%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: n9.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips n9.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.03.29:1506 %%BeginProcSet: texc.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 round sub abs 0.00001 lt{round}if} forall round exch round exch]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 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /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 userdict /eop-hook known{eop-hook}if showpage}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 newpath 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 (n9.dvi) @start /Fa 2 2 df<120112021206120CA21218A21230A21270A3126012E0AC12601270 A31230A21218A2120CA212061202120108267B8010>0 D<1280124012601230A21218A2 120CA2120EA312061207AC1206120EA3120CA21218A21230A212601240128008267E8010 >I E /Fb 6 112 df<00FEEB03F8001E14C000171305A338138009A23811C011A33810E0 21A2EB7041A3EB3881A2EB1D01A2130EA2123839FE040FF81D177F9621>77 D103 D<12FF1218AF12FF08117F900A>105 D<12FE1230A91304A3130C13081338EAFFF80E117F9011>108 D<38F00F803838070013 02122C12261223A2EA2182EA20C21362A21332131A130EA2EA7006EAF80211117F9014> 110 DI E /Fc 1 81 df80 D E /Fd 3 74 df<12021206120FEA1F80EA7FE0EAE670EA0600AC0C127F8D0F>34 D<1218A31230A31260A312C0A2050B7E8B09>48 D73 D E /Fe 3 82 df28 D77 D81 D E /Ff 6 118 df83 D<38FF03F0383C00C0001C1380A2121E380E 0100A2EA0702A3EA0384A2EA01C8A3EA00F0A31360A214147F9318>86 D<5AEA0380A3EA04C0A3EA0860A2EA1FF0EA1030A2EA20181270EAF87E0F0F7F8E12>97 D112 D114 D117 D E /Fg 1 22 df21 D E /Fh 1 79 df<39FFE00FF8393010032039080801 4090380400C0120CEA0E02EA0D01380C8080EB4040A2EB2020EB1010EB0808EB0404A2EB 0202EB0101EB0080EC40401420A2141014081404140200121301123339FFC000C0C81240 1D1D809B20>78 D E /Fi 6 111 df<1204A2EA0F80121BEA30005AA2123FA212605AA3 12F8123FEA0780EA0100120F09127E8D0E>24 D26 DI<1208A21200A41270129812B01230A21260126412681270060F7D 8E0B>105 D<12381218A35A13C0EA3360EA3440EA7800127E12631320EAC340EAC1800B 0E7E8D10>107 D 110 D E /Fj 2 100 df28 D99 D E /Fk 3 116 df101 D<12E0B3A503177E9608>108 D<121FEA7FC012E01300A27E127FEA3F80EA0FC0EA01E0128012C0EAE1C0127FEA1F000B 0F7F8E0E>115 D E /Fl 6 55 df<1218127812981218AC12FF08107D8F0F>49 D<121FEA6180EA40C0EA806012C01200A213C0EA0180EA030012065AEA10201220EA7FC0 12FF0B107F8F0F>I<121FEA2180EA60C0A212001380EA0100121FEA00801340136012C0 A2EA8040EA6080EA1F000B107F8F0F>IIII E /Fm 1 49 df<1207EA0F80A3EA1F00A2121E123E123CA25AA2127012F01260090F7F8F 0B>48 D E /Fn 30 122 df12 D<127812FCA4127806067D850D >46 D48 D<1360EA01E0120F12FF12F31203 B3A2387FFF80A2111B7D9A18>IIII< 38380180383FFF005B5B5B13C00030C7FCA4EA31F8EA361E38380F80EA3007000013C014 E0A3127812F8A214C012F038600F8038381F00EA1FFEEA07F0131B7E9A18>I<137EEA03 FF38078180380F03C0EA1E07123C387C03800078C7FCA212F813F8EAFB0E38FA0780EAFC 0314C000F813E0A41278A214C0123CEB0780381E0F00EA07FEEA03F8131B7E9A18>I<12 60387FFFE0A214C01480A238E00300EAC0065B5BC65AA25B13E0A212015B1203A41207A6 6C5A131C7D9B18>III<90381FE0209038FFF8E03803F80F3807 C003380F800148C7FC123E1560127E127C00FC1400A8007C1460127E123E15C07E390F80 01803907C003003803F80E3800FFFCEB1FE01B1C7D9B22>67 DII76 D<007FB512E0A238781F81007013800060146000E0147000C01430A400001400B03807FF FEA21C1C7E9B21>84 D97 D101 D104 D<121E123FA4121EC7FCA6127FA2121FAEEAFFC0A20A1E7F9D0E>I108 D<39FF0FC07E903831E18F3A1F40F20780D980FC13 C0A2EB00F8AB3AFFE7FF3FF8A225127F9128>I<38FF0FC0EB31E0381F40F0EB80F8A213 00AB38FFE7FFA218127F911B>II<38FF3F80EB E1E0381F80F0EB0078147C143C143EA6143C147C1478EB80F0EBC1E0EB3F0090C7FCA6EA FFE0A2171A7F911B>I114 D<1203A45AA25AA2EA3FFC12FFEA1F00A9130CA4EA0F08EA0798EA03F00E1A7F 9913>116 D<38FFC7FCA2381F81C0380F83803807C700EA03EEEA01FC5B1200137C13FE EA01DF38039F80EA070F380607C0380C03E038FF07FCA216127F9119>120 D<38FFC1FCA2381F00601380000F13C0A23807C180A23803E300A213F7EA01F613FE6C5A A21378A21330A25B1270EAF8E05BEAF9800073C7FC123E161A7F9119>I E /Fo 2 50 df<121EEA61801240EAC0C0A7EA40801261EA1E000A0D7E8C0E>48 D<121812F81218AA12FF080D7D8C0E>I E /Fp 14 121 df<121FEA7FC01261EA40E012 001207123F127812E0A212F1127F123E0B0D7F8C0F>97 D<12E0A712EFEAFFC0EAF1E012 E01370A513E012F1EAFFC0EAEF000C147E9310>II<1370A7120FEA3FF01278EA707012E0A51270 EA78F0123FEA0F700C147F9310>I<120FEA3F80EA71C0EA70E0EAE060EAFFE0A2EAE000 A21270EA7860EA3FE0EA0F800B0D7F8C0E>I103 D<38E7C3E038FFEFF038F87C38EAF078EAE070A9150D7E8C1A>109 D<120FEA3FC0EA70E0A2EAE070A5EA70E01279EA3FC0EA0F000C0D7F8C0F>111 D<12EFEAFFC0EAF1E012E01370A513E012F1EAFFC0EAEF0012E0A60C137E8C10>I<12E7 12EF12F812F0A212E0A8080D7E8C0B>114 D<123EEA7F8012E1EAE000A212FE127FEA1F 801203128312C3EAFF00123E090D7F8C0C>I<1270A412FFA21270A81271127F123C0811 7F900B>I<38E1E0C013E1A2EAE36138737380A2133300321300A21236EA3E1FEA1C1EA2 120D7F8C15>119 DI E /Fq 9 122 df<13FCEA0386EA0707000F1380121E A3383C0F00A2130E5BEA7C38EA7BE00078C7FCA25AA3126011137F8C14>26 D<3801FFF0120F5A383E1E00EA380F1278A2EAF01EA2EA701C5B6C5AEA0FC0140D7F8C16 >I76 D<3907F007F813F80001EB018013BCA2139E39031F0300130FEB0783A2380603 C614E61301EB00F64813FC147CA2143CB41318A21D147E931F>78 D<3807FFF814FE3801E01F801580A23903C01F00A2141E147C3807FFF00180C7FCA348C8 FCA4EAFFC0A219147E9317>80 D97 D<123FA2121EA45AEA3DF0EA3E18EA3C1CEA781EA4EA F03CA2EA70381370EA38E0EA1F800F147F9311>I120 D<381F0180383783C0126738C78780EACF07120FA2381E0F00A4EA0E3EEA03FEEA001EEA 383C12785BEA30E0EA1F8012137F8C14>I E /Fr 41 123 df12 DI<1238127C12FE12FFA2127F123B1203A31206A2120C121C121812 70122008117C8610>44 DI<1238127C12FEA3127C123807077C 8610>I<13181378EA01F812FFA21201B3A7387FFFE0A213207C9F1C>49 DI<13FE3807FFC0380F07E0381E03F0123FEB81F8A3EA1F 0314F0120014E0EB07C0EB1F803801FE007F380007C0EB01F014F8EB00FCA2003C13FE12 7EB4FCA314FCEA7E01007813F8381E07F0380FFFC03801FE0017207E9F1C>I<14E01301 1303A21307130F131FA21337137713E7EA01C71387EA03071207120E120C121812381270 12E0B6FCA2380007E0A790B5FCA218207E9F1C>I<00301320383E01E0383FFFC0148014 005B13F8EA33C00030C7FCA4EA31FCEA37FF383E0FC0383807E0EA3003000013F0A214F8 A21238127C12FEA200FC13F0A2387007E0003013C0383C1F80380FFF00EA03F815207D9F 1C>II67 DII73 D78 D80 D82 D<007FB61280A2397E03F80F0078140700 7014030060140100E015C0A200C01400A400001500B3A248B512F0A222227EA127>84 D97 DIII<13FE3807FF80380F87C0381E01 E0003E13F0EA7C0014F812FCA2B5FCA200FCC7FCA3127CA2127E003E13186C1330380FC0 703803FFC0C6130015167E951A>II104 D<121C123E127FA3123E12 1CC7FCA7B4FCA2121FB2EAFFE0A20B247EA310>I108 D<3AFF07F007F090391FFC1FFC3A1F303E303E01401340496C487EA201001300 AE3BFFE0FFE0FFE0A22B167E9530>I<38FF07E0EB1FF8381F307CEB403CEB803EA21300 AE39FFE1FFC0A21A167E951F>I<13FE3807FFC0380F83E0381E00F0003E13F848137CA3 00FC137EA7007C137CA26C13F8381F01F0380F83E03807FFC03800FE0017167E951C>I< 38FF0FE0EB3FF8381FE07CEB803E497E1580A2EC0FC0A8EC1F80A29038803F00EBC03EEB E0FCEB3FF8EB0FC090C8FCA8EAFFE0A21A207E951F>I114 DI<487EA41203A21207A2120F123FB5FCA2EA0F 80ABEB8180A5EB8300EA07C3EA03FEEA00F811207F9F16>I<38FF01FEA2381F003EAF14 7E14FE380F81BE3907FF3FC0EA01FC1A167E951F>I<39FFE01FE0A2391F800700000F13 06EBC00E0007130C13E000035BA26C6C5AA26C6C5AA2EB7CC0A2137F6D5AA26DC7FCA213 0EA21B167F951E>I<3AFFE3FF07F8A23A1F007800C09038807C01000F1580A23A07C07E 030014DE5D3903E1DF06148FD801F1138CEBF307A2D800FF13D8EBFE0315F890387C01F0 A2013C5BEB3800A225167F9528>I<39FFE07FC0A2390F801C006C6C5A6C6C5AEBF0606C 6C5A3800F980137F6DC7FC7F80497E1337EB63E0EBC1F03801C0F848487E3807007E000E 133E39FF80FFE0A21B167F951E>I<39FFE01FE0A2391F800700000F1306EBC00E000713 0C13E000035BA26C6C5AA26C6C5AA2EB7CC0A2137F6D5AA26DC7FCA2130EA2130CA25B12 78EAFC3813305BEA69C0EA7F80001FC8FC1B207F951E>I<387FFFF0A2387C03E0387007 C0EA600F38E01F8000C01300133E137EC65A5B485A00031330EA07E013C0380F8070121F 383F0060003E13E0EA7C03B5FCA214167E9519>I E /Fs 1 22 df21 D E /Ft 53 124 df<903801F03C9038071C 47010C13C7EC19C690381C0180140313181338A2EC0700A20003B512F03900700700A314 0EA213E0A35CA2EA01C0A35CA2EA0380A21430EB0070A248136038C630E038E638C038CC 3180D8781EC7FC2025819C19>11 D<14FE90380301801306EB0C03EB1C0191C7FC131813 38A43803FFFE3800700EA35CA213E0A25CA3EA01C01472A438038034141891C7FC90C8FC A25A12C612E65A12781925819C17>I<1218123CA31204A21208A2121012201240128006 0C779C0D>39 D<13031306130813181330136013C0A2EA0180EA0300A21206A25AA2121C 1218A212381230A21270A21260A412E0A51260A51220123012107EA2102A7B9E11>I<13 10A21308130C13041306A51307A51306A4130EA2130CA2131C1318A213381330A21360A2 13C0A2EA0180EA0300A212065A5A121012605A102A809E11>I<12181238127812381208 A21210A212201240A21280050C7D830D>44 DI<1230127812F0 126005047C830D>I<1304130C131813381378EA07B8EA0070A413E0A4EA01C0A4EA0380 A4EA0700A45AEAFFF00E1C7B9B15>49 D<133EEB4180EB80C0EA0100000213E0EA0440A2 1208A3381081C0A238110380000E1300EA00065B5B136013800003C7FC12044813404813 805AEB0100EA7F07EA43FEEA81FCEA8078131D7D9B15>I53 DI<133E13E13801 8180380300C01206120E120C121CA213011238A31303001813801307EA080B380C3300EA 03C7EA0007130E130C131C1318EAE0305BEA80C0EAC180003EC7FC121D7C9B15>57 D<1418A21438A21478A214B8EB0138A2EB023C141C1304130C13081310A21320A2EB7FFC EBC01C1380EA0100141E0002130EA25A120C001C131EB4EBFFC01A1D7E9C1F>65 D<48B5FC39003C038090383801C0EC00E0A35B1401A2EC03C001E01380EC0F00141EEBFF FC3801C00E801580A2EA0380A43907000F00140E141E5C000E13F0B512C01B1C7E9B1D> I<903803F02090381E0C6090383002E09038E003C03801C001EA038048C7FC000E148012 1E121C123C15005AA35AA41404A35C12705C6C5B00185B6C485AD80706C7FCEA01F81B1E 7A9C1E>I<48B5FC39003C03C090383800E0A21570A24913781538A215785BA4484813F0 A315E03803800115C0140315803907000700140E5C5C000E13E0B512801D1C7E9B1F>I< 48B512F038003C00013813301520A35BA214081500495AA21430EBFFF03801C020A43903 8040801400A2EC0100EA07005C14021406000E133CB512FC1C1C7E9B1C>I<48B512F038 003C00013813301520A35BA214081500495AA21430EBFFF03801C020A448485A91C7FCA3 48C8FCA45AEAFFF01C1C7E9B1B>I<3A01FFC3FF803A003C00780001381370A4495BA449 485AA390B5FC3901C00380A4484848C7FCA43807000EA448131E39FFE1FFC0211C7E9B1F >72 DI<3801FFC038003C001338A45BA45BA4485AA438038002A31404EA0700140C 14181438000E13F0B5FC171C7E9B1A>76 D II<3801FFFE39003C038090383801C0EC00E0A3EB7001A315C0EBE0031580EC07 00141C3801FFF001C0C7FCA3485AA448C8FCA45AEAFFE01B1C7E9B1C>80 D83 D<001FB512C0381C070138300E0000201480126012405B1280A200001400 5BA45BA45BA4485AA41203EA7FFE1A1C799B1E>I97 D<123F1207A2120EA45AA4EA39E0EA3A18EA3C0C12381270130EA3EAE01CA31318133813 301360EA60C0EA3180EA1E000F1D7C9C13>I<13F8EA0304120EEA1C0EEA181CEA300012 70A25AA51304EA60081310EA3060EA0F800F127C9113>II<13F8EA0704120CEA1802EA38041230EA7008EA7FF0EAE000A5EA6004 1308EA30101360EA0F800F127C9113>IIIII107 DI<391C1E078039266318C0394683A0E0384703C0008E1380A212 0EA2391C0701C0A3EC0380D8380E1388A2EC0708151039701C032039300C01C01D127C91 22>II<13F8EA030CEA0E06487E1218123000701380A238 E00700A3130EA25BEA60185BEA30E0EA0F8011127C9115>I<380387803804C860EBD030 13E0EA09C014381201A238038070A31460380700E014C0EB0180EB8300EA0E86137890C7 FCA25AA45AB4FC151A809115>IIII<12035AA3120EA4EAFFE0EA1C00A35AA45AA4EAE080A2EAE100 A2126612380B1A7C990E>I<381C0180EA2E03124EA2388E0700A2121CA2EA380EA43830 1C80A3EA383C38184D00EA0F8611127C9116>II<381E018338270387 1247148338870701A2120EA2381C0E02A31404EA180C131C1408EA1C1E380C26303807C3 C018127C911C>I<38038780380CC840380870E012103820E0C014001200A2485AA4EA03 811263EAE38212C5EA8584EA787813127E9113>I<381C0180EA2E03124EA2388E0700A2 121CA2EA380EA4EA301CA3EA383CEA1878EA0FB8EA003813301370EAE0605BEA81800043 C7FC123C111A7C9114>III E /Fu 11 90 df<132013401380EA01005A1206A25AA25AA212381230A21270A312 6012E0AD12601270A31230A212381218A27EA27EA27E7EEA0080134013200B317A8113> 0 D<7E12407E7E12187EA27EA27EA213801201A213C0A3120013E0AD13C01201A31380A2 12031300A21206A25AA25A12105A5A5A0B317F8113>I<141C143C14F8EB01E0EB03C0EB 0780EB0F00130E131E5BA35BB3B3A25BA3485AA2485A5B48C7FC120E5A127812E0A21278 121C7E7E6C7E7F6C7EA26C7EA31378B3B3A27FA37F130E130FEB0780EB03C0EB01E0EB00 F8143C141C167C7B8121>40 D<1318137813F0EA01E0EA03C0EA0780EA0F005A121E123E 123C127CA2127812F8B3A50D25707E25>56 D<12F8B3A51278127CA2123C123E121E121F 7EEA0780EA03C0EA01E0EA00F0137813180D25708025>58 D<137CB3A613F8A313F01201 13E0120313C0EA07801300120E5A5A12F012C012F012387E7E7E1380EA03C013E0120113 F0120013F8A3137CB3A60E4D798025>60 D<12F8AE050E708025>62 D80 D<1303801307A2497EA3EB1CE0A3EB3870A3497EA3497EA348487EA3 48487EA33907000380A2000EEB01C0A348EB00E0A3481470A3481438A348141CA248140C 1E2A7E7F23>86 D88 DI E /Fv 6 122 df101 D<3803E3C03807F7E0EA0FFF38 1C1CC038380E00A56C5AEA0FF8485AEA1BE00038C7FC1218EA1FFC13FF481380387003C0 38E000E0A4387001C0EA7C07383FFF80380FFE00EA03F8131C7F9116>103 D<38F9C38038FFEFC0EBFFE0EA3C78A2EA3870AA38FE7CF8A31512809116>109 DI115 D<387F1FC038FF9FE0387F1FC0381C0700120E 130EA212075BA2EA039CA21398EA01B8A2EA00F0A35BA3485A1279127BEA7F8090C7FC12 3C131B7F9116>121 D E /Fw 26 122 df70 D80 D83 D<00F0EB0380A2007814005CA26C 130EA2123E001E5BA26C5BA36C6C5AA23803C06014E0A26C6C5A13E1A23800F18013F301 73C7FCA2137B133EA2131C191D7F9C1C>86 D<00F8EB01E0007C14C06CEB0380001E1307 001F1400380F800E0007131EEBC01C3803E03C000113386D5A000013F0EB78E0EB7DC013 3F6D5A91C7FC7FAC1B1D809C1C>89 D97 D<12E0ABEAE3E0EAEFF8EAFFFCEAF83EEAE01E130E1307A6130EEAF01EEAF83CEAFFF8EA EFF0EAE3E0101D7D9C15>II<1307ABEA07C7EA1FF7EA3FFFEA3C1FEA 7807127012E0A61270EA780FEA3C1FEA3FFFEA1FF7EA07C7101D7F9C15>II<13FC12011203EA0700120EA7EAFFE0A2EA0E00B00E1D809C0D>I<3807 C3C0EA0FFF5A38383800487EA56C5AEA3FF05BEA77C00070C7FCA2EA3FFC13FF481380EA 700738E001C0A3EAF003387C0F80383FFF006C5AEA07F8121B7F9115>I<12E0ABEAE3E0 EAEFF0EAFFF8EAF83CEAF01C12E0AD0E1D7D9C15>I<12F0A41200A71270B2041D7E9C0A> I<12E0B3AB031D7D9C0A>108 D<38E3F03F39EFF8FF80D8FFFD13C039F81F81E038F00F 00EAE00EAD1B127D9122>II< EA03F0EA0FFC487EEA3C0F38780780EA700338E001C0A5EAF00300701380EA7807383C0F 00EA1FFE6C5AEA03F012127F9115>II114 DI<121C A6EAFFE0A2EA1C00AC1320EA1FF0120FEA07C00C187F970F>III<3870038038780700EA3C0EEA1C1C120E6C5AEA03 F06C5A5B7F487EEA0738EA0618EA0E1C487E487E3870038000F013C01212809113>120 DI E /Fx 13 123 df26 D<90387FFF800003B5FC5A481400381F03E0123EEA7C01A2EAF803A35CEAF0075C49C7FC EA780EEA3C3CEA0FE019127E911C>I<3A01FFF80FFC5A3A003F000180ED0700150C1538 017E5B15C0EC03800206C7FCEBFC1C143C147EEBFDFE3801FB7FEBFE3F01FC7FEBF81FD8 03F07F140F811407D807E07F14038114013AFFFE0FFF8013FC261C7E9B28>75 D<3801FFFC5AD8003FC7FCA4137EA45BA4485AA31560484813C0A2140115803807E00314 07EC0F00143FB512FEA21B1C7E9B1F>I<48B46CEB07FE48150FD8003FEC1FC0EB37C016 3716670167ECCF80A2903963E0018FED030F01C3EC1F001506150CEBC1F0D80181EB183E 1530A2156048C66C485AECF980A2ECFB00000601FE5B147E147C1478D8FFE090381FFF80 D9C07014002F1C7E9B2F>I<3A01FF8007FE48EBC00F3A003FE000C01337801333903963 F80180136114FCEB60FE9039C07E0300147F143F15833901801F8615C6140F15E6390300 07FC1403A2140100065C1400A21578D8FFE013305B271C7E9B27>I<0003B512E015FC39 003F007E153FA3137EA3153E49137E157C15F8EC03F048B512C001F8C7FCA3485AA4485A A4EAFFFEA2201C7E9B1E>80 DI97 DI<3807E1E038187B1838 303E3800601378EB7CF812C00000137014005BA3003013083879F01812F9143000F31320 386270C0383C1F0015127E911B>120 D<380F801C3819C03EEA31E012630043137C12C3 EA07C0A214F8EA0F80A3EB01F0A31303380787E0EA01FBEA000314C0EA1E07003E1380EB 0F00EA3C1EEA1878EA0FE0171A7F9119>I<3801E030EA07F8380FFEE0EBFFC03819F180 38000300130E13185B13E0EA01803803002000061360380C3FE0381FFFC0003913803860 7F00EAC03E14127F9117>I E /Fy 14 95 df<1202120412081218121012301220126012 40A212C0AA1240A212601220123012101218120812041202071E7D950D>40 D<1280124012201230121012181208120C1204A21206AA1204A2120C1208121812101230 122012401280071E7E950D>I<1360AAB512F0A238006000AA14167E9119>43 D<120FEA30C0EA6060A2EA4020EAC030A9EA4020EA6060A2EA30C0EA0F000C137E9211> 48 D<120C121C12EC120CAFEAFFC00A137D9211>I<121FEA60C01360EAF07013301260EA 0070A2136013C012011380EA02005AEA08101210EA2020EA7FE012FF0C137E9211>II<136013E0A2EA016012021206120C120812101220126012 C0EAFFFCEA0060A5EA03FC0E137F9211>III61 D<12FCA212C0B3A712FCA2061D7E9509>91 D<12FCA2120CB3A712FCA2061D809509>93 D<1218123C1266128108047C9311>I E /Fz 12 80 df0 D2 D<1204A3EAC460EAF5E0EA3F80EA0E00EA3F 80EAF5E0EAC460EA0400A30B0D7E8D11>I<14101418A280A28080B612E0A2C7EA030014 065CA25CA214101B107E8E21>33 D<5A5AA2EA0780EA0FC0EA1FE0EA7B78EAE31CEA8304 EA0300B10E1A7F9311>I<1204120EA2121CA31238A212301270A21260A212C0A2070F7F 8F0A>48 D<000F131E393BC06180396060804038403100D8801A1320130EA3130B394011 8040903820C0C03930C07B80390F001E001B0D7E8C21>II<12C0AA020A7E8B00> 55 D<134013C0B1B512E0A213147D931A>63 D<3803FFE0000F13C0381018001220485A 120013701360A213E0A25BA212015BA2EA0301EA0203EA7FFEEAFFF81314809313>73 D79 D E /FA 24 123 df11 D<1218A31230A412601261A212C212C41278080D7F 8C0C>19 D<121C120612077EA2EA0180A213C01200A213E01201EA0370EA0630120CEA18 181230EA601CEAC00CEA800E0F147E9314>21 D<5AA213F01206EA0C005A5AA31210EA1F C0A2EA30005AA25AA37E1278123FEA0780EA01C012001208EA07800C1A7F930F>24 D26 DII< 124012E0124003037D820A>58 D<124012E012601220A31240A2128003097D820A>I<38 07FFE03800E0703801C018140CA2140EEA0380A43807001CA31438000E1330147014E0EB 01C0381C0700EAFFFC17147F931B>68 D77 D<3807FFE03800E0703801C018141CA338038038A21470EB 81C03807FF0090C7FCA3120EA45AB47E16147F9315>80 D<133F3801C1C0380300E00006 137048133048133812385AA3481370A314E014C0EA60013871C380383A2600EA1C3C3807 F040EA003014C0EB3180133FEB1F00130E151A7E931A>I97 D99 D<133C130C1318A41330EA 07B0EA0C701210EA30601260A3EAC0C013C8A21241EA62D0EA3C700E147E9311>I<1206 120712061200A41238124CA2128C12981218A212301232A21264A2123808147F930C> 105 D<121E12065AA45A1338135C139CEA3118EA36001238EA3F80EA61C0EA60C8A3EAC0 D013600E147F9312>107 D110 D<1207EA1880EA19C0EA3180EA3800121E7EEA0380124112 E1EAC1001282127C0A0D7E8C10>115 D<38381820004C13701420EA8C3012981218A238 306040A314803818B100EA0F1E140D7F8C18>119 DII< EA0610EA1FA0EA10E0EA00401380EA01001202120CEA10201220EA3840EA4FC0EA83000C 0D7F8C10>I E /FB 26 107 df0 D<126012F0A2126004047C8B 0C>I<0040132000C01360006013C03830018038180300EA0C066C5A6C5AEA01B0EA00E0 A2EA01B0EA0318EA060C487E487E38300180386000C04813600040132013147A9320>I< EA03C0EA0FF0EA1C38EA300CEA6006A2EAC003A4EA6006A2EA300CEA1C38EA0FF0EA03C0 10107E9115>14 DI<90387FFF800003B5FCD80780C7FC000CC8FC5A5AA25A A25AA81260A27EA27E120E6C7E0001B512806C7E90C8FCA8007FB51280A219247D9920> 18 D20 D<12C012F0123C120FEA03C0EA00F01338130E6D7EEB 01E0EB0078141EEC0780A2EC1E001478EB01E0EB0780010EC7FC133813F0EA03C0000FC8 FC123C127012C0C9FCA8007FB5FCB6128019247D9920>I<153081A381A281811680ED00 C0B712F8A2C912C0ED0380160015065DA25DA35D25167E942A>33 D<134013C0A3487EA2487EEA06D8EA0CCCEA38C738F0C3C0EAC0C000001300B3A712257F 9C15>I<13C0B3A700C013C0EAF0C33838C700EA0CCCEA06D8EA03F06C5AA26C5AA31340 12257F9C15>I50 D<1460A214C0A2EB0180 A3EB0300A21306A25BA25BA35BA25BA25BA2485AA248C7FCA31206A25AA25AA25AA35AA2 5A124013287A9D00>54 D<12C0A612E0A212C0A6030E7E9000>I<0040130400C0130C00 601318A36C1330A36C1360A2381FFFE06C13C0EA0C00A238060180A238030300A3EA0186 A3EA00CCA31378A31330A2161E809C17>II<1304130CEA03CCEA0C38EA1818EA301C133CEA703EEA60361366A2EAE067 A213C7A3EAE187A3EAE307A312E6A3EA6606126CEA7C0EEA3C0C1238EA1818EA1C30EA33 C0EA3000A210237E9F15>59 D<1304130CB3A7B612E0A21B1C7D9B21>63 D66 D<90383FFF8048B51200380700C0EA0C01381C03801230D82007C7FC12005BA2130EA213 1EA2131CA2133C1338A213781370A25B140C48485A495A387FFFE0B51280191C819B17> 73 D<90380C1F809038107FE090382087F0EBC38138018700D8030613F8EA060ED80E1C 1378121C5BEA382090C7FC127800701470A200F014F015E0A2140115C015806C1303EC07 0000781306007C5B007E13106C5B381FC1C06CB4C7FCEA03F81D1E7E9C21>79 D<133C13E0EA01C013801203AD13005A121C12F0121C12077E1380AD120113C0EA00E013 3C0E297D9E15>102 D<12F0121C12077E1380AD120113C0EA00E0133C13E0EA01C01380 1203AD13005A121C12F00E297D9E15>I<134013C0EA0180A3EA0300A21206A35AA25AA3 5AA25AA35AA21260A37EA27EA37EA27EA37EA2EA0180A3EA00C013400A2A7D9E10>I<12 C0A21260A37EA27EA37EA27EA37EA2EA0180A3EA00C0A2EA0180A3EA0300A21206A35AA2 5AA35AA25AA35AA20A2A7E9E10>I<12C0B3B3A502297B9E0C>I E /FC 46 122 df<13FCEA0782EA0E07121C130290C7FCA4B5FCEA1C07AC387F1FC0121780 9614>12 D<9038FC7F80380703C3380E0783001C1303A6B6FC381C0703AC397F1FCFE01B 1780961D>15 D<120112021204120C1218A21230A212701260A312E0AA1260A312701230 A21218A2120C12041202120108227D980E>40 D<12801240122012301218A2120CA2120E 1206A31207AA1206A3120E120CA21218A2123012201240128008227E980E>I<126012F0 A212701210A21220A21240A2040A7D830A>44 DI<126012F0A2 126004047D830A>I48 D<1206120E12FE120EB1EAFFE00B157D9412>I< EA0F80EA30E0EA4070EA8030EAC03812E0124012001370A2136013C0EA0180EA03001206 EA0C081208EA1018EA3FF0127F12FF0D157E9412>II<1330A2137013F012011370120212041208121812101220124012C0EAFFFE EA0070A5EA03FE0F157F9412>I61 D70 D73 D76 D<00FEEB03F8001E14C000171305A338 138009A23811C011A33810E021A2EB7041A3EB3881A2EB1D01A2130EA2123839FE040FF8 1D177F9620>I83 D<387FFFF83860381800401308A200801304A300001300AF3803FF8016177F9619>I<3A FF07FC3F803A3C00E00E00001C1404A2EB0170000E5CA2EB023800075CA2EB041CD80384 5BA2EB880ED801C85BA2EBD007D800F05BA3EBE003016090C7FCA221177F9624>87 D<12FCA212C0B3AB12FCA206217D980A>91 D<12FCA2120CB3AB12FCA2062180980A>93 D97 D<12F81238A8EA39F0EA3E0CEA380613077F1480A414005B1306EA361C EA21F011177F9614>II<133E130EA8EA07CEEA1C3EEA300E1270126012E0A412601270 EA301EEA182E3807CF8011177F9614>IIII<12F81238A813F8EA3B1CEA3C0E1238AA38FE3F8011177F9614>I<12301278A212 301200A512F81238AC12FE07177F960A>I<1203EA0780A2EA0300C7FCA5EA1F801203AF 1243EAE30012E7127C091D82960B>I<12F81238A8133E13381330134013801239EA3FC0 EA39E0123813F01378133CA2EAFE7F10177F9613>I<12F81238B3A312FE07177F960A>I< 38F8F83E383B1CC7393C0F0380EA380EAA39FE3F8FE01B0E7F8D1E>IIII114 DI<1208A31218A21238EAFFC0EA3800A71340A4EA1C 80EA0F000A147F930E>III<38 FEFE7C383838381410133C001C1320134C381E4E60380ECE401387000713801303A20003 1300EA0201160E7F8D19>III E /FD 3 50 df<120CA2EACCC012EDEA7F80EA0C00EA7F80EAEDC012CCEA0C00A20A0B7D 8B10>3 D<5A5AA2EA0780EA0FC0EA7B78EAF33CEA0300AF0E167F9010>34 D<001F13F8383381843840E2023880640113381318131C132638404702382181CC381F00 F8180B7D8A1E>49 D E /FE 43 128 df45 D<14801301EB0300A31306A35BA25BA35BA25BA35BA2485AA348C7FCA31206A25AA35AA2 5AA35AA35AA211297D9E17>47 D<1303A3497EA2497E130BA2EB11E0A2EB31F01320A2EB 4078A3497EA23801003EEBFFFEEB001E00027FA348EB0780A2000C14C0121E39FF803FFC 1E1D7E9C22>65 DI<90380FE02090387018 603801C00439030003E000061301000E13004814605A15201278127000F01400A8007014 2012781238A26C14407E000614806CEB01003801C00638007018EB0FE01B1E7D9C21>I< B512C0380F0078141C1407A2EC0380EC01C0A215E0140015F0A815E0A2140115C0EC0380 A2EC0700141E1478B512E01C1C7D9B22>III<90381FC04090387030C03801C00C38030003000E1301120C001C1300 5A15401278127000F01400A6EC7FF8EC07C00070130312781238A27E120C120E00031305 3801C008390070304090381FC0001D1E7D9C23>I<39FFF0FFF0390F000F00AC90B5FCEB 000FAD39FFF0FFF01C1C7D9B22>I<3803FFC038003E00131EB3127012F8A2131CEA703C EA4038EA3070EA0FC0121D7E9B18>74 D76 DI80 D82 D<3803F040380C0CC0EA1002EA3001EA600012E01440A36C13007E127EEA7FE0EA3F FC6CB4FC00071380EA007FEB07C0EB03E0130113007EA36C13C0A238E0018038D00300EA CE06EA81F8131E7D9C19>I<39FFE003FC001FC712F06C146015406D13C000071480A239 03C00100A23801E002A213F000005BA2EB7808A2137CEB3C10A26D5AA2131F6D5AA26D5A A36DC7FCA21E1D7E9B22>86 D<13201370A313B8A3EA011CA2EA031EEA020EA2487EEA07 FFEA040738080380A2001813C01301123838FC07F815157F9419>97 DIIIIII<38FF8FF8381C01C0A9EA1FFFEA1C01A938FF8FF815157F9419>III<38FF81F8381C01E01480140013025B5B5B1330137013B8EA1D1C121EEA1C0E7F 14801303EB01C014E014F038FF83FC16157F941A>II<00FEEB0FE0001E140000171317A338138027A23811C047A3 3810E087A2EB7107A3133AA2131CA2123839FE083FE01B157F941F>I<38FC03F8381E00 E014401217EA138013C01211EA10E01370A21338131CA2130E130714C013031301130012 3800FE134015157F9419>I IIIII<387FFFF03860703000401310A200801308A300001300ADEA07FF15157F9419>I<38FF 83F8381C00E01440AE000C13C0000E138038060100EA0386EA00FC15157F9419>I<38FF 01F8383C0070001C13601440A26C1380A238070100A3EA0382A2EA01C4A3EA00E8A21370 A3132015157F9419>I<39FF07F87E393C01E03C0038EBC018391C02E010A3390E047020 A33907083840A33903901C80A33901E00F00A33800C006A31F157F9423>I<38FF07F038 1E03C0000E1300EA0F02EA0706EA038413C8EA01D0EA00F0A21370137813BCEA011C487E EA020F487E380C0380000813C0003C13E038FE07F815157F9419>I<38FF80FE381E0038 000E1320000F13606C13403803808013C03801C10013E212001374137C1338A848B4FC17 15809419>I127 D E /FF 50 123 df<13F8EA030C380E0604EA 1C07383803080030138800701390A200E013A0A214C01480A3EA6007EB0B883830719038 0F80E016127E911B>11 DI<1338137FEB87803801030090C7 FC7FA27F12007FA2137013F8EA03B8EA063CEA0C1C121812381270A212E0A413181338EA 6030EA70606C5AEA0F80111E7F9D12>14 D17 D<1206120EA35AA45AA35A1340A2EAE080A2EA6300123C0A127E910F>19 D21 D<1310A3EB1F8013F03801CF00EA038048C7FC120EA6EA06FCEA0384EA06FC0008C7FC12 185A12201260A212E0A31270127CEA3F80EA1FE0EA03F8C67E131E130E130CEA0108EA00 F01125809C12>24 D<380FFFF85A5A386084001241EA81041201EA030CA212021206A212 0E120CEA1C0EA21238EA180615127E9118>I<131EEB7180EBC0C0EA01801203EB00E05A A2380E01C0A31480EA1C0314001306EA1E0CEA3A18EA39E00038C7FCA25AA45AA25A131B 7F9115>I<3801FFF85A120F381E1E00EA180EEA38061270A2EAE00EA3130C131C13185B EA60606C5A001FC7FC15127E9118>I<380FFFE05A5A3860C0001240485A12001201A348 C7FCA35AA3120E120613127E9112>I<153081A281A2811507ED0380ED01C0ED00E0B712 F8A2250C7E942A>42 D<126012F0A2126004047C830C>58 D<126012F0A212701210A412 20A212401280040C7C830C>I<130113031306A3130CA31318A31330A31360A213C0A3EA 0180A3EA0300A31206A25AA35AA35AA35AA35AA210297E9E15>61 D<12C012F0123C120FEA03C0EA00F01338130E6D7EEB01E0EB0078141EEC0780A2EC1E00 1478EB01E0EB0780010EC7FC133813F0EA03C0000FC8FC123C12F012C0191A7D9620>I< 140CA2141CA2143C145CA2149E148EEB010E1302A21304A213081310A2497EEB3FFFEB40 071380A2EA0100A212025AA2001C148039FF803FF01C1D7F9C1F>65 D<48B5FC39003C01C090383800E015F01570A25B15F0A2EC01E09038E003C0EC0780EC1F 00EBFFFC3801C00FEC0780EC03C0A2EA0380A439070007801500140E5C000E1378B512C0 1C1C7E9B1F>I<48B5128039003C01E090383800701538153C151C5B151EA35BA4484813 3CA3153848481378157015F015E039070001C0EC0380EC0700141C000E1378B512C01F1C 7E9B22>68 D<48B512F839003C0078013813181510A35BA214081500495AA21430EBFFF0 3801C020A4390380404014001580A23907000100A25C1406000E133EB512FC1D1C7E9B1F >I<48B512F038003C00013813301520A35BA214081500495AA21430EBFFF03801C020A4 48485A91C7FCA348C8FCA45AEAFFF01C1C7E9B1B>I<3A01FFC07F803A003C001E000138 131815205D5DD97002C7FC5C5C5CEBE04014C0EBE1E013E23801C47013D0EBE03813C0EA 038080A280EA0700A280A2488039FFE03FF0211C7E9B23>75 D<3801FFE038003C001338 A45BA45BA4485AA438038002A31404EA0700140C14181438000E13F0B5FC171C7E9B1C> III<48B5FC39003C03C090383800 E015F01570A24913F0A315E0EBE001EC03C0EC0700141E3801FFF001C0C7FCA3485AA448 C8FCA45AEAFFE01C1C7E9B1B>80 DI<3801FFFE39003C03C090383800E015F01570A24913F0A3EC01E001E0 13C0EC0780EC1E00EBFFF03801C038140C140EA2EA0380A43807001E1508A2151048130F D8FFE01320C7EA03C01D1D7E9B20>I<001FB512F0391C03807039300700300020142012 601240130E1280A2000014005BA45BA45BA45BA41201EA7FFF1C1C7F9B18>84 D97 D<123F1207A2120EA45AA4EA39E0EA3A30EA3C1812381270131CA3EA E038A313301370136013C01261EA2300121E0E1D7E9C12>IIIIIIII107 DI< EA3C1F384E6180384681C0EA4701128F128E120EA2381C0380A3EB070000381310A2130E 1420387006403830038014127E9119>110 D<380787803809C8603808D03013E0EA11C0 14381201A238038070A31460380700E014C0EB0180EB8300EA0E86137890C7FCA25AA412 3CB4FC151A819115>112 D114 DI<13C012 01A3EA0380A4EAFFF0EA0700A3120EA45AA4EA3820A21340A2EA1880EA0F000C1A80990F >I<001C13C0EA27011247A238870380A2120EA2381C0700A438180E20A3EA1C1E380C26 403807C38013127E9118>II<380787803808C8403810F0C03820F1E0 EBE3C03840E1803800E000A2485AA43863808012F3EB810012E5EA84C6EA787813127E91 18>120 D<001C13C0EA27011247A238870380A2120EA2381C0700A4EA180EA3EA1C1EEA 0C3CEA07DCEA001C1318EA6038EAF0305B485AEA4180003EC7FC121A7E9114>II E /FG 87 128 df<130CA3131EA3133F132FA2EB4F 801347A2EB87C01383A2380103E01301A200027F1300A2487F1478A248137C143C121800 3C133E39FF01FFC01A1D7F9C1D>3 D 5 D11 D<137E3801C180EA0301380703C0120EEB018090C7FCA5 B512C0EA0E01B0387F87F8151D809C17>II<90383F07E03901C09C18380380F0D80701133C000E13 E00100131892C7FCA5B612FC390E00E01CB03A7FC7FCFF80211D809C23>I<120EA2121E 1238127012E012800707779C15>19 D<126012F0A71260AD1200A5126012F0A21260041E 7C9D0C>33 DI<126012F012F812681208A31210A2122012401280050C7C9C0C>39 D<1380EA0100120212065AA25AA25AA35AA412E0AC1260A47EA37EA27EA27E12027EEA00 80092A7C9E10>I<7E12407E12307EA27EA27EA37EA41380AC1300A41206A35AA25AA25A 12205A5A092A7E9E10>I<1306ADB612E0A2D80006C7FCAD1B1C7E9720>43 D<126012F0A212701210A41220A212401280040C7C830C>II<12 6012F0A2126004047C830C>I<130113031306A3130CA31318A31330A31360A213C0A3EA 0180A3EA0300A31206A25AA35AA35AA35AA35AA210297E9E15>II<5A1207123F12C71207B3A5EAFFF80D1C7C9B15>III<130CA2131C133CA2135C13DC139C EA011C120312021204120C1208121012301220124012C0B512C038001C00A73801FFC012 1C7F9B15>II<13F0EA03 0CEA0404EA0C0EEA181E1230130CEA7000A21260EAE3E0EAE430EAE818EAF00C130EEAE0 061307A51260A2EA7006EA300E130CEA1818EA0C30EA03E0101D7E9B15>I<1240387FFF 801400A2EA4002485AA25B485AA25B1360134013C0A212015BA21203A41207A66CC7FC11 1D7E9B15>III<126012F0A212601200AA126012F0A2126004127C910C>I<126012F0A2 12601200AA126012F0A212701210A41220A212401280041A7C910C>I61 D63 D<1306A3130FA3EB1780A2EB37C01323A2EB43E01341A2EB80F0A338010078A2EBFFF838 02003CA3487FA2000C131F80001E5BB4EBFFF01C1D7F9C1F>65 DI<90381F8080EBE0613801801938070007000E13035A14015A00781300A212 7000F01400A8007014801278A212386CEB0100A26C13026C5B380180083800E030EB1FC0 191E7E9C1E>IIII<90381F8080EBE0613801801938070007000E13035A14015A007813 00A2127000F01400A6ECFFF0EC0F80007013071278A212387EA27E6C130B380180113800 E06090381F80001C1E7E9C21>I<39FFF0FFF0390F000F00AC90B5FCEB000FAD39FFF0FF F01C1C7F9B1F>II<3807FF8038007C00133CB3 127012F8A21338EA7078EA4070EA30E0EA0F80111D7F9B15>I<39FFF01FE0390F000780 EC060014045C5C5C5C5C49C7FC13021306130FEB17801327EB43C0EB81E013016D7E1478 A280143E141E80158015C039FFF03FF01C1C7F9B20>IIIIII82 D<3807E080EA1C19EA30051303EA600112E01300 A36C13007E127CEA7FC0EA3FF8EA1FFEEA07FFC61380130FEB07C0130313011280A300C0 1380A238E00300EAD002EACC0CEA83F8121E7E9C17>I<007FB512C038700F0100601300 00401440A200C014201280A300001400B1497E3803FFFC1B1C7F9B1E>I<39FFF01FF039 0F000380EC0100B3A26C1302138000035BEA01C03800E018EB7060EB0F801C1D7F9B1F> I<39FFE00FF0391F0003C0EC01806C1400A238078002A213C000035BA2EBE00C00011308 A26C6C5AA213F8EB7820A26D5AA36D5AA2131F6DC7FCA21306A31C1D7F9B1F>I<3AFFE1 FFC0FF3A1F003E003C001E013C13186C6D1310A32607801F1320A33A03C0278040A33A01 E043C080A33A00F081E100A39038F900F3017913F2A2017E137E013E137CA2013C133C01 1C1338A20118131801081310281D7F9B2B>I<39FFF003FC390F8001E00007EB00C06D13 800003EB01006D5A000113026C6C5A13F8EB7808EB7C18EB3C10EB3E20131F6D5A14C06D 5AABEB7FF81E1C809B1F>89 D<387FFFF0EA7C01007013E0386003C0A238400780130F14 00131E12005B137C13785BA2485A1203EBC010EA0780A2EA0F00481330001E13205A1460 4813E0EAF803B5FC141C7E9B19>I<12FEA212C0B3B312FEA207297C9E0C>II<12FEA21206 B3B312FEA20729809E0C>I<120C12121221EA4080EA80400A057B9B15>I97 D<12FC121CAA137CEA1D87381E0180381C00C014E014601470A61460 14E014C0381E018038190700EA10FC141D7F9C17>IIII<13F8EA018CEA071E1206EA0E0C1300A6EA FFE0EA0E00B0EA7FE00F1D809C0D>II<12FC121CAA137C1387EA1D03 001E1380121CAD38FF9FF0141D7F9C17>I<1218123CA21218C7FCA712FC121CB0EAFF80 091D7F9C0C>I<13C0EA01E0A2EA00C01300A7EA07E01200B3A21260EAF0C012F1EA6180 EA3E000B25839C0D>I<12FC121CAAEB0FE0EB0780EB06005B13105B5B13E0121DEA1E70 EA1C781338133C131C7F130F148038FF9FE0131D7F9C16>I<12FC121CB3A9EAFF80091D 7F9C0C>I<39FC7E07E0391C838838391D019018001EEBE01C001C13C0AD3AFF8FF8FF80 21127F9124>IIII<3803E080EA0E19EA1805 EA3807EA7003A212E0A61270A2EA38071218EA0E1BEA03E3EA0003A7EB1FF0141A7F9116 >III<1204A4120CA2121C123CEAFFE0EA1C00A91310A5120CEA0E20EA03 C00C1A7F9910>I<38FC1F80EA1C03AD1307120CEA0E1B3803E3F014127F9117>I<38FF07 E0383C0380381C0100A2EA0E02A2EA0F06EA0704A2EA0388A213C8EA01D0A2EA00E0A313 4013127F9116>I<39FF3FC7E0393C0703C0001CEB01801500130B000E1382A213110007 13C4A213203803A0E8A2EBC06800011370A2EB8030000013201B127F911E>I<38FF0FE0 381E0700EA1C06EA0E046C5AEA039013B0EA01E012007F12011338EA021C1204EA0C0E48 7E003C138038FE1FF014127F9116>I<38FF07E0383C0380381C0100A2EA0E02A2EA0F06 EA0704A2EA0388A213C8EA01D0A2EA00E0A31340A25BA212F000F1C7FC12F31266123813 1A7F9116>III127 D E /FH 29 123 df45 D<130E131E137EEA07FE12FFA212F81200B3ABB512FEA317277BA622>49 DII65 D73 D78 D82 D<3803FF80000F13F0381F01FC383F80FE14 7F801580EA1F00C7FCA4EB3FFF3801FC3FEA0FE0EA1F80EA3F00127E5AA4145F007E13DF 393F839FFC381FFE0F3803FC031E1B7E9A21>97 DIIIII104 D<1207EA0F80EA1FC0EA3FE0A3 EA1FC0EA0F80EA0700C7FCA7EAFFE0A3120FB3A3EAFFFEA30F2B7EAA12>I108 D<26FFC07FEB1FC0903AC1FFC07FF0903AC307E0C1 F8D80FC49038F101FC9039C803F20001D801FE7F01D05BA201E05BB03CFFFE3FFF8FFFE0 A3331B7D9A38>I<38FFC07E9038C1FF809038C30FC0D80FC413E0EBC80701D813F013D0 A213E0B039FFFE3FFFA3201B7D9A25>II<38FFE1FE9038EFFF809038FE0F E0390FF803F09038F001F801E013FC140015FEA2157FA8157E15FEA215FC140101F013F8 9038F807F09038FC0FE09038EFFF809038E1FC0001E0C7FCA9EAFFFEA320277E9A25>I< 38FFC1F0EBC7FCEBC63E380FCC7F13D813D0A2EBF03EEBE000B0B5FCA3181B7F9A1B> 114 D<3803FE30380FFFF0EA3E03EA7800127000F01370A27E00FE1300EAFFE06CB4FC14 C06C13E06C13F0000713F8C6FCEB07FC130000E0137C143C7E14387E6C137038FF01E038 E7FFC000C11300161B7E9A1B>I<13E0A41201A31203A21207120F381FFFE0B5FCA2380F E000AD1470A73807F0E0000313C03801FF8038007F0014267FA51A>I<39FFE07FF0A300 0F1307B2140FA2000713173903F067FF3801FFC738007F87201B7D9A25>I<39FFFC03FF A3390FF000F0000714E07F0003EB01C0A2EBFC0300011480EBFE070000140013FFEB7F0E A2149EEB3F9C14FC6D5AA26D5AA36D5AA26D5AA2201B7F9A23>I<3BFFFC7FFC1FFCA33B 0FE00FE001C02607F007EB0380A201F8EBF00700031600EC0FF801FC5C0001150EEC1FFC 2600FE1C5B15FE9039FF387E3C017F1438EC787F6D486C5A16F0ECE01F011F5CA26D486C 5AA2EC800701075CA22E1B7F9A31>I<39FFFC03FFA3390FF000F0000714E07F0003EB01 C0A2EBFC0300011480EBFE070000140013FFEB7F0EA2149EEB3F9C14FC6D5AA26D5AA36D 5AA26D5AA25CA21307003890C7FCEA7C0FEAFE0E131E131C5BEA74F0EA3FE0EA0F802027 7F9A23>121 D<003FB5FCA2EB00FEEA3C01383803FC007813F8EB07F0EA700F14E0EB1F C0EA003F1480EB7F005B5B3801FC07120313F8EA07F0000F130F13E0381FC00E003F131E 387F803EEB00FEB5FCA2181B7E9A1E>I E /FI 1 22 df<120FEA03807F1201A27F1200 A27F1370A213781338A2133C137C13DEEA018EEA030EEA060F487E121C00381380EA7003 12E038C001C0121A7E9916>21 D E /FJ 35 123 df<13FEEA038138060180EA0E03381C 010090C7FCA5B51280EA1C03AE38FF8FF0141A809915>12 D<90387E1F803901C1704039 0703C0600006EB80E0000E14401500A5B612E0380E0380AE397F8FE3FC1E1A809920>14 D<1380EA010012025A120C120812185AA35AA412E0AA1260A47EA37E1208120C12047E7E EA008009267D9B0F>40 D<7E12407E7E12181208120C7EA37EA41380AA1300A41206A35A 1208121812105A5A5A09267E9B0F>I<126012F0A212701210A31220A21240A2040B7D83 0B>44 DI<126012F0A2126004047D830B>I53 D<007FB5FC38701C0700401301A200C0148000801300A3 00001400B13803FFE0191A7F991C>84 D<3AFF81FF07F03A3C007801C0001CEC0080A36C 90389C0100A33907010E02A33903830F04EB8207A2150C3901C40388A33900E801D0A390 387000E0A301305B01201340241A7F9927>87 D<12FEA212C0B3AF12FEA207257D9B0B> 91 D<12FEA21206B3AF12FEA20725809B0B>93 D97 D<12FC121CA913FCEA1D07381E0380381C01C0130014E0A6EB01C01480381E0300EA1906 EA10F8131A809915>II<133F1307A9EA03E7EA0C17EA180F487E127012E0A6126012706C5A EA1C373807C7E0131A7F9915>IIII<12FC121CA9137CEA1D87381E0380A2121CAB38FF9FF0141A809915>I<1218 123CA212181200A612FC121CAE12FF081A80990A>I<12FC121CB3A6EAFF80091A80990A> 108 D<38FC7C1F391D8E6380391E0781C0A2001C1301AB39FF9FE7F81D107F8F20>IIII114 DI<1208A41218A21238EAFFC0EA3800A81320A41218EA1C40 EA07800B177F960F>I<38FC1F80EA1C03AB1307120CEA0E0B3803F3F01410808F15>I<38 FF0F80383C0700EA1C061304A26C5AA26C5AA3EA03A0A2EA01C0A36C5A11107F8F14>I< 39FE7F1F8039381C0700003C1306381C0C04130E380E16081317A238072310149013A338 03C1A014E0380180C0A319107F8F1C>I<38FE3F80383C1E00EA1C086C5AEA0F306C5A6C 5A12017F1203EA0270487E1208EA181CEA381E38FC3FC012107F8F14>I<38FF0F80383C 0700EA1C061304A26C5AA26C5AA3EA03A0A2EA01C0A36C5AA248C7FCA212E112E212E412 7811177F8F14>II E /FK 7 117 df<1303497EA2497EA3EB1BE0A2 EB3BF01331A2EB60F8A2EBE0FCEBC07CA248487EEBFFFE487FEB001F4814800006130FA2 48EB07C039FF803FFCA21E1A7F9921>65 D97 D<12FCA2123CA713FE38 3F8780383E01C0003C13E0EB00F0A214F8A514F0A2EB01E0003E13C0383B07803830FE00 151A7E9919>II114 DI<1206A4120EA2121EEA3FF012 FFEA1E00A81318A5EA0F30EA03E00D187F9711>I E /FL 4 74 df<1202A3EAC218EAF2 78EA3AE0EA0F80A2EA3AE0EAF278EAC218EA0200A30D0E7E8E12>3 D 35 D<13201360B3B512F8A215167D951B>63 D<3801FFF8000F13F038180C00EA3008EA 6018EA00381330A21370A21360A213E0A25BA212015BA23803008038020100EA7FFE485A 1517809613>73 D E /FM 27 120 df<127012F812FCA212741204A41208A21210A21220 1240060F7C840E>44 D48 D<13801203120F12F31203B3A9EA07C0EAFFFE0F217CA018>II57 D66 D69 D<39FFFC3FFF390FC003F039078001E0AE90B5FCEB8001AF39 0FC003F039FFFC3FFF20227EA125>72 D77 D<3803F020380C0C60EA1802383001E0EA70000060136012E0A21420A36C1300A2127812 7FEA3FF0EA1FFE6C7E0003138038003FC0EB07E01301EB00F0A214707EA46C1360A26C13 C07E38C8018038C60700EA81FC14247DA21B>83 D<39FFFC07FF390FC000F86C48137015 20B3A5000314407FA2000114806C7E9038600100EB3006EB1C08EB03F020237EA125>85 D97 D<120E12FE121E120EAB131FEB61C0 EB8060380F0030000E1338143C141C141EA7141C143C1438000F1370380C8060EB41C038 083F0017237FA21B>II<14E0130F13011300AB EA01F8EA0704EA0C02EA1C01EA38001278127012F0A7127012781238EA1801EA0C023807 0CF03801F0FE17237EA21B>I I<14703803F198380E1E18EA1C0E38380700A200781380A400381300A2EA1C0EEA1E1CEA 33F00020C7FCA212301238EA3FFE381FFFC06C13E0383000F0481330481318A400601330 A2003813E0380E03803803FE0015217F9518>103 D<120E12FE121E120EABEB1F80EB60 C0EB80E0380F0070A2120EAF38FFE7FF18237FA21B>I<121C123EA3121CC7FCA8120E12 7E121E120EB1EAFFC00A227FA10E>I<120E12FE121E120EB3ADEAFFE00B237FA20E>108 D<390E1FC07F3AFE60E183803A1E807201C03A0F003C00E0A2000E1338AF3AFFE3FF8FFE 27157F942A>I<380E1F8038FE60C0381E80E0380F0070A2120EAF38FFE7FF18157F941B> I114 DI<1202A41206A3120E121E123E EAFFFCEA0E00AB1304A6EA07081203EA01F00E1F7F9E13>I<000E137038FE07F0EA1E00 000E1370AD14F0A238060170380382783800FC7F18157F941B>I<39FF8FF87F393E01E0 3C001CEBC01814E0000E1410EB0260147000071420EB04301438D803841340EB8818141C D801C81380EBD00C140E3900F00F00497EA2EB6006EB400220157F9423>119 D E /FN 18 123 df<007FB712E0A23A7E000F8007007815010070150000601660004016 20A200C01630A2481610A6C71500B3AC4A7E010FB57EA22C317EB030>84 D<13FE380303C0380C00E00010137080003C133C003E131C141EA21208C7FCA3EB0FFEEB FC1EEA03E0EA0F80EA1F00123E123C127C481404A3143EA21278007C135E6CEB8F08390F 0307F03903FC03E01E1F7D9E21>97 DI101 DI<15F090387F03083901C1C41C380380E8390700700848EB78 00001E7FA2003E133EA6001E133CA26C5B6C13706D5A3809C1C0D8087FC7FC0018C8FCA5 121C7E380FFFF86C13FF6C1480390E000FC00018EB01E048EB00F000701470481438A500 701470A26C14E06CEB01C00007EB07003801C01C38003FE01E2F7E9F21>I<120FEA1F80 A4EA0F00C7FCABEA0780127FA2120F1207B3A6EA0FC0EAFFF8A20D307EAF12>105 D108 D<260780FEEB1FC03BFF83078060F0903A8C03C180783B0F9001E2003CD807A013E4DA00 F47F01C013F8A2495BB3A2486C486C133F3CFFFC1FFF83FFF0A2341F7E9E38>I<380780 FE39FF83078090388C03C0390F9001E0EA07A06E7E13C0A25BB3A2486C487E3AFFFC1FFF 80A2211F7E9E25>II<380783E038FF8418EB887CEA0F 90EA07A01438EBC000A35BB3487EEAFFFEA2161F7E9E19>114 D<1340A513C0A31201A2 12031207120F381FFFE0B5FC3803C000B01410A80001132013E000001340EB78C0EB1F00 142C7FAB19>116 DII<3BFFF03FF80FFCA23B0F80 07E003F0913903C001C00007ED0080A201C0EBE00100031600140401E06D5A00011502EC 087001F0EB780600001504EC10380178EB3C08A2EC201C013CEB1E10A2EC400E011EEB0F 20A2EC8007010F14C0A2EC00036D5CA201061301010291C7FC2E1F7F9E30>I121 D<003FB5FC383E001E12 380030133C00201378A2006013F0384001E0A2EB03C0EB07801200EB0F00131EA25B5BA2 EBF001EA01E0A2EA03C0EA07801403380F0002121E14065A48130E147EB512FE181F7E9E 1D>I E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 296 436 a FN(T)-6 b(erm)22 b(rewriting)h(for)e(normalization)g (b)n(y)g(ev)l(aluation)347 556 y FM(Ulric)o(h)15 b(Berger,)g(Matthias)h (Eb)q(erl)g(and)h(Helm)o(ut)d(Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg)1602 538 y FL(\003)810 654 y FM(Marc)o(h)h(29,)i(2001)889 845 y FK(Abstract)419 910 y FJ(W)m(e)g(extend)h(normalization)j(b)o(y)c (ev)n(aluation)j(\(\014rst)d(presen)o(ted)h(in)g([5]\))f(from)361 956 y(the)e(pure)g(t)o(yp)q(ed)g FI(\025)p FJ(-calculus)h(to)f(general) g(higher)h(t)o(yp)q(e)f(term)f(rewriting)i(systems)361 1001 y(and)c(pro)o(v)o(e)g(its)h(correctness)f(w.r.t.)e(a)i (domain-theoretic)i(mo)q(del.)j(W)m(e)12 b(distinguish)361 1047 y(b)q(et)o(w)o(een)k(computational)j(rules)e(and)g(prop)q(er)g (rewrite)f(rules.)27 b(The)16 b(former)g(is)h(a)361 1093 y(rather)c(restricted)g(class)h(of)e(rules,)h(whic)o(h,)g(ho)o(w)o(ev)o (er,)f(allo)o(ws)i(for)e(a)g(more)h(e\016cien)o(t)361 1138 y(implemen)o(tation.)257 1276 y FH(1)67 b(In)n(tro)r(duction)257 1366 y FG(It)18 b(is)f(w)o(ell)f(kno)o(wn)h(that)g(implemen)o(ting)d (normalization)g(of)j FF(\025)p FG(-terms)g(in)f(the)i(usual)f(re-)257 1416 y(cursiv)o(e)f(fashion)e(is)h(quite)g(ine\016cien)o(t.)22 b(Ho)o(w)o(ev)o(er,)15 b(it)f(is)h(p)q(ossible)h(to)e(compute)h(the)g (long)257 1466 y(normal)9 b(form)f(of)i(a)g FF(\025)p FG(-term)g(b)o(y)g(ev)n(aluating)f(it)g(in)h(an)g(appropriate)h(mo)q (del)d(\(cf.)j([5)o(]\).)16 b(When)257 1516 y(using)11 b(for)f(that)h(purp)q(ose)g(the)g(built-in)f(ev)n(aluation)f(mec)o (hanism)f(of)i(e.g.)g FE(Scheme)h FG(\(a)g(pure)257 1566 y FE(Lisp)j FG(dialect\))f(one)h(obtains)f(an)g(amazingly)d(fast)k (algorithm)c(called)j(\\normalization)e(b)o(y)257 1616 y(ev)n(aluation")g(or)i(NbE)g(for)g(short.)18 b(In)13 b(the)g(con)o(text)g(of)g(t)o(yp)q(e-directed)h(partial)e(ev)n (aluation)257 1665 y([8])i(it)f(has)i(b)q(een)h(analyzed)e(in)g(what)g (sense)i(NbE)f(is)f(more)f(e\016cien)o(t,)i(and)f(wh)o(y;)g(a)g(punc-) 257 1715 y(tual)e(comparison)e(b)q(et)o(w)o(een)j(NbE)f(and)g(a)f(naiv) o(e,)g(sym)o(b)q(olic)f(normalizer)g(can)i(b)q(e)h(found)e(in)257 1765 y([4,)j(section)i(5].)k(The)15 b(essen)o(tial)g(idea)g(is)g(to)f (\014nd)h(an)g(in)o(v)o(erse)h(to)e(ev)n(aluation,)f(con)o(v)o(erting) 257 1815 y(a)i(seman)o(tic)g(ob)r(ject)h(in)o(to)e(a)h(syn)o(tactic)h (term.)21 b(This)15 b(normalization)d(pro)q(cedure)18 b(is)d(used)257 1865 y(and)d(tested)i(in)e(the)h(pro)q(of)f(system)g FE(Minlog)g FG(dev)o(elop)q(ed)h(in)f(Munic)o(h)g(\(cf.)g([2)o(]\).)18 b({)12 b(Notice,)257 1914 y(ho)o(w)o(ev)o(er,)g(that)g(once)g(NbE)g(is) g(expressed)i(in)d(a)h(functional)e(programming)e(language,)j(the)257 1964 y(ev)n(aluation)h(order)h(of)g(this)f(language)g(\(call-b)o(y-v)n (alue)f(for)i FE(Scheme)p FG(\))g(determines)h(the)f(re-)257 2014 y(duction)g(order)h(of)e(NbE)h(\(applicativ)o(e)f(order)h(for)f(a) h(call-b)o(y-v)n(alue)e(language\).)17 b(It)12 b(is)h(th)o(us)257 2064 y(easy)k(to)f(defeat)h(NbE)g(in)e FE(Scheme)i FG(b)o(y)f (normalizing)e(the)i(application)f(of)h(a)g(non-strict)257 2114 y(function)h(to)h(an)f(expression)h(that)f(is)h(exp)q(ensiv)o(e)g (to)f(normalize.)27 b(F)m(or)17 b(suc)o(h)h(a)f(term,)g(a)257 2163 y(sym)o(b)q(olic)d(normalizer)g(follo)o(wing)f(a)j(normal)d(order) j(reduction)h(strategy)f(can)g(easily)f(b)q(e)257 2213 y(more)e(e\016cien)o(t.)320 2263 y(Ob)o(viously)m(,)d(for)i (applications)f(pure)i(t)o(yp)q(ed)g FF(\025)p FG(-terms)f(are)g(not)h (su\016cien)o(t;)f(one)h(clearly)257 2313 y(needs)19 b(constan)o(ts)e(as)g(w)o(ell.)27 b(In)16 b([4])g(NbE)h(has)g(b)q(een)h (extended)h(to)d(term)h(systems)g(with)257 2363 y(higher)e(order)h (term)e(rewrite)h(rules.)22 b(The)15 b(presen)o(t)h(pap)q(er)f(adds)g (a)g(distinction)f(b)q(et)o(w)o(een)p 257 2397 573 2 v 303 2424 a FD(\003)321 2436 y FC(The)e(hospitalit)o(y)c(of)k(the)f (Mittag-Le\017er)f(Institute)f(in)i(the)g(spring)g(of)g(2001)f(is)i (gratefully)d(ac)o(kno)o(w-)257 2475 y(ledged.)963 2628 y FG(1)p eop %%Page: 2 2 2 1 bop 257 262 a FG(what)15 b(w)o(e)g(call)f(computational)f(rules)i (and)g(\(prop)q(er\))h(rewrite)g(rules;)f(NbE)g(seems)g(to)g(b)q(e)257 311 y(m)o(uc)o(h)10 b(more)g(e\016cien)o(t)h(for)g(the)g(former)f(than) h(for)f(the)i(latter.)17 b(In)11 b(our)f(implemen)o(tation)e(\(in)257 361 y(the)15 b FE(Minlog)f FG(system\))g(w)o(e)g(therefore)h(use)g (computational)c(rules)k(whenev)o(er)g(p)q(ossible.)320 411 y(A)g(related)h(approac)o(h)g(\(using)f(a)g(glueing)g (construction\))h(is)g(elab)q(orated)g(b)o(y)f(T.)g FE(Co-)257 461 y(quand)24 b FG(and)e(P)m(.)g FE(D)o(ybjer)h FG(in)f([6)o(].)43 b(Another)23 b(related)g(pap)q(er)g(is)f(T.)f FE(Al)m(tenkir)o(ch)p FG(,)257 511 y(M.)10 b FE(Hofmann)j FG(and)d(T.)g FE(Streicher)g FG([1)o(];)h(there)h(a)e(cartesian)h(closed)g(category)g(is)f (de\014ned)257 560 y(whic)o(h)j(has)h(the)g(prop)q(ert)o(y)g(that)f (the)h(in)o(terpretation)g(of)e(the)i(simply)d(t)o(yp)q(ed)j(lam)o(b)q (da)d(cal-)257 610 y(culus)j(in)f(it)f(yields)h(the)h(reduction-free)h (normalization)10 b(algorithm)h(from)g([5)o(],)i(as)g(w)o(ell)f(as)257 660 y(its)f(correctness.)20 b(Moreo)o(v)o(er,)11 b(O.)f FE(D)o(anvy)j FG(\(cf.)e(e.g.)e([8]\))h(has)g(successfully)i(used)f (this)g(algo-)257 710 y(rithm)i(\(or)h(more)f(precisely)i(its)f(call-b) o(y-v)n(alue)e(coun)o(terpart\))j(in)f(the)g(con)o(text)h(of)e(partial) 257 760 y(ev)n(aluation.)18 b(A.)c FE(Filinski)g FG([10)o(])g(also)f (treats)j(NbE)f(for)e(an)i(extension)g(of)e(the)i FF(\025)p FG(-calculus)257 809 y(b)o(y)f(constan)o(ts,)g(where)h(non-termination) c(is)i(allo)o(w)o(ed.)k(Ho)o(w)o(ev)o(er,)d(he)g(do)q(es)g(not)g (consider)257 859 y(constan)o(ts)k(whose)g(meaning)d(is)h(only)g(giv)o (en)h(op)q(erationally)m(,)e(i.e.)h(b)o(y)h(arbitrary)f(rewrite)257 909 y(rules.)j(Therefore)14 b(the)f(normal)e(pro)q(of)i(tec)o(hnique)h (emplo)o(ying)c(the)k(logical)d(relation)h(\\the)257 959 y(v)n(alue)18 b(of)f(expression)i FF(e)f FG(in)f(en)o(vironmen)o(t) g FF(\016)j FG(is)e FF(a)p FG(")f(is)h(a)o(v)n(ailable)e(in)h(his)h (case,)h(whereas)257 1009 y(in)14 b(ours)h(it)f(is)g(more)f(con)o(v)o (enien)o(t)i(to)f(follo)o(w)f(a)h(di\013eren)o(t)h(approac)o(h,)f(via)f (an)h(appropriate)257 1059 y(inductiv)o(e)g(generation)g(of)g(the)g (reducibilit)o(y)f(relation.)320 1108 y(Wh)o(y)g(should)h(one)h(b)q(e)g (in)o(terested)h(in)e(the)g(correctness)k(of)13 b(NbE)i(for)f(general)g (rewrite)257 1158 y(rules,)j(where)h(neither)f(termination)e(nor)h(ev)o (en)h(con\015uence)h(is)e(assumed?)25 b(One)17 b(reason)257 1208 y(is)j(that)f(in)g(an)g(in)o(teractiv)o(e)h(pro)q(of)f(dev)o (elopmen)o(t)f(system)h(\()p FE(Minlog)h FG(in)f(our)g(case\))i(it)257 1258 y(is)c(con)o(v)o(enien)o(t)f(not)h(ha)o(ving)e(to)h(deal)g (explicit)o(y)g(with)g(equalit)o(y)f(axioms,)g(but)h(rather)i(to)257 1308 y(iden)o(tify)c(terms)g(with)f(the)i(same)e(normal)f(form,)g(mo)q (dulo)g(a)i(giv)o(en)g(set)h(of)e(rewrite)i(rules.)257 1357 y(Then)e(an)e(e\016cien)o(t)h(normalization)d(algorithm)g(lik)o(e) i(NbE)h(to)g(test)h(for)e(equalit)o(y)g(clearly)h(is)257 1407 y(useful.)18 b(Ho)o(w)o(ev)o(er,)13 b(one)g(do)q(es)g(not)g(w)o (an)o(t)f(to)h(ha)o(v)o(e)f(the)h(obligation)e(to)h(pro)o(v)o(e)h (termination)257 1457 y(and)h(con\015uence)i(of)d(the)h(whole)g(set)h (of)e(rewrite)i(rules)g(whenev)o(er)g(a)e(new)i(one)f(is)g(added.)320 1507 y(The)i(aim)e(of)h(the)h(presen)o(t)i(pap)q(er)e(is)g(to)g(dev)o (elop)g(the)g(theory)g(of)g(normalization)d(b)o(y)257 1557 y(ev)n(aluation)e(from)f(scratc)o(h,)j(up)f(to)g(and)g(including)f (\(some)g(generalizations)h(of)s(\))f FE(G)1566 1554 y(\177)1564 1557 y(odel)p FG('s)257 1606 y(system)20 b FF(T)26 b FG(of)20 b(higher)g(order)h(primitiv)o(e)d(recursion.)38 b(In)20 b(fact,)h(w)o(e)g(will)d(treat)j(almost)257 1656 y(arbitrary)14 b(rewrite)h(systems.)320 1706 y(Let)f(us)g(b)q(egin)g (with)g(a)f(short)i(explanation)e(of)g(the)i(essence)h(of)d(the)i (metho)q(d)e(for)h(nor-)257 1756 y(malizing)c(t)o(yp)q(ed)j FF(\025)p FG(-terms)f(b)o(y)g(means)f(of)h(an)g(ev)n(aluation)e(pro)q (cedure)15 b(of)c(some)g(functional)257 1806 y(programming)i(language)h (suc)o(h)j(as)f FE(Scheme)p FG(.)24 b(F)m(or)16 b(simplicit)o(y)d(w)o (e)j(return)h(to)f(the)g(sim-)257 1856 y(plest)f(case,)f(simply)e(t)o (yp)q(ed)i FF(\025)p FG(-calculus)h(without)e(constan)o(ts.)320 1905 y(Simple)h(t)o(yp)q(es)j(are)f(built)f(from)g(ground)h(t)o(yp)q (es)h FF(\034)j FG(b)o(y)c FF(\032)g FB(!)e FF(\033)k FG(\(later)e(also)f(pro)q(ducts)257 1955 y FF(\032)f FB(\002)g FF(\033)21 b FG(will)d(b)q(e)j(included\).)37 b(The)20 b(set)h(\003)f(of)g(terms)f(is)h(giv)o(en)g(b)o(y)f FF(x)1383 1940 y FA(\033)1406 1955 y FG(,)i(\()p FF(\025x)1503 1940 y FA(\032)1522 1955 y FF(M)1567 1940 y FA(\033)1589 1955 y FG(\))1605 1940 y FA(\032)p Fz(!)p FA(\033)1678 1955 y FG(,)257 2005 y(\()p FF(M)318 1990 y FA(\032)p Fz(!)p FA(\033)391 2005 y FF(N)429 1990 y FA(\032)448 2005 y FG(\))464 1990 y FA(\033)487 2005 y FG(;)12 b(let)h(\003)599 2011 y FA(\032)631 2005 y FG(denote)h(the)f(set)h(of)e(all)g(terms)g (of)g(t)o(yp)q(e)i FF(\032)p FG(.)k(The)13 b(set)h FE(Lnf)f FG(of)g(terms)257 2055 y(in)h(long)f(normal)e(form)i(\(i.e.)g(normal)e (w.r.t.)i FF(\014)r FG(-reduction)i(and)f FF(\021)q FG(-expansion\))g (is)f(de\014ned)257 2105 y(inductiv)o(ely)19 b(b)o(y)g(\()p FF(xM)619 2111 y Fy(1)645 2105 y FF(:)7 b(:)g(:)e(M)740 2111 y FA(n)763 2105 y FG(\))779 2090 y FA(\034)800 2105 y FG(,)20 b FF(\025xM)25 b FG(\(w)o(e)19 b(abbreviate)h FF(xM)1301 2111 y Fy(1)1327 2105 y FF(:)7 b(:)g(:)e(M)1422 2111 y FA(n)1464 2105 y FG(b)o(y)19 b FF(x)p Fx(M)25 b FG(and)257 2154 y(similar)13 b(a)h(list)h FF(M)541 2160 y Fy(1)567 2154 y FF(:)7 b(:)g(:)e(M)662 2160 y FA(n)700 2154 y FG(b)o(y)14 b Fx(M)6 b FG(\).)21 b(By)15 b Fw(nf)s FG(\()p FF(M)5 b FG(\))15 b(w)o(e)g(denote)h(the)f(long)f (normal)f(form)g(of)257 2204 y FF(M)5 b FG(,)14 b(i.e.)e(the)j(unique)f (term)f(in)h(long)e(normal)g(form)g FF(\014)r(\021)q FG(-equal)j(to)e FF(M)5 b FG(.)320 2254 y(No)o(w)17 b(w)o(e)h(ha)o(v)o (e)f(to)h(c)o(ho)q(ose)g(our)g(mo)q(del.)27 b(A)18 b(simple)e(solution) h(is)h(to)f(tak)o(e)h(terms)f(of)257 2304 y(ground)d(t)o(yp)q(e)f(as)h (ground)f(t)o(yp)q(e)h(ob)r(jects,)g(and)f(all)f(functions)h(as)g(p)q (ossible)h(function)f(t)o(yp)q(e)257 2354 y(ob)r(jects:)456 2428 y([)-7 b([)p FF(\034)5 b FG(])-7 b(])10 b(:=)h(\003)607 2434 y FA(\034)628 2428 y FF(;)48 b FG([)-7 b([)p FF(\032)11 b FB(!)g FF(\033)q FG(])-7 b(])11 b(:=)g([)-7 b([)p FF(\033)q FG(])g(])957 2411 y Fy([)h([)p FA(\032)p Fy(])g(])1043 2428 y FG(\(the)15 b(full)d(function)i(space\))q FF(:)257 2503 y FG(It)e(is)g(crucial)g(that)g(all)e(terms)i(\(of)f(ground)h(t)o (yp)q(e\))g(are)g(presen)o(t,)i(not)e(just)g(the)g(closed)g(ones.)963 2628 y(2)p eop %%Page: 3 3 3 2 bop 257 262 a FG(Next)18 b(w)o(e)f(need)h(an)f(assignmen)o(t)f FB(")h FG(lifting)e(a)i(v)n(ariable)f(to)h(an)g(ob)r(ject,)h(and)f(a)g (function)257 311 y FB(#)i FG(giving)e(us)j(a)e(normal)f(term)h(from)g (an)g(ob)r(ject.)34 b(They)19 b(should)g(meet)g(the)g(follo)o(wing)257 361 y(condition,)13 b(whic)o(h)h(migh)o(t)e(b)q(e)i(called)g (\\correctness)i(of)e(normalization)d(b)o(y)i(ev)n(aluation")808 452 y FB(#)p FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])924 458 y Fz(")941 452 y FG(\))12 b(=)g Fw(nf)s FG(\()p FF(M)5 b FG(\))p FF(;)257 544 y FG(where)16 b([)-7 b([)p FF(M)440 529 y FA(\032)459 544 y FG(])g(])476 550 y Fz(")507 544 y FB(2)13 b FG([)-7 b([)p FF(\032)p FG(])g(])13 b(denotes)j(the)g(v)n (alue)e(of)g FF(M)20 b FG(under)c(the)f(assignmen)o(t)f FB(")o FG(.)21 b(Tw)o(o)15 b(suc)o(h)257 594 y(functions)h FB(#)f FG(and)h FB(")f FG(can)h(b)q(e)g(de\014ned)h(sim)o(ultaneously)m (,)c(b)o(y)i(induction)g(on)h(the)g(t)o(yp)q(e.)23 b(It)257 643 y(is)16 b(con)o(v)o(enien)o(t)g(to)g(de\014ne)h FB(")f FG(on)f(all)g(terms)h(\(not)g(just)g(on)f(v)n(ariables\).)24 b(Hence)17 b(for)f(ev)o(ery)257 693 y(t)o(yp)q(e)f FF(\032)f FG(w)o(e)g(de\014ne)h FB(#)588 703 y FA(\032)612 693 y FG(:)e([)-7 b([)p FF(\032)p FG(])g(])10 b FB(!)h FG(\003)784 699 y FA(\032)817 693 y FG(and)j FB(")919 703 y FA(\032)943 693 y FG(:)f(\003)997 699 y FA(\032)1028 693 y FB(!)e FG([)-7 b([)p FF(\032)p FG(])g(])12 b(\(called)i(reify)g(and)g (re\015ect\))h(b)o(y)361 790 y FB(#)381 800 y FA(\034)402 790 y FG(\()p FF(M)5 b FG(\))12 b(:=)f FF(M)r(;)604 b FB(")1225 800 y FA(\034)1245 790 y FG(\()p FF(M)5 b FG(\))12 b(:=)f FF(M)r(;)332 852 y FB(#)352 862 y FA(\032)p Fz(!)p FA(\033)425 852 y FG(\()p FF(a)p FG(\))h(:=)f FF(\025x)p FB(#)615 862 y FA(\033)637 852 y FG(\()p FF(a)p FG(\()p FB(")712 862 y FA(\032)731 852 y FG(\()p FF(x)p FG(\)\)\))42 b(\\)p FF(x)14 b FG(new")p FF(;)74 b FB(")1118 862 y FA(\032)p Fz(!)p FA(\033)1191 852 y FG(\()p FF(M)5 b FG(\)\()p FF(a)p FG(\))12 b(:=)f FB(")1410 862 y FA(\033)1432 852 y FG(\()p FF(M)5 b FB(#)1514 862 y FA(\032)1533 852 y FG(\()p FF(a)p FG(\)\))p FF(:)257 943 y FG(Here)16 b(a)f(little)f(di\016cult)o(y)g(app)q(ears:)20 b(what)15 b(do)q(es)h(it)e(mean)g(that)h FF(x)f FG(is)h(new?)21 b(This)15 b(clearly)257 993 y(is)h(not)f(a)h(problem)e(for)h(an)h (implem)o(en)o(tation,)d(where)j(w)o(e)g(ha)o(v)o(e)g(an)f(op)q (erational)g(under-)257 1043 y(standing)10 b(and)g(ma)o(y)e(use)j (something)e(lik)o(e)g Fv(gensym)p FG(,)g(but)h(it)g(is)g(for)g(a)g (mathematical)c(mo)q(del.)257 1093 y(W)m(e)14 b(will)f(solv)o(e)g(this) h(problem)f(b)o(y)h(sligh)o(tly)f(mo)q(difying)e(the)j(mo)q(del)f(and)h (de\014ning)f([)-7 b([)p FF(\034)5 b FG(])-7 b(])13 b(to)257 1142 y(b)q(e)18 b(the)f(set)h(of)e(families)e(of)i(terms)h(of)f(t)o(yp) q(e)h FF(\034)22 b FG(\(instead)17 b(of)f(single)h(terms\))f(and)h (setting)257 1192 y FB(#)278 1202 y FA(\032)p Fz(!)p FA(\033)351 1192 y FG(\()p FF(a)p FG(\)\()p FF(k)q FG(\))12 b(:=)f FF(\025x)575 1198 y FA(k)595 1192 y FG(\()p FB(#)632 1202 y FA(\033)655 1159 y Fu(\000)674 1192 y FF(a)p FG(\()p FB(")733 1202 y FA(\032)752 1192 y FG(\()p FF(x)792 1177 y Fz(1)792 1204 y FA(k)827 1192 y FG(\)\))859 1159 y Fu(\001)878 1192 y FG(\()p FF(k)d FG(+)f(1\)\),)13 b(where)h FF(x)1184 1177 y Fz(1)1184 1204 y FA(k)1231 1192 y FG(is)f(the)g (constan)o(t)g(family)d FF(x)1658 1198 y FA(k)1678 1192 y FG(.)257 1242 y(The)17 b(de\014nition)e(of)h FB(")602 1252 y FA(\032)p Fz(!)p FA(\033)690 1242 y FG(has)g(to)g(b)q(e)h(mo)q (di\014ed)d(accordingly)m(.)24 b(This)16 b(idea)f(corresp)q(onds)257 1292 y(to)j(a)f(represen)o(tation)i(of)e(terms)g(in)g(the)h(st)o(yle)g (of)i FE(de)g(Br)o(uijn)d FG([9)o(].)29 b(An)17 b(adv)n(an)o(tage)g(of) 257 1342 y(this)f(approac)o(h)g(is)g(that)g(the)h(NbE)f(program)e(is)i (purely)g(functional)f(and)h(hence)i(can)e(b)q(e)257 1391 y(v)o(eri\014ed)g(relativ)o(ely)f(easily)m(.)22 b(If)15 b(side)g(e\013ects)j(w)o(ere)e(in)o(v)o(olv)o(ed)e(the)i(v)o (eri\014cation)f(w)o(ould)g(b)q(e)257 1441 y(m)o(uc)o(h)e(more)g (complicated.)320 1491 y(The)i(pro)q(of)g(of)g(correctness)k(is)c(easy) h(\(ignoring)e(the)i(problem)e(with)h(the)h(\\new)g(v)n(ari-)257 1541 y(able"\):)k(Since)15 b(for)f(the)h(t)o(yp)q(ed)g(lam)o(b)q(da)e (calculus)i(without)f(constan)o(ts)h(w)o(e)g(ha)o(v)o(e)g(preser-)257 1591 y(v)n(ation)e(of)h(v)n(alues,)f(i.e.)g([)-7 b([)p FF(M)5 b FG(])-7 b(])712 1597 y FA(\030)741 1591 y FG(=)12 b([)-7 b([)p Fw(nf)r FG(\()p FF(M)5 b FG(\)])-7 b(])932 1597 y FA(\030)964 1591 y FG(for)13 b(all)g(terms)h FF(M)19 b FG(and)14 b(en)o(vironmen)o(ts)f FF(\030)r FG(,)h(w)o(e)257 1641 y(only)j(ha)o(v)o(e)h(to)f(v)o(erify)h FB(#)o FG(\([)-7 b([)p FF(N)5 b FG(])-7 b(])734 1647 y Fz(")752 1641 y FG(\))18 b(=)g FF(N)23 b FG(for)17 b(terms)g FF(N)23 b FG(in)17 b(long)g(normal)f(form,)g(whic)o(h)i(is)257 1690 y(straigh)o(tforw)o(ard,)13 b(b)o(y)h(induction)f(on)h FF(N)5 b FG(:)320 1740 y Ft(Case)17 b FF(x)448 1725 y FA(\032)p Fz(!)p FA(\034)518 1740 y FF(N)556 1725 y FA(\032)589 1740 y FG(\(w.l.o.g.\))487 1831 y FB(#)508 1842 y FA(\034)529 1798 y Fu(\000)548 1831 y FG([)-7 b([)p FF(xN)5 b FG(])-7 b(])644 1837 y Fz(")661 1798 y Fu(\001)691 1831 y FG(=)12 b FB(")756 1842 y FA(\032)p Fz(!)p FA(\034)827 1831 y FG(\()p FF(x)p FG(\))883 1798 y Fu(\000)902 1831 y FG([)-7 b([)p FF(N)5 b FG(])-7 b(])974 1837 y Fz(")991 1798 y Fu(\001)1022 1831 y FG(=)11 b FB(")1086 1842 y FA(\034)1107 1798 y Fu(\000)1126 1831 y FF(x)c FB(#)1177 1842 y FA(\032)1197 1798 y Fu(\000)1216 1831 y FG([)-7 b([)p FF(N)5 b FG(])-7 b(])1288 1837 y Fz(")1305 1798 y Fu(\001\001)1354 1831 y FG(=)12 b FF(xN)320 1923 y Ft(Case)17 b FF(\025y)q(N)537 2014 y FB(#)558 2024 y FA(\032)p Fz(!)p FA(\033)631 1981 y Fu(\000)650 2014 y FG([)-7 b([)p FF(\025y)q(N)5 b FG(])-7 b(])767 2020 y Fz(")785 1981 y Fu(\001)816 2014 y FG(=)11 b FF(\025x)c FB(#)935 2024 y FA(\033)957 1981 y Fu(\000)976 2014 y FG([)-7 b([)p FF(\025y)q(N)5 b FG(])-7 b(])1093 2020 y Fz(")1112 2014 y FG(\()p FB(")1149 2024 y FA(\032)1168 2014 y FG(\()p FF(x)p FG(\)\))1240 1981 y Fu(\001)1301 2014 y FF(x)13 b FG(new)816 2082 y(=)e FF(\025x)c FB(#)935 2092 y FA(\033)957 2048 y Fu(\000)976 2082 y FG([)-7 b([)p FF(N)1026 2088 y FA(y)1046 2082 y FG([)p FF(x)p FG(])o(])g(])1110 2088 y Fz(")1128 2048 y Fu(\001)816 2144 y FG(=)11 b FF(\025xN)940 2150 y FA(y)961 2144 y FG([)p FF(x)p FG(])292 b(b)o(y)13 b(IH)816 2206 y(=)848 2212 y FA(\013)883 2206 y FF(\025y)q(N)257 2298 y FG(Notice)e(that)f (this)g(is)g(a)g(correctness)j(pro)q(of)c(in)h(the)h(st)o(yle)f(of)f ([5].)16 b(The)10 b(situation)g(is)f(di\013eren)o(t)257 2347 y(when)17 b(w)o(e)f(add)g(constan)o(ts)h(together)g(with)e (rewrite)i(rules,)g(since)g(then)f(preserv)n(ation)h(of)257 2397 y(v)n(alues)10 b(\(in)f(our)h(mo)q(del\))f(is)g(false)h(in)f (general)h(\(cf.)g(examples)f(20)g(and)h(19)f(b)q(elo)o(w\).)16 b(Ho)o(w)o(ev)o(er,)257 2447 y(correctness)k(of)c(normalization)e(b)o (y)j(ev)n(aluation)e(still)h(holds,)h(but)g(needs)h(to)f(b)q(e)g(pro)o (v)o(en)257 2497 y(b)o(y)d(a)g(di\013eren)o(t)h(metho)q(d.)i(It)d(migh) o(t)e(b)q(e)j(w)o(orth)f(noting)f(that)h(in)g(the)g(sp)q(ecial)g(case)h (where)963 2628 y(3)p eop %%Page: 4 4 4 3 bop 257 262 a FG(no)15 b(rewrite)h(or)e(computation)f(rules)j(are)f (presen)o(t)h(our)f(pro)q(of)f(b)q(elo)o(w)h(b)q(oils)f(do)o(wn)g(to)h (the)257 311 y(simple)e(correctness)k(pro)q(of)c(sk)o(etc)o(hed)j(ab)q (o)o(v)o(e.)320 361 y(The)e(structure)j(of)c(the)i(pap)q(er)g(is)f(as)g (follo)o(ws.)k(In)c(section)h(2)f(w)o(e)g(presen)o(t)i(the)e(simply)257 411 y(t)o(yp)q(ed)i FF(\025)p FG(-calculus)f(with)g(constan)o(ts)g(and) g(pairing)f(and)h(giv)o(e)f(some)g(examples)g(of)g(higher)257 461 y(order)e(rewrite)g(systems.)17 b(W)m(e)11 b(also)f(in)o(tro)q (duce)i(the)f(distinction)g(b)q(et)o(w)o(een)h(computational)257 511 y(and)f(\(prop)q(er\))h(rewrite)f(rules.)18 b(Then)11 b(w)o(e)g(inductiv)o(ely)f(de\014ne)h(a)f(relation)h FF(M)16 b FB(\000)-7 b(!)11 b FF(Q)p FG(,)g(with)257 560 y(the)18 b(in)o(tended)g(meaning)e(that)h FF(M)22 b FG(is)17 b(normalizable)e(with)i(long)f(normal)g(form)f FF(Q)p FG(,)j(and)257 610 y(pro)o(v)o(e)d(in)g(section)h(3.6)e(the)h (correctness)j(of)d(normalization)d(b)o(y)j(ev)n(aluation)e(b)o(y)i (sho)o(wing)257 660 y(that)i FF(M)22 b FB(\000)-7 b(!)17 b FF(Q)g FG(\(essen)o(tially\))g(implies)e FB(#)p FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])1040 666 y Fz(")1057 660 y FG(\))17 b(=)g FF(Q)p FG(.)27 b(Hence)19 b(the)f(mapping)c FF(M)22 b FB(7!)257 710 y(#)p FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])373 716 y Fz(")390 710 y FG(\))14 b(is)g(a)g (normalization)d(function.)18 b(In)c(order)h(to)e(de\014ne)i(the)g (seman)o(tics)e([)-7 b([)p FF(M)5 b FG(])-7 b(])12 b(of)i(a)257 760 y(term)e FF(M)18 b FG(prop)q(erly)13 b(w)o(e)g(use)g(domain)d (theory)m(.)18 b(This)13 b(is)f(describ)q(ed)j(brie\015y)d(in)h (section)g(3.1.)320 809 y(Note)h(that)g(w)o(e)h(pro)o(v)o(e)f (correctness)j(of)d(NbE)g(w.r.t.)f(a)h(denotational)f(seman)o(tics,)g (but)257 859 y(do)f(not)f(attempt)g(to)h(pro)o(v)o(e)g(op)q(erational)e (correctness,)15 b(i.e.)c(the)h(fact)f(that)h(the)g(functional)257 909 y(program)h(formalizing)f(NbE)j(when)g(called)f(with)h(a)f(term)g FF(M)19 b FG(suc)o(h)d(that)e FF(M)k FB(\000)-7 b(!)13 b FF(Q)h FG(will)257 959 y(terminate)j(with)g FF(Q)g FG(as)g(output.)28 b(In)17 b(order)h(to)f(obtain)g(op)q(erational)f (correctness)k(from)257 1009 y(denotational)f(correctness)j(one)e (needs)g(a)f(suitable)h(adequacy)f(result)h(\023)-21 b(a)19 b(la)g FE(Plotkin)257 1059 y FG([13)o(])14 b(relating)g(the)h (denotational)e(and)h(the)h(op)q(erational)e(seman)o(tics.)19 b FE(Plotkin)p FG('s)c(result)257 1108 y(cannot)h(b)q(e)h(applied)e (here)i(b)q(ecause)h(it)d(refers)j(to)d(a)h(call-b)o(y-name)d(op)q (erational)i(seman-)257 1158 y(tics,)f(whereas)h(w)o(e)f(are)g(in)o (terested)h(in)e(a)g(call-b)o(y-v)n(alue)f(seman)o(tics)h(in)g(order)h (to)g(obtain)f(a)257 1208 y(correctness)18 b(result)d(for)f(our)h (implemen)o(tatio)o(n)d(of)i(NbE)h(in)f(the)h(call-b)o(y-v)n(alue)d (language)257 1258 y FE(Scheme)p FG(.)26 b(F)m(urthermore)15 b FE(Plotkin)i FG(only)e(considers)i(the)g(in)o(tegers)g(and)f(the)g(b) q(o)q(oleans)257 1308 y(as)e(base)f(t)o(yp)q(es,)h(whereas)g(w)o(e)g (need)g(complex)d(recursiv)o(ely)k(de\014ned)f(t)o(yp)q(es)g(as)f(base) h(t)o(yp)q(es)257 1357 y(\(see)g(section)e(3.2\).)17 b(W)m(e)11 b(lea)o(v)o(e)h(the)g(problem)f(of)g(pro)o(ving)g(adequacy)h (of)g(our)f(denotational)257 1407 y(seman)o(tics)h(for)g(a)g(fragmen)o (t)e(of)i(a)f(call-b)o(y-v)n(alue)g(language)g(suitable)h(for)f (formalizing)e(our)257 1457 y(extension)15 b(of)e(NbE)h(to)g(future)h (w)o(ork.)257 1594 y FH(2)67 b(A)22 b(simply)j(t)n(yp)r(ed)e Fs(\025)p FH(-calculus)h(with)f(constan)n(ts)257 1694 y Fr(2.1)56 b(T)n(yp)r(es,)18 b(terms,)e(rewrite)h(rules)257 1770 y FG(W)m(e)12 b(start)g(from)f(a)g(giv)o(en)h(set)h(of)e Ft(gr)n(ound)j(typ)n(es)p FG(.)j Ft(T)m(yp)n(es)e FG(are)e(inductiv)o (ely)e(generated)i(from)257 1820 y(ground)h(t)o(yp)q(es)h FF(\034)k FG(b)o(y)13 b FF(\032)f FB(!)f FF(\033)k FG(and)f FF(\032)c FB(\002)f FF(\033)q FG(.)18 b Ft(T)m(erms)f FG(are)587 1900 y FF(x)611 1885 y FA(\032)1074 1900 y FG(t)o(yp)q(ed)d(v)n(ariables,)587 1950 y FF(c)605 1935 y FA(\032)1074 1950 y FG(constan)o(ts,)587 2000 y(\()p FF(\025x)651 1985 y FA(\032)670 2000 y FF(M)715 1985 y FA(\033)738 2000 y FG(\))754 1985 y FA(\032)p Fz(!)p FA(\033)1074 2000 y FG(abstractions,)587 2050 y(\()p FF(M)648 2035 y FA(\032)p Fz(!)p FA(\033)721 2050 y FF(N)759 2035 y FA(\032)778 2050 y FG(\))794 2035 y FA(\033)1074 2050 y FG(applications,)587 2100 y FB(h)p FF(M)648 2080 y FA(\032)643 2111 y Fy(0)667 2100 y FF(;)7 b(M)731 2084 y FA(\033)726 2110 y Fy(1)753 2100 y FB(i)769 2084 y FA(\032)p Fz(\002)p FA(\033)1074 2100 y FG(pairing,)587 2149 y FF(\031)611 2155 y Fy(0)629 2149 y FG(\()p FF(M)690 2134 y FA(\032)p Fz(\002)p FA(\033)756 2149 y FG(\))772 2134 y FA(\032)791 2149 y FG(,)14 b FF(\031)841 2155 y Fy(1)859 2149 y FG(\()p FF(M)920 2134 y FA(\032)p Fz(\002)p FA(\033)985 2149 y FG(\))1001 2134 y FA(\033)1074 2149 y FG(pro)r(jections.)257 2230 y(T)o(yp)q(e)19 b(indices)f(will)e(b)q(e) j(omitted)d(whenev)o(er)k(they)e(are)h(inessen)o(tial)f(or)f(clear)i (from)d(the)257 2280 y(con)o(text.)29 b(Also,)17 b FF(\025x)g FG(binds)g(tigh)o(ter)h(than)f(application)f(and)h(pairing;)g(ho)o(w)o (ev)o(er,)h(a)f(dot)257 2330 y(after)d FF(\025x)g FG(means)e(that)i (the)g(scop)q(e)h(extends)g(as)e(far)g(as)h(allo)o(w)o(ed)e(b)o(y)h (the)i(paren)o(theses.)20 b(So)257 2380 y FF(\025xM)5 b(N)19 b FG(means)13 b(\()p FF(\025xM)5 b FG(\))p FF(N)g FG(,)13 b(but)h FF(\025x:M)5 b(N)18 b FG(means)13 b FF(\025x)p FG(\()p FF(M)5 b(N)g FG(\).)320 2430 y(Ground)18 b(t)o(yp)q(es)h(will)e (alw)o(a)o(ys)h(b)q(e)h(denoted)g(b)o(y)g FF(\034)5 b FG(.)31 b(W)m(e)18 b(sometimes)f(write)i FF(M)5 b FG(0)18 b(for)257 2480 y FF(\031)281 2486 y Fy(0)300 2480 y FG(\()p FF(M)5 b FG(\))15 b(and)g FF(M)5 b FG(1)15 b(for)g FF(\031)644 2486 y Fy(1)662 2480 y FG(\()p FF(M)5 b FG(\).)23 b(Tw)o(o)14 b(terms)h FF(M)21 b FG(and)15 b FF(N)20 b FG(are)c(called)f FF(\013)p Ft(-e)n(qual)k FG({)c(written)963 2628 y(4)p eop %%Page: 5 5 5 4 bop 257 262 a FF(M)22 b FG(=)351 268 y FA(\013)392 262 y FF(N)f FG({)c(if)f(they)i(are)f(equal)g(up)g(to)g(renaming)e(of)h (b)q(ound)i(v)n(ariables.)26 b(\003)1517 268 y FA(\032)1553 262 y FG(denotes)257 311 y(the)16 b(set)g(of)f(all)e(terms)i(of)g(t)o (yp)q(e)g FF(\032)h FG(\()p FF(\013)p FG(-equal)e(terms)h(are)h Ft(not)j FG(iden)o(ti\014ed\).)j FF(M)5 b Fx(N)20 b FG(denotes)257 361 y(\()p FF(:)7 b(:)g(:)f FG(\()p FF(M)f(N)423 367 y Fy(1)442 361 y FG(\))p FF(N)491 367 y Fy(2)517 361 y FF(:)i(:)g(:)e FG(\))p FF(N)621 367 y FA(n)644 361 y FG(,)18 b(where)g(some)e(of)h(the)h FF(N)1063 367 y FA(i)1077 361 y FG('s)f(ma)o(y)e(b)q(e)j(0)f(or)g(1.)28 b(By)17 b Fw(FV)p FG(\()p FF(M)5 b FG(\))17 b(w)o(e)257 411 y(denote)e(a)e(list)h(of)f(v)n(ariables)g(o)q(ccurring)h(free)h(in) e FF(M)5 b FG(.)18 b(By)c FF(M)1205 417 y FA(x)1226 411 y FG([)p FF(N)5 b FG(])13 b(w)o(e)h(mean)e(substitution)257 461 y(of)17 b(ev)o(ery)h(free)g(o)q(ccurrence)i(of)c FF(x)h FG(in)g FF(M)22 b FG(b)o(y)17 b FF(N)5 b FG(,)17 b(renaming)f(b)q(ound)h(v)n(ariables)g(if)f(neces-)257 511 y(sary)m(.)31 b(Similarly)15 b FF(M)593 517 y Fq(x)616 511 y FG([)p Fx(N)5 b FG(])18 b(denotes)h(sim)o(ultaneous)e (substitution.)30 b FF(\025)p Fx(x)q FF(M)23 b FG(abbreviates)257 560 y FF(\025x)305 566 y Fy(1)331 560 y FF(:)7 b(:)g(:)e(\025x)434 566 y FA(n)457 560 y FF(M)g FG(.)23 b(If)16 b FF(M)5 b Fx(N)21 b FG(is)16 b(of)f(t)o(yp)q(e)h FF(\033)q FG(,)g FF(N)960 566 y FA(i)990 560 y FG(of)f(t)o(yp)q(e)i FF(\032)1156 566 y FA(i)1170 560 y FG(,)f(then)h(w)o(e)f(call)f Fx(\032)g FB(!)f FF(\033)j FG(a)f(t)o(yp)q(e)257 610 y(information)9 b(for)j FF(M)5 b FG(.)17 b(Here)c Fx(\032)f FG(is)g(a)f(list)g(of)g(t)o (yp)q(es,)i(0's)e(or)h(1's)g(indicating)e(the)j(left)e(or)h(righ)o(t) 257 660 y(part)e(of)g(a)f(pro)q(duct)i(t)o(yp)q(e.)17 b(So)10 b(e.g.)f(a)g(term)h FF(M)k FG(of)9 b(t)o(yp)q(e)i FF(\032)h FG(=)g(\()p FF(\034)k FB(!)11 b FF(\034)17 b FB(!)11 b FF(\034)5 b FG(\))q FB(\002)q FG(\()p FF(\034)17 b FB(!)11 b FG(\()p FF(\034)6 b FB(\002)q FF(\034)f FG(\)\))257 710 y(has)18 b(\(0)p FF(;)7 b(\034)e FG(\))16 b FB(!)h FG(\()p FF(\034)22 b FB(!)17 b FF(\034)5 b FG(\))17 b(or)h(\(1)p FF(;)7 b(\034)r(;)g FG(0\))16 b FB(!)h FF(\034)22 b FG(as)17 b(a)g(t)o(yp)q(e)h(information.)25 b(If)17 b(there)i(are)f(no)257 760 y(pro)q(duct)d(t)o(yp)q(es)g Fx(\032)d FB(!)f FF(\033)k FG(simply)d(abbreviates)i(\()p FF(\032)1042 766 y Fy(1)1073 760 y FB(!)d FG(\()p FF(\032)1163 766 y Fy(2)1189 760 y FB(\001)c(\001)g(\001)j(!)h FG(\()p FF(\032)1339 766 y FA(n)1374 760 y FB(!)g FF(\033)q FG(\))c FF(:)g(:)g(:)f FG(\)\).)320 809 y(F)m(or)13 b(the)h(constan)o(ts)g FF(c)666 794 y FA(\032)699 809 y FG(w)o(e)f(assume)g(that)h(some)e(rewrite)i (rules)g(of)f(the)h(form)e FF(c)p Fx(K)j FB(7\000)-7 b(!)257 859 y FF(N)25 b FG(are)20 b(giv)o(en,)g(where)h Fw(FV)p FG(\()p FF(N)5 b FG(\))22 b FB(\022)f Fw(FV)p FG(\()p Fx(K)s FG(\))f(and)g FF(c)p Fx(K)s FG(,)h FF(N)j FG(ha)o(v)o(e)c(the)g(same)f(t)o(yp)q(e)h(\(not)257 909 y(necessarily)d(a)e(ground)h(t)o(yp)q(e\).)23 b(Moreo)o(v)o(er,)17 b(for)e(an)o(y)g(t)o(yp)q(e)h(information)d FF(\032)1460 915 y Fy(1)1479 909 y FF(;)7 b(:)g(:)g(:)t(\032)1573 915 y FA(n)1611 909 y FB(!)14 b FF(\034)257 959 y FG(for)c FF(c)f FG(\()p FF(\034)15 b FG(a)10 b(ground)f(t)o(yp)q(e\),)i(w)o(e)f (require)g(that)g(there)h(is)f(a)f(\014xed)i(length)e FF(k)k FB(\024)e FF(n)f FG(of)f(argumen)o(ts)257 1009 y(for)k(the)h(rewrite)g(rules,)g(i.e.)e FF(c)p Fx(M)17 b FB(7\000)-7 b(!)11 b FF(N)18 b FG(implies)12 b(that)h Fx(M)18 b FG(has)c(length)f FF(k)q FG(,)g(pro)o(vided)g(the)257 1059 y(pro)r(jection)f(mark)o(ers)e(in)h Fx(M)16 b FG(and)11 b(in)g FF(\032)860 1065 y Fy(1)879 1059 y FF(;)c(:)g(:)g(:)e(\032)974 1065 y FA(k)1006 1059 y FG(coincide.)17 b(If)11 b(no)g(rewrite)h(rule)f (of)g(the)g(form)257 1108 y FF(c)p Fx(M)18 b FB(7\000)-6 b(!)12 b FF(N)19 b FG(\(1)13 b FB(\024)i FG(length)g(of)f Fx(M)j FB(\024)d FF(n)p FG(\))g(applies,)g(then)i(this)e(\014xed)h (length)g(is)f(stipulated)257 1158 y(to)f(b)q(e)h FF(n)p FG(.)k(W)m(e)13 b(write)g FF(c)612 1143 y Fq(\032)p Fz(!)p FA(\033)700 1158 y FG(to)g(indicate)h(that)f(w)o(e)g(only)g(consider)h FF(c)f FG(with)f(argumen)o(t)h(lists)257 1208 y Fx(K)19 b FG(with)d(these)h(pro)r(jection)f(mark)o(ers;)f(the)h(notation)f FF(c)p Fx(M)5 b(N)21 b FG(is)16 b(used)g(to)g(indicate)f(that)257 1258 y Fx(M)20 b FG(are)c(the)f(\014xed)h(argumen)o(ts)e(for)h(the)g (rewrite)h(rules)g(of)e FF(c)p FG(.)21 b(In)15 b(particular.)21 b(if)14 b(there)j(is)257 1308 y(no)d(rewrite)h(rule)f(for)g FF(c)p FG(,)f(then)h Fx(N)20 b FG(is)13 b(empt)o(y)g(and)h FF(c)p Fx(M)19 b FG(is)14 b(of)f(ground)h(t)o(yp)q(e.)320 1357 y(F)m(or)e(example,)g(if)g FF(c)i FG(is)f(of)f(t)o(yp)q(e)i(\()p FF(\034)i FB(!)c FF(\034)k FB(!)11 b FF(\034)5 b FG(\))j FB(\002)g FG(\()p FF(\034)17 b FB(!)11 b FF(\034)5 b FG(\),)12 b(then)i(the)g(rules)g FF(c)p FG(0)p FF(xx)d FB(7\000)-7 b(!)257 1407 y FF(a)21 b FG(and)f FF(c)p FG(1)i FB(7\000)-7 b(!)22 b FF(b)f FG(are)f(admitted,)h(and)f FF(c)954 1392 y Fy(0)p FA(;\034)r(;\034)s Fz(!)p FA(\034)1100 1407 y FG(indicates)h(that)f(w)o(e)h(only)f(consider)257 1457 y(argumen)o(t)13 b(lists)h(of)f(the)i(form)d(0)p FF(;)7 b(x;)g(y)q FG(.)257 1573 y Fr(2.2)56 b(Computation)18 b(rules)257 1650 y FG(Giv)o(en)12 b(a)h(set)g(of)f(rewrite)i(rules,)f (w)o(e)g(w)o(an)o(t)f(to)h(treat)g(some)f(rules)h(-)g(whic)o(h)f(w)o(e) h(call)f Ft(c)n(ompu-)257 1700 y(tation)j(rules)i FG(-)d(in)f(a)h (di\013eren)o(t,)g(more)f(e\016cien)o(t)h(w)o(a)o(y)m(.)j(The)d(idea)g (is)g(that)g(a)f(computation)257 1749 y(rule)i(can)g(b)q(e)g(understo)q (o)q(d)h(as)f(a)f(description)i(of)e(a)g(computation)f(in)h(a)h (suitable)f Ft(seman-)257 1799 y(tic)n(al)j FG(mo)q(del,)11 b(pro)o(vided)h(the)i(syn)o(tactic)f(constructors)i(corresp)q(ond)f(to) f(seman)o(tic)f(ones)h(in)257 1849 y(the)i(mo)q(del,)d(whereas)j(the)f (other)h(rules)f(describ)q(e)i Ft(syntactic)g FG(transformations.)320 1899 y(A)f(constan)o(t)h FF(c)g FG(is)f(called)g(a)g Ft(c)n(onstructor)20 b FG(if)15 b(there)h(is)g(no)f(rule)h(of)e(the)j (form)c FF(c)p Fx(K)18 b FB(7\000)-7 b(!)257 1949 y FF(N)5 b FG(.)38 b(F)m(or)20 b(instance)h(in)f(the)h(examples)f(of)g(section)h (2.3)e(the)i(constan)o(ts)h(0,)f Fw(S)g FG(and)f FB(9)1662 1934 y Fy(+)257 1999 y FG(are)15 b(constructors.)21 b Ft(Constructor)15 b(p)n(atterns)i FG(are)e(sp)q(ecial)f(terms)g (de\014ned)i(inductiv)o(ely)d(as)257 2048 y(follo)o(ws.)320 2131 y FB(\017)20 b FG(Ev)o(ery)15 b(v)n(ariable)d(is)i(a)g (constructor)h(pattern.)320 2214 y FB(\017)20 b FG(If)13 b FF(c)g FG(is)g(a)f(constructor)j(and)e FF(P)833 2220 y Fy(1)851 2214 y FF(;)7 b(:)g(:)g(:)e(;)i(P)971 2220 y FA(n)1006 2214 y FG(are)13 b(constructor)i(patterns)f(or)f(pro)r (jection)361 2264 y(mark)o(ers)h(0)g(or)g(1,)g(suc)o(h)h(that)g FF(c)p Fx(P)21 b FG(is)14 b(of)g(ground)g(t)o(yp)q(e,)g(then)i FF(c)p Fx(P)k FG(is)15 b(a)f(constructor)361 2314 y(pattern.)257 2397 y(F)m(rom)d(the)i(giv)o(en)f(set)i(of)e(rewrite)h(rules)h(w)o(e)e (c)o(ho)q(ose)i(a)e(subset)i FE(Comp)f FG(with)f(the)h(follo)o(wing)257 2447 y(prop)q(erties.)963 2628 y(5)p eop %%Page: 6 6 6 5 bop 320 262 a FB(\017)20 b FG(If)13 b FF(c)p Fx(P)18 b FB(7\000)-6 b(!)11 b FF(Q)g FB(2)g FE(Comp)p FG(,)i(then)h FF(P)889 268 y Fy(1)907 262 y FF(;)7 b(:)g(:)g(:)e(;)i(P)1027 268 y FA(n)1062 262 y FG(are)14 b(constructor)h(patterns)f(or)g(pro)r (jec-)361 311 y(tion)f(mark)o(ers.)320 389 y FB(\017)20 b FG(The)15 b(rules)g(are)g(left-linear,)e(i.e.)g(if)h FF(c)p Fx(P)19 b FB(7\000)-7 b(!)12 b FF(Q)g FB(2)g FE(Comp)p FG(,)i(then)h(ev)o(ery)h(v)n(ariable)d(in)361 439 y FF(c)p Fx(P)21 b FG(o)q(ccurs)15 b(only)e(once)i(in)e FF(c)p Fx(P)7 b FG(.)320 517 y FB(\017)20 b FG(The)i(rules)f(are)g(non-o)o(v)o (erlapping,)g(i.e.)f(for)h(di\013eren)o(t)h(rules)f FF(c)p Fx(K)27 b FB(7\000)-7 b(!)23 b FF(M)j FG(and)361 567 y FF(c)p Fx(L)12 b FB(7\000)-7 b(!)11 b FF(N)19 b FG(in)14 b FE(Comp)g FG(the)g(left)g(hand)f(sides)i FF(c)p Fx(K)i FG(and)d FF(c)p Fx(L)g FG(are)g(non-uni\014able.)257 640 y(W)m(e)20 b(write)g FF(c)p Fx(M)26 b FB(7\000)-7 b(!)605 646 y Fp(comp)698 640 y FF(Q)19 b FG(to)h(indicate)g(that)f (the)i(rule)f(is)f(in)h FE(Comp)p FG(.)35 b(The)20 b(set)h(of)257 690 y(constructors)c(app)q(earing)d(in)g(the)h(constructor)h(patterns)g (is)e(denoted)h(b)o(y)g FE(Constr)p FG(.)k(All)257 740 y(other)c(rules)f(will)f(b)q(e)h(called)g(\(prop)q(er\))h(rewrite)g (rules,)f(written)g FF(c)p Fx(M)j FB(7\000)-7 b(!)1429 746 y Fp(rew)1488 740 y FF(K)s FG(.)320 789 y(In)18 b(our)g(reduction)g (strategy)h(b)q(elo)o(w)e(computation)g(rules)h(will)f(alw)o(a)o(ys)g (b)q(e)h(applied)257 839 y(\014rst,)f(and)f(since)h(they)f(are)h(non-o) o(v)o(erlapping,)d(this)i(part)g(of)g(the)g(reduction)h(is)f(unique.) 257 889 y(Ho)o(w)o(ev)o(er,)h(since)g(w)o(e)f(allo)o(w)o(ed)f(almost)g (arbitrary)h(rewrite)h(rules,)g(it)e(ma)o(y)g(happ)q(en)h(that)257 939 y(in)e(case)i(no)e(computation)f(rule)h(applies)g(a)g(term)g(ma)o (y)f(b)q(e)i(rewritten)g(b)o(y)f(di\013eren)o(t)i(rules)262 989 y FF(=)-26 b FB(2)12 b FE(Comp)p FG(.)17 b(In)12 b(order)h(to)f(obtain)f(a)h(deterministic)f(pro)q(cedure)j(w)o(e)e (assume)g(that)g(for)g(ev)o(ery)257 1039 y(constan)o(t)19 b FF(c)447 1023 y Fq(\032)p Fz(!)p FA(\033)540 1039 y FG(w)o(e)g(are)f(giv)o(en)g(a)g(function)g Fw(sel)1042 1045 y FA(c)1077 1039 y FG(computing)f(from)f Fx(M)23 b FG(either)c(a)f(rule)257 1088 y FF(c)p Fx(K)g FB(7\000)-7 b(!)400 1094 y Fp(rew)462 1088 y FF(N)5 b FG(,)16 b(in)f(whic)o(h)g (case)i Fx(M)k FG(is)15 b(an)g(instance)i(of)e Fx(K)s FG(,)h(i.e.)e Fx(M)20 b FG(=)14 b Fx(K)1467 1094 y Fq(x)1490 1088 y FG([)p Fx(L)p FG(],)h(or)h(else)257 1138 y(the)g(message)e(\\)p Fw(no)p FG(-)p Fw(match)p FG(",)g(in)h(whic)o(h)f(case)i Fx(M)k FG(do)q(esn't)c(matc)o(h)d(an)o(y)i(rewrite)h(rule,)e(i.e.)257 1188 y(there)k(is)f(no)g(rule)g FF(c)p Fx(K)i FB(7\000)-6 b(!)703 1194 y Fp(rew)766 1188 y FF(N)22 b FG(suc)o(h)c(that)e Fx(M)22 b FG(is)17 b(an)f(instance)i(of)e Fx(K)s FG(.)27 b(Clearly)16 b Fw(sel)1673 1194 y FA(c)257 1238 y FG(should)d(b)q(e)h (compatible)d(with)h FF(\013)p FG(-equalit)o(y)m(,)f(and)i(should)g (satisfy)f(an)h(ob)o(vious)f Ft(uniformity)257 1288 y FG(prop)q(ert)o(y)m(,)i(i.e.)e(whenev)o(er)i Fx(M)k FG(and)13 b Fx(M)880 1269 y Fz(0)904 1288 y FG(are)h(v)n(arian)o(ts)e(\(i.e.)h (can)g(b)q(e)h(obtained)f(from)e(eac)o(h)257 1337 y(other)k(b)o(y)f(an) f(in)o(v)o(ertible)h(substitution\),)f(then)i Fw(sel)1062 1343 y FA(c)1079 1337 y FG(\()p Fx(M)5 b FG(\))12 b(=)g Fw(sel)1263 1343 y FA(c)1280 1337 y FG(\()p Fx(M)1348 1319 y Fz(0)1360 1337 y FG(\).)320 1387 y(Often)18 b(the)h(rewrite)g (rules)g(will)e(b)q(e)i(left-linear)e(\(i.e.)h(no)g(v)n(ariable)f(o)q (ccurs)i(t)o(wice)g(in)257 1437 y(the)e(left)f(hand)h(side)f(of)g(a)g (rule\);)h(then)g(is)f(is)h(reasonable)f(to)g(require)i(that)e(ev)o (ery)h(select)257 1487 y(function)h Fw(sel)468 1493 y FA(c)503 1487 y FG(is)f Ft(str)n(ongly)h(uniform)j FG(in)c(the)i(sense) g(that)f(for)f(all)g(instances)i(\(with)f(not)257 1537 y(necessarily)d(distinct)f(v)n(ariables)g Fx(z)q FG(\))g(w)o(e)h(ha)o (v)o(e)e Fw(sel)1043 1543 y FA(c)1059 1537 y FG(\()p Fx(M)6 b FG(\))11 b(=)h Fw(sel)1243 1543 y FA(c)1260 1537 y FG(\()p FF(M)1316 1543 y Fq(x)1340 1537 y FG([)p Fx(z)r FG(])o(\).)257 1651 y Fr(2.3)56 b(Examples)257 1727 y FG(\(a\))15 b(Usually)f(w)o(e)h(ha)o(v)o(e)g(the)g(ground)f(t)o (yp)q(e)i FF(\023)e FG(of)g(natural)g(n)o(um)o(b)q(ers)h(a)o(v)n (ailable,)d(with)i(con-)257 1785 y(structors)i(0)452 1770 y FA(\023)466 1785 y FG(,)d Fw(S)514 1770 y FA(\023)p Fz(!)p FA(\023)587 1785 y FG(and)g Ft(r)n(e)n(cursion)h(op)n(er)n (ators)j FF(R)1062 1764 y FA(\023)p Fz(!)p FA(\032)p Fz(!)p Fy(\()p FA(\023)p Fz(!)p FA(\032)p Fz(!)p FA(\032)p Fy(\))p Fz(!)p FA(\032)1062 1790 y(\032)1349 1785 y FG(.)g(The)d (rewrite)h(rules)257 1835 y(for)f FF(R)g FG(are)798 1914 y FF(R)p FG(0)d FB(7\000)-6 b(!)11 b FF(\025y)q(z)r(:y)q(;)740 1976 y(R)p FG(\()p Fw(S)p FF(x)p FG(\))g FB(7\000)-6 b(!)11 b FF(\025y)q(z)r(:z)r(x)p FG(\()p FF(Rxy)q(z)r FG(\))p FF(:)257 2055 y FG(The)k(reason)f(for)g(writing)f(the)i(rules)g (in)e(this)h(w)o(a)o(y)m(,)f(and)g(not)h(in)g(the)h(more)e(famili)o(ar) e(form)257 2105 y FF(R)p FG(0)p FF(y)q(z)j FB(7\000)-6 b(!)11 b FF(y)q FG(,)j FF(R)p FG(\()p Fw(S)p FF(x)p FG(\))p FF(y)q(z)g FB(7\000)-7 b(!)11 b FF(z)r(x)p FG(\()p FF(Rxy)q(z)r FG(\),)j(will)e(b)q(ecome)h(clear)h(later)f(\(see)i(example)d(16)h(in) 257 2155 y(section)i(3.5\).)i(A)d(simpli\014ed)e(sc)o(heme)i(of)f(a)h (similar)d(form)i(giv)o(es)g(a)h(cases)h(construct.)872 2233 y Fw(if)9 b FG(0)i FB(7\000)-6 b(!)11 b FF(\025y)q(z)r(:y)q(;)813 2296 y Fw(if)f FG(\()p Fw(S)p FF(x)p FG(\))h FB(7\000)-6 b(!)11 b FF(\025y)q(z)r(:z)r(:)257 2374 y FG(Moreo)o(v)o(er)17 b(w)o(e)e(can)h(write)g(do)o(wn)f(rules)i(according)e(to)h(the)g(usual) f(recursiv)o(e)i(de\014nitions)257 2424 y(of)d(addition)e(and)i(m)o (ultiplication,)c(e.g.)694 2503 y Fw(mult)o FG(\()p Fw(S)p FF(x)p FG(\))i FB(7\000)-7 b(!)11 b FF(\025z)r(:)p Fw(add)d FF(z)r FG(\()p Fw(mult)e FF(xz)r FG(\))963 2628 y(6)p eop %%Page: 7 7 7 6 bop 257 262 a FG(Sim)o(ultaneous)12 b(recursion)j(ma)o(y)d(b)q(e)j (treated)g(as)f(w)o(ell,)f(e.g.)554 347 y Fw(o)q(dd)8 b FG(0)j FB(7\000)-6 b(!)11 b Fw(S)p FG(0)361 b Fw(even)8 b FG(0)k FB(7\000)-7 b(!)11 b FG(0)p FF(;)503 409 y Fw(o)q(dd)q FG(\()p Fw(S)p FF(x)p FG(\))g FB(7\000)-6 b(!)11 b Fw(even)d FF(x)245 b Fw(even)r FG(\()p Fw(S)p FF(x)p FG(\))12 b FB(7\000)-7 b(!)11 b Fw(o)q(dd)d FF(x:)257 495 y FG(All)15 b(these)j(rules)e(are)g(p)q(ossible)h(computation)d(rules,)i(whereas)h (the)g(next)f(t)o(w)o(o)g(are)g(rules)257 544 y(are)f(not)e(\(since)i Fw(if)24 b FG(and)13 b Fw(add)i FG(are)f(no)g(constructors\).)641 630 y Fw(if)9 b FG(\()p Fw(if)h FF(xy)q(z)r FG(\))i FB(7\000)-7 b(!)11 b FF(\025uv)q(:)p Fw(if)f FF(x)p FG(\()p Fw(if)f FF(y)q(uv)q FG(\)\()p Fw(if)i FF(z)r(uv)q FG(\))p FF(;)257 715 y FG(\(a)j(rewrite)h(rule)f(due)h(to)e FE(McCar)m(thy)i FG([12)o(]\),)e(or)587 801 y Fw(mult)o FG(\()p Fw(add)8 b FF(xy)q FG(\))k FB(7\000)-6 b(!)11 b FF(\025z)r(:)p Fw(add)q FG(\()p Fw(mult)6 b FF(xz)r FG(\)\()p Fw(mult)g FF(y)q(z)r FG(\))p FF(:)320 886 y FG(\(b\))15 b(W)m(e)g(can)h(also)f (deal)g(with)g(in\014nitely)g(branc)o(hing)g(trees)i(lik)o(e)e(the)h FE(Br)o(ouwer)g FG(or-)257 936 y(dinals)h(of)g(t)o(yp)q(e)h FB(O)q FG(.)29 b(There)18 b(are)g(constructors)i(0)1061 921 y Fz(O)1107 936 y FG(and)d FE(Sup)1265 921 y Fy(\()p FA(\023)p Fz(!O)q Fy(\))p Fz(!)p FA(\023)1411 936 y FG(,)h(and)f (recursion)257 992 y(constan)o(ts)e FE(Rec)522 970 y Fz(O)q(!)p FA(\032)p Fz(!)p Fy(\(\()p FA(\023)p Fz(!O)q Fy(\))p Fz(!)p Fy(\()p FA(\023)p Fz(!)p FA(\032)p Fy(\))p Fz(!)p FA(\032)p Fy(\))p Fz(!)p FA(\032)522 996 y(\032)982 992 y FG(.)j(The)c(rewrite)h(rules)f(for)g FE(Rec)g FG(are)711 1077 y FE(Rec)e FG(0)f FB(7\000)-7 b(!)11 b FF(\025y)q(z)r(:y)q(;)607 1139 y FE(Rec)o FG(\()p FE(Sup)e FF(x)p FG(\))i FB(7\000)-7 b(!)11 b FF(\025y)q(z)r(:z)r(x)p FG(\()p FF(\025u)c FE(Rec)q FG(\()p FF(xu)p FG(\))p FF(y)q(z)r FG(\))p FF(:)320 1225 y FG(\(c\))18 b(It)f(is)h(w)o(ell)f(kno)o(wn)g(that)g(b)o(y)h(the)g FE(Curr)m(y)q FG(-)p FE(Ho)o(w)l(ard)g FG(corresp)q(ondence)j(natural) 257 1275 y(deduction)f(pro)q(ofs)f(can)h(b)q(e)f(written)h(as)f FF(\025)p FG(-terms)g(with)g(form)o(ulas)e(as)i(t)o(yp)q(es.)35 b(T)m(o)19 b(use)257 1324 y(normalization)e(b)o(y)j(ev)n(aluation)e (for)h(normalizing)e(pro)q(ofs)j(w)o(e)g(ma)o(y)d(also)i(in)o(tro)q (duce)i(a)257 1374 y(ground)14 b(t)o(yp)q(e)h Fw(ex)f FG(with)g(constructors)h(and)f(destructors)514 1460 y(\()p FB(9)553 1442 y Fy(+)553 1470 y FA(\032)570 1474 y Fo(0)587 1470 y FA(;\032)614 1474 y Fo(1)632 1460 y FG(\))648 1442 y FA(\032)665 1446 y Fo(0)682 1442 y Fz(!)p FA(\032)732 1446 y Fo(1)748 1442 y Fz(!)p Fp(ex)853 1460 y FG(and)41 b(\()p FB(9)1000 1442 y Fz(\000)1000 1470 y FA(\032)1017 1474 y Fo(0)1034 1470 y FA(;\032)1061 1474 y Fo(1)1077 1470 y FA(;\033)1109 1460 y FG(\))1125 1442 y Fp(ex)p Fz(!)p Fy(\()p FA(\032)1216 1446 y Fo(0)1233 1442 y Fz(!)p FA(\032)1283 1446 y Fo(1)1299 1442 y Fz(!)p FA(\033)q Fy(\))p Fz(!)p FA(\033)1421 1460 y FG(;)257 1545 y(these)16 b(are)e(called)g Ft(existential)g(c)n(onstants)p FG(.)19 b(The)14 b(rewrite)h(rule)f(for)g FB(9)1342 1530 y Fz(\000)1383 1545 y FG(is)731 1630 y FB(9)754 1613 y Fz(\000)782 1630 y FG(\()p FB(9)821 1613 y Fy(+)849 1630 y FF(x)873 1636 y Fy(0)891 1630 y FF(x)915 1636 y Fy(1)934 1630 y FG(\))e FB(7\000)-7 b(!)11 b FF(\025y)q(:y)q(x)1142 1636 y Fy(0)1162 1630 y FF(x)1186 1636 y Fy(1)1204 1630 y FF(:)257 1716 y FG(The)i(\(constructiv)o(e\))i(existen)o(tial)d(quan)o(ti\014er)h (can)g(then)g(b)q(e)g(dealt)g(with)f(con)o(v)o(enien)o(tly)h(b)o(y)257 1766 y(means)g(of)h(axioms)539 1851 y FB(9)562 1834 y Fy(+)594 1851 y FG(:)f FB(8)p FF(x)p FG(\()p FF(A)f FB(!)f(9)p FF(xA)p FG(\))p FF(;)539 1918 y FB(9)562 1901 y Fz(\000)594 1918 y FG(:)j FB(9)p FF(xA)d FB(!)g(8)p FF(x)p FG(\()p FF(A)h FB(!)f FF(B)r FG(\))h FB(!)f FF(B)44 b FG(with)14 b FF(x)h(=)-25 b FB(2)11 b Fw(FV)p FG(\()p FF(B)r FG(\))q FF(:)257 2003 y FG(If)h FF(x)g FG(has)h(t)o(yp)q(e)g FF(\032)519 2009 y Fy(0)550 2003 y FG(and)f(the)h(form)o(ulas)e FF(A)h FG(and)g FF(B)j FG(are)e(asso)q(ciated)g(with)f(the)h(t)o(yp)q (es)g FF(\032)1591 2009 y Fy(1)1623 2003 y FG(and)257 2053 y FF(\033)q FG(,)f(resp)q(ectiv)o(ely)m(,)h(the)f(rewrite)h(rule)f (ab)q(o)o(v)o(e)f(is)h(clear.)18 b(It)12 b(seems)f(that)h(the)h (existen)o(tial)e(t)o(yp)q(e)257 2103 y Fw(ex)g FG(could)g(b)q(e)g (replaced)g(b)o(y)f FF(\032)701 2109 y Fy(0)723 2103 y FB(\002)r FF(\032)778 2109 y Fy(1)808 2103 y FG(and)g(the)h(constan)o (ts)h FB(9)1157 2088 y Fy(+)1157 2113 y FA(\032)1174 2117 y Fo(0)1190 2113 y FA(;\032)1217 2117 y Fo(1)1246 2103 y FG(and)e FB(9)1346 2088 y Fz(\000)1346 2113 y FA(\032)1363 2117 y Fo(0)1380 2113 y FA(;\032)1407 2117 y Fo(1)1423 2113 y FA(;\033)1465 2103 y FG(b)o(y)h(the)g(terms)257 2153 y FF(\025x)305 2159 y Fy(0)324 2153 y FF(\025x)372 2159 y Fy(1)390 2153 y FB(h)p FF(x)430 2159 y Fy(0)449 2153 y FF(;)c(x)492 2159 y Fy(1)510 2153 y FB(i)16 b FG(and)h FF(\025z)r(\025f)t FG(\()p FF(f)t(\031)783 2159 y Fy(0)803 2153 y FG(\()p FF(z)r FG(\))p FF(\031)880 2159 y Fy(1)899 2153 y FG(\()p FF(z)r FG(\)\))g(resp)q(ectiv)o(ely)m(.) 27 b(Ho)o(w)o(ev)o(er,)17 b(the)g(latter)g(term)257 2203 y(do)q(es)e(not)e(corresp)q(ond)j(to)d(a)h(deriv)n(ation)f(in)g (\014rst)h(order)h(logic,)d(since)j(it)e(is)h(imp)q(ossible)e(to)257 2252 y(pass)j(from)d(an)i(arbitrary)g(deriv)n(ation)f FF(d)g FG(\(p)q(ossibly)h(with)g(free)g(assumptions\))f(of)h FB(9)p FF(xA)g FG(to)257 2302 y(a)g(term)f FF(\031)415 2308 y Fy(0)434 2302 y FG(\()p FF(d)p FG(\))g(and)h(a)g(deriv)n(ation)f FF(\031)836 2308 y Fy(1)854 2302 y FG(\()p FF(d)p FG(\))h(of)f FF(A)1000 2308 y FA(x)1021 2302 y FG([)p FF(\031)1057 2308 y Fy(0)1075 2302 y FG(\()p FF(d)p FG(\)])o(.)320 2352 y(One)j(can)f(easily)g(form)o(ulate)e(rules)j(for)f Ft(p)n(ermutative)h(c)n(onversions)p FG(,)f(whic)o(h)g(p)q(erm)o(ute) 257 2402 y(an)f(application)e(of)i(an)f FB(9)p FG(-elimination)e(rule)j (with)g(other)g(elimination)d(rules,)j(e.g.)564 2487 y FB(9)587 2470 y Fz(\000)587 2497 y FA(\032)604 2501 y Fo(0)621 2497 y FA(;\032)648 2501 y Fo(1)664 2497 y FA(;\033)693 2501 y Fo(0)709 2497 y Fz(!)p FA(\033)761 2501 y Fo(1)780 2487 y FF(p)d FB(7\000)-6 b(!)11 b FF(\025z)r(v)q(:)p FB(9)992 2470 y Fz(\000)992 2497 y FA(\032)1009 2501 y Fo(0)1026 2497 y FA(;\032)1053 2501 y Fo(1)1069 2497 y FA(;\033)1098 2501 y Fo(1)1116 2487 y FF(p)p FG(\()p FF(\025xy)q(:)p FG(\()p FF(z)r(xy)q(v)q FG(\)\))p FF(:)963 2628 y FG(7)p eop %%Page: 8 8 8 7 bop 257 262 a Fr(2.4)56 b(Normalizable)16 b(terms)h(and)i(their)e (normal)h(forms)257 338 y FG(W)m(e)13 b(inductiv)o(ely)f(de\014ne)i(a)f (relation)f FF(M)17 b FB(\000)-7 b(!)11 b FF(Q)i FG(for)g(terms)f FF(M)r(;)7 b(Q)p FG(.)18 b(The)13 b(in)o(tended)h(mean-)257 388 y(ing)d(of)h FF(M)k FB(\000)-7 b(!)11 b FF(Q)h FG(is)g(that)f FF(M)17 b FG(is)12 b(normalizable)d(with)j(\(long\))f(normal)e(form)h FF(Q)p FG(.)17 b(Ho)o(w)o(ev)o(er,)257 438 y(it)e(is)g(necessary)h(to)f (split)g(up)f FB(\000)-6 b(!)14 b FG(in)o(to)g(t)o(w)o(o)h(relations:)k (a)c(\\w)o(eak")f(one)h FB(\000)-6 b(!)1493 444 y FA(w)1534 438 y FG(in)o(tended)257 488 y(to)15 b(un)o(wrap)g(the)g(outer)g (constructor)h(form,)d(follo)o(w)o(ed)g(b)o(y)i(a)f(\\strong")g(one)h FB(\000)-6 b(!)1540 494 y FA(s)1557 488 y FG(,)14 b(where)257 537 y(w)o(e)g(assume)g(that)g(it)f(is)h(applied)f(to)h(terms)g FF(M)k FG(irreducible)d(w.r.t.)e FB(\000)-7 b(!)1389 543 y FA(w)1415 537 y FG(.)320 587 y(Lo)q(oking)12 b(at)i(the)h(form)d (of)h(a)h(term)f(w)o(e)h(will)e(em)o(bark)h(on)h(the)g(follo)o(wing)d (strategy:)320 666 y FB(\017)20 b FF(\014)r FG(-redexes)e(\()p FF(\025xM)5 b FG(\))p FF(N)21 b FG(and)16 b(computation)e(rules)i FF(c)p Fx(M)5 b(N)21 b FG(are)c(reduced)g(promptly)m(,)361 715 y(i.e.)c(w)o(e)h(use)h(call-b)o(y-name)c(here.)320 796 y FB(\017)20 b FG(If)15 b(no)g(rule)h(applies)f(to)g FF(c)p Fx(M)5 b(N)21 b FG(one)16 b(\014rst)g(tries)g(to)f(\014nd)h(out) f(whether)i Fx(M)j FG(can)c(b)q(e)361 846 y(reduced)i(to)f Fx(P)23 b FG(suc)o(h)17 b(that)f FF(c)p Fx(P)23 b FG(matc)o(hes)16 b(a)g(computation)f(rule.)26 b(This)16 b(do)q(es)i(not)361 896 y(require)d(reducing)g(eac)o(h)f FF(M)803 902 y FA(i)831 896 y FG(to)g(normal)e(form,)g(it)h(su\016ces)i(to)f(\014nd)g(out)g (the)h(outer)361 946 y(pattern)k(of)f FF(M)605 952 y FA(i)638 946 y FG(\(let)h(us)g(call)e(it)h(for)h(no)o(w)f (\\constructor)i(normal)c(form"\).)30 b(The)361 995 y(reductions)14 b(for)d(doing)h(so)g(will)f(b)q(e)i(called)f(\\w)o(eak")g(and)g(w)o(e)g (write)h FB(\000)-7 b(!)1485 1001 y FA(w)1524 995 y FG(for)12 b(them.)320 1076 y FB(\017)20 b FG(If)d(in)f FF(c)p Fx(M)5 b(N)22 b FG(all)16 b Fx(M)21 b FG(are)d(already)e(in)g(constructor)j (normal)c(form)g(and)h(no)h(com-)361 1126 y(putation)d(rule)g(applies,) g(then)h(in)e(a)h(second)h(step)g(one)g(reduces)h(all)d Fx(M)19 b FG(and)14 b Fx(N)20 b FG(to)361 1176 y(normal)12 b(form)h(\(if)h(it)g(exists\))h(and)f(tries)i(to)e(apply)g(a)g(prop)q (er)h(rewrite)g(rule,)g(i.e.)e(w)o(e)361 1226 y(use)i(call-b)o(y-v)n (alue)d(at)i(this)g(p)q(oin)o(t.)320 1304 y(Let)i Fx(M)j FB(\000)-7 b(!)14 b Fx(M)595 1286 y Fz(0)622 1304 y FG(abbreviate)i FF(M)868 1310 y Fy(1)900 1304 y FB(\000)-6 b(!)13 b FF(M)1026 1289 y Fz(0)1021 1314 y Fy(1)1040 1304 y FG(,)j FF(:)7 b(:)g(:)e FG(,)15 b FF(M)1190 1310 y FA(n)1227 1304 y FB(\000)-7 b(!)14 b FF(M)1353 1289 y Fz(0)1348 1314 y FA(n)1386 1304 y FG(and)h(similarly)e(for)257 1354 y(other)i (relations,)e(and)h FB(\000)-7 b(!)694 1339 y Fz(\003)694 1364 y FA(w)734 1354 y FG(b)q(e)15 b(the)f(re\015exiv)o(e)h(and)f (transitiv)o(e)g(closure)g(of)g FB(\000)-7 b(!)1544 1360 y FA(w)1570 1354 y FG(.)257 1432 y Fn(De\014nition)12 b(1.)21 b FE(Split)p FG(.)764 1465 y FF(M)16 b FB(\000)-7 b(!)887 1450 y Fz(\003)887 1475 y FA(w)925 1465 y FF(N)47 b(N)16 b FB(\000)-6 b(!)1122 1471 y FA(s)1150 1465 y FF(Q)p 764 1483 420 2 v 890 1522 a(M)16 b FB(\000)-7 b(!)11 b FF(Q)320 1584 y FE(Et)m(a)p FG(.)451 1652 y FF(M)5 b(y)13 b FB(\000)-6 b(!)11 b FF(Q)p 394 1670 305 2 v 394 1708 a(M)439 1696 y FA(\032)p Fz(!)p FA(\033)523 1708 y FB(\000)-7 b(!)590 1714 y FA(s)619 1708 y FF(\025y)q(Q)744 1680 y FG(for)14 b FF(y)k(=)-26 b FB(2)12 b Fw(FV)o FG(\()p FF(M)5 b FG(\))1098 1652 y FF(M)g FG(0)11 b FB(\000)-7 b(!)11 b FF(Q)1286 1658 y Fy(0)1346 1652 y FF(M)5 b FG(1)11 b FB(\000)-7 b(!)11 b FF(Q)1534 1658 y Fy(1)p 1098 1670 456 2 v 1139 1708 a FF(M)1184 1696 y FA(\032)p Fz(\002)p FA(\033)1261 1708 y FB(\000)-6 b(!)1329 1714 y FA(s)1357 1708 y FB(h)p FF(Q)1406 1714 y Fy(0)1425 1708 y FF(;)7 b(Q)1477 1714 y Fy(1)1495 1708 y FB(i)320 1787 y FE(V)-5 b(arApp)p FG(.)550 1861 y Fx(M)16 b FB(\000)-6 b(!)11 b Fx(M)744 1843 y Fz(0)p 517 1880 272 2 v 517 1919 a FF(x)p Fx(M)17 b FB(\000)-7 b(!)672 1925 y FA(s)701 1919 y FF(x)p Fx(M)777 1901 y Fz(0)793 1889 y FF(;)48 b FG(pro)o(vided)14 b FF(x)p Fx(M)19 b FG(is)13 b(of)h(ground)g(t)o(yp)q(e.)320 1994 y FE(Bet)m(a)p FG(.)339 2079 y(\()p FF(\025xM)5 b FG(\))p FF(N)g Fx(P)18 b FB(\000)-6 b(!)618 2085 y FA(w)656 2079 y FF(M)696 2085 y FA(x)717 2079 y FG([)p FF(N)5 b FG(])o Fx(P)89 b FB(h)p FF(M)953 2085 y Fy(0)972 2079 y FF(;)7 b(M)1031 2085 y Fy(1)1050 2079 y FB(i)p FF(i)p Fx(P)18 b FB(\000)-6 b(!)1196 2085 y FA(w)1234 2079 y FF(M)1274 2085 y FA(i)1288 2079 y Fx(P)48 b FG(for)14 b FF(i)d FB(2)h(f)p FG(0)p FF(;)7 b FG(1)p FB(g)p FG(.)320 2165 y FE(Comp)p FG(.)579 2215 y FF(c)p Fx(P)634 2221 y Fq(x)657 2215 y FG([)p Fx(L)p FG(])p Fx(N)16 b FB(\000)-6 b(!)835 2221 y FA(w)873 2215 y FF(Q)906 2221 y Fq(x)929 2215 y FG([)p Fx(L)p FG(])o Fx(N)47 b FG(if)13 b FF(c)p Fx(P)18 b FB(7\000)-6 b(!)1241 2221 y Fp(comp)1323 2215 y FF(Q)p FG(.)257 2286 y(F)m(or)16 b(the)g(next)g(three)h(rules)f (assume)f(that)h FF(c)p Fx(M)21 b FG(is)15 b(not)h(an)f(instance)i(of)e (a)g(computation)257 2336 y(rule.)320 2385 y FE(Ar)o(g)p FG(.)385 2460 y Fx(M)h FB(\000)-6 b(!)516 2445 y Fz(\003)516 2470 y FA(w)554 2460 y Fx(M)606 2442 y Fz(0)p 322 2478 358 2 v 322 2518 a FF(c)p Fx(M)6 b(N)16 b FB(\000)-6 b(!)516 2524 y FA(w)554 2518 y FF(c)p Fx(M)624 2499 y Fz(0)635 2518 y Fx(N)726 2488 y FG(with)14 b(at)f(least)h(one)g FB(\000)-6 b(!)1112 2494 y FA(w)1138 2488 y FG(-reduction)15 b(in)e Fx(M)k FB(\000)-7 b(!)1516 2471 y Fz(\003)1516 2498 y FA(w)1554 2488 y Fx(M)1606 2470 y Fz(0)1618 2488 y FG(.)963 2628 y(8)p eop %%Page: 9 9 9 8 bop 257 262 a FG(The)16 b(\014nal)f(t)o(w)o(o)g(rules)h(ha)o(v)o(e) f(premises)h Fx(M)j FB(\000)-7 b(!)1020 268 y FA(s)1052 262 y Fx(M)1104 243 y Fz(0)1116 262 y FG(.)22 b(Note)16 b(that)g(b)o(y)f(lemma)d(2)j(b)q(elo)o(w,)257 311 y FF(c)p Fx(M)327 296 y Fm(0)356 311 y FG(cannot)g(b)q(e)g(an)g(instance)h(of)e (a)h(computation)e(rule,)i(for)g(then)g(also)g FF(c)p Fx(M)20 b FG(w)o(ould)14 b(b)q(e)257 361 y(one.)320 411 y FE(Rew)p FG(.)425 489 y Fx(M)i FB(\000)-6 b(!)556 495 y FA(s)585 489 y Fx(M)637 471 y Fz(0)p 344 508 387 2 v 344 546 a FF(c)p Fx(M)5 b(N)16 b FB(\000)-6 b(!)537 552 y FA(w)575 546 y FF(Q)608 552 y Fq(x)631 546 y FG([)p Fx(L)p FG(])o Fx(N)776 517 y FG(if)13 b Fw(sel)858 523 y FA(c)875 517 y FG(\()p Fx(M)943 500 y Fm(0)957 517 y FG(\))f(=)f FF(c)p Fx(K)k FB(7\000)-6 b(!)1169 523 y Fp(rew)1227 517 y FF(Q)14 b FG(and)g Fx(M)1407 500 y Fm(0)1432 517 y FG(=)e Fx(K)1519 523 y Fq(x)1542 517 y FG([)p Fx(L)p FG(].)320 628 y FE(P)l(assApp)p FG(.)266 706 y Fx(M)17 b FB(\000)-7 b(!)397 712 y FA(s)426 706 y Fx(M)478 688 y Fz(0)531 706 y Fx(N)17 b FB(\000)-7 b(!)11 b Fx(N)710 688 y Fz(0)p 266 725 456 2 v 314 764 a FF(c)p Fx(M)5 b(N)17 b FB(\000)-7 b(!)507 770 y FA(s)536 764 y FF(c)p Fx(M)606 746 y Fz(0)618 764 y Fx(N)662 746 y Fz(0)768 734 y FG(if)13 b Fw(sel)850 740 y FA(c)867 734 y FG(\()p Fx(M)935 717 y Fm(0)949 734 y FG(\))e(=)h Fw(no)p FG(-)p Fw(match)i FG(and)f FF(c)p Fx(M)6 b(N)19 b FG(of)13 b(ground)h(t)o(yp)q(e.)257 843 y(In)j(case)h(the)g(constan)o (t)f FF(c)g FG(in)f(the)i(rules)f FE(Ar)o(g)f FG(and)h FE(P)l(assApp)h FG(is)e(a)h(constructor,)i Fx(N)j FG(is)257 893 y(required)15 b(to)f(b)q(e)g(empt)o(y)m(.)320 974 y(F)m(or)d(readabilit)o(y)f(w)o(e)h(will)f(often)i(write)f FE(Rew)h FG(in)f(the)h(follo)o(wing)d(form,)h(assuming)f(that)257 1024 y FF(c)p Fx(K)15 b FB(7\000)-7 b(!)397 1030 y Fp(rew)456 1024 y FF(Q)14 b FG(is)g(the)g(selected)i(rule.)320 1074 y FE(Rew)p FG(.)672 1109 y Fx(M)g FB(\000)-6 b(!)803 1115 y FA(s)831 1109 y Fx(K)875 1115 y Fq(x)898 1109 y FG([)p Fx(L)p FG(])p 619 1128 387 2 v 619 1166 a FF(c)p Fx(M)5 b(N)17 b FB(\000)-7 b(!)812 1172 y FA(w)850 1166 y FF(Q)883 1172 y Fq(x)906 1166 y FG([)p Fx(L)p FG(])p Fx(N)1052 1137 y FG(if)13 b FF(c)p Fx(K)h FB(7\000)-6 b(!)1230 1143 y Fp(rew)1289 1137 y FF(Q)o(:)320 1233 y FG(F)m(or)13 b(the)i(de\014nition)e(ab)q(o)o(v)o(e)h(to)f(mak)o(e)g (sense)i(w)o(e)g(pro)o(v)o(e)e(the)i(follo)o(wing.)257 1314 y Fn(Lemma)h(2.)21 b Ft(If)16 b FF(M)k FB(\000)-7 b(!)657 1320 y FA(s)690 1314 y FF(M)735 1299 y Fz(0)763 1314 y Ft(and)17 b FF(M)890 1299 y Fz(0)919 1314 y Ft(is)f(an)h (instanc)n(e)h(of)e(a)h(c)n(onstructor)g(p)n(attern)f FF(P)6 b Ft(,)257 1364 y(then)16 b(also)f FF(M)k Ft(is)c(an)g(instanc)n (e)g(of)g FF(P)6 b Ft(.)257 1446 y(Pr)n(o)n(of.)20 b FG(By)14 b(induction)e(on)h FF(P)6 b FG(.)17 b(If)c FF(P)19 b FG(is)13 b(a)f(v)n(ariable)g(the)i(claim)d(is)i(trivial,)e(so)i(let)h FF(P)j FG(=)11 b FF(c)p Fx(P)c FG(.)257 1496 y(Then)15 b FF(M)411 1480 y Fz(0)435 1496 y FG(=)e FF(c)p Fx(K)541 1477 y Fz(0)567 1496 y FG(and)h Fx(K)691 1477 y Fz(0)718 1496 y FG(is)g(an)g(instance)h(of)f Fx(P)7 b FG(.)19 b(Moreo)o(v)o(er,)c(the)g(only)e(p)q(ossibilit)o(y)h(to)257 1545 y(infer)i FF(M)k FB(\000)-7 b(!)482 1551 y FA(s)514 1545 y FF(M)559 1530 y Fz(0)586 1545 y FG(=)15 b FF(c)p Fx(K)694 1527 y Fz(0)721 1545 y FG(is)h(b)o(y)g FE(P)l(assApp)p FG(.)25 b(Th)o(us)16 b FF(M)k FG(=)15 b FF(c)p Fx(K)s FG(,)h Fx(K)i FB(\000)-7 b(!)1460 1551 y FA(s)1492 1545 y Fx(K)1536 1527 y Fz(0)1563 1545 y FG(and)16 b(b)o(y)257 1595 y(induction)11 b(h)o(yp)q(othesis)g(\(IH\))g Fx(K)j FG(is)c(an)h(instance)g(of)f Fx(P)d FG(.)17 b(Since)11 b Fx(P)17 b FG(is)11 b(linear)f(w)o(e)h(ev)o(en)o(tually)257 1645 y(get)k(that)e FF(c)p Fx(K)18 b FG(is)13 b(an)h(instance)h(of)e FF(c)p Fx(P)6 b FG(.)p 1672 1620 18 2 v 1672 1643 2 24 v 1688 1643 V 1672 1645 18 2 v 257 1728 a Fn(De\014nition)12 b(3.)21 b FG(The)15 b(set)g FE(Lnf)g FG(of)e(terms)h(in)f(long)g (normal)f(form)g(is)i(de\014ned)h(as)g(follo)o(ws.)257 1777 y FF(\025xM)5 b FG(,)16 b FB(h)p FF(M)r(;)7 b(N)e FB(i)p FG(,)16 b(\()p FF(x)p Fx(M)5 b FG(\))645 1762 y FA(\034)682 1777 y FG(and)16 b(\()p FF(c)p Fx(M)6 b(N)f FG(\))912 1762 y FA(\034)949 1777 y FG(are)16 b(in)g FE(Lnf)h FG(if)e FF(M)r(;)7 b(N)r(;)g Fx(M)e FF(;)i Fx(N)21 b FG(are,)c(pro)o(vided)257 1827 y(that)d FF(c)p Fx(M)19 b FG(is)14 b(not)g(an)f(instance)i(of)e(an)o(y)h(computation)e(or)i (rewrite)h(rule.)320 1909 y(F)m(or)e(example,)f(the)i FF(\021)q Ft(-exp)n(ansion)19 b Fw(exp)q FG(\()p FF(x)p FG(\))13 b(of)h(a)f(v)n(ariable)g FF(x)g FG(is)h(in)f(long)g(normal)f (form;)257 1959 y(it)i(is)h(de\014ned)g(using)g(induction)f(on)g(t)o (yp)q(es)h(b)o(y)f(\(e.g.)g(for)g(pure)i FB(!)p FG(-t)o(yp)q(es\))e Fw(exp)q FG(\()p FF(x)1539 1944 y FA(\034)1560 1959 y FG(\))f(=)f FF(x)1657 1944 y FA(\034)1678 1959 y FG(,)257 2008 y Fw(exp)q FG(\()p FF(x)356 1993 y Fq(\032)p Fz(!)p FA(\034)430 2008 y FG(\))g(=)f FF(\025)p Fx(y)552 1993 y Fq(\032)573 2008 y FF(:x)c Fw(exp)q FG(\()p Fx(y)717 1993 y Fq(\032)739 2008 y FG(\).)257 2090 y Fn(Lemma)16 b(4.)21 b Ft(If)14 b FF(M)i FB(\000)-6 b(!)11 b FF(Q)k Ft(or)f FF(M)i FB(\000)-6 b(!)888 2096 y FA(s)917 2090 y FF(Q)p Ft(,)14 b(then)h FF(Q)g Ft(is)f(in)h(long)h(normal)e(form.)257 2171 y(Pr)n(o)n(of.)20 b FG(By)h(sim)o(ultaneous)e(induction)h(on)g FF(M)27 b FB(\000)-7 b(!)22 b FF(Q)e FG(and)g FF(M)27 b FB(\000)-7 b(!)1399 2177 y FA(s)1439 2171 y FF(Q)p FG(.)37 b(The)20 b(only)257 2221 y(in)o(teresting)12 b(case)h(is)e FE(P)l(assApp)p FG(,)h(where)g(w)o(e)g(ha)o(v)o(e)f(to)g (sho)o(w)h(that)f FF(c)p Fx(M)1353 2203 y Fz(0)1376 2221 y FG(is)g(not)h(an)f(instance)257 2271 y(of)j(a)g(computation)f(rule.) 20 b(But)15 b(if)f FF(c)p Fx(M)876 2253 y Fz(0)902 2271 y FG(w)o(ould)g(b)q(e)h(suc)o(h)g(an)g(instance,)g(b)o(y)f(the)h (previous)257 2321 y(lemma)c FF(c)p Fx(M)19 b FG(w)o(ould)13 b(also)g(b)q(e,)h(con)o(tradicting)g(the)h(assumption.)p 1672 2296 V 1672 2319 2 24 v 1688 2319 V 1672 2321 18 2 v 320 2403 a(F)m(urthermore)e(it)g(can)g(b)q(e)h(sho)o(wn)g(easily)e (that)i(if)e FF(M)k FB(\000)-6 b(!)11 b FF(Q)p FG(,)i FF(M)j FB(\000)-7 b(!)1414 2409 y FA(w)1452 2403 y FF(Q)13 b FG(or)h FF(M)i FB(\000)-7 b(!)1672 2409 y FA(s)257 2453 y FF(Q)p FG(,)13 b(then)i FF(M)j FG(reduces)e(to)e FF(Q)f FG(in)h(the)g(usual)f(sense)j(w.r.t.)c FF(\014)r FG(-reduction,)j FF(\021)q FG(-expansion)f(and)257 2503 y(the)i(computation)e(and)h(rewrite)h(rules)g(for)f(the)h(constan)o (ts.)23 b(Ho)o(w)o(ev)o(er,)16 b(the)g(con)o(v)o(erse)h(is)963 2628 y(9)p eop %%Page: 10 10 10 9 bop 257 262 a FG(not)12 b(true)g(in)f(general.)17 b(F)m(or)11 b(a)h(coun)o(terexample,)f(consider)h(the)g (non-terminating)d(rewrite)257 311 y(rules)17 b Fw(mult)6 b FF(x)469 296 y FA(\023)483 311 y FG(0)15 b FB(7\000)-7 b(!)586 317 y Fp(rew)648 311 y FG(0)16 b(and)f FB(?)800 296 y FA(\023)829 311 y FB(7\000)-7 b(!)896 317 y Fp(rew)958 311 y FB(?)p FG(.)23 b(Then)17 b(0)e(is)h(a)g(normal)d(form)h(of)i Fw(mult)o FB(?)p FG(0,)257 361 y(but)f(w)o(e)f(cannot)g(ha)o(v)o(e)g Fw(mult)o FB(?)p FG(0)d FB(\000)-6 b(!)11 b FF(Q)j FG(for)f(an)o(y)h FF(Q)p FG(.)k(T)m(o)c(see)h(this,)e(note)i(that)f(w)o(e)g(cannot)257 411 y(ha)o(v)o(e)20 b FB(?)i(\000)-7 b(!)481 417 y FA(s)520 411 y FF(N)25 b FG(for)20 b(an)o(y)f FF(N)25 b FG(\(since)c FB(?)h(7\000)-7 b(!)1036 417 y Fp(rew)1105 411 y FB(?)p FG(\),)21 b(hence)g(w)o(e)g(also)e(cannot)i(ha)o(v)o(e)257 461 y Fw(mult)p FB(?)p FG(0)15 b FB(\000)-6 b(!)474 467 y FA(s)507 461 y FF(Q)17 b FG(for)f(an)o(y)h FF(Q)p FG(.)26 b(Since)17 b FB(?)p FF(;)7 b FG(0)16 b(are)h FB(\000)-7 b(!)1115 446 y Fz(\003)1115 471 y FA(w)1142 461 y FG(-reducible)17 b(only)f(to)h(themselv)o(es,)257 511 y(the)e(claim)e(follo)o(ws.)18 b({)c(But)h(under)g(the)g(h)o(yp)q(othesis)g(that)g FF(M)k FG(is)14 b Ft(str)n(ongly)k FG(normalizable)257 560 y(the)d(con)o(v)o (erse)g(is)f(true.)257 639 y Fn(Lemma)i(5.)21 b Ft(If)14 b FF(M)19 b Ft(is)14 b(str)n(ongly)g(normalizable)g(w.r.t.)f(these)i(r) n(e)n(ductions)f(\(i.e.)g(every)g(r)n(e-)257 689 y(duction)i(se)n (quenc)n(e)g(terminates\),)e(then)h FF(M)h FB(\000)-6 b(!)11 b FF(Q)j Ft(for)h(some)g FF(Q)p Ft(.)257 768 y(Pr)n(o)n(of.)20 b FG(F)m(or)14 b(simplicit)o(y)e(w)o(e)j(consider)g(pure)g FB(!)p FG(-t)o(yp)q(es)g(only;)e(the)i(extension)g(to)g(pro)q(duct)257 818 y(t)o(yp)q(es)21 b(is)f(immediate.)34 b(W)m(e)20 b(will)e(pro)o(v)o(e)i(the)h(claim)d(b)o(y)i(induction)g(on)f FF(h)1476 824 y FA(M)1533 818 y FG(and)h(side)257 868 y(induction)15 b(on)g Fw(ht)p FG(\()p FF(M)5 b FG(\),)15 b(where)h FF(h)787 874 y FA(M)839 868 y FG(denotes)g(the)f(heigh)o(t)g (of)g(the)g(reduction)h(tree)g(for)f FF(M)257 918 y FG(and)f Fw(ht)p FG(\()p FF(M)5 b FG(\))14 b(is)f(the)h(heigh)o(t)f(of)g FF(M)5 b FG(.)18 b(Note)c(that)f(if)g FF(M)j FB(\000)-7 b(!)1174 924 y FA(w)1212 918 y FF(Q)14 b FG(then)g FF(M)k FG(reduces)d(to)f FF(Q)f FG(in)257 967 y(at)h(least)g(one)g(step,)h (hence)g FF(h)720 973 y FA(M)768 967 y FF(>)d(h)836 973 y FA(Q)864 967 y FG(.)320 1017 y Ft(Case)17 b FF(\025y)q(M)5 b FG(.)21 b(W)m(e)14 b(ha)o(v)o(e)g(\()p FF(\025y)q(M)5 b FG(\))p FF(y)16 b FB(\000)-7 b(!)940 1023 y FA(w)979 1017 y FF(M)18 b FB(\000)-7 b(!)12 b FF(Q)i FG(b)o(y)h FE(Bet)m(a)g FG(and)f(the)i(side)e(induc-)257 1067 y(tion)g(h)o(yp)q (othesis)g(\(SIH\),)g(hence)h FF(\025y)q(M)j FB(\000)-7 b(!)11 b FF(\025y)q(Q)k FG(b)o(y)f FE(Et)m(a)p FG(.)320 1117 y Ft(Case)19 b FF(M)h FG(has)c(a)g(t)o(yp)q(e)g FF(\032)f FB(!)g FF(\033)q FG(,)h(but)g(is)f(not)h(an)g(abstraction.)24 b(Then)16 b FF(M)21 b(\021)q FG(-expands)257 1167 y(to)d FF(\025y)q(:M)5 b(y)21 b FG(where)e FF(y)h FG(is)e(a)f(new)i(v)n (ariable)e(of)h(t)o(yp)q(e)g FF(\032)p FG(,)h(hence)h FF(h)1298 1173 y FA(M)1353 1167 y FF(>)f(h)1428 1173 y FA(\025y)q(:M)s(y)1548 1167 y FB(\025)g FF(h)1623 1173 y FA(M)s(y)1678 1167 y FG(.)257 1217 y(Therefore)d FF(M)5 b(y)13 b FB(\000)-7 b(!)11 b FF(Q)j FG(b)o(y)f(IH.)h(Hence)h FF(M)i FB(\000)-7 b(!)11 b FF(\025y)q(Q)k FG(b)o(y)f FE(Et)m(a)p FG(.)320 1266 y(It)g(remains)e(to)i(consider)h(terms)f(of)f (ground)h(t)o(yp)q(e.)320 1316 y Ft(Case)j FF(x)p Fx(M)t FG(.)h(Ob)o(vious,)c(using)f(the)i(SIH)f(and)f(rule)i FE(V)-5 b(arApp)p FG(.)320 1366 y Ft(Case)17 b FG(\()p FF(\025xM)5 b FG(\))p FF(N)g Fx(P)h FG(.)19 b(Then)c(\()p FF(\025xM)5 b FG(\))p FF(N)g Fx(P)18 b FB(\000)-7 b(!)1041 1372 y FA(w)1080 1366 y FF(M)1120 1372 y FA(x)1141 1366 y FG([)p FF(N)5 b FG(])o Fx(P)18 b FB(\000)-6 b(!)11 b FF(Q)j FG(b)o(y)g FE(Bet)m(a)h FG(and)f(the)257 1416 y(IH.)320 1466 y Ft(Case)j FF(c)p Fx(P)478 1472 y Fq(x)502 1466 y FG([)p Fx(L)p FG(])o Fx(N)j FG(with)13 b FF(c)p Fx(P)19 b FB(7\000)-7 b(!)843 1472 y Fp(comp)926 1466 y FF(Q)p FG(.)19 b(Then)14 b FF(c)p Fx(P)1153 1472 y Fq(x)1176 1466 y FG([)p Fx(L)p FG(])p Fx(N)j FB(\000)-7 b(!)1354 1472 y FA(w)1392 1466 y FF(Q)1425 1472 y Fq(x)1449 1466 y FG([)p Fx(L)p FG(])o Fx(N)17 b FB(\000)-6 b(!)11 b FF(Q)1671 1472 y Fy(1)257 1515 y FG(b)o(y)k FE(Comp)f FG(and)f(the)i(IH.)320 1565 y Ft(Case)f FF(c)p Fx(M)5 b(N)16 b FG(with)11 b FF(c)p Fx(M)16 b FG(not)10 b(an)h(instance)h(of)e (a)h(computation)e(rule.)18 b(By)11 b(SIH)g Fx(M)17 b FB(\000)-7 b(!)257 1615 y Fx(M)310 1597 y Fz(0)321 1615 y FG(.)32 b(If)18 b(at)g(least)h(one)g FF(M)689 1621 y FA(i)721 1615 y FG(is)g FB(\000)-7 b(!)835 1621 y FA(w)861 1615 y FG(-reduced,)21 b(the)e(claim)d(follo)o(ws)h(from)g(the)i(IH)g (and)257 1665 y FE(Ar)o(g)p FG(.)37 b(Otherwise)21 b(w)o(e)g(ha)o(v)o (e)f Fx(M)27 b FB(\000)-7 b(!)904 1671 y FA(s)944 1665 y Fx(M)996 1647 y Fz(0)1007 1665 y FG(.)37 b(No)o(w)20 b(if)f Fw(sel)1246 1671 y FA(c)1263 1665 y FG(\()p Fx(M)1331 1650 y Fm(0)1344 1665 y FG(\))j(=)h FF(c)p Fx(K)i FB(7\000)-7 b(!)1587 1671 y Fp(rew)1657 1665 y FF(Q)257 1715 y FG(and)20 b Fx(M)396 1700 y Fm(0)431 1715 y FG(=)i Fx(K)528 1721 y Fq(x)552 1715 y FG([)p Fx(L)p FG(])o(,)f(the)g(claim)d(follo)o(ws)g (from)g(the)j(IH)f(for)f FF(Q)1326 1721 y Fq(x)1350 1715 y FG([)p Fx(L)p FG(])o Fx(N)5 b FG(.)37 b(If)19 b(ho)o(w)o(ev)o(er)257 1764 y Fw(sel)302 1770 y FA(c)319 1764 y FG(\()p Fx(M)387 1749 y Fm(0)400 1764 y FG(\))14 b(=)h Fw(no)p FG(-)p Fw(match)p FG(,)g(then)h(pro)q(ceed)h(as)f(in)f(case)h FF(x)p Fx(M)5 b FG(,)15 b(using)h FE(P)l(assApp)g FG(instead)g(of)257 1814 y FE(V)-5 b(arApp)p FG(.)p 1672 1789 18 2 v 1672 1813 2 24 v 1688 1813 V 1672 1815 18 2 v 320 1896 a(Moreo)o(v)o(er,)19 b(the)g(relation)e FF(M)24 b FB(\000)-7 b(!)18 b FF(Q)g FG(clearly)g(is)g(not)g(closed)h(under)g(substitution.)257 1946 y(Ho)o(w)o(ev)o(er,)g(it)e(is)h(closed)g(under)h(substitution)e (of)g(v)n(ariables,)h(pro)o(vided)f(the)i(result)f(is)g(a)257 1996 y(v)n(arian)o(t)13 b(of)g FF(M)5 b FG(.)257 2075 y Fn(Lemma)16 b(6.)21 b Ft(L)n(et)15 b FB(!2)e(f\000)-7 b(!)p FF(;)7 b FB(\000)-7 b(!)814 2081 y FA(w)840 2075 y FF(;)7 b FB(\000)-7 b(!)926 2081 y FA(s)943 2075 y FB(g)p Ft(.)22 b(If)16 b FF(M)j FB(!)13 b FF(Q)p Ft(,)j(then)g FF(M)1353 2081 y Fq(x)1377 2075 y FG([)p Fx(z)q FG(])e FB(!)f FF(Q)1527 2081 y Fq(x)1550 2075 y FG([)p Fx(z)r FG(])i Ft(with)257 2125 y(a)g(derivation)g(of)g(the)g(same)g(height,)g (pr)n(ovide)n(d)g Fx(z)h Ft(ar)n(e)f(distinct)f(variables)19 b FF(=)-25 b FB(2)11 b Fw(FV)p FG(\()p FF(M)5 b FG(\))1612 2110 y Fy(1)1631 2125 y Ft(.)257 2204 y(Pr)n(o)n(of.)20 b FG(W)m(e)13 b(use)i(induction)e(on)g(the)h(heigh)o(t)f(of)g(the)h (deriv)n(ation)f(of)f FF(M)17 b FB(!)11 b FF(Q)p FG(.)17 b(Clearly)c(w)o(e)257 2254 y(ma)o(y)f(assume)i Fx(x)e FB(2)f Fw(FV)p FG(\()p FF(M)5 b FG(\).)320 2303 y Ft(Case)17 b FE(Et)m(a)p FG(.)720 2336 y FF(M)5 b(y)13 b FB(\000)-7 b(!)11 b FF(Q)p 662 2354 305 2 v 662 2392 a(M)707 2380 y FA(\032)p Fz(!)p FA(\033)791 2392 y FB(\000)-6 b(!)859 2398 y FA(s)888 2392 y FF(\025y)q(Q)1013 2364 y FG(for)13 b FF(y)18 b(=)-25 b FB(2)11 b Fw(FV)p FG(\()p FF(M)5 b FG(\).)p 257 2425 573 2 v 304 2452 a Fl(1)321 2464 y FC(If)11 b(w)o(e)h(assume)e(strong)g(uniformit)o(y)f(of)i(all)g Fk(sel)939 2468 y Fj(c)955 2464 y FC(-functions)e(\(as)i(w)o(e)h (implicitly)c(did)j(in)h([4)o(]\),)f(then)f(the)257 2503 y(pro)o(viso)g(is)i(not)e(necessary)m(.)953 2628 y FG(10)p eop %%Page: 11 11 11 10 bop 257 262 a FG(Recall)363 380 y(\()p FF(\025y)q(Q)p FG(\))473 386 y Fq(x)497 380 y FG([)p Fx(z)r FG(])11 b(:=)612 309 y Fu(\()646 352 y FF(\025uQ)727 358 y Fq(x)p FA(;y)775 362 y Fi(i)790 352 y FG([)p Fx(z)q FF(;)c(u)p FG(])40 b(if)13 b FF(y)h FG(=)e FF(z)1056 358 y FA(i)1084 352 y FG(for)h(some)g FF(i)h FG(with)g FF(x)1398 358 y FA(i)1423 352 y FB(2)d Fw(FV)p FG(\()p FF(Q)p FG(\))646 411 y FF(\025y)q(Q)724 417 y Fq(x)748 411 y FG([)p Fx(z)q FG(])125 b(otherwise)257 506 y(with)14 b(a)g(new)g(v)n(ariable)f FF(u)p FG(.)320 556 y Ft(Sub)n(c)n(ase)i FG(1.)i FF(y)c FG(=)f FF(z)617 562 y FA(i)643 556 y FG(for)f(some)g FF(i)g FG(with)h FF(x)948 562 y FA(i)973 556 y FB(2)f Fw(FV)p FG(\()p FF(Q)p FG(\).)17 b(By)12 b(IH)g FF(M)1319 562 y Fq(x)1342 556 y FG([)p Fx(z)r FG(])o FF(u)f FB(\000)-6 b(!)11 b FF(Q)1537 562 y Fq(x)p FA(;y)1588 556 y FG([)p Fx(z)q FF(;)c(u)p FG(])o(,)257 606 y(hence)16 b FF(M)413 612 y Fq(x)436 606 y FG([)p Fx(z)r FG(])11 b FB(\000)-7 b(!)563 612 y FA(s)592 606 y FF(\025uQ)673 612 y Fq(x)p FA(;y)724 606 y FG([)p Fx(z)q FF(;)7 b(u)p FG(])13 b(b)o(y)h FE(Et)m(a)p FG(.)320 656 y Ft(Sub)n(c)n(ase)19 b FG(2.)k FF(y)18 b FG(not)e(in)f Fx(z)r FG(.)23 b(Because)18 b(of)d Fx(x)g FB(2)f Fw(FV)p FG(\()p FF(M)5 b FG(\))15 b(w)o(e)h(ha)o(v)o(e)g FF(y)h FG(not)f(in)f Fx(x)q FG(.)23 b(Hence)257 705 y FF(M)297 711 y Fq(x)321 705 y FG([)p Fx(z)q FG(])p FF(y)13 b FG(=)f(\()p FF(M)5 b(y)q FG(\))544 711 y Fq(x)569 705 y FG([)p Fx(z)q FG(])11 b FB(\000)-7 b(!)11 b FF(Q)739 711 y Fq(x)763 705 y FG([)p Fx(z)q FG(])i(b)o(y)h(IH)g(and)g(the)g (claim)e(follo)o(ws)g(b)o(y)i FE(Et)m(a)p FG(.)320 755 y Ft(Sub)n(c)n(ase)f FG(3.)k FF(y)c FG(=)f FF(z)615 761 y FA(i)639 755 y FG(for)e(some)f FF(i)h FG(with)f FF(x)937 761 y FA(i)967 755 y FF(=)-26 b FB(2)12 b Fw(FV)o FG(\()p FF(Q)p FG(\).)17 b(Let)1220 754 y(^)1218 755 y Fx(z)11 b FG(b)q(e)g Fx(z)g FG(without)f FF(z)1509 761 y FA(i)1533 755 y FG(and)1613 754 y(^)1610 755 y Fx(x)g FG(b)q(e)257 805 y Fx(x)16 b FG(without)f FF(x)480 811 y FA(i)494 805 y FG(.)23 b(Then)16 b FF(Q)672 811 y Fq(x)695 805 y FG([)p Fx(z)r FG(])d(=)i FF(Q)839 811 y Fy(^)-19 b Fq(x)860 805 y FG([)874 804 y(^)872 805 y Fx(z)r FG(])o(,)16 b(hence)h FF(\025y)q(Q)1133 811 y Fy(^)-19 b Fq(x)1155 805 y FG([)1169 804 y(^)1167 805 y Fx(z)q FG(])14 b(=)h(\()p FF(\025y)q(Q)p FG(\))1376 811 y Fy(^)-19 b Fq(x)1398 805 y FG([)1412 804 y(^)1410 805 y Fx(z)r FG(])13 b(=)i(\()p FF(\025y)q(Q)p FG(\))1617 811 y Fq(x)1642 805 y FG([)p Fx(z)q FG(])257 855 y(and)f(the)h(claim)c(follo)o(ws)i(as)h(in)f(sub)q (case)i(2.)320 905 y Ft(Case)i FE(Rew)p FG(.)425 985 y Fx(M)g FB(\000)-7 b(!)556 991 y FA(s)585 985 y Fx(M)637 967 y Fz(0)p 344 1003 386 2 v 344 1041 a FF(c)p Fx(M)5 b(N)17 b FB(\000)-7 b(!)537 1047 y FA(w)575 1041 y FF(Q)608 1047 y Fq(y)631 1041 y FG([)p Fx(L)p FG(])o Fx(N)776 1013 y FG(if)13 b Fw(sel)858 1019 y FA(c)875 1013 y FG(\()p Fx(M)943 996 y Fm(0)957 1013 y FG(\))f(=)f FF(c)p Fx(K)k FB(7\000)-6 b(!)1169 1019 y Fp(rew)1227 1013 y FF(Q)14 b FG(and)g Fx(M)1407 996 y Fm(0)1432 1013 y FG(=)e Fx(K)1519 1019 y Fq(y)1542 1013 y FG([)p Fx(L)p FG(])o(.)257 1130 y(Let)j Fx(z)h FG(b)q(e)f(distinct)f(v)n(ariables)k FF(=)-26 b FB(2)12 b Fw(FV)p FG(\()p FF(c)p Fx(M)5 b(N)g FG(\).)19 b(W)m(e)14 b(w)o(an)o(t)f(to)h(deriv)o(e)h(\()p FF(c)p Fx(M)5 b(N)g FG(\))1512 1136 y Fq(x)1536 1130 y FG([)p Fx(z)q FG(])12 b FB(\000)-7 b(!)1663 1136 y FA(w)257 1180 y FG(\()p FF(Q)306 1186 y Fq(y)329 1180 y FG([)p Fx(L)p FG(])p Fx(N)5 b FG(\))444 1186 y Fq(x)467 1180 y FG([)p Fx(z)r FG(])12 b(b)o(y)h FE(Rew)g FG(again.)k(Clearly)12 b(w)o(e)h(ma)o(y)e(assume)h Fx(x)p FF(;)7 b Fx(z)18 b FF(=)-26 b FB(2)11 b Fw(FV)p FG(\()p Fx(K)s FF(;)c(Q)p FG(\).)17 b(By)c(IH)257 1230 y Fx(M)310 1236 y Fq(x)333 1230 y FG([)p Fx(z)q FG(])g FB(\000)-6 b(!)462 1236 y FA(s)493 1230 y Fx(M)545 1211 y Fz(0)545 1240 y Fq(x)568 1230 y FG([)p Fx(z)r FG(])14 b(and)h Fx(z)20 b FF(=)-26 b FB(2)14 b Fw(FV)o FG(\()p Fx(M)913 1211 y Fz(0)924 1230 y FG(\))i(\(since)g Fx(M)i FB(\000)-6 b(!)1208 1236 y FA(s)1238 1230 y Fx(M)1291 1211 y Fz(0)1317 1230 y FG(implies)13 b(that)i Fx(M)1603 1211 y Fz(0)1629 1230 y FG(has)257 1279 y(no)g(more)e(free)j(v)n(ariables)d(than)i Fx(M)5 b FG(\).)20 b(No)o(w)14 b Fw(sel)1011 1285 y FA(c)1027 1279 y FG(\()p Fx(M)1096 1261 y Fz(0)1096 1290 y Fq(x)1119 1279 y FG([)p Fx(z)r FG(])o(\))f(=)g Fw(sel)1285 1285 y FA(c)1302 1279 y FG(\()p Fx(M)1370 1261 y Fz(0)1382 1279 y FG(\))g(=)g FF(c)p Fx(K)i FB(7\000)-6 b(!)1597 1285 y Fp(rew)1657 1279 y FF(Q)257 1329 y FG(b)o(y)15 b(uniformit)o(y)d(of)i Fw(sel)613 1335 y FA(c)630 1329 y FG(,)h(and)f Fx(M)791 1311 y Fz(0)791 1339 y Fq(x)814 1329 y FG([)p Fx(z)q FG(])f(=)g Fx(K)963 1335 y Fq(y)986 1329 y FG([)p Fx(L)1029 1335 y Fq(x)1053 1329 y FG([)p Fx(z)q FG(]])o(.)21 b(An)15 b(application)e(of)k FE(Rew)f FG(yields)257 1379 y(\()p FF(c)p Fx(M)6 b(N)f FG(\))404 1385 y Fq(x)427 1379 y FG([)p Fx(z)r FG(])11 b(=)h FF(c)p Fx(M)601 1385 y Fq(x)624 1379 y FG([)p Fx(z)r FG(])o Fx(N)716 1385 y Fq(x)740 1379 y FG([)p Fx(z)q FG(])f FB(\000)-6 b(!)867 1385 y FA(w)904 1379 y FF(Q)937 1385 y Fq(y)960 1379 y FG([)p Fx(L)1003 1385 y Fq(x)1027 1379 y FG([)p Fx(z)q FG(])o(])p Fx(N)1130 1385 y Fq(x)1154 1379 y FG([)p Fx(z)q FG(])11 b(=)h(\()p FF(Q)1306 1385 y Fq(y)1329 1379 y FG([)p Fx(L)o FG(])p Fx(N)5 b FG(\))1443 1385 y Fq(x)1467 1379 y FG([)p Fx(z)q FG(].)320 1429 y(The)10 b(simpli\014cations)e(in)h(case)i(w)o(e)g(assume)e(strongly)h (uniformit)o(y)d(of)j(all)e Fw(sel)1494 1435 y FA(c)1511 1429 y FG(-functions)257 1479 y(are)15 b(ob)o(vious.)p 1672 1454 18 2 v 1672 1477 2 24 v 1688 1477 V 1672 1479 18 2 v 257 1562 a Fn(Example)g(7.)21 b FG(In)13 b(the)g(examples)f(in)g (section)h(2.3,)f(man)o(y)e(of)i(the)h(\(prop)q(er\))h(rewrite)g(rules) 257 1611 y(ha)o(v)o(e)h(b)q(een)h(written)f(as)g(\\higher)g(t)o(yp)q(e) g(rules",)g(i.e.)e(in)i(the)g(form)e FF(c)p Fx(M)18 b FB(7\000)-6 b(!)12 b FF(\025xN)20 b FG(rather)257 1661 y(than)15 b FF(c)p Fx(M)5 b FF(x)13 b FB(7\000)-7 b(!)13 b FF(N)5 b FG(.)20 b(This)15 b(is)g(preferable)g(from)e(a)i(seman)o (tical)e(p)q(oin)o(t)h(of)g(view,)h(b)q(ecause)257 1711 y(the)k(latter)e(form)f(ma)o(y)g(cause)i(unnecessary)i(calculations)d (\(cf.)g(the)h(de\014nition)g(of)f FB(I)s FG(\()p FF(c)p FG(\))257 1761 y(in)d(section)h(3.5\).)j(Note)d(also)f(that)g(in)g(the) h(presence)i(of)c(non-terminating)g(rewrite)i(rules)257 1811 y(b)q(oth)10 b(v)o(ersions)h(can)f(lead)g(to)g(di\013eren)o(t)h (sets)g(of)e(normalizing)f(terms.)16 b(T)m(o)9 b(see)j(that,)e(assume) 257 1861 y(there)j(is)e(a)g(term)g FF(M)16 b FG(that)11 b(has)h(no)f(normal)e(form,)h(so)h(no)g FF(N)16 b FG(exists)c(suc)o(h)g (that)g FF(M)k FB(\000)-7 b(!)1611 1867 y FA(s)1640 1861 y FF(N)5 b FG(.)257 1910 y(Then)15 b(for)g(a)f(prop)q(er)h(rewrite)h (rule)f FF(cy)f FB(7\000)-6 b(!)12 b FF(d)i FG(the)h(term)f FF(cM)19 b FG(has)c(no)f(normal)f(form,)f(but)257 1960 y(for)i FF(c)d FB(7\000)-6 b(!)11 b FF(\025y)q(d)j FG(it)g(has,)f (namely)g FF(d)p FG(.)320 2010 y(Also)h(for)g(terminating)f(but)h (non-con\015uen)o(t)h(rewrite)h(rules)f(b)q(oth)f(v)o(ersions)h(can)g (lead)257 2060 y(to)20 b(di\013eren)o(t)h(normal)c(forms.)34 b(Here)21 b(is)e(an)h(example,)f(with)h(all)e(rules)i(considered)h(as) 257 2110 y(prop)q(er)15 b(rewrite)g(rules.)557 2201 y FF(d)p FG(2)p FF(x)c FB(7\000)-7 b(!)11 b FG(2)253 b FF(d)13 b FG(of)h(t)o(yp)q(e)g FF(\034)i FB(!)c FF(\034)k FB(!)11 b FF(\034)557 2263 y(dx)p FG(3)g FB(7\000)-7 b(!)11 b FG(3)h(=)g Fw(sel)834 2269 y FA(d)853 2263 y FG(\(2)p FF(;)7 b FG(3\))584 2325 y FF(cy)13 b FB(7\000)-7 b(!)11 b FF(y)q FG(3)232 b FF(c)14 b FG(of)f(t)o(yp)q(e)h(\()p FF(\034)j FB(!)11 b FF(\034)5 b FG(\))12 b FB(!)f FF(\034)953 2628 y FG(11)p eop %%Page: 12 12 12 11 bop 257 262 a FG(Then)715 292 y(2)11 b FB(\000)-6 b(!)815 298 y FA(s)843 292 y FG(2)93 b FF(x)12 b FB(\000)-7 b(!)1060 298 y FA(s)1089 292 y FF(x)p 698 309 431 2 v 1142 322 a FE(Rew)748 346 y FF(d)p FG(2)p FF(x)10 b FB(\000)-6 b(!)893 352 y FA(w)931 346 y FG(2)11 b FB(\000)-7 b(!)1030 352 y FA(s)1059 346 y FG(2)p 731 362 366 2 v 1109 375 a FE(Split)825 399 y FF(d)p FG(2)p FF(x)11 b FB(\000)-7 b(!)11 b FG(2)p 788 409 252 2 v 1052 422 a FE(Et)m(a)804 446 y FF(d)p FG(2)g FB(\000)-6 b(!)926 452 y FA(s)954 446 y FF(\025x)p FG(2)p 599 462 631 2 v 1241 476 a FE(Rew)615 502 y FF(c)p FG(\()p FF(d)p FG(2\))11 b FB(\000)-6 b(!)787 508 y FA(w)825 502 y FG(\()p FF(\025x)p FG(2\)3)11 b FB(\000)-7 b(!)1025 508 y FA(w)1063 502 y FG(2)11 b FB(\000)-6 b(!)1163 508 y FA(s)1192 502 y FG(2)p 599 522 V 1241 535 a FE(Split)812 562 y FF(c)p FG(\()p FF(d)p FG(2\))11 b FB(\000)-6 b(!)11 b FG(2)257 628 y(If,)i(ho)o(w)o(ev)o(er,)h(the)h (last)e(rule)h(is)g(replaced)h(b)o(y)864 715 y FF(c)c FB(7\000)-7 b(!)12 b FF(\025y)q(:y)q FG(3)p FF(;)257 803 y FG(w)o(e)i(obtain)g FF(c)p FG(\()p FF(d)p FG(2\))d FB(\000)-7 b(!)619 809 y FA(w)657 803 y FG(\()p FF(\025y)q(:y)q FG(3\)\()p FF(d)p FG(2\))15 b(b)o(y)f FE(Rew)p FG(,)g(hence)483 886 y FF(c)p FG(\()p FF(d)p FG(2\))d FB(\000)-7 b(!)654 892 y FA(w)692 886 y FG(\()p FF(\025y)q(:y)q FG(3\)\()p FF(d)p FG(2\))13 b FB(\000)-7 b(!)978 892 y FA(w)1016 886 y FF(d)p FG(23)11 b FB(\000)-7 b(!)1158 892 y FA(w)1196 886 y FG(3)11 b FB(\000)-7 b(!)1295 892 y FA(s)1324 886 y FG(3)p 466 907 896 2 v 1374 920 a FE(Split)812 946 y FF(c)p FG(\()p FF(d)p FG(2\))11 b FB(\000)-6 b(!)11 b FG(3)257 1027 y Fn(Example)k(8.)21 b FG(F)m(ormally)12 b(it)j(is)g(p)q(ossible)g(to)g(add)g(a)f(\014xp)q(oin)o(t)h(op)q (erator)g Fw(Y)h FG(with)e(rewrite)257 1077 y(rule)i Fw(Y)f FB(7\000)-7 b(!)14 b FF(\025x:x)p FG(\()p Fw(Y)q FF(x)p FG(\))g(or)i Fw(Y)q FF(x)d FB(7\000)-6 b(!)13 b FF(x)p FG(\()p Fw(Y)q FF(x)p FG(\).)23 b(But)16 b(if)e(one)i(tries)g (to)f(de\014ne)i(e.g.)e(addition)257 1127 y Fw(add)g FG(b)o(y)f(a)f(\014xp)q(oin)o(t)h(op)q(erator,)g Fw(add)7 b FF(xy)16 b FG(w)o(ould)d(ha)o(v)o(e)h(no)f(normal)f(form:)540 1215 y Fw(add)h FG(:=)e FF(\025x:)p Fw(Y)q FF(M)18 b FG(where)d FF(M)h FG(:=)c FF(\025z)r(y)q(:)p Fw(if)e FF(y)q(x)p FG(\()p Fw(S)p FG(\()p FF(z)r FG(\()p Fw(P)p FF(y)q FG(\)\)\))p FF(:)257 1302 y FG(If)15 b Fw(add)7 b FF(xy)17 b FG(w)o(ould)d(ha)o(v)o(e)g(a)g(normal)f(form)g(with)h (resp)q(ect)j(to)e(our)g(rules,)g(there)h(m)o(ust)d(b)q(e)j(a)257 1352 y(term)e FF(N)k FG(suc)o(h)d(that)799 1402 y(\()p FF(\025x:)p Fw(Y)q FF(M)5 b FG(\))p FF(xy)13 b FB(\000)-7 b(!)11 b FF(N)r(:)257 1474 y FG(T)m(o)j(deriv)o(e)h(this)f(relation)g (it)g(is)h(necessary)h(to)e(sho)o(w)h Fw(Y)q FF(M)5 b(y)14 b FB(\000)-7 b(!)12 b FF(N)5 b FG(,)14 b(and)g(b)o(y)g(the)h(rewrite) 257 1524 y(rule)g(for)e(the)i(\014xp)q(oin)o(t)e(op)q(erator)h(w)o(e)g (w)o(ould)g(ha)o(v)o(e)f(to)h(sho)o(w)g FF(M)5 b FG(\()p Fw(Y)q FF(M)g FG(\))p FF(y)13 b FB(\000)-7 b(!)11 b FF(N)5 b FG(,)13 b(hence)745 1612 y Fw(if)c FF(y)q(x)p FG(\()p Fw(S)p FG(\()p Fw(Y)q FF(M)c FG(\()p Fw(P)p FF(y)q FG(\)\)\))13 b FB(\000)-6 b(!)11 b FF(N)r(:)257 1699 y FG(But)i(b)o(y)f(the)h(rule)f FE(Ar)o(g)g FG(w)o(e)g(no)o(w)g(need)h(to)f(ha)o(v)o(e)g(a)g(normal)e (form)h(for)h Fw(Y)q FF(M)5 b FG(\()p Fw(P)p FF(y)q FG(\),)12 b(and)g(th)o(us)257 1749 y(for)i Fw(Y)q FF(M)5 b FG(\()p Fw(P)p FG(\()p Fw(P)p FF(y)q FG(\)\),)14 b(and)g(so)g(on.)19 b({)14 b(Ho)o(w)o(ev)o(er,)g(the)h(term)e Fw(add)8 b FF(M)d(N)19 b FG(with)14 b(n)o(umerals)f FF(M)r(;)7 b(N)257 1799 y FG(reduces)16 b(to)e(a)g(n)o(umeral.)257 1915 y Fr(2.5)56 b(T)-5 b(erm)17 b(families)257 1991 y FG(Since)d (normalization)c(b)o(y)j(ev)n(aluation)f(needs)j(to)e(create)i(b)q (ound)e(v)n(ariables)f(when)i(\\reify-)257 2041 y(ing")f(abstract)i(ob) r(jects)g(of)f(higher)g(t)o(yp)q(e,)g(it)f(is)h(useful)g(to)g(follo)o (w)f FE(de)j(Br)o(uijn)p FG('s)e([9)o(])f(st)o(yle)257 2091 y(of)h(represen)o(ting)i(b)q(ound)e(v)n(ariables)g(in)g(terms.)19 b(This)14 b(is)g(done)h(here)g({)f(as)h(in)e([5,)g(10])h({)g(b)o(y)257 2141 y(means)d(of)h Ft(term)g(families)p FG(.)k(A)c(term)f(family)e(is) j(a)f(parametrized)h(v)o(ersion)g(of)f(a)h(giv)o(en)f(term)257 2190 y FF(M)5 b FG(.)18 b(The)12 b(idea)g(is)g(that)g(the)h(term)f (family)d(of)i FF(M)17 b FG(at)12 b(index)g FF(k)h FG(repro)q(duces)i FF(M)i FG(with)11 b(b)q(ound)257 2240 y(v)n(ariables)j(renamed)f (starting)h(at)g FF(k)q FG(.)j(F)m(or)d(example,)e(for)697 2328 y FF(M)17 b FG(:=)11 b FF(\025u\025v)q(:c)p FG(\()p FF(\025x:v)q(x)p FG(\)\()p FF(\025y)q(\025z)r(:z)r(u)p FG(\))257 2415 y(the)k(asso)q(ciated)f(term)g(family)d FF(M)798 2400 y Fz(1)846 2415 y FG(at)j(index)g(3)g(yields)558 2503 y FF(M)603 2486 y Fz(1)638 2503 y FG(\(3\))e(:=)f FF(\025x)806 2509 y Fy(3)825 2503 y FF(\025x)873 2509 y Fy(4)891 2503 y FF(:c)p FG(\()p FF(\025x)985 2509 y Fy(5)1003 2503 y FF(:x)1039 2509 y Fy(4)1057 2503 y FF(x)1081 2509 y Fy(5)1099 2503 y FG(\)\()p FF(\025x)1179 2509 y Fy(5)1198 2503 y FF(\025x)1246 2509 y Fy(6)1265 2503 y FF(:x)1301 2509 y Fy(6)1319 2503 y FF(x)1343 2509 y Fy(3)1361 2503 y FG(\))p FF(:)953 2628 y FG(12)p eop %%Page: 13 13 13 12 bop 257 262 a FG(W)m(e)14 b(denote)h(terms)e(b)o(y)h FF(M)r(;)7 b(N)r(;)g(K)q(;)g(:)g(:)g(:)e FG(,)13 b(and)h(term)f (families)e(b)o(y)j FF(r)o(;)7 b(s;)g(t;)g(:)g(:)g(:)t FG(.)320 311 y(T)m(o)13 b(ev)o(ery)h(term)g FF(M)635 296 y FA(\032)668 311 y FG(w)o(e)g(assign)f(a)h(term)f(family)e FF(M)1157 296 y Fz(1)1197 311 y FG(:)i Fh(N)d FB(!)h FG(\003)1346 317 y FA(\032)1379 311 y FG(b)o(y)379 400 y FF(x)403 383 y Fz(1)437 400 y FG(\()p FF(k)q FG(\))h(:=)f FF(x;)384 462 y(c)402 445 y Fz(1)437 462 y FG(\()p FF(k)q FG(\))h(:=)f FF(c;)279 525 y FG(\()p FF(\025y)q(M)5 b FG(\))401 508 y Fz(1)437 525 y FG(\()p FF(k)q FG(\))12 b(:=)f FF(\025x)607 531 y FA(k)628 525 y FG(\()p FF(M)684 531 y FA(y)704 525 y FG([)p FF(x)740 531 y FA(k)760 525 y FG(])771 504 y Fz(1)807 525 y FG(\()p FF(k)f FG(+)f(1\)\))p FF(;)41 b FB(h)p FF(M)1058 531 y Fy(0)1078 525 y FF(;)7 b(M)1137 531 y Fy(1)1155 525 y FB(i)1171 508 y Fz(1)1206 525 y FG(\()p FF(k)q FG(\))12 b(:=)f FB(h)p FF(M)1389 508 y Fz(1)1384 535 y Fy(0)1424 525 y FG(\()p FF(k)q FG(\))p FF(;)c(M)1543 508 y Fz(1)1538 535 y Fy(1)1578 525 y FG(\()p FF(k)q FG(\))p FB(i)p FF(;)287 587 y FG(\()p FF(M)e(N)g FG(\))402 570 y Fz(1)437 587 y FG(\()p FF(k)q FG(\))12 b(:=)f FF(M)604 570 y Fz(1)639 587 y FG(\()p FF(k)q FG(\))p FF(N)732 570 y Fz(1)768 587 y FG(\()p FF(k)q FG(\))p FF(;)221 b(\031)1080 593 y FA(i)1094 587 y FG(\()p FF(M)5 b FG(\))1171 570 y Fz(1)1206 587 y FG(\()p FF(k)q FG(\))12 b(:=)f FF(\031)1352 593 y FA(i)1366 587 y FG(\()p FF(M)1427 570 y Fz(1)1462 587 y FG(\()p FF(k)q FG(\)\))p FF(:)257 676 y FG(Application)j(of)h(a)g(term)f(family)e FF(r)6 b FG(:)13 b Fh(N)f FB(!)h FG(\003)974 682 y FA(\032)p Fz(!)p FA(\033)1061 676 y FG(to)i(a)g(term)f(family)e FF(s)5 b FG(:)14 b Fh(N)e FB(!)h FG(\003)1555 682 y FA(\032)1589 676 y FG(is)i(the)257 726 y(family)h FF(r)q(s)5 b FG(:)14 b Fh(N)j FB(!)h FG(\003)597 732 y FA(\033)638 726 y FG(de\014ned)h(b)o (y)f(\()p FF(r)q(s)p FG(\)\()p FF(k)q FG(\))h(:=)g FF(r)q FG(\()p FF(k)q FG(\))p FF(s)p FG(\()p FF(k)q FG(\),)g(and)g(similarly)c (for)j(pairing)257 775 y FB(h)p FF(r)292 781 y Fy(0)311 775 y FF(;)7 b(r)349 781 y Fy(1)367 775 y FB(i)p FG(\()p FF(k)q FG(\))20 b(:=)g FB(h)p FF(r)557 781 y Fy(0)575 775 y FG(\()p FF(k)q FG(\))p FF(;)7 b(r)668 781 y Fy(1)686 775 y FG(\()p FF(k)q FG(\))p FB(i)19 b FG(and)g(pro)r(jections)h FF(\031)1105 781 y FA(i)1118 775 y FG(\()p FF(r)q FG(\)\()p FF(k)q FG(\))g(:=)g FF(\031)1333 781 y FA(i)1346 775 y FG(\()p FF(r)q FG(\()p FF(k)q FG(\)\).)34 b(Hence)20 b(e.g.)257 825 y(\()p FF(M)5 b(N)g FG(\))372 810 y Fz(1)419 825 y FG(=)12 b FF(M)508 810 y Fz(1)543 825 y FF(N)581 810 y Fz(1)616 825 y FG(.)320 875 y(W)m(e)i(let)g FF(k)f(>)g Fw(FV)p FG(\()p FF(M)5 b FG(\))14 b(mean)f(that)h FF(k)i FG(is)e(greater)h(than)f(all)f FF(i)i FG(suc)o(h)g(that)f FF(x)1489 855 y FA(\032)1489 887 y(i)1521 875 y FB(2)e Fw(FV)o FG(\()p FF(M)5 b FG(\))257 925 y(for)14 b(some)f(t)o(yp)q(e)h FF(\032)p FG(.)257 1006 y Fn(Lemma)i(9.)70 b Ft(a.)20 b(If)15 b FF(M)h FG(=)720 1012 y FA(\013)756 1006 y FF(N)5 b Ft(,)14 b(then)h FF(M)958 991 y Fz(1)1005 1006 y FG(=)d FF(N)1087 991 y Fz(1)1122 1006 y Ft(.)309 1088 y(b.)20 b(If)15 b FF(k)d(>)g Fw(FV)p FG(\()p FF(M)5 b FG(\))p Ft(,)15 b(then)g FF(M)777 1073 y Fz(1)812 1088 y FG(\()p FF(k)q FG(\))d(=)911 1094 y FA(\013)946 1088 y FF(M)5 b Ft(.)257 1169 y(Pr)n(o)n(of.)20 b FG(a.)e(Induction)c(on)g(the)g (heigh)o(t)g Fw(ht)p FG(\()p FF(M)5 b FG(\))14 b(of)g FF(M)5 b FG(.)17 b(Only)d(the)g(case)h(where)g FF(M)k FG(and)14 b FF(N)257 1219 y FG(are)h(abstractions)f(is)g(critical.)j (So)d(assume)f FF(\025y)1002 1204 y FA(\032)1022 1219 y FF(M)k FG(=)1111 1225 y FA(\013)1146 1219 y FF(\025z)1191 1204 y FA(\032)1211 1219 y FF(N)5 b FG(.)18 b(Then)c FF(M)1427 1225 y FA(y)1447 1219 y FG([)p FF(P)6 b FG(])k(=)1546 1225 y FA(\013)1582 1219 y FF(N)1615 1225 y FA(z)1634 1219 y FG([)p FF(P)c FG(])257 1269 y(for)16 b(all)f(terms)h FF(P)534 1254 y FA(\032)553 1269 y FG(.)24 b(In)17 b(particular)e FF(M)877 1275 y FA(y)898 1269 y FG([)p FF(x)934 1275 y FA(k)953 1269 y FG(])g(=)1012 1275 y FA(\013)1051 1269 y FF(N)1084 1275 y FA(z)1104 1269 y FG([)p FF(x)1140 1275 y FA(k)1160 1269 y FG(])g(for)h(arbitrary)g FF(k)g FB(2)f Fh(N)p FG(.)23 b(Hence)257 1318 y FF(M)297 1324 y FA(y)318 1318 y FG([)p FF(x)354 1324 y FA(k)373 1318 y FG(])385 1298 y Fz(1)420 1318 y FG(\()p FF(k)10 b FG(+)g(1\))h(=)h FF(N)635 1324 y FA(z)654 1318 y FG([)p FF(x)690 1324 y FA(k)710 1318 y FG(])722 1298 y Fz(1)757 1318 y FG(\()p FF(k)e FG(+)g(1\),)j(b)o(y)h(IH.)f(Therefore)289 1407 y(\()p FF(\025y)q(M)5 b FG(\))411 1390 y Fz(1)447 1407 y FG(\()p FF(k)q FG(\))12 b(=)g FF(\025x)606 1413 y FA(k)626 1407 y FG(\()p FF(M)682 1413 y FA(y)702 1407 y FG([)p FF(x)738 1413 y FA(k)758 1407 y FG(])770 1386 y Fz(1)805 1407 y FG(\()p FF(k)e FG(+)g(1\)\))h(=)h FF(\025x)1051 1413 y FA(k)1071 1407 y FG(\()p FF(N)1120 1413 y FA(z)1140 1407 y FG([)p FF(x)1176 1413 y FA(k)1196 1407 y FG(])1207 1386 y Fz(1)1242 1407 y FG(\()p FF(k)f FG(+)e(1\)\))j(=)g(\()p FF(\025z)r(N)5 b FG(\))1556 1390 y Fz(1)1591 1407 y FG(\()p FF(k)q FG(\))p FF(:)257 1496 y FG(b.)35 b(Induction)20 b(on)f Fw(ht)p FG(\()p FF(M)5 b FG(\).)35 b(W)m(e)19 b(only)g(consider)h(the)g(case)g FF(\025y)q(M)5 b FG(.)36 b(The)20 b(assumption)257 1546 y FF(k)c(>)g Fw(FV)p FG(\()p FF(\025y)q(M)5 b FG(\))17 b(implies)d FF(x)701 1552 y FA(k)741 1546 y FF(=)-26 b FB(2)15 b Fw(FV)p FG(\()p FF(\025y)q(M)5 b FG(\))17 b(and)f(hence)i FF(\025y)q(M)j FG(=)1309 1552 y FA(\013)1348 1546 y FF(\025x)1396 1552 y FA(k)1416 1546 y FG(\()p FF(M)1472 1552 y FA(y)1493 1546 y FG([)p FF(x)1529 1552 y FA(k)1548 1546 y FG(]\).)k(F)m(ur-)257 1596 y(thermore)16 b FF(k)c FG(+)f(1)k FF(>)h Fw(FV)o FG(\()p FF(M)705 1602 y FA(y)726 1596 y FG([)p FF(x)762 1602 y FA(k)781 1596 y FG(]\),)g(and)g(hence)i FF(M)1078 1602 y FA(y)1098 1596 y FG([)p FF(x)1134 1602 y FA(k)1154 1596 y FG(])1165 1575 y Fz(1)1200 1596 y FG(\()p FF(k)12 b FG(+)f(1\))k(=)1377 1602 y FA(\013)1416 1596 y FF(M)1456 1602 y FA(y)1477 1596 y FG([)p FF(x)1513 1602 y FA(k)1532 1596 y FG(],)h(b)o(y)g(IH.)257 1645 y(Therefore)401 1734 y(\()p FF(\025y)q(M)5 b FG(\))523 1717 y Fz(1)559 1734 y FG(\()p FF(k)q FG(\))12 b(=)g FF(\025x)718 1740 y FA(k)738 1734 y FG(\()p FF(M)794 1740 y FA(y)815 1734 y FG([)p FF(x)851 1740 y FA(k)870 1734 y FG(])882 1713 y Fz(1)917 1734 y FG(\()p FF(k)e FG(+)g(1\)\))h(=)1103 1740 y FA(\013)1139 1734 y FF(\025x)1187 1740 y FA(k)1207 1734 y FG(\()p FF(M)1263 1740 y FA(y)1283 1734 y FG([)p FF(x)1319 1740 y FA(k)1339 1734 y FG(]\))g(=)1410 1740 y FA(\013)1446 1734 y FF(\025y)q(M)r(:)p 1672 1798 18 2 v 1672 1822 2 24 v 1688 1822 V 1672 1824 18 2 v 320 1906 a FG(Let)k Fw(ext)p FG(\()p FF(r)q FG(\))e(:=)g FF(r)q FG(\()p FF(k)q FG(\),)h(where)i FF(k)f FG(is)g(the)g(least)g(n)o(um)o(b)q(er)f (greater)h(than)g(all)e FF(i)i FG(suc)o(h)g(that)257 1956 y(some)e(v)n(ariable)g(of)g(the)i(form)d FF(x)758 1936 y FA(\032)758 1967 y(i)791 1956 y FG(o)q(ccurs)j(\(free)g(or)f(b)q (ound\))g(in)f FF(r)q FG(\(0\).)257 2037 y Fn(Lemma)j(10.)k Fw(ext)561 2003 y Fu(\000)580 2037 y FF(M)625 2021 y Fz(1)660 2003 y Fu(\001)691 2037 y FG(=)723 2043 y FA(\013)758 2037 y FF(M)5 b Ft(.)257 2122 y(Pr)n(o)n(of.)20 b Fw(ext)440 2088 y Fu(\000)459 2122 y FF(M)504 2107 y Fz(1)539 2088 y Fu(\001)578 2122 y FG(=)f FF(M)674 2107 y Fz(1)709 2122 y FG(\()p FF(k)q FG(\))g(for)f(the)i(least)f FF(k)h(>)g(i)f FG(for)f(all)f FF(i)i FG(suc)o(h)h(that)e FF(x)1537 2102 y FA(\032)1537 2133 y(i)1575 2122 y FG(o)q(ccurs)257 2172 y(\(free)g(or)e(b)q(ound\))h(in)f FF(M)652 2156 y Fz(1)687 2172 y FG(\(0\),)h(hence)h FF(k)f(>)f Fw(FV)p FG(\()p FF(M)5 b FG(\).)26 b(No)o(w)17 b(use)g(part)g(b)f(of)g(the)i (lemma)257 2221 y(ab)q(o)o(v)o(e.)p 1672 2196 V 1672 2220 2 24 v 1688 2220 V 1672 2222 18 2 v 320 2304 a(F)m(or)d(our)i(in)o (terpretation)f(of)g(t)o(yp)q(es)h(in)e(section)i(3.2)e(w)o(e)i(will)d (also)i(ha)o(v)o(e)g(to)g(consider)257 2354 y(partial)d(term)f (families)f FF(r)6 b FG(:)13 b Fh(N)c FF(*)i FG(\003)813 2360 y FA(\032)832 2354 y FG(.)18 b(W)m(e)13 b(extend)h(application)e (of)g(term)h(families,)d FF(r)q(s)p FG(,)j(as)257 2403 y(w)o(ell)i(as)h(the)g(op)q(eration)g Fw(ext)g FG(to)f(partial)g(term)g (families)e(in)i(the)h(ob)o(vious)f(w)o(a)o(y)g(follo)o(wing)257 2453 y(the)d(principle)g(that)f(all)g(syn)o(tactic)h(op)q(erations)g (are)g(strict,)g(i.e.)f(unde\014ned)h(whenev)o(er)h(one)257 2503 y(argumen)o(t)g(is)h(unde\014ned.)953 2628 y(13)p eop %%Page: 14 14 14 13 bop 257 262 a FH(3)67 b(Normalization)24 b(b)n(y)f(ev)l(aluation) 257 361 y Fr(3.1)56 b(Domain)17 b(theoretic)g(seman)n(tics)g(of)i (simply)e(t)n(yp)r(ed)h Fg(\025)p Fr(-calculi)257 437 y FG(In)13 b(this)g(section,)h(w)o(e)f(shall)f(discuss)i(the)g(domain)c (theoretic)k(seman)o(tics)f(of)f(simply)f(t)o(yp)q(ed)257 487 y(lam)o(b)q(da)i(calculi)h(in)g(general.)20 b(Although)14 b(the)i(constructions)g(b)q(elo)o(w)e(are)h(standard)g(\(see)257 537 y(e.g.)e(the)h(b)q(o)q(oks)f(of)j FE(Lambek/Scott)f FG([11)o(])d(or)i FE(Cr)o(ole)g FG([7)o(]\),)f(w)o(e)g(discuss)i(them)d (in)h(some)257 587 y(detail)j(in)f(order)i(to)f(mak)o(e)e(the)j(pap)q (er)g(accessible)g(also)e(for)h(readers)h(not)f(familiar)d(with)257 637 y(this)18 b(sub)r(ject.)32 b(Most)18 b(constructions)i(mak)o(e)c (sense)k(in)e(an)f(arbitrary)h(cartesian)h(closed)257 686 y(category)c(\(ccc\).)k(Ho)o(w)o(ev)o(er)14 b(w)o(e)g(will)f (con\014ne)h(ourselv)o(es)h(to)e(the)i(domain)c(seman)o(tics)j(and)257 736 y(will)f(only)g(o)q(ccasionally)g(commen)o(t)e(on)j(the)h (categorical)e(asp)q(ects.)320 786 y(It)c(is)h(w)o(ell-kno)o(wn)e(that) i FE(Scott)p FG(-domai)o(ns)d(with)j(con)o(tin)o(uous)f(functions)h (form)d(a)j(carte-)257 836 y(sian)i(closed)h(category)g FE(Dom)p FG(.)18 b(The)12 b(pro)q(duct)h FF(D)7 b FB(\002)e FF(E)15 b FG(is)c(the)i(set-theoretic)h(pro)q(duct)f(with)257 886 y(comp)q(onen)o(t-wise)18 b(ordering.)32 b(The)19 b(exp)q(onen)o(tial)f([)p FF(D)i FB(!)e FF(E)r FG(])g(is)g(the)h(con)o (tin)o(uous)f(func-)257 935 y(tion)c(space)h(with)f(p)q(oin)o(t)o(wise) g(ordering.)20 b(The)14 b(terminal)f(ob)r(ject)i(is)f(the)h(one)g(p)q (oin)o(t)e(space)257 985 y Fn(1)20 b FG(:=)f FB(f?g)e FG(\(there)j(is)f(no)f(initial)f(ob)r(ject)i(and)g(there)h(are)f(no)f (copro)q(ducts\).)34 b(In)19 b(order)257 1035 y(to)d(cop)q(e)h(with)f (the)h(categorical)e(in)o(terpretation,)i(w)o(e)f(will)f(iden)o(tify)g (an)h(elemen)o(t)g FF(x)f FG(of)h(a)257 1085 y(domain)c FF(D)j FG(with)f(the)g(mapping)e(from)g Fn(1)i FG(to)g FF(D)h FG(with)f(v)n(alue)f FF(x)p FG(.)320 1135 y(Besides)19 b(the)g(cartesian)g(closedness,)h(w)o(e)f(also)e(use)i(the)g(fact)f (that)g FE(Dom)h FG(is)f(closed)257 1185 y(under)i(in\014nite)g(pro)q (ducts)g(and)f(that)g(there)i(is)e(a)g(\014xed)h(p)q(oin)o(t)f(op)q (erator)g FE(Fix)5 b FG(:)15 b(\()p FF(D)22 b FB(!)257 1234 y FF(D)q FG(\))13 b FB(!)e FF(D)16 b FG(assigning)d(to)h(ev)o(ery) h(con)o(tin)o(uous)f(function)g FF(f)9 b FG(:)14 b FF(D)f FB(!)e FF(D)16 b FG(its)e(least)g(\014xed)h(p)q(oin)o(t)257 1284 y FE(Fix)p FG(\()p FF(f)t FG(\))21 b FB(2)f FF(D)q FG(.)33 b(F)m(urthermore)19 b(w)o(e)g(will)f(use)h(that)g(partial)f (families)f(of)h(terms)h(form)e(a)257 1334 y(domain)7 b(and)j(some)e(basic)h(op)q(erations)h(on)f(terms)g(and)g(term)g (families)e(are)i(con)o(tin)o(uous)h(and)257 1384 y(hence)16 b(exist)f(as)g(morphisms)d(in)i(the)h(category)m(.)20 b(An)o(y)14 b(other)h(ccc)h(with)e(these)i(prop)q(erties)257 1434 y(w)o(ould)d(do)h(as)g(w)o(ell.)257 1499 y Ft(Notation.)21 b FG(Elemen)o(ts)12 b(of)g(a)g(pro)q(duct)h(domain)d FF(D)1040 1505 y Fy(1)1065 1499 y FB(\002)c(\001)h(\001)g(\001)e(\002)h FF(D)1230 1505 y FA(n)1266 1499 y FG(are)13 b(written)g([)p FF(a)1512 1505 y Fy(1)1530 1499 y FF(;)7 b(:)g(:)g(:)t(;)g(a)1644 1505 y FA(n)1666 1499 y FG(].)257 1549 y(If)14 b FF(f)j FB(2)12 b FG([)p FF(D)422 1555 y Fy(1)452 1549 y FB(!)g FG([)p FF(D)552 1555 y Fy(2)582 1549 y FB(!)g FF(:)7 b(:)g(:)e FG([)p FF(D)737 1555 y FA(n)772 1549 y FB(!)11 b FF(E)r FG(])c FF(:)g(:)g(:)e FG(]])13 b(and)h FF(a)1072 1555 y FA(i)1098 1549 y FB(2)e FF(D)1172 1555 y FA(i)1186 1549 y FG(,)i(then)g FF(f)t FG(\()p FF(a)1368 1555 y Fy(1)1388 1549 y FF(;)7 b(:)g(:)g(:)e(;)i(a)1503 1555 y FA(n)1525 1549 y FG(\))14 b(or)g FF(f)t FG(\()p Fx(a)q FG(\))257 1599 y(stands)h(for)f FF(f)t FG(\()p FF(a)513 1605 y Fy(1)532 1599 y FG(\))7 b FF(:)g(:)g(:)e FG(\()p FF(a)648 1605 y FA(n)671 1599 y FG(\).)320 1664 y(An)k Ft(interpr)n(etation)j FG(for)d(a)h(giv)o(en)f(system)g(of)g(ground)h (t)o(yp)q(es)g(is)g(a)f(mapping)e FB(I)12 b FG(assigning)257 1714 y(to)i(ev)o(ery)h(ground)g(t)o(yp)q(e)f FF(\034)19 b FG(a)14 b(domain)e FB(I)s FG(\()p FF(\034)5 b FG(\).)19 b(Giv)o(en)13 b(suc)o(h)i(an)f(in)o(terpretation)h(w)o(e)f(de\014ne)257 1764 y(domains)e([)-7 b([)p FF(\032)p FG(])g(])476 1749 y Fz(I)512 1764 y FG(for)13 b(ev)o(ery)i(t)o(yp)q(e)f FF(\032)h FG(b)o(y)343 1852 y([)-7 b([)p FF(\034)5 b FG(])-7 b(])400 1835 y Fz(I)432 1852 y FG(:=)12 b FB(I)s FG(\()p FF(\034)5 b FG(\))p FF(;)47 b FG([)-7 b([)p FF(\032)11 b FB(!)g FF(\033)q FG(])-7 b(])772 1835 y Fz(I)806 1852 y FG(:=)11 b([[)-7 b([)p FF(\032)p FG(])g(])928 1835 y Fz(I)961 1852 y FB(!)11 b FG([)-7 b([)p FF(\033)q FG(])g(])1073 1835 y Fz(I)1094 1852 y FG(])p FF(;)48 b FG([)-7 b([)p FF(\032)9 b FB(\002)g FF(\033)q FG(])-7 b(])1296 1835 y Fz(I)1330 1852 y FG(:=)11 b([)-7 b([)p FF(\032)p FG(])g(])1440 1835 y Fz(I)1471 1852 y FB(\002)9 b FG([)-7 b([)p FF(\033)q FG(])g(])1571 1835 y Fz(I)1593 1852 y FF(:)257 1941 y FG(W)m(e)16 b(write)g([)-7 b([)p FF(\032)477 1947 y Fy(1)495 1941 y FF(;)7 b(:)g(:)g(:)e(;)i(\032)609 1947 y FA(n)631 1941 y FG(])-7 b(])648 1926 y Fz(I)685 1941 y FG(:=)14 b([)-7 b([)p FF(\032)781 1947 y Fy(1)809 1941 y FB(\002)11 b(\001)c(\001)g(\001)i(\002)i FF(\032)975 1947 y FA(n)998 1941 y FG(])-7 b(])1015 1926 y Fz(I)1051 1941 y FG(=)15 b([)-7 b([)p FF(\032)1136 1947 y Fy(1)1154 1941 y FG(])g(])1171 1926 y Fz(I)1203 1941 y FB(\002)11 b(\001)c(\001)g(\001)i(\002)i FG([)-7 b([)p FF(\032)1386 1947 y FA(n)1408 1941 y FG(])g(])1425 1926 y Fz(I)1461 1941 y FG(=:)15 b([)-7 b([)p Fx(\032)o FG(])g(])1578 1926 y Fz(I)1600 1941 y FG(.)23 b(An)257 1991 y Ft(interpr)n(etation)13 b(of)g(a)g(typ)n(e)n(d)g(lamb)n(da)g(c)n (alculus)j FG(\(sp)q(eci\014ed)d(b)o(y)f(a)g(set)h(of)e(ground)h(t)o (yp)q(es)h(and)257 2041 y(a)i(set)i(of)d(constan)o(ts\))j(is)e(a)g (mapping)e FB(I)18 b FG(assigning)c(to)h(ev)o(ery)i(ground)e(t)o(yp)q (e)h FF(\034)k FG(a)15 b(domain)257 2091 y FB(I)s FG(\()p FF(\034)5 b FG(\))20 b(\(hence)h FB(I)i FG(is)c(an)h(in)o(terpretation) g(of)f(ground)h(t)o(yp)q(es\),)i(and)d(assigning)g(to)h(ev)o(ery)257 2140 y(constan)o(t)15 b FF(c)443 2125 y FA(\032)476 2140 y FG(a)e(v)n(alue)h FB(I)s FG(\()p FF(c)p FG(\))d FB(2)g FG([)-7 b([)p FF(\032)p FG(])g(])799 2125 y Fz(I)835 2140 y FG(\(i.e.)13 b(a)g(morphism)e(from)i Fn(1)g FG(to)h([)-7 b([)p FF(\032)p FG(])g(])1388 2125 y Fz(I)1409 2140 y FG(\).)320 2190 y(In)15 b(order)h(to)f(extend)h(suc)o(h)g(an)g(in)o (terpretation)f(to)g(all)f(terms)i(w)o(e)f(use)h(the)g(follo)o(wing)257 2240 y(con)o(tin)o(uous)11 b(functions,)h(i.e.)e(morphisms)f(\(in)i (the)g(sequel)h(a)f(con)o(tin)o(uous)g(function)g(will)f(b)q(e)257 2290 y(called)j(morphism)d(if)i(its)h(role)g(as)g(a)f(morphism)e(in)j (the)g(ccc)h FE(Dom)g FG(is)f(to)g(b)q(e)g(emphasized\).)373 2379 y(!)385 2385 y FA(D)419 2379 y FG(:)g FF(D)g FB(!)f Fn(1)p FF(;)47 b FG(!)640 2385 y FA(D)670 2379 y FG(\()p FF(d)p FG(\))11 b(:=)h FB(?)373 2441 y FF(\031)397 2447 y FA(i)415 2441 y FG(:)h FF(D)474 2447 y Fy(1)502 2441 y FB(\002)d(\001)d(\001)g(\001)h(\002)h FF(D)677 2447 y FA(n)712 2441 y FB(!)i FF(D)799 2447 y FA(i)813 2441 y FF(;)48 b(\031)897 2447 y FA(i)910 2441 y FG(\([)p Fx(a)p FG(]\))11 b(:=)h FF(a)1081 2447 y FA(i)1095 2441 y FF(;)373 2503 y Fw(curry)6 b FG(:)14 b([)p FF(D)c FB(\002)g FF(E)j FB(!)e FF(F)6 b FG(])11 b FB(!)g FG([)p FF(D)h FB(!)f FG([)p FF(E)i FB(!)e FF(F)6 b FG(]])p FF(;)47 b Fw(curry)r FG(\()p FF(f)r(;)7 b(a;)g(b)p FG(\))k(:=)g FF(f)t FG(\([)p FF(a;)c(b)p FG(]\))p FF(;)953 2628 y FG(14)p eop %%Page: 15 15 15 14 bop 373 262 a Fw(eval)5 b FG(:)13 b([)p FF(D)g FB(!)e FF(E)r FG(])e FB(\002)g FF(D)k FB(!)e FF(E)r(;)48 b Fw(eval)q FG(\([)p FF(f)r(;)7 b(a)p FG(]\))j(:=)i FF(f)t FG(\()p FF(a)p FG(\))p FF(:)257 353 y FG(F)m(urthermore)f(w)o(e)g(use)g (the)h(fact)e(that)h(morphisms)d(are)k(closed)f(under)g(comp)q(osition) e FB(\016)i FG(and)257 403 y(\(since)i FE(Dom)f FG(is)f(a)g(ccc\))i (under)f(pairing)f FB(h)p FF(:;)c(:)p FB(i)p FG(,)j(where)i(for)f FF(f)f FG(:)j FF(D)g FB(!)e FF(E)i FG(and)f FF(g)6 b FG(:)13 b FF(D)g FB(!)e FF(F)17 b FG(the)257 452 y(function)d FB(h)p FF(f)r(;)7 b(g)q FB(i)e FG(:)14 b FF(D)f FB(!)f FF(E)f FB(\002)f FF(F)19 b FG(maps)13 b FF(a)h FG(to)g([)p FF(f)t FG(\()p FF(a)p FG(\))p FF(;)7 b(g)q FG(\()p FF(a)p FG(\)].)19 b(F)m(or)14 b(ev)o(ery)h(t)o(yp)q(e)f FF(\032)h FG(and)f(ev)o(ery)257 502 y(list)g(of)f(distinct)h(v)n(ariables)f Fx(x)723 487 y Fq(\032)756 502 y FG(=)f FF(x)824 482 y FA(\032)841 486 y Fo(1)824 513 y Fy(1)859 502 y FF(;)7 b(:)g(:)g(:)e(;)i(x)976 487 y FA(\032)993 491 y Fi(n)976 513 y FA(n)1029 502 y FG(w)o(e)14 b(let)g(\003)1179 508 y FA(\032)1198 502 y FG(\()p Fx(x)p FG(\))g(denote)h(the)f(set)h(of)e (terms)257 552 y(of)h(t)o(yp)q(e)h FF(\032)g FG(with)f(free)i(v)n (ariables)d(among)g FB(f)p Fx(x)p FB(g)p FG(.)19 b(Let)c FB(I)i FG(b)q(e)e(an)g(in)o(terpretation.)20 b(Then)15 b(for)257 602 y(ev)o(ery)g FF(M)h FB(2)c FG(\003)492 608 y FA(\032)511 602 y FG(\()p Fx(x)554 587 y Fq(\032)576 602 y FG(\))i(w)o(e)g(de\014ne)h(a)f(morphism)d([)-7 b([)p FF(M)5 b FG(])-7 b(])1095 587 y Fz(I)1095 612 y Fq(x)1121 602 y FG(:)13 b([)-7 b([)p Fx(\032)p FG(])g(])10 b FB(!)i FG([)-7 b([)p FF(\032)p FG(])g(])12 b(b)o(y)779 693 y([)-7 b([)p FF(c)p FG(])g(])831 676 y Fz(I)831 703 y Fq(x)865 693 y FG(:=)11 b FB(I)s FG(\()p FF(c)p FG(\))p FB(\016)p FG(!)1029 700 y Fy([)-6 b([)p Fq(\032)p Fy(])g(])1076 693 y FF(;)760 762 y FG([)f([)p FF(x)801 768 y FA(i)813 762 y FG(])g(])830 745 y Fz(I)830 772 y Fq(x)865 762 y FG(:=)11 b FF(\031)944 768 y FA(i)957 762 y FF(;)704 830 y FG([)-7 b([)p FF(\025xM)5 b FG(])-7 b(])831 813 y Fz(I)831 841 y Fq(x)865 830 y FG(:=)11 b Fw(curry)r FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])1103 813 y Fz(I)1103 841 y Fq(x)o FA(;x)1153 830 y FG(\))p FF(;)715 900 y FG([)g([)p FF(M)5 b(N)g FG(])-7 b(])832 883 y Fz(I)832 910 y Fq(x)865 900 y FG(:=)11 b Fw(eval)f FB(\016)f FG([[)-7 b([)p FF(M)5 b FG(])-7 b(])1118 883 y Fz(I)1118 910 y Fq(x)1139 900 y FF(;)7 b FG([)-7 b([)p FF(N)5 b FG(])-7 b(])1230 883 y Fz(I)1230 910 y Fq(x)1251 900 y FG(])p FF(;)666 968 y FG([)g([)p FB(h)p FF(M)r(;)7 b(N)e FB(i)p FG(])-7 b(])831 951 y Fz(I)831 979 y Fq(x)865 968 y FG(:=)11 b([[)-7 b([)p FF(M)5 b FG(])-7 b(])1011 951 y Fz(I)1011 979 y Fq(x)1032 968 y FF(;)7 b FG([)-7 b([)p FF(N)5 b FG(])-7 b(])1123 951 y Fz(I)1123 979 y Fq(x)1144 968 y FG(])p FF(;)683 1037 y FG([)g([)p FF(\031)724 1043 y FA(i)736 1037 y FG(\()p FF(M)5 b FG(\)])-7 b(])830 1020 y Fz(I)830 1047 y Fq(x)865 1037 y FG(:=)11 b FF(\031)944 1043 y FA(i)967 1037 y FB(\016)e FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1076 1020 y Fz(I)1076 1047 y Fq(x)1097 1037 y FF(:)257 1128 y FG(This)16 b(de\014nition)g(w)o(orks)g(in)f(an)o (y)h(ccc.)25 b(F)m(or)15 b(our)h(purp)q(oses)i(it)d(will)g(b)q(e)h (more)f(con)o(v)o(enien)o(t)257 1178 y(to)f(ev)n(aluate)g(a)f(term)h (in)f(a)h(global)e(en)o(vironmen)o(t)h(and)g(not)h(in)g(a)f(lo)q(cal)g (con)o(text.)19 b(Let)724 1275 y FE(Env)12 b FG(:=)904 1236 y Fu(Y)872 1325 y FA(x)891 1317 y Fi(\033)911 1325 y Fz(2)p Ff(V)m(ar)989 1275 y FG([)-7 b([)p FF(\033)q FG(])g(])1048 1258 y Fz(I)1081 1275 y FB(2)11 b FE(Dom)q FF(:)257 1408 y FG(F)m(or)j(ev)o(ery)h(term)e FF(M)j FB(2)11 b FG(\003)665 1414 y FA(\032)684 1408 y FG(\()p FF(x)724 1414 y Fy(1)743 1408 y FF(;)c(:)g(:)g(:)e(;)i(x)860 1414 y FA(n)882 1408 y FG(\))13 b(w)o(e)i(de\014ne)g(a)e(con)o(tin)o (uous)h(function)457 1499 y([)-7 b([)p FF(M)5 b FG(])-7 b(])536 1482 y Fz(I)562 1499 y FG(:)13 b FE(Env)f FB(!)f FG([)-7 b([)p FF(\032)p FG(])g(])788 1482 y Fz(I)810 1499 y FF(;)48 b FG([)-7 b([)p FF(M)5 b FG(])-7 b(])949 1482 y Fz(I)949 1510 y FA(\030)981 1499 y FG(:=)12 b([)-7 b([)p FF(M)5 b FG(])-7 b(])1116 1482 y Fz(I)1116 1510 y Fq(x)1137 1499 y FG(\([)p FF(\030)r FG(\()p FF(x)1225 1505 y Fy(1)1243 1499 y FG(\))p FF(;)7 b(:)g(:)g(:)e(;)i(\030)r FG(\()p FF(x)1412 1505 y FA(n)1434 1499 y FG(\)]\))p FF(:)257 1591 y FG(F)m(ormally)14 b(this)i(de\014nition)g(dep)q(ends)i (on)e(a)h(particular)f(c)o(hoice)h(of)e(the)i(list)f(of)g(v)n(ariables) 257 1641 y FF(x)281 1647 y Fy(1)300 1641 y FF(;)7 b(:)g(:)g(:)e(;)i(x) 417 1647 y FA(n)438 1641 y FG(.)20 b(Ho)o(w)o(ev)o(er,)15 b(b)q(ecause)h(of)e(the)h(w)o(ell-kno)o(wn)e(coincidence)j(prop)q(ert)o (y)f(in)f(fact)h(it)257 1690 y(do)q(es)g(not.)320 1740 y(F)m(rom)d(this)i(w)o(e)g(easily)f(get)h(the)h(famili)o(ar)c (equations)847 1832 y([)-7 b([)p FF(c)p FG(])g(])899 1814 y Fz(I)899 1842 y FA(\030)932 1832 y FG(=)12 b FB(I)s FG(\()p FF(c)p FG(\))p FF(;)842 1901 y FG([)-7 b([)p FF(x)p FG(])g(])900 1884 y Fz(I)900 1911 y FA(\030)932 1901 y FG(=)12 b FF(\030)r FG(\()p FF(x)p FG(\))p FF(;)718 1971 y FG([)-7 b([)p FF(\025xM)5 b FG(])-7 b(])845 1953 y Fz(I)845 1981 y FA(\030)866 1971 y FG(\()p FF(a)p FG(\))12 b(=)g([)-7 b([)p FF(M)5 b FG(])-7 b(])1055 1953 y Fz(I)1055 1982 y FA(\030)q Fy([)p FA(x)p Fz(7!)p FA(a)p Fy(])1160 1971 y FF(;)783 2043 y FG([)g([)p FF(M)5 b(N)g FG(])-7 b(])900 2026 y Fz(I)900 2053 y FA(\030)932 2043 y FG(=)12 b([)-7 b([)p FF(M)5 b FG(])-7 b(])1055 2026 y Fz(I)1055 2053 y FA(\030)1076 2009 y Fu(\000)1095 2043 y FG([)g([)p FF(N)5 b FG(])-7 b(])1167 2026 y Fz(I)1167 2053 y FA(\030)1188 2009 y Fu(\001)1207 2043 y FF(;)734 2112 y FG([)g([)p FB(h)p FF(M)r(;)7 b(N)e FB(i)p FG(])-7 b(])899 2095 y Fz(I)899 2123 y FA(\030)932 2112 y FG(=)12 b([[)-7 b([)p FF(M)5 b FG(])-7 b(])1067 2095 y Fz(I)1067 2123 y FA(\030)1087 2112 y FF(;)7 b FG([)-7 b([)p FF(N)5 b FG(])-7 b(])1178 2095 y Fz(I)1178 2123 y FA(\030)1198 2112 y FG(])p FF(;)751 2182 y FG([)g([)p FF(\031)792 2188 y FA(i)804 2182 y FG(\()p FF(M)5 b FG(\)])-7 b(])898 2165 y Fz(I)898 2192 y FA(\030)932 2182 y FG(=)12 b FF(\031)1000 2188 y FA(i)1013 2182 y FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])1108 2165 y Fz(I)1108 2192 y FA(\030)1129 2182 y FG(\))p FF(:)257 2273 y FG(In)20 b(man)o(y)e(cases)j(the)f(in)o(terpretation)g FB(I)i FG(of)d(the)h(constan)o(ts)h(will)d(ha)o(v)o(e)i(to)f(b)q(e)h (de\014ned)257 2323 y(recursiv)o(ely)m(,)j(b)o(y)d(e.g.)g(referring)h (to)f([)-7 b([)p FF(M)5 b FG(])-7 b(])938 2308 y Fz(I)980 2323 y FG(for)20 b(sev)o(eral)h(terms)f FF(M)5 b FG(.)38 b(This)21 b(causes)h(no)257 2373 y(problem,)16 b(since)h(the)g (functionals)f([)-7 b([)p FF(M)888 2358 y FA(\032)906 2373 y FG(])g(])923 2358 y Fz(I)950 2373 y FG(:)14 b FE(Env)j FB(!)e FG([)-7 b([)p FF(\032)p FG(])g(])15 b(dep)q(end)j(con)o (tin)o(uously)e(on)g FB(I)s FG(,)257 2423 y(where)g FB(I)i FG(is)c(to)h(b)q(e)g(considered)h(as)f(an)f(elemen)o(t)h(of)f(the)h (in\014nite)g(pro)q(duct)g(\005)1488 2429 y FA(c)1503 2420 y Fi(\032)1522 2423 y FG([)-7 b([)p FF(\032)p FG(])g(].)20 b(This)257 2472 y(can)d(b)q(e)g(seen)h(as)f(follo)o(ws.)24 b(Lo)q(oking)16 b(at)g(their)h(de\014nitions)g(w)o(e)g(see)h(that)e (the)h(functions)953 2628 y(15)p eop %%Page: 16 16 16 15 bop 257 262 a FG([)p FB(I)s FF(;)7 b Fx(a)o FG(])k FB(7!)g FG([)-7 b([)p FF(M)5 b FG(])-7 b(])494 246 y Fz(I)494 272 y Fq(x)516 262 y FG(\()p Fx(a)p FG(\))14 b(are)h(built)e(b)o(y)g(comp)q(osition)f(from)h(the)h(con)o(tin)o(uous) g(functions)583 353 y FF(\031)607 359 y FA(c)622 351 y Fi(\033)647 353 y FG(:)g(\005)704 359 y FA(c)719 351 y Fi(\032)738 353 y FG([)-7 b([)p FF(\032)p FG(])g(])10 b FB(!)h FG([)-7 b([)p FF(\033)q FG(])g(])p FF(;)47 b(\031)998 359 y FA(c)1013 351 y Fi(\033)1034 353 y FG(\()p FB(I)s FG(\))12 b(:=)f FB(I)s FG(\()p FF(c)p FG(\))p FF(;)592 415 y FB(\001)e(\016)e(\001)e FG(:)12 b([)p FF(E)h FB(!)e FF(F)6 b FG(])i FB(\002)i FG([)p FF(D)i FB(!)f FF(E)r FG(])g FB(!)g FG([)p FF(D)i FB(!)e FF(F)6 b FG(])p FF(;)583 477 y FB(h\001)p FF(;)h FB(\001i)e FG(:)11 b([)p FF(D)i FB(!)e FF(E)r FG(])e FB(\002)g FG([)p FF(D)k FB(!)e FF(F)6 b FG(])k FB(!)h FG([)p FF(D)i FB(!)e FF(E)g FB(\002)f FF(F)c FG(])p FF(;)257 569 y FG(as)21 b(w)o(ell)f(as)h(the)h(functions) e(!)740 575 y FA(D)770 569 y FF(;)7 b(\031)813 575 y FA(i)826 569 y FF(;)g Fw(curry)22 b FG(and)f Fw(eval)g FG(listed)g(ab)q(o)o(v)o(e.)38 b(Hence)23 b([)p FB(I)s FF(;)7 b Fx(a)n FG(])23 b FB(7!)257 619 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])336 603 y Fz(I)336 629 y Fq(x)358 619 y FG(\()p Fx(a)p FG(\))10 b(is)g(con)o(tin)o(uous.)16 b(But)10 b(then)h(also)e([)p FB(I)s FF(;)e(\030)r FG(])i FB(7!)j FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1165 603 y Fz(I)1165 630 y FA(\030)1195 619 y FG(is)9 b(con)o(tin)o(uous,)h(since) h([)-7 b([)p FF(M)5 b FG(])-7 b(])1625 603 y Fz(I)1625 630 y FA(\030)1657 619 y FG(=)257 675 y([)g([)p FF(M)5 b FG(])-7 b(])336 660 y Fz(I)336 685 y Fq(x)358 675 y FG(\([)p FF(\031)410 681 y FA(x)429 685 y Fo(1)446 675 y FG(\()p FF(\030)r FG(\))p FF(;)7 b(:)g(:)g(:)f(;)h(\031)616 681 y FA(x)635 685 y Fi(n)656 675 y FG(\()p FF(\030)r FG(\)]\),)13 b(where)i FF(\031)905 681 y FA(x)924 673 y Fi(\032)948 675 y FG(:)e FE(Env)f FB(!)f FG([)-7 b([)p FF(\032)p FG(])g(],)12 b FF(\031)1222 681 y FA(x)1243 675 y FG(\()p FF(\030)r FG(\))g(:=)f FF(\030)r FG(\()p FF(x)p FG(\).)320 725 y(Hence)17 b(the)f(v)n(alue)e FB(I)s FG(\()p FF(c)p FG(\))i(ma)o(y)d(b)q(e)j(de\014ned)h(as)e(a)g(least)h (\014xed)g(p)q(oin)o(t)f(of)f(a)i(con)o(tin)o(uous)257 775 y(function)f(on)f(the)i(domain)c(\005)731 781 y FA(c)746 773 y Fi(\032)766 775 y FG([)-7 b([)p FF(\032)p FG(])g(].)19 b({)c(In)f(the)i(sequel)f(w)o(e)g(will)f(omit)e(the)k(sup)q(erscript)h FB(I)257 824 y FG(when)e(it)e(is)h(clear)g(from)e(the)j(con)o(text.)320 874 y(The)f(follo)o(wing)d(facts)j(hold)g(in)f(an)o(y)h(ccc.)257 957 y Fn(Lemma)i(11.)416 1049 y FG([)-7 b([)p FF(M)473 1055 y Fq(x)496 1049 y FG([)p Fx(N)5 b FG(]])-7 b(])581 1055 y FA(\030)609 1049 y FG(=)12 b([)-7 b([)p FF(M)5 b FG(])-7 b(])732 1056 y FA(\030)q Fy([)p Fq(x)o Fz(7!)p Fy([)h([)p Fq(N)t Fy(])g(])869 1060 y Fi(\030)887 1056 y Fy(])1261 1049 y Ft(\(substitution)15 b(lemma\))385 1116 y FG([)-7 b([\()p FF(\025xM)5 b FG(\))p FF(N)g FG(])-7 b(])582 1122 y FA(\030)609 1116 y FG(=)12 b([)-7 b([)p FF(M)710 1122 y FA(x)731 1116 y FG([)p FF(N)5 b FG(])o(])-7 b(])809 1122 y FA(\030)1261 1116 y Ft(\(b)n(eta)15 b(1\))309 1178 y FG([)-7 b([)p FF(\031)350 1184 y FA(i)363 1178 y FG(\()p FB(h)p FF(M)435 1184 y Fy(0)454 1178 y FF(;)7 b(M)513 1184 y Fy(1)531 1178 y FB(i)p FG(\)])-7 b(])580 1184 y FA(\030)609 1178 y FG(=)12 b([)-7 b([)p FF(M)710 1184 y FA(i)724 1178 y FG(])g(])741 1184 y FA(\030)1261 1178 y Ft(\(b)n(eta)15 b(2\))503 1241 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])582 1247 y FA(\030)609 1241 y FG(=)12 b([)-7 b([)p FF(\025y)q FG(\()p FF(M)5 b(y)q FG(\)])-7 b(])830 1247 y FA(\030)891 1241 y Ft(\()p FF(y)929 1223 y FA(\032)965 1241 y FF(=)-25 b FB(2)11 b Fw(FV)p FG(\()p FF(M)1113 1223 y FA(\032)p Fz(!)p FA(\033)1186 1241 y FG(\))p Ft(\))42 b(\(eta)15 b(1\))503 1303 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])582 1309 y FA(\030)609 1303 y FG(=)12 b([)-7 b([)p FB(h)p FF(\031)710 1309 y Fy(0)728 1303 y FG(\()p FF(M)5 b FG(\))p FF(;)i(\031)848 1309 y Fy(1)866 1303 y FG(\()p FF(M)e FG(\))p FB(i)p FG(])-7 b(])976 1309 y FA(\030)1261 1303 y Ft(\(eta)15 b(2\))257 1394 y Fn(Lemma)h(12.)k Ft(If)h FG([)-7 b([)p FF(P)6 b FG(])-7 b(])625 1400 y FA(\030)662 1394 y FG(=)22 b([)-7 b([)p FF(Q)p FG(])g(])783 1400 y FA(\030)820 1394 y Ft(for)20 b(al)r(l)g(envir)n(onments)h FF(\030)r Ft(,)g(and)h FF(M)j Ft(is)20 b(tr)n(ansforme)n(d)257 1444 y(into)c FF(N)21 b Ft(by)c(r)n(eplacing)e(an)i(o)n(c)n(curr)n(enc)n(e)f(of)g FF(P)21 b Ft(in)16 b FF(M)21 b Ft(by)16 b FF(Q)p Ft(,)g(then)g FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1397 1450 y FA(\030)1427 1444 y FG(=)14 b([)-7 b([)p FF(N)5 b FG(])-7 b(])1545 1450 y FA(\030)1577 1444 y Ft(for)16 b(al)r(l)257 1494 y(envir)n(onments)g FF(\030)r Ft(.)257 1577 y(Pr)n(o)n(of.)k FG(Induction)14 b(on)g FF(M)5 b FG(.)p 1672 1552 18 2 v 1672 1575 2 24 v 1688 1575 V 1672 1577 18 2 v 257 1660 a Fn(Lemma)16 b(13.)k Ft(If)d FF(M)23 b Ft(r)n(e)n(duc)n(es)17 b(to)g FF(N)23 b Ft(by)17 b FF(\014)r Ft(-r)n(e)n(duction)h(or)f FF(\021)q Ft(-exp)n(ansion,)i(then)f FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1625 1666 y FA(\030)1657 1660 y FG(=)257 1710 y([)g([)p FF(N)5 b FG(])-7 b(])329 1716 y FA(\030)346 1710 y Ft(.)p 1672 1685 V 1672 1708 2 24 v 1688 1708 V 1672 1710 18 2 v 257 1826 a Fr(3.2)56 b(In)n(terpretation)18 b(of)g(the)h(t)n(yp)r(es)257 1902 y FG(W)m(e)14 b(no)o(w)g(consider)g (a)g(sp)q(ecial)g(mo)q(del,)e(whose)j(ground)f(t)o(yp)q(e)g(ob)r(jects) h(con)o(tain)f(syn)o(tactic)257 1952 y(material.)27 b(W)m(e)17 b(let)h Fh(N)e FF(*)h FG(\003)723 1958 y FA(\032)759 1952 y FG(denote)i(the)f(set)g(of)f(partial)g(term)g(famili)o(es,)f (i.e.)g(partial)257 2002 y(functions)j(from)d(the)j(in)o(tegers)g(to)f (the)g(set)h(of)f(terms)g(of)f(t)o(yp)q(e)i FF(\032)p FG(.)31 b Fh(N)17 b FF(*)h FG(\003)1499 2008 y FA(\032)1536 2002 y FG(partially)257 2052 y(ordered)d(b)o(y)f(inclusion)f(of)g (graphs)i(is)e(a)h(domain.)i(W)m(e)d(will)f(in)o(terpret)j(the)g (ground)f(t)o(yp)q(es)257 2102 y(in)g(suc)o(h)g(a)g(w)o(a)o(y)f(that)h (w)o(e)g(ha)o(v)o(e)g(functions)464 2193 y FB(#)484 2203 y FA(\034)510 2193 y FG(:)f FB(I)s FG(\()p FF(\034)5 b FG(\))12 b FB(!)f FG(\()p Fh(N)e FF(*)i FG(\003)820 2199 y FA(\034)841 2193 y FG(\))83 b(and)g FB(")1111 2203 y FA(\034)1136 2193 y FG(:)13 b(\()p Fh(N)d FF(*)h FG(\003)1301 2199 y FA(\034)1322 2193 y FG(\))g FB(!)g(I)s FG(\()p FF(\034)5 b FG(\))257 2284 y(satisfying)847 2376 y FB(#)868 2386 y FA(\034)889 2376 y FG(\()p FB(")926 2386 y FA(\034)947 2376 y FG(\()p FF(r)q FG(\)\))11 b(=)h FF(r)o(:)536 b FG(\(1\))257 2467 y(This)14 b(sho)o(ws)g(that)g(there)i (is)d(an)h(em)o(b)q(edding)f(of)g(the)i(term)e(famili)o(es)f Fh(N)e FF(*)h FG(\003)1470 2473 y FA(\034)1504 2467 y FG(in)o(to)i FB(I)s FG(\()p FF(\034)5 b FG(\).)953 2628 y(16)p eop %%Page: 17 17 17 16 bop 320 262 a FG(Recall)15 b(that)h FE(Constr)g FG(is)g(the)h(set)g(of)e(all)g(constructors)j(used)f(in)f(the)g (computation)257 311 y(rules.)39 b(W)m(e)20 b(de\014ne)576 296 y Fy(2)617 311 y FG(the)h(in)o(terpretation)g(of)f(the)h(ground)f (t)o(yp)q(es)i(in)e(a)g(w)o(a)o(y)g(that)h(all)257 361 y(syn)o(tactic)15 b(constructors)h FF(c)11 b FB(2)g FE(Constr)j FG(ha)o(v)o(e)g(seman)o(tical)e(coun)o(terparts.)528 458 y FB(I)s FG(\()p FF(\034)5 b FG(\))11 b(=)664 419 y Fu(X)724 458 y FG(\()c([)-7 b([)p Fx(\032)p FG(])g(])806 441 y Fz(I)839 458 y FB(j)11 b FF(c)880 441 y Fq(\032)p Fz(!)p FA(\034)965 458 y FB(2)g FE(Constr)c FG(\))j(+)f(\()p Fh(N)h FF(*)h FG(\003)1371 464 y FA(\034)1391 458 y FG(\))p FF(:)257 530 y Fu(P)315 561 y FG(and)j(+)g(denote)h(the)f (domain-theoretic)f(separated)1140 546 y Fy(3)1174 561 y FG(sum)f(and)387 652 y([)-7 b([)p FB(;)p FG(])g(])10 b(:=)h Fn(1)h FG(:=)f FB(f?g)p FF(;)47 b FG([)-7 b([)p Fx(\032)o FF(\033)q FG(])g(])11 b(:=)h([)-7 b([)p Fx(\032)o FG(])g(])9 b FB(\002)g FG([)-7 b([)p FF(\033)q FG(])g(])p FF(;)46 b FG([)-7 b([)p Fx(\032)p FF(i)p FG(])g(])11 b(:=)g([)-7 b([)p Fx(\032)p FG(])g(])13 b(for)g FF(i)f FB(2)f(f)p FG(0)p FF(;)c FG(1)p FB(g)p FF(:)257 743 y FG(So)19 b(for)f(a)h(giv)o(en)f(ground)h(t)o(yp)q(e)g FF(\034)5 b FG(,)19 b(its)g(constructors)i FF(c)e FG(together)g(with)g (term)f(families)257 793 y Fh(N)10 b FF(*)h FG(\003)381 799 y FA(\034)415 793 y FG(freely)j(generate)i(the)e(in)o(terpretation) g(of)g FF(\034)5 b FG(,)13 b(i.e.)g(there)i(are)f(injections)545 885 y Fw(in)577 867 y Fq(\032)577 895 y FA(c)603 885 y FG(:)g([)-7 b([)p Fx(\032)o FG(])g(])11 b FB(!)g FG([)-7 b([)p FF(\034)5 b FG(])-7 b(])158 b(for)14 b(ev)o(ery)g FF(c)1157 867 y Fq(\032)p Fz(!)p FA(\034)1242 885 y FB(2)e FE(Constr)512 947 y Fw(fam)578 953 y FA(\034)603 947 y FG(:)i(\()p Fh(N)9 b FF(*)i FG(\003)768 953 y FA(\034)789 947 y FG(\))g FB(!)h FG([)-7 b([)p FF(\034)5 b FG(])-7 b(])257 1038 y(suc)o(h)13 b(that)e(ev)o(ery)i FF(a)e FB(2)h FG([)-7 b([)p FF(\034)5 b FG(])-7 b(])9 b(is)j(either)g FB(?)p FG(,)f(or)h(else)g(can)g(b)q(e)g(written)g(uniquely)f(as)h FF(a)g FG(=)f Fw(in)1614 1020 y Fq(\032)1614 1048 y FA(c)1636 1038 y FG(\()p Fx(b)o FG(\))257 1088 y(or)j FF(a)e FG(=)g Fw(fam)451 1094 y FA(\034)472 1088 y FG(\()p FF(r)q FG(\).)18 b(F)m(or)13 b(example)g(\(cf.)h(section)g(2.3\))f(w)o(e)h(ha)o(v)o(e) 558 1179 y FB(I)s FG(\()p FF(\023)p FG(\))d(=)h Fn(1)d FG(+)h FB(I)s FG(\()p FF(\023)p FG(\))f(+)g(\()p Fh(N)h FF(*)h FG(\003)1024 1185 y FA(\023)1038 1179 y FG(\))p FF(;)539 1242 y FB(I)s FG(\()p FB(O)q FG(\))g(=)h Fn(1)d FG(+)h([)p FB(I)s FG(\()p FF(\023)p FG(\))g FB(!)h(I)s FG(\()p FB(O)q FG(\)])e(+)h(\()p Fh(N)g FF(*)h FG(\003)1204 1248 y Fz(O)1233 1242 y FG(\))p FF(;)535 1317 y FB(I)s FG(\()p Fw(ex)q FG(\))g(=)686 1277 y Fu(X)746 1317 y FG(\()c([)-7 b([)p FF(\032)p FG(])g(])8 b FB(\002)i FG([)-7 b([)p FF(\033)q FG(])g(])10 b FB(j)h FF(\032;)c(\033)15 b FG(t)o(yp)q(es)8 b(\))h(+)g(\()p Fh(N)h FF(*)h FG(\003)1354 1323 y Fp(ex)1384 1317 y FG(\))p FF(:)257 1433 y Fr(3.3)56 b(Rei\014cation)17 b(and)j(re\015ection)257 1509 y FG(The)15 b(con)o(tin)o(uous)e(functions)357 1601 y FB(#)378 1611 y FA(\032)402 1601 y FG(:)h([)-7 b([)p FF(\032)p FG(])g(])10 b FB(!)h FG(\()p Fh(N)e FF(*)j FG(\003)686 1607 y FA(\032)705 1601 y FG(\))41 b(\(\\reify"\))83 b FB(")1021 1611 y FA(\032)1045 1601 y FG(:)13 b(\()p Fh(N)d FF(*)h FG(\003)1210 1607 y FA(\032)1229 1601 y FG(\))g FB(!)h FG([)-7 b([)p FF(\032)p FG(])g(])40 b(\(\\re\015ect"\))257 1692 y(are)15 b(de\014ned)g(sim)o(ultaneously)c(b)o(y)j(recursion)972 1677 y Fy(4)992 1692 y FG(.)331 1783 y FB(#)351 1794 y FA(\034)372 1783 y FG(\()p Fw(in)420 1789 y FA(c)437 1783 y FG(\()p Fx(b)o FG(\)\))e(:=)g FF(c)592 1766 y Fz(1)627 1783 y FB(#)o FG(\()p Fx(b)p FG(\))p FF(;)497 b FB(")1231 1794 y FA(\034)1251 1783 y FG(\()p FF(r)q FG(\))12 b(:=)f Fw(fam)1436 1789 y FA(\034)1457 1783 y FG(\()p FF(r)q FG(\))p FF(;)294 1846 y FB(#)315 1856 y FA(\034)336 1846 y FG(\()p Fw(fam)418 1852 y FA(\034)438 1846 y FG(\()p FF(r)q FG(\)\))h(:=)g FF(r)o(;)401 1908 y FB(#)421 1918 y FA(\034)442 1908 y FG(\()p FB(?)p FG(\))f(:=)h FB(?)p FF(;)304 1975 y FB(#)325 1985 y FA(\032)p Fz(!)p FA(\033)397 1975 y FG(\()p FF(a)p FG(\)\()p FF(k)q FG(\))g(:=)g FF(\025x)622 1955 y FA(\032)622 1987 y(k)642 1941 y Fu(\000)661 1975 y FB(#)682 1985 y FA(\033)704 1941 y Fu(\000)723 1975 y FF(a)p FG(\()p FB(")782 1985 y FA(\032)801 1975 y FG(\()p FF(x)841 1958 y Fz(1)841 1985 y FA(k)876 1975 y FG(\)\))908 1941 y Fu(\001)928 1975 y FG(\()p FF(k)q FG(+1\))1036 1941 y Fu(\001)1055 1975 y FF(;)41 b FB(")1129 1985 y FA(\032)p Fz(!)p FA(\033)1201 1975 y FG(\()p FF(r)q FG(\)\()p FF(b)p FG(\))12 b(:=)f FB(")1391 1985 y FA(\033)1414 1975 y FG(\()p FF(r)d FB(#)1477 1985 y FA(\032)1497 1975 y FG(\()p FF(b)p FG(\)\))p FF(;)307 2042 y FB(#)328 2052 y FA(\032)p Fz(\002)p FA(\033)393 2042 y FG(\([)p FF(a;)f(b)p FG(]\))j(:=)i FB(h#)610 2052 y FA(\032)630 2042 y FG(\()p FF(a)p FG(\))p FF(;)7 b FB(#)723 2052 y FA(\033)746 2042 y FG(\()p FF(b)p FG(\))p FB(i)p FF(;)341 b FB(")1186 2052 y FA(\032)p Fz(\002)p FA(\033)1251 2042 y FG(\()p FF(r)q FG(\))12 b(:=)f([)p FB(")1403 2052 y FA(\032)1422 2042 y FG(\()p FF(r)q FG(0\))p FF(;)c FB(")1534 2052 y FA(\033)1557 2042 y FG(\()p FF(r)q FG(1\)])p FF(:)257 2133 y FG(Note)15 b(that)f(in)f FB(#)517 2144 y FA(\034)538 2133 y FG(\()p Fw(in)585 2139 y FA(c)602 2133 y FG(\()p Fx(b)p FG(\)\))f(:=)f FF(c)757 2118 y Fz(1)792 2133 y FB(#)p FG(\()p Fx(b)p FG(\))j(w)o(e)g(need)h(to)e(refer)i(to)f FB(#)g FG(at)f(higher)h(t)o(yp)q(es.)p 257 2170 573 2 v 304 2197 a Fl(2)321 2209 y FC(This)h(is)g(a)h(recursiv)o(e)d (de\014nition)f(of)j(a)g(family)f(of)h(domains)e(\()p FL(I)r FC(\()p Fe(\034)t FC(\)\))1262 2213 y Fj(\034)1296 2209 y FC(i.e.)i(a)g(least)f(\014xed)g(p)q(oin)o(t)g(of)257 2248 y(certain)e(con)o(tin)o(uous)g(functions.)20 b(The)14 b(theory)e(of)i(\(con)o(tin)o(uou)o(s\))d(families)h(of)i(domains)e (and)h(recursiv)o(e)257 2288 y(de\014nitions)c(thereof)h(is)h(dev)o (elop)q(ed)e(in)i(detail)f(in)h([3].)304 2316 y Fl(3)321 2327 y FC(F)m(rom)g(a)h(mathemat)o(ical)c(p)q(oin)o(t)j(of)g(view)h(it) g(is)g(also)f(p)q(ossible)f(to)h(tak)o(e)g(the)g(coalesced)f(sum,)g (but)h(the)257 2367 y(iden)o(ti\014cation)e(of)j(an)f(unde\014ned)f(ob) r(ject)g(with)j(the)e(total)g(unde\014ned)f(term)h(family)f(is)i (computation)o(aly)257 2406 y(doubtful.)304 2434 y Fl(4)321 2446 y FC(It)h(is)g(easy)f(to)h(c)o(hec)o(k)f(that)g(the)g(term)g (families)f(stemming)g(from)h FL(#)h FC(are)f(total)g(or)h(the)f(empt)o (y)g(term)257 2485 y(family)e FL(?)p FC(.)16 b(So)11 b(if)g FL(?)i FC(app)q(ears)c(in)i(an)g(application,)e(this)h(should)g (alw)o(a)o(ys)h(b)q(e)g FL(?)h FC(again.)953 2628 y FG(17)p eop %%Page: 18 18 18 17 bop 320 262 a FG(In)15 b(the)h(sequel)g(w)o(e)f(use)h(\(similar)d (to)i(our)g(syn)o(tactic)h(con)o(v)o(en)o(tion\))f(the)h(abbreviation) 257 311 y FF(ai)g FG(for)f FF(\031)398 317 y FA(i)412 311 y FG(\()p FF(a)p FG(\),)g FF(i)f FB(2)g(f)p FG(0)p FF(;)7 b FG(1)p FB(g)p FG(.)21 b(W)m(e)15 b(write)h(successiv)o(e)h (applications)e(of)g FB(")o FG(\()p FF(r)q FG(\))h(to)f(a)g(sequence) 257 361 y Fx(a)d FG(=)g FF(a)361 367 y Fy(1)380 361 y FF(;)7 b(:)g(:)g(:)t(;)g(a)494 367 y FA(n)530 361 y FG(\()p FF(a)568 367 y FA(i)594 361 y FB(2)k FG([)-7 b([)p FF(\032)671 367 y FA(i)684 361 y FG(])g(])13 b(or)h(pro)r(jection)g(mark)o(ers)g(0) f(or)h(1\))g(shortly)g(as)785 440 y FB(")p FG(\()p FF(r)q FG(\)\()p Fx(a)p FG(\))e(=)g FB(")993 450 y FA(\032)1012 440 y FG(\()p FF(r)c FB(#)p FG(\()p Fx(a)p FG(\)\))p FF(:)474 b FG(\(2\))257 519 y(In)16 b(particular,)g(if)f FF(\032)h FG(is)g(a)g(ground)f(t)o(yp)q(e,)i FB(")p FG(\()p FF(r)q FG(\)\()p Fx(a)p FG(\))e(=)g Fw(fam)1184 525 y FA(\032)1203 519 y FG(\()p FF(r)8 b FB(#)p FG(\()p Fx(a)p FG(\)\))17 b(and)e(therefore)j FB(")o FG(\()p FF(r)q FG(\))257 568 y(can)c(b)q(e)h(understo)q(o)q(d)g(as)f(a)g(\\self-ev)n (aluating")e(in)o(terpretation)i(of)f FF(r)q FG(.)320 618 y(Without)f(computation)f(rules)i(the)g(de\014nition)f(w)o(ould)g (b)q(e)i(m)o(uc)o(h)d(simpler.)16 b(It)d(is)g(then)257 668 y(p)q(ossible)20 b(to)g(de\014ne)g FB(I)s FG(\()p FF(\034)5 b FG(\))21 b(:=)g Fh(N)e FF(*)h FG(\003)910 674 y FA(\034)931 668 y FG(,)g(and)g(the)g(functions)g FB(#)1332 678 y FA(\034)1373 668 y FG(and)f FB(")1480 678 y FA(\034)1520 668 y FG(w)o(ould)g(b)q(e)257 718 y(iden)o(tities.)f(Then)12 b(the)g(de\014nition)g(of)f FB(#)g FG(and)h FB(")f FG(b)q(ecomes)h(an)f(inductiv)o(e)h (de\014nition)f(on)h(the)257 768 y(t)o(yp)q(es)j(\(see)g([4]\).)320 817 y(W)m(e)f(will)e(need)k(these)g(functions)e(to)g(de\014ne)i(an)e (in)o(terpretation)g(of)g(the)h(constan)o(ts)g(as)257 867 y(w)o(ell)f(as)g(normalization)d(b)o(y)i(ev)n(aluation)g(itself.) 257 981 y Fr(3.4)56 b(Predecessor)18 b(functions)257 1058 y FG(In)d(this)f(section)h(w)o(e)f(de\014ne)h(for)f(a)g (constructor)i(pattern)f FF(P)1211 1043 y FA(\032)1244 1058 y FG(with)f Fw(FV)p FG(\()p FF(P)6 b FG(\))11 b(=)i Fx(x)1539 1043 y Fq(\033)1579 1058 y FG(gener-)257 1108 y(alized)g(predecessor)i(functions)e Fw(gp)o(red)865 1118 y FA(P)898 1108 y FG(:)g([)-7 b([)p FF(\032)p FG(])g(])10 b FB(!)h FG([)-7 b([)p Fx(\033)q FG(])g(].)17 b(They)c(are)g(used)h (for)e(the)i(in)o(terpre-)257 1158 y(tation)g(of)f(the)h(constan)o(ts)h (in)f(the)g(presence)i(of)e(computation)e(rules.)320 1207 y(F)m(or)h(analyzing)f(elemen)o(ts)h(of)g([)-7 b([)p FF(\034)5 b FG(])-7 b(])12 b(w)o(e)h(de\014ne)i(b)q(o)q(olean)e (functions)g Fw(inst)p FG(?)1458 1213 y FA(P)1491 1207 y FG(:)g([)-7 b([)p FF(\032)p FG(])g(])10 b FB(!)h(B)1661 1213 y Fz(?)257 1257 y FG(\(where)16 b FB(B)d FG(=)e FB(f)p FG(t)-7 b(t)q FF(;)7 b FG(f)l(f)r FB(g)p FG(\))14 b(for)f(ev)o(ery)i(constructor)g(pattern)g FF(P)1191 1242 y FA(\032)1221 1257 y FB(2)d FE(Constr)1414 1263 y FA(\032)1433 1257 y FG(.)515 1336 y Fw(inst)p FG(?)597 1342 y FA(x)618 1336 y FG(\()p FF(a)p FG(\))g(:=)f(t)-7 b(t)413 1497 y Fw(inst)p FG(?)495 1503 y FA(cP)531 1507 y Fo(1)547 1503 y FA(:::P)598 1507 y Fi(k)618 1497 y FG(\()p FF(a)p FG(\))12 b(:=)739 1375 y Fu(8)739 1412 y(>)739 1425 y(>)739 1437 y(>)739 1450 y(<)739 1524 y(>)739 1537 y(>)739 1549 y(>)739 1562 y(:)776 1378 y(V)-28 b(V)825 1409 y Fw(inst)p FG(?)906 1415 y Fq(P)937 1409 y FG(\()p Fx(b)p FG(\))41 b(if)13 b FF(a)f FG(=)g Fw(in)1179 1415 y FA(c)1196 1409 y FG(\()p Fx(b)p FG(\))776 1469 y(f)l(f)234 b(if)13 b FF(a)f FG(=)g Fw(in)1179 1475 y FA(d)1198 1469 y FG(\()p Fx(b)p FG(\))i(for)g(some)f FF(d)e FB(6)p FG(=)h FF(c)1032 1529 y FG(or)i FF(a)e FG(=)g Fw(fam)1226 1535 y FA(\034)1247 1529 y FG(\()p FF(r)q FG(\))i(for)f(some)g FF(r)776 1589 y FB(?)223 b FG(otherwise)q FF(:)257 1671 y FG(Here)355 1640 y Fu(V)-28 b(V)404 1671 y Fw(inst)p FG(?)486 1677 y Fq(P)516 1671 y FG(\()p Fx(b)p FG(\))14 b(is)626 1640 y Fu(V)660 1683 y FA(i)681 1671 y Fw(inst)p FG(?)763 1677 y FA(P)784 1681 y Fi(i)799 1671 y FG(\()p FF(b)833 1677 y FA(i)847 1671 y FG(\))g(where)997 1640 y Fu(V)1046 1671 y FG(denotes)h(strict)f(b)q(o)q(olean)g(conjunction.) 257 1744 y Fn(Lemma)i(14.)k Ft(L)n(et)d FF(P)q(;)7 b(P)662 1729 y Fz(0)688 1744 y Ft(b)n(e)17 b(c)n(onstructor)f(p)n(atterns)g (and)i(let)e FF(a)e FB(2)g FG([)-7 b([)p FF(\032)p FG(])g(])p Ft(.)23 b(If)16 b Fw(inst)p FG(?)1561 1750 y FA(P)1589 1744 y FG(\()p FF(a)p FG(\))e(=)257 1794 y Fw(inst)p FG(?)339 1800 y FA(P)364 1791 y Fd(0)378 1794 y FG(\()p FF(a)p FG(\))e(=)g(t)-7 b(t)p Ft(,)15 b(then)g FF(P)20 b Ft(and)c FF(P)794 1778 y Fz(0)820 1794 y Ft(ar)n(e)e(uni\014able.)257 1867 y(Pr)n(o)n(of.)20 b FG(Induction)14 b(on)g FF(P)6 b FG(.)p 1672 1842 18 2 v 1672 1865 2 24 v 1688 1865 V 1672 1867 18 2 v 320 1947 a(The)14 b(generalized)h(predecessor)h (functions)807 2026 y Fw(gp)o(red)902 2036 y FA(P)934 2026 y FG(:)e([)-7 b([)p FF(\032)p FG(])g(])10 b FB(!)h FG([)-7 b([)p Fx(\033)q FG(])g(])257 2104 y(are)11 b(de\014ned)g (inductiv)o(ely)e(for)h(ev)o(ery)h(constructor)g(pattern)g FF(P)1229 2089 y FA(\032)1258 2104 y FG(with)e Fw(FV)p FG(\()p FF(P)d FG(\))12 b(=)f Fx(x)1548 2089 y Fq(\033)1583 2104 y FG(where)257 2154 y(the)k(v)n(ariables)d(are)i(listed)g(from)e (left)h(to)h(righ)o(t)f(in)g(the)h(order)h(of)e(their)h(o)q(ccurrences) i(in)e FF(P)6 b FG(.)446 2233 y Fw(gp)o(red)541 2243 y FA(x)562 2233 y FG(\()p FF(a)p FG(\))11 b(:=)h FF(a;)421 2337 y Fw(gp)o(red)516 2347 y FA(c)p Fq(P)562 2337 y FG(\()p FF(a)p FG(\))f(:=)683 2266 y Fu(\()716 2309 y Fw(gp)o(red)811 2319 y Fq(P)842 2309 y FG(\()p Fx(a)p FG(\))42 b(if)13 b Fw(inst)p FG(?)1062 2315 y FA(c)p Fq(P)1107 2309 y FG(\()p FF(a)p FG(\))f(=)g(t)-7 b(t)14 b(and)g FF(a)d FG(=)h Fw(in)1445 2315 y FA(c)1462 2309 y FG(\()p Fx(a)q FG(\))716 2369 y FB(?)p FF(;)7 b(:)g(:)g(:)t(;)g FB(?)68 b FG(otherwise)q FF(:)257 2453 y FG(Here)12 b Fw(gp)o(red)447 2463 y Fq(P)477 2453 y FG(\()p Fx(a)q FG(\))e(denotes)h(the)g(concatenation)f(of)g(the)g(lists)g Fw(gp)o(red)1309 2463 y FA(P)1330 2467 y Fi(i)1346 2453 y FG(\()p FF(b)1380 2459 y FA(i)1394 2453 y FG(\),)g(and)g(the)g (length)257 2503 y(of)k FB(?)p FF(;)7 b(:)g(:)g(:)t(;)g FB(?)12 b FG(is)i(the)h(n)o(um)o(b)q(er)e(of)g(v)n(ariables)g(in)h FF(c)p Fx(P)6 b FG(.)953 2628 y(18)p eop %%Page: 19 19 19 18 bop 257 262 a Fr(3.5)56 b(In)n(terpretation)18 b(of)g(the)h(constan)n(ts)257 338 y FG(No)o(w)h(w)o(e)g(are)h(able)e (to)h(in)o(terpret)h(the)g(constan)o(ts.)37 b(Notice)21 b(that)f FB(")f FG(giv)o(es)h(rise)h(to)f(an)257 388 y(en)o(vironmen)o(t)13 b(b)o(y)h FF(x)d FB(7!)g(")p FG(\()p FF(x)703 373 y Fz(1)738 388 y FG(\).)18 b(Let)624 472 y FB(I)s FG(\()p FF(c)p FG(\)\()p Fx(a)p FG(\))12 b(:=)f Fw(in)856 454 y Fq(\032)856 482 y FA(c)878 472 y FG(\()p Fx(a)q FG(\))41 b(if)13 b FF(c)1034 455 y Fq(\032)p Fz(!)p FA(\034)1119 472 y FB(2)f FE(Constr)p FF(:)257 557 y FG(Otherwise)18 b FB(I)s FG(\()p FF(c)p FG(\))e(is)g(de\014ned)i (recursiv)o(ely)f(as)f(follo)o(ws.)24 b(If)15 b(for)h(some)g (computation)e(rule)257 606 y FF(c)p Fx(P)19 b FB(7\000)-7 b(!)391 612 y Fp(comp)474 606 y FF(Q)13 b FG(w)o(e)h(ha)o(v)o(e)677 575 y Fu(V)-28 b(V)726 606 y Fw(inst)p FG(?)807 612 y Fq(P)838 606 y FG(\()p Fx(a)p FG(\))12 b(=)g(t)-7 b(t,)14 b(then)528 691 y FB(I)s FG(\()p FF(c)p FG(\)\()p Fx(a)p FG(\))e(:=)f([)-7 b([)p FF(Q)p FG(])g(])796 673 y Fz(I)796 702 y Fy([)p Fq(x)o Fz(7!)p Fp(gp)o(red)929 710 y Fc(P)954 702 y Fy(\()p Fq(a)p Fy(\)])1013 691 y FF(;)48 b FG(where)15 b Fx(x)d FG(:=)f Fw(FV)p FG(\()p Fx(P)c FG(\).)257 779 y(If)14 b(for)f(all)g(computation)f(rules)j FF(c)p Fx(P)j FB(7\000)-7 b(!)895 785 y Fp(comp)978 779 y FF(Q)14 b FG(w)o(e)g(ha)o(v)o(e)1181 748 y Fu(V)-28 b(V)1230 779 y Fw(inst)p FG(?)1312 785 y Fq(P)1342 779 y FG(\()p Fx(a)q FG(\))11 b(=)h(f)l(f)s(,)h(then)429 936 y FB(I)s FG(\()p FF(c)p FG(\)\()p Fx(a)p FG(\))f(:=)630 838 y Fu(8)630 876 y(>)630 888 y(<)630 963 y(>)630 975 y(:)667 872 y FG([)-7 b([)p FF(N)5 b FG(])-7 b(])739 857 y Fz(I)739 887 y Fy([)p Fq(x)o Fz(7!)p Fy([)h([)p Fq(L)q Fy(])g(])851 877 y Fd(I)851 898 y(")872 887 y Fy(])925 872 y FG(if)13 b Fw(sel)1008 878 y FA(c)1024 872 y FG(\()p Fw(ext)q FG(\()p FB(#)p FG(\()p Fx(a)q FG(\)\)\))e(=)h FF(c)p Fx(K)j FB(7\000)-7 b(!)1416 878 y Fp(rew)1475 872 y FF(N)925 943 y FG(and)14 b Fw(ext)q FG(\()p FB(#)o FG(\()p Fx(a)q FG(\)\))e(=)f Fx(K)1269 949 y Fq(x)1292 943 y FG([)p Fx(L)p FG(])667 1003 y FB(")p FG(\()p FF(c)722 988 y Fz(1)757 1003 y FB(#)p FG(\()p Fx(a)p FG(\)\))73 b(if)13 b Fw(sel)1008 1009 y FA(c)1024 1003 y FG(\()p Fw(ext)q FG(\()p FB(#)p FG(\()p Fx(a)q FG(\)\)\))e(=)h Fw(no)p FG(-)p Fw(match)p FG(.)257 1091 y(In)i(all)f(other)h(cases,)h FB(I)s FG(\()p FF(c)p FG(\)\()p Fx(a)q FG(\))c(:=)h FB(?)p FG(.)320 1141 y(Since)e(in)g(the)h(case)g(of)f(computation)f(rules)i(w) o(e)f(assumed)g(the)h(left)f(hand)g(sides)h(of)f(these)257 1191 y(rules)15 b(to)f(b)q(e)g(non-uni\014able,)f(lemma)d(14)k(guaran)o (tees)h(that)f(this)f(is)h(a)g(sound)g(de\014nition.)320 1240 y(The)j(usefulness)g(of)f(the)h(computation)e(rules)i(is)g(due)g (to)f(the)h(fact)g(that)f(it)g(is)h(m)o(uc)o(h)257 1290 y(simpler)c(to)g(compute)g Fw(gp)o(red)716 1300 y Fq(P)747 1290 y FG(\()p Fx(a)p FG(\))h(than)f(to)g(compute)g([)-7 b([)p Fx(L)p FG(])g(])1198 1296 y Fz(")1216 1290 y FG(,)13 b(where)i Fw(ext)p FG(\()p FB(#)p FG(\()p Fx(a)q FG(\)\))d(=)f Fx(K)t FG([)p Fx(L)o FG(].)257 1340 y(This)j(can)g(b)q(e)h(seen)g(from) d(the)j(follo)o(wing)c(examples.)257 1417 y Fn(Example)k(15.)21 b FG(Let)16 b(us)g(compare)f(the)h(t)o(w)o(o)g(p)q(ossibilities)f(for)g (in)o(terpreting)h(constan)o(ts,)257 1467 y(either)h(as)f(computation)f (rules)h(or)g(else)h(as)f(rewrite)h(rules.)26 b(T)m(o)15 b(this)h(end)g(w)o(e)h(in)o(tro)q(duce)257 1517 y Ft(iter)n(ation)e (typ)n(es)j FF(n)p FG(,)c(b)o(y)h(0)d(:=)h FF(\023)h FG(and)g FF(n)c FG(+)g(1)i(:=)h FF(n)g FB(!)f FF(n)p FG(.)20 b(Let)15 b Fw(abst)1306 1523 y FA(n)1343 1517 y FG(b)q(e)h(a)e(constructor)i(of)257 1567 y(t)o(yp)q(e)f FF(n)e FB(!)f FG(0,)i(and)h FF(x)596 1573 y FA(n)632 1567 y FG(b)q(e)h(a)e(v)n(ariable)f(of)h(iteration)g(t)o(yp)q(e)i FF(n)p FG(,)e(e.g.)f FF(x)1344 1573 y Fy(3)1367 1567 y FG(:)h(\(\()p FF(\023)f FB(!)f FF(\023)p FG(\))g FB(!)g FG(\()p FF(\023)h FB(!)257 1617 y FF(\023)p FG(\)\))f FB(!)f FG(\()p FF(\023)g FB(!)g FF(\023)p FG(\))h FB(!)f FF(\023)g FB(!)g FF(\023)p FG(.)18 b(W)m(e)13 b(add)h(sp)q(ecial)g (constan)o(ts)h FF(c)1173 1623 y FA(n)1209 1617 y FG(with)f(the)g(rule) 828 1701 y FF(c)846 1707 y FA(n)869 1701 y FG(\()p Fw(abst)957 1707 y FA(n)980 1701 y FF(x)1004 1707 y FA(n)1026 1701 y FG(\))e(=)g(0)517 b(\(3\))257 1785 y(\(0)10 b(a)f(constan)o(t)g(of)g (t)o(yp)q(e)h(0\).)16 b(The)10 b(in)o(ten)o(tion)f(again)f(is)h(to)g (sho)o(w)h(the)g(di\013erence)h(in)e(e\016ciency)257 1835 y(b)q(et)o(w)o(een)17 b(viewing)f(\(3\))g(as)f(computation)f(or)i (as)g(rewrite)h(rule.)23 b(T)m(o)15 b(see)i(this)f(di\013erence,)257 1885 y(consider)f(the)g(term)e FF(c)608 1891 y FA(n)630 1885 y FG(\()p Fw(abst)719 1891 y FA(n)742 1885 y FF(x)766 1891 y FA(n)788 1885 y FG(\))h(with)f(a)h(v)n(ariable)f FF(x)1127 1891 y FA(n)1149 1885 y FG(.)320 1935 y(If)i(w)o(e)g(view)h (\(3\))f(as)g(a)g(rewrite)i(rule,)e([)-7 b([)p FF(c)953 1941 y FA(n)975 1935 y FG(])g(])14 b(gets)i FF(a)e FG(:=)g([)-7 b([)p Fw(abst)1275 1941 y FA(n)1298 1935 y FF(x)1322 1941 y FA(n)1344 1935 y FG(])g(])1361 1941 y Fz(")1395 1935 y FG(as)15 b(an)g(argumen)o(t.)257 1984 y(It)i(computes)f Fw(ext)p FG(\()p FB(#)p FG(\()p FF(a)p FG(\)\),)h(i.e.)e(the)i(term)e Fw(abst)996 1990 y FA(n)1019 1984 y FG(\()p Fw(nf)s FG(\()p FF(x)1112 1990 y FA(n)1135 1984 y FG(\)\).)25 b(Then)16 b(it)g(\014nds)h FF(L)e FG(:=)g Fw(nf)s FG(\()p FF(x)1639 1990 y FA(n)1662 1984 y FG(\),)257 2034 y(and)g(computes)f([)-7 b([)p Fw(nf)r FG(\()p FF(x)617 2040 y FA(n)639 2034 y FG(\)])g(])672 2040 y Fz(")691 2034 y FG(.)19 b(Finally)13 b(in)h(the)h(en)o(vironmen)o(t)e(assigning)h(this)g(v)n(alue)g(to)g FF(x)1667 2040 y FA(n)257 2084 y FG(it)g(computes)g([)-7 b([0])g(].)320 2134 y(If)18 b(ho)o(w)o(ev)o(er)g(w)o(e)h(view)g(\(3\))f (as)h(a)f(computation)f(rule,)i(then)g([)-7 b([)p FF(c)1343 2140 y FA(n)1365 2134 y FG(])g(])17 b(again)h(gets)h FF(a)g FG(:=)257 2184 y([)-7 b([)p Fw(abst)346 2190 y FA(n)368 2184 y FF(x)392 2190 y FA(n)415 2184 y FG(])g(])432 2190 y Fz(")463 2184 y FG(as)14 b(an)f(argumen)o(t.)j(Since)e Fw(inst)p FG(?)962 2190 y Fp(abst)1016 2194 y Fi(n)1036 2190 y FA(x)1055 2194 y Fi(n)1078 2184 y FG(\()p FF(a)p FG(\))d(=)h(t)-7 b(t,)13 b(w)o(e)h(only)f(need)h(to)f(compute)257 2233 y([)-7 b([0])g(])13 b(in)i(the)g(en)o(vironmen)o(t)f(assigning)g Fw(gp)o(red)961 2244 y FA(a)995 2233 y FG(=)f FB(")p FG(\()p FF(x)1101 2218 y Fz(1)1101 2244 y FA(n)1136 2233 y FG(\))i(to)g FF(x)1243 2239 y FA(n)1265 2233 y FG(.)21 b(This)15 b(do)q(es)h(not)f(in)o(v)o(olv)o(e)257 2283 y(computation)e(of)g(the)h(long)f(normal)f(form)g(of)h(an)o(y)h(v)n (ariable.)320 2333 y(The)20 b(example)e(sho)o(ws)i(that)f(normalizing)e (via)i(the)h(rewrite)h(rule)f(ma)o(y)d(tak)o(e)j(time)257 2383 y(exp)q(onen)o(tial)13 b(in)g(the)h(t)o(yp)q(e)g(lev)o(el)f FF(n)g FG(of)g FF(x)894 2389 y FA(n)916 2383 y FG(,)g(whereas)h (normalizing)d(via)h(the)i(computation)257 2433 y(rule)19 b(tak)o(es)g(linear)g(time)659 2418 y Fy(5)676 2433 y FG(.)32 b(The)19 b(reason)h(is)e(simply)f(that)i(the)g(length)f FF(l)1443 2439 y FA(n)1485 2433 y FG(of)g(the)h(long)p 257 2465 573 2 v 304 2491 a Fl(5)321 2503 y FC(W)m(e)12 b(ha)o(v)o(e)e(v)o(eri\014ed)f(this)i(in)g(the)g Fb(Minlog)i FC(system)953 2628 y FG(19)p eop %%Page: 20 20 20 19 bop 257 262 a FG(normal)17 b(form)g Fw(nf)s FG(\()p FF(x)583 268 y FA(n)605 262 y FG(\))i(of)f(a)h(v)n(ariable)e FF(x)916 268 y FA(n)957 262 y FG(of)i(iteration)f(t)o(yp)q(e)h FF(n)g FG(is)f FB(\025)i FG(2)1444 246 y FA(n)1467 262 y FG(.)32 b(\(Pro)q(of)19 b(b)o(y)257 311 y(induction)14 b(on)g FF(n)p FG(.)j(Step:)i FF(l)677 317 y FA(n)p Fy(+1)754 311 y FB(\025)11 b FG(1)e(+)869 280 y Fu(P)913 290 y FA(n)913 324 y(i)p Fy(=0)976 311 y FF(l)988 317 y FA(i)1013 311 y FB(\025)j FG(1)d(+)1129 280 y Fu(P)1172 290 y FA(n)1172 324 y(i)p Fy(=0)1235 311 y FG(2)1256 296 y FA(i)1281 311 y FG(=)j(2)1346 296 y FA(n)p Fy(+1)1410 311 y FG(\).)p 1672 286 18 2 v 1672 310 2 24 v 1688 310 V 1672 312 18 2 v 320 391 a(The)17 b(di\013erence)i(b)q(et)o(w)o(een)f(the)g(t)o(w)o (o)f(p)q(ossibilities)f(for)h(in)o(terpreting)g(constan)o(ts)h(can)257 441 y(also)c(b)q(e)i(seen)g(in)e(the)h(tree)h(example)d(in)h(2.3\(b\).) 20 b(In)15 b(the)g(case)h(of)e(a)g(prop)q(er)i(rewrite)f(rule)257 491 y(w)o(e)f(ha)o(v)o(e)502 578 y FB(I)s FG(\()p FB(O)q FG(\))e(=)g Fh(N)d FF(*)i FG(\003)773 584 y Fz(O)802 578 y FF(;)408 640 y FB(I)s FG(\()p FE(Sup)q FG(\)\()p FF(a)p FG(\))h(=)g Fw(fam)715 646 y FA(\023)730 640 y FG(\()p FE(Sup)820 623 y Fz(1)855 640 y FB(#)p FG(\()p FF(a)p FG(\)\))291 861 y FB(I)s FG(\()p FE(Rec)p FG(\)\()p FF(a;)7 b(b)505 867 y Fy(0)523 861 y FF(;)g(b)560 867 y Fy(1)578 861 y FG(\))12 b(=)650 677 y Fu(8)650 714 y(>)650 727 y(>)650 739 y(>)650 751 y(>)650 764 y(>)650 776 y(>)650 789 y(>)650 801 y(>)650 814 y(<)650 888 y(>)650 901 y(>)650 913 y(>)650 926 y(>)650 938 y(>)650 951 y(>)650 963 y(>)650 976 y(>)650 988 y(:)687 714 y FF(b)705 720 y Fy(0)1082 714 y FG(if)h Fw(ext)q FG(\()p FB(#)o FG(\()p FF(a)p FG(\)\))f(=)g(0)687 773 y FF(b)705 779 y Fy(1)723 773 y FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])818 779 y Fz(")835 773 y FF(;)7 b(g)q FG(\))191 b(if)13 b Fw(ext)q FG(\()p FB(#)o FG(\()p FF(a)p FG(\)\))f(=)g FE(Sup)q FF(M)19 b FG(and)1082 833 y FF(g)q FG(\()p FF(e)p FG(\))12 b(:=)g FB(I)s FG(\()p FE(Rec)o FG(\))q(\([)-7 b([)p FF(M)5 b FG(])-7 b(])1456 839 y Fz(")1473 833 y FG(\()p FF(e)p FG(\))p FF(;)7 b(b)1561 839 y Fy(0)1579 833 y FF(;)g(b)1616 839 y Fy(1)1634 833 y FG(\))687 893 y FB(")o FG(\()p FE(Rec)804 878 y Fz(1)840 893 y FB(#)o FG(\()p FF(a;)g(b)935 899 y Fy(0)953 893 y FF(;)g(b)990 899 y Fy(1)1008 893 y FG(\)\))42 b(if)13 b Fw(ext)q FG(\()p FB(#)o FG(\()p FF(a)p FG(\)\))i(is)e(de\014ned)1082 953 y(but)h(not)g(0)f(or)h FE(Sup)q FF(N)687 1012 y FB(?)362 b FG(otherwise)257 1101 y(and)14 b(in)g(case)g(of)g(a)f(computation)f(rule:)538 1188 y FB(I)s FG(\()p FB(O)q FG(\))g(=)f Fn(1)e FG(+)h([)p FB(I)s FG(\()p FF(\023)p FG(\))h FB(!)g(I)s FG(\()p FB(O)q FG(\)])e(+)g(\()p Fh(N)h FF(*)h FG(\003)1203 1194 y Fz(O)1232 1188 y FG(\))p FF(;)444 1250 y FB(I)s FG(\()p FE(Sup)q FG(\)\()p FF(a)p FG(\))h(=)f FF(in)724 1256 y Ff(Sup)778 1250 y FG(\()p FF(a)p FG(\))327 1442 y FB(I)s FG(\()p FE(Rec)p FG(\)\()p FF(a;)c(b)541 1448 y Fy(0)559 1442 y FF(;)g(b)596 1448 y Fy(1)614 1442 y FG(\))12 b(=)685 1282 y Fu(8)685 1319 y(>)685 1332 y(>)685 1344 y(>)685 1357 y(>)685 1369 y(>)685 1381 y(>)685 1394 y(<)685 1469 y(>)685 1481 y(>)685 1494 y(>)685 1506 y(>)685 1518 y(>)685 1531 y(>)685 1543 y(:)722 1324 y FF(b)740 1330 y Fy(0)1118 1324 y FG(if)h FF(a)e FG(=)h FF(in)1272 1330 y Fy(0)1291 1324 y FG(\()p FB(?)p FG(\))722 1383 y FF(b)740 1389 y Fy(1)759 1383 y FG(\()p FF(f)r(;)7 b(g)q FG(\))265 b(if)13 b FF(a)e FG(=)h FF(in)1272 1389 y Ff(Sup)1326 1383 y FG(\()p FF(f)t FG(\))j(and)1118 1443 y FF(g)q FG(\()p FF(e)p FG(\))d(:=)f FB(I)s FG(\()p FE(Rec)p FG(\)\()p FF(f)t FG(\()p FF(e)p FG(\))p FF(;)c(b)1524 1449 y Fy(0)1544 1443 y FF(;)g(b)1581 1449 y Fy(1)1599 1443 y FG(\))722 1503 y FB(")p FG(\()p FE(Rec)840 1488 y Fz(1)875 1503 y FB(#)p FG(\()p FF(a;)g(b)971 1509 y Fy(0)989 1503 y FF(;)g(b)1026 1509 y Fy(1)1044 1503 y FG(\)\))42 b(if)13 b FF(a)g FG(is)h(a)g(term)f(family)722 1563 y FB(?)363 b FG(otherwise)q FF(:)257 1651 y Fn(Example)15 b(16.)21 b FG(As)e(men)o(tioned)f(at)g(the)h(end)g(of)f(section)h(2.4,)g(it)f (mak)o(es)f(a)i(di\013erence)257 1701 y(whether)d(w)o(e)e(use)h(the)g FF(c)p Fx(M)5 b FF(x)12 b FB(7\000)-6 b(!)11 b FF(N)19 b FG(or)14 b(the)h FF(c)p Fx(M)j FB(7\000)-7 b(!)12 b FF(\025xN)18 b FG(v)o(ersion)d(of)e(a)h(rewrite)i(rule.)257 1751 y(No)o(w)g(w)o(e)g(see)h(that)e(in)h(the)g(former)f(case)h(the)h (last)e(argumen)o(t)g FF(a)g FG(forces)i(us)f(to)f(calculate)257 1801 y Fw(ext)q FG(\()p FB(#)p FG(\()p FF(a)p FG(\)\))e(and)g(then)g (in)o(terprete)i(this)e(term)f(again,)g(whic)o(h)g(ma)o(y)f(b)q(e)j (cum)o(b)q(ersome.)j(This)257 1850 y(can)k(also)f(b)q(e)g(v)o (eri\014ed)h(easily)f(b)o(y)g(an)g(example)f(similar)f(to)i(the)h(one)g (ab)q(o)o(v)o(e:)30 b(Let)21 b(us)257 1900 y(compare)16 b(the)h(t)o(w)o(o)f(rules)h FF(c)p FG(0)p FF(x)746 1906 y FA(n)784 1900 y FB(7\000)-7 b(!)15 b FG(0)i(and)f FF(c)p FG(0)f FB(7\000)-6 b(!)15 b FF(\025x)1172 1906 y FA(n)1194 1900 y FG(0.)26 b(Consider)17 b(the)g(term)e FF(c)p FG(0)p FF(x)1667 1906 y FA(n)257 1950 y FG(with)20 b(a)g(v)n(ariable)e FF(x)584 1956 y FA(n)627 1950 y FG(of)h(iteration)g(t)o(yp)q(e)i FF(n)p FG(.)36 b(NbE)20 b(via)f(the)i(\014rst)f(form)e(pro)q(ceeds)k (as)257 2000 y(follo)o(ws.)17 b([)-7 b([)p FF(c)p FG(])g(])13 b(gets)h FF(a)582 2006 y Fy(1)613 2000 y FG(:=)d([)-7 b([0])g(])12 b(and)i FF(a)838 2006 y Fy(2)868 2000 y FG(:=)e([)-7 b([)p FF(x)965 2006 y FA(n)987 2000 y FG(])g(])1004 2006 y Fz(")1034 2000 y FG(=)12 b FB(")o FG(\()p FF(x)1138 1985 y Fz(1)1138 2010 y FA(n)1173 2000 y FG(\))j(as)f(argumen)o(ts.)j (It)e(computes)257 2050 y Fw(ext)q FG(\()p FB(#)p FG(\()p FF(a)385 2056 y Fy(2)404 2050 y FG(\)\),)f(i.e.)g(the)h(long)e(normal)g (form)g(of)g FF(x)1004 2056 y FA(n)1027 2050 y FG(.)20 b(If)14 b(ho)o(w)o(ev)o(er)h(w)o(e)g(emplo)o(y)d(NbE)j(via)f(the)257 2099 y(second)k(form,)c(then)j([)-7 b([)p FF(c)p FG(])g(])15 b(only)g(gets)i FF(a)873 2105 y Fy(1)907 2099 y FG(:=)e([)-7 b([0])g(].)23 b(It)17 b(computes)f([)-7 b([)p FF(\025x)p FG(0])g(])14 b(and)i(applies)g(the)257 2149 y(result)d(to)e FB(")p FG(\()p FF(x)480 2134 y Fz(1)480 2160 y FA(n)515 2149 y FG(\);)h(this)g(do)q(es)g(not)f(in)o(v)o(olv)o(e)g(computation)e (of)i(an)o(y)h(long)e(normal)g(form.)p 1672 2124 V 1672 2148 2 24 v 1688 2148 V 1672 2150 18 2 v 257 2229 a Fn(Lemma)16 b(17.)k Ft(L)n(et)12 b FF(P)610 2214 y FA(\032)641 2229 y Ft(b)n(e)h(a)f(c)n(onstructor)g(p)n(attern)g(with)f Fw(FV)p FG(\()p FF(P)6 b FG(\))11 b(=)h Fx(x)1366 2214 y Fq(\033)1391 2229 y Ft(,)h(and)g(let)e FF(a)h FB(2)f FG([)-7 b([)p FF(\032)p FG(])g(])p Ft(.)257 2279 y(Then)16 b(for)e(al)r(l)g Fx(b)d FB(2)h FG([)-7 b([)p Fx(\033)q FG(])g(])492 2366 y FF(a)11 b FG(=)h([)-7 b([)p FF(P)6 b FG(])-7 b(])636 2373 y Fy([)p Fq(x)o Fz(7!)p Fq(b)p Fy(])769 2366 y Ft(i\013)42 b Fw(inst)p FG(?)931 2372 y FA(P)959 2366 y FG(\()p FF(a)p FG(\))12 b(=)g(t)-7 b(t)15 b Ft(and)g Fw(gp)o(red)1285 2376 y FA(P)1312 2366 y FG(\()p FF(a)p FG(\))d(=)g Fx(b)o FF(:)257 2453 y Ft(Pr)n(o)n(of.)20 b FG(Induction)14 b(on)f FF(P)6 b FG(.)18 b(If)13 b FF(P)19 b FG(is)13 b(a)g(v)n(ariable,)f(then)j(b)q(oth)e(sides)i(are)f(equiv)n (alen)o(t)e(to)i(the)257 2503 y(statemen)o(t)h(\\)p FF(a)d FG(=)h FF(b)p FG(".)19 b(No)o(w)14 b(let)h FF(P)j FG(=)12 b FF(c)p Fx(P)7 b FG(.)20 b(Then)15 b([)-7 b([)p FF(P)6 b FG(])-7 b(])1129 2510 y Fy([)p Fq(x)n Fz(!)p Fq(b)p Fy(])1231 2503 y FG(=)13 b Fw(in)1307 2509 y FA(c)1324 2503 y FG(\([)-7 b([)p Fx(P)6 b FG(])-7 b(])1410 2510 y Fy([)p Fq(x)p Fz(!)p Fq(b)p Fy(])1502 2503 y FG(\).)20 b(Assume)953 2628 y(20)p eop %%Page: 21 21 21 20 bop 257 262 a FF(a)18 b FG(=)g([)-7 b([)p FF(P)6 b FG(])-7 b(])414 269 y Fy([)p Fq(x)o Fz(7!)p Fq(b)p Fy(])504 262 y FG(.)30 b(Then)18 b FF(a)g FG(=)g Fw(in)779 268 y FA(c)796 262 y FG(\()p Fx(a)p FG(\))g(where)h Fx(a)f FG(=)g([)-7 b([)p Fx(P)6 b FG(])-7 b(])1160 269 y Fy([)p Fq(x)p Fz(!)p Fq(b)p Fy(])1252 262 y FG(.)29 b(Hence)1420 230 y Fu(V)-28 b(V)1468 262 y Fw(inst)p FG(?)1550 268 y Fq(P)1581 262 y FG(\()p Fx(a)p FG(\))18 b(=)257 311 y(t)-7 b(t)21 b(and)g Fw(gp)o(red)486 321 y Fq(P)517 311 y FG(\()p Fx(a)p FG(\))i(=)g Fx(b)p FG(,)f(b)o(y)e(IH.)g(Therefore) i Fw(inst)p FG(?)1128 317 y FA(c)p Fq(P)1174 311 y FG(\()p FF(a)p FG(\))h(=)g(t)-7 b(t)21 b(and)g Fw(gp)o(red)1535 321 y FA(c)p Fq(P)1580 311 y FG(\()p FF(a)p FG(\))i(=)257 361 y Fw(gp)o(red)352 371 y Fq(P)383 361 y FG(\()p Fx(a)p FG(\))16 b(=)g Fx(b)p FG(.)25 b(F)m(or)16 b(the)h(con)o(v)o(erse)h (assume)d Fw(inst)p FG(?)1110 367 y FA(c)p Fq(P)1156 361 y FG(\()p FF(a)p FG(\))g(=)h(t)-7 b(t)17 b(and)f Fw(gp)o(red)1493 371 y FA(c)p Fq(P)1539 361 y FG(\()p FF(a)p FG(\))f(=)h Fx(b)p FG(.)257 411 y(Then)g FF(a)d FG(=)g Fw(in)479 417 y FA(c)495 411 y FG(\()p Fx(a)q FG(\),)581 380 y Fu(V)-28 b(V)629 411 y Fw(inst)p FG(?)711 417 y Fq(P)741 411 y FG(\()p Fx(a)q FG(\))13 b(=)h(t)-7 b(t)15 b(and)f(moreo)o(v)o(er)g Fw(gp)o(red)1255 421 y Fq(P)1286 411 y FG(\()p Fx(a)p FG(\))g(=)f Fw(gp)o(red)1498 421 y FA(c)p Fq(P)1544 411 y FG(\()p FF(a)p FG(\))g(=)g Fx(b)p FG(.)257 461 y(Hence)j([)-7 b([)p Fx(P)6 b FG(])-7 b(])451 468 y Fy([)p Fq(x)p Fz(!)p Fq(b)p Fy(])554 461 y FG(=)11 b Fx(a)q FG(,)i(b)o(y)h(IH.)f(Hence)j FF(a)11 b FG(=)h Fw(in)1010 467 y FA(c)1027 461 y FG(\()p Fx(a)q FG(\))f(=)h([)-7 b([)p FF(P)6 b FG(])-7 b(])1208 468 y Fy([)p Fq(x)o Fz(!)p Fq(b)p Fy(])1298 461 y FG(.)p 1672 436 18 2 v 1672 459 2 24 v 1688 459 V 1672 461 18 2 v 257 543 a Fn(Lemma)16 b(18.)k Ft(F)m(or)15 b(every)g(rule)f FF(c)p Fx(P)k FB(7\000)-6 b(!)914 549 y Fp(comp)996 543 y FF(Q)15 b Ft(and)h(every)e(envir)n(onment)i FF(\030)846 630 y FG([)-7 b([)p FF(c)p Fx(P)6 b FG(])-7 b(])934 636 y FA(\030)962 630 y FG(=)12 b([)-7 b([)p FF(Q)p FG(])g(])1073 636 y FA(\030)1090 630 y FF(:)257 716 y Ft(Pr)n(o)n(of.)20 b FG(Let)h Fw(FV)p FG(\()p Fx(P)7 b FG(\))23 b(=)g Fx(x)p FG(.)38 b(By)21 b(the)g(previous)g(lemma,)1219 685 y Fu(V)-28 b(V)1268 716 y Fw(inst)p FG(?)1349 722 y Fq(P)1380 716 y FG(\([)-7 b([)p Fx(P)6 b FG(])-7 b(])1466 722 y FA(\030)1483 716 y FG(\))23 b(=)g(t)-7 b(t)21 b(and)257 766 y Fw(gp)o(red)352 776 y Fq(P)383 766 y FG(\([)-7 b([)p Fx(P)6 b FG(])-7 b(])469 772 y FA(\030)486 766 y FG(\))12 b(=)g FF(\030)r FG(\()p Fx(x)p FG(\).)17 b(Hence,)12 b(since)f FF(c)f FG(is)f(not)h(a)g(constructor,)i([)-7 b([)p FF(c)p Fx(P)6 b FG(])-7 b(])1378 772 y FA(\030)1407 766 y FG(=)11 b FB(I)s FG(\()p FF(c)p FG(\)\([)-7 b([)p Fx(P)6 b FG(])-7 b(])1612 772 y FA(\030)1630 766 y FG(\))11 b(=)257 816 y([)-7 b([)p FF(Q)p FG(])g(])324 822 y FA(\030)341 816 y FG(.)p 1672 791 V 1672 814 2 24 v 1688 814 V 1672 816 18 2 v 257 931 a Fr(3.6)56 b(Correctness)18 b(of)h(normalization)d (b)n(y)j(ev)m(aluation)257 1008 y FG(W)m(e)f(sa)o(y)g(that)h (normalization)c(b)o(y)j(ev)n(aluation)f(is)h(correct)i(if)d FF(M)24 b FB(\000)-7 b(!)18 b FF(Q)g FG(implies)e(that)257 1058 y FB(#)278 1024 y Fu(\000)297 1058 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])376 1064 y Fz(")393 1024 y Fu(\001)412 1058 y FG(\()p FF(k)q FG(\))15 b(=)514 1064 y FA(\013)552 1058 y FF(Q)h FG(for)f FF(k)g(>)g Fw(FV)p FG(\()p FF(M)5 b FG(\).)23 b(If)16 b(w)o(e)f(ha)o(v)o(e)h(preserv)n(ation)g(of)f(v)n (alues)h(\(i.e.)e(that)257 1108 y(con)o(v)o(ersion)j(do)q(es)g(not)f(c) o(hange)h(the)g(meaning)d(\(i.e.)i(seman)o(tics\))g(of)f(a)h(term\),)g (then)h(this)257 1157 y(is)h(easy)g(to)f(pro)o(v)o(e,)h(ev)o(en)g(in)f (the)h(stronger)h(from)c FB(#)1093 1124 y Fu(\000)1112 1157 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1191 1163 y Fz(")1208 1124 y Fu(\001)1245 1157 y FG(=)18 b FF(Q)1328 1142 y Fz(1)1380 1157 y FG(\(b)o(y)g(induction)f(on)257 1207 y(terms)h FF(M)k FG(in)17 b(long)g(normal)e(form\).)27 b(But)19 b(in)e(general,)h(preserv)n(ation)g(of)f(v)n(alues)g(is)h(not) 257 1257 y(guaran)o(teed.)257 1336 y Fn(Example)d(19.)21 b FG(In)11 b(the)g(presence)i(of)d(higher-t)o(yp)q(e)h(v)n(ariables)f (it)g(is)h(p)q(ossible)g(that)f(rewrite)257 1386 y(rules)16 b(do)e(not)h(preserv)o(e)i(the)e(v)n(alue.)20 b(Let)15 b FF(c)g FG(b)q(e)g(a)g(constan)o(t)g(of)f(t)o(yp)q(e)h(\()p FF(\034)j FB(!)13 b FF(\034)5 b FG(\))13 b FB(!)f FF(\034)18 b FB(!)13 b FF(\034)257 1436 y FG(with)20 b(the)g(single)f(rule)h FF(cx)h FB(7\000)-7 b(!)20 b FF(x)p FG(,)h(considered)g(as)e(a)h(prop)q (er)g(rewrite)h(rule.)35 b(Cho)q(ose)257 1486 y Fw(id)289 1467 y Fz(0)316 1486 y FB(2)15 b FG([)-7 b([)p FF(\034)19 b FB(!)c FF(\034)5 b FG(])-7 b(])510 1471 y Fz(I)547 1486 y FG(=)16 b([)p FB(I)s FG(\()p FF(\034)5 b FG(\))14 b FB(!)h(I)s FG(\()p FF(\034)5 b FG(\)])16 b(with)f Fw(id)996 1467 y Fz(0)1023 1486 y FB(6)p FG(=)g Fw(id)i FG(but)f Fw(id)1228 1467 y Fz(0)1239 1486 y FG(\()p Fw(fam)1321 1492 y FA(\034)1342 1486 y FG(\()p FF(x)1382 1471 y Fz(1)1417 1486 y FG(\)\))f(=)h Fw(fam)1578 1492 y FA(\034)1598 1486 y FG(\()p FF(x)1638 1471 y Fz(1)1673 1486 y FG(\))257 1536 y(for)c(ev)o(ery)h(v)n(ariable)e FF(x)g FB(2)g FG(\003)684 1542 y FA(\034)705 1536 y FG(.)17 b(No)o(w)12 b(preserv)n(ation)h(of)e (v)n(alue)h(w)o(ould)f(require)i([)-7 b([)p FF(cx)p FG(])g(])1544 1542 y FA(\030)1571 1536 y FG(=)12 b([)-7 b([)p FF(x)p FG(])g(])1673 1542 y FA(\030)257 1585 y FG(for)18 b(ev)o(ery)h(en)o (vironmen)o(t)e FF(\030)r FG(,)i(so)f(in)g(particular)g(for)f FF(\030)r FG(\()p FF(x)p FG(\))i(=)g Fw(id)1281 1567 y Fz(0)1293 1585 y FG(.)31 b(But)18 b(b)q(ecause)i(of)e(the)257 1635 y(reference)g(to)d FB(#)506 1645 y FA(\034)s Fz(!)p FA(\034)594 1635 y FG(in)g(the)h(de\014nition)f(of)g FB(I)s FG(\()p FF(c)p FG(\))g(w)o(e)g(ha)o(v)o(e)g FB(I)s FG(\()p FF(c)p FG(\)\()p Fw(id)1326 1617 y Fz(0)1338 1635 y FG(\))f(=)g Fw(id)p FG(.)23 b(T)m(o)14 b(see)j(this,)257 1685 y(observ)o(e)22 b(that)f FB(#)529 1695 y FA(\034)s Fz(!)p FA(\034)602 1685 y FG(\()p Fw(id)649 1667 y Fz(0)661 1685 y FG(\)\()p FF(k)q FG(\))i(=)g FF(\025x)858 1691 y FA(k)879 1685 y FF(:)p FG(\()p Fw(id)938 1667 y Fz(0)949 1685 y FG(\()p FB(")p FG(\()p Fw(fam)1068 1691 y FA(\034)1089 1685 y FG(\()p FF(x)1129 1670 y Fz(1)1129 1697 y FA(k)1164 1685 y FG(\)\)\)\()p FF(k)15 b FG(+)f(1\)\))23 b(=)g FF(\025x)1490 1691 y FA(k)1510 1685 y FF(x)1534 1691 y FA(k)1554 1685 y FG(,)f(hence)257 1735 y Fw(ext)q FG(\()p FB(#)p FG(\()p Fw(id)394 1716 y Fz(0)406 1735 y FG(\)\))12 b(=)g FF(\025x)542 1741 y Fy(1)560 1735 y FF(x)584 1741 y Fy(1)616 1735 y FG(and)i(therefore)h FB(I)s FG(\()p FF(c)p FG(\)\()p Fw(id)995 1716 y Fz(0)1007 1735 y FG(\))c(=)h([)-7 b([)p FF(\025x)1143 1741 y Fy(1)1161 1735 y FF(x)1185 1741 y Fy(1)1203 1735 y FG(])g(])1220 1720 y Fz(I)1253 1735 y FG(=)12 b Fw(id)q FG(.)p 1672 1710 V 1672 1733 2 24 v 1688 1733 V 1672 1735 18 2 v 257 1814 a Fn(Example)j(20.)21 b FG(In)14 b(the)g(non-con\015uen)o(t)g(rewrite)g(system)f(considered)i (in)e(example)f(7)h(w)o(e)257 1864 y(had)h FF(d)p FG(2)d FB(\000)-7 b(!)11 b FF(\025x)p FG(2.)17 b(Ho)o(w)o(ev)o(er,)d(for)f (the)h(in)o(terpretation)g(of)f FF(d)p FG(2)g(and)g FF(\025x)p FG(2)g(w)o(e)h(get)f(di\013eren)o(t)257 1914 y(v)n(alues:)31 b([)-7 b([)p FF(d)p FG(2])g(]\(3)524 1899 y Fz(1)556 1914 y FG(\))22 b(=)h FB(I)s FG(\()p FF(d)p FG(\)\(2)766 1899 y Fz(1)800 1914 y FG(\)\(3)853 1899 y Fz(1)888 1914 y FG(\))f(=)h(3)1002 1899 y Fz(1)1057 1914 y FG(since)e Fw(sel)1209 1920 y FA(d)1228 1914 y FG(\(2)p FF(;)7 b FG(3\))21 b(=)i FF(dx)p FG(3)e FB(7\000)-7 b(!)21 b FG(3,)g(but)257 1964 y([)-7 b([)p FF(\025x)p FG(2])g(]\(3)397 1949 y Fz(1)430 1964 y FG(\))12 b(=)g(2)523 1949 y Fz(1)558 1964 y FG(.)p 1672 1939 V 1672 1962 2 24 v 1688 1962 V 1672 1964 18 2 v 320 2043 a(Ho)o(w)o(ev)o(er,)h(w)o(e)i(still)e(ha)o (v)o(e)g(correctness)k(of)c(normalization)e(b)o(y)j(ev)n(aluation.)257 2122 y Fn(Theorem)h(21.)70 b Ft(a.)21 b(If)14 b FF(M)j FB(\000)-7 b(!)11 b FF(Q)p Ft(,)j(then)i FB(#)997 2089 y Fu(\000)1016 2122 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1095 2128 y Fz(")1113 2089 y Fu(\001)1132 2122 y FG(\()p FF(k)q FG(\))11 b(=)h FF(Q)1275 2107 y Fz(1)1310 2122 y FG(\()p FF(k)q FG(\))j Ft(for)f FF(k)f(>)f Fw(FV)p FG(\()p FF(M)5 b FG(\))p Ft(.)309 2208 y(b.)20 b(Assume)d FF(M)j FB(\000)-7 b(!)645 2214 y FA(s)677 2208 y FF(Q)p Ft(.)24 b(Then)17 b FB(#)878 2174 y Fu(\000)897 2208 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])976 2214 y Fz(")993 2174 y Fu(\001)1012 2208 y FG(\()p FF(k)q FG(\))15 b(=)h FF(Q)1163 2193 y Fz(1)1198 2208 y FG(\()p FF(k)q FG(\))h Ft(for)f FF(k)g(>)f Fw(FV)p FG(\()p FF(M)5 b FG(\))p Ft(,)17 b(and)g(if)361 2258 y Fw(inst)p FG(?)443 2264 y FA(P)471 2258 y FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])566 2264 y Fz(")583 2258 y FG(\))12 b(=)f(t)-7 b(t)14 b Ft(for)f(some)h(c)n (onstructor)f(p)n(attern)h FF(P)6 b Ft(,)13 b(then)h(ther)n(e)f(ar)n(e) g(terms)g Fx(L)361 2307 y Ft(such)j(that)e FF(M)j FG(=)12 b FF(P)668 2313 y Fq(x)691 2307 y FG([)p Fx(L)p FG(])i Ft(and)i Fw(gp)o(red)936 2317 y FA(P)963 2307 y FG(\([)-7 b([)p FF(M)5 b FG(])-7 b(])1058 2313 y Fz(")1076 2307 y FG(\))11 b(=)h([)-7 b([)p Fx(L)p FG(])g(])1212 2313 y Fz(")1230 2292 y Fy(6)1249 2307 y Ft(.)309 2389 y(c.)20 b(If)15 b FF(M)h FB(\000)-7 b(!)528 2395 y FA(w)566 2389 y FF(Q)p Ft(,)15 b(then)g FG([)-7 b([)p FF(M)5 b FG(])-7 b(])798 2395 y Fz(")827 2389 y FG(=)12 b([)-7 b([)p FF(Q)p FG(])g(])938 2395 y Fz(")955 2389 y Ft(.)p 257 2425 573 2 v 304 2452 a Fl(6)321 2464 y FC(Moreo)o(v)o(er,)13 b(if)h(w)o(e)h(assume)e(strong)f(uniformit)o(y)g(of)i(all)f Fk(sel)1123 2468 y Fj(c)1139 2464 y FC(-functions)f(\(as)i(w)o(e)g (implicitely)e(did)h(in)257 2503 y([4]\),)e(then)f FL(#)428 2476 y Fa(\000)444 2503 y FC([)-6 b([)p Fe(M)t FC(])g(])510 2509 y FD(")527 2476 y Fa(\001)553 2503 y FC(=)10 b Fe(Q)618 2491 y FD(1)662 2503 y FC(holds)h(in)g(parts)f(a)h(and)g(b.)953 2628 y FG(21)p eop %%Page: 22 22 22 21 bop 257 262 a Fn(Example)15 b(22.)21 b FG(Without)16 b(strong)i(uniformit)o(y)c(of)i(the)i(select)g(functions)f(the)g (stronger)257 311 y(claim)c FB(#)389 278 y Fu(\000)408 311 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])487 317 y Fz(")505 278 y Fu(\001)536 311 y FG(=)13 b FF(Q)614 296 y Fz(1)663 311 y FG(for)h FF(M)j FB(\000)-6 b(!)12 b FF(Q)i FG(is)g(not)h(true.)20 b(F)m(or)14 b(a)g(coun)o(terexample)h(consider)257 361 y(the)f(single)e(rewrite)i(rule)f FF(cxx)e FB(7\000)-7 b(!)808 367 y Fp(rew)867 361 y FF(d)12 b FG(with)g FF(x;)7 b(d)12 b FG(of)g(ground)h(t)o(yp)q(e)g FF(\034)5 b FG(.)17 b(Then)c FF(cx)1550 367 y Fy(0)1569 361 y FF(x)1593 367 y Fy(1)1623 361 y FB(\000)-7 b(!)257 411 y FF(cx)299 417 y Fy(0)318 411 y FF(x)342 417 y Fy(1)374 411 y FG(b)o(y)14 b FE(P)l(assApp)p FG(,)g(hence)h FF(cx)785 417 y Fy(0)815 411 y FB(\000)-6 b(!)11 b FF(\025x)942 417 y Fy(1)960 411 y FF(:cx)1014 417 y Fy(0)1032 411 y FF(x)1056 417 y Fy(1)1088 411 y FG(b)o(y)k FE(Et)m(a)p FG(.)j(W)m(e)13 b(obtain)475 500 y FB(#)495 467 y Fu(\000)514 500 y FG([)-7 b([)p FF(cx)573 506 y Fy(0)591 500 y FG(])g(])608 506 y Fz(")626 467 y Fu(\001)645 500 y FG(\(0\))12 b(=)f FF(\025x)801 506 y Fy(0)820 467 y Fu(\000)839 500 y FB(#)860 510 y FA(\034)880 467 y Fu(\000)900 500 y FG([)-7 b([)p FF(cx)959 506 y Fy(0)976 500 y FG(])g(])993 506 y Fz(")1011 500 y FG(\()p FB(")1048 510 y FA(\034)1069 500 y FG(\()p FF(x)1109 483 y Fz(1)1109 510 y Fy(0)1144 500 y FG(\)\))1176 467 y Fu(\001)1195 500 y FG(\(1\))1248 467 y Fu(\001)710 567 y FG(=)11 b FF(\025x)801 573 y Fy(0)820 533 y Fu(\000)839 567 y FB(#)860 577 y FA(\034)880 533 y Fu(\000)900 567 y FB(I)s FG(\()p FF(c)p FG(\)\()p Fw(fam)1057 573 y FA(\034)1078 567 y FG(\()p FF(x)1118 550 y Fz(1)1118 577 y Fy(0)1153 567 y FG(\))p FF(;)c Fw(fam)1253 573 y FA(\034)1274 567 y FG(\()p FF(x)1314 550 y Fz(1)1314 577 y Fy(0)1349 567 y FG(\)\))1381 533 y Fu(\001)1400 567 y FG(\(1\))1453 533 y Fu(\001)257 669 y FG(No)o(w)14 b Fw(ext)q FG(\()p FB(#)441 679 y FA(\034)462 669 y FG(\()p Fw(fam)544 675 y FA(\034)565 669 y FG(\()p FF(x)605 654 y Fz(1)605 679 y Fy(0)640 669 y FG(\)\)\))e(=)f Fw(ext)q FG(\()p FF(x)836 654 y Fz(1)836 679 y Fy(0)871 669 y FG(\))h(=)g FF(x)967 675 y Fy(0)999 669 y FG(and)h Fw(sel)1124 675 y FA(c)1141 669 y FG(\()p FF(x)1181 675 y Fy(0)1199 669 y FF(;)7 b(x)1242 675 y Fy(0)1260 669 y FG(\))12 b(=)f FF(cxx)g FB(7\000)-6 b(!)1476 675 y Fp(rew)1535 669 y FF(d)p FG(,)13 b(hence)710 770 y(=)e FF(\025x)801 776 y Fy(0)820 737 y Fu(\000)839 770 y FB(#)860 780 y FA(\034)880 737 y Fu(\000)900 770 y FB(I)s FG(\()p FF(d)p FG(\)\))996 737 y Fu(\001)1014 770 y FG(\(1\))1067 737 y Fu(\001)710 837 y FG(=)g FF(\025x)801 843 y Fy(0)820 803 y Fu(\000)839 837 y FB(#)860 847 y FA(\034)880 803 y Fu(\000)900 837 y Fw(fam)965 843 y FA(\034)986 837 y FG(\()p FF(d)1024 820 y Fz(1)1059 837 y FG(\))1075 803 y Fu(\001)1094 837 y FG(\(1\))1147 803 y Fu(\001)710 899 y FG(=)g FF(\025x)801 905 y Fy(0)820 899 y FF(d:)257 989 y FG(Ho)o(w)o(ev)o(er,)j(\()p FF(\025x)500 995 y Fy(1)519 989 y FF(:cx)573 995 y Fy(0)591 989 y FF(x)615 995 y Fy(1)633 989 y FG(\))649 974 y Fz(1)684 989 y FG(\(0\))e(=)g FF(\025x)841 995 y Fy(0)859 989 y FF(:cx)913 995 y Fy(0)931 989 y FF(x)955 995 y Fy(0)974 989 y FG(.)p 1672 964 18 2 v 1672 987 2 24 v 1688 987 V 1672 989 18 2 v 257 1070 a Ft(Pr)n(o)n(of)j(of)g(the)g(the)n(or)n (em.)20 b FG(By)11 b(sim)o(ultaneous)e(induction)h(on)h(the)g(heigh)o (t)g(of)f(the)h(deriv)n(ation)257 1120 y(of)i FF(M)j FB(\000)-7 b(!)11 b FF(Q)i FG(resp.)h FF(M)i FB(\000)-7 b(!)706 1126 y FA(w)744 1120 y FF(Q)13 b FG(and)f FF(M)17 b FB(\000)-7 b(!)993 1126 y FA(s)1022 1120 y FF(Q)p FG(.)17 b(F)m(or)c(brevit)o(y)g(w)o(e)g(lea)o(v)o(e)g(out)f(the)i(rules)257 1170 y(concerning)21 b(pro)q(duct)g(t)o(yp)q(es,)h(since)e(their)h (treatmen)o(t)f(do)q(es)g(not)g(bring)g(up)g(an)o(y)f(new)257 1219 y(issues.)34 b(Note)19 b(that)g(the)g(second)h(statemen)o(t)e(ab)q (out)h FB(\000)-7 b(!)1207 1225 y FA(s)1243 1219 y FG(holds)18 b(if)g FF(P)24 b FG(is)19 b(a)f(v)n(ariable,)257 1269 y(therefore)f(w)o(e)f(assume)f(in)f(the)i(sequel)g(that)g FF(P)k FG(is)c(not)f(a)g(v)n(ariable)f(and)h(th)o(us)h(of)f(ground)257 1319 y(t)o(yp)q(e.)320 1369 y Ft(Case)i FE(Split)p FG(.)764 1402 y FF(M)f FB(\000)-7 b(!)887 1386 y Fz(\003)887 1412 y FA(w)925 1402 y FF(N)47 b(N)16 b FB(\000)-6 b(!)1122 1408 y FA(s)1150 1402 y FF(Q)p 764 1420 420 2 v 890 1458 a(M)16 b FB(\000)-7 b(!)11 b FF(Q)257 1525 y FG(By)j(IH)g(for)g(the)g (left)f(premise)h([)-7 b([)p FF(M)5 b FG(])-7 b(])821 1531 y Fz(")849 1525 y FG(=)12 b([)-7 b([)p FF(N)5 b FG(])-7 b(])965 1531 y Fz(")983 1525 y FG(,)13 b(th)o(us)h(w)o(e)g(can) g(apply)f(the)h(IH)g(to)g(the)g(righ)o(t)257 1575 y(premise)g(to)g (infer)f FB(#)577 1541 y Fu(\000)596 1575 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])675 1581 y Fz(")692 1541 y Fu(\001)711 1575 y FG(\()p FF(k)q FG(\))12 b(=)g FB(#)843 1541 y Fu(\000)862 1575 y FG([)-7 b([)p FF(N)5 b FG(])-7 b(])934 1581 y Fz(")951 1541 y Fu(\001)970 1575 y FG(\()p FF(k)q FG(\))12 b(=)g FF(Q)1114 1560 y Fz(1)1149 1575 y FG(\()p FF(k)q FG(\))i(for)f FF(k)g(>)e Fw(FV)p FG(\()p FF(M)5 b FG(\).)320 1625 y Ft(Case)17 b FE(Et)m(a)p FG(.)720 1657 y FF(M)5 b(y)13 b FB(\000)-7 b(!)11 b FF(Q)p 699 1676 232 2 v 699 1714 a(M)16 b FB(\000)-7 b(!)822 1720 y FA(s)851 1714 y FF(\025y)q(Q)977 1685 y FG(for)13 b FF(y)18 b(=)-26 b FB(2)12 b Fw(FV)p FG(\()p FF(M)5 b FG(\).)257 1781 y(Let)17 b FF(k)f(>)g Fw(FV)p FG(\()p FF(M)5 b FG(\).)25 b(Then)16 b(b)o(y)g(lemma)d(6)j FF(M)5 b(x)998 1787 y FA(k)1033 1781 y FB(\000)-6 b(!)14 b FF(Q)1148 1787 y FA(y)1168 1781 y FG([)p FF(x)1204 1787 y FA(k)1224 1781 y FG(])h(with)h(a)g(deriv)n(ation)f(of)h(the)257 1831 y(same)d(heigh)o(t.)18 b(Hence)277 1920 y FB(#)298 1886 y Fu(\000)317 1920 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])396 1926 y Fz(")413 1886 y Fu(\001)432 1920 y FG(\()p FF(k)q FG(\))12 b(=)f FF(\025x)590 1926 y FA(k)611 1886 y Fu(\000)630 1920 y FB(#)651 1886 y Fu(\000)670 1920 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])749 1926 y Fz(")766 1920 y FG(\()p FB(")p FG(\()p FF(x)843 1903 y Fz(1)843 1930 y FA(k)878 1920 y FG(\)\))910 1886 y Fu(\001)929 1920 y FG(\()p FF(k)10 b FG(+)g(1\))1056 1886 y Fu(\001)499 1987 y FG(=)h FF(\025x)590 1993 y FA(k)611 1953 y Fu(\000)630 1987 y FB(#)651 1953 y Fu(\000)670 1987 y FG([)-7 b([)p FF(M)5 b(x)756 1993 y FA(k)775 1987 y FG(])-7 b(])792 1993 y Fz(")810 1953 y Fu(\001)829 1987 y FG(\()p FF(k)10 b FG(+)g(1\))956 1953 y Fu(\001)499 2053 y FG(=)h FF(\025x)590 2059 y FA(k)611 2020 y Fu(\000)630 2053 y FF(Q)663 2059 y FA(y)683 2053 y FG([)p FF(x)719 2059 y FA(k)738 2053 y FG(])750 2033 y Fz(1)785 2053 y FG(\()p FF(k)f FG(+)g(1\))912 2020 y Fu(\001)1116 2053 y FG(b)o(y)k(IH,)f(since)i FF(k)10 b FG(+)g(1)h FF(>)h Fw(FV)p FG(\()p FF(M)5 b(x)1634 2059 y FA(k)1654 2053 y FG(\))499 2116 y(=)11 b(\()p FF(\025y)q(Q)p FG(\))652 2099 y Fz(1)689 2116 y FG(\()p FF(k)q FG(\))p FF(:)257 2205 y FG(The)k(additional)d(claim)f(holds)j(since)h FF(M)j FG(is)c(not)g(of)f(ground)h(t)o(yp)q(e.)320 2255 y Ft(Case)j FE(V)-5 b(arApp)p FG(.)870 2293 y Fx(M)17 b FB(\000)-6 b(!)11 b Fx(M)1065 2275 y Fz(0)p 838 2312 272 2 v 838 2351 a FF(x)p Fx(M)16 b FB(\000)-6 b(!)993 2357 y FA(s)1021 2351 y FF(x)p Fx(M)1097 2333 y Fz(0)257 2414 y FG(with)14 b FF(x)p Fx(M)19 b FG(of)13 b(ground)h(t)o(yp)q(e.)k (W)m(e)c(ha)o(v)o(e)552 2503 y([)-7 b([)p FF(x)p Fx(M)t FG(])g(])661 2509 y Fz(")691 2503 y FG(=)12 b FB(")755 2469 y Fu(\000)774 2503 y FF(x)798 2486 y Fz(1)833 2469 y Fu(\001\000)871 2503 y FG([)-7 b([)p Fx(M)5 b FG(])-7 b(])957 2509 y Fz(")975 2469 y Fu(\001)1005 2503 y FG(=)12 b Fw(fam)1115 2509 y FA(\034)1136 2469 y Fu(\000)1155 2503 y FF(x)1179 2486 y Fz(1)1213 2503 y FB(#)1234 2469 y Fu(\000)1253 2503 y FG([)-7 b([)p Fx(M)5 b FG(])-7 b(])1339 2509 y Fz(")1357 2469 y Fu(\001\001)953 2628 y FG(22)p eop %%Page: 23 23 23 22 bop 257 262 a FG(b)o(y)15 b(\(2\),)e(hence)402 353 y FB(#)423 319 y Fu(\000)442 353 y FG([)-7 b([)p FF(x)p Fx(M)t FG(])g(])551 359 y Fz(")569 319 y Fu(\001)588 353 y FG(\()p FF(k)q FG(\))12 b(=)699 319 y Fu(\000)718 353 y FF(x)742 336 y Fz(1)777 353 y FB(#)798 319 y Fu(\000)817 353 y FG([)-7 b([)p Fx(M)t FG(])g(])902 359 y Fz(")920 319 y Fu(\001\001)958 353 y FG(\()p FF(k)q FG(\))12 b(=)g FF(x)p FG(\()p Fx(M)1161 335 y Fz(0)1172 353 y FG(\))1188 336 y Fz(1)1224 353 y FG(\()p FF(k)q FG(\))g(=)f(\()p FF(x)p Fx(M)1426 335 y Fz(0)1438 353 y FG(\))1454 336 y Fz(1)1489 353 y FG(\()p FF(k)q FG(\))257 444 y(b)o(y)j(IH.)g(The)g (second)h(claim)d(holds)h(since)522 535 y Fw(inst)p FG(?)603 541 y FA(P)631 535 y FG(\([)-7 b([)p FF(x)p Fx(M)t FG(])g(])756 541 y Fz(")775 535 y FG(\))11 b(=)h Fw(inst)p FG(?)928 541 y FA(P)956 535 y FG(\()p Fw(fam)1037 541 y FA(\034)1058 502 y Fu(\000)1077 535 y FF(x)1101 518 y Fz(1)1136 535 y FB(#)1157 502 y Fu(\000)1176 535 y FG([)-7 b([)p Fx(M)t FG(])g(])1261 541 y Fz(")1279 502 y Fu(\001\001)1317 535 y FG(\))12 b(=)g(f)l(f)s FF(:)320 627 y Ft(Case)17 b FE(Bet)m(a)p FG(.)736 677 y(\()p FF(\025xM)5 b FG(\))p FF(N)g Fx(P)18 b FB(\000)-7 b(!)1014 683 y FA(w)1052 677 y FF(M)1092 683 y FA(x)1113 677 y FG([)p FF(N)5 b FG(])o Fx(P)257 751 y FG(Use)14 b([)-7 b([\()p FF(\025xM)5 b FG(\))p FF(N)g Fx(P)h FG(])-7 b(])569 757 y Fz(")598 751 y FG(=)12 b([)-7 b([)p FF(M)699 757 y FA(x)720 751 y FG([)p FF(N)5 b FG(])o Fx(P)h FG(])-7 b(])834 757 y Fz(")852 751 y FG(,)13 b(whic)o(h)g(holds)f(in)g(ev)o(ery)i(mo)q(del)d (of)h(the)i FF(\025)p FG(-calculus.)320 801 y Ft(Case)j FE(Comp)p FG(.)579 892 y FF(c)p Fx(P)634 898 y Fq(x)657 892 y FG([)p Fx(L)p FG(])p Fx(N)f FB(\000)-6 b(!)835 898 y FA(w)873 892 y FF(Q)906 898 y Fq(x)929 892 y FG([)p Fx(L)p FG(])o Fx(N)47 b FG(if)13 b FF(c)p Fx(P)18 b FB(7\000)-6 b(!)1241 898 y Fp(comp)1323 892 y FF(Q)p FG(.)257 984 y(Then)408 1075 y([)f([)p FF(c)p Fx(P)479 1081 y Fq(x)502 1075 y FG([)p Fx(L)p FG(])o Fx(N)6 b FG(])-7 b(])618 1081 y Fz(")647 1075 y FG(=)12 b([)-7 b([)p FF(c)p Fx(P)762 1081 y Fq(x)785 1075 y FG([)p Fx(L)p FG(]])g(])857 1081 y Fz(")875 1075 y FG([)g([)p Fx(N)t FG(])g(])952 1081 y Fz(")647 1137 y FG(=)12 b([)-7 b([)p FF(c)p Fx(P)6 b FG(])-7 b(])779 1144 y Fy([)p Fq(x)p Fz(7!)p Fy([)h([)p Fq(L)q Fy(])g(])892 1148 y Fd(")910 1144 y Fy(])922 1137 y FG([)f([)p Fx(N)t FG(])g(])999 1143 y Fz(")1059 1137 y FG(b)o(y)14 b(the)g(substitution)g(lemma)647 1205 y(=)e([)-7 b([)p FF(Q)p FG(])g(])758 1212 y Fy([)p Fq(x)o Fz(7!)p Fy([)h([)p Fq(L)q Fy(])g(])870 1216 y Fd(")889 1212 y Fy(])900 1205 y FG([)f([)p Fx(N)t FG(])g(])977 1211 y Fz(")1059 1205 y FG(b)o(y)14 b(lemma)c(18)647 1272 y(=)i([)-7 b([)p FF(Q)741 1278 y Fq(x)764 1272 y FG([)p Fx(L)p FG(])o Fx(N)5 b FG(])-7 b(])879 1278 y Fz(")898 1272 y FF(:)320 1364 y Ft(Case)17 b FE(Ar)o(g)p FG(.)871 1396 y Fx(M)g FB(\000)-7 b(!)1002 1381 y Fz(\003)1002 1407 y FA(w)1040 1396 y Fx(Q)p 809 1415 330 2 v 809 1453 a FF(c)p Fx(M)5 b(N)17 b FB(\000)-7 b(!)1002 1459 y FA(w)1040 1453 y FF(c)p Fx(QN)257 1521 y FG(Then)15 b([)-7 b([)p FF(c)p Fx(M)t(N)5 b FG(])-7 b(])513 1527 y Fz(")543 1521 y FG(=)12 b FB(I)s FG(\()p FF(c)p FG(\)[)-7 b([)p Fx(M)t FG(])g(])748 1527 y Fz(")766 1521 y FG([)g([)p Fx(N)5 b FG(])-7 b(])844 1527 y Fz(")873 1521 y FG(=)12 b FB(I)s FG(\()p FF(c)p FG(\)[)-7 b([)p Fx(Q)o FG(])g(])1062 1527 y Fz(")1080 1521 y FG([)g([)p Fx(N)5 b FG(])-7 b(])1158 1527 y Fz(")1188 1521 y FG(=)11 b([)-7 b([)p FF(c)p Fx(QN)5 b FG(])-7 b(])1363 1527 y Fz(")1395 1521 y FG(b)o(y)13 b(IH.)320 1571 y Ft(Case)k FE(Rew)p FG(.)677 1606 y Fx(M)g FB(\000)-7 b(!)808 1612 y FA(s)837 1606 y Fx(K)880 1612 y Fq(x)904 1606 y FG([)p Fx(L)p FG(])p 625 1625 387 2 v 625 1663 a FF(c)p Fx(M)5 b(N)17 b FB(\000)-7 b(!)818 1669 y FA(w)856 1663 y FF(Q)889 1669 y Fq(x)912 1663 y FG([)p Fx(L)p FG(])p Fx(N)1057 1634 y FG(if)13 b FF(c)p Fx(K)i FB(7\000)-7 b(!)1235 1640 y Fp(rew)1294 1634 y FF(Q)257 1733 y FG(where)13 b Fw(sel)419 1739 y FA(c)436 1733 y FG(\()p Fx(K)495 1739 y Fq(x)519 1733 y FG([)p Fx(L)p FG(])o(\))f(=)g FF(c)p Fx(K)i FB(7\000)-6 b(!)785 1739 y Fp(rew)844 1733 y FF(Q)11 b FG(and)g FF(c)p Fx(M)16 b FG(is)11 b(not)g(the)h(instance)g (of)e(a)h(computation)257 1783 y(rule.)320 1833 y(W)m(e)17 b(sho)o(w)g(that)595 1802 y Fu(V)-28 b(V)643 1833 y Fw(inst)p FG(?)725 1839 y Fq(P)756 1833 y FG(\([)-7 b([)p Fx(M)t FG(])g(])857 1839 y Fz(")875 1833 y FG(\))18 b(=)g(f)l(f)i(for)d(all)g (computation)f(rules)i FF(c)p Fx(P)24 b FB(7\000)-7 b(!)1618 1839 y Fp(comp)257 1883 y FF(Q)p FG(.)27 b(So)17 b(assume)f(\014rst)i (that)718 1852 y Fu(V)-28 b(V)767 1883 y Fw(inst)p FG(?)848 1889 y Fq(P)879 1883 y FG(\([)-7 b([)p Fx(M)t FG(])g(])980 1889 y Fz(")999 1883 y FG(\))16 b(=)h(t)-7 b(t)q(.)27 b(Then)17 b(b)o(y)g(IH)g(there)h(are)f Fx(L)h FG(suc)o(h)257 1933 y(that)g Fx(M)24 b FG(=)19 b Fx(P)509 1939 y Fq(x)533 1933 y FG([)p Fx(L)p FG(])e(\(note)i(that)f Fx(P)25 b FG(is)17 b(linear\),)i(so)f FF(c)p Fx(M)23 b FG(w)o(ould)17 b(b)q(e)i(an)f(instance)g(of)g(a)257 1982 y(computation)d(rule,)h(whic) o(h)g(con)o(tradicts)h(the)f(assumption.)23 b(It)17 b(remains)e(to)g (sho)o(w)h(that)257 2032 y Fw(inst)p FG(?)339 2038 y FA(P)360 2042 y Fi(i)376 2032 y FG(\([)-7 b([)p FF(M)449 2038 y FA(i)462 2032 y FG(])g(])479 2038 y Fz(")497 2032 y FG(\))20 b FB(6)p FG(=)g FB(?)e FG(for)g(all)g FF(i)p FG(.)32 b(But)20 b(if)e(w)o(e)g(had)h(=)h FB(?)e FG(for)g(some)g FF(i)p FG(,)i(then)f([)-7 b([)p FF(M)1589 2038 y FA(i)1602 2032 y FG(])g(])1619 2038 y Fz(")1657 2032 y FG(=)257 2082 y FB(?)20 b FG(in)h(con)o(trast)g(to)f FB(#)p FG(\([)-7 b([)p Fx(M)5 b FG(])-7 b(])713 2088 y Fz(")731 2082 y FG(\)\()p FF(k)q FG(\))23 b(=)g Fx(K)923 2088 y Fq(x)947 2082 y FG([)p Fx(L)p FG(])1001 2061 y Fz(1)1037 2082 y FG(\()p FF(k)q FG(\),)f(whic)o(h)e(holds)h(b)o(y)f(IH.)g(Moreo)o(v)o (er,)257 2132 y Fw(sel)302 2138 y FA(c)319 2132 y FG(\()p Fw(ext)p FG(\()p FB(#)p FG(\([)-7 b([)p Fx(M)5 b FG(])-7 b(])526 2138 y Fz(")544 2132 y FG(\)\)\))18 b(=)h Fw(sel)705 2138 y FA(c)722 2132 y FG(\()p Fx(K)781 2138 y Fq(x)805 2132 y FG([)p Fx(L)p FG(])o(\))f(=)h FF(c)p Fx(K)i FB(7\000)-7 b(!)18 b FF(Q)p FG(.)29 b(This)18 b(giv)o(es)g(us)g(the)g(necessary)257 2182 y(information)11 b(ab)q(out)j FB(I)s FG(\()p FF(c)p FG(\).)432 2273 y([)-7 b([)p FF(c)p Fx(M)5 b(N)g FG(])-7 b(])580 2279 y Fz(")610 2273 y FG(=)11 b FB(I)s FG(\()p FF(c)p FG(\)\([)-7 b([)p Fx(M)5 b FG(])-7 b(])831 2279 y Fz(")849 2273 y FG(\)\([)g([)p Fx(N)5 b FG(])-7 b(])959 2279 y Fz(")977 2273 y FG(\))610 2335 y(=)11 b([)-7 b([)p FF(Q)p FG(])g(])720 2342 y Fy([)p Fq(x)o Fz(7!)p Fy([)h([)p Fq(L)q Fy(])g(])832 2346 y Fd(")851 2342 y Fy(])862 2335 y FG(\([)f([)p Fx(N)5 b FG(])-7 b(])956 2341 y Fz(")974 2335 y FG(\))44 b(b)o(y)14 b(de\014nition)g(of)f FB(I)s FG(\()p FF(c)p FG(\))610 2403 y(=)e([)-7 b([)p FF(Q)703 2409 y Fq(x)726 2403 y FG([)p Fx(L)p FG(])o(])g(])797 2409 y Fz(")815 2403 y FG(\([)g([)p Fx(N)5 b FG(])-7 b(])909 2409 y Fz(")927 2403 y FG(\))91 b(b)o(y)14 b(the)h (substitution)f(lemma)610 2465 y(=)d([)-7 b([)p FF(Q)703 2471 y Fq(x)726 2465 y FG([)p Fx(L)p FG(])o Fx(N)5 b FG(])-7 b(])841 2471 y Fz(")860 2465 y FF(:)953 2628 y FG(23)p eop %%Page: 24 24 24 23 bop 257 262 a Ft(Case)17 b FE(P)l(assApp)p FG(.)746 300 y Fx(M)g FB(\000)-7 b(!)877 306 y FA(s)906 300 y Fx(M)958 282 y Fz(0)1011 300 y Fx(N)17 b FB(\000)-7 b(!)11 b Fx(N)1189 282 y Fz(0)p 746 319 456 2 v 794 358 a FF(c)p Fx(M)5 b(N)16 b FB(\000)-6 b(!)987 364 y FA(s)1016 358 y FF(c)p Fx(M)1086 340 y Fz(0)1097 358 y Fx(N)1142 340 y Fz(0)257 424 y FG(where)16 b FF(c)p Fx(M)21 b FG(is)14 b(not)h(an)g(instance)h(of)e(a)h(computation)e(rule,)i Fw(sel)1270 430 y FA(c)1287 424 y FG(\()p Fx(M)1355 409 y Fm(0)1369 424 y FG(\))e(=)h Fw(no)p FG(-)p Fw(match)h FG(and)257 474 y FF(c)p Fx(M)5 b(N)20 b FG(of)13 b(ground)h(t)o(yp)q (e.)320 524 y(Let)j(us)g(\014rst)g(consider)g(the)g(case)h(where)f FF(c)g FG(is)f(a)g(constructor)j(\(then)e Fx(N)k FG(is)c(empt)o(y\).) 257 574 y(W)m(e)d(obtain)654 624 y([)-7 b([)p FF(c)p Fx(M)t FG(])g(])757 630 y Fz(")787 624 y FG(=)12 b FB(I)s FG(\()p FF(c)p FG(\)\([)-7 b([)p Fx(M)t FG(])g(])1008 630 y Fz(")1026 624 y FG(\))12 b(=)g Fw(in)1129 630 y FA(c)1146 624 y FG(\([)-7 b([)p Fx(M)t FG(])g(])1247 630 y Fz(")1266 624 y FG(\))p FF(;)257 698 y FG(hence)16 b(b)o(y)d(de\014nition)h(of)f FB(#)h FG(and)f(the)i(IH)541 790 y FB(#)p FG(\([)-7 b([)p FF(c)p Fx(M)t FG(])g(])681 796 y Fz(")699 790 y FG(\)\()p FF(k)q FG(\))12 b(=)826 756 y Fu(\000)845 790 y FF(c)863 773 y Fz(1)898 790 y FB(#)p FG(\([)-7 b([)p Fx(M)t FG(])g(])1020 796 y Fz(")1039 790 y FG(\))1055 756 y Fu(\001)1074 790 y FG(\()p FF(k)q FG(\))11 b(=)1184 756 y Fu(\000)1203 790 y FF(c)p Fx(M)1273 771 y Fz(0)1285 756 y Fu(\001)1304 765 y Fz(1)1339 790 y FG(\()p FF(k)q FG(\))p FF(:)257 881 y FG(This)17 b(is)g(the)h (\014rst)g(claim.)26 b(No)o(w)17 b(assume)g FF(P)22 b FG(=)17 b FF(c)p Fx(P)24 b FG(and)17 b Fw(inst)p FG(?)1282 887 y FA(P)1310 881 y FG(\([)-7 b([)p FF(c)p Fx(M)t FG(])g(])1429 887 y Fz(")1447 881 y FG(\))17 b(=)h(t)-7 b(t.)28 b(Then)257 900 y Fu(V)-28 b(V)306 931 y Fw(inst)p FG(?)388 937 y Fq(P)418 931 y FG(\([)-7 b([)p Fx(M)5 b FG(])-7 b(])520 937 y Fz(")538 931 y FG(\))13 b(=)h(t)-7 b(t)15 b(and)g(b)o(y)f(IH)h (there)i(are)e Fx(L)g FG(suc)o(h)h(that)f Fx(M)j FG(=)c Fx(P)1411 937 y Fq(x)1435 931 y FG([)p Fx(L)p FG(])g(\(note)h(that)257 981 y FF(c)p Fx(P)21 b FG(is)14 b(linear\),)f(hence)i FF(c)p Fx(M)i FG(=)11 b FF(c)p Fx(P)806 987 y Fq(x)830 981 y FG([)p Fx(L)o FG(].)320 1030 y(If)20 b FF(c)g FG(is)g(not)h(a)f (constructor,)j(w)o(e)e(ha)o(v)o(e)985 999 y Fu(V)-28 b(V)1033 1030 y Fw(inst)p FG(?)1115 1036 y Fq(P)1146 1030 y FG(\([)-7 b([)p Fx(M)t FG(])g(])1247 1036 y Fz(")1265 1030 y FG(\))23 b(=)g(f)l(f)g(\(with)d(the)h(same)257 1080 y(argumen)o(t)13 b(as)h(in)g FE(Rew)p FG(\))g(and)407 1172 y Fw(sel)451 1178 y FA(c)468 1172 y FG(\()p Fw(ext)q FG(\()p FB(#)p FG(\([)-7 b([)p Fx(M)t FG(])g(])675 1178 y Fz(")693 1172 y FG(\)\)\))12 b(=)g Fw(sel)842 1178 y FA(c)858 1172 y FG(\()p Fw(ext)927 1138 y Fu(\000)946 1172 y FG(\()p Fx(M)1015 1153 y Fz(0)1026 1172 y FG(\))1042 1154 y Fz(1)1077 1138 y Fu(\001)1108 1172 y FG(=)g Fw(sel)1196 1178 y FA(c)1213 1172 y FG(\()p Fx(M)1281 1153 y Fz(0)1293 1172 y FG(\))g(=)f Fw(no)p FG(-)p Fw(match)p FF(;)257 1267 y FG(since)18 b(b)o(y)e(IH)g FB(#)505 1233 y Fu(\000)524 1267 y FG([)-7 b([)p Fx(M)t FG(])g(])609 1273 y Fz(")628 1233 y Fu(\001)647 1267 y FG(\()p FF(k)q FG(\))16 b(=)g(\()p Fx(M)834 1249 y Fz(0)846 1267 y FG(\))862 1252 y Fz(1)897 1267 y FG(\()p FF(k)q FG(\))h(for)f FF(k)g(>)h Fw(FV)o FG(\()p Fx(M)6 b FG(\).)26 b(No)o(w)16 b(w)o(e)g(can)h(compute)257 1317 y([)-7 b([)p FF(c)p Fx(M)5 b(N)g FG(])-7 b(])405 1323 y Fz(")423 1317 y FG(.)583 1408 y([)g([)p FF(c)p Fx(M)t(N)5 b FG(])-7 b(])730 1414 y Fz(")760 1408 y FG(=)12 b FB(I)s FG(\()p FF(c)p FG(\)\([)-7 b([)p Fx(M)t FG(])g(])981 1414 y Fz(")999 1408 y FG(\)\([)g([)p Fx(N)5 b FG(])-7 b(])1109 1414 y Fz(")1127 1408 y FG(\))760 1475 y(=)12 b FB(")824 1441 y Fu(\000)843 1475 y FF(c)861 1458 y Fz(1)897 1475 y FB(#)917 1441 y Fu(\000)936 1475 y FG([)-7 b([)p Fx(M)5 b FG(])-7 b(])1022 1481 y Fz(")1040 1441 y Fu(\001\001\000)1097 1475 y FG([)g([)p Fx(N)t FG(])g(])1174 1481 y Fz(")1192 1441 y Fu(\001)760 1542 y FG(=)12 b Fw(fam)869 1548 y FA(\034)890 1508 y Fu(\000)909 1542 y FF(c)927 1525 y Fz(1)962 1542 y FB(#)983 1508 y Fu(\000)1002 1542 y FG([)-7 b([)p Fx(M)t FF(;)7 b Fx(N)e FG(])-7 b(])1150 1548 y Fz(")1168 1508 y Fu(\001\001)1253 1542 y FG(b)o(y)14 b(\(2\))257 1633 y(hence)545 1725 y FB(#)o FG(\([)-7 b([)p FF(c)p Fx(M)5 b(N)g FG(])-7 b(])729 1731 y Fz(")747 1725 y FG(\)\()p FF(k)q FG(\))12 b(=)874 1691 y Fu(\000)893 1725 y FF(c)911 1707 y Fz(1)946 1725 y FB(#)967 1691 y Fu(\000)986 1725 y FG([)-7 b([)p Fx(M)t FF(;)7 b Fx(N)e FG(])-7 b(])1134 1731 y Fz(")1152 1691 y Fu(\001\001)1190 1725 y FG(\()p FF(k)q FG(\))830 1794 y(=)874 1760 y Fu(\000)893 1794 y FF(c)p Fx(M)963 1775 y Fz(0)975 1794 y Fx(N)1019 1775 y Fz(0)1031 1760 y Fu(\001)1050 1769 y Fz(1)1085 1794 y FG(\()p FF(k)q FG(\))147 b(b)o(y)14 b(IH.)257 1885 y(The)h(second)g(claim)c(holds)j(trivially)e(again.)p 1672 1860 18 2 v 1672 1883 2 24 v 1688 1883 V 1672 1885 18 2 v 257 1968 a Fn(Corollary)j(23.)20 b Ft(If)12 b FF(M)k Ft(is)11 b(str)n(ongly)g(normalizable,)g(then)h FF(M)k FB(\000)-7 b(!)12 b Fw(nf)s FG(\()p FF(M)5 b FG(\))p Ft(,)11 b(and)h(ther)n(efor)n(e)257 2018 y(the)j(\(long\))g(normal)g (form)f Fw(nf)s FG(\()p FF(M)5 b FG(\))15 b Ft(of)g FF(M)k Ft(c)n(an)c(b)n(e)g(obtaine)n(d)h(as)e FB(#)1294 1984 y Fu(\000)1313 2018 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1392 2024 y Fz(")1409 1984 y Fu(\001)1428 2018 y FG(\()p FF(k)q FG(\))12 b(=)1527 2024 y FA(\013)1563 2018 y Fw(nf)s FG(\()p FF(M)5 b FG(\))p Ft(,)257 2068 y(for)15 b(any)g FF(k)e(>)e Fw(FV)p FG(\()p FF(M)5 b FG(\))p Ft(.)257 2151 y(Pr)n(o)n(of.)20 b FG(Let)12 b FF(M)j FG(b)q(e)c(strongly)g(normalizable.)j(Then)e FF(M)k FB(\000)-7 b(!)11 b FF(Q)g FG(for)f(some)g FF(Q)g FG(b)o(y)h(lemma)c (5,)257 2200 y(and)20 b FF(Q)g FG(is)g(the)g(long)f(normal)f(form)g Fw(nf)s FG(\()p FF(M)5 b FG(\))20 b(of)f FF(M)25 b FG(b)o(y)20 b(lemma)c(4.)36 b(By)20 b(the)h(theorem)257 2250 y FB(#)278 2217 y Fu(\000)297 2250 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])376 2256 y Fz(")393 2217 y Fu(\001)412 2250 y FG(\()p FF(k)q FG(\))12 b(=)g Fw(nf)s FG(\()p FF(M)5 b FG(\))637 2235 y Fz(1)672 2250 y FG(\()p FF(k)q FG(\))14 b(for)g(an)o(y)f FF(k)g(>)e Fw(FV)p FG(\()p FF(M)5 b FG(\).)18 b(No)o(w)c(use)h(lemma)10 b(9b.)p 1672 2225 V 1672 2249 2 24 v 1688 2249 V 1672 2251 18 2 v 320 2333 a(Another)18 b(immedia)o(te)d(corollary)i(is)g (that)h(the)g FB(\000)-7 b(!)p FG(-relation)17 b(is)g(indeed)h(a)f (\(partial\))257 2383 y(function:)k(If)15 b FF(M)k FB(\000)-7 b(!)14 b FF(Q)h FG(and)g FF(M)k FB(\000)-7 b(!)14 b FF(P)6 b FG(,)15 b(then)h FF(P)1081 2368 y Fz(1)1115 2383 y FG(\()p FF(k)q FG(\))e(=)h FB(#)1252 2349 y Fu(\000)1271 2383 y FG([)-7 b([)p FF(M)5 b FG(])-7 b(])1350 2389 y Fz(")1367 2349 y Fu(\001)1386 2383 y FG(\()p FF(k)q FG(\))14 b(=)g FF(Q)1534 2368 y Fz(1)1569 2383 y FG(\()p FF(k)q FG(\))i(for)257 2433 y FF(k)d(>)f Fw(FV)p FG(\()p FF(M)5 b FG(\),)13 b(hence)i FF(P)i FG(=)681 2439 y FA(\013)717 2433 y FF(Q)c FG(b)o(y)h(lemma)d(10.)953 2628 y(24)p eop %%Page: 25 25 25 24 bop 257 262 a Ft(A)n(cknow)r(le)n(dgements.)21 b FG(The)13 b(presen)o(t)h(w)o(ork)e(has)h(b)q(ene\014tted)h (considerably)f(from)d(ideas)j(of)257 311 y FE(Felix)j(Jo)o(a)o (chimski)e FG(and)f FE(Ralph)k(Ma)m(tthes)d FG(concerning)h(strategies) g(for)e(normaliza-)257 361 y(tion)k(pro)q(ofs,)g(including)f FF(\021)q FG(-expansion)h(and)g(primitiv)o(e)e(recursion.)29 b(In)17 b(particular,)g(the)257 411 y(idea)f(to)h(emplo)o(y)d(the)j (inductiv)o(e)g(de\014nition)f(of)f(the)i(relation)f FF(M)21 b FB(\000)-7 b(!)15 b FF(Q)h FG(is)h(essen)o(tially)257 461 y(due)11 b(to)g(them.)16 b(W)m(e)10 b(also)g(w)o(an)o(t)g(to)h (thank)f FE(Holger)j(Benl)e FG(for)f(illuminating)d(discussions,)257 511 y(and)14 b(an)g(anon)o(ymous)e(referee)k(for)d(his)h(helpful)f (commen)o(ts.)257 648 y FH(References)278 739 y FG([1])20 b(Thorsten)e(Altenkirc)o(h,)g(Martin)g(Hofmann,)d(and)i(Thomas)g (Streic)o(her.)31 b(Categor-)343 789 y(ical)16 b(reconstruction)i(of)e (a)h(reduction)g(free)h(normalization)c(pro)q(of.)26 b(In)17 b Ft(CTCS'95,)343 838 y(Cambridge)p FG(,)d(v)o(olume)g(953)g (of)h Ft(LNCS)p FG(,)g(pages)h(182{199.)d(Springer)i(V)m(erlag,)g (Berlin,)343 888 y(Heidelb)q(erg,)f(New)g(Y)m(ork,)f(1995.)278 971 y([2])20 b(Holger)14 b(Benl,)g(Ulric)o(h)f(Berger,)j(Helm)o(ut)c (Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg,)j(Monik)n(a)e(Seisen)o(b)q(erger,) 343 1021 y(and)i(W)m(olfgang)e(Zub)q(er.)24 b(Pro)q(of)15 b(theory)h(at)g(w)o(ork:)e(Program)g(dev)o(elopmen)o(t)h(in)g(the)343 1071 y(Minlog)d(system.)18 b(In)c(W.)f(Bib)q(el)i(and)e(P)m(.H.)g(Sc)o (hmitt,)g(editors,)h Ft(A)o(utomate)n(d)h(De)n(duc-)343 1121 y(tion)i({)h(A)f(Basis)g(for)g(Applic)n(ations)p FG(,)f(v)o(olume)f(I)q(I:)h(Systems)g(and)h(Implemen)o(tatio)o(n)343 1170 y(T)m(ec)o(hniques)h(of)e Ft(Applie)n(d)i(L)n(o)n(gic)g(Series)p FG(,)e(pages)i(41{71.)d(Klu)o(w)o(er)j(Academic)e(Pub-)343 1220 y(lishers,)e(Dordrec)o(h)o(t,)g(1998.)278 1303 y([3])20 b(Ulric)o(h)14 b(Berger.)22 b(Con)o(tin)o(uous)14 b(functionals)g(of)g (dep)q(enden)o(t)i(and)f(trans\014nite)g(t)o(yp)q(es.)343 1353 y(Habilitationssc)o(hrift,)22 b(Mathematisc)o(hes)j(Institut)f (der)h(Univ)o(ersit\177)-21 b(at)25 b(M)q(\177)-22 b(unc)o(hen,)343 1403 y(1997.)278 1486 y([4])20 b(Ulric)o(h)11 b(Berger,)i(Matthias)e (Eb)q(erl,)g(and)h(Helm)o(ut)e(Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg.)15 b(Normalization)343 1536 y(b)o(y)9 b(ev)n(aluation.)g(In)h(B.)f(M\177) -21 b(oller)9 b(and)h(J.V.)f(T)m(uc)o(k)o(er,)g(editors,)h Ft(Pr)n(osp)n(e)n(cts)g(for)h(Har)n(dwar)n(e)343 1586 y(F)m(oundations)p FG(,)f(v)o(olume)f(1546)f(of)i Ft(LNCS)p FG(,)f(pages)i(117{137.)c(Springer)k(V)m(erlag,)e(Berlin,)343 1635 y(Heidelb)q(erg,)14 b(New)g(Y)m(ork,)f(1998.)278 1718 y([5])20 b(Ulric)o(h)d(Berger)i(and)f(Helm)o(ut)e(Sc)o(h)o(wic)o (h)o(ten)o(b)q(erg.)30 b(An)18 b(in)o(v)o(erse)h(of)e(the)h(ev)n (aluation)343 1768 y(functional)h(for)i(t)o(yp)q(ed)g FF(\025)p FG(-calculus.)39 b(In)21 b(R.)e(V)m(em)o(uri,)g(editor,)h Ft(Pr)n(o)n(c)n(e)n(e)n(dings)i(6'th)343 1818 y(Symp)n(osium)12 b(on)h(L)n(o)n(gic)f(in)g(Computer)g(Scienc)n(e)h(\(LICS'91\))p FG(,)d(pages)h(203{211.)e(IEEE)343 1868 y(Computer)k(So)q(ciet)o(y)h (Press,)h(Los)f(Alamitos,)d(1991.)278 1951 y([6])20 b(Thierry)12 b(Co)q(quand)f(and)g(P)o(eter)i(Dyb)r(jer.)h(In)o(tuitionistic)d(mo)q (del)f(constructions)j(and)343 2001 y(normalization)c(pro)q(ofs.)16 b Ft(Mathematic)n(al)d(Structur)n(es)g(in)g(Computer)g(Scienc)n(e)p FG(,)g(7:73{)343 2051 y(94,)g(1997.)278 2134 y([7])20 b(Ro)o(y)13 b(L.)g(Crole.)18 b Ft(Cate)n(gories)c(for)g(T)m(yp)n(es)p FG(.)k(Cam)o(bridge)12 b(Univ)o(ersit)o(y)i(Press,)h(1993.)278 2217 y([8])20 b(Olivier)9 b(Dan)o(vy)m(.)i(Pragmatics)e(of)g(t)o(yp)q (e-directed)j(partial)e(ev)n(aluation.)g(In)g(O.)g(Dan)o(vy)m(,)343 2266 y(R.)16 b(Gl)q(\177)-22 b(uc)o(k,)17 b(and)h(P)m(.)f(Thiemann,)e (editors,)j Ft(Partial)g(Evaluation)p FG(,)g(v)o(olume)d(1110)i(of)343 2316 y Ft(LNCS)p FG(,)c(pages)h(73{94.)e(Springer)j(V)m(erlag,)d (Berlin,)i(Heidelb)q(erg,)g(New)h(Y)m(ork,)e(1996.)278 2399 y([9])20 b(Nicolaas)12 b(G.)h(de)h(Bruijn.)j(Lam)o(b)q(da)12 b(calculus)h(notation)g(with)g(nameless)g(dummies,)343 2449 y(a)e(to)q(ol)f(for)h(automatic)f(form)o(ula)e(manipulation,)g (with)j(application)f(to)h(the)h(Ch)o(urc)o(h-)343 2499 y(Rosser)i(theorem.)k Ft(Indagationes)e(Math.)p FG(,)e(34:381{392,)c (1972.)953 2628 y(25)p eop %%Page: 26 26 26 25 bop 257 262 a FG([10])20 b(Andrzej)d(Filinski.)24 b(A)16 b(seman)o(tic)g(accoun)o(t)g(of)g(t)o(yp)q(e-directed)i(partial) d(ev)n(aluation.)343 311 y(In)f Ft(Principles)g(and)i(Pr)n(actic)n(e)e (of)h(De)n(clar)n(ative)f(Pr)n(o)n(gr)n(amming)g(1999)p FG(,)g(v)o(olume)e(1702)343 361 y(of)k Ft(LNCS)p FG(,)g(pages)h (378{395.)e(Springer)i(V)m(erlag,)f(Berlin,)h(Heidelb)q(erg,)g(New)g(Y) m(ork,)343 411 y(1999.)g(h)o(ttp://www.brics.dk/)12 b(andrzej/pap)q (ers/.)257 494 y([11])20 b(Jim)12 b(Lam)o(b)q(ek)g(and)h(Phil)g(Scott.) 18 b Ft(Intr)n(o)n(duction)d(to)g(higher)f(or)n(der)g(c)n(ate)n(goric)n (al)g(lo)n(gic)p FG(,)343 544 y(v)o(olume)c(7)i(of)f Ft(Cambridge)i(Studies)h(in)f(A)n(dvanc)n(e)n(d)h(Mathematics)p FG(.)i(Cam)o(bridge)10 b(Uni-)343 594 y(v)o(ersit)o(y)k(Press,)h(1986.) 257 677 y([12])20 b(John)10 b(McCarth)o(y)m(.)i(Recursiv)o(e)f (functions)f(of)g(sym)o(b)q(olic)e(expressions)k(and)e(their)h(com-)343 726 y(putation)i(b)o(y)h(mac)o(hine.)i Ft(Communic)n(ations)g(of)e(the) h(A)o(CM)p FG(,)e(3\(4\):184{195,)e(1960.)257 809 y([13])20 b(Gordon)14 b(D.)f(Plotkin.)19 b(LCF)14 b(considered)i(as)f(a)f (programmi)o(ng)d(language.)19 b Ft(The)n(or)n(e-)343 859 y(tic)n(al)14 b(Computer)g(Scienc)n(e)p FG(,)h(5:223{255,)10 b(1977.)953 2628 y(26)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF