| 14 |      1 | %!PS-Adobe-2.0
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|  |      2 | %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software
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|  |      3 | %%Title: main.dvi
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|  |      4 | %%Pages: 15
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|  |      5 | %%PageOrder: Ascend
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|  |     19 | 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
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|  |     21 | mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
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|  |     22 | matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
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|  |     37 | /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
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|  |     38 | }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
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|  |     39 | bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
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|  |     45 | /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
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|  |     46 | /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
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|  |     54 | delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
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|  |     56 | p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
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|  |     59 | %%EndProcSet
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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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|  |    205 | %%EndProcSet
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|  |    209 | index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll
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|  |    210 | exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0
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|  |    211 | ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{
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|  |    212 | pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get
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|  |    213 | div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type
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|  |    214 | /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end
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|  |    215 | definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup
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|  |    216 | sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll
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|  |    217 | mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[
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|  |    218 | exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if}
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|  |    219 | forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def
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|  |    224 | %!
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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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|  |    229 | /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
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|  |    230 | X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
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|  |    231 | /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
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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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|  |    237 | itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
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|  |    238 | transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
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|  |    239 | curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
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|  |    240 | pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
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|  |    241 | if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
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|  |    242 | -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
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|  |    244 | yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
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|  |    245 | neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
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|  |    248 | neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
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|  |    250 | 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
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|  |    256 | currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
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|  |    257 | /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
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|  |    258 | psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
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|  |    259 | psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
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|  |    260 | TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def
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|  |    261 | @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll
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|  |    262 | newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto
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|  |    263 | closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N
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|  |    264 | /@beginspecial{SDict begin/SpecialSave save N gsave normalscale
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|  |    265 | currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}
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|  |    266 | N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs
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|  |    267 | neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate
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|  |    268 | rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse
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|  |    269 | scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg
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|  |    270 | lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx
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|  |    271 | ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N
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|  |    272 | /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{
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|  |    273 | pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave
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|  |    274 | restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B
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|  |    275 | /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1
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|  |    276 | setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY
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|  |    277 | moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix
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|  |    278 | matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc
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|  |    279 | savematrix setmatrix}N end
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|  |    281 | %%EndProcSet
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|  |    283 | %!
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|  |    285 | setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll
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|  |    286 | }repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def
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|  |    287 | /TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{
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|  |    288 | setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{
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|  |    289 | /currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch
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|  |    290 | known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC
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|  |   2331 | -166 a
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|  |   2332 | SDict begin H.R end
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|  |   2333 |  291 -166 a 291 -166 a
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|  |   2334 | SDict begin [ /View [/XYZ H.V] /Dest (page.2) cvn H.B /DEST pdfmark
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|  |   2335 | end
 | 
|  |   2336 |  291 -166 a Black 1173 w
 | 
|  |   2337 | Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(on)e
 | 
|  |   2338 | (\223paper\224)i(seems)f(to)f(be)i(quite)e(rob)n(ust,)h(in)g
 | 
|  |   2339 | (particular)g(issues)f(arising)g(from)h(binders)f(and)291
 | 
|  |   2340 | 203 y(alpha-equi)n(v)n(alence)22 b(seem)i(to)f(cause)h(little)f
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|  |   2341 | (problems)f(and)i(introduce)f(almost)g(no)g(o)o(v)o(erhead.)291
 | 
|  |   2342 | 324 y(\(The)33 b(point)f(of)h(formalising)f(lambda-calculi)g(is)h(to)f
 | 
|  |   2343 | (achie)n(v)o(e)h(100\045)f(correctness,)k(to)c(pro-)291
 | 
|  |   2344 | 444 y(vide)c(easy)i(maintenance)e(of)h(proofs)g(and)g(to)g(allo)n(w)f
 | 
|  |   2345 | (for)h(proofs)g(about)g(languages)f(where)i(a)291 565
 | 
|  |   2346 | y(human)c(reasoner)h(is)g(o)o(v)o(erwhelmed)e(by)i(the)g(sheer)g
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|  |   2347 | (number)g(of)g(cases)g(and)g(subtleties)f(to)g(be)291
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|  |   2349 | SDict begin H.S end
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|  |   2350 |  780
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|  |   2351 | 686 a 0 0 1 TeXcolorrgb -1 x Fu(3)p 0 0 1 TeXcolorrgb
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|  |   2352 | 830 617 a
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|  |   2353 | SDict begin H.R end
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|  |   2354 |  830 617 a 830 685 a
 | 
|  |   2355 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
 | 
|  |   2356 | (cite.challenge05) cvn H.B /ANN pdfmark end
 | 
|  |   2357 |  830 685 a Black Fu(].\))437
 | 
|  |   2358 | 814 y(When)31 b(engineering)g(a)g(formal)g(proof)g(in)g(a)g(theorem)g
 | 
|  |   2359 | (pro)o(v)o(er)l(,)h(blindly)e(applying)g(auto-)291 934
 | 
|  |   2360 | y(matic)23 b(proof)h(tools)e(often)i(leads)g(to)f(a)i(dead)f(end.)30
 | 
|  |   2361 | b(Usually)23 b(more)h(successful)f(is)h(the)f(strate)o(gy)291
 | 
|  |   2362 | 1054 y(to)35 b(start)g(with)g(a)i(rough)e(sk)o(etch)g(containing)g(a)h
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|  |   2363 | (proof)g(idea,)i(and)e(then)f(to)h(try)f(to)h(translate)291
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|  |   2364 | 1175 y(this)c(idea)i(into)e(actual)i(proof)f(steps)g(in)g(the)g
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|  |   2365 | (theorem)g(pro)o(v)o(er)-5 b(.)56 b(This)33 b(style)f(of)i(formalising)
 | 
|  |   2366 | 291 1295 y(proofs)d(is)f(v)o(ery)h(much)g(encouraged)h(by)f(the)g(Isar)
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|  |   2368 | 1296 a
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|  |   2369 | SDict begin H.S end
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|  |   2373 |  3069 1227 a 3069 1295 a
 | 
|  |   2374 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
 | 
|  |   2375 | (cite.Wenzel99) cvn H.B /ANN pdfmark end
 | 
|  |   2376 |  3069
 | 
|  |   2377 | 1295 a Black Fu(].)51 b(In)31 b(case)h(of)291 1416 y(the)24
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|  |   2378 | b(substitution)e(lemma)i(in)h(the)f(lambda-calculus)p
 | 
|  |   2379 | Black Black 541 1595 a Fn(Substitution)i(Lemma:)53 b
 | 
|  |   2380 | Fu(If)26 b Fm(x)i Fl(6\021)g Fm(y)g Fu(and)d Fm(x)j Fl(62)g
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|  |   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
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|  |   2387 | (gi)n(v)o(en)g(by)g(Barendre)o(gt)h([)p 0 0 1 TeXcolorrgb
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|  |   2388 | 3146 2043 a
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|  |   2389 | SDict begin H.S end
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|  |   2391 | TeXcolorrgb 3196 1975 a
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|  |   2392 | SDict begin H.R end
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|  |   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
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|  |   2439 | (the)g(datatype)291 4712 y
 | 
|  |   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
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|  |   2443 |  291
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|  |   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
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|  |   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
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|  |   2451 | 291 5226 a(where)j(the)g(type)f Fm(name)i Fu(can,)h(for)f(e)o(xample,)f
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|  |   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
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|  |   2456 | eop end
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|  |   2464 | 291 -166 a
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|  |   2467 | -166 a
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|  |   2468 | SDict begin H.R end
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|  |   2470 | SDict begin [ /View [/XYZ H.V] /Dest (page.3) cvn H.B /DEST pdfmark
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|  |   2471 | end
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|  |   2472 |  291 -166 a Black 1173 w
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|  |   2473 | Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(modulo)23
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|  |   2474 | b(an)i(e)o(xplicit)e(notion)h(of)h(alpha-equi)n(v)n(alence,)e(that)i
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|  |   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
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|  |   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
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|  |   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
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|  |   2488 | 396 a
 | 
|  |   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
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|  |   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
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|  |   2497 | SDict begin [ /View [/XYZ H.V] /Dest (equation.2) cvn H.B /DEST pdfmark
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|  |   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
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|  |   2501 | Fm(x)27 b Fk(:=)h Fm(N)10 b Fk(])1773 606 y Ff(def)1778
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|  |   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
 | 
|  |   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(\()p Fm(M)2229 958 y Fj(1)2269 943 y Fk([)p
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|  |   2509 | Fm(x)28 b Fk(:=)g Fm(N)10 b Fk(]\))28 b(\()p Fm(M)2823
 | 
|  |   2510 | 958 y Fj(2)2863 943 y Fk([)p Fm(x)g Fk(:=)g Fm(N)10 b
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|  |   2511 | Fk(]\))870 1123 y(\()p Fh(L)-5 b(am)35 b Fm(x)28 b(M)1306
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|  |   2512 | 1138 y Fj(1)1346 1123 y Fk(\)[)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
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|  |   2514 | b(am)35 b Fm(x)28 b(M)2289 1138 y Fj(1)874 1304 y Fk(\()p
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|  |   2515 | Fh(L)-5 b(am)35 b Fm(y)c(M)1306 1319 y Fj(1)1346 1304
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|  |   2516 | y Fk(\)[)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
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|  |   2545 | 2364 y Fm(N)1860 2323 y Fd(0)1909 2364 y Fu(then)25 b
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|  |   2546 | Fm(M)10 b Fk([)p Fm(x)29 b Fk(:=)e Fm(N)10 b Fk(])29
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|  |   2547 | b Fl(\031)2672 2379 y Fg(\013)2749 2364 y Fm(M)2853 2323
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|  |   2548 | y Fd(0)2877 2364 y Fk([)p Fm(x)f Fk(:=)g Fm(N)3206 2323
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|  |   2549 | y Fd(0)3230 2364 y Fk(])291 2525 y Fu(in)35 b(order)i(to)e(manually)g
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|  |   2550 | (massage)h(the)g(lambda-terms)f(to)h(a)g(suitable)f(form.)64
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|  |   2560 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2561 | (cite.CurryFeys58) cvn H.B /ANN pdfmark end
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|  |   2562 |  2779 2645 a Black Fu(,)h(P)o(ages)g(94\226104].)29
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|  |   2563 | b(As)291 2766 y(can)e(be)h(easily)f(imagined,)g(implementing)e(these)i
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|  |   2575 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2576 | (cite.Leroy06) cvn H.B /ANN pdfmark end
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|  |   2577 |  2321 3007 a Black
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|  |   2578 | Fu(,)j(P)o(age)d(53].)54 b(One)32 b(reason)h(for)g(the)291
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|  |   2579 | 3127 y(dif)n(\002culties)g(is)i(the)g(f)o(act)h(that)e(Curry')-5
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|  |   2589 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2590 | (cite.Pitts03) cvn H.B /ANN pdfmark end
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|  |   2591 |  2150 3247 a Black Fu(].)437
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|  |   2592 | 3376 y(The)33 b(main)e(point)h(of)h(de)f(Bruijn)h(indices)f(and)g
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|  |   2604 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2605 | (section.0.2) cvn H.B /ANN pdfmark end
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|  |   2606 |  3056
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|  |   2607 | 3617 a Black 29 w Fu(we)h(illustrate)291 3737 y(this)g(in)g(the)h
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|  |   2617 | (section.0.3) cvn H.B /ANN pdfmark end
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|  |   2618 |  2527 3737 a Black 30 w Fu(contains)29 b(a)h(brief)h(sk)o
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|  |   2619 | (etch)e(of)291 3858 y(the)k(formalisations)f(for)i(the)f(narro)n(wing)g
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|  |   2629 | (cite.challenge05) cvn H.B /ANN pdfmark end
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|  |   2630 | 
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|  |   2631 | 1284 3978 a Black Fu(].)31 b(Section)p 0 0 1 TeXcolorrgb
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|  |   2638 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2639 | (section.0.4) cvn H.B /ANN pdfmark end
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|  |   2640 |  1747
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|  |   2641 | 3978 a Black 24 w Fu(dra)o(ws)25 b(some)f(conclusions.)291
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|  |   2646 |  291 4133 a 291 4133 a
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|  |   2649 |  291
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|  |   2656 | a
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|  |   2659 |  291 4361 a 126 x Fc(2.1)99 b(V)-11 b(er)o(sion)24 b(using)f(de)i
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|  |   2662 | y
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|  |   2675 |  2437 4674 a 2437 4674 a
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|  |   2678 |  2437
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|  |   2679 | 4674 a Black 30 w Fu(since)g(the)o(y)f(are)i(a)g(v)o(ery)f(useful)291
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|  |   2680 | 4795 y(implementation)c(technique,)i(b)n(ut)g(are)i(often)e(dismissed)f
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|  |   2681 | (as)h(being)g(un\002t)h(for)f(consumption)291 4915 y(of)h(a)h(human)f
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|  |   2697 |  374 5347 a Black Fs(personal)19 b(communication)e(with)j
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|  |   2698 | (N.)h(G.)f(de)g(Bruijn)p Black Black 1897 5513 a Fu(3)p
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|  |   2715 |  291 -166 a Black 1173 w
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|  |   2716 | Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(theorem)25
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|  |   2717 | b(pro)o(ving)g(e)o(xperts.)35 b(In)26 b(this)f(section,)h(for)h(the)f
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|  |   2718 | (bene\002t)h(of)f(casual)g(users)g(of)h(theorem)291 203
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|  |   2719 | y(pro)o(v)o(ers,)41 b(we)e(w)o(ant)g(to)f(study)g(in)h(minutiae)e
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|  |   2720 | (detail)i(a)g(formalisation)e(of)i(the)g(substitution)291
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|  |   2721 | 324 y(lemma)24 b(using)f(this)h(formalisation)g(technique.)437
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|  |   2722 | 452 y(W)-8 b(e)30 b(assume)g(the)g(reader)h(is)e(f)o(amiliar)h(with)f
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|  |   2724 | 573 y(using)23 b(for)i(e)o(xample)f(the)h(datatype:)p
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|  |   2725 | Black Black 933 737 a Fn(datatype)56 b Fh(dB)65 b Fk(=)56
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|  |   2726 | b Fh(V)-7 b(ar)38 b Fm(nat)56 b Fl(j)f Fh(App)34 b(dB)j(dB)65
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|  |   2727 | b Fl(j)55 b Fh(L)-5 b(am)35 b(dB)291 910 y Fu(One)f(central)h(notion)e
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|  |   2730 | 994 y Fg(n)657 1057 y(k)739 1031 y Fu(where)i Fm(n)f
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|  |   2731 | Fu(is)g(an)h(of)n(fset)f(by)g(which)g(the)g(indices)g(greater)h(or)g
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|  |   2732 | (equal)f(than)g Fm(k)k Fu(are)291 1151 y(incremented;)e
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|  |   2733 | Fm(k)h Fu(is)c(the)g(upper)g(bound)g(of)g(indices)g(that)f(are)i(re)o
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|  |   2734 | (garded)f(as)h Fc(locally)e(bound)p Fu(.)291 1271 y(This)24
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|  |   2735 | b(operation)g(can)h(be)g(de\002ned)g(as:)p Black Black
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|  |   2736 | 1057 1550 a Fl(")1107 1514 y Fg(n)1107 1576 y(k)1181
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|  |   2737 | 1550 y Fk(\()p Fh(V)-7 b(ar)36 b Fm(i)p Fk(\))1791 1494
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|  |   2738 | y Ff(def)1796 1550 y Fk(=)1909 1345 y Fe(8)1909 1435
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|  |   2739 | y(<)1909 1614 y(:)2015 1482 y Fh(V)-7 b(ar)35 b Fm(i)354
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|  |   2740 | b Fu(if)25 b Fm(i)j(<)g(k)2015 1639 y Fh(V)-7 b(ar)35
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|  |   2741 | b Fk(\()p Fm(i)22 b Fk(+)g Fm(n)p Fk(\))100 b Fu(otherwise)1057
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|  |   2742 | 1830 y Fl(")1107 1794 y Fg(n)1107 1856 y(k)1181 1830
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|  |   2743 | y Fk(\()p Fh(App)34 b Fm(M)1519 1845 y Fj(1)1586 1830
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|  |   2744 | y Fm(M)1680 1845 y Fj(2)1720 1830 y Fk(\))1791 1775 y
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|  |   2745 | Ff(def)1796 1830 y Fk(=)j Fh(App)d Fk(\()p Fl(")2203
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|  |   2746 | 1794 y Fg(n)2203 1856 y(k)2278 1830 y Fm(M)2372 1845
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|  |   2747 | y Fj(1)2411 1830 y Fk(\))28 b(\()p Fl(")2565 1794 y Fg(n)2565
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|  |   2748 | 1856 y(k)2639 1830 y Fm(M)2733 1845 y Fj(2)2773 1830
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|  |   2749 | y Fk(\))1057 2011 y Fl(")1107 1975 y Fg(n)1107 2037 y(k)1181
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|  |   2750 | 2011 y Fk(\()p Fh(L)-5 b(am)35 b Fm(M)1534 2026 y Fj(1)1574
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|  |   2751 | 2011 y Fk(\))1791 1955 y Ff(def)1796 2011 y Fk(=)i Fh(L)-5
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|  |   2752 | b(am)35 b Fk(\()p Fl(")2218 1975 y Fg(n)2218 2037 y(k)r
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|  |   2753 | Fj(+1)2378 2011 y Fm(M)2472 2026 y Fj(1)2512 2011 y Fk(\))291
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|  |   2755 | b Fu(for)28 b(a)g(v)n(ariable)g(with)f(inde)o(x)g Fm(k)s
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|  |   2756 | Fu(,)h(written)f(as)h Fk([)p Fm(k)37 b Fk(:=)c Fm(N)10
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|  |   2757 | b Fk(])p Fu(,)29 b(can)291 2321 y(then)24 b(be)h(de\002ned)g(as)g
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|  |   2758 | (follo)n(ws:)p Black Black 1085 2647 a Fk(\()p Fh(V)-7
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|  |   2759 | b(ar)36 b Fm(i)p Fk(\)[)p Fm(k)31 b Fk(:=)c Fm(N)10 b
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|  |   2760 | Fk(])1774 2591 y Ff(def)1779 2647 y Fk(=)1892 2383 y
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|  |   2761 | Fe(8)1892 2473 y(>)1892 2503 y(>)1892 2533 y(<)1892 2712
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|  |   2762 | y(>)1892 2742 y(>)1892 2772 y(:)1997 2501 y Fh(V)-7 b(ar)36
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|  |   2763 | b Fm(i)346 b Fu(if)25 b Fm(i)j(<)f(k)1997 2658 y Fl(")2047
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|  |   2764 | 2622 y Fg(k)2047 2683 y Fj(0)2117 2658 y Fm(N)373 b Fu(if)25
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|  |   2765 | b Fm(i)j Fk(=)f Fm(k)1997 2815 y Fh(V)-7 b(ar)36 b Fk(\()p
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|  |   2766 | Fm(i)22 b Fl(\000)h Fk(1\))99 b Fu(if)25 b Fm(i)j(>)f(k)809
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|  |   2767 | 2996 y Fk(\()p Fh(App)34 b Fm(M)1147 3011 y Fj(1)1214
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|  |   2768 | 2996 y Fm(M)1308 3011 y Fj(2)1348 2996 y Fk(\)[)p Fm(k)d
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|  |   2769 | Fk(:=)c Fm(N)10 b Fk(])1774 2940 y Ff(def)1779 2996 y
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|  |   2770 | Fk(=)37 b Fh(App)d Fk(\()p Fm(M)2230 3011 y Fj(1)2270
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|  |   2771 | 2996 y Fk([)p Fm(k)c Fk(:=)e Fm(N)10 b Fk(]\))28 b(\()p
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|  |   2772 | Fm(M)2822 3011 y Fj(2)2862 2996 y Fk([)p Fm(k)j Fk(:=)d
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|  |   2773 | Fm(N)10 b Fk(]\))984 3176 y(\()p Fh(L)-5 b(am)35 b Fm(M)10
 | 
|  |   2774 | b Fk(\)[)p Fm(k)32 b Fk(:=)27 b Fm(N)10 b Fk(])1774 3121
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|  |   2775 | y Ff(def)1779 3176 y Fk(=)37 b Fh(L)-5 b(am)35 b Fk(\()p
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|  |   2776 | Fm(M)10 b Fk([)p Fm(k)26 b Fk(+)c(1)27 b(:=)h Fm(N)10
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|  |   2777 | b Fk(]\))291 3361 y Fu(Since)32 b(the)g(type)g Fh(dB)42
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|  |   2778 | b Fu(is)32 b(a)h(completely)e(standard)h(datatype,)h(both)f
 | 
|  |   2779 | (de\002nitions)f(can)h(be)h(im-)291 3481 y(plemented)22
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|  |   2780 | b(by)h(primiti)n(v)o(e)e(recursion.)30 b(The)24 b(substitution)d(lemma)
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|  |   2781 | h(then)i(tak)o(es)f(the)g(follo)n(wing)291 3601 y(form:)p
 | 
|  |   2782 | Black Black 541 3780 a Fn(Substitution)35 b(Lemma)f(with)g(de)h(Bruijn)
 | 
|  |   2783 | g(Indices:)79 b Fu(F)o(or)33 b(all)h(indices)f Fm(i)p
 | 
|  |   2784 | Fu(,)j Fm(j)6 b Fu(,)541 3901 y(with)24 b Fm(i)k Fl(\024)g
 | 
|  |   2785 | Fm(j)j Fu(we)25 b(ha)n(v)o(e)f(that)777 4081 y Fm(M)10
 | 
|  |   2786 | b Fk([)p Fm(i)29 b Fk(:=)e Fm(N)10 b Fk(][)p Fm(j)35
 | 
|  |   2787 | b Fk(:=)27 b Fm(L)p Fk(])i(=)e Fm(M)10 b Fk([)p Fm(j)29
 | 
|  |   2788 | b Fk(+)22 b(1)28 b(:=)f Fm(L)p Fk(][)p Fm(i)i Fk(:=)e
 | 
|  |   2789 | Fm(N)10 b Fk([)p Fm(j)29 b Fl(\000)23 b Fm(i)28 b Fk(:=)f
 | 
|  |   2790 | Fm(L)p Fk(]])i Fu(.)291 4255 y(Note)24 b(that)h(one)f(pro)o(v)o(es)g
 | 
|  |   2791 | (an)h(equation,)f(rather)h(than)f(an)h(alpha-equi)n(v)n(alence.)30
 | 
|  |   2792 | b(Because)c(equa-)291 4375 y(tional)i(reasoning)g(is)h(usually)f(much)g
 | 
|  |   2793 | (better)h(supported)f(by)h(theorem)f(pro)o(v)o(ers)g(or)h(is)g(e)n(v)o
 | 
|  |   2794 | (en)f(a)291 4496 y(basic)19 b(notion)g(in)h(their)g(logics,)g(the)g(de)
 | 
|  |   2795 | g(Bruijn)g(indices)f(v)o(ersion)g(a)n(v)n(oids)g(the)h(manual)g
 | 
|  |   2796 | (massag-)291 4616 y(ing)25 b(of)g(terms)h(with)e(respect)i(to)g
 | 
|  |   2797 | (alpha-equiv)n(alence)f(needed)h(in)f(the)g(v)o(ersion)g(with)g
 | 
|  |   2798 | (concrete)291 4736 y(names.)50 b(This)30 b(f)o(act)i(alone)f(already)h
 | 
|  |   2799 | (relie)n(v)o(es)e(one)i(of)f(much)g(w)o(ork)g(when)g(formalising)f
 | 
|  |   2800 | (this)291 4857 y(lemma.)48 b(Notice)30 b(also)g(that)h(the)f(condition)
 | 
|  |   2801 | g Fm(i)39 b Fl(\024)g Fm(j)e Fu(is)30 b(necessary)-6
 | 
|  |   2802 | b(,)32 b(otherwise)e(the)h(equation)291 4977 y(does)24
 | 
|  |   2803 | b(not)g(hold)g(in)h(general.)437 5106 y(Lik)o(e)34 b(the)h(informal)f
 | 
|  |   2804 | (proof)g(by)h(Barendre)o(gt,)i(the)e(formalised)f(proof)g(proceeds)h
 | 
|  |   2805 | (by)g(in-)291 5226 y(duction)i(on)g(the)h(structure)g(of)g
 | 
|  |   2806 | Fm(M)10 b Fu(.)71 b(Unlik)o(e)38 b(the)g(informal)f(proof,)k(ho)n(we)n
 | 
|  |   2807 | (v)o(er)l(,)f(the)e(induc-)291 5347 y(tion)27 b(hypothesis)e(needs)j
 | 
|  |   2808 | (to)f(be)h(strengthened)f(to)h(quantify)f(o)o(v)o(er)g
 | 
|  |   2809 | Fc(all)g Fu(indices)g Fm(i)h Fu(and)g Fm(j)6 b Fu(.)40
 | 
|  |   2810 | b(This)p Black 1897 5513 a(4)p Black eop end
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|  |   2824 | SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark
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|  |   2826 |  291 -166 a Black 1173 w
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|  |   2827 | Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(strengthening)h
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|  |   2828 | (is)i(necessary)g(in)g(the)g(de)h(Bruijn)e(v)o(ersion)g(in)h(order)h
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|  |   2829 | (to)f(get)g(the)g Fh(L)-5 b(am)7 b Fu(-case)291 203 y(through.)55
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|  |   2830 | b(W)l(ith)33 b(this)f(strengthening)g(the)h Fh(L)-5 b(am)41
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|  |   2831 | b Fu(and)33 b Fh(App)40 b Fu(case)34 b(are)g(completely)e(routine.)291
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|  |   2832 | 324 y(The)h Fc(non)p Fu(-routine)g(case)i(in)e(the)h(de)g(Bruijn)g(v)o
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|  |   2833 | (ersion)e(is)i(the)f Fh(V)-7 b(ar)11 b Fu(-case)35 b(where)f(we)g(ha)n
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|  |   2834 | (v)o(e)g(to)291 444 y(sho)n(w)23 b(that)698 444 y
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|  |   2835 | SDict begin H.S end
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|  |   2836 |  698
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|  |   2837 | 444 a 698 444 a
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|  |   2838 | SDict begin 14.5 H.A end
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|  |   2839 |  698 444 a 698 444 a
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|  |   2842 |  698 444 a Black Black
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|  |   2843 | 657 605 a Fk(\()p Fh(V)-7 b(ar)36 b Fm(n)p Fk(\)[)p Fm(i)28
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|  |   2844 | b Fk(:=)f Fm(N)10 b Fk(][)p Fm(j)35 b Fk(:=)27 b Fm(L)p
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|  |   2845 | Fk(])i(=)e(\()p Fh(V)-7 b(ar)36 b Fm(n)p Fk(\)[)p Fm(j)28
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|  |   2846 | b Fk(+)22 b(1)27 b(:=)h Fm(L)p Fk(][)p Fm(i)h Fk(:=)e
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|  |   2847 | Fm(N)10 b Fk([)p Fm(j)29 b Fl(\000)23 b Fm(i)k Fk(:=)h
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|  |   2848 | Fm(L)p Fk(]])p Black -3115 w Fu(\(3\))p Black 291 775
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|  |   2849 | a(holds)h(for)h(an)g(arbitrary)h Fm(n)p Fu(.)47 b(Lik)o(e)30
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|  |   2850 | b(in)f(the)h(informal)g(proof,)h(we)f(need)h(to)e(distinguish)f(cases)
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|  |   2851 | 291 895 y(so)d(that)h(we)g(can)g(apply)g(the)g(de\002nition)f(of)h
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|  |   2852 | (substitution.)31 b(There)c(are)f(se)n(v)o(eral)g(w)o(ays)f(to)h(order)
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|  |   2853 | 291 1016 y(the)f(cases;)h(belo)n(w)e(we)i(ha)n(v)o(e)f(gi)n(v)o(en)f
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|  |   2854 | (the)h(cases)h(as)f(the)o(y)g(are)h(suggested)e(by)i(the)f
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|  |   2855 | (de\002nition)f(of)291 1136 y(substitution)e(\(namely)i
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|  |   2856 | Fm(n)11 b(<)28 b(i)p Fu(,)d Fm(n)j Fk(=)f Fm(i)e Fu(and)g
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|  |   2857 | Fm(n)11 b(>)28 b(i)p Fu(\):)p Black 291 1330 a Fl(\017)p
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|  |   2858 | Black 49 w Fu(Case)i Fm(n)20 b(<)36 b(i)p Fu(:)k(W)-8
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|  |   2859 | b(e)30 b(kno)n(w)e(by)h(the)h(assumption)d Fm(i)36 b
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|  |   2860 | Fl(\024)h Fm(j)e Fu(that)29 b(also)g Fm(n)20 b(<)36 b(j)g
 | 
|  |   2861 | Fu(and)29 b Fm(n)20 b(<)36 b(j)31 b Fk(+)26 b(1)p Fu(.)390
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|  |   2862 | 1450 y(Therefore)g(both)e(sides)g(of)h(\()p 0 0 1 TeXcolorrgb
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|  |   2863 | 1375 1451 a
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|  |   2864 | SDict begin H.S end
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|  |   2865 |  1375 1451 a 0 0 1 TeXcolorrgb -1 x Fu(3)p
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|  |   2866 | 0 0 1 TeXcolorrgb 1425 1382 a
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|  |   2867 | SDict begin H.R end
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|  |   2868 |  1425 1382 a 1425 1450 a
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|  |   2869 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2870 | (equation.3) cvn H.B /ANN pdfmark end
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|  |   2871 | 
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|  |   2872 | 1425 1450 a Black Fu(\))g(are)h(equal)e(to)h Fh(V)-7
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|  |   2873 | b(ar)35 b Fm(n)p Fu(.)p Black 291 1600 a Fl(\017)p Black
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|  |   2874 | 49 w Fu(Case)30 b Fm(n)36 b Fk(=)f Fm(i)p Fu(:)40 b(The)29
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|  |   2875 | b(left-hand)g(side)g(of)g(\()p 0 0 1 TeXcolorrgb 1827
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|  |   2876 | 1601 a
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|  |   2877 | SDict begin H.S end
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|  |   2878 |  1827 1601 a 0 0 1 TeXcolorrgb -1 x Fu(3)p 0 0 1
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|  |   2879 | TeXcolorrgb 1877 1532 a
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|  |   2880 | SDict begin H.R end
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|  |   2881 |  1877 1532 a 1877 1600 a
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|  |   2882 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2883 | (equation.3) cvn H.B /ANN pdfmark end
 | 
|  |   2884 |  1877
 | 
|  |   2885 | 1600 a Black Fu(\))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(\()p 0 0 1 TeXcolorrgb 2097 4163 a
 | 
|  |   2963 | SDict begin H.S end
 | 
|  |   2964 |  2097 4163 a
 | 
|  |   2965 | 0 0 1 TeXcolorrgb -1 x Fu(3)p 0 0 1 TeXcolorrgb 2147
