Publications/nomu-tcs.ps
changeset 14 680070975206
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13:0b4a5595cbe4 14:680070975206
       
     1 %!PS-Adobe-2.0
       
     2 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software
       
     3 %%Title: nomu-tcs.dvi
       
     4 %%Pages: 29
       
     5 %%PageOrder: Ascend
       
     6 %%BoundingBox: 0 0 596 842
       
     7 %%DocumentFonts: Times-Bold Times-Roman CMR10 CMR8 Times-Italic CMMI10
       
     8 %%+ CMMI12 CMSY10 CMTT12 CMR12 CMSY8 Courier CMBSY10 CMMI8 CMTI12 CMEX10
       
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    10 %%EndComments
       
    11 %DVIPSWebPage: (www.radicaleye.com)
       
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    14 %DVIPSSource:  TeX output 2004.04.09:1442
       
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    74 % File  8r.enc as of 2002-03-12 for PSNFSS 9
       
    75 %
       
    76 % This is the encoding vector for Type1 and TrueType fonts to be used
       
    77 % with TeX.  This file is part of the PSNFSS bundle, version 9
       
    78 % 
       
    79 % Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt
       
    80 %
       
    81 % Idea is to have all the characters normally included in Type 1 fonts
       
    82 % available for typesetting. This is effectively the characters in Adobe
       
    83 % Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro.
       
    84 % 
       
    85 % Character code assignments were made as follows:
       
    86 % 
       
    87 % (1) the Windows ANSI characters are almost all in their Windows ANSI
       
    88 % positions, because some Windows users cannot easily reencode the
       
    89 % fonts, and it makes no difference on other systems. The only Windows
       
    90 % ANSI characters not available are those that make no sense for
       
    91 % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
       
    92 % (173). quotesingle and grave are moved just because it's such an
       
    93 % irritation not having them in TeX positions.
       
    94 % 
       
    95 % (2) Remaining characters are assigned arbitrarily to the lower part
       
    96 % of the range, avoiding 0, 10 and 13 in case we meet dumb software.
       
    97 % 
       
    98 % (3) Y&Y Lucida Bright includes some extra text characters; in the
       
    99 % hopes that other PostScript fonts, perhaps created for public
       
   100 % consumption, will include them, they are included starting at 0x12.
       
   101 % 
       
   102 % (4) Remaining positions left undefined are for use in (hopefully)
       
   103 % upward-compatible revisions, if someday more characters are generally
       
   104 % available.
       
   105 % 
       
   106 % (5) hyphen appears twice for compatibility with both ASCII and Windows.
       
   107 %
       
   108 % (6) /Euro is assigned to 128, as in Windows ANSI
       
   109 % 
       
   110 /TeXBase1Encoding [
       
   111 % 0x00 (encoded characters from Adobe Standard not in Windows 3.1)
       
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   115   /breve /minus /.notdef 
       
   116 % These are the only two remaining unencoded characters, so may as
       
   117 % well include them.
       
   118   /Zcaron /zcaron 
       
   119 % 0x10
       
   120  /caron /dotlessi 
       
   121 % (unusual TeX characters available in, e.g., Lucida Bright)
       
   122  /dotlessj /ff /ffi /ffl 
       
   123  /.notdef /.notdef /.notdef /.notdef
       
   124  /.notdef /.notdef /.notdef /.notdef
       
   125  % very contentious; it's so painful not having quoteleft and quoteright
       
   126  % at 96 and 145 that we move the things normally found there down to here.
       
   127  /grave /quotesingle 
       
   128 % 0x20 (ASCII begins)
       
   129  /space /exclam /quotedbl /numbersign
       
   130  /dollar /percent /ampersand /quoteright
       
   131  /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
       
   132 % 0x30
       
   133  /zero /one /two /three /four /five /six /seven
       
   134  /eight /nine /colon /semicolon /less /equal /greater /question
       
   135 % 0x40
       
   136  /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O
       
   137 % 0x50
       
   138  /P /Q /R /S /T /U /V /W
       
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   140 % 0x60
       
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   142 % 0x70
       
   143  /p /q /r /s /t /u /v /w
       
   144  /x /y /z /braceleft /bar /braceright /asciitilde
       
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   146 % 0x80
       
   147  /Euro /.notdef /quotesinglbase /florin
       
   148  /quotedblbase /ellipsis /dagger /daggerdbl
       
   149  /circumflex /perthousand /Scaron /guilsinglleft
       
   150  /OE /.notdef /.notdef /.notdef
       
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   152  /.notdef /.notdef /.notdef /quotedblleft
       
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   154  /tilde /trademark /scaron /guilsinglright
       
   155  /oe /.notdef /.notdef /Ydieresis
       
   156 % 0xA0
       
   157  /.notdef % nobreakspace
       
   158  /exclamdown /cent /sterling
       
   159  /currency /yen /brokenbar /section
       
   160  /dieresis /copyright /ordfeminine /guillemotleft
       
   161  /logicalnot
       
   162  /hyphen % Y&Y (also at 45); Windows' softhyphen
       
   163  /registered
       
   164  /macron
       
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   166  /degree /plusminus /twosuperior /threesuperior
       
   167  /acute /mu /paragraph /periodcentered
       
   168  /cedilla /onesuperior /ordmasculine /guillemotright
       
   169  /onequarter /onehalf /threequarters /questiondown
       
   170 % 0xC0
       
   171  /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla
       
   172  /Egrave /Eacute /Ecircumflex /Edieresis
       
   173  /Igrave /Iacute /Icircumflex /Idieresis
       
   174 % 0xD0
       
   175  /Eth /Ntilde /Ograve /Oacute
       
   176  /Ocircumflex /Otilde /Odieresis /multiply
       
   177  /Oslash /Ugrave /Uacute /Ucircumflex
       
   178  /Udieresis /Yacute /Thorn /germandbls
       
   179 % 0xE0
       
   180  /agrave /aacute /acircumflex /atilde
       
   181  /adieresis /aring /ae /ccedilla
       
   182  /egrave /eacute /ecircumflex /edieresis
       
   183  /igrave /iacute /icircumflex /idieresis
       
   184 % 0xF0
       
   185  /eth /ntilde /ograve /oacute
       
   186  /ocircumflex /otilde /odieresis /divide
       
   187  /oslash /ugrave /uacute /ucircumflex
       
   188  /udieresis /yacute /thorn /ydieresis
       
   189 ] def
       
   190 
       
   191 %%EndProcSet
       
   192 %%BeginProcSet: f7b6d320.enc
       
   193 % Thomas Esser, Dec 2002. public domain
       
   194 %
       
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   229 
       
   230 %%EndProcSet
       
   231 %%BeginProcSet: aae443f0.enc
       
   232 % Thomas Esser, Dec 2002. public domain
       
   233 %
       
   234 % Encoding for:
       
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   236 %
       
   237 /TeXaae443f0Encoding [
       
   238 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega
       
   239 /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa
       
   240 /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi
       
   241 /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf
       
   242 /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft
       
   243 /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle
       
   244 /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle
       
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   246 /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N
       
   247 /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow
       
   248 /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p
       
   249 /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector
       
   250 /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
       
   251 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
       
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   253 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
       
   254 /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi
       
   255 /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta
       
   256 /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon
       
   257 /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
       
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   269 % Thomas Esser, Dec 2002. public domain
       
   270 %
       
   271 % Encoding for:
       
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   273 %
       
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   276 /plusminus /minusplus /circleplus /circleminus /circlemultiply
       
   277 /circledivide /circledot /circlecopyrt /openbullet /bullet
       
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   280 /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows
       
   281 /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast
       
   282 /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup
       
   283 /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional
       
   284 /prime /infinity /element /owner /triangle /triangleinv /negationslash
       
   285 /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur
       
   286 /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K
       
   287 /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection
       
   288 /unionmulti /logicaland /logicalor /turnstileleft /turnstileright
       
   289 /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright
       
   290 /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv
       
   291 /backslash /wreathproduct /radical /coproduct /nabla /integral
       
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   299 /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef
       
   300 /circlemultiply /circledivide /circledot /circlecopyrt /openbullet
       
   301 /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset
       
   302 /lessequal /greaterequal /precedesequal /followsequal /similar
       
   303 /approxequal /propersubset /propersuperset /lessmuch /greatermuch
       
   304 /precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef
       
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   316 % Thomas Esser, Dec 2002. public domain
       
   317 %
       
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   614 %!PS-AdobeFont-1.1: CMSY6 1.0
       
   615 %%CreationDate: 1991 Aug 15 07:21:34
       
   616 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   617 11 dict begin
       
   618 /FontInfo 7 dict dup begin
       
   619 /version (1.0) readonly def
       
   620 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
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   680 %%BeginFont: CMMI6
       
   681 %!PS-AdobeFont-1.1: CMMI6 1.100
       
   682 %%CreationDate: 1996 Jul 23 07:53:52
       
   683 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   684 11 dict begin
       
   685 /FontInfo 7 dict dup begin
       
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   687 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
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   773 %%EndFont 
       
   774 %%BeginFont: CMR6
       
   775 %!PS-AdobeFont-1.1: CMR6 1.0
       
   776 %%CreationDate: 1991 Aug 20 16:39:02
       
   777 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   778 11 dict begin
       
   779 /FontInfo 7 dict dup begin
       
   780 /version (1.0) readonly def
       
   781 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   782 /FullName (CMR6) readonly def
       
   783 /FamilyName (Computer Modern) readonly def
       
   784 /Weight (Medium) readonly def
       
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   861 %%EndFont 
       
   862 %%BeginFont: EUSM10
       
   863 %!PS-AdobeFont-1.1: EUSM10 2.1
       
   864 %%CreationDate: 1992 Nov 20 17:36:44
       
   865 % Euler fonts were designed by Hermann Zapf.
       
   866 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   867 11 dict begin
       
   868 /FontInfo 7 dict dup begin
       
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   870 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   871 /FullName (EUSM10) readonly def
       
   872 /FamilyName (Euler) readonly def
       
   873 /Weight (Medium) readonly def
       
   874 /ItalicAngle 0 def
       
   875 /isFixedPitch false def
       
   876 end readonly def
       
   877 /FontName /EUSM10 def
       
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   880 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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   883 dup 85 /U put
       
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   885 /FontBBox{-8 -194 963 738}readonly def
       
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   887 currentdict end
       
   888 currentfile eexec
       
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   938 %%EndFont 
       
   939 %%BeginFont: CMMIB10
       
   940 %!PS-AdobeFont-1.1: CMMIB10 1.100
       
   941 %%CreationDate: 1996 Jul 23 07:54:00
       
   942 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   943 11 dict begin
       
   944 /FontInfo 7 dict dup begin
       
   945 /version (1.100) readonly def
       
   946 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   947 /FullName (CMMIB10) readonly def
       
   948 /FamilyName (Computer Modern) readonly def
       
   949 /Weight (Bold) readonly def
       
   950 /ItalicAngle -14.04 def
       
   951 /isFixedPitch false def
       
   952 end readonly def
       
   953 /FontName /CMMIB10 def
       
   954 /PaintType 0 def
       
   955 /FontType 1 def
       
   956 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
   957 /Encoding 256 array
       
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   959 dup 0 /.notdef put
       
   960 readonly def
       
   961 /FontBBox{-15 -250 1216 750}readonly def
       
   962 /UniqueID 5087392 def
       
   963 currentdict end
       
   964 currentfile eexec
       
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  1009 %%EndFont 
       
  1010 %%BeginFont: CMTI10
       
  1011 %!PS-AdobeFont-1.1: CMTI10 1.00B
       
  1012 %%CreationDate: 1992 Feb 19 19:56:16
       
  1013 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1014 11 dict begin
       
  1015 /FontInfo 7 dict dup begin
       
  1016 /version (1.00B) readonly def
       
  1017 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1018 /FullName (CMTI10) readonly def
       
  1019 /FamilyName (Computer Modern) readonly def
       
  1020 /Weight (Medium) readonly def
       
  1021 /ItalicAngle -14.04 def
       
  1022 /isFixedPitch false def
       
  1023 end readonly def
       
  1024 /FontName /CMTI10 def
       
  1025 /PaintType 0 def
       
  1026 /FontType 1 def
       
  1027 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1028 /Encoding 256 array
       
  1029 0 1 255 {1 index exch /.notdef put} for
       
  1030 dup 0 /.notdef put
       
  1031 readonly def
       
  1032 /FontBBox{-163 -250 1146 969}readonly def
       
  1033 /UniqueID 5000828 def
       
  1034 currentdict end
       
  1035 currentfile eexec
       
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  1086 cleartomark
       
  1087 %%EndFont 
       
  1088 %%BeginFont: CMR10
       
  1089 %!PS-AdobeFont-1.1: CMR10 1.00B
       
  1090 %%CreationDate: 1992 Feb 19 19:54:52
       
  1091 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1092 11 dict begin
       
  1093 /FontInfo 7 dict dup begin
       
  1094 /version (1.00B) readonly def
       
  1095 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1096 /FullName (CMR10) readonly def
       
  1097 /FamilyName (Computer Modern) readonly def
       
  1098 /Weight (Medium) readonly def
       
  1099 /ItalicAngle 0 def
       
  1100 /isFixedPitch false def
       
  1101 end readonly def
       
  1102 /FontName /CMR10 def
       
  1103 /PaintType 0 def
       
  1104 /FontType 1 def
       
  1105 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1106 /Encoding 256 array
       
  1107 0 1 255 {1 index exch /.notdef put} for
       
  1108 dup 0 /.notdef put
       
  1109 readonly def
       
  1110 /FontBBox{-251 -250 1009 969}readonly def
       
  1111 /UniqueID 5000793 def
       
  1112 currentdict end
       
  1113 currentfile eexec
       
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  1207 cleartomark
       
  1208 %%EndFont 
       
  1209 %%BeginFont: CMSY10
       
  1210 %!PS-AdobeFont-1.1: CMSY10 1.0
       
  1211 %%CreationDate: 1991 Aug 15 07:20:57
       
  1212 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1213 11 dict begin
       
  1214 /FontInfo 7 dict dup begin
       
  1215 /version (1.0) readonly def
       
  1216 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1217 /FullName (CMSY10) readonly def
       
  1218 /FamilyName (Computer Modern) readonly def
       
  1219 /Weight (Medium) readonly def
       
  1220 /ItalicAngle -14.035 def
       
  1221 /isFixedPitch false def
       
  1222 end readonly def
       
  1223 /FontName /CMSY10 def
       
  1224 /PaintType 0 def
       
  1225 /FontType 1 def
       
  1226 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1227 /Encoding 256 array
       
  1228 0 1 255 {1 index exch /.notdef put} for
       
  1229 dup 0 /.notdef put
       
  1230 readonly def
       
  1231 /FontBBox{-29 -960 1116 775}readonly def
       
  1232 /UniqueID 5000820 def
       
  1233 currentdict end
       
  1234 currentfile eexec
       
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  1356 cleartomark
       
  1357 %%EndFont 
       
  1358 %%BeginFont: CMBSY10
       
  1359 %!PS-AdobeFont-1.1: CMBSY10 1.00
       
  1360 %%CreationDate: 1992 Jul 23 21:21:18
       
  1361 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1362 11 dict begin
       
  1363 /FontInfo 7 dict dup begin
       
  1364 /version (1.00) readonly def
       
  1365 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1366 /FullName (CMBSY10) readonly def
       
  1367 /FamilyName (Computer Modern) readonly def
       
  1368 /Weight (Bold) readonly def
       
  1369 /ItalicAngle -14.035 def
       
  1370 /isFixedPitch false def
       
  1371 end readonly def
       
  1372 /FontName /CMBSY10 def
       
  1373 /PaintType 0 def
       
  1374 /FontType 1 def
       
  1375 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1376 /Encoding 256 array
       
  1377 0 1 255 {1 index exch /.notdef put} for
       
  1378 dup 0 /.notdef put
       
  1379 readonly def
       
  1380 /FontBBox{-27 -940 1332 825}readonly def
       
  1381 /UniqueID 5000762 def
       
  1382 currentdict end
       
  1383 currentfile eexec
       
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  1424 cleartomark
       
  1425 %%EndFont 
       
  1426 %%BeginFont: CMEX10
       
  1427 %!PS-AdobeFont-1.1: CMEX10 1.00
       
  1428 %%CreationDate: 1992 Jul 23 21:22:48
       
  1429 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1430 11 dict begin
       
  1431 /FontInfo 7 dict dup begin
       
  1432 /version (1.00) readonly def
       
  1433 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1434 /FullName (CMEX10) readonly def
       
  1435 /FamilyName (Computer Modern) readonly def
       
  1436 /Weight (Medium) readonly def
       
  1437 /ItalicAngle 0 def
       
  1438 /isFixedPitch false def
       
  1439 end readonly def
       
  1440 /FontName /CMEX10 def
       
  1441 /PaintType 0 def
       
  1442 /FontType 1 def
       
  1443 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1444 /Encoding 256 array
       
  1445 0 1 255 {1 index exch /.notdef put} for
       
  1446 dup 56 /bracelefttp put
       
  1447 dup 57 /bracerighttp put
       
  1448 dup 58 /braceleftbt put
       
  1449 dup 59 /bracerightbt put
       
  1450 dup 60 /braceleftmid put
       
  1451 dup 61 /bracerightmid put
       
  1452 dup 62 /braceex put
       
  1453 dup 88 /summationdisplay put
       
  1454 readonly def
       
  1455 /FontBBox{-24 -2960 1454 772}readonly def
       
  1456 /UniqueID 5000774 def
       
  1457 currentdict end
       
  1458 currentfile eexec
       
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  1528 cleartomark
       
  1529 %%EndFont 
       
  1530 %%BeginFont: CMTI12
       
  1531 %!PS-AdobeFont-1.1: CMTI12 1.0
       
  1532 %%CreationDate: 1991 Aug 18 21:06:53
       
  1533 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1534 11 dict begin
       
  1535 /FontInfo 7 dict dup begin
       
  1536 /version (1.0) readonly def
       
  1537 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1538 /FullName (CMTI12) readonly def
       
  1539 /FamilyName (Computer Modern) readonly def
       
  1540 /Weight (Medium) readonly def
       
  1541 /ItalicAngle -14.04 def
       
  1542 /isFixedPitch false def
       
  1543 end readonly def
       
  1544 /FontName /CMTI12 def
       
  1545 /PaintType 0 def
       
  1546 /FontType 1 def
       
  1547 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1548 /Encoding 256 array
       
  1549 0 1 255 {1 index exch /.notdef put} for
       
  1550 dup 0 /.notdef put
       
  1551 readonly def
       
  1552 /FontBBox{-36 -251 1103 750}readonly def
       
  1553 /UniqueID 5000829 def
       
  1554 currentdict end
       
  1555 currentfile eexec
       
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  1660 cleartomark
       
  1661 %%EndFont 
       
  1662 %%BeginFont: CMMI8
       
  1663 %!PS-AdobeFont-1.1: CMMI8 1.100
       
  1664 %%CreationDate: 1996 Jul 23 07:53:54
       
  1665 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1666 11 dict begin
       
  1667 /FontInfo 7 dict dup begin
       
  1668 /version (1.100) readonly def
       
  1669 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1670 /FullName (CMMI8) readonly def
       
  1671 /FamilyName (Computer Modern) readonly def
       
  1672 /Weight (Medium) readonly def
       
  1673 /ItalicAngle -14.04 def
       
  1674 /isFixedPitch false def
       
  1675 end readonly def
       
  1676 /FontName /CMMI8 def
       
  1677 /PaintType 0 def
       
  1678 /FontType 1 def
       
  1679 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1680 /Encoding 256 array
       
  1681 0 1 255 {1 index exch /.notdef put} for
       
  1682 dup 0 /.notdef put
       
  1683 readonly def
       
  1684 /FontBBox{-24 -250 1110 750}readonly def
       
  1685 /UniqueID 5087383 def
       
  1686 currentdict end
       
  1687 currentfile eexec
       
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  1827 %%EndFont 
       
  1828 %%BeginFont: CMSY8
       
  1829 %!PS-AdobeFont-1.1: CMSY8 1.0
       
  1830 %%CreationDate: 1991 Aug 15 07:22:10
       
  1831 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1832 11 dict begin
       
  1833 /FontInfo 7 dict dup begin
       
  1834 /version (1.0) readonly def
       
  1835 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1836 /FullName (CMSY8) readonly def
       
  1837 /FamilyName (Computer Modern) readonly def
       
  1838 /Weight (Medium) readonly def
       
  1839 /ItalicAngle -14.035 def
       
  1840 /isFixedPitch false def
       
  1841 end readonly def
       
  1842 /FontName /CMSY8 def
       
  1843 /PaintType 0 def
       
  1844 /FontType 1 def
       
  1845 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1846 /Encoding 256 array
       
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  1852 currentdict end
       
  1853 currentfile eexec
       
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  1942 %%EndFont 
       
  1943 %%BeginFont: CMR12
       
  1944 %!PS-AdobeFont-1.1: CMR12 1.0
       
  1945 %%CreationDate: 1991 Aug 20 16:38:05
       
  1946 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1947 11 dict begin
       
  1948 /FontInfo 7 dict dup begin
       
  1949 /version (1.0) readonly def
       
  1950 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1951 /FullName (CMR12) readonly def
       
  1952 /FamilyName (Computer Modern) readonly def
       
  1953 /Weight (Medium) readonly def
       
  1954 /ItalicAngle 0 def
       
  1955 /isFixedPitch false def
       
  1956 end readonly def
       
  1957 /FontName /CMR12 def
       
  1958 /PaintType 0 def
       
  1959 /FontType 1 def
       
  1960 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1961 /Encoding 256 array
       
  1962 0 1 255 {1 index exch /.notdef put} for
       
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  1964 readonly def
       
  1965 /FontBBox{-34 -251 988 750}readonly def
       
  1966 /UniqueID 5000794 def
       
  1967 currentdict end
       
  1968 currentfile eexec
       
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  2058 cleartomark
       
  2059 %%EndFont 
       
  2060 %%BeginFont: CMTT12
       
  2061 %!PS-AdobeFont-1.1: CMTT12 1.0
       
  2062 %%CreationDate: 1991 Aug 20 16:45:46
       
  2063 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2064 11 dict begin
       
  2065 /FontInfo 7 dict dup begin
       
  2066 /version (1.0) readonly def
       
  2067 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2068 /FullName (CMTT12) readonly def
       
  2069 /FamilyName (Computer Modern) readonly def
       
  2070 /Weight (Medium) readonly def
       
  2071 /ItalicAngle 0 def
       
  2072 /isFixedPitch true def
       
  2073 end readonly def
       
  2074 /FontName /CMTT12 def
       
  2075 /PaintType 0 def
       
  2076 /FontType 1 def
       
  2077 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  2078 /Encoding 256 array
       
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  2081 readonly def
       
  2082 /FontBBox{-1 -234 524 695}readonly def
       
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  2084 currentdict end
       
  2085 currentfile eexec
       
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  2223 cleartomark
       
  2224 %%EndFont 
       
  2225 %%BeginFont: CMMI12
       
  2226 %!PS-AdobeFont-1.1: CMMI12 1.100
       
  2227 %%CreationDate: 1996 Jul 27 08:57:55
       
  2228 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2229 11 dict begin
       
  2230 /FontInfo 7 dict dup begin
       
  2231 /version (1.100) readonly def
       
  2232 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2233 /FullName (CMMI12) readonly def
       
  2234 /FamilyName (Computer Modern) readonly def
       
  2235 /Weight (Medium) readonly def
       
  2236 /ItalicAngle -14.04 def
       
  2237 /isFixedPitch false def
       
  2238 end readonly def
       
  2239 /FontName /CMMI12 def
       
  2240 /PaintType 0 def
       
  2241 /FontType 1 def
       
  2242 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  2243 /Encoding 256 array
       
  2244 0 1 255 {1 index exch /.notdef put} for
       
  2245 dup 0 /.notdef put
       
  2246 readonly def
       
  2247 /FontBBox{-30 -250 1026 750}readonly def
       
  2248 /UniqueID 5087386 def
       
  2249 currentdict end
       
  2250 currentfile eexec
       
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  2542 cleartomark
       
  2543 %%EndFont 
       
  2544 %%BeginFont: CMMI10
       
  2545 %!PS-AdobeFont-1.1: CMMI10 1.100
       
  2546 %%CreationDate: 1996 Jul 23 07:53:57
       
  2547 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2548 11 dict begin
       
  2549 /FontInfo 7 dict dup begin
       
  2550 /version (1.100) readonly def
       
  2551 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2552 /FullName (CMMI10) readonly def
       
  2553 /FamilyName (Computer Modern) readonly def
       
  2554 /Weight (Medium) readonly def
       
  2555 /ItalicAngle -14.04 def
       
  2556 /isFixedPitch false def
       
  2557 end readonly def
       
  2558 /FontName /CMMI10 def
       
  2559 /PaintType 0 def
       
  2560 /FontType 1 def
       
  2561 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  2562 /Encoding 256 array
       
  2563 0 1 255 {1 index exch /.notdef put} for
       
  2564 dup 0 /.notdef put
       
  2565 readonly def
       
  2566 /FontBBox{-32 -250 1048 750}readonly def
       
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  2568 currentdict end
       
  2569 currentfile eexec
       
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  2666 %%EndFont 
       
  2667 %%BeginFont: CMR8
       
  2668 %!PS-AdobeFont-1.1: CMR8 1.0
       
  2669 %%CreationDate: 1991 Aug 20 16:39:40
       
  2670 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2671 11 dict begin
       
  2672 /FontInfo 7 dict dup begin
       
  2673 /version (1.0) readonly def
       
  2674 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2675 /FullName (CMR8) readonly def
       
  2676 /FamilyName (Computer Modern) readonly def
       
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  2678 /ItalicAngle 0 def
       
  2679 /isFixedPitch false def
       
  2680 end readonly def
       
  2681 /FontName /CMR8 def
       
  2682 /PaintType 0 def
       
  2683 /FontType 1 def
       
  2684 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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  2687 dup 0 /.notdef put
       
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  2691 currentdict end
       
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  2832 cleartomark
       
  2833 %%EndFont 
       
  2834 TeXDict begin 39158280 55380996 1000 600 600 (nomu-tcs.dvi)
       
  2835 @start /Fa 254[23 1[{}1 66.4176 /CMBSY7 rf /Fb 137[35
       
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  3079 1310 334 1001 4 v 1369 425 a Fv(let)18 b Fz(X)24 b Fu(\()p
       
  3080 Fz(\025a:F)30 b(a)p Fu(\))56 b Fw(+)f Fz(V)1109 b FA(\(4\))166
       
  3081 647 y(or)l(,)25 b(modulo)e Fz(\021)t FA(-equi)n(v)n(alence)1371
       
  3082 853 y Fv(app)17 b Fu(\()p Fv(fn)g Fz(F)d Fu(\))j Fz(X)63
       
  3083 b Fw(+)55 b Fz(V)p 1371 898 798 4 v 1468 984 a Fv(let)18
       
  3084 b Fz(X)24 b(F)69 b Fw(+)56 b Fz(V)2224 901 y FA(.)3338
       
  3085 984 y(\(5\))166 1206 y(No)n(w)26 b(goal)h(\(3\))g(becomes)g
       
  3086 Fv(let)17 b Fu(1)g Fz(\025b:b)32 b Fw(+)g Fu(1)26 b FA(and)h(there)g
       
  3087 (is)g(no)g(problem)f(unifying)f(it)i(with)f(the)166 1326
       
  3088 y(conclusion)i(of)h(\(5\))h(via)f(a)g(capture-a)n(v)n(oiding)g
       
  3089 (substitution)d(of)j Fu(1)h FA(for)f Fz(X)8 b FA(,)29
       
  3090 b Fz(\025c:c)g FA(for)h Fz(F)43 b FA(and)29 b Fu(1)166
       
  3091 1447 y FA(for)c Fz(V)d FA(.)166 1668 y(This)36 b(is)h(all)f(v)o(ery)h
       
  3092 (\002ne,)g(b)n(ut)f(the)h(situation)e(is)i(not)f(as)h(pleasant)f(as)h
       
  3093 (for)g(\002rst-order)h(terms:)166 1788 y(higher)n(-order)43
       
  3094 b(uni\002cation)g(problems)f(can)h(be)h(undecidable,)e(decidable)h(b)n
       
  3095 (ut)g(lack)g(most)166 1909 y(general)h(uni\002ers,)f(or)g(ha)n(v)o(e)g
       
  3096 (such)g(uni\002ers)g(only)g(by)g(imposing)e(some)i(restrictions)f([3];)
       
  3097 166 2029 y(see)36 b([4])g(for)g(a)g(surv)o(e)o(y)f(of)g(higher)n
       
  3098 (-order)h(uni\002cation.)f(W)-8 b(e)36 b(started)g(out)f(w)o(anting)g
       
  3099 (to)g(com-)166 2149 y(pute)42 b(with)g(binders)f(modulo)g
       
  3100 Fz(\013)q FA(-equi)n(v)n(alence,)g(and)h(someho)n(w)f(the)h(process)g
       
  3101 (of)h(making)166 2270 y(possibly-capturing)31 b(substitution)f
       
  3102 (respectable)k(has)f(led)g(to)f(function)h(v)n(ariables,)f(applica-)166
       
  3103 2390 y(tion,)24 b(capture-a)n(v)n(oiding)g(substitution)e(and)j
       
  3104 Fz(\014)6 b(\021)t FA(-equi)n(v)n(alence.)23 b(Does)h(it)h(ha)n(v)o(e)f
       
  3105 (to)h(be)g(so?)f(No!)166 2611 y(F)o(or)43 b(one)g(thing,)f(se)n(v)o
       
  3106 (eral)g(authors)h(ha)n(v)o(e)g(already)g(noted)g(that)f(one)h(can)h
       
  3107 (mak)o(e)f(sense)g(of)166 2731 y(possibly-capturing)35
       
  3108 b(substitution)f(modulo)i Fz(\013)q FA(-equi)n(v)n(alence)f(by)i(using)
       
  3109 f Fx(e)n(xplicit)h(substitu-)166 2852 y(tions)21 b FA(in)h(the)h(term)f
       
  3110 (representation)g(language:)g(see)g([1,5\2269].)g(Compared)h(with)e
       
  3111 (those)h(w)o(orks,)166 2972 y(we)36 b(mak)o(e)g(a)h(number)f(of)g
       
  3112 (simpli\002cations.)e(First,)i(we)g(\002nd)g(that)g(we)g(do)g(not)g
       
  3113 (need)g(to)g(use)166 3093 y(function)24 b(v)n(ariables,)g(application)f
       
  3114 (or)i Fz(\014)6 b(\021)t FA(-equi)n(v)n(alence)23 b(in)h(our)g
       
  3115 (representation)g(language\227)166 3213 y(lea)n(ving)d(just)f(binders)g
       
  3116 (and)i Fz(\013)q FA(-equi)n(v)n(alence.)d(Secondly)-6
       
  3117 b(,)21 b(instead)f(of)i(using)e(e)o(xplicit)g(substitu-)166
       
  3118 3333 y(tions)26 b(of)h(names)g(for)h(names,)e(we)i(use)f(only)f(the)h
       
  3119 (special)g(case)h(of)f Fx(e)n(xplicit)g(permutations)e
       
  3120 FA(of)166 3454 y(names.)33 b(The)g(idea)g(of)g(using)f
       
  3121 (name-permutations,)f(and)i(in)f(particular)h(name-sw)o(appings,)166
       
  3122 3574 y(when)g(dealing)g(with)g Fz(\013)q FA(-con)l(v)o(ersion)f(w)o(as)
       
  3123 i(described)f(in)g([10])h(and)f(there)h(is)f(gro)n(wing)f(e)n(vi-)166
       
  3124 3694 y(dence)k(of)f(its)f(usefulness)g(\(see)i([11\22613],)f(for)g(e)o
       
  3125 (xample\).)f(When)i(a)f(name)g(substitution)d(is)166
       
  3126 3815 y(actually)f(a)g(permutation,)f(the)h(function)f(it)h(induces)g
       
  3127 (from)g(terms)g(to)f(terms)h(is)g(a)g(bijection;)166
       
  3128 3935 y(this)g(bijecti)n(vity)f(gi)n(v)o(es)g(the)i(operation)g(of)g
       
  3129 (permuting)e(names)i(v)o(ery)g(good)f(logical)h(proper)n(-)166
       
  3130 4056 y(ties)21 b(compared)g(with)f(name)h(substitution.)d(Consider)j
       
  3131 (for)h(e)o(xample)e(the)h Fz(\013)q FA(-equi)n(v)n(alent)e(terms)166
       
  3132 4176 y Fv(fn)e Fz(a:b)j FA(and)g Fv(fn)d Fz(c:b)p FA(,)j(where)g
       
  3133 Fz(a)p FA(,)f Fz(b)h FA(and)g Fz(c)f FA(are)h(distinct.)e(If)i(we)f
       
  3134 (apply)g(the)h(substitution)c Fu([)p Fz(b)3141 4170 y
       
  3135 Ft(7!)3212 4176 y Fz(a)p Fu(])k FA(\(re-)166 4296 y(naming)f(all)h
       
  3136 (free)h(occurrences)f(of)h Fz(b)f FA(to)g(be)g Fz(a)p
       
  3137 FA(\))g(to)g(them)f(we)i(get)e Fv(fn)f Fz(a:a)i FA(and)g
       
  3138 Fv(fn)d Fz(c:a)p FA(,)j(which)g(are)166 4417 y(no)27
       
  3139 b(longer)g Fz(\013)q FA(-equi)n(v)n(alent.)f(Thus)g(renaming)h
       
  3140 (substitutions)d(do)k(not)e(respect)i Fz(\013)q FA(-equi)n(v)n(alence)
       
  3141 166 4537 y(in)20 b(general,)h(and)g(an)o(y)f(uni\002cation)h(algorithm)
       
  3142 e(using)h(them)g(needs)h(to)f(tak)o(e)h(e)o(xtra)f(precautions)166
       
  3143 4658 y(to)j(not)f(inadv)o(ertently)f(change)i(the)g(intended)f(meaning)
       
  3144 h(of)g(terms.)f(The)h(traditional)f(solution)166 4778
       
  3145 y(for)27 b(this)f(problem)g(is)g(to)g(introduce)g(a)h(more)g
       
  3146 (complicated)e(form)i(of)g(renaming)f(substitution)166
       
  3147 4898 y(that)e(a)n(v)n(oids)f(capture)h(of)h(names)e(by)h(binders.)g(In)
       
  3148 g(contrast,)f(the)h(simple)f(operation)h(of)g(name-)166
       
  3149 5019 y(permutation)29 b(respects)h Fz(\013)q FA(-equi)n(v)n(alence;)f
       
  3150 (for)i(e)o(xample,)e(applying)g(the)h(name-permutation)166
       
  3151 5139 y Fu(\()p Fz(a)17 b(b)p Fu(\))30 b FA(that)e(sw)o(aps)h(all)g
       
  3152 (occurrences)h(of)g Fz(a)f FA(and)g Fz(b)h FA(\(be)g(the)o(y)e(free,)i
       
  3153 (bound)f(or)g(binding\))f(to)h(the)166 5259 y(terms)f(abo)o(v)o(e)g(gi)
       
  3154 n(v)o(es)f Fv(fn)17 b Fz(b:a)29 b FA(and)g Fv(fn)17 b
       
  3155 Fz(c:a)p FA(,)29 b(which)f(are)h(still)f Fz(\013)q FA(-equi)n(v)n
       
  3156 (alent.)e(W)-8 b(e)29 b(e)o(xploit)e(such)166 5380 y(good)38
       
  3157 b(properties)g(of)h(name-permutations)d(to)i(gi)n(v)o(e)g(a)h
       
  3158 (conceptually)e(simple)h(uni\002cation)1773 5712 y(3)p
       
  3159 eop end
       
  3160 %%Page: 4 4
       
  3161 TeXDict begin 4 3 bop 166 83 a FA(algorithm.)166 303
       
  3162 y(In)22 b(addition)e(to)h(the)g(use)g(of)h(e)o(xplicit)e
       
  3163 (name-permutations,)g(we)h(also)g(compute)g(symbolically)166
       
  3164 423 y(with)26 b(predicates)g(e)o(xpressing)f Fx(fr)l(eshness)g
       
  3165 FA(of)h(names)g(for)h(terms.)e(Such)i(predicates)f(certainly)166
       
  3166 544 y(feature)43 b(in)e(pre)n(vious)g(w)o(ork)h(on)f(binding)g(\(for)h
       
  3167 (e)o(xample,)f(Qu-Prolog')-5 b(s)41 b Fs(not)p 3030 544
       
  3168 30 4 v 35 w(free)p 3305 544 V 35 w(in)166 664 y FA(predicate)26
       
  3169 b([8],)g(the)f(notion)f(of)i(\223algebraic)g(independence\224)g(in)f
       
  3170 ([14,)g(De\002nition)g(3],)g(and)h(the)166 785 y
       
  3171 (\223non-occurrence\224)31 b(predicates)f(of)g([15]\).)g(But)h(once)f
       
  3172 (again,)f(the)h(use)g(of)g(such)g(a)g(freshness)166 905
       
  3173 y(predicate)d(based)g(upon)f(name)h Fx(swapping)f FA(rather)h(than)f
       
  3174 (renaming,)g(which)h(arises)g(naturally)166 1025 y(from)33
       
  3175 b(the)g(w)o(ork)g(reported)g(in)g([10,16],)f(gi)n(v)o(es)g(us)h(a)g
       
  3176 (simpler)f(theory)h(with)f(good)h(algorith-)166 1146
       
  3177 y(mic)28 b(properties.)f(It)h(is)g(easy)g(to)f(see)i(why)e(there)h(is)g
       
  3178 (a)g(need)g(for)h(computing)d(with)h(freshness,)166 1266
       
  3179 y(gi)n(v)o(en)i(that)h(we)h(tak)o(e)g(a)g(\223nominal\224)f(approach)h
       
  3180 (to)f(binders.)g(\(In)h(other)f(w)o(ords)g(we)h(use)g(con-)166
       
  3181 1386 y(crete)25 b(v)o(ersions)e(of)i(binding)e(and)h
       
  3182 Fz(\013)q FA(-equi)n(v)n(alence)g(in)g(which)g(bound)f(entities)h(are)h
       
  3183 (named)f(e)o(x-)166 1507 y(plicitly)-6 b(,)26 b(rather)i(than)g(using)e
       
  3184 (de)i(Bruijn-style)f(representations,)g(as)h(for)g(e)o(xample)f(in)h
       
  3185 ([1,7].\))166 1627 y(A)i(basic)h(instance)e(of)i(our)f(generalised)g
       
  3186 (form)g(of)h Fz(\013)q FA(-equi)n(v)n(alence)e(identi\002es)h
       
  3187 Fv(fn)25 b Fz(a:X)39 b FA(with)166 1748 y Fv(fn)25 b
       
  3188 Fz(b:)p Fu(\()p Fz(a)17 b(b)p Fu(\))p Fr(\001)q Fz(X)33
       
  3189 b FA(pro)o(vided)25 b Fz(b)h Fx(is)e(fr)l(esh)i(for)h
       
  3190 Fz(X)8 b FA(,)25 b(where)h(the)f(subterm)f Fu(\()p Fz(a)17
       
  3191 b(b)p Fu(\))p Fr(\001)p Fz(X)34 b FA(indicates)24 b(an)i(e)o(x-)166
       
  3192 1868 y(plicit)h(permutation\227namely)e(the)i(sw)o(apping)g(of)h
       
  3193 Fz(a)f FA(and)h Fz(b)p FA(\227w)o(aiting)f(to)g(be)h(applied)f(to)g
       
  3194 Fz(X)8 b FA(.)166 1988 y(W)-8 b(e)30 b(write)f(\223)p
       
  3195 Fz(b)g FA(is)g(fresh)g(for)h Fz(X)8 b FA(\224)29 b(symbolically)e(as)i
       
  3196 Fz(b)36 b Fu(#)g Fz(X)8 b FA(;)29 b(the)g(intended)f(meaning)h(of)g
       
  3197 (this)166 2109 y(relation)g(is)g(that)g Fz(b)h FA(does)f(not)g(occur)g
       
  3198 (free)i(in)e(an)o(y)f(\(ground\))h(term)g(that)g(may)g(be)h
       
  3199 (substituted)166 2229 y(for)23 b Fz(X)8 b FA(.)23 b(If)g(we)g(kno)n(w)f
       
  3200 (more)h(about)f Fz(X)31 b FA(we)23 b(may)f(be)h(able)g(to)f(eliminate)g
       
  3201 (the)h(e)o(xplicit)e(permuta-)166 2350 y(tion)26 b(in)g
       
  3202 Fu(\()p Fz(a)17 b(b)p Fu(\))p Fr(\001)p Fz(X)8 b FA(;)26
       
  3203 b(for)h(e)o(xample,)e(if)i(we)f(kne)n(w)g(that)g Fz(a)31
       
  3204 b Fu(#)g Fz(X)k FA(holds)25 b(as)i(well)f(as)h Fz(b)k
       
  3205 Fu(#)g Fz(X)8 b FA(,)26 b(then)166 2470 y Fu(\()p Fz(a)17
       
  3206 b(b)p Fu(\))p Fr(\001)p Fz(X)33 b FA(can)25 b(be)g(replaced)g(by)g
       
  3207 Fz(X)8 b FA(.)166 2690 y(It)24 b(should)e(already)i(be)g(clear)h(from)e
       
  3208 (these)h(simple)e(e)o(xamples)h(that)g(in)h(our)g(setting)e(the)i
       
  3209 (appro-)166 2810 y(priate)c(notion)f(of)h(term-equality)f(is)h(not)g(a)
       
  3210 g(bare)h(equation,)e Fz(t)28 b Fw(\031)g Fz(t)2430 2774
       
  3211 y Ft(0)2454 2810 y FA(,)20 b(b)n(ut)g(rather)g(a)h(hypothetical)166
       
  3212 2931 y(judgement)j(of)g(the)h(form)1597 3051 y Fw(r)j(`)f
       
  3213 Fz(t)h Fw(\031)g Fz(t)1999 3010 y Ft(0)3338 3051 y FA(\(6\))166
       
  3214 3200 y(where)37 b Fw(r)g FA(is)g(a)g Fx(fr)l(eshness)f(en)l(vir)l
       
  3215 (onment)r FA(\227a)f(\002nite)i(set)f Fw(f)p Fz(a)2317
       
  3216 3215 y FG(1)2407 3200 y Fu(#)50 b Fz(X)2619 3215 y FG(1)2658
       
  3217 3200 y Fz(;)17 b(:)g(:)g(:)f(;)h(a)2928 3215 y Fq(n)3025
       
  3218 3200 y Fu(#)50 b Fz(X)3237 3215 y Fq(n)3284 3200 y Fw(g)37
       
  3219 b FA(of)166 3320 y(freshness)25 b(assumptions.)d(F)o(or)j(e)o(xample)
       
  3220 1055 3539 y Fw(f)p Fz(a)j Fu(#)g Fz(X)r(;)17 b(b)28 b
       
  3221 Fu(#)g Fz(X)8 b Fw(g)27 b(`)h Fv(fn)d Fz(a:X)36 b Fw(\031)28
       
  3222 b Fv(fn)d Fz(b:X)782 b FA(\(7\))166 3758 y(is)27 b(a)g(v)n(alid)f
       
  3223 (judgement)f(of)i(our)g Fx(nominal)f(equational)f(lo)o(gic)p
       
  3224 FA(.)h(Similarly)-6 b(,)26 b(judgements)f(about)166 3879
       
  3225 y(freshness)g(itself)f(will)g(tak)o(e)h(the)f(form)1571
       
  3226 4098 y Fw(r)k(`)g Fz(a)f Fu(#)h Fz(t)g(:)1290 b FA(\(8\))166
       
  3227 4317 y(T)-8 b(w)o(o)34 b(e)o(xamples)f(of)i(v)n(alid)e(freshness)h
       
  3228 (judgements)f(are)i Fw(f)p Fz(a)46 b Fu(#)f Fz(X)8 b
       
  3229 Fw(g)45 b(`)h Fz(a)f Fu(#)h Fv(fn)25 b Fz(b:X)60 b FA(and)166
       
  3230 4438 y Fw(;)28 b(`)f Fz(a)h Fu(#)g Fv(fn)d Fz(a:X)8 b
       
  3231 FA(.)166 4658 y(The)20 b(freshness)f(en)l(vironment)f
       
  3232 Fw(r)i FA(in)f(judgements)f(of)i(the)f(form)g(\(6\))h(and)g(\(8\))f(e)o
       
  3233 (xpresses)g(fresh-)166 4778 y(ness)28 b(conditions)f(that)h(an)o(y)g
       
  3234 (te)o(xtual)f(substitution)f(of)j(terms)f(for)g(v)n(ariables)g(must)f
       
  3235 (respect)i(in)166 4898 y(order)k(for)f(the)g(right-hand)g(side)g(of)g
       
  3236 (the)g(judgement)f(to)h(be)g(v)n(alid)f(after)i(substitution.)d(This)
       
  3237 166 5019 y(e)o(xplicit)37 b(use)h(of)g(freshness)f(mak)o(es)h(the)g
       
  3238 (operation)f(of)i(te)o(xtual)d(substitution)g(respect)i(our)166
       
  3239 5139 y(generalised)29 b(form)g(of)h Fz(\013)q FA(-equi)n(v)n(alence.)e
       
  3240 (F)o(or)h(e)o(xample,)g(if)g(we)h(were)g(na)m(\250)-30
       
  3241 b(\021v)o(ely)28 b(to)h(re)o(gard)g(the)166 5259 y(terms)23
       
  3242 b Fv(fn)i Fz(a:X)32 b FA(and)23 b Fv(fn)j Fz(b:X)32 b
       
  3243 FA(as)23 b Fz(\013)q FA(-equi)n(v)n(alent,)f(then)h(applying)f(for)i(e)
       
  3244 o(xample)e(the)h(capturing)166 5380 y(substitution)j
       
  3245 Fu([)p Fz(X)44 b Fu(:=)35 b Fz(a)p Fu(])30 b FA(or)f
       
  3246 Fu([)p Fz(X)44 b Fu(:=)35 b Fz(b)p Fu(])30 b FA(results)e(into)h(tw)o
       
  3247 (o)f(terms)h(that)g(are)g Fx(not)i Fz(\013)q FA(-equi)n(v)n(alent)1773
       
  3248 5712 y(4)p eop end
       
  3249 %%Page: 5 5
       
  3250 TeXDict begin 5 4 bop 166 83 a FA(an)o(ymore.)22 b(\(A)h(similar)e
       
  3251 (observ)n(ation)g(partly)h(moti)n(v)n(ates)f(the)h(w)o(ork)h(in)f
       
  3252 ([17].\))h(Ho)n(we)n(v)o(er)l(,)e(if)i(we)166 203 y(assume)e
       
  3253 Fz(a)27 b Fu(#)h Fz(X)h FA(and)22 b Fz(b)28 b Fu(#)g
       
  3254 Fz(X)h FA(as)21 b(in)g(\(7\),)g(then)g(all)g(problematic)f
       
  3255 (substitutions)e(are)k(ruled)e(out.)166 324 y(In)k(this)e(w)o(ay)i(we)g
       
  3256 (obtain)f(a)g(v)o(ersion)g(of)h Fz(\013)q FA(-equi)n(v)n(alence)e
       
  3257 (between)i(terms)f(with)f(v)n(ariables)h(that)166 444
       
  3258 y(is)g(respected)g(by)g(te)o(xtual)e(substitutions)f(\(see)k(Lemma)e
       
  3259 (2.14)h(belo)n(w\),)f(unlik)o(e)g(the)h(traditional)166
       
  3260 565 y(notion)h(of)g Fz(\013)q FA(-equi)n(v)n(alence.)166
       
  3261 897 y Fx(Summary)166 1237 y FA(W)-8 b(e)26 b(will)f(represent)h
       
  3262 (languages)g(in)l(v)n(olving)e(binders)h(using)g(the)g(usual)h(notion)e
       
  3263 (of)i(\002rst-order)166 1357 y(terms)34 b(o)o(v)o(er)f(a)h(man)o
       
  3264 (y-sorted)f(signature,)g(b)n(ut)h(with)f(certain)h(distinguished)e
       
  3265 (constants)h(and)166 1478 y(function)26 b(symbols.)f(These)h(gi)n(v)o
       
  3266 (e)f(us)i(terms)f(with:)f(distinguished)f(constants)i(naming)f(bind-)
       
  3267 166 1598 y(able)35 b(entities,)f(that)g(we)i(call)e Fx(atoms)p
       
  3268 FA(;)g(terms)h Fz(a:t)g FA(e)o(xpressing)f(a)h(generic)g(form)g(of)g
       
  3269 Fx(binding)166 1719 y FA(of)g(an)f(atom)g Fz(a)h FA(in)f(a)h(term)f
       
  3270 Fz(t)p FA(;)g(and)h(terms)f Fz(\031)t Fr(\001)o Fz(X)43
       
  3271 b FA(representing)34 b(an)g(e)o(xplicit)f Fx(permutation)g
       
  3272 Fz(\031)166 1839 y FA(of)d(atoms)e(w)o(aiting)h(to)g(be)g(applied)g(to)
       
  3273 h(whate)n(v)o(er)e(term)h(is)h(substituted)d(for)j(the)f(v)n(ariable)g
       
  3274 Fz(X)8 b FA(.)166 1959 y(Section)20 b(2)h(presents)e(this)h
       
  3275 (term-language)g(together)f(with)h(a)h(syntax-directed)e(inducti)n(v)o
       
  3276 (e)f(def-)166 2080 y(inition)25 b(of)i(the)g(pro)o(v)n(able)e
       
  3277 (judgements)h(of)h(the)f(form)h(\(6\))g(and)g(\(8\))g(which)f(for)h
       
  3278 Fx(gr)l(ound)h FA(terms)166 2200 y(\(i.e.)36 b(ones)g(with)f(no)g(v)n
       
  3279 (ariables\))h(agrees)g(with)g(the)f(usual)h(notions)e(of)i
       
  3280 Fz(\013)q FA(-equi)n(v)n(alence)f(and)166 2320 y(\223not)26
       
  3281 b(a)g(free)h(v)n(ariable)e(of)5 b(\224.)26 b(Ho)n(we)n(v)o(er)l(,)f(on)
       
  3282 h(open)f(terms)h(our)f(judgements)g(dif)n(fer)g(from)h(these)166
       
  3283 2441 y(standard)h(notions.)f(Section)i(3)f(considers)g(uni\002cation)g
       
  3284 (in)g(this)g(setting.)f(Solving)h(equalities)166 2561
       
  3285 y(between)h(abstractions)f Fz(a:t)34 b Fw(\031)q Fu(?)f
       
  3286 Fz(a)1385 2525 y Ft(0)1409 2561 y Fz(:t)1471 2525 y Ft(0)1522
       
  3287 2561 y FA(entails)28 b(solving)e(both)h(equalities)g
       
  3288 Fz(t)34 b Fw(\031)q Fu(?)f(\()p Fz(a)17 b(a)3130 2525
       
  3289 y Ft(0)3153 2561 y Fu(\))p Fr(\001)p Fz(t)3258 2525 y
       
  3290 Ft(0)3310 2561 y FA(and)166 2682 y(freshness)37 b(problems)f
       
  3291 Fz(a)51 b Fu(#?)g Fz(t)1297 2645 y Ft(0)1320 2682 y FA(.)37
       
  3292 b(Therefore)h(our)f(general)h(form)f(of)g Fx(nominal)f(uni\002cation)
       
  3293 166 2802 y(pr)l(oblem)23 b FA(is)h(a)g(\002nite)h(collection)e(of)h
       
  3294 (indi)n(vidual)e(equality)i(and)g(freshness)g(problems.)f(Such)h(a)166
       
  3295 2922 y(problem)e Fz(P)37 b FA(is)23 b(solv)o(ed)f(by)h(pro)o(viding)e
       
  3296 (not)i(only)f(a)i(substitution)c Fz(\033)28 b FA(\(of)23
       
  3297 b(terms)g(for)g(v)n(ariables\),)166 3043 y(b)n(ut)31
       
  3298 b(also)g(a)h(freshness)g(en)l(vironment)e Fw(r)i FA(\(as)g(abo)o(v)o
       
  3299 (e\),)f(which)g(together)g(ha)n(v)o(e)h(the)f(property)166
       
  3300 3163 y(that)25 b Fw(r)i(`)h Fz(\033)t Fu(\()p Fz(t)p
       
  3301 Fu(\))g Fw(\031)h Fz(\033)t Fu(\()p Fz(t)976 3127 y Ft(0)999
       
  3302 3163 y Fu(\))c FA(and)g Fw(r)j(`)g Fz(a)g Fu(#)g Fz(\033)t
       
  3303 Fu(\()p Fz(t)1751 3127 y Ft(00)1794 3163 y Fu(\))d FA(hold)f(for)h
       
  3304 (each)h(indi)n(vidual)d(equality)h Fz(t)k Fw(\031)p Fu(?)g
       
  3305 Fz(t)3430 3127 y Ft(0)166 3284 y FA(and)i(freshness)g
       
  3306 Fz(a)38 b Fu(#?)g Fz(t)1030 3247 y Ft(00)1102 3284 y
       
  3307 FA(in)30 b(the)g(problem)f Fz(P)14 b FA(.)30 b(Our)g(main)f(result)h
       
  3308 (with)f(respect)i(to)e(uni\002ca-)166 3404 y(tion)20
       
  3309 b(is)h(that)g Fx(solvability)e(is)i(decidable)g(and)g(that)f(solvable)g
       
  3310 (pr)l(oblems)g(possess)g(most)h(g)o(ener)o(al)166 3524
       
  3311 y(solutions)30 b FA(\(for)i(a)g(reasonably)g(ob)o(vious)d(notion)i(of)h
       
  3312 (\223most)f(general\224\).)h(The)g(proof)g(is)f(via)g(a)166
       
  3313 3645 y(uni\002cation)i(algorithm)f(that)h(is)g(v)o(ery)g(similar)g(to)g
       
  3314 (the)h(\002rst-order)g(algorithm)e(gi)n(v)o(en)g(in)h(the)166
       
  3315 3765 y(no)n(w-common)23 b(transformational)g(style)h([18].)h(\(See)h
       
  3316 ([19,)f(Sect.)g(2.6])g(or)g([20,)g(Sect.)g(4.6])g(for)166
       
  3317 3885 y(e)o(xpositions)k(of)j(this.\))f(Section)h(4)f(considers)h(the)f
       
  3318 (relationship)g(of)g(our)h(v)o(ersion)f(of)h(\223uni\002-)166
       
  3319 4006 y(cation)26 b(modulo)f Fz(\013)q FA(-equi)n(v)n(alence\224)g(to)g
       
  3320 (e)o(xisting)g(approaches.)h(Section)g(5)g(assesses)g(what)g(has)166
       
  3321 4126 y(been)f(achie)n(v)o(ed)f(and)h(the)f(prospects)h(for)g
       
  3322 (applications.)166 4458 y Fx(Quiz)166 4799 y FA(T)-8
       
  3323 b(o)29 b(appreciate)i(the)e(kind)g(of)h(problem)e(that)h(nominal)g
       
  3324 (uni\002cation)g(solv)o(es,)f(you)h(might)f(lik)o(e)166
       
  3325 4919 y(to)h(try)g(the)g(follo)n(wing)f(quiz)h(about)g(the)g
       
  3326 Fz(\025)p FA(-calculus)g([21])h(before)f(we)h(apply)f(our)g(algorithm)
       
  3327 166 5039 y(to)24 b(solv)o(e)g(it)g(at)h(the)g(end)g(of)g(Section)f(3.)
       
  3328 166 5259 y Fx(Assuming)i Fz(a)i Fx(and)g Fz(b)g Fx(ar)l(e)g(distinct)e
       
  3329 (variables)p FA(,)g(is)i(it)f(possible)f(to)i(\002nd)f
       
  3330 Fz(\025)p FA(-terms)h Fz(M)3062 5274 y FG(1)3101 5259
       
  3331 y Fz(;)17 b(:)g(:)g(:)f(;)h(M)3414 5274 y FG(7)166 5380
       
  3332 y FA(that)24 b(mak)o(e)h(the)g(follo)n(wing)e(pairs)h(of)h(terms)f
       
  3333 Fz(\013)q FA(-equi)n(v)n(alent?)1773 5712 y(5)p eop end
       
  3334 %%Page: 6 6
       
  3335 TeXDict begin 6 5 bop 206 83 a FA(\(1\))50 b Fz(\025a:\025b:)p
       
  3336 Fu(\()p Fz(M)764 98 y FG(1)821 83 y Fz(b)p Fu(\))125
       
  3337 b FA(and)f Fz(\025b:\025a:)p Fu(\()p Fz(a)17 b(M)1753
       
  3338 98 y FG(1)1794 83 y Fu(\))206 203 y FA(\(2\))50 b Fz(\025a:\025b:)p
       
  3339 Fu(\()p Fz(M)764 218 y FG(2)821 203 y Fz(b)p Fu(\))125
       
  3340 b FA(and)f Fz(\025b:\025a:)p Fu(\()p Fz(a)17 b(M)1753
       
  3341 218 y FG(3)1794 203 y Fu(\))206 324 y FA(\(3\))50 b Fz(\025a:\025b:)p
       
  3342 Fu(\()p Fz(b)17 b(M)822 339 y FG(4)862 324 y Fu(\))125
       
  3343 b FA(and)f Fz(\025b:\025a:)p Fu(\()p Fz(a)17 b(M)1753
       
  3344 339 y FG(5)1794 324 y Fu(\))206 444 y FA(\(4\))50 b Fz(\025a:\025b:)p
       
  3345 Fu(\()p Fz(b)17 b(M)822 459 y FG(6)862 444 y Fu(\))125
       
  3346 b FA(and)f Fz(\025a:\025a:)p Fu(\()p Fz(a)17 b(M)1763
       
  3347 459 y FG(7)1803 444 y Fu(\))166 679 y FA(If)28 b(it)g(is)f(possible)g
       
  3348 (to)g(\002nd)h(a)g(solution)e(for)j(an)o(y)e(of)h(these)f(four)h
       
  3349 (problems,)f(can)h(you)g(describe)166 800 y(what)36 b(all)g(possible)e
       
  3350 (solutions)h(for)h(that)f(problem)h(are)h(lik)o(e?)e(\(The)i(answers)f
       
  3351 (are)h(gi)n(v)o(en)d(in)166 920 y(Example)24 b(3.8.\))166
       
  3352 1423 y FB(2)99 b(Nominal)25 b(equational)g(logic)166
       
  3353 1779 y FA(W)-8 b(e)31 b(tak)o(e)g(a)g(concrete)g(approach)g(to)f(the)h
       
  3354 (syntax)e(of)i(binders)f(in)g(which)h(bound)f(entities)f(are)166
       
  3355 1899 y(e)o(xplicitly)d(named.)i(Furthermore)h(we)g(do)f(not)g(assume)g
       
  3356 (that)f(the)i(names)f(of)g(bound)g(entities)166 2019
       
  3357 y(are)23 b(necessarily)f(v)n(ariables)f(\(things)g(that)h(may)f(be)h
       
  3358 (substituted)e(for\),)j(in)f(order)g(to)g(encompass)166
       
  3359 2140 y(e)o(xamples)29 b(lik)o(e)g(the)h Fz(\031)t FA(-calculus)f([22],)
       
  3360 h(in)f(which)h(the)f(restriction)g(operator)h(binds)f(channel)166
       
  3361 2260 y(names)g(and)h(these)f(are)i(quite)e(dif)n(ferent)g(from)h(names)
       
  3362 f(of)h(unkno)n(wn)e(processes.)h(Names)h(of)166 2380
       
  3363 y(bound)25 b(entities)g(will)g(be)h(called)g Fx(atoms)p
       
  3364 FA(.)f(This)h(is)f(partly)h(for)g(historical)f(reasons)h(\(stemming)166
       
  3365 2501 y(from)32 b(the)f(w)o(ork)h(by)f(the)h(second)g(tw)o(o)f(authors)g
       
  3366 ([10]\))h(and)g(partly)f(to)h(indicate)f(that)g(the)h(in-)166
       
  3367 2621 y(ternal)26 b(structure)h(of)f(such)g(names)g(is)g(irrele)n(v)n
       
  3368 (ant)g(to)g(us:)g(all)g(we)h(care)g(about)f(is)g(their)g(identity)166
       
  3369 2742 y(\(i.e.)i(whether)g(or)g(not)f(one)h(atom)g(is)f(the)h(same)g(as)
       
  3370 g(another\))g(and)g(that)f(the)h(supply)f(of)h(atoms)166
       
  3371 2862 y(is)c(ine)o(xhaustible.)166 3097 y(Although)i(there)h(are)i(se)n
       
  3372 (v)o(eral)d(general)i(frame)n(w)o(orks)f(in)f(the)i(literature)f(for)g
       
  3373 (specifying)g(lan-)166 3217 y(guages)j(with)f(binders,)g(not)h(all)f
       
  3374 (of)h(them)g(meet)f(the)h(requirements)f(mentioned)g(in)h(the)f(pre-)
       
  3375 166 3338 y(vious)g(paragraph.)h(Use)g(of)h(the)e(simply)g(typed)g
       
  3376 Fz(\025)p FA(-calculus)h(for)g(this)f(purpose)h(is)f(common;)166
       
  3377 3458 y(b)n(ut)35 b(as)g(discussed)f(in)h(the)g(Introduction,)f(it)h
       
  3378 (leads)g(to)f(a)i(problematic)e(uni\002cation)h(theory)-6
       
  3379 b(.)166 3579 y(Among)29 b Fx(\002r)o(st-or)l(der)j FA(frame)n(w)o
       
  3380 (orks,)d(Plotkin')-5 b(s)29 b(notion)g(of)h Fx(binding)f(signatur)l(e)g
       
  3381 FA([23,24],)g(be-)166 3699 y(ing)f(unsorted,)h(equates)f(names)h(used)g
       
  3382 (in)f(binding)g(with)g(names)h(of)g(v)n(ariables)f(standing)g(for)166
       
  3383 3819 y(unkno)n(wn)22 b(terms;)h(so)h(it)f(is)g(not)g(suf)n(\002ciently)
       
  3384 g(general)h(for)g(us.)g(A)g(\002rst-order)g(frame)n(w)o(ork)f(that)166
       
  3385 3940 y(does)h(meet)f(our)h(requirements)f(is)g(the)h(notion)e(of)i
       
  3386 Fx(nominal)e(alg)o(ebr)o(as)g FA(in)i([15].)f(The)h Fx(nominal)166
       
  3387 4060 y(signatur)l(es)19 b FA(that)i(we)g(use)g(in)f(this)g(paper)h(are)
       
  3388 h(a)f(mild)e(\(b)n(ut)i(practically)f(useful\))h(generalisation)166
       
  3389 4181 y(of)27 b(nominal)e(algebras)i(in)g(which)f(name-abstraction)g
       
  3390 (and)h(pairing)f(can)h(be)g(mix)o(ed)f(freely)h(in)166
       
  3391 4301 y(arities)h(\(rather)g(than)g(insisting)e(as)i(in)g([15])g(that)g
       
  3392 (the)g(ar)n(gument)g(sort)f(of)i(a)f(function)f(symbol)166
       
  3393 4421 y(be)e(normalised)f(to)g(a)h(tuple)f(of)h(abstractions\).)166
       
  3394 4656 y FB(De\002nition)f(2.1.)40 b FA(A)23 b Fx(nominal)f(signatur)l(e)
       
  3395 h FA(is)g(speci\002ed)g(by:)g(a)h(set)g(of)f Fx(sorts)g(of)g(atoms)f
       
  3396 FA(\(typical)166 4777 y(symbol)28 b Fz(\027)6 b FA(\);)31
       
  3397 b(a)f(disjoint)e(set)i(of)g Fx(sorts)e(of)i(data)f FA(\(typical)g
       
  3398 (symbol)g Fz(\016)t FA(\);)g(and)h(a)h(set)e(of)h Fx(function)166
       
  3399 4897 y(symbols)25 b FA(\(typical)h(symbol)f Fz(f)11 b
       
  3400 FA(\),)26 b(each)h(of)g(which)f(has)g(an)g Fx(arity)g
       
  3401 FA(of)g(the)h(form)f Fz(\034)42 b Fw(!)30 b Fz(\016)t
       
  3402 FA(.)c(Here)h Fz(\034)166 5018 y FA(ranges)d(o)o(v)o(er)f(\(compound\))
       
  3403 g Fx(sorts)g FA(gi)n(v)o(en)g(by)g(the)h(grammar)g Fz(\034)39
       
  3404 b Fu(::=)28 b Fz(\027)34 b Fw(j)27 b Fz(\016)32 b Fw(j)27
       
  3405 b Fu(1)h Fw(j)f Fz(\034)k Fw(\002)19 b Fz(\034)39 b Fw(j)28
       
  3406 b(h)p Fz(\027)6 b Fw(i)p Fz(\034)39 b FA(.)166 5138 y(Sorts)26
       
  3407 b(of)g(the)h(form)f Fw(h)p Fz(\027)6 b Fw(i)p Fz(\034)37
       
  3408 b FA(classify)26 b(terms)g(that)f(are)i(binding)e(abstractions)g(of)i
       
  3409 (atoms)e(of)h(sort)166 5258 y Fz(\027)31 b FA(o)o(v)o(er)24
       
  3410 b(terms)g(of)g(sort)g Fz(\034)11 b FA(.)25 b(W)-8 b(e)25
       
  3411 b(will)e(e)o(xplain)h(the)g(syntax)g(and)g(properties)g(of)g(such)h
       
  3412 (terms)e(in)h(a)166 5379 y(moment.)1773 5712 y(6)p eop
       
  3413 end
       
  3414 %%Page: 7 7
       
  3415 TeXDict begin 7 6 bop 166 83 a FB(Example)31 b(2.2.)43
       
  3416 b FA(Here)31 b(is)f(a)g(nominal)f(signature)g(for)h(e)o(xpressions)f
       
  3417 (in)h(a)g(small)f(fragment)h(of)166 203 y(ML)24 b([25]:)638
       
  3418 416 y(sort)g(of)h(atoms:)267 b Fp(vid)638 530 y FA(sort)24
       
  3419 b(of)h(data:)340 b Fp(exp)638 645 y FA(function)24 b(symbols:)97
       
  3420 b Fv(vr)29 b Fu(:)e Fp(vid)38 b Fw(!)27 b Fp(exp)1454
       
  3421 759 y Fv(app)i Fu(:)f Fp(exp)f Fw(\002)c Fp(exp)33 b
       
  3422 Fw(!)27 b Fp(exp)1454 874 y Fv(fn)i Fu(:)e Fw(h)p Fp(vid)10
       
  3423 b Fw(i)p Fp(exp)33 b Fw(!)27 b Fp(exp)1454 988 y Fv(lv)i
       
  3424 Fu(:)e Fp(exp)h Fw(\002)23 b(h)p Fp(vid)9 b Fw(i)p Fp(exp)33
       
  3425 b Fw(!)28 b Fp(exp)1454 1102 y Fv(lf)h Fu(:)e Fw(h)p
       
  3426 Fp(vid)10 b Fw(i)p Fu(\(\()p Fw(h)p Fp(vid)f Fw(i)p Fp(exp)c
       
  3427 Fu(\))23 b Fw(\002)f Fp(exp)6 b Fu(\))27 b Fw(!)h Fp(exp)33
       
  3428 b FA(.)166 1322 y(The)27 b(function)f(symbol)f Fv(vr)j
       
  3429 FA(constructs)e(terms)g(of)h(sort)g Fp(exp)32 b FA(representing)27
       
  3430 b(v)n(alue)f(identi\002ers)166 1442 y(\(named)e(by)g(atoms)f(of)h(sort)
       
  3431 f Fp(vid)10 b FA(\);)24 b Fv(app)h FA(constructs)e(application)g(e)o
       
  3432 (xpressions)f(from)i(pairs)g(of)166 1562 y(e)o(xpressions;)29
       
  3433 b Fv(fn)p FA(,)h Fv(lv)h FA(and)g Fv(lf)g FA(construct)e(terms)h
       
  3434 (representing)g(respecti)n(v)o(ely)f(function)g(ab-)166
       
  3435 1683 y(stractions)k(\()p Fv(fn)25 b(x)j Fu(=)p Fz(>)g
       
  3436 Fv(e)p FA(\),)34 b(local)f(v)n(alue)g(declarations)h(\()p
       
  3437 Fv(let)26 b(val)f(x)j Fu(=)g Fv(e1)d(in)h(e2)f(end)q
       
  3438 FA(\))34 b(and)166 1803 y(local)28 b(recursi)n(v)o(e)e(function)h
       
  3439 (declarations)g(\()p Fv(let)f(fun)g(f)f(x)j Fu(=)g Fv(e1)d(in)h(e2)f
       
  3440 (end)q FA(\).)j(The)f(arities)h(of)166 1924 y(the)f(function)g(symbols)
       
  3441 e(specify)j(which)f(are)h(binders)f(and)g(in)g(which)g(w)o(ay)g(their)g
       
  3442 (ar)n(guments)166 2044 y(are)32 b(bound.)f(F)o(or)h(e)o(xample,)f(in)g
       
  3443 (the)h(e)o(xpression)e(\()p Fv(let)c(fun)g(f)f(x)j Fu(=)f
       
  3444 Fv(e1)f(in)f(e2)h(end)p FA(\))32 b(there)g(is)166 2164
       
  3445 y(a)h(binding)e(occurrence)i(of)g(the)f(v)n(alue)g(identi\002er)g
       
  3446 Fv(f)h FA(whose)f(scope)h(is)f(both)f(of)i Fv(e1)g FA(and)f
       
  3447 Fv(e2)q FA(;)166 2285 y(and)i(a)g(binding)e(occurrence)i(of)g(the)g(v)n
       
  3448 (alue)f(identi\002er)g Fv(x)h FA(whose)f(scope)h(is)f(just)g
       
  3449 Fv(e1)p FA(.)h(These)166 2405 y(binding)27 b(scopes)h(are)i
       
  3450 (re\003ected)f(by)g(the)f(ar)n(gument)g(sort)h(of)f(the)h(function)e
       
  3451 (symbol)g Fv(lf)q FA(.)i(This)166 2525 y(kind)i(of)h(speci\002cation)g
       
  3452 (of)g(binding)e(scopes)i(is)f(of)h(course)g(a)g(feature)h(of)f
       
  3453 Fx(higher)n(-or)l(der)f(ab-)166 2646 y(str)o(act)24 b(syntax)h
       
  3454 FA([26],)g(using)g(function)f(types)h Fz(\027)6 b Fw(!)p
       
  3455 Fz(\034)37 b FA(in)25 b(simply)e(typed)i Fz(\025)p FA(-calculus)g
       
  3456 (where)h(we)166 2766 y(use)g(abstraction)g(sorts)g Fw(h)p
       
  3457 Fz(\027)6 b Fw(i)p Fz(\034)11 b FA(.)26 b(W)-8 b(e)27
       
  3458 b(shall)f(see)h(that)f(the)g(latter)g(ha)n(v)o(e)g(much)g(more)g
       
  3459 (elementary)166 2887 y(\(indeed,)f(\002rst-order\))g(properties)f
       
  3460 (compared)h(with)f(the)h(former)-5 b(.)166 3116 y FB(De\002nition)40
       
  3461 b(2.3.)49 b FA(Gi)n(v)o(en)38 b(a)i(nominal)e(signature,)h(we)g(assume)
       
  3462 g(that)g(there)h(are)g(countably)166 3236 y(in\002nite)21
       
  3463 b(and)h(pairwise)f(disjoint)e(sets)j(of)f Fx(atoms)g
       
  3464 FA(\(typical)g(symbol)f Fz(a)p FA(\))i(for)g(each)g(sort)f(of)h(atoms)
       
  3465 166 3357 y Fz(\027)6 b FA(,)27 b(and)f Fx(variables)g
       
  3466 FA(\(typical)g(symbol)f Fz(X)8 b FA(\))26 b(for)h(each)g(sort)f(of)h
       
  3467 (atoms)e Fz(\027)34 b FA(and)26 b(each)h(sort)f(of)h(data)166
       
  3468 3477 y Fz(\016)t FA(.)36 b(The)h Fx(terms)e FA(o)o(v)o(er)h(a)g
       
  3469 (nominal)f(signature)h(and)g(their)g(sorts)f(are)i(inducti)n(v)o(ely)d
       
  3470 (de\002ned)j(as)166 3597 y(follo)n(ws,)23 b(where)j(we)f(write)f
       
  3471 Fz(t)k Fu(:)g Fz(\034)36 b FA(to)25 b(indicate)f(that)g(a)i(term)e
       
  3472 Fz(t)h FA(has)g(sort)f Fz(\034)11 b FA(.)166 3826 y FB(Unit)25
       
  3473 b(v)o(alue)50 b Fw(hi)27 b Fu(:)h(1)p FA(.)166 3947 y
       
  3474 FB(P)o(airs)49 b Fw(h)p Fz(t)510 3962 y FG(1)550 3947
       
  3475 y Fz(;)17 b(t)629 3962 y FG(2)668 3947 y Fw(i)27 b Fu(:)h
       
  3476 Fz(\034)831 3962 y FG(1)893 3947 y Fw(\002)23 b Fz(\034)1035
       
  3477 3962 y FG(2)1075 3947 y FA(,)h(if)h Fz(t)1245 3962 y
       
  3478 FG(1)1312 3947 y Fu(:)j Fz(\034)1409 3962 y FG(1)1474
       
  3479 3947 y FA(and)d Fz(t)1678 3962 y FG(2)1745 3947 y Fu(:)j
       
  3480 Fz(\034)1842 3962 y FG(2)1882 3947 y FA(.)166 4067 y
       
  3481 FB(Data)50 b Fz(f)27 b(t)h Fu(:)g Fz(\016)t FA(,)c(if)h
       
  3482 Fz(f)36 b FA(is)24 b(a)h(function)f(symbol)f(of)i(arity)g
       
  3483 Fz(\034)39 b Fw(!)27 b Fz(\016)i FA(and)c Fz(t)j Fu(:)g
       
  3484 Fz(\034)11 b FA(.)166 4188 y FB(Atoms)50 b Fz(a)27 b
       
  3485 Fu(:)h Fz(\027)6 b FA(,)25 b(if)g Fz(a)g FA(is)g(an)g(atom)f(of)h(sort)
       
  3486 f Fz(\027)6 b FA(.)166 4308 y FB(Atom-abstraction)50
       
  3487 b Fz(a:t)29 b Fu(:)e Fw(h)p Fz(\027)6 b Fw(i)p Fz(\034)11
       
  3488 b FA(,)26 b(if)e Fz(a)i FA(is)e(an)h(atom)f(of)h(sort)f
       
  3489 Fz(\027)32 b FA(and)24 b Fz(t)k Fu(:)g Fz(\034)11 b FA(.)166
       
  3490 4428 y FB(Suspension)51 b Fz(\031)t Fr(\001)p Fz(X)35
       
  3491 b Fu(:)28 b Fz(\034)11 b FA(,)21 b(if)g Fz(\031)32 b
       
  3492 Fu(=)27 b(\()p Fz(a)1414 4443 y FG(1)1471 4428 y Fz(b)1512
       
  3493 4443 y FG(1)1551 4428 y Fu(\)\()p Fz(a)1678 4443 y FG(2)1735
       
  3494 4428 y Fz(b)1776 4443 y FG(2)1815 4428 y Fu(\))17 b Fw(\001)g(\001)g
       
  3495 (\001)e Fu(\()p Fz(a)2092 4443 y Fq(n)2156 4428 y Fz(b)2197
       
  3496 4443 y Fq(n)2244 4428 y Fu(\))21 b FA(is)f(a)h(\002nite)g(list)f(whose)
       
  3497 h(elements)266 4549 y Fu(\()p Fz(a)355 4564 y Fq(i)400
       
  3498 4549 y Fz(b)441 4564 y Fq(i)469 4549 y Fu(\))27 b FA(are)h(pairs)f(of)g
       
  3499 (atoms,)f(with)g Fz(a)1559 4564 y Fq(i)1614 4549 y FA(and)h
       
  3500 Fz(b)1826 4564 y Fq(i)1882 4549 y FA(of)g(the)g(same)f(sort,)h(and)g
       
  3501 Fz(X)35 b FA(is)26 b(a)h(v)n(ariable)g(of)266 4669 y(sort)d
       
  3502 Fz(\034)11 b FA(,)25 b(where)h Fz(\034)36 b FA(is)24
       
  3503 b(either)h(a)g(sort)g(of)f(data)h(or)g(a)g(sort)g(of)g(atoms)f(\(i.e.)g
       
  3504 Fz(\034)40 b Fu(::=)27 b Fz(\027)34 b Fw(j)28 b Fz(\016)t
       
  3505 FA(\).)166 4898 y(Recall)20 b(that)f(e)n(v)o(ery)g(\002nite)g
       
  3506 (permutation)f(can)i(be)f(e)o(xpressed)g(as)h(a)f(composition)f(of)h
       
  3507 (sw)o(appings)166 5019 y Fu(\()p Fz(a)255 5034 y Fq(i)300
       
  3508 5019 y Fz(b)341 5034 y Fq(i)370 5019 y Fu(\))p FA(;)28
       
  3509 b(the)g(list)f Fz(\031)33 b FA(of)28 b(pairs)h(of)f(atoms)g(occurring)g
       
  3510 (in)g(a)h(suspension)e(term)h Fz(\031)t Fr(\001)p Fz(X)36
       
  3511 b FA(speci\002es)28 b(a)166 5139 y(\002nite)34 b(permutation)f(of)h
       
  3512 (atoms)g(w)o(aiting)f(to)h(be)g(applied)g(once)g(we)h(kno)n(w)e(more)h
       
  3513 (about)g(the)166 5259 y(v)n(ariable)e Fz(X)40 b FA(\(by)32
       
  3514 b(substituting)d(for)k(it,)e(for)i(e)o(xample\).)e(W)-8
       
  3515 b(e)33 b(represent)f(\002nite)g(permutations)166 5380
       
  3516 y(in)24 b(this)g(w)o(ay)h(because)g(it)f(is)g(really)g(the)h(operation)
       
  3517 f(of)g(sw)o(apping)g(which)g(plays)g(a)h(fundamen-)1773
       
  3518 5712 y(7)p eop end
       
  3519 %%Page: 8 8
       
  3520 TeXDict begin 8 7 bop 166 83 a FA(tal)33 b(r)8 b(\210)-41
       
  3521 b(ole)33 b(in)g(the)f(theory)-6 b(.)33 b(Since,)g(semantically)f
       
  3522 (speaking,)g(sw)o(apping)g(commutes)g(with)g(all)166
       
  3523 203 y(term-forming)f(operations,)h(we)h(can)g(normalise)e(terms)h(in)l
       
  3524 (v)n(olving)f(an)i(e)o(xplicit)e(sw)o(apping)166 324
       
  3525 y(operation)24 b(by)h(pushing)e(the)i(sw)o(ap)f(in)h(as)g(f)o(ar)g(as)g
       
  3526 (it)f(will)g(go,)h(until)e(it)h(reaches)i(a)f(v)n(ariable)f(\(ap-)166
       
  3527 444 y(plying)g(the)h(sw)o(apping)g(to)f(atoms)h(that)g(it)g(meets)f(on)
       
  3528 h(the)h(w)o(ay\);)f(the)g(terms)g(in)g(De\002nition)f(2.3)166
       
  3529 565 y(are)38 b(all)f(normalised)e(in)i(this)f(w)o(ay)-6
       
  3530 b(,)37 b(with)f(e)o(xplicit)f(permutations)h(\223piled)g(up\224)h(in)g
       
  3531 (front)g(of)166 685 y(v)n(ariables)30 b(gi)n(ving)g(what)h(we)g(ha)n(v)
       
  3532 o(e)g(called)g Fx(suspensions)p FA(.)f(In)h(case)h(the)f(permutation)e
       
  3533 Fz(\031)36 b FA(in)30 b(a)166 805 y(suspension)23 b(is)i(the)f(empty)g
       
  3534 (list,)g(we)h(just)f(write)g Fz(X)33 b FA(for)25 b Fz(\031)t
       
  3535 Fr(\001)p Fz(X)8 b FA(.)166 1038 y FB(De\002nition)38
       
  3536 b(2.4.)48 b FA(The)37 b Fx(permutation)e(action)p FA(,)h
       
  3537 Fz(\031)t Fr(\001)p Fz(t)p FA(,)h(of)g(a)h(\002nite)f(permutation)e(of)
       
  3538 j(atoms)e Fz(\031)166 1159 y FA(on)31 b(a)h(term)f Fz(t)h
       
  3539 FA(is)f(de\002ned)h(as)f(in)g(Figure)h(1,)f(making)f(use)i(of)f(the)h
       
  3540 (follo)n(wing)d(notations.)h(The)166 1279 y(composition)36
       
  3541 b(of)i(a)h(permutation)d Fz(\031)42 b FA(follo)n(wed)37
       
  3542 b(by)h(a)h(sw)o(ap)f Fu(\()p Fz(a)17 b(b)p Fu(\))38 b
       
  3543 FA(is)g(gi)n(v)o(en)e(by)i(list-cons,)166 1399 y(written)25
       
  3544 b Fu(\()p Fz(a)17 b(b)p Fu(\))30 b(::)f Fz(\031)t FA(.)d(\(Note)f(that)
       
  3545 h(we)g(apply)f(permutations)f(to)h(terms)g(on)h(the)f(left,)h(and)f
       
  3546 (hence)166 1520 y(the)20 b(order)g(of)g(the)g(composition)e(is)h(from)h
       
  3547 (right)f(to)h(left.\))g(The)g(composition)d(of)j Fz(\031)k
       
  3548 FA(follo)n(wed)19 b(by)166 1640 y(another)j(permutation)f
       
  3549 Fz(\031)1050 1604 y Ft(0)1096 1640 y FA(is)h(gi)n(v)o(en)f(by)h
       
  3550 (list-concatenation,)f(written)h(as)g Fz(\031)2772 1604
       
  3551 y Ft(0)2795 1640 y Fu(@)p Fz(\031)t FA(.)g(The)h Fx(identity)166
       
  3552 1761 y FA(permutation)j(is)g(gi)n(v)o(en)g(by)h(the)g(empty)f(list)g
       
  3553 Fu([])p FA(;)h(and)g(the)g Fx(in)l(ver)o(se)g FA(of)g(a)g(permutation)f
       
  3554 (is)g(gi)n(v)o(en)166 1881 y(by)f(list)e(re)n(v)o(ersal,)h(written)h
       
  3555 (as)g Fz(\031)1273 1845 y Ft(\000)p FG(1)1367 1881 y
       
  3556 FA(.)166 2114 y(Permutation)h(actions)g(ha)n(v)o(e)g(e)o(xcellent)g
       
  3557 (logical)g(properties)g(\(stemming)f(from)h(the)g(f)o(act)h(that)166
       
  3558 2234 y(the)o(y)f(are)i(bijections\).)e(W)-8 b(e)28 b(e)o(xploit)d
       
  3559 (these)i(properties)g(in)g(our)g(de\002nition)f(of)h
       
  3560 Fz(\013)q FA(-equi)n(v)n(alence)166 2355 y(for)g(terms)e(o)o(v)o(er)h
       
  3561 (a)h(nominal)e(signature,)g(which)h(is)g(respected)h(by)f(substitution)
       
  3562 d(of)k(terms)f(for)166 2475 y(v)n(ariables)33 b(e)n(v)o(en)h(though)f
       
  3563 (the)h(latter)g(may)g(in)l(v)n(olv)o(e)f(capture)h(of)g(atoms)g(by)g
       
  3564 (binders.)f(T)-8 b(o)34 b(do)166 2595 y(so)e(we)h(will)f(need)h(to)f
       
  3565 (mak)o(e)h(use)g(of)f(an)h(auxiliary)f(relation)g(of)h
       
  3566 Fx(fr)l(eshness)f FA(between)h(atoms)166 2716 y(and)k(terms,)f(whose)g
       
  3567 (intended)g(meaning)g(is)g(that)g(the)g(atom)g(does)h(not)f(occur)h
       
  3568 (free)g(in)f(an)o(y)166 2836 y(substitution)23 b(instance)i(of)h(the)f
       
  3569 (term.)g(As)h(discussed)e(in)h(the)h(Introduction,)e(our)h(judgements)
       
  3570 166 2957 y(about)j(term)f(equi)n(v)n(alence)g(\()p Fz(t)34
       
  3571 b Fw(\031)h Fz(t)1374 2920 y Ft(0)1397 2957 y FA(\))29
       
  3572 b(need)f(to)g(contain)f(hypotheses)f(about)i(the)g(freshness)g(of)166
       
  3573 3077 y(atoms)k(with)g(respect)h(to)f(v)n(ariables)g(\()p
       
  3574 Fz(a)43 b Fu(#)g Fz(X)8 b FA(\);)32 b(and)h(the)g(same)f(goes)h(for)g
       
  3575 (our)f(judgements)166 3197 y(about)25 b(freshness)g(itself)g(\()p
       
  3576 Fz(a)30 b Fu(#)f Fz(t)p FA(\).)d(Figure)f(2)h(gi)n(v)o(es)e(a)i
       
  3577 (syntax-directed)e(inducti)n(v)o(e)g(de\002nition)166
       
  3578 3318 y(of)h(equi)n(v)n(alence)f(and)g(freshness)h(using)f(judgements)f
       
  3579 (of)i(the)g(form)1115 3564 y Fw(r)i(`)h Fz(t)g Fw(\031)g
       
  3580 Fz(t)1517 3523 y Ft(0)1740 3564 y FA(and)199 b Fw(r)28
       
  3581 b(`)f Fz(a)h Fu(#)g Fz(t)166 3810 y FA(where)40 b Fz(t)g
       
  3582 FA(and)g Fz(t)743 3773 y Ft(0)806 3810 y FA(are)g(terms)f(of)h(the)f
       
  3583 (same)h(sort)f(o)o(v)o(er)f(a)i(gi)n(v)o(en)e(nominal)h(signature,)f
       
  3584 Fz(a)i FA(is)166 3930 y(an)34 b(atom,)g(and)g(the)g Fx(fr)l(eshness)f
       
  3585 (en)l(vir)l(onment)i Fw(r)f FA(is)f(a)i(\002nite)f(set)g(of)g
       
  3586 Fx(fr)l(eshness)f(constr)o(aints)166 4050 y Fz(a)c Fu(#)g
       
  3587 Fz(X)8 b FA(,)26 b(each)g(speci\002ed)g(by)f(an)g(atom)g(and)h(a)g(v)n
       
  3588 (ariable.)e(Rule)i(\()p Fw(\031)p FA(-suspension\))f(in)g(Figure)h(2)
       
  3589 166 4171 y(mak)o(es)e(use)h(of)g(the)g(follo)n(wing)e(de\002nition.)166
       
  3590 4404 y FB(De\002nition)36 b(2.5.)46 b FA(The)35 b Fx(disa)o(gr)l
       
  3591 (eement)f(set)i FA(of)f(tw)o(o)f(permutations)f Fz(\031)39
       
  3592 b FA(and)34 b Fz(\031)2909 4368 y Ft(0)2967 4404 y FA(is)h(the)f(set)h
       
  3593 (of)166 4540 y(atoms)24 b Fp(ds)8 b Fu(\()p Fz(\031)t(;)17
       
  3594 b(\031)727 4504 y Ft(0)750 4540 y Fu(\))815 4487 y Fo(def)820
       
  3595 4540 y Fu(=)32 b Fw(f)p Fz(a)27 b Fw(j)h Fz(\031)t Fr(\001)o
       
  3596 Fz(a)g Fw(6)p Fu(=)g Fz(\031)1444 4504 y Ft(0)1467 4540
       
  3597 y Fr(\001)p Fz(a)p Fw(g)p FA(.)166 4773 y(Note)38 b(that)f(e)n(v)o(ery)
       
  3598 g(disagreement)g(set)h Fp(ds)8 b Fu(\()p Fz(\031)t(;)17
       
  3599 b(\031)1856 4737 y Ft(0)1879 4773 y Fu(\))38 b FA(is)f(a)h(subset)f(of)
       
  3600 h(the)g Fx(\002nite)f FA(set)h(of)g(atoms)166 4893 y(occurring)29
       
  3601 b(in)f(either)h(of)g(the)g(lists)f Fz(\031)33 b FA(and)c
       
  3602 Fz(\031)1712 4857 y Ft(0)1735 4893 y FA(,)g(because)h(if)e
       
  3603 Fz(a)i FA(does)e(not)h(occur)g(in)g(those)f(lists,)166
       
  3604 5014 y(then)i(from)g(Figure)h(1)f(we)h(get)g Fz(\031)t
       
  3605 Fr(\001)o Fz(a)39 b Fu(=)f Fz(a)g Fu(=)g Fz(\031)1819
       
  3606 4977 y Ft(0)1842 5014 y Fr(\001)p Fz(a)p FA(.)31 b(T)-8
       
  3607 b(o)30 b(illustrate)f(the)i(use)f(of)h(disagreement)166
       
  3608 5134 y(sets,)24 b(consider)h(the)f(judgement)909 5380
       
  3609 y Fw(f)p Fz(a)k Fu(#)g Fz(X)8 b(;)17 b(c)27 b Fu(#)h
       
  3610 Fz(X)8 b Fw(g)27 b(`)h Fu(\()p Fz(a)17 b(c)p Fu(\)\()p
       
  3611 Fz(a)g(b)p Fu(\))p Fr(\001)o Fz(X)36 b Fw(\031)28 b Fu(\()p
       
  3612 Fz(b)17 b(c)p Fu(\))p Fr(\001)p Fz(X)57 b(:)1773 5712
       
  3613 y FA(8)p eop end
       
  3614 %%Page: 9 9
       
  3615 TeXDict begin 9 8 bop 166 3 3288 4 v 166 1022 4 1020
       
  3616 v 791 332 a Fu([])p Fr(\001)p Fz(a)1028 280 y Fo(def)1032
       
  3617 332 y Fu(=)104 b Fz(a)327 628 y Fu(\(\()p Fz(a)454 643
       
  3618 y FG(1)510 628 y Fz(a)561 643 y FG(2)601 628 y Fu(\))27
       
  3619 b(::)h Fz(\031)t Fu(\))p Fr(\001)p Fz(a)1028 576 y Fo(def)1032
       
  3620 628 y Fu(=)1212 404 y Fn(8)1212 479 y(>)1212 504 y(>)1212
       
  3621 528 y(<)1212 678 y(>)1212 703 y(>)1212 728 y(:)1286 488
       
  3622 y Fz(a)1337 503 y FG(1)1528 488 y FA(if)c Fz(\031)t Fr(\001)p
       
  3623 Fz(a)k Fu(=)g Fz(a)1938 503 y FG(2)1286 632 y Fz(a)1337
       
  3624 647 y FG(2)1528 632 y FA(if)c Fz(\031)t Fr(\001)p Fz(a)k
       
  3625 Fu(=)g Fz(a)1938 647 y FG(1)1286 777 y Fz(\031)t Fr(\001)p
       
  3626 Fz(a)100 b FA(otherwise)2430 188 y Fz(\031)t Fr(\001)p
       
  3627 Fw(hi)2698 135 y Fo(def)2702 188 y Fu(=)k Fw(hi)2237
       
  3628 368 y Fz(\031)t Fr(\001)o Fw(h)p Fz(t)2401 383 y FG(1)2441
       
  3629 368 y Fz(;)17 b(t)2520 383 y FG(2)2559 368 y Fw(i)2698
       
  3630 316 y Fo(def)2702 368 y Fu(=)104 b Fw(h)p Fz(\031)t Fr(\001)p
       
  3631 Fz(t)3047 383 y FG(1)3086 368 y Fz(;)17 b(\031)t Fr(\001)p
       
  3632 Fz(t)3256 383 y FG(2)3295 368 y Fw(i)2321 549 y Fz(\031)t
       
  3633 Fr(\001)p Fu(\()p Fz(f)27 b(t)p Fu(\))2698 496 y Fo(def)2702
       
  3634 549 y Fu(=)104 b Fz(f)27 b Fu(\()p Fz(\031)t Fr(\001)p
       
  3635 Fz(t)p Fu(\))2318 730 y Fz(\031)t Fr(\001)p Fu(\()p Fz(a:t)p
       
  3636 Fu(\))2698 677 y Fo(def)2702 730 y Fu(=)104 b(\()p Fz(\031)t
       
  3637 Fr(\001)p Fz(a)p Fu(\))p Fz(:)p Fu(\()p Fz(\031)t Fr(\001)p
       
  3638 Fz(t)p Fu(\))2229 910 y Fz(\031)t Fr(\001)o Fu(\()p Fz(\031)2416
       
  3639 874 y Ft(0)2439 910 y Fr(\001)p Fz(X)8 b Fu(\))2698 858
       
  3640 y Fo(def)2702 910 y Fu(=)104 b(\()p Fz(\031)t Fu(@)p
       
  3641 Fz(\031)3114 874 y Ft(0)3137 910 y Fu(\))p Fr(\001)p
       
  3642 Fz(X)p 3450 1022 V 166 1025 3288 4 v 1075 1162 a FD(Fig.)22
       
  3643 b(1.)h(Permutation)i(action)g(on)f(terms,)f FC(\031)s
       
  3644 Fm(\001)q FC(t)p FD(.)p 166 1267 V 166 3748 4 2482 v
       
  3645 637 1441 445 4 v 637 1525 a Fl(r)i(`)f(hi)i(\031)f(hi)1123
       
  3646 1464 y FD(\()p Fl(\031)p FD(-unit\))1653 1395 y Fl(r)g(`)g
       
  3647 FC(t)1868 1409 y FG(1)1932 1395 y Fl(\031)g FC(t)2061
       
  3648 1362 y Ft(0)2061 1419 y FG(1)2191 1395 y Fl(r)g(`)g FC(t)2406
       
  3649 1409 y FG(2)2471 1395 y Fl(\031)g FC(t)2600 1362 y Ft(0)2600
       
  3650 1419 y FG(2)p 1653 1439 987 4 v 1739 1525 a Fl(r)g(`)f(h)p
       
  3651 FC(t)1988 1539 y FG(1)2028 1525 y FC(;)15 b(t)2101 1539
       
  3652 y FG(2)2141 1525 y Fl(i)25 b(\031)g(h)p FC(t)2365 1492
       
  3653 y Ft(0)2365 1550 y FG(1)2405 1525 y FC(;)15 b(t)2478
       
  3654 1492 y Ft(0)2478 1550 y FG(2)2518 1525 y Fl(i)2680 1460
       
  3655 y FD(\()p Fl(\031)p FD(-pair\))494 1693 y Fl(r)25 b(`)g
       
  3656 FC(t)g Fl(\031)g FC(t)863 1660 y Ft(0)p 424 1713 531
       
  3657 4 v 424 1800 a Fl(r)g(`)g FC(f)g(t)g Fl(\031)g FC(f)f(t)932
       
  3658 1767 y Ft(0)997 1734 y FD(\()p Fl(\031)p FD(-function)i(symbol\))2052
       
  3659 1693 y Fl(r)f(`)g FC(t)g Fl(\031)g FC(t)2421 1660 y Ft(0)p
       
  3660 1979 1713 539 4 v 1979 1800 a Fl(r)g(`)g FC(a:t)g Fl(\031)g
       
  3661 FC(a:t)2494 1767 y Ft(0)2559 1736 y FD(\()p Fl(\031)p
       
  3662 FD(-abstraction-1\))741 1968 y FC(a)g Fl(6)p Fk(=)g FC(a)958
       
  3663 1935 y Ft(0)1072 1968 y Fl(r)g(`)g FC(t)g Fl(\031)g Fk(\()p
       
  3664 FC(a)15 b(a)1554 1935 y Ft(0)1578 1968 y Fk(\))p Fm(\001)p
       
  3665 FC(t)1675 1935 y Ft(0)1789 1968 y Fl(r)25 b(`)g FC(a)g
       
  3666 Fk(#)g FC(t)2178 1935 y Ft(0)p 741 2010 1461 4 v 1190
       
  3667 2097 a Fl(r)g(`)g FC(a:t)g Fl(\031)g FC(a)1647 2064 y
       
  3668 Ft(0)1671 2097 y FC(:t)1729 2064 y Ft(0)2243 2033 y FD(\()p
       
  3669 Fl(\031)p FD(-abstraction-2\))p 338 2314 400 4 v 338
       
  3670 2394 a Fl(r)g(`)g FC(a)g Fl(\031)g FC(a)779 2337 y FD(\()p
       
  3671 Fl(\031)p FD(-atom\))1350 2265 y Fk(\()p FC(a)g Fk(#)g
       
  3672 FC(X)7 b Fk(\))26 b Fl(2)f(r)50 b FD(for)24 b(all)52
       
  3673 b FC(a)25 b Fl(2)g Fj(ds)8 b Fk(\()p FC(\031)s(;)15 b(\031)2620
       
  3674 2232 y Ft(0)2644 2265 y Fk(\))p 1350 2307 1330 4 v 1685
       
  3675 2394 a Fl(r)25 b(`)g FC(\031)s Fm(\001)p FC(X)32 b Fl(\031)25
       
  3676 b FC(\031)2209 2361 y Ft(0)2233 2394 y Fm(\001)p FC(X)2721
       
  3677 2329 y FD(\()p Fl(\031)p FD(-suspension\))p 659 2687
       
  3678 427 4 v 659 2772 a Fl(r)g(`)g FC(a)h Fk(#)f Fl(hi)1128
       
  3679 2710 y FD(\()p Fk(#)p FD(-unit\))1663 2650 y Fl(r)g(`)g
       
  3680 FC(a)g Fk(#)g FC(t)2052 2664 y FG(1)2182 2650 y Fl(r)g(`)g
       
  3681 FC(a)h Fk(#)f FC(t)2572 2664 y FG(2)p 1663 2687 948 4
       
  3682 v 1831 2772 a Fl(r)g(`)g FC(a)g Fk(#)g Fl(h)p FC(t)2255
       
  3683 2786 y FG(1)2295 2772 y FC(;)15 b(t)2368 2786 y FG(2)2408
       
  3684 2772 y Fl(i)2652 2709 y FD(\()p Fk(#)p FD(-pair\))1215
       
  3685 2933 y Fl(r)25 b(`)f FC(a)i Fk(#)f FC(t)p 1180 2971 459
       
  3686 4 v 1180 3050 a Fl(r)g(`)g FC(a)g Fk(#)g FC(f)f(t)1680
       
  3687 2992 y FD(\()p Fk(#)p FD(-function)i(symbol\))p 422 3263
       
  3688 463 4 v 422 3343 a Fl(r)f(`)g FC(a)g Fk(#)g FC(a:t)926
       
  3689 3286 y FD(\()p Fk(#)p FD(-abstraction-1\))1794 3218 y
       
  3690 FC(a)h Fl(6)p Fk(=)f FC(a)2012 3185 y Ft(0)2126 3218
       
  3691 y Fl(r)g(`)g FC(a)g Fk(#)g FC(t)p 1794 3256 721 4 v 1912
       
  3692 3343 a Fl(r)g(`)g FC(a)g Fk(#)g FC(a)2316 3310 y Ft(0)2339
       
  3693 3343 y FC(:t)2556 3279 y FD(\()p Fk(#)p FD(-abstraction-2\))731
       
  3694 3515 y FC(a)g Fl(6)p Fk(=)g FC(a)948 3482 y Ft(0)p 638
       
  3695 3552 428 4 v 638 3639 a Fl(r)g(`)f FC(a)i Fk(#)f FC(a)1042
       
  3696 3606 y Ft(0)1107 3575 y FD(\()p Fk(#)p FD(-atom\))1682
       
  3697 3516 y Fk(\()p FC(\031)1772 3483 y Ft(\000)p FG(1)1867
       
  3698 3516 y Fm(\001)p FC(a)g Fk(#)g FC(X)7 b Fk(\))26 b Fl(2)f(r)p
       
  3699 1682 3559 693 4 v 1767 3639 a(r)g(`)g FC(a)g Fk(#)g FC(\031)s
       
  3700 Fm(\001)q FC(X)2417 3581 y FD(\()p Fk(#)p FD(-suspension\))p
       
  3701 3450 3748 4 2482 v 166 3751 3288 4 v 1092 3889 a(Fig.)e(2.)g(Inducti)n
       
  3702 (v)o(e)i(de\002nition)g(of)f Fl(\031)e FD(and)i Fk(#)p
       
  3703 FD(.)166 4073 y FA(This)g(holds)f(by)i(applying)e(rule)i(\()p
       
  3704 Fw(\031)p FA(-suspension\))f(in)g(Figure)h(2,)f(since)h(the)f
       
  3705 (disagreement)g(set)166 4193 y(of)h(the)g(permutations)e
       
  3706 Fu(\()p Fz(a)17 b(c)p Fu(\)\()p Fz(a)g(b)p Fu(\))24 b
       
  3707 FA(and)h Fu(\()p Fz(b)17 b(c)p Fu(\))25 b FA(is)f Fw(f)p
       
  3708 Fz(a;)17 b(c)p Fw(g)p FA(.)166 4417 y FB(Remark)32 b(2.6)f(\(Fr)n
       
  3709 (eshness)h(en)l(vir)n(onments\).)46 b FA(Note)31 b(that)g(the)f
       
  3710 (freshness)h(en)l(vironment)f(on)166 4537 y(the)d(left-hand)f(side)h
       
  3711 (of)g(judgements)e(in)i(the)g(rules)f(in)h(Figure)g(2)g(does)f(not)h
       
  3712 (change)g(from)f(hy-)166 4658 y(potheses)d(to)f(conclusion.)g(So)i(in)f
       
  3713 (the)g(same)g(w)o(ay)g(that)g(we)h(assume)e(v)n(ariables)h(ha)n(v)o(e)g
       
  3714 (attached)166 4778 y(sorting)i(information,)f(we)i(could)f(dispense)g
       
  3715 (with)g(the)g(use)h(of)g(freshness)f(en)l(vironments)f(en-)166
       
  3716 4898 y(tirely)d(by)g(attaching)f(the)h(freshness)g(information)f
       
  3717 (directly)h(to)g(v)n(ariables.)f(Ho)n(we)n(v)o(er)l(,)g(we)i(\002nd)166
       
  3718 5019 y(the)31 b(use)g(of)g(freshness)g(en)l(vironments)f(more)h(ele)o
       
  3719 (gant)f(\(for)i(one)f(thing,)f(without)g(them)g(tw)o(o)166
       
  3720 5139 y(v)n(ariables)24 b(with)g(the)h(same)g(name)g(b)n(ut)f(dif)n
       
  3721 (ferent)g(freshness)h(information)f(w)o(ould)g(ha)n(v)o(e)g(to)h(be)166
       
  3722 5259 y(re)o(garded)30 b(as)h(dif)n(ferent\).)f(The)o(y)g(also)g(mak)o
       
  3723 (e)g(life)h(simpler)e(when)i(we)f(come)h(on)f(to)g(nominal)166
       
  3724 5380 y(uni\002cation)24 b(problems)g(and)h(their)f(solutions)f(in)h
       
  3725 (the)h(ne)o(xt)f(section.)1773 5712 y(9)p eop end
       
  3726 %%Page: 10 10
       
  3727 TeXDict begin 10 9 bop 166 83 a FA(Belo)n(w)27 b(we)g(sk)o(etch)f(a)i
       
  3728 (proof)e(that)h Fw(\031)g FA(is)g(an)g(equi)n(v)n(alence)e(relation.)i
       
  3729 (At)f(\002rst)h(sight)f(this)g(prop-)166 203 y(erty)k(might)e(be)i
       
  3730 (surprising)e(considering)g(the)i(\223unsymmetric\224)e(de\002nition)h
       
  3731 (of)h(the)f(rule)h(\()p Fw(\031)p FA(-)166 324 y(abstraction-2\).)25
       
  3732 b(Ho)n(we)n(v)o(er)e(it)i(holds)f(because)i(of)f(the)g(good)g(logical)f
       
  3733 (properties)h(of)g(the)g(rela-)166 444 y(tion)30 b Fw(\031)h
       
  3734 FA(with)f(respect)h(to)f(permutation)f(actions.)h(Although)f(reasoning)
       
  3735 h(about)g Fw(\031)h FA(is)f(rather)166 565 y(pleasant)k(once)h(equi)n
       
  3736 (v)n(alence)e(is)h(pro)o(v)o(ed,)f(establishing)f(it)i(\002rst)h(is)f
       
  3737 (rather)g(trick)o(y\227mainly)166 685 y(because)29 b(of)f(the)h(lar)n
       
  3738 (ge)g(number)f(of)g(cases,)h(b)n(ut)f(also)g(because)h(se)n(v)o(eral)f
       
  3739 (f)o(acts)g(needed)h(in)f(the)166 805 y(proof)21 b(are)i
       
  3740 (interdependent.)1180 769 y FG(1)1258 805 y FA(W)-8 b(e)22
       
  3741 b(\002rst)g(sho)n(w)e(that)h(permutations)f(can)i(be)g(mo)o(v)o(ed)e
       
  3742 (from)h(one)166 926 y(side)28 b(of)g(the)g(freshness)g(relation)g(to)g
       
  3743 (the)g(other)g(by)g(forming)f(the)i(in)l(v)o(erse)e(permutation,)g(and)
       
