%!PS-Adobe-2.0 %%Creator: dvips 5.516 Copyright 1986, 1993 Radical Eye Software %%Title: paper.dvi %%CreationDate: Mon Jun 12 11:28:06 1995 %%Pages: 41 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o paper.ps paper.dvi %DVIPSSource: TeX output 1995.06.12:1124 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D }B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale false def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (/amnt/fauna/users/c1t0d0s6/pistaff/enno/TEX/JSC/paper.dvi) @start /Fa 5 83 df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b 1 51 df<7FFF00FFFF80C00180C00180C00180C00180C00180C00180C00180C0 0180C00180C00180C00180C00180C00180FFFF80FFFF8011117D9217>50 D E /Fc 2 70 df0 D<03F00C381010300030003C000F801800 20004000C000C020E0C07F000D0E7E8D10>69 D E /Fd 55 122 df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e 1 51 df<7FFFFFC0FFFFFFE0C0000060C00000 60C0000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060C00000 60C0000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060C00000 60C0000060C0000060C0000060FFFFFFE0FFFFFFE01B1B7B9E25>50 D E /Ff 2 51 df<18F818181818181818181818FF080D7D8C0E>49 D<3E00418080C0C0C000C000C0018003000400084030407F80FF800A0D7E8C0E>I E /Fg 9 93 df<0000700001F00003C0000780000E00001C0000380000700000700000F0 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00001C00001C00001C0000380000700000600 000E0000380000700000C000007000003800000E000006000007000003800001C00001C0 0001C00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00000E00000F000007000007000003800001C 00000E000007800003C00001F000007014637B811F>26 D<0018007800F001E003C00780 0F001F001E003E003C007C007C007800F800F800F800F800F800F800F800F800F800F800 F800F800F800F800F800F800F800F800F800F800F800F800F8000D25707E25>56 D58 D<007C007C007C007C007C007C007C007C007C007C007C007C00 7C007C007C007C007C007C007C007C007C007C007C007C00F800F800F800F001F001E003 E003C0078007000E001C003800F000C000F00038001C000E000700078003C003E001E001 F000F000F800F800F8007C007C007C007C007C007C007C007C007C007C007C007C007C00 7C007C007C007C007C007C007C007C007C007C007C0E4D798025>60 D62 D83 D<001FE000007FF80001FFFE0007E01F800F 8007C01E0001E01C0000E038000070380000707000003870000038E000001CE000001CE0 00001CE000001CE000001CE000001CE000001CE000001CE000001CE000001CE000001CE0 00001CE000001CE000001CE000001CE000001CE000001CE000001CE000001CE000001CE0 00001CE000001CE000001CE000001CE000001CE000001CE000001CE000001CE000001CE0 00001CE000001C1E2A7E7F23>I91 D<0000FF8000000007FFF00000001FFFFC0000007F007F000000F8 000F800003E00003E00007C00001F0000700000070000E00000038001E0000003C001C00 00001C00380000000E00380000000E00700000000700700000000700700000000700E000 00000380E00000000380E00000000380E00000000380E00000000380E00000000380E000 00000380E00000000380E00000000380E00000000380E00000000380E00000000380E000 00000380E00000000380E00000000380E00000000380E00000000380E00000000380E000 00000380E00000000380E00000000380E00000000380E00000000380E00000000380E000 00000380E00000000380E00000000380E00000000380E00000000380E00000000380E000 00000380E00000000380E00000000380E00000000380E00000000380E00000000380E000 00000380E00000000380E00000000380E00000000380E00000000380E00000000380293A 7E7F2E>I E /Fh 16 118 df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i 4 28 df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j 2 51 df<0C003C00CC000C000C000C000C000C000C000C000C000C000C000C00 0C00FF8009107E8F0F>49 D<1F00618040C08060C0600060006000C00180030006000C00 102020207FC0FFC00B107F8F0F>I E /Fk 37 122 df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l 10 115 df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m 1 79 df78 D E /Fn 16 94 df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o 2 60 df<7FFFFCFFFFFEC00006C00006C00006C0 0006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C0 0006C00006C00006C00006FFFFFEFFFFFE17177C991F>50 D<0000000080000000004007 C00000400FF00000201838000010700C000008E0060007FF4003001FFF00018038080000 E0E01000007FC02000001F004000000000400000000080280E81902A>59 D E /Fp 27 120 df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q 5 122 df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r 10 62 df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s 40 123 df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t 43 122 df<400020C000606000C03001801803000C0600060C0003180001B00000E00000 E00001B000031800060C000C06001803003001806000C0C0006040002013147A9320>2 D<01800180018001804182F18F399C0FF003C003C00FF0399CF18F418201800180018001 8010127E9215>I<03C00FF01C38300C60066006C003C003C003C00360066006300C1C38 0FF003C010107E9115>14 D<007FFF8001FFFF80078000000E0000001800000030000000 300000006000000060000000C0000000C0000000C0000000C0000000C0000000C0000000 C000000060000000600000003000000030000000180000000E0000000780000001FFFF80 007FFF80000000000000000000000000000000000000000000000000000000007FFFFF80 7FFFFF8019227D9920>18 D<7FFF00007FFFC0000000F0000000380000000C0000000600 000006000000030000000300000001800000018000000180000001800000018000000180 000001800000030000000300000006000000060000000C00000038000000F0007FFFC000 7FFF000000000000000000000000000000000000000000000000000000000000FFFFFF00 FFFFFF0019227D9920>I<000001800000078000001E00000078000001E0000007800000 1E00000078000001E00000078000001E00000078000000E0000000780000001E00000007 80000001E0000000780000001E0000000780000001E0000000780000001E000000078000 000180000000000000000000000000000000000000000000000000000000007FFFFF00FF FFFF8019227D9920>II<007FFF8003FFFF80078000000C0000001800000030000000300000 006000000060000000C0000000C0000000C0000000C0000000C0000000C0000000C00000 00C000000060000000600000003000000030000000180000000E0000000780000001FFFF 80007FFF80191A7D9620>26 D<80000000C0000000C00000006000000060000000600000 0030000000180000000E0000000780000001E00000007F0000001FFF80001FFF80007F00 0001E00000078000000E00000018000000300000006000000060000000C0000000C00000 00C000000019197D9520>31 D<0200000000040000000004000000000800000000100000 00002000000000FFFFFFFFF0FFFFFFFFF020000000001000000000080000000004000000 0004000000000200000000240E7D902A>I<000000040000000002000000000200000000 010000000000800000000040FFFFFFFFF8FFFFFFFFF80000000040000000008000000001 00000000020000000002000000000400250E7E902A>I<03000300030003000300030003 000300030003000300030003000300030003000300030003000300030003000300030003 0003000300030003008304631813200B4007800300030001000E257D9C15>35 D<020000080004000004000400000400080000020010000001002000000080FFFFFFFFF0 FFFFFFFFF020000000801000000100080000020004000004000400000400020000080024 0E7D902A>I<00060000000006000000000C000000000C00000000180000000030000000 007FFFFFF000FFFFFFF8038000000007000000003C00000000F0000000003C000000000E 00000000030000000001C000000000FFFFFFF8007FFFFFF000300000000018000000000C 000000000C0000000006000000000600000025187E952A>40 D<00000300000000030000 00000180000000018000000000C00000000060007FFFFFF000FFFFFFF8000000000E0000 0000070000000001E0000000007800000001E0000000038000000006000000001C00FFFF FFF8007FFFFFF0000000006000000000C000000001800000000180000000030000000003 000025187E952A>I<030180030180030180030180030180030180030180030180030180 030180030180030180030180030180030180030180030180030180030180030180030180 030180C30186E3018E3B01B81F01F00701C003018001830000C60000C600006C00006C00 00380000380000100000100017257F9C19>43 D<07C001E01FF0078030780C00201C1800 400E3000400760008007E0008003C0008003C0008001E0008001E0008003F00040037000 40063800600C1C0020180F0018F007E007C001E01B127E9120>47 D<07E0003F000FF800F180183E018040200F03002040078400104003CC00108001D80008 8000F000088000F00008800078000880007800088000DC000840019E001040010F001020 06078020100C03E0C00C7800FF8007E0003F0025127E912A>49 D<007FF801FFF8078000 0E0000180000300000300000600000600000C00000C00000C00000FFFFF8FFFFF8C00000 C00000C000006000006000003000003000001800000E000007800001FFF8007FF8151A7D 961C>I<0000600000600000C00000C00001800001800001800003000003000006000006 00000C00000C0000180000180000180000300000300000600000600000C00000C0000180 000180000300000300000300000600000600000C00000C00001800001800003000003000 00300000600000600000C0000040000013287A9D00>54 DI<0008001803D80C301838303C303C706E606660666066E0C7E0C7E0 C7E0C7E187E187E187E187E187E307E307E307E307E60766066606760E3C0C3C0C1C180C 301BC01800180010237E9F15>59 D<00000020000000E0000001E0000001E0000003E000 0003E0000005E0000005E0000009E0000009E0000011E0000031E0000021E0000061E000 0041E0000081E0000181E0000101E0000301E0000601E0000401E0000FFFF0001FFFF000 3000F0003000F0006000F060C000F0718000F87F8000F8FF00007E7E00007C3C0000001F 20809D21>65 D<00603F0001E1FF8007E30FC001EC07C001D803C001F803C003F0038003 E0038003E0070003E0040003C0180007C060000783C000078FF8000793FC0007007E000F 003E000F001F000F001F000E000F000E000F001E000F001C000E001C000E003C001C0039 8018003BC0200077E0C00063FF0000C1FC00001A1E7F9C1B>I<0003F0000FF80030F800 40780180780300700700700E00E00E00E01C01801C000038000038000078000078000070 0000700000F00000F00000F00000F00000F00000F00020F80060F801C07C01807E03003F 04001FF8000FE000151E7F9C16>I<003FFE0001FFFFC0071E1FE0081C03F0381C01F870 3C00F8603C007CC03C007C003C003C0038003C0038003C0078003C0078003C0078003800 7000380070007800F0007000E0006000E000E001E001C001C0018001C0020003C0040003 80180003806000070380000FFE00001FF000001E1C809B20>I<0001FFFF000FFFFF0038 E00E0041E00C01C1E0000381C0000301C0000603C0000003C00000038000000380000007 80000007000000070000000FFFC0000FFF00000E0000001C0000001C0000003C00000038 000000380000007000000070000030E0000070C00000FD800000FF0000003C000000201D 7F9B1E>70 D<0001800000100003800000300003800000300003800000600007C00000E0 0007C00001E00005C00003E00005C00007E00005C00007E00005E0000DC00009E00019C0 0008E00033C00008E00073C00010F000E3C00010F001C3C00010700383C00020700303C0 0020780603800020780C038000403818078000403C38078000403C70078000801CE00780 00801FC0078001001F80078001000F00078062000E000780FE00040007C0FC00000007F0 FC00000003C07800000000002C1F7F9C32>77 D<003FFF0001FFFFC0071E0FE0081C03F0 381C01F8703C00F8603C0078C03C0078003C00780038007000380070007800E0007800E0 007801C0007001800070020000F00C0000E0300000E7C00001EF000001E0000001C00000 01C0000003800000038000000780000007000000070000000E0000000C0000001D1E809B 1D>80 D<003FFF000001FFFFE000071E0FF000081C01F800381C00F800703C007800603C 007800C03C007800003C0070000038007000003800E000007800C0000078018000007802 000000700C00000071F0000000F3F0000000E1F0000000E0F8000001E0F8000001C07C00 0001C07C000003C03C000003803E018003803E070007801F060007001F880006000FF000 0C0007C000211D809B23>82 D<0001FC000007FF0000083F8000300F800070078000E007 8000E0060001E0040001E0000001F0000001F8000000FC0000007E0000003F8000001FE0 000007F0000001F8000000FC0008007C0038003C0070003C0070003C00F0003800F00038 00F8007000F80060007C0080007F0300003FFC00000FF00000191E7F9C19>I<00000004 00FFFFF803FFFFE00400E0001C00E0003801C0003001C0007003C000E003C000C0038000 0003800000078000000780000007000000070000000F0000000E0000000E0000001E0000 001E0000001C0000001C00000038000000380000007000000060000000C0000007FF0000 0FFC00001E1D7F9C17>I<78000CFC001E3E001F1E001F0F000F0F000707000307000207 80020780020380040380040380080380180380300380300380600380C003818003838003 8700038E00039C0003B80003F00003E00007C000078000060000040000181E7E9B19>86 D<400002C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C000 06C00006C00006C00006C00006C00006C00006C0000660000C60000C3000181C00700F01 E003FF8000FE00171A7E981C>91 D<00FE0003FF800F01E01C007030001860000C60000C C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006 C00006C00006C00006C00006C00006C00006400002171A7E981C>I<400002C00006C000 06C00006C00006C01806C01806C01806C01806C01806C01806C7FFC6C7FFC6C01806C018 06C01806C01806C01806C0180660000C60000C3000181C00700F01E003FF8000FE00171A 7E981C>I<003C00E001C001800380038003800380038003800380038003800380038003 800380030007001C00F0001C000700030003800380038003800380038003800380038003 80038003800380018001C000E0003C0E297D9E15>102 DI<008001800300030003000600060006000C000C000C00180018001800300030 003000600060006000C000C0006000600060003000300030001800180018000C000C000C 0006000600060003000300030001800080092A7C9E10>III 110 D<02000700070007000700070007000200020002007270FFF8727002000200070007 000700070007000700070007000700070007000700070007000700070007000200020002 000200020002000D267E9C12>121 D E /Fu 46 123 df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v 36 122 df<78FCFCFCFC7806067D850D>46 D<00600001E0000FE000FFE000F3E00003E00003E00003E00003E00003E00003E00003E0 0003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E0 0003E0007FFF807FFF80111B7D9A18>49 D<07F8001FFE00383F80780FC0FC07C0FC07E0 FC03E0FC03E07803E00007E00007C00007C0000F80001F00001E0000380000700000E000 0180600300600600600800E01FFFC03FFFC07FFFC0FFFFC0FFFFC0131B7E9A18>I<03F8 001FFE003C1F003C0F807C07C07E07C07C07C03807C0000F80000F80001E00003C0003F8 00001E00000F800007C00007C00007E03007E07807E0FC07E0FC07E0FC07C0780F80781F 001FFE0007F800131B7E9A18>I<000180000380000780000F80001F80003F80006F8000 CF80008F80018F80030F80060F800C0F80180F80300F80600F80C00F80FFFFF8FFFFF800 0F80000F80000F80000F80000F80000F8001FFF801FFF8151B7F9A18>I<1801801FFF00 1FFE001FFC001FF8001FC00018000018000018000018000019F8001E0E00180F80100780 0007C00007E00007E00007E07807E0F807E0F807E0F807C0F007C0600F80381F001FFE00 07F000131B7E9A18>I<007E0003FF000781800F03C01E07C03C07C03C03807800007800 00F80000F8F800FB0E00FA0780FC0380FC03C0F803E0F803E0F803E0F803E07803E07803 E07803C03C03C03C07801E0F0007FE0003F800131B7E9A18>I<6000007FFFE07FFFE07F FFC07FFF807FFF80E00300C00600C00C00C0180000300000300000600000E00000E00001 E00001C00003C00003C00003C00003C00007C00007C00007C00007C00007C00007C00003 8000131C7D9B18>I<78FCFCFCFC7800000000000078FCFCFCFC7806127D910D>58 D<00038000000380000007C0000007C0000007C000000FE000000FE000001FF000001BF0 00001BF0000031F8000031F8000061FC000060FC0000E0FE0000C07E0000C07E0001803F 0001FFFF0003FFFF8003001F8003001F8006000FC006000FC00E000FE00C0007E0FFC07F FEFFC07FFE1F1C7E9B24>65 D<001FE02000FFF8E003F80FE007C003E00F8001E01F0000 E03E0000E03E0000607E0000607C000060FC000000FC000000FC000000FC000000FC0000 00FC000000FC000000FC0000007C0000607E0000603E0000603E0000C01F0000C00F8001 8007C0030003F80E0000FFFC00001FE0001B1C7D9B22>67 D73 D77 D80 D82 D86 D<0FF8001C1E003E0F803E07803E07C01C07C00007C0007FC007E7C0 1F07C03C07C07C07C0F807C0F807C0F807C0780BC03E13F80FE1F815127F9117>97 DI<03FC000E0E001C1F003C1F0078 1F00780E00F80000F80000F80000F80000F80000F800007800007801803C01801C03000E 0E0003F80011127E9115>I<000FF0000FF00001F00001F00001F00001F00001F00001F0 0001F00001F00001F001F9F00F07F01C03F03C01F07801F07801F0F801F0F801F0F801F0 F801F0F801F0F801F07801F07801F03C01F01C03F00F0FFE03F9FE171D7E9C1B>I<01FC 000F07001C03803C01C07801C07801E0F801E0F801E0FFFFE0F80000F80000F800007800 007C00603C00601E00C00F038001FC0013127F9116>I<007F0001E38003C7C00787C00F 87C00F83800F80000F80000F80000F80000F8000FFF800FFF8000F80000F80000F80000F 80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80007FF8007F F800121D809C0F>I<03F8F00E0F381E0F381C07303C07803C07803C07803C07801C0700 1E0F000E0E001BF8001000001800001800001FFF001FFFC00FFFE01FFFF07801F8F00078 F00078F000787000707800F01E03C007FF00151B7F9118>I<1E003F003F003F003F001E 00000000000000000000000000FF00FF001F001F001F001F001F001F001F001F001F001F 001F001F001F001F00FFE0FFE00B1E7F9D0E>105 D107 DIII<01FC000F07801C01C03C01E07800F07800F0F8 00F8F800F8F800F8F800F8F800F8F800F87800F07800F03C01E01E03C00F078001FC0015 127F9118>II114 D<1FD830786018E018E018F000FF807FE07FF01FF807FC007CC01C C01CE01CE018F830CFC00E127E9113>I<0300030003000300070007000F000F003FFCFF FC1F001F001F001F001F001F001F001F001F001F0C1F0C1F0C1F0C0F08079803F00E1A7F 9913>II 119 D121 D E /Fw 76 128 df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x 86 127 df<000300000003000000078000000780 00000FC000000BC0000013E0000011E0000021F0000020F0000040F8000040780000807C 0000803C0001003E0001001E0002001F0002000F0004000F8004000780080007C0080003 C0100003E0100001E0200000F0200000F07FFFFFF8FFFFFFFCFFFFFFFC1E1D7E9C23>1 D<03FFF000003F0000001E0000001E0000001E000000FF8000039EE0000E1E38001C1E1C 00381E0E00781E0F00F01E0780F01E0780F01E0780F01E0780F01E0780F01E0780781E0F 00381E0E001C1E1C000E1E3800039EE00000FF8000001E0000001E0000001E0000003F00 0003FFF000191C7E9B1E>8 D<007E1F0001C1B1800303E3C00703C3C00E03C1800E01C0 000E01C0000E01C0000E01C0000E01C0000E01C000FFFFFC000E01C0000E01C0000E01C0 000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0 000E01C0000E01C0000E01C0000E01C0007F87FC001A1D809C18>11 D<007E0001C1800301800703C00E03C00E01800E00000E00000E00000E00000E0000FFFF C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E01C00E01C00E01C00E01C07F87F8151D809C17>I<007FC001C1C00303C00703C00E 01C00E01C00E01C00E01C00E01C00E01C00E01C0FFFFC00E01C00E01C00E01C00E01C00E 01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C07F CFF8151D809C17>I<003F07E00001C09C18000380F018000701F03C000E01E03C000E00 E018000E00E000000E00E000000E00E000000E00E000000E00E00000FFFFFFFC000E00E0 1C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C 000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C00 0E00E01C007FC7FCFF80211D809C23>I<1C1C3C3870C0800607779C15>19 D22 D<60F0F0F0F0F0F0F06060606060606060606060606000000000 0060F0F060041E7C9D0C>33 D<6060F0F0F8F86868080808080808101010102020404080 800D0C7F9C15>I<60F0F8680808081010204080050C7C9C0C>39 D<004000800100020006000C000C0018001800300030007000600060006000E000E000E0 00E000E000E000E000E000E000E000E000E000600060006000700030003000180018000C 000C00060002000100008000400A2A7D9E10>I<800040002000100018000C000C000600 060003000300038001800180018001C001C001C001C001C001C001C001C001C001C001C0 01C0018001800180038003000300060006000C000C00180010002000400080000A2A7E9E 10>I<000600000006000000060000000600000006000000060000000600000006000000 06000000060000000600000006000000060000FFFFFFE0FFFFFFE0000600000006000000 060000000600000006000000060000000600000006000000060000000600000006000000 060000000600001B1C7E9720>43 D<60F0F0701010101020204080040C7C830C>II<60F0F06004047C830C>I<03C00C301818300C300C700E60066006E0 07E007E007E007E007E007E007E007E007E007E007E007E00760066006700E300C300C18 180C3007E0101D7E9B15>48 D<030007003F00C700070007000700070007000700070007 00070007000700070007000700070007000700070007000700070007000F80FFF80D1C7C 9B15>I<07C01830201C400C400EF00FF80FF807F8077007000F000E000E001C001C0038 0070006000C00180030006010C01180110023FFE7FFEFFFE101C7E9B15>I<07E0183020 1C201C781E780E781E381E001C001C00180030006007E00030001C001C000E000F000F70 0FF80FF80FF80FF00E401C201C183007E0101D7E9B15>I<000C00000C00001C00003C00 003C00005C0000DC00009C00011C00031C00021C00041C000C1C00081C00101C00301C00 201C00401C00C01C00FFFFC0001C00001C00001C00001C00001C00001C00001C0001FFC0 121C7F9B15>I<300C3FF83FF03FC020002000200020002000200023E024302818301C20 0E000E000F000F000F600FF00FF00FF00F800E401E401C2038187007C0101D7E9B15>I< 00F0030C06040C0E181E301E300C700070006000E3E0E430E818F00CF00EE006E007E007 E007E007E007600760077006300E300C18180C3003E0101D7E9B15>I<4000007FFF807F FF007FFF0040020080040080040080080000100000100000200000600000400000C00000 C00001C00001800001800003800003800003800003800007800007800007800007800007 8000078000030000111D7E9B15>I<03E00C301008200C20066006600660067006780C3E 083FB01FE007F007F818FC307E601E600FC007C003C003C003C00360026004300C1C1007 E0101D7E9B15>I<03C00C301818300C700C600EE006E006E007E007E007E007E0076007 700F300F18170C2707C700060006000E300C780C78187010203030C00F80101D7E9B15> I<60F0F0600000000000000000000060F0F06004127C910C>I<60F0F060000000000000 0000000060F0F0701010101020204080041A7C910C>I<7FFFFFC0FFFFFFE00000000000 000000000000000000000000000000000000000000000000000000FFFFFFE07FFFFFC01B 0C7E8F20>61 D<0FE03038401CE00EF00EF00EF00E000C001C0030006000C00080018001 00010001000100010001000000000000000000000003000780078003000F1D7E9C14>63 D<000600000006000000060000000F0000000F0000000F00000017800000178000001780 000023C0000023C0000023C0000041E0000041E0000041E0000080F0000080F0000180F8 000100780001FFF80003007C0002003C0002003C0006003E0004001E0004001E000C001F 001E001F00FF80FFF01C1D7F9C1F>65 DI< 001F808000E0618001801980070007800E0003801C0003801C0001803800018078000080 7800008070000080F0000000F0000000F0000000F0000000F0000000F0000000F0000000 F0000000700000807800008078000080380000801C0001001C0001000E00020007000400 0180080000E03000001FC000191E7E9C1E>IIII<001F8080 00E0618001801980070007800E0003801C0003801C000180380001807800008078000080 70000080F0000000F0000000F0000000F0000000F0000000F0000000F000FFF0F0000F80 700007807800078078000780380007801C0007801C0007800E00078007000B8001801180 00E06080001F80001C1E7E9C21>III<1FFF00F8007800780078007800780078007800 78007800780078007800780078007800780078007800787078F878F878F878F0F040E021 C01F00101D7F9B15>IIIII<003F800000E0E0000380380007001C000E00 0E001C0007003C00078038000380780003C0780003C0700001C0F00001E0F00001E0F000 01E0F00001E0F00001E0F00001E0F00001E0F00001E0700001C0780003C0780003C03800 03803C0007801C0007000E000E0007001C000380380000E0E000003F80001B1E7E9C20> II82 D<07E0801C1980300580700380600180E00180E00080E00080E00080F00000F800007C00 007FC0003FF8001FFE0007FF0000FF80000F800007C00003C00001C08001C08001C08001 C0C00180C00180E00300D00200CC0C0083F800121E7E9C17>I<7FFFFFC0700F01C0600F 00C0400F0040400F0040C00F0020800F0020800F0020800F0020000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000001F800003FFFC001B1C7F9B1E> IIII91 D<08081010202040404040808080808080B0B0F8F8 787830300D0C7A9C15>II<0C0012002100408080400A057B 9B15>I<1FC000307000783800781C00301C00001C00001C0001FC000F1C00381C00701C 00601C00E01C40E01C40E01C40603C40304E801F870012127E9115>97 DI<07E00C301878307870306000E0 00E000E000E000E000E00060007004300418080C3007C00E127E9112>I<003F00000700 00070000070000070000070000070000070000070000070000070003E7000C1700180F00 300700700700600700E00700E00700E00700E00700E00700E00700600700700700300700 180F000C370007C7E0131D7E9C17>I<03E00C301818300C700E6006E006FFFEE000E000 E000E00060007002300218040C1803E00F127F9112>I<00F8018C071E061E0E0C0E000E 000E000E000E000E00FFE00E000E000E000E000E000E000E000E000E000E000E000E000E 000E000E000E007FE00F1D809C0D>I<00038003C4C00C38C01C3880181800381C00381C 00381C00381C001818001C38000C300013C0001000003000001800001FF8001FFF001FFF 803003806001C0C000C0C000C0C000C06001803003001C0E0007F800121C7F9215>II<18003C003C00180000000000000000 00000000000000FC001C001C001C001C001C001C001C001C001C001C001C001C001C001C 001C001C00FF80091D7F9C0C>I<00C001E001E000C00000000000000000000000000000 0FE000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E0 00E000E000E060E0F0C0F1C061803E000B25839C0D>IIIII<03F0000E1C00180600300300700380600180E001 C0E001C0E001C0E001C0E001C0E001C06001807003803003001806000E1C0003F0001212 7F9115>II<03C1000C3300180B00300F00700700700700 E00700E00700E00700E00700E00700E00700600700700700300F00180F000C370007C700 000700000700000700000700000700000700000700003FE0131A7E9116>II<1F9030704030C010C010E010F8007F803FE00FF000F880388018C018C018E010D060 8FC00D127F9110>I<04000400040004000C000C001C003C00FFE01C001C001C001C001C 001C001C001C001C001C101C101C101C101C100C100E2003C00C1A7F9910>IIII<7F8FF00F03800F030007020003840001C80001 D80000F00000700000780000F800009C00010E00020E000607000403801E07C0FF0FF815 12809116>II<7FFC70386038407040F040E041C003C003 8007000F040E041C043C0C380870087038FFF80E127F9112>III<1C043F0843F080E00E047D9B15>126 D E /Fy 24 122 df<00003FF001800003FFFE0380000FFFFF8780003FF007DF8000FF80 01FF8001FE00007F8003FC00003F8007F000001F800FF000000F801FE0000007801FE000 0007803FC0000007803FC0000003807FC0000003807F80000003807F8000000000FF8000 000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000 000000FF8000000000FF80000000007F80000000007F80000000007FC0000003803FC000 0003803FC0000003801FE0000003801FE0000007000FF00000070007F000000E0003FC00 001E0001FE00003C0000FF8000F800003FF007E000000FFFFFC0000003FFFF000000003F F8000029297CA832>67 D77 D80 D82 D<007F806003FFF0E007FFF9E00F80 7FE01F001FE03E0007E07C0003E07C0001E0FC0001E0FC0001E0FC0000E0FE0000E0FE00 00E0FF000000FFC000007FFE00007FFFE0003FFFFC001FFFFE000FFFFF8007FFFFC003FF FFE000FFFFE00007FFF000007FF000000FF8000007F8000003F8600001F8E00001F8E000 01F8E00001F8F00001F0F00001F0F80003F0FC0003E0FF0007C0FFE01F80F3FFFF00E0FF FE00C01FF0001D297CA826>I<7FFFFFFFFFC07FFFFFFFFFC07FFFFFFFFFC07F803FC03F C07E003FC007C078003FC003C078003FC003C070003FC001C0F0003FC001E0F0003FC001 E0E0003FC000E0E0003FC000E0E0003FC000E0E0003FC000E0E0003FC000E000003FC000 0000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC000 0000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC000 0000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC000 0000003FC0000000003FC0000000003FC00000007FFFFFE000007FFFFFE000007FFFFFE0 002B287EA730>I<01FF800007FFF0000F81F8001FC07E001FC07E001FC03F000F803F80 07003F8000003F8000003F8000003F80000FFF8000FFFF8007FC3F800FE03F803F803F80 3F003F807F003F80FE003F80FE003F80FE003F80FE003F807E007F807F00DF803F839FFC 0FFF0FFC01FC03FC1E1B7E9A21>97 DI<00003FF80000003FF80000003FF800000003F8 00000003F800000003F800000003F800000003F800000003F800000003F800000003F800 000003F800000003F800000003F800000003F800001FE3F80000FFFBF80003F03FF80007 E00FF8000FC007F8001F8003F8003F8003F8007F0003F8007F0003F8007F0003F800FF00 03F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F8007F0003 F8007F0003F8007F0003F8003F8003F8001F8003F8000F8007F80007C00FF80003F03BFF 8000FFF3FF80003FC3FF80212A7EA926>100 D<003FE00001FFF80003F07E0007C01F00 0F801F801F800F803F800FC07F000FC07F0007C07F0007E0FF0007E0FF0007E0FFFFFFE0 FFFFFFE0FF000000FF000000FF0000007F0000007F0000007F0000003F8000E01F8000E0 0FC001C007E0038003F81F0000FFFE00001FF0001B1B7E9A20>I<0007F0003FFC00FE3E 01F87F03F87F03F07F07F07F07F03E07F00007F00007F00007F00007F00007F00007F000 FFFFC0FFFFC0FFFFC007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 7FFF807FFF807FFF80182A7EA915>I<00FF81F003FFE7F80FC1FE7C1F80FC7C1F007C38 3F007E107F007F007F007F007F007F007F007F007F007F007F007F003F007E001F007C00 1F80FC000FC1F8001FFFE00018FF800038000000380000003C0000003E0000003FFFF800 1FFFFF001FFFFF800FFFFFC007FFFFE01FFFFFF03E0007F07C0001F8F80000F8F80000F8 F80000F8F80000F87C0001F03C0001E01F0007C00FC01F8003FFFE00007FF0001E287E9A 22>I<07000F801FC03FE03FE03FE01FC00F8007000000000000000000000000000000FF E0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00F E00FE00FE00FE00FE00FE0FFFEFFFEFFFE0F2B7DAA14>105 D108 DII<003FE00001FFFC0003F07E 000FC01F801F800FC03F800FE03F0007E07F0007F07F0007F07F0007F0FF0007F8FF0007 F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F87F0007F07F0007F03F800F E03F800FE01F800FC00FC01F8007F07F0001FFFC00003FE0001D1B7E9A22>II114 D<03FE300FFFF01E03F03800F0700070F00070F00070F80070FC0000FFE0007FFE007FFF 803FFFE01FFFF007FFF800FFF80003FC0000FC60007CE0003CF0003CF00038F80038FC00 70FF01E0F7FFC0C1FF00161B7E9A1B>I<00700000700000700000700000F00000F00000 F00001F00003F00003F00007F0001FFFF0FFFFF0FFFFF007F00007F00007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F03807F03807F03807 F03807F03807F03803F03803F87001F86000FFC0001F8015267FA51B>II119 D121 D E /Fz 1 50 df<00C003C0FFC0FFC003C003C003C003C003C003C003C0 03C003C003C003C003C003C003C003C003C003C003C07FFE7FFE0F187D9716>49 D E /FA 8 124 df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df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end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 100 197 a FB(J.)13 b(Symb)n(olic)f(Computation)h FA(\(1995\))h Fz(11)p FA(,)f(1{000)275 473 y Fy(Mo)r(dular)23 b(Prop)r(erties)f(of)h(Comp)r(osable)f(T)-6 b(erm)614 540 y(Rewriting)20 b(Systems)697 648 y Fx(ENNO)15 b(OHLEBUSCH)168 714 y FB(T)m(e)n(chnische)c(F)m(akult\177)-20 b(at,)11 b(Universit\177)-20 b(at)11 b(Bielefeld,)g(Postfach)h(100131,)f(33501)g (Bielefeld,)g(Germany)692 798 y(\(R)n(e)n(c)n(eive)n(d)g(12)i(June)f (1995\))p 224 897 1345 2 v 224 955 a Fw(In)e(this)g(pap)q(er)f(w)o(e)j (pro)o(v)o(e)d(sev)o(eral)g(new)h(mo)q(dularit)o(y)e(results)h(for)h (unconditiona)o(l)e(and)i(conditiona)o(l)224 997 y(term)j(rewriting)f (systems.)g(Most)h(of)h(the)f(kno)o(wn)g(mo)q(dularit)o(y)d(results)j (for)g(the)g(former)f(systems)224 1038 y(hold)f(for)h(disjoin)o(t)e(or) i(constructor)o(-sha)o(rin)o(g)d(com)o(binations)o(.)h(Here)h(w)o(e)i (fo)q(cus)e(on)h(a)f(more)g(general)224 1080 y(kind)h(of)i(com)o (binatio)o(n:)c(so-called)i(comp)q(osabl)o(e)f(systems.)g(As)k(far)d (as)i(conditiona)o(l)d(term)h(rewrit-)224 1121 y(ing)h(systems)e(are)i (concerned)o(,)e(all)h(kno)o(wn)h(mo)q(dularit)n(y)d(result)i(but)h (one)f(apply)f(only)i(to)f(disjoin)o(t)224 1163 y(systems.)h(Here)i(w)o (e)h(in)o(v)o(estigate)c(conditional)g(systems)h(whic)o(h)h(ma)o(y)g (share)g(constructors.)e(F)m(ur-)224 1204 y(thermore,)d(w)o(e)j(refute) e(a)h(conjecture)e(of)i(Middeldorp)e(\(1990,)h(1993\).)p 224 1246 V 735 1470 a Fv(1.)24 b(In)o(tro)q(ducti)o(on)100 1544 y Fx(T)m(erm)15 b(rewriting)h(has)h(applications)e(in)i(v)n (arious)e(\014elds)i(of)f(computer)g(science)j(suc)o(h)e(as)f(sym)o(b)q (olic)100 1594 y(computation,)d(functional)i(programming,)d(abstract)17 b(data)e(t)o(yp)q(e)i(sp)q(eci\014cations,)f(program)f(v)o(eri-)100 1644 y(\014cation,)f(program)h(syn)o(thesis,)h(and)f(automated)f (theorem)i(pro)o(ving.)e(In)i(an)f(outstanding)g(pap)q(er,)100 1694 y(Kn)o(uth)g(and)g(Bendix)h(\(1970\))e(describ)q(e)j(a)e (completion)e(pro)q(cedure)k(whic)o(h)e(can)g(often)h(b)q(e)f(success-) 100 1743 y(fully)f(used)j(to)f(transform)e(a)i(giv)o(en)f(set)i(of)e (equations)h(in)o(to)f(a)h(complete)f(term)g(rewriting)h(system)100 1793 y(\(TRS\))j(whic)o(h)h(de\014nes)h(the)g(same)d(equational)h (theory)m(.)g(Th)o(us)i(TRSs)e(pro)o(vide)h(an)f(op)q(erational)100 1843 y(mo)q(del)c(of)i(algebraic)f(sp)q(eci\014cations)j(of)d(abstract) i(data)f(t)o(yp)q(es.)h(Large)f(sp)q(eci\014cations,)h(ho)o(w)o(ev)o (er,)100 1893 y(m)o(ust)11 b(b)q(e)h(written)h(in)e(a)h(mo)q(dular)e(w) o(a)o(y)i(according)g(to)g(the)g(one)h(page)f(principle)f(of)h(Mark)g (Ardis:)g(\\A)100 1943 y(sp)q(eci\014cation)i(that)f(will)f(not)i (\014t)f(on)g(one)h(page)f(of)g(8)p Fu(:)p Fx(5)8 b Ft(\002)g Fx(11)13 b(inc)o(h)g(pap)q(er)h(cannot)g(b)q(e)g(understo)q(o)q(d".)141 1993 y(Mo)q(dularit)o(y)c(is)i(a)e(w)o(ell-kno)o(wn)g(programming)e (paradigm)h(in)i(computer)g(science.)h(Programmers)100 2042 y(should)j(design)g(their)g(programs)f(in)h(a)g(mo)q(dular)e(w)o (a)o(y)m(,)g(that)i(is,)g(as)g(a)g(com)o(bination)d(of)j(small)e(pro-) 100 2092 y(grams.)g(These)k(so-called)e(mo)q(dules)g(are)g(implemen)o (ted)f(separately)i(and)f(are)h(then)g(in)o(tegrated)g(to)100 2142 y(form)d(the)j(whole)f(program.)e(Since)j(TRSs)f(ha)o(v)o(e)g(imp) q(ortan)o(t)e(applications)i(in)f(computer)h(science,)100 2192 y(it)e(is)h({)g(not)f(only)h(from)e(a)h(theoretical)i(viewp)q(oin) o(t)e(but)h(also)f(from)g(a)g(practical)h(p)q(oin)o(t)g(of)f(view)h({)f (of)100 2242 y(utmost)d(imp)q(ortance)h(to)g(kno)o(w)g(under)h(whic)o (h)g(conditions)f(a)g(com)o(bined)f(system)h(inherits)h(desirable)100 2291 y(prop)q(erties)j(from)d(its)h(constituen)o(t)i(systems.)e(F)m(or) h(this)g(reason)g(mo)q(dular)e(asp)q(ects)j(of)e(term)g(rewrit-)100 2341 y(ing)18 b(ha)o(v)o(e)h(b)q(een)h(receiving)f(increasing)g(atten)o (tion.)f(A)h(prop)q(ert)o(y)h Ft(P)i Fx(of)c(TRSs)h(\(lik)o(e)f (con\015uence,)100 2391 y(termination)12 b(etc.\))j(is)g(called)f Fs(mo)n(dular)g Fx(if)g(whenev)o(er)i Ft(R)999 2397 y Fr(1)1032 2391 y Fx(and)e Ft(R)1148 2397 y Fr(2)1181 2391 y Fx(are)h(TRSs)g(b)q(oth)f(satisfying)g Ft(P)s Fx(,)100 2441 y(then)h(their)g(com)o(bined)e(system)h Ft(R)654 2447 y Fr(1)682 2441 y Ft([)c(R)755 2447 y Fr(2)788 2441 y Fx(also)k(satis\014es)h Ft(P)s Fx(.)f(The)h(kno)o(wledge)f(that) h(\(p)q(erhaps)h(un-)100 2491 y(der)e(certain)h(conditions\))f(a)f (prop)q(ert)o(y)i Ft(P)i Fx(is)d(mo)q(dular)e(facilitates)h(program)f (syn)o(thesis)j(b)q(ecause)h(it)100 2540 y(allo)o(ws)c(an)h(incremen)o (tal)f(dev)o(elopmen)o(t)g(of)h(programs.)e(On)j(the)g(other)f(hand,)g (it)g(pro)o(vides)g(a)g(divide)100 2590 y(and)g(conquer)h(approac)o(h)f (to)h(establishing)f(prop)q(erties)i(of)d(TRSs.)h(If)g(one)h(w)o(an)o (ts)f(to)g(kno)o(w)g(whether)100 2640 y(a)k(large)h(TRS)f(has)i(a)e (certain)i(mo)q(dular)d(prop)q(ert)o(y)j Ft(P)s Fx(,)e(then)i(this)f (system)g(can)g(b)q(e)g(decomp)q(osed)100 2715 y Fw(0747{7171)o(/90)o (/00)o(000)o(0)9 b(+)j(00)17 b($03.00/0)1199 2714 y(c)1190 2715 y Fq(\015)12 b Fw(1995)e(Academic)g(Press)h(Limited)p eop %%Page: 2 2 2 1 bop 100 197 a Fw(2)105 b(E.)12 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fx(in)o(to)f(small)f(subsystems)i(and)g(one)g (merely)f(has)h(to)g(c)o(hec)o(k)h(whether)h(eac)o(h)e(of)f(these)j (subsystems)e(has)100 349 y(prop)q(ert)o(y)i Ft(P)s Fx(.)141 399 y(As)g(all)e(in)o(teresting)i(prop)q(erties)h(are)f(in)e(general)i (not)f(mo)q(dular,)e(the)j(starting-p)q(oin)o(t)f(of)g(researc)o(h)100 448 y(w)o(ere)20 b Fs(disjoint)g(unions)p Fx(,)h(com)o(binations)c(of)j (TRSs)f(ha)o(ving)g(no)h(function)f(sym)o(b)q(ols)g(in)g(common.)100 498 y(T)m(o)o(y)o(ama)f(\(1987)p Fs(a)p Fx(\))i(pro)o(v)o(ed)h(that)g (con\015uence)i(is)e(mo)q(dular)e(for)i(disjoin)o(t)f(systems.)h(In)g (con)o(trast)100 548 y(to)d(that,)g(termination)e(and)j(completeness)g (lac)o(k)f(a)g(mo)q(dular)e(b)q(eha)o(vior)i(\(see)i(T)m(o)o(y)o(ama,) 15 b(1987)p Fs(b)p Fx(\).)100 598 y(Kurihara)g(and)h(Oh)o(uc)o(hi)f (\(1992\))g(in)o(v)o(estigated)h Fs(c)n(onstructor-sharing)g(systems)p Fx(;)f(constructors)i(are)100 648 y(function)i(sym)o(b)q(ols)e(that)j (do)f(not)g(o)q(ccur)h(at)f(the)h(ro)q(ot)f(p)q(osition)g(of)f(the)i (left-hand)f(side)g(of)g(an)o(y)100 697 y(rewrite)g(rule,)f(the)g (others)h(are)g(called)f(de\014ned)h(sym)o(b)q(ols.)d(Among)g(other)j (things,)f(they)g(sho)o(w)o(ed)100 747 y(that)12 b(con\015uence)i(is)e (not)g(mo)q(dular)e(for)i(constructor-sharing)h(systems.)f(Middeldorp)g (and)g(T)m(o)o(y)o(ama)100 797 y(\(1993\))k(in)o(tro)q(duced)i Fs(c)n(omp)n(osable)g(systems)e Fx(whic)o(h)h(ha)o(v)o(e)g(to)g(con)o (tain)f(all)g(rewrite)i(rules)f(that)g(de-)100 847 y(\014ne)e(a)f (de\014ned)i(sym)o(b)q(ol)c(whenev)o(er)k(that)f(sym)o(b)q(ol)d(is)j (shared.)g(The)g(authors,)f(ho)o(w)o(ev)o(er,)h(restricted)100 897 y(their)e(in)o(v)o(estigations)g(to)g(constructor)i(systems)f (\(where)h(no)e(prop)q(er)h(subterm)f(of)g(a)g(left-hand)g(side)100 946 y(of)j(a)h(rewrite)h(rule)g(is)f(allo)o(w)o(ed)e(to)i(con)o(tain)g (de\014ned)h(sym)o(b)q(ols\).)e(Their)h(main)e(result)j(states)h(that) 100 996 y(completeness)d(is)g(mo)q(dular)e(for)i(comp)q(osable)f (constructor)i(systems.)f(W)m(e)f(drop)h(the)h(constructor)100 1046 y(system)10 b(requiremen)o(t,)f(so)i(the)g(comp)q(osable)e (systems)h(w)o(e)h(consider)g(are)f(a)g(prop)q(er)i(generalization)d (of)100 1096 y(constructor-sharing)15 b(systems.)e(It)h(is)f(w)o(orth)o (while)g(to)h(in)o(v)o(estigate)f(com)o(binations)f(of)h(comp)q(osable) 100 1146 y(systems)j(b)q(ecause)i(they)f(corresp)q(ond)h(to)f(the)g (union)f(of)f(sp)q(eci\014cations)j(with)e(common)e(subparts)100 1196 y(whic)o(h)f(exist)i(in)e(most)g(sp)q(eci\014cation)h(languages.) 141 1245 y(The)e(title)g(of)f(this)h(pap)q(er)g(re\015ects)i(that)e (the)h(com)o(bination)c(of)i(comp)q(osable)g(systems)h(is)f(the)i(most) 100 1295 y(general)c(kind)g(of)g(com)o(bination)e(whic)o(h)j(will)e(b)q (e)i(in)o(v)o(estigated)f(here.)i(It)e(will)f(b)q(e)j(sho)o(wn)e(that)h (for)f(those)100 1345 y(systems)15 b(semi-completeness)g(is)g(mo)q (dular,)e(termination)h(is)h(mo)q(dular)f(for)h(la)o(y)o(er-preserving) h(and)100 1395 y(for)e(non-duplicating)g(systems,)g(completeness)i(is)f (mo)q(dular)e(for)i(o)o(v)o(erla)o(y)f(systems,)g(and)h(that)g(the)100 1445 y(simplifyi)o(ng)10 b(prop)q(ert)o(y)k(is)f(mo)q(dular)e(as)i(w)o (ell.)e(W)m(e)i(stress)i(the)e(fact)g(that)g(it)f(is)h(p)q(ossible)g (to)g(compute)100 1494 y(in)j(the)i(com)o(bined)e(system)h(of)f (pairwise)h(comp)q(osable)f(complete)h(systems.)f(More)i(precisely)m(,) f(the)100 1544 y(unique)i(normal)e(form)g(of)h(a)h(term)f(can)h(b)q(e)h (obtained)f(b)o(y)f(an)o(y)h(innermost)f(reduction)i(strategy)m(.)100 1594 y(Then)11 b(conditional)f(term)h(rewriting)h(systems)f(\(CTRSs\))h (are)f(studied.)h(The)g(rewrite)g(rules)g(of)f(those)100 1644 y(systems)e(ma)o(y)f(p)q(ossess)j(conditions,)e(and)g(suc)o(h)i(a) e(conditional)f(rewrite)i(rule)g(is)g(only)e(applicable)h(if)g(its)100 1694 y(conditions)h(are)h(ful\014lled.)e(W)m(e)h(fo)q(cus)h(on)f(the)h (most)f(prominen)o(t)f(kind)h(of)g(CTRSs,)g(the)h(so-called)g(join)100 1743 y(or)e(standard)h(systems.)f(It)g(should)h(b)q(e)g(p)q(oin)o(ted)f (out)g(that)h(conditional)e(term)h(rewriting)g(is)g(inheren)o(tly)100 1793 y(more)15 b(complicated)g(than)h(unconditional)f(term)h (rewriting.)g(So)g(it)g(is)g(not)g(surprising)g(that)h(most)100 1843 y(of)c(our)h(results)i(apply)d(only)g(to)h(constructor-sharing)i (systems)e(|)g(as)g(a)g(matter)f(of)h(fact,)f(up)h(un)o(til)100 1893 y(no)o(w)h(no)h(p)q(ositiv)o(e)f(mo)q(dularit)o(y)e(result)k(is)e (kno)o(wn)h(for)f(the)i(com)o(bination)c(of)i(comp)q(osable)g(CTRSs)100 1943 y(whic)o(h)c(ma)o(y)f(ha)o(v)o(e)h(extra)h(v)n(ariables)f(in)g (their)h(conditions.)e(Middeldorp)i(\(1990,)e(1993\))g(w)o(as)i(the)g (\014rst)100 1993 y(to)j(in)o(v)o(estigate)h(mo)q(dular)d(prop)q (erties)18 b(of)d(\(disjoin)o(t\))g(CTRSs.)g(Among)f(other)i(things,)f (he)h(sho)o(w)o(ed)100 2042 y(that)h(for)h(disjoin)o(t)f(conditional)f (term)h(rewriting)h(systems)g(con\015uence)i(and)d(semi-completeness) 100 2092 y(are)12 b(mo)q(dular)e(whereas)k(lo)q(cal)d(con\015uence)j (and)e(normalization)d(lac)o(k)j(a)f(mo)q(dular)g(b)q(eha)o(vior.)g(So) h(the)100 2142 y(b)q(est)18 b(one)g(can)g(hop)q(e)g(for)g(when)g (considering)g(constructor-sharing)h(CTRSs)e(is)h(the)g(mo)q(dularit)o (y)100 2192 y(of)12 b(semi-completeness)g(\(all)g(other)h(ab)q(o)o(v)o (e-men)o(tioned)f(prop)q(erties)i(cannot)f(b)q(e)g(mo)q(dular)e(for)i (those)100 2242 y(systems)e(since)h(they)g(already)e(fail)g(to)h(b)q(e) h(mo)q(dular)d(for)i(more)f(restricted)j(systems\).)f(W)m(e)e(pro)o(v)o (e)i(that)100 2291 y(semi-completeness)19 b(is)g(indeed)h(mo)q(dular)d (for)j(constructor-sharing)g(CTRSs.)f(Middeldorp)g(has)100 2341 y(also)12 b(sho)o(wn)h(that)g(termination)f(is)h(mo)q(dular)e(for) i(non-collapsing,)e(and)i(completeness)h(is)e(mo)q(dular)100 2391 y(for)g(non-duplicating)f(disjoin)o(t)g(CTRSs.)h(F)m(urthermore,)g (he)h(conjectured)h(\(see)g(Middeldorp,)e(1990,)100 2441 y(1993\))e(that)h(the)h(disjoin)o(t)e(union)g(of)g(t)o(w)o(o)h (terminating)e(join)h(CTRSs)h(is)g(terminating)e(if)i(one)g(of)f(them) 100 2491 y(con)o(tains)g(neither)i(collapsing)e(nor)h(duplicating)f (rules)h(and)g(the)g(other)h(is)e(con\015uen)o(t.)i(W)m(e)e(will)g (refute)100 2540 y(this)15 b(conjecture)h(b)o(y)f(a)g(simple)e(coun)o (terexample.)h(Moreo)o(v)o(er,)h(it)g(will)e(b)q(e)j(sho)o(wn)f(that)g (his)f(results)100 2590 y(also)g(hold,)h(m)o(utatis)f(m)o(utandis,)f (in)i(the)h(presence)i(of)d(shared)h(constructors.)h(W)m(e)e(p)q(oin)o (t)g(out)h(that)100 2640 y(our)e(pro)q(of)f(\(though)h(based)h(on)f (the)h(ideas)f(of)f(Middeldorp,)h(1993\))f(is)h(considerably)g(simpler) f(than)p eop %%Page: 3 3 3 2 bop 636 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)g(of)h(Comp)q (osable)e(T)m(erm)i(Rewriting)f(Systems)104 b(3)p 100 224 1595 2 v 100 299 a Fx(that)11 b(of)g(Middeldorp)h(\(1993\).)e(Then) i(w)o(e)g(in)o(v)o(estigate)g(\014nite)g(and)f(decreasing)i(CTRSs.)e (Since)h(these)100 349 y(systems)18 b(exactly)f(capture)i(the)g (\014niteness)g(of)e(recursiv)o(e)i(ev)n(aluation)d(of)i(conditions,)f (they)h(ha)o(v)o(e)100 399 y(b)q(een)d(studied)g(b)o(y)e(man)o(y)g (authors.)g(Our)i(main)d(result)j(in)e(this)i(con)o(text)f(states)i (that)e(it)f(is)h(p)q(ossible)100 448 y(to)h(compute)g(in)g(the)i(com)o (bined)d(system)i(of)f(decreasing,)h(con\015uen)o(t,)g(and)g(pairwise)f (constructor-)100 498 y(sharing)i(CTRSs.)h(Finally)m(,)d(it)i(is)h(sho) o(wn)g(that)g(the)h(related)f(simplifying)c(prop)q(ert)o(y)19 b(is)f(mo)q(dular,)100 548 y(ev)o(en)c(for)g(comp)q(osable)e(CTRSs.)141 598 y(Since)e(v)o(ery)h(man)o(y)c(new)k(mo)q(dularit)o(y)c(results)k (ha)o(v)o(e)e(b)q(een)i(published)f(recen)o(tly)m(,)g(w)o(e)g(cannot)g (render)100 648 y(a)g(detailed)h(accoun)o(t)h(of)e(those)i(here.)g (Instead,)f(the)h(reader)g(is)f(referred)h(to)f(Marc)o(hiori)g (\(1995\),)e(Ohle-)100 697 y(busc)o(h)16 b(\(1993,1995)p Fs(a,)e(b)p Fx(\))h(and)h(Gramlic)o(h)d(\(1994)p Fs(c)p Fx(\))i(for)g(recen)o(t)i(results)f(on)g Fs(disjoint)f Fx(\(conditional\))100 747 y(TRSs,)i(to)h(Dersho)o(witz)h(\(1994\),)e (Gramlic)o(h)e(\(1994)p Fs(a,)j(b)p Fx(\),)g(and)g(Ohlebusc)o(h)h (\(1994)p Fs(a)p Fx(\))f(whic)o(h)g(deal)100 797 y(with)d Fs(c)n(onstructor-sharing)g Fx(TRSs,)g(and)h(to)g(Kurihara)f(and)h(Oh)o (uc)o(hi)g(\(1995\))f(as)g(w)o(ell)g(as)h(Middel-)100 847 y(dorp)e(\(1994)p Fs(a)p Fx(\))g(whic)o(h)g(con)o(tain)f(results)j (for)e Fs(c)n(omp)n(osable)g Fx(\(conditional\))f(TRSs.)h(In)g(this)g (pap)q(er)h(w)o(e)100 897 y(do)c(not)h(in)o(v)o(estigate)g(so-called)f Fs(hier)n(ar)n(chic)n(al)h(c)n(ombinations)h Fx(of)e(rewrite)i(systems) f(but)g(refer)h(to)f(Sec-)100 946 y(tion)g(7)h(for)g(a)g(brief)g (discussion)h(of)e(this)i(related)g(w)o(ork.)e(The)i(pap)q(er)g(is)f (organized)g(as)g(follo)o(ws.)e(First)100 996 y(w)o(e)16 b(brie\015y)g(recall)f(the)i(basic)e(notions)h(of)f(term)g(rewriting.)g (In)h(Section)g(3)g(w)o(e)g(sp)q(ecify)g(the)g(di\013er-)100 1046 y(en)o(t)d(kinds)g(of)g(com)o(bination.)d(Then)j(the)h(basic)f (notions)g(of)g(comp)q(osable)f(systems)h(are)g(in)o(tro)q(duced.)100 1096 y(Section)f(5)f(con)o(tains)h(our)g(results)h(ab)q(out)f(comp)q (osable)f(TRSs,)g(while)g(Section)i(6)e(is)h(concerned)i(with)100 1146 y(constructor-sharing)19 b(CTRSs.)e(The)i(pap)q(er)f(is)g (concluded)h(with)f(a)f(brief)h(discussion)h(of)e(related)100 1196 y(w)o(ork)c(and)h(op)q(en)g(problems.)728 1320 y Fv(2.)24 b(Prelimi)o(nari)o(es)141 1395 y Fx(This)15 b(section)g(con)o(tains)g(a)f(concise)i(in)o(tro)q(duction)e(to)h(term) f(rewriting.)g(The)h(reader)h(is)f(referred)100 1445 y(to)e(the)i(surv)o(eys)g(of)e(Dersho)o(witz)i(and)e(Jouannaud)h (\(1990\))f(and)h(Klop)f(\(1992\))g(for)h(more)f(detail.)141 1494 y(A)18 b Fs(signatur)n(e)f Fx(is)g(a)h(coun)o(table)f(set)i Ft(F)i Fx(of)c Fs(function)i(symb)n(ols)e Fx(or)g Fs(op)n(er)n(ators)p Fx(,)g(where)h(ev)o(ery)h Fu(f)j Ft(2)100 1544 y(F)e Fx(is)d(asso)q(ciated)h(with)e(a)h(natural)f(n)o(um)o(b)q(er)g (denoting)h(its)g(arit)o(y)m(.)e(Nullary)h(op)q(erators)i(are)f(called) 100 1594 y Fs(c)n(onstants)p Fx(.)h(The)h(set)g Ft(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))19 b(of)f Fs(terms)f Fx(built)h(from)e(a)j(signature)f Ft(F)k Fx(and)d(a)f(coun)o(table)g (set)h(of)100 1644 y Fs(variables)14 b Ft(V)k Fx(with)c Ft(F)g(\\)9 b(V)17 b Fx(=)c Ft(;)h Fx(is)h(the)g(smallest)e(set)j(suc)o (h)f(that)g Ft(V)h(\022)e(T)c Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))15 b(and)f(if)g Fu(f)k Ft(2)12 b(F)18 b Fx(has)100 1694 y(arit)o(y)11 b Fu(n)g Fx(and)h Fu(t)327 1700 y Fr(1)345 1694 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)453 1700 y Fp(n)487 1694 y Ft(2)k(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(,)k(then)h Fu(f)t Fx(\()p Fu(t)843 1700 y Fr(1)863 1694 y Fu(;)7 b(:)g(:)g(:)t(;)g(t)970 1700 y Fp(n)992 1694 y Fx(\))12 b Ft(2)f(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\).)k(W)m(e)g(write)h Fu(f)17 b Fx(instead)12 b(of)f Fu(f)t Fx(\()i(\))100 1743 y(whenev)o(er)f Fu(f)17 b Fx(is)11 b(a)g(constan)o(t.)g(The)h(set)g(of)f(v)n(ariables)g(app)q (earing)g(in)g(a)g(term)g Fu(t)g Ft(2)g(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\))k(is)h(denoted)100 1793 y(b)o(y)i Ft(V)s Fu(ar)q Fx(\()p Fu(t)p Fx(\))q(.)g(F)m(or)g Fu(t)f Ft(2)g(T)d Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\),)14 b(w)o(e)h(de\014ne)h Fu(r)q(oot)p Fx(\()p Fu(t)p Fx(\))f(b)o(y)f Fu(r)q(oot)p Fx(\()p Fu(t)p Fx(\))g(=)f Fu(t)h Fx(if)g Fu(t)f Ft(2)f(V)s Fx(,)j(and)g Fu(r)q(oot)p Fx(\()p Fu(t)p Fx(\))e(=)g Fu(f)19 b Fx(if)100 1843 y Fu(t)11 b Fx(=)h Fu(f)t Fx(\()p Fu(t)225 1849 y Fr(1)245 1843 y Fu(;)7 b(:)g(:)g(:)t(;)g(t)352 1849 y Fp(n)374 1843 y Fx(\).)13 b Ft(j)p Fu(t)p Ft(j)e Fx(denotes)j(the)g Fs(size)e Fx(of)g Fu(t)p Fx(,)g(i.e.)g Ft(j)p Fu(t)p Ft(j)f Fx(=)g(1)i(if)f Fu(t)f Ft(2)g(V)s Fx(,)i(and)g Ft(j)p Fu(t)p Ft(j)d Fx(=)i(1)7 b(+)g Ft(j)p Fu(t)1463 1849 y Fr(1)1481 1843 y Ft(j)g Fx(+)g Fu(:)g(:)g(:)e Fx(+)i Ft(j)p Fu(t)1660 1849 y Fp(n)1682 1843 y Ft(j)100 1893 y Fx(if)13 b Fu(t)e Fx(=)h Fu(f)t Fx(\()p Fu(t)263 1899 y Fr(1)283 1893 y Fu(;)7 b(:)g(:)g(:)t(;)g(t)390 1899 y Fp(n)412 1893 y Fx(\).)141 1943 y(A)13 b Fs(substitution)g Fu(\033)h Fx(is)e(a)h(mapping)d(from)h Ft(V)17 b Fx(to)c Ft(T)d Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))13 b(suc)o(h)h(that)e Ft(f)p Fu(x)f Ft(2)h(V)k(j)d Fu(\033)q Fx(\()p Fu(x)p Fx(\))p Ft(6)p Fx(=)p Fu(x)p Ft(g)f Fx(is)h(\014nite.)100 1993 y(This)18 b(set)i(is)e(called)g(the)i Fs(domain)f Fx(of)f Fu(\033)h Fx(and)g(will)e(b)q(e)i(denoted)h(b)o(y)e Ft(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))q(.)g(Occasionally)m(,)f(w)o(e) 100 2042 y(presen)o(t)i(a)f(substitution)g Fu(\033)h Fx(as)f Ft(f)p Fu(x)g Ft(7!)g Fu(\033)q Fx(\()p Fu(x)p Fx(\))g Ft(j)g Fu(x)g Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))q Ft(g)p Fx(.)f(The)i(substitution)f(with)g(empt)o(y)100 2092 y(domain)11 b(will)g(b)q(e)j(denoted)g(b)o(y)f Fu(\017)p Fx(.)g(Substitutions)g(extend)h(uniquely)f(to)g(morphisms)e(from)g Ft(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))100 2142 y(to)j Ft(T)g Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\),)k(that)f(is,)g Fu(\033)q Fx(\()p Fu(f)t Fx(\()p Fu(t)548 2148 y Fr(1)568 2142 y Fu(;)d(:)g(:)g(:)e(;)i(t)676 2148 y Fp(n)698 2142 y Fx(\)\))12 b(=)g Fu(f)t Fx(\()p Fu(\033)q Fx(\()p Fu(t)882 2148 y Fr(1)901 2142 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i(\033)q Fx(\()p Fu(t)1066 2148 y Fp(n)1089 2142 y Fx(\)\))k(for)f(ev)o(ery)h Fu(n)p Fx(-ary)f(function)g(sym)o(b)q(ol)100 2192 y Fu(f)18 b Fx(and)c(terms)g Fu(t)350 2198 y Fr(1)368 2192 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)476 2198 y Fp(n)498 2192 y Fx(.)13 b(W)m(e)h(call)f Fu(\033)q Fx(\()p Fu(t)p Fx(\))h(an)g Fs(instanc)n(e)g Fx(of)g Fu(t)p Fx(.)f(W)m(e)g(also)h(write)g Fu(t\033)h Fx(instead)f(of)f Fu(\033)q Fx(\()p Fu(t)p Fx(\).)141 2242 y(Let)j Fo(2)g Fx(b)q(e)h(a)e(sp)q(ecial)h(constan)o (t.)g(A)g Fs(c)n(ontext)g Fx(is)g(a)f(term)g(in)h Ft(T)10 b Fx(\()p Ft(F)k([)c(f)p Fo(2)p Ft(g)p Fu(;)d Ft(V)s Fx(\).)16 b Fu(C)s Fx([)p Fu(;)7 b(:)g(:)g(:)t(;)g Fx(])14 b(denotes)100 2291 y(a)g(con)o(text)h(whic)o(h)f(con)o(tains)g(at)g (least)h(one)f(o)q(ccurrence)j(of)d Fo(2)g Fx(and)g(ma)o(y)f(b)q(e)i (equal)f(to)g Fo(2)p Fx(,)f Fu(C)s Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)100 2341 y Fx(stands)16 b(for)f(a)g(con)o(text)h(whic)o(h)f(con)o (tains)g(zero)h(or)f(more)g(o)q(ccurrence)j(of)c Fo(2)h Fx(and)h(ma)o(y)d(b)q(e)j(equal)f(to)100 2391 y Fo(2)p Fx(,)d(while)g Fu(C)s Ft(f)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(g)12 b Fx(denotes)i(a)e(con)o(text)i(whic)o(h)f(con)o(tains)g(zero)g (or)g(more)f(o)q(ccurrence)k(of)c Fo(2)h Fx(and)g(is)100 2441 y(di\013eren)o(t)f(from)e Fo(2)p Fx(.)g(If)h Fu(C)s Fx([)p Fu(;)c(:)g(:)g(:)t(;)g Fx(])k(is)g(a)g(con)o(text)h(with)f Fu(n)g Fx(o)q(ccurrences)k(of)10 b Fo(2)i Fx(and)f Fu(t)1354 2447 y Fr(1)1372 2441 y Fu(;)c(:)g(:)g(:)e(;)i(t)1480 2447 y Fp(n)1514 2441 y Fx(are)k(terms,)100 2491 y(then)16 b Fu(C)s Fx([)p Fu(t)256 2497 y Fr(1)274 2491 y Fu(;)7 b(:)g(:)g(:)t(;)g(t)381 2497 y Fp(n)403 2491 y Fx(])15 b(is)h(the)g(result)g(of)f(replacing)g(from)f(left)h(to)g(righ)o(t)g (the)h(o)q(ccurrences)j(of)c Fo(2)g Fx(with)100 2540 y Fu(t)115 2546 y Fr(1)133 2540 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)241 2546 y Fp(n)263 2540 y Fx(.)12 b(A)h(con)o(text)h(con)o(taining)e (precisely)i(one)f(o)q(ccurrence)i(of)d Fo(2)h Fx(is)g(denoted)h(b)o(y) e Fu(C)s Fx([)g(].)g(A)h(term)100 2590 y Fu(t)j Fx(is)f(a)h Fs(subterm)g Fx(of)f(a)h(term)f Fu(s)h Fx(if)f(there)j(exists)e(a)g (con)o(text)h Fu(C)s Fx([)e(])g(suc)o(h)i(that)f Fu(s)f Fx(=)h Fu(C)s Fx([)p Fu(t)p Fx(].)e(A)i(subterm)100 2640 y Fu(t)g Fx(of)f Fu(s)h Fx(is)g Fs(pr)n(op)n(er)p Fx(,)f(denoted)i(b)o (y)f Fu(s)p 661 2642 3 25 v 20 w(>)11 b(t)p Fx(,)k(if)g Fu(s)h Ft(6)p Fx(=)f Fu(t)p Fx(.)g(By)i(abuse)f(of)g(notation)f(w)o(e)h (write)g Ft(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))17 b(for)p eop %%Page: 4 4 4 3 bop 100 197 a Fw(4)105 b(E.)12 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Ft(T)e Fx(\()p Ft(F)k([)c(f)p Fo(2)p Ft(g)p Fu(;)d Ft(V)s Fx(\),)15 b(in)o(terpreting)h Fo(2)g Fx(as)f(a)h(sp)q(ecial)f(constan)o(t)h(whic)o(h)g(is)f(alw)o(a)o (ys)g(a)o(v)n(ailable)e(but)j(used)100 349 y(only)d(for)g(the)i (aforemen)o(tioned)d(purp)q(ose.)141 399 y(Let)21 b Ft(!)e Fx(b)q(e)i(a)f(binary)g(relation)f(on)h(terms,)g(i.e.)f Ft(!)h(\022)i(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))14 b Ft(\002)g(T)c Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(.)19 b(The)i(re\015exiv)o(e)100 448 y(transitiv)o(e)11 b(closure)i(of)e Ft(!)g Fx(is)h(denoted)g(b)o(y)g Ft(!)809 433 y Fn(\003)827 448 y Fx(.)f(If)h Fu(s)g Ft(!)963 433 y Fn(\003)993 448 y Fu(t)p Fx(,)f(w)o(e)h(sa)o(y)f(that)h Fu(s)g Fs(r)n(e)n(duc)n(es)g Fx(to)f Fu(t)h Fx(and)f(w)o(e)h(call)100 498 y Fu(t)h Fx(a)h Fs(r)n(e)n(duct)f Fx(of)g Fu(s)p Fx(.)h(W)m(e)f(write)h Fu(s)e Ft( )f Fu(t)j Fx(if)f Fu(t)e Ft(!)g Fu(s)p Fx(;)j(lik)o(ewise)f (for)g Fu(s)1084 483 y Fn(\003)1101 498 y Ft( )h Fu(t)p Fx(.)f(The)h(transitiv)o(e)g(closure)g(of)f Ft(!)100 548 y Fx(is)h(denoted)h(b)o(y)g Ft(!)400 533 y Fr(+)427 548 y Fx(,)f(and)g Ft($)g Fx(denotes)h(the)g(symmetric)e(closure)i(of)f Ft(!)g Fx(\(i.e.)g Ft($)g Fx(=)g Ft(!)e([)h( )p Fx(\).)g(The)100 598 y(re\015exiv)o(e)h(transitiv)o(e)f(closure)i(of)d Ft($)h Fx(is)h(called)f Fs(c)n(onversion)h Fx(and)f(denoted)h(b)o(y)f Ft($)1385 583 y Fn(\003)1404 598 y Fx(.)g(If)g Fu(s)f Ft($)1543 583 y Fn(\003)1573 598 y Fu(t)p Fx(,)h(then)100 648 y Fu(s)g Fx(and)g Fu(t)h Fx(are)f Fs(c)n(onvertible)p Fx(.)g(Tw)o(o)f(terms)h Fu(t)749 654 y Fr(1)768 648 y Fu(;)7 b(t)802 654 y Fr(2)833 648 y Fx(are)14 b Fs(joinable)p Fx(,)f(denoted)h(b)o(y)f Fu(t)1295 654 y Fr(1)1325 648 y Ft(#)e Fu(t)1372 654 y Fr(2)1391 648 y Fx(,)i(if)f(there)j(exists)f (a)100 697 y(term)c Fu(t)211 703 y Fr(3)241 697 y Fx(suc)o(h)i(that)f Fu(t)434 703 y Fr(1)464 697 y Ft(!)506 682 y Fn(\003)536 697 y Fu(t)551 703 y Fr(3)602 682 y Fn(\003)619 697 y Ft( )24 b Fu(t)700 703 y Fr(2)719 697 y Fx(.)10 b(Suc)o(h)i(a)f(term)g Fu(t)983 703 y Fr(3)1012 697 y Fx(is)g(called)g(a)g Fs(c)n(ommon)i(r)n (e)n(duct)e Fx(of)g Fu(t)1541 703 y Fr(1)1570 697 y Fx(and)h Fu(t)1664 703 y Fr(2)1682 697 y Fx(.)100 747 y(The)f(relation)f Ft(#)g Fx(is)h(called)f Fs(joinability)p Fx(.)g(A)h(term)f Fu(s)h Fx(is)g(a)f Fs(normal)i(form)e Fx(w.r.t.)f Ft(!)i Fx(if)e(there)j(is)f(no)g(term)f Fu(t)100 797 y Fx(suc)o(h)i(that)g Fu(s)g Ft(!)f Fu(t)p Fx(.)g(A)h(term)f Fu(s)i Fx(has)f(a)f(normal)f (form)g(if)h Fu(s)h Ft(!)1021 782 y Fn(\003)1051 797 y Fu(t)g Fx(for)f(some)g(normal)f(form)g Fu(t)p Fx(.)h(The)h(set)h(of) 100 847 y(all)d(normal)f(forms)h(of)g Ft(!)h Fx(is)g(denoted)i(b)o(y)e Fu(N)5 b(F)h Fx(\()p Ft(!)p Fx(\).)j(The)j(relation)f Ft(!)g Fx(is)g Fs(normalizing)g Fx(if)f(ev)o(ery)i(term)100 897 y(has)h(a)g(normal)e(form;)g(it)i(is)g Fs(terminating)p Fx(,)f(if)g(there)j(is)e(no)g(in\014nite)g(reduction)g(sequence)j Fu(t)1524 903 y Fr(1)1554 897 y Ft(!)11 b Fu(t)1622 903 y Fr(2)1652 897 y Ft(!)100 946 y Fu(t)115 952 y Fr(3)145 946 y Ft(!)g Fu(:)c(:)g(:)n Fx(.)13 b(In)h(the)g(literature,)g(the)g (terminology)e Fs(we)n(akly)i(normalizing)f Fx(and)h Fs(str)n(ongly)g(normalizing)100 996 y Fx(is)19 b(often)g(used)h (instead)f(of)g(normalizing)d(and)j(terminating,)e(resp)q(ectiv)o(ely)m (.)i(The)h(relation)e Ft(!)h Fx(is)100 1046 y Fs(c)n(on\015uent)e Fx(if)d(for)i(all)e(terms)i Fu(s;)7 b(t)615 1052 y Fr(1)633 1046 y Fu(;)g(t)667 1052 y Fr(2)701 1046 y Fx(with)15 b Fu(t)812 1052 y Fr(1)851 1031 y Fn(\003)868 1046 y Ft( )29 b Fu(s)15 b Ft(!)1015 1031 y Fn(\003)1048 1046 y Fu(t)1063 1052 y Fr(2)1097 1046 y Fx(w)o(e)h(ha)o(v)o(e)g Fu(t)1273 1052 y Fr(1)1306 1046 y Ft(#)e Fu(t)1356 1052 y Fr(2)1375 1046 y Fx(.)h(It)g(is)h(w)o(ell-kno)o(wn)100 1096 y(that)c Ft(!)g Fx(is)h(con\015uen)o(t)g(if)f(and)g(only)g(if)g (ev)o(ery)h(pair)f(of)g(con)o(v)o(ertible)h(terms)f(is)g(joinable.)f (The)i(relation)100 1146 y Ft(!)j Fx(is)h Fs(lo)n(c)n(al)r(ly)g(c)n (on\015uent)i Fx(if)d(for)h(all)f(terms)g Fu(s;)7 b(t)854 1152 y Fr(1)873 1146 y Fu(;)g(t)907 1152 y Fr(2)942 1146 y Fx(with)17 b Fu(t)1055 1152 y Fr(1)1090 1146 y Ft( )g Fu(s)g Ft(!)f Fu(t)1258 1152 y Fr(2)1294 1146 y Fx(w)o(e)h(ha)o(v)o(e)g Fu(t)1472 1152 y Fr(1)1508 1146 y Ft(#)f Fu(t)1560 1152 y Fr(2)1579 1146 y Fx(.)g(If)h Ft(!)100 1196 y Fx(is)d(con\015uen)o(t)h (and)f(terminating,)e(it)i(is)g(called)g Fs(c)n(omplete)g Fx(or)g Fs(c)n(onver)n(gent)p Fx(.)h(The)f(famous)f(Newman's)100 1245 y(Lemma)e(states)k(that)f(termination)e(and)i(lo)q(cal)f (con\015uence)i(imply)d(con\015uence.)j(If)f Ft(!)f Fx(is)h(con\015uen) o(t)100 1295 y(and)c(normalizing,)e(then)k(it)e(is)h(called)g Fs(semi-c)n(omplete)p Fx(.)f(Sometimes)f(this)i(prop)q(ert)o(y)g(is)g (called)g Fs(unique)100 1345 y(normalization)19 b Fx(b)q(ecause)j(it)d (is)g(equiv)n(alen)o(t)g(to)h(the)g(prop)q(ert)o(y)h(that)f(ev)o(ery)g (term)f(has)h(a)g(unique)100 1395 y(normal)11 b(form.)141 1445 y(A)j Fs(term)h(r)n(ewriting)e(system)h Fx(\(TRS\))g(is)g(a)g (pair)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(consisting)g(of)f(a)h (signature)h Ft(F)j Fx(and)c(a)g(set)100 1494 y Ft(R)d(\032)h(T)e Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))h Ft(\002)f(T)j Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))14 b(of)e Fs(r)n(ewrite)g(rules)g Fx(or)h Fs(r)n(e)n(duction)h(rules)p Fx(.)e(Ev)o(ery)h(rewrite)h(rule)f (\()p Fu(l)q(;)7 b(r)q Fx(\))13 b(m)o(ust)100 1544 y(satisfy)j(the)i (follo)o(wing)c(t)o(w)o(o)j(constrain)o(ts:)g(\(i\))g(the)g(left-hand)g (side)g Fu(l)i Fx(is)d(not)h(a)g(v)n(ariable,)e(and)i(\(ii\))100 1594 y(v)n(ariables)f(o)q(ccurring)h(in)f(the)h(righ)o(t-hand)f(side)h Fu(r)h Fx(also)e(o)q(ccur)i(in)e Fu(l)q Fx(.)g(Rewrite)h(rules)g(\()p Fu(l)q(;)7 b(r)q Fx(\))17 b(will)e(b)q(e)100 1644 y(denoted)g(b)o(y)g Fu(l)f Ft(!)f Fu(r)q Fx(.)h(An)h(instance)g(of)f(a)h(left-hand)f(side)h (of)g(a)f(rewrite)i(rule)f(is)g(a)f Fs(r)n(e)n(dex)h Fx(\(reducible)100 1694 y(expression\).)f(The)f(rewrite)i(rules)f(of)e (a)h(TRS)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))13 b(de\014ne)i(a)e Fs(r)n(ewrite)f(r)n(elation)h Ft(!)1447 1700 y Fn(R)1490 1694 y Fx(on)g Ft(T)d Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))100 1743 y(as)k(follo)o(ws:)f Fu(s)i Ft(!)367 1749 y Fn(R)408 1743 y Fu(t)g Fx(if)f(there)h(exists)h(a)e(rewrite)h(rule)g Fu(l)h Ft(!)e Fu(r)h Fx(in)f Ft(R)p Fx(,)h(a)f(substitution)h Fu(\033)g Fx(and)g(a)f(con)o(text)100 1793 y Fu(C)s Fx([)h(])h(suc)o(h) i(that)e Fu(s)f Fx(=)g Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])h(and)g Fu(t)e Fx(=)h Fu(C)s Fx([)p Fu(r)q(\033)q Fx(].)g(W)m(e)h(sa)o(y)h (that)f Fu(s)h Fx(rewrites)h(to)e Fu(t)h Fx(b)o(y)f Fs(c)n(ontr)n (acting)h Fx(redex)100 1843 y Fu(l)q(\033)q Fx(.)g(W)m(e)f(call)g Fu(s)g Ft(!)385 1849 y Fn(R)426 1843 y Fu(t)h Fx(a)g Fs(r)n(ewrite)f(step)h Fx(or)g Fs(r)n(e)n(duction)h(step)p Fx(.)f(A)g(TRS)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))13 b(has)i(one)f(of)f(the)i(ab)q(o)o(v)o(e)100 1893 y(prop)q(erties)d (\(e.g.)e(termination\))g(if)g(its)h(rewrite)h(relation)e(has)h(the)h (resp)q(ectiv)o(e)h(prop)q(ert)o(y)m(.)d(Let)i(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))100 1943 y(b)q(e)17 b(an)f(arbitrary)g(TRS.)f(A)i (function)f(sym)o(b)q(ol)e Fu(f)21 b Ft(2)15 b(F)20 b Fx(is)c(called)g(a)g Fs(de\014ne)n(d)i(symb)n(ol)f Fx(if)e(there)j(is)e (a)100 1993 y(rewrite)g(rule)g Fu(l)g Ft(!)d Fu(r)j Ft(2)e(R)h Fx(suc)o(h)i(that)e Fu(f)k Fx(=)c Fu(r)q(oot)p Fx(\()p Fu(l)q Fx(\).)h(F)m(unction)f(sym)o(b)q(ols)f(from)g Ft(F)19 b Fx(whic)o(h)d(are)g(not)100 2042 y(de\014ned)g(sym)o(b)q(ols) e(are)i(called)f Fs(c)n(onstructors)p Fx(.)g(The)h(set)g(of)f(normal)e (forms)h(of)h(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))15 b(will)f(also)h(b)q (e)100 2092 y(denoted)c(b)o(y)g Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)p Fx(\).)i(W)m(e)h(often)h(simply)d(write)j Ft(R)g Fx(instead)g(of)f(\()p Ft(F)t Fu(;)d Ft(R)p Fx(\))j(if)g(there)i (is)e(no)h(am)o(biguit)o(y)100 2142 y(ab)q(out)16 b(the)i(underlying)f (signature)g Ft(F)t Fx(.)f(A)h(rewrite)h(rule)f Fu(l)h Ft(!)e Fu(r)i Fx(of)f(a)f(TRS)h Ft(R)g Fx(is)g Fs(c)n(ol)r(lapsing)g Fx(if)f Fu(r)100 2192 y Fx(is)f(a)g(v)n(ariable,)f(and)i Fs(duplic)n(ating)g Fx(if)e Fu(r)j Fx(con)o(tains)e(more)g(o)q (ccurrences)k(of)14 b(some)h(v)n(ariable)g(than)g Fu(l)q Fx(.)h(A)100 2242 y(TRS)11 b Ft(R)h Fx(is)f Fs(non-duplic)n(ating)i Fx(\(non-collapsing,)c(resp)q(ectiv)o(ely\))14 b(if)c(it)i(do)q(es)g (not)g(con)o(tain)f(duplicating)100 2291 y(\(collapsing,)h(resp)q (ectiv)o(ely\))j(rewrite)g(rules.)141 2341 y(In)e(a)f Fs(join)i(c)n(onditional)g(term)f(r)n(ewriting)f(system)h Fx(\(CTRS)f(for)h(short\))g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\),)12 b(the)h(rewrite)h(rules)100 2391 y(of)e Ft(R)h Fx(ha)o(v)o(e)g(the)g(form)f Fu(l)g Ft(!)f Fu(r)i Ft(\()e Fu(s)638 2397 y Fr(1)669 2391 y Ft(#)g Fu(t)716 2397 y Fr(1)734 2391 y Fu(;)c(:)g(:)g(:)e(;)i(s)846 2397 y Fp(n)880 2391 y Ft(#)12 b Fu(t)928 2397 y Fp(n)963 2391 y Fx(with)h Fu(l)q(;)7 b(r)o(;)g(s)1145 2397 y Fr(1)1162 2391 y Fu(;)g(:)g(:)g(:)e(;)i(s)1274 2397 y Fp(n)1297 2391 y Fx(,)12 b Fu(t)1336 2397 y Fr(1)1355 2391 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1463 2397 y Fp(n)1496 2391 y Ft(2)k(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\).)100 2441 y Fu(s)119 2447 y Fr(1)147 2441 y Ft(#)i Fu(t)192 2447 y Fr(1)211 2441 y Fu(;)e(:)g(:)g(:)e(;)i(s)323 2447 y Fp(n)354 2441 y Ft(#)j Fu(t)400 2447 y Fp(n)431 2441 y Fx(are)g(the)g Fs(c)n(onditions)g Fx(of)f(the)h(rewrite)g(rule.)g(If)f(a)g(rewrite)h (rule)g(has)f(no)g(conditions,)100 2491 y(w)o(e)20 b(write)g Fu(l)i Ft(!)f Fu(r)q Fx(.)e(W)m(e)g(imp)q(ose)f(the)j(same)d (restrictions)j(on)f(conditional)e(rewrite)j(rules)f(as)g(on)100 2540 y(unconditional)10 b(rewrite)j(rules.)e(That)h(is,)f(w)o(e)h(allo) o(w)e Fs(extr)n(a)i(variables)g Fx(in)f(the)h(conditions)f(but)h(not)g (on)100 2590 y(righ)o(t-hand)i(sides)i(of)f(rewrite)h(rules.)f(The)h (rewrite)g(relation)f(asso)q(ciated)h(with)e(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))15 b(is)h(de\014ned)100 2640 y(b)o(y:)c Fu(s)g Ft(!)241 2646 y Fn(R)283 2640 y Fu(t)i Fx(if)e(there)j(exists)f (a)g(rewrite)g(rule)g Fu(l)f Ft(!)e Fu(r)h Ft(\()f Fu(s)1008 2646 y Fr(1)1039 2640 y Ft(#)g Fu(t)1086 2646 y Fr(1)1105 2640 y Fu(;)c(:)g(:)g(:)t(;)g(s)1216 2646 y Fp(n)1251 2640 y Ft(#)k Fu(t)1298 2646 y Fp(n)1334 2640 y Fx(in)i Ft(R)p Fx(,)g(a)g(substitution)p eop %%Page: 5 5 5 4 bop 636 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)g(of)h(Comp)q (osable)e(T)m(erm)i(Rewriting)f(Systems)104 b(5)p 100 224 1595 2 v 100 299 a Fu(\033)13 b Fx(:)f Ft(V)k(!)c(T)e Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(,)13 b(and)i(a)f(con)o(text)h Fu(C)s Fx([)e(])h(suc)o(h)h(that)f Fu(s)f Fx(=)g Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])p Fu(;)7 b(t)k Fx(=)h Fu(C)s Fx([)p Fu(r)q(\033)q Fx(],)h(and)h Fu(s)1449 305 y Fp(j)1467 299 y Fu(\033)h Ft(#)1527 309 y Fn(R)1572 299 y Fu(t)1587 305 y Fp(j)1604 299 y Fu(\033)h Fx(for)100 349 y(all)c Fu(j)i Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)12 b(F)m(or)i(ev)o(ery)g(CTRS)g Ft(R)p Fx(,)f(w)o(e)i(inductiv)o (ely)e(de\014ne)i(TRSs)f Ft(R)1348 355 y Fp(i)1362 349 y Fx(,)f Fu(i)f Ft(2)f Fm(N)p Fx(,)g(b)o(y:)400 425 y Ft(R)435 431 y Fr(0)495 425 y Fx(=)h Ft(f)p Fu(l)g Ft(!)f Fu(r)k Ft(j)e Fu(l)g Ft(!)e Fu(r)i Ft(2)e(Rg)362 475 y(R)398 481 y Fp(i)p Fr(+1)495 475 y Fx(=)h Ft(f)p Fu(l)q(\033)g Ft(!)f Fu(r)q(\033)k Ft(j)f Fu(l)e Ft(!)g Fu(r)g Ft(\()f Fu(s)928 481 y Fr(1)959 475 y Ft(#)g Fu(t)1006 481 y Fr(1)1024 475 y Fu(;)c(:)g(:)g(:)e(;)i(s)1136 481 y Fp(n)1170 475 y Ft(#)12 b Fu(t)1218 481 y Fp(n)1252 475 y Ft(2)f(R)j Fx(and)802 525 y Fu(s)821 531 y Fp(j)839 525 y Fu(\033)h Ft(#)899 535 y Fn(R)928 539 y Fl(i)956 525 y Fu(t)971 531 y Fp(j)989 525 y Fu(\033)g Fx(for)e(all)g Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(gg)p Fu(:)100 600 y Fx(Note)14 b(that)g Ft(R)325 606 y Fp(i)350 600 y Ft(\022)e(R)429 606 y Fp(i)p Fr(+1)499 600 y Fx(for)h(all)g Fu(i)f Ft(2)f Fm(N)p Fx(.)g(F)m(urthermore,)i Fu(s)f Ft(!)1065 606 y Fn(R)1107 600 y Fu(t)i Fx(if)f(and)g(only)h(if)e Fu(s)g Ft(!)1456 606 y Fn(R)1485 610 y Fl(i)1511 600 y Fu(t)i Fx(for)f(some)100 650 y Fu(i)g Ft(2)f Fm(N)p Fx(.)h(The)i Fs(depth)g Fx(of)f(a)h(rewrite)g(step)h Fu(s)d Ft(!)809 656 y Fn(R)853 650 y Fu(t)h Fx(is)h(de\014ned)h(to)e(b) q(e)i(the)f(minim)o(al)c Fu(i)k Fx(with)g Fu(s)e Ft(!)1611 656 y Fn(R)1640 660 y Fl(i)1667 650 y Fu(t)p Fx(.)100 700 y(Depths)e(of)g(reduction)g(sequences)j Fu(s)e Ft(!)724 684 y Fn(\003)724 711 y(R)766 700 y Fu(t)p Fx(,)e(con)o(v)o(ersions)i Fu(s)g Ft($)1092 684 y Fn(\003)1092 711 y(R)1133 700 y Fu(t)p Fx(,)f(and)f(v)n(alleys)h Fu(s)g Ft(#)1431 710 y Fn(R)1472 700 y Fu(t)g Fx(are)h(de\014ned)100 749 y(analogously)m(.)f (All)i(notions)g(de\014ned)i(previously)f(for)g(TRSs)f(extend)i(to)f (CTRSs.)141 799 y(A)i Fs(p)n(artial)h(or)n(dering)e Fx(\()p Fu(A;)7 b(>)p Fx(\))17 b(is)f(a)g(pair)f(consisting)h(of)g(a)g(set)h Fu(A)f Fx(and)g(a)g(binary)g(irre\015exiv)o(e)h(and)100 849 y(transitiv)o(e)11 b(relation)f Fu(>)i Fx(on)f Fu(A)p Fx(.)f(A)i(partial)e(ordering)h(is)g(called)g Fs(wel)r(l-founde)n(d)g Fx(if)f(there)i(are)g(no)f(in\014nite)100 899 y(sequences)19 b Fu(a)313 905 y Fr(1)348 899 y Fu(>)e(a)419 905 y Fr(2)455 899 y Fu(>)f(a)525 905 y Fr(3)561 899 y Fu(>)h(:)7 b(:)g(:)15 b Fx(of)h(elemen)o(ts)h(from)e Fu(A)p Fx(.)i(A)g Fs(multiset)f Fx(is)h(a)g(collection)f(in)h(whic)o(h)100 949 y(elemen)o(ts)g(are)i (allo)o(w)o(ed)d(to)i(o)q(ccur)h(more)d(than)i(once.)g(If)g Fu(A)g Fx(is)f(a)h(set,)g(then)g(the)h(set)g(of)e(all)f(\014nite)100 999 y(m)o(ultisets)10 b(o)o(v)o(er)i Fu(A)g Fx(is)f(denoted)i(b)o(y)e Ft(M)p Fx(\()p Fu(A)p Fx(\).)g(The)h Fs(multiset)g(extension)h Fx(of)e(a)g(partial)g(ordering)g(\()p Fu(A;)c(>)p Fx(\))100 1048 y(is)i(the)h(partial)f(ordering)g(\()p Ft(M)p Fx(\()p Fu(A)p Fx(\))p Fu(;)e(>)673 1033 y Fp(mul)735 1048 y Fx(\))j(de\014ned)g(as)g(follo)o(ws:)d Fu(M)1129 1054 y Fr(1)1160 1048 y Fu(>)1192 1033 y Fp(mul)1265 1048 y Fu(M)1305 1054 y Fr(2)1334 1048 y Fx(if)h Fu(M)1407 1054 y Fr(2)1438 1048 y Fx(=)j(\()p Fu(M)1537 1054 y Fr(1)1557 1048 y Ft(n)p Fu(X)s Fx(\))p Ft([)p Fu(Y)100 1098 y Fx(for)h(some)f(m)o(ultisets)h Fu(X)q(;)7 b(Y)21 b Ft(2)11 b(M)p Fx(\()p Fu(A)p Fx(\))i(that)f(satisfy)g(\(i\))h Ft(;)e(6)p Fx(=)h Fu(X)j Ft(\022)d Fu(M)1186 1104 y Fr(1)1217 1098 y Fx(and)h(\(ii\))e(for)i(all)e Fu(y)i Ft(2)e Fu(Y)22 b Fx(there)100 1148 y(exists)13 b(an)f Fu(x)f Ft(2)g Fu(X)16 b Fx(suc)o(h)d(that)f Fu(x)f(>)h(y)q Fx(.)h(Dersho)o(witz)f (and)g(Manna)g(\(1979\))g(pro)o(v)o(ed)g(that)g(the)h(m)o(ultiset)100 1198 y(extension)h(of)f(a)h(w)o(ell-founded)f(partial)g(ordering)h(is)g (a)f(w)o(ell-founded)h(partial)e(ordering.)141 1248 y(A)f Fs(simpli\014c)n(ation)i(or)n(dering)d Fu(>)i Fx(is)f(a)g(partial)f (ordering)i(on)f(terms)g(whic)o(h)g(\(i\))g(is)g Fs(close)n(d)h(under)h (c)n(on-)100 1298 y(texts)f Fx(\(i.e.)h Fu(s)f(>)f(t)i Fx(implies)e Fu(C)s Fx([)p Fu(s)p Fx(])g Fu(>)h(C)s Fx([)p Fu(t)p Fx(])f(for)i(ev)o(ery)h(con)o(text)g Fu(C)s Fx([)e(]\),)g (\(ii\))g Fs(close)n(d)i(under)h(substitutions)100 1348 y Fx(\(i.e.)f Fu(s)h(>)f(t)h Fx(implies)f Fu(s\033)h(>)g(t\033)h Fx(for)f(ev)o(ery)h(substitution)g Fu(\033)q Fx(\),)f(and)h(\(iii\))e (has)h(the)h Fs(subterm)g(pr)n(op)n(erty)100 1397 y Fx(\(i.e.)d Fu(C)s Fx([)p Fu(t)p Fx(])d Fu(>)i(t)i Fx(for)f(all)g(con)o(texts)i Fu(C)s Fx([)e(])e Ft(6)p Fx(=)h Fo(2)p Fx(\).)642 1535 y Fv(3.)24 b(V)l(arious)15 b(com)o(bination)o(s)141 1610 y Fx(V)m(ery)k(simple)d(examples)i(sho)o(w)g(that)g(in)g(general)h(all) e(in)o(teresting)h(prop)q(erties)i(are)f(lost)f(under)100 1660 y(arbitrary)i(com)o(binations)e(of)h(TRSs.)h(Th)o(us,)g(sev)o (eral)g(restricted)i(kinds)e(of)g(com)o(binations)e(ha)o(v)o(e)100 1710 y(b)q(een)d(prop)q(osed)f(in)g(the)g(literature.)g(The)h(next)f (de\014nition)f(sp)q(eci\014es)j(these)g(com)o(binations.)100 1815 y Fk(Definition)g(3.1.)21 b Fx(Let)16 b(\()p Ft(F)550 1821 y Fr(1)569 1815 y Fu(;)7 b Ft(R)623 1821 y Fr(1)641 1815 y Fx(\))17 b(and)f(\()p Ft(F)803 1821 y Fr(2)821 1815 y Fu(;)7 b Ft(R)875 1821 y Fr(2)893 1815 y Fx(\))17 b(b)q(e)f(TRSs.)g(Let)h Ft(D)1222 1821 y Fr(1)1257 1815 y Fx(and)f Ft(D)1373 1821 y Fr(2)1408 1815 y Fx(denote)h(their)f(re-) 100 1864 y(sp)q(ectiv)o(e)i(sets)f(of)f(de\014ned)i(sym)o(b)q(ols)d (and)i(let)f Ft(C)872 1870 y Fr(1)907 1864 y Fx(and)g Ft(C)1014 1870 y Fr(2)1050 1864 y Fx(denote)h(their)g(resp)q(ectiv)o(e) i(sets)f(of)e(con-)100 1914 y(structors.)g(Their)f Fs(c)n(ombine)n(d)h (system)f Fx(is)g(their)g(union)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))13 b(=)g(\()p Ft(F)1201 1920 y Fr(1)1230 1914 y Ft([)c(F)1297 1920 y Fr(2)1316 1914 y Fu(;)e Ft(R)1370 1920 y Fr(1)1398 1914 y Ft([)j(R)1471 1920 y Fr(2)1489 1914 y Fx(\).)15 b(Its)g(set)h(of)100 1964 y(de\014ned)f(sym)o(b)q(ols) d(is)i(ob)o(viously)e Ft(D)h Fx(=)f Ft(D)748 1970 y Fr(1)776 1964 y Ft([)d(D)846 1970 y Fr(2)878 1964 y Fx(and)14 b(its)g(set)h(of)e(constructors)j(is)e Ft(C)g Fx(=)d Ft(F)j(n)9 b(D)q Fx(.)125 2068 y(\(1\))21 b(\()p Ft(F)249 2074 y Fr(1)268 2068 y Fu(;)7 b Ft(R)322 2074 y Fr(1)340 2068 y Fx(\))16 b(and)f(\()p Ft(F)504 2074 y Fr(2)523 2068 y Fu(;)7 b Ft(R)576 2074 y Fr(2)595 2068 y Fx(\))16 b(are)f Fs(disjoint)h Fx(if)e(they)i(do)f(not)h(share)g(function)f(sym) o(b)q(ols,)f(that)h(is,)199 2118 y Ft(F)233 2124 y Fr(1)261 2118 y Ft(\\)9 b(F)332 2124 y Fr(2)362 2118 y Fx(=)j Ft(;)h Fx(\(or)h(equiv)n(alen)o(tly)f Ft(C)760 2124 y Fr(1)788 2118 y Ft(\\)c(C)847 2124 y Fr(2)877 2118 y Fx(=)j Ft(C)943 2124 y Fr(1)970 2118 y Ft(\\)d(D)1039 2124 y Fr(2)1069 2118 y Fx(=)j Ft(D)1145 2124 y Fr(1)1173 2118 y Ft(\\)d(C)1232 2124 y Fr(2)1262 2118 y Fx(=)j Ft(D)1338 2124 y Fr(1)1366 2118 y Ft(\\)c(D)1434 2124 y Fr(2)1465 2118 y Fx(=)j Ft(;)p Fx(\).)199 2168 y(In)f(the)g (literature,)f(\()p Ft(F)t Fu(;)e Ft(R)p Fx(\))j(is)f(sometimes)f (called)h(the)h Fs(dir)n(e)n(ct)g(sum)f Fx(of)g(\()p Ft(F)1332 2174 y Fr(1)1351 2168 y Fu(;)e Ft(R)1405 2174 y Fr(1)1423 2168 y Fx(\))j(and)f(\()p Ft(F)1575 2174 y Fr(2)1594 2168 y Fu(;)e Ft(R)1647 2174 y Fr(2)1666 2168 y Fx(\).)125 2222 y(\(2\))21 b(\()p Ft(F)249 2228 y Fr(1)268 2222 y Fu(;)7 b Ft(R)322 2228 y Fr(1)340 2222 y Fx(\))15 b(and)g(\()p Ft(F)503 2228 y Fr(2)522 2222 y Fu(;)7 b Ft(R)576 2228 y Fr(2)594 2222 y Fx(\))15 b(are)h Fs(c)n(onstructor-sharing)f Fx(if)f(they)h(at)g(most)f(share)i (constructors,)199 2272 y(i.e.,)d Ft(F)312 2278 y Fr(1)339 2272 y Ft(\\)c(F)410 2278 y Fr(2)440 2272 y Ft(\022)j(C)k Fx(\(or)e(equiv)n(alen)o(tly)f Ft(C)842 2278 y Fr(1)870 2272 y Ft(\\)c(D)939 2278 y Fr(2)969 2272 y Fx(=)j Ft(D)1045 2278 y Fr(1)1072 2272 y Ft(\\)d(C)1131 2278 y Fr(2)1161 2272 y Fx(=)j Ft(D)1238 2278 y Fr(1)1266 2272 y Ft(\\)d(D)1336 2278 y Fr(2)1366 2272 y Fx(=)j Ft(;)p Fx(\).)125 2326 y(\(3\))21 b(\()p Ft(F)249 2332 y Fr(1)268 2326 y Fu(;)7 b Ft(R)322 2332 y Fr(1)340 2326 y Fx(\))15 b(and)f(\()p Ft(F)502 2332 y Fr(2)521 2326 y Fu(;)7 b Ft(R)575 2332 y Fr(2)593 2326 y Fx(\))15 b(are)g Fs(c)n(omp)n(osable)g Fx(if)f Ft(C)969 2332 y Fr(1)997 2326 y Ft(\\)9 b(D)1066 2332 y Fr(2)1097 2326 y Fx(=)k Ft(D)1174 2332 y Fr(1)1203 2326 y Ft(\\)c(C)1262 2332 y Fr(2)1293 2326 y Fx(=)k Ft(;)h Fx(and)h(b)q(oth)f(systems)199 2376 y(con)o(tain)20 b(all)f(rewrite)j(rules)f(that)f(de\014ne)i(a)e(de\014ned)h(sym)o(b)q (ol)e(whenev)o(er)j(that)e(sym)o(b)q(ol)f(is)199 2426 y(shared,)d(more)f(precisely)m(,)h Ft(S)h Fx(=)e Ft(f)p Fu(l)h Ft(!)e Fu(r)h Ft(2)f(R)i(j)f Fu(r)q(oot)p Fx(\()p Fu(l)q Fx(\))g Ft(2)f(D)1189 2432 y Fr(1)1218 2426 y Ft(\\)c(D)1289 2432 y Fr(2)1308 2426 y Ft(g)k(\022)h(R)1425 2432 y Fr(1)1454 2426 y Ft(\\)10 b(R)1527 2432 y Fr(2)1546 2426 y Fx(.)15 b(In)h(this)199 2476 y(situation,)d(the)h(set)h Ft(S)i Fx(is)d(said)g(to)f(b)q(e)i(the)f(set)h(of)e Fs(shar)n(e)n(d)i (rules)f Fx(of)f Ft(R)1291 2482 y Fr(1)1324 2476 y Fx(and)g Ft(R)1439 2482 y Fr(2)1458 2476 y Fx(.)100 2578 y(The)h(di\013eren)o(t) h(kinds)f(of)f(com)o(binations)f(are)i(illustrated)g(in)f(Figure)h(1.)p eop %%Page: 6 6 6 5 bop 100 197 a Fw(6)105 b(E.)12 b(Ohlebusc)o(h)p 100 224 1595 2 v 343 376 a Fq(D)370 381 y Fj(1)305 427 y Fi(\032\031)305 311 y(\033)q(\030)347 518 y Fq(C)366 523 y Fj(1)305 569 y Fi(\032\031)305 452 y(\033)q(\030)477 518 y Fq(C)496 523 y Fj(2)435 569 y Fi(\032\031)435 452 y(\033)q(\030)473 376 y Fq(D)500 381 y Fj(2)435 427 y Fi(\032\031)435 311 y(\033)q(\030)413 671 y Fw(\(1\))804 376 y Fq(D)831 381 y Fj(1)766 427 y Fi(\032\031)766 311 y(\033\030)820 518 y Fq(C)839 523 y Fj(1)790 569 y Fi(\032\031)790 452 y(\033\030)926 518 y Fq(C)945 523 y Fj(2)872 569 y Fi(\032\031)872 452 y(\033)q(\030)933 376 y Fq(D)960 381 y Fj(2)896 427 y Fi(\032\031)896 311 y(\033\030)874 671 y Fw(\(2\))1276 376 y Fq(D)1303 381 y Fj(1)1250 427 y Fi(\032\031)1250 311 y(\033\030)1280 518 y Fq(C)1299 523 y Fj(1)1250 569 y Fi(\032\031)1250 452 y(\033\030)1343 375 y Fq(\003)1386 518 y(C)1405 523 y Fj(2)1333 569 y Fi(\032\031)1333 452 y(\033\030)1382 376 y Fq(D)1409 381 y Fj(2)1333 427 y Fi(\032\031)1333 311 y(\033\030)1335 671 y Fw(\(3\))636 800 y Fh(Figure)g(1)i Fw(Di\013eren)o(t)c(com)o (binatio)o(ns.)100 935 y Fk(Definition)16 b(3.2.)21 b Fx(A)16 b(prop)q(ert)o(y)i Ft(P)h Fx(is)e Fs(mo)n(dular)g(for)g(c)n (omp)n(osable)g(TRSs)g Fx(if,)e(for)h(all)g(comp)q(osable)100 985 y(TRSs)e(\()p Ft(F)264 991 y Fr(1)282 985 y Fu(;)7 b Ft(R)336 991 y Fr(1)355 985 y Fx(\))14 b(and)g(\()p Ft(F)516 991 y Fr(2)535 985 y Fu(;)7 b Ft(R)589 991 y Fr(2)607 985 y Fx(\),)14 b(their)g(union)g(\()p Ft(F)910 991 y Fr(1)938 985 y Ft([)9 b(F)1005 991 y Fr(2)1023 985 y Fu(;)e Ft(R)1077 991 y Fr(1)1105 985 y Ft([)i(R)1177 991 y Fr(2)1196 985 y Fx(\))14 b(has)g(the)h(prop)q(ert)o(y)g Ft(P)i Fx(if)d(and)100 1034 y(only)f(if)g(b)q(oth)h(\()p Ft(F)377 1040 y Fr(1)396 1034 y Fu(;)7 b Ft(R)450 1040 y Fr(1)468 1034 y Fx(\))14 b(and)f(\()p Ft(F)629 1040 y Fr(2)647 1034 y Fu(;)7 b Ft(R)701 1040 y Fr(2)719 1034 y Fx(\))14 b(ha)o(v)o(e)g(the)h(prop)q(ert)o(y)f Ft(P)s Fx(.)141 1135 y(W)m(e)i(will)f(also)h(use)h(the)g(phrases)g Fs(mo)n(dular)g(for)g(c)n(onstructor-sharing)g(TRSs)f Fx(and)g Fs(mo)n(dular)h(for)100 1185 y(disjoint)h(TRSs)p Fx(.)f(The)h(meanings)e(of)h(these)i(phrases)g(are)f(ob)o(vious.)f(W)m (e)g(are)h(of)f(course)i(not)e(only)100 1235 y(in)o(terested)g(in)f (the)h(com)o(bination)c(of)j(t)o(w)o(o)f(TRSs.)h(It)g(should)g(also)f (b)q(e)i(p)q(ossible)f(to)g(deal)g(with)g(sit-)100 1285 y(uations)g(where)i(more)e(than)h(t)o(w)o(o)g(systems)g(are)g(com)o (bined.)e(The)j(next)f(prop)q(osition)f(sho)o(ws)i(that)100 1335 y(the)f(com)o(bination)e(of)h Fu(n)h Fx(TRSs,)g Fu(n)g Ft(\025)g Fx(2,)f(can)i(b)q(e)g(reduced)g(to)f(the)h(case)g Fu(n)f Fx(=)g(2)g(b)o(y)g(successiv)o(ely)100 1384 y(com)o(bining)11 b(t)o(w)o(o)j(systems.)f(The)h(simple)f(pro)q(of)g(is)h(omitted.)100 1485 y Fk(Pr)o(oposition)i(3.3.)21 b Fx(Let)14 b(\()p Ft(F)584 1491 y Fr(1)603 1485 y Fu(;)7 b Ft(R)657 1491 y Fr(1)675 1485 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)834 1491 y Fp(n)856 1485 y Fu(;)g Ft(R)910 1491 y Fp(n)933 1485 y Fx(\))14 b(b)q(e)g Fu(n)p Fx(,)g Fu(n)e Ft(\025)g Fx(2,)h(pairwise)h(comp)q(osable)f(TRSs.)100 1539 y(Then)h(the)h(term)e(rewriting)h(systems)g(\()727 1508 y Fg(S)761 1518 y Fp(n)p Fn(\000)p Fr(1)761 1552 y Fp(j)r Fr(=1)833 1539 y Ft(F)867 1545 y Fp(j)885 1539 y Fu(;)904 1508 y Fg(S)938 1518 y Fp(n)p Fn(\000)p Fr(1)938 1552 y Fp(j)r Fr(=1)1010 1539 y Ft(R)1045 1545 y Fp(j)1063 1539 y Fx(\))g(and)f(\()p Ft(F)1223 1545 y Fp(n)1246 1539 y Fu(;)7 b Ft(R)1300 1545 y Fp(n)1322 1539 y Fx(\))14 b(are)g(comp)q(osable.)501 1667 y Fv(4.)24 b(Comp)q(osable)14 b(systems:)h(basic)g(notions)141 1742 y Fx(In)f(this)g(section,)g(the)g (basic)g(notions)g(concerning)g(the)h(com)o(bination)c(of)i(t)o(w)o(o)h (comp)q(osable)e(term)100 1792 y(rewriting)18 b(systems)h(\()p Ft(F)491 1798 y Fr(1)509 1792 y Fu(;)7 b Ft(R)563 1798 y Fr(1)582 1792 y Fx(\))18 b(and)g(\()p Ft(F)751 1798 y Fr(2)770 1792 y Fu(;)7 b Ft(R)824 1798 y Fr(2)842 1792 y Fx(\))19 b(are)g(in)o(tro)q(duced.)f(These)i(notions)e(will)f(easily) h(b)q(e)100 1842 y(iden)o(ti\014ed)13 b(with)h(those)g(already)f(in)o (tro)q(duced)i(for)e(disjoin)o(t)g(systems)h(\(see)h(e.g.)e (Middeldorp,)f(1990\))100 1892 y(and)i(constructor-sharing)i(systems)f (\(see)i(e.g.)d(Kurihara)g(and)h(Oh)o(uc)o(hi,)f(1992\).)g(So)g(from)f (no)o(w)i(on)100 1941 y(w)o(e)d(tacitly)f(assume)g(that)h(\()p Ft(F)566 1947 y Fr(1)584 1941 y Fu(;)7 b Ft(R)638 1947 y Fr(1)657 1941 y Fx(\))12 b(and)f(\()p Ft(F)813 1947 y Fr(2)832 1941 y Fu(;)c Ft(R)885 1947 y Fr(2)904 1941 y Fx(\))12 b(are)g(t)o(w)o(o)f(comp)q(osable)g(TRSs)g(and)h(that)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))100 1991 y(denotes)20 b(their)g(com)o (bined)e(system.)h(In)h(the)g(sequel)g Ft(!)f Fx(=)g Ft(!)1124 1997 y Fn(R)1174 1991 y Fx(=)h Ft(!)1268 1997 y Fn(R)1297 2001 y Ff(1)1312 1997 y Fn([R)1363 2001 y Ff(2)1381 1991 y Fx(.)f(First)h(of)f(all,)e(w)o(e)100 2041 y(in)o(tro)q(duce)d(the)h(c)o(hromatic)d(terminology)g(whic)o(h)i (is)f(no)o(w)h(common.)100 2142 y Fk(Definition)i(4.1.)21 b Fx(The)13 b(set)h Ft(F)610 2148 y Fr(1)636 2142 y Ft(\\)7 b(F)704 2148 y Fr(2)736 2142 y Fx(of)12 b Fs(shar)n(e)n(d)i(function)g (symb)n(ols)p Fx(,)f(i.e.)f(function)g(sym)o(b)q(ols)g(that)100 2192 y(o)q(ccur)i(in)f Fs(b)n(oth)h Fx(signatures,)g(is)f(denoted)i(b)o (y)e Ft(B)r Fx(.)g Ft(A)896 2198 y Fr(1)926 2192 y Fx(=)f Ft(F)1004 2198 y Fr(1)1031 2192 y Ft(n)c(B)15 b Fx(is)f(called)f(the)h (set)h(of)e Fs(alien)h(function)100 2242 y(symb)n(ols)g Fx(for)h Ft(F)349 2248 y Fr(2)383 2242 y Fx(and)f Ft(B)j Fx(b)q(ecause)f Ft(A)695 2248 y Fr(1)724 2242 y Ft(\\)9 b(F)795 2248 y Fr(2)827 2242 y Fx(=)k Ft(;)h Fx(and)h Ft(A)1022 2248 y Fr(1)1050 2242 y Ft(\\)10 b(B)k Fx(=)f Ft(;)p Fx(.)h Ft(A)1255 2248 y Fr(2)1288 2242 y Fx(is)h(de\014ned)g (analogously)m(.)100 2291 y(Note)c(that)g Ft(F)k Fx(=)d Ft(A)406 2297 y Fr(1)428 2291 y Ft(])s(A)492 2297 y Fr(2)513 2291 y Ft(])s(B)r Fx(.)e(In)h(order)g(to)g(enhance)h(readabilit)o(y)m (,)d(function)h(sym)o(b)q(ols)g(from)f Ft(A)1609 2297 y Fr(1)1638 2291 y Fx(are)100 2341 y(called)15 b Fs(black)p Fx(,)g(those)h(from)e Ft(A)578 2347 y Fr(2)612 2341 y Fs(white)p Fx(,)h(and)g(shared)h(function)f(sym)o(b)q(ols)f(as)i(w)o (ell)f(as)g(v)n(ariables)g(are)100 2391 y(called)f Fs(tr)n(ansp)n(ar)n (ent)p Fx(.)g(A)h(term)g Fu(s)g Fx(is)g(called)f Fs(top)i(black)g (\(top)g(white,)f(top)h(tr)n(ansp)n(ar)n(ent\))f Fx(if)f Fu(r)q(oot)p Fx(\()p Fu(s)p Fx(\))i(is)100 2441 y(blac)o(k)d(\(white,)g (transparen)o(t\).)i(In)f(a)f(term)g(ev)o(ery)i(transparen)o(t)g(sym)o (b)q(ol)d Fp(C)j Fx(acts)g(lik)o(e)e(a)g(c)o(hameleon,)100 2491 y(that)e(is,)g(it)g(c)o(hanges)h(its)g(color)f(to)g(matc)o(h)f (the)i(surrounding:)f(If)g(there)i(is)e(no)h(blac)o(k)e(or)i(white)f (sym)o(b)q(ol)100 2540 y(ab)q(o)o(v)o(e)k Fp(C)i Fx(\(there)g(is)e(no)g (surrounding)h(so)f(to)g(sp)q(eak\),)h(then)g(it)f(remains)g (transparen)o(t.)h(Otherwise,)100 2590 y(its)g(color)h(is)f(the)h(same) f(as)h(the)g(color)g(of)f(its)g(paren)o(t)h(\(the)h(de\014nition)e (applies)h(recursiv)o(ely)g(if)f(the)100 2640 y(paren)o(t)c(is)f(a)g (shared)h(sym)o(b)q(ol\).)e(If)h(a)g(term)g(do)q(es)h(not)f(con)o(tain) g(white)h(\(blac)o(k\))f(function)g(sym)o(b)q(ols,)e(w)o(e)p eop %%Page: 7 7 7 6 bop 636 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)g(of)h(Comp)q (osable)e(T)m(erm)i(Rewriting)f(Systems)104 b(7)p 100 224 1595 2 v 100 299 a Fx(sp)q(eak)13 b(of)f(a)h Fs(black)h(\(white\))f (term)p Fx(.)f(A)h(term)f(con)o(taining)g(b)q(oth)h(blac)o(k)g(and)f (white)h(function)g(sym)o(b)q(ols)100 349 y(is)e(called)h(a)f Fs(mixe)n(d)i(term)p Fx(.)d(A)i(term)f(is)h(said)f(to)g(b)q(e)i Fs(tr)n(ansp)n(ar)n(ent)e Fx(if)g(it)g(only)g(con)o(tains)g(shared)i (function)100 399 y(sym)o(b)q(ols)f(and)i(v)n(ariables.)141 498 y(Please)k(notice)g(a)f(subtlet)o(y)g(in)g(the)h(preceding)g (de\014nition:)f(A)g(transparen)o(t)h(term)f(ma)o(y)e(b)q(e)j(re-)100 548 y(garded)c(as)g(blac)o(k)f(or)h(white;)g(this)f(is)h(v)o(ery)g(con) o(v)o(enien)o(t)h(for)e(later)h(purp)q(oses.)100 647 y Fk(Example)j(4.2.)k Fx(Let)14 b Ft(F)497 653 y Fr(1)527 647 y Fx(=)e Ft(f)p Fu(add;)7 b(mul)q(t;)g(S;)g Fx(0)p Ft(g)12 b Fx(and)427 797 y Ft(R)462 803 y Fr(1)492 797 y Fx(=)536 687 y Fg(8)536 724 y(>)536 737 y(>)536 749 y(<)536 824 y(>)536 836 y(>)536 849 y(:)593 722 y Fu(add)p Fx(\(0)p Fu(;)7 b(x)p Fx(\))125 b Ft(!)41 b Fu(x)593 772 y(add)p Fx(\()p Fu(S)r Fx(\()p Fu(x)p Fx(\))p Fu(;)7 b(y)q Fx(\))66 b Ft(!)41 b Fu(S)r Fx(\()p Fu(add)p Fx(\()p Fu(x;)7 b(y)q Fx(\)\))593 821 y Fu(mul)q(t)p Fx(\(0)p Fu(;)g(x)p Fx(\))103 b Ft(!)41 b Fx(0)593 871 y Fu(mul)q(t)p Fx(\()p Fu(S)r Fx(\()p Fu(x)p Fx(\))p Fu(;)7 b(y)q Fx(\))44 b Ft(!)d Fu(add)p Fx(\()p Fu(mul)q(t)p Fx(\()p Fu(x;)7 b(y)q Fx(\))p Fu(;)g(y)q Fx(\))p Fu(:)100 946 y Fx(Moreo)o(v)o(er,)14 b(let)g Ft(F)387 952 y Fr(2)418 946 y Fx(=)d Ft(f)p Fu(add;)c(f)t(ib;)g (S;)g Fx(0)p Ft(g)13 b Fx(and)415 1121 y Ft(R)450 1127 y Fr(2)480 1121 y Fx(=)524 986 y Fg(8)524 1023 y(>)524 1035 y(>)524 1048 y(>)524 1060 y(>)524 1073 y(<)524 1148 y(>)524 1160 y(>)524 1172 y(>)524 1185 y(>)524 1197 y(:)582 1021 y Fu(add)p Fx(\(0)p Fu(;)7 b(x)p Fx(\))112 b Ft(!)41 b Fu(x)582 1070 y(add)p Fx(\()p Fu(S)r Fx(\()p Fu(x)p Fx(\))p Fu(;)7 b(y)q Fx(\))53 b Ft(!)41 b Fu(S)r Fx(\()p Fu(add)p Fx(\()p Fu(x;)7 b(y)q Fx(\)\))582 1120 y Fu(f)t(ib)p Fx(\(0\))165 b Ft(!)41 b Fx(0)582 1170 y Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\(0\)\))106 b Ft(!)41 b Fu(S)r Fx(\(0\))582 1220 y Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(S)r Fx(\()p Fu(x)p Fx(\)\)\))j Ft(!)d Fu(add)p Fx(\()p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(x)p Fx(\)\))p Fu(;)7 b(f)t(ib)p Fx(\()p Fu(x)p Fx(\)\))p Fu(:)100 1295 y Fx(It)k(is)g(apparen)o(t)g (that)h(\()p Ft(F)490 1301 y Fr(1)508 1295 y Fu(;)7 b Ft(R)562 1301 y Fr(1)581 1295 y Fx(\))k(and)g(\()p Ft(F)736 1301 y Fr(2)755 1295 y Fu(;)c Ft(R)808 1301 y Fr(2)827 1295 y Fx(\))k(are)h(comp)q(osable)e(systems.)g Fu(mul)q(t)i Fx(is)f(the)h(only)e(blac)o(k)100 1345 y(sym)o(b)q(ol,)k Fu(f)t(ib)j Fx(is)g(the)g(only)f(white)g(sym)o(b)q(ol,)e(and)j(the)g (sym)o(b)q(ols)e Fu(add;)7 b(S;)g Fx(0)15 b(are)i(transparen)o(t.)g (Con-)100 1394 y(sider)f(the)g(mixed)e(term)h Fu(s)g Fx(=)g Fu(add)p Fx(\(0)p Fu(;)7 b(add)p Fx(\()p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(mul)q(t)p Fx(\(0)p Fu(;)g Fx(0\)\)\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)\)\).)100 1444 y(Figure)16 b(2)h(sho)o(ws)g(ho)o(w)f Fu(s)h Fx(can)g(b)q(e)g(decomp)q (osed)g(in)o(to)f(an)g(outer)i(transparen)o(t)f(and)g(further)g(inner) 100 1494 y(blac)o(k)c(and)h(white)g(parts.)g(W)m(e)f(will)g(next)h(sp)q (ecify)g(this)g(decomp)q(osition.)563 2199 y @beginspecial 132 @llx 236 @lly 480 @urx 555 @ury 1630 @rwi @setspecial %%BeginDocument: rank.ps /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def end /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 0 setlinecap 0 setlinejoin -27.0 595.5 translate 0.900 -0.900 scale 0.500 setlinewidth n 443 233 m 413 260 l gs 0.90 setgray fill gr gs col-1 s gr n 452 233 m 497 257 l gs 0.90 setgray fill gr gs col-1 s gr n 449 197 m 359 284 l 554 284 l 449 197 l gs 0.75 setgray fill gr gs col-1 s gr n 290 236 m 290 260 l gs 0.90 setgray fill gr gs col-1 s gr n 269 209 m 269 293 l 314 293 l 314 209 l 269 209 l gs col-1 s gr n 290 98 m 245 137 l gs col-1 s gr n 307 99 m 349 135 l gs col-1 s gr n 290 311 m 200 398 l 395 398 l 290 311 l gs 0.75 setgray fill gr gs col-1 s gr n 307 352 m 337 373 l gs col-1 s gr n 290 290 m 290 314 l gs 0.90 setgray fill gr gs col-1 s gr n 440 314 m 440 398 l 485 398 l 485 314 l 440 314 l gs col-1 s gr n 461 341 m 461 365 l gs 0.90 setgray fill gr gs col-1 s gr n 518 314 m 518 398 l 563 398 l 563 314 l 518 314 l gs col-1 s gr n 539 341 m 539 365 l gs 0.90 setgray fill gr gs col-1 s gr n 458 236 m 488 257 l gs col-1 s gr n 443 236 m 416 260 l gs col-1 s gr n 282 351 m 255 375 l gs col-1 s gr n 497 278 m 464 317 l gs col-1 s gr n 512 278 m 542 317 l gs col-1 s gr n 179 164 m 419 164 l 299 44 l 176 164 l 179 164 l gs 0.95 setgray fill gr gs col-1 s gr n 287 95 m 242 137 l gs col-1 s gr n 359 158 m 290 212 l gs col-1 s gr n 374 158 m 449 209 l gs col-1 s gr n 305 95 m 356 137 l gs col-1 s gr /Times-Roman findfont 18.00 scalefont setfont 401 275 m gs 1 -1 scale (0) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 281 230 m gs 1 -1 scale (fib) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 287 284 m gs 1 -1 scale (S) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 287 89 m gs 1 -1 scale (add) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 434 230 m gs 1 -1 scale (mult) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 230 152 m gs 1 -1 scale (0) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 353 155 m gs 1 -1 scale (add) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 344 389 m gs 1 -1 scale (0) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 278 344 m gs 1 -1 scale (mult) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 458 389 m gs 1 -1 scale (0) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 455 335 m gs 1 -1 scale (fib) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 536 389 m gs 1 -1 scale (0) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 530 335 m gs 1 -1 scale (fib) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 488 275 m gs 1 -1 scale (mult) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 242 389 m gs 1 -1 scale (0) col-1 show gr showpage $F2psEnd %%EndDocument @endspecial 694 2278 a Fh(Figure)f(2)h Fw(A)e(colored)e(term.)100 2451 y Fk(Lemma)16 b(4.3.)21 b Fx(Ev)o(ery)15 b(term)e Fu(s)f Ft(2)f(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\))14 b(has)g(unique)g(represen)o(tations)189 2576 y Fu(s)e Fx(=)264 2490 y Fg(8)264 2528 y(<)264 2603 y(:)322 2525 y Fu(C)355 2510 y Fp(t)369 2525 y Ft(h)p Fu(s)404 2531 y Fr(1)423 2525 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)535 2531 y Fp(l)548 2525 y Ft(i)p Fu(;)72 b Fx(where)42 b Fu(C)828 2510 y Fp(t)843 2525 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)k(2)g(T)f Fx(\()p Ft(B)q Fu(;)d Ft(V)s Fx(\))15 b Fu(;)20 b(r)q(oot)p Fx(\()p Fu(s)1315 2531 y Fp(j)1334 2525 y Fx(\))11 b Ft(2)g(A)1433 2531 y Fr(1)1461 2525 y Ft(])e(A)1531 2531 y Fr(2)322 2575 y Fu(C)355 2560 y Fp(b)371 2575 y Ft(h)p Fu(t)402 2581 y Fr(1)421 2575 y Fu(;)e(:)g(:)g(:)e(;)i(t)529 2581 y Fp(m)560 2575 y Ft(i)p Fu(;)60 b Fx(where)42 b Fu(C)828 2560 y Fp(b)845 2575 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)k(2)g(T)f Fx(\()p Ft(A)1102 2581 y Fr(1)1130 2575 y Ft(])f(B)r Fu(;)e Ft(V)s Fx(\))14 b Fu(;)20 b(r)q(oot)p Fx(\()p Fu(t)1411 2581 y Fp(j)1429 2575 y Fx(\))12 b Ft(2)f(A)1529 2581 y Fr(2)322 2625 y Fu(C)355 2610 y Fp(w)381 2625 y Ft(h)q Fu(u)422 2631 y Fr(1)440 2625 y Fu(;)c(:)g(:)g(:)e(;)i(u)557 2631 y Fp(n)578 2625 y Ft(i)q Fu(;)41 b Fx(where)h Fu(C)828 2610 y Fp(w)855 2625 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)k(2)g(T)f Fx(\()p Ft(A)1113 2631 y Fr(2)1140 2625 y Ft(])f(B)r Fu(;)e Ft(V)s Fx(\))14 b Fu(;)20 b(r)q(oot)p Fx(\()p Fu(u)1430 2631 y Fp(j)1448 2625 y Fx(\))12 b Ft(2)f(A)1548 2631 y Fr(1)1567 2625 y Fu(:)p eop %%Page: 8 8 8 7 bop 100 197 a Fw(8)105 b(E.)12 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fk(Pr)o(oof.)22 b Fx(Routine.)12 b Fe(2)100 398 y Fk(Definition)k(4.4.)21 b Fx(In)12 b(the)g(situation)f (of)g(Lemma)e(4.3,)i(w)o(e)h(will)e(use)j(the)f(follo)o(wing)d (shorthands)k(for)100 447 y(the)h(unique)g(represen)o(tations)i(of)d (the)i(term)e Fu(s)p Fx(:)666 569 y Fu(s)f Fx(=)741 484 y Fg(8)741 521 y(<)741 596 y(:)799 518 y Fu(C)832 503 y Fp(t)846 518 y Ft(h)-7 b(h)p Fu(s)890 524 y Fr(1)909 518 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1021 524 y Fp(l)1034 518 y Ft(i)-7 b(i)799 569 y Fu(C)832 554 y Fp(b)848 569 y Ft(h)g(h)p Fu(t)888 575 y Fr(1)907 569 y Fu(;)7 b(:)g(:)g(:)e(;)i(t) 1015 575 y Fp(m)1046 569 y Ft(i)-7 b(i)799 618 y Fu(C)832 603 y Fp(w)858 618 y Ft(h)g(h)p Fu(u)907 624 y Fr(1)926 618 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1043 624 y Fp(n)1064 618 y Ft(i)-7 b(i)q Fu(:)100 693 y Fx(If)13 b(w)o(e)h(ha)o(v)o(e)g (some)f(more)g(information)e(ab)q(out)j(a)f(con)o(text)i Fu(C)s Ft(h)-7 b(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)-7 b(i)p Fx(,)13 b(then)h(w)o(e)h(also)e(use)i(di\013eren)o(t)100 742 y(notations.)e(If)h(w)o(e)h(kno)o(w)f(that)h Fu(C)s Ft(h)-7 b(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)-7 b(i)12 b(6)p Fx(=)h Fo(2)p Fx(,)h(then)i(w)o(e)e(write)h Fu(C)s Ft(f)-14 b(f)p Fu(;)7 b(:)g(:)g(:)t(;)g Ft(g)-14 b(g)p Fx(.)14 b(If)g(it)g(is)g(kno)o(wn)h(that)100 792 y(furthermore)i Fu(C)s Ft(f)-14 b(f)o Fu(;)7 b(:)g(:)g(:)e(;)i Ft(g)-14 b(g)17 b Fx(con)o(tains)g(at)g(least)h(one)g(o)q(ccurrence)i(of)d Fo(2)p Fx(,)f(then)j(w)o(e)e(write)h Fu(C)s Fx([)-7 b([)o Fu(;)7 b(:)g(:)g(:)e(;)i Fx(])-7 b(])n(.)100 842 y(Moreo)o(v)o(er,)14 b(w)o(e)g(de\014ne)304 916 y Fu(S)329 922 y Fr(1)348 916 y Fx(\()p Fu(s)p Fx(\))e(=)g([)p Fu(s)486 922 y Fr(1)505 916 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)616 922 y Fp(n)639 916 y Fx(])p Fu(;)40 b(S)730 899 y Fp(w)728 926 y(P)758 916 y Fx(\()p Fu(s)p Fx(\))12 b(=)g([)p Fu(t)892 922 y Fr(1)910 916 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1018 922 y Fp(m)1049 916 y Fx(])p Fu(;)41 b(S)1141 899 y Fp(b)1139 926 y(P)1167 916 y Fx(\()p Fu(s)p Fx(\))12 b(=)g([)p Fu(u)1310 922 y Fr(1)1328 916 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1445 922 y Fp(n)1466 916 y Fx(])p Fu(:)100 990 y(t)115 996 y Fr(1)133 990 y Fu(;)g(:)g(:)g(:)e(;)i(t)241 996 y Fp(m)290 990 y Fx(\()p Fu(u)330 996 y Fr(1)348 990 y Fu(;)g(:)g(:)g(:)e(;)i(u)465 996 y Fp(n)487 990 y Fx(,)17 b(resp)q(ectiv)o(ely\))i(are)g(called)e Fs(white)h(\(black,)g(r)n(esp)n(e)n(ctively\))g(princip)n(al)f(sub-)100 1039 y(terms)12 b Fx(of)g Fu(s)p Fx(.)g(The)h Fs(topmost)h(black)g (homo)n(gene)n(ous)h(p)n(art)d Fu(top)1034 1024 y Fp(b)1051 1039 y Fx(\()p Fu(s)p Fx(\))h(of)f Fu(s)i Fx(is)e(the)h(term)f Fu(C)1435 1024 y Fp(b)1452 1039 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)12 b Fx(in)g(the)100 1089 y(unique)h(represen)o(tation)h Fu(s)e Fx(=)g Fu(C)612 1074 y Fp(b)629 1089 y Ft(h)-7 b(h)p Fu(t)669 1095 y Fr(1)688 1089 y Fu(;)7 b(:)g(:)g(:)t(;)g(t)795 1095 y Fp(m)826 1089 y Ft(i)-7 b(i)q Fx(.)12 b(The)i Fs(topmost)g(white)g(homo)n(gene)n(ous)h(p)n(art)e Fu(top)1615 1074 y Fp(w)1642 1089 y Fx(\()p Fu(s)p Fx(\))100 1139 y(and)g(the)i Fs(topmost)g(tr)n(ansp)n(ar)n(ent)f(homo)n(gene)n(ous)j (p)n(art)c Fu(top)1017 1124 y Fp(t)1032 1139 y Fx(\()p Fu(s)p Fx(\))h(of)g Fu(s)g Fx(are)g(de\014ned)h(analogously)m(.)100 1238 y Fk(Example)i(4.5.)k Fx(In)14 b(the)g(situation)f(of)h(Example)e (4.2,)g Fu(s)j Fx(has)f(represen)o(tations)120 1309 y Fu(C)153 1294 y Fp(t)168 1309 y Fx([)-7 b([)p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(mul)q(t)p Fx(\(0)p Fu(;)7 b Fx(0\)\)\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)])-7 b(])p Fu(;)19 b(C)1130 1294 y Fp(t)1145 1309 y Fx([)p Fu(;)7 b(:)g(:)g(:)t(;)g Fx(])j(=)i Fu(add)p Fx(\(0)p Fu(;)7 b(add)p Fx(\()p Fo(2)p Fu(;)g Fo(2)p Fx(\)\))120 1409 y Fu(C)153 1394 y Fp(b)170 1409 y Fx([)-7 b([)p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(mul)q(t)p Fx(\(0)p Fu(;)7 b Fx(0\)\)\))p Fu(;)g(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)])-7 b(])p Fu(;)19 b(C)852 1394 y Fp(b)868 1409 y Fx([)p Fu(;)7 b(:)g(:)g(:)t(;)g Fx(])k(=)h Fu(add)p Fx(\(0)p Fu(;)7 b(add)p Fx(\()p Fo(2)p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fo(2)p Fu(;)g Fo(2)p Fx(\)\)\)\))120 1508 y Fu(C)153 1493 y Fp(w)180 1508 y Fx([)-7 b([)p Fu(mul)q(t)p Fx(\(0)p Fu(;)7 b Fx(0\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)])-7 b(])p Fu(;)17 b(C)993 1493 y Fp(w)1020 1508 y Fx([)p Fu(;)7 b(:)g(:)g(:)t(;)g Fx(])k(=)g Fu(add)p Fx(\(0)p Fu(;)c(add)p Fx(\()p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fo(2)p Fx(\)\))p Fu(;)g Fo(2)p Fx(\)\))p Fu(:)100 1583 y Fx(T)m(o)12 b(exemplify)e(the)k(ab)q(o)o(v)o (e,)e(w)o(e)h(ha)o(v)o(e)f(for)h(instance)g Fu(S)949 1568 y Fp(w)947 1594 y(P)977 1583 y Fx(\()p Fu(s)p Fx(\))f(=)g([)p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(mul)q(t)p Fx(\(0)p Fu(;)7 b Fx(0\)\)\))p Fu(;)g(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)])100 1632 y(and)13 b Fu(top)236 1617 y Fp(b)253 1632 y Fx(\()p Fu(s)p Fx(\))f(=)g Fu(add)p Fx(\(0)p Fu(;)7 b(add)p Fx(\()p Fo(2)p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fo(2)p Fu(;)g Fo(2)p Fx(\)\)\)\).)141 1682 y(The)23 b(term)f Fu(t)k Fx(=)h Fu(mul)q(t)p Fx(\(0)p Fu(;)7 b Fx(0\))22 b(has)h(represen)o(tations)h Fu(t)i Fx(=)h Fu(C)1158 1667 y Fp(t)1172 1682 y Ft(h)-7 b(h)p Fu(t)p Ft(i)g(i)q Fx(,)22 b Fu(t)k Fx(=)g Fu(C)1404 1667 y Fp(w)1431 1682 y Ft(h)-7 b(h)p Fu(t)p Ft(i)g(i)q Fx(,)22 b(and)g Fu(t)k Fx(=)100 1732 y Fu(C)133 1717 y Fp(b)149 1732 y Ft(h)-7 b(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)-7 b(i)p Fx(,)14 b(where)j Fu(C)473 1717 y Fp(t)487 1732 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)13 b Fx(=)g Fu(C)703 1717 y Fp(w)730 1732 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)13 b Fx(=)g Fo(2)i Fx(and)g Fu(C)1074 1717 y Fp(b)1090 1732 y Ft(h)q Fu(;)7 b(:)g(:)g(:)t(;)g Ft(i)15 b Fx(con)o(tains)g(no)f(o)q (ccurrence)k(of)100 1782 y Fo(2)p Fx(,)13 b(i.e.)g(is)h(equal)g(to)g Fu(mul)q(t)p Fx(\(0)p Fu(;)7 b Fx(0\).)13 b(W)m(e)g(ha)o(v)o(e)h(for)g (instance)h Fu(S)1052 1767 y Fp(w)1050 1793 y(P)1080 1782 y Fx(\()p Fu(t)p Fx(\))d(=)g([)h(],)g Fu(S)1272 1767 y Fp(b)1270 1793 y(P)1298 1782 y Fx(\()p Fu(t)p Fx(\))f(=)h([)p Fu(t)p Fx(],)f Fu(top)1521 1767 y Fp(w)1548 1782 y Fx(\()p Fu(t)p Fx(\))g(=)g Fo(2)p Fx(,)100 1832 y(and)h Fu(top)236 1817 y Fp(b)253 1832 y Fx(\()p Fu(t)p Fx(\))f(=)g Fu(mul)q(t)p Fx(\(0)p Fu(;)7 b Fx(0\).)141 1882 y(The)19 b(term)f Fu(u)i Fx(=)g Fu(add)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)7 b(f)t(ib)p Fx(\(0\)\))19 b(has)g(represen)o(tations)h Fu(u)g Fx(=)g Fu(C)1285 1866 y Fp(t)1299 1882 y Ft(h)-7 b(h)p Fu(f)t(ib)p Fx(\(0\))p Fu(;)7 b(f)t(ib)p Fx(\(0\))p Ft(i)-7 b(i)r Fx(,)18 b Fu(u)h Fx(=)100 1931 y Fu(C)133 1916 y Fp(b)149 1931 y Ft(h)-7 b(h)p Fu(f)t(ib)p Fx(\(0\))p Fu(;)7 b(f)t(ib)p Fx(\(0\))p Ft(i)-7 b(i)r Fx(,)14 b(and)g Fu(u)e Fx(=)h Fu(C)659 1916 y Fp(w)685 1931 y Ft(h)-7 b(h)q Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)-7 b(i)o Fx(,)14 b(where)i Fu(C)1008 1916 y Fp(t)1022 1931 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)12 b Fx(=)h Fu(C)1237 1916 y Fp(b)1253 1931 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)12 b Fx(=)h Fu(add)p Fx(\()p Fo(2)p Fu(;)7 b Fo(2)p Fx(\))13 b(and)100 1981 y Fu(C)133 1966 y Fp(w)159 1981 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)j Fx(con)o(tains)g(no)g(o)q(ccurrence)j(of)d Fo(2)p Fx(,)g(i.e.)f(is)h(equal)g(to)g Fu(u)p Fx(.)f(W)m(e)h(ha)o(v)o(e)g(for) g(instance)h Fu(S)1511 1966 y Fp(b)1509 1993 y(P)1537 1981 y Fx(\()p Fu(u)p Fx(\))h(=)g([)e(],)100 2031 y Fu(S)127 2016 y Fp(w)125 2042 y(P)154 2031 y Fx(\()p Fu(u)p Fx(\))i(=)g([)p Fu(f)t(ib)p Fx(\(0\))p Fu(;)7 b(f)t(ib)p Fx(\(0\)],)14 b(and)f Fu(top)689 2016 y Fp(w)716 2031 y Fx(\()p Fu(u)p Fx(\))f(=)g Fu(u)p Fx(.)100 2130 y Fk(Definition)k(4.6.)21 b Fx(The)14 b Fs(r)n(ank)g Fx(of)f(a)h(term)f Fu(t)f Ft(2)f(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))14 b(is)g(de\014ned)h(as) f(follo)o(ws.)100 2179 y(If)f Fu(t)h Fx(is)g(a)f(top)h(blac)o(k)f(or)h (top)g(white)g(term,)f(then)100 2276 y Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))e(=)292 2218 y Fg(\032)344 2251 y Fx(1)613 b(if)13 b Fu(t)e Ft(2)g(T)g Fx(\()p Ft(A)1164 2257 y Fr(1)1192 2251 y Ft(])e(B)q Fu(;)e Ft(V)s Fx(\))j Ft([)f(T)h Fx(\()p Ft(A)1449 2257 y Fr(2)1477 2251 y Ft(])f(B)q Fu(;)e Ft(V)t Fx(\))344 2301 y(1)i(+)g Fu(max)p Ft(f)p Fu(r)q(ank)q Fx(\()p Fu(t)639 2307 y Fp(j)656 2301 y Fx(\))21 b Ft(j)g Fx(1)11 b Ft(\024)h Fu(j)i Ft(\024)e Fu(n)p Ft(g)55 b Fx(if)13 b Fu(t)e Fx(=)h Fu(C)1119 2286 y Fp(b)1135 2301 y Fx([)-7 b([)p Fu(t)1167 2307 y Fr(1)1185 2301 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1293 2307 y Fp(n)1315 2301 y Fx(])-7 b(])13 b(or)h Fu(t)d Fx(=)h Fu(C)1499 2286 y Fp(w)1526 2301 y Fx([)-7 b([)p Fu(t)1558 2307 y Fr(1)1575 2301 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1683 2307 y Fp(n)1705 2301 y Fx(])-7 b(])o Fu(:)100 2373 y Fx(If)13 b Fu(t)h Fx(is)g(a)f(top)h(transparen)o(t)h(term,)e(then)289 2469 y Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))e(=)481 2410 y Fg(\032)533 2444 y Fx(0)553 b(if)13 b Fu(t)e Ft(2)h(T)e Fx(\()p Ft(B)q Fu(;)d Ft(V)s Fx(\))533 2493 y Fu(max)p Ft(f)p Fu(r)q(ank)q Fx(\()p Fu(t)757 2499 y Fp(j)774 2493 y Fx(\))21 b Ft(j)f Fx(1)12 b Ft(\024)f Fu(j)j Ft(\024)e Fu(m)p Ft(g)56 b Fx(if)13 b Fu(t)e Fx(=)h Fu(C)1248 2478 y Fp(t)1262 2493 y Fx([)-7 b([)p Fu(t)1294 2499 y Fr(1)1312 2493 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1420 2499 y Fp(m)1451 2493 y Fx(])-7 b(])o Fu(:)141 2590 y Fx(The)13 b(term)f Fu(s)h Fx(of)f(Example)f(4.5)g(has)i(rank)g(2.)e(Sev)o(eral)i (de\014nitions)g(and)f(considerations)h(are)g(sym-)100 2640 y(metrical)c(in)i(the)g(colors)g(blac)o(k)g(and)g(white.)f (Therefore,)i(w)o(e)f(often)g(state)h(the)g(resp)q(ectiv)o(e)h (de\014nitions)p eop %%Page: 9 9 9 8 bop 636 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)g(of)h(Comp)q (osable)e(T)m(erm)i(Rewriting)f(Systems)104 b(9)p 100 224 1595 2 v 100 299 a Fx(and)15 b(considerations)h(only)e(for)h(one)g (color)g(\(the)h(same)f(applies)g(m)o(utatis)f(m)o(utandis)f(for)i(the) h(other)100 349 y(color\).)100 447 y Fk(Definition)g(4.7.)21 b Fx(Let)14 b Fu(s)e Ft(!)f Fu(t)j Fx(b)o(y)g(an)f(application)g(of)g (a)h(rewrite)h(rule)f Fu(l)e Ft(!)f Fu(r)i Ft(2)e(R)p Fx(.)100 497 y(If)i Fu(s)f Fx(=)g Fu(C)249 482 y Fp(b)265 497 y Fx([)-7 b([)p Fu(s)301 503 y Fr(1)319 497 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)431 503 y Fp(n)454 497 y Fx(])-7 b(])13 b(is)h(a)f(top)h(blac)o(k)f(term,)g(then)i(w)o(e)f(write)199 594 y Fu(s)e Ft(!)272 579 y Fp(i)297 594 y Fu(t)i Fx(if)f Fu(t)f Fx(=)f Fu(C)467 579 y Fp(b)484 594 y Fx([)p Fu(s)515 600 y Fr(1)533 594 y Fu(;)c(:)g(:)g(:)e(;)i(s)645 600 y Fp(j)r Fn(\000)p Fr(1)705 594 y Fu(;)g(t)739 600 y Fp(j)756 594 y Fu(;)g(s)794 600 y Fp(j)r Fr(+1)853 594 y Fu(;)g(:)g(:)g(:)e(;)i(s)965 600 y Fp(n)988 594 y Fx(])13 b(and)h Fu(s)1113 600 y Fp(j)1142 594 y Ft(!)d Fu(t)1210 600 y Fp(j)1242 594 y Fx(for)i(some)g Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)199 642 y Fu(s)15 b Ft(!)275 627 y Fp(o)275 653 y Fn(A)302 657 y Ff(1)333 642 y Fu(t)f Fx(otherwise.)100 740 y(If)f Fu(s)f Fx(=)g Fu(C)249 725 y Fp(t)263 740 y Fx([)-7 b([)p Fu(s)299 746 y Fr(1)317 740 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)429 746 y Fp(n)452 740 y Fx(])-7 b(])13 b(is)g(a)h(top)g(transparen)o(t)h (term,)e(then)h(w)o(e)g(write)199 838 y Fu(s)e Ft(!)272 823 y Fp(i)297 838 y Fu(t)i Fx(if)f Fu(t)f Fx(=)f Fu(C)467 823 y Fp(t)482 838 y Fx([)-7 b([)p Fu(s)518 844 y Fr(1)536 838 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)648 844 y Fp(j)r Fn(\000)p Fr(1)707 838 y Fu(;)g(t)741 844 y Fp(j)758 838 y Fu(;)g(s)796 844 y Fp(j)r Fr(+1)856 838 y Fu(;)g(:)g(:)g(:)e(;)i(s)968 844 y Fp(n)990 838 y Fx(])-7 b(])13 b(and)h Fu(s)1120 844 y Fp(j)1149 838 y Ft(!)1191 823 y Fp(i)1216 838 y Fu(t)1231 844 y Fp(j)1262 838 y Fx(for)g(some)f Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)199 885 y Fu(s)15 b Ft(!)275 870 y Fp(o)275 897 y Fn(A)302 901 y Ff(1)333 885 y Fu(t)f Fx(if)f Fu(t)e Fx(=)h Fu(C)503 870 y Fp(t)517 885 y Fx([)p Fu(s)548 891 y Fr(1)567 885 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)679 891 y Fp(j)r Fn(\000)p Fr(1)739 885 y Fu(;)g(t)773 891 y Fp(j)790 885 y Fu(;)g(s)828 891 y Fp(j)r Fr(+1)887 885 y Fu(;)g(:)g(:)g(:)e(;)i(s)999 891 y Fp(n)1021 885 y Fx(])14 b(and)f Fu(s)1146 891 y Fp(j)1178 885 y Ft(!)1220 870 y Fp(o)1220 897 y Fn(A)1247 901 y Ff(1)1278 885 y Fu(t)1293 891 y Fp(j)1325 885 y Fx(for)g(a)h(top)g(blac)o(k)f Fu(s)1624 891 y Fp(j)1642 885 y Fx(.)199 933 y Fu(s)i Ft(!)275 918 y Fp(o)275 944 y Fn(A)302 948 y Ff(2)333 933 y Fu(t)f Fx(if)f Fu(t)e Fx(=)h Fu(C)503 918 y Fp(t)517 933 y Fx([)p Fu(s)548 939 y Fr(1)567 933 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)679 939 y Fp(j)r Fn(\000)p Fr(1)739 933 y Fu(;)g(t)773 939 y Fp(j)790 933 y Fu(;)g(s)828 939 y Fp(j)r Fr(+1)887 933 y Fu(;)g(:)g(:)g(:)e(;)i(s)999 939 y Fp(n)1021 933 y Fx(])14 b(and)f Fu(s)1146 939 y Fp(j)1178 933 y Ft(!)1220 918 y Fp(o)1220 944 y Fn(A)1247 948 y Ff(2)1278 933 y Fu(t)1293 939 y Fp(j)1325 933 y Fx(for)g(a)h(top)g(white)g Fu(s)1629 939 y Fp(j)1647 933 y Fx(.)199 980 y Fu(s)e Ft(!)272 965 y Fp(t)298 980 y Fu(t)i Fx(otherwise.)100 1079 y(The)h(relations)f Ft(!)397 1064 y Fp(i)411 1079 y Fx(,)28 b Ft(!)493 1064 y Fp(o)493 1090 y Fn(A)520 1094 y Ff(1)551 1079 y Fx(,)g Ft(!)633 1064 y Fp(o)633 1090 y Fn(A)660 1094 y Ff(2)691 1079 y Fx(,)15 b Ft(!)760 1064 y Fp(o)792 1079 y Fx(=)27 b Ft(!)893 1064 y Fp(o)893 1090 y Fn(A)920 1094 y Ff(1)961 1079 y Ft([)d(!)1055 1064 y Fp(o)1055 1090 y Fn(A)1082 1094 y Ff(2)1113 1079 y Fx(,)14 b(and)h Ft(!)1263 1064 y Fp(t)1292 1079 y Fx(are)g(called)f Fs(inner,)i(black)100 1129 y(outer,)h(white)f(outer,)h(outer)p Fx(,)f(and)h Fs(tr)n(ansp)n(ar)n(ent)f Fx(reduction,)h(resp)q(ectiv)o (ely)m(.)g(W)m(e)f(will)g(also)g(use)h(the)100 1178 y(abbreviations)27 b Ft(!)413 1159 y Fp(t;o)413 1191 y Fn(A)440 1195 y Ff(1)484 1178 y Fx(=)15 b Ft(!)573 1163 y Fp(t)599 1178 y Ft([)f(!)683 1163 y Fp(o)683 1190 y Fn(A)710 1194 y Ff(1)741 1178 y Fx(,)28 b Ft(!)823 1159 y Fp(t;o)823 1191 y Fn(A)850 1195 y Ff(2)893 1178 y Fx(=)15 b Ft(!)982 1163 y Fp(t)1008 1178 y Ft([)f(!)1092 1163 y Fp(o)1092 1190 y Fn(A)1119 1194 y Ff(2)1150 1178 y Fx(,)g(and)g Ft(!)1299 1163 y Fp(t;o)1366 1178 y Fx(=)h Ft(!)1455 1163 y Fp(t)1481 1178 y Ft([)d(!)1563 1163 y Fp(o)1581 1178 y Fx(.)i(Note)100 1228 y(that)g(ev)o(ery)g(reduction)h(step)g Fu(s)d Ft(!)f Fu(t)j Fx(is)f(classi\014ed)i(b)o(y)e(the)i(ab)q(o)o(v)o(e)f (de\014nition:)f(it)g(is)h(either)h(an)e(inner)100 1278 y(or)i(a)g(blac)o(k)g(outer)h(or)f(a)g(white)h(outer)g(or)f(a)g (transparen)o(t)i(reduction)f(step.)g(Moreo)o(v)o(er,)f(if)g Fu(s)g Fx(is)h(top)100 1328 y(blac)o(k)11 b(\(top)g(white\),)h(then)g (the)g(reduction)g(step)h(cannot)e(b)q(e)i(a)e(transparen)o(t)h(or)g (white)g(\(blac)o(k\))f(outer)100 1378 y(reduction)j(step.)141 1476 y(F)m(or)19 b(disjoin)o(t)e(and)i(constructor-sharing)h(systems,)f (resp)q(ectiv)o(ely)m(,)g(w)o(e)g(will)e(use)j(the)f(common)113 1526 y Ft(!)155 1511 y Fp(o)155 1537 y Fn(R)184 1541 y Ff(1)232 1526 y Fx(instead)d(of)30 b Ft(!)483 1511 y Fp(o)483 1537 y Fn(A)510 1541 y Ff(1)557 1526 y Fx(b)q(ecause)18 b(for)e(those)h(system)f(the)h(reduction)g(in)f(a)g(\\blac)o(k)f(part") h(of)g(a)100 1576 y(term)e(implies)f(that)j(the)f(applied)g(rule)g (stems)g(exclusiv)o(ely)h(from)d Ft(R)1198 1582 y Fr(1)1217 1576 y Fx(.)h(As)i(to)f(comp)q(osable)f(TRSs,)100 1626 y(this)g(is)h(not)g(true)g(in)f(general.)h(If)f Fu(s)h Fx(is)g(for)f(instance)i(a)e(top)h(blac)o(k)f(term,)g(then)h(the)h (topmost)d(blac)o(k)100 1675 y(homogeneous)f(part)i(ma)o(y)d(v)o(ery)j (w)o(ell)f(b)q(e)h(reduced)h(b)o(y)e(a)g(rule)h(from)e Ft(R)1224 1681 y Fr(2)1256 1675 y Fx(\(but)i(nev)o(er)g(at)f(the)h(ro)q (ot!\).)100 1725 y(Ho)o(w)o(ev)o(er,)h(since)h Ft(R)418 1731 y Fr(1)452 1725 y Fx(and)g Ft(R)570 1731 y Fr(2)604 1725 y Fx(are)g(comp)q(osable,)d(w)o(e)j(kno)o(w)f(that)g(this)h(rule)g (is)f(also)g(con)o(tained)g(in)100 1775 y Ft(R)135 1781 y Fr(1)153 1775 y Fx(.)g(So)f(if)g(the)h(topmost)f(blac)o(k)g (homogeneous)g(part)h(is)g(reduced,)h(w)o(e)f(ma)o(y)d(w.l.o.g.)g (assume)j(that)100 1825 y(the)i(applied)g(rule)h(stems)f(from)e Ft(R)665 1831 y Fr(1)701 1825 y Fx(\(not)o(withstanding)i(the)g(fact)h (that)f(it)g(ma)o(y)e(also)i(stem)f(from)100 1875 y Ft(R)135 1881 y Fr(2)153 1875 y Fx(\).)100 1973 y Fk(Example)h(4.8.)k Fx(Once)g(again,)e(consider)i(the)g(TRSs)f Ft(R)1031 1979 y Fr(1)1070 1973 y Fx(and)g Ft(R)1193 1979 y Fr(2)1232 1973 y Fx(as)g(w)o(ell)g(as)g(the)h(term)e Fu(s)i Fx(of)100 2023 y(Example)12 b(4.2.)238 2094 y Fu(s)55 b Fx(=)110 b Fu(add)p Fx(\(0)p Fu(;)7 b(add)p Fx(\()p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(mul)q(t)p Fx(\(0)p Fu(;)g Fx(0\)\)\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)\)\))312 2144 y Ft(!)354 2129 y Fp(t)454 2144 y Fu(add)p Fx(\()p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\()p Fu(mul)q(t)p Fx(\(0)p Fu(;)g Fx(0\)\)\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)\))312 2194 y Ft(!)354 2178 y Fp(i)454 2194 y Fu(add)p Fx(\()p Fu(f)t(ib)p Fx(\()p Fu(S)r Fx(\(0\)\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)\))312 2243 y Ft(!)354 2228 y Fp(o)354 2255 y Fn(A)381 2259 y Ff(2)454 2243 y Fu(add)p Fx(\()p Fu(S)r Fx(\(0\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\()p Fu(f)t(ib)p Fx(\(0\))p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)\))312 2294 y Ft(!)354 2279 y Fp(i)454 2294 y Fu(add)p Fx(\()p Fu(S)r Fx(\(0\))p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(mul)q(t)p Fx(\(0)p Fu(;)g(f)t(ib)p Fx(\(0\)\)\)\))312 2344 y Ft(!)354 2329 y Fp(o)354 2355 y Fn(A)381 2359 y Ff(1)454 2344 y Fu(add)p Fx(\()p Fu(S)r Fx(\(0\))p Fu(;)g Fx(0\))p Fu(:)100 2441 y Fk(Definition)16 b(4.9.)21 b Fx(Let)12 b Fu(s)h Fx(b)q(e)f(a)g(top)g(blac)o(k)f(term.)g(A)h(rewrite)h(step)g Fu(s)f Ft(!)f Fu(t)h Fx(is)g Fs(destructive)h(at)g(level)f(1)100 2491 y Fx(if)g(the)i(ro)q(ot)g(sym)o(b)q(ols)e(of)g Fu(s)i Fx(and)f Fu(t)h Fx(ha)o(v)o(e)f(di\013eren)o(t)h(colors)g(\(that)f(is)h (to)f(sa)o(y)m(,)f Fu(r)q(oot)p Fx(\()p Fu(t)p Fx(\))g Ft(2)f(A)1501 2497 y Fr(2)1528 2491 y Ft(])d(B)i(])e(V)s Fx(\).)100 2540 y(It)19 b(is)h Fs(destructive)f(at)h(level)f Fu(m)14 b Fx(+)f(1,)19 b Fu(m)i Ft(\025)g Fx(1,)e(if)f Fu(s)k Ft(!)1000 2525 y Fp(i)1034 2540 y Fu(t)p Fx(,)d(where)h Fu(s)i Fx(=)f Fu(C)1332 2525 y Fp(b)1348 2540 y Fx([)-7 b([)p Fu(s)1384 2546 y Fr(1)1402 2540 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1514 2546 y Fp(j)1532 2540 y Fu(;)g(:)g(:)g(:)t(;)g(s)1643 2546 y Fp(n)1666 2540 y Fx(])-7 b(],)100 2590 y Fu(t)11 b Fx(=)h Fu(C)203 2575 y Fp(b)219 2590 y Fx([)p Fu(s)250 2596 y Fr(1)269 2590 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)377 2596 y Fp(j)394 2590 y Fu(;)g(:)g(:)g(:)t(;)g(s)505 2596 y Fp(n)528 2590 y Fx(],)h(and)i Fu(s)656 2596 y Fp(j)685 2590 y Ft(!)h Fu(t)753 2596 y Fp(j)780 2590 y Fx(is)f(destructiv)o(e)h (at)e(lev)o(el)g Fu(m)p Fx(.)h(F)m(or)f(a)g(top)h(transparen)o(t)g (term)100 2640 y Fu(s)p Fx(,)h(a)g(rewrite)h(step)g Fu(s)g Ft(!)f Fu(t)h Fx(is)f Fs(destructive)h(at)g(level)g(0)g Fx(if)e(the)i(ro)q(ot)f(sym)o(b)q(ols)f(of)h Fu(s)g Fx(and)g Fu(t)h Fx(ha)o(v)o(e)f(di\013eren)o(t)p eop %%Page: 10 10 10 9 bop 100 197 a Fw(10)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fx(colors)k(\(that)g(is)g(to)g(sa)o(y)m(,)f Fu(r)q(oot)p Fx(\()p Fu(t)p Fx(\))g Ft(2)f(A)712 305 y Fr(1)741 299 y Ft(])d(A)812 305 y Fr(2)831 299 y Fx(\).)k(It)i(is)f Fs(destructive)g(at)h(level)g(m)f Fx(for)f(some)h Fu(m)f Ft(\025)g Fx(1)g(if)100 349 y(it)f(has)g(a)h(represen)o(tation)h(of)e (the)h(form)d Fu(s)h Fx(=)g Fu(C)843 334 y Fp(t)857 349 y Fx([)-7 b([)p Fu(s)893 355 y Fr(1)912 349 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)1023 355 y Fp(j)1041 349 y Fu(;)g(:)g(:)g(:)e(;)i (s)1153 355 y Fp(n)1175 349 y Fx(])-7 b(])11 b Ft(!)g Fu(C)1289 334 y Fp(t)1303 349 y Fx([)p Fu(s)1334 355 y Fr(1)1353 349 y Fu(;)c(:)g(:)g(:)e(;)i(t)1461 355 y Fp(j)1477 349 y Fu(;)g(:)g(:)g(:)e(;)i(s)1589 355 y Fp(n)1612 349 y Fx(])k(=)h Fu(t)100 399 y Fx(with)h Fu(s)213 405 y Fp(j)243 399 y Ft(!)e Fu(t)311 405 y Fp(j)342 399 y Fx(destructiv)o(e)16 b(at)d(lev)o(el)h Fu(m)p Fx(.)141 500 y(In)19 b(the)g(reduction)h(sequence)g(of)f(Example)e(4.8,)g(the)i (third)g(and)g(the)g(last)f(rewrite)i(steps)g(are)100 550 y(destructiv)o(e)15 b(at)f(lev)o(el)f(1,)h(whereas)h(the)f(second)h (and)f(fourth)g(are)g(destructiv)o(e)i(at)d(lev)o(el)h(2.)100 652 y Fk(Lemma)i(4.10.)21 b Fx(Let)15 b Fu(s)d Ft(!)f Fu(t)j Fx(b)o(y)f(an)h(application)e(of)i(some)f(rule)h Fu(l)f Ft(!)e Fu(r)h Ft(2)f(R)p Fx(.)125 754 y(\(1\))21 b(If)14 b Fu(s)e Ft(!)f Fu(t)j Fx(is)f(destructiv)o(e)j(at)e(lev)o(el)f (0,)g(then)i Fu(l)e Ft(!)e Fu(r)j Fx(is)g(a)g(shared)g(collapsing)f (rule.)125 806 y(\(2\))21 b(If)14 b Fu(s)e Ft(!)f Fu(t)j Fx(is)f(destructiv)o(e)j(at)d(lev)o(el)h Fu(m)e(>)g Fx(0,)h(then)h Fu(r)q(oot)p Fx(\()p Fu(l)q Fx(\))f Ft(2)e(A)1190 812 y Fr(1)1218 806 y Ft(])d(A)1288 812 y Fr(2)1320 806 y Fx(and)14 b Fu(r)q(oot)p Fx(\()p Fu(r)q Fx(\))e Ft(2)f(B)f(])f(V)t Fx(.)100 907 y Fk(Pr)o(oof.)22 b Fx(Straigh)o(tforw)o(ard.)12 b Fe(2)141 1008 y Fx(In)h(order)g(to)g(co)q(de)g(certain)g(sp)q(ecial)g (subterms)g(b)o(y)f(v)n(ariables)g(and)h(to)f(cop)q(e)i(with)e (transparen)o(t)i(or)100 1058 y(outer)g(rewrite)h(steps)g(using)f (non-left-linear)f(rules,)h(the)g(follo)o(wing)d(notation)j(is)f(con)o (v)o(enien)o(t.)100 1160 y Fk(Definition)j(4.11.)21 b Fx(Let)13 b Fu(s)543 1166 y Fr(1)562 1160 y Fu(;)7 b(:)g(:)g(:)e(;)i(s) 674 1166 y Fp(n)709 1160 y Fx(and)13 b Fu(t)804 1166 y Fr(1)822 1160 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)930 1166 y Fp(n)965 1160 y Fx(b)q(e)13 b(sequences)i(of)e(terms)f(from)f Ft(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(.)12 b(W)m(e)100 1210 y(write)e Fu(s)221 1216 y Fr(1)241 1210 y Fu(;)d(:)g(:)g(:)t(;)g (s)352 1216 y Fp(n)386 1210 y Ft(/)12 b Fu(t)445 1216 y Fr(1)464 1210 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)572 1216 y Fp(n)604 1210 y Fx(if)j Fu(s)658 1216 y Fp(i)684 1210 y Fx(=)h Fu(s)746 1216 y Fp(j)775 1210 y Fx(implies)d Fu(t)927 1216 y Fp(i)953 1210 y Fx(=)j Fu(t)1011 1216 y Fp(j)1039 1210 y Fx(for)g(all)e(1)i Ft(\024)h Fu(i)g(<)g(j)i Ft(\024)d Fu(n)p Fx(.)f(If)g(w)o(e)h(ha)o(v)o(e)f(b)q(oth)100 1260 y Fu(s)119 1266 y Fr(1)138 1260 y Fu(;)d(:)g(:)g(:)e(;)i(s)250 1266 y Fp(n)284 1260 y Ft(/)k Fu(t)342 1266 y Fr(1)361 1260 y Fu(;)c(:)g(:)g(:)e(;)i(t)469 1266 y Fp(n)505 1260 y Fx(and)13 b Fu(t)600 1266 y Fr(1)619 1260 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)727 1266 y Fp(n)760 1260 y Ft(/)12 b Fu(s)823 1266 y Fr(1)842 1260 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)954 1266 y Fp(n)977 1260 y Fx(,)13 b(then)h(w)o(e)h(write)f Fu(s)1283 1266 y Fr(1)1302 1260 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1414 1266 y Fp(n)1450 1260 y Ft(1)13 b Fu(t)1520 1266 y Fr(1)1539 1260 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1647 1266 y Fp(n)1669 1260 y Fx(.)141 1362 y(W)m(e)14 b(omit)d(the)k(simple)d(pro)q(ofs)i(of) g(the)g(follo)o(wing)d(lemmata.)100 1464 y Fk(Lemma)16 b(4.12.)21 b Fx(The)15 b(pair)e(\()p Ft(B)r Fu(;)7 b Ft(S)s Fx(\))14 b(is)f(a)h(term)f(rewriting)h(system.)100 1565 y Fk(Lemma)i(4.13.)21 b Fx(Let)11 b Fu(s;)c(t)j Fx(b)q(e)h(terms)g(suc)o(h)g(that)f Fu(s)i Ft(!)f Fu(t)f Fx(b)o(y)h(an)f(application)f(of)g(some)h(rule)h Fu(l)h Ft(!)f Fu(r)i Ft(2)e(R)p Fx(.)100 1615 y(Then)k Fu(s)f Ft(2)e(T)e Fx(\()p Ft(B)q Fu(;)d Ft(V)s Fx(\))16 b(implies)d Fu(l)h Ft(!)f Fu(r)h Ft(2)f(S)18 b Fx(and)c Fu(t)g Ft(2)e(T)f Fx(\()p Ft(B)q Fu(;)c Ft(V)s Fx(\))q(.)14 b(Moreo)o(v)o(er,)h(the)g (restrictions)h(of)f Ft(!)1670 1621 y Fn(S)100 1665 y Fx(and)e Ft(!)222 1671 y Fn(R)266 1665 y Fx(to)h Ft(T)c Fx(\()p Ft(B)q Fu(;)d Ft(V)s Fx(\))15 b(coincide.)141 1767 y(The)e(follo)o(wing)c(facts)j(will)f(b)q(e)i(hea)o(vily)e(used)i (in)e(the)i(sequel)g(\(sometimes)d(without)i(b)q(eing)g(explic-)100 1817 y(itly)h(men)o(tioned\).)100 1919 y Fk(Lemma)j(4.14.)21 b Fx(If)15 b Fu(s)f Ft(!)488 1904 y Fp(t)516 1919 y Fu(t)p Fx(,)h(then)h Fu(s)e Fx(=)g Fu(C)766 1904 y Fp(t)780 1919 y Ft(f)-14 b(f)p Fu(s)827 1925 y Fr(1)846 1919 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)958 1925 y Fp(n)980 1919 y Ft(g)-14 b(g)p Fu(;)7 b(t)13 b Fx(=)1111 1908 y(^)1101 1919 y Fu(C)1134 1904 y Fp(t)1149 1919 y Ft(h)-7 b(h)p Fu(s)1193 1925 y Fp(i)1205 1929 y Ff(1)1223 1919 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1335 1925 y Fp(i)1347 1929 y Fl(m)1376 1919 y Ft(i)-7 b(i)16 b Fx(for)f(some)f(trans-)100 1968 y(paren)o(t)h(con)o(texts)g Fu(C)428 1953 y Fp(t)442 1968 y Ft(f)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(g)p Fx(,)611 1958 y(^)602 1968 y Fu(C)635 1953 y Fp(t)649 1968 y Ft(h)p Fu(;)g(:)g(:)g(:)e(;)i Ft(i)p Fx(,)14 b Fu(i)814 1974 y Fr(1)833 1968 y Fu(;)7 b(:)g(:)g(:)t(;)g(i)939 1974 y Fp(m)983 1968 y Ft(2)12 b(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(,)13 b(and)h(terms)g Fu(s)1445 1974 y Fr(1)1464 1968 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1576 1974 y Fp(n)1613 1968 y Fx(with)100 2018 y Fu(r)q(oot)p Fx(\()p Fu(s)210 2024 y Fp(j)228 2018 y Fx(\))16 b Ft(2)f(A)336 2024 y Fr(1)366 2018 y Ft(])10 b(A)438 2024 y Fr(2)456 2018 y Fx(.)16 b(If)g Fu(s)g Ft(!)605 2003 y Fp(t)635 2018 y Fu(t)g Fx(is)h(not)f(destructiv)o(e)i(at)e(lev)o(el)g(0,)g(then)h Fu(t)f Fx(=)1386 2008 y(^)1377 2018 y Fu(C)1410 2003 y Fp(t)1424 2018 y Ft(f)-14 b(f)p Fu(s)1471 2024 y Fp(i)1483 2028 y Ff(1)1501 2018 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1613 2024 y Fp(i)1625 2028 y Fl(m)1654 2018 y Ft(g)-14 b(g)p Fx(.)100 2074 y(If)14 b Fu(C)175 2059 y Fp(t)189 2074 y Ft(f)-14 b(f)p Fu(s)236 2080 y Fr(1)255 2074 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)367 2080 y Fp(n)389 2074 y Ft(g)-14 b(g)13 b(!)472 2059 y Fp(t)508 2063 y Fx(^)499 2074 y Fu(C)532 2059 y Fp(t)546 2074 y Ft(h)-7 b(h)p Fu(s)590 2080 y Fp(i)602 2084 y Ff(1)621 2074 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)733 2080 y Fp(i)745 2084 y Fl(m)774 2074 y Ft(i)-7 b(i)p Fx(,)14 b(b)o(y)h(application)e(of)h(some)g(rule)h Fu(l)f Ft(!)e Fu(r)i Ft(2)f(R)p Fx(,)h(then)h(w)o(e)100 2123 y(also)f(ha)o(v)o(e)g Fu(C)313 2108 y Fp(t)327 2123 y Ft(f)p Fu(t)363 2129 y Fr(1)382 2123 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)490 2129 y Fp(n)512 2123 y Ft(g)12 b(!)609 2113 y Fx(^)600 2123 y Fu(C)633 2108 y Fp(t)647 2123 y Ft(h)p Fu(t)678 2129 y Fp(i)690 2133 y Ff(1)708 2123 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)816 2129 y Fp(i)828 2133 y Fl(m)857 2123 y Ft(i)15 b Fx(b)o(y)f(an)g (application)g(of)g(the)h(same)f(rule)h Fu(l)f Ft(!)e Fu(r)k Fx(for)100 2173 y(all)d(terms)h Fu(t)289 2179 y Fr(1)308 2173 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)416 2179 y Fp(n)452 2173 y Fx(with)14 b Fu(s)566 2179 y Fr(1)585 2173 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)697 2179 y Fp(n)747 2173 y Ft(/)27 b Fu(t)821 2179 y Fr(1)839 2173 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)947 2179 y Fp(n)969 2173 y Fx(.)14 b(Moreo)o(v)o(er,)h Fu(l)f Ft(!)e Fu(r)h Ft(2)f(S)s Fx(.)i(Please)i (note)f(that)100 2223 y(analogous)d(statemen)o(ts)i(hold)g(for)f Fu(s)i Ft(!)732 2203 y Fp(t;o)732 2235 y Fn(A)759 2239 y Ff(1)790 2223 y Fu(t)f Fx(and)f Fu(s)i Ft(!)975 2203 y Fp(t;o)975 2235 y Fn(A)1002 2239 y Ff(2)1033 2223 y Fu(t)p Fx(.)100 2331 y Fk(Lemma)h(4.15.)21 b Fx(Let)12 b Fu(s)g Fx(=)g Fu(C)550 2316 y Fp(b)566 2331 y Fx([)-7 b([)p Fu(s)602 2337 y Fr(1)621 2331 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)732 2337 y Fp(n)755 2331 y Fx(])-7 b(])o(.)11 b(If)g Fu(s)j Ft(!)908 2316 y Fp(o)908 2342 y Fn(A)935 2346 y Ff(2)967 2331 y Fu(t)d Fx(or)g Fu(s)h Ft(!)1114 2316 y Fp(i)1139 2331 y Fu(t)p Fx(,)f(then)h(there)h(is)e(a)g Fu(j)j Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)100 2386 y Fx(suc)o(h)16 b(that)g Fu(s)306 2392 y Fp(j)338 2386 y Ft(!)e Fu(t)409 2392 y Fp(j)442 2386 y Fx(for)h(some)g(term)g Fu(t)729 2392 y Fp(j)746 2386 y Fx(.)g(That)h(is,)f Fu(t)f Fx(=)h Fu(C)1043 2371 y Fp(b)1059 2386 y Fx([)p Fu(s)1090 2392 y Fr(1)1109 2386 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1217 2392 y Fp(j)1234 2386 y Fu(;)g(:)g(:)g(:)t(;)g(s)1345 2392 y Fp(n)1368 2386 y Fx(].)14 b(If)i(the)g(reduction)100 2436 y(step)e(is)g(non-destructiv)o(e,)h(then)g Fu(t)c Fx(=)h Fu(C)732 2421 y Fp(b)748 2436 y Fx([)-7 b([)p Fu(s)784 2442 y Fr(1)802 2436 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)910 2442 y Fp(j)927 2436 y Fu(;)g(:)g(:)g(:)e(;)i(s)1039 2442 y Fp(n)1061 2436 y Fx(])-7 b(].)141 2537 y(No)o(w)15 b(it)g(is)g(p)q(ossible)g(to)g(pro)o(v)o(e)g(that)h(the)f(rank)h(of)e (a)h(term)f(is)h(nev)o(er)i(increased)f(b)o(y)f(a)g(reduction)100 2587 y(step)j Fu(s)g Ft(!)e Fu(t)p Fx(.)h(This)h(can)f(b)q(e)h(done)g (b)o(y)f(induction)g(on)g Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))g(and)h(further)g(distinguishing)e(the)100 2637 y(cases)f Fu(s)d Ft(!)277 2622 y Fp(i)302 2637 y Fu(t)p Fx(,)h Fu(s)f Ft(!)415 2622 y Fp(o)415 2649 y Fn(A)442 2653 y Fl(j)470 2637 y Fu(t)p Fx(,)i(and)f Fu(s)f Ft(!)664 2622 y Fp(t)690 2637 y Fu(t)p Fx(.)p eop %%Page: 11 11 11 10 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(11)p 100 224 1595 2 v 100 299 a Fk(Pr)o(oposition)16 b(4.16.)21 b Fx(If)13 b Fu(s)f Ft(!)597 284 y Fn(\003)627 299 y Fu(t)p Fx(,)i(then)g Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))e Ft(\025)g Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\).)100 402 y Fk(Definition)k(4.17.)21 b Fx(Let)d Fu(\033)h Fx(and)f Fu(\034)23 b Fx(b)q(e)18 b(substitutions.)g(W)m(e)g(write)g Fu(\033)i Ft(/)e Fu(\034)23 b Fx(if)17 b Fu(x\033)i Fx(=)g Fu(y)q(\033)h Fx(implies)100 452 y Fu(x\034)e Fx(=)c Fu(y)q(\034)21 b Fx(for)15 b(all)f Fu(x;)7 b(y)15 b Ft(2)e(V)s Fx(.)j(The)g(notation)e Fu(\033)h Ft(!)900 437 y Fn(\003)933 452 y Fu(\034)20 b Fx(is)15 b(used)h(if)e Fu(x\033)h Ft(!)1254 437 y Fn(\003)1287 452 y Fu(x\034)20 b Fx(for)14 b(all)g Fu(x)g Ft(2)g(V)s Fx(.)h(Note)100 502 y(that)g Fu(\033)h Ft(!)273 487 y Fn(\003)306 502 y Fu(\034)k Fx(implies)13 b Fu(t\033)j Ft(!)583 487 y Fn(\003)616 502 y Fu(t\034)k Fx(for)15 b(all)f Fu(t)h Ft(2)f(T)c Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\).)15 b(Moreo)o(v)o(er,)h Fu(\033)h Fx(is)e(said)g(to)h(b)q(e)g(in)f Fs(normal)100 552 y(form)f Fx(or)g Ft(!)h Fs(normalize)n(d)f Fx(if)g Fu(x\033)g Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(!)p Fx(\))13 b(for)i(ev)o(ery)g Fu(x)e Ft(2)g(V)t Fx(.)h(A)h(substitution)g Fu(\033)g Fx(is)g(called)g Fs(black)100 602 y Fx(if)f Fu(x\033)h Fx(is)g(blac)o(k)f(for)h(all)f Fu(x)e Ft(2)h(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))j(and)f(it)f(is)h(said)g(to)f(b)q(e)i Fs(top)g(black)f Fx(if)f Fu(x\033)h Fx(is)g(top)g(blac)o(k)f(for)h(all) 100 652 y Fu(x)c Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))q(.)100 755 y Fk(Pr)o(oposition)16 b(4.18.)21 b Fx(Ev)o(ery)15 b(substitution)g Fu(\033)h Fx(can)f(b)q(e)g(decomp)q (osed)g(in)o(to)f Fu(\033)1351 761 y Fr(2)1379 755 y Ft(\016)9 b Fu(\033)1433 761 y Fr(1)1466 755 y Fx(suc)o(h)16 b(that)e Fu(\033)1675 761 y Fr(1)100 805 y Fx(is)f(blac)o(k)h(and)f Fu(\033)354 811 y Fr(2)387 805 y Fx(is)g(top)h(white)g(and)g Fu(\033)720 811 y Fr(2)750 805 y Ft(/)e Fu(\017)h Fx(\(recall)h(that)g Fu(\017)g Fx(denotes)h(the)f(empt)o(y)f(substitution\).)100 907 y Fk(Pr)o(oof.)22 b Fx(Essen)o(tially)13 b(the)i(same)e(as)h(for)f (disjoin)o(t)g(systems,)h(see)h(Middeldorp)e(\(1990,)g(1993\).)f Fe(2)418 1043 y Fv(5.)24 b(Mo)q(dular)15 b(prop)q(erties)e(of)i(comp)q (osable)f(systems)656 1118 y Fk(5.1.)24 b(semi-completeness)141 1223 y Fx(Our)12 b(\014rst)g(result)g(is)f(the)g(mo)q(dularit)o(y)e(of) h(semi-completeness)h(for)g(comp)q(osable)f(TRSs.)g(F)m(rom)g(our)100 1273 y(p)q(oin)o(t)i(of)g(view,)g(this)h(result)g(is)g(v)o(ery)g(imp)q (ortan)o(t)e(b)q(ecause)j(semi-completeness)e(is)h(one)f(of)h(the)g (most)100 1323 y(desirable)c(prop)q(erties)i(of)e(TRSs.)g(Let)h(us)g (mak)o(e)e(this)h(more)g(precise.)h(A)g(TRS)e(is)i(a)f(kind)g(of)g (applicativ)o(e)100 1373 y(program)h(that)h(computes)h(b)o(y)f (reducing)h(terms)g(to)f(other)h(terms.)f(The)h(p)q(oin)o(t)f(of)g(a)g (computation)f(is,)100 1423 y(of)h(course,)i(its)f(result)h(whic)o(h)e (consists)i(of)f(an)f(irreducible)i(term.)e(If)h(the)g(TRS)g(under)g (consideration)100 1473 y(is)i(con\015uen)o(t,)i(then)f(w)o(e)g(kno)o (w)f(that)h(a)g(computed)f(result)i(is)e(uniquely)h(determined.)f(That) h(is,)f(the)100 1522 y(normal)f(form)h(obtained)h(is)g(indep)q(enden)o (t)i(of)e(the)h(strategy)g(used)g(to)f(compute)g(it.)g(If)g(the)h(TRS)f (is)100 1572 y(also)e(normalizing,)d(then)15 b(w)o(e)f(kno)o(w)f(in)g (addition)g(that)h(ev)o(ery)g(term)f(has)h(a)g(normal)d(form.)h(Th)o (us,)h(if)100 1622 y(the)g(TRS)g(is)g(semi-complete,)e(then)j(ev)o(ery) g(term)f(has)g(a)g(unique)h(normal)d(form)g(and)i(all)f(w)o(e)i (further)100 1672 y(need)k(is)g(a)f(\(go)q(o)q(d\))h(normalizing)d (reduction)k(strategy)f(to)g(compute)f(that)h(unique)g(normal)d(form) 100 1722 y(\(e\016cien)o(tly\).)10 b(On)i(the)f(other)h(hand,)e(there)j (is)e(hardly)f(a)h(metho)q(d)f(to)h(pro)o(v)o(e)g(semi-completeness)f (of)h(a)100 1771 y(TRS.)f(In)g(practise,)i(one)f(alw)o(a)o(ys)f(tries)h (to)g(apply)f(the)h(follo)o(wing)d(tec)o(hnique)k(to)f(pro)o(v)o(e)g Fs(c)n(ompleteness)p Fx(:)100 1821 y(A)o(t)i(\014rst,)g(termination)f (of)g(the)i(TRS)f(is)g(pro)o(v)o(ed)g(\(mostly)e(b)o(y)i(some)f (simpli\014cation)f(ordering\),)i(and)100 1871 y(then)20 b(con)o(v)o(ergence)h(of)e(all)f(critical)h(pairs)h(is)f(c)o(hec)o(k)o (ed.)i(Moreo)o(v)o(er,)e(man)o(y)f(complete)h(TRSs)g(are)100 1921 y(obtained)g(via)h(Kn)o(uth-Bendix)g(completion.)e(Since)j (completeness)f(is)g(not)g(mo)q(dular)e(\(ev)o(en)j(for)100 1971 y(disjoin)o(t)11 b(systems\),)h(the)h(com)o(bination)c(of)j (pairwise)g(comp)q(osable)f(complete)h(TRSs)g(do)q(es)h(not)f(yield)100 2020 y(a)i(complete)g(system.)g(Ho)o(w)o(ev)o(er,)h(it)f(yields)h(a)f (semi-complete)f(and)h(innermost)g(terminating)f(TRS)100 2070 y(\(see)i(b)q(elo)o(w\).)f(So)g(in)f(this)i(v)o(ery)f(imp)q(ortan) o(t)e(case)k(the)e(unique)h(normal)d(form)g(w.r.t.)h(the)i(com)o(bined) 100 2120 y(system)e(can)h(b)q(e)h(obtained)f(b)o(y)f(an)o(y)h (innermost)f(reduction)h(strategy)m(.)100 2224 y Fk(Lemma)i(5.1.)21 b Fx(Let)15 b(\()p Ft(F)468 2230 y Fr(1)486 2224 y Fu(;)7 b Ft(R)540 2230 y Fr(1)559 2224 y Fx(\))14 b(and)f(\()p Ft(F)715 2230 y Fr(2)734 2224 y Fu(;)7 b Ft(R)788 2230 y Fr(2)806 2224 y Fx(\))14 b(b)q(e)h(comp)q(osable)d(TRSs.)125 2328 y(\(1\))21 b(If)14 b(one)g(of)f(the)i(systems)f(is)f(con\015uen)o (t,)i(then)f(\()p Ft(B)q Fu(;)7 b Ft(S)s Fx(\))15 b(is)e(con\015uen)o (t.)125 2381 y(\(2\))21 b(If)14 b(one)g(of)f(the)i(systems)f(is)f (normalizing,)e(then)k(\()p Ft(B)q Fu(;)7 b Ft(S)s Fx(\))14 b(is)g(normalizing.)125 2435 y(\(3\))21 b(If)14 b(one)g(of)f(the)i (systems)f(is)f(semi-complete,)f(then)j(\()p Ft(B)q Fu(;)7 b Ft(S)s Fx(\))14 b(is)g(semi-complete.)100 2536 y Fk(Pr)o(oof.)22 b Fx(Routine.)12 b Fe(2)141 2640 y Fx(The)19 b(basic)g(pro)q(of)f(idea) g(of)g(Theorem)g(5.2)g(is)g(illustrated)h(in)f(Figure)h(3.)e(F)m(or)i (eac)o(h)g(term)f(in)g(a)p eop %%Page: 12 12 12 11 bop 100 197 a Fw(12)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 281 872 a @beginspecial 58 @llx 271 @lly 555 @urx 520 @ury 2982 @rwi @setspecial %%BeginDocument: figure1.ps /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix } def end /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 0 setlinecap 0 setlinejoin -95.5 524.5 translate 0.900 -0.900 scale 0.500 setlinewidth [4.000000] 0 setdash n 199 119 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash [4.000000] 0 setdash n 279 79 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash [4.000000] 0 setdash n 359 44 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash [4.000000] 0 setdash n 439 79 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash [4.000000] 0 setdash n 519 119 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash [4.000000] 0 setdash n 599 159 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash n 279 274 5 5 0 360 DrawEllipse gs col-1 s gr n 199 274 5 5 0 360 DrawEllipse gs col-1 s gr n 360 274 5 5 0 360 DrawEllipse gs col-1 s gr n 439 274 5 5 0 360 DrawEllipse gs col-1 s gr n 519 274 5 5 0 360 DrawEllipse gs col-1 s gr n 599 274 5 5 0 360 DrawEllipse gs col-1 s gr n 354 44 m 289 74 l gs col-1 s gr n 354 44 m 289 74 l gs col-1 s gr n 354 44 m 289 74 l gs col-1 s gr n 297.102 72.463 m 289.000 74.000 l 295.426 68.832 l gs 2 setlinejoin col-1 s gr n 274 84 m 209 114 l gs col-1 s gr n 217.102 112.463 m 209.000 114.000 l 215.426 108.832 l gs 2 setlinejoin col-1 s gr n 364 44 m 429 74 l gs col-1 s gr n 422.574 68.832 m 429.000 74.000 l 420.898 72.463 l gs 2 setlinejoin col-1 s gr n 448 86 m 513 116 l gs col-1 s gr n 506.574 110.832 m 513.000 116.000 l 504.898 114.463 l gs 2 setlinejoin col-1 s gr n 524 123 m 589 153 l gs col-1 s gr n 582.574 147.832 m 589.000 153.000 l 580.898 151.463 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 279 89 m 279 264 l gs 0.50 setgray fill gr gs col-1 s gr [] 0 setdash n 281.000 256.000 m 279.000 264.000 l 277.000 256.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 359 49 m 359 264 l gs 0.50 setgray fill gr gs col-1 s gr [] 0 setdash n 361.000 256.000 m 359.000 264.000 l 357.000 256.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 439 89 m 439 264 l gs 0.50 setgray fill gr gs col-1 s gr [] 0 setdash n 441.000 256.000 m 439.000 264.000 l 437.000 256.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 519 129 m 519 264 l gs 0.50 setgray fill gr gs col-1 s gr [] 0 setdash n 521.000 256.000 m 519.000 264.000 l 517.000 256.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 599 164 m 599 264 l gs 0.50 setgray fill gr gs col-1 s gr [] 0 setdash n 601.000 256.000 m 599.000 264.000 l 597.000 256.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 199 129 m 199 264 l gs 0.50 setgray fill gr gs col-1 s gr [] 0 setdash n 201.000 256.000 m 199.000 264.000 l 197.000 256.000 l gs 2 setlinejoin col-1 s gr /Symbol findfont 12.00 scalefont setfont 269 264 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 349 264 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 429 264 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 509 264 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 589 264 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 189 264 m gs 1 -1 scale (*) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 356 32 m gs 1 -1 scale (t) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 170 122 m gs 1 -1 scale (t) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 611 161 m gs 1 -1 scale (t) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 179 125 m gs 1 -1 scale (1) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 620 164 m gs 1 -1 scale (2) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 233 281 m gs 1 -1 scale (=) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 314 281 m gs 1 -1 scale (=) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 395 281 m gs 1 -1 scale (=) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 473 281 m gs 1 -1 scale (=) col-1 show gr /Times-Roman findfont 24.00 scalefont setfont 554 281 m gs 1 -1 scale (=) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 620 281 m gs 1 -1 scale (normal forms) col-1 show gr showpage $F2psEnd %%EndDocument @endspecial 574 951 a Fh(Figure)h(3)i Fw(The)e(pro)q(of)e(idea)h(of)g (Theorem)f(5.2.)100 1086 y Fx(con)o(v)o(ersion)18 b Fu(t)321 1092 y Fr(1)360 1071 y Fn(\003)377 1086 y Ft( )13 b Fu(t)19 b Ft(!)508 1071 y Fn(\003)545 1086 y Fu(t)560 1092 y Fr(2)596 1086 y Fx(w)o(e)g(construct)g(a)f(normal)e(form)1118 1071 y Ft(y)1155 1086 y Fx(and)i(then)g(sho)o(w)g(that)g(all)f(these) 100 1136 y(normal)10 b(forms)h(are)h(iden)o(tical.)f(Hence)j(ev)o(ery)f (term)f Fu(t)g Fx(has)g(a)g(unique)h(normal)d(form.)g(The)i (simpli\014ed)100 1186 y(pro)q(of)k(of)h(the)h(mo)q(dularit)o(y)d(of)h (con\015uence)j(for)e(disjoin)o(t)f(TRSs)h(giv)o(en)g(b)o(y)g(Klop)g Fs(et)h(al.)e Fx(\(1994\))h(is)100 1236 y(based)c(on)g(a)f(similar)e (idea.)i(There,)i(ev)o(ery)f(term)f(in)h(a)f(con)o(v)o(ersion)h Fu(t)1171 1242 y Fr(1)1211 1221 y Fn(\003)1227 1236 y Ft( )h Fu(t)d Ft(!)1351 1221 y Fn(\003)1381 1236 y Fu(t)1396 1242 y Fr(2)1428 1236 y Fx(is)h(\014rst)i(reduced)100 1286 y(to)c(a)h(so-called)f(witness)h(and)g(then)g(it)g(is)f(sho)o(wn)h (that)f(these)j(witnesses)f(ha)o(v)o(e)e(a)h(common)d(reduct.)j(As)100 1336 y(a)g(matter)h(of)f(fact,)h(their)g(approac)o(h)h(has)f(b)q(een)h (extended)h(to)e(comp)q(osable)f(systems.)g(In)h(Ohlebusc)o(h)100 1385 y(\(1994)p Fs(b)p Fx(\),)k(it)h(is)g(sho)o(wn)g(that)g (con\015uence)i(is)f(mo)q(dular)d(for)i(comp)q(osable)f(systems)h(pro)o (vided)h(that)100 1435 y(a)d(certain)i(collapsing)e(reduction)i (relation)e Ft(!)854 1441 y Fp(c)887 1435 y Fx(is)g(normalizing.)e(In)j (the)h(case)g(of)e(semi-complete)100 1485 y(constructor-sharing)i (TRSs,)f(w)o(e)h(kno)o(w)f(that)h Ft(!)905 1491 y Fp(c)938 1485 y Fx(is)f(normalizing,)d(see)18 b(Ohlebusc)o(h)g(\(1994)p Fs(a,)e(b)p Fx(\).)100 1535 y(Ho)o(w)o(ev)o(er,)g(it)g(is)h(unkno)o(wn) f(whether)i Ft(!)754 1541 y Fp(c)787 1535 y Fx(is)e(also)g(normalizing) e(for)i(semi-complete)f(comp)q(osable)100 1585 y(TRSs.)e(Th)o(us)h(w)o (e)g(use)h(a)f(di\013eren)o(t)g(approac)o(h.)100 1686 y Fk(Theorem)i(5.2.)21 b Fx(Semi-completeness)13 b(is)h(a)g(mo)q(dular) e(prop)q(ert)o(y)i(of)g(comp)q(osable)e(TRSs.)100 1787 y Fk(Pr)o(oof.)22 b Fx(Let)16 b(\()p Ft(F)386 1793 y Fr(1)405 1787 y Fu(;)7 b Ft(R)459 1793 y Fr(1)477 1787 y Fx(\))16 b(and)g(\()p Ft(F)638 1793 y Fr(2)657 1787 y Fu(;)7 b Ft(R)711 1793 y Fr(2)729 1787 y Fx(\))17 b(b)q(e)f(t)o(w)o (o)g(comp)q(osable)f(TRSs.)h(It)g(has)g(to)g(b)q(e)h(sho)o(wn)g(that) 100 1836 y(their)e(com)o(bined)e(system)h(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(is)h(semi-complete)d(if)i(and)h(only)e(if)h (\()p Ft(F)1268 1842 y Fr(1)1287 1836 y Fu(;)7 b Ft(R)1341 1842 y Fr(1)1359 1836 y Fx(\))14 b(and)h(\()p Ft(F)1517 1842 y Fr(2)1535 1836 y Fu(;)7 b Ft(R)1589 1842 y Fr(2)1608 1836 y Fx(\))14 b(are)100 1886 y(semi-complete.)i(The)i(only-if)f (direction)i(is)f(straigh)o(tforw)o(ard,)f(so)i(supp)q(ose)g(that)g(\() p Ft(F)t Fu(;)7 b Ft(R)o Fx(\))19 b(is)f(the)100 1936 y(com)o(bined)g(system)h(of)g(t)o(w)o(o)g(semi-complete)e(comp)q (osable)h(TRSs)i(\()p Ft(F)1240 1942 y Fr(1)1258 1936 y Fu(;)7 b Ft(R)1312 1942 y Fr(1)1330 1936 y Fx(\))20 b(and)f(\()p Ft(F)1498 1942 y Fr(2)1517 1936 y Fu(;)7 b Ft(R)1571 1942 y Fr(2)1589 1936 y Fx(\).)19 b(W)m(e)100 1986 y(sho)o(w)14 b(b)o(y)f(induction)h(on)f Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))f(=)g Fu(k)j Fx(that)f(ev)o(ery)g(term)g Fu(t)d Ft(2)h(T)e Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))14 b(has)g(a)g(unique)g(norm)e(form)100 2036 y(w.r.t.)j Ft(R)p Fx(.)h(In)g(the)h(base)g(case,)f Fu(k)h Fx(=)f(0)g(implies)e Fu(t)h Ft(2)g(T)c Fx(\()p Ft(B)q Fu(;)c Ft(V)s Fx(\))q(.)15 b(Here)j(the)f(claim)d(follo)o(ws)h(from)f(the)100 2086 y(semi-completeness)j(of)g(\()p Ft(B)q Fu(;)7 b Ft(S)s Fx(\))18 b(and)f(Lemma)e(4.13.)h(So)h(let)h Fu(k)h Ft(\025)f Fx(1)f(and)h(consider)g(a)f(con)o(v)o(ersion)100 2135 y Fu(t)115 2141 y Fr(1)154 2120 y Fn(\003)171 2135 y Ft( )c Fu(t)f Ft(!)295 2120 y Fn(\003)325 2135 y Fu(t)340 2141 y Fr(2)358 2135 y Fx(.)100 2235 y Fs(Case)i(\(i\):)g Fu(t)g Fx(is)f(top)h(blac)o(k.)100 2285 y(Let)i Fu(u)g Fx(b)q(e)h(an)o(y)e(term)h(in)g(the)g(con)o(v)o(ersion)h Fu(t)801 2291 y Fr(1)840 2270 y Fn(\003)857 2285 y Ft( )c Fu(t)i Ft(!)984 2270 y Fn(\003)1018 2285 y Fu(t)1033 2291 y Fr(2)1052 2285 y Fx(.)h(With)f Fu(u)h Fx(w)o(e)g(asso)q(ciate)h (terms)i(~)-24 b Fu(u)16 b Fx(and)102 2335 y(^)-23 b Fu(u)17 b Fx(whic)o(h)h(are)h(de\014ned)g(as)f(follo)o(ws.)e(If)i Fu(r)q(ank)q Fx(\()p Fu(u)p Fx(\))g Fu(<)h(k)q Fx(,)e(then)i Fu(u)f Fx(has)g(a)g(unique)g(normal)e(form)g Fu(u)p Ft(#)100 2384 y Fx(according)g(to)g(the)h(induction)f(h)o(yp)q(othesis)h(and)f (w)o(e)g(set)k(~)-24 b Fu(u)16 b Fx(=)i(^)-24 b Fu(u)16 b Fx(=)f Fu(u)p Ft(#)p Fx(.)h(If)f Fu(r)q(ank)q Fx(\()p Fu(u)p Fx(\))h(=)f Fu(k)q Fx(,)h(then)h Fu(u)100 2434 y Fx(cannot)12 b(b)q(e)h(top)f(white,)f(hence)j(it)e(can)g(b)q(e)h (written)f(as)h Fu(u)e Fx(=)h Fu(C)1071 2419 y Fp(b)1087 2434 y Ft(f)-14 b(f)p Fu(s)1134 2440 y Fr(1)1153 2434 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1265 2440 y Fp(n)1287 2434 y Ft(g)-14 b(g)p Fx(.)11 b(Since)i Fu(r)q(ank)q Fx(\()p Fu(s)1570 2440 y Fp(j)1588 2434 y Fx(\))e Fu(<)h(k)q Fx(,)100 2484 y(it)e(follo)o(ws)f(from)h(the)h(induction)g(h)o(yp)q (othesis)g(that,)g(for)f(ev)o(ery)i Fu(j)i Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(,)j(the)h(white)g (principal)100 2534 y(subterm)k Fu(s)282 2540 y Fp(j)316 2534 y Fx(has)g(a)h(unique)f(normal)f(form)g Fu(s)826 2540 y Fp(j)844 2534 y Ft(#)p Fx(.)h(The)h(result)g(of)f(replacing)h (eac)o(h)g(white)f(principal)135 2628 y Fq(y)170 2640 y Fw(One)d(of)f(the)g(referees)e(has)i(observ)o(ed)f(that)g(a)h (similar)f(construction)e(app)q(eared)h(in)j(Middeldorp)d(\(1994)p Fd(a)p Fw(\).)p eop %%Page: 13 13 13 12 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(13)p 100 224 1595 2 v 100 299 a Fx(subterm)16 b(with)g(its)g(unique)h(normal)d (form)h(is)h(denoted)h(b)o(y)i(~)-24 b Fu(u)p Fx(,)16 b(i.e.)i(~)-24 b Fu(u)16 b Fx(=)g Fu(C)1308 284 y Fp(b)1324 299 y Ft(f)p Fu(s)1364 305 y Fr(1)1383 299 y Ft(#)p Fu(;)7 b(:)g(:)g(:)t(;)g(s)1515 305 y Fp(n)1538 299 y Ft(#g)p Fx(.)15 b(Note)100 349 y(that)i Fu(u)g Ft(!)276 334 y Fn(\003)315 349 y Fx(~)-23 b Fu(u)o Fx(.)17 b(Moreo)o(v)o(er,)k(~)-24 b Fu(u)17 b Fx(has)h(a)f(unique)g(represen)o(tation)22 b(~)-23 b Fu(u)17 b Fx(=)1234 338 y(~)1225 349 y Fu(C)1258 334 y Fp(b)1274 349 y Ft(f)-14 b(f)p Fu(u)1326 355 y Fr(1)1344 349 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1461 355 y Fp(m)1492 349 y Ft(g)-14 b(g)17 b Fx(in)g(whic)o(h)100 399 y(the)e Fu(u)196 405 y Fp(i)209 399 y Fx(,)f Fu(i)f Ft(2)f(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(m)p Ft(g)p Fx(,)13 b(are)i(top)f(white)h(normal)d(forms.)h(Cho)q(ose)h(v)n (ariables)g Fu(x)1385 405 y Fr(1)1403 399 y Fu(;)7 b(:)g(:)g(:)e(;)i(x) 1520 405 y Fp(m)1565 399 y Fx(not)14 b(o)q(c-)100 448 y(curring)i(in)i(~)-24 b Fu(u)16 b Fx(satisfying)f Fu(u)546 454 y Fr(1)564 448 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)681 454 y Fp(m)727 448 y Ft(1)16 b Fu(x)809 454 y Fr(1)827 448 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)944 454 y Fp(m)974 448 y Fx(.)16 b(Since)1121 438 y(~)1112 448 y Fu(C)1145 433 y Fp(b)1161 448 y Ft(f)p Fu(x)1206 454 y Fr(1)1224 448 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1341 454 y Fp(m)1372 448 y Ft(g)14 b(2)h(T)10 b Fx(\()p Ft(F)1529 454 y Fr(1)1548 448 y Fu(;)d Ft(V)s Fx(\))16 b(and)100 498 y(the)d(TRS)e(\()p Ft(F)316 504 y Fr(1)334 498 y Fu(;)c Ft(R)388 504 y Fr(1)406 498 y Fx(\))13 b(is)f(semi-complete,)e(it)i(follo)o(ws)f(that)1028 488 y(~)1018 498 y Fu(C)1051 483 y Fp(b)1068 498 y Ft(f)p Fu(x)1113 504 y Fr(1)1131 498 y Fu(;)c(:)g(:)g(:)e(;)i(x)1248 504 y Fp(m)1278 498 y Ft(g)12 b Fx(rewrites)i(to)e(its)h(unique)100 548 y(\()p Ft(F)150 554 y Fr(1)168 548 y Fu(;)7 b Ft(R)222 554 y Fr(1)241 548 y Fx(\))16 b(normal)e(form)526 537 y(^)516 548 y Fu(C)549 533 y Fp(b)566 548 y Ft(h)p Fu(x)606 554 y Fp(i)618 558 y Ff(1)636 548 y Fu(;)7 b(:)g(:)g(:)t(;)g(x)752 554 y Fp(i)764 558 y Fl(l)777 548 y Ft(i)p Fx(.)16 b(W)m(e)g(set)k(^) -24 b Fu(u)15 b Fx(=)1058 537 y(^)1049 548 y Fu(C)1082 533 y Fp(b)1098 548 y Ft(h)p Fu(u)1138 554 y Fp(i)1150 558 y Ff(1)1168 548 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1285 554 y Fp(i)1297 558 y Fl(l)1310 548 y Ft(i)p Fx(.)16 b(It)g(is)g(easy)h(to)f(v)o(erify)100 598 y(that)g(^)-23 b Fu(u)11 b Ft(2)g Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)p Fx(\).)13 b(Observ)o(e)i(that)i(~)-24 b Fu(u)11 b Ft(!)806 583 y Fn(\003)806 609 y(R)835 613 y Ff(1)867 598 y Fx(^)-24 b Fu(u)14 b Fx(and)f(hence)j Fu(u)11 b Ft(!)1175 583 y Fn(\003)1208 598 y Fx(^)-24 b Fu(u)p Fx(.)141 648 y(Let)18 b Fu(u)243 654 y Fr(1)278 648 y Ft(!)e Fu(u)360 654 y Fr(2)395 648 y Fx(b)q(e)i(a)f(step)h(in)e(the)i (con)o(v)o(ersion)f Fu(t)930 654 y Fr(1)970 633 y Fn(\003)987 648 y Ft( )c Fu(t)k Ft(!)1116 633 y Fn(\003)1151 648 y Fu(t)1166 654 y Fr(2)1185 648 y Fx(.)f(W)m(e)h(sho)o(w)g(that)28 b(^)-32 b Fu(u)1511 654 y Fr(1)1546 648 y Fx(=)28 b(^)-32 b Fu(u)1619 654 y Fr(2)1637 648 y Fx(.)17 b(If)100 698 y Fu(r)q(ank)q Fx(\()p Fu(u)230 704 y Fr(1)248 698 y Fx(\))12 b Fu(<)g(k)q Fx(,)i(then)g Fu(r)q(ank)q Fx(\()p Fu(u)593 704 y Fr(2)612 698 y Fx(\))e Fu(<)g(k)j Fx(as)f(w)o(ell.)f (Hence)18 b(^)-23 b Fu(u)1016 704 y Fr(1)1046 698 y Fx(=)12 b Fu(u)1114 704 y Fr(1)1133 698 y Ft(#)f Fx(=)i Fu(u)1234 704 y Fr(2)1252 698 y Ft(#)f Fx(=)j(^)-24 b Fu(u)1353 704 y Fr(2)1371 698 y Fx(.)14 b(If)g Fu(r)q(ank)q Fx(\()p Fu(u)1569 704 y Fr(1)1587 698 y Fx(\))e(=)g Fu(k)q Fx(,)100 747 y(then)j Fu(u)219 753 y Fr(1)252 747 y Fx(is)g(a)g(top)g(blac)o(k)f (or)h(top)g(transparen)o(t)h(term,)e(i.e.)g Fu(u)1070 753 y Fr(1)1101 747 y Fx(=)g Fu(C)1180 732 y Fp(b)1177 758 y Fr(1)1196 747 y Ft(f)-14 b(f)p Fu(s)1243 753 y Fr(1)1262 747 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1374 753 y Fp(n)1396 747 y Ft(g)-14 b(g)p Fx(.)14 b(Here)j(w)o(e)e(ha)o(v)o(e)100 797 y(the)f(follo)o(wing)d(sub)q(cases.)125 901 y(\(a\))21 b(If)11 b Fu(u)262 907 y Fr(1)294 901 y Ft(!)336 881 y Fp(t;o)336 913 y Fn(A)363 917 y Ff(1)394 901 y Fu(u)418 907 y Fr(2)437 901 y Fx(,)f(then)i Fu(u)575 907 y Fr(2)604 901 y Fx(can)g(b)q(e)f(written)h(as)f Fu(u)946 907 y Fr(2)976 901 y Fx(=)h Fu(C)1053 886 y Fp(b)1050 911 y Fr(2)1069 901 y Ft(h)-7 b(h)p Fu(s)1113 907 y Fp(i)1125 911 y Ff(1)1144 901 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1256 907 y Fp(i)1268 911 y Fl(m)1297 901 y Ft(i)-7 b(i)p Fx(.)11 b(It)g(follo)o(ws)e(that)14 b(~)-24 b Fu(u)1631 907 y Fr(1)1661 901 y Fx(=)199 957 y Fu(C)232 942 y Fp(b)229 967 y Fr(1)249 957 y Ft(f)p Fu(s)289 963 y Fr(1)307 957 y Ft(#)p Fu(;)7 b(:)g(:)g(:)e(;)i(s)440 963 y Fp(n)462 957 y Ft(#g)18 b Fx(and)i(~)-24 b Fu(u)630 963 y Fr(2)667 957 y Fx(=)18 b Fu(C)750 942 y Fp(b)747 967 y Fr(2)766 957 y Ft(h)p Fu(s)801 963 y Fp(i)813 967 y Ff(1)832 957 y Ft(#)o Fu(;)7 b(:)g(:)g(:)e(;)i(s)964 963 y Fp(i)976 967 y Fl(l)990 957 y Ft(#i)p Fx(.)17 b(W)m(e)g(obtain)j(~)-24 b Fu(u)1288 963 y Fr(1)1321 957 y Ft(!)1363 963 y Fn(R)1392 967 y Ff(1)1425 957 y Fx(~)h Fu(u)1447 963 y Fr(2)1483 957 y Fx(\(cf.)17 b(Lemma)199 1006 y(4.14\).)c(Ev)o(ery)h Fu(s)453 1012 y Fp(j)471 1006 y Ft(#)g Fx(has)g(a)f(represen)o(tation)j Fu(s)905 1012 y Fp(j)923 1006 y Ft(#)11 b Fx(=)1008 996 y(\026)999 1006 y Fu(C)1032 991 y Fp(b)1029 1017 y(j)1048 1006 y Ft(h)-7 b(h)p Fu(u)1097 986 y Fp(j)1097 1017 y Fr(1)1116 1006 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1233 991 y Fp(j)1233 1017 y(m)1262 1021 y Fl(j)1279 1006 y Ft(i)-7 b(i)p Fx(.)14 b(Hence)439 1093 y(~)-24 b Fu(u)460 1099 y Fr(1)490 1093 y Fx(=)12 b Fu(C)567 1076 y Fp(b)564 1103 y Fr(1)583 1093 y Ft(f)614 1083 y Fx(\026)604 1093 y Fu(C)637 1076 y Fp(b)634 1103 y Fr(1)653 1093 y Ft(h)-7 b(h)p Fu(u)702 1076 y Fr(1)702 1103 y(1)721 1093 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)838 1076 y Fr(1)838 1103 y Fp(m)867 1107 y Ff(1)885 1093 y Ft(i)-7 b(i)p Fu(;)7 b(:)g(:)g(:)e(;)1012 1083 y Fx(\026)1003 1093 y Fu(C)1036 1076 y Fp(b)1033 1103 y(n)1055 1093 y Ft(h)-7 b(h)p Fu(u)1104 1076 y Fp(n)1104 1103 y Fr(1)1127 1093 y Fu(;)7 b(:)g(:)g(:)t(;)g(u)1243 1076 y Fp(n)1243 1103 y(m)1272 1107 y Fl(n)1294 1093 y Ft(i)-7 b(ig)14 b(!)1396 1099 y Fn(R)1425 1103 y Ff(1)470 1191 y Fu(C)503 1174 y Fp(b)500 1201 y Fr(2)519 1191 y Ft(h)545 1180 y Fx(\026)535 1191 y Fu(C)568 1174 y Fp(b)565 1201 y(i)577 1205 y Ff(1)595 1191 y Ft(h)-7 b(h)p Fu(u)644 1172 y Fp(i)656 1176 y Ff(1)644 1202 y Fr(1)674 1191 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)791 1174 y Fp(i)803 1178 y Ff(1)791 1201 y Fp(m)820 1205 y Fl(i)831 1211 y Ff(1)851 1191 y Ft(i)-7 b(i)p Fu(;)7 b(:)g(:)g(:)e(;)978 1180 y Fx(\026)969 1191 y Fu(C)1002 1174 y Fp(b)999 1201 y(i)1011 1205 y Fl(l)1024 1191 y Ft(h)-7 b(h)p Fu(u)1073 1172 y Fp(i)1085 1176 y Fl(l)1073 1202 y Fr(1)1099 1191 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1216 1174 y Fp(i)1228 1178 y Fl(l)1216 1201 y Fp(m)1245 1205 y Fl(i)1256 1212 y(l)1272 1191 y Ft(i)-7 b(ii)12 b Fx(=)j(~)-24 b Fu(u)1393 1197 y Fr(2)1411 1191 y Fu(:)199 1277 y Fx(Cho)q(ose)15 b(fresh)h(v)n (ariables)d Fu(x)642 1262 y Fr(1)642 1288 y(1)661 1277 y Fu(;)7 b(:)g(:)g(:)t(;)g(x)777 1262 y Fp(n)777 1287 y(m)806 1291 y Fl(n)843 1277 y Fx(satisfying)14 b Fu(u)1053 1262 y Fr(1)1053 1288 y(1)1071 1277 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1188 1262 y Fp(n)1188 1287 y(m)1217 1291 y Fl(n)1253 1277 y Ft(1)14 b Fu(x)1333 1262 y Fr(1)1333 1288 y(1)1352 1277 y Fu(;)7 b(:)g(:)g(:)t(;)g(x)1468 1262 y Fp(n)1468 1287 y(m)1497 1291 y Fl(n)1534 1277 y Fx(and)14 b(note)199 1333 y(that)f(this)g(implies)e Fu(u)532 1315 y Fp(i)544 1319 y Ff(1)532 1345 y Fr(1)561 1333 y Fu(;)c(:)g(:)g(:)e(;)i(u)678 1318 y Fp(i)690 1322 y Fl(l)678 1344 y Fp(m)707 1348 y Fl(i)718 1355 y(l)747 1333 y Ft(1)12 b Fu(x)825 1315 y Fp(i)837 1319 y Ff(1)825 1345 y Fr(1)855 1333 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)972 1318 y Fp(i)984 1322 y Fl(l)972 1344 y Fp(m)1001 1348 y Fl(i)1012 1355 y(l)1028 1333 y Fx(.)12 b(Another)h(application)f(of)g(Lemma)e(4.14)199 1388 y(yields)512 1452 y Fu(C)545 1435 y Fp(b)542 1463 y Fr(1)561 1452 y Ft(f)591 1442 y Fx(\026)582 1452 y Fu(C)615 1435 y Fp(b)612 1463 y Fr(1)631 1452 y Ft(h)p Fu(x)671 1435 y Fr(1)671 1463 y(1)690 1452 y Fu(;)d(:)g(:)g(:)t(;)g(x) 806 1435 y Fr(1)806 1463 y Fp(m)835 1467 y Ff(1)853 1452 y Ft(i)p Fu(;)g(:)g(:)g(:)e(;)971 1442 y Fx(\026)962 1452 y Fu(C)995 1435 y Fp(b)992 1463 y(n)1014 1452 y Ft(h)p Fu(x)1054 1435 y Fp(n)1054 1463 y Fr(1)1077 1452 y Fu(;)i(:)g(:)g(:)t(;)g(x)1193 1435 y Fp(n)1193 1463 y(m)1222 1467 y Fl(n)1244 1452 y Ft(ig)12 b(!)1335 1458 y Fn(R)1364 1462 y Ff(1)537 1550 y Fu(C)570 1533 y Fp(b)567 1560 y Fr(2)587 1550 y Ft(h)612 1539 y Fx(\026)603 1550 y Fu(C)636 1533 y Fp(b)633 1560 y(i)645 1564 y Ff(1)663 1550 y Ft(h)p Fu(x)703 1532 y Fp(i)715 1536 y Ff(1)703 1561 y Fr(1)732 1550 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)849 1533 y Fp(i)861 1537 y Ff(1)849 1560 y Fp(m)878 1564 y Fl(i)889 1570 y Ff(1)909 1550 y Ft(i)p Fu(;)g(:)g(:)g(:)e(;)1027 1539 y Fx(\026)1018 1550 y Fu(C)1051 1533 y Fp(b)1048 1560 y(i)1060 1564 y Fl(l)1073 1550 y Ft(h)p Fu(x)1113 1531 y Fp(i)1125 1535 y Fl(l)1113 1561 y Fr(1)1139 1550 y Fu(;)i(:)g(:)g(:)e(;)i(x)1256 1533 y Fp(i)1268 1537 y Fl(l)1256 1560 y Fp(m)1285 1564 y Fl(i)1296 1571 y(l)1312 1550 y Ft(ii)p Fu(:)199 1634 y Fx(Since)18 b(b)q(oth)f(terms)f(are)i (trivially)c(joinable,)i(they)h(reduce)h(to)f(the)h(same)e(unique)h(\() p Ft(F)1587 1640 y Fr(1)1605 1634 y Fu(;)7 b Ft(R)1659 1640 y Fr(1)1678 1634 y Fx(\))199 1684 y(normal)26 b(form)475 1673 y(^)465 1684 y Fu(C)498 1669 y Fp(b)515 1684 y Ft(h)p Fu(y)551 1690 y Fr(1)570 1684 y Fu(;)7 b(:)g(:)g(:)e(;)i(y)683 1690 y Fp(p)702 1684 y Ft(i)p Fx(,)27 b(where)h Fu(y)910 1690 y Fr(1)929 1684 y Fu(;)7 b(:)g(:)g(:)e(;)i(y)1042 1690 y Fp(p)1095 1684 y Ft(2)34 b(f)p Fu(x)1202 1669 y Fr(1)1202 1694 y(1)1220 1684 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1337 1669 y Fp(n)1337 1694 y(m)1366 1698 y Fl(n)1388 1684 y Ft(g)p Fx(.)27 b(Hence)k(^)-23 b Fu(u)1609 1690 y Fr(1)1661 1684 y Fx(=)199 1741 y Fu(\033)q Fx(\()250 1731 y(^)240 1741 y Fu(C)273 1726 y Fp(b)290 1741 y Ft(h)p Fu(y)326 1747 y Fr(1)345 1741 y Fu(;)7 b(:)g(:)g(:)e(;)i(y)458 1747 y Fp(p)477 1741 y Ft(i)p Fx(\))12 b(=)i(^)-24 b Fu(u)588 1747 y Fr(2)621 1741 y Fx(where)15 b Fu(\033)d Fx(=)g Ft(f)p Fu(x)866 1722 y Fp(j)866 1753 y(i)895 1741 y Ft(7!)f Fu(u)972 1722 y Fp(j)972 1753 y(i)1003 1741 y Ft(j)i Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fu(;)g(i)j Ft(2)h(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(m)1532 1747 y Fp(j)1549 1741 y Ft(gg)p Fx(.)123 1845 y(\(b\))21 b(If)26 b Fu(u)277 1851 y Fr(1)310 1845 y Ft(!)352 1830 y Fp(o)352 1856 y Fn(A)379 1860 y Ff(2)410 1845 y Fu(u)434 1851 y Fr(2)479 1845 y Fx(or)g Fu(u)566 1851 y Fr(1)617 1845 y Ft(!)659 1830 y Fp(i)705 1845 y Fu(u)729 1851 y Fr(2)747 1845 y Fx(,)g(then)h(w)o(e)g(ha)o(v)o(e)f Fu(u)1098 1851 y Fr(1)1149 1845 y Fx(=)32 b Fu(C)1246 1830 y Fp(b)1243 1855 y Fr(1)1263 1845 y Ft(f)-14 b(f)o Fu(s)1309 1851 y Fr(1)1329 1845 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)1440 1851 y Fp(j)1458 1845 y Fu(;)g(:)g(:)g(:)e(;)i(s)1570 1851 y Fp(n)1592 1845 y Ft(g)-14 b(g)32 b(!)199 1900 y Fu(C)232 1885 y Fp(b)229 1910 y Fr(1)249 1900 y Ft(f)p Fu(s)289 1906 y Fr(1)307 1900 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)419 1885 y Fn(0)419 1911 y Fp(j)437 1900 y Fu(;)g(:)g(:)g(:)t(;)g(s)548 1906 y Fp(n)571 1900 y Ft(g)21 b Fx(=)h Fu(u)691 1906 y Fr(2)709 1900 y Fx(,)d(where)i Fu(s)885 1906 y Fp(j)924 1900 y Ft(!)g Fu(s)1006 1885 y Fn(0)1006 1911 y Fp(j)1024 1900 y Fx(.)e(If)h(the)g(rewrite)h(step)g(is)e(not)h(destruc-)199 1950 y(tiv)o(e,)f(then)g(w)o(e)h(conclude)f(from)f Fu(s)761 1956 y Fp(j)778 1950 y Ft(#)i Fx(=)h Fu(s)891 1935 y Fn(0)891 1961 y Fp(j)909 1950 y Ft(#)d Fx(that)k(~)-24 b Fu(u)1067 1956 y Fr(1)1106 1950 y Fx(=)23 b(~)-24 b Fu(u)1182 1956 y Fr(2)1219 1950 y Fx(and)19 b(th)o(us)j(^)-23 b Fu(u)1426 1956 y Fr(1)1464 1950 y Fx(=)23 b(^)-23 b Fu(u)1541 1956 y Fr(2)1559 1950 y Fx(.)18 b(If)h(the)199 2006 y(rewrite)j(step)g(is)f(destructiv)o(e,)h(then)f Fu(s)842 1990 y Fn(0)842 2016 y Fp(j)881 2006 y Fx(has)g(a)g(represen)o (tation)h Fu(s)1301 1990 y Fn(0)1301 2016 y Fp(j)1342 2006 y Fx(=)i Fu(C)1431 1990 y Fn(0)p Fp(b)1428 2016 y(j)1456 2006 y Ft(f)-14 b(f)p Fu(v)1504 2012 y Fr(1)1523 2006 y Fu(;)7 b(:)g(:)g(:)e(;)i(v)1636 2012 y Fp(p)1654 2006 y Ft(g)-14 b(g)p Fx(.)199 2061 y(Clearly)m(,)22 b Fu(s)380 2046 y Fn(0)380 2072 y Fp(j)424 2061 y Ft(!)466 2046 y Fn(\003)512 2061 y Fu(C)545 2046 y Fn(0)p Fp(b)542 2072 y(j)571 2061 y Ft(f)p Fu(v)612 2067 y Fr(1)630 2061 y Ft(#)p Fu(;)7 b(:)g(:)g(:)e(;)i(v)764 2067 y Fp(p)782 2061 y Ft(#g)p Fx(.)22 b(F)m(urthermore,)h Fu(C)1153 2046 y Fn(0)p Fp(b)1150 2072 y(j)1179 2061 y Ft(f)p Fu(v)1220 2067 y Fr(1)1238 2061 y Ft(#)p Fu(;)7 b(:)g(:)g(:)e(;)i(v)1372 2067 y Fp(p)1391 2061 y Ft(#)o(g)23 b Fx(can)g(b)q(e)h(writ-)199 2120 y(ten)19 b(as)340 2110 y(~)331 2120 y Fu(C)364 2105 y Fp(b)361 2131 y(j)380 2120 y Ft(f)-14 b(f)p Fu(u)432 2105 y Fn(0)432 2131 y Fr(1)450 2120 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)567 2105 y Fn(0)567 2130 y Fp(q)585 2120 y Ft(g)-14 b(g)p Fx(,)17 b(where)776 2110 y(~)767 2120 y Fu(C)800 2105 y Fp(b)797 2131 y(j)816 2120 y Ft(f)p Fu(:)7 b(:)g(:)e(;)i Ft(g)18 b Fx(is)g(a)h(blac)o(k)f(con)o(text)h(and)f Fu(u)1410 2105 y Fn(0)1410 2131 y Fp(i)1442 2120 y Fx(are)h(top)f(white)199 2170 y(terms)e(in)h Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\).)16 b(No)o(w)g(w)o(e)h(kno)o(w)e(from)g(the)i(induction)f (h)o(yp)q(othesis)h(that)g Fu(s)1539 2176 y Fp(j)1573 2170 y Fx(and)g Fu(s)1676 2155 y Fn(0)1676 2181 y Fp(j)199 2238 y Fx(m)o(ust)h(ha)o(v)o(e)g(the)h(same)e(normal)g(form)g Fu(s)859 2244 y Fp(j)876 2238 y Ft(#)p Fx(.)h(Hence)1064 2228 y(~)1055 2238 y Fu(C)1088 2223 y Fp(b)1085 2249 y(j)1104 2238 y Ft(f)-14 b(f)p Fu(u)1156 2223 y Fn(0)1156 2248 y Fr(1)1174 2238 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1291 2223 y Fn(0)1291 2248 y Fp(q)1309 2238 y Ft(g)-14 b(g)1364 2213 y Fp(t;o)14 b Fn(\003)1371 2238 y Ft(!)1417 2244 y Fn(A)1444 2248 y Ff(1)1476 2238 y Fu(s)1495 2244 y Fp(j)1512 2238 y Ft(#)p Fx(.)k(Cho)q(ose)199 2288 y(fresh)i(v)n (ariables)f Fu(y)503 2294 y Fr(1)521 2288 y Fu(;)7 b(:)g(:)g(:)e(y)615 2294 y Fp(q)653 2288 y Fx(suc)o(h)20 b(that)f Fu(u)871 2273 y Fn(0)871 2298 y Fr(1)889 2288 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1006 2273 y Fn(0)1006 2298 y Fp(q)1043 2288 y Ft(1)18 b Fu(y)1123 2294 y Fr(1)1142 2288 y Fu(;)7 b(:)g(:)g(:)e(;)i(y)1255 2294 y Fp(q)1273 2288 y Fx(.)19 b(Rep)q(eated)h(application)199 2346 y(of)d(Lemma)d(4.14)h(yields)620 2336 y(~)611 2346 y Fu(C)644 2331 y Fp(b)641 2357 y(j)660 2346 y Ft(f)p Fu(y)701 2352 y Fr(1)720 2346 y Fu(;)7 b(:)g(:)g(:)e(;)i(y)833 2352 y Fp(q)851 2346 y Ft(g)14 b(!)928 2331 y Fn(\003)928 2358 y(R)957 2362 y Ff(1)997 2336 y Fx(\026)988 2346 y Fu(C)1021 2331 y Fp(b)1018 2357 y(j)1037 2346 y Ft(h)p Fu(y)1073 2352 y Fp(i)1085 2356 y Ff(1)1104 2346 y Fu(;)7 b(:)g(:)g(:)t(;)g(y)1216 2352 y Fp(i)1228 2356 y Fl(l)1242 2346 y Ft(i)17 b Fx(for)g(some)f(blac)o(k)g(con)o(text)209 2392 y(\026)199 2402 y Fu(C)232 2387 y Fp(b)229 2413 y(j)249 2402 y Ft(h)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(i)13 b Fx(as)i(w)o(ell)e(as)i Fu(s)594 2408 y Fp(j)612 2402 y Ft(#)d Fx(=)699 2392 y(\026)689 2402 y Fu(C)722 2387 y Fp(b)719 2413 y(j)738 2402 y Ft(h)-7 b(h)p Fu(u)787 2387 y Fn(0)787 2413 y Fp(i)799 2417 y Ff(1)818 2402 y Fu(;)7 b(:)g(:)g(:)t(;)g(u)934 2387 y Fn(0)934 2413 y Fp(i)946 2417 y Fl(l)959 2402 y Ft(i)-7 b(i)p Fx(.)15 b(Ev)o(ery)f Fu(s)1149 2408 y Fp(i)1164 2402 y Ft(#)p Fx(,)f Fu(i)g Ft(6)p Fx(=)f Fu(j)r Fx(,)j(has)f(a)g(represen)o(tation) 199 2458 y Fu(s)218 2464 y Fp(i)233 2458 y Ft(#)d Fx(=)318 2447 y(\026)309 2458 y Fu(C)342 2443 y Fp(b)339 2469 y(i)358 2458 y Ft(h)-7 b(h)p Fu(u)407 2443 y Fp(i)407 2468 y Fr(1)426 2458 y Fu(;)7 b(:)g(:)g(:)t(;)g(u)542 2443 y Fp(i)542 2468 y(m)571 2472 y Fl(i)586 2458 y Ft(i)-7 b(i)p Fx(.)14 b(W)m(e)g(obtain)f(the)h(follo)o(wing)e(reduction)i (sequence.)252 2542 y(~)-23 b Fu(u)274 2548 y Fr(2)303 2542 y Fx(=)12 b Fu(C)380 2524 y Fp(b)377 2552 y Fr(1)397 2542 y Ft(f)427 2531 y Fx(\026)418 2542 y Fu(C)451 2524 y Fp(b)448 2552 y Fr(1)467 2542 y Ft(h)-7 b(h)p Fu(u)516 2524 y Fr(1)516 2552 y(1)534 2542 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)651 2524 y Fr(1)651 2552 y Fp(m)680 2556 y Ff(1)698 2542 y Ft(i)-7 b(i)p Fu(;)7 b(:)g(:)g(:)e(;)825 2531 y Fx(~)816 2542 y Fu(C)849 2524 y Fp(b)846 2552 y(j)865 2542 y Ft(f)-14 b(f)p Fu(u)917 2524 y Fn(0)917 2552 y Fr(1)935 2542 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1052 2524 y Fn(0)1052 2552 y Fp(q)1069 2542 y Ft(g)-14 b(g)p Fu(;)7 b(:)g(:)g(:)e(;)1199 2531 y Fx(\026)1190 2542 y Fu(C)1223 2524 y Fp(b)1220 2552 y(n)1242 2542 y Ft(h)-7 b(h)p Fu(u)1291 2524 y Fp(n)1291 2552 y Fr(1)1313 2542 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1430 2524 y Fp(n)1430 2552 y(m)1459 2556 y Fl(n)1481 2542 y Ft(i)-7 b(ig)14 b(!)1583 2526 y Fn(\003)1583 2553 y(R)1612 2557 y Ff(1)305 2640 y Fu(C)338 2623 y Fp(b)335 2650 y Fr(1)354 2640 y Ft(f)384 2630 y Fx(\026)375 2640 y Fu(C)408 2623 y Fp(b)405 2650 y Fr(1)424 2640 y Ft(h)-7 b(h)p Fu(u)473 2623 y Fr(1)473 2650 y(1)492 2640 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)609 2623 y Fr(1)609 2650 y Fp(m)638 2654 y Ff(1)656 2640 y Ft(i)-7 b(i)p Fu(;)7 b(:)g(:)g(:)e(;)783 2630 y Fx(\026)774 2640 y Fu(C)807 2623 y Fp(b)804 2650 y(j)823 2640 y Ft(h)-7 b(h)p Fu(u)872 2623 y Fn(0)872 2650 y Fp(i)884 2654 y Ff(1)902 2640 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1019 2623 y Fn(0)1019 2650 y Fp(i)1031 2654 y Fl(l)1044 2640 y Ft(i)-7 b(i)p Fu(;)7 b(:)g(:)g(:)e(;)1171 2630 y Fx(\026)1162 2640 y Fu(C)1195 2623 y Fp(b)1192 2650 y(n)1214 2640 y Ft(h)-7 b(h)p Fu(u)1263 2623 y Fp(n)1263 2650 y Fr(1)1285 2640 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1402 2623 y Fp(n)1402 2650 y(m)1431 2654 y Fl(n)1453 2640 y Ft(i)-7 b(i)12 b Fx(=)j(~)-24 b Fu(u)1558 2646 y Fr(1)1576 2640 y Fu(:)p eop %%Page: 14 14 14 13 bop 100 197 a Fw(14)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 199 299 a Fx(Cho)q(ose)k(fresh)f(v)n(ariables)f Fu(x)640 284 y Fr(1)640 309 y(1)659 299 y Fu(;)7 b(:)g(:)g(:)e(;)i(x) 776 284 y Fp(j)r Fn(\000)p Fr(1)776 309 y Fp(m)805 313 y Fl(j)q Fc(\000)p Ff(1)858 299 y Fu(;)g(y)897 305 y Fr(1)916 299 y Fu(;)g(:)g(:)g(:)e(;)i(y)1029 305 y Fp(q)1047 299 y Fu(;)g(x)1090 279 y Fp(j)r Fr(+1)1090 310 y(1)1148 299 y Fu(;)g(:)g(:)g(:)e(;)i(x)1265 284 y Fp(n)1265 309 y(m)1294 313 y Fl(n)1330 299 y Fx(satisfying)203 384 y Fu(u)227 367 y Fr(1)227 394 y(1)245 384 y Fu(;)g(:)g(:)g(:)e(;)i(u) 362 367 y Fp(j)r Fn(\000)p Fr(1)362 394 y Fp(m)391 398 y Fl(j)q Fc(\000)p Ff(1)445 384 y Fu(;)g(u)488 367 y Fn(0)488 394 y Fr(1)506 384 y Fu(;)g(:)g(:)g(:)t(;)g(u)622 367 y Fn(0)622 394 y Fp(q)640 384 y Fu(;)g(u)683 364 y Fp(j)r Fr(+1)683 395 y(1)742 384 y Fu(;)g(:)g(:)g(:)t(;)g(u)858 367 y Fp(n)858 394 y(m)887 398 y Fl(n)923 384 y Ft(1)14 b Fu(x)1003 367 y Fr(1)1003 394 y(1)1021 384 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1138 367 y Fp(j)r Fn(\000)p Fr(1)1138 394 y Fp(m)1167 398 y Fl(j)q Fc(\000)p Ff(1)1221 384 y Fu(;)g(y)1260 390 y Fr(1)1278 384 y Fu(;)g(:)g(:)g(:)e(;)i(y)1391 390 y Fp(q)1409 384 y Fu(;)g(x)1452 364 y Fp(j)r Fr(+1)1452 395 y(1)1511 384 y Fu(;)g(:)g(:)g(:)e(;)i(x)1628 367 y Fp(n)1628 394 y(m)1657 398 y Fl(n)1679 384 y Fu(:)199 457 y Fx(Note)15 b(that)f(this)f(implies)199 530 y Fu(u)223 513 y Fr(1)223 540 y(1)242 530 y Fu(;)7 b(:)g(:)g(:)t(;)g(u)358 513 y Fp(j)r Fn(\000)p Fr(1)358 540 y Fp(m)387 544 y Fl(j)q Fc(\000)p Ff(1)441 530 y Fu(;)g(u)484 513 y Fn(0)484 540 y Fp(i)496 544 y Ff(1)514 530 y Fu(;)g(:)g(:)g(:)t(;)g(u)630 513 y Fn(0)630 540 y Fp(i)642 544 y Fl(l)655 530 y Fu(;)g(u)698 510 y Fp(j)r Fr(+1)698 541 y(1)757 530 y Fu(;)g(:)g(:)g(:)e(;)i(u)874 513 y Fp(n)874 540 y(m)903 544 y Fl(n)934 530 y Ft(1)i Fu(x)1009 513 y Fr(1)1009 540 y(1)1027 530 y Fu(;)e(:)g(:)g(:)e(;)i(x) 1144 513 y Fp(j)r Fn(\000)p Fr(1)1144 540 y Fp(m)1173 544 y Fl(j)q Fc(\000)p Ff(1)1227 530 y Fu(;)g(y)1266 536 y Fp(i)1278 540 y Ff(1)1296 530 y Fu(;)g(:)g(:)g(:)e(;)i(y)1409 536 y Fp(i)1421 540 y Fl(l)1434 530 y Fu(;)g(x)1477 510 y Fp(j)r Fr(+1)1477 541 y(1)1536 530 y Fu(;)g(:)g(:)g(:)e(;)i(x)1653 513 y Fp(n)1653 540 y(m)1682 544 y Fl(n)1704 530 y Fu(:)199 614 y Fx(W)m(e)14 b(deriv)o(e)g(from)501 603 y(~)491 614 y Fu(C)524 599 y Fp(b)521 625 y(j)541 614 y Ft(f)p Fu(y)582 620 y Fr(1)600 614 y Fu(;)7 b(:)g(:)g(:)e(;)i(y)713 620 y Fp(q)731 614 y Ft(g)14 b(!)808 599 y Fn(\003)808 625 y(R)837 629 y Ff(1)877 603 y Fx(\026)868 614 y Fu(C)901 599 y Fp(b)898 625 y(j)917 614 y Ft(h)p Fu(y)953 620 y Fp(i)965 624 y Ff(1)984 614 y Fu(;)7 b(:)g(:)g(:)e(;)i(y)1097 620 y Fp(i)1109 624 y Fl(l)1122 614 y Ft(i)14 b Fx(that)335 696 y Fu(C)368 679 y Fp(b)365 706 y Fr(1)385 696 y Ft(f)415 685 y Fx(\026)406 696 y Fu(C)439 679 y Fp(b)436 706 y Fr(1)455 696 y Ft(h)p Fu(x)495 679 y Fr(1)495 706 y(1)513 696 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)630 679 y Fr(1)630 706 y Fp(m)659 710 y Ff(1)677 696 y Ft(i)p Fu(;)g(:)g(:)g(:)e(;)795 685 y Fx(~)786 696 y Fu(C)819 679 y Fp(b)816 706 y(j)835 696 y Ft(f)p Fu(y)876 702 y Fr(1)894 696 y Fu(;)i(:)g(:)g(:)e(;)i(y) 1007 702 y Fp(q)1025 696 y Ft(g)p Fu(;)g(:)g(:)g(:)e(;)1148 685 y Fx(\026)1139 696 y Fu(C)1172 679 y Fp(b)1169 706 y(n)1191 696 y Ft(h)p Fu(x)1231 679 y Fp(n)1231 706 y Fr(1)1253 696 y Fu(;)i(:)g(:)g(:)e(;)i(x)1370 679 y Fp(n)1370 706 y(m)1399 710 y Fl(n)1421 696 y Ft(ig)k(!)1511 679 y Fn(\003)1511 706 y(R)1540 710 y Ff(1)375 789 y Fu(C)408 772 y Fp(b)405 799 y Fr(1)424 789 y Ft(f)454 779 y Fx(\026)445 789 y Fu(C)478 772 y Fp(b)475 799 y Fr(1)494 789 y Ft(h)p Fu(x)534 772 y Fr(1)534 799 y(1)552 789 y Fu(;)c(:)g(:)g(:)e(;)i(x)669 772 y Fr(1)669 799 y Fp(m)698 803 y Ff(1)716 789 y Ft(i)p Fu(;)g(:)g(:)g(:)e(;)834 779 y Fx(\026)825 789 y Fu(C)858 772 y Fp(b)855 799 y(j)874 789 y Ft(h)p Fu(y)910 795 y Fp(i)922 799 y Ff(1)940 789 y Fu(;)i(:)g(:)g(:)e(;)i(y)1053 795 y Fp(i)1065 799 y Fl(l)1079 789 y Ft(i)p Fu(;)g(:)g(:)g(:)e(;)1197 779 y Fx(\026)1188 789 y Fu(C)1221 772 y Fp(b)1218 799 y(n)1240 789 y Ft(h)p Fu(x)1280 772 y Fp(n)1280 799 y Fr(1)1302 789 y Fu(;)i(:)g(:)g(:)e(;)i(x)1419 772 y Fp(n)1419 799 y(m)1448 803 y Fl(n)1470 789 y Ft(ig)p Fu(:)199 862 y Fx(Since)19 b(b)q(oth)f(terms)g(are)h(joinable,)d(they)j(reduce)h(to) e(the)h(same)e(unique)h(\()p Ft(F)1442 868 y Fr(1)1460 862 y Fu(;)7 b Ft(R)1514 868 y Fr(1)1532 862 y Fx(\))19 b(normal)199 912 y(from)306 901 y(^)296 912 y Fu(C)329 897 y Fp(b)346 912 y Ft(h)p Fu(z)381 918 y Fr(1)400 912 y Fu(;)7 b(:)g(:)g(:)e(;)i(z)512 918 y Fp(r)530 912 y Ft(i)13 b Fx(where)h Fu(z)697 918 y Fr(1)716 912 y Fu(;)7 b(:)g(:)g(:)e(;)i(z)828 918 y Fp(r)857 912 y Ft(2)12 b(f)p Fu(x)942 897 y Fr(1)942 922 y(1)960 912 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1077 897 y Fp(j)r Fn(\000)p Fr(1)1077 922 y Fp(m)1106 926 y Fl(j)q Fc(\000)p Ff(1)1159 912 y Fu(;)g(y)1198 918 y Fr(1)1217 912 y Fu(;)g(:)g(:)g(:)e(;)i(y)1330 918 y Fp(q)1348 912 y Fu(;)g(x)1391 892 y Fp(j)r Fr(+1)1391 923 y(1)1449 912 y Fu(;)g(:)g(:)g(:)e(;)i(x)1566 897 y Fp(n)1566 922 y(m)1595 926 y Fl(n)1617 912 y Ft(g)p Fx(.)13 b(It)199 962 y(follo)o(ws)g(as)h(ab)q(o)o(v)o(e)f(that)k(^)-24 b Fu(u)618 968 y Fr(1)648 962 y Fx(=)14 b(^)-23 b Fu(u)716 968 y Fr(2)734 962 y Fx(.)100 1060 y(All)13 b(in)g(all,)286 1052 y(^)285 1060 y Fu(t)f Fx(=)362 1052 y(^)356 1060 y Fu(t)371 1066 y Fr(1)401 1060 y Fx(=)451 1052 y(^)445 1060 y Fu(t)460 1066 y Fr(2)492 1060 y Fx(is)i(the)g(unique)g(normal)e (form)g(of)h Fu(t)h Fx(w.r.t.)f(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\).)100 1159 y Fs(Case)14 b(\(ii\):)f Fu(t)h Fx(is)g(top)g(white.)f(The)i(assertion)f(follo)o(ws)f(from)f(similar)f (argumen)o(ts)i(as)h(in)g(case)g(\(i\).)100 1259 y Fs(Case)19 b(\(iii\):)f Fu(t)h Fx(is)g(top)g(transparen)o(t,)h(i.e.)e Fu(t)j Fx(=)f Fu(C)903 1244 y Fp(t)917 1259 y Fx([)-7 b([)p Fu(s)953 1265 y Fr(1)972 1259 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)1083 1265 y Fp(n)1106 1259 y Fx(])-7 b(])o(.)19 b(W)m(e)f(pro)q(ceed)j(in)e (a)g(similar)d(w)o(a)o(y)100 1309 y(as)c(ab)q(o)o(v)o(e.)g(Let)h Fu(u)f Fx(b)q(e)h(an)o(y)f(term)g(in)g(the)h(con)o(v)o(ersion)g Fu(t)949 1315 y Fr(1)988 1294 y Fn(\003)1005 1309 y Ft( )h Fu(t)d Ft(!)1129 1294 y Fn(\003)1159 1309 y Fu(t)1174 1315 y Fr(2)1193 1309 y Fx(.)h(With)g Fu(u)g Fx(w)o(e)h(asso)q(ciate)g (terms)102 1359 y(~)-23 b Fu(u)14 b Fx(and)j(^)-23 b Fu(u)14 b Fx(whic)o(h)h(are)g(de\014ned)h(as)f(follo)o(ws.)d(If)j Fu(r)q(ank)q Fx(\()p Fu(u)p Fx(\))d Fu(<)i(k)q Fx(,)g(then)h Fu(u)g Fx(has)f(a)h(unique)g(normal)d(form)100 1408 y Fu(u)p Ft(#)i Fx(according)h(to)f(the)h(induction)g(h)o(yp)q(othesis)g (and)g(w)o(e)g(set)j(~)-24 b Fu(u)13 b Fx(=)j(^)-24 b Fu(u)13 b Fx(=)g Fu(u)p Ft(#)o Fx(.)i(If)f Fu(r)q(ank)q Fx(\()p Fu(u)p Fx(\))f(=)g Fu(k)i Fx(and)g Fu(u)100 1458 y Fx(is)d(top)h(blac)o(k)f(or)h(top)g(white,)g(then)g Fu(u)g Fx(has)g(a)f(unique)h(normal)e(form)g Fu(u)p Ft(#)h Fx(according)h(to)g(cases)h(\(i\))f(and)100 1508 y(\(ii\).)g(Again,)h (w)o(e)h(set)j(~)-23 b Fu(u)13 b Fx(=)j(^)-24 b Fu(u)13 b Fx(=)g Fu(u)p Ft(#)p Fx(.)h(If)h Fu(r)q(ank)q Fx(\()p Fu(u)p Fx(\))e(=)g Fu(k)j Fx(and)e Fu(u)h Fx(is)g(top)f(transparen)o (t,)i(then)f(it)g(can)g(b)q(e)100 1558 y(written)e(as)g Fu(u)e Fx(=)h Fu(C)406 1543 y Fp(t)420 1558 y Fx([)-7 b([)o Fu(s)455 1564 y Fr(1)474 1558 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)586 1564 y Fp(n)609 1558 y Fx(])-7 b(])o(.)12 b(It)h(follo)o(ws)e(from)g (the)j(foregoing)e(that)h(ev)o(ery)g Fu(s)1390 1564 y Fp(j)1408 1558 y Fx(,)f Fu(j)i Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(,)100 1608 y(has)17 b(a)f(unique)h (normal)e(form)g Fu(s)616 1614 y Fp(j)634 1608 y Ft(#)o Fx(.)i(The)g(result)h(of)e(replacing)g(eac)o(h)i Fu(s)1236 1614 y Fp(j)1271 1608 y Fx(with)e(its)h(unique)g(normal)100 1657 y(form)e(is)i(denoted)h(b)o(y)h(~)-23 b Fu(u)o Fx(,)17 b(i.e.)i(~)-24 b Fu(u)16 b Fx(=)h Fu(C)711 1642 y Fp(t)726 1657 y Fx([)p Fu(s)757 1663 y Fr(1)775 1657 y Ft(#)p Fu(;)7 b(:)g(:)g(:)e(;)i(s)908 1663 y Fp(n)930 1657 y Ft(#)p Fx(].)16 b(Note)i(that)f Fu(u)f Ft(!)1270 1642 y Fn(\003)1308 1657 y Fx(~)-23 b Fu(u)o Fx(.)17 b(Moreo)o(v)o(er,)j(~) -24 b Fu(u)17 b Fx(has)g(a)100 1707 y(represen)o(tation)g(~)-24 b Fu(u)12 b Fx(=)459 1697 y(~)449 1707 y Fu(C)482 1692 y Fp(t)497 1707 y Ft(f)-14 b(f)p Fu(u)549 1713 y Fr(1)567 1707 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)684 1713 y Fp(m)714 1707 y Ft(g)-14 b(g)13 b Fx(in)g(whic)o(h)g(the)h Fu(u)1016 1713 y Fp(i)1029 1707 y Fx(,)f Fu(i)f Ft(2)f(f)p Fx(1)p Fu(;)c(:)g(:)g(:)t(;)g(m)p Ft(g)p Fx(,)12 b(are)i(top)f(blac)o(k)g(or)g (top)100 1757 y(white)i(normal)d(forms)i(w.r.t.)f(\()p Ft(F)5 b Fu(;)i Ft(R)o Fx(\).)15 b(Cho)q(ose)g(v)n(ariables)f Fu(x)1071 1763 y Fr(1)1089 1757 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1206 1763 y Fp(m)1252 1757 y Fx(not)14 b(o)q(ccurring)i(in)h(~)-24 b Fu(u)15 b Fx(satis-)100 1807 y(fying)f Fu(u)229 1813 y Fr(1)248 1807 y Fu(;)7 b(:)g(:)g(:)t(;)g(u)364 1813 y Fp(m)411 1807 y Ft(1)15 b Fu(x)492 1813 y Fr(1)510 1807 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)627 1813 y Fp(m)658 1807 y Fx(.)15 b(Since)805 1796 y(~)795 1807 y Fu(C)828 1792 y Fp(t)842 1807 y Ft(f)p Fu(x)887 1813 y Fr(1)906 1807 y Fu(;)7 b(:)g(:)g(:)t(;)g(x)1022 1813 y Fp(m)1053 1807 y Ft(g)14 b(2)h(T)10 b Fx(\()p Ft(B)q Fu(;)d Ft(V)s Fx(\))17 b(and)e(the)h(TRS)g(\()p Ft(B)q Fu(;)7 b Ft(S)s Fx(\))16 b(is)100 1857 y(semi-complete,)c(it)i(follo)o(ws)e(that)660 1846 y(~)651 1857 y Fu(C)684 1842 y Fp(t)698 1857 y Ft(f)p Fu(x)743 1863 y Fr(1)761 1857 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)878 1863 y Fp(m)909 1857 y Ft(g)14 b Fx(rewrites)h(to)g(its)f(unique)g(\()p Ft(B)r Fu(;)7 b Ft(S)s Fx(\))14 b(normal)e(form)109 1896 y(^)100 1906 y Fu(C)133 1891 y Fp(t)147 1906 y Ft(h)p Fu(x)187 1912 y Fp(i)199 1916 y Ff(1)217 1906 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)334 1912 y Fp(i)346 1916 y Fl(l)359 1906 y Ft(i)16 b(2)f Fu(N)5 b(F)h Fx(\()p Ft(B)q Fu(;)h Ft(S)s Fx(\))16 b(=)g Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)o Fx(\))k Ft(\\)g(T)f Fx(\()p Ft(B)q Fu(;)d Ft(V)s Fx(\))q(.)16 b(W)m(e)h(set)j(^)-24 b Fu(u)16 b Fx(=)1324 1896 y(^)1315 1906 y Fu(C)1348 1891 y Fp(t)1362 1906 y Ft(h)p Fu(u)1402 1912 y Fp(i)1414 1916 y Ff(1)1432 1906 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1549 1912 y Fp(i)1561 1916 y Fl(l)1574 1906 y Ft(i)p Fx(.)16 b(It)h(is)100 1956 y(easy)d(to)g(v)o(erify)f(that)k(^)-24 b Fu(u)11 b Ft(2)h Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\).)13 b(Observ)o(e)j(that)g(~)-23 b Fu(u)11 b Ft(!)1064 1941 y Fn(\003)1097 1956 y Fx(^)-24 b Fu(u)14 b Fx(and)f(hence)j Fu(u)11 b Ft(!)1405 1941 y Fn(\003)1438 1956 y Fx(^)-24 b Fu(u)p Fx(.)141 2006 y(Let)14 b Fu(u)239 2012 y Fr(1)269 2006 y Ft(!)d Fu(u)346 2012 y Fr(2)377 2006 y Fx(b)q(e)j(a)f(step)h(in) f(the)h(con)o(v)o(ersion)g Fu(t)890 2012 y Fr(1)929 1991 y Fn(\003)946 2006 y Ft( )f Fu(t)f Ft(!)1070 1991 y Fn(\003)1100 2006 y Fu(t)1115 2012 y Fr(2)1133 2006 y Fx(.)h(Again,)f(w)o(e)h(sho)o (w)h(that)24 b(^)-32 b Fu(u)1566 2012 y Fr(1)1596 2006 y Fx(=)23 b(^)-32 b Fu(u)1664 2012 y Fr(2)1682 2006 y Fx(.)100 2056 y(If)11 b Fu(r)q(ank)q Fx(\()p Fu(u)269 2062 y Fr(1)287 2056 y Fx(\))g Fu(<)h(k)q Fx(,)f(then)h Fu(r)q(ank)q Fx(\()p Fu(u)626 2062 y Fr(2)644 2056 y Fx(\))g Fu(<)g(k)g Fx(as)f(w)o(ell.)f(Hence)16 b(^)-24 b Fu(u)1036 2062 y Fr(1)1066 2056 y Fx(=)12 b Fu(u)1134 2062 y Fr(1)1152 2056 y Ft(#)g Fx(=)f Fu(u)1252 2062 y Fr(2)1271 2056 y Ft(#)g Fx(=)k(^)-24 b Fu(u)1371 2062 y Fr(2)1389 2056 y Fx(.)11 b(If)g Fu(r)q(ank)q Fx(\()p Fu(u)1581 2062 y Fr(1)1599 2056 y Fx(\))h(=)g Fu(k)100 2106 y Fx(and)j Fu(u)206 2112 y Fr(1)239 2106 y Fx(is)g(top)g(blac)o(k) g(or)g(top)g(white,)g(then)27 b(^)-32 b Fu(u)840 2112 y Fr(1)872 2106 y Fx(=)14 b Fu(u)942 2112 y Fr(1)961 2106 y Ft(#)h Fx(is)g(the)h(unique)f(normal)e(form)h(of)g Fu(u)1563 2112 y Fr(1)1597 2106 y Fx(w.r.t.)100 2156 y(\()p Ft(F)t Fu(;)7 b Ft(R)o Fx(\).)13 b(Since)h Fu(u)376 2162 y Fr(1)406 2156 y Ft(!)d Fu(u)483 2162 y Fr(2)513 2156 y Ft(!)555 2140 y Fn(\003)596 2156 y Fx(^)-32 b Fu(u)609 2162 y Fr(2)639 2156 y Ft(2)11 b Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\),)13 b(it)g(follo)o(ws)22 b(^)-32 b Fu(u)1093 2162 y Fr(1)1123 2156 y Fx(=)23 b(^)-32 b Fu(u)1191 2162 y Fr(2)1209 2156 y Fx(.)13 b(If)g Fu(r)q(ank)q Fx(\()p Fu(u)1405 2162 y Fr(1)1423 2156 y Fx(\))e(=)h Fu(k)i Fx(and)f Fu(u)1634 2162 y Fr(1)1666 2156 y Fx(is)100 2205 y(top)g(transparen)o(t,)i(then)g Fu(u)527 2211 y Fr(1)556 2205 y Fx(=)d Fu(C)633 2190 y Fp(t)630 2216 y Fr(1)649 2205 y Fx([)-7 b([)o Fu(s)684 2211 y Fr(1)703 2205 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)815 2211 y Fp(n)837 2205 y Fx(])-7 b(])o(.)14 b(Consider)g(the)g(follo)o(wing)e(sub)q (cases.)125 2302 y(\(a\))21 b(If)14 b Fu(u)265 2308 y Fr(1)295 2302 y Ft(!)337 2287 y Fp(t)362 2302 y Fu(u)386 2308 y Fr(2)405 2302 y Fx(,)f(then)i(the)f(assertion)h(follo)o(ws)d(as) i(in)f(case)i(\(i\))f(\(a\).)123 2348 y(\(b\))21 b(If)14 b Fu(u)265 2354 y Fr(1)295 2348 y Ft(!)337 2333 y Fp(o)366 2348 y Fu(u)390 2354 y Fr(2)409 2348 y Fx(,)f(then)h(the)h(assertion)g (follo)o(ws)d(as)i(in)f(case)i(\(i\))f(\(b\).)128 2394 y(\(c\))21 b(If)12 b Fu(u)263 2400 y Fr(1)293 2394 y Ft(!)335 2379 y Fp(i)360 2394 y Fu(u)384 2400 y Fr(2)402 2394 y Fx(,)g(then)h Fu(u)543 2400 y Fr(1)573 2394 y Fx(=)f Fu(C)650 2379 y Fp(t)664 2394 y Fx([)-7 b([)o Fu(s)699 2400 y Fr(1)718 2394 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)830 2400 y Fp(j)847 2394 y Fu(;)g(:)g(:)g(:)e(;)i(s)959 2400 y Fp(n)982 2394 y Fx(])-7 b(])11 b(and)h Fu(u)1113 2400 y Fr(2)1143 2394 y Fx(=)g Fu(C)1220 2379 y Fp(t)1234 2394 y Fx([)-7 b([)o Fu(s)1269 2400 y Fr(1)1289 2394 y Fu(;)7 b(:)g(:)g(:)t(;)g(t)1396 2400 y Fp(j)1413 2394 y Fu(;)g(:)g(:)g(:)e(;)i(s)1525 2400 y Fp(n)1548 2394 y Fx(])-7 b(])o(,)11 b(where)199 2444 y Fu(s)218 2450 y Fp(j)248 2444 y Ft(!)290 2429 y Fp(i)315 2444 y Fu(t)330 2450 y Fp(j)347 2444 y Fx(.)i(Hence)18 b(^)-23 b Fu(u)520 2450 y Fr(1)550 2444 y Fx(=)12 b Fu(u)618 2450 y Fr(1)636 2444 y Ft(#)f Fx(=)h Fu(u)736 2450 y Fr(2)754 2444 y Ft(#)g Fx(=)i(^)-23 b Fu(u)855 2450 y Fr(2)873 2444 y Fx(.)100 2542 y(Again,)233 2535 y(^)232 2542 y Fu(t)12 b Fx(=)309 2535 y(^)303 2542 y Fu(t)318 2548 y Fr(1)348 2542 y Fx(=)398 2535 y(^)392 2542 y Fu(t)407 2548 y Fr(2)439 2542 y Fx(is)i(the)g(unique)g(normal)e(form)g(of)h Fu(t)h Fx(w.r.t.)f Ft(R)p Fx(.)h(This)f(concludes)i(the)g(pro)q(of.)e Fe(2)141 2640 y Fx(It)i(has)g(already)g(b)q(een)h(men)o(tioned)e(that)h (giv)o(en)g(a)g(semi-complete)e(TRS)i(\()p Ft(F)t Fu(;)7 b Ft(R)o Fx(\),)15 b(w)o(e)g(also)g(ha)o(v)o(e)p eop %%Page: 15 15 15 14 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(15)p 100 224 1595 2 v 100 299 a Fx(to)12 b(solv)o(e)h(the)g(problem)e(ho)o(w)i (to)f(\014nd)h(the)g(unique)g(normal)e(form)g(of)h(a)g(term)g Fu(t)p Fx(.)h(If)f(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))12 b(is)h(\014nitely)100 349 y(branc)o(hing,)j(then)h(the)g(normal)d(form) h(of)h Fu(t)h Fx(can)f(alw)o(a)o(ys)g(b)q(e)h(obtained)f(b)o(y)h(tra)o (v)o(ersing)f(the)i(reduc-)100 399 y(tion)c(graph)g(of)g Fu(t)g Fx(breadth)i(\014rst.)f(This)f(is)h(w)o(ell-kno)o(wn)e(from)g (logic)g(programming.)e(F)m(or)j(e\016ciency)100 448 y(reasons,)i(ho)o(w)o(ev)o(er,)g(the)h(searc)o(hing)f(strategy)h(of)e (almost)f(all)h(Prolog)g(implemen)o(tations)e(is)j(depth)100 498 y(\014rst.)f(F)m(ortunately)m(,)f(in)g(the)i(aforemen)o(tioned)e(v) o(ery)i(imp)q(ortan)o(t)d(sp)q(ecial)i(case,)h(w)o(e)f(can)g(use)h(an)f (in-)100 548 y(nermost)f(reduction)h(strategy)m(.)f(A)g(reduction)h (step)g Fu(s)e Ft(!)1006 554 y Fn(R)1048 548 y Fu(t)i Fx(is)f Fs(innermost)g Fx(if)f(no)h(prop)q(er)i(subterm)100 598 y(of)11 b(the)h(con)o(tracted)h(redex)f(is)g(itself)f(a)g(redex.)h (An)g Fs(innermost)h(r)n(e)n(duction)g(se)n(quenc)n(e)f Fx(consists)g(only)f(of)100 648 y(innermost)j(reduction)h(steps.)h(The) f(TRS)f(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(is)h Fs(innermost)h (normalizing)e Fx(if,)f(for)i(ev)o(ery)g(term)100 697 y Fu(s)p Fx(,)h(there)i(is)e(an)g(innermost)g(reduction)h(sequence)h Fu(s)c Ft(!)993 682 y Fn(\003)993 709 y(R)1037 697 y Fu(t)j Fx(so)f(that)h Fu(t)e Ft(2)h Fu(N)5 b(F)h Fx(\()p Ft(!)1418 703 y Fn(R)1447 697 y Fx(\).)16 b(It)h(is)f Fs(inner-)100 747 y(most)h(terminating)g Fx(if)f(there)i(is)f(no)f (in\014nite)h(innermost)f(reduction)i(sequence.)g(The)g(notions)e(are) 100 797 y(related)e(as)g(follo)o(ws:)e(termination)g Ft(\))i Fx(innermost)f(termination)f Ft(\))i Fx(innermost)f (normalization)e Ft(\))100 847 y Fx(normalization.)100 946 y Fk(Pr)o(oposition)16 b(5.3.)21 b Fx(The)14 b(follo)o(wing)d(prop) q(erties)16 b(are)e(mo)q(dular)e(for)i(comp)q(osable)e(TRSs:)125 1045 y(\(1\))21 b(Lo)q(cal)14 b(con\015uence.)125 1095 y(\(2\))21 b(Normalization.)125 1144 y(\(3\))g(Innermost)14 b(normalization.)125 1194 y(\(4\))21 b(Innermost)14 b(termination.)100 1293 y Fk(Pr)o(oof.)47 b Fx(\(1\))21 b(In)16 b(essence,)h(this)f(follo) o(ws)d(from)h(the)i(Critical)f(P)o(air)g(Lemma)d(since)17 b(the)f(set)g(of)f(all)199 1343 y(critical)d(pairs)g(of)g Ft(R)h Fx(coincides)g(with)f(the)h(union)e(of)h(the)h(sets)h(of)e(all)f (critical)h(pairs)g(of)g Ft(R)1596 1349 y Fr(1)1627 1343 y Fx(and)199 1393 y Ft(R)234 1399 y Fr(2)267 1393 y Fx(\(cf.)i (Middeldorp,)f(1990,)f(and)i(Ohlebusc)o(h,)g(1994)p Fs(b)p Fx(\).)125 1442 y(\(2\))21 b(Ev)o(ery)14 b(term)e Fu(t)f Ft(2)g(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\))13 b(can)g(b)q(e)g (rewritten)h(to)f(normal)e(form)g(reducing)i(la)o(y)o(er)f(b)o(y)h(la)o (y)o(er)f(in)199 1492 y(a)h(b)q(ottom)e(up)h(fashion.)g(That)g(is,)g (\014rst)h(the)h(b)q(ottom)d(la)o(y)o(er)h(of)g Fu(t)g Fx(is)h(reduced)h(to)e(normal)f(form,)199 1542 y(then)j(the)f(same)f (is)h(done)g(with)g(the)h(la)o(y)o(er)e(ab)q(o)o(v)o(e)h(the)g(b)q (ottom)f(la)o(y)o(er)g(and)h(so)g(on.)f(Ev)o(en)o(tually)199 1592 y(the)17 b(top)f(la)o(y)o(er)f(is)h(reduced)h(to)f(normal)e(form;) g(the)i(term)f(obtained)h(is)g(a)f(normal)f(form)g(of)i Fu(t)199 1641 y Fx(\(cf.)h(Ohlebusc)o(h,)h(1994)p Fs(b)p Fx(,)e(and)h(Middeldorp,)g(1990\).)f(Note)i(that)f(ev)o(en)h(if)f(the)h (top)f(la)o(y)o(er)g(is)199 1691 y(transparen)o(t,)e(it)e(can)h(b)q(e)h (normalized)d(b)o(y)i(\()p Ft(B)r Fu(;)7 b Ft(S)r Fx(\))14 b(according)g(to)g(Lemma)d(5.1.)125 1741 y(\(3\))21 b(Analogous)13 b(to)h(\(2\).)125 1790 y(\(4\))21 b(It)16 b(is)f(not)h(to)q(o)f (di\016cult)g(to)g(pro)o(v)o(e)h(this)f(b)o(y)g(structural)i(induction) e(\(see)i(Ohlebusc)o(h,)f(1994)p Fs(b)p Fx(;)199 1840 y(cf.)e(also)f(Gramlic)o(h,)e(1994)p Fs(b)p Fx(,)h(Krishna)j(Rao,)d (1993\).)141 1890 y Fe(2)100 1989 y Fk(Cor)o(ollar)m(y)18 b(5.4.)j Fx(The)c(com)o(bined)f(system)g Ft(R)h Fx(of)g(t)o(w)o(o)f (complete)g(comp)q(osable)g(TRSs)h Ft(R)1591 1995 y Fr(1)1627 1989 y Fx(and)100 2039 y Ft(R)135 2045 y Fr(2)167 2039 y Fx(is)d(semi-complete)e(and)i(innermost)f(terminating.)722 2138 y Fk(5.2.)24 b(termina)m(tion)141 2238 y Fx(As)12 b(far)g(as)g(termination)e(is)i(concerned,)h(the)g(\014rst)f(mo)q (dularit)o(y)e(results)j(w)o(ere)g(obtained)e(b)o(y)h(in)o(v)o(es-)100 2288 y(tigating)h(the)i(distribution)f(of)g(collapsing)f(and)i (duplicating)e(rules)i(among)e(the)i(TRSs.)f(Rusino)o(w-)100 2337 y(itc)o(h)h(\(1987\))g(sho)o(w)o(ed)h(that)f(termination)f(is)i (mo)q(dular)d(for)j(non-collapsing)e(and)h(non-duplicating)100 2387 y(disjoin)o(t)e(TRSs,)h(resp)q(ectiv)o(ely)m(.)g(F)m(urthermore,)g (Middeldorp)g(\(1989\))g(pro)o(v)o(ed)g(that)g(termination)f(is)100 2437 y(preserv)o(ed)g(under)f(disjoin)o(t)e(union)h(if)f(one)i(of)f (the)g(systems)h(con)o(tains)f(neither)h(collapsing)e(nor)i(dupli-)100 2487 y(cating)g(rules.)g(A)g(simple)f(pro)q(of)h(for)g(all)f Fs(thr)n(e)n(e)h Fx(results)h(can)f(b)q(e)h(found)f(in)g(Ohlebusc)o(h)h (\(1993\).)e(These)100 2545 y(results)j(extend,)g Fs(mutatis)g (mutandis)p Fx(,)g(to)f(constructor-sharing)i(TRSs,)e(see)h(Ohlebusc)o (h)h(\(1995)p Fs(a)p Fx(\))1662 2530 y Ft(y)1682 2545 y Fx(.)135 2628 y Fq(y)170 2640 y Fw(A)d(similar)e(pro)q(of)g(sk)o(etc) o(h)g(w)o(as)i(giv)o(en)e(indep)q(enden)o(tl)o(y)f(in)i(Dersho)o(witz)f (\(1994\).)p eop %%Page: 16 16 16 15 bop 100 197 a Fw(16)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fx(The)j(basic)g(underlying)g(idea)g(of)f (Ohlebusc)o(h)i(\(1993,)e(1995)p Fs(a)p Fx(\))g(can)i(b)q(e)f(used)h (to)f(establish)g(the)h(next)100 349 y(result,)i(namely)e(the)j (generalization)e(of)g(the)i(ab)q(o)o(v)o(e-men)o(tioned)d(results)j (to)f(comp)q(osable)f(TRSs.)100 399 y(First,)d(w)o(e)i(need)f(a)g(few)g (prerequisites.)100 498 y Fk(Definition)i(5.5.)21 b Fx(Let)16 b(\()p Ft(F)554 504 y Fr(1)573 498 y Fu(;)7 b Ft(R)627 504 y Fr(1)645 498 y Fx(\))16 b(and)f(\()p Ft(F)809 504 y Fr(2)828 498 y Fu(;)7 b Ft(R)882 504 y Fr(2)900 498 y Fx(\))16 b(b)q(e)g(comp)q(osable)e(TRSs.)i(Let)g Fu(j)g Ft(2)f(f)p Fx(1)p Fu(;)7 b Fx(2)p Ft(g)p Fx(.)13 b(The)100 548 y(system)j(\()p Ft(F)290 554 y Fp(j)308 548 y Fu(;)7 b Ft(R)361 554 y Fp(j)379 548 y Fx(\))16 b(is)h(called)f Fs(layer-pr)n(eserving)p Fx(,)f(if)h(for)g(all)f Fu(l)j Ft(!)d Fu(r)i Ft(2)f(R)1260 554 y Fp(j)1294 548 y Fx(w)o(e)h(ha)o(v)o (e)f Fu(r)q(oot)p Fx(\()p Fu(r)q Fx(\))g Ft(2)g(A)1676 554 y Fp(j)100 597 y Fx(whenev)o(er)f Fu(r)q(oot)p Fx(\()p Fu(l)q Fx(\))d Ft(2)f(A)486 603 y Fp(j)504 597 y Fx(.)141 697 y(Disjoin)o(t)g(TRSs)i(are)g(la)o(y)o(er-preserving)g(if)f(and)h (only)f(if)g(they)h(are)g(non-collapsing.)e(Constructor-)100 746 y(sharing)20 b(TRSs)h(are)h(la)o(y)o(er-preserving)f(if)f(and)h (only)f(if)g(they)i(con)o(tain)e(neither)i(collapsing)e(nor)100 796 y(constructor-lifting)e(rules)i(\(constructor-lifting)e(rules)i (are)f(rewrite)h(rules)f(in)f(whic)o(h)h(the)g(righ)o(t-)100 846 y(hand)13 b(side)i(has)f(a)f(shared)i(constructor)h(at)d(its)h(ro)q (ot)g(p)q(osition\).)100 945 y Fk(Lemma)i(5.6.)21 b Fx(Let)d Fu(s;)7 b(t)17 b Ft(2)g(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))18 b(suc)o(h)g(that)f Fu(s)h Ft(!)973 930 y Fp(t)1004 945 y Fu(t)f Fx(is)g(a)g(non-duplicating)f(reduction)i (step.)100 995 y(Then)c Fu(S)233 1001 y Fr(1)252 995 y Fx(\()p Fu(t)p Fx(\))e Ft(\022)g Fu(S)380 1001 y Fr(1)399 995 y Fx(\()p Fu(s)p Fx(\).)i(In)g(particular,)f Fu(S)758 980 y Fp(w)756 1006 y(P)786 995 y Fx(\()p Fu(t)p Fx(\))e Ft(\022)h Fu(S)915 980 y Fp(w)913 1006 y(P)943 995 y Fx(\()p Fu(s)p Fx(\))j(and)e Fu(S)1116 980 y Fp(b)1114 1006 y(P)1142 995 y Fx(\()p Fu(t)p Fx(\))f Ft(\022)g Fu(S)1272 980 y Fp(b)1270 1006 y(P)1298 995 y Fx(\()p Fu(s)p Fx(\).)100 1094 y Fk(Pr)o(oof.)22 b Fx(According)e(to)h(Lemma)d (4.14,)h Fu(s)k Fx(=)g Fu(C)916 1079 y Fp(t)930 1094 y Fx([)-7 b([)p Fu(s)966 1100 y Fr(1)984 1094 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1096 1100 y Fp(n)1118 1094 y Fx(])-7 b(])22 b Ft(!)1199 1079 y Fp(t)1236 1094 y Fu(t)h Fx(=)g Fu(C)1362 1079 y Fp(t)1376 1094 y Ft(h)-7 b(h)p Fu(s)1420 1100 y Fp(i)1432 1104 y Ff(1)1451 1094 y Fu(;)7 b(:)g(:)g(:)e(;)i(s) 1563 1100 y Fp(i)1575 1104 y Fl(m)1604 1094 y Ft(i)-7 b(i)21 b Fx(b)o(y)100 1144 y(some)c(rule)i Fu(l)h Ft(!)e Fu(r)i Ft(2)f(S)s Fx(.)f(Let)h Fu(l)h Fx(=)g Fu(C)s Fx([)p Fu(x)764 1150 y Fr(1)781 1144 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)898 1150 y Fp(k)917 1144 y Fx(])18 b(with)g(all)f(v)n(ariables)h(displa)o (y)o(ed.)f(Then)i Fu(r)g Fx(has)100 1194 y(a)c(represen)o(tation)i Fu(r)f Fx(=)e Fu(C)523 1179 y Fn(0)535 1194 y Fx([)p Fu(x)571 1200 y Fp(i)583 1204 y Ff(1)600 1194 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)717 1200 y Fp(i)729 1204 y Fl(j)745 1194 y Fx(].)15 b(The)h(m)o(ultiset)e(inclusion)h([)p Fu(x)1244 1200 y Fp(i)1256 1204 y Ff(1)1274 1194 y Fu(;)7 b(:)g(:)g(:)t(;)g(x)1390 1200 y Fp(i)1402 1204 y Fl(j)1419 1194 y Fx(])14 b Ft(\022)h Fx([)p Fu(x)1528 1200 y Fr(1)1546 1194 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1663 1200 y Fp(k)1682 1194 y Fx(])100 1244 y(holds)15 b(b)q(ecause)j(the)f(rule)f(is)g (non-duplicating.)f(It)h(is)g(easy)h(to)f(v)o(erify)f(that)h(this)h (implies)d Fu(S)1580 1250 y Fr(1)1599 1244 y Fx(\()p Fu(t)p Fx(\))h Ft(\022)100 1293 y Fu(S)125 1299 y Fr(1)144 1293 y Fx(\()p Fu(s)p Fx(\).)20 b(Consequen)o(tly)m(,)g(w)o(e)g(also)f (ha)o(v)o(e)h Fu(S)784 1278 y Fp(w)782 1304 y Fr(1)812 1293 y Fx(\()p Fu(t)p Fx(\))i Ft(\022)g Fu(S)962 1278 y Fp(w)960 1304 y Fr(1)990 1293 y Fx(\()p Fu(s)p Fx(\))e(and)g Fu(S)1175 1278 y Fp(b)1173 1304 y Fr(1)1193 1293 y Fx(\()p Fu(t)p Fx(\))i Ft(\022)g Fu(S)1343 1278 y Fp(b)1341 1304 y Fr(1)1360 1293 y Fx(\()p Fu(s)p Fx(\).)f(No)o(w)e(w)o(e)i(infer)100 1343 y Fu(S)127 1328 y Fp(w)125 1354 y Fr(2)154 1343 y Fx(\()p Fu(t)p Fx(\))i Ft(\022)f Fu(S)305 1328 y Fp(w)303 1354 y Fr(2)333 1343 y Fx(\()p Fu(s)p Fx(\))f(from)e Fu(S)537 1328 y Fp(b)535 1354 y Fr(1)554 1343 y Fx(\()p Fu(t)p Fx(\))k Ft(\022)f Fu(S)705 1328 y Fp(b)703 1354 y Fr(1)723 1343 y Fx(\()p Fu(s)p Fx(\).)e(All)g(in)f(all,)g(it)h(follo) o(ws)f Fu(S)1229 1328 y Fp(w)1227 1355 y(P)1256 1343 y Fx(\()p Fu(t)p Fx(\))k(=)f Fu(S)1407 1328 y Fp(w)1405 1354 y Fr(1)1435 1343 y Fx(\()p Fu(t)p Fx(\))14 b Ft([)f Fu(S)1564 1328 y Fp(w)1562 1354 y Fr(2)1592 1343 y Fx(\()p Fu(t)p Fx(\))22 b Ft(\022)100 1393 y Fu(S)127 1378 y Fp(w)125 1403 y Fr(1)154 1393 y Fx(\()p Fu(s)p Fx(\))11 b Ft([)f Fu(S)281 1378 y Fp(w)279 1403 y Fr(2)309 1393 y Fx(\()p Fu(s)p Fx(\))15 b(=)g Fu(S)449 1378 y Fp(w)447 1404 y(P)476 1393 y Fx(\()p Fu(s)p Fx(\).)h(The)g(remaining)e (inclusion)h Fu(S)1040 1378 y Fp(b)1038 1404 y(P)1066 1393 y Fx(\()p Fu(t)p Fx(\))f Ft(\022)h Fu(S)1201 1378 y Fp(b)1199 1404 y(P)1227 1393 y Fx(\()p Fu(s)p Fx(\))h(is)g(pro)o(v)o (ed)f(analogously)m(.)100 1443 y Fe(2)100 1542 y Fk(Lemma)h(5.7.)21 b Fx(Let)15 b Fu(s)f Ft(!)497 1522 y Fp(t;o)497 1554 y Fn(A)524 1558 y Ff(1)555 1542 y Fu(t)g Fx(b)q(e)h(a)e (non-duplicating)g(reduction)h(step.)h(Then)f Fu(S)1392 1527 y Fp(w)1390 1553 y(P)1420 1542 y Fx(\()p Fu(t)p Fx(\))e Ft(\022)g Fu(S)1550 1527 y Fp(w)1548 1553 y(P)1577 1542 y Fx(\()p Fu(s)p Fx(\).)100 1641 y Fk(Pr)o(oof.)22 b Fx(Similar)11 b(to)i(the)i(pro)q(of)e(of)h(Lemma)d(5.6.)h Fe(2)100 1740 y Fk(Pr)o(oposition)k(5.8.)21 b Fx(Let)12 b Ft(R)567 1746 y Fr(1)598 1740 y Fx(and)f Ft(R)711 1746 y Fr(2)742 1740 y Fx(b)q(e)h(t)o(w)o(o)f(terminating)f(comp)q(osable)h (TRSs)h(suc)o(h)g(that)g(their)100 1790 y(com)o(bined)i(system)i Ft(R)g Fx(do)q(es)h(not)f(terminate.)f(Then)i(either)g(statemen)o(t)f (\(i\))f(holds)h(or,)g(if)f(\(i\))h(do)q(es)100 1840 y(not)d(hold,)g(then)i(statemen)o(t)f(\(ii\))f(m)o(ust)g(hold.)100 1940 y(\(i\))19 b(There)i(is)e(an)h(in\014nite)f Ft(R)h Fx(deriv)n(ation)f Fu(D)i Fx(starting)e(from)f(a)h(non-top-transparen)o (t,)h(sa)o(y)g(top)100 1990 y(blac)o(k,)13 b(term)g(suc)o(h)h(that:)125 2088 y(\(1\))21 b(There)15 b(is)f(no)g(top)f(white)h(term)g(in)f Fu(D)q Fx(.)125 2137 y(\(2\))21 b(There)15 b(are)g(in\014nitely)e(man)o (y)26 b Ft(!)732 2117 y Fp(t;o)732 2149 y Fn(A)759 2153 y Ff(1)804 2137 y Fx(reduction)14 b(steps)i(in)d Fu(D)q Fx(.)125 2186 y(\(3\))21 b(There)15 b(are)f(in\014nitely)f(man)o(y)f Ft(!)717 2192 y Fn(R)746 2196 y Ff(2)777 2186 y Fx(reduction)i(steps)h (in)e Fu(D)i Fx(whic)o(h)f(are)g(destructiv)o(e)h(at)f(lev)o(el)199 2236 y(1)g(or)g(lev)o(el)f(2.)125 2285 y(\(4\))21 b(There)15 b(are)g(in\014nitely)e(man)o(y)f(duplicating)26 b Ft(!)948 2265 y Fp(t;o)948 2297 y Fn(A)975 2301 y Ff(1)1020 2285 y Fx(reduction)15 b(steps)g(in)f Fu(D)q Fx(.)100 2384 y(\(ii\))f(There)i(is)f(an)f(in\014nite)h Ft(R)g Fx(deriv)n(ation)f Fu(D)i Fx(suc)o(h)g(that:)125 2483 y(\(1\))21 b Fu(D)16 b Fx(consists)f(solely)e(of)g(top)h(transparen)o(t)h(terms.)125 2532 y(\(2\))21 b(There)15 b(are)g(in\014nitely)e(man)o(y)f Ft(!)718 2517 y Fp(t)746 2532 y Fx(reduction)i(steps)i(in)d Fu(D)q Fx(.)125 2581 y(\(3\))21 b(There)15 b(are)g(in\014nitely)e(man)o (y)f Ft(!)718 2566 y Fp(o)750 2581 y Fx(reduction)i(steps)h(in)f Fu(D)h Fx(destructiv)o(e)h(at)d(lev)o(el)h(1.)125 2630 y(\(4\))21 b(There)15 b(are)g(in\014nitely)e(man)o(y)f(duplicating)g Ft(!)934 2615 y Fp(t)962 2630 y Fx(reduction)j(steps)g(in)f Fu(D)q Fx(.)p eop %%Page: 17 17 17 16 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(17)p 100 224 1595 2 v 100 299 a Fk(Pr)o(oof.)22 b Fx(\(i\))13 b(Supp)q(ose)i(there)h(is)e(an)g(in\014nite)g Ft(R)g Fx(deriv)n(ation)f Fu(D)j Fx(starting)e(from)e(a)i(top)g(blac)o(k)g (term.)100 349 y(Let)f(the)h(rank)f(of)g(a)g(deriv)n(ation)f Fu(D)h Fx(:)e Fu(s)703 355 y Fr(1)734 349 y Ft(!)g Fu(s)806 355 y Fr(2)837 349 y Ft(!)g Fu(s)909 355 y Fr(3)939 349 y Ft(!)g Fu(:)c(:)g(:)12 b Fx(b)q(e)i(de\014ned)g(b)o(y)f Fu(r)q(ank)q Fx(\()p Fu(D)q Fx(\))f(=)g Fu(r)q(ank)q Fx(\()p Fu(s)1647 355 y Fr(1)1666 349 y Fx(\).)100 399 y(W.l.o.g.)o(,)i(w)o(e)k(ma)o(y)d(assume)i(that)g Fu(D)i Fx(is)e(of)g(minim)o(al)d(rank.)i(In)i(other)g(w)o(ords,)f(if)f Fu(r)q(ank)q Fx(\()p Fu(D)q Fx(\))i(=)f Fu(k)q Fx(,)100 448 y(then)g Ft(!)239 454 y Fn(R)286 448 y Fx(is)g(terminating)e(on)h Ft(T)654 433 y Fp()g Fx(=)g(\()p Ft(!)438 1230 y Fn(R)487 1224 y Ft([)p 527 1226 3 25 v 9 w Fu(>)p Fx(\))572 1209 y Fr(+)600 1224 y Fx(.)e(Since)i Ft(!)784 1230 y Fn(R)833 1224 y Fx(is)f(closed)h(under)g(con)o(texts,)g(it)f(is)g(not)g (to)q(o)g(di\016cult)199 1273 y(to)c(pro)o(v)o(e)g(that)g(\()p Ft(T)501 1258 y Fp()p Fx(\))14 b(is)g(a)f(w)o(ell-founded)g(ordering.)g(Let)i(\()p Ft(M)p Fx(\()p Ft(T)1311 1258 y Fp()1424 1258 y Fp(mul)1486 1273 y Fx(\))14 b(denote)h(its)199 1323 y(w)o(ell-founded)g(m)o(ultiset)e(extension.)i(Note)h(that)f(ev)o (ery)h(m)o(ultiset)d Fu(S)1292 1308 y Fp(w)1290 1335 y(P)1320 1323 y Fx(\()p Fu(s)1355 1329 y Fp(j)1373 1323 y Fx(\))i(is)g(an)g(elemen)o(t)f(of)199 1373 y Ft(M)p Fx(\()p Ft(T)298 1358 y Fp()7 b(w)13 b Fx(for)f(an)o(y)g (principal)g(subterm)274 1859 y Fu(w)g Ft(2)g Fu(S)383 1844 y Fp(w)381 1871 y(P)410 1859 y Fx(\()p Fu(v)q Fx(\))j(that)f Fu(u)d(>)h(w)j Fx(for)e(an)o(y)h Fu(w)e Ft(2)f Fu(S)942 1844 y Fp(w)940 1871 y(P)970 1859 y Fx(\()p Fu(v)q Fx(\).)j(Therefore)h Fu(S)1264 1844 y Fp(w)1262 1871 y(P)1292 1859 y Fx(\()p Fu(s)1327 1865 y Fp(j)1345 1859 y Fx(\))d Fu(>)1405 1844 y Fp(mul)1479 1859 y Fu(S)1506 1844 y Fp(w)1504 1871 y(P)1533 1859 y Fx(\()p Fu(s)1568 1865 y Fp(j)r Fr(+1)1628 1859 y Fx(\).)199 1936 y(W)m(e)g(conclude)g(from)e(the)j(w)o (ell-foundedness)f(of)f(\()p Ft(M)p Fx(\()p Ft(T)1077 1921 y Fp()1191 1921 y Fp(mul)1252 1936 y Fx(\))12 b(that)g(only)f(\014nitely)g(man)o(y)213 1986 y Ft(!)255 1971 y Fp(o)255 1997 y Fn(A)282 2001 y Ff(2)327 1986 y Fx(and)j Ft(!)450 1971 y Fp(i)477 1986 y Fx(steps)h(can)f(o)q(ccur)h(in)f(the)g(deriv)n(ation)f Fu(D)q Fx(.)h(This)g(con)o(tradicts)h(\(3\).)100 2086 y(\(ii\))e(Supp)q(ose)i(that)f(there)h(are)f(no)g(in\014nite)f(rewrite) i(deriv)n(ations)f(starting)g(from)e(top)h(blac)o(k)h(or)g(top)100 2136 y(white)g(terms.)f(Let)h Fu(D)i Fx(b)q(e)e(an)g(in\014nite)g Ft(R)g Fx(deriv)n(ation.)e(Clearly)m(,)h(this)h(implies)e(that)i(ev)o (ery)g(blac)o(k)g(or)100 2186 y(white)f(principal)g(subterm)h(o)q (ccurring)h(in)e Fu(D)i Fx(is)f(terminating.)d(W.l.o.g.,)f(w)o(e)k(ma)o (y)e(assume)h(that)h Fu(D)100 2236 y Fx(is)f(of)h(minim)o(al)c(rank)k Fu(k)q Fx(.)125 2337 y(\(1\))21 b(If)13 b(there)i(w)o(as)e(a)h(top)f (blac)o(k)g(or)g(top)h(white)f(term)g(in)g Fu(D)q Fx(,)g(then)i(there)f (w)o(ould)f(b)q(e)h(an)f(in\014nite)g Ft(R)199 2387 y Fx(deriv)n(ation)g(starting)h(from)e(a)i(top)g(blac)o(k)f(or)h(top)g (white)g(term)f({)h(a)f(con)o(tradiction.)125 2439 y(\(2\))21 b(Supp)q(ose)12 b(that)g(there)g(are)g(only)e(\014nitely)h(man)o(y)e Ft(!)994 2424 y Fp(t)1019 2439 y Fx(reduction)j(steps)g(in)f Fu(D)q Fx(.)g(W.l.o.g.)d(w)o(e)k(ma)o(y)199 2489 y(assume)i(that)h Fu(D)h Fx(con)o(tains)e(no)h Ft(!)748 2474 y Fp(t)776 2489 y Fx(reduction)g(steps)h(at)f(all.)e(Th)o(us,)h(if)g Fu(s)1363 2495 y Fr(1)1394 2489 y Fx(=)f Fu(C)1472 2474 y Fp(t)1486 2489 y Fx([)-7 b([)p Fu(t)1518 2495 y Fr(1)1536 2489 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1644 2495 y Fp(n)1666 2489 y Fx(])-7 b(],)199 2539 y(then)17 b(there)g(m)o(ust)d(b)q(e)j(an)e (in\014nite)h(rewrite)g(deriv)n(ation)f(starting)h(from)e(some)h(top)g (blac)o(k)h(or)199 2588 y(top)e(white)g(term)f Fu(t)500 2594 y Fp(l)525 2588 y Ft(2)e Fu(S)589 2594 y Fr(1)608 2588 y Fx(\()p Fu(s)643 2594 y Fr(1)662 2588 y Fx(\))j(whic)o(h)g(is)g (a)f(con)o(tradiction)h(to)g(\(1\).)125 2640 y(\(3\))21 b(Supp)q(ose)14 b(that)g(there)g(is)f(no)g Ft(!)696 2625 y Fp(o)727 2640 y Fx(reduction)h(step)h(in)d Fu(D)j Fx(whic)o(h)e(is)g (destructiv)o(e)i(at)e(lev)o(el)g(1.)g(In)p eop %%Page: 18 18 18 17 bop 100 197 a Fw(18)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 199 299 a Fx(this)g(case)g(w)o(e)f(ha)o(v)o(e)g(for)g (an)o(y)g(reduction)h(step)g Fu(s)929 305 y Fp(j)959 299 y Ft(!)1001 284 y Fp(t)1027 299 y Fu(s)1046 305 y Fp(j)r Fr(+1)1116 299 y Fx(in)f Fu(D)h Fx(that)g Fu(top)1349 284 y Fp(t)1363 299 y Fx(\()p Fu(s)1398 305 y Fp(j)1416 299 y Fx(\))h Ft(!)1486 284 y Fp(t)1512 299 y Fu(top)1568 284 y Fp(t)1582 299 y Fx(\()p Fu(s)1617 305 y Fp(j)r Fr(+1)1678 299 y Fx(\))199 349 y(using)20 b(the)h(same)f(rule)g(from)f Ft(S)k Fx(and)d(for)g(ev)o(ery)h Fu(s)1037 355 y Fp(j)1077 349 y Ft(!)1119 334 y Fp(o)1160 349 y Fu(s)1179 355 y Fp(j)r Fr(+1)1259 349 y Fx(or)f Fu(s)1335 355 y Fp(j)1375 349 y Ft(!)1417 334 y Fp(i)1453 349 y Fu(s)1472 355 y Fp(j)r Fr(+1)1552 349 y Fx(step)h(w)o(e)199 399 y(ha)o(v)o(e)d Fu(top)355 383 y Fp(t)370 399 y Fx(\()p Fu(s)405 405 y Fp(j)423 399 y Fx(\))h(=)h Fu(top)566 383 y Fp(t)580 399 y Fx(\()p Fu(s)615 405 y Fp(j)r Fr(+1)675 399 y Fx(\).)e(Hence)i(w) o(e)f(conclude)g(that)g Ft(S)i Fx(is)d(non-terminating)f(whic)o(h)199 448 y(con)o(tradicts)e(the)f(termination)f(of)g Ft(R)793 454 y Fr(1)812 448 y Fx(.)125 500 y(\(4\))21 b(Let)d Fu(T)23 b Fx(=)17 b Ft(f)p Fu(t)g Ft(2)g(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))18 b Ft(j)e Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))h(=)h Fu(k)c Fx(and)g Fu(r)q(oot)p Fx(\()p Fu(t)p Fx(\))j Ft(2)g(A)1201 506 y Fr(1)1231 500 y Ft(])11 b(A)1304 506 y Fr(2)1336 500 y Fx(or)j Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))j Fu(<)g(k)q Ft(g)p Fx(.)g(It)199 550 y(follo)o(ws)c(from) f(our)j(assumptions)e(that)h Ft(!)877 556 y Fn(R)921 550 y Fx(is)h(terminating)d(on)i Fu(T)6 b Fx(.)14 b(Let)g Fu(>)h Fx(=)d(\()p Ft(!)1527 556 y Fn(R)1570 550 y Ft([)p 1610 552 3 25 v 8 w Fu(>)p Fx(\))1654 535 y Fr(+)1682 550 y Fx(.)199 600 y(Again,)f(\()p Fu(T)s(;)c(>)p Fx(\))12 b(is)g(a)f(w)o(ell-founded)g(ordering.)g(Let)h(\()p Ft(M)p Fx(\()p Fu(T)6 b Fx(\))p Fu(;)h(>)1188 585 y Fp(mul)1250 600 y Fx(\))12 b(denote)h(its)e(w)o(ell-founded)199 650 y(m)o(ultiset)k(extension.)h(Note)g(that)g Fu(S)778 656 y Fr(1)797 650 y Fx(\()p Fu(s)832 656 y Fp(j)850 650 y Fx(\))f Ft(2)f(M)p Fx(\()p Fu(T)6 b Fx(\))p Fu(:)15 b Fx(Again,)g(w)o(e)h(ma)o(y)d(supp)q(ose)k(that)f(there)199 699 y(is)e(no)f(duplicating)f Ft(!)556 684 y Fp(t)584 699 y Fx(reduction)i(step)h(in)e Fu(D)i Fx(at)f(all.)d(W)m(e)j (distinguish)f(b)q(et)o(w)o(een)i(t)o(w)o(o)e(cases:)274 776 y(If)k Fu(s)338 782 y Fp(j)374 776 y Ft(!)416 761 y Fp(t)447 776 y Fu(s)466 782 y Fp(j)r Fr(+1)526 776 y Fx(,)g(then)h(b)o(y)g(Lemma)d(5.6)h Fu(S)962 782 y Fr(1)981 776 y Fx(\()p Fu(s)1016 782 y Fp(j)r Fr(+1)1076 776 y Fx(\))i Ft(\022)g Fu(S)1185 782 y Fr(1)1204 776 y Fx(\()p Fu(s)1239 782 y Fp(j)1257 776 y Fx(\))g(b)q(ecause)h(the)f (reduction)274 826 y(step)d(is)f(non-duplicating.)e(Clearly)m(,)g(this) i(implies)e Fu(S)1112 832 y Fr(1)1131 826 y Fx(\()p Fu(s)1166 832 y Fp(j)1184 826 y Fx(\))g Ft(\025)1244 811 y Fp(mul)1317 826 y Fu(S)1342 832 y Fr(1)1361 826 y Fx(\()p Fu(s)1396 832 y Fp(j)r Fr(+1)1456 826 y Fx(\).)274 925 y(If)19 b Fu(s)340 931 y Fp(j)378 925 y Ft(!)420 910 y Fp(o)458 925 y Fu(s)477 931 y Fp(j)r Fr(+1)556 925 y Fx(or)g Fu(s)631 931 y Fp(j)669 925 y Ft(!)711 910 y Fp(i)744 925 y Fu(s)763 931 y Fp(j)r Fr(+1)823 925 y Fx(,)g(then)g(there)i(is)e(a)f(blac)o(k)h (or)g(white)g(principal)f(sub-)274 975 y(term)g Fu(u)h Ft(2)g Fu(S)493 981 y Fr(1)512 975 y Fx(\()p Fu(s)547 981 y Fp(j)565 975 y Fx(\))g(suc)o(h)g(that)g Fu(u)g Ft(!)g Fu(v)h Fx(for)e(some)g Fu(v)q Fx(,)h(i.e.)e Fu(s)1256 981 y Fp(j)1293 975 y Fx(=)j Fu(C)1378 960 y Fp(t)1392 975 y Fx([)-7 b([)p Fu(;)7 b(:)g(:)g(:)t(;)g(u;)g(:)g(:)g(:)t(;)g Fx(])-7 b(])18 b Ft(!)274 1025 y Fu(C)307 1010 y Fp(t)321 1025 y Fx([)p Fu(;)7 b(:)g(:)g(:)e(;)i(v)q(;)g(:)g(:)g(:)t(;)g Fx(])12 b(=)g Fu(s)626 1031 y Fp(j)r Fr(+1)686 1025 y Fx(.)i(Th)o(us)g(w)o(e)h(ha)o(v)o(e)f Fu(S)1000 1031 y Fr(1)1019 1025 y Fx(\()p Fu(s)1054 1031 y Fp(j)r Fr(+1)1114 1025 y Fx(\))e(=)h(\()p Fu(S)1228 1031 y Fr(1)1247 1025 y Fx(\()p Fu(s)1282 1031 y Fp(j)1300 1025 y Fx(\))d Ft(n)f Fx([)p Fu(u)p Fx(]\))g Ft([)g Fu(S)1491 1031 y Fr(1)1510 1025 y Fx(\()p Fu(v)q Fx(\).)14 b(It)h(fol-)274 1075 y(lo)o(ws)f(from)f Fu(u)g Ft(!)f Fu(v)17 b Fx(in)d(conjunction)g(with)h Fu(v)f Fx(=)g Fu(w)h Fx(or)g Fu(v)p 1184 1077 V 20 w(>)10 b(w)16 b Fx(for)e(an)o(y)g(term)g Fu(w)g Ft(2)f Fu(S)1621 1081 y Fr(1)1640 1075 y Fx(\()p Fu(v)q Fx(\))274 1125 y(that)h Fu(u)d(>)h(w)j Fx(for)e(an)o(y)h Fu(w)e Ft(2)f Fu(S)736 1131 y Fr(1)755 1125 y Fx(\()p Fu(v)q Fx(\).)k(Therefore)g Fu(S)1048 1131 y Fr(1)1067 1125 y Fx(\()p Fu(s)1102 1131 y Fp(j)1120 1125 y Fx(\))d Fu(>)1180 1110 y Fp(mul)1253 1125 y Fu(S)1278 1131 y Fr(1)1297 1125 y Fx(\()p Fu(s)1332 1131 y Fp(j)r Fr(+1)1392 1125 y Fx(\).)199 1201 y(W)m(e)17 b(conclude)g(from)e(the)i(w)o(ell-foundedness)h(of)e(\()p Ft(M)p Fx(\()p Fu(T)6 b Fx(\))p Fu(;)h(>)1171 1186 y Fp(mul)1233 1201 y Fx(\))16 b(that)h(only)f(\014nitely)g(man)o(y)199 1251 y Ft(!)241 1236 y Fp(o)273 1251 y Fx(and)e Ft(!)396 1236 y Fp(i)423 1251 y Fx(steps)h(can)f(o)q(ccur)h(in)f(the)g(deriv)n (ation)f Fu(D)q Fx(.)h(This)g(con)o(tradicts)g(\(3\).)100 1352 y Fe(2)141 1453 y Fx(The)g(follo)o(wing)e(example)g(illustrates)i (that)g(case)h(\(ii\))e(of)h(Prop)q(osition)f(5.8)g(ma)o(y)f(o)q(ccur.) 100 1555 y Fk(Example)17 b(5.9.)k Fx(Let)16 b Ft(R)501 1561 y Fr(1)535 1555 y Fx(=)f Ft(f)p Fp(F)5 b Fx(\()p Fu(x;)i Fp(C)713 1561 y Fr(1)731 1555 y Fu(;)g Fp(C)776 1561 y Fr(2)794 1555 y Fu(;)g(y)q(;)g Fp(D)881 1561 y Fr(1)900 1555 y Fu(;)g Fp(D)946 1561 y Fr(2)965 1555 y Fx(\))15 b Ft(!)g Fp(F)5 b Fx(\()p Fu(x;)i(x;)g(x;)g(y)q(;)g(y)q(;)g (y)q Fx(\))p Fu(;)g(A)14 b Ft(!)h Fp(C)1488 1561 y Fr(1)1507 1555 y Fu(;)7 b(A)15 b Ft(!)f Fp(C)1654 1561 y Fr(2)1673 1555 y Ft(g)100 1605 y Fx(and)k Ft(R)221 1611 y Fr(2)259 1605 y Fx(=)j Ft(f)p Fp(F)t Fx(\()p Fu(x;)7 b Fp(C)442 1611 y Fr(1)461 1605 y Fu(;)g Fp(C)505 1611 y Fr(2)524 1605 y Fu(;)g(y)q(;)g Fp(D)610 1611 y Fr(1)629 1605 y Fu(;)g Fp(D)675 1611 y Fr(2)694 1605 y Fx(\))20 b Ft(!)g Fp(F)t Fx(\()p Fu(x;)7 b(x;)g(x;)g(y)q(;)g(y)q(;)g(y)q Fx(\))p Fu(;)g(b)19 b Ft(!)g Fp(D)1224 1611 y Fr(1)1243 1605 y Fu(;)7 b(b)19 b Ft(!)h Fp(D)1389 1611 y Fr(2)1407 1605 y Ft(g)p Fx(.)f(W)m(e)f(ha)o(v)o(e)h(the)100 1655 y(cyclic)14 b(deriv)n(ation)177 1731 y Fu(t)e Fx(=)g Fp(F)t Fx(\()p Fu(A;)7 b Fp(C)365 1737 y Fr(1)383 1731 y Fu(;)g Fp(C)428 1737 y Fr(2)446 1731 y Fu(;)g(b;)g Fp(D)529 1737 y Fr(1)548 1731 y Fu(;)g Fp(D)594 1737 y Fr(2)613 1731 y Fx(\))k Ft(!)682 1713 y Fp(t)708 1731 y(F)5 b Fx(\()p Fu(A;)i(A;)g(A;)g(b;)g(b;)g(b)p Fx(\))1046 1707 y Fp(o)15 b Fn(\003)1035 1731 y Ft(!)1077 1737 y Fn(A)1104 1741 y Ff(1)1135 1731 y Fp(F)5 b Fx(\()p Fu(A;)i Fp(C)1253 1737 y Fr(1)1271 1731 y Fu(;)g Fp(C)1315 1737 y Fr(2)1334 1731 y Fu(;)g(b;)g(b;)g(b)p Fx(\))1501 1707 y Fp(o)14 b Fn(\003)1490 1731 y Ft(!)1531 1737 y Fn(A)1558 1741 y Ff(2)1590 1731 y Fu(t:)100 1832 y Fk(Theorem)i(5.10.)21 b Fx(Let)14 b Ft(R)527 1838 y Fr(1)558 1832 y Fx(and)f Ft(R)673 1838 y Fr(2)705 1832 y Fx(b)q(e)h(t)o(w)o(o)e(terminating)f (comp)q(osable)h(TRSs.)h(Their)g(com)o(bined)100 1882 y(system)18 b Ft(R)h Fx(=)h Ft(R)383 1888 y Fr(1)414 1882 y Ft([)12 b(R)489 1888 y Fr(2)526 1882 y Fx(is)19 b(terminating)e(pro)o(vided)h(that)h(one)f(of)g(the)i(follo)o(wing)c (conditions)i(is)100 1932 y(satis\014ed:)125 2034 y(\(1\))j(Both)14 b Ft(R)337 2040 y Fr(1)370 2034 y Fx(and)g Ft(R)486 2040 y Fr(2)518 2034 y Fx(are)h(la)o(y)o(er-preserving.)125 2085 y(\(2\))21 b(Both)14 b Ft(R)337 2091 y Fr(1)370 2085 y Fx(and)g Ft(R)486 2091 y Fr(2)518 2085 y Fx(are)h (non-duplicating.)125 2137 y(\(3\))21 b(One)15 b(of)e(the)i(systems)f (is)f(b)q(oth)h(la)o(y)o(er-preserving)h(and)f(non-duplicating.)100 2238 y Fk(Pr)o(oof.)47 b Fx(\(1\))21 b(If)d(b)q(oth)h(systems)f(are)h (la)o(y)o(er-preserving,)g(then)g(there)g(can)g(b)q(e)g(no)f(rewrite)i (step)199 2288 y(whic)o(h)15 b(is)g(destructiv)o(e)i(at)d(lev)o(el)h(1) g(or)g(lev)o(el)f(2;)h(so)g(neither)h(case)g(\(i\))e(\(3\))h(nor)g (case)h(\(ii\))f(\(3\))g(is)199 2337 y(p)q(ossible.)125 2389 y(\(2\))21 b(If)c(b)q(oth)h(systems)f(are)h(non-duplicating,)e (then)i(neither)g(case)g(\(i\))g(\(4\))f(nor)g(case)i(\(ii\))d(\(4\))i (is)199 2439 y(p)q(ossible.)125 2491 y(\(3\))j(Let)12 b Ft(R)307 2497 y Fr(1)337 2491 y Fx(b)q(e)g(la)o(y)o(er-preserving)g (and)f(non-duplicating.)f(The)i(existence)h(of)e(an)h(in\014nite)f (deriv)n(a-)199 2540 y(tion)f(starting)g(from)e(a)i(top)g(blac)o(k)g (term)f(is)h(ruled)h(out)f(b)o(y)g(\(i\))g(\(4\).)f(Also,)h(no)g (in\014nite)f(deriv)n(ation)199 2590 y(starting)k(from)e(a)i(top)f (white)h(term)g(is)f(p)q(ossible)h(b)q(ecause)i(of)d(\(the)i(adjusted)f (v)o(ersion)g(of)s(\))g(case)199 2640 y(\(i\))g(\(3\).)g(Th)o(us,)g(if) f Ft(R)h Fx(w)o(ere)h(not)f(terminating,)e(then)j(there)h(w)o(ould)d(b) q(e)i(an)f(in\014nite)f(deriv)n(ation)p eop %%Page: 19 19 19 18 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(19)p 100 224 1595 2 v 199 299 a Fx(starting)15 b(from)f(a)h(top)g(transparen)o (t)h(term.)e(Ho)o(w)o(ev)o(er,)h(this)g(p)q(ossibilit)o(y)f(is)h (excluded)h(b)o(y)f(\(ii\))199 349 y(\(4\).)141 399 y Fe(2)141 504 y Fx(An)e(equiv)n(alen)o(t)e(form)o(ulation)e(of)j (Theorem)g(5.10)f(reads)i(as)f(follo)o(ws:)e(If)i Ft(R)1308 510 y Fr(1)1339 504 y Fx(and)g Ft(R)1453 510 y Fr(2)1484 504 y Fx(are)h(t)o(w)o(o)f(ter-)100 554 y(minating)c(comp)q(osable)i (TRSs)h(suc)o(h)h(that)g(their)f(com)o(bined)f(system)h Ft(R)1231 560 y Fr(1)1254 554 y Ft([)t(R)1321 560 y Fr(2)1350 554 y Fx(is)g(non-terminating,)100 604 y(then)j Ft(R)229 610 y Fr(1)262 604 y Fx(is)g(duplicating)e(and)i Ft(R)636 610 y Fr(2)669 604 y Fx(is)f(not)h(la)o(y)o(er-preserving)h(or)e(vice)i (v)o(ersa.)707 721 y Fk(5.3.)23 b(completeness)141 829 y Fx(The)14 b(next)g(theorem)f(is)h(due)g(to)f(Gramlic)o(h)e(\(1994)p Fs(b)p Fx(\).)i(His)g(original)f(pro)q(of)h(w)o(as,)g(ho)o(w)o(ev)o (er,)h(rather)100 879 y(complicated.)f(In)i(the)g(mean)o(time,)d(sev)o (eral)k(authors)f(ha)o(v)o(e)g(indep)q(enden)o(tly)g(giv)o(en)g (simpler)f(pro)q(ofs)100 929 y(for)d(this)i(theorem)f(\(whic)o(h)g (resem)o(ble)g(one)g(another\),)h(see)g(Dersho)o(witz)g(and)f(Ho)q(ot)g (\(1994\),)f(Middel-)100 979 y(dorp)16 b(\(1994)p Fs(b)p Fx(\))g(and)g(Ohlebusc)o(h)h(\(1994)p Fs(b)p Fx(\).)e(Thereb)o(y)m(,)i (a)f(TRS)g Ft(R)g Fx(is)g(called)g(an)g Fs(overlay)h(system)f Fx(if)100 1028 y(ev)o(ery)e(critical)g(pair)g(b)q(et)o(w)o(een)h(rules) g(of)e Ft(R)i Fx(is)f(obtained)f(b)o(y)h(o)o(v)o(erlapping)f(left-hand) h(sides)h(of)e(rules)100 1078 y(at)g(ro)q(ot)h(p)q(ositions.)100 1184 y Fk(Theorem)i(5.11.)21 b Fx(An)14 b(o)o(v)o(erla)o(y)g(system)f (is)h(complete)g(if)f(and)h(only)f(if)h(it)f(is)h(lo)q(cally)f (con\015uen)o(t)i(and)100 1234 y(innermost)e(terminating.)100 1339 y Fk(Cor)o(ollar)m(y)18 b(5.12.)j Fx(Completeness)14 b(is)f(mo)q(dular)f(for)i(comp)q(osable)f(o)o(v)o(erla)o(y)g(systems.) 100 1442 y Fk(Pr)o(oof.)22 b Fx(Immedia)o(te)12 b(consequence)k(of)d (Theorem)h(5.11)f(and)g(Prop)q(osition)h(5.3.)e Fe(2)141 1545 y Fx(The)20 b(main)e(result)j(of)e(Middeldorp)g(and)h(T)m(o)o(y)o (ama)d(\(1993\))i(stating)g(that)h(completeness)g(is)g(a)100 1595 y(mo)q(dular)9 b(prop)q(ert)o(y)i(of)g(comp)q(osable)f (constructor)i(systems)f(follo)o(ws)e(from)h(Corollary)f(5.12)h(b)q (ecause)100 1645 y(a)k(constructor)j(system)d(is)h(an)g(o)o(v)o(erla)o (y)f(system)g(and)h(the)h(com)o(bined)d(system)i(of)f(t)o(w)o(o)h(comp) q(osable)100 1695 y(constructor)i(systems)f(is)f(again)g(a)g (constructor)i(system.)e(Also,)g(mo)q(dularit)o(y)f(of)h(termination)e (\(or)100 1744 y(equiv)n(alen)o(tly)f(completeness\))j(for)f(non-o)o(v) o(erlapping)f(TRSs)g(is)h(a)g(consequence)j(of)c(Corollary)g(5.12)100 1794 y(b)q(ecause)19 b(those)f(systems)g(are)g(lo)q(cally)e(con\015uen) o(t)j(o)o(v)o(erla)o(y)d(systems)i(and)f(the)h(com)o(bined)f(system)100 1844 y(of)i(t)o(w)o(o)g(non-o)o(v)o(erlapping)f(comp)q(osable)g (systems)i(is)g(non-o)o(v)o(erlapping.)d(The)j(same)f(is)g(true)i(for) 100 1894 y(orthogonal)12 b(systems)i(whic)o(h)g(are)g(\(left-linear)g (and\))f(non-o)o(v)o(erlapping.)585 2011 y Fk(5.4.)23 b(the)16 b(simplifying)f(pr)o(oper)m(ty)141 2119 y Fx(A)h(TRS)g(is)g (called)f Fs(simplifying)g Fx(if)g(its)h(rewrite)h(relation)f(is)g(con) o(tained)g(in)f(some)g(simpli\014cation)100 2169 y(ordering.)d(This)g (prop)q(ert)o(y)h(is)f(imp)q(ortan)o(t)f(b)q(ecause)j(ev)o(ery)f (\014nite)f(simplifying)d(TRS)j(is)g(terminating)100 2219 y(\(cf.)d(Dersho)o(witz,)g(1982\))g(and)g(virtually)f(all)h (termination)e(pro)q(ofs)j(are)g(based)g(on)f(this)h(fact.)f(Kurihara) 100 2268 y(and)i(Oh)o(uc)o(hi)i(\(1992\))e(ha)o(v)o(e)h(pro)o(v)o(ed)g (that)g(the)g(simplifying)d(prop)q(ert)o(y)k(is)e(mo)q(dular)f(for)i (constructor-)100 2318 y(sharing)17 b(TRSs)h(\(please)g(note)g(that)g (they)g(used)g(the)h(phrase)f(\\simply)e(terminating")f(instead)j(of) 100 2368 y(\\simplifyi)o(ng"\).)9 b(W)m(e)k(will)f(next)i(generalize)f (their)h(result)g(to)f(comp)q(osable)f(systems)h(b)o(y)g(com)o(bining) 100 2426 y(the)k(tec)o(hniques)h(of)d(Kurihara)i(and)f(Oh)o(uc)o(hi)h (\(1992\))f(and)g(Gramlic)o(h)e(\(1994)p Fs(a)p Fx(\))1375 2411 y Ft(y)1395 2426 y Fx(.)i(Again)g(w)o(e)h(need)100 2476 y(some)c(preparatory)h(lemmata.)135 2591 y Fq(y)170 2603 y Fw(This)8 b(generalizatio)o(n)d(has)j(b)q(een)f(claimed)f(indep) q(ende)o(n)o(tly)f(b)o(y)i(Krishna)g(Rao)h(\(1994\),)e(it)i(is)g (stated)e(there)h(without)100 2640 y(pro)q(of.)p eop %%Page: 20 20 20 19 bop 100 197 a Fw(20)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fk(Definition)16 b(5.13.)21 b Fx(Let)14 b Ft(F)k Fx(b)q(e)c(a)g(signature.)g(The)g(TRS)f Ft(F)1072 281 y Fp(ar)q(g)1140 299 y Fx(consists)i(of)e(all)g(rewite)h (rules)719 373 y Fu(f)t Fx(\()p Fu(x)783 379 y Fr(1)802 373 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)919 379 y Fp(n)941 373 y Fx(\))12 b Ft(!)f Fu(x)1046 379 y Fp(j)1063 373 y Fu(;)100 446 y Fx(where)k Fu(f)h Ft(2)11 b(F)18 b Fx(is)c(a)f(function)h(sym)o (b)q(ol)e(of)h(arit)o(y)h Fu(n)d Ft(\025)h Fx(1)h(and)h Fu(j)g Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)100 545 y Fk(Lemma)16 b(5.14.)21 b Fx(A)14 b(TRS)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(is)f(simplifying)e(if)i(and)g(only)h (if)f Ft(!)1191 527 y Fr(+)1191 557 y Fn(R[F)1269 549 y Fl(ar)q(g)1332 545 y Fx(is)g(irre\015exiv)o(e.)100 644 y Fk(Pr)o(oof.)22 b Fx(See)14 b(Kurihara)g(and)g(Oh)o(uc)o(hi)g (\(1992\).)e Fe(2)100 742 y Fk(Lemma)k(5.15.)21 b Fx(Let)14 b(\()p Ft(F)494 748 y Fr(1)512 742 y Fu(;)7 b Ft(R)566 748 y Fr(1)585 742 y Fx(\))12 b(and)h(\()p Ft(F)743 748 y Fr(2)762 742 y Fu(;)7 b Ft(R)816 748 y Fr(2)834 742 y Fx(\))13 b(b)q(e)g(comp)q(osable)f(TRSs.)g(If)g(one)h(of)g(them)f(is) g(simpli-)100 792 y(fying,)g(then)i(\()p Ft(B)r Fu(;)7 b Ft(S)s Fx(\))14 b(is)g(simplifyi)o(ng.)100 891 y Fk(Pr)o(oof.)22 b Fx(Let)13 b Fu(>)f Fx(b)q(e)i(a)e(simpli\014cation)e(ordering)j(on)f Ft(T)e Fx(\()p Ft(F)1023 897 y Fp(j)1041 891 y Fu(;)d Ft(V)s Fx(\),)12 b Fu(j)i Ft(2)d(f)p Fx(1)p Fu(;)c Fx(2)p Ft(g)p Fx(,)k(suc)o(h)i(that)g Ft(!)1547 897 y Fn(R)1576 901 y Fl(j)1605 891 y Ft(\022)g Fu(>)p Fx(.)100 941 y(The)h (restriction)h Fu(>)5 b Ft(j)433 948 y Fn(T)i Fr(\()p Fn(B)q Fp(;)p Fn(V)r Fr(\))557 941 y Fx(of)27 b Fu(>)i Fx(to)13 b Ft(T)e Fx(\()p Ft(B)q Fu(;)c Ft(V)s Fx(\))15 b(is)f(a)f(simpli\014cation)f(ordering)i(and)g(the)g(inclusion)113 990 y Ft(!)155 996 y Fn(S)186 990 y Ft(j)198 997 y Fn(T)7 b Fr(\()p Fn(B)q Fp(;)p Fn(V)r Fr(\))334 990 y Ft(\022)14 b Fu(>)5 b Ft(j)429 997 y Fn(T)i Fr(\()p Fn(B)q Fp(;)p Fn(V)q Fr(\))552 990 y Fx(holds.)14 b(This)f(means)g(that)h(\()p Ft(B)r Fu(;)7 b Ft(S)s Fx(\))14 b(is)f(simplifying.)d Fe(2)100 1089 y Fk(Theorem)16 b(5.16.)21 b Fx(The)15 b(simplifyi)o(ng)c(prop)q(ert)o(y)k(is)e(a)h(mo)q(dular)e(prop)q(ert)o (y)j(of)e(comp)q(osable)g(TRSs.)100 1188 y Fk(Pr)o(oof.)22 b Fx(Let)16 b(\()p Ft(F)390 1194 y Fr(1)408 1188 y Fu(;)7 b Ft(R)462 1194 y Fr(1)481 1188 y Fx(\))16 b(and)f(\()p Ft(F)645 1194 y Fr(2)664 1188 y Fu(;)7 b Ft(R)718 1194 y Fr(2)736 1188 y Fx(\))16 b(b)q(e)g(comp)q(osable)f(TRSs)h(and)f(let)h (\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))16 b(b)q(e)g(their)g(com-)100 1238 y(bined)e(system.)g(It)g(has)h(to)f(b)q(e)h(pro)o(v)o(ed)f(that)h (\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(is)g(simplifying)d(if)i(and)h (only)g(if)f(\()p Ft(F)1505 1244 y Fr(1)1524 1238 y Fu(;)7 b Ft(R)1578 1244 y Fr(1)1596 1238 y Fx(\))15 b(and)100 1287 y(\()p Ft(F)150 1293 y Fr(2)168 1287 y Fu(;)7 b Ft(R)222 1293 y Fr(2)241 1287 y Fx(\))14 b(are)g(simplifying)o(.)100 1387 y(\\only-if)s(":)g(Let)k(\()p Ft(T)11 b Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\))p Fu(;)g(>)p Fx(\))17 b(b)q(e)h(a)f (simpli\014cation)e(ordering)i(with)31 b Ft(!)1292 1393 y Fn(R)1353 1387 y Ft(\022)18 b Fu(>)p Fx(.)f(It)g(is)h(not)f(to)q(o) 100 1437 y(di\016cult)11 b(to)h(pro)o(v)o(e)f(that)h(\()p Ft(T)f Fx(\()p Ft(F)592 1443 y Fp(j)609 1437 y Fu(;)c Ft(V)s Fx(\))q Fu(;)18 b(>)5 b Ft(j)752 1444 y Fn(T)i Fr(\()p Fn(F)807 1454 y Fl(j)823 1444 y Fp(;)p Fn(V)r Fr(\))871 1437 y Fx(\))12 b(is)g(a)f(simpli\014cation)e(ordering)j(and) g(that)g(further-)100 1493 y(more)h Ft(!)246 1499 y Fn(R)275 1503 y Fl(j)296 1493 y Ft(j)308 1500 y Fn(T)7 b Fr(\()p Fn(F)363 1510 y Fl(j)379 1500 y Fp(;)p Fn(V)r Fr(\))452 1493 y Ft(\022)14 b Fu(>)5 b Ft(j)547 1500 y Fn(T)i Fr(\()p Fn(F)603 1510 y Fl(j)618 1500 y Fp(;)p Fn(V)r Fr(\))667 1493 y Fx(.)13 b(In)h(other)g(w)o(ords,)g(\()p Ft(F)1027 1499 y Fp(j)1045 1493 y Fu(;)7 b Ft(R)1099 1499 y Fp(j)1116 1493 y Fx(\))14 b(is)g(simplifying)o(.)100 1593 y(\\if)s(":)k(First)j (of)f(all,)f(note)h(that)h Ft(R)668 1599 y Fr(1)700 1593 y Ft([)13 b(F)775 1573 y Fp(ar)q(g)771 1604 y Fr(1)849 1593 y Fx(and)21 b Ft(R)972 1599 y Fr(2)1004 1593 y Ft([)13 b(F)1079 1573 y Fp(ar)q(g)1075 1604 y Fr(2)1153 1593 y Fx(are)21 b(comp)q(osable)e(systems.)h(Ac-)100 1642 y(cording)14 b(to)h(Lemma)d(5.14,)h(it)h(m)o(ust)f(b)q(e)j(sho)o(wn)e (that)29 b Ft(!)1025 1625 y Fr(+)1025 1655 y Fn(R[F)1103 1646 y Fl(ar)q(g)1181 1642 y Fx(is)14 b(irre\015exiv)o(e.)h(Assuming)f (that)113 1692 y Ft(!)155 1674 y Fr(+)155 1704 y Fn(R[F)233 1696 y Fl(ar)q(g)313 1692 y Fx(is)i(not)g(irre\015exiv)o(e,)g(w)o(e)h (will)d(deriv)o(e)j(a)f(con)o(tradiction.)g(So)g(supp)q(ose)h(that)f (there)i(is)e(a)100 1742 y(cyclic)e(deriv)n(ation)501 1804 y Fu(D)f Fx(:)39 b Fu(t)11 b Fx(=)h Fu(t)684 1810 y Fr(1)717 1804 y Ft(!)759 1810 y Fn(R[F)837 1802 y Fl(ar)q(g)907 1804 y Fu(:)7 b(:)g(:)19 b Ft(!)1018 1810 y Fn(R[F)1096 1802 y Fl(ar)q(g)1173 1804 y Fu(t)1188 1810 y Fp(n)1222 1804 y Fx(=)12 b Fu(t;)100 1878 y(n)g(>)g Fx(1,)i(of)f(terms)h Fu(t)406 1884 y Fp(j)436 1878 y Ft(2)e(T)e Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(,)13 b Fu(j)i Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)13 b(W.l.o.g.)e(w)o(e)k(ma)o (y)d(assume)i(that)g Fu(z)i Fx(is)f(the)f(only)100 1928 y(v)n(ariable)e(o)q(ccurring)i(in)g Fu(D)q Fx(.)f(W)m(e)g(ma)o(y)f (further)i(assume)g(that)f Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))f(=)g Fu(k)i Fx(is)f(minimal)o(,)d(i.e.,)i(there)100 1978 y(is)g(no)g(cyclic)h(deriv)n(ation)e Fu(s)h Fx(=)g Fu(s)596 1984 y Fr(1)629 1978 y Ft(!)671 1984 y Fn(R[F)749 1976 y Fl(ar)q(g)819 1978 y Fu(:)7 b(:)g(:)19 b Ft(!)930 1984 y Fn(R[F)1008 1976 y Fl(ar)q(g)1084 1978 y Fu(s)1103 1984 y Fp(m)1146 1978 y Fx(=)12 b Fu(s;)20 b(m)11 b(>)h Fx(1,)g(with)g(rank\()p Fu(s)p Fx(\))g Fu(<)g(k)q Fx(.)100 2027 y(Consequen)o(tly)m(,)27 b Ft(!)421 2010 y Fr(+)421 2040 y Fn(R[F)499 2031 y Fl(ar)q(g)576 2027 y Fx(is)14 b(irre\015exiv)o(e)g(on)g Ft(T)899 2012 y Fp()g Fx(1)i(b)o(y)f(Lemma)f(5.15.)100 2127 y Fs(Case)k(\(i\):)e Fu(t)i Fx(is)f(top)g(blac)o(k.)f(Ob)o (viously)m(,)g(ev)o(ery)i(term)f(in)g Fu(D)i Fx(m)o(ust)d(ha)o(v)o(e)h (rank)g Fu(k)q Fx(.)g(Therefore,)h(eac)o(h)100 2177 y(term)d(in)g Fu(D)j Fx(is)d(either)i(top)f(blac)o(k)f(or)h(top)g(transparen)o(t.)h (Let)314 2251 y Ft(T)331 2257 y Fp(D)373 2251 y Fx(=)c Ft(f)p Fu(s)h Ft(2)f(T)f Fx(\()p Ft(F)5 b Fu(;)i Ft(f)p Fu(z)r Ft(g)p Fx(\))13 b Ft(j)g Fu(s)h Fx(is)g(a)g(subterm)f(of)h(a)f (term)h(o)q(ccurring)g(in)g Fu(D)q Ft(g)p Fu(:)100 2324 y Fx(Note)e(that)g Ft(T)303 2330 y Fp(D)345 2324 y Fx(is)g(\014nite.)f (Let)i Fu(C)s(ons)f Fx(b)q(e)g(a)g(new)g(binary)f(function)h(sym)o(b)q (ol)e(not)i(o)q(ccurring)h(in)e Ft(F)16 b Fx(and)100 2374 y(let)11 b Ft(C)179 2380 y Fn(E)213 2374 y Fx(=)h Ft(f)p Fu(C)s(ons)p Fx(\()p Fu(x;)7 b(y)q Fx(\))12 b Ft(!)f Fu(x;)17 b(C)s(ons)p Fx(\()p Fu(x;)7 b(y)q Fx(\))12 b Ft(!)f Fu(y)q Ft(g)p Fx(.)h(The)g(pro)q(of)f(idea)g(is)g(to)h (de\014ne)g(a)g(transformation)100 2424 y(function)h(\010)292 2409 y Fp(D)292 2436 y(b)334 2424 y Fx(:)e Ft(T)374 2430 y Fp(D)415 2424 y Ft(!)h(T)e Fx(\()p Ft(F)548 2430 y Fr(1)576 2424 y Ft(])e(f)p Fu(C)s(ons)p Ft(g)p Fu(;)f Ft(f)p Fu(z)r Ft(g)p Fx(\))13 b(suc)o(h)i(that)181 2498 y(\010)211 2481 y Fp(D)211 2508 y(b)241 2498 y Fx(\()p Fu(D)q Fx(\))d(:)39 b(\010)401 2481 y Fp(D)401 2508 y(b)431 2498 y Fx(\()p Fu(t)p Fx(\))12 b(=)f(\010)563 2481 y Fp(D)563 2508 y(b)594 2498 y Fx(\()p Fu(t)625 2504 y Fr(1)643 2498 y Fx(\))j Ft(!)715 2483 y Fn(\003)715 2512 y Fr(\()p Fn(R)757 2516 y Ff(1)772 2512 y Fn([F)821 2498 y Fl(ar)q(g)818 2522 y Ff(1)870 2512 y Fr(\))p Fn(]C)923 2516 y Fc(E)966 2498 y Fu(:)7 b(:)g(:)19 b Ft(!)1077 2483 y Fn(\003)1077 2512 y Fr(\()p Fn(R)1119 2516 y Ff(1)1134 2512 y Fn([F)1183 2498 y Fl(ar)q(g)1180 2522 y Ff(1)1231 2512 y Fr(\))p Fn(]C)1284 2516 y Fc(E)1320 2498 y Fx(\010)1350 2481 y Fp(D)1350 2508 y(b)1380 2498 y Fx(\()p Fu(t)1411 2504 y Fp(n)1434 2498 y Fx(\))12 b(=)f(\010)1535 2481 y Fp(D)1535 2508 y(b)1565 2498 y Fx(\()p Fu(t)p Fx(\))100 2582 y(is)17 b(a)g(non-empt)o(y)f(cyclic)i(deriv)n(ation)e(of)h(terms)g (from)f Ft(T)10 b Fx(\()p Ft(F)1061 2588 y Fr(1)1091 2582 y Ft(])i(f)p Fu(C)s(ons)p Ft(g)p Fu(;)7 b Ft(f)p Fu(z)r Ft(g)p Fx(\).)15 b(This)j(con)o(tradicts)100 2631 y(the)13 b(irre\015exivit)o(y)f(of)26 b Ft(!)492 2614 y Fr(+)492 2647 y Fn(R)521 2651 y Ff(1)537 2647 y Fn([F)586 2633 y Fl(ar)q(g)583 2657 y Ff(1)662 2631 y Fx(b)q(ecause)15 b(one)e(can)g(pro)o(v)o(e)g(that)26 b Ft(!)1220 2614 y Fr(+)1220 2647 y(\()p Fn(R)1262 2651 y Ff(1)1278 2647 y Fn([F)1327 2633 y Fl(ar)q(g)1324 2657 y Ff(1)1375 2647 y Fr(\))p Fn(]C)1428 2651 y Fc(E)1476 2631 y Fx(is)13 b(irre\015exiv)o(e)p eop %%Page: 21 21 21 20 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(21)p 100 224 1595 2 v 100 299 a Fx(on)13 b Ft(T)e Fx(\()p Ft(F)241 305 y Fr(1)269 299 y Ft(])e(f)p Fu(C)s(ons)p Ft(g)p Fu(;)e Ft(f)p Fu(z)r Ft(g)p Fx(\))13 b(if)g(and)h(only)f(if)27 b Ft(!)860 281 y Fr(+)860 314 y Fn(R)889 318 y Ff(1)904 314 y Fn([F)953 300 y Fl(ar)q(g)950 324 y Ff(1)1032 299 y Fx(is)14 b(irre\015exiv)o(e)g(on)g Ft(T)c Fx(\()p Ft(F)1405 305 y Fr(1)1424 299 y Fu(;)d Ft(f)p Fu(z)r Ft(g)p Fx(\).)13 b(In)h(order)100 363 y(to)g(de\014ne)h(\010)301 348 y Fp(D)301 375 y(b)345 363 y Fx(w)o(e)g(need)g(the)g(follo)o(wing)c (de\014nitions.)j(The)h Fs(inner)g(subterm)f(o)n(c)n(curr)n(enc)n(es)h (of)h Fu(D)f Fx(are)100 413 y(those)e(terms)g(whic)o(h)f(are)i (subterms)f(of)f(a)g(white)h(principal)f(subterm)h(o)q(ccurring)g(in)g Fu(D)q Fx(.)f(The)i(others)100 463 y(are)i(called)f Fs(outer)i(subterm) f(o)n(c)n(curr)n(enc)n(es)h(of)f Fu(D)q Fx(.)g(Let)g Fu(O)1005 448 y Fp(D)1004 475 y(b)1051 463 y Fx(denote)g(the)h(set)g (of)e(all)f(outer)j(subterm)100 513 y(o)q(ccurrences)f(of)c Fu(D)q Fx(.)h(F)m(urthermore,)g(let)g Fu(S)766 498 y Fp(w)764 524 y(P)793 513 y Fx(\()p Fu(D)q Fx(\))h(denote)g(the)g(set)g (of)e(all)g(white)h(principal)f(subterms)100 563 y(app)q(earing)g(in)g Fu(D)q Fx(.)g(Observ)o(e)i(that)f(b)q(oth)f(sets)i(are)e(\014nite)h (and)f(that)h(ev)o(ery)g(elemen)o(t)f(of)g Fu(S)1493 547 y Fp(w)1491 574 y(P)1520 563 y Fx(\()p Fu(D)q Fx(\))h(has)g(a)100 612 y(rank)d(less)i(than)f Fu(k)q Fx(.)f(Moreo)o(v)o(er,)h(for)g Fu(s)h Ft(2)f Fu(S)754 597 y Fp(w)752 624 y(P)782 612 y Fx(\()p Fu(D)q Fx(\),)g(w)o(e)g(de\014ne)h(\001)1082 597 y Fp(D)1082 624 y(b)1112 612 y Fx(\()p Fu(s)p Fx(\))g(=)g Ft(f)p Fu(u)f Ft(2)g Fu(O)1347 597 y Fp(D)1346 624 y(b)1388 612 y Ft(j)f Fu(s)15 b Ft(!)1486 595 y Fr(+)1486 625 y Fn(R[F)1564 616 y Fl(ar)q(g)1638 612 y Fu(u)p Ft(g)p Fx(.)100 662 y(It)c(is)f(imp)q(ortan)o(t)g(to)g(notice)i(that)e(\001) 662 647 y Fp(D)662 674 y(b)692 662 y Fx(\()p Fu(s)p Fx(\))i(is)f (\014nite)g(for)f(an)o(y)h Fu(s)h Ft(2)f Fu(S)1130 647 y Fp(w)1128 674 y(P)1158 662 y Fx(\()p Fu(D)q Fx(\).)g(Let)g Ft(\037)g Fx(b)q(e)h(a)e(total)h(ordering)100 712 y(on)i Ft(T)d Fx(\()p Ft(F)k(])9 b(f)p Fu(C)s(ons)p Ft(g)p Fu(;)e Ft(f)p Fu(z)r Ft(g)p Fx(\).)12 b(Let)293 835 y(\010)323 818 y Fp(D)323 846 y(b)353 835 y Fx(\()p Fu(s)p Fx(\))g(=)460 750 y Fg(8)460 788 y(<)460 862 y(:)518 785 y Fu(C)551 770 y Fp(b)567 785 y Ft(f)p Fx(\010)618 770 y Fp(D)618 797 y(b)648 785 y Fx(\()p Fu(s)683 791 y Fr(1)702 785 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i Fx(\010)841 770 y Fp(D)841 797 y(b)870 785 y Fx(\()p Fu(s)905 791 y Fp(m)938 785 y Fx(\))p Ft(g)101 b Fx(if)13 b Fu(s)f Fx(=)g Fu(C)1222 770 y Fp(b)1238 785 y Ft(f)-14 b(f)p Fu(s)1285 791 y Fr(1)1304 785 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1416 791 y Fp(m)1447 785 y Ft(g)-14 b(g)518 885 y Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fx(\010)667 870 y Fp(D)667 897 y(b)697 885 y Fx(\()p Fu(u)p Fx(\))14 b Ft(j)g Fu(u)d Ft(2)g Fx(\001)902 870 y Fp(D)902 897 y(b)932 885 y Fx(\()p Fu(s)p Fx(\))p Ft(g)p Fx(\))56 b(if)13 b Fu(r)q(oot)p Fx(\()p Fu(s)p Fx(\))f Ft(2)g(A)1325 891 y Fr(2)1343 885 y Fu(;)100 959 y Fx(where)j Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fu(t)354 965 y Fr(1)373 959 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)481 965 y Fp(n)503 959 y Ft(g)p Fx(\))12 b(=)g Ft(h)p Fu(t)627 966 y Fp(\031)q Fr(\(1\))692 959 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)800 966 y Fp(\031)q Fr(\()p Fp(n)p Fr(\))868 959 y Ft(i)15 b Fx(suc)o(h)g(that)f Fu(t)1098 966 y Fp(\031)q Fr(\()p Fp(j)r Fr(\))1168 959 y Ft(\037)8 b Fu(t)1223 966 y Fp(\031)q Fr(\()p Fp(j)r Fr(+1\))1343 959 y Fx(for)13 b(1)f Ft(\024)g Fu(j)j(<)d(n)p Fx(.)h(Here)100 1008 y Ft(h)p Fu(t)131 1015 y Fp(\031)q Fr(\(1\))196 1008 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)304 1015 y Fp(\031)q Fr(\()p Fp(n)p Fr(\))372 1008 y Ft(i)j Fx(stands)h(for)f(the)h(term)f Fu(C)s(ons)p Fx(\()p Fu(t)876 1015 y Fp(\031)q Fr(\(1\))941 1008 y Fu(;)d(C)s(ons)p Fx(\()p Fu(t)1088 1015 y Fp(\031)q Fr(\(2\))1153 1008 y Fu(;)g(:)g(:)g(:)t(;)g(C)s(ons)p Fx(\()p Fu(t)1373 1015 y Fp(\031)q Fr(\()p Fp(n)p Fr(\))1442 1008 y Fu(;)g(z)r Fx(\))g Fu(:)g(:)g(:)n Fx(\)\).)j(Note)100 1058 y(that)k Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fx(\010)339 1043 y Fp(D)339 1070 y(b)369 1058 y Fx(\()p Fu(u)p Fx(\))g Ft(j)g Fu(u)d Ft(2)g Fx(\001)574 1043 y Fp(D)574 1070 y(b)604 1058 y Fx(\()p Fu(s)p Fx(\))p Ft(g)p Fx(\))h(=)g Fu(z)k Fx(if)d Fu(r)q(oot)p Fx(\()p Fu(s)p Fx(\))g Ft(2)e(A)1032 1064 y Fr(2)1064 1058 y Fx(and)j(\001)1180 1043 y Fp(D)1180 1070 y(b)1210 1058 y Fx(\()p Fu(s)p Fx(\))e(=)g Ft(;)p Fx(.)h(It)h(is)g(easy)g(to)g(v)o(erify)100 1108 y(that)g(the)g (transformation)e(function)i(\010)737 1093 y Fp(D)737 1120 y(b)781 1108 y Fx(is)f(w)o(ell-de\014ned.)100 1208 y(W)m(e)20 b(sho)o(w)g(next)h(that)g Fu(t)499 1214 y Fp(j)530 1208 y Ft(!)572 1214 y Fn(R[F)650 1206 y Fl(ar)q(g)734 1208 y Fu(t)749 1214 y Fp(j)r Fr(+1)828 1208 y Fx(implies)e(\010)1006 1193 y Fp(D)1006 1219 y(b)1036 1208 y Fx(\()p Fu(t)1067 1214 y Fp(j)1085 1208 y Fx(\))14 b Ft(!)1157 1193 y Fn(\003)1157 1222 y Fr(\()p Fn(R)1199 1226 y Ff(1)1214 1222 y Fn([F)1263 1208 y Fl(ar)q(g)1260 1232 y Ff(1)1311 1222 y Fr(\))p Fn(]C)1364 1226 y Fc(E)1400 1208 y Fx(\010)1430 1193 y Fp(D)1430 1219 y(b)1460 1208 y Fx(\()p Fu(t)1491 1214 y Fp(j)r Fr(+1)1551 1208 y Fx(\),)20 b(using)100 1265 y Ft(!)14 b Fx(as)h(a)g(shorthand)h(for)28 b Ft(!)562 1271 y Fn(R[F)640 1263 y Fl(ar)q(g)717 1265 y Fx(=)15 b Ft(!)806 1274 y Fr(\()p Fn(R)848 1278 y Ff(1)863 1274 y Fn([F)912 1260 y Fl(ar)q(g)909 1284 y Ff(1)960 1274 y Fr(\))989 1265 y Ft([)e(!)1072 1274 y Fr(\()p Fn(R)1114 1278 y Ff(2)1129 1274 y Fn([F)1178 1260 y Fl(ar)q(g)1175 1284 y Ff(2)1226 1274 y Fr(\))1241 1265 y Fx(.)i(There)h(are)g(the)f (follo)o(wing)100 1315 y(cases.)125 1412 y(\(a\))21 b(If)e Fu(t)261 1418 y Fp(j)293 1412 y Ft(!)335 1392 y Fp(t;o)335 1424 y Fn(A)362 1428 y Ff(1)393 1412 y Fu(t)408 1418 y Fp(j)r Fr(+1)487 1412 y Fx(b)o(y)g(some)f(rewrite)j(rule)f Fu(l)h Ft(!)g Fu(r)q Fx(,)d(then)j(w)o(e)e(ha)o(v)o(e)g Fu(l)j Ft(!)f Fu(r)g Ft(2)g(R)1530 1418 y Fr(1)1558 1412 y Ft([)9 b(F)1629 1392 y Fp(ar)q(g)1625 1423 y Fr(1)1682 1412 y Fx(,)199 1472 y Fu(t)214 1478 y Fp(j)243 1472 y Fx(=)j Fu(C)320 1456 y Fp(b)336 1472 y Fx([)-7 b([)p Fu(s)372 1478 y Fr(1)391 1472 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)502 1478 y Fp(m)534 1472 y Fx(])-7 b(])o(,)9 b(and)h Fu(t)663 1478 y Fp(j)r Fr(+1)734 1472 y Fx(=)787 1461 y(^)778 1472 y Fu(C)811 1456 y Fp(b)827 1472 y Ft(h)-7 b(h)p Fu(s)871 1478 y Fp(i)883 1482 y Ff(1)902 1472 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1014 1478 y Fp(i)1026 1482 y Fl(l)1039 1472 y Ft(i)-7 b(i)q Fx(.)9 b(Applying)g(\010)1292 1456 y Fp(D)1292 1483 y(b)1322 1472 y Fx(,)g(w)o(e)h(obtain)f(\010)1555 1456 y Fp(D)1555 1483 y(b)1585 1472 y Fx(\()p Fu(t)1616 1478 y Fp(j)1634 1472 y Fx(\))i(=)199 1527 y Fu(C)232 1512 y Fp(b)249 1527 y Fx([\010)291 1512 y Fp(D)291 1539 y(b)320 1527 y Fx(\()p Fu(s)355 1533 y Fr(1)374 1527 y Fx(\))p Fu(;)c(:)g(:)g(:)e(;)i Fx(\010)513 1512 y Fp(D)513 1539 y(b)543 1527 y Fx(\()p Fu(s)578 1533 y Fp(m)610 1527 y Fx(\)])13 b(and)f(\010)760 1512 y Fp(D)760 1539 y(b)790 1527 y Fx(\()p Fu(t)821 1533 y Fp(j)r Fr(+1)881 1527 y Fx(\))g(=)962 1516 y(^)952 1527 y Fu(C)985 1512 y Fp(b)1002 1527 y Ft(h)p Fx(\010)1048 1512 y Fp(D)1048 1539 y(b)1078 1527 y Fx(\()p Fu(s)1113 1533 y Fp(i)1125 1537 y Ff(1)1144 1527 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i Fx(\010)1283 1512 y Fp(D)1283 1539 y(b)1312 1527 y Fx(\()p Fu(s)1347 1533 y Fp(i)1359 1537 y Fl(l)1374 1527 y Fx(\))p Ft(i)p Fx(.)12 b(It)h(follo)o(ws)e(from)199 1577 y Fu(s)218 1583 y Fr(1)237 1577 y Fu(;)c(:)g(:)g(:)e(;)i(s)349 1583 y Fp(m)405 1577 y Ft(/)26 b Fx(\010)493 1562 y Fp(D)493 1588 y(b)523 1577 y Fx(\()p Fu(s)558 1583 y Fr(1)577 1577 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i Fx(\010)716 1562 y Fp(D)716 1588 y(b)746 1577 y Fx(\()p Fu(s)781 1583 y Fp(m)813 1577 y Fx(\))13 b(that)h(the)g(rule)f Fu(l)g Ft(!)e Fu(r)j Fx(that)g(reduced)h Fu(t)1455 1583 y Fp(j)1486 1577 y Fx(to)e Fu(t)1551 1583 y Fp(j)r Fr(+1)1624 1577 y Fx(also)199 1627 y(reduces)j(\010)377 1611 y Fp(D)377 1638 y(b)407 1627 y Fx(\()p Fu(t)438 1633 y Fp(j)456 1627 y Fx(\))e(to)g(\010)567 1611 y Fp(D)567 1638 y(b)597 1627 y Fx(\()p Fu(t)628 1633 y Fp(j)r Fr(+1)687 1627 y Fx(\))g(\(cf.)g(Lemma)d(4.14\).)123 1723 y(\(b\))21 b(If)f Fu(t)262 1729 y Fp(j)293 1723 y Ft(!)335 1708 y Fp(o)335 1735 y Fn(A)362 1739 y Ff(2)393 1723 y Fu(t)408 1729 y Fp(j)r Fr(+1)488 1723 y Fx(or)g Fu(t)560 1729 y Fp(j)599 1723 y Ft(!)641 1708 y Fp(i)676 1723 y Fu(t)691 1729 y Fp(j)r Fr(+1)750 1723 y Fx(,)g(then)g(w)o(e)g(ha)o(v)o(e)g Fu(t)1066 1729 y Fp(j)1105 1723 y Fx(=)i Fu(C)1192 1708 y Fp(b)1208 1723 y Fx([)-7 b([)p Fu(s)1244 1729 y Fr(1)1263 1723 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)1374 1729 y Fp(l)1387 1723 y Fu(;)g(:)g(:)g(:)e(;)i(s)1499 1729 y Fp(m)1530 1723 y Fx(])-7 b(])19 b(as)h(w)o(ell)199 1778 y(as)d Fu(t)268 1784 y Fp(j)r Fr(+1)343 1778 y Fx(=)e Fu(C)423 1763 y Fp(b)440 1778 y Fx([)p Fu(s)471 1784 y Fr(1)489 1778 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)601 1763 y Fn(0)601 1790 y Fp(l)614 1778 y Fu(;)g(:)g(:)g(:)e(;)i(s)726 1784 y Fp(m)757 1778 y Fx(])16 b(for)g(some)f Fu(l)i Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(m)p Ft(g)16 b Fx(and)g(some)g(term)f Fu(s)1546 1763 y Fn(0)1546 1790 y Fp(l)1560 1778 y Fx(,)g(where)199 1828 y Fu(s)218 1834 y Fp(l)253 1828 y Ft(!)21 b Fu(s)335 1813 y Fn(0)335 1840 y Fp(l)349 1828 y Fx(.)e(Clearly)m(,)g(\010)569 1813 y Fp(D)569 1840 y(b)599 1828 y Fx(\()p Fu(t)630 1834 y Fp(j)647 1828 y Fx(\))j(=)g Fu(C)772 1813 y Fp(b)788 1828 y Fx([\010)830 1813 y Fp(D)830 1840 y(b)860 1828 y Fx(\()p Fu(s)895 1834 y Fr(1)914 1828 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i Fx(\010)1053 1813 y Fp(D)1053 1840 y(b)1083 1828 y Fx(\()p Fu(s)1118 1834 y Fp(m)1150 1828 y Fx(\)].)19 b(W)m(e)h(consider)h(the)f(follo)o(wing)199 1878 y(sub)q(cases.)177 1950 y(\(b1\))h(If)16 b Fu(r)q(oot)p Fx(\()p Fu(s)428 1935 y Fn(0)428 1961 y Fp(l)441 1950 y Fx(\))g Ft(2)e(A)549 1956 y Fr(2)567 1950 y Fx(,)i(then)h Fu(t)707 1956 y Fp(j)r Fr(+1)782 1950 y Fx(has)f(a)g(represen)o(tation) i Fu(t)1184 1956 y Fp(j)r Fr(+1)1259 1950 y Fx(=)d Fu(C)1339 1935 y Fp(b)1356 1950 y Fx([)-7 b([)o Fu(s)1391 1956 y Fr(1)1410 1950 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1522 1935 y Fn(0)1522 1961 y Fp(l)1534 1950 y Fu(;)g(:)g(:)g(:)e(;)i(s)1646 1956 y Fp(m)1678 1950 y Fx(])-7 b(])274 1999 y(and)16 b(\010)387 1984 y Fp(D)387 2011 y(b)417 1999 y Fx(\()p Fu(t)448 2005 y Fp(j)r Fr(+1)508 1999 y Fx(\))g(=)h Fu(C)622 1984 y Fp(b)638 1999 y Fx([\010)680 1984 y Fp(D)680 2011 y(b)710 1999 y Fx(\()p Fu(s)745 2005 y Fr(1)764 1999 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i Fx(\010)903 1984 y Fp(D)903 2011 y(b)932 1999 y Fx(\()p Fu(s)967 1984 y Fn(0)967 2011 y Fp(l)981 1999 y Fx(\))p Fu(;)g(:)g(:)g(:)e Fx(\010)1101 1984 y Fp(D)1101 2011 y(b)1131 1999 y Fx(\()p Fu(s)1166 2005 y Fp(m)1198 1999 y Fx(\)].)16 b(Therefore,)h(it)f(is)h (su\016cen)o(t)274 2049 y(to)e(sho)o(w)f(\010)459 2034 y Fp(D)459 2061 y(b)489 2049 y Fx(\()p Fu(s)524 2055 y Fp(l)538 2049 y Fx(\))g Ft(!)610 2034 y Fn(\003)610 2064 y Fr(\()p Fn(R)652 2068 y Ff(1)667 2064 y Fn([F)716 2050 y Fl(ar)q(g)713 2074 y Ff(1)764 2064 y Fr(\))p Fn(]C)817 2068 y Fc(E)853 2049 y Fx(\010)883 2034 y Fp(D)883 2061 y(b)913 2049 y Fx(\()p Fu(s)948 2034 y Fn(0)948 2061 y Fp(l)961 2049 y Fx(\).)h(No)o(w)f(it)g(follo)o(ws)f(from)g(\001)1412 2034 y Fp(D)1412 2061 y(b)1442 2049 y Fx(\()p Fu(s)1477 2034 y Fn(0)1477 2061 y Fp(l)1490 2049 y Fx(\))g Ft(\022)g Fx(\001)1599 2034 y Fp(D)1599 2061 y(b)1629 2049 y Fx(\()p Fu(s)1664 2055 y Fp(l)1678 2049 y Fx(\))274 2107 y(that)621 2158 y(\010)651 2143 y Fp(D)651 2170 y(b)681 2158 y Fx(\()p Fu(s)716 2164 y Fp(l)729 2158 y Fx(\))42 b(=)12 b Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fx(\010)980 2143 y Fp(D)980 2170 y(b)1011 2158 y Fx(\()p Fu(u)p Fx(\))i Ft(j)f Fu(u)e Ft(2)h Fx(\001)1216 2143 y Fp(D)1216 2170 y(b)1245 2158 y Fx(\()p Fu(s)1280 2164 y Fp(l)1294 2158 y Fx(\))p Ft(g)p Fx(\))787 2208 y(=)g Ft(h)p Fu(u)871 2214 y Fr(1)889 2208 y Fu(;)7 b(:)g(:)g(:)e(;)i (u)1006 2214 y Fp(n)1028 2208 y Ft(i)787 2258 y(!)829 2243 y Fn(\003)829 2269 y(C)847 2273 y Fc(E)880 2258 y Ft(h)p Fu(u)920 2264 y Fp(i)932 2268 y Ff(1)950 2258 y Fu(;)g(:)g(:)g(:)e(;)i(u)1067 2264 y Fp(i)1079 2268 y Fl(p)1097 2258 y Ft(i)787 2309 y Fx(=)12 b Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fx(\010)980 2293 y Fp(D)980 2320 y(b)1011 2309 y Fx(\()p Fu(u)p Fx(\))i Ft(j)f Fu(u)e Ft(2)h Fx(\001)1216 2293 y Fp(D)1216 2320 y(b)1245 2309 y Fx(\()p Fu(s)1280 2293 y Fn(0)1280 2320 y Fp(l)1294 2309 y Fx(\))p Ft(g)p Fx(\))787 2358 y(=)g(\010)861 2343 y Fp(D)861 2370 y(b)891 2358 y Fx(\()p Fu(s)926 2343 y Fn(0)926 2370 y Fp(l)939 2358 y Fx(\))p Fu(;)274 2430 y Fx(where)j Fu(u)418 2436 y Fp(i)430 2440 y Ff(1)448 2430 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)565 2436 y Fp(i)577 2440 y Fl(p)609 2430 y Fx(is)14 b(a)f(subsequence)k(of) c Fu(u)991 2436 y Fr(1)1009 2430 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1126 2436 y Fp(n)1148 2430 y Fx(.)177 2529 y(\(b2\))21 b(If)26 b Fu(r)q(oot)p Fx(\()p Fu(s)438 2514 y Fn(0)438 2541 y Fp(l)451 2529 y Fx(\))32 b Ft(62)f(A)591 2535 y Fr(2)610 2529 y Fx(,)25 b(then)i Fu(s)773 2514 y Fn(0)773 2541 y Fp(l)817 2529 y Fx(=)891 2519 y(\026)881 2529 y Fu(C)914 2514 y Fp(b)931 2529 y Ft(f)-14 b(f)o Fu(u)982 2535 y Fr(1)1001 2529 y Fu(;)7 b(:)g(:)g(:)t(;)g(u)1117 2535 y Fp(p)1136 2529 y Ft(g)-14 b(g)25 b Fx(and)h(w)o(e)g(ha)o(v)o(e)g (\010)1493 2514 y Fp(D)1493 2541 y(b)1523 2529 y Fx(\()p Fu(t)1554 2535 y Fp(j)r Fr(+1)1614 2529 y Fx(\))31 b(=)283 2574 y(^)274 2585 y Fu(C)307 2570 y Fp(b)323 2585 y Ft(f)p Fx(\010)374 2570 y Fp(D)374 2597 y(b)404 2585 y Fx(\()p Fu(s)439 2591 y Fr(1)458 2585 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i Fx(\010)597 2570 y Fp(D)597 2597 y(b)627 2585 y Fx(\()p Fu(s)662 2591 y Fp(l)p Fn(\000)p Fr(1)718 2585 y Fx(\))p Fu(;)g Fx(\010)783 2570 y Fp(D)783 2597 y(b)812 2585 y Fx(\()p Fu(u)852 2591 y Fr(1)871 2585 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;) i Fx(\010)1010 2570 y Fp(D)1010 2597 y(b)1039 2585 y Fx(\()p Fu(u)1079 2591 y Fp(p)1098 2585 y Fx(\))p Fu(;)g Fx(\010)1163 2570 y Fp(D)1163 2597 y(b)1193 2585 y Fx(\()p Fu(s)1228 2591 y Fp(l)p Fr(+1)1284 2585 y Fx(\))p Fu(;)g(:)g(:)g(:)t(;) g Fx(\010)1422 2570 y Fp(D)1422 2597 y(b)1452 2585 y Fx(\()p Fu(s)1487 2591 y Fp(m)1519 2585 y Fx(\))p Ft(g)p Fx(,)19 b(where)283 2630 y(^)274 2640 y Fu(C)307 2625 y Fp(b)323 2640 y Ft(f)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(g)15 b Fx(=)h Fu(C)554 2625 y Fp(b)570 2640 y Fx([)p Fu(;)7 b(:)g(:)g(:)e(;)683 2630 y Fx(\026)675 2640 y Fu(C)708 2625 y Fp(b)723 2640 y Ft(f)p Fu(;)i(:)g(:)g(:)n Ft(g)p Fu(;)g(:)g(:)g(:)e(;)i Fx(].)14 b(No)o(w)j(it)f(is)g(a)g(consequence)j (of)d Fu(s)1492 2625 y Fn(0)1492 2652 y Fp(l)1521 2640 y Ft(2)f Fx(\001)1599 2625 y Fp(D)1599 2652 y(b)1629 2640 y Fx(\()p Fu(s)1664 2646 y Fp(l)1678 2640 y Fx(\))p eop %%Page: 22 22 22 21 bop 100 197 a Fw(22)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 274 299 a Fx(that)j(\010)394 284 y Fp(D)394 311 y(b)424 299 y Fx(\()p Fu(s)459 284 y Fn(0)459 311 y Fp(l)472 299 y Fx(\))g(o)q(ccurs)i(in)d(the)i(term)e(\010)880 284 y Fp(D)880 311 y(b)910 299 y Fx(\()p Fu(s)945 305 y Fp(l)958 299 y Fx(\))h(and)g(hence)621 372 y(\010)651 357 y Fp(D)651 384 y(b)681 372 y Fx(\()p Fu(s)716 378 y Fp(l)729 372 y Fx(\))42 b(=)12 b Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fx(\010)980 357 y Fp(D)980 384 y(b)1011 372 y Fx(\()p Fu(u)p Fx(\))i Ft(j)f Fu(u)e Ft(2)h Fx(\001)1216 357 y Fp(D)1216 384 y(b)1245 372 y Fx(\()p Fu(s)1280 378 y Fp(l)1294 372 y Fx(\))p Ft(g)p Fx(\))787 422 y(=)g Ft(h)p Fu(:)7 b(:)g(:)e(;)i Fx(\010)951 407 y Fp(D)951 434 y(b)981 422 y Fx(\()p Fu(s)1016 407 y Fn(0)1016 434 y Fp(l)1029 422 y Fx(\))p Fu(;)g(:)g(:)g(:)n Ft(i)787 472 y(!)829 455 y Fr(+)829 485 y Fn(C)847 489 y Fc(E)880 472 y Fx(\010)910 457 y Fp(D)910 484 y(b)940 472 y Fx(\()p Fu(s)975 457 y Fn(0)975 484 y Fp(l)988 472 y Fx(\))787 524 y(=)840 514 y(\026)831 524 y Fu(C)864 509 y Fp(b)880 524 y Ft(f)p Fx(\010)931 509 y Fp(D)931 536 y(b)961 524 y Fx(\()p Fu(u)1001 530 y Fr(1)1020 524 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;) i Fx(\010)1159 509 y Fp(D)1159 536 y(b)1188 524 y Fx(\()p Fu(u)1228 530 y Fp(p)1247 524 y Fx(\))p Ft(g)p Fu(:)199 600 y Fx(All)13 b(in)h(all,)e(\010)415 585 y Fp(D)415 612 y(b)445 600 y Fx(\()p Fu(t)476 606 y Fp(j)494 600 y Fx(\))f Ft(!)563 585 y Fn(\003)563 612 y(C)581 616 y Fc(E)614 600 y Fx(\010)644 585 y Fp(D)644 612 y(b)674 600 y Fx(\()p Fu(t)705 606 y Fp(j)r Fr(+1)765 600 y Fx(\))j(whic)o(h)g (concludes)h(case)g(\(b\).)100 697 y(Since)g(there)h(m)o(ust)e(b)q(e)i (at)e(least)i(one)f Ft(!)747 682 y Fp(t)776 697 y Fx(or)28 b Ft(!)883 682 y Fp(o)883 709 y Fn(A)910 713 y Ff(1)957 697 y Fx(step)15 b(in)g Fu(D)q Fx(,)g(w)o(e)g(obtain)f(a)h(non-empt)o (y)e(cyclic)100 752 y(deriv)n(ation)e(\010)323 737 y Fp(D)323 764 y(b)353 752 y Fx(\()p Fu(D)q Fx(\))i(of)f(terms)h(from)d Ft(T)g Fx(\()p Ft(F)769 758 y Fr(1)794 752 y Ft(])c(f)p Fu(C)s(ons)p Ft(g)p Fu(;)h Ft(f)p Fu(z)r Ft(g)p Fx(\).)k(This)i(is)f(a) g(con)o(tradiction)g(to)h(the)g(fact)100 802 y(that)h Ft(R)225 808 y Fr(1)257 802 y Fx(is)g(simplifying,)c(so)k(case)h(\(i\)) e(is)h(pro)o(v)o(ed.)100 901 y Fs(Case)22 b(\(ii\):)e Fu(t)i Fx(is)f(top)h(white.)f(Here)i(the)f(ab)q(o)o(v)o(e)g(pro)q(of)f (applies)g(with)g(appropriate)h(notational)100 951 y(c)o(hanges.)100 1051 y Fs(Case)14 b(\(iii\):)f Fu(t)h Fx(is)g(top)g(transparen)o(t.)g (If)g(one)g(of)f(the)h(terms)g(in)250 1123 y Fu(D)f Fx(:)39 b Fu(t)11 b Fx(=)h Fu(t)433 1129 y Fr(1)465 1123 y Ft(!)507 1129 y Fn(R[F)585 1121 y Fl(ar)q(g)655 1123 y Fu(:)7 b(:)g(:)20 b Ft(!)767 1129 y Fn(R[F)845 1121 y Fl(ar)q(g)922 1123 y Fu(t)937 1129 y Fp(j)968 1123 y Ft(!)1010 1129 y Fn(R[F)1088 1121 y Fl(ar)q(g)1158 1123 y Fu(:)7 b(:)g(:)19 b Ft(!)1269 1129 y Fn(R[F)1347 1121 y Fl(ar)q(g)1424 1123 y Fu(t)1439 1129 y Fp(n)1473 1123 y Fx(=)12 b Fu(t;)100 1195 y Fx(is)29 b(top)h(blac)o(k,)f(sa)o(y)h Fu(t)486 1201 y Fp(j)503 1195 y Fx(,)f(then)i(there)g(exists)g(a)e(non-empt)o(y) g(cyclic)h(reduction)g(deriv)n(ation)100 1245 y Fu(t)115 1251 y Fp(j)146 1245 y Ft(!)188 1251 y Fn(R[F)266 1242 y Fl(ar)q(g)336 1245 y Fu(:)7 b(:)g(:)19 b Ft(!)447 1251 y Fn(R[F)525 1242 y Fl(ar)q(g)588 1245 y Fu(t)603 1251 y Fp(j)637 1245 y Fx(starting)c(from)g(the)h(top)g(blac)o(k)f(term)h Fu(t)1270 1251 y Fp(j)1303 1245 y Fx(and)g(the)g(assertion)h(fol-)100 1294 y(lo)o(ws)d(from)f(case)j(\(i\).)f(So)f(w)o(e)i(ma)o(y)d(assume)h (that)h(ev)o(ery)h(term)f(in)f Fu(D)j Fx(is)e(top)f(transparen)o(t.)i (W)m(e)f(use)100 1344 y(the)h(same)f(pro)q(of)h(idea)f(as)h(in)g (\(i\).)f(Since)h(the)h(pro)q(ofs)e(are)i(v)o(ery)f(m)o(uc)o(h)f(alik)o (e,)f(w)o(e)i(only)f(sk)o(etc)o(h)i(the)100 1394 y(construction.)g(No)o (w)f(the)h Fs(inner)g(subterm)g(o)n(c)n(curr)n(enc)n(es)g(of)h Fu(D)g Fx(are)e(those)i(terms)e(whic)o(h)g(are)h(sub-)100 1444 y(terms)d(of)f(a)h(blac)o(k)g(or)g(white)h(principal)e(subterm)h (o)q(ccurring)h(in)f Fu(D)q Fx(.)g(The)h(others)g(are)g(called)f Fs(outer)100 1494 y(subterm)g(o)n(c)n(curr)n(enc)n(es)g(of)g Fu(D)q Fx(.)g(Let)f Fu(O)692 1479 y Fp(D)691 1504 y(t)736 1494 y Fx(denote)h(the)g(set)g(of)f(all)f(outer)i(subterm)f(o)q (ccurrences)k(of)c Fu(D)q Fx(.)100 1543 y(F)m(urthermore,)e(let)g Fu(S)432 1549 y Fp(P)460 1543 y Fx(\()p Fu(D)q Fx(\))i(denote)f(the)g (set)h(of)e(all)f(blac)o(k)h(or)h(white)f(principal)g(subterms)h(app)q (earing)100 1593 y(in)h Fu(D)q Fx(.)h(Moreo)o(v)o(er,)g(for)f Fu(s)f Ft(2)g Fu(S)562 1599 y Fp(P)590 1593 y Fx(\()p Fu(D)q Fx(\),)i(de\014ne)h(\001)838 1578 y Fp(D)838 1603 y(t)868 1593 y Fx(\()p Fu(s)p Fx(\))d(=)g Ft(f)p Fu(u)f Ft(2)g Fu(O)1103 1578 y Fp(D)1102 1603 y(t)1147 1593 y Ft(j)i Fu(s)h Ft(!)1247 1575 y Fr(+)1247 1605 y Fn(R[F)1325 1597 y Fl(ar)q(g)1402 1593 y Fu(u)p Ft(g)p Fx(.)f(Let)294 1717 y(\010)324 1700 y Fp(D)324 1727 y(t)354 1717 y Fx(\()p Fu(s)p Fx(\))f(=)461 1632 y Fg(8)461 1669 y(<)461 1744 y(:)519 1667 y Fu(C)552 1652 y Fp(t)566 1667 y Ft(f)p Fx(\010)617 1652 y Fp(D)617 1677 y(t)647 1667 y Fx(\()p Fu(s)682 1673 y Fr(1)701 1667 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i Fx(\010)840 1652 y Fp(D)840 1677 y(t)869 1667 y Fx(\()p Fu(s)904 1673 y Fp(m)937 1667 y Fx(\))p Ft(g)103 b Fx(if)13 b Fu(s)f Fx(=)g Fu(C)1223 1652 y Fp(t)1237 1667 y Ft(f)-14 b(f)p Fu(s)1284 1673 y Fr(1)1303 1667 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1415 1673 y Fp(m)1446 1667 y Ft(g)-14 b(g)519 1766 y Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fx(\010)668 1751 y Fp(D)668 1777 y(t)698 1766 y Fx(\()p Fu(u)p Fx(\))14 b Ft(j)g Fu(u)d Ft(2)g Fx(\001)903 1751 y Fp(D)903 1777 y(t)933 1766 y Fx(\()p Fu(s)p Fx(\))p Ft(g)p Fx(\))56 b(if)13 b Fu(r)q(oot)p Fx(\()p Fu(s)p Fx(\))f Ft(2)g(A)1326 1772 y Fr(1)1354 1766 y Ft(])c(A)1424 1772 y Fr(2)100 1843 y Fx(where)17 b Fu(S)r(or)q(t)g Fx(is)e(de\014ned)j(as)e(in)f(case)i(\(i\).)f(Again,) e(note)j(that)f Fu(S)r(or)q(t)p Fx(\()p Ft(f)p Fx(\010)1245 1827 y Fp(D)1245 1853 y(t)1276 1843 y Fx(\()p Fu(u)p Fx(\))g Ft(j)f Fu(u)g Ft(2)g Fx(\001)1492 1827 y Fp(D)1492 1853 y(t)1521 1843 y Fx(\()p Fu(s)p Fx(\))p Ft(g)p Fx(\))h(=)g Fu(z)100 1892 y Fx(if)g Fu(r)q(oot)p Fx(\()p Fu(s)p Fx(\))h Ft(2)f(A)361 1898 y Fr(1)390 1892 y Ft(])11 b(A)462 1898 y Fr(2)498 1892 y Fx(and)16 b(\001)616 1877 y Fp(D)616 1903 y(t)646 1892 y Fx(\()p Fu(s)p Fx(\))h(=)f Ft(;)p Fx(.)g(It)h(follo)o(ws)e(from)g(similar)f(argumen)o(ts)i(as)h(ab)q(o)o (v)o(e)g(that)100 1942 y Fu(t)115 1948 y Fp(j)146 1942 y Ft(!)188 1948 y Fn(R[F)266 1940 y Fl(ar)q(g)344 1942 y Fu(t)359 1948 y Fp(j)r Fr(+1)434 1942 y Fx(implies)c(\010)606 1927 y Fp(D)606 1952 y(t)636 1942 y Fx(\()p Fu(t)667 1948 y Fp(j)685 1942 y Fx(\))h Ft(!)757 1927 y Fn(\003)757 1956 y Fr(\()p Fn(S)r([B)837 1947 y Fl(ar)q(g)885 1956 y Fr(\))p Fn(]C)938 1960 y Fc(E)974 1942 y Fx(\010)1004 1927 y Fp(D)1004 1952 y(t)1034 1942 y Fx(\()p Fu(t)1065 1948 y Fp(j)r Fr(+1)1125 1942 y Fx(\).)g(W)m(e)h(just)h(ha)o(v)o(e)f (to)g(consider)h(the)100 1998 y(follo)o(wing)11 b(cases)k(\(using)f Ft(!)f Fx(as)h(a)g(shorthand)g(for)28 b Ft(!)961 2004 y Fn(R[F)1039 1996 y Fl(ar)q(g)1102 1998 y Fx(\).)125 2092 y(\(a\))21 b(If)e Fu(t)261 2098 y Fp(j)298 2092 y Ft(!)340 2077 y Fp(t)374 2092 y Fu(t)389 2098 y Fp(j)r Fr(+1)448 2092 y Fx(,)g(then)g(sho)o(w)g(\010)716 2077 y Fp(D)716 2103 y(t)746 2092 y Fx(\()p Fu(t)777 2098 y Fp(j)794 2092 y Fx(\))h Ft(!)872 2098 y Fn(S)r([B)939 2090 y Fl(ar)q(g)1009 2092 y Fx(\010)1039 2077 y Fp(D)1039 2103 y(t)1069 2092 y Fx(\()p Fu(t)1100 2098 y Fp(j)r Fr(+1)1160 2092 y Fx(\))f(using)f(argumen)o(ts)g(similar)f(to)199 2142 y(those)e(giv)o(en)e(in)h(case)h(\(i\))e(\(a\).)123 2236 y(\(b\))21 b(If)e Fu(t)261 2242 y Fp(j)299 2236 y Ft(!)i Fu(t)377 2242 y Fp(j)r Fr(+1)456 2236 y Fx(is)e(not)g(a)g (transparen)o(t)i(rewrite)f(step,)g(then)g Fu(t)1217 2242 y Fp(j)1255 2236 y Fx(=)h Fu(C)1341 2221 y Fp(t)1356 2236 y Fx([)-7 b([)o Fu(s)1391 2242 y Fr(1)1410 2236 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1522 2242 y Fp(l)1534 2236 y Fu(;)g(:)g(:)g(:)e(;)i(s)1646 2242 y Fp(m)1678 2236 y Fx(])-7 b(])199 2286 y(and)15 b Fu(t)296 2292 y Fp(j)r Fr(+1)368 2286 y Fx(=)e Fu(C)446 2271 y Fp(t)460 2286 y Fx([)p Fu(s)491 2292 y Fr(1)509 2286 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)621 2271 y Fn(0)621 2298 y Fp(l)634 2286 y Fu(;)g(:)g(:)g(:)e(;)i(s)746 2292 y Fp(m)777 2286 y Fx(])14 b(for)g(some)g Fu(l)g Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(m)p Ft(g)14 b Fx(and)h(some)e(term)h Fu(s)1548 2271 y Fn(0)1548 2298 y Fp(l)1561 2286 y Fx(,)g(where)199 2336 y Fu(s)218 2342 y Fp(l)250 2336 y Ft(!)k Fu(s)329 2321 y Fn(0)329 2348 y Fp(l)342 2336 y Fx(.)g(No)o(w)g(c)o(hec)o(k)h(whether)g Fu(r)q(oot)p Fx(\()p Fu(s)860 2321 y Fn(0)860 2348 y Fp(l)874 2336 y Fx(\))f Ft(2)h(A)988 2342 y Fr(1)1018 2336 y Ft(])12 b(A)1091 2342 y Fr(2)1128 2336 y Fx(holds.)17 b(If)h(so,)g(then)h(case)g(\(i\))f(\(b1\))199 2386 y(applies,)f(if)h (not,)f(then)h(case)h(\(i\))f(\(b2\))g(applies,)f(making)f(only)h (notational)f(c)o(hanges.)i(Th)o(us)199 2435 y(\010)229 2420 y Fp(D)229 2446 y(t)259 2435 y Fx(\()p Fu(t)290 2441 y Fp(j)308 2435 y Fx(\))12 b Ft(!)378 2420 y Fn(\003)378 2447 y(C)396 2451 y Fc(E)428 2435 y Fx(\010)458 2420 y Fp(D)458 2446 y(t)489 2435 y Fx(\()p Fu(t)520 2441 y Fp(j)r Fr(+1)579 2435 y Fx(\).)100 2532 y(Since)i(there)h(m)o(ust)e (b)q(e)i(at)f(least)g(one)g Ft(!)740 2517 y Fp(t)768 2532 y Fx(step)h(in)f Fu(D)q Fx(,)g(w)o(e)g(obtain)f(a)h(non-empt)o(y)e (cyclic)i(deriv)n(ation)100 2582 y(\010)130 2567 y Fp(D)130 2592 y(t)160 2582 y Fx(\()p Fu(D)q Fx(\))f(of)f(terms)h(from)e Ft(T)f Fx(\()p Ft(B)f(])d(f)p Fu(C)s(ons)p Ft(g)p Fu(;)h Ft(f)p Fu(z)r Ft(g)p Fx(\).)k(This)i(is)g(a)f(con)o(tradiction)h(to)f (the)i(fact)e(that)h(\()p Ft(B)r Fu(;)7 b Ft(S)s Fx(\))100 2632 y(is)13 b(simplifying)e(according)j(to)f(Lemma)f(5.15.)g Fe(2)p eop %%Page: 23 23 23 22 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(23)p 100 224 1595 2 v 488 299 a Fv(6.)16 b(Conditional)c(term)j(rewriting)e (systems)1286 284 y Ft(y)141 374 y Fx(When)21 b(dealing)e(with)h(the)h (com)o(bination)c Ft(R)k Fx(of)e(comp)q(osable)h(TRSs)g Ft(R)1333 380 y Fr(1)1372 374 y Fx(and)g Ft(R)1494 380 y Fr(2)1513 374 y Fx(,)f(w)o(e)i(ha)o(v)o(e)100 424 y(tacitly)14 b(used)i(the)g(fundamen)o(tal)d(prop)q(ert)o(y)j(that)f Fu(s)f Ft(!)978 430 y Fn(R)1022 424 y Fu(t)h Fx(implies)e Fu(s)h Ft(!)1269 430 y Fn(R)1298 434 y Ff(1)1329 424 y Fu(t)h Fx(or)g Fu(s)f Ft(!)1486 430 y Fn(R)1515 434 y Ff(2)1546 424 y Fu(t)p Fx(.)g(It)h(has)100 473 y(b)q(een)j(stressed)i (b)o(y)d(Middeldorp)g(\(1990,)f(1993\))g(that)i(this)f(basic)h(prop)q (ert)o(y)g(do)q(es)g(not)f(hold)g(true)100 523 y(for)f(CTRSs.)g (Consequen)o(tly)m(,)f(it)i(is)f(m)o(uc)o(h)f(more)g(subtle)i(to)g(pro) o(v)o(e)f(the)h(mo)q(dularit)o(y)d(of)i(a)g(certain)100 573 y(prop)q(ert)o(y)e(for)g(CTRSs.)141 623 y(If)f(\()p Ft(F)232 629 y Fr(1)251 623 y Fu(;)7 b Ft(R)304 629 y Fr(1)323 623 y Fx(\))13 b(and)g(\()p Ft(F)482 629 y Fr(2)501 623 y Fu(;)7 b Ft(R)554 629 y Fr(2)573 623 y Fx(\))13 b(are)h(comp)q(osable)d(or)j(constructor-sharing)g(CTRSs,)e(then)i(\()p Ft(F)t Fu(;)7 b Ft(R)1659 629 y Fr(1)1678 623 y Fx(\))100 673 y(and)13 b(\()p Ft(F)t Fu(;)7 b Ft(R)284 679 y Fr(2)303 673 y Fx(\))13 b(are)h(also)g(CTRSs,)f(where)h Ft(F)i Fx(=)c Ft(F)883 679 y Fr(1)910 673 y Ft([)d(F)981 679 y Fr(2)999 673 y Fx(.)14 b(In)f(order)i(to)e(a)o(v)o(oid)g (misunderstandings,)100 722 y(w)o(e)h(write)h Ft(\))310 728 y Fn(R)339 732 y Fl(i)368 722 y Fx(for)f(the)h(rewrite)h(relation)e (asso)q(ciated)h(with)g(\()p Ft(F)1137 728 y Fp(i)1151 722 y Fu(;)7 b Ft(R)1205 728 y Fp(i)1218 722 y Fx(\))15 b(and)f Ft(!)1372 728 y Fn(R)1401 732 y Fl(i)1430 722 y Fx(for)g(the)i(rewrite)100 772 y(relation)f(asso)q(ciated)h(with)f (\()p Ft(F)5 b Fu(;)i Ft(R)653 778 y Fp(i)667 772 y Fx(\),)15 b(where)i Fu(i)e Ft(2)f(f)p Fx(1)p Fu(;)7 b Fx(2)p Ft(g)p Fx(.)14 b(If)h Fu(s;)7 b(t)14 b Ft(2)g(T)c Fx(\()p Ft(F)1262 778 y Fp(i)1276 772 y Fu(;)d Ft(V)s Fx(\))16 b(and)f Fu(s)g Ft(\))1513 778 y Fn(R)1542 782 y Fl(i)1571 772 y Fu(t)p Fx(,)g(then)100 822 y(w)o(e)k(clearly)f(ha)o(v)o(e)h Fu(s)h Ft(!)485 828 y Fn(R)514 832 y Fl(i)548 822 y Fu(t)p Fx(.)e(A)h(priori,)f(it)g(is)h(not)g(clear)g(at)f(all)g(whether)i(the)f (con)o(v)o(erse)i(is)d(also)100 872 y(true.)d(F)m(or,)f(if)f Fu(s)h Ft(!)400 878 y Fn(R)429 882 y Fl(i)456 872 y Fu(t)p Fx(,)g(then)h(there)h(exists)g(a)e(rewrite)i(rule)f Fu(l)e Ft(!)e Fu(r)h Ft(\()f Fu(s)1257 878 y Fr(1)1288 872 y Ft(#)g Fu(t)1335 878 y Fr(1)1354 872 y Fu(;)c(:)g(:)g(:)t(;)g(s)1465 878 y Fp(n)1499 872 y Ft(#)12 b Fu(t)1547 878 y Fp(n)1584 872 y Fx(in)i Ft(R)1668 878 y Fp(i)1682 872 y Fx(,)100 922 y(a)g(substitution)h Fu(\033)f Fx(:)f Ft(V)k(!)c(T)d Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\),)15 b(and)f(a)h(con)o(text)g Fu(C)s Fx([)f(])g(suc)o(h)i(that)f Fu(s)e Fx(=)h Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])p Fu(;)7 b(t)k Fx(=)j Fu(C)s Fx([)p Fu(r)q(\033)q Fx(],)f(and)100 972 y Fu(s)119 978 y Fp(j)137 972 y Fu(\033)h Ft(#)196 982 y Fn(R)225 986 y Fl(i)253 972 y Fu(t)268 978 y Fp(j)285 972 y Fu(\033)h Fx(for)e(all)f Fu(j)i Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)12 b(And)h Fu(\033)g Fx(:)e Ft(V)16 b(!)11 b(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))14 b(ma)o(y)d(substitute)k (mixed)d(terms)h(for)100 1021 y(extra-v)n(ariables)g(o)q(ccurring)i(in) e(the)i(conditions.)656 1127 y Fk(6.1.)24 b(semi-completeness)141 1229 y Fx(Our)18 b(next)f(goal)f(is)h(to)g(sho)o(w)g(that)g (semi-completeness)g(is)f(mo)q(dular)g(for)g(constructor-sharing)100 1279 y(CTRSs.)g(So)h(in)f(this)h(subsection)h(w)o(e)f(tacitly)g(assume) f(that)h(\()p Ft(F)1145 1285 y Fr(1)1164 1279 y Fu(;)7 b Ft(R)1217 1285 y Fr(1)1236 1279 y Fx(\))17 b(and)g(\()p Ft(F)1403 1285 y Fr(2)1422 1279 y Fu(;)7 b Ft(R)1475 1285 y Fr(2)1494 1279 y Fx(\))17 b(are)g(giv)o(en)100 1329 y(constructor-sharing)i(CTRSs.)f(W)m(e)g(use)h(the)g(structure)h (and)e(the)h(ideas)g(of)e(the)i(pro)q(of)f(sho)o(wing)100 1379 y(that)d(con\015uence)i(is)e(mo)q(dular)e(for)i(disjoin)o(t)f (CTRSs,)h(see)h(Middeldorp)f(\(1990,)f(1993\).)g(The)h(basic)100 1428 y(pro)q(of)j(idea)g(is)g(to)g(construct)i(t)o(w)o(o)e(rewrite)i (relations)32 b Ft(!)1050 1434 y Fr(1)1101 1428 y Fx(and)g Ft(!)1241 1434 y Fr(2)1292 1428 y Fx(on)18 b Ft(T)11 b Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\))18 b(suc)o(h)i(that)100 1478 y(their)14 b(union)f(is)h(semi-complete,)d(and)j(reduction)h(in)e (the)i(com)o(bined)d(system)i(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(corresp)q(onds)100 1528 y(to)i(joinabilit)o(y)e(with)i(resp) q(ect)j(to)d Ft(!)690 1534 y Fr(1)p Fp(;)p Fr(2)751 1528 y Fx(=)30 b Ft(!)855 1534 y Fr(1)898 1528 y Ft([)25 b(!)992 1534 y Fr(2)1025 1528 y Fx(.)16 b(F)m(rom)f(these)j(t)o(w)o(o)e(prop)q (erties)i(and)e(the)100 1578 y(equalit)o(y)e Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\))13 b(=)h Fu(N)5 b(F)h Fx(\()p Ft(!)637 1584 y Fr(1)p Fp(;)p Fr(2)681 1578 y Fx(\),)14 b(the)i(mo)q(dularit)o(y)c(of)i(semi-completeness)h(for)f (CTRSs)h(with)100 1628 y(shared)f(constructors)i(follo)o(ws.)100 1729 y Fk(Definition)g(6.1.)21 b Fx(The)12 b(rewrite)h(relation)25 b Ft(!)855 1735 y Fr(1)900 1729 y Fx(is)11 b(de\014ned)i(b)o(y:)e Fu(s)k Ft(!)1222 1735 y Fr(1)1255 1729 y Fu(t)c Fx(if)g(there)j(exists) e(a)g(rewrite)100 1779 y(rule)d Fu(l)k Ft(!)e Fu(r)i Ft(\()e Fu(s)360 1785 y Fr(1)390 1779 y Ft(#)h Fu(t)438 1785 y Fr(1)456 1779 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)568 1785 y Fp(n)602 1779 y Ft(#)k Fu(t)649 1785 y Fp(n)681 1779 y Fx(in)f Ft(R)761 1785 y Fr(1)779 1779 y Fx(,)f(a)h(substitution) f Fu(\033)k Fx(:)e Ft(V)16 b(!)11 b(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\),)i(and)h(a)f(con)o(text)h Fu(C)s Fx([)f(])100 1829 y(suc)o(h)16 b(that)g Fu(s)f Fx(=)f Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])p Fu(;)7 b(t)13 b Fx(=)i Fu(C)s Fx([)p Fu(r)q(\033)q Fx(],)f(and)h Fu(s)785 1835 y Fp(j)803 1829 y Fu(\033)i Ft(#)865 1810 y Fp(o)865 1839 y Fr(1)899 1829 y Fu(t)914 1835 y Fp(j)932 1829 y Fu(\033)f Fx(for)g Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)14 b(Here)j(the)f(sup)q(erscript)100 1879 y Fu(o)h Fx(in)h Fu(s)209 1885 y Fp(j)227 1879 y Fu(\033)g Ft(#)290 1860 y Fp(o)290 1889 y Fr(1)326 1879 y Fu(t)341 1885 y Fp(j)359 1879 y Fu(\033)h Fx(means)e(that)g Fu(s)645 1885 y Fp(j)663 1879 y Fu(\033)i Fx(and)f Fu(t)806 1885 y Fp(j)823 1879 y Fu(\033)h Fx(are)f(joinable)f(using)g(only)g Fs(outer)31 b Ft(!)1473 1885 y Fr(1)1523 1879 y Fx(reduction)100 1929 y(steps.)15 b(The)f(relation)28 b Ft(!)509 1935 y Fr(2)556 1929 y Fx(is)14 b(de\014ned)h(analogously)m(.)d(The)i(union) g(of)27 b Ft(!)1279 1935 y Fr(1)1326 1929 y Fx(and)g Ft(!)1462 1935 y Fr(2)1509 1929 y Fx(is)14 b(denoted)100 1978 y(b)o(y)27 b Ft(!)213 1984 y Fr(1)p Fp(;)p Fr(2)271 1978 y Fx(.)100 2080 y Fk(Example)17 b(6.2.)k Fx(Let)d Ft(R)502 2086 y Fr(1)539 2080 y Fx(=)h Ft(f)p Fu(F)6 b Fx(\()p Fu(x;)h Fp(C)p Fx(\))18 b Ft(!)g Fu(G)p Fx(\()p Fu(x)p Fx(\))g Ft(\()g Fu(x)f Ft(#)h Fp(C)r Ft(g)f Fx(and)h Ft(R)1252 2086 y Fr(2)1289 2080 y Fx(=)h Ft(f)p Fu(a)f Ft(!)f Fp(C)r Ft(g)p Fx(.)g(W)m(e)h(ha)o(v)o(e)100 2130 y Fu(F)6 b Fx(\()p Fu(a;)h Fp(C)p Fx(\))14 b Ft(!)286 2136 y Fn(R)330 2130 y Fu(G)p Fx(\()p Fu(a)p Fx(\))c(but)g(neither)h Fu(F)6 b Fx(\()p Fu(a;)h Fp(C)q Fx(\))14 b Ft(!)823 2136 y Fr(1)855 2130 y Fu(G)p Fx(\()p Fu(a)p Fx(\))c(nor)g Fu(F)c Fx(\()p Fu(a;)h Fp(C)q Fx(\))14 b Ft(!)1208 2136 y Fr(2)1241 2130 y Fu(G)p Fx(\()p Fu(a)p Fx(\).)9 b(Ho)o(w)o(ev)o(er,)h (the)h(terms)100 2180 y(are)j(joinable)f(with)g(resp)q(ect)j(to)28 b Ft(!)667 2186 y Fr(1)p Fp(;)p Fr(2)726 2180 y Fx(:)14 b Fu(F)6 b Fx(\()p Fu(a;)h Fp(C)p Fx(\))14 b Ft(!)938 2186 y Fr(2)970 2180 y Fu(F)6 b Fx(\()p Fp(C)r Fu(;)h Fp(C)q Fx(\))14 b Ft(!)1161 2186 y Fr(1)1193 2180 y Fu(G)p Fx(\()p Fp(C)r Fx(\))d Ft( )1337 2186 y Fr(2)1367 2180 y Fu(G)p Fx(\()p Fu(a)p Fx(\))p Fu(:)141 2281 y Fx(The)j(simple)f(pro)q (ofs)h(of)f(the)h(next)h(t)o(w)o(o)e(lemmata)e(are)j(omitted.)100 2383 y Fk(Lemma)i(6.3.)21 b Fx(If)14 b Fu(s)g Ft(!)464 2389 y Fr(1)p Fp(;)p Fr(2)523 2383 y Fu(t)p Fx(,)f(then)i Fu(s)f Ft(!)733 2389 y Fn(R)777 2383 y Fu(t)p Fx(.)100 2484 y Fk(Lemma)i(6.4.)21 b Fx(Let)f Fu(s)g Fx(b)q(e)g(a)f(blac)o(k)g (term)f(and)i(let)f Fu(\033)h Fx(b)q(e)g(a)f(top)h(white)f (substitution)g(suc)o(h)i(that)100 2534 y Fu(s\033)13 b Ft(!)198 2519 y Fp(o)198 2544 y Fr(1)227 2534 y Fu(t)p Fx(.)h(Then)g(there)h(is)f(a)g(blac)o(k)f(term)g Fu(u)h Fx(suc)o(h)h(that)f Fu(t)d Fx(=)h Fu(u\033)q Fx(.)135 2628 y Fq(y)170 2640 y Fw(P)o(arts)f(of)g(the)g(material)e(presen)o (ted)g(in)i(this)g(section)f(originate)f(from)h(Ohlebusc)o(h)g(\(1994)p Fd(c)p Fw(\).)p eop %%Page: 24 24 24 23 bop 100 197 a Fw(24)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fk(Lemma)16 b(6.5.)21 b Fx(Let)12 b Fu(s;)7 b(t)k Fx(b)q(e)h(blac)o(k)f(terms)f(and)h(let)h Fu(\033)g Fx(b)q(e)g(a)f(top)g(white)g(substitution)g(with)g Fu(s\033)i Ft(!)1612 284 y Fp(o)1612 309 y Fr(1)1642 299 y Fu(t\033)q Fx(.)100 349 y(If)g Fu(\034)19 b Fx(is)14 b(a)f(substitution)h(with)g Fu(\033)f Ft(/)e Fu(\034)5 b Fx(,)14 b(then)g Fu(s\034)j Ft(!)899 334 y Fp(o)899 359 y Fr(1)929 349 y Fu(t\034)5 b Fx(.)100 448 y Fk(Pr)o(oof.)22 b Fx(The)14 b(lemma)e(is)i(pro)o(v)o(ed)h(b)o(y)f(induction)g(on)g(the) h(depth)h(of)d Fu(s\033)i Ft(!)1301 433 y Fp(o)1301 458 y Fr(1)1331 448 y Fu(t\033)q Fx(.)f(The)h(case)h(of)e(zero)100 498 y(depth)e(is)f(straigh)o(tforw)o(ard.)g(Let)h(the)g(depth)g(of)f Fu(s\033)i Ft(!)955 483 y Fp(o)955 508 y Fr(1)985 498 y Fu(t\033)f Fx(equal)f Fu(d)t Fx(+)t(1,)h Fu(d)f Ft(\025)h Fx(0.)f(There)h(is)g(a)f(con)o(text)100 547 y Fu(C)s Fx([)f(],)f(a)i(substitution)g Fu(\032)h Fx(:)f Ft(V)k(!)c(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\),)j(and)h(a)g(rewrite)g(rule)g Fu(l)i Ft(!)e Fu(r)i Ft(\()e Fu(s)1275 553 y Fr(1)1304 547 y Ft(#)g Fu(t)1351 553 y Fr(1)1370 547 y Fu(;)c(:)g(:)g(:)t(;)g(s) 1481 553 y Fp(n)1515 547 y Ft(#)j Fu(t)1561 553 y Fp(n)1594 547 y Fx(in)h Ft(R)1675 553 y Fr(1)100 597 y Fx(suc)o(h)j(that)f Fu(s\033)h Fx(=)d Fu(C)s Fx([)p Fu(l)q(\032)p Fx(],)i Fu(t\033)f Fx(=)g Fu(C)s Fx([)p Fu(r)q(\032)p Fx(])h(and)g Fu(s)803 603 y Fp(j)821 597 y Fu(\032)h Ft(#)877 579 y Fp(o)877 608 y Fr(1)909 597 y Fu(t)924 603 y Fp(j)941 597 y Fu(\032)g Fx(is)f(of)g(depth)h Ft(\024)e Fu(d)h Fx(for)g(ev)o(ery)h Fu(j)g Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g (n)p Ft(g)p Fx(.)100 647 y(According)15 b(to)f(Prop)q(osition)g(4.18,)f Fu(\032)i Fx(has)f(a)g(decomp)q(osition)g Fu(\032)f Fx(=)f Fu(\032)1192 653 y Fr(2)1221 647 y Ft(\016)d Fu(\032)1272 653 y Fr(1)1306 647 y Fx(suc)o(h)15 b(that)g Fu(\032)1512 653 y Fr(1)1545 647 y Fx(is)g(blac)o(k,)100 697 y Fu(\032)121 703 y Fr(2)160 697 y Fx(is)k(top)h(white,)g(and)g Fu(\032)526 703 y Fr(2)566 697 y Ft(/)i Fu(\017)p Fx(.)e(W)m(e)f(de\014ne)i(a)f (substitution)g Fu(\032)1171 682 y Fn(0)1203 697 y Fx(b)o(y)g Fu(\032)1288 682 y Fn(0)1300 697 y Fx(\()p Fu(x)p Fx(\))h(=)h Fu(\034)5 b Fx(\()p Fu(y)q Fx(\))21 b(for)f(ev)o(ery)100 747 y Fu(x)11 b Ft(2)h(D)q Fu(om)p Fx(\()p Fu(\032)301 753 y Fr(2)321 747 y Fx(\))i(and)g Fu(y)f Ft(2)f(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))j(satisfying)e Fu(\032)872 753 y Fr(2)891 747 y Fx(\()p Fu(x)p Fx(\))f(=)g Fu(\033)q Fx(\()p Fu(y)q Fx(\).)j Fu(\032)1129 732 y Fn(0)1155 747 y Fx(is)f(w)o(ell-de\014ned)g(b)q(ecause)i Fu(\033)d Ft(/)f Fu(\034)5 b Fx(.)100 797 y(It)13 b(follo)o(ws)e(from)h Fu(\032)398 803 y Fr(2)428 797 y Ft(/)g Fu(\017)h Fx(and)g Fu(\017)e Ft(/)h Fu(\032)675 781 y Fn(0)700 797 y Fx(that)h Fu(\032)810 803 y Fr(2)841 797 y Ft(/)f Fu(\032)906 781 y Fn(0)918 797 y Fx(.)g(By)i(Lemma)c(6.4,)i(for)h(an)o(y)f Fu(j)i Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(,)k(w)o(e)100 846 y(ma)o(y)h(write)192 921 y Fu(\032)213 927 y Fr(2)232 921 y Fx(\()p Fu(\032)269 927 y Fr(1)288 921 y Fx(\()p Fu(s)323 927 y Fp(j)341 921 y Fx(\)\))g(=)g Fu(\032)450 927 y Fr(2)469 921 y Fx(\()p Fu(u)509 927 y Fr(1)528 921 y Fx(\))f Ft(!)597 903 y Fp(o)597 931 y Fr(1)627 921 y Fu(:)c(:)g(:)j Ft(!)729 903 y Fp(o)729 931 y Fr(1)759 921 y Fu(\032)780 927 y Fr(2)799 921 y Fx(\()p Fu(u)839 927 y Fp(k)859 921 y Fx(\))i(=)g Fu(\032)952 927 y Fr(2)971 921 y Fx(\()p Fu(v)1007 927 y Fp(l)1020 921 y Fx(\))1050 903 y Fp(o)1050 931 y Fr(1)1066 921 y Ft( )f Fu(:)c(:)g(:)1188 903 y Fp(o)1188 931 y Fr(1)1204 921 y Ft( )12 b Fu(\032)1279 927 y Fr(2)1298 921 y Fx(\()p Fu(v)1334 927 y Fr(1)1353 921 y Fx(\))f(=)h Fu(\032)1445 927 y Fr(2)1464 921 y Fx(\()p Fu(\032)1501 927 y Fr(1)1520 921 y Fx(\()p Fu(t)1551 927 y Fp(j)1569 921 y Fx(\)\))100 995 y(for)18 b(some)g(blac)o(k)g(terms)g Fu(u)534 1001 y Fr(1)553 995 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)670 1001 y Fp(k)689 995 y Fu(;)g(v)728 1001 y Fr(1)746 995 y Fu(;)g(:)g(:)g(:)e (;)i(v)859 1001 y Fp(l)871 995 y Fx(.)19 b(No)o(w)f(rep)q(eated)i (application)e(of)g(the)h(induction)100 1045 y(h)o(yp)q(othesis)14 b(yields)213 1119 y Fu(\032)234 1102 y Fn(0)246 1119 y Fx(\()p Fu(\032)283 1125 y Fr(1)302 1119 y Fx(\()p Fu(s)337 1125 y Fp(j)355 1119 y Fx(\)\))e(=)g Fu(\032)464 1102 y Fn(0)476 1119 y Fx(\()p Fu(u)516 1125 y Fr(1)535 1119 y Fx(\))f Ft(!)604 1102 y Fp(o)604 1129 y Fr(1)634 1119 y Fu(:)c(:)g(:)j Ft(!)736 1102 y Fp(o)736 1129 y Fr(1)766 1119 y Fu(\032)787 1102 y Fn(0)799 1119 y Fx(\()p Fu(u)839 1125 y Fp(k)859 1119 y Fx(\))i(=)g Fu(\032)952 1102 y Fn(0)964 1119 y Fx(\()p Fu(v)1000 1125 y Fp(l)1013 1119 y Fx(\))1043 1102 y Fp(o)1043 1129 y Fr(1)1059 1119 y Ft( )f Fu(:)c(:)g(:)1181 1102 y Fp(o)1181 1129 y Fr(1)1198 1119 y Ft( )k Fu(\032)1272 1102 y Fn(0)1284 1119 y Fx(\()p Fu(v)1320 1125 y Fr(1)1339 1119 y Fx(\))g(=)h Fu(\032)1431 1102 y Fn(0)1443 1119 y Fx(\()p Fu(\032)1480 1125 y Fr(1)1499 1119 y Fx(\()p Fu(t)1530 1125 y Fp(j)1548 1119 y Fx(\)\))100 1199 y(Th)o(us)f Fu(\032)223 1183 y Fn(0)235 1199 y Fx(\()p Fu(\032)272 1205 y Fr(1)291 1199 y Fx(\()p Fu(l)q Fx(\)\))h Ft(!)406 1183 y Fp(o)406 1209 y Fr(1)436 1199 y Fu(\032)457 1183 y Fn(0)469 1199 y Fx(\()p Fu(\032)506 1205 y Fr(1)525 1199 y Fx(\()p Fu(r)q Fx(\)\).)f(Let)696 1188 y(^)687 1199 y Fu(C)r Fx([)f(])h(b)q(e)g(the)g(con)o(text)h(obtained)e(from)f Fu(C)s Fx([)h(])g(b)o(y)g(replacing)h(ev)o(ery)100 1248 y(white)j(principal)g(subterm)h(whic)o(h)f(m)o(ust)f(b)q(e)j(of)e(the)h (form)e Fu(\033)q Fx(\()p Fu(x)p Fx(\))h(for)h(some)e(v)n(ariable)h Fu(x)e Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))100 1298 y(b)o(y)j(the)i(corresp)q(onding)g Fu(\034)5 b Fx(\()p Fu(x)p Fx(\).)15 b(\(This)h(is)g(a)g(sligh)o(t)f(abuse)h(of)g(notation) f(b)q(ecause)1416 1288 y(^)1407 1298 y Fu(C)r Fx([)h(])f(con)o(tains)h (in)100 1348 y(general)c(more)e(that)i(one)g(o)q(ccurrence)i(of)e Fo(2)p Fx(.\))f(It)h(is)f(fairly)g(simple)f(to)h(v)o(erify)h(that)f Fu(s\034)17 b Fx(=)1496 1337 y(^)1487 1348 y Fu(C)s Fx([)p Fu(\032)1553 1333 y Fn(0)1564 1348 y Fx(\()p Fu(\032)1601 1354 y Fr(1)1620 1348 y Fx(\()p Fu(l)q Fx(\)\)])100 1398 y(and)c Fu(t\034)k Fx(=)283 1387 y(^)274 1398 y Fu(C)r Fx([)p Fu(\032)339 1383 y Fn(0)351 1398 y Fx(\()p Fu(\032)388 1404 y Fr(1)407 1398 y Fx(\()p Fu(r)q Fx(\)\)].)c(Hence)j Fu(s\034)g Ft(!)731 1383 y Fp(o)731 1408 y Fr(1)761 1398 y Fu(t\034)5 b Fx(.)13 b Fe(2)100 1497 y Fk(Lemma)j(6.6.)21 b Fx(The)15 b(restriction)g(of)e Ft(!)721 1503 y Fr(1)753 1497 y Fx(to)h Ft(T)c Fx(\()p Ft(F)883 1503 y Fr(1)901 1497 y Fu(;)d Ft(V)s Fx(\))j Ft(\002)f(T)i Fx(\()p Ft(F)1095 1503 y Fr(1)1113 1497 y Fu(;)c Ft(V)s Fx(\))15 b(and)e Ft(\))1313 1503 y Fn(R)1342 1507 y Ff(1)1373 1497 y Fx(coincide.)100 1596 y Fk(Pr)o(oof.)22 b Fx(\\)p Ft(\023)p Fx(")13 b(T)m(rivial.)100 1646 y(\\)p Ft(\022)p Fx(")20 b(Let)g Fu(s;)7 b(t)22 b Ft(2)g(T)10 b Fx(\()p Ft(F)478 1652 y Fr(1)497 1646 y Fu(;)d Ft(V)s Fx(\))21 b(with)e Fu(s)k Ft(!)765 1652 y Fr(1)805 1646 y Fu(t)p Fx(.)d(In)g(order)h(to)f(sho)o(w)h(that)f Fu(s)i Ft(\))1371 1652 y Fn(R)1400 1656 y Ff(1)1440 1646 y Fu(t)p Fx(,)e(w)o(e)g(pro)q(ceed)100 1696 y(b)o(y)f(induction)g(on)h (the)g(depth)g(of)f Fu(s)j Ft(!)753 1681 y Fp(o)753 1706 y Fr(1)792 1696 y Fu(t)p Fx(.)d(The)h(case)h(of)e(zero)h(depth)h(is)e (straigh)o(tforw)o(ard.)g(So)100 1745 y(supp)q(ose)e(that)e(the)h (depth)h(of)e Fu(s)g Ft(!)667 1730 y Fp(o)667 1756 y Fr(1)699 1745 y Fu(t)h Fx(equals)g Fu(d)10 b Fx(+)g(1,)15 b Fu(d)f Ft(\025)h Fx(0.)g(Then)h(there)h(exists)f(a)g(rewrite)g(rule) 100 1795 y Fu(l)c Ft(!)f Fu(r)i Ft(\()e Fu(s)281 1801 y Fr(1)311 1795 y Ft(#)h Fu(t)359 1801 y Fr(1)377 1795 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)489 1801 y Fp(n)523 1795 y Ft(#)k Fu(t)570 1801 y Fp(n)609 1795 y Fx(in)16 b Ft(R)695 1801 y Fr(1)713 1795 y Fx(,)g(a)g(substitution)g Fu(\033)g Fx(:)f Ft(V)k(!)14 b(T)d Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\),)16 b(and)f(a)h(con)o(text)h Fu(C)s Fx([)e(])100 1845 y(suc)o(h)20 b(that)g Fu(s)h Fx(=)g Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])p Fu(;)7 b(t)20 b Fx(=)h Fu(C)s Fx([)p Fu(r)q(\033)q Fx(],)d(and)h Fu(s)827 1851 y Fp(j)845 1845 y Fu(\033)i Ft(#)911 1827 y Fp(o)911 1855 y Fr(1)949 1845 y Fu(t)964 1851 y Fp(j)982 1845 y Fu(\033)f Fx(with)g(depth)g Ft(\024)h Fu(d)e Fx(for)h Fu(j)j Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)100 1895 y(According)15 b(to)f(Prop)q(osition)g(4.18,)f Fu(\033)j Fx(can)e(b)q(e)i(decomp)q (osed)e(in)o(to)g Fu(\033)1188 1901 y Fr(2)1216 1895 y Ft(\016)c Fu(\033)1271 1901 y Fr(1)1303 1895 y Fx(suc)o(h)16 b(that)e Fu(\033)1512 1901 y Fr(1)1545 1895 y Fx(is)h(blac)o(k,)100 1945 y Fu(\033)124 1951 y Fr(2)160 1945 y Fx(is)j(top)g(white,)g(and)g Fu(\033)522 1951 y Fr(2)558 1945 y Ft(/)h Fu(\017)p Fx(.)f(Induction)g (on)g(the)g(n)o(um)o(b)q(er)g(of)f(rewrite)i(steps)h(in)d Fu(s)1517 1951 y Fp(j)1535 1945 y Fu(\033)j Ft(#)1599 1926 y Fp(o)1599 1955 y Fr(1)1636 1945 y Fu(t)1651 1951 y Fp(j)1668 1945 y Fu(\033)100 1995 y Fx(in)c(com)o(bination)e(with)i (Lemma)e(6.5)h(yields)i Fu(\033)851 2001 y Fr(1)869 1995 y Fx(\()p Fu(s)904 2001 y Fp(j)922 1995 y Fx(\))g Ft(#)975 1976 y Fp(o)975 2005 y Fr(1)1011 1995 y Fu(\033)1035 2001 y Fr(1)1053 1995 y Fx(\()p Fu(t)1084 2001 y Fp(j)1102 1995 y Fx(\))f(for)g Fu(j)j Ft(2)c(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)15 b(Since)i(ev)o(ery)100 2044 y(term)11 b(in)h(the)h(con)o(v)o(ersion)g Fu(\033)539 2050 y Fr(1)557 2044 y Fx(\()p Fu(s)592 2050 y Fp(j)610 2044 y Fx(\))g Ft(#)659 2026 y Fp(o)659 2055 y Fr(1)690 2044 y Fu(\033)714 2050 y Fr(1)733 2044 y Fx(\()p Fu(t)764 2050 y Fp(j)781 2044 y Fx(\))g(is)f(blac)o(k,)f(w)o(e)i(obtain)f Fu(\033)1180 2050 y Fr(1)1198 2044 y Fx(\()p Fu(s)1233 2050 y Fp(j)1251 2044 y Fx(\))g Ft(+)1304 2050 y Fn(R)1333 2054 y Ff(1)1362 2044 y Fu(\033)1386 2050 y Fr(1)1405 2044 y Fx(\()p Fu(t)1436 2050 y Fp(j)1453 2044 y Fx(\))h(b)o(y)f(rep)q (eated)100 2094 y(application)j(of)i(the)g(induction)g(h)o(yp)q (othesis.)g(Consequen)o(tly)m(,)f(w)o(e)i(ha)o(v)o(e)e Fu(\033)1304 2100 y Fr(1)1323 2094 y Fx(\()p Fu(l)q Fx(\))h Ft(\))1427 2100 y Fn(R)1456 2104 y Ff(1)1490 2094 y Fu(\033)1514 2100 y Fr(1)1532 2094 y Fx(\()p Fu(r)q Fx(\).)g(No)o(w)100 2144 y Fu(s)e Ft(\))176 2150 y Fn(R)205 2154 y Ff(1)238 2144 y Fu(t)h Fx(follo)o(ws)f(from)f Fu(s)i Fx(=)f Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])g(=)g Fu(C)s Fx([)p Fu(l)q(\033)829 2150 y Fr(1)847 2144 y Fx(])g(and)h Fu(t)g Fx(=)f Fu(C)s Fx([)p Fu(r)q(\033)q Fx(])f(=)i Fu(C)s Fx([)p Fu(r)q(\033)1288 2150 y Fr(1)1305 2144 y Fx(])g(b)q(ecause)i Fu(s)e Fx(and)g Fu(t)g Fx(are)100 2194 y(blac)o(k.)d Fe(2)100 2293 y Fk(Pr)o(oposition)j(6.7.)21 b Fx(If)12 b(\()p Ft(F)550 2299 y Fr(1)568 2293 y Fu(;)7 b Ft(R)622 2299 y Fr(1)641 2293 y Fx(\))12 b(and)g(\()p Ft(F)798 2299 y Fr(2)817 2293 y Fu(;)7 b Ft(R)871 2299 y Fr(2)889 2293 y Fx(\))13 b(are)g(semi-complete,)d(then)j(the)g(relation)25 b Ft(!)1635 2299 y Fr(1)p Fp(;)p Fr(2)100 2343 y Fx(is)13 b(semi-complete)f(as)i(w) o(ell.)100 2442 y Fk(Pr)o(oof.)22 b Fx(W)m(e)13 b(de\014ne)i(t)o(w)o(o) e(unconditional)g(TRSs)h(\()p Ft(F)958 2448 y Fr(1)976 2442 y Fu(;)7 b Ft(S)1020 2448 y Fr(1)1039 2442 y Fx(\))14 b(and)f(\()p Ft(F)1199 2448 y Fr(2)1218 2442 y Fu(;)7 b Ft(S)1262 2448 y Fr(2)1280 2442 y Fx(\))14 b(b)o(y)337 2516 y Ft(S)362 2522 y Fp(i)387 2516 y Fx(=)e Ft(f)p Fu(u)f Ft(!)g Fu(v)k Ft(j)f Fu(u;)7 b(v)12 b Ft(2)f(T)f Fx(\()p Ft(F)794 2522 y Fp(i)808 2516 y Fu(;)d Ft(V)s Fx(\))p Fu(;)g(r)q(oot)p Fx(\()p Fu(u)p Fx(\))12 b Ft(62)f(C)1094 2522 y Fr(1)1122 2516 y Ft(\\)e(C)1183 2522 y Fr(2)1215 2516 y Fx(and)14 b Fu(u)g Ft(!)1375 2522 y Fp(i)1403 2516 y Fu(v)q Ft(g)p Fu(:)100 2590 y Fx(First)d(of)g(all)f(note)h(that) h(\()p Ft(F)525 2596 y Fr(1)543 2590 y Fu(;)7 b Ft(S)587 2596 y Fr(1)606 2590 y Fx(\))k(and)g(\()p Ft(F)761 2596 y Fr(2)780 2590 y Fu(;)c Ft(S)824 2596 y Fr(2)842 2590 y Fx(\))k(are)h(constructor-sharing)g(TRSs.)f(By)h(Lemma)c(6.6,)100 2640 y(the)14 b(restriction)i(of)27 b Ft(!)473 2646 y Fp(i)515 2640 y Fx(to)14 b Ft(T)c Fx(\()p Ft(F)639 2646 y Fp(i)654 2640 y Fu(;)d Ft(V)s Fx(\))i Ft(\002)h(T)g Fx(\()p Ft(F)841 2646 y Fp(i)855 2640 y Fu(;)d Ft(V)s Fx(\))15 b(and)f Ft(\))1056 2646 y Fn(R)1085 2650 y Fl(i)1113 2640 y Fx(coincide.)g(It)h(is)f(easy)g(to)g(sho)o(w)h(that)p eop %%Page: 25 25 25 24 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(25)p 100 224 1595 2 v 113 299 a Ft(!)155 305 y Fn(S)175 309 y Fl(i)218 299 y Fx(and)13 b(the)i(restriction)f(of)27 b Ft(!)671 305 y Fp(i)712 299 y Fx(to)13 b Ft(T)e Fx(\()p Ft(F)836 305 y Fp(i)850 299 y Fu(;)c Ft(V)s Fx(\))i Ft(\002)g(T)h Fx(\()p Ft(F)1036 305 y Fp(i)1050 299 y Fu(;)d Ft(V)s Fx(\))14 b(are)g(also)f(the)h(same.)f(Hence)29 b Ft(!)1644 305 y Fn(S)1664 309 y Fl(i)100 349 y Fx(and)10 b Ft(\))219 355 y Fn(R)248 359 y Fl(i)273 349 y Fx(coincide)h(on)f Ft(T)h Fx(\()p Ft(F)557 355 y Fp(i)571 349 y Fu(;)c Ft(V)s Fx(\))s Ft(\002)s(T)j Fx(\()p Ft(F)745 355 y Fp(i)760 349 y Fu(;)d Ft(V)s Fx(\).)j(In)h(particular,)f(the)h(TRS)f(\()p Ft(F)1301 355 y Fp(i)1315 349 y Fu(;)d Ft(S)1359 355 y Fp(i)1372 349 y Fx(\))k(is)f(semi-complete)100 399 y(b)q(ecause)k(\()p Ft(F)297 405 y Fp(i)310 399 y Fu(;)7 b Ft(R)364 405 y Fp(i)378 399 y Fx(\))13 b(is)g(semi-complete.)d(It)j (follo)o(ws)e(from)g(Theorem)i(5.2)f(that)h(\()p Ft(F)1380 405 y Fr(1)1405 399 y Ft([)7 b(F)1470 405 y Fr(2)1489 399 y Fu(;)g Ft(S)1533 405 y Fr(1)1558 399 y Ft([)g(S)1618 405 y Fr(2)1637 399 y Fx(\))13 b(is)100 448 y(also)g(semi-complete.)141 499 y(W)m(e)c(next)h(sho)o(w)f(that)g(the)h(relations)23 b Ft(!)767 505 y Fn(S)787 509 y Fl(i)825 499 y Fx(and)g Ft(!)957 505 y Fp(i)994 499 y Fx(are)9 b(also)g(the)h(same)e(on)h Ft(T)i Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\))p Ft(\002T)k Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\).)100 549 y(\\)p Ft(\022)p Fx(")13 b(Straigh)o(tforw)o(ard.)100 598 y(\\)p Ft(\023)p Fx(")19 b(Without)f(loss)h(of)g(generalit)o(y)m(,)f(let)h Fu(i)i Fx(=)f(1.)f(If)f Fu(s)j Ft(!)1039 604 y Fr(1)1078 598 y Fu(t)p Fx(,)d(then)i(there)h(exist)e(a)g(rewrite)h(rule)100 648 y Fu(l)12 b Ft(!)f Fu(r)i Ft(\()e Fu(s)281 654 y Fr(1)311 648 y Ft(#)h Fu(t)359 654 y Fr(1)377 648 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)489 654 y Fp(n)523 648 y Ft(#)k Fu(t)570 654 y Fp(n)609 648 y Fx(in)16 b Ft(R)695 654 y Fr(1)713 648 y Fx(,)g(a)g(substitution)g Fu(\033)g Fx(:)f Ft(V)k(!)14 b(T)d Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\),)16 b(and)f(a)h(con)o(text)h Fu(C)s Fx([)e(])100 698 y(suc)o(h)f(that)h Fu(s)d Fx(=)g Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])p Fu(;)7 b(t)k Fx(=)h Fu(C)s Fx([)p Fu(r)q(\033)q Fx(],)g(and)i Fu(s)769 704 y Fp(j)787 698 y Fu(\033)i Ft(#)847 679 y Fp(o)847 708 y Fr(1)880 698 y Fu(t)895 704 y Fp(j)913 698 y Fu(\033)f Fx(for)f Fu(j)g Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)13 b(Note)h(that)g(particularly)100 748 y Fu(l)q(\033)h Ft(!)194 754 y Fr(1)225 748 y Fu(r)q(\033)q Fx(.)g(According)g(to)g(Prop)q(osition)g(4.18,)e Fu(\033)j Fx(has)f(a)g(decomp)q(osition)f Fu(\033)h Fx(=)e Fu(\033)1406 754 y Fr(2)1435 748 y Ft(\016)c Fu(\033)1489 754 y Fr(1)1523 748 y Fx(suc)o(h)16 b(that)100 798 y Fu(\033)124 804 y Fr(1)157 798 y Fx(is)e(blac)o(k,)g Fu(\033)344 804 y Fr(2)377 798 y Fx(is)g(top)h(white,)f(and)h Fu(\033)725 804 y Fr(2)756 798 y Ft(/)e Fu(\017)p Fx(.)h(No)o(w)h(w)o(e)g(apply)f (Lemma)e(6.5:)h Fu(\033)1369 804 y Fr(1)1387 798 y Fx(\()p Fu(l)q Fx(\))i(and)g Fu(\033)1553 804 y Fr(1)1571 798 y Fx(\()p Fu(r)q Fx(\))g(are)100 847 y(blac)o(k)g(terms)h(and)h Fu(\033)436 853 y Fr(2)470 847 y Fx(is)f(a)g(top)h(white)f (substitution)g(with)g Fu(\033)1097 853 y Fr(2)1116 847 y Fx(\()p Fu(\033)1156 853 y Fr(1)1174 847 y Fx(\()p Fu(l)q Fx(\)\))g Ft(!)1293 853 y Fr(1)1327 847 y Fu(\033)1351 853 y Fr(2)1369 847 y Fx(\()p Fu(\033)1409 853 y Fr(1)1428 847 y Fx(\()p Fu(r)q Fx(\)\))h(and)f Fu(\017)g Fx(is)g(a)100 897 y(substitution)e(with)g Fu(\033)450 903 y Fr(2)480 897 y Ft(/)e Fu(\017)p Fx(.)i(Consequen)o(tly)m(,)f(w)o(e)i(obtain)e Fu(\033)1047 903 y Fr(1)1065 897 y Fx(\()p Fu(l)q Fx(\))g(=)f Fu(\017)p Fx(\()p Fu(\033)1224 903 y Fr(1)1242 897 y Fx(\()p Fu(l)q Fx(\)\))h Ft(!)1358 903 y Fr(1)1388 897 y Fu(\017)p Fx(\()p Fu(\033)1445 903 y Fr(1)1463 897 y Fx(\()p Fu(r)q Fx(\)\))f(=)h Fu(\033)1612 903 y Fr(1)1630 897 y Fx(\()p Fu(r)q Fx(\).)100 947 y(Since)g Fu(\033)231 953 y Fr(1)250 947 y Fx(\()p Fu(l)q Fx(\))g(and)g Fu(\033)412 953 y Fr(1)430 947 y Fx(\()p Fu(r)q Fx(\))g(are)h(blac)o(k)f(terms)f (and)h Fu(r)q(oot)p Fx(\()p Fu(\033)981 953 y Fr(1)1000 947 y Fx(\()p Fu(l)q Fx(\)\))f(=)g Fu(r)q(oot)p Fx(\()p Fu(l)q Fx(\))g Ft(62)f(C)1312 953 y Fr(1)1339 947 y Ft(\\)c(C)1398 953 y Fr(2)1417 947 y Fx(,)12 b(it)h(follo)o(ws)f(that)100 997 y Fu(\033)124 1003 y Fr(1)142 997 y Fx(\()p Fu(l)q Fx(\))g Ft(!)f Fu(\033)276 1003 y Fr(1)294 997 y Fx(\()p Fu(r)q Fx(\))j(is)g(a)g(rewrite)h(rule)f(of)f Ft(S)736 1003 y Fr(1)755 997 y Fx(.)g(Th)o(us)h Fu(s)e Fx(=)g Fu(C)s Fx([)p Fu(\033)1029 1003 y Fr(2)1047 997 y Fx(\()p Fu(\033)1087 1003 y Fr(1)1105 997 y Fx(\()p Fu(l)q Fx(\)\)])g Ft(!)1232 1003 y Fn(S)1252 1007 y Ff(1)1282 997 y Fu(C)s Fx([)p Fu(\033)1351 1003 y Fr(2)1368 997 y Fx(\()p Fu(\033)1408 1003 y Fr(1)1427 997 y Fx(\()p Fu(r)q Fx(\)\)])f(=)h Fu(t)p Fx(.)141 1047 y(With)h(the)i(ab)q(o)o(v)o(e)e(results,)i(it)f (further)g(follo)o(ws)f(from)445 1128 y Ft(!)487 1134 y Fn(S)507 1138 y Ff(1)523 1134 y Fn([S)565 1138 y Ff(2)610 1128 y Fx(=)25 b Ft(!)709 1134 y Fn(S)729 1138 y Ff(1)771 1128 y Ft([)d(!)863 1134 y Fn(S)883 1138 y Ff(2)927 1128 y Fx(=)k Ft(!)1026 1134 y Fr(1)1068 1128 y Ft([)c(!)1160 1134 y Fr(2)1204 1128 y Fx(=)k Ft(!)1303 1134 y Fr(1)p Fp(;)p Fr(2)100 1208 y Fx(that)h Ft(!)245 1214 y Fr(1)p Fp(;)p Fr(2)318 1208 y Fx(is)13 b(semi-complete)f(on)i Ft(T)c Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))j Ft(\002)g(T)g Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\).)14 b Fe(2)141 1314 y Fx(Note)j(that)g(the)g(ab)q(o)o(v)o(e)f(approac)o(h)h(fails)e(for)i (comp)q(osable)e(CTRSs)h(b)q(ecause)j(\(the)e(accordingly)100 1364 y(de\014ned\))e(sets)g Ft(S)368 1370 y Fr(1)401 1364 y Fx(and)e Ft(S)510 1370 y Fr(2)542 1364 y Fx(are)h(in)g(general)g (not)g(comp)q(osable.)100 1476 y Fk(Definition)i(6.8.)21 b Fx(If)13 b Ft(!)511 1482 y Fr(1)p Fp(;)p Fr(2)570 1476 y Fx(is)g(semi-complete,)f(then)i(ev)o(ery)h(term)e Fu(t)h Fx(has)g(a)g(unique)g(normal)d(form)100 1526 y(w.r.t.)h Ft(!)252 1532 y Fr(1)p Fp(;)p Fr(2)296 1526 y Fx(.)h(In)g(the)g (sequel,)h(this)f(normal)e(form)g(will)h(b)q(e)h(denoted)h(b)o(y)f Fu(t)1255 1510 y Fn(!)1290 1526 y Fx(.)g(F)m(urthermore,)f(for)h(an)o (y)100 1575 y(substitution)h Fu(\033)q Fx(,)f Fu(\033)406 1560 y Fn(!)455 1575 y Fx(denotes)j(the)e(substitution)g Ft(f)p Fu(x)d Ft(7!)g Fu(\033)q Fx(\()p Fu(x)p Fx(\))1098 1560 y Fn(!)1147 1575 y Ft(j)j Fu(x)d Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))r Ft(g)p Fx(.)100 1687 y Fk(Lemma)16 b(6.9.)21 b Fx(Let)14 b Ft(!)463 1693 y Fr(1)p Fp(;)p Fr(2)521 1687 y Fx(b)q(e)f(semi-complete.)e(If)i Fu(s)g Fx(and)g Fu(t)g Fx(are)g(blac)o(k)g(terms)g(and)g Fu(\033)h Fx(is)f(a)g(top)f(white)100 1737 y Ft(!)142 1743 y Fr(1)p Fp(;)p Fr(2)200 1737 y Fx(normalized)g(substitution)i(suc)o(h)h(that)f Fu(s\033)h Ft(#)906 1747 y Fr(1)p Fp(;)p Fr(2)965 1737 y Fu(t\033)q Fx(,)f(then)g Fu(s\033)i Ft(#)1204 1718 y Fp(o)1204 1747 y Fr(1)1237 1737 y Fu(t\033)q Fx(.)100 1843 y Fk(Pr)o(oof.)22 b Fx(W)m(e)e(sho)o(w)h(that)h Fu(s\033)j Ft(!)660 1849 y Fr(1)p Fp(;)p Fr(2)728 1843 y Fu(u)c Fx(implies)e Fu(s\033)26 b Ft(!)1032 1828 y Fp(o)1032 1853 y Fr(1)1073 1843 y Fu(u)p Fx(.)21 b(Since)h Fu(u)h Fx(=)h Fu(v)q(\033)f Fx(for)e(some)f(blac)o(k)100 1893 y(term)d Fu(v)i Fx(b)o(y)f(Lemma)d(6.4,)h(the)i(lemma)d(then)k (follo)o(ws)d(b)o(y)h(a)h(straigh)o(tforw)o(ard)f(induction)g(on)h(the) 100 1943 y(length)i(of)g(the)i(v)n(alley)m(.)c(In)j(order)g(to)g(pro)o (v)o(e)f(the)i(claim,)c(w)o(e)i(use)i(induction)e(on)h(the)g(depth)g (of)100 1993 y Fu(s\033)d Ft(!)203 1999 y Fr(1)p Fp(;)p Fr(2)265 1993 y Fu(u)p Fx(.)f(The)g(case)i(of)d(zero)i(depth)g(is)g (trivial.)d(So)i(supp)q(ose)h(that)g(the)g(depth)g(of)e Fu(s\033)j Ft(!)1608 1999 y Fr(1)p Fp(;)p Fr(2)1670 1993 y Fu(u)100 2042 y Fx(equals)e Fu(d)10 b Fx(+)i(1,)k Fu(d)g Ft(\025)h Fx(0.)f(Since)i Fu(\033)g Fx(is)f(a)f(top)h(white)g Ft(!)961 2048 y Fr(1)p Fp(;)p Fr(2)1023 2042 y Fx(normalized)e (substitution,)i(there)h(exists)100 2092 y(a)e(rewrite)i(rule)f Fu(l)c Ft(!)e Fu(r)h Ft(\()f Fu(s)548 2098 y Fr(1)579 2092 y Ft(#)g Fu(t)626 2098 y Fr(1)645 2092 y Fu(;)c(:)g(:)g(:)e(;)i(s) 757 2098 y Fp(n)791 2092 y Ft(#)k Fu(t)838 2098 y Fp(n)877 2092 y Fx(in)17 b Ft(R)964 2098 y Fr(1)983 2092 y Fx(,)f(a)g (substitution)i Fu(\032)e Fx(:)g Ft(V)21 b(!)16 b(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\),)17 b(and)100 2142 y(a)g(con)o(text)h Fu(C)s Fx([)f(])g(suc)o(h)h(that)g Fu(s\033)h Fx(=)f Fu(C)s Fx([)p Fu(l)q(\032)p Fx(],)f Fu(u)g Fx(=)h Fu(C)s Fx([)p Fu(r)q(\032)p Fx(],)e(and)i Fu(s)1124 2148 y Fp(j)1142 2142 y Fu(\032)g Ft(#)1201 2152 y Fr(1)p Fp(;)p Fr(2)1264 2142 y Fu(t)1279 2148 y Fp(j)1296 2142 y Fu(\032)g Fx(with)g(depth)g Ft(\024)g Fu(d)f Fx(for)100 2192 y Fu(j)23 b Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)18 b(By)i(Prop)q(osition)f (4.18,)e Fu(\032)j Fx(can)f(b)q(e)h(decomp)q(osed)g(in)o(to)e Fu(\032)1334 2198 y Fr(2)1366 2192 y Ft(\016)13 b Fu(\032)1421 2198 y Fr(1)1459 2192 y Fx(suc)o(h)20 b(that)g Fu(\032)1675 2198 y Fr(1)100 2242 y Fx(is)f(blac)o(k,)f Fu(\032)293 2248 y Fr(2)331 2242 y Fx(is)h(top)g(white,)g(and)g Fu(\032)694 2248 y Fr(2)734 2242 y Ft(/)h Fu(\017)p Fx(.)f(Note)g(that)h(for)e(ev)o (ery)i Fu(x)h Ft(2)f(D)q Fu(om)p Fx(\()p Fu(\032)1437 2248 y Fr(2)1457 2242 y Fx(\))12 b Ft(\\)h(V)s Fu(ar)q Fx(\()p Fu(l)q(\032)1646 2248 y Fr(1)1666 2242 y Fx(\),)100 2291 y(w)o(e)k(ha)o(v)o(e)h Fu(\032)285 2297 y Fr(2)304 2291 y Fx(\()p Fu(x)p Fx(\))g Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(!)552 2297 y Fr(1)p Fp(;)p Fr(2)596 2291 y Fx(\).)18 b(Nev)o(ertheless,)h(w)o(e)f(do)f(not)h(ha)o(v)o(e)f Fu(\032)1220 2297 y Fr(2)1239 2291 y Fx(\()p Fu(x)p Fx(\))h Ft(2)g Fu(N)5 b(F)h Fx(\()p Ft(!)1488 2297 y Fr(1)p Fp(;)p Fr(2)1532 2291 y Fx(\))17 b(in)h(gen-)100 2341 y(eral)k(b)q(ecause)i (of)e(p)q(ossible)h(extra)g(v)n(ariables.)e(Since)i Ft(!)1039 2347 y Fr(1)p Fp(;)p Fr(2)1106 2341 y Fx(is)g(semi-complete,)d Fu(\032)1468 2347 y Fr(2)1513 2341 y Ft(!)1555 2326 y Fn(\003)1555 2352 y Fr(1)p Fp(;)p Fr(2)1625 2341 y Fu(\032)1646 2326 y Fn(!)1646 2352 y Fr(2)1682 2341 y Fx(.)100 2391 y(Th)o(us)e Fu(\032)230 2376 y Fn(!)230 2401 y Fr(2)266 2391 y Fx(\()p Fu(\032)303 2397 y Fr(1)322 2391 y Fx(\()p Fu(s)357 2397 y Fp(j)375 2391 y Fx(\)\))446 2376 y Fn(\003)446 2401 y Fr(1)p Fp(;)p Fr(2)489 2391 y Ft( )13 b Fu(s)563 2397 y Fp(j)581 2391 y Fu(\032)18 b Ft(#)641 2401 y Fr(1)p Fp(;)p Fr(2)704 2391 y Fu(t)719 2397 y Fp(j)737 2391 y Fu(\032)h Ft(!)819 2376 y Fn(\003)819 2401 y Fr(1)p Fp(;)p Fr(2)882 2391 y Fu(\032)903 2376 y Fn(!)903 2401 y Fr(2)938 2391 y Fx(\()p Fu(\032)975 2397 y Fr(1)994 2391 y Fx(\()p Fu(t)1025 2397 y Fp(j)1043 2391 y Fx(\)\).)f(The)g (con\015uence)i(of)e Ft(!)1493 2397 y Fr(1)p Fp(;)p Fr(2)1555 2391 y Fx(guaran-)100 2441 y(tees)c Fu(\032)203 2426 y Fn(!)203 2451 y Fr(2)238 2441 y Fx(\()p Fu(\032)275 2447 y Fr(1)295 2441 y Fx(\()p Fu(s)330 2447 y Fp(j)348 2441 y Fx(\)\))f Ft(#)413 2451 y Fr(1)p Fp(;)p Fr(2)471 2441 y Fu(\032)492 2426 y Fn(!)492 2451 y Fr(2)528 2441 y Fx(\()p Fu(\032)565 2447 y Fr(1)584 2441 y Fx(\()p Fu(t)615 2447 y Fp(j)633 2441 y Fx(\)\))g(for)f(ev)o(ery)i Fu(j)g Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)k(By)i(Prop)q(osition)g(4.18,)e Fu(\032)1528 2426 y Fn(!)1528 2451 y Fr(2)1576 2441 y Fx(can)i(b)q(e)100 2491 y(decomp)q(osed)j(in)o(to)g Fu(\032)442 2497 y Fr(4)472 2491 y Ft(\016)11 b Fu(\032)525 2497 y Fr(3)561 2491 y Fx(suc)o(h)17 b(that)g Fu(\032)771 2497 y Fr(3)806 2491 y Fx(is)g(blac)o(k,)e Fu(\032)994 2497 y Fr(4)1030 2491 y Fx(is)h(top)h(white,)f(and)h Fu(\032)1383 2497 y Fr(4)1418 2491 y Ft(/)f Fu(\017)p Fx(.)g(Eviden)o(tly)m(,)100 2540 y Fu(\032)121 2546 y Fr(3)140 2540 y Fx(\()p Fu(\032)177 2546 y Fr(1)196 2540 y Fx(\()p Fu(s)231 2546 y Fp(j)249 2540 y Fx(\)\),)11 b Fu(\032)325 2546 y Fr(3)344 2540 y Fx(\()p Fu(\032)381 2546 y Fr(1)400 2540 y Fx(\()p Fu(t)431 2546 y Fp(j)449 2540 y Fx(\)\))g(are)h(blac)o(k)e(terms)h(and) g Fu(\032)876 2546 y Fr(4)906 2540 y Fx(is)g(a)g(top)g(white)g Ft(!)1200 2546 y Fr(1)p Fp(;)p Fr(2)1255 2540 y Fx(normalized)f (substitution.)100 2590 y(Hence)17 b(the)g(induction)e(h)o(yp)q (othesis)i(yields)e Fu(\032)831 2596 y Fr(4)850 2590 y Fx(\()p Fu(\032)887 2596 y Fr(3)907 2590 y Fx(\()p Fu(\032)944 2596 y Fr(1)963 2590 y Fx(\()p Fu(s)998 2596 y Fp(j)1016 2590 y Fx(\)\)\))h Ft(#)1101 2572 y Fp(o)1101 2601 y Fr(1)1135 2590 y Fu(\032)1156 2596 y Fr(4)1176 2590 y Fx(\()p Fu(\032)1213 2596 y Fr(3)1232 2590 y Fx(\()p Fu(\032)1269 2596 y Fr(1)1288 2590 y Fx(\()p Fu(t)1319 2596 y Fp(j)1337 2590 y Fx(\)\)\).)f(In)h(other)h(w)o(ords,)100 2640 y Fu(\032)121 2625 y Fn(!)121 2650 y Fr(2)156 2640 y Fx(\()p Fu(\032)193 2646 y Fr(1)212 2640 y Fx(\()p Fu(s)247 2646 y Fp(j)266 2640 y Fx(\)\))d Ft(#)333 2622 y Fp(o)333 2650 y Fr(1)366 2640 y Fu(\032)387 2625 y Fn(!)387 2650 y Fr(2)423 2640 y Fx(\()p Fu(\032)460 2646 y Fr(1)479 2640 y Fx(\()p Fu(t)510 2646 y Fp(j)528 2640 y Fx(\)\),)g(and)g(w)o(e)h(obtain)f(as)g(a)h(consequence)i(that)d Fu(\032)1292 2625 y Fn(!)1292 2650 y Fr(2)1328 2640 y Fx(\()p Fu(\032)p Fx(\()p Fu(l)q Fx(\)\))g Ft(!)1482 2625 y Fp(o)1482 2650 y Fr(1)1512 2640 y Fu(\032)1533 2625 y Fn(!)1533 2650 y Fr(2)1569 2640 y Fx(\()p Fu(\032)1606 2646 y Fr(1)1625 2640 y Fx(\()p Fu(r)q Fx(\)\))p eop %%Page: 26 26 26 25 bop 100 197 a Fw(26)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fx(and)16 b Fu(C)s Fx([)p Fu(\032)249 284 y Fn(!)249 309 y Fr(2)284 299 y Fx(\()p Fu(\032)321 305 y Fr(1)340 299 y Fx(\()p Fu(l)q Fx(\)\)])h Ft(!)472 284 y Fp(o)472 309 y Fr(1)507 299 y Fu(C)s Fx([)p Fu(\032)573 284 y Fn(!)573 309 y Fr(2)607 299 y Fx(\()p Fu(\032)644 305 y Fr(1)664 299 y Fx(\()p Fu(r)q Fx(\)\)].)f(Clearly)m (,)f Fu(s\033)j Fx(=)f Fu(C)s Fx([)p Fu(\032)1103 284 y Fn(!)1103 309 y Fr(2)1138 299 y Fx(\()p Fu(\032)1175 305 y Fr(1)1194 299 y Fx(\()p Fu(l)q Fx(\)\)])g(and)g Fu(u)f Fx(=)h Fu(C)s Fx([)p Fu(\032)1523 284 y Fn(!)1523 309 y Fr(2)1558 299 y Fx(\()p Fu(\032)1595 305 y Fr(1)1614 299 y Fx(\()p Fu(r)q Fx(\)\)])100 349 y(b)q(ecause)e Fu(\032)274 355 y Fr(2)293 349 y Fx(\()p Fu(x)p Fx(\))d Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(!)529 355 y Fr(1)p Fp(;)p Fr(2)573 349 y Fx(\))14 b(for)f(ev)o(ery)i Fu(x)c Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\032)976 355 y Fr(2)996 349 y Fx(\))f Ft(\\)e(V)s Fu(ar)q Fx(\()p Fu(l)q(\032)1178 355 y Fr(1)1198 349 y Fx(\))q(.)13 b(This)h(pro)o(v)o(es)g(the)g(claim.)d Fe(2)100 451 y Fk(Lemma)16 b(6.10.)21 b Fx(Let)14 b Ft(!)486 457 y Fr(1)p Fp(;)p Fr(2)544 451 y Fx(b)q(e)g(semi-complete)d(and)i (let)g Fu(s)1027 457 y Fr(1)1047 451 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)1158 457 y Fp(n)1181 451 y Fu(;)g(t)1215 457 y Fr(1)1233 451 y Fu(;)g(:)g(:)g(:)e(;)i(t)1341 457 y Fp(n)1376 451 y Fx(b)q(e)14 b(blac)o(k)e(terms.)h(If)100 501 y Fu(\033)i Fx(is)e(a)h(substitution)f(with)h Fu(s)559 507 y Fp(j)577 501 y Fu(\033)h Ft(#)636 511 y Fr(1)p Fp(;)p Fr(2)695 501 y Fu(t)710 507 y Fp(j)728 501 y Fu(\033)g Fx(for)e(ev)o(ery)h Fu(j)g Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(,)12 b(then)j Fu(\033)1334 486 y Fn(!)1369 501 y Fx(\()p Fu(s)1404 507 y Fp(j)1422 501 y Fx(\))f Ft(#)1473 483 y Fp(o)1473 512 y Fr(1)1505 501 y Fu(\033)1530 486 y Fn(!)1565 501 y Fx(\()p Fu(t)1596 507 y Fp(j)1614 501 y Fx(\))g(for)100 551 y(ev)o(ery)g Fu(j)g Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)100 654 y Fk(Pr)o(oof.)22 b Fx(W)m(e)11 b(ha)o(v)o(e)g Fu(\033)451 639 y Fn(!)487 654 y Fx(\()p Fu(s)522 660 y Fp(j)540 654 y Fx(\))577 639 y Fn(\003)577 664 y Fr(1)p Fp(;)p Fr(2)619 654 y Ft( )j Fu(s)694 660 y Fp(j)712 654 y Fu(\033)f Ft(#)769 664 y Fr(1)p Fp(;)p Fr(2)826 654 y Fu(t)841 660 y Fp(j)859 654 y Fu(\033)f Ft(!)937 639 y Fn(\003)937 664 y Fr(1)p Fp(;)p Fr(2)993 654 y Fu(\033)1018 639 y Fn(!)1054 654 y Fx(\()p Fu(t)1085 660 y Fp(j)1102 654 y Fx(\).)g(The)g(con\015uence)h(of)e Ft(!)1510 660 y Fr(1)p Fp(;)p Fr(2)1567 654 y Fx(implies)100 704 y Fu(\033)125 689 y Fn(!)160 704 y Fx(\()p Fu(s)195 710 y Fp(j)213 704 y Fx(\))h Ft(#)262 714 y Fr(1)p Fp(;)p Fr(2)319 704 y Fu(\033)344 689 y Fn(!)380 704 y Fx(\()p Fu(t)411 710 y Fp(j)428 704 y Fx(\).)g(Prop)q(osition)g(4.18)f(yields)h(a)g(decomp)q (osition)f(of)g Fu(\033)1266 689 y Fn(!)1314 704 y Fx(in)o(to)g Fu(\033)1420 710 y Fr(2)1444 704 y Ft(\016)6 b Fu(\033)1495 710 y Fr(1)1526 704 y Fx(suc)o(h)13 b(that)100 754 y Fu(\033)124 760 y Fr(1)160 754 y Fx(is)k(blac)o(k)g(and)h Fu(\033)426 760 y Fr(2)462 754 y Fx(is)f(top)h(white.)f(Eviden)o(tly)m (,)f Fu(\033)937 760 y Fr(1)955 754 y Fx(\()p Fu(s)990 760 y Fp(j)1009 754 y Fx(\),)h Fu(\033)1078 760 y Fr(1)1096 754 y Fx(\()p Fu(t)1127 760 y Fp(j)1145 754 y Fx(\))g(are)i(blac)o(k)e (terms)g(and)g Fu(\033)1591 760 y Fr(2)1627 754 y Fx(is)h(a)100 804 y(top)c(white)h Ft(!)330 810 y Fr(1)p Fp(;)p Fr(2)389 804 y Fx(normalized)f(substitution.)g(According)h(to)g(Lemma)d(6.9,)h (w)o(e)i(ev)o(en)o(tually)f(deriv)o(e)100 853 y Fu(\033)125 838 y Fn(!)160 853 y Fx(\()p Fu(s)195 859 y Fp(j)213 853 y Fx(\))e(=)g Fu(\033)309 859 y Fr(2)327 853 y Fx(\()p Fu(\033)367 859 y Fr(1)385 853 y Fx(\()p Fu(s)420 859 y Fp(j)439 853 y Fx(\)\))i Ft(#)505 835 y Fp(o)505 864 y Fr(1)538 853 y Fu(\033)562 859 y Fr(2)580 853 y Fx(\()p Fu(\033)620 859 y Fr(1)639 853 y Fx(\()p Fu(t)670 859 y Fp(j)687 853 y Fx(\)\))e(=)g Fu(\033)800 838 y Fn(!)835 853 y Fx(\()p Fu(t)866 859 y Fp(j)884 853 y Fx(\).)h Fe(2)100 956 y Fk(Pr)o(oposition)j(6.11.)21 b Fx(If)j(\()p Ft(F)585 962 y Fr(1)604 956 y Fu(;)7 b Ft(R)657 962 y Fr(1)676 956 y Fx(\))25 b(and)f(\()p Ft(F)858 962 y Fr(2)877 956 y Fu(;)7 b Ft(R)931 962 y Fr(2)949 956 y Fx(\))25 b(are)g(semi-complete)e(and)i Fu(s)14 b Ft(!)1518 962 y Fn(R)1562 956 y Fu(t)p Fx(,)24 b(then)100 1006 y Fu(s)14 b Ft(#)154 1016 y Fr(1)p Fp(;)p Fr(2)213 1006 y Fu(t)p Fx(.)100 1109 y Fk(Pr)o(oof.)22 b Fx(W)m(e)c(pro)q(ceed)j(b)o(y)e (induction)g(on)f(the)i(depth)g(of)f Fu(s)h Ft(!)g Fu(t)p Fx(.)f(The)g(case)h(of)f(zero)h(depth)g(is)100 1159 y(trivial.)12 b(So)j(supp)q(ose)g(that)g(the)g(depth)g(of)f Fu(s)f Ft(!)f Fu(t)i Fx(equals)h Fu(d)9 b Fx(+)h(1,)k Fu(d)e Ft(\025)h Fx(0.)h(Then)h(there)g(is)g(a)f(con)o(text)100 1209 y Fu(C)s Fx([)h(],)h(a)g(substitution)g Fu(\033)h Fx(:)e Ft(V)20 b(!)15 b(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(,)16 b(and)g(a)g(rewrite)i(rule)e Fu(l)h Ft(!)e Fu(r)i Ft(\()e Fu(s)1362 1215 y Fr(1)1398 1209 y Ft(#)h Fu(t)1450 1215 y Fr(1)1468 1209 y Fu(;)7 b(:)g(:)g(:)e(;)i(s) 1580 1215 y Fp(n)1619 1209 y Ft(#)16 b Fu(t)1671 1215 y Fp(n)100 1258 y Fx(in)f Ft(R)h Fx(suc)o(h)g(that)g Fu(s)f Fx(=)g Fu(C)s Fx([)p Fu(l)q(\033)q Fx(],)g Fu(t)f Fx(=)h Fu(C)s Fx([)p Fu(r)q(\033)q Fx(],)f(and)i Fu(s)897 1264 y Fp(j)915 1258 y Fu(\033)h Ft(#)e Fu(t)1007 1264 y Fp(j)1025 1258 y Fu(\033)i Fx(is)e(of)h(depth)g(less)g(than)g(or)g (equal)f(to)h Fu(d)100 1308 y Fx(for)f(ev)o(ery)h Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)14 b(Figure)h(4)h(depicts)g(ho)o(w)g(the)g(induction)f(h)o(yp)q (othesis)h(and)g(con\015uence)100 1358 y(of)f Ft(!)191 1364 y Fr(1)p Fp(;)p Fr(2)252 1358 y Fx(yield)g Fu(s)373 1364 y Fp(j)391 1358 y Fu(\033)j Ft(#)453 1368 y Fr(1)p Fp(;)p Fr(2)515 1358 y Fu(t)530 1364 y Fp(j)547 1358 y Fu(\033)f Fx(for)f(ev)o(ery)h Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)15 b Fx(\(where)j(\(1\))e(signals)g (an)f(application)g(of)100 1408 y(the)g(induction)g(h)o(yp)q(othesis)h (and)e(\(2\))h(stands)h(for)f(an)g(application)e(of)i(Prop)q(osition)f (6.7\).)g(W.l.o.g.)100 1458 y(w)o(e)h(ma)o(y)e(assume)i(that)g(the)h (applied)f(rewrite)h(rule)g(stems)f(from)e Ft(R)1187 1464 y Fr(1)1206 1458 y Fx(.)i(By)g(Lemma)e(6.10,)g(w)o(e)j(ha)o(v)o(e) 100 1508 y Fu(\033)125 1492 y Fn(!)160 1508 y Fx(\()p Fu(s)195 1514 y Fp(j)213 1508 y Fx(\))11 b Ft(#)261 1489 y Fp(o)261 1518 y Fr(1)291 1508 y Fu(\033)316 1492 y Fn(!)351 1508 y Fx(\()p Fu(t)382 1514 y Fp(j)400 1508 y Fx(\))g(for)g Fu(j)j Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)e(;)i(n)p Ft(g)j Fx(and)h(th)o(us)g Fu(\033)940 1492 y Fn(!)976 1508 y Fx(\()p Fu(l)q Fx(\))h Ft(!)1075 1514 y Fr(1)1104 1508 y Fu(\033)1129 1492 y Fn(!)1165 1508 y Fx(\()p Fu(r)q Fx(\).)f(Finally)m(,)d(w)o(e)k(obtain)e Fu(s)i Ft(#)1622 1518 y Fr(1)p Fp(;)p Fr(2)1679 1508 y Fu(t)100 1557 y Fx(from)g Fu(s)g Fx(=)g Fu(C)s Fx([)p Fu(l)q(\033)q Fx(])e Ft(!)420 1542 y Fn(\003)420 1568 y Fr(1)p Fp(;)p Fr(2)476 1557 y Fu(C)s Fx([)p Fu(\033)546 1542 y Fn(!)581 1557 y Fx(\()p Fu(l)q Fx(\)])h Ft(!)691 1563 y Fr(1)721 1557 y Fu(C)s Fx([)p Fu(\033)791 1542 y Fn(!)826 1557 y Fx(\()p Fu(r)q Fx(\)])903 1542 y Fn(\003)903 1568 y Fr(1)p Fp(;)p Fr(2)946 1557 y Ft( )g Fu(C)s Fx([)p Fu(r)q(\033)q Fx(])f(=)i Fu(t)p Fx(.)h Fe(2)218 2332 y @beginspecial 0 @llx 0 @lly 448 @urx 212 @ury 3158 @rwi @setspecial %%BeginDocument: ../diss/Figures/CTRS1.ps /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix } def end /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 0 setlinecap 0 setlinejoin -137.0 440.0 translate 0.900 -0.900 scale 0.500 setlinewidth n 239 319 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 319 319 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 399 319 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 479 319 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 559 319 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 359 439 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 279 274 5 5 0 360 DrawEllipse gs col-1 s gr n 199 274 5 5 0 360 DrawEllipse gs col-1 s gr n 360 274 5 5 0 360 DrawEllipse gs col-1 s gr n 439 274 5 5 0 360 DrawEllipse gs col-1 s gr n 519 274 5 5 0 360 DrawEllipse gs col-1 s gr n 599 274 5 5 0 360 DrawEllipse gs col-1 s gr n 279 359 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 319 399 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 404 479 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr n 404 479 5 5 0 360 DrawEllipse gs 0.50 setgray fill gr gs col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 244 324 m 274 354 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 269.757 346.929 m 274.000 354.000 l 266.929 349.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 284 364 m 314 394 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 309.757 386.929 m 314.000 394.000 l 306.929 389.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 324 404 m 354 434 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 349.757 426.929 m 354.000 434.000 l 346.929 429.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 364 444 m 394 474 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 389.757 466.929 m 394.000 474.000 l 386.929 469.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 314 324 m 284 354 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 291.071 349.757 m 284.000 354.000 l 288.243 346.929 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 394 324 m 324 394 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 331.071 389.757 m 324.000 394.000 l 328.243 386.929 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 474 324 m 364 429 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 371.168 424.923 m 364.000 429.000 l 368.406 422.029 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 554 324 m 409 469 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 416.071 464.757 m 409.000 469.000 l 413.243 461.929 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 206 281 m 236 311 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 231.757 303.929 m 236.000 311.000 l 228.929 306.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 367 280 m 397 310 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 392.757 302.929 m 397.000 310.000 l 389.929 305.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 445 280 m 475 310 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 470.757 302.929 m 475.000 310.000 l 467.929 305.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 527 281 m 557 311 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 552.757 303.929 m 557.000 311.000 l 549.929 306.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 273 280 m 243 310 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 250.071 305.757 m 243.000 310.000 l 247.243 302.929 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 513 280 m 483 310 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 490.071 305.757 m 483.000 310.000 l 487.243 302.929 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 592 281 m 562 311 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 569.071 306.757 m 562.000 311.000 l 566.243 303.929 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 354 280 m 324 310 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 331.071 305.757 m 324.000 310.000 l 328.243 302.929 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 286 280 m 316 310 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 311.757 302.929 m 316.000 310.000 l 308.929 305.757 l gs 2 setlinejoin col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash n 433 278 m 403 308 l gs 0.00 setgray fill gr gs col-1 s gr [] 0 setdash 0 setlinecap n 410.071 303.757 m 403.000 308.000 l 407.243 300.929 l gs 2 setlinejoin col-1 s gr n 209 275 m 263 275 l gs col-1 s gr n 255.000 273.000 m 263.000 275.000 l 255.000 277.000 l gs 2 setlinejoin col-1 s gr n 293 275 m 347 275 l gs col-1 s gr n 339.000 273.000 m 347.000 275.000 l 339.000 277.000 l gs 2 setlinejoin col-1 s gr n 371 275 m 425 275 l gs col-1 s gr n 417.000 273.000 m 425.000 275.000 l 417.000 277.000 l gs 2 setlinejoin col-1 s gr n 506 275 m 452 275 l gs col-1 s gr n 460.000 277.000 m 452.000 275.000 l 460.000 273.000 l gs 2 setlinejoin col-1 s gr n 587 275 m 533 275 l gs col-1 s gr n 541.000 277.000 m 533.000 275.000 l 541.000 273.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 479 479 m 521 479 l gs col-1 s gr [] 0 setdash n 513.000 477.000 m 521.000 479.000 l 513.000 481.000 l gs 2 setlinejoin col-1 s gr n 554 479 m 596 479 l gs col-1 s gr n 588.000 477.000 m 596.000 479.000 l 588.000 481.000 l gs 2 setlinejoin col-1 s gr /Symbol findfont 12.00 scalefont setfont 384 479 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 344 439 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 304 399 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 264 359 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 224 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 304 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 384 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 464 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 544 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 334 399 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 419 474 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 374 434 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 254 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 334 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 414 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 574 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 489 319 m gs 1 -1 scale (*) col-1 show gr /Symbol findfont 12.00 scalefont setfont 294 359 m gs 1 -1 scale (*) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 533 485 m gs 1 -1 scale (=) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 599 488 m gs 1 -1 scale (1,2) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 311 293 m gs 1 -1 scale (\(1\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 392 293 m gs 1 -1 scale (\(1\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 473 293 m gs 1 -1 scale (\(1\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 554 293 m gs 1 -1 scale (\(1\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 353 317 m gs 1 -1 scale (\(2\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 512 317 m gs 1 -1 scale (\(2\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 275 317 m gs 1 -1 scale (\(2\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 431 317 m gs 1 -1 scale (\(2\)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 230 293 m gs 1 -1 scale (\(1\)) col-1 show gr /Symbol findfont 18.00 scalefont setfont 152 278 m gs 1 -1 scale (s) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 167 278 m gs 1 -1 scale (\(s \)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 179 284 m gs 1 -1 scale (j) col-1 show gr /Symbol findfont 18.00 scalefont setfont 611 278 m gs 1 -1 scale (s) col-1 show gr /Times-Roman findfont 18.00 scalefont setfont 626 278 m gs 1 -1 scale (\(t \)) col-1 show gr /Times-Roman findfont 12.00 scalefont setfont 638 281 m gs 1 -1 scale (j) col-1 show gr $F2psEnd %%EndDocument @endspecial 544 2411 a Fh(Figure)g(4)h Fw(The)e(pro)q(of)e(idea)g(of)h (Prop)q(osition)e(6.11.)100 2590 y Fk(Pr)o(oposition)16 b(6.12.)21 b Fx(If)12 b(\()p Ft(F)573 2596 y Fr(1)592 2590 y Fu(;)7 b Ft(R)645 2596 y Fr(1)664 2590 y Fx(\))13 b(and)f(\()p Ft(F)822 2596 y Fr(2)841 2590 y Fu(;)7 b Ft(R)895 2596 y Fr(2)913 2590 y Fx(\))13 b(are)g(semi-complete,)d(then) k(the)f(relations)g Ft($)1664 2575 y Fn(\003)1664 2602 y(R)100 2640 y Fx(and)27 b Ft(#)215 2650 y Fr(1)p Fp(;)p Fr(2)288 2640 y Fx(coincide.)p eop %%Page: 27 27 27 26 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(27)p 100 224 1595 2 v 100 299 a Fk(Pr)o(oof.)22 b Fx(This)13 b(is)h(a)g (consequence)i(of)d(Lemma)e(6.3)i(and)h(Prop)q(ositions)g(6.11)e(and)i (6.7.)f Fe(2)100 399 y Fk(Lemma)j(6.13.)21 b Fx(If)14 b(\()p Ft(F)462 405 y Fr(1)481 399 y Fu(;)7 b Ft(R)534 405 y Fr(1)553 399 y Fx(\))14 b(and)g(\()p Ft(F)714 405 y Fr(2)732 399 y Fu(;)7 b Ft(R)786 405 y Fr(2)805 399 y Fx(\))14 b(are)g(semi-complete,)d(then)125 498 y(\(1\))21 b Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)o Fx(\))12 b(=)g Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)619 504 y Fr(1)638 498 y Fx(\))i Ft(\\)g Fu(N)c(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)874 504 y Fr(2)892 498 y Fx(\).)125 548 y(\(2\))21 b Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)o Fx(\))12 b(=)g Fu(N)5 b(F)h Fx(\()p Ft(!)574 554 y Fr(1)p Fp(;)p Fr(2)618 548 y Fx(\).)100 648 y Fk(Pr)o(oof.)22 b Fx(W)m(e)10 b(will)g(only)g(pro)o(v)o(e)h(the)g(\014rst)h(statemen)o(t)f(b)q (ecause)i(the)e(pro)q(of)g(of)f(the)i(second)g(statemen)o(t)100 697 y(is)h(essen)o(tially)h(the)h(same.)100 747 y(\\)p Ft(\022)p Fx(")e(T)m(rivial.)100 797 y(\\)p Ft(\023)p Fx(")e(If)g Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)398 803 y Fr(1)417 797 y Fx(\))t Ft(\\)t Fu(N)e(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)643 803 y Fr(2)662 797 y Fx(\))k Ft(6\022)h Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)o Fx(\),)k(then)h(there)h(is)e (a)g(term)g Fu(s)h Fx(with)f Fu(s)h Ft(62)f Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)p Fx(\))100 847 y(and)12 b Fu(s)g Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)423 853 y Fr(1)441 847 y Fx(\))f Ft(\\)g Fu(N)f(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)670 853 y Fr(2)689 847 y Fx(\).)12 b(W.l.o.g.)d(w)o(e)j(ma)o(y)e(assume)i(that)g Fu(s)g Fx(is)g(of)g(minim)o(al)c(size)13 b(\(i.e.,)100 897 y Ft(j)p Fu(s)p Ft(j)g Fx(is)i(minim)o(al)o(\).)c(Hence)16 b Fu(s)f Fx(is)f(a)g(redex)h(and)g(ev)o(ery)g(prop)q(er)g(subterm)f(of) g Fu(s)h Fx(is)f(irreducible)h(b)o(y)f Ft(!)1652 903 y Fn(R)1682 897 y Fx(.)100 946 y(Therefore,)e(there)h(exists)g(a)f (rewrite)g(rule)h Fu(l)f Ft(!)f Fu(r)i Ft(\()e Fu(s)948 952 y Fr(1)978 946 y Ft(#)h Fu(t)1026 952 y Fr(1)1044 946 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1156 952 y Fp(n)1190 946 y Ft(#)k Fu(t)1237 952 y Fp(n)1272 946 y Fx(in)g Ft(R)h Fx(and)g(a)f(substitution)100 996 y Fu(\033)22 b Fx(:)e Ft(V)25 b(!)20 b(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))20 b(suc)o(h)g(that)g Fu(s)h Fx(=)g Fu(l)q(\033)o(;)7 b(t)20 b Fx(=)h Fu(r)q(\033)q Fx(,)e(and)h Fu(s)1069 1002 y Fp(j)1087 996 y Fu(\033)i Ft(#)1154 1002 y Fn(R)1205 996 y Fu(t)1220 1002 y Fp(j)1237 996 y Fu(\033)f Fx(for)e(all)f Fu(j)23 b Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)100 1046 y(Note)16 b(that)g(for)f(ev)o(ery)i(v)n(ariable)e Fu(x)f Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))e Ft(\\)e(V)s Fu(ar)q Fx(\()p Fu(l)q Fx(\))r(,)15 b(w)o(e)h(ha)o(v)o(e)g Fu(\033)q Fx(\()p Fu(x)p Fx(\))e Ft(2)h Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\))16 b(b)q(ecause)100 1096 y Fu(\033)q Fx(\()p Fu(x)p Fx(\))e(is)h(a)f(prop)q(er)i(subterm)f (of)f Fu(s)p Fx(.)g(W.l.o.g.)e(w)o(e)j(ma)o(y)e(further)i(assume)f (that)h(the)g(applied)g(rewrite)100 1146 y(rule)h(originates)g(from)e Ft(R)513 1152 y Fr(1)531 1146 y Fx(.)i(By)g(Prop)q(osition)g(6.12,)f Fu(s)973 1152 y Fp(j)991 1146 y Fu(\033)h Ft(#)1052 1152 y Fr(1)p Fp(;)p Fr(2)1129 1146 y Fu(t)1144 1152 y Fp(j)1161 1146 y Fu(\033)h Fx(whic)o(h,)f(in)g(conjunction)g(with)100 1196 y(Lemma)c(6.10,)h(yields)i Fu(\033)492 1180 y Fn(!)527 1196 y Fx(\()p Fu(s)562 1202 y Fp(j)580 1196 y Fx(\))28 b Ft(#)645 1180 y Fp(o)645 1206 y Fr(1)692 1196 y Fu(\033)717 1180 y Fn(!)752 1196 y Fx(\()p Fu(t)783 1202 y Fp(j)801 1196 y Fx(\).)14 b(It)h(follo)o(ws)f Fu(s)f Fx(=)h Fu(\033)q Fx(\()p Fu(l)q Fx(\))g(=)f Fu(\033)1259 1180 y Fn(!)1294 1196 y Fx(\()p Fu(l)q Fx(\))h Ft(!)1395 1180 y Fp(o)1395 1206 y Fr(1)1427 1196 y Fu(\033)1452 1180 y Fn(!)1487 1196 y Fx(\()p Fu(r)q Fx(\))h(b)q(ecause)100 1245 y Fu(\033)q Fx(\()p Fu(x)p Fx(\))c(=)h Fu(\033)261 1230 y Fn(!)297 1245 y Fx(\()p Fu(x)p Fx(\))g(for)g(ev)o(ery)h Fu(x)f Ft(2)f(V)s Fu(ar)q Fx(\()p Fu(l)q Fx(\))q(.)h(This)g(means)g(that)h Fu(s)e Ft(62)h Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)1302 1251 y Fr(1)1320 1245 y Fx(\),)12 b(a)h(con)o(tradiction.)e Fe(2)100 1345 y Fk(Theorem)16 b(6.14.)21 b Fx(Semi-completeness)13 b(is)h(mo)q(dular)e(for)h(constructor-sharing)j(CTRSs.)100 1445 y Fk(Pr)o(oof.)22 b Fx(Let)e(\()p Ft(F)390 1451 y Fr(1)408 1445 y Fu(;)7 b Ft(R)462 1451 y Fr(1)480 1445 y Fx(\))20 b(and)g(\()p Ft(F)649 1451 y Fr(2)667 1445 y Fu(;)7 b Ft(R)721 1451 y Fr(2)740 1445 y Fx(\))19 b(b)q(e)h(CTRSs)g (with)f(shared)i(constructors.)g(W)m(e)e(ha)o(v)o(e)h(to)100 1494 y(sho)o(w)c(that)h(their)g(com)o(bined)e(system)h(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))17 b(is)f(semi-complete)f(if)g(and)i(only) e(if)h(b)q(oth)h(\()p Ft(F)1587 1500 y Fr(1)1605 1494 y Fu(;)7 b Ft(R)1659 1500 y Fr(1)1678 1494 y Fx(\))100 1544 y(and)15 b(\()p Ft(F)228 1550 y Fr(2)247 1544 y Fu(;)7 b Ft(R)301 1550 y Fr(2)319 1544 y Fx(\))16 b(are)g (semi-complete.)e(In)i(order)g(to)g(sho)o(w)f(the)i(if)e(case,)h(w)o(e) g(consider)h(a)e(con)o(v)o(ersion)100 1594 y Fu(t)115 1600 y Fr(1)154 1579 y Fn(\003)154 1605 y(R)182 1594 y Ft( )27 b Fu(s)15 b Ft(!)327 1579 y Fn(\003)327 1605 y(R)370 1594 y Fu(t)385 1600 y Fr(2)404 1594 y Fx(.)g(According)g(to)g (Prop)q(osition)g(6.12)f(w)o(e)h(ha)o(v)o(e)g Fu(t)1169 1600 y Fr(1)1203 1594 y Ft(#)1224 1604 y Fr(1)p Fp(;)p Fr(2)1284 1594 y Fu(t)1299 1600 y Fr(2)1318 1594 y Fx(.)f(Since)i Ft(!)1496 1600 y Fr(1)p Fp(;)p Fr(2)1556 1594 y Fx(is)f(semi-)100 1644 y(complete,)g Fu(t)304 1650 y Fr(1)339 1644 y Ft(!)381 1629 y Fn(\003)381 1654 y Fr(1)p Fp(;)p Fr(2)442 1644 y Fu(t)457 1650 y Fr(3)493 1644 y Fx(and)i Fu(t)592 1650 y Fr(2)627 1644 y Ft(!)669 1629 y Fn(\003)669 1654 y Fr(1)p Fp(;)p Fr(2)730 1644 y Fu(t)745 1650 y Fr(3)763 1644 y Fx(,)g(where)h Fu(t)930 1650 y Fr(3)965 1644 y Fx(is)f(the)g(unique)g(normal)e(form)g(of)h Fu(s)p Fx(,)h Fu(t)1580 1650 y Fr(1)1598 1644 y Fx(,)g(and)100 1694 y Fu(t)115 1700 y Fr(2)133 1694 y Fx(.)f(No)o(w)h(Lemma)d(6.3)h (implies)g Fu(t)638 1700 y Fr(1)670 1694 y Ft(!)712 1679 y Fn(\003)712 1705 y(R)756 1694 y Fu(t)771 1700 y Fr(3)811 1679 y Fn(\003)811 1705 y(R)839 1694 y Ft( )27 b Fu(t)923 1700 y Fr(2)942 1694 y Fx(.)16 b(Th)o(us)h(\()p Ft(F)t Fu(;)7 b Ft(R)o Fx(\))17 b(is)f(con\015uen)o(t.)h(It)g(remains)f(to)100 1743 y(sho)o(w)i(normalization)d(of)i Ft(!)567 1749 y Fn(R)597 1743 y Fx(.)g(Let)i Fu(s)f Ft(2)g(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(.)17 b(Since)i Ft(!)1119 1749 y Fr(1)p Fp(;)p Fr(2)1181 1743 y Fx(is)f(normalizing,)d Fu(s)j Ft(!)1548 1728 y Fn(\003)1548 1754 y Fr(1)p Fp(;)p Fr(2)1611 1743 y Fu(t)g Fx(for)100 1793 y(some)g Fu(t)j Ft(2)g Fu(N)5 b(F)h Fx(\()p Ft(!)423 1799 y Fr(1)p Fp(;)p Fr(2)467 1793 y Fx(\).)19 b(By)h(Lemma)c(6.3,)i Fu(s)k Ft(!)905 1778 y Fn(\003)905 1805 y(R)956 1793 y Fu(t)p Fx(.)d(It)g(follo)o(ws)f(from)g(Lemma)f(6.13)h(\(2\))i(that)100 1843 y Fu(t)d Ft(2)g Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\).)17 b(Hence)h(\()p Ft(F)5 b Fu(;)i Ft(R)o Fx(\))18 b(is)f(also)f(normalizing.)e(This)k(all)e(pro)o(v)o(es)h(that) h(\()p Ft(F)t Fu(;)7 b Ft(R)o Fx(\))18 b(is)f(semi-)100 1893 y(complete.)12 b(The)j(only-if)d(case)j(follo)o(ws)d(straigh)o (tforw)o(ardly)h(from)f(Lemma)f(6.15.)h Fe(2)100 1993 y Fk(Lemma)k(6.15.)21 b Fx(Let)f(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))18 b(b)q(e)h(the)h(com)o(bined)d(system)h(of)h(t)o(w)o(o)f (constructor-sharing)i(CTRSs)100 2042 y(\()p Ft(F)146 2048 y Fr(1)164 2042 y Fu(;)7 b Ft(R)218 2048 y Fr(1)236 2042 y Fx(\))12 b(and)f(\()p Ft(F)388 2048 y Fr(2)406 2042 y Fu(;)c Ft(R)460 2048 y Fr(2)478 2042 y Fx(\))12 b(suc)o(h)g(that)f(\()p Ft(F)t Fu(;)c Ft(R)p Fx(\))k(is)g (semi-complete.)d(If)j Fu(s)h Fx(is)f(a)f(blac)o(k)h(term)g(and)g Fu(s)h Ft(!)1626 2048 y Fn(R)1667 2042 y Fu(t)p Fx(,)100 2092 y(then)i Fu(s)e Ft(\))267 2098 y Fn(R)296 2102 y Ff(1)325 2092 y Fu(t)p Fx(.)100 2192 y Fk(Pr)o(oof.)22 b Fx(W)m(e)10 b(sho)o(w)g(the)h(follo)o(wing)d(stronger)j(claim,)d (where)j(the)g(rewrite)h(relation)e(asso)q(ciated)h(with)100 2242 y(\()p Ft(F)150 2248 y Fr(1)178 2242 y Ft([)d(f)p Fo(2)p Ft(g)p Fu(;)f Ft(R)341 2248 y Fr(1)359 2242 y Fx(\))14 b(is)g(also)f(denoted)i(b)o(y)f Ft(\))771 2248 y Fn(R)800 2252 y Ff(1)817 2242 y Fx(.)100 2341 y Fs(Claim:)f Fx(If)i Fu(s)g Fx(is)g(a)g(blac)o(k)f(term)h(and)g Fu(\033)h Fx(is)f(a)f(top)h(white)g Ft(!)1032 2347 y Fn(R)1077 2341 y Fx(normalized)f(substitution)h(suc)o(h)h(that)100 2391 y Fu(s\033)d Ft(!)198 2397 y Fn(R)239 2391 y Fu(t\033)q Fx(,)h(then)h Fu(s\033)444 2376 y Fb(2)481 2391 y Ft(\))523 2397 y Fn(R)552 2401 y Ff(1)581 2391 y Fu(t\033)621 2376 y Fb(2)647 2391 y Fx(,)e(where)i Fu(\033)817 2376 y Fb(2)854 2391 y Fx(=)d Ft(f)p Fu(x)f Ft(7!)g Fo(2)j Ft(j)f Fu(x)e Ft(2)h(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))q Ft(g)p Fx(.)141 2441 y(Since)h Ft(R)g Fx(is)f(semi-complete,)f(ev)o(ery)i(term)f Fu(t)g Fx(has)h(a)f(unique)h(normal)d(form)h Fu(t)p Ft(#)h Fx(w.r.t.)g Ft(R)p Fx(.)g(F)m(urther-)100 2491 y(more,)g(for)h(an)o(y)g (substitution)g Fu(\033)q Fx(,)g(let)h Fu(\033)q Ft(#)f Fx(denote)i(the)f(substitution)f Ft(f)p Fu(x)e Ft(7!)g Fu(\033)q Fx(\()p Fu(x)p Fx(\))p Ft(#)j(j)f Fu(x)e Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))q Ft(g)p Fx(.)100 2540 y(The)17 b(claim)d(is)j(pro)o(v)o(ed)g(b)o(y)g(induction)f(on)h(the)g (depth)h(of)e Fu(s\033)i Ft(!)e Fu(t\033)q Fx(.)g(The)i(case)f(of)g (zero)g(depth)h(is)100 2590 y(straigh)o(tforw)o(ard.)8 b(Let)i(the)h(depth)f(of)f Fu(s\033)k Ft(!)e Fu(t\033)g Fx(equal)e Fu(d)q Fx(+)q(1,)g Fu(d)i Ft(\025)h Fx(0.)d(There)h(is)g(a)f (con)o(text)i Fu(C)s Fx([)d(],)h(a)g(sub-)100 2640 y(stitution)i Fu(\032)h Fx(:)f Ft(V)k(!)c(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(,)j(and)h(a)g(rewrite)h(rule)g Fu(l)h Ft(!)e Fu(r)h Ft(\()f Fu(s)1094 2646 y Fr(1)1124 2640 y Ft(#)g Fu(t)1171 2646 y Fr(1)1190 2640 y Fu(;)c(:)g(:)g(:)e(;)i(s) 1302 2646 y Fp(n)1335 2640 y Ft(#)k Fu(t)1382 2646 y Fp(n)1416 2640 y Fx(in)g Ft(R)1497 2646 y Fr(1)1527 2640 y Fx(suc)o(h)h(that)p eop %%Page: 28 28 28 27 bop 100 197 a Fw(28)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fu(s\033)i Fx(=)f Fu(C)s Fx([)p Fu(l)q(\032)p Fx(],)d Fu(t\033)j Fx(=)g Fu(C)s Fx([)p Fu(r)q(\032)p Fx(])d(and)h Fu(s)610 305 y Fp(j)628 299 y Fu(\032)h Ft(#)f Fu(t)706 305 y Fp(j)723 299 y Fu(\032)h Fx(is)f(of)g(depth)g Ft(\024)i Fu(d)e Fx(for)g(ev)o(ery)h Fu(j)j Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)i(According)i(to)100 349 y(Prop)q(osition)e(4.18,)e Fu(\032)j Fx(can)g(b)q(e)g(decomp)q(osed)f(in)o(to)g Fu(\032)896 355 y Fr(2)915 349 y Ft(\016)p Fu(\032)957 355 y Fr(1)986 349 y Fx(suc)o(h)h(that)f Fu(\032)1181 355 y Fr(1)1210 349 y Fx(is)g(blac)o(k)g(and)g Fu(\032)1448 355 y Fr(2)1476 349 y Fx(is)h(top)f(white.)100 399 y(Note)15 b(that)g(for)g(ev)o(ery)g(v)n(ariable)f Fu(x)f Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\032)828 405 y Fr(2)848 399 y Fx(\))d Ft(\\)g(V)s Fu(ar)q Fx(\()p Fu(l)q(\032)1032 405 y Fr(1)1052 399 y Fx(\),)15 b(w)o(e)g(ha)o(v)o(e)g Fu(\032)1275 405 y Fr(2)1294 399 y Fx(\()p Fu(x)p Fx(\))e Ft(2)g Fu(N)5 b(F)h Fx(\()p Ft(!)p Fx(\).)14 b(Nev)o(er-)100 448 y(theless,)i(w)o(e)g (do)g(not)g(ha)o(v)o(e)f Fu(\032)564 454 y Fr(2)583 448 y Fx(\()p Fu(x)p Fx(\))g Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(!)p Fx(\))15 b(in)h(general)g(b)q(ecause)h(of)e(p)q(ossible)i(extra) f(v)n(ariables.)100 498 y(Since)g Ft(!)f Fx(is)h(semi-complete,)d Fu(\032)615 504 y Fr(2)649 498 y Ft(!)691 483 y Fn(\003)724 498 y Fu(\032)745 504 y Fr(2)764 498 y Ft(#)p Fx(.)i(Th)o(us)h Fu(\032)940 504 y Fr(2)959 498 y Ft(#)p Fx(\()p Fu(\032)1017 504 y Fr(1)1036 498 y Fx(\()p Fu(s)1071 504 y Fp(j)1089 498 y Fx(\)\))1158 483 y Fn(\003)1175 498 y Ft( )d Fu(s)1249 504 y Fp(j)1267 498 y Fu(\032)j Ft(#)f Fu(t)1355 504 y Fp(j)1373 498 y Fu(\032)g Ft(!)1451 483 y Fn(\003)1484 498 y Fu(\032)1505 504 y Fr(2)1524 498 y Ft(#)p Fx(\()p Fu(\032)1582 504 y Fr(1)1601 498 y Fx(\()p Fu(t)1632 504 y Fp(j)1650 498 y Fx(\)\).)100 548 y(The)20 b(con\015uence)i(of)d Ft(!)h Fx(guaran)o(tees)h Fu(\032)747 554 y Fr(2)766 548 y Ft(#)o Fx(\()p Fu(\032)823 554 y Fr(1)843 548 y Fx(\()p Fu(s)878 554 y Fp(j)896 548 y Fx(\)\))f Ft(#)40 b Fu(\032)1030 554 y Fr(2)1049 548 y Ft(#)p Fx(\()p Fu(\032)1107 554 y Fr(1)1126 548 y Fx(\()p Fu(t)1157 554 y Fp(j)1174 548 y Fx(\)\))21 b(for)e(ev)o(ery)i Fu(j)j Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)100 598 y(By)20 b(Prop)q(osition)g(4.18,)e Fu(\032)527 604 y Fr(2)546 598 y Ft(#)i Fx(can)g(b)q(e)h(decomp)q(osed)f(in)o(to)f Fu(\032)1081 604 y Fr(4)1114 598 y Ft(\016)13 b Fu(\032)1169 604 y Fr(3)1208 598 y Fx(suc)o(h)21 b(that)f Fu(\032)1425 604 y Fr(3)1464 598 y Fx(is)g(blac)o(k)g(and)100 648 y Fu(\032)121 654 y Fr(4)164 648 y Fx(is)k(top)g(white.)g(Eviden)o(tly) m(,)f Fu(\032)663 654 y Fr(3)682 648 y Fx(\()p Fu(\032)719 654 y Fr(1)739 648 y Fx(\()p Fu(s)774 654 y Fp(j)792 648 y Fx(\)\))h(and)g Fu(\032)960 654 y Fr(3)979 648 y Fx(\()p Fu(\032)1016 654 y Fr(1)1036 648 y Fx(\()p Fu(t)1067 654 y Fp(j)1084 648 y Fx(\)\))h(are)f(blac)o(k)g(terms)g(and) g Fu(\032)1577 654 y Fr(4)1621 648 y Fx(is)g(a)100 697 y(top)19 b(white)g Ft(!)g Fx(normalized)e(substitution.)i(Rep)q(eated)i (application)d(of)g(the)i(induction)f(h)o(yp)q(oth-)100 747 y(esis)25 b(yields)f Fu(\032)335 732 y Fb(2)335 758 y Fr(4)361 747 y Fx(\()p Fu(\032)398 753 y Fr(3)417 747 y Fx(\()p Fu(\032)454 753 y Fr(1)473 747 y Fx(\()p Fu(s)508 753 y Fp(j)526 747 y Fx(\)\)\))54 b Ft(+)653 753 y Fn(R)682 757 y Ff(1)753 747 y Fu(\032)774 732 y Fb(2)774 758 y Fr(4)800 747 y Fx(\()p Fu(\032)837 753 y Fr(3)856 747 y Fx(\()p Fu(\032)893 753 y Fr(1)912 747 y Fx(\()p Fu(t)943 753 y Fp(j)961 747 y Fx(\)\)\).)24 b(W)m(e)g(obtain)f(as)i(a)f (consequence)j(that)100 797 y Fu(\032)121 782 y Fb(2)121 807 y Fr(4)146 797 y Fx(\()p Fu(\032)183 803 y Fr(3)203 797 y Fx(\()p Fu(\032)240 803 y Fr(1)259 797 y Fx(\()p Fu(l)q Fx(\)\)\))19 b Ft(\))397 803 y Fn(R)426 807 y Ff(1)462 797 y Fu(\032)483 782 y Fb(2)483 807 y Fr(4)509 797 y Fx(\()p Fu(\032)546 803 y Fr(3)565 797 y Fx(\()p Fu(\032)602 803 y Fr(1)621 797 y Fx(\()p Fu(r)q Fx(\)\)\).)f(Clearly)m (,)f Fu(s\033)j Fx(=)f Fu(C)s Fx([)p Fu(\032)1072 803 y Fr(2)1090 797 y Ft(#)p Fx(\()p Fu(\032)1148 803 y Fr(1)1167 797 y Fx(\()p Fu(l)q Fx(\)\)])f(and)g Fu(t\033)i Fx(=)f Fu(C)s Fx([)p Fu(\032)1519 803 y Fr(2)1537 797 y Ft(#)p Fx(\()p Fu(\032)1595 803 y Fr(1)1614 797 y Fx(\()p Fu(r)q Fx(\)\)])100 847 y(b)q(ecause)h Fu(\032)279 853 y Fr(2)298 847 y Fx(\()p Fu(x)p Fx(\))e Ft(2)h Fu(N)5 b(F)h Fx(\()p Ft(!)p Fx(\))17 b(for)h(ev)o(ery)h Fu(x)g Ft(2)f(D)q Fu(om)p Fx(\()p Fu(\032)978 853 y Fr(2)998 847 y Fx(\))13 b Ft(\\)e(V)s Fu(ar)q Fx(\()p Fu(l)q(\032)1186 853 y Fr(1)1206 847 y Fx(\))q(.)17 b(Let)1340 836 y(^)1331 847 y Fu(C)s Fx([)g(])h(b)q(e)h(the)g(con)o(text)100 897 y(obtained)f(from)f Fu(C)s Fx([)h(])g(b)o(y)g(replacing)g(ev)o(ery) i(white)e(principal)g(subterm)h(whic)o(h)f(m)o(ust)g(b)q(e)h(of)f(the) 100 946 y(form)h Fu(\033)q Fx(\()p Fu(x)p Fx(\))i(for)g(some)f(v)n (ariable)g Fu(x)j Ft(2)g(D)q Fu(om)p Fx(\()p Fu(\033)q Fx(\))g(with)d Fo(2)p Fx(.)h(It)g(is)g(fairly)f(simple)f(to)i(v)o (erify)g(that)100 996 y Fu(s\033)144 981 y Fb(2)188 996 y Fx(=)249 986 y(^)239 996 y Fu(C)s Fx([)p Fu(\032)305 981 y Fb(2)305 1007 y Fr(4)330 996 y Fx(\()p Fu(\032)367 1002 y Fr(3)386 996 y Fx(\()p Fu(\032)423 1002 y Fr(1)443 996 y Fx(\()p Fu(l)q Fx(\)\)\)])d(and)g Fu(t\033)675 981 y Fb(2)719 996 y Fx(=)780 986 y(^)770 996 y Fu(C)s Fx([)p Fu(\032)836 981 y Fb(2)836 1007 y Fr(4)861 996 y Fx(\()p Fu(\032)898 1002 y Fr(3)917 996 y Fx(\()p Fu(\032)954 1002 y Fr(1)974 996 y Fx(\()p Fu(r)q Fx(\)\)\)].)f(Th)o(us)i Fu(s\033)1253 981 y Fb(2)1297 996 y Ft(\))1339 1002 y Fn(R)1368 1006 y Ff(1)1404 996 y Fu(t\033)1444 981 y Fb(2)1488 996 y Fx(and)f(w)o(e)g(are)100 1046 y(done.)13 b Fe(2)526 1147 y Fk(6.2.)24 b(termina)m(tion)16 b(and)g(completeness) 141 1252 y Fx(Middeldorp)g(\(1990,)f(1993\))g(conjectured)j(that)e(the) h(disjoin)o(t)e(union)h(of)f(t)o(w)o(o)h(terminating)f(join)100 1302 y(CTRSs)c(is)f(terminating)g(if)g(one)h(of)f(them)h(con)o(tains)g (neither)g(collapsing)f(nor)h(duplicating)f(rules)i(and)100 1352 y(the)g(other)h(is)f(con\015uen)o(t.)g(The)g(next)h(example)d (dispro)o(v)o(es)j(this)f(conjecture.)h(The)f(function)g(sym)o(b)q(ols) 100 1401 y(ha)o(v)o(e)h(b)q(een)i(c)o(hosen)g(in)f(resem)o(blance)g(to) f(other)i(kno)o(wn)e(coun)o(terexamples.)100 1505 y Fk(Example)k(6.16.) k Fx(Let)346 1658 y Ft(R)381 1664 y Fr(1)411 1658 y Fx(=)455 1573 y Fg(8)455 1610 y(<)455 1685 y(:)505 1613 y Fx(0)168 b(1)500 1709 y Fu(A)600 1708 y Fx(2)688 1709 y Fu(B)p 515 1671 2 48 v 516 1671 a Fa(?)539 1665 y(@)545 1671 y(@)-42 b(R)640 1665 y(\000)634 1671 y(\000)g(\011)p 704 1671 V 29 w(?)879 1658 y Fu(F)6 b Fx(\()p Fu(x)p Fx(\))11 b Ft(!)g Fu(F)6 b Fx(\()p Fu(x)p Fx(\))11 b Ft(\()g Fu(x)j Ft(#)f Fu(A;)21 b(x)13 b Ft(#)h Fu(B)100 1781 y Fx(and)671 1860 y Ft(R)706 1866 y Fr(2)736 1860 y Fx(=)780 1801 y Fg(\032)832 1834 y Fu(g)8 b Fx(\()p Fu(x;)f(y)q(;)g(y)q Fx(\))12 b Ft(!)f Fu(x)832 1884 y(g)d Fx(\()p Fu(y)q(;)f(y)q(;)g(x)p Fx(\))12 b Ft(!)f Fu(x:)100 1986 y Fx(Clearly)m(,)f Ft(R)286 1992 y Fr(1)316 1986 y Fx(is)j(non-collapsing,)d(non-duplicating,)g(and)i(terminating)f (\(there)j(is)e(no)g Fu(t)f Ft(2)g(T)g Fx(\()p Ft(F)1612 1992 y Fr(1)1630 1986 y Fu(;)c Ft(V)s Fx(\))100 2036 y(whic)o(h)15 b(rewrites)i(to)f(b)q(oth)f Fu(A)h Fx(and)f Fu(B)r Fx(\).)h(Note)g(that)g Ft(R)966 2042 y Fr(1)1000 2036 y Fx(is)g(not)f(con\015uen)o(t.)h(Moreo)o(v)o(er,)g(the)g(CTRS)100 2086 y Ft(R)135 2092 y Fr(2)169 2086 y Fx(is)f(eviden)o(tly)h (terminating)d(and)j(con\015uen)o(t.)f(Ho)o(w)o(ev)o(er,)h(their)g (disjoin)o(t)e(union)h Ft(R)f Fx(=)h Ft(R)1573 2092 y Fr(1)1602 2086 y Ft(])10 b(R)1675 2092 y Fr(2)100 2135 y Fx(is)j(not)h(terminating.)e(Since)321 2212 y Fu(B)368 2218 y Fn(R)397 2212 y Ft( )h Fx(1)487 2218 y Fn(R)515 2212 y Ft( )g Fu(g)q Fx(\(0)p Fu(;)7 b Fx(0)p Fu(;)g Fx(1\))j Ft(!)776 2218 y Fn(R)818 2212 y Fu(g)q Fx(\(0)p Fu(;)d Fx(2)p Fu(;)g Fx(1\))j Ft(!)1024 2218 y Fn(R)1066 2212 y Fu(g)q Fx(\(0)p Fu(;)d Fx(2)p Fu(;)g Fx(2\))j Ft(!)1272 2218 y Fn(R)1314 2212 y Fx(0)h Ft(!)1388 2218 y Fn(R)1430 2212 y Fu(A;)100 2288 y Fx(there)k(is)f(the)g(cyclic)g (reduction)h("sequence")g Fu(F)6 b Fx(\()p Fu(g)q Fx(\(0)p Fu(;)h Fx(0)p Fu(;)g Fx(1\)\))13 b Ft(!)1105 2294 y Fn(R)1149 2288 y Fu(F)6 b Fx(\()p Fu(g)q Fx(\(0)p Fu(;)h Fx(0)p Fu(;)g Fx(1\)\).)141 2391 y(Note)20 b(that)g(the)g(ab)q(o)o(v)o(e)f (example)f(also)h(sho)o(ws)h(\(the)g(kno)o(wn)f(fact\))h(that)g (termination)d(is)j(not)100 2441 y(mo)q(dular)8 b(for)i (non-duplicating)e(disjoin)o(t)h(CTRSs.)h(Middeldorp)f(\(1990,)g (1993\))g(has)i(giv)o(en)e(su\016cien)o(t)100 2491 y(conditions)14 b(for)g(the)h(mo)q(dularit)o(y)d(of)h(termination.)f(It)j(will)e(next)i (b)q(e)g(sho)o(wn)f(that)g(his)h(results)g(also)100 2540 y(hold,)f Fs(mutatis)i(mutandis)p Fx(,)f(in)g(the)h(presence)i(of)c (shared)i(constructors.)h(W)m(e)e(emphasize)g(that)h(our)100 2590 y(pro)q(of)d(is)h(considerably)g(simpler)f(than)g(that)h(of)g (Middeldorp)f(\(1990,)g(1993\).)141 2640 y(As)j(in)g(the)g(previous)g (subsection,)h(let)f(\()p Ft(F)822 2646 y Fr(1)840 2640 y Fu(;)7 b Ft(R)894 2646 y Fr(1)913 2640 y Fx(\))16 b(and)f(\()p Ft(F)1077 2646 y Fr(2)1096 2640 y Fu(;)7 b Ft(R)1150 2646 y Fr(2)1168 2640 y Fx(\))16 b(b)q(e)h(constructor-sharing)g(join)p eop %%Page: 29 29 29 28 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(29)p 100 224 1595 2 v 100 299 a Fx(CTRSs.)15 b(It)g(is)g(not)h(di\016cult)e(to)i (v)o(erify)f(that)g(the)h(CTRS)f(\()p Ft(F)1083 305 y Fp(i)1107 299 y Ft([)10 b(f)p Fo(2)p Ft(g)p Fu(;)d Ft(R)1271 305 y Fp(i)1285 299 y Fx(\))15 b(is)g(terminating)f(if)h(and)100 349 y(only)c(if)g(\()p Ft(F)270 355 y Fp(i)283 349 y Fu(;)c Ft(R)337 355 y Fp(i)351 349 y Fx(\))12 b(is)g(terminating.)e (Again,)g(w)o(e)j(also)e(denote)i(the)g(rewrite)g(relation)e(asso)q (ciated)i(with)100 399 y(\()p Ft(F)144 405 y Fp(i)167 399 y Ft([)c(f)p Fo(2)p Ft(g)p Fu(;)e Ft(R)330 405 y Fp(i)344 399 y Fx(\))14 b(b)o(y)g Ft(\))474 405 y Fn(R)503 409 y Fl(i)531 399 y Fx(\(b)o(y)f(abuse)i(of)e(notation\).)100 497 y Fk(Pr)o(oposition)j(6.17.)21 b Fx(Let)14 b(\()p Ft(F)607 503 y Fr(2)626 497 y Fu(;)7 b Ft(R)679 503 y Fr(2)698 497 y Fx(\))14 b(b)q(e)h(la)o(y)o(er-preserving.)125 593 y(\(1\))21 b(If)14 b Fu(s)e Ft(!)314 578 y Fp(o)343 593 y Fu(t)i Fx(b)o(y)g(some)f(rule)h(from)e Ft(R)750 599 y Fr(1)769 593 y Fx(,)h(then)i Fu(top)945 578 y Fp(b)961 593 y Fx(\()p Fu(s)p Fx(\))e Ft(\))1067 599 y Fn(R)1096 603 y Ff(1)1125 593 y Fu(top)1181 578 y Fp(b)1197 593 y Fx(\()p Fu(t)p Fx(\).)125 640 y(\(2\))21 b(If)14 b Fu(s)e Ft(!)314 625 y Fp(o)343 640 y Fu(t)i Fx(b)o(y)g(some)f(rule)h (from)e Ft(R)750 646 y Fr(2)783 640 y Fx(or)i Fu(s)d Ft(!)906 625 y Fp(i)931 640 y Fu(t)p Fx(,)j(then)g Fu(top)1122 625 y Fp(b)1139 640 y Fx(\()p Fu(s)p Fx(\))e(=)g Fu(top)1302 625 y Fp(b)1319 640 y Fx(\()p Fu(t)p Fx(\).)100 738 y Fk(Pr)o(oof.)22 b Fx(W)m(e)c(pro)q(ceed)j(b)o(y)e(induction)g(on)f(the) i(depth)g(of)f Fu(s)h Ft(!)g Fu(t)p Fx(.)f(The)g(case)h(of)f(zero)h (depth)g(is)100 788 y(straigh)o(tforw)o(ard.)13 b(So)h(supp)q(ose)i (that)f(the)g(depth)g(of)f Fu(s)g Ft(!)e Fu(t)i Fx(equals)h Fu(d)9 b Fx(+)h(1,)k Fu(d)e Ft(\025)h Fx(0.)h(The)h(induction)100 837 y(h)o(yp)q(othesis)c(co)o(v)o(ers)h(the)g(statemen)o(t)f(that)g Fu(u)g Ft(!)g Fu(v)i Fx(implies)c Fu(top)1083 822 y Fp(b)1099 837 y Fx(\()p Fu(u)p Fx(\))j Ft(\))1209 822 y Fn(\003)1209 849 y(R)1238 853 y Ff(1)1267 837 y Fu(top)1323 822 y Fp(b)1339 837 y Fx(\()p Fu(v)q Fx(\))g(whenev)o(er)h Fu(u)e Ft(!)g Fu(v)100 887 y Fx(is)i(of)h(depth)g(less)h(than)f(or)g (equal)f(to)h Fu(d)p Fx(.)125 984 y(\(1\))21 b(If)11 b Fu(s)h Ft(!)311 969 y Fp(o)340 984 y Fu(t)f Fx(b)o(y)g(some)f(rule)i (from)d Ft(R)733 990 y Fr(1)752 984 y Fx(,)h(then)i Fu(s)g Fx(=)g Fu(C)974 969 y Fp(b)990 984 y Ft(f)-14 b(f)p Fu(u)1042 990 y Fr(1)1060 984 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1177 990 y Fp(p)1195 984 y Ft(g)-14 b(g)11 b Fx(and)g Fu(t)g Fx(=)1391 973 y(^)1382 984 y Fu(C)1415 969 y Fp(b)1431 984 y Ft(h)-7 b(h)q Fu(u)1481 990 y Fp(i)1493 994 y Ff(1)1510 984 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)1627 990 y Fp(i)1639 994 y Fl(q)1657 984 y Ft(i)-7 b(i)p Fx(,)199 1034 y(where)22 b Fu(i)340 1040 y Fr(1)359 1034 y Fu(;)7 b(:)g(:)g(:)e(;)i(i)466 1040 y Fp(q)507 1034 y Ft(2)23 b(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)e(;)i(p) p Ft(g)p Fx(.)19 b(Moreo)o(v)o(er,)i(there)i(is)d(a)h(con)o(text)h Fu(C)s Fx([)e(],)g(a)g(substitution)199 1083 y Fu(\033)g Fx(and)f(a)g(rewrite)h(rule)f Fu(l)j Ft(!)d Fu(r)j Ft(\()d Fu(s)819 1089 y Fr(1)857 1083 y Ft(#)g Fu(t)912 1089 y Fr(1)931 1083 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)1042 1089 y Fp(n)1084 1083 y Ft(#)19 b Fu(t)1139 1089 y Fp(n)1181 1083 y Ft(2)h(R)1264 1089 y Fr(1)1302 1083 y Fx(suc)o(h)g(that)f Fu(s)i Fx(=)f Fu(C)s Fx([)p Fu(l)q(\033)q Fx(],)199 1133 y Fu(t)12 b Fx(=)g Fu(C)s Fx([)p Fu(r)q(\033)q Fx(])f(and)i Fu(s)482 1139 y Fp(j)500 1133 y Fu(\033)h Ft(#)f Fu(t)587 1139 y Fp(j)604 1133 y Fu(\033)i Fx(is)d(of)h(depth)g(less)h(than)f(or) g(equal)g(to)g Fu(d)f Fx(for)h(ev)o(ery)g Fu(j)h Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)199 1183 y(W)m(e)23 b(\014rst)i(sho)o(w)e(that)h Fu(top)644 1168 y Fp(b)661 1183 y Fx(\()p Fu(s)696 1189 y Fp(j)714 1183 y Fu(\033)q Fx(\))k Ft(+)808 1189 y Fn(R)837 1193 y Ff(1)883 1183 y Fu(top)939 1168 y Fp(b)955 1183 y Fx(\()p Fu(t)986 1189 y Fp(j)1004 1183 y Fu(\033)q Fx(\))c(for)f(ev)o(ery)i Fu(j)30 b Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)23 b(Fix)g Fu(j)r Fx(.)199 1233 y(Let)e Fu(w)h Fx(b)q(e)f(the)g(common)d(reduct)k(of)e Fu(s)855 1239 y Fp(j)873 1233 y Fu(\033)h Fx(and)g Fu(t)1021 1239 y Fp(j)1038 1233 y Fu(\033)q Fx(.)g(Clearly)m(,)e(it)h(su\016ces)i(to)e (sho)o(w)h(that)199 1283 y Fu(top)255 1268 y Fp(b)272 1283 y Fx(\()p Fu(s)307 1289 y Fp(j)325 1283 y Fu(\033)q Fx(\))15 b Ft(\))423 1268 y Fn(\003)423 1294 y(R)452 1298 y Ff(1)483 1283 y Fu(top)539 1268 y Fp(b)556 1283 y Fx(\()p Fu(w)q Fx(\))h(and)f Fu(top)773 1268 y Fp(b)790 1283 y Fx(\()p Fu(t)821 1289 y Fp(j)838 1283 y Fu(\033)q Fx(\))g Ft(\))936 1268 y Fn(\003)936 1294 y(R)965 1298 y Ff(1)997 1283 y Fu(top)1053 1268 y Fp(b)1069 1283 y Fx(\()p Fu(w)q Fx(\).)h(W.l.o.g.)c(w)o(e)k(consider)g(only)f(the)199 1332 y(former)h(claim.)e(The)j(claim)e(is)i(pro)o(v)o(ed)g(b)o(y)f (induction)h(on)f(the)i(length)e(of)h Fu(s)1427 1338 y Fp(j)1444 1332 y Fu(\033)h Ft(!)1528 1317 y Fn(\003)1563 1332 y Fu(w)q Fx(.)e(The)199 1382 y(case)21 b(of)d(zero)j(length)e(is)g (trivial,)f(so)h(let)h Fu(s)900 1388 y Fp(j)918 1382 y Fu(\033)i Ft(!)e Fu(v)i Ft(!)1110 1367 y Fp(l)1143 1382 y Fu(w)e Fx(with)g Fu(l)h Ft(\025)h Fx(0.)c(The)i(induction)199 1432 y(h)o(yp)q(othesis)15 b(\(on)g Fu(l)q Fx(\))g(yields)f Fu(top)696 1417 y Fp(b)713 1432 y Fx(\()p Fu(v)q Fx(\))f Ft(\))821 1417 y Fn(\003)821 1444 y(R)850 1448 y Ff(1)880 1432 y Fu(top)936 1417 y Fp(b)953 1432 y Fx(\()p Fu(w)q Fx(\).)h(F)m(urthermore,)g(the)h(induction)f(h)o(yp)q(oth-)199 1487 y(esis)i(\(on)e Fu(d)p Fx(\))g(yields)h Fu(top)577 1472 y Fp(b)593 1487 y Fx(\()p Fu(s)628 1493 y Fp(j)647 1487 y Fu(\033)q Fx(\))e Ft(\))743 1472 y Fn(\003)743 1498 y(R)772 1502 y Ff(1)802 1487 y Fu(top)858 1472 y Fp(b)874 1487 y Fx(\()p Fu(v)q Fx(\).)i(This)g(pro)o(v)o(es)g(the)g (claim.)d(Th)o(us)j Fu(top)1536 1472 y Fp(b)1553 1487 y Fx(\()p Fu(w)q Fx(\))f(is)h(a)199 1542 y(common)f(reduct)19 b(of)d Fu(top)607 1527 y Fp(b)624 1542 y Fx(\()p Fu(s)659 1548 y Fp(j)677 1542 y Fu(\033)q Fx(\))h(and)g Fu(top)875 1527 y Fp(b)892 1542 y Fx(\()p Fu(t)923 1548 y Fp(j)941 1542 y Fu(\033)q Fx(\))g(w.r.t.)f Ft(\))1155 1548 y Fn(R)1184 1552 y Ff(1)1201 1542 y Fx(.)h(According)g(to)g(Prop)q(osition)199 1592 y(4.18,)12 b Fu(\033)h Fx(=)f Fu(\033)403 1598 y Fr(2)430 1592 y Ft(\016)c Fu(\033)483 1598 y Fr(1)502 1592 y Fx(,)13 b(where)h Fu(\033)670 1598 y Fr(1)702 1592 y Fx(is)g(a)f(blac)o(k)g(substitution)h(and)g Fu(\033)1222 1598 y Fr(2)1253 1592 y Fx(is)g(top)f(white.)h(Recall)f(that)199 1642 y Fu(\033)224 1627 y Fb(2)223 1652 y Fr(2)265 1642 y Fx(denotes)k(the)e(substitution)h Ft(f)p Fu(x)d Ft(7!)g Fo(2)j Ft(j)e Fu(x)g Ft(2)f(D)q Fu(om)p Fx(\()p Fu(\033)1116 1648 y Fr(2)1135 1642 y Fx(\))q Ft(g)p Fx(.)h(It)h(is)h(clear)f(that)g Fu(top)1536 1627 y Fp(b)1553 1642 y Fx(\()p Fu(s)1588 1648 y Fp(j)1606 1642 y Fu(\033)q Fx(\))f(=)199 1691 y Fu(\033)224 1676 y Fb(2)223 1702 y Fr(2)250 1691 y Fx(\()p Fu(\033)290 1697 y Fr(1)308 1691 y Fx(\()p Fu(s)343 1697 y Fp(j)361 1691 y Fx(\)\))g(and)f Fu(top)543 1676 y Fp(b)560 1691 y Fx(\()p Fu(t)591 1697 y Fp(j)609 1691 y Fu(\033)q Fx(\))e(=)h Fu(\033)730 1676 y Fb(2)729 1702 y Fr(2)756 1691 y Fx(\()p Fu(\033)796 1697 y Fr(1)814 1691 y Fx(\()p Fu(t)845 1697 y Fp(j)863 1691 y Fx(\)\).)h(Hence)i Fu(\033)1068 1676 y Fb(2)1067 1702 y Fr(2)1094 1691 y Fx(\()p Fu(\033)1134 1697 y Fr(1)1152 1691 y Fx(\()p Fu(s)1187 1697 y Fp(j)1205 1691 y Fx(\)\))d Ft(+)1274 1697 y Fn(R)1303 1701 y Ff(1)1333 1691 y Fu(\033)1358 1676 y Fb(2)1357 1702 y Fr(2)1383 1691 y Fx(\()p Fu(\033)1423 1697 y Fr(1)1442 1691 y Fx(\()p Fu(t)1473 1697 y Fp(j)1490 1691 y Fx(\)\))i(and)f(th)o(us)199 1747 y Fu(\033)224 1732 y Fb(2)223 1757 y Fr(2)250 1747 y Fx(\()p Fu(\033)290 1753 y Fr(1)308 1747 y Fx(\()p Fu(l)q Fx(\)\))g Ft(\))424 1753 y Fn(R)453 1757 y Ff(1)482 1747 y Fu(\033)507 1732 y Fb(2)506 1757 y Fr(2)533 1747 y Fx(\()p Fu(\033)573 1753 y Fr(1)591 1747 y Fx(\()p Fu(r)q Fx(\)\).)h(Let)769 1736 y(^)759 1747 y Fu(C)s Fx([)g(])f(b)q(e)i(the)f(con)o(text)h (obtained)f(from)e Fu(C)s Fx([)h(])h(b)o(y)g(replacing)199 1797 y(all)g(white)i(principal)e(subterms)i(with)f Fo(2)p Fx(.)g(No)o(w)g(\(1\))g(follo)o(ws)f(from)g Fu(top)1339 1782 y Fp(b)1355 1797 y Fx(\()p Fu(s)p Fx(\))h(=)1477 1786 y(^)1467 1797 y Fu(C)s Fx([)p Fu(\033)1537 1782 y Fb(2)1536 1807 y Fr(2)1562 1797 y Fx(\()p Fu(\033)1602 1803 y Fr(1)1620 1797 y Fx(\()p Fu(l)q Fx(\)\)])199 1846 y(and)f Fu(top)336 1831 y Fp(b)353 1846 y Fx(\()p Fu(t)p Fx(\))d(=)465 1836 y(^)455 1846 y Fu(C)s Fx([)p Fu(\033)525 1831 y Fb(2)524 1857 y Fr(2)550 1846 y Fx(\()p Fu(\033)590 1852 y Fr(1)609 1846 y Fx(\()p Fu(r)q Fx(\)\)].)125 1893 y(\(2\))21 b(Let)16 b Fu(s)d Ft(!)349 1878 y Fp(o)381 1893 y Fu(t)i Fx(b)o(y)f(some)g(rule)i(from)d Ft(R)793 1899 y Fr(2)827 1893 y Fx(or)i Fu(s)e Ft(!)953 1878 y Fp(i)980 1893 y Fu(t)p Fx(.)h(Since)i Ft(R)1166 1899 y Fr(2)1200 1893 y Fx(is)e(la)o(y)o(er-preserving,)h(w)o(e)h(ma)o(y)199 1943 y(write)g Fu(s)e Fx(=)h Fu(C)420 1928 y Fp(b)436 1943 y Ft(h)-7 b(h)p Fu(u)485 1949 y Fr(1)504 1943 y Fu(;)7 b(:)g(:)g(:)e(;)i(u)621 1949 y Fp(j)637 1943 y Fu(;)g(:)g(:)g(:)e(;)i(u)754 1949 y Fp(p)772 1943 y Ft(i)-7 b(i)16 b Fx(and)f Fu(t)f Fx(=)h Fu(C)1004 1928 y Fp(b)1020 1943 y Ft(h)-7 b(h)p Fu(u)1069 1949 y Fr(1)1088 1943 y Fu(;)7 b(:)g(:)g(:)t(;)g(u)1204 1928 y Fn(0)1204 1954 y Fp(j)1221 1943 y Fu(;)g(:)g(:)g(:)e(;)i(u)1338 1949 y Fp(p)1356 1943 y Ft(i)-7 b(i)q Fx(,)15 b(where)h Fu(u)1554 1949 y Fp(j)1585 1943 y Ft(!)e Fu(u)1665 1928 y Fn(0)1665 1954 y Fp(j)1682 1943 y Fx(.)199 1999 y(Hence)i Fu(top)379 1984 y Fp(b)395 1999 y Fx(\()p Fu(s)p Fx(\))d(=)e Fu(top)558 1984 y Fp(b)575 1999 y Fx(\()p Fu(t)p Fx(\).)100 2097 y Fe(2)141 2195 y Fx(In)i(the)h(preceding)g(prop)q(osition,)f(the)g (assumption)f(that)i(\()p Ft(F)1108 2201 y Fr(2)1126 2195 y Fu(;)7 b Ft(R)1180 2201 y Fr(2)1199 2195 y Fx(\))13 b(has)g(to)g(b)q(e)h(la)o(y)o(er-preserving)100 2245 y(cannot)g(b)q(e)g(dropp)q(ed,)g(as)g(is)g(witnessed)h(b)o(y)f(the)h (next)f(example)f(\(cf.)g(Middeldorp,)g(1990,)g(1993\).)100 2343 y Fk(Example)k(6.18.)k Fx(Let)e Ft(R)526 2349 y Fr(1)565 2343 y Fx(=)h Ft(f)p Fu(F)6 b Fx(\()p Fu(x)p Fx(\))19 b Ft(!)h Fu(G)p Fx(\()p Fu(x)p Fx(\))f Ft(\()h Fu(x)e Ft(#)h Fu(A)p Ft(g)g Fx(and)f Ft(R)1251 2349 y Fr(2)1290 2343 y Fx(=)i Ft(f)p Fu(h)p Fx(\()p Fu(x)p Fx(\))g Ft(!)f Fu(x)p Ft(g)p Fx(.)f(Then)100 2393 y Fu(F)6 b Fx(\()p Fu(h)p Fx(\()p Fu(A)p Fx(\)\))17 b Ft(!)311 2378 y Fp(o)347 2393 y Fu(G)p Fx(\()p Fu(h)p Fx(\()p Fu(A)p Fx(\)\))h(b)o(y)f(the)i(only)e(rule)g(of)g Ft(R)922 2399 y Fr(1)958 2393 y Fx(but)h Fu(top)1094 2378 y Fp(b)1110 2393 y Fx(\()p Fu(F)6 b Fx(\()p Fu(h)p Fx(\()p Fu(a)p Fx(\)\)\)\))18 b(=)h Fu(F)6 b Fx(\()p Fo(2)p Fx(\))17 b(is)g(a)h(normal)100 2442 y(form)12 b(w.r.t.)h Ft(\))351 2448 y Fn(R)380 2452 y Ff(1)397 2442 y Fx(.)141 2540 y(Our)i(next)g(goal)e(is)i(to)f(sho)o(w)g(an)g(analogous)g(statemen)o (t)g(to)g(Prop)q(osition)g(6.17)f(\(1\))i(without)f(the)100 2590 y(la)o(y)o(er-preservingness)i(requiremen)o(t)e(on)g(\()p Ft(F)814 2596 y Fr(2)832 2590 y Fu(;)7 b Ft(R)886 2596 y Fr(2)905 2590 y Fx(\))14 b(but)g(under)i(the)e(additional)f (assumption)g(that)100 2640 y Ft(!)142 2646 y Fr(1)p Fp(;)p Fr(2)200 2640 y Fx(is)h(semi-complete.)p eop %%Page: 30 30 30 29 bop 100 197 a Fw(30)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fk(Definition)16 b(6.19.)21 b Fx(Let)16 b(the)h(rewrite)f(relation)g Ft(!)940 305 y Fr(1)p Fp(;)p Fr(2)1000 299 y Fx(b)q(e)h(semi-complete.)c(F)m(or)j (ev)o(ery)g(term)g Fu(t)e Fx(=)100 349 y Fu(C)133 334 y Fp(b)149 349 y Ft(h)-7 b(h)p Fu(t)189 355 y Fr(1)208 349 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)316 355 y Fp(m)347 349 y Ft(i)-7 b(i)p Fx(,)13 b(w)o(e)i(de\014ne)f Fu(top)634 334 y Fp(b)634 359 y Fn(!)670 349 y Fx(\()p Fu(t)p Fx(\))g(b)o(y:)610 425 y Fu(top)666 408 y Fp(b)666 435 y Fn(!)701 425 y Fx(\()p Fu(t)p Fx(\))e(=)g Fu(top)860 408 y Fp(b)877 425 y Fx(\()p Fu(C)926 408 y Fp(b)942 425 y Ft(h)p Fu(t)973 408 y Fn(!)973 435 y Fr(1)1008 425 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)1116 408 y Fn(!)1116 435 y Fp(m)1151 425 y Ft(i)p Fx(\))141 527 y(In)15 b(other)h(w)o(ords,)e(\014rst)i(the)g(white)f(principal)f (subterms)h(in)f Fu(t)h Fx(are)h(replaced)g(with)e(their)i(unique)100 577 y Ft(!)142 583 y Fr(1)p Fp(;)p Fr(2)201 577 y Fx(normal)c(form,)g (and)i(then)i(the)f(topmost)e(blac)o(k)h(homogeneous)f(part)i(of)f(the) h(term)f(obtained)100 627 y(is)f(tak)o(en.)100 730 y Fk(Lemma)j(6.20.)21 b Fx(Let)d Ft(!)490 736 y Fr(1)p Fp(;)p Fr(2)551 730 y Fx(b)q(e)f(semi-complete.)e(If)h Fu(s;)7 b(t)17 b Fx(are)g(blac)o(k)f(terms)g(and)h Fu(\033)h Fx(is)e(a)h(top)f(white)100 779 y(substitution)e(suc)o(h)g(that)g Fu(s\033)f Ft(!)612 764 y Fp(o)642 779 y Fu(t\033)i Fx(b)o(y)f(some)f (rule)h(from)e Ft(R)1074 785 y Fr(1)1093 779 y Fx(,)h(then)i Fu(\033)1238 764 y Fn(!)1273 779 y Fx(\()p Fu(s)p Fx(\))d Ft(!)1378 764 y Fp(o)1378 790 y Fr(1)1408 779 y Fu(\033)1433 764 y Fn(!)1468 779 y Fx(\()p Fu(t)p Fx(\).)100 881 y Fk(Pr)o(oof.)22 b Fx(There)i(is)g(a)f(con)o(text)i Fu(C)s Fx([)d(],)h(a)h(substitution)f Fu(\032)29 b Fx(:)e Ft(V)32 b(!)27 b(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))25 b(and)e(a)h(rewrite)100 930 y(rule)h Fu(l)32 b Ft(!)f Fu(r)h Ft(\()e Fu(s)453 936 y Fr(1)498 930 y Ft(#)25 b Fu(t)559 936 y Fr(1)577 930 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)689 936 y Fp(n)737 930 y Ft(#)25 b Fu(t)798 936 y Fp(n)852 930 y Ft(2)30 b(R)945 936 y Fr(1)990 930 y Fx(suc)o(h)c(that)f Fu(s\033)33 b Fx(=)e Fu(C)s Fx([)p Fu(l)q(\032)p Fx(],)24 b Fu(t\033)32 b Fx(=)g Fu(C)s Fx([)p Fu(r)q(\032)p Fx(])100 980 y(and)27 b Fu(s)213 986 y Fp(j)230 980 y Fu(\032)h Ft(#)f Fu(t)342 986 y Fp(j)359 980 y Fu(\032)h Fx(for)e Fu(j)36 b Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)e(;)i(n)p Ft(g)p Fx(.)25 b(Fix)i Fu(j)r Fx(.)g(F)m(rom)e(Prop)q(osition)i(6.12)f (w)o(e)h(kno)o(w)g(that)100 1030 y Fu(s)119 1036 y Fp(j)137 1030 y Fu(\032)21 b Ft(#)199 1040 y Fr(1)p Fp(;)p Fr(2)265 1030 y Fu(t)280 1036 y Fp(j)297 1030 y Fu(\032)p Fx(.)g(According)g(to) f(Prop)q(osition)g(4.18,)f Fu(\032)i Fx(can)g(b)q(e)g(decomp)q(osed)f (in)o(to)g Fu(\032)1486 1036 y Fr(2)1519 1030 y Ft(\016)13 b Fu(\032)1574 1036 y Fr(1)1614 1030 y Fx(suc)o(h)100 1080 y(that)22 b Fu(\032)219 1086 y Fr(1)260 1080 y Fx(is)g(blac)o(k)f (and)h Fu(\032)536 1086 y Fr(2)577 1080 y Fx(is)g(top)g(white.)f(Since) i Ft(!)1000 1086 y Fr(1)p Fp(;)p Fr(2)1066 1080 y Fx(is)f (semi-complete,)e(it)i(follo)o(ws)e(as)i(in)100 1130 y(the)g(pro)q(of)g(of)f(Lemma)f(6.9)h(that)h Fu(\032)702 1115 y Fn(!)702 1140 y Fr(2)738 1130 y Fx(\()p Fu(\032)775 1136 y Fr(1)794 1130 y Fx(\()p Fu(s)829 1136 y Fp(j)847 1130 y Fx(\)\))g Ft(#)922 1140 y Fr(1)p Fp(;)p Fr(2)989 1130 y Fu(\032)1010 1115 y Fn(!)1010 1140 y Fr(2)1046 1130 y Fx(\()p Fu(\032)1083 1136 y Fr(1)1102 1130 y Fx(\()p Fu(t)1133 1136 y Fp(j)1151 1130 y Fx(\)\).)f(Applying)h(Lemma)d(6.10)i (to)100 1180 y(the)k(blac)o(k)f(terms)g Fu(\032)448 1186 y Fr(1)467 1180 y Fx(\()p Fu(s)502 1186 y Fr(1)521 1180 y Fx(\))p Fu(;)7 b(:)g(:)g(:)e(;)i(\032)651 1186 y Fr(1)670 1180 y Fx(\()p Fu(s)705 1186 y Fp(n)728 1180 y Fx(\))p Fu(;)g(\032)784 1186 y Fr(1)802 1180 y Fx(\()p Fu(t)833 1186 y Fr(1)852 1180 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i(\032)982 1186 y Fr(1)1001 1180 y Fx(\()p Fu(t)1032 1186 y Fp(n)1054 1180 y Fx(\))25 b(and)f(the)h(substitution)g Fu(\032)1531 1164 y Fn(!)1531 1190 y Fr(2)1591 1180 y Fx(yields)100 1229 y Fu(\032)121 1214 y Fn(!)121 1240 y Fr(2)156 1229 y Fx(\()p Fu(\032)193 1235 y Fr(1)212 1229 y Fx(\()p Fu(s)247 1235 y Fp(j)266 1229 y Fx(\)\))19 b Ft(#)337 1211 y Fp(o)337 1240 y Fr(1)375 1229 y Fu(\032)396 1214 y Fn(!)396 1240 y Fr(2)431 1229 y Fx(\()p Fu(\032)468 1235 y Fr(1)488 1229 y Fx(\()p Fu(t)519 1235 y Fp(j)536 1229 y Fx(\)\).)g(Therefore,)g Fu(\032)824 1214 y Fn(!)824 1240 y Fr(2)860 1229 y Fx(\()p Fu(\032)897 1235 y Fr(1)916 1229 y Fx(\()p Fu(l)q Fx(\)\))h Ft(!)1039 1214 y Fp(o)1039 1240 y Fr(1)1096 1229 y Fu(\032)1117 1214 y Fn(!)1117 1240 y Fr(2)1153 1229 y Fx(\()p Fu(\032)1190 1235 y Fr(1)1209 1229 y Fx(\()p Fu(r)q Fx(\)\).)e(Let)1396 1219 y(^)1387 1229 y Fu(C)r Fx([)h(])f(b)q(e)h(the)h(con-)100 1279 y(text)c(obtained)g(from)e Fu(C)s Fx([)h(])g(b)o(y)h(replacing)f(all)g (white)h(principal)f(subterms)h(with)f(their)i(resp)q(ectiv)o(e)100 1329 y Ft(!)142 1335 y Fr(1)p Fp(;)p Fr(2)204 1329 y Fx(normal)f(form.)g(It)i(is)f(clear)i(that)f Fu(\033)780 1314 y Fn(!)815 1329 y Fx(\()p Fu(s)p Fx(\))h(=)945 1318 y(^)936 1329 y Fu(C)r Fx([)p Fu(\032)1001 1314 y Fn(!)1001 1339 y Fr(2)1037 1329 y Fx(\()p Fu(\032)1074 1335 y Fr(1)1093 1329 y Fx(\()p Fu(l)q Fx(\)\)])f(and)g Fu(\033)1294 1314 y Fn(!)1329 1329 y Fx(\()p Fu(t)p Fx(\))h(=)1455 1318 y(^)1445 1329 y Fu(C)s Fx([)p Fu(\032)1511 1314 y Fn(!)1511 1339 y Fr(2)1546 1329 y Fx(\()p Fu(\032)1583 1335 y Fr(1)1602 1329 y Fx(\()p Fu(r)q Fx(\)\)].)100 1379 y(Th)o(us)14 b Fu(\033)230 1364 y Fn(!)265 1379 y Fx(\()p Fu(s)p Fx(\))e Ft(!)370 1364 y Fp(o)370 1389 y Fr(1)400 1379 y Fu(\033)425 1364 y Fn(!)460 1379 y Fx(\()p Fu(t)p Fx(\).)i Fe(2)100 1480 y Fk(Pr)o(oposition)i(6.21.)21 b Fx(Let)16 b Ft(!)601 1486 y Fr(1)p Fp(;)p Fr(2)661 1480 y Fx(b)q(e)g(semi-complete.)d(If)i Fu(s)f Ft(!)1120 1465 y Fp(o)1153 1480 y Fu(t)h Fx(b)o(y)g(some)g(rule) g(from)f Ft(R)1567 1486 y Fr(1)1586 1480 y Fx(,)h(then)100 1530 y Fu(top)156 1515 y Fp(b)156 1540 y Fn(!)191 1530 y Fx(\()p Fu(s)p Fx(\))d Ft(\))296 1536 y Fn(R)325 1540 y Ff(1)354 1530 y Fu(top)410 1515 y Fp(b)410 1540 y Fn(!)445 1530 y Fx(\()p Fu(t)p Fx(\).)100 1631 y Fk(Pr)o(oof.)22 b Fx(W)m(e)c(ma)o(y)e(write)j Fu(s)h Fx(=)f Fu(C)668 1616 y Fp(b)684 1631 y Ft(f)-14 b(f)p Fu(s)731 1637 y Fr(1)750 1631 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)862 1637 y Fp(n)884 1631 y Ft(g)-14 b(g)18 b Fx(and)h Fu(t)g Fx(=)1111 1621 y(^)1101 1631 y Fu(C)1134 1616 y Fp(b)1151 1631 y Ft(h)-7 b(h)p Fu(s)1195 1637 y Fp(i)1207 1641 y Ff(1)1226 1631 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)1337 1637 y Fp(i)1349 1641 y Fl(m)1379 1631 y Ft(i)-7 b(i)18 b Fx(for)h(some)e(blac)o(k)100 1681 y(con)o(texts)i Fu(C)301 1666 y Fp(b)318 1681 y Ft(f)p Fu(;)7 b(:)g(:)g(:)t(;)g Ft(g)p Fx(,)491 1670 y(^)481 1681 y Fu(C)514 1666 y Fp(b)531 1681 y Ft(h)p Fu(;)g(:)g(:)g(:)e(;)i Ft(i)o Fx(,)18 b(and)h Fu(i)785 1687 y Fr(1)804 1681 y Fu(;)7 b(:)g(:)g(:)t(;)g(i)910 1687 y Fp(m)961 1681 y Ft(2)19 b(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n) p Ft(g)p Fx(.)18 b(Let)h Fu(x)1321 1687 y Fr(1)1339 1681 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1456 1687 y Fp(n)1496 1681 y Fx(b)q(e)19 b(distinct)100 1731 y(fresh)i(v)n(ariables)f(and)g (de\014ne)i Fu(\033)i Fx(=)f Ft(f)p Fu(x)748 1737 y Fp(j)787 1731 y Ft(7!)f Fu(s)870 1737 y Fp(j)909 1731 y Ft(j)e Fx(1)i Ft(\024)h Fu(j)j Ft(\024)d Fu(n)p Ft(g)p Fx(,)c Fu(s)1233 1716 y Fn(0)1268 1731 y Fx(=)k Fu(C)1356 1716 y Fp(b)1372 1731 y Ft(f)p Fu(x)1417 1737 y Fr(1)1435 1731 y Fu(;)7 b(:)g(:)g(:)e(;)i(x)1552 1737 y Fp(n)1574 1731 y Ft(g)p Fx(,)20 b(and)100 1786 y Fu(t)115 1771 y Fn(0)146 1786 y Fx(=)208 1776 y(^)198 1786 y Fu(C)231 1771 y Fp(b)248 1786 y Ft(h)p Fu(x)288 1792 y Fp(i)300 1796 y Ff(1)318 1786 y Fu(;)7 b(:)g(:)g(:)t(;)g(x)434 1792 y Fp(i)446 1796 y Fl(m)475 1786 y Ft(i)p Fx(.)19 b(Since)g Fu(\033)h Fx(is)f(top)g(white,)f(w)o(e)h(obtain)g Fu(\033)1160 1771 y Fn(!)1195 1786 y Fx(\()p Fu(s)1230 1771 y Fn(0)1242 1786 y Fx(\))h Ft(!)1320 1771 y Fp(o)1320 1796 y Fr(1)1358 1786 y Fu(\033)1383 1771 y Fn(!)1419 1786 y Fx(\()p Fu(t)1450 1771 y Fn(0)1462 1786 y Fx(\))f(b)o(y)f(Lemma) 100 1836 y(6.20.)c(According)j(to)g(Prop)q(osition)f(4.18,)e Fu(\033)807 1821 y Fn(!)859 1836 y Fx(has)j(a)f(decomp)q(osition)f Fu(\033)1274 1821 y Fn(!)1325 1836 y Fx(=)h Fu(\033)1397 1842 y Fr(2)1427 1836 y Ft(\016)10 b Fu(\033)1482 1842 y Fr(1)1501 1836 y Fx(,)16 b(where)h Fu(\033)1675 1842 y Fr(1)100 1886 y Fx(is)e(blac)o(k)g(and)g Fu(\033)359 1892 y Fr(2)393 1886 y Fx(is)h(top)f(white.)g(It)h(follo)o(ws)e(from)f (Lemma)g(6.5)i(that)g Fu(\033)1257 1871 y Fb(2)1256 1896 y Fr(2)1283 1886 y Fx(\()p Fu(\033)1323 1892 y Fr(1)1341 1886 y Fx(\()p Fu(s)1376 1871 y Fn(0)1389 1886 y Fx(\)\))f Ft(!)1477 1871 y Fp(o)1477 1896 y Fr(1)1509 1886 y Fu(\033)1534 1871 y Fb(2)1533 1896 y Fr(2)1560 1886 y Fx(\()p Fu(\033)1600 1892 y Fr(1)1619 1886 y Fx(\()p Fu(t)1650 1871 y Fn(0)1661 1886 y Fx(\)\))100 1935 y(b)q(ecause)19 b Fu(\033)281 1941 y Fr(2)317 1935 y Ft(/)f Fu(\033)392 1920 y Fb(2)391 1946 y Fr(2)418 1935 y Fx(.)f(T)m(o)f(v)o(erify)h(that)h Fu(\033)749 1920 y Fb(2)748 1946 y Fr(2)775 1935 y Fx(\()p Fu(\033)815 1941 y Fr(1)833 1935 y Fx(\()p Fu(s)868 1920 y Fn(0)880 1935 y Fx(\)\))g Ft(\))972 1941 y Fn(R)1001 1945 y Ff(1)1036 1935 y Fu(\033)1061 1920 y Fb(2)1060 1946 y Fr(2)1087 1935 y Fx(\()p Fu(\033)1127 1941 y Fr(1)1145 1935 y Fx(\()p Fu(t)1176 1920 y Fn(0)1188 1935 y Fx(\)\))g(is)f (relativ)o(ely)g(simple.)f(No)o(w)100 1985 y Fu(top)156 1970 y Fp(b)156 1996 y Fn(!)191 1985 y Fx(\()p Fu(s)p Fx(\))c Ft(\))296 1991 y Fn(R)325 1995 y Ff(1)354 1985 y Fu(top)410 1970 y Fp(b)410 1996 y Fn(!)445 1985 y Fx(\()p Fu(t)p Fx(\))i(is)g(a)g(consequence)i(of)165 2061 y Fu(top)221 2044 y Fp(b)221 2072 y Fn(!)256 2061 y Fx(\()p Fu(s)p Fx(\))c(=)g Fu(top)419 2044 y Fp(b)435 2061 y Fx(\()p Fu(C)484 2044 y Fp(b)501 2061 y Ft(f)p Fu(s)541 2044 y Fn(!)541 2072 y Fr(1)576 2061 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)688 2044 y Fn(!)688 2072 y Fp(n)723 2061 y Ft(g)p Fx(\))12 b(=)f Fu(top)871 2044 y Fp(b)888 2061 y Fx(\()p Fu(\033)929 2044 y Fn(!)965 2061 y Fx(\()p Fu(s)1000 2044 y Fn(0)1012 2061 y Fx(\)\))h(=)f Fu(top)1155 2044 y Fp(b)1172 2061 y Fx(\()p Fu(\033)1212 2067 y Fr(2)1231 2061 y Fx(\()p Fu(\033)1271 2067 y Fr(1)1289 2061 y Fx(\()p Fu(s)1324 2044 y Fn(0)1336 2061 y Fx(\)\)\))h(=)g Fu(\033)1465 2044 y Fb(2)1464 2072 y Fr(2)1491 2061 y Fx(\()p Fu(\033)1531 2067 y Fr(1)1549 2061 y Fx(\()p Fu(s)1584 2044 y Fn(0)1596 2061 y Fx(\)\))100 2137 y(and)h Fu(top)236 2122 y Fp(b)236 2148 y Fn(!)272 2137 y Fx(\()p Fu(t)p Fx(\))e(=)h Fu(\033)399 2122 y Fb(2)398 2148 y Fr(2)425 2137 y Fx(\()p Fu(\033)465 2143 y Fr(1)483 2137 y Fx(\()p Fu(t)514 2122 y Fn(0)526 2137 y Fx(\)\).)i Fe(2)141 2239 y Fx(With)e(the)h(ab)q(o)o(v)o(e)f (preparatory)g(considerations,)h(w)o(e)f(are)h(no)o(w)f(able)g(to)g (pro)o(v)o(e)g(one)h(of)e(the)i(ma)r(jor)100 2288 y(results)18 b(of)f(this)g(subsection.)h(In)f(Theorem)g(6.22,)f(statemen)o(t)h (\(3\))g(is)g(the)h(in)o(teresting)g(new)g(part.)100 2338 y(F)m(or)d(disjoin)o(t)f(unions,)h(statemen)o(ts)g(\(1\))g(and)h (\(2\))f(w)o(ere)h(already)f(pro)o(v)o(ed)h(in)f(Gramlic)o(h)d (\(1993\).)j(In)100 2388 y(the)i(con)o(text)h(of)e(Theorem)g(6.22)g(|)g (but)h(only)f(for)g(\014nite)h(disjoin)o(t)f(unions)h(|)f(Gramlic)o(h)e (\(1993\))100 2438 y(sho)o(w)o(ed)g(furthermore)f(that)h(the)g(system)g Ft(R)806 2444 y Fp(d)839 2438 y Fx(cannot)g(b)q(e)g Ft(C)1053 2444 y Fn(E)1076 2438 y Fx(-terminating,)d(i.e.)i(the)h(system)g Ft(R)1638 2444 y Fp(d)1666 2438 y Ft(])100 2488 y(f)p Fu(C)s(ons)p Fx(\()p Fu(x;)7 b(y)q Fx(\))k Ft(!)g Fu(x;)c(C)s(ons)p Fx(\()p Fu(x;)g(y)q Fx(\))12 b Ft(!)f Fu(y)q Ft(g)j Fx(m)o(ust)e(b)q(e) i(non-terminating.)e(The)i(\014niteness)h(requiremen)o(t)100 2537 y(results)g(from)d(the)i(sp)q(ecial)g(transformation)e(pro)q(of)i (tec)o(hnique)h(used)f(in)g(Gramlic)o(h)d(\(1993\).)100 2640 y Fk(Theorem)16 b(6.22.)21 b Fx(Let)c Ft(R)530 2646 y Fr(1)565 2640 y Fx(and)g Ft(R)684 2646 y Fr(2)719 2640 y Fx(b)q(e)g(terminating)e(constructor-sharing)j(CTRSs)e(suc)o(h)i (that)p eop %%Page: 31 31 31 30 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(31)p 100 224 1595 2 v 100 299 a Fx(their)15 b(com)o(bined)e(system)h Ft(R)f Fx(=)g Ft(R)652 305 y Fr(1)680 299 y Ft([)d(R)753 305 y Fr(2)786 299 y Fx(is)k(not)h(terminating.)d(Then)j(the)g(follo)o (wing)d(statemen)o(ts)100 349 y(hold)h(\(where)i Fu(d;)375 338 y Fx(\026)369 349 y Fu(d)10 b Ft(2)h(f)p Fx(1)p Fu(;)c Fx(2)p Ft(g)13 b Fx(with)g Fu(d)e Ft(6)p Fx(=)735 338 y(\026)727 349 y Fu(d)p Fx(\):)125 451 y(\(1\))21 b(There)c(exists)g (an)e(in\014nite)h Ft(R)f Fx(rewrite)i(deriv)n(ation)e Fu(D)h Fx(:)e Fu(s)1125 457 y Fr(1)1159 451 y Ft(!)g Fu(s)1234 457 y Fr(2)1268 451 y Ft(!)g Fu(s)1343 457 y Fr(3)1377 451 y Ft(!)g Fu(:)7 b(:)g(:)14 b Fx(of)h(minim)o(al)199 501 y(rank)j(suc)o(h)h(that)f Fu(D)h Fx(con)o(tains)f(in\014nitely)f (man)o(y)f Fu(s)1025 507 y Fp(j)1061 501 y Ft(!)1103 486 y Fp(o)1139 501 y Fu(s)1158 507 y Fp(j)r Fr(+1)1236 501 y Fx(reduction)i(steps)i(where)f Fu(s)1676 507 y Fp(j)199 550 y Fx(reduces)d(to)e Fu(s)417 556 y Fp(j)r Fr(+1)491 550 y Fx(b)o(y)f(some)g(rule)i(from)d Ft(R)869 556 y Fp(d)888 550 y Fx(.)125 603 y(\(2\))21 b Ft(R)240 605 y Fr(\026)234 613 y Fp(d)268 603 y Fx(is)13 b(not)h(la)o(y)o (er-preserving.)125 655 y(\(3\))21 b(If)16 b(b)q(oth)f(systems)h(are)g (con\015uen)o(t,)g(then)g Fu(D)i Fx(con)o(tains)d(in\014nitely)g(man)o (y)f(duplicating)g Fu(s)1601 661 y Fp(j)1634 655 y Ft(!)1676 640 y Fp(o)199 705 y Fu(s)218 711 y Fp(j)r Fr(+1)292 705 y Fx(reduction)h(steps)g(suc)o(h)g(that)e Fu(s)783 711 y Fp(j)815 705 y Fx(reduces)j(to)e Fu(s)1033 711 y Fp(j)r Fr(+1)1107 705 y Fx(b)o(y)f(some)g(rule)i(from)d Ft(R)1485 711 y Fp(d)1504 705 y Fx(.)100 805 y Fk(Pr)o(oof.)22 b Fx(Let)16 b Fu(D)h Fx(b)q(e)g(an)f(in\014nite)f Ft(R)h Fx(rewrite)h(deriv)n(ation)e(of)h(minim)o(al)c(rank,)k(sa)o(y)g Fu(r)q(ank)q Fx(\()p Fu(D)q Fx(\))f(=)g Fu(k)q Fx(.)100 855 y(Then)j Fu(r)q(ank)q Fx(\()p Fu(s)337 861 y Fp(j)355 855 y Fx(\))g(=)g Fu(r)q(ank)q Fx(\()p Fu(D)q Fx(\))g(for)g(all)e (indices)j Fu(j)r Fx(.)e(Moreo)o(v)o(er,)h Ft(!)1172 861 y Fn(R)1220 855 y Fx(is)g(terminating)e(on)h Ft(T)1590 840 y Fp()g Fx(=)46 b(\()p Ft(!)497 1872 y Fn(R)573 1866 y Ft([)p 613 1868 3 25 v 8 w Fu(>)q Fx(\))658 1851 y Fr(+)685 1866 y Fx(.)34 b(Then)h(\()p Ft(T)909 1851 y Fp()p Fx(\))34 b(is)g(a)g(w)o(ell-founded)g(ordering.)g(Let)199 1915 y(\()p Ft(M)p Fx(\()p Ft(T)315 1900 y Fp()428 1900 y Fp(mul)490 1915 y Fx(\))16 b(denote)h(its)f(m)o(ultiset)f(extension.)i(Note)g(that)f Fu(S)1301 1900 y Fp(w)1299 1927 y(P)1329 1915 y Fx(\()p Fu(s)1364 1921 y Fp(j)1382 1915 y Fx(\))f Ft(2)h(M)p Fx(\()p Ft(T)1556 1900 y Fp()441 2376 y Fp(mul)515 2391 y Fu(S)542 2376 y Fp(w)540 2402 y(P)570 2391 y Fx(\()p Fu(s)605 2397 y Fp(j)r Fr(+1)665 2391 y Fx(\).)274 2441 y(If)21 b Fu(s)342 2447 y Fp(j)384 2441 y Ft(!)i Fu(s)468 2447 y Fp(j)r Fr(+1)550 2441 y Fx(b)o(y)e(some)f(rule)i(from)e Ft(R)958 2447 y Fr(2)976 2441 y Fx(,)h(then)h(there)h(is)e(a)g(white)h(principal)e(sub-)274 2491 y(term)d Fu(u)h Ft(2)g Fu(S)492 2476 y Fp(w)490 2502 y(P)520 2491 y Fx(\()p Fu(s)555 2497 y Fp(j)573 2491 y Fx(\))g(suc)o(h)h(that)f Fu(u)g Ft(!)g Fu(v)h Fx(for)f(some)f Fu(v)q Fx(,)h(i.e.)f Fu(s)1257 2497 y Fp(j)1293 2491 y Fx(=)i Fu(C)1377 2476 y Fp(b)1393 2491 y Fx([)-7 b([)p Fu(;)7 b(:)g(:)g(:)t(;)g(u;)g(:)g(:)g(:)t(;)g Fx(])-7 b(])17 b Ft(!)274 2540 y Fu(C)307 2525 y Fp(b)323 2540 y Fx([)p Fu(;)7 b(:)g(:)g(:)e(;)i(v)q(;)g(:)g(:)g(:)t(;)g Fx(])k(=)h Fu(s)627 2546 y Fp(j)r Fr(+1)687 2540 y Fx(.)f(Th)o(us)h(w)o (e)g(ha)o(v)o(e)g Fu(S)993 2525 y Fp(w)991 2552 y(P)1021 2540 y Fx(\()p Fu(s)1056 2546 y Fp(j)r Fr(+1)1116 2540 y Fx(\))g(=)f(\()p Fu(S)1230 2525 y Fp(w)1228 2552 y(P)1258 2540 y Fx(\()p Fu(s)1293 2546 y Fp(j)1311 2540 y Fx(\))5 b Ft(n)g Fx([)p Fu(u)p Fx(]\))g Ft([)g Fu(S)1487 2525 y Fp(w)1485 2552 y(P)1515 2540 y Fx(\()p Fu(v)q Fx(\).)12 b(It)g(fol-)274 2590 y(lo)o(ws)g(from)f Fu(u)g Ft(!)g Fu(v)j Fx(in)e(conjunction)g(with)h Fu(v)g Fx(=)e Fu(w)j Fx(or)e Fu(v)p 1157 2592 V 17 w(>)7 b(w)13 b Fx(for)f(an)o(y)g (principal)g(subterm)274 2640 y Fu(w)g Ft(2)g Fu(S)383 2625 y Fp(w)381 2652 y(P)410 2640 y Fx(\()p Fu(v)q Fx(\))j(that)f Fu(u)d(>)h(w)j Fx(for)e(an)o(y)h Fu(w)e Ft(2)f Fu(S)942 2625 y Fp(w)940 2652 y(P)970 2640 y Fx(\()p Fu(v)q Fx(\).)j(Therefore)h Fu(S)1264 2625 y Fp(w)1262 2652 y(P)1292 2640 y Fx(\()p Fu(s)1327 2646 y Fp(j)1345 2640 y Fx(\))d Fu(>)1405 2625 y Fp(mul)1479 2640 y Fu(S)1506 2625 y Fp(w)1504 2652 y(P)1533 2640 y Fx(\()p Fu(s)1568 2646 y Fp(j)r Fr(+1)1628 2640 y Fx(\).)p eop %%Page: 32 32 32 31 bop 100 197 a Fw(32)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 199 299 a Fx(W)m(e)h(conclude)h(from)e(the)i(w)o (ell-foundedness)g(of)f(\()p Ft(M)p Fx(\()p Ft(T)1081 284 y Fp()1194 284 y Fp(mul)1256 299 y Fx(\))13 b(that)f(only)g(a)g(\014nite)g(n)o(um-)199 349 y(b)q(er)k(of)e(inner)h(reduction)h(steps)g(as)f(w)o(ell)f(as)h (reduction)h(steps)g(using)e(a)h(rule)g(from)e Ft(R)1562 355 y Fr(2)1596 349 y Fx(o)q(ccur)199 399 y(in)k Fu(D)q Fx(.)g(W.l.o.g.)c(w)o(e)k(ma)o(y)e(supp)q(ose)j(that)f(there)i(are)e (no)g(reduction)g(steps)h(of)f(that)g(kind)f(in)199 448 y Fu(D)q Fx(.)h(Consequen)o(tly)m(,)f(for)g(all)f Fu(j)j Ft(2)d Fm(N)p Fx(,)f(w)o(e)j(ha)o(v)o(e)f Fu(s)974 454 y Fp(j)1007 448 y Ft(!)1049 433 y Fp(o)1083 448 y Fu(s)1102 454 y Fp(j)r Fr(+1)1178 448 y Fx(b)o(y)h(some)e(rule)i(from)d Ft(R)1566 454 y Fr(1)1585 448 y Fx(.)i(No)o(w)199 498 y Ft(!)241 504 y Fr(1)p Fp(;)p Fr(2)304 498 y Fx(is)i(semi-complete)f (b)q(ecause)j(\()p Ft(F)833 504 y Fr(1)852 498 y Fu(;)7 b Ft(R)905 504 y Fr(1)924 498 y Fx(\))18 b(and)h(\()p Ft(F)1094 504 y Fr(2)1112 498 y Fu(;)7 b Ft(R)1166 504 y Fr(2)1185 498 y Fx(\))18 b(are)h(complete.)e(Prop)q(osition)199 548 y(6.21)e(yields)i Fu(top)465 533 y Fp(b)465 558 y Fn(!)500 548 y Fx(\()p Fu(s)535 554 y Fp(j)553 548 y Fx(\))f Ft(\))627 554 y Fn(R)656 558 y Ff(1)689 548 y Fu(top)745 533 y Fp(b)745 558 y Fn(!)780 548 y Fx(\()p Fu(s)815 554 y Fp(j)r Fr(+1)875 548 y Fx(\))h(for)f(ev)o(ery)h Fu(j)h Ft(2)e Fm(N)p Fx(.)e(This)i(is)g(a)h(con)o(tradiction)f(to)199 598 y(the)f(termination)d(of)h Ft(\))586 604 y Fn(R)615 608 y Ff(1)632 598 y Fx(.)100 697 y Fe(2)100 797 y Fk(Cor)o(ollar)m(y) 18 b(6.23.)j Fx(If)12 b Ft(R)532 803 y Fr(1)563 797 y Fx(and)g Ft(R)677 803 y Fr(2)708 797 y Fx(are)h(terminating)e(CTRSs)h (with)h(shared)g(constructors,)h(then)100 847 y(their)g(com)o(bined)e (system)i Ft(R)f Fx(is)h(terminating)e(pro)o(vided)h(that)h(one)g(of)f (the)h(follo)o(wing)d(conditions)j(is)100 897 y(satis\014ed:)125 996 y(\(1\))21 b(Neither)15 b Ft(R)383 1002 y Fr(1)416 996 y Fx(nor)f Ft(R)525 1002 y Fr(2)557 996 y Fx(con)o(tain)g(either)g (collapsing)f(or)h(constructor-lifting)g(rules.)125 1046 y(\(2\))21 b(Both)14 b(systems)h(are)f(con\015uen)o(t)h(and)e (non-duplicating.)125 1096 y(\(3\))21 b(Both)g(systems)f(are)g (con\015uen)o(t)h(and)e(one)i(of)e(the)i(systems)f(con)o(tains)f (neither)i(collapsing,)199 1146 y(constructor-lifting,)13 b(nor)h(duplicating)f(rules.)100 1245 y Fk(Pr)o(oof.)22 b Fx(This)13 b(is)h(an)g(imm)o(ediate)e(consequence)k(of)d(Theorem)h (6.22.)e Fe(2)100 1345 y Fk(Cor)o(ollar)m(y)18 b(6.24.)125 1445 y Fx(\(1\))j(T)m(ermination)12 b(is)h(mo)q(dular)f(for)i(la)o(y)o (er-preserving)g(constructor-sharing)i(CTRSs.)125 1494 y(\(2\))21 b(Completeness)14 b(is)g(mo)q(dular)e(for)i(la)o(y)o (er-preserving)g(constructor-sharing)h(CTRSs.)125 1544 y(\(3\))21 b(Completeness)14 b(is)g(mo)q(dular)e(for)i(non-duplicating) e(constructor-sharing)j(CTRSs.)100 1644 y Fk(Pr)o(oof.)22 b Fx(\(1\))e(is)g(an)g(immediate)e(consequence)k(of)e(Corollary)f (6.23.)g(\(2\))h(and)h(\(3\))f(follo)o(w)e(from)100 1694 y(Theorem)13 b(6.14)g(in)g(conjunction)h(with)f(Corollary)g(6.23.)f Fe(2)141 1793 y Fx(Clearly)m(,)e(it)g(also)h(follo)o(ws)e(from)h(the)h (aforemen)o(tioned)f(that)h Ft(C)1104 1799 y Fn(E)1127 1793 y Fx(-termination)e(is)i(a)f(mo)q(dular)g(prop-)100 1843 y(ert)o(y)k(of)f(\014nite)h(disjoin)o(t)f(CTRSs;)h(see)h(Gramlic)o (h)c(\(1993\).)533 1943 y Fk(6.3.)23 b(combining)16 b(decreasing)f (systems)141 2042 y Fx(Simple)e(coun)o(terexamples)i(sho)o(w)g(that)h (innermost)e(termination)f(is)i(not)g(mo)q(dular)e(for)i(disjoin)o(t) 100 2092 y(CTRSs.)20 b(So)h(in)f(con)o(trast)h(to)g(the)h (unconditional)d(case)j(\(see)g(Corollary)e(5.4\),)f(it)i(is)f(not)h (clear)100 2142 y(ho)o(w)d(the)h(unique)g(normal)e(form)g(of)h(a)g (term)g(w.r.t.)g(the)h(com)o(bined)f(system)g(of)g(complete)g(pair-)100 2192 y(wise)g(constructor-sharing)h(CTRSs)e(can)h(b)q(e)g(obtained.)f (W)m(e)h(will)e(sho)o(w)h(next)i(ho)o(w)e(this)h(unique)100 2242 y(normal)11 b(form)i(can)h(b)q(e)g(computed)g(for)f(\014nite,)h (decreasing)h(CTRSs.)e(Note)i(that)e(decreasingness)k(is)100 2291 y(not)g(mo)q(dular,)d(ev)o(en)k(for)f(disjoin)o(t)f(CTRSs.)h(The)g (coun)o(terexample)g(of)g(T)m(o)o(y)o(ama)d(\(1987)p Fs(b)p Fx(\))i(to)h(the)100 2341 y(mo)q(dularit)o(y)11 b(of)h(termination)g(for)h(disjoin)o(t)f(TRSs)i(applies)f(b)q(ecause)i (ev)o(ery)f(terminating)d(TRS)i(can)100 2391 y(b)q(e)h(regarded)h(as)f (a)g(decreasing)h(CTRS.)100 2491 y Fk(Definition)h(6.25.)21 b Fx(A)10 b(CTRS)g Ft(R)h Fx(is)g Fs(de)n(cr)n(e)n(asing)f Fx(if)g(there)i(exists)f(a)f(w)o(ell-founded)g(partial)g(ordering)100 2540 y Fu(>)i Fx(p)q(ossessing)i(the)e(subterm)g(prop)q(ert)o(y)h(suc)o (h)g(that)g Fu(>)f Fx(con)o(tains)26 b Ft(!)1180 2546 y Fn(R)1236 2540 y Fx(and)12 b(for)g(ev)o(ery)h(rewrite)g(rule)100 2590 y Fu(l)f Ft(!)f Fu(r)i Ft(\()e Fu(s)281 2596 y Fr(1)311 2590 y Ft(#)h Fu(t)359 2596 y Fr(1)377 2590 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)489 2596 y Fp(n)523 2590 y Ft(#)k Fu(t)570 2596 y Fp(n)609 2590 y Ft(2)16 b(R)h Fx(and)f(ev)o(ery)h (substitution)g Fu(\033)h Fx(w)o(e)f(ha)o(v)o(e)f Fu(l)q(\033)i(>)e(s) 1460 2596 y Fp(i)1474 2590 y Fu(\033)i Fx(as)f(w)o(ell)f(as)100 2640 y Fu(l)q(\033)d(>)e(t)208 2646 y Fp(i)222 2640 y Fu(\033)q Fx(,)j(where)h(1)c Ft(\024)h Fu(i)g Ft(\024)g Fu(n)p Fx(.)p eop %%Page: 33 33 33 32 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(33)p 100 224 1595 2 v 141 299 a Fx(Note)11 b(that)g(decreasing)h(systems)f(do)g (not)f(allo)o(w)g(extra)h(v)n(ariables)f(in)g(the)i(conditions.)e (Decreasing)100 349 y(\(\014nite\))15 b(CTRSs)f(ha)o(v)o(e)h(b)q(een)h (in)o(v)o(estigated)e(b)o(y)h(man)o(y)e(researc)o(hers)k(b)q(ecause)g (all)c(basic)i(prop)q(erties)100 399 y(\(lik)o(e)i(reducibilit)o(y)g (for)g(instance\))i(are)f(decidable)g(and)f(a)h(critical)f(pair)g (lemma)e(holds)i(for)h(those)100 448 y(systems)c(\(cf.)f(Dersho)o(witz) i Fs(et)g(al.)e Fx(1988\).)g(In)h(order)h(to)e(sho)o(w)h(ho)o(w)g (\(unique\))g(normal)e(forms)h(w.r.t.)100 498 y(the)h(com)o(bined)f (system)g(of)g Fu(n)h Fx(\014nite,)f(decreasing,)i(con\015uen)o(t,)f (and)g(pairwise)f(constructor-sharing)100 548 y(CTRSs)d(can)h(b)q(e)g (obtained,)f(w)o(e)h(recall)g(the)g(mo)q(dular)d(reduction)k(relation)e (in)o(tro)q(duced)h(b)o(y)g(Kurihara)100 598 y(and)i(Oh)o(uc)o(hi)h (\(1991\).)100 702 y Fk(Definition)i(6.26.)21 b Fx(Let)c(\()p Ft(F)578 708 y Fr(1)597 702 y Fu(;)7 b Ft(R)651 708 y Fr(1)669 702 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)828 708 y Fp(n)850 702 y Fu(;)g Ft(R)904 708 y Fp(n)926 702 y Fx(\))17 b(b)q(e)h(pairwise)f(constructor-sharing)h(CTRSs.)100 752 y(Let)j Ft(F)26 b Fx(=)292 721 y Fg(S)327 731 y Fp(n)327 765 y(j)r Fr(=1)393 752 y Ft(F)427 758 y Fp(j)444 752 y Fx(.)20 b(F)m(or)h Fu(s;)7 b(t)22 b Ft(2)g(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))21 b(de\014ne)h Fu(s)h Fo(;)1061 758 y Fn(R)1090 762 y Fl(j)1129 752 y Fu(t)e Fx(if)e(and)i(only)e(if)h Fu(s)j Ft(!)1523 734 y Fr(+)1523 764 y Fn(R)1552 768 y Fl(j)1591 752 y Fu(t)e Fx(and)100 808 y Fu(t)12 b Ft(2)h Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)342 814 y Fp(j)359 808 y Fx(\),)14 b(where)i Fu(j)f Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)13 b(Moreo)o(v)o(er,)i(de\014ne)g Fu(s)f Fo(;)e Fu(t)i Fx(if)g(and)g(only)g(if)g Fu(s)f Fo(;)1556 814 y Fn(R)1585 818 y Fl(j)1614 808 y Fu(t)i Fx(for)100 858 y(some)e Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)t(;)g (n)p Ft(g)p Fx(.)13 b Fo(;)g Fx(is)h(called)f Fs(mo)n(dular)i(r)n(e)n (duction)g(r)n(elation)p Fx(.)141 962 y(Roughly)c(sp)q(eaking,)h (reduction)h(steps)h(\(including)d(the)i(ev)n(aluation)e(of)h(the)g (conditions\))h(ha)o(v)o(e)f(to)100 1012 y(b)q(e)i(p)q(erformed)g (using)f(the)i(same)e(constituen)o(t)i(CTRS)e Ft(R)1017 1018 y Fp(j)1049 1012 y Fx(for)g(as)h(long)f(as)h(p)q(ossible.)100 1116 y Fk(Theorem)i(6.27.)21 b Fx(If)10 b(\()p Ft(F)506 1122 y Fr(1)525 1116 y Fu(;)d Ft(R)579 1122 y Fr(1)597 1116 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)750 1122 y Fp(n)773 1116 y Fu(;)g Ft(R)827 1122 y Fp(n)849 1116 y Fx(\))j(are)h(pairwise)f(constructor-sharing)i(CTRSs,)e(then)100 1166 y(the)k(mo)q(dular)e(reduction)j(relation)e Fo(;)g Fx(is)h(terminating)e(on)i Ft(T)c Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(.)100 1268 y Fk(Pr)o(oof.)22 b Fx(The)14 b(same)f(as)h(for)f(unconditional)g(TRSs;)g(see)i(Kurihara)f(and)g(Oh)o (uc)o(hi)g(\(1991\).)f Fe(2)141 1370 y Fx(The)g(pro)q(ofs)f(of)g(the)h (follo)o(wing)c(results)k(hea)o(vily)f(dep)q(end)h(on)f(the)h(fact)f (that)g(w)o(e)h(are)g(dealing)e(with)100 1420 y(constructor-sharing)16 b(systems)e(instead)h(of)f(disjoin)o(t)g(unions.)g(This)g(is)h (probably)e(the)j(reason)f(wh)o(y)100 1470 y(no)e(suc)o(h)i(results)g (had)f(b)q(een)h(ac)o(hiev)o(ed)f(in)f(the)i(in)o(v)o(estigation)d(of)i (disjoin)o(t)e(unions)i(of)f(CTRSs.)100 1574 y Fk(Lemma)j(6.28.)21 b Fx(Let)12 b(\()p Ft(F)492 1580 y Fr(1)510 1574 y Fu(;)7 b Ft(R)564 1580 y Fr(1)582 1574 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)735 1580 y Fp(n)758 1574 y Fu(;)g Ft(R)812 1580 y Fp(n)834 1574 y Fx(\),)j Fu(n)i Ft(\025)g Fx(2,)e(b)q(e)h (semi-complete)e(pairwise)h(constructor-)100 1624 y(sharing)j(CTRSs.)h (Then)g Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\))12 b(=)755 1593 y Fg(T)789 1603 y Fp(n)789 1637 y(j)r Fr(=1)856 1624 y Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)1030 1630 y Fp(j)1047 1624 y Fx(\).)100 1726 y Fk(Pr)o(oof.)22 b Fx(\\)p Ft(\022)p Fx(")13 b(T)m(rivial.)100 1776 y(\\)p Ft(\023)p Fx(")f(W)m(e)h(use)h(induction)f(on)f(the)i(n)o(um)o(b)q(er)e Fu(n)h Fx(of)g(CTRSs.)f(The)i(case)g Fu(n)d Fx(=)h(2)h(is)f(co)o(v)o (ered)j(b)o(y)d(Lemma)100 1826 y(6.13)e(\(1\).)h(So)g(supp)q(ose)i Fu(n)e(>)h Fx(2.)f(First)g(of)g(all,)f(b)o(y)h(rep)q(eated)i (application)d(of)h(Theorem)g(6.14,)f(w)o(e)i(infer)100 1880 y(that)g(the)h(CTRS)f(\()400 1849 y Fg(S)435 1859 y Fp(n)p Fn(\000)p Fr(1)435 1893 y Fp(j)r Fr(=1)507 1880 y Ft(F)535 1886 y Fp(j)553 1880 y Fu(;)572 1849 y Fg(S)606 1859 y Fp(n)p Fn(\000)p Fr(1)606 1893 y Fp(j)r Fr(=1)678 1880 y Ft(R)713 1886 y Fp(j)731 1880 y Fx(\))g(is)h(semi-complete.)d (It)i(is)h(imm)o(ediately)d(ob)o(vious)h(that)i(the)100 1942 y(systems)j(\()272 1911 y Fg(S)307 1921 y Fp(n)p Fn(\000)p Fr(1)307 1955 y Fp(j)r Fr(=1)379 1942 y Ft(F)407 1948 y Fp(j)424 1942 y Fu(;)443 1911 y Fg(S)477 1921 y Fp(n)p Fn(\000)p Fr(1)477 1955 y Fp(j)r Fr(=1)549 1942 y Ft(R)585 1948 y Fp(j)602 1942 y Fx(\))g(and)g(\()p Ft(F)762 1948 y Fp(n)784 1942 y Fu(;)7 b Ft(R)838 1948 y Fp(n)861 1942 y Fx(\))16 b(are)g(constructor-sharing;)h(th)o(us,)f (using)g(Lemma)100 2004 y(6.13)c(\(1\),)i(w)o(e)g(deriv)o(e)g Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)o Fx(\))12 b(=)g Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)834 1973 y Fg(S)868 1983 y Fp(n)p Fn(\000)p Fr(1)868 2016 y Fp(j)r Fr(=1)940 2004 y Ft(R)975 2010 y Fp(j)993 2004 y Fx(\))j Ft(\\)g Fu(N)c(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)1229 2010 y Fp(n)1252 2004 y Fx(\).)13 b(The)i(equalit)o(y)591 2128 y Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)731 2076 y Fp(n)p Fn(\000)p Fr(1)739 2089 y Fg([)733 2177 y Fp(j)r Fr(=1)800 2128 y Ft(R)835 2134 y Fp(j)853 2128 y Fx(\))11 b(=)924 2076 y Fp(n)p Fn(\000)p Fr(1)933 2089 y Fg(\\)927 2177 y Fp(j)r Fr(=1)994 2128 y Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)1168 2134 y Fp(j)1186 2128 y Fx(\))100 2258 y(remains)k(to)h(b)q(e)g(sho)o (wn.)g(Set)h Ft(F)596 2240 y Fn(0)619 2258 y Fx(=)f Ft(F)e(n)c Fx(\()746 2227 y Fg(S)780 2237 y Fp(n)p Fn(\000)p Fr(1)780 2270 y Fp(j)r Fr(=1)852 2258 y Ft(F)880 2264 y Fp(j)898 2258 y Fx(\).)11 b(It)i(is)f(not)g(di\016cult)f(to)h(v)o(erify)g(that)g (the)h(CTRSs)100 2314 y(\()p Ft(F)150 2320 y Fr(1)173 2314 y Ft(])5 b(F)239 2296 y Fn(0)251 2314 y Fu(;)i Ft(R)304 2320 y Fr(1)323 2314 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)476 2320 y Fp(n)p Fn(\000)p Fr(1)546 2314 y Ft(])e(F)612 2296 y Fn(0)623 2314 y Fu(;)i Ft(R)677 2320 y Fp(n)p Fn(\000)p Fr(1)742 2314 y Fx(\))12 b(are)g(semi-complete) d(and)j(pairwise)f(constructor-sharing)100 2364 y(b)q(ecause)i(\()p Ft(F)301 2370 y Fr(1)320 2364 y Fu(;)7 b Ft(R)374 2370 y Fr(1)392 2364 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)545 2370 y Fp(n)p Fn(\000)p Fr(1)610 2364 y Fu(;)g Ft(R)664 2370 y Fp(n)p Fn(\000)p Fr(1)729 2364 y Fx(\))12 b(are)g(semi-complete)e(and)i(pairwise)g(constructor-sharing.)100 2414 y(An)i(application)e(of)h(the)i(induction)e(h)o(yp)q(othesis)i (yields)100 2528 y Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)239 2476 y Fp(n)p Fn(\000)p Fr(1)247 2488 y Fg([)241 2577 y Fp(j)r Fr(=1)308 2528 y Ft(R)344 2534 y Fp(j)361 2528 y Fx(\))12 b(=)g Fu(N)5 b(F)h Fx(\()520 2476 y Fp(n)p Fn(\000)p Fr(1)528 2488 y Fg([)523 2577 y Fp(j)r Fr(=1)582 2528 y Fx(\()p Ft(F)627 2534 y Fp(j)644 2528 y Ft(]F)706 2510 y Fn(0)717 2528 y Fx(\))p Fu(;)752 2476 y Fp(n)p Fn(\000)p Fr(1)760 2488 y Fg([)755 2577 y Fp(j)r Fr(=1)822 2528 y Ft(R)857 2534 y Fp(j)874 2528 y Fx(\))12 b(=)946 2476 y Fp(n)p Fn(\000)p Fr(1)954 2488 y Fg(\\)949 2577 y Fp(j)r Fr(=1)1016 2528 y Fu(N)5 b(F)h Fx(\()1103 2476 y Fp(n)p Fn(\000)p Fr(1)1111 2488 y Fg([)1108 2577 y Fp(i)p Fr(=1)1165 2528 y Fx(\()p Ft(F)1210 2534 y Fp(i)1224 2528 y Ft(]F)1285 2510 y Fn(0)1297 2528 y Fx(\))p Fu(;)h Ft(R)1367 2534 y Fp(j)1384 2528 y Fx(\))12 b(=)1456 2476 y Fp(n)p Fn(\000)p Fr(1)1464 2488 y Fg(\\)1458 2577 y Fp(j)r Fr(=1)1526 2528 y Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)1700 2534 y Fp(j)1717 2528 y Fx(\))p Fu(:)100 2640 y Fe(2)p eop %%Page: 34 34 34 33 bop 100 197 a Fw(34)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fk(Pr)o(oposition)16 b(6.29.)21 b Fx(If)d(\()p Ft(F)580 305 y Fr(1)598 299 y Fu(;)7 b Ft(R)652 305 y Fr(1)670 299 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)823 305 y Fp(n)846 299 y Fu(;)g Ft(R)899 305 y Fp(n)922 299 y Fx(\))19 b(are)h(semi-complete)d(pairwise)i (constructor-)100 349 y(sharing)13 b(CTRSs,)h(then)g Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)o Fx(\))12 b(=)g Fu(N)5 b(F)h Fx(\()p Fo(;)p Fx(\).)100 448 y Fk(Pr)o(oof.)22 b Fx(\\)p Ft(\022)p Fx(")13 b(T)m(rivial.)100 497 y(\\)p Ft(\023)p Fx(")h(Let)g Fu(t)f Ft(2)e Fu(N)5 b(F)h Fx(\()p Fo(;)p Fx(\))14 b(and)g(supp)q(ose)h Fu(t)d Ft(62)g Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)966 503 y Fp(j)984 497 y Fx(\))14 b(for)g(some)f Fu(j)i Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)13 b(According)i(to)100 547 y(Theorem)f(6.14,)f(\()p Ft(F)t Fu(;)7 b Ft(R)480 553 y Fp(j)497 547 y Fx(\))15 b(is)g(normalizing.)d(Hence)k(there)g(is) f(a)g(term)f Fu(t)1235 532 y Fn(0)1260 547 y Ft(2)e Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)1474 553 y Fp(j)1492 547 y Fx(\))15 b(suc)o(h)h(that)100 597 y Fu(t)11 b Ft(!)168 579 y Fr(+)168 609 y Fn(R)197 613 y Fl(j)225 597 y Fu(t)240 582 y Fn(0)252 597 y Fx(.)g(It)i(follo)o(ws)d Fu(t)i Fo(;)522 603 y Fn(R)551 607 y Fl(j)579 597 y Fu(t)594 582 y Fn(0)617 597 y Fx(whic)o(h)g(con)o(tradicts)h(the)g(assumption)e Fu(t)g Ft(2)g Fu(N)5 b(F)h Fx(\()p Fo(;)p Fx(\).)11 b(W)m(e)h(conclude) 100 653 y Fu(t)k Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)348 659 y Fp(j)366 653 y Fx(\))16 b(for)h(all)e Fu(j)j Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)16 b(Finally)m(,)e(it)i(is)g(a)g(consequence)j(of)d(Lemma)e (6.28)i(that)100 703 y Fu(t)11 b Ft(2)g Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)o Fx(\).)14 b Fe(2)100 802 y Fk(Theorem)i(6.30.)21 b Fx(If)32 b(\()p Ft(F)528 808 y Fr(1)547 802 y Fu(;)7 b Ft(R)600 808 y Fr(1)619 802 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)772 808 y Fp(n)794 802 y Fu(;)g Ft(R)848 808 y Fp(n)870 802 y Fx(\))33 b(are)f(semi-complete)e(pairwise)i(constructor-)100 851 y(sharing)13 b(CTRSs,)h(then)g Fo(;)f Fx(is)h(complete.)100 950 y Fk(Pr)o(oof.)22 b Fx(According)14 b(to)g(Theorem)f(6.27,)f Fo(;)h Fx(is)h(terminating.)e(Th)o(us)i(it)f(su\016ces)j(to)d(sho)o(w)h (that)g Fo(;)100 1000 y Fx(has)c(unique)h(normal)e(forms)g(w.r.t.)h (reduction.)h(Consider)g Fu(t)1037 1006 y Fr(1)1066 985 y Fn(\003)1120 1000 y gsave currentpoint currentpoint translate -1 1 scale neg exch neg exch translate 1120 1000 a Fo(;)1120 1000 y currentpoint grestore moveto 1120 1000 a 10 w Fu(t)h Fo(;)1199 985 y Fn(\003)1229 1000 y Fu(t)1244 1006 y Fr(2)1263 1000 y Fx(,)e(where)i Fu(t)1417 1006 y Fr(1)1435 1000 y Fu(;)7 b(t)1469 1006 y Fr(2)1499 1000 y Ft(2)k Fu(N)5 b(F)h Fx(\()p Fo(;)p Fx(\).)100 1050 y(By)19 b(the)h(preceding)g(prop)q(osition,)f Fu(t)690 1056 y Fr(1)708 1050 y Fu(;)7 b(t)742 1056 y Fr(2)781 1050 y Ft(2)20 b Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\).)19 b(No)o(w)g(since)h Fo(;)f Ft(\022)g(!)1411 1035 y Fn(\003)1411 1061 y(R)1441 1050 y Fx(,)g(w)o(e)g(obtain)g(a)100 1100 y(con)o(v)o(ersion)e Fu(t)320 1106 y Fr(1)352 1085 y Fn(\003)352 1111 y(R)380 1100 y Ft( )d Fu(t)f Ft(!)506 1085 y Fn(\003)506 1111 y(R)550 1100 y Fu(t)565 1106 y Fr(2)601 1100 y Fx(b)q(et)o(w)o(een)18 b(the)f(t)o(w)o(o)f(normal)f (forms)g Fu(t)1195 1106 y Fr(1)1230 1100 y Fx(and)h Fu(t)1328 1106 y Fr(2)1347 1100 y Fx(.)g(It)h(follo)o(ws)e Fu(t)1577 1106 y Fr(1)1612 1100 y Fx(=)h Fu(t)1675 1106 y Fr(2)100 1150 y Fx(b)q(ecause)f(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(is)g(con\015uen)o(t)g(\(indeed)h(semi-complete\))d(b)o(y)i(Theorem)f (6.14.)f Fe(2)100 1249 y Fk(Cor)o(ollar)m(y)18 b(6.31.)j Fx(Let)f(\()p Ft(F)588 1255 y Fr(1)606 1249 y Fu(;)7 b Ft(R)660 1255 y Fr(1)678 1249 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)831 1255 y Fp(n)854 1249 y Fu(;)g Ft(R)907 1255 y Fp(n)930 1249 y Fx(\))20 b(b)q(e)h(semi-complete)e(pairwise)h (constructor-)100 1298 y(sharing)g(CTRSs)h(and)f(let)h Fu(t)i Ft(2)f(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))q(.)20 b(The)h(unique)g(normal)d(form)h Fu(t)p Ft(#)i Fx(of)f Fu(t)g Fx(w.r.t.)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))100 1348 y(coincides)14 b(with)g(the)g(unique)g(normal)e(form)g Fu(t)845 1316 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 845 1316 a 15 x Fo(;)845 1357 y currentpoint grestore moveto 845 1357 a 859 1348 a Fx(of)h Fu(t)h Fx(w.r.t.)f Fo(;)p Fx(.)100 1447 y Fk(Pr)o(oof.)22 b Fx(Clearly)m(,)13 b Fu(t)h Fo(;)488 1432 y Fn(\003)521 1447 y Fu(t)551 1415 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 551 1415 a 15 x Fo(;)551 1456 y currentpoint grestore moveto 551 1456 a 566 1447 a Fx(implies)f Fu(t)h Ft(!)779 1432 y Fn(\003)779 1459 y(R)823 1447 y Fu(t)853 1415 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 853 1415 a 15 x Fo(;)853 1456 y currentpoint grestore moveto 853 1456 a -9 x Fx(.)h(F)m(urthermore,)g Fu(t)1164 1415 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 1164 1415 a 15 x Fo(;)1164 1456 y currentpoint grestore moveto 1164 1456 a 1178 1447 a Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\))15 b(b)o(y)g(Prop)q(osition)100 1497 y(6.29.)d(It)h(follo)o(ws)f(from)g (the)i(semi-completeness)f(of)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))14 b(that)f(the)i(normal)c(form)h Fu(t)p Ft(#)h Fx(of)g Fu(t)h Fx(w.r.t.)100 1547 y(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))13 b(is)h(unique.)g(Th)o(us)g Fu(t)p Ft(#)d Fx(=)h Fu(t)648 1514 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 648 1514 a 15 x Fo(;)648 1556 y currentpoint grestore moveto 648 1556 a -9 x Fx(.)h Fe(2)141 1646 y Fx(In)d(order)g(to)g(pro) o(v)o(e)f(the)i(principal)d(theorem)i(of)f(this)g(subsection,)i(w)o(e)f (ha)o(v)o(e)f(to)h(sho)o(w)f(that)h(decreas-)100 1696 y(ingness)16 b(is)h(conserv)o(ed)g(under)g(signature)g(extensions.)g (This)f(is)g(b)o(y)g(no)g(means)g(trivial.)e(Gramlic)o(h)100 1745 y(\(1994)p Fs(c)p Fx(\))9 b(sho)o(w)o(ed)i(that)f(sev)o(eral)g (prop)q(erties)i(\(lik)o(e)d(normalization,)e(for)j(instance\))h(are)g (not)f(preserv)o(ed)100 1795 y(under)k(signature)h(extensions.)141 1845 y(F)m(rom)j(no)o(w)i(on,)f(w)o(e)h(assume)g(that)g(the)g(CTRS)f (\()p Ft(F)998 1851 y Fr(1)1016 1845 y Fu(;)7 b Ft(R)1070 1851 y Fr(1)1088 1845 y Fx(\))20 b(is)g(decreasing)h(w.r.t.)e(the)i (partial)100 1895 y(ordering)11 b Fu(>)293 1901 y Fr(1)324 1895 y Ft(\022)h(T)e Fx(\()p Ft(F)447 1901 y Fr(1)465 1895 y Fu(;)d Ft(V)s Fx(\))e Ft(\002)g(T)11 b Fx(\()p Ft(F)650 1901 y Fr(1)668 1895 y Fu(;)c Ft(V)s Fx(\))q(.)k(It)g(will)g (b)q(e)h(sho)o(wn)f(that)h(the)g(CTRS)f(\()p Ft(F)1384 1901 y Fr(1)1407 1895 y Ft(])5 b(F)1473 1877 y Fn(0)1485 1895 y Fu(;)i Ft(R)1538 1901 y Fr(1)1557 1895 y Fx(\))12 b(is)f(also)100 1945 y(decreasing)h(for)f(an)o(y)g(signature)h Ft(F)647 1926 y Fn(0)670 1945 y Fx(with)f Ft(F)796 1951 y Fr(1)819 1945 y Ft(\\)t(F)885 1926 y Fn(0)908 1945 y Fx(=)h Ft(;)p Fx(.)f(First)g(w)o(e)h(sho)o(w)g(that)f Fu(>)1373 1951 y Fr(1)1403 1945 y Fx(can)h(b)q(e)g(extended)100 1994 y(to)h(a)h(partial)e(ordering)i Fu(>)514 2000 y Fr(2)546 1994 y Fx(on)g Ft(T)c Fx(\()p Ft(F)687 2000 y Fr(1)715 1994 y Ft(])e(F)785 1976 y Fn(0)797 1994 y Fu(;)f Ft(V)s Fx(\))14 b(whic)o(h)f(has)h(almost)e(all)g(the)i(prop)q (erties)i(necessary)100 2044 y(for)10 b(sho)o(wing)h(that)g(\()p Ft(F)453 2050 y Fr(1)475 2044 y Ft(])t(F)540 2026 y Fn(0)552 2044 y Fu(;)c Ft(R)605 2050 y Fr(1)624 2044 y Fx(\))k(is)g(decreasing.) h(Hereb)o(y)m(,)f(\()p Ft(F)1097 2050 y Fr(1)1119 2044 y Ft(])t(F)1184 2026 y Fn(0)1196 2044 y Fu(;)c Ft(R)1249 2050 y Fr(1)1268 2044 y Fx(\))k(is)g(considered)h(to)f(b)q(e)h(the)100 2094 y(disjoin)o(t)i(union)g(of)g(the)i(CTRSs)f(\()p Ft(F)681 2100 y Fr(1)700 2094 y Fu(;)7 b Ft(R)753 2100 y Fr(1)772 2094 y Fx(\))15 b(and)g(\()p Ft(F)935 2076 y Fn(0)947 2094 y Fu(;)7 b Ft(;)p Fx(\).)14 b(In)h(particular,)f (function)h(sym)o(b)q(ols)e(from)100 2144 y Ft(F)134 2150 y Fr(1)166 2144 y Fx(are)h(blac)o(k)f(and)h(those)g(from)e Ft(F)664 2126 y Fn(0)690 2144 y Fx(are)i(white.)f(In)h(the)g(disjoin)o (t)f(union)g(case,)i(it)e(is)h(con)o(v)o(enien)o(t)g(to)100 2194 y(use)g(the)h(follo)o(wing)c(notation.)i(Let)h Fu(t)e Ft(2)f(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\).)566 2293 y Fu(S)591 2299 y Fr(2)610 2293 y Fx(\()p Fu(t)p Fx(\))12 b(=)713 2234 y Fg(\032)765 2267 y Fu(S)792 2252 y Fp(w)790 2279 y(P)819 2267 y Fx(\()p Fu(t)p Fx(\))42 b(if)13 b Fu(t)h Fx(is)g(top)g(blac)o(k.)765 2317 y Fu(S)792 2302 y Fp(b)790 2329 y(P)818 2317 y Fx(\()p Fu(t)p Fx(\))43 b(if)13 b Fu(t)h Fx(is)g(top)g(white.)100 2420 y Fk(Definition)i(6.32.) 21 b Fx(W)m(e)13 b(de\014ne)i Fu(>)674 2426 y Fr(2)707 2420 y Fx(on)f Ft(T)c Fx(\()p Ft(F)848 2426 y Fr(1)876 2420 y Ft(])f(F)946 2402 y Fn(0)958 2420 y Fu(;)e Ft(V)s Fx(\))14 b(b)o(y:)g Fu(s)e(>)1168 2426 y Fr(2)1198 2420 y Fu(t)i Fx(if)f(either)125 2518 y(\(1\))21 b Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))12 b Fu(>)g(r)q(ank)q Fx(\()p Fu(t)p Fx(\),)h(or)125 2567 y(\(2\))21 b Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))12 b(=)g Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))i(and)f(either) 200 2640 y(\(a\))21 b Fu(top)330 2625 y Fp(b)347 2640 y Fx(\()p Fu(s)p Fx(\))12 b Fu(>)442 2646 y Fr(1)472 2640 y Fu(top)528 2625 y Fp(b)545 2640 y Fx(\()p Fu(t)p Fx(\),)i(or)p eop %%Page: 35 35 35 34 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(35)p 100 224 1595 2 v 198 299 a Fx(\(b\))21 b Fu(top)330 284 y Fp(b)347 299 y Fx(\()p Fu(s)p Fx(\))12 b(=)g Fu(top)510 284 y Fp(b)526 299 y Fx(\()p Fu(t)p Fx(\))j(and)e Fu(S)693 305 y Fr(2)712 299 y Fx(\()p Fu(s)p Fx(\))g Fu(>)808 284 y Fp(mul)808 309 y Fr(2)881 299 y Fu(S)906 305 y Fr(2)925 299 y Fx(\()p Fu(t)p Fx(\))p Fu(:)100 401 y Fk(Lemma)j(6.33.)21 b Fx(The)15 b(relation)e Fu(>)640 407 y Fr(2)673 401 y Fx(is)h(w)o(ell-founded)f(on)g Ft(T)d Fx(\()p Ft(F)1098 407 y Fr(1)1125 401 y Ft(])f(F)1196 383 y Fn(0)1208 401 y Fu(;)e Ft(V)s Fx(\).)100 503 y Fk(Pr)o(oof.)22 b Fx(W)m(e)13 b(sho)o(w)h(b)o(y)g(induction)f(on)h Fu(r)q(ank)q Fx(\()p Fu(s)863 509 y Fr(1)882 503 y Fx(\))g(the)g(imp)q (ossibilit)o(y)d(of)i(an)h(in\014nite)f(sequence)662 580 y Fu(D)g Fx(:)25 b Fu(s)765 586 y Fr(1)796 580 y Fu(>)828 586 y Fr(2)858 580 y Fu(s)877 586 y Fr(2)908 580 y Fu(>)940 586 y Fr(2)970 580 y Fu(s)989 586 y Fr(3)1020 580 y Fu(>)1052 586 y Fr(2)1082 580 y Fu(:)7 b(:)g(:)100 656 y Fx(If)12 b Fu(r)q(ank)q Fx(\()p Fu(s)265 662 y Fr(1)284 656 y Fx(\))g(=)f(0,)h(then)i Fu(s)513 662 y Fr(1)544 656 y Fx(is)f(a)g(v)n(ariable)e(and)i(there)h(is)e(nothing)g (to)h(sho)o(w.)f(If)h Fu(r)q(ank)q Fx(\()p Fu(s)1478 662 y Fr(1)1496 656 y Fx(\))f(=)g(1,)g(then)100 706 y Fu(r)q(ank)q Fx(\()p Fu(s)225 712 y Fp(j)242 706 y Fx(\))k(=)f(1)h(for) f(an)o(y)h Fu(j)h Ft(2)e Fm(N)p Fx(,)f(and)i(either)g Fu(s)860 712 y Fp(j)893 706 y Ft(2)f(T)10 b Fx(\()p Ft(F)1015 712 y Fr(1)1034 706 y Fu(;)d Ft(V)s Fx(\))16 b(or)g Fu(s)1185 712 y Fp(j)1218 706 y Ft(2)f(T)10 b Fx(\()p Ft(F)1344 688 y Fn(0)1356 706 y Fu(;)d Ft(V)s Fx(\).)16 b(In)g(the)h(former)100 756 y(case,)d(w)o(e)g(deriv)o(e)538 820 y Fu(top)594 803 y Fp(b)610 820 y Fx(\()p Fu(s)645 826 y Fr(1)665 820 y Fx(\))d Fu(>)724 826 y Fr(1)755 820 y Fu(top)811 803 y Fp(b)827 820 y Fx(\()p Fu(s)862 826 y Fr(2)882 820 y Fx(\))g Fu(>)941 826 y Fr(1)972 820 y Fu(top)1028 803 y Fp(b)1044 820 y Fx(\()p Fu(s)1079 826 y Fr(2)1099 820 y Fx(\))g Fu(>)1158 826 y Fr(1)1189 820 y Fu(:)c(:)g(:)e(;)100 897 y Fx(whic)o(h)11 b(con)o(tradicts)h(the)g(w)o(ell-foundedness)h(of) d Fu(>)887 903 y Fr(1)906 897 y Fx(,)h(and)h(the)g(latter)g(case)g(is)f (ob)o(viously)g(imp)q(ossible.)100 947 y(Therefore,)h(let)g Fu(r)q(ank)q Fx(\()p Fu(s)480 953 y Fr(1)499 947 y Fx(\))f(=)h Fu(k)h(>)e Fx(1.)g(The)h(induction)g(h)o(yp)q(othesis)g(states)h(that)e Fu(>)1395 953 y Fr(2)1426 947 y Fx(is)g(w)o(ell-founded)100 997 y(on)18 b Ft(T)195 978 y Fp()1482 1031 y Fp(mul)1482 1057 y Fr(2)1557 1046 y Fx(of)e Fu(>)1634 1052 y Fr(2)1666 1046 y Fx(is)100 1096 y(w)o(ell-founded)i(on)h Ft(M)p Fx(\()p Ft(T)499 1078 y Fp()490 1152 y Fr(1)523 1146 y Fu(top)579 1131 y Fp(b)596 1146 y Fx(\()p Fu(s)631 1152 y Fp(j)r Fr(+1)691 1146 y Fx(\))g(steps)i(in)e Fu(D)i Fx(\(due)g(to)e(the)h(w)o (ell-foundedness)g(of)f Fu(>)1550 1152 y Fr(1)1569 1146 y Fx(\).)g(If)h Fu(s)1675 1152 y Fr(1)100 1196 y Fx(is)d(top)g(white,)h (then)g(it)f(follo)o(ws)f(that)i Fu(top)756 1181 y Fp(b)772 1196 y Fx(\()p Fu(s)807 1202 y Fp(j)825 1196 y Fx(\))e(=)g Fo(2)f Fx(=)h Fu(top)1039 1181 y Fp(b)1056 1196 y Fx(\()p Fu(s)1091 1202 y Fp(j)r Fr(+1)1151 1196 y Fx(\))i(for)f(ev)o(ery)h Fu(j)g Ft(2)d Fm(N)p Fx(.)g(Hence)k(there)100 1246 y(m)o(ust)d(b)q(e)j (an)f(index)f Fu(m)f Ft(2)g Fm(N)f Fx(suc)o(h)k(that)464 1322 y Fu(S)489 1328 y Fr(2)508 1322 y Fx(\()p Fu(s)543 1328 y Fp(m)575 1322 y Fx(\))c Fu(>)634 1305 y Fp(mul)634 1333 y Fr(2)708 1322 y Fu(S)733 1328 y Fr(2)752 1322 y Fx(\()p Fu(s)787 1328 y Fp(m)p Fr(+1)861 1322 y Fx(\))h Fu(>)921 1305 y Fp(mul)921 1333 y Fr(2)995 1322 y Fu(S)1020 1328 y Fr(2)1039 1322 y Fx(\()p Fu(s)1074 1328 y Fp(m)p Fr(+2)1148 1322 y Fx(\))g Fu(>)1208 1305 y Fp(mul)1208 1333 y Fr(2)1281 1322 y Fu(:)7 b(:)g(:)100 1399 y Fx(is)j(in\014nite.)g (This)g(con)o(tradicts)h(the)g(w)o(ell-foundedness)g(of)f Fu(>)1045 1384 y Fp(mul)1045 1409 y Fr(2)1118 1399 y Fx(on)g Ft(M)p Fx(\()p Ft(T)1271 1381 y Fp()1201 1557 y Fr(2)1231 1551 y Fu(t)p Fx(.)100 1653 y Fk(Pr)o(oof.)22 b Fx(The)16 b(lemma)d(will)i(b)q(e)h(established)h(b)o(y)f(induction)g(on)g Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\).)g(If)f Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))h(=)g(0,)f(then)100 1703 y Fu(s)d Ft(2)f(V)16 b Fx(and)c(the)h(lemma)c(holds)j(v)n(acuously)m(.)f(So)h (let)h Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))f(=)f Fu(k)i Ft(\025)f Fx(1.)f(If)h Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))g Fu(<)g(k)q Fx(,)f(then)i(there)100 1752 y(is)g(nothing)h(to)f(sho)o(w.) h(Th)o(us)g(assume)f Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))f(=)g Fu(k)q Fx(.)h(W)m(e)g(distinguish)h(the)g(cases:)100 1852 y Fs(Case)g(\(i\):)g Fu(s)g Fx(is)g(top)g(blac)o(k.)100 1902 y(It)j(follo)o(ws)e(from)g(Prop)q(osition)h(6.17)g(that)h Fu(s)g Ft(!)876 1887 y Fp(o)876 1913 y Fn(R)905 1917 y Ff(1)938 1902 y Fu(t)g Fx(implies)e Fu(top)1170 1887 y Fp(b)1187 1902 y Fx(\()p Fu(s)p Fx(\))i Ft(\))1297 1908 y Fn(R)1326 1912 y Ff(1)1360 1902 y Fu(top)1416 1887 y Fp(b)1432 1902 y Fx(\()p Fu(t)p Fx(\).)g(Therefore,)100 1957 y Fu(top)156 1942 y Fp(b)172 1957 y Fx(\()p Fu(s)p Fx(\))22 b Fu(>)277 1963 y Fr(1)316 1957 y Fu(top)372 1942 y Fp(b)389 1957 y Fx(\()p Fu(t)p Fx(\))e(and)f(further)h Fu(s)i(>)760 1963 y Fr(2)799 1957 y Fu(t)p Fx(.)d(If)g(on)g(the)h (other)g(hand)g Fu(s)h Ft(!)1338 1942 y Fp(i)1338 1968 y Fn(R)1367 1972 y Ff(1)1405 1957 y Fu(t)p Fx(,)e(then)h(w)o(e)g(ma)o (y)100 2012 y(write)15 b Fu(s)e Fx(=)g Fu(C)317 1997 y Fp(b)333 2012 y Fx([)-7 b([)o Fu(s)368 2018 y Fr(1)387 2012 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)499 2018 y Fp(j)516 2012 y Fu(;)g(:)g(:)g(:)e(;)i(s)628 2018 y Fp(n)651 2012 y Fx(])-7 b(])11 b Ft(!)721 2018 y Fn(R)750 2022 y Ff(1)780 2012 y Fu(C)813 1997 y Fp(b)830 2012 y Fx([)-7 b([)o Fu(s)865 2018 y Fr(1)884 2012 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)996 1997 y Fn(0)996 2023 y Fp(j)1013 2012 y Fu(;)g(:)g(:)g(:)e(;)i(s)1125 2018 y Fp(n)1147 2012 y Fx(])-7 b(])12 b(=)h Fu(t)p Fx(.)h(The)h (induction)f(h)o(yp)q(othe-)100 2068 y(sis)g(yields)g Fu(s)294 2074 y Fp(j)324 2068 y Fu(>)356 2074 y Fr(2)387 2068 y Fu(s)406 2053 y Fn(0)406 2079 y Fp(j)438 2068 y Fx(from)e(whic)o(h)i(w)o(e)g(immediately)d(get)j Fu(S)1047 2074 y Fr(2)1066 2068 y Fx(\()p Fu(s)p Fx(\))f Fu(>)1162 2053 y Fp(mul)1162 2078 y Fr(2)1236 2068 y Fu(S)1261 2074 y Fr(2)1280 2068 y Fx(\()p Fu(t)p Fx(\).)h(No)o(w)g Fu(s)e(>)1511 2074 y Fr(2)1542 2068 y Fu(t)i Fx(follo)o(ws)100 2124 y(b)q(ecause)h Fu(top)309 2109 y Fp(b)326 2124 y Fx(\()p Fu(s)p Fx(\))d(=)g Fu(top)489 2109 y Fp(b)506 2124 y Fx(\()p Fu(t)p Fx(\).)100 2223 y Fs(Case)i(\(ii\):)f Fu(s)i Fx(is)e(top)h(white.)100 2273 y(Here)19 b Fu(s)g Fx(=)g Fu(C)324 2258 y Fp(w)350 2273 y Fx([)-7 b([)o Fu(s)385 2279 y Fr(1)404 2273 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)516 2279 y Fp(j)534 2273 y Fu(;)g(:)g(:)g(:)t(;)g(s)645 2279 y Fp(n)668 2273 y Fx(])-7 b(])18 b Ft(!)745 2279 y Fn(R)774 2283 y Ff(1)809 2273 y Fu(C)842 2258 y Fp(w)869 2273 y Fx([)p Fu(s)900 2279 y Fr(1)918 2273 y Fu(;)7 b(:)g(:)g(:)e(;)i(s) 1030 2258 y Fn(0)1030 2284 y Fp(j)1048 2273 y Fu(;)g(:)g(:)g(:)t(;)g(s) 1159 2279 y Fp(n)1182 2273 y Fx(])18 b(=)h Fu(t)p Fx(.)e(If)h(the)h (rewrite)g(step)g(is)100 2323 y(non-destructiv)o(e,)f(then)g Fu(t)g Fx(is)f(indeed)h(equal)g(to)f Fu(C)920 2308 y Fp(w)947 2323 y Fx([)-7 b([)o Fu(s)982 2329 y Fr(1)1001 2323 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1113 2308 y Fn(0)1113 2334 y Fp(j)1130 2323 y Fu(;)g(:)g(:)g(:)e(;)i(s)1242 2329 y Fp(n)1264 2323 y Fx(])-7 b(])17 b(and)g(the)i(assertion)f(fol-) 100 2383 y(lo)o(ws)g(as)h(ab)q(o)o(v)o(e.)f(Otherwise)j Fu(s)607 2368 y Fn(0)607 2394 y Fp(j)645 2383 y Fx(=)706 2372 y(^)697 2383 y Fu(C)730 2368 y Fp(w)757 2383 y Ft(f)-14 b(f)o Fu(t)799 2389 y Fr(1)818 2383 y Fu(;)7 b(:)g(:)g(:)e(;)i(t)926 2389 y Fp(m)957 2383 y Ft(g)-14 b(g)o Fx(.)19 b(Since)g Fu(r)q(ank)q Fx(\()p Fu(s)1253 2389 y Fp(j)1271 2383 y Fx(\))h Fu(>)h(r)q(ank)q Fx(\()p Fu(t)1481 2389 y Fp(i)1494 2383 y Fx(\))e(for)g(ev)o(ery)100 2439 y Fu(i)11 b Ft(2)h(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(m)p Ft(g)p Fx(,)k(w)o(e)i(infer)f(that) h([)p Fu(s)653 2445 y Fp(j)670 2439 y Fx(])e Fu(>)725 2424 y Fp(mul)725 2449 y Fr(2)799 2439 y Fx([)p Fu(t)826 2445 y Fr(1)844 2439 y Fu(;)c(:)g(:)g(:)e(;)i(t)952 2445 y Fp(m)983 2439 y Fx(].)k(Again,)g(w)o(e)i(conclude)g Fu(S)1403 2445 y Fr(2)1422 2439 y Fx(\()p Fu(s)p Fx(\))f Fu(>)1517 2424 y Fp(mul)1517 2449 y Fr(2)1591 2439 y Fu(S)1616 2445 y Fr(2)1635 2439 y Fx(\()p Fu(t)p Fx(\).)100 2488 y(No)o(w)h Fu(s)f(>)257 2494 y Fr(2)288 2488 y Fu(t)h Fx(follo)o(ws)g(from)f Fu(top)607 2473 y Fp(b)624 2488 y Fx(\()p Fu(s)p Fx(\))g(=)g Fo(2)f Fx(=)h Fu(top)873 2473 y Fp(b)890 2488 y Fx(\()p Fu(t)p Fx(\).)h Fe(2)100 2590 y Fk(Lemma)j(6.35.)21 b Fx(If)10 b Fu(l)j Ft(!)e Fu(r)i Ft(\()e Fu(s)590 2596 y Fr(1)620 2590 y Ft(#)h Fu(t)668 2596 y Fr(1)686 2590 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)798 2596 y Fp(n)832 2590 y Ft(#)k Fu(t)879 2596 y Fp(n)912 2590 y Fx(is)f(a)h(rule)f(from)f Ft(R)1191 2596 y Fr(1)1220 2590 y Fx(and)h Fu(\033)j Fx(:)e Ft(V)k(!)d(T)e Fx(\()p Ft(F)1534 2596 y Fr(1)1555 2590 y Ft(])r(F)1619 2572 y Fn(0)1630 2590 y Fu(;)d Ft(V)t Fx(\))100 2640 y(is)13 b(a)h(substitution,)g(then)g Fu(l)q(\033)f(>)595 2646 y Fr(2)626 2640 y Fu(s)645 2646 y Fp(j)662 2640 y Fu(\033)j Fx(and)d Fu(l)q(\033)g(>)864 2646 y Fr(2)895 2640 y Fu(t)910 2646 y Fp(j)927 2640 y Fu(\033)i Fx(for)f(all)e Fu(j)i Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)p eop %%Page: 36 36 36 35 bop 100 197 a Fw(36)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fk(Pr)o(oof.)22 b Fx(According)15 b(to)g(Prop)q(osition)g(4.18,)f Fu(\033)i Fx(can)g(b)q(e)f(decomp)q (osed)h(in)o(to)e Fu(\033)1358 305 y Fr(2)1387 299 y Ft(\016)c Fu(\033)1442 305 y Fr(1)1460 299 y Fx(,)15 b(where)h Fu(\033)1632 305 y Fr(1)1666 299 y Fx(is)100 349 y(blac)o(k)f(and)h Fu(\033)317 355 y Fr(2)351 349 y Fx(is)f(top)h(white.)g(Since)g(\()p Ft(F)757 355 y Fr(1)776 349 y Fu(;)7 b Ft(R)829 355 y Fr(1)848 349 y Fx(\))16 b(is)g(decreasing)h(w.r.t.)d Fu(>)1271 355 y Fr(1)1290 349 y Fx(,)i(w)o(e)g(ha)o(v)o(e)f Fu(l)q(\033)1515 355 y Fr(1)1549 349 y Fu(>)1581 355 y Fr(1)1614 349 y Fu(s)1633 355 y Fp(j)1651 349 y Fu(\033)1675 355 y Fr(1)100 399 y Fx(and)i Fu(l)q(\033)221 405 y Fr(1)257 399 y Fu(>)289 405 y Fr(1)326 399 y Fu(t)341 405 y Fp(j)358 399 y Fu(\033)382 405 y Fr(1)418 399 y Fx(for)g(all)f Fu(j)k Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)16 b(Since)i Ft(V)s Fu(ar)q Fx(\()p Fu(s)1054 405 y Fp(j)1073 399 y Fx(\))g Ft(\022)g(V)s Fu(ar)q Fx(\()p Fu(l)q Fx(\))g(and)g Ft(V)s Fu(ar)q Fx(\()p Fu(t)1476 405 y Fp(j)1494 399 y Fx(\))g Ft(\022)f(V)s Fu(ar)q Fx(\()p Fu(l)q Fx(\))100 448 y(for)c(an)o(y)h Fu(j)g Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(,)13 b(it)h(follo)o(ws)e Fu(r)q(ank)q Fx(\()p Fu(l)q(\033)q Fx(\))g Ft(\025)g Fu(r)q(ank)q Fx(\()p Fu(s)1037 454 y Fp(j)1055 448 y Fu(\033)q Fx(\))i(and)g Fu(r)q(ank)q Fx(\()p Fu(l)q(\033)q Fx(\))f Ft(\025)f Fu(r)q(ank)q Fx(\()p Fu(t)1529 454 y Fp(j)1546 448 y Fu(\033)q Fx(\).)i(No)o(w)100 498 y(it)h(follo)o(ws)f (from)h Fu(top)438 483 y Fp(b)454 498 y Fx(\()p Fu(l)q(\033)q Fx(\))h(=)f Fu(l)q(\033)624 504 y Fr(1)657 498 y Fu(>)689 504 y Fr(1)723 498 y Fu(s)742 504 y Fp(j)760 498 y Fu(\033)784 504 y Fr(1)817 498 y Fx(=)g Fu(top)920 483 y Fp(b)937 498 y Fx(\()p Fu(s)972 504 y Fp(j)990 498 y Fu(\033)q Fx(\))h(and)g Fu(top)1186 483 y Fp(b)1203 498 y Fx(\()p Fu(l)q(\033)q Fx(\))f Fu(>)1320 504 y Fr(1)1354 498 y Fu(top)1410 483 y Fp(b)1427 498 y Fx(\()p Fu(t)1458 504 y Fp(j)1475 498 y Fu(\033)q Fx(\))h(that)g(also)100 548 y Fu(l)q(\033)d(>)182 554 y Fr(2)212 548 y Fu(s)231 554 y Fp(j)249 548 y Fu(\033)i Fx(and)f Fu(l)q(\033)f(>)451 554 y Fr(2)481 548 y Fu(t)496 554 y Fp(j)514 548 y Fu(\033)i Fx(for)e(all)g Fu(j)h Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)13 b Fe(2)141 648 y Fx(The)g(preceding)h(lemmata)c(sho)o(w) j(that)f Fu(>)803 654 y Fr(2)835 648 y Fx(has)h(three)h(out)f(of)f(the) h(four)g(prop)q(erties)h(required)g(for)100 698 y(sho)o(wing)e(that)i (the)f(CTRS)g(\()p Ft(F)594 704 y Fr(1)621 698 y Ft(])7 b(F)690 679 y Fn(0)702 698 y Fu(;)g Ft(R)756 704 y Fr(1)774 698 y Fx(\))14 b(is)f(decreasing.)h(Unfortunately)m(,)e(it)h(lac)o(ks)g (the)h(subterm)100 747 y(prop)q(ert)o(y)m(.)e(If,)g(for)g(example,)f Fu(g)q(;)c(a)k Ft(2)g(F)709 729 y Fn(0)721 747 y Fx(,)h(then)h Fu(g)q Fx(\()p Fu(a)p Fx(\))f Ft(6)p Fu(>)957 753 y Fr(2)988 747 y Fu(a)p Fx(.)g(Ho)o(w)o(ev)o(er,)g(w)o(e)h(can)g(extend)g Fu(>)1511 753 y Fr(2)1543 747 y Fx(with)f(the)100 797 y(subterm)f(prop)q(ert)o(y)m(.)h(T)m(o)e(b)q(e)j(exact,)f(w)o(e)f (de\014ne)i Fu(>)875 803 y Fr(3)905 797 y Fx(=)f(\()p Fu(>)997 803 y Fr(2)1028 797 y Ft([)p 1068 799 3 25 v 8 w Fu(>)q Fx(\))1113 782 y Fr(+)1140 797 y Fx(.)f Fu(>)1195 803 y Fr(3)1226 797 y Fx(is)g(a)h(relation)f(whic)o(h)g(has)h(the)100 847 y(subterm)j(prop)q(ert)o(y)h(and)f(ob)o(viously)f(Lemmata)e(6.34)i (and)h(6.35)f(also)g(hold)h(when)h Fu(>)1467 853 y Fr(2)1501 847 y Fx(is)f(replaced)100 897 y(with)i Fu(>)230 903 y Fr(3)249 897 y Fx(.)h(But)h(it)e(is)h(not)h(ob)o(vious)e(that)h Fu(>)816 903 y Fr(3)853 897 y Fx(is)g(a)g(w)o(ell-founded)f(partial)h (ordering)g(since)h Fu(>)1629 903 y Fr(2)1666 897 y Fx(is)100 947 y(not)12 b(closed)h(under)h(con)o(texts.)f(In)g(order)g(to)g(pro)o (v)o(e)g(this,)f(it)g(su\016ces)i(to)f(pro)o(v)o(e)f(its)h(w)o (ell-foundedness)100 996 y(b)q(ecause)i Fu(>)285 1002 y Fr(3)318 996 y Fx(is)f(transitiv)o(e)g(b)o(y)f(de\014nition.)100 1096 y Fk(Lemma)j(6.36.)21 b Fx(The)15 b(relation)e Fu(>)640 1102 y Fr(3)673 1096 y Fx(=)f(\()p Fu(>)765 1102 y Fr(2)795 1096 y Ft([)p 835 1098 V 9 w Fu(>)p Fx(\))880 1081 y Fr(+)922 1096 y Fx(is)h(w)o(ell-founded)g(on)h Ft(T)c Fx(\()p Ft(F)1346 1102 y Fr(1)1374 1096 y Ft(])f(F)1445 1078 y Fn(0)1457 1096 y Fu(;)e Ft(V)s Fx(\).)100 1196 y Fk(Pr)o(oof.)22 b Fx(Let)14 b Fu(s;)7 b(t)k Ft(2)h(T)e Fx(\()p Ft(F)525 1202 y Fr(1)553 1196 y Ft(])f(F)623 1178 y Fn(0)635 1196 y Fu(;)e Ft(V)s Fx(\).)14 b(W)m(e)f(\014rst)i(sho) o(w:)125 1296 y(\(1\))21 b(If)14 b Fu(s)g Fx(is)g(top)g(blac)o(k)f(and) h Fu(s)p 620 1298 V 18 w(>)c(t)p Fx(,)j(then)i Fu(s)d(>)856 1302 y Fr(2)886 1296 y Fu(t)p Fx(.)125 1346 y(\(2\))21 b(If)14 b Fu(s)g Fx(is)g(top)g(white)g(and)f Fu(s)p 624 1348 V 19 w(>)c(t)p Fx(,)14 b(then)g Fu(s)e Ft(\025)860 1352 y Fr(2)891 1346 y Fu(t)p Fx(.)100 1446 y(\(1\))i(If)h Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))e Fu(>)g(r)q(ank)q Fx(\()p Fu(t)p Fx(\),)h(then)h(the)h(claim)c(is)j(true.)g(Otherwise)h Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))d(=)g Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))i(and)f(it)h(is)100 1496 y(clear)h(that)f Fu(s)g Fx(=)g Fu(C)406 1480 y Fp(b)422 1496 y Ft(f)-14 b(f)p Fu(s)469 1502 y Fr(1)488 1496 y Fu(;)7 b(:)g(:)g(:)t(;)g(s)599 1502 y Fp(n)622 1496 y Ft(g)-14 b(g)15 b Fx(and)h Fu(t)e Fx(=)833 1485 y(^)823 1496 y Fu(C)856 1480 y Fp(b)873 1496 y Ft(f)-14 b(f)o Fu(s)919 1502 y Fp(i)934 1496 y Fu(;)7 b(s)972 1502 y Fp(i)p Fr(+1)1028 1496 y Fu(;)g(:)g(:)g(:)t(;)g (s)1139 1502 y Fp(m)p Fn(\000)p Fr(1)1213 1496 y Fu(;)g(s)1251 1502 y Fp(m)1283 1496 y Ft(g)-14 b(g)o Fx(,)15 b(where)i Fu(C)1492 1480 y Fp(b)1508 1496 y Ft(f)p Fu(;)7 b(:)g(:)g(:)e(;)i Ft(g)p 1665 1498 V 18 w Fu(>)109 1535 y Fx(^)100 1545 y Fu(C)133 1530 y Fp(b)149 1545 y Ft(f)p Fu(;)g(:)g(:)g(:)t(;)g Ft(g)15 b Fx(and)h(1)e Ft(\024)g Fu(i)h Ft(\024)f Fu(m)h Ft(\024)g Fu(n)p Fx(.)g(Therefore,)h Fu(top)944 1530 y Fp(b)961 1545 y Fx(\()p Fu(s)p Fx(\))f(=)f Fu(C)1106 1530 y Fp(b)1123 1545 y Ft(f)p Fu(;)7 b(:)g(:)g(:)t(;)g Ft(g)p 1279 1547 V 19 w Fu(>)1328 1535 y Fx(^)1318 1545 y Fu(C)1351 1530 y Fp(b)1368 1545 y Ft(f)p Fu(;)g(:)g(:)g(:)t(;)g Ft(g)14 b Fx(=)g Fu(top)1618 1530 y Fp(b)1635 1545 y Fx(\()p Fu(t)p Fx(\).)100 1595 y(Since)g Fu(>)240 1601 y Fr(1)273 1595 y Fx(has)g(the)g(subterm)g(prop)q(ert)o(y)m(,)g(it)f (follo)o(ws)g Fu(top)992 1580 y Fp(b)1008 1595 y Fx(\()p Fu(s)p Fx(\))g Fu(>)1104 1601 y Fr(1)1134 1595 y Fu(top)1190 1580 y Fp(b)1207 1595 y Fx(\()p Fu(t)p Fx(\).)g(Th)o(us)i Fu(s)d(>)1448 1601 y Fr(2)1478 1595 y Fu(t)p Fx(.)100 1645 y(\(2\))j(As)h(in)f(case)i(\(1\),)e(w)o(e)h(ma)o(y)d(assume)i Fu(r)q(ank)q Fx(\()p Fu(s)p Fx(\))g(=)g Fu(r)q(ank)q Fx(\()p Fu(t)p Fx(\))g(and)h Fu(s)e Fx(=)h Fu(C)1301 1630 y Fp(w)1327 1645 y Ft(f)-14 b(f)p Fu(s)1374 1651 y Fr(1)1393 1645 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)1505 1651 y Fp(n)1527 1645 y Ft(g)-14 b(g)16 b Fx(as)f(w)o(ell)100 1695 y(as)f Fu(t)f Fx(=)234 1684 y(^)224 1695 y Fu(C)257 1680 y Fp(w)284 1695 y Ft(f)-14 b(f)p Fu(s)331 1701 y Fp(i)345 1695 y Fu(;)7 b(s)383 1701 y Fp(i)p Fr(+1)439 1695 y Fu(;)g(:)g(:)g(:)t(;)g(s)550 1701 y Fp(m)p Fn(\000)p Fr(1)624 1695 y Fu(;)g(s)662 1701 y Fp(m)694 1695 y Ft(g)-14 b(g)o Fx(,)15 b(where)g Fu(C)901 1680 y Fp(w)928 1695 y Ft(f)p Fu(;)7 b(:)g(:)g(:)t(;)g Ft(g)p 1084 1697 V 18 w Fu(>)1132 1684 y Fx(^)1122 1695 y Fu(C)1155 1680 y Fp(w)1182 1695 y Ft(f)p Fu(;)g(:)g(:)g(:)t(;)g Ft(g)14 b Fx(and)h(1)d Ft(\024)h Fu(i)h Ft(\024)f Fu(m)g Ft(\024)g Fu(n)p Fx(.)100 1745 y(Clearly)m(,)f Fu(top)308 1730 y Fp(b)325 1745 y Fx(\()p Fu(s)p Fx(\))g(=)g Fo(2)f Fx(=)h Fu(top)574 1730 y Fp(b)591 1745 y Fx(\()p Fu(t)p Fx(\))i(and)g Fu(S)758 1751 y Fr(2)777 1745 y Fx(\()p Fu(t)p Fx(\))d Ft(\022)h Fu(S)904 1751 y Fr(2)923 1745 y Fx(\()p Fu(s)p Fx(\).)i(No)o(w)g Fu(s)e Ft(\025)1158 1751 y Fr(2)1188 1745 y Fu(t)i Fx(is)g(a)g(direct)g(consequence.)141 1794 y(No)o(w)g(supp)q(ose)h(that)f(there)h(is)f(an)f(in\014nite)h(sequence) 662 1869 y Fu(D)f Fx(:)25 b Fu(s)765 1875 y Fr(1)796 1869 y Fu(>)828 1875 y Fr(3)858 1869 y Fu(s)877 1875 y Fr(2)908 1869 y Fu(>)940 1875 y Fr(3)970 1869 y Fu(s)989 1875 y Fr(3)1020 1869 y Fu(>)1052 1875 y Fr(3)1082 1869 y Fu(:)7 b(:)g(:)100 1944 y Fx(It)14 b(follo)o(ws)e(immedia)o(tely)f (from)h(the)j(ab)q(o)o(v)o(e)f(claim)d(that)662 2019 y Fu(D)i Fx(:)25 b Fu(s)765 2025 y Fr(1)796 2019 y Ft(\025)828 2025 y Fr(2)858 2019 y Fu(s)877 2025 y Fr(2)908 2019 y Ft(\025)940 2025 y Fr(2)970 2019 y Fu(s)989 2025 y Fr(3)1020 2019 y Ft(\025)1052 2025 y Fr(2)1082 2019 y Fu(:)7 b(:)g(:)100 2094 y Fx(If)15 b(there)h(w)o(ere)h(only)d (\014nitely)h(man)o(y)f Fu(s)715 2100 y Fp(j)746 2094 y Fu(>)778 2100 y Fr(2)811 2094 y Fu(s)830 2100 y Fp(j)r Fr(+1)906 2094 y Fx(steps)i(in)f Fu(D)q Fx(,)h(then)g(there)g(w)o(ould) f(b)q(e)h(an)f(in\014nite)100 2144 y(subsequence)645 2206 y Fu(D)680 2189 y Fn(0)704 2206 y Fx(:)25 b Fu(s)760 2212 y Fp(l)p 794 2208 V 791 2206 a Fu(>)9 b(s)851 2212 y Fp(l)p Fr(+1)p 928 2208 V 925 2206 a Fu(>)g(s)985 2212 y Fp(l)p Fr(+2)p 1062 2208 V 1058 2206 a Fu(>)h(:)d(:)g(:)100 2281 y Fx(in)17 b(con)o(trast)h(to)g(the)g(w)o(ell-foundedness)g(of)p 829 2283 V 26 w Fu(>)p Fx(.)f(Hence)i(there)g(are)f(in\014nitely)f(man) o(y)e Fu(s)1510 2287 y Fp(j)1546 2281 y Fu(>)1578 2287 y Fr(2)1615 2281 y Fu(s)1634 2287 y Fp(j)r Fr(+1)100 2331 y Fx(steps)g(in)e Fu(D)j Fx(whic)o(h)d(con)o(tradicts)i(the)g(w)o (ell-foundedness)f(of)f Fu(>)1101 2337 y Fr(2)1120 2331 y Fx(.)h Fe(2)100 2430 y Fk(Pr)o(oposition)i(6.37.)21 b Fx(If)10 b(the)h(CTRS)g(\()p Ft(F)764 2436 y Fr(1)782 2430 y Fu(;)c Ft(R)836 2436 y Fr(1)855 2430 y Fx(\))k(is)f(decreasing,) h(then)h(\()p Ft(F)1271 2436 y Fr(1)1293 2430 y Ft(])s(F)1357 2412 y Fn(0)1369 2430 y Fu(;)7 b Ft(R)1422 2436 y Fr(1)1441 2430 y Fx(\))k(is)g(decreasing)100 2480 y(for)i(an)o(y)h Ft(F)276 2462 y Fn(0)301 2480 y Fx(with)g Ft(F)430 2486 y Fr(1)457 2480 y Ft(\\)9 b(F)528 2462 y Fn(0)551 2480 y Fx(=)j Ft(;)p Fx(.)100 2580 y Fk(Pr)o(oof.)22 b Fx(Let)10 b(\()p Ft(F)384 2586 y Fr(1)403 2580 y Fu(;)d Ft(R)457 2586 y Fr(1)475 2580 y Fx(\))k(b)q(e)g(decreasing)g(w.r.t.)e(the)i (partial)f(ordering)g Fu(>)1250 2586 y Fr(1)1279 2580 y Ft(\022)i(T)e Fx(\()p Ft(F)1402 2586 y Fr(1)1421 2580 y Fu(;)d Ft(V)s Fx(\))r Ft(\002)r(T)k Fx(\()p Ft(F)1600 2586 y Fr(1)1619 2580 y Fu(;)c Ft(V)s Fx(\).)100 2630 y(De\014ne)14 b Fu(>)260 2636 y Fr(3)293 2630 y Fx(on)g Ft(T)c Fx(\()p Ft(F)434 2636 y Fr(1)462 2630 y Ft(])f(F)533 2612 y Fn(0)544 2630 y Fu(;)e Ft(V)t Fx(\))13 b(as)h(ab)q(o)o(v)o(e.)g (Then)g(\()p Ft(F)960 2636 y Fr(1)988 2630 y Ft(])9 b(F)1059 2612 y Fn(0)1070 2630 y Fu(;)e Ft(R)1124 2636 y Fr(1)1143 2630 y Fx(\))14 b(is)f(decreasing)i(w.r.t.)e Fu(>)1558 2636 y Fr(3)1577 2630 y Fx(.)g Fe(2)p eop %%Page: 37 37 37 36 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(37)p 100 224 1595 2 v 100 299 a Fk(Pr)o(oposition)16 b(6.38.)21 b Fx(Let)14 b(\()p Ft(F)607 305 y Fr(1)626 299 y Fu(;)7 b Ft(R)680 305 y Fr(1)698 299 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)851 305 y Fp(n)874 299 y Fu(;)g Ft(R)927 305 y Fp(n)950 299 y Fx(\))14 b(b)q(e)h(pairwise)e(constructor-sharing) j(CTRSs)100 349 y(suc)o(h)f(that)g(ev)o(ery)g Ft(R)430 355 y Fp(j)462 349 y Fx(is)g(\014nite.)f(If)h(the)g(systems)g(are)g (decreasing,)g(then)g(the)h(function)e Fu(n)-7 b(f)20 b Fx(de\014ned)100 399 y(b)o(y)495 464 y Fu(n)-7 b(f)t Fx(\()p Fu(s)p Fx(\))13 b(=)f Ft(f)p Fu(t)f Ft(2)g(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\))14 b Ft(j)f Fu(s)f Fo(;)990 447 y Fn(\003)1020 464 y Fu(t;)21 b(t)11 b Ft(2)g Fu(N)5 b(F)h Fx(\()p Fo(;)p Fx(\))p Ft(g)100 542 y Fx(is)13 b(computable.)100 645 y Fk(Pr)o(oof.)22 b Fx(By)17 b(Theorem)g(6.27,)e Fo(;)i Fx(is)g(terminating.)e(The)i(computabilit)o(y)e(of)h(the)i (function)f Fu(n)-7 b(f)22 b Fx(is)100 695 y(sho)o(wn)10 b(b)o(y)g(induction)f(on)h(the)h(w)o(ell-founded)e(partial)g(ordering)i (\()p Ft(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))p Fu(;)g Fo(;)1330 680 y Fr(+)1357 695 y Fx(\).)j(Let)g Fu(s)i Ft(2)f(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\),)100 745 y(and)13 b(let)188 823 y Fu(n)-7 b(f)p 219 823 13 2 v 232 829 a Fp(j)249 823 y Fx(\()p Fu(s)p Fx(\))13 b(=)e Ft(f)p Fu(t)h Ft(2)f(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))14 b Ft(j)g Fu(s)e Ft(!)702 805 y Fr(+)702 835 y Fn(R)731 839 y Fl(j)759 823 y Fu(t;)20 b(t)12 b Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)1046 829 y Fp(j)1063 823 y Fx(\))p Ft(g)12 b Fx(=)g Ft(f)p Fu(t)f Ft(2)g(T)f Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))15 b Ft(j)e Fu(s)f Fo(;)1501 829 y Fn(R)1530 833 y Fl(j)1558 823 y Fu(t)p Ft(g)p Fu(:)100 907 y Fx(Note)h(that)f(the)h(CTRSs)g(\()p Ft(F)t Fu(;)7 b Ft(R)603 913 y Fr(1)622 907 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)s Fu(;)g Ft(R)834 913 y Fp(n)857 907 y Fx(\))12 b(are)h(decreasing)h(b)o(y)e(Prop)q (osition)g(6.37.)f(Th)o(us,)h(for)100 957 y(an)o(y)j Fu(j)j Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(,)15 b(the)i(\014nite)f(set)i Fu(n)-7 b(f)p 746 957 V 759 963 a Fp(j)776 957 y Fx(\()p Fu(s)p Fx(\))17 b(is)f(computable,)e(see)k(Dersho)o(witz)e Fs(et)h(al.)f Fx(\(1988\).)f(If)100 1006 y Fu(n)-7 b(f)p 131 1006 V 144 1012 a Fp(j)161 1006 y Fx(\()p Fu(s)p Fx(\))18 b(is)e(empt)o(y)g (for)h(all)f Fu(j)j Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(,)16 b(then)i Fu(s)f Ft(2)f Fu(N)5 b(F)h Fx(\()p Ft(F)t Fu(;)h Ft(R)1172 1012 y Fp(j)1190 1006 y Fx(\))17 b(for)f(all)g Fu(j)j Ft(2)d(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)e(;)i(n)p Ft(g)16 b Fx(and)100 1056 y(th)o(us)h Fu(s)g Ft(2)f Fu(N)5 b(F)h Fx(\()p Fo(;)p Fx(\).)16 b(In)h(this)g(case)g Fu(n)-7 b(f)t Fx(\()p Fu(s)p Fx(\))19 b(=)d Ft(f)p Fu(s)p Ft(g)h Fx(and)g(w)o(e)g(are)g(done.)g(Otherwise,)h(the)g(\014nite)f(set)100 1106 y Fu(R)p Fx(\()p Fu(s)p Fx(\))12 b(=)239 1075 y Fg(S)273 1085 y Fp(n)273 1118 y(j)r Fr(=1)340 1106 y Fu(n)-7 b(f)p 371 1106 V 384 1112 a Fp(j)401 1106 y Fx(\()p Fu(s)p Fx(\))13 b(of)e(all)g(one)h(step)h(reducts)h(of)d Fu(s)i Fx(w.r.t.)e Fo(;)g Fx(is)h(not)g(empt)o(y)m(.)e(Let)i Fu(t)g Ft(2)f Fu(R)p Fx(\()p Fu(s)p Fx(\).)h(Since)100 1156 y Fu(s)j Fo(;)g Fu(t)p Fx(,)g(it)h(follo)o(ws)e(from)g(the)j (induction)e(h)o(yp)q(othesis)i(that)f(the)g(\014nite)h(set)f Fu(n)-7 b(f)t Fx(\()p Fu(t)p Fx(\))17 b(is)f(computable.)100 1206 y(Hence)f(the)f(\014nite)h(set)f Fu(n)-7 b(f)t Fx(\()p Fu(s)p Fx(\))13 b(=)615 1174 y Fg(S)650 1218 y Fp(t)p Fn(2)p Fp(R)p Fr(\()p Fp(s)p Fr(\))761 1206 y Fu(n)-7 b(f)t Fx(\()p Fu(t)p Fx(\))14 b(is)g(computable.)e Fe(2)100 1314 y Fk(Theorem)k(6.39.)21 b Fx(Let)28 b(\()p Ft(F)556 1320 y Fr(1)574 1314 y Fu(;)7 b Ft(R)628 1320 y Fr(1)647 1314 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)799 1320 y Fp(n)822 1314 y Fu(;)g Ft(R)876 1320 y Fp(n)898 1314 y Fx(\))27 b(b)q(e)h(pairwise)f(constructor-sharing)h(CTRSs)100 1364 y(suc)o(h)20 b(that)h(ev)o(ery)f Ft(R)447 1370 y Fp(j)484 1364 y Fx(is)g(\014nite.)g(If)g(the)g(systems)g(are)h (decreasing)g(and)f(con\015uen)o(t,)g(then)h(their)100 1414 y(com)o(bined)d(system)i(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))20 b(is)f(semi-complete)f(and)i(the)h(unique)e(normal)f(form)g Fu(s)p Ft(#)i Fx(of)g(a)f(term)100 1463 y Fu(s)c Ft(2)g(T)10 b Fx(\()p Ft(F)t Fu(;)d Ft(V)s Fx(\))17 b(with)f(resp)q(ect)i(to)e(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))16 b(is)g(computable)f(b)o(y)h(computing)f (the)h(normal)e(form)h(of)g Fu(s)100 1513 y Fx(with)e(resp)q(ect)j(to)e Fo(;)p Fx(.)100 1617 y Fk(Pr)o(oof.)22 b Fx(Since)17 b(the)h(CTRSs)g(\()p Ft(F)648 1623 y Fr(1)666 1617 y Fu(;)7 b Ft(R)720 1623 y Fr(1)739 1617 y Fx(\))p Fu(;)g(:)g(:)g(:)e(;)i Fx(\()p Ft(F)892 1623 y Fp(n)914 1617 y Fu(;)g Ft(R)968 1623 y Fp(n)990 1617 y Fx(\))18 b(are)g(decreasing,)g(they)g(are)f (particularly)100 1666 y(terminating.)12 b(Hence)k(they)f(are)g (complete.)f(Semi-compl)o(eteness)g(of)g(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))15 b(is)f(a)g(consequence)j(of)100 1716 y(Theorem)11 b(6.14.)f(It)i(remains)f(to)g(pro)o(v)o(e)h(the)h (computabilit)o(y)c(of)i(the)i(function)e(whic)o(h)h(calculates)g(the) 100 1766 y(unique)g(normal)e(form)h Fu(s)p Ft(#)h(2)f Fu(N)5 b(F)h Fx(\()p Ft(F)s Fu(;)h Ft(R)p Fx(\))13 b(of)e(a)h(giv)o(en) g(term)g Fu(s)g Ft(2)f(T)g Fx(\()p Ft(F)t Fu(;)c Ft(V)s Fx(\).)12 b(According)h(to)f(Theorem)100 1816 y(6.30,)f Fo(;)h Fx(is)g(complete.)g(Moreo)o(v)o(er,)g(b)o(y)h(Prop)q(osition)f (6.38,)f(the)i(unique)g(normal)e(form)g Fu(s)1522 1783 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 1522 1783 a 16 x Fo(;)1522 1825 y currentpoint grestore moveto 1522 1825 a 1535 1816 a Fx(of)h Fu(s)h Fx(with)100 1866 y(resp)q(ect)j(to)e Fo(;)f Fx(is)h(computable.)e(By)i(Corollary)f(6.31,)f Fu(s)p Ft(#)f Fx(=)h Fu(s)1103 1833 y gsave currentpoint currentpoint translate 90 rotate neg exch neg exch translate 1103 1833 a 15 x Fo(;)1103 1875 y currentpoint grestore moveto 1103 1875 a 1117 1866 a Fx(whic)o(h)i(concludes)h (the)f(pro)q(of.)f Fe(2)585 1969 y Fk(6.4.)23 b(the)16 b(simplifying)f(pr)o(oper)m(ty)141 2078 y Fx(In)c(the)g(preceding)g (subsection,)g(w)o(e)g(ha)o(v)o(e)f(seen)i(that)e(decreasing)i(CTRSs)e (b)q(eha)o(v)o(e)h(quite)f(\\nicely")100 2128 y(w.r.t.)g(com)o (binations)g(with)h(shared)i(constructors.)h(The)e(ob)r(jectiv)o(e)g (of)f(this)h(subsection)h(is)f(to)g(pro)o(v)o(e)100 2178 y(that)e(the)h(related)f(simplifying)d(prop)q(ert)o(y)k(b)q(eha)o(v)o (es)g(ev)o(en)g(nicer.)g(That)f(is)g(to)g(sa)o(y)m(,)f(it)h(is)g(mo)q (dular)e(ev)o(en)100 2228 y(for)14 b(comp)q(osable)f(CTRSs.)h(This)g (will)f(b)q(e)i(pro)o(v)o(en)g(b)o(y)f(a)g(straigh)o(tforw)o(ard)g (reduction)h(to)f(Theorem)100 2278 y(5.16.)100 2384 y Fk(Definition)i(6.40.)21 b Fx(A)f(CTRS)f Ft(R)i Fx(is)f Fs(simplifying)f Fx(\(Kaplan,)g(1987\))g(if)g(there)j(exists)e(a)g (simpli-)100 2434 y(\014cation)g(ordering)g Fu(>)h Fx(with)f Fu(l)j(>)g(r)o(;)7 b(l)22 b(>)h(s)813 2440 y Fp(j)831 2434 y Fx(,)c(and)i Fu(l)i(>)g(t)1055 2440 y Fp(j)1072 2434 y Fx(,)d(for)g(eac)o(h)h(rewrite)g(rule)g Fu(l)i Ft(!)f Fu(r)h Ft(\()100 2484 y Fu(s)119 2490 y Fr(1)152 2484 y Ft(#)13 b Fu(t)201 2490 y Fr(1)220 2484 y Fu(;)7 b(:)g(:)g(:)e(;)i(s)332 2490 y Fp(n)368 2484 y Ft(#)13 b Fu(t)417 2490 y Fp(n)454 2484 y Fx(of)g Ft(R)h Fx(and)g(ev)o(ery)h (index)e Fu(j)h Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(g)p Fx(.)141 2590 y(If)13 b(a)f(\014nite)i(CTRS)e(is)h(simplifying)o (,)d(then)j(it)g(is)g(also)f(decreasing.)i(The)f(con)o(v)o(erse)h(is)f (not)g(true;)g(see)100 2640 y(Dersho)o(witz)h Fs(et)h(al.)e Fx(\(1988\).)p eop %%Page: 38 38 38 37 bop 100 197 a Fw(38)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fk(Definition)16 b(6.41.)21 b Fx(Let)16 b(\()p Ft(F)t Fu(;)7 b Ft(R)o Fx(\))16 b(b)q(e)g(a)f(CTRS)g (without)h(extra)f(v)n(ariables.)g(With)g(\()p Ft(F)t Fu(;)7 b Ft(R)o Fx(\))16 b(w)o(e)g(as-)100 349 y(so)q(ciate)e(the)h (unconditional)d(TRS)i(\()p Ft(F)t Fu(;)7 b Ft(R)770 331 y Fp(u)792 349 y Fx(\),)14 b(where)259 424 y Ft(R)294 407 y Fp(u)327 424 y Fx(=)111 b Ft(f)p Fu(l)13 b Ft(!)e Fu(r)k Ft(j)e Fu(l)g Ft(!)e Fu(r)h Ft(\()f Fu(s)809 430 y Fr(1)840 424 y Ft(#)g Fu(t)887 430 y Fr(1)906 424 y Fu(;)c(:)g(:)g(:)e(;)i(s)1018 430 y Fp(n)1052 424 y Ft(#)k Fu(t)1099 430 y Fp(n)1133 424 y Ft(2)g(Rg)401 486 y([)41 b(f)p Fu(l)13 b Ft(!)e Fu(s)588 492 y Fp(j)620 486 y Ft(j)i Fu(l)g Ft(!)e Fu(r)h Ft(\()f Fu(s)826 492 y Fr(1)857 486 y Ft(#)g Fu(t)904 492 y Fr(1)923 486 y Fu(;)c(:)g(:)g(:)e(;)i(s) 1035 492 y Fp(n)1069 486 y Ft(#)k Fu(t)1116 492 y Fp(n)1150 486 y Ft(2)g(R)p Fx(;)21 b Fu(j)14 b Ft(2)d(f)p Fx(1)p Fu(;)c(:)g(:)g(:)t(;)g(n)p Ft(gg)401 549 y([)41 b(f)p Fu(l)13 b Ft(!)e Fu(t)584 555 y Fp(j)615 549 y Ft(j)j Fu(l)e Ft(!)f Fu(r)i Ft(\()e Fu(s)822 555 y Fr(1)853 549 y Ft(#)g Fu(t)900 555 y Fr(1)918 549 y Fu(;)c(:)g(:)g(:)e(;)i(s) 1030 555 y Fp(n)1064 549 y Ft(#)12 b Fu(t)1112 555 y Fp(n)1146 549 y Ft(2)f(R)p Fx(;)20 b Fu(j)14 b Ft(2)e(f)p Fx(1)p Fu(;)7 b(:)g(:)g(:)t(;)g(n)p Ft(gg)p Fu(:)141 650 y Fx(W)m(e)14 b(omit)d(the)k(simple)d(pro)q(ofs)i(of)g(the)g(follo) o(wing)d(t)o(w)o(o)j(lemmata.)100 751 y Fk(Lemma)i(6.42.)21 b Fx(Let)f(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))19 b(b)q(e)h(a)g(CTRS)e (without)h(extra)h(v)n(ariables)f(and)g(let)h(\()p Ft(F)t Fu(;)7 b Ft(R)1530 733 y Fp(u)1552 751 y Fx(\))20 b(b)q(e)g(its)100 801 y(asso)q(ciated)14 b(TRS.)f(Then)i(\()p Ft(F)t Fu(;)7 b Ft(R)o Fx(\))14 b(is)g(simplifying)c(if)k(and)f(only)g(if)g(\()p Ft(F)t Fu(;)7 b Ft(R)1254 783 y Fp(u)1276 801 y Fx(\))14 b(is)g(simplifyi)o(ng.)100 902 y Fk(Lemma)i(6.43.)21 b Fx(Let)12 b(\()p Ft(F)492 908 y Fr(1)510 902 y Fu(;)7 b Ft(R)564 908 y Fr(1)583 902 y Fx(\))j(and)h(\()p Ft(F)737 908 y Fr(2)756 902 y Fu(;)c Ft(R)809 908 y Fr(2)828 902 y Fx(\))k(b)q(e)g(comp)q(osable)f(CTRSs)g(without)h(extra)g(v)n (ariables.)100 952 y(Then)j(their)g(asso)q(ciated)h(unconditional)e (TRSs)g(\()p Ft(F)929 958 y Fr(1)948 952 y Fu(;)7 b Ft(R)1002 934 y Fp(u)1002 962 y Fr(1)1023 952 y Fx(\))14 b(and)g(\()p Ft(F)1184 958 y Fr(2)1203 952 y Fu(;)7 b Ft(R)1256 934 y Fp(u)1256 962 y Fr(2)1278 952 y Fx(\))14 b(are)g(comp)q(osable.)100 1053 y Fk(Theorem)i(6.44.)21 b Fx(The)15 b(simplifyi)o(ng)c(prop)q(ert) o(y)k(is)e(mo)q(dular)f(for)i(comp)q(osable)f(CTRSs.)100 1153 y Fk(Pr)o(oof.)22 b Fx(Let)15 b(\()p Ft(F)385 1159 y Fr(1)403 1153 y Fu(;)7 b Ft(R)457 1159 y Fr(1)476 1153 y Fx(\))14 b(and)h(\()p Ft(F)634 1159 y Fr(2)653 1153 y Fu(;)7 b Ft(R)707 1159 y Fr(2)725 1153 y Fx(\))15 b(b)q(e)g(comp)q (osable)f(CTRSs.)g(W)m(e)g(ha)o(v)o(e)h(to)g(sho)o(w)f(that)h(their)100 1203 y(com)o(bined)c(system)h(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))12 b(is)g(simplifying)d(if)i(and)h(only)f(if)h(b)q(oth)g(\()p Ft(F)1186 1209 y Fr(1)1205 1203 y Fu(;)7 b Ft(R)1259 1209 y Fr(1)1277 1203 y Fx(\))12 b(and)g(\()p Ft(F)1430 1209 y Fr(2)1449 1203 y Fu(;)7 b Ft(R)1503 1209 y Fr(2)1521 1203 y Fx(\))13 b(are)f(sim-)100 1253 y(plifying.)e(The)j(only-if)e (case)j(is)e(straigh)o(tforw)o(ard.)g(So)h(let)f(\()p Ft(F)1063 1259 y Fr(1)1082 1253 y Fu(;)7 b Ft(R)1136 1259 y Fr(1)1154 1253 y Fx(\))13 b(and)g(\()p Ft(F)1309 1259 y Fr(2)1327 1253 y Fu(;)7 b Ft(R)1381 1259 y Fr(2)1399 1253 y Fx(\))13 b(b)q(e)h(simplifyi)o(ng.)100 1303 y(W)m(e)c(ha)o(v)o (e)g(to)h(sho)o(w)g(that)f(\()p Ft(F)5 b Fu(;)i Ft(R)o Fx(\))k(is)g(simplifyi)o(ng,)c(or)k(equiv)n(alen)o(tly)e(b)o(y)i(Lemma) d(6.42,)h(that)i(the)g(TRS)100 1353 y(\()p Ft(F)t Fu(;)c Ft(R)203 1335 y Fp(u)225 1353 y Fx(\))15 b(asso)q(ciated)g(with)f(\()p Ft(F)t Fu(;)7 b Ft(R)p Fx(\))15 b(is)f(simplifying.)d(By)k(Lemma)c (6.43,)i(the)j(TRSs)e(\()p Ft(F)1502 1359 y Fr(1)1521 1353 y Fu(;)7 b Ft(R)1574 1335 y Fp(u)1574 1363 y Fr(1)1596 1353 y Fx(\))15 b(and)100 1402 y(\()p Ft(F)150 1408 y Fr(2)168 1402 y Fu(;)7 b Ft(R)222 1384 y Fp(u)222 1413 y Fr(2)244 1402 y Fx(\))13 b(are)h(comp)q(osable.)e(Hence)j(their)f (com)o(bined)f(TRS)g(\()p Ft(F)1131 1408 y Fr(1)1158 1402 y Ft([)8 b(F)1228 1408 y Fr(2)1247 1402 y Fu(;)f Ft(R)1301 1384 y Fp(u)1301 1413 y Fr(1)1331 1402 y Ft([)h(R)1402 1384 y Fp(u)1402 1413 y Fr(2)1424 1402 y Fx(\))14 b(is)f(simplifying) 100 1452 y(according)h(to)h(Theorem)f(5.16.)f(No)o(w)i(the)g(equalit)o (y)f(\()p Ft(F)t Fu(;)7 b Ft(R)1046 1434 y Fp(u)1067 1452 y Fx(\))13 b(=)h(\()p Ft(F)1192 1458 y Fr(1)1220 1452 y Ft([)c(F)1292 1458 y Fr(2)1310 1452 y Fu(;)d Ft(R)1364 1434 y Fp(u)1364 1463 y Fr(1)1395 1452 y Ft([)j(R)1468 1434 y Fp(u)1468 1463 y Fr(2)1490 1452 y Fx(\))k(concludes)100 1502 y(the)g(pro)q(of.)f Fe(2)519 1631 y Fv(7.)24 b(Related)15 b(w)o(ork)g(and)g(op)q(en)g(problems)141 1706 y Fx(Another)j(extension) g(of)f(com)o(binations)e(with)i(shared)h(constructors)i(are)d(hierarc)o (hical)h(com)o(bi-)100 1756 y(nations.)e(In)i(a)f(hierarc)o(hical)g (com)o(bination)e(one)j(of)e(the)i(systems)g(ma)o(y)d(use)k(de\014ned)f (sym)o(b)q(ols)e(of)100 1805 y(the)g(other)g(in)f(the)i(righ)o(t-hand)e (sides)h(of)f(its)h(rewrite)h(rules)f(without)f(imp)q(orting)f(the)i (rules)g(de\014n-)100 1855 y(ing)d(those)j(sym)o(b)q(ols)d(\(a)h (precise)i(de\014nition)e(can)h(b)q(e)g(found)g(in)f(Ohlebusc)o(h,)h (1994)p Fs(b)p Fx(,)e(for)h(instance\).)100 1905 y(The)f(standard)g (example)e(of)i(a)f(hierarc)o(hical)h(com)o(bination)d(is)j(the)g (follo)o(wing)d(one,)j(where)h(the)f(base)100 1955 y(system)474 2040 y Ft(R)509 2046 y Fr(1)539 2040 y Fx(=)583 1982 y Fg(\032)635 2015 y Fx(0)c(+)g Fu(x)103 b Ft(!)41 b Fu(x)635 2065 y(S)r Fx(\()p Fu(x)p Fx(\))10 b(+)f Fu(y)44 b Ft(!)d Fu(S)r Fx(\()p Fu(x)10 b Fx(+)f Fu(y)q Fx(\))100 2138 y(is)k(extended)j(with)468 2216 y Ft(R)503 2222 y Fr(2)533 2216 y Fx(=)577 2157 y Fg(\032)629 2191 y Fx(0)9 b Ft(\003)g Fu(x)102 b Ft(!)41 b Fx(0)629 2240 y Fu(S)r Fx(\()p Fu(x)p Fx(\))10 b Ft(\003)f Fu(y)43 b Ft(!)e Fx(\()p Fu(x)9 b Ft(\003)g Fu(y)q Fx(\))h(+)g Fu(y)q(:)100 2341 y Fx(Here)16 b(the)f(de\014ned)h(sym)o(b)q(ol)d(+)i (o)q(ccurs)h(as)f(a)g(constructor)h(in)e(the)i(righ)o(t-hand)e(side)h (of)f(the)h(second)100 2391 y(rule)h(of)f Ft(R)269 2397 y Fr(2)303 2391 y Fx(and)h Ft(\003)f Fx(do)q(es)i(not)e(app)q(ear)i(in) e Ft(R)818 2397 y Fr(1)837 2391 y Fx(.)g(Clearly)m(,)f Ft(R)1053 2397 y Fr(1)1088 2391 y Fx(and)h Ft(R)1205 2397 y Fr(2)1240 2391 y Fx(are)h(complete)f(constructor)100 2441 y(systems.)10 b(Using)h(standard)g(tec)o(hniques,)h(their)f (hierarc)o(hical)g(com)o(bination)e Ft(R)i Fx(=)h Ft(R)1433 2447 y Fr(1)1455 2441 y Ft([)s(R)1522 2447 y Fr(2)1551 2441 y Fx(can)f(also)100 2491 y(easily)16 b(b)q(e)h(sho)o(wn)f(to)h(b)q (e)g(a)f(complete)g(constructor)i(system.)e(But)h(with)f(our)h(former)e (results,)i(w)o(e)100 2540 y(cannot)e(infer)f(completeness)i(of)e Ft(R)h Fx(from)e(the)j(completeness)f(of)g(its)g(constituen)o(ts.)g(On) g(the)h(other)100 2590 y(hand,)f(the)i(com)o(bination)c(of)i Ft(R)614 2572 y Fn(0)614 2601 y Fr(1)647 2590 y Fx(=)g Ft(f)p Fu(a)g Ft(!)g Fu(b)p Ft(g)g Fx(and)h Ft(R)981 2572 y Fn(0)981 2601 y Fr(2)1015 2590 y Fx(=)f Ft(f)p Fu(F)6 b Fx(\()p Fu(b)p Fx(\))14 b Ft(!)h Fu(F)6 b Fx(\()p Fu(a)p Fx(\))p Ft(g)15 b Fx(sho)o(ws)h(that)g(almost)100 2640 y(all)e(in)o(teresting)i(prop)q(erties)g(are)g(destro)o(y)o(ed)g (under)h(hierarc)o(hical)e(com)o(binations.)e(So)i(what)g(is)g(the)p eop %%Page: 39 39 39 38 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(39)p 100 224 1595 2 v 100 299 a Fx(di\013erence)15 b(b)q(et)o(w)o(een)g(the)f(t) o(w)o(o)f(examples,)f(that)h(is,)g(wh)o(y)g(is)h(the)g(former)e(so)i(b) q(enign)f(and)g(the)h(latter)100 349 y(so)h(malignan)o(t?)d(In)k (essence,)i(this)d(is)h(due)g(to)f(the)h(fact)g(that)f(in)h(the)g(righ) o(t-hand)f(side)h(\()p Fu(x)10 b Ft(\003)g Fu(y)q Fx(\))h(+)g Fu(y)100 399 y Fx(the)16 b(function)g(sym)o(b)q(ol)f(+)h(from)f Ft(D)665 405 y Fr(1)700 399 y Fx(o)q(ccurs)i(ab)q(o)o(v)o(e)f(the)h (function)f(sym)o(b)q(ol)e Ft(\003)i Fx(from)f Ft(D)1504 405 y Fr(2)1523 399 y Fx(,)g(whereas)100 448 y(in)h(the)i(righ)o (t-hand)e(side)i Fu(F)6 b Fx(\()p Fu(a)p Fx(\))16 b(the)i(function)f (sym)o(b)q(ol)e Fu(a)i Fx(from)e Ft(D)1182 430 y Fn(0)1182 459 y Fr(1)1218 448 y Fx(o)q(ccurs)j(b)q(elo)o(w)f(the)h(function)100 498 y(sym)o(b)q(ol)13 b Fu(F)21 b Fx(from)13 b Ft(D)423 480 y Fn(0)423 508 y Fr(2)442 498 y Fx(.)h(This)h(fact)g(has)h(b)q(een) g(observ)o(ed)g(indep)q(enden)o(tly)g(and)f(con)o(temp)q(oraneously)100 548 y(b)o(y)10 b(Dersho)o(witz)h(\(1994\))f(and)g(Krishna)h(Rao)f (\(1993\).)f(Other)j(hierarc)o(hical)e(com)o(binations)f(for)h(whic)o (h)100 598 y(termination)i(is)i(mo)q(dular)f(are)h(describ)q(ed)j(in)c (F)m(ern\023)-21 b(andez)16 b(and)e(Jouannaud)g(\(1994\))g(\(their)g (results)100 648 y(are)g(based)h(on)f(the)h(new)f(notion)g(of)f (\\alien-decreasingness"\).)i(W)m(e)f(will)e(not)i(go)g(in)o(to)f (details)h(here)100 697 y(but)g(just)g(relate)g(other)h(kno)o(wn)e (results)i(to)f(ours.)141 747 y(W)m(e)22 b(ha)o(v)o(e)f(sho)o(wn)h (that)g(semi-completeness)f(is)h(mo)q(dular)e(for)i(comp)q(osable)e (TRSs)i(and)g(for)100 797 y(constructor-sharing)15 b(CTRSs.)e(V)m(ery)h (recen)o(tly)m(,)g(Krishna)g(Rao)e(\(1995\))h(pro)o(vided)h(a)f (su\016cien)o(t)i(cri-)100 847 y(terion)i(for)g(the)i(mo)q(dularit)o(y) c(of)h(semi-completeness)h(for)h(hierarc)o(hical)f(com)o(binations)e (of)i(TRSs.)100 897 y(His)c(pro)q(of)h(tec)o(hnique)h(is)e(di\013eren)o (t)i(from)d(ours.)i(Moreo)o(v)o(er,)g(w)o(e)g(p)q(oin)o(t)f(out)h(that) f(there)i(are)g(closely)100 946 y(related)g(results)h(obtained)f(b)o(y) g(Middeldorp)g(\(1994)p Fs(a)p Fx(\).)f(He)h(pro)o(v)o(ed)g(that)g (semi-completeness)g(and)100 996 y(completeness)e(are)g(mo)q(dular)d (for)j(comp)q(osable)e(conditional)g(constructor)j(systems)f(without)f (extra)100 1046 y(v)n(ariables.)j(It)i(is)g(y)o(et)g(unkno)o(wn)f(if)g (the)i(same)e(is)g(true)i(when)f(extra)g(v)n(ariables)f(in)h (conditions)f(are)100 1096 y(allo)o(w)o(ed.)10 b(As)i(a)g(matter)f(of)h (fact,)f(Middeldorp)h(\(1994)p Fs(a)p Fx(\))g(conjectured)h(that)f (this)g(is)g(the)h(case.)f(As)h(far)100 1146 y(as)f(mo)q(dularit)o(y)f (of)h(semi-completeness)g(of)g(CTRSs)h(is)f(concerned,)j(it)d(is)h (de\014nitely)g(w)o(orth)o(while)f(to)100 1196 y(try)i(to)g(extend)h (the)f(aforemen)o(tioned)f(result)i(to)f(the)h(whole)e(class)i(of)e (comp)q(osable)g(CTRSs.)h(Note,)100 1245 y(ho)o(w)o(ev)o(er,)h(that)g (the)g(pro)q(of)g(presen)o(ted)i(in)d(this)h(pap)q(er)h(do)q(es)g(not)f (carry)g(o)o(v)o(er)g(to)g(comp)q(osable)f(sys-)100 1295 y(tems.)f(By)h(the)g(w)o(a)o(y)m(,)f(the)h(last)g(t)o(w)o(o)f(statemen) o(ts)h(also)g(apply)f(to)h(the)g(mo)q(dularit)o(y)e(of)h(completeness) 100 1345 y(for)g(non-duplicating)g(CTRSs.)141 1395 y(Sev)o(eral)e (recen)o(t)i(pap)q(ers)e(deal)g(with)f(the)i(mo)q(dularit)o(y)c(of)i (completeness)i(for)e(constructor)i(systems)100 1445 y(and)e(the)i(more)e(general)h(class)h(of)e(o)o(v)o(erla)o(y)g (systems,)h(resp)q(ectiv)o(ely)m(.)h(The)f(in)o(v)o(estigation)f(of)g (construc-)100 1494 y(tor)j(systems)h(started)h(with)e(the)i(w)o(ork)e (of)g(Middeldorp)h(and)f(T)m(o)o(y)o(ama)e(\(1993\).)h(Their)i(main)e (result)100 1544 y(has)g(b)q(een)i(extended)g(to)f(certain)g(classes)h (of)e(hierarc)o(hical)h(com)o(binations)d(b)o(y)j(Krishna)g(Rao)e (\(1993\))100 1594 y(and)16 b(Dersho)o(witz)h(\(1994\).)e(Using)h(the)h (strong)f(Theorem)g(5.11,)f(Gramlic)o(h)e(\(1994)p Fs(b)p Fx(\))j(pro)o(v)o(ed)h(that)100 1644 y(completeness)g(is)f(mo)q(dular)f (for)i(constructor-sharing)g(o)o(v)o(erla)o(y)f(systems.)h(Corollary)e (5.12)g(sho)o(ws)100 1694 y(that)j(this)f(is)h(true)h(ev)o(en)f(for)g (comp)q(osable)f(TRSs,)g(so)h(the)g(result)h(of)e(Middeldorp)h(and)g(T) m(o)o(y)o(ama)100 1743 y(\(1993\))11 b(is)h(actually)g(a)g(sp)q(ecial)h (case)g(thereof.)g(Lately)m(,)e(Gramlic)o(h)f(\(1994)p Fs(c)p Fx(\))h(sho)o(w)o(ed)i(that)f(complete-)100 1793 y(ness)i(is)e(mo)q(dular)f(for)i(the)g(class)h(of)e(disjoin)o(t)g (conditional)f(o)o(v)o(erla)o(y)h(systems)h(with)f(joinable)g(critical) 100 1843 y(pairs.)j(The)h(question)g(whether)h(this)e(result)i(extends) g(to)e(more)g(general)h(com)o(binations)d(has)j(v)o(ery)100 1893 y(recen)o(tly)f(also)f(b)q(een)h(answ)o(ered)h(a\016rmativ)o(ely) 11 b(b)o(y)j(Gramlic)o(h)e(\(1995\).)h(Using)h(our)h(Theorem)e(6.14,) 100 1943 y(he)h(w)o(as)g(able)f(to)h(extend)h(the)g(result)f(to)g (constructor-sharing)h(CTRSs.)141 1993 y(Finally)m(,)j(generalizing)i (a)g(result)h(of)e(Kurihara)h(and)g(Oh)o(uc)o(hi)h(\(1992\),)e(w)o(e)h (ha)o(v)o(e)g(sho)o(wn)h(that)100 2042 y(the)16 b(simplifying)c(prop)q (ert)o(y)17 b(is)e(mo)q(dular)f(for)h(comp)q(osable)g(CTRSs.)g(Their)h (result)h(has)e(also)g(b)q(een)100 2092 y(extended)g(b)o(y)e(Krishna)g (Rao)g(\(1994\))f(to)i(a)f(certain)h(class)g(of)e(hierarc)o(hical)i (TRSs.)f(In)g(this)g(con)o(text,)100 2142 y(the)18 b(reader)h(is)e (also)g(referred)j(to)d(Gramlic)o(h)e(\(1994)p Fs(a)p Fx(\),)i(Ohlebusc)o(h)i(\(1995)p Fs(a)p Fx(\))e(and)g(Kurihara)h(and) 100 2192 y(Oh)o(uc)o(hi)c(\(1995\))f(for)g(related)i(results.)141 2242 y(Up)c(un)o(til)f(no)o(w,)g(nob)q(o)q(dy)h(has)g(studied)h (hierarc)o(hical)f(com)o(binations)e(of)h(CTRSs.)h(It)g(go)q(es)g (without)100 2291 y(sa)o(ying)18 b(that)h(it)g(should)g(also)g(b)q(e)h (in)o(v)o(estigated)f(whic)o(h)g(of)g(the)h(kno)o(wn)f(mo)q(dularit)o (y)e(results)j(for)100 2341 y(hierarc)o(hical)g(com)o(binations)f(of)h (unconditional)f(systems)i(can)g(in)f(some)f(w)o(a)o(y)h(b)q(e)i (extended)g(to)100 2391 y(conditional)12 b(systems.)141 2441 y(In)h(a)f(di\013eren)o(t)h(con)o(text,)g(Raoult)e(and)h(V)m (uillemin)e(\(1980\))i(sho)o(w)o(ed)g(that)h(con\015uence)h(is)e(mo)q (dular)100 2491 y(for)f(left-linear)h(TRSs)g(whic)o(h)f(are)i (orthogonal)e(to)h(eac)o(h)g(other.)g(Tw)o(o)g(TRSs)g Ft(R)1347 2497 y Fr(1)1378 2491 y Fx(and)g Ft(R)1492 2497 y Fr(2)1522 2491 y Fx(are)h(called)100 2540 y Fs(ortho)n(gonal)k (to)g(e)n(ach)h(other)p Fx(,)d(if)h(there)h(is)f(no)g(o)o(v)o(erlap)g (b)q(et)o(w)o(een)i(a)e(rule)g(from)f Ft(R)1394 2546 y Fr(1)1429 2540 y Fx(and)h(one)g(of)g Ft(R)1675 2546 y Fr(2)100 2590 y Fx(\(cf.)c(Klop,)f(1992\).)g(Note)i(that)f(this)h (de\014nition)f(do)q(es)h(not)f(exclude)h(the)g(existence)h(of)e (critical)g(pairs.)100 2640 y(There)19 b(ma)o(y)d(b)q(e)j(critical)f (pairs)g(due)h(to)f(o)o(v)o(erlaps)g(b)q(et)o(w)o(een)h(rules)g(of)f Ft(R)1286 2646 y Fr(1)1322 2640 y Fx(or)h(rules)f(of)g Ft(R)1568 2646 y Fr(2)1587 2640 y Fx(.)f(It)i(is)p eop %%Page: 40 40 40 39 bop 100 197 a Fw(40)105 b(E.)11 b(Ohlebusc)o(h)p 100 224 1595 2 v 100 299 a Fx(easy)j(to)f(see)i(that)e(t)o(w)o(o)g (constructor-sharing)i(TRSs)e Ft(R)988 305 y Fr(1)1020 299 y Fx(and)h Ft(R)1136 305 y Fr(2)1168 299 y Fx(are)g(orthogonal)e (to)h(eac)o(h)h(other.)100 349 y(Tw)o(o)j(comp)q(osable)g(systems)h Ft(R)617 355 y Fr(1)654 349 y Fx(and)g Ft(R)774 355 y Fr(2)810 349 y Fx(are,)g(ho)o(w)o(ev)o(er,)g(in)g(general)g(not)g (orthogonal)e(to)i(eac)o(h)100 399 y(other.)d(Ov)o(erlaps)i(b)q(et)o(w) o(een)g(rules)f(from)e Ft(R)795 405 y Fr(1)829 399 y Fx(and)i(rules)g(from)e Ft(R)1148 405 y Fr(2)1182 399 y Fx(ma)o(y)g(o)q(ccur,)i(not)o(withstanding)100 448 y(the)j(fact)g(that)f(these)i(o)o(v)o(erlaps)f(do)f(only)g(create)i (critical)f(pairs)f(already)g(con)o(tained)h(in)f(the)i(set)100 498 y Fu(C)s(P)6 b Fx(\()p Ft(R)216 504 y Fr(1)235 498 y Fx(\))11 b Ft([)g Fu(C)s(P)6 b Fx(\()p Ft(R)417 504 y Fr(2)436 498 y Fx(\))16 b(of)g(all)g(critical)g(pairs)g(b)q(et)o(w)o (een)i(rules)g(from)c Ft(R)1226 504 y Fr(1)1261 498 y Fx(and)i(b)q(et)o(w)o(een)i(rules)f(from)100 548 y Ft(R)135 554 y Fr(2)153 548 y Fx(.)c(The)g(reader)h(is)f(in)o(vited)f(to)h(c)o (hec)o(k)h(that)f(in)f(consequence)j(of)d(this)h(subtle)h(di\013erence) h(the)e(pro)q(of)100 598 y(of)h(Raoult)g(and)h(V)m(uillemin)d(\(1980\)) i(do)q(es)i(not)f(extend)h(to)f(comp)q(osable)f(systems.)h(Th)o(us)g (it)g(is)g(still)100 648 y(op)q(en)f(whether)h(con\015uence)h(is)e(a)f (mo)q(dular)f(prop)q(ert)o(y)j(of)e(left-linear)g(comp)q(osable)g (TRSs.)141 697 y(A)g(somewhat)f(di\013eren)o(t)h(approac)o(h)g(to)f(mo) q(dularit)o(y)e(of)i(TRSs)h(has)g(b)q(een)h(presen)o(ted)g(in)e (Prehofer)100 747 y(\(1994\).)g(This)i(pap)q(er)g(deals)g(with)f(a)h (prop)q(ert)o(y)g(called)g("mo)q(dular)e(normalization")o(,)f(meaning)h (that)100 797 y(ev)o(ery)18 b Ft(R)g Fx(=)g Ft(R)351 803 y Fr(1)381 797 y Ft([)12 b(R)456 803 y Fr(2)492 797 y Fx(normal)j(form)h(of)h(some)g(term)g Fu(s)h Fx(can)g(b)q(e)g (obtained)f(b)o(y)h(\014rst)g(reducing)g Fu(s)100 847 y Fx(to)f(an)g Ft(R)250 853 y Fr(1)286 847 y Fx(normal)f(form)f Fu(s)p Ft(#)572 857 y Fn(R)601 861 y Ff(1)637 847 y Fx(and)i(then)h (reducing)g Fu(s)p Ft(#)1031 857 y Fn(R)1060 861 y Ff(1)1095 847 y Fx(to)g(an)f Ft(R)1246 853 y Fr(2)1282 847 y Fx(normal)e(form.)g (Prehofer)100 897 y(dev)o(elop)q(ed)h(su\016cien)o(t)g(criteria)f(for)g (this)h(prop)q(ert)o(y)g(whic)o(h)f(also)g(co)o(v)o(er)h(non-complete)f (TRSs)g(\(the)100 946 y(main)g(restriction)j(b)q(eing)f(that)g(the)h (system)f Ft(R)868 952 y Fr(1)904 946 y Fx(is)g(required)h(to)f(b)q(e)h (left-linear)f(and)g(complete\).)100 996 y(One)i(of)f(the)h(giv)o(en)f (in)o(teresting)h(applications)f(of)g(mo)q(dular)e(normalization)g(is)i (a)h(new)f(mo)q(dular)100 1046 y(narro)o(wing)13 b(strategy)m(.)698 1173 y Fv(Ac)o(kno)o(wledgemen)o(ts)141 1248 y Fx(I)h(am)e(grateful)i (to)f(Aart)h(Middeldorp)g(and)g(the)g(referees)j(for)c(their)h(commen)o (ts)e(on)i(the)h(pap)q(er.)785 1375 y Fv(References)100 1438 y Fw(Dersho)o(witz,)f(N.)i(\(1982\).)26 b(Orderings)14 b(for)h(term-rewriting)e(systems.)26 b Fd(The)n(or)n(etic)n(al)18 b(Computer)f(Scienc)n(e)i Fh(17)p Fd(\(3\))p Fw(,)174 1475 y(279{301.)100 1513 y(Dersho)o(witz,)10 b(N.)j(\(1994\).)i (Hierarc)o(hical)10 b(terminatio)o(n.)k(In)e Fd(Pr)n(o)n(c)n(e)n(e)n (dings)k(of)d(the)h(4th)g(International)g(Workshop)f(on)174 1550 y(Conditional)g(T)m(erm)g(R)n(ewriting)h(Systems)p Fw(.)h(T)m(o)d(app)q(ear.)100 1588 y(Dersho)o(witz,)d(N.,)j(Ho)q(ot,)f (C.)h(\(1994\).)i(Natural)c(termination)o(.)j Fd(The)n(or)n(etic)n(al)h (Computer)g(Scienc)n(e)p Fw(.)j(T)m(o)12 b(app)q(ear.)100 1626 y(Dersho)o(witz,)6 b(N.,)j(Jouannaud,)c(J.-P)m(.)j(\(1990\).)h (Rewrite)e(Systems.)h(In)g(J.)h(v)n(an)e(Leeu)o(w)o(en,)g(editor,)g Fd(F)m(ormal)k(Mo)n(dels)f(and)174 1663 y(Semantics,)k(Handb)n(o)n(ok)h (of)e(The)n(or)n(etic)n(al)h(Computer)g(Scienc)n(e)p Fw(,)f(v)o(olume)c(B,)j(243{320.)d(Amsterdam:)g(Elsevier.)100 1701 y(Dersho)o(witz,)f(D.,)h(Manna,)g(Z.)h(\(1979\).)h(Pro)o(ving)d (termination)e(with)k(m)o(ultiset)e(orderings.)i Fd(Communic)n(ations)i (of)f(the)174 1739 y(A)o(CM)i Fh(22)p Fd(\(8\))p Fw(,)f(465{476.)100 1776 y(Dersho)o(witz,)7 b(N.,)j(Ok)n(ada,)e(M.,)h(Siv)n(akumar,)d(G.)j (\(1988\).)h(Canonical)d(conditional)f(rewrite)i(systems.)i(In)f Fd(Pr)n(o)n(c)n(e)n(e)n(dings)174 1814 y(of)k(the)f(9th)g(Confer)n(enc) n(e)i(on)e(A)o(utomate)n(d)i(De)n(duction)p Fw(,)e(538{549.)c(Lecture)i (Notes)g(in)g(Computer)f(Science)g Fh(310)p Fw(,)174 1851 y(Berlin:)i(Springer)e(V)m(erlag.)100 1889 y(F)m(ern\023)-18 b(andez,)10 b(M.,)j(Jouannaud,)e(J.-P)m(.)h(\(1994\).)19 b(Mo)q(dular)11 b(termination)f(of)j(term)f(rewriting)f(systems)h (revisited.)17 b(In)174 1926 y Fd(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)d (the)g(10th)g(Workshop)g(on)g(Sp)n(e)n(ci\014c)n(ation)i(of)e(A)o(bstr) n(act)h(Data)f(T)m(yp)n(es)p Fw(.)j(T)m(o)c(app)q(ear.)100 1964 y(Gramlic)o(h,)6 b(B.)i(\(1993\).)h(Su\016cien)o(t)e(conditions)f (for)i(mo)q(dular)f(termination)e(of)k(conditiona)o(l)d(term)h (rewriting)h(systems.)174 2002 y(In)13 b Fd(Pr)n(o)n(c)n(e)n(e)n(dings) j(of)f(the)f(3r)n(d)h(International)f(Workshop)h(on)f(Conditional)g(T)m (erm)h(R)n(ewriting)f(Systems)g(1992)p Fw(,)174 2039 y(128{142,)9 b(Lecture)h(Notes)h(in)h(Computer)d(Science)h Fh(656)p Fw(,)i(Berlin:)f(Springer)e(V)m(erlag.)100 2077 y(Gramlic)o(h,)e(B.)j(\(1994)p Fd(a)p Fw(\).)i(Generalized)7 b(su\016cien)o(t)i(conditions)e(for)j(mo)q(dular)e(termination)f(of)i (rewriting.)j Fd(Applic)n(able)174 2114 y(A)o(lgebr)n(a)j(in)e(Engine)n (ering,)i(Communic)n(ation)e(and)h(Computing)f Fh(5)p Fw(,)f(131{158.)100 2152 y(Gramlic)o(h,)f(B.)k(\(1994)p Fd(b)p Fw(\).)21 b(Abstract)13 b(relations)f(b)q(et)o(w)o(een)h (restricted)f(termination)f(and)i(con\015uence)f(prop)q(erties)f(of)174 2189 y(rewrite)g(systems.)j Fd(F)m(undamenta)g(Informatic)n(ae)p Fw(.)j(T)m(o)12 b(app)q(ear.)100 2227 y(Gramlic)o(h,)f(B.)j(\(1994)p Fd(c)p Fw(\).)22 b(On)14 b(mo)q(dularit)o(y)d(of)j(termination)c(and)k (con\015uence)d(prop)q(erties)g(of)j(conditional)d(rewrite)174 2265 y(systems.)i(In)d Fd(Pr)n(o)n(c)n(e)n(e)n(dings)15 b(of)d(the)h(4th)f(International)h(Confer)n(enc)n(e)g(on)g(A)o(lgebr)n (aic)h(and)f(L)n(o)n(gic)h(Pr)n(o)n(gr)n(amming)p Fw(,)174 2302 y(186{203.)9 b(Lecture)h(Notes)h(in)h(Computer)d(Science)h Fh(850)p Fw(,)i(Berlin:)f(Springer)e(V)m(erlag.)100 2340 y(Gramlic)o(h,)d(B.)j(\(1995\).)g(On)g(terminatio)o(n)d(and)i (con\015uence)e(prop)q(erties)g(of)i(disjoin)o(t)f(and)h(constructor-s) o(har)o(ing)e(condi-)174 2377 y(tional)k(rewrite)f(systems.)k(Extended) 8 b(v)o(ersion)h(of)h(Gramlic)o(h)f(\(1994c\).)f(Submitted)g(to)i (Theoretical)f(Computer)174 2414 y(Science.)100 2452 y(Kaplan,)h(S.)g(\(1987\).)j(Simplifying)8 b(conditional)g(term)i (rewriting)g(systems:)f(Uni\014cation,)g(termination)f(and)i(con\015u-) 174 2490 y(ence.)k Fd(J.)f(Symb)n(olic)h(Computation)f Fh(4)p Fd(\(3\))p Fw(,)f(295{334.)100 2528 y(Kn)o(uth,)f(D.E.,)h (Bendix,)e(P)m(.)j(\(1970\).)i(Simple)10 b(w)o(ord)i(problems)e(in)i (univ)o(ersal)e(algebra.)15 b(In)d(J.)h(Leec)o(h,)e(editor,)f Fd(Com-)174 2565 y(putational)k(Pr)n(oblems)g(in)f(A)o(bstr)n(act)g(A)o (lgebr)n(a)p Fw(,)h(263{297.)9 b(Oxford:)i(P)o(ergamon)e(Press.)100 2603 y(Klop,)k(J.W.)g(\(1992\).)20 b(T)m(erm)13 b(Rewriting)f(Systems.) 20 b(In)13 b(S.)g(Abramsky)m(,)e(D.)j(Gabba)o(y)m(,)d(and)i(T.)h (Maibaum,)e(editors,)174 2640 y Fd(Handb)n(o)n(ok)k(of)d(L)n(o)n(gic)h (in)f(Computer)g(Scienc)n(e)p Fw(,)g(v)o(olume)d(2,)h(1{116.)f(Oxford:) g(Oxford)h(Univ)o(ersit)o(y)f(Press.)p eop %%Page: 41 41 41 40 bop 619 197 a Fw(Mo)q(dular)10 b(Prop)q(erties)f(of)i(Comp)q (osable)e(T)m(erm)i(Rewriting)g(Systems)103 b(41)p 100 224 1595 2 v 100 299 a(Klop,)12 b(J.W.,)i(Middeldorp,)c(A.,)k(T)m(o)o (y)o(ama,)e(Y.,)h(de)g(V)m(rijer,)g(R.)g(\(1994\).)18 b(Mo)q(dularit)o(y)11 b(of)i(con\015uence:)e(A)i(simpli\014ed)174 336 y(pro)q(of.)h Fd(Information)f(Pr)n(o)n(c)n(essing)h(L)n(etters)g Fh(49)p Fw(,)e(101{109.)100 374 y(Kurihara,)d(M.,)i(Oh)o(uc)o(hi,)e(A.) j(\(1991\).)g(Mo)q(dular)e(term)f(rewriting)h(systems)f(with)i(shared)e (constructors.)i Fd(Journal)h(of)174 411 y(Information)h(Pr)n(o)n(c)n (essing)h Fh(14)p Fd(\(3\),)f(IPS)g(of)g(Jap)n(an)p Fw(,)f(357{358.)100 448 y(Kurihara,)i(M.,)i(Oh)o(uc)o(hi,)e(A.)j(\(1992\).)26 b(Mo)q(dularit)o(y)14 b(of)h(simple)g(terminatio)o(n)e(of)j(term)e (rewriting)h(systems)f(with)174 486 y(shared)c(constructors.)i Fd(The)n(or)n(etic)n(al)j(Computer)e(Scienc)n(e)i Fh(103)p Fw(,)d(273{282.)100 523 y(Kurihara,)h(M.,)j(Oh)o(uc)o(hi,)e(A.)h (\(1995\).)25 b(Decomp)q(osable)12 b(termination)g(of)j(comp)q(osable)d (term)i(rewriting)g(systems.)174 560 y Fd(IEICE)j(T)m(r)n(ansactions)h (on)g(Information)h(and)f(Systems,)g(V)m(ol.E78-D,)h(No.4,)f (Information)g(and)h(Systems)174 598 y(So)n(ciety)14 b(of)f(Jap)n(an)p Fw(,)f(314{320.)100 635 y(Krishna)f(Rao,)g(M.R.K.)h (\(1993\).)k(Completeness)9 b(of)j(hierarc)o(hica)o(l)d(com)o (binations)g(of)j(term)f(rewriting)f(systems.)15 b(In)174 672 y Fd(Pr)n(o)n(c)n(e)n(e)n(dings)k(of)e(the)f(13th)h(Confer)n(enc)n (e)h(on)e(the)h(F)m(oundations)g(of)g(Softwar)n(e)f(T)m(e)n(chnolo)n (gy)j(and)e(The)n(or)n(etic)n(al)174 710 y(Computer)d(Scienc)n(e)p Fw(,)f(125{139.)c(Lecture)h(Notes)h(in)g(Computer)f(Science)f Fh(761)p Fw(,)k(Berlin:)d(Springer)f(V)m(erlag.)100 747 y(Krishna)f(Rao,)g(M.R.K.)i(\(1994\).)g(Simple)d(termination)f(of)j (hierarc)o(hica)o(l)e(com)o(binati)o(ons)f(of)j(term)f(rewriting)g (systems.)174 785 y(In)i Fd(Pr)n(o)n(c)n(e)n(e)n(dings)k(of)e(the)g (International)g(Symp)n(osium)g(on)g(The)n(or)n(etic)n(al)h(Asp)n(e)n (cts)g(of)e(Computer)h(Softwar)n(e)p Fw(,)f(203{)174 822 y(223.)g(Lecture)f(Notes)h(in)g(Computer)f(Science)f Fh(789)p Fw(,)k(Berlin:)d(Springer)f(V)m(erlag.)100 859 y(Krishna)h(Rao,)h(M.R.K.)i(\(1995\).)h(Semi-comp)o(lete)o(ness)8 b(of)k(hierarc)o(hical)c(and)j(sup)q(er-hierarc)o(h)o(ica)o(l)e(com)o (bination)o(s)g(of)174 897 y(term)i(rewriting)g(systems.)k(In)c Fd(Pr)n(o)n(c)n(e)n(e)n(dings)16 b(of)d(the)h(20th)g(Col)r(lo)n(quium)g (on)f(T)m(r)n(e)n(es)h(in)f(A)o(lgebr)n(a)i(and)f(Pr)n(o)n(gr)n(am-)174 934 y(ming)p Fw(,)e(379{393.)d(Lecture)h(Notes)i(in)f(Computer)e (Science)h Fh(915)p Fw(,)i(Berlin:)f(Springer)e(V)m(erlag.)100 971 y(Marc)o(hiori,)k(M.)i(\(1995\).)23 b(Mo)q(dularit)o(y)13 b(of)i(completene)o(ss)d(revisited.)23 b(In)15 b Fd(Pr)n(o)n(c)n(e)n(e) n(dings)j(of)e(the)h(6th)f(International)174 1009 y(Confer)n(enc)n(e)g (on)f(R)n(ewriting)g(T)m(e)n(chniques)h(and)f(Applic)n(ations)p Fw(,)f(2{10.)e(Lecture)g(Notes)h(in)g(Computer)e(Science)174 1046 y Fh(914)p Fw(,)i(Berlin:)d(Springer)f(V)m(erlag.)100 1083 y(Middeldorp,)i(A.)j(\(1989\).)20 b(A)14 b(su\016cien)o(t)e (condition)f(for)i(the)g(termination)d(of)k(the)f(direct)f(sum)h(of)g (term)f(rewriting)174 1121 y(systems.)28 b(In)16 b Fd(Pr)n(o)n(c)n(e)n (e)n(dings)k(of)d(the)h(4th)f(IEEE)g(Symp)n(osium)h(on)g(L)n(o)n(gic)h (in)e(Computer)g(Scienc)n(e)p Fw(,)h(396{401.)174 1158 y(W)m(ashington,)9 b(D.C.:)j(IEEE)f(Computer)f(So)q(ciet)o(y)g(Press.) 100 1196 y(Middeldorp,)f(A.)j(\(1990\).)h Fd(Mo)n(dular)h(pr)n(op)n (erties)g(of)f(term)g(r)n(ewriting)g(systems)p Fw(.)i(PhD)c(thesis,)g (V)m(rije)g(Univ)o(ersiteit)e(te)174 1233 y(Amsterdam.)100 1270 y(Middeldorp,)j(A.)k(\(1993\).)25 b(Mo)q(dular)14 b(prop)q(erties)f(of)i(conditiona)o(l)e(term)h(rewriting)g(systems.)25 b Fd(Information)16 b(and)174 1308 y(Computation)d Fh(104)p Fd(\(1\))p Fw(,)f(110{158.)100 1345 y(Middeldorp,)c(A.)k(\(1994)p Fd(a)p Fw(\).)h(Completeness)c(of)i(com)o(binatio)o(ns)d(of)j (conditional)d(constructor)h(systems.)k Fd(J.)f(Symb)n(olic)174 1382 y(Computation)h Fh(17)p Fw(,)g(3{21.)100 1420 y(Middeldorp,)c(A.)j (\(1994)p Fd(b)p Fw(\).)h(A)g(simple)d(pro)q(of)g(to)h(a)g(result)g(of) g(Bernhard)e(Gramlic)o(h.)k(Unpublished)c(note.)100 1457 y(Middeldorp,)f(A.,)k(T)m(o)o(y)o(ama,)e(Y.)h(\(1993\).)i(Completeness) c(of)i(com)o(binatio)o(ns)d(of)j(constructor)e(systems.)k Fd(J.)f(Symb)n(olic)174 1494 y(Computation)h Fh(15)p Fd(\(3\))p Fw(,)f(331{348.)100 1532 y(Ohlebusc)o(h,)f(E.)i(\(1993\).)18 b(A)c(simple)d(pro)q(of)h(of)h(su\016cien)o(t)e(conditions)g(for)h(the) g(termination)e(of)j(the)f(disjoin)o(t)f(union)174 1569 y(of)g(term)f(rewriting)f(systems.)k Fd(Bul)r(letin)f(of)g(the)h(Eur)n (op)n(e)n(an)h(Asso)n(ciation)f(for)f(The)n(or)n(etic)n(al)i(Computer)f (Scienc)n(e)174 1606 y Fh(49)p Fw(,)g(178{183.)100 1644 y(Ohlebusc)o(h,)c(E.)j(\(1994)p Fd(a)p Fw(\).)h(On)f(the)e(mo)q (dularit)o(y)f(of)i(con\015uence)d(of)j(constructor-sh)o(arin)o(g)e (term)h(rewriting)g(systems.)174 1681 y(In)15 b Fd(Pr)n(o)n(c)n(e)n(e)n (dings)j(of)e(the)g(19th)h(Col)r(lo)n(quium)f(on)h(T)m(r)n(e)n(es)f(in) g(A)o(lgebr)n(a)i(and)e(Pr)n(o)n(gr)n(amming)p Fw(,)h(261{275.)12 b(Lecture)174 1719 y(Notes)g(in)f(Computer)e(Science)h Fh(787)p Fw(,)i(Berlin:)f(Springer)e(V)m(erlag.)100 1756 y(Ohlebusc)o(h,)d(E.)i(\(1994)p Fd(b)p Fw(\).)g Fd(Mo)n(dular)i(pr)n (op)n(erties)h(of)e(c)n(omp)n(osable)j(term)e(r)n(ewriting)g(systems)p Fw(.)f(PhD)e(thesis,)g(Univ)o(ersit\177)-18 b(at)174 1793 y(Bielefeld,)10 b(German)o(y)m(.)100 1831 y(Ohlebusc)o(h,)h(E.)i (\(1994)p Fd(c)p Fw(\).)19 b(Mo)q(dular)12 b(prop)q(erties)f(of)i (constructor-)o(sha)o(ring)d(conditiona)o(l)h(term)h(rewriting)g (systems.)174 1868 y(In)j Fd(Pr)n(o)n(c)n(e)n(e)n(dings)k(of)d(the)g (4th)g(International)h(Workshop)f(on)h(Conditional)f(T)m(erm)g(R)n (ewriting)h(Systems)p Fw(.)25 b(T)m(o)174 1905 y(app)q(ear.)100 1943 y(Ohlebusc)o(h,)9 b(E.)j(\(1995)p Fd(a)p Fw(\).)i(On)e(the)f(mo)q (dularit)o(y)e(of)i(termination)d(of)k(term)e(rewriting)g(systems.)k Fd(The)n(or)n(etic)n(al)h(Com-)174 1980 y(puter)f(Scienc)n(e)g Fh(136)p Fw(,)f(333{360.)100 2017 y(Ohlebusc)o(h,)g(E.)j(\(1995)p Fd(b)p Fw(\).)27 b(T)m(ermination)13 b(is)j(not)f(mo)q(dular)f(for)h (con\015uen)o(t)e(v)n(ariable-prese)o(rvi)o(ng)g(term)h(rewriting)174 2055 y(systems.)g Fd(Information)f(Pr)n(o)n(c)n(essing)h(L)n(etters)f Fh(53)p Fw(,)g(223{228.)100 2092 y(Prehofer,)8 b(C.)j(\(1994\).)h(On)f (mo)q(dularit)o(y)d(in)i(term)f(rewriting)g(and)h(narro)o(wing.)i(In)e Fd(Pr)n(o)n(c)n(e)n(e)n(dings)15 b(of)d(the)g(1st)g(Interna-)174 2130 y(tional)j(Confer)n(enc)n(e)i(on)d(Constr)n(aints)g(in)h (Computational)g(L)n(o)n(gics)p Fw(,)g(253{268,)c(Lecture)g(Notes)j(in) f(Computer)174 2167 y(Science)d Fh(845)p Fw(,)i(Berlin:)f(Springer)e(V) m(erlag.)100 2204 y(Raoult,)h(J.-C.,)i(V)m(uillemin,)e(J.)i(\(1980\).)k (Op)q(erational)9 b(and)i(seman)o(tic)f(equiv)n(alence)f(b)q(et)o(w)o (een)i(recursiv)o(e)e(programs.)174 2242 y Fd(Journal)14 b(of)e(the)i(A)o(CM)e Fh(27)p Fd(\(4\))p Fw(,)g(772{796.)100 2279 y(Rusino)o(witc)o(h,)h(M.)k(\(1987\).)26 b(On)16 b(termination)c(of)k(the)f(direct)f(sum)h(of)h(term)f(rewriting)f (systems.)26 b Fd(Information)174 2316 y(Pr)n(o)n(c)n(essing)14 b(L)n(etters)g Fh(26)p Fw(,)e(65{70.)100 2354 y(T)m(o)o(y)o(ama,)i(Y.)i (\(1987)p Fd(a)p Fw(\).)26 b(On)16 b(the)f(Ch)o(urc)o(h-Rosser)f(prop)q (ert)o(y)f(for)i(the)g(direct)g(sum)f(of)i(term)e(rewriting)h(systems.) 174 2391 y Fd(Journal)f(of)e(the)i(A)o(CM)e Fh(34)p Fd(\(1\))p Fw(,)g(128{143.)100 2428 y(T)m(o)o(y)o(ama,)i(Y.)h(\(1987)p Fd(b)p Fw(\).)26 b(Coun)o(terexam)o(ples)12 b(to)j(termination)d(for)j (the)f(direct)g(sum)h(of)g(term)f(rewriting)g(systems.)174 2466 y Fd(Information)f(Pr)n(o)n(c)n(essing)h(L)n(etters)g Fh(25)p Fw(,)e(141{143.)100 2503 y(T)m(o)o(y)o(ama,)g(Y.,)j(Klop,)f (J.W.,)g(Barendregt,)d(H.P)m(.)k(\(1989\).)21 b(T)m(ermination)12 b(for)h(the)h(direct)e(sum)i(of)g(left-linear)d(term)174 2540 y(rewriting)d(systems.)i(In)f Fd(Pr)n(o)n(c)n(e)n(e)n(dings)k(of)e (the)g(3r)n(d)g(International)h(Confer)n(enc)n(e)g(on)f(R)n(ewriting)g (T)m(e)n(chniques)h(and)174 2578 y(Applic)n(ations)p Fw(,)h(477{491.)c(Lecture)h(Notes)h(in)h(Computer)d(Science)h Fh(355)p Fw(,)i(Berlin:)e(Springer)g(V)m(erlag.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF