Publications/alpha-tlca.ps
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     2 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software
       
     3 %%Title: alpha-tlca.dvi
       
     4 %%Pages: 15
       
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   389 %%EndFont 
       
   390 %%BeginFont: CMR5
       
   391 %!PS-AdobeFont-1.1: CMR5 1.00B
       
   392 %%CreationDate: 1992 Feb 19 19:55:02
       
   393 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   394 11 dict begin
       
   395 /FontInfo 7 dict dup begin
       
   396 /version (1.00B) readonly def
       
   397 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   398 /FullName (CMR5) readonly def
       
   399 /FamilyName (Computer Modern) readonly def
       
   400 /Weight (Medium) readonly def
       
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   403 end readonly def
       
   404 /FontName /CMR5 def
       
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   414 currentdict end
       
   415 currentfile eexec
       
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   459 %%EndFont 
       
   460 %%BeginFont: CMMI5
       
   461 %!PS-AdobeFont-1.1: CMMI5 1.100
       
   462 %%CreationDate: 1996 Aug 02 08:21:10
       
   463 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   464 11 dict begin
       
   465 /FontInfo 7 dict dup begin
       
   466 /version (1.100) readonly def
       
   467 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   468 /FullName (CMMI5) readonly def
       
   469 /FamilyName (Computer Modern) readonly def
       
   470 /Weight (Medium) readonly def
       
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   473 end readonly def
       
   474 /FontName /CMMI5 def
       
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   477 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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   485 currentdict end
       
   486 currentfile eexec
       
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   543 %%EndFont 
       
   544 %%BeginFont: CMBSY7
       
   545 %!PS-AdobeFont-1.1: CMBSY7 001.000
       
   546 %%CreationDate: 1992 Oct 22 12:18:11
       
   547 % Computer Modern fonts were designed by Donald E. Knuth.
       
   548 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   549 11 dict begin
       
   550 /FontInfo 7 dict dup begin
       
   551 /version (001.000) readonly def
       
   552 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   553 /FullName (CMBSY7) readonly def
       
   554 /FamilyName (Computer Modern) readonly def
       
   555 /Weight (Bold) readonly def
       
   556 /ItalicAngle -14.035 def
       
   557 /isFixedPitch false def
       
   558 end readonly def
       
   559 /FontName /CMBSY7 def
       
   560 /PaintType 0 def
       
   561 /FontType 1 def
       
   562 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
   563 /Encoding 256 array
       
   564 0 1 255 {1 index exch /.notdef put} for
       
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   566 readonly def
       
   567 /FontBBox{0 -927 1542 750}readonly def
       
   568 /UniqueID 5032008 def
       
   569 currentdict end
       
   570 currentfile eexec
       
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   611 cleartomark
       
   612 %%EndFont 
       
   613 %%BeginFont: CMTT8
       
   614 %!PS-AdobeFont-1.1: CMTT8 1.0
       
   615 %%CreationDate: 1991 Aug 20 16:46:05
       
   616 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   617 11 dict begin
       
   618 /FontInfo 7 dict dup begin
       
   619 /version (1.0) readonly def
       
   620 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   621 /FullName (CMTT8) readonly def
       
   622 /FamilyName (Computer Modern) readonly def
       
   623 /Weight (Medium) readonly def
       
   624 /ItalicAngle 0 def
       
   625 /isFixedPitch true def
       
   626 end readonly def
       
   627 /FontName /CMTT8 def
       
   628 /PaintType 0 def
       
   629 /FontType 1 def
       
   630 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
   631 /Encoding 256 array
       
   632 0 1 255 {1 index exch /.notdef put} for
       
   633 dup 108 /l put
       
   634 dup 118 /v put
       
   635 readonly def
       
   636 /FontBBox{-5 -232 545 699}readonly def
       
   637 /UniqueID 5000830 def
       
   638 currentdict end
       
   639 currentfile eexec
       
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   670 EA9ABE3BB3E35DDCCE817CA242E9F1E5F8B0F3F9D71F6B2D1B4367E5E16705C5
       
   671 3DF729ADEA1CB20FC61E742C3130769F9BD97AE48C672A691664FD17B3B1720B
       
   672 4AF8CD579A8584A6464966DA6D22F8B3DE2DB9C3585F2F6F7990855831D925D7
       
   673 19E2E27DF9730D560CB7A933F6A9922C3A5489EA7FC9CC2FCB4BDA7331110A27
       
   674 7326B81B798C6585ACDBBAC073D6C161B8228111207EB6309FD058D574635EE4
       
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   685 cleartomark
       
   686 %%EndFont 
       
   687 %%BeginFont: MSBM7
       
   688 %!PS-AdobeFont-1.1: MSBM7 2.1
       
   689 %%CreationDate: 1992 Oct 17 08:30:50
       
   690 % Math Symbol fonts were designed by the American Mathematical Society.
       
   691 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   692 11 dict begin
       
   693 /FontInfo 7 dict dup begin
       
   694 /version (2.1) readonly def
       
   695 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   696 /FullName (MSBM7) readonly def
       
   697 /FamilyName (Euler) readonly def
       
   698 /Weight (Medium) readonly def
       
   699 /ItalicAngle 0 def
       
   700 /isFixedPitch false def
       
   701 end readonly def
       
   702 /FontName /MSBM7 def
       
   703 /PaintType 0 def
       
   704 /FontType 1 def
       
   705 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
   706 /Encoding 256 array
       
   707 0 1 255 {1 index exch /.notdef put} for
       
   708 dup 63 /emptyset put
       
   709 readonly def
       
   710 /FontBBox{0 -504 2615 1004}readonly def
       
   711 /UniqueID 5032014 def
       
   712 currentdict end
       
   713 currentfile eexec
       
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   715 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
       
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   721 D10F04E2CB7C8461B10646CD63AFEB7608468CA0FCFC4D3458FB43D22879B515
       
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   737 6B2CC834894FDCA891643D9417369458A630FD498794823FFA55705315F0687E
       
   738 7592A5DFC8B8D6FE2F3C64B4A4F9D37F5F2200BAA277F2E0BA8E5A84ABC10319
       
   739 7BA47E1E8D80405541FC64FF71B84C0010920AAA4DD4BEC0D71E447A356CE717
       
   740 6FCD3B21A07E2B32DF50D54BDBD96C634197E502FCA75F122E91314F241D6D4F
       
   741 C28E2867440D324E80504245CDF43A7106CFF64E8BA7809A292EF017796AE18E
       
   742 F2CD78CCF84F82685BB3A147E4A5B7350541966264D191C1BEC22F6F0FF82DA8
       
   743 1D43CD551DF0F9CCC2142D03BE81DFBE02CA779D7E6E51CF2F1D7D48A788F42C
       
   744 908AF42E7A50C5BC65AE2FDDBA97B01E600D24882D98FFEA0EC3807B36688149
       
   745 C199CF15A74FA8DBB731CA8FD16C13491E39309AC7C8C47FEF5AC2CE02A2652A
       
   746 7416CBB2486D61DA08323C6BB1BCE4D3CC421DEB14276C7F1294CC3B5BA2BAFA
       
   747 C9053BE2615B7F28FF7418CAC67326884A2CC3402BFD71495D6BB7EFC64450D1
       
   748 C94E41977843140A6FD0340C39C88E86CC0C058B87234810E1E900ADFB99C232
       
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   758 cleartomark
       
   759 %%EndFont 
       
   760 %%BeginFont: CMMIB10
       
   761 %!PS-AdobeFont-1.1: CMMIB10 1.100
       
   762 %%CreationDate: 1996 Jul 23 07:54:00
       
   763 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   764 11 dict begin
       
   765 /FontInfo 7 dict dup begin
       
   766 /version (1.100) readonly def
       
   767 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   768 /FullName (CMMIB10) readonly def
       
   769 /FamilyName (Computer Modern) readonly def
       
   770 /Weight (Bold) readonly def
       
   771 /ItalicAngle -14.04 def
       
   772 /isFixedPitch false def
       
   773 end readonly def
       
   774 /FontName /CMMIB10 def
       
   775 /PaintType 0 def
       
   776 /FontType 1 def
       
   777 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
   778 /Encoding 256 array
       
   779 0 1 255 {1 index exch /.notdef put} for
       
   780 dup 25 /pi put
       
   781 readonly def
       
   782 /FontBBox{-15 -250 1216 750}readonly def
       
   783 /UniqueID 5087392 def
       
   784 currentdict end
       
   785 currentfile eexec
       
   786 D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE
       
   787 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B
       
   788 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470
       
   789 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B
       
   790 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE
       
   791 D919C2DDD26BDC0D99398B9F4D004B836D34E88C25F6CE738846C8E2E59A2BCF
       
   792 4ACF80A26D78872E9343A0537BC3BD7715F32ACD958D5AAED865BFE129278935
       
   793 063A31C2634DE2F9077E0AAAAEB224466B779096D8E3FF0A12AD5157F6603DED
       
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   798 CEE3671644A75F1E696DA906B4C66870DBE87F5B4A176920C078ADBE24F55C09
       
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   800 A5CF4F56C5E39E2726106E9713E638775E606464CD674E5DC25CE9A696A65806
       
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   802 B523A55ACA6A7A2A0193A536E590291ED9D577B527CAD0372E05BFCA1829FED1
       
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   804 B4F4771F80D1F7AAD9024868C30DAD5CF728DB2A71D86D53B0E674996E8C01F7
       
   805 EF97B225A28872F8AD4752A466E078C2B020EB832F237CB9B5631EB2D2EDDB00
       
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   807 5557A602E9E3C204909D84DB988F0BAB914E87CD685C7DA55C5E0B9F0176184F
       
   808 FC39B570873BBF346A0D1DE3942DA05434949A65CE64D8BAB0A091C40F7FF47A
       
   809 4FC57CB4420221C7B3EB8B891044B5FB0227009F0F6028D3F28545E63C975F11
       
   810 6BCCB67C58A544AE706BA5309FE392AA911EEFA38C1AB8172C910B2666991114
       
   811 2EA6274F510B58B9882853EE8CE5A60D55C7A7A75B71EC91C5EFC4C06F478F23
       
   812 D3795EEA771506A0487A1BAEA09569AD4CD5B5679EA872DB15D1A4D375930255
       
   813 1FE5F7444E33451AB0B79E2679C21F5DBF213E4DFDE5244366686676C8C2AB88
       
   814 0F03A53A1D3B534C4217B645D997826F519AF3EDB78AD6597E1FBCDC4D50C91E
       
   815 2DF53CB322E8ADB668FE485B24ACEB8D95E93AF897368CED4D8518EC15936B5C
       
   816 A8105D497DB8DDBAB0D252F285272B073EB50445B665F931590330585800038C
       
   817 ECF547ADA8DDBEE045A224C673F63EF681EE9EF694233772179D2A13C8EB29BC
       
   818 0C5CCB82D33D081C46CBB947AA42E240335622B6E4A20B7E0F7A98CE101F72E7
       
   819 676E3A23CA38A1CF96618C1EE0CF08D7A4BA86ADBA4C84D881DBA6B558AAF4
       
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   828 cleartomark
       
   829 %%EndFont 
       
   830 %%BeginFont: CMSY5
       
   831 %!PS-AdobeFont-1.1: CMSY5 1.0
       
   832 %%CreationDate: 1991 Aug 15 07:21:16
       
   833 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   834 11 dict begin
       
   835 /FontInfo 7 dict dup begin
       
   836 /version (1.0) readonly def
       
   837 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
   838 /FullName (CMSY5) readonly def
       
   839 /FamilyName (Computer Modern) readonly def
       
   840 /Weight (Medium) readonly def
       
   841 /ItalicAngle -14.035 def
       
   842 /isFixedPitch false def
       
   843 end readonly def
       
   844 /FontName /CMSY5 def
       
   845 /PaintType 0 def
       
   846 /FontType 1 def
       
   847 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
   848 /Encoding 256 array
       
   849 0 1 255 {1 index exch /.notdef put} for
       
   850 dup 0 /minus put
       
   851 dup 48 /prime put
       
   852 dup 102 /braceleft put
       
   853 dup 103 /braceright put
       
   854 readonly def
       
   855 /FontBBox{21 -944 1448 791}readonly def
       
   856 /UniqueID 5000815 def
       
   857 currentdict end
       
   858 currentfile eexec
       
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   860 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4
       
   861 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85
       
   862 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A
       
   863 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C
       
   864 F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5
       
   865 283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833
       
   866 94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6
       
   867 801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403
       
   868 BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614
       
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   873 6EB47369D84838AECC135C8E249DA6EC0A0E8881C0D98346182E1B3F29007291
       
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   875 1CEFD3E9FF22ABA54D04A0C14D31DB1221D4AC9A546879C50C891D2C645C0348
       
   876 D8DEE53C5A21BE5B80A96D5C851A8B3557ED7B44CEB5C4BDAFBD84A7524034D0
       
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   878 5C7AC8D81933EC095A71518032570DA435C86BA274CE71FE795D4312979A5682
       
   879 B5FE665E75C19EFF4AA80D2365AD710E2076C2D48291816A4880E89D1D5C1113
       
   880 49F95607D91577CE5D0FD2B524263B727432F90D7E90D168E7A6991289004BE9
       
   881 DCD198EE46889FEC4D0A2DDC4AD566B4A79883F2E4CC93AD1C2523FEF2158505
       
   882 744947864C82EBA6654A0A000A2A88B67FB8346E4169D151BA47C2934D571FA5
       
   883 A78CAAC84AC767D76C77668895067398459F089D20103F80805AD2F23B1F6D2E
       
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   906 % Math Symbol fonts were designed by the American Mathematical Society.
       
   907 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
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   979 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
   980 11 dict begin
       
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  1221 %%BeginFont: TeX-cmex9
       
  1222 %!PS-AdobeFont-1.0: TeX-cmex9 001.001
       
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  1224 %%EndComments
       
  1225 13 dict dup begin
       
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  1228 /FamilyName (TeX\040cmex9) def
       
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  1233 /UnderlineThickness 50 def
       
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  1237 end readonly def
       
  1238 /FontName /TeX-cmex9 def
       
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  1241 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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  1244 dup 40 /parenleft put
       
  1245 dup 56 /eight put
       
  1246 dup 58 /colon put
       
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  1252 currentdict end
       
  1253 currentfile eexec
       
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  1318 cleartomark
       
  1319 %%EndFont 
       
  1320 %%BeginFont: CMSY6
       
  1321 %!PS-AdobeFont-1.1: CMSY6 1.0
       
  1322 %%CreationDate: 1991 Aug 15 07:21:34
       
  1323 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1324 11 dict begin
       
  1325 /FontInfo 7 dict dup begin
       
  1326 /version (1.0) readonly def
       
  1327 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1328 /FullName (CMSY6) readonly def
       
  1329 /FamilyName (Computer Modern) readonly def
       
  1330 /Weight (Medium) readonly def
       
  1331 /ItalicAngle -14.035 def
       
  1332 /isFixedPitch false def
       
  1333 end readonly def
       
  1334 /FontName /CMSY6 def
       
  1335 /PaintType 0 def
       
  1336 /FontType 1 def
       
  1337 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1338 /Encoding 256 array
       
  1339 0 1 255 {1 index exch /.notdef put} for
       
  1340 dup 0 /minus put
       
  1341 dup 15 /bullet put
       
  1342 dup 25 /approxequal put
       
  1343 dup 27 /propersuperset put
       
  1344 dup 41 /arrowdblright put
       
  1345 dup 48 /prime put
       
  1346 dup 56 /universal put
       
  1347 dup 62 /latticetop put
       
  1348 dup 66 /B put
       
  1349 dup 91 /union put
       
  1350 dup 96 /turnstileleft put
       
  1351 dup 102 /braceleft put
       
  1352 dup 103 /braceright put
       
  1353 dup 104 /angbracketleft put
       
  1354 dup 105 /angbracketright put
       
  1355 dup 114 /nabla put
       
  1356 readonly def
       
  1357 /FontBBox{-4 -948 1329 786}readonly def
       
  1358 /UniqueID 5000816 def
       
  1359 currentdict end
       
  1360 currentfile eexec
       
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  1455 cleartomark
       
  1456 %%EndFont 
       
  1457 %%BeginFont: CMTI10
       
  1458 %!PS-AdobeFont-1.1: CMTI10 1.00B
       
  1459 %%CreationDate: 1992 Feb 19 19:56:16
       
  1460 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1461 11 dict begin
       
  1462 /FontInfo 7 dict dup begin
       
  1463 /version (1.00B) readonly def
       
  1464 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1465 /FullName (CMTI10) readonly def
       
  1466 /FamilyName (Computer Modern) readonly def
       
  1467 /Weight (Medium) readonly def
       
  1468 /ItalicAngle -14.04 def
       
  1469 /isFixedPitch false def
       
  1470 end readonly def
       
  1471 /FontName /CMTI10 def
       
  1472 /PaintType 0 def
       
  1473 /FontType 1 def
       
  1474 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1475 /Encoding 256 array
       
  1476 0 1 255 {1 index exch /.notdef put} for
       
  1477 dup 88 /X put
       
  1478 dup 97 /a put
       
  1479 dup 98 /b put
       
  1480 dup 99 /c put
       
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  1483 readonly def
       
  1484 /FontBBox{-163 -250 1146 969}readonly def
       
  1485 /UniqueID 5000828 def
       
  1486 currentdict end
       
  1487 currentfile eexec
       
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  1561 cleartomark
       
  1562 %%EndFont 
       
  1563 %%BeginFont: CMTI9
       
  1564 %!PS-AdobeFont-1.1: CMTI9 1.0
       
  1565 %%CreationDate: 1991 Aug 18 21:08:07
       
  1566 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1567 11 dict begin
       
  1568 /FontInfo 7 dict dup begin
       
  1569 /version (1.0) readonly def
       
  1570 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1571 /FullName (CMTI9) readonly def
       
  1572 /FamilyName (Computer Modern) readonly def
       
  1573 /Weight (Medium) readonly def
       
  1574 /ItalicAngle -14.04 def
       
  1575 /isFixedPitch false def
       
  1576 end readonly def
       
  1577 /FontName /CMTI9 def
       
  1578 /PaintType 0 def
       
  1579 /FontType 1 def
       
  1580 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1581 /Encoding 256 array
       
  1582 0 1 255 {1 index exch /.notdef put} for
       
  1583 dup 88 /X put
       
  1584 dup 97 /a put
       
  1585 dup 98 /b put
       
  1586 dup 99 /c put
       
  1587 dup 100 /d put
       
  1588 dup 115 /s put
       
  1589 readonly def
       
  1590 /FontBBox{-35 -250 1148 750}readonly def
       
  1591 /UniqueID 5000827 def
       
  1592 currentdict end
       
  1593 currentfile eexec
       
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  1669 cleartomark
       
  1670 %%EndFont 
       
  1671 %%BeginFont: CMBSY10
       
  1672 %!PS-AdobeFont-1.1: CMBSY10 1.00
       
  1673 %%CreationDate: 1992 Jul 23 21:21:18
       
  1674 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1675 11 dict begin
       
  1676 /FontInfo 7 dict dup begin
       
  1677 /version (1.00) readonly def
       
  1678 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1679 /FullName (CMBSY10) readonly def
       
  1680 /FamilyName (Computer Modern) readonly def
       
  1681 /Weight (Bold) readonly def
       
  1682 /ItalicAngle -14.035 def
       
  1683 /isFixedPitch false def
       
  1684 end readonly def
       
  1685 /FontName /CMBSY10 def
       
  1686 /PaintType 0 def
       
  1687 /FontType 1 def
       
  1688 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1689 /Encoding 256 array
       
  1690 0 1 255 {1 index exch /.notdef put} for
       
  1691 dup 1 /periodcentered put
       
  1692 readonly def
       
  1693 /FontBBox{-27 -940 1332 825}readonly def
       
  1694 /UniqueID 5000762 def
       
  1695 currentdict end
       
  1696 currentfile eexec
       
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  1737 cleartomark
       
  1738 %%EndFont 
       
  1739 %%BeginFont: CMSY7
       
  1740 %!PS-AdobeFont-1.1: CMSY7 1.0
       
  1741 %%CreationDate: 1991 Aug 15 07:21:52
       
  1742 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1743 11 dict begin
       
  1744 /FontInfo 7 dict dup begin
       
  1745 /version (1.0) readonly def
       
  1746 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1747 /FullName (CMSY7) readonly def
       
  1748 /FamilyName (Computer Modern) readonly def
       
  1749 /Weight (Medium) readonly def
       
  1750 /ItalicAngle -14.035 def
       
  1751 /isFixedPitch false def
       
  1752 end readonly def
       
  1753 /FontName /CMSY7 def
       
  1754 /PaintType 0 def
       
  1755 /FontType 1 def
       
  1756 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1757 /Encoding 256 array
       
  1758 0 1 255 {1 index exch /.notdef put} for
       
  1759 dup 0 /minus put
       
  1760 dup 48 /prime put
       
  1761 dup 66 /B put
       
  1762 dup 102 /braceleft put
       
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  1764 readonly def
       
  1765 /FontBBox{-15 -951 1252 782}readonly def
       
  1766 /UniqueID 5000817 def
       
  1767 currentdict end
       
  1768 currentfile eexec
       
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  1818 cleartomark
       
  1819 %%EndFont 
       
  1820 %%BeginFont: CMTT10
       
  1821 %!PS-AdobeFont-1.1: CMTT10 1.00B
       
  1822 %%CreationDate: 1992 Apr 26 10:42:42
       
  1823 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1824 11 dict begin
       
  1825 /FontInfo 7 dict dup begin
       
  1826 /version (1.00B) readonly def
       
  1827 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1828 /FullName (CMTT10) readonly def
       
  1829 /FamilyName (Computer Modern) readonly def
       
  1830 /Weight (Medium) readonly def
       
  1831 /ItalicAngle 0 def
       
  1832 /isFixedPitch true def
       
  1833 end readonly def
       
  1834 /FontName /CMTT10 def
       
  1835 /PaintType 0 def
       
  1836 /FontType 1 def
       
  1837 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  1838 /Encoding 256 array
       
  1839 0 1 255 {1 index exch /.notdef put} for
       
  1840 dup 102 /f put
       
  1841 readonly def
       
  1842 /FontBBox{-4 -235 731 800}readonly def
       
  1843 /UniqueID 5000832 def
       
  1844 currentdict end
       
  1845 currentfile eexec
       
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  1888 cleartomark
       
  1889 %%EndFont 
       
  1890 %%BeginFont: CMTT9
       
  1891 %!PS-AdobeFont-1.1: CMTT9 1.0
       
  1892 %%CreationDate: 1991 Aug 20 16:46:24
       
  1893 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1894 11 dict begin
       
  1895 /FontInfo 7 dict dup begin
       
  1896 /version (1.0) readonly def
       
  1897 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1898 /FullName (CMTT9) readonly def
       
  1899 /FamilyName (Computer Modern) readonly def
       
  1900 /Weight (Medium) readonly def
       
  1901 /ItalicAngle 0 def
       
  1902 /isFixedPitch true def
       
  1903 end readonly def
       
  1904 /FontName /CMTT9 def
       
  1905 /PaintType 0 def
       
  1906 /FontType 1 def
       
  1907 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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  1911 dup 102 /f put
       
  1912 dup 108 /l put
       
  1913 dup 109 /m put
       
  1914 dup 114 /r put
       
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  1916 readonly def
       
  1917 /FontBBox{-6 -233 542 698}readonly def
       
  1918 /UniqueID 5000831 def
       
  1919 currentdict end
       
  1920 currentfile eexec
       
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  1984 cleartomark
       
  1985 %%EndFont 
       
  1986 %%BeginFont: CMMI7
       
  1987 %!PS-AdobeFont-1.1: CMMI7 1.100
       
  1988 %%CreationDate: 1996 Jul 23 07:53:53
       
  1989 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  1990 11 dict begin
       
  1991 /FontInfo 7 dict dup begin
       
  1992 /version (1.100) readonly def
       
  1993 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  1994 /FullName (CMMI7) readonly def
       
  1995 /FamilyName (Computer Modern) readonly def
       
  1996 /Weight (Medium) readonly def
       
  1997 /ItalicAngle -14.04 def
       
  1998 /isFixedPitch false def
       
  1999 end readonly def
       
  2000 /FontName /CMMI7 def
       
  2001 /PaintType 0 def
       
  2002 /FontType 1 def
       
  2003 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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  2008 dup 65 /A put
       
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  2012 dup 97 /a put
       
  2013 dup 105 /i put
       
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  2018 /FontBBox{0 -250 1171 750}readonly def
       
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  2020 currentdict end
       
  2021 currentfile eexec
       
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  2129 cleartomark
       
  2130 %%EndFont 
       
  2131 %%BeginFont: CMSY9
       
  2132 %!PS-AdobeFont-1.1: CMSY9 1.0
       
  2133 %%CreationDate: 1991 Aug 15 07:22:27
       
  2134 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2135 11 dict begin
       
  2136 /FontInfo 7 dict dup begin
       
  2137 /version (1.0) readonly def
       
  2138 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2139 /FullName (CMSY9) readonly def
       
  2140 /FamilyName (Computer Modern) readonly def
       
  2141 /Weight (Medium) readonly def
       
  2142 /ItalicAngle -14.035 def
       
  2143 /isFixedPitch false def
       
  2144 end readonly def
       
  2145 /FontName /CMSY9 def
       
  2146 /PaintType 0 def
       
  2147 /FontType 1 def
       
  2148 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  2149 /Encoding 256 array
       
  2150 0 1 255 {1 index exch /.notdef put} for
       
  2151 dup 0 /minus put
       
  2152 dup 14 /openbullet put
       
  2153 dup 25 /approxequal put
       
  2154 dup 27 /propersuperset put
       
  2155 dup 33 /arrowright put
       
  2156 dup 41 /arrowdblright put
       
  2157 dup 50 /element put
       
  2158 dup 54 /negationslash put
       
  2159 dup 56 /universal put
       
  2160 dup 62 /latticetop put
       
  2161 dup 66 /B put
       
  2162 dup 94 /logicaland put
       
  2163 dup 95 /logicalor put
       
  2164 dup 96 /turnstileleft put
       
  2165 dup 102 /braceleft put
       
  2166 dup 103 /braceright put
       
  2167 dup 104 /angbracketleft put
       
  2168 dup 105 /angbracketright put
       
  2169 dup 106 /bar put
       
  2170 dup 110 /backslash put
       
  2171 dup 114 /nabla put
       
  2172 readonly def
       
  2173 /FontBBox{-30 -958 1146 777}readonly def
       
  2174 /UniqueID 5000819 def
       
  2175 currentdict end
       
  2176 currentfile eexec
       
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  2290 cleartomark
       
  2291 %%EndFont 
       
  2292 %%BeginFont: CMR9
       
  2293 %!PS-AdobeFont-1.1: CMR9 1.0
       
  2294 %%CreationDate: 1991 Aug 20 16:39:59
       
  2295 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2296 11 dict begin
       
  2297 /FontInfo 7 dict dup begin
       
  2298 /version (1.0) readonly def
       
  2299 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2300 /FullName (CMR9) readonly def
       
  2301 /FamilyName (Computer Modern) readonly def
       
  2302 /Weight (Medium) readonly def
       
  2303 /ItalicAngle 0 def
       
  2304 /isFixedPitch false def
       
  2305 end readonly def
       
  2306 /FontName /CMR9 def
       
  2307 /PaintType 0 def
       
  2308 /FontType 1 def
       
  2309 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  2310 /Encoding 256 array
       
  2311 0 1 255 {1 index exch /.notdef put} for
       
  2312 dup 35 /numbersign put
       
  2313 dup 40 /parenleft put
       
  2314 dup 41 /parenright put
       
  2315 dup 58 /colon put
       
  2316 dup 59 /semicolon put
       
  2317 dup 61 /equal put
       
  2318 dup 64 /at put
       
  2319 dup 91 /bracketleft put
       
  2320 dup 93 /bracketright put
       
  2321 readonly def
       
  2322 /FontBBox{-39 -250 1036 750}readonly def
       
  2323 /UniqueID 5000792 def
       
  2324 currentdict end
       
  2325 currentfile eexec
       
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  2403 cleartomark
       
  2404 %%EndFont 
       
  2405 %%BeginFont: CMR10
       
  2406 %!PS-AdobeFont-1.1: CMR10 1.00B
       
  2407 %%CreationDate: 1992 Feb 19 19:54:52
       
  2408 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2409 11 dict begin
       
  2410 /FontInfo 7 dict dup begin
       
  2411 /version (1.00B) readonly def
       
  2412 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2413 /FullName (CMR10) readonly def
       
  2414 /FamilyName (Computer Modern) readonly def
       
  2415 /Weight (Medium) readonly def
       
  2416 /ItalicAngle 0 def
       
  2417 /isFixedPitch false def
       
  2418 end readonly def
       
  2419 /FontName /CMR10 def
       
  2420 /PaintType 0 def
       
  2421 /FontType 1 def
       
  2422 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  2423 /Encoding 256 array
       
  2424 0 1 255 {1 index exch /.notdef put} for
       
  2425 dup 35 /numbersign put
       
  2426 dup 40 /parenleft put
       
  2427 dup 41 /parenright put
       
  2428 dup 49 /one put
       
  2429 dup 50 /two put
       
  2430 dup 58 /colon put
       
  2431 dup 59 /semicolon put
       
  2432 dup 61 /equal put
       
  2433 dup 63 /question put
       
  2434 dup 64 /at put
       
  2435 dup 91 /bracketleft put
       
  2436 dup 93 /bracketright put
       
  2437 readonly def
       
  2438 /FontBBox{-251 -250 1009 969}readonly def
       
  2439 /UniqueID 5000793 def
       
  2440 currentdict end
       
  2441 currentfile eexec
       
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  2546 cleartomark
       
  2547 %%EndFont 
       
  2548 %%BeginFont: CMSY10
       
  2549 %!PS-AdobeFont-1.1: CMSY10 1.0
       
  2550 %%CreationDate: 1991 Aug 15 07:20:57
       
  2551 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2552 11 dict begin
       
  2553 /FontInfo 7 dict dup begin
       
  2554 /version (1.0) readonly def
       
  2555 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2556 /FullName (CMSY10) readonly def
       
  2557 /FamilyName (Computer Modern) readonly def
       
  2558 /Weight (Medium) readonly def
       
  2559 /ItalicAngle -14.035 def
       
  2560 /isFixedPitch false def
       
  2561 end readonly def
       
  2562 /FontName /CMSY10 def
       
  2563 /PaintType 0 def
       
  2564 /FontType 1 def
       
  2565 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  2566 /Encoding 256 array
       
  2567 0 1 255 {1 index exch /.notdef put} for
       
  2568 dup 0 /minus put
       
  2569 dup 1 /periodcentered put
       
  2570 dup 14 /openbullet put
       
  2571 dup 15 /bullet put
       
  2572 dup 25 /approxequal put
       
  2573 dup 27 /propersuperset put
       
  2574 dup 33 /arrowright put
       
  2575 dup 41 /arrowdblright put
       
  2576 dup 50 /element put
       
  2577 dup 54 /negationslash put
       
  2578 dup 56 /universal put
       
  2579 dup 62 /latticetop put
       
  2580 dup 66 /B put
       
  2581 dup 94 /logicaland put
       
  2582 dup 95 /logicalor put
       
  2583 dup 96 /turnstileleft put
       
  2584 dup 102 /braceleft put
       
  2585 dup 103 /braceright put
       
  2586 dup 104 /angbracketleft put
       
  2587 dup 105 /angbracketright put
       
  2588 dup 106 /bar put
       
  2589 dup 110 /backslash put
       
  2590 dup 114 /nabla put
       
  2591 dup 116 /unionsq put
       
  2592 dup 117 /intersectionsq put
       
  2593 readonly def
       
  2594 /FontBBox{-29 -960 1116 775}readonly def
       
  2595 /UniqueID 5000820 def
       
  2596 currentdict end
       
  2597 currentfile eexec
       
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  2720 cleartomark
       
  2721 %%EndFont 
       
  2722 %%BeginFont: CMMI10
       
  2723 %!PS-AdobeFont-1.1: CMMI10 1.100
       
  2724 %%CreationDate: 1996 Jul 23 07:53:57
       
  2725 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  2726 11 dict begin
       
  2727 /FontInfo 7 dict dup begin
       
  2728 /version (1.100) readonly def
       
  2729 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  2730 /FullName (CMMI10) readonly def
       
  2731 /FamilyName (Computer Modern) readonly def
       
  2732 /Weight (Medium) readonly def
       
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  2734 /isFixedPitch false def
       
  2735 end readonly def
       
  2736 /FontName /CMMI10 def
       
  2737 /PaintType 0 def
       
  2738 /FontType 1 def
       
  2739 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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  2782 currentdict end
       
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  3051 cleartomark
       
  3052 %%EndFont 
       
  3053 %%BeginFont: CMMI9
       
  3054 %!PS-AdobeFont-1.1: CMMI9 1.100
       
  3055 %%CreationDate: 1996 Jul 23 07:53:55
       
  3056 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  3057 11 dict begin
       
  3058 /FontInfo 7 dict dup begin
       
  3059 /version (1.100) readonly def
       
  3060 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  3061 /FullName (CMMI9) readonly def
       
  3062 /FamilyName (Computer Modern) readonly def
       
  3063 /Weight (Medium) readonly def
       
  3064 /ItalicAngle -14.04 def
       
  3065 /isFixedPitch false def
       
  3066 end readonly def
       
  3067 /FontName /CMMI9 def
       
  3068 /PaintType 0 def
       
  3069 /FontType 1 def
       
  3070 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  3071 /Encoding 256 array
       
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  3073 dup 0 /Gamma put
       
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  3113 currentdict end
       
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  3384 cleartomark
       
  3385 %%EndFont 
       
  3386 %%BeginFont: CMR6
       
  3387 %!PS-AdobeFont-1.1: CMR6 1.0
       
  3388 %%CreationDate: 1991 Aug 20 16:39:02
       
  3389 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  3390 11 dict begin
       
  3391 /FontInfo 7 dict dup begin
       
  3392 /version (1.0) readonly def
       
  3393 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  3394 /FullName (CMR6) readonly def
       
  3395 /FamilyName (Computer Modern) readonly def
       
  3396 /Weight (Medium) readonly def
       
  3397 /ItalicAngle 0 def
       
  3398 /isFixedPitch false def
       
  3399 end readonly def
       
  3400 /FontName /CMR6 def
       
  3401 /PaintType 0 def
       
  3402 /FontType 1 def
       
  3403 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
  3404 /Encoding 256 array
       
  3405 0 1 255 {1 index exch /.notdef put} for
       
  3406 dup 40 /parenleft put
       
  3407 dup 41 /parenright put
       
  3408 dup 49 /one put
       
  3409 dup 50 /two put
       
  3410 dup 51 /three put
       
  3411 dup 52 /four put
       
  3412 dup 58 /colon put
       
  3413 dup 61 /equal put
       
  3414 dup 64 /at put
       
  3415 dup 91 /bracketleft put
       
  3416 dup 93 /bracketright put
       
  3417 readonly def
       
  3418 /FontBBox{-20 -250 1193 750}readonly def
       
  3419 /UniqueID 5000789 def
       
  3420 currentdict end
       
  3421 currentfile eexec
       
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  3506 %%EndFont 
       
  3507 %%BeginFont: CMR7
       
  3508 %!PS-AdobeFont-1.1: CMR7 1.0
       
  3509 %%CreationDate: 1991 Aug 20 16:39:21
       
  3510 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
       
  3511 11 dict begin
       
  3512 /FontInfo 7 dict dup begin
       
  3513 /version (1.0) readonly def
       
  3514 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
       
  3515 /FullName (CMR7) readonly def
       
  3516 /FamilyName (Computer Modern) readonly def
       
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  3519 /isFixedPitch false def
       
  3520 end readonly def
       
  3521 /FontName /CMR7 def
       
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  3523 /FontType 1 def
       
  3524 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
       
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  3527 dup 40 /parenleft put
       
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  3538 currentdict end
       
  3539 currentfile eexec
       
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  3605 cleartomark
       
  3606 %%EndFont 
       
  3607 TeXDict begin 39139632 55387786 1000 600 600 (alpha-tlca.dvi)
       
  3608 @start /Fa 139[17 20 15[23 23 25 8[36 88[{}6 41.511 /CMTI7
       
  3609 rf /Fb 206[28 49[{}1 41.511 /CMR5 rf /Fc 167[46 28[19
       
  3610 59[{}2 41.511 /CMMI5 rf /Fd 254[22 1[{}1 58.1154 /CMBSY7
       
  3611 rf /Fe 137[26 9[26 108[{}2 49.8132 /CMTT8 rf /Ff 139[20
       
  3612 24 15[27 27 30 8[43 88[{}6 49.8132 /CMTI7 rf /Fg 192[48
       
  3613 63[{}1 49.8132 /MSBM7 rf /Fh 230[68 25[{}1 99.6264 /CMMIB10
       
  3614 rf /Fi 134[42 4[28 32 37 1[46 42 46 69 23 2[23 46 2[37
       
  3615 3[42 12[55 3[51 3[55 7[60 60 14[42 42 42 42 2[21 43[46
       
  3616 2[{TeXBase1Encoding ReEncodeFont}24 83.022 /Times-Bold
       
  3617 rf /Fj 177[42 78[{TeXBase1Encoding ReEncodeFont}1 58.1154
       
  3618 /Helvetica rf /Fk 254[21 1[{}1 49.8132 /CMBSY7 rf /Fl
       
  3619 152[31 31 53[18 47[45{}4 41.511 /CMSY5 rf /Fm 192[65
       
  3620 63[{}1 83.022 /MSBM10 rf /Fn 133[30 32 35 3[24 29 2[32
       
  3621 6[22 3[29 32 28 27 34 8[50 3[36 1[46 4[58 42 4[47 2[50
       
  3622 2[46 3[32 1[19 19 32[36 25[{}24 49.8132 /CMMI6 rf /Fo
       
  3623 193[68 1[68 1[68 1[68 15[62 40[{}5 74.7198 /TeX-cmex9
       
  3624 rf /Fp 141[51 8[25 25 32 32 5[38 4[42 24[41 3[48 5[35
       
  3625 7[18 6[61 13[48 1[48 9[32 14[48{}16 49.8132 /CMSY6 rf
       
  3626 /Fq 192[60 63[{}1 74.7198 /MSBM10 rf /Fr 139[28 34 15[38
       
  3627 38 42 8[62 88[{}6 83.022 /CMTI10 rf /Fs 139[23 28 15[32
       
  3628 32 35 8[50 88[{}6 58.1154 /CMTI7 rf /Ft 140[31 14[39
       
  3629 35 35 39 8[57 88[{}6 74.7198 /CMTI9 rf /Fu 254[27 1[{}1
       
  3630 83.022 /CMBSY10 rf /Fv 152[34 34 35[44 17[19 47[52{}5
       
  3631 58.1154 /CMSY7 rf /Fw 153[44 102[{}1 83.022 /CMTT10 rf
       
  3632 /Fx 137[39 3[39 4[39 39 5[39 4[39 97[{}6 74.7198 /CMTT9
       
  3633 rf /Fy 203[25 25 51[{TeXBase1Encoding ReEncodeFont}2
       
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  3636 58.1154 /Times-Roman rf /FB 177[54 78[{TeXBase1Encoding ReEncodeFont}1
       
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  3639 rf /FD 177[60 78[{TeXBase1Encoding ReEncodeFont}1 83.022
       
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  3645 1[60 10[38 13[60{}21 74.7198 /CMSY9 rf /FG 162[21 1[21
       
  3646 26[60 2[60 1[21 21 16[30 30 4[64 35[{}9 74.7198 /CMR9
       
  3647 rf /FH 162[23 1[23 26[65 39 1[65 1[23 23 7[42 42 7[32
       
  3648 32 4[69 35[{}12 83.022 /CMR10 rf /FI 138[55 55 1[69 3[42
       
  3649 3[23 32 32 42 42 5[51 55 55 27[55 3[65 5[46 1[0 3[55
       
  3650 8[83 7[83 5[65 1[65 9[42 42 12[23 65{}25 83.022 /CMSY10
       
  3651 rf /FJ 134[50 50 1[50 50 50 50 50 50 50 1[50 50 50 2[50
       
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  3693 rf end
       
  3694 %%EndProlog
       
  3695 %%BeginSetup
       
  3696 %%Feature: *Resolution 600dpi
       
  3697 TeXDict begin
       
  3698 %%PaperSize: A4
       
  3699  end
       
  3700 %%EndSetup
       
  3701 %%Page: 1 1
       
  3702 TeXDict begin 1 0 bop 958 448 a FU(A)-12 b(v)o(oiding)31
       
  3703 b(Equi)o(v)o(ariance)g(in)f(Alpha-Pr)n(olog)1342 737
       
  3704 y FT(Christian)20 b(Urban)1876 707 y FS(1)1932 737 y
       
  3705 FT(and)g(James)h(Chene)o(y)2551 707 y FS(2)565 879 y
       
  3706 FR(1)637 911 y FQ(Ludwig-Maximilians-Uni)n(v)o(ersity)f(Munich)g(\()p
       
  3707 FP(urban@mathematik.uni-muenchen.de)p FQ(\))1124 971
       
  3708 y FR(2)1196 1002 y FQ(Edinb)o(ur)o(gh)f(Uni)n(v)o(ersity)g(\()p
       
  3709 FP(jcheney@inf.ed.ac.uk)p FQ(\))759 1302 y FO(Abstract.)41
       
  3710 b FN(\013)p FQ(Prolog)30 b(is)f(a)g(logic)h(programming)h(language)g
       
  3711 (which)e(is)g(well-suited)h(for)759 1393 y(rapid)25 b(prototyping)h(of)
       
  3712 f(type)g(systems)g(and)g(operational)h(semantics)f(of)f(typed)i
       
  3713 FN(\025)p FQ(-calculi)759 1485 y(and)21 b(man)o(y)g(other)g(languages)g
       
  3714 (in)m(v)o(olving)g(bound)h(names.)f(In)f FN(\013)p FQ(Prolog,)g(the)g
       
  3715 (nominal)h(uni-)759 1576 y(\002cation)f(algorithm)g(of)f(Urban,)h
       
  3716 (Pitts)e(and)j(Gabbay)f(is)g(used)g(instead)g(of)g(\002rst-order)f
       
  3717 (uni\002-)759 1667 y(cation.)f(Ho)n(we)n(v)o(er)m(,)f(although)i
       
  3718 FN(\013)p FQ(Prolog)e(can)h(be)g(vie)n(wed)g(as)f(Horn-clause)h(logic)g
       
  3719 (program-)759 1759 y(ming)e(in)f(Pitts')f FM(nominal)i(lo)o(gic)p
       
  3720 FQ(,)g(proof)g(search)g(using)g(nominal)g(uni\002cation)g(is)f
       
  3721 (incomplete)759 1850 y(in)26 b(nominal)h(logic.)e(Because)i(of)f
       
  3722 (nominal)g(logic')l(s)g FM(equivariance)i(principle)p
       
  3723 FQ(,)e(complete)759 1941 y(proof)j(search)g(w)o(ould)g(require)g
       
  3724 (solving)g(NP-hard)g FM(equivariant)g(uni\002cation)h
       
  3725 FQ(problems.)759 2033 y(Ne)n(v)o(ertheless,)18 b(the)g
       
  3726 FN(\013)p FQ(Prolog)g(programs)h(we)e(studied)i(run)f(correctly)g
       
  3727 (without)f(equi)n(v)n(ariant)759 2124 y(uni\002cation.)j(In)g(this)g
       
  3728 (paper)m(,)g(we)g(gi)n(v)o(e)h(se)n(v)o(eral)f(e)o(xamples)h(of)f
       
  3729 FN(\013)p FQ(Prolog)g(programs)h(that)f(do)759 2215 y(not)i(require)f
       
  3730 (equi)n(v)n(ariant)h(uni\002cation,)f(de)n(v)o(elop)i(a)e(test)f(for)h
       
  3731 (identifying)h(such)g(programs,)759 2307 y(and)e(pro)o(v)o(e)f(the)g
       
  3732 (correctness)h(of)f(this)g(test)f(via)h(a)g(proof-theoretic)h(ar)o
       
  3733 (gument.)523 2570 y FL(1)99 b(Intr)n(oduction)523 2755
       
  3734 y FT(Logic)19 b(programming)e(is)k(particularly)e(suited)h(for)f
       
  3735 (implementing)f(inference)h(rules)h(de\002ning)f(re-)523
       
  3736 2855 y(lations)f(o)o(v)o(er)e(terms.)i(Man)o(y)f(interesting)f(e)o
       
  3737 (xamples)h(of)h(such)f(inference)f(rules,)i(ho)n(we)n(v)o(er)m(,)d(in)m
       
  3738 (v)n(olv)o(e)523 2954 y(terms)28 b(with)g(binders)f(and)h
       
  3739 FK(\013)p FT(-equi)n(v)n(alence,)e(for)h(which)h(Prolog,)f(for)g(e)o
       
  3740 (xample,)f(pro)o(vides)h(little)523 3054 y(assistance.)h(In)f([3])f(we)
       
  3741 i(presented)e FK(\013)p FT(Prolog)o(,)h(which)g(is)h(designed)e(to)h
       
  3742 (simplify)g(programming)523 3153 y(with)g(binders.)f(F)o(or)g
       
  3743 (instance,)g(the)h(operation)e(of)i(capture-a)n(v)n(oiding)c
       
  3744 (substitution)j(for)g FK(\025)p FT(-terms)523 3253 y(can)20
       
  3745 b(be)g(implemented)e(in)j FK(\013)p FT(Prolog)e(as)i(follo)n(ws:)529
       
  3746 3395 y FP(id\(X,X\).)529 3510 y(subst\(var\(X\),X,T,T\).)529
       
  3747 3601 y(subst\(var\(X\),Y,T,var\(X\)\))253 b(:-)23 b(not\(id\(X,Y\)\).)
       
  3748 529 3693 y(subst\(app\(M,N\),X,T,app\(M',N'\)\))-17 b(:-)23
       
  3749 b(subst\(M,X,T,M'\),)10 b(subst\(N,X,T,N'\).)529 3784
       
  3750 y(subst\(lam\(a.M\),X,T,lam\(a.M'\)\))28 b(:-)23 b(a)13
       
  3751 b(#)f(T,)g(a)g(#)h(X,)f(subst\(M,X,T,M'\).)523 3928 y
       
  3752 FT(where)21 b(the)h(terms)g FJ(var\(X\))p FT(,)f FJ(app\(M,N\))g
       
  3753 FT(and)g FJ(lam\(a.M\))g FT(encode)g(v)n(ariables,)g(applications)523
       
  3754 4028 y(and)26 b FK(\025)p FT(-abstractions.)f(The)g(predicate)g
       
  3755 FJ(subst\(E,X,T,E'\))f FT(de\002ned)h(by)h(the)g(clauses)g(holds)523
       
  3756 4127 y(only)f(if)h FJ(E')g FT(contains)f(the)h(result)f(of)h(the)f
       
  3757 (usual)h(capture-a)n(v)n(oiding)d(substitution)h FJ(E[X:=T])h
       
  3758 FT(in)523 4227 y(the)20 b FK(\025)p FT(-calculus.)648
       
  3759 4327 y(T)-7 b(w)o(o)29 b(features)g(of)h FK(\013)p FT(Prolog)f(are)g
       
  3760 (immediately)f(visible)i(to)g(the)f(user)-5 b(.)30 b(First,)g(the)g
       
  3761 (term)f(lan-)523 4426 y(guage)c(includes)h(the)h(term-constructor)c
       
  3762 FI(\000)p FK(:)p FI(\000)j FT(for)g(forming)e(abstractions,)i(which)g
       
  3763 (are)g(used)g(to)523 4526 y(encode)20 b(binding.)f(Second,)h
       
  3764 FK(\013)p FT(Prolog)h(has)g(a)h(freshness-predicate,)c(written)j(as)h
       
  3765 FI(\000)p FJ(#)p FI(\000)p FT(,)e(b)n(uilt)i(into)523
       
  3766 4625 y(the)29 b(language;)e(this)j(predicate)e(ensures)g(that)h(a)g
       
  3767 (name)g(does)f(not)h(occur)f(freely)g(in)h(a)h(term)e(\(by)523
       
  3768 4725 y(a)h(name)f(we)h(mean)f(lo)n(wer)n(-case)g(symbols,)g(for)g
       
  3769 (instance)g FJ(a)h FT(in)f(the)h(e)o(xpression)e FJ(lam\(a.M\))p
       
  3770 FT(\).)523 4825 y(In)e(this)h FJ(subst)p FT(-program,)21
       
  3771 b(the)k(freshness-predicate)e(is)j(used)f(to)h(mak)o(e)e(sure)h(that)h
       
  3772 (no)f(v)n(ariable)523 4924 y(capture)19 b(occurs)h(inside)g(the)g(term)
       
  3773 g(being)f(substituted.)p eop end
       
  3774 %%Page: 2 2
       
  3775 TeXDict begin 2 1 bop 648 448 a FT(T)-7 b(o)23 b(illustrate)g(ho)n(w)g
       
  3776 (the)g FJ(subst)p FT(-program)d(calculates)j(the)g(result)h(of)e(the)i
       
  3777 (capture-a)n(v)n(oiding)523 548 y(substitution)19 b FH(\()p
       
  3778 FK(\025b:a)24 b(b)p FH(\)[)p FK(a)f FH(:=)f FK(b)p FH(])p
       
  3779 FT(,)f(we)f(consider)f(the)h(query:)1023 732 y FP
       
  3780 (subst\(lam\(b.app\(var\(a\),var\(b\)\),a,var)o(\(b\),R\))396
       
  3781 b FT(\(1\))523 916 y(T)-7 b(o)20 b(solv)o(e)g(this)h(query)-5
       
  3782 b(,)18 b FK(\013)p FT(Prolog)i(uni\002es)g(it)h(with)f(the)g(head)g(of)
       
  3783 g(the)g(fourth)f FJ(subst)p FT(-clause)530 1100 y FP(subst\(lam\(a)1025
       
  3784 1108 y FR(1)1057 1100 y FP(.M)1147 1108 y FR(1)1182 1100
       
  3785 y FP(\))-12 b(,X)1305 1108 y FR(1)1338 1100 y FP(,)g(T)1416
       
  3786 1108 y FR(1)1450 1100 y FP(,)g(lam\(a)1708 1108 y FR(1)1741
       
  3787 1100 y FP(.M')1850 1108 y FR(1)1885 1100 y FP(\)\))g(:-)18
       
  3788 b(a)2116 1108 y FR(1)2151 1100 y FP(#)12 b(T)2253 1108
       
  3789 y FR(1)2287 1100 y FP(,a)2377 1108 y FR(1)2412 1100 y
       
  3790 FP(#)g(X)2514 1108 y FR(1)2548 1100 y FP(,subst\(M)2908
       
  3791 1108 y FR(1)2942 1100 y FP(,X)3032 1108 y FR(1)3066 1100
       
  3792 y FP(,)-12 b(T)3144 1108 y FR(1)3178 1100 y FP(,M')3287
       
  3793 1108 y FR(1)3321 1100 y FP(\))g(.)523 1284 y FT(where,)15
       
  3794 b(as)j(in)e(Prolog,)f(the)h(v)n(ariables)g FJ(M)p FT(,)g
       
  3795 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
       
  3796 (with)h(fresh)g(v)n(ariables)523 1384 y(\(indicated)i(by)h(the)g
       
  3797 (subscript\),)f(and)h(also)h(the)f(name)g FJ(a)g FT(has)h(been)f
       
  3798 (freshened)e(\(we)i(shall)h(return)e(to)523 1484 y(the)f(dif)n(ference)
       
  3799 f(between)g(v)n(ariables)h(and)g(names)g(later\).)g(The)g(uni\002er)g
       
  3800 (that)g FK(\013)p FT(Prolog)g(calculates)h(is)523 1583
       
  3801 y FJ(app\(var\(a\),var\(a)1323 1595 y FS(1)1357 1583
       
  3802 y FJ(\)\))h FT(for)e FJ(M)1641 1595 y FS(1)1678 1583
       
  3803 y FT(,)h FJ(a)h FT(for)e FJ(X)1951 1595 y FS(1)1988 1583
       
  3804 y FT(,)i FJ(var\(b\))e FT(for)h FJ(T)2511 1595 y FS(1)2567
       
  3805 1583 y FT(and)f FJ(lam\(a)2955 1595 y FS(1)2991 1583
       
  3806 y FJ(.M')3113 1595 y FS(1)3151 1583 y FJ(\))h FT(for)f
       
  3807 FJ(R)p FT(.)523 1683 y(Ne)o(xt,)22 b FK(\013)p FT(Prolog)f(checks)h
       
  3808 (that)g(the)g(freshness-predicates)e FJ(a)2290 1695 y
       
  3809 FS(1)2327 1683 y FJ(#)i(var\(b\))f FT(and)h FJ(a)2913
       
  3810 1695 y FS(1)2950 1683 y FJ(#)g(a)g FT(hold,)f(and)523
       
  3811 1783 y(continues)14 b(unifying)f(the)i(ne)n(w)g(query)f
       
  3812 FJ(subst)-14 b(\(app\(var\(a\),)g(var\(a)2700 1795 y
       
  3813 FS(1)2720 1783 y FJ(\)\))g(,a,)g(var)g(\(b\),M')3350
       
  3814 1795 y FS(1)3388 1783 y FJ(\))523 1882 y FT(with)25 b(the)g(third)g
       
  3815 FJ(subst)p FT(-clause.)f(Then)g(it)i(uses)f(the)h(\002rst)f(and)g
       
  3816 (second)f FJ(subst)p FT(-clause)g(and)h(af-)523 1982
       
  3817 y(ter)17 b(the)o(y)f(succeed,)g FK(\013)p FT(Prolog)g(returns)g
       
  3818 FJ(lam\(a)1878 1994 y FS(1)1914 1982 y FJ(.app\(var\(b\),var\(a)2764
       
  3819 1994 y FS(1)2798 1982 y FJ(\)\)\))h FT(as)g(the)g(answer)523
       
  3820 2082 y(for)j FJ(R)p FT(.)648 2173 y(Another)g(e)o(xample,)h(which)g
       
  3821 (illustrates)i(ho)n(w)e(easily)i(inference)d(rules)i(can)g(be)g
       
  3822 (implemented)523 2265 y(in)e FK(\013)p FT(Prolog)o(,)h(is)g(the)f
       
  3823 (follo)n(wing)f(program)548 2441 y FP(mem\(X,[X|T]\).)548
       
  3824 2532 y(mem\(X,[Y|T]\))43 b(:-)h(mem\(X,T\).)548 2647
       
  3825 y(type\(Gamma,var\(X\),T\))87 b(:-)23 b(mem\(\(X,T\),Gamma\).)548
       
  3826 2739 y(type\(Gamma,app\(M,N\),T\))m(:-)g(type\(Gamma,M,arr\(S,T\)\),)16
       
  3827 b(type\(Gamma,N,S\))-12 b(.)548 2830 y
       
  3828 (type\(Gamma,lam\(x.M\),arr\(S,T\)\))-6 b(:-)6 b(x)h(#)12
       
  3829 b(Gamma,)g(type\([\(x,S\)|Gamma],M,T\))-12 b(.)523 3009
       
  3830 y FT(implementing)18 b(the)i(usual)g(inference)f(rules)h(for)g
       
  3831 (inferring)e(the)i(types)g(of)g FK(\025)p FT(-terms.)535
       
  3832 3249 y FN(x)h FG(:)g FN(T)32 b FF(2)22 b FN(\000)p 535
       
  3833 3273 315 4 v 545 3340 a(\000)28 b(.)17 b(x)k FG(:)h FN(T)891
       
  3834 3289 y(v)s(ar)1103 3253 y(\000)27 b(.)18 b(M)e FG(:)22
       
  3835 b FN(S)k FF(!)21 b FN(T)85 b(\000)28 b(.)17 b(N)g FG(:)22
       
  3836 b FN(S)p 1103 3273 867 4 v 1341 3340 a(\000)28 b(.)18
       
  3837 b(M)8 b(N)17 b FG(:)k FN(T)2010 3281 y(app)2223 3233
       
  3838 y(x)f FG(:)i FN(S;)13 b(\000)28 b(.)17 b(M)g FG(:)22
       
  3839 b FN(T)85 b FG(\()p FN(x)21 b FF(62)g FN(F)11 b(V)17
       
  3840 b FG(\()p FN(\000)11 b FG(\)\))p 2223 3272 1019 4 v 2432
       
  3841 3340 a FN(\000)28 b(.)17 b(\025x:M)g FG(:)k FN(S)26 b
       
  3842 FF(!)21 b FN(T)3282 3298 y(l)q(am)523 3550 y FT(Note)j(that,)f(in)h
       
  3843 (contrast)f(to)h(for)e(e)o(xample)h FK(\025)p FT(Prolog,)f
       
  3844 (abstractions)h(in)h FK(\013)p FT(Prolog)f(bind)g(a)h(concrete)523
       
  3845 3650 y(name)i(which)h(is)g FE(not)h FT(restricted)f(to)g(the)f(scope)h
       
  3846 (of)f(the)h(abstractions.)f(Therefore)e(it)k(is)g(possible)523
       
  3847 3749 y(in)23 b FK(\013)p FT(Prolog)g(to)g(use)h(a)g(name)e(of)h(a)h
       
  3848 (binder)e(in)h(the)g(body)f(of)h(the)g(clause,)g(for)g(instance)g(to)g
       
  3849 (append)523 3849 y FJ(\(x,S\))f FT(to)g(the)g(conte)o(xt)e
       
  3850 FJ(Gamma)i FT(in)g(the)g(third)f FJ(type)p FT(-clause.)f(The)i
       
  3851 (implicit)g(side-condition)d(in)523 3949 y(the)i(rule)g
       
  3852 FK(l)r(am)g FT(requiring)f(that)h FK(\000)33 b FT(has)22
       
  3853 b(no)f(type-assignment)e(for)i FK(x)h FT(is)g(implemented)e(in)h
       
  3854 FK(\013)p FT(Prolog)523 4048 y(by)f(the)g(freshness-predicate)e
       
  3855 FJ(x#Gamma)p FT(.)648 4140 y(W)-7 b(e)23 b(ha)n(v)o(e)e(implemented)f
       
  3856 (a)i(lar)o(ge)f(number)f(of)i(such)g FK(\025)p FT(-calculus)f(e)o
       
  3857 (xamples,)g(including)f(type)523 4232 y(systems)26 b(and)e(operational)
       
  3858 f(semantics)i(for)g(System)g(F)-7 b(,)26 b FK(\025\026)g
       
  3859 FT(and)f(linear)f FK(\025)p FT(-calculi.)h(Our)g(e)o(xperi-)523
       
  3860 4323 y(ence)d(from)f(these)h(e)o(xamples)g(suggests)g(that)g(the)g
       
  3861 (combination)e(of)i(concrete)f(names)h(in)g(abstrac-)523
       
  3862 4414 y(tions)32 b(and)f(the)g(freshness-predicate)e(is)k(v)o(ery)d
       
  3863 (useful)h(for)g(programming)e(with)i(binders.)g(One)523
       
  3864 4505 y(question,)21 b(ho)n(we)n(v)o(er)m(,)f(might)i(arise:)h(what)g
       
  3865 (are)f(the)h(adv)n(antages)e(of)h FK(\013)p FT(Prolog)g(relati)n(v)o(e)
       
  3866 g(to,)g(for)g(e)o(x-)523 4597 y(ample,)29 b FK(\025)p
       
  3867 FT(Prolog)f([7],)h(which)g(has)g(both)g FK(\013)p FT(-equi)n(v)n
       
  3868 (alence)e(and)i(capture-a)n(v)n(oiding)d(substitution)523
       
  3869 4688 y(b)n(uilt-in)20 b(and)f(the)h(typing)f(rules)i(can)f(be)g
       
  3870 (correctly)f(implemented)f(by)i(the)g(tw)o(o)g(clauses:)577
       
  3871 4847 y FP(\(type)44 b(\(app)h(M)f(N\))h(T\))f(:-)h(type)f(M)h(\(arr)f
       
  3872 (S)g(T\),)h(type)f(N)h(S.)577 4938 y(\(type)f(\(lam)h(M\))f(\(arr)g(S)h
       
  3873 (T\)\))f(:-)h(\(pi)f(x)p FF(n)p FP(\(type)g(x)h(S)g(=>)f(type)g(\(M)h
       
  3874 (x\))f(T\)\).)p eop end
       
  3875 %%Page: 3 3
       
  3876 TeXDict begin 3 2 bop 523 448 a FT(\(Notice)19 b(that)g(in)h(this)g
       
  3877 (program)d(the)i(typing-conte)o(xt)d(is)21 b(implicitly)e(gi)n(v)o(en)f
       
  3878 (by)h(the)g(\223surrounding\224)523 540 y(program-conte)o(xt.)c(This)20
       
  3879 b(program-conte)o(xt)c(can)j(be)h(modi\002ed)e(using)i(the)f(uni)n(v)o
       
  3880 (ersal)g(quanti\002ca-)523 631 y(tion)25 b(\(i.e.)f FJ(pi)50
       
  3881 b(x)p FI(n)p FJ(...)p FT(\))24 b(and)g(implications)g(in)h
       
  3882 (goal-formulae.)d(Therefore)h(there)h(is)i(no)e(clause)523
       
  3883 722 y(for)h(the)g(v)n(ariable)f(case.\))h(W)-7 b(e)26
       
  3884 b(\002nd)f(the)g(most)g(important)f(reason)g(in)i(f)o(a)n(v)n(our)e(of)
       
  3885 h FK(\013)p FT(Prolog)f(is)i(that)523 814 y(by)g(ha)n(ving)f(concrete)f
       
  3886 (names)i(\(namely)f FJ(x)h FT(in)g(the)g FJ(type)p FT(-e)o(xample\))d
       
  3887 (and)j(freshness-predicates)523 905 y(one)e(can)g(almost)h(directly)e
       
  3888 (translate)i(the)f(three)g(typing)f(rules)i(into)f(three)g(clauses)h
       
  3889 (and)e(obtain)h(a)523 996 y(correct)19 b(implementation.)e(This)j
       
  3890 (should)f(be)h(seen)g(in)g(the)g(conte)o(xt)e(that,)i(despite)g(the)g
       
  3891 (ele)o(gance)e(of)523 1088 y FK(\025)p FT(Prolog,)g(some)h(recent)f(te)
       
  3892 o(xtbooks)f(use)i(\(standard\))e(Prolog)h(for)h(implementing)d
       
  3893 (inference)i(rules)523 1179 y(o)o(v)o(er)25 b FK(\025)p
       
  3894 FT(-terms.)h(F)o(or)g(e)o(xample)e(one)i(of)g(them)f(presents)h(the)g
       
  3895 (follo)n(wing)f(implementation)f(of)h(the)523 1270 y(typing)19
       
  3896 b(rules:)548 1436 y FP(mem\(X,[X|T]\).)548 1527 y(mem\(X,[Y|T]\))43
       
  3897 b(:-)h(mem\(X,T\).)548 1642 y(type\(Gamma,var\(X\),T\))87
       
  3898 b(:-)23 b(mem\(\(X,T\),Gamma\).)548 1733 y(type\(Gamma,app\(M,N\),T\))m
       
  3899 (:-)g(type\(Gamma,M,arr\(S,T\)\),)16 b(type\(Gamma,N,S\))-12
       
  3900 b(.)548 1825 y(type\(Gamma,lam\(X,M\),arr\(S,T\)\))-6
       
  3901 b(:-)6 b(type\([\(X)o(,S\)|Ga)o(mma],)o(M,T\))-13 b(.)523
       
  3902 1996 y FT(which)21 b(calculates)g(the)h FE(wr)l(ong)f
       
  3903 FT(type)g(for)g FK(\025)p FT(-terms)g(such)g(as)h FK(\025x:\025x:)p
       
  3904 FH(\()p FK(x)28 b(x)p FH(\))p FT(.)23 b(Although)c(this)j(prob-)523
       
  3905 2096 y(lem)h(can)g(be)f(\002x)o(ed)h(by)f(judicious)g(use)h(of)g(cut)g
       
  3906 (or)f(side-ef)n(fects,)g(\002rst-order)f(terms)i(of)g(Prolog)f(are)523
       
  3907 2195 y(unwieldy)j(for)g(implementing)f(relations)h(o)o(v)o(er)f(syntax)
       
  3908 i(with)g(binders)f(correctly)-5 b(.)23 b(On)j(the)g(other)523
       
  3909 2295 y(hand,)20 b FK(\025)p FT(Prolog)h(does)g(not)h(allo)n(w)f
       
  3910 (concrete)f(names)h(as)i(binders)d(and)h(therefore)f(operations)g(such)
       
  3911 523 2394 y(as)g(adding)e(the)i(type)f(for)g FJ(x)g FT(to)h(the)f
       
  3912 (typing-conte)o(xt)d(need)j(to)h(be)f(encoded)f(using)h(uni)n(v)o
       
  3913 (ersal)f(quan-)523 2494 y(ti\002cation,)i(implications)f(in)h(goals)g
       
  3914 (and)g(beta-reduction.)648 2596 y(The)30 b FK(\013)p
       
  3915 FT(Prolog)h(language)e(is)j(based)f(on)f(nominal)g(terms)h(and)g(uses)g
       
  3916 (the)h(nominal)d(uni\002ca-)523 2696 y(tion)f(algorithm)e(of)i(Urban,)f
       
  3917 (Pitts,)i(and)e(Gabbay)g([9],)g(which)h(calculates)g(\(most)g
       
  3918 (general\))e(uni-)523 2795 y(\002ers)j(modulo)d FK(\013)p
       
  3919 FT(-equi)n(v)n(alence.)h(F)o(or)g(e)o(xample,)g(the)h(query)f
       
  3920 FJ(?-)14 b(id\(a.a,b.X\))27 b FT(is)i(solv)o(ed)e(in)523
       
  3921 2895 y FK(\013)p FT(Prolog)e(by)g(the)h(capturing)e(substitution)g
       
  3922 FH([)p FJ(X)33 b FH(:=)g FJ(b)p FH(])26 b FT(since)g
       
  3923 FJ(a.a)f FT(and)g FJ(b.b)h FT(are)f FK(\013)p FT(-equi)n(v)n(alent.)523
       
  3924 2995 y(Ho)n(we)n(v)o(er)m(,)e(nominal)g(uni\002cation)h(is)i(not)e
       
  3925 (enough)f(to)i(mak)o(e)g(the)g(programs)e(gi)n(v)o(en)g(earlier)i
       
  3926 (func-)523 3094 y(tion)e(as)g(intended.)e(F)o(or)h(this)i
       
  3927 FK(\013)p FT(Prolog)e(generates)g(fresh)g(names)g(during)f
       
  3928 (proof-search.)f(As)j(seen)523 3194 y(abo)o(v)o(e,)28
       
  3929 b(before)f(a)j(query)e(is)j(uni\002ed)d(with)i(the)f(fourth)f
       
  3930 FJ(subst)p FT(-clause,)g FK(\013)p FT(Prolog)h(generates)f(a)523
       
  3931 3293 y(fresh)17 b(name)h(for)f FJ(a)p FT(.)h(This)g(ensures)g(that)f
       
  3932 (substitutions)h(can)f(al)o(w)o(ays)i(be)f(\223pushed\224)e(under)g(a)j
       
  3933 (binder)523 3393 y(without)h(risk)g(of)g(capture.)648
       
  3934 3487 y(While)25 b(in)g([3])f(we)h(ha)n(v)o(e)f(described)f(our)h
       
  3935 (implementation)f(of)h FK(\013)p FT(Prolog)o(,)h(its)h(beha)n(viour)d
       
  3936 (can)523 3578 y(be)18 b(justi\002ed)g(in)g(terms)f(of)h(nominal)e
       
  3937 (logic)i([8,)13 b(2].)k(F)o(or)g(instance,)g(the)h(generation)e(of)i(a)
       
  3938 g(fresh)f(name)523 3669 y(can)23 b(be)g(e)o(xpressed)e(in)i(terms)g(of)
       
  3939 g(the)1686 3669 y
       
  3940  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  3941  1686 3669 a FD(N)1686 3669 y
       
  3942  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  3943 neg exch translate
       
  3944  1686 3669
       
  3945 a FT(-quanti\002er)f(of)g(nominal)g(logic,)g(and)g(an)h
       
  3946 FK(\013)p FT(Prolog)f(clause)523 3761 y FJ(A)50 b(:-)f(B1,..,Bn)28
       
  3947 b FT(can)15 b(vie)n(wed)g(as)h(the)f(formula)2125 3761
       
  3948 y
       
  3949  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  3950  2125 3761 a FD(N)2125 3761 y
       
  3951  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  3952 neg exch translate
       
  3953  2125 3761 a FK(a)2169
       
  3954 3773 y FS(1)2207 3761 y FK(::a)2297 3773 y FC(n)2342
       
  3955 3761 y FK(:)p FI(8)p FK(X)2480 3773 y FS(1)2517 3761
       
  3956 y FK(::X)2632 3773 y FC(n)2677 3761 y FK(:B)2763 3773
       
  3957 y FS(1)2800 3761 y FI(^)q(\001\001^)p FK(B)3020 3773
       
  3958 y FC(n)3065 3761 y FI(\033)p FK(A)p FT(,)h(where)523
       
  3959 3852 y(the)24 b FK(X)718 3864 y FC(i)769 3852 y FT(and)f
       
  3960 FK(a)957 3864 y FC(i)1009 3852 y FT(are)g(the)g(v)n(ariables,)g
       
  3961 (respecti)n(v)o(ely)f(names,)h(in)g(the)h(clause.)f(The)g(problem)f(is)
       
  3962 i(that)523 3943 y(the)k(generation)f(of)h(fresh)f(names)h(is)i(more)d
       
  3963 (subtle)h(in)h FK(\013)p FT(Prolog)f(than)f(the)i(usual)f
       
  3964 (\223freshening\224)523 4035 y(of)c(v)n(ariables)f(when)g(backchaining)
       
  3965 f(a)i(Prolog-clause.)e(The)i(reason)f(is)i(that)f(distinct)g(names)g
       
  3966 (are)523 4126 y(al)o(w)o(ays)29 b(considered)d(to)j(denote)e(dif)n
       
  3967 (ferent)g(v)n(alues.)g(Consider)h(the)g(clause)h FI(8)p
       
  3968 FJ(X.p\(X\))f FT(and)f(the)523 4217 y(query)19 b FJ(p\(b\))h
       
  3969 FT(written)g(as)h(a)f(sequent)g(as)h(follo)n(ws)1650
       
  3970 4386 y FF(8)p FP(X.p\(X\))44 b FF(`)g FP(p\(b\))1040
       
  3971 b FQ(\(2\))523 4555 y FT(When)27 b(constructing)f(a)h(proof)f(for)h
       
  3972 (\(2\),)f(Prolog)g(generates)h(a)h(fresh)e(name)h(for)g(the)g(v)n
       
  3973 (ariable)f FJ(X)p FT(,)523 4646 y(say)h FJ(X')p FT(,)g(and)g(then)f
       
  3974 (uni\002es)i FJ(p\(X'\))e FT(and)h FJ(p\(b\))g FT(gi)n(ving)e(the)i
       
  3975 (solution)g FJ([X':=b])p FT(.)f(A)h(similar)523 4737
       
  3976 y FK(\013)p FT(Prolog)o(-clause)21 b(that)g(has)h(a)g(name)f(in)g
       
  3977 (place)g(of)g(the)h(v)n(ariable)e(beha)n(v)o(es)g(dif)n(ferently:)g(if)
       
  3978 h(we)h(ha)n(v)o(e)523 4829 y(the)e(sequent)1699 4924
       
  3979 y
       
  3980  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  3981  1699 4924 a FB(N)1699 4924 y
       
  3982  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  3983 neg exch translate
       
  3984  1699 4924 a FP(a.p\(a\))43
       
  3985 b FF(`)i FP(p\(b\))1034 b FQ(\(3\))p eop end
       
  3986 %%Page: 4 4
       
  3987 TeXDict begin 4 3 bop 523 448 a FT(with)22 b(the)h(clause)1108
       
  3988 448 y
       
  3989  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  3990  1108 448 a FD(N)1108 448 y
       
  3991  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  3992 neg exch translate
       
  3993  1108 448 a FJ(a.p\(a\))p
       
  3994 FT(,)e(then)h(\223freshening\224)e FJ(a)j FT(to)f FJ(a')g
       
  3995 FT(leads)h(to)f(the)g(uni\002cation)f(problem)523 540
       
  3996 y FJ(p\(a'\))p FI(\031)o FH(?)p FJ(p\(b\))p FT(.)15 b(Since)h(nominal)e
       
  3997 (uni\002cation)h(treats)h(names)f(as)h(distinct)g(constants,)f(this)h
       
  3998 (prob-)523 631 y(lem)24 b(is)h(unsolv)n(able.)c(\(T)m(reating)i(names)g
       
  3999 (as)h(distinct)g(constants)g(is)g(important,)e(because)h(treating)523
       
  4000 722 y(them)c(as)i(substitutable)d(entities)i(w)o(ould)f(break)g(the)h
       
  4001 (most-general)d(uni\002er)i(property)e(of)j(nominal)523
       
  4002 814 y(uni\002cation,)c(see)i([9].\))e(On)i(the)f(other)g(hand,)f(\(3\))
       
  4003 h FE(is)h FT(pro)o(v)n(able)d(in)j(nominal)e(logic.)h(This)h(is)g
       
  4004 (because)523 905 y(after)26 b(freshening)d FJ(a)k FT(to)f
       
  4005 FJ(a')g FT(one)f(can)h(in)g(nominal)e(logic)i(apply)e(the)i(equi)n(v)n
       
  4006 (ariance)e(principle\227)523 996 y(e)o(xpressed)19 b(as)i(an)f
       
  4007 (inference)f(rule)1519 966 y FA(3)1718 1085 y FN(\031)s
       
  4008 Fz(\001)o FN(B)t(;)13 b(\000)32 b FF(\))21 b FN(C)p 1718
       
  4009 1120 405 4 v 1753 1187 a(B)t(;)13 b(\000)32 b FF(\))21
       
  4010 b FN(C)2164 1136 y(\031)523 1337 y FT(where)26 b FK(\031)k
       
  4011 FT(is)d(a)g(permutation)d(of)i(names,)f FK(B)t FT(,)i
       
  4012 FK(C)33 b FT(stand)26 b(for)g(formulae)e(and)i FK(\000)38
       
  4013 b FT(for)26 b(a)g(multiset)h(of)523 1437 y(formulae.)18
       
  4014 b(This)i(means)f(if)h(the)g FE(full)g FT(Horn-fragment)c(of)j(nominal)g
       
  4015 (logic)g(were)g(used)h(as)g(the)g(basis)523 1537 y(of)d
       
  4016 FK(\013)p FT(Prolog)o(,)h(then)f(we)h(need)f(equi)n(v)n(ariant)e
       
  4017 (uni\002cation)i(for)f(complete)h(proof)f(search.)h(Equi)n(v)n(ariant)
       
  4018 523 1636 y(uni\002cation)30 b(solv)o(es)h(a)h(problem)d(not)i(just)h
       
  4019 (by)f(\002nding)f(a)h(substitution)g(b)n(ut)g(also)g(by)g(\002nding)f
       
  4020 (a)523 1736 y(permutation;)18 b(for)i(e)o(xample)e(in)j(\(3\))e(the)i
       
  4021 (identity)e(substitution)g(and)h(the)g(permutation)e
       
  4022 FJ(\(a')49 b(b\))p FT(.)648 1841 y(The)31 b(second)h(author)f(has)h
       
  4023 (sho)n(wn)g(in)g([1])g(that)g(equi)n(v)n(ariant)e(uni\002cation)h(and)h
       
  4024 (equi)n(v)n(ariant)523 1941 y(matching)17 b(problems)f(are)i(NP-hard.)e
       
  4025 (F)o(or)i(proof-search)d(in)j FK(\013)p FT(Prolog)f(this)i(means)e
       
  4026 (that)i(one)e(needs)523 2040 y(to)g(guess)g(which)f(permutation)e
       
  4027 FK(\031)21 b FT(leads)c(to)f(a)h(proof.)e(Ho)n(we)n(v)o(er)m(,)g(in)i
       
  4028 (e)o(xperimenting)c(with)k FK(\013)p FT(Prolog)523 2140
       
  4029 y(we)29 b(found)e(that)h(such)g(guessing)g(is)h(ne)n(v)o(er)f(needed)f
       
  4030 (in)h(the)h(programs)d(we)j(considered.)e(In)h(this)523
       
  4031 2240 y(paper)e(we)i(identify)e(a)h(class)h(of)f(nominal)f(Horn-clause)f
       
  4032 (programs)h(for)g(which)h(the)g FK(\031)s FT(-rule)g(can)523
       
  4033 2339 y(be)k(eliminated)f(from)f(deductions)h(\(this)h(is)g(the)g(place)
       
  4034 g(where)f(equi)n(v)n(ariant)f(uni\002cation)g(prob-)523
       
  4035 2439 y(lems)20 b(arise\),)e(and)h(thus)g(nominal)f(uni\002cation)f(is)k
       
  4036 (complete)d(for)g(proof-search.)e(In)j(order)f(to)h(sho)n(w)523
       
  4037 2539 y(this)30 b(result,)f(we)g(introduce)e(a)j FE(well-formedness)f
       
  4038 FT(condition)e(which)i(guarantees)f(that)h(nominal)523
       
  4039 2638 y(uni\002cation-based)18 b(proof)h(search)g(is)j(complete.)d(This)
       
  4040 h(condition)f(roughly)f(says)j(that)f(a)h(clause)g(is)523
       
  4041 2738 y(\223insensiti)n(v)o(e\224)e(to)i(the)f(particular)f(choice)g(of)
       
  4042 h(names)g(occurring)e(in)i(it.)648 2843 y(Some)c(programs)g(do)g(not)h
       
  4043 (satisfy)g(this)h(condition.)d(F)o(or)i(e)o(xample,)e(in)i(the)g(follo)
       
  4044 n(wing)f(program)523 2943 y(calculating)j(a)i(list)g(of)f(bound)e(v)n
       
  4045 (ariables)i(of)g(a)g FK(\025)p FT(-term,)g(the)g(last)h(clause)f(is)i
       
  4046 FE(not)f FT(well-formed.)659 3100 y FP(bv\(var\(X\),[]\).)659
       
  4047 3191 y(bv\(app\(E1,E2\),L\))110 b(:-)23 b(bv\(E1,L1\),)43
       
  4048 b(bv\(E2,L2\),)h(append\(L1,L2,L\).)659 3283 y(bv\(lam\(x.E\),[x|L]\))
       
  4049 20 b(:-)j(bv\(E,L\).)523 3442 y FT(In)30 b(the)g(last)g(clause,)g(the)g
       
  4050 (result)g(accumulated)e(in)i(the)g(second)f(ar)o(gument)e(depends)i(on)
       
  4051 g(which)523 3542 y(name)j(is)i(chosen)e(for)h(the)g(binder)e
       
  4052 FJ(x)p FT(.)j(In)e(contrast,)g(the)h(names)g(chosen)f(in)h(the)g
       
  4053 FJ(subst)g FT(and)523 3641 y FJ(type)21 b FT(e)o(xample)e(do)i(not)f
       
  4054 (matter)h(\(up)f(to)h FK(\013)p FT(-equi)n(v)n(alence\))d(and)i
       
  4055 (therefore)f(will)j(satisfy)f(our)f(well-)523 3741 y(formedness)f
       
  4056 (condition.)648 3846 y(The)25 b(e)o(xistence)g(of)g(a)h(tri)n(vial)f
       
  4057 (syntactic)h(criterion)e(for)h(deciding)f(when)h(a)h(clause)g(is)g
       
  4058 (\223insen-)523 3946 y(siti)n(v)o(e\224)c(to)g(the)h(choice)e(of)h(a)g
       
  4059 (name)g(seems)g(unlik)o(ely)-5 b(.)21 b(Consider)m(,)g(for)g(e)o
       
  4060 (xample,)g(allo)n(wing)g(names)523 4046 y(to)27 b(only)f(occur)g(bound)
       
  4061 g(or)g(in)i(binding)d(position\227then)g(the)i FJ(type)p
       
  4062 FT(-program)d(w)o(ould)i(be)h(ruled)523 4145 y(out)c(since)g
       
  4063 FJ(x)g FT(occurs)f(free)h(in)g(the)g(body)f(of)g(the)h(clause.)g
       
  4064 (Restricting)g(free)g(names)f(to)h(occur)f(only)523 4245
       
  4065 y(in)i(the)f(body)f(of)h(a)h(clause)f(w)o(ould)g(permit)f(the)i(clause)
       
  4066 f FJ(r\(X\):-id\(X,a\))f FT(which)g FE(is)j FT(sensiti)n(v)o(e)523
       
  4067 4344 y(to)h(the)g(choice)f(of)h FJ(a)h FT(since)f FJ(id)g
       
  4068 FT(\223propagates\224)d(the)j(choice)g(for)f(the)h(name)f
       
  4069 FJ(a)i FT(back)e(to)h(the)g(head)523 4444 y(of)20 b(the)h(clause)g(\()p
       
  4070 FJ(id)f FT(is)i(de\002ned)d(in)i(the)g FJ(subst)p FT(-e)o(xample\).)c
       
  4071 (Our)j(well-formedness)f(condition)g(is)523 4544 y(therefore)h(more)h
       
  4072 (subtle;)h(it)g(is)h(a)f(test)g(whether)f(a)h(certain)g(matching)e
       
  4073 (problem)g(deri)n(v)o(ed)g(from)g(the)523 4643 y(clause)26
       
  4074 b(is)h(solv)n(able.)d(Despite)i(being)f(technically)g(relati)n(v)o(ely)
       
  4075 f(comple)o(x,)g(well-formedness)g(can)523 4743 y(be)c(automatically)f
       
  4076 (v)o(eri\002ed.)p 523 4837 473 4 v 558 4893 a Fy(3)606
       
  4077 4924 y FQ(The)g(corresponding)i(right-rule)e(has)g(been)h(sho)n(wn)g
       
  4078 (to)f(be)g(admissible)g(in)g(nominal)h(logic)f(in)g([5].)p
       
  4079 eop end
       
  4080 %%Page: 5 5
       
  4081 TeXDict begin 5 4 bop 648 448 a FT(The)31 b(paper)f(is)i(or)o(ganised)d
       
  4082 (as)j(follo)n(ws:)f(Section)g(2)h(describes)f(nominal)f(terms,)h
       
  4083 (formulae)523 548 y(and)f(the)h(inference)e(rules)i(of)g
       
  4084 FK(\013)p FT(Prolog)o(')-5 b(s)31 b(proof-search)d(procedure.)g
       
  4085 (Section)j(3)g(introduces)e(a)523 648 y(well-formedness)j(condition)f
       
  4086 (for)i(clauses)h(and)f(sho)n(ws)h(that)g(the)f FK(\031)s
       
  4087 FT(-rule)g(can)h(be)f(eliminated)523 747 y(from)f(proofs)f(in)m(v)n
       
  4088 (olving)f(only)i(well-formed)e(clauses.)j(Section)f(4)h(describes)f(ho)
       
  4089 n(w)g(the)g(well-)523 847 y(formedness)19 b(condition)h(can)g(be)h
       
  4090 (automatically)f(v)o(eri\002ed.)f(Section)i(5)g(concludes)f(and)g
       
  4091 (describes)523 946 y(future)f(w)o(ork.)523 1212 y FL(2)99
       
  4092 b(T)-9 b(erms,)26 b(F)n(ormulae)e(and)i(Pr)n(oof-Sear)n(ch)g(Rules)523
       
  4093 1404 y FT(The)e(terms)h(used)g(in)g FK(\013)p FT(Prolog)f(are)h
       
  4094 FE(nominal)e(terms)j FT(\(see)f([9])f(for)g(more)g(details\))h(as)g
       
  4095 (de\002ned)f(by)523 1495 y(the)c(grammar:)1414 1586 y
       
  4096 FN(t)h FG(::=)h FN(a)f FF(j)h FN(\031)s Fz(\001)o FN(X)28
       
  4097 b FF(j)21 b(hi)g(j)h(h)p FN(t;)13 b(t)p FF(i)21 b(j)g
       
  4098 FN(a:t)g FF(j)h Fx(f)p FG(\()p FN(t)p FG(\))523 1719
       
  4099 y FT(where)i FK(a)i FT(is)g(a)f(name,)f FK(X)32 b FT(a)25
       
  4100 b(v)n(ariable,)f Fw(f)h FT(a)g(function)f(symbol)g(and)g
       
  4101 FK(\031)29 b FT(a)c(permutation)e(e)o(xpressed)523 1819
       
  4102 y(as)g(a)f(list)h(of)e(sw)o(appings)g FH(\()p FK(a)1337
       
  4103 1831 y FS(1)1389 1819 y FK(b)1425 1831 y FS(1)1462 1819
       
  4104 y FH(\))14 b FI(\001)g(\001)g(\001)f FH(\()p FK(a)1694
       
  4105 1831 y FC(n)1754 1819 y FK(b)1790 1831 y FC(n)1834 1819
       
  4106 y FH(\))p FT(.)23 b(W)-7 b(e)23 b(ha)n(v)o(e)e(the)h(operations)e
       
  4107 FI(\000)p FH(@)p FI(\000)h FT(and)g FH(\()p FI(\000)p
       
  4108 FH(\))3196 1789 y Fv(\000)p FS(1)3308 1819 y FT(for)523
       
  4109 1919 y(composing)g(\(list)i(concatenation\))e(tw)o(o)i(permutations)e
       
  4110 (and)h(in)m(v)o(erting)f(\(list)i(re)n(v)o(ersal\))f(a)h(permu-)523
       
  4111 2018 y(tation,)30 b(respecti)n(v)o(ely)-5 b(.)28 b(Constants)i(are)h
       
  4112 (encoded)d(as)j(function)e(symbols)h(with)g(unit)h(ar)o(guments)523
       
  4113 2118 y Fw(f)p FH(\()p FI(hi)p FH(\))p FT(,)22 b(and)e
       
  4114 FK(n)p FT(-tuples)f(are)i(encoded)e(by)h(iterated)g(pairs)g
       
  4115 FI(h)p FK(t)2214 2130 y FS(1)2252 2118 y FK(;)14 b FI(\001)g(\001)g
       
  4116 (\001)f(h)p FK(t)2461 2130 y FC(n)p Fv(\000)p FS(1)2592
       
  4117 2118 y FK(;)h(t)2659 2130 y FC(n)2704 2118 y FI(ii)p
       
  4118 FT(.)21 b(F)o(ollo)n(wing)e([9],)h(we)523 2217 y(refer)f(to)h(terms)h
       
  4119 (of)e(the)h(form)f FK(\031)s Fu(\001)p FK(X)27 b FT(as)21
       
  4120 b FE(suspensions)p FT(,)e(because)g(the)h(permutation)e
       
  4121 FK(\031)24 b FT(is)d(suspended)523 2317 y(in)f(front)g(of)f(a)i(v)n
       
  4122 (ariable)e(w)o(aiting)h(to)g(be)h(applied)e(to)h(a)h(term)f
       
  4123 (substituted)f(for)h FK(X)7 b FT(.)648 2408 y(F)o(ormulae)28
       
  4124 b(are)i(di)n(vided)f(into)h(goal)g(formulae)e FK(G)k
       
  4125 FT(and)d(de\002nite)h(\(or)g(program\))d(clauses)k FK(D)523
       
  4126 2500 y FT(de\002ned)19 b(as)738 2662 y FN(G)j FG(::=)f
       
  4127 FN(p)p FG(\()p FN(t)p FG(\))g FF(j)g FN(G)p FF(^)q FN(G)g
       
  4128 FF(j)h FN(G)p FF(_)p FN(G)g FF(j)f(>)173 b FN(B)25 b
       
  4129 FG(::=)d FN(G)f FF(\033)g FN(p)p FG(\()p FN(t)p FG(\))172
       
  4130 b FN(D)24 b FG(::=)2769 2662 y
       
  4131  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4132  2769 2662 a FB(N)2769
       
  4133 2662 y
       
  4134  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4135 neg exch translate
       
  4136  2769 2662 a Ft(as)7 b FN(:)p FF(8)p Ft(Xs)f FN(:)p
       
  4137 FF(r)p FN(=B)523 2832 y FT(where)31 b FK(p)p FH(\()p
       
  4138 FK(t)p FH(\))i FT(stands)f(for)f(an)h(atomic)f(predicate)g(with)h(the)f
       
  4139 (ar)o(gument)f FK(t)i FT(\(we)g(shall)g(also)g(write)523
       
  4140 2932 y FK(A)27 b FT(for)e(such)h(formulae)e(whene)n(v)o(er)g(the)j(ar)o
       
  4141 (gument)c(is)k(unimportant\);)d FI(>)p FK(;)14 b FI(^)o
       
  4142 FK(;)g FI(_)q FK(;)g FI(\033)26 b FT(are)g(standard)523
       
  4143 3031 y(connecti)n(v)o(es;)16 b(and)i FI(r)g FT(is)h(a)f(set)h(of)f
       
  4144 (freshness)f(constraints)g(of)h(the)g(form)f FK(a)2644
       
  4145 3043 y FS(1)2704 3031 y FH(#)23 b FK(X)2865 3043 y FS(1)2902
       
  4146 3031 y FK(;)14 b(:)g(:)g(:)g(;)g(a)3131 3043 y FC(n)3199
       
  4147 3031 y FH(#)23 b FK(X)3360 3043 y FC(n)523 3131 y FT(\()p
       
  4148 FK(X)620 3143 y FC(i)676 3131 y FT(and)29 b FK(a)870
       
  4149 3143 y FC(i)927 3131 y FT(being)e(v)n(ariables)h(and)h(names,)f
       
  4150 (respecti)n(v)o(ely\).)e(The)j(intended)e(meaning)g(of)i
       
  4151 FI(r)g FT(in)523 3231 y FK(D)r FT(-formulae)22 b(is)j(that)g(a)g
       
  4152 (clause)f(is)h(applicable)e(only)h(if)g(its)h(freshness)f(constraints)g
       
  4153 (are)g(satis\002ed.)523 3330 y(F)o(or)c(freshness)f(constraints)h(and)g
       
  4154 (quanti\002er)n(-free)e(formulae)g(we)i(shall)h(use)f(the)h(notation)d
       
  4155 FK(Q)3230 3342 y Fs(as)5 b FC(;)p Fs(Xs)523 3430 y FT(\()p
       
  4156 FK(Q)22 b FH(::=)h FI(rj)p FK(G)p FI(j)p FK(B)t FT(\))c(to)f(indicate)f
       
  4157 (that)h(the)f(terms)h(of)g FK(Q)g FT(are)f(b)n(uilt)h(up)g(from)e
       
  4158 (names)i Fr(as)25 b FT(and)17 b(v)n(ariables)523 3530
       
  4159 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
       
  4160 (that)j Fr(as)31 b FT(stands)24 b(for)f(lists)i(of)e(names)g(and)g
       
  4161 Fr(Xs)31 b FT(for)23 b(lists)i(of)523 3629 y(v)n(ariables;)e(similarly)
       
  4162 1247 3629 y
       
  4163  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4164  1247 3629 a FD(N)1247 3629 y
       
  4165  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4166 neg exch translate
       
  4167  1247 3629 a
       
  4168 Fr(as)31 b FT(stands)24 b(for)1766 3629 y
       
  4169  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4170  1766 3629 a
       
  4171 FD(N)1766 3629 y
       
  4172  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4173 neg exch translate
       
  4174  1766 3629 a FK(a)1810 3641 y FS(1)1861
       
  4175 3629 y FK(:)14 b(:)g(:)2032 3629 y
       
  4176  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4177  2032 3629 a FD(N)2032
       
  4178 3629 y
       
  4179  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4180 neg exch translate
       
  4181  2032 3629 a FK(a)2076 3641 y FC(n)2145 3629 y
       
  4182 FT(and)23 b FI(8)p Fr(Xs)31 b FT(for)22 b FI(8)p FK(X)2697
       
  4183 3641 y FS(1)2748 3629 y FK(:)14 b(:)g(:)g FI(8)p FK(X)2974
       
  4184 3641 y FC(n)3019 3629 y FT(\).)24 b(W)-7 b(e)24 b(call)g(a)523
       
  4185 3729 y FK(D)r FT(-formula)16 b FE(closed)j FT(when)e(it)i(has)e(no)g
       
  4186 (free)h(v)n(ariables)e(and)h(free)h(names,)f(that)g(is)i(the)e(formula)
       
  4187 f(must)523 3828 y(be)24 b(of)g(the)h(form)1093 3828 y
       
  4188  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4189 
       
  4190 1093 3828 a FD(N)1093 3828 y
       
  4191  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4192 neg exch translate
       
  4193  1093 3828 a Fr(as)7 b FK(:)p
       
  4194 FI(8)p Fr(Xs)g FK(:)p FI(r)1440 3840 y Fs(as)e FC(;)p
       
  4195 Fs(Xs)1615 3828 y FK(=B)1720 3840 y Fs(as)t FC(;)p Fs(Xs)1894
       
  4196 3828 y FT(.)25 b(Fig.)f(1)h(sho)n(ws)f(tw)o(o)h(e)o(xamples)e
       
  4197 (illustrating)h(ho)n(w)523 3928 y FK(D)r FT(-formulae)18
       
  4198 b(relate)i(to)h(the)f FK(\013)p FT(Prolog)o(-clauses)g(gi)n(v)o(en)f
       
  4199 (at)h(the)h(be)o(ginning.)648 4028 y(There)j(is)j(a)e(delicate)h(point)
       
  4200 e(with)i(respect)f(to)h(binding:)d(while)j(in)f(nominal)f(terms)i(the)f
       
  4201 (con-)523 4127 y(structor)f FK(a:)p FH(\()p FI(\000)p
       
  4202 FH(\))h FT(is)g FE(not)g FT(a)g(binder)e(in)i(the)f(traditional)g
       
  4203 (sense)g(\(it)h(only)f FE(acts)g FT(as)h(a)g(binder\),)e(in)h(for)n(-)
       
  4204 523 4227 y(mulae)19 b(the)g(quanti\002ers)1297 4227 y
       
  4205  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4206 
       
  4207 1297 4227 a FD(N)1297 4227 y
       
  4208  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4209 neg exch translate
       
  4210  1297 4227 a FK(a:)p FH(\()p
       
  4211 FI(\000)p FH(\))g FT(and)g FI(8)p FK(X)r(:)p FH(\()p
       
  4212 FI(\000)p FH(\))g FT(do)g(bind)f FK(a)h FT(and)g FK(X)7
       
  4213 b FT(,)19 b(respecti)n(v)o(ely)-5 b(.)16 b(Therefore)h(we)523
       
  4214 4327 y(ha)n(v)o(e)24 b(the)g(usual)g(con)m(v)o(ention)d(that)j
       
  4215 (formulae)f(are)h(identi\002ed)g(if)g(the)o(y)g(only)f(dif)n(fer)g(in)i
       
  4216 (the)f(names)523 4426 y(of)g(binders)f(\(i.e.)h FI(8)p
       
  4217 FK(X)r(:)p FH(\()p FI(\000)p FH(\))g FT(and)1538 4426
       
  4218 y
       
  4219  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4220  1538 4426 a FD(N)1538 4426 y
       
  4221  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4222 neg exch translate
       
  4223  1538 4426 a FK(a:)p FH(\()p
       
  4224 FI(\000)p FH(\))p FT(\),)g(and)g(operations)e(on)i(formulae)e(need)i
       
  4225 (to)g(respect)g(this)523 4526 y(con)m(v)o(ention.)17
       
  4226 b(As)k(a)g(result)f(the)h(de\002nition)e(of)h(the)g(permutation)e
       
  4227 (operation)h(introduced)f(for)i(nom-)523 4625 y(inal)k(terms)g(in)g
       
  4228 ([9])g(needs)f(to)h(be)g(e)o(xtended.)e(W)-7 b(e)25 b(de\002ne)f(a)g
       
  4229 (generalised)f(permutation)e(operation)523 4725 y FK(\031)s
       
  4230 Fu(\001)600 4737 y Fv(B)649 4725 y FH(\()p FI(\000)p
       
  4231 FH(\))27 b FT(that)f(depends)e(on)h(a)i(set)f(of)g(v)n(ariables)f
       
  4232 FI(B)s FT(.)g(The)g(permutation)f FK(\031)29 b FT(only)c(acts)i(upon)d
       
  4233 (v)n(ari-)523 4825 y(ables)i FE(not)i FT(in)e FI(B)s
       
  4234 FT(.)f(Whene)n(v)o(er)g(a)h(permutation)e(is)j(\223pushed\224)e(under)f
       
  4235 (a)j FI(8)p FK(X)r(:)p FH(\()p FI(\000)p FH(\))p FT(-quanti\002er)m(,)d
       
  4236 (then)523 4924 y FK(X)k FT(is)21 b(added)f(to)h(the)g(set)g(of)g(v)n
       
  4237 (ariables)f(the)g(permutation)f(does)i(not)f(af)n(fect.)g(The)g
       
  4238 (de\002nition)g(of)g(the)p eop end
       
  4239 %%Page: 6 6
       
  4240 TeXDict begin 6 5 bop 523 369 2882 4 v 523 807 4 439
       
  4241 v 600 457 a FP(subst\(var\(X\),X,T,T\).)600 549 y FF(8)p
       
  4242 FN(X)r(;)12 b(T)6 b(:)22 b Fq(?)13 b FN(=)g FF(>)21 b(\033)g
       
  4243 FN(s)p FG(\()p Fx(var)p FG(\()p FN(X)6 b FG(\))p FN(;)14
       
  4244 b(X)r(;)f(T)6 b(;)13 b(T)e FG(\))600 664 y FP
       
  4245 (subst\(lam\(a.M\),X,T,lam\(a.M'\)\))39 b(:-)45 b(a)12
       
  4246 b(#)h(T,)f(a)g(#)g(X,)g(subst\(M,X,T,M'\).)653 755 y
       
  4247  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4248 
       
  4249 653 755 a FB(N)653 755 y
       
  4250  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4251 neg exch translate
       
  4252  653 755 a FN(a:)p FF(8)p FN(M)t(;)h(X)r(;)g(T)
       
  4253 6 b(;)14 b(M)1137 723 y Fp(0)1159 755 y FN(:)22 b FF(f)p
       
  4254 FN(a)f FG(#)9 b FN(T)d(;)13 b(a)21 b FG(#)9 b FN(X)d
       
  4255 FF(g)13 b FN(=)h(s)p FG(\()p FN(M)t(;)e(X)r(;)h(T)6 b(;)14
       
  4256 b(M)2211 723 y Fp(0)2233 755 y FG(\))f FF(\033)g FN(s)p
       
  4257 FG(\()p Fx(lam)p FG(\()p FN(a:M)8 b FG(\))p FN(;)13 b(X)r(;)g(T)6
       
  4258 b(;)14 b Fx(lam)p FG(\()p FN(a:M)3246 723 y Fp(0)3269
       
  4259 755 y FG(\)\))p 3402 807 4 439 v 523 810 2882 4 v 523
       
  4260 953 a FO(Fig)o(.)e(1.)j FQ(T)-6 b(w)o(o)15 b(e)o(xamples)g(sho)n(wing)i
       
  4261 (ho)n(w)e FN(\013)p FQ(Prolog)q(-clauses)g(relate)g(to)g
       
  4262 FN(D)r FQ(-formulae)g(\()p FN(s)g FQ(is)f(a)h(predicate)h(symbol)523
       
  4263 1045 y(standing)k(for)f FP(subst)p FQ(\).)e(W)-6 b(e)19
       
  4264 b(ha)o(v)o(e)g(the)g(usual)g(con)m(v)o(ention)i(that)d(clauses)i(stand)
       
  4265 f(for)g(closed)h FN(D)r FQ(-formulae.)p 523 1122 V 523
       
  4266 2297 4 1175 v 567 1211 a FO(T)-7 b(erms:)1069 1234 y
       
  4267 FG([])q Fz(\001)1136 1242 y Fp(B)1182 1234 y FN(a)1246
       
  4268 1191 y FA(def)1253 1234 y FG(=)30 b FN(a)697 1433 y FG(\(\()p
       
  4269 FN(a)798 1441 y FR(1)845 1433 y FN(a)886 1441 y FR(2)920
       
  4270 1433 y FG(\))21 b(::)h FN(\031)s FG(\))p Fz(\001)1136
       
  4271 1441 y Fp(B)1182 1433 y FN(a)1246 1390 y FA(def)1253
       
  4272 1433 y FG(=)1343 1257 y Fo(8)1343 1325 y(>)1343 1347
       
  4273 y(<)1343 1481 y(>)1343 1504 y(:)1411 1327 y FN(a)1452
       
  4274 1335 y FR(1)1644 1327 y FQ(if)c FN(\031)s Fz(\001)1779
       
  4275 1335 y Fp(B)1825 1327 y FN(a)j FG(=)g FN(a)2009 1335
       
  4276 y FR(2)1411 1436 y FN(a)1452 1444 y FR(2)1644 1436 y
       
  4277 FQ(if)d FN(\031)s Fz(\001)1779 1444 y Fp(B)1825 1436
       
  4278 y FN(a)j FG(=)g FN(a)2009 1444 y FR(1)1411 1546 y FN(\031)s
       
  4279 Fz(\001)1483 1554 y Fp(B)1528 1546 y FN(a)75 b FQ(otherwise)1036
       
  4280 1733 y FN(\031)s Fz(\001)1107 1741 y Fp(B)1153 1733 y
       
  4281 FN(X)1246 1690 y FA(def)1253 1733 y FG(=)1343 1605 y
       
  4282 Fo(\()1405 1681 y FN(X)152 b FQ(if)19 b FN(X)27 b FF(2)22
       
  4283 b(B)1405 1791 y FN(\031)s Fz(\001)o FN(X)81 b FQ(otherwise)2285
       
  4284 1234 y FN(\031)s Fz(\001)2356 1242 y Fp(B)2401 1234 y
       
  4285 FG(\()p FN(\031)2478 1203 y Fp(0)2500 1234 y Fz(\001)p
       
  4286 FN(X)6 b FG(\))2647 1191 y FA(def)2654 1234 y FG(=)31
       
  4287 b FN(\031)s FG(@)p FN(\031)2899 1203 y Fp(0)2920 1234
       
  4288 y Fz(\001)p FN(X)2388 1352 y(\031)s Fz(\001)2459 1360
       
  4289 y Fp(B)2505 1352 y FG(\()p FF(hi)p FG(\))2647 1310 y
       
  4290 FA(def)2654 1352 y FG(=)g FF(hi)2229 1470 y FN(\031)s
       
  4291 Fz(\001)2300 1478 y Fp(B)2346 1470 y FG(\()p FF(h)p FN(t)2434
       
  4292 1478 y FR(1)2468 1470 y FN(;)13 b(t)2530 1478 y FR(2)2564
       
  4293 1470 y FF(i)p FG(\))2647 1428 y FA(def)2654 1470 y FG(=)31
       
  4294 b FF(h)p FN(\031)s Fz(\001)2846 1478 y Fp(B)2891 1470
       
  4295 y FN(t)2919 1478 y FR(1)2954 1470 y FN(;)13 b(\031)s
       
  4296 Fz(\001)3059 1478 y Fp(B)3105 1470 y FN(t)3133 1478 y
       
  4297 FR(2)3167 1470 y FF(i)2321 1589 y FN(\031)s Fz(\001)2392
       
  4298 1597 y Fp(B)2438 1589 y FG(\()p Fx(f)p FG(\()p FN(t)p
       
  4299 FG(\)\))2647 1546 y FA(def)2654 1589 y FG(=)31 b Fx(f)p
       
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  4301 y FN(t)p FG(\))2358 1707 y FN(\031)s Fz(\001)2429 1715
       
  4302 y Fp(B)2475 1707 y FG(\()p FN(a:t)p FG(\))2647 1664 y
       
  4303 FA(def)2654 1707 y FG(=)g(\()p FN(\031)s Fz(\001)2846
       
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  4305 FN(\031)s Fz(\001)3084 1715 y Fp(B)3130 1707 y FN(t)p
       
  4306 FG(\))567 1890 y FO(F)n(ormulae:)996 1914 y FN(\031)s
       
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  4310 FN(t)p FG(\)\))1255 1989 y FA(def)1263 2032 y FG(=)g
       
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  4312 2032 y FN(t)p FG(\))794 2150 y FN(\031)s Fz(\001)865
       
  4313 2158 y Fp(B)910 2150 y FG(\()p FN(G)1000 2158 y FR(1)1052
       
  4314 2150 y FN(?)18 b(G)1168 2158 y FR(2)1202 2150 y FG(\))1255
       
  4315 2107 y FA(def)1263 2150 y FG(=)30 b(\()p FN(\031)s Fz(\001)1454
       
  4316 2158 y Fp(B)1500 2150 y FN(G)1560 2158 y FR(1)1594 2150
       
  4317 y FG(\))17 b FN(?)h FG(\()p FN(\031)s Fz(\001)1798 2158
       
  4318 y Fp(B)1844 2150 y FN(G)1904 2158 y FR(2)1938 2150 y
       
  4319 FG(\))1353 2241 y FQ(for)40 b FN(?)22 b FG(::=)f FF(^)q(j_)2200
       
  4320 1914 y FN(\031)s Fz(\001)2271 1922 y Fp(B)2317 1914 y
       
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  4322 1914 y FG(=)30 b(\()p FN(\031)s Fz(\001)2776 1922 y Fp(B)2822
       
  4323 1914 y FN(G)p FG(\))12 b FF(\033)h FG(\()p FN(\031)s
       
  4324 Fz(\001)3098 1922 y Fp(B)3144 1914 y FN(A)p FG(\))2213
       
  4325 2032 y FN(\031)s Fz(\001)2285 2040 y Fp(B)2330 2032 y
       
  4326 FG(\()p FF(r)p FN(=B)t FG(\))2577 1989 y FA(def)2585
       
  4327 2032 y FG(=)30 b FF(r)p FN(=)p FG(\()p FN(\031)s Fz(\001)2878
       
  4328 2040 y Fp(B)2924 2032 y FN(B)t FG(\))2196 2150 y FN(\031)s
       
  4329 Fz(\001)2267 2158 y Fp(B)2313 2150 y FG(\()2397 2150
       
  4330 y
       
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  4338  2729 2150
       
  4339 a FB(N)2729 2150 y
       
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  4341 neg exch translate
       
  4342  2729 2150 a FN(a:\031)s Fz(\001)2862
       
  4343 2158 y Fp(B)2908 2150 y FN(D)2183 2268 y(\031)s Fz(\001)2254
       
  4344 2276 y Fp(B)2300 2268 y FG(\()p FF(8)p FN(X)r(:D)r FG(\))2577
       
  4345 2225 y FA(def)2585 2268 y FG(=)g FF(8)p FN(X)r(:\031)s
       
  4346 Fz(\001)2875 2280 y Fp(f)p Fn(X)t Fp(g[B)3080 2268 y
       
  4347 FN(D)p 3402 2297 4 1175 v 523 2300 2882 4 v 523 2447
       
  4348 a FO(Fig)o(.)12 b(2.)20 b FQ(De\002nition)g(of)g(the)g(permutation)h
       
  4349 (operation)g FN(\031)s Fz(\001)2004 2455 y Fp(B)2049
       
  4350 2447 y FG(\()p FF(\000)p FG(\))f FQ(for)g(terms)g(and)g(formulae.)h(In)
       
  4351 f(the)g(clause)h(for)523 2539 y(the)16 b(ne)n(w-quanti\002er)m(,)h(it)e
       
  4352 (is)h(assumed)i(that)e FN(a)f FQ(is)h(renamed,)h(so)g(that)f(the)g
       
  4353 (permutation)h FN(\031)i FQ(can)d(safely)h(be)f(pushed)523
       
  4354 2630 y(under)k(the)f(binder)g(without)g(capture.)523
       
  4355 2917 y FT(permutation)d(operation)g(is)k(gi)n(v)o(en)c(in)j(Fig.)f(2.)g
       
  4356 (W)-7 b(e)19 b(use)g(the)f(shorthand)e(notation)h FK(\031)s
       
  4357 Fu(\001)p FH(\()p FI(\000)p FH(\))i FT(in)f(case)h FI(B)523
       
  4358 3016 y FT(is)f(the)f(empty)f(set.)h(This)g(is)h(a)f(generalisation)e
       
  4359 (of)i(the)f(permutation)f(action)h(gi)n(v)o(en)g(in)h([9];)f(ho)n(we)n
       
  4360 (v)o(er)m(,)523 3116 y(when)25 b(a)g(permutation)e(acts)j(on)e(a)i
       
  4361 (formula)d(with)j(quanti\002ers,)e(it)h(acts)h(only)e(on)h(the)g(free)g
       
  4362 (names)523 3215 y(and)20 b(free)f(v)n(ariables.)648 3307
       
  4363 y(Similar)j(problems)e(arise)i(in)h(the)f(de\002nition)e(of)i(the)g
       
  4364 (substitution)f(operation\227with)f(respect)523 3398
       
  4365 y(to)g(the)f(abstractions)g FK(a:)p FH(\()p FI(\000)p
       
  4366 FH(\))h FT(substitution)f(is)h(possibly-capturing,)c(whereas)j(with)h
       
  4367 (respect)f(to)g(the)583 3489 y
       
  4368  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4369  583 3489 a FD(N)583 3489
       
  4370 y
       
  4371  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4372 neg exch translate
       
  4373  583 3489 a FT(-)29 b(and)e FI(8)p FT(-quanti\002er)g(it)i(must)g(be)f
       
  4374 (capture-a)n(v)n(oiding.)d(F)o(or)j(terms)g(we)h(can)f(use)g(the)h
       
  4375 (de\002nition)523 3581 y(gi)n(v)o(en)e(in)i([9]:)e(a)i(substitution)f
       
  4376 FK(\033)k FT(is)d(a)g(function)e(from)g(v)n(ariables)h(to)g(terms)h
       
  4377 (with)f(the)h(property)523 3672 y(that)e FK(\033)s FH(\()p
       
  4378 FK(X)7 b FH(\))34 b(=)g FK(X)g FT(for)25 b(all)i(b)n(ut)g(\002nitely)f
       
  4379 (man)o(y)f(v)n(ariables)h FK(X)7 b FT(.)26 b(If)g(the)g(domain)f(of)i
       
  4380 FK(\033)j FT(consists)d(of)523 3763 y(distinct)e(v)n(ariables)f
       
  4381 FK(X)1185 3775 y FS(1)1222 3763 y FK(;)14 b(:)g(:)g(:)g(;)g(X)1476
       
  4382 3775 y FC(n)1546 3763 y FT(and)24 b FK(\033)s FH(\()p
       
  4383 FK(X)1842 3775 y FC(i)1871 3763 y FH(\))32 b(=)f FK(t)2061
       
  4384 3775 y FC(i)2114 3763 y FT(for)25 b FK(i)31 b FH(=)g(1)14
       
  4385 b FK(:)g(:)g(:)f(n)p FT(,)25 b(we)h(sometimes)e(write)i
       
  4386 FK(\033)523 3855 y FT(as)20 b FH([)p FK(X)704 3867 y
       
  4387 FS(1)764 3855 y FH(:=)j FK(t)905 3867 y FS(1)942 3855
       
  4388 y FK(;)14 b(:)g(:)g(:)g(;)g(X)1196 3867 y FC(n)1264 3855
       
  4389 y FH(:=)23 b FK(t)1405 3867 y FC(n)1450 3855 y FH(])p
       
  4390 FT(.)d(Moreo)o(v)o(er)m(,)c(we)k(shall)g(write)f FK(\033)s
       
  4391 FH(\()p FK(t)p FH(\))i FT(for)e(the)h(result)f(of)g(applying)f(a)523
       
  4392 3946 y(substitution)h FK(\033)24 b FT(to)c(a)g(term)f
       
  4393 FK(t)p FT(;)i(this)f(is)h(the)f(term)f(obtained)g(from)f
       
  4394 FK(t)j FT(by)e(replacing)f(each)i(v)n(ariable)f FK(X)523
       
  4395 4037 y FT(by)24 b(the)g(term)f FK(\033)s FH(\()p FK(X)7
       
  4396 b FH(\))25 b FT(and)e(each)h(suspension)f FK(\031)s Fu(\001)p
       
  4397 FK(X)30 b FT(in)25 b FK(t)f FT(by)g(the)g(term)f FK(\031)s
       
  4398 Fu(\001)p FK(\033)s FH(\()p FK(X)7 b FH(\))25 b FT(got)e(by)h(letting)g
       
  4399 FK(\031)523 4129 y FT(act)d(on)e(the)i(term)f FK(\033)s
       
  4400 FH(\()p FK(X)7 b FH(\))p FT(.)20 b(This)h(de\002nition)e(is)i(e)o
       
  4401 (xtended)d(to)i(formulae)f(as)i(follo)n(ws:)1160 4327
       
  4402 y FN(\033)s FG(\()p FF(>)p FG(\))1349 4284 y FA(def)1356
       
  4403 4327 y FG(=)30 b FF(>)1093 4445 y FN(\033)s FG(\()p FN(p)p
       
  4404 FG(\()p FN(t)p FG(\)\))1349 4402 y FA(def)1356 4445 y
       
  4405 FG(=)g FN(p)p FG(\()p FN(\033)s FG(\()p FN(t)p FG(\)\))957
       
  4406 4563 y FN(\033)s FG(\()p FN(G)1094 4571 y FR(1)1146 4563
       
  4407 y FN(?)17 b(G)1261 4571 y FR(2)1296 4563 y FG(\))1349
       
  4408 4520 y FA(def)1356 4563 y FG(=)30 b FN(\033)s FG(\()p
       
  4409 FN(G)1583 4571 y FR(1)1618 4563 y FG(\))17 b FN(?)g(\033)s
       
  4410 FG(\()p FN(G)1857 4571 y FR(2)1891 4563 y FG(\))1520
       
  4411 4654 y FQ(for)i FN(?)i FG(::=)h FF(^j_)2150 4313 y FN(\033)s
       
  4412 FG(\()p FN(G)p FF(\033)p FN(A)p FG(\))2457 4271 y FA(def)2465
       
  4413 4313 y FG(=)30 b FN(\033)s FG(\()p FN(G)p FG(\))12 b
       
  4414 FF(\033)h FN(\033)s FG(\()p FN(A)p FG(\))2164 4431 y
       
  4415 FN(\033)s FG(\()p FF(r)p FN(=B)t FG(\))2457 4389 y FA(def)2465
       
  4416 4431 y FG(=)30 b FN(\033)s FG(\()p FF(r)p FG(\))p FN(=\033)s
       
  4417 FG(\()p FN(B)t FG(\))2146 4549 y FN(\033)s FG(\()2277
       
  4418 4549 y
       
  4419  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4420  2277 4549 a FB(N)2277 4549 y
       
  4421  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4422 neg exch translate
       
  4423  2277 4549 a FN(a:D)r
       
  4424 FG(\))2457 4507 y FA(def)2465 4549 y FG(=)2609 4549 y
       
  4425  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4426 
       
  4427 2609 4549 a FB(N)2609 4549 y
       
  4428  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4429 neg exch translate
       
  4430  2609 4549 a FN(a:\033)s
       
  4431 FG(\()p FN(D)r FG(\))2133 4668 y FN(\033)s FG(\()p FF(8)p
       
  4432 FN(X)r(:D)r FG(\))2457 4625 y FA(def)2465 4668 y FG(=)g
       
  4433 FF(8)p FN(X)r(:\033)s FG(\()p FN(D)r FG(\))523 4825 y
       
  4434 FT(with)c(the)h(pro)o(viso)d(that)j(the)f(quanti\002ed)f(names)h(and)g
       
  4435 (v)n(ariables)f(are)h(suitably)g(renamed)f(so)h(that)523
       
  4436 4924 y(no)33 b(capturing)e(is)j(possible.)e(F)o(or)g(e)o(xample,)g(if)h
       
  4437 FK(\033)50 b FH(=)c([)p FK(X)53 b FH(:=)47 b FI(h)p FK(a;)14
       
  4438 b(Y)k FI(i)p FH(])34 b FT(and)f FK(t)46 b FH(=)g FK(a:X)7
       
  4439 b FT(,)33 b(then)p eop end
       
  4440 %%Page: 7 7
       
  4441 TeXDict begin 7 6 bop 523 369 2882 4 v 523 1776 4 1408
       
  4442 v 551 489 376 4 v 551 562 a FF(r)22 b(`)f(hi)g(\031)g(hi)943
       
  4443 508 y FG(\()p FF(\031)p FQ(-unit)p FG(\))1261 453 y FF(r)h(`)f
       
  4444 FN(t)1443 461 y FR(1)1499 453 y FF(\031)g FN(t)1608 421
       
  4445 y Fp(0)1608 466 y FR(1)1717 453 y FF(r)g(`)g FN(t)1898
       
  4446 461 y FR(2)1954 453 y FF(\031)g FN(t)2063 421 y Fp(0)2063
       
  4447 466 y FR(2)p 1261 486 836 4 v 1333 562 a FF(r)g(`)g(h)p
       
  4448 FN(t)1544 570 y FR(1)1579 562 y FN(;)13 b(t)1641 570
       
  4449 y FR(2)1675 562 y FF(i)21 b(\031)g(h)p FN(t)1865 530
       
  4450 y Fp(0)1865 575 y FR(1)1900 562 y FN(;)13 b(t)1962 530
       
  4451 y Fp(0)1962 575 y FR(2)1996 562 y FF(i)2113 505 y FG(\()p
       
  4452 FF(\031)p FQ(-pair)p FG(\))2484 466 y FF(r)21 b(`)g FN(t)h
       
  4453 FF(\031)f FN(t)2796 434 y Fp(0)p 2432 486 438 4 v 2432
       
  4454 562 a FF(r)g(`)g Fx(f)13 b FN(t)22 b FF(\031)f Fx(f)13
       
  4455 b FN(t)2848 530 y Fp(0)2886 505 y FG(\()p FF(\031)p FQ(-fun.)18
       
  4456 b(symbol)p FG(\))750 703 y FF(r)j(`)g FN(t)g FF(\031)h
       
  4457 FN(t)1062 671 y Fp(0)p 688 723 458 4 v 688 799 a FF(r)f(`)g
       
  4458 FN(a:t)g FF(\031)h FN(a:t)1124 767 y Fp(0)1187 742 y
       
  4459 FG(\()p FF(\031)p FQ(-abs-1)p FG(\))1661 684 y FN(a)g
       
  4460 FF(6)p FG(=)f FN(a)1846 653 y Fp(0)1942 684 y FF(r)h(`)f
       
  4461 FN(t)g FF(\031)g FG(\()p FN(a)12 b(a)2350 653 y Fp(0)2372
       
  4462 684 y FG(\))p Fz(\001)p FN(t)2455 653 y Fp(0)2552 684
       
  4463 y FF(r)21 b(`)g FN(a)g FG(#)h FN(t)2881 653 y Fp(0)p
       
  4464 1661 723 1242 4 v 2042 799 a FF(r)f(`)h FN(a:t)f FF(\031)g
       
  4465 FN(a)2429 767 y Fp(0)2451 799 y FN(:t)2500 767 y Fp(0)2944
       
  4466 742 y FG(\()p FF(\031)p FQ(-abs-2)p FG(\))p 726 967 338
       
  4467 4 v 726 1036 a FF(r)g(`)g FN(a)g FF(\031)h FN(a)1105
       
  4468 986 y FG(\()p FF(\031)p FQ(-name)p FG(\))1621 921 y(\()p
       
  4469 FN(a)e FG(#)i FN(X)6 b FG(\))21 b FF(2)h(r)f FQ(for)e(all)i
       
  4470 FN(a)g FF(2)g Ft(ds)7 b FG(\()p FN(\031)s(;)13 b(\031)2647
       
  4471 889 y Fp(0)2669 921 y FG(\))p 1621 960 1079 4 v 1880
       
  4472 1036 a FF(r)21 b(`)g FN(\031)s Fz(\001)p FN(X)27 b FF(\031)21
       
  4473 b FN(\031)2323 1004 y Fp(0)2345 1036 y Fz(\001)p FN(X)2740
       
  4474 978 y FG(\()p FF(\031)p FQ(-suspension)p FG(\))p 598
       
  4475 1185 361 4 v 598 1258 a FF(r)h(`)f FN(a)g FG(#)g FF(hi)975
       
  4476 1204 y FG(\(#)p FQ(-unit)p FG(\))1298 1150 y FF(r)g(`)g
       
  4477 FN(a)g FG(#)h FN(t)1627 1158 y FR(1)1736 1150 y FF(r)f(`)g
       
  4478 FN(a)g FG(#)h FN(t)2065 1158 y FR(2)p 1298 1185 802 4
       
  4479 v 1439 1258 a FF(r)f(`)g FN(a)g FG(#)h FF(h)p FN(t)1798
       
  4480 1266 y FR(1)1832 1258 y FN(;)13 b(t)1894 1266 y FR(2)1928
       
  4481 1258 y FF(i)2115 1204 y FG(\(#)p FQ(-pair)p FG(\))2464
       
  4482 1155 y FF(r)21 b(`)g FN(a)g FG(#)h FN(t)p 2438 1189 381
       
  4483 4 v 2438 1258 a FF(r)f(`)g FN(a)g FG(#)h Fx(f)13 b FN(t)2834
       
  4484 1208 y FG(\(#)p FQ(-fun.)19 b(symbol)p FG(\))p 1012 1422
       
  4485 391 4 v 1012 1490 a FF(r)j(`)f FN(a)g FG(#)g FN(a:t)1444
       
  4486 1440 y FG(\(#)p FQ(-abs-1)p FG(\))1965 1380 y FN(a)g
       
  4487 FF(6)p FG(=)g FN(a)2149 1348 y Fp(0)2245 1380 y FF(r)h(`)f
       
  4488 FN(a)g FG(#)g FN(t)p 1965 1414 610 4 v 2063 1490 a FF(r)g(`)g
       
  4489 FN(a)g FG(#)h FN(a)2405 1459 y Fp(0)2427 1490 y FN(:t)2616
       
  4490 1433 y FG(\(#)p FQ(-abs-2)p FG(\))1032 1614 y FN(a)f
       
  4491 FF(6)p FG(=)g FN(a)1216 1582 y Fp(0)p 953 1648 364 4
       
  4492 v 953 1724 a FF(r)h(`)f FN(a)g FG(#)g FN(a)1295 1692
       
  4493 y Fp(0)1359 1667 y FG(\(#)p FQ(-name)p FG(\))1879 1617
       
  4494 y(\()p FN(\031)1956 1585 y Fp(\000)p FR(1)2038 1617 y
       
  4495 Fz(\001)p FN(a)g FG(#)g FN(X)6 b FG(\))21 b FF(2)h(r)p
       
  4496 1879 1656 588 4 v 1952 1724 a(r)g(`)f FN(a)g FG(#)g FN(\031)s
       
  4497 Fz(\001)o FN(X)2508 1674 y FG(\(#)p FQ(-suspension)p
       
  4498 FG(\))p 3402 1776 4 1408 v 523 1779 2882 4 v 668 1923
       
  4499 a FO(Fig)o(.)12 b(3.)19 b FQ(Inducti)n(v)o(e)h(de\002nitions)f(for)g
       
  4500 FF(\031)f FQ(and)i FG(#)p FQ(.)e(The)h(reader)g(is)g(referred)g(to)g
       
  4501 ([9])g(for)f(more)i(details.)523 2215 y FK(\033)s FH(\()p
       
  4502 FK(t)p FH(\))32 b(=)f FK(a:)p FI(h)p FK(a;)14 b(Y)19
       
  4503 b FI(i)p FT(,)25 b(b)n(ut)g(if)g FK(D)33 b FH(=)1584
       
  4504 2215 y
       
  4505  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4506  1584 2215 a FD(N)1584 2215 y
       
  4507  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4508 neg exch translate
       
  4509  1584 2215 a FK(a:)p
       
  4510 FI(8)p FK(Y)5 b(:)p Fm(?)p FK(=)p FI(>)14 b(\033)e FK(A)p
       
  4511 FH(\()p FK(a;)i(Y)5 b(;)14 b(X)7 b FH(\))25 b FT(then)f(forming)f
       
  4512 FK(\033)s FH(\()p FK(D)r FH(\))j FT(gi)n(v)o(es)e(the)523
       
  4513 2314 y(formula)866 2314 y
       
  4514  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4515  866 2314 a FD(N)866 2314 y
       
  4516  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4517 neg exch translate
       
  4518 
       
  4519 866 2314 a FK(a)910 2284 y Fv(0)934 2314 y FK(:)p FI(8)p
       
  4520 FK(Y)1069 2284 y Fv(0)1093 2314 y FK(:)p Fm(?)p FK(=)p
       
  4521 FI(>)14 b(\033)e FK(A)p FH(\()p FK(a)1517 2284 y Fv(0)1540
       
  4522 2314 y FK(;)i(Y)1644 2284 y Fv(0)1667 2314 y FK(;)g FI(h)p
       
  4523 FK(a;)g(Y)19 b FI(i)p FH(\))p FT(.)i(W)-7 b(e)21 b(use)f(the)h
       
  4524 (notation)d FK(\033)s FH(\()p FI(r)p FH(\))k FT(to)e(mean)g(that)g(e)n
       
  4525 (v-)523 2414 y(ery)g(freshness)f(constraint)h FK(a)j
       
  4526 FH(#)g FK(X)k FT(in)20 b FI(r)h FT(is)g(replaced)e(by)h
       
  4527 FK(a)j FH(#)g FK(\033)s FH(\()p FK(X)7 b FH(\))p FT(.)648
       
  4528 2516 y(It)34 b(is)i(crucial)e(for)f(programming)e(in)k
       
  4529 FK(\013)p FT(Prolog)f(that)g(abstractions)g FK(a:)p FH(\()p
       
  4530 FI(\000)p FH(\))h FT(ha)n(v)o(e)f(concrete)523 2615 y(names.)d(This)i
       
  4531 (allo)n(ws)f(us)g(to)g(formulate)f(the)h FJ(type)p FT(-clause)f(for)g
       
  4532 (lambda-abstractions)e(in)j(the)523 2715 y(usual)25 b(f)o(ashion)f
       
  4533 (whereby)g(the)h(abstracted)f(name)h FJ(x)g FT(and)g(its)h(type)f(is)h
       
  4534 (just)f(added)f(to)i(the)f(conte)o(xt)523 2815 y FJ(Gamma)p
       
  4535 FT(.)c(Furthermore,)d(the)j(w)o(ork)g(reported)e(in)j([9])e(pro)o
       
  4536 (vides)g(us)h(with)h(a)f(simple)g(algorithm)f(for)523
       
  4537 2914 y(unifying)g(nominal)h(terms.)h(This)h(uni\002cation)e(algorithm)g
       
  4538 (does)h(not)g(calculate)g(uni\002ers)f(to)i(mak)o(e)523
       
  4539 3014 y(nominal)29 b(terms)i(syntactically)f(equal,)g(b)n(ut)g(equal)g
       
  4540 (modulo)f(an)i(equi)n(v)n(alence)d(relation)i FI(\031)p
       
  4541 FT(.)h(F)o(or)523 3114 y(e)o(xample)23 b(when)h(unifying)e(the)i(tw)o
       
  4542 (o)h(terms)f FK(a:a)31 b FI(\031)o FH(?)g FK(b:X)7 b
       
  4543 FT(,)24 b(the)g(nominal)f(uni\002cation)g(algorithm)523
       
  4544 3213 y(produces)i(the)h(uni\002er)f FH([)p FK(X)41 b
       
  4545 FH(:=)34 b FK(b)p FH(])p FT(.)26 b(While)h(the)f(relation)f
       
  4546 FI(\031)i FT(is)g(intended)e(to)h(capture)f(the)h(\(tradi-)523
       
  4547 3313 y(tional\))h(notion)f(of)i FK(\013)p FT(-equi)n(v)n(alence,)d(it)j
       
  4548 (is)h(in)f(f)o(act)f(a)h(more)f(general)g(relation.)f(F)o(or)h(e)o
       
  4549 (xample,)f FI(\031)523 3413 y FT(is)g(not)f(just)g(a)g(relation)f
       
  4550 (between)h(tw)o(o)g(nominal)e(terms,)i(b)n(ut)g(a)g(relation)g(that)g
       
  4551 (depends)e(on)i(some)523 3512 y(freshness)18 b(constraints)g
       
  4552 FI(r)p FT(.)h(Figure)f(3)h(gi)n(v)o(es)f(a)h(syntax-directed)d(inducti)
       
  4553 n(v)o(e)h(de\002nition)h(for)g(judge-)523 3612 y(ments)i(of)h(the)f
       
  4554 (form)f FI(r)24 b(`)f FH(\()p FI(\000)p FH(\))h FI(\031)f
       
  4555 FH(\()p FI(\000)p FH(\))p FT(,)e(which)f(asserts)h(that)g(tw)o(o)g
       
  4556 (terms)f(are)g FI(\031)p FT(-equal)f(under)g(the)523
       
  4557 3711 y(hypotheses)h FI(r)p FT(;)i(the)g(de\002nition)e(depends)h(on)g
       
  4558 (the)h(auxiliary)e(relation)h FI(r)26 b(`)f FH(\()p FI(\000)p
       
  4559 FH(\))h(#)g(\()p FI(\000)p FH(\))p FT(,)c(which)523 3811
       
  4560 y(de\002nes)17 b(when)g(a)h(name)f(is)i FE(fr)m(esh)f
       
  4561 FT(for)f(a)h(term)f(under)g(some)g(hypotheses.)f(This)h(de\002nition)g
       
  4562 (depends)523 3911 y(on)k(the)g(auxiliary)e(notion)h(of)h(a)g
       
  4563 (disagreement)e(set,)j FK(ds)p FT(,)f(between)f(tw)o(o)h(permutations)e
       
  4564 (\(the)i(set)h(of)523 4010 y(names)e(on)g(which)f(the)i(permutations)d
       
  4565 (disagree\))h(gi)n(v)o(en)g(by:)h FI(f)p FK(a)i FI(j)h
       
  4566 FK(\031)2515 4022 y FS(1)2553 4010 y Fu(\001)o FK(a)g
       
  4567 FI(6)p FH(=)g FK(\031)2781 4022 y FS(2)2818 4010 y Fu(\001)p
       
  4568 FK(a)p FI(g)p FT(.)648 4104 y(W)-7 b(e)21 b(can)f(e)o(xtend)f
       
  4569 FI(\031)h FT(to)g FE(quanti\002er)n(-fr)m(ee)f FK(G)p
       
  4570 FT(-formulae)f(as)j(follo)n(ws:)p 610 4292 376 4 v 610
       
  4571 4360 a FF(r)h(`)f(>)g(\031)g(>)1252 4264 y(r)g(`)h FN(t)f
       
  4572 FF(\031)g FN(t)1564 4233 y Fp(0)p 1154 4284 531 4 v 1154
       
  4573 4360 a FF(r)g(`)g FN(p)p FG(\()p FN(t)p FG(\))g FF(\031)g
       
  4574 FN(p)p FG(\()p FN(t)1633 4329 y Fp(0)1654 4360 y FG(\))1852
       
  4575 4264 y FF(r)h(`)f FN(G)2066 4272 y FR(1)2122 4264 y FF(\031)g
       
  4576 FN(G)2263 4272 y FR(3)2372 4264 y FF(r)h(`)f FN(G)2586
       
  4577 4272 y FR(2)2642 4264 y FF(\031)g FN(G)2783 4272 y FR(4)p
       
  4578 1852 4292 966 4 v 1945 4360 a FF(r)g(`)h FN(G)2159 4368
       
  4579 y FR(1)2211 4360 y FN(?)17 b(G)2326 4368 y FR(2)2382
       
  4580 4360 y FF(\031)k FN(G)2523 4368 y FR(3)2575 4360 y FN(?)c(G)2690
       
  4581 4368 y FR(4)2902 4310 y FQ(for)22 b FN(?)f FG(::=)h FF(^j_)523
       
  4582 4526 y FT(The)28 b(adv)n(antage)e(of)i(setting)g(up)g(the)h(formalism)e
       
  4583 (in)h(this)h(w)o(ay)f(is)h(that)g(the)f FI(\031)p FT(-equi)n(v)n
       
  4584 (alence)d(has)523 4625 y(a)h(number)d(of)i(good)f(properties,)f(which)i
       
  4585 (will)h(play)f(an)g(important)f(rle)h(in)g(our)g(proof)f(for)g(sho)n
       
  4586 (w-)523 4725 y(ing)30 b(that)g(the)h FK(\031)s FT(-rule)e(can)h(be)h
       
  4587 (eliminated.)e(F)o(or)h(e)o(xample,)e FI(\031)j FT(is)g(preserv)o(ed)d
       
  4588 (under)h(\(possibly-)523 4825 y(capturing\))f(substitutions)i(and)g
       
  4589 (beha)n(v)o(es)f(well)i(with)g(respect)f(to)g(the)g(permutation)f
       
  4590 (operation.)523 4924 y(This)20 b(is)i(made)d(precise)h(in)g(the)h
       
  4591 (follo)n(wing)d(lemma.)p eop end
       
  4592 %%Page: 8 8
       
  4593 TeXDict begin 8 7 bop 523 369 2882 4 v 523 1244 4 875
       
  4594 v 551 521 335 4 v 551 590 a FF(r)p FG(;)13 b FN(\000)32
       
  4595 b FF(\))21 b(>)902 543 y(>)962 551 y Fn(R)1070 479 y
       
  4596 FF(r)p FG(;)13 b FN(\000)32 b FF(\))21 b FN(G)75 b FF(r)p
       
  4597 FG(;)13 b FN(\000)32 b FF(\))21 b FN(G)1815 448 y Fp(0)p
       
  4598 1070 514 768 4 v 1220 590 a FF(r)p FG(;)13 b FN(\000)32
       
  4599 b FF(\))21 b FN(G)p FF(^)p FN(G)1666 558 y Fp(0)1854
       
  4600 530 y FF(^)1905 539 y Fn(R)2090 488 y FF(r)p FG(;)14
       
  4601 b FN(\000)31 b FF(\))22 b FN(G)2426 496 y Fn(i)p 2013
       
  4602 522 516 4 v 2013 590 a FF(r)p FG(;)13 b FN(\000)32 b
       
  4603 FF(\))21 b FN(G)2348 598 y FR(1)2383 590 y FF(_)p FN(G)2494
       
  4604 598 y FR(2)2545 539 y FF(_)2596 547 y Fn(R)t(i)2731 479
       
  4605 y FF(r)p FG(;)13 b FN(D)r(;)h(\000)3027 436 y Fn(D)3008
       
  4606 479 y FF(\000)-36 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))p
       
  4607 2731 517 525 4 v 2743 590 a FF(r)p FG(;)13 b FN(D)r(;)h(\000)32
       
  4608 b FF(\))21 b FN(p)p FG(\()p FN(t)p FG(\))3297 542 y FQ(Sel)1006
       
  4609 760 y FF(r)g(`)g FN(t)1187 729 y Fp(0)1231 760 y FF(\031)g
       
  4610 FN(t)p 923 780 501 4 v 923 898 a FF(r)p FG(;)13 b FN(\000)1119
       
  4611 849 y Fn(p)p FR(\()p Fn(t)1199 828 y Fl(0)1221 849 y
       
  4612 FR(\))1100 898 y FF(\000)-33 b(\000)-17 b(\000)-31 b(!)21
       
  4613 b FN(p)p FG(\()p FN(t)p FG(\))1464 805 y FQ(Ax)1649 742
       
  4614 y FF(r)g(`)h(r)1867 710 y Fp(0)1964 742 y FF(r)p FG(;)13
       
  4615 b FN(\000)32 b FF(\))21 b FN(G)75 b FF(r)p FG(;)13 b
       
  4616 FN(\000)2570 693 y Fn(p)p FR(\()p Fn(t)2650 672 y Fl(0)2672
       
  4617 693 y FR(\))2551 742 y FF(\000)-33 b(\000)-17 b(\000)-31
       
  4618 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))p 1649 780 1225
       
  4619 4 v 1911 898 a FF(r)p FG(;)13 b FN(\000)2107 849 y Fp(r)2158
       
  4620 828 y Fl(0)2181 849 y Fn(=G)p Fp(\033)p Fn(p)p FR(\()p
       
  4621 Fn(t)2388 828 y Fl(0)2410 849 y FR(\))2088 898 y FF(\000)-39
       
  4622 b(\000)-17 b(\000)g(\000)f(\000)h(\000)g(\000)f(\000)-37
       
  4623 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))2915 797 y FF(\033)2975
       
  4624 805 y Fn(L)728 1050 y FF(r)p FG(;)13 b FN(\000)924 1002
       
  4625 y Fn(D)r FR([)p Fn(X)t FR(:=)p Fn(t)1137 981 y Fl(0)1157
       
  4626 1002 y FR(])905 1050 y FF(\000)-34 b(\000)-18 b(\000)h(\000)g(\000)f
       
  4627 (\000)-33 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))p 728
       
  4628 1089 625 4 v 775 1191 a FF(r)p FG(;)13 b FN(\000)971
       
  4629 1149 y Fp(8)p Fn(X)q(:D)952 1191 y FF(\000)-39 b(\000)-17
       
  4630 b(\000)g(\000)-38 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))1394
       
  4631 1111 y FF(8)1437 1119 y Fn(L)1576 1043 y FN(b)g FG(#)g
       
  4632 Ft(Xs)7 b FN(;)13 b FF(r)p FG(;)g FN(\000)2041 994 y
       
  4633 FR(\()p Fn(a)d(b)p FR(\))p Fk(\001)p Fn(D)2021 1043 y
       
  4634 FF(\000)-21 b(\000)k(\000)f(\000)e(!)21 b FN(p)p FG(\()p
       
  4635 FN(t)p FG(\))p 1576 1082 834 4 v 1733 1191 a FF(r)p FG(;)13
       
  4636 b FN(\000)1971 1149 y
       
  4637  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4638  1971 1149 a Fj(N)1971 1149 y
       
  4639  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4640 neg exch translate
       
  4641  1971
       
  4642 1149 a Fn(a:D)1910 1191 y FF(\000)-23 b(\000)-17 b(\000)-22
       
  4643 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))2505 1104 y
       
  4644  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4645  2505
       
  4646 1104 a FB(N)2505 1104 y
       
  4647  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4648 neg exch translate
       
  4649  2505 1104 a 9 x Fn(L)2644 1051
       
  4650 y FF(r)p FG(;)14 b FN(\000)2841 1008 y Fn(\031)r Fk(\001)o
       
  4651 Fn(D)2821 1051 y FF(\000)-19 b(\000)h(!)21 b FN(p)p FG(\()p
       
  4652 FN(t)p FG(\))p 2644 1090 484 4 v 2674 1191 a FF(r)p FG(;)13
       
  4653 b FN(\000)2870 1149 y Fn(D)2851 1191 y FF(\000)-36 b(!)21
       
  4654 b FN(p)p FG(\()p FN(t)p FG(\))3170 1106 y FN(\031)p 3402
       
  4655 1244 4 875 v 523 1247 2882 4 v 523 1390 a FO(Fig)o(.)12
       
  4656 b(4.)19 b FQ(Proof-search)g(rules)g(of)g FN(\013)p FQ(Prolog.)f(In)h
       
  4657 (the)g(ne)n(w-left)g(rule)g(it)f(is)g(assumed)i(that)f
       
  4658 FN(b)g FQ(is)f(a)h(fresh)g(name)g(not)523 1481 y(occurring)h(in)f(the)g
       
  4659 (conclusion)h(and)g Ft(Xs)25 b FQ(are)19 b(all)f(free)h(v)n(ariables)h
       
  4660 (in)e FN(\000)30 b FQ(and)19 b FN(p)p FG(\()p FN(t)p
       
  4661 FG(\))p FQ(.)523 1762 y Fi(Lemma)i(1.)41 b FE(The)20
       
  4662 b(permutation)f(and)g(substitution)g(oper)o(ations)g(pr)m(eserve)h
       
  4663 FI(\031)h FE(in)f(the)g(sense)h(that)544 1929 y(\(i\))41
       
  4664 b(if)21 b FI(r)i(`)g FK(t)g FI(\031)g FK(t)1068 1899
       
  4665 y Fv(0)1112 1929 y FE(then)d FI(r)j(`)g FK(\031)s Fu(\001)o
       
  4666 FK(t)g FI(\031)g FK(\031)s Fu(\001)p FK(t)1766 1899 y
       
  4667 Fv(0)1810 1929 y FE(for)e(all)f(permutations)f FK(\031)24
       
  4668 b FE(and)521 2070 y(\(ii\))41 b(if)28 b FI(r)37 b(`)f
       
  4669 FK(t)h FI(\031)f FK(t)1129 2040 y Fv(0)1152 2070 y FE(,)28
       
  4670 b(then)f FI(r)1441 2040 y Fv(0)1501 2070 y FI(`)36 b
       
  4671 FK(\033)s FH(\()p FK(t)p FH(\))h FI(\031)f FK(\033)s
       
  4672 FH(\()p FK(t)1982 2040 y Fv(0)2006 2070 y FH(\))29 b
       
  4673 FE(for)f(all)f(substitutions)g FK(\033)32 b FE(with)c
       
  4674 FI(r)3074 2040 y Fv(0)3134 2070 y FI(`)36 b FK(\033)s
       
  4675 FH(\()p FI(r)p FH(\))664 2170 y FE(\(wher)m(eby)20 b
       
  4676 FI(r)1060 2140 y Fv(0)1106 2170 y FI(`)j FK(\033)s FH(\()p
       
  4677 FI(r)p FH(\))f FE(means)e(that)g FI(r)1837 2140 y Fv(0)1883
       
  4678 2170 y FI(`)j FK(a)g FH(#)g FK(\033)s FH(\()p FK(X)7
       
  4679 b FH(\))21 b FE(holds)f(for)h(eac)o(h)e FH(\()p FK(a)k
       
  4680 FH(#)g FK(X)7 b FH(\))23 b FI(2)h(r)p FE(\).)523 2340
       
  4681 y FT(The)19 b(proof)e(of)h(these)h(tw)o(o)h(f)o(acts)f(are)g(a)g(minor)
       
  4682 f(e)o(xtension)f(of)i(the)g(proofs)e(gi)n(v)o(en)h(for)g([9];)g(the)o
       
  4683 (y)g(hold)523 2439 y(because)24 b(permutations)f(are)i(bijections)g(on)
       
  4684 f(names,)h(and)f(substitutions)g(act)i(on)e(v)n(ariables)g(only)523
       
  4685 2539 y(\(not)k(names\).)g(The)h(properties)f(stated)h(in)g(Lemma)f(1)i
       
  4686 (should)e(be)h(compared)d(with)k(the)f(notion)523 2639
       
  4687 y(of)j FK(\013)p FT(-equi)n(v)n(alence)e(we)i(imposed)f(\(on)g(the)h
       
  4688 (meta-le)n(v)o(el\))f(on)g(quanti\002ed)g FK(D)r FT(-formulae.)f
       
  4689 (There,)523 2738 y(whene)n(v)o(er)f(a)j(permutation)d(or)i
       
  4690 (substitution)f(is)i(pushed)e(under)g(a)h(binder)m(,)f(we)h(might)g(ha)
       
  4691 n(v)o(e)f(to)523 2838 y(rename)19 b(its)i(binder)e(in)i(order)d(to)j(a)
       
  4692 n(v)n(oid)f(possible)g(capture.)648 2929 y(Ne)o(xt)26
       
  4693 b(we)h(introduce)e(the)h(inference)f(rules)i(on)f(which)g(proof-search)
       
  4694 e(is)j(based)f(in)h FK(\013)p FT(Prolog)523 3051 y(\(see)f(Figure)g
       
  4695 (4\).)f(Sequents)h(are)f(of)h(the)g(form)f FI(r)p FH(;)14
       
  4696 b FK(\000)46 b FI(\))34 b FK(G)27 b FT(or)e FI(r)p FH(;)14
       
  4697 b FK(\000)2640 3003 y FC(D)2619 3051 y FI(\000)-39 b(!)34
       
  4698 b FK(p)p FH(\()p FK(t)p FH(\))27 b FT(where)e(the)h(for)n(-)523
       
  4699 3142 y(mer)18 b(models)g FE(goal-dir)m(ected)e(pr)l(oof-sear)m(c)o(h)h
       
  4700 FT(and)g(the)i(latter)f(models)g FE(focused)g(bac)n(kc)o(haining)d
       
  4701 FT(\(the)523 3233 y(formula)21 b(abo)o(v)o(e)g(the)i(sequent)f(arro)n
       
  4702 (w)g(is)h(usually)f(called)h(the)g FE(stoup)p FT(-formula\).)c(These)j
       
  4703 (inference)523 3324 y(rules)j(are)f(adapted)f(from)h(a)h(standard)e
       
  4704 (focusing)g(approach)g(to)i(\002rst-order)e(logic)h(programming)523
       
  4705 3416 y(\(for)j(e)o(xample)g([4]\).)1132 3386 y FA(4)1192
       
  4706 3416 y FT(The)h(main)g(no)o(v)o(elty)e(of)i(these)g(rules)g(is)h(the)f
       
  4707 (presence)f(of)h(the)g(freshness-)523 3507 y(constraints)20
       
  4708 b FI(r)p FT(.)g(T)m(raditionally)f(axiom)g(rules)h(are)g(formulated)e
       
  4709 (as)p 893 3660 486 4 v 893 3736 a FN(p)p FG(\()p FN(t)990
       
  4710 3704 y Fp(0)1012 3736 y FG(\))p FN(;)13 b(\000)31 b FF(\))22
       
  4711 b FN(p)p FG(\()p FN(t)p FG(\))1420 3685 y FQ(Ax)1511
       
  4712 3736 y FN(;)88 b FT(or)20 b(in)g(focusing)f(proofs)g(as)p
       
  4713 2479 3619 402 4 v 75 w FN(\000)2577 3687 y Fn(p)p FR(\()p
       
  4714 Fn(t)2657 3667 y Fl(0)2679 3687 y FR(\))2558 3736 y FF(\000)-33
       
  4715 b(\000)-17 b(\000)-31 b(!)21 b FN(p)p FG(\()p FN(t)p
       
  4716 FG(\))2922 3644 y FQ(Ax)3014 3736 y FN(;)523 3887 y FT(where)h(the)h
       
  4717 (terms)g FK(t)g FT(and)f FK(t)1308 3857 y Fv(0)1355 3887
       
  4718 y FT(need)g(to)h(be)f(syntactically)g(equal.)g(In)h FK(\013)p
       
  4719 FT(Prolog)f(this)h(requirement)e(is)523 3979 y(relax)o(ed:)j(terms)h
       
  4720 (only)g(need)f(to)i(be)f(being)f(equal)h(modulo)e FI(\031)p
       
  4721 FT(.)i(Ho)n(we)n(v)o(er)m(,)e FI(\031)j FT(only)e(mak)o(es)h(sense)523
       
  4722 4070 y(in)d(the)f(conte)o(xt)f(of)h(some)g(freshness)g(constraints.)f
       
  4723 (Consequently)-5 b(,)19 b(in)j FK(\013)p FT(Prolog)o(,)f(the)h
       
  4724 (axiom-rule)523 4161 y(tak)o(es)f(the)f(form)1731 4230
       
  4725 y FF(r)h(`)g FN(t)1912 4198 y Fp(0)1956 4230 y FF(\031)g
       
  4726 FN(t)p 1648 4250 501 4 v 1648 4368 a FF(r)p FG(;)13 b
       
  4727 FN(\000)1844 4319 y Fn(p)p FR(\()p Fn(t)1924 4298 y Fl(0)1946
       
  4728 4319 y FR(\))1825 4368 y FF(\000)-33 b(\000)-17 b(\000)-32
       
  4729 b(!)22 b FN(p)p FG(\()p FN(t)p FG(\))2189 4275 y FQ(Ax)523
       
  4730 4495 y FT(where)g(the)h(conte)o(xt)f FI(r)h FT(e)o(xplicitly)f(records)
       
  4731 g(all)h(freshness)g(constraints)f(in)h(a)g(sequent.)f(The)g(only)523
       
  4732 4595 y(inference)i(rule)g(which)h(adds)g(ne)n(w)f
       
  4733 (freshness-constraints)f(to)j(this)f(conte)o(xt)f(is)i(the)3075
       
  4734 4595 y
       
  4735  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4736  3075 4595 a FD(N)3075 4595 y
       
  4737  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4738 neg exch translate
       
  4739  3075 4595 a FT(-rule;)e(that)p
       
  4740 523 4654 473 4 v 558 4710 a Fy(4)606 4742 y FQ(The)h(question)g(of)g
       
  4741 (establishing)g(the)g(precise)f(relation)h(between)g(the)g(inference)g
       
  4742 (rules)g(gi)n(v)o(en)g(here)g(and)606 4833 y(nominal)j(logic)f
       
  4743 (introduced)h(in)f([8])g(is)f(be)o(yond)j(the)e(scope)g(of)g(this)g
       
  4744 (paper)m(,)g(b)o(ut)g(will)f(appear)i(in)e(a)h(full)606
       
  4745 4924 y(v)o(ersion)20 b(\(some)f(results)g(concerning)h(this)f(question)
       
  4746 h(ha)o(v)o(e)f(already)h(been)f(presented)h(in)f([5]\).)p
       
  4747 eop end
       
  4748 %%Page: 9 9
       
  4749 TeXDict begin 9 8 bop 523 448 a FT(is)33 b(whene)n(v)o(er)e(a)1093
       
  4750 448 y
       
  4751  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4752  1093 448 a FD(N)1093 448 y
       
  4753  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4754 neg exch translate
       
  4755  1093 448 a FT(-quanti\002er)g(is)i
       
  4756 (analysed,)e(a)i(ne)n(w)f(name)g(is)h(chosen)e(and)h(some)g(freshness-)
       
  4757 523 548 y(constraints)20 b(are)g(added)f(to)h FI(r)h
       
  4758 FT(in)f(order)f(to)i(enforce)d(the)i(\223freshness\224)g(of)g(this)g
       
  4759 (name.)648 648 y(The)h FI(\033)864 660 y FC(L)914 648
       
  4760 y FT(-rule)g(includes)g(the)h(judgement)e FI(r)27 b(`)f(r)2131
       
  4761 617 y Fv(0)2177 648 y FT(where)21 b FI(r)2471 617 y Fv(0)2517
       
  4762 648 y FT(is)i(the)f(set)h(of)f(freshness)f(con-)523 747
       
  4763 y(straints)29 b(associated)g(with)f(the)h FK(D)r FT(-formula)e(in)i
       
  4764 (the)f(stoup.)g(This)h(judgement)e(requires)g(that)i(all)523
       
  4765 847 y(constraints)22 b(in)g FI(r)1062 817 y Fv(0)1109
       
  4766 847 y FT(\(being)f(of)h(the)g(form)g FK(a)27 b FH(#)g
       
  4767 FK(t)p FT(\))c(are)f(satis\002ed)h(by)f(the)h FI(r)p
       
  4768 FT(,)g(that)f(is)i(for)d(all)i FK(a)28 b FH(#)f FK(t)523
       
  4769 946 y FT(the)20 b(judgement)f FI(r)k(`)g FK(a)g FH(#)g
       
  4770 FK(t)d FT(de\002ned)g(in)g(Fig.)g(3)h(holds.)648 1046
       
  4771 y(Of)15 b(most)h(interest)g(in)f(this)h(paper)f(is)h(the)g
       
  4772 FK(\031)s FT(-rule.)f(In)g(a)h(\223root-\002rst\224)f(proof-search,)d
       
  4773 (this)k(rule)g(is)g(a)523 1178 y(source)h(of)g(non-determinism.)d(F)o
       
  4774 (or)j(e)o(xample,)f(if)i(we)g(w)o(ant)f(to)h(pro)o(v)o(e)d(the)j
       
  4775 (sequent)f Fm(?)p FH(;)3065 1124 y FC(p)p FS(\()p FC(a)p
       
  4776 FS(\))3044 1178 y FI(\000)-19 b(\000)g(!)23 b FK(p)p
       
  4777 FH(\()p FK(b)p FH(\))p FT(,)523 1278 y(we)32 b(need)f(the)h
       
  4778 FK(\031)s FT(-rule)g(in)g(order)e(to)i(mak)o(e)g(the)g(terms)f
       
  4779 FK(a)i FT(and)e FK(b)h FI(\031)p FT(-equi)n(v)n(alent\227in)d(this)j
       
  4780 (case,)523 1378 y(only)19 b(after)h(applying)e(a)i(permutation)e(such)i
       
  4781 (as)g FH(\()p FK(a)14 b(b)p FH(\))21 b FT(to)f FK(a)g
       
  4782 FT(may)g(the)g(axiom-rule)d(be)j(used.)g(Prima)523 1477
       
  4783 y(f)o(acie)d(the)g FK(\031)s FT(-rule)g(is)h(innocuous,)d(ho)n(we)n(v)o
       
  4784 (er)m(,)f(the)j(problem)f(of)g(simultaneously)g(unifying)f(nominal)523
       
  4785 1577 y(terms)j(and)g(\002nding)f(a)i FK(\031)j FT(is,)d(as)h(mentioned)
       
  4786 c(earlier)m(,)h(an)i(NP-hard)e(decision)g(problem.)g(In)h(the)g(ne)o
       
  4787 (xt)523 1677 y(section)g(we)g(shall)h(sho)n(w)f(that)g(such)g(problems)
       
  4788 e(ne)n(v)o(er)h(need)h(to)g(be)g(solv)o(ed)f(pro)o(vided)f(the)i
       
  4789 (program)523 1776 y(clauses)j(are)f(well-formed.)523
       
  4790 2015 y FL(3)99 b(Elimination)25 b(of)g(the)h Fh(\031)t
       
  4791 FL(-Rule)523 2180 y FT(W)-7 b(e)18 b(implemented)c FK(\013)p
       
  4792 FT(Prolog)i(using)g(the)g(nominal)g(uni\002cation)f(algorithm.)f(W)m
       
  4793 (ith)j(this)g(implemen-)523 2271 y(tation)22 b(we)h(were)g(able)f(to)h
       
  4794 (calculate)f(the)h(e)o(xpected)e(results)i(for)e(programs)g(such)i(as)g
       
  4795 FJ(subst)f FT(and)523 2362 y FJ(type)p FT(.)16 b(The)f(reason)h(for)f
       
  4796 (this)i(is,)g(roughly)d(speaking,)g(that)j(the)f(name)f(we)i(used)e
       
  4797 (for)h(specifying)e(the)523 2454 y(clauses)26 b(dealing)f(with)h
       
  4798 FK(\025)p FT(-abstractions)f(does)h(not)g(matter)-5 b(.)25
       
  4799 b(When)h(using)f(nominal)g(uni\002cation,)523 2545 y(the)20
       
  4800 b(follo)n(wing)f(renamed)f(clauses)j(\(where)e FJ(a)i
       
  4801 FT(and)e FJ(x)i FT(are)f(renamed)f(to)h FJ(b)h FT(and)e
       
  4802 FJ(y)p FT(,)i(respecti)n(v)o(ely\))527 2675 y FP
       
  4803 (subst\(lam\(b.M\),X,T,lam\(b.M'\)\))-16 b(:-)23 b(b)12
       
  4804 b(#)g(T,)g(b)h(#)f(X,)g(subst\(M,X,T,M'\).)527 2790 y
       
  4805 (type\(Gamma,lam\(y.M\),arr\(S,T\)\))-16 b(:-)23 b(y)12
       
  4806 b(#)g(Gamma,)g(type\([\(y,S\)|Gamma],M,T\).)523 2922
       
  4807 y FT(beha)n(v)o(e)25 b(just)h(the)g(same)g(as)g(the)g(original)f
       
  4808 (clauses,)h(in)g(the)f(sense)i(that)f(all)g(queries)f(successfully)523
       
  4809 3013 y(solv)o(ed)30 b(by)h(the)g(original)f(v)o(ersions)g(are)h(solv)o
       
  4810 (ed)f(by)g(the)h(renamed)f(v)o(ersions.)g(In)g(contrast,)g(the)523
       
  4811 3105 y(name)19 b FJ(a)i FT(in)f(the)g(clauses)g FJ(p\(a\))13
       
  4812 b FT(,)44 b FJ(q\(a.X,X\))e FT(and)19 b FJ(r\(X\))13
       
  4813 b(:-)h(id\(X,a\))19 b FT(determines)g(which)523 3196
       
  4814 y(queries)k(can)h(be)g(solv)o(ed)g(successfully)f(using)h(nominal)f
       
  4815 (uni\002cation)f(and)i(which)g(cannot:)f(gi)n(v)o(en)523
       
  4816 3287 y(our)k(inference)g(rules,)h(which)g(choose)f(a)h(fresh)g(name)g
       
  4817 (for)f FJ(a)p FT(,)i(there)e(are)h(some)g(queries)g(whose)523
       
  4818 3379 y(answers)18 b(can)g(only)g(be)g(found)e(using)i(the)g
       
  4819 FK(\031)s FT(-rule,)g(and)f(this)i(means)f(the)o(y)f(cannot)h(be)g
       
  4820 (solv)o(ed)f(using)523 3470 y(nominal)i(uni\002cation.)g(Consider)g
       
  4821 (for)h(e)o(xample)f(the)h(follo)n(wing)e(deduction.)1500
       
  4822 3730 y FN(:)13 b(:)g(:)1856 3592 y Fq(?)21 b FF(`)h FN(c)f
       
  4823 FF(\031)g FN(c)p 1813 3619 405 4 v 1813 3730 a Fq(?)p
       
  4824 FG(;)1927 3681 y Fn(p)p FR(\()p Fn(c)p FR(\))1907 3730
       
  4825 y FF(\000)-20 b(\000)h(!)21 b FN(p)p FG(\()p FN(c)p FG(\))2259
       
  4826 3644 y FQ(Ax)p 1500 3769 718 4 v 1558 3879 a Fq(?)p FG(;)1671
       
  4827 3831 y Fg(?)p Fn(=)p Fp(>)11 b(\033)f Fn(p)p FR(\()p
       
  4828 Fn(c)p FR(\))1652 3879 y FF(\000)-29 b(\000)-17 b(\000)g(\000)f(\000)h
       
  4829 (\000)g(\000)-28 b(!)21 b FN(p)p FG(\()p FN(c)p FG(\))2259
       
  4830 3785 y FF(\033)2319 3793 y Fn(L)p 1558 3918 601 4 v 1558
       
  4831 4029 a Fq(?)p FG(;)1672 3980 y Fg(?)p Fn(=)p Fp(>)10
       
  4832 b(\033)g Fn(p)p FR(\()p Fn(b)p FR(\))1652 4029 y FF(\000)-29
       
  4833 b(\000)-17 b(\000)g(\000)f(\000)h(\000)g(\000)-28 b(!)21
       
  4834 b FN(p)p FG(\()p FN(c)p FG(\))2200 3937 y FN(\031)46
       
  4835 b FG(\()p FN(c)13 b(b)p FG(\))p 1508 4068 702 4 v 1508
       
  4836 4183 a Fq(?)p FG(;)1663 4134 y
       
  4837  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4838  1663 4134 a Fj(N)1663
       
  4839 4134 y
       
  4840  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4841 neg exch translate
       
  4842  1663 4134 a Fn(a:)p Fg(?)p Fn(=)p Fp(>)d(\033)h
       
  4843 Fn(p)p FR(\()p Fn(a)p FR(\))1602 4183 y FF(\000)-21 b(\000)j(\000)h
       
  4844 (\000)g(\000)f(\000)h(\000)g(\000)f(\000)e(!)22 b FN(p)p
       
  4845 FG(\()p FN(c)p FG(\))2305 4090 y
       
  4846  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4847  2305 4090 a FB(N)2305
       
  4848 4090 y
       
  4849  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4850 neg exch translate
       
  4851  2305 4090 a 9 x Fn(L)523 4327 y FT(In)e(this)g(deduction)e(the)i
       
  4852 FK(\031)s FT(-rule,)f(applying)f(the)i(permutation)e
       
  4853 FH(\()p FK(c)c(b)p FH(\))20 b FT(\(annotated)e(to)i(the)g
       
  4854 FK(\031)s FT(-rule\),)f(is)523 4426 y(crucial)j(for)g(the)h(sequent)f
       
  4855 (being)f(pro)o(v)n(able)g(and)h(it)h(will)h(turn)e(out)g(that)h(it)g
       
  4856 (is)h(impossible)e(to)h(elim-)523 4526 y(inate)e(it)h(from)f(this)g
       
  4857 (deduction.)e(Consequently)-5 b(,)19 b(a)j(proof-search)c(procedure)h
       
  4858 (based)i(on)g(nominal)523 4625 y(uni\002cation)e(will)i(not)f(\002nd)g
       
  4859 (this)g(proof.)648 4725 y(If)i(we)h(impose)f(the)h(follo)n(wing)e
       
  4860 (well-formedness)g(condition)g(on)h FK(D)r FT(-formulae,)e(we)j(can)g
       
  4861 (en-)523 4825 y(sure)18 b(that)h(the)f FK(\031)s FT(-rule)g(can)h(al)o
       
  4862 (w)o(ays)f(be)h(eliminated)e(from)h(corresponding)d(deductions)h(and)i
       
  4863 (hence)523 4924 y(the)i(nominal)f(uni\002cation)g(algorithm)g(alone)g
       
  4864 (is)i(suf)n(\002cient)f(for)g(solving)f(queries.)p eop
       
  4865 end
       
  4866 %%Page: 10 10
       
  4867 TeXDict begin 10 9 bop 523 448 a Fi(De\002nition)20 b(1.)41
       
  4868 b FE(A)25 b(closed)f FK(D)r FE(-formula)1757 448 y
       
  4869  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4870  1757
       
  4871 448 a FD(N)1757 448 y
       
  4872  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4873 neg exch translate
       
  4874  1757 448 a Fr(as)7 b FK(:)p FI(8)p
       
  4875 Fr(Xs)g FK(:)p FI(r)2104 460 y Fs(as)e FC(;)p Fs(Xs)2279
       
  4876 448 y FK(=G)2386 460 y Fs(as)t FC(;)p Fs(Xs)2574 448
       
  4877 y FI(\033)14 b FK(A)2715 460 y Fs(as)5 b FC(;)p Fs(Xs)2914
       
  4878 448 y FE(is)26 b FT(well-formed)523 540 y FE(if)21 b(ther)m(e)f(e)n
       
  4879 (xists)i(a)e(substitution)f FK(\033)25 b FE(and)19 b(a)h(permutation)f
       
  4880 FK(\031)24 b FE(suc)o(h)19 b(that)552 679 y FM(\(i\))41
       
  4881 b Ft(bs)28 b FG(#)22 b Ft(Xs)6 b FN(;)13 b FF(r)1036
       
  4882 687 y Ff(as)t Fn(;)p Ff(Xs)1210 679 y FF(`)21 b FN(\033)s
       
  4883 FG(\()p FN(A)1413 688 y Ff(bs)s Fn(;)p Ff(Xs)1562 679
       
  4884 y FG(\))g FF(\031)g FN(A)1752 687 y Ff(as)t Fn(;)p Ff(Xs)1923
       
  4885 679 y FM(and)531 766 y(\(ii\))41 b Ft(bs)28 b FG(#)22
       
  4886 b Ft(Xs)6 b FN(;)13 b FF(r)1036 774 y Ff(as)t Fn(;)p
       
  4887 Ff(Xs)1210 766 y FF(`)21 b FN(\033)s FG(\()p FN(\031)s
       
  4888 Fz(\001)o FN(G)1486 775 y Ff(bs)t Fn(;)p Ff(Xs)1635 766
       
  4889 y FG(\))h FF(\031)f FN(G)1828 774 y Ff(as)t Fn(;)p Ff(Xs)523
       
  4890 918 y FE(wher)m(e)g(the)f Fr(bs)28 b FE(ar)m(e)20 b(some)g(fr)m(esh)h
       
  4891 (names)f(\(dif)o(fer)m(ent)f(fr)l(om)i Fr(as)7 b FE(\).)523
       
  4892 1065 y FT(Let)20 b(us)h(illustrate)f(this)h(condition)d(with)i(some)g
       
  4893 (e)o(xamples.)f(Clauses)h(without)g(names)g(clearly)f(sat-)523
       
  4894 1156 y(isfy)h(the)h(condition.)d(F)o(or)i(e)o(xample)e(the)j(\002rst)f
       
  4895 FJ(subst)p FT(-clause)f(in)i(Fig.)f(1)691 1299 y Fq(?)h
       
  4896 FF(`)g FN(\033)s FG(\()p FN(S)t FG(\()p Fx(var)p FG(\()p
       
  4897 FN(X)6 b FG(\))p FN(;)13 b(X)r(;)h(T)6 b(;)13 b(T)e FG(\)\))42
       
  4898 b FF(\031)g FN(S)t FG(\()p Fx(var)q FG(\()p FN(X)6 b
       
  4899 FG(\))p FN(;)13 b(X)r(;)g(T)6 b(;)14 b(T)d FG(\))84 b
       
  4900 FT(and)h Fq(?)21 b FF(`)g FN(\033)s FG(\()p FN(\031)s
       
  4901 Fz(\001)o FF(>)p FG(\))42 b FF(\031)g(>)523 1442 y FT(tri)n(vially)27
       
  4902 b(satis\002es)i(the)f(condition)e(by)h(taking)f(for)h
       
  4903 FK(\033)32 b FT(the)27 b(identity)g(substitution)g(and)g(for)g
       
  4904 FK(\031)k FT(the)523 1533 y(empty)17 b(permutation.)f(More)i
       
  4905 (complicated)e(is)j(the)g(case)f(of)g(the)g(second)g
       
  4906 FJ(subst)p FT(-clause)f(in)h(Fig.)g(1)789 1675 y FF(r)j(`)h
       
  4907 FN(\033)s FG(\()p FN(S)t FG(\()p Fx(lam)p FG(\()p FN(b:M)8
       
  4908 b FG(\))p FN(;)13 b(X)r(;)g(T)6 b(;)14 b Fx(lam)p FG(\()p
       
  4909 FN(b:M)1916 1643 y Fp(0)1939 1675 y FG(\)\))42 b FF(\031)h
       
  4910 FN(S)t FG(\()p Fx(lam)p FG(\()p FN(a:M)8 b FG(\))p FN(;)14
       
  4911 b(X)r(;)f(T)6 b(;)13 b Fx(lam)q FG(\()p FN(a:M)3057 1643
       
  4912 y Fp(0)3079 1675 y FG(\)\))789 1766 y FF(r)21 b(`)h FN(\033)s
       
  4913 FG(\()p FN(\031)s Fz(\001)n FN(S)t FG(\()p FN(M)t(;)13
       
  4914 b(X)r(;)h(T)6 b(;)13 b(M)1550 1735 y Fp(0)1572 1766 y
       
  4915 FG(\)\))43 b FF(\031)f FN(S)t FG(\()p FN(M)t(;)13 b(X)r(;)h(T)6
       
  4916 b(;)13 b(M)2237 1735 y Fp(0)2260 1766 y FG(\))523 1916
       
  4917 y FT(where)20 b FI(r)h FT(is)g FI(f)p FK(b)h FH(#)h FK(M)t(;)14
       
  4918 b(b)23 b FH(#)g FK(X)r(;)14 b(b)22 b FH(#)h FK(T)7 b(;)14
       
  4919 b(b)22 b FH(#)i FK(M)1970 1886 y Fv(0)1993 1916 y FK(;)14
       
  4920 b(a)22 b FH(#)i FK(X)r(;)14 b(a)22 b FH(#)h FK(T)12 b
       
  4921 FI(g)p FT(.)20 b(In)f(this)i(case)651 2072 y FN(\033)j
       
  4922 FG(=)d([)p FN(M)30 b FG(:=)21 b(\()p FN(a)12 b(b)p FG(\))p
       
  4923 Fz(\001)p FN(M)t(;)h(X)27 b FG(:=)22 b(\()p FN(a)12 b(b)p
       
  4924 FG(\))p Fz(\001)o FN(X)r(;)i(T)31 b FG(:=)22 b(\()p FN(a)12
       
  4925 b(b)p FG(\))p Fz(\001)o FN(T)6 b(;)14 b(M)2290 2040 y
       
  4926 Fp(0)2334 2072 y FG(:=)21 b(\()p FN(a)13 b(b)p FG(\))p
       
  4927 Fz(\001)o FN(M)2689 2040 y Fp(0)2711 2072 y FG(])66 b
       
  4928 FT(and)d FN(\031)24 b FG(=)d(\()p FN(a)13 b(b)p FG(\))523
       
  4929 2236 y FT(v)o(erify)19 b(that)h(the)g(clause)h(is)g(well-formed.)648
       
  4930 2327 y(Before)k(we)h(formally)e(sho)n(w)i(that)g(all)g
       
  4931 FK(\031)s FT(-rules)g(can)f(be)h(eliminated)f(from)f(deductions)g(con-)
       
  4932 523 2419 y(sisting)31 b(of)f(well-formed)e(clauses)i(only)-5
       
  4933 b(,)29 b(we)h(outline)g(our)f(proof-plan)e(with)k(some)f(e)o(xamples.)
       
  4934 523 2510 y(Consider)25 b(the)g(follo)n(wing)e(deduction,)g(which)i(has)
       
  4935 g(a)h FK(\031)s FT(-rule)e(on)h(the)g(top)g(right-hand)e(side.)i(The)
       
  4936 523 2601 y(corresponding)17 b(permutation)h FH(\()p FK(e)c(d)p
       
  4937 FH(\))21 b FT(transforms)e FK(p)p FH(\()p FK(b:d)p FH(\))i
       
  4938 FT(into)f FK(p)p FH(\()p FK(b:e)p FH(\))h FT(so)g(that)f(the)g
       
  4939 (axiom-rule)f(is)523 2692 y(applicable.)1349 3051 y FN(:)13
       
  4940 b(:)g(:)1743 2771 y FF(`)21 b FN(b:e)h FF(\031)f FN(d:e)p
       
  4941 1664 2791 516 4 v 1664 2901 a Fq(?)p FG(;)1777 2853 y
       
  4942 Fn(p)p FR(\()p Fn(b:e)p FR(\))1758 2901 y FF(\000)-39
       
  4943 b(\000)-17 b(\000)f(\000)-38 b(!)21 b FN(p)p FG(\()p
       
  4944 FN(d:e)p FG(\))2221 2816 y FQ(Ax)p 1663 2940 518 4 v
       
  4945 1663 3051 a Fq(?)p FG(;)1776 3002 y Fn(p)p FR(\()p Fn(b:d)p
       
  4946 FR(\))1757 3051 y FF(\000)-38 b(\000)-17 b(\000)f(\000)-36
       
  4947 b(!)21 b FN(p)p FG(\()p FN(d:e)p FG(\))2222 2959 y FN(\031)j
       
  4948 FG(\()p FN(e)13 b(d)p FG(\))p 1349 3089 832 4 v 1418
       
  4949 3200 a Fq(?)p FG(;)1531 3152 y Fg(?)p Fn(=)p Fp(>\033)p
       
  4950 Fn(p)p FR(\()p Fn(b:d)p FR(\))1512 3200 y FF(\000)-35
       
  4951 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)-34
       
  4952 b(!)21 b FN(p)p FG(\()p FN(d:e)p FG(\))2222 3106 y FF(\033)2282
       
  4953 3114 y Fn(L)p 1355 3239 820 4 v 1355 3350 a Fq(?)p FG(;)1468
       
  4954 3301 y Fp(8)p Fn(X)q(:)p Fg(?)p Fn(=)p Fp(>\033)o Fn(p)p
       
  4955 FR(\()p Fn(b:X)t FR(\))1449 3350 y FF(\000)-36 b(\000)-18
       
  4956 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)-35
       
  4957 b(!)21 b FN(p)p FG(\()p FN(d:e)p FG(\))2216 3258 y FF(8)2259
       
  4958 3266 y Fn(L)2348 3258 y FG([)p FN(X)28 b FG(:=)21 b FN(d)p
       
  4959 FG(])p 1305 3388 921 4 v 1305 3504 a Fq(?)p FG(;)1460
       
  4960 3455 y
       
  4961  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4962  1460 3455 a Fj(N)1460 3455 y
       
  4963  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4964 neg exch translate
       
  4965  1460 3455 a Fn(a:)p
       
  4966 Fp(8)p Fn(X)q(:)p Fg(?)p Fn(=)p Fp(>\033)o Fn(p)p FR(\()p
       
  4967 Fn(a:X)t FR(\))1398 3504 y FF(\000)-28 b(\000)-17 b(\000)f(\000)h(\000)
       
  4968 g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)-27
       
  4969 b(!)21 b FN(p)p FG(\()p FN(d:e)p FG(\))2321 3407 y
       
  4970  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4971  2321
       
  4972 3407 a FB(N)2321 3407 y
       
  4973  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4974 neg exch translate
       
  4975  2321 3407 a 8 x Fn(L)2388 3407
       
  4976 y FG(\()p FN(a)13 b(b)p FG(\))523 3629 y FT(Observ)o(e)21
       
  4977 b(that)i(the)f(\223choice\224)g(of)g(the)g(fresh)g(name)g(\(namely)f
       
  4978 FK(b)p FT(\))h(introduced)e(by)i(the)3065 3629 y
       
  4979  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  4980  3065
       
  4981 3629 a FD(N)3065 3629 y
       
  4982  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  4983 neg exch translate
       
  4984  3065 3629 a 12 x FC(L)3115 3629
       
  4985 y FT(-rule)f(has)523 3729 y(no)i(ef)n(fect)f(on)h(whether)f(this)i
       
  4986 (sequent)e(is)i(deri)n(v)n(able,)e(since)h(this)h(binder)e(will)h(not)g
       
  4987 (bind)g(an)o(ything)523 3828 y(inside)g(the)h(abstraction.)e(The)h
       
  4988 (annotated)f(substitution)h FH([)p FK(X)35 b FH(:=)29
       
  4989 b FK(d)p FH(])24 b FT(ho)n(we)n(v)o(er)e(is)i(important)e(with)523
       
  4990 3928 y(respect)h(to)h(the)f FK(\031)s FT(-rule)g(we)h(are)g(trying)e
       
  4991 (to)i(eliminate.)e(If)i(we)f(had)g(instead)h(substituted)e
       
  4992 FK(e)i FT(for)f FK(X)7 b FT(,)523 4028 y(then)20 b(the)g(axiom)f(is)i
       
  4993 (applicable)e FE(without)j FT(the)e FK(\031)s FT(-rule.)648
       
  4994 4127 y(Note,)31 b(ho)n(we)n(v)o(er)m(,)e(that)i(changing)f(the)h
       
  4995 (instantiation)g(of)g FI(8)p FT(-quanti\002ers)f(might)h(ha)n(v)o(e)g
       
  4996 (some)523 4227 y(\223non-local\224)18 b(consequences)g(in)j
       
  4997 (deductions.)d(Consider)i(for)f(e)o(xample)g(the)i(deduction)d(in)i
       
  4998 (Fig.)h(5.)523 4327 y(In)j(this)i(deduction,)c(the)j
       
  4999 FK(\031)s FT(-rule)f(\(mark)o(ed)f(by)i FI(\017)p FT(\))f(sw)o(aps)i
       
  5000 (the)e(names)h FK(z)j FT(and)c FK(y)s FT(.)h(If)g(we)g(eliminate)523
       
  5001 4426 y(this)e FK(\031)s FT(-rule)g(by)f(applying)f(the)i(sw)o(apping)f
       
  5002 (to)g(the)h(terms)g(instantiated)f(for)g(the)h(v)n(ariables)f
       
  5003 FK(M)9 b FT(,)23 b FK(X)7 b FT(,)523 4526 y FK(T)29 b
       
  5004 FT(and)17 b FK(M)829 4496 y Fv(0)852 4526 y FT(,)h(then)f(the)h
       
  5005 FK(\031)s FT(-rule)f(is)i(not)e(needed,)f(b)n(ut)i(at)g(the)g(same)g
       
  5006 (time)f(the)h(subgoal)f(\(mark)o(ed)f(by)h FK(?)p FT(\))523
       
  5007 4625 y(is)26 b(changed.)d(The)i(well-formedness)e(condition)h(ensures)g
       
  5008 (that)i(the)f(modi\002cation)e(of)i(the)g(terms)523 4725
       
  5009 y(introduced)18 b(by)i(the)g FI(8)1171 4737 y FC(L)1221
       
  5010 4725 y FT(-rules)f(does)h(not)g(af)n(fect)g(the)g(pro)o(v)n(ability)e
       
  5011 (of)i(the)g(sequent.)648 4825 y(T)-7 b(o)18 b(sho)n(w)f(that)h
       
  5012 FK(\031)s FT(-rules)g(can)f(be)h(eliminated)f(from)g(deri)n(v)n(ations)
       
  5013 f(in)m(v)n(olving)g(well-formed)g(pro-)523 4924 y(grams,)j(we)i
       
  5014 (\002rst)g(pro)o(v)o(e)d(some)i(auxiliary)f(f)o(acts.)p
       
  5015 eop end
       
  5016 %%Page: 11 11
       
  5017 TeXDict begin 11 10 bop 523 369 2882 4 v 523 1436 4 1068
       
  5018 v 616 752 a FN(::)973 611 y FQ(.)973 644 y(.)973 677
       
  5019 y(.)p 708 698 548 4 v 708 752 a Fp(\))p Fn(s)p FR(\()p
       
  5020 Fe(v)p FR(\()p Fn(z)r FR(\))p Fn(;z)r(;)p Fe(v)p FR(\()p
       
  5021 Fn(y)r FR(\))p Fn(;)p Fe(v)p FR(\()p Fn(y)r FR(\)\))1272
       
  5022 715 y FN(?)1491 459 y Fn(::)p Fp(`h)p Fe(l)p FR(\()p
       
  5023 Fn(b:)p Fe(v)p FR(\()p Fn(y)r FR(\)\))p Fn(;y)r(;)p Fe(v)p
       
  5024 FR(\()p Fn(z)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(b:)p
       
  5025 Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fp(i\031h)p Fe(l)p FR(\()p
       
  5026 Fn(x:)p Fe(v)p FR(\()p Fn(y)r FR(\)\))p Fn(;y)r(;)p Fe(v)p
       
  5027 FR(\()p Fn(z)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(x:)p
       
  5028 Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fp(i)p 1334 492 1858
       
  5029 4 v 1334 603 a FN(::)p FG(;)1430 554 y Fn(s)p FR(\()p
       
  5030 Fe(l)p FR(\()p Fn(b:)p Fe(v)p FR(\()p Fn(y)r FR(\)\))p
       
  5031 Fn(;y)r(;)p Fe(v)p FR(\()p Fn(z)r FR(\))p Fn(;)p Fe(l)p
       
  5032 FR(\()p Fn(b:)p Fe(v)p FR(\()p Fn(z)r FR(\)\)\))1410
       
  5033 603 y FF(\000)-29 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)
       
  5034 f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)f(\000)-29
       
  5035 b(!)22 b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p
       
  5036 FG(\()p FN(y)s FG(\)\))p FN(;)12 b(y)s(;)g Fx(v)p FG(\()p
       
  5037 FN(z)s FG(\))p FN(;)h Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
       
  5038 FN(z)s FG(\)\)\))3194 517 y FQ(Ax)p 1334 641 V 1334 752
       
  5039 a FN(::)p FG(;)1430 703 y Fn(s)p FR(\()p Fe(l)p FR(\()p
       
  5040 Fn(b:)p Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fn(;z)r(;)p Fe(v)p
       
  5041 FR(\()p Fn(y)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(b:)p
       
  5042 Fe(v)p FR(\()p Fn(y)r FR(\)\)\))1410 752 y FF(\000)-29
       
  5043 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
       
  5044 (\000)f(\000)h(\000)g(\000)f(\000)h(\000)f(\000)-29 b(!)22
       
  5045 b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
       
  5046 FN(y)s FG(\)\))p FN(;)12 b(y)s(;)g Fx(v)p FG(\()p FN(z)s
       
  5047 FG(\))p FN(;)h Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
       
  5048 FN(z)s FG(\)\)\))3207 669 y FN(\031)3254 637 y Fp(\017)p
       
  5049 616 791 2575 4 v 666 901 a FN(::)p FG(;)762 853 y Fp(r)p
       
  5050 Fn(=s)p FR(\()p Fe(v)p FR(\()p Fn(z)r FR(\))p Fn(;z)r(;)p
       
  5051 Fe(v)p FR(\()p Fn(y)r FR(\))p Fn(;)p Fe(v)p FR(\()p Fn(y)r
       
  5052 FR(\)\))p Fp(\033)o Fn(s)p FR(\()p Fe(l)p FR(\()p Fn(b:)p
       
  5053 Fe(v)p FR(\()p Fn(z)r FR(\)\))p Fn(;z)r(;)p Fe(v)p FR(\()p
       
  5054 Fn(y)r FR(\))p Fn(;)p Fe(l)p FR(\()p Fn(b:)p Fe(v)p FR(\()p
       
  5055 Fn(y)r FR(\)\)\))743 901 y FF(\000)-20 b(\000)j(\000)f(\000)h(\000)g
       
  5056 (\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
       
  5057 (\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
       
  5058 (\000)g(\000)f(\000)h(\000)g(\000)f(\000)g(!)21 b FN(s)p
       
  5059 FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p FN(y)s
       
  5060 FG(\)\))p FN(;)12 b(y)s(;)h Fx(v)p FG(\()p FN(z)s FG(\))p
       
  5061 FN(;)g Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p FN(z)s FG(\)\)\))3194
       
  5062 807 y FF(\033)3254 815 y Fn(L)p 666 940 2476 4 v 1509
       
  5063 1014 a FQ(.)1509 1047 y(.)1509 1080 y(.)284 b FG([)p
       
  5064 FN(M)29 b FG(:=)22 b Fx(v)p FG(\()p FN(z)s FG(\))p FN(;)13
       
  5065 b(X)28 b FG(:=)21 b FN(z)s(;)13 b(T)32 b FG(:=)21 b Fx(v)p
       
  5066 FG(\()p FN(y)s FG(\))p FN(;)13 b(M)2912 1049 y Fp(0)2956
       
  5067 1080 y FG(:=)21 b Fx(v)p FG(\()p FN(y)s FG(\)])3157 962
       
  5068 y FF(8)3200 970 y Fn(L)p 638 1119 2532 4 v 638 1236 a
       
  5069 FN(::)p FG(;)712 1188 y Fp(8)p Fn(M)r(;X)q(;T)5 b(;M)1019
       
  5070 1167 y Fl(0)1031 1188 y Fn(:)p Fp(r)-11 b Fn(=s)p FR(\()p
       
  5071 Fn(M)r(;X)q(;T)5 b(;M)1447 1167 y Fl(0)1470 1188 y FR(\))p
       
  5072 Fp(\033)p Fn(s)p FR(\()p Fe(l)p FR(\()p Fn(b:M)g FR(\))p
       
  5073 Fn(;X)q(;T)g(;)p Fe(l)p FR(\()p Fn(b:M)2086 1167 y Fl(0)2110
       
  5074 1188 y FR(\)\))715 1236 y FF(\000)-35 b(\000)-17 b(\000)g(\000)f(\000)h
       
  5075 (\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
       
  5076 (\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
       
  5077 (\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)-33
       
  5078 b(!)21 b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p
       
  5079 FG(\()p FN(y)s FG(\)\))p FN(;)12 b(y)s(;)h Fx(v)p FG(\()p
       
  5080 FN(z)s FG(\))p FN(;)g Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
       
  5081 FN(z)s FG(\)\)\))3185 1141 y FF(8)3228 1149 y Fn(L)p
       
  5082 590 1275 2629 4 v 590 1393 a FN(::)p FG(;)696 1344 y
       
  5083  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5084 
       
  5085 696 1344 a Fj(N)696 1344 y
       
  5086  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5087 neg exch translate
       
  5088  696 1344 a Fn(a:)p Fp(8)p
       
  5089 Fn(M)r(;X)q(;T)5 b(;M)1056 1323 y Fl(0)1067 1344 y Fn(:)p
       
  5090 Fp(r)-11 b Fn(=s)p FR(\()p Fn(M)r(;X)q(;T)5 b(;M)1483
       
  5091 1323 y Fl(0)1506 1344 y FR(\))p Fp(\033)p Fn(s)p FR(\()p
       
  5092 Fe(l)p FR(\()p Fn(a:M)g FR(\))p Fn(;X)q(;T)g(;)p Fe(l)p
       
  5093 FR(\()p Fn(a:M)2136 1323 y Fl(0)2158 1344 y FR(\)\))666
       
  5094 1393 y FF(\000)-28 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g
       
  5095 (\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
       
  5096 (\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
       
  5097 (\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)-28
       
  5098 b(!)21 b FN(s)p FG(\()p Fx(l)p FG(\()p FN(x:)p Fx(v)p
       
  5099 FG(\()p FN(y)s FG(\)\))p FN(;)12 b(y)s(;)h Fx(v)p FG(\()p
       
  5100 FN(z)s FG(\))p FN(;)g Fx(l)p FG(\()p FN(x:)p Fx(v)p FG(\()p
       
  5101 FN(z)s FG(\)\)\))3263 1292 y
       
  5102  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5103  3263 1292 a Fj(N)3263 1292
       
  5104 y
       
  5105  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5106 neg exch translate
       
  5107  3263 1292 a 8 x Fn(L)p 3402 1436 4 1068 v 523 1439
       
  5108 2882 4 v 523 1582 a FO(Fig)o(.)f(5.)31 b FQ(Deduction)g(pro)o(ving)h
       
  5109 (the)f(f)o(act)g FN(s)p FP(\(l\(x.v\(y\)\),y,v\(z\),l\(x.v\(z\)\)\))26
       
  5110 b FQ(where)31 b FP(l)g FQ(and)g FP(v)523 1674 y FQ(stand)19
       
  5111 b(for)g(lambda-abstractions)i(and)e(v)n(ariables,)h(respecti)n(v)o(ely)
       
  5112 -5 b(.)523 1971 y Fi(Lemma)21 b(2.)41 b FE(F)-9 b(or)26
       
  5113 b(all)h(permutations)e FK(\031)s FE(,)i(the)f(sequent)g
       
  5114 FI(r)p FH(;)14 b FK(\000)46 b FI(\))34 b FK(\031)s Fu(\001)p
       
  5115 FK(G)27 b FE(is)g(derivable)f(only)f(if)i(the)523 2071
       
  5116 y(sequent)19 b FI(r)p FH(;)14 b FK(\031)953 2041 y Fv(\000)p
       
  5117 FS(1)1043 2071 y Fu(\001)p FK(\000)34 b FI(\))23 b FK(G)f
       
  5118 FE(is)f(derivable)e(\(wher)m(e)h(we)h(use)g(the)f(notation)f
       
  5119 FK(\031)s Fu(\001)p FK(\000)32 b FE(to)20 b(indicate)g(that)g
       
  5120 FK(\031)523 2171 y FE(is)h(applied)e(to)h(e)o(very)h(formula)e(in)i
       
  5121 FK(\000)12 b FE(\).)523 2364 y(Pr)l(oof)o(.)40 b FT(By)25
       
  5122 b(induction)d(on)h(the)h(structure)f(of)h(deductions.)e(It)i(mak)o(es)f
       
  5123 (use)i(of)e(the)h(property)e(of)h FI(\031)523 2464 y
       
  5124 FT(that)c FI(r)k(`)g FK(t)g FI(\031)g FK(\031)s Fu(\001)p
       
  5125 FK(t)1081 2433 y Fv(0)1123 2464 y FT(holds)c(only)f(if)h
       
  5126 FI(r)k(`)g FK(\031)1776 2433 y Fv(\000)p FS(1)1865 2464
       
  5127 y Fu(\001)p FK(t)g FI(\031)g FK(t)2063 2433 y Fv(0)2105
       
  5128 2464 y FT(holds.)18 b(By)i(inspection)e(we)h(can)g(further)e(see)523
       
  5129 2563 y(that)j(no)g(additional)f FK(\031)s FT(-rule)h(is)h(necessary)e
       
  5130 (to)i(sho)n(w)f(the)g(pro)o(v)n(ability)e(in)i(both)f(directions.)191
       
  5131 b FI(u)-55 b(t)523 2757 y FT(The)31 b(follo)n(wing)f(corollary)g(is)j
       
  5132 (a)f(simple)f(consequence)e(of)j(this)g(lemma)f(by)g(the)h(f)o(act)f
       
  5133 (that)h(for)523 2856 y(closed)20 b FK(D)r FT(-formulae)e
       
  5134 FK(\031)s Fu(\001)p FK(D)25 b FH(=)e FK(D)f FT(holds.)523
       
  5135 3055 y Fi(Cor)o(ollary)c(1.)41 b FE(F)-9 b(or)25 b(all)g(permutations)e
       
  5136 FK(\031)29 b FE(and)23 b(conte)n(xts)i FK(\000)37 b FE(consisting)23
       
  5137 b(of)i(closed)f FK(D)r FE(-formulae)523 3154 y(only)-5
       
  5138 b(,)20 b FI(r)p FH(;)14 b FK(\000)35 b FI(\))23 b FK(\031)s
       
  5139 Fu(\001)p FK(G)e FE(is)g(derivable)e(only)h(if)h FI(r)p
       
  5140 FH(;)14 b FK(\000)34 b FI(\))24 b FK(G)d FE(is)g(derivable)o(.)523
       
  5141 3348 y Fi(Lemma)g(3.)41 b FE(If)23 b(the)h(sequent)e
       
  5142 FI(r)p FH(;)14 b FK(\000)41 b FI(\))28 b FK(G)d FE(is)f(derivable)e
       
  5143 (and)h FI(r)29 b(`)f FK(G)i FI(\031)e FK(G)2791 3317
       
  5144 y Fv(0)2814 3348 y FE(,)c(then)f(the)g(sequent)523 3447
       
  5145 y FI(r)p FH(;)14 b FK(\000)35 b FI(\))23 b FK(G)886 3417
       
  5146 y Fv(0)930 3447 y FE(is)e(derivable)o(.)523 3641 y(Pr)l(oof)o(.)40
       
  5147 b FT(Since)21 b FI(r)i(`)g FK(G)g FI(\031)g FK(G)1382
       
  5148 3611 y Fv(0)1426 3641 y FT(is)e(inducti)n(v)o(ely)d(de\002ned)i(e)o
       
  5149 (xtending)e(the)i FI(\031)p FT(-equality)e(of)i(the)g(terms)523
       
  5150 3740 y(occurring)i(in)i FK(G)h FT(and)f FK(G)1255 3710
       
  5151 y Fv(0)1279 3740 y FT(,)g(we)h(can)f(pro)o(v)o(e)e(this)j(lemma)e(by)h
       
  5152 (inspection)f(of)h(the)g(inference-rules,)523 3840 y(noting)c(that)h
       
  5153 (in)g(the)g FH(\()p FI(\000)p FH(\))1241 3852 y FC(L)1291
       
  5154 3840 y FT(-rules)f(the)h(right-hand)e(side)i(of)f(sequents)h(is)h(al)o
       
  5155 (w)o(ays)f(of)g(the)g(form)f FK(p)p FH(\()p FK(t)p FH(\))523
       
  5156 3940 y FT(and)g(the)g(lemma)g(for)f(axioms)h(follo)n(ws)g(from)f(the)h
       
  5157 (transiti)n(vity)f(of)h FI(\031)p FT(.)648 4133 y(F)o(or)25
       
  5158 b(sho)n(wing)f(our)h(main)h(result,)f(it)i(is)f(con)m(v)o(enient)d(to)j
       
  5159 (restrict)g(attention)f(to)h(some)f(speci\002c)523 4233
       
  5160 y(instances)17 b(of)f(the)g FK(\031)s FT(-rule.)g(The)g(ne)o(xt)g
       
  5161 (lemma)g(sho)n(ws)h(that)g(we)g(only)e(need)h(to)h(consider)e(unmo)o(v)
       
  5162 n(able)523 4332 y(instances)20 b(of)g(the)g FK(\031)s
       
  5163 FT(-rule.)523 4530 y Fi(De\002nition)g(2.)41 b FE(A)19
       
  5164 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
       
  5165 (is)h(not)f(dir)m(ectly)g(under)f(an)h(axiom,)f(otherwise)523
       
  5166 4630 y(it)j(is)g(said)f(to)h(be)f FT(unmo)o(v)n(able)p
       
  5167 FE(.)523 4824 y Fi(Lemma)h(4.)41 b FE(All)33 b(mo)o(vable)f(instances)h
       
  5168 (of)g(the)g FK(\031)s FE(-rules)g(can)g(be)g(r)m(eplaced)f(by)h(unmo)o
       
  5169 (vable)e(in-)523 4923 y(stances.)p eop end
       
  5170 %%Page: 12 12
       
  5171 TeXDict begin 12 11 bop 523 448 a FE(Pr)l(oof)o(.)40
       
  5172 b FT(W)-7 b(e)25 b(call)f(a)g(deri)n(v)n(ation)d FK(\031)s
       
  5173 FT(-normalised)h(if)i(all)f(instances)h(of)f(the)g FK(\031)s
       
  5174 FT(-rule)g(are)h(unmo)o(v)n(able.)523 569 y(W)-7 b(e)26
       
  5175 b(\002rst)f(sho)n(w)g(that)g(if)g FK(\000)1351 522 y
       
  5176 FC(\031)r Fd(\001)o FC(D)1330 569 y FI(\000)-21 b(\000)g(!)32
       
  5177 b FK(A)26 b FT(has)f(a)g FK(\031)s FT(-normalised)e(deri)n(v)n(ation,)g
       
  5178 (then)h(we)h(can)g(construct)523 684 y(a)i FK(\031)s
       
  5179 FT(-normalised)e(deri)n(v)n(ation)g(of)h FK(\000)1636
       
  5180 637 y FC(D)1615 684 y FI(\000)-39 b(!)35 b FK(A)p FT(.)27
       
  5181 b(Using)g(this)g(construction,)e(we)i(can)f(eliminate)g(the)523
       
  5182 784 y(mo)o(v)n(able)18 b FK(\031)s FT(-rules)i(from)g(an)o(y)f(deri)n
       
  5183 (v)n(ation)f(one)i(at)h(a)f(time.)648 883 y(There)f(is)i(one)f(case)g
       
  5184 (for)g(each)g(left-rule.)f(F)o(or)g FK(Ax)p FT(,)j(we)e(ha)n(v)o(e)p
       
  5185 1217 1092 482 4 v 1217 1194 a FF(r)p FG(;)14 b FN(\000)1414
       
  5186 1151 y Fn(\031)r Fk(\001)o Fn(A)1394 1194 y FF(\000)-21
       
  5187 b(\000)g(!)21 b FN(\031)s Fz(\001)o FN(A)1740 1118 y(Ax)1864
       
  5188 1123 y FF(\000)-13 b(!)p 2059 956 V 2059 1058 a(r)p FG(;)13
       
  5189 b FN(\000)2255 1015 y Fn(\031)r Fk(\001)p Fn(A)2236 1058
       
  5190 y FF(\000)-22 b(\000)h(!)21 b FN(\031)s Fz(\001)o FN(A)2581
       
  5191 981 y(Ax)p 2059 1092 V 2088 1194 a FF(r)p FG(;)13 b FN(\000)2284
       
  5192 1151 y Fn(A)2265 1194 y FF(\000)-41 b(!)21 b FN(\031)s
       
  5193 Fz(\001)o FN(A)2581 1124 y(\031)2628 1092 y Fp(\000)p
       
  5194 FR(1)523 1327 y FT(since)k FK(\031)769 1296 y Fv(\000)p
       
  5195 FS(1)858 1327 y Fu(\001)p FK(\031)s Fu(\001)p FK(A)31
       
  5196 b FH(=)g FK(A)p FT(.)25 b(A)h FK(\031)s FT(-normalised)d(deri)n(v)n
       
  5197 (ation)g(ending)g(in)i(a)g FK(\031)2628 1296 y Fv(0)2651
       
  5198 1327 y FT(-rule)f(must)h(be)g(immedi-)523 1426 y(ately)20
       
  5199 b(follo)n(wed)f(by)h FK(Ax)p FT(,)h(we)g(can)f(deri)n(v)o(e)p
       
  5200 1007 1505 656 4 v 1007 1617 a FF(r)p FG(;)13 b FN(\000)1203
       
  5201 1574 y Fn(\031)1241 1553 y Fl(0)1264 1574 y Fk(\001)o
       
  5202 Fn(\031)r Fk(\001)p Fn(A)1184 1617 y FF(\000)-24 b(\000)-18
       
  5203 b(\000)h(\000)-23 b(!)21 b FN(\031)1487 1585 y Fp(0)1509
       
  5204 1617 y Fz(\001)p FN(\031)s Fz(\001)o FN(A)1704 1531 y(Ax)p
       
  5205 1007 1651 V 1047 1753 a FF(r)p FG(;)13 b FN(\000)1244
       
  5206 1710 y Fn(\031)r Fk(\001)o Fn(A)1224 1753 y FF(\000)-22
       
  5207 b(\000)i(!)21 b FN(\031)1447 1721 y Fp(0)1469 1753 y
       
  5208 Fz(\001)o FN(\031)s Fz(\001)o FN(A)1704 1681 y(\031)1751
       
  5209 1649 y Fp(0)1828 1682 y FF(\000)-13 b(!)p 2023 1505 682
       
  5210 4 v 2023 1617 a(r)p FG(;)13 b FN(\000)2219 1574 y Fn(\031)2257
       
  5211 1553 y Fl(0)2279 1574 y FR(@)p Fn(\031)r Fk(\001)p Fn(A)2200
       
  5212 1617 y FF(\000)-33 b(\000)-17 b(\000)g(\000)f(\000)-31
       
  5213 b(!)21 b FN(\031)2529 1585 y Fp(0)2551 1617 y Fz(\001)p
       
  5214 FN(\031)s Fz(\001)o FN(A)2746 1531 y(Ax)p 2023 1651 V
       
  5215 2105 1753 a FF(r)p FG(;)13 b FN(\000)2302 1710 y Fn(A)2282
       
  5216 1753 y FF(\000)-41 b(!)21 b FN(\031)2446 1721 y Fp(0)2468
       
  5217 1753 y Fz(\001)p FN(\031)s Fz(\001)o FN(A)2746 1681 y(\031)2793
       
  5218 1649 y Fp(0)2815 1681 y FG(@)p FN(\031)523 1881 y FT(since)h
       
  5219 FK(\031)766 1851 y Fv(0)790 1881 y Fu(\001)o FK(\031)s
       
  5220 Fu(\001)p FK(A)k FH(=)g FK(\031)1122 1851 y Fv(0)1146
       
  5221 1881 y FH(@)p FK(\031)s Fu(\001)o FK(A)p FT(.)d(F)o(or)e
       
  5222 FI(8)1575 1893 y FC(L)1625 1881 y FT(,)h(since)g FK(\031)s
       
  5223 Fu(\001)p FI(8)p FK(X)r(:D)28 b FH(=)e FI(8)p FK(X)r(:\031)s
       
  5224 Fu(\001)2482 1896 y Fv(f)p FC(X)5 b Fv(g)2613 1881 y
       
  5225 FK(D)25 b FT(and)c FH(\()p FK(\031)s Fu(\001)2958 1896
       
  5226 y Fv(f)p FC(X)5 b Fv(g)3089 1881 y FK(D)r FH(\)[)p FK(X)33
       
  5227 b FH(:=)523 1980 y FK(t)p FH(])23 b(=)g FK(\031)s Fu(\001)p
       
  5228 FH(\()p FK(D)r FH([)p FK(X)29 b FH(:=)23 b FK(\031)1149
       
  5229 1950 y Fv(\000)p FS(1)1238 1980 y Fu(\001)p FK(t)p FH(]\))p
       
  5230 FT(,)e(so)f(we)h(ha)n(v)o(e)1013 2143 y FF(r)p FG(;)13
       
  5231 b FN(\000)1209 2082 y FR(\()p Fn(\031)r Fk(\001)1292
       
  5232 2096 y Fl(f)p Fc(X)s Fl(g)1406 2082 y Fn(D)r FR(\)[)p
       
  5233 Fn(X)t FR(:=)p Fn(t)p FR(])1190 2143 y FF(\000)-20 b(\000)i(\000)h
       
  5234 (\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)e(!)21
       
  5235 b FN(A)p 1013 2177 755 4 v 1073 2297 a FF(r)p FG(;)13
       
  5236 b FN(\000)1269 2237 y Fp(8)p Fn(X)q(:\031)r Fk(\001)1432
       
  5237 2251 y Fl(f)p Fc(X)s Fl(g)1547 2237 y Fn(D)1250 2297
       
  5238 y FF(\000)-38 b(\000)-17 b(\000)g(\000)f(\000)h(\000)g(\000)f(\000)-36
       
  5239 b(!)21 b FN(A)1809 2199 y FF(8)1852 2207 y Fn(L)1921
       
  5240 2226 y FF(\000)-13 b(!)2115 2160 y(r)p FG(;)13 b FN(\000)2312
       
  5241 2112 y Fn(D)r FR([)p Fn(X)t FR(:=)p Fn(\031)2539 2091
       
  5242 y Fl(\000)p Fb(1)2614 2112 y Fk(\001)o Fn(t)p FR(])2292
       
  5243 2160 y FF(\000)-20 b(\000)j(\000)g(\000)f(\000)h(\000)g(\000)f(\000)f
       
  5244 (!)21 b FN(A)p 2115 2195 670 4 v 2219 2297 a FF(r)p FG(;)13
       
  5245 b FN(\000)2416 2254 y Fp(8)p Fn(X)q(:D)2396 2297 y FF(\000)-38
       
  5246 b(\000)-18 b(\000)h(\000)-38 b(!)21 b FN(A)2826 2217
       
  5247 y FF(8)2869 2225 y Fn(L)523 2478 y FT(where)f(by)g(induction)f
       
  5248 FI(r)p FH(;)14 b FK(\000)1399 2423 y FC(D)r FS([)p FC(X)5
       
  5249 b FS(:=)p FC(\031)1644 2398 y Fl(\000)p Fb(1)1721 2423
       
  5250 y Fd(\001)o FC(t)p FS(])1378 2478 y FI(\000)-24 b(\000)-19
       
  5251 b(\000)g(\000)g(\000)g(\000)g(\000)g(\000)-25 b(!)24
       
  5252 b FK(A)d FT(has)g(a)g FK(\031)s FT(-normalised)e(deri)n(v)n(ation)g
       
  5253 (obtained)g(from)523 2619 y(that)28 b(of)g FI(r)p FH(;)14
       
  5254 b FK(\000)1000 2565 y FC(\031)r Fd(\001)p FC(D)r FS([)p
       
  5255 FC(X)5 b FS(:=)p FC(\031)1308 2540 y Fl(\000)p Fb(1)1385
       
  5256 2565 y Fd(\001)o FC(t)p FS(])979 2619 y FI(\000)-39 b(\000)-19
       
  5257 b(\000)g(\000)g(\000)g(\000)h(\000)f(\000)g(\000)g(\000)-40
       
  5258 b(!)37 b FK(A)29 b FT(.)f(The)g(cases)g(for)f FI(^)2201
       
  5259 2631 y FC(L)2279 2619 y FT(and)2487 2619 y
       
  5260  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5261  2487 2619
       
  5262 a FD(N)2487 2619 y
       
  5263  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5264 neg exch translate
       
  5265  2487 2619 a 12 x FC(L)2565 2619 y
       
  5266 FT(are)h(straightforw)o(ard)d(since)523 2719 y FK(\031)s
       
  5267 Fu(\001)600 2731 y Fv(B)649 2719 y FH(\()p FI(\000)p
       
  5268 FH(\))c FT(commutes)e(with)i FI(^)g FT(and)1606 2719
       
  5269 y
       
  5270  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5271  1606 2719 a FD(N)1606 2719 y
       
  5272  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5273 neg exch translate
       
  5274  1606 2719 a FT(.)g(F)o(or)f
       
  5275 FI(\033)1847 2731 y FC(L)1918 2719 y FT(we)g(ha)n(v)o(e)567
       
  5276 2860 y FF(r)h(`)g(r)75 b(r)p FG(;)13 b FN(\000)32 b FF(\))21
       
  5277 b FN(\031)s Fz(\001)o FN(G)75 b FF(r)p FG(;)13 b FN(\000)1537
       
  5278 2817 y Fn(\031)r Fk(\001)o Fn(A)1517 2860 y FF(\000)-22
       
  5279 b(\000)h(!)22 b FN(A)1751 2828 y Fp(0)p 567 2894 1206
       
  5280 4 v 787 3005 a FF(r)p FG(;)13 b FN(\000)983 2957 y Fp(r)p
       
  5281 Fn(=)p FR(\()p Fn(\031)r Fk(\001)p Fn(G)p FR(\))p Fp(\033)p
       
  5282 FR(\()p Fn(\031)r Fk(\001)p Fn(A)p FR(\))964 3005 y FF(\000)-26
       
  5283 b(\000)-17 b(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)-25
       
  5284 b(!)21 b FN(A)1530 2974 y Fp(0)1814 2911 y FF(\033)1874
       
  5285 2919 y Fn(L)1943 2934 y FF(\000)-13 b(!)2138 2860 y(r)21
       
  5286 b(`)g(r)75 b(r)p FG(;)13 b FN(\000)32 b FF(\))21 b FN(G)75
       
  5287 b FF(r)p FG(;)13 b FN(\000)3036 2817 y Fn(A)3017 2860
       
  5288 y FF(\000)-41 b(!)21 b FN(A)3192 2828 y Fp(0)p 2138 2894
       
  5289 1077 4 v 2399 3005 a FF(r)p FG(;)14 b FN(\000)2596 2957
       
  5290 y Fp(r)p Fn(=G)p Fp(\033)p Fn(A)2577 3005 y FF(\000)-26
       
  5291 b(\000)-18 b(\000)h(\000)g(\000)-25 b(!)21 b FN(A)2930
       
  5292 2974 y Fp(0)3255 2911 y FF(\033)3315 2919 y Fn(L)523
       
  5293 3127 y FT(using)i(Lemma)f(3)h(to)h(deri)n(v)o(e)e FI(r)p
       
  5294 FH(;)14 b FK(\000)40 b FI(\))29 b FK(G)24 b FT(from)e
       
  5295 FI(r)p FH(;)14 b FK(\000)40 b FI(\))29 b FK(\031)s Fu(\001)p
       
  5296 FK(G)24 b FT(and)f(the)g(induction)e(hypothesis)523 3249
       
  5297 y(to)f(obtain)g(a)g FK(\031)s FT(-normalised)f(deri)n(v)n(ation)f(of)i
       
  5298 FI(r)p FH(;)14 b FK(\000)2018 3202 y FC(A)1997 3249 y
       
  5299 FI(\000)-45 b(!)23 b FK(A)2185 3219 y Fv(0)2229 3249
       
  5300 y FT(from)c(that)i(of)e FI(r)p FH(;)14 b FK(\000)2859
       
  5301 3202 y FC(\031)r Fd(\001)o FC(A)2838 3249 y FI(\000)-23
       
  5302 b(\000)e(!)24 b FK(A)3089 3219 y Fv(0)3112 3249 y FT(.)523
       
  5303 3384 y Fi(Theor)o(em)c(1.)41 b FE(If)22 b FK(\000)34
       
  5304 b FE(consists)22 b(of)h(well-formed)e(clauses)h(and)f(the)h(sequent)f
       
  5305 FI(r)p FH(;)14 b FK(\000)38 b FI(\))26 b FK(G)d FE(is)g(deriv-)523
       
  5306 3484 y(able)o(,)d(then)f(it)i(is)g(derivable)f(without)g(using)f(the)i
       
  5307 FK(\031)s FE(-rule)o(.)523 3611 y(Pr)l(oof)o(.)40 b FT(Since)26
       
  5308 b FK(\000)39 b FT(consists)26 b(of)g(well-formed)e(clauses)i(only)-5
       
  5309 b(,)24 b(all)j FK(\000)12 b FT(')-5 b(s)26 b(in)g(the)g(deduction)e
       
  5310 (consist)523 3703 y(of)29 b(well-formed)e(clauses)i(\(formulae)e(on)h
       
  5311 (the)h(left-hand)f(side)h(are)g(analysed)f(only)g(if)h(the)o(y)g(are)
       
  5312 523 3794 y(selected)e(to)g(be)g(in)h(the)f(stoup-position\).)d(By)j
       
  5313 (Lemma)g(4,)g(we)g(can)g(replace)f(this)i(deduction)d(by)523
       
  5314 3885 y(one)j(in)h(which)g(all)g FK(\031)s FT(-rules)g(are)g(unmo)o(v)n
       
  5315 (able.)d(So)j(we)g(need)g(to)g(consider)e(ho)n(w)i(unmo)o(v)n(able)d
       
  5316 FK(\031)s FT(-)523 3977 y(rules)20 b(can)g(be)g(eliminated.)f(Recall)i
       
  5317 (that)f(unmo)o(v)n(able)e FK(\031)s FT(-rules)i(occur)f(in)h(se)o
       
  5318 (gments)g(of)g(the)g(form)1061 4339 y FG(:)847 4415 y
       
  5319 FF(r)911 4383 y Fp(00)911 4431 y Ff(bs)971 4415 y FN(;)13
       
  5320 b FF(r)1069 4383 y Fp(0)1112 4415 y FF(`)22 b(r)1245
       
  5321 4423 y Ff(ts)1703 4339 y FG(:)1393 4415 y FF(r)1457 4383
       
  5322 y Fp(00)1457 4431 y Ff(bs)1516 4415 y FN(;)13 b FF(r)1614
       
  5323 4383 y Fp(0)1637 4415 y FG(;)g FN(\000)32 b FF(\))21
       
  5324 b FN(G)1908 4424 y Ff(bs)t Fn(;)p Ff(ts)2197 4102 y FF(r)2261
       
  5325 4070 y Fp(00)2261 4117 y Ff(bs)2320 4102 y FN(;)14 b
       
  5326 FF(r)2419 4070 y Fp(0)2462 4102 y FF(`)22 b FN(\031)s
       
  5327 Fz(\001)o FN(s)2638 4111 y Ff(bs)t Fn(;)p Ff(ts)2785
       
  5328 4102 y FF(\031)f FN(t)p 2130 4140 831 4 v 2130 4258 a
       
  5329 FF(r)2194 4227 y Fp(00)2194 4274 y Ff(bs)2254 4258 y
       
  5330 FN(;)13 b FF(r)2352 4227 y Fp(0)2374 4258 y FG(;)g FN(\000)2507
       
  5331 4203 y Fn(\031)r Fk(\001)o Fn(p)p FR(\()p Fn(s)2650 4215
       
  5332 y Fa(bs)s Fc(;)p Fa(ts)2759 4203 y FR(\))2487 4258 y
       
  5333 FF(\000)-21 b(\000)k(\000)f(\000)h(\000)g(\000)d(!)21
       
  5334 b FN(p)p FG(\()p FN(t)p FG(\))2989 4165 y FQ(Ax)p 2130
       
  5335 4297 V 2159 4415 a FF(r)2223 4383 y Fp(00)2223 4431 y
       
  5336 Ff(bs)2283 4415 y FN(;)13 b FF(r)2381 4383 y Fp(0)2403
       
  5337 4415 y FG(;)g FN(\000)2536 4359 y Fn(p)p FR(\()p Fn(s)2621
       
  5338 4371 y Fa(bs)s Fc(;)p Fa(ts)2730 4359 y FR(\))2516 4415
       
  5339 y FF(\000)-29 b(\000)-17 b(\000)g(\000)f(\000)-28 b(!)21
       
  5340 b FN(p)p FG(\()p FN(t)p FG(\))2989 4313 y FN(\031)p 847
       
  5341 4454 2085 4 v 1338 4572 a FF(r)1402 4540 y Fp(00)1402
       
  5342 4587 y Ff(bs)1461 4572 y FN(;)13 b FF(r)1559 4540 y Fp(0)1581
       
  5343 4572 y FG(;)h FN(\000)1714 4516 y Fp(r)1765 4524 y Fa(ts)1809
       
  5344 4516 y Fn(=G)1888 4528 y Fa(bs)s Fc(;)p Fa(ts)1998 4516
       
  5345 y Fp(\033)p Fn(p)p FR(\()p Fn(s)2131 4528 y Fa(bs)s Fc(;)p
       
  5346 Fa(ts)2240 4516 y FR(\))1694 4572 y FF(\000)-33 b(\000)-18
       
  5347 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
       
  5348 (\000)-33 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))2973 4470
       
  5349 y FF(\033)3033 4478 y Fn(L)1810 4663 y FG(:)473 b FF(g)2409
       
  5350 4663 y
       
  5351  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5352  2409 4663 a FB(N)2409 4663 y
       
  5353  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5354 neg exch translate
       
  5355  2409 4663 a FF(8)p
       
  5356 1260 4702 1259 4 v 1260 4824 a(r)1324 4792 y Fp(0)1347
       
  5357 4824 y FG(;)13 b FN(\000)1521 4769 y
       
  5358  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5359  1521 4769 a Fj(N)1521
       
  5360 4769 y
       
  5361  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5362 neg exch translate
       
  5363  1521 4769 a Fn(as:)p Fp(8)p Ff(Xs)t Fn(:)p Fp(r)1779
       
  5364 4780 y Fa(Xs)1843 4769 y Fn(=G)1922 4780 y Fa(as)s Fc(;)p
       
  5365 Fa(Xs)2053 4769 y Fp(\033)p Fn(p)p FR(\()p Fn(s)2186
       
  5366 4780 y Fa(as)s Fc(;)p Fa(Xs)2317 4769 y FR(\))1460 4824
       
  5367 y FF(\000)-27 b(\000)-18 b(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f
       
  5368 (\000)h(\000)g(\000)f(\000)h(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
       
  5369 (\000)f(\000)-26 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))p
       
  5370 1260 4862 1259 4 v 1678 4938 a FF(r)1742 4907 y Fp(0)1764
       
  5371 4938 y FG(;)13 b FN(\000)32 b FF(\))21 b FN(p)p FG(\()p
       
  5372 FN(t)p FG(\))2560 4887 y FQ(Sel)3318 4523 y(\(4\))p eop
       
  5373 end
       
  5374 %%Page: 13 13
       
  5375 TeXDict begin 13 12 bop 523 448 a FT(where)23 b(the)936
       
  5376 448 y
       
  5377  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5378  936 448 a FD(N)936 448 y
       
  5379  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5380 neg exch translate
       
  5381  936 448 a FT(-quanti\002er)f(introduces)
       
  5382 g(the)i(names)g Fr(bs)31 b FT(and)23 b(the)h FI(8)p FT(-quanti\002ers)e
       
  5383 (replace)h(the)h(v)n(ari-)523 548 y(ables)g Fr(Xs)31
       
  5384 b FT(with)24 b(the)g(terms)g Fr(ts)7 b FT(.)24 b(W)-7
       
  5385 b(e)25 b(indicate)f(this)g(by)g(using)f(the)h(notation)f
       
  5386 FK(G)2831 560 y Fs(as)5 b FC(;)p Fs(Xs)3030 548 y FT(and)24
       
  5387 b FK(G)3240 560 y Fs(bs)t FC(;)p Fs(ts)3384 548 y FT(.)523
       
  5388 648 y(The)19 b(freshness)f(constraints)g FI(r)1448 617
       
  5389 y Fv(00)1448 671 y Fs(bs)1537 648 y FT(stand)g(for)h(the)g(constraints)
       
  5390 f(introduced)f(by)h(the)3006 648 y
       
  5391  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5392  3006 648 a FD(N)3006
       
  5393 648 y
       
  5394  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5395 neg exch translate
       
  5396  3006 648 a FT(-quanti\002ers,)523 747 y(that)27
       
  5397 b(is)h FK(b)36 b FH(#)g FK(F)12 b(V)19 b FH(\()p FK(t)p
       
  5398 FH(\))28 b FT(for)f(each)g FK(b)g FT(in)h Fr(bs)7 b FT(.)27
       
  5399 b(Let)h FK(\033)j FT(be)c(the)g(substitution)f(of)h(the)h(terms)f
       
  5400 Fr(ts)34 b FT(for)26 b(the)523 847 y(v)n(ariables)19
       
  5401 b Fr(Xs)7 b FT(,)20 b(that)h(is)g(the)f(terms)g(introduced)e(by)i(the)g
       
  5402 FI(8)p FT(-quanti\002ers.)648 939 y(Belo)n(w)31 b(we)g(gi)n(v)o(e)g(a)g
       
  5403 (deduction)f(without)g(the)h FK(\031)s FT(-rule)g(where)f(the)i
       
  5404 Fr(bs)38 b FT(and)31 b Fr(ts)38 b FT(are)31 b(suitably)523
       
  5405 1030 y(changed.)17 b(F)o(or)i(this)h(we)g(choose)e(\002rst)i(some)f
       
  5406 (fresh)g(names)g Fr(cs)27 b FT(with)19 b(the)h(pro)o(viso)d(that)j
       
  5407 FK(\031)s Fu(\001)p Fr(cs)30 b FH(=)22 b FK(cs)p FT(,)523
       
  5408 1121 y(which)29 b(means)h(the)o(y)f(are)g(unaf)n(fected)f(by)h(the)h
       
  5409 (permutation)d(introduced)h(by)h(the)h FK(\031)s FT(-rule)f(\(such)523
       
  5410 1213 y(fresh)15 b(names)h(al)o(w)o(ays)g(e)o(xist\).)f(From)h(the)f
       
  5411 (well-formedness)f(of)i(the)f(clause)h(in)g(the)g(stoup-position,)523
       
  5412 1304 y(we)21 b(kno)n(w)e(that)h(there)g(is)h(a)f(substitution)g
       
  5413 FK(\033)1768 1274 y Fv(0)1812 1304 y FT(and)g(a)h(permutation)d
       
  5414 FK(\031)2483 1274 y Fv(0)2527 1304 y FT(such)i(that)1296
       
  5415 1466 y Ft(cs)28 b FG(#)22 b Ft(Xs)6 b FN(;)13 b FF(r)1668
       
  5416 1474 y Ff(Xs)1765 1466 y FF(`)21 b FN(\033)1880 1434
       
  5417 y Fp(0)1902 1466 y FG(\()p FN(p)p FG(\()p FN(s)2037 1474
       
  5418 y Ff(cs)t Fn(;)p Ff(Xs)2186 1466 y FG(\)\))g FF(\031)g
       
  5419 FN(p)p FG(\()p FN(s)2453 1475 y Ff(bs)s Fn(;)p Ff(Xs)2602
       
  5420 1466 y FG(\))1296 1557 y Ft(cs)28 b FG(#)22 b Ft(Xs)6
       
  5421 b FN(;)13 b FF(r)1668 1565 y Ff(Xs)1765 1557 y FF(`)21
       
  5422 b FN(\033)1880 1525 y Fp(0)1902 1557 y FG(\()p FN(\031)1979
       
  5423 1525 y Fp(0)2001 1557 y Fz(\001)p FN(G)2086 1565 y Ff(cs)t
       
  5424 Fn(;)p Ff(Xs)2235 1557 y FG(\))h FF(\031)f FN(G)2428
       
  5425 1566 y Ff(bs)t Fn(;)p Ff(Xs)3318 1512 y FQ(\(5\))523
       
  5426 1723 y FT(hold)g(where)h(we)g(use)h(the)f(short-hand)e(notation)h
       
  5427 Fr(cs)33 b FH(#)27 b Fr(Xs)j FT(to)22 b(refer)f(the)i(sets)g(of)f
       
  5428 (freshness)f(con-)523 1814 y(straints)h FK(c)821 1826
       
  5429 y FC(i)875 1814 y FH(#)k FK(X)1039 1826 y FS(1)1077 1814
       
  5430 y FK(;)14 b(:)g(:)g(:)f(;)h(c)1297 1826 y FC(i)1351 1814
       
  5431 y FH(#)26 b FK(X)1515 1826 y FC(n)1583 1814 y FT(for)21
       
  5432 b(all)h(names)g FK(c)2078 1826 y FC(i)2128 1814 y FT(in)g
       
  5433 Fr(cs)7 b FT(.)22 b(By)g(Lemma)f(1)p FE(\(ii\))p FT(,)h
       
  5434 FI(\031)g FT(is)g(preserv)o(ed)523 1905 y(under)d(substitutions,)g(so)i
       
  5435 (we)f(can)g(infer)g(from)f(\(5\))g(that)1247 2064 y Ft(cs)28
       
  5436 b FG(#)22 b Ft(Xs)27 b FF(`)22 b FN(\033)d FF(\016)f
       
  5437 FN(\033)1777 2032 y Fp(0)1799 2064 y FG(\()p FN(p)p FG(\()p
       
  5438 FN(s)1934 2072 y Ff(cs)s Fn(;)p Ff(Xs)2082 2064 y FG(\)\))j
       
  5439 FF(\031)g FN(\033)s FG(\()p FN(p)p FG(\()p FN(s)2426
       
  5440 2073 y Ff(bs)s Fn(;)p Ff(Xs)2575 2064 y FG(\)\))1247
       
  5441 2155 y Ft(cs)28 b FG(#)22 b Ft(Xs)27 b FF(`)22 b FN(\033)d
       
  5442 FF(\016)f FN(\033)1777 2124 y Fp(0)1799 2155 y FG(\()p
       
  5443 FN(\031)1876 2124 y Fp(0)1898 2155 y Fz(\001)o FN(G)1982
       
  5444 2163 y Ff(cs)t Fn(;)p Ff(Xs)2132 2155 y FG(\))j FF(\031)g
       
  5445 FN(\033)s FG(\()p FN(G)2401 2164 y Ff(bs)t Fn(;)p Ff(Xs)2550
       
  5446 2155 y FG(\))3318 2110 y FQ(\(6\))523 2326 y FT(hold)16
       
  5447 b(where)f(the)i(right-hand)d(sides)j(are)f FK(p)p FH(\()p
       
  5448 FK(s)1803 2338 y Fs(bs)t FC(;)p Fs(ts)1947 2326 y FH(\))h
       
  5449 FT(and)f FK(G)2198 2338 y Fs(bs)5 b FC(;)p Fs(ts)2342
       
  5450 2326 y FT(,)17 b(respecti)n(v)o(ely)-5 b(.)14 b(Note)j(that)f(the)g
       
  5451 FI(r)3317 2338 y Fs(Xs)523 2417 y FT(\223v)n(anish\224)j(because)h(we)g
       
  5452 (ha)n(v)o(e)g(that)g Fr(cs)30 b FH(#)23 b Fr(Xs)30 b
       
  5453 FI(`)23 b FK(\033)s FH(\()p FI(r)2098 2429 y Fs(Xs)2186
       
  5454 2417 y FH(\))p FT(.)e(From)e(\(6\))h(we)g(can)g(further)f(infer)h(that)
       
  5455 1257 2580 y Ft(cs)29 b FG(#)21 b Ft(Xs)28 b FF(`)21 b
       
  5456 FN(\031)s Fz(\001)o FN(\033)f FF(\016)d FN(\033)1858
       
  5457 2544 y Fp(0)1880 2580 y FG(\()p FN(p)p FG(\()p FN(s)2015
       
  5458 2588 y Ff(cs)s Fn(;)p Ff(Xs)2164 2580 y FG(\)\))k FF(\031)g
       
  5459 FN(\031)s Fz(\001)o FG(\()p FN(p)p FG(\()p FN(s)2532
       
  5460 2589 y Ff(bs)s Fn(;)p Ff(ts)2657 2580 y FG(\)\))601 b
       
  5461 FQ(\(7\))1257 2696 y Ft(cs)29 b FG(#)21 b Ft(Xs)28 b
       
  5462 FF(`)21 b FN(\031)s Fz(\001)o FN(\033)f FF(\016)d FN(\033)1858
       
  5463 2660 y Fp(0)1880 2696 y FG(\()p FN(\031)1957 2660 y Fp(0)1979
       
  5464 2696 y Fz(\001)p FN(G)2064 2704 y Ff(cs)t Fn(;)p Ff(Xs)2213
       
  5465 2696 y FG(\))k FF(\031)g FN(\031)s Fz(\001)p FG(\()p
       
  5466 FN(G)2507 2705 y Ff(bs)t Fn(;)p Ff(ts)2633 2696 y FG(\))655
       
  5467 b FQ(\(8\))523 2860 y FT(hold)25 b(by)g(Lemma)g(1)p FE(\(i\))g
       
  5468 FT(asserting)g(that)h FI(\031)f FT(is)i(preserv)o(ed)d(under)g
       
  5469 (permutations.)f(Recall)j(that)g(we)523 2951 y(chosen)20
       
  5470 b(the)g Fr(cs)28 b FT(so)21 b(that)g FK(\031)j FT(does)d(not)f(af)n
       
  5471 (fect)g(them.)g(So)g(if)h(we)g(apply)f(the)g(substitution)g
       
  5472 FK(\033)i FI(\016)c FK(\033)3240 2921 y Fv(0)3285 2951
       
  5473 y FT(and)523 3042 y(the)i(permutation)e FK(\031)25 b
       
  5474 FT(to)20 b(the)g(left-hand)f(side)h(of)g(\(7\))g(we)h(ha)n(v)o(e)e
       
  5475 FK(\031)s Fu(\001)p FK(\033)j FI(\016)c FK(\033)2577
       
  5476 3012 y Fv(0)2601 3042 y FH(\()p FK(p)p FH(\()p FK(s)2746
       
  5477 3054 y Fs(cs)5 b FC(;)p Fs(Xs)2917 3042 y FH(\)\))24
       
  5478 b(=)f FK(p)p FH(\()p FK(s)3206 3054 y Fs(cs)t FC(;)p
       
  5479 Fs(ts)3346 3038 y Fl(0)3373 3042 y FH(\))523 3134 y FT(for)f(some)g
       
  5480 (terms)h Fr(ts)1117 3103 y Fv(0)1140 3134 y FT(.)g(Moreo)o(v)o(er)d(we)
       
  5481 j(ha)n(v)o(e)f Fr(cs)34 b FH(#)28 b Fr(Xs)33 b FI(`)27
       
  5482 b FK(s)2278 3146 y Fs(cs)5 b FC(;)p Fs(ts)2418 3129 y
       
  5483 Fl(0)2472 3134 y FI(\031)27 b FK(\031)s Fu(\001)p FK(s)2680
       
  5484 3146 y Fs(bs)t FC(;)p Fs(ts)2847 3134 y FT(which)22 b(means)g(we)523
       
  5485 3225 y(can)e(replace)f(in)g(the)h(deduction)e(\(4\))h
       
  5486 FK(\031)s Fu(\001)p FK(s)1706 3237 y Fs(bs)t FC(;)p Fs(ts)1870
       
  5487 3225 y FT(by)h FK(s)2013 3237 y Fs(cs)t FC(;)p Fs(ts)2153
       
  5488 3221 y Fl(0)2200 3225 y FT(and)f(get)h(by)f(transiti)n(vity)g(of)g
       
  5489 FI(\031)h FT(a)g(correct)523 3316 y(instance)g(of)g(the)g(axiom.)f
       
  5490 (Thus)h(we)g(can)g(form)g(the)g(deduction:)1056 3479
       
  5491 y FQ(.)1056 3513 y(.)1056 3546 y(.)829 3623 y FF(r)893
       
  5492 3591 y Fp(00)893 3636 y Ff(cs)952 3623 y FN(;)14 b FF(r)1051
       
  5493 3591 y Fp(0)1094 3623 y FF(`)22 b(r)1227 3635 y Ff(ts)1274
       
  5494 3620 y Fl(0)1720 3479 y FQ(.)1720 3513 y(.)1720 3546
       
  5495 y(.)1397 3623 y FF(r)1461 3591 y Fp(00)1461 3636 y Ff(cs)1520
       
  5496 3623 y FN(;)14 b FF(r)1619 3591 y Fp(0)1641 3623 y FG(;)f
       
  5497 FN(\000)32 b FF(\))21 b FN(G)1912 3635 y Ff(cs)t Fn(;)p
       
  5498 Ff(ts)2034 3621 y Fl(0)2230 3457 y FF(r)2294 3425 y Fp(00)2294
       
  5499 3470 y Ff(cs)2353 3457 y FN(;)14 b FF(r)2452 3425 y Fp(0)2495
       
  5500 3457 y FF(`)22 b FN(s)2600 3469 y Ff(cs)t Fn(;)p Ff(ts)2721
       
  5501 3455 y Fl(0)2769 3457 y FF(\031)f FN(t)p 2157 3498 795
       
  5502 4 v 2157 3623 a FF(r)2221 3591 y Fp(00)2221 3636 y Ff(cs)2280
       
  5503 3623 y FN(;)13 b FF(r)2378 3591 y Fp(0)2401 3623 y FG(;)g
       
  5504 FN(\000)2533 3560 y Fn(p)p FR(\()p Fn(s)2618 3579 y Fa(cs)s
       
  5505 Fc(;)p Fa(ts)2723 3567 y Fl(0)2750 3560 y FR(\))2514
       
  5506 3623 y FF(\000)-39 b(\000)-18 b(\000)h(\000)g(\000)f(\000)-38
       
  5507 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))2980 3523 y FQ(Ax)p
       
  5508 829 3664 2123 4 v 1305 3789 a FF(r)1369 3757 y Fp(00)1369
       
  5509 3802 y Ff(cs)1428 3789 y FN(;)13 b FF(r)1526 3757 y Fp(0)1549
       
  5510 3789 y FG(;)g FN(\000)1681 3726 y Fp(r)1732 3744 y Fa(ts)1772
       
  5511 3732 y Fl(0)1799 3726 y Fn(=G)1878 3744 y Fa(cs)s Fc(;)p
       
  5512 Fa(ts)1983 3732 y Fl(0)2010 3726 y Fp(\033)p Fn(p)p FR(\()p
       
  5513 Fn(s)2143 3744 y Fa(cs)s Fc(;)p Fa(ts)2248 3732 y Fl(0)2274
       
  5514 3726 y FR(\))1662 3789 y FF(\000)-22 b(\000)-17 b(\000)f(\000)h(\000)g
       
  5515 (\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)d(!)21
       
  5516 b FN(p)p FG(\()p FN(t)p FG(\))2993 3680 y FF(\033)3053
       
  5517 3689 y Fn(L)1811 3880 y FG(:)472 b FF(g)2410 3880 y
       
  5518  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5519  2410
       
  5520 3880 a FB(N)2410 3880 y
       
  5521  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5522 neg exch translate
       
  5523  2410 3880 a FF(8)p 1261 3919
       
  5524 1259 4 v 1261 4041 a(r)1325 4009 y Fp(0)1348 4041 y FG(;)13
       
  5525 b FN(\000)1522 3985 y
       
  5526  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5527  1522 3985 a Fj(N)1522 3985 y
       
  5528  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5529 neg exch translate
       
  5530  1522
       
  5531 3985 a Fn(as:)p Fp(8)p Ff(Xs)t Fn(:)p Fp(r)1780 3996
       
  5532 y Fa(Xs)1844 3985 y Fn(=G)1923 3996 y Fa(as)s Fc(;)p
       
  5533 Fa(Xs)2054 3985 y Fp(\033)p Fn(p)p FR(\()p Fn(s)2187
       
  5534 3996 y Fa(as)s Fc(;)p Fa(Xs)2318 3985 y FR(\))1461 4041
       
  5535 y FF(\000)-28 b(\000)-17 b(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h
       
  5536 (\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g(\000)f(\000)h(\000)g
       
  5537 (\000)f(\000)-26 b(!)21 b FN(p)p FG(\()p FN(t)p FG(\))p
       
  5538 1261 4079 1259 4 v 1678 4155 a FF(r)1742 4123 y Fp(0)1765
       
  5539 4155 y FG(;)13 b FN(\000)32 b FF(\))21 b FN(p)p FG(\()p
       
  5540 FN(t)p FG(\))2561 4104 y FQ(Sel)523 4327 y FT(without)k(the)g
       
  5541 FK(\031)s FT(-rule.)g(W)-7 b(e)26 b(still)h(need)e(to)g(ensure)g(that)h
       
  5542 FI(r)2205 4296 y Fv(00)2205 4347 y Fs(cs)2273 4327 y
       
  5543 FK(;)14 b FI(r)2379 4296 y Fv(0)2403 4327 y FH(;)g FK(\000)44
       
  5544 b FI(\))33 b FK(G)2716 4339 y Fs(cs)t FC(;)p Fs(ts)2856
       
  5545 4322 y Fl(0)2908 4327 y FT(and)25 b FI(r)3123 4296 y
       
  5546 Fv(00)3123 4347 y Fs(cs)3192 4327 y FK(;)14 b FI(r)3298
       
  5547 4296 y Fv(0)3354 4327 y FI(`)523 4426 y(r)592 4438 y
       
  5548 Fs(ts)648 4422 y Fl(0)699 4426 y FT(are)23 b(deri)n(v)n(able.)f(The)h
       
  5549 (second)g(sequent)g(is)i(deri)n(v)n(able)d(because)g
       
  5550 Fr(cs)37 b FH(#)29 b Fr(Xs)36 b FI(`)29 b FK(\033)s FH(\()p
       
  5551 FI(r)3125 4438 y Fs(Xs)3214 4426 y FH(\))p FT(.)24 b(F)o(or)523
       
  5552 4526 y(the)j(\002rst)h(sequent)f(we)g(can)g(infer)g(from)f(the)i
       
  5553 (original)e(\(sub\)deduction)e FI(r)2743 4496 y Fv(00)2743
       
  5554 4549 y Fs(bs)2811 4526 y FK(;)14 b FI(r)2917 4496 y Fv(0)2941
       
  5555 4526 y FH(;)g FK(\000)47 b FI(\))36 b FK(G)3260 4538
       
  5556 y Fs(bs)5 b FC(;)p Fs(ts)523 4625 y FT(by)28 b(Corollary)f(1)h(that)h
       
  5557 FI(r)1275 4595 y Fv(00)1275 4649 y Fs(bs)1343 4625 y
       
  5558 FK(;)14 b FI(r)1449 4595 y Fv(0)1473 4625 y FH(;)g FK(\000)49
       
  5559 b FI(\))38 b FK(\031)s Fu(\001)p FK(G)1873 4637 y Fs(bs)5
       
  5560 b FC(;)p Fs(ts)2046 4625 y FT(is)30 b(deri)n(v)n(able)c(\(this)j
       
  5561 (deduction)d(does)i(not)g(in-)523 4725 y(troduce)33 b(an)o(y)h(ne)n(w)h
       
  5562 FK(\031)s FT(-rules\).)f(In)h(\(8\))f(we)h(can)f(pull)h(out)f(the)h
       
  5563 (permutation)e FK(\031)2905 4695 y Fv(0)2964 4725 y FT(and)h(we)h(ha)n
       
  5564 (v)o(e)523 4825 y FK(\031)s Fu(\001)p FK(\033)28 b FI(\016)d
       
  5565 FK(\033)792 4795 y Fv(0)816 4825 y FH(\()p FK(\031)898
       
  5566 4795 y Fv(0)922 4825 y Fu(\001)o FK(G)1013 4837 y Fs(cs)5
       
  5567 b FC(;)p Fs(Xs)1185 4825 y FH(\))40 b(=)f FK(\031)s FH(@)p
       
  5568 FK(\031)1526 4795 y Fv(0)1550 4825 y Fu(\001)o FH(\()p
       
  5569 FK(\033)29 b FI(\016)c FK(\033)1801 4795 y Fv(0)1825
       
  5570 4825 y FH(\()p FK(G)1922 4837 y Fs(cs)5 b FC(;)p Fs(Xs)2094
       
  5571 4825 y FH(\)\))p FT(.)30 b(Therefore)d(applying)g(the)i(substitution)
       
  5572 523 4924 y(to)22 b FK(G)675 4936 y Fs(cs)5 b FC(;)p Fs(Xs)869
       
  5573 4924 y FT(gi)n(v)o(es)21 b FK(\031)s FH(@)p FK(\031)1228
       
  5574 4894 y Fv(0)1252 4924 y Fu(\001)o FH(\()p FK(\033)j FI(\016)19
       
  5575 b FK(\033)1492 4894 y Fv(0)1516 4924 y FH(\()p FK(G)1613
       
  5576 4936 y Fs(cs)5 b FC(;)p Fs(Xs)1785 4924 y FH(\)\))26
       
  5577 b(=)g FK(\031)s FH(@)p FK(\031)2131 4894 y Fv(0)2154
       
  5578 4924 y Fu(\001)p FK(G)2246 4936 y Fs(cs)5 b FC(;)p Fs(ts)2386
       
  5579 4920 y Fl(0)2435 4924 y FT(\(taking)21 b(the)h Fr(ts)2885
       
  5580 4894 y Fv(0)2931 4924 y FT(we)g(introduced)p eop end
       
  5581 %%Page: 14 14
       
  5582 TeXDict begin 14 13 bop 523 448 a FT(for)21 b FK(s)681
       
  5583 460 y Fs(cs)t FC(;)p Fs(ts)820 444 y Fl(0)869 448 y FT(earlier\).)f
       
  5584 (Thus)h(by)f(Lemma)h(3)g(we)h(can)f(sho)n(w)f(that)i
       
  5585 FI(r)2448 418 y Fv(00)2448 469 y Fs(cs)2516 448 y FK(;)14
       
  5586 b FI(r)2622 418 y Fv(0)2646 448 y FH(;)g FK(\000)36 b
       
  5587 FI(\))25 b FK(G)2943 460 y Fs(cs)5 b FC(;)p Fs(ts)3083
       
  5588 444 y Fl(0)3132 448 y FT(is)22 b(deri)n(v-)523 548 y(able.)648
       
  5589 651 y(Each)16 b(transformation)f(decreases)i(the)g(number)e(of)i
       
  5590 FK(\031)s FT(-rules)g(in)h(a)f(deduction)e(by)i(one)g(and)g(thus)523
       
  5591 751 y(by)j(repeated)f(application)f(we)j(will)g(e)n(v)o(entually)d(end)
       
  5592 i(up)g(with)g(a)h FK(\031)s FT(-free)e(proof.)500 b FI(u)-55
       
  5593 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
       
  5594 (all)i(the)f(formulas)e(in)i FK(\000)39 b FT(are)27 b(well-formed,)d(e)
       
  5595 n(v)o(ery)i(deduction)f(of)523 1039 y FK(\000)57 b FI(\))45
       
  5596 b FK(G)33 b FT(containing)d FK(\031)s FT(-rules)i(can)g(be)g(replaced)f
       
  5597 (by)h(one)g(without)f FK(\031)s FT(-rules.)h(Consequently)-5
       
  5598 b(,)523 1139 y(nominal)19 b(uni\002cation)g(is)i(suf)n(\002cient)f(for)
       
  5599 f(e)o(x)o(ecuting)f(well-formed)g FK(\013)p FT(Prolog)o(-programs.)523
       
  5600 1423 y FL(4)99 b(V)-10 b(eri\002cation)26 b(of)f(W)-6
       
  5601 b(ell-F)n(ormedness)24 b(Using)g(Nominal)h(Matching)523
       
  5602 1632 y FT(In)19 b(this)h(section)f(we)g(consider)f(the)h(question)f(of)
       
  5603 h(ho)n(w)g(to)g(v)o(erify)f(the)h(well-formedness)e(condition)523
       
  5604 1724 y(gi)n(v)o(en)k(in)g(De\002nition)g(1.)h(F)o(or)f(a)i(clause)1742
       
  5605 1724 y
       
  5606  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5607  1742 1724 a FD(N)1742 1724 y
       
  5608  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5609 neg exch translate
       
  5610  1742 1724 a Fr(as)8
       
  5611 b FK(:)p FI(8)p Fr(Xs)e FK(:)p FI(r)2089 1736 y Fs(as)f
       
  5612 FC(;)p Fs(Xs)2264 1724 y FK(=G)2371 1736 y Fs(as)g FC(;)p
       
  5613 Fs(Xs)2546 1724 y FI(\033)o FK(A)2672 1736 y Fs(as)h
       
  5614 FC(;)p Fs(Xs)2848 1724 y FT(,)22 b(we)g(need)f(to)h(\002nd)523
       
  5615 1815 y(a)f(substitution)e FK(\033)24 b FT(and)c(permutation)e
       
  5616 FK(\031)24 b FT(which)19 b(mak)o(e)h(the)g(tw)o(o)h(judgements)541
       
  5617 1987 y Ft(bs)28 b FG(#)21 b Ft(Xs)6 b FN(;)14 b FF(r)913
       
  5618 1995 y Ff(as)t Fn(;)p Ff(Xs)1086 1987 y FF(`)22 b FN(\033)s
       
  5619 FG(\()p FN(A)1290 1996 y Ff(bs)s Fn(;)p Ff(Xs)1438 1987
       
  5620 y FG(\))f FF(\031)g FN(A)1628 1995 y Ff(as)t Fn(;)p Ff(Xs)1866
       
  5621 1987 y FT(and)84 b Ft(bs)28 b FG(#)22 b Ft(Xs)6 b FN(;)13
       
  5622 b FF(r)2443 1995 y Ff(as)t Fn(;)p Ff(Xs)2617 1987 y FF(`)21
       
  5623 b FN(\033)s FG(\()p FN(\031)s Fz(\001)o FN(G)2893 1996
       
  5624 y Ff(bs)t Fn(;)p Ff(Xs)3043 1987 y FG(\))g FF(\031)g
       
  5625 FN(G)3235 1995 y Ff(as)t Fn(;)p Ff(Xs)523 2164 y FT(hold.)f(F)o(or)h
       
  5626 (the)g(\002rst)h(judgement,)d FK(\033)25 b FT(can)c(be)g(found)f(by)h
       
  5627 (nominal)e(matching.)h(But)i(for)e(the)h(second)523 2264
       
  5628 y(judgement,)j(\002nding)i(both)f(substitution)h FK(\033)k
       
  5629 FT(and)c(permutation)e FK(\031)30 b FT(requires)c(solving)f
       
  5630 (\(NP-hard\))523 2363 y(equi)n(v)n(ariant)13 b(matching)h(problems.)g
       
  5631 (This)h(seemingly)f(ne)o(gati)n(v)o(e)f(result)j(should,)e(ho)n(we)n(v)
       
  5632 o(er)m(,)e(be)j(seen)523 2463 y(in)29 b(the)f(conte)o(xt)g(that)g
       
  5633 (well-formedness)f(only)g(needs)h(to)h(be)g(v)o(eri\002ed)e(once)h(per)
       
  5634 g(clause,)g(rather)523 2563 y(than)20 b(repeatedly)e(during)h
       
  5635 (proof-search.)e(Thus,)i(the)h(one-time)f(cost)i(of)e(performing)f
       
  5636 (equi)n(v)n(ariant)523 2662 y(uni\002cation)h(in)i(checking)e
       
  5637 (well-formedness)f(is)j(ne)o(gligible)e(compared)g(to)h(the)h(cost)g
       
  5638 (of)f(perform-)523 2762 y(ing)j(equi)n(v)n(ariant)f(uni\002cation)h
       
  5639 (throughout)d(computation.)i(Furthermore,)f(as)j(can)g(be)g(seen)g
       
  5640 (from)523 2862 y(the)h(e)o(xamples,)e(the)i(number)e(of)i(names)f(in)h
       
  5641 (a)g(clause)g(is)h(usually)e(small.)h(T)-7 b(aking)24
       
  5642 b(the)h(follo)n(wing)523 2961 y(proposition)18 b(\(whose)i(proof)e(we)j
       
  5643 (omit\))523 3154 y Fi(Pr)o(oposition)e(1.)41 b FE(If)20
       
  5644 b FK(G)1193 3166 y Fs(bs)t FC(;)p Fs(Xs)1385 3154 y FE(equivariantly)e
       
  5645 (matc)o(hes)h(with)i FK(G)2368 3166 y Fs(as)5 b FC(;)p
       
  5646 Fs(Xs)2543 3154 y FE(,)20 b(then)f(a)h(matc)o(hing)e(e)n(xists)j(in)523
       
  5647 3254 y(whic)o(h)f(the)g(permutation)f FK(\031)24 b FE(consists)c(of)h
       
  5648 (swappings)e FH(\()p FK(a)2177 3266 y FC(i)2219 3254
       
  5649 y FK(b)2255 3266 y FC(i)2282 3254 y FH(\))i FE(only)-5
       
  5650 b(.)523 3442 y FT(into)27 b(account,)e(we)j(can)f(just)g(enumerate)f
       
  5651 (all)h(possible)g(cases)h(\()p FH(2)2458 3412 y FC(n)2530
       
  5652 3442 y FT(gi)n(v)o(en)e FK(n)h FT(names\))f(and)h(solv)o(e)523
       
  5653 3542 y(each)22 b(of)f(the)h(nominal)e(matching)h(problems.)f(If)i(one)f
       
  5654 (problem)f(can)i(be)g(solv)o(ed,)e(then)i(we)g(ha)n(v)o(e)f(a)523
       
  5655 3642 y FK(\033)j FT(and)c(a)g FK(\031)25 b FT(as)20 b(required)f(by)h
       
  5656 (the)g(condition.)523 3926 y FL(5)99 b(Conclusion)523
       
  5657 4122 y FT(W)-7 b(e)22 b(ha)n(v)o(e)f(sho)n(wn)g(that)g(for)g
       
  5658 (well-formed)e FK(\013)p FT(Prolog)h(programs,)g(all)h(instances)h(of)f
       
  5659 (the)g FK(\031)s FT(-rule)g(can)523 4221 y(be)j(remo)o(v)o(ed)d(from)i
       
  5660 (deductions.)f(As)i(a)h(result,)e(proof)g(search)g(using)g(only)g
       
  5661 (nominal)g(uni\002cation)523 4321 y(is)k(complete)e(for)h(such)g
       
  5662 (programs,)e(which)h(coincides)h(with)g(our)g(e)o(xperimental)d
       
  5663 (results)k(gained)523 4421 y(from)17 b(our)g(implementation)f(of)h
       
  5664 FK(\013)p FT(Prolog)o(.)h(This)g(is)h(a)g(signi\002cant)e(result,)h
       
  5665 (because)f(the)h(alternati)n(v)o(e)523 4520 y(is)j(to)g(use)f(an)g
       
  5666 (NP-hard)f(equi)n(v)n(ariant)f(uni\002cation)h(algorithm)g(for)g(proof)
       
  5667 g(search.)648 4615 y(In)e(order)g(to)h(be)g(well-formed,)d(the)j
       
  5668 FJ(type)p FT(-program)d(gi)n(v)o(en)h(in)i(the)g(Introduction)d(needs)j
       
  5669 (to)g(be)523 4706 y(stated)i(as)h(follo)n(ws)800 4861
       
  5670 y FP(type\(Gamma,lam\(x.M\),arr\(S,T\)\))40 b(:-)1272
       
  5671 4953 y(x)13 b(#)f(Gamma,)f(x#S,)h(x#T,)g(type\([\(x,S\)|Gamma],M,T\).)p
       
  5672 eop end
       
  5673 %%Page: 15 15
       
  5674 TeXDict begin 15 14 bop 523 448 a FT(e)o(xplicitly)24
       
  5675 b(gi)n(ving)g(the)i(freshness)e(constraints)h FJ(x#S)g
       
  5676 FT(and)g FJ(x#T)p FT(.)g(These)g(constraints)g(do)g(not)g(af-)523
       
  5677 548 y(fect)f(the)f(meaning)f(of)h(the)h(program)d(because)i(term)h(v)n
       
  5678 (ariables)e(are)i(e)o(xpected)e(\(by)h(programmer)523
       
  5679 648 y(con)m(v)o(ention\))e(not)i(to)i(appear)e(in)h(types.)f(In)h(f)o
       
  5680 (act,)g(our)g(implementation)d(of)j FK(\013)p FT(Prolog)g(is)h
       
  5681 (strongly)523 747 y(typed)i(and)h(therefore)e(can)i(determine)e
       
  5682 (automatically)h(from)g(type)g(information)f(that)i(lambda-)523
       
  5683 847 y(term)19 b(v)n(ariables)f(can)i(ne)n(v)o(er)d(occur)i(in)g(types.)
       
  5684 g(Thus,)f(our)h(analysis)g(could)g(be)g(made)f(more)h(precise)523
       
  5685 946 y(by)h(taking)f(type)h(information)e(into)i(account.)648
       
  5686 1046 y(Let)i(us)i(brie\003y)e(mention)f(whether)h(our)f(result)i(can)g
       
  5687 (be)f(strengthened.)e(The)j(logic)f(program-)523 1146
       
  5688 y(ming)j(language)f FK(\025)p FT(Prolog)h([7])g(has)h(con)m(vincingly)d
       
  5689 (demonstrated)h(the)i(usefulness)f(of)g(implica-)523
       
  5690 1245 y(tions)f(in)h FK(G)p FT(-formulae)e(\(that)h(is)h(e)o(xtending)d
       
  5691 (logic)i(programming)d(to)k(the)f(setting)h(of)f(Hereditary)523
       
  5692 1345 y(Harrop)j(formulae\).)f(In)h FK(\013)p FT(Prolog)h(we)g(w)o(ould)
       
  5693 g(lik)o(e)g(to)g(allo)n(w)g(implications)f(in)i FK(G)p
       
  5694 FT(-formulae)d(as)523 1445 y(well.)20 b(Whether)f(our)f(result)i(e)o
       
  5695 (xtends)e(to)i(such)f(formulae)f(is)i(still)h(open.)d(It)i(seems)g
       
  5696 (that)f(our)g(de\002ni-)523 1544 y(tion)f(of)f FK(G)p
       
  5697 FT(-formulae)f(can)i(be)g(e)o(xtended)e(to)i(include)f(e)o(xistential)g
       
  5698 (and)h(uni)n(v)o(ersal)e(formulae.)g(Ho)n(w-)523 1644
       
  5699 y(e)n(v)o(er)m(,)f(our)h(pro)o(ving)e(technique)h(for)h(sho)n(wing)f
       
  5700 (this)i(w)o(ould)f(require)f(some)h(subtle)h(modi\002cations\227)523
       
  5701 1743 y(for)k(e)o(xample)e(we)j(w)o(ould)e(need)h(to)g(de\002ne)f(when)h
       
  5702 (tw)o(o)g(formulae)f(with)h(quanti\002ers)f(are)h FI(\031)p
       
  5703 FT(-equal,)523 1843 y(which)j(is)i(non-tri)n(vial.)d(Ho)n(we)n(v)o(er)m
       
  5704 (,)f(we)j(e)o(xpect)f(that)h(this)h(can)e(be)h(done.)e(What)j(is)f
       
  5705 (impossible)g(is)523 1943 y(to)20 b(allo)n(w)871 1943
       
  5706 y
       
  5707  currentpoint currentpoint translate -1 1 scale neg exch neg exch translate
       
  5708  871 1943 a FD(N)871 1943 y
       
  5709  currentpoint currentpoint translate 1 -1 div 1 1 div scale neg exch
       
  5710 neg exch translate
       
  5711  871 1943 a FT(-quanti\002ers)f(in)h
       
  5712 (goal-formulae.)d(Such)j(formulae)e(really)i(need)f(equi)n(v)n(ariant)f
       
  5713 (uni\002ca-)523 2042 y(tion.)523 2150 y FO(Ackno)o(wledgements:)28
       
  5714 b FQ(This)14 b(research)h(w)o(as)g(supported)h(by)f(a)f(fello)n(wship)h
       
  5715 (for)f(Urban)h(from)f(the)h(Ale)o(xander)o(-)523 2249
       
  5716 y(v)o(on-Humboldt)20 b(foundation.)523 2515 y FL(Refer)n(ences)523
       
  5717 2706 y FQ(1.)42 b(J.)22 b(Chene)o(y)-5 b(.)41 b(The)22
       
  5718 b(comple)o(xity)i(of)f(equi)n(v)n(ariant)h(uni\002cation.)40
       
  5719 b(In)23 b FM(Pr)m(oc.)f(of)h(International)h(Colloquium)621
       
  5720 2797 y(on)19 b(A)o(utomata,)f(Langua)o(g)o(es)k(and)d(Pr)m(o)o(gr)o
       
  5721 (amming)p FQ(,)h(v)o(olume)f(3142)h(of)f FM(LNCS)p FQ(,)f(pages)i
       
  5722 (332\226344,)h(2004.)523 2889 y(2.)42 b(J.)18 b(Chene)o(y)-5
       
  5723 b(.)28 b FM(Nominal)19 b(Lo)o(gic)g(Pr)m(o)o(gr)o(amming)p
       
  5724 FQ(.)28 b(PhD)18 b(thesis,)h(Cornell)g(Uni)n(v)o(ersity)-5
       
  5725 b(,)19 b(Ithaca,)g(NY)-10 b(,)18 b(2004.)523 2980 y(3.)42
       
  5726 b(J.)18 b(Chene)o(y)i(and)g(C.)e(Urban.)28 b(Alpha-prolog:)20
       
  5727 b(A)f(logic)h(programming)g(language)h(with)d(names,)i(binding,)621
       
  5728 3071 y(and)e FN(\013)p FQ(-equi)n(v)n(alence.)26 b(In)17
       
  5729 b(B.)g(Demoen)i(and)f(V)-10 b(.)17 b(Lifschitz,)g(editors,)g
       
  5730 FM(Pr)m(oc.)g(of)h(International)g(Confer)m(ence)621
       
  5731 3163 y(on)h(Lo)o(gic)g(Pr)m(o)o(gr)o(amming)p FQ(,)h(v)o(olume)f(3132)h
       
  5732 (of)f FM(LNCS)p FQ(,)f(pages)i(269\226283,)h(2004.)523
       
  5733 3254 y(4.)42 b(R.)14 b(Dyckhof)n(f)j(and)f(L.)e(Pinto.)19
       
  5734 b(Proof)c(Search)g(in)g(Constructi)n(v)o(e)h(Logic.)j(In)d(S.)e(Barry)h
       
  5735 (Cooper)h(and)g(John)g(K.)621 3345 y(T)m(russ,)28 b(editors,)g
       
  5736 FM(Pr)m(oc.)g(of)h(the)f(Lo)o(gic)h(Colloquium)g(1997)p
       
  5737 FQ(,)g(v)o(olume)g(258)h(of)e FM(London)i(Mathematical)621
       
  5738 3437 y(Society)19 b(Lectur)m(e)g(Note)g(Series)p FQ(,)g(pages)h
       
  5739 (53\22665.)g(Cambridge)g(Uni)n(v)o(ersity)f(Press,)f(1997.)523
       
  5740 3528 y(5.)42 b(M.)25 b(J.)h(Gabbay)h(and)f(J.)g(Chene)o(y)-5
       
  5741 b(.)50 b(A)25 b(proof)i(theory)f(for)g(nominal)h(logic.)49
       
  5742 b(In)26 b FM(Pr)m(oc.)f(of)h(Annual)g(IEEE)621 3619 y(Symposium)20
       
  5743 b(on)f(Lo)o(gic)g(in)g(Computer)h(Science)p FQ(,)f(pages)h(139\226148,)
       
  5744 h(2004.)523 3710 y(6.)42 b(J.)18 b(C.)g(Mitchell.)27
       
  5745 b FM(Concepts)20 b(in)f(Pr)m(o)o(gr)o(amming)g(Langua)o(g)o(es)p
       
  5746 FQ(.)29 b(CUP)18 b(Press,)g(2003.)523 3802 y(7.)42 b(G.)22
       
  5747 b(Nadathur)i(and)f(D.)g(Miller)l(.)39 b(Higher)o(-order)23
       
  5748 b(logic)g(programming.)42 b(In)23 b(D.)f(M.)h(Gabbay)-5
       
  5749 b(,)24 b(C.)e(J.)g(Hog-)621 3893 y(ger)m(,)28 b(and)g(J.)g(A.)g
       
  5750 (Robinson,)h(editors,)f FM(Handbook)i(of)e(Lo)o(gics)h(for)f
       
  5751 (Arti\002cial)f(Intellig)o(ence)i(and)g(Lo)o(gic)621
       
  5752 3984 y(Pr)m(o)o(gr)o(amming)p FQ(,)19 b(v)o(olume)g(5,)g(pages)h
       
  5753 (499\226590.)h(Clarendon)f(Press,)e(1998.)523 4076 y(8.)42
       
  5754 b(A.)28 b(M.)i(Pitts.)59 b(Nominal)29 b(logic,)g(a)h(\002rst)e(order)i
       
  5755 (theory)g(of)f(names)h(and)g(binding.)62 b FM(Information)30
       
  5756 b(and)621 4167 y(Computation)p FQ(,)19 b(186:165\226193,)j(2003.)523
       
  5757 4258 y(9.)42 b(C.)16 b(Urban,)h(A.)f(M.)g(Pitts,)g(and)h(M.)g(J.)f
       
  5758 (Gabbay)-5 b(.)24 b(Nominal)17 b(uni\002cation.)22 b
       
  5759 FM(Theor)m(etical)c(Computer)f(Science)p FQ(,)621 4350
       
  5760 y(323\(1-2\):473\226497,)22 b(2004.)p eop end
       
  5761 %%Trailer
       
  5762 
       
  5763 userdict /end-hook known{end-hook}if
       
  5764 %%EOF