%!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: flops_04.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -o flops_04.ps flops_04.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.08.13:1112 %%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 false[(Display)(NeXT) (LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{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 600 600 (flops_04.dvi) @start %DVIPSBitmapFont: Fa cmti9 9 36 /Fa 36 122 df<923803FF80031F13F092383F00F803F8133C4A48133E4A48137E17FE4A 5A17FC17384A481300A3141F92C8FCA55C143E011FB612E0A217C09039007E0007147C16 0F1780A214FC4A131F1700A301015C4A133EA3167E0103147C5C1718EEFC1CEEF83C0107 15385C1778177016F0010F15F04AEBF8E01679EE3FC0011FEC0F0093C7FC91C9FCA3133E A21238EA7E3C137CEAFE7812FC485AEA79E0EA3FC0000FCAFC2F4582B42B>12 D45 D<010614C090380FC00F91B51280160015FC4913F0 15C0D91CFEC7FC91C8FC133C1338A313781370A313F0EBE0FE9038E3FF809038EF03C039 01FC01E001F87FEBF000497F485A5BC8FCA41401A4003C130300FC5CA34A5A5A00E0495A A24A5A4AC7FC6C137E00705B387801F8383E07F0381FFFC06C90C8FCEA03F8223478B127 >53 DI<161C163CA2167C16FCA21501821503A2ED077E150F150E151CA21538A215 7015F015E0EC01C0A2913803807F82EC0700A2140E141E141C5CA25CA25C49B6FCA25B91 3880003F49C7EA1F80A2130E131E131C133C13385B13F05B12011203D80FF0EC3FC0D8FF FE903807FFFEA32F367BB539>65 D67 D<0107B712E05B18C0903A003F80003F170F170792C7FC17035C1880147EA214FEA25C16 1C0101EC3C07043813004A91C7FCA20103147816704A13F0150349B5FCA25EECE003130F 6F5A14C0A2011F13035E1480A2013F90C9FCA291CAFCA25BA2137EA213FEA25B1201387F FFFCB5FCA233337CB232>70 D<010FB51280A216009038003FC05DA292C7FCA25CA2147E A214FEA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C8 FCA25BA2137EA213FEA25B1201B512F8A25C21337BB21E>73 D<91381FFFFE5C16FC9138 003F80A31600A25D157EA315FE5DA314015DA314035DA314075DA3140F5DA3141F5DA314 3F92C7FCA2121C007E5B00FE137EA214FE485BEAF80100E05B495A387007E038780FC06C 48C8FCEA1FFCEA07F0273579B228>I<0107B512C05BA29026003FC0C7FC5DA292C8FCA2 5CA2147EA214FEA25CA21301A25CA21303A25CA21307A25CA2130FA25C17E0011F140117 C05C1603013F1580160791C7FCEE0F005B5E017E143EA201FE5CED01FC4913030001EC1F F8007FB6FCB7FC5E2B337CB230>76 D<902607FFC0ED7FFC4917FF81D9003F4B13006118 03023BED077CA2027BED0EFC610273151C1838DAF1F01439F071F014E118E10101ED01C3 6102C1EC0383EF070301031607050E5BEC80F8171C0107ED380F6102001470A249EDE01F DC01C090C7FC130EEE0380011E017C5C933807003E011C140EA2013C4A137E187C01385C 5E017816FC6F485B1370ED3FC001F0EC80016000011500D807F81503277FFF803E90B512 C0B5EB3C01151C46337BB245>I<902607FF8090383FFFC0496D5BA2D9001F913803F800 4A6C6D5A6060EC3BF0027B140360EC71F8A202F11407DAF0FC91C7FC14E0A20101017E5B 170E14C0810103151EEE801CEC801FA20107ECC03C030F1338140016E049010713781770 010E14F01503011E15F0705A011C1301A2013C14FD03005B133816FF0178147F5F017014 3FA213F070C8FC1201EA07F8267FFF807FB5140EA23A337BB239>I<0107B612C04915F8 83903A003F8001FEEE003FEF1F8092C713C0170F5C18E0147EA214FEEF1FC05CA2010116 80173F4A1500177E010315FE5F4AEB03F8EE07E00107EC3FC091B6C7FC16F802E0C9FC13 0FA25CA2131FA25CA2133FA291CAFCA25BA2137EA213FEA25B1201387FFFF0B5FCA23333 7CB234>80 D<0107B512FE49ECFFC017F0903A003F8007F8EE01FCEE007E92C7127F835C 1880147EA214FEEF7F005CA2010115FE5F4A13015F01034A5AEE0FC04A495A04FEC7FC49 B512F016C09138E003E0ED01F8010F6D7E167C4A137EA2131FA25CA2013F14FEA291C7FC A24913015E137EEF01C001FE150318805B00011607277FFFF0001400B5ECFE0EEE7E1CC9 EA1FF8EE07E032357BB238>82 D<913901FC018091380FFF03023F13C791387E07EF903A 01F801FF0049487E4A7F495A4948133E131F91C7FC5B013E143CA3137E1638A293C7FC13 7FA26D7E14E014FE90381FFFC06D13F86D7F01017F6D6C7E020F7F1400153F6F7E150FA4 120EA2001E5D121CA2151F003C92C7FCA2003E143E5D127E007F5C6D485A9038C007E039 F3F80FC000F0B5C8FC38E03FFC38C00FF029377AB42B>I<0003B812C05A1880903AF800 FC003F260FC001141F0180150F01005B001EEE07001403121C003C4A5BA200380107140E 127800705CA2020F141E00F0161CC74990C7FCA2141FA25DA2143FA292C9FCA25CA2147E A214FEA25CA21301A25CA21303A25CA21307A25C497E001FB512F05AA2323374B237>I< EB03F0EB0FF890383E1C6090387C0FF0EBF807EA01F0EA03E00007EB03E0EA0FC0A2381F 800715C0EA3F00A2140F481480127EA2141F00FE14005A1506EC3F07EC3E0F150E147E00 7C141EECFE1CEB01FCD83C03133C393E07BE38391F0E1E783907FC0FF03901F003C02022 78A027>97 D<137EEA0FFE121F5B1200A35BA21201A25BA21203A25BA21207A2EBC3E0EB CFF8380FDC3EEBF81F497E01E01380EA1FC0138015C013005AA2123EA2007E131F158012 7CA2143F00FC14005AA2147EA25CA2387801F85C495A6C485A495A6C48C7FCEA0FFCEA03 F01A3578B323>I<14FCEB07FF90381F078090383E03C0EBFC013801F8033803F0073807 E00F13C0120F391F80070091C7FC48C8FCA35A127EA312FE5AA4007C14C0EC01E0A2EC03 C06CEB0F80EC1F006C137C380F81F03803FFC0C648C7FC1B2278A023>III<151FED7FC0EDF0E0020113F0EC03E3A2EC 07C316E0EDC1C091380FC0005DA4141F92C7FCA45C143E90381FFFFEA3D9007EC7FC147C A414FC5CA513015CA413035CA413075CA3130FA25CA3131F91C8FCA35B133E1238EA7E3C A2EAFE7812FC485AEA78E0EA3FC0000FC9FC244582B418>I<143FECFF80903803E1E690 3807C0FF90380F807FEB1F00133E017E133F49133EA24848137EA24848137CA215FC1207 4913F8A21401A2D80FC013F0A21403120715E01407140F141F3903E03FC00001137FEBF0 FF38007FCF90381F0F801300141FA21500A25C143E1238007E137E5C00FE5B48485A3878 03E0387C0F80D81FFFC7FCEA07F820317CA023>III<133FEA07FF5A13FEEA007EA3137CA213FCA213F8A21201A213F0A21203A213E0A2 1207A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2127C1318EAFC1C133CEA F838A21378137012F013F0EAF8E01279EA3FC0EA0F00103579B314>108 D<2703C003F8137F3C0FF00FFE01FFC03C1E783C1F07C1E03C1C7CF00F8F01F03B3C3DE0 079E0026383FC001FC7FD97F805B007001005B5E137ED8F0FC90380FC00100E05FD860F8 148012000001021F130360491400A200034A13076049013E130FF081800007027EEC83C0 051F138049017C1403A2000F02FC1407053E130049495CEF1E0E001F01015D183C010049 EB0FF0000E6D48EB03E03A227AA03F>I<3903C007F0390FF01FFC391E787C1E391C7CF0 1F393C3DE00F26383FC01380EB7F8000781300EA707EA2D8F0FC131F00E01500EA60F812 0000015C153E5BA20003147E157C4913FCEDF8180007153C0201133801C013F0A2000F15 78EDE070018014F016E0001FECE1C015E390C7EAFF00000E143E26227AA02B>I<14FCEB 07FF90381F07C090383E03E09038FC01F0EA01F83903F000F8485A5B120F484813FCA248 C7FCA214014814F8127EA2140300FE14F05AA2EC07E0A2007CEB0FC01580141FEC3F006C 137E5C381F01F0380F83E03803FF80D800FCC7FC1E2278A027>I<011E137C90387F81FF 9039F3C387C09039E3EF03E03901E1FE01D9C1FC13F0EBC3F8000313F0018314F814E0EA 07871307000313C01200010F130316F01480A2011F130716E01400A249EB0FC0A2013EEB 1F80A2017EEB3F00017F133E5D5D9038FF81F09038FDC3E09038F8FF80027EC7FC000190 C8FCA25BA21203A25BA21207A25BB5FCA325307FA027>I<3903C00FC0390FF03FF0391E 78F078391C7DE03C393C3FC0FC00381380EB7F00007814F8D8707E13701500EAF0FC12E0 EA60F812001201A25BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1E227A A020>114 DI<1303EB0F80 A3131FA21400A25BA2133EA2137EA2137C387FFFF8A2B5FC3800F800A21201A25BA21203 A25BA21207A25BA2120FA25B1460001F13F014E01300130114C01303001E1380EB07005B EA0F1EEA07F8EA01E015307AAE19>II119 D<13F0D803FC1307D80F1E130F000E14 1F121C123C0038143FD8783E133E1270A2017E137ED8F07C137CEA60FCC65A15FC000114 F85BA21401000314F013E0A2140315E0EA07C0A20003130715C0EBE00F141F0001133F90 38F07F8038007FEFEB1F8FEB001F1500A25C003E133E007E137E147C5C007C5BEA700149 5A38380780D83C1FC7FCEA0FFCEA07F020317AA025>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmr5 5 2 /Fb 2 51 df<1360EA01E0120F12FF12F11201B3A3387FFF80A2111C7B9B1C>49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc msbm10 10 1 /Fc 1 86 df<007FB500F090387FFF80B691B5FC6C822603C07CC7381F7C0000010170EC 077018F0715AA3715AB3AD1703A201E05E120017071478D9F03892C7FC01705D1378D938 3C141ED93C1C5CD91E1E147CD90F0E5CD907CFEB03F0903A03F7800FE0903A01FFE07FC0 6D6CB55A021F49C8FC020313F0393B7EB839>85 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmr6 6 2 /Fd 2 51 df<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmsy5 5 1 /Fe 1 49 df48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi5 5 3 /Ff 3 118 df113 D<137E3801FF80EA0381380703C0380E0780EB0300EA0F80EA07F86CB4FC6C1380EA000F EA3003127812F8EB0700EAF00EEA7FFCEA1FF012127C911C>115 D<380F800C381FC01EEA39E012615C12C1EA03C0A25CEA0780A21540ECF0C0A213819038 83F1803903FE7F003800F81E1A127D9123>117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmex10 10 5 /Fg 5 84 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmbx10 10 35 /Fh 35 120 df<913803FFC0027F13F00103B512FC010FEB00FED93FF8133FD97FE0EBFF 8049485A5A1480484A13C04A6C1380A36F1300167E93C7FCA592383FFFC0B8FCA4000390 C7FCB3ABB5D8FC3F13FFA4303A7EB935>12 D46 D<49B4FC010F13E0017F13FC9038FF83FE4848C67E48 48EB7F804848EB3FC04848EB1FE0A2001F15F0A24848EB0FF8A3007F15FCA500FF15FEB3 007F15FCA4003F15F8A26D131F001F15F0A2000F15E06D133F000715C06C6CEB7F806C6C EBFF003900FF83FE6DB45A011F13F0010190C7FC27387CB630>48 D<141E143E14FE1307133FB5FCA313CFEA000FB3B3A6007FB61280A4213779B630>IIII<001C15C0D81F801307 01F8137F90B61280A216005D5D15F05D15804AC7FC14F090C9FCA8EB07FE90383FFFE090 B512F89038FC07FC9038E003FFD98001138090C713C0120EC813E0157F16F0A216F8A212 06EA3F80EA7FE012FF7FA44914F0A26C4813FF90C713E0007C15C06C5B6C491380D9C007 1300390FF01FFE6CB512F8000114E06C6C1380D90FF8C7FC25387BB630>II<123C123EEA3FE090B71280A41700485D5E5E5EA2 5E007CC7EA0FC000784A5A4BC7FC00F8147E48147C15FC4A5A4A5AC7485A5D140F4A5A14 3F92C8FC5C147E14FE1301A2495AA31307A2130F5CA2131FA5133FA96D5A6D5A6D5A293A 7BB830>I57 DI65 D67 DI76 D80 D<003FB91280A4D9F800EBF003D87FC09238007FC049161F 007EC7150FA2007C1707A200781703A400F818E0481701A4C892C7FCB3AE010FB7FCA43B 387DB742>84 D97 D<903801FFC0010F13FC017F13FFD9FF8013802603FE00 13C048485AEA0FF8121F13F0123F6E13804848EB7F00151C92C7FC12FFA9127FA27F123F ED01E06C7E15036C6CEB07C06C6C14806C6C131FC69038C07E006DB45A010F13F0010113 8023257DA42A>99 DI<903803FF80011F13F0017F13FC3901FF83FE3A03FE007F804848133F484814C0 001FEC1FE05B003FEC0FF0A2485A16F8150712FFA290B6FCA301E0C8FCA4127FA36C7E16 78121F6C6C14F86D14F000071403D801FFEB0FE06C9038C07FC06DB51200010F13FC0101 13E025257DA42C>I<161FD907FEEBFFC090387FFFE348B6EAEFE02607FE07138F260FF8 01131F48486C138F003F15CF4990387FC7C0EEC000007F81A6003F5DA26D13FF001F5D6C 6C4890C7FC3907FE07FE48B512F86D13E0261E07FEC8FC90CAFCA2123E123F7F6C7E90B5 12F8EDFF8016E06C15F86C816C815A001F81393FC0000F48C8138048157F5A163FA36C15 7F6C16006D5C6C6C495AD81FF0EB07FCD807FEEB3FF00001B612C06C6C91C7FC010713F0 2B377DA530>103 D<13FFB5FCA412077EAFED7FC0913803FFF8020F13FE91381F03FFDA 3C01138014784A7E4A14C05CA25CA291C7FCB3A3B5D8FC3F13FFA4303A7DB935>II<13FFB5FCA412077EAF92380FFFE0A4923803FC0016F0ED0FE0ED1F804BC7 FC157E5DEC03F8EC07E04A5A141FEC7FE04A7E8181A2ECCFFEEC0FFF496C7F806E7F6E7F 82157F6F7E6F7E82150F82B5D8F83F13F8A42D3A7EB932>107 D<13FFB5FCA412077EB3 B3ACB512FCA4163A7DB91B>I<01FED97FE0EB0FFC00FF902601FFFC90383FFF80020701 FF90B512E0DA1F81903983F03FF0DA3C00903887801F000749DACF007F00034914DE6D48 D97FFC6D7E4A5CA24A5CA291C75BB3A3B5D8FC1FB50083B512F0A44C257DA451>I<01FE EB7FC000FF903803FFF8020F13FE91381F03FFDA3C011380000713780003497E6D4814C0 5CA25CA291C7FCB3A3B5D8FC3F13FFA430257DA435>I<903801FFC0010F13F8017F13FF D9FF807F3A03FE003FE048486D7E48486D7E48486D7EA2003F81491303007F81A300FF16 80A9007F1600A3003F5D6D1307001F5DA26C6C495A6C6C495A6C6C495A6C6C6CB45A6C6C B5C7FC011F13FC010113C029257DA430>I<9039FF01FF80B5000F13F0023F13FC9138FE 07FFDAF00113800003496C13C00280EB7FE091C713F0EE3FF8A2EE1FFCA3EE0FFEAA17FC 161FA217F8163F17F06E137F6E14E06EEBFFC0DAF00313809139FC07FE0091383FFFF802 0F13E0020390C7FC91C9FCACB512FCA42F357EA435>I<9038FE03F000FFEB0FFEEC3FFF 91387C7F809138F8FFC000075B6C6C5A5CA29138807F80ED3F00150C92C7FC91C8FCB3A2 B512FEA422257EA427>114 D<90383FF0383903FFFEF8000F13FF381FC00F383F000300 7E1301007C130012FC15787E7E6D130013FCEBFFE06C13FCECFF806C14C06C14F06C14F8 1203C614FC131F9038007FFE140700F0130114007E157E7E157C6C14FC6C14F8EB800190 38F007F090B512C000F8140038E01FF81F257DA426>I<130FA55BA45BA25B5BA25A1207 001FEBFFE0B6FCA3000390C7FCB21578A815F86CEB80F014816CEBC3E090383FFFC06D13 80903803FE001D357EB425>I119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmsy7 7 10 /Fi 10 107 df0 D<1338A50060130C00F8133E00FC137E00FE 13FE383FBBF83807FFC000011300EA007C48B4FC000713C0383FBBF838FE38FE00FC137E 00F8133E0060130C00001300A517197B9A22>3 D<160E163E16FEED03F8ED0FE0ED3F80 EDFE00EC03F8EC0FE0EC3F8002FEC7FCEB03F8EB0FE0EB3F8001FEC8FCEA03F8EA0FE0EA 3F8000FEC9FC12F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F 80EC0FE0EC03F8EC00FEED3F80ED0FE0ED03F8ED00FE163E160E1600AB007FB612FCB712 FEA227357AA734>20 D<13E0EA01F0EA03F8A3EA07F0A313E0A2120F13C0A3EA1F80A213 00A25A123EA35AA3127812F8A25A12100D1E7D9F13>48 D<49B5FC130F133F01FFC7FCEA 01F8EA03E0EA078048C8FC121E121C123C123812781270A212F05AA2B7FCA300E0C8FCA2 7E1270A212781238123C121C121E7E6C7EEA03E0EA01F86CB4FC013FB5FC130F13012027 7AA12D>50 D69 D<0207B612C0023F15E091B7FC903A01E07C 0007D90780EC03C090260F00781400011E01F890C7FC5B137C01785BEB70011300A25D14 03A25D1407A292B512C05C5FDB0006C7FC4A90C8FCA2141E143E143C147C147814F85C13 015CEA180300785BEAFC0700FE5BD8FF8FCAFCEA7FFEEA3FF8EA0FE0332A7EA730>I<01 03B512F8013FECFF8090B712E02703F078007F260F80F8EB0FF8D81E001403481501007C 1500127812F812E0C7485C010114015F16035F4A495A01034AC7FC161E1678ED03F09138 C0FFC0902607C3FEC8FCA2EC80FF010F7F6F7E151F02007F496D7E131E013E6D7E150301 3C6E1370017C010114F09338FC01E00178903900FE038001F89138FF0F0049EC7FFE4848 EC3FF849EC1FC034297EA739>82 D<0060153000E01570B3A76C15F0007015E000781401 6CEC03C0001FEC0F80D80FC0EB3F003907F801FE0001B512F86C6C13E0010F90C7FC2424 7CA22D>91 D<12E0B3B3B3A5033B78AB14>106 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmr7 7 7 /Fj 7 62 df<1306130C13181330136013E0EA01C0EA0380A2EA07005A120E121EA2121C 123CA35AA512F85AAB7E1278A57EA3121C121EA2120E120F7EEA0380A2EA01C0EA00E013 6013301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01C013E0 120013F0A213701378A3133CA5133E131EAB133E133CA51378A3137013F0A213E0120113 C0EA0380A2EA0700120E120C5A5A5A5A0F3B7DAB1A>I<140EB3A2B812E0A3C7000EC8FC B3A22B2B7DA333>43 D<13381378EA01F8121F12FE12E01200B3AB487EB512F8A215267B A521>49 D<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80A412 7CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA0180 390300030012065A001FB5FC5A485BB5FCA219267DA521>I<13FF000313E0380F01F838 1C007C0030137E003C133E007E133FA4123CC7123E147E147C5C495AEB07E03801FF8091 C7FC380001E06D7E147C80143F801580A21238127C12FEA21500485B0078133E00705B6C 5B381F01F03807FFC0C690C7FC19277DA521>I 61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk lasy10 10 1 /Fk 1 51 df<003FB712FEB9FCA300F0C9120FB3B3A4B9FCA4303079B43E>50 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl msam10 10 1 /Fl 1 67 df<126012F812FE6C7E13E013F8EAF3FE38F0FF80EB3FE0EB0FF8EB03FE9038 00FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80EE3FE0EE0F F8EE03FE933800FF80EF3FC0171FEF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7FCED 07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC0D8F1 FFCAFCEAF7FCEAFFF013C090CBFC12FC1270323279AC41>66 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmti10 10 54 /Fm 54 122 df12 D<150C151C153815F0EC01E0EC03C0EC0780EC0F00141E5C147C5C5C495A1303 495A5C130F49C7FCA2133EA25BA25BA2485AA212035B12075BA2120F5BA2121FA290C8FC A25AA2123EA2127EA2127CA412FC5AAD1278A57EA3121C121EA2120E7EA26C7E6C7EA212 001E5274BD22>40 D<140C140E80EC0380A2EC01C015E0A2140015F0A21578A4157C153C AB157CA715FCA215F8A21401A215F0A21403A215E0A21407A215C0140F1580A2141F1500 A2143EA25CA25CA2495AA2495A5C1307495A91C7FC5B133E133C5B5B485A12035B48C8FC 120E5A12785A12C01E527FBD22>I44 D<387FFFF8A2B5FCA214F0150579941E>I<120EEA3F80127F12FFA31300127E123C0909 778819>I48 D<15181538157815F0140114031407EC0FE0141F147FEB03FF90383FEFC0148FEB1C 1F13001580A2143FA21500A25CA2147EA214FEA25CA21301A25CA21303A25CA21307A25C A2130FA25CA2131FA25CA2133FA291C7FC497EB61280A31D3877B72A>III<010314186E13F8903907F007F091B512E016C01600495B15F8010E 13E0020CC7FC011EC8FC131CA3133C1338A313781370A2147F9038F3FFC09038EF83E090 38FC01F0496C7E485A497F49137CC8FC157EA315FEA41401000C5C123F5A1403485C5A4A 5A12F800E05C140F4A5A5D6C49C7FC0070137E00785B387C01F8383E07F0381FFFC06C90 C8FCEA01F8253A77B72A>53 D56 DI<133C137E13FF5AA313FE13FCEA00701300B2120EEA3F80127F12 FFA31300127E123C102477A319>II65 D67 D<0103B612FEEFFFC018F0903B0007F8000FF84BEB03FCEF00FE020F157FF03F804B141F 19C0021F150F19E05D1807143F19F05DA2147FA292C8FCA25C180F5CA2130119E04A151F A2130319C04A153FA201071780187F4A1600A2010F16FEA24A4A5A60011F15034D5A4A5D 4D5A013F4B5A173F4A4AC7FC17FC017FEC03F84C5A91C7EA1FC04949B45A007F90B548C8 FCB712F016803C397CB83F>I<0107B8FCA3903A000FF000034BEB007F183E141F181E5D A2143FA25D181C147FA29238000380A24A130718004A91C7FC5E13015E4A133E167E49B5 12FEA25EECF8000107147C163C4A1338A2010F147818E04A13701701011F16C016004A14 031880013F150718004A5CA2017F151E173E91C8123C177C4915FC4C5A4914070001ED7F F0B8FCA25F38397BB838>I<0107B712FEA3903A000FF000074B1300187C021F153CA25D A2143FA25D1838147FA292C8FCEE03804A130718004A91C7FCA201015CA24A131E163E01 0314FE91B5FC5EA2903807F800167C4A1378A2130FA24A1370A2011F14F0A24A90C8FCA2 133FA25CA2137FA291CAFCA25BA25B487EB6FCA337397BB836>II<0103B512F8A3 90390007F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA21301 A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC497E B6FCA25C25397CB820>73 D<0107B512FCA25E9026000FF8C7FC5D5D141FA25DA2143FA2 5DA2147FA292C8FCA25CA25CA21301A25CA21303A25CA21307A25CA2130F170C4A141CA2 011F153C17384A1478A2013F157017F04A14E01601017F140317C091C71207160F49EC1F 80163F4914FF000102071300B8FCA25E2E397BB834>76 D<902607FFF8923807FFF0614F 13E0D9000FEFF0004F5AA2021F167FF1EFC0141DDA1CFCEC01CF023C16DF9538039F8002 38ED071FA20278ED0E3F97C7FC0270151CA202F04B5AF0707E14E0037E14E0010117FE4D 485A02C0EC0380A20103ED0701610280140EA20107ED1C0305385B14006F137049160705 E05B010EEC01C0A2011E913803800F61011CEC0700A2013C020E131F4C5C1338ED1FB801 78163F04F091C8FC01705CA201F04A5B187E00015DD807F816FEB500C09039007FFFFC15 1E150E4C397AB84A>I<0107B612F817FF1880903B000FF0003FE04BEB0FF0EF03F8141F EF01FC5DA2023F15FEA25DA2147FEF03FC92C7FCA24A15F817074A15F0EF0FE01301EF1F C04AEC3F80EFFE0001034A5AEE0FF091B612C04CC7FCD907F8C9FCA25CA2130FA25CA213 1FA25CA2133FA25CA2137FA291CAFCA25BA25B1201B512FCA337397BB838>80 D<0103B612F017FEEFFF80903B0007F8003FC04BEB0FF01707020FEC03F8EF01FC5DA202 1F15FEA25DA2143FEF03FC5DA2027FEC07F818F092C7120F18E04AEC1FC0EF3F004A14FE EE01F80101EC0FE091B6128004FCC7FC9138FC003F0103EC0F80834A6D7E8301071403A2 5C83010F14075F5CA2011F140FA25CA2133F161F4AECE007A2017F160F180E91C7FC4902 0F131C007F01FE153CB5913807F078040313F0CAEAFFE0EF3F80383B7CB83D>82 D<92383FC00E913901FFF01C020713FC91391FC07E3C91393F001F7C027CEB0FF84A1307 49481303495A4948EB01F0A2495AA2011F15E091C7FCA34915C0A36E90C7FCA2806D7E14 FCECFF806D13F015FE6D6D7E6D14E0010080023F7F14079138007FFC150F15031501A215 00A2167C120EA3001E15FC5EA3003E4A5AA24B5AA2007F4A5A4B5A6D49C7FC6D133ED8F9 F013FC39F8FC03F839F07FFFE0D8E01F138026C003FCC8FC2F3D7ABA2F>I<0007B812E0 A25AD9F800EB001F01C049EB07C0485AD900011403121E001C5C003C1780140312380078 5C00701607140700F01700485CA2140FC792C7FC5DA2141FA25DA2143FA25DA2147FA292 C9FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CEB3FF0007FB512F8B6FC A2333971B83B>I87 D<14F8EB07FE90381F871C90383E03FE137CEBF801120148486C5A485A120FEBC001001F 5CA2EA3F801403007F5C1300A21407485C5AA2140F5D48ECC1C0A2141F15831680143F15 87007C017F1300ECFF076C485B9038038F8E391F0F079E3907FE03FC3901F000F0222677 A42A>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0F8EBE7FE 9038EF0F80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F1300A214075A 127EA2140F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F8007C5B383C03 E0383E07C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FFC090380FC1E0 90381F0070017E13784913383901F801F83803F003120713E0120FD81FC013F091C7FC48 5AA2127F90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003EEB03C0EC0F80 6CEB3E00380F81F83803FFE0C690C7FC1D2677A426>II<147F903803FFC090380F C1E090383F00F0017E13785B485A485A485A120F4913F8001F14F0383F8001EC07E0EC1F 80397F81FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E14F0003EEB01 E0EC03C06CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426>IIIII<150E153F157FA3157E151C1500ABEC1F80EC7FC0ECF1F0EB01 C090380380F813071401130F130E131EEB1C03133C013813F0A2EB0007A215E0A2140FA2 15C0A2141FA21580A2143FA21500A25CA2147EA214FEA25CA21301A25CA213035C121C38 7E07E0A238FE0FC05C49C7FCEAF83EEA787CEA3FF0EA0FC0204883B619>IIIII<147F903803FFC090380FC1F090 381F00F8017E137C5B4848137E4848133E0007143F5B120F485AA2485A157F127F90C7FC A215FF5A4814FEA2140115FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F80003EEB 3F00147E6C13F8380F83F03803FFC0C648C7FC202677A42A>I<9039078007C090391FE0 3FF090393CF0787C903938F8E03E9038787FC00170497EECFF00D9F0FE148013E05CEA01 E113C15CA2D80003143FA25CA20107147FA24A1400A2010F5C5E5C4B5A131F5EEC80035E 013F495A6E485A5E6E48C7FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA25BA21201 A25BA21203A25B1207B512C0A3293580A42A>II<3903C003F0390FF01FFC391E783C0F381C7C703A3C3EE03F8038383FC0EB7F8000 78150000701300151CD8F07E90C7FCEAE0FE5BA2120012015BA312035BA312075BA3120F 5BA3121F5BA3123F90C9FC120E212679A423>I<14FE903807FF8090380F83C090383E00 E04913F00178137001F813F00001130313F0A215E00003EB01C06DC7FC7FEBFFC06C13F8 14FE6C7F6D13807F010F13C01300143F141F140F123E127E00FE1480A348EB1F0012E06C 133E00705B6C5B381E03E06CB45AD801FEC7FC1C267AA422>II<13F8D803FEEB01C0D8078FEB03E0390E0F800712 1E121C0038140F131F007815C01270013F131F00F0130000E015805BD8007E133FA201FE 14005B5D120149137EA215FE120349EBFC0EA20201131E161C15F813E0163CD9F0031338 14070001ECF07091381EF8F03A00F83C78E090393FF03FC090390FC00F00272679A42D> I<01F0130ED803FC133FD8071EEB7F80EA0E1F121C123C0038143F49131F0070140FA25B D8F07E140000E08013FEC6485B150E12015B151E0003141C5BA2153C000714385B5DA35D A24A5A140300035C6D48C7FC0001130E3800F83CEB7FF8EB0FC0212679A426>I<01F015 07D803FC903903801F80D8071E903907C03FC0D80E1F130F121C123C0038021F131F49EC 800F00701607A249133FD8F07E168000E0ED000313FEC64849130718000001147E5B03FE 5B0003160E495BA2171E00070101141C01E05B173C1738A217781770020314F05F000301 0713016D486C485A000190391E7C07802800FC3C3E0FC7FC90393FF81FFE90390FE003F0 322679A437>I<903907E007C090391FF81FF89039787C383C9038F03E703A01E01EE0FE 3803C01F018013C0D8070014FC481480000E1570023F1300001E91C7FC121CA2C75AA214 7EA214FEA25CA21301A24A1370A2010314F016E0001C5B007E1401010714C000FEEC0380 010F1307010EEB0F0039781CF81E9038387C3C393FF03FF03907C00FC027267CA427>I< 13F0D803FCEB01C0D8071EEB03E0D80E1F1307121C123C0038140F4914C01270A249131F D8F07E148012E013FEC648133F160012015B5D0003147E5BA215FE00075C5BA214015DA3 14035D14070003130FEBF01F3901F87FE038007FF7EB1FC7EB000F5DA2141F003F5C4813 3F92C7FC147E147C007E13FC387001F8EB03E06C485A383C1F80D80FFEC8FCEA03F02336 79A428>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmmi10 10 35 /Fn 35 123 df11 D<15FE913803FF8091380F83E091383E01F091387C00F85C494813FC01 03147C4948137E5C130F495AA249C7FC16FE5B137EA2150113FE4914FCA20001140316F8 5BED07F01203ED0FE04914C0151F000715806DEB3F00157E6D5B390FEE01F09038E707E0 9038C3FF80D9C0FCC7FC001F90C8FCA25BA2123FA290C9FCA25AA2127EA212FEA25AA212 7027377EA42B>26 D<027FB512C00103B612E0130F5B017F15C09026FF81FEC7FC3901FC 007E48487F485A497F484880485AA248C7FCA2127EA2153F00FE92C7FC5AA25D157E5A5D A24A5AA24A5A007C495A5D003C495A003E013FC8FC6C137C380F81F83803FFE0C66CC9FC 2B257DA32F>I39 D<121C127FEAFF80A5EA7F00121C0909798817>58 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A 12600A19798817>I I<1760177017F01601A21603A21607160FA24C7EA216331673166316C3A2ED0183A2ED03 03150683150C160115181530A21560A215C014011580DA03007FA202061300140E140C5C 021FB5FC5CA20260C7FC5C83495A8349C8FC1306A25BA25B13385B01F01680487E000716 FFB56C013F13FF5EA2383C7DBB3E>65 D<9339FF8001C0030F13E0037F9038F80380913A 01FF807E07913A07F8000F0FDA1FE0EB079FDA3F80903803BF0002FFC76CB4FCD901FC80 495A4948157E495A495A4948153E017F163C49C9FC5B1201484816385B1207485A183012 1F4993C7FCA2485AA3127F5BA312FF90CCFCA41703A25F1706A26C160E170C171C5F6C7E 5F001F5E6D4A5A6C6C4A5A16076C6C020EC8FC6C6C143C6C6C5C6CB4495A90393FE00FC0 010FB5C9FC010313FC9038007FC03A3D7CBA3B>67 D<0103B612F849EDFF8018E0903B00 07F8001FF84BEB03FCEF00FE020F157FA24BEC3F80A2021F16C0A25DA2143FF07F805DA2 027FEDFF006092C7485A4D5A4A4A5A4D5A4AEC1F80057FC7FC0101EC07F891B612E094C8 FC9139FC000FC00103EC03F0707E4A6D7E831307177E5C177F010F5D5F5CA2011F1401A2 5CA2133F16034A4A1360A2017F17E019C091C71401496C01011480B61503933900FE0700 EF7E0ECAEA1FFCEF07F03B3B7DB83F>82 D<003FB56C48B51280485DA226007F80C7381F F00091C8EA07C0604993C7FCA2491506A20001160E170C5BA20003161C17185BA2000716 3817305BA2000F167017605BA2001F16E05F5BA2003F15015F5BA2007F150394C8FC90C8 FCA25E4815065A160E160C161C161816385E127E5E4B5A6C4A5A4BC9FC6C6C131E6C6C5B 6C6C13F83903F807E06CB55A6C6C48CAFCEB0FF0393B7BB839>85 D<147E903803FF8090390FC1C38090391F00EFC0017E137F49133F485A4848EB1F801207 5B000F143F48481400A2485A5D007F147E90C7FCA215FE485C5AA214015D48150CA21403 EDF01C16181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83C0F9C0 3A03FF007F80D800FCEB1F0026267DA42C>97 D<133FEA1FFFA3C67E137EA313FE5BA312 015BA312035BA31207EBE0FCEBE3FF9038E707C0390FFE03E09038F801F001F013F8EBE0 00485A15FC5BA2123F90C7FCA214015A127EA2140312FE4814F8A2140715F05AEC0FE0A2 15C0EC1F80143F00781400007C137E5C383C01F86C485A380F07C06CB4C7FCEA01FC1E3B 7CB924>II<163FED1FFFA3ED007F167EA216FEA216FCA21501A216F8A21503A216F0A2 1507A2027E13E0903803FF8790380FC1CF90381F00EF017EEB7FC049133F485A4848131F 000715805B000F143F485A1600485A5D127F90C7127EA215FE5A485CA21401A248ECF80C A21403161CEDF0181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83 C0F9C03A03FF007F80D800FCEB1F00283B7DB92B>II<16F8ED03FEED0F879238 1F0F80ED3E3F167F157CA215FC1700161C4A48C7FCA414035DA414075DA20107B512F0A3 9026000FE0C7FC5DA4141F5DA4143F92C8FCA45C147EA514FE5CA413015CA4495AA45C13 07A25C121E123F387F8F80A200FF90C9FC131E12FEEA7C3CEA7878EA1FF0EA07C0294C7C BA29>III<14E0EB03F8A21307A314F0EB01C090C7 FCAB13F8EA03FEEA070F000E1380121C121812381230EA701F1260133F00E0130012C05B EA007EA213FE5B1201A25B12035BA20007131813E01438000F133013C01470EB806014E0 14C01381EB838038078700EA03FEEA00F815397EB71D>I<150FED3F80A2157FA3160015 1C92C7FCABEC0F80EC3FE0ECF0F0903801C0F849487E14005B130E130C131CEB18011338 01305BA2EB0003A25DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25CA2147E A214FEA25CA21301001E5B123F387F83F0A238FF87E0495A00FE5BD87C1FC8FCEA707EEA 3FF8EA0FC0214981B722>IIIIII<02FC13C0903803FF0190380F8383 90383F01C790397E00EF8049137F485A4848133F000715005B485A001F5C157E485AA200 7F14FE90C75AA3481301485CA31403485CA314075D140F127C141F007E495A003E137F38 1F01EF380F839F3903FF1F80EA00FC1300143F92C7FCA35C147EA314FE5C130190387FFF F0A322357DA425>113 D<3903E001F83907F807FE390E3C1E07391C3E381F3A183F703F 800038EBE07F0030EBC0FF00705B00601500EC007E153CD8E07F90C7FCEAC07EA2120013 FE5BA312015BA312035BA312075BA3120F5BA3121F5B0007C9FC21267EA425>I<14FF01 0313C090380F80F090383E00380178131C153C4913FC0001130113E0A33903F000F06D13 007F3801FFE014FC14FF6C14806D13C0011F13E013039038003FF014071403001E130112 7FA24814E0A348EB03C012F800E0EB07800070EB0F006C133E001E13F83807FFE0000190 C7FC1E267CA427>II<13 F8D803FE1438D8070F147C000E6D13FC121C1218003814011230D8701F5C12601503EAE0 3F00C001005B5BD8007E1307A201FE5C5B150F1201495CA2151F120349EC80C0A2153F16 81EE0180A2ED7F0303FF130012014A5B3A00F8079F0E90397C0E0F1C90393FFC07F89039 07F001F02A267EA430>I<01F8EB03C0D803FEEB07E0D8070F130F000E018013F0121C12 180038140700301403D8701F130112601500D8E03F14E000C090C7FC5BEA007E16C013FE 5B1501000115805B150316001203495B1506150E150C151C151815385D00015C6D485A6C 6C485AD97E0FC7FCEB1FFEEB07F024267EA428>I<903907E001F090391FF807FC903978 3E0E0F9039E01F1C1FD801C09038383F803A03800FF07F0100EBE0FF5A000E4A1300000C 157E021F133C001C4AC7FC1218A2C7123FA292C8FCA25CA2147EA214FEA24A130CA20101 141C001E1518003F5BD87F81143801835C00FF1560010714E03AFE0E7C01C0D87C1C495A 2778383E0FC7FC391FF00FFC3907C003F029267EA42F>120 D<13F8D803FE1470D8070F 14F8000EEB8001121C121800381403003015F0EA701F1260013F130700E0010013E012C0 5BD8007E130F16C013FE5B151F000115805BA2153F000315005BA25D157EA315FE5D1401 000113033800F80790387C1FF8EB3FF9EB0FE1EB00035DA2000E1307D83F805B007F495A A24A5A92C7FCEB003E007C5B00705B6C485A381E07C06CB4C8FCEA01FC25367EA429>I< D901E01360D90FF813E0496C13C090383FFE0190397FFF038090B5EA07009038F81FFF39 01E003FE9038C0001C495B5DC85A4A5A4A5A4AC7FC140E5C5C14F0495AEB038049C8FC13 0E5B4913035B495B484813064848130E48C75AD80FFC137C391FFF81F8381E0FFFD83807 5B486C5B00605CD8E00190C7FC38C0007C23267DA427>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmmi7 7 18 /Fo 18 122 df11 D<14FCEB03FF903807878090381E03C0EB3C01017813E0A213F0000114F013E01203A239 07C003E0A4390F8007C0A21580EC0F00EA1F00141E6D5A6D5A383EE1F0EB7FC0011FC7FC 90C8FC5AA45AA45A5A1C267D9922>26 D<1238127C12FE12FFA2127F123B1203A31206A3 120C121812381270122008127A8614>59 D<3B7FFFE003FFF0A2D803F8C7EA3E0049143C 16180007153816305BA2000F157016605BA2001F15E05E5BA2003F14015E90C7FCA24814 0393C7FC127EA200FE5C15065AA2150E150C151C5D007C5C5D6C495A003F495A261F800F C8FC3807C07C3801FFF038007F802C297BA72D>85 D98 D<15F8141FA2EC01F0A21403A215E0A21407A215C0A2140FEB1F8F90387FCF80 EBF0EF3803C03FEA0780390F001F00A2001E5B123E003C133E127C147E5A147CA214FC5A ECF830A3903801F060A2EA7803010E13C0393C1CF980381FF07F3907C01E001D297CA723 >100 D102 D<130E131F5BA2133E131C90C7 FCA7EA03E0487EEA0C78EA187C1230A212605B12C0A2EA01F0A3485AA2485AA2EBC180EA 0F81A2381F0300A213066C5A131CEA07F06C5A11287DA617>105 D<1407EC0F80141FA21500140E91C7FCA7EB03E0EB07F8EB0C3C1318EB303E136013C0A2 48485AA2C7FCA25CA4495AA4495AA4495AA4495AA21238D87C1FC7FC12FC133E485AEA70 F8EA7FE0EA1F80193380A61B>I<133EEA07FEA2EA007CA213FCA25BA21201A25BA21203 EC07809038E01FC0EC38600007EB61E014C3EBC187EBC307D80FC613C09038CC038001B8 C7FC13E0487E13FEEB3F80EB0FC0486C7E1303003E1460A2127EECC0C0127CECC18012FC 903801E30038F800FE0070137C1B297CA723>I<3B07801FC007E03B0FE07FF01FF83B18 F0E0F8783C3B30F1807CE03E903AFB007D801ED860FEEB3F005B49133E00C14A133E5B12 01A24848495BA35F4848485A1830EE01F0A23C0F8003E003E060A218C0933801E180271F 0007C013E3933800FF00000E6D48137C341B7D993B>109 D<3907801FC0390FE07FF039 18F0E0F83930F1807CEBFB00D860FE133C5B5B00C1147C5B1201A248485BA34A5AEA07C0 1660EC03E0A23A0F8007C0C0A2EDC180913803C300D81F0013C7EC01FE000EEB00F8231B 7D9929>I113 D115 D<131C133EA25BA45BA4485AB512E0A23801F000485AA4485AA4485AA448 C7FC1460A214C0123EEB0180EB0300EA1E06EA1F1CEA0FF8EA03E013267EA419>II<90387C03C03901FF0FF03907079C30390E03B078000CEBF0F8001813 E1123015F0396007C0E015001200A2495AA449C7FC15301238007C1460EAFC3E15C0EAF8 7E39F06F03803970C70700383F83FE381F01F81D1B7D9926>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmsy10 10 30 /Fp 30 111 df<121C127FEAFF80A5EA7F00121C0909799917>1 D<020FB6128091B712C01303010F1680D91FF8C9FCEB7F8001FECAFCEA01F8485A485A48 5A5B48CBFCA2123EA25AA2127812F8A25AA87EA21278127CA27EA27EA26C7E7F6C7E6C7E 6C7EEA00FEEB7F80EB1FF86DB71280010316C01300020F158091CAFCAE001FB812804817 C0A26C1780324479B441>18 D20 D<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC6 6C7EEB3FE0EB0FF8EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8 ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FC0171FEF7F80933801FF00EE 07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948 C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007F B81280B912C0A26C1780324479B441>I<126012F0A37EA21278127CA27EA27E7F6C7E6C 7E6C7EEA01FC6CB4FCEB3FC0EB1FF8903807FF80010113F89039007FFFF8020F90B51280 020015C0A2020F1580027F01F8C7FC902601FFF8C8FC01071380D91FF8C9FCEB3FC001FF CAFCEA01FCEA03F0485A485A485A90CBFC123EA25AA2127812F8A25AA31260CCFCAE007F B81280B912C0A26C1780324479B441>23 D<126012F0A37EA21278127CA27EA27E7F6C7E 6C7E6C7EEA01FC6CB4FCEB3FC0EB1FF8903807FF80010113F89039007FFFF8020F90B512 80020015C0A2020F1580027F01F8C7FC902601FFF8C8FC01071380D91FF8C9FCEB3FC001 FFCAFCEA01FCEA03F0485A485A485A90CBFC123EA25AA2127812F8A25AA31260323279AC 41>31 D<1478A414F85CA213015C1303495AA2495A49CCFC5B137E5B485A485AEA0FE000 3FBA12FEBCFCA2003F19FED80FE0CCFCEA03F06C7E6C7E137E7F7F6D7E6D7EA26D7E1301 801300A2801478A4482C7BAA53>I<181EA4181F84A285180785727EA2727E727E85197E 85F11F80F10FC0F107F0007FBA12FCBCFCA26C19FCCCEA07F0F10FC0F11F80F13F00197E 61614E5A4E5AA24E5A61180F96C7FCA260181EA4482C7BAA53>I<14301478B3B3AD00C0 150C00F8157C00FEEC01FCD8FF801307D83FE0EB1FF0D807F0EB3F80D801F8EB7E00D800 FC5B90383E79F090381F7BE06DB45A6D5BA26D90C7FC6D5AA26D5AA21478A31430A3264A 7EB92A>35 D<153CA2157C157815F85D14014A5A5D14074A5A4ACBFC143E027FB812FE91 BAFC5B4918FED90FC0CBFC495A017FCCFCEA01FCEA07F8EA1FE0EAFF80A2EA1FE0EA07F8 EA01FCEA007FEB1F806D7E0103B912FE6D18FF7F6E17FE023ECBFC806E7E6E7E1403816E 7E1400811578157C153CA248307BAC53>40 D<173CA2173E171E171F8384717E17038471 7E717E187C007FB812FEBAFC856C84CBEA03F0727EF000FEF13F80F11FE0F107F8F101FF A2F107F8F11FE0F13F80F1FE00F001F84E5A007FB912C0BA5A96C7FC6C5FCB127C604D5A 4D5A6017074D5A95C8FC5F171E173E173CA248307BAC53>I<91381FFFFE91B6FC130301 0F14FED91FF0C7FCEB7F8001FEC8FCEA01F8485A485A485A5B48C9FCA2123EA25AA21278 12F8A25AA2B712FE16FFA216FE00F0C9FCA27EA21278127CA27EA27EA26C7E7F6C7E6C7E 6C7EEA00FEEB7F80EB1FF06DB512FE010314FF1300021F13FE283279AD37>50 D54 D<156015F0A21401EB07F190383FFFE0EB7C 1FEBF00748486C5AD803C07F4848487ED80F007FA248497E001E14BC153C003E143E141F A248EB1E1F143EA2143CA2147C00FC1580147814F8A214F0A21301A214E01303A214C0A2 1307A21480A2130FA214005B007C1500131EA2D87E3E5BA2D83E3C133E137CA21378001F 5C13F8000F14784913F800075C0003495AEBE0033901F007802603FC1FC7FCEBFFFEEBC7 F0D807C0C8FCA25BA26CC9FC21477CBF2A>59 D67 D<0203B512F0027F14FF49B712E0 010F16F890273FC3F00713FED978039038007FFF2601E007020F1380D803C0030313C0D8 0780030013E0000F177FD81F00EE3FF048EF1FF8003E4A140F5A0078EF07FC00C0010F15 03C7FCA24B1401A3141F5DA3023F16F8A292C8FCF003F0A25C027EED07E0A219C04A150F 1980F01F00495A183E6049481578604D5A49484A5A4D5A050EC7FC4948143C5FEE01E049 48EB07C0043FC8FC91380001FC49EB3FF049B5128048B500FCC9FC4814E04801FCCAFC3E 397FB840>II<0307B612FE 033FEDFF804AB812C0140791260F807EC7FC91263C00FEEC3F004A161E4A491418010194 C7FC495A01071301A2D90FC05B148014000118130390C75BA34B5AA3150F5EA34B5AA293 B512FC4B5C604B14C0037ECAFCA25DA25D1401A24A5AA25D14075D140F5D141F92CBFC5C 0006133E003E137E007E137CB413FC6D5AEBC1F0EBF1E06CB45A6C90CCFC6C5AEA07F042 3C7EB83C>I<0203B512F8027FECFF8049B712F0010F8290273FC3F00313FED978039038 003FFF2601E00702071380D803C06F13C0D807801500000F177FD81F00EE3FE0484A141F 123E5A0078010F150F12C0C7FC4B15C0A3021FED1F80A24B1500183EA2023F5D6092C85A 4D5A4D5A4A4A5A027E020EC7FC173C17F84AEB03E0EE3F80DB1FFEC8FC0101EB7FF89138 F8FFC0DAF9FCC9FC02F8CAFC495AA3495AA3495AA3495AA291CBFC5BA2137EA35B13F013 C03B3D7FB83A>80 D<0203B512FE027FECFFF049B712FC010F16FF90273FC3F00080D978 0302077F2601E0071401D803C06F6C7ED80780163F000F171FEA1F00484A140F123E5A00 78010F5E12C0C7FC4B4A5AA296C7FC021F5D183E4B5C187860023F4A5A4D5A92C7000FC8 FC173EEE03F84AEBFFE0DA7E0313804B48C9FC4B7EECFC036F7F6F7F0101147F4A80163F 707E495A707EA249481307830403151049486E14F0F101E04A6D6CEB03C0011F93388007 8070EC0F0049C8EBC01E716C5A013E92383FF0F0017EEEFFE0017C6F1380496F48C7FC01 E0ED07F0443B7FB846>82 D<1A801907F10F00023FB712FE49B85A010F17F0013F17C049 4CC7FC2801E00003F0C9FC48481307485A120F48C7485A5A5AA200FE4A5A5A12F01280C8 485AA44BCAFCA415FEA44A5AA44A5AA44A5AA4140F5DA35D141FA25D143FA292CBFC5CA2 147E14FE5CA2495A5C495A5C0102CCFC41427DBB2D>84 D86 D<0060161800F0163CB3B26C167CA2007C16F8A26CED01F0003F15036C6CEC07E06C6CEC 0FC0D807F0EC3F80D803FE903801FF003A00FFC00FFC6DB55A011F14E0010391C7FC9038 007FF82E347CB137>91 DI102 D<12FCEAFFC0EA07F0EA01FCEA007E7F80131F80130FB3A7801307806D7E6D7EEB007EEC 1FF0EC07F8EC1FF0EC7E00495A495A495A5C130F5CB3A7131F5C133F91C7FC137E485AEA 07F0EAFFC000FCC8FC1D537ABD2A>I<14C0EB01E01303A214C01307A21480130FA2EB1F 00A2131E133EA25BA2137813F8A2485AA25B1203A25B1207A2485AA290C7FC5AA2123EA2 123C127CA2127812F8A41278127CA2123C123EA27EA27E7FA26C7EA212037FA212017FA2 6C7EA21378137CA27FA2131E131FA2EB0F80A2130714C0A2130314E0A21301EB00C01352 78BD20>I<126012F07EA21278127CA2123C123EA27EA27E7FA26C7EA212037FA26C7EA2 12007FA21378137CA27FA2131E131FA2EB0F80A2130714C0A2130314E0A414C01307A214 80130FA2EB1F00A2131E133EA25BA2137813F8A25B1201A2485AA25B1207A2485AA290C7 FC5AA2123EA2123C127CA2127812F8A25A126013527CBD20>I<126012F0B3B3B3B3A912 60045377BD17>I<126012F07EA21278127CA2123C123EA2121E121FA27E7FA212077FA2 12037FA212017FA212007FA21378137CA2133C133EA2131E131FA27F80A2130780A26D7E A2130180A2130080A21478147CA2143C143EA2141E141FA2801580A2140715C0A2140315 E0A2140115F0A2140015F8A21578157CA2153C153EA2151E150C1F537BBD2A>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmbx12 12 36 /Fq 36 122 df45 D49 DII<163FA25E5E5D5DA25D5D5D5DA25D92B5FCEC01F7EC03E7140715C7EC0F 87EC1F07143E147E147C14F8EB01F0EB03E0130714C0EB0F80EB1F00133E5BA25B485A48 5A485A120F5B48C7FC123E5A12FCB91280A5C8000F90C7FCAC027FB61280A531417DC038 >I<0007150301E0143F01FFEB07FF91B6FC5E5E5E5E5E16804BC7FC5D15E092C8FC01C0 C9FCAAEC3FF001C1B5FC01C714C001DF14F09039FFE03FFC9138000FFE01FC6D7E01F06D 13804915C0497F6C4815E0C8FC6F13F0A317F8A4EA0F80EA3FE0487E12FF7FA317F05B5D 6C4815E05B007EC74813C0123E003F4A1380D81FC0491300D80FF0495AD807FEEBFFFC6C B612F0C65D013F1480010F01FCC7FC010113C02D427BC038>I<4AB47E021F13F0027F13 FC49B6FC01079038807F8090390FFC001FD93FF014C04948137F4948EBFFE048495A5A14 00485A120FA248486D13C0EE7F80EE1E00003F92C7FCA25B127FA2EC07FC91381FFF8000 FF017F13E091B512F89039F9F01FFC9039FBC007FE9039FF8003FF17804A6C13C05B6F13 E0A24915F0A317F85BA4127FA5123FA217F07F121FA2000F4A13E0A26C6C15C06D491380 6C018014006C6D485A6C9038E01FFC6DB55A011F5C010714C0010191C7FC9038003FF02D 427BC038>I67 DI73 D77 D80 D<923807FFC092B512FE0207ECFFC0021F15F09126 7FFE0013FC902601FFF0EB1FFF010701C0010713C04990C700017F49486E7F49486F7E49 486F7E49486F7E48496F7E48496F1380A248496F13C0A24819E091C97E4819F0A2484870 13F8A3007F19FCA249177FA300FF19FEAD007F19FCA36D17FF003F19F8A3001F19F06D5E A26C19E06E01FE5B6C912603FF8014C06C6D486D4813804B13E06C9028E01F83F00F1300 6C903BF01E00F81FFE90267FF83E90387C3FFC90263FFC3C6D485AD91FFE91381EFFF0D9 0FFF021F5B6D01FE5D010194C7FC6D6D6CB45A023F90B512F8020703E0130202006F1307 030713C792C7EA07F8716C130F72131F9538FF80FF96B5FC7114FEA3831AFCA27213F81A F0847213E07213C0721300F001FC48587AC454>III<003FBA12E0A59026FE000FEB8003D87FE09338003FF049171F90C71607A2 007E1803007C1801A300781800A400F819F8481978A5C81700B3B3A20107B8FCA545437C C24E>I87 D<903801FFE0011F13FE01 7F6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA 00F090C7FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E0000F1380381FFE00 485A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007 EBFFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 D99 DII II<137C48B4FC4813804813C0A24813E0A56C13C0A26C13806C1300EA007C90C7FC AAEB7FC0EA7FFFA512037EB3AFB6FCA518467CC520>105 D107 DI<90277F8007FEEC0FFCB590263FFFC090387FFF8092B5D8 F001B512E002816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F801FFC0003D99F0090 26FF3E007F6C019E6D013C130F02BC5D02F86D496D7EA24A5D4A5DA34A5DB3A7B60081B6 0003B512FEA5572D7CAC5E>I<90397F8007FEB590383FFF8092B512E0028114F8913987 F03FFC91388F801F000390399F000FFE6C139E14BC02F86D7E5CA25CA35CB3A7B60083B5 12FEA5372D7CAC3E>II<90 397FC00FF8B590B57E02C314E002CF14F89139DFC03FFC9139FF001FFE000301FCEB07FF 6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318F017FFA24C13 E06E15C06E5B6E4913806E4913006E495A9139DFC07FFC02CFB512F002C314C002C091C7 FCED1FF092C9FCADB67EA536407DAC3E>I<90387F807FB53881FFE0028313F0028F13F8 ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED01E092C7FCA35C B3A5B612E0A5272D7DAC2E>114 D<90391FFC038090B51287000314FF120F381FF00338 3FC00049133F48C7121F127E00FE140FA215077EA27F01E090C7FC13FE387FFFF014FF6C 14C015F06C14FC6C800003806C15806C7E010F14C0EB003F020313E0140000F0143FA26C 141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B55A00FC5CD8F0 3F13E026E007FEC7FC232F7CAD2C>IIII121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmbx9 9 8 /Fr 8 117 df<120FEA3FC0EA7FE0EAFFF0A6EA7FE0EA3FC0EA0F000C0C7A8B19>46 D65 D97 DI<903807FF80013F13F090B512FC3903FE01FE4848487EEA0FF8EA 1FF0EA3FE0A2007F6D5A496C5A153000FF91C7FCA9127F7FA2003FEC07807F6C6C130F00 0FEC1F00D807FE133E3903FF80FCC6EBFFF8013F13E0010790C7FC21217DA027>I<3901 F81F8000FFEB7FF0ECFFF89038F9E3FC9038FBC7FE380FFF876C1307A213FEEC03FCEC01 F8EC0060491300B1B512F0A41F217EA024>114 D<9038FFE1C0000713FF5A383F803F38 7E000F14075A14037EA26C6CC7FC13FCEBFFE06C13FC806CEBFF80000F14C06C14E0C6FC 010F13F0EB007F140F00F0130714037EA26C14E06C13076CEB0FC09038C01F8090B51200 00F913FC38E03FE01C217DA023>I<133CA5137CA313FCA21201A212031207001FB51280 B6FCA3D807FCC7FCB0EC03C0A79038FE078012033901FF0F006C13FEEB3FFCEB0FF01A2F 7EAE22>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmtt9 9 20 /Fs 20 112 df<007FB512F8B612FCA46C14F81E067C9927>45 D<121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A0A728927>I64 D<007FB5FCB612C08115F87E3907E003FCEC00FE157E157F81A6157EA25D1403EC0FF890 B55A15C015F081819038E000FE157FED3F80151FA2ED0FC0A6151F1680153FED7F004A5A 007FB55AB65A5D15E06C1480222E7FAD27>66 D<387FFFFC14FFB612C06C80813907E00F F81407EC01FC6E7EA2157E157F811680151FA316C0150FABED1F80A3153F1600A25D15FE A24A5A4A5A140F007FB55A5DB65A6C91C7FC14FC222E7FAD27>68 D<007FB61280B712C0A37E3907E0000FA6ED078092C7FCA4EC07804A7EA390B5FCA5EBE0 0FA36E5A91C8FCA4ED03C0ED07E0A7007FB6FCB7FCA36C15C0232E7FAD27>I<007FB612 80B712C0A37E3907E0000FA6ED078092C7FCA4EC07804A7EA390B5FCA5EBE00FA36E5A91 C8FCAC387FFF80B57EA36C5B222E7EAD27>I<007FB61280B712C0A439FC03F00FA60078 EC0780000091C7FCB3AB90B512C04880A36C5C222E7EAD27>84 D<3A7FFE01FFF8B54813 FCA36C486C13F83A07E0001F80B3AB6D133F00031500A26D5B0001147E6D13FE6C6C485A 90387F87F814FF6D5B010F13C06D5BD901FEC7FC262F80AD27>I<3803FFC0000F13F048 13FC4813FF811380EC1FC0381F000F000480C71207A2EB0FFF137F0003B5FC120F5A383F FC07EA7FC0130012FE5AA46C130F007F131FEBC0FF6CB612806C15C07E000313F1C69038 807F8022207C9F27>97 D99 DIII104 D<130F497E497EA46D5A6DC7FC90C8FCA7383FFF80487FA37EEA000FB3A4007F B512F0B6FC15F815F07E1D2F7BAE27>I107 D<387FFF80B57EA37EEA000FB3B2007FB512F8B612FCA3 6C14F81E2E7CAD27>I<387FE07F39FFF1FFC001F713F090B5FC6C80000313C1EC01FCEB FE005B5BA25BB03A7FFF83FFE0B500C713F0A36C018313E024207F9F27>110 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr9 9 74 /Ft 74 128 df12 DI16 D<137813FCA212011203EA07 F813E0EA0FC0EA1F801300123C5A5A12400E0E71B326>19 D<147CEB01FEEB07C790380F 8380EB1F0181EB3E00A2137EA2137C137EA214015D140392C9FC5C140E6D5A1418143802 F090380FFFF05C6D5A04011300EE00FC6D6C1470011F1560013F15E0D977F0495AD9E3F8 5CD801C31403260381FC91C7FC00075D48C66C130E486D130C486D131C003E6D6C5A007E ECC03091381FE07000FE010F5B6F5AEC07F96EB45A6C6D90C712306E5A157F6C6C6D6C13 604B6C13E03A3FC001EFE03C1FE003C7F803C03C0FF01F83FE0F802707FFFE00B5120000 0101F8EB3FFE26003FC0EB07F034387DB53C>38 D<14C01301EB0380EB0F00130E5B133C 5B5BA2485A485AA212075B120F90C7FC5AA2121E123EA3123C127CA55AB0127CA5123C12 3EA3121E121FA27E7F12077F1203A26C7E6C7EA213787F131C7F130FEB0380EB01C01300 124A79B71E>40 D<12C07E1270123C121C7E120F6C7E6C7EA26C7E6C7EA27F1378137C13 3C133EA2131E131FA37F1480A5EB07C0B0EB0F80A514005BA3131E133EA2133C137C1378 13F85BA2485A485AA2485A48C7FC120E5A123C12705A5A124A7CB71E>I<123C127EB4FC A21380A2127F123D1201A412031300A25A1206120E120C121C5A5A126009177A8715>44 DI<123C127E12FFA4127E123C08087A8715>I48 D<13075B5B137FEA07FFB5FC13BFEAF83F 1200B3B3A2497E007FB51280A319327AB126>IIII<000C14C0380FC00F90B5 128015005C5C14F014C0D80C18C7FC90C8FCA9EB0FC0EB7FF8EBF07C380FC03F9038001F 80EC0FC0120E000CEB07E0A2C713F01403A215F8A41218127E12FEA315F0140712F80060 14E01270EC0FC06C131F003C14806CEB7F00380F80FE3807FFF8000113E038003F801D34 7CB126>I<14FE903807FF80011F13E090383F00F0017C13703901F801F8EBF003EA03E0 1207EA0FC0EC01F04848C7FCA248C8FCA35A127EEB07F0EB1FFC38FE381F9038700F8090 38E007C039FFC003E0018013F0EC01F8130015FC1400A24814FEA5127EA4127F6C14FCA2 6C1301018013F8000F14F0EBC0030007EB07E03903E00FC03901F81F806CB51200EB3FFC EB0FE01F347DB126>I<1230123C003FB6FCA34814FEA215FC0070C71238006014301570 15E04814C01401EC0380C7EA07001406140E5C141814385CA25CA2495A1303A3495AA213 0FA3131F91C7FCA25BA55BA9131C20347CB126>III<123C127E12FFA4127E123C1200B012 3C127E12FFA4127E123C08207A9F15>I<123C127E12FFA4127E123C1200B0123C127E12 FE12FFA3127F123F1203A412071206A3120E120C121C1238123012701260082F7A9F15> I<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC187FA34A6C7EA202707FEC601FA202E0 7FECC00FA2D901807F1507A249486C7EA301066D7EA2010E80010FB5FCA249800118C77E A24981163FA2496E7EA3496E7EA20001821607487ED81FF04A7ED8FFFE49B512E0A33336 7DB53A>65 DIIIIIIII<017FB5FCA3 9038003FE0EC1FC0B3B1127EB4FCA4EC3F805A0060140000705B6C13FE6C485A380F03F0 3803FFC0C690C7FC20357DB227>IIIIIII82 D<90381FE00390387FFC0748B5FC3907F01FCF390F8003FF 48C7FC003E80814880A200788000F880A46C80A27E92C7FC127F13C0EA3FF013FF6C13F0 6C13FF6C14C06C14F0C680013F7F01037F9038003FFF140302001380157F153FED1FC015 0F12C0A21507A37EA26CEC0F80A26C15006C5C6C143E6C147E01C05B39F1FC03F800E0B5 12E0011F138026C003FEC7FC22377CB42B>I<007FB712FEA390398007F001D87C00EC00 3E0078161E0070160EA20060160600E01607A3481603A6C71500B3AB4A7E011FB512FCA3 30337DB237>IIII<003FB612FCA39039F80007 F813C090C7EA0FF0003EEC1FE0123C0038EC3FC00078EC7F801270EDFF004A5AA2006049 5AA24A5A4A5AC7FC4A5A4A5AA24A5A4AC7FCA2495A495AA2495A495AA24948130C495AA2 495A49C7FCA24848141CA2485A485A1638485A4848147816F84848130148481307153FB7 FCA326337CB22F>90 DI93 D97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3F FCEBFC0F3901F003BF3907E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA2 7E6C6C5BA26C6C5B6C6C4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB3 2B>III<151F90391FC07F809039FFF8E3C03901F07FC739 07E03F033A0FC01F83809039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F 6C6C48C7FCEBF07C380EFFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F0 6C14FC4880381F0001003EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C39 0FC001F83903F007E0C6B51280D91FFCC7FC22337EA126>III 107 DI<2703F01FE013FF 00FF90267FF80313C0903BF1E07C0F03E0903BF3803E1C01F02807F7003F387FD803FE14 70496D486C7EA2495CA2495CB3486C496C487EB53BC7FFFE3FFFF0A33C217EA041>I<39 03F01FC000FFEB7FF09038F1E0FC9038F3807C3907F7007EEA03FE497FA25BA25BB3486C EB7F80B538C7FFFCA326217EA02B>II<3903F03F8000FFEBFFE09038F3C0F89038F7007ED807FE7F6C48EB1F804914 C049130F16E0ED07F0A3ED03F8A9150716F0A216E0150F16C06D131F6DEB3F80160001FF 13FC9038F381F89038F1FFE0D9F07FC7FC91C8FCAA487EB512C0A325307EA02B>I<9038 07F00390383FFC07EBFC0F3901F8038F3807E001000F14DF48486CB4FC497F123F90C77E 5AA25A5AA9127FA36C6C5B121F6D5B000F5B3907E003BF3903F0073F3800F81EEB3FF8EB 0FE090C7FCAAED7F8091380FFFFCA326307DA029>I<3803E07C38FFE1FF9038E38F8090 38E71FC0EA07EEEA03ECA29038FC0F8049C7FCA35BB2487EB512E0A31A217FA01E>II<1330A51370A313F0A21201A212 031207381FFFFEB5FCA23803F000AF1403A814073801F806A23800FC0EEB7E1CEB1FF8EB 07E0182F7FAD1E>IIIII<3A7FFF807FF8A33A07F8001FC00003EC0F 800001EC070015066C6C5BA26D131C017E1318A26D5BA2EC8070011F1360ECC0E0010F5B A2903807E180A214F3010390C7FC14FBEB01FEA26D5AA31478A21430A25CA214E05CA249 5A1278D8FC03C8FCA21306130EEA701CEA7838EA1FF0EA0FC025307F9F29>I<003FB512 F0A2EB000F003C14E00038EB1FC00030EB3F800070137F1500006013FE495A13035CC648 5A495AA2495A495A49C7FC153013FE485A12035B48481370485A001F14604913E0485A38 7F000348130F90B5FCA21C207E9F22>II<001C1370387F01FC00 FF13FEA4007F13FC381C0070170879B226>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr10 10 77 /Fu 77 123 df11 DIII<001C131C007F137F39FF80FF80A26D13C0A3007F137F001C 131C00001300A40001130101801380A20003130301001300485B00061306000E130E485B 485B485B006013601A197DB92A>34 D<121C127FEAFF80A213C0A3127F121C1200A41201 1380A2120313005A1206120E5A5A5A12600A1979B917>39 D<146014E0EB01C0EB0380EB 0700130E131E5B5BA25B485AA2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2 127CA67EA3121EA2121F7EA27F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB 01C0EB00E01460135278BD20>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378 A2137C133C133E131EA2131F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A2 5B131EA2133E133C137C1378A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD 20>I<15301578B3A6007FB812F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41> 43 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A 5A5A12600A19798817>II<121C127FEAFF80A5EA7F00121C0909 798817>I48 DIII<15 38A2157815F8A2140114031407A2140F141F141B14331473146314C313011483EB030313 071306130C131C131813301370136013C01201EA038013005A120E120C5A123812305A12 E0B712F8A3C73803F800AB4A7E0103B512F8A325397EB82A>I<0006140CD80780133C90 38F003F890B5FC5D5D158092C7FC14FC38067FE090C9FCABEB07F8EB3FFE9038780F8039 07E007E090388003F0496C7E12066E7EC87EA28181A21680A4123E127F487EA490C71300 485C12E000605C12700030495A00385C6C1303001E495A6C6C485A3907E03F800001B5C7 FC38007FFCEB1FE0213A7CB72A>II<12301238123E 003FB612E0A316C05A168016000070C712060060140E5D151800E01438485C5D5DC71201 4A5A92C7FC5C140E140C141C5CA25CA214F0495AA21303A25C1307A2130FA3495AA3133F A5137FA96DC8FC131E233B7BB82A>III<121C127FEAFF80A5EA7F00121CC7FCB2121C127FEAFF80A5EA7F00121C092479A3 17>I<121C127FEAFF80A5EA7F00121CC7FCB2121C127F5A1380A4127F121D1201A41203 1300A25A1206A2120E5A121812385A1260093479A317>I<007FB812F8B912FCA26C17F8 CCFCAE007FB812F8B912FCA26C17F836167B9F41>61 D<1538A3157CA315FEA34A7EA34A 6C7EA202077FEC063FA2020E7FEC0C1FA2021C7FEC180FA202387FEC3007A202707FEC60 03A202C07F1501A2D901807F81A249C77F167FA20106810107B6FCA24981010CC7121FA2 496E7EA3496E7EA3496E7EA213E0707E1201486C81D80FFC02071380B56C90B512FEA337 3C7DBB3E>65 DI<913A01FF800180020FEBE003027F13F8903A01FF 807E07903A03FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F01FE153F12 014848151F4848150FA248481507A2485A1703123F5B007F1601A35B00FF93C7FCAD127F 6DED0180A3123F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D00001618017F1538 6D6C5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFFFC020F13F0 02011380313D7BBA3C>IIIIIII75 DIIIIIIII<003FB8 12E0A3D9C003EB001F273E0001FE130348EE01F00078160000701770A300601730A400E0 1738481718A4C71600B3B0913807FF80011FB612E0A335397DB83C>IIII< 003FB7FCA39039FC0001FE01C0130349495A003EC7FC003C4A5A5E0038141F00784A5A12 704B5A5E006014FF4A90C7FCA24A5A5DC712074A5AA24A5A5D143F4A5AA24A5A92C8FC5B 495AA2495A5C130F4948EB0180A2495A5C137F495A16034890C7FC5B1203485AEE070048 5A495C001F5D48485C5E4848495A49130FB8FCA329397BB833>90 DI<390180018000031303390700 0700000E130E485B0018131800381338003013300070137000601360A200E013E0485BA4 00CE13CE39FF80FF806D13C0A3007F137FA2393F803F80390E000E001A1974B92A>II97 DIIII<147E903803FF8090380FC1E0EB1F8790383F0FF0137EA213 FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387FFFF8A31C3B7FBA19>I< ED03F090390FF00FF890393FFC3C3C9039F81F707C3901F00FE03903E007C03A07C003E0 10000FECF000A248486C7EA86C6C485AA200075C6C6C485A6D485A6D48C7FC38073FFC38 060FF0000EC9FCA4120FA213C06CB512C015F86C14FE6CECFF804815C03A0F80007FE048 C7EA0FF0003E140348140116F8481400A56C1401007C15F06CEC03E0003F1407D80F80EB 0F80D807E0EB3F003901FC01FC39007FFFF0010790C7FC26387EA52A>IIIIII<2703F00FF0EB1FE000FFD93FFCEB7FF8913AF03F01E07E903BF1C01F8380 3F3D0FF3800FC7001F802603F70013CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A348 6C496CEB1FE0B500C1B50083B5FCA340257EA445>I<3903F00FF000FFEB3FFCECF03F90 39F1C01F803A0FF3800FC03803F70013FE496D7EA25BA35BB3A3486C497EB500C1B51280 A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E9039F3801F 803A07F7000FC0D803FEEB07E049EB03F04914F849130116FC150016FEA3167FAA16FEA3 ED01FCA26DEB03F816F06D13076DEB0FE001F614C09039F7803F009038F1E07E9038F0FF F8EC1FC091C8FCAB487EB512C0A328357EA42E>II<3807E01F00FFEB7FC09038E1E3 E09038E387F0380FE707EA03E613EE9038EC03E09038FC0080491300A45BB3A2487EB512 F0A31C257EA421>II<1318A51338A31378A313F8120112031207001FB5FCB6FCA2D801F8C7FCB215 C0A93800FC011580EB7C03017E13006D5AEB0FFEEB01F81A347FB220>IIIIII<003FB5 12FCA2EB8003D83E0013F8003CEB07F00038EB0FE012300070EB1FC0EC3F800060137F15 0014FE495AA2C6485A495AA2495A495A495AA290387F000613FEA2485A485A0007140E5B 4848130C4848131CA24848133C48C7127C48EB03FC90B5FCA21F247EA325>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmbx12 14.4 26 /Fv 26 122 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 698 448 a Fv(On)45 b(Quasi-Reductiv)l(e)h(and)f (Quasi-Simplifying)684 598 y(Deterministic)i(Conditional)g(Rewrite)f (Systems)1667 886 y Fu(Enno)27 b(Ohlebusc)n(h)586 1060 y Ft(Univ)n(ersit)n(y)e(of)h(Bielefeld,)i(F)-6 b(acult)n(y)25 b(of)i(T)-6 b(ec)n(hnology)g(,)26 b(P)-6 b(.O.)26 b(Bo)n(x)f(10)i(01)f (31,)h(33501)h(Bielefeld,)1108 1152 y(German)n(y)-6 b(,)24 b(email:)i Fs(enno@TechFak.Uni-Bielefeld.)q(DE)759 1476 y Fr(Abstract.)43 b Ft(Deterministic)23 b(conditional)h(rewrite)g (systems)f(p)r(ermit)f(extra)h(v)l(ari-)759 1567 y(ables)38 b(on)f(the)f(righ)n(t-hand)g(sides)i(of)g(the)e(rules.)i(If)f(suc)n(h)f (a)h(system)f(is)i(quasi-)759 1658 y(reductiv)n(e)e(or)h (quasi-simplifying,)g(then)f(it)h(is)g(terminating)f(and)h(has)g(a)g (com-)759 1750 y(putable)c(rewrite)h(relation.)h(This)e(pap)r(er)g(pro) n(vides)g(new)g(criteria)i(for)f(sho)n(wing)759 1841 y(quasi-reductivit)n(y)23 b(and)g(quasi-simplifyingness.)i(In)e(this)h (con)n(text,)g(another)g(cri-)759 1932 y(terion)39 b(from)f([ALS94)q(]) g(will)i(b)r(e)e(recti\014ed)h(and)f(a)h(claim)f(in)g([Mar96)r(])h (will)h(b)r(e)759 2024 y(refuted.)27 b(Moreo)n(v)n(er,)h(w)n(e)f(will)h (in)n(v)n(estigate)g(under)e(whic)n(h)h(conditions)g(the)f(prop-)759 2115 y(erties)h(exhibit)e(a)h(mo)r(dular)f(b)r(eha)n(vior.)523 2389 y Fq(1)112 b(In)m(tro)s(duction)523 2583 y Fu(Conditional)24 b(term)h(rewriting)f(systems)h(\(CTRSs\))g(are)f(the)h(basis)g(for)f (the)h(in)n(tegration)f(of)523 2682 y(the)34 b(functional)g(and)f (logic)g(programming)e(paradigms;)h(see)h([Han94)o(])h(for)f(an)g(o)n (v)n(erview)523 2782 y(of)d(this)h(\014eld.)f(In)h(these)f(systems)f(v) -5 b(ariables)29 b(on)h(the)h(righ)n(t-hand)e(side)h(of)g(a)g(rewrite)f (rule)523 2882 y(whic)n(h)24 b(do)h(not)f(o)r(ccur)g(on)g(the)h (left-hand)f(side)h(are)e(problematic)h(b)r(ecause)g(it)g(is)h(in)f (general)523 2981 y(not)29 b(clear)f(ho)n(w)h(to)g(instan)n(tiate)g (them.)g(On)g(the)h(other)e(hand,)i(a)e(restricted)h(use)g(of)g(these) 523 3081 y(extra)d(v)-5 b(ariables)26 b(enables)h(a)g(more)f(natural)h (and)g(e\016cien)n(t)g(w)n(a)n(y)g(of)g(writing)g(programs.)e(A)523 3181 y(paradigmatic)h(example)h(is)h(the)g(Fib)r(onacci)f(system)g Fp(R)2302 3193 y Fo(f)7 b(ib)1343 3358 y Fn(f)i(ib)p Fu(\(0\))22 b Fp(!)h(h)p Fu(0)p Fn(;)14 b(s)p Fu(\(0\))p Fp(i)1233 3458 y Fn(f)9 b(ib)p Fu(\()p Fn(s)p Fu(\()p Fn(x)p Fu(\)\))24 b Fp(!)f(h)p Fn(z)t(;)14 b(y)20 b Fu(+)e Fn(z)t Fp(i)23 b(\()g Fn(f)9 b(ib)p Fu(\()p Fn(x)p Fu(\))23 b Fp(!)g(h)p Fn(y)s(;)14 b(z)t Fp(i)523 3629 y Fu(whic)n(h)35 b(has)f(extra)g(v)-5 b(ariables)34 b(on)h(the)g(righ)n(t-hand)f(side)g (of)h(the)h(last)e(rule.)h(The)g(rewrite)523 3729 y(relation)h(induced) h(b)n(y)g(the)g(ab)r(o)n(v)n(e)f(CTRS)h(is)g(e\013ectiv)n(ely)g (terminating)f(\(that)i(is,)f(com-)523 3828 y(putable)30 b(and)g(terminating\))g(b)r(ecause)g(the)h(system)f(is)g(a)f Fm(quasi-r)l(e)l(ductive)k(deterministic)523 3928 y Fu(CTRS.)23 b(This)f(class)g(of)h(CTRSs)f(w)n(as)g(in)n(tro)r(duced)g(b)n(y)g (Ganzinger)g([Gan91)o(])g(in)h(order)f(to)g(ef-)523 4028 y(\014cien)n(tly)j(translate)e(order-sorted)g(sp)r(eci\014cations)h(in) n(to)g(conditional)g(man)n(y-sorted)f(equa-)523 4127 y(tions.)j(Quasi-reductivit)n(y)f(is)i(in)g(general)e(undecidable)h (but)h(su\016cien)n(t)g(criteria)e(to)i(c)n(hec)n(k)523 4227 y(quasi-reductivit)n(y)j(are)g(kno)n(wn)h([Gan91)n(,ALS94].)g(The) h(criterion)e(in)h([ALS94])g(con)n(tains)523 4327 y(a)38 b(\015a)n(w)h(whic)n(h)f(will)h(b)r(e)g(recti\014ed)g(and)g(the)g (recti\014ed)f(criterion)g(sho)n(ws)g(in)h(fact)g Fm(quasi-)523 4426 y(simplifyingness)30 b Fu(\(a)d(stronger)f(prop)r(ert)n(y)h(than)g (quasi-reductivit)n(y\).)648 4526 y(Similar)e(to)g(the)h(approac)n(h)e (of)i(Marc)n(hiori)d([Mar96)o(],)i(w)n(e)h(will)f(sho)n(w)g(ho)n(w)g (ev)n(ery)g(deter-)523 4625 y(ministic)c(CTRS)g Fp(R)g Fu(can)g(b)r(e)g(transformed)f(in)n(to)g(an)g(unconditional)h(TRS)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\))21 b(suc)n(h)g(that)523 4725 y(\(simple\))35 b(termination)f(of)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\))35 b(implies)f(quasi-reductivit)n(y)f (\(quasi-simplifyingness\))523 4825 y(of)e Fp(R)p Fu(.)g(\(A)g(coun)n (terexample)e(will)i(sho)n(w)f(that)h(quasi-reductivit)n(y)e(of)i Fp(R)g Fu(do)r(es)f(not)h(imply)523 4924 y(termination)i(of)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\),)34 b(ev)n(en)f(if)h Fp(R)f Fu(is)g(left-linear)g(and)g(con\015uen)n(t.\))h(By)f(means)g(of)g(this) p eop %%Page: 2 2 2 1 bop 523 448 a Fu(transformational)28 b(approac)n(h,)f(standard)i (metho)r(ds)h(for)f(pro)n(ving)f(\(simple\))i(termination)523 548 y(of)25 b(TRSs)g(can)g(no)n(w)f(b)r(e)h(emplo)n(y)n(ed)f(to)h (infer)g(quasi-reductivit)n(y)f(\(quasi-simplifyingness\).)523 648 y(Due)g(to)g(the)g(fact)g(that)h(p)r(o)n(w)n(erful)e(tec)n(hniques) g(for)h(sho)n(wing)e(termination)i(lik)n(e)f(simpli\014ca-)523 747 y(tion)c(orderings)e([Der87)o(])i(and)g(dep)r(endency)h(pairs)e([A) n(G99)o(])h(are)f(amenable)h(to)g(automation,)523 847 y(our)30 b(new)h(criteria)e(no)n(w)h(allo)n(w)g(us)g(to)h(infer)g (quasi-reductivit)n(y)e(\(quasi-simplifyingness\))523 946 y(automatically)-7 b(.)27 b(This)g(is)h(a)f(ma)5 b(jor)26 b(impro)n(v)n(emen)n(t)h(on)g(the)h(kno)n(wn)f(criteria.)648 1048 y(Since)e(b)r(oth)g(simple)g(termination)g(of)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\))26 b(and)e(the)i(recti\014ed)f(criterion) f(of)h([ALS94)o(])523 1147 y(pro)n(v)n(e)19 b(quasi-simplifyingness)g (of)i Fp(R)p Fu(,)g(w)n(e)f(will)h(in)n(v)n(estigate)e(the)i (relationship)f(b)r(et)n(w)n(een)h(the)523 1247 y(t)n(w)n(o)j (criteria.)h(It)g(will)g(b)r(e)h(sho)n(wn)e(that)i(none)f(of)g(the)h (criteria)e(is)h(subsumed)g(b)n(y)g(the)h(other.)523 1347 y(En)j(passan)n(t,)f(a)g(claim)h(in)g([Mar96)n(])g(will)h(b)r(e)f (refuted:)g(Simplifyingness)g(of)g(a)g(join)g(CTRS)523 1446 y Fp(R)e Fu(without)f(extra)g(v)-5 b(ariables)25 b(do)r(es)h Fm(not)34 b Fu(imply)26 b(simple)h(termination)e(of)i(its)f (transformed)523 1546 y(unconditional)h(TRS.)648 1647 y(Finally)-7 b(,)25 b(w)n(e)h(will)g(address)f(the)h(problem)f(of)h(mo) r(dularit)n(y)-7 b(.)25 b(W)-7 b(e)27 b(will)f(sho)n(w)f(that)h(quasi-) 523 1747 y(simplifyingness)35 b(is)h(not)f(mo)r(dular,)g(whereas)f (quasi-reductivit)n(y)g(is)i(mo)r(dular)f(for)f(non-)523 1846 y(o)n(v)n(erlapping)d(syn)n(tactically)h(deterministic)h(CTRSs.)h (Under)f(certain)g(\(natural\))g(condi-)523 1946 y(tions,)28 b(it)g(is)f(mo)r(dular)g(ev)n(en)g(for)g(hierarc)n(hical)f(com)n (binations)g(of)i(these)f(systems.)648 2047 y(The)19 b(material)f(presen)n(ted)h(in)h(this)f(pap)r(er)g(complemen)n(ts)g (results)g(rep)r(orted)f(in)i([Ohl99)o(].)523 2321 y Fq(2)112 b(Preliminaries)523 2528 y Fu(The)29 b(reader)f(is)h(assumed)g (to)g(b)r(e)g(familiar)g(with)h(the)f(basic)g(concepts)g(of)g(term)g (rewriting)523 2628 y(whic)n(h)e(can)f(for)h(instance)g(b)r(e)g(found)g (in)g(the)h(textb)r(o)r(ok)e(of)h(Baader)f(and)h(Nipk)n(o)n(w)f([BN98)o (].)523 2728 y(Here)h(w)n(e)h(will)f(only)h(recall)e(the)i (de\014nitions)g(whic)n(h)f(are)g(crucial)g(to)g(this)h(pap)r(er.)648 2829 y(A)39 b Fm(r)l(ewrite)i(r)l(elation)47 b Fn(R)40 b Fu(is)g(a)f(binary)g(relation)f(on)h(terms)g(whic)n(h)h(is)f Fm(close)l(d)j(under)523 2929 y(c)l(ontexts)36 b Fu(\(i.e.,)31 b(if)g Fn(s)f(R)i(t)p Fu(,)e(then)h Fn(C)6 b Fu([)p Fn(s)p Fu(])31 b Fn(R)g(C)6 b Fu([)p Fn(t)p Fu(])31 b(for)f(all)g(con)n(texts) g Fn(C)6 b Fu([)31 b(]\))f(and)h Fm(close)l(d)i(under)523 3028 y(substitutions)e Fu(\(i.e.,)25 b(if)g Fn(s)g(R)h(t)p Fu(,)f(then)g Fn(s\033)k(R)c(t\033)k Fu(for)24 b(all)g(substitutions)i Fn(\033)s Fu(\).)f(A)h Fm(r)l(ewrite)h(or)l(der)523 3128 y Fu(is)j(a)g(rewrite)f(relation)g(whic)n(h)h(is)g(also)f(a)h(partial)f (order.)g(A)h(w)n(ell-founded)g(rewrite)f(order)523 3227 y(is)35 b(called)h Fm(r)l(e)l(duction)h(or)l(der)p Fu(.)f(A)g Fm(simpli\014c)l(ation)j(or)l(der)45 b Fp(\037)35 b Fu(is)h(a)f (reduction)g(order)f(whic)n(h)523 3327 y(con)n(tains)c(the)h(prop)r(er) e(subterm)i(relation)e Fl(B)p Fu(,)i(i.e.,)f Fn(C)6 b Fu([)p Fn(t)p Fu(])29 b Fp(\037)e Fn(t)k Fu(for)f(all)g(con)n(texts)g Fn(C)6 b Fu([)31 b(])d Fp(6)p Fu(=)g Fk(2)523 3427 y Fu(and)f(terms)h Fn(t)p Fu(.)648 3528 y(In)33 b(a)f(CTRS)h(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))34 b(rules)e(ha)n(v)n(e)g(the)i(form)e Fn(l)h Fp(!)f Fn(r)j Fp(\()d Fn(s)2514 3540 y Fj(1)2583 3528 y Fu(=)g Fn(t)2710 3540 y Fj(1)2747 3528 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)2971 3540 y Fo(k)3043 3528 y Fu(=)32 b Fn(t)3170 3540 y Fo(k)3243 3528 y Fu(with)523 3628 y Fn(l)r(;)14 b(r)n(;)g(s)698 3640 y Fj(1)735 3628 y Fn(;)g(:)g(:)g(:)f(;)h(s)958 3640 y Fo(k)999 3628 y Fu(,)30 b Fn(t)1082 3640 y Fj(1)1120 3628 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)1334 3640 y Fo(k)1403 3628 y Fp(2)28 b(T)21 b Fu(\()p Fp(F)8 b Fn(;)14 b Fp(V)7 b Fu(\).)31 b Fn(l)h Fu(ma)n(y)e(not)g(b)r(e)h(a)f (v)-5 b(ariable.)30 b(W)-7 b(e)30 b(frequen)n(tly)h(ab-)523 3727 y(breviate)j(the)h(conditional)f(part)g(of)h(the)g(rule)f(b)n(y)h Fn(c)p Fu(.)g(If)g(a)f(rule)g(has)h(no)f(conditions,)g(w)n(e)523 3827 y(write)23 b Fn(l)h Fp(!)g Fn(r)r Fu(,)g(demand)f(that)g Fp(V)7 b Fn(ar)r Fu(\()p Fn(r)r Fu(\))25 b Fp(\022)e(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\),)23 b(and)g(call)g Fn(l)i Fp(!)e Fn(r)j Fu(an)d(unconditional)f(rule.)523 3926 y(The)27 b(=)g(sym)n(b)r(ol)g(in)g(the)h(conditions)f(can)f(b)r(e)i(in) n(terpreted)f(in)g(di\013eren)n(t)h(w)n(a)n(ys)d(whic)n(h)i(lead)523 4026 y(to)33 b(di\013eren)n(t)g(rewrite)f(relations)g(asso)r(ciated)g (with)h Fp(R)p Fu(.)h(F)-7 b(or)32 b(instance,)h(in)h(a)e Fm(join)41 b Fu(CTRS)523 4126 y(the)31 b(=)g(sym)n(b)r(ol)f(stands)h (for)f(joinabilit)n(y)h(\()p Fp(#)1917 4138 y Fi(R)1978 4126 y Fu(\).)g(This)g(pap)r(er)g(deals)f(with)i(\014nite)f Fm(oriente)l(d)523 4225 y Fu(CTRSs)26 b(in)g(whic)n(h)f(the)h(equalit)n (y)f(signs)g(are)g(in)n(terpreted)g(as)g(reac)n(habilit)n(y)f(\()p Fp(!)3021 4195 y Fi(\003)3021 4248 y(R)3082 4225 y Fu(\).)i(A)g Fm(nor-)523 4325 y(mal)g Fu(CTRS)f(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))26 b(is)f(an)g(orien)n(ted)g(CTRS)g(in)h(whic)n(h)f(the) g(rewrite)g(rules)g(are)f(sub)5 b(ject)25 b(to)523 4425 y(the)j(additional)g(constrain)n(t)f(that)h(ev)n(ery)f Fn(t)1880 4437 y Fo(j)1943 4425 y Fu(is)h(a)f(ground)g(normal)g(form)h (with)g(resp)r(ect)g(to)523 4524 y Fp(R)593 4536 y Fo(u)637 4524 y Fu(,)g(where)f Fp(R)998 4536 y Fo(u)1064 4524 y Fu(=)c Fp(f)p Fn(l)h Fp(!)f Fn(r)31 b Fp(j)c Fn(l)e Fp(!)e Fn(r)j Fp(\()d Fn(c)g Fp(2)g(Rg)p Fu(.)648 4625 y(F)-7 b(or)30 b(ev)n(ery)h(rule)g Fn(\032)e Fu(:)g Fn(l)i Fp(!)f Fn(r)i Fp(\()d Fn(c)p Fu(,)j(the)g(set)f(of)g(v)-5 b(ariables)31 b(o)r(ccurring)f(in)h Fn(\032)h Fu(is)f(denoted)523 4725 y(b)n(y)26 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(\032)p Fu(\))26 b(and)g(the)h(set)e(of)h(extra)f(v)-5 b(ariables)25 b(in)h Fn(\032)g Fu(is)g Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(\032)p Fu(\))24 b(=)f Fp(V)7 b Fn(ar)r Fu(\()p Fn(\032)p Fu(\))16 b Fp(n)e(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\).)26 b(A)h(1-)523 4825 y(CTRS)22 b(has)g(no)f(extra)g(v)-5 b(ariables,)21 b(a)h(2-CTRS)f(has)h(no)f(extra)g(v)-5 b(ariables)21 b(on)h(the)g(righ)n(t-hand)523 4924 y(sides)27 b(of)g(the)h(rules,)f(and)g(a)g(3-CTRS)g(ma)n(y)g(con)n(tain)g(extra)f (v)-5 b(ariables)26 b(on)h(the)h(righ)n(t-hand)p eop %%Page: 3 3 3 2 bop 523 448 a Fu(sides)23 b(of)g(the)h(rules)e(pro)n(vided)g(that)i (these)f(also)f(o)r(ccur)g(in)i(the)f(corresp)r(onding)f(conditional) 523 548 y(part)27 b(\(i.e.,)h Fp(V)7 b Fn(ar)r Fu(\()p Fn(r)r Fu(\))25 b Fp(\022)e(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))18 b Fp([)h(V)7 b Fn(ar)r Fu(\()p Fn(c)p Fu(\)\).)523 805 y Fq(3)112 b(Quasi-Reductiv)m(e)37 b(Deterministic)d(3-CTRSs)523 995 y Fu(First)27 b(of)g(all,)f(w)n(e)h(will)g(review)f(the)i (de\014nition)f(of)g(deterministic)g(systems)f(from)h([Gan91)o(].)523 1159 y Fh(De\014nition)k(1.)41 b Fm(A)n(n)36 b(oriente)l(d)h(3-CTRS)g Fp(R)g Fm(is)h(c)l(al)t(le)l(d)47 b Fu(deterministic)37 b Fm(if)g(\(after)h(appr)l(o-)523 1258 y(priately)45 b(changing)g(the)e(or)l(der)h(of)h(the)e(c)l(onditions)i(in)e(the)h(r)l (ewrite)g(rules\))f(for)h(every)523 1358 y Fn(l)33 b Fp(!)e Fn(r)j Fp(\()d Fn(s)919 1370 y Fj(1)987 1358 y Fp(!)h Fn(t)1132 1370 y Fj(1)1169 1358 y Fn(;)14 b(:)g(:)g(:)f(;)h(s) 1392 1370 y Fo(k)1464 1358 y Fp(!)31 b Fn(t)1608 1370 y Fo(k)1683 1358 y Fm(in)k Fp(R)f Fm(and)h(every)g Fu(1)c Fp(\024)g Fn(i)f Fp(\024)h Fn(k)s Fm(,)k(we)f(have)i Fp(V)7 b Fn(ar)r Fu(\()p Fn(s)3249 1370 y Fo(i)3277 1358 y Fu(\))31 b Fp(\022)523 1467 y(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))30 b Fp([)870 1405 y Fg(S)939 1425 y Fo(i)p Fi(\000)p Fj(1)939 1492 y Fo(j)s Fj(=1)1072 1467 y Fp(V)7 b Fn(ar)r Fu(\()p Fn(t)1275 1479 y Fo(j)1311 1467 y Fu(\))p Fm(.)46 b(In)f(the)g(fol)t(lowing,)k(we)c(wil)t(l)i(fr)l(e)l(quently)e (use)g(the)h(notation)523 1592 y Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)777 1604 y Fo(i)806 1592 y Fu(\))23 b(=)g Fp(V)7 b Fn(ar)r Fu(\()p Fn(t)1152 1604 y Fo(i)1180 1592 y Fu(\))19 b Fp(n)e Fu(\()p Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))20 b Fp([)1647 1529 y Fg(S)1717 1550 y Fo(i)p Fi(\000)p Fj(1)1717 1617 y Fo(j)s Fj(=1)1849 1592 y Fp(V)7 b Fn(ar)r Fu(\()p Fn(t)2052 1604 y Fo(j)2088 1592 y Fu(\)\))p Fm(.)648 1755 y Fu(The)29 b(rewrite)f(relation)g Fp(!)1492 1767 y Fi(R)1582 1755 y Fu(asso)r(ciated)g(with)h(an)g(orien)n(ted)f (deterministic)i(3-CTRS)523 1855 y Fp(R)41 b Fu(is)g(de\014ned)f(b)n (y:)h Fn(s)j Fp(!)1347 1867 y Fi(R)1453 1855 y Fn(t)d Fu(if)g(and)f(only)h(if)g(there)f(exists)g(a)g(rewrite)g(rule)g Fn(\032)45 b Fu(:)g Fn(l)h Fp(!)523 1954 y Fn(r)37 b Fp(\()d Fn(s)753 1966 y Fj(1)824 1954 y Fp(!)g Fn(t)971 1966 y Fj(1)1008 1954 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)1232 1966 y Fo(k)1306 1954 y Fp(!)34 b Fn(t)1453 1966 y Fo(k)1528 1954 y Fu(in)h Fp(R)p Fu(,)f(a)g(substitution)h Fn(\033)i Fu(:)d Fp(V)7 b Fn(ar)r Fu(\()p Fn(\032)p Fu(\))35 b Fp(!)f(T)21 b Fu(\()p Fp(F)8 b Fn(;)14 b Fp(V)7 b Fu(\),)34 b(and)g(a)523 2054 y(con)n(text)c Fn(C)6 b Fu([)31 b(])g(suc)n(h)f (that)h Fn(s)d Fu(=)f Fn(C)6 b Fu([)p Fn(l)r(\033)s Fu(])p Fn(;)14 b(t)28 b Fu(=)g Fn(C)6 b Fu([)p Fn(r)r(\033)s Fu(],)32 b(and)e Fn(s)2360 2066 y Fo(i)2388 2054 y Fn(\033)h Fp(!)2549 2024 y Fi(\003)2549 2077 y(R)2638 2054 y Fn(t)2668 2066 y Fo(i)2696 2054 y Fn(\033)j Fu(for)c(all)g(1)d Fp(\024)h Fn(i)g Fp(\024)f Fn(k)s Fu(.)523 2154 y(W)-7 b(e)37 b(stress)e(the)i(fact)f(that)h Fn(\033)i Fu(instan)n(tiates)d (ev)n(ery)f(v)-5 b(ariable)35 b(in)i Fn(\032)f Fu(and)g(not)g(only)g (those)523 2253 y(v)-5 b(ariables)23 b(o)r(ccurring)f(in)i Fn(l)r Fu(;)f(for)g(an)h(extra)f(v)-5 b(ariable)22 b Fn(x)p Fu(,)j Fn(x\033)i Fu(is)d(determined)g(as)f(follo)n(ws.)g(The) 523 2353 y(conditions)h(are)f(ev)-5 b(aluated)24 b(from)g(left-to-righ) n(t.)f(Since)i Fn(s)2333 2365 y Fj(1)2394 2353 y Fu(con)n(tains)f(only) f(v)-5 b(ariables)23 b(from)523 2452 y Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\),)28 b(the)f(v)-5 b(ariables)26 b(in)h Fp(V)7 b Fn(ar)r Fu(\()p Fn(s)1600 2464 y Fj(1)1638 2452 y Fu(\))27 b(ha)n(v)n(e)f(a)h(binding.)g(Then)g Fn(s)2534 2464 y Fj(1)2572 2452 y Fn(\033)j Fu(is)d(rewritten)g(un)n(til)g Fn(t)3317 2464 y Fj(1)3355 2452 y Fn(\033)523 2552 y Fu(matc)n(hes)e(a)h(reduct.)g(The)g(term)g Fn(t)1586 2564 y Fj(1)1623 2552 y Fn(\033)k Fu(ma)n(y)25 b(con)n(tain)h(extra)f (v)-5 b(ariables)25 b(but)h(all)g(of)g(these)g(are)523 2652 y(b)r(ound)e(during)f(the)i(matc)n(h.)e(No)n(w)h Fn(s)1669 2664 y Fj(2)1730 2652 y Fu(con)n(tains)e(only)i(v)-5 b(ariables)22 b(whic)n(h)i(already)e(o)r(ccurred)523 2751 y(to)34 b(its)h(left)g(\(in)g Fn(l)h Fu(and)e Fn(t)1301 2763 y Fj(1)1338 2751 y Fu(\))h(and)f(are)f(th)n(us)i(b)r(ound.)g(The)f (instan)n(tiated)g(term)h Fn(s)3082 2763 y Fj(2)3153 2751 y Fu(is)f(then)523 2851 y(reduced)25 b(un)n(til)i(the)f (\(partially\))f(instan)n(tiated)h(term)f Fn(t)2247 2863 y Fj(2)2310 2851 y Fu(matc)n(hes)g(a)h(reduct)f(and)h(so)f(on.)h(If)523 2951 y(all)h(the)h(conditions)f(are)f(satis\014ed,)h(then)h(all)f(v)-5 b(ariables)26 b(in)h(the)h(conditions)f(are)f(b)r(ound)i(in)523 3050 y(the)j(pro)r(cess)e(of)i(ev)-5 b(aluating)30 b(the)h(conditions.) f(Hence)h(the)f(reduct)h(of)f Fn(l)r(\033)k Fu(is)c(w)n(ell-de\014ned) 523 3150 y(\(but)d(in)g(general)e(not)i(unique\))g(b)r(ecause)f Fn(r)j Fu(con)n(tains)d(only)g(v)-5 b(ariables)25 b(whic)n(h)i(also)e (app)r(ear)523 3249 y(in)j(the)g(conditions)f(or)g(in)h Fn(l)r Fu(.)648 3349 y(The)20 b(next)i(de\014nition)f(is)g(based)f(on)g (the)i(w)n(ell-kno)n(wn)d(fact)i(that)g(if)h Fp(\037)e Fu(is)h(a)f(w)n(ell-founded)523 3449 y(partial)30 b(order)f(whic)n(h)i (is)f(closed)g(under)h(con)n(texts,)f(then)h(the)g(order)e Fp(\037)2805 3461 y Fo(st)2865 3449 y Fu(=)f(\()p Fp(\037)g([)g Fl(B)p Fu(\))3263 3419 y Fj(+)3349 3449 y Fu(is)523 3548 y(also)f(w)n(ell-founded)g(\()p Fl(B)g Fu(denotes)h(the)g(prop)r(er)e (subterm)i(relation\).)523 3712 y Fh(De\014nition)j(2.)41 b Fm(A)25 b(deterministic)i(3-CTRS)f Fu(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))26 b Fm(is)g(c)l(al)t(le)l(d)36 b Fu(quasi-reductiv)n(e)24 b Fm(if)j(ther)l(e)523 3811 y(is)38 b(an)g(extension)g Fp(F)1191 3781 y Fi(0)1252 3811 y Fm(of)h(the)f(signatur)l(e)f Fp(F)46 b Fm(\(so)38 b Fp(F)46 b(\022)37 b(F)2399 3781 y Fi(0)2422 3811 y Fm(\))h(and)h(a)f(r)l(e)l(duction)g(or)l(der)g Fp(\037)523 3911 y Fm(on)e Fp(T)21 b Fu(\()p Fp(F)814 3881 y Fi(0)838 3911 y Fn(;)14 b Fp(V)7 b Fu(\))36 b Fm(which,)i(for)f(every)g(rule)f Fn(l)g Fp(!)e Fn(r)k Fp(\()c Fn(s)2215 3923 y Fj(1)2287 3911 y Fp(!)g Fn(t)2434 3923 y Fj(1)2472 3911 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2695 3923 y Fo(k)2770 3911 y Fp(!)35 b Fn(t)2918 3923 y Fo(k)2993 3911 y Fp(2)g(R)p Fm(,)i(every)523 4010 y(substitution)29 b Fn(\033)12 b Fu(:)28 b Fp(V)i(!)23 b(T)e Fu(\()p Fp(F)1434 3980 y Fi(0)1458 4010 y Fn(;)14 b Fp(V)7 b Fu(\))p Fm(,)30 b(and)g(every)h Fu(0)22 b Fp(\024)h Fn(i)g(<)f(k)33 b Fm(satis\014es:)555 4165 y(1.)42 b(if)31 b Fn(s)784 4177 y Fo(j)819 4165 y Fn(\033)26 b Fp(\027)d Fn(t)1010 4177 y Fo(j)1045 4165 y Fn(\033)33 b Fm(for)d(every)h Fu(1)23 b Fp(\024)f Fn(j)28 b Fp(\024)23 b Fn(i)p Fm(,)30 b(then)f Fn(l)r(\033)d Fp(\037)2212 4177 y Fo(st)2295 4165 y Fn(s)2334 4177 y Fo(i)p Fj(+1)2446 4165 y Fn(\033)s Fm(,)555 4263 y(2.)42 b(if)31 b Fn(s)784 4275 y Fo(j)819 4263 y Fn(\033)26 b Fp(\027)d Fn(t)1010 4275 y Fo(j)1045 4263 y Fn(\033)33 b Fm(for)d(every)h Fu(1)23 b Fp(\024)f Fn(j)28 b Fp(\024)23 b Fn(k)s Fm(,)30 b(then)g Fn(l)r(\033)25 b Fp(\037)e Fn(r)r(\033)s Fm(.)523 4426 y Fu(Quasi-reductiv)n(e)30 b(deterministic)i(3-CTRSs)f(w)n(ere)f (in)n(tro)r(duced)i(b)n(y)f(Ganzinger)g([Gan91)o(,)523 4526 y(Def.)d(4.2])f(without)h(men)n(tioning)f(that)h(the)g(original)e (signature)g(can)h(b)r(e)h(extended.)g(This,)523 4625 y(ho)n(w)n(ev)n(er,)c(is)i(crucial)f(b)r(ecause)h(otherwise)f(Prop)r (ositions)f(4.3)h(and)h(4.4)f(in)h([Gan91)o(])g(w)n(ould)523 4725 y(b)r(e)33 b(incorrect.)f(Finite)h(quasi-reductiv)n(e)e (deterministic)i(3-CTRSs)f(ha)n(v)n(e)f(a)h(terminating)523 4825 y(and)27 b(computable)h(rewrite)f(relation)f([Gan91)o(,ALS94].)648 4924 y(T)-7 b(o)24 b(start)g(with,)h(there)f(is)g(the)h(follo)n(wing)f (su\016cien)n(t)g(condition)g(for)g(quasi-reductivit)n(y)-7 b(.)p eop %%Page: 4 4 4 3 bop 523 448 a Fh(De\014nition)31 b(3.)41 b Fm(Given)34 b(a)h(deterministic)g(3-CTRS)f Fp(R)p Fm(,)h(we)g(de\014ne)f(a)h (deterministic)g(3-)523 548 y(CTRS)d Fp(R)851 560 y Fo(q)919 548 y Fm(with)h Fp(R)26 b(\022)h(R)1360 560 y Fo(q)1428 548 y Fm(as)32 b(fol)t(lows:)i Fp(R)1909 560 y Fo(q)1973 548 y Fu(=)2064 486 y Fg(S)2133 573 y Fo(\032)p Fi(2R)2287 548 y Fn(q)s Fu(\()p Fn(\032)p Fu(\))p Fm(,)f(wher)l(e)f(the)g(tr)l (ansformation)523 648 y Fn(q)h Fm(on)d(a)g(rule)f Fn(\032)23 b Fu(:)h Fn(l)g Fp(!)f Fn(r)j Fp(\()d Fn(s)1428 660 y Fj(1)1488 648 y Fp(!)g Fn(t)1624 660 y Fj(1)1661 648 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)1885 660 y Fo(k)1949 648 y Fp(!)23 b Fn(t)2085 660 y Fo(k)2155 648 y Fm(is)31 b(de\014ne)l(d)f(by)988 811 y Fn(q)s Fu(\()p Fn(\032)p Fu(\))23 b(=)g Fp(f)g Fn(l)h Fp(!)f Fn(s)1505 823 y Fj(1)1311 911 y Fn(l)h Fp(!)f Fn(s)1505 923 y Fj(2)1569 911 y Fp(\()g Fn(s)1714 923 y Fj(1)1775 911 y Fp(!)g Fn(t)1911 923 y Fj(1)1311 1010 y Fn(:)14 b(:)g(:)1311 1110 y(l)24 b Fp(!)f Fn(s)1505 1122 y Fo(k)1569 1110 y Fp(\()g Fn(s)1714 1122 y Fj(1)1775 1110 y Fp(!)g Fn(t)1911 1122 y Fj(1)1948 1110 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2171 1122 y Fo(k)q Fi(\000)p Fj(1)2320 1110 y Fp(!)23 b Fn(t)2456 1122 y Fo(k)q Fi(\000)p Fj(1)1311 1209 y Fn(l)h Fp(!)f Fn(r)66 b Fp(\()23 b Fn(s)1714 1221 y Fj(1)1775 1209 y Fp(!)g Fn(t)1911 1221 y Fj(1)1948 1209 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2171 1221 y Fo(k)q Fi(\000)p Fj(1)2320 1209 y Fp(!)23 b Fn(t)2456 1221 y Fo(k)q Fi(\000)p Fj(1)2582 1209 y Fn(;)14 b(s)2658 1221 y Fo(k)2722 1209 y Fp(!)23 b Fn(t)2858 1221 y Fo(k)2899 1209 y Fp(g)523 1374 y Fh(Prop)s(osition)30 b(4.)41 b Fm(A)30 b(deterministic)i(3-CTRS)f Fp(R)f Fm(is)h(quasi-r)l(e)l (ductive)h(if)f(the)g(determin-)523 1474 y(istic)f(3-CTRS)g Fp(R)1101 1486 y Fo(q)1168 1474 y Fm(is)g(terminating.)523 1635 y(Pr)l(o)l(of.)43 b Fu(Since)29 b Fp(R)1071 1647 y Fo(q)1138 1635 y Fu(is)g(terminating,)f(the)i(relation)e Fp(!)2233 1599 y Fj(+)2233 1659 y Fi(R)2290 1667 y Ff(q)2356 1635 y Fu(is)h(a)g(reduction)g(order)f(and)h(it)g(is)523 1734 y(easy)e(to)g(see)g(that)h Fp(R)g Fu(is)g(quasi-reductiv)n(e)e (w.r.t.)h(this)h(order.)648 1895 y(The)22 b(follo)n(wing)g(example)h (tak)n(en)f(from)g([Mar95)n(])h(sho)n(ws)f(that)h(the)g(con)n(v)n(erse) e(of)i(Prop)r(o-)523 1994 y(sition)28 b(4)f(do)r(es)g(not)h(hold.)523 2141 y Fm(Example)j(5.)42 b Fu(The)28 b(normal)e(1-CTRS)h Fp(R)1478 2304 y Fn(a)d Fp(!)f Fn(c)791 b(a)23 b Fp(!)g Fn(d)1487 2404 y(b)g Fp(!)g Fn(c)799 b(b)23 b Fp(!)g Fn(d)1486 2504 y(c)h Fp(!)f Fn(e)796 b(c)23 b Fp(!)g Fn(l)1477 2603 y(k)j Fp(!)d Fn(l)800 b(k)26 b Fp(!)d Fn(m)1479 2703 y(d)h Fp(!)f Fn(m)1460 2802 y(A)h Fp(!)f Fn(h)p Fu(\()p Fn(f)9 b Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))1278 2902 y Fn(h)p Fu(\()p Fn(x;)14 b(x)p Fu(\))25 b Fp(!)e Fn(g)s Fu(\()p Fn(x;)14 b(x;)g(f)9 b Fu(\()p Fn(k)s Fu(\)\))1203 3002 y Fn(g)s Fu(\()p Fn(d;)14 b(x;)g(x)p Fu(\))25 b Fp(!)e Fn(A)1361 3101 y(f)9 b Fu(\()p Fn(x)p Fu(\))24 b Fp(!)f Fn(x)433 b Fp(\()23 b Fn(x)h Fp(!)f Fn(e)523 3271 y Fu(can)32 b(b)r(e)g(sho)n(wn)f(quasi-reductiv)n(e.)g(The)h(system)g Fp(R)2181 3283 y Fo(q)2248 3271 y Fu(=)e Fp(R)22 b([)g(f)p Fn(f)9 b Fu(\()p Fn(x)p Fu(\))30 b Fp(!)h Fn(x)p Fp(g)p Fu(,)h(ho)n(w)n(ev)n(er,)e(is)523 3371 y(not)e(terminating)f(b)r (ecause)g(there)g(is)h(the)g(follo)n(wing)e(cyclic)i(deriv)-5 b(ation)809 3536 y Fn(A)24 b Fp(!)978 3548 y Fi(R)1035 3556 y Ff(q)1095 3536 y Fn(h)p Fu(\()p Fn(f)9 b Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))23 b Fp(!)1658 3500 y Fj(+)1658 3560 y Fi(R)1715 3568 y Ff(q)1776 3536 y Fn(h)p Fu(\()p Fn(f)9 b Fu(\()p Fn(d)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(d)p Fu(\)\))24 b Fp(!)2346 3548 y Fi(R)2403 3556 y Ff(q)2463 3536 y Fn(g)s Fu(\()p Fn(f)9 b Fu(\()p Fn(d)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(d)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(k)s Fu(\)\))895 3648 y Fp(!)978 3660 y Fi(R)1035 3668 y Ff(q)1095 3648 y Fn(g)s Fu(\()p Fn(d;)14 b(f)9 b Fu(\()p Fn(d)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(k)s Fu(\)\))24 b Fp(!)1743 3612 y Fj(+)1743 3672 y Fi(R)1800 3680 y Ff(q)1860 3648 y Fn(g)s Fu(\()p Fn(d;)14 b(f)9 b Fu(\()p Fn(m)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(m)p Fu(\)\))23 b Fp(!)2564 3660 y Fi(R)2621 3668 y Ff(q)2682 3648 y Fn(A:)648 3828 y Fu(Ganzinger)28 b([Gan91)o(,)i(Prop.)e(4.3])h(pro)n(vided)g(the)h (follo)n(wing)e(su\016cien)n(t)i(condition)g(for)523 3928 y(quasi-reductivit)n(y:)23 b(Let)i Fp(F)1394 3898 y Fi(0)1442 3928 y Fu(b)r(e)g(an)f(enric)n(hmen)n(t)g(of)h(the)g (original)e(signature)g Fp(F)33 b Fu(suc)n(h)24 b(that)523 4028 y(the)g(order)e Fp(\037)h Fu(can)h(b)r(e)g(extended)g(to)f(a)g (reduction)g(order)g(o)n(v)n(er)e Fp(T)h Fu(\()p Fp(F)2651 3998 y Fi(0)2674 4028 y Fn(;)14 b Fp(V)7 b Fu(\).)24 b(A)g(deterministic)523 4127 y(rule)30 b Fn(l)g Fp(!)e Fn(r)j Fp(\()d Fn(s)1076 4139 y Fj(1)1141 4127 y Fp(!)h Fn(t)1283 4139 y Fj(1)1320 4127 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)1543 4139 y Fo(k)1612 4127 y Fp(!)28 b Fn(t)1753 4139 y Fo(k)1825 4127 y Fu(is)i(quasi-reductiv)n(e)f(if)j(there)e(exists)g(a)h(sequence) 523 4227 y Fn(h)571 4239 y Fo(i)599 4227 y Fu(\()p Fn(x)p Fu(\))20 b(of)g(terms)f(in)h Fp(T)i Fu(\()p Fp(F)1296 4197 y Fi(0)1319 4227 y Fn(;)14 b Fp(V)7 b Fu(\),)20 b(where)f Fn(x)24 b Fp(2)f(V)7 b Fu(,)20 b(suc)n(h)f(that)h Fn(l)k Fp(\037)f Fn(h)2507 4239 y Fj(1)2544 4227 y Fu(\()p Fn(s)2615 4239 y Fj(1)2653 4227 y Fu(\))p Fn(;)14 b(h)2770 4239 y Fo(i)2797 4227 y Fu(\()p Fn(t)2859 4239 y Fo(i)2887 4227 y Fu(\))24 b Fp(\027)e Fn(h)3078 4239 y Fo(i)p Fj(+1)3190 4227 y Fu(\()p Fn(s)3261 4239 y Fo(i)p Fj(+1)3373 4227 y Fu(\))523 4327 y(for)27 b(ev)n(ery)f(1)d Fp(\024)g Fn(i)f(<)h(k)s(;)k Fu(and)h Fn(h)1467 4339 y Fo(k)1508 4327 y Fu(\()p Fn(t)1570 4339 y Fo(k)1611 4327 y Fu(\))23 b Fp(\027)g Fn(r)r Fu(.)648 4426 y(This)k(criterion,)f(ho)n(w)n(ev)n (er,)g(do)r(es)h(not)g(tell)h(us)f(ho)n(w)g(the)h(terms)f Fn(h)2712 4438 y Fo(i)2740 4426 y Fu(\()p Fn(x)p Fu(\))h(should)f(b)r (e)h(c)n(ho-)523 4526 y(sen.)34 b(A)h(systematic)e(w)n(a)n(y)g(of)h (sho)n(wing)f(quasi-reductivit)n(y)g(consists)h(of)g(transforming)e(a) 523 4625 y(deterministic)22 b(3-CTRS)g Fp(R)h Fu(in)n(to)f(an)f (unconditional)h(TRS)h Fn(U)9 b Fu(\()p Fp(R)p Fu(\))23 b(and)f(sho)n(wing)f(termina-)523 4725 y(tion)28 b(of)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\).)29 b(F)-7 b(or)28 b(normal)f(1-CTRSs,)g (a)g(similar)h(transformation)e(w)n(as)h(already)g(giv)n(en)523 4825 y(in)h([BK86)n(,)g(Def.)g(2.5.1].)e(Marc)n(hiori)g([Mar96)n (,Mar95)n(])i(studied)f(suc)n(h)h(transformations)d(of)523 4924 y(1-CTRSs)i(\(whic)n(h)h(he)f(called)h Fm(unr)l(avelings)p Fu(\))g(in)g(detail.)p eop %%Page: 5 5 5 4 bop 523 448 a Fh(De\014nition)31 b(6.)41 b Fm(L)l(et)22 b Fp(R)i Fm(b)l(e)f(a)g(deterministic)h(3-CTRS)g(over)f(the)h(signatur) l(e)e Fp(F)8 b Fm(.)24 b(F)-6 b(or)23 b(every)523 548 y(r)l(ewrite)40 b(rule)f Fn(\032)h Fu(:)g Fn(l)i Fp(!)e Fn(r)j Fp(\()e Fn(c)f Fp(2)g(R)p Fm(,)g(let)g Fp(j)p Fn(\032)p Fp(j)f Fm(denote)h(the)f(numb)l(er)f(of)j(c)l(onditions)f(in) 523 648 y Fn(\032)p Fm(.)i(In)f(the)h(tr)l(ansformation,)h(we)f(ne)l(e) l(d)f Fp(j)p Fn(\032)p Fp(j)h Fm(fr)l(esh)g(function)g(symb)l(ols)g Fn(U)2903 608 y Fo(\032)2894 670 y Fj(1)2941 648 y Fn(;)14 b(:)g(:)g(:)g(;)g(U)3192 608 y Fo(\032)3183 676 y Fi(j)p Fo(\032)p Fi(j)3302 648 y Fm(for)523 761 y(every)29 b(c)l(onditional)h (rule)f Fn(\032)23 b Fp(2)g(R)p Fm(.)29 b(Mor)l(e)l(over,)i(by)d(abuse) h(of)g(notation,)h Fp(V)7 b Fn(ar)30 b Fm(\(r)l(esp.)f Fp(E)7 b(V)g Fn(ar)r Fm(\))523 861 y(denotes)39 b(a)f(function)g(which) i(assigns)f(the)f(se)l(quenc)l(e)g(of)h(the)f(variables)j(\(in)d(some)g (\014xe)l(d)523 960 y(or)l(der\))c(in)f(the)h(set)f Fp(V)7 b Fn(ar)r Fu(\()p Fn(t)p Fu(\))34 b Fm(\(r)l(esp.)g Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)p Fu(\))p Fm(;)35 b(cf.)f(Def.)g(1\))g(to)f (a)h(term)f Fn(t)p Fm(.)g(We)h(tr)l(ansform)523 1060 y Fn(\032)h Fu(:)h Fn(l)h Fp(!)e Fn(r)j Fp(\()e Fn(s)1073 1072 y Fj(1)1145 1060 y Fp(!)g Fn(t)1294 1072 y Fj(1)1331 1060 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)1554 1075 y Fi(j)p Fo(\032)p Fi(j)1668 1060 y Fp(!)35 b Fn(t)1816 1075 y Fi(j)p Fo(\032)p Fi(j)1931 1060 y Fm(into)h(a)h(set)f Fn(U)9 b Fu(\()p Fn(\032)p Fu(\))37 b Fm(of)h Fp(j)p Fn(\032)p Fp(j)23 b Fu(+)g(1)36 b Fm(unc)l(onditional)523 1159 y(r)l(ewrite)30 b(rules)g(as)g(fol)t(lows:)1719 1335 y Fn(l)24 b Fp(!)g Fn(U)1941 1295 y Fo(\032)1932 1357 y Fj(1)1979 1335 y Fu(\()p Fn(s)2050 1347 y Fj(1)2087 1335 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\)\))1240 1434 y Fn(U)1306 1394 y Fo(\032)1297 1457 y Fj(1)1344 1434 y Fu(\()p Fn(t)1406 1446 y Fj(1)1444 1434 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\)\))23 b Fp(!)h Fn(U)1941 1394 y Fo(\032)1932 1457 y Fj(2)1979 1434 y Fu(\()p Fn(s)2050 1446 y Fj(2)2087 1434 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))p Fn(;)14 b Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)2647 1446 y Fj(1)2686 1434 y Fu(\)\))879 1534 y Fn(U)945 1494 y Fo(\032)936 1556 y Fj(2)983 1534 y Fu(\()p Fn(t)1045 1546 y Fj(2)1082 1534 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))p Fn(;)14 b Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)1642 1546 y Fj(1)1681 1534 y Fu(\)\))23 b Fp(!)h Fn(U)1941 1494 y Fo(\032)1932 1556 y Fj(3)1979 1534 y Fu(\()p Fn(s)2050 1546 y Fj(3)2087 1534 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))p Fn(;)14 b Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)2647 1546 y Fj(1)2686 1534 y Fu(\))p Fn(;)14 b Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)3009 1546 y Fj(2)3047 1534 y Fu(\)\))1768 1634 y Fn(:)14 b(:)g(:)817 1733 y(U)883 1693 y Fo(\032)874 1762 y Fi(j)p Fo(\032)p Fi(j)951 1733 y Fu(\()p Fn(t)1013 1748 y Fi(j)p Fo(\032)p Fi(j)1092 1733 y Fn(;)g Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))p Fn(;)14 b Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)1652 1745 y Fj(1)1690 1733 y Fu(\))p Fn(;)23 b(:)14 b(:)g(:)g(;)g Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)2170 1748 y Fi(j)p Fo(\032)p Fi(j\000)p Fj(1)2334 1733 y Fu(\)\))24 b Fp(!)f Fn(r)523 1928 y Fm(Sinc)l(e)32 b Fp(R)h Fm(is)g(deterministic,)g(the)g(system)f Fn(U)9 b Fu(\()p Fp(R)p Fu(\))28 b(=)2206 1865 y Fg(S)2275 1953 y Fo(\032)p Fi(2R)2415 1928 y Fp(f)p Fn(U)9 b Fu(\()p Fn(\032)p Fu(\))p Fp(g)32 b Fm(is)h(an)f(unc)l(onditional)523 2037 y(TRS)f(over)h(the)f(extende)l(d)g(signatur)l(e)g Fp(F)1811 2007 y Fi(0)1860 2037 y Fu(=)25 b Fp(F)i([)2112 1975 y Fg(S)2181 2062 y Fo(\032)p Fi(2R)p Fo(;)p Fj(1)p Fi(\024)p Fo(i)p Fi(\024j)p Fo(\032)p Fi(j)2589 2037 y Fn(U)2655 1997 y Fo(\032)2646 2060 y(i)2725 2037 y Fm(\(that)k(is,)h Fp(V)7 b Fn(ar)r Fu(\()p Fn(r)3258 2007 y Fi(0)3282 2037 y Fu(\))26 b Fp(\022)523 2147 y(V)7 b Fn(ar)r Fu(\()p Fn(l)723 2117 y Fi(0)746 2147 y Fu(\))30 b Fm(holds)i(for)e(every)h(r)l(ewrite)f(rule)f Fn(l)1848 2117 y Fi(0)1894 2147 y Fp(!)23 b Fn(r)2039 2117 y Fi(0)2086 2147 y Fp(2)h Fn(U)9 b Fu(\()p Fp(R)p Fu(\))p Fm(\).)523 2304 y Fu(F)-7 b(or)27 b(example,)g(the)h(transformation)e(of)i(the)g (system)f Fp(R)2312 2316 y Fo(f)7 b(ib)2436 2304 y Fu(yields)27 b(the)h(TRS)1660 2486 y Fn(f)9 b(ib)p Fu(\(0\))22 b Fp(!)i(h)p Fu(0)p Fn(;)14 b(s)p Fu(\(0\))p Fp(i)1551 2586 y Fn(f)9 b(ib)p Fu(\()p Fn(s)p Fu(\()p Fn(x)p Fu(\)\))23 b Fp(!)h Fn(U)2067 2598 y Fj(1)2103 2586 y Fu(\()p Fn(f)9 b(ib)p Fu(\()p Fn(x)p Fu(\))p Fn(;)14 b(x)p Fu(\))1450 2686 y Fn(U)1507 2698 y Fj(1)1544 2686 y Fu(\()p Fp(h)p Fn(y)s(;)g(z)t Fp(i)p Fn(;)g(x)p Fu(\))23 b Fp(!)h(h)p Fn(z)t(;)14 b(y)20 b Fu(+)e Fn(z)t Fp(i)523 2867 y Fu(It)j(turns)f(out)g(that)g (termination)g(of)h Fn(U)9 b Fu(\()p Fp(R)p Fu(\))21 b(is)f(a)g(su\016cien)n(t)g(but)h(not)f(a)g(necessary)e(condition)523 2967 y(for)27 b(quasi-reductivit)n(y)f(of)i Fp(R)p Fu(.)523 3123 y Fh(Prop)s(osition)i(7.)41 b Fm(If)30 b Fn(U)9 b Fu(\()p Fp(R)p Fu(\))31 b Fm(is)f(terminating,)g(then)g Fp(R)g Fm(is)g(quasi-r)l(e)l(ductive.)523 3279 y Fu(The)35 b(pro)r(of)f(of)h(the)g(prop)r(osition)f(can)g(b)r(e)h(found)g(in)g ([Ohl99)o(])g(and)g(a)f(similar)g(result)h(for)523 3379 y(normal)23 b(1-CTRSs)g(app)r(eared)f(in)i([Mar96)n(].)g(In)g(our)f (Fib)r(onacci)g(example,)h(termination)f(of)523 3479 y(the)k(transformed)f(system)h Fn(U)9 b Fu(\()p Fp(R)1572 3491 y Fo(f)e(ib)1668 3479 y Fu(\))27 b(can)g(b)r(e)g(sho)n(wn)f(b)n(y) h(rp)r(o.)f(Th)n(us)h(the)g(system)g Fp(R)3226 3491 y Fo(f)7 b(ib)3349 3479 y Fu(is)523 3578 y(quasi-reductiv)n(e)26 b(b)n(y)h(Prop)r(osition)f(7.)648 3678 y(Example)h(5)h(can)g(b)r(e)h (used)f(to)g(sho)n(w)g(that)g(the)h(con)n(v)n(erse)d(of)i(Prop)r (osition)f(7)h(do)r(es)g(not)523 3777 y(hold.)39 b(T)-7 b(o)38 b(b)r(e)h(precise,)f(it)h(can)f(b)r(e)h(sho)n(wn)f(that)h Fn(h)p Fu(\()p Fn(f)9 b Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))42 b Fp(!)2723 3742 y Fj(+)2723 3806 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))2929 3777 y Fn(A)39 b Fu(and)f(hence)523 3891 y Fn(U)9 b Fu(\()p Fp(R)p Fu(\))32 b(is)f(not)h(terminating;)f(see)g([Mar95)n(].)h(In)f (Example)g(5,)g(ho)n(w)n(ev)n(er,)e(the)j(CTRS)f Fp(R)h Fu(is)523 3990 y(neither)d(left-linear)f(nor)g(con\015uen)n(t)h(and)f (one)g(migh)n(t)h(w)n(onder)f(whether)g(this)h(is)g(essen)n(tial)523 4090 y(\(for)22 b(1-CTRSs)f(non-left-linearit)n(y)f(is)i(crucial;)f (see)h([Mar96)n(,)g(Thm.)g(6.12]\).)f(The)h(follo)n(wing)523 4190 y(example)i(sho)n(ws)g(that)g(left-linearit)n(y)g(and)h (con\015uence)f(of)h(a)f(quasi-reductiv)n(e)f(3-CTRS)h Fp(R)523 4289 y Fu(are)j(not)g(su\016cien)n(t)h(to)f(ensure)g (termination)h(of)f Fn(U)9 b Fu(\()p Fp(R)p Fu(\).)523 4462 y Fm(Example)31 b(8.)42 b Fu(Let)31 b Fp(R)g Fu(con)n(tain)f(the)g (rule)h Fn(g)s Fu(\()p Fn(x)p Fu(\))d Fp(!)g Fn(k)s Fu(\()p Fn(y)s Fu(\))g Fp(\()g Fn(h)p Fu(\()p Fn(x)p Fu(\))h Fp(!)f Fn(d;)14 b(h)p Fu(\()p Fn(x)p Fu(\))28 b Fp(!)g Fn(c)p Fu(\()p Fn(y)s Fu(\))j(and)523 4561 y(the)d(follo)n(wing)f (unconditional)g(rules)1405 4737 y Fn(h)p Fu(\()p Fn(d)p Fu(\))d Fp(!)f Fn(c)p Fu(\()p Fn(a)p Fu(\))648 b Fn(h)p Fu(\()p Fn(d)p Fu(\))24 b Fp(!)f Fn(c)p Fu(\()p Fn(b)p Fu(\))1517 4836 y Fn(a)g Fp(!)g Fn(e)873 b(b)23 b Fp(!)g Fn(e)1025 4936 y(f)9 b Fu(\()p Fn(k)s Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(k)s Fu(\()p Fn(b)p Fu(\))p Fn(;)g(x)p Fu(\))24 b Fp(!)f Fn(f)9 b Fu(\()p Fn(x;)14 b(x;)g(x)p Fu(\))297 b Fn(f)9 b Fu(\()p Fn(x;)14 b(y)s(;)g(z)t Fu(\))23 b Fp(!)g Fn(e)p eop %%Page: 6 6 6 5 bop 523 448 a Fu(W)-7 b(e)33 b(ha)n(v)n(e)e Fp(R)937 460 y Fo(q)1005 448 y Fu(=)g Fp(R)22 b([)g(f)p Fn(g)s Fu(\()p Fn(x)p Fu(\))32 b Fp(!)f Fn(h)p Fu(\()p Fn(x)p Fu(\))p Fn(;)14 b(g)s Fu(\()p Fn(x)p Fu(\))33 b Fp(!)e Fn(h)p Fu(\()p Fn(x)p Fu(\))h Fp(\()f Fn(h)p Fu(\()p Fn(x)p Fu(\))h Fp(!)f Fn(d)p Fp(g)p Fu(.)i(Let)f Fp(R)3083 418 y Fi(0)3140 448 y Fu(con)n(tain)523 548 y(the)j(rule)f Fn(g)s Fu(\()p Fn(x)p Fu(\))h Fp(!)g Fn(h)p Fu(\()p Fn(x)p Fu(\))g(and)g(the)f(unconditional)h(rules)e(of)i Fp(R)p Fu(.)g(The)f(rewrite)g(relations)523 648 y Fp(!)606 660 y Fi(R)663 643 y Fe(0)715 648 y Fu(and)25 b Fp(!)957 660 y Fi(R)1014 668 y Ff(q)1077 648 y Fu(coincide)g(b)r(ecause)g(the)h (rules)f Fn(g)s Fu(\()p Fn(x)p Fu(\))e Fp(!)g Fn(k)s Fu(\()p Fn(y)s Fu(\))h Fp(\()f Fn(h)p Fu(\()p Fn(x)p Fu(\))h Fp(!)f Fn(d;)14 b(h)p Fu(\()p Fn(x)p Fu(\))24 b Fp(!)f Fn(c)p Fu(\()p Fn(y)s Fu(\))523 747 y(and)k Fn(g)s Fu(\()p Fn(x)p Fu(\))d Fp(!)f Fn(h)p Fu(\()p Fn(x)p Fu(\))h Fp(\()f Fn(h)p Fu(\()p Fn(x)p Fu(\))h Fp(!)f Fn(d)k Fu(are)g(nev)n(er)f(applicable)h(\(there)g(is)g(no)g(term)g Fn(t)h Fu(suc)n(h)f(that)523 847 y Fn(h)p Fu(\()p Fn(t)p Fu(\))c Fp(!)771 817 y Fi(\003)771 870 y(R)856 847 y Fn(d)p Fu(\).)j(It)h(can)e(b)r(e)i(sho)n(wn)e(that)h Fp(R)1829 817 y Fi(0)1879 847 y Fu(is)g(terminating.)f(Hence)h Fp(R)h Fu(is)f(quasi-reductiv)n(e)523 946 y(b)n(y)g(Prop)r(osition)f (4.)h Fp(R)h Fu(is)g(also)e(con\015uen)n(t)i(b)r(ecause)f(ev)n(ery)f (critical)h(pair)g(is)h(joinable.)f(The)523 1046 y(transformed)h (system)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\))28 b(consists)f(of)1737 1217 y Fn(g)s Fu(\()p Fn(x)p Fu(\))d Fp(!)f Fn(U)2078 1229 y Fj(1)2115 1217 y Fu(\()p Fn(h)p Fu(\()p Fn(x)p Fu(\))p Fn(;)14 b(x)p Fu(\))1605 1316 y Fn(U)1662 1328 y Fj(1)1699 1316 y Fu(\()p Fn(d;)g(x)p Fu(\))25 b Fp(!)e Fn(U)2078 1328 y Fj(2)2115 1316 y Fu(\()p Fn(h)p Fu(\()p Fn(x)p Fu(\))p Fn(;)14 b(x)p Fu(\))1504 1416 y Fn(U)1561 1428 y Fj(2)1598 1416 y Fu(\()p Fn(c)p Fu(\()p Fn(y)s Fu(\))p Fn(;)g(x)p Fu(\))25 b Fp(!)e Fn(k)s Fu(\()p Fn(y)s Fu(\))523 1588 y(and)33 b(the)g(unconditional)f(rules)g(of)h Fp(R)p Fu(.)h Fn(U)9 b Fu(\()p Fp(R)p Fu(\))33 b(is)g(not)g (terminating)f(b)r(ecause)g(of)h(the)g(fol-)523 1688 y(lo)n(wing)27 b(cyclic)g(deriv)-5 b(ation)690 1854 y Fn(f)9 b Fu(\()p Fn(k)s Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(k)s Fu(\()p Fn(b)p Fu(\))p Fn(;)g(U)1203 1866 y Fj(2)1240 1854 y Fu(\()p Fn(h)p Fu(\()p Fn(d)p Fu(\))p Fn(;)g(d)p Fu(\)\))25 b Fp(!)1679 1869 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))1867 1854 y Fn(f)j Fu(\()p Fn(U)2006 1866 y Fj(2)2043 1854 y Fu(\()p Fn(h)p Fu(\()p Fn(d)p Fu(\))p Fn(;)14 b(d)p Fu(\))p Fn(;)g(U)2436 1866 y Fj(2)2474 1854 y Fu(\()p Fn(h)p Fu(\()p Fn(d)p Fu(\))p Fn(;)g(d)p Fu(\))p Fn(;)g(U)2867 1866 y Fj(2)2905 1854 y Fu(\()p Fn(h)p Fu(\()p Fn(d)p Fu(\))p Fn(;)g(d)p Fu(\)\))1596 1955 y Fp(!)1679 1920 y Fj(+)1679 1984 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))1867 1955 y Fn(f)j Fu(\()p Fn(U)2006 1967 y Fj(2)2043 1955 y Fu(\()p Fn(c)p Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(d)p Fu(\))p Fn(;)g(U)2425 1967 y Fj(2)2463 1955 y Fu(\()p Fn(c)p Fu(\()p Fn(b)p Fu(\))p Fn(;)g(d)p Fu(\))p Fn(;)g(U)2837 1967 y Fj(2)2874 1955 y Fu(\()p Fn(h)p Fu(\()p Fn(d)p Fu(\))p Fn(;)g(d)p Fu(\)\))1596 2070 y Fp(!)1679 2034 y Fj(+)1679 2098 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))1867 2070 y Fn(f)j Fu(\()p Fn(k)s Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(k)s Fu(\()p Fn(b)p Fu(\))p Fn(;)g(U)2380 2082 y Fj(2)2417 2070 y Fu(\()p Fn(h)p Fu(\()p Fn(d)p Fu(\))p Fn(;)g(d)p Fu(\)\))p Fn(:)523 2250 y Fu(Observ)n(e)30 b(that)h(the)h(preceding)e (deriv)-5 b(ation)31 b(is)g(not)h(innermost)e(and)i(it)f(has)g(b)r(een) h(sho)n(wn)523 2350 y(in)c([Ohl99)o(])f(that)h(this)g(is)g(essen)n (tial.)523 2499 y Fh(Theorem)i(9.)41 b Fm(If)29 b Fp(R)g Fm(is)g(a)g(quasi-r)l(e)l(ductive)g(deterministic)h(3-CTRS,)f(then)f Fn(U)9 b Fu(\()p Fp(R)p Fu(\))29 b Fm(is)g(in-)523 2598 y(nermost)g(terminating.)523 2747 y Fu(Another)22 b(su\016cien)n(t)h (criterion)e(for)h(quasi-reductivit)n(y)f(is)h(giv)n(en)g(in)h([ALS94)o (,)g(Lemma)f(3.1].)523 2847 y(In)28 b(order)e(to)i(form)n(ulate)e(it)i (\(claim)g(b)r(elo)n(w\),)g(w)n(e)f(need)h(the)g(follo)n(wing)e (de\014nition.)523 2995 y Fh(De\014nition)31 b(10.)40 b Fm(L)l(et)33 b Fn(\032)d Fu(:)g Fn(l)h Fp(!)f Fn(r)i Fp(\()e Fn(s)1795 3007 y Fj(1)1862 2995 y Fp(!)g Fn(t)2005 3007 y Fj(1)2042 2995 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)2266 3007 y Fo(k)2336 2995 y Fp(!)30 b Fn(t)2479 3007 y Fo(k)2553 2995 y Fm(b)l(e)k(a)g(c)l(onditional)h(rule)e(of)523 3095 y(the)e(deterministic)i(3-CTRS)e Fp(R)p Fm(.)h(The)g(tr)l (ansforme)l(d)g(rule)p 2423 3049 43 4 v 31 w Fn(\032)f Fm(is)h(de\014ne)l(d)f(as)h(fol)t(lows.)i(F)-6 b(or)523 3195 y Fn(x)32 b Fp(2)g(E)7 b(V)g Fn(ar)r Fu(\()p Fn(\032)p Fu(\))36 b Fm(let)f Fn(\013)p Fu(\()p Fn(x)p Fu(\))g Fm(b)l(e)g(the)g(smal)t(lest)g Fn(i)p Fm(,)g Fu(1)c Fp(\024)g Fn(i)h Fp(\024)f Fn(k)s Fm(,)k(such)f(that)h Fn(x)d Fp(2)g(V)7 b Fn(ar)r Fu(\()p Fn(t)3178 3207 y Fo(i)3207 3195 y Fu(\))34 b Fm(and)523 3294 y(de\014ne)863 3460 y Fn(')917 3472 y Fj(1)978 3460 y Fu(=)23 b Fn(id)788 3560 y(')842 3572 y Fo(i)p Fj(+1)978 3560 y Fu(=)g Fp(f)p Fn(x)g Fp( )p 1284 3514 39 4 v 23 w Fn(s)1323 3575 y Fo(\013)p Fj(\()p Fo(x)p Fj(\))1489 3560 y Fp(j)30 b Fn(x)23 b Fp(2)h(V)7 b Fn(ar)r Fu(\()p Fn(t)1894 3572 y Fj(1)1932 3560 y Fn(;)14 b(:)g(:)g(:)f(;)h(t)2146 3572 y Fo(i)2174 3560 y Fu(\))k Fp(\\)h(E)7 b(V)g Fn(ar)r Fu(\()p Fn(\032)p Fu(\))p Fp(g)31 b Fm(for)f Fu(1)23 b Fp(\024)f Fn(i)h Fp(\024)g Fn(k)p 888 3614 V 888 3660 a(s)927 3672 y Fo(i)978 3660 y Fu(=)g Fn(')1120 3672 y Fo(i)1147 3660 y Fu(\()p Fn(s)1218 3672 y Fo(i)1246 3660 y Fu(\))523 3827 y Fm(Then)31 b(the)g(b)l(ackwar)l(d)i (substitute)l(d)c(rule)p 1814 3782 43 4 v 31 w Fn(\032)h Fm(is)i Fn(l)26 b Fp(!)p 2137 3782 40 4 v 25 w Fn(r)h Fp(\()p 2309 3782 39 4 v 25 w Fn(s)2348 3839 y Fj(1)2410 3827 y Fp(!)e Fn(c;)14 b(:)g(:)g(:)g(;)p 2739 3782 V 14 w(s)2778 3839 y Fo(k)2843 3827 y Fp(!)25 b Fn(c)p Fm(,)31 b(wher)l(e)h Fn(c)f Fm(is)523 3927 y(a)f(new)g(c)l(onstant)f (and)p 1257 3881 40 4 v 30 w Fn(r)d Fu(=)d Fn(')1462 3939 y Fo(k)q Fj(+1)1587 3927 y Fu(\()p Fn(r)r Fu(\))p Fm(.)523 4076 y Fh(Claim:)f Fu(Let)j Fp(\037)g Fu(b)r(e)g(a)f (reduction)g(order)g(and)g(let)h Fp(R)h Fu(b)r(e)f(a)f(deterministic)h (3-CTRS.)f(If,)i(for)523 4175 y(ev)n(ery)i(rule)h Fn(\032)g Fu(in)g Fp(R)p Fu(,)h(the)f(bac)n(kw)n(ard)e(substituted)j(rule)p 2326 4130 43 4 v 29 w Fn(\032)f Fu(satis\014es)f Fn(l)f Fp(\037)2822 4187 y Fo(st)p 2908 4130 39 4 v 2908 4175 a Fn(s)2947 4187 y Fo(i)3002 4175 y Fu(for)g(1)e Fp(\024)g Fn(i)g Fp(\024)523 4275 y Fn(k)31 b Fu(and)c Fn(l)d Fp(\037)p 895 4229 40 4 v 23 w Fn(r)s Fu(,)k(then)g Fp(R)g Fu(is)f (quasi-reductiv)n(e)f(w.r.t.)i Fp(\037)p Fu(.)523 4422 y(The)g(next)f(example,)h(ho)n(w)n(ev)n(er,)d(refutes)j(the)g(claim.) 523 4571 y Fm(Example)j(11.)43 b Fu(Consider)26 b(the)i(deterministic)g (3-CTRS)1375 4837 y Fp(R)23 b Fu(=)1556 4667 y Fg(8)1556 4742 y(<)1556 4891 y(:)1763 4737 y Fn(b)g Fp(!)g Fn(g)s Fu(\()p Fn(d)p Fu(\))1641 4836 y Fn(f)9 b Fu(\()p Fn(d)p Fu(\))24 b Fp(!)f Fn(f)9 b Fu(\()p Fn(a)p Fu(\))1755 4936 y Fn(a)23 b Fp(!)g Fn(y)141 b Fp(\()23 b Fn(b)f Fp(!)i Fn(g)s Fu(\()p Fn(y)s Fu(\))p eop %%Page: 7 7 7 6 bop 523 448 a Fu(Its)27 b(bac)n(kw)n(ard)e(substituted)j(system)p 1725 382 71 4 v 27 w Fp(R)g Fu(consists)e(of)h(the)h(t)n(w)n(o)e (unconditional)h(rules)f(of)i Fp(R)523 548 y Fu(and)i(the)h (conditional)e(rule)h Fn(a)e Fp(!)f Fn(b)h Fp(\()f Fn(b)g Fp(!)h Fn(c)p Fu(.)i(The)h(unconditional)f(TRS)p 2947 481 V 30 w Fp(R)3018 560 y Fo(u)3091 548 y Fu(obtained)523 648 y(from)p 723 581 V 31 w Fp(R)i Fu(b)n(y)g(dropping)e(the)i (conditions)f(is)h(terminating.)f(Hence)h(the)g(order)e Fp(\037)f Fu(=)g Fp(!)3304 612 y Fj(+)p 3304 638 58 3 v 3304 686 a Fi(R)3362 694 y Ff(u)523 760 y Fu(is)39 b(a)g(reduction)f(order)g(whic)n(h)h(satis\014es)f(the)i(ab)r(o)n(v)n (e)d(claim.)i(The)g(original)f(system)h Fp(R)p Fu(,)523 860 y(ho)n(w)n(ev)n(er,)29 b(is)j(not)f(ev)n(en)g(terminating)g(b)r (ecause)g(there)h(is)f(the)h(in\014nite)g(rewrite)f(sequence)523 960 y Fn(f)9 b Fu(\()p Fn(d)p Fu(\))p Fp(!)763 972 y Fi(R)825 960 y Fn(f)g Fu(\()p Fn(a)p Fu(\))p Fp(!)1066 972 y Fi(R)1127 960 y Fn(f)g Fu(\()p Fn(d)p Fu(\))p Fp(!)1367 972 y Fi(R)1442 960 y Fn(:)14 b(:)g(:)g Fu(.)523 1120 y(The)28 b(criterion)e(will)i(b)r(e)g(recti\014ed)g(in)f(the)h(next)g (section.)523 1390 y Fq(4)112 b(Quasi-Simplifying)35 b(Deterministic)f(3-CTRSs)523 1593 y Fh(De\014nition)d(12.)40 b Fm(We)26 b(c)l(al)t(l)h Fp(R)f Fu(quasi-simplifying)e Fm(if)j(it)e(is)h(quasi-r)l(e)l(ductive)g(w.r.t.)h(a)f(sim-)523 1693 y(pli\014c)l(ation)31 b(or)l(der)g Fp(\037)e Fm(\(note)g(that)h (in)g(this)g(c)l(ase)g Fp(\037)23 b Fu(=)g Fp(\037)2288 1705 y Fo(st)2348 1693 y Fm(\).)523 1870 y Fu(By)k(de\014nition,)h (quasi-simplifyingness)e(implies)i(quasi-reductivit)n(y)-7 b(.)26 b(W)-7 b(e)28 b(shall)f(see)g(later)523 1970 y(that)f(the)g(con) n(v)n(erse)d(is)j(not)f(true.)h(In)f(con)n(trast)g(to)g (quasi-reductivit)n(y)-7 b(,)24 b(quasi-simplifying-)523 2070 y(ness)37 b(is)f(indep)r(enden)n(t)i(of)f(signature)f(extensions)g (in)i(the)f(follo)n(wing)f(sense:)g(if)i(a)f(deter-)523 2169 y(ministic)c(3-CTRS)g(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))33 b(is)g(quasi-simplifying)f(w.r.t.)h(a)g(simpli\014cation)f (order)g Fp(\037)g Fu(on)523 2269 y Fp(T)21 b Fu(\()p Fp(F)689 2239 y Fi(0)713 2269 y Fn(;)14 b Fp(V)7 b Fu(\),)31 b(where)f Fp(F)36 b(\022)28 b(F)1394 2239 y Fi(0)1417 2269 y Fu(,)j(then)h(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))31 b(is)g(quasi-simplifying)f(w.r.t.)h(the)g(restriction)523 2369 y(of)d Fp(\037)f Fu(on)g Fp(T)21 b Fu(\()p Fp(F)8 b Fn(;)14 b Fp(V)7 b Fu(\))q(.)27 b(The)h(simple)g(pro)r(of)f(of)g (this)h(fact)g(is)f(left)i(to)e(the)h(reader.)648 2469 y(Quasi-simplifyingness)23 b(is)i(closely)f(related)g(to)h (simplifyingness.)g(A)g(join)g(1-CTRS)g Fp(R)523 2569 y Fu(is)c Fm(simplifying)31 b Fu(if)22 b(there)f(is)g(a)g (simpli\014cation)g(order)f Fp(\037)h Fu(suc)n(h)g(that)g Fn(l)k Fp(\037)d Fn(r)r Fu(,)g Fn(l)j Fp(\037)e Fn(s)2992 2581 y Fo(j)3048 2569 y Fu(and)e Fn(l)j Fp(\037)f Fn(t)3370 2581 y Fo(j)523 2668 y Fu(for)29 b(ev)n(ery)f Fn(l)g Fp(!)e Fn(r)k Fp(\()c Fn(s)1250 2680 y Fj(1)1313 2668 y Fp(#)g Fn(t)1411 2680 y Fj(1)1448 2668 y Fn(;)14 b(:)g(:)g(:)g(;)g(s) 1672 2680 y Fo(k)1739 2668 y Fp(#)26 b Fn(t)1837 2680 y Fo(k)1904 2668 y Fp(2)h(R)p Fu(;)j(see)f([Kap87)n(].)h(F)-7 b(or)29 b(orien)n(ted)g(1-CTRSs,)523 2768 y(the)e(condition)g Fn(l)e Fp(\037)d Fn(t)1196 2780 y Fo(j)1258 2768 y Fu(can)27 b(b)r(e)h(dropp)r(ed)e(b)r(ecause)h(the)g(term)g Fn(t)2524 2780 y Fo(j)2587 2768 y Fu(is)f(nev)n(er)h(reduced.)f(Th)n(us)523 2867 y(an)34 b(orien)n(ted)g(1-CTRS)g Fp(R)h Fu(is)f Fm(simplifying)45 b Fu(if)35 b(there)f(is)h(a)f(simpli\014cation)g (order)f Fp(\037)h Fu(suc)n(h)523 2967 y(that)c Fn(l)e Fp(\037)f Fn(r)32 b Fu(and)e Fn(l)e Fp(\037)f Fn(s)1267 2979 y Fo(j)1332 2967 y Fu(for)i(ev)n(ery)g Fn(l)f Fp(!)f Fn(r)i Fp(\()e Fn(s)2061 2979 y Fj(1)2125 2967 y Fp(!)g Fn(t)2265 2979 y Fj(1)2302 2967 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)2526 2979 y Fo(k)2593 2967 y Fp(!)27 b Fn(t)2733 2979 y Fo(k)2804 2967 y Fu(in)j Fp(R)p Fu(.)g(Ob)n(viously)-7 b(,)523 3067 y(ev)n(ery)26 b(simplifying)i(orien)n(ted)f(1-CTRS)g(is)g (quasi-simplifying.)648 3167 y(Next)j(w)n(e)f(will)i(pro)n(vide)d(sev)n (eral)h(criteria)f(whic)n(h)i(guaran)n(tee)e(quasi-simplifyingness.)523 3267 y(T)-7 b(o)27 b(this)h(end,)g(w)n(e)f(need)h(the)g(follo)n(wing)f (lemmata.)523 3428 y Fh(Lemma)j(13.)40 b Fm(A)n(n)34 b(unc)l(onditional)i(TRS)e Fp(R)h Fm(over)h(the)f(\(\014nite\))f (signatur)l(e)h Fp(F)42 b Fm(is)36 b(simply)523 3528 y(terminating)24 b(if)h(and)g(only)f(if)h Fp(R)6 b([)g(E)h Fn(mb)p Fu(\()p Fp(F)h Fu(\))25 b Fm(is)g(terminating.)f(The)h(TRS)f Fp(E)7 b Fn(mb)p Fu(\()p Fp(F)h Fu(\))24 b Fm(c)l(onsists)523 3627 y(of)k(al)t(l)g(rules)e Fn(f)9 b Fu(\()p Fn(x)1062 3639 y Fj(1)1100 3627 y Fn(;)14 b(:)g(:)g(:)f(;)h(x)1331 3639 y Fo(n)1377 3627 y Fu(\))23 b Fp(!)g Fn(x)1585 3639 y Fo(j)1647 3627 y Fm(wher)l(e)28 b Fn(f)k Fp(2)23 b(F)35 b Fm(has)27 b(arity)h Fn(n)23 b Fp(\025)f Fu(1)p Fm(,)27 b Fn(j)h Fp(2)23 b(f)p Fu(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(n)p Fp(g)p Fm(,)26 b(and)523 3727 y(the)k(variables)i Fn(x)1053 3739 y Fj(1)1090 3727 y Fn(;)14 b(:)g(:)g(:)g(;)g(x)1322 3739 y Fo(n)1397 3727 y Fm(ar)l(e)30 b(p)l(airwise)i(distinct.)523 3887 y Fu(A)h(pro)r(of)f(of)g(the)h(preceding)f(lemma)h(can)f(b)r(e)h (found)g(in)g([Zan94)n(])g(and)f(the)h(pro)r(of)f(of)h(the)523 3987 y(next)28 b(lemma)f(is)h(straigh)n(tforw)n(ard;)d(cf.)j([Ohl94)n (,)g(Lemma)f(8.1.9].)523 4148 y Fh(Lemma)j(14.)40 b Fm(A)n(n)29 b(oriente)l(d)h(1-CTRS)g Fp(R)g Fm(is)h(simplifying)h(if)e(and)h(only)f (if)h(the)f(TRS)732 4332 y Fp(R)802 4344 y Fo(s)860 4332 y Fu(=)23 b Fp(R)1018 4344 y Fo(u)1080 4332 y Fp([)c(f)p Fn(l)24 b Fp(!)f Fn(s)1390 4344 y Fo(j)1455 4332 y Fp(j)30 b Fn(l)24 b Fp(!)f Fn(r)j Fp(\()d Fn(s)1871 4344 y Fj(1)1931 4332 y Fp(!)g Fn(t)2067 4344 y Fj(1)2105 4332 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2328 4344 y Fo(k)2392 4332 y Fp(!)23 b Fn(t)2528 4344 y Fo(k)2592 4332 y Fp(2)g(R)p Fu(;)44 b(1)23 b Fp(\024)f Fn(j)28 b Fp(\024)23 b Fn(k)s Fp(g)523 4516 y Fm(is)30 b(simply)h(terminating.)f(A)n(nalo)l(gously,)h(a)f (join)g(1-CTRS)g Fp(R)g Fm(is)g(simplifying)i(if)e(and)g(only)523 4616 y(if)h(the)f(fol)t(lowing)i(TRS)d(is)h(simply)h(terminating:)866 4795 y Fp(R)936 4807 y Fo(u)1003 4795 y Fp([)f(f)p Fn(l)24 b Fp(!)f Fn(s)1324 4807 y Fo(j)1389 4795 y Fp(j)30 b Fn(l)24 b Fp(!)f Fn(r)j Fp(\()d Fn(s)1805 4807 y Fj(1)1865 4795 y Fp(#)g Fn(t)1960 4807 y Fj(1)1997 4795 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)2221 4807 y Fo(k)2285 4795 y Fp(#)22 b Fn(t)2379 4807 y Fo(k)2443 4795 y Fp(2)h(R)p Fu(;)44 b(1)23 b Fp(\024)g Fn(j)28 b Fp(\024)22 b Fn(k)s Fp(g)1003 4894 y([)30 b(f)p Fn(l)24 b Fp(!)f Fn(t)1315 4906 y Fo(j)1380 4894 y Fp(j)30 b Fn(l)24 b Fp(!)f Fn(r)j Fp(\()d Fn(s)1796 4906 y Fj(1)1857 4894 y Fp(#)f Fn(t)1951 4906 y Fj(1)1988 4894 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)2212 4906 y Fo(k)2276 4894 y Fp(#)22 b Fn(t)2370 4906 y Fo(k)2434 4894 y Fp(2)h(R)p Fu(;)44 b(1)23 b Fp(\024)g Fn(j)28 b Fp(\024)22 b Fn(k)s Fp(g)p Fn(:)p eop %%Page: 8 8 8 7 bop 523 448 a Fu(The)29 b(\014rst)f(su\016cien)n(t)h(criterion)e (uses)i(the)g(CTRS)g Fp(R)2198 460 y Fo(q)2263 448 y Fu(from)g(De\014nition)g(3.)f(It)h(is)g(actually)523 548 y(a)e(c)n(haracterization)e(of)j(quasi-simplifyingness.)523 694 y Fh(Prop)s(osition)i(15.)41 b Fm(A)i(deterministic)j(3-CTRS)e Fp(R)g Fm(over)h(a)f(\(\014nite\))g(signatur)l(e)g Fp(F)52 b Fm(is)523 794 y(quasi-simplifying)36 b(if)d(and)g(only)h(if)f(the)g (deterministic)h(3-CTRS)f Fp(R)2724 806 y Fo(q)2781 794 y Fp([)21 b(E)7 b Fn(mb)p Fu(\()p Fp(F)h Fu(\))33 b Fm(is)g(ter-)523 893 y(minating.)523 1053 y(Pr)l(o)l(of.)43 b Fu(\\if)6 b(":)22 b(It)g(is)f(not)g(di\016cult)i(to)e(pro)n(v)n(e)f(that)i Fp(!)2137 1018 y Fj(+)2137 1082 y Fi(R)2194 1090 y Ff(q)2227 1082 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))2531 1053 y Fu(is)22 b(a)f(simpli\014cation)g(order)3369 1023 y Fj(1)523 1164 y Fu(and)27 b(that)h Fp(R)g Fu(is)g(quasi-simplifying)f (w.r.t.)g(that)h(order.)523 1264 y(\\only-if)6 b(":)25 b(Let)h Fp(R)g Fu(b)r(e)h(quasi-simplifying)e(w.r.t.)h(the)g (simpli\014cation)g(order)e Fp(\037)p Fu(.)i(W)-7 b(e)26 b(sho)n(w)523 1363 y Fp(!)606 1328 y Fj(+)606 1392 y Fi(R)663 1400 y Ff(q)696 1392 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))1006 1363 y Fp(\022)27 b(\037)p Fu(.)j(Clearly)-7 b(,)29 b(it)i(is)f(su\016cien)n(t)g(to)g(sho)n(w)g(that)g Fn(s)d Fp(!)2690 1378 y Fi(R)2747 1386 y Ff(q)2780 1378 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))3091 1363 y Fn(t)30 b Fu(implies)523 1479 y Fn(s)38 b Fp(\037)f Fn(t)p Fu(.)g(If)g Fn(s)h Fp(!)1044 1494 y Fi(E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))1320 1479 y Fn(t)p Fu(,)37 b(that)f(is,)h Fn(s)h Fu(=)f Fn(C)6 b Fu([)p Fn(f)j Fu(\()p Fn(u)2111 1491 y Fj(1)2148 1479 y Fn(;)14 b(:)g(:)g(:)g(;)g(u)2381 1491 y Fo(n)2425 1479 y Fu(\)])37 b(and)g Fn(t)h Fu(=)f Fn(C)6 b Fu([)p Fn(u)2994 1491 y Fo(j)3029 1479 y Fu(])37 b(for)f(some)523 1578 y Fn(f)c Fp(2)23 b(F)8 b Fu(,)25 b(then)g(the)f(assertion)f(follo)n(ws)h(from)g(the)h(fact)f(that)h Fp(\037)f Fu(has)g(the)h(subterm)f(prop)r(ert)n(y)523 1678 y(and)h(is)h(closed)e(under)i(con)n(texts.)e(W)-7 b(e)26 b(pro)n(v)n(e)e(b)n(y)h(induction)h(on)f(the)h(depth)g(of)f(the) h(rewrite)523 1777 y(step)33 b(that)f Fn(s)f Fp(!)1041 1792 y Fi(R)1098 1800 y Ff(q)1131 1792 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))1445 1777 y Fn(t)33 b Fu(also)e(implies)i Fn(s)e Fp(\037)f Fn(t)p Fu(.)j(Consider)e(the)i(reduction)f(step)g Fn(s)f Fu(=)523 1877 y Fn(C)6 b Fu([)p Fn(l)r(\033)s Fu(])33 b Fp(!)827 1892 y Fi(R)884 1900 y Ff(q)917 1892 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))1233 1877 y Fn(C)g Fu([)p Fn(s)1360 1889 y Fo(i)p Fj(+1)1472 1877 y Fn(\033)s Fu(])33 b(=)f Fn(t)p Fu(,)i(where)f(the)g(rewrite)g(rule)g Fn(l)h Fp(!)f Fn(s)2830 1889 y Fo(i)p Fj(+1)2974 1877 y Fp(\()g Fn(s)3129 1889 y Fj(1)3199 1877 y Fp(!)g Fn(t)3345 1889 y Fj(1)3382 1877 y Fu(,)523 1977 y Fn(:)14 b(:)g(:)g(;)g(s)710 1989 y Fo(i)760 1977 y Fp(!)23 b Fn(t)896 1989 y Fo(i)947 1977 y Fu(is)g(used)g(\(let)h Fn(s)1398 1989 y Fo(k)q Fj(+1)1546 1977 y Fu(=)f Fn(r)r Fu(\),)h(so)e Fn(s)1888 1989 y Fo(j)1923 1977 y Fn(\033)27 b Fp(!)2080 1947 y Fi(\003)2080 2003 y(R)2137 2011 y Ff(q)2170 2003 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))2476 1977 y Fn(t)2506 1989 y Fo(j)2541 1977 y Fn(\033)26 b Fu(for)d(1)g Fp(\024)f Fn(j)28 b Fp(\024)23 b Fn(i)p Fu(.)g(One)g(has)523 2086 y Fn(s)562 2098 y Fo(j)597 2086 y Fn(\033)j Fp(\027)d Fn(t)788 2098 y Fo(j)823 2086 y Fn(\033)i Fu(b)n(y)c(the)h(inductiv)n (e)g(h)n(yp)r(othesis.)f(It)h(then)g(follo)n(ws)e(from)h (quasi-simplifyingness)523 2185 y(that)28 b Fn(l)r(\033)e Fp(\037)c Fn(s)929 2197 y Fo(i)p Fj(+1)1041 2185 y Fn(\033)s Fu(.)28 b(W)-7 b(e)28 b(ev)n(en)n(tually)f(infer)g Fn(s)c Fp(\037)g Fn(t)28 b Fu(b)r(ecause)f Fp(\037)g Fu(is)h(closed)f(under)g (con)n(texts.)523 2345 y(The)h(second)f(su\016cien)n(t)g(criterion)g (uses)g(the)h(transformation)e(of)i(De\014nition)g(6.)523 2492 y Fh(Prop)s(osition)i(16.)41 b Fm(If)30 b Fn(U)9 b Fu(\()p Fp(R)p Fu(\))30 b Fm(is)g(simply)h(terminating,)g(then)e Fp(R)h Fm(is)g(quasi-simplifying.)523 2652 y(Pr)l(o)l(of.)43 b Fu(By)25 b(Lemma)g(13,)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\))15 b Fp([)f(E)7 b Fn(mb)p Fu(\()p Fp(F)1881 2621 y Fi(0)1905 2652 y Fu(\))25 b(is)h(terminating.)f(W)-7 b(e)26 b(sho)n(w)e Fp(!)2944 2667 y Fi(R)3001 2675 y Ff(q)3034 2667 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))3340 2652 y Fp(\022)523 2763 y(!)606 2727 y Fj(+)606 2791 y Fo(U)g Fj(\()p Fi(R)p Fj(\))p Fi([E)f Fo(mb)p Fj(\()p Fi(F)1020 2774 y Fe(0)1042 2791 y Fj(\))1072 2763 y Fu(.)24 b(The)h(prop)r(osition)e(then)i(follo) n(ws)e(from)h(Prop)r(osition)f(15.)g(If)i Fn(s)e Fp(!)3167 2778 y Fi(E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))523 2876 y Fn(t)p Fu(,)33 b(then)h(w)n(e)f(ha)n(v)n(e)e Fn(s)i Fp(!)1283 2891 y Fi(E)5 b Fo(mb)p Fj(\()p Fi(F)1492 2875 y Fe(0)1513 2891 y Fj(\))1576 2876 y Fn(t)33 b Fu(b)r(ecause)f Fp(F)40 b(\022)32 b(F)2216 2846 y Fi(0)2239 2876 y Fu(.)i(W)-7 b(e)33 b(pro)n(v)n(e)f(b)n(y)g(induction)i(on)f(the)523 2986 y(depth)28 b(of)g(the)g(rewrite)f(step)h(that)g Fn(s)23 b Fp(!)1779 3001 y Fi(R)1836 3009 y Ff(q)1869 3001 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))2175 2986 y Fn(t)28 b Fu(implies)g Fn(s)23 b Fp(!)2660 2950 y Fj(+)2660 3014 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))p Fi([E)f Fo(mb)p Fj(\()p Fi(F)3074 2997 y Fe(0)3096 3014 y Fj(\))3149 2986 y Fn(t)p Fu(.)28 b(Con-)523 3099 y(sider)g(the)h(reduction)f(step) g Fn(s)d Fu(=)f Fn(C)6 b Fu([)p Fn(l)r(\033)s Fu(])24 b Fp(!)1861 3114 y Fi(R)1918 3122 y Ff(q)1951 3114 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))2259 3099 y Fn(C)g Fu([)p Fn(s)2386 3111 y Fo(i)p Fj(+1)2498 3099 y Fn(\033)s Fu(])25 b(=)f Fn(t)p Fu(,)k(where)g(the)h(rewrite)523 3199 y(rule)39 b Fn(l)45 b Fp(!)e Fn(s)936 3211 y Fo(i)p Fj(+1)1091 3199 y Fp(\()h Fn(s)1257 3211 y Fj(1)1337 3199 y Fp(!)g Fn(t)1494 3211 y Fj(1)1531 3199 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)1754 3211 y Fo(i)1825 3199 y Fp(!)43 b Fn(t)1981 3211 y Fo(i)2049 3199 y Fu(is)d(used,)f(so)g Fn(s)2522 3211 y Fo(j)2557 3199 y Fn(\033)47 b Fp(!)2734 3168 y Fi(\003)2734 3225 y(R)2791 3233 y Ff(q)2824 3225 y Fi([E)5 b Fo(mb)p Fj(\()p Fi(F)h Fj(\))3150 3199 y Fn(t)3180 3211 y Fo(j)3215 3199 y Fn(\033)44 b Fu(for)523 3308 y(1)23 b Fp(\024)f Fn(j)28 b Fp(\024)23 b Fn(i)p Fu(.)k(One)g(has)h Fn(s)1267 3320 y Fo(j)1301 3308 y Fn(\033)f Fp(!)1458 3278 y Fi(\003)1458 3334 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))p Fi([E)f Fo(mb)p Fj(\()p Fi(F)1872 3318 y Fe(0)1894 3334 y Fj(\))1947 3308 y Fn(t)1977 3320 y Fo(j)2012 3308 y Fn(\033)31 b Fu(b)n(y)c(the)h(inductiv)n(e)g(h)n(yp) r(othesis.)f(Th)n(us,)846 3496 y Fn(l)r(\033)f Fp(!)1029 3511 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))1518 3496 y Fn(U)1584 3456 y Fo(\032)1575 3518 y Fj(1)1623 3496 y Fu(\()p Fn(s)1694 3508 y Fj(1)1731 3496 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\)\))p Fn(\033)946 3595 y Fp(!)1029 3565 y Fi(\003)1029 3622 y Fo(U)f Fj(\()p Fi(R)p Fj(\))p Fi([E)f Fo(mb)p Fj(\()p Fi(F)1443 3605 y Fe(0)1465 3622 y Fj(\))1518 3595 y Fn(U)1584 3555 y Fo(\032)1575 3617 y Fj(1)1623 3595 y Fu(\()p Fn(t)1685 3607 y Fj(1)1722 3595 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\)\))p Fn(\033)946 3706 y Fp(!)1029 3721 y Fo(U)f Fj(\()p Fi(R)p Fj(\))1518 3706 y Fn(U)1584 3666 y Fo(\032)1575 3728 y Fj(2)1623 3706 y Fu(\()p Fn(s)1694 3718 y Fj(2)1731 3706 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))p Fn(;)14 b Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)2291 3718 y Fj(1)2329 3706 y Fu(\)\))p Fn(\033)946 3806 y(:)14 b(:)g(:)946 3906 y Fp(!)1029 3875 y Fi(\003)1029 3932 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))p Fi([E)f Fo(mb)p Fj(\()p Fi(F)1443 3916 y Fe(0)1465 3932 y Fj(\))1518 3906 y Fn(U)1584 3866 y Fo(\032)1575 3929 y(i)1623 3906 y Fu(\()p Fn(t)1685 3918 y Fo(i)1712 3906 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))p Fn(;)14 b Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)2272 3918 y Fj(1)2311 3906 y Fu(\))p Fn(;)14 b(:)g(:)g(:)g(;)g Fp(E)7 b(V)f Fn(ar)r Fu(\()p Fn(t)2781 3918 y Fo(i)p Fi(\000)p Fj(1)2895 3906 y Fu(\)\))p Fn(\033)946 4017 y Fp(!)1029 4032 y Fo(U)g Fj(\()p Fi(R)p Fj(\))1518 4017 y Fn(U)1584 3977 y Fo(\032)1575 4040 y(i)p Fj(+1)1687 4017 y Fu(\()p Fn(s)1758 4029 y Fo(i)p Fj(+1)1869 4017 y Fn(;)14 b Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\))p Fn(;)14 b Fp(E)7 b(V)h Fn(ar)r Fu(\()p Fn(t)2430 4029 y Fj(1)2468 4017 y Fu(\))p Fn(;)14 b(:)g(:)g(:)g(;)g Fp(E)7 b(V)f Fn(ar)r Fu(\()p Fn(t)2938 4029 y Fo(i)2967 4017 y Fu(\)\))p Fn(\033)946 4116 y Fp(!)1029 4131 y Fi(E)f Fo(mb)p Fj(\()p Fi(F)1238 4115 y Fe(0)1260 4131 y Fj(\))1518 4116 y Fn(s)1557 4128 y Fo(i)p Fj(+1)1669 4116 y Fn(\033)n(:)523 4285 y Fu(W)-7 b(e)30 b(ha)n(v)n(e)e(already)g(seen)i(that)f Fn(U)9 b Fu(\()p Fp(R)1688 4297 y Fo(f)e(ib)1784 4285 y Fu(\))30 b(is)g(simply)f(terminating,)g(hence)h Fp(R)2979 4297 y Fo(f)7 b(ib)3105 4285 y Fu(is)29 b(quasi-)523 4385 y(simplifying)f(according)e(to)h(Prop)r(osition)f(16.)648 4485 y(In)h(order)g(to)g(rectify)h([ALS94,)f(Lemma)h(3.1],)f(it)h(is)g (su\016cien)n(t)g(to)f(replace)g(\\reduction)523 4584 y(order")g(with)i(\\simpli\014cation)f(order".)f(This)i(yields)g(the)g (third)g(su\016cien)n(t)g(condition)g(for)523 4684 y (quasi-simplifyingness.)p 523 4748 473 4 v 546 4801 a Fd(1)606 4833 y Ft(This)35 b(is)f(true)g(b)r(ecause)h(w)n(e)f(consider) h(\014nite)e(signatures)i(only;)g(see)f([MZ97)r(])g(for)h(details)g(on) 606 4924 y(in\014nite)25 b(signatures.)p eop %%Page: 9 9 9 8 bop 523 448 a Fh(Prop)s(osition)30 b(17.)41 b Fm(L)l(et)26 b Fu(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))27 b Fm(b)l(e)f(a)h (deterministic)g(3-CTRS.)g(If)g(its)g(b)l(ackwar)l(d)h(substi-)523 548 y(tute)l(d)h(system)g Fu(\()p Fp(F)e([)19 b(f)p Fn(c)p Fp(g)p Fn(;)p 1350 481 71 4 v 14 w Fp(R)o Fu(\))30 b Fm(is)g(simplifying,)j(then)c Fu(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))31 b Fm(is)f(quasi-simplifying.)523 716 y(Pr)l(o)l(of.)43 b Fu(If)28 b(\()p Fp(F)d([)18 b(f)p Fn(c)p Fp(g)p Fn(;)p 1213 649 V 14 w Fp(R)p Fu(\))27 b(is)g(simplifying,)g(then)h(there)f(is)g(a)g(simpli\014cation)g(order) f Fp(\037)g Fu(suc)n(h)523 816 y(that)32 b Fn(l)f Fp(\037)p 857 770 40 4 v 29 w Fn(r)f Fu(and)e Fn(l)j Fp(\037)p 1236 770 39 4 v 29 w Fn(s)1275 828 y Fo(i)1330 816 y Fu(for)c(1)i Fp(\024)g Fn(i)h Fp(\024)f Fn(k)34 b Fu(for)d(ev)n(ery)f (bac)n(kw)n(ard)g(substituted)i(rule)p 3190 770 43 4 v 31 w Fn(\032)g Fu(of)f(a)523 915 y(rule)c Fn(\032)c Fu(=)g Fn(l)h Fp(!)f Fn(r)j Fp(\()d Fn(s)1206 927 y Fj(1)1267 915 y Fp(!)g Fn(t)1403 927 y Fj(1)1440 915 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)1664 927 y Fo(k)1727 915 y Fp(!)23 b Fn(t)1863 927 y Fo(k)1932 915 y Fu(from)k Fp(R)p Fu(.)648 1015 y(W)-7 b(e)30 b(will)f(only)h(sho)n(w)f(that)g(the)i(\014rst)e (condition)g(of)h(De\014nition)g(12)f(\(if)i Fn(s)2958 1027 y Fo(j)2992 1015 y Fn(\033)f Fp(\027)c Fn(t)3190 1027 y Fo(j)3225 1015 y Fn(\033)34 b Fu(for)523 1115 y(ev)n(ery)22 b(1)h Fp(\024)f Fn(j)28 b Fp(\024)23 b Fn(i)p Fu(,)g(then)h Fn(l)r(\033)h Fp(\037)e Fn(s)1526 1127 y Fo(i)p Fj(+1)1638 1115 y Fn(\033)s Fu(\))h(is)f(satis\014ed.)g (The)g(second)g(condition)g(\(if)h Fn(s)3094 1127 y Fo(j)3128 1115 y Fn(\033)j Fp(\027)c Fn(t)3320 1127 y Fo(j)3355 1115 y Fn(\033)523 1214 y Fu(for)i(ev)n(ery)f(1)e Fp(\024)h Fn(j)28 b Fp(\024)23 b Fn(k)s Fu(,)i(then)h Fn(l)r(\033)g Fp(\037)c Fn(r)r(\033)s Fu(\))27 b(follo)n(ws)e(b)n(y)g(similar)f (reasoning.)g(It)h(will)h(b)r(e)g(sho)n(wn)523 1314 y(b)n(y)34 b(induction)g(on)f Fn(i)h Fu(that)g Fn(s)1429 1326 y Fo(j)1464 1314 y Fn(\033)j Fp(\027)c Fn(t)1676 1326 y Fo(j)1711 1314 y Fn(\033)k Fu(for)d(ev)n(ery)e(1)h Fp(\024)g Fn(j)39 b Fp(\024)33 b Fn(i)g Fu(implies)p 2848 1268 39 4 v 34 w Fn(s)2887 1326 y Fo(i)p Fj(+1)2999 1314 y Fn(\033)k Fp(\027)c Fn(s)3220 1326 y Fo(i)p Fj(+1)3331 1314 y Fn(\033)s Fu(.)523 1413 y(This)38 b(is)f(su\016cien)n(t)h(b)r (ecause)f Fn(l)42 b Fp(\037)p 1663 1368 V 39 w Fn(s)1702 1425 y Fo(i)p Fj(+1)1852 1413 y Fu(further)37 b(implies)h Fn(l)r(\033)43 b Fp(\037)p 2655 1368 V 40 w Fn(s)2693 1425 y Fo(i)p Fj(+1)2805 1413 y Fn(\033)h Fp(\027)39 b Fn(s)3039 1425 y Fo(i)p Fj(+1)3151 1413 y Fn(\033)s Fu(.)f(The)523 1513 y(base)f(case)f Fn(i)j Fu(=)g(0)e(holds)h(as)p 1493 1467 V 36 w Fn(s)1532 1525 y Fj(1)1609 1513 y Fu(=)h Fn(s)1752 1525 y Fj(1)1789 1513 y Fu(.)f(In)f(order)f(to)i(sho)n(w)e (the)i(inductiv)n(e)f(step,)h(note)523 1613 y(that)p 706 1567 V 31 w Fn(s)745 1625 y Fo(i)p Fj(+1)857 1613 y Fn(\033)31 b Fu(=)d Fn(')1082 1625 y Fo(i)p Fj(+1)1194 1613 y Fu(\()p Fn(s)1265 1625 y Fo(i)p Fj(+1)1377 1613 y Fu(\))p Fn(\033)s Fu(.)k(Let)f Fn(y)g Fp(2)e(V)7 b Fn(ar)r Fu(\()p Fn(t)2025 1625 y Fj(1)2062 1613 y Fn(;)14 b(:)g(:)g(:)g(;)g(t)2277 1625 y Fo(i)2304 1613 y Fu(\))21 b Fp(\\)g(E)7 b(V)g Fn(ar)r Fu(\()p Fn(\032)p Fu(\).)32 b(According)e(to)h(the)523 1712 y(inductiv)n(e)25 b(h)n(yp)r(othesis,)p 1310 1667 V 25 w Fn(s)1349 1727 y Fo(\013)p Fj(\()p Fo(y)r Fj(\))1484 1712 y Fn(\033)i Fp(\027)22 b Fn(s)1684 1727 y Fo(\013)p Fj(\()p Fo(y)r Fj(\))1819 1712 y Fn(\033)s Fu(.)k(Therefore,)p 2315 1667 V 24 w Fn(s)2354 1727 y Fo(\013)p Fj(\()p Fo(y)r Fj(\))2489 1712 y Fn(\033)g Fp(\027)d Fn(s)2689 1727 y Fo(\013)p Fj(\()p Fo(y)r Fj(\))2824 1712 y Fn(\033)j Fp(\027)d Fn(t)3015 1727 y Fo(\013)p Fj(\()p Fo(y)r Fj(\))3150 1712 y Fn(\033)j Fp(\027)d Fn(y)s(\033)523 1812 y Fu(and)36 b(hence)h Fn(')987 1824 y Fo(i)p Fj(+1)1099 1812 y Fu(\()p Fn(y)s Fu(\))p Fn(\033)k Fu(=)p 1397 1766 V 37 w Fn(s)1436 1827 y Fo(\013)p Fj(\()p Fo(y)r Fj(\))1571 1812 y Fn(\033)g Fp(\027)c Fn(y)s(\033)s Fu(.)g(No)n(w)p 2112 1766 V 35 w Fn(s)2151 1824 y Fo(i)p Fj(+1)2263 1812 y Fn(\033)k Fp(\027)c Fn(s)2492 1824 y Fo(i)p Fj(+1)2604 1812 y Fn(\033)j Fu(is)c(a)g(consequence)f(of)523 1912 y(the)40 b(follo)n(wing)f(observ)-5 b(ation:)38 b(If)i Fn(u)1660 1924 y Fj(1)1740 1912 y Fp(\027)i Fn(v)1887 1924 y Fj(1)1925 1912 y Fn(;)14 b(:)g(:)g(:)g(;)g(u)2158 1924 y Fo(n)2245 1912 y Fp(\027)43 b Fn(v)2393 1924 y Fo(n)2438 1912 y Fu(,)d(then)g Fn(C)6 b Fu([)p Fn(u)2838 1924 y Fj(1)2875 1912 y Fn(;)14 b(u)2960 1924 y Fj(2)2997 1912 y Fn(;)g(:)g(:)g(:)g(;)g(u)3230 1924 y Fo(n)3274 1912 y Fu(])43 b Fp(\027)523 2011 y Fn(C)6 b Fu([)p Fn(v)651 2023 y Fj(1)689 2011 y Fn(;)14 b(u)774 2023 y Fj(2)811 2011 y Fn(;)g(:)g(:)g(:)f(;)h(u)1043 2023 y Fo(n)1088 2011 y Fu(])23 b Fp(\027)g(\001)14 b(\001)g(\001)22 b(\027)h Fn(C)6 b Fu([)p Fn(v)1557 2023 y Fj(1)1595 2011 y Fn(;)14 b(:)g(:)g(:)f(;)h(v)1819 2023 y Fo(n)1865 2011 y Fu(])28 b(since)f Fp(\037)g Fu(is)h(closed)f(under)g(con)n(texts.)523 2179 y(Prop)r(osition)d(17)i(can)f(also)g(b)r(e)i(used)f(to)g(sho)n(w)f (quasi-simplifyingness)g(of)h Fp(R)2947 2191 y Fo(f)7 b(ib)3043 2179 y Fu(.)26 b(This)g(can)523 2279 y(b)r(e)i(seen)f(as)g (follo)n(ws.)p 1214 2212 71 4 v 27 w Fp(R)1284 2291 y Fo(f)7 b(ib)1408 2279 y Fu(consists)27 b(of)g(the)h(rules)1144 2457 y Fn(f)9 b(ib)p Fu(\(0\))22 b Fp(!)h(h)p Fu(0)p Fn(;)14 b(s)p Fu(\(0\))p Fp(i)1034 2557 y Fn(f)9 b(ib)p Fu(\()p Fn(s)p Fu(\()p Fn(x)p Fu(\)\))24 b Fp(!)f(h)p Fn(f)9 b(ib)p Fu(\()p Fn(x)p Fu(\))p Fn(;)14 b(f)9 b(ib)p Fu(\()p Fn(x)p Fu(\))18 b(+)g Fn(f)9 b(ib)p Fu(\()p Fn(x)p Fu(\))p Fp(i)24 b(\()f Fn(f)9 b(ib)p Fu(\()p Fn(x)p Fu(\))23 b Fp(!)g Fn(c)523 2745 y Fu(By)h(Lemma)g(14,)f(simplifyingness)h(of)p 1735 2679 V 25 w Fp(R)1805 2757 y Fo(f)7 b(ib)1925 2745 y Fu(is)24 b(equiv)-5 b(alen)n(t)24 b(to)g(simple)h(termination)e(of)i (the)523 2845 y(TRS)j(\()p 750 2778 V Fp(R)821 2857 y Fo(f)7 b(ib)916 2845 y Fu(\))948 2857 y Fo(s)1012 2845 y Fu(and)27 b(simple)h(termination)f(of)h(this)g(TRS)f(can)h(easily)e (b)r(e)i(sho)n(wn)f(b)n(y)h(rp)r(o.)648 2945 y(Since)36 b(b)r(oth)g(Prop)r(osition)f(16)g(and)g(Prop)r(osition)g(17)g(are)g (su\016cien)n(t)h(conditions)g(for)523 3044 y(pro)n(ving)30 b(quasi-simplifyingness,)g(it)h(is)h(natural)e(to)h(ask)g(whether)g (one)g(is)g(subsumed)g(b)n(y)523 3144 y(the)d(other.)f(This)h(is)f(not) h(the)g(case)e(as)h(the)h(follo)n(wing)f(examples)g(will)h(sho)n(w.)523 3297 y Fm(Example)j(18.)43 b Fu(Let)22 b Fp(R)h Fu(=)g Fp(f)p Fn(f)9 b Fu(\()p Fn(s)p Fu(\()p Fn(x)p Fu(\)\))23 b Fp(!)g Fn(f)9 b Fu(\()p Fn(s)p Fu(\()p Fn(y)s Fu(\)\))24 b Fp(\()f Fn(f)9 b Fu(\()p Fn(x)p Fu(\))24 b Fp(!)f Fn(f)9 b Fu(\()p Fn(s)p Fu(\()p Fn(y)s Fu(\)\))p Fp(g)p Fu(.)22 b(It)g(is)g(fairly)f(simple)523 3396 y(to)31 b(sho)n(w)g(that)g Fn(U)9 b Fu(\()p Fp(R)p Fu(\))32 b(is)f(simply)h(terminating.)f(T)-7 b(o)31 b(v)n(erify)f(that)i(Prop)r(osition)d(17)i(is)g(not)523 3496 y(applicable)c(is)h(equally)f(simple;)g(see)h([ALS94)o(].)523 3664 y Fm(Example)j(19.)43 b Fu(Consider)26 b(the)i(orien)n(ted)f (1-CTRS)g(\()p Fp(F)8 b Fn(;)14 b Fp(R)p Fu(\))28 b(consisting)f(of)h (the)g(rules)1490 3839 y Fn(a)23 b Fp(!)h Fn(d)719 b(a)23 b Fp(!)g Fn(e)1499 3939 y(b)f Fp(!)i Fn(d)727 b(b)23 b Fp(!)g Fn(e)1472 4039 y(A)g Fp(!)h Fn(h)p Fu(\()p Fn(f)9 b Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))1290 4138 y Fn(h)p Fu(\()p Fn(x;)14 b(x)p Fu(\))24 b Fp(!)g Fn(g)s Fu(\()p Fn(x;)14 b(x)p Fu(\))1308 4238 y Fn(g)s Fu(\()p Fn(d;)g(e)p Fu(\))23 b Fp(!)h Fn(A)1373 4338 y(f)9 b Fu(\()p Fn(x)p Fu(\))23 b Fp(!)h Fn(x)433 b Fp(\()23 b Fn(x)h Fp(!)f Fn(d)523 4526 y Fu(Its)35 b(bac)n(kw)n(ard)d(substituted)k(system)p 1755 4459 V 34 w Fp(R)f Fu(o)n(v)n(er)p 2045 4459 68 4 v 33 w Fp(F)43 b Fu(=)34 b Fp(F)d([)24 b(f)p Fn(c)p Fp(g)33 b Fu(is)i(obtained)f(from) g Fp(R)h Fu(b)n(y)523 4625 y(replacing)27 b(its)i(last)g(rule)f(with)h Fn(f)9 b Fu(\()p Fn(x)p Fu(\))25 b Fp(!)g Fn(x)g Fp(\()g Fn(x)g Fp(!)g Fn(c)p Fu(.)k(W)-7 b(e)29 b(claim)f(that)p 2801 4559 71 4 v 29 w Fp(R)h Fu(is)g(simplifying.)523 4725 y(By)k(Lemmata)g(13)f(and)h(14,)f(it)i(su\016ces)f(to)g(sho)n(w)f (that)p 2337 4658 V 33 w Fp(R)2408 4737 y Fo(s)2465 4725 y Fp([)23 b(E)7 b Fn(mb)p Fu(\()p 2735 4658 68 4 v Fp(F)h Fu(\))33 b(is)g(terminating.)523 4825 y(F)-7 b(or)23 b(an)h(indirect)g(pro)r(of)g(of)g(the)g(claim,)g(supp)r(ose)f(that)i (there)f(is)f(an)h(in\014nite)p 2929 4758 71 4 v 25 w Fp(R)2999 4837 y Fo(s)3046 4825 y Fp([)11 b(E)c Fn(mb)p Fu(\()p 3304 4758 68 4 v Fp(F)i Fu(\))523 4924 y(reduction)21 b(sequence.)f(It)h(is)g(not)g(di\016cult)g(to)g(see)g(that)g(in)g(this) g(case)f(there)h(m)n(ust)g(b)r(e)g(a)f(cycle)p eop %%Page: 10 10 10 9 bop 523 448 a Fn(A)40 b Fp(!)p 708 425 58 3 v 26 x Fi(R)765 482 y Ff(s)797 474 y Fi([E)5 b Fo(mb)p Fj(\()p 997 425 54 3 v Fi(F)g Fj(\))1120 448 y Fn(h)p Fu(\()p Fn(f)k Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))40 b Fp(!)1700 413 y Fj(+)p 1700 439 58 3 v 1700 487 a Fi(R)1757 495 y Ff(s)1789 487 y Fi([E)5 b Fo(mb)p Fj(\()p 1989 439 54 3 v Fi(F)g Fj(\))2112 448 y Fn(A)p Fu(.)38 b(If)g Fn(h)p Fu(\()p Fn(f)9 b Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))40 b Fp(!)2908 413 y Fj(+)p 2908 439 58 3 v 2908 487 a Fi(R)2965 495 y Ff(s)2997 487 y Fi([E)5 b Fo(mb)p Fj(\()p 3197 439 54 3 v Fi(F)g Fj(\))3320 448 y Fn(A)p Fu(,)523 567 y(then)28 b(there)g(m)n(ust)f(b)r(e)h(a)f(term)h Fn(t)g Fu(suc)n(h)f(that)523 747 y Fn(h)p Fu(\()p Fn(f)9 b Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))23 b Fp(!)1086 712 y Fj(+)p 1086 737 58 3 v 1086 786 a Fi(R)1143 794 y Ff(s)1175 786 y Fi([E)5 b Fo(mb)p Fj(\()p 1375 737 54 3 v Fi(F)g Fj(\))1481 747 y Fn(h)p Fu(\()p Fn(t;)14 b(t)p Fu(\))24 b Fp(!)p 1797 724 58 3 v 25 x Fi(R)1854 781 y Ff(s)1913 747 y Fn(g)s Fu(\()p Fn(t;)14 b(t)p Fu(\))23 b Fp(!)2223 712 y Fj(+)p 2223 737 V 2223 786 a Fi(R)2280 794 y Ff(s)2312 786 y Fi([E)5 b Fo(mb)p Fj(\()p 2512 737 54 3 v Fi(F)g Fj(\))2618 747 y Fn(g)s Fu(\()p Fn(d;)14 b(e)p Fu(\))23 b Fp(!)p 2950 724 58 3 v 25 x Fi(R)3007 781 y Ff(s)3039 772 y Fi([E)5 b Fo(mb)p Fj(\()p 3239 724 54 3 v Fi(F)g Fj(\))3345 747 y Fn(A:)523 951 y Fu(So)28 b Fn(t)g Fu(m)n(ust)g(b)r(e)h(a)e(common)h(reduct)g(of)g Fn(f)9 b Fu(\()p Fn(a)p Fu(\))28 b(and)g Fn(f)9 b Fu(\()p Fn(b)p Fu(\))28 b(and)g Fn(t)g Fu(m)n(ust)h(rewrite)e(to)h Fn(d)g Fu(and)g Fn(e)p Fu(.)523 1051 y(The)f(common)g(reducts)g(of)h Fn(f)9 b Fu(\()p Fn(a)p Fu(\))27 b(and)h Fn(f)9 b Fu(\()p Fn(b)p Fu(\))27 b(are)f Fn(f)9 b Fu(\()p Fn(d)p Fu(\),)28 b Fn(f)9 b Fu(\()p Fn(e)p Fu(\),)27 b Fn(d)p Fu(,)h(and)f Fn(e)p Fu(.)h(Neither)f(of)g(them)523 1150 y(reduces)j(to)h(b)r(oth)g Fn(d)g Fu(and)g Fn(e)p Fu(.)g(W)-7 b(e)31 b(conclude)f(that)p 2131 1084 71 4 v 31 w Fp(R)2202 1162 y Fo(s)2258 1150 y Fp([)21 b(E)7 b Fn(mb)p Fu(\()p 2526 1084 68 4 v Fp(F)h Fu(\))31 b(is)g(terminating.)f(Th)n(us)p 523 1183 71 4 v 523 1250 a Fp(R)e Fu(is)g(simplifying)f(and,)h(according)e(to)h (Prop)r(osition)f(17,)h Fp(R)h Fu(is)f(quasi-simplifying.)648 1350 y Fn(U)9 b Fu(\()p Fp(R)p Fu(\))29 b(is)g(obtained)f(from)h Fp(R)g Fu(b)n(y)g(replacing)e(the)j(conditional)e(rewrite)g(rule)g (with)i(the)523 1449 y(unconditional)39 b(rewrite)f(rules)h Fn(f)9 b Fu(\()p Fn(x)p Fu(\))43 b Fp(!)g Fn(U)9 b Fu(\()p Fn(x;)14 b(x)p Fu(\))40 b(and)f Fn(U)9 b Fu(\()p Fn(d;)14 b(x)p Fu(\))43 b Fp(!)g Fn(x)p Fu(.)d(The)f(follo)n(wing)523 1549 y(cyclic)27 b(deriv)-5 b(ation)27 b(sho)n(ws)g(that)h Fn(U)9 b Fu(\()p Fp(R)p Fu(\))28 b(is)f(not)h(simply)g(terminating.)619 1730 y Fn(A)23 b Fp(!)787 1745 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))975 1730 y Fn(h)p Fu(\()p Fn(f)j Fu(\()p Fn(a)p Fu(\))p Fn(;)14 b(f)9 b Fu(\()p Fn(b)p Fu(\)\))23 b Fp(!)1538 1694 y Fj(+)1538 1758 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))1725 1730 y Fn(h)p Fu(\()p Fn(U)j Fu(\()p Fn(a;)14 b(a)p Fu(\))p Fn(;)g(U)9 b Fu(\()p Fn(b;)14 b(b)p Fu(\)\))23 b Fp(!)2474 1694 y Fj(+)2474 1758 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))2662 1730 y Fn(h)p Fu(\()p Fn(U)j Fu(\()p Fn(d;)14 b(e)p Fu(\))p Fn(;)g(U)9 b Fu(\()p Fn(d;)14 b(e)p Fu(\)\))704 1844 y Fp(!)787 1859 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))975 1844 y Fn(g)s Fu(\()p Fn(U)j Fu(\()p Fn(d;)14 b(e)p Fu(\))p Fn(;)g(U)9 b Fu(\()p Fn(d;)14 b(e)p Fu(\)\))23 b Fp(!)1723 1808 y Fj(+)1723 1872 y Fi(E)5 b Fo(mb)p Fj(\()p Fi(F)1932 1855 y Fe(0)1954 1872 y Fj(\))2007 1844 y Fn(g)s Fu(\()p Fn(d;)14 b(e)p Fu(\))23 b Fp(!)2339 1859 y Fo(U)6 b Fj(\()p Fi(R)p Fj(\))2526 1844 y Fn(A:)523 2033 y Fu(The)39 b(preceding)e (example)i(is)f(also)g(in)n(teresting)f(b)r(ecause)i(it)g(refutes)f (the)h(claim)f(b)r(elo)n(w)523 2132 y(whic)n(h)22 b(is)g(a)f(reform)n (ulation)f(of)i([Mar96)n(,)g(Lemma)g(5.6].)f(Let)h(us)g(\014rst)f (review)g(the)i(de\014nition)523 2232 y(of)31 b(the)g(transformation)e Fc(U)34 b Fu(as)c(giv)n(en)g(in)h([Mar96)n(,)g(Def.)h(4.1].)e(Giv)n(en) g(a)g(join)h(1-CTRS)g Fp(R)523 2332 y Fu(and)f(a)f(rule)h Fn(\032)d Fu(:)g Fn(l)h Fp(!)f Fn(r)j Fp(\()d Fn(s)1426 2344 y Fj(1)1490 2332 y Fp(#)f Fn(t)1588 2344 y Fj(1)1625 2332 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)1849 2344 y Fo(k)1916 2332 y Fp(#)27 b Fn(t)2015 2344 y Fo(k)2083 2332 y Fp(2)g(R)p Fu(,)j(the)h(transformation)d Fc(U)p Fu(\()p Fn(\032)p Fu(\))33 b(yields)523 2431 y(the)28 b(set)g(whic)n(h)f(con)n(tains)g (the)h(t)n(w)n(o)f(unconditional)g(rules)1363 2590 y Fn(l)d Fp(!)f Fn(U)1575 2602 y Fo(\032)1614 2590 y Fu(\()p Fn(s)1685 2602 y Fj(1)1722 2590 y Fn(;)14 b(t)1789 2602 y Fj(1)1826 2590 y Fn(;)g(:)g(:)g(:)g(;)g(s)2050 2602 y Fo(k)2091 2590 y Fn(;)g(t)2158 2602 y Fo(k)2198 2590 y Fn(;)g Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\)\))1363 2689 y Fn(U)1420 2701 y Fo(\032)1458 2689 y Fu(\()p Fn(x)1537 2701 y Fj(1)1575 2689 y Fn(;)14 b(x)1659 2701 y Fj(1)1697 2689 y Fn(;)g(:)g(:)g(:)f(;)h(x)1928 2701 y Fo(k)1969 2689 y Fn(;)g(x)2053 2701 y Fo(k)2095 2689 y Fn(;)g Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\)\))23 b Fp(!)g Fn(r)523 2870 y Fu(where)37 b Fn(x)820 2882 y Fj(1)858 2870 y Fn(;)14 b(:)g(:)g(:)f(;)h(x)1089 2882 y Fo(k)1168 2870 y Fu(are)36 b(fresh)h(and)h(pairwise)e(distinct)i(v)-5 b(ariables.)36 b(Moreo)n(v)n(er,)f Fc(U)p Fu(\()p Fp(R)p Fu(\))41 b(is)523 2970 y(de\014ned)28 b(as)f(usual:)g Fc(U)p Fu(\()p Fp(R)p Fu(\))g(=)1457 2907 y Fg(S)1526 2994 y Fo(\032)p Fi(2R)1680 2970 y Fc(U)p Fu(\()p Fn(\032)p Fu(\).)523 3169 y Fh(Claim:)36 b Fu(If)i(a)f(join)i(1-CTRS)e Fp(R)h Fu(is)g(simplifying,)g(then)h(the)f(transformed)f(TRS)h Fc(U)p Fu(\()p Fp(R)p Fu(\))523 3268 y(is)28 b(simply)f(terminating.) 523 3468 y(If)43 b(w)n(e)f(view)g(the)h(system)f Fp(R)h Fu(from)f(Example)g(19)g(as)f(a)h(join)h(CTRS,)g(then)g Fc(U)p Fu(\()p Fp(R)p Fu(\))51 b(=)523 3567 y Fp(R)593 3537 y Fi(0)639 3567 y Fp([)22 b(f)p Fn(f)9 b Fu(\()p Fn(x)p Fu(\))32 b Fp(!)h Fn(U)9 b Fu(\()p Fn(x;)14 b(d;)g(x)p Fu(\))p Fn(;)g(U)9 b Fu(\()p Fn(x)1590 3579 y Fj(1)1629 3567 y Fn(;)14 b(x)1713 3579 y Fj(1)1750 3567 y Fn(;)g(x)p Fu(\))33 b Fp(!)f Fn(x)p Fp(g)p Fu(,)h(where)g Fp(R)2475 3537 y Fi(0)2532 3567 y Fu(consists)f(of)h(the)g(uncondi-)523 3667 y(tional)19 b(rules)g(of)g Fp(R)p Fu(.)g(As)h(in)f(Example)g(19,)f (it)h(can)g(b)r(e)h(sho)n(wn)e(that)i(the)f(system)g Fp(R)3013 3637 y Fi(0)3039 3667 y Fp([)r(f)p Fn(f)9 b Fu(\()p Fn(x)p Fu(\))23 b Fp(!)523 3767 y Fn(x;)14 b(f)9 b Fu(\()p Fn(x)p Fu(\))26 b Fp(!)f Fn(d)p Fp(g)k Fu(is)f(simply)h (terminating.)f(Th)n(us)h(the)g(join)g(1-CTRS)f Fp(R)h Fu(is)g(simplifying)g(ac-)523 3866 y(cording)f(to)h(Lemma)f(14.)g(On)h (the)g(other)g(hand,)g(the)g(transformed)f(system)g Fc(U)p Fu(\()p Fp(R)p Fu(\))33 b(is)c(not)523 3966 y(simply)f(terminating)f(b) r(ecause)g(there)h(is)f(a)g(cyclic)h(deriv)-5 b(ation)26 b(as)h(in)h(Example)f(19.)523 4229 y Fq(5)112 b(Mo)s(dularit)m(y)523 4426 y Fu(In)26 b(this)g(section,)g(w)n(e)f(will)h(in)n(v)n(estigate)f (under)h(whic)n(h)f(conditions)h(quasi-reductivit)n(y)e(and)523 4526 y(quasi-simplifyingness)c(are)h(mo)r(dular.)g(The)h(reader)e(is)i (assumed)f(to)g(b)r(e)h(familiar)f(with)i(the)523 4625 y(concepts)31 b(of)g(the)h(\014eld)f(of)h(mo)r(dularit)n(y)-7 b(.)30 b(Details)i(can)e(b)r(e)i(found)g(e.g.)f(in)g([Ohl94)o(,KR95)o (].)523 4725 y(Let)i Fp(R)g Fu(b)r(e)g(a)f(CTRS)h(o)n(v)n(er)e(the)i (signature)e Fp(F)8 b Fu(.)33 b(A)g(function)g(sym)n(b)r(ol)f Fn(f)40 b Fp(2)32 b(F)41 b Fu(is)32 b(called)g(a)523 4825 y Fm(de\014ne)l(d)i(symb)l(ol)e Fu(if)g(there)f(is)g(a)h(rewrite)e (rule)h Fn(l)g Fp(!)f Fn(r)i Fp(\()e Fn(c)f Fp(2)h(R)i Fu(suc)n(h)f(that)h Fn(f)38 b Fu(=)29 b Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\).)523 4924 y(F)-7 b(unction)25 b(sym)n(b)r(ols)e (from)h Fp(F)32 b Fu(whic)n(h)24 b(are)f(not)h(de\014ned)g(sym)n(b)r (ols)g(are)f(called)h Fm(c)l(onstructors)p Fu(.)p eop %%Page: 11 11 11 10 bop 523 448 a Fu(If)26 b Fp(R)674 460 y Fj(1)736 448 y Fu(and)f Fp(R)965 460 y Fj(2)1028 448 y Fu(are)f(CTRSs)h(o)n(v)n (er)e(the)j(signatures)d Fp(F)2215 460 y Fj(1)2277 448 y Fu(and)i Fp(F)2496 460 y Fj(2)2533 448 y Fu(,)g(resp)r(ectiv)n(ely)-7 b(,)24 b(then)i(their)523 548 y Fm(c)l(ombine)l(d)h(system)d Fu(is)g(their)f(union)h Fp(R)f Fu(=)g Fp(R)1898 560 y Fj(1)1946 548 y Fp([)11 b(R)2082 560 y Fj(2)2144 548 y Fu(o)n(v)n(er)22 b(the)i(signature)f Fp(F)31 b Fu(=)22 b Fp(F)3052 560 y Fj(1)3100 548 y Fp([)11 b(F)3226 560 y Fj(2)3263 548 y Fu(.)24 b(Its)523 648 y(set)29 b(of)f(de\014ned)h (sym)n(b)r(ols)f(is)g Fp(D)f Fu(=)d Fp(D)1682 660 y Fj(1)1739 648 y Fp([)19 b(D)1877 660 y Fj(2)1943 648 y Fu(and)29 b(its)f(set)h(of)f(constructors)f(is)i Fp(C)g Fu(=)24 b Fp(F)j(n)19 b(D)r Fu(,)523 747 y(where)27 b Fp(D)827 759 y Fo(i)882 747 y Fu(\()p Fp(C)958 759 y Fo(i)986 747 y Fu(\))h(denotes)f(the)h(de\014ned)g(sym)n(b)r(ols)f (\(constructors\))g(in)g Fp(R)2795 759 y Fo(i)2823 747 y Fu(.)517 915 y(\(1\))41 b Fp(R)734 927 y Fj(1)799 915 y Fu(and)28 b Fp(R)1031 927 y Fj(2)1096 915 y Fu(are)f Fm(disjoint)i Fu(if)f Fp(F)1668 927 y Fj(1)1723 915 y Fp(\\)19 b(F)1857 927 y Fj(2)1917 915 y Fu(=)k Fp(;)p Fu(.)517 1012 y(\(2\))41 b Fp(R)734 1024 y Fj(1)799 1012 y Fu(and)28 b Fp(R)1031 1024 y Fj(2)1096 1012 y Fu(are)f Fm(c)l(onstructor-sharing)h Fu(if)g Fp(F)2100 1024 y Fj(1)2155 1012 y Fp(\\)19 b(F)2289 1024 y Fj(2)2349 1012 y Fu(=)k Fp(C)2481 1024 y Fj(1)2536 1012 y Fp(\\)c(C)2654 1024 y Fj(2)2714 1012 y Fu(\()p Fp(\022)k(C)5 b Fu(\).)517 1109 y(\(3\))41 b Fp(R)734 1121 y Fj(1)799 1109 y Fu(and)28 b Fp(R)1031 1121 y Fj(2)1096 1109 y Fu(form)f(a)g(hierarc)n(hical)e (com)n(bination)i(of)g(base)g Fp(R)2628 1121 y Fj(1)2693 1109 y Fu(and)g(extension)g Fp(R)3291 1121 y Fj(2)3356 1109 y Fu(if)664 1209 y Fp(C)708 1221 y Fj(1)764 1209 y Fp(\\)18 b(D)901 1221 y Fj(2)962 1209 y Fu(=)k Fp(D)1113 1221 y Fj(1)1169 1209 y Fp(\\)d(D)1307 1221 y Fj(2)1367 1209 y Fu(=)k Fp(;)p Fu(.)523 1379 y(A)j(prop)r(ert)n(y)f Fp(P)32 b Fu(is)26 b Fm(mo)l(dular)g Fu(for)g(a)f(certain)g(class)g(of) g(CTRSs)h(if,)h(for)e(all)g(CTRSs)h(\()p Fp(F)3191 1391 y Fj(1)3228 1379 y Fn(;)14 b Fp(R)3335 1391 y Fj(1)3373 1379 y Fu(\))523 1479 y(and)29 b(\()p Fp(F)778 1491 y Fj(2)815 1479 y Fn(;)14 b Fp(R)922 1491 y Fj(2)960 1479 y Fu(\))29 b(b)r(elonging)g(to)g(that)g(class)f(and)h(ha)n(ving)f(prop) r(ert)n(y)g Fp(P)7 b Fu(,)29 b(their)g(union)g(\()p Fp(F)3293 1491 y Fj(1)3350 1479 y Fp([)523 1579 y(F)583 1591 y Fj(2)620 1579 y Fn(;)14 b Fp(R)727 1591 y Fj(1)783 1579 y Fp([)19 b(R)927 1591 y Fj(2)964 1579 y Fu(\))28 b(also)f(b)r(elongs)g (to)g(that)h(class)f(and)g(has)g(the)h(prop)r(ert)n(y)f Fp(P)7 b Fu(.)648 1678 y(It)29 b(is)h(w)n(ell)f(kno)n(wn)g(that)g (simple)h(termination)f(is)g(mo)r(dular)g(for)g(constructor-sharing)523 1778 y(TRSs;)39 b(see)f([K)n(O92)n(,MZ97)o(].)h(It)g(readily)f(follo)n (ws)g(from)h(Lemma)f(14)g(that)h(simplifying-)523 1878 y(ness)26 b(is)g(also)f(mo)r(dular)h(for)f(\014nite)i (constructor-sharing)c(1-CTRSs.)i(Therefore,)g(if)i(quasi-)523 1977 y(simplifyingness)21 b(of)h(t)n(w)n(o)f(constructor-sharing)d (3-CTRSs)j Fp(R)2475 1989 y Fj(1)2534 1977 y Fu(and)g Fp(R)2759 1989 y Fj(2)2818 1977 y Fu(can)h(b)r(e)g(sho)n(wn)e(b)n(y)523 2077 y(Prop)r(osition)25 b(17,)h(then)h Fp(R)1360 2089 y Fj(1)1415 2077 y Fp([)17 b(R)1557 2089 y Fj(2)1622 2077 y Fu(is)26 b(also)g(quasi-simplifying.)g(This)h(is)g(b)r(ecause)f (the)h(sim-)523 2176 y(plifying)c(bac)n(kw)n(ard)e(substituted)i (1-CTRSs)p 1966 2110 71 4 v 22 w Fp(R)2037 2188 y Fj(1)2097 2176 y Fu(and)p 2253 2110 V 22 w Fp(R)2324 2188 y Fj(2)2384 2176 y Fu(are)f(also)f(constructor-sharing.)523 2276 y(Hence)33 b(it)h(is)f(a)g(bit)g(surprising)f(that)h (quasi-simplifyingness)f(is)h(in)h(general)d Fm(not)41 b Fu(mo)r(du-)523 2376 y(lar)32 b(for)g(disjoin)n(t)h(deterministic)g (3-CTRSs.)f(The)h(next)g(example)g(whic)n(h)f(is)h(tak)n(en)f(from)523 2475 y([Mid93)o(])c(is)g(a)f(coun)n(terexample.)523 2623 y Fm(Example)k(20.)43 b Fu(Consider)26 b(the)i(1-CTRS)1288 2794 y Fp(R)1358 2806 y Fj(1)1418 2794 y Fu(=)23 b Fp(f)p Fn(f)9 b Fu(\()p Fn(x)p Fu(\))23 b Fp(!)g Fn(f)9 b Fu(\()p Fn(x)p Fu(\))24 b Fp(\()f Fn(x)h Fp(!)f Fn(a;)14 b(x)23 b Fp(!)g Fn(b)p Fp(g)523 2964 y Fu(o)n(v)n(er)36 b(the)i(signature)f Fp(F)1295 2976 y Fj(1)1372 2964 y Fu(=)j Fp(f)p Fn(f)t(;)14 b(a;)g(b)p Fp(g)p Fu(.)36 b(It)j(is)e(not)h(di\016cult)h(to)f(sho)n(w)f (that)h(the)g(system)523 3064 y Fp(f)p Fn(f)9 b Fu(\()p Fn(x)p Fu(\))30 b Fp(!)f Fn(x;)14 b(f)9 b Fu(\()p Fn(x)p Fu(\))31 b Fp(!)e Fn(x)i Fp(\()e Fn(x)h Fp(!)g Fn(a;)14 b(f)9 b Fu(\()p Fn(x)p Fu(\))30 b Fp(!)f Fn(f)9 b Fu(\()p Fn(x)p Fu(\))31 b Fp(\()e Fn(x)h Fp(!)g Fn(a;)14 b(x)30 b Fp(!)f Fn(b)p Fp(g)i Fu(is)g(terminating)523 3163 y(b)r(ecause)37 b(the)g(last)g(rule)g(can)g(nev)n(er)f(b)r(e)i(applied.)f(Hence)g Fp(R)2500 3175 y Fj(1)2575 3163 y Fu(is)g(quasi-simplifying)f(b)n(y)523 3263 y(Prop)r(osition)26 b(15.)h(The)g(TRS)1582 3478 y Fp(R)1652 3490 y Fj(2)1713 3478 y Fu(=)1801 3361 y Fg(\032)1875 3428 y Fn(or)r Fu(\()p Fn(x;)14 b(y)s Fu(\))24 b Fp(!)g Fn(x)1875 3527 y(or)r Fu(\()p Fn(x;)14 b(y)s Fu(\))24 b Fp(!)g Fn(y)523 3694 y Fu(is)39 b(ob)n(viously)f (quasi-simplifying,)h(to)r(o.)g(The)h(com)n(bined)f(system)g Fp(R)2775 3706 y Fj(1)2839 3694 y Fp([)27 b(R)2991 3706 y Fj(2)3028 3694 y Fu(,)40 b(ho)n(w)n(ev)n(er,)523 3794 y(is)d(not)g(ev)n(en)g(terminating:)g Fn(f)9 b Fu(\()p Fn(or)r Fu(\()p Fn(a;)14 b(b)p Fu(\)\))40 b Fp(!)1954 3806 y Fi(R)2011 3814 y Fb(1)2043 3806 y Fi([R)2145 3814 y Fb(2)2221 3794 y Fn(f)9 b Fu(\()p Fn(or)r Fu(\()p Fn(a;)14 b(b)p Fu(\)\))38 b(is)f(a)g(cyclic)g(deriv)-5 b(ation)523 3893 y(b)r(ecause)27 b Fn(or)r Fu(\()p Fn(a;)14 b(b)p Fu(\))24 b Fp(!)1197 3905 y Fi(R)1254 3913 y Fb(2)1314 3893 y Fn(a)j Fu(and)h Fn(or)r Fu(\()p Fn(a;)14 b(b)p Fu(\))24 b Fp(!)1914 3905 y Fi(R)1971 3913 y Fb(2)2031 3893 y Fn(b)p Fu(.)523 4041 y(Quasi-simplifyingness)i(of)i Fp(R)1497 4053 y Fj(1)1562 4041 y Fu(cannot)f(b)r(e)h(pro)n(v)n(en)e(b) n(y)h(Prop)r(osition)f(16)h(b)r(ecause)1383 4306 y Fn(U)9 b Fu(\()p Fp(R)1551 4318 y Fj(1)1589 4306 y Fu(\))23 b(=)1732 4136 y Fg(8)1732 4211 y(<)1732 4360 y(:)1817 4206 y Fn(f)9 b Fu(\()p Fn(x)p Fu(\))149 b Fp(!)23 b Fn(U)2290 4218 y Fj(1)2327 4206 y Fu(\()p Fn(x;)14 b(x)p Fu(\))1817 4306 y Fn(U)1874 4318 y Fj(1)1911 4306 y Fu(\()p Fn(a;)g(x)p Fu(\))24 b Fp(!)f Fn(U)2290 4318 y Fj(2)2327 4306 y Fu(\()p Fn(x;)14 b(x)p Fu(\))1817 4405 y Fn(U)1874 4417 y Fj(2)1911 4405 y Fu(\()p Fn(b;)g(x)p Fu(\))32 b Fp(!)23 b Fn(f)9 b Fu(\()p Fn(x)p Fu(\))523 4572 y(is)28 b(not)f(simply)h(terminating)f(as)g(the)h(follo)n(wing)e(cyclic)i (deriv)-5 b(ation)27 b(sho)n(ws:)782 4737 y Fn(f)9 b Fu(\()p Fn(U)921 4749 y Fj(1)957 4737 y Fu(\()p Fn(a;)14 b(b)p Fu(\)\))24 b Fp(!)1277 4752 y Fo(U)6 b Fj(\()p Fi(R)1411 4760 y Fb(1)1444 4752 y Fj(\))1497 4737 y Fn(U)1554 4749 y Fj(1)1591 4737 y Fu(\()p Fn(U)1680 4749 y Fj(1)1717 4737 y Fu(\()p Fn(a;)14 b(b)p Fu(\))p Fn(;)g(U)1992 4749 y Fj(1)2029 4737 y Fu(\()p Fn(a;)g(b)p Fu(\)\))23 b Fp(!)2348 4752 y Fi(E)5 b Fo(mb)p Fj(\()p Fi(F)2557 4735 y Fe(0)2579 4752 y Fj(\))2632 4737 y Fn(U)2689 4749 y Fj(1)2726 4737 y Fu(\()p Fn(a;)14 b(U)2896 4749 y Fj(1)2933 4737 y Fu(\()p Fn(a;)g(b)p Fu(\)\))1194 4836 y Fp(!)1277 4851 y Fo(U)6 b Fj(\()p Fi(R)1411 4859 y Fb(1)1444 4851 y Fj(\))1497 4836 y Fn(U)1554 4848 y Fj(2)1591 4836 y Fu(\()p Fn(U)1680 4848 y Fj(1)1717 4836 y Fu(\()p Fn(a;)14 b(b)p Fu(\))p Fn(;)g(U)1992 4848 y Fj(1)2029 4836 y Fu(\()p Fn(a;)g(b)p Fu(\)\))23 b Fp(!)2348 4851 y Fi(E)5 b Fo(mb)p Fj(\()p Fi(F)2557 4835 y Fe(0)2579 4851 y Fj(\))2632 4836 y Fn(U)2689 4848 y Fj(2)2726 4836 y Fu(\()p Fn(b;)14 b(U)2888 4848 y Fj(1)2925 4836 y Fu(\()p Fn(a;)g(b)p Fu(\)\))1194 4936 y Fp(!)1277 4951 y Fo(U)6 b Fj(\()p Fi(R)1411 4959 y Fb(1)1444 4951 y Fj(\))1497 4936 y Fn(f)j Fu(\()p Fn(U)1636 4948 y Fj(1)1673 4936 y Fu(\()p Fn(a;)14 b(b)p Fu(\)\))p Fn(:)p eop %%Page: 12 12 12 11 bop 523 448 a Fu(This)30 b(is)g(not)h(surprising)d(b)r(ecause)i (the)h(com)n(bined)f(system)g(of)g(t)n(w)n(o)f(constructor-sharing)523 548 y(deterministic)22 b(3-CTRSs)f Fp(R)1437 560 y Fj(1)1496 548 y Fu(and)g Fp(R)1721 560 y Fj(2)1781 548 y Fu(is)g (quasi-simplifying)g(if)h(b)r(oth)g Fn(U)9 b Fu(\()p Fp(R)2920 560 y Fj(1)2958 548 y Fu(\))22 b(and)f Fn(U)9 b Fu(\()p Fp(R)3335 560 y Fj(2)3373 548 y Fu(\))523 648 y(are)25 b(simply)h(terminating.)g(This)f(fact)i(is)e(an)h(easy)f (consequence)g(of)h(the)g(follo)n(wing)f(simple)523 747 y(generic)i(prop)r(osition.)523 894 y Fh(Prop)s(osition)j(21.)41 b Fm(L)l(et)d Fp(R)1425 906 y Fj(1)1501 894 y Fm(and)h Fp(R)1741 906 y Fj(2)1817 894 y Fm(b)l(e)g(deterministic)h(3-CTRSs.)f (Their)h(c)l(ombine)l(d)523 994 y(system)30 b Fp(R)865 1006 y Fj(1)921 994 y Fp([)18 b(R)1064 1006 y Fj(2)1132 994 y Fm(is)30 b(quasi-simplifying)i(if)555 1147 y(1.)42 b(b)l(oth)30 b Fn(U)9 b Fu(\()p Fp(R)1008 1159 y Fj(1)1046 1147 y Fu(\))30 b Fm(and)g Fn(U)9 b Fu(\()p Fp(R)1437 1159 y Fj(2)1475 1147 y Fu(\))30 b Fm(ar)l(e)g(simply)h(terminating,)f (and)555 1244 y(2.)42 b Fn(U)9 b Fu(\()p Fp(R)832 1256 y Fj(1)870 1244 y Fu(\))34 b Fm(and)f Fn(U)9 b Fu(\()p Fp(R)1268 1256 y Fj(2)1306 1244 y Fu(\))34 b Fm(b)l(elong)g(to)f(a)h (class)g(of)h(TRSs)e(for)h(which)h(simple)g(termination)664 1343 y(is)30 b(mo)l(dular.)523 1504 y(Pr)l(o)l(of.)43 b Fu(Since)38 b(simple)h(termination)e(is)h(a)g(mo)r(dular)f(prop)r (ert)n(y)-7 b(,)37 b(the)i(com)n(bined)e(system)523 1604 y Fn(U)9 b Fu(\()p Fp(R)691 1616 y Fj(1)729 1604 y Fu(\))i Fp([)g Fn(U)e Fu(\()p Fp(R)1006 1616 y Fj(2)1044 1604 y Fu(\))24 b(is)g(simply)g(terminating.)g(No)n(w)f(the)i(prop)r (osition)d(follo)n(ws)h(from)h Fn(U)9 b Fu(\()p Fp(R)3301 1616 y Fj(1)3350 1604 y Fp([)523 1703 y(R)593 1715 y Fj(2)631 1703 y Fu(\))23 b(=)g Fn(U)9 b Fu(\()p Fp(R)942 1715 y Fj(1)979 1703 y Fu(\))19 b Fp([)g Fn(U)9 b Fu(\()p Fp(R)1272 1715 y Fj(2)1309 1703 y Fu(\))28 b(in)g(conjunction)g(with)g (Prop)r(osition)e(16.)648 1864 y(A)35 b(similar)g(result)g(can)g(of)g (course)f(b)r(e)i(stated)f(for)g(quasi-reductivit)n(y)f(\(just)i (replace)523 1964 y(simple)31 b(termination)f(with)h(termination\).)g (Ho)n(w)n(ev)n(er,)d(b)r(etter)j(results)f(can)g(b)r(e)h(obtained)523 2064 y(b)n(y)39 b(taking)g(adv)-5 b(an)n(tage)38 b(of)i(the)g (implications)f Fn(U)9 b Fu(\()p Fp(R)p Fu(\))40 b(is)g(terminating)f Fp(\))h(R)g Fu(is)f(quasi-)523 2163 y(reductiv)n(e)20 b(\(Prop)r(osition)f(7\),)h Fp(R)g Fu(is)h(quasi-reductiv)n(e)d Fp(\))i Fn(U)9 b Fu(\()p Fp(R)p Fu(\))21 b(is)g(innermost)e (terminating)523 2263 y(\(Theorem)25 b(9\),)h(and)g(the)g(fact)g(that)g (termination)f(and)h(innermost)f(termination)g(coincide)523 2363 y(for)i(non-o)n(v)n(erlapping)e(TRSs;)j(see)f([Gra95)n(,)h(Thm.)g (3.23].)523 2510 y Fh(Prop)s(osition)i(22.)41 b Fm(L)l(et)i Fp(R)1430 2522 y Fj(1)1511 2510 y Fm(and)i Fp(R)1757 2522 y Fj(2)1838 2510 y Fm(b)l(e)f(quasi-r)l(e)l(ductive)g (deterministic)h(3-CTRSs.)523 2609 y(Their)31 b(c)l(ombine)l(d)g (system)e Fp(R)1455 2621 y Fj(1)1512 2609 y Fp([)18 b(R)1655 2621 y Fj(2)1723 2609 y Fm(is)30 b(quasi-r)l(e)l(ductive)g(if)555 2762 y(1.)42 b Fn(U)9 b Fu(\()p Fp(R)832 2774 y Fj(1)870 2762 y Fu(\))22 b Fm(and)h Fn(U)9 b Fu(\()p Fp(R)1246 2774 y Fj(2)1284 2762 y Fu(\))22 b Fm(b)l(elong)h(to)f(a)h(class)g(of)g (TRSs)f(for)h(which)g(innermost)g(termination)664 2862 y(is)30 b(mo)l(dular,)h(and)555 2959 y(2.)42 b Fn(U)9 b Fu(\()p Fp(R)832 2971 y Fj(1)888 2959 y Fp([)19 b(R)1032 2971 y Fj(2)1070 2959 y Fu(\))30 b Fm(is)g(non-overlapping.)523 3120 y(Pr)l(o)l(of.)43 b Fu(Since)28 b Fp(R)1070 3132 y Fj(1)1136 3120 y Fu(and)f Fp(R)1367 3132 y Fj(2)1433 3120 y Fu(are)g(quasi-reductiv)n(e,)f(the)i(transformed)f(TRSs)g Fn(U)9 b Fu(\()p Fp(R)3173 3132 y Fj(1)3211 3120 y Fu(\))28 b(and)523 3219 y Fn(U)9 b Fu(\()p Fp(R)691 3231 y Fj(2)729 3219 y Fu(\))33 b(are)f(innermost)g(terminating)h(according)e(to)i (Theorem)f(9.)g(Their)h(com)n(bination)523 3319 y Fn(U)9 b Fu(\()p Fp(R)691 3331 y Fj(1)729 3319 y Fu(\))16 b Fp([)f Fn(U)9 b Fu(\()p Fp(R)1015 3331 y Fj(2)1053 3319 y Fu(\))23 b(=)g Fn(U)9 b Fu(\()p Fp(R)1364 3331 y Fj(1)1417 3319 y Fp([)16 b(R)1558 3331 y Fj(2)1596 3319 y Fu(\))26 b(is)g(also)f(innermost)h(terminating)g(b)r(ecause)f(innermost)523 3418 y(termination)h(is)h(mo)r(dular.)f(Therefore,)g(termination)g(of)g Fn(U)9 b Fu(\()p Fp(R)2520 3430 y Fj(1)2575 3418 y Fp([)17 b(R)2717 3430 y Fj(2)2754 3418 y Fu(\))27 b(is)g(a)f(consequence)523 3518 y(of)37 b(its)g(non-o)n(v)n(erlappingness;)c(see)k([Gra95)n(,)g (Thm.)g(3.23].)e(No)n(w)i(the)g(assertion)e(follo)n(ws)523 3618 y(from)27 b(Prop)r(osition)f(7.)523 3779 y(Ho)n(w)n(ev)n(er,)e (non-o)n(v)n(erlappingness)e(of)k Fn(U)9 b Fu(\()p Fp(R)p Fu(\))26 b(is)g(not)f(implied)h(b)n(y)g(non-o)n(v)n(erlappingness)c(of) 523 3878 y Fp(R)p Fu(.)37 b(F)-7 b(or)36 b(example,)g(the)h(system)g Fp(R)h Fu(=)g Fp(f)p Fn(a)f Fp(!)h Fn(b)g Fp(\()g Fn(b)g Fp(!)g Fn(a)p Fp(g)e Fu(is)h(non-o)n(v)n(erlapping)c(but)523 3978 y Fn(U)9 b Fu(\()p Fp(R)p Fu(\))24 b(=)e Fp(f)p Fn(a)h Fp(!)g Fn(U)9 b Fu(\()p Fn(b)p Fu(\))p Fn(;)14 b(U)9 b Fu(\()p Fn(a)p Fu(\))23 b Fp(!)g Fn(b)p Fp(g)i Fu(is)g(not.)g(The)g(situation)g(is)g(di\013eren)n(t)g(for)g(syn)n (tactically)523 4078 y(deterministic)j(3-CTRSs.)523 4225 y Fh(De\014nition)j(23.)40 b Fm(A)27 b(deterministic)h(3-CTRS)f Fp(R)g Fm(is)g(c)l(al)t(le)l(d)h Fu(syn)n(tactically)c(deterministic) 523 4324 y Fm(if,)33 b(for)f(every)g Fn(l)27 b Fp(!)f Fn(r)j Fp(\()d Fn(s)1361 4336 y Fj(1)1424 4324 y Fp(!)f Fn(t)1562 4336 y Fj(1)1600 4324 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)1823 4336 y Fo(k)1890 4324 y Fp(!)26 b Fn(t)2029 4336 y Fo(k)2101 4324 y Fm(in)31 b Fp(R)p Fm(,)h(every)g(term)f Fn(t)2782 4336 y Fo(i)2810 4324 y Fm(,)h Fu(1)25 b Fp(\024)g Fn(i)h Fp(\024)f Fn(k)s Fm(,)32 b(is)f(a)523 4424 y(c)l(onstructor)e(term)1129 4394 y Fj(2)1196 4424 y Fm(or)h(a)g(gr)l(ound)g Fp(R)1720 4436 y Fo(u)1764 4424 y Fm(-normal)g(form.)523 4571 y Fu(Syn)n(tactically)18 b(deterministic)g(CTRSs)h(are)e(a)h(natural)g (generalization)e(of)i(normal)g(CTRSs.)523 4671 y(F)-7 b(or)22 b(example,)f(the)i(Fib)r(onacci)f(system)g Fp(R)1861 4683 y Fo(f)7 b(ib)1979 4671 y Fu(is)22 b(syn)n(tactically)f (deterministic.)h(The)h(pro)r(of)523 4770 y(of)28 b(the)g(next)f(lemma) h(is)f(straigh)n(tforw)n(ard;)e(see)i([Ohl99)o(].)p 523 4839 473 4 v 546 4893 a Fd(2)606 4924 y Ft(A)e Fa(c)l(onstructor)31 b(term)26 b Ft(is)h(a)f(term)e(without)i(de\014ned)f(sym)n(b)r(ols.)p eop %%Page: 13 13 13 12 bop 523 448 a Fh(Lemma)30 b(24.)40 b Fm(The)35 b(tr)l(ansforme)l(d)f(system)f Fn(U)9 b Fu(\()p Fp(R)p Fu(\))35 b Fm(of)f(a)g(syntactic)l(al)t(ly)h(deterministic)g(3-)523 548 y(CTRS)30 b Fp(R)g Fm(is)g(non-overlapping)i(if)e Fp(R)h Fm(is)f(non-overlapping.)523 712 y Fu(Lemma)25 b(24)f(can)h(b)r(e)h(re\014ned)f(to)g(demand)g(only)g(exactly)f(what)i (is)f(required)f(b)n(y)h(the)h(pro)r(of.)523 812 y(F)-7 b(or)32 b(instance,)h(the)g(3-CTRS)g Fp(R)g Fu(need)g(not)g(b)r(e)h (syn)n(tactically)e(deterministic;)h(it)g(is)g(suf-)523 911 y(\014cien)n(t)e(to)f(demand)g(that)h(no)f(left-hand)h(side)f Fn(l)2044 923 y Fj(1)2112 911 y Fu(of)g(a)g(rule)g(from)g Fp(R)h Fu(o)n(v)n(erlaps)d(a)i(term)g Fn(t)3377 923 y Fo(i)523 1011 y Fu(of)k(another)f(rule)g Fn(l)1132 1023 y Fj(2)1202 1011 y Fp(!)h Fn(r)1356 1023 y Fj(2)1427 1011 y Fp(\()f Fn(s)1582 1023 y Fj(1)1653 1011 y Fp(!)g Fn(t)1799 1023 y Fj(1)1837 1011 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2060 1023 y Fo(k)2134 1011 y Fp(!)34 b Fn(t)2281 1023 y Fo(k)2355 1011 y Fu(from)g Fp(R)p Fu(.)g(On)g(the)g(other)f(hand,)523 1111 y(the)j(example)g(b)r(efore)g(Def.)g(23)f(sho)n(ws)g(that)h(the)h (lemma)f(do)r(es)f(not)h(hold)g(for)f(strongly)523 1210 y(deterministic)28 b(CTRSs)f(\(see)h([ALS94)o(])g(for)f(a)g (de\014nition)h(of)g(this)g(notion\).)648 1312 y(Next)18 b(w)n(e)g(will)h(pro)n(v)n(e)e(a)h(mo)r(dularit)n(y)f(result)h(for)g (hierarc)n(hical)e(com)n(binations)i(of)g(CTRSs)523 1412 y(with)24 b(extra)e(v)-5 b(ariables)22 b(on)h(the)h(righ)n(t-hand)e (sides)h(of)h(the)f(rules.)g(Let)h(us)f(\014rst)g(review)g(some)523 1511 y(formal)31 b(de\014nitions.)h(Let)g Fp(R)g Fu(b)r(e)g(a)g(CTRS)g (and)f Fp(D)k Fu(b)r(e)d(the)g(set)g(of)g(its)g(de\014ned)g(sym)n(b)r (ols.)523 1611 y(As)g(de\014ned)h(in)g([KR95)n(,Mar95)o(],)f(the)h(dep) r(endency)g(relation)e Fp(\027)2588 1623 y Fo(d)2659 1611 y Fu(on)h Fp(D)j Fu(is)d(the)g(smallest)523 1710 y(quasi-order)19 b(satisfying)h Fn(f)32 b Fp(\027)1450 1722 y Fo(d)1511 1710 y Fn(g)24 b Fu(whenev)n(er)c(there)g(is)h(a)g (rule)g Fn(l)j Fp(!)f Fn(r)j Fp(\()d Fn(s)2801 1722 y Fj(1)2861 1710 y Fp(!)g Fn(t)2997 1722 y Fj(1)3035 1710 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)3258 1722 y Fo(k)3322 1710 y Fp(!)523 1810 y Fn(t)553 1822 y Fo(k)617 1810 y Fp(2)23 b(R)29 b Fu(suc)n(h)e(that)h Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))c(=)f Fn(f)36 b Fu(and)28 b Fn(g)d Fp(2)f(D)30 b Fu(o)r(ccurs)d(in)h(one)f(of)g(the)h(terms)g Fn(s)3004 1822 y Fj(1)3041 1810 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)3265 1822 y Fo(k)3305 1810 y Fn(;)g(r)r Fu(.)523 1910 y(If)30 b Fp(R)678 1922 y Fj(1)746 1910 y Fu(and)f Fp(R)979 1922 y Fj(2)1047 1910 y Fu(form)g(a)h(hierarc)n(hical)d(com)n(bination,)j (then)g(the)g(set)g(of)g(de\014ned)g(sym)n(b)r(ols)523 2009 y Fp(D)587 2021 y Fj(2)655 2009 y Fu(of)h Fp(R)823 2021 y Fj(2)891 2009 y Fu(is)g(split)g(in)n(to)f(t)n(w)n(o)g(sets)h Fp(D)1729 1979 y Fj(1)1727 2030 y(2)1794 2009 y Fu(=)d Fp(f)p Fn(f)22 b Fp(j)14 b Fn(f)37 b Fp(2)28 b(D)2254 2021 y Fj(2)2292 2009 y Fn(;)14 b(f)36 b Fp(\027)2471 2021 y Fo(d)2538 2009 y Fn(g)30 b Fu(for)d(some)g Fn(g)k Fp(2)e(D)3162 2021 y Fj(1)3199 2009 y Fp(g)h Fu(and)523 2109 y Fp(D)589 2079 y Fj(2)587 2130 y(2)650 2109 y Fu(=)22 b Fp(D)f(n)d(D)948 2079 y Fj(1)946 2130 y(2)986 2109 y Fu(.)27 b(Krishna)g(Rao)g([KR95)n(])h(pro)n(v)n(ed)e(the)i(follo)n (wing)f(theorem.)523 2275 y Fh(Theorem)j(25.)41 b Fm(L)l(et)25 b Fp(R)1301 2287 y Fj(1)1364 2275 y Fm(and)i Fp(R)1592 2287 y Fj(2)1655 2275 y Fm(form)f(a)g(hier)l(ar)l(chic)l(al)j(c)l (ombination)d(of)h(unc)l(onditional)523 2375 y(TRSs.)38 b(Supp)l(ose)h Fp(R)1182 2387 y Fj(2)1258 2375 y