gt)h(con)l(v)o(e)o(ying)e(v)o(ery)h(well)g(the)h |
|
3525 (proof)f(idea,)291 1416 y(for)k(the)f(formalisation)f(of)i(the)g |
|
3526 (substitution)d(lemma)i(we)h(need)g(to)f(supply)g(some)g(details)g |
|
3527 (that)291 1536 y(are)k(left)f(out)f(in)h(his)g(notes.)34 |
|
3528 b(F)o(or)26 b(e)o(xample)f(in)h(Case)h(1.2)f(the)g(details)f(are)i |
|
3529 (left)f(out)g(for)g(ho)n(w)g(to)291 1656 y(pro)o(v)o(e)d(the)i |
|
3530 (property)f(of)291 1678 y |
|
3531 SDict begin H.S end |
|
3532 291 1678 a 291 1678 a |
|
3533 SDict begin 14.5 H.A end |
|
3534 291 1678 |
|
3535 a 291 1678 a |
|
3536 SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark |
|
3537 end |
|
3538 291 1678 a Black Black 1312 1815 a Fm(x)k |
|
3539 Fk(#)g Fm(L)e Fu(implies)d(that)h Fm(L)p Fk([)p Fm(x)29 |
|
3540 b Fk(:=)f Fm(P)14 b Fk(])27 b(=)g Fm(L)h(:)p Black -2460 |
|
3541 w Fu(\(8\))p Black 291 1982 a(where)19 b Fm(x)28 b Fk(#)g |
|
3542 Fm(L)20 b Fu(stands)e(for)i Fm(x)28 b Fl(62)g Fm(F)14 |
|
3543 b(V)21 b Fk(\()p Fm(L)p Fk(\))p Fu(.)29 b(This)19 b(f)o(act)g(can)h(be) |
|
3544 f(pro)o(v)o(ed)f(by)h(an)g(induction)f(o)o(v)o(er)g Fm(L)291 |
|
3545 2102 y Fu(using)k(the)i(strong)f(induction)f(principle.)29 |
|
3546 b(F)o(or)24 b(this)f(we)g(mak)o(e)h(the)f(follo)n(wing)f |
|
3547 (instantiations:)p Black Black 1116 2297 a Fm(P)46 b |
|
3548 Fl(7!)33 b Fm(\025L:\025)p Fk(\()p Fm(x;)17 b(P)d Fk(\))p |
|
3549 Fm(:)55 b(x)29 b Fk(#)e Fm(L)56 b Fl(\))f Fm(L)p Fk([)p |
|
3550 Fm(x)29 b Fk(:=)e Fm(P)14 b Fk(])28 b(=)f Fm(L)1087 2454 |
|
3551 y(M)44 b Fl(7!)33 b Fm(L)1150 2611 y(c)g Fl(7!)g Fk(\()p |
|
3552 Fm(x;)17 b(P)d Fk(\))291 2788 y Fu(As)21 b(a)g(result,)g(the)g(v)n |
|
3553 (ariable)g(and)g(application)f(case)i(are)g(completely)e(routine.)29 |
|
3554 b(In)21 b(the)g(lambda-)291 2908 y(case)39 b(we)g(ha)n(v)o(e)g(to)f |
|
3555 (sho)n(w)g(that)g Fm(x)55 b Fk(#)f(\()p Fh(L)-5 b(am)46 |
|
3556 b Fm(z)d(L)2100 2923 y Fj(1)2140 2908 y Fk(\))c Fu(implies)f |
|
3557 Fk(\()p Fh(L)-5 b(am)46 b Fm(z)d(L)2974 2923 y Fj(1)3014 |
|
3558 2908 y Fk(\)[)p Fm(x)54 b Fk(:=)g Fm(P)14 b Fk(])53 b(=)291 |
|
3559 3028 y(\()p Fh(L)-5 b(am)32 b Fm(z)d(L)687 3043 y Fj(1)727 |
|
3560 3028 y Fk(\))c Fu(with)f(the)h(assumption)d(that)j Fm(z)32 |
|
3561 b Fk(#)c(\()p Fm(x;)17 b(P)d Fk(\))24 b Fu(and)h(the)g(induction)e |
|
3562 (hypothesis)p Black Black 1343 3187 a Fl(8)p Fm(x;)17 |
|
3563 b(L:)29 b(x)f Fk(#)g Fm(L)53 b Fl(\))27 b Fm(L)p Fk([)p |
|
3564 Fm(x)i Fk(:=)f Fm(P)14 b Fk(])27 b(=)g Fm(L)h(:)291 3354 |
|
3565 y Fu(From)g(the)h(assumption)e(that)i Fm(z)k Fu(is)c(not)f(equal)h(to)g |
|
3566 Fm(x)g Fu(and)g(not)f(free)i(in)f Fm(L)p Fu(,)h(we)f(can)g(infer)h |
|
3567 (from)291 3474 y Fm(x)k Fk(#)g(\()p Fh(L)-5 b(am)35 b |
|
3568 Fm(z)e(L)898 3489 y Fj(1)938 3474 y Fk(\))28 b Fu(that)g |
|
3569 Fm(x)34 b Fk(#)h Fm(L)1453 3489 y Fj(1)1521 3474 y Fu(holds)27 |
|
3570 b(and)h(by)g(applying)f(the)h(de\002nition)f(of)h(substitution)291 |
|
3571 3595 y(that)f Fk(\()p Fh(L)-5 b(am)35 b Fm(z)e(L)871 |
|
3572 3610 y Fj(1)911 3595 y Fk(\)[)p Fm(x)h Fk(:=)f Fm(P)14 |
|
3573 b Fk(])33 b(=)h Fh(L)-5 b(am)35 b Fm(z)e Fk(\()p Fm(L)1851 |
|
3574 3610 y Fj(1)1891 3595 y Fk([)p Fm(x)h Fk(:=)f Fm(P)14 |
|
3575 b Fk(]\))p Fu(.)40 b(No)n(w)27 b(we)h(just)f(need)h(to)g(apply)g(the) |
|
3576 291 3715 y(induction)23 b(hypothesis)f(and)j(are)h(done.)437 |
|
3577 3844 y(Although)k(not)h(ob)o(vious)f(from)i(\002rst)f(glance,)j(also)d |
|
3578 (in)h(Case)g(2,)h(in)f(the)f(last)g(step)h(of)g(the)291 |
|
3579 3964 y(calculation)f(where)i(the)f(substitution)d(is)j(pulled)f(back)h |
|
3580 (from)g(under)h(the)f(binder)f Fm(\025z)t Fu(,)k(there)291 |
|
3581 4085 y(are)e(some)f(details)g(missing)f(from)h(Barendre)o(gt')-5 |
|
3582 b(s)32 b(informal)g(proof.)54 b(In)33 b(order)g(to)f(get)h(from)291 |
|
3583 4205 y Fh(L)-5 b(am)35 b Fm(z)e Fk(\()p Fm(M)722 4220 |
|
3584 y Fj(1)762 4205 y Fk([)p Fm(y)k Fk(:=)d Fm(L)p Fk(][)p |
|
3585 Fm(x)h Fk(:=)f Fm(N)10 b Fk([)p Fm(y)38 b Fk(:=)c Fm(L)p |
|
3586 Fk(]]\))29 b Fu(to)f Fk(\()p Fh(L)-5 b(am)35 b Fm(z)f(M)2421 |
|
3587 4220 y Fj(1)2460 4205 y Fk(\)[)p Fm(y)k Fk(:=)c Fm(L)p |
|
3588 Fk(][)p Fm(x)h Fk(:=)f Fm(N)10 b Fk([)p Fm(y)38 b Fk(:=)c |
|
3589 Fm(L)p Fk(]])p Fu(,)291 4325 y(we)25 b(need)g(the)f(property)h(that:) |
|
3590 1362 4325 y |
|
3591 SDict begin H.S end |
|
3592 1362 4325 a 1362 4325 a |
|
3593 SDict begin 14.5 H.A end |
|
3594 1362 4325 a 1362 |
|
3595 4325 a |
|
3596 SDict begin [ /View [/XYZ H.V] /Dest (equation.9) cvn H.B /DEST pdfmark |
|
3597 end |
|
3598 1362 4325 a Black Black 1155 4484 a Fu(if)g Fm(z)33 |
|
3599 b Fk(#)28 b Fm(N)35 b Fu(and)25 b Fm(z)32 b Fk(#)c Fm(L)d |
|
3600 Fu(then)g Fm(z)32 b Fk(#)c(\()p Fm(N)10 b Fk([)p Fm(y)32 |
|
3601 b Fk(:=)27 b Fm(L)p Fk(]\))17 b Fm(:)p Black -2617 w |
|
3602 Fu(\(9\))p Black 291 4651 a(where)24 b(the)f(preconditions)f(are)i(gi)n |
|
3603 (v)o(en)e(by)i(his)f(use)g(of)h(the)f(v)n(ariable)g(con)l(v)o(ention.) |
|
3604 29 b(This)22 b(prop-)291 4771 y(erty)-6 b(,)27 b(too,)h(can)g(be)g |
|
3605 (easily)g(pro)o(v)o(ed)e(by)i(strong)e(induction)h(o)o(v)o(er)g(the)g |
|
3606 (structure)h(of)f Fm(N)10 b Fu(.)40 b(In)28 b(this)291 |
|
3607 4892 y(induction)c(we)i(instantiate)e(the)i(induction)e(conte)o(xt)h |
|
3608 (with)g Fm(c)30 b Fl(7!)f Fk(\()p Fm(z)t(;)17 b(y)t(;)g(L)p |
|
3609 Fk(\))p Fu(,)25 b(because)i(then)e(we)291 5012 y(can)i(in)g(the)f |
|
3610 Fh(L)-5 b(am)8 b Fu(-case,)28 b(say)f(instantiated)e(as)i |
|
3611 Fk(\()p Fh(L)-5 b(am)34 b Fm(x)28 b(N)2335 5027 y Fj(1)2374 |
|
3612 5012 y Fk(\))p Fu(,)g(mo)o(v)o(e)d(the)i(substitution)d(under)p |
|
3613 Black 291 5156 299 4 v 291 5217 a Ft(4)p 0 TeXcolorgray |
|
3614 374 5247 a |
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3615 SDict begin H.S end |
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3616 374 5247 a 0 TeXcolorgray 0 TeXcolorgray 374 |
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3617 5247 a |
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3618 SDict begin H.R end |
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3619 374 5247 a 374 5247 a |
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3620 SDict begin [ /View [/XYZ H.V] /Dest (Hfootnote.2) cvn H.B /DEST pdfmark |
|
3621 end |
|
3622 374 5247 a Black Fs(An)f(aspect)h(we)f(do) |
|
3623 g(not)g(dwell)h(on)f(here)g(is)h(the)g(f)o(act)f(that)h(the)f |
|
3624 (induction)f(conte)o(xt)g(must)i(al)o(w)o(ays)g(be)f(\002nitely)291 |
|
3625 5347 y(supported,)18 b(i.e.)i(mentions)f(only)g(\002nitely)h(man)o(y)f |
|
3626 (free)h(names,)g(see)g([)p 0 0 1 TeXcolorrgb 2293 5348 |
|
3627 a |
|
3628 SDict begin H.S end |
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3629 2293 5348 a 0 0 1 TeXcolorrgb -1 x Fs(10)p 0 0 1 TeXcolorrgb |
|
3630 2376 5290 a |
|
3631 SDict begin H.R end |
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3632 2376 5290 a 2376 5347 a |
|
3633 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3634 (cite.Pitts03) cvn H.B /ANN pdfmark end |
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3635 2376 5347 a Black |
|
3636 Fs(,)p 0 0 1 TeXcolorrgb 2397 5348 a |
|
3637 SDict begin H.S end |
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3638 2397 5348 a 0 0 1 |
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3639 TeXcolorrgb -1 x Fs(15)p 0 0 1 TeXcolorrgb 2480 5290 |
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3640 a |
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3641 SDict begin H.R end |
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3645 2480 5347 a Black Fs(].)p Black |
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3646 Black 1897 5513 a Fu(8)p Black eop end |
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3657 -166 a |
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3659 291 -166 a 291 -166 a |
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3660 SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark |
|
3661 end |
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3662 291 -166 a Black 1173 w |
|
3663 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(the)i(binder)g |
|
3664 Fm(x)i Fu(and)e(also)h(infer)f(from)h(the)f(assumption)f |
|
3665 Fm(z)49 b Fk(#)44 b(\()p Fh(L)-5 b(am)41 b Fm(x)34 b(N)2939 |
|
3666 98 y Fj(1)2979 83 y Fk(\))g Fu(that)f Fm(z)39 b Fu(is)33 |
|
3667 b(also)291 203 y(fresh)c(for)h Fm(N)743 218 y Fj(1)812 |
|
3668 203 y Fu(\(this)e(reasoning)h(step)g(depends)g(on)g Fm(z)41 |
|
3669 b Fl(6)p Fk(=)36 b Fm(x)p Fu(\).)45 b(Consequently)29 |
|
3670 b(we)g(can)h(apply)291 324 y(the)c(induction)f(hypothesis)g(and)h |
|
3671 (infer)h(that)g Fm(z)35 b Fk(#)d Fm(N)2128 339 y Fj(1)2167 |
|
3672 324 y Fk([)p Fm(y)j Fk(:=)c Fm(L)p Fk(])c Fu(holds.)36 |
|
3673 b(Again)25 b(since)i Fm(z)36 b Fl(6)p Fk(=)31 b Fm(x)p |
|
3674 Fu(,)291 444 y(also)24 b Fm(z)32 b Fk(#)c(\()p Fh(L)-5 |
|
3675 b(am)32 b Fm(x)26 b(N)1077 459 y Fj(1)1116 444 y Fk([)p |
|
3676 Fm(y)31 b Fk(:=)d Fm(L)p Fk(]\))d Fu(holds)f(and)h(we)g(are)g(done.)437 |
|
3677 573 y(The)g(formalisation)e(of)i(the)f(substitution)e(lemma)p |
|
3678 Black Black 541 752 a Fn(Substitution)k(Lemma)f(with)h(Names:)52 |
|
3679 b Fu(If)25 b Fm(x)j Fl(6)p Fk(=)g Fm(y)g Fu(and)d Fm(x)j |
|
3680 Fk(#)g Fm(L)d Fu(then)777 932 y Fm(M)10 b Fk([)p Fm(x)29 |
|
3681 b Fk(:=)f Fm(N)10 b Fk(][)p Fm(y)31 b Fk(:=)d Fm(L)p |
|
3682 Fk(])g(=)f Fm(M)10 b Fk([)p Fm(y)32 b Fk(:=)c Fm(L)p |
|
3683 Fk(][)p Fm(x)g Fk(:=)g Fm(N)10 b Fk([)p Fm(y)31 b Fk(:=)d |
|
3684 Fm(L)p Fk(]])g Fu(.)291 1106 y(no)n(w)k(follo)n(ws)g(almost)h(to)g(the) |
|
3685 g(w)o(ord)g(Barendre)o(gt')-5 b(s)33 b(informal)g(proof.)57 |
|
3686 b(The)33 b(v)n(ariable-case,)291 1226 y(say)f(with)f(the)h |
|
3687 (instantiation)e Fk(\()p Fh(V)-7 b(ar)43 b Fm(z)t Fk(\))p |
|
3688 Fu(,)34 b(proceeds)e(by)g(a)h(case-analysis)f(with)f |
|
3689 Fm(z)46 b Fk(=)41 b Fm(x)p Fu(,)34 b Fm(z)46 b Fl(6)p |
|
3690 Fk(=)291 1347 y Fm(x)29 b Fl(^)h Fm(z)50 b Fk(=)c Fm(y)37 |
|
3691 b Fu(and)e Fm(z)50 b Fl(6)p Fk(=)c Fm(x)29 b Fl(^)h Fm(z)50 |
|
3692 b Fl(6)p Fk(=)c Fm(y)t Fu(.)59 b(The)34 b(calculations)g(in)l(v)n(olv)o |
|
3693 (ed)e(are)j(routine)f(using)g(in)291 1467 y(the)24 b(second)h(case)g |
|
3694 (the)g(property)f(in)g(\()p 0 0 1 TeXcolorrgb 1579 1468 |
|
3695 a |
|
3696 SDict begin H.S end |
|
3697 1579 1468 a 0 0 1 TeXcolorrgb -1 x Fu(8)p 0 0 1 TeXcolorrgb |
|
3698 1629 1399 a |
|
3699 SDict begin H.R end |
|
3700 1629 1399 a 1629 1467 a |
|
3701 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3702 (equation.8) cvn H.B /ANN pdfmark end |
|
3703 1629 1467 a Black |
|
3704 Fu(\).)31 b(The)25 b(application)f(case)h(does)f(not)g(need)h(an)o(y)g |
|
3705 (special)291 1588 y(attention.)49 b(The)31 b(lambda-case,)i(too,)g(is)e |
|
3706 (relati)n(v)o(ely)e(easy:)44 b(by)31 b(instantiating)f(the)h(induction) |
|
3707 291 1708 y(conte)o(xt)d(with)h Fm(c)36 b Fl(7!)g Fk(\()p |
|
3708 Fm(x;)17 b(y)t(;)g(N)5 b(;)17 b(L)p Fk(\))p Fu(,)30 b(the)g(strong)e |
|
3709 (induction)g(principle)h(allo)n(ws)g(us)g(to)g(assume)291 |
|
3710 1828 y(that)23 b(the)h(binder)g(is)f(not)h(equal)g(to)g |
|
3711 Fm(x)g Fu(and)g Fm(y)t Fu(,)g(and)g(is)f(not)h(free)h(in)e |
|
3712 Fm(N)10 b Fu(and)25 b Fm(L)p Fu(.)31 b(Consequently)23 |
|
3713 b(we)291 1949 y(can)i(reason)g(lik)o(e)f(Barendre)o(gt:)p |
|
3714 Black Black 495 2148 a Fk(\()p Fh(L)-5 b(am)32 b Fm(z)e(M)920 |
|
3715 2163 y Fj(1)960 2148 y Fk(\)[)p Fm(x)e Fk(:=)g Fm(N)10 |
|
3716 b Fk(][)p Fm(y)31 b Fk(:=)d Fm(L)p Fk(])47 b(=)h Fh(L)-5 |
|
3717 b(am)32 b Fm(z)d Fk(\()p Fm(M)2279 2163 y Fj(1)2319 2148 |
|
3718 y Fk([)p Fm(x)f Fk(:=)g Fm(N)10 b Fk(][)p Fm(y)31 b Fk(:=)d |
|
3719 Fm(L)p Fk(]\))1731 2329 y(=)48 b Fh(L)-5 b(am)32 b Fm(z)d |
|
3720 Fk(\()p Fm(M)2279 2344 y Fj(1)2319 2329 y Fk([)p Fm(y)i |
|
3721 Fk(:=)d Fm(L)p Fk(][)p Fm(x)g Fk(:=)g Fm(N)10 b Fk([)p |
|
3722 Fm(y)31 b Fk(:=)d Fm(L)p Fk(]]\))1731 2509 y(=)48 b(\()p |
|
3723 Fh(L)-5 b(am)32 b Fm(z)d(M)2279 2524 y Fj(1)2319 2509 |
|
3724 y Fk(\)[)p Fm(y)i Fk(:=)c Fm(L)p Fk(][)p Fm(x)i Fk(:=)f |
|
3725 Fm(N)10 b Fk([)p Fm(y)31 b Fk(:=)d Fm(L)p Fk(]])291 2683 |
|
3726 y Fu(where,)d(as)g(mentioned)e(earlier)l(,)i(in)g(the)g(last)f |
|
3727 (equation)g(we)h(mak)o(e)g(use)f(the)h(property)f(in)h(\()p |
|
3728 0 0 1 TeXcolorrgb 3401 2685 a |
|
3729 SDict begin H.S end |
|
3730 3401 2685 a 0 0 1 TeXcolorrgb |
|
3731 -2 x Fu(9)p 0 0 1 TeXcolorrgb 3451 2615 a |
|
3732 SDict begin H.R end |
|
3733 3451 2615 a |
|
3734 3451 2683 a |
|
3735 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3736 (equation.9) cvn H.B /ANN pdfmark end |
|
3737 3451 2683 a Black Fu(\).)437 2812 y(The)f(resulting)f |
|
3738 (formalised)h(proof)g(is)g(quite)g(simple:)29 b(one)24 |
|
3739 b(only)g(has)g(to)g(manually)f(set)i(up)291 2932 y(the)k(induction)g |
|
3740 (and)g(supply)g(the)h(properties)f(\()p 0 0 1 TeXcolorrgb |
|
3741 1931 2933 a |
|
3742 SDict begin H.S end |
|
3743 1931 2933 a 0 0 1 TeXcolorrgb -1 x Fu(8)p |
|
3744 0 0 1 TeXcolorrgb 1981 2864 a |
|
3745 SDict begin H.R end |
|
3746 1981 2864 a 1981 2932 a |
|
3747 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3748 (equation.8) cvn H.B /ANN pdfmark end |
|
3749 |
|
3750 1981 2932 a Black Fu(\))h(and)g(\()p 0 0 1 TeXcolorrgb |
|
3751 2251 2934 a |
|
3752 SDict begin H.S end |
|
3753 2251 2934 a 0 0 1 TeXcolorrgb -2 x Fu(9)p |
|
3754 0 0 1 TeXcolorrgb 2301 2864 a |
|
3755 SDict begin H.R end |
|
3756 2301 2864 a 2301 2932 a |
|
3757 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3758 (equation.9) cvn H.B /ANN pdfmark end |
|
3759 |
|
3760 2301 2932 a Black Fu(\))g(to)g(the)f(automatic)g(pro)o(ving)f(tools)291 |
|
3761 3052 y(for)23 b(which)g(it)g(is)g(a)g(straightforw)o(ard)g(task)g(to)g |
|
3762 (complete)f(the)i(proof)f(\(similar)f(for)i(the)f(tw)o(o)g(side)291 |
|
3763 3173 y(lemmas\).)63 b(W)-8 b(e)37 b(tak)o(e)f(this)f(as)h(an)h |
|
3764 (indicator)e(that)h(the)g(formalised)f(proof)h(using)f(names)h(is)291 |
|
3765 3293 y(\223simpler\224)24 b(than)g(the)h(one)g(based)f(on)h(de)g |
|
3766 (Bruijn)f(indices.)291 3469 y |
|
3767 SDict begin H.S end |
|
3768 291 3469 a 291 3469 a |
|
3769 SDict begin 14.5 H.A end |
|
3770 291 |
|
3771 3469 a 291 3469 a |
|
3772 SDict begin [ /View [/XYZ H.V] /Dest (section.0.3) cvn H.B /DEST pdfmark |
|
3773 end |
|
3774 291 3469 a 136 x Fv(3)119 b(T)-9 b(ransiti)o(vity)29 |
|
3775 b(and)i(Narr)n(o)o(wing)f(f)m(or)g(Subtyping)291 3828 |
|
3776 y Fu(Another)i(proof)h(where)h(we)f(can)h(compare)f(names)g(and)g(de)h |
|
3777 (Bruijn)e(indices)h(is)g(the)g(transi-)291 3948 y(ti)n(vity)26 |
|
3778 b(and)j(narro)n(wing)e(proof)i(for)g(the)f(subtyping)f(relation)h |
|
3779 (described)g(in)g(the)h(POPLmark-)291 4068 y(Challenge.)h(This)23 |
|
3780 b(proof)g(is)h(quite)f(trick)o(y)f(in)l(v)n(olving)g(a)i(simultaneous)e |
|
3781 (outer)h(induction)f(o)o(v)o(er)291 4189 y(a)34 b(type)g(and)h(tw)o(o)f |
|
3782 (inner)g(inductions)f(on)h(the)g(de\002nition)g(of)g(the)h(subtyping)d |
|
3783 (relation.)59 b(The)291 4309 y(\223rough)31 b(notes\224)h(from)f(which) |
|
3784 h(we)g(can)g(start)g(the)f(formalisations)f(are)j(gi)n(v)o(en)d(in)i([) |
|
3785 p 0 0 1 TeXcolorrgb 3210 4310 a |
|
3786 SDict begin H.S end |
|
3787 3210 4310 a 0 0 1 TeXcolorrgb |
|
3788 -1 x Fu(3)p 0 0 1 TeXcolorrgb 3260 4241 a |
|
3789 SDict begin H.R end |
|
3790 3260 4241 a |
|
3791 3260 4309 a |
|
3792 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3793 (cite.challenge05) cvn H.B /ANN pdfmark end |
|
3794 3260 4309 a Black Fu(])g(by)g(the)291 4429 |
|
3795 y(authors)24 b(of)h(this)f(challenge.)291 4605 y |
|
3796 SDict begin H.S end |
|
3797 291 |
|
3798 4605 a 291 4605 a |
|
3799 SDict begin 14.5 H.A end |
|
3800 291 4605 a 291 4605 a |
|
3801 SDict begin [ /View [/XYZ H.V] /Dest (subsection.0.3.1) cvn H.B /DEST |
|
3802 pdfmark end |
|
3803 291 4605 a 107 |
|
3804 x Fc(3.1)99 b(V)-11 b(er)o(sion)24 b(using)f(the)i(Nominal)f(Datatype)g |
|
3805 (P)-8 b(ac)n(ka)o(g)o(e)291 4903 y Fu(Using)23 b(the)i(nominal)f |
|
3806 (datatype)g(package)h(the)g(types)f(can)h(be)g(de\002ned)h(as)p |
|
3807 Black Black 440 5061 a Fn(nominal)p 795 5061 30 4 v 35 |
|
3808 w(datatype)57 b Fh(ty)c Fk(=)44 b Fh(Tvar)38 b Fm(name)57 |
|
3809 b Fl(j)27 b Fh(T)-7 b(op)60 b Fl(j)28 b Fh(F)-7 b(un)34 |
|
3810 b(ty)j(ty)65 b Fl(j)27 b Fh(A)n(l)5 b(l)38 b(ty)3169 |
|
3811 5048 y Fb(h)-11 b(h)3209 5061 y Fh(name)3446 5048 y Fb(i)g(i)3486 |
|
3812 5061 y Fh(ty)291 5226 y Fu(with)30 b(typing)g(conte)o(xts)g(being)h |
|
3813 (lists)f(of)i(pairs)f(consisting)f(of)h(a)h(name)f(and)g(a)h(type.)50 |
|
3814 b(A)32 b(type)291 5347 y Fm(T)46 b Fu(is)32 b Fc(well-formed)j |
|
3815 Fu(w)-6 b(.r)h(.t.)31 b(a)h(typing)g(conte)o(xt)f Fk(\000)p |
|
3816 Fu(,)j(written)e Fk(\000)42 b Fl(`)g Fm(T)14 b Fu(,)34 |
|
3817 b(pro)o(vided)d Fk(\()p Fh(supp)38 b Fm(T)14 b Fk(\))42 |
|
3818 b Fl(\022)p Black 1897 5513 a Fu(9)p Black eop end |
|
3819 %%Page: 10 10 |
|
3820 TeXDict begin 10 9 bop 0 0 a |
|
3821 SDict begin /product where{pop product(Distiller)search{pop pop pop |
|
3822 version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto |
|
3823 closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show |
|
3824 grestore}if}{pop}ifelse}{pop}ifelse}if end |
|
3825 0 0 a Black 0 TeXcolorgray |
|
3826 291 -166 a |
|
3827 SDict begin H.S end |
|
3828 291 -166 a 0 TeXcolorgray 0 TeXcolorgray 291 |
|
3829 -166 a |
|
3830 SDict begin H.R end |
|
3831 291 -166 a 291 -166 a |
|
3832 SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark |
|
3833 end |
|
3834 291 -166 a Black 1173 w |
|
3835 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fk(\()p |
|
3836 Fh(dom)37 b Fk(\000\))p Fu(\227that)29 b(means)h(all)g(free)h(names)f |
|
3837 (of)h Fm(T)14 b Fu(,)31 b(i.e.)f(its)g Fc(support)h Fu([)p |
|
3838 0 0 1 TeXcolorrgb 2697 84 a |
|
3839 SDict begin H.S end |
|
3840 2697 84 a 0 0 1 TeXcolorrgb |
|
3841 -1 x Fu(10)p 0 0 1 TeXcolorrgb 2797 15 a |
|
3842 SDict begin H.R end |
|
3843 2797 15 a 2797 |
|
3844 83 a |
|
3845 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest |
|
3846 (cite.Pitts03) cvn H.B /ANN pdfmark end |
|
3847 2797 83 a Black Fu(],)g(must)f(be)g(included)291 |
|
3848 203 y(in)f(the)g(domain)g(of)g(the)h(typing)e(conte)o(xt)h |
|
3849 Fk(\000)p Fu(.)45 b(A)29 b Fc(valid)j Fu(typing)c(conte)o(xt,)i |
|
3850 (written)f Fh(valid)39 b Fk(\000)p Fu(,)31 b(is)291 324 |
|
3851 y(de\002ned)25 b(inducti)n(v)o(ely)d(by:)p Black Black |
|
3852 1014 516 289 4 v 1014 608 a Fh(valid)34 b Fk([])1539 |
|
3853 472 y Fh(valid)g Fk(\000)100 b Fm(X)35 b Fk(#)28 b(\()p |
|
3854 Fh(dom)k Fk(\000\))99 b(\000)28 b Fl(`)f Fm(T)p 1539 |
|
3855 516 1317 4 v 1836 608 a Fh(valid)35 b Fk(\(\()p Fm(X)r(;)17 |
|
3856 b(T)d Fk(\))d(::)g(\000\))291 783 y Fu(The)33 b(subtyping)e(relation,)k |
|
3857 (written)e Fk(\000)43 b Fl(`)g Fm(S)49 b(<)p Fk(:)43 |
|
3858 b Fm(Q)p Fu(,)36 b(can)e(then)f(be)g(inducti)n(v)o(ely)e(de\002ned)i |
|
3859 (as)291 903 y(follo)n(ws:)p Black Black 901 1075 a Fh(valid)h |
|
3860 Fk(\000)100 b(\000)28 b Fl(`)f Fm(S)p 901 1095 639 4 |
|
3861 v 935 1181 a Fk(\000)h Fl(`)f Fm(S)34 b(<)p Fk(:)28 b |
|
3862 Fh(T)-7 b(op)1580 1120 y(T)g(op)1968 1051 y(valid)35 |
|
3863 b Fk(\000)99 b Fm(X)36 b Fl(2)28 b Fk(\()p Fh(dom)k Fk(\000\))p |
|
3864 1948 1095 996 4 v 1948 1181 a(\000)27 b Fl(`)h Fh(Tvar)35 |
|
3865 b Fm(X)g(<)p Fk(:)28 b Fh(Tvar)35 b Fm(X)2985 1120 y |
|
3866 Fh(R)-5 b(e\015)584 1355 y Fk(\()p Fm(X)r(;)17 b(S)6 |
|
3867 b Fk(\))28 b Fl(2)g Fk(\000)99 b(\000)28 b Fl(`)f Fm(S)34 |
|
3868 b(<)p Fk(:)27 b Fm(T)p 584 1400 1024 4 v 728 1486 a Fk(\000)g |
|
3869 Fl(`)h Fh(Tvar)35 b Fm(X)g(<)p Fk(:)28 b Fm(T)1649 1434 |
|
3870 y Fh(T)-7 b(r)i(ans)2102 1365 y Fk(\000)28 b Fl(`)f Fm(T)2336 |
|
3871 1380 y Fj(1)2403 1365 y Fm(<)p Fk(:)h Fm(S)2594 1380 |
|
3872 y Fj(1)2733 1365 y Fk(\000)g Fl(`)f Fm(S)2970 1380 y |
|
3873 Fj(2)3037 1365 y Fm(<)p Fk(:)h Fm(T)3225 1380 y Fj(2)p |
|
3874 2099 1400 1169 4 v 2099 1486 a Fk(\000)g Fl(`)f Fh(F)-7 |
|
3875 b(un)32 b Fm(S)2532 1501 y Fj(1)2596 1486 y Fm(S)2656 |
|
3876 1501 y Fj(2)2723 1486 y Fm(<)p Fk(:)c Fh(F)-7 b(un)31 |
|
3877 b Fm(T)3106 1501 y Fj(1)3171 1486 y Fm(T)3228 1501 y |
|
3878 Fj(2)3309 1434 y Fh(F)-7 b(un)981 1659 y Fk(\000)28 b |
|
3879 Fl(`)g Fm(T)1216 1674 y Fj(1)1283 1659 y Fm(<)p Fk(:)f |
|
3880 Fm(S)1473 1674 y Fj(1)1613 1659 y Fm(X)35 b Fk(#)28 b(\000)100 |
|
3881 b(\()p Fm(X)r(;)17 b(T)2221 1674 y Fj(1)2260 1659 y Fk(\))11 |
|
3882 b(::)g(\000)28 b Fl(`)g Fm(S)2612 1674 y Fj(2)2679 1659 |
|
3883 y Fm(<)p Fk(:)f Fm(T)2866 1674 y Fj(2)p 981 1704 1925 |
|
3884 4 v 1281 1790 a Fk(\000)h Fl(`)g Fh(A)n(l)5 b(l)34 b |
|
3885 Fm(S)1678 1805 y Fj(1)1743 1790 y Fm(X)f(S)1917 1805 |
|
3886 y Fj(2)1984 1790 y Fm(<)p Fk(:)27 b Fh(A)n(l)5 b(l)35 |
|
3887 b Fm(T)2331 1805 y Fj(1)2396 1790 y Fm(X)d(T)2566 1805 |
|
3888 y Fj(2)2947 1739 y Fh(A)n(l)5 b(l)291 1962 y Fu(These)32 |
|
3889 b(de\002nitions)f(are)i(quite)e(close)h(to)g(the)g(\223rough)g |
|
3890 (notes\224)f(from)h(the)g(POPLmark-Chal-)291 2082 y(lenge;)23 |
|
3891 b(the)h(only)f(dif)n(ference)i(is)e(that)h(we)g(had)g(to)f(ensure)i(v)n |
|
3892 (alidity)d(of)i(the)f(typing)g(conte)o(xts)g(in)291 2202 |
|
3893 y(the)k(lea)n(v)o(es)f(and)i(to)f(e)o(xplicitly)e(require)i(that)g(the) |
|
3894 g(binder)g Fm(X)35 b Fu(is)27 b(fresh)g(for)h Fk(\000)f |
|
3895 Fu(in)g(the)g Fh(A)n(l)5 b(l)10 b Fu(-rule.)291 2323 |
|
3896 y(The)24 b(transiti)n(vity)f(and)h(narro)n(wing)g(lemma)g(can)i(then)e |
|
3897 (be)h(stated)f(as)p Black Black 375 2522 a Fn(T)-7 b(ransiti)o(vity)25 |
|
3898 b(and)g(Narr)n(o)o(wing)g(with)g(Names:)53 b Fu(F)o(or)24 |
|
3899 b(all)h Fk(\000)p Fu(,)g Fm(S)6 b Fu(,)24 b Fm(T)14 b |
|
3900 Fu(,)25 b Fk(\001)p Fu(,)g Fm(X)8 b Fu(,)25 b Fm(P)14 |
|
3901 b Fu(,)24 b Fm(M)10 b Fu(,)25 b Fm(N)10 b Fu(:)493 2703 |
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3902 y Fl(\017)28 b Fk(\000)g Fl(`)f Fm(S)34 b(<)p Fk(:)27 |
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3903 b Fm(Q)e Fu(and)g Fk(\000)j Fl(`)f Fm(Q)h(<)p Fk(:)g |
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3904 Fm(T)39 b Fu(implies)23 b Fk(\000)k Fl(`)h Fm(S)34 b(<)p |
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3905 Fk(:)27 b Fm(T)14 b Fu(,)25 b(and)493 2883 y Fl(\017)j |
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3906 Fk(\(\001@\()p Fm(X)r(;)17 b(Q)p Fk(\)@\000\))28 b Fl(`)f |
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3907 Fm(M)39 b(<)p Fk(:)27 b Fm(N)36 b Fu(and)25 b Fk(\000)i |
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3908 Fl(`)h Fm(P)41 b(<)p Fk(:)27 b Fm(Q)588 3040 y Fu(implies)c |
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3909 Fk(\(\001@\()p Fm(X)r(;)17 b(P)d Fk(\)@\000\))27 b Fl(`)h |
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3910 Fm(M)38 b(<)p Fk(:)28 b Fm(N)38 b Fu(.)291 3214 y(About)24 |
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3911 b(the)g(proof)h(of)g(this)f(lemma)g(the)g(POPLmark-paper)i(states:)p |
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3912 Black Black 390 3413 a Fc(\223)f(The)c(two)f(parts)f(ar)l(e)h(pr)l(o)o |
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3913 (ved)g(simultaneously)-5 b(,)18 b(by)i(induction)f(on)h(the)g(size)f |
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3914 (of)h Fm(Q)p Fc(.)30 b(The)470 3534 y(ar)l(gument)g(for)g(part)g(\(2\)) |
|
3915 h(assumes)e(that)h(part)g(\(1\))h(has)f(been)g(established)f(alr)l |
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3916 (eady)470 3654 y(for)24 b(the)h Fm(Q)g Fc(in)g(question;)e(part)h |
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3917 (\(1\))h(uses)g(part)f(\(2\))h(only)f(for)h(strictly)e(smaller)h |
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3918 Fm(Q)p Fc(.)-14 b(\224)291 3828 y Fu(The)20 b(main)g(point)g(we)h(w)o |
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3919 (ant)g(to)f(mak)o(e)h(here)g(is)f(that)g(the)h(formal)f(proof)h(using)f |
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3920 (names)g(proceeds)291 3948 y(e)o(xactly)30 b(as)h(stated,)h(while)e(as) |
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3921 h(we)g(shall)f(see)i(later)f(this)f(is)g Fc(not)i Fu(the)f(case)h(for)f |
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3922 (the)g(de)g(Bruijn)291 4069 y(indices)19 b(v)o(ersion.)29 |
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3923 b(The)20 b(main)g(incon)l(v)o(enience)f(with)h(the)g(named)g(approach)h |
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3924 (is,)g(ho)n(we)n(v)o(er)l(,)f(that)291 4189 y(the)27 |
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3925 b(proof)g(then)g(proceeds)g(by)g(tw)o(o)g(inner)g(inductions)e(on)i |
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3926 (the)g(de\002nition)g(of)g(the)g(subtyping)291 4309 y(relation)18 |
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3927 b(and)h(in)g(order)h(to)f(follo)n(w)e(the)j(reasoning)e(on)h |
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3928 (\223paper\224)h(one)f(has)g(to)g(pro)o(vide)f Fc(manually)291 |
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3929 4430 y Fu(a)36 b(strong)g(v)o(ersion)f(of)i(the)f(induction)f |
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3930 (principle)g(for)i(subtyping.)64 b(This)35 b(strong)h(induction)291 |
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3931 4550 y(principle)24 b(has)g(the)h(form)g(\(sho)n(wing)e(only)h(the)g |
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3932 (premise)h(for)g(the)g Fh(A)n(l)5 b(l)10 b Fu(-inference)25 |
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3933 b(rule\):)p Black Black 500 4739 a Fm(:)17 b(:)g(:)500 |
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3934 4896 y Fl(8)p Fk(\000)g Fm(X)24 b(S)798 4911 y Fj(1)854 |
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3935 4896 y Fm(S)914 4911 y Fj(2)970 4896 y Fm(T)1027 4911 |
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3936 y Fj(1)1083 4896 y Fm(T)1140 4911 y Fj(2)1196 4896 y |
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3937 Fm(c:)56 b(X)35 b Fk(#)28 b(\()p Fm(c;)17 b Fk(\000)p |
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3938 Fm(;)g(T)1832 4911 y Fj(1)1871 4896 y Fm(;)g(S)1975 4911 |
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3939 y Fj(1)2014 4896 y Fk(\))50 b Fl(^)g Fk(\000)28 b Fl(`)f |
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3940 Fm(T)2452 4911 y Fj(1)2520 4896 y Fm(<)p Fk(:)g Fm(S)2710 |
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3941 4911 y Fj(1)2777 4896 y Fl(^)1326 5042 y Fk(\()p Fl(8)p |
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3942 Fm(d:)h(P)i Fk(\000)17 b Fm(T)1753 5057 y Fj(1)1809 5042 |
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3943 y Fm(S)1869 5057 y Fj(1)1925 5042 y Fm(d)p Fk(\))49 b |
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3944 Fl(^)i Fk(\000)27 b Fl(`)h Fm(S)2417 5057 y Fj(2)2484 |
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3945 5042 y Fm(<)p Fk(:)g Fm(T)2672 5057 y Fj(2)2761 5042 |
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3946 y Fl(^)50 b Fk(\()p Fl(8)p Fm(d:)28 b(P)i Fk(\000)17 |
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3947 b Fm(S)3307 5057 y Fj(2)3363 5042 y Fm(T)3420 5057 y |
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3948 Fj(2)3476 5042 y Fm(d)p Fk(\))1326 5187 y Fl(\))55 b |
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3949 Fm(P)30 b Fk(\000)17 b(\()p Fh(A)n(l)5 b(l)35 b Fm(S)1910 |
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3950 5202 y Fj(1)1974 5187 y Fm(X)e(S)2148 5202 y Fj(2)2187 |
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3951 5187 y Fk(\))17 b(\()p Fh(A)n(l)5 b(l)35 b Fm(T)2497 |
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3952 5202 y Fj(1)2561 5187 y Fm(X)e(T)2732 5202 y Fj(2)2772 |
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3953 5187 y Fk(\))17 b Fm(c)p 500 5261 3065 4 v 1527 5347 |
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3954 a Fk(\000)28 b Fl(`)g Fm(S)33 b(<)p Fk(:)28 b Fm(T)41 |
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3955 b Fl(\))27 b Fm(P)j Fk(\000)17 b Fm(S)22 b(T)31 b(c)p |
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3956 Black 1872 5513 a Fu(10)p Black eop end |
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3980 y(strong)25 b(structural)g(induction)f(principle)i(that)f(comes)h(with) |
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3998 291 1008 a 107 x Fc(3.2)99 b(V)-11 b(er)o(sion)24 |
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3999 b(using)f(de)i(Bruijn)f(Indices)291 1311 y Fu(T)-8 b(w)o(o)33 |
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4001 (solv)o(e)f Fc(all)g Fu(theorem)h(pro)o(ving)291 1431 |
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4020 2611 1431 a Black 31 w Fu(The)h(solution)e(of)i(the)g |
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4021 (\002rst)291 1552 y(author)f(de\002nes)h(types)f(as:)p |
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4022 Black Black 640 1714 a Fn(datatype)56 b Fh(dbT)h Fk(=)44 |
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4023 b Fh(Tvar)38 b Fm(nat)56 b Fl(j)27 b Fh(T)-7 b(op)61 |
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4024 b Fl(j)27 b Fh(F)-7 b(un)34 b(dbT)41 b(dbT)68 b Fl(j)27 |
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4025 b Fh(A)n(l)5 b(l)38 b(dbT)i(dbT)291 1884 y Fu(with)24 |
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4026 b(the)g(lifting)g(operation)g(gi)n(v)o(en)f(by:)p Black |
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4027 Black 1104 2160 a Fl(")1154 2124 y Fg(n)1154 2186 y(k)1228 |
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4028 2160 y Fk(\()p Fh(Tvar)36 b Fm(i)p Fk(\))1695 2104 y |
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4029 Ff(def)1699 2160 y Fk(=)1813 1956 y Fe(8)1813 2046 y(<)1813 |
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4030 2225 y(:)1918 2093 y Fh(Tvar)f Fm(i)354 b Fu(if)25 b |
|
4031 Fm(i)j(<)f(k)1918 2250 y Fh(Tvar)35 b Fk(\()p Fm(i)22 |
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4032 b Fk(+)g Fm(n)p Fk(\))100 b Fu(otherwise)1104 2441 y |
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4033 Fl(")1154 2405 y Fg(n)1154 2467 y(k)1228 2441 y Fh(T)-7 |
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4034 b(op)1695 2385 y Ff(def)1699 2441 y Fk(=)38 b Fh(T)-7 |
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4035 b(op)1104 2622 y Fl(")1154 2585 y Fg(n)1154 2647 y(k)1228 |
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4036 2622 y Fk(\()p Fh(F)g(un)32 b Fm(S)f(T)14 b Fk(\))1695 |
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4037 2566 y Ff(def)1699 2622 y Fk(=)38 b Fh(F)-7 b(un)31 b |
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4038 Fk(\()p Fl(")2096 2585 y Fg(n)2096 2647 y(k)2171 2622 |
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4039 y Fm(S)6 b Fk(\)\()p Fl(")2363 2585 y Fg(n)2363 2647 |
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4040 y(k)2437 2622 y Fm(T)14 b Fk(\))1104 2802 y Fl(")1154 |
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4041 2766 y Fg(n)1154 2828 y(k)1228 2802 y Fk(\()p Fh(A)n(l)5 |
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4042 b(l)35 b Fm(S)c(T)14 b Fk(\))1695 2746 y Ff(def)1699 |
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4043 2802 y Fk(=)38 b Fh(A)n(l)5 b(l)35 b Fk(\()p Fl(")2061 |
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4044 2766 y Fg(n)2061 2828 y(k)2135 2802 y Fm(S)6 b Fk(\))25 |
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4045 b(\()p Fl(")2352 2766 y Fg(n)2352 2828 y(k)r Fj(+1)2512 |
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4046 2802 y Fm(T)14 b Fk(\))291 2984 y Fu(Note)27 b(that)h(the)g(lifting)e |
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4047 (operation)i(preserv)o(es)g(the)f(size)h(of)g(a)h Fh(dbT)13 |
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4048 b Fu(-type.)40 b(This)27 b(often)h(allo)n(ws)291 3104 |
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4049 y(one)20 b(to)h(establish)e(f)o(acts)i(in)l(v)n(olving)e(lifting)h |
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4050 (using)g(inductions)f(o)o(v)o(er)h(the)h(size,)g(if)g(an)g(induction) |
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4051 291 3225 y(o)o(v)o(er)j(the)g(structure)h(is)f(not)g(strong)g(enough.) |
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4052 437 3353 y(T)-8 b(yping)23 b(conte)o(xts)h(are)h(lists)e(of)i(types)f |
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4053 (and)h(the)f(predicate)h(for)g(v)n(alid)e(conte)o(xts)h(is)g(de\002ned) |
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4054 291 3474 y(lik)o(e)31 b(in)h(the)f(named)h(v)n(ariant,)h(e)o(xcept)e |
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4055 (that)h(we)g(do)g(not)f(need)h(freshness)g(constraints)e(when)291 |
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4056 3594 y(w)o(orking)c(with)g(de)g(Bruijn)h(indices.)36 |
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4057 b(One)27 b(w)o(ay)g(for)g(de\002ning)f(when)h(a)g(type)f(is)h |
|
4058 (well-formed)291 3714 y(is)d(by)g(using)g(the)h(function)p |
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4059 Black Black 976 3990 a Fh(fr)-5 b(e)g(es)33 b Fm(j)e |
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4060 Fk(\()p Fh(Tvar)k(i)10 b Fk(\))1737 3934 y Ff(def)1741 |
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4061 3990 y Fk(=)1855 3786 y Fe(8)1855 3876 y(<)1855 4055 |
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4062 y(:)1960 3923 y Fl(;)350 b Fu(if)25 b Fm(i)j(<)g(j)1960 |
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4063 4080 y Fl(f)p Fm(i)22 b Fl(\000)h Fm(j)6 b Fl(g)99 b |
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4064 Fu(otherwise)976 4271 y Fh(fr)-5 b(e)g(es)33 b Fm(j)e |
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4065 Fk(\()p Fh(T)-7 b(op)5 b Fk(\))1737 4215 y Ff(def)1741 |
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4066 4271 y Fk(=)38 b Fl(;)976 4452 y Fh(fr)-5 b(e)g(es)33 |
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4067 b Fm(j)e Fk(\()p Fh(F)-7 b(un)31 b Fm(S)g(T)14 b Fk(\))1737 |
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4068 4396 y Ff(def)1741 4452 y Fk(=)38 b(\()p Fh(fr)-5 b(e)g(es)32 |
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4069 b Fm(j)f(S)6 b Fk(\))22 b Fl([)h Fk(\()p Fh(fr)-5 b(e)g(es)32 |
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4070 b Fm(j)f(T)14 b Fk(\))976 4632 y Fh(fr)-5 b(e)g(es)33 |
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4071 b Fm(j)e Fk(\()p Fh(A)n(l)5 b(l)35 b Fm(S)30 b(T)14 b |
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4072 Fk(\))1737 4576 y Ff(def)1741 4632 y Fk(=)38 b(\()p Fh(fr)-5 |
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4073 b(e)g(es)32 b Fm(j)f(S)6 b Fk(\))22 b Fl([)h Fk(\()p |
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4074 Fh(fr)-5 b(e)g(es)32 b Fk(\()p Fm(j)c Fk(+)22 b(1\))j |
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4075 Fm(T)14 b Fk(\))291 4843 y Fu(and)34 b(then)g(de\002ne)g(the)g |
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4076 (well-formedness)g(judgement)f Fk(\000)45 b Fl(`)g Fm(T)j |
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4077 Fu(as)34 b(the)g(proposition)e Fk(\()p Fl(8)p Fm(i)46 |
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4078 b Fl(2)291 4964 y Fk(\()p Fh(fr)-5 b(e)g(es)41 b Fk(0)34 |
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4079 b Fm(T)14 b Fk(\))p Fm(:)44 b(i)g(<)g Fl(j)p Fk(\000)p |
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4080 Fl(j)p Fk(\))33 b Fu(where)i Fl(j)p Fk(\000)p Fl(j)e |
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4081 Fu(stands)g(for)h(the)f(length)g(of)h(the)f(list)g Fk(\000)p |
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4082 Fu(.)58 b(The)33 b(look-up)291 5084 y(function)28 b(for)i(typing)e |
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4083 (conte)o(xt)g(is)h(written)f Fk(\000\()p Fm(i)p Fk(\))i |
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4084 Fu(and)f(returns)g(the)g(type)g(on)g(the)g Fm(i)p Fu(th)g(place)h(in)p |
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4112 291 -166 a Black 1173 w |
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4113 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(the)25 |
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4114 b(list)g Fk(\000)p Fu(.)34 b(The)25 b(inducti)n(v)o(e)f(de\002nition)h |
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4117 Black Black 949 370 a Fh(valid)35 b Fk(\000)99 b(\000)28 |
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4119 Fm(S)33 b(<)p Fk(:)28 b Fh(T)-7 b(op)1629 414 y(T)g(op)2015 |
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4123 424 y Fh(Tvar)502 660 y Fk(\000\()p Fm(i)p Fk(\))28 b(=)g |
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4124 Fm(S)105 b Fk(\000)28 b Fl(`)f Fk(\()p Fl(")1234 619 |
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4125 y Fg(i)p Fj(+1)1234 684 y(0)1380 660 y Fm(S)6 b Fk(\))27 |
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4126 b Fm(<)p Fk(:)h Fm(T)p 502 705 1211 4 v 767 790 a Fk(\000)g |
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4127 Fl(`)f Fh(Tvar)36 b Fm(i)27 b(<)p Fk(:)h Fm(T)1754 739 |
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4128 y Fh(T)-7 b(r)i(ans)2208 670 y Fk(\000)27 b Fl(`)h Fm(T)2442 |
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4129 685 y Fj(1)2509 670 y Fm(<)p Fk(:)g Fm(S)2700 685 y Fj(1)2839 |
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4154 Fu(,)32 b Fk(\001)423 1816 y Fm(P)14 b Fu(,)24 b Fm(M)10 |
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4155 b Fu(,)25 b Fm(N)10 b Fu(:)541 1997 y Fl(\017)27 b Fk(\000)h |
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4156 Fl(`)f Fm(S)34 b(<)p Fk(:)27 b Fm(Q)f Fu(and)e Fk(\000)k |
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4157 Fl(`)g Fm(Q)g(<)p Fk(:)f Fm(T)39 b Fu(implies)23 b Fk(\000)28 |
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4158 b Fl(`)f Fm(S)34 b(<)p Fk(:)27 b Fm(T)14 b Fu(,)25 b(and)541 |
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4160 Fm(M)38 b(<)p Fk(:)28 b Fm(N)35 b Fu(and)25 b Fk(\000)i |
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4161 Fl(`)h Fm(P)41 b(<)p Fk(:)28 b Fm(Q)635 2334 y Fu(implies)c |
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4162 Fk(\(\001@)p Fm(P)14 b Fk(@\000\))26 b Fl(`)i Fm(M)38 |
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4176 b Fl(\))27 b Fk(\000)h Fl(`)f Fm(S)34 b(<)p Fk(:)27 b |
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4177 Fm(T)291 3313 y Fu(rather)e(the)f(strengthened)g(property)p |
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4178 Black Black 702 3507 a Fl(8)p Fm(Q)834 3471 y Fd(0)874 |
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4180 b Fm(Q)p Fk(\))c(=)f(\()p Fh(size)32 b Fm(Q)1960 3471 |
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4181 y Fd(0)1984 3507 y Fk(\))49 b Fl(^)i Fk(\000)27 b Fl(`)h |
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4193 (are)g(as)g(follo)n(ws:)291 4313 y(the)i(statement)g(is)p |
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4198 b Fk(@\000)27 b Fl(`)g Fm(M)39 b(<)p Fk(:)28 b Fm(N)291 |
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4301 b(e)26 b(ha)n(v)o(e)g(studied)g(formalisations)e(based)j(on)f(de)g |
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4302 (Bruijn)g(indices)g(and)g(on)g(names)g(from)h(the)291 |
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4309 y(POPLmark-Challenge,)26 b(it)f(seems)g(that)g(all)g(problems)f |
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4315 (numbers\))g(has)g(been)g(part)g(of)h(theorem)e(pro)o(v)o(ers,)h(for)h |
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4316 (e)o(xample)e(Coq)291 2312 y(and)29 b(Isabelle/HOL,)f(for)i(a)g(long)e |
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4317 (time.)43 b(In)30 b(contrast,)g(the)f(nominal)f(datatype)h(package)g |
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4318 (has)291 2433 y(been)e(implemented)e(in)h(Isabelle/HOL,)g(only)-6 |
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4329 478 2553 a Black Fu(],)i(there)f(is)g(little)e |
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4330 (w)o(ork)i(about)g(replicating)f(our)h(results)f(in)g(non-HOL-based)h |
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4331 (theorem)291 2673 y(pro)o(v)o(ers.)437 2802 y(Another)i(adv)n(antage)f |
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4332 (of)h(de)h(Bruijn)f(indices)f(is)h(that)f(the)o(y)h(do)g(not)f |
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4336 (places)h(classical)f(reasoning)g(principles.)29 b(It)23 |
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4337 b(is)f(currently)g(un-)291 3163 y(kno)n(wn)28 b(whether)h(a)g |
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4338 (constructi)n(v)o(e)f(v)n(ariant)g(of)h(the)g(nominal)f(datatype)h |
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4339 (package)h(that)e(of)n(fers)291 3284 y(the)f(same)h(con)l(v)o(enience)g |
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4340 (is)f(attainable.)40 b(Connected)28 b(with)f(the)h(aspect)g(of)g |
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4341 (constructi)n(vity)e(is)291 3404 y(the)31 b(infrastructure)g(to)f(e)o |
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4342 (xtract)h(programs)g(from)g(proof,)i(which)d(e)o(xists)g(in)h(Isabelle) |
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4345 (nominal)291 3645 y(datatype)f(package.)437 3773 y(The)33 |
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4346 b(biggest)f(disadv)n(antage)g(we)i(see)f(with)f(using)h(the)g(nominal)e |
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4347 (datatype)i(package)h(is)291 3894 y(the)g(amount)g(of)h(infrastructure) |
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4348 g(that)f(needs)h(to)f(be)i(implemented.)59 b(So)35 b(f)o(ar)l(,)j(this) |
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4349 c(package)291 4014 y(supports)22 b(only)h(single)g(binders)g |
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4350 (\(although)g(iteration)f(is)i(possible)e(and)i(the)o(y)f(can)h(occur)g |
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4351 (an)o(y-)291 4134 y(where)34 b(in)g(a)g(term-constructor\).)58 |
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4638 291 2087 a Black 131 x Fw([14])p Black 26 |
|
4639 w(Urban,)d(C.)e(and)i(M.)e(Norrish,)i Fy(A)e(Formal)i(Tr)m(eatment)g |
|
4640 (of)f(the)h(Bar)m(endr)m(e)l(gt)i(Variable)f(Con)l(vention)467 |
|
4641 2330 y(in)27 b(Rule)g(Inductions)p Fw(,)i(in:)e Fy(Pr)l(oc.)g(of)g(the) |
|
4642 g(3r)m(d)g(International)k(A)m(CM)25 b(W)-8 b(orkshop)28 |
|
4643 b(on)f(Mec)o(hanized)467 2443 y(Reasoning)h(about)g(Langua)o(g)o(es)h |
|
4644 (with)d(V)-10 b(ariable)28 b(Binding)g(and)f(Names)f(\(MERLIN\))p |
|
4645 Fw(,)e(2005,)k(pp.)467 2556 y(25\22632.)291 2594 y |
|
4646 SDict begin H.S end |
|
4647 291 |
|
4648 2594 a 291 2594 a |
|
4649 SDict begin 13.6 H.A end |
|
4650 291 2594 a 291 2594 a |
|
4651 SDict begin [ /View [/XYZ H.V] /Dest (cite.UrbanTasson05) cvn H.B /DEST |
|
4652 pdfmark end |
|
4653 291 2594 a Black |
|
4654 149 x Fw([15])p Black 26 w(Urban,)f(C.)f(and)h(C.)f(T)-7 |
|
4655 b(asson,)27 b Fy(Nominal)h(Tec)o(hniques)h(in)e(Isabelle/HOL)p |
|
4656 Fw(,)g(in:)h Fy(Pr)l(oc.)e(of)h(the)h(20th)467 2856 y(International)36 |
|
4657 b(Confer)m(ence)d(on)f(A)n(utomated)h(Deduction)h(\(CADE\))p |
|
4658 Fw(,)29 b(LNCS)h Fx(3632)p Fw(,)i(2005,)h(pp.)467 2968 |
|
4659 y(38\22653.)291 3006 y |
|
4660 SDict begin H.S end |
|
4661 291 3006 a 291 3006 a |
|
4662 SDict begin 13.6 H.A end |
|
4663 291 3006 |
|
4664 a 291 3006 a |
|
4665 SDict begin [ /View [/XYZ H.V] /Dest (cite.Wenzel99) cvn H.B /DEST |
|
4666 pdfmark end |
|
4667 291 3006 a Black 149 x Fw([16])p Black 26 |
|
4668 w(W)-7 b(enzel,)35 b(M.,)f Fy(Isar)i(\227)e(A)f(Generic)j(Interpr)m |
|
4669 (etative)j(Appr)l(oac)o(h)e(to)e(Readable)h(Formal)f(Pr)l(oof)467 |
|
4670 3268 y(Documents)p Fw(,)h(in:)g Fy(Pr)l(oc.)f(of)g(the)h(12th)g |
|
4671 (International)j(Confer)m(ence)e(on)f(Theor)m(em)f(Pr)l(o)o(ving)i(in) |
|
4672 467 3381 y(Higher)24 b(Or)m(der)g(Lo)o(gics)g(\(TPHOLs\))p |
|
4673 Fw(,)d(number)k(1690)f(in)g(LNCS,)d(1999,)j(pp.)f(167\226184.)291 |
|
4674 3436 y |
|
4675 SDict begin H.S end |
|
4676 291 3436 a 291 3436 a |
|
4677 SDict begin 13.6 H.A end |
|
4678 291 3436 a 291 3436 a |
|
4679 SDict begin [ /View [/XYZ H.V] /Dest (cite.Wenzel06) cvn H.B /DEST |
|
4680 pdfmark end |
|
4681 291 |
|
4682 3436 a Black 131 x Fw([17])p Black 26 w(W)-7 b(enzel,)54 |
|
4683 b(M.,)e Fy(Structur)m(ed)57 b(Induction)f(Pr)l(oofs)e(in)g |
|
4684 (Isabelle/Isar)p Fw(,)j(in:)d Fy(Pr)l(oc.)f(of)h(the)g(5th)467 |
|
4685 3680 y(International)38 b(Confer)m(ence)d(on)f(Mathematical)i(Knowledg) |
|
4686 o(e)f(Mana)o(g)o(ement)g(\(MKM\))p Fw(,)d(LN)m(AI)467 |
|
4687 3793 y Fx(4108)p Fw(,)24 b(2006.)p Black 1872 5513 a |
|
4688 Fu(15)p Black eop end |
|
4689 %%Trailer |
|
4690 |
|
4691 userdict /end-hook known{end-hook}if |
|
4692 %%EOF |