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cleartomark
%%EndFont
%%BeginFont: CMR5
%!PS-AdobeFont-1.1: CMR5 1.00B
%%CreationDate: 1992 Feb 19 19:55:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 49 /one put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 59 /comma put
dup 88 /X put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMBSY7
%!PS-AdobeFont-1.1: CMBSY7 001.000
%%CreationDate: 1992 Oct 22 12:18:11
% Computer Modern fonts were designed by Donald E. Knuth.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (001.000) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMBSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 1 /periodcentered put
readonly def
/FontBBox{0 -927 1542 750}readonly def
/UniqueID 5032008 def
currentdict end
currentfile eexec
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F00FCB77E47839B2E76E72333A9E4BB4734434BC32E2263F7B27289D59C17B58
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cleartomark
%%EndFont
%%BeginFont: CMTT8
%!PS-AdobeFont-1.1: CMTT8 1.0
%%CreationDate: 1991 Aug 20 16:46:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 108 /l put
dup 118 /v put
readonly def
/FontBBox{-5 -232 545 699}readonly def
/UniqueID 5000830 def
currentdict end
currentfile eexec
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41B35C8AEFE73C4CD6A591AAE4F45762EBD6D3636C03F08C552BBFD0A13D11D5
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7CA27B7A0CE3FA7EBDAA87A600E906E14DB6FC1033BC8782EA666B4CEF4330F7
4590527700DC34426815EEEDD533F622923068DC1E881D2EE5A0897EF2C5FB6D
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9178D8AAA44409E3F0C436F0760A01BE68C163488EC4D9BAF5BF4A8B90ABC6DB
352D4AE54F4A189878CBE8CB879B591241F574384CA243E8A9217A3C4A992727
1E6DE28DD3992E8ECD9FD192881DFA393199064066D8BC4DBDEF03E4B8F416FD
EA9ABE3BB3E35DDCCE817CA242E9F1E5F8B0F3F9D71F6B2D1B4367E5E16705C5
3DF729ADEA1CB20FC61E742C3130769F9BD97AE48C672A691664FD17B3B1720B
4AF8CD579A8584A6464966DA6D22F8B3DE2DB9C3585F2F6F7990855831D925D7
19E2E27DF9730D560CB7A933F6A9922C3A5489EA7FC9CC2FCB4BDA7331110A27
7326B81B798C6585ACDBBAC073D6C161B8228111207EB6309FD058D574635EE4
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cleartomark
%%EndFont
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 63 /emptyset put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
/UniqueID 5032014 def
currentdict end
currentfile eexec
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016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
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D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758
469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8
2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D
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27DD9CCF44C2BFCD95A4DE911E4915FBC02335E9999FD9B546134081D6DA3792
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cleartomark
%%EndFont
%%BeginFont: CMMIB10
%!PS-AdobeFont-1.1: CMMIB10 1.100
%%CreationDate: 1996 Jul 23 07:54:00
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMIB10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMIB10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 25 /pi put
readonly def
/FontBBox{-15 -250 1216 750}readonly def
/UniqueID 5087392 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY5
%!PS-AdobeFont-1.1: CMSY5 1.0
%%CreationDate: 1991 Aug 15 07:21:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 48 /prime put
dup 102 /braceleft put
dup 103 /braceright put
readonly def
/FontBBox{21 -944 1448 791}readonly def
/UniqueID 5000815 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964
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cleartomark
%%EndFont
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 63 /emptyset put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 25 /pi put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 65 /A put
dup 68 /D put
dup 71 /G put
dup 76 /L put
dup 77 /M put
dup 82 /R put
dup 84 /T put
dup 88 /X put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 105 /i put
dup 112 /p put
dup 115 /s put
dup 116 /t put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: TeX-cmex9
%!PS-AdobeFont-1.0: TeX-cmex9 001.001
% Filtered by type1fix.pl 0.05
%%EndComments
13 dict dup begin
/FontInfo 16 dict dup begin
/Copyright (see\040copyright\040of\040original\040TeX\040font) def
/FamilyName (TeX\040cmex9) def
/FullName (TeX\040cmex9\040Regular) def
/ItalicAngle 0 def
/Notice (converted\040after\040April\0402001) def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
/Weight (Regular) def
/isFixedPitch false def
/version (001.001) def
end readonly def
/FontName /TeX-cmex9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 56 /eight put
dup 58 /colon put
dup 60 /less put
dup 62 /greater put
readonly def
/FontBBox {-26 -2961 1503 773} readonly def
/UniqueID 4314405 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 15 /bullet put
dup 25 /approxequal put
dup 27 /propersuperset put
dup 41 /arrowdblright put
dup 48 /prime put
dup 56 /universal put
dup 62 /latticetop put
dup 66 /B put
dup 91 /union put
dup 96 /turnstileleft put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 114 /nabla put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueID 5000816 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 88 /X put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTI9
%!PS-AdobeFont-1.1: CMTI9 1.0
%%CreationDate: 1991 Aug 18 21:08:07
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 88 /X put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 115 /s put
readonly def
/FontBBox{-35 -250 1148 750}readonly def
/UniqueID 5000827 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMBSY10
%!PS-AdobeFont-1.1: CMBSY10 1.00
%%CreationDate: 1992 Jul 23 21:21:18
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMBSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 1 /periodcentered put
readonly def
/FontBBox{-27 -940 1332 825}readonly def
/UniqueID 5000762 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 48 /prime put
dup 66 /B put
dup 102 /braceleft put
dup 103 /braceright put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTT10
%!PS-AdobeFont-1.1: CMTT10 1.00B
%%CreationDate: 1992 Apr 26 10:42:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 102 /f put
readonly def
/FontBBox{-4 -235 731 800}readonly def
/UniqueID 5000832 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTT9
%!PS-AdobeFont-1.1: CMTT9 1.0
%%CreationDate: 1991 Aug 20 16:46:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 97 /a put
dup 102 /f put
dup 108 /l put
dup 109 /m put
dup 114 /r put
dup 118 /v put
readonly def
/FontBBox{-6 -233 542 698}readonly def
/UniqueID 5000831 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 25 /pi put
dup 59 /comma put
dup 65 /A put
dup 68 /D put
dup 76 /L put
dup 88 /X put
dup 97 /a put
dup 105 /i put
dup 110 /n put
dup 112 /p put
dup 116 /t put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY9
%!PS-AdobeFont-1.1: CMSY9 1.0
%%CreationDate: 1991 Aug 15 07:22:27
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 14 /openbullet put
dup 25 /approxequal put
dup 27 /propersuperset put
dup 33 /arrowright put
dup 41 /arrowdblright put
dup 50 /element put
dup 54 /negationslash put
dup 56 /universal put
dup 62 /latticetop put
dup 66 /B put
dup 94 /logicaland put
dup 95 /logicalor put
dup 96 /turnstileleft put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 110 /backslash put
dup 114 /nabla put
readonly def
/FontBBox{-30 -958 1146 777}readonly def
/UniqueID 5000819 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 35 /numbersign put
dup 40 /parenleft put
dup 41 /parenright put
dup 58 /colon put
dup 59 /semicolon put
dup 61 /equal put
dup 64 /at put
dup 91 /bracketleft put
dup 93 /bracketright put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueID 5000792 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 35 /numbersign put
dup 40 /parenleft put
dup 41 /parenright put
dup 49 /one put
dup 50 /two put
dup 58 /colon put
dup 59 /semicolon put
dup 61 /equal put
dup 63 /question put
dup 64 /at put
dup 91 /bracketleft put
dup 93 /bracketright put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 14 /openbullet put
dup 15 /bullet put
dup 25 /approxequal put
dup 27 /propersuperset put
dup 33 /arrowright put
dup 41 /arrowdblright put
dup 50 /element put
dup 54 /negationslash put
dup 56 /universal put
dup 62 /latticetop put
dup 66 /B put
dup 94 /logicaland put
dup 95 /logicalor put
dup 96 /turnstileleft put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 110 /backslash put
dup 114 /nabla put
dup 116 /unionsq put
dup 117 /intersectionsq put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /Gamma put
dup 11 /alpha put
dup 21 /lambda put
dup 22 /mu put
dup 25 /pi put
dup 27 /sigma put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 63 /star put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 77 /M put
dup 81 /Q put
dup 84 /T put
dup 86 /V put
dup 88 /X put
dup 89 /Y put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 115 /s put
dup 116 /t put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI9
%!PS-AdobeFont-1.1: CMMI9 1.100
%%CreationDate: 1996 Jul 23 07:53:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /Gamma put
dup 11 /alpha put
dup 21 /lambda put
dup 25 /pi put
dup 27 /sigma put
dup 46 /triangleright put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 63 /star put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 77 /M put
dup 78 /N put
dup 83 /S put
dup 84 /T put
dup 86 /V put
dup 88 /X put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 108 /l put
dup 109 /m put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 118 /v put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-29 -250 1075 750}readonly def
/UniqueID 5087384 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 41 /parenright put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 58 /colon put
dup 61 /equal put
dup 64 /at put
dup 91 /bracketleft put
dup 93 /bracketright put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
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TeXDict begin 1 0 bop 958 448 a FU(A)-12 b(v)o(oiding)31
b(Equi)o(v)o(ariance)g(in)f(Alpha-Pr)n(olog)1342 737
y FT(Christian)20 b(Urban)1876 707 y FS(1)1932 737 y
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FR(1)637 911 y FQ(Ludwig-Maximilians-Uni)n(v)o(ersity)f(Munich)g(\()p
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y FR(2)1196 1002 y FQ(Edinb)o(ur)o(gh)f(Uni)n(v)o(ersity)g(\()p
FP(jcheney@inf.ed.ac.uk)p FQ(\))759 1302 y FO(Abstract.)41
b FN(\013)p FQ(Prolog)30 b(is)f(a)g(logic)h(programming)h(language)g
(which)e(is)g(well-suited)h(for)759 1393 y(rapid)25 b(prototyping)h(of)
f(type)g(systems)g(and)g(operational)h(semantics)f(of)f(typed)i
FN(\025)p FQ(-calculi)759 1485 y(and)21 b(man)o(y)g(other)g(languages)g
(in)m(v)o(olving)g(bound)h(names.)f(In)f FN(\013)p FQ(Prolog,)g(the)g
(nominal)h(uni-)759 1576 y(\002cation)f(algorithm)g(of)f(Urban,)h
(Pitts)e(and)j(Gabbay)f(is)g(used)g(instead)g(of)g(\002rst-order)f
(uni\002-)759 1667 y(cation.)f(Ho)n(we)n(v)o(er)m(,)f(although)i
FN(\013)p FQ(Prolog)e(can)h(be)g(vie)n(wed)g(as)f(Horn-clause)h(logic)g
(program-)759 1759 y(ming)e(in)f(Pitts')f FM(nominal)i(lo)o(gic)p
FQ(,)g(proof)g(search)g(using)g(nominal)g(uni\002cation)g(is)f
(incomplete)759 1850 y(in)26 b(nominal)h(logic.)e(Because)i(of)f
(nominal)g(logic')l(s)g FM(equivariance)i(principle)p
FQ(,)e(complete)759 1941 y(proof)j(search)g(w)o(ould)g(require)g
(solving)g(NP-hard)g FM(equivariant)g(uni\002cation)h
FQ(problems.)759 2033 y(Ne)n(v)o(ertheless,)18 b(the)g
FN(\013)p FQ(Prolog)g(programs)h(we)e(studied)i(run)f(correctly)g
(without)f(equi)n(v)n(ariant)759 2124 y(uni\002cation.)j(In)g(this)g
(paper)m(,)g(we)g(gi)n(v)o(e)h(se)n(v)o(eral)f(e)o(xamples)h(of)f
FN(\013)p FQ(Prolog)g(programs)h(that)f(do)759 2215 y(not)i(require)f
(equi)n(v)n(ariant)h(uni\002cation,)f(de)n(v)o(elop)i(a)e(test)f(for)h
(identifying)h(such)g(programs,)759 2307 y(and)e(pro)o(v)o(e)f(the)g
(correctness)h(of)f(this)g(test)f(via)h(a)g(proof-theoretic)h(ar)o
(gument.)523 2570 y FL(1)99 b(Intr)n(oduction)523 2755
y FT(Logic)19 b(programming)e(is)k(particularly)e(suited)h(for)f
(implementing)f(inference)h(rules)h(de\002ning)f(re-)523
2855 y(lations)f(o)o(v)o(er)e(terms.)i(Man)o(y)f(interesting)f(e)o
(xamples)h(of)h(such)f(inference)f(rules,)i(ho)n(we)n(v)o(er)m(,)d(in)m
(v)n(olv)o(e)523 2954 y(terms)28 b(with)g(binders)f(and)h
FK(\013)p FT(-equi)n(v)n(alence,)e(for)h(which)h(Prolog,)f(for)g(e)o
(xample,)f(pro)o(vides)h(little)523 3054 y(assistance.)h(In)f([3])f(we)
i(presented)e FK(\013)p FT(Prolog)o(,)h(which)g(is)h(designed)e(to)h
(simplify)g(programming)523 3153 y(with)g(binders.)f(F)o(or)g
(instance,)g(the)h(operation)e(of)i(capture-a)n(v)n(oiding)c
(substitution)j(for)g FK(\025)p FT(-terms)523 3253 y(can)20
b(be)g(implemented)e(in)j FK(\013)p FT(Prolog)e(as)i(follo)n(ws:)529
3395 y FP(id\(X,X\).)529 3510 y(subst\(var\(X\),X,T,T\).)529
3601 y(subst\(var\(X\),Y,T,var\(X\)\))253 b(:-)23 b(not\(id\(X,Y\)\).)
529 3693 y(subst\(app\(M,N\),X,T,app\(M',N'\)\))-17 b(:-)23
b(subst\(M,X,T,M'\),)10 b(subst\(N,X,T,N'\).)529 3784
y(subst\(lam\(a.M\),X,T,lam\(a.M'\)\))28 b(:-)23 b(a)13
b(#)f(T,)g(a)g(#)h(X,)f(subst\(M,X,T,M'\).)523 3928 y
FT(where)21 b(the)h(terms)g FJ(var\(X\))p FT(,)f FJ(app\(M,N\))g
FT(and)g FJ(lam\(a.M\))g FT(encode)g(v)n(ariables,)g(applications)523
4028 y(and)26 b FK(\025)p FT(-abstractions.)f(The)g(predicate)g
FJ(subst\(E,X,T,E'\))f FT(de\002ned)h(by)h(the)g(clauses)g(holds)523
4127 y(only)f(if)h FJ(E')g FT(contains)f(the)h(result)f(of)h(the)f
(usual)h(capture-a)n(v)n(oiding)d(substitution)h FJ(E[X:=T])h
FT(in)523 4227 y(the)20 b FK(\025)p FT(-calculus.)648
4327 y(T)-7 b(w)o(o)29 b(features)g(of)h FK(\013)p FT(Prolog)f(are)g
(immediately)f(visible)i(to)g(the)f(user)-5 b(.)30 b(First,)g(the)g
(term)f(lan-)523 4426 y(guage)c(includes)h(the)h(term-constructor)c
FI(\000)p FK(:)p FI(\000)j FT(for)g(forming)e(abstractions,)i(which)g
(are)g(used)g(to)523 4526 y(encode)20 b(binding.)f(Second,)h
FK(\013)p FT(Prolog)h(has)g(a)h(freshness-predicate,)c(written)j(as)h
FI(\000)p FJ(#)p FI(\000)p FT(,)e(b)n(uilt)i(into)523
4625 y(the)29 b(language;)e(this)j(predicate)e(ensures)g(that)h(a)g
(name)g(does)f(not)h(occur)f(freely)g(in)h(a)h(term)e(\(by)523
4725 y(a)h(name)f(we)h(mean)f(lo)n(wer)n(-case)g(symbols,)g(for)g
(instance)g FJ(a)h FT(in)f(the)h(e)o(xpression)e FJ(lam\(a.M\))p
FT(\).)523 4825 y(In)e(this)h FJ(subst)p FT(-program,)21
b(the)k(freshness-predicate)e(is)j(used)f(to)h(mak)o(e)e(sure)h(that)h
(no)f(v)n(ariable)523 4924 y(capture)19 b(occurs)h(inside)g(the)g(term)
g(being)f(substituted.)p eop end
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TeXDict begin 2 1 bop 648 448 a FT(T)-7 b(o)23 b(illustrate)g(ho)n(w)g
(the)g FJ(subst)p FT(-program)d(calculates)j(the)g(result)h(of)e(the)i
(capture-a)n(v)n(oiding)523 548 y(substitution)19 b FH(\()p
FK(\025b:a)24 b(b)p FH(\)[)p FK(a)f FH(:=)f FK(b)p FH(])p
FT(,)f(we)f(consider)f(the)h(query:)1023 732 y FP
(subst\(lam\(b.app\(var\(a\),var\(b\)\),a,var)o(\(b\),R\))396
b FT(\(1\))523 916 y(T)-7 b(o)20 b(solv)o(e)g(this)h(query)-5
b(,)18 b FK(\013)p FT(Prolog)i(uni\002es)g(it)h(with)f(the)g(head)g(of)
g(the)g(fourth)f FJ(subst)p FT(-clause)530 1100 y FP(subst\(lam\(a)1025
1108 y FR(1)1057 1100 y FP(.M)1147 1108 y FR(1)1182 1100
y FP(\))-12 b(,X)1305 1108 y FR(1)1338 1100 y FP(,)g(T)1416
1108 y FR(1)1450 1100 y FP(,)g(lam\(a)1708 1108 y FR(1)1741
1100 y FP(.M')1850 1108 y FR(1)1885 1100 y FP(\)\))g(:-)18
b(a)2116 1108 y FR(1)2151 1100 y FP(#)12 b(T)2253 1108
y FR(1)2287 1100 y FP(,a)2377 1108 y FR(1)2412 1100 y
FP(#)g(X)2514 1108 y FR(1)2548 1100 y FP(,subst\(M)2908
1108 y FR(1)2942 1100 y FP(,X)3032 1108 y FR(1)3066 1100
y FP(,)-12 b(T)3144 1108 y FR(1)3178 1100 y FP(,M')3287
1108 y FR(1)3321 1100 y FP(\))g(.)523 1284 y FT(where,)15
b(as)j(in)e(Prolog,)f(the)h(v)n(ariables)g FJ(M)p FT(,)g
FJ(X)p FT(,)h FJ(T)f FT(and)g FJ(M')h FT(ha)n(v)o(e)e(been)h(replaced)f
(with)h(fresh)g(v)n(ariables)523 1384 y(\(indicated)i(by)h(the)g
(subscript\),)f(and)h(also)h(the)f(name)g FJ(a)g FT(has)h(been)f
(freshened)e(\(we)i(shall)h(return)e(to)523 1484 y(the)f(dif)n(ference)
f(between)g(v)n(ariables)h(and)g(names)g(later\).)g(The)g(uni\002er)g
(that)g FK(\013)p FT(Prolog)g(calculates)h(is)523 1583
y FJ(app\(var\(a\),var\(a)1323 1595 y FS(1)1357 1583
y FJ(\)\))h FT(for)e FJ(M)1641 1595 y FS(1)1678 1583
y FT(,)h FJ(a)h FT(for)e FJ(X)1951 1595 y FS(1)1988 1583
y FT(,)i FJ(var\(b\))e FT(for)h FJ(T)2511 1595 y FS(1)2567
1583 y FT(and)f FJ(lam\(a)2955 1595 y FS(1)2991 1583
y FJ(.M')3113 1595 y FS(1)3151 1583 y FJ(\))h FT(for)f
FJ(R)p FT(.)523 1683 y(Ne)o(xt,)22 b FK(\013)p FT(Prolog)f(checks)h
(that)g(the)g(freshness-predicates)e FJ(a)2290 1695 y
FS(1)2327 1683 y FJ(#)i(var\(b\))f FT(and)h FJ(a)2913
1695 y FS(1)2950 1683 y FJ(#)g(a)g FT(hold,)f(and)523
1783 y(continues)14 b(unifying)f(the)i(ne)n(w)g(query)f
FJ(subst)-14 b(\(app\(var\(a\),)g(var\(a)2700 1795 y
FS(1)2720 1783 y FJ(\)\))g(,a,)g(var)g(\(b\),M')3350
1795 y FS(1)3388 1783 y FJ(\))523 1882 y FT(with)25 b(the)g(third)g
FJ(subst)p FT(-clause.)f(Then)g(it)i(uses)f(the)h(\002rst)f(and)g
(second)f FJ(subst)p FT(-clause)g(and)h(af-)523 1982
y(ter)17 b(the)o(y)f(succeed,)g FK(\013)p FT(Prolog)g(returns)g
FJ(lam\(a)1878 1994 y FS(1)1914 1982 y FJ(.app\(var\(b\),var\(a)2764
1994 y FS(1)2798 1982 y FJ(\)\)\))h FT(as)g(the)g(answer)523
2082 y(for)j FJ(R)p FT(.)648 2173 y(Another)g(e)o(xample,)h(which)g
(illustrates)i(ho)n(w)e(easily)i(inference)d(rules)i(can)g(be)g
(implemented)523 2265 y(in)e FK(\013)p FT(Prolog)o(,)h(is)g(the)f
(follo)n(wing)f(program)548 2441 y FP(mem\(X,[X|T]\).)548
2532 y(mem\(X,[Y|T]\))43 b(:-)h(mem\(X,T\).)548 2647
y(type\(Gamma,var\(X\),T\))87 b(:-)23 b(mem\(\(X,T\),Gamma\).)548
2739 y(type\(Gamma,app\(M,N\),T\))m(:-)g(type\(Gamma,M,arr\(S,T\)\),)16
b(type\(Gamma,N,S\))-12 b(.)548 2830 y
(type\(Gamma,lam\(x.M\),arr\(S,T\)\))-6 b(:-)6 b(x)h(#)12
b(Gamma,)g(type\([\(x,S\)|Gamma],M,T\))-12 b(.)523 3009
y FT(implementing)18 b(the)i(usual)g(inference)f(rules)h(for)g
(inferring)e(the)i(types)g(of)g FK(\025)p FT(-terms.)535
3249 y FN(x)h FG(:)g FN(T)32 b FF(2)22 b FN(\000)p 535
3273 315 4 v 545 3340 a(\000)28 b(.)17 b(x)k FG(:)h FN(T)891
3289 y(v)s(ar)1103 3253 y(\000)27 b(.)18 b(M)e FG(:)22
b FN(S)k FF(!)21 b FN(T)85 b(\000)28 b(.)17 b(N)g FG(:)22
b FN(S)p 1103 3273 867 4 v 1341 3340 a(\000)28 b(.)18
b(M)8 b(N)17 b FG(:)k FN(T)2010 3281 y(app)2223 3233
y(x)f FG(:)i FN(S;)13 b(\000)28 b(.)17 b(M)g FG(:)22
b FN(T)85 b FG(\()p FN(x)21 b FF(62)g FN(F)11 b(V)17
b FG(\()p FN(\000)11 b FG(\)\))p 2223 3272 1019 4 v 2432
3340 a FN(\000)28 b(.)17 b(\025x:M)g FG(:)k FN(S)26 b
FF(!)21 b FN(T)3282 3298 y(l)q(am)523 3550 y FT(Note)j(that,)f(in)h
(contrast)f(to)h(for)e(e)o(xample)h FK(\025)p FT(Prolog,)f
(abstractions)h(in)h FK(\013)p FT(Prolog)f(bind)g(a)h(concrete)523
3650 y(name)i(which)h(is)g FE(not)h FT(restricted)f(to)g(the)f(scope)h
(of)f(the)h(abstractions.)f(Therefore)e(it)k(is)g(possible)523
3749 y(in)23 b FK(\013)p FT(Prolog)g(to)g(use)h(a)g(name)e(of)h(a)h
(binder)e(in)h(the)g(body)f(of)h(the)g(clause,)g(for)g(instance)g(to)g
(append)523 3849 y FJ(\(x,S\))f FT(to)g(the)g(conte)o(xt)e
FJ(Gamma)i FT(in)g(the)g(third)f FJ(type)p FT(-clause.)f(The)i
(implicit)g(side-condition)d(in)523 3949 y(the)i(rule)g
FK(l)r(am)g FT(requiring)f(that)h FK(\000)33 b FT(has)22
b(no)f(type-assignment)e(for)i FK(x)h FT(is)g(implemented)e(in)h
FK(\013)p FT(Prolog)523 4048 y(by)f(the)g(freshness-predicate)e
FJ(x#Gamma)p FT(.)648 4140 y(W)-7 b(e)23 b(ha)n(v)o(e)e(implemented)f
(a)i(lar)o(ge)f(number)f(of)i(such)g FK(\025)p FT(-calculus)f(e)o
(xamples,)g(including)f(type)523 4232 y(systems)26 b(and)e(operational)
f(semantics)i(for)g(System)g(F)-7 b(,)26 b FK(\025\026)g
FT(and)f(linear)f FK(\025)p FT(-calculi.)h(Our)g(e)o(xperi-)523
4323 y(ence)d(from)f(these)h(e)o(xamples)g(suggests)g(that)g(the)g
(combination)e(of)i(concrete)f(names)h(in)g(abstrac-)523
4414 y(tions)32 b(and)f(the)g(freshness-predicate)e(is)k(v)o(ery)d
(useful)h(for)g(programming)e(with)i(binders.)g(One)523
4505 y(question,)21 b(ho)n(we)n(v)o(er)m(,)f(might)i(arise:)h(what)g
(are)f(the)h(adv)n(antages)e(of)h FK(\013)p FT(Prolog)g(relati)n(v)o(e)
g(to,)g(for)g(e)o(x-)523 4597 y(ample,)29 b FK(\025)p
FT(Prolog)f([7],)h(which)g(has)g(both)g FK(\013)p FT(-equi)n(v)n
(alence)e(and)i(capture-a)n(v)n(oiding)d(substitution)523
4688 y(b)n(uilt-in)20 b(and)f(the)h(typing)f(rules)i(can)f(be)g
(correctly)f(implemented)f(by)i(the)g(tw)o(o)g(clauses:)577
4847 y FP(\(type)44 b(\(app)h(M)f(N\))h(T\))f(:-)h(type)f(M)h(\(arr)f
(S)g(T\),)h(type)f(N)h(S.)577 4938 y(\(type)f(\(lam)h(M\))f(\(arr)g(S)h
(T\)\))f(:-)h(\(pi)f(x)p FF(n)p FP(\(type)g(x)h(S)g(=>)f(type)g(\(M)h
(x\))f(T\)\).)p eop end
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TeXDict begin 3 2 bop 523 448 a FT(\(Notice)19 b(that)g(in)h(this)g
(program)d(the)i(typing-conte)o(xt)d(is)21 b(implicitly)e(gi)n(v)o(en)f
(by)h(the)g(\223surrounding\224)523 540 y(program-conte)o(xt.)c(This)20
b(program-conte)o(xt)c(can)j(be)h(modi\002ed)e(using)i(the)f(uni)n(v)o
(ersal)g(quanti\002ca-)523 631 y(tion)25 b(\(i.e.)f FJ(pi)50
b(x)p FI(n)p FJ(...)p FT(\))24 b(and)g(implications)g(in)h
(goal-formulae.)d(Therefore)h(there)h(is)i(no)e(clause)523
722 y(for)h(the)g(v)n(ariable)f(case.\))h(W)-7 b(e)26
b(\002nd)f(the)g(most)g(important)f(reason)g(in)i(f)o(a)n(v)n(our)e(of)
h FK(\013)p FT(Prolog)f(is)i(that)523 814 y(by)g(ha)n(ving)f(concrete)f
(names)i(\(namely)f FJ(x)h FT(in)g(the)g FJ(type)p FT(-e)o(xample\))d
(and)j(freshness-predicates)523 905 y(one)e(can)g(almost)h(directly)e
(translate)i(the)f(three)g(typing)f(rules)i(into)f(three)g(clauses)h
(and)e(obtain)h(a)523 996 y(correct)19 b(implementation.)e(This)j
(should)f(be)h(seen)g(in)g(the)g(conte)o(xt)e(that,)i(despite)g(the)g
(ele)o(gance)e(of)523 1088 y FK(\025)p FT(Prolog,)g(some)h(recent)f(te)
o(xtbooks)f(use)i(\(standard\))e(Prolog)h(for)h(implementing)d
(inference)i(rules)523 1179 y(o)o(v)o(er)25 b FK(\025)p
FT(-terms.)h(F)o(or)g(e)o(xample)e(one)i(of)g(them)f(presents)h(the)g
(follo)n(wing)f(implementation)f(of)h(the)523 1270 y(typing)19
b(rules:)548 1436 y FP(mem\(X,[X|T]\).)548 1527 y(mem\(X,[Y|T]\))43
b(:-)h(mem\(X,T\).)548 1642 y(type\(Gamma,var\(X\),T\))87
b(:-)23 b(mem\(\(X,T\),Gamma\).)548 1733 y(type\(Gamma,app\(M,N\),T\))m
(:-)g(type\(Gamma,M,arr\(S,T\)\),)16 b(type\(Gamma,N,S\))-12
b(.)548 1825 y(type\(Gamma,lam\(X,M\),arr\(S,T\)\))-6
b(:-)6 b(type\([\(X)o(,S\)|Ga)o(mma],)o(M,T\))-13 b(.)523
1996 y FT(which)21 b(calculates)g(the)h FE(wr)l(ong)f
FT(type)g(for)g FK(\025)p FT(-terms)g(such)g(as)h FK(\025x:\025x:)p
FH(\()p FK(x)28 b(x)p FH(\))p FT(.)23 b(Although)c(this)j(prob-)523
2096 y(lem)h(can)g(be)f(\002x)o(ed)h(by)f(judicious)g(use)h(of)g(cut)g
(or)f(side-ef)n(fects,)g(\002rst-order)f(terms)i(of)g(Prolog)f(are)523
2195 y(unwieldy)j(for)g(implementing)f(relations)h(o)o(v)o(er)f(syntax)
i(with)g(binders)f(correctly)-5 b(.)23 b(On)j(the)g(other)523
2295 y(hand,)20 b FK(\025)p FT(Prolog)h(does)g(not)h(allo)n(w)f
(concrete)f(names)h(as)i(binders)d(and)h(therefore)f(operations)g(such)
523 2394 y(as)g(adding)e(the)i(type)f(for)g FJ(x)g FT(to)h(the)f
(typing-conte)o(xt)d(need)j(to)h(be)f(encoded)f(using)h(uni)n(v)o
(ersal)f(quan-)523 2494 y(ti\002cation,)i(implications)f(in)h(goals)g
(and)g(beta-reduction.)648 2596 y(The)30 b FK(\013)p
FT(Prolog)h(language)e(is)j(based)f(on)f(nominal)g(terms)h(and)g(uses)g
(the)h(nominal)d(uni\002ca-)523 2696 y(tion)f(algorithm)e(of)i(Urban,)f
(Pitts,)i(and)e(Gabbay)g([9],)g(which)h(calculates)g(\(most)g
(general\))e(uni-)523 2795 y(\002ers)j(modulo)d FK(\013)p
FT(-equi)n(v)n(alence.)h(F)o(or)g(e)o(xample,)g(the)h(query)f
FJ(?-)14 b(id\(a.a,b.X\))27 b FT(is)i(solv)o(ed)e(in)523
2895 y FK(\013)p FT(Prolog)e(by)g(the)h(capturing)e(substitution)g
FH([)p FJ(X)33 b FH(:=)g FJ(b)p FH(])26 b FT(since)g
FJ(a.a)f FT(and)g FJ(b.b)h FT(are)f FK(\013)p FT(-equi)n(v)n(alent.)523
2995 y(Ho)n(we)n(v)o(er)m(,)e(nominal)g(uni\002cation)h(is)i(not)e
(enough)f(to)i(mak)o(e)g(the)g(programs)e(gi)n(v)o(en)g(earlier)i
(func-)523 3094 y(tion)e(as)g(intended.)e(F)o(or)h(this)i
FK(\013)p FT(Prolog)e(generates)g(fresh)g(names)g(during)f
(proof-search.)f(As)j(seen)523 3194 y(abo)o(v)o(e,)28
b(before)f(a)j(query)e(is)j(uni\002ed)d(with)i(the)f(fourth)f
FJ(subst)p FT(-clause,)g FK(\013)p FT(Prolog)h(generates)f(a)523
3293 y(fresh)17 b(name)h(for)f FJ(a)p FT(.)h(This)g(ensures)g(that)f
(substitutions)h(can)f(al)o(w)o(ays)i(be)f(\223pushed\224)e(under)g(a)j
(binder)523 3393 y(without)h(risk)g(of)g(capture.)648
3487 y(While)25 b(in)g([3])f(we)h(ha)n(v)o(e)f(described)f(our)h
(implementation)f(of)h FK(\013)p FT(Prolog)o(,)h(its)h(beha)n(viour)d
(can)523 3578 y(be)18 b(justi\002ed)g(in)g(terms)f(of)h(nominal)e
(logic)i([8,)13 b(2].)k(F)o(or)g(instance,)g(the)h(generation)e(of)i(a)
g(fresh)f(name)523 3669 y(can)23 b(be)g(e)o(xpressed)e(in)i(terms)g(of)
g(the)1686 3669 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
1686 3669 a FD(N)1686 3669 y
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1686 3669
a FT(-quanti\002er)f(of)g(nominal)g(logic,)g(and)g(an)h
FK(\013)p FT(Prolog)f(clause)523 3761 y FJ(A)50 b(:-)f(B1,..,Bn)28
b FT(can)15 b(vie)n(wed)g(as)h(the)f(formula)2125 3761
y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
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2125 3761 a FK(a)2169
3773 y FS(1)2207 3761 y FK(::a)2297 3773 y FC(n)2342
3761 y FK(:)p FI(8)p FK(X)2480 3773 y FS(1)2517 3761
y FK(::X)2632 3773 y FC(n)2677 3761 y FK(:B)2763 3773
y FS(1)2800 3761 y FI(^)q(\001\001^)p FK(B)3020 3773
y FC(n)3065 3761 y FI(\033)p FK(A)p FT(,)h(where)523
3852 y(the)24 b FK(X)718 3864 y FC(i)769 3852 y FT(and)f
FK(a)957 3864 y FC(i)1009 3852 y FT(are)g(the)g(v)n(ariables,)g
(respecti)n(v)o(ely)f(names,)h(in)g(the)h(clause.)f(The)g(problem)f(is)
i(that)523 3943 y(the)k(generation)f(of)h(fresh)f(names)h(is)i(more)d
(subtle)h(in)h FK(\013)p FT(Prolog)f(than)f(the)i(usual)f
(\223freshening\224)523 4035 y(of)c(v)n(ariables)f(when)g(backchaining)
f(a)i(Prolog-clause.)e(The)i(reason)f(is)i(that)f(distinct)g(names)g
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(ferent)g(v)n(alues.)g(Consider)h(the)g(clause)h FI(8)p
FJ(X.p\(X\))f FT(and)f(the)523 4217 y(query)19 b FJ(p\(b\))h
FT(written)g(as)h(a)f(sequent)g(as)h(follo)n(ws)1650
4386 y FF(8)p FP(X.p\(X\))44 b FF(`)g FP(p\(b\))1040
b FQ(\(2\))523 4555 y FT(When)27 b(constructing)f(a)h(proof)f(for)h
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(ariable)f FJ(X)p FT(,)523 4646 y(say)h FJ(X')p FT(,)g(and)g(then)f
(uni\002es)i FJ(p\(X'\))e FT(and)h FJ(p\(b\))g FT(gi)n(ving)e(the)i
(solution)g FJ([X':=b])p FT(.)f(A)h(similar)523 4737
y FK(\013)p FT(Prolog)o(-clause)21 b(that)g(has)h(a)g(name)f(in)g
(place)g(of)g(the)h(v)n(ariable)e(beha)n(v)o(es)g(dif)n(ferently:)g(if)
h(we)h(ha)n(v)o(e)523 4829 y(the)e(sequent)1699 4924
y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
1699 4924 a FB(N)1699 4924 y
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1699 4924 a FP(a.p\(a\))43
b FF(`)i FP(p\(b\))1034 b FQ(\(3\))p eop end
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1108 448 a FJ(a.p\(a\))p
FT(,)e(then)h(\223freshening\224)e FJ(a)j FT(to)f FJ(a')g
FT(leads)h(to)f(the)g(uni\002cation)f(problem)523 540
y FJ(p\(a'\))p FI(\031)o FH(?)p FJ(p\(b\))p FT(.)15 b(Since)h(nominal)e
(uni\002cation)h(treats)h(names)f(as)h(distinct)g(constants,)f(this)h
(prob-)523 631 y(lem)24 b(is)h(unsolv)n(able.)c(\(T)m(reating)i(names)g
(as)h(distinct)g(constants)g(is)g(important,)e(because)h(treating)523
722 y(them)c(as)i(substitutable)d(entities)i(w)o(ould)f(break)g(the)h
(most-general)d(uni\002er)i(property)e(of)j(nominal)523
814 y(uni\002cation,)c(see)i([9].\))e(On)i(the)f(other)g(hand,)f(\(3\))
h FE(is)h FT(pro)o(v)n(able)d(in)j(nominal)e(logic.)h(This)h(is)g
(because)523 905 y(after)26 b(freshening)d FJ(a)k FT(to)f
FJ(a')g FT(one)f(can)h(in)g(nominal)e(logic)i(apply)e(the)i(equi)n(v)n
(ariance)e(principle\227)523 996 y(e)o(xpressed)19 b(as)i(an)f
(inference)f(rule)1519 966 y FA(3)1718 1085 y FN(\031)s
Fz(\001)o FN(B)t(;)13 b(\000)32 b FF(\))21 b FN(C)p 1718
1120 405 4 v 1753 1187 a(B)t(;)13 b(\000)32 b FF(\))21
b FN(C)2164 1136 y(\031)523 1337 y FT(where)26 b FK(\031)k
FT(is)d(a)g(permutation)d(of)i(names,)f FK(B)t FT(,)i
FK(C)33 b FT(stand)26 b(for)g(formulae)e(and)i FK(\000)38
b FT(for)26 b(a)g(multiset)h(of)523 1437 y(formulae.)18
b(This)i(means)f(if)h(the)g FE(full)g FT(Horn-fragment)c(of)j(nominal)g
(logic)g(were)g(used)h(as)g(the)g(basis)523 1537 y(of)d
FK(\013)p FT(Prolog)o(,)h(then)f(we)h(need)f(equi)n(v)n(ariant)e
(uni\002cation)i(for)f(complete)h(proof)f(search.)h(Equi)n(v)n(ariant)
523 1636 y(uni\002cation)30 b(solv)o(es)h(a)h(problem)d(not)i(just)h
(by)f(\002nding)f(a)h(substitution)g(b)n(ut)g(also)g(by)g(\002nding)f
(a)523 1736 y(permutation;)18 b(for)i(e)o(xample)e(in)j(\(3\))e(the)i
(identity)e(substitution)g(and)h(the)g(permutation)e
FJ(\(a')49 b(b\))p FT(.)648 1841 y(The)31 b(second)h(author)f(has)h
(sho)n(wn)g(in)g([1])g(that)g(equi)n(v)n(ariant)e(uni\002cation)h(and)h
(equi)n(v)n(ariant)523 1941 y(matching)17 b(problems)f(are)i(NP-hard.)e
(F)o(or)i(proof-search)d(in)j FK(\013)p FT(Prolog)f(this)i(means)e
(that)i(one)e(needs)523 2040 y(to)g(guess)g(which)f(permutation)e
FK(\031)21 b FT(leads)c(to)f(a)h(proof.)e(Ho)n(we)n(v)o(er)m(,)g(in)i
(e)o(xperimenting)c(with)k FK(\013)p FT(Prolog)523 2140
y(we)29 b(found)e(that)h(such)g(guessing)g(is)h(ne)n(v)o(er)f(needed)f
(in)h(the)h(programs)d(we)j(considered.)e(In)h(this)523
2240 y(paper)e(we)i(identify)e(a)h(class)h(of)f(nominal)f(Horn-clause)f
(programs)h(for)g(which)h(the)g FK(\031)s FT(-rule)g(can)523
2339 y(be)k(eliminated)f(from)f(deductions)h(\(this)h(is)g(the)g(place)
g(where)f(equi)n(v)n(ariant)f(uni\002cation)g(prob-)523
2439 y(lems)20 b(arise\),)e(and)h(thus)g(nominal)f(uni\002cation)f(is)k
(complete)d(for)g(proof-search.)e(In)j(order)f(to)h(sho)n(w)523
2539 y(this)30 b(result,)f(we)g(introduce)e(a)j FE(well-formedness)f
FT(condition)e(which)i(guarantees)f(that)h(nominal)523
2638 y(uni\002cation-based)18 b(proof)h(search)g(is)j(complete.)d(This)
h(condition)f(roughly)f(says)j(that)f(a)h(clause)g(is)523
2738 y(\223insensiti)n(v)o(e\224)e(to)i(the)f(particular)f(choice)g(of)
h(names)g(occurring)e(in)i(it.)648 2843 y(Some)c(programs)g(do)g(not)h
(satisfy)g(this)h(condition.)d(F)o(or)i(e)o(xample,)e(in)i(the)g(follo)
n(wing)f(program)523 2943 y(calculating)j(a)i(list)g(of)f(bound)e(v)n
(ariables)i(of)g(a)g FK(\025)p FT(-term,)g(the)g(last)h(clause)f(is)i
FE(not)f FT(well-formed.)659 3100 y FP(bv\(var\(X\),[]\).)659
3191 y(bv\(app\(E1,E2\),L\))110 b(:-)23 b(bv\(E1,L1\),)43
b(bv\(E2,L2\),)h(append\(L1,L2,L\).)659 3283 y(bv\(lam\(x.E\),[x|L]\))
20 b(:-)j(bv\(E,L\).)523 3442 y FT(In)30 b(the)g(last)g(clause,)g(the)g
(result)g(accumulated)e(in)i(the)g(second)f(ar)o(gument)e(depends)i(on)
g(which)523 3542 y(name)j(is)i(chosen)e(for)h(the)g(binder)e
FJ(x)p FT(.)j(In)e(contrast,)g(the)h(names)g(chosen)f(in)h(the)g
FJ(subst)g FT(and)523 3641 y FJ(type)21 b FT(e)o(xample)e(do)i(not)f
(matter)h(\(up)f(to)h FK(\013)p FT(-equi)n(v)n(alence\))d(and)i
(therefore)f(will)j(satisfy)f(our)f(well-)523 3741 y(formedness)f
(condition.)648 3846 y(The)25 b(e)o(xistence)g(of)g(a)h(tri)n(vial)f
(syntactic)h(criterion)e(for)h(deciding)f(when)h(a)h(clause)g(is)g
(\223insen-)523 3946 y(siti)n(v)o(e\224)c(to)g(the)h(choice)e(of)h(a)g
(name)g(seems)g(unlik)o(ely)-5 b(.)21 b(Consider)m(,)g(for)g(e)o
(xample,)g(allo)n(wing)g(names)523 4046 y(to)27 b(only)f(occur)g(bound)
g(or)g(in)i(binding)d(position\227then)g(the)i FJ(type)p
FT(-program)d(w)o(ould)i(be)h(ruled)523 4145 y(out)c(since)g
FJ(x)g FT(occurs)f(free)h(in)g(the)g(body)f(of)g(the)h(clause.)g
(Restricting)g(free)g(names)f(to)h(occur)f(only)523 4245
y(in)i(the)f(body)f(of)h(a)h(clause)f(w)o(ould)g(permit)f(the)i(clause)
f FJ(r\(X\):-id\(X,a\))f FT(which)g FE(is)j FT(sensiti)n(v)o(e)523
4344 y(to)h(the)g(choice)f(of)h FJ(a)h FT(since)f FJ(id)g
FT(\223propagates\224)d(the)j(choice)g(for)f(the)h(name)f
FJ(a)i FT(back)e(to)h(the)g(head)523 4444 y(of)20 b(the)h(clause)g(\()p
FJ(id)f FT(is)i(de\002ned)d(in)i(the)g FJ(subst)p FT(-e)o(xample\).)c
(Our)j(well-formedness)f(condition)g(is)523 4544 y(therefore)h(more)h
(subtle;)h(it)g(is)h(a)f(test)g(whether)f(a)h(certain)g(matching)e
(problem)g(deri)n(v)o(ed)g(from)g(the)523 4643 y(clause)26
b(is)h(solv)n(able.)d(Despite)i(being)f(technically)g(relati)n(v)o(ely)
f(comple)o(x,)g(well-formedness)g(can)523 4743 y(be)c(automatically)f
(v)o(eri\002ed.)p 523 4837 473 4 v 558 4893 a Fy(3)606
4924 y FQ(The)g(corresponding)i(right-rule)e(has)g(been)h(sho)n(wn)g
(to)f(be)g(admissible)g(in)g(nominal)h(logic)f(in)g([5].)p
eop end
%%Page: 5 5
TeXDict begin 5 4 bop 648 448 a FT(The)31 b(paper)f(is)i(or)o(ganised)d
(as)j(follo)n(ws:)f(Section)g(2)h(describes)f(nominal)f(terms,)h
(formulae)523 548 y(and)f(the)h(inference)e(rules)i(of)g
FK(\013)p FT(Prolog)o(')-5 b(s)31 b(proof-search)d(procedure.)g
(Section)j(3)g(introduces)e(a)523 648 y(well-formedness)j(condition)f
(for)i(clauses)h(and)f(sho)n(ws)h(that)g(the)f FK(\031)s
FT(-rule)g(can)h(be)f(eliminated)523 747 y(from)f(proofs)f(in)m(v)n
(olving)f(only)i(well-formed)e(clauses.)j(Section)f(4)h(describes)f(ho)
n(w)g(the)g(well-)523 847 y(formedness)19 b(condition)h(can)g(be)h
(automatically)f(v)o(eri\002ed.)f(Section)i(5)g(concludes)f(and)g
(describes)523 946 y(future)f(w)o(ork.)523 1212 y FL(2)99
b(T)-9 b(erms,)26 b(F)n(ormulae)e(and)i(Pr)n(oof-Sear)n(ch)g(Rules)523
1404 y FT(The)e(terms)h(used)g(in)g FK(\013)p FT(Prolog)f(are)h
FE(nominal)e(terms)j FT(\(see)f([9])f(for)g(more)g(details\))h(as)g
(de\002ned)f(by)523 1495 y(the)c(grammar:)1414 1586 y
FN(t)h FG(::=)h FN(a)f FF(j)h FN(\031)s Fz(\001)o FN(X)28
b FF(j)21 b(hi)g(j)h(h)p FN(t;)13 b(t)p FF(i)21 b(j)g
FN(a:t)g FF(j)h Fx(f)p FG(\()p FN(t)p FG(\))523 1719
y FT(where)i FK(a)i FT(is)g(a)f(name,)f FK(X)32 b FT(a)25
b(v)n(ariable,)f Fw(f)h FT(a)g(function)f(symbol)g(and)g
FK(\031)29 b FT(a)c(permutation)e(e)o(xpressed)523 1819
y(as)g(a)f(list)h(of)e(sw)o(appings)g FH(\()p FK(a)1337
1831 y FS(1)1389 1819 y FK(b)1425 1831 y FS(1)1462 1819
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1831 y FC(n)1754 1819 y FK(b)1790 1831 y FC(n)1834 1819
y FH(\))p FT(.)23 b(W)-7 b(e)23 b(ha)n(v)o(e)e(the)h(operations)e
FI(\000)p FH(@)p FI(\000)h FT(and)g FH(\()p FI(\000)p
FH(\))3196 1789 y Fv(\000)p FS(1)3308 1819 y FT(for)523
1919 y(composing)g(\(list)i(concatenation\))e(tw)o(o)i(permutations)e
(and)h(in)m(v)o(erting)f(\(list)i(re)n(v)o(ersal\))f(a)h(permu-)523
2018 y(tation,)30 b(respecti)n(v)o(ely)-5 b(.)28 b(Constants)i(are)h
(encoded)d(as)j(function)e(symbols)h(with)g(unit)h(ar)o(guments)523
2118 y Fw(f)p FH(\()p FI(hi)p FH(\))p FT(,)22 b(and)e
FK(n)p FT(-tuples)f(are)i(encoded)e(by)h(iterated)g(pairs)g
FI(h)p FK(t)2214 2130 y FS(1)2252 2118 y FK(;)14 b FI(\001)g(\001)g
(\001)f(h)p FK(t)2461 2130 y FC(n)p Fv(\000)p FS(1)2592
2118 y FK(;)h(t)2659 2130 y FC(n)2704 2118 y FI(ii)p
FT(.)21 b(F)o(ollo)n(wing)e([9],)h(we)523 2217 y(refer)f(to)h(terms)h
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b FE(suspensions)p FT(,)e(because)g(the)h(permutation)e
FK(\031)24 b FT(is)d(suspended)523 2317 y(in)f(front)g(of)f(a)i(v)n
(ariable)e(w)o(aiting)h(to)g(be)h(applied)e(to)h(a)h(term)f
(substituted)f(for)h FK(X)7 b FT(.)648 2408 y(F)o(ormulae)28
b(are)i(di)n(vided)f(into)h(goal)g(formulae)e FK(G)k
FT(and)d(de\002nite)h(\(or)g(program\))d(clauses)k FK(D)523
2500 y FT(de\002ned)19 b(as)738 2662 y FN(G)j FG(::=)f
FN(p)p FG(\()p FN(t)p FG(\))g FF(j)g FN(G)p FF(^)q FN(G)g
FF(j)h FN(G)p FF(_)p FN(G)g FF(j)f(>)173 b FN(B)25 b
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2769 2662 a Ft(as)7 b FN(:)p FF(8)p Ft(Xs)f FN(:)p
FF(r)p FN(=B)523 2832 y FT(where)31 b FK(p)p FH(\()p
FK(t)p FH(\))i FT(stands)f(for)f(an)h(atomic)f(predicate)g(with)h(the)f
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FK(;)g FI(_)q FK(;)g FI(\033)26 b FT(are)g(standard)523
3031 y(connecti)n(v)o(es;)16 b(and)i FI(r)g FT(is)h(a)f(set)h(of)f
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3043 y FS(1)2704 3031 y FH(#)23 b FK(X)2865 3043 y FS(1)2902
3031 y FK(;)14 b(:)g(:)g(:)g(;)g(a)3131 3043 y FC(n)3199
3031 y FH(#)23 b FK(X)3360 3043 y FC(n)523 3131 y FT(\()p
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3143 y FC(i)927 3131 y FT(being)e(v)n(ariables)h(and)h(names,)f
(respecti)n(v)o(ely\).)e(The)j(intended)e(meaning)g(of)i
FI(r)g FT(in)523 3231 y FK(D)r FT(-formulae)22 b(is)j(that)g(a)g
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y Fr(Xs)31 b FT(\(we)23 b(ha)n(v)o(e)g(the)h(usual)f(con)m(v)o(ention)e
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Fr(Xs)31 b FT(for)23 b(lists)i(of)523 3629 y(v)n(ariables;)e(similarly)
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2032 3629 a FK(a)2076 3641 y FC(n)2145 3629 y
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3641 y FC(n)3019 3629 y FT(\).)24 b(W)-7 b(e)24 b(call)g(a)523
3729 y FK(D)r FT(-formula)16 b FE(closed)j FT(when)e(it)i(has)e(no)g
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FI(8)p Fr(Xs)g FK(:)p FI(r)1440 3840 y Fs(as)e FC(;)p
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e(with)i(respect)f(to)h(binding:)d(while)j(in)f(nominal)f(terms)i(the)f
(con-)523 4127 y(structor)f FK(a:)p FH(\()p FI(\000)p
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b FT(,)19 b(respecti)n(v)o(ely)-5 b(.)16 b(Therefore)h(we)523
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y
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FI(\000)p FH(\))p FT(\),)g(and)g(operations)e(on)i(formulae)e(need)i
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b(As)k(a)g(result)f(the)h(de\002nition)e(of)h(the)g(permutation)e
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(v)n(ari-)523 4825 y(ables)i FE(not)i FT(in)e FI(B)s
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%%Page: 6 6
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FN(D)r FQ(-formulae)g(\()p FN(s)g FQ(is)f(a)h(predicate)h(symbol)523
1045 y(standing)k(for)f FP(subst)p FQ(\).)e(W)-6 b(e)19
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f(for)g(closed)h FN(D)r FQ(-formulae.)p 523 1122 V 523
2297 4 1175 v 567 1211 a FO(T)-7 b(erms:)1069 1234 y
FG([])q Fz(\001)1136 1242 y Fp(B)1182 1234 y FN(a)1246
1191 y FA(def)1253 1234 y FG(=)30 b FN(a)697 1433 y FG(\(\()p
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1441 y Fp(B)1182 1433 y FN(a)1246 1390 y FA(def)1253
1433 y FG(=)1343 1257 y Fo(8)1343 1325 y(>)1343 1347
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1335 y FR(1)1644 1327 y FQ(if)c FN(\031)s Fz(\001)1779
1335 y Fp(B)1825 1327 y FN(a)j FG(=)g FN(a)2009 1335
y FR(2)1411 1436 y FN(a)1452 1444 y FR(2)1644 1436 y
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FN(X)1246 1690 y FA(def)1253 1733 y FG(=)1343 1605 y
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y Fz(\001)p FN(X)2388 1352 y(\031)s Fz(\001)2459 1360
y Fp(B)2505 1352 y FG(\()p FF(hi)p FG(\))2647 1310 y
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1597 y Fp(B)2438 1589 y FG(\()p Fx(f)p FG(\()p FN(t)p
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y FN(t)p FG(\))2358 1707 y FN(\031)s Fz(\001)2429 1715
y Fp(B)2475 1707 y FG(\()p FN(a:t)p FG(\))2647 1664 y
FA(def)2654 1707 y FG(=)g(\()p FN(\031)s Fz(\001)2846
1715 y Fp(B)2891 1707 y FN(a)p FG(\))p FN(:)p FG(\()p
FN(\031)s Fz(\001)3084 1715 y Fp(B)3130 1707 y FN(t)p
FG(\))567 1890 y FO(F)n(ormulae:)996 1914 y FN(\031)s
Fz(\001)1067 1922 y Fp(B)1113 1914 y FG(\()p FF(>)p FG(\))1255
1871 y FA(def)1263 1914 y FG(=)f FF(>)930 2032 y FN(\031)s
Fz(\001)1001 2040 y Fp(B)1046 2032 y FG(\()p FN(p)p FG(\()p
FN(t)p FG(\)\))1255 1989 y FA(def)1263 2032 y FG(=)g
FN(p)p FG(\()p FN(\031)s Fz(\001)1492 2040 y Fp(B)1538
2032 y FN(t)p FG(\))794 2150 y FN(\031)s Fz(\001)865
2158 y Fp(B)910 2150 y FG(\()p FN(G)1000 2158 y FR(1)1052
2150 y FN(?)18 b(G)1168 2158 y FR(2)1202 2150 y FG(\))1255
2107 y FA(def)1263 2150 y FG(=)30 b(\()p FN(\031)s Fz(\001)1454
2158 y Fp(B)1500 2150 y FN(G)1560 2158 y FR(1)1594 2150
y FG(\))17 b FN(?)h FG(\()p FN(\031)s Fz(\001)1798 2158
y Fp(B)1844 2150 y FN(G)1904 2158 y FR(2)1938 2150 y
FG(\))1353 2241 y FQ(for)40 b FN(?)22 b FG(::=)f FF(^)q(j_)2200
1914 y FN(\031)s Fz(\001)2271 1922 y Fp(B)2317 1914 y
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1914 y FN(G)p FG(\))12 b FF(\033)h FG(\()p FN(\031)s
Fz(\001)3098 1922 y Fp(B)3144 1914 y FN(A)p FG(\))2213
2032 y FN(\031)s Fz(\001)2285 2040 y Fp(B)2330 2032 y
FG(\()p FF(r)p FN(=B)t FG(\))2577 1989 y FA(def)2585
2032 y FG(=)30 b FF(r)p FN(=)p FG(\()p FN(\031)s Fz(\001)2878
2040 y Fp(B)2924 2032 y FN(B)t FG(\))2196 2150 y FN(\031)s
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y
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2397 2150 a FB(N)2397 2150 y
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2397 2150 a FN(a:D)r FG(\))2577
2107 y FA(def)2585 2150 y FG(=)2729 2150 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
2729 2150
a FB(N)2729 2150 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
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2729 2150 a FN(a:\031)s Fz(\001)2862
2158 y Fp(B)2908 2150 y FN(D)2183 2268 y(\031)s Fz(\001)2254
2276 y Fp(B)2300 2268 y FG(\()p FF(8)p FN(X)r(:D)r FG(\))2577
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Fz(\001)2875 2280 y Fp(f)p Fn(X)t Fp(g[B)3080 2268 y
FN(D)p 3402 2297 4 1175 v 523 2300 2882 4 v 523 2447
a FO(Fig)o(.)12 b(2.)20 b FQ(De\002nition)g(of)g(the)g(permutation)h
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2447 y FG(\()p FF(\000)p FG(\))f FQ(for)g(terms)g(and)g(formulae.)h(In)
f(the)g(clause)h(for)523 2539 y(the)16 b(ne)n(w-quanti\002er)m(,)h(it)e
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2630 y(under)k(the)f(binder)g(without)g(capture.)523
2917 y FT(permutation)d(operation)g(is)k(gi)n(v)o(en)c(in)j(Fig.)f(2.)g
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3016 y FT(is)f(the)f(empty)f(set.)h(This)g(is)h(a)f(generalisation)e
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(formula)d(with)j(quanti\002ers,)e(it)h(acts)h(only)e(on)h(the)g(free)g
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y(Similar)j(problems)e(arise)i(in)h(the)f(de\002nition)e(of)i(the)g
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y(to)g(the)f(abstractions)g FK(a:)p FH(\()p FI(\000)p
FH(\))h FT(substitution)f(is)h(possibly-capturing,)c(whereas)j(with)h
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currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
583 3489 a FD(N)583 3489
y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
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583 3489 a FT(-)29 b(and)e FI(8)p FT(-quanti\002er)g(it)i(must)g(be)f
(capture-a)n(v)n(oiding.)d(F)o(or)j(terms)g(we)h(can)f(use)g(the)h
(de\002nition)523 3581 y(gi)n(v)o(en)e(in)i([9]:)e(a)i(substitution)f
FK(\033)k FT(is)d(a)g(function)e(from)g(v)n(ariables)h(to)g(terms)h
(with)f(the)h(property)523 3672 y(that)e FK(\033)s FH(\()p
FK(X)7 b FH(\))34 b(=)g FK(X)g FT(for)25 b(all)i(b)n(ut)g(\002nitely)f
(man)o(y)f(v)n(ariables)h FK(X)7 b FT(.)26 b(If)g(the)g(domain)f(of)i
FK(\033)j FT(consists)d(of)523 3763 y(distinct)e(v)n(ariables)f
FK(X)1185 3775 y FS(1)1222 3763 y FK(;)14 b(:)g(:)g(:)g(;)g(X)1476
3775 y FC(n)1546 3763 y FT(and)24 b FK(\033)s FH(\()p
FK(X)1842 3775 y FC(i)1871 3763 y FH(\))32 b(=)f FK(t)2061
3775 y FC(i)2114 3763 y FT(for)25 b FK(i)31 b FH(=)g(1)14
b FK(:)g(:)g(:)f(n)p FT(,)25 b(we)h(sometimes)e(write)i
FK(\033)523 3855 y FT(as)20 b FH([)p FK(X)704 3867 y
FS(1)764 3855 y FH(:=)j FK(t)905 3867 y FS(1)942 3855
y FK(;)14 b(:)g(:)g(:)g(;)g(X)1196 3867 y FC(n)1264 3855
y FH(:=)23 b FK(t)1405 3867 y FC(n)1450 3855 y FH(])p
FT(.)d(Moreo)o(v)o(er)m(,)c(we)k(shall)g(write)f FK(\033)s
FH(\()p FK(t)p FH(\))i FT(for)e(the)h(result)f(of)g(applying)f(a)523
3946 y(substitution)h FK(\033)24 b FT(to)c(a)g(term)f
FK(t)p FT(;)i(this)f(is)h(the)f(term)f(obtained)g(from)f
FK(t)j FT(by)e(replacing)f(each)i(v)n(ariable)f FK(X)523
4037 y FT(by)24 b(the)g(term)f FK(\033)s FH(\()p FK(X)7
b FH(\))25 b FT(and)e(each)h(suspension)f FK(\031)s Fu(\001)p
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Fu(\001)p FK(\033)s FH(\()p FK(X)7 b FH(\))25 b FT(got)e(by)h(letting)g
FK(\031)523 4129 y FT(act)d(on)e(the)i(term)f FK(\033)s
FH(\()p FK(X)7 b FH(\))p FT(.)20 b(This)h(de\002nition)e(is)i(e)o
(xtended)d(to)i(formulae)f(as)i(follo)n(ws:)1160 4327
y FN(\033)s FG(\()p FF(>)p FG(\))1349 4284 y FA(def)1356
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FG(\()p FN(t)p FG(\)\))1349 4402 y FA(def)1356 4445 y
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4654 y FQ(for)i FN(?)i FG(::=)h FF(^j_)2150 4313 y FN(\033)s
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FF(\033)h FN(\033)s FG(\()p FN(A)p FG(\))2164 4431 y
FN(\033)s FG(\()p FF(r)p FN(=B)t FG(\))2457 4389 y FA(def)2465
4431 y FG(=)30 b FN(\033)s FG(\()p FF(r)p FG(\))p FN(=\033)s
FG(\()p FN(B)t FG(\))2146 4549 y FN(\033)s FG(\()2277
4549 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
2277 4549 a FB(N)2277 4549 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
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2277 4549 a FN(a:D)r
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2609 4549 a FB(N)2609 4549 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
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2609 4549 a FN(a:\033)s
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FF(8)p FN(X)r(:\033)s FG(\()p FN(D)r FG(\))523 4825 y
FT(with)c(the)h(pro)o(viso)d(that)j(the)f(quanti\002ed)f(names)h(and)g
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%%Page: 7 7
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866 2314 a FK(a)910 2284 y Fv(0)934 2314 y FK(:)p FI(8)p
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FI(>)14 b(\033)e FK(A)p FH(\()p FK(a)1517 2284 y Fv(0)1540
2314 y FK(;)i(Y)1644 2284 y Fv(0)1667 2314 y FK(;)g FI(h)p
FK(a;)g(Y)19 b FI(i)p FH(\))p FT(.)i(W)-7 b(e)21 b(use)f(the)h
(notation)d FK(\033)s FH(\()p FI(r)p FH(\))k FT(to)e(mean)g(that)g(e)n
(v-)523 2414 y(ery)g(freshness)f(constraint)h FK(a)j
FH(#)g FK(X)k FT(in)20 b FI(r)h FT(is)g(replaced)e(by)h
FK(a)j FH(#)g FK(\033)s FH(\()p FK(X)7 b FH(\))p FT(.)648
2516 y(It)34 b(is)i(crucial)e(for)f(programming)e(in)k
FK(\013)p FT(Prolog)f(that)g(abstractions)g FK(a:)p FH(\()p
FI(\000)p FH(\))h FT(ha)n(v)o(e)f(concrete)523 2615 y(names.)d(This)i
(allo)n(ws)f(us)g(to)g(formulate)f(the)h FJ(type)p FT(-clause)f(for)g
(lambda-abstractions)e(in)j(the)523 2715 y(usual)25 b(f)o(ashion)f
(whereby)g(the)h(abstracted)f(name)h FJ(x)g FT(and)g(its)h(type)f(is)h
(just)f(added)f(to)i(the)f(conte)o(xt)523 2815 y FJ(Gamma)p
FT(.)c(Furthermore,)d(the)j(w)o(ork)g(reported)e(in)j([9])e(pro)o
(vides)g(us)h(with)h(a)f(simple)g(algorithm)f(for)523
2914 y(unifying)g(nominal)h(terms.)h(This)h(uni\002cation)e(algorithm)g
(does)h(not)g(calculate)g(uni\002ers)f(to)i(mak)o(e)523
3014 y(nominal)29 b(terms)i(syntactically)f(equal,)g(b)n(ut)g(equal)g
(modulo)f(an)i(equi)n(v)n(alence)d(relation)i FI(\031)p
FT(.)h(F)o(or)523 3114 y(e)o(xample)23 b(when)h(unifying)e(the)i(tw)o
(o)h(terms)f FK(a:a)31 b FI(\031)o FH(?)g FK(b:X)7 b
FT(,)24 b(the)g(nominal)f(uni\002cation)g(algorithm)523
3213 y(produces)i(the)h(uni\002er)f FH([)p FK(X)41 b
FH(:=)34 b FK(b)p FH(])p FT(.)26 b(While)h(the)f(relation)f
FI(\031)i FT(is)g(intended)e(to)h(capture)f(the)h(\(tradi-)523
3313 y(tional\))h(notion)f(of)i FK(\013)p FT(-equi)n(v)n(alence,)d(it)j
(is)h(in)f(f)o(act)f(a)h(more)f(general)g(relation.)f(F)o(or)h(e)o
(xample,)f FI(\031)523 3413 y FT(is)g(not)f(just)g(a)g(relation)f
(between)h(tw)o(o)g(nominal)e(terms,)i(b)n(ut)g(a)g(relation)g(that)g
(depends)e(on)i(some)523 3512 y(freshness)18 b(constraints)g
FI(r)p FT(.)h(Figure)f(3)h(gi)n(v)o(es)f(a)h(syntax-directed)d(inducti)
n(v)o(e)h(de\002nition)h(for)g(judge-)523 3612 y(ments)i(of)h(the)f
(form)f FI(r)24 b(`)f FH(\()p FI(\000)p FH(\))h FI(\031)f
FH(\()p FI(\000)p FH(\))p FT(,)e(which)f(asserts)h(that)g(tw)o(o)g
(terms)f(are)g FI(\031)p FT(-equal)f(under)g(the)523
3711 y(hypotheses)h FI(r)p FT(;)i(the)g(de\002nition)e(depends)h(on)g
(the)h(auxiliary)e(relation)h FI(r)26 b(`)f FH(\()p FI(\000)p
FH(\))h(#)g(\()p FI(\000)p FH(\))p FT(,)c(which)523 3811
y(de\002nes)17 b(when)g(a)h(name)f(is)i FE(fr)m(esh)f
FT(for)f(a)h(term)f(under)g(some)g(hypotheses.)f(This)h(de\002nition)g
(depends)523 3911 y(on)k(the)g(auxiliary)e(notion)h(of)h(a)g
(disagreement)e(set,)j FK(ds)p FT(,)f(between)f(tw)o(o)h(permutations)e
(\(the)i(set)h(of)523 4010 y(names)e(on)g(which)f(the)i(permutations)d
(disagree\))h(gi)n(v)o(en)g(by:)h FI(f)p FK(a)i FI(j)h
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FI(6)p FH(=)g FK(\031)2781 4022 y FS(2)2818 4010 y Fu(\001)p
FK(a)p FI(g)p FT(.)648 4104 y(W)-7 b(e)21 b(can)f(e)o(xtend)f
FI(\031)h FT(to)g FE(quanti\002er)n(-fr)m(ee)f FK(G)p
FT(-formulae)f(as)j(follo)n(ws:)p 610 4292 376 4 v 610
4360 a FF(r)h(`)f(>)g(\031)g(>)1252 4264 y(r)g(`)h FN(t)f
FF(\031)g FN(t)1564 4233 y Fp(0)p 1154 4284 531 4 v 1154
4360 a FF(r)g(`)g FN(p)p FG(\()p FN(t)p FG(\))g FF(\031)g
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4264 y FF(r)h(`)f FN(G)2066 4272 y FR(1)2122 4264 y FF(\031)g
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4368 y FR(4)2902 4310 y FQ(for)22 b FN(?)f FG(::=)h FF(^j_)523
4526 y FT(The)28 b(adv)n(antage)e(of)i(setting)g(up)g(the)h(formalism)e
(in)h(this)h(w)o(ay)f(is)h(that)g(the)f FI(\031)p FT(-equi)n(v)n
(alence)d(has)523 4625 y(a)h(number)d(of)i(good)f(properties,)f(which)i
(will)h(play)f(an)g(important)f(rle)h(in)g(our)g(proof)f(for)g(sho)n
(w-)523 4725 y(ing)30 b(that)g(the)h FK(\031)s FT(-rule)e(can)h(be)h
(eliminated.)e(F)o(or)h(e)o(xample,)e FI(\031)j FT(is)g(preserv)o(ed)d
(under)h(\(possibly-)523 4825 y(capturing\))f(substitutions)i(and)g
(beha)n(v)o(es)f(well)i(with)g(respect)f(to)g(the)g(permutation)f
(operation.)523 4924 y(This)20 b(is)i(made)d(precise)h(in)g(the)h
(follo)n(wing)d(lemma.)p eop end
%%Page: 8 8
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b(permutation)f(and)g(substitution)g(oper)o(ations)g(pr)m(eserve)h
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FE(for)f(all)f(substitutions)g FK(\033)32 b FE(with)c
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f(e)o(xtension)f(of)i(the)g(proofs)e(gi)n(v)o(en)h(for)g([9];)g(the)o
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y(of)j FK(\013)p FT(-equi)n(v)n(alence)e(we)i(imposed)f(\(on)g(the)h
(meta-le)n(v)o(el\))f(on)g(quanti\002ed)g FK(D)r FT(-formulae.)f
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n(v)n(oid)f(possible)g(capture.)648 2929 y(Ne)o(xt)26
b(we)h(introduce)e(the)h(inference)f(rules)i(on)f(which)g(proof-search)
e(is)j(based)f(in)h FK(\013)p FT(Prolog)523 3051 y(\(see)f(Figure)g
(4\).)f(Sequents)h(are)f(of)h(the)g(form)f FI(r)p FH(;)14
b FK(\000)46 b FI(\))34 b FK(G)27 b FT(or)e FI(r)p FH(;)14
b FK(\000)2640 3003 y FC(D)2619 3051 y FI(\000)-39 b(!)34
b FK(p)p FH(\()p FK(t)p FH(\))27 b FT(where)e(the)h(for)n(-)523
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FT(and)g(the)i(latter)f(models)g FE(focused)g(bac)n(kc)o(haining)d
FT(\(the)523 3233 y(formula)21 b(abo)o(v)o(e)g(the)i(sequent)f(arro)n
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(inference)523 3324 y(rules)j(are)f(adapted)f(from)h(a)h(standard)e
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3416 y(\(for)j(e)o(xample)g([4]\).)1132 3386 y FA(4)1192
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b(\000)-17 b(\000)-31 b(!)21 b FN(p)p FG(\()p FN(t)p
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y FT(need)g(to)h(be)f(syntactically)g(equal.)g(In)h FK(\013)p
FT(Prolog)f(this)h(requirement)e(is)523 3979 y(relax)o(ed:)j(terms)h
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FK(t)p FT(\))c(are)f(satis\002ed)h(by)f(the)h FI(r)p
FT(,)g(that)f(is)i(for)d(all)i FK(a)28 b FH(#)f FK(t)523
946 y FT(the)20 b(judgement)f FI(r)k(`)g FK(a)g FH(#)g
FK(t)d FT(de\002ned)g(in)g(Fig.)g(3)h(holds.)648 1046
y(Of)15 b(most)h(interest)g(in)f(this)h(paper)f(is)h(the)g
FK(\031)s FT(-rule.)f(In)g(a)h(\223root-\002rst\224)f(proof-search,)d
(this)k(rule)g(is)g(a)523 1178 y(source)h(of)g(non-determinism.)d(F)o
(or)j(e)o(xample,)f(if)i(we)g(w)o(ant)f(to)h(pro)o(v)o(e)d(the)j
(sequent)f Fm(?)p FH(;)3065 1124 y FC(p)p FS(\()p FC(a)p
FS(\))3044 1178 y FI(\000)-19 b(\000)g(!)23 b FK(p)p
FH(\()p FK(b)p FH(\))p FT(,)523 1278 y(we)32 b(need)f(the)h
FK(\031)s FT(-rule)g(in)g(order)e(to)i(mak)o(e)g(the)g(terms)f
FK(a)i FT(and)e FK(b)h FI(\031)p FT(-equi)n(v)n(alent\227in)d(this)j
(case,)523 1378 y(only)19 b(after)h(applying)e(a)i(permutation)e(such)i
(as)g FH(\()p FK(a)14 b(b)p FH(\))21 b FT(to)f FK(a)g
FT(may)g(the)g(axiom-rule)d(be)j(used.)g(Prima)523 1477
y(f)o(acie)d(the)g FK(\031)s FT(-rule)g(is)h(innocuous,)d(ho)n(we)n(v)o
(er)m(,)f(the)j(problem)f(of)g(simultaneously)g(unifying)f(nominal)523
1577 y(terms)j(and)g(\002nding)f(a)i FK(\031)j FT(is,)d(as)h(mentioned)
c(earlier)m(,)h(an)i(NP-hard)e(decision)g(problem.)g(In)h(the)g(ne)o
(xt)523 1677 y(section)g(we)g(shall)h(sho)n(w)f(that)g(such)g(problems)
e(ne)n(v)o(er)h(need)h(to)g(be)g(solv)o(ed)f(pro)o(vided)f(the)i
(program)523 1776 y(clauses)j(are)f(well-formed.)523
2015 y FL(3)99 b(Elimination)25 b(of)g(the)h Fh(\031)t
FL(-Rule)523 2180 y FT(W)-7 b(e)18 b(implemented)c FK(\013)p
FT(Prolog)i(using)g(the)g(nominal)g(uni\002cation)f(algorithm.)f(W)m
(ith)j(this)g(implemen-)523 2271 y(tation)22 b(we)h(were)g(able)f(to)h
(calculate)f(the)h(e)o(xpected)e(results)i(for)e(programs)g(such)i(as)g
FJ(subst)f FT(and)523 2362 y FJ(type)p FT(.)16 b(The)f(reason)h(for)f
(this)i(is,)g(roughly)d(speaking,)g(that)j(the)f(name)f(we)i(used)e
(for)h(specifying)e(the)523 2454 y(clauses)26 b(dealing)f(with)h
FK(\025)p FT(-abstractions)f(does)h(not)g(matter)-5 b(.)25
b(When)h(using)f(nominal)g(uni\002cation,)523 2545 y(the)20
b(follo)n(wing)f(renamed)f(clauses)j(\(where)e FJ(a)i
FT(and)e FJ(x)i FT(are)f(renamed)f(to)h FJ(b)h FT(and)e
FJ(y)p FT(,)i(respecti)n(v)o(ely\))527 2675 y FP
(subst\(lam\(b.M\),X,T,lam\(b.M'\)\))-16 b(:-)23 b(b)12
b(#)g(T,)g(b)h(#)f(X,)g(subst\(M,X,T,M'\).)527 2790 y
(type\(Gamma,lam\(y.M\),arr\(S,T\)\))-16 b(:-)23 b(y)12
b(#)g(Gamma,)g(type\([\(y,S\)|Gamma],M,T\).)523 2922
y FT(beha)n(v)o(e)25 b(just)h(the)g(same)g(as)g(the)g(original)f
(clauses,)h(in)g(the)f(sense)i(that)f(all)g(queries)f(successfully)523
3013 y(solv)o(ed)30 b(by)h(the)g(original)f(v)o(ersions)g(are)h(solv)o
(ed)f(by)g(the)h(renamed)f(v)o(ersions.)g(In)g(contrast,)g(the)523
3105 y(name)19 b FJ(a)i FT(in)f(the)g(clauses)g FJ(p\(a\))13
b FT(,)44 b FJ(q\(a.X,X\))e FT(and)19 b FJ(r\(X\))13
b(:-)h(id\(X,a\))19 b FT(determines)g(which)523 3196
y(queries)k(can)h(be)g(solv)o(ed)g(successfully)f(using)h(nominal)f
(uni\002cation)f(and)i(which)g(cannot:)f(gi)n(v)o(en)523
3287 y(our)k(inference)g(rules,)h(which)g(choose)f(a)h(fresh)g(name)g
(for)f FJ(a)p FT(,)i(there)e(are)h(some)g(queries)g(whose)523
3379 y(answers)18 b(can)g(only)g(be)g(found)e(using)i(the)g
FK(\031)s FT(-rule,)g(and)f(this)i(means)f(the)o(y)f(cannot)h(be)g
(solv)o(ed)f(using)523 3470 y(nominal)i(uni\002cation.)g(Consider)g
(for)h(e)o(xample)f(the)h(follo)n(wing)e(deduction.)1500
3730 y FN(:)13 b(:)g(:)1856 3592 y Fq(?)21 b FF(`)h FN(c)f
FF(\031)g FN(c)p 1813 3619 405 4 v 1813 3730 a Fq(?)p
FG(;)1927 3681 y Fn(p)p FR(\()p Fn(c)p FR(\))1907 3730
y FF(\000)-20 b(\000)h(!)21 b FN(p)p FG(\()p FN(c)p FG(\))2259
3644 y FQ(Ax)p 1500 3769 718 4 v 1558 3879 a Fq(?)p FG(;)1671
3831 y Fg(?)p Fn(=)p Fp(>)11 b(\033)f Fn(p)p FR(\()p
Fn(c)p FR(\))1652 3879 y FF(\000)-29 b(\000)-17 b(\000)g(\000)f(\000)h
(\000)g(\000)-28 b(!)21 b FN(p)p FG(\()p FN(c)p FG(\))2259
3785 y FF(\033)2319 3793 y Fn(L)p 1558 3918 601 4 v 1558
4029 a Fq(?)p FG(;)1672 3980 y Fg(?)p Fn(=)p Fp(>)10
b(\033)g Fn(p)p FR(\()p Fn(b)p FR(\))1652 4029 y FF(\000)-29
b(\000)-17 b(\000)g(\000)f(\000)h(\000)g(\000)-28 b(!)21
b FN(p)p FG(\()p FN(c)p FG(\))2200 3937 y FN(\031)46
b FG(\()p FN(c)13 b(b)p FG(\))p 1508 4068 702 4 v 1508
4183 a Fq(?)p FG(;)1663 4134 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
1663 4134 a Fj(N)1663
4134 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
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1663 4134 a Fn(a:)p Fg(?)p Fn(=)p Fp(>)d(\033)h
Fn(p)p FR(\()p Fn(a)p FR(\))1602 4183 y FF(\000)-21 b(\000)j(\000)h
(\000)g(\000)f(\000)h(\000)g(\000)f(\000)e(!)22 b FN(p)p
FG(\()p FN(c)p FG(\))2305 4090 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
2305 4090 a FB(N)2305
4090 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
2305 4090 a 9 x Fn(L)523 4327 y FT(In)e(this)g(deduction)e(the)i
FK(\031)s FT(-rule,)f(applying)f(the)i(permutation)e
FH(\()p FK(c)c(b)p FH(\))20 b FT(\(annotated)e(to)i(the)g
FK(\031)s FT(-rule\),)f(is)523 4426 y(crucial)j(for)g(the)h(sequent)f
(being)f(pro)o(v)n(able)g(and)h(it)h(will)h(turn)e(out)g(that)h(it)g
(is)h(impossible)e(to)h(elim-)523 4526 y(inate)e(it)h(from)f(this)g
(deduction.)e(Consequently)-5 b(,)19 b(a)j(proof-search)c(procedure)h
(based)i(on)g(nominal)523 4625 y(uni\002cation)e(will)i(not)f(\002nd)g
(this)g(proof.)648 4725 y(If)i(we)h(impose)f(the)h(follo)n(wing)e
(well-formedness)g(condition)g(on)h FK(D)r FT(-formulae,)e(we)j(can)g
(en-)523 4825 y(sure)18 b(that)h(the)f FK(\031)s FT(-rule)g(can)h(al)o
(w)o(ays)f(be)h(eliminated)e(from)h(corresponding)d(deductions)h(and)i
(hence)523 4924 y(the)i(nominal)f(uni\002cation)g(algorithm)g(alone)g
(is)i(suf)n(\002cient)f(for)g(solving)f(queries.)p eop
end
%%Page: 10 10
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y FI(\033)14 b FK(A)2715 460 y Fs(as)5 b FC(;)p Fs(Xs)2914
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FT(and)h Fq(?)21 b FF(`)g FN(\033)s FG(\()p FN(\031)s
Fz(\001)o FF(>)p FG(\))42 b FF(\031)g(>)523 1442 y FT(tri)n(vially)27
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FK(\033)32 b FT(the)27 b(identity)g(substitution)g(and)g(for)g
FK(\031)k FT(the)523 1533 y(empty)17 b(permutation.)f(More)i
(complicated)e(is)j(the)g(case)f(of)g(the)g(second)g
FJ(subst)p FT(-clause)f(in)h(Fig.)g(1)789 1675 y FF(r)j(`)h
FN(\033)s FG(\()p FN(S)t FG(\()p Fx(lam)p FG(\()p FN(b:M)8
b FG(\))p FN(;)13 b(X)r(;)g(T)6 b(;)14 b Fx(lam)p FG(\()p
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b(X)r(;)h(T)6 b(;)13 b(M)1550 1735 y Fp(0)1572 1766 y
FG(\)\))43 b FF(\031)f FN(S)t FG(\()p FN(M)t(;)13 b(X)r(;)h(T)6
b(;)13 b(M)2237 1735 y Fp(0)2260 1766 y FG(\))523 1916
y FT(where)20 b FI(r)h FT(is)g FI(f)p FK(b)h FH(#)h FK(M)t(;)14
b(b)23 b FH(#)g FK(X)r(;)14 b(b)22 b FH(#)h FK(T)7 b(;)14
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b(a)22 b FH(#)i FK(X)r(;)14 b(a)22 b FH(#)h FK(T)12 b
FI(g)p FT(.)20 b(In)f(this)i(case)651 2072 y FN(\033)j
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Fz(\001)p FN(M)t(;)h(X)27 b FG(:=)22 b(\()p FN(a)12 b(b)p
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Fz(\001)o FN(M)2689 2040 y Fp(0)2711 2072 y FG(])66 b
FT(and)d FN(\031)24 b FG(=)d(\()p FN(a)13 b(b)p FG(\))523
2236 y FT(v)o(erify)19 b(that)h(the)g(clause)h(is)g(well-formed.)648
2327 y(Before)k(we)h(formally)e(sho)n(w)i(that)g(all)g
FK(\031)s FT(-rules)g(can)f(be)h(eliminated)f(from)f(deductions)g(con-)
523 2419 y(sisting)31 b(of)f(well-formed)e(clauses)i(only)-5
b(,)29 b(we)h(outline)g(our)f(proof-plan)e(with)k(some)f(e)o(xamples.)
523 2510 y(Consider)25 b(the)g(follo)n(wing)e(deduction,)g(which)i(has)
g(a)h FK(\031)s FT(-rule)e(on)h(the)g(top)g(right-hand)e(side.)i(The)
523 2601 y(corresponding)17 b(permutation)h FH(\()p FK(e)c(d)p
FH(\))21 b FT(transforms)e FK(p)p FH(\()p FK(b:d)p FH(\))i
FT(into)f FK(p)p FH(\()p FK(b:e)p FH(\))h FT(so)g(that)f(the)g
(axiom-rule)f(is)523 2692 y(applicable.)1349 3051 y FN(:)13
b(:)g(:)1743 2771 y FF(`)21 b FN(b:e)h FF(\031)f FN(d:e)p
1664 2791 516 4 v 1664 2901 a Fq(?)p FG(;)1777 2853 y
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currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
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2321 3407 a 8 x Fn(L)2388 3407
y FG(\()p FN(a)13 b(b)p FG(\))523 3629 y FT(Observ)o(e)21
b(that)i(the)f(\223choice\224)g(of)g(the)g(fresh)g(name)g(\(namely)f
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3065
3629 a FD(N)3065 3629 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
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3065 3629 a 12 x FC(L)3115 3629
y FT(-rule)f(has)523 3729 y(no)i(ef)n(fect)f(on)h(whether)f(this)i
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(bind)g(an)o(ything)523 3828 y(inside)g(the)h(abstraction.)e(The)h
(annotated)f(substitution)h FH([)p FK(X)35 b FH(:=)29
b FK(d)p FH(])24 b FT(ho)n(we)n(v)o(er)e(is)i(important)e(with)523
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(to)i(eliminate.)e(If)i(we)f(had)g(instead)h(substituted)e
FK(e)i FT(for)f FK(X)7 b FT(,)523 4028 y(then)20 b(the)g(axiom)f(is)i
(applicable)e FE(without)j FT(the)e FK(\031)s FT(-rule.)648
4127 y(Note,)31 b(ho)n(we)n(v)o(er)m(,)e(that)i(changing)f(the)h
(instantiation)g(of)g FI(8)p FT(-quanti\002ers)f(might)h(ha)n(v)o(e)g
(some)523 4227 y(\223non-local\224)18 b(consequences)g(in)j
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FK(M)9 b FT(,)23 b FK(X)7 b FT(,)523 4526 y FK(T)29 b
FT(and)17 b FK(M)829 4496 y Fv(0)852 4526 y FT(,)h(then)f(the)h
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(time)f(the)h(subgoal)f(\(mark)o(ed)f(by)h FK(?)p FT(\))523
4625 y(is)26 b(changed.)d(The)i(well-formedness)e(condition)h(ensures)g
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y(introduced)18 b(by)i(the)g FI(8)1171 4737 y FC(L)1221
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f(in)m(v)n(olving)g(well-formed)g(pro-)523 4924 y(grams,)j(we)i
(\002rst)g(pro)o(v)o(e)d(some)i(auxiliary)f(f)o(acts.)p
eop end
%%Page: 11 11
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Fe(v)p FR(\()p Fn(z)r FR(\))p Fn(;z)r(;)p Fe(v)p FR(\()p
Fn(y)r FR(\))p Fn(;)p Fe(v)p FR(\()p Fn(y)r FR(\)\))1272
715 y FN(?)1491 459 y Fn(::)p Fp(`h)p Fe(l)p FR(\()p
Fn(b:)p Fe(v)p FR(\()p Fn(y)r FR(\)\))p Fn(;y)r(;)p Fe(v)p
FR(\()p Fn(z)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(b:)p
Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fp(i\031h)p Fe(l)p FR(\()p
Fn(x:)p Fe(v)p FR(\()p Fn(y)r FR(\)\))p Fn(;y)r(;)p Fe(v)p
FR(\()p Fn(z)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(x:)p
Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fp(i)p 1334 492 1858
4 v 1334 603 a FN(::)p FG(;)1430 554 y Fn(s)p FR(\()p
Fe(l)p FR(\()p Fn(b:)p Fe(v)p FR(\()p Fn(y)r FR(\)\))p
Fn(;y)r(;)p Fe(v)p FR(\()p Fn(z)r FR(\))p Fn(;)p Fe(l)p
FR(\()p Fn(b:)p Fe(v)p FR(\()p Fn(z)r FR(\)\)\))1410
603 y FF(\000)-29 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)
f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)f(\000)-29
b(!)22 b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p
FG(\()p FN(y)s FG(\)\))p FN(;)12 b(y)s(;)g Fx(v)p FG(\()p
FN(z)s FG(\))p FN(;)h Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
FN(z)s FG(\)\)\))3194 517 y FQ(Ax)p 1334 641 V 1334 752
a FN(::)p FG(;)1430 703 y Fn(s)p FR(\()p Fe(l)p FR(\()p
Fn(b:)p Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fn(;z)r(;)p Fe(v)p
FR(\()p Fn(y)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(b:)p
Fe(v)p FR(\()p Fn(y)r FR(\)\)\))1410 752 y FF(\000)-29
b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
(\000)f(\000)h(\000)g(\000)f(\000)h(\000)f(\000)-29 b(!)22
b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
FN(y)s FG(\)\))p FN(;)12 b(y)s(;)g Fx(v)p FG(\()p FN(z)s
FG(\))p FN(;)h Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
FN(z)s FG(\)\)\))3207 669 y FN(\031)3254 637 y Fp(\017)p
616 791 2575 4 v 666 901 a FN(::)p FG(;)762 853 y Fp(r)p
Fn(=s)p FR(\()p Fe(v)p FR(\()p Fn(z)r FR(\))p Fn(;z)r(;)p
Fe(v)p FR(\()p Fn(y)r FR(\))p Fn(;)p Fe(v)p FR(\()p Fn(y)r
FR(\)\))p Fp(\033)o Fn(s)p FR(\()p Fe(l)p FR(\()p Fn(b:)p
Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fn(;z)r(;)p Fe(v)p FR(\()p
Fn(y)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(b:)p Fe(v)p FR(\()p
Fn(y)r FR(\)\)\))743 901 y FF(\000)-20 b(\000)j(\000)f(\000)h(\000)g
(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
(\000)g(\000)f(\000)h(\000)g(\000)f(\000)g(!)21 b FN(s)p
FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p FN(y)s
FG(\)\))p FN(;)12 b(y)s(;)h Fx(v)p FG(\()p FN(z)s FG(\))p
FN(;)g Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p FN(z)s FG(\)\)\))3194
807 y FF(\033)3254 815 y Fn(L)p 666 940 2476 4 v 1509
1014 a FQ(.)1509 1047 y(.)1509 1080 y(.)284 b FG([)p
FN(M)29 b FG(:=)22 b Fx(v)p FG(\()p FN(z)s FG(\))p FN(;)13
b(X)28 b FG(:=)21 b FN(z)s(;)13 b(T)32 b FG(:=)21 b Fx(v)p
FG(\()p FN(y)s FG(\))p FN(;)13 b(M)2912 1049 y Fp(0)2956
1080 y FG(:=)21 b Fx(v)p FG(\()p FN(y)s FG(\)])3157 962
y FF(8)3200 970 y Fn(L)p 638 1119 2532 4 v 638 1236 a
FN(::)p FG(;)712 1188 y Fp(8)p Fn(M)r(;X)q(;T)5 b(;M)1019
1167 y Fl(0)1031 1188 y Fn(:)p Fp(r)-11 b Fn(=s)p FR(\()p
Fn(M)r(;X)q(;T)5 b(;M)1447 1167 y Fl(0)1470 1188 y FR(\))p
Fp(\033)p Fn(s)p FR(\()p Fe(l)p FR(\()p Fn(b:M)g FR(\))p
Fn(;X)q(;T)g(;)p Fe(l)p FR(\()p Fn(b:M)2086 1167 y Fl(0)2110
1188 y FR(\)\))715 1236 y FF(\000)-35 b(\000)-17 b(\000)g(\000)f(\000)h
(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)-33
b(!)21 b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p
FG(\()p FN(y)s FG(\)\))p FN(;)12 b(y)s(;)h Fx(v)p FG(\()p
FN(z)s FG(\))p FN(;)g Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
FN(z)s FG(\)\)\))3185 1141 y FF(8)3228 1149 y Fn(L)p
590 1275 2629 4 v 590 1393 a FN(::)p FG(;)696 1344 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
696 1344 a Fj(N)696 1344 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
696 1344 a Fn(a:)p Fp(8)p
Fn(M)r(;X)q(;T)5 b(;M)1056 1323 y Fl(0)1067 1344 y Fn(:)p
Fp(r)-11 b Fn(=s)p FR(\()p Fn(M)r(;X)q(;T)5 b(;M)1483
1323 y Fl(0)1506 1344 y FR(\))p Fp(\033)p Fn(s)p FR(\()p
Fe(l)p FR(\()p Fn(a:M)g FR(\))p Fn(;X)q(;T)g(;)p Fe(l)p
FR(\()p Fn(a:M)2136 1323 y Fl(0)2158 1344 y FR(\)\))666
1393 y FF(\000)-28 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g
(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)-28
b(!)21 b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p
FG(\()p FN(y)s FG(\)\))p FN(;)12 b(y)s(;)h Fx(v)p FG(\()p
FN(z)s FG(\))p FN(;)g Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
FN(z)s FG(\)\)\))3263 1292 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
3263 1292 a Fj(N)3263 1292
y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
3263 1292 a 8 x Fn(L)p 3402 1436 4 1068 v 523 1439
2882 4 v 523 1582 a FO(Fig)o(.)f(5.)31 b FQ(Deduction)g(pro)o(ving)h
(the)f(f)o(act)g FN(s)p FP(\(l\(x.v\(y\)\),y,v\(z\),l\(x.v\(z\)\)\))26
b FQ(where)31 b FP(l)g FQ(and)g FP(v)523 1674 y FQ(stand)19
b(for)g(lambda-abstractions)i(and)e(v)n(ariables,)h(respecti)n(v)o(ely)
-5 b(.)523 1971 y Fi(Lemma)21 b(2.)41 b FE(F)-9 b(or)26
b(all)h(permutations)e FK(\031)s FE(,)i(the)f(sequent)g
FI(r)p FH(;)14 b FK(\000)46 b FI(\))34 b FK(\031)s Fu(\001)p
FK(G)27 b FE(is)g(derivable)f(only)f(if)i(the)523 2071
y(sequent)19 b FI(r)p FH(;)14 b FK(\031)953 2041 y Fv(\000)p
FS(1)1043 2071 y Fu(\001)p FK(\000)34 b FI(\))23 b FK(G)f
FE(is)f(derivable)e(\(wher)m(e)h(we)h(use)g(the)f(notation)f
FK(\031)s Fu(\001)p FK(\000)32 b FE(to)20 b(indicate)g(that)g
FK(\031)523 2171 y FE(is)h(applied)e(to)h(e)o(very)h(formula)e(in)i
FK(\000)12 b FE(\).)523 2364 y(Pr)l(oof)o(.)40 b FT(By)25
b(induction)d(on)h(the)h(structure)f(of)h(deductions.)e(It)i(mak)o(es)f
(use)i(of)e(the)h(property)e(of)h FI(\031)523 2464 y
FT(that)c FI(r)k(`)g FK(t)g FI(\031)g FK(\031)s Fu(\001)p
FK(t)1081 2433 y Fv(0)1123 2464 y FT(holds)c(only)f(if)h
FI(r)k(`)g FK(\031)1776 2433 y Fv(\000)p FS(1)1865 2464
y Fu(\001)p FK(t)g FI(\031)g FK(t)2063 2433 y Fv(0)2105
2464 y FT(holds.)18 b(By)i(inspection)e(we)h(can)g(further)e(see)523
2563 y(that)j(no)g(additional)f FK(\031)s FT(-rule)h(is)h(necessary)e
(to)i(sho)n(w)f(the)g(pro)o(v)n(ability)e(in)i(both)f(directions.)191
b FI(u)-55 b(t)523 2757 y FT(The)31 b(follo)n(wing)f(corollary)g(is)j
(a)f(simple)f(consequence)e(of)j(this)g(lemma)f(by)g(the)h(f)o(act)f
(that)h(for)523 2856 y(closed)20 b FK(D)r FT(-formulae)e
FK(\031)s Fu(\001)p FK(D)25 b FH(=)e FK(D)f FT(holds.)523
3055 y Fi(Cor)o(ollary)c(1.)41 b FE(F)-9 b(or)25 b(all)g(permutations)e
FK(\031)29 b FE(and)23 b(conte)n(xts)i FK(\000)37 b FE(consisting)23
b(of)i(closed)f FK(D)r FE(-formulae)523 3154 y(only)-5
b(,)20 b FI(r)p FH(;)14 b FK(\000)35 b FI(\))23 b FK(\031)s
Fu(\001)p FK(G)e FE(is)g(derivable)e(only)h(if)h FI(r)p
FH(;)14 b FK(\000)34 b FI(\))24 b FK(G)d FE(is)g(derivable)o(.)523
3348 y Fi(Lemma)g(3.)41 b FE(If)23 b(the)h(sequent)e
FI(r)p FH(;)14 b FK(\000)41 b FI(\))28 b FK(G)d FE(is)f(derivable)e
(and)h FI(r)29 b(`)f FK(G)i FI(\031)e FK(G)2791 3317
y Fv(0)2814 3348 y FE(,)c(then)f(the)g(sequent)523 3447
y FI(r)p FH(;)14 b FK(\000)35 b FI(\))23 b FK(G)886 3417
y Fv(0)930 3447 y FE(is)e(derivable)o(.)523 3641 y(Pr)l(oof)o(.)40
b FT(Since)21 b FI(r)i(`)g FK(G)g FI(\031)g FK(G)1382
3611 y Fv(0)1426 3641 y FT(is)e(inducti)n(v)o(ely)d(de\002ned)i(e)o
(xtending)e(the)i FI(\031)p FT(-equality)e(of)i(the)g(terms)523
3740 y(occurring)i(in)i FK(G)h FT(and)f FK(G)1255 3710
y Fv(0)1279 3740 y FT(,)g(we)h(can)f(pro)o(v)o(e)e(this)j(lemma)e(by)h
(inspection)f(of)h(the)g(inference-rules,)523 3840 y(noting)c(that)h
(in)g(the)g FH(\()p FI(\000)p FH(\))1241 3852 y FC(L)1291
3840 y FT(-rules)f(the)h(right-hand)e(side)i(of)f(sequents)h(is)h(al)o
(w)o(ays)f(of)g(the)g(form)f FK(p)p FH(\()p FK(t)p FH(\))523
3940 y FT(and)g(the)g(lemma)g(for)f(axioms)h(follo)n(ws)g(from)f(the)h
(transiti)n(vity)f(of)h FI(\031)p FT(.)648 4133 y(F)o(or)25
b(sho)n(wing)f(our)h(main)h(result,)f(it)i(is)f(con)m(v)o(enient)d(to)j
(restrict)g(attention)f(to)h(some)f(speci\002c)523 4233
y(instances)17 b(of)f(the)g FK(\031)s FT(-rule.)g(The)g(ne)o(xt)g
(lemma)g(sho)n(ws)h(that)g(we)g(only)e(need)h(to)h(consider)e(unmo)o(v)
n(able)523 4332 y(instances)20 b(of)g(the)g FK(\031)s
FT(-rule.)523 4530 y Fi(De\002nition)g(2.)41 b FE(A)19
b FK(\031)s FE(-rule)h(is)g FT(mo)o(v)n(able)d FE(pr)l(o)o(vided)h(it)h
(is)h(not)f(dir)m(ectly)g(under)f(an)h(axiom,)f(otherwise)523
4630 y(it)j(is)g(said)f(to)h(be)f FT(unmo)o(v)n(able)p
FE(.)523 4824 y Fi(Lemma)h(4.)41 b FE(All)33 b(mo)o(vable)f(instances)h
(of)g(the)g FK(\031)s FE(-rules)g(can)g(be)g(r)m(eplaced)f(by)h(unmo)o
(vable)e(in-)523 4923 y(stances.)p eop end
%%Page: 12 12
TeXDict begin 12 11 bop 523 448 a FE(Pr)l(oof)o(.)40
b FT(W)-7 b(e)25 b(call)f(a)g(deri)n(v)n(ation)d FK(\031)s
FT(-normalised)h(if)i(all)f(instances)h(of)f(the)g FK(\031)s
FT(-rule)g(are)h(unmo)o(v)n(able.)523 569 y(W)-7 b(e)26
b(\002rst)f(sho)n(w)g(that)g(if)g FK(\000)1351 522 y
FC(\031)r Fd(\001)o FC(D)1330 569 y FI(\000)-21 b(\000)g(!)32
b FK(A)26 b FT(has)f(a)g FK(\031)s FT(-normalised)e(deri)n(v)n(ation,)g
(then)h(we)h(can)g(construct)523 684 y(a)i FK(\031)s
FT(-normalised)e(deri)n(v)n(ation)g(of)h FK(\000)1636
637 y FC(D)1615 684 y FI(\000)-39 b(!)35 b FK(A)p FT(.)27
b(Using)g(this)g(construction,)e(we)i(can)f(eliminate)g(the)523
784 y(mo)o(v)n(able)18 b FK(\031)s FT(-rules)i(from)g(an)o(y)f(deri)n
(v)n(ation)f(one)i(at)h(a)f(time.)648 883 y(There)f(is)i(one)f(case)g
(for)g(each)g(left-rule.)f(F)o(or)g FK(Ax)p FT(,)j(we)e(ha)n(v)o(e)p
1217 1092 482 4 v 1217 1194 a FF(r)p FG(;)14 b FN(\000)1414
1151 y Fn(\031)r Fk(\001)o Fn(A)1394 1194 y FF(\000)-21
b(\000)g(!)21 b FN(\031)s Fz(\001)o FN(A)1740 1118 y(Ax)1864
1123 y FF(\000)-13 b(!)p 2059 956 V 2059 1058 a(r)p FG(;)13
b FN(\000)2255 1015 y Fn(\031)r Fk(\001)p Fn(A)2236 1058
y FF(\000)-22 b(\000)h(!)21 b FN(\031)s Fz(\001)o FN(A)2581
981 y(Ax)p 2059 1092 V 2088 1194 a FF(r)p FG(;)13 b FN(\000)2284
1151 y Fn(A)2265 1194 y FF(\000)-41 b(!)21 b FN(\031)s
Fz(\001)o FN(A)2581 1124 y(\031)2628 1092 y Fp(\000)p
FR(1)523 1327 y FT(since)k FK(\031)769 1296 y Fv(\000)p
FS(1)858 1327 y Fu(\001)p FK(\031)s Fu(\001)p FK(A)31
b FH(=)g FK(A)p FT(.)25 b(A)h FK(\031)s FT(-normalised)d(deri)n(v)n
(ation)g(ending)g(in)i(a)g FK(\031)2628 1296 y Fv(0)2651
1327 y FT(-rule)f(must)h(be)g(immedi-)523 1426 y(ately)20
b(follo)n(wed)f(by)h FK(Ax)p FT(,)h(we)g(can)f(deri)n(v)o(e)p
1007 1505 656 4 v 1007 1617 a FF(r)p FG(;)13 b FN(\000)1203
1574 y Fn(\031)1241 1553 y Fl(0)1264 1574 y Fk(\001)o
Fn(\031)r Fk(\001)p Fn(A)1184 1617 y FF(\000)-24 b(\000)-18
b(\000)h(\000)-23 b(!)21 b FN(\031)1487 1585 y Fp(0)1509
1617 y Fz(\001)p FN(\031)s Fz(\001)o FN(A)1704 1531 y(Ax)p
1007 1651 V 1047 1753 a FF(r)p FG(;)13 b FN(\000)1244
1710 y Fn(\031)r Fk(\001)o Fn(A)1224 1753 y FF(\000)-22
b(\000)i(!)21 b FN(\031)1447 1721 y Fp(0)1469 1753 y
Fz(\001)o FN(\031)s Fz(\001)o FN(A)1704 1681 y(\031)1751
1649 y Fp(0)1828 1682 y FF(\000)-13 b(!)p 2023 1505 682
4 v 2023 1617 a(r)p FG(;)13 b FN(\000)2219 1574 y Fn(\031)2257
1553 y Fl(0)2279 1574 y FR(@)p Fn(\031)r Fk(\001)p Fn(A)2200
1617 y FF(\000)-33 b(\000)-17 b(\000)g(\000)f(\000)-31
b(!)21 b FN(\031)2529 1585 y Fp(0)2551 1617 y Fz(\001)p
FN(\031)s Fz(\001)o FN(A)2746 1531 y(Ax)p 2023 1651 V
2105 1753 a FF(r)p FG(;)13 b FN(\000)2302 1710 y Fn(A)2282
1753 y FF(\000)-41 b(!)21 b FN(\031)2446 1721 y Fp(0)2468
1753 y Fz(\001)p FN(\031)s Fz(\001)o FN(A)2746 1681 y(\031)2793
1649 y Fp(0)2815 1681 y FG(@)p FN(\031)523 1881 y FT(since)h
FK(\031)766 1851 y Fv(0)790 1881 y Fu(\001)o FK(\031)s
Fu(\001)p FK(A)k FH(=)g FK(\031)1122 1851 y Fv(0)1146
1881 y FH(@)p FK(\031)s Fu(\001)o FK(A)p FT(.)d(F)o(or)e
FI(8)1575 1893 y FC(L)1625 1881 y FT(,)h(since)g FK(\031)s
Fu(\001)p FI(8)p FK(X)r(:D)28 b FH(=)e FI(8)p FK(X)r(:\031)s
Fu(\001)2482 1896 y Fv(f)p FC(X)5 b Fv(g)2613 1881 y
FK(D)25 b FT(and)c FH(\()p FK(\031)s Fu(\001)2958 1896
y Fv(f)p FC(X)5 b Fv(g)3089 1881 y FK(D)r FH(\)[)p FK(X)33
b FH(:=)523 1980 y FK(t)p FH(])23 b(=)g FK(\031)s Fu(\001)p
FH(\()p FK(D)r FH([)p FK(X)29 b FH(:=)23 b FK(\031)1149
1950 y Fv(\000)p FS(1)1238 1980 y Fu(\001)p FK(t)p FH(]\))p
FT(,)e(so)f(we)h(ha)n(v)o(e)1013 2143 y FF(r)p FG(;)13
b FN(\000)1209 2082 y FR(\()p Fn(\031)r Fk(\001)1292
2096 y Fl(f)p Fc(X)s Fl(g)1406 2082 y Fn(D)r FR(\)[)p
Fn(X)t FR(:=)p Fn(t)p FR(])1190 2143 y FF(\000)-20 b(\000)i(\000)h
(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)e(!)21
b FN(A)p 1013 2177 755 4 v 1073 2297 a FF(r)p FG(;)13
b FN(\000)1269 2237 y Fp(8)p Fn(X)q(:\031)r Fk(\001)1432
2251 y Fl(f)p Fc(X)s Fl(g)1547 2237 y Fn(D)1250 2297
y FF(\000)-38 b(\000)-17 b(\000)g(\000)f(\000)h(\000)g(\000)f(\000)-36
b(!)21 b FN(A)1809 2199 y FF(8)1852 2207 y Fn(L)1921
2226 y FF(\000)-13 b(!)2115 2160 y(r)p FG(;)13 b FN(\000)2312
2112 y Fn(D)r FR([)p Fn(X)t FR(:=)p Fn(\031)2539 2091
y Fl(\000)p Fb(1)2614 2112 y Fk(\001)o Fn(t)p FR(])2292
2160 y FF(\000)-20 b(\000)j(\000)g(\000)f(\000)h(\000)g(\000)f(\000)f
(!)21 b FN(A)p 2115 2195 670 4 v 2219 2297 a FF(r)p FG(;)13
b FN(\000)2416 2254 y Fp(8)p Fn(X)q(:D)2396 2297 y FF(\000)-38
b(\000)-18 b(\000)h(\000)-38 b(!)21 b FN(A)2826 2217
y FF(8)2869 2225 y Fn(L)523 2478 y FT(where)f(by)g(induction)f
FI(r)p FH(;)14 b FK(\000)1399 2423 y FC(D)r FS([)p FC(X)5
b FS(:=)p FC(\031)1644 2398 y Fl(\000)p Fb(1)1721 2423
y Fd(\001)o FC(t)p FS(])1378 2478 y FI(\000)-24 b(\000)-19
b(\000)g(\000)g(\000)g(\000)g(\000)g(\000)-25 b(!)24
b FK(A)d FT(has)g(a)g FK(\031)s FT(-normalised)e(deri)n(v)n(ation)g
(obtained)g(from)523 2619 y(that)28 b(of)g FI(r)p FH(;)14
b FK(\000)1000 2565 y FC(\031)r Fd(\001)p FC(D)r FS([)p
FC(X)5 b FS(:=)p FC(\031)1308 2540 y Fl(\000)p Fb(1)1385
2565 y Fd(\001)o FC(t)p FS(])979 2619 y FI(\000)-39 b(\000)-19
b(\000)g(\000)g(\000)g(\000)h(\000)f(\000)g(\000)g(\000)-40
b(!)37 b FK(A)29 b FT(.)f(The)g(cases)g(for)f FI(^)2201
2631 y FC(L)2279 2619 y FT(and)2487 2619 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
2487 2619
a FD(N)2487 2619 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
2487 2619 a 12 x FC(L)2565 2619 y
FT(are)h(straightforw)o(ard)d(since)523 2719 y FK(\031)s
Fu(\001)600 2731 y Fv(B)649 2719 y FH(\()p FI(\000)p
FH(\))c FT(commutes)e(with)i FI(^)g FT(and)1606 2719
y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
1606 2719 a FD(N)1606 2719 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
1606 2719 a FT(.)g(F)o(or)f
FI(\033)1847 2731 y FC(L)1918 2719 y FT(we)g(ha)n(v)o(e)567
2860 y FF(r)h(`)g(r)75 b(r)p FG(;)13 b FN(\000)32 b FF(\))21
b FN(\031)s Fz(\001)o FN(G)75 b FF(r)p FG(;)13 b FN(\000)1537
2817 y Fn(\031)r Fk(\001)o Fn(A)1517 2860 y FF(\000)-22
b(\000)h(!)22 b FN(A)1751 2828 y Fp(0)p 567 2894 1206
4 v 787 3005 a FF(r)p FG(;)13 b FN(\000)983 2957 y Fp(r)p
Fn(=)p FR(\()p Fn(\031)r Fk(\001)p Fn(G)p FR(\))p Fp(\033)p
FR(\()p Fn(\031)r Fk(\001)p Fn(A)p FR(\))964 3005 y FF(\000)-26
b(\000)-17 b(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)-25
b(!)21 b FN(A)1530 2974 y Fp(0)1814 2911 y FF(\033)1874
2919 y Fn(L)1943 2934 y FF(\000)-13 b(!)2138 2860 y(r)21
b(`)g(r)75 b(r)p FG(;)13 b FN(\000)32 b FF(\))21 b FN(G)75
b FF(r)p FG(;)13 b FN(\000)3036 2817 y Fn(A)3017 2860
y FF(\000)-41 b(!)21 b FN(A)3192 2828 y Fp(0)p 2138 2894
1077 4 v 2399 3005 a FF(r)p FG(;)14 b FN(\000)2596 2957
y Fp(r)p Fn(=G)p Fp(\033)p Fn(A)2577 3005 y FF(\000)-26
b(\000)-18 b(\000)h(\000)g(\000)-25 b(!)21 b FN(A)2930
2974 y Fp(0)3255 2911 y FF(\033)3315 2919 y Fn(L)523
3127 y FT(using)i(Lemma)f(3)h(to)h(deri)n(v)o(e)e FI(r)p
FH(;)14 b FK(\000)40 b FI(\))29 b FK(G)24 b FT(from)e
FI(r)p FH(;)14 b FK(\000)40 b FI(\))29 b FK(\031)s Fu(\001)p
FK(G)24 b FT(and)f(the)g(induction)e(hypothesis)523 3249
y(to)f(obtain)g(a)g FK(\031)s FT(-normalised)f(deri)n(v)n(ation)f(of)i
FI(r)p FH(;)14 b FK(\000)2018 3202 y FC(A)1997 3249 y
FI(\000)-45 b(!)23 b FK(A)2185 3219 y Fv(0)2229 3249
y FT(from)c(that)i(of)e FI(r)p FH(;)14 b FK(\000)2859
3202 y FC(\031)r Fd(\001)o FC(A)2838 3249 y FI(\000)-23
b(\000)e(!)24 b FK(A)3089 3219 y Fv(0)3112 3249 y FT(.)523
3384 y Fi(Theor)o(em)c(1.)41 b FE(If)22 b FK(\000)34
b FE(consists)22 b(of)h(well-formed)e(clauses)h(and)f(the)h(sequent)f
FI(r)p FH(;)14 b FK(\000)38 b FI(\))26 b FK(G)d FE(is)g(deriv-)523
3484 y(able)o(,)d(then)f(it)i(is)g(derivable)f(without)g(using)f(the)i
FK(\031)s FE(-rule)o(.)523 3611 y(Pr)l(oof)o(.)40 b FT(Since)26
b FK(\000)39 b FT(consists)26 b(of)g(well-formed)e(clauses)i(only)-5
b(,)24 b(all)j FK(\000)12 b FT(')-5 b(s)26 b(in)g(the)g(deduction)e
(consist)523 3703 y(of)29 b(well-formed)e(clauses)i(\(formulae)e(on)h
(the)h(left-hand)f(side)h(are)g(analysed)f(only)g(if)h(the)o(y)g(are)
523 3794 y(selected)e(to)g(be)g(in)h(the)f(stoup-position\).)d(By)j
(Lemma)g(4,)g(we)g(can)g(replace)f(this)i(deduction)d(by)523
3885 y(one)j(in)h(which)g(all)g FK(\031)s FT(-rules)g(are)g(unmo)o(v)n
(able.)d(So)j(we)g(need)g(to)g(consider)e(ho)n(w)i(unmo)o(v)n(able)d
FK(\031)s FT(-)523 3977 y(rules)20 b(can)g(be)g(eliminated.)f(Recall)i
(that)f(unmo)o(v)n(able)e FK(\031)s FT(-rules)i(occur)f(in)h(se)o
(gments)g(of)g(the)g(form)1061 4339 y FG(:)847 4415 y
FF(r)911 4383 y Fp(00)911 4431 y Ff(bs)971 4415 y FN(;)13
b FF(r)1069 4383 y Fp(0)1112 4415 y FF(`)22 b(r)1245
4423 y Ff(ts)1703 4339 y FG(:)1393 4415 y FF(r)1457 4383
y Fp(00)1457 4431 y Ff(bs)1516 4415 y FN(;)13 b FF(r)1614
4383 y Fp(0)1637 4415 y FG(;)g FN(\000)32 b FF(\))21
b FN(G)1908 4424 y Ff(bs)t Fn(;)p Ff(ts)2197 4102 y FF(r)2261
4070 y Fp(00)2261 4117 y Ff(bs)2320 4102 y FN(;)14 b
FF(r)2419 4070 y Fp(0)2462 4102 y FF(`)22 b FN(\031)s
Fz(\001)o FN(s)2638 4111 y Ff(bs)t Fn(;)p Ff(ts)2785
4102 y FF(\031)f FN(t)p 2130 4140 831 4 v 2130 4258 a
FF(r)2194 4227 y Fp(00)2194 4274 y Ff(bs)2254 4258 y
FN(;)13 b FF(r)2352 4227 y Fp(0)2374 4258 y FG(;)g FN(\000)2507
4203 y Fn(\031)r Fk(\001)o Fn(p)p FR(\()p Fn(s)2650 4215
y Fa(bs)s Fc(;)p Fa(ts)2759 4203 y FR(\))2487 4258 y
FF(\000)-21 b(\000)k(\000)f(\000)h(\000)g(\000)d(!)21
b FN(p)p FG(\()p FN(t)p FG(\))2989 4165 y FQ(Ax)p 2130
4297 V 2159 4415 a FF(r)2223 4383 y Fp(00)2223 4431 y
Ff(bs)2283 4415 y FN(;)13 b FF(r)2381 4383 y Fp(0)2403
4415 y FG(;)g FN(\000)2536 4359 y Fn(p)p FR(\()p Fn(s)2621
4371 y Fa(bs)s Fc(;)p Fa(ts)2730 4359 y FR(\))2516 4415
y FF(\000)-29 b(\000)-17 b(\000)g(\000)f(\000)-28 b(!)21
b FN(p)p FG(\()p FN(t)p FG(\))2989 4313 y FN(\031)p 847
4454 2085 4 v 1338 4572 a FF(r)1402 4540 y Fp(00)1402
4587 y Ff(bs)1461 4572 y FN(;)13 b FF(r)1559 4540 y Fp(0)1581
4572 y FG(;)h FN(\000)1714 4516 y Fp(r)1765 4524 y Fa(ts)1809
4516 y Fn(=G)1888 4528 y Fa(bs)s Fc(;)p Fa(ts)1998 4516
y Fp(\033)p Fn(p)p FR(\()p Fn(s)2131 4528 y Fa(bs)s Fc(;)p
Fa(ts)2240 4516 y FR(\))1694 4572 y FF(\000)-33 b(\000)-18
b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
(\000)-33 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))2973 4470
y FF(\033)3033 4478 y Fn(L)1810 4663 y FG(:)473 b FF(g)2409
4663 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
2409 4663 a FB(N)2409 4663 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
2409 4663 a FF(8)p
1260 4702 1259 4 v 1260 4824 a(r)1324 4792 y Fp(0)1347
4824 y FG(;)13 b FN(\000)1521 4769 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
1521 4769 a Fj(N)1521
4769 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
1521 4769 a Fn(as:)p Fp(8)p Ff(Xs)t Fn(:)p Fp(r)1779
4780 y Fa(Xs)1843 4769 y Fn(=G)1922 4780 y Fa(as)s Fc(;)p
Fa(Xs)2053 4769 y Fp(\033)p Fn(p)p FR(\()p Fn(s)2186
4780 y Fa(as)s Fc(;)p Fa(Xs)2317 4769 y FR(\))1460 4824
y FF(\000)-27 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
(\000)h(\000)g(\000)f(\000)h(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
(\000)f(\000)-26 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))p
1260 4862 1259 4 v 1678 4938 a FF(r)1742 4907 y Fp(0)1764
4938 y FG(;)13 b FN(\000)32 b FF(\))21 b FN(p)p FG(\()p
FN(t)p FG(\))2560 4887 y FQ(Sel)3318 4523 y(\(4\))p eop
end
%%Page: 13 13
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448 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
936 448 a FD(N)936 448 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
936 448 a FT(-quanti\002er)f(introduces)
g(the)i(names)g Fr(bs)31 b FT(and)23 b(the)h FI(8)p FT(-quanti\002ers)e
(replace)h(the)h(v)n(ari-)523 548 y(ables)g Fr(Xs)31
b FT(with)24 b(the)g(terms)g Fr(ts)7 b FT(.)24 b(W)-7
b(e)25 b(indicate)f(this)g(by)g(using)f(the)h(notation)f
FK(G)2831 560 y Fs(as)5 b FC(;)p Fs(Xs)3030 548 y FT(and)24
b FK(G)3240 560 y Fs(bs)t FC(;)p Fs(ts)3384 548 y FT(.)523
648 y(The)19 b(freshness)f(constraints)g FI(r)1448 617
y Fv(00)1448 671 y Fs(bs)1537 648 y FT(stand)g(for)h(the)g(constraints)
f(introduced)f(by)h(the)3006 648 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
3006 648 a FD(N)3006
648 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
3006 648 a FT(-quanti\002ers,)523 747 y(that)27
b(is)h FK(b)36 b FH(#)g FK(F)12 b(V)19 b FH(\()p FK(t)p
FH(\))28 b FT(for)f(each)g FK(b)g FT(in)h Fr(bs)7 b FT(.)27
b(Let)h FK(\033)j FT(be)c(the)g(substitution)f(of)h(the)h(terms)f
Fr(ts)34 b FT(for)26 b(the)523 847 y(v)n(ariables)19
b Fr(Xs)7 b FT(,)20 b(that)h(is)g(the)f(terms)g(introduced)e(by)i(the)g
FI(8)p FT(-quanti\002ers.)648 939 y(Belo)n(w)31 b(we)g(gi)n(v)o(e)g(a)g
(deduction)f(without)g(the)h FK(\031)s FT(-rule)g(where)f(the)i
Fr(bs)38 b FT(and)31 b Fr(ts)38 b FT(are)31 b(suitably)523
1030 y(changed.)17 b(F)o(or)i(this)h(we)g(choose)e(\002rst)i(some)f
(fresh)g(names)g Fr(cs)27 b FT(with)19 b(the)h(pro)o(viso)d(that)j
FK(\031)s Fu(\001)p Fr(cs)30 b FH(=)22 b FK(cs)p FT(,)523
1121 y(which)29 b(means)h(the)o(y)f(are)g(unaf)n(fected)f(by)h(the)h
(permutation)d(introduced)h(by)h(the)h FK(\031)s FT(-rule)f(\(such)523
1213 y(fresh)15 b(names)h(al)o(w)o(ays)g(e)o(xist\).)f(From)h(the)f
(well-formedness)f(of)i(the)f(clause)h(in)g(the)g(stoup-position,)523
1304 y(we)21 b(kno)n(w)e(that)h(there)g(is)h(a)f(substitution)g
FK(\033)1768 1274 y Fv(0)1812 1304 y FT(and)g(a)h(permutation)d
FK(\031)2483 1274 y Fv(0)2527 1304 y FT(such)i(that)1296
1466 y Ft(cs)28 b FG(#)22 b Ft(Xs)6 b FN(;)13 b FF(r)1668
1474 y Ff(Xs)1765 1466 y FF(`)21 b FN(\033)1880 1434
y Fp(0)1902 1466 y FG(\()p FN(p)p FG(\()p FN(s)2037 1474
y Ff(cs)t Fn(;)p Ff(Xs)2186 1466 y FG(\)\))g FF(\031)g
FN(p)p FG(\()p FN(s)2453 1475 y Ff(bs)s Fn(;)p Ff(Xs)2602
1466 y FG(\))1296 1557 y Ft(cs)28 b FG(#)22 b Ft(Xs)6
b FN(;)13 b FF(r)1668 1565 y Ff(Xs)1765 1557 y FF(`)21
b FN(\033)1880 1525 y Fp(0)1902 1557 y FG(\()p FN(\031)1979
1525 y Fp(0)2001 1557 y Fz(\001)p FN(G)2086 1565 y Ff(cs)t
Fn(;)p Ff(Xs)2235 1557 y FG(\))h FF(\031)f FN(G)2428
1566 y Ff(bs)t Fn(;)p Ff(Xs)3318 1512 y FQ(\(5\))523
1723 y FT(hold)g(where)h(we)g(use)h(the)f(short-hand)e(notation)h
Fr(cs)33 b FH(#)27 b Fr(Xs)j FT(to)22 b(refer)f(the)i(sets)g(of)f
(freshness)f(con-)523 1814 y(straints)h FK(c)821 1826
y FC(i)875 1814 y FH(#)k FK(X)1039 1826 y FS(1)1077 1814
y FK(;)14 b(:)g(:)g(:)f(;)h(c)1297 1826 y FC(i)1351 1814
y FH(#)26 b FK(X)1515 1826 y FC(n)1583 1814 y FT(for)21
b(all)h(names)g FK(c)2078 1826 y FC(i)2128 1814 y FT(in)g
Fr(cs)7 b FT(.)22 b(By)g(Lemma)f(1)p FE(\(ii\))p FT(,)h
FI(\031)g FT(is)g(preserv)o(ed)523 1905 y(under)d(substitutions,)g(so)i
(we)f(can)g(infer)g(from)f(\(5\))g(that)1247 2064 y Ft(cs)28
b FG(#)22 b Ft(Xs)27 b FF(`)22 b FN(\033)d FF(\016)f
FN(\033)1777 2032 y Fp(0)1799 2064 y FG(\()p FN(p)p FG(\()p
FN(s)1934 2072 y Ff(cs)s Fn(;)p Ff(Xs)2082 2064 y FG(\)\))j
FF(\031)g FN(\033)s FG(\()p FN(p)p FG(\()p FN(s)2426
2073 y Ff(bs)s Fn(;)p Ff(Xs)2575 2064 y FG(\)\))1247
2155 y Ft(cs)28 b FG(#)22 b Ft(Xs)27 b FF(`)22 b FN(\033)d
FF(\016)f FN(\033)1777 2124 y Fp(0)1799 2155 y FG(\()p
FN(\031)1876 2124 y Fp(0)1898 2155 y Fz(\001)o FN(G)1982
2163 y Ff(cs)t Fn(;)p Ff(Xs)2132 2155 y FG(\))j FF(\031)g
FN(\033)s FG(\()p FN(G)2401 2164 y Ff(bs)t Fn(;)p Ff(Xs)2550
2155 y FG(\))3318 2110 y FQ(\(6\))523 2326 y FT(hold)16
b(where)f(the)i(right-hand)d(sides)j(are)f FK(p)p FH(\()p
FK(s)1803 2338 y Fs(bs)t FC(;)p Fs(ts)1947 2326 y FH(\))h
FT(and)f FK(G)2198 2338 y Fs(bs)5 b FC(;)p Fs(ts)2342
2326 y FT(,)17 b(respecti)n(v)o(ely)-5 b(.)14 b(Note)j(that)f(the)g
FI(r)3317 2338 y Fs(Xs)523 2417 y FT(\223v)n(anish\224)j(because)h(we)g
(ha)n(v)o(e)g(that)g Fr(cs)30 b FH(#)23 b Fr(Xs)30 b
FI(`)23 b FK(\033)s FH(\()p FI(r)2098 2429 y Fs(Xs)2186
2417 y FH(\))p FT(.)e(From)e(\(6\))h(we)g(can)g(further)f(infer)h(that)
1257 2580 y Ft(cs)29 b FG(#)21 b Ft(Xs)28 b FF(`)21 b
FN(\031)s Fz(\001)o FN(\033)f FF(\016)d FN(\033)1858
2544 y Fp(0)1880 2580 y FG(\()p FN(p)p FG(\()p FN(s)2015
2588 y Ff(cs)s Fn(;)p Ff(Xs)2164 2580 y FG(\)\))k FF(\031)g
FN(\031)s Fz(\001)o FG(\()p FN(p)p FG(\()p FN(s)2532
2589 y Ff(bs)s Fn(;)p Ff(ts)2657 2580 y FG(\)\))601 b
FQ(\(7\))1257 2696 y Ft(cs)29 b FG(#)21 b Ft(Xs)28 b
FF(`)21 b FN(\031)s Fz(\001)o FN(\033)f FF(\016)d FN(\033)1858
2660 y Fp(0)1880 2696 y FG(\()p FN(\031)1957 2660 y Fp(0)1979
2696 y Fz(\001)p FN(G)2064 2704 y Ff(cs)t Fn(;)p Ff(Xs)2213
2696 y FG(\))k FF(\031)g FN(\031)s Fz(\001)p FG(\()p
FN(G)2507 2705 y Ff(bs)t Fn(;)p Ff(ts)2633 2696 y FG(\))655
b FQ(\(8\))523 2860 y FT(hold)25 b(by)g(Lemma)g(1)p FE(\(i\))g
FT(asserting)g(that)h FI(\031)f FT(is)i(preserv)o(ed)d(under)g
(permutations.)f(Recall)j(that)g(we)523 2951 y(chosen)20
b(the)g Fr(cs)28 b FT(so)21 b(that)g FK(\031)j FT(does)d(not)f(af)n
(fect)g(them.)g(So)g(if)h(we)g(apply)f(the)g(substitution)g
FK(\033)i FI(\016)c FK(\033)3240 2921 y Fv(0)3285 2951
y FT(and)523 3042 y(the)i(permutation)e FK(\031)25 b
FT(to)20 b(the)g(left-hand)f(side)h(of)g(\(7\))g(we)h(ha)n(v)o(e)e
FK(\031)s Fu(\001)p FK(\033)j FI(\016)c FK(\033)2577
3012 y Fv(0)2601 3042 y FH(\()p FK(p)p FH(\()p FK(s)2746
3054 y Fs(cs)5 b FC(;)p Fs(Xs)2917 3042 y FH(\)\))24
b(=)f FK(p)p FH(\()p FK(s)3206 3054 y Fs(cs)t FC(;)p
Fs(ts)3346 3038 y Fl(0)3373 3042 y FH(\))523 3134 y FT(for)f(some)g
(terms)h Fr(ts)1117 3103 y Fv(0)1140 3134 y FT(.)g(Moreo)o(v)o(er)d(we)
j(ha)n(v)o(e)f Fr(cs)34 b FH(#)28 b Fr(Xs)33 b FI(`)27
b FK(s)2278 3146 y Fs(cs)5 b FC(;)p Fs(ts)2418 3129 y
Fl(0)2472 3134 y FI(\031)27 b FK(\031)s Fu(\001)p FK(s)2680
3146 y Fs(bs)t FC(;)p Fs(ts)2847 3134 y FT(which)22 b(means)g(we)523
3225 y(can)e(replace)f(in)g(the)h(deduction)e(\(4\))h
FK(\031)s Fu(\001)p FK(s)1706 3237 y Fs(bs)t FC(;)p Fs(ts)1870
3225 y FT(by)h FK(s)2013 3237 y Fs(cs)t FC(;)p Fs(ts)2153
3221 y Fl(0)2200 3225 y FT(and)f(get)h(by)f(transiti)n(vity)g(of)g
FI(\031)h FT(a)g(correct)523 3316 y(instance)g(of)g(the)g(axiom.)f
(Thus)h(we)g(can)g(form)g(the)g(deduction:)1056 3479
y FQ(.)1056 3513 y(.)1056 3546 y(.)829 3623 y FF(r)893
3591 y Fp(00)893 3636 y Ff(cs)952 3623 y FN(;)14 b FF(r)1051
3591 y Fp(0)1094 3623 y FF(`)22 b(r)1227 3635 y Ff(ts)1274
3620 y Fl(0)1720 3479 y FQ(.)1720 3513 y(.)1720 3546
y(.)1397 3623 y FF(r)1461 3591 y Fp(00)1461 3636 y Ff(cs)1520
3623 y FN(;)14 b FF(r)1619 3591 y Fp(0)1641 3623 y FG(;)f
FN(\000)32 b FF(\))21 b FN(G)1912 3635 y Ff(cs)t Fn(;)p
Ff(ts)2034 3621 y Fl(0)2230 3457 y FF(r)2294 3425 y Fp(00)2294
3470 y Ff(cs)2353 3457 y FN(;)14 b FF(r)2452 3425 y Fp(0)2495
3457 y FF(`)22 b FN(s)2600 3469 y Ff(cs)t Fn(;)p Ff(ts)2721
3455 y Fl(0)2769 3457 y FF(\031)f FN(t)p 2157 3498 795
4 v 2157 3623 a FF(r)2221 3591 y Fp(00)2221 3636 y Ff(cs)2280
3623 y FN(;)13 b FF(r)2378 3591 y Fp(0)2401 3623 y FG(;)g
FN(\000)2533 3560 y Fn(p)p FR(\()p Fn(s)2618 3579 y Fa(cs)s
Fc(;)p Fa(ts)2723 3567 y Fl(0)2750 3560 y FR(\))2514
3623 y FF(\000)-39 b(\000)-18 b(\000)h(\000)g(\000)f(\000)-38
b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))2980 3523 y FQ(Ax)p
829 3664 2123 4 v 1305 3789 a FF(r)1369 3757 y Fp(00)1369
3802 y Ff(cs)1428 3789 y FN(;)13 b FF(r)1526 3757 y Fp(0)1549
3789 y FG(;)g FN(\000)1681 3726 y Fp(r)1732 3744 y Fa(ts)1772
3732 y Fl(0)1799 3726 y Fn(=G)1878 3744 y Fa(cs)s Fc(;)p
Fa(ts)1983 3732 y Fl(0)2010 3726 y Fp(\033)p Fn(p)p FR(\()p
Fn(s)2143 3744 y Fa(cs)s Fc(;)p Fa(ts)2248 3732 y Fl(0)2274
3726 y FR(\))1662 3789 y FF(\000)-22 b(\000)-17 b(\000)f(\000)h(\000)g
(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)d(!)21
b FN(p)p FG(\()p FN(t)p FG(\))2993 3680 y FF(\033)3053
3689 y Fn(L)1811 3880 y FG(:)472 b FF(g)2410 3880 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
2410
3880 a FB(N)2410 3880 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
2410 3880 a FF(8)p 1261 3919
1259 4 v 1261 4041 a(r)1325 4009 y Fp(0)1348 4041 y FG(;)13
b FN(\000)1522 3985 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
1522 3985 a Fj(N)1522 3985 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
1522
3985 a Fn(as:)p Fp(8)p Ff(Xs)t Fn(:)p Fp(r)1780 3996
y Fa(Xs)1844 3985 y Fn(=G)1923 3996 y Fa(as)s Fc(;)p
Fa(Xs)2054 3985 y Fp(\033)p Fn(p)p FR(\()p Fn(s)2187
3996 y Fa(as)s Fc(;)p Fa(Xs)2318 3985 y FR(\))1461 4041
y FF(\000)-28 b(\000)-17 b(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
(\000)f(\000)-26 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))p
1261 4079 1259 4 v 1678 4155 a FF(r)1742 4123 y Fp(0)1765
4155 y FG(;)13 b FN(\000)32 b FF(\))21 b FN(p)p FG(\()p
FN(t)p FG(\))2561 4104 y FQ(Sel)523 4327 y FT(without)k(the)g
FK(\031)s FT(-rule.)g(W)-7 b(e)26 b(still)h(need)e(to)g(ensure)g(that)h
FI(r)2205 4296 y Fv(00)2205 4347 y Fs(cs)2273 4327 y
FK(;)14 b FI(r)2379 4296 y Fv(0)2403 4327 y FH(;)g FK(\000)44
b FI(\))33 b FK(G)2716 4339 y Fs(cs)t FC(;)p Fs(ts)2856
4322 y Fl(0)2908 4327 y FT(and)25 b FI(r)3123 4296 y
Fv(00)3123 4347 y Fs(cs)3192 4327 y FK(;)14 b FI(r)3298
4296 y Fv(0)3354 4327 y FI(`)523 4426 y(r)592 4438 y
Fs(ts)648 4422 y Fl(0)699 4426 y FT(are)23 b(deri)n(v)n(able.)f(The)h
(second)g(sequent)g(is)i(deri)n(v)n(able)d(because)g
Fr(cs)37 b FH(#)29 b Fr(Xs)36 b FI(`)29 b FK(\033)s FH(\()p
FI(r)3125 4438 y Fs(Xs)3214 4426 y FH(\))p FT(.)24 b(F)o(or)523
4526 y(the)j(\002rst)h(sequent)f(we)g(can)g(infer)g(from)f(the)i
(original)e(\(sub\)deduction)e FI(r)2743 4496 y Fv(00)2743
4549 y Fs(bs)2811 4526 y FK(;)14 b FI(r)2917 4496 y Fv(0)2941
4526 y FH(;)g FK(\000)47 b FI(\))36 b FK(G)3260 4538
y Fs(bs)5 b FC(;)p Fs(ts)523 4625 y FT(by)28 b(Corollary)f(1)h(that)h
FI(r)1275 4595 y Fv(00)1275 4649 y Fs(bs)1343 4625 y
FK(;)14 b FI(r)1449 4595 y Fv(0)1473 4625 y FH(;)g FK(\000)49
b FI(\))38 b FK(\031)s Fu(\001)p FK(G)1873 4637 y Fs(bs)5
b FC(;)p Fs(ts)2046 4625 y FT(is)30 b(deri)n(v)n(able)c(\(this)j
(deduction)d(does)i(not)g(in-)523 4725 y(troduce)33 b(an)o(y)h(ne)n(w)h
FK(\031)s FT(-rules\).)f(In)h(\(8\))f(we)h(can)f(pull)h(out)f(the)h
(permutation)e FK(\031)2905 4695 y Fv(0)2964 4725 y FT(and)h(we)h(ha)n
(v)o(e)523 4825 y FK(\031)s Fu(\001)p FK(\033)28 b FI(\016)d
FK(\033)792 4795 y Fv(0)816 4825 y FH(\()p FK(\031)898
4795 y Fv(0)922 4825 y Fu(\001)o FK(G)1013 4837 y Fs(cs)5
b FC(;)p Fs(Xs)1185 4825 y FH(\))40 b(=)f FK(\031)s FH(@)p
FK(\031)1526 4795 y Fv(0)1550 4825 y Fu(\001)o FH(\()p
FK(\033)29 b FI(\016)c FK(\033)1801 4795 y Fv(0)1825
4825 y FH(\()p FK(G)1922 4837 y Fs(cs)5 b FC(;)p Fs(Xs)2094
4825 y FH(\)\))p FT(.)30 b(Therefore)d(applying)g(the)i(substitution)
523 4924 y(to)22 b FK(G)675 4936 y Fs(cs)5 b FC(;)p Fs(Xs)869
4924 y FT(gi)n(v)o(es)21 b FK(\031)s FH(@)p FK(\031)1228
4894 y Fv(0)1252 4924 y Fu(\001)o FH(\()p FK(\033)j FI(\016)19
b FK(\033)1492 4894 y Fv(0)1516 4924 y FH(\()p FK(G)1613
4936 y Fs(cs)5 b FC(;)p Fs(Xs)1785 4924 y FH(\)\))26
b(=)g FK(\031)s FH(@)p FK(\031)2131 4894 y Fv(0)2154
4924 y Fu(\001)p FK(G)2246 4936 y Fs(cs)5 b FC(;)p Fs(ts)2386
4920 y Fl(0)2435 4924 y FT(\(taking)21 b(the)h Fr(ts)2885
4894 y Fv(0)2931 4924 y FT(we)g(introduced)p eop end
%%Page: 14 14
TeXDict begin 14 13 bop 523 448 a FT(for)21 b FK(s)681
460 y Fs(cs)t FC(;)p Fs(ts)820 444 y Fl(0)869 448 y FT(earlier\).)f
(Thus)h(by)f(Lemma)h(3)g(we)h(can)f(sho)n(w)f(that)i
FI(r)2448 418 y Fv(00)2448 469 y Fs(cs)2516 448 y FK(;)14
b FI(r)2622 418 y Fv(0)2646 448 y FH(;)g FK(\000)36 b
FI(\))25 b FK(G)2943 460 y Fs(cs)5 b FC(;)p Fs(ts)3083
444 y Fl(0)3132 448 y FT(is)22 b(deri)n(v-)523 548 y(able.)648
651 y(Each)16 b(transformation)f(decreases)i(the)g(number)e(of)i
FK(\031)s FT(-rules)g(in)h(a)f(deduction)e(by)i(one)g(and)g(thus)523
751 y(by)j(repeated)f(application)f(we)j(will)g(e)n(v)o(entually)d(end)
i(up)g(with)g(a)h FK(\031)s FT(-free)e(proof.)500 b FI(u)-55
b(t)523 940 y FT(W)-7 b(e)28 b(ha)n(v)o(e)e(sho)n(wn)g(that)h(when)f
(all)i(the)f(formulas)e(in)i FK(\000)39 b FT(are)27 b(well-formed,)d(e)
n(v)o(ery)i(deduction)f(of)523 1039 y FK(\000)57 b FI(\))45
b FK(G)33 b FT(containing)d FK(\031)s FT(-rules)i(can)g(be)g(replaced)f
(by)h(one)g(without)f FK(\031)s FT(-rules.)h(Consequently)-5
b(,)523 1139 y(nominal)19 b(uni\002cation)g(is)i(suf)n(\002cient)f(for)
f(e)o(x)o(ecuting)f(well-formed)g FK(\013)p FT(Prolog)o(-programs.)523
1423 y FL(4)99 b(V)-10 b(eri\002cation)26 b(of)f(W)-6
b(ell-F)n(ormedness)24 b(Using)g(Nominal)h(Matching)523
1632 y FT(In)19 b(this)h(section)f(we)g(consider)f(the)h(question)f(of)
h(ho)n(w)g(to)g(v)o(erify)f(the)h(well-formedness)e(condition)523
1724 y(gi)n(v)o(en)k(in)g(De\002nition)g(1.)h(F)o(or)f(a)i(clause)1742
1724 y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
1742 1724 a FD(N)1742 1724 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
1742 1724 a Fr(as)8
b FK(:)p FI(8)p Fr(Xs)e FK(:)p FI(r)2089 1736 y Fs(as)f
FC(;)p Fs(Xs)2264 1724 y FK(=G)2371 1736 y Fs(as)g FC(;)p
Fs(Xs)2546 1724 y FI(\033)o FK(A)2672 1736 y Fs(as)h
FC(;)p Fs(Xs)2848 1724 y FT(,)22 b(we)g(need)f(to)h(\002nd)523
1815 y(a)f(substitution)e FK(\033)24 b FT(and)c(permutation)e
FK(\031)24 b FT(which)19 b(mak)o(e)h(the)g(tw)o(o)h(judgements)541
1987 y Ft(bs)28 b FG(#)21 b Ft(Xs)6 b FN(;)14 b FF(r)913
1995 y Ff(as)t Fn(;)p Ff(Xs)1086 1987 y FF(`)22 b FN(\033)s
FG(\()p FN(A)1290 1996 y Ff(bs)s Fn(;)p Ff(Xs)1438 1987
y FG(\))f FF(\031)g FN(A)1628 1995 y Ff(as)t Fn(;)p Ff(Xs)1866
1987 y FT(and)84 b Ft(bs)28 b FG(#)22 b Ft(Xs)6 b FN(;)13
b FF(r)2443 1995 y Ff(as)t Fn(;)p Ff(Xs)2617 1987 y FF(`)21
b FN(\033)s FG(\()p FN(\031)s Fz(\001)o FN(G)2893 1996
y Ff(bs)t Fn(;)p Ff(Xs)3043 1987 y FG(\))g FF(\031)g
FN(G)3235 1995 y Ff(as)t Fn(;)p Ff(Xs)523 2164 y FT(hold.)f(F)o(or)h
(the)g(\002rst)h(judgement,)d FK(\033)25 b FT(can)c(be)g(found)f(by)h
(nominal)e(matching.)h(But)i(for)e(the)h(second)523 2264
y(judgement,)j(\002nding)i(both)f(substitution)h FK(\033)k
FT(and)c(permutation)e FK(\031)30 b FT(requires)c(solving)f
(\(NP-hard\))523 2363 y(equi)n(v)n(ariant)13 b(matching)h(problems.)g
(This)h(seemingly)f(ne)o(gati)n(v)o(e)f(result)j(should,)e(ho)n(we)n(v)
o(er)m(,)e(be)j(seen)523 2463 y(in)29 b(the)f(conte)o(xt)g(that)g
(well-formedness)f(only)g(needs)h(to)h(be)g(v)o(eri\002ed)e(once)h(per)
g(clause,)g(rather)523 2563 y(than)20 b(repeatedly)e(during)h
(proof-search.)e(Thus,)i(the)h(one-time)f(cost)i(of)e(performing)f
(equi)n(v)n(ariant)523 2662 y(uni\002cation)h(in)i(checking)e
(well-formedness)f(is)j(ne)o(gligible)e(compared)g(to)h(the)h(cost)g
(of)f(perform-)523 2762 y(ing)j(equi)n(v)n(ariant)f(uni\002cation)h
(throughout)d(computation.)i(Furthermore,)f(as)j(can)g(be)g(seen)g
(from)523 2862 y(the)h(e)o(xamples,)e(the)i(number)e(of)i(names)f(in)h
(a)g(clause)g(is)h(usually)e(small.)h(T)-7 b(aking)24
b(the)h(follo)n(wing)523 2961 y(proposition)18 b(\(whose)i(proof)e(we)j
(omit\))523 3154 y Fi(Pr)o(oposition)e(1.)41 b FE(If)20
b FK(G)1193 3166 y Fs(bs)t FC(;)p Fs(Xs)1385 3154 y FE(equivariantly)e
(matc)o(hes)h(with)i FK(G)2368 3166 y Fs(as)5 b FC(;)p
Fs(Xs)2543 3154 y FE(,)20 b(then)f(a)h(matc)o(hing)e(e)n(xists)j(in)523
3254 y(whic)o(h)f(the)g(permutation)f FK(\031)24 b FE(consists)c(of)h
(swappings)e FH(\()p FK(a)2177 3266 y FC(i)2219 3254
y FK(b)2255 3266 y FC(i)2282 3254 y FH(\))i FE(only)-5
b(.)523 3442 y FT(into)27 b(account,)e(we)j(can)f(just)g(enumerate)f
(all)h(possible)g(cases)h(\()p FH(2)2458 3412 y FC(n)2530
3442 y FT(gi)n(v)o(en)e FK(n)h FT(names\))f(and)h(solv)o(e)523
3542 y(each)22 b(of)f(the)h(nominal)e(matching)h(problems.)f(If)i(one)f
(problem)f(can)i(be)g(solv)o(ed,)e(then)i(we)g(ha)n(v)o(e)f(a)523
3642 y FK(\033)j FT(and)c(a)g FK(\031)25 b FT(as)20 b(required)f(by)h
(the)g(condition.)523 3926 y FL(5)99 b(Conclusion)523
4122 y FT(W)-7 b(e)22 b(ha)n(v)o(e)f(sho)n(wn)g(that)g(for)g
(well-formed)e FK(\013)p FT(Prolog)h(programs,)g(all)h(instances)h(of)f
(the)g FK(\031)s FT(-rule)g(can)523 4221 y(be)j(remo)o(v)o(ed)d(from)i
(deductions.)f(As)i(a)h(result,)e(proof)g(search)g(using)g(only)g
(nominal)g(uni\002cation)523 4321 y(is)k(complete)e(for)h(such)g
(programs,)e(which)h(coincides)h(with)g(our)g(e)o(xperimental)d
(results)k(gained)523 4421 y(from)17 b(our)g(implementation)f(of)h
FK(\013)p FT(Prolog)o(.)h(This)g(is)h(a)g(signi\002cant)e(result,)h
(because)f(the)h(alternati)n(v)o(e)523 4520 y(is)j(to)g(use)f(an)g
(NP-hard)f(equi)n(v)n(ariant)f(uni\002cation)h(algorithm)g(for)g(proof)
g(search.)648 4615 y(In)e(order)g(to)h(be)g(well-formed,)d(the)j
FJ(type)p FT(-program)d(gi)n(v)o(en)h(in)i(the)g(Introduction)d(needs)j
(to)g(be)523 4706 y(stated)i(as)h(follo)n(ws)800 4861
y FP(type\(Gamma,lam\(x.M\),arr\(S,T\)\))40 b(:-)1272
4953 y(x)13 b(#)f(Gamma,)f(x#S,)h(x#T,)g(type\([\(x,S\)|Gamma],M,T\).)p
eop end
%%Page: 15 15
TeXDict begin 15 14 bop 523 448 a FT(e)o(xplicitly)24
b(gi)n(ving)g(the)i(freshness)e(constraints)h FJ(x#S)g
FT(and)g FJ(x#T)p FT(.)g(These)g(constraints)g(do)g(not)g(af-)523
548 y(fect)f(the)f(meaning)f(of)h(the)h(program)d(because)i(term)h(v)n
(ariables)e(are)i(e)o(xpected)e(\(by)h(programmer)523
648 y(con)m(v)o(ention\))e(not)i(to)i(appear)e(in)h(types.)f(In)h(f)o
(act,)g(our)g(implementation)d(of)j FK(\013)p FT(Prolog)g(is)h
(strongly)523 747 y(typed)i(and)h(therefore)e(can)i(determine)e
(automatically)h(from)g(type)g(information)f(that)i(lambda-)523
847 y(term)19 b(v)n(ariables)f(can)i(ne)n(v)o(er)d(occur)i(in)g(types.)
g(Thus,)f(our)h(analysis)g(could)g(be)g(made)f(more)h(precise)523
946 y(by)h(taking)f(type)h(information)e(into)i(account.)648
1046 y(Let)i(us)i(brie\003y)e(mention)f(whether)h(our)f(result)i(can)g
(be)f(strengthened.)e(The)j(logic)f(program-)523 1146
y(ming)j(language)f FK(\025)p FT(Prolog)h([7])g(has)h(con)m(vincingly)d
(demonstrated)h(the)i(usefulness)f(of)g(implica-)523
1245 y(tions)f(in)h FK(G)p FT(-formulae)e(\(that)h(is)h(e)o(xtending)d
(logic)i(programming)d(to)k(the)f(setting)h(of)f(Hereditary)523
1345 y(Harrop)j(formulae\).)f(In)h FK(\013)p FT(Prolog)h(we)g(w)o(ould)
g(lik)o(e)g(to)g(allo)n(w)g(implications)f(in)i FK(G)p
FT(-formulae)d(as)523 1445 y(well.)20 b(Whether)f(our)f(result)i(e)o
(xtends)e(to)i(such)f(formulae)f(is)i(still)h(open.)d(It)i(seems)g
(that)f(our)g(de\002ni-)523 1544 y(tion)f(of)f FK(G)p
FT(-formulae)f(can)i(be)g(e)o(xtended)e(to)i(include)f(e)o(xistential)g
(and)h(uni)n(v)o(ersal)e(formulae.)g(Ho)n(w-)523 1644
y(e)n(v)o(er)m(,)f(our)h(pro)o(ving)e(technique)h(for)h(sho)n(wing)f
(this)i(w)o(ould)f(require)f(some)h(subtle)h(modi\002cations\227)523
1743 y(for)k(e)o(xample)e(we)j(w)o(ould)e(need)h(to)g(de\002ne)f(when)h
(tw)o(o)g(formulae)f(with)h(quanti\002ers)f(are)h FI(\031)p
FT(-equal,)523 1843 y(which)j(is)i(non-tri)n(vial.)d(Ho)n(we)n(v)o(er)m
(,)f(we)j(e)o(xpect)f(that)h(this)h(can)e(be)h(done.)e(What)j(is)f
(impossible)g(is)523 1943 y(to)20 b(allo)n(w)871 1943
y
currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
871 1943 a FD(N)871 1943 y
currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
neg exch translate
871 1943 a FT(-quanti\002ers)f(in)h
(goal-formulae.)d(Such)j(formulae)e(really)i(need)f(equi)n(v)n(ariant)f
(uni\002ca-)523 2042 y(tion.)523 2150 y FO(Ackno)o(wledgements:)28
b FQ(This)14 b(research)h(w)o(as)g(supported)h(by)f(a)f(fello)n(wship)h
(for)f(Urban)h(from)f(the)h(Ale)o(xander)o(-)523 2249
y(v)o(on-Humboldt)20 b(foundation.)523 2515 y FL(Refer)n(ences)523
2706 y FQ(1.)42 b(J.)22 b(Chene)o(y)-5 b(.)41 b(The)22
b(comple)o(xity)i(of)f(equi)n(v)n(ariant)h(uni\002cation.)40
b(In)23 b FM(Pr)m(oc.)f(of)h(International)h(Colloquium)621
2797 y(on)19 b(A)o(utomata,)f(Langua)o(g)o(es)k(and)d(Pr)m(o)o(gr)o
(amming)p FQ(,)h(v)o(olume)f(3142)h(of)f FM(LNCS)p FQ(,)f(pages)i
(332\226344,)h(2004.)523 2889 y(2.)42 b(J.)18 b(Chene)o(y)-5
b(.)28 b FM(Nominal)19 b(Lo)o(gic)g(Pr)m(o)o(gr)o(amming)p
FQ(.)28 b(PhD)18 b(thesis,)h(Cornell)g(Uni)n(v)o(ersity)-5
b(,)19 b(Ithaca,)g(NY)-10 b(,)18 b(2004.)523 2980 y(3.)42
b(J.)18 b(Chene)o(y)i(and)g(C.)e(Urban.)28 b(Alpha-prolog:)20
b(A)f(logic)h(programming)g(language)h(with)d(names,)i(binding,)621
3071 y(and)e FN(\013)p FQ(-equi)n(v)n(alence.)26 b(In)17
b(B.)g(Demoen)i(and)f(V)-10 b(.)17 b(Lifschitz,)g(editors,)g
FM(Pr)m(oc.)g(of)h(International)g(Confer)m(ence)621
3163 y(on)h(Lo)o(gic)g(Pr)m(o)o(gr)o(amming)p FQ(,)h(v)o(olume)f(3132)h
(of)f FM(LNCS)p FQ(,)f(pages)i(269\226283,)h(2004.)523
3254 y(4.)42 b(R.)14 b(Dyckhof)n(f)j(and)f(L.)e(Pinto.)19
b(Proof)c(Search)g(in)g(Constructi)n(v)o(e)h(Logic.)j(In)d(S.)e(Barry)h
(Cooper)h(and)g(John)g(K.)621 3345 y(T)m(russ,)28 b(editors,)g
FM(Pr)m(oc.)g(of)h(the)f(Lo)o(gic)h(Colloquium)g(1997)p
FQ(,)g(v)o(olume)g(258)h(of)e FM(London)i(Mathematical)621
3437 y(Society)19 b(Lectur)m(e)g(Note)g(Series)p FQ(,)g(pages)h
(53\22665.)g(Cambridge)g(Uni)n(v)o(ersity)f(Press,)f(1997.)523
3528 y(5.)42 b(M.)25 b(J.)h(Gabbay)h(and)f(J.)g(Chene)o(y)-5
b(.)50 b(A)25 b(proof)i(theory)f(for)g(nominal)h(logic.)49
b(In)26 b FM(Pr)m(oc.)f(of)h(Annual)g(IEEE)621 3619 y(Symposium)20
b(on)f(Lo)o(gic)g(in)g(Computer)h(Science)p FQ(,)f(pages)h(139\226148,)
h(2004.)523 3710 y(6.)42 b(J.)18 b(C.)g(Mitchell.)27
b FM(Concepts)20 b(in)f(Pr)m(o)o(gr)o(amming)g(Langua)o(g)o(es)p
FQ(.)29 b(CUP)18 b(Press,)g(2003.)523 3802 y(7.)42 b(G.)22
b(Nadathur)i(and)f(D.)g(Miller)l(.)39 b(Higher)o(-order)23
b(logic)g(programming.)42 b(In)23 b(D.)f(M.)h(Gabbay)-5
b(,)24 b(C.)e(J.)g(Hog-)621 3893 y(ger)m(,)28 b(and)g(J.)g(A.)g
(Robinson,)h(editors,)f FM(Handbook)i(of)e(Lo)o(gics)h(for)f
(Arti\002cial)f(Intellig)o(ence)i(and)g(Lo)o(gic)621
3984 y(Pr)m(o)o(gr)o(amming)p FQ(,)19 b(v)o(olume)g(5,)g(pages)h
(499\226590.)h(Clarendon)f(Press,)e(1998.)523 4076 y(8.)42
b(A.)28 b(M.)i(Pitts.)59 b(Nominal)29 b(logic,)g(a)h(\002rst)e(order)i
(theory)g(of)f(names)h(and)g(binding.)62 b FM(Information)30
b(and)621 4167 y(Computation)p FQ(,)19 b(186:165\226193,)j(2003.)523
4258 y(9.)42 b(C.)16 b(Urban,)h(A.)f(M.)g(Pitts,)g(and)h(M.)g(J.)f
(Gabbay)-5 b(.)24 b(Nominal)17 b(uni\002cation.)22 b
FM(Theor)m(etical)c(Computer)f(Science)p FQ(,)621 4350
y(323\(1-2\):473\226497,)22 b(2004.)p eop end
%%Trailer
userdict /end-hook known{end-hook}if
%%EOF