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|  |   2966 | 4094 a
 | 
|  |   2967 | SDict begin H.R end
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|  |   2968 |  2147 4094 a 2147 4162 a
 | 
|  |   2969 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   2970 | (equation.3) cvn H.B /ANN pdfmark end
 | 
|  |   2971 |  2147 4162 a Black Fu(\))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
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|  |   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
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|  |   2999 | b(we)h(can)f(calculate)g(that)g(the)g(left-hand)f(side)h(of)g(\()p
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|  |   3001 | SDict begin H.S end
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|  |   3007 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3008 | (equation.3) cvn H.B /ANN pdfmark end
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|  |   3009 |  3059 4945 a Black Fu(\))g(equals)g Fl(")3445
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|  |   3010 | 4897 y Fg(j)3445 4969 y Fj(0)3512 4945 y Fm(L)468 5065
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|  |   3011 | y Fu(and)30 b(the)f(right-hand)g(side)h(to)g Fk(\()p
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|  |   3012 | Fl(")1625 5018 y Fg(j)t Fj(+1)1625 5089 y(0)1788 5065
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|  |   3013 | y Fm(L)p Fk(\)[)p Fm(i)38 b Fk(:=)f Fm(N)10 b Fk([)p
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|  |   3014 | Fm(j)33 b Fl(\000)26 b Fm(i)38 b Fk(:=)f Fm(L)p Fk(]])p
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|  |   3015 | Fu(.)46 b(T)-8 b(o)30 b(sho)n(w)f(that)h(these)468 5185
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|  |   3016 | y(terms)24 b(are)i(equal)e(we)h(need)g(the)g(lemma)1898
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|  |   3017 | 5185 y
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|  |   3018 | SDict begin H.S end
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|  |   3021 |  1898 5185 a 1898 5185
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|  |   3023 | SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark
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|  |   3024 | end
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|  |   3025 |  1898 5185 a Black Black 845 5347 a Fl(8)p Fm(k)s(;)17
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|  |   3027 | Fm(i)j(<)g(k)d Fk(+)d(\()p Fm(j)28 b Fk(+)22 b(1\))j
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|  |   3028 | Fu(then)f Fk(\()p Fl(")2525 5299 y Fg(j)t Fj(+1)2525
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|  |   3029 | 5374 y Fg(k)2679 5347 y Fm(L)p Fk(\)[)p Fm(i)k Fk(:=)g
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|  |   3030 | Fm(P)14 b Fk(])27 b(=)p Fl(")3259 5299 y Fg(j)3259 5374
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|  |   3031 | y(k)3329 5347 y Fm(L)p Black -2927 w Fu(\(6\))p Black
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|  |   3032 | Black 1897 5513 a(5)p Black eop end
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|  |   3052 | b Fm(i)42 b Fl(7!)f Fm(i)32 b Fu(and)468 203 y(using)25
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|  |   3053 | b(the)i(assumption)d Fm(i)32 b Fl(\024)f Fm(j)6 b Fu(,)27
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|  |   3055 | 324 y(hold)e(and)g(thus)g(can)i(conclude)e(that)h Fk(\()p
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|  |   3056 | Fl(")1840 277 y Fg(j)t Fj(+1)1840 348 y(0)1994 324 y
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|  |   3057 | Fm(L)p Fk(\)[)p Fm(i)j Fk(:=)g Fm(N)10 b Fk([)p Fm(j)28
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|  |   3058 | b Fl(\000)23 b Fm(i)28 b Fk(:=)g Fm(L)p Fk(]])g(=)p Fl(")3066
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|  |   3062 | (kno)n(w)g(that)h Fm(n)11 b(>)28 b(j)g Fk(+)22 b(1)p
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|  |   3063 | Fu(.)31 b(These)25 b(inequalities)468 565 y(in)f(turn)h(imply)e(that)h
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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
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|  |   3073 |  1804 565 a Black Fu(\))g(are)h(equal)f
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|  |   3074 | (to)f Fh(V)-7 b(ar)36 b Fk(\()p Fm(n)22 b Fl(\000)h Fk(2\))p
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|  |   3075 | Fu(.)291 728 y(This)h(concludes)g(the)h(proof)f(of)h(the)g
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|  |   3076 | (substitution)d(lemma.)1159 b Fi(2)291 917 y Fu(In)28
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|  |   3077 | b(this)f(formalisation)g(considerable)g(ingenuity)g(is)h(needed)g(when)
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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
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|  |   3087 |  374
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|  |   3088 | 1037 a Black Fu(\),)35 b(\()p 0 0 1 TeXcolorrgb 500 1038
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|  |   3089 | a
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|  |   3090 | SDict begin H.S end
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|  |   3095 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3096 | (equation.5) cvn H.B /ANN pdfmark end
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|  |   3097 |  550 1037 a Black Fu(\))e(and)h(\()p
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|  |   3099 | SDict begin H.S end
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|  |   3102 | SDict begin H.R end
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|  |   3103 |  877 969 a 877
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|  |   3104 | 1037 a
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|  |   3105 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3106 | (equation.6) cvn H.B /ANN pdfmark end
 | 
|  |   3107 |  877 1037 a Black Fu(\).)56 b(Also)32 b(the)o(y)g(are)i(quite)f
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|  |   3108 | (\223brittle\224\227in)f(the)h(sense)g(that)g(the)o(y)f(seem)h(to)g(go)
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|  |   3109 | 291 1158 y(through)d(just)h(in)g(the)h(form)f(stated.)51
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|  |   3110 | b(T)-8 b(o)31 b(\002nd)h(them)f(can)h(be)g(a)g(daunting)e(task)h(for)h
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|  |   3111 | (an)g(ine)o(x-)291 1278 y(perienced)27 b(user)g(of)h(theorem)f(pro)o(v)
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|  |   3112 | o(ers)f(\(the)o(y)g(are)i(only)f(in)g(little)f(part)h(inspired)f(by)h
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|  |   3113 | (the)g(f)o(acts)291 1399 y(needed)h(in)g(the)g(main)g(proof\).)41
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|  |   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
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|  |   3115 | (fe)n(w)h(problems,)291 1519 y(because)23 b(the)o(y)f(\223carry)i(o)o
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|  |   3116 | (v)o(er\224)f(from)g(language)g(to)f(language,)h(and)g(hence)h(one)f
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|  |   3117 | (does)g(not)f(need)291 1639 y(to)g(\223in)l(v)o(ent)g(the)g(wheel\224)h
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|  |   3118 | (again)f(for)h(a)g(ne)n(w)f(language.)30 b(Theorem)23
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|  |   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
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|  |   3123 | (minimally)d(adapted)j(to)g(System)291 2000 y(F)346 2015
 | 
|  |   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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|  |   3129 | SDict begin H.R end
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|  |   3130 |  1351 1932 a
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|  |   3131 | 1351 2000 a
 | 
|  |   3132 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
 | 
|  |   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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|  |   3138 | SDict begin H.S end
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|  |   3141 | SDict begin H.R end
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|  |   3142 |  2808
 | 
|  |   3143 | 2053 a 2808 2121 a
 | 
|  |   3144 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
 | 
|  |   3145 | (cite.rasmussen95churchrosser) cvn H.B /ANN pdfmark end
 | 
|  |   3146 |  2808 2121 a Black Fu(])g(who)h(w)o(ork)o(ed)f(with)
 | 
|  |   3147 | 291 2241 y(Isabelle/ZF)-8 b(.)24 b(Nipk)o(o)n(w)g(wrote)g([)p
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|  |   3149 | SDict begin H.S end
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|  |   3152 | SDict begin H.R end
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|  |   3153 |  1463 2173 a
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|  |   3154 | 1463 2241 a
 | 
|  |   3155 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
 | 
|  |   3156 | (cite.Nipkow-JAR01) cvn H.B /ANN pdfmark end
 | 
|  |   3157 |  1463 2241 a Black Fu(,)h(P)o(age)g(57]:)p
 | 
|  |   3158 | Black Black 390 2441 a Fc(\223)g(Initially)30 b(I)g(tried)g(to)h
 | 
|  |   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
 | 
|  |   3163 | b(I)g(did)f(not)h(have)g(to)f(\002nd)g(this)g(collection)g(of)h
 | 
|  |   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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|  |   3167 | 0 0 1 TeXcolorrgb 3387 3037 a
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|  |   3168 | SDict begin H.S end
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|  |   3169 |  3387 3037 a 0 0 1 TeXcolorrgb
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|  |   3170 | -1 x Fu(7)p 0 0 1 TeXcolorrgb 3437 2969 a
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|  |   3171 | SDict begin H.R end
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|  |   3172 |  3437 2969 a
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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
 | 
|  |   3176 |  3437 3036 a Black Fu(])g(in)291 3156 y(Coq.)437
 | 
|  |   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
 | 
|  |   3179 | 3405 y(might)c(be)i(surprising)f(that)g(one)h(does)g(not)f(end)h(up)g
 | 
|  |   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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|  |   3183 | (lines\227)291 3646 y(similar)e(numbers)h(for)h(the)f(lemmas)g
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|  |   3184 | (corresponding)f(to)i(\()p 0 0 1 TeXcolorrgb 2284 3646
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|  |   3186 | SDict begin H.S end
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|  |   3189 | SDict begin H.R end
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|  |   3190 |  2334 3578 a 2334 3646 a
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|  |   3191 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3192 | (equation.4) cvn H.B /ANN pdfmark end
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|  |   3193 |  2334 3646 a Black
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|  |   3194 | Fu(\),)g(\()p 0 0 1 TeXcolorrgb 2446 3647 a
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|  |   3195 | SDict begin H.S end
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|  |   3196 |  2446 3647
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|  |   3197 | a 0 0 1 TeXcolorrgb -1 x Fu(5)p 0 0 1 TeXcolorrgb 2496
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|  |   3198 | 3578 a
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|  |   3199 | SDict begin H.R end
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|  |   3200 |  2496 3578 a 2496 3646 a
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|  |   3201 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3202 | (equation.5) cvn H.B /ANN pdfmark end
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|  |   3203 |  2496 3646 a Black Fu(\))g(and)g(\()p
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|  |   3204 | 0 0 1 TeXcolorrgb 2748 3647 a
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|  |   3205 | SDict begin H.S end
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|  |   3206 |  2748 3647 a 0 0 1 TeXcolorrgb
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|  |   3208 | SDict begin H.R end
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|  |   3209 |  2798 3578 a
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|  |   3210 | 2798 3646 a
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|  |   3211 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3212 | (equation.6) cvn H.B /ANN pdfmark end
 | 
|  |   3213 |  2798 3646 a Black Fu(\).)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
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|  |   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
 | 
|  |   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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|  |   3228 | SDict begin H.S end
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|  |   3229 |  2524 4488
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|  |   3230 | a 0 0 1 TeXcolorrgb Fu(4)p 0 0 1 TeXcolorrgb 2574 4420
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|  |   3231 | a
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|  |   3232 | SDict begin H.R end
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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
 | 
|  |   3235 | (equation.4) cvn H.B /ANN pdfmark end
 | 
|  |   3236 |  2574 4488 a Black Fu(\),)k(\()p
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|  |   3244 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3245 | (equation.5) cvn H.B /ANN pdfmark end
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|  |   3246 |  2752 4488 a Black Fu(\))e(and)f(\()p 0 0 1
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|  |   3247 | TeXcolorrgb 3031 4489 a
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|  |   3248 | SDict begin H.S end
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|  |   3251 | SDict begin H.R end
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|  |   3252 |  3081 4420 a
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|  |   3254 | SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3255 | (equation.6) cvn H.B /ANN pdfmark end
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|  |   3256 |  3081 4488 a Black Fu(\).)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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|  |   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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|  |   3266 | Black Black 878 5347 a Fm(M)10 b Fk([)p Fm(j)29 b Fk(+)22
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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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|  |   3320 | (cite.UrbanTasson05) cvn H.B /ANN pdfmark end
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|  |   3322 | (to)h(reason)g(with)f(\223named\224)i(alpha-equi-)291
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|  |   3331 |  291
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|  |   3332 | 811 a Black Black 496 946 a Fn(nominal)p 851 946 30 4
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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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|  |   3337 | Black 291 1110 a(where)27 b Fm(name)g Fu(is)f(a)h(type)f(representing)g
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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
 | 
|  |   3368 | (package)291 2273 y(one)24 b(can)i(de\002ne)f(this)f(operation)g(as)p
 | 
|  |   3369 | Black Black 798 2543 a Fk(\()p Fh(V)-7 b(ar)39 b Fm(y)t
 | 
|  |   3370 | Fk(\)[)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
 | 
|  |   3372 | 2607 y(:)1763 2475 y Fm(N)264 b Fu(if)25 b Fm(x)j Fl(\021)g
 | 
|  |   3373 | Fm(y)1763 2632 y Fh(V)-7 b(ar)36 b Fm(y)102 b Fu(otherwise)543
 | 
|  |   3374 | 2824 y Fk(\()p Fh(App)33 b Fm(M)880 2839 y Fj(1)948 2824
 | 
|  |   3375 | y Fm(M)1042 2839 y Fj(2)1082 2824 y Fk(\)[)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(\()p Fm(M)1965 2839 y Fj(1)2005
 | 
|  |   3378 | 2824 y Fk([)p Fm(x)28 b Fk(:=)g Fm(N)10 b Fk(]\))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(\()p Fh(L)-5 b(am)35 b Fm(y)c(M)1042
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|  |   3381 | 3019 y Fj(1)1082 3004 y Fk(\)[)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(\()p Fm(M)2059 3019 y Fj(1)2099 3004
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|  |   3384 | y Fk([)p Fm(x)d Fk(:=)g Fm(N)10 b Fk(]\))199 b Fu(pro)o(vided)24
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|  |   3385 | b Fm(y)31 b Fk(#)d(\()p Fm(x;)17 b(N)10 b Fk(\))291 3185
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|  |   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
 | 
|  |   3388 | b Fl(6)p Fk(=)e Fm(x)c Fu(and)g Fm(y)j Fu(not)c(free)h(in)g
 | 
|  |   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)
 | 
|  |   3390 | h(one)f(has)g(indeed)h(de\002ned)g(a)g(function,)h(one)e(needs)h(to)f
 | 
|  |   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(\(see)g([)p
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|  |   3394 | SDict begin H.S end
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|  |   3398 |  2960 3358 a 2960
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|  |   3399 | 3426 a
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|  |   3400 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3401 | (cite.Pitts05) cvn H.B /ANN pdfmark end
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|  |   3402 |  2960 3426 a Black Fu(,)p 0 0 1 TeXcolorrgb 2985
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|  |   3403 | 3427 a
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|  |   3404 | SDict begin H.S end
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|  |   3405 |  2985 3427 a 0 0 1 TeXcolorrgb -1 x Fu(13)p 0 0 1
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|  |   3407 | SDict begin H.R end
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|  |   3408 |  3085 3358 a 3085 3426 a
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|  |   3409 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3410 | (cite.UrbanBerghofer06) cvn H.B /ANN pdfmark end
 | 
|  |   3411 |  3085
 | 
|  |   3412 | 3426 a Black Fu(])g(for)g(the)f(de-)291 3546 y(tails\).)62
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|  |   3413 | b(This)35 b(requires)g(some)g(small)g(proofs)g(that)g(ha)n(v)o(e)g(no)g
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|  |   3414 | (counterpart)h(in)f(the)g(informal)291 3667 y(proof)29
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|  |   3415 | b(and)h(in)f(the)h(formalisation)e(based)h(on)h(de)g(Bruijn)f(indices.)
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|  |   3416 | 45 b(This)29 b(need)h(of)g(v)o(erifying)291 3787 y(some)d(properties)h
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|  |   3417 | (arises)g(whene)n(v)o(er)g(a)g(function)g(is)f(de\002ned)i(by)f
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|  |   3418 | (recursion)g(o)o(v)o(er)f(the)h(struc-)291 3908 y(ture)c(of)h
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|  |   3419 | (alpha-equated)g(lambda-terms.)437 4036 y(W)l(ith)36
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|  |   3420 | b(the)g(de\002nition)f(of)h(the)h(nominal)e(datatype)g
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|  |   3421 | Fh(lam)44 b Fu(comes)36 b(the)g(follo)n(wing)e Fc(str)l(ong)291
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|  |   3422 | 4157 y Fu(structural)24 b(induction)f(principle)h([)p
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|  |   3428 |  1605 4089 a 1605
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|  |   3429 | 4157 a
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|  |   3430 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3431 | (cite.UrbanNorrish05) cvn H.B /ANN pdfmark end
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|  |   3432 |  1605 4157 a Black Fu(,)p 0 0 1 TeXcolorrgb 1630
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|  |   3433 | 4158 a
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|  |   3438 |  1729 4089 a 1729 4157 a
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|  |   3439 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3440 | (cite.UrbanTasson05) cvn H.B /ANN pdfmark end
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|  |   3441 |  1729
 | 
|  |   3442 | 4157 a Black Fu(]:)p Black Black 690 4348 a Fl(8)p Fm(c)17
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|  |   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(\()p
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|  |   3446 | Fl(8)p Fm(d:)28 b(P)41 b(M)1568 4520 y Fj(1)1635 4505
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|  |   3447 | y Fm(d)p Fk(\))50 b Fl(^)g Fk(\()p Fl(8)p Fm(d:)28 b(P)41
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|  |   3448 | b(M)2287 4520 y Fj(2)2354 4505 y Fm(d)p Fk(\))55 b Fl(\))28
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|  |   3449 | b Fm(P)41 b Fk(\()p Fh(App)31 b Fm(M)3065 4520 y Fj(1)3132
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|  |   3450 | 4505 y Fm(M)3226 4520 y Fj(2)3266 4505 y Fk(\))d Fm(c)690
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|  |   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(\()p Fl(8)p Fm(d:)28 b(P)41 b(M)d(d)p
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|  |   3453 | Fk(\))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
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|  |   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
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|  |   3457 | Fm(P)48 b Fu(for)34 b(all)291 5106 y(lambda-terms)29
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|  |   3458 | b Fm(M)10 b Fu(,)33 b(then,)f(as)e(e)o(xpected,)i(one)e(has)h(to)f(pro)
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|  |   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
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|  |   3465 | Black 1897 5513 a(7)p Black eop end
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|  |   3468 | SDict begin /product where{pop product(Distiller)search{pop pop pop
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|  |   3469 | version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto
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|  |   3470 | closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show
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|  |   3471 | grestore}if}{pop}ifelse}{pop}ifelse}if end
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|  |   3472 |  0 0 a Black 0 TeXcolorgray
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|  |   3473 | 291 -166 a
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|  |   3474 | SDict begin H.S end
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|  |   3475 |  291 -166 a 0 TeXcolorgray 0 TeXcolorgray 291
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|  |   3476 | -166 a
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|  |   3477 | SDict begin H.R end
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|  |   3478 |  291 -166 a 291 -166 a
 | 
|  |   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
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|  |   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
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|  |   3500 | SDict begin H.S end
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|  |   3501 |  990 444 a 1006 408 a Fj(4)1062
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|  |   3502 | 444 y
 | 
|  |   3503 | SDict begin 14.5 H.L end
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|  |   3504 |  1062 444 a 1062 444 a
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|  |   3505 | SDict begin [ /Subtype /Link /Dest (Hfootnote.2) cvn /H /I /Border
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|  |   3506 | [0 0 0] /Color [1 0 0] H.B /ANN pdfmark end
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|  |   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
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|  |   3532 |  291 1678 a 291 1678 a
 | 
|  |   3533 | SDict begin 14.5 H.A end
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|  |   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
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|  |   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
 | 
|  |   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
 | 
|  |   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
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|  |   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
 | 
|  |   3629 |  2293 5348 a 0 0 1 TeXcolorrgb -1 x Fs(10)p 0 0 1 TeXcolorrgb
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|  |   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
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|  |   3634 | (cite.Pitts03) cvn H.B /ANN pdfmark end
 | 
|  |   3635 |  2376 5347 a Black
 | 
|  |   3636 | Fs(,)p 0 0 1 TeXcolorrgb 2397 5348 a
 | 
|  |   3637 | SDict begin H.S end
 | 
|  |   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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|  |   3642 |  2480 5290 a 2480 5347 a
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|  |   3643 | SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
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|  |   3644 | (cite.UrbanTasson05) cvn H.B /ANN pdfmark end
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|  |   3645 |  2480 5347 a Black Fs(].)p Black
 | 
|  |   3646 | Black 1897 5513 a Fu(8)p Black eop end
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|  |   3647 | %%Page: 9 9
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|  |   3648 | TeXDict begin 9 8 bop 0 0 a
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|  |   3649 | SDict begin /product where{pop product(Distiller)search{pop pop pop
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|  |   3650 | version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto
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|  |   3651 | closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show
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|  |   3652 | grestore}if}{pop}ifelse}{pop}ifelse}if end
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|  |   3653 |  0 0 a Black 0 TeXcolorgray
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|  |   3654 | 291 -166 a
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|  |   3655 | SDict begin H.S end
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|  |   3656 |  291 -166 a 0 TeXcolorgray 0 TeXcolorgray 291
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|  |   3657 | -166 a
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|  |   3658 | SDict begin H.R end
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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
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|  |   3661 | end
 | 
|  |   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
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|  |   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
 | 
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|  |   4183 | y Fl(^)51 b Fk(\000)27 b Fl(`)h Fm(Q)3110 3471 y Fd(0)3161
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|  |   4276 | b Fl(`)f Fm(P)45 b(<)p Fk(:)31 b Fm(Q)c Fu(by)f(assumption)e(and)j
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|  |   4277 | (hence)291 444 y Fk(\001@)p Fm(P)14 b Fk(@\000)27 b Fl(`)g
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|  |   4278 | Fk(\()p Fl(")865 403 y Fg(i)p Fj(+1)865 469 y(0)1011
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|  |   4279 | 444 y Fm(P)14 b Fk(\))27 b Fm(<)p Fk(:)h(\()p Fl(")1372
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|  |   4280 | 403 y Fg(i)p Fj(+1)1372 469 y(0)1517 444 y Fm(Q)p Fk(\))c
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|  |   4281 | Fu(by)f(weak)o(ening.)30 b(Since)23 b Fm(S)34 b Fk(=)27
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|  |   4300 |  291 989 a 136 x Fv(4)119 b(Conclusion)291 1349 y Fu(W)-8
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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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|  |   4303 | 1469 y(nominal)d(logic)g(w)o(ork.)33 b(The)25 b(former)h(approach)g(is)
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|  |   4307 | 1710 y(datatype)30 b(package.)48 b(Extrapolating)29 b(an)i(amazing)f
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|  |   4308 | (amount)g(from)g(the)g(submissions)e(to)j(the)291 1831
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|  |   4310 | (occurring)h(in)g(programming)f(meta-)291 1951 y(theory)38
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|  |   4311 | b(can,)43 b(in)38 b(principle,)k(be)d(solv)o(ed)f(by)g(theorem)h(pro)o
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|  |   4538 | version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto
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|  |   4539 | closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show
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|  |   4540 | grestore}if}{pop}ifelse}{pop}ifelse}if end
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|  |   4541 |  0 0 a Black 0 TeXcolorgray
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|  |   4542 | 291 -166 a
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|  |   4543 | SDict begin H.S end
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|  |   4544 |  291 -166 a 0 TeXcolorgray 0 TeXcolorgray 291
 | 
|  |   4545 | -166 a
 | 
|  |   4546 | SDict begin H.R end
 | 
|  |   4547 |  291 -166 a 291 -166 a
 | 
|  |   4548 | SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark
 | 
|  |   4549 | end
 | 
|  |   4550 |  291 -166 a Black 1173 w
 | 
|  |   4551 | Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 0 a
 | 
|  |   4552 | SDict begin H.S end
 | 
|  |   4553 |  291
 | 
|  |   4554 | 0 a 291 0 a
 | 
|  |   4555 | SDict begin 13.6 H.A end
 | 
|  |   4556 |  291 0 a 291 0 a
 | 
|  |   4557 | SDict begin [ /View [/XYZ H.V] /Dest (cite.Leroy06) cvn H.B /DEST pdfmark
 | 
|  |   4558 | end
 | 
|  |   4559 |  291 0 a Black 336 83 a Fw([8])p
 | 
|  |   4560 | Black 26 w(Lero)o(y)-6 b(,)18 b(X.,)f Fy(Formal)h(Certi\002cation)j(of)
 | 
|  |   4561 | d(a)g(Compiler)h(Bac)n(k-End,)g(or:)g(Pr)l(o)o(gr)o(amming)g(a)f
 | 
|  |   4562 | (Compiler)467 196 y(with)27 b(a)g(Pr)l(oof)h(Assistant)p
 | 
|  |   4563 | Fw(,)h(in:)f Fy(Pr)l(oc.)f(of)g(the)h(33r)m(d)g(A)m(CM)e(SIGPLAN-SIGA)m
 | 
|  |   4564 | (CT)f(Symposium)j(on)467 309 y(Principles)d(of)f(Pr)l(o)o(gr)o(amming)g
 | 
|  |   4565 | (Langua)o(g)o(es)i(\(POPL\))c Fw(\(2006\),)j(pp.)e(42\22654.)291
 | 
|  |   4566 | 365 y
 | 
|  |   4567 | SDict begin H.S end
 | 
|  |   4568 |  291 365 a 291 365 a
 | 
|  |   4569 | SDict begin 13.6 H.A end
 | 
|  |   4570 |  291 365 a 291 365 a
 | 
|  |   4571 | SDict begin [ /View [/XYZ H.V] /Dest (cite.Nipkow-JAR01) cvn H.B /DEST
 | 
|  |   4572 | pdfmark end
 | 
|  |   4573 |  291 365
 | 
|  |   4574 | a Black 336 495 a Fw([9])p Black 26 w(Nipk)o(o)n(w)-6
 | 
|  |   4575 | b(,)38 b(T)-7 b(.,)37 b Fy(Mor)m(e)h(Chur)m(c)o(h-Rosser)i(Pr)l(oofs)e
 | 
|  |   4576 | (\(in)h(Isabelle/HOL\))p Fw(,)g(Journal)h(of)e(Automated)467
 | 
|  |   4577 | 608 y(Reasoning)25 b Fx(26)f Fw(\(2001\),)h(pp.)e(51\22666.)291
 | 
|  |   4578 | 664 y
 | 
|  |   4579 | SDict begin H.S end
 | 
|  |   4580 |  291 664 a 291 664 a
 | 
|  |   4581 | SDict begin 13.6 H.A end
 | 
|  |   4582 |  291 664 a 291 664 a
 | 
|  |   4583 | SDict begin [ /View [/XYZ H.V] /Dest (cite.Pitts03) cvn H.B /DEST pdfmark
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|  |   4584 | end
 | 
|  |   4585 |  291 664
 | 
|  |   4586 | a Black 131 x Fw([10])p Black 26 w(Pitts,)e(A.)e(M.,)h
 | 
|  |   4587 | Fy(Nominal)h(Lo)o(gic,)g(A)f(Fir)o(st)h(Or)m(der)g(Theory)h(of)f(Names)
 | 
|  |   4588 | g(and)g(Binding)p Fw(,)h(Information)467 907 y(and)i(Computation)h
 | 
|  |   4589 | Fx(186)g Fw(\(2003\),)f(pp.)g(165\226193.)291 963 y
 | 
|  |   4590 | SDict begin H.S end
 | 
|  |   4591 |  291
 | 
|  |   4592 | 963 a 291 963 a
 | 
|  |   4593 | SDict begin 13.6 H.A end
 | 
|  |   4594 |  291 963 a 291 963 a
 | 
|  |   4595 | SDict begin [ /View [/XYZ H.V] /Dest (cite.Pitts05) cvn H.B /DEST pdfmark
 | 
|  |   4596 | end
 | 
|  |   4597 |  291 963 a Black 131
 | 
|  |   4598 | x Fw([11])p Black 26 w(Pitts,)47 b(A.)f(M.,)g Fy(Alpha-Structur)o(al)52
 | 
|  |   4599 | b(Recur)o(sion)d(and)f(Induction)j(\(Extended)e(Abstr)o(act\))p
 | 
|  |   4600 | Fw(,)f(in:)467 1207 y Fy(Pr)l(oc.)40 b(of)f(the)i(18th)g(International)
 | 
|  |   4601 | j(Confer)m(ence)e(on)e(Theor)m(em)g(Pr)l(o)o(ving)h(in)f(Higher)g(Or)m
 | 
|  |   4602 | (der)467 1320 y(Lo)o(gics)24 b(\(TPHOLs\))p Fw(,)d(LNCS)h
 | 
|  |   4603 | Fx(3603)p Fw(,)i(2005,)g(pp.)f(17\22634.)291 1375 y
 | 
|  |   4604 | SDict begin H.S end
 | 
|  |   4605 |  291
 | 
|  |   4606 | 1375 a 291 1375 a
 | 
|  |   4607 | SDict begin 13.6 H.A end
 | 
|  |   4608 |  291 1375 a 291 1375 a
 | 
|  |   4609 | SDict begin [ /View [/XYZ H.V] /Dest (cite.rasmussen95churchrosser)
 | 
|  |   4610 | cvn H.B /DEST pdfmark end
 | 
|  |   4611 |  291 1375 a Black
 | 
|  |   4612 | 131 x Fw([12])p Black 26 w(Rasmussen,)f(O.,)d Fy(The)h(Chur)m(c)o
 | 
|  |   4613 | (h-Rosser)j(Theor)m(em)e(in)f(Isabelle:)j(A)d(Pr)l(oof)h(Porting)g
 | 
|  |   4614 | (Experiment)p Fw(,)467 1619 y(T)-6 b(echnical)25 b(Report)f(364,)g
 | 
|  |   4615 | (Cambridge)g(Uni)n(v)o(ersity)h(\(1995\).)291 1675 y
 | 
|  |   4616 | SDict begin H.S end
 | 
|  |   4617 | 
 | 
|  |   4618 | 291 1675 a 291 1675 a
 | 
|  |   4619 | SDict begin 13.6 H.A end
 | 
|  |   4620 |  291 1675 a 291 1675 a
 | 
|  |   4621 | SDict begin [ /View [/XYZ H.V] /Dest (cite.UrbanBerghofer06) cvn H.B
 | 
|  |   4622 | /DEST pdfmark end
 | 
|  |   4623 |  291 1675
 | 
|  |   4624 | a Black 130 x Fw([13])p Black 26 w(Urban,)55 b(C.)e(and)i(S.)e(Ber)n
 | 
|  |   4625 | (ghofer)l(,)j Fy(A)d(Recur)o(sion)k(Combinator)f(for)f(Nominal)g
 | 
|  |   4626 | (Datatypes)467 1918 y(Implemented)29 b(in)e(Isabelle/HOL)p
 | 
|  |   4627 | Fw(,)h(in:)g Fy(Pr)l(oc.)f(of)g(the)g(3r)m(d)h(International)j(J)n
 | 
|  |   4628 | (oint)d(Confer)m(ence)h(on)467 2031 y(A)n(utomated)c(Reasoning)g
 | 
|  |   4629 | (\(IJCAR\))p Fw(,)e(LN)m(AI)e Fx(4130)p Fw(,)k(2006,)f(pp.)f
 | 
|  |   4630 | (498\226512.)291 2087 y
 | 
|  |   4631 | SDict begin H.S end
 | 
|  |   4632 |  291 2087 a 291 2087 a
 | 
|  |   4633 | SDict begin 13.6 H.A end
 | 
|  |   4634 |  291 2087
 | 
|  |   4635 | a 291 2087 a
 | 
|  |   4636 | SDict begin [ /View [/XYZ H.V] /Dest (cite.UrbanNorrish05) cvn H.B
 | 
|  |   4637 | /DEST pdfmark end
 | 
|  |   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
 |