Fm(is)g(a)f(pr)l(op)l (er)i(extension)d(of)j Fp(R)2254 2387 y Fj(1)2291 2375 y Fm(,)f(i.e.,)h(every)f(rule)g Fn(l)h Fp(!)e Fn(r)j Fp(2)e(R)3367 2387 y Fj(2)523 2474 y Fm(satis\014es)34 b(the)f(fol)t(lowing)j(c)l(ondition:)f(F)-6 b(or)34 b(every)g(subterm)f Fn(t)g Fm(of)i Fn(r)r Fm(,)f(if)h Fn(r)r(oot)p Fu(\()p Fn(t)p Fu(\))c Fp(2)f(D)3202 2444 y Fj(1)3200 2495 y(2)3273 2474 y Fm(and)523 2574 y Fn(r)r(oot)p Fu(\()p Fn(t)p Fu(\))f Fp(\027)860 2586 y Fo(d)926 2574 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))p Fm(,)34 b(then)e Fn(t)g Fm(do)l(es)h(not)f(c)l (ontain)h(symb)l(ols)g(fr)l(om)f Fp(D)2674 2586 y Fj(1)2732 2574 y Fp([)21 b(D)2874 2544 y Fj(1)2872 2595 y(2)2943 2574 y Fm(exc)l(ept)32 b(at)h(the)523 2674 y(r)l(o)l(ot)j(p)l(osition.) i(If)e(b)l(oth)g Fp(R)1387 2686 y Fj(1)1461 2674 y Fm(and)h Fp(R)1699 2686 y Fj(2)1772 2674 y Fm(ar)l(e)g(innermost)e(terminating,) i(then)f Fp(R)3063 2686 y Fj(1)3124 2674 y Fp([)23 b(R)3272 2686 y Fj(2)3346 2674 y Fm(is)523 2773 y(innermost)30 b(terminating)f(as)i(wel)t(l.)523 2937 y Fu(The)e(com)n(bination)f(of)h (Lemma)g(24,)f(Prop)r(osition)f(22,)h(and)h(Theorem)f(25)g(yields)h (the)g(fol-)523 3037 y(lo)n(wing)e(theorem.)523 3222 y Fh(Theorem)j(26.)41 b Fm(L)l(et)31 b Fp(R)1307 3234 y Fj(1)1377 3222 y Fm(and)h Fp(R)1610 3234 y Fj(2)1680 3222 y Fm(b)l(e)g(quasi-r)l(e)l(ductive)g(non-overlapping)i(syntactic)l (al)t(ly)523 3321 y(deterministic)45 b(3-CTRSs.)g(Their)h(hier)l(ar)l (chic)l(al)h(c)l(ombination)e Fp(R)2697 3333 y Fj(1)2764 3321 y Fp([)29 b(R)2918 3333 y Fj(2)3000 3321 y Fm(is)45 b(a)g(quasi-)523 3421 y(r)l(e)l(ductive)37 b(non-overlapping)h (syntactic)l(al)t(ly)g(deterministic)f(3-CTRS)g(as)g(wel)t(l)g(pr)l (ovide)l(d)523 3521 y(that)30 b(every)g(rule)g Fn(l)25 b Fp(!)e Fn(r)i Fp(\()e Fn(s)1444 3533 y Fj(1)1505 3521 y Fp(!)g Fn(t)1641 3533 y Fj(1)1678 3521 y Fn(;)14 b(:)g(:)g(:)g(;)g(s) 1902 3533 y Fo(k)1965 3521 y Fp(!)23 b Fn(t)2101 3533 y Fo(k)2172 3521 y Fm(in)30 b Fp(R)2344 3533 y Fj(2)2411 3521 y Fm(satis\014es:)555 3695 y(1.)42 b(neither)30 b Fn(l)i Fm(nor)d(one)h(of)h(the)f(terms)f Fn(t)1812 3707 y Fj(1)1849 3695 y Fn(;)14 b(:)g(:)g(:)g(;)g(t)2064 3707 y Fo(k)2134 3695 y Fm(c)l(ontains)30 b(a)g(symb)l(ol)h(fr)l(om)f Fp(D)3065 3707 y Fj(1)3102 3695 y Fm(,)523 3868 y(If)g(some)g Fn(s)861 3880 y Fo(j)896 3868 y Fm(,)g Fu(1)23 b Fp(\024)g Fn(j)28 b Fp(\024)22 b Fn(k)g Fu(+)c(1)p Fm(,)29 b(wher)l(e)i Fn(s)1771 3880 y Fo(k)q Fj(+1)1919 3868 y Fu(=)22 b Fn(r)r Fm(,)31 b(c)l(ontains)f(a)g(symb)l(ol)h(fr)l(om)f Fp(D)3032 3880 y Fj(1)3088 3868 y Fp([)18 b(D)3227 3837 y Fj(1)3225 3888 y(2)3265 3868 y Fm(,)555 4042 y(2.)42 b(then)28 b(every)g(subterm)f Fn(t)g Fm(of)h Fn(s)1568 4054 y Fo(j)1631 4042 y Fm(with)g Fn(r)r(oot)p Fu(\()p Fn(t)p Fu(\))d Fp(2)e(D)2221 4012 y Fj(1)2219 4063 y(2)2286 4042 y Fm(and)28 b Fn(r)r(oot)p Fu(\()p Fn(t)p Fu(\))d Fp(\027)2778 4054 y Fo(d)2839 4042 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))k Fm(do)l(es)f(not)664 4142 y(c)l(ontain)i(symb)l(ols)g(fr)l(om)h Fp(D)1523 4154 y Fj(1)1578 4142 y Fp([)19 b(D)1718 4112 y Fj(1)1716 4162 y(2)1786 4142 y Fm(exc)l(ept)29 b(at)h(the)g(r)l(o)l (ot)f(p)l(osition,)j(and)555 4243 y(3.)42 b(none)30 b(of)g(the)g(terms) g Fn(s)1375 4255 y Fo(j)s Fj(+1)1494 4243 y Fn(;)14 b(:)g(:)g(:)f(;)h (s)1717 4255 y Fo(k)1758 4243 y Fn(;)g(s)1834 4255 y Fo(k)q Fj(+1)1988 4243 y Fm(c)l(ontain)30 b(an)g Fn(f)i Fp(2)23 b(D)2619 4213 y Fj(1)2617 4264 y(2)2686 4243 y Fm(with)31 b Fn(f)g Fp(\027)3004 4255 y Fo(d)3066 4243 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))p Fm(.)523 4426 y(Pr)l(o)l(of.)43 b Fu(The)28 b(com)n(bined)f(system)g Fn(U)9 b Fu(\()p Fp(R)1768 4438 y Fj(1)1806 4426 y Fu(\))18 b Fp([)g Fn(U)9 b Fu(\()p Fp(R)2097 4438 y Fj(2)2135 4426 y Fu(\))23 b(=)g Fn(U)9 b Fu(\()p Fp(R)2446 4438 y Fj(1)2502 4426 y Fp([)18 b(R)2645 4438 y Fj(2)2683 4426 y Fu(\))28 b(is)f(non-o)n(v)n (erlapping)523 4526 y(b)r(ecause)37 b(the)h(TRSs)g Fn(U)9 b Fu(\()p Fp(R)1399 4538 y Fj(1)1436 4526 y Fu(\))38 b(and)g Fn(U)9 b Fu(\()p Fp(R)1846 4538 y Fj(2)1883 4526 y Fu(\))38 b(are)f(non-o)n(v)n(erlapping)e(b)n(y)i(Lemma)g(24)g(and)523 4625 y(ev)n(ery)c(rule)g Fn(l)i Fp(!)e Fn(r)j Fp(\()e Fn(s)1326 4637 y Fj(1)1396 4625 y Fp(!)f Fn(t)1542 4637 y Fj(1)1580 4625 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)1803 4637 y Fo(k)1877 4625 y Fp(!)34 b Fn(t)2024 4637 y Fo(k)2098 4625 y Fu(in)g Fp(R)2271 4637 y Fj(2)2343 4625 y Fu(satis\014es:)f (neither)g Fn(l)j Fu(nor)d(one)g(of)523 4725 y(the)h(terms)f Fn(t)939 4737 y Fj(1)976 4725 y Fn(;)14 b(:)g(:)g(:)g(t)1154 4737 y Fo(k)1228 4725 y Fu(con)n(tains)33 b(a)g(sym)n(b)r(ol)f(from)i Fp(D)2191 4737 y Fj(1)2228 4725 y Fu(.)f(F)-7 b(or)33 b(the)h(same)f(reason,)f Fp(R)3160 4737 y Fj(1)3220 4725 y Fp([)22 b(R)3367 4737 y Fj(2)523 4825 y Fu(is)37 b(non-o)n(v)n (erlapping)c(and)k(syn)n(tactically)f(deterministic.)h(W)-7 b(e)37 b(claim)f(that)h Fn(U)9 b Fu(\()p Fp(R)3164 4837 y Fj(2)3202 4825 y Fu(\))37 b(is)f(a)523 4924 y(prop)r(er)30 b(extension)g(of)g Fn(U)9 b Fu(\()p Fp(R)1427 4936 y Fj(1)1465 4924 y Fu(\).)31 b(Since)g(innermost)f(termination)g(is)h(mo) r(dular)f(for)g(prop)r(er)p eop %%Page: 14 14 14 13 bop 523 448 a Fu(extensions)28 b(b)n(y)g(Theorem)g(25,)g(it)h (then)g(follo)n(ws)e(from)h(Prop)r(osition)f(22)h(that)h Fp(R)3082 460 y Fj(1)3138 448 y Fp([)20 b(R)3283 460 y Fj(2)3349 448 y Fu(is)523 548 y(quasi-reductiv)n(e.)648 649 y(In)42 b(order)f(to)h(pro)n(v)n(e)e(the)j(claim,)f(consider)f Fn(\032)47 b Fu(:)h Fn(l)g Fp(!)g Fn(r)i Fp(\()d Fn(s)2728 661 y Fj(1)2813 649 y Fp(!)g Fn(t)2973 661 y Fj(1)3010 649 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)3234 661 y Fo(k)3322 649 y Fp(!)523 749 y Fn(t)553 761 y Fo(k)626 749 y Fp(2)33 b(R)784 761 y Fj(2)855 749 y Fu(and)g(its)g(transformation)f Fn(U)9 b Fu(\()p Fn(\032)p Fu(\).)33 b(If)h Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))f Fp(2)g(D)2460 719 y Fj(2)2458 770 y(2)2498 749 y Fu(,)g(then)h(none)f(of)g(the)g(terms)523 849 y Fn(s)562 861 y Fj(1)599 849 y Fn(;)14 b(:)g(:)g(:)g(;)g(s)823 861 y Fo(k)863 849 y Fn(;)g(s)939 861 y Fo(k)q Fj(+1)1064 849 y Fu(\(=)41 b Fn(r)r Fu(\))e(con)n(tain)e(a)h(sym)n(b)r(ol)g(from)f Fp(D)2260 861 y Fj(1)2323 849 y Fp([)26 b(D)2470 819 y Fj(1)2468 869 y(2)2546 849 y Fu(and)38 b(ev)n(ery)f(rule)g(in)i Fn(U)9 b Fu(\()p Fn(\032)p Fu(\))523 948 y(satis\014es)31 b(the)g(prop)r(er)g(extension)g(condition.)g(Th)n(us,)g(supp)r(ose)g Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))f Fp(2)g(D)2985 918 y Fj(1)2983 969 y(2)3054 948 y Fu(and)h(let)h Fn(j)5 b Fu(,)523 1048 y(1)23 b Fp(\024)f Fn(j)28 b Fp(\024)23 b Fn(k)9 b Fu(+)d(1,)21 b(b)r(e)h(the)g(smallest)f(index)g(suc)n(h)g (that)h Fn(s)2202 1060 y Fo(j)2259 1048 y Fu(con)n(tains)e(a)h(sym)n(b) r(ol)g(from)g Fp(D)3173 1060 y Fj(1)3217 1048 y Fp([)6 b(D)3344 1018 y Fj(1)3342 1069 y(2)3382 1048 y Fu(.)523 1148 y(Ev)n(ery)35 b(rule)h Fn(l)i Fp(!)g Fn(U)1196 1108 y Fo(\032)1187 1170 y Fj(1)1234 1148 y Fu(\()p Fn(s)1305 1160 y Fj(1)1343 1148 y Fn(;)14 b(:)g(:)g(:)f Fu(\),)37 b Fn(U)1648 1108 y Fo(\032)1639 1170 y Fj(1)1686 1148 y Fu(\()p Fn(t)1748 1160 y Fj(1)1786 1148 y Fn(;)14 b(:)g(:)g(:)f Fu(\))38 b Fp(!)g Fn(U)2190 1108 y Fo(\032)2181 1170 y Fj(2)2228 1148 y Fu(\()p Fn(s)2299 1160 y Fj(2)2336 1148 y Fn(;)14 b(:)g(:)g(:)g Fu(\),)37 b Fn(:)14 b(:)g(:)f Fu(,)37 b Fn(U)2812 1108 y Fo(\032)2803 1171 y(j)s Fi(\000)p Fj(2)2922 1148 y Fu(\()p Fn(t)2984 1160 y Fo(j)s Fi(\000)p Fj(2)3105 1148 y Fn(;)14 b(:)g(:)g(:)f Fu(\))38 b Fp(!)523 1247 y Fn(U)589 1207 y Fo(\032)580 1270 y(j)s Fi(\000)p Fj(1)700 1247 y Fu(\()p Fn(s)771 1259 y Fo(j)s Fi(\000)p Fj(1)891 1247 y Fn(;)14 b(:)g(:)g(:)f Fu(\))27 b(v)-5 b(acuously)25 b(satis\014es)g(the)i(prop)r(er)e(extension)g(condition.) h(Observ)n(e)f(that)523 1347 y Fn(U)589 1307 y Fo(\032)580 1370 y(j)665 1347 y Fp(\027)730 1359 y Fo(d)806 1347 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))38 b(can)e(only)f(hold)i(if)g(one)f (of)g(the)h(terms)f Fn(s)2409 1359 y Fo(j)s Fj(+1)2528 1347 y Fn(;)14 b(:)g(:)g(:)f(;)h(s)2751 1359 y Fo(k)2792 1347 y Fn(;)g(s)2868 1359 y Fo(k)q Fj(+1)3029 1347 y Fu(con)n(tains)35 b(a)523 1457 y(sym)n(b)r(ol)d Fn(f)42 b Fu(with)33 b Fn(f)40 b Fp(\027)1235 1469 y Fo(d)1305 1457 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))34 b(\(whic)n(h)f(further)f (implies)h Fn(f)40 b Fp(2)32 b(D)2659 1427 y Fj(1)2657 1478 y(2)2697 1457 y Fu(\).)h(This,)f(ho)n(w)n(ev)n(er,)f(is)523 1557 y(imp)r(ossible)i(b)r(ecause)f(of)h(assumption)f(\(3\).)h(No)n(w)f (the)h(rewrite)f(rule)h Fn(U)2818 1517 y Fo(\032)2809 1580 y(j)s Fi(\000)p Fj(1)2928 1557 y Fu(\()p Fn(t)2990 1569 y Fo(j)s Fi(\000)p Fj(1)3110 1557 y Fn(;)14 b(:)g(:)g(:)g Fu(\))32 b Fp(!)523 1656 y Fn(U)589 1616 y Fo(\032)580 1680 y(j)627 1656 y Fu(\()p Fn(s)698 1668 y Fo(j)734 1656 y Fn(;)14 b(:)g(:)g(:)f Fu(\))34 b(satis\014es)e(the)i(prop)r(er)e (extension)h(condition)g(b)n(y)f(assumption)h(\(2\))g(b)r(ecause)523 1767 y(for)h(ev)n(ery)g(subterm)h Fn(t)g Fu(of)f Fn(U)1447 1727 y Fo(\032)1438 1790 y(j)1486 1767 y Fu(\()p Fn(s)1557 1779 y Fo(j)1592 1767 y Fn(;)14 b(:)g(:)g(:)f Fu(\))36 b(with)f Fn(r)r(oot)p Fu(\()p Fn(t)p Fu(\))i Fp(2)e(D)2439 1737 y Fj(1)2437 1787 y(2)2512 1767 y Fu(and)g Fn(r)r(oot)p Fu(\()p Fn(t)p Fu(\))h Fp(\027)3025 1779 y Fo(d)3099 1767 y Fn(U)3165 1727 y Fo(\032)3156 1790 y(j)s Fi(\000)p Fj(1)3310 1767 y Fu(w)n(e)523 1866 y(ha)n(v)n(e)27 b Fn(r)r(oot)p Fu(\()p Fn(t)p Fu(\))f Fp(\027)1049 1878 y Fo(d)1111 1866 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\).)j(The)f (remaining)f(rules)h Fn(U)2228 1826 y Fo(\032)2219 1889 y(j)2266 1866 y Fu(\()p Fn(t)2328 1878 y Fo(j)2363 1866 y Fn(;)14 b(:)g(:)g(:)g Fu(\))24 b Fp(!)g Fn(U)2740 1826 y Fo(\032)2731 1889 y(j)s Fj(+1)2850 1866 y Fu(\()p Fn(s)2921 1878 y Fo(j)s Fj(+1)3040 1866 y Fn(;)14 b(:)g(:)g(:)g Fu(\),)28 b Fn(:)14 b(:)g(:)g Fu(,)523 1966 y Fn(U)589 1926 y Fo(\032)580 1991 y(k)627 1966 y Fu(\()p Fn(t)689 1978 y Fo(k)730 1966 y Fn(;)g(:)g(:)g(:)g Fu(\))36 b Fp(!)g Fn(r)i Fu(satisfy)c(the)i(prop)r(er)e(extension)h(condition)g(b) r(ecause)f(of)i(assumption)523 2066 y(\(3\))28 b(and)f(the)h(resultan)n (t)f(fact)h(that)g Fn(U)1720 2026 y Fo(\032)1711 2089 y(i)1781 2066 y Fp(6\027)1846 2078 y Fo(d)1907 2066 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\))h(for)e(ev)n(ery)g Fn(j)c Fu(+)18 b(1)23 b Fp(\024)f Fn(i)h Fp(\024)g Fn(k)s Fu(.)523 2247 y(As)c(an)g(example,)f(consider)g(the)h(quasi-reductiv)n (e)f(systems)g Fp(R)2468 2259 y Fj(+)2547 2247 y Fu(=)k Fp(f)p Fu(0)q(+)q Fn(y)j Fp(!)e Fn(y)s(;)14 b(s)p Fu(\()p Fn(x)p Fu(\))q(+)q Fn(y)26 b Fp(!)523 2347 y Fn(s)p Fu(\()p Fn(x)20 b Fu(+)f Fn(y)s Fu(\))p Fp(g)29 b Fu(and)g Fp(R)1125 2359 y Fo(f)7 b(ib)1221 2347 y Fu(.)30 b(Since)f Fp(R)1562 2359 y Fj(+)1647 2347 y Fu(and)g Fp(R)1880 2359 y Fo(f)7 b(ib)2005 2347 y Fu(meet)30 b(the)f(requiremen)n(ts)f(of)h(Theorem)g (26,)523 2447 y(their)f(hierarc)n(hical)d(com)n(bination)i Fp(R)1710 2459 y Fj(+)1784 2447 y Fp([)19 b(R)1928 2459 y Fo(f)7 b(ib)2051 2447 y Fu(is)28 b(quasi-reductiv)n(e)e(as)h(w)n (ell.)648 2548 y(Condition)h(\(1\))h(in)g(Theorem)g(26)f(guaran)n(tees) e(that)j(the)h(system)e Fn(U)9 b Fu(\()p Fp(R)2918 2560 y Fj(1)2956 2548 y Fu(\))20 b Fp([)f Fn(U)9 b Fu(\()p Fp(R)3250 2560 y Fj(2)3288 2548 y Fu(\))29 b(is)523 2648 y(non-o)n(v)n(erlapping.)h(The)j(follo)n(wing)g(example)f(sho)n(ws)g (that)i(condition)e(\(2\))i(is)f(necessary)-7 b(.)523 2747 y(The)27 b(quasi-reductiv)n(e)e(non-o)n(v)n(erlapping)g(syn)n (tactically)g(deterministic)j(3-CTRSs)e Fp(R)3280 2759 y Fj(1)3340 2747 y Fu(=)523 2847 y Fp(f)p Fn(a)c Fp(!)i Fn(b)p Fp(g)19 b Fu(and)i Fp(R)1060 2859 y Fj(2)1120 2847 y Fu(=)i Fp(f)p Fn(f)9 b Fu(\()p Fn(x;)14 b(x)p Fu(\))23 b Fp(!)g Fn(c)g Fp(\()h Fn(f)9 b Fu(\()p Fn(a;)14 b(b)p Fu(\))22 b Fp(!)h Fn(c)p Fp(g)d Fu(form)g(a)g(hierarc)n(hical)f (com)n(bination.)523 2947 y(If)g Fp(R)667 2959 y Fj(1)705 2947 y Fp([R)830 2959 y Fj(2)886 2947 y Fu(w)n(ere)f(quasi-reductiv)n (e)f(w.r.t.)h(an)g(order)g Fp(\037)p Fu(,)g(then)h Fn(f)9 b Fu(\()p Fn(b;)14 b(b)p Fu(\))22 b Fp(\037)2758 2959 y Fo(st)2841 2947 y Fn(f)9 b Fu(\()p Fn(a;)14 b(b)p Fu(\))23 b Fp(\037)g Fn(f)9 b Fu(\()p Fn(b;)14 b(b)p Fu(\))523 3046 y(w)n(ould)33 b(hold,)h(but)g(this)f(con)n(tradicts)g(the)h (irre\015exivit)n(y)e(of)h Fp(\037)2507 3058 y Fo(st)2567 3046 y Fu(.)h(Finally)-7 b(,)34 b(w)n(e)f(exemplify)523 3146 y(the)g(necessit)n(y)e(of)i(condition)f(\(3\).)g Fp(R)1722 3158 y Fj(1)1792 3146 y Fu(and)g Fp(R)2028 3158 y Fj(3)2097 3146 y Fu(=)f Fp(f)p Fn(c)f Fp(!)h Fn(d)g Fp(\()g Fn(a)g Fp(!)g Fn(b;)14 b(c)30 b Fp(!)h Fn(d)p Fp(g)h Fu(form)g(a)523 3245 y(hierarc)n(hical)f(com)n(bination.)i(Here) g Fn(s)1717 3257 y Fj(1)1787 3245 y Fu(=)g Fn(a)g Fu(con)n(tains)f(a)h (sym)n(b)r(ol)g(from)h Fp(D)2925 3257 y Fj(1)2995 3245 y Fu(and)g Fn(s)3202 3257 y Fj(2)3272 3245 y Fu(=)e Fn(c)523 3345 y Fu(con)n(tains)i Fn(c)i Fp(2)f(D)1084 3315 y Fj(1)1082 3366 y(2)1157 3345 y Fu(with)h Fn(c)f Fp(\027)1490 3357 y Fo(d)1564 3345 y Fn(r)r(oot)p Fu(\()p Fn(l)r Fu(\).)i(If)e Fp(R)2024 3357 y Fj(1)2085 3345 y Fp([)24 b(R)2234 3357 y Fj(3)2307 3345 y Fu(w)n(ere)34 b(quasi-reductiv)n(e)f(w.r.t.)i(an)523 3445 y(order)26 b Fp(\037)p Fu(,)i(then)g Fn(c)23 b Fp(\037)1169 3457 y Fo(st)1252 3445 y Fn(c)k Fu(w)n(ould)h(hold,)f(but)h Fp(\037)1982 3457 y Fo(st)2070 3445 y Fu(is)g(irre\015exiv)n(e.)523 3720 y Fq(6)112 b(Related)37 b(W)-9 b(ork)523 3928 y Fu(One)26 b(of)g(the)h(anon)n(ymous)e(referees)g(p)r(oin)n(ted)h(out)h (that)f(a)g(transformation)f(similar)g(to)i(the)523 4028 y(one)i(from)g(De\014nition)h(6)f(w)n(as)f(indep)r(enden)n(tly)i(found) g(b)n(y)f(Marc)n(hiori)e([Mar97)n(,)j(Def.)g(4.1].)523 4127 y(Our)20 b(transformation)f(di\013ers)i(only)f(sligh)n(tly)g(from) h(Marc)n(hiori's:)d(in)j(the)g(sequence)g Fp(V)7 b Fn(ar)r Fu(\()p Fn(l)r Fu(\),)523 4227 y Fp(E)g(V)g Fn(ar)r Fu(\()p Fn(t)777 4239 y Fj(1)815 4227 y Fu(\))p Fn(;)14 b(:)g(:)g(:)g(;)g Fp(E)7 b(V)g Fn(ar)r Fu(\()p Fn(t)1286 4239 y Fo(i)1314 4227 y Fu(\))29 b(ev)n(ery)e(v)-5 b(ariable)28 b(o)r(ccurs)f(exactly)h (once)g(whic)n(h)g(is)h(not)f(the)h(case)523 4327 y(in)21 b(the)g(sequence)g(V)-9 b(AR\()p Fn(l)r(;)14 b(t)1389 4339 y Fj(1)1426 4327 y Fn(;)g(:)g(:)g(:)g(;)g(t)1641 4339 y Fo(i)1668 4327 y Fu(\))21 b(from)g([Mar97)n(,)g(Def.)h(4.1].)e (The)h(results)f(of)h(the)g(pap)r(er)523 4426 y(at)32 b(hand,)f(ho)n(w)n(ev)n(er,)f(are)g(completely)i(di\013eren)n(t)f(from) h(the)g(ones)f(rep)r(orted)f(in)i(the)g(tec)n(h-)523 4526 y(nical)20 b(rep)r(ort)f([Mar97)n(],)h(except)g(for)f(one:)h(Prop) r(osition)e(7)i(is)g(akin)f(to)h([Mar97)n(,)g(Lemma)g(4.6].)523 4725 y Fh(Ac)m(kno)m(wledgemen)m(ts:)28 b Fu(I)j(thank)e(Mic)n(hael)h (Han)n(us)g(for)f(the)h(\(email\))h(discussion)e(whic)n(h)523 4825 y(led)40 b(to)g(the)h(dev)n(elopmen)n(t)e(of)h(the)h (transformation)d Fn(U)9 b Fu(.)40 b(I)g(am)g(also)f(grateful)g(to)h (Aart)523 4924 y(Middeldorp)27 b(and)h(the)g(anon)n(ymous)e(referees)g (for)h(their)h(commen)n(ts.)p eop %%Page: 15 15 15 14 bop 523 448 a Fq(References)523 637 y Ft([A)n(G99])54 b(T.)28 b(Arts)f(and)f(J.)i(Giesl.)40 b(T)-6 b(ermination)27 b(of)h(term)e(rewriting)j(using)e(dep)r(endency)f(pairs.)811 728 y Fa(The)l(or)l(etic)l(al)j(Computer)g(Scienc)l(e)p Ft(,)e(1999.)36 b(T)-6 b(o)26 b(app)r(ear.)523 819 y([ALS94])42 b(J.)28 b(Av)n(enhaus)e(and)g(C.)i(Lor)-9 b(\023)-30 b(\020a-S\023)-38 b(aenz.)39 b(On)27 b(conditional)h(rewrite)g(systems) e(with)h(extra)811 910 y(v)l(ariables)e(and)f(deterministic)h(logic)h (programs.)32 b(In)24 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)29 b(of)d(the)h(5th)g(Interna-)811 1002 y(tional)32 b(Confer)l(enc)l(e)h (on)g(L)l(o)l(gic)f(Pr)l(o)l(gr)l(amming)h(and)f(A)n(utomate)l(d)i(R)l (e)l(asoning)p Ft(,)e(v)n(olume)811 1093 y(822)j(of)h Fa(L)l(e)l(ctur)l(e)h(Notes)g(in)e(A)n(rti\014cial)h(Intel)t(ligenc)l (e)p Ft(,)f(pages)g(215{229,)i(Berlin,)f(1994.)811 1184 y(Springer-V)-6 b(erlag.)523 1275 y([BK86])56 b(J.A.)28 b(Bergstra)g(and)f(J.W.)h(Klop.)40 b(Conditional)28 b(rewrite)g(rules:) g(Con\015uence)f(and)g(ter-)811 1367 y(mination.)34 b Fa(Journal)28 b(of)f(Computer)i(and)f(System)h(Scienc)l(es)p Ft(,)e(32\(3\):323{362,)j(1986.)523 1457 y([BN98])58 b(F.)25 b(Baader)h(and)e(T.)i(Nipk)n(o)n(w.)33 b Fa(T)-6 b(erm)27 b(R)l(ewriting)g(and)g(A)n(l)t(l)f(That)p Ft(.)33 b(Cam)n(bridge)25 b(Univ)n(er-)811 1549 y(sit)n(y)g(Press,)i(1998.)523 1640 y([Der87])47 b(N.)28 b(Dersho)n(witz.)43 b(T)-6 b(ermination)28 b(of)h(rewriting.)44 b Fa(Journal)31 b(of)e(Symb)l(olic)h(Computation)p Ft(,)811 1731 y(3\(1\):69{116,)f (1987.)523 1822 y([Gan91])43 b(H.)32 b(Ganzinger.)53 b(Order-sorted)31 b(completion:)h(The)g(man)n(y-sorted)e(w)n(a)n(y)-6 b(.)52 b Fa(The)l(or)l(etic)l(al)811 1913 y(Computer)29 b(Scienc)l(e)p Ft(,)e(89:3{32,)h(1991.)523 2004 y([Gra95])44 b(B.)19 b(Gramlic)n(h.)24 b(Abstract)19 b(relations)h(b)r(et)n(w)n(een) f(restricted)h(termination)f(and)f(con\015uence)811 2095 y(prop)r(erties)26 b(of)h(rewrite)f(systems.)34 b Fa(F)-6 b(undamenta)29 b(Informatic)l(ae)p Ft(,)e(24:3{23,)h(1995.)523 2186 y([Han94])43 b(M.)25 b(Han)n(us.)31 b(The)25 b(in)n(tegration)g (of)g(functions)g(in)n(to)f(logic)i(programming:)e(F)-6 b(rom)23 b(theory)811 2277 y(to)j(practice.)35 b Fa(The)28 b(Journal)g(of)f(L)l(o)l(gic)h(Pr)l(o)l(gr)l(amming)p Ft(,)f(19&20:583{628,)k(1994.)523 2368 y([Kap87])43 b(S.)29 b(Kaplan.)45 b(Simplifying)29 b(conditional)h(term)e(rewriting)j (systems:)e(Uni\014cation,)g(ter-)811 2460 y(mination)37 b(and)f(con\015uence.)68 b Fa(Journal)39 b(of)e(Symb)l(olic)h (Computation)p Ft(,)h(4\(3\):295{334,)811 2551 y(1987.)523 2642 y([K)n(O92])52 b(M.)22 b(Kurihara)g(and)g(A.)g(Oh)n(uc)n(hi.)27 b(Mo)r(dularit)n(y)22 b(of)h(simple)e(termination)h(of)h(term)d (rewrit-)811 2733 y(ing)h(systems)f(with)h(shared)g(constructors.)28 b Fa(The)l(or)l(etic)l(al)c(Computer)h(Scienc)l(e)p Ft(,)d(103:273{)811 2824 y(282,)27 b(1992.)523 2915 y([KR95])53 b(M.R.K.)32 b(Krishna)f(Rao.)50 b(Mo)r(dular)32 b(pro)r(ofs)h(for)f(completeness)f (of)g(hierarc)n(hical)i(term)811 3007 y(rewriting)27 b(systems.)34 b Fa(The)l(or)l(etic)l(al)29 b(Computer)f(Scienc)l(e)p Ft(,)f(151:487{512,)k(1995.)523 3097 y([Mar95])44 b(M.)24 b(Marc)n(hiori.)32 b(Unra)n(v)n(elings)24 b(and)f(ultra-prop)r(erties.) 31 b(T)-6 b(ec)n(hnical)25 b(Rep)r(ort)e(8,)h(Dept.)f(of)811 3189 y(Pure)j(and)f(Applied)g(Mathematics,)h(Univ)n(ersit)n(y)f(of)h(P) n(ado)n(v)l(a,)h(Italy)-6 b(,)25 b(1995.)523 3280 y([Mar96])44 b(M.)35 b(Marc)n(hiori.)62 b(Unra)n(v)n(elings)34 b(and)g(ultra-prop)r (erties.)61 b(In)34 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)f(the)g(5th) 811 3371 y(International)c(Confer)l(enc)l(e)g(on)f(A)n(lgebr)l(aic)h (and)f(L)l(o)l(gic)g(Pr)l(o)l(gr)l(amming)p Ft(,)f(v)n(olume)e(1139)811 3462 y(of)e Fa(L)l(e)l(ctur)l(e)i(Notes)h(in)d(Computer)i(Scienc)l(e)p Ft(,)e(pages)g(107{121,)i(Berlin,)e(1996.)g(Springer-)811 3554 y(V)-6 b(erlag.)523 3644 y([Mar97])44 b(M.)32 b(Marc)n(hiori.)54 b(On)31 b(deterministic)g(conditional)i(rewriting.)54 b(Computation)31 b(Struc-)811 3736 y(tures)26 b(Group,)f(Memo)h(405,)h (MIT)f(Lab)r(oratory)h(for)f(Computer)f(Science,)h(1997.)523 3827 y([Mid93])43 b(A.)35 b(Middeldorp.)61 b(Mo)r(dular)36 b(prop)r(erties)f(of)h(conditional)g(term)d(rewriting)j(systems.)811 3918 y Fa(Information)27 b(and)h(Computation)p Ft(,)g (104\(1\):110{158,)i(1993.)523 4009 y([MZ97])53 b(A.)24 b(Middeldorp)h(and)f(H.)g(Zan)n(tema.)32 b(Simple)24 b(termination)g(of)h(rewrite)g(systems.)32 b Fa(The-)811 4100 y(or)l(etic)l(al)c(Computer)h(Scienc)l(e)p Ft(,)e (175\(1\):127{158,)j(1997.)523 4191 y([Ohl94])46 b(E.)33 b(Ohlebusc)n(h.)56 b Fa(Mo)l(dular)34 b(Pr)l(op)l(erties)j(of)d(Comp)l (osable)h(T)-6 b(erm)34 b(R)l(ewriting)h(Systems)p Ft(.)811 4282 y(PhD)25 b(thesis,)i(Univ)n(ersit\177)-38 b(at)25 b(Bielefeld,)j(1994.)523 4373 y([Ohl99])46 b(E.)27 b(Ohlebusc)n(h.)36 b(T)-6 b(ransforming)27 b(conditional)h(rewrite)f(systems)f(with)h (extra)f(v)l(ariables)811 4464 y(in)n(to)32 b(unconditional)g(systems.) 52 b(In)31 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)36 b(of)d(the)h(6th)g (International)g(Confer-)811 4556 y(enc)l(e)e(on)f(L)l(o)l(gic)h(for)f (Pr)l(o)l(gr)l(amming)h(and)f(A)n(utomate)l(d)i(R)l(e)l(asoning)p Ft(,)e(Lecture)e(Notes)h(in)811 4647 y(Arti\014cial)c(In)n(telligence,) h(Berlin,)g(1999.)g(Springer-V)-6 b(erlag.)35 b(T)-6 b(o)26 b(app)r(ear.)523 4738 y([Zan94])43 b(H.)25 b(Zan)n(tema.)33 b(T)-6 b(ermination)25 b(of)h(term)e(rewriting:)j(In)n(terpretation)e (and)g(t)n(yp)r(e)f(elimina-)811 4829 y(tion.)35 b Fa(Journal)28 b(of)f(Symb)l(olic)h(Computation)p Ft(,)f(17:23{50,)i(1994.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF