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/BlueViolet{0.86 0.91 0 0.04 setcmykcolor}DC/Periwinkle{0.57 0.55 0 0 setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC /CornflowerBlue{0.65 0.13 0 0 setcmykcolor}DC/MidnightBlue{0.98 0.13 0 0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1 0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11 0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0 setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0 0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC /Aquamarine{0.82 0 0.30 0 setcmykcolor}DC/BlueGreen{0.85 0 0.33 0 setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0 0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0 1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC /PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0 setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{ 0.26 0 0.76 0 setcmykcolor}DC/OliveGreen{0.64 0 0.95 0.40 setcmykcolor} DC/RawSienna{0 0.72 1 0.45 setcmykcolor}DC/Sepia{0 0.83 1 0.70 setcmykcolor}DC/Brown{0 0.81 1 0.60 setcmykcolor}DC/Tan{0.14 0.42 0.56 0 setcmykcolor}DC/Gray{0 0 0 0.50 setcmykcolor}DC/Black{0 0 0 1 setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end %%EndProcSet %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /prime put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSS9 %!PS-AdobeFont-1.1: CMSS9 1.0 %%CreationDate: 1991 Aug 20 17:34:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSS9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 101 /e put dup 114 /r put dup 116 /t put
dup 117 /u put readonly def /FontBBox{-63 -250 1027 760}readonly def /UniqueID 5000802 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BDCEEA888D07B439DBD4E8B4C9 D198C0387612A3C669D6A672BF4726B90213C2EC063317A043B6447FEC896BD0 598D0AA1DDB874B7D72F3E89529FFB544C16C41F04AB6972093A91C201589D61 A8FE417702E200FC26E47359B881D02F7CB42A5B21F90424F82FEFEEC5ED59F6 C48895523098D73036537A54639526839F785D7157E08DDBCD96AEF023667FC0 1921070EA097CAAAE9E4E9D5D21DB9363555AD932858E23A534F7DA9F2F0E79E 9D46C0D5EDC452179B6A328A52412F2FC548B03A056663DCE243A441EADC0D63 8FF6D19EEC71606A368EC3EF1DB8FA9117DB6169ED3BD792D9AB2811378E50A8 535232683CFB2C5E0F3C350CD294AF9804B7B88FC848D962819940AC88247756 BD1EDE31C397076351447974F91BA99BF34081C1A6C65879C36934E1494108BF FA61B57E3A8154D7CBB6AC9B671700FE227AECF9160145957A2E629C3931CA4D 42147A6CA42FE5E0947DD2EEEB0A640EF9C73800A59F0F0B6B75C3C9C863ADED 9627DEBC218141FA414913611061B3EBB46AA393CDB074A40492FDEAD1207838 A13CA709C2CDE567B9688DC5D4D622ACD752A7D6DE3F79002ACDA6AE41EEC084 BB1D3C962725E1AB05922698C598FEAB0D02ACFD97CD8BAD811193660043AEEE 659C18DADD0E614A401FA0F0A485AD4A47DB73FDB3D2B294145FFBCCB994FD2E 68B4ECB3364B377FF6243ED93921F3B3DFDAC3CE9CB3E2B23115BD45FF1F6734 70DDC37F712D78543A8A8A561DAF3303DD8F3742EB700043207D04A782D94952 1E57EC7E5858AA91F07524F2EF2A129D38CAF8698126B8981537CD4DC45AB99E EE571322DA22745F552675C785D0A8898ADEDD5DA7388C11DF0FF177141F1A21 46B93D36F6683C8669896AD45F68544CB9D098057BF0A54A5465FEF047CCDAD9 A05EA8E336711F7ABC015CE3579143DD98536E0FC91F9E102CD7959C85349916 5E7D0F0964096CE93F978979FAE9C6DCCE505931D6CA68831D0214C9F414D66C EBAE04BD69B92A8352C702237CCA9EAF537867922C599D301E2E158D13DC11BA F6C1B23D6EDB92E1AD151C80A5CF3071CF2A1702ED85ABF05F7D3591E57DF679 0F211C3B56B0A3D977D9CB43AC1A0BC4FA10713999D00204C8065DBB9F9D528D 179E5A0A3CE96789FC9BAB472AF5191D07F60264E5312F77AA0C082AC6168038 20D44299FF0F0C33448EE1E90F16115124DDA4EBDDA0E6CE307262E4E1FDD0A0 8D082F033EE3C20AB26FCA9BD9B8B5A96DFB8AA616E79EFF87DA0297D9906FA7 502313644721FFD761D497312D46A0512BBAF36CF6173CDC72C61E62EAE0AC58 C5556D2DD14B9DBB4C00222AE717E360DCF036C680DBF87B6D5BC2CF99A50C1C B77AA5DA50A22BA49A74F48D082FDC85D9FB7CFF9F72EDB6E1B5A77B11EA8CDA 824211502E8641568824AA0F18FC3B8B67724FB64EE2A8E1D200D9D0FC914026 A7DEA173597927DC5E14CEEF5A3D18C122B0ADB0AF4D830ECF684BA64574B30E F7B53EF4FB3D11ACF4D1C9960C98E11DD8C586149E24AA601D1A170EA0BCA619 06D8BA83206D42D506C6B06CA268837488B2729273CEC4462AAF8E680DFAE953 8090187B7FD3433957 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9
%!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /periodcentered put dup 27 /propersuperset put dup 95 /logicalor put dup 96 /turnstileleft put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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
0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 73 /I put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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cleartomark %%EndFont
%%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 58 /colon put dup 61 /equal put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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
409A5D417DA6730A69E310FA6A4229FC7E4F620B0FC4C63C50E99E179EB51E4C 4BC45217722F1E8E40F1E1428E792EAFE05C5A50D38C52114DFCD24D54027CBF 2512DD116F0463DE4052A7AD53B641A27E81E481947884CE35661B49153FA19E 0A2A860C7B61558671303DE6AE06A80E4E450E17067676E6BBB42A9A24ACBC3E B0CA7B7A3BFEA84FED39CCFB6D545BB2BCC49E5E16976407AB9D94556CD4F008 24EF579B6800B6DC3AAF840B3FC6822872368E3B4274DD06CA36AF8F6346C11B 43C772CC242F3B212C4BD7018D71A1A74C9A94ED0093A5FB6557F4E0751047AF D72098ECA301B8AE68110F983796E581F106144951DF5B750432A230FDA3B575 5A38B5E7972AABC12306A01A99FCF8189D71B8DBF49550BAEA9CF1B97CBFC7CC 96498ECC938B1A1710B670657DE923A659DB8757147B140A48067328E7E3F9C3 7D1888B284904301450CE0BC15EEEA00E48CCD6388F3FC3C8578EF9A20A0E06E 4F7ADDAF0E7D1E182D115BF1AD931977325AD391E72E2B13CC108E3726C11099 E2000623188AAAC9F3E233EB253BDD8B0A4759A66A113E066238B0086AC1B634 5ABFF90E4B5ED3FA69C22541981B2BFC9710AEF6B50A8BB53431C7B4D380D721 639E005D6B4688EE16BFF48443E7C9E5FB5BC5883E271CB03428966C96B6988B 2C9127404E8C64B122D405613C972C5D49D75D6B9D16DFE9F4499B055104E3C7 5B11E20D6ADC5F5DFEBD8D0ED2170A9E1FA5A579A77664988E6746B4C60DC2C7 8F7ECEAA5816111C8B055490D1B769CC771D790B6A52403ED5436F15A5EC7B24 FAEE4BA8F72B656D2FEF73A6A14FA643DF9E00F9D4C724D36560761224DF6C5D 3A29195024BEBAF8271736464448C050B2AB0D73CFE1B49904CC95E9D8AB5407 C63FF25AB2BD2B48265ADCF84D30BA159297CA560B82ED9861836D9DB3C718E4 8A4B43453F6CA8203985BC90D6529A750B5A735C3863ACCA6E1C5FD4D1FCAC46 3BB7CF3FB91D74F45E44FF6F5DB9F1D2CE262ED3F719C1233E50D14B85517D58 30A49CFD32FE3EBEC09BDEAF4F16B42F057F7ECD8A92CD20B8A0236885FDA431 5FA29FAC39410C580A49BA703739384ABBEE371D7AFEC546D5B809E7F20AFBC8 3706A4001087EF91C5C7A6482B98EF8D20CFEB323C3184A81A817ED263BFA175 A1B7CAEAB959FBEF1EED8A2950E23DB6D8119A09D29DA52E90596E5FC9EC5F67 ABE68B11864BD446276749384200A71026702BE0D7DEEDD1527982D25C3D7276 65FA902B47133126C22AB6849F1FAFF0BA70D2C3AE54272342A54FD3694FB680 BDEA3FD9BBAC5A6EEEB7AD178C31EA107F1A3357B7D1C472AD51F4A25D3EE0FA 41DB85B922B627940D98341E99D20560584ED26592B51698A7A697058E7159CA EF1705386B24BD83964E62C5B3C87FE9FC43E528C1AC664499A9EF5AC8853D0E 9DD7F813C5E61D5D51C551504296C7685E21FDA73E61CD71BBED732057305B6D 19628DA1E95A508312276F130D7E4346F28DD7CF94B13F696A8A0B34B22C6DD5 9F4C95C32DBA43AA91DD364FDF1A640466D856B56942B9D8C3270BEA447C4E32 ED5DD9D9F82FA08B1D29F9EE674C78C391EF10D2AE0317F501B304C5C59D6CEA 997E157A276D06CC76652CF11E998B6F13615CE215F9EFFC868EC786D74A5927 534C3BB9747C6DC7F6AE8622464692DD7B984D50250AF579E086083B9B6D4396 F23E1DD313019DE2DFDBE5022C6C81FA9C7C42EC6FE6E76CE596FE5A82401F2A 2B69BBC9A4C34695533EFEA7C687E7331781B3A1391F7167C66BF4C54DDB94C6 71EF0599826AB6A72589058C6B9F1E12D56EB1FE955C97C131FF8E16 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved)
readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put dup 110 /n put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /periodcentered put dup 2 /multiply put dup 17 /equivalence put dup 27 /propersuperset put dup 33 /arrowright put dup 41 /arrowdblright put dup 55 /mapsto put
dup 58 /logicalnot put dup 62 /latticetop put dup 63 /perpendicular put dup 68 /D put dup 69 /E put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 102 /braceleft put dup 103 /braceright put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSS10 %!PS-AdobeFont-1.1: CMSS10 1.0 %%CreationDate: 1991 Aug 20 17:33:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin
/version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def /FontBBox{-61 -250 999 759}readonly def /UniqueID 5000803 def currentdict end currentfile eexec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(small)i(ambiguity)e(her)n(e:)50 b(ther)n(e)33 b(may)i(be)f(several)482 3552 y(hypotheses)15 b Fu(A)25 b Ft(true)20 b Fv(and)f(fr)n(om)g(the)g (form)g(of)g(the)f(r)o(ule)h(above)g(we)g(cannot)g(tell)g(which)482 3665 y(one)i(was)i(meant.)28 b(In)23 b(or)n(der)e(to)i(make)f(this)h (unambiguous)f(we)g(r)n(ecor)n(d)h(the)f(number)482 3778 y(of)g(the)g(assumption)g(that)g(was)h(used.)482 3936 y Fo(Hypothesis)f(rule.)p 1044 4098 1632 4 v 1044 4195 a Fu(A)1112 4209 y Fn(1)1177 4195 y Ft(true)q Fu(;)15 b(:)g(:)g(:)h(;)f(A)1598 4209 y Fm(i)1652 4195 y Ft(true)p Fu(;)g(:)g(:)g(:)i(;)e(A)2073 4209 y Fm(n)2146 4195 y Ft(true)25 b Fs(`)g Fu(A)2471 4209 y Fm(i)2525 4195 y Ft(true)2718 4126 y Fr(Hyp)2879 4089 y Fm(i)482 4424 y Fo(Substitution)c(principle.)p Black Black 709 4600 a Fv(If)h Fu(H)868 4614 y Fn(1)933 4600 y Fs(`)j Fu(A)g Ft(true)e Fv(and)g Fu(H)1534 4614 y Fn(1)1573 4600 y Fu(;)15 b(A)25 b Ft(true)q Fu(;)15 b(H)1974 4614 y Fn(2)2038 4600 y Fs(`)25 b Fu(C)32 b Ft(true)23 b Fv(then)f Fu(H)2668 4614 y Fn(1)2707 4600 y Fu(;)15 b(H)2823 4614 y Fn(2)2887 4600 y Fs(`)25 b Fu(C)32 b Ft(true)p Fv(.)623 4775 y(The)e(latter)h(is) h(called)g(a)g(substitution)e(principle)h(because)g(in)h(or)n(der)e(to) h(obtain)482 4888 y(evidence)c(for)h Fu(C)42 b Ft(true)29 b Fv(we)f(substitute)e(derivations)i(of)g Fu(A)36 b Ft(true)28 b Fv(for)h(uses)d(of)j(the)e(as-)482 5001 y(sumption)21 b Fu(A)26 b Ft(true)p Fv(.)p Black 484 5250 a Fp(L)t Fq(E)t(C)t(T)t(U)t(R)t(E)e Fp(N)t Fq(O)t(T)t(E)t(S)1598 b Fp(N)t Fq(O)t(V)t(E)t(M)t(B)t(E)t(R)25 b Fp(1)t(8)t(,)f(2)t(0)t(0)t (4)p Black eop end %%Page: 3 3 TeXDict begin 3 2 bop Black 490 265 a Fo(The)22 b(Curry-Howard)g (Isomorphism)1400 b(L23.3)p 490 302 2989 4 v Black 631 598 a Fv(Now)25 b(we)h(have)g(the)f(concepts)g(in)i(place)f(to)g(be)g (able)h(to)e(de\002ne)g(implication)j(by)490 711 y(its)22 b(intr)n(oduction)g(r)o(ule.)1545 784 y Fu(H)r(;)15 b(A)26 b Ft(true)f Fs(`)g Fu(B)30 b Ft(true)p 1545 838 720 4 v 1590 934 a Fu(H)j Fs(`)24 b Fu(A)i Fs(\033)f Fu(B)30 b Ft(true)2306 867 y Fs(\033)o Fu(I)490 1094 y Fv(The)k(elimination)i (r)o(ule)f(is)g(based)f(on)g(the)g(substitution)f(principle.)65 b(Assume)33 b(we)490 1207 y(know)26 b(that)g Fu(A)33 b Fs(\033)g Fu(B)k Ft(true)p Fv(.)k(By)26 b(the)f(above)j(r)o(ule)e (this)h(means)f Fu(B)38 b Ft(true)27 b Fv(under)e(the)h(as-)490 1319 y(sumption)21 b(that)i Fu(A)i Ft(true)q Fv(.)i(Now)-8 b(,)22 b(if)h(we)f(had)h(a)g(pr)n(oof)f(of)h Fu(A)i Ft(true)e Fv(we)f(could)h(substitute)490 1432 y(it)g(for)f(uses)f(of)i(the)f (assumption)f(in)j(the)d(pr)n(oof)i(of)f Fu(B)30 b Ft(true)p Fv(.)e(As)22 b(a)h(r)o(ule:)1315 1615 y Fu(H)33 b Fs(`)24 b Fu(A)i Fs(\033)f Fu(B)30 b Ft(true)91 b Fu(H)32 b Fs(`)25 b Fu(A)g Ft(true)p 1315 1655 1154 4 v 1672 1752 a Fu(H)33 b Fs(`)24 b Fu(B)30 b Ft(true)2510 1684 y Fs(\033)p Fu(E)631 1948 y Fv(Now)22 b(we)g(can)h(pr)n(ove,)f(for)h(example,)f(that)g Fl(\()p Fu(A)f Fs(^)f Fu(B)5 b Fl(\))25 b Fs(\033)g Fl(\()p Fu(B)g Fs(^)20 b Fu(A)p Fl(\))26 b Ft(true)p Fv(.)p 752 2120 945 4 v 752 2216 a Fu(A)20 b Fs(^)g Fu(B)30 b Ft(true)25 b Fs(`)g Fu(A)20 b Fs(^)g Fu(B)30 b Ft(true)1738 2147 y Fr(Hyp)1899 2110 y Fn(1)p 752 2252 V 836 2349 a Fu(A)21 b Fs(^)e Fu(B)30 b Ft(true)c Fs(`)f Fu(B)k Ft(true)1738 2277 y Fs(^)o Fu(E)1865 2291 y Fn(2)p 2030 2120 V 2030 2216 a Fu(A)20 b Fs(^)g Fu(B)30 b Ft(true)25 b Fs(`)g Fu(A)20 b Fs(^)g Fu(B)30 b Ft(true)3016 2147 y Fr(Hyp)3177 2110 y Fn(1)p 2030 2252 V 2117 2349 a Fu(A)20 b Fs(^)g
Fu(B)30 b Ft(true)25 b Fs(`)g Fu(A)h Ft(true)3016 2277 y Fs(^)o Fu(E)3143 2291 y Fn(1)p 836 2385 2051 4 v 1389 2482 a Fu(A)21 b Fs(^)f Fu(B)29 b Ft(true)d Fs(`)f Fu(B)f Fs(^)c Fu(A)26 b Ft(true)2928 2416 y Fs(^)p Fu(I)p 1334 2518 1056 4 v 1334 2620 a Fs(\001)f(`)g Fl(\()p Fu(A)c Fs(^)f Fu(B)5 b Fl(\))25 b Fs(\033)g Fl(\()p Fu(B)g Fs(^)19 b Fu(A)p Fl(\))26 b Ft(true)2431 2547 y Fs(\033)p Fu(I)490 2814 y Fv(Note)17 b(that)h(this)g(is)h(holds)f(for)g(any)h(pr)n (opositions)d Fu(A)j Fv(and)g Fu(B)5 b Fv(,)18 b(that)g(is,)i(it)e(is)h (a)g(schematic)490 2926 y(derivation)j(just)g(like)h(infer)n(ence)g(r)o (ules)f(ar)n(e)h(schematic.)631 3039 y(W)-8 b(e)21 b(continue)g(our)h (analysis)g(of)g(logical)i(connectives)d(with)h(disjunction)f Fu(A)d Fs(_)f Fu(B)5 b Fv(.)490 3152 y(The)21 b(disjunction)g(is)g(tr)o (ue)g(if)i(either)d(of)i(the)f(disjuncts)f(is)i(tr)o(ue.)27 b(This)21 b(means)g(we)g(have)490 3265 y(two)h(intr)n(oduction)g(r)o (ules.)1145 3424 y Fu(H)33 b Fs(`)24 b Fu(A)i Ft(true)p 1058 3460 609 4 v 1058 3557 a Fu(H)32 b Fs(`)25 b Fu(A)c Fs(_)e Fu(B)30 b Ft(true)1708 3485 y Fs(_)o Fu(I)1808 3499 y Fn(1)2205 3424 y Fu(H)i Fs(`)25 b Fu(B)30 b Ft(true)p 2121 3460 V 2121 3557 a Fu(H)i Fs(`)25 b Fu(A)20 b Fs(_)g Fu(B)30 b Ft(true)2770 3485 y Fs(_)p Fu(I)2871 3499 y Fn(2)490 3751 y Fv(In)g(or)n(der)f(to)h(determine)e(the)i(elimination)i (r)o(ule,)g(we)d(must)h(consider)f(how)h(to)g(use)490 3864 y(the)d(knowledge)e(that)j Fu(A)c Fs(_)g Fu(B)39 b Ft(true)p Fv(.)k(Clearly)-10 b(,)30 b(we)d(do)g(not)g(know)g(which)i (of)e Fu(A)35 b Ft(true)490 3976 y Fv(or)24 b Fu(B)34 b Ft(true)25 b Fv(holds.)33 b(This)25 b(means)f(if)i(we)e(ar)n(e)h (trying)f(to)g(pr)n(ove)g Fu(C)36 b Ft(true)25 b Fv(and)g(we)f(know)490 4089 y Fu(A)h Fs(_)f Fu(B)41 b Ft(true)p Fv(,)30 b(we)e(must)g(be)h (able)g(to)f(show)g Fu(C)43 b Ft(true)29 b Fv(no)f(matter)g(whether)f Fu(A)37 b Ft(true)29 b Fv(or)490 4202 y Fu(B)h Ft(true)p Fv(.)d(In)22 b(other)e(wor)n(ds,)g(we)h(must)g(pr)n(oceed)f(with)i(a)g (pr)n(oof)f(by)h(cases.)27 b(In)21 b(the)g(form)490 4315 y(of)h(an)i(elimination)f(r)o(ule:)782 4474 y Fu(H)32 b Fs(`)25 b Fu(A)c Fs(_)e Fu(B)30 b Ft(true)91 b Fu(H)r(;)15 b(A)26 b Ft(true)g Fs(`)e Fu(C)32 b Ft(true)91 b Fu(H)r(;)15 b(B)30 b Ft(true)c Fs(`)f Fu(C)31 b Ft(true)p 782 4528 2230 4 v 1678 4624 a Fu(H)i Fs(`)25 b Fu(C)31 b Ft(true)3053 4559 y Fs(_)p Fu(E)631 4818 y Fv(As)22 b(a)h(sample)g(pr)n(oof,)f (consider)f(the)h(statement)p Black Black 717 5001 a(If)h Fu(A)g Fv(or)f Fu(B)27 b Fv(implies)c Fu(C)7 b Fv(,)22 b(then)g Fu(A)h Fv(implies)g Fu(C)7 b Fv(.)p Black 492 5250 a Fp(L)t Fq(E)t(C)t(T)t(U)t(R)t(E)24 b Fp(N)t Fq(O)t(T)t(E)t(S) 1598 b Fp(N)t Fq(O)t(V)t(E)t(M)t(B)t(E)t(R)25 b Fp(1)t(8)t(,)f(2)t(0)t (0)t(4)p Black eop end %%Page: 4 4 TeXDict begin 4 3 bop Black 482 265 a Fo(L23.4)1401 b(The)22 b(Curry-Howard)g(Isomorphism)p 482 302 2989 4 v Black 482 598 a Fv(Formally:)1373 711 y Fl(\(\()p Fu(A)f Fs(_)f Fu(B)5 b Fl(\))25 b Fs(\033)g Fu(C)7 b Fl(\))25 b Fs(\033)g Fl(\()p Fu(A)h Fs(\033)f Fu(C)7 b Fl(\))25 b Ft(true)482 856 y Fv(The)c(pr)n(oof:)p 597 998 1114 4 v 597 1087 a Fk(\()p Fj(A)16 b Fi(_)h Fj(B)t Fk(\))k Fi(\033)g Fj(C)27 b Fh(true)p Fj(;)13 b(A)21 b Fh(true)g Fi(`)g Fj(A)g Fh(true)1751 1019 y Fg(Hyp)1884 988 y Ff(2)p 523 1142 1261 4 v 523 1232 a Fk(\()p Fj(A)c Fi(_)g Fj(B)t Fk(\))k Fi(\033)g Fj(C)26 b Fh(true)p Fj(;)13 b(A)21 b Fh(true)g Fi(`)h Fj(A)16 b Fi(_)h Fj(B)26 b Fh(true)1825 1164 y Fi(_)p Fj(I)1910 1172 y Ff(1)p 2019 1142 1483 4 v 2019
1232 a Fk(\()p Fj(A)16 b Fi(_)i Fj(B)t Fk(\))j Fi(\033)g Fj(C)26 b Fh(true)p Fj(;)13 b(A)21 b Fh(true)g Fi(`)g Fk(\()p Fj(A)c Fi(_)g Fj(B)t Fk(\))k Fi(\033)g Fj(C)27 b Fh(true)3543 1164 y Fg(Hyp)3676 1132 y Ff(1)p 523 1287 2979 4 v 1454 1376 a Fk(\()p Fj(A)17 b Fi(_)g Fj(B)t Fk(\))k Fi(\033)g Fj(C)27 b Fh(true)p Fj(;)13 b(A)21 b Fh(true)g Fi(`)g Fj(C)26 b Fh(true)3543 1310 y Fi(\033)p Fj(E)p 1454 1431 1116 4 v 1495 1520 a Fk(\()p Fj(A)16 b Fi(_)i Fj(B)t Fk(\))j Fi(\033)g Fj(C)26 b Fh(true)21 b Fi(`)h Fj(A)e Fi(\033)i Fj(C)k Fh(true)2612 1455 y Fi(\033)o Fj(I)p 1448 1576 1129 4 v 1448 1665 a Fi(\001)21 b(`)h Fk(\(\()p Fj(A)16 b Fi(_)h Fj(B)t Fk(\))k Fi(\033)g Fj(C)5 b Fk(\))22 b Fi(\033)f Fk(\()p Fj(A)g Fi(\033)g Fj(C)5 b Fk(\))21 b Fh(true)2618 1599 y Fi(\033)p Fj(I)623 1880 y Fv(Next)i(we)g(look)h(at)g(some)g(degenerate)d(cases.)32 b(Consider)24 b(tr)o(uth)f(\()p Fs(>)p Fv(\))i(as)f(a)h(logical)482 1992 y(constant.)i(It)22 b(should)f(be)i(pr)n(ovable)g(no)f(matter)h (what)f(assumptions)f(we)h(have.)p 1666 2154 462 4 v 1666 2250 a Fu(H)32 b Fs(`)25 b(>)50 b Ft(true)2169 2185 y Fs(>)p Fu(I)482 2449 y Fv(Because)21 b(we)g(put)g(no)h(information)h (into)f(the)f(pr)n(oof)h(of)g Fs(>)p Fv(,)g(we)f(can)i(obtain)g(no)f (infor)n(-)482 2562 y(mation)27 b(out.)41 b(Ther)n(efor)n(e,)27 b(ther)n(e)f(is)i(no)f(elimination)h(r)o(ule)f(for)h Fs(>)p Fv(.)41 b(W)-8 b(e)26 b(can)j(observe)482 2675 y(that)23 b Fs(>)g Fv(is)h(like)f(a)h(0-ary)h(version)e(of)g (conjunction:)30 b Fs(^)o Fu(I)h Fv(has)24 b(two)e(pr)n(emises)h(and)g (con-)482 2788 y(sequently)h(we)i(have)g(two)g(elimination)h(r)o(ules)f (\()p Fs(^)p Fu(E)2315 2802 y Fn(1)2381 2788 y Fv(and)g Fs(^)p Fu(E)2689 2802 y Fn(2)2728 2788 y Fv(\),)i(while)e Fs(>)p Fu(I)33 b Fv(has)27 b(no)482 2901 y(pr)n(emises)21 b(and)i(consequently)d(no)j(elimination)g(r)o(ules.)623 3014 y(Now)k(consider)g(falsehood)h(\()p Fs(?)p Fv(\).)45 b(It)27 b(r)n(epr)n(esents)f(a)j(contradiction)f(and)g(should)482 3127 y(ther)n(efor)n(e)j(not)i(be)g(pr)n(ovable.)60 b(In)33 b(other)f(wor)n(ds,)i(ther)n(e)e(is)h(no)g(intr)n(oduction)g(r)o(ule.) 482 3239 y(Conversely)-10 b(,)21 b(if)i(we)f(know)g Fs(?)j Ft(true)e Fv(we)f(should)g(be)g(able)i(to)e(conclude)g(anything.)1653 3427 y Fu(H)32 b Fs(`)25 b(?)50 b Ft(true)p 1653 3464 463 4 v 1653 3560 a Fu(H)32 b Fs(`)25 b Fu(C)57 b Ft(true)2156 3495 y Fs(?)p Fu(E)482 3759 y Fv(W)-8 b(e)25 b(can)i(r)n(ecognize)e (falsehood)h(as)g(a)g(disjunction)g(of)g(zer)n(o)g(alternatives.)38 b(Wher)n(eas)482 3872 y(ther)n(e)26 b(ar)n(e)i(two)e(intr)n(oduction)h (r)o(ules)g(for)g Fs(_)h Fv(and)f(ther)n(efor)n(e)f(two)h(cases)g(to)f (consider)482 3985 y(in)f(the)g(elimination)i(r)o(ule,)f(ther)n(e)e(ar) n(e)i(no)f(intr)n(oduction)g(r)o(ules)g(for)h Fs(?)f Fv(and)h(ther)n(efor)n(e)482 4098 y(no)c(branches)g(in)i(the)d (elimination)j(r)o(ule.)623 4211 y(Figur)n(e)39 b(1)h(summarizes)g(the) f(r)o(ules,)k(adding)c(hypotheses)d(to)j(the)g(\002rst)g(r)o(ules)482 4324 y(about)f(conjunction)h(in)g(the)f(straightforwar)n(d)g(way)-10 b(.)75 b(W)-8 b(e)38 b(list)h(the)f(intr)n(oduction)482 4437 y(r)o(ules)21 b(in)h(the)f(left)h(column)g(and)g(elimination)h(r)o (ules)e(in)i(the)e(right)g(column.)28 b(W)-8 b(e)21 b(have)482 4550 y(stacked)g(the)h(pr)n(emises)f(of)i(the)f Fs(_)o Fu(E)28 b Fv(r)o(ules)23 b(pur)n(ely)f(for)g(typographical)h(r)n (easons.)623 4663 y(A)i(natural)i(question)e(is)h(if)g(these)e(ar)n(e)j (all)g(the)e(logical)j(connectives)d(we)g(may)i(be)482 4775 y(inter)n(ested)f(in,)k(and)f(if)g(the)f(given)g(r)o(ules)h (de\002ne)e(logical)j(r)n(easoning)e(completely)g(if)482 4888 y(r)n(estricted)g(to)h(the)h(consider)n(ed)e(connectives.)49
b(If)30 b(we)g(ignor)n(e)f(universal)i(and)f(exis-)482 5001 y(tential)23 b(quanti\002cation,)i(then)d(the)h(main)i(missing)e (connectives)g(ar)n(e)h(logical)h(equiv-)p Black 484 5250 a Fp(L)t Fq(E)t(C)t(T)t(U)t(R)t(E)f Fp(N)t Fq(O)t(T)t(E)t(S)1598 b Fp(N)t Fq(O)t(V)t(E)t(M)t(B)t(E)t(R)25 b Fp(1)t(8)t(,)f(2)t(0)t(0)t (4)p Black eop end %%Page: 5 5 TeXDict begin 5 4 bop Black 490 265 a Fo(The)22 b(Curry-Howard)g (Isomorphism)1400 b(L23.5)p 490 302 2989 4 v Black Black 1053 659 1632 4 v 1053 755 a Fu(A)1121 769 y Fn(1)1186 755 y Ft(true)p Fu(;)15 b(:)g(:)g(:)h(;)f(A)1606 769 y Fm(i)1661 755 y Ft(true)p Fu(;)g(:)g(:)g(:)h(;)f(A)2081 769 y Fm(n)2154 755 y Ft(true)26 b Fs(`)f Fu(A)2480 769 y Fm(i)2533 755 y Ft(true)2726 687 y Fr(Hyp)2888 650 y Fm(i)717 1023 y Fu(H)33 b Fs(`)24 b Fu(A)i Ft(true)91 b Fu(H)32 b Fs(`)25 b Fu(B)30 b Ft(true)p 717 1059 964 4 v 895 1156 a Fu(H)i Fs(`)25 b Fu(A)c Fs(^)e Fu(B)30 b Ft(true)1723 1090 y Fs(^)o Fu(I)2174 943 y(H)i Fs(`)25 b Fu(A)20 b Fs(^)g Fu(B)30 b Ft(true)p 2174 980 609 4 v 2261 1076 a Fu(H)i Fs(`)25 b Fu(A)g Ft(true)2823 1004 y Fs(^)p Fu(E)2951 1018 y Fn(1)2174 1219 y Fu(H)32 b Fs(`)25 b Fu(A)20 b Fs(^)g Fu(B)30 b Ft(true)p 2174 1255 V 2258 1352 a Fu(H)j Fs(`)24 b Fu(B)30 b Ft(true)2823 1279 y Fs(^)p Fu(E)2951 1293 y Fn(2)p 964 1499 462 4 v 964 1595 a Fu(H)i Fs(`)25 b(>)50 b Ft(true)1467 1530 y Fs(>)p Fu(I)2357 1595 y Fv(no)22 b Fs(>)p Fu(E)28 b Fv(r)o(ule)835 1783 y Fu(H)r(;)15 b(A)25 b Ft(true)h Fs(`)f Fu(B)30 b Ft(true)p 835 1837 720 4 v 880 1933 a Fu(H)i Fs(`)25 b Fu(A)g Fs(\033)g Fu(B)30 b Ft(true)1595 1866 y Fs(\033)p Fu(I)1913 1797 y(H)j Fs(`)24 b Fu(A)i Fs(\033)f Fu(B)30 b Ft(true)91 b Fu(H)32 b Fs(`)25 b Fu(A)g Ft(true)p 1913 1837 1154 4 v 2270 1933 a Fu(H)33 b Fs(`)24 b Fu(B)30 b Ft(true)3108 1866 y Fs(\033)p Fu(E)966 2121 y(H)i Fs(`)25 b Fu(A)h Ft(true)p 879 2158 609 4 v 879 2254 a Fu(H)32 b Fs(`)25 b Fu(A)20 b Fs(_)g Fu(B)30 b Ft(true)1529 2182 y Fs(_)o Fu(I)1629 2196 y Fn(1)964 2397 y Fu(H)i Fs(`)25 b Fu(B)k Ft(true)p 879 2433 V 879 2530 a Fu(H)j Fs(`)25 b Fu(A)20 b Fs(_)g Fu(B)30 b Ft(true)1529 2458 y Fs(_)o Fu(I)1629 2472 y Fn(2)2134 2137 y Fu(H)i Fs(`)25 b Fu(A)20 b Fs(_)g Fu(B)30 b Ft(true)2134 2250 y Fu(H)r(;)15 b(A)25 b Ft(true)h Fs(`)f Fu(C)32 b Ft(true)2134 2363 y Fu(H)r(;)15 b(B)30 b Ft(true)25 b Fs(`)g Fu(C)32 b Ft(true)p 2092 2433 806 4 v 2276 2530 a Fu(H)h Fs(`)24 b Fu(C)32 b Ft(true)2939 2465 y Fs(_)p Fu(E)1061 2927 y Fv(no)22 b Fs(?)p Fu(I)30 b Fv(r)o(ule)2272 2794 y Fu(H)i Fs(`)25 b(?)g Ft(true)p 2271 2831 438 4 v 2271 2927 a Fu(H)33 b Fs(`)24 b Fu(C)32 b Ft(true)2750 2862 y Fs(?)p Fu(E)p Black 959 3210 a Fv(Figur)n(e)22 b(1:)29 b(Rules)21 b(of)i(Intuitionistic)f(Pr)n(opositional)h(Logic)p Black Black 490 3501 a(alence)k Fu(A)34 b Fs(\021)f Fu(B)e Fv(and)c(negation)g Fs(:)p Fu(A)p Fv(.)41 b(These)25 b(can)j(be)f(easily)g(consider)n(ed)e(abbr)n(evia-)490 3614 y(tions,)c(using)1350 3740 y Fu(A)26 b Fs(\021)f Fu(B)87 b Fl(=)c(\()p Fu(A)26 b Fs(\033)f Fu(B)5 b Fl(\))20 b Fs(^)g Fl(\()p Fu(B)30 b Fs(\033)25 b Fu(A)p Fl(\))1484 3852 y Fs(:)p Fu(A)83 b Fl(=)g Fu(A)25 b Fs(\033)g(?)631 4034 y Fv(On)d(the)f(question)g(of)h(the)f(completeness)f(of)i(these)e (r)o(ules,)h(a)i(debate)e(is)h(possible.)490 4146 y(W)-5 b(ith)23 b(the)f(right)h(pr)n(oof-theor)n(etic)f(analysis)h(we)g(can)g
(show)f(that)h Fu(A)e Fs(_)f(:)p Fu(A)26 b Ft(true)d Fv(is)g Fr(not)490 4259 y Fv(pr)n(ovable)k(for)g(an)g(arbitrary)g Fu(A)g Fv(in)g(this)g(logic,)g(essentially)f(because)h(we)f(can)h(pr)n (ove)490 4372 y(neither)h Fu(A)37 b Ft(true)29 b Fv(nor)g Fs(:)p Fu(A)37 b Ft(true)p Fv(,)30 b(which)g(ar)n(e)f(the)g(two)f (possibilities)g(if)i(we)f(consider)490 4485 y(the)22 b(intr)n(oduction)g(r)o(ules)g(for)g(disjunction)h(\()p Fs(_)p Fv(\).)631 4602 y(The)j(logic)h(we)f(have)h(developed)d(so)i (far)h(is,)g(for)g(historical)g(r)n(easons,)f(called)i Fr(in-)490 4715 y(tuitionistic)23 b(logic)p Fv(.)29 b(If)21 b(we)g(also)g(allow)i(arbitrary)f(instances)e(of)i(the)e(axiom)i (schema)g(of)490 4828 y Fr(excluded)i(middle)h Fv(\()p Fr(XM)o Fv(\),)p 1556 4905 664 4 v 1556 5001 a Fu(H)32 b Fs(`)25 b Fu(A)20 b Fs(_)g(:)p Fu(A)25 b Ft(true)2261 4936 y Fr(XM)p Black 492 5250 a Fp(L)t Fq(E)t(C)t(T)t(U)t(R)t(E)f Fp(N)t Fq(O)t(T)t(E)t(S)1598 b Fp(N)t Fq(O)t(V)t(E)t(M)t(B)t(E)t(R)25 b Fp(1)t(8)t(,)f(2)t(0)t(0)t(4)p Black eop end %%Page: 6 6 TeXDict begin 6 5 bop Black 482 265 a Fo(L23.6)1401 b(The)22 b(Curry-Howard)g(Isomorphism)p 482 302 2989 4 v Black 482 598 a Fv(we)28 b(obtain)i(what)g(is)f(called)i Fr(classical)h (logic)p Fv(.)49 b(Note)28 b(how)h(classical)j(logic,)f(in)f(a)g(r)o (ule)482 711 y(rather)24 b(dif)n(\002cult)j(to)e(motivate,)h(destr)n (oys)d(the)h(design)g(principles)i(and)f(attempts)f(at)482 824 y(explaining)d(the)f(meaning)h(of)g(the)f(connectives.)26 b(For)20 b(example,)h(to)f(understand)f(the)482 936 y(meaning)j(of)g (disjunction)f(it)i(is)f(no)f(longer)g(suf)n(\002cient)h(to)g (understand)e(its)h(intr)n(oduc-)482 1049 y(tion)f(r)o(ules,)g(but)g (we)g(must)g(also)g(understand)f(the)g(law)j(of)e(excluded)f(middle,)i (which)482 1162 y(contains)k(and)h(appeal)g(to)f(negation)g(and)h (falsehood.)37 b(All)26 b(is)g(not)f(lost,)h(but)f(we)h(can)482 1275 y(say)c(that)h(the)g(Curry-Howar)n(d)g(isomorphism)f(\(the)h (subject)g(of)g(this)g(lectur)n(e)g(whose)482 1388 y(explanation)i(is)h (yet)f(to)g(come\))h(will)h(fail)g(in)f(the)f(pr)n(esence)g(of)g(the)g (law)i(of)f(excluded)482 1501 y(middle.)623 1614 y(For)20 b(the)h(r)n(est)f(of)h(this)g(lectur)n(e)g(we)g(will)i(only)e(be)g (inter)n(ested)e(in)i(the)g(intuitionistic)482 1727 y(logic)30 b(as)f(de\002ned)f(with)i(the)f(r)o(ules)g(in)h(Figur)n(e)f(1.)49 b(The)29 b(\002rst)g(observation)g(is)h(that)f(a)482 1840 y(derivable)24 b(judgment)e Fu(H)34 b Fs(`)26 b Fu(A)i Ft(true)c Fv(does)e(not)g(contain)i(any)g(information)g(about)g (its)482 1953 y(derivation.)37 b(When)26 b(we)f(assert)g Fu(H)38 b Fs(`)31 b Fu(A)h Ft(true)26 b Fv(is)g(derivable)g(we)g(mean)g (that)g(it)g(has)g(a)482 2066 y(derivation,)e(but)h(we)f(do)f(not)h (exhibit)h(such)f(a)h(derivation.)34 b(This)24 b(makes)g(it)g(dif)n (\002cult)482 2178 y(to)f(convince)i(someone)e(else)g(of)i(the)e(tr)o (uth)h(of)g Fu(A)h Fv(under)e(assumptions)g Fu(H)7 b Fv(.)33 b(So)24 b(what)482 2291 y(we)k(would)g(like)g(to)g(do)g(is)h (to)f(enrich)h(the)f(judgment)f(with)h(a)i Fr(pr)n(oof)h(term)e Fu(M)39 b Fv(which)482 2404 y(contains)22 b(enough)f(information)j(to)e (r)n(econstr)o(uct)f(the)h(derivation.)623 2517 y(So)g(we)h(would)g (like)g(to)g(uniformly)g(r)n(eplace)h(the)f(judgment)e Fu(A)27 b Ft(true)d Fv(with)f Fu(M)36 b Fl(:)27 b Fu(A)482 2630 y Fv(\(r)n(ead:)38 b Fu(M)g Fr(is)28 b(a)h(pr)n(oof)h(of)e Fu(A)p Fv(\).)44 b(For)27 b(assumptions)f Fu(A)35 b Ft(true)29 b Fv(we)e(do)g(not)g(actually)i(have)482 2743 y(a)k(pr)n(oof)f(of)h Fu(A)p Fv(,)i(we)e Fr(assume)g Fv(that)g(ther)n(e)f(is)h(a)g(pr)n(oof.) 58 b(W)-8 b(e)32 b(model)g(this)h(by)g(using)f(a)482 2856 y Fr(variable)p Fv(,)25 b(wher)n(e)c(each)i(assumption)f(is)h
(labeled)g(by)f(a)h(distinct)g(variable.)623 2969 y(So)f(we)g(want)g (to)g(translate)h(a)g(judgment)1412 3173 y Fu(A)1480 3187 y Fn(1)1545 3173 y Ft(true)p Fu(;)15 b(:)g(:)g(:)h(;)f(A)1965 3187 y Fm(n)2038 3173 y Ft(true)26 b Fs(`)f Fu(A)g Ft(true)482 3377 y Fv(to)d(the)f(form)1469 3490 y Fu(x)1521 3504 y Fn(1)1560 3490 y Fl(:)q Fu(A)1654 3504 y Fn(1)1693 3490 y Fu(;)15 b(:)g(:)g(:)i(;)e(x)1947 3504 y Fm(n)1994 3490 y Fl(:)q Fu(A)2088 3504 y Fm(n)2160 3490 y Fs(`)25 b Fu(M)35 b Fl(:)26 b Fu(A)482 3657 y Fv(in)20 b(such)g(a)h(why)e(that) h(the)f(derivation)i(of)f(the)f(\002rst)g(judgment)g(can)i(be)f(r)n (econstr)o(ucted)482 3770 y(dir)n(ectly)i(fr)n(om)h Fu(M)10 b Fv(.)28 b(First,)22 b(the)g(hypothesis)e(r)o(ule)i(is)h (straightforwar)n(d:)p 1044 3948 1632 4 v 1044 4044 a Fu(A)1112 4058 y Fn(1)1177 4044 y Ft(true)q Fu(;)15 b(:)g(:)g(:)h(;)f (A)1598 4058 y Fm(i)1652 4044 y Ft(true)p Fu(;)g(:)g(:)g(:)i(;)e(A)2073 4058 y Fm(n)2146 4044 y Ft(true)25 b Fs(`)g Fu(A)2471 4058 y Fm(i)2525 4044 y Ft(true)2718 3976 y Fr(Hyp)2879 3939 y Fm(i)482 4244 y Fv(becomes)p 1161 4291 1428 4 v 1161 4387 a Fu(x)1213 4401 y Fn(1)1252 4387 y Fl(:)p Fu(A)1345 4401 y Fn(1)1385 4387 y Fu(;)15 b(:)g(:)g(:)h(;)f(x)1638 4401 y Fm(i)1667 4387 y Fl(:)p Fu(A)1760 4401 y Fm(i)1789 4387 y Fu(;)g(:)g(:)g(:)h(;)f(x)2042 4401 y Fm(n)2089 4387 y Fl(:)q Fu(A)2183 4401 y Fm(n)2255 4387 y Fs(`)25 b Fu(x)2388 4401 y Fm(i)2441 4387 y Fl(:)h Fu(A)2560 4401 y Fm(i)2630 4310 y Fr(Hyp)482 4550 y Fv(Since)e(the)f(assumptions) f(ar)n(e)i(labeled)h(by)f(distinct)f(variables,)j(we)d(no)h(longer)f (need)482 4663 y(to)f(annotate)f(the)h(justi\002cation)h(with)g(an)g (integer)e(and)i(we)f(just)g(write)g Fr(Hyp)p Fv(.)623 4775 y(The)16 b(r)n(emaining)i(r)o(ules)f(mention)g(a)g(collection)h (of)f(hypotheses)d Fu(H)32 b Fl(=)25 b(\()p Fu(A)3132 4789 y Fn(1)3197 4775 y Ft(true)q Fu(;)15 b(:)g(:)g(:)h(;)f(A)3618 4789 y Fm(n)3691 4775 y Ft(true)p Fl(\))482 4888 y Fv(which)27 b(we)g(annotate)g(uniformly)h(with)f(distinct)g(variables,)j(leading)d (to)g(a)h(context)482 5001 y Fl(\000)d(=)g(\()p Fu(x)747 5015 y Fn(1)786 5001 y Fl(:)q Fu(A)880 5015 y Fn(1)919 5001 y Fu(;)15 b(:)g(:)g(:)i(;)e(x)1173 5015 y Fm(n)1220 5001 y Fl(:)p Fu(A)1313 5015 y Fm(n)1361 5001 y Fl(\))p Fv(.)p Black 484 5250 a Fp(L)t Fq(E)t(C)t(T)t(U)t(R)t(E)24 b Fp(N)t Fq(O)t(T)t(E)t(S)1598 b Fp(N)t Fq(O)t(V)t(E)t(M)t(B)t(E)t(R)25 b Fp(1)t(8)t(,)f(2)t(0)t(0)t(4)p Black eop end %%Page: 7 7 TeXDict begin 7 6 bop Black 490 265 a Fo(The)22 b(Curry-Howard)g (Isomor...