  3744 166 1046 y(that)d(the)h(freshness)g(relation)f(is)g(preserv)o(ed)h
       
  3745 (under)g(permutation)e(actions.)166 1266 y FB(Lemma)i(2.7.)206
       
  3746 1486 y Fx(\(1\))50 b(If)41 b Fw(r)28 b(`)g Fz(a)f Fu(#)h
       
  3747 Fz(\031)t Fr(\001)p Fz(t)42 b Fx(then)f Fw(r)28 b(`)f
       
  3748 Fz(\031)1500 1450 y Ft(\000)p FG(1)1594 1486 y Fr(\001)p
       
  3749 Fz(a)h Fu(#)g Fz(t)g Fx(.)206 1606 y(\(2\))50 b(If)41
       
  3750 b Fw(r)28 b(`)g Fz(\031)t Fr(\001)o Fz(a)g Fu(#)g Fz(t)42
       
  3751 b Fx(then)f Fw(r)28 b(`)f Fz(a)h Fu(#)g Fz(\031)1688
       
  3752 1570 y Ft(\000)p FG(1)1782 1606 y Fr(\001)p Fz(t)g Fx(.)206
       
  3753 1727 y(\(3\))50 b(If)41 b Fw(r)28 b(`)g Fz(a)f Fu(#)h
       
  3754 Fz(t)42 b Fx(then)f Fw(r)28 b(`)g Fz(\031)t Fr(\001)o
       
  3755 Fz(a)g Fu(#)g Fz(\031)t Fr(\001)p Fz(t)g Fx(.)166 2059
       
  3756 y FB(PR)m(OOF)-11 b(.)49 b FA(\(1\))27 b(and)g(\(2\))h(are)g(by)f
       
  3757 (routine)g(inductions)f(on)h(the)g(structure)g(of)g Fz(t)p
       
  3758 FA(,)h(using)e(the)h(f)o(act)166 2179 y(that)g Fz(\031)t
       
  3759 Fr(\001)p Fz(a)33 b Fu(=)g Fz(b)28 b FA(if)n(f)f Fz(a)34
       
  3760 b Fu(=)e Fz(\031)1067 2143 y Ft(\000)p FG(1)1162 2179
       
  3761 y Fr(\001)o Fz(b)d FA(;)e(\(3\))h(is)f(a)h(consequence)g(of)g(\(2\))g
       
  3762 (and)f(the)h(f)o(act)g(that)f(permuta-)166 2300 y(tions)d(are)h
       
  3763 (bijections)f(on)g(atoms.)p 3382 2300 4 68 v 3386 2236
       
  3764 60 4 v 3386 2300 V 3445 2300 4 68 v 166 2632 a(According)h(to)h(the)f
       
  3765 (de\002nition)g(of)h(the)f(permutation)g(action)g(gi)n(v)o(en)f(in)h
       
  3766 (Figure)h(1,)g(if)g(we)g(push)166 2752 y(a)38 b(permutation)f(inside)g
       
  3767 (a)i(term,)e(we)i(need)f(to)g(apply)f(the)h(permutation)f(to)h(all)f
       
  3768 (atoms)h(we)166 2873 y(meet)e(on)g(the)g(w)o(ay)-6 b(.)35
       
  3769 b(Suppose)h(we)h(apply)e(tw)o(o)h(distinct)e(permutations,)h(say)h
       
  3770 Fz(\031)k FA(and)c Fz(\031)3292 2836 y Ft(0)3315 2873
       
  3771 y FA(,)g(to)166 2993 y(a)d(term)f Fz(t)p FA(,)g(then)g(in)g(general)h
       
  3772 Fz(\031)t Fr(\001)o Fz(t)g FA(and)f Fz(\031)1583 2957
       
  3773 y Ft(0)1607 2993 y Fr(\001)o Fz(t)h FA(are)g(not)f Fz(\013)q
       
  3774 FA(-equi)n(v)n(alent\227the)d(disagreement)j(set)166
       
  3775 3113 y Fz(ds)p Fu(\()p Fz(\031)t(;)17 b(\031)463 3077
       
  3776 y Ft(0)485 3113 y Fu(\))36 b FA(characterises)g(all)f(atoms)g(which)g
       
  3777 (potentially)f(lead)h(to)h(dif)n(ferences.)f(Ho)n(we)n(v)o(er)l(,)166
       
  3778 3234 y(if)30 b(we)g(assume)g(that)g(all)f(atoms)h(in)f
       
  3779 Fz(ds)p Fu(\()p Fz(\031)t(;)17 b(\031)1709 3198 y Ft(0)1732
       
  3780 3234 y Fu(\))30 b FA(are)h(fresh)f(for)g Fz(t)p FA(,)h(then)e(we)i(can)
       
  3781 f(infer)g(that)g(the)166 3354 y(permutation)19 b(actions)g(produce)h
       
  3782 (equi)n(v)n(alent)e(terms.)h(This)h(is)f(made)h(precise)g(in)g(the)g
       
  3783 (follo)n(wing)166 3474 y(lemma.)166 3694 y FB(Lemma)31
       
  3784 b(2.8.)44 b Fx(Given)30 b(any)h Fz(\031)j Fx(and)c Fz(\031)1488
       
  3785 3658 y Ft(0)1511 3694 y Fx(,)h(if)f Fw(r)38 b(`)g Fz(a)g
       
  3786 Fu(#)g Fz(t)31 b Fx(holds)e(for)h(all)g Fz(a)38 b Fw(2)g
       
  3787 Fz(ds)p Fu(\()p Fz(\031)t(;)17 b(\031)3166 3658 y Ft(0)3189
       
  3788 3694 y Fu(\))p Fx(,)30 b(then)166 3815 y Fw(r)e(`)f Fz(\031)t
       
  3789 Fr(\001)p Fz(t)h Fw(\031)g Fz(\031)683 3779 y Ft(0)706
       
  3790 3815 y Fr(\001)p Fz(t)p Fx(.)166 4147 y FB(PR)m(OOF)-11
       
  3791 b(.)49 b FA(By)25 b(induction)g(on)g(the)g(structure)h(of)f
       
  3792 Fz(t)p FA(,)h(for)g(all)f Fz(\031)30 b FA(and)25 b Fz(\031)2541
       
  3793 4111 y Ft(0)2590 4147 y FA(simultaneously)-6 b(,)23 b(using)166
       
  3794 4267 y(the)e(f)o(act)h(about)f(disagreement)g(sets)g(that)g(for)h(all)f
       
  3795 (atoms)f Fz(a;)d(b)p FA(,)22 b(if)f Fz(a)28 b Fw(2)g
       
  3796 Fz(ds)p Fu(\()p Fz(\031)t(;)17 b Fu(\()p Fz(\031)t Fr(\001)o
       
  3797 Fz(b)51 b(\031)3090 4231 y Ft(0)3113 4267 y Fr(\001)p
       
  3798 Fz(b)p Fu(\))28 b(::)g Fz(\031)3393 4231 y Ft(0)3416
       
  3799 4267 y Fu(\))166 4388 y FA(then)d Fz(a)i Fw(2)h Fz(ds)p
       
  3800 Fu(\()p Fz(\031)t(;)17 b(\031)832 4352 y Ft(0)855 4388
       
  3801 y Fu(\))p FA(.)p 3382 4388 V 3386 4324 60 4 v 3386 4388
       
  3802 V 3445 4388 4 68 v 166 4720 a(An)27 b(e)o(xample)g(of)g(this)g(lemma)f
       
  3803 (is)h(that)g Fw(r)33 b(`)f Fz(\031)t Fr(\001)p Fu(\()p
       
  3804 Fz(a)17 b(b)p Fu(\))p Fr(\001)p Fz(t)32 b Fw(\031)h Fu(\()p
       
  3805 Fz(\031)t Fr(\001)p Fz(a)50 b(\031)t Fr(\001)p Fz(b)p
       
  3806 Fu(\))p Fr(\001)p Fz(\031)t Fr(\001)p Fz(t)27 b FA(is)g(a)h(v)n(alid)e
       
  3807 (judge-)166 4840 y(ment,)e(because)h(the)g(disagreement)f(set)h
       
  3808 Fz(ds)p Fu(\()p Fz(\031)t Fu(@\()p Fz(a)17 b(b)p Fu(\))g
       
  3809 Fz(;)32 b Fu(\()p Fz(\031)t Fr(\001)p Fz(a)50 b(\031)t
       
  3810 Fr(\001)p Fz(b)p Fu(\))28 b(::)f Fz(\031)t Fu(\))e FA(is)g(empty)-6
       
  3811 b(.)p 166 4943 299 4 v 166 5007 a FG(1)257 5040 y FD(In)107
       
  3812 b(addition)j(some)d(further)i(simple)g(properties)h(of)d(permutations)j
       
  3813 (and)e(dis-)166 5153 y(agreement)j(sets)g(need)f(to)f(be)h(established)
       
  3814 j(\002rst.)c(A)f(machine-check)o(ed)114 b(proof)166 5266
       
  3815 y(of)i FF(all)g FD(results)h(using)g(the)g(theorem)f(pro)o(v)o(er)h
       
  3816 (Isabelle)h(can)e(be)g(found)h(at)166 5379 y Fi(http://www.cl.c)o(am)o
       
  3817 (.a)o(c.)o(uk/)o(us)o(er)o(s/)o(cu)o(200)o(/U)o(ni)o(fi)o(ca)o(tio)o(n)
       
  3818 p FD(.)1748 5712 y FA(10)p eop end
       
  3819 %%Page: 11 11
       
  3820 TeXDict begin 11 10 bop 166 83 a FA(The)25 b(ne)o(xt)f(lemma)g(sho)n
       
  3821 (ws)f(that)i Fw(\031)g FA(respects)g(the)g(freshness)f(relation.)166
       
  3822 317 y FB(Lemma)h(2.9.)41 b Fx(If)25 b Fw(r)j(`)f Fz(a)h
       
  3823 Fu(#)g Fz(t)d Fx(and)g Fw(r)i(`)h Fz(t)g Fw(\031)g Fz(t)1818
       
  3824 281 y Ft(0)1842 317 y Fx(,)c(then)h Fw(r)j(`)f Fz(a)h
       
  3825 Fu(#)g Fz(t)2510 281 y Ft(0)2533 317 y Fx(.)166 770 y
       
  3826 FB(PR)m(OOF)-11 b(.)49 b FA(Routine)24 b(induction)f(on)i(the)f
       
  3827 (de\002nition)g(of)h Fw(\031)h FA(using)d(Lemma)i(2.7.)p
       
  3828 3382 770 4 68 v 3386 706 60 4 v 3386 770 V 3445 770 4
       
  3829 68 v 166 1224 a(F)o(or)j(sho)n(wing)d(transiti)n(vity)g(of)j(the)f
       
  3830 (relation)g Fw(\031)p FA(,)h(it)f(will)f(be)i(necessary)g(to)f
       
  3831 (de\002ne)h(a)g(measure)166 1344 y(that)c(counts)g(all)h(term)f
       
  3832 (constructors)g(occurring)h(in)f(a)i(term.)166 1578 y
       
  3833 FB(De\002nition)g(2.10.)40 b FA(The)25 b Fx(size)g FA(of)f(a)i(term)e
       
  3834 Fz(t)h FA(is)g(the)f(natural)h(number)f Fw(j)p Fz(t)p
       
  3835 Fw(j)h FA(de\002ned)g(by:)1184 1844 y Fw(j)p Fz(\031)t
       
  3836 Fr(\001)p Fz(X)8 b Fw(j)p Fz(;)17 b Fw(j)p Fz(a)p Fw(j)p
       
  3837 Fz(;)g Fw(jhij)1774 1792 y Fo(def)1778 1844 y Fu(=)32
       
  3838 b(1)1368 2012 y Fw(j)p Fz(a:t)p Fw(j)p Fz(;)17 b Fw(j)p
       
  3839 Fz(f)26 b(t)p Fw(j)1774 1959 y Fo(def)1778 2012 y Fu(=)32
       
  3840 b(1)22 b(+)g Fw(j)p Fz(t)p Fw(j)1420 2179 y(jh)p Fz(t)1522
       
  3841 2194 y FG(1)1561 2179 y Fz(;)17 b(t)1640 2194 y FG(2)1680
       
  3842 2179 y Fw(ij)1774 2126 y Fo(def)1778 2179 y Fu(=)32 b(1)22
       
  3843 b(+)g Fw(j)p Fz(t)2118 2194 y FG(1)2158 2179 y Fw(j)f
       
  3844 Fu(+)h Fw(j)p Fz(t)2368 2194 y FG(2)2408 2179 y Fw(j)166
       
  3845 2539 y FA(Notice)d(that)g(the)g(size)g(of)h(a)f(term)g(is)g(preserv)o
       
  3846 (ed)g(under)h(permutation)d(actions)i(\(i.e.)g Fw(j)p
       
  3847 Fz(\031)t Fr(\001)p Fz(t)p Fw(j)27 b Fu(=)h Fw(j)p Fz(t)p
       
  3848 Fw(j)p FA(\))166 2660 y(and)d(respected)g(by)f(the)h(relation)f
       
  3849 Fw(\031)i FA(in)e(the)h(sense)f(that)h(if)g Fw(r)i(`)h
       
  3850 Fz(t)g Fw(\031)g Fz(t)2591 2624 y Ft(0)2639 2660 y FA(then)d
       
  3851 Fw(j)p Fz(t)p Fw(j)i Fu(=)h Fw(j)p Fz(t)3121 2624 y Ft(0)3144
       
  3852 2660 y Fw(j)p FA(.)166 2894 y FB(Theor)n(em)f(2.11)d(\(Equi)o(v)o
       
  3853 (alence\).)43 b Fw(r)28 b(`)f(\000)h(\031)g(\000)e Fx(is)e(an)h
       
  3854 (equivalence)f(r)l(elation.)166 3347 y FB(PR)m(OOF)-11
       
  3855 b(.)49 b FA(Re\003e)o(xi)n(vity)19 b(is)h(by)h(a)g(simple)e(induction)h
       
  3856 (on)g(the)h(structure)f(of)h(terms.)f(T)m(ransiti)n(vity)166
       
  3857 3467 y(is)k(by)h(an)g(induction)e(on)h(the)h(size)g(of)g(terms:)e(a)i
       
  3858 (slight)f(complication)f(is)h(that)g(man)o(y)g(subcases)166
       
  3859 3588 y(need)e(to)f(be)h(analysed)f(\(for)h(e)o(xample)f(\002)n(v)o(e)g
       
  3860 (subcases)h(when)f(dealing)g(with)g(abstractions\))g(and)166
       
  3861 3708 y(also)26 b(that)g(transiti)n(vity)e(needs)j(to)f(be)g(sho)n(wn)g
       
  3862 (by)g(mutual)f(induction)g(with)h(the)g(f)o(act)h(that)f
       
  3863 Fw(\031)h FA(is)166 3829 y(preserv)o(ed)e(under)f(permutation)g
       
  3864 (actions,)g(that)g(is)837 4076 y(gi)n(v)o(en)f(an)o(y)k
       
  3865 Fz(\031)t(;)45 b FA(if)27 b Fw(r)h(`)f Fz(t)h Fw(\031)g
       
  3866 Fz(t)1870 4035 y Ft(0)1922 4076 y FA(then)f Fw(r)g(`)h
       
  3867 Fz(\031)t Fr(\001)p Fz(t)g Fw(\031)g Fz(\031)t Fr(\001)p
       
  3868 Fz(t)2705 4035 y Ft(0)2756 4076 y Fz(:)555 b FA(\(9\))166
       
  3869 4323 y(W)-8 b(e)36 b(illustrate)e(the)i(proof)f(of)h(transiti)n(vity)d
       
  3870 (for)j(the)f(case)i(when)e Fw(r)48 b(`)f Fz(a)2767 4338
       
  3871 y FG(1)2807 4323 y Fz(:t)2869 4338 y FG(1)2957 4323 y
       
  3872 Fw(\031)h Fz(a)3133 4338 y FG(2)3172 4323 y Fz(:t)3234
       
  3873 4338 y FG(2)3310 4323 y FA(and)166 4444 y Fw(r)28 b(`)f
       
  3874 Fz(a)416 4459 y FG(2)456 4444 y Fz(:t)518 4459 y FG(2)585
       
  3875 4444 y Fw(\031)i Fz(a)742 4459 y FG(3)781 4444 y Fz(:t)843
       
  3876 4459 y FG(3)908 4444 y FA(hold,)24 b(with)g Fz(a)1388
       
  3877 4459 y FG(1)1428 4444 y FA(,)g Fz(a)1528 4459 y FG(2)1593
       
  3878 4444 y FA(and)h Fz(a)1813 4459 y FG(3)1877 4444 y FA(all)g(distinct)e
       
  3879 (atoms,)h(and)h(we)g(ha)n(v)o(e)f(to)h(pro)o(v)o(e)166
       
  3880 4564 y Fw(r)j(`)f Fz(a)416 4579 y FG(1)456 4564 y Fz(:t)518
       
  3881 4579 y FG(1)585 4564 y Fw(\031)i Fz(a)742 4579 y FG(3)781
       
  3882 4564 y Fz(:t)843 4579 y FG(3)883 4564 y FA(.)20 b(By)h(the)f(\()p
       
  3883 Fw(\031)p FA(-abstraction-2\))h(rule)g(we)f(can)h(infer)g(from)f(the)g
       
  3884 (assumptions)166 4685 y(the)25 b(follo)n(wing)e(f)o(acts:)803
       
  3885 4954 y(\(i\))155 b Fw(r)27 b(`)h Fz(t)1286 4969 y FG(1)1353
       
  3886 4954 y Fw(\031)g Fu(\()p Fz(a)1547 4969 y FG(1)1604 4954
       
  3887 y Fz(a)1655 4969 y FG(2)1694 4954 y Fu(\))p Fr(\001)p
       
  3888 Fz(t)1799 4969 y FG(2)2075 4954 y FA(\(ii\))119 b Fw(r)28
       
  3889 b(`)f Fz(a)2566 4969 y FG(1)2633 4954 y Fu(#)h Fz(t)2777
       
  3890 4969 y FG(2)803 5135 y FA(\(iii\))99 b Fw(r)27 b(`)h
       
  3891 Fz(t)1286 5150 y FG(2)1353 5135 y Fw(\031)g Fu(\()p Fz(a)1547
       
  3892 5150 y FG(2)1604 5135 y Fz(a)1655 5150 y FG(3)1694 5135
       
  3893 y Fu(\))p Fr(\001)p Fz(t)1799 5150 y FG(3)2075 5135 y
       
  3894 FA(\(i)n(v\))99 b Fw(r)28 b(`)f Fz(a)2566 5150 y FG(2)2633
       
  3895 5135 y Fu(#)h Fz(t)2777 5150 y FG(3)166 5380 y FA(Belo)n(w)d(we)g(gi)n
       
  3896 (v)o(e)e(the)i(steps)f(that)g(pro)o(v)o(e)g Fw(r)k(`)f
       
  3897 Fz(a)1826 5395 y FG(1)1866 5380 y Fz(:t)1928 5395 y FG(1)1995
       
  3898 5380 y Fw(\031)i Fz(a)2152 5395 y FG(3)2191 5380 y Fz(:t)2253
       
  3899 5395 y FG(3)2293 5380 y FA(.)1748 5712 y(11)p eop end
       
  3900 %%Page: 12 12
       
  3901 TeXDict begin 12 11 bop 166 126 a FA(\(a\))106 b Fw(r)28
       
  3902 b(`)f Fu(\()p Fz(a)670 141 y FG(1)726 126 y Fz(a)777
       
  3903 141 y FG(2)817 126 y Fu(\))p Fr(\001)p Fz(t)922 141 y
       
  3904 FG(2)989 126 y Fw(\031)h Fu(\()p Fz(a)1183 141 y FG(1)1239
       
  3905 126 y Fz(a)1290 141 y FG(2)1330 126 y Fu(\)\()p Fz(a)1457
       
  3906 141 y FG(2)1513 126 y Fz(a)1564 141 y FG(3)1604 126 y
       
  3907 Fu(\))p Fr(\001)p Fz(t)1709 141 y FG(3)2740 126 y FA(by)c(\(iii\))h
       
  3908 (and)g(IH)g(\(9\))166 307 y(\(b\))100 b Fw(r)28 b(`)f
       
  3909 Fz(t)616 322 y FG(1)683 307 y Fw(\031)i Fu(\()p Fz(a)878
       
  3910 322 y FG(1)934 307 y Fz(a)985 322 y FG(2)1024 307 y Fu(\)\()p
       
  3911 Fz(a)1151 322 y FG(2)1208 307 y Fz(a)1259 322 y FG(3)1298
       
  3912 307 y Fu(\))p Fr(\001)p Fz(t)1403 322 y FG(3)2255 307
       
  3913 y FA(by)c(\(i\),)g(\(a\))g(and)g(IH)g(\(transiti)n(vity\))166
       
  3914 488 y(\(c\))106 b Fz(ds)p Fu(\(\()p Fz(a)606 503 y FG(1)662
       
  3915 488 y Fz(a)713 503 y FG(2)752 488 y Fu(\)\()p Fz(a)879
       
  3916 503 y FG(2)935 488 y Fz(a)986 503 y FG(3)1026 488 y Fu(\))17
       
  3917 b Fz(;)33 b Fu(\()p Fz(a)1230 503 y FG(1)1286 488 y Fz(a)1337
       
  3918 503 y FG(3)1377 488 y Fu(\)\))27 b(=)h Fw(f)p Fz(a)1685
       
  3919 503 y FG(1)1724 488 y Fz(;)17 b(a)1819 503 y FG(2)1859
       
  3920 488 y Fw(g)1038 b FA(by)25 b(de\002nition)166 668 y(\(d\))100
       
  3921 b Fw(r)28 b(`)f Fz(a)632 683 y FG(1)699 668 y Fu(#)h(\()p
       
  3922 Fz(a)897 683 y FG(2)954 668 y Fz(a)1005 683 y FG(3)1044
       
  3923 668 y Fu(\))p Fr(\001)p Fz(t)1149 683 y FG(3)2361 668
       
  3924 y FA(by)c(\(ii\),)h(\(iii\))f(and)h(Lemma)f(2.9)166 849
       
  3925 y(\(e\))106 b Fw(r)28 b(`)f Fz(a)632 864 y FG(1)699 849
       
  3926 y Fu(#)h Fz(t)843 864 y FG(3)1804 849 y FA(by)c Fu(\()p
       
  3927 Fz(a)2017 864 y FG(2)2073 849 y Fz(a)2124 864 y FG(3)2164
       
  3928 849 y Fu(\))p Fr(\001)p Fz(a)2285 864 y FG(1)2352 849
       
  3929 y Fu(=)k Fz(a)2507 864 y FG(1)2546 849 y FA(,)d(\(d\))g(and)g(Lemma)f
       
  3930 (2.7\(i\))166 1029 y(\(f\))117 b Fw(r)28 b(`)f Fu(\()p
       
  3931 Fz(a)670 1044 y FG(1)726 1029 y Fz(a)777 1044 y FG(2)817
       
  3932 1029 y Fu(\)\()p Fz(a)944 1044 y FG(2)1000 1029 y Fz(a)1051
       
  3933 1044 y FG(3)1091 1029 y Fu(\))p Fr(\001)o Fz(t)1195 1044
       
  3934 y FG(3)1263 1029 y Fw(\031)h Fu(\()p Fz(a)1457 1044 y
       
  3935 FG(1)1513 1029 y Fz(a)1564 1044 y FG(3)1604 1029 y Fu(\))p
       
  3936 Fr(\001)p Fz(t)1709 1044 y FG(3)2220 1029 y FA(by)c(\(c\),)h(\(i)n
       
  3937 (v\),)g(\(e\))g(and)g(Lemma)f(2.8)166 1210 y(\(g\))100
       
  3938 b Fw(r)28 b(`)f Fz(t)616 1225 y FG(1)683 1210 y Fw(\031)i
       
  3939 Fu(\()p Fz(a)878 1225 y FG(1)934 1210 y Fz(a)985 1225
       
  3940 y FG(3)1024 1210 y Fu(\))p Fr(\001)p Fz(t)1129 1225 y
       
  3941 FG(3)2244 1210 y FA(by)c(\(b\),)g(\(f\))g(and)g(IH)g(\(transiti)n
       
  3942 (vity\))166 1390 y(\(h\))100 b Fw(r)28 b(`)f Fz(a)632
       
  3943 1405 y FG(1)672 1390 y Fz(:t)734 1405 y FG(1)801 1390
       
  3944 y Fw(\031)h Fz(a)957 1405 y FG(3)997 1390 y Fz(:t)1059
       
  3945 1405 y FG(3)2162 1390 y FA(by)c(\(e\),)i(\(g\))f(and)g(\()p
       
  3946 Fw(\031)p FA(-abstraction-2\))166 1621 y(The)g(other)g(cases)g(are)h
       
  3947 (by)f(similar)e(ar)n(guments.)i(Symmetry)f(is)g(then)h(by)g(a)g
       
  3948 (routine)f(induction)166 1741 y(on)h(the)f(de\002nition)g(of)h
       
  3949 Fw(\031)g FA(using)f(Lemma)g(2.8)h(and)g(transiti)n(vity)-6
       
  3950 b(.)p 3382 1741 4 68 v 3386 1677 60 4 v 3386 1741 V 3445
       
  3951 1741 4 68 v 166 2082 a(No)n(w)23 b(it)h(is)f(relati)n(v)o(ely)g
       
  3952 (straightforw)o(ard)g(to)h(obtain)f(the)h(follo)n(wing)e(properties)i
       
  3953 (of)g(our)g(equi)n(v-)166 2202 y(alence)h(relation)g(with)f(respect)h
       
  3954 (to)f(permutation)g(actions.)166 2422 y FB(Cor)n(ollary)g(2.12.)206
       
  3955 2643 y Fx(\(1\))50 b Fw(r)28 b(`)f Fz(t)h Fw(\031)g Fz(\031)798
       
  3956 2606 y Ft(\000)p FG(1)893 2643 y Fr(\001)o Fz(\031)t
       
  3957 Fr(\001)p Fz(t)1050 2606 y Ft(0)1126 2643 y Fx(if)c(and)h(only)f(if)52
       
  3958 b Fw(r)28 b(`)g Fz(t)f Fw(\031)i Fz(t)2088 2606 y Ft(0)2139
       
  3959 2643 y Fx(.)206 2763 y(\(2\))50 b Fw(r)28 b(`)f Fz(t)h
       
  3960 Fw(\031)g Fz(\031)t Fr(\001)p Fz(t)865 2727 y Ft(0)941
       
  3961 2763 y Fx(if)c(and)h(only)f(if)52 b Fw(r)28 b(`)f Fz(\031)1758
       
  3962 2727 y Ft(\000)p FG(1)1853 2763 y Fr(\001)p Fz(t)g Fw(\031)i
       
  3963 Fz(t)2088 2727 y Ft(0)2139 2763 y Fx(.)206 2883 y(\(3\))50
       
  3964 b(Given)32 b(any)h Fz(\031)j Fx(and)c Fz(\031)1151 2847
       
  3965 y Ft(0)1174 2883 y Fx(,)h(if)48 b Fw(r)42 b(`)f Fz(\031)t
       
  3966 Fr(\001)p Fz(t)h Fw(\031)g Fz(\031)1909 2847 y Ft(0)1932
       
  3967 2883 y Fr(\001)p Fz(t)50 b Fx(then)32 b(for)f(all)h Fz(a)h
       
  3968 Fx(in)f Fz(ds)p Fu(\()p Fz(\031)t(;)17 b(\031)3030 2847
       
  3969 y Ft(0)3052 2883 y Fu(\))32 b Fx(we)i(have)372 3004 y
       
  3970 Fw(r)28 b(`)f Fz(a)h Fu(#)g Fz(t)p Fx(.)166 3345 y FB(PR)m(OOF)-11
       
  3971 b(.)49 b FA(\(i\))38 b(follo)n(ws)f(immediately)g(from)h(Lemma)g(2.8)h
       
  3972 (and)f(transiti)n(vity;)e(\(ii\))i(follo)n(ws)166 3465
       
  3973 y(from)i(\(9\))h(and)f(\(i\);)g(and)g(\(iii\))g(is)g(by)g(a)g(routine)g
       
  3974 (induction)f(on)h(the)g(structure)g(of)g Fz(t)h FA(using)166
       
  3975 3585 y(Lemma)24 b(2.9.)p 3382 3585 V 3386 3522 60 4 v
       
  3976 3386 3585 V 3445 3585 4 68 v 166 3927 a(The)33 b(main)f(reason)h(for)g
       
  3977 (using)f(suspensions)f(in)h(the)h(syntax)f(of)g(terms)h(is)f(to)g
       
  3978 (enable)h(a)g(def-)166 4047 y(inition)d(of)j Fx(substitution)c(of)i
       
  3979 (terms)h(for)f(variables)g FA(that)h(allo)n(ws)f(capture)h(of)g(free)h
       
  3980 (atoms)e(by)166 4167 y(atom-abstractions)i(while)i(still)e(respecting)i
       
  3981 Fz(\013)q FA(-equi)n(v)n(alence.)e(The)i(follo)n(wing)e(lemma)i(es-)166
       
  3982 4288 y(tablishes)24 b(this.)f(First)i(we)g(gi)n(v)o(e)f(some)g
       
  3983 (terminology)f(and)i(notation)e(for)i(term-substitution.)166
       
  3984 4508 y FB(De\002nition)37 b(2.13.)47 b FA(A)36 b Fx(substitution)d
       
  3985 Fz(\033)40 b FA(is)c(a)g(sort-respecting)f(function)g(from)h(v)n
       
  3986 (ariables)f(to)166 4628 y(terms)28 b(with)h(the)f(property)h(that)f
       
  3987 Fz(\033)t Fu(\()p Fz(X)8 b Fu(\))35 b(=)g Fz(X)i FA(for)29
       
  3988 b(all)g(b)n(ut)f(\002nitely)h(man)o(y)f(v)n(ariables)g
       
  3989 Fz(X)8 b FA(.)28 b(W)-8 b(e)166 4749 y(write)20 b Fp(dom)6
       
  3990 b Fu(\()p Fz(\033)t Fu(\))20 b FA(for)g(the)g(\002nite)f(set)h(of)g(v)n
       
  3991 (ariables)f Fz(X)27 b FA(satisfying)19 b Fz(\033)t Fu(\()p
       
  3992 Fz(X)8 b Fu(\))27 b Fw(6)p Fu(=)g Fz(X)8 b FA(.)20 b(If)g
       
  3993 Fp(dom)7 b Fu(\()p Fz(\033)t Fu(\))20 b FA(con-)166 4869
       
  3994 y(sists)28 b(of)h(distinct)f(v)n(ariables)g Fz(X)1269
       
  3995 4884 y FG(1)1308 4869 y Fz(;)17 b(:)g(:)g(:)f(;)h(X)1608
       
  3996 4884 y Fq(n)1684 4869 y FA(and)29 b Fz(\033)t Fu(\()p
       
  3997 Fz(X)2035 4884 y Fq(i)2063 4869 y Fu(\))36 b(=)f Fz(t)2283
       
  3998 4884 y Fq(i)2341 4869 y FA(for)29 b Fz(i)36 b Fu(=)g(1)p
       
  3999 Fz(::n)p FA(,)29 b(we)h(sometimes)166 4989 y(write)25
       
  4000 b Fz(\033)k FA(as)1204 5110 y Fz(\033)j Fu(=)27 b([)p
       
  4001 Fz(X)1502 5125 y FG(1)1569 5110 y Fu(:=)h Fz(t)1735 5125
       
  4002 y FG(1)1775 5110 y Fz(;)17 b(:)g(:)g(:)e(;)i(X)2074 5125
       
  4003 y Fq(n)2149 5110 y Fu(:=)27 b Fz(t)2314 5125 y Fq(n)2362
       
  4004 5110 y Fu(])p Fz(:)872 b FA(\(10\))166 5259 y(W)-8 b(e)37
       
  4005 b(write)f Fz(\033)t Fu(\()p Fz(t)p Fu(\))g FA(for)h(the)f(result)g(of)g
       
  4006 Fx(applying)f(a)h(substitution)e Fz(\033)40 b FA(to)c(a)h(term)f
       
  4007 Fz(t)p FA(;)g(this)f(is)h(the)166 5380 y(term)g(obtained)g(from)g
       
  4008 Fz(t)h FA(by)f(replacing)g(each)h(suspension)e Fz(\031)t
       
  4009 Fr(\001)p Fz(X)44 b FA(in)36 b Fz(t)h FA(\(as)g Fz(X)44
       
  4010 b FA(ranges)37 b(o)o(v)o(er)1748 5712 y(12)p eop end
       
  4011 %%Page: 13 13
       
  4012 TeXDict begin 13 12 bop 166 83 a Fp(dom)7 b Fu(\()p Fz(\033)t
       
  4013 Fu(\))p FA(\))39 b(by)h(the)f(term)g Fz(\031)t Fr(\001)p
       
  4014 Fz(\033)t Fu(\()p Fz(X)8 b Fu(\))39 b FA(got)g(by)g(letting)f
       
  4015 Fz(\031)43 b FA(act)d(on)f(the)g(term)g Fz(\033)t Fu(\()p
       
  4016 Fz(X)8 b Fu(\))40 b FA(using)e(the)166 203 y(de\002nition)27
       
  4017 b(in)g(Figure)h(1.)g(F)o(or)g(e)o(xample,)e(if)i Fz(\033)37
       
  4018 b Fu(=)c([)p Fz(X)41 b Fu(:=)33 b Fw(h)p Fz(b;)17 b(Y)22
       
  4019 b Fw(i)p Fu(])27 b FA(and)h Fz(t)33 b Fu(=)g Fz(a:)p
       
  4020 Fu(\()p Fz(a)17 b(b)p Fu(\))p Fr(\001)q Fz(X)8 b FA(,)27
       
  4021 b(then)166 324 y Fz(\033)t Fu(\()p Fz(t)p Fu(\))h(=)g
       
  4022 Fz(a:)p Fw(h)p Fz(a;)17 b Fu(\()p Fz(a)g(b)p Fu(\))p
       
  4023 Fr(\001)p Fz(Y)k Fw(i)p FA(.)k(Gi)n(v)o(en)e(substitutions)f
       
  4024 Fz(\033)29 b FA(and)c Fz(\033)2169 288 y Ft(0)2193 324
       
  4025 y FA(,)g(and)f(freshness)h(en)l(vironments)f Fw(r)166
       
  4026 444 y FA(and)h Fw(r)418 408 y Ft(0)441 444 y FA(,)g(we)g(write)869
       
  4027 747 y(\(a\))100 b Fw(r)1162 705 y Ft(0)1213 747 y Fw(`)28
       
  4028 b Fz(\033)t Fu(\()p Fw(r)p Fu(\))199 b FA(and)g(\(b\))100
       
  4029 b Fw(r)27 b(`)h Fz(\033)k Fw(\031)c Fz(\033)2728 705
       
  4030 y Ft(0)3288 747 y FA(\(11\))166 1049 y(to)i(mean,)f(for)i(\(a\),)f
       
  4031 (that)g Fw(r)1119 1013 y Ft(0)1179 1049 y Fw(`)37 b Fz(a)g
       
  4032 Fu(#)h Fz(\033)t Fu(\()p Fz(X)8 b Fu(\))30 b FA(holds)f(for)h(each)g
       
  4033 Fu(\()p Fz(a)38 b Fu(#)f Fz(X)8 b Fu(\))37 b Fw(2)h(r)30
       
  4034 b FA(and,)f(for)i(\(b\),)166 1169 y(that)24 b Fw(r)k(`)g
       
  4035 Fz(\033)t Fu(\()p Fz(X)8 b Fu(\))27 b Fw(\031)h Fz(\033)955
       
  4036 1133 y Ft(0)978 1169 y Fu(\()p Fz(X)8 b Fu(\))25 b FA(holds)f(for)h
       
  4037 (all)f Fz(X)36 b Fw(2)28 b Fp(dom)7 b Fu(\()p Fz(\033)t
       
  4038 Fu(\))22 b Fw([)g Fp(dom)7 b Fu(\()p Fz(\033)2601 1133
       
  4039 y Ft(0)2624 1169 y Fu(\))p FA(.)166 1430 y FB(Lemma)28
       
  4040 b(2.14)g(\(Substitution\).)44 b Fx(Substitution)25 b(commutes)j(with)f
       
  4041 (the)h(permutation)d(action:)166 1551 y Fz(\033)t Fu(\()p
       
  4042 Fz(\031)t Fr(\001)p Fz(t)p Fu(\))j(=)f Fz(\031)t Fr(\001)p
       
  4043 Fu(\()p Fz(\033)t Fu(\()p Fz(t)p Fu(\)\))p Fx(.)e(Substitution)d(also)i
       
  4044 (pr)l(eserves)h Fw(\031)g Fx(and)f Fu(#)i Fx(in)e(the)h(following)e
       
  4045 (sense:)206 1812 y(\(1\))50 b(if)24 b Fw(r)535 1776 y
       
  4046 Ft(0)586 1812 y Fw(`)k Fz(\033)t Fu(\()p Fw(r)p Fu(\))d
       
  4047 Fx(and)f Fw(r)k(`)f Fz(t)h Fw(\031)g Fz(t)1494 1776 y
       
  4048 Ft(0)1518 1812 y Fx(,)d(then)f Fw(r)1847 1776 y Ft(0)1898
       
  4049 1812 y Fw(`)k Fz(\033)t Fu(\()p Fz(t)p Fu(\))f Fw(\031)i
       
  4050 Fz(\033)t Fu(\()p Fz(t)2422 1776 y Ft(0)2445 1812 y Fu(\))p
       
  4051 Fx(;)206 1932 y(\(2\))50 b(if)24 b Fw(r)535 1896 y Ft(0)586
       
  4052 1932 y Fw(`)k Fz(\033)t Fu(\()p Fw(r)p Fu(\))d Fx(and)f
       
  4053 Fw(r)k(`)f Fz(a)h Fu(#)g Fz(t)p Fx(,)d(then)g Fw(r)1844
       
  4054 1896 y Ft(0)1895 1932 y Fw(`)i Fz(a)h Fu(#)g Fz(\033)t
       
  4055 Fu(\()p Fz(t)p Fu(\))p Fx(.)166 2612 y FB(PR)m(OOF)-11
       
  4056 b(.)49 b FA(The)19 b(\002rst)h(sentence)g(follo)n(ws)e(by)i(a)g
       
  4057 (routine)f(induction)f(on)i(the)f(structure)h(of)g Fz(t)p
       
  4058 FA(.)g(The)166 2733 y(second)25 b(follo)n(ws)e(by)h(induction)g(on)g
       
  4059 (the)h(de\002nition)f(of)h Fw(\031)g FA(and)g Fu(#)g
       
  4060 FA(using)f(Lemma)g(2.8.)p 3382 2733 4 68 v 3386 2669
       
  4061 60 4 v 3386 2733 V 3445 2733 4 68 v 166 3413 a(W)-8 b(e)38
       
  4062 b(claim)f(that)g(the)g(relation)g Fw(\031)h FA(de\002ned)g(in)f(Figure)
       
  4063 h(2)g(gi)n(v)o(es)d(the)j(correct)g(notion)e(of)i Fz(\013)q
       
  4064 FA(-)166 3533 y(equi)n(v)n(alence)d(for)h(terms)f(o)o(v)o(er)g(a)h
       
  4065 (nominal)e(signature.)h(This)g(is)h(reasonable,)g(gi)n(v)o(en)e(Theo-)
       
  4066 166 3654 y(rem)24 b(2.11)g(and)g(the)g(f)o(act)g(that,)g(by)f
       
  4067 (de\002nition,)g(it)h(satis\002es)g(rules)g(\()p Fw(\031)p
       
  4068 FA(-abstraction-1\))g(and)g(\()p Fw(\031)p FA(-)166 3774
       
  4069 y(abstraction-2\).)34 b(Further)i(e)n(vidence)f(is)f(pro)o(vided)g(by)h
       
  4070 (the)g(follo)n(wing)e(proposition,)g(which)166 3894 y(sho)n(ws)d(that)i
       
  4071 (for)g(ground)f(terms)g Fw(\031)h FA(agrees)g(with)f(the)h(follo)n
       
  4072 (wing)e(more)h(traditional)g(de\002ni-)166 4015 y(tion)24
       
  4073 b(of)h Fz(\013)q FA(-equi)n(v)n(alence.)166 4276 y FB(De\002nition)c
       
  4074 (2.15)f(\(Na)954 4275 y(\250)957 4276 y(\021v)o(e)h Fh(\013)p
       
  4075 FB(-equi)o(v)o(alence\).)37 b FA(De\002ne)21 b(the)f(binary)h(relation)
       
  4076 f Fz(t)28 b Fu(=)2966 4291 y Fq(\013)3043 4276 y Fz(t)3078
       
  4077 4240 y Ft(0)3122 4276 y FA(between)166 4396 y(terms)h(o)o(v)o(er)f(a)i
       
  4078 (nominal)e(signature)h(to)g(be)h(the)f(least)g(sort-respecting)g
       
  4079 (congruence)g(relation)166 4517 y(satisfying)i Fz(a:t)43
       
  4080 b Fu(=)818 4532 y Fq(\013)909 4517 y Fz(b:)p Fu([)p Fz(a)1055
       
  4081 4511 y Ft(7!)1127 4517 y Fz(b)p Fu(])p Fz(t)33 b FA(whene)n(v)o(er)f
       
  4082 Fz(b)h FA(is)f(an)h(atom)f(\(of)h(the)f(same)h(sort)f(as)g
       
  4083 Fz(a)p FA(\))h(not)f(oc-)166 4637 y(curring)j(at)h(all)f(in)g(the)h
       
  4084 (term)f Fz(t)p FA(.)h(Here)g Fu([)p Fz(a)1624 4631 y
       
  4085 Ft(7!)1695 4637 y Fz(b)p Fu(])p Fz(t)g FA(indicates)f(the)h(result)f
       
  4086 (of)g(replacing)h(all)f(free)166 4757 y(occurrences)26
       
  4087 b(of)f Fz(a)g FA(with)f Fz(b)h FA(in)g Fz(t)p FA(.)166
       
  4088 5019 y FB(Pr)n(oposition)39 b(2.16)f(\(Adequacy\).)50
       
  4089 b Fx(If)39 b Fz(t)g Fx(and)f Fz(t)1872 4983 y Ft(0)1934
       
  4090 5019 y Fx(ar)l(e)h(gr)l(ound)e(terms)h FA(\(i.e.)h(terms)f(with)g(no)
       
  4091 166 5139 y(v)n(ariables)d(and)g(hence)h(no)f(suspensions\))f
       
  4092 Fx(o)o(ver)i(a)f(nominal)f(signatur)l(e)o(,)g(then)i(the)f(r)l(elation)
       
  4093 166 5259 y Fz(t)d Fu(=)309 5274 y Fq(\013)390 5259 y
       
  4094 Fz(t)425 5223 y Ft(0)476 5259 y Fx(of)26 b(De\002nition)g(2.15)g(holds)
       
  4095 g(if)g(and)h(only)g(if)f Fw(;)31 b(`)h Fz(t)g Fw(\031)g
       
  4096 Fz(t)2375 5223 y Ft(0)2426 5259 y Fx(is)26 b(pr)l(o)o(vable)g(fr)l(om)g
       
  4097 (the)g(rules)166 5380 y(in)31 b(F)l(igur)l(e)f(2.)h(Furthermor)l(e)o(,)
       
  4098 f Fw(;)39 b(`)h Fz(a)f Fu(#)h Fz(t)31 b Fx(is)g(pr)l(o)o(vable)f(if)h
       
  4099 (and)f(only)h(if)g Fz(a)g Fx(is)g(not)f(in)h(the)g(set)1748
       
  4100 5712 y FA(13)p eop end
       
  4101 %%Page: 14 14
       
  4102 TeXDict begin 14 13 bop 166 83 a Fp(F)-10 b(A)p Fu(\()p
       
  4103 Fz(t)p Fu(\))25 b Fx(of)g(fr)l(ee)g(atoms)f(of)g Fz(t)p
       
  4104 Fx(,)h(de\002ned)g(by:)1366 325 y Fp(F)-10 b(A)q Fu(\()p
       
  4105 Fw(hi)p Fu(\))1674 266 y Fg(def)1676 325 y Fu(=)30 b
       
  4106 Fw(;)1173 498 y Fp(F)-10 b(A)q Fu(\()p Fw(h)p Fz(t)1412
       
  4107 513 y FG(1)1451 498 y Fz(;)17 b(t)1530 513 y FG(2)1569
       
  4108 498 y Fw(i)p Fu(\))1674 439 y Fg(def)1676 498 y Fu(=)30
       
  4109 b Fp(F)-10 b(A)p Fu(\()p Fz(t)1981 513 y FG(1)2021 498
       
  4110 y Fu(\))22 b Fw([)h Fp(F)-10 b(A)p Fu(\()p Fz(t)2369
       
  4111 513 y FG(2)2408 498 y Fu(\))1333 672 y Fp(F)g(A)p Fu(\()p
       
  4112 Fz(f)28 b(t)p Fu(\))1674 613 y Fg(def)1676 672 y Fu(=)i
       
  4113 Fp(F)-10 b(A)p Fu(\()p Fz(t)p Fu(\))1393 846 y Fp(F)g(A)p
       
  4114 Fu(\()p Fz(a)p Fu(\))1674 787 y Fg(def)1676 846 y Fu(=)30
       
  4115 b Fw(f)p Fz(a)p Fw(g)1330 1019 y Fp(F)-10 b(A)q Fu(\()p
       
  4116 Fz(a:t)p Fu(\))1674 960 y Fg(def)1676 1019 y Fu(=)30
       
  4117 b Fp(F)-10 b(A)p Fu(\()p Fz(t)p Fu(\))23 b Fw(\000)f(f)p
       
  4118 Fz(a)p Fw(g)p Fz(:)166 1357 y FB(PR)m(OOF)-11 b(.)49
       
  4119 b FA(The)24 b(proof)h(is)f(similar)g(to)g(the)h(proof)g(of)g([10,)f
       
  4120 (Proposition)g(2.2].)p 3382 1357 4 68 v 3386 1293 60
       
  4121 4 v 3386 1357 V 3445 1357 4 68 v 166 1694 a(F)o(or)33
       
  4122 b(non-ground)f(terms,)h(the)g(relations)g Fu(=)1722 1709
       
  4123 y Fq(\013)1804 1694 y FA(and)h Fw(\031)f FA(dif)n(fer!)h(F)o(or)f(e)o
       
  4124 (xample)f Fz(a:X)52 b Fu(=)3204 1709 y Fq(\013)3296 1694
       
  4125 y Fz(b:X)166 1815 y FA(al)o(w)o(ays)24 b(holds,)f(whereas)i
       
  4126 Fw(;)j(`)f Fz(a:X)36 b Fw(\031)28 b Fz(b:X)33 b FA(is)24
       
  4127 b(not)f(pro)o(v)n(able)g(unless)h Fz(a)k Fu(=)f Fz(b)p
       
  4128 FA(.)e(This)f(disagree-)166 1935 y(ment)h(is)g(to)g(be)g(e)o(xpected,)g
       
  4129 (since)h(we)f(noted)g(in)g(the)g(Introduction)g(that)f
       
  4130 Fu(=)2746 1950 y Fq(\013)2821 1935 y FA(is)h Fx(not)i
       
  4131 FA(preserv)o(ed)166 2056 y(by)e(substitution,)c(whereas)26
       
  4132 b(from)e(Lemma)h(2.14)f(we)h(kno)n(w)f(that)g Fw(\031)h
       
  4133 FA(is.)166 2453 y FB(3)99 b(Uni\002cation)166 2794 y
       
  4134 FA(Gi)n(v)o(en)37 b(terms)h Fz(t)h FA(and)g Fz(t)996
       
  4135 2758 y Ft(0)1058 2794 y FA(of)g(the)f(same)h(sort)f(o)o(v)o(er)g(a)h
       
  4136 (nominal)e(signature,)h(can)h(we)g(decide)166 2914 y(whether)21
       
  4137 b(or)h(not)e(there)i(is)f(a)h(substitution)c(of)j(terms)g(for)h(the)f
       
  4138 (v)n(ariables)g(in)f Fz(t)i FA(and)f Fz(t)2984 2878 y
       
  4139 Ft(0)3029 2914 y FA(that)g(mak)o(es)166 3034 y(them)k(equal)g(in)g(the)
       
  4140 g(sense)h(of)f(the)g(relation)g Fw(\031)h FA(introduced)f(in)g(the)g
       
  4141 (pre)n(vious)f(section?)h(Since)166 3155 y(instances)30
       
  4142 b(of)g Fw(\031)h FA(are)h(established)d(modulo)g(freshness)h
       
  4143 (constraints,)g(it)f(mak)o(es)i(more)f(sense)166 3275
       
  4144 y(to)c(ask)g(whether)g(or)g(not)g(there)g(is)g(both)f(a)i(substitution)
       
  4145 c Fz(\033)30 b FA(and)c(a)h(freshness)f(en)l(vironment)f
       
  4146 Fw(r)166 3396 y FA(for)k(which)f Fw(r)34 b(`)h Fz(\033)t
       
  4147 Fu(\()p Fz(t)p Fu(\))f Fw(\031)h Fz(\033)t Fu(\()p Fz(t)1244
       
  4148 3359 y Ft(0)1268 3396 y Fu(\))28 b FA(holds.)g(As)g(for)h(ordinary)f
       
  4149 (\002rst-order)h(uni\002cation,)e(solving)166 3516 y(such)h(an)f
       
  4150 (equational)g(problem)g(may)h(thro)n(w)f(up)g Fx(se)o(ver)o(al)g
       
  4151 FA(equational)g(subproblems;)e(b)n(ut)j(an)166 3636 y(added)21
       
  4152 b(complication)f(here)i(is)f(that)g(because)h(of)f(rule)h(\()p
       
  4153 Fw(\031)p FA(-abstraction-2\))g(in)f(Figure)g(2,)g(equa-)166
       
  4154 3757 y(tional)30 b(problems)g(may)h(generate)g Fx(fr)l(eshness)g
       
  4155 FA(problems,)f(i.e.)h(ones)f(in)l(v)n(olving)f(the)i(relation)166
       
  4156 3877 y Fu(#)p FA(.)c(W)-8 b(e)28 b(are)g(thus)e(led)h(to)f(the)h(follo)
       
  4157 n(wing)e(de\002nition)h(of)i(uni\002cation)e(problems)g(for)h(nominal)
       
  4158 166 3998 y(equational)d(logic.)166 4218 y FB(De\002nition)k(3.1.)42
       
  4159 b FA(A)27 b Fx(uni\002cation)f(pr)l(oblem)f Fz(P)41 b
       
  4160 FA(o)o(v)o(er)26 b(a)i(nominal)d(signature)i(is)f(a)i(\002nite)f(set)g
       
  4161 (of)166 4338 y(atomic)32 b(problems,)f(each)i(of)g(which)f(is)g(either)
       
  4162 g(an)h Fx(equational)e(pr)l(oblem)g Fz(t)42 b Fw(\031)p
       
  4163 Fu(?)g Fz(t)3086 4302 y Ft(0)3143 4338 y FA(where)32
       
  4164 b Fz(t)166 4458 y FA(and)27 b Fz(t)372 4422 y Ft(0)422
       
  4165 4458 y FA(are)g(terms)f(of)h(the)f(same)h(sort)f(o)o(v)o(er)f(the)i
       
  4166 (signature,)f(or)g(a)h Fx(fr)l(eshness)f(pr)l(oblem)f
       
  4167 Fz(a)31 b Fu(#?)g Fz(t)166 4579 y FA(where)e Fz(a)g FA(is)f(an)g(atom)g
       
  4168 (and)g Fz(t)h FA(a)g(term)f(o)o(v)o(er)f(the)i(signature.)e(A)i
       
  4169 Fx(solution)d FA(for)j Fz(P)42 b FA(consists)27 b(of)h(a)166
       
  4170 4699 y(pair)d Fu(\()p Fw(r)p Fz(;)17 b(\033)t Fu(\))24
       
  4171 b FA(where)i Fw(r)f FA(is)f(a)h(freshness)g(en)l(vironment)e(and)i
       
  4172 Fz(\033)k FA(is)24 b(a)h(substitution)d(satisfying)166
       
  4173 4919 y Fw(\017)50 b(r)27 b(`)h Fz(a)g Fu(#)g Fz(\033)t
       
  4174 Fu(\()p Fz(t)p Fu(\))52 b FA(for)26 b(each)53 b Fu(\()p
       
  4175 Fz(a)27 b Fu(#?)h Fz(t)p Fu(\))g Fw(2)g Fz(P)66 b FA(and)166
       
  4176 5039 y Fw(\017)50 b(r)27 b(`)h Fz(\033)t Fu(\()p Fz(t)p
       
  4177 Fu(\))g Fw(\031)g Fz(\033)t Fu(\()p Fz(t)900 5003 y Ft(0)923
       
  4178 5039 y Fu(\))53 b FA(for)25 b(each)53 b Fu(\()p Fz(t)28
       
  4179 b Fw(\031)p Fu(?)g Fz(t)1677 5003 y Ft(0)1701 5039 y
       
  4180 Fu(\))f Fw(2)h Fz(P)14 b FA(.)166 5259 y(W)-8 b(e)24
       
  4181 b(write)f Ff(U)p Fu(\()p Fz(P)14 b Fu(\))22 b FA(for)i(the)f(set)g(of)g
       
  4182 (all)g(solutions)e(of)i(a)g(problem)g Fz(P)14 b FA(.)22
       
  4183 b Fu(\()p Fw(r)p Fz(;)17 b(\033)t Fu(\))28 b Fw(2)g Ff(U)p
       
  4184 Fu(\()p Fz(P)14 b Fu(\))22 b FA(is)h(a)h Fx(most)166
       
  4185 5380 y(g)o(ener)o(al)i FA(solution)g(for)i Fz(P)40 b
       
  4186 FA(if)28 b(gi)n(v)o(en)e(an)o(y)g(other)h(solution)f
       
  4187 Fu(\()p Fw(r)2296 5344 y Ft(0)2319 5380 y Fz(;)17 b(\033)2422
       
  4188 5344 y Ft(0)2445 5380 y Fu(\))33 b Fw(2)f Ff(U)p Fu(\()p
       
  4189 Fz(P)14 b Fu(\))p FA(,)27 b(then)g(there)h(is)e(a)1748
       
  4190 5712 y(14)p eop end
       
  4191 %%Page: 15 15
       
  4192 TeXDict begin 15 14 bop 166 83 a FA(substitution)18 b
       
  4193 Fz(\033)711 47 y Ft(00)775 83 y FA(satisfying)i Fw(r)1267
       
  4194 47 y Ft(0)1318 83 y Fw(`)28 b Fz(\033)1466 47 y Ft(00)1508
       
  4195 83 y Fu(\()p Fw(r)p Fu(\))21 b FA(and)h Fw(r)1937 47
       
  4196 y Ft(0)1988 83 y Fw(`)27 b Fz(\033)2135 47 y Ft(00)2187
       
  4197 83 y Fw(\016)9 b Fz(\033)31 b Fw(\031)d Fz(\033)2496
       
  4198 47 y Ft(0)2520 83 y FA(.)21 b(\(Here)h(we)g(are)g(using)e(the)166
       
  4199 203 y(notation)25 b(of)h(\(11\);)f(and)h Fz(\033)1076
       
  4200 167 y Ft(00)1142 203 y Fw(\016)d Fz(\033)30 b FA(denotes)25
       
  4201 b(the)h Fx(substitution)d(composition)h FA(of)i Fz(\033)k
       
  4202 FA(follo)n(wed)25 b(by)166 339 y Fz(\033)225 303 y Ft(00)267
       
  4203 339 y FA(,)j(gi)n(v)o(en)e(by)h Fu(\()p Fz(\033)789 303
       
  4204 y Ft(00)855 339 y Fw(\016)d Fz(\033)t Fu(\)\()p Fz(X)8
       
  4205 b Fu(\))1223 287 y Fo(def)1227 339 y Fu(=)37 b Fz(\033)1399
       
  4206 303 y Ft(00)1441 339 y Fu(\()p Fz(\033)t Fu(\()p Fz(X)8
       
  4207 b Fu(\)\))p FA(.\))27 b(A)g(solution)f Fu(\()p Fw(r)p
       
  4208 Fz(;)17 b(\033)t Fu(\))32 b Fw(2)g Ff(U)p Fu(\()p Fz(P)14
       
  4209 b Fu(\))27 b FA(is)g Fx(idempotent)166 460 y FA(pro)o(vided)d
       
  4210 Fw(r)j(`)h Fz(\033)e Fw(\016)c Fz(\033)32 b Fw(\031)c
       
  4211 Fz(\033)t FA(.)166 683 y(W)-8 b(e)31 b(describe)g(an)g(algorithm)e
       
  4212 (which,)h(gi)n(v)o(en)f(an)o(y)h(nominal)f(uni\002cation)h(problem,)g
       
  4213 (decides)166 803 y(whether)i(or)h(not)e(it)h(has)g(a)h(solution)d(and)j
       
  4214 (if)f(it)g(does,)g(returns)g(a)g(most)f(general)i(\(and)f(idem-)166
       
  4215 923 y(potent\))23 b(solution.)e(The)j(algorithm)e(uses)h(labelled)g
       
  4216 (transformations,)f(directly)h(generalising)166 1044
       
  4217 y(the)31 b(presentation)f(of)i(\002rst-order)f(uni\002cation)g(in)g
       
  4218 ([19,)g(Sect.)h(2.6])f(which)f(in)h(turn)g(is)g(based)166
       
  4219 1164 y(upon)g(the)g(approach)h(in)g([18].)f(\(See)i(also)e([20,)h
       
  4220 (Sect.)g(4.6])f(for)h(a)g(detailed)f(e)o(xposition,)e(b)n(ut)166
       
  4221 1284 y(not)d(using)f(labels.\))h(W)-8 b(e)26 b(use)g(tw)o(o)g(types)f
       
  4222 (of)i(labelled)e(transformation)g(between)i(uni\002cation)166
       
  4223 1405 y(problems,)d(namely)1248 1547 y Fz(P)1410 1494
       
  4224 y Fq(\033)1352 1547 y Fu(=)-17 b Fw(\))28 b Fz(P)1616
       
  4225 1506 y Ft(0)1738 1547 y FA(and)99 b Fz(P)2136 1494 y
       
  4226 Ft(r)2086 1547 y Fu(=)-17 b Fw(\))27 b Fz(P)2349 1506
       
  4227 y Ft(0)166 1702 y FA(where)i(the)g(substitution)d Fz(\033)33
       
  4228 b FA(is)c(either)g(the)f(identity)g Fz(")p FA(,)g(or)h(a)g(single)f
       
  4229 (replacement)h Fu([)p Fz(X)43 b Fu(:=)36 b Fz(t)p Fu(])p
       
  4230 FA(;)166 1822 y(and)24 b(where)h(the)f(freshness)g(en)l(vironment)e
       
  4231 Fw(r)j FA(is)e(either)h(empty)f Fw(;)p FA(,)h(or)h(a)f(singleton)f
       
  4232 Fw(f)p Fz(a)k Fu(#)h Fz(X)8 b Fw(g)p FA(.)166 1943 y(The)21
       
  4233 b(le)o(gal)f(transformations)g(are)h(gi)n(v)o(en)f(in)h(Figure)g(3.)g
       
  4234 (This)f(\002gure)i(uses)f(the)g(notation)e Fz(P)j Fw(])8
       
  4235 b Fz(P)3430 1906 y Ft(0)166 2063 y FA(to)26 b(indicate)f(the)h(union)g
       
  4236 (of)g(problems)f Fz(P)40 b FA(and)26 b Fz(P)1869 2027
       
  4237 y Ft(0)1918 2063 y FA(that)g(are)g(disjoint)f(\()p Fz(P)36
       
  4238 b Fw(\\)24 b Fz(P)2866 2027 y Ft(0)2919 2063 y Fu(=)30
       
  4239 b Fw(;)p FA(\);)c(and)g(the)166 2183 y(notation)j Fz(\033)t(P)44
       
  4240 b FA(to)30 b(indicate)g(the)g(problem)f(resulting)g(from)h(applying)f
       
  4241 (the)h(substitution)e Fz(\033)34 b FA(to)166 2304 y(all)25
       
  4242 b(the)f(terms)g(occurring)h(in)g(the)f(problem)g Fz(P)14
       
  4243 b FA(.)166 2526 y FB(Algorithm.)33 b FA(Gi)n(v)o(en)18
       
  4244 b(a)h(uni\002cation)f(problem)g Fz(P)c FA(,)19 b(the)g(algorithm)e
       
  4245 (proceeds)i(in)g(tw)o(o)g(phases.)3398 2490 y FG(2)166
       
  4246 2647 y FA(In)j(the)g(\002rst)g(phase)g(it)f(applies)g(as)h(man)o(y)1627
       
  4247 2594 y Fq(\033)1569 2647 y Fu(=)-17 b Fw(\))22 b FA(transformations)e
       
  4248 (as)i(possible)f(\(non-determin-)166 2767 y(istically\).)30
       
  4249 b(If)i(this)f(results)g(in)g(a)h(problem)f(containing)f(no)i
       
  4250 (equational)e(subproblems,)g(then)166 2888 y(it)d(proceeds)h(to)f(the)g
       
  4251 (second)g(phase;)g(otherwise)g(it)g(halts)g(signalling)f(f)o(ailure.)h
       
  4252 (In)h(the)f(second)166 3024 y(phase)22 b(it)f(applies)g(as)g(man)o(y)
       
  4253 1192 2971 y Ft(r)1142 3024 y Fu(=)-17 b Fw(\))21 b FA(transformations)f
       
  4254 (as)i(possible)e(\(non-deterministically\).)f(If)166
       
  4255 3145 y(this)24 b(does)h(not)f(result)g(in)h(the)g(empty)f(problem,)g
       
  4256 (then)g(it)g(halts)h(signalling)e(f)o(ailure;)h(otherwise)166
       
  4257 3265 y(o)o(v)o(erall)g(it)g(has)h(constructed)f(a)h(transformation)e
       
  4258 (sequence)i(of)g(the)g(form)1152 3509 y Fz(P)1298 3454
       
  4259 y Fq(\033)1338 3463 y Fe(1)1256 3509 y Fu(=)-17 b Fw(\))28
       
  4260 b(\001)17 b(\001)g(\001)1625 3455 y Fq(\033)1665 3463
       
  4261 y Fd(n)1587 3509 y Fu(=)-17 b Fw(\))27 b Fz(P)1850 3468
       
  4262 y Ft(0)1933 3454 y(r)1992 3463 y Fe(1)1901 3509 y Fu(=)-17
       
  4263 b Fw(\))27 b(\001)17 b(\001)g(\001)2252 3455 y Ft(r)2311
       
  4264 3463 y Fd(m)2231 3509 y Fu(=)-17 b Fw(\))28 b(;)820 b
       
  4265 FA(\(12\))166 3734 y(\(where)26 b Fz(P)545 3698 y Ft(0)593
       
  4266 3734 y FA(does)f(not)g(contain)f(an)o(y)h(equational)f(subproblems\))g
       
  4267 (and)h(the)g(algorithm)f(returns)166 3854 y(the)h(solution)e
       
  4268 Fu(\()p Fw(r)780 3869 y FG(1)841 3854 y Fw([)g(\001)17
       
  4269 b(\001)g(\001)j([)j(r)1240 3869 y Fq(m)1306 3854 y Fz(;)34
       
  4270 b(\033)1422 3869 y Fq(n)1491 3854 y Fw(\016)22 b(\001)17
       
  4271 b(\001)g(\001)j(\016)i Fz(\033)1828 3869 y FG(1)1868
       
  4272 3854 y Fu(\))p FA(.)166 4077 y(T)-8 b(o)31 b(sho)n(w)g(the)g
       
  4273 (correctness)h(of)g(this)f(algorithm,)f(we)i(\002rst)f(establish)g
       
  4274 (that)g(all)g(sequences)h(of)166 4198 y(uni\002cation)24
       
  4275 b(transitions)f(must)h(terminate.)166 4420 y FB(Lemma)h(3.2.)41
       
  4276 b Fx(Ther)l(e)26 b(is)e(no)h(in\002nite)f(series)g(of)g(uni\002cation)f
       
  4277 (tr)o(ansitions.)166 4783 y FB(PR)m(OOF)-11 b(.)49 b
       
  4278 FA(Since)28 b(e)n(v)o(ery)f(reduction)f(sequence)i(consists)e(of)i(tw)o
       
  4279 (o)f(\(possibly)f(empty\))h(subse-)166 4920 y(quences,)j(namely)g(one)h
       
  4280 (containing)e(only)1757 4867 y Fq(\033)1699 4920 y Fu(=)-17
       
  4281 b Fw(\))p FA(-steps)30 b(and)h(the)f(other)g(only)2941
       
  4282 4867 y Ft(r)2891 4920 y Fu(=)-17 b Fw(\))p FA(-steps,)30
       
  4283 b(we)166 5040 y(can)23 b(sho)n(w)f(termination)f(for)i(both)f
       
  4284 (subsequences)g(separately)-6 b(.)22 b(F)o(or)h(e)n(v)o(ery)f
       
  4285 (uni\002cation)g(prob-)166 5161 y(lem)g Fz(P)36 b FA(we)23
       
  4286 b(de\002ne)h(a)f(measure)f(of)h(the)g(size)f(of)h Fz(P)36
       
  4287 b FA(to)22 b(be)h(the)g(le)o(xicographically)e(ordered)i(pair)p
       
  4288 166 5282 299 4 v 166 5346 a FG(2)257 5379 y FD(See)g(Remark)g(3.9)h
       
  4289 (for)g(discussion)i(of)e(this)g(use)g(of)f(tw)o(o)g(phases.)1748
       
  4290 5712 y FA(15)p eop end
       
  4291 %%Page: 16 16
       
  4292 TeXDict begin 16 15 bop 166 3 3288 4 v 166 3123 4 3121
       
  4293 v 204 154 a FD(\()p Fl(\031)p Fk(?)p FD(-unit\))798 b
       
  4294 Fw(fhi)27 b(\031)p Fu(?)h Fw(hig)21 b(])i Fz(P)2063 101
       
  4295 y Fq(")1999 154 y Fu(=)-17 b Fw(\))34 b Fz(P)204 335
       
  4296 y FD(\()p Fl(\031)p Fk(?)p FD(-pair\))412 b Fw(fh)p Fz(t)1084
       
  4297 350 y FG(1)1123 335 y Fz(;)17 b(t)1202 350 y FG(2)1241
       
  4298 335 y Fw(i)28 b(\031)p Fu(?)g Fw(h)p Fz(t)1533 298 y
       
  4299 Ft(0)1533 359 y FG(1)1572 335 y Fz(;)17 b(t)1651 298
       
  4300 y Ft(0)1651 359 y FG(2)1691 335 y Fw(ig)k(])i Fz(P)2063
       
  4301 281 y Fq(")1999 335 y Fu(=)-17 b Fw(\))34 b(f)p Fz(t)2277
       
  4302 350 y FG(1)2344 335 y Fw(\031)p Fu(?)28 b Fz(t)2530 298
       
  4303 y Ft(0)2530 359 y FG(1)2570 335 y Fz(;)17 b(t)2649 350
       
  4304 y FG(2)2716 335 y Fw(\031)p Fu(?)28 b Fz(t)2902 298 y
       
  4305 Ft(0)2902 359 y FG(2)2942 335 y Fw(g)21 b([)i Fz(P)204
       
  4306 515 y FD(\()p Fl(\031)p Fk(?)p FD(-function)j(symbol\))256
       
  4307 b Fw(f)p Fz(f)27 b(t)h Fw(\031)q Fu(?)f Fz(f)h(t)1706
       
  4308 479 y Ft(0)1729 515 y Fw(g)22 b(])h Fz(P)2063 462 y Fq(")1999
       
  4309 515 y Fu(=)-17 b Fw(\))34 b(f)p Fz(t)27 b Fw(\031)q Fu(?)g
       
  4310 Fz(t)2490 479 y Ft(0)2514 515 y Fw(g)22 b([)h Fz(P)204
       
  4311 696 y FD(\()p Fl(\031)p Fk(?)p FD(-abstraction-1\))372
       
  4312 b Fw(f)p Fz(a:t)28 b Fw(\031)q Fu(?)f Fz(a:t)1705 660
       
  4313 y Ft(0)1729 696 y Fw(g)22 b(])h Fz(P)2063 643 y Fq(")1999
       
  4314 696 y Fu(=)-17 b Fw(\))34 b(f)p Fz(t)27 b Fw(\031)q Fu(?)g
       
  4315 Fz(t)2490 660 y Ft(0)2514 696 y Fw(g)22 b([)h Fz(P)204
       
  4316 876 y FD(\()p Fl(\031)p Fk(?)p FD(-abstraction-2\))349
       
  4317 b Fw(f)p Fz(a:t)28 b Fw(\031)p Fu(?)g Fz(a)1620 840 y
       
  4318 Ft(0)1644 876 y Fz(:t)1706 840 y Ft(0)1729 876 y Fw(g)22
       
  4319 b(])h Fz(P)2063 823 y Fq(")1999 876 y Fu(=)-17 b Fw(\))34
       
  4320 b(f)p Fz(t)27 b Fw(\031)q Fu(?)g(\()p Fz(a)17 b(a)2612
       
  4321 840 y Ft(0)2636 876 y Fu(\))p Fr(\001)p Fz(t)2741 840
       
  4322 y Ft(0)2764 876 y Fz(;)g(a)28 b Fu(#?)g Fz(t)3077 840
       
  4323 y Ft(0)3100 876 y Fw(g)22 b([)h Fz(P)2781 1010 y FA(pro)o(vided)h
       
  4324 Fz(a)k Fw(6)p Fu(=)f Fz(a)3392 973 y Ft(0)204 1143 y
       
  4325 FD(\()p Fl(\031)p Fk(?)p FD(-atom\))809 b Fw(f)p Fz(a)28
       
  4326 b Fw(\031)p Fu(?)g Fz(a)p Fw(g)22 b(])h Fz(P)2063 1090
       
  4327 y Fq(")1999 1143 y Fu(=)-17 b Fw(\))34 b Fz(P)204 1324
       
  4328 y FD(\()p Fl(\031)p Fk(?)p FD(-suspension\))315 b Fw(f)p
       
  4329 Fz(\031)t Fr(\001)p Fz(X)35 b Fw(\031)q Fu(?)27 b Fz(\031)1585
       
  4330 1287 y Ft(0)1609 1324 y Fr(\001)p Fz(X)8 b Fw(g)21 b(])i
       
  4331 Fz(P)2063 1270 y Fq(")1999 1324 y Fu(=)-17 b Fw(\))34
       
  4332 b(f)p Fz(a)27 b Fu(#?)h Fz(X)36 b Fw(j)27 b Fz(a)h Fw(2)g
       
  4333 Fp(ds)8 b Fu(\()p Fz(\031)t(;)17 b(\031)3118 1287 y Ft(0)3141
       
  4334 1324 y Fu(\))p Fw(g)22 b([)g Fz(P)204 1583 y FD(\()p
       
  4335 Fl(\031)p Fk(?)p FD(-v)n(ariable\))1259 1504 y Fw(f)p
       
  4336 Fz(t)27 b Fw(\031)q Fu(?)g Fz(\031)t Fr(\001)p Fz(X)8
       
  4337 b Fw(g)22 b(])h Fz(P)1259 1685 y Fw(f)p Fz(\031)t Fr(\001)o
       
  4338 Fz(X)36 b Fw(\031)p Fu(?)28 b Fz(t)p Fw(g)22 b(])h Fz(P)1939
       
  4339 1384 y Fn(9)1939 1459 y(>)1939 1484 y(=)1939 1633 y(>)1939
       
  4340 1658 y(;)2058 1530 y Fq(\033)1999 1583 y Fu(=)-17 b Fw(\))34
       
  4341 b Fz(\033)t(P)160 b FA(with)52 b Fz(\033)31 b Fu(=)d([)p
       
  4342 Fz(X)36 b Fu(:=)27 b Fz(\031)3227 1547 y Ft(\000)p FG(1)3321
       
  4343 1583 y Fr(\001)p Fz(t)p Fu(])2181 1724 y FA(pro)o(vided)c
       
  4344 Fz(X)33 b FA(does)25 b(not)f(occur)h(in)f Fz(t)204 1928
       
  4345 y FD(\()p Fk(#?)p FD(-unit\))815 b Fw(f)p Fz(a)28 b Fu(#?)g
       
  4346 Fw(hig)21 b(])i Fz(P)2061 1875 y Ft(;)1999 1928 y Fu(=)-17
       
  4347 b Fw(\))34 b Fz(P)204 2108 y FD(\()p Fk(#?)p FD(-pair\))622
       
  4348 b Fw(f)p Fz(a)28 b Fu(#?)g Fw(h)p Fz(t)1533 2123 y FG(1)1572
       
  4349 2108 y Fz(;)17 b(t)1651 2123 y FG(2)1691 2108 y Fw(ig)k(])i
       
  4350 Fz(P)2061 2055 y Ft(;)1999 2108 y Fu(=)-17 b Fw(\))34
       
  4351 b(f)p Fz(a)27 b Fu(#?)h Fz(t)2510 2123 y FG(1)2550 2108
       
  4352 y Fz(;)17 b(a)28 b Fu(#?)g Fz(t)2863 2123 y FG(2)2902
       
  4353 2108 y Fw(g)22 b([)h Fz(P)204 2289 y FD(\()p Fk(#?)p
       
  4354 FD(-function)j(symbol\))330 b Fw(f)p Fz(a)28 b Fu(#?)g
       
  4355 Fz(f)f(t)p Fw(g)22 b(])h Fz(P)2061 2236 y Ft(;)1999 2289
       
  4356 y Fu(=)-17 b Fw(\))34 b(f)p Fz(a)27 b Fu(#?)h Fz(t)p
       
  4357 Fw(g)22 b([)h Fz(P)204 2469 y FD(\()p Fk(#?)p FD(-abstraction-1\))449
       
  4358 b Fw(f)p Fz(a)28 b Fu(#?)g Fz(a:t)p Fw(g)22 b(])h Fz(P)2061
       
  4359 2416 y Ft(;)1999 2469 y Fu(=)-17 b Fw(\))34 b Fz(P)204
       
  4360 2650 y FD(\()p Fk(#?)p FD(-abstraction-2\))426 b Fw(f)p
       
  4361 Fz(a)27 b Fu(#?)h Fz(a)1643 2614 y Ft(0)1667 2650 y Fz(:t)p
       
  4362 Fw(g)22 b(])h Fz(P)2061 2597 y Ft(;)1999 2650 y Fu(=)-17
       
  4363 b Fw(\))34 b(f)p Fz(a)27 b Fu(#?)h Fz(t)p Fw(g)22 b([)h
       
  4364 Fz(P)47 b FA(pro)o(vided)24 b Fz(a)k Fw(6)p Fu(=)f Fz(a)3392
       
  4365 2614 y Ft(0)204 2831 y FD(\()p Fk(#?)p FD(-atom\))777
       
  4366 b Fw(f)p Fz(a)28 b Fu(#?)g Fz(a)1706 2794 y Ft(0)1729
       
  4367 2831 y Fw(g)22 b(])h Fz(P)2061 2777 y Ft(;)1999 2831
       
  4368 y Fu(=)-17 b Fw(\))34 b Fz(P)526 b FA(pro)o(vided)24
       
  4369 b Fz(a)k Fw(6)p Fu(=)f Fz(a)3392 2794 y Ft(0)204 3011
       
  4370 y FD(\()p Fk(#?)p FD(-suspension\))458 b Fw(f)p Fz(a)28
       
  4371 b Fu(#?)g Fz(\031)t Fr(\001)p Fz(X)8 b Fw(g)21 b(])i
       
  4372 Fz(P)2049 2958 y Ft(r)1999 3011 y Fu(=)-17 b Fw(\))34
       
  4373 b Fz(P)113 b FA(with)24 b Fw(r)j Fu(=)h Fw(f)p Fz(\031)2893
       
  4374 2975 y Ft(\000)p FG(1)2987 3011 y Fr(\001)p Fz(a)g Fu(#)g
       
  4375 Fz(X)8 b Fw(g)p 3450 3123 V 166 3126 3288 4 v 1214 3263
       
  4376 a FD(Fig.)23 b(3.)g(Labelled)h(transformations.)166 3422
       
  4377 y FA(of)29 b(natural)f(numbers)f Fu(\()p Fz(n)1050 3437
       
  4378 y FG(1)1090 3422 y Fz(;)17 b(n)1192 3437 y FG(2)1231
       
  4379 3422 y Fu(\))p FA(,)29 b(where)g Fz(n)1653 3437 y FG(1)1721
       
  4380 3422 y FA(is)f(the)g(number)g(of)g(dif)n(ferent)g(v)n(ariables)g(used)g
       
  4381 (in)166 3542 y Fz(P)14 b FA(,)24 b(and)h Fz(n)519 3557
       
  4382 y FG(2)584 3542 y FA(is)f(the)h(size)f(\(see)i(De\002nition)e(2.10\))g
       
  4383 (of)h(all)g(equational)f(problems)f(in)i Fz(P)14 b FA(,)24
       
  4384 b(that)g(is)1334 3767 y Fz(n)1392 3782 y FG(2)1459 3714
       
  4385 y Fo(def)1464 3767 y Fu(=)1671 3684 y Fn(X)1572 3872
       
  4386 y FG(\()p Fq(t)p Ft(\031)q FG(?)p Fq(t)1738 3853 y Fc(0)1761
       
  4387 3872 y FG(\))p Ft(2)p Fq(P)1906 3767 y Fw(j)p Fz(t)p
       
  4388 Fw(j)e Fu(+)g Fw(j)p Fz(t)2180 3726 y Ft(0)2203 3767
       
  4389 y Fw(j)28 b Fz(:)166 4084 y FA(In)34 b(e)n(v)o(ery)592
       
  4390 4031 y Fq(\033)533 4084 y Fu(=)-17 b Fw(\))p FA(-step)34
       
  4391 b(this)e(measure)i(decreases:)g(the)f(\()p Fw(\031)p
       
  4392 FA(?-v)n(ariable\))h(transition)e(eliminates)166 4205
       
  4393 y(\(completely\))20 b(one)g(v)n(ariable)h(from)f(the)g(uni\002cation)g
       
  4394 (problem,)g(and)h(therefore)g Fz(n)2984 4220 y FG(1)3044
       
  4395 4205 y FA(decreases;)166 4325 y(the)j(\()p Fw(\031)p
       
  4396 FA(?-suspension\))f(transition)g(may)g(eliminate)g(a)h(v)n(ariable)f
       
  4397 (and)h(also)g(decreases)g(the)g(size)166 4445 y Fz(n)224
       
  4398 4460 y FG(2)264 4445 y FA(;)i(all)h(other)f(transitions)f(lea)n(v)o(e)i
       
  4399 (the)g(number)f(of)h(v)n(ariables)f(unchanged,)g(b)n(ut)g(decrease)i
       
  4400 Fz(n)3389 4460 y FG(2)3429 4445 y FA(.)166 4582 y(F)o(or)d(the)525
       
  4401 4529 y Ft(r)474 4582 y Fu(=)-17 b Fw(\))p FA(-steps)25
       
  4402 b(the)f(size)1696 4619 y Fn(X)1601 4808 y FG(\()p Fq(a)p
       
  4403 FG(#?)q Fq(t)p FG(\))p Ft(2)p Fq(P)1929 4702 y Fw(j)p
       
  4404 Fz(t)p Fw(j)166 4927 y FA(decreases)30 b(in)e(e)n(v)o(ery)g(step.)h(T)
       
  4405 -8 b(aking)28 b(both)g(f)o(acts)h(together)f(means)h(that)f(e)n(v)o
       
  4406 (ery)g(reduction)h(se-)166 5047 y(quence)c(must)f(terminate.)p
       
  4407 3382 5047 4 68 v 3386 4983 60 4 v 3386 5047 V 3445 5047
       
  4408 4 68 v 166 5380 a(The)e(follo)n(wing)f(lemmas)g(help)h(us)g(to)g(sho)n
       
  4409 (w)f(that)h(the)g(algorithm)f(gi)n(v)o(es)g(correct)i(results)e(upon)
       
  4410 1748 5712 y(16)p eop end
       
  4411 %%Page: 17 17
       
  4412 TeXDict begin 17 16 bop 166 83 a FA(termination.)166
       
  4413 306 y FB(Lemma)25 b(3.3.)41 b Fx(If)25 b Fw(r)j(`)f Fz(\033)t
       
  4414 Fu(\()p Fz(\031)t Fr(\001)p Fz(X)8 b Fu(\))27 b Fw(\031)i
       
  4415 Fz(\033)t Fu(\()p Fz(t)p Fu(\))c Fx(then)f Fw(r)k(`)f
       
  4416 Fz(\033)g Fw(\016)22 b Fu([)p Fz(X)35 b Fu(:=)28 b Fz(\031)2518
       
  4417 270 y Ft(\000)p FG(1)2612 306 y Fr(\001)p Fz(t)p Fu(])g
       
  4418 Fw(\031)g Fz(\033)t Fx(.)166 669 y FB(PR)m(OOF)-11 b(.)49
       
  4419 b FA(W)-8 b(e)32 b(ha)n(v)o(e)f(to)h(pro)o(v)o(e)e(that)i(both)f
       
  4420 (substitutions)d(agree)33 b(\(modulo)e Fw(\031)p FA(\))h(on)g(all)f(v)n
       
  4421 (ari-)166 790 y(ables)f(in)f Fz(dom)p Fu(\()p Fz(\033)t
       
  4422 Fu(\))d Fw([)g(f)p Fz(X)8 b Fw(g)p FA(.)30 b(The)g(only)f(interesting)g
       
  4423 (case)h(is)g(for)g(the)g(substitutions)c(applied)166
       
  4424 910 y(to)35 b Fz(X)8 b FA(,)34 b(when)h(we)g(need)g(to)g(sho)n(w)f
       
  4425 (that)g Fw(r)46 b(`)h Fz(\033)t Fu(\()p Fz(\031)1985
       
  4426 874 y Ft(\000)p FG(1)2079 910 y Fr(\001)p Fz(t)p Fu(\))f
       
  4427 Fw(\031)h Fz(\033)t Fu(\()p Fz(X)8 b Fu(\))p FA(.)35
       
  4428 b(By)g(Lemma)f(2.14)h(we)166 1030 y(can)i(commute)e(the)h(permutation)f
       
  4429 (to)h(the)g(outside)f(and)h(mo)o(v)o(e)f(it)h(to)g(the)g(other)g(side)g
       
  4430 (of)g Fw(\031)166 1151 y FA(by)h(Lemma)g(2.12\227this)e(gi)n(v)o(es)h
       
  4431 Fw(r)51 b(`)g Fz(\033)t Fu(\()p Fz(t)p Fu(\))g Fw(\031)g
       
  4432 Fz(\031)t Fr(\001)p Fz(\033)t Fu(\()p Fz(X)8 b Fu(\))p
       
  4433 FA(.)37 b(The)g(case)h(then)f(follo)n(ws)f(from)166 1271
       
  4434 y(the)27 b(assumptions)e(by)h(symmetry)g(and)h(commuting)e(the)i
       
  4435 (permutation)f(inside)g(the)h(substitu-)166 1392 y(tion.)p
       
  4436 3382 1392 4 68 v 3386 1328 60 4 v 3386 1392 V 3445 1392
       
  4437 4 68 v 166 1653 a FB(Lemma)g(3.4.)41 b Fx(Given)26 b(a)g(uni\002cation)
       
  4438 e(pr)l(oblem)h Fz(P)14 b Fx(,)25 b Fu(\()p Fw(r)p Fz(;)17
       
  4439 b(\033)t Fu(\))29 b Fw(2)h Ff(U)p Fu(\()p Fz(\033)2538
       
  4440 1616 y Ft(0)2562 1653 y Fz(P)14 b Fu(\))25 b Fx(holds)g(if)g(and)h
       
  4441 (only)f(if)166 1773 y Fu(\()p Fw(r)p Fz(;)17 b(\033)26
       
  4442 b Fw(\016)c Fz(\033)543 1737 y Ft(0)566 1773 y Fu(\))28
       
  4443 b Fw(2)g Ff(U)p Fu(\()p Fz(P)14 b Fu(\))p Fx(.)166 2136
       
  4444 y FB(PR)m(OOF)-11 b(.)49 b FA(Simple)24 b(calculation)g(using)g(the)g
       
  4445 (f)o(act)i(that)e Fz(\033)t Fu(\()p Fz(\033)2236 2100
       
  4446 y Ft(0)2259 2136 y Fu(\()p Fz(t)p Fu(\)\))k(=)f(\()p
       
  4447 Fz(\033)f Fw(\016)c Fz(\033)2789 2100 y Ft(0)2813 2136
       
  4448 y Fu(\)\()p Fz(t)p Fu(\))p FA(.)p 3382 2136 V 3386 2073
       
  4449 60 4 v 3386 2136 V 3445 2136 4 68 v 166 2500 a(The)31
       
  4450 b(follo)n(wing)e(tw)o(o)i(lemmas)f(sho)n(w)g(that)h(the)g
       
  4451 (uni\002cation)g(transformations)e(can)j(be)f(used)166
       
  4452 2620 y(to)d(determine)g(whether)h(or)f(not)g(solutions)f(e)o(xists)g
       
  4453 (and)h(to)h(describe)f(all)g(of)h(them)f(if)g(the)o(y)g(do)166
       
  4454 2741 y(e)o(xist.)166 2964 y FB(Lemma)d(3.5.)228 3186
       
  4455 y Fx(\(i\))50 b(If)25 b Fu(\()p Fw(r)579 3150 y Ft(0)602
       
  4456 3186 y Fz(;)17 b(\033)705 3150 y Ft(0)728 3186 y Fu(\))37
       
  4457 b Fw(2)g Ff(U)p Fu(\()p Fz(P)14 b Fu(\))24 b Fx(and)h
       
  4458 Fz(P)1500 3133 y Fq(\033)1442 3186 y Fu(=)-17 b Fw(\))37
       
  4459 b Fz(P)1715 3150 y Ft(0)1738 3186 y Fz(;)17 b Fx(then)24
       
  4460 b Fu(\()p Fw(r)2099 3150 y Ft(0)2122 3186 y Fz(;)17 b(\033)2225
       
  4461 3150 y Ft(0)2248 3186 y Fu(\))37 b Fw(2)g Ff(U)p Fu(\()p
       
  4462 Fz(P)2612 3150 y Ft(0)2635 3186 y Fu(\))25 b Fx(and)f
       
  4463 Fw(r)2955 3150 y Ft(0)3015 3186 y Fw(`)37 b Fz(\033)3172
       
  4464 3150 y Ft(0)3205 3186 y Fw(\016)25 b Fz(\033)41 b Fw(\031)372
       
  4465 3307 y Fz(\033)431 3271 y Ft(0)454 3307 y Fz(:)200 3427
       
  4466 y Fx(\(ii\))50 b(If)25 b Fu(\()p Fw(r)579 3391 y Ft(0)602
       
  4467 3427 y Fz(;)17 b(\033)705 3391 y Ft(0)728 3427 y Fu(\))28
       
  4468 b Fw(2)g Ff(U)p Fu(\()p Fz(P)14 b Fu(\))24 b Fx(and)g
       
  4469 Fz(P)1465 3374 y Ft(r)1415 3427 y Fu(=)-17 b Fw(\))27
       
  4470 b Fz(P)1678 3391 y Ft(0)1701 3427 y Fz(;)17 b Fx(then)24
       
  4471 b Fu(\()p Fw(r)2062 3391 y Ft(0)2085 3427 y Fz(;)17 b(\033)2188
       
  4472 3391 y Ft(0)2211 3427 y Fu(\))28 b Fw(2)g Ff(U)p Fu(\()p
       
  4473 Fz(P)2557 3391 y Ft(0)2580 3427 y Fu(\))d Fx(and)f Fw(r)2900
       
  4474 3391 y Ft(0)2951 3427 y Fw(`)j Fz(\033)3098 3391 y Ft(0)3122
       
  4475 3427 y Fu(\()p Fw(r)p Fu(\))p Fz(:)166 3791 y FB(PR)m(OOF)-11
       
  4476 b(.)49 b FA(W)-8 b(e)27 b(just)e(gi)n(v)o(e)h(the)g(details)g(for)h(tw)
       
  4477 o(o)f(uni\002cation)g(transitions:)f(the)i(case)g(for)g(\()p
       
  4478 Fw(\031)p Fu(?)p FA(-)166 3911 y(suspension\))h(follo)n(ws)f(from)i
       
  4479 (Lemma)g(2.12\(iii\);)f(and)h(the)g(\()p Fw(\031)p Fu(?)p
       
  4480 FA(-v)n(ariable\))h(case)f(is)g(a)h(conse-)166 4031 y(quence)25
       
  4481 b(of)g(Lemmas)f(3.3)g(and)h(3.4.)p 3382 4031 V 3386 3967
       
  4482 60 4 v 3386 4031 V 3445 4031 4 68 v 166 4292 a FB(Lemma)g(3.6.)228
       
  4483 4515 y Fx(\(i\))50 b(If)25 b Fu(\()p Fw(r)579 4479 y
       
  4484 Ft(0)602 4515 y Fz(;)17 b(\033)705 4479 y Ft(0)728 4515
       
  4485 y Fu(\))28 b Fw(2)g Ff(U)p Fu(\()p Fz(P)1074 4479 y Ft(0)1097
       
  4486 4515 y Fu(\))c Fx(and)h Fz(P)1496 4462 y Fq(\033)1438
       
  4487 4515 y Fu(=)-17 b Fw(\))27 b Fz(P)1701 4479 y Ft(0)1724
       
  4488 4515 y Fz(;)17 b Fx(then)24 b Fu(\()p Fw(r)2085 4479
       
  4489 y Ft(0)2109 4515 y Fz(;)17 b(\033)2212 4479 y Ft(0)2257
       
  4490 4515 y Fw(\016)22 b Fz(\033)t Fu(\))27 b Fw(2)h Ff(U)p
       
  4491 Fu(\()p Fz(P)14 b Fu(\))p Fz(:)200 4655 y Fx(\(ii\))50
       
  4492 b(If)25 b Fu(\()p Fw(r)579 4619 y Ft(0)602 4655 y Fz(;)17
       
  4493 b(\033)705 4619 y Ft(0)728 4655 y Fu(\))28 b Fw(2)g Ff(U)p
       
  4494 Fu(\()p Fz(P)1074 4619 y Ft(0)1097 4655 y Fu(\))p Fz(;)44
       
  4495 b(P)1360 4602 y Ft(r)1310 4655 y Fu(=)-17 b Fw(\))27
       
  4496 b Fz(P)1573 4619 y Ft(0)1621 4655 y Fx(and)d Fw(r)1878
       
  4497 4619 y Ft(00)1949 4655 y Fw(`)j Fz(\033)2096 4619 y Ft(0)2119
       
  4498 4655 y Fu(\()p Fw(r)p Fu(\))p Fz(;)17 b Fx(then)25 b
       
  4499 Fu(\()p Fw(r)2640 4619 y Ft(0)2685 4655 y Fw([)d(r)2856
       
  4500 4619 y Ft(00)2899 4655 y Fz(;)17 b(\033)3002 4619 y Ft(0)3025
       
  4501 4655 y Fu(\))28 b Fw(2)g Ff(U)p Fu(\()p Fz(P)14 b Fu(\))p
       
  4502 Fz(:)166 5019 y FB(PR)m(OOF)-11 b(.)49 b FA(Once)33 b(again,)f(we)h
       
  4503 (just)f(gi)n(v)o(e)g(the)g(details)g(for)i(tw)o(o)e(uni\002cation)g
       
  4504 (transitions:)f(the)166 5139 y(\()p Fw(\031)p Fu(?)p
       
  4505 FA(-suspension\))23 b(case)h(follo)n(ws)e(from)h(Lemma)f(2.8;)h(and)g
       
  4506 (the)h(\()p Fw(\031)p Fu(?)p FA(-v)n(ariable\))f(case)h(follo)n(ws)166
       
  4507 5259 y(from)34 b(Lemma)f(3.4)h(and)g(the)g(f)o(act)h(that)e
       
  4508 Fz(t)p Fu([)p Fz(X)53 b Fu(:=)45 b Fz(\031)1994 5223
       
  4509 y Ft(\000)p FG(1)2088 5259 y Fr(\001)p Fz(t)p Fu(])g(=)g
       
  4510 Fz(t)p FA(,)35 b(which)e(holds)g(by)h(the)g(side-)166
       
  4511 5380 y(condition)23 b(on)i(the)g(\()p Fw(\031)p Fu(?)p
       
  4512 FA(-v)n(ariable\))g(transition)e(about)h(the)h(non-occurrence)h(of)e
       
  4513 Fz(X)33 b FA(in)25 b Fz(t)p FA(.)p 3382 5380 V 3386 5316
       
  4514 60 4 v 3386 5380 V 3445 5380 4 68 v 1748 5712 a(17)p
       
  4515 eop end
       
  4516 %%Page: 18 18
       
  4517 TeXDict begin 18 17 bop 166 83 a FA(The)26 b(follo)n(wing)d(theorem)j
       
  4518 (establishes)e(the)i(correctness)f(of)h(the)g(nominal)e(uni\002cation)h
       
  4519 (algo-)166 203 y(rithm)f(and)h(is)f(the)h(central)g(result)f(of)h(the)f
       
  4520 (paper)-5 b(.)166 432 y FB(Theor)n(em)27 b(3.7)d(\(Corr)n(ectness\).)43
       
  4521 b Fx(Given)25 b(a)f(uni\002cation)g(pr)l(oblem)f Fz(P)228
       
  4522 660 y Fx(\(i\))50 b(if)24 b(the)h(algorithm)e(fails)g(on)i
       
  4523 Fz(P)14 b Fx(,)24 b(then)h Fz(P)38 b Fx(has)24 b(no)h(solution;)e(and)
       
  4524 200 780 y(\(ii\))50 b(if)31 b(the)h(algorithm)e(succeeds)i(on)f
       
  4525 Fz(P)14 b Fx(,)32 b(then)f(the)h(r)l(esult)f(it)g(pr)l(oduces)g(is)g
       
  4526 (an)h(idempotent)372 900 y(most)24 b(g)o(ener)o(al)g(solution.)166
       
  4527 1308 y FB(PR)m(OOF)-11 b(.)49 b FA(When)37 b(f)o(ailure)g(happens)g(it)
       
  4528 f(is)h(because)h(of)f(certain)g(subproblems)f(that)g(mani-)166
       
  4529 1428 y(festly)41 b(ha)n(v)o(e)g(no)g(solution)f(\(namely)h(in)g(the)g
       
  4530 (\002rst)h(phase,)f Fz(a)58 b Fw(\031)q Fu(?)g Fz(a)2641
       
  4531 1392 y Ft(0)2706 1428 y FA(with)41 b Fz(a)59 b Fw(6)p
       
  4532 Fu(=)f Fz(a)3220 1392 y Ft(0)3243 1428 y FA(,)42 b(and)166
       
  4533 1548 y Fz(\031)t Fr(\001)p Fz(X)i Fw(\031)p Fu(?)37 b
       
  4534 Fz(f)27 b(t)j FA(or)g Fz(f)d(t)37 b Fw(\031)p Fu(?)g
       
  4535 Fz(\031)t Fr(\001)o Fz(X)h FA(with)28 b Fz(X)38 b FA(occurring)29
       
  4536 b(in)g Fz(t)p FA(;)h(in)f(the)g(second)h(phase,)f Fz(a)37
       
  4537 b Fu(#?)f Fz(a)p FA(\).)166 1669 y(Therefore)30 b(part)e(\(i\))h(is)g
       
  4538 (a)g(consequence)g(of)f(Lemma)h(3.5.)f(F)o(or)h(part)f(\(ii\))h(one)g
       
  4539 (gets)f(that)g(a)i(se-)166 1789 y(quence)22 b(lik)o(e)g(\(12\))g(e)o
       
  4540 (xists,)f(and)h(thus)g Fu(\()p Fw(r)p Fz(;)17 b(\033)t
       
  4541 Fu(\))27 b(=)g(\()p Fw(r)1970 1804 y FG(1)2022 1789 y
       
  4542 Fw([)12 b(\001)17 b(\001)g(\001)12 b([)g(r)2391 1804
       
  4543 y Fq(m)2458 1789 y Fz(;)34 b(\033)2574 1804 y Fq(n)2633
       
  4544 1789 y Fw(\016)12 b(\001)17 b(\001)g(\001)11 b(\016)h
       
  4545 Fz(\033)2941 1804 y FG(1)2982 1789 y Fu(\))22 b FA(is)f(in)h
       
  4546 Ff(U)p Fu(\()p Fz(P)14 b Fu(\))166 1910 y FA(by)19 b(Lemma)g(3.6)g(and)
       
  4547 h(the)f(f)o(act)h(that)f Fu(\()p Fw(;)p Fz(;)e(")p Fu(\))26
       
  4548 b Fw(2)i Ff(U)p Fu(\()p Fw(;)p Fu(\))p FA(.)20 b(Furthermore)f(from)g
       
  4549 (Lemma)g(3.5,)g(we)h(get)166 2030 y(that)k(an)o(y)g(other)h(solution)d
       
  4550 Fu(\()p Fw(r)1203 1994 y Ft(0)1227 2030 y Fz(;)17 b(\033)1330
       
  4551 1994 y Ft(0)1353 2030 y Fu(\))27 b Fw(2)h Ff(U)p Fu(\()p
       
  4552 Fz(P)14 b Fu(\))24 b FA(satis\002es)h Fw(r)2184 1994
       
  4553 y Ft(0)2235 2030 y Fw(`)i Fz(\033)2382 1994 y Ft(0)2405
       
  4554 2030 y Fu(\()p Fw(r)p Fu(\))e FA(and)f Fw(r)2840 1994
       
  4555 y Ft(0)2891 2030 y Fw(`)k Fz(\033)3039 1994 y Ft(0)3084
       
  4556 2030 y Fw(\016)21 b Fz(\033)31 b Fw(\031)d Fz(\033)3405
       
  4557 1994 y Ft(0)3429 2030 y FA(,)166 2150 y(so)c(that)h Fu(\()p
       
  4558 Fw(r)p Fz(;)17 b(\033)t Fu(\))24 b FA(is)g(indeed)h(a)g(most)e(general)
       
  4559 i(solution.)e(Since)i(one)g(of)g(those)f(solutions)f(is)h(the)166
       
  4560 2271 y(most)31 b(general)h(solution)f Fu(\()p Fw(r)p
       
  4561 Fz(;)17 b(\033)t Fu(\))p FA(,)31 b(we)i(also)f(kno)n(w)f(that)g
       
  4562 Fw(r)42 b(`)f Fz(\033)31 b Fw(\016)d Fz(\033)45 b Fw(\031)c
       
  4563 Fz(\033)c FA(and)32 b(hence)g(that)166 2391 y Fu(\()p
       
  4564 Fw(r)p Fz(;)17 b(\033)t Fu(\))25 b FA(is)f(idempotent.)p
       
  4565 3382 2391 4 68 v 3386 2327 60 4 v 3386 2391 V 3445 2391
       
  4566 4 68 v 166 2798 a(W)-8 b(e)30 b(no)n(w)f(apply)g(the)g(nominal)f
       
  4567 (uni\002cation)h(algorithm)f(to)h(solv)o(e)g(the)g(quiz)g(questions)f
       
  4568 (from)166 2919 y(the)d(Introduction.)166 3147 y FB(Example)33
       
  4569 b(3.8.)45 b FA(Using)32 b(the)g(\002rst)h(three)f(function)g(symbols)f
       
  4570 (of)h(the)h(nominal)e(signature)h(of)166 3267 y(Example)23
       
  4571 b(2.2)f(to)h(represent)h Fz(\025)p FA(-terms,)e(the)i(Quiz)f(at)g(the)g
       
  4572 (end)g(of)g(the)h(Introduction)d(translates)166 3388
       
  4573 y(into)26 b(the)g(follo)n(wing)f(four)i(uni\002cation)f(problems)f(o)o
       
  4574 (v)o(er)h(that)g(signature,)g(where)h Fz(a)g FA(and)g
       
  4575 Fz(b)g FA(are)166 3508 y(distinct)c(atoms)h(of)h(sort)f
       
  4576 Fp(vid)35 b FA(and)25 b Fz(X)1439 3523 y FG(1)1478 3508
       
  4577 y Fz(;)17 b(:)g(:)g(:)f(;)h(X)1778 3523 y FG(7)1842 3508
       
  4578 y FA(are)26 b(distinct)d(v)n(ariables)h(of)h(sort)f Fp(exp)6
       
  4579 b FA(:)587 3770 y Fz(P)650 3785 y FG(1)717 3717 y Fo(def)722
       
  4580 3770 y Fu(=)29 b Fw(f)p Fv(fn)17 b Fz(a:)p Fv(fn)h Fz(b:)p
       
  4581 Fv(app)q Fw(h)p Fz(X)1536 3785 y FG(1)1575 3770 y Fz(;)f
       
  4582 Fv(vr)g Fz(b)p Fw(i)28 b(\031)p Fu(?)g Fv(fn)17 b Fz(b:)p
       
  4583 Fv(fn)h Fz(a:)p Fv(app)q Fw(h)p Fv(vr)f Fz(a;)g(X)2870
       
  4584 3785 y FG(1)2909 3770 y Fw(ig)p FA(,)587 3950 y Fz(P)650
       
  4585 3965 y FG(2)717 3898 y Fo(def)722 3950 y Fu(=)29 b Fw(f)p
       
  4586 Fv(fn)17 b Fz(a:)p Fv(fn)h Fz(b:)p Fv(app)q Fw(h)p Fz(X)1536
       
  4587 3965 y FG(2)1575 3950 y Fz(;)f Fv(vr)g Fz(b)p Fw(i)28
       
  4588 b(\031)p Fu(?)g Fv(fn)17 b Fz(b:)p Fv(fn)h Fz(a:)p Fv(app)q
       
  4589 Fw(h)p Fv(vr)f Fz(a;)g(X)2870 3965 y FG(3)2909 3950 y
       
  4590 Fw(ig)p FA(,)587 4131 y Fz(P)650 4146 y FG(3)717 4079
       
  4591 y Fo(def)722 4131 y Fu(=)29 b Fw(f)p Fv(fn)17 b Fz(a:)p
       
  4592 Fv(fn)h Fz(b:)p Fv(app)q Fw(h)p Fv(vr)f Fz(b;)g(X)1740
       
  4593 4146 y FG(4)1779 4131 y Fw(i)28 b(\031)p Fu(?)g Fv(fn)17
       
  4594 b Fz(b:)p Fv(fn)h Fz(a:)p Fv(app)q Fw(h)p Fv(vr)f Fz(a;)g(X)2870
       
  4595 4146 y FG(5)2909 4131 y Fw(ig)p FA(,)587 4312 y Fz(P)650
       
  4596 4327 y FG(4)717 4259 y Fo(def)722 4312 y Fu(=)29 b Fw(f)p
       
  4597 Fv(fn)17 b Fz(a:)p Fv(fn)h Fz(b:)p Fv(app)q Fw(h)p Fv(vr)f
       
  4598 Fz(b;)g(X)1740 4327 y FG(6)1779 4312 y Fw(i)28 b(\031)p
       
  4599 Fu(?)g Fv(fn)17 b Fz(a:)p Fv(fn)h Fz(a:)p Fv(app)q Fw(h)p
       
  4600 Fv(vr)f Fz(a;)g(X)2880 4327 y FG(7)2919 4312 y Fw(ig)p
       
  4601 FA(.)166 4550 y(Applying)23 b(the)i(nominal)e(uni\002cation)i
       
  4602 (algorithm)e(described)i(abo)o(v)o(e,)e(we)i(\002nd)g(that)166
       
  4603 4778 y Fw(\017)50 b Fz(P)329 4793 y FG(1)393 4778 y FA(has)25
       
  4604 b(no)f(solution;)166 4898 y Fw(\017)50 b Fz(P)329 4913
       
  4605 y FG(2)397 4898 y FA(has)29 b(a)h(most)e(general)h(solution)f(gi)n(v)o
       
  4606 (en)f(by)i Fw(r)1987 4913 y FG(2)2062 4898 y Fu(=)36
       
  4607 b Fw(;)29 b FA(and)g Fz(\033)2481 4913 y FG(2)2557 4898
       
  4608 y Fu(=)35 b([)p Fz(X)2776 4913 y FG(2)2852 4898 y Fu(:=)g
       
  4609 Fv(vr)17 b Fz(b;)g(X)3275 4913 y FG(3)3351 4898 y Fu(:=)266
       
  4610 5019 y Fv(vr)g Fz(a)p Fu(])p FA(;)166 5139 y Fw(\017)50
       
  4611 b Fz(P)329 5154 y FG(3)393 5139 y FA(has)25 b(a)g(most)e(general)j
       
  4612 (solution)d(gi)n(v)o(en)g(by)h Fw(r)1952 5154 y FG(3)2019
       
  4613 5139 y Fu(=)k Fw(;)d FA(and)g Fz(\033)2422 5154 y FG(3)2489
       
  4614 5139 y Fu(=)j([)p Fz(X)2701 5154 y FG(4)2768 5139 y Fu(:=)f(\()p
       
  4615 Fz(a)17 b(b)p Fu(\))p Fr(\001)p Fz(X)3196 5154 y FG(5)3236
       
  4616 5139 y Fu(])p FA(;)166 5259 y Fw(\017)50 b Fz(P)329 5274
       
  4617 y FG(4)403 5259 y FA(has)35 b(a)g(most)f(general)i(solution)d(gi)n(v)o
       
  4618 (en)g(by)i Fw(r)2034 5274 y FG(4)2120 5259 y Fu(=)47
       
  4619 b Fw(f)p Fz(b)f Fu(#)h Fz(X)2589 5274 y FG(7)2629 5259
       
  4620 y Fw(g)35 b FA(and)g Fz(\033)2948 5274 y FG(3)3034 5259
       
  4621 y Fu(=)47 b([)p Fz(X)3265 5274 y FG(6)3351 5259 y Fu(:=)266
       
  4622 5380 y(\()p Fz(b)17 b(a)p Fu(\))p Fr(\001)p Fz(X)564
       
  4623 5395 y FG(7)603 5380 y Fu(])p FA(.)1748 5712 y(18)p eop
       
  4624 end
       
  4625 %%Page: 19 19
       
  4626 TeXDict begin 19 18 bop 166 3 3288 4 v 166 2372 4 2369
       
  4627 v 283 252 a Fq(P)328 261 y Fe(1)438 210 y Fd(")395 252
       
  4628 y FG(=)-12 b Ft(\))33 b(f)p Fb(fn)13 b Fq(b:)p Fb(app)p
       
  4629 Ft(h)p Fq(X)900 261 y Fe(1)936 252 y Fq(;)e Fb(vr)h Fq(b)p
       
  4630 Ft(i)20 b(\031)p FG(?)g Fb(fn)13 b Fq(b:)p Fb(app)p Ft(h)p
       
  4631 Fb(vr)g Fq(b;)f FG(\()p Fq(a)g(b)p FG(\))p Fa(\001)p
       
  4632 Fq(X)1858 261 y Fe(1)1893 252 y Ft(i)p Fq(;)24 b(a)c
       
  4633 FG(#?)g Fb(fn)12 b Fq(a:)p Fb(app)q Ft(h)p Fb(vr)h Fq(a;)e(X)2614
       
  4634 261 y Fe(1)2649 252 y Ft(ig)34 b Fo(\()p Ft(\031)p FG(?)p
       
  4635 Fo(-abstraction-2\))438 329 y Fd(")395 370 y FG(=)-12
       
  4636 b Ft(\))33 b(f)p Fb(app)q Ft(h)p Fq(X)768 379 y Fe(1)803
       
  4637 370 y Fq(;)12 b Fb(vr)g Fq(b)p Ft(i)20 b(\031)p FG(?)g
       
  4638 Fb(app)q Ft(h)p Fb(vr)13 b Fq(b;)e FG(\()p Fq(a)i(b)p
       
  4639 FG(\))p Fa(\001)p Fq(X)1594 379 y Fe(1)1629 370 y Ft(i)p
       
  4640 Fq(;)31 b(a)20 b FG(#?)g Fb(fn)12 b Fq(a:)p Fb(app)q
       
  4641 Ft(h)p Fb(vr)h Fq(a;)f(X)2358 379 y Fe(1)2393 370 y Ft(ig)290
       
  4642 b Fo(\()p Ft(\031)p FG(?)p Fo(-abstraction-1\))405 489
       
  4643 y Ft(\001)11 b(\001)h(\001)54 b(\001)12 b(\001)f(\001)2120
       
  4644 b(\001)11 b(\001)h(\001)438 542 y Fd(")395 583 y FG(=)-12
       
  4645 b Ft(\))33 b(f)p Fq(X)635 592 y Fe(1)690 583 y Ft(\031)p
       
  4646 FG(?)20 b Fb(vr)12 b Fq(b;)31 b Fb(vr)12 b Fq(b)20 b
       
  4647 Ft(\031)p FG(?)g(\()p Fq(a)13 b(b)p FG(\))p Fa(\001)p
       
  4648 Fq(X)1416 592 y Fe(1)1450 583 y Fq(;)31 b(a)21 b FG(#?)e
       
  4649 Fb(fn)13 b Fq(a:)p Fb(app)q Ft(h)p Fb(vr)f Fq(a;)g(X)2152
       
  4650 592 y Fe(1)2187 583 y Ft(ig)496 b Fo(\()p Ft(\031)p FG(?)p
       
  4651 Fo(-pair\))433 660 y Fd(\033)395 702 y FG(=)-12 b Ft(\))33
       
  4652 b(f)p Fb(vr)13 b Fq(b)20 b Ft(\031)o FG(?)g Fb(vr)13
       
  4653 b Fq(a;)31 b(a)20 b FG(#?)g Fb(fn)12 b Fq(a:)p Fb(app)q
       
  4654 Ft(h)p Fb(vr)h Fq(a;)f Fb(vr)g Fq(b)p Ft(ig)40 b Fo(with)21
       
  4655 b Fq(\033)h FG(=)e([)p Fq(X)2148 711 y Fe(1)2202 702
       
  4656 y FG(:=)f Fb(vr)12 b Fq(b)p FG(])317 b Fo(\()p Ft(\031)p
       
  4657 FG(?)p Fo(-v)n(ariable\))438 778 y Fd(")395 820 y FG(=)-12
       
  4658 b Ft(\))33 b(f)p Fq(b)20 b Ft(\031)p FG(?)g Fq(a;)32
       
  4659 b(a)20 b FG(#?)f Fb(fn)13 b Fq(a:)p Fb(app)q Ft(h)p Fb(vr)f
       
  4660 Fq(a;)g Fb(vr)h Fq(b)p Ft(ig)1154 b Fo(\()p Ft(\031)p
       
  4661 FG(?)p Fo(-function)19 b(symbol\))542 938 y FG(F)-8 b(AIL)283
       
  4662 1128 y Fq(P)328 1137 y Fe(4)438 1086 y Fd(")395 1128
       
  4663 y FG(=)c Ft(\))33 b(f)p Fb(fn)13 b Fq(b:)p Fb(app)p Ft(h)p
       
  4664 Fb(vr)g Fq(b;)f(X)1045 1137 y Fe(6)1079 1128 y Ft(i)20
       
  4665 b(\031)p FG(?)g Fb(fn)13 b Fq(a:)p Fb(app)q Ft(h)p Fb(vr)f
       
  4666 Fq(a;)g(X)1716 1137 y Fe(7)1751 1128 y Ft(ig)932 b Fo(\()p
       
  4667 Ft(\031)p FG(?)p Fo(-abstraction-1\))438 1204 y Fd(")395
       
  4668 1246 y FG(=)-12 b Ft(\))33 b(f)p Fb(app)q Ft(h)p Fb(vr)13
       
  4669 b Fq(b;)f(X)913 1255 y Fe(6)947 1246 y Ft(i)20 b(\031)p
       
  4670 FG(?)g Fb(app)q Ft(h)p Fb(vr)13 b Fq(b;)e FG(\()p Fq(b)i(a)p
       
  4671 FG(\))p Fa(\001)p Fq(X)1594 1255 y Fe(7)1629 1246 y Ft(i)p
       
  4672 Fq(;)31 b(b)20 b FG(#?)g Fb(app)q Ft(h)p Fb(vr)12 b Fq(a;)g(X)2211
       
  4673 1255 y Fe(7)2246 1246 y Ft(ig)437 b Fo(\()p Ft(\031)p
       
  4674 FG(?)p Fo(-abstraction-2\))405 1364 y Ft(\001)11 b(\001)h(\001)54
       
  4675 b(\001)12 b(\001)f(\001)2120 b(\001)11 b(\001)h(\001)438
       
  4676 1417 y Fd(")395 1459 y FG(=)-12 b Ft(\))33 b(f)p Fq(b)20
       
  4677 b Ft(\031)p FG(?)g Fq(b;)31 b(X)874 1468 y Fe(6)928 1459
       
  4678 y Ft(\031)p FG(?)20 b(\()p Fq(b)13 b(a)p FG(\))p Fa(\001)p
       
  4679 Fq(X)1251 1468 y Fe(7)1286 1459 y Fq(;)31 b(b)20 b FG(#?)f
       
  4680 Fb(app)q Ft(h)p Fb(vr)13 b Fq(a;)f(X)1841 1468 y Fe(7)1875
       
  4681 1459 y Ft(ig)808 b Fo(\()p Ft(\031)p FG(?)p Fo(-function)19
       
  4682 b(symbol\))438 1535 y Fd(")395 1577 y FG(=)-12 b Ft(\))33
       
  4683 b(f)p Fq(X)635 1586 y Fe(6)690 1577 y Ft(\031)p FG(?)20
       
  4684 b(\()p Fq(b)12 b(a)p FG(\))p Fa(\001)p Fq(X)1012 1586
       
  4685 y Fe(7)1047 1577 y Fq(;)31 b(b)20 b FG(#?)g Fb(app)p
       
  4686 Ft(h)p Fb(vr)13 b Fq(a;)f(X)1602 1586 y Fe(7)1637 1577
       
  4687 y Ft(ig)1046 b Fo(\()p Ft(\031)p FG(?)p Fo(-atom\))433
       
  4688 1654 y Fd(\033)395 1695 y FG(=)-12 b Ft(\))33 b(f)p Fq(b)20
       
  4689 b FG(#?)g Fb(app)q Ft(h)p Fb(vr)13 b Fq(a;)e(X)1081 1704
       
  4690 y Fe(7)1116 1695 y Ft(ig)67 b Fo(with)21 b Fq(\033)h
       
  4691 FG(=)e([)p Fq(X)1598 1704 y Fe(6)1652 1695 y FG(:=)f(\()p
       
  4692 Fq(b)12 b(a)p FG(\))p Fa(\001)q Fq(X)1961 1704 y Fe(7)1995
       
  4693 1695 y FG(])730 b Fo(\()p Ft(\031)p FG(?)p Fo(-v)n(ariable\))436
       
  4694 1777 y Fc(;)395 1818 y FG(=)-12 b Ft(\))33 b(f)p Fq(b)20
       
  4695 b FG(#?)g Ft(h)p Fb(vr)13 b Fq(a;)f(X)976 1827 y Fe(7)1010
       
  4696 1818 y Ft(ig)1673 b Fo(\()p FG(#?)p Fo(-function)19 b(symbol\))405
       
  4697 1937 y Ft(\001)11 b(\001)h(\001)54 b(\001)12 b(\001)f(\001)2120
       
  4698 b(\001)11 b(\001)h(\001)436 1994 y Fc(;)395 2036 y FG(=)-12
       
  4699 b Ft(\))33 b(f)p Fq(b)20 b FG(#?)g Fq(a;)31 b(b)20 b
       
  4700 FG(#?)g Fq(X)1047 2045 y Fe(7)1082 2036 y Ft(g)1628 b
       
  4701 Fo(\()p FG(#?)p Fo(-function)19 b(symbol\))436 2117 y
       
  4702 Fc(;)395 2159 y FG(=)-12 b Ft(\))33 b(f)p Fq(b)20 b FG(#?)g
       
  4703 Fq(X)797 2168 y Fe(7)832 2159 y Ft(g)1878 b Fo(\()p FG(#?)p
       
  4704 Fo(-atom\))427 2237 y Fc(r)395 2278 y FG(=)-12 b Ft(\))33
       
  4705 b(;)67 b Fo(with)21 b Ft(r)e FG(=)g Ft(f)p Fq(b)i FG(#)e
       
  4706 Fq(X)1156 2287 y Fe(7)1191 2278 y Ft(g)1519 b Fo(\()p
       
  4707 FG(#?)p Fo(-suspension\))p 3450 2372 V 166 2375 3288
       
  4708 4 v 1308 2512 a FD(Fig.)23 b(4.)g(Example)h(deri)n(v)n(ations)166
       
  4709 2705 y FA(Deri)n(v)n(ations)j(for)j Fz(P)864 2720 y FG(1)933
       
  4710 2705 y FA(and)f Fz(P)1169 2720 y FG(4)1237 2705 y FA(are)h(sk)o(etched)
       
  4711 f(in)g(Figure)h(4.)f(Using)f(the)h(Adequac)o(y)g(property)166
       
  4712 2826 y(of)j(Proposition)e(2.16,)i(one)f(can)i(interpret)e(these)h
       
  4713 (solutions)e(as)i(the)f(follo)n(wing)f(statements)166
       
  4714 2946 y(about)24 b(the)h Fz(\025)p FA(-terms)f(mentioned)g(in)g(the)h
       
  4715 (quiz.)p 166 3085 V 166 4368 4 1284 v 227 3212 a FB(Quiz)g(answers)267
       
  4716 3332 y FA(\(1\))50 b(There)26 b(is)e(no)h Fz(\025)p FA(-term)f
       
  4717 Fz(M)1298 3347 y FG(1)1363 3332 y FA(making)g(the)g(\002rst)h(pair)g
       
  4718 (of)g(terms)f Fz(\013)q FA(-equi)n(v)n(alent.)267 3453
       
  4719 y(\(2\))50 b(The)25 b(only)f(solution)f(for)i(the)g(second)g(problem)f
       
  4720 (is)g(to)g(tak)o(e)h Fz(M)2585 3468 y FG(2)2653 3453
       
  4721 y Fu(=)i Fz(b)f FA(and)e Fz(M)3085 3468 y FG(3)3153 3453
       
  4722 y Fu(=)j Fz(a)p FA(.)267 3573 y(\(3\))50 b(F)o(or)24
       
  4723 b(the)g(third)g(problem)f(we)h(can)g(tak)o(e)g Fz(M)1894
       
  4724 3588 y FG(5)1958 3573 y FA(to)g(be)g(an)o(y)f Fz(\025)p
       
  4725 FA(-term,)h(so)g(long)f(as)h(we)g(tak)o(e)433 3693 y
       
  4726 Fz(M)527 3708 y FG(4)592 3693 y FA(to)h(be)g(the)f(result)g(of)h(sw)o
       
  4727 (apping)f(all)h(occurrences)g(of)g Fz(a)g FA(and)g Fz(b)g
       
  4728 FA(throughout)f Fz(M)3321 3708 y FG(5)3360 3693 y FA(.)267
       
  4729 3814 y(\(4\))50 b(F)o(or)21 b(the)f(last)g(problem,)f(we)i(can)g(tak)o
       
  4730 (e)f Fz(M)1844 3829 y FG(7)1904 3814 y FA(to)g(be)h(an)o(y)f
       
  4731 Fz(\025)p FA(-term)g(that)g Fx(does)g(not)f(contain)433
       
  4732 3934 y(fr)l(ee)30 b(occurr)l(ences)f(of)g Fz(b)p FA(,)h(so)e(long)h(as)
       
  4733 g(we)g(tak)o(e)h Fz(M)2199 3949 y FG(6)2268 3934 y FA(to)e(be)i(the)f
       
  4734 (result)f(of)h(sw)o(apping)433 4055 y(all)k(occurrences)g(of)g
       
  4735 Fz(b)g FA(and)f Fz(a)h FA(throughout)e Fz(M)2088 4070
       
  4736 y FG(7)2128 4055 y FA(,)h(or)h(equi)n(v)n(alently)d(\(since)j
       
  4737 Fz(b)g FA(is)f(not)433 4175 y(free)c(in)e Fz(M)813 4190
       
  4738 y FG(7)852 4175 y FA(\),)h(taking)e Fz(M)1306 4190 y
       
  4739 FG(6)1373 4175 y FA(to)h(be)g(the)g(result)g(of)h(replacing)f(all)g
       
  4740 (free)h(occurrences)g(of)433 4295 y Fz(a)f FA(in)e Fz(M)706
       
  4741 4310 y FG(7)771 4295 y FA(with)g Fz(b)p FA(.)p 3450 4368
       
  4742 V 166 4371 3288 4 v 166 4554 a FB(Remark)34 b(3.9)f(\(Separation)i(of)f
       
  4743 (the)g(algorithm)f(into)h(tw)o(o)g(phases\).)46 b FA(W)-8
       
  4744 b(e)34 b(or)n(ganised)f(the)166 4675 y(algorithm)f(into)g(tw)o(o)h
       
  4745 (phases:)g(equation-solving)e(follo)n(wed)h(by)h(freshness-solving.)e
       
  4746 (Note)166 4795 y(that)22 b(the)f(second)h(phase)g(is)g(crucial)g(for)g
       
  4747 (the)g(soundness)f(of)h(the)f(algorithm.)g(Consider)h(for)g(e)o(x-)166
       
  4748 4915 y(ample)i(the)h(uni\002cation)f(problem)g(consisting)f(of)i(tw)o
       
  4749 (o)f(terms)h(which)f(are)i Fx(not)g Fz(\013)q FA(-equi)n(v)n(alent:)
       
  4750 1551 5148 y Fw(f)p Fz(a:b)i Fw(\031)q Fu(?)f Fz(b:a)p
       
  4751 Fw(g)1220 b FA(\(13\))166 5380 y(After)29 b(applying)d(the)i
       
  4752 (transformation)f(\()p Fw(\031)q Fu(?)p FA(-abstraction-2\))h(one)g
       
  4753 (needs)g(to)g(solv)o(e)f(the)h(prob-)1748 5712 y(19)p
       
  4754 eop end
       
  4755 %%Page: 20 20
       
  4756 TeXDict begin 20 19 bop 166 83 a FA(lem)29 b Fw(f)p Fz(a)37
       
  4757 b Fw(\031)p Fu(?)g Fz(a;)17 b(a)36 b Fu(#?)h Fz(a)p Fw(g)p
       
  4758 FA(,)30 b(whose)f(\002rst)h(component)e(is)h(solv)o(ed)g(by)g(\()p
       
  4759 Fw(\031)q Fu(?)p FA(-atom\).)g(F)o(ailure)g(is)166 203
       
  4760 y(only)g(signalled)g(by)g(the)h(algorithm)e(in)i(the)g(second)f(phase)h
       
  4761 (when)g(attempting)e(to)h(solv)o(e)g(the)166 324 y(unsolv)n(able)d
       
  4762 (freshness)h(problem)g Fw(f)p Fz(a)33 b Fu(#?)g Fz(a)p
       
  4763 Fw(g)p FA(.)28 b(The)g(second)f(phase,)h(i.)f(e.)h(solving)e(all)h
       
  4764 (fresh-)166 444 y(ness)e(problems,)f(ensures)h(that)f(the)h(uni\002ers)
       
  4765 g(calculated)g(by)g(the)g(algorithm)f(are)h(sound)g(with)166
       
  4766 565 y(respect)g(to)g(our)f(notion)g(of)h Fz(\013)q FA(-equi)n(v)n
       
  4767 (alence.)166 955 y(W)-8 b(e)27 b(used)f(this)g(separation)g(of)h(the)f
       
  4768 (algorithm)f(into)h(tw)o(o)g(phases)g(in)g(order)h(to)f(mak)o(e)h(the)f
       
  4769 (cor)n(-)166 1075 y(rectness)c(proof)f(easier)-5 b(.)22
       
  4770 b(More)f(ef)n(\002cient)h(algorithms)e(w)o(ould)h(seek)g(to)h(minimise)
       
  4771 d(the)j(amount)166 1196 y(of)29 b(redundant)g(calculations)f(before)i
       
  4772 (f)o(ailures)f(are)h(signalled,)d(by)i(solving)f(freshness)h(prob-)166
       
  4773 1316 y(lems)g(more)g(eagerly)-6 b(.)29 b(Ho)n(we)n(v)o(er)l(,)f(care)i
       
  4774 (needs)f(then)g(to)g(be)g(tak)o(en)g(to)g(not)g(remo)o(v)o(e)f
       
  4775 (freshness)166 1436 y(constraints)h(from)g(problems)g(too)h(early)-6
       
  4776 b(.)29 b(F)o(or)h(e)o(xample,)f(consider)h(the)f(follo)n(wing)f
       
  4777 (uni\002ca-)166 1557 y(tion)c(problem,)g(which)g(has)h(no)f(solution.)
       
  4778 1420 2117 y Fw(f)p Fz(a)k Fu(#?)g Fz(X)r(;)17 b(a)28
       
  4779 b Fw(\031)p Fu(?)g Fz(X)8 b Fw(g)1088 b FA(\(14\))166
       
  4780 2677 y(If)40 b(one)g(applies)f(\002rst)h(\()p Fu(#?)p
       
  4781 FA(-suspension\))f(follo)n(wed)f(by)i(\()p Fw(\031)p
       
  4782 Fu(?)q FA(-v)n(ariable\),)f(then)g(one)h(gets)f(a)166
       
  4783 2798 y Fx(wr)l(ong)31 b FA(result,)g(namely)g Fu(\()p
       
  4784 Fw(f)p Fz(a)40 b Fu(#)g Fz(X)8 b Fw(g)p Fz(;)17 b Fu([)p
       
  4785 Fz(X)48 b Fu(:=)40 b Fz(a)p Fu(]\))p FA(.)32 b(The)f(problem)g(is)g
       
  4786 (that)g(the)h(substitution)166 2918 y Fu([)p Fz(X)46
       
  4787 b Fu(:=)38 b Fz(a)p Fu(])31 b FA(has)f(not)g(been)g(properly)g
       
  4788 (propagated)g(to)g(the)g(freshness)g(constraint)g Fz(a)38
       
  4789 b Fu(#)g Fz(X)8 b FA(.)30 b(If)166 3039 y(freshness)h(problems)g(are)h
       
  4790 (solv)o(ed)e(more)i(eagerly)-6 b(,)31 b(then)g(proper)h(propagation)e
       
  4791 (of)i(substitu-)166 3159 y(tions)24 b(into)g(freshness)g(constraints)g
       
  4792 (needs)h(to)f(be)h(tak)o(en)g(into)f(account.)166 3549
       
  4793 y FB(Remark)34 b(3.10)f(\(Atoms)g(ar)n(e)h(not)f(v)o(ariables\).)46
       
  4794 b FA(Nominal)32 b(uni\002cation)h(uni\002es)g(v)n(ariables,)166
       
  4795 3670 y(b)n(ut)e(it)f(does)h(not)g(unify)f(atoms.)h(Indeed)g(the)g
       
  4796 (operation)f(of)i(identifying)d(tw)o(o)i(atoms)f(by)h(re-)166
       
  4797 3790 y(naming)h(one)i(of)f(them)g(to)g(be)h(the)f(other)g(does)g(not)g
       
  4798 (necessarily)g(preserv)o(e)h(the)f(v)n(alidity)f(of)166
       
  4799 3910 y(the)k(judgements)f(in)h(Figure)g(2.)g(F)o(or)h(e)o(xample,)e
       
  4800 Fw(;)49 b(`)g Fz(a:b)g Fw(\031)h Fz(c:b)37 b FA(holds)e(if)h
       
  4801 Fz(b)50 b Fw(6)p Fu(=)e Fz(a;)17 b(c)p FA(;)36 b(b)n(ut)166
       
  4802 4031 y(renaming)30 b Fz(b)g FA(to)g(be)g Fz(a)h FA(in)f(this)f
       
  4803 (judgement)g(we)h(get)g Fw(;)37 b(`)h Fz(a:a)g Fw(\031)g
       
  4804 Fz(c:a)p FA(,)30 b(which)g(does)g(not)g(hold)166 4151
       
  4805 y(so)g(long)f(as)i Fz(a)38 b Fw(6)p Fu(=)f Fz(c)p FA(.)31
       
  4806 b(Referring)g(to)f(De\002nition)f(2.3,)h(you)g(will)f(see)i(that)e(we)i
       
  4807 (do)f(allo)n(w)f(v)n(ari-)166 4271 y(ables)e(ranging)f(o)o(v)o(er)g
       
  4808 (sorts)h(of)g(atoms;)f(and)h(such)f(v)n(ariables)h(can)g(be)g
       
  4809 (uni\002ed)g(lik)o(e)g(an)o(y)f(other)166 4392 y(v)n(ariables.)e(Ho)n
       
  4810 (we)n(v)o(er)l(,)g(if)h Fz(A)g FA(is)f(such)h(a)g(v)n(ariable,)f(then)h
       
  4811 (it)f(cannot)h(appear)g(in)g(abstraction)f(po-)166 4512
       
  4812 y(sition,)19 b(i.e.)h(as)h Fz(A:t)p FA(.)g(This)e(is)h(because)h(we)g
       
  4813 (speci\002cally)f(restricted)g(abstraction)g(to)g(range)h(o)o(v)o(er)
       
  4814 166 4633 y(atoms,)27 b(rather)h(than)g(o)o(v)o(er)f(arbitrary)h(terms)f
       
  4815 (of)h(atom)f(sort.)g(Such)i(a)f(restriction)f(seems)g(nec-)166
       
  4816 4753 y(essary)35 b(to)g(obtain)f(single,)g(most)g(general,)h(solutions)
       
  4817 e(to)i(nominal)f(uni\002cation)g(problems.)166 4873 y(F)o(or)i(without)
       
  4818 f(such)h(a)h(restriction,)e(because)i(of)f(rule)g(\()p
       
  4819 Fw(\031)p FA(-abstraction-2\))h(in)f(Figure)g(2)h(we)166
       
  4820 4994 y(w)o(ould)27 b(also)h(ha)n(v)o(e)f(to)h(allo)n(w)f(v)n(ariables)g
       
  4821 (to)g(appear)i(on)e(the)h(left-hand)g(side)f(of)h(freshness)g(re-)166
       
  4822 5114 y(lations)c(and)i(in)f(suspended)f(permutations.)g(So)i(then)f(we)
       
  4823 g(w)o(ould)g(get)g(uni\002cation)g(problems)166 5235
       
  4824 y(lik)o(e)i Fw(f)p Fu(\()p Fz(A)17 b(B)5 b Fu(\))p Fr(\001)p
       
  4825 Fz(C)40 b Fw(\031)p Fu(?)33 b Fz(C)7 b Fw(g)p FA(,)28
       
  4826 b(where)g Fz(A)p FA(,)g Fz(B)33 b FA(and)28 b Fz(C)34
       
  4827 b FA(are)29 b(v)n(ariables)e(of)h(atom)f(sort;)g(this)g(has)g(tw)o(o)
       
  4828 166 5355 y(incomparable)d(solutions,)f(namely)h Fu(\()p
       
  4829 Fw(;)p Fz(;)17 b Fu([)p Fz(A)27 b Fu(:=)h Fz(B)5 b Fu(]\))25
       
  4830 b FA(and)g Fu(\()p Fw(f)p Fz(A)i Fu(#)h Fz(C)r(;)17 b(B)32
       
  4831 b Fu(#)c Fz(C)7 b Fw(g)p Fz(;)17 b(")p Fu(\))p FA(.)1748
       
  4832 5712 y(20)p eop end
       
  4833 %%Page: 21 21
       
  4834 TeXDict begin 21 20 bop 166 83 a FB(4)99 b(Related)26
       
  4835 b(w)o(ork)166 424 y Fx(Higher)n(-or)l(der)e(pattern)g(uni\002cation)166
       
  4836 765 y FA(Most)32 b(pre)n(vious)h(w)o(ork)g(on)g(uni\002cation)g(for)g
       
  4837 (languages)g(with)g(binders)g(is)g(based)g(on)g(forms)166
       
  4838 885 y(of)h(higher)n(-order)g(uni\002cation,)f(i.e.)g(solving)f
       
  4839 (equations)h(between)h Fz(\025)p FA(-terms)f(modulo)g
       
  4840 Fz(\013)q(\014)6 b(\021)t FA(-)166 1005 y(equi)n(v)n(alence)40
       
  4841 b(\()p Fu(=)787 1020 y Fq(\013\014)s(\021)917 1005 y
       
  4842 FA(\))i(by)e(capture-a)n(v)n(oiding)h(substitution)d(of)j(terms)g(for)g
       
  4843 (function)f(v)n(ari-)166 1126 y(ables.)33 b(Notable)f(among)g(that)h(w)
       
  4844 o(ork)g(is)f(Miller')-5 b(s)32 b Fx(higher)n(-or)l(der)g(pattern)g
       
  4845 (uni\002cation)f FA(used)166 1246 y(in)22 b(his)f Fz(L)470
       
  4846 1261 y Fq(\025)538 1246 y FA(logic)h(programming)f(language)h([3].)g
       
  4847 (This)f(kind)h(of)g(uni\002cation)g(retains)f(the)h(good)166
       
  4848 1367 y(properties)j(of)h(\002rst-order)g(uni\002cation:)e(a)i(linear)n
       
  4849 (-time)f(decision)g(procedure)h(and)g(e)o(xistence)166
       
  4850 1487 y(of)34 b(most)f(general)h(uni\002ers.)g(This)f(good)g(beha)n
       
  4851 (viour)g(of)h(higher)n(-order)g(pattern)g(uni\002cation)166
       
  4852 1607 y(is)c(the)h(result)f(of)h(equations)f(being)g(solv)o(ed)f(only)h
       
  4853 (modulo)g Fu(=)2356 1622 y Fq(\013\014)2441 1631 y Fe(0)2475
       
  4854 1622 y Fq(\021)2548 1607 y FA(\(where)h Fz(\014)2910
       
  4855 1622 y FG(0)2950 1607 y FA(-equi)n(v)n(alence)166 1728
       
  4856 y(is)26 b(the)g(restricted)g(form)g(of)g Fz(\014)6 b
       
  4857 FA(-equi)n(v)n(alence)25 b(that)g(identi\002es)h Fu(\()p
       
  4858 Fz(\025x:M)10 b Fu(\))p Fz(y)30 b FA(and)d Fz(M)10 b
       
  4859 Fu([)p Fz(y)t(=x)p Fu(])26 b FA(with)g Fz(y)166 1848
       
  4860 y FA(being)i(a)i(v)n(ariable\))e(and)h(of)g Fz(\025)p
       
  4861 FA(-terms)f(being)g(restricted)h(such)g(that)f(function)g(v)n(ariables)
       
  4862 g(may)166 1969 y(only)f(be)h(applied)f(to)g(distinct)f(bound)h(v)n
       
  4863 (ariables.)g(An)h(empirical)f(study)f(by)i(Michaylo)o(v)e(and)166
       
  4864 2089 y(Pfenning)31 b([27])g(suggests)e(that)i(most)f(uni\002cations)g
       
  4865 (arising)g(dynamically)f(in)i(higher)n(-order)166 2209
       
  4866 y(logic)k(programming)g(satisfy)g(Miller')-5 b(s)35 b(restrictions,)g
       
  4867 (b)n(ut)g(that)h(it)f(rules)h(out)g(some)f(useful)166
       
  4868 2330 y(programming)23 b(idioms.)166 2550 y(The)i(main)g(dif)n(ference)g
       
  4869 (between)g(higher)n(-order)g(pattern)g(uni\002cation)g(and)g(nominal)e
       
  4870 (uni\002ca-)166 2671 y(tion)29 b(is)h(that)g(the)g(former)g(solv)o(es)f
       
  4871 (a)h(set)g(of)g(equations)f(by)h(calculating)g(a)g Fx(captur)l
       
  4872 (e-avoiding)166 2791 y FA(substitution,)23 b(while)j(the)g(latter)h
       
  4873 (calculates)f(a)h Fx(possibly-capturing)c FA(substitution)h
       
  4874 Fx(and)k FA(some)166 2911 y(freshness)33 b(constraints.)f(Moreo)o(v)o
       
  4875 (er)l(,)g(uni\002ers)h(in)f(higher)n(-order)i(pattern)e(uni\002cation)h
       
  4876 (solv)o(e)166 3032 y(equations)40 b(with)h(respect)h(to)f
       
  4877 Fu(=)1327 3047 y Fq(\013\014)1412 3056 y Fe(0)1446 3047
       
  4878 y Fq(\021)1488 3032 y FA(;)g(whereas)h(in)f(nominal)f(uni\002cation,)h
       
  4879 (uni\002ers)g(solv)o(e)166 3152 y(equations)f(with)h(respect)g(to)g
       
  4880 (the)g(equi)n(v)n(alence)f Fw(\031)i FA(de\002ned)f(in)g(Figure)h(2,)f
       
  4881 (which)f(agrees)166 3272 y(with)29 b Fz(\013)q FA(-equi)n(v)n(alence)f
       
  4882 (on)i(ground)f(terms)g(\(see)h(Proposition)e(2.16\),)h(b)n(ut)g(dif)n
       
  4883 (fers)h(from)f(it)g(on)166 3393 y(open)g(terms,)f(since)h(unlik)o(e)g
       
  4884 Fz(\013)q FA(-equi)n(v)n(alence,)f(it)g(is)h(respected)g(by)g
       
  4885 (possibly-capturing)e(sub-)166 3513 y(stitutions)c(\(see)j(Lemma)f
       
  4886 (2.14\).)g(F)o(or)g(us,)g(the)g(main)g(disadv)n(antage)f(of)i(higher)n
       
  4887 (-order)f(pattern)166 3634 y(uni\002cation)k(is)h(the)f(one)h(common)f
       
  4888 (to)g(most)g(approaches)h(based)f(on)h(higher)n(-order)g(abstract)166
       
  4889 3754 y(syntax)23 b(that)h(w)o(as)g(discussed)f(in)g(the)h
       
  4890 (Introduction:)e(one)i(cannot)g Fx(dir)l(ectly)g FA(e)o(xpress)f(the)h
       
  4891 (com-)166 3874 y(mon)33 b(idiom)g(of)h(possibly-capturing)d
       
  4892 (substitution)h(of)i(terms)f(for)h(meta)n(v)n(ariables.)f(Instead)166
       
  4893 3995 y(one)26 b(has)f(to)g(encode)h(meta)n(v)n(ariables)f
       
  4894 Fz(X)33 b FA(as)26 b(function)e(v)n(ariables)h(applied)g(to)g(distinct)
       
  4895 f(lists)h(of)166 4115 y(\(bound\))d(v)n(ariables,)g Fz(X)i(x)1069
       
  4896 4130 y FG(1)1126 4115 y Fz(:)17 b(:)g(:)f(x)1312 4130
       
  4897 y Fq(n)1359 4115 y FA(,)23 b(and)f(use)h(capture-a)n(v)n(oiding)e
       
  4898 (substitution.)f(At)i(\002rst)h(sight,)166 4236 y(there)e(seems)f(to)h
       
  4899 (be)g(a)g(simple)f(encoding)g(for)h(doing)f(that.)g(Consider)h(for)g(e)
       
  4900 o(xample)f(the)g(purely)166 4356 y(equational)k(nominal)f
       
  4901 (uni\002cation)i(problem)1582 4577 y Fz(a:X)36 b Fw(\031)p
       
  4902 Fu(?)28 b Fz(b:b)1251 b FA(\(15\))166 4798 y(which)29
       
  4903 b(is)g(solv)o(ed)f(by)i Fu(\()p Fw(;)p Fz(;)17 b Fu([)p
       
  4904 Fz(X)43 b Fu(:=)37 b Fz(a)p Fu(]\))p FA(.)29 b(The)h(literal)f
       
  4905 (encoding)g(as)g(the)h(higher)n(-order)f(pattern)166
       
  4906 4918 y(uni\002cation)23 b(problem)h Fz(\025a:X)35 b Fu(=)1300
       
  4907 4933 y Fq(\013\014)1385 4942 y Fe(0)1420 4933 y Fq(\021)1462
       
  4908 4918 y Fu(?)p Fz(\025b:b)25 b FA(does)e(not)h(w)o(ork)g(of)g(course,)g
       
  4909 (because)g(there)h(is)e(no)166 5039 y(capture-a)n(v)n(oiding)33
       
  4910 b(substitution)e(that)i(solv)o(es)f(this)h(problem.)f(Ho)n(we)n(v)o(er)
       
  4911 l(,)h Fz(X)41 b FA(can)34 b(be)g(made)166 5159 y(dependent)24
       
  4912 b(on)h Fz(a)g FA(yielding)f(the)g(uni\002cation)g(problem)1381
       
  4913 5380 y Fz(\025a:)p Fu(\()p Fz(X)8 b(a)p Fu(\))28 b(=)1836
       
  4914 5395 y Fq(\013\014)1921 5404 y Fe(0)1956 5395 y Fq(\021)1998
       
  4915 5380 y Fu(?)f Fz(\025b:b)1051 b FA(\(16\))1748 5712 y(21)p
       
  4916 eop end
       
  4917 %%Page: 22 22
       
  4918 TeXDict begin 22 21 bop 166 83 a FA(which)30 b(is)g(solv)o(ed)g(by)g
       
  4919 (the)h(capture-a)n(v)n(oiding)e(substitution)f(of)j Fz(\025c:c)g
       
  4920 FA(for)g Fz(X)8 b FA(.)30 b(If)h(one)g(further)166 203
       
  4921 y(applies)d(to)h Fz(\025c:c)g FA(the)g(atom)f Fz(a)h
       
  4922 FA(used)g(by)g(the)f(encoding,)h(then)f(one)h(can)h(read)f(back)g(the)g
       
  4923 (orig-)166 324 y(inal)f(solution)f Fu([)p Fz(X)43 b Fu(:=)35
       
  4924 b Fz(a)p Fu(])29 b FA(by)g(applying)f(some)g Fz(\014)6
       
  4925 b FA(-reductions.)27 b(There)j(are)f(ho)n(we)n(v)o(er)f(se)n(v)o(eral)
       
  4926 166 444 y(problems)j(with)h(this)f(encoding.)g(First,)h(the)g(encoding)
       
  4927 g(in)g(general)g(results)g(in)f(a)i(quadratic)166 565
       
  4928 y(blo)n(w-up)24 b(in)g(the)h(size)f(of)h(terms.)f(F)o(or)h(e)o(xample)f
       
  4929 (the)h(nominal)e(uni\002cation)h(problem)1325 861 y Fz(a:b:)p
       
  4930 Fw(h)p Fz(X)r(;)17 b(Y)22 b Fw(i)28 b(\031)p Fu(?)g Fz(a:b:)p
       
  4931 Fw(h)p Fz(a;)17 b(b)p Fw(i)994 b FA(\(17\))166 1158 y(solv)o(ed)29
       
  4932 b(by)g(the)h(uni\002er)g Fu(\()p Fw(;)p Fz(;)17 b Fu([)p
       
  4933 Fz(X)44 b Fu(:=)37 b Fz(a;)17 b(Y)58 b Fu(:=)36 b Fz(b)p
       
  4934 Fu(]\))31 b FA(needs)e(to)h(be)g(encoded)f(so)h(that)f
       
  4935 Fz(X)38 b FA(and)29 b Fz(Y)166 1278 y FA(depend)c(on)f(both)g
       
  4936 Fz(a)h FA(and)g Fz(b)p FA(.)g(This)f(gi)n(v)o(es)g(the)g(higher)n
       
  4937 (-order)h(pattern)g(problem)975 1575 y Fz(\025a:\025b:)p
       
  4938 Fw(h)p Fz(X)g(a)17 b(b;)g(Y)38 b(a)17 b(b)p Fw(i)28 b
       
  4939 Fu(=)1880 1590 y Fq(\013\014)1965 1599 y Fe(0)1999 1590
       
  4940 y Fq(\021)2041 1575 y Fu(?)g Fz(\025a:\025b:)p Fw(h)p
       
  4941 Fz(a;)17 b(b)p Fw(i)28 b Fz(:)644 b FA(\(18\))166 1871
       
  4942 y(In)35 b(the)g(general)h(case,)g(the)f(encoding)f(needs)i(to)e(mak)o
       
  4943 (e)i(meta)n(v)n(ariables)e(dependent)h(on)g Fx(all)166
       
  4944 1992 y FA(atoms)23 b(occurring)g(in)h(a)g(nominal)e(uni\002cation)h
       
  4945 (problem,)g(re)o(gardless)f(of)i(whether)g(the)o(y)f(actu-)166
       
  4946 2112 y(ally)f(occur)g(in)g(an)h(indi)n(vidual)d(equational)h(problem.)h
       
  4947 (F)o(or)g(e)o(xample,)f(if)h Fz(X)31 b FA(occurs)22 b(else)n(where)166
       
  4948 2232 y(within)31 b(the)i(scope)f(of)h(abstractions)e(of)i
       
  4949 Fz(c)p FA(,)f Fz(d)p FA(,)g Fz(e)h FA(and)f Fz(f)11 b
       
  4950 FA(,)33 b(then)f Fz(X)40 b FA(needs)33 b(to)f(be)g(encoded)h(as)166
       
  4951 2353 y Fu(\()p Fz(X)24 b(a)17 b(b)g(c)g(d)g(e)g(f)11
       
  4952 b Fu(\))26 b FA(e)n(v)o(en)h(though)f(an)i(indi)n(vidual)d(equational)i
       
  4953 (problem)g(might)f(contain)h(only)g Fz(a)166 2473 y FA(and)d
       
  4954 Fz(b)p FA(.)h(Secondly)-6 b(,)23 b(and)h(more)g(importantly)-6
       
  4955 b(,)22 b(we)i(cannot)g(see)h(ho)n(w)e(to)h(encode)g(our)g(freshness)166
       
  4956 2594 y(constraints)35 b(using)h(this)g(kind)f(of)i(higher)n(-order)g
       
  4957 (patterns.)f(\(Note)g(that)g(in)h(nominal)e(uni\002-)166
       
  4958 2714 y(cation,)28 b(freshness)g(constraints)f(do)h(not)g(necessarily)g
       
  4959 (come)g(from)g(analysing)f(abstractions,)166 2834 y(rather)e(the)o(y)f
       
  4960 (can)h(be)g(chosen)g(arbitrarily)-6 b(.\))166 3093 y(A)34
       
  4961 b(more)g(promising)e(tar)n(get)j(for)f(a)h(reduction)e(of)h(nominal)f
       
  4962 (uni\002cation)h(to)f(some)h(form)g(of)166 3213 y(higher)n(-order)28
       
  4963 b(pattern)g(uni\002cation)f(is)h Fz(\025\033)t FA(,)g(a)g
       
  4964 Fz(\025)p FA(-calculus)g(with)f(de-Bruijn)i(indices)e(and)h(e)o(x-)166
       
  4965 3333 y(plicit)d(substitutions.)e(Do)n(wek)i Fx(et)h(al)g
       
  4966 FA([28])g(present)g(a)g(v)o(ersion)f(of)h(higher)n(-order)g(pattern)g
       
  4967 (uni-)166 3454 y(\002cation)f(for)f Fz(\025\033)29 b
       
  4968 FA(in)24 b(which)g(uni\002cation)g(problems)f(are)j(solv)o(ed,)d(as)h
       
  4969 (in)g(nominal)g(uni\002cation,)166 3574 y(by)i(te)o(xtual)e
       
  4970 (replacements)i(of)g(terms)g(for)g(v)n(ariables;)f(ho)n(we)n(v)o(er)f
       
  4971 (a)j(\223pre-cooking\224)e(operation)166 3694 y(ensures)30
       
  4972 b(that)f(the)h(te)o(xtual)f(replacements)h(can)g(be)g(f)o(aithfully)f
       
  4973 (related)h(to)g(capture-a)n(v)n(oiding)166 3815 y(substitutions.)24
       
  4974 b(It)k(seems)f(possible)f(that)g(the)i(freshness)f(\(as)h(well)f(as)g
       
  4975 (the)g(equational\))g(prob-)166 3935 y(lems)34 b(of)g(nominal)f
       
  4976 (uni\002cation)h(can)h(be)g(encoded)f(into)g(higher)n(-order)g(pattern)
       
  4977 h(uni\002cation)166 4056 y(problems)21 b(o)o(v)o(er)f
       
  4978 Fz(\025\033)t FA(,)i(using)f(a)h(non-tri)n(vial)d(translation)i(in)l(v)
       
  4979 n(olving)e(the)j(use)f(of)h(the)g(shift)e(oper)n(-)166
       
  4980 4176 y(ator)k(and)f(the)g(introduction)f(of)h(fresh)h(uni\002cation)f
       
  4981 (v)n(ariables.)f(The)i(details)e(of)i(this)e(encoding)166
       
  4982 4296 y(still)32 b(remain)i(to)f(be)h(in)l(v)o(estigated.)d
       
  4983 (Furthermore,)j(it)f(is)h(not)f(clear)h(to)g(us)f(ho)n(w)g(to)g
       
  4984 (translate)166 4417 y(solutions)k(obtained)i(via)g(the)g(encoding)f
       
  4985 (back)h(into)g(solutions)e(of)i(the)g(original)f(nominal)166
       
  4986 4537 y(uni\002cation)33 b(problem.)f(But)h(e)n(v)o(en)f(if)i(it)e
       
  4987 (turns)h(out)f(that)h(it)g(is)g(possible)f(to)g(reduce)i(nominal)166
       
  4988 4658 y(uni\002cation)24 b(to)g(the)g(algorithm)f(of)i(Do)n(wek)f
       
  4989 Fx(et)g(al)p FA(,)g(the)h(calculations)e(in)l(v)n(olv)o(ed)g(in)h
       
  4990 (translating)166 4778 y(our)h(terms)g(into)f Fz(\025\033)29
       
  4991 b FA(patterns)c(and)g(then)g(using)f(higher)n(-order)h(pattern)g
       
  4992 (uni\002cation)f(seem)h(f)o(ar)166 4898 y(more)j(intricate)f(than)g
       
  4993 (our)h(simple)e(algorithm)h(that)g(solv)o(es)f(nominal)h(uni\002cation)
       
  4994 g(problems)166 5019 y(directly)-6 b(.)27 b(The)h(conclusion)f(we)h(dra)
       
  4995 o(w)g(is)g(that)f(an)h(encoding)g(of)g(nominal)f(uni\002cation)g(prob-)
       
  4996 166 5139 y(lems)34 b(into)g(higher)n(-order)h(pattern)g(uni\002cation)g
       
  4997 (problems)e(\(using)i(de)g(Bruijn)g(indices)f(and)166
       
  4998 5259 y(e)o(xplicit)c(substitutions\))f(might)i(be)g(possible,)g(b)n(ut)
       
  4999 g(such)h(an)f(encoding)h(is)f(no)g(substitute)f(in)166
       
  5000 5380 y(practice)25 b(for)g(ha)n(ving)f(the)h(simple,)f(direct)g
       
  5001 (algorithm)g(we)h(presented)f(here.)1748 5712 y(22)p
       
  5002 eop end
       
  5003 %%Page: 23 23
       
  5004 TeXDict begin 23 22 bop 166 83 a Fx(Hamana')l(s)24 b
       
  5005 Fz(\014)657 98 y FG(0)697 83 y Fx(-uni\002cation)f(of)h
       
  5006 Fz(\025)p Fx(-terms)h(with)f(\223holes\224)166 433 y
       
  5007 FA(Hamana)31 b([5,29])f(manages)h(to)f(add)h(possibly-capturing)d
       
  5008 (substitution)g(to)j(a)g(language)f(lik)o(e)166 554 y(Miller')-5
       
  5009 b(s)20 b Fz(L)569 569 y Fq(\025)615 554 y FA(.)i(This)f(is)g(achie)n(v)
       
  5010 o(ed)g(by)h(adding)f(syntax)g(for)h(e)o(xplicit)e(renaming)h
       
  5011 (operations)g(and)166 674 y(by)31 b(recording)g(implicit)f
       
  5012 (dependencies)h(of)g(v)n(ariables)g(upon)g(bindable)f(names)h(in)g(a)h
       
  5013 (typing)166 795 y(conte)o(xt.)20 b(The)h(mathematical)f(foundation)f
       
  5014 (for)j(Hamana')-5 b(s)20 b(system)g(is)g(the)h(model)f(of)h(binding)166
       
  5015 915 y(syntax)29 b(of)h(Fiore)g Fx(et)f(al)h FA([24].)f(The)h
       
  5016 (mathematical)e(foundation)h(for)h(our)f(w)o(ork)h(appeared)g(at)166
       
  5017 1035 y(the)25 b(same)f(time)g(\(see)i([10]\))f(and)g(is)f(in)g(a)i
       
  5018 (sense)e(complementary)-6 b(.)23 b(F)o(or)i(in)g(Hamana')-5
       
  5019 b(s)24 b(system)166 1156 y(the)g(typing)e(conte)o(xt)h(restricts)g
       
  5020 (which)h(terms)f(may)h(be)g(substituted)d(for)j(a)h(v)n(ariable)e(by)h
       
  5021 (gi)n(ving)166 1276 y(a)39 b(\002nite)f(set)g(of)g(names)g(that)g
       
  5022 Fx(must)g(contain)f FA(the)h(free)i(names)e(of)g(such)g(a)h(term;)e
       
  5023 (whereas)166 1396 y(we)29 b(gi)n(v)o(e)e(a)i(\002nite)f(set)g(of)h
       
  5024 (names)f(which)g(the)h(term')-5 b(s)27 b(free)j(v)n(ariables)d
       
  5025 Fx(must)h(avoid)p FA(.)g(Since)h Fz(\013)q FA(-)166 1517
       
  5026 y(con)l(v)o(ersion)c(is)i(phrased)f(in)g(terms)g(of)h(a)n(v)n(oidance,)
       
  5027 f(i.e.)h(freshness)f(of)h(names,)f(our)h(approach)166
       
  5028 1637 y(seems)g(more)h(natural)g(if)g(one)f(w)o(ants)h(to)f(compute)g
       
  5029 Fz(\013)q FA(-equi)n(v)n(alences)g(concretely)-6 b(.)27
       
  5030 b(On)g(top)h(of)166 1758 y(that,)d(our)g(use)g(of)h(name)f
       
  5031 (permutations,)f(rather)i(than)f(arbitrary)g(renaming)g(functions,)f
       
  5032 (leads)166 1878 y(to)31 b(technical)f(simpli\002cations.)f(In)i(an)o(y)
       
  5033 f(case,)i(the)f(bottom)e(line)i(is)f(that)h(Hamana')-5
       
  5034 b(s)30 b(system)166 1998 y(seems)k(more)h(complicated)e(than)h(the)h
       
  5035 (one)f(presented)h(here)g(and)f(does)g(not)g(possess)g(most)166
       
  5036 2119 y(general)25 b(uni\002ers.)166 2512 y Fx(Qu-Pr)l(olo)o(g)166
       
  5037 2863 y FA(The)h(w)o(ork)g([8,9])g(on)g(uni\002cation)g(in)g(Qu-Prolog)g
       
  5038 (is)f(most)g(closely)h(related)g(to)g(that)g(reported)166
       
  5039 2983 y(here.)32 b(Qu-Prolog)g(is)g(a)g(mature)g(logic)f(programming)f
       
  5040 (language)i(addressing)f(man)o(y)g(prob-)166 3103 y(lems)g(we)i(set)f
       
  5041 (out)f(in)h(the)g(Introduction.)e(T)-8 b(o)32 b(be)o(gin)f(with,)g
       
  5042 (Qu-Prolog')-5 b(s)32 b(uni\002cation)f(algo-)166 3224
       
  5043 y(rithm)j(uni\002es)g(terms)g(modulo)g Fz(\013)q FA(-equi)n(v)n(alence)
       
  5044 f(and)i(may)f(produce)h(solutions)d(that,)i(as)h(in)166
       
  5045 3344 y(nominal)18 b(uni\002cation,)h(depend)g(on)g(freshness)g
       
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  5048 Fs(not)p 1845 3465 30 4 v 35 w(free)p 2120 3465 V 35
       
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  5064 f(and)g(as)g(a)h(result)e(the)h(algorithm)f(emplo)o(yed)g(in)h
       
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  5075 g(most)166 5150 y(general)h(solutions)e(may)h(not)h(e)o(xist.)166
       
  5076 5380 y(Another)j(dif)n(ference)h(arises)f(from)g(the)h(f)o(act)f(that)g
       
  5077 (in)g(Qu-Prolog)h(binders)e(are)i(renamed)g(via)1748
       
  5078 5712 y(23)p eop end
       
  5079 %%Page: 24 24
       
  5080 TeXDict begin 24 23 bop 166 83 a FA(capture-a)n(v)n(oiding)26
       
  5081 b(substitutions.)d(This)j(means)g(that)h(fresh)f(names)h(need)g(to)f
       
  5082 (be)h(introduced)166 203 y(during)c(uni\002cation)g(in)h(order)g(to)g
       
  5083 (respect)g Fz(\013)q FA(-equi)n(v)n(alence.)f(This)g(is)g(not)h
       
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  5085 e(permutation)f(operation)h(already)h(respects)g Fz(\013)q
       
  5086 FA(-equi)n(v)n(alence.)166 444 y(In)k(f)o(act)g(the)f(introduction)f
       
  5087 (of)i(fresh)g(atoms)f(during)g(uni\002cation)g(leads)g(to)g(a)h(more)g
       
  5088 (compli-)166 565 y(cated)g(notion)f(of)h(most)f(general)h(solution.)e
       
  5089 (Consider)h(the)h(follo)n(wing)e(v)n(ariant)h(of)h(the)g(\()p
       
  5090 Fw(\031)p Fu(?)p FA(-)166 685 y(abstraction-2\))24 b(transformation:)
       
  5091 166 921 y Fu(\()p Fw(\031)p Fu(?)p FA(-abstraction-2)881
       
  5092 878 y Ft(0)904 921 y Fu(\))k Fw(f)p Fz(a:t)g Fw(\031)p
       
  5093 Fu(?)g Fz(a)1363 880 y Ft(0)1386 921 y Fz(:t)1448 880
       
  5094 y Ft(0)1472 921 y Fw(g])p Fz(P)1756 868 y Fq(")1692 921
       
  5095 y Fu(=)-17 b Fw(\))28 b(f)p Fu(\()p Fz(a)17 b(b)p Fu(\))p
       
  5096 Fr(\001)p Fz(t)28 b Fw(\031)p Fu(?)g(\()p Fz(a)2449 880
       
  5097 y Ft(0)2489 921 y Fz(b)p Fu(\))p Fr(\001)p Fz(t)2635
       
  5098 880 y Ft(0)2659 921 y Fz(;)17 b(b)28 b Fu(#?)g Fz(t;)17
       
  5099 b(b)28 b Fu(#?)g Fz(t)3265 880 y Ft(0)3288 921 y Fw(g[)p
       
  5100 Fz(P)166 1157 y FA(which)d(is)g(applicable)g(pro)o(vided)f
       
  5101 Fz(a)29 b Fw(6)p Fu(=)g Fz(a)1576 1121 y Ft(0)1625 1157
       
  5102 y FA(and)d Fz(b)g FA(is)f(a)g(fresh)h(atom,)f(not)g(occurring)g(else)n
       
  5103 (where)166 1277 y(in)39 b(the)g(problem.)g(This)f(rule)i(is)f
       
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  5105 (ab-)166 1398 y(stracted)24 b(terms)f(in)g(Qu-Prolog)h(\(see)g([8,)g(P)
       
  5106 o(age)g(105]\).)g(If)g(we)g(were)g(to)g(use)f(\()p Fw(\031)p
       
  5107 Fu(?)p FA(-abstraction-)166 1518 y(2)216 1482 y Ft(0)239
       
  5108 1518 y FA(\))31 b(instead)g(of)g(\()p Fw(\031)p Fu(?)p
       
  5109 FA(-abstraction-2\))g(in)f(our)h(nominal)f(uni\002cation)g(algorithm,)f
       
  5110 (then)i(when)166 1639 y(applied)24 b(to)h(the)f(problem)1486
       
  5111 1775 y Fw(f)j Fz(a:X)36 b Fw(\031)q Fu(?)27 b Fz(b:Y)50
       
  5112 b Fw(g)1154 b FA(\(19\))166 1941 y(it)29 b(w)o(ould)g(produce)h(the)g
       
  5113 (solution)e Fu(\()p Fw(f)p Fz(a)37 b Fu(#)g Fz(Y)5 b(;)17
       
  5114 b(c)37 b Fu(#)g Fz(Y)21 b Fw(g)p Fz(;)c Fu([)p Fz(X)45
       
  5115 b Fu(:=)37 b(\()p Fz(a)17 b(c)p Fu(\)\()p Fz(b)g(c)p
       
  5116 Fu(\))p Fr(\001)o Fz(Y)k Fu(]\))p FA(.)30 b(While)g(this)166
       
  5117 2061 y(solution)e(solv)o(es)g(the)i(problem,)e(it)h(is)h(not)f(a)h
       
  5118 (most)e(general)i(solution)e(according)h(to)g(De\002ni-)166
       
  5119 2181 y(tion)h(3.1\227we)h(lost)g(the)g(information)f(that)g
       
  5120 Fz(c)i FA(is)f(a)g(completely)g(fresh)g(atom.)g(On)g(the)g(other)166
       
  5121 2302 y(hand,)21 b(applying)f(transformation)g(\()p Fw(\031)p
       
  5122 Fu(?)p FA(-abstraction-2\))i(to)e(\(19\))i(leads)f(to)g
       
  5123 Fu(\()p Fw(f)p Fz(a)27 b Fu(#)h Fz(Y)22 b Fw(g)p Fz(;)17
       
  5124 b Fu([)p Fz(X)35 b Fu(:=)166 2422 y(\()p Fz(a)17 b(b)p
       
  5125 Fu(\))p Fr(\001)p Fz(Y)k Fu(]\))p FA(\227a)k(most)f(general)h
       
  5126 (solution.)166 2650 y(Ov)o(erall,)i(the)h(theory)g(of)g(Qu-Prolog')-5
       
  5127 b(s)27 b(uni\002cation)h(is)f(more)h(comple)o(x)f(than)h(that)g(of)g
       
  5128 (nomi-)166 2770 y(nal)22 b(uni\002cation:)g(in)g(nominal)f
       
  5129 (uni\002cation)h(we)g(do)h(not)e(need)i(to)f(resort)h(to)f(a)g
       
  5130 (semantic)g(notion)166 2891 y(of)28 b Fz(\013)q FA(-equi)n(v)n(alence)e
       
  5131 (in)h(order)h(to)g(sho)n(w)e(the)i(correctness)g(of)f(the)h(nominal)e
       
  5132 (uni\002cation)h(algo-)166 3011 y(rithm;)h(and)i(the)g(use)f(of)h
       
  5133 (permutations)e(mak)o(es)i(our)f Fw(\031)p FA(-relation)h(much)f
       
  5134 (simpler)g(compared)166 3132 y(with)f(Qu-Prolog')-5 b(s)28
       
  5135 b(use)g(of)h(the)f(traditional)f(notion)g(of)i Fz(\013)q
       
  5136 FA(-equi)n(v)n(alence)e(e)o(xtended)h(to)g(terms)166
       
  5137 3252 y(with)c(meta)n(v)n(ariables.)166 3707 y FB(5)99
       
  5138 b(Conclusion)25 b(and)h(Futur)n(e)g(W)-7 b(ork)166 4056
       
  5139 y FA(In)19 b(this)f(paper)h(we)g(ha)n(v)o(e)g(proposed)f(a)h(ne)n(w)f
       
  5140 (solution)f(to)i(the)g(problem)f(of)g(computing)g(possibly-)166
       
  5141 4176 y(capturing)i(substitutions)d(that)j(unify)g(terms)g(in)l(v)n
       
  5142 (olving)e(binders)i(up)g(to)g Fz(\013)q FA(-con)l(v)o(ersion.)f(T)-8
       
  5143 b(o)20 b(do)166 4296 y(so)30 b(we)h(considered)f(a)h(man)o(y-sorted)e
       
  5144 (\002rst-order)i(term)g(language)f(with)g(distinguished)e(col-)166
       
  5145 4417 y(lections)h(of)h(constants)f(called)h Fx(atoms)f
       
  5146 FA(and)h(with)f Fx(atom-abstr)o(action)e FA(operations)i(for)h(bind-)
       
  5147 166 4537 y(ing)35 b(atoms)f(in)i(terms.)e(This)h(pro)o(vides)f(a)i
       
  5148 (simple,)e(b)n(ut)h(\003e)o(xible,)g(frame)n(w)o(ork)g(for)h(specify-)
       
  5149 166 4658 y(ing)28 b(binding)g(operations)g(and)h(their)f(scopes,)h(in)f
       
  5150 (which)h(the)f(bound)g(entities)g(are)i(e)o(xplicitly)166
       
  5151 4778 y(named.)37 b(By)i(using)d(v)n(ariables)h(pre\002x)o(ed)h(with)f
       
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  5153 4898 y(substitution)27 b(of)j(terms)f(for)h(v)n(ariables)f(both)g(allo)
       
  5154 n(w)f(capture)i(of)g(atoms)f(by)g(binders)g(and)h(re-)166
       
  5155 5019 y(spect)i Fz(\013)q FA(-equi)n(v)n(alence)f(\(renaming)h(of)h
       
  5156 (bound)e(atoms\).)h(The)g(de\002nition)g(of)g Fz(\013)q
       
  5157 FA(-equi)n(v)n(alence)166 5139 y(for)d(the)g(term)g(language)f(mak)o
       
  5158 (es)h(use)g(of)g(an)g(auxiliary)f Fx(fr)l(eshness)g FA(relation)h
       
  5159 (between)g(atoms)166 5259 y(and)i(terms)g(which)g(generalises)h(the)f
       
  5160 (\223not)g(a)h(free)g(atom)f(of)5 b(\224)32 b(relation)f(from)h(ground)
       
  5161 e(terms)166 5380 y(to)h(terms)f(with)h(v)n(ariables;)f(furthermore,)h
       
  5162 (because)g(v)n(ariables)g(stand)f(for)i(unkno)n(wn)d(terms,)1748
       
  5163 5712 y(24)p eop end
       
  5164 %%Page: 25 25
       
  5165 TeXDict begin 25 24 bop 166 3 3288 4 v 166 1030 4 1027
       
  5166 v 227 127 a Fs(type)59 b(Gamma)g(\(var)g(X\))g(A)h(:-)f(mem)g(\(pair)g
       
  5167 (X)g(A\))h(Gamma.)227 248 y(type)f(Gamma)g(\(app)g(M)g(N\))h(B)f(:-)h
       
  5168 (type)f(Gamma)f(M)i(\(arrow)e(A)i(B\),)1782 368 y(type)f(Gamma)f(N)i
       
  5169 (A.)227 489 y(type)f(Gamma)g(\(lam)g(x.M\))g(\(arrow)f(A)i(B\))f(/)h
       
  5170 (x#Gamma)e(:-)1782 609 y(type)h(\(pair)f(x)i(A\)::Gamma)d(M)j(B.)227
       
  5171 850 y(mem)g(A)f(A::Tail.)227 970 y(mem)h(A)f(B::Tail)f(:-)i(mem)f(A)g
       
  5172 (Tail.)p 3450 1030 V 166 1033 3288 4 v 1134 1170 a FD(Fig.)22
       
  5173 b(5.)h(An)g(e)o(xample)h FC(\013)p FD(Prolog)h(program)166
       
  5174 1431 y FA(hence)c(with)g(unkno)n(wn)e(free)j(atoms,)e(it)g(is)g
       
  5175 (necessary)i(to)e(mak)o(e)h(hypotheses)e(about)i(the)f(fresh-)166
       
  5176 1551 y(ness)33 b(of)h(atoms)f(for)h(v)n(ariables)e(in)i(judgements)e
       
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  5178 1671 y(This)e(reliance)h(on)f(\223freshness\224,)g(coupled)g(with)g
       
  5179 (name-sw)o(apping)g(rather)g(than)h(renaming,)166 1792
       
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  5182 1912 y(lence)31 b(and)f(freshness)g(ha)n(v)o(e)g(to)g(be)g(solv)o(ed)f
       
  5183 (by)h(gi)n(ving)f(term-substitutions)e(and)j(\(possibly\))166
       
  5184 2032 y(freshness)h(conditions)f(on)h(v)n(ariables)g(in)g(the)h
       
  5185 (solution.)d(W)-8 b(e)32 b(sho)n(wed)f(that)g(this)g(uni\002cation)166
       
  5186 2153 y(problem)24 b(is)g(decidable)h(and)g(unitary)-6
       
  5187 b(.)166 2396 y(Chene)o(y)g(,)29 b(Gabbay)h(and)g(Urban)h([30,31])e(are)
       
  5188 i(in)l(v)o(estigating)c(the)j(e)o(xtent)f(to)h(which)f(nominal)166
       
  5189 2516 y(uni\002cation)k(can)h(be)g(used)f(in)g(resolution-based)g(proof)
       
  5190 g(search)h(for)g(a)g(form)g(of)f(\002rst-order)166 2637
       
  5191 y(logic)38 b(programming)f(for)i(languages)f(with)g(binders)f(\(with)h
       
  5192 (a)h(vie)n(w)f(to)g(pro)o(viding)f(better)166 2757 y
       
  5193 (machine-assistance)h(for)i(structural)e(operational)h(semantics\).)f
       
  5194 (Such)h(a)h(logic)e(program-)166 2877 y(ming)28 b(language)g(should)g
       
  5195 (permit)g(a)h(concrete,)h(\223nominal\224)e(approach)h(to)f(bound)g
       
  5196 (entities)g(in)166 2998 y(programs)j(while)f(ensuring)g(that)h
       
  5197 (computation)e(\(which)i(in)g(this)f(case)i(is)f(the)g(computation)166
       
  5198 3118 y(of)38 b(answers)f(to)h(queries\))f(respects)h
       
  5199 Fz(\013)q FA(-equi)n(v)n(alence)e(between)i(terms.)f(This)g(is)g
       
  5200 (illustrated)166 3238 y(with)25 b(the)h(Prolog-lik)o(e)g(program)f(in)h
       
  5201 (Figure)g(5,)g(which)g(implements)e(a)i(simple)f(typing)g(algo-)166
       
  5202 3359 y(rithm)d(for)i Fz(\025)p FA(-terms.)f(The)g(third)f(clause)i(is)e
       
  5203 (the)i(interesting)e(one.)h(First,)g(note)g(the)g(term)g
       
  5204 Fs(\(lam)166 3479 y(x.M\))p FA(,)29 b(which)g(uni\002es)g(with)g(an)o
       
  5205 (y)g Fz(\025)p FA(-abstraction.)g(The)h(binder)f Fs(x)p
       
  5206 FA(,)g(roughly)g(speaking,)g(has)166 3600 y(in)24 b(the)h
       
  5207 (\223nominal\224)f(approach)h(a)g(v)n(alue)g(which)f(can)h(be)g(used)g
       
  5208 (in)f(the)h(body)f(of)h(the)g(clause,)g(for)166 3720
       
  5209 y(e)o(xample)h(for)h(adding)f Fs(\(pair)59 b(x)h(A\))26
       
  5210 b FA(to)h(the)f(conte)o(xt)g Fs(Gamma)p FA(.)g(Secondly)-6
       
  5211 b(,)26 b(the)h(freshness)166 3840 y(constraint)h Fs(x)16
       
  5212 b(#)h(Gamma)27 b FA(ensures)i(that)f Fs(Gamma)g FA(cannot)g(be)h
       
  5213 (replaced)g(by)g(a)g(term)f(that)h(con-)166 3961 y(tains)f
       
  5214 Fs(x)h FA(freely)-6 b(.)28 b(Since)h(this)f(clause)h(is)f(intended)g
       
  5215 (to)h(implement)e(the)h(usual)g(rule)h(for)g(typing)166
       
  5216 4081 y Fz(\025)p FA(-abstractions)1361 4211 y Fw(f)p
       
  5217 Fz(x)f Fu(:)g Fz(A)p Fw(g)22 b([)h Fu(\000)49 b Fz(.)h(M)38
       
  5218 b Fu(:)28 b Fz(B)p 1361 4256 898 4 v 1399 4342 a Fu(\000)50
       
  5219 b Fz(.)g(\025x:M)39 b Fu(:)27 b Fz(A)h Fw(\033)g Fz(B)166
       
  5220 4537 y FA(its)f(operational)h(beha)n(viour)f(is)h(gi)n(v)o(en)f(by:)g
       
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  5225 y(goal)19 b(formula,)h(apply)f(the)g(resulting)g(uni\002er)h(to)f(the)h
       
  5226 (body)f(of)h(the)f(clause)h(and)g(mak)o(e)f(sure)h(that)166
       
  5227 4898 y Fs(Gamma)f FA(is)h(not)f(replaced)i(by)f(a)g(term)g(that)g
       
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  5229 5019 y(for)31 b Fs(x)p FA(.)g(Similar)f(f)o(acilities)g(for)h
       
  5230 Fx(functional)e(pr)l(o)o(gr)o(amming)f FA(already)j(e)o(xist)f(in)g
       
  5231 (the)h(FreshML)166 5139 y(language,)38 b(b)n(uilt)f(upon)h(the)h(same)f
       
  5232 (foundations:)f(see)h([13])h(and)f Fv(www)p Fz(:)p Fv(freshml)p
       
  5233 Fz(:)p Fv(org)t FA(.)g(W)-8 b(e)166 5259 y(are)27 b(also)f(interested)g
       
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  5235 (its)g(application)f(to)166 5380 y(term-re)n(writing)e(modulo)h
       
  5236 Fz(\013)q FA(-equi)n(v)n(alence.)1748 5712 y(25)p eop
       
  5237 end
       
  5238 %%Page: 26 26
       
  5239 TeXDict begin 26 25 bop 166 83 a Fx(A)25 b(note)f(on)h(comple)n(xity)
       
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  5441 (T)l(uri,)g(Abstract)i(syntax)h(and)e(v)n(ariable)i(binding,)g(in:)e
       
  5442 (14th)342 4964 y(Annual)f(Symposium)f(on)g(Logic)g(in)f(Computer)i
       
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  5446 b(R.)f(Harper)l(,)i(D.)e(MacQueen,)i(The)f(De\002nition)h(of)g
       
  5447 (Standard)g(ML)342 5380 y(\(Re)n(vised\),)25 b(MIT)d(Press,)i(1997.)
       
  5448 1748 5712 y FA(28)p eop end
       
  5449 %%Page: 29 29
       
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  5452 (in:)g(Proc.)e(A)l(CM-SIGPLAN)342 196 y(Conference)d(on)d(Programming)h
       
  5453 (Language)h(Design)e(and)h(Implementation,)h(A)l(CM)d(Press,)342
       
  5454 309 y(1988,)25 b(pp.)e(199\226208.)166 495 y([27])j(S.)59
       
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  5458 (\(Ed.\),)f(Proc.)f(W)-7 b(orkshop)42 b(on)f(the)f FC(\025)p
       
  5459 FD(Prolog)342 721 y(Programming)61 b(Language,)g(Uni)n(v)o(ersity)f(of)
       
  5460 g(Pennsylv)n(ania,)i(1992,)e(pp.)f(257\226271,)i(CIS)342
       
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  5462 1020 y([28])i(G.)i(Do)n(wek,)g(T)-7 b(.)27 b(Hardin,)i(C.)e(Kirchner)l
       
  5463 (,)j(F)-7 b(.)27 b(Pfenning,)j(Higher)n(-order)i(uni\002cation)e(via)f
       
  5464 (e)o(xplicit)342 1133 y(substitutions:)36 b(the)31 b(case)h(of)f
       
  5465 (higher)n(-order)k(patterns,)d(in:)g(Proc.)e(of)h(JICSLP)-10
       
  5466 b(,)29 b(1996,)j(pp.)f(259\226)342 1246 y(273.)166 1433
       
  5467 y([29])26 b(M.)33 b(Hamana,)h(Simple)f FC(\014)1179 1447
       
  5468 y FG(0)1219 1433 y FD(-uni\002cation)j(for)e(terms)g(with)g(conte)o(xt)
       
  5469 h(holes,)g(in:)f(C.)e(Ringeissen,)342 1545 y(C.)d(T)m(inelli,)i(R.)e(T)
       
  5470 m(reinen,)h(R.)f(M.)g(V)-10 b(erma)30 b(\(Eds.\),)g(Proc.)f(of)i(UNIF)d
       
  5471 (2002,)j(2002,)g(unpublished)342 1658 y(proceedings.)166
       
  5472 1845 y([30])26 b(J.)37 b(Chene)o(y)-6 b(,)37 b(C.)e(Urban,)j
       
  5473 FC(\013)p FD(Prolog,)f(a)g(fresh)h(approach)h(to)e(logic)h(programming)
       
  5474 h(modulo)f FC(\013)p FD(-)342 1958 y(equi)n(v)n(alence,)32
       
  5475 b(in:)d(J.)g(Le)n(vy)-6 b(,)28 b(M.)g(K)m(ohlhase,)j(J.)d(Niehren,)i
       
  5476 (M.)e(V)-5 b(illaret)29 b(\(Eds.\),)g(Proc.)g(of)g(UNIF)342
       
  5477 2071 y(2003,)j(no.)e(DSIC-II/12/03)i(in)f(Departamento)i(de)d(Sistemas)
       
  5478 h(Inform)5 b(\264)-35 b(aticos)34 b(y)c(Computaci)8 b(\264)-38
       
  5479 b(on)342 2183 y(T)-6 b(echnical)25 b(Report)f(Series,)g(Uni)n(v)o
       
  5480 (ersidad)h(Polit)5 b(\264)-35 b(ecnica)26 b(de)e(V)-10
       
  5481 b(alencia,)24 b(2003,)g(pp.)g(15\22619.)166 2370 y([31])i(M.)41
       
  5482 b(Gabbay)-6 b(,)42 b(J.)f(Chene)o(y)-6 b(,)42 b(A)f(proof)i(theory)g
       
  5483 (for)f(nominal)h(logic,)f(in:)g(Nineteenth)i(Annual)342
       
  5484 2483 y(IEEE)38 b(Symposium)j(on)e(Logic)h(in)g(Computer)h(Science,)f
       
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  5486 b(2004.)166 2782 y([32])g(M.)40 b(S.)g(P)o(aterson,)h(M.)f(N.)g(W)-7
       
  5487 b(e)o(gman,)40 b(Linear)i(uni\002cation,)g(Journal)h(of)e(Computer)h
       
  5488 (System)342 2895 y(Sciences)25 b(16)f(\(2\))g(\(1978\))h(158\226167.)
       
  5489 166 3081 y([33])h(A.)j(Martelli,)h(U.)f(Montanari,)i(An)e(ef)n
       
  5490 (\002cient)i(uni\002cation)h(algorithm,)f(A)l(CM)d(T)m(ransactions)33
       
  5491 b(on)342 3194 y(Programming)25 b(Languages)h(and)e(Systems)f(4)h
       
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  5501 (in)g(Computer)h(Science,)g(Springer)n(-V)-10 b(erlag,)342
       
  5502 4019 y(Berlin,)24 b(2003,)g(pp.)g(513\226527.)1748 5712
       
  5503 y FA(29)p eop end
       
  5504 %%Trailer
       
  5505 
       
  5506 userdict /end-hook known{end-hook}if
       
  5507 %%EOF