%!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: k.dvi %%CreationDate: Thu Nov 28 11:56:42 1996 %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips k.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1996.11.28:1155 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D }B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (/tmp_mnt/home/math63/schwicht/wwwpublic/papers/kreisel93/k.dvi) @start /Fa 2 106 df<0FC3E003018003820001C40000C80000F00000600000F00001B8 00031800041C00080C00100600FC1F80130E7E8D17>88 D<0808000000007098B0303060 646870060F7D8E0B>105 D E /Fb 2 49 df<0C000C00EDC07F801E007F80EDC00C000C 000A097E890F>3 D<181818303030606060C0C0050B7E8B09>48 D E /Fc 2 67 df<001C0000001C0000003E0000003E0000007F0000006F000000EF8000 00C7800000C780000183C0000183C0000301E0000301E00007FFF00007FFF0000E00F800 0C0078000C007800FF01FF80FF01FF8019147F931C>65 DI E /Fd 9 117 df<1E0061004180FF80C000 C000400020801F0009097E880D>101 D<060B181818FE181818181818187E080E7F8D09> I<1DE022206300630022003C0060003F003F8040C0C0C0C0C061803F000B0E7E880E>I< F0003000300030003000378038C030C030C030C030C030C030C0FDF00C0E7E8D10>I<20 7020000000F030303030303030FC060F7F8E08>I108 D110 D114 D<08081838FE181818181919190E080D7F8C 0B>116 D E /Fe 5 92 df<0000700001F00003C0000780000E00001C00003800007000 00700000F00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00001C00001C00001C000038000 0700000600000E0000380000700000C000007000003800000E0000060000070000038000 01C00001C00001C00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000F000007000007000 003800001C00000E000007800003C00001F000007014637B811F>26 D70 DI83 D91 D E /Ff 21 123 df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g 8 107 df0 D<8002C006600C301818300C6006C0 0380038006C00C6018303018600CC00680020F107B8E1A>2 D<040004000400C460E4E0 3F800E003F80E4E0C4600400040004000B0D7E8D11>I<03FFC00FFFC01C000030000060 0000600000C00000C00000C00000C00000C00000C00000C00000C0000060000060000030 00001C00000FFFC003FFC00000000000000000000000000000007FFFC07FFFC0121B7D93 1A>18 D<00000400000004000000020000000100FFFFFFE0FFFFFFE00000010000000200 00000400000004001B0A7E8B21>33 D<040E0E1C1C1C38383070706060C0C0070F7F8F0A >48 D<03FC0FFC1C003000600060006000C000C000FFFCFFFCC000C00060006000600030 001C000FFC03FC0E147D9016>50 D106 D E /Fh 1 4 df3 D E /Fi 17 117 df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j 43 124 df<0007E1F0001C173800703C3800E07C7801C0783001C0380001C038 0001C038000380700003807000038070003FFFFF0003807000038070000700E0000700E0 000700E0000700E0000700E0000700E0000E01C0000E01C0000E01C0000E01C0000E01C0 000E01C0001C0380001C03C000FF8FF8001D1D7F9C18>11 D<0007E0001C1000703800E0 7801C07801C03001C00001C0000380000380000380003FFFF00380F00380700700E00700 E00700E00700E00700E00700E00E01C00E01C00E01C00E01C00E01C00E01C01C03801C03 C0FF8FF0151D7F9C17>I<183C7C340404080810106080060C799C0C>39 D<000400080030006000C0008001800300060006000C000C001800180038003000300070 007000600060006000E000E000E000E000E000E000E000E0006000600060006000200030 003000180008000C00040002000E2A7C9E10>I<01000080004000600020003000100018 00180018001C000C000C000C000C000C000C000C000C001C001C001C0018001800380038 003000300070006000E000C001C001800300030006000C0018003000600080000E2A809E 10>I<387878380808101020204080050C7D830C>44 D<7FF0FFE00C027F890E>I<7070F0 6004047C830C>I<060F0E0E00000000000000000000387878380808101020204080081A 7D910C>59 D<000020000000300000007000000070000000F0000000F0000001F0000003 7800000278000006780000047800000C780000087C0000183C0000103C0000203C000020 3C0000403C0000403E0000FFFE0000801E0001001E0001001E0002001E0002000F000400 0F0004000F001E001F00FF80FFF01C1D7F9C1F>65 D<0007F010001C0C300070026000C0 01E0038000E0070000E00E0000600E0000601C0000403C00004038000040780000007800 000078000000F0000000F0000000F0000000F0000000F0000000F0000080F00001007000 01007000010038000200380004001C0004000C001800060020000380C000007F00001C1E 7C9C1E>67 D<0FFFFC0000F8078000F001C000F000E000F0007000F0007000F0007801E0 003801E0003801E0003801E0003801E0003C01E0003803C0003803C0007803C0007803C0 007803C0007003C000F0078000E0078000E0078001C0078003800780078007800E000F00 1C000F007000FFFFC0001E1C7E9B20>I<0FFFFFC000F803C000F001C000F0008000F000 8000F0008000F0008001E0008001E0408001E0400001E0400001E0C00001E1C00003FF80 0003C1800003C0800003C0800003C0800003C08000078000000780000007800000078000 0007800000078000000F0000000F800000FFF800001A1C7E9B1B>70 D<0FFF8000F80000F00000F00000F00000F00000F00001E00001E00001E00001E00001E0 0001E00003C00003C00003C00003C00003C00003C0000780000780000780000780000780 000780000F00000F8000FFF800111C7F9B0F>73 D<0FFFC000F80000F00000F00000F000 00F00000F00001E00001E00001E00001E00001E00001E00003C00003C00003C00003C000 03C00003C0040780040780040780040780080780080780180F00380F00F0FFFFF0161C7E 9B1A>76 D<003F0400C0CC0180380300380600180E00180E00180E00181E00101E00001F 00000F80000FF80007FF0003FF8001FFC0003FE00003E00001E00000E00000E04000E040 00E04000E04000C06001C0E00180F00300CC0E0083F800161E7E9C17>83 D<1FFFFFF03C07C0F03007803020078020600780204007802040078020400F0020800F00 20000F0000000F0000000F0000000F0000001E0000001E0000001E0000001E0000001E00 00001E0000003C0000003C0000003C0000003C0000003C0000003C000000780000007C00 001FFFE0001C1C7C9B1E>I87 D<07F0001C18001E0C001C0E00180E00000E00000E0001 FE000F0E001C1C00301C00701C00E01C40E01C40E03C40E05C80709D803F0E0012127D91 15>97 D<3F00000F00000E00000E00000E00000E00000E00000E00001C00001C00001C00 001C78001D86001E03003C03803801803801803801C03801C03801C07003807003807003 80700300700700700E00F00C00CC300083C000121D7C9C17>I<01F8071C0C1E181C3818 3000700070007000E000E000E000600060047008301018200FC00F127D9112>I<0003F0 0000F00000E00000E00000E00000E00000E00000E00001C00001C00001C000F1C0030DC0 0C03C01C0380380380300380700380700380700380E00700E00700E00700600700600700 700F00301E00186F00078FC0141D7D9C17>I<01F8070C0C061C073803300370037FFF70 00E000E000E00060006002300430081C3007C010127E9112>I<000F800039C00061C000 E3C001C18001C00001C00001C0000380000380000380003FF80003800003800007000007 00000700000700000700000700000E00000E00000E00000E00000E00000E00001C00001E 0000FFC000121D7F9C0D>I<000038003CCC00C69C018308038380070380070380070380 070380070700030600038C0004F0000400000C00000C00000FFE0007FF800FFFC01801C0 2000C06000E0C000C0C000C0600180200300180E0007F000161C809215>I<07E00001E0 0001C00001C00001C00001C00001C00001C000038000038000038000038F8003B0C003C0 E00780E00780E00700E00700E00700E00700E00E01C00E01C00E01C00E01C00E01C00E01 C01C03801E03C0FF9FF0141D7F9C17>I<00C001C001C001800000000000000000000000 0000001F80078003800700070007000700070007000E000E000E000E000E000E001C001E 00FF800A1D7F9C0C>I<0003000700070006000000000000000000000000000000FE001E 000E001C001C001C001C001C001C00380038003800380038003800700070007000700070 007060E0F0E0F1C063803E001025849C0D>I<07E001E001C001C001C001C001C001C003 80038003800380038003800700070007000700070007000E000E000E000E000E000E001C 001E00FF800B1D7F9C0C>108 D<1F8FC0FC00079061060003E076070007807807000780 78070007007007000700700700070070070007007007000E00E00E000E00E00E000E00E0 0E000E00E00E000E00E00E000E00E00E001C01C01C001E01E01E00FF8FF8FF8021127F91 24>I<1F8F8007B0C003C0E00780E00780E00700E00700E00700E00700E00E01C00E01C0 0E01C00E01C00E01C00E01C01C03801E03C0FF9FF014127F9117>I<00FC000307000E01 801C01C03800C03000C07000E07000E07000E0E001C0E001C0E001C06001806003807007 00380E001C180007E00013127E9115>I<0FC78003D86001E03003C03803803803801803 801C03801C03801C0700380700380700380700700700700700E00F01C00EC3000E3C000E 00000E00000E00001C00001C00001C00001C0000FF8000161A809117>I<00F0400308C0 0E05C01C0380380380380380700380700380700380E00700E00700E00700600700700700 700F00301E00186E00078E00000E00000E00000E00001C00001C00001C00001C0000FF80 121A7D9116>I<1F9C07EE03CF078E078C07000700070007000E000E000E000E000E000E 001C001E00FFC010127F9110>I<03F20C0E18061004300438043E001FE00FF007F8003C 401C400C400C6018E010D0608FC00F127F9110>I<020002000200060006000C001C003C 00FFE01C001C00380038003800380038003800700070407040704070407080708031001E 000B1A7C9910>IIII<1FE3FC07C1E003 818001C10001C20000E40000EC00007800007000003800007800009C00011E00020E0004 07000C07003C0780FE1FF016127F9116>I<0FF0FE03C03801C03001C02001C06001C040 01E08000E08000E10000E10000E200007200007400007C00007800007000003000002000 002000004000004000708000F10000F10000E60000780000171A809116>I123 D E /Fk 47 122 df<003FE3F801F03F1C03C03E3E07C07C3E0F807C3E0F 807C1C0F807C000F807C000F807C000F807C000F807C00FFFFFFC0FFFFFFC00F807C000F 807C000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F 807C000F807C000F807C000F807C007FE1FFC07FE1FFC01F1D809C1C>11 D<003FC00001F0300003C0380007C07C000F807C000F807C000F8038000F8000000F8000 000F8000000F800000FFFFFC00FFFFFC000F807C000F807C000F807C000F807C000F807C 000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F807C 007FE1FF807FE1FF80191D809C1B>I<0020004001800380030006000E001C001C003C00 38003800780078007800F800F000F000F000F000F000F000F000F000F000F80078007800 7800380038003C001C001C000E000600030003800180004000200B297C9E13>40 D<800040003000380018000C000E000700070007800380038003C003C003C003E001E001 E001E001E001E001E001E001E001E003E003C003C003C0038003800780070007000E000C 00180038003000400080000B297D9E13>I<78FCFCFCFC7806067D850D>46 D<03F8000F1E001C07003C07803803807803C07803C07803C0F803E0F803E0F803E0F803 E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E07803C07803C03803803C07 801C07000F1E0003F800131B7E9A18>48 D<00600001E0000FE000FFE000F3E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E0007FFF807FFF80111B7D9A18>I<07F8 001FFE00383F80780FC0FC07C0FC07E0FC03E0FC03E07803E00007E00007C00007C0000F 80001F00001E0000380000700000E0000180600300600600600800E01FFFC03FFFC07FFF C0FFFFC0FFFFC0131B7E9A18>I<03F8001FFE003C1F003C0F807C07C07E07C07C07C038 07C0000F80000F80001E00003C0003F800001E00000F800007C00007C00007E03007E078 07E0FC07E0FC07E0FC07C0780F80781F001FFE0007F800131B7E9A18>I<000180000380 000780000F80001F80003F80006F8000CF80008F80018F80030F80060F800C0F80180F80 300F80600F80C00F80FFFFF8FFFFF8000F80000F80000F80000F80000F80000F8001FFF8 01FFF8151B7F9A18>I<1801801FFF001FFE001FFC001FF8001FC0001800001800001800 0018000019F8001E0E00180F801007800007C00007E00007E00007E07807E0F807E0F807 E0F807C0F007C0600F80381F001FFE0007F000131B7E9A18>I<007E0003FF000781800F 03C01E07C03C07C03C0380780000780000F80000F8F800FB0E00FA0780FC0380FC03C0F8 03E0F803E0F803E0F803E07803E07803E07803C03C03C03C07801E0F0007FE0003F80013 1B7E9A18>I<6000007FFFE07FFFE07FFFC07FFF807FFF80E00300C00600C00C00C01800 00300000300000600000E00000E00001E00001C00003C00003C00003C00003C00007C000 07C00007C00007C00007C00007C000038000131C7D9B18>I<03F8000FFE001E0F803807 803803C07803C07803C07E03C07F83807FC7003FFE001FFC000FFE0007FF801DFF80387F C0781FE0F007E0F003E0F001E0F001E0F001E07801C07803803E07801FFE0003F800131B 7E9A18>I<03F8000FFE001E0F003C07807807807803C0F803C0F803C0F803E0F803E0F8 03E0F803E07807E03807E03C0BE00E1BE003E3E00003E00003C00003C03807C07C07807C 0700780F00383C001FF8000FE000131B7E9A18>I<00038000000380000007C0000007C0 000007C000000FE000000FE000001FF000001BF000001BF0000031F8000031F8000061FC 000060FC0000E0FE0000C07E0000C07E0001803F0001FFFF0003FFFF8003001F8003001F 8006000FC006000FC00E000FE00C0007E0FFC07FFEFFC07FFE1F1C7E9B24>65 DI<001FE02000FFF8E003F80FE007C003E00F8001E01F0000E0 3E0000E03E0000607E0000607C000060FC000000FC000000FC000000FC000000FC000000 FC000000FC000000FC0000007C0000607E0000603E0000603E0000C01F0000C00F800180 07C0030003F80E0000FFFC00001FE0001B1C7D9B22>IIII73 D76 DII80 D82 D<7FFFFFE07FFFFFE0781F81E0701F80E0601F8060E01F8070C01F8030C01F8030C01F80 30C01F8030001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F80 00001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F800007FFFE 0007FFFE001C1C7E9B21>84 D<0FF8001C1E003E0F803E07803E07C01C07C00007C0007F C007E7C01F07C03C07C07C07C0F807C0F807C0F807C0780BC03E13F80FE1F815127F9117 >97 D<03FC000E0E001C1F003C1F00781F00780E00F80000F80000F80000F80000F80000 F800007800007801803C01801C03000E0E0003F80011127E9115>99 D<000FF0000FF00001F00001F00001F00001F00001F00001F00001F00001F00001F001F9 F00F07F01C03F03C01F07801F07801F0F801F0F801F0F801F0F801F0F801F0F801F07801 F07801F03C01F01C03F00F0FFE03F9FE171D7E9C1B>I<01FC000F07001C03803C01C078 01C07801E0F801E0F801E0FFFFE0F80000F80000F800007800007C00603C00601E00C00F 038001FC0013127F9116>I<007F0001E38003C7C00787C00F87C00F83800F80000F8000 0F80000F80000F8000FFF800FFF8000F80000F80000F80000F80000F80000F80000F8000 0F80000F80000F80000F80000F80000F80000F80007FF8007FF800121D809C0F>I<03F8 F00E0F381E0F381C07303C07803C07803C07803C07801C07001E0F000E0E001BF8001000 001800001800001FFF001FFFC00FFFE01FFFF07801F8F00078F00078F000787000707800 F01E03C007FF00151B7F9118>II< 1E003F003F003F003F001E00000000000000000000000000FF00FF001F001F001F001F00 1F001F001F001F001F001F001F001F001F001F00FFE0FFE00B1E7F9D0E>I108 DII<01FC000F07801C01C03C01E07800F07800F0F800F8F800F8F800F8F800F8F800 F8F800F87800F07800F03C01E01E03C00F078001FC0015127F9118>II114 D<1FD830786018E018E018F000FF807FE07FF01FF807FC007CC01CC01CE01CE018F830CF C00E127E9113>I<0300030003000300070007000F000F003FFCFFFC1F001F001F001F00 1F001F001F001F001F001F0C1F0C1F0C1F0C0F08079803F00E1A7F9913>III121 D E /Fl 47 127 df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m 26 107 df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n 53 128 df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o 79 128 df<007E1F0001C1B180 0303E3C00703C3C00E03C1800E01C0000E01C0000E01C0000E01C0000E01C0000E01C000 FFFFFC000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C000 0E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0007F87FC00 1A1D809C18>11 D<007E0001C1800301800703C00E03C00E01800E00000E00000E00000E 00000E0000FFFFC00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E 01C00E01C00E01C00E01C00E01C00E01C00E01C07F87F8151D809C17>I<007FC001C1C0 0303C00703C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0FFFFC00E01C00E01C0 0E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C07FCFF8151D809C17>I<003F07E00001C09C18000380F018000701F03C00 0E01E03C000E00E018000E00E000000E00E000000E00E000000E00E000000E00E00000FF FFFFFC000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00 E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E0 1C000E00E01C000E00E01C007FC7FCFF80211D809C23>I<60F0F8680808081010204080 050C7C9C0C>39 D<004000800100020006000C000C001800180030003000700060006000 6000E000E000E000E000E000E000E000E000E000E000E000E00060006000600070003000 3000180018000C000C00060002000100008000400A2A7D9E10>I<800040002000100018 000C000C000600060003000300038001800180018001C001C001C001C001C001C001C001 C001C001C001C001C0018001800180038003000300060006000C000C0018001000200040 0080000A2A7E9E10>I<01800180018001804182F18F399C0FF003C003C00FF0399CF18F 4182018001800180018010127E9E15>I<00060000000600000006000000060000000600 000006000000060000000600000006000000060000000600000006000000060000FFFFFF E0FFFFFFE000060000000600000006000000060000000600000006000000060000000600 0000060000000600000006000000060000000600001B1C7E9720>I<60F0F07010101010 20204080040C7C830C>II<60F0F06004047C830C>I<03C00C30 1818300C300C700E60066006E007E007E007E007E007E007E007E007E007E007E007E007 E00760066006700E300C300C18180C3007E0101D7E9B15>48 D<030007003F00C7000700 070007000700070007000700070007000700070007000700070007000700070007000700 0700070007000F80FFF80D1C7C9B15>I<07C01830201C400C400EF00FF80FF807F80770 07000F000E000E001C001C00380070006000C00180030006010C01180110023FFE7FFEFF FE101C7E9B15>I<07E01830201C201C781E780E781E381E001C001C00180030006007E0 0030001C001C000E000F000F700FF80FF80FF80FF00E401C201C183007E0101D7E9B15> I<000C00000C00001C00003C00003C00005C0000DC00009C00011C00031C00021C00041C 000C1C00081C00101C00301C00201C00401C00C01C00FFFFC0001C00001C00001C00001C 00001C00001C00001C0001FFC0121C7F9B15>I<300C3FF83FF03FC02000200020002000 2000200023E024302818301C200E000E000F000F000F600FF00FF00FF00F800E401E401C 2038187007C0101D7E9B15>I<00F0030C06040C0E181E301E300C700070006000E3E0E4 30E818F00CF00EE006E007E007E007E007E007600760077006300E300C18180C3003E010 1D7E9B15>I<4000007FFF807FFF007FFF00400200800400800400800800001000001000 00200000600000400000C00000C00001C000018000018000038000038000038000038000 078000078000078000078000078000078000030000111D7E9B15>I<03E00C301008200C 20066006600660067006780C3E083FB01FE007F007F818FC307E601E600FC007C003C003 C003C00360026004300C1C1007E0101D7E9B15>I<03C00C301818300C700C600EE006E0 06E007E007E007E007E0076007700F300F18170C2707C700060006000E300C780C781870 10203030C00F80101D7E9B15>I<60F0F0600000000000000000000060F0F06004127C91 0C>I<60F0F0600000000000000000000060F0F0701010101020204080041A7C910C>I<7F FFFFC0FFFFFFE00000000000000000000000000000000000000000000000000000000000 000000FFFFFFE07FFFFFC01B0C7E8F20>61 D<0FE03038401CE00EF00EF00EF00E000C00 1C0030006000C00080018001000100010001000100010000000000000000000000030007 80078003000F1D7E9C14>63 D<000600000006000000060000000F0000000F0000000F00 000017800000178000001780000023C0000023C0000023C0000041E0000041E0000041E0 000080F0000080F0000180F8000100780001FFF80003007C0002003C0002003C0006003E 0004001E0004001E000C001F001E001F00FF80FFF01C1D7F9C1F>65 DI<001F808000E061800180198007000780 0E0003801C0003801C00018038000180780000807800008070000080F0000000F0000000 F0000000F0000000F0000000F0000000F0000000F0000000700000807800008078000080 380000801C0001001C0001000E000200070004000180080000E03000001FC000191E7E9C 1E>IIII<001F808000E0618001801980070007800E000380 1C0003801C00018038000180780000807800008070000080F0000000F0000000F0000000 F0000000F0000000F0000000F000FFF0F0000F8070000780780007807800078038000780 1C0007801C0007800E00078007000B800180118000E06080001F80001C1E7E9C21>II I75 DIII<003F800000E0E0000380380007001C000E000E001C0007003C00 078038000380780003C0780003C0700001C0F00001E0F00001E0F00001E0F00001E0F000 01E0F00001E0F00001E0F00001E0700001C0780003C0780003C0380003803C0007801C00 07000E000E0007001C000380380000E0E000003F80001B1E7E9C20>II82 D<07E0801C1980300580700380600180E00180E00080E00080E00080F00000F800007C00 007FC0003FF8001FFE0007FF0000FF80000F800007C00003C00001C08001C08001C08001 C0C00180C00180E00300D00200CC0C0083F800121E7E9C17>I<7FFFFFC0700F01C0600F 00C0400F0040400F0040C00F0020800F0020800F0020800F0020000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000001F800003FFFC001B1C7F9B1E> IIII89 D91 D93 D<0810204040808080B0F87830050C7D9C0C>96 D<1FC000307000783800781C00301C00001C00001C0001FC000F1C00381C00701C00601C 00E01C40E01C40E01C40603C40304E801F870012127E9115>II<07E00C301878307870306000E000E000E000E000E000E000 60007004300418080C3007C00E127E9112>I<003F000007000007000007000007000007 0000070000070000070000070000070003E7000C1700180F00300700700700600700E007 00E00700E00700E00700E00700E00700600700700700300700180F000C370007C7E0131D 7E9C17>I<03E00C301818300C700E6006E006FFFEE000E000E000E00060007002300218 040C1803E00F127F9112>I<00F8018C071E061E0E0C0E000E000E000E000E000E00FFE0 0E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E007FE00F1D 809C0D>I<00038003C4C00C38C01C3880181800381C00381C00381C00381C001818001C 38000C300013C0001000003000001800001FF8001FFF001FFF803003806001C0C000C0C0 00C0C000C06001803003001C0E0007F800121C7F9215>II<18003C003C0018000000000000000000000000000000FC001C00 1C001C001C001C001C001C001C001C001C001C001C001C001C001C001C00FF80091D7F9C 0C>I<00C001E001E000C000000000000000000000000000000FE000E000E000E000E000 E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E060E0F0C0F1 C061803E000B25839C0D>IIIII<03F0000E1C00180600300300700380600180E001C0E001C0E001C0E001C0E0 01C0E001C06001807003803003001806000E1C0003F00012127F9115>II<03C1000C3300180B00300F00700700700700E00700E00700E00700E007 00E00700E00700600700700700300F00180F000C370007C7000007000007000007000007 00000700000700000700003FE0131A7E9116>II<1F9030704030C010C0 10E010F8007F803FE00FF000F880388018C018C018E010D0608FC00D127F9110>I<0400 0400040004000C000C001C003C00FFE01C001C001C001C001C001C001C001C001C001C10 1C101C101C101C100C100E2003C00C1A7F9910>IIII<7F8FF00F03800F030007020003840001C80001D80000F000007000007800 00F800009C00010E00020E000607000403801E07C0FF0FF81512809116>II<7FFC70386038407040F040E041C003C0038007000F040E041C043C0C 380870087038FFF80E127F9112>II<6060F0F0F0F060600C047C9C 15>127 D E /Fp 16 122 df68 D<03FF80000FFFF0001F01FC003F80FE003F807F003F 803F003F803F801F003F8000003F8000003F8000003F8000003F80003FFF8001FC3F800F E03F801F803F803F003F807E003F80FC003F80FC003F80FC003F80FC003F80FC005F807E 00DF803F839FFC1FFE0FFC03F803FC1E1B7E9A21>97 D<003FF00001FFFC0003F03E000F C07F001F807F003F007F003F007F007F003E007E0000007E000000FE000000FE000000FE 000000FE000000FE000000FE000000FE0000007E0000007E0000007F0000003F0003803F 8003801F8007000FE00E0003F83C0001FFF800003FC000191B7E9A1E>99 D<00007FF000007FF000007FF0000007F0000007F0000007F0000007F0000007F0000007 F0000007F0000007F0000007F0000007F0000007F0000007F0003F87F001FFF7F007F03F F00FC00FF01F8007F03F0007F03F0007F07E0007F07E0007F07E0007F0FE0007F0FE0007 F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F07E0007F07E0007F03F0007 F03F0007F01F800FF00FC01FF007E07FFF01FFE7FF007F87FF202A7EA925>I<003FC000 01FFF00003E07C000F803E001F801F001F001F003F000F807E000F807E000FC07E000FC0 FE0007C0FE0007C0FFFFFFC0FFFFFFC0FE000000FE000000FE0000007E0000007E000000 7F0000003F0001C01F0001C00F80038007C0070003F01E0000FFFC00003FE0001A1B7E9A 1F>I<0007F8003FFC007E3E01FC7F03F87F03F07F07F07F07F03E07F00007F00007F000 07F00007F00007F00007F000FFFFC0FFFFC0FFFFC007F00007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F0007FFF807FFF807FFF80182A7EA915>I104 D<07000F801FC03FE03FE03F E01FC00F8007000000000000000000000000000000FFE0FFE0FFE00FE00FE00FE00FE00F E00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFEFF FEFFFE0F2B7EAA12>I108 D110 D<003FE00001FFFC0003F07E000FC01F801F800FC03F0007E03F00 07E07E0003F07E0003F07E0003F0FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8FE00 03F8FE0003F8FE0003F87E0003F07E0003F03F0007E03F0007E01F800FC00FC01F8007F0 7F0001FFFC00003FE0001D1B7E9A22>I114 D<03FE300FFFF03E03F07800F07000F0F00070F00070F80070FE0000FFE0007FFF007FFF C03FFFE01FFFF007FFF800FFF80007FC0000FCE0007CE0003CF0003CF00038F80038FC00 70FF01E0E7FFC0C1FF00161B7E9A1B>I<00700000700000700000700000F00000F00000 F00001F00003F00003F00007F0001FFFE0FFFFE0FFFFE007F00007F00007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F07007F07007F07007 F07007F07007F07007F07003F0E001F8C000FFC0003F0014267FA51A>I I121 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 216 42 a Fp(Densit)n(y)23 b(and)g(c)n(hoice)g(for)f(total)h (con)n(tin)n(uous)g(functionals)p Fo(*)762 137 y(Helm)o(ut)12 b(Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg)520 187 y(Mathematisc)o(hes)i (Institut)g(der)h(Univ)o(ersit\177)-21 b(at)14 b(M)q(\177)-22 b(unc)o(hen)0 329 y(In)17 b(his)h(seminal)d(pap)q(er)j Fn(Interpr)n(etation)g(of)g(analysis)g(by)g(me)n(ans)g(of)g(c)n (onstructive)g(functionals)h(of)e(\014nite)i(typ)n(es)i Fo(in)16 b(the)0 379 y(v)o(olume)11 b(`Constructivit)o(y)i(in)g (Mathematics')f(edited)h(b)o(y)g(A.)g(Heyting)g(in)g(1959)f(Georg)h (Kreisel)h(has)f(studied)h(partial)e(and)0 429 y(total)h(con)o(tin)o (uous)h(higher)g(order)h(functionals.)i(The)d(pap)q(er)h(states)g(t)o (w)o(o)e(imp)q(ortan)o(t)f(theorems)i(on)g(these)h(notions:)42 478 y Fm(\017)20 b Fo(the)15 b(densit)o(y)f(theorem,)f(whic)o(h)h(sa)o (ys)g(that)g(an)o(y)f(\014nite)h(functional)f(can)h(b)q(e)h(extended)g (to)f(a)f(total)g(one,)h(and)42 528 y Fm(\017)20 b Fo(the)c(c)o(hoice)f (principle)g(for)f(total)g(con)o(tin)o(uous)h(functionals,)f(whic)o(h)h (sa)o(ys)g(that)g(whenev)o(er)h(for)f(an)o(y)f(total)g Fl(x)h Fo(there)h(is)83 578 y(a)f(total)f Fl(y)j Fo(suc)o(h)e(that)h Fl(Rxy)q Fo(,)f(where)h(the)f(relation)g Fl(R)g Fo(is)f(giv)o(en)h(b)o (y)g(a)f(total)h(b)q(o)q(olean{v)n(alued)e(functional,)h(then)i(this)83 628 y(dep)q(endency)g(can)e(b)q(e)h(realized)f(b)o(y)g(a)f(total)g (functional)g Fl(f)19 b Fo(\(whic)o(h)14 b(can)g(b)q(e)g(found)g (uniformly)d(in)i Fl(R)p Fo(\).)0 678 y(Both)h(of)f(these)j(theorems)e (also)f(ha)o(v)o(e)h(e\013ectiv)o(e)h(v)o(ersions,)f(in)f(a)h(natural)f (sense)j(of)d(the)i(w)o(ord.)83 728 y(Kreisel's)k(pap)q(er)g(and)g(the) g(one)g(b)o(y)f(Kleene)i([Kle59])d(on)h Fn(Countable)i(functionals)i Fo(in)c(the)h(same)f(v)o(olume,)f(whic)o(h)0 777 y(indep)q(enden)o(tly) g(in)o(tro)q(duced)h(equiv)n(alen)o(t)e(concepts,)i(ha)o(v)o(e)e(giv)o (en)h(rise)g(to)f(a)g(lot)g(of)g(activit)o(y)g(aimed)f(at)h (understanding)0 827 y(and)e(dev)o(eloping)g(the)g(mathematical)d (notions)j(in)o(v)o(olv)o(ed.)j(First)e(of)f(all)e(there)k(is)e(the)h (unpublished)f(w)o(ork)g(of)g(T)m(ait)f(in)g(the)0 877 y(so{called)f(Stanford)h(Rep)q(ort,)f(whic)o(h)h(is)f(based)h(directly) g(on)g(Kreisel's)g(w)o(ork.)k(Another)d(early)e(con)o(tribution)g(of)g (essen)o(tial)0 927 y(impact)h(w)o(as)h(Platek's)g(thesis)h([Pla66],)c (who)j(had)g(started)i(his)e(w)o(ork)f(with)h(Kreisel)h(and)f(later)g (also)g(w)o(ork)o(ed)g(with)g(Dana)0 977 y(Scott.)30 b(P)o(articularly)16 b(Scott,)j(in)e(a)g(n)o(um)o(b)q(er)g(of)g (attempts)g([Sco70,)h(Sco82],)f(con)o(tributed)h(a)g(lot)f(to)g(the)h (sub)r(ject.)31 b(His)0 1026 y(w)o(ork)17 b(in\015uenced)h(Y)m(uri)f (Ersho)o(v)h(to)f(come)f(up)i(with)f(an)g(elab)q(orated)g(theory)m(,)h (with)f(a)g(somewhat)f(top)q(ological)g(\015a)o(v)o(our)0 1076 y([Ers74,)h(Ers75,)g(Ers77].)27 b(Man)o(y)16 b(other)h(p)q(eople)h (ha)o(v)o(e)e(also)g(made)g(essen)o(tial)h(con)o(tributions,)g(among)e (them)h(F)m(eferman,)0 1126 y(Gandy)m(,)f(Hyland,)h(Normann)f(and)h (Berger,)i(mostly)c(again)h(in)h(a)g(rather)h(general)f(con)o(text.)26 b(In)16 b(particular)g(Normann's)0 1176 y(w)o(ork)e([Nor93])e(can)i(b)q (e)h(view)o(ed)f(as)g(establishing)f(the)i(densit)o(y)f(theorem)f(in)h (a)g(more)e(general)j(setting.)83 1226 y(My)d(aim)e(in)i(this)h(pap)q (er)g(is)f(to)g(giv)o(e)g(complete)f(pro)q(ofs)i(of)e(the)i(densit)o(y) g(theorem)f(and)g(the)h(c)o(hoice)g(principle)f(for)g(total)0 1275 y(con)o(tin)o(uous)19 b(functionals)f(in)g(the)i(natural)e(and)h (concrete)i(con)o(text)e(of)g(the)g(partial)f(con)o(tin)o(uous)h (functionals)f([Ers77],)0 1325 y(essen)o(tially)12 b(b)o(y)g(sp)q (ecializing)f(more)g(general)h(treatmen)o(ts)h(in)e(the)i(literature.) 18 b(The)12 b(pro)q(ofs)g(obtained)g(are)g(relativ)o(ely)f(short)0 1375 y(and)17 b(hop)q(efully)f(p)q(erspicious,)j(and)d(ma)o(y)g(con)o (tribute)h(to)g(redirect)i(atten)o(tion)e(to)g(the)g(fundamen)o(tal)e (questions)j(Kreisel)0 1425 y(originally)11 b(w)o(as)j(in)o(terested)i (in.)83 1475 y(Ob)o(viously)11 b(this)i(w)o(ork)f(o)o(w)o(es)g(m)o(uc)o (h)f(to)h(other)h(sources.)19 b(In)12 b(particular)g(I)g(ha)o(v)o(e)g (made)f(use)i(of)f(w)o(ork)g(b)o(y)f(Scott)i([Sco82])0 1525 y(\(whose)j(notion)f(of)g(an)g(information)e(system)i(is)g(tak)o (en)h(as)g(a)f(basis)g(to)h(in)o(tro)q(duce)g(domains\),)e(Rosco)q(e)i ([Ros87],)d(Larsen)0 1574 y(and)h(Winsk)o(el)f([L)-5 b(W84])12 b(and)i(Berger)h([Ber93].)83 1624 y(The)i(pap)q(er)g(is)f (organized)g(as)h(follo)o(ws.)23 b(Section)17 b(1)f(treats)i (information)13 b(systems,)k(and)f(in)g(section)h(2)f(it)g(is)g(sho)o (wn)0 1674 y(that)i(the)g(partial)f(orders)h(de\014ned)h(b)o(y)f(them)e (are)j(exactly)e(the)i(\(Scott\))f(domains)e(with)h(coun)o(table)h (basis.)29 b(Section)18 b(3)0 1724 y(giv)o(es)d(a)f(c)o (haracterization)i(of)e(the)i(con)o(tin)o(uous)f(functions)g(b)q(et)o (w)o(een)h(domains,)d(in)i(terms)f(of)h(appro)o(ximable)d(mappings.)0 1774 y(In)i(section)h(4)f(cartesian)h(pro)q(ducts)g(and)f(function)g (spaces)i(of)d(domains)f(and)i(information)d(systems)k(are)f(in)o(tro)q (duced.)20 b(In)0 1823 y(section)13 b(5)f(the)h(partial)e(and)h(total)g (con)o(tin)o(uous)g(functionals)g(are)h(de\014ned.)18 b(Section)13 b(6)f(\014nally)f(con)o(tains)h(the)h(pro)q(ofs)g(of)e (the)0 1873 y(t)o(w)o(o)i(theorems)h(ab)q(o)o(v)o(e;)f(it)h(will)e(b)q (e)j(clear)f(that)g(the)h(same)e(pro)q(ofs)g(also)h(yield)f(e\013ectiv) o(e)i(v)o(ersions)g(of)e(these)i(theorems.)0 1969 y Fk(1.)21 b(Information)13 b(systems)83 2080 y Fo(The)g(basic)g(idea)g(of)f (information)e(systems)j(is)g(to)g(pro)o(vide)f(an)h(axiomatic)d (setting)k(for)e(to)h(describ)q(e)i(appro)o(ximations)0 2130 y(of)h(abstract)i(ob)r(jects)g(\(lik)o(e)f(functions)g(or)g (functionals\))f(b)o(y)h(concrete,)i(\014nite)e(ones.)28 b(W)m(e)17 b(do)f(not)h(attempt)f(to)h(analyse)0 2180 y(the)d(notion)f(of)f(`concreteness')k(or)e(\014niteness)g(here,)h(but) e(rather)i(tak)o(e)e(an)g(arbitrary)h(coun)o(table)f(set)h Fl(A)g Fo(of)f(`data)f(ob)r(jects')0 2230 y(or)19 b(`tok)o(ens')g(as)h (a)f(basic)g(notion)g(to)g(b)q(e)h(explained)f(axiomatically)l(.)32 b(In)19 b(order)h(to)f(use)i(suc)o(h)f(data)f(ob)r(jects)i(to)e(build)0 2279 y(appro)o(ximations)12 b(of)h(abstract)i(ob)r(jects,)g(w)o(e)g (certainly)f(need)h(a)f(notion)g(of)f(`consistency',)i(whic)o(h)f (determines)h(when)f(the)0 2329 y(elemen)o(ts)e(of)f(a)h(\014nite)g (set)h(of)e(data)h(ob)r(jects)h(are)f(consisten)o(t)h(with)f(eac)o(h)h (other.)18 b(A)o(t)12 b(our)g(presen)o(t)h(lev)o(el)f(of)f(generalit)o (y)h(there)0 2379 y(clearly)k(is)g(no)g(w)o(a)o(y)f(to)h(de\014ne)h (this)g(notion)e(concretely)m(,)i(hence)h(it)d(again)g(m)o(ust)h(b)q(e) g(describ)q(ed)i(axiomaticall)o(y)m(.)k(Finally)0 2429 y(w)o(e)14 b(need)h(an)e(`en)o(tailmen)o(t)f(relation')h(b)q(et)o(w)o (een)i(consisten)o(t)g(sets)g Fl(X)j Fo(of)13 b(data)g(ob)r(jects)i (and)f(single)f(data)h(ob)r(jects)h Fl(a)p Fo(,)e(whic)o(h)0 2479 y(expresses)h(the)d(fact)g(that)g(the)h(information)c(con)o (tained)j(in)f Fl(X)15 b Fo(sup)q(ersedes)e(the)f(information)c(in)i Fl(a)p Fo(.)17 b(The)12 b(follo)o(wing)c(axioms)0 2529 y(on)14 b(these)h(notions)e(ha)o(v)o(e)h(b)q(een)h(giv)o(en)f(b)o(y)f (Scott)i(in)e([Sco82].)p 0 2574 600 2 v 42 2620 a(*)20 b(Published)14 b(in:)k Fn(Kr)n(eiseliana.)h(A)o(b)n(out)c(and)h(A)o(r)n (ound)f(Ge)n(or)n(g)g(Kr)n(eisel)i Fo(\(ed.)d(P)m(.)f(Odifreddi\),)h (A.K.)f(P)o(eters,)j(W)m(ellesley)m(,)0 2670 y(Massac)o(h)o(usetts)g (1996,)c(pages)j(335{362)965 2770 y(1)p eop %%Page: 2 2 2 1 bop 0 42 a Fk(1.1)16 b(De\014niti)o(on.)21 b Fj(An)14 b Fn(information)g(system)j Fj(is)c(de\014ned)i(to)e(b)q(e)i(a)e (structure)j Fo(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))14 b Fj(where)h Fl(A)e Fj(is)h(a)f(coun)o(table)h(set)0 91 y(\(the)g Fn(tokens)s Fj(\),)f Fo(Con)g Fj(is)f(a)h(nonempt)o(y)e (set)j(of)e(\014nite)h(subsets)i(of)d Fl(A)h Fj(\(the)h Fn(c)n(onsistent)i Fj(sets\),)e(and)f Fm(`)g Fj(is)g(a)f(subset)j(of)d Fo(Con)7 b Fm(\002)g Fl(A)0 141 y Fj(\(the)15 b Fn(entailment)f(r)n (elation)s Fj(\))g(whic)o(h)g(satisfy)18 193 y(\(i\))21 b Fl(X)15 b Fm(\022)d Fl(Y)21 b Fm(2)11 b Fo(Con)j Fj(implies)e Fl(X)j Fm(2)c Fo(Con)p Fj(;)7 245 y(\(ii\))20 b Fl(a)11 b Fm(2)h Fl(A)i Fj(implies)e Fm(f)p Fl(a)p Fm(g)f(2)g Fo(Con)p Fj(;)-5 296 y(\(iii\))20 b Fl(X)15 b Fm(`)d Fl(a)i Fj(implies)e Fl(X)h Fm([)c(f)p Fl(a)p Fm(g)i(2)g Fo(Con)p Fj(;)-3 348 y(\(iv\))20 b Fl(X)15 b Fm(2)c Fo(Con)j Fj(and)g Fl(a)d Fm(2)g Fl(X)18 b Fj(implies)12 b Fl(X)j Fm(`)d Fl(a)p Fj(;)8 400 y(\(v\))21 b(\()p Fl(X)q(;)7 b(Y)21 b Fm(2)11 b Fo(Con)j Fj(and)f Fo(\()p Fm(8)p Fl(b)f Fm(2)f Fl(Y)e Fo(\))p Fl(X)16 b Fm(`)c Fl(b)h Fj(and)h Fl(Y)21 b Fm(`)12 b Fl(c)p Fj(\))i(implies)e Fl(X)j Fm(`)d Fl(c)p Fj(.)83 481 y Fo(An)o(y)i(coun)o(table)h(set)g Fl(A)f Fo(can)h(b)q(e)g(turned)g(in)o(to)e(an)h(information)e(system)i Fk(A)g Fo(b)o(y)g(letting)g(the)h(set)g(of)f(data)g(ob)r(jects)h(b)q(e) 0 530 y Fl(A)p Fo(,)f(Con)d(=)h Fm(f;g)c([)h(ff)p Fl(a)p Fm(g)i Fo(:)g Fl(a)g Fm(2)h Fl(A)p Fm(g)h Fo(and)h Fl(X)h Fm(`)d Fl(a)23 b Fm(\()-7 b(\))22 b Fl(a)12 b Fm(2)f Fl(X)s Fo(.)0 611 y Fk(1.2)17 b(De\014nitio)o(n.)k Fj(The)15 b Fn(elements)j Fj(or)d Fn(ide)n(als)j Fj(of)c(an)h(information)d (system)i Fk(A)f Fo(=)h(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))15 b Fj(are)g(de\014ned)h(to)f(b)q(e)g(those)0 661 y(subsets)h Fl(z)g Fj(of)d Fl(A)h Fj(whic)o(h)g(satisfy)18 712 y(\(i\))21 b Fl(X)15 b Fm(\022)164 697 y Fi(\014n)215 712 y Fl(z)h Fj(implies)c Fl(X)j Fm(2)c Fo(Con)j Fj(\()p Fl(z)i Fj(is)e Fn(c)n(onsistent)t Fj(\);)7 764 y(\(ii\))20 b Fl(X)15 b Fm(\022)164 749 y Fi(\014n)215 764 y Fl(z)h Fj(and)e Fl(X)h Fm(`)d Fl(a)i Fj(implies)e Fl(a)f Fm(2)g Fl(z)16 b Fj(\()p Fl(z)g Fj(is)e Fn(de)n(ductively)h(close)n(d)t Fj(\).)83 845 y Fo(The)i(set)h(of)e(all)g(elemen)o(ts)g(of)g Fk(A)h Fo(is)g(written)g Fm(j)p Fk(A)p Fm(j)p Fo(.)26 b(It)16 b(clearly)h(follo)o(ws)e(that)i(the)g(in)o(tersection)h(of)e (an)o(y)h(n)o(um)o(b)q(er)f(of)0 895 y(ideals)e(is)f(an)h(ideal)f (again.)0 975 y Fk(1.3)j(Lemma.)23 b Fj(Supp)q(ose)15 b Fk(A)d Fo(=)g(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))14 b Fj(is)g(an)f(information)e(system.)18 b(Then)18 1027 y(\(i\))j Fm(;)11 b(2)h Fo(Con)o Fj(;)7 1079 y(\(ii\))20 b(\()p Fl(X)15 b Fm(2)d Fo(Con)h Fj(and)h Fl(Y)21 b Fm(\022)12 b Fl(X)17 b Fj(and)d Fl(Y)21 b Fm(`)12 b Fl(a)p Fj(\))h(implies)f Fl(X)k Fm(`)11 b Fl(a)p Fj(.)0 1159 y Fk(Pro)q(of.)17 b Fo(\(i\))c(follo)o(ws)e(b)q(ecause)k(Con)e(is)g(b)o(y)f(assumption)g (nonempt)o(y)m(,)f(and)i(axiom)d(\(i\).)18 b(\(ii\))12 b(holds)g(b)o(y)h(axiom)e(\(v\))i(since,)g(b)o(y)0 1209 y(axiom)e(\(iv\),)i Fl(X)j Fm(`)c Fl(b)h Fo(for)h(eac)o(h)g Fl(b)d Fm(2)h Fl(Y)d Fo(.)1315 b Fh(\003)83 1261 y Fo(Supp)q(ose)16 b Fk(A)e Fo(=)g(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))15 b(is)g(an)g(information)e(system)i(and)g Fl(B)h Fm(\022)e Fl(A)p Fo(.)22 b(W)m(e)15 b(de\014ne)p 1459 1228 34 2 v 16 w Fl(B)r Fo(,)g(the)h Fn(de)n(ductive)h(closur)n(e)h Fo(of)0 1311 y Fl(B)r Fo(,)c(b)o(y)p 607 1333 V 607 1366 a Fl(B)g Fo(=)e Fm(f)p Fl(a)f Fm(2)h Fl(A)f Fo(:)g Fl(X)k Fm(`)d Fl(a)i Fo(for)g(some)f Fl(X)i Fm(\022)1226 1351 y Fi(\014n)1277 1366 y Fl(B)r Fm(g)p Fl(:)p 0 1418 V 0 1451 a(B)j Fo(is)e(the)g(set)g(of)f(tok)o(ens)h(deducible)h(from)d Fl(B)r Fo(.)23 b(Note)16 b(that)g(b)o(y)f(part)h(\(ii\))f(of)g(the)h (previous)g(Lemma,)d(if)i Fl(X)j Fm(2)c Fo(Con,)h(then)p 0 1468 38 2 v 0 1501 a Fl(X)g Fo(=)d Fm(f)p Fl(a)f Fm(2)g Fl(A)h Fo(:)f Fl(X)k Fm(`)d Fl(a)p Fm(g)p Fo(.)0 1581 y Fk(1.4)h(Lemma.)24 b Fj(Supp)q(ose)12 b Fk(A)g Fo(=)g(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))12 b Fj(is)f(an)g(information)e (system,)i Fl(X)k Fm(2)d Fo(Con)f Fj(and)g Fl(Y)21 b Fj(a)11 b(\014nite)h(subset)h(of)e Fl(A)p Fj(.)17 b(Then)18 1633 y(\(i\))k(if)13 b Fl(X)i Fm(`)d Fl(a)i Fj(for)f(ev)o(ery)i Fl(a)d Fm(2)f Fl(Y)e Fj(,)k(then)i Fl(Y)21 b Fm(2)11 b Fo(Con)p Fj(;)7 1685 y(\(ii\))p 83 1652 V 20 w Fl(X)k Fm(2)c(j)p Fk(A)p Fm(j)p Fj(;)-5 1737 y(\(iii\))p 83 1701 21 2 v 20 w Fm(;)g(\022)h Fl(z)k Fj(for)e(ev)o(ery)g Fl(z)g Fm(2)d(j)p Fk(A)p Fm(j)p Fj(.)0 1817 y Fk(Pro)q(of.)17 b Fo(\(i\))11 b(W)m(e)h(pro)o(v)o(e)f Fl(X)e Fm([)c Fl(Y)20 b Fm(2)12 b Fo(Con)f(b)o(y)g(induction)h(on)f(the)i(size)f(of)f Fl(Y)e Fo(;)j(the)h(result)f(then)g(follo)o(ws)f(b)o(y)g(axiom)e (\(i\).)17 b(In)12 b(case)0 1867 y Fl(Y)22 b Fo(=)13 b Fm(;)h Fo(there)i(is)e(nothing)g(to)g(pro)o(v)o(e,)g(so)h(supp)q(ose) h Fl(Y)21 b Fo(=)13 b Fl(Y)936 1852 y Fg(0)957 1867 y Fm([)c(f)p Fl(a)p Fm(g)14 b Fo(and)g Fl(X)g Fm([)9 b Fl(Y)1271 1852 y Fg(0)1295 1867 y Fm(2)k Fo(Con)o(.)20 b(Since)15 b Fl(X)h Fm(`)d Fl(a)i Fo(b)o(y)f(assumption,)0 1917 y(w)o(e)g(ha)o(v)o(e)g Fl(X)f Fm([)c Fl(Y)274 1902 y Fg(0)297 1917 y Fm(`)j Fl(a)i Fo(b)o(y)f(Lemma)f(1.3\(ii\))g(and)i (hence)h Fl(X)e Fm([)c Fl(Y)20 b Fm(2)12 b Fo(Con)h(b)o(y)h(axiom)d (\(iii\).)83 1969 y(\(ii\))i(If)g Fl(Y)23 b Fo(is)14 b(a)f(\014nite)h(subset)h(of)p 596 1935 38 2 v 13 w Fl(X)t Fo(,)e(then)h Fl(Y)21 b Fm(2)11 b Fo(Con)j(b)o(y)f(\(i\);)g(hence)p 1166 1935 V 15 w Fl(X)18 b Fo(is)13 b(consisten)o(t.)19 b(If)14 b Fl(Y)22 b Fo(is)14 b(a)f(\014nite)h(subset)h(of)p 1912 1935 V 13 w Fl(X)0 2019 y Fo(and)f Fl(Y)21 b Fm(`)11 b Fl(a)p Fo(,)j(then)g Fl(X)i Fm(`)11 b Fl(a)j Fo(b)o(y)g(the)g (de\014nition)g(of)p 787 1985 V 13 w Fl(X)k Fo(and)13 b(axiom)f(\(v\);)h(hence)p 1238 1985 V 16 w Fl(X)k Fo(is)d(deductiv)o (ely)g(closed.)83 2070 y(\(iii\))f(Let)h Fl(z)g Fm(2)d(j)p Fk(A)p Fm(j)i Fo(and)h Fl(a)d Fm(2)p 536 2034 21 2 v 11 w(;)p Fo(,)i(i.e.)18 b Fm(;)11 b(`)h Fl(a)p Fo(.)18 b(Since)c Fl(z)i Fo(is)e(deductiv)o(ely)g(closed,)g(it)g(follo)o(ws)e (that)i Fl(a)e Fm(2)f Fl(z)r Fo(.)230 b Fh(\003)83 2122 y Fo(In)13 b(the)h(ligh)o(t)e(of)g(\(i\))h(it)f(is)h(con)o(v)o(enien)o (t)h(informall)o(y)c(to)j(allo)o(w)e(`)p Fm(`)p Fo(')i(to)g(b)q(e)g (used)h(as)f(a)g(relation)g(b)q(et)o(w)o(een)h(Con)f(and)g(Con)o(.)0 2172 y(So)h(from)e(no)o(w)h(on)h(`)p Fl(X)h Fm(`)d Fl(Y)d Fo(')k(will)g(mean)f(`)p Fl(X)j Fm(`)d Fl(a)i Fo(for)f(eac)o(h)i Fl(a)c Fm(2)g Fl(Y)f Fo('.)0 2252 y Fk(1.5)16 b(Lemma.)23 b Fj(If)14 b Fk(A)e Fo(=)f(\()p Fl(A;)c Fo(Con)p Fl(;)g Fm(`)p Fo(\))14 b Fj(is)f(an)h(information)d(system)j(and)f Fl(U)k Fm(\022)12 b Fl(A)i Fj(is)f(consisten)o(t,)i(then)p 1630 2219 33 2 v 14 w Fl(U)i Fm(2)11 b(j)p Fk(A)p Fm(j)p Fj(.)0 2333 y Fk(Pro)q(of.)22 b Fo(W)m(e)15 b(\014rst)h(sho)o(w)f(that) p 508 2300 V 16 w Fl(U)20 b Fo(is)15 b(consisten)o(t.)23 b(If)15 b Fl(Y)23 b Fo(=)15 b Fm(f)p Fl(a)992 2339 y Fi(1)1010 2333 y Fl(;)7 b(a)1051 2339 y Fi(2)1069 2333 y Fl(;)g(:)g(:)g(:)e(;)i(a)1184 2339 y Ff(n)1206 2333 y Fm(g)15 b Fo(is)g(an)o(y)g(\014nite)g(subset)i(of)p 1650 2300 V 15 w Fl(U)t Fo(,)f(then)f(for)g(eac)o(h)0 2383 y Fl(i)d Fm(2)f(f)p Fo(1)p Fl(;)c Fo(2)p Fl(;)g(:)g(:)g(:)t(;)g(n) p Fm(g)k Fo(there)j(exists)g(a)e(\014nite)h(subset)h Fl(Z)807 2389 y Ff(i)834 2383 y Fo(of)e Fl(U)17 b Fo(suc)o(h)d(that)e Fl(Z)1134 2389 y Ff(i)1160 2383 y Fm(`)g Fl(a)p Fo(.)18 b(But)13 b Fl(Z)i Fo(=)c Fl(Z)1444 2389 y Fi(1)1470 2383 y Fm([)c(\001)g(\001)g(\001)e([)i Fl(Z)1623 2389 y Ff(n)1658 2383 y Fo(is)12 b(a)h(\014nite)g(subset)0 2433 y(of)g Fl(U)5 b Fo(,)14 b(and)f(so)h(is)g(in)f(Con.)18 b(No)o(w)c Fl(Z)h Fm(`)c Fl(a)627 2439 y Ff(i)655 2433 y Fo(for)j(eac)o(h)g Fl(i)g Fo(b)o(y)g(Lemma)d(1.3\(ii\).)16 b(Lemma)11 b(1.4\(i\))i(then)i (tells)f(us)g(that)g Fl(Y)21 b Fm(2)11 b Fo(Con.)83 2484 y(T)m(o)i(sho)o(w)h(deductiv)o(e)h(closure,)f(supp)q(ose)h(that)f Fl(Y)23 b Fo(is)14 b(a)g(\014nite)g(subset)h(of)p 1234 2451 V 13 w Fl(U)k Fo(and)14 b(that)g Fl(Y)21 b Fm(`)12 b Fl(b)p Fo(.)17 b(Exactly)d(as)g(ab)q(o)o(v)o(e)g(w)o(e)0 2534 y(can)g(\014nd)g(a)g(\014nite)g(subset)h Fl(Z)i Fo(of)d Fl(U)k Fo(suc)o(h)d(that)f Fl(Z)h Fm(`)d Fl(Y)d Fo(.)18 b(But)c(then)h Fl(Z)g Fm(`)d Fl(b)h Fo(b)o(y)h(axiom)d(\(v\),)j (so)g Fl(b)d Fm(2)p 1561 2501 V 11 w Fl(U)19 b Fo(as)14 b(required.)99 b Fh(\003)965 2770 y Fo(2)p eop %%Page: 3 3 3 2 bop 0 42 a Fk(2.)21 b(Complete)14 b(partial)g(orders)g(and)i (domains)83 152 y Fo(Let)e(\()p Fl(I)s(;)7 b Fm(\024)p Fo(\))14 b(b)q(e)g(a)f(partial)g(order,)g(i.e.)18 b Fm(\024)13 b Fo(is)h(a)f(re\015exiv)o(e)h(and)f(transitiv)o(e)h(subset)h(of)d Fl(I)g Fm(\002)d Fl(I)17 b Fo(satisfying)12 b Fl(i)g Fm(\024)g Fl(j)i Fm(\024)e Fl(i)23 b Fo(=)-7 b Fm(\))0 202 y Fl(i)12 b Fo(=)h Fl(j)j Fo(\(an)o(tisymmetry\).)h(W)m(e)d(sa)o(y) g(that)g Fl(I)k Fo(is)c Fn(dir)n(e)n(cte)n(d)29 b Fo(if)13 b(it)h(is)g(nonempt)o(y)f(and)h(for)g(an)o(y)g Fl(i;)7 b(j)14 b Fm(2)e Fl(I)17 b Fo(there)f(is)e(a)g Fl(k)f Fm(2)f Fl(I)17 b Fo(suc)o(h)0 252 y(that)e Fl(i)f Fm(\024)g Fl(k)i Fo(and)f Fl(j)h Fm(\024)e Fl(k)q Fo(.)22 b(A)15 b(partial)g(order)h(\()p Fl(D)q(;)7 b Fm(\024)p Fo(\))15 b(ha)o(ving)f(a)h(least)h(elemen)o(t)e Fm(?)h Fo(is)g(said)g(to)g(b)q (e)g Fn(c)n(omplete)k Fo(\(and)c(w)o(e)g(sa)o(y)0 301 y(that)h Fl(D)h Fo(is)f(a)g Fn(c)n(omplete)g(p)n(artial)g(or)n(der)t Fo(,)g(abbreviated)g(to)g Fn(cp)n(o)s Fo(\))g(if)f(ev)o(ery)i(directed) g(subset)h Fl(M)h Fm(\022)d Fl(D)h Fo(has)f(a)g(least)g(upp)q(er)0 351 y(b)q(ound)130 320 y Fe(F)171 351 y Fl(M)5 b Fo(.)23 b(A)16 b(p)q(oin)o(t)f Fl(x)g Fo(of)g(a)h(cp)q(o)g Fl(D)h Fo(is)e(said)h(to)f(b)q(e)h Fn(c)n(omp)n(act)k Fo(or)c Fn(\014nite)j Fo(if,)c(for)g(ev)o(ery)h(directed)h(collection)f Fl(M)j Fm(\022)14 b Fl(D)0 401 y Fo(suc)o(h)g(that)f Fl(x)f Fm(\024)261 370 y Fe(F)303 401 y Fl(M)5 b Fo(,)13 b(there)h(is)f(a)g Fl(y)g Fm(2)f Fl(M)18 b Fo(suc)o(h)c(that)f Fl(x)e Fm(\024)h Fl(y)q Fo(.)19 b(Let)13 b Fk(B)1104 407 y Ff(D)1148 401 y Fo(denote)h(the)g(collection)e(of)h(compact)f (elemen)o(ts)i(of)0 451 y Fl(D)q Fo(;)f Fk(B)94 457 y Ff(D)138 451 y Fo(is)g(called)g(the)h Fn(b)n(asis)i Fo(of)d Fl(D)q Fo(.)18 b(The)c(cp)q(o)f Fl(D)i Fo(is)e Fn(algebr)n(aic)i Fo(if,)d(for)h(ev)o(ery)h Fl(x)e Fm(2)f Fl(D)q Fo(,)i(the)h(set)g Fl(M)j Fo(=)11 b Fm(f)p Fl(x)1640 457 y Fi(0)1670 451 y Fm(2)g Fk(B)1743 457 y Ff(D)1785 451 y Fo(:)g Fl(x)1832 457 y Fi(0)1862 451 y Fm(\024)h Fl(x)p Fm(g)0 501 y Fo(is)i(directed)h (and)f Fl(x)e Fo(=)362 470 y Fe(F)403 501 y Fl(M)5 b Fo(.)18 b(A)c(cp)q(o)h Fl(D)g Fo(is)f Fn(b)n(ounde)n(d)i(c)n(omplete)h Fo(or)d Fn(c)n(onsistently)h(c)n(omplete)i Fo(if)d(ev)o(ery)g(b)q (ounded)h(subset)h(of)0 551 y Fl(D)h Fo(\(or)f(equiv)n(alen)o(tly)e(ev) o(ery)i(b)q(ounded)g(\014nite)g(subset)h(of)e Fl(D)q Fo(\))h(has)f(a)h(least)f(upp)q(er)i(b)q(ound.)23 b(W)m(e)15 b(call)f(b)q(ounded)j(complete)0 600 y(algebraic)c(cp)q(o's)i Fn(Sc)n(ott)g(domains)i Fo(or)d(just)g Fn(domains)s Fo(.)83 650 y(With)j(an)o(y)g(information)d(system)j Fk(A)h Fo(=)f(\()p Fl(A;)7 b Fo(Con)p Fl(;)g Fm(`)p Fo(\))17 b(w)o(e)h(can)f(asso)q(ciate) h(the)g(partial)f(order)h(\()p Fm(j)p Fk(A)p Fm(j)p Fl(;)7 b Fm(\022)p Fo(\).)27 b(Our)18 b(aim)0 700 y(in)e(this)g(section)h(is)f (to)g(sho)o(w)h(that)f(the)h(partial)e(orders)i(obtained)g(in)e(this)i (w)o(a)o(y)e(are)i(exactly)f(the)h(Scott)g(domains)d(with)0 750 y(coun)o(table)g(basis.)0 821 y Fk(2.1)20 b(Theorem.)j Fj(Supp)q(ose)18 b Fk(A)f Fo(=)h(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))17 b Fj(is)h(an)f(information)d(system.)28 b(Then)18 b Fo(\()p Fm(j)p Fk(A)p Fm(j)p Fl(;)7 b Fm(\022)p Fo(\))16 b Fj(is)h(a)g(domain)e(with)i(the)0 871 y(coun)o(table)d (basis)g Fm(f)p 311 838 38 2 v Fl(X)h Fo(:)c Fl(X)k Fm(2)c Fo(Con)p Fm(g)p Fj(.)0 943 y Fk(Pro)q(of.)36 b Fo(\()p Fm(j)p Fk(A)p Fm(j)p Fl(;)7 b Fm(\022)p Fo(\))20 b(clearly)g(is)g(a)g (partial)f(order.)37 b(It)21 b(has)p 972 907 21 2 v 20 w Fm(;)f Fo(as)g(its)g(least)h(elemen)o(t)e(b)o(y)h(Lemma)e (1.4\(iii\).)34 b(T)m(o)20 b(pro)o(v)o(e)0 992 y(completeness,)e(let)f Fl(M)k Fm(\022)c(j)p Fk(A)p Fm(j)f Fo(b)q(e)h(directed.)29 b(Then)869 961 y Fe(S)911 992 y Fl(M)21 b Fo(is)c(consisten)o(t)h(and)f (deductiv)o(ely)g(closed)h(and)e(hence)j(is)d(the)0 1042 y(least)e(upp)q(er)h(b)q(ound)f(of)f Fl(M)5 b Fo(.)83 1092 y(W)m(e)19 b(no)o(w)g(sho)o(w)g(that,)i(for)e(an)o(y)g Fl(z)j Fm(2)f(j)p Fk(A)p Fm(j)p Fo(,)e(that)h Fl(z)h Fo(is)e(compact)g(if)f(and)i(only)e(if)h Fl(z)k Fo(=)p 1522 1059 38 2 v 20 w Fl(X)g Fo(for)c(some)g Fl(X)24 b Fm(2)d Fo(Con)o(.)0 1142 y(First)g(assume)e(that)i Fl(z)h Fo(is)e(compact.)36 b(Clearly)20 b Fl(M)27 b Fo(:=)22 b Fm(f)p 949 1109 V Fl(X)j Fo(:)d Fl(X)k Fm(\022)1134 1127 y Fi(\014n)1196 1142 y Fl(z)r Fm(g)20 b Fo(is)g(directed)h(and)f Fl(z)25 b Fo(=)1656 1111 y Fe(S)1698 1142 y Fl(M)5 b Fo(.)36 b(Since)21 b(b)o(y)0 1192 y(assumption)15 b Fl(z)k Fo(is)e(compact,)f(there)i(is)f(a)f(\014nite)h(subset)i Fl(X)h Fo(of)c Fl(z)j Fo(suc)o(h)f(that)e Fl(z)j Fm(\022)p 1335 1158 V 17 w Fl(X)s Fo(,)e(hence)h Fl(z)h Fo(=)p 1606 1158 V 17 w Fl(X)s Fo(.)27 b(Con)o(v)o(ersely)m(,)17 b(let)0 1242 y Fl(X)i Fo(=)e Fm(f)p Fl(a)145 1248 y Fi(1)163 1242 y Fl(;)7 b(a)204 1248 y Fi(2)222 1242 y Fl(;)g(:)g(:)g(:)e(;)i(a) 337 1248 y Ff(n)359 1242 y Fm(g)15 b(2)h Fo(Con)g(and)g Fl(M)21 b Fm(\022)16 b(j)p Fk(A)p Fm(j)g Fo(directed)h(suc)o(h)h(that)p 1147 1208 V 16 w Fl(X)i Fm(\022)1249 1210 y Fe(S)1290 1242 y Fl(M)5 b Fo(.)26 b(Then)17 b Fl(a)1506 1248 y Ff(i)1535 1242 y Fm(2)f Fl(z)1598 1248 y Ff(i)1628 1242 y Fo(for)h(some)e Fl(z)1820 1248 y Ff(i)1850 1242 y Fm(2)h Fl(M)5 b Fo(.)0 1291 y(Since)16 b Fl(M)k Fo(is)15 b(directed,)i(there)f (is)g Fl(z)558 1276 y Fg(\003)591 1291 y Fm(2)d Fl(M)21 b Fo(suc)o(h)16 b(that)f Fl(z)898 1297 y Ff(i)926 1291 y Fm(\022)f Fl(z)993 1276 y Fg(\003)1013 1291 y Fo(,)h(hence)i Fl(X)g Fm(\022)e Fl(z)1276 1276 y Fg(\003)1310 1291 y Fo(and)g(therefore)p 1568 1258 V 17 w Fl(X)j Fm(\022)c Fl(z)1687 1276 y Fg(\003)1706 1291 y Fo(.)23 b(This)15 b(means)0 1341 y(that)p 90 1308 V 14 w Fl(X)i Fo(is)d(compact.)83 1391 y(The)h(algebraicit)o(y)e(of)h(\()p Fm(j)p Fk(A)p Fm(j)p Fl(;)7 b Fm(\022)p Fo(\))14 b(no)o(w)g(is)g(ob)o(vious)g(since,) h(for)f(ev)o(ery)h Fl(z)g Fm(2)d(j)p Fk(A)p Fm(j)p Fo(,)h Fm(f)p 1346 1358 V Fl(X)j Fo(:)c Fl(X)k Fm(2)c Fo(Con)j(and)p 1680 1358 V 14 w Fl(X)h Fm(\022)d Fl(z)r Fm(g)h Fo(clearly)0 1441 y(is)g(directed)h(and)f(has)g Fl(z)i Fo(as)e(its)g(least)g(upp)q (er)h(b)q(ound.)83 1491 y(Finally)10 b(ev)o(ery)i(b)q(ounded)f(subset)i Fl(M)j Fm(\022)c(j)p Fk(A)p Fm(j)f Fo(has)p 862 1454 87 2 v 862 1459 a Fe(S)904 1491 y Fl(M)16 b Fo(as)11 b(its)g(least)h(upp)q(er)g(b)q(ound,)f(and)g(hence)i(\()p Fm(j)p Fk(A)p Fm(j)p Fl(;)7 b Fm(\022)p Fo(\))j(is)h(b)q(ounded)0 1540 y(complete.)1744 b Fh(\003)83 1590 y Fo(W)m(e)18 b(no)o(w)g(pro)o(v)o(e)h(that)f(an)o(y)g(domain)e(with)i(coun)o(table)h (basis)f(can)h(in)f(fact)g(b)q(e)h(represen)o(ted)i(in)d(this)h(w)o(a)o (y)m(,)f(as)g(the)0 1640 y(domain)c(of)i(an)g(appropriate)h (information)d(system)i(asso)q(ciated)h(with)f(it.)26 b(T)m(o)16 b(obtain)f(this)i(result)g(w)o(e)g(need)g(a)g(Lemma)0 1690 y(sa)o(ying)c(that)h(in)f(a)h(cp)q(o)g(least)g(upp)q(er)h(b)q (ounds)g(of)e(\014nite)h(sets)h(of)e(compact)g(elemen)o(ts)h(are)h (compact,)d(when)i(they)h(exist.)0 1761 y Fk(2.2)h(Lemma.)23 b Fj(Supp)q(ose)15 b Fo(\()p Fl(D)q(;)7 b Fm(\024)p Fo(\))15 b Fj(is)e(a)h(cp)q(o,)g Fl(X)h Fm(\022)809 1746 y Fi(\014n)860 1761 y Fk(B)894 1767 y Ff(D)938 1761 y Fj(and)1019 1730 y Fe(F)1060 1761 y Fl(X)j Fj(exists.)g(Then)1351 1730 y Fe(F)1392 1761 y Fl(X)d Fm(2)d Fk(B)1515 1767 y Ff(D)1545 1761 y Fj(.)0 1833 y Fk(Pro)q(of.)30 b Fo(Supp)q(ose)19 b Fl(M)k Fm(\022)c Fl(D)h Fo(is)e(directed)h(and)789 1802 y Fe(F)830 1833 y Fl(X)j Fm(\024)937 1802 y Fe(F)979 1833 y Fl(M)5 b Fo(.)30 b(Let)19 b Fl(X)j Fo(=)c Fm(f)p Fl(e)1291 1839 y Fi(1)1310 1833 y Fl(;)7 b(e)1348 1839 y Fi(2)1367 1833 y Fl(;)g(:)g(:)g(:)t(;)g(e)1478 1839 y Ff(n)1501 1833 y Fm(g)p Fo(.)30 b(Since)18 b Fl(e)1695 1839 y Ff(i)1728 1833 y Fm(\024)1779 1802 y Fe(F)1820 1833 y Fl(M)23 b Fo(and)0 1883 y Fl(e)19 1889 y Ff(i)45 1883 y Fm(2)11 b Fk(B)118 1889 y Ff(D)162 1883 y Fo(there)k(is)f(a)g Fl(x)369 1889 y Ff(i)394 1883 y Fm(2)d Fl(M)19 b Fo(with)13 b Fl(e)605 1889 y Ff(i)631 1883 y Fm(\024)f Fl(x)699 1889 y Ff(i)712 1883 y Fo(.)19 b(Since)14 b Fl(M)19 b Fo(is)13 b(directed,)i(there)g(is)f(a)g Fl(x)1329 1868 y Fg(\003)1359 1883 y Fm(2)e Fl(M)18 b Fo(suc)o(h)d(that)f Fl(x)1665 1889 y Ff(i)1690 1883 y Fm(\024)e Fl(x)1758 1868 y Fg(\003)1791 1883 y Fo(for)h(ev)o(ery)0 1933 y Fl(i)f Fm(2)f(f)p Fo(1)p Fl(;)c Fo(2)p Fl(;)g(:)g(:)g(:)t(;)g(n)p Fm(g)p Fo(.)17 b(Hence)437 1901 y Fe(F)479 1933 y Fl(X)e Fm(\024)d Fl(x)596 1917 y Fg(\003)614 1933 y Fo(.)1292 b Fh(\003)0 2004 y Fk(2.3)17 b(De\014niti)o(on.)k Fj(Let)15 b Fl(D)h Fj(b)q(e)f(a)g(domain)d(with)i(coun)o(table)h(basis)g Fk(B)1109 2010 y Ff(D)1139 2004 y Fj(.)20 b(De\014ne)15 b Fl(I)s(D)g Fo(=)e(\()p Fk(B)1465 2010 y Ff(D)1495 2004 y Fl(;)7 b Fo(Con)p Fl(;)g Fm(`)p Fo(\))p Fj(,)14 b(where)i Fo(Con)e Fj(and)0 2054 y Fm(`)g Fj(are)h(de\014ned)f(b)o(y)478 2101 y Fl(X)h Fm(2)d Fo(Con)53 b Fm(\()-7 b(\))52 b Fl(X)15 b Fm(\022)904 2086 y Fi(\014n)955 2101 y Fk(B)989 2107 y Ff(D)1033 2101 y Fj(and)e Fl(X)18 b Fj(is)c(b)q(ounded)g(in)g Fl(D)q Fj(;)478 2164 y Fl(X)h Fm(`)d Fl(e)111 b Fm(\()-7 b(\))52 b Fl(X)15 b Fm(2)c Fo(Con)j Fj(and)f Fl(e)f Fm(\024)1154 2132 y Fe(F)1196 2164 y Fl(X)s Fj(.)83 2236 y Fo(Let)j Fl(D)i Fo(and)e Fl(E)i Fo(b)q(e)e(partial)f(orders.)23 b(A)15 b(function)f Fl(f)t Fo(:)7 b Fl(D)16 b Fm(!)c Fl(E)17 b Fo(is)e Fn(monotone)k Fo(if)c Fl(d)e Fm(\024)g Fl(d)1460 2221 y Fg(0)1486 2236 y Fo(implies)g Fl(f)t Fo(\()p Fl(d)p Fo(\))h Fm(\024)g Fl(f)t Fo(\()p Fl(d)1828 2221 y Fg(0)1840 2236 y Fo(\),)h(and)0 2286 y Fn(or)n(derpr)n(eserving) k Fo(if)c Fl(d)g Fm(\024)h Fl(d)445 2271 y Fg(0)488 2286 y Fm(\()-7 b(\))30 b Fl(f)t Fo(\()p Fl(d)p Fo(\))16 b Fm(\024)g Fl(f)t Fo(\()p Fl(d)799 2271 y Fg(0)811 2286 y Fo(\).)26 b(A)16 b(bijectiv)o(e)g(orderpreserving)i(function)e Fl(f)t Fo(:)7 b Fl(D)18 b Fm(!)d Fl(E)j Fo(is)e(called)g(an)0 2336 y Fn(isomorphism)s Fo(.)h(No)o(w)11 b(let)h Fl(D)h Fo(and)e Fl(E)j Fo(b)q(e)e(cp)q(o's.)17 b Fl(f)f Fo(is)c Fn(c)n(ontinuous)k Fo(if)10 b(it)i(is)f(monotone)f(and)h(if)g(for)g(an) o(y)g(directed)i(set)f Fl(M)17 b Fm(\022)11 b Fl(D)705 2427 y(f)t Fo(\()745 2388 y Fe(G)799 2427 y Fl(M)5 b Fo(\))12 b(=)915 2388 y Fe(G)961 2427 y Fm(f)p Fl(f)t Fo(\()p Fl(d)p Fo(\))g(:)f Fl(d)h Fm(2)f Fl(M)5 b Fm(g)p Fl(:)0 2521 y Fo(Note)15 b(that)f(from)f(the)i(monotonicit)o(y)d(of)i Fl(f)19 b Fo(the)c(directedness)i(of)d Fm(f)p Fl(f)t Fo(\()p Fl(d)p Fo(\))e(:)g Fl(d)g Fm(2)g Fl(M)5 b Fm(g)14 b Fo(follo)o(ws,)f(and)h(hence)i(the)f(suprem)o(um)0 2570 y(on)g(the)h(righ)o(t)f(hand)h(side)f(exists)h(in)f Fl(E)r Fo(.)23 b(Note)16 b(also)f(that)g(an)o(y)g(surjectiv)o(e)i (orderpreserving)g(function)e Fl(f)t Fo(:)7 b Fl(D)16 b Fm(!)e Fl(E)j Fo(m)o(ust)0 2620 y(b)q(e)e(injectiv)o(e,)e(hence)i(an) f(isomorphism)d(and)i(therefore)j(also)d(con)o(tin)o(uous.)83 2670 y(It)h(is)g(easy)g(to)g(see)h(that)f(the)g(concatenation)h(of)e(t) o(w)o(o)g(con)o(tin)o(uous)h(functions)g(is)g(con)o(tin)o(uous)g (again.)965 2770 y(3)p eop %%Page: 4 4 4 3 bop 0 42 a Fk(2.4)16 b(Lemma.)24 b Fj(Supp)q(ose)15 b Fl(D)q Fj(,)g Fl(E)h Fj(and)e Fl(F)20 b Fj(are)15 b(cp)q(o's.)k(Let)c Fl(f)t Fo(:)7 b Fl(D)14 b Fm(!)e Fl(E)k Fj(and)e Fl(g)q Fo(:)7 b Fl(E)14 b Fm(!)e Fl(F)20 b Fj(b)q(e)15 b(con)o(tin)o(uous)f (functions.)19 b(Then)0 91 y Fl(g)11 b Fm(\016)d Fl(f)t Fo(:)f Fl(D)14 b Fm(!)d Fl(F)19 b Fj(is)14 b(con)o(tin)o(uous.)0 169 y Fk(Pro)q(of.)k Fo(Clearly)13 b Fl(g)e Fm(\016)e Fl(f)19 b Fo(is)14 b(monotone.)j(F)m(urthermore)c(w)o(e)i(ha)o(v)o(e)e (that)1130 138 y Fe(F)1164 181 y Ff(d)p Fg(2)p Ff(M)1248 169 y Fl(g)q Fo(\()p Fl(f)t Fo(\()p Fl(d)p Fo(\)\))f(=)h Fl(g)q Fo(\()1473 138 y Fe(F)1508 181 y Ff(d)p Fg(2)p Ff(M)1591 169 y Fl(f)t Fo(\()p Fl(d)p Fo(\)\))f(=)h Fl(g)q Fo(\()p Fl(f)t Fo(\()1819 138 y Fe(F)1861 169 y Fl(M)5 b Fo(\)\).)0 219 y Fh(\003)0 296 y Fk(2.5)17 b(Theorem.)23 b Fj(Let)15 b Fl(D)h Fj(b)q(e)g(a)e(domain)f(with)h(coun)o(table)h (basis.)21 b(Then)15 b Fl(I)s(D)i Fj(is)e(an)f(information)e(system)j (with)f(domain)0 346 y(of)f(elemen)o(ts)h Fm(j)p Fl(I)s(D)q Fm(j)g Fj(isomorphic)e(to)i Fl(D)q Fj(.)k(The)d(isomorphism)10 b(pair)k(is)517 448 y Fl(')p Fo(:)7 b Fl(D)13 b Fm(!)e(j)p Fl(I)s(D)q Fm(j)42 b Fj(giv)o(en)13 b(b)o(y)41 b Fl(')p Fo(\()p Fl(d)p Fo(\))12 b(=)g Fm(f)p Fl(e)f Fm(2)g Fk(B)1239 454 y Ff(D)1281 448 y Fo(:)g Fl(e)h Fm(\024)g Fl(d)p Fm(g)p Fl(;)632 561 y( )q Fo(:)7 b Fm(j)p Fl(I)s(D)q Fm(j)12 b(!)f Fl(D)43 b Fj(giv)o(en)13 b(b)o(y)42 b Fl( )q Fo(\()p Fl(z)r Fo(\))12 b(=)1232 522 y Fe(G)1285 561 y Fl(z)r(:)0 650 y Fk(Pro)q(of.)18 b Fo(It)c(is)f(easy)i(to)e(see)j (that)d Fl(I)s(D)j Fo(is)e(an)g(information)d(system.)83 701 y Fl(')17 b Fo(is)g(w)o(ell{de\014ned:)24 b(W)m(e)17 b(ha)o(v)o(e)f(to)h(sho)o(w)g(that)g Fl(')p Fo(\()p Fl(d)p Fo(\))g Fm(2)f(j)p Fl(I)s(D)q Fm(j)p Fo(.)27 b(Let)17 b(us)h(\014rst)f(pro)o(v)o(e)g(that)g Fl(')p Fo(\()p Fl(d)p Fo(\))g(is)g(consisten)o(t.)28 b(So)0 750 y(let)15 b Fl(e)80 756 y Fi(1)99 750 y Fl(;)7 b(e)137 756 y Fi(2)156 750 y Fl(;)g(:)g(:)g(:)t(;)g(e)267 756 y Ff(n)304 750 y Fm(\024)14 b Fl(d)p Fo(.)21 b(W)m(e)15 b(ha)o(v)o(e)g(to)g(sho)o(w)g (that)g Fm(f)p Fl(e)862 756 y Fi(1)881 750 y Fl(;)7 b(e)919 756 y Fi(2)938 750 y Fl(;)g(:)g(:)g(:)t(;)g(e)1049 756 y Ff(n)1072 750 y Fm(g)13 b(2)h Fo(Con)o(.)22 b(But)16 b(this)f(is)g(clear,)h(since)g Fm(f)p Fl(e)1720 756 y Fi(1)1739 750 y Fl(;)7 b(e)1777 756 y Fi(2)1795 750 y Fl(;)g(:)g(:)g(:)e(;)i(e)1907 756 y Ff(n)1929 750 y Fm(g)0 800 y Fo(is)17 b(b)q(ounded)g(b)o(y)g Fl(d)p Fo(.)27 b(W)m(e)17 b(no)o(w)f(pro)o(v)o(e)i(that)f Fl(')p Fo(\()p Fl(d)p Fo(\))g(is)g(deductiv)o(ely)g(closed.)28 b(Cho)q(ose)17 b(again)f Fl(e)1506 806 y Fi(1)1525 800 y Fl(;)7 b(e)1563 806 y Fi(2)1582 800 y Fl(;)g(:)g(:)g(:)t(;)g(e)1693 806 y Ff(n)1732 800 y Fm(\024)17 b Fl(d)g Fo(and)g(let)0 850 y Fl(e)12 b Fm(\024)75 819 y Fe(F)109 850 y Fm(f)p Fl(e)149 856 y Fi(1)168 850 y Fl(;)7 b(e)206 856 y Fi(2)224 850 y Fl(;)g(:)g(:)g(:)e(;)i(e)336 856 y Ff(n)359 850 y Fm(g)p Fo(.)17 b(W)m(e)d(ha)o(v)o(e)f(to)h(sho)o(w)g Fl(e)e Fm(2)f Fl(')p Fo(\()p Fl(d)p Fo(\),)j(i.e.)j Fl(e)12 b Fm(\024)g Fl(d)p Fo(.)17 b(But)e(this)f(follo)o(ws)e(from)g Fl(e)1521 856 y Fi(1)1540 850 y Fl(;)7 b(e)1578 856 y Fi(2)1596 850 y Fl(;)g(:)g(:)g(:)e(;)i(e)1708 856 y Ff(n)1742 850 y Fm(\024)12 b Fl(d)p Fo(.)83 901 y Fl( )k Fo(is)e(w)o (ell{de\014ned:)k(Let)d Fl(z)f Fm(2)e(j)p Fl(I)s(D)q Fm(j)p Fo(.)19 b(By)14 b(Theorem)g(2.1)f(it)h(su\016ces)h(to)f(sho)o(w) g(that)h Fl(z)h Fo(is)e(directed.)20 b(So)14 b(let)g Fl(X)i Fm(\022)1866 886 y Fi(\014n)1917 901 y Fl(z)r Fo(.)0 951 y(Then)f Fl(X)g Fm(\022)d Fk(B)236 957 y Ff(D)280 951 y Fo(and)i Fl(X)h Fm(2)c Fo(Con,)j(hence)h Fl(X)j Fo(is)c(b)q(ounded)g(in)g Fl(D)q Fo(,)g(hence)1152 919 y Fe(F)1194 951 y Fl(X)j Fo(exists.)i(No)o(w)14 b(b)o(y)g(Lemma)d(2.2) 1744 919 y Fe(F)1786 951 y Fl(X)k Fm(2)c Fk(B)1908 957 y Ff(D)1938 951 y Fo(.)0 1000 y(Therefore)188 969 y Fe(F)230 1000 y Fl(X)k Fm(2)c Fl(z)16 b Fo(b)o(y)e(the)g(deductiv)o(e)h(closure) g(of)e Fl(z)r Fo(.)83 1051 y Fl( )e Fm(\016)e Fl(')i Fo(=)h(id)268 1057 y Ff(D)298 1051 y Fo(:)18 b(Since)c Fl(D)i Fo(is)d(algebraic,)g(w)o(e)h(ha)o(v)o(e)g(that)g Fl(')p Fo(\()p Fl(d)p Fo(\))e(=)g Fm(f)p Fl(e)f Fm(2)g Fk(B)1222 1057 y Ff(D)1264 1051 y Fo(:)g Fl(e)h Fm(\024)g Fl(d)p Fm(g)h Fo(is)h(directed)h(and)f Fl(d)d Fo(=)1777 1020 y Fe(F)1818 1051 y Fl(')p Fo(\()p Fl(d)p Fo(\).)83 1102 y Fl(')h Fm(\016)f Fl( )19 b Fo(=)f(id)285 1109 y Fg(j)p Ff(I)r(D)q Fg(j)352 1102 y Fo(:)25 b(W)m(e)17 b(ha)o(v)o(e)g(to)h(sho)o(w)f(that)h Fl(')p Fo(\()861 1071 y Fe(F)902 1102 y Fl(z)r Fo(\))g(=)g Fl(z)i Fo(for)d(ev)o(ery)h Fl(z)i Fm(2)d(j)p Fl(I)s(D)q Fm(j)p Fo(.)29 b(T)m(o)17 b(pro)o(v)o(e)g Fm(\023)p Fo(,)i(let)e Fl(z)j Fm(\022)e Fk(B)1860 1108 y Ff(D)1907 1102 y Fo(b)q(e)0 1152 y(consisten)o(t)d (and)g(deductiv)o(ely)f(closed.)20 b(Recall)13 b(that)i Fl(')p Fo(\()894 1120 y Fe(F)936 1152 y Fl(z)r Fo(\))d(=)g Fm(f)p Fl(e)h Fm(2)f Fk(B)1156 1158 y Ff(D)1198 1152 y Fo(:)g Fl(e)g Fm(\024)1298 1120 y Fe(F)1339 1152 y Fl(z)r Fm(g)p Fo(.)19 b(No)o(w)14 b(let)g Fl(e)f Fm(2)f Fl(z)r Fo(.)19 b(Then)c Fl(e)d Fm(\024)1876 1120 y Fe(F)1917 1152 y Fl(z)r Fo(,)0 1201 y(hence)j Fl(e)d Fm(2)g Fl(')p Fo(\()229 1170 y Fe(F)271 1201 y Fl(z)r Fo(\).)18 b(T)m(o)13 b(pro)o(v)o(e)h Fm(\022)p Fo(,)g(let)g Fl(z)g Fm(2)d(j)p Fl(I)s(D)q Fm(j)j Fo(and)g Fl(e)e Fm(2)f Fl(')p Fo(\()989 1170 y Fe(F)1031 1201 y Fl(z)r Fo(\),)j(i.e.)k Fl(e)12 b Fm(\024)1240 1170 y Fe(F)1282 1201 y Fl(z)r Fo(.)18 b(W)m(e)c(m)o(ust)f(sho)o(w)h Fl(e)e Fm(2)f Fl(z)r Fo(.)18 b(No)o(w)c(since)h Fl(z)0 1251 y Fo(is)f(directed,)i(from)c Fl(e)h Fm(\024)389 1220 y Fe(F)430 1251 y Fl(z)k Fo(and)d Fl(e)f Fm(2)f Fk(B)653 1257 y Ff(D)698 1251 y Fo(w)o(e)i(can)h (conclude)g Fl(e)e Fm(\024)g Fl(e)1103 1236 y Fg(0)1129 1251 y Fo(for)h(some)g Fl(e)1317 1236 y Fg(0)1341 1251 y Fm(2)e Fl(z)r Fo(.)20 b(Since)15 b Fl(z)h Fo(is)f(deductiv)o(ely)f (closed)0 1301 y(and)g Fl(e)e Fm(\024)155 1270 y Fe(F)190 1301 y Fm(f)p Fl(e)230 1286 y Fg(0)242 1301 y Fm(g)p Fo(,)h(it)g(follo)o(ws)g(that)h Fl(e)d Fm(2)h Fl(z)r Fo(.)83 1352 y Fl(')p Fo(,)i Fl( )h Fo(clearly)f(are)g(monotone.)i (Hence)g Fl(')e Fo(and)g Fl( )h Fo(are)f(con)o(tin)o(uous)g(and)g (therefore)h(form)d(an)i(isomorphism)d(pair.)28 b Fh(\003)0 1456 y Fk(3.)21 b(Mappings)14 b(on)i(informati)o(on)d(systems)83 1571 y Fo(Let)f Fk(A)g Fo(=)g(\()p Fl(A;)7 b Fo(Con)386 1577 y Ff(A)413 1571 y Fl(;)g Fm(`)457 1577 y Ff(A)484 1571 y Fo(\))12 b(and)g Fk(B)f Fo(=)h(\()p Fl(B)r(;)7 b Fo(Con)822 1577 y Ff(B)851 1571 y Fl(;)g Fm(`)895 1577 y Ff(B)923 1571 y Fo(\))12 b(b)q(e)g(information)d(systems.)18 b(W)m(e)11 b(w)o(an)o(t)g(to)h(study)g(`information)0 1620 y(resp)q(ecting')j(mappings)d(from)h Fk(A)h Fo(in)o(to)f Fk(B)p Fo(.)18 b(Suc)o(h)d(a)f(mapping)d(is)j(giv)o(en)g(b)o(y)g(a)f (relation)h Fl(r)h Fo(b)q(et)o(w)o(een)g(Con)1663 1626 y Ff(A)1704 1620 y Fo(and)f Fl(B)r Fo(,)g(where)0 1670 y Fl(X)s(r)q(b)j Fo(in)o(tuitiv)o(ely)f(means)g(that)h(whenev)o(er)h(w) o(e)f(are)h(giv)o(en)e(the)i(information)c Fl(X)20 b Fm(2)c Fo(Con)1425 1676 y Ff(A)1469 1670 y Fo(on)g(the)i(argumen)o(t,)e (then)i(w)o(e)0 1720 y(kno)o(w)13 b(that)h(at)g(least)g(the)h(data)e (ob)r(ject)i Fl(b)f Fo(b)q(elongs)g(to)f(the)i(v)n(alue.)0 1797 y Fk(3.1)24 b(De\014niti)o(on.)d Fj(Let)g Fk(A)h Fo(=)h(\()p Fl(A;)7 b Fo(Con)663 1803 y Ff(A)690 1797 y Fl(;)g Fm(`)734 1803 y Ff(A)761 1797 y Fo(\))20 b Fj(and)g Fk(B)j Fo(=)g(\()p Fl(B)r(;)7 b Fo(Con)1138 1803 y Ff(B)1166 1797 y Fl(;)g Fm(`)1210 1803 y Ff(B)1239 1797 y Fo(\))20 b Fj(b)q(e)h(information)d(systems.)38 b(A)21 b(relation)0 1847 y Fl(r)12 b Fm(\022)g Fo(Con)149 1853 y Ff(A)185 1847 y Fm(\002)e Fl(B)16 b Fj(is)e(an)g Fn(appr)n(oximable)h(mapping)j Fj(if)13 b(it)g(satis\014es)18 1898 y(\(i\))21 b Fl(X)s(r)q(b)158 1904 y Ff(i)186 1898 y Fj(for)14 b(all)e Fl(i)g Fm(2)f(f)p Fo(1)p Fl(;)c Fo(2)p Fl(;)g(:)g(:)g(:)t(;)g(n)p Fm(g)13 b Fj(implies)f Fm(f)p Fl(b)785 1904 y Fi(1)803 1898 y Fl(;)7 b(b)840 1904 y Fi(2)858 1898 y Fl(;)g(:)g(:)g(:)e(;)i(b)969 1904 y Ff(n)990 1898 y Fm(g)12 b(2)f Fo(Con)1136 1904 y Ff(B)1164 1898 y Fj(;)7 1949 y(\(ii\))20 b Fl(X)s(r)q(b)158 1955 y Ff(i)186 1949 y Fj(for)14 b(all)e Fl(i)g Fm(2)f(f)p Fo(1)p Fl(;)c Fo(2)p Fl(;)g(:)g(:)g(:)t(;)g(n)p Fm(g)13 b Fj(and)g Fm(f)p Fl(b)724 1955 y Fi(1)743 1949 y Fl(;)7 b(b)780 1955 y Fi(2)798 1949 y Fl(;)g(:)g(:)g(:)t(;)g(b)908 1955 y Ff(n)930 1949 y Fm(g)k(`)987 1955 y Ff(B)1028 1949 y Fl(b)j Fj(implies)d Fl(X)s(r)q(b)p Fj(;)-5 1999 y(\(iii\))20 b Fl(X)15 b Fm(`)157 2005 y Ff(A)196 1999 y Fl(X)233 1984 y Fg(0)259 1999 y Fj(and)f Fl(X)377 1984 y Fg(0)389 1999 y Fl(r)q(b)g Fj(implies)e Fl(X)s(r)q(b)p Fj(.)0 2077 y Fo(W)m(e)h(write)i Fl(r)q Fo(:)7 b Fk(A)k Fm(!)g Fk(B)i Fo(to)h(mean)f(that)h Fl(r)h Fo(is)e(an)h(appro)o (ximable)d(mapping)h(from)g Fk(A)i Fo(to)g Fk(B)p Fo(.)0 2154 y Fk(3.2)j(Theorem.)23 b Fj(Let)15 b Fk(A)e Fo(=)h(\()p Fl(A;)7 b Fo(Con)611 2160 y Ff(A)638 2154 y Fl(;)g Fm(`)682 2160 y Ff(A)709 2154 y Fo(\))15 b Fj(and)g Fk(B)e Fo(=)g(\()p Fl(B)r(;)7 b Fo(Con)1056 2160 y Ff(B)1085 2154 y Fl(;)g Fm(`)1129 2160 y Ff(B)1157 2154 y Fo(\))15 b Fj(b)q(e)h(information)c (systems.)21 b(With)14 b(an)o(y)g(appro-)0 2204 y(ximable)e(mapping)f Fl(s)p Fo(:)c Fk(A)12 b Fm(!)f Fk(B)j Fj(w)o(e)g(can)g(asso)q(ciate)g (a)g(con)o(tin)o(uous)g(function)f Fm(j)p Fl(s)p Fm(j)p Fo(:)7 b Fm(j)p Fk(A)p Fm(j)j(!)h(j)p Fk(B)p Fm(j)i Fj(b)o(y)594 2306 y Fm(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\))e(:=)h Fm(f)p Fl(b)f Fm(2)g Fl(B)j Fo(:)d Fl(X)s(sb)k Fj(for)e(some)g Fl(X)j Fm(\022)1252 2289 y Fi(\014n)1303 2306 y Fl(z)r Fm(g)p Fl(:)0 2409 y Fj(Con)o(v)o(ersely)m(,)10 b(with)h(an)o(y)e(con)o (tin)o(uous)i(function)f Fl(f)t Fo(:)d Fm(j)p Fk(A)p Fm(j)k(!)g(j)p Fk(B)p Fm(j)e Fj(w)o(e)i(can)f(asso)q(ciate)h(an)g (appro)o(ximable)d(mapping)g Fl(r)1776 2415 y Ff(f)1797 2409 y Fo(:)f Fk(A)k Fm(!)g Fk(B)0 2459 y Fj(b)o(y)773 2511 y Fl(X)s(r)829 2517 y Ff(f)851 2511 y Fl(b)g Fo(:)g Fm(\()-7 b(\))22 b Fl(b)12 b Fm(2)f Fl(f)t Fo(\()p 1111 2478 38 2 v Fl(X)t Fo(\))p Fl(:)0 2593 y Fj(These)k(assignmen)o(ts)e (are)i(in)o(v)o(erse)f(to)g(eac)o(h)g(other,)g(i.e.)k Fl(f)e Fo(=)c Fm(j)p Fl(r)1004 2599 y Ff(f)1025 2593 y Fm(j)h Fj(and)h Fl(s)e Fo(=)g Fl(r)1225 2600 y Fg(j)p Ff(s)p Fg(j)1262 2593 y Fj(.)0 2670 y Fk(Pro)q(of.)18 b Fo(W)m(e)13 b(\014rst)i(sho)o(w)f(that)g Fm(j)p Fl(s)p Fm(j)f Fo(is)h(w)o(ell{de\014ned.)k(So)c(let)g Fl(z)f Fm(2)e(j)p Fk(A)p Fm(j)p Fo(.)965 2770 y(4)p eop %%Page: 5 5 5 4 bop 83 42 a Fm(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\))16 b(is)g(consisten)o(t.)26 b(Let)17 b Fl(b)550 48 y Fi(1)568 42 y Fl(;)7 b(b)605 48 y Fi(2)623 42 y Fl(;)g(:)g(:)g(:)e(;)i(b)734 48 y Ff(n)771 42 y Fm(2)15 b(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\).)25 b(Then)17 b(there)g(are)g Fl(X)1272 48 y Fi(1)1291 42 y Fl(;)7 b(X)1344 48 y Fi(2)1362 42 y Fl(;)g(:)g(:)g(:)e(;)i(X)1489 48 y Ff(n)1527 42 y Fm(\022)1559 26 y Fi(\014n)1614 42 y Fl(z)18 b Fo(suc)o(h)f(that)f Fl(X)1873 48 y Ff(i)1887 42 y Fl(sb)1924 48 y Ff(i)1938 42 y Fo(.)0 91 y(Hence)h Fl(X)h Fo(:=)c Fl(X)269 97 y Fi(1)298 91 y Fm([)c Fl(X)370 97 y Fi(2)399 91 y Fm([)g(\001)d(\001)g (\001)i([)h Fl(X)568 97 y Ff(n)605 91 y Fm(\022)637 76 y Fi(\014n)691 91 y Fl(z)17 b Fo(and)e Fl(X)s(sb)883 97 y Ff(i)914 91 y Fo(b)o(y)g(De\014nition)g(3.1\(iii\).)20 b(No)o(w)15 b(from)f(De\014nition)h(3.1\(i\))f(w)o(e)i(can)0 141 y(conclude)f(that)f Fm(f)p Fl(b)300 147 y Fi(1)318 141 y Fl(;)7 b(b)355 147 y Fi(2)373 141 y Fl(;)g(:)g(:)g(:)t(;)g(b)483 147 y Ff(n)505 141 y Fm(g)12 b(2)f Fo(Con)651 147 y Ff(B)679 141 y Fo(.)83 191 y Fm(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\))g(is)g(deductiv)o(ely)h(closed.)17 b(Let)12 b Fl(b)674 197 y Fi(1)692 191 y Fl(;)7 b(b)729 197 y Fi(2)747 191 y Fl(;)g(:)g(:)g(:)e(;)i(b)858 197 y Ff(n)891 191 y Fm(2)12 b(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\))f(and)g Fm(f)p Fl(b)1155 197 y Fi(1)1173 191 y Fl(;)c(b)1210 197 y Fi(2)1228 191 y Fl(;)g(:)g(:)g(:)t(;)g(b)1338 197 y Ff(n)1360 191 y Fm(g)12 b(`)1418 197 y Ff(B)1458 191 y Fl(b)p Fo(.)17 b(W)m(e)11 b(m)o(ust)f(sho)o(w)h Fl(b)g Fm(2)h(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\).)0 241 y(As)17 b(b)q(efore)h(w)o(e)f (\014nd)g Fl(X)i Fm(\022)427 226 y Fi(\014n)483 241 y Fl(z)g Fo(suc)o(h)e(that)g Fl(X)s(sb)784 247 y Ff(i)799 241 y Fo(.)27 b(No)o(w)16 b(from)f(De\014nition)h(3.1\(ii\))f(w)o(e)i (can)g(conclude)g Fl(X)s(sb)h Fo(and)f(hence)0 291 y Fl(b)11 b Fm(2)h(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\).)83 340 y(The)j(monotonicit)o(y)c(of)j Fm(j)p Fl(s)p Fm(j)g Fo(clearly)g(follo)o(ws)e(from)h(the)i(de\014nition.)j(T)m(o)c(see)h (that)f Fm(j)p Fl(s)p Fm(j)p Fo(:)7 b Fm(j)p Fk(A)p Fm(j)j(!)i(j)p Fk(B)p Fm(j)h Fo(is)h(con)o(tin)o(uous,)g(let)0 390 y Fl(M)i Fm(\022)c(j)p Fk(A)p Fm(j)h Fo(b)q(e)i(directed.)k(Then)14 b(w)o(e)h(ha)o(v)o(e)544 466 y Fm(j)p Fl(s)p Fm(j)p Fo(\()603 426 y Fe([)656 466 y Fl(M)5 b Fo(\))11 b(=)h Fm(f)p Fl(b)f Fm(2)g Fl(B)j Fo(:)e(\()p Fm(9)p Fl(X)j Fm(\022)1050 449 y Fi(\014n)1101 426 y Fe([)1154 466 y Fl(M)5 b Fo(\))p Fl(X)s(sb)p Fm(g)728 543 y Fo(=)12 b Fm(f)p Fl(b)f Fm(2)g Fl(B)j Fo(:)e(\()p Fm(9)p Fl(z)h Fm(2)f Fl(M)5 b Fo(\)\()p Fm(9)p Fl(X)15 b Fm(\022)1222 525 y Fi(\014n)1273 543 y Fl(z)r Fo(\))p Fl(X)s(sb)p Fm(g)198 b Fo(since)14 b Fl(M)19 b Fo(is)14 b(directed)728 613 y(=)786 574 y Fe([)772 663 y Ff(z)q Fg(2)p Ff(M)846 613 y Fm(f)p Fl(b)d Fm(2)h Fl(B)i Fo(:)d(\()p Fm(9)p Fl(X)k Fm(\022)1124 596 y Fi(\014n)1175 613 y Fl(z)r Fo(\))p Fl(X)s(sb)p Fm(g)728 729 y Fo(=)786 690 y Fe([)772 779 y Ff(z)q Fg(2)p Ff(M)853 729 y Fm(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\))p Fl(:)83 838 y Fo(No)o(w)h(let)g Fl(f)t Fo(:)7 b Fm(j)p Fk(A)p Fm(j)15 b(!)g(j)p Fk(B)p Fm(j)g Fo(b)q(e)h(con)o(tin)o(uous.)25 b(It)16 b(is)g(easy)h(to)f(v)o (erify)g(that)g Fl(r)1246 844 y Ff(f)1283 838 y Fo(is)g(indeed)h(an)f (appro)o(ximable)e(mapping.)0 888 y(F)m(urthermore)633 955 y Fl(f)t Fo(\()p Fl(z)r Fo(\))f(=)f Fl(f)t Fo(\()838 916 y Fe([)807 1007 y Ff(X)r Fg(\022)862 999 y Fd(fin)898 1007 y Ff(z)p 922 922 38 2 v 922 955 a Fl(X)t Fo(\))596 b(since)15 b Fm(j)p Fk(A)p Fm(j)e Fo(is)g(algebraic)723 1077 y(=)797 1038 y Fe([)767 1130 y Ff(X)r Fg(\022)822 1121 y Fd(fin)857 1130 y Ff(z)881 1077 y Fl(f)t Fo(\()p 921 1044 V Fl(X)5 b Fo(\))610 b(since)15 b(f)e(is)h(con)o(tin)o(uous) 723 1193 y(=)e Fm(f)p Fl(b)f Fm(2)g Fl(B)j Fo(:)d(\()p Fm(9)p Fl(X)16 b Fm(\022)1045 1176 y Fi(\014n)1096 1193 y Fl(z)r Fo(\))p Fl(b)11 b Fm(2)g Fl(f)t Fo(\()p 1241 1160 V Fl(X)5 b Fo(\))p Fm(g)723 1255 y Fo(=)12 b Fm(f)p Fl(b)f Fm(2)g Fl(B)j Fo(:)d(\()p Fm(9)p Fl(X)16 b Fm(\022)1045 1238 y Fi(\014n)1096 1255 y Fl(z)r Fo(\))p Fl(X)s(r)1189 1261 y Ff(f)1211 1255 y Fl(b)p Fm(g)723 1318 y Fo(=)c Fm(j)p Fl(r)798 1324 y Ff(f)818 1318 y Fm(j)p Fo(\()p Fl(z)r Fo(\))p Fl(:)0 1385 y Fo(Finally)m(,)f(for)j(an)o(y)f(appro)o (ximable)f(mapping)f Fl(s)p Fo(:)c Fk(A)12 b Fm(!)f Fk(B)j Fo(w)o(e)g(ha)o(v)o(e)731 1452 y Fl(X)s(sb)24 b Fm(\()-7 b(\))22 b Fo(\()p Fm(9)p Fl(Y)f Fm(\022)1044 1435 y Fi(\014n)p 1095 1419 V 1095 1452 a Fl(X)t Fo(\))p Fl(Y)9 b(sb)553 b Fo(b)o(y)14 b(3)p Fl(:)p Fo(1\(iii)n(\))829 1514 y Fm(\()-7 b(\))22 b Fl(b)11 b Fm(2)g(j)p Fl(s)p Fm(j)p Fo(\()p 1055 1481 V Fl(X)t Fo(\))829 1577 y Fm(\()-7 b(\))22 b Fl(X)s(r)984 1584 y Fg(j)p Ff(s)p Fg(j)1022 1577 y Fl(b:)866 b Fh(\003)0 1679 y Fk(4.)21 b(Pro)q(ducts)14 b(and)i(function)d(spaces)83 1788 y Fo(W)m(e)h(discuss)h(some)e (canonical)g(w)o(a)o(ys)g(to)h(construct)i(new)e(cp)q(o's)g(from)e(giv) o(en)i(ones:)42 1838 y Fm(\017)20 b Fo(the)14 b(cartesian)g(pro)q(duct) h Fl(D)10 b Fm(\002)f Fl(E)r Fo(,)k(consisting)g(of)g(all)f(pairs)i(\() p Fl(d;)7 b(e)p Fo(\))13 b(with)g Fl(d)e Fm(2)h Fl(D)j Fo(and)e Fl(e)f Fm(2)f Fl(E)k Fo(under)g(the)f(comp)q(onen)o(t{)83 1887 y(wise)g(order,)g(and)42 1937 y Fm(\017)20 b Fo(the)f(function)e (space)i Fl(D)h Fm(!)e Fl(E)r Fo(,)g(consisting)g(of)g(all)f(con)o(tin) o(uous)g(functions)h(from)f Fl(D)i Fo(to)f Fl(E)i Fo(under)f(the)f(p)q (oin)o(t)o(wise)83 1987 y(order.)0 2037 y(W)m(e)13 b(in)o(tro)q(duce)i (these)g(t)o(w)o(o)f(constructs,)h(and)f(de\014ne)h(op)q(erations)f(on) g(information)d(systems)j(corresp)q(onding)h(to)e(them.)83 2087 y(The)e Fn(c)n(artesian)h(pr)n(o)n(duct)i Fo(of)c(t)o(w)o(o)g(cp)q (o's)h(\()p Fl(D)q(;)c Fm(\024)809 2093 y Ff(D)840 2087 y Fo(\))k(and)f(\()p Fl(E)r(;)d Fm(\024)1044 2093 y Ff(E)1072 2087 y Fo(\))k(is)f(de\014ned)i(to)e(b)q(e)i(\()p Fl(D)t Fm(\002)s Fl(E)r(;)7 b Fm(\024)p Fo(\))j(with)h(the)g Fn(c)n(omp)n(onent{)0 2136 y(wise)g(or)n(der)k Fm(\024)p Fo(,)c(i.e.)16 b(\()p Fl(d;)7 b(e)p Fo(\))k Fm(\024)h Fo(\()p Fl(d)508 2121 y Fg(0)519 2136 y Fl(;)7 b(e)557 2121 y Fg(0)569 2136 y Fo(\))23 b Fm(\()-7 b(\))22 b Fl(d)11 b Fm(\024)772 2142 y Ff(D)814 2136 y Fl(d)836 2121 y Fg(0)858 2136 y Fo(and)g Fl(e)g Fm(\024)998 2142 y Ff(E)1038 2136 y Fl(e)1057 2121 y Fg(0)1069 2136 y Fo(.)17 b(W)m(e)10 b(w)o(an)o(t)h(to)f(sho)o(w)h(that)f(\()p Fl(D)t Fm(\002)s Fl(E)r(;)d Fm(\024)p Fo(\))k(is)f(a)h(cp)q(o)g(again.) 0 2186 y(F)m(or)g(the)g(pro)q(of)g(w)o(e)g(need)h(to)e(kno)o(w)h(that)g (the)g(suprem)o(um)f(of)g(a)h(directed)h(subset)g Fl(M)17 b Fm(\022)11 b Fl(D)t Fm(\002)s Fl(E)k Fo(op)q(erates)d(comp)q(onen)o (t{wise.)0 2236 y(Denoting)h(the)i(left)e(and)h(righ)o(t)g(pro)r (jection)g(functions)g(b)o(y)f Fl(\031)962 2242 y Fi(left)1025 2236 y Fo(and)h Fl(\031)1130 2242 y Fi(righ)o(t)1215 2236 y Fo(w)o(e)g(ha)o(v)o(e)0 2304 y Fk(4.1)j(Lemma.)23 b Fj(Supp)q(ose)16 b Fl(D)g Fj(and)e Fl(E)j Fj(are)e(cp)q(o's.)21 b(Let)15 b Fl(M)i Fm(\022)d Fl(D)d Fm(\002)f Fl(E)16 b Fj(b)q(e)g(directed.)21 b(Then)15 b Fl(\031)1485 2310 y Fi(left)1535 2304 y Fo(\()p Fl(M)5 b Fo(\))14 b Fj(and)h Fl(\031)1732 2310 y Fi(righ)o(t)1803 2304 y Fo(\()p Fl(M)5 b Fo(\))15 b Fj(are)0 2353 y(directed)g(to)q(o,)e(and)324 2322 y Fe(F)366 2353 y Fl(M)j Fo(=)c(\()482 2322 y Fe(F)523 2353 y Fl(\031)547 2359 y Fi(left)597 2353 y Fo(\()p Fl(M)5 b Fo(\))p Fl(;)693 2322 y Fe(F)734 2353 y Fl(\031)758 2359 y Fi(righ)o(t)829 2353 y Fo(\()p Fl(M)g Fo(\)\))p Fj(.)0 2421 y Fk(Pro)q(of.)31 b Fo(W)m(e)18 b(\014rst)h(sho)o(w)f(that) h(e.g.)31 b Fl(\031)647 2427 y Fi(left)696 2421 y Fo(\()p Fl(M)5 b Fo(\))18 b(is)h(directed.)32 b(So)18 b(let)h Fl(d;)7 b(d)1216 2406 y Fg(0)1245 2421 y Fm(2)19 b Fl(\031)1316 2427 y Fi(left)1365 2421 y Fo(\()p Fl(M)5 b Fo(\).)31 b(Then)19 b(there)h(are)f Fl(e;)7 b(e)1840 2406 y Fg(0)1870 2421 y Fm(2)19 b Fl(E)0 2471 y Fo(suc)o(h)d(that)g(\()p Fl(d;)7 b(e)p Fo(\))p Fl(;)g Fo(\()p Fl(d)336 2456 y Fg(0)347 2471 y Fl(;)g(e)385 2456 y Fg(0)396 2471 y Fo(\))15 b Fm(2)f Fl(M)5 b Fo(.)23 b(Since)16 b Fl(M)k Fo(is)c(directed,)h(w)o (e)f(ha)o(v)o(e)f(\()p Fl(d;)7 b(e)p Fo(\))p Fl(;)g Fo(\()p Fl(d)1245 2456 y Fg(0)1256 2471 y Fl(;)g(e)1294 2456 y Fg(0)1305 2471 y Fo(\))15 b Fm(\024)f Fo(\()p Fl(d)1420 2456 y Fg(00)1441 2471 y Fl(;)7 b(e)1479 2456 y Fg(00)1500 2471 y Fo(\))16 b(for)f(some)g(\()p Fl(d)1741 2456 y Fg(00)1762 2471 y Fl(;)7 b(e)1800 2456 y Fg(00)1821 2471 y Fo(\))15 b Fm(2)f Fl(M)5 b Fo(.)0 2521 y(Therefore)15 b Fl(d;)7 b(d)251 2505 y Fg(0)273 2521 y Fm(\024)305 2527 y Ff(D)347 2521 y Fl(d)369 2505 y Fg(00)401 2521 y Fm(2)12 b Fl(\031)465 2527 y Fi(left)514 2521 y Fo(\()p Fl(M)5 b Fo(\).)83 2570 y(Hence)19 b(\()226 2539 y Fe(F)268 2570 y Fl(\031)292 2576 y Fi(left)341 2570 y Fo(\()p Fl(M)5 b Fo(\))p Fl(;)437 2539 y Fe(F)478 2570 y Fl(\031)502 2576 y Fi(righ)o(t)573 2570 y Fo(\()p Fl(M)g Fo(\)\))18 b(exists.)30 b(Clearly)17 b(this)h(pair)f(is)h(an)f(upp)q(er)i(b)q (ound)f(of)f Fl(M)5 b Fo(.)29 b(It)18 b(is)f(also)g(a)h(least)0 2620 y(upp)q(er)c(b)q(ound.)j(T)m(o)12 b(see)i(this)f(supp)q(ose)h (that)f(\()p Fl(d)749 2605 y Fg(0)760 2620 y Fl(;)7 b(e)798 2605 y Fg(0)810 2620 y Fo(\))13 b(is)f(another)h(upp)q(er)h(b)q(ound,)f (i.e.)k(\()p Fl(d;)7 b(e)p Fo(\))k Fm(\024)h Fo(\()p Fl(d)1543 2605 y Fg(0)1554 2620 y Fl(;)7 b(e)1592 2605 y Fg(0)1604 2620 y Fo(\))12 b(for)h(all)e(\()p Fl(d;)c(e)p Fo(\))k Fm(2)h Fl(M)5 b Fo(.)0 2670 y(But)14 b(then)h Fl(d)c Fm(\024)242 2676 y Ff(D)284 2670 y Fl(d)306 2655 y Fg(0)331 2670 y Fo(for)j(all)e Fl(d)f Fm(2)h Fl(\031)549 2676 y Fi(left)598 2670 y Fo(\()p Fl(M)5 b Fo(\),)13 b(i.e.)772 2639 y Fe(F)813 2670 y Fl(\031)837 2676 y Fi(left)886 2670 y Fo(\()p Fl(M)5 b Fo(\))12 b Fm(\024)1007 2676 y Ff(D)1049 2670 y Fl(d)1071 2655 y Fg(0)1082 2670 y Fo(.)824 b Fh(\003)965 2770 y Fo(5)p eop %%Page: 6 6 6 5 bop 0 42 a Fk(4.2)15 b(Corollary)l(.)23 b Fj(Supp)q(ose)14 b Fl(D)h Fj(and)e Fl(E)i Fj(are)e(cp)q(o's.)19 b(Then)13 b(the)h(set)g Fl(D)c Fm(\002)e Fl(E)15 b Fj(equipp)q(ed)f(with)f(the)h (comp)q(onen)o(t{wise)e(order)0 91 y(is)i(a)f(cp)q(o)i(again.)1654 b Fh(\003)83 175 y Fo(Note)14 b(that)g(the)h(con)o(tin)o(uit)o(y)e(of)g Fl(\031)612 181 y Fi(left)675 175 y Fo(and)h Fl(\031)780 181 y Fi(righ)o(t)865 175 y Fo(also)f(follo)o(ws)f(from)g(Lemma)g(4.1.) 83 228 y(W)m(e)i(no)o(w)f(de\014ne)i(a)f(corresp)q(onding)h(op)q (eration)f(on)f(information)e(systems,)j(assigning)f(to)h(an)o(y)g Fk(A)d Fo(=)h(\()p Fl(A;)7 b Fo(Con)1836 234 y Ff(A)1863 228 y Fl(;)g Fm(`)1907 234 y Ff(A)1934 228 y Fo(\))0 278 y(and)14 b Fk(B)e Fo(=)h(\()p Fl(B)r(;)7 b Fo(Con)314 284 y Ff(B)342 278 y Fl(;)g Fm(`)386 284 y Ff(B)415 278 y Fo(\))14 b(an)g Fk(A)c Fm(\002)f Fk(B)k Fo(=)f(\()p Fl(C)q(;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))14 b(suc)o(h)h(that)f Fm(j)p Fk(A)9 b Fm(\002)h Fk(B)p Fm(j)1235 267 y(\030)1235 280 y Fo(=)1279 278 y Fm(j)p Fk(A)p Fm(j)f(\002)h(j)p Fk(B)p Fm(j)p Fo(.)18 b(Without)13 b(loss)i(of)e(generalit)o(y)0 328 y(w)o(e)19 b(ma)o(y)e(assume)i(that)f Fl(A)h Fo(and)g Fl(B)j Fo(are)d(disjoin)o(t.)32 b(But)19 b(then)h(an)o(y)e(pair)h(\()p Fl(x;)7 b(y)q Fo(\))19 b(of)f(elemen)o(ts)h Fl(x)g Fm(2)h(j)p Fk(A)p Fm(j)e Fo(and)h Fl(y)i Fm(2)f(j)p Fk(B)p Fm(j)0 378 y Fo(can)f(b)q(e)g(appro)o(ximated)d(in)i(eac)o(h)h(comp)q(onen)o (t)f(separately)m(.)32 b(Hence)20 b(w)o(e)f(c)o(ho)q(ose)g(as)g(data)f (ob)r(jects)i(in)e Fk(A)12 b Fm(\002)h Fk(B)18 b Fo(simply)0 427 y(the)f(union)g Fl(C)i Fo(:=)d Fl(A)11 b Fm([)g Fl(B)r Fo(.)28 b(Consistency)17 b(and)g(en)o(tailmen)o(t)e(then)j(is)f (inherited)g(in)f(the)i(ob)o(vious)e(w)o(a)o(y)g(from)f Fk(A)i Fo(and)g Fk(B)p Fo(:)0 477 y Fl(X)h Fm(2)c Fo(Con)28 b Fm(\()-7 b(\))27 b Fl(X)14 b Fm(\\)c Fl(A)k Fm(2)g Fo(Con)547 483 y Ff(A)589 477 y Fo(and)i Fl(X)e Fm(\\)c Fl(B)16 b Fm(2)e Fo(Con)921 483 y Ff(A)948 477 y Fo(,)h(and)h Fl(X)h Fm(`)e Fl(c)28 b Fm(\()-7 b(\))28 b Fo(\()p Fl(c)14 b Fm(2)g Fl(A)28 b Fo(=)-7 b Fm(\))28 b Fl(X)14 b Fm(\\)c Fl(A)15 b Fm(`)1701 483 y Ff(A)1742 477 y Fl(c)p Fo(\))h(and)f(\()p Fl(c)f Fm(2)0 527 y Fl(B)30 b Fo(=)-7 b Fm(\))27 b Fl(X)13 b Fm(\\)d Fl(B)16 b Fm(`)312 533 y Ff(B)355 527 y Fl(c)p Fo(\).)22 b(The)15 b(elemen)o(ts)g(of)g Fk(A)10 b Fm(\002)g Fk(B)15 b Fo(then)h(are)f(exactly)h(the)f(unions)g(of)g(the)g(elemen)o (ts)g(of)g Fk(A)g Fo(and)g(of)g Fk(B)p Fo(:)0 577 y Fm(j)p Fk(A)9 b Fm(\002)g Fk(B)p Fm(j)i Fo(=)h Fm(f)p Fl(x)d Fm([)g Fl(y)k Fo(:)e Fl(x)g Fm(2)h(j)p Fk(A)p Fm(j)h Fo(and)g Fl(y)h Fm(2)d(j)p Fk(B)p Fm(jg)p Fo(.)0 661 y Fk(4.3)16 b(Lemma.)24 b Fj(If)13 b Fk(A)f Fo(=)g(\()p Fl(A;)7 b Fo(Con)541 667 y Ff(A)568 661 y Fl(;)g Fm(`)612 667 y Ff(A)639 661 y Fo(\))14 b Fj(and)g Fk(B)d Fo(=)h(\()p Fl(B)r(;)7 b Fo(Con)981 667 y Ff(B)1010 661 y Fl(;)g Fm(`)1054 667 y Ff(B)1082 661 y Fo(\))15 b Fj(are)f(information)d (systems)j(with)g Fl(A)c Fm(\\)f Fl(B)14 b Fo(=)e Fm(;)p Fj(,)h(then)0 711 y Fk(A)c Fm(\002)h Fk(B)k Fj(de\014ned)h(as)f(ab)q(o) o(v)o(e)f(is)h(an)g(information)d(system,)i(and)496 819 y Fm(j)p Fk(A)c Fm(\002)g Fk(B)p Fm(j)i Fo(=)h Fm(f)p Fl(x)d Fm([)g Fl(y)k Fo(:)e Fl(x)g Fm(2)g(j)p Fk(A)p Fm(j)i Fj(and)h Fl(y)g Fm(2)d(j)p Fk(B)p Fm(jg)1232 808 y(\030)1232 822 y Fo(=)1276 819 y Fm(j)p Fk(A)p Fm(j)d(\002)i(j)p Fk(B)p Fm(j)p Fl(:)462 b Fh(\003)0 931 y Fk(4.4)18 b(Corollary)l(.)k Fj(Supp)q(ose)16 b Fl(D)h Fj(and)e Fl(E)j Fj(are)e(domains)d(with)i (coun)o(table)g(bases.)24 b(Then)16 b(the)f(set)i Fl(D)11 b Fm(\002)g Fl(E)17 b Fj(equipp)q(ed)f(with)0 981 y(the)e(comp)q(onen)o (t{wise)g(order)g(is)g(again)f(a)g(domain)f(with)i(coun)o(table)g (basis.)0 1065 y Fk(Pro)q(of.)k Fl(D)10 b Fm(\002)g Fl(E)278 1054 y Fm(\030)278 1067 y Fo(=)322 1065 y Fm(j)p Fl(I)s(D)q Fm(j)g(\002)f(j)p Fl(I)s(E)r Fm(j)542 1054 y(\030)542 1067 y Fo(=)586 1065 y Fm(j)p Fl(I)s(D)i Fm(\002)e Fl(I)s(E)r Fm(j)p Fo(.)1135 b Fh(\003)83 1118 y Fo(The)16 b Fn(function)i(sp)n(ac) n(e)h Fo(of)c(t)o(w)o(o)h(cp)q(o's)g(\()p Fl(D)q(;)7 b Fm(\024)787 1124 y Ff(D)818 1118 y Fo(\))16 b(and)g(\()p Fl(E)r(;)7 b Fm(\024)1033 1124 y Ff(E)1061 1118 y Fo(\))16 b(is)f(de\014ned)j(to)d(b)q(e)i(\()p Fl(D)g Fm(!)d Fl(E)r(;)7 b Fm(\024)p Fo(\),)16 b(where)h Fl(D)g Fm(!)d Fl(E)k Fo(is)0 1168 y(the)g(set)h(of)e(all)g(con)o(tin)o(uous)h(functions)g (from)e Fl(D)j Fo(to)e Fl(E)r Fo(,)i(and)e(the)i(order)f Fm(\024)g Fo(is)g(de\014ned)h(p)q(oin)o(t)o(wise,)f(i.e.)29 b Fl(f)23 b Fm(\024)18 b Fl(g)38 b Fm(\()-7 b(\))0 1218 y(8)p Fl(d)16 b(f)t Fo(\()p Fl(d)p Fo(\))f Fm(\024)g Fl(g)q Fo(\()p Fl(d)p Fo(\).)25 b(W)m(e)15 b(w)o(an)o(t)h(to)f(sho)o(w) h(that)g(\()p Fl(D)h Fm(!)d Fl(E)r(;)7 b Fm(\024)p Fo(\))16 b(is)g(a)f(cp)q(o)i(again.)22 b(This)16 b(will)f(follo)o(w)f(from)g (the)i(observ)n(ation)0 1267 y(that)f(the)h(suprem)o(um)e(of)h(a)g (directed)h(subset)h(of)e Fl(D)g Fm(!)f Fl(E)j Fo(is)e(formed)g(p)q (oin)o(t)o(wise.)22 b(T)m(o)14 b(pro)o(v)o(e)i(this)f(w)o(e)h(will)e (mak)o(e)f(use)k(of)0 1317 y(the)c(fact)g(that)g(the)g(order)g(of)f (forming)e(least)j(upp)q(er)h(b)q(ounds)f(is)f(irrelev)n(an)o(t)h(pro)o (vided)f(all)g(relev)n(an)o(t)g(suprema)h(exist.)k(More)0 1367 y(precisely)e(w)o(e)f(ha)o(v)o(e)0 1451 y Fk(4.5)i(Lemma.)23 b Fj(Supp)q(ose)15 b Fl(D)h Fj(is)d(a)h(cp)q(o)g(and)g Fl(d)736 1457 y Ff(ij)776 1451 y Fm(2)d Fl(D)16 b Fj(for)d(all)g Fl(i)f Fm(2)f Fl(I)17 b Fj(and)d Fl(j)g Fm(2)d Fl(J)t Fj(.)18 b(Assume)c(that)18 1519 y(\(i\))83 1488 y Fe(F)118 1532 y Ff(i)p Fg(2)p Ff(I)178 1519 y Fl(d)200 1525 y Ff(ij)242 1519 y Fj(exists)h(for)f(an)o(y)f Fl(j)h Fm(2)d Fl(J)t Fj(,)7 1572 y(\(ii\))83 1541 y Fe(F)118 1585 y Ff(j)r Fg(2)p Ff(J)186 1572 y Fl(d)208 1578 y Ff(ij)250 1572 y Fj(exists)k(for)e(an)o(y)h Fl(i)e Fm(2)f Fl(I)s Fj(,)j(and)-5 1625 y(\(iii\))83 1594 y Fe(F)118 1638 y Ff(i)p Fg(2)p Ff(I)178 1594 y Fe(F)212 1638 y Ff(j)r Fg(2)p Ff(J)280 1625 y Fl(d)302 1631 y Ff(ij)345 1625 y Fj(exists.)0 1694 y(Then)g(also)192 1662 y Fe(F)226 1706 y Ff(j)r Fg(2)p Ff(J)294 1662 y Fe(F)329 1706 y Ff(i)p Fg(2)p Ff(I)389 1694 y Fl(d)411 1700 y Ff(ij)453 1694 y Fj(exists,)h(and)e(w)o(e)h(ha)o(v)o(e)817 1662 y Fe(F)852 1706 y Ff(i)p Fg(2)p Ff(I)912 1662 y Fe(F)946 1706 y Ff(j)r Fg(2)p Ff(J)1014 1694 y Fl(d)1036 1700 y Ff(ij)1077 1694 y Fo(=)1120 1662 y Fe(F)1155 1706 y Ff(j)r Fg(2)p Ff(J)1223 1662 y Fe(F)1258 1706 y Ff(i)p Fg(2)p Ff(I)1318 1694 y Fl(d)1340 1700 y Ff(ij)1368 1694 y Fj(.)0 1778 y Fk(Pro)q(of.)23 b Fo(W)m(e)15 b(\014rst)i(pro)o(v)o(e)e (that)519 1746 y Fe(F)554 1790 y Ff(i)p Fg(2)p Ff(I)614 1746 y Fe(F)648 1790 y Ff(j)r Fg(2)p Ff(J)716 1778 y Fl(d)738 1784 y Ff(ij)783 1778 y Fo(is)g(an)h(upp)q(er)g(b)q(ound)g(of) f(all)1245 1746 y Fe(F)1279 1790 y Ff(i)p Fg(2)p Ff(I)1339 1778 y Fl(d)1361 1784 y Ff(ij)1390 1778 y Fo(.)23 b(But)17 b(this)e(is)h(clear,)g(since)g Fl(d)1874 1784 y Ff(ij)1918 1778 y Fm(\024)0 1796 y Fe(F)35 1840 y Ff(j)r Fg(2)p Ff(J)102 1827 y Fl(d)124 1833 y Ff(ij)167 1827 y Fo(for)e(all)e Fl(i)g Fm(2)f Fl(I)18 b Fo(and)13 b(hence)585 1796 y Fe(F)619 1840 y Ff(i)p Fg(2)p Ff(I)679 1827 y Fl(d)701 1833 y Ff(ij)742 1827 y Fm(\024)786 1796 y Fe(F)820 1840 y Ff(i)p Fg(2)p Ff(I)880 1796 y Fe(F)915 1840 y Ff(j)r Fg(2)p Ff(J)983 1827 y Fl(d)1005 1833 y Ff(ij)1034 1827 y Fo(.)83 1880 y(It)k(remains)f(to)h(pro)o(v)o(e)g(that)551 1849 y Fe(F)585 1893 y Ff(i)p Fg(2)p Ff(I)645 1849 y Fe(F)680 1893 y Ff(j)r Fg(2)p Ff(J)748 1880 y Fl(d)770 1886 y Ff(ij)816 1880 y Fo(is)f(the)i(least)f(upp)q(er)h(b)q(ound)f(of) f(all)1399 1849 y Fe(F)1433 1893 y Ff(i)p Fg(2)p Ff(I)1494 1880 y Fl(d)1516 1886 y Ff(ij)1544 1880 y Fo(.)27 b(So)17 b(assume)g(that)g Fl(d)f Fo(is)0 1930 y(some)f(upp)q(er)i(b)q(ound.)25 b(W)m(e)16 b(m)o(ust)f(sho)o(w)662 1899 y Fe(F)697 1943 y Ff(i)p Fg(2)p Ff(I)757 1899 y Fe(F)791 1943 y Ff(j)r Fg(2)p Ff(J)859 1930 y Fl(d)881 1936 y Ff(ij)925 1930 y Fm(\024)h Fl(d)p Fo(.)24 b(But)17 b(this)f(is)g(clear,)g(since)h Fl(d)1482 1936 y Ff(ij)1526 1930 y Fm(\024)1574 1899 y Fe(F)1608 1943 y Ff(i)p Fg(2)p Ff(I)1668 1930 y Fl(d)1690 1936 y Ff(ij)1735 1930 y Fm(\024)e Fl(d)h Fo(for)f(an)o(y)0 1980 y Fl(i)d Fm(2)f Fl(I)17 b Fo(and)d Fl(j)g Fm(2)d Fl(J)18 b Fo(b)o(y)c(assumption.)1349 b Fh(\003)0 2064 y Fk(4.6)16 b(Lemma.)24 b Fj(Supp)q(ose)15 b Fl(D)h Fj(and)e Fl(E)i Fj(are)e(cp)q(o's.)20 b(Let)14 b Fm(f)p Fl(f)916 2070 y Ff(i)942 2064 y Fo(:)e Fl(i)g Fm(2)g Fl(I)s Fm(g)g(\022)h Fl(D)g Fm(!)f Fl(E)k Fj(b)q(e)f(directed.)20 b(Then)15 b Fl(f)t Fo(\()p Fl(d)p Fo(\))d(:=)1767 2033 y Fe(F)1802 2076 y Ff(i)p Fg(2)p Ff(I)1862 2064 y Fl(f)1882 2070 y Ff(i)1896 2064 y Fo(\()p Fl(d)p Fo(\))0 2114 y Fj(is)i(a)f(w)o (ell{de\014ned)h(con)o(tin)o(uous)g(function)g(from)e Fl(D)j Fj(to)f Fl(E)r Fj(,)f(and)h Fl(f)19 b Fj(is)13 b(the)i(least)f(upp)q(er)h(b)q(ound)f(of)f Fm(f)p Fl(f)1602 2120 y Ff(i)1628 2114 y Fo(:)e Fl(i)h Fm(2)f Fl(I)s Fm(g)p Fj(.)0 2198 y Fk(Pro)q(of.)20 b Fo(Supp)q(ose)15 b Fl(d)e Fm(2)f Fl(D)q Fo(.)21 b(Then)15 b Fm(f)p Fl(f)606 2204 y Ff(i)620 2198 y Fo(\()p Fl(d)p Fo(\))e(:)f Fl(i)h Fm(2)g Fl(I)s Fm(g)h Fo(is)h(directed)h(since)f Fm(f)p Fl(f)1181 2204 y Ff(i)1208 2198 y Fo(:)d Fl(i)i Fm(2)e Fl(I)s Fm(g)j Fo(is,)f(hence)1527 2166 y Fe(F)1562 2210 y Ff(i)p Fg(2)p Ff(I)1622 2198 y Fl(f)1642 2204 y Ff(i)1656 2198 y Fo(\()p Fl(d)p Fo(\))g(exists,)h(i.e.)20 b Fl(f)0 2247 y Fo(is)14 b(w)o(ell{de\014ned.)k Fl(f)h Fo(is)13 b(con)o(tin)o(uous,)h(for)f(if)g Fm(f)p Fl(d)736 2253 y Ff(j)765 2247 y Fo(:)e Fl(j)j Fm(2)d Fl(J)t Fm(g)i Fo(is)h(directed,)h(w)o(e)f(ha)o(v)o(e)495 2366 y Fl(f)t Fo(\()541 2327 y Fe(G)535 2416 y Ff(j)r Fg(2)p Ff(J)602 2366 y Fl(d)624 2372 y Ff(j)641 2366 y Fo(\))d(=)715 2327 y Fe(G)712 2416 y Ff(i)p Fg(2)p Ff(I)777 2327 y Fe(G)770 2416 y Ff(j)r Fg(2)p Ff(J)836 2366 y Fl(f)856 2372 y Ff(i)870 2366 y Fo(\()p Fl(d)908 2372 y Ff(j)925 2366 y Fo(\))h(=)1003 2327 y Fe(G)997 2416 y Ff(j)r Fg(2)p Ff(J)1065 2327 y Fe(G)1063 2416 y Ff(i)p Fg(2)p Ff(I)1121 2366 y Fl(f)1141 2372 y Ff(i)1155 2366 y Fo(\()p Fl(d)1193 2372 y Ff(j)1210 2366 y Fo(\))g(=)1288 2327 y Fe(G)1282 2416 y Ff(j)r Fg(2)p Ff(J)1347 2366 y Fl(f)t Fo(\()p Fl(d)1409 2372 y Ff(j)1427 2366 y Fo(\))p Fl(;)0 2517 y Fo(where)j(the)g(middle)d(equalit)o(y)h(follo)o(ws)f (from)g(Lemma)f(4.5.)83 2570 y(It)16 b(remains)e(to)i(b)q(e)g(sho)o(wn) g(that)f Fl(f)21 b Fo(is)15 b(the)h(least)g(upp)q(er)h(b)q(ound)e(of)g Fm(f)p Fl(f)1212 2576 y Ff(i)1241 2570 y Fo(:)e Fl(i)i Fm(2)f Fl(I)s Fm(g)p Fo(.)24 b(Clearly)15 b Fl(f)20 b Fo(is)15 b(an)h(upp)q(er)g(b)q(ound,)0 2620 y(since)g Fl(f)123 2626 y Ff(i)137 2620 y Fo(\()p Fl(d)p Fo(\))d Fm(\024)249 2589 y Fe(F)284 2633 y Ff(i)p Fg(2)p Ff(I)344 2620 y Fl(f)364 2626 y Ff(i)378 2620 y Fo(\()p Fl(d)p Fo(\))g(=)g Fl(f)t Fo(\()p Fl(d)p Fo(\))i(for)g(an)o(y)f Fl(d)f Fm(2)g Fl(D)q Fo(.)21 b(It)15 b(is)f(also)g(the)i(least)f(upp)q (er)h(b)q(ound,)e(for)h(if)f Fl(g)i Fo(is)e(an)o(y)h(other)g(upp)q(er)0 2670 y(b)q(ound,)f(w)o(e)g(ha)o(v)o(e)f Fl(f)316 2676 y Ff(i)330 2670 y Fo(\()p Fl(d)p Fo(\))f Fm(\024)g Fl(g)q Fo(\()p Fl(d)p Fo(\))i(and)f(hence)j Fl(f)t Fo(\()p Fl(d)p Fo(\))c(=)859 2639 y Fe(F)893 2682 y Ff(i)p Fg(2)p Ff(I)953 2670 y Fl(f)973 2676 y Ff(i)988 2670 y Fo(\()p Fl(d)p Fo(\))f Fm(\024)h Fl(g)q Fo(\()p Fl(d)p Fo(\))i(for)f(an)o(y)h Fl(d)d Fm(2)g Fl(D)q Fo(.)471 b Fh(\003)965 2770 y Fo(6)p eop %%Page: 7 7 7 6 bop 0 42 a Fk(4.7)16 b(Corollary)l(.)23 b Fl(D)13 b Fm(!)e Fl(E)k Fj(is)f(a)g(cp)q(o)g(if)f Fl(D)i Fj(and)f Fl(E)i Fj(are.)1029 b Fh(\003)83 118 y Fo(W)m(e)14 b(no)o(w)f(de\014ne) i(a)f(corresp)q(onding)h(op)q(eration)f(on)f(information)e(systems,)j (assigning)f(to)h(an)o(y)g Fk(A)d Fo(=)h(\()p Fl(A;)7 b Fo(Con)1836 124 y Ff(A)1863 118 y Fl(;)g Fm(`)1907 124 y Ff(A)1934 118 y Fo(\))0 168 y(and)14 b Fk(B)f Fo(=)g(\()p Fl(B)r(;)7 b Fo(Con)315 174 y Ff(B)344 168 y Fl(;)g Fm(`)388 174 y Ff(B)416 168 y Fo(\))15 b(an)f Fk(A)f Fm(!)f Fk(B)h Fo(=)g(\()p Fl(C)q(;)7 b Fo(Con)n Fl(;)g Fm(`)p Fo(\))15 b(suc)o(h)g(that)g Fm(j)p Fk(A)d Fm(!)g Fk(B)p Fm(j)1270 157 y(\030)1270 170 y Fo(=)1315 168 y Fm(j)p Fk(A)p Fm(j)g(!)h(j)p Fk(B)p Fm(j)p Fo(.)19 b(W)m(e)14 b(\014rst)h(ha)o(v)o(e)g(to)f(decide)0 218 y(on)f(the)h(set)g(of)e(data)h(ob)r(jects.)19 b(The)13 b(simplest)f(\014nite)i(piece)g(of)e(information)e(on)j(a)g(con)o(tin)o (uous)g(function)g(from)e Fm(j)p Fk(A)p Fm(j)h Fo(to)h Fm(j)p Fk(B)p Fm(j)0 268 y Fo(is)g(to)g(sa)o(y)g(that)g(if)f(w)o(e)h (are)h(giv)o(en)e(the)i(information)c Fl(X)15 b Fm(2)c Fo(Con)981 274 y Ff(A)1021 268 y Fo(on)i(the)h(argumen)o(t,)d(then)j (this)f(su\016ces)h(to)f(kno)o(w)g(at)g(least)0 318 y(the)i (information)c Fl(b)g Fm(2)g Fl(B)17 b Fo(on)d(the)g(v)n(alue.)k(Hence) d(w)o(e)f(let)g Fl(C)h Fo(:=)c(Con)1084 324 y Ff(A)1120 318 y Fm(\002)f Fl(B)r Fo(.)18 b(No)o(w)c(when)g(a)g(set)h Fm(f)p Fo(\()p Fl(X)1599 324 y Fi(1)1618 318 y Fl(;)7 b(b)1655 324 y Fi(1)1673 318 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)1832 324 y Ff(n)1854 318 y Fl(;)g(b)1891 324 y Ff(n)1913 318 y Fo(\))p Fm(g)0 367 y Fo(of)12 b(suc)o(h)i(data)e (ob)r(jects)i(is)f(to)g(b)q(e)g(called)g(consisten)o(t?)18 b(Clearly)13 b(just)g(in)f(the)h(case)h(when)f(for)g(an)o(y)f(subset)j Fl(I)g Fm(\022)c(f)p Fo(1)p Fl(;)c Fo(2)p Fl(;)g(:)g(:)g(:)t(;)g(n)p Fm(g)0 417 y Fo(suc)o(h)19 b(that)g(the)g(left)f(hand)h(sides)559 386 y Fe(S)594 417 y Fm(f)p Fl(X)649 423 y Ff(i)682 417 y Fo(:)g Fl(i)g Fm(2)g Fl(I)s Fm(g)g Fo(are)g(consisten)o(t,)h(i.e.)32 b(ma)o(y)16 b(b)q(e)k(view)o(ed)e(as)h(appro)o(ximations)d(to)i(a)0 467 y(common)c(argumen)o(t,)j(then)h(the)g(corresp)q(onding)g(righ)o(t) e(hand)i(sides)g Fm(f)p Fl(b)1167 473 y Ff(i)1197 467 y Fo(:)f Fl(i)g Fm(2)g Fl(I)s Fm(g)g Fo(should)g(b)q(e)h(consisten)o(t) g(to)q(o,)g(i.e.)27 b(it)0 517 y(should)13 b(b)q(e)i(p)q(ossible)e(to)h (view)f(them)g(as)h(appro)o(ximations)d(to)i(a)g(single)h(v)n(alue.)j (Therefore)e(Con)e(is)h(de\014ned)g(to)g(b)q(e)g(the)g(set)0 567 y(of)f(all)g Fm(f)p Fo(\()p Fl(X)176 573 y Fi(1)195 567 y Fl(;)7 b(b)232 573 y Fi(1)250 567 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;) i Fo(\()p Fl(X)409 573 y Ff(n)431 567 y Fl(;)g(b)468 573 y Ff(n)490 567 y Fo(\))p Fm(g)12 b(\022)f Fl(C)17 b Fo(suc)o(h)d(that)461 677 y Fm(8)p Fl(I)h Fm(\022)d(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)t(;)g(n)p Fm(g)p Fl(:)762 637 y Fe([)760 726 y Ff(i)p Fg(2)p Ff(I)817 677 y Fl(X)851 683 y Ff(i)877 677 y Fm(2)k Fo(Con)990 683 y Ff(A)1040 677 y Fo(=)-7 b Fm(\))23 b(f)p Fl(b)1169 683 y Ff(i)1194 677 y Fo(:)11 b Fl(i)h Fm(2)f Fl(I)s Fm(g)h(2)f Fo(Con)1449 683 y Ff(B)1477 677 y Fl(:)0 818 y Fo(F)m(or)17 b(the)i(de\014nition)e (of)h(the)g(en)o(tailmen)o(t)e(relation)h Fm(`)i Fo(it)e(is)h(helpful)f (to)h(\014rst)g(de\014ne)h(the)f(notion)g(of)f(an)g Fn(applic)n(ation) 22 b Fo(of)0 867 y Fl(W)c Fo(:=)11 b Fm(f)p Fo(\()p Fl(X)183 873 y Fi(1)202 867 y Fl(;)c(b)239 873 y Fi(1)257 867 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)416 873 y Ff(n)438 867 y Fl(;)g(b)475 873 y Ff(n)497 867 y Fo(\))p Fm(g)k(2)h Fo(Con)h(to)h Fl(X)h Fm(2)c Fo(Con)885 873 y Ff(A)912 867 y Fo(:)571 969 y Fm(f)p Fo(\()p Fl(X)642 975 y Fi(1)661 969 y Fl(;)c(b)698 975 y Fi(1)716 969 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)875 975 y Ff(n)898 969 y Fl(;)g(b)935 975 y Ff(n)957 969 y Fo(\))p Fm(g)p Fl(X)15 b Fo(:=)c Fm(f)p Fl(b)1137 975 y Ff(i)1162 969 y Fo(:)g Fl(X)k Fm(`)1259 975 y Ff(A)1298 969 y Fl(X)1332 975 y Ff(i)1346 969 y Fm(g)p Fl(:)0 1071 y Fo(F)m(rom)d(the)i (de\014nition)f(of)g(Con)g(w)o(e)h(kno)o(w)f(that)g(this)h(set)g(is)g (in)f(Con)1065 1077 y Ff(B)1093 1071 y Fo(.)18 b(Clearly)13 b(application)f(is)h Fn(monotone)j(in)f(the)f(se)n(c)n(ond)0 1121 y(ar)n(gument)t Fo(,)f(in)h(the)g(sense)i(that)705 1173 y Fl(X)f Fm(`)779 1179 y Ff(A)818 1173 y Fl(X)855 1155 y Fg(0)890 1173 y Fo(=)-7 b Fm(\))23 b Fl(W)6 b(X)15 b Fm(`)1099 1179 y Ff(B)1140 1173 y Fl(W)6 b(X)1222 1155 y Fg(0)1234 1173 y Fl(:)0 1254 y Fo(In)14 b(fact)g(w)o(e)g(ev)o(en)g (ha)o(v)o(e)g Fl(W)6 b(X)467 1238 y Fg(0)491 1254 y Fm(\022)12 b Fl(W)6 b(X)17 b Fo(if)c Fl(X)i Fm(`)743 1260 y Ff(A)782 1254 y Fl(X)819 1238 y Fg(0)832 1254 y Fo(.)j(No)o(w)13 b(de\014ne)i Fl(W)j Fm(`)12 b Fo(\()p Fl(X)q(;)7 b(b)p Fo(\))13 b(b)o(y)715 1355 y Fl(W)k Fm(`)12 b Fo(\()p Fl(X)q(;)7 b(b)p Fo(\))12 b(:)f Fm(\()-7 b(\))22 b Fl(W)6 b(X)15 b Fm(`)1165 1361 y Ff(B)1206 1355 y Fl(b:)0 1458 y Fk(4.8)h(Lemma.)23 b Fj(If)14 b Fk(A)g Fj(and)g Fk(B)f Fj(are)i(information)c(systems,)i(then)i(so)f(is)f Fk(A)f Fm(!)f Fk(B)j Fj(de\014ned)h(as)f(ab)q(o)o(v)o(e.)0 1535 y Fk(Pro)q(of.)37 b Fo(Let)20 b Fk(A)j Fo(=)f(\()p Fl(A;)7 b Fo(Con)499 1541 y Ff(A)526 1535 y Fl(;)g Fm(`)570 1541 y Ff(A)597 1535 y Fo(\))21 b(and)f Fk(B)i Fo(=)g(\()p Fl(B)r(;)7 b Fo(Con)973 1541 y Ff(B)1002 1535 y Fl(;)g Fm(`)1046 1541 y Ff(B)1074 1535 y Fo(\).)37 b(The)21 b(axioms)d(1.1\(i\),)i(\(ii\))g(and)g(\(iv\))f(are)i(clearly)0 1584 y(satis\014ed.)37 b(F)m(or)19 b(\(iii\),)h(supp)q(ose)h Fm(f)p Fo(\()p Fl(X)607 1590 y Fi(1)626 1584 y Fl(;)7 b(b)663 1590 y Fi(1)681 1584 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)840 1590 y Ff(n)863 1584 y Fl(;)g(b)900 1590 y Ff(n)922 1584 y Fo(\))p Fm(g)21 b(`)h Fo(\()p Fl(X)q(;)7 b(b)p Fo(\),)21 b(i.e.)36 b Fm(f)p Fl(b)1293 1590 y Ff(j)1331 1584 y Fo(:)22 b Fl(X)j Fm(`)1449 1590 y Ff(A)1498 1584 y Fl(X)1532 1590 y Ff(j)1550 1584 y Fm(g)c(`)1617 1590 y Ff(B)1668 1584 y Fl(b)p Fo(.)36 b(W)m(e)19 b(ha)o(v)o(e)h(to)0 1634 y(sho)o(w)c(that)g Fm(f)p Fo(\()p Fl(X)268 1640 y Fi(1)287 1634 y Fl(;)7 b(b)324 1640 y Fi(1)342 1634 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)501 1640 y Ff(n)523 1634 y Fl(;)g(b)560 1640 y Ff(n)582 1634 y Fo(\))p Fl(;)g Fo(\()p Fl(X)q(;)g(b)p Fo(\))p Fm(g)14 b(2)h Fo(Con.)24 b(So)16 b(let)g Fl(I)i Fm(\022)d(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)e(;)i(n)p Fm(g)15 b Fo(and)g(supp)q(ose)j Fl(X)c Fm([)1637 1603 y Fe(S)1672 1647 y Ff(i)p Fg(2)p Ff(I)1732 1634 y Fl(X)1766 1640 y Ff(i)1795 1634 y Fm(2)h Fo(Con)1911 1640 y Ff(A)1938 1634 y Fl(:)0 1684 y Fo(W)m(e)20 b(m)o(ust)f(sho)o(w)g(that)h Fl(b)14 b Fm([)f(f)p Fl(b)503 1690 y Ff(i)538 1684 y Fo(:)21 b Fl(i)h Fm(2)f Fl(I)s Fm(g)h(2)f Fo(Con)843 1690 y Ff(B)872 1684 y Fo(.)36 b(Let)20 b Fl(J)26 b Fm(\022)c(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)t(;)g(n)p Fm(g)19 b Fo(consist)i(of)e(those)i Fl(j)i Fo(with)c Fl(X)26 b Fm(`)1838 1690 y Ff(A)1887 1684 y Fl(X)1921 1690 y Ff(j)1938 1684 y Fo(.)0 1734 y(Then)19 b(also)f Fl(X)e Fm([)291 1703 y Fe(S)326 1746 y Ff(i)p Fg(2)p Ff(I)386 1734 y Fl(X)420 1740 y Ff(i)447 1734 y Fm([)487 1703 y Fe(S)521 1746 y Ff(j)r Fg(2)p Ff(J)589 1734 y Fl(X)623 1740 y Ff(j)660 1734 y Fm(2)k Fo(Con)781 1740 y Ff(A)808 1734 y Fo(.)33 b(Since)966 1703 y Fe(S)1000 1746 y Ff(i)p Fg(2)p Ff(I)1060 1734 y Fl(X)1094 1740 y Ff(i)1121 1734 y Fm([)1161 1703 y Fe(S)1196 1746 y Ff(j)r Fg(2)p Ff(J)1264 1734 y Fl(X)1298 1740 y Ff(j)1335 1734 y Fm(2)19 b Fo(Con)1456 1740 y Ff(A)1483 1734 y Fo(,)g(from)e(the)i(consistency)i(of)0 1784 y Fm(f)p Fo(\()p Fl(X)71 1790 y Fi(1)90 1784 y Fl(;)7 b(b)127 1790 y Fi(1)145 1784 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)304 1790 y Ff(n)326 1784 y Fl(;)g(b)363 1790 y Ff(n)385 1784 y Fo(\))p Fm(g)15 b Fo(w)o(e)g(can)g(conclude)h (that)f Fm(f)p Fl(b)878 1790 y Ff(i)905 1784 y Fo(:)e Fl(i)g Fm(2)g Fl(I)s Fm(g)d([)g(f)p Fl(b)1127 1790 y Ff(j)1157 1784 y Fo(:)j Fl(j)j Fm(2)d Fl(J)t Fm(g)g(2)g Fo(Con)1432 1790 y Ff(B)1460 1784 y Fo(.)22 b(But)15 b Fm(f)p Fl(b)1616 1790 y Ff(j)1647 1784 y Fo(:)d Fl(j)k Fm(2)d Fl(J)t Fm(g)g(`)1831 1790 y Ff(B)1873 1784 y Fl(b)i Fo(b)o(y)0 1833 y(assumption.)i(Hence)e Fm(f)p Fl(b)398 1839 y Ff(i)423 1833 y Fo(:)c Fl(i)h Fm(2)f Fl(I)s Fm(g)f([)e(f)p Fl(b)638 1839 y Ff(j)667 1833 y Fo(:)j Fl(j)j Fm(2)d Fl(J)t Fm(g)e([)g(f)p Fl(b)p Fm(g)i(2)g Fo(Con)1038 1839 y Ff(B)1067 1833 y Fo(.)83 1884 y(F)m(or)16 b(\(v\),)h(supp)q(ose)g Fl(W)22 b Fm(`)16 b(f)p Fo(\()p Fl(X)574 1890 y Fi(1)593 1884 y Fl(;)7 b(b)630 1890 y Fi(1)648 1884 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)807 1890 y Ff(n)830 1884 y Fl(;)g(b)867 1890 y Ff(n)889 1884 y Fo(\))p Fm(g)15 b(`)h Fo(\()p Fl(X)q(;)7 b(b)p Fo(\).)26 b(W)m(e)16 b(ha)o(v)o(e)g(to)h (sho)o(w)f(that)g Fl(W)6 b(X)20 b Fm(`)1671 1890 y Ff(B)1716 1884 y Fl(b)p Fo(.)25 b(Note)17 b(that)0 1934 y(from)c Fl(X)k Fm(`)175 1940 y Ff(A)216 1934 y Fl(X)250 1940 y Ff(i)279 1934 y Fo(w)o(e)e(can)h(conclude)f Fl(W)6 b(X)17 b Fm(`)711 1940 y Ff(B)754 1934 y Fl(W)6 b(X)833 1940 y Ff(i)862 1934 y Fo(b)o(y)14 b(the)i(monotonicit)o(y)c(of)j (application)e(in)i(the)g(second)i(argumen)o(t.)0 1984 y(Hence)e Fl(W)6 b(X)16 b Fm(`)243 1990 y Ff(B)283 1952 y Fe(S)318 1984 y Fm(f)p Fl(W)6 b(X)418 1990 y Ff(i)443 1984 y Fo(:)11 b Fl(X)k Fm(`)540 1990 y Ff(A)579 1984 y Fl(X)613 1990 y Ff(i)627 1984 y Fm(g)d(`)685 1990 y Ff(B)725 1984 y Fm(f)p Fl(b)764 1990 y Ff(i)789 1984 y Fo(:)f Fl(X)k Fm(`)886 1990 y Ff(A)925 1984 y Fl(X)959 1990 y Ff(i)974 1984 y Fm(g)c(`)1031 1990 y Ff(B)1071 1984 y Fl(b)829 b Fh(\003)83 2034 y Fo(Note)12 b(that)g(with)f(the)h (ab)q(o)o(v)o(e)f(de\014nitions)h(of)f(the)h(en)o(tailmen)o(t)e (relation)h Fm(`)h Fo(in)f Fk(A)h Fm(!)f Fk(B)g Fo(application)f(is)i (also)f Fn(monotone)0 2084 y(in)k(the)g(\014rst)f(ar)n(gument)t Fo(,)g(i.e.)712 2136 y Fl(W)j Fm(`)12 b Fl(W)850 2119 y Fg(0)885 2136 y Fo(=)-7 b Fm(\))23 b Fl(W)6 b(X)15 b Fm(`)1094 2142 y Ff(B)1135 2136 y Fl(W)1180 2119 y Fg(0)1191 2136 y Fl(X)q(:)0 2217 y Fo(T)m(o)f(see)i(this)f(let)f Fl(W)317 2202 y Fg(0)342 2217 y Fo(=)f Fm(f)p Fo(\()p Fl(X)458 2223 y Fi(1)477 2217 y Fl(;)7 b(b)514 2223 y Fi(1)532 2217 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)691 2223 y Ff(n)713 2217 y Fl(;)g(b)750 2223 y Ff(n)772 2217 y Fo(\))p Fm(g)15 b Fo(and)f(observ)o(e)i(that)f Fl(W)6 b(X)16 b Fm(`)1265 2223 y Ff(B)1307 2186 y Fe(S)1342 2217 y Fm(f)p Fl(W)6 b(X)1442 2223 y Ff(i)1468 2217 y Fo(:)13 b Fl(X)j Fm(`)1568 2223 y Ff(A)1609 2217 y Fl(X)1643 2223 y Ff(i)1657 2217 y Fm(g)d(`)1716 2223 y Ff(B)1757 2217 y Fm(f)p Fl(b)1796 2223 y Ff(i)1823 2217 y Fo(:)f Fl(X)17 b Fm(`)1923 2223 y Ff(A)0 2267 y Fl(X)34 2273 y Ff(i)48 2267 y Fm(g)11 b Fo(=)h Fl(W)169 2252 y Fg(0)181 2267 y Fl(X)s Fo(.)0 2343 y Fk(4.9)k(Lemma.)24 b Fj(Let)15 b Fk(A)f Fj(and)g Fk(B)h Fj(b)q(e)g(information)c(systems.)20 b(Then)15 b(the)g(elemen)o(ts)f(of)g Fk(A)e Fm(!)g Fk(B)i Fj(are)h(exactly)g(the)g(appro)o(x-)0 2393 y(imable)d(mappings)g(from)g Fk(A)i Fj(to)g Fk(B)p Fj(.)0 2470 y Fk(Pro)q(of.)25 b Fo(Let)17 b Fk(A)e Fo(=)h(\()p Fl(A;)7 b Fo(Con)470 2476 y Ff(A)497 2470 y Fl(;)g Fm(`)541 2476 y Ff(A)568 2470 y Fo(\))17 b(and)f Fk(B)f Fo(=)h(\()p Fl(B)r(;)7 b Fo(Con)923 2476 y Ff(B)952 2470 y Fl(;)g Fm(`)996 2476 y Ff(B)1024 2470 y Fo(\).)25 b(Supp)q(ose)17 b Fl(r)g Fm(2)e(j)p Fk(A)g Fm(!)g Fk(B)p Fm(j)p Fo(.)25 b(Then)16 b Fl(r)h Fm(\022)f Fo(Con)1791 2476 y Ff(A)1829 2470 y Fm(\002)11 b Fl(B)19 b Fo(is)0 2520 y(consisten)o(t)d(and)f(deductiv)o(ely)g (closed.)22 b(W)m(e)15 b(ha)o(v)o(e)g(to)f(sho)o(w)h(that)g Fl(r)h Fo(satis\014es)g(the)g(axioms)d(3.1\(i\){\(iii\))f(of)i(appro)o (ximable)0 2570 y(mappings.)83 2620 y(\(i\).)k(Let)c Fl(X)s(r)q(b)306 2626 y Ff(i)334 2620 y Fo(for)g Fl(i)e Fm(2)f(f)p Fo(1)p Fl(;)c(:)g(:)g(:)t(;)g(n)p Fm(g)p Fo(.)17 b(W)m(e)d(m)o(ust)f(sho)o(w)h(that)f Fm(f)p Fl(b)1078 2626 y Fi(1)1097 2620 y Fl(;)7 b(:)g(:)g(:)t(;)g(b)1207 2626 y Ff(n)1229 2620 y Fm(g)12 b(2)f Fo(Con)1375 2626 y Ff(B)1403 2620 y Fo(.)18 b(But)d(this)f(clearly)f(follo)o(ws)g(from)0 2670 y(the)h(consistency)i(of)d Fl(r)q Fo(.)965 2770 y(7)p eop %%Page: 8 8 8 7 bop 83 42 a Fo(\(ii\).)19 b(Let)14 b Fl(X)s(r)q(b)319 48 y Ff(i)348 42 y Fo(for)g Fl(i)e Fm(2)g(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)e(;)i(n)p Fm(g)13 b Fo(and)h Fm(f)p Fl(b)792 48 y Fi(1)810 42 y Fl(;)7 b(:)g(:)g(:)e(;)i(b)921 48 y Ff(n)943 42 y Fm(g)12 b(`)1001 48 y Ff(B)1042 42 y Fl(b)p Fo(.)19 b(W)m(e)14 b(m)o(ust)f(sho)o(w)i(that)f Fl(X)s(r)q(b)p Fo(.)19 b(But)c(b)o(y)f(the)h(de\014nition)0 91 y(of)e Fm(`)h Fo(in)g Fk(A)e Fm(!)f Fk(B)i Fo(w)o(e)h(ha)o(v)o(e)g Fm(f)p Fo(\()p Fl(X)q(;)7 b(b)549 97 y Fi(1)567 91 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)q(;)g(b)764 97 y Ff(n)786 91 y Fo(\))p Fm(g)k(`)h Fo(\()p Fl(X)q(;)7 b(b)p Fo(\),)14 b(hence)h Fl(X)s(r)q(b)f Fo(b)o(y)g(the)g(deductiv)o(e) h(closure)g(of)e Fl(r)q Fo(.)83 142 y(\(iii\).)20 b(Let)15 b Fl(X)i Fm(`)334 148 y Ff(A)374 142 y Fl(X)411 127 y Fg(0)439 142 y Fo(and)d Fl(X)557 127 y Fg(0)570 142 y Fl(r)q(b)p Fo(.)20 b(W)m(e)15 b(m)o(ust)e(sho)o(w)i(that)g Fl(X)s(r)q(b)p Fo(.)21 b(But)16 b Fm(f)p Fo(\()p Fl(X)1277 127 y Fg(0)1289 142 y Fl(;)7 b(b)p Fo(\))p Fm(g)12 b(`)i Fo(\()p Fl(X)q(;)7 b(b)p Fo(\))14 b(since)i Fm(f)p Fo(\()p Fl(X)1709 127 y Fg(0)1721 142 y Fl(;)7 b(b)p Fo(\))p Fm(g)p Fl(X)16 b Fo(=)e Fm(f)p Fl(b)p Fm(g)0 192 y Fo(\(whic)o(h)g (follo)o(ws)e(from)g Fl(X)k Fm(`)444 198 y Ff(A)482 192 y Fl(X)519 177 y Fg(0)532 192 y Fo(\),)d(hence)j(again)c Fl(X)s(r)q(b)i Fo(b)o(y)g(the)h(deductiv)o(e)f(closure)h(of)e Fl(r)q Fo(.)83 243 y(F)m(or)19 b(the)h(other)g(direction)g(supp)q(ose)h (that)e Fl(r)q Fo(:)7 b Fk(A)21 b Fm(!)f Fk(B)g Fo(is)f(an)g(appro)o (ximable)e(mapping.)33 b(W)m(e)19 b(m)o(ust)f(sho)o(w)i(that)0 293 y Fl(r)12 b Fm(2)g(j)p Fk(A)f Fm(!)g Fk(B)p Fm(j)p Fo(.)83 343 y(Consistency)19 b(of)f Fl(r)q Fo(.)32 b(Supp)q(ose)19 b Fl(X)634 349 y Ff(i)648 343 y Fl(r)q(b)686 349 y Ff(i)718 343 y Fo(for)f(all)f Fl(i)j Fm(2)e(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)e(;)i(n)p Fm(g)17 b Fo(and)i Fl(X)j Fo(:=)1331 312 y Fe(S)1366 343 y Fm(f)p Fl(X)1421 349 y Ff(i)1454 343 y Fo(:)c Fl(i)i Fm(2)e Fl(I)s Fm(g)i(2)f Fo(Con)1746 349 y Ff(A)1792 343 y Fo(for)f(some)0 393 y Fl(I)e Fm(\022)d(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)t(;)g(n)p Fm(g)p Fo(.)19 b(W)m(e)14 b(m)o(ust)f(sho)o(w)i(that)f Fm(f)p Fl(b)698 399 y Ff(i)724 393 y Fo(:)e Fl(i)h Fm(2)f Fl(I)s Fm(g)h(2)f Fo(Con)984 399 y Ff(B)1013 393 y Fo(.)19 b(No)o(w)14 b(from)f Fl(X)1272 399 y Ff(i)1286 393 y Fl(r)q(b)1324 399 y Ff(i)1352 393 y Fo(and)i Fl(X)h Fm(`)1509 399 y Ff(A)1549 393 y Fl(X)1583 399 y Ff(i)1611 393 y Fo(w)o(e)f(obtain)f Fl(X)s(r)q(b)1878 399 y Ff(i)1906 393 y Fo(b)o(y)0 443 y(3.1\(iii\))e(for)h(an)o(y)h Fl(i)e Fm(2)f Fl(I)s Fo(,)j(and)f(hence)j Fm(f)p Fl(b)623 449 y Ff(i)648 443 y Fo(:)11 b Fl(i)g Fm(2)h Fl(I)s Fm(g)f(2)h Fo(Con)902 449 y Ff(B)945 443 y Fo(b)o(y)i(3.1\(i\).)83 494 y(Deductiv)o(e)h(closure)f(of)g Fl(r)q Fo(.)k(Supp)q(ose)d Fl(X)712 500 y Ff(i)726 494 y Fl(r)q(b)764 500 y Ff(i)791 494 y Fo(for)f(all)e Fl(i)g Fm(2)g(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)t(;)g(n)p Fm(g)13 b Fo(and)h Fl(W)j Fo(:=)12 b Fm(f)p Fo(\()p Fl(X)1435 500 y Fi(1)1454 494 y Fl(;)7 b(b)1491 500 y Fi(1)1509 494 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)1668 500 y Ff(n)1690 494 y Fl(;)g(b)1727 500 y Ff(n)1749 494 y Fo(\))p Fm(g)12 b(`)g Fo(\()p Fl(X)q(;)7 b(b)p Fo(\).)0 544 y(W)m(e)15 b(m)o(ust)f(sho)o(w)i Fl(X)s(r)q(b)p Fo(.)23 b(By)15 b(the)h(de\014nition)f(of)g Fm(`)h Fo(for)f Fk(A)f Fm(!)f Fk(B)j Fo(w)o(e)f(ha)o(v)o(e)g Fl(W)6 b(X)18 b Fm(`)1308 550 y Ff(B)1351 544 y Fl(b)p Fo(,)d(whic)o(h)g(is)h Fm(f)p Fl(b)1599 550 y Ff(i)1626 544 y Fo(:)e Fl(X)j Fm(`)1728 550 y Ff(A)1770 544 y Fl(X)1804 550 y Ff(i)1818 544 y Fm(g)d(`)1878 550 y Ff(B)1921 544 y Fl(b)p Fo(.)0 593 y(F)m(urther)h(b)o(y)e(assumption)g Fl(X)462 599 y Ff(i)476 593 y Fl(r)q(b)514 599 y Ff(i)541 593 y Fo(w)o(e)i(kno)o(w)e Fl(X)s(r)q(b)786 599 y Ff(i)814 593 y Fo(b)o(y)h(3.1\(iii\))d(for)j (all)f Fl(i)h Fo(with)f Fl(X)i Fm(`)1323 599 y Ff(A)1362 593 y Fl(X)1396 599 y Ff(i)1411 593 y Fo(.)j(Hence)d Fl(X)s(r)q(b)f Fo(b)o(y)g(3.1\(ii\).)85 b Fh(\003)0 671 y Fk(4.10)16 b(Theorem.)23 b Fj(Let)14 b Fk(A)e Fo(=)g(\()p Fl(A;)7 b Fo(Con)630 677 y Ff(A)657 671 y Fl(;)g Fm(`)701 677 y Ff(A)728 671 y Fo(\))14 b Fj(and)g Fk(B)d Fo(=)h(\()p Fl(B)r(;)7 b Fo(Con)1070 677 y Ff(B)1098 671 y Fl(;)g Fm(`)1142 677 y Ff(B)1171 671 y Fo(\))14 b Fj(b)q(e)g(information)d (systems.)19 b(Then)772 774 y Fm(j)p Fk(A)11 b Fm(!)h Fk(B)p Fm(j)942 763 y(\030)942 776 y Fo(=)985 774 y Fm(j)p Fk(A)p Fm(j)f(!)g(j)p Fk(B)p Fm(j)p Fl(:)0 876 y Fj(The)j(isomorphism)d (pair)i(is)523 979 y Fl(')p Fo(:)7 b Fm(j)p Fk(A)k Fm(!)g Fk(B)p Fm(j)g(!)g Fo(\()p Fm(j)p Fk(A)p Fm(j)f(!)i(j)p Fk(B)p Fm(j)p Fo(\))40 b Fj(giv)o(en)14 b(b)o(y)41 b Fl(')p Fo(\()p Fl(s)p Fo(\))12 b(=)g Fm(j)p Fl(s)p Fm(j)p Fl(;)520 1085 y( )q Fo(:)7 b(\()p Fm(j)p Fk(A)p Fm(j)k(!)g(j)p Fk(B)p Fm(j)p Fo(\))f Fm(!)h(j)p Fk(A)h Fm(!)f Fk(B)p Fm(j)41 b Fj(giv)o(en)13 b(b)o(y)41 b Fl( )q Fo(\()p Fl(f)t Fo(\))14 b(=)d Fl(r)1397 1091 y Ff(f)1419 1085 y Fl(:)0 1167 y Fk(Pro)q(of.)18 b Fl( )10 b Fm(\016)f Fl(')j Fo(=)g(id)332 1174 y Fg(j)p Fc(A)p Fg(!)p Fc(B)p Fg(j)456 1167 y Fo(and)i Fl(')9 b Fm(\016)g Fl( )k Fo(=)f(id)721 1174 y Fg(j)p Fc(A)p Fg(j!j)p Fc(B)p Fg(j)865 1167 y Fo(has)i(b)q(een)h(sho)o(wn)f(in)f(Theorem)g(3.2.)83 1218 y Fl(')f Fo(is)g(monotone.)k(Supp)q(ose)d Fl(s;)7 b(s)587 1203 y Fg(0)611 1218 y Fm(2)k(j)p Fk(A)g Fm(!)g Fk(B)p Fm(j)g Fo(and)h Fl(s)g Fm(\022)g Fl(s)992 1203 y Fg(0)1004 1218 y Fo(.)18 b(W)m(e)11 b(m)o(ust)g(sho)o(w)h(that)g Fm(j)p Fl(s)p Fm(j)f(\024)h(j)p Fl(s)1522 1203 y Fg(0)1534 1218 y Fm(j)f Fo(in)h(the)g(p)q(oin)o(t)o(wise)g(order)0 1268 y(of)h(the)i(con)o(tin)o(uous)f(functions)g(in)f Fm(j)p Fk(A)p Fm(j)e(!)g(j)p Fk(B)p Fm(j)p Fo(.)17 b(So)c(let)h Fl(z)g Fm(2)d(j)p Fk(A)p Fm(j)p Fo(.)17 b(Then)e(w)o(e)f(ha)o(v)o(e)600 1373 y Fm(j)p Fl(s)p Fm(j)p Fo(\()p Fl(z)r Fo(\))d(=)h Fm(f)p Fl(b)f Fm(2)g Fl(B)j Fo(:)d Fl(X)s(sb)k Fo(for)f(some)f Fl(X)i Fm(\022)1246 1356 y Fi(\014n)1297 1373 y Fl(z)r Fm(g)707 1441 y(\022)d(f)p Fl(b)f Fm(2)g Fl(B)j Fo(:)d Fl(X)s(s)964 1424 y Fg(0)977 1441 y Fl(b)j Fo(for)f(some)g Fl(X)i Fm(\022)1257 1424 y Fi(\014n)1309 1441 y Fl(z)r Fm(g)707 1503 y Fo(=)d Fm(j)p Fl(s)782 1486 y Fg(0)794 1503 y Fm(j)p Fo(\()p Fl(z)r Fo(\))p Fl(:)83 1607 y( )k Fo(is)e(monotone.)j(Supp)q(ose)e Fl(f)r(;)7 b(f)604 1592 y Fg(0)617 1607 y Fo(:)g Fm(j)p Fk(A)p Fm(j)j(!)i(j)p Fk(B)p Fm(j)h Fo(are)i(con)o(tin)o(uous)f(and)g Fl(f)t Fo(\()p Fl(z)r Fo(\))f Fm(\022)f Fl(f)1348 1592 y Fg(0)1361 1607 y Fo(\()p Fl(z)r Fo(\))i(for)g(all)f Fl(z)h Fm(2)e(j)p Fk(A)p Fm(j)p Fo(.)18 b(Then)d(for)e(an)o(y)0 1657 y Fl(X)i Fm(2)c Fo(Con)162 1663 y Ff(A)203 1657 y Fo(and)j Fl(b)d Fm(2)g Fl(B)17 b Fo(w)o(e)d(ha)o(v)o(e)783 1759 y Fl(X)s(r)839 1765 y Ff(f)861 1759 y Fl(b)23 b Fm(\()-7 b(\))22 b Fl(b)11 b Fm(2)h Fl(f)t Fo(\()p 1110 1726 38 2 v Fl(X)t Fo(\))902 1822 y(=)-7 b Fm(\))23 b Fl(b)11 b Fm(2)g Fl(f)1084 1805 y Fg(0)1097 1822 y Fo(\()p 1113 1788 V Fl(X)t Fo(\))902 1884 y Fm(\()-7 b(\))22 b Fl(X)s(r)1057 1890 y Ff(f)1076 1882 y Fb(0)1090 1884 y Fl(b:)798 b Fh(\003)0 1988 y Fk(4.11)12 b(Corollary)l(.)23 b Fj(Supp)q(ose)11 b Fl(D)h Fj(and)e Fl(E)j Fj(are)e(domains)e(with)h(coun)o(table)h (bases.)17 b(Then)11 b(the)h(set)f Fl(D)i Fm(!)e Fl(E)i Fj(of)d(all)f(con)o(tin)o(uous)0 2037 y(functions)14 b(from)e Fl(D)j Fj(to)f Fl(E)i Fj(equipp)q(ed)f(with)e(the)i(p)q(oin)o (t)o(wise)e(order)i(is)f(again)e(a)i(domain)d(with)j(coun)o(table)g (basis.)0 2115 y Fk(Pro)q(of.)k Fl(D)13 b Fm(!)e Fl(E)292 2104 y Fm(\030)292 2117 y Fo(=)336 2115 y Fm(j)p Fl(I)s(D)q Fm(j)h(!)f(j)p Fl(I)s(E)r Fm(j)570 2104 y(\030)570 2117 y Fo(=)614 2115 y Fm(j)p Fl(I)s(D)i Fm(!)e Fl(I)s(E)r Fm(j)p Fo(.)1093 b Fh(\003)83 2166 y Fo(The)16 b(cp)q(o's)g(together)g (with)f(the)h(con)o(tin)o(uous)g(functions)f(b)q(et)o(w)o(een)i(them)e (form)e(a)i(cartesian)i(closed)f(category)m(.)22 b(W)m(e)0 2216 y(only)11 b(need)i(here)f(the)h(con)o(tin)o(uit)o(y)e(of)g (application,)f(hence)j(w)o(e)f(restrict)h(ourselv)o(es)g(to)e(a)h(pro) q(of)f(of)g(this)h(fact.)17 b(W)m(e)11 b(\014rst)i(pro)o(v)o(e)0 2266 y(the)i(w)o(ell{kno)o(wn)e(univ)o(ersal)h(prop)q(ert)o(y)h(of)f (cartesian)h(pro)q(ducts.)20 b(Another)15 b(w)o(a)o(y)f(to)g(put)g (this)h(is)f(to)g(sa)o(y)g(that)g(a)g(function)0 2315 y(in)o(to)f(a)h(pro)q(duct)g(is)g(con)o(tin)o(uous)f(if)g(its)h(comp)q (onen)o(t)f(functions)h(are)g(con)o(tin)o(uous.)k(Note)c(that)g(the)g (con)o(v)o(erse)i(follo)o(ws)c(from)0 2365 y(Lemma)f(2.4)i(and)h(the)g (con)o(tin)o(uit)o(y)f(of)h(the)g(pro)r(jections.)0 2443 y Fk(4.12)h(Lemma.)23 b Fj(Supp)q(ose)14 b Fl(F)6 b Fj(,)12 b Fl(D)j Fj(and)e Fl(E)i Fj(are)e(cp)q(o's.)18 b(Let)c Fl(f)t Fo(:)7 b Fl(F)17 b Fm(!)11 b Fl(D)j Fj(and)f Fl(g)q Fo(:)7 b Fl(F)17 b Fm(!)11 b Fl(E)k Fj(b)q(e)f(con)o(tin)o(uous)f (functions.)k(Then)0 2493 y(there)e(is)f(a)g(unique)g(con)o(tin)o(uous) h(function)e Fl(h)p Fo(:)7 b Fl(F)17 b Fm(!)12 b Fl(D)f Fm(\002)e Fl(E)16 b Fj(suc)o(h)f(that)f Fl(\031)1171 2499 y Fi(left)1230 2493 y Fm(\016)9 b Fl(h)j Fo(=)g Fl(f)19 b Fj(and)14 b Fl(\031)1484 2499 y Fi(righ)o(t)1565 2493 y Fm(\016)9 b Fl(h)j Fo(=)g Fl(g)q Fj(.)19 b Fl(h)14 b Fj(is)g(denoted)0 2542 y(b)o(y)g Fl(f)g Fm(\002)9 b Fl(g)q Fj(.)0 2620 y Fk(Pro)q(of.)18 b Fo(Supp)q(ose)c Fl(h)g Fo(is)g(suc)o(h)g(a)g(function.)k(Then)c(for)f(an)o(y)h Fl(x)d Fm(2)g Fl(F)19 b(h)p Fo(\()p Fl(x)p Fo(\))12 b(=)g(\()p Fl(f)t Fo(\()p Fl(x)p Fo(\))p Fl(;)7 b(g)q Fo(\()p Fl(x)p Fo(\)\),)14 b(so)g Fl(h)f Fo(is)h(uniquely)f(determined.)0 2670 y(F)m(or)20 b(existence,)j(de\014ne)e Fl(h)g Fo(b)o(y)f Fl(h)p Fo(\()p Fl(x)p Fo(\))i(=)g(\()p Fl(f)t Fo(\()p Fl(x)p Fo(\))p Fl(;)7 b(g)q Fo(\()p Fl(x)p Fo(\)\).)38 b(Then)21 b Fl(h)f Fo(satis\014es)h(the)g(equations)g(ab)q(o)o(v)o(e,)g (and)f(clearly)g Fl(h)g Fo(is)965 2770 y(8)p eop %%Page: 9 9 9 8 bop 0 42 a Fo(monotone.)17 b(It)d(remains)e(to)i(pro)o(v)o(e)g (that)g Fl(h)g Fo(is)g(con)o(tin)o(uous.)j(So)d(let)g Fl(M)i Fm(\022)c Fl(F)20 b Fo(b)q(e)14 b(directed.)19 b(Then)c(w)o(e)f(ha)o(v)o(e)574 140 y Fl(h)p Fo(\()614 100 y Fe(G)668 140 y Fl(M)5 b Fo(\))11 b(=)h(\()p Fl(f)t Fo(\()840 100 y Fe(G)894 140 y Fl(M)5 b Fo(\))p Fl(;)i(g)q Fo(\()1011 100 y Fe(G)1064 140 y Fl(M)e Fo(\)\))740 223 y(=)12 b(\()800 183 y Fe(G)853 223 y Fl(f)t Fo(\()p Fl(M)5 b Fo(\))p Fl(;)973 183 y Fe(G)1026 223 y Fl(g)q Fo(\()p Fl(M)g Fo(\)\))740 306 y(=)12 b(\()800 266 y Fe(G)853 306 y Fl(\031)877 312 y Fi(left)926 306 y Fo(\()p Fl(h)p Fo(\()p Fl(M)5 b Fo(\)\))p Fl(;)1078 266 y Fe(G)1131 306 y Fl(\031)1155 312 y Fi(righ)o(t)1226 306 y Fo(\()p Fl(h)p Fo(\()p Fl(M)g Fo(\)\)\))740 389 y(=)784 349 y Fe(G)837 389 y Fl(h)p Fo(\()p Fl(M)g Fo(\))695 b(b)o(y)14 b(Lemma)d(4.1.)i Fh(\003)83 494 y Fo(The)19 b(next)g(thing)g(to)f(do)h (is)f(to)h(sho)o(w)f(that)h(the)h(sections)f Fl(\023)1058 478 y Ff(e)1058 504 y Fi(1)1095 494 y Fo(and)g Fl(\023)1196 478 y Ff(d)1196 504 y Fi(2)1234 494 y Fo(of)f(the)h(iden)o(tit)o(y)f (function)h(on)f Fl(D)c Fm(\002)f Fl(E)21 b Fo(are)0 543 y(con)o(tin)o(uous.)d(By)c(these)h(w)o(e)f(mean)e(the)i(functions)g Fl(\023)831 528 y Ff(e)831 554 y Fi(1)849 543 y Fo(:)7 b Fl(D)13 b Fm(!)e Fl(D)f Fm(\002)f Fl(E)16 b Fo(giv)o(en)d(b)o(y)g Fl(\023)1280 528 y Ff(e)1280 554 y Fi(1)1299 543 y Fo(\()p Fl(d)p Fo(\))e(=)h(\()p Fl(d;)7 b(e)p Fo(\))13 b(for)h(an)o(y)f (\014xed)h Fl(e)e Fm(2)f Fl(E)r Fo(,)i(and)0 593 y Fl(\023)15 578 y Ff(d)15 603 y Fi(2)34 593 y Fo(:)7 b Fl(E)16 b Fm(!)e Fl(D)e Fm(\002)f Fl(E)18 b Fo(giv)o(en)d(b)o(y)h Fl(\023)479 578 y Ff(d)479 603 y Fi(2)498 593 y Fo(\()p Fl(e)p Fo(\))f(=)g(\()p Fl(d;)7 b(e)p Fo(\))16 b(for)f(an)o(y)g (\014xed)i Fl(d)d Fm(2)g Fl(D)q Fo(.)24 b(Clearly)16 b(e.g.)23 b Fl(\023)1364 578 y Ff(e)1364 603 y Fi(1)1398 593 y Fo(is)16 b(monotone.)22 b(T)m(o)15 b(see)i(that)f(it)f(is)0 643 y(con)o(tin)o(uous,)10 b(let)h Fl(M)16 b Fm(\022)c Fl(D)g Fo(b)q(e)e(directed.)19 b(Then)10 b(clearly)g Fm(f)p Fo(\()p Fl(d;)d(e)p Fo(\))k(:)g Fl(d)g Fm(2)h Fl(M)5 b Fm(g)k Fo(is)h(directed)i(to)q(o,)e(and)1526 612 y Fe(F)1561 655 y Ff(d)p Fg(2)p Ff(M)1637 643 y Fo(\()p Fl(d;)d(e)p Fo(\))12 b(=)f(\()1800 612 y Fe(F)1842 643 y Fl(M)r(;)c(e)p Fo(\).)83 693 y(No)o(w)13 b(w)o(e)h(can)g(sho)o(w)g (that)g(a)g(function)f(from)f(a)h(pro)q(duct)i Fl(D)c Fm(\002)e Fl(E)16 b Fo(is)d(con)o(tin)o(uous)h(if)f(its)h(sections)h (are)f(con)o(tin)o(uous,)f(or)0 743 y(as)i(one)g(migh)o(t)e(sa)o(y)i (if)f(it)g(is)h(con)o(tin)o(uous)g(in)f(eac)o(h)i(comp)q(onen)o(t)e (separately)m(.)21 b(Note)15 b(that)g(the)h(con)o(v)o(erse)g(follo)o (ws)d(from)g(the)0 792 y(con)o(tin)o(uit)o(y)g(of)g(the)i(functions)f Fl(\023)509 777 y Ff(e)509 803 y Fi(1)541 792 y Fo(and)g Fl(\023)637 777 y Ff(d)637 803 y Fi(2)656 792 y Fo(.)0 866 y Fk(4.13)k(Lemma.)24 b Fj(Supp)q(ose)16 b Fl(D)q Fj(,)h Fl(E)h Fj(and)d Fl(F)22 b Fj(are)16 b(cp)q(o's.)24 b(Then)17 b(a)e(function)h Fl(f)t Fo(:)7 b Fl(D)12 b Fm(\002)f Fl(E)17 b Fm(!)d Fl(F)21 b Fj(is)16 b(con)o(tin)o(uous)g(if)f (and)h(only)0 915 y(if)e(it)g(is)g(con)o(tin)o(uous)h(in)f(eac)o(h)h (comp)q(onen)o(t)f(separately)m(,)g(i.e.)20 b(if)13 b(all)h(sections)h Fl(f)1244 900 y Ff(e)1240 926 y Fi(1)1263 915 y Fo(:)7 b Fl(D)14 b Fm(!)e Fl(F)20 b Fj(giv)o(en)14 b(b)o(y)g Fl(f)1622 900 y Ff(e)1618 926 y Fi(1)1641 915 y Fo(\()p Fl(d)p Fo(\))e(=)h Fl(f)t Fo(\()p Fl(d;)7 b(e)p Fo(\))15 b Fj(and)0 965 y Fl(f)24 950 y Ff(d)20 975 y Fi(2)44 965 y Fo(:)7 b Fl(E)13 b Fm(!)e Fl(F)20 b Fj(giv)o(en)13 b(b)o(y)h Fl(f)397 950 y Ff(d)393 975 y Fi(2)417 965 y Fo(\()p Fl(e)p Fo(\))e(=)g Fl(f)t Fo(\()p Fl(d;)7 b(e)p Fo(\))14 b Fj(are)g(con)o(tin)o(uous.)0 1038 y Fk(Pro)q(of.)22 b Fo(If)14 b Fl(f)20 b Fo(is)15 b(con)o(tin)o(uous,)g(then)h(its)f (sections)i Fl(f)834 1023 y Ff(e)830 1049 y Fi(1)866 1038 y Fo(=)d Fl(f)h Fm(\016)10 b Fl(\023)993 1023 y Ff(e)993 1049 y Fi(1)1026 1038 y Fo(and)15 b Fl(f)1132 1023 y Ff(d)1128 1049 y Fi(2)1166 1038 y Fo(=)f Fl(f)h Fm(\016)10 b Fl(\023)1293 1023 y Ff(d)1293 1049 y Fi(2)1327 1038 y Fo(are)15 b(con)o(tin)o(uous)h(b)o(y)e(Lemma)f(2.4)h(and)0 1088 y(the)g(con)o(tin)o(uit)o(y)g(of)f(the)h(functions)g Fl(\023)580 1073 y Ff(e)580 1098 y Fi(1)612 1088 y Fo(and)g Fl(\023)708 1073 y Ff(d)708 1098 y Fi(2)727 1088 y Fo(.)83 1138 y(Con)o(v)o(ersely)m(,)h(w)o(e)g(\014rst)g(sho)o(w)g(that)g Fl(f)20 b Fo(is)14 b(monotone.)20 b(So)15 b(let)g(\()p Fl(d;)7 b(e)p Fo(\))12 b Fm(\024)i Fo(\()p Fl(d)1247 1123 y Fg(0)1258 1138 y Fl(;)7 b(e)1296 1123 y Fg(0)1308 1138 y Fo(\).)21 b(Then)15 b Fl(f)t Fo(\()p Fl(d;)7 b(e)p Fo(\))14 b(=)f Fl(f)1665 1123 y Ff(e)1661 1148 y Fi(1)1684 1138 y Fo(\()p Fl(d)p Fo(\))g Fm(\024)g Fl(f)1820 1123 y Ff(e)1816 1148 y Fi(1)1839 1138 y Fo(\()p Fl(d)1877 1123 y Fg(0)1888 1138 y Fo(\))h(=)0 1188 y Fl(f)t Fo(\()p Fl(d)62 1173 y Fg(0)74 1188 y Fl(;)7 b(e)p Fo(\))13 b(=)f Fl(f)209 1173 y Ff(d)226 1160 y Fb(0)205 1198 y Fi(2)241 1188 y Fo(\()p Fl(e)p Fo(\))h Fm(\024)g Fl(f)374 1173 y Ff(d)391 1160 y Fb(0)370 1198 y Fi(2)405 1188 y Fo(\()p Fl(e)440 1173 y Fg(0)452 1188 y Fo(\))g(=)g Fl(f)t Fo(\()p Fl(d)588 1173 y Fg(0)600 1188 y Fl(;)7 b(e)638 1173 y Fg(0)649 1188 y Fo(\).)20 b(No)o(w)14 b(assume)g(that)h Fl(M)i Fm(\022)c Fl(D)e Fm(\002)f Fl(E)16 b Fo(is)f(directed.)21 b(W)m(e)14 b(m)o(ust)f(sho)o(w)i Fl(f)t Fo(\()1802 1157 y Fe(F)1844 1188 y Fl(M)5 b Fo(\))13 b(=)0 1206 y Fe(F)42 1237 y Fl(f)t Fo(\()p Fl(M)5 b Fo(\).)22 b(Clearly)15 b Fl(f)t Fo(\()363 1206 y Fe(F)405 1237 y Fl(M)5 b Fo(\))16 b(is)f(an)g(upp)q(er)h(b)q(ound)f(of)g Fl(f)t Fo(\()p Fl(M)5 b Fo(\).)23 b(No)o(w)15 b(supp)q(ose)h Fl(u)f Fo(is)g(some)f(upp)q(er)j(b)q(ound)e(of)f Fl(f)t Fo(\()p Fl(M)5 b Fo(\).)23 b(W)m(e)0 1287 y(m)o(ust)13 b(sho)o(w)h Fl(f)t Fo(\()246 1256 y Fe(F)288 1287 y Fl(M)5 b Fo(\))12 b Fm(\024)f Fl(u)p Fo(.)18 b(No)o(w)c(b)o(y)f(Lemma)f(4.1)825 1256 y Fe(F)867 1287 y Fl(\031)891 1293 y Fi(left)940 1287 y Fo(\()p Fl(M)5 b Fo(\))14 b(and)1112 1256 y Fe(F)1153 1287 y Fl(\031)1177 1293 y Fi(righ)o(t)1248 1287 y Fo(\()p Fl(M)5 b Fo(\))12 b(=:)f Fl(e)1411 1272 y Fg(\003)1445 1287 y Fo(exist)j(and)f(w)o(e)i(ha)o(v)o(e)618 1389 y Fl(f)t Fo(\()658 1350 y Fe(G)712 1389 y Fl(M)5 b Fo(\))11 b(=)h Fl(f)t Fo(\()868 1350 y Fe(G)922 1389 y Fl(\031)946 1395 y Fi(left)995 1389 y Fo(\()p Fl(M)5 b Fo(\))p Fl(;)1091 1350 y Fe(G)1144 1389 y Fl(\031)1168 1395 y Fi(righ)o(t)1239 1389 y Fo(\()p Fl(M)g Fo(\)\))784 1472 y(=)887 1433 y Fe(G)828 1524 y Ff(d)p Fg(2)p Ff(\031)886 1528 y Fd(left)931 1524 y Fi(\()p Ff(M)s Fi(\))998 1472 y Fl(f)t Fo(\()p Fl(d;)1079 1433 y Fe(G)1132 1472 y Fl(\031)1156 1478 y Fi(righ)o(t)1227 1472 y Fo(\()p Fl(M)g Fo(\)\))219 b(since)15 b Fl(f)1665 1457 y Ff(e)1681 1445 y Fb(\003)1661 1483 y Fi(1)1714 1472 y Fo(is)f(con)o(tin)o(uous)784 1596 y(=)887 1556 y Fe(G)828 1647 y Ff(d)p Fg(2)p Ff(\031)886 1651 y Fd(left)931 1647 y Fi(\()p Ff(M)s Fi(\))1065 1556 y Fe(G)998 1647 y Ff(e)p Fg(2)p Ff(\031)1055 1651 y Fd(righ)o(t)1118 1647 y Fi(\()p Ff(M)s Fi(\))1185 1596 y Fl(f)t Fo(\()p Fl(d;)7 b(e)p Fo(\))242 b(since)15 b Fl(f)1669 1580 y Ff(d)1665 1606 y Fi(2)1703 1596 y Fo(is)f(con)o(tin)o(uous.)0 1737 y(Let)j Fl(d)d Fm(2)h Fl(\031)180 1743 y Fi(left)229 1737 y Fo(\()p Fl(M)5 b Fo(\))17 b(and)e Fl(e)h Fm(2)f Fl(\031)507 1743 y Fi(righ)o(t)578 1737 y Fo(\()p Fl(M)5 b Fo(\).)25 b(Then)16 b(\()p Fl(d;)7 b(e)878 1722 y Fg(0)889 1737 y Fo(\))16 b Fm(2)f Fl(M)20 b Fo(for)c(some)f Fl(e)1215 1722 y Fg(0)1243 1737 y Fm(2)f Fl(E)19 b Fo(and)c(\()p Fl(d)1455 1722 y Fg(0)1467 1737 y Fl(;)7 b(e)p Fo(\))15 b Fm(2)g Fl(M)21 b Fo(for)15 b(some)h Fl(d)1834 1722 y Fg(0)1860 1737 y Fm(2)f Fl(D)q Fo(.)0 1787 y(Since)f Fl(M)k Fo(is)12 b(directed,)j(there)f(is)f(some)f(\()p Fl(d)664 1772 y Fg(00)685 1787 y Fl(;)7 b(e)723 1772 y Fg(00)744 1787 y Fo(\))12 b Fm(2)f Fl(M)18 b Fo(with)13 b Fl(d;)7 b(d)1026 1772 y Fg(0)1048 1787 y Fm(\024)12 b Fl(d)1114 1772 y Fg(00)1147 1787 y Fo(and)h Fl(e;)7 b(e)1284 1772 y Fg(0)1308 1787 y Fm(\024)k Fl(e)1370 1772 y Fg(00)1392 1787 y Fo(.)18 b(Hence)c(from)e(the)h(monotonicit)o (y)0 1837 y(of)g Fl(f)19 b Fo(w)o(e)14 b(get)g Fl(f)t Fo(\()p Fl(d;)7 b(e)p Fo(\))12 b Fm(\024)g Fl(f)t Fo(\()p Fl(d)450 1822 y Fg(00)471 1837 y Fl(;)7 b(e)509 1822 y Fg(00)530 1837 y Fo(\))12 b Fm(\024)g Fl(u)p Fo(.)18 b(Since)c(this)g(holds)f(for)h(an)o(y)f Fl(e)f Fm(2)f Fl(\031)1189 1843 y Fi(righ)o(t)1260 1837 y Fo(\()p Fl(M)5 b Fo(\),)14 b(w)o(e)g(ha)o(v)o(e)1519 1806 y Fe(F)1554 1850 y Ff(e)p Fg(2)p Ff(\031)1611 1854 y Fd(righ)o(t)1673 1850 y Fi(\()p Ff(M)s Fi(\))1743 1837 y Fl(f)t Fo(\()p Fl(d;)7 b(e)p Fo(\))12 b Fm(\024)g Fl(u)p Fo(,)0 1893 y(and)i(since)g(this)g(again)f(holds)h(for)f(an)o(y)h Fl(d)d Fm(2)g Fl(\031)721 1899 y Fi(left)770 1893 y Fo(\()p Fl(M)5 b Fo(\),)14 b(w)o(e)g(get)1003 1862 y Fe(F)1037 1906 y Ff(d)p Fg(2)p Ff(\031)1095 1910 y Fd(left)1140 1906 y Fi(\()p Ff(M)s Fi(\))1210 1862 y Fe(F)1244 1906 y Ff(e)p Fg(2)p Ff(\031)1301 1910 y Fd(righ)o(t)1364 1906 y Fi(\()p Ff(M)s Fi(\))1433 1893 y Fl(f)t Fo(\()p Fl(d;)7 b(e)p Fo(\))12 b Fm(\024)g Fl(u)p Fo(.)265 b Fh(\003)p Fo(.)83 1943 y(No)o(w)14 b(w)o(e)g(can)g(sho)o(w)g(that)g (application)e(is)i(con)o(tin)o(uous.)0 2016 y Fk(4.14)20 b(Lemma.)k Fj(Let)18 b Fl(D)i Fj(and)d Fl(E)j Fj(b)q(e)f(cp)q(o's.)30 b(Then)18 b Fo(apply:)7 b(\()p Fl(D)19 b Fm(!)f Fl(E)r Fo(\))12 b Fm(\002)g Fl(D)20 b Fm(!)d Fl(E)j Fj(giv)o(en)e(b)o(y)f Fo(apply\()p Fl(f)r(;)7 b(d)p Fo(\))18 b(=)h Fl(f)t Fo(\()p Fl(d)p Fo(\))f Fj(is)0 2066 y(con)o(tin)o(uous.)0 2139 y Fk(Pro)q(of.)k Fo(By)16 b(Lemma)d(4.13)h(it)h(su\016ces)i(to)e(pro)o (v)o(e)g(that)h(apply)f(is)g(con)o(tin)o(uous)g(in)g(eac)o(h)h(argumen) o(t)e(separately)m(.)23 b(F)m(or)15 b(the)0 2189 y(second)f(argumen)o (t)e(this)h(is)g(trivial,)f(and)h(monotonicit)o(y)d(in)j(the)h(\014rst) g(argumen)o(t)e(follo)o(ws)f(from)h(the)i(p)q(oin)o(t)o(wise)e (de\014nition)0 2239 y(of)h Fm(\024)h Fo(in)g Fl(D)f Fm(!)e Fl(E)r Fo(.)18 b(So)c(assume)f Fm(f)p Fl(f)547 2245 y Ff(i)572 2239 y Fo(:)f Fl(i)f Fm(2)h Fl(I)s Fm(g)f(\022)h Fl(D)h Fm(!)e Fl(E)16 b Fo(is)e(directed.)19 b(Then)c(w)o(e)f(ha)o(v)o (e)676 2339 y(apply\()795 2300 y Fe(G)793 2389 y Ff(i)p Fg(2)p Ff(I)851 2339 y Fl(f)871 2345 y Ff(i)885 2339 y Fl(;)7 b(d)p Fo(\))k(=)h(\()1015 2300 y Fe(G)1013 2389 y Ff(i)p Fg(2)p Ff(I)1071 2339 y Fl(f)1091 2345 y Ff(i)1105 2339 y Fo(\)\()p Fl(d)p Fo(\))953 2455 y(=)999 2416 y Fe(G)997 2505 y Ff(i)p Fg(2)p Ff(I)1055 2455 y Fl(f)1075 2461 y Ff(i)1089 2455 y Fo(\()p Fl(d)p Fo(\))548 b(b)o(y)14 b(Lemma)d(4.6)953 2572 y(=)999 2532 y Fe(G)997 2621 y Ff(i)p Fg(2)p Ff(I)1055 2572 y Fo(apply\()p Fl(f)1192 2578 y Ff(i)1206 2572 y Fl(;)c(d)p Fo(\))p Fl(:)643 b Fh(\003)965 2770 y Fo(9)p eop %%Page: 10 10 10 9 bop 0 42 a Fk(5.)21 b(P)o(artial)14 b(con)o(tin)o(uo)o(us)f (functional)o(s)83 153 y Fo(W)m(e)c(no)o(w)g(construct)j(the)e(partial) f(con)o(tin)o(uous)g(functionals.)16 b(First)10 b(w)o(e)g(de\014ne)h (for)e(an)o(y)g(simple)f(t)o(yp)q(e)i Fl(\045)g Fo(an)g(information)0 203 y(system)k Fk(C)172 209 y Ff(\045)204 203 y Fo(=)f(\()p Fl(C)295 209 y Ff(\045)314 203 y Fl(;)7 b Fo(Con)406 209 y Ff(\045)426 203 y Fl(;)g Fm(`)470 209 y Ff(\045)489 203 y Fo(\))14 b(b)o(y)h(letting)f Fk(C)744 209 y Fi(nat)806 203 y Fo(:=)f Fk(N)p Fo(,)h(i.e.)19 b(the)c(information)d(system)i (built)g(from)e(the)j(set)h(of)d(natural)0 253 y(n)o(um)o(b)q(ers)h(as) g(describ)q(ed)h(after)f(De\014nition)f(1.1,)g(and)795 344 y Fk(C)829 350 y Ff(\045)p Fg(!)p Ff(\033)913 344 y Fo(:=)f Fk(C)1003 350 y Ff(\045)1034 344 y Fm(!)f Fk(C)1121 350 y Ff(\033)1144 344 y Fl(;)802 407 y Fk(C)836 413 y Ff(\045)p Fg(\002)p Ff(\033)913 407 y Fo(:=)h Fk(C)1003 389 y Fg(0)1003 417 y Ff(\045)1032 407 y Fm(\002)d Fk(C)1107 389 y Fg(00)1107 417 y Ff(\033)1130 407 y Fl(;)0 502 y Fo(where)18 b Fk(C)157 487 y Fg(0)157 512 y Ff(\045)176 502 y Fo(,)f Fk(C)239 487 y Fg(00)239 512 y Ff(\033)279 502 y Fo(are)f(v)n(arian)o(ts)g(of)g Fk(C)594 508 y Ff(\045)614 502 y Fo(,)h Fk(C)677 508 y Ff(\033)716 502 y Fo(with)f(disjoin)o(t)g (sets)i(of)e(data)g(ob)r(jects)i(\(e.g.)25 b Fl(C)1477 487 y Fg(0)1474 512 y Ff(\045)1509 502 y Fo(=)17 b Fm(f)p Fo(\(0)p Fl(;)7 b(a)p Fo(\))15 b(:)g Fl(a)h Fm(2)g Fl(C)1827 508 y Ff(\045)1846 502 y Fm(g)g Fo(and)0 552 y Fl(C)33 537 y Fg(00)30 562 y Ff(\033)65 552 y Fo(=)c Fm(f)p Fo(\(1)p Fl(;)7 b(a)p Fo(\))k(:)g Fl(a)g Fm(2)h Fl(C)361 558 y Ff(\033)383 552 y Fm(g)p Fo(\).)83 602 y(The)h(information)c(systems)j Fk(C)576 608 y Ff(\045)608 602 y Fo(ha)o(v)o(e)g(a)g(rather)h(pleasan)o (t)g(prop)q(ert)o(y)m(,)f(whic)o(h)g(amoun)o(ts)f(to)h(the)h(p)q (ossibilit)o(y)e(to)h(lo)q(cate)0 652 y(inconsistencies)19 b(in)f(t)o(w)o(o{elemen)o(t)f(sets)i(of)e(data)h(ob)r(jects;)i(this)e (prop)q(ert)o(y)h(has)f(b)q(een)h(called)e(coheren)o(t)j(b)o(y)d (Plotkin)g(in)0 701 y([Plo78],)12 b(p.210.)18 b(It)d(is)f(de\014ned)h (as)g(follo)o(ws.)j(An)c(information)d(system)k Fk(A)d Fo(=)h(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))14 b(is)h(called)f Fn(c)n(oher)n(ent)k Fo(if)13 b(for)h(an)o(y)0 751 y(set)h Fl(X)g Fm(\022)146 736 y Fi(\014n)197 751 y Fl(A)f Fo(w)o(e)g(ha)o(v)o(e)614 801 y Fl(X)h Fm(2)c Fo(Con)23 b Fm(\()-7 b(\))22 b Fo(\()p Fm(8)p Fl(a;)7 b(b)k Fm(2)g Fl(X)s Fo(\))p Fm(f)p Fl(a;)c(b)p Fm(g)12 b(2)f Fo(Con)p Fl(:)0 877 y Fo(Clearly)i Fk(C)178 883 y Fi(nat)240 877 y Fo(=)f Fk(N)i Fo(is)g(coheren)o(t,)h(and)f (furthermore)f(w)o(e)h(ha)o(v)o(e)0 951 y Fk(Lemma)h(5.1.)24 b Fj(Let)15 b Fk(A)f Fj(and)f Fk(B)h Fj(b)q(e)h(information)c(systems.) 18 1001 y(\(i\))21 b(If)14 b Fk(B)f Fj(is)h(coheren)o(t,)h(then)f(so)g (is)g Fk(A)e Fm(!)f Fk(B)p Fj(.)7 1051 y(\(ii\))20 b(If)14 b Fk(A)f Fj(and)h Fk(B)g Fj(are)g(coheren)o(t,)h(then)g(so)f(is)f Fk(A)d Fm(\002)f Fk(B)p Fj(.)0 1124 y Fk(Pro)q(of.)18 b Fo(Let)c Fk(A)e Fo(=)f(\()p Fl(A;)c Fo(Con)453 1130 y Ff(A)480 1124 y Fl(;)g Fm(`)524 1130 y Ff(A)551 1124 y Fo(\))14 b(and)g Fk(B)d Fo(=)h(\()p Fl(B)r(;)7 b Fo(Con)893 1130 y Ff(B)921 1124 y Fl(;)g Fm(`)965 1130 y Ff(B)994 1124 y Fo(\).)83 1174 y(\(i\)Let)14 b Fm(f)p Fo(\()p Fl(X)272 1180 y Fi(1)291 1174 y Fl(;)7 b(b)328 1180 y Fi(1)346 1174 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)505 1180 y Ff(n)528 1174 y Fl(;)g(b)565 1180 y Ff(n)586 1174 y Fo(\))p Fm(g)12 b(\022)g Fo(Con)752 1180 y Ff(A)789 1174 y Fm(\002)d Fl(B)17 b Fo(and)c(assume)545 1268 y Fm(8)p Fl(i;)7 b(j:)p Fo(1)k Fm(\024)h Fl(i)g(<)g(j)i Fm(\024)e Fl(n)f Fm(!)g(f)p Fo(\()p Fl(X)1011 1274 y Ff(i)1025 1268 y Fl(;)c(b)1062 1274 y Ff(i)1075 1268 y Fo(\))p Fl(;)g Fo(\()p Fl(X)1160 1274 y Ff(j)1178 1268 y Fl(;)g(b)1215 1274 y Ff(j)1232 1268 y Fo(\))p Fm(g)k(2)g Fo(Con)p Fl(:)492 b Fo(\()p Fm(\003)p Fo(\))0 1362 y(W)m(e)15 b(ha)o(v)o(e)f(to)h(sho)o(w)g(that)g Fm(f)p Fo(\()p Fl(X)487 1368 y Fi(1)506 1362 y Fl(;)7 b(b)543 1368 y Fi(1)561 1362 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)720 1368 y Ff(n)743 1362 y Fl(;)g(b)780 1368 y Ff(n)802 1362 y Fo(\))p Fm(g)13 b(2)g Fo(Con.)21 b(So)15 b(assume)g Fl(I)i Fm(\022)c(f)p Fo(1)p Fl(;)7 b(:)g(:)g(:)e(;)i(n)p Fm(g)14 b Fo(and)1561 1331 y Fe(S)1595 1374 y Ff(i)p Fg(2)p Ff(I)1655 1362 y Fl(X)1689 1368 y Ff(i)1717 1362 y Fm(2)f Fo(Con)1832 1368 y Ff(A)1859 1362 y Fo(.)21 b(W)m(e)0 1412 y(m)o(ust)c(sho)o(w)i(that)f Fm(f)p Fl(b)348 1418 y Ff(i)381 1412 y Fo(:)g Fl(i)i Fm(2)e Fl(I)s Fm(g)i(2)e Fo(Con)673 1418 y Ff(B)702 1412 y Fo(.)31 b(No)o(w)18 b(since)h Fk(B)g Fo(is)f(coheren)o(t)i(b)o(y)e(assumption,)g(it)g (su\016ces)h(to)g(sho)o(w)f(that)0 1462 y Fm(f)p Fl(b)39 1468 y Ff(i)52 1462 y Fl(;)7 b(b)89 1468 y Ff(j)106 1462 y Fm(g)14 b(2)f Fo(Con)256 1468 y Ff(B)300 1462 y Fo(for)i(all)f Fl(i;)7 b(j)16 b Fm(2)e Fl(I)s Fo(.)23 b(So)15 b(let)g Fl(i;)7 b(j)16 b Fm(2)e Fl(I)s Fo(.)23 b(By)15 b(assumption)f(w)o(e)i (ha)o(v)o(e)f Fl(X)1353 1468 y Ff(i)1378 1462 y Fm([)9 b Fl(X)1449 1468 y Ff(j)1481 1462 y Fm(2)14 b Fo(Con)1597 1468 y Ff(A)1639 1462 y Fo(and)h(hence)i(b)o(y)e(\()p Fm(\003)p Fo(\))0 1511 y(and)f(the)g(de\014nition)g(of)f(Con)h(also)f Fm(f)p Fl(b)594 1517 y Ff(i)607 1511 y Fl(;)7 b(b)644 1517 y Ff(j)661 1511 y Fm(g)k(2)h Fo(Con)806 1517 y Ff(B)835 1511 y Fo(.)83 1561 y(\(ii\))20 b(W)m(e)g(ma)o(y)f(assume)h(that)h Fl(A)g Fo(and)f Fl(B)j Fo(are)e(disjoin)o(t.)38 b(Let)21 b Fl(a)1138 1567 y Fi(1)1157 1561 y Fl(;)7 b(:)g(:)g(:)t(;)g(a)1271 1567 y Ff(n)1316 1561 y Fm(2)23 b Fl(A)d Fo(and)h Fl(b)1524 1567 y Fi(1)1542 1561 y Fl(;)7 b(:)g(:)g(:)e(;)i(b)1653 1567 y Ff(m)1707 1561 y Fm(2)22 b Fl(B)i Fo(and)c(as-)0 1611 y(sume)15 b(that)g(an)o(y)g(t)o(w)o(o{elemen)o(t)f(subset)j(of)d Fm(f)p Fl(a)739 1617 y Fi(1)757 1611 y Fl(;)7 b(:)g(:)g(:)e(;)i(a)872 1617 y Ff(n)894 1611 y Fl(;)g(b)931 1617 y Fi(1)949 1611 y Fl(;)g(:)g(:)g(:)e(;)i(b)1060 1617 y Ff(m)1091 1611 y Fm(g)15 b Fo(is)g(consisten)o(t.)23 b(Then)16 b Fm(f)p Fl(a)1536 1617 y Fi(1)1554 1611 y Fl(;)7 b(:)g(:)g(:)e(;)i(a)1669 1617 y Ff(n)1691 1611 y Fm(g)14 b(2)f Fo(Con)1841 1617 y Ff(A)1883 1611 y Fo(and)0 1661 y Fm(f)p Fl(b)39 1667 y Fi(1)57 1661 y Fl(;)7 b(:)g(:)g(:)e(;)i(b)168 1667 y Ff(m)199 1661 y Fm(g)12 b(2)h Fo(Con)346 1667 y Ff(B)389 1661 y Fo(since)j Fk(A)e Fo(and)h Fk(B)f Fo(are)h(coheren)o(t.)22 b(Therefore)15 b Fm(f)p Fl(a)1159 1667 y Fi(1)1178 1661 y Fl(;)7 b(:)g(:)g(:)e(;)i(a)1293 1667 y Ff(n)1315 1661 y Fl(;)g(b)1352 1667 y Fi(1)1369 1661 y Fl(;)g(:)g(:)g(:)e(;)i(b)1480 1667 y Ff(m)1511 1661 y Fm(g)12 b(2)h Fo(Con)h(b)o(y)g(de\014nition)h (of)0 1711 y(Con.)1832 b Fh(\003)0 1784 y Fk(Corollary)14 b(5.2.)24 b Fj(The)15 b(information)c(systems)j Fk(C)809 1790 y Ff(\045)842 1784 y Fj(are)h(all)d(coheren)o(t.)785 b Fh(\003)83 1858 y Fo(W)m(e)18 b(no)o(w)g(let)h(Con)o(t)403 1864 y Ff(\045)442 1858 y Fo(:=)g Fm(j)p Fk(C)551 1864 y Ff(\045)570 1858 y Fm(j)f Fo(b)q(e)h(the)g(domain)d(of)i(elemen)o(ts) h(of)e Fm(j)p Fk(C)1213 1864 y Ff(\045)1233 1858 y Fm(j)p Fo(;)j(these)f(elemen)o(ts)g(are)g(called)f(the)h Fn(p)n(artial)0 1907 y(c)n(ontinuous)d(functionals)h Fo(of)c(t)o(yp)q(e)g Fl(\045)p Fo(.)19 b(The)14 b(partial)e(con)o(tin)o(uous)h(functionals)g (are)h(partial)e(in)h(the)i(sense)g(that)e(the)h(ob)r(ject)0 1957 y(unde\014ned,)k(i.e.)25 b Fm(?)15 b Fo(:=)p 390 1921 21 2 v 15 w Fm(;)p Fo(,)h(is)g(a)g(p)q(ossible)h(v)n(alue.)25 b(W)m(e)16 b(no)o(w)g(can)g(in)o(tro)q(duce)h(easily)f(the)h(total)f (con)o(tin)o(uous)g(functionals,)0 2007 y(i.e.)22 b(those)16 b(whic)o(h)f(for)g(total)g(argumen)o(ts)f(only)h(pro)q(duce)h(total)f (v)n(alues;)g(this)h(idea)f(clearly)g(can)h(b)q(e)g(turned)g(in)o(to)e (a)i(v)n(alid)0 2057 y(inductiv)o(e)g(de\014nition.)24 b(W)m(e)16 b(will)e(sho)o(w)j(that)f(the)g(total)g(functionals)f(are)i (dense)g(in)f(the)g(sense)i(that)e(an)o(y)g(\014nite)g(partial)0 2107 y(con)o(tin)o(uous)e(functional)f(\(i.e.)k(an)o(y)p 569 2073 38 2 v 14 w Fl(X)g Fo(with)d Fl(X)h Fm(2)c Fo(Con)877 2113 y Ff(\045)896 2107 y Fo(\))j(can)g(b)q(e)h(extended)g(to)f(a)g (total)f(functional.)83 2157 y(The)h(set)h Fl(G)266 2163 y Ff(\045)299 2157 y Fo(of)e(the)i Fn(total)i Fo(functionals)d(of)f(t)o (yp)q(e)h Fl(\045)h Fo(is)e(the)i(subset)g(of)e(Con)o(t)1280 2163 y Ff(\045)1313 2157 y Fo(de\014ned)i(b)o(y)491 2252 y Fl(G)524 2258 y Fi(nat)585 2252 y Fo(:=)d Fm(f)p Fl(z)h Fm(2)e Fo(Con)o(t)822 2258 y Fi(nat)884 2252 y Fo(:)g Fl(z)i Fm(6)p Fo(=)f Fm(;g)f Fo(=)h Fm(ff)p Fl(n)p Fm(g)f Fo(:)g Fl(n)g Fm(2)g Fk(N)p Fm(g)p Fl(;)469 2314 y(G)502 2320 y Ff(\045)p Fg(!)p Ff(\033)585 2314 y Fo(:=)h Fm(f)p Fl(z)h Fm(2)e Fo(Con)o(t)822 2320 y Ff(\045)p Fg(!)p Ff(\033)906 2314 y Fo(:)g Fm(8)p Fl(x)h Fm(2)f Fl(G)1060 2320 y Ff(\045)1079 2314 y Fl(:z)r(x)g Fm(2)g Fl(G)1219 2320 y Ff(\033)1241 2314 y Fm(g)p Fl(;)476 2377 y(G)509 2383 y Ff(\045)p Fg(\002)p Ff(\033)585 2377 y Fo(=)h Fm(f)p Fl(z)i Fm(2)d Fo(Con)o(t)811 2383 y Ff(\045)p Fg(\002)p Ff(\033)888 2377 y Fo(:)g Fl(z)g Fm(\\)e Fl(C)1008 2383 y Ff(\045)1038 2377 y Fm(2)j Fl(G)1111 2383 y Ff(\045)1143 2377 y Fo(and)i Fl(z)e Fm(\\)c Fl(C)1321 2383 y Ff(\033)1355 2377 y Fm(2)j Fl(G)1427 2383 y Ff(\033)1449 2377 y Fm(g)p Fl(:)0 2471 y Fo(Here)17 b(w)o(e)f(ha)o(v)o(e)f(written)h Fl(z)r(x)f Fo(instead)h(of)f(the)h(more)f(correct)i Fm(j)p Fl(z)r Fm(j)p Fo(\()p Fl(x)p Fo(\))e(\(cf.)23 b(Theorem)15 b(4.10\).)22 b(W)m(e)15 b(will)f(con)o(tin)o(ue)i(to)g(do)f(so)0 2521 y(b)q(elo)o(w.)83 2570 y(W)m(e)g(no)o(w)g(de\014ne)i(a)e(relation) g Fm(\030)588 2576 y Ff(\045)623 2570 y Fo(on)g Fl(G)715 2576 y Ff(\045)734 2570 y Fo(.)23 b(F)m(or)15 b Fl(z)r(;)7 b(z)906 2555 y Fg(0)931 2570 y Fm(2)14 b Fl(G)1006 2576 y Fi(nat)1071 2570 y Fo(let)h Fl(z)i Fm(\030)1200 2576 y Fi(nat)1264 2570 y Fl(z)1285 2555 y Fg(0)1325 2570 y Fm(\()-7 b(\))28 b Fl(z)16 b Fo(=)e Fl(z)1532 2555 y Fg(0)1544 2570 y Fo(.)23 b(F)m(or)15 b Fl(z)r(;)7 b(z)1716 2555 y Fg(0)1741 2570 y Fm(2)14 b Fl(G)1816 2576 y Ff(\045)p Fg(!)p Ff(\033)1904 2570 y Fo(let)0 2620 y Fl(z)i Fm(\030)67 2626 y Ff(\045)p Fg(!)p Ff(\033)154 2620 y Fl(z)175 2605 y Fg(0)214 2620 y Fm(\()-7 b(\))27 b(8)p Fl(x)13 b Fm(2)g Fl(G)452 2626 y Ff(\045)471 2620 y Fl(:z)r(x)h Fm(\030)574 2626 y Ff(\033)610 2620 y Fl(z)631 2605 y Fg(0)643 2620 y Fl(x)p Fo(.)21 b(F)m(or)15 b Fl(z)r(;)7 b(z)837 2605 y Fg(0)862 2620 y Fm(2)14 b Fl(G)937 2626 y Ff(\045)p Fg(\002)p Ff(\033)1017 2620 y Fo(let)h Fl(z)h Fm(\030)1145 2626 y Ff(\045)p Fg(\002)p Ff(\033)1225 2620 y Fl(z)1246 2605 y Fg(0)1285 2620 y Fm(\()-7 b(\))27 b Fl(\031)1413 2626 y Fi(left)1462 2620 y Fl(z)16 b Fm(\030)1529 2626 y Ff(\045)1562 2620 y Fl(\031)1586 2626 y Fi(left)1635 2620 y Fl(z)1656 2605 y Fg(0)1683 2620 y Fo(and)f Fl(\031)1789 2626 y Fi(righ)o(t)1860 2620 y Fl(z)h Fm(\030)1927 2626 y Ff(\033)0 2670 y Fl(\031)24 2676 y Fi(righ)o(t)95 2670 y Fl(z)116 2655 y Fg(0)128 2670 y Fo(.)i(Clearly)13 b Fm(\030)334 2676 y Ff(\045)367 2670 y Fo(is)h(an)g(equiv)n(alence)g (relation.)954 2770 y(10)p eop %%Page: 11 11 11 10 bop 83 42 a Fo(W)m(e)16 b(ob)o(viously)f(w)o(an)o(t)h(to)h(kno)o (w)f(that)g Fm(\030)736 48 y Ff(\045)772 42 y Fo(is)h(compatible)d (with)j(application.)24 b(The)17 b(only)f(non)o(trivial)e(part)j(of)f (this)0 91 y(argumen)o(t)g(is)g(to)h(sho)o(w)f(that)h Fl(x)f Fm(\030)557 97 y Ff(\045)593 91 y Fl(y)34 b Fo(=)-7 b Fm(\))32 b Fl(z)r(x)16 b Fm(\030)839 97 y Ff(\033)878 91 y Fl(z)r(y)q Fo(.)27 b(This)17 b(has)g(\014rst)g(b)q(een)h(pro)o(v)o (ed)f(b)o(y)f(Y)m(uri)g(Ersho)o(v)i([Ers75],)e(as)0 141 y(an)e(application)f(of)h(the)h(densit)o(y)g(theorem.)k(Ho)o(w)o(ev)o (er,)14 b(a)g(simpler)g(pro)q(of)g(has)g(later)h(b)q(een)g(found)f(b)o (y)g(Longo)g(and)g(Moggi)0 191 y([LM84],)g(and)h(w)o(e)g(presen)o(t)i (that)e(one.)23 b(First)15 b(w)o(e)h(need)g(some)e(Lemmata.)19 b(The)d(\014rst)g(one)f(notes)h(a)f(rather)h(ob)o(vious)f(fact)0 241 y(on)f(total)f(functionals.)0 317 y Fk(5.3)j(Lemma.)23 b Fj(If)14 b Fl(z)g Fm(2)d Fl(G)414 323 y Ff(\045)433 317 y Fj(,)i Fl(z)479 302 y Fg(0)502 317 y Fm(2)f Fo(Con)o(t)630 323 y Ff(\045)664 317 y Fj(and)h Fl(z)h Fm(\022)e Fl(z)842 302 y Fg(0)853 317 y Fj(,)i(then)g Fl(z)994 302 y Fg(0)1018 317 y Fm(2)d Fl(G)1090 323 y Ff(\045)1109 317 y Fj(.)0 394 y Fk(Pro)q(of.)18 b Fo(By)c(induction)f(on)h Fl(\045)p Fo(.)k(F)m(or)c(the)g(ground)g(t)o(yp)q(e)h(nat)e(the)i(claim)d(is)h (ob)o(vious.)83 444 y Fl(\045)j Fm(!)e Fl(\033)q Fo(:)23 b(Assume)16 b Fl(z)h Fm(2)e Fl(G)504 450 y Ff(\045)p Fg(!)p Ff(\033)592 444 y Fo(and)h Fl(z)h Fm(\022)f Fl(z)780 429 y Fg(0)792 444 y Fo(.)24 b(W)m(e)16 b(m)o(ust)f(sho)o(w)h Fl(z)1133 429 y Fg(0)1160 444 y Fm(2)f Fl(G)1236 450 y Ff(\045)p Fg(!)p Ff(\033)1308 444 y Fo(.)24 b(So)16 b(let)g Fl(x)f Fm(2)g Fl(G)1581 450 y Ff(\045)1600 444 y Fo(.)25 b(W)m(e)15 b(ha)o(v)o(e)h(to)g(sho)o(w)0 494 y Fl(z)21 479 y Fg(0)33 494 y Fl(x)11 b Fm(2)g Fl(G)140 500 y Ff(\033)162 494 y Fo(.)18 b(But)d Fl(z)296 479 y Fg(0)307 494 y Fl(x)d Fm(\023)g Fl(z)r(x)f Fm(2)g Fl(G)515 500 y Ff(\033)537 494 y Fo(,)i(so)h(the)h(claim)d(follo)o(ws)g(b)o(y)i (induction)f(h)o(yp)q(othesis.)83 544 y Fl(\045)d Fm(\002)f Fl(\033)q Fo(:)18 b(The)d(claim)c(follo)o(ws)i(easily)g(from)f(the)j (monotonicit)o(y)c(of)i(the)i(pro)r(jection)f(functions.)347 b Fh(\003)0 621 y Fk(5.4)16 b(Lemma.)23 b Fj(F)m(or)14 b(an)o(y)f Fl(z)439 627 y Fi(1)458 621 y Fl(;)7 b(z)496 627 y Fi(2)526 621 y Fm(2)k Fo(Con)o(t)654 627 y Ff(\045)p Fg(!)p Ff(\033)741 621 y Fj(and)i Fl(x)f Fm(2)f Fo(Con)o(t)985 627 y Ff(\045)1018 621 y Fj(w)o(e)j(ha)o(v)o(e)g Fo(\()p Fl(z)1210 627 y Fi(1)1238 621 y Fm(\\)9 b Fl(z)1294 627 y Fi(2)1313 621 y Fo(\))p Fl(x)i Fo(=)h Fl(z)1427 627 y Fi(1)1446 621 y Fl(x)d Fm(\\)g Fl(z)1535 627 y Fi(2)1554 621 y Fl(x)p Fj(.)0 697 y Fk(Pro)q(of.)18 b Fo(By)c(the)g(de\014nition) g(of)f Fm(j)p Fl(r)q Fm(j)g Fo(in)g(Theorem)h(3.2)f(w)o(e)h(ha)o(v)o(e) 206 800 y Fm(j)p Fl(z)237 806 y Fi(1)265 800 y Fm(\\)9 b Fl(z)321 806 y Fi(2)340 800 y Fm(j)p Fl(x)i Fo(=)g Fm(f)p Fl(b)h Fm(2)f Fl(C)550 806 y Ff(\033)583 800 y Fo(:)g(\()p Fm(9)p Fl(X)16 b Fm(\022)727 783 y Fi(\014n)778 800 y Fl(x)p Fo(\)\()p Fl(X)q(;)7 b(b)p Fo(\))k Fm(2)g Fl(z)991 806 y Fi(1)1019 800 y Fm(\\)e Fl(z)1075 806 y Fi(2)1094 800 y Fm(g)387 869 y Fo(=)i Fm(f)p Fl(b)h Fm(2)f Fl(C)550 875 y Ff(\033)583 869 y Fo(:)g(\()p Fm(9)p Fl(X)679 875 y Fi(1)710 869 y Fm(\022)742 852 y Fi(\014n)793 869 y Fl(x)p Fo(\)\()p Fl(X)883 875 y Fi(1)902 869 y Fl(;)c(b)p Fo(\))k Fm(2)g Fl(z)1024 875 y Fi(1)1043 869 y Fm(g)e(\\)g(f)p Fl(b)i Fm(2)g Fl(C)1229 875 y Ff(\033)1263 869 y Fo(:)g(\()p Fm(9)p Fl(X)1359 875 y Fi(2)1390 869 y Fm(\022)1422 852 y Fi(\014n)1473 869 y Fl(x)p Fo(\)\()p Fl(X)1563 875 y Fi(1)1582 869 y Fl(;)c(b)p Fo(\))k Fm(2)g Fl(z)1704 875 y Fi(2)1723 869 y Fm(g)387 931 y Fo(=)g Fm(j)p Fl(z)461 937 y Fi(1)480 931 y Fm(j)p Fl(x)d Fm(\\)h(j)p Fl(z)592 937 y Fi(2)611 931 y Fm(j)p Fl(x)0 1037 y Fo(The)18 b(part)h Fm(\022)f Fo(of)f(the)i(middle)d(equalit)o(y)h(is)h(ob)o (vious.)30 b(F)m(or)17 b Fm(\023)p Fo(,)i(let)f Fl(X)1127 1043 y Ff(i)1160 1037 y Fm(\022)1192 1022 y Fi(\014n)1249 1037 y Fl(x)g Fo(with)g(\()p Fl(X)1440 1043 y Ff(i)1454 1037 y Fl(;)7 b(b)p Fo(\))18 b Fm(2)g Fl(z)1590 1043 y Ff(i)1622 1037 y Fo(b)q(e)h(giv)o(en.)29 b(Cho)q(ose)0 1086 y Fl(X)15 b Fo(=)d Fl(X)127 1092 y Fi(1)155 1086 y Fm([)d Fl(X)226 1092 y Fi(2)245 1086 y Fo(.)18 b(Then)d(clearly)e(\() p Fl(X)q(;)7 b(b)p Fo(\))12 b Fm(2)f Fl(z)690 1092 y Ff(i)718 1086 y Fo(\(as)j Fm(f)p Fo(\()p Fl(X)856 1092 y Ff(i)870 1086 y Fl(;)7 b(b)p Fo(\))p Fm(g)k(`)h Fo(\()p Fl(X)q(;)7 b(b)p Fo(\))13 b(and)h Fl(z)1209 1092 y Ff(i)1237 1086 y Fo(is)g(deductiv)o(ely)g(closed\).)283 b Fh(\003)0 1163 y Fk(5.5)16 b(Lemma.)23 b Fj(F)m(or)14 b(an)o(y)f Fl(z)r(;)7 b(z)481 1148 y Fg(0)504 1163 y Fm(2)k Fl(G)576 1169 y Ff(\045)609 1163 y Fj(w)o(e)j(ha)o(v)o(e)g Fl(z)g Fm(\030)831 1169 y Ff(\045)862 1163 y Fl(z)883 1148 y Fg(0)918 1163 y Fm(\()-7 b(\))22 b Fl(z)11 b Fm(\\)e Fl(z)1105 1148 y Fg(0)1128 1163 y Fm(2)j Fl(G)1201 1169 y Ff(\045)1219 1163 y Fj(.)0 1239 y Fk(Pro)q(of.)18 b Fo(By)c(induction)f(on)h Fl(\045)p Fo(.)k(F)m(or)c(the)g(ground)g(t)o (yp)q(e)h(nat)e(the)i(claim)d(is)h(ob)o(vious.)83 1290 y Fl(\045)f Fm(!)f Fl(\033)q Fo(:)358 1339 y Fl(z)j Fm(\030)423 1345 y Ff(\045)p Fg(!)p Ff(\033)507 1339 y Fl(z)528 1322 y Fg(0)563 1339 y Fm(\()-7 b(\))22 b(8)p Fl(x)11 b Fm(2)h Fl(G)793 1345 y Ff(\045)811 1339 y Fl(:z)r(x)f Fm(\030)911 1345 y Ff(\033)946 1339 y Fl(z)967 1322 y Fg(0)978 1339 y Fl(x)563 1401 y Fm(\()-7 b(\))22 b(8)p Fl(x)11 b Fm(2)h Fl(G)793 1407 y Ff(\045)811 1401 y Fl(:z)r(x)d Fm(\\)g Fl(z)935 1384 y Fg(0)947 1401 y Fl(x)i Fm(2)g Fl(G)1054 1407 y Ff(\033)1159 1401 y Fo(b)o(y)j(induction)f(h)o(yp)q(othesis)563 1463 y Fm(\()-7 b(\))22 b(8)p Fl(x)11 b Fm(2)h Fl(G)793 1469 y Ff(\045)811 1463 y Fl(:)p Fo(\()p Fl(z)g Fm(\\)c Fl(z)927 1446 y Fg(0)939 1463 y Fo(\))p Fl(x)k Fm(2)f Fl(G)1063 1469 y Ff(\033)1168 1463 y Fo(b)o(y)i(Lemma)f(5.4)563 1526 y Fm(\()-7 b(\))22 b Fl(z)11 b Fm(\\)e Fl(z)750 1509 y Fg(0)774 1526 y Fm(2)i Fl(G)846 1532 y Ff(\045)p Fg(!)p Ff(\033)83 1600 y Fl(\045)f Fm(\002)f Fl(\033)q Fo(:)315 1701 y Fl(z)k Fm(\030)379 1707 y Ff(\045)p Fg(\002)p Ff(\033)457 1701 y Fl(z)478 1684 y Fg(0)512 1701 y Fm(\()-7 b(\))23 b Fl(\031)636 1707 y Fi(left)685 1701 y Fl(z)13 b Fm(\030)749 1707 y Ff(\045)780 1701 y Fl(\031)804 1707 y Fi(left)854 1701 y Fl(z)875 1684 y Fg(0)928 1701 y Fo(and)41 b Fl(\031)1060 1707 y Fi(righ)o(t)1131 1701 y Fl(z)14 b Fm(\030)1196 1707 y Ff(\033)1230 1701 y Fl(\031)1254 1707 y Fi(righ)o(t)1325 1701 y Fl(z)1346 1684 y Fg(0)512 1764 y Fm(\()-7 b(\))23 b Fo(\()p Fl(\031)652 1770 y Fi(left)701 1764 y Fl(z)r Fo(\))9 b Fm(\\)g Fo(\()p Fl(\031)824 1770 y Fi(left)874 1764 y Fl(z)895 1746 y Fg(0)906 1764 y Fo(\))j Fm(2)f Fl(G)1006 1770 y Ff(\045)1067 1764 y Fo(and)41 b(\()p Fl(\031)1215 1770 y Fi(righ)o(t)1286 1764 y Fl(z)r Fo(\))10 b Fm(\\)f Fo(\()p Fl(\031)1410 1770 y Fi(righ)o(t)1481 1764 y Fl(z)1502 1746 y Fg(0)1514 1764 y Fo(\))i Fm(2)g Fl(G)1613 1770 y Ff(\033)512 1826 y Fm(\()-7 b(\))23 b Fl(\031)636 1832 y Fi(left)685 1826 y Fo(\()p Fl(z)11 b Fm(\\)e Fl(z)789 1809 y Fg(0)801 1826 y Fo(\))j Fm(2)f Fl(G)901 1832 y Ff(\045)961 1826 y Fo(and)42 b Fl(\031)1094 1832 y Fi(righ)o(t)1165 1826 y Fo(\()p Fl(z)11 b Fm(\\)e Fl(z)1269 1809 y Fg(0)1281 1826 y Fo(\))i Fm(2)h Fl(G)1381 1832 y Ff(\033)512 1888 y Fm(\()-7 b(\))23 b Fl(z)11 b Fm(\\)e Fl(z)700 1871 y Fg(0)723 1888 y Fm(2)i Fl(G)795 1894 y Ff(\045)p Fg(\002)p Ff(\033)1918 1888 y Fh(\003)0 1990 y Fk(5.6)16 b(Theorem.)23 b Fj(F)m(or)13 b(an)o(y)h Fl(x;)7 b(y)12 b Fm(2)f Fl(G)601 1996 y Ff(\045)634 1990 y Fj(and)j Fl(z)g Fm(2)d Fl(G)820 1996 y Ff(\045)p Fg(!)p Ff(\033)906 1990 y Fj(w)o(e)j(ha)o(v)o(e)g Fl(x)d Fm(\030)1130 1996 y Ff(\045)1161 1990 y Fl(y)25 b Fo(=)-7 b Fm(\))23 b Fl(z)r(x)11 b Fm(\030)1384 1996 y Ff(\033)1419 1990 y Fl(z)r(y)q Fj(.)0 2067 y Fk(Pro)q(of.)18 b Fo(Since)c Fl(x)d Fm(\030)323 2073 y Ff(\045)354 2067 y Fl(y)16 b Fo(w)o(e)e(ha)o(v)o(e)f Fl(x)c Fm(\\)g Fl(y)k Fm(2)e Fl(G)721 2073 y Ff(\045)754 2067 y Fo(b)o(y)i(Lemma)f(5.5.)k(No) o(w)e Fl(z)r(x;)7 b(z)r(y)13 b Fm(\023)f Fl(z)r Fo(\()p Fl(x)d Fm(\\)f Fl(y)q Fo(\))15 b(b)o(y)e(Theorem)h(3.2,)e(and)i(hence)0 2116 y Fl(z)r(x)9 b Fm(\\)g Fl(z)r(y)k Fm(2)f Fl(G)218 2122 y Ff(\033)254 2116 y Fo(b)o(y)h(Lemma)e(5.3.)17 b(But)e(this)f(implies)e Fl(z)r(x)f Fm(\030)935 2122 y Ff(\033)969 2116 y Fl(z)r(y)16 b Fo(again)d(b)o(y)g(Lemma)f(5.5.)509 b Fh(\003)83 2167 y Fo(W)m(e)13 b(no)o(w)f(pro)o(v)o(e)i(the)f(densit)o (y)h(theorem,)e(whic)o(h)h(sa)o(ys)g(that)g(an)o(y)g(\014nite)g (partial)f(con)o(tin)o(uous)h(functional)f(\(i.e.)18 b(an)o(y)p 1912 2133 38 2 v 12 w Fl(X)0 2217 y Fo(with)13 b Fl(X)i Fm(2)c Fo(Con)256 2223 y Ff(\045)275 2217 y Fo(\))i(can)h(b)q(e)f(extended)i(to)e(a)g(total)f(functional.)17 b(This)c(result)h(is)f(essen)o(tially)g(due)g(to)g(Kreisel)h([Kre59].)k (The)0 2266 y(pro)q(of)e(w)o(e)g(giv)o(e)g(here)h(has)f(b)q(een)i (obtained)e(b)o(y)g(sp)q(ecialization)f(of)h(a)g(pro)q(of)g(for)f(a)h (more)f(general)i(densit)o(y)f(theorem)g(\(for)0 2316 y(arbitrary)c Fl(f)195 2322 y Fi(0)214 2316 y Fo({spaces\))i(giv)o(en)e (b)o(y)g(Berger)i(in)e([Ber90,)g(Ber93])g(\(this)h(pro)q(of)f(also)g (app)q(ears)h(in)f([SHGL94]\).)k(Berger's)d(pro)q(of)0 2366 y(in)h(turn)h(has)f(b)q(een)i(obtained)e(b)o(y)g(abstracting)h (some)e(general)i(features)g(from)e(a)h(pro)q(of)g(of)g(a)g(densit)o(y) g(theorem)g(giv)o(en)g(b)o(y)0 2416 y(Ersho)o(v)g(in)g([Ers74,)f (Ers75].)0 2492 y Fk(5.7)j(Densit)o(y)d(Theorem.)23 b Fj(F)m(or)14 b(an)o(y)f Fl(X)i Fm(2)d Fo(Con)791 2498 y Ff(\045)824 2492 y Fj(w)o(e)j(can)f(\014nd)g(an)f Fl(x)f Fm(2)f Fl(G)1210 2498 y Ff(\045)1243 2492 y Fj(suc)o(h)j(that)g Fl(X)h Fm(\022)d Fl(x)p Fj(.)0 2569 y Fk(Pro)q(of.)18 b Fo(Call)12 b(a)i(t)o(yp)q(e)g Fl(\045)h Fn(dense)i Fo(if)c Fm(8)p Fl(X)j Fm(2)11 b Fo(Con)739 2575 y Ff(\045)758 2569 y Fm(9)p Fl(x)g Fm(2)g Fl(G)888 2575 y Ff(\045)907 2569 y Fl(:X)k Fm(\022)d Fl(x)p Fo(.)18 b(F)m(urthermore)13 b(call)g(a)h(t)o(yp)q(e)g Fl(\045)h Fn(sep)n(ar)n(ating)i Fo(if)380 2670 y Fm(8)p Fl(X)437 2676 y Fi(1)456 2670 y Fl(;)7 b(X)509 2676 y Fi(2)539 2670 y Fm(2)k Fo(Con)652 2676 y Ff(\045)671 2670 y Fo(\()p Fl(X)721 2676 y Fi(1)749 2670 y Fm([)e Fl(X)820 2676 y Fi(2)855 2670 y Fl(=)-25 b Fm(2)11 b Fo(Con)964 2676 y Ff(\045)995 2670 y Fm(!)g(9)o Fl(~)-20 b(z)13 b Fm(2)f Fl(G:)p Fm(;)e(6)p Fo(=)p 1263 2637 54 2 v 12 w Fl(X)1297 2676 y Fi(1)1315 2670 y Fl(~)-20 b(z)14 b Fm(6)p Fo(=)p 1392 2637 V 11 w Fl(X)1426 2676 y Fi(2)1445 2670 y Fl(~)-21 b(z)14 b Fm(6)p Fo(=)e Fm(;)p Fo(\))p Fl(:)954 2770 y Fo(11)p eop %%Page: 12 12 12 11 bop 0 42 a Fo(Here)14 b Fl(~)-20 b(z)14 b Fm(2)d Fl(G)i Fo(means)h(that)f Fl(~)-21 b(z)16 b Fo(is)e(a)g(sequence)i(of)d (total)g Fl(z)882 48 y Ff(i)910 42 y Fo(or)h(of)f(0)h(or)g(1)f(suc)o(h) i(that)f Fl(X)1346 48 y Ff(j)1363 42 y Fl(~)-20 b(z)16 b Fo(is)e(of)f(t)o(yp)q(e)h(nat.)83 91 y(W)m(e)i(pro)o(v)o(e)g(b)o(y)f (sim)o(ultaneous)g(induction)g(on)h Fl(\045)g Fo(that)g(an)o(y)f(t)o (yp)q(e)i Fl(\045)f Fo(is)g(dense)h(and)f(separating.)24 b(This)16 b(extension)h(of)0 141 y(our)d(claim)e(is)h(helpful)h(to)f (get)h(the)h(induction)e(through.)83 191 y(F)m(or)h(the)g(ground)g(t)o (yp)q(e)g(nat)g(b)q(oth)g(claims)e(are)i(ob)o(vious.)83 241 y Fl(\045)e Fm(!)f Fl(\033)j Fo(is)f(separating:)18 b(This)13 b(will)f(follo)o(w)f(from)h(the)h(inductiv)o(e)h(h)o(yp)q (otheses)g(that)g Fl(\045)f Fo(is)g(dense)i(and)e Fl(\033)h Fo(is)f(separating.)0 291 y(So)18 b(let)g Fl(W)o(;)7 b(W)228 276 y Fg(0)257 291 y Fm(2)18 b Fo(Con)376 297 y Ff(\045)p Fg(!)p Ff(\033)467 291 y Fo(suc)o(h)g(that)g Fl(W)g Fm([)12 b Fl(W)800 276 y Fg(0)834 291 y Fl(=)-26 b Fm(2)18 b Fo(Con)949 297 y Ff(\045)p Fg(!)p Ff(\033)1022 291 y Fo(.)30 b(Since)18 b Fk(C)1210 297 y Ff(\045)1247 291 y Fo(is)g(coheren)o(t)h(b)o(y)f(Corollary)f(5.2)f(there)k(are)0 340 y(\()p Fl(X)q(;)7 b(a)p Fo(\))16 b Fm(2)f Fl(W)23 b Fo(and)16 b(\()p Fl(X)365 325 y Fg(0)377 340 y Fl(;)7 b(a)418 325 y Fg(0)429 340 y Fo(\))16 b Fm(2)g Fl(W)550 325 y Fg(0)578 340 y Fo(suc)o(h)h(that)f Fl(X)f Fm([)10 b Fl(X)890 325 y Fg(0)919 340 y Fm(2)15 b Fo(Con)1036 346 y Ff(\045)1071 340 y Fo(but)i Fm(f)p Fl(a;)7 b(a)1234 325 y Fg(0)1245 340 y Fm(g)20 b Fl(=)-26 b Fm(2)16 b Fo(Con)1399 346 y Ff(\033)1421 340 y Fo(.)26 b(Since)16 b Fl(\045)h Fo(is)g(dense)g(w)o(e)g(ha)o(v)o(e)f(a)0 390 y Fl(z)e Fm(2)d Fl(G)105 396 y Ff(\045)138 390 y Fo(suc)o(h)j(that)g Fl(X)f Fm([)c Fl(X)442 375 y Fg(0)466 390 y Fm(\022)j Fl(z)r Fo(.)18 b(Hence)744 484 y Fl(a)11 b Fm(2)p 817 451 45 2 v 12 w Fl(W)6 b(z)43 b Fo(and)f Fl(a)1055 467 y Fg(0)1078 484 y Fm(2)p 1117 451 57 2 v 11 w Fl(W)1162 472 y Fg(0)1173 484 y Fl(z)r(:)0 578 y Fo(No)o(w)14 b(since)g Fl(\033)h Fo(is)f(separating)g(there)h(are)f Fl(~)-20 b(z)13 b Fm(2)e Fl(G)j Fo(suc)o(h)g(that)775 672 y Fm(;)d(6)p Fo(=)p 851 636 64 2 v 12 w Fm(f)p Fl(a)p Fm(g)o Fl(~)-20 b(z)13 b Fm(6)p Fo(=)p 991 636 76 2 v 12 w Fm(f)p Fl(a)1034 660 y Fg(0)1045 672 y Fm(g)o Fl(~)-20 b(z)14 b Fm(6)p Fo(=)e Fm(;)p Fl(;)0 766 y Fo(hence)j(also)772 816 y Fm(;)d(6)p Fo(=)p 849 782 45 2 v 12 w Fl(W)5 b(z)q(~)-20 b(z)14 b Fm(6)p Fo(=)p 991 782 57 2 v 12 w Fl(W)1036 804 y Fg(0)1048 816 y Fl(z)q(~)-20 b(z)13 b Fm(6)p Fo(=)f Fm(;)p Fl(:)0 892 y Fo(This)i(concludes)h(the)f(pro)q(of)g(that)g Fl(\045)e Fm(!)f Fl(\033)k Fo(is)f(separating.)83 942 y Fl(\045)f Fm(!)f Fl(\033)k Fo(is)e(dense:)20 b(This)15 b(will)e(follo)o(w)f(from)h(the)i(inductiv)o(e)g(h)o(yp)q(otheses)h (that)e Fl(\045)h Fo(is)f(separating)h(and)f Fl(\033)i Fo(is)e(dense.)21 b(So)0 991 y(let)14 b Fl(W)k Fo(=)12 b Fm(f)p Fo(\()p Fl(X)232 997 y Ff(i)246 991 y Fl(;)7 b(a)287 997 y Ff(i)300 991 y Fo(\))12 b(:)g Fl(i)g Fm(2)g Fl(I)s Fm(g)g(2)f Fo(Con)585 997 y Ff(\045)p Fg(!)p Ff(\033)658 991 y Fo(.)18 b(Consider)d Fl(i;)7 b(j)16 b Fo(suc)o(h)f(that)f Fm(f)p Fl(a)1155 997 y Ff(i)1169 991 y Fl(;)7 b(a)1210 997 y Ff(j)1226 991 y Fm(g)17 b Fl(=)-26 b Fm(2)12 b Fo(Con)1372 997 y Ff(\033)1395 991 y Fo(.)18 b(Then)d Fl(X)1568 997 y Ff(i)1591 991 y Fm([)9 b Fl(X)1662 997 y Ff(j)1697 991 y Fl(=)-26 b Fm(2)12 b Fo(Con)1806 997 y Ff(\045)1825 991 y Fo(.)18 b(Since)0 1041 y Fl(\045)c Fo(is)g(separating,)g(there)h(are)e Fl(~)-20 b(z)484 1047 y Ff(ij)525 1041 y Fm(2)11 b Fl(G)j Fo(and)g Fl(k)714 1047 y Ff(ij)742 1041 y Fo(,)g Fl(l)780 1047 y Ff(ij)823 1041 y Fo(suc)o(h)h(that)700 1135 y Fm(f)p Fl(k)743 1141 y Ff(ij)771 1135 y Fm(g)d Fo(=)p 848 1102 49 2 v 12 w Fl(X)882 1141 y Ff(i)895 1135 y Fl(~)-20 b(z)915 1141 y Ff(ij)956 1135 y Fm(6)p Fo(=)p 1000 1102 52 2 v 12 w Fl(X)1034 1141 y Ff(j)1051 1135 y Fl(~)g(z)1071 1141 y Ff(ij)1112 1135 y Fo(=)11 b Fm(f)p Fl(l)1188 1141 y Ff(ij)1218 1135 y Fm(g)p Fl(:)0 1229 y Fo(W)m(e)i(clearly)h(ma)o(y)e (assume)i(that)f Fl(~)-20 b(z)547 1235 y Ff(ij)588 1229 y Fo(=)11 b Fl(~)-21 b(z)650 1235 y Ff(j)r(i)694 1229 y Fo(and)14 b(\()p Fl(k)813 1235 y Ff(ij)842 1229 y Fl(;)7 b(l)873 1235 y Ff(ij)902 1229 y Fo(\))k(=)h(\()p Fl(l)1001 1235 y Ff(j)r(i)1031 1229 y Fl(;)7 b(k)1072 1235 y Ff(j)r(i)1100 1229 y Fo(\).)83 1279 y(No)o(w)14 b(de\014ne)h(for)f(an)o(y)g Fl(X)h Fm(2)d Fo(Con)604 1285 y Ff(\045)638 1279 y Fo(a)h(set)j Fl(I)756 1285 y Ff(X)802 1279 y Fo(of)d(indices)i Fl(i)d Fm(2)g Fl(I)18 b Fo(suc)o(h)d(that)f(`)p Fl(X)j Fo(b)q(eha)o(v)o(es)e (as)g Fl(X)1576 1285 y Ff(i)1604 1279 y Fo(with)f(resp)q(ect)i(to)e (the)-1 1329 y Fl(~)-20 b(z)19 1335 y Ff(ij)49 1329 y Fo('.)17 b(More)d(precisely)m(,)g(let)513 1378 y Fl(I)531 1384 y Ff(X)575 1378 y Fo(:=)d Fm(f)p Fl(i)h Fm(2)f Fl(I)k Fo(:)c Fm(8)p Fl(j:)p Fm(f)p Fl(a)867 1384 y Ff(i)880 1378 y Fl(;)c(a)921 1384 y Ff(j)938 1378 y Fm(g)16 b Fl(=)-26 b Fm(2)12 b Fo(Con)1083 1384 y Ff(\033)1117 1378 y Fm(!)p 1170 1345 38 2 v 11 w Fl(X)s(~)-20 b(z)1227 1384 y Ff(ij)1268 1378 y Fo(=)12 b Fm(f)p Fl(k)1355 1384 y Ff(ij)1384 1378 y Fm(gg)p Fl(:)0 1455 y Fo(W)m(e)h(\014rst)i(sho)o(w) f(that)782 1505 y Fm(f)p Fl(a)825 1511 y Ff(i)850 1505 y Fo(:)e Fl(i)f Fm(2)h Fl(I)957 1511 y Ff(X)988 1505 y Fm(g)g(2)f Fo(Con)1134 1511 y Ff(\033)1156 1505 y Fl(:)729 b Fo(\(1\))0 1581 y(Since)16 b Fk(C)144 1587 y Ff(\033)182 1581 y Fo(is)f(coheren)o(t)i(it)e(su\016ces)h(to)f(sho)o(w)h(that)f Fm(f)p Fl(a)870 1587 y Ff(i)884 1581 y Fl(;)7 b(a)925 1587 y Ff(j)942 1581 y Fm(g)13 b(2)h Fo(Con)1092 1587 y Ff(\033)1129 1581 y Fo(for)h(all)f Fl(i;)7 b(j)16 b Fm(2)e Fl(I)1379 1587 y Ff(X)1411 1581 y Fo(.)22 b(So)15 b(let)g Fl(i;)7 b(j)17 b Fm(2)c Fl(I)1691 1587 y Ff(X)1738 1581 y Fo(and)i(assume)0 1631 y Fm(f)p Fl(a)43 1637 y Ff(i)57 1631 y Fl(;)7 b(a)98 1637 y Ff(j)114 1631 y Fm(g)16 b Fl(=)-25 b Fm(2)11 b Fo(Con)260 1637 y Ff(\033)282 1631 y Fo(.)18 b(Then)c(w)o(e)h(ha)o(v)o(e)p 754 1647 V 754 1681 a Fl(X)s(~)-20 b(z)811 1687 y Ff(ij)852 1681 y Fo(=)12 b Fm(f)p Fl(k)939 1687 y Ff(ij)968 1681 y Fm(g)41 b Fo(as)14 b Fl(i)e Fm(2)f Fl(I)1164 1687 y Ff(X)0 1757 y Fo(and)p 746 1773 V 746 1807 a Fl(X)s(~)-20 b(z)803 1813 y Ff(j)r(i)844 1807 y Fo(=)12 b Fm(f)p Fl(k)931 1813 y Ff(j)r(i)959 1807 y Fm(g)41 b Fo(as)14 b Fl(j)g Fm(2)e Fl(I)1161 1813 y Ff(X)1192 1807 y Fl(;)0 1883 y Fo(and)18 b(b)q(ecause)i(of)e Fl(~)-21 b(z)314 1889 y Ff(ij)363 1883 y Fo(=)19 b Fl(~)-21 b(z)433 1889 y Ff(j)r(i)481 1883 y Fo(and)19 b Fl(k)589 1889 y Ff(ij)637 1883 y Fm(6)p Fo(=)g Fl(l)700 1889 y Ff(ij)749 1883 y Fo(=)g Fl(k)822 1889 y Ff(j)r(i)870 1883 y Fo(w)o(e)f(could)h(conclude) g(that)p 1320 1850 V 18 w Fl(X)s(~)-20 b(z)1377 1889 y Ff(ij)1425 1883 y Fo(w)o(ould)18 b(b)q(e)h(inconsisten)o(t.)32 b(This)0 1933 y(con)o(tradiction)14 b(pro)o(v)o(es)g Fm(f)p Fl(a)424 1939 y Ff(i)437 1933 y Fl(;)7 b(a)478 1939 y Ff(j)495 1933 y Fm(g)12 b(2)f Fo(Con)641 1939 y Ff(\033)677 1933 y Fo(and)j(hence)h(\(1\).)83 1983 y(Since)g(\(1\))f(holds)h(and)f Fl(\033)i Fo(is)e(dense)i(b)o(y)e (induction)g(h)o(yp)q(othesis,)h(w)o(e)f(can)h(\014nd)g(a)f Fl(y)1383 1989 y Ff(I)1398 1993 y Fa(X)1439 1983 y Fm(2)e Fl(G)1512 1989 y Ff(\033)1549 1983 y Fo(suc)o(h)j(that)g Fl(a)1756 1989 y Ff(i)1782 1983 y Fm(2)d Fl(y)1842 1989 y Ff(I)1857 1993 y Fa(X)1900 1983 y Fo(for)0 2032 y(all)h Fl(i)e Fm(2)h Fl(I)141 2038 y Ff(X)172 2032 y Fo(.)18 b(No)o(w)c(de\014ne)h(a)f(relation)f Fl(r)f Fm(\022)g Fo(Con)753 2038 y Ff(\045)781 2032 y Fm(\002)e Fl(C)853 2038 y Ff(\033)889 2032 y Fo(b)o(y)436 2153 y Fl(X)s(r)q(a)i Fo(:)f Fm(\()-7 b(\))649 2094 y Fe(\032)687 2131 y Fl(a)12 b Fm(2)f Fl(y)780 2137 y Ff(I)795 2141 y Fa(X)824 2131 y Fl(;)224 b Fo(if)p 1098 2097 V 13 w Fl(X)17 b Fo(is)d(de\014ned)h(on) e(all)f Fl(~)-20 b(z)1468 2137 y Ff(ij)1498 2131 y Fo(;)687 2180 y Fm(f)p Fl(a)730 2186 y Ff(i)755 2180 y Fo(:)11 b Fl(i)h Fm(2)f Fl(I)861 2186 y Ff(X)893 2180 y Fm(g)g(`)950 2186 y Ff(\033)985 2180 y Fl(a;)41 b Fo(otherwise,)0 2271 y(where)15 b(`)p Fl(x)e Fo(is)h(de\014ned)h(on)e Fl(~)-20 b(z)9 b Fo(')k(means)g Fl(x)o(~)-20 b(z)13 b Fm(6)p Fo(=)f Fm(;)p Fo(.)18 b(W)m(e)13 b(will)g(sho)o(w)h(that)g Fl(r)e Fm(2)f Fl(G)1189 2277 y Ff(\045)p Fg(!)p Ff(\033)1275 2271 y Fo(and)j Fl(W)k Fm(\022)11 b Fl(r)q Fo(.)83 2321 y(F)m(or)19 b Fl(W)26 b Fm(\022)20 b Fl(r)g Fo(w)o(e)g(ha)o(v)o(e)f(to) g(sho)o(w)g Fl(X)685 2327 y Ff(i)699 2321 y Fl(r)q(a)741 2327 y Ff(i)774 2321 y Fo(for)f(all)g Fl(i)j Fm(2)f Fl(I)s Fo(.)34 b(But)20 b(this)f(holds,)g(since)h(clearly)f Fl(i)i Fm(2)f Fl(I)1701 2327 y Ff(X)1728 2331 y Fa(i)1763 2321 y Fo(and)f(hence)0 2371 y Fm(f)p Fl(a)43 2377 y Ff(j)72 2371 y Fo(:)11 b Fl(j)j Fm(2)d Fl(I)183 2377 y Ff(X)210 2381 y Fa(i)226 2371 y Fm(g)g(`)283 2377 y Ff(\033)317 2371 y Fl(a)339 2377 y Ff(i)367 2371 y Fo(and)j(also)f Fl(a)553 2377 y Ff(i)578 2371 y Fm(2)e Fl(y)637 2377 y Ff(I)652 2381 y Fa(X)675 2388 y(i)693 2371 y Fo(.)83 2421 y(W)m(e)k(no)o(w)h(sho)o(w)f(that)h Fl(r)h Fo(is)e(an)h(appro)o (ximable)d(mapping;)h(b)o(y)i(Lemma)d(4.9)h(this)i(means)f Fl(r)h Fm(2)e Fo(Con)o(t)1681 2427 y Ff(\045)p Fg(!)p Ff(\033)1753 2421 y Fo(.)24 b(T)m(o)15 b(pro)o(v)o(e)0 2471 y(this)f(w)o(e)g(ha)o(v)o(e)g(to)g(v)o(erify)f(conditions)g(\(i\)) h(to)g(\(iii\))f(from)f(De\014nition)h(3.1.)83 2521 y(\(i\))k Fl(X)s(r)q(b)219 2527 y Fi(1)238 2521 y Fl(;)7 b(:)g(:)g(:)e(;)i(X)s(r) q(b)406 2527 y Ff(n)464 2521 y Fo(=)-7 b Fm(\))36 b(f)p Fl(b)606 2527 y Fi(1)624 2521 y Fl(;)7 b(:)g(:)g(:)e(;)i(b)735 2527 y Ff(n)757 2521 y Fm(g)17 b(2)h Fo(Con)915 2527 y Ff(\033)937 2521 y Fo(.)30 b(If)p 1024 2487 V 17 w Fl(X)21 b Fo(is)d(de\014ned)g(on)g(all)e Fl(~)-21 b(z)1413 2527 y Ff(ij)1461 2521 y Fo(the)18 b(claim)d(follo)o(ws)h(from)g(the)0 2570 y(consistency)f(of)f Fl(y)286 2576 y Ff(I)301 2580 y Fa(X)330 2570 y Fo(.)k(If)13 b(not,)g(the)i(claim)d(follo)o(ws)g (from)g(the)i(prop)q(erties)i(of)d Fm(`)1242 2576 y Ff(\033)1265 2570 y Fo(.)83 2620 y(\(ii\))i Fl(X)s(r)q(b)229 2626 y Fi(1)248 2620 y Fl(;)7 b(:)g(:)g(:)e(;)i(X)s(r)q(b)416 2626 y Ff(n)438 2620 y Fl(;)g Fm(f)p Fl(b)496 2626 y Fi(1)514 2620 y Fl(;)g(:)g(:)g(:)t(;)g(b)624 2626 y Ff(n)646 2620 y Fm(g)14 b(`)706 2626 y Ff(\033)743 2620 y Fl(a)28 b Fo(=)-7 b Fm(\))29 b Fl(X)s(r)q(a)p Fo(.)23 b(If)p 1046 2587 V 15 w Fl(X)c Fo(is)c(de\014ned)i(on)e(all)f Fl(~)-21 b(z)1424 2626 y Ff(ij)1469 2620 y Fo(the)16 b(claim)e(follo)o(ws)g(from)f(the)0 2670 y(deductiv)o(e)i(closure)g(of) e Fl(y)393 2676 y Ff(I)408 2680 y Fa(X)437 2670 y Fo(.)18 b(If)13 b(not,)h(the)g(claim)e(follo)o(ws)g(from)g(the)j(prop)q(erties) g(of)e Fm(`)1349 2676 y Ff(\033)1372 2670 y Fo(.)954 2770 y(12)p eop %%Page: 13 13 13 12 bop 83 42 a Fo(\(iii\))16 b Fl(X)k Fm(`)246 48 y Ff(\045)283 42 y Fl(X)320 26 y Fg(0)332 42 y Fl(;)7 b(X)388 26 y Fg(0)400 42 y Fl(r)q(a)33 b Fo(=)-7 b Fm(\))34 b Fl(X)s(r)q(a)p Fo(.)28 b(Case)17 b(1.)p 858 8 50 2 v 28 w Fl(X)895 30 y Fg(0)924 42 y Fo(is)g(de\014ned)h(on)f(all)e Fl(~)-20 b(z)1256 48 y Ff(ij)1285 42 y Fo(.)28 b(Then)17 b(also)p 1523 8 38 2 v 17 w Fl(X)j Fo(is)d(de\014ned)h(on)f(all)e Fl(~)-20 b(z)1909 48 y Ff(ij)1938 42 y Fo(.)0 91 y(F)m(rom)15 b Fl(X)149 76 y Fg(0)161 91 y Fl(r)q(a)h Fo(w)o(e)h(get)g Fl(a)f Fm(2)g Fl(y)457 97 y Ff(I)472 105 y Fa(X)497 99 y Fb(0)512 91 y Fo(.)26 b(W)m(e)16 b(ha)o(v)o(e)h(to)f(sho)o(w)h(that)f Fl(a)g Fm(2)g Fl(y)1076 97 y Ff(I)1091 101 y Fa(X)1120 91 y Fo(.)26 b(No)o(w)16 b(since)i Fl(X)i Fo(and)c Fl(X)1534 76 y Fg(0)1563 91 y Fo(are)h(de\014ned)h(on)e(all)f Fl(~)-21 b(z)1920 97 y Ff(ij)0 141 y Fo(and)15 b Fl(X)i Fm(`)158 147 y Ff(\045)191 141 y Fl(X)228 126 y Fg(0)241 141 y Fo(,)e(they)g(m)o(ust)f(ha)o(v)o(e)h(the)h(same)e(v)n(alues)h(on)g(the) g Fl(~)-20 b(z)1017 147 y Ff(ij)1046 141 y Fo(,)15 b(hence)i Fl(I)1208 147 y Ff(X)1237 139 y Fb(0)1264 141 y Fo(=)d Fl(I)1328 147 y Ff(X)1375 141 y Fo(and)h(therefore)h Fl(y)1652 147 y Ff(I)1667 155 y Fa(X)1692 149 y Fb(0)1721 141 y Fo(=)e Fl(y)1787 147 y Ff(I)1802 151 y Fa(X)1831 141 y Fo(.)21 b(Case)0 191 y(2.)j(Otherwise.)i(F)m(rom)14 b Fl(X)424 176 y Fg(0)436 191 y Fl(r)q(a)i Fo(w)o(e)g(get)h Fm(f)p Fl(a)672 197 y Ff(i)700 191 y Fo(:)e Fl(i)g Fm(2)g Fl(I)817 197 y Ff(X)846 189 y Fb(0)860 191 y Fm(g)f(`)920 197 y Ff(\033)958 191 y Fl(a)p Fo(.)24 b(No)o(w)16 b(from)e Fl(X)19 b Fm(`)1291 197 y Ff(\045)1326 191 y Fl(X)1363 176 y Fg(0)1391 191 y Fo(w)o(e)d(can)g(conclude)h Fl(I)1723 197 y Ff(X)1752 189 y Fb(0)1781 191 y Fm(\022)e Fl(I)1846 197 y Ff(X)1878 191 y Fo(,)h(b)o(y)0 241 y(the)g(de\014nition)f(of)g Fl(I)326 247 y Ff(X)357 241 y Fo(.)23 b(Hence)16 b Fm(f)p Fl(a)559 247 y Ff(i)587 241 y Fo(:)d Fl(i)h Fm(2)g Fl(I)700 247 y Ff(X)732 241 y Fm(g)f(`)791 247 y Ff(\033)828 241 y Fl(a)p Fo(,)i(and)g(also)g Fl(a)f Fm(2)f Fl(y)1141 247 y Ff(I)1156 251 y Fa(X)1200 241 y Fo(\(since)k Fl(a)1342 247 y Ff(i)1369 241 y Fm(2)d Fl(y)1431 247 y Ff(I)1446 251 y Fa(X)1490 241 y Fo(for)h(all)f Fl(i)g Fm(2)g Fl(I)1702 247 y Ff(X)1734 241 y Fo(,)h(and)g Fl(y)1863 247 y Ff(I)1878 251 y Fa(X)1922 241 y Fo(is)0 291 y(deductiv)o(ely)f(closed\).)19 b(Therefore)c Fl(X)s(r)q(a)p Fo(.)83 340 y(This)f(concludes)h(the)f (pro)q(of)g(that)g Fl(r)g Fo(is)g(an)g(appro)o(ximable)d(mapping.)83 390 y(It)j(remains)f(to)h(sho)o(w)f(that)h Fl(r)f Fm(2)e Fl(G)630 396 y Ff(\045)p Fg(!)p Ff(\033)702 390 y Fo(.)18 b(So)c(let)g Fl(x)d Fm(2)g Fl(G)957 396 y Ff(\045)976 390 y Fo(.)18 b(W)m(e)c(m)o(ust)f(sho)o(w)604 487 y Fm(j)p Fl(r)q Fm(j)p Fo(\()p Fl(x)p Fo(\))d(=)i Fm(f)p Fl(a)f Fm(2)h Fl(C)882 493 y Ff(\033)915 487 y Fo(:)f Fm(9)p Fl(X)k Fm(\022)1042 470 y Fi(\014n)1093 487 y Fl(x:X)s(r)q(a)p Fm(g)c(2)h Fl(G)1313 493 y Ff(\033)1335 487 y Fl(:)0 585 y Fo(No)o(w)j(for)g(an)o(y)g Fl(i;)7 b(j)16 b Fm(2)d Fl(I)19 b Fo(w)o(e)d(ha)o(v)o(e)f Fl(x)o(~)-20 b(z)588 591 y Ff(ij)631 585 y Fm(6)p Fo(=)14 b Fm(;)p Fo(,)h(hence)i(there)f (is)g(some)e Fl(X)1132 591 y Ff(ij)1176 585 y Fm(\022)1208 570 y Fi(\014n)1261 585 y Fl(x)h Fo(suc)o(h)h(that)p 1486 551 64 2 v 15 w Fl(X)1520 591 y Ff(ij)1549 585 y Fl(~)-20 b(z)1569 591 y Ff(ij)1612 585 y Fm(6)p Fo(=)15 b Fm(;)p Fo(.)22 b(Let)15 b Fl(X)j Fm(\022)1873 570 y Fi(\014n)1926 585 y Fl(x)0 635 y Fo(b)q(e)e(the)f(union)g(of)f(all)g (the)h Fl(X)460 641 y Ff(ij)490 635 y Fo(.)21 b(Then)15 b(clearly)p 766 601 38 2 v 15 w Fl(X)k Fo(is)c(de\014ned)h(on)e(all)f Fl(~)-20 b(z)1142 641 y Ff(ij)1186 635 y Fo(and)15 b(hence)i Fl(X)s(r)q(a)e Fo(for)g(all)e Fl(a)h Fm(2)f Fl(y)1699 641 y Ff(I)1714 645 y Fa(X)1742 635 y Fo(.)22 b(Therefore)0 684 y Fl(y)20 690 y Ff(I)35 694 y Fa(X)75 684 y Fm(\022)12 b(j)p Fl(r)q Fm(j)p Fo(\()p Fl(x)p Fo(\))h(and)h(hence)h Fm(j)p Fl(r)q Fm(j)p Fo(\()p Fl(x)p Fo(\))c Fm(2)g Fl(G)611 690 y Ff(\033)647 684 y Fo(b)o(y)i(Lemma)e(5.3.)83 734 y(This)j(concludes)h(the)f(pro)q(of)g(that)g Fl(\045)e Fm(!)f Fl(\033)k Fo(is)f(dense.)83 784 y Fl(\045)8 b Fm(\002)f Fl(\033)14 b Fo(is)f(dense)h(and)e(separating:)18 b(This)13 b(follo)o(ws)e(easily)h(from)f(the)j(induction)e(h)o(yp)q (otheses)j(that)e Fl(\045)g Fo(and)f Fl(\033)i Fo(are)g(dense)0 834 y(and)g(separating.)1638 b Fh(\003)83 884 y Fo(As)15 b(an)f(application)f(of)g(the)i(Densit)o(y)g(Theorem)e(w)o(e)i(pro)o(v) o(e)f(a)g(c)o(hoice)h(principle)f(for)g(total)g(con)o(tin)o(uous)g (functionals.)0 933 y(This)g(result)g(w)o(as)g(\014rst)h(pro)o(v)o(ed)f (b)o(y)g(Kreisel)g(in)g([Kre59];)f(our)h(pro)q(of)f(essen)o(tially)h (follo)o(ws)e(Berger)k([Ber93].)0 1008 y Fk(5.8)k(Theorem)g(\(Choice)f (Principl)o(e)e(for)j(T)l(otal)f(Con)o(tin)o(uous)e(F)l(unctionals\).)k Fj(W)m(e)c(can)h(construct)i(an)d(ob)r(ject)0 1057 y Fl(r)12 b Fm(2)g Fo(Con)o(t)159 1064 y Fi(\()p Ff(\045)p Fg(!)p Ff(\033)q Fg(!)p Fi(nat)q(\))p Fg(!)p Ff(\045)p Fg(!)p Ff(\033)456 1057 y Fj(suc)o(h)j(that)f(for)g(an)o(y)f Fl(F)k Fm(2)11 b Fl(G)898 1063 y Ff(\045)p Fg(!)p Ff(\033)q Fg(!)p Fi(nat)1065 1057 y Fj(satisfying)695 1155 y Fo(\()p Fm(8)p Fl(x)g Fm(2)h Fl(G)842 1161 y Ff(\045)860 1155 y Fo(\)\()p Fm(9)p Fl(y)i Fm(2)d Fl(G)1021 1161 y Ff(\033)1043 1155 y Fo(\))p Fl(F)6 b(xy)13 b Fo(=)f Fm(f)p Fo(0)p Fm(g)0 1252 y Fj(w)o(e)i(ha)o(v)o(e)g Fl(r)q(F)j Fm(2)11 b Fl(G)293 1258 y Ff(\045)p Fg(!)p Ff(\033)379 1252 y Fj(and)737 1302 y Fo(\()p Fm(8)p Fl(x)h Fm(2)f Fl(G)884 1308 y Ff(\045)903 1302 y Fo(\))p Fl(F)6 b(x)p Fo(\()p Fl(r)q(F)g(x)p Fo(\))k(=)i Fm(f)p Fo(0)p Fm(g)p Fl(:)0 1380 y Fk(Pro)q(of.)20 b Fo(Let)15 b Fl(Y)249 1386 y Fi(0)268 1380 y Fl(;)7 b(Y)311 1386 y Fi(1)329 1380 y Fl(;)g(Y)372 1386 y Fi(2)390 1380 y Fl(;)g(:)g(:)g(:)12 b Fo(b)q(e)k(an)e(en)o(umeration)g(of)g(Con)948 1386 y Ff(\033)971 1380 y Fo(.)20 b(By)15 b(the)g(Densit)o(y)g(Theorem)f (5.7)f(w)o(e)i(can)g(\014nd)g(for)f(an)o(y)g Fl(Y)1927 1386 y Ff(n)0 1430 y Fo(a)g Fl(y)55 1436 y Ff(n)89 1430 y Fm(2)d Fl(G)161 1436 y Ff(\033)197 1430 y Fo(suc)o(h)k(that)f Fl(Y)405 1436 y Ff(n)439 1430 y Fm(\022)e Fl(y)503 1436 y Ff(n)526 1430 y Fo(.)18 b(De\014ne)c(for)g(an)o(y)f Fl(W)18 b Fm(2)11 b Fo(Con)996 1436 y Ff(\045)p Fg(!)p Ff(\033)q Fg(!)p Fi(nat)1149 1430 y Fo(,)j Fl(X)h Fm(2)c Fo(Con)1337 1436 y Ff(\045)1370 1430 y Fo(and)j Fl(a)d Fm(2)g Fl(C)1553 1436 y Ff(\033)187 1534 y Fl(W)6 b(r)q Fo(\()p Fl(X)q(;)h(a)p Fo(\))23 b Fm(\()-7 b(\))22 b Fo(\()p Fm(9)p Fl(m;)574 1523 y(~)576 1534 y(k)q(:)p Fo(\()p Fm(8)p Fl(i)12 b(<)g(m)p Fo(\))p 772 1500 45 2 v Fl(W)p 824 1500 38 2 v 13 w(X)t(y)882 1540 y Ff(i)907 1534 y Fo(=)g Fm(f)p Fl(k)994 1540 y Ff(i)1017 1534 y Fo(+)d(1)p Fm(g)g(^)p 1146 1500 45 2 v 9 w Fl(W)p 1198 1500 38 2 v 13 w(X)s(y)1255 1540 y Ff(m)1299 1534 y Fo(=)j Fm(f)p Fo(0)p Fm(g)c(^)h Fl(a)i Fm(2)h Fl(y)1544 1540 y Ff(m)1576 1534 y Fo(\))d Fm(_)g(;)i(`)1695 1540 y Ff(\033)1729 1534 y Fl(a:)0 1631 y Fo(W)m(e)h(\014rst)h(sho)o(w)g(that)f Fl(r)i Fo(is)e(an)g(appro)o(ximable)e(mapping.)16 b(T)m(o)11 b(pro)o(v)o(e)i(this)f(w)o(e)h(ha)o(v)o(e)f(to)h(v)o(erify)f (conditions)g(\(i\))g(to)g(\(iii\))g(from)0 1681 y(De\014nition)h(3.1.) 83 1731 y(\(i\))18 b Fl(W)6 b(r)q Fo(\()p Fl(X)260 1737 y Fi(1)279 1731 y Fl(;)h(a)320 1737 y Fi(1)338 1731 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i(W)f(r)q Fo(\()p Fl(X)562 1737 y Ff(n)585 1731 y Fl(;)h(a)626 1737 y Ff(n)647 1731 y Fo(\))39 b(=)-7 b Fm(\))37 b(f)p Fo(\()p Fl(X)877 1737 y Fi(1)896 1731 y Fl(;)7 b(a)937 1737 y Fi(1)955 1731 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)1114 1737 y Ff(n)1137 1731 y Fl(;)g(a)1178 1737 y Ff(n)1200 1731 y Fo(\))p Fm(g)19 b(2)f Fo(Con)1376 1737 y Ff(\045)p Fg(!)p Ff(\033)1449 1731 y Fo(.)31 b(Assume)18 b(the)h(premise)f(and)0 1780 y Fl(X)k Fo(:=)117 1749 y Fe(S)152 1793 y Ff(i)p Fg(2)p Ff(I)212 1780 y Fl(X)246 1786 y Ff(i)278 1780 y Fm(2)c Fo(Con)398 1786 y Ff(\045)417 1780 y Fo(.)29 b(W)m(e)18 b(m)o(ust)e(sho)o(w)i(that)g Fm(f)p Fl(a)884 1786 y Ff(i)915 1780 y Fo(:)g Fl(i)g Fm(2)g Fl(I)s Fm(g)g(2)f Fo(Con)1202 1786 y Ff(\033)1225 1780 y Fo(.)29 b(If)17 b Fm(;)h(`)1375 1786 y Ff(\033)1416 1780 y Fl(a)1438 1786 y Ff(i)1469 1780 y Fo(for)g(all)e Fl(i)i Fm(2)g Fl(I)j Fo(w)o(e)d(are)g(done.)0 1830 y(Otherwise)f(for)e(all)f Fl(i)h Fm(2)f Fl(I)19 b Fo(suc)o(h)d(that)f Fm(;)f(6`)676 1836 y Ff(\033)713 1830 y Fl(a)735 1836 y Ff(i)764 1830 y Fo(the)i(n)o(um)o(b)q(ers)f Fl(m)1042 1836 y Ff(i)1072 1830 y Fo(in)g(the)h(de\014nition)f(of)g Fl(W)6 b(r)q Fo(\()p Fl(X)1545 1836 y Ff(i)1559 1830 y Fl(;)h(a)1600 1836 y Ff(i)1614 1830 y Fo(\))15 b(are)h(all)e(the)i(same,)0 1880 y(=)c Fl(m)i Fo(sa)o(y)m(.)k(Hence)d Fm(f)p Fl(a)345 1886 y Ff(i)370 1880 y Fo(:)c Fl(i)h Fm(2)f Fl(I)s Fm(g)h(\022)g Fl(y)576 1886 y Ff(m)608 1880 y Fo(,)h(and)h(the)g(claim)e(follo)o(ws)g (from)g(the)j(consistency)g(of)e Fl(y)1487 1886 y Ff(m)1519 1880 y Fo(.)83 1930 y(\(ii\))18 b Fl(W)6 b(r)q Fo(\()p Fl(X)272 1936 y Fi(1)291 1930 y Fl(;)h(a)332 1936 y Fi(1)350 1930 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i(W)f(r)q Fo(\()p Fl(X)574 1936 y Ff(n)597 1930 y Fl(;)h(a)638 1936 y Ff(n)660 1930 y Fo(\))p Fl(;)g Fm(f)p Fo(\()p Fl(X)766 1936 y Fi(1)784 1930 y Fl(;)g(a)825 1936 y Fi(1)843 1930 y Fo(\))p Fl(;)g(:)g(:)g(:)e(;)i Fo(\()p Fl(X)1002 1936 y Ff(n)1025 1930 y Fl(;)g(a)1066 1936 y Ff(n)1088 1930 y Fo(\))p Fm(g)19 b(`)1169 1936 y Ff(\045)p Fg(!)p Ff(\033)1262 1930 y Fo(\()p Fl(X)q(;)7 b(a)p Fo(\))40 b(=)-7 b Fm(\))39 b Fl(W)6 b(r)q Fo(\()p Fl(X)q(;)h(a)p Fo(\).)33 b(Assume)18 b(the)0 1980 y(premise)f(and)f Fl(I)k Fo(:=)c Fm(f)p Fl(i)h Fo(:)f Fl(X)k Fm(`)496 1986 y Ff(\045)532 1980 y Fl(X)566 1986 y Ff(i)580 1980 y Fm(g)p Fo(.)27 b(W)m(e)16 b(kno)o(w)h(that)g Fm(f)p Fl(a)962 1986 y Ff(i)992 1980 y Fo(:)f Fl(i)g Fm(2)g Fl(I)s Fm(g)h(`)1178 1986 y Ff(\033)1217 1980 y Fl(a)p Fo(.)27 b(If)17 b Fm(;)f(`)1385 1986 y Ff(\033)1424 1980 y Fl(a)1446 1986 y Ff(i)1477 1980 y Fo(for)g(all)g Fl(i)g Fm(2)h Fl(I)j Fo(w)o(e)d(are)g(done.)0 2029 y(Otherwise)g(for)e(all)f Fl(i)h Fm(2)f Fl(I)19 b Fo(suc)o(h)d(that)f Fm(;)f(6`)676 2035 y Ff(\033)713 2029 y Fl(a)735 2035 y Ff(i)764 2029 y Fo(the)i(n)o(um)o(b)q(ers)f Fl(m)1042 2035 y Ff(i)1072 2029 y Fo(in)g(the)h(de\014nition)f(of)g Fl(W)6 b(r)q Fo(\()p Fl(X)1545 2035 y Ff(i)1559 2029 y Fl(;)h(a)1600 2035 y Ff(i)1614 2029 y Fo(\))15 b(are)h(all)e(the)i (same,)0 2079 y(=)c Fl(m)i Fo(sa)o(y)m(.)k(Hence)d Fm(f)p Fl(a)345 2085 y Ff(i)370 2079 y Fo(:)c Fl(i)h Fm(2)f Fl(I)s Fm(g)h(\022)g Fl(y)576 2085 y Ff(m)608 2079 y Fo(,)h(and)h(the)g(claim)e(follo)o(ws)g(from)g(the)j(deductiv)o(e)g (closure)f(of)g Fl(y)1595 2085 y Ff(m)1626 2079 y Fo(.)83 2129 y(\(iii\))h Fl(W)20 b Fm(`)250 2135 y Ff(\045)p Fg(!)p Ff(\033)q Fg(!)p Fi(nat)419 2129 y Fl(W)464 2114 y Fg(0)476 2129 y Fl(;)7 b(W)540 2114 y Fg(0)551 2129 y Fl(r)q Fo(\()p Fl(X)q(;)g(a)p Fo(\))29 b(=)-7 b Fm(\))30 b Fl(W)6 b(r)q Fo(\()p Fl(X)q(;)h(a)p Fo(\).)23 b(This)16 b(clearly)g(follo)o(ws)e(from)g(the)i(de\014nition)g(of)f Fl(r)q Fo(;)h(the)h Fl(m)0 2179 y Fo(from)12 b Fl(W)143 2164 y Fg(0)155 2179 y Fl(r)q Fo(\()p Fl(X)q(;)7 b(a)p Fo(\))13 b(can)i(b)q(e)f(used)h(for)e Fl(W)6 b(r)q Fo(\()p Fl(X)q(;)h(a)p Fo(\).)83 2229 y(W)m(e)14 b(\014nally)e(sho)o(w)i(that)g (for)g(all)e Fl(F)17 b Fm(2)11 b Fl(G)710 2235 y Ff(\045)p Fg(!)p Ff(\033)q Fg(!)p Fi(nat)877 2229 y Fo(satisfying)695 2326 y(\()p Fm(8)p Fl(x)g Fm(2)h Fl(G)842 2332 y Ff(\045)860 2326 y Fo(\)\()p Fm(9)p Fl(y)i Fm(2)d Fl(G)1021 2332 y Ff(\033)1043 2326 y Fo(\))p Fl(F)6 b(xy)13 b Fo(=)f Fm(f)p Fo(0)p Fm(g)0 2423 y Fo(and)i(for)h(all)e Fl(x)g Fm(2)f Fl(G)314 2429 y Ff(\045)347 2423 y Fo(w)o(e)j(ha)o(v)o(e)g Fl(r)q(F)6 b(x)11 b Fm(2)i Fl(G)668 2429 y Ff(\033)704 2423 y Fo(and)i Fl(F)6 b(x)p Fo(\()p Fl(r)q(F)g(x)p Fo(\))11 b(=)i Fm(f)p Fo(0)p Fm(g)p Fo(.)20 b(So)14 b(let)h Fl(F)20 b Fo(and)14 b Fl(x)g Fo(with)h(these)h(prop)q(erties)g(b)q(e)f(giv)o (en.)0 2473 y(By)i(assumption)e(there)i(is)g(a)f Fl(y)h Fm(2)f Fl(G)594 2479 y Ff(\033)632 2473 y Fo(suc)o(h)h(that)g Fl(F)6 b(xy)17 b Fo(=)f Fm(f)p Fo(0)p Fm(g)p Fo(.)24 b(Hence)18 b(b)o(y)e(the)h(de\014nition)f(of)g(application)f(there)j (is)e(a)0 2523 y Fl(Y)24 2529 y Ff(n)60 2523 y Fm(2)e Fo(Con)175 2529 y Ff(\033)213 2523 y Fo(suc)o(h)i(that)f Fl(F)6 b(x)p 456 2490 47 2 v(Y)480 2529 y Ff(n)515 2523 y Fo(=)14 b Fm(f)p Fo(0)p Fm(g)p Fo(.)21 b(Since)16 b Fl(Y)791 2529 y Ff(n)827 2523 y Fm(\022)e Fl(y)893 2529 y Ff(n)931 2523 y Fo(w)o(e)i(also)e(ha)o(v)o(e)h Fl(F)6 b(xy)1252 2529 y Ff(n)1288 2523 y Fo(=)14 b Fm(f)p Fo(0)p Fm(g)p Fo(.)21 b(Clearly)14 b(w)o(e)i(ma)o(y)d(assume)i(here)0 2573 y(that)f Fl(n)g Fo(is)f(minima)o(l)e(with)i(this)h(prop)q(ert)o(y) m(,)g(i.e.)k(that)573 2670 y Fl(F)6 b(xy)650 2676 y Fi(0)680 2670 y Fo(=)11 b Fm(f)p Fl(k)766 2676 y Fi(0)794 2670 y Fo(+)e(1)p Fm(g)p Fl(;)e(:)g(:)g(:)t(;)g(F)f(xy)1046 2676 y Ff(n)p Fg(\000)p Fi(1)1122 2670 y Fo(=)12 b Fm(f)p Fl(k)1209 2676 y Ff(n)p Fg(\000)p Fi(1)1283 2670 y Fo(+)d(1)p Fm(g)p Fl(:)954 2770 y Fo(13)p eop %%Page: 14 14 14 13 bop 0 42 a Fo(W)m(e)13 b(sho)o(w)h(that)g Fl(r)q(F)6 b(x)11 b Fm(\023)h Fl(y)416 48 y Ff(n)452 42 y Fo(or)i(more)f (precisely)i Fm(jj)p Fl(r)q Fm(j)p Fo(\()p Fl(F)6 b Fo(\))p Fm(j)p Fo(\()p Fl(x)p Fo(\))j Fm(\023)j Fl(y)1039 48 y Ff(n)1062 42 y Fo(;)i(this)f(su\016ces)j(b)o(y)d(Lemma)e(5.3.)17 b(Recall)d(that)478 141 y Fm(j)p Fl(r)q Fm(j)p Fo(\()p Fl(F)6 b Fo(\))k(=)i Fm(f)p Fo(\()p Fl(X)q(;)7 b(a)p Fo(\))k Fm(2)h Fo(Con)894 147 y Ff(\045)923 141 y Fm(\002)d Fl(C)994 147 y Ff(\033)1028 141 y Fo(:)i(\()p Fm(9)p Fl(W)18 b Fm(\022)1179 124 y Fi(\014n)1230 141 y Fl(F)6 b Fo(\))p Fl(W)g(r)q Fo(\()p Fl(X)q(;)h(a)p Fo(\))p Fm(g)0 241 y Fo(and)442 340 y Fm(jj)p Fl(r)q Fm(j)p Fo(\()p Fl(F)f Fo(\))p Fm(j)p Fo(\()p Fl(x)p Fo(\))j(=)j Fm(f)p Fl(a)f Fm(2)h Fl(C)808 346 y Ff(\033)841 340 y Fo(:)f(\()p Fm(9)p Fl(X)16 b Fm(\022)985 323 y Fi(\014n)1036 340 y Fl(x)p Fo(\)\()p Fl(X)q(;)7 b(a)p Fo(\))k Fm(2)g(j)p Fl(r)q Fm(j)p Fo(\()p Fl(F)6 b Fo(\))p Fm(g)640 403 y Fo(=)12 b Fm(f)p Fl(a)f Fm(2)h Fl(C)808 409 y Ff(\033)841 403 y Fo(:)f(\()p Fm(9)p Fl(X)16 b Fm(\022)985 386 y Fi(\014n)1036 403 y Fl(x)p Fo(\)\()p Fm(9)p Fl(W)h Fm(\022)1203 386 y Fi(\014n)1254 403 y Fl(F)6 b Fo(\))p Fl(W)g(r)q Fo(\()p Fl(X)q(;)h(a)p Fo(\))p Fm(g)p Fl(:)0 502 y Fo(No)o(w)14 b(let)g Fl(a)d Fm(2)g Fl(y)247 508 y Ff(n)270 502 y Fo(.)18 b(By)c(the)h(c)o(hoice)f(of)f Fl(n)h Fo(w)o(e)g(then)h(get)f Fl(X)h Fm(\022)952 487 y Fi(\014n)1003 502 y Fl(x)f Fo(and)f Fl(W)18 b Fm(\022)1210 487 y Fi(\014n)1261 502 y Fl(F)h Fo(suc)o(h)c(that)489 602 y(\()p Fm(8)p Fl(i)d(<)g(n)p Fo(\))p 639 569 45 2 v Fl(W)p 691 569 38 2 v 13 w(X)s(y)748 608 y Ff(i)774 602 y Fo(=)g Fm(f)p Fl(k)861 608 y Ff(i)883 602 y Fo(+)e(1)p Fm(g)82 b Fo(and)p 1199 569 45 2 v 83 w Fl(W)p 1251 569 38 2 v 13 w(X)t(y)1309 608 y Ff(n)1343 602 y Fo(=)12 b Fm(f)p Fo(0)p Fm(g)p Fl(:)0 702 y Fo(Therefore)j Fl(W)6 b(r)q Fo(\()p Fl(X)q(;)h(a)p Fo(\))14 b(and)g(hence)h Fl(a)c Fm(2)h(jj)p Fl(r)q Fm(j)p Fo(\()p Fl(F)6 b Fo(\))p Fm(j)p Fo(\()p Fl(x)p Fo(\).)1073 b Fh(\003)83 751 y Fo(W)m(e)14 b(\014nally)f(commen)o(t)e(of)j(the)g(notion)g(of)f (e\013ectivit)o(y)i(in)e(this)h(con)o(text.)20 b(In)14 b(particular)g(w)o(e)g(w)o(an)o(t)f(to)h(de\014ne)h(what)f(it)0 801 y(means)f(for)h(a)f(partial)g(con)o(tin)o(uous)h(functional)f(of)g (an)h(arbitrary)f(simple)g(t)o(yp)q(e)h Fl(\045)h Fo(to)e(b)q(e)i (computable.)83 851 y(An)k(information)e(system)i(\()p Fl(A;)7 b Fo(Con)o Fl(;)g Fm(`)p Fo(\))20 b(is)f(called)g Fn(e\013e)n(ctive)k Fo(if)18 b(the)i(\(coun)o(table\))g(set)g Fl(A)p Fo(,)g(the)g(set)g(Con)f(and)g(the)0 901 y(relation)e Fm(`)g Fo(are)h(all)e(decidable.)29 b(Since)17 b(decidabilit)o(y)f(alw) o(a)o(ys)h(means)f(decidabilit)o(y)g(relativ)o(e)h(to)g(a)g(giv)o(en)g (set)h(w)o(e)g(m)o(ust)0 951 y(assume)d(here)h(that)g(the)g(data)f(ob)r (jects)h(are)g(tak)o(en)g(from)d(a)i(\014xed)h(giv)o(en)f(set,)h(e.g.) 22 b(the)16 b(set)g(of)f(natural)g(n)o(um)o(b)q(ers)g(or)g(the)0 1000 y(set)h(of)e(all)f(strings)j(o)o(v)o(er)f(some)e(\014nite)i (alphab)q(et.)21 b(It)15 b(is)g(easy)g(to)f(see)j(that)d(all)g(the)h (op)q(erations)g(on)g(information)d(systems)0 1050 y(in)o(tro)q(duced)j (in)e(section)i(4)e(preserv)o(e)j(e\013ectivit)o(y)m(.)83 1100 y(In)g(an)g(e\013ectiv)o(e)i(information)13 b(system)k(it)e(mak)o (es)h(sense)i(to)e(talk)f(ab)q(out)h(a)g(recursiv)o(ely)i(en)o (umerable)d(\(r.e.\))26 b(ideal,)0 1150 y(since)19 b(an)o(y)e(ideal)g (is)h(a)f(set)i(of)e(data)h(ob)r(jects.)31 b(The)18 b Fn(c)n(omputable)j Fo(elemen)o(ts)d(of)f(an)h(e\013ectiv)o(e)h (information)c(system)j(are)0 1200 y(de\014ned)d(to)f(b)q(e)g(its)g (r.e.)19 b(ideals.)e(Clearly)d(application)e(do)q(es)j(not)f(carry)g (us)g(out)g(of)g(the)g(realm)f(of)g(computable)g(elemen)o(ts,)0 1249 y(i.e.)29 b(if)17 b Fl(r)i Fm(2)e(j)p Fk(A)h Fm(!)f Fk(B)p Fm(j)h Fo(and)f Fl(x)h Fm(2)f(j)p Fk(B)p Fm(j)g Fo(are)h(computable,)f(then)i(so)f(is)f Fl(r)q(x)g Fo(\(or)h(more)f (precisely)i Fm(j)p Fl(r)q Fm(j)p Fo(\()p Fl(x)p Fo(\)\).)28 b(Also)18 b(w)o(e)g(can)0 1299 y(compute)13 b(an)h(r.e.)k(index)c(of)f Fm(j)p Fl(r)q Fm(j)p Fl(x)g Fo(from)f(those)j(of)e Fl(r)i Fo(and)e Fl(x)p Fo(.)83 1349 y(Let)18 b(us)g(no)o(w)g(lo)q(ok)f (somewhat)g(closer)h(at)g(our)g(particular)f(information)e(systems)j Fk(C)1469 1355 y Ff(\045)1489 1349 y Fo(,)g(whose)g(elemen)o(ts)g(are)g (the)0 1399 y(partial)c(con)o(tin)o(uous)g(functionals.)19 b(A)c Fn(c)n(omputable)j Fo(functional)13 b(of)h(t)o(yp)q(e)h Fl(\045)g Fo(is)g(de\014ned)g(to)g(b)q(e)g(a)f(computable)g(elemen)o(t) g(of)0 1449 y Fk(C)34 1455 y Ff(\045)54 1449 y Fo(.)k(The)c(set)h(of)e (all)g(computable)g(functionals)g(of)g(t)o(yp)q(e)h Fl(\045)h Fo(is)e(denoted)i(b)o(y)f(Comp)1314 1459 y Ff(\045)1334 1449 y Fo(.)0 1523 y Fk(5.9)g(E\013ectiv)o(e)f(Densit)o(y)f(Theorem.)23 b Fj(F)m(or)12 b(an)o(y)g Fl(X)j Fm(2)c Fo(Con)979 1529 y Ff(\045)1011 1523 y Fj(w)o(e)h(can)h(\014nd)g(e\013ectiv)o(ely)g(an)f Fl(x)f Fm(2)h Fl(G)1583 1529 y Ff(\045)1608 1523 y Fm(\\)6 b Fo(Comp)1750 1534 y Ff(\045)1782 1523 y Fj(suc)o(h)13 b(that)0 1573 y Fl(X)i Fm(\022)d Fl(x)p Fj(.)0 1648 y Fk(Pro)q(of.)22 b Fo(By)16 b(insp)q(ection)g(of)f(the)h(pro)q(of)f(of)g (the)h(Densit)o(y)g(Theorem)f(5.7.)22 b(T)m(o)14 b(see)j(that)f Fl(r)g Fo(\(in)f(the)h(pro)q(of)f(that)h Fl(\045)f Fm(!)e Fl(\033)k Fo(is)0 1698 y(dense\))e(is)f(e\013ectiv)o(e)h(one)g(needs)g (that)42 1773 y({)20 b Fm(f)p Fl(a)126 1779 y Ff(i)151 1773 y Fo(:)11 b Fl(i)h Fm(2)f Fl(I)257 1779 y Ff(X)289 1773 y Fm(g)g(`)346 1779 y Ff(\033)381 1773 y Fl(a)i Fo(implies)f Fl(a)g Fm(2)f Fl(y)650 1779 y Ff(I)665 1783 y Fa(X)708 1773 y Fo(for)i(all)g Fl(X)k Fo(and)d Fl(a)p Fo(,)f(since)i(b)o(y)f(de\014nition)f Fl(a)1374 1779 y Ff(i)1399 1773 y Fm(2)f Fl(y)1459 1779 y Ff(I)1474 1783 y Fa(X)1516 1773 y Fo(for)i(all)f Fl(i)e Fm(2)h Fl(I)1721 1779 y Ff(X)1752 1773 y Fo(.)18 b(Hence)331 1872 y Fl(X)s(r)q(a)23 b Fm(\()-7 b(\))22 b(f)p Fl(a)575 1878 y Ff(i)600 1872 y Fo(:)12 b Fl(i)f Fm(2)h Fl(I)707 1878 y Ff(X)738 1872 y Fm(g)g(`)796 1878 y Ff(\033)830 1872 y Fl(a)41 b Fo(or)h(\()p Fl(a)12 b Fm(2)f Fl(y)1081 1878 y Ff(I)1096 1882 y Fa(X)1166 1872 y Fo(and)p 1275 1839 V 42 w Fl(X)17 b Fo(is)d(de\014ned)h(on)e(all)24 b Fl(~)-32 b(z)1645 1878 y Ff(ij)1675 1872 y Fo(\))p Fl(:)42 1984 y Fo({)20 b(If)p 123 1951 V 12 w Fl(X)15 b Fo(is)d(de\014ned)h(on)f(all)22 b Fl(~)-32 b(z)484 1990 y Ff(ij)514 1984 y Fo(,)12 b(one)g(can)g(actually)f(compute)h Fl(I)1027 1990 y Ff(X)1071 1984 y Fo(\(and)g(not)g(only)f(an)h(en)o (umeration)f(pro)q(cedure)j(for)e Fl(I)1891 1990 y Ff(X)1922 1984 y Fo(\).)83 2034 y Fh(\003)0 2109 y Fk(5.10)18 b(Theorem)e (\(E\013ectiv)o(e)g(Choice)h(Principle\))o(.)k Fj(There)c(is)e(an)g Fl(r)h Fm(2)d Fo(Comp)1335 2119 y Fi(\()p Ff(\045)p Fg(!)p Ff(\033)q Fg(!)p Fi(nat\))p Fg(!)p Ff(\045)p Fg(!)p Ff(\033)1634 2109 y Fj(suc)o(h)j(that)f(for)g(an)o(y)0 2159 y Fl(F)i Fm(2)11 b Fl(G)116 2165 y Ff(\045)p Fg(!)p Ff(\033)q Fg(!)p Fi(nat)283 2159 y Fj(satisfying)695 2208 y Fo(\()p Fm(8)p Fl(x)g Fm(2)h Fl(G)842 2214 y Ff(\045)860 2208 y Fo(\)\()p Fm(9)p Fl(y)i Fm(2)d Fl(G)1021 2214 y Ff(\033)1043 2208 y Fo(\))p Fl(F)6 b(xy)13 b Fo(=)f Fm(f)p Fo(0)p Fm(g)0 2287 y Fj(w)o(e)i(ha)o(v)o(e)g Fl(r)q(F)j Fm(2)11 b Fl(G)293 2293 y Ff(\045)p Fg(!)p Ff(\033)379 2287 y Fj(and)737 2337 y Fo(\()p Fm(8)p Fl(x)h Fm(2)f Fl(G)884 2343 y Ff(\045)903 2337 y Fo(\))p Fl(F)6 b(x)p Fo(\()p Fl(r)q(F)g(x)p Fo(\))k(=)i Fm(f)p Fo(0)p Fm(g)p Fl(:)0 2416 y Fk(Pro)q(of.)18 b Fo(Immediate)11 b(from)h(the)j(pro)q(of)e(of)g (the)i(c)o(hoice)f(principle)g(for)g(total)f(con)o(tin)o(uous)h (functionals)f(in)g(5.8.)177 b Fh(\003)954 2770 y Fo(14)p eop %%Page: 15 15 15 14 bop 0 42 a Fk(References)0 118 y Fo([Ber90])69 b(Ulric)o(h)15 b(Berger.)23 b Fn(T)m(otale)16 b(Objekte)g(und)h(Mengen) g(in)f(der)g(Ber)n(eichsthe)n(orie)p Fo(.)21 b(PhD)15 b(thesis,)h(Mathematisc)o(hes)198 168 y(Institut)e(der)h(Univ)o (ersit\177)-21 b(at)14 b(M)q(\177)-22 b(unc)o(hen,)14 b(1990.)0 245 y([Ber93])69 b(Ulric)o(h)19 b(Berger.)37 b(T)m(otal)18 b(sets)j(and)f(ob)r(jects)g(in)g(domain)d(theory)m(.)35 b Fn(A)o(nnals)21 b(of)f(Pur)n(e)g(and)h(Applie)n(d)f(L)n(o)n(gic)p Fo(,)198 294 y(60:91{117,)11 b(1993.)0 371 y([Ers74])72 b(Y)m(uri)13 b(L.)h(Ersho)o(v.)k(The)c(mo)q(del)f Fl(G)g Fo(of)h(the)g(theory)g Fl(B)r(R)p Fo(.)19 b Fn(Soviet)c(Math.)g (Doklady)p Fo(,)f(15\(4\):1158{1160,)c(1974.)0 448 y([Ers75])72 b(Y)m(uri)13 b(L.)h(Ersho)o(v.)19 b(Theorie)14 b(der)h(Numerierungen)f (I)q(I.)k Fn(Zeitschrift)c(f)q(\177)-22 b(ur)15 b(Mathematische)g(L)n (o)n(gik)g(und)h(Grund-)198 497 y(lagen)f(der)g(Mathematik)p Fo(,)f(21:473{584,)c(1975.)0 574 y([Ers77])72 b(Y)m(uri)12 b(L.)h(Ersho)o(v.)k(Mo)q(del)c Fl(C)i Fo(of)e(partial)f(con)o(tin)o (uous)h(functionals.)i(In)e(R.)f(Gandy)h(and)f(M.)h(Hyland,)f(editors,) 198 624 y Fn(L)n(o)n(gic)j(Col)r(lo)n(quium)f(1976)p Fo(,)h(pages)f(455{467.)d(North{Holland,)i(Amsterdam,)f(1977.)0 700 y([Kle59])70 b(Stephen)14 b(C.)e(Kleene.)18 b(Coun)o(table)12 b(functionals.)j(In)e(A.)g(Heyting,)f(editor,)g Fn(Constructivity)h(in) h(Mathematics)p Fo(,)198 750 y(pages)g(81{100.)e(North{Holland,)g (Amsterdam,)g(1959.)0 827 y([Kre59])66 b(Georg)14 b(Kreisel.)21 b(In)o(terpretation)15 b(of)f(analysis)g(b)o(y)g(means)f(of)h (constructiv)o(e)i(functionals)e(of)g(\014nite)g(t)o(yp)q(es.)21 b(In)198 877 y(A.)15 b(Heyting,)g(editor,)h Fn(Constructivity)g(in)g (Mathematics)p Fo(,)g(pages)g(101{128.)d(North{Holland,)h(Amsterdam,) 198 927 y(1959.)0 1003 y([LM84])68 b(G.)10 b(Longo)f(and)i(E.)f(Moggi.) i(The)f(hereditary)g(partial)f(e\013ectiv)o(e)i(functionals)e(and)g (recursion)i(theory)f(in)f(higher)198 1053 y(t)o(yp)q(es.)19 b Fn(The)c(Journal)g(of)f(Symb)n(olic)h(L)n(o)n(gic)p Fo(,)f(49\(4\):1319{1332)o(,)d(Decem)o(b)q(er)j(1984.)0 1130 y([L)-5 b(W84])68 b(K.G.)17 b(Larsen)i(and)f(G.)g(Winsk)o(el.)30 b(Using)18 b(information)e(systems)i(to)g(solv)o(e)g(recursiv)o(e)i (domain)c(equations)198 1179 y(e\013ectiv)o(ely)m(.)32 b(In)18 b Fn(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)g(Confer)n(enc)n(e)g (on)g(A)o(bstr)n(act)f(Datatyp)n(es,)h(Sophia{A)o(ntip)n(olis,)h(F)m(r) n(anc)n(e)p Fo(,)198 1229 y(pages)11 b(109{129,)e(Berlin,)i(Heidelb)q (erg,)g(New)g(Y)m(ork,)f(1984.)g(Springer.)j(Lecture)g(Notes)e(in)g (Computer)f(Science,)198 1279 y(V)m(ol.)i(173.)0 1356 y([Nor93])64 b(Dag)9 b(Normann.)g(Closing)g(the)h(gap)f(b)q(et)o(w)o (een)i(the)f(con)o(tin)o(uous)g(functionals)f(and)g(recursion)i(in)1686 1341 y Fi(3)1705 1356 y Fl(E)r Fo(.)g(Submitted)198 1405 y(to)j(the)g(pro)q(ceedings)h(of)f(the)g(Sac)o(ks)g(conference,)i(MIT,) d(Cam)o(bridge,)f(1993.)0 1482 y([Pla66])71 b(Ric)o(hard)16 b(A.)g(Platek.)26 b Fn(F)m(oundations)18 b(of)f(r)n(e)n(cursion)g(the)n (ory)p Fo(.)25 b(PhD)17 b(thesis,)f(Departmen)o(t)g(of)g(Mathematics,) 198 1532 y(Stanford)e(Univ)o(ersit)o(y)m(,)f(1966.)0 1609 y([Plo77])71 b(Gordon)13 b(D.)g(Plotkin.)k(LCF)d(considered)h(as)f (a)f(programmi)o(ng)e(language.)17 b Fn(The)n(or)n(etic)n(al)c (Computer)i(Scienc)n(e)p Fo(,)198 1658 y(5:223{255,)c(1977.)0 1735 y([Plo78])71 b(Gordon)20 b(D.)f(Plotkin.)36 b Fk(T)633 1720 y Ff(!)678 1735 y Fo(as)20 b(a)g(univ)o(ersal)g(domain.)35 b Fn(Journal)20 b(of)h(Computer)f(and)i(System)f(Scienc)n(es)p Fo(,)198 1785 y(17:209{236,)11 b(1978.)0 1861 y([Ros87])64 b(A.W.)13 b(Rosco)q(e.)18 b(Notes)d(on)f(domain)d(theory)m(,)j(1987.)j (Unpublished)d(notes,)g(Oxford)g(Univ)o(ersit)o(y)m(.)0 1938 y([Sc)o(h91])69 b(Helm)o(ut)14 b(Sc)o(h)o(wic)o(h)o(ten)o(b)q (erg.)25 b(Primitiv)o(e)13 b(recursion)k(on)e(the)h(partial)f(con)o (tin)o(uous)h(functionals.)22 b(In)16 b(M.)f(Bro)o(y)m(,)198 1988 y(editor,)f Fn(Informatik)g(und)i(Mathematik)p Fo(,)d(pages)i (251{269.)c(Springer,)j(Berlin,)g(1991.)0 2064 y([Sco70])70 b(Dana)16 b(Scott.)27 b(Outline)16 b(of)g(a)h(mathematical)c(theory)k (of)f(computation.)24 b(T)m(ec)o(hnical)17 b(Monograph)f(PR)o(G{2,)198 2114 y(Oxford)e(Univ)o(ersit)o(y)g(Computing)d(Lab)q(oratory)m(,)i (1970.)0 2191 y([Sco82])70 b(Dana)19 b(Scott.)36 b(Domains)18 b(for)h(denotational)g(seman)o(tics.)35 b(In)20 b(E.)g(Nielsen)g(and)g (E.M.)f(Sc)o(hmidt,)f(editors,)198 2241 y Fn(A)o(utomata,)12 b(L)n(anguages)i(and)f(Pr)n(o)n(gr)n(amming)p Fo(,)d(v)o(olume)f(140)h (of)g Fn(L)n(e)n(ctur)n(e)i(Notes)g(in)g(Computer)g(Scienc)n(e)p Fo(,)g(pages)198 2291 y(577{613.)e(Springer)j(V)m(erlag,)f(Berlin,)g (Heidelb)q(erg,)h(New)g(Y)m(ork,)e(1982.)k(A)e(corrected)h(and)f (expanded)g(v)o(ersion)198 2340 y(of)g(a)h(pap)q(er)g(prepared)i(for)d (ICALP'82,)g(Aarh)o(us,)h(Denmark.)0 2417 y([SHGL94])19 b(Viggo)14 b(Stolten)o(b)q(erg-Hansen,)i(Edw)o(ard)f(Gri\013or,)g(and)g (Ingrid)f(Lindstr\177)-21 b(om.)21 b Fn(Mathematic)n(al)16 b(The)n(ory)g(of)g(Do-)198 2467 y(mains)p Fo(.)i(Cam)o(bridge)12 b(T)m(racts)j(in)e(Theoretical)h(Computer)f(Science.)i(Cam)o(bridge)d (Univ)o(ersit)o(y)i(Press,)h(1994.)954 2770 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF