%!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: most.dvi %%Pages: 32 1 %%BoundingBox: 0 0 596 843 %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@letter{/vsize 10 N}B /@landscape{/isls true N /vsize -1 N}B /@a4{/vsize 10.6929133858 N}B /@a3{ /vsize 15.5531 N}B /@ledger{/vsize 16 N}B /@legal{/vsize 13 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}N /eop{clear SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if /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}N /p /show load 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 statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval(Display)eq}{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 /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{clear SS restore}B end %%EndProcSet TeXDict begin 1000 300 300 @start /Fa 8 40 df<7FC0000000000000FFFF000000000000 FFFFE000000000007FFFFC0000000000007FFF80000000000001FFE00000000000003FF8000000 00000007FE00000000000000FF800000000000003FC00000000000000FF000000000000003F800 000000000000FC000000000000007F000000000000001F800000000000000FC000000000000007 E000000000000001F000000000000000F8000000000000007C000000000000003E000000000000 001F000000000000000F80000000000000078000000000000007C000000000000003E000000000 000001F000000000000000F000000000000000F8000000000000007C000000000000003C000000 000000003E000000000000001E000000000000001F000000000000000F000000000000000F8000 0000000000078000000000000007C000000000000003C000000000000003C000000000000003E0 00000000000001E000000000000001E000000000000001F000000000000000F000000000000000 F000000000000000F8000000000000007800000000000000780000000000000078000000000000 00780000000000000078000000000000007C000000000000003C000000000000003C0000000000 00003C000000000000003C000000000000003C000000000000003C000000000000003C00000000 0000003C00000000000000183E3EBA8374>24 D<0000000000000018000000000000003C000000 000000003C000000000000003C000000000000003C000000000000003C000000000000003C0000 00000000003C000000000000003C000000000000007C0000000000000078000000000000007800 000000000000780000000000000078000000000000007800000000000000F800000000000000F0 00000000000000F000000000000001F000000000000001E000000000000001E000000000000003 E000000000000003C000000000000003C000000000000007C00000000000000780000000000000 0F800000000000000F000000000000001F000000000000001E000000000000003E000000000000 003C000000000000007C00000000000000F800000000000000F000000000000001F00000000000 0003E000000000000007C000000000000007800000000000000F800000000000001F0000000000 00003E000000000000007C00000000000000F800000000000001F000000000000007E000000000 00000FC00000000000001F800000000000007F00000000000000FC00000000000003F800000000 00000FF00000000000003FC0000000000000FF80000000000007FE0000000000003FF800000000 0001FFE000000000007FFF80000000007FFFFC0000000000FFFFE00000000000FFFF0000000000 007FC00000000000003E3EBABD74>I<6000000000000000F000000000000000F0000000000000 00F000000000000000F000000000000000F000000000000000F000000000000000F00000000000 0000F000000000000000F800000000000000780000000000000078000000000000007800000000 000000780000000000000078000000000000007C000000000000003C000000000000003C000000 000000003E000000000000001E000000000000001E000000000000001F000000000000000F0000 00000000000F000000000000000F80000000000000078000000000000007C000000000000003C0 00000000000003E000000000000001E000000000000001F000000000000000F000000000000000 F8000000000000007C000000000000003C000000000000003E000000000000001F000000000000 000F80000000000000078000000000000007C000000000000003E000000000000001F000000000 000000F8000000000000007C000000000000003E000000000000001F800000000000000FC00000 0000000007E000000000000003F800000000000000FC000000000000007F000000000000003FC0 0000000000000FF000000000000007FC00000000000001FF800000000000007FF0000000000000 1FFE00000000000007FFF8000000000000FFFFF800000000001FFFFC000000000003FFFC000000 0000000FF83E3E80BD74>I<0000000000000FF8000000000003FFFC00000000001FFFFC000000 0000FFFFF80000000007FFF800000000001FFE0000000000007FF0000000000001FF8000000000 0007FC0000000000000FF00000000000003FC00000000000007F00000000000000FC0000000000 0003F800000000000007E00000000000000FC00000000000001F800000000000003E0000000000 00007C00000000000000F800000000000001F000000000000003E000000000000007C000000000 000007800000000000000F800000000000001F000000000000003E000000000000003C00000000 0000007C00000000000000F800000000000000F000000000000001F000000000000001E0000000 00000003E000000000000003C000000000000007C000000000000007800000000000000F800000 000000000F000000000000000F000000000000001F000000000000001E000000000000001E0000 00000000003E000000000000003C000000000000003C000000000000007C000000000000007800 0000000000007800000000000000780000000000000078000000000000007800000000000000F8 00000000000000F000000000000000F000000000000000F000000000000000F000000000000000 F000000000000000F000000000000000F000000000000000F00000000000000060000000000000 003E3E808374>I<7FE0000000000000000000FFFFC00000000000000000FFFFFC000000000000 00007FFFFFC000000000000000003FFFF80000000000000000007FFF00000000000000000007FF C00000000000000000007FF80000000000000000000FFE00000000000000000001FF8000000000 00000000007FE00000000000000000000FF000000000000000000003FC00000000000000000000 FE000000000000000000003F800000000000000000001FC000000000000000000007F000000000 000000000003F800000000000000000000FC000000000000000000007F00000000000000000000 1F800000000000000000000FC000000000000000000007E000000000000000000001F000000000 000000000000F8000000000000000000007C000000000000000000003E00000000000000000000 1F000000000000000000000F8000000000000000000007C000000000000000000003E000000000 000000000001F000000000000000000000F000000000000000000000F800000000000000000000 7C000000000000000000003E000000000000000000001E000000000000000000001F0000000000 00000000000F80000000000000000000078000000000000000000007C000000000000000000003 E000000000000000000001E000000000000000000001F000000000000000000000F80000000000 000000000078000000000000000000007C000000000000000000003C000000000000000000003E 000000000000000000001E000000000000000000001F000000000000000000000F000000000000 000000000F000000000000000000000F80000000000000000000078000000000000000000007C0 00000000000000000003C000000000000000000003C000000000000000000003E0000000000000 00000001E000000000000000000001E000000000000000000001F000000000000000000000F000 000000000000000000F000000000000000000000F000000000000000000000F800000000000000 0000007800000000000000000000780000000000000000000078000000000000000000007C0000 00000000000000003C000000000000000000003C000000000000000000003C0000000000000000 00003C000000000000000000003C000000000000000000003C000000000000000000003E000000 000000000000001E000000000000000000001E000000000000000000001E000000000000000000 001E000000000000000000001E000000000000000000001E000000000000000000001E00000000 0000000000001E000000000000000000001E000000000000000000000C5757D383A6>36 D<000000000000000000000C000000000000000000001E000000000000000000001E0000000000 00000000001E000000000000000000001E000000000000000000001E000000000000000000001E 000000000000000000001E000000000000000000001E000000000000000000001E000000000000 000000003E000000000000000000003C000000000000000000003C000000000000000000003C00 0000000000000000003C000000000000000000003C000000000000000000003C00000000000000 0000007C0000000000000000000078000000000000000000007800000000000000000000780000 0000000000000000F800000000000000000000F000000000000000000000F00000000000000000 0000F000000000000000000001F000000000000000000001E000000000000000000001E0000000 00000000000003E000000000000000000003C000000000000000000003C0000000000000000000 07C000000000000000000007800000000000000000000F800000000000000000000F0000000000 00000000000F000000000000000000001F000000000000000000001E000000000000000000003E 000000000000000000003C000000000000000000007C0000000000000000000078000000000000 00000000F800000000000000000001F000000000000000000001E000000000000000000003E000 000000000000000007C000000000000000000007800000000000000000000F8000000000000000 00001F000000000000000000001E000000000000000000003E000000000000000000007C000000 00000000000000F800000000000000000000F000000000000000000001F0000000000000000000 03E000000000000000000007C00000000000000000000F800000000000000000001F0000000000 00000000003E000000000000000000007C00000000000000000000F800000000000000000001F0 00000000000000000007E00000000000000000000FC00000000000000000001F80000000000000 0000007F00000000000000000000FC00000000000000000003F800000000000000000007F00000 000000000000001FC00000000000000000003F80000000000000000000FE000000000000000000 03FC0000000000000000000FF00000000000000000007FE0000000000000000001FF8000000000 000000000FFE0000000000000000007FF8000000000000000007FFC000000000000000007FFF00 000000000000003FFFF8000000000000007FFFFFC000000000000000FFFFFC0000000000000000 FFFFC000000000000000007FE00000000000000000005757D3D6A6>I<60000000000000000000 00F000000000000000000000F000000000000000000000F000000000000000000000F000000000 000000000000F000000000000000000000F000000000000000000000F000000000000000000000 F000000000000000000000F000000000000000000000F800000000000000000000780000000000 000000000078000000000000000000007800000000000000000000780000000000000000000078 0000000000000000000078000000000000000000007C000000000000000000003C000000000000 000000003C000000000000000000003C000000000000000000003E000000000000000000001E00 0000000000000000001E000000000000000000001E000000000000000000001F00000000000000 0000000F000000000000000000000F000000000000000000000F80000000000000000000078000 0000000000000000078000000000000000000007C000000000000000000003C000000000000000 000003E000000000000000000001E000000000000000000001E000000000000000000001F00000 0000000000000000F000000000000000000000F800000000000000000000780000000000000000 00007C000000000000000000003C000000000000000000003E000000000000000000001F000000 000000000000000F000000000000000000000F8000000000000000000007C00000000000000000 0003C000000000000000000003E000000000000000000001F000000000000000000000F0000000 00000000000000F8000000000000000000007C000000000000000000003E000000000000000000 001E000000000000000000001F000000000000000000000F8000000000000000000007C0000000 00000000000003E000000000000000000001F000000000000000000000F8000000000000000000 007C000000000000000000003E000000000000000000001F000000000000000000000FC0000000 00000000000007E000000000000000000003F000000000000000000001FC000000000000000000 007E000000000000000000003F800000000000000000001FC000000000000000000007F0000000 00000000000003F800000000000000000000FE000000000000000000007F800000000000000000 001FE00000000000000000000FFC00000000000000000003FF00000000000000000000FFE00000 000000000000003FFC00000000000000000007FFC0000000000000000001FFFC00000000000000 00003FFFF8000000000000000007FFFFFC00000000000000007FFFFE000000000000000007FFFE 0000000000000000000FFC575780D6A6>I<0000000000000000000FFC000000000000000007FF FE00000000000000007FFFFE0000000000000007FFFFFC000000000000003FFFF8000000000000 0001FFFC000000000000000007FFC000000000000000003FFC000000000000000000FFE0000000 000000000003FF0000000000000000000FFC0000000000000000001FE00000000000000000007F 80000000000000000000FE00000000000000000003F800000000000000000007F0000000000000 0000001FC00000000000000000003F800000000000000000007E00000000000000000001FC0000 0000000000000003F000000000000000000007E00000000000000000000FC00000000000000000 001F000000000000000000003E000000000000000000007C00000000000000000000F800000000 000000000001F000000000000000000003E000000000000000000007C00000000000000000000F 800000000000000000001F000000000000000000001E000000000000000000003E000000000000 000000007C00000000000000000000F800000000000000000000F000000000000000000001F000 000000000000000003E000000000000000000003C000000000000000000007C000000000000000 00000F800000000000000000000F000000000000000000001F000000000000000000003E000000 000000000000003C000000000000000000007C0000000000000000000078000000000000000000 00F800000000000000000000F000000000000000000001F000000000000000000001E000000000 000000000001E000000000000000000003E000000000000000000003C000000000000000000007 C0000000000000000000078000000000000000000007800000000000000000000F800000000000 000000000F000000000000000000000F000000000000000000001F000000000000000000001E00 0000000000000000001E000000000000000000001E000000000000000000003E00000000000000 0000003C000000000000000000003C000000000000000000003C000000000000000000007C0000 000000000000000078000000000000000000007800000000000000000000780000000000000000 0000780000000000000000000078000000000000000000007800000000000000000000F8000000 00000000000000F000000000000000000000F000000000000000000000F0000000000000000000 00F000000000000000000000F000000000000000000000F000000000000000000000F000000000 000000000000F000000000000000000000F0000000000000000000006000000000000000000000 57578083A6>I E /Fb 2 70 df<081C1C3838383070706060C0C0060D7E8D09>48 D<01F8061C0C0C0808180018001E0007C00C00100020004000C000C010C020F0C03F000E117E90 11>69 D E /Fc 2 79 df73 D78 D E /Fd 37 122 df<0001FF803FE000001FFFE3FFF800007F80FFF01E 0000FE007F801F0001F800FF003F0003F801FF007F8007F001FE007F8007F001FE007F8007F001 FE007F8007F000FE003F0007F000FE001E0007F000FE00000007F000FE00000007F000FE000000 07F000FE000000FFFFFFFFFFFF80FFFFFFFFFFFF80FFFFFFFFFFFF8007F000FE003F8007F000FE 003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007 F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE00 3F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F000FE003F8007F0 00FE003F8007F000FE003F8007F000FE003F807FFF0FFFE3FFF87FFF0FFFE3FFF87FFF0FFFE3FF F8352A7FA939>14 D<1C003E007F00FF80FF80FF807F003E001C0009097B8813>46 D<000E00001E00007E0007FE00FFFE00FFFE00F8FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE007FFFFE7FFFFE7FFF FE17277BA622>49 D<00FF800003FFF0000FFFFC001F03FE003800FF007C007F80FE003FC0FF00 3FC0FF003FE0FF001FE0FF001FE07E001FE03C003FE000003FE000003FC000003FC000007F8000 007F000000FE000000FC000001F8000003F0000003E00000078000000F0000001E0000003C00E0 007000E000E000E001C001C0038001C0070001C00FFFFFC01FFFFFC03FFFFFC07FFFFFC0FFFFFF 80FFFFFF80FFFFFF801B277DA622>I<007F800003FFF00007FFFC000F81FE001F00FF003F80FF 003F807F803F807F803F807F801F807F800F007F800000FF000000FF000000FE000001FC000001 F8000007F00000FFC00000FFF0000001FC0000007E0000007F0000007F8000003FC000003FC000 003FE000003FE03C003FE07E003FE0FF003FE0FF003FE0FF003FC0FF007FC07E007F807C007F00 3F01FE001FFFFC0007FFF00000FF80001B277DA622>I<00000E0000001E0000003E0000007E00 0000FE000000FE000001FE000003FE0000077E00000E7E00000E7E00001C7E0000387E0000707E 0000E07E0000E07E0001C07E0003807E0007007E000E007E000E007E001C007E0038007E007000 7E00E0007E00FFFFFFF8FFFFFFF8FFFFFFF80000FE000000FE000000FE000000FE000000FE0000 00FE000000FE000000FE00007FFFF8007FFFF8007FFFF81D277EA622>I<0C0003000F803F000F FFFE000FFFFC000FFFF8000FFFF0000FFFE0000FFFC0000FFE00000E0000000E0000000E000000 0E0000000E0000000E0000000E7FC0000FFFF8000F80FC000E003E000C003F0000001F8000001F C000001FC000001FE000001FE018001FE07C001FE0FE001FE0FE001FE0FE001FE0FE001FC0FC00 1FC078003F8078003F803C007F001F01FE000FFFF80003FFF00000FF80001B277DA622>I66 D<00003FF001800003FFFE0380000FFFFF8780003FF0 07DF8000FF8001FF8001FE00007F8003FC00003F8007F000001F800FF000000F801FE000000780 1FE0000007803FC0000007803FC0000003807FC0000003807F80000003807F8000000000FF8000 000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000 FF8000000000FF80000000007F80000000007F80000000007FC0000003803FC0000003803FC000 0003801FE0000003801FE0000007000FF00000070007F000000E0003FC00001E0001FE00003C00 00FF8000F800003FF007E000000FFFFFC0000003FFFF000000003FF8000029297CA832>II 77 DI80 D82 D<007F806003FFF0E007FFF9E00F807FE01F00 1FE03E0007E07C0003E07C0001E0FC0001E0FC0001E0FC0000E0FE0000E0FE0000E0FF000000FF C000007FFE00007FFFE0003FFFFC001FFFFE000FFFFF8007FFFFC003FFFFE000FFFFE00007FFF0 00007FF000000FF8000007F8000003F8600001F8E00001F8E00001F8E00001F8F00001F0F00001 F0F80003F0FC0003E0FF0007C0FFE01F80F3FFFF00E0FFFE00C01FF0001D297CA826>I<7FFFFF FFFFC07FFFFFFFFFC07FFFFFFFFFC07F803FC03FC07E003FC007C078003FC003C078003FC003C0 70003FC001C0F0003FC001E0F0003FC001E0E0003FC000E0E0003FC000E0E0003FC000E0E0003F C000E0E0003FC000E000003FC0000000003FC0000000003FC0000000003FC0000000003FC00000 00003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003F C0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC00000 00003FC0000000003FC0000000003FC0000000003FC00000007FFFFFE000007FFFFFE000007FFF FFE0002B287EA730>II<01FF800007FFF0000F81 F8001FC07E001FC07E001FC03F000F803F8007003F8000003F8000003F8000003F80000FFF8000 FFFF8007FC3F800FE03F803F803F803F003F807F003F80FE003F80FE003F80FE003F80FE003F80 7E007F807F00DF803F839FFC0FFF0FFC01FC03FC1E1B7E9A21>97 D<001FF80000FFFE0003F01F 0007E03F800FC03F801F803F803F801F007F800E007F0000007F000000FF000000FF000000FF00 0000FF000000FF000000FF000000FF0000007F0000007F0000007F8000003F8001C01F8001C00F C0038007E0070003F01E0000FFFC00001FE0001A1B7E9A1F>99 D<00003FF80000003FF8000000 3FF800000003F800000003F800000003F800000003F800000003F800000003F800000003F80000 0003F800000003F800000003F800000003F800000003F800001FE3F80000FFFBF80003F03FF800 07E00FF8000FC007F8001F8003F8003F8003F8007F0003F8007F0003F8007F0003F800FF0003F8 00FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F8007F0003F8007F0003 F8007F0003F8003F8003F8001F8003F8000F8007F80007C00FF80003F03BFF8000FFF3FF80003F C3FF80212A7EA926>I<003FE00001FFF80003F07E0007C01F000F801F801F800F803F800FC07F 000FC07F0007C07F0007E0FF0007E0FF0007E0FFFFFFE0FFFFFFE0FF000000FF000000FF000000 7F0000007F0000007F0000003F8000E01F8000E00FC001C007E0038003F81F0000FFFE00001FF0 001B1B7E9A20>I<0007F0003FFC00FE3E01F87F03F87F03F07F07F07F07F03E07F00007F00007 F00007F00007F00007F00007F000FFFFC0FFFFC0FFFFC007F00007F00007F00007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007 F00007F00007F0007FFF807FFF807FFF80182A7EA915>I<00FF81F003FFE7F80FC1FE7C1F80FC 7C1F007C383F007E107F007F007F007F007F007F007F007F007F007F007F007F003F007E001F00 7C001F80FC000FC1F8001FFFE00018FF800038000000380000003C0000003E0000003FFFF8001F FFFF001FFFFF800FFFFFC007FFFFE01FFFFFF03E0007F07C0001F8F80000F8F80000F8F80000F8 F80000F87C0001F03C0001E01F0007C00FC01F8003FFFE00007FF0001E287E9A22>II<07000F801FC03FE03FE03FE01FC00F800700000000 0000000000000000000000FFE0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFEFFFEFFFE0F2B7DAA14>I<000700000F80 001FC0003FE0003FE0003FE0001FC0000F80000700000000000000000000000000000000000000 00000001FFE001FFE001FFE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0 000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0 000FE0000FE0000FE0000FE0000FE0000FE07C0FE0FE0FE0FE0FC0FE1F80FE1F007C3E003FFC00 0FF000133784AA15>I108 DII<003FE00001FFFC0003F07E000FC01F801F800FC03F800FE03F0007 E07F0007F07F0007F07F0007F0FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF00 07F8FF0007F87F0007F07F0007F03F800FE03F800FE01F800FC00FC01F8007F07F0001FFFC0000 3FE0001D1B7E9A22>II114 D<03FE300FFFF01E03F038 00F0700070F00070F00070F80070FC0000FFE0007FFE007FFF803FFFE01FFFF007FFF800FFF800 03FC0000FC60007CE0003CF0003CF00038F80038FC0070FF01E0F7FFC0C1FF00161B7E9A1B>I< 00700000700000700000700000F00000F00000F00001F00003F00003F00007F0001FFFF0FFFFF0 FFFFF007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F03807F03807F03807F03807F03807F03803F03803F87001F86000FFC0001F8015267F A51B>II119 D121 D E /Fe 7 115 df<072018E0306060606060C0C0C0C0C0C841C8 62D03C700D0B7E8A11>97 D<007800180018003000300030073018E0306060606060C0C0C0C0C0 C841C862D03C700D117E9010>100 D<03900C7018303030303060606060606020E031C01EC000 C000C04180C3007E000C107E8A0F>103 D<7818183030303060606060C0C0C8C8D07005117E90 0A>108 D<71F1F09A1A189C1C1898181818181830303030303030303230306260606460603817 0B7E8A1B>I<71F09A189C18981818183030303030323062606460380F0B7E8A13>I<73C09C2098 60980018003000300030003000600060000B0B7E8A0E>114 D E /Ff 1 51 df<7FFFFFC0FFFFFFE0C0000060C0000060C0000060C0000060C0000060C0000060C0000060 C0000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060C0000060C00000 60C0000060C0000060C0000060C0000060C0000060C0000060FFFFFFE0FFFFFFE01B1B7B9E25> 50 D E /Fg 11 126 df<00001C00003C0000F80001E00003C0000780000F00000E00001E0000 3C00003C00003C0000780000780000780000780000780000780000780000780000780000780000 780000780000780000780000780000780000780000780000780000780000780000780000780000 780000780000780000780000780000780000780000780000780000780000780000780000780000 780000780000F00000F00000F00001E00001E00003C0000380000700000E00001C0000780000E0 0000E000007800001C00000E000007000003800003C00001E00001E00000F00000F00000F00000 780000780000780000780000780000780000780000780000780000780000780000780000780000 780000780000780000780000780000780000780000780000780000780000780000780000780000 7800007800007800007800007800007800007800007800007800007800007800007800003C0000 3C00003C00001E00000E00000F000007800003C00001E00000F800003C00001C167C7B8121>40 D<0018007800F001E003C007800F001F001E003E003C007C007C007800F800F800F800F800F800 F800F800F800F800F800F800F800F800F800F800F800F800F800F800F800F800F800F8000D2570 7E25>56 D58 D<007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C 007C007C007C007C007C007C007C007C007C00F800F800F800F001F001E003E003C0078007000E 001C003800F000C000F00038001C000E000700078003C003E001E001F000F000F800F800F8007C 007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C00 7C007C007C007C0E4D798025>60 D62 D83 D91 D<0001F8000FF8003FF800FFF801 FFF803FE0007E0000F80001E0000380000700000600000E00000C00000150E818413>122 DII<0000180000380000300000700000E00003C0 000F80003F0003FE00FFFC00FFF800FFE000FF8000FC0000150E818D13>I E /Fh 23 120 df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i 53 123 df<00000FE0000030180000E01C0001C03C0001803C000380380003800000038000 0007000000070000000700000007000000070000000E000000FFFFE0000E00E0000E00E0000E01 C0001C01C0001C01C0001C01C0001C0380001C0380003803800038038000380700003807000038 07000070070800700E1000700E1000700E1000700E2000E0062000E003C000E0000000E0000000 C0000001C0000001C0000071800000F1800000F3000000620000003C0000001E2D82A21B>12 D<00000FF3800000303F000000607F000000C07F000001C037000001800E000003800E00000380 0E000003800E000003801C000007001C000007001C000007001C00000700380000FFFFF800000E 003800000E003800000E007000000E007000000E007000001C007000001C00E000001C00E00000 1C00E000001C00E000003801C000003801C000003801C000003801C20000380384000070038400 007003840000700388000070018800007000F00000E000000000E000000000E000000000C00000 0000C00000007180000000F180000000F10000000062000000003C00000000212D82A21D>I<00 008000010000020000040000080000100000300000600000C00000C00001800003000003000006 00000600000E00000C00001C000018000018000038000030000030000070000070000060000060 0000E00000E00000E00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C0 0000C00000C00000C00000400000600000600000200000300000100000080000113278A414>40 D<0008000004000006000002000003000003000001000001800001800001800001800001800001 800001800001800001800001800001800001800001800003800003800003800003000003000007 00000700000600000600000E00000C00000C00001C000018000038000030000030000060000060 0000C000018000018000030000060000040000080000100000200000400000800000113280A414 >I<0E1E1E1E1E02020404080810204080070F7D840F>44 DI<70F8 F8F0E005057A840F>I<0001000300030006001E002E03CE001C001C001C001C00380038003800 38007000700070007000E000E000E000E001C001C001C001C003800380038003800780FFFC1021 7AA019>49 D<000FC000307000C01801001C02001C04000C04401C08201C08201C08201C084038 08C0380700700000600001C000070000FC000007000003800003800001C00001C00001C00003C0 6003C0F003C0F00380E00780800700800E00801C0040380020F0001F800016227BA019>51 D<0000180000380000380000700000700000700000E00000E00000E00000C00001C00001800003 80000300000300000600000600000C00000C000018000010000031800061C00043800083800183 800303800207000407000807003FC700403E00800FF0000E00000E00001C00001C00001C00001C 00003800003800003800003000152B7EA019>I<00400400703800FFF000FFE000BF8000800001 0000010000010000010000020000020000023E0002C3000501800601C00401C00001E00001E000 01E00001E00001E00001E07003C0F003C0F003C0E00780800700800F00800E00401C0040380030 E0000F800016227BA019>I<000FC000306000401000801801000803000C03000C060018070018 07001007003007C06007E0C003F18001FE0000FC0000FE00033F00061F800C07C01803C03001C0 6001C06000C0C000C0C000C0C00080C00180C00100C00200C006006008003030000FC00016227B A019>56 D<0000030000000300000007000000070000000F0000000F0000001F0000002F000000 2F0000004F0000004F800000878000008780000107800002078000020780000407800004078000 08078000080780001007800030078000200780007FFF80004007C0008007C0008003C0010003C0 030003C0020003C0040003C0040003C00C0003C03C0007C0FF003FFC1E237DA224>65 D<00FFFFE0000F0038000F001C000F001E001E000E001E000F001E000F001E000F003C000E003C 001E003C001E003C003C00780078007800F0007801E00078078000FFFF8000F001E000F000F000 F0007801E0007801E0003801E0003C01E0003C03C0007803C0007803C0007803C000F0078000F0 078001E0078003C0078007000F801E00FFFFF00020227DA122>I<00007F00800003808100000E 00630000380027000070001F0000E0000E0001C0000E000380000E000700000E000F000004000E 000004001E000004003C000004003C00000800780000000078000000007800000000F000000000 F000000000F000000000F000000000F000000000E000000000E000002000E000002000E0000040 00E000004000F00000800070000080007000010000380002000018000400001C00080000060030 00000381C0000000FE000000212479A223>I<00FFFFFF80000F000780000F000180000F000180 001E000180001E000180001E000100001E000100003C000100003C000100003C010100003C0100 0000780200000078020000007806000000780E000000FFFC000000F00C000000F00C000000F00C 000001E008000001E008000001E008040001E000080003C000080003C000080003C000100003C0 00100007800020000780006000078000C000078001C0000F8007C000FFFFFF800021227DA121> 69 D<00FFF87FFC000F000780000F000780000F000780001E000F00001E000F00001E000F0000 1E000F00003C001E00003C001E00003C001E00003C001E000078003C000078003C000078003C00 0078003C0000FFFFF80000F000780000F000780000F000780001E000F00001E000F00001E000F0 0001E000F00003C001E00003C001E00003C001E00003C001E000078003C000078003C000078003 C000078003C0000F8007C000FFF87FFC0026227DA124>72 D<00FFF8000F00000F00000F00001E 00001E00001E00001E00003C00003C00003C00003C0000780000780000780000780000F00000F0 0000F00000F00001E00001E00001E00001E00003C00003C00003C00003C0000780000780000780 000780000F8000FFF80015227DA113>I<0007FFC000003C0000003C0000003C00000078000000 780000007800000078000000F0000000F0000000F0000000F0000001E0000001E0000001E00000 01E0000003C0000003C0000003C0000003C00000078000000780000007800000078000000F0000 000F0000380F0000780F0000F81E0000F81E0000F03C0000403800004070000021E000001F8000 001A237CA11A>I<00FFFC00000F8000000F0000000F0000001E0000001E0000001E0000001E00 00003C0000003C0000003C0000003C00000078000000780000007800000078000000F0000000F0 000000F0000000F0000001E0000001E0000001E0002001E0002003C0004003C0004003C0008003 C0008007800180078001000780030007800F000F803E00FFFFFE001B227DA11F>76 D<00FF800007FC000F80000F80000F80001780000F80001780001780002F000013C0002F000013 C0004F000013C0008F000023C0009E000023C0011E000023C0011E000023C0021E000043C0043C 000043C0043C000043C0083C000041E0083C000081E01078000081E02078000081E02078000081 E04078000101E040F0000101E080F0000101E100F0000101E100F0000200F201E0000200F201E0 000200F401E0000200F801E0000400F803C0000400F003C0000400F003C0000C00E003C0001E00 C007C000FFC0C07FFC002E227DA12C>I<00FFFFE0000F0038000F001E000F000E001E0007001E 0007001E0007001E0007003C000F003C000F003C000F003C001E0078001E0078003C0078007800 7800E000F003C000FFFE0000F0000000F0000001E0000001E0000001E0000001E0000003C00000 03C0000003C0000003C00000078000000780000007800000078000000F800000FFF8000020227D A121>80 D<00FFFFC0000F0070000F003C000F001C001E000E001E000E001E000F001E000F003C 001E003C001E003C001E003C003C0078003800780070007801E00078078000FFFC0000F00E0000 F0070000F0038001E003C001E003C001E003C001E003C003C0078003C0078003C0078003C00780 07800F0007800F0107800F01078007020F800702FFF8038C000000F020237DA124>82 D<0001F020000E0C40001802C0003001C0006001C000E0018000C0018001C0018001C0018003C0 010003C0010003C0000003C0000003E0000001F8000001FF000000FFE000007FF000001FF80000 03FC0000007C0000003C0000001E0000001E0000001E0020001C0020001C0020001C0020001800 6000380060003000700060007000C000C8018000C607000081FC00001B247DA21B>I<1FFFFFF8 1E03C0381803C0183003C01820078018200780184007801040078010400F0010800F0010800F00 10000F0000001E0000001E0000001E0000001E0000003C0000003C0000003C0000003C00000078 000000780000007800000078000000F0000000F0000000F0000000F0000001E0000001E0000001 E0000001E0000003E00000FFFF00001D2277A123>I86 DI<00F8C0 0185C00705C00E03800E03801C03803C0380380700780700780700780700F00E00F00E00F00E00 F00E10F01C20701C20703C20305C40308C400F078014157B9419>97 D<03C03F80038003800380 07000700070007000E000E000E000E001C001CF81D0C1E0E3C0638073807380F700F700F700F70 0FE01EE01EE01EE03CE038E038607060E031C01F0010237BA216>I<007E0001C1000301800703 800E07801C07803C0000380000780000780000780000F00000F00000F00000F00000F001007001 00700200300C001830000FC00011157B9416>I<00003C0003F800003800003800003800007000 00700000700000700000E00000E00000E00000E00001C000F9C00185C00705C00E03800E03801C 03803C0380380700780700780700780700F00E00F00E00F00E00F00E10F01C20701C20703C2030 5C40308C400F078016237BA219>I<00F803840E021C023C0238027804F018FFE0F000F000E000 E000E000E000E002E0026004701830600F800F157A9416>I<00003E0000470000CF00018F0001 86000380000380000380000700000700000700000700000700000E0000FFF0000E00000E00000E 00001C00001C00001C00001C00001C000038000038000038000038000038000070000070000070 0000700000700000E00000E00000E00000E00000C00001C00001C000718000F18000F300006200 003C0000182D82A20F>I<001F180030B800E0B801C07001C0700380700780700700E00F00E00F 00E00F00E01E01C01E01C01E01C01E01C01E03800E03800E0780060B8006170001E70000070000 0700000E00000E00000E00701C00F01800F0300060E0003F8000151F7E9416>I<00F0000FE000 00E00000E00000E00001C00001C00001C00001C000038000038000038000038000070000071F00 07218007C0C00F00E00F00E00E00E00E00E01C01C01C01C01C01C01C01C0380380380380380380 380704700708700E08700E10700610E006206003C016237DA219>I<00C001E001C001C0000000 000000000000000000000000001C002300430043008700870087000E000E001C001C001C003800 38003840708070807080710032001C000B217BA00F>I<0000E00001E00001E00000C000000000 0000000000000000000000000000000000000000001E0000230000438000838000838001038001 0380000700000700000700000700000E00000E00000E00000E00001C00001C00001C00001C0000 380000380000380000380000700000700000700070E000F0C000F180006300003C0000132B82A0 0F>I<00F0000FE00000E00000E00000E00001C00001C00001C00001C000038000038000038000 0380000700000701E0070210070C700E10F00E10F00E20600E40001D80001E00001FC0001C7000 383800383800381C00381C20703840703840703840701880E01880600F0014237DA216>I<01E0 1FC001C001C001C0038003800380038007000700070007000E000E000E000E001C001C001C001C 0038003800380038007000700070007100E200E200E200E200640038000B237CA20C>I<1C0F80 F8002610C10C00476066060087807807008780780700870070070087007007000E00E00E000E00 E00E000E00E00E000E00E00E001C01C01C001C01C01C001C01C01C001C01C03820380380384038 038070403803807080380380308070070031003003001E0023157B9428>I<1C0F002631C04740 C08780E08780E08700E08700E00E01C00E01C00E01C00E01C01C03801C03801C03801C07043807 08380E08380E103806107006203003C016157B941B>I<007E0001C3000381800701C00E01C01C 01E03C01E03801E07801E07801E07801E0F003C0F003C0F00380F00780700700700E00700C0030 180018700007C00013157B9419>I<01C1F002621804741C08780C08700E08700E08701E00E01E 00E01E00E01E00E01E01C03C01C03C01C03C01C07803807003807003C0E003C1C0072380071E00 0700000700000E00000E00000E00000E00001C00001C00001C0000FFC000171F7F9419>I<00F8 400184C00705C00E03800E03801C03803C0380380700780700780700780700F00E00F00E00F00E 00F00E00F01C00701C00703C00305C0030B8000F38000038000038000070000070000070000070 0000E00000E00000E0000FFE00121F7B9416>I<1C1F002620804741C08783C08703C087018087 00000E00000E00000E00000E00001C00001C00001C00001C000038000038000038000038000070 000030000012157B9415>I<00FC000183000200800401800C03800C03000C00000F00000FF000 07FC0003FE00003E00000F00000700700700F00600F00600E004004008002030001FC00011157D 9414>I<00C001C001C001C001C003800380038003800700FFF8070007000E000E000E000E001C 001C001C001C003800380038003810702070207040708031001E000D1F7C9E10>I<1E00602300 E04380E04381C08381C08701C08701C00703800E03800E03800E03801C07001C07001C07001C07 081C0E10180E101C0E101C1E200C262007C3C015157B941A>I<1E03802307C04387C04383C083 81C08700C08700C00700800E00800E00800E00801C01001C01001C01001C02001C02001C04001C 08001C08000C300003C00012157B9416>I<1E0060E02300E1F04380E1F04381C0F08381C07087 01C0308701C030070380200E0380200E0380200E0380201C0700401C0700401C0700401C070080 1C0700801C0701001C0F01000C0F020006138C0003E0F0001C157B9420>I<03C1E00462100834 70103CF02038F020386020380000700000700000700000700000E00000E00000E00000E02061C0 40F1C040F1C080E2C100446200383C0014157D9416>I<1E00302300704380704380E08380E087 00E08700E00701C00E01C00E01C00E01C01C03801C03801C03801C03801C07001C07001C07001C 0F000C3E0003CE00000E00000E00001C00601C00F03800F03000E0600080C0004380003E000014 1F7B9418>I<01E02003F06007F8C0041F80080100080200000400000800001000002000004000 0080000100000200000400800801001003003F060061FC0040F80080700013157D9414>I E /Fj 3 51 df22 D<0C003C00CC000C000C000C000C000C000C000C000C00 0C000C000C000C00FF8009107E8F0F>49 D<1F00618040C08060C0600060006000C00180030006 000C00102020207FC0FFC00B107F8F0F>I E /Fk 42 122 df<000FF000007FFC0001F80E0003 E01F0007C03F000F803F000F803F000F801E000F800C000F8000000F8000000F8000000F800000 FFFFFF00FFFFFF000F801F000F801F000F801F000F801F000F801F000F801F000F801F000F801F 000F801F000F801F000F801F000F801F000F801F000F801F000F801F000F801F000F801F000F80 1F007FF0FFE07FF0FFE01B237FA21F>12 D<387CFEFEFE7C3807077C8610>46 D<00FE0007FFC00F83E01F01F03E00F83E00F87C007C7C007C7C007CFC007CFC007EFC007EFC00 7EFC007EFC007EFC007EFC007EFC007EFC007EFC007EFC007EFC007EFC007E7C007C7C007C7C00 7C3E00F83E00F81F01F00F83E007FFC000FE0017207E9F1C>48 D<00180000780001F800FFF800 FFF80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 7FFFE07FFFE013207C9F1C>I<03FC000FFF003C1FC07007E07C07F0FE03F0FE03F8FE03F8FE01 F87C01F83803F80003F80003F00003F00007E00007C0000F80001F00003E0000380000700000E0 1801C0180380180700180E00380FFFF01FFFF03FFFF07FFFF0FFFFF0FFFFF015207D9F1C>I<00 FE0007FFC00F07E01E03F03F03F03F81F83F81F83F81F81F03F81F03F00003F00003E00007C000 1F8001FE0001FF000007C00001F00001F80000FC0000FC3C00FE7E00FEFF00FEFF00FEFF00FEFF 00FC7E01FC7801F81E07F00FFFC001FE0017207E9F1C>I<0000E00001E00003E00003E00007E0 000FE0001FE0001FE00037E00077E000E7E001C7E00187E00307E00707E00E07E00C07E01807E0 3807E07007E0E007E0FFFFFEFFFFFE0007E00007E00007E00007E00007E00007E00007E000FFFE 00FFFE17207E9F1C>I<1000201E01E01FFFC01FFF801FFF001FFE001FF8001BC0001800001800 0018000018000019FC001FFF001E0FC01807E01803E00003F00003F00003F80003F83803F87C03 F8FE03F8FE03F8FC03F0FC03F07007E03007C01C1F800FFF0003F80015207D9F1C>I<001F8000 FFE003F07007C0F00F01F81F01F83E01F83E01F87E00F07C00007C0000FC0800FC7FC0FCFFE0FD 80F0FF00F8FE007CFE007CFC007EFC007EFC007EFC007E7C007E7C007E7C007E3C007C3E007C1E 00F80F00F00783E003FFC000FF0017207E9F1C>I<6000007800007FFFFE7FFFFE7FFFFC7FFFF8 7FFFF87FFFF0E00060E000C0C00180C00300C00300000600000C00001C00001800003800007800 00780000F00000F00000F00001F00001F00001F00003F00003F00003F00003F00003F00003F000 03F00001E00017227DA11C>I<00FE0003FFC00703E00E00F01C00F01C00783C00783E00783F00 783F80783FE0F01FF9E01FFFC00FFF8007FFC003FFE007FFF01E7FF83C1FFC7807FC7801FEF000 FEF0003EF0001EF0001EF0001CF8001C7800383C00381F01F00FFFC001FF0017207E9F1C>I<01 FE0007FF800F83E01E01F03E00F07C00F87C0078FC007CFC007CFC007CFC007EFC007EFC007EFC 007E7C00FE7C00FE3E01FE1E037E0FFE7E07FC7E00207E00007C00007C1E007C3F00F83F00F83F 00F03F01E01E03C01C0F800FFE0003F80017207E9F1C>I<387CFEFEFE7C380000000000000000 387CFEFEFE7C3807167C9510>I<000070000000007000000000F800000000F800000000F80000 0001FC00000001FC00000003FE00000003FE00000003FE00000006FF000000067F0000000E7F80 00000C3F8000000C3F800000183FC00000181FC00000381FE00000300FE00000300FE00000600F F000006007F00000E007F80000FFFFF80000FFFFF800018001FC00018001FC00038001FE000300 00FE00030000FE000600007F000600007F00FFE00FFFF8FFE00FFFF825227EA12A>65 D<0003FE0080001FFF818000FF01E38001F8003F8003E0001F8007C0000F800F800007801F8000 07803F000003803F000003807F000001807E000001807E00000180FE00000000FE00000000FE00 000000FE00000000FE00000000FE00000000FE00000000FE000000007E000000007E000001807F 000001803F000001803F000003801F800003000F8000030007C000060003F0000C0001F8003800 00FF00F000001FFFC0000003FE000021227DA128>67 DII76 D80 D<7FFFFFFF807FFFFFFF807E03F80F807803F807807003F803806003F80180E003 F801C0E003F801C0C003F800C0C003F800C0C003F800C0C003F800C00003F800000003F8000000 03F800000003F800000003F800000003F800000003F800000003F800000003F800000003F80000 0003F800000003F800000003F800000003F800000003F800000003F800000003F800000003F800 000003F800000003F8000003FFFFF80003FFFFF80022227EA127>84 D<07FC001FFF803F07C03F 03E03F01E03F01F01E01F00001F00001F0003FF003FDF01FC1F03F01F07E01F0FC01F0FC01F0FC 01F0FC01F07E02F07E0CF81FF87F07E03F18167E951B>97 D<00FF8007FFE00F83F01F03F03E03 F07E03F07C01E07C0000FC0000FC0000FC0000FC0000FC0000FC00007C00007E00007E00003E00 301F00600FC0E007FF8000FE0014167E9519>99 D<0001FE000001FE0000003E0000003E000000 3E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0001FC3E0007 FFBE000F81FE001F007E003E003E007E003E007C003E00FC003E00FC003E00FC003E00FC003E00 FC003E00FC003E00FC003E00FC003E007C003E007C003E003E007E001E00FE000F83BE0007FF3F C001FC3FC01A237EA21F>I<00FE0007FF800F87C01E01E03E01F07C00F07C00F8FC00F8FC00F8 FFFFF8FFFFF8FC0000FC0000FC00007C00007C00007E00003E00181F00300FC07003FFC000FF00 15167E951A>I<003F8000FFC001E3E003C7E007C7E00F87E00F83C00F80000F80000F80000F80 000F80000F8000FFFC00FFFC000F80000F80000F80000F80000F80000F80000F80000F80000F80 000F80000F80000F80000F80000F80000F80000F80000F80000F80007FF8007FF80013237FA211 >I<03FC1E0FFF7F1F0F8F3E07CF3C03C07C03E07C03E07C03E07C03E07C03E03C03C03E07C01F 0F801FFF0013FC003000003000003800003FFF801FFFF00FFFF81FFFFC3800FC70003EF0001EF0 001EF0001EF0001E78003C7C007C3F01F80FFFE001FF0018217E951C>II<1C003E007F007F007F003E001C0000000000 00000000000000000000FF00FF001F001F001F001F001F001F001F001F001F001F001F001F001F 001F001F001F001F001F00FFE0FFE00B247EA310>I<0038007C00FE00FE00FE007C0038000000 000000000000000000000003FE03FE003E003E003E003E003E003E003E003E003E003E003E003E 003E003E003E003E003E003E003E003E003E003E003E783EFC3EFC3CFC7C78F87FE01F800F2E83 A311>IIIII<00FE0007FFC0 0F83E01E00F03E00F87C007C7C007C7C007CFC007EFC007EFC007EFC007EFC007EFC007EFC007E 7C007C7C007C3E00F81F01F00F83E007FFC000FE0017167E951C>II114 D<0FF3003FFF00781F00600700E00300E00300F00300FC00007FE0007FF8003FFE000FFF 0001FF00000F80C00780C00380E00380E00380F00700FC0E00EFFC00C7F00011167E9516>I<01 80000180000180000180000380000380000780000780000F80003F8000FFFF00FFFF000F80000F 80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F81800F81800F81800F 81800F81800F830007C30003FE0000F80011207F9F16>II119 DII E /Fl 44 123 df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m 15 62 df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n 43 111 df0 D<800002C0000660000C3000181800300C 00600600C003018001830000C600006C00003800003800006C0000C6000183000301800600C00C 006018003030001860000CC000068000021718789727>2 D<000FF00000708E00018081800200 804004008020080080101000800810008008200080042000800440008002400080024000800280 0080018000800180008001FFFFFFFF800080018000800180008001800080014000800240008002 40008002200080042000800410008008100080080800801004008020020080400180818000708E 00000FF00020227D9C27>8 D<03F0000FFC001FFE003FFF007FFF807FFF80FFFFC0FFFFC0FFFF C0FFFFC0FFFFC0FFFFC0FFFFC0FFFFC07FFF807FFF803FFF001FFE000FFC0003F00012147D9519 >15 D17 D<000FFFFC007FFFFC01F00000038000000600 00000C0000001800000030000000300000006000000060000000C0000000C0000000C0000000C0 000000C0000000C0000000C000000060000000600000003000000030000000180000000C000000 060000000380000001F00000007FFFFC000FFFFC00000000000000000000000000000000000000 000000000000000000000000007FFFFFFC7FFFFFFC1E277C9F27>I<0000000C0000003C000000 F0000003C000000F0000003C000000F0000007C000001F00000078000001E00000078000001E00 000078000000E0000000780000001E0000000780000001E0000000780000001F00000007C00000 00F00000003C0000000F00000003C0000000F00000003C0000000C000000000000000000000000 00000000000000000000000000000000000000007FFFFFF8FFFFFFFC1E277C9F27>20 DI<000FFFFC007FFFFC01F0000003800000060000000C00000018000000300000 00300000006000000060000000C0000000C0000000C0000000C0000000C0000000C0000000C000 0000C000000060000000600000003000000030000000180000000C000000060000000380000001 E00000007FFFFC001FFFFC1E1E7C9A27>26 D<80000000C0000000C0000000C000000060000000 600000003000000030000000180000000C0000000700000003C0000000FC0000003FE0000003FF FC0003FFFC003FE00000FC000003C00000070000000C0000001800000030000000600000006000 000060000000C0000000C0000000C00000001E1D7C9A27>31 D<01800000000003000000000003 00000000000600000000000600000000000C0000000000180000000000700000000000FFFFFFFF FFC0FFFFFFFFFFC07000000000001800000000000C000000000006000000000006000000000003 00000000000300000000000180000000002A127C9432>I<000000006000000000003000000000 003000000000001800000000001800000000000C00000000000600000000000380FFFFFFFFFFE0 FFFFFFFFFFC0000000000380000000000600000000000C00000000001800000000001800000000 0030000000000030000000000060002B127D9432>I<00C00000C00000C00000C00000C00000C0 0000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C0 0000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C0 0000C00000C00000C000C0C0C0F0C3C038C7001CCE0006D80003F00001E00001E00000C0000040 00122D7DA219>35 D<01800000C000030000006000030000006000060000003000060000003000 0C0000001800180000000C00700000000700FFFFFFFFFFC0FFFFFFFFFF80700000000700180000 000C000C000000180006000000300006000000300003000000600003000000600001800000C000 2A127C9432>I<0000C00000000000C00000000001800000000003000000000003000000000006 00000000000C00000000001800000000003FFFFFFFE0007FFFFFFFE001C0000000000700000000 003E0000000000F800000000003C00000000000F000000000003800000000000C000000000007F FFFFFFE0003FFFFFFFE0001C00000000000C000000000006000000000003000000000001800000 000001800000000000C00000000000C00000002B1C7D9932>40 D<000000600000000000600000 0000003000000000001800000000001800000000000C0000000000060000000000030000FFFFFF FF8000FFFFFFFFC000000000007000000000001C00000000000F800000000003E0000000000780 000000001E00000000003800000000006000FFFFFFFFC000FFFFFFFF8000000000070000000000 0600000000000C0000000000180000000000300000000000300000000000600000000000600000 2B1C7D9932>I<0080200001803000018030000180300001803000018030000180300001803000 018030000180300001803000018030000180300001803000018030000180300001803000018030 00018030000180300001803000018030000180300001803000018030000180300001803000C180 3060E18030E0398033801D80370007803C00038038000180300000C060000060C0000060C00000 31800000110000001B0000000A0000000E0000000E000000040000000400001B2D7FA21E>43 D<000600030000000600030000000C00018000000C0001800000180000C00000180000C0000030 0000600000600000300000FFFFFFF80001FFFFFFFC000700000007000E00000003803800000000 E0F000000000783800000000E01C00000001C0070000000700038000000E0001FFFFFFFC0000FF FFFFF80000600000300000300000600000180000C00000180000C000000C000180000006000300 000006000300000006000300002D1C7E9932>I<03E0000F0FFC0038181E00E0300F0180200383 004003C6004001E4008000EC008000F800800078008000380080003C0080003E0080006E004000 4F004000C78060018380200301E0100E00F00E38007F03E0000F20157D9427>47 D<03F80001F80007FE000FFE001E3F801C0300380FC03001802003E06000804001F0C000404000 F9800040C0007F00002080003F00002080003E00002080001F00002080000F80002080001F8000 2080001FC00060400033E00040400061F000402000C0F800803001807E03801807003F8F000FFE 000FFC0003F00003F8002B157D9432>49 D<001FFF007FFF01E0000380000600000C0000180000 300000300000600000600000600000C00000C00000FFFFFFFFFFFFC00000C00000600000600000 6000003000003000001800000C000006000003800001E000007FFF001FFF181E7C9A21>I<0000 0300000300000600000600000C00000C0000180000180000300000300000600000600000C00000 C00000C0000180000180000300000300000600000600000C00000C000018000018000030000030 0000600000600000C00000C0000180000180000300000300000300000600000600000C00000C00 00180000180000300000300000600000600000C00000400000183079A300>54 DI<00020000060001E600061C000C 0C00180E00300F00301B00301B00701B80601980603180603180E031C0E031C0E061C0E061C0E0 61C0E061C0E0C1C0E0C1C0E0C1C0E0C1C0E181C0E181C0E181C0E181C0E301C0E301C063018063 01807603807603803603003E07001C06000C0C000E18000FF000180000180000180000122A7DA5 19>59 D<00000006000000000E000000001E000000003E000000003E000000007E000000005E00 000000DE000000009E000000019E000000011E000000031E000000021E000000061E0000000C1E 0000000C1E000000181E000000181E000000301E000000601E000000601E000000C01E000001C0 1E000001801E000003001E0000077FFF000007FFFF00000FFFFF00001C000F000038000F004070 000F0040F0000F80E1E0000F80FFC0000780FF800007F8FF000007F07E000003C03C0000000025 267EA328>65 D<00003F000001FF800007FFC0000E07C0003003C00060038000C0038001800780 03000700070006000E000E000E0018001C0010001C0000003C0000003800000038000000780000 007000000070000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F800 0300F80006007C000C007C0018003E0030003F80C0001FFF80000FFE000003F800001A2480A21A >67 D<000FFFE000007FFFFC0001FFFFFF00038780FF800C07801FC018078007E038070003F070 070001F0600F0000F8C00F0000F8000F000078000F000078000E000078000E000078001E000078 001E000070001E000070001C0000F0001C0000E0003C0000E000380001C000380001C000780003 800070000300007000060000F0000C0000E000180000E000200001E000C00001C007000001C07E 000003FFF800000FFFC000001FFC00000025227FA126>I<00001FFFFE0000FFFFFF0003FFFFFF 000F1E000C00181E000800701E000000E01C000000E01C000001C03C0000010038000000003800 0000007800000000780000000070000000007000000000F000000000FFFF000000FFFE000001FF FC000001C000000003C000000003800000000380000000070000000007000000000F000000000E 000000000E000000001C000000201C0000007038000000F838000000FC700000007FC00000003F 800000001E0000000028247FA124>70 D<00001800000010000078000000300000780000006000 00F8000000600000F8000000E00000F8000001E00000FC000003E00000FC000007C00000BC0000 0FC000013C00000FC000013C00001FC000013C00003BC000013E000073C000023E0000E3C00002 1E0001E38000021E0003C38000041E0003878000041F0007078000041F000E078000080F001C07 8000080F0038078000080F8078070000100F80F0070000100781E00F0000100783C00F00002007 C7800F00002007CF000F00004003FE000F00004003FC000F0000C001F8000F00008001F0000F00 618000E0000F007F0000C0000F00FF000000000FF0FE000000000FE0FE0000000007803C000000 00000034257EA23C>77 D<000FFFE000007FFFFC0001FFFFFF00038780FF800C07801FC0180780 07C038070003E070070003E0600F0001E0C00F0001E0000F0001E0000F0001C0000E0001C0000E 000380001E000380001E000700001E000600001C000C00001C001000003C006000003803800000 387F00000079FC0000007BE000000070000000007000000000F000000000E000000000E0000000 01E000000001C000000001C0000000038000000003800000000700000000060000000023247FA1 23>80 D<000FFFF800007FFFFF0001FFFFFF800387803FC00C078007E018078003E038070001E0 70070001E0600F0001E0C00F0001C0000F0001C0000F000380000E000300000E000600001E000C 00001E001800001E00E000001C1FC000001C3F0000003C7F000000381F000000380F800000780F 8000007007C000007007C00000F003E00000E003E00000E001F00001E001F00301C000F80601C0 00F80C03C0007C180380007FE00300003FC00400001F0028237FA12A>82 D<0000000018001FFFFFE000FFFFFFC001FFFFFF0006001E00000E001E00001C003C00003C003C 000038003C000078007800007000780000C0007800000000F800000000F000000000F000000000 F000000001F000000001E000000001E000000001E000000003E000000003C000000003C0000000 03C00000000780000000078000000007000000000F000000000E000000001C0000000018000000 003000000001FFE0000003FFC000000FFF00000025237FA21B>84 D<7C0000C0FF0000F03F8001 F00F8001F807C001F807C0007803E0003803E0001801E0001001E0001001F0001001F0003000F0 002000F0006000F0004000F000C000F0018000F0038000F0030000F0070000F00E0000F01C0000 F0380000F0780000F0F00000E1E00000E3C00001E7800001EF000001FE000001FC000001F80000 01F0000001E0000003800000030000001D247EA11F>86 D<40000040C00000C0C00000C0C00000 C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C000 00C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C060 0001806000018030000300180006000E001C000780780001FFE000007F80001A1F7D9D21>91 D<007F800001FFE000078078000E001C0018000600300003006000018060000180C00000C0C000 00C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C0 0000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0C00000C0 C00000C0400000401A1F7D9D21>I<40000040C00000C0C00000C0C00000C0C00000C0C00000C0 C00C00C0C00C00C0C00C00C0C00C00C0C00C00C0C00C00C0C00C00C0C00C00C0C7FFF8C0C7FFF8 C0C00C00C0C00C00C0C00C00C0C00C00C0C00C00C0C00C00C0C00C00C060000180600001803000 0300180006000E001C000780780001FFE000007F80001A1F7D9D21>I<000F0038007000E001C0 01C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C0038007001E00F0 001E000700038001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C0 01C000E000700038000F10317CA419>102 DI<004000C000C0018001800180030003000300060006000C000C000C0018001800 1800300030003000600060006000C000C000C000C0006000600060003000300030001800180018 000C000C000C000600060003000300030001800180018000C000C000400A327BA413>III<0004000000040000000E0000000E0000001B000000318000003180000060C00000C060 0001803000038038000F803E003D803780F18031E0C18030600180300001803000018030000180 300001803000018030000180300001803000018030000180300001803000018030000180300001 80300001803000018030000180300001803000018030000180300001803000C1803060F18031E0 3D8037800F803E00038038000180300000C060000060C0000031800000318000001B0000000E00 00000E000000040000000400001B337FA51E>109 DI E /Fo 36 122 df<0001E0000003E000000FE0 00007FE0001FFFE000FFFFE000FFBFE000E03FE000003FE000003FE000003FE000003FE000003F E000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE00000 3FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000 003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE0 00003FE0007FFFFFF07FFFFFF07FFFFFF01C2E7AAD29>49 D<003FF00001FFFE0007FFFF800FC0 7FE01E001FF03C000FF87F0007FC7F8007FEFFC007FEFFC003FEFFC003FFFFC003FF7F8003FF7F 8003FF3F0003FF000003FF000003FE000003FE000007FC000007FC00000FF800000FF000001FE0 00001FC000003F8000007F000000FE000001F8000001F0000003E00000078007000F0007001E00 07003C000F0078000E00F0000E01C0001E03FFFFFE07FFFFFE0FFFFFFE1FFFFFFE3FFFFFFE7FFF FFFCFFFFFFFCFFFFFFFCFFFFFFFC202E7CAD29>I<000FFC0000007FFF800001F01FE00003C00F F000070007F8000FE007FC000FF007FC001FF007FE001FF807FE001FF807FE001FF807FE001FF8 07FE000FF007FC0007E007FC00018007FC0000000FF80000000FF00000001FE00000001FC00000 007F8000001FFE0000001FFC0000001FFF800000001FF000000007F800000003FC00000003FE00 000003FF00000001FF80000001FF800E0001FFC03F8001FFC07FC001FFC07FC001FFC0FFE001FF C0FFE001FFC0FFE001FF80FFE001FF80FFC003FF007F8003FF003F0003FE001F0007FC000FE01F F80007FFFFE00001FFFF8000001FFC0000222E7DAD29>I<0000007800000000F800000001F800 000003F800000007F800000007F80000000FF80000001FF80000003FF80000007FF800000077F8 000000F7F8000001E7F8000003C7F800000787F800000707F800000F07F800001E07F800003C07 F800007807F800007007F80000F007F80001E007F80003C007F800078007F8000F0007F8000F00 07F8001E0007F8003C0007F800780007F800F00007F800FFFFFFFFF0FFFFFFFFF0FFFFFFFFF000 000FF80000000FF80000000FF80000000FF80000000FF80000000FF80000000FF80000000FF800 00000FF800000FFFFFF0000FFFFFF0000FFFFFF0242E7EAD29>I<0C0000380FC003F80FFFFFF8 0FFFFFF00FFFFFE00FFFFFC00FFFFF800FFFFE000FFFFC000FFFF0000FFF00000F0000000F0000 000F0000000F0000000F0000000F0000000F0000000F0FF8000F7FFF000FFFFFC00FF01FE00F80 0FF00F0007F80E0007FC000003FC000003FE000003FE000003FF000003FF1E0003FF3F0003FF7F 8003FFFF8003FFFFC003FFFFC003FEFF8003FEFF8003FE7F0007FC7C0007F83C000FF01E001FE0 0FC07FC007FFFF8001FFFE00003FE000202E7CAD29>I<00007F80000007FFF000001FC0780000 7F001C0000FC001E0001F8007E0003F800FF0007F001FF000FF001FF000FE001FF001FE001FF00 3FE000FE003FE0007C003FC00000007FC00000007FC00000007FC0000000FFC3FF8000FFC7FFE0 00FFCFBFF000FFDC03F800FFF801FC00FFF001FE00FFF000FF00FFE000FF80FFE000FF80FFE000 FF80FFC000FFC0FFC000FFC0FFC000FFC07FC000FFC07FC000FFC07FC000FFC07FC000FFC03FC0 00FFC03FC000FF803FC000FF801FE000FF801FE000FF000FE001FE0007F001FC0003F803F80001 FC0FF00000FFFFE000003FFF80000007FC0000222E7DAD29>I<38000000003E000000003FFFFF FFC03FFFFFFFC03FFFFFFFC03FFFFFFF807FFFFFFF007FFFFFFE007FFFFFFC007FFFFFF8007FFF FFF000780001F000700003E000700007C000F0000F8000E0000F0000E0001E0000E0003E000000 007C00000000F800000000F800000001F000000001F000000003F000000003E000000007E00000 0007E00000000FE00000000FE00000001FC00000001FC00000001FC00000003FC00000003FC000 00003FC00000003FC00000003FC00000007FC00000007FC00000007FC00000007FC00000007FC0 0000007FC00000007FC00000007FC00000003F800000003F800000000E00000022307BAF29>I< 000003FF80018000003FFFF003800001FFFFFC0F800007FF007F1F80001FF8000FBF80003FE000 03FF8000FF800000FF8001FF0000007F8003FE0000003F8007FC0000003F8007FC0000001F800F F80000001F801FF80000000F801FF00000000F803FF000000007803FF000000007807FF0000000 07807FE000000007807FE000000000007FE00000000000FFE00000000000FFE00000000000FFE0 0000000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE000000000 00FFE000000000007FE000000000007FE000000000007FE000000000007FF000000003803FF000 000003803FF000000003801FF000000003801FF800000007800FF8000000070007FC0000000700 07FC0000000E0003FE0000001E0001FF0000003C0000FF8000007800003FE00000F000001FF800 03E0000007FF003F80000001FFFFFE000000003FFFF80000000003FF80000031317BB03C>67 D73 D77 D<00000FFE0000000000FFFFE000000007FFFFFC0000001FFC07FF0000003FE000FF800000FF80 003FE00001FF00001FF00003FE00000FF80007FC000007FC0007FC000007FC000FF8000003FE00 1FF8000003FF001FF0000001FF003FF0000001FF803FF0000001FF803FF0000001FF807FE00000 00FFC07FE0000000FFC07FE0000000FFC0FFE0000000FFE0FFE0000000FFE0FFE0000000FFE0FF E0000000FFE0FFE0000000FFE0FFE0000000FFE0FFE0000000FFE0FFE0000000FFE0FFE0000000 FFE0FFE0000000FFE0FFE0000000FFE07FE0000000FFC07FE0000000FFC07FF0000001FFC07FF0 000001FFC03FF0000001FF803FF0000001FF801FF8000003FF001FF8000003FF000FFC000007FE 000FFC000007FE0007FE00000FFC0003FF00001FF80001FF80003FF00000FFC0007FE000003FE0 00FF8000001FFC07FF00000007FFFFFC00000000FFFFE0000000000FFE00000033317BB03E>79 DI82 D<001FF0018000FFFF038003FFFFC78007F00FFF800F8001FF801F00007F803F00001F803E0000 0F807E00000F807E00000780FE00000780FE00000780FE00000380FF00000380FF00000380FF80 000000FFE00000007FFC0000007FFFE000007FFFFE00003FFFFFC0001FFFFFF0001FFFFFF8000F FFFFFC0003FFFFFE0001FFFFFF00007FFFFF80001FFFFF800000FFFFC0000007FFC0000000FFE0 0000003FE00000003FE00000001FE06000001FE0E000000FE0E000000FE0E000000FE0E000000F C0F000000FC0F000000FC0F800001F80FC00001F80FF00003F00FFC0007E00FFFC01FC00F1FFFF F800E03FFFE000C007FF000023317BB02E>I<3FFFFFFFFFFF003FFFFFFFFFFF003FFFFFFFFFFF 003FE00FFC01FF007F000FFC003F807E000FFC001F807C000FFC000F8078000FFC00078078000F FC00078070000FFC00038070000FFC00038070000FFC00038070000FFC000380E0000FFC0001C0 E0000FFC0001C0E0000FFC0001C0E0000FFC0001C000000FFC00000000000FFC00000000000FFC 00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000 000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00 000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC0000000000 0FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC0000 0000000FFC00000000000FFC00000000000FFC000000007FFFFFFF8000007FFFFFFF8000007FFF FFFF800032307DAF39>I<007FF8000003FFFF000007FFFFC0000FE01FE0001FF007F0001FF003 F8001FF003FC001FF001FE000FE001FE0007C001FE00010001FE00000001FE00000001FE000001 FFFE00003FFFFE0001FFF1FE0007FE01FE000FF001FE001FC001FE003F8001FE007F8001FE00FF 0001FE00FF0001FE00FF0001FE00FF0001FE00FF0003FE007F8003FE007FC00EFE003FF03CFF00 0FFFF87FF807FFF03FF800FF800FF825207E9F28>97 D<01F8000000FFF8000000FFF8000000FF F80000000FF800000007F800000007F800000007F800000007F800000007F800000007F8000000 07F800000007F800000007F800000007F800000007F800000007F800000007F800000007F80FF0 0007F87FFE0007F9FFFF8007FFE03FC007FF000FE007FE0007F007F80003F807F80003FC07F800 03FC07F80001FE07F80001FE07F80001FE07F80001FF07F80001FF07F80001FF07F80001FF07F8 0001FF07F80001FF07F80001FF07F80001FF07F80001FE07F80001FE07F80001FE07F80003FC07 F80003FC07FC0007F807FE0007F007F7001FE007E3E07FC007C1FFFF0007807FFE0007001FE000 28327EB12E>I<0007FF00007FFFE000FFFFF003FC03F807F007FC0FE007FC1FE007FC3FC007FC 3FC003F87FC001F07F8000407F800000FF800000FF800000FF800000FF800000FF800000FF8000 00FF800000FF8000007F8000007FC000007FC000003FC0000E3FE0000E1FE0001C0FF0001C07F8 007803FF01F000FFFFE0007FFF800007FC001F207D9F25>I<00000007E0000003FFE0000003FF E0000003FFE00000003FE00000001FE00000001FE00000001FE00000001FE00000001FE0000000 1FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE0000F F81FE0007FFF1FE001FFFFDFE003FE03FFE007F800FFE00FE0003FE01FE0001FE03FC0001FE03F C0001FE07F80001FE07F80001FE07F80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0 FF80001FE0FF80001FE0FF80001FE0FF80001FE07F80001FE07F80001FE07F80001FE03FC0001F E03FC0001FE01FC0003FE00FE0007FE007F001FFE003FC07DFF001FFFF9FFF007FFE1FFF000FF0 1FFF28327DB12E>I<0007FC0000003FFF800000FFFFE00003FC07F00007F801F8000FE000FC00 1FE0007E003FC0007E003FC0003F007FC0003F007F80003F007F80003F80FF80003F80FF80003F 80FFFFFFFF80FFFFFFFF80FFFFFFFF80FF80000000FF80000000FF800000007F800000007F8000 00003FC00000003FC00003801FC00003801FE00007800FF0000F0007F8001E0003FE00FC0000FF FFF800003FFFE0000003FF000021207E9F26>I<0000FF000007FFC0001FFFE0003FC7F0007F0F F800FE0FF801FE0FF801FC0FF803FC07F003FC03E003FC01C003FC000003FC000003FC000003FC 000003FC000003FC000003FC0000FFFFF800FFFFF800FFFFF80003FC000003FC000003FC000003 FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC0000 03FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC00 0003FC000003FC000003FC00007FFFF0007FFFF0007FFFF0001D327EB119>I<001FF007E000FF FE3FF001FFFF7FF807F83FF1F80FE00FE1F80FE00FE0F01FC007F0601FC007F0003FC007F8003F C007F8003FC007F8003FC007F8003FC007F8001FC007F0001FC007F0000FE00FE0000FE00FE000 07F83FC00007FFFF000006FFFE00000E1FF000000E000000001E000000001E000000001F000000 001F800000001FFFFFC0000FFFFFF8000FFFFFFE0007FFFFFF0003FFFFFF8007FFFFFFC01FFFFF FFE03F00007FE07E00000FF0FC000007F0FC000003F0FC000003F0FC000003F0FC000003F07E00 0007E03F00000FC01FC0003F800FF801FF0007FFFFFE0000FFFFF000001FFF8000252F7E9F29> I<01F800000000FFF800000000FFF800000000FFF8000000000FF80000000007F80000000007F8 0000000007F80000000007F80000000007F80000000007F80000000007F80000000007F8000000 0007F80000000007F80000000007F80000000007F80000000007F80000000007F807F8000007F8 3FFF000007F87FFF800007F8F03FC00007F9C01FE00007FB000FE00007FE000FF00007FE000FF0 0007FC000FF00007FC000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8 000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF0 0007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8 000FF00007F8000FF000FFFFC1FFFF80FFFFC1FFFF80FFFFC1FFFF8029327DB12E>I<01C00007 F0000FF8000FF8001FFC001FFC001FFC000FF8000FF80007F00001C00000000000000000000000 000000000000000000000000000001F800FFF800FFF800FFF8000FF80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F800FFFF80FFFF80FFFF8011337DB217>I< 01F8000000FFF8000000FFF8000000FFF80000000FF800000007F800000007F800000007F80000 0007F800000007F800000007F800000007F800000007F800000007F800000007F800000007F800 000007F800000007F800000007F8007FFC07F8007FFC07F8007FFC07F8001FC007F8001F0007F8 003E0007F800780007F801F00007F803E00007F807800007F81F000007F83E000007F87C000007 F9FE000007FBFF000007FFFF800007FF7FC00007FE3FE00007F81FE00007F01FF00007F00FF800 07F007FC0007F003FE0007F001FF0007F000FF0007F000FF8007F0007FC007F0003FE007F0003F F0FFFF80FFFFFFFF80FFFFFFFF80FFFF28327EB12C>107 D<01F800FFF800FFF800FFF8000FF8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F800FFFFC0FFFFC0FFFFC012327DB117>I<03F007F8000FF000FFF03FFF00 7FFE00FFF07FFF80FFFF00FFF0F03FC1E07F800FF1C01FE3803FC007F3000FE6001FC007F6000F FC001FE007FE000FFC001FE007FC000FF8001FE007FC000FF8001FE007F8000FF0001FE007F800 0FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8 000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007 F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE0 07F8000FF0001FE007F8000FF0001FE007F8000FF0001FE0FFFFC1FFFF83FFFFFFFFC1FFFF83FF FFFFFFC1FFFF83FFFF40207D9F45>I<03F007F80000FFF03FFF0000FFF07FFF8000FFF0F03FC0 000FF1C01FE00007F3000FE00007F6000FF00007FE000FF00007FC000FF00007FC000FF00007F8 000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF0 0007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8 000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF000FFFFC1FFFF 80FFFFC1FFFF80FFFFC1FFFF8029207D9F2E>I<0007FE0000003FFFC00000FFFFF00003FC03FC 0007F000FE000FE0007F001FC0003F803FC0003FC03FC0003FC07F80001FE07F80001FE07F8000 1FE0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80 001FF07F80001FE07F80001FE07F80001FE03FC0003FC03FC0003FC01FE0007F800FE0007F0007 F801FE0003FE07FC0001FFFFF800003FFFC0000007FE000024207E9F29>I<01F80FF000FFF87F FE00FFF9FFFF80FFFFE07FC00FFF001FE007FE000FF007F80007F807F80007FC07F80003FC07F8 0003FE07F80003FE07F80001FE07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07 F80001FF07F80001FF07F80001FF07F80001FE07F80003FE07F80003FE07F80003FC07F80007FC 07FC0007F807FE000FF007FF001FE007FBE07FC007F9FFFF0007F87FFE0007F81FE00007F80000 0007F800000007F800000007F800000007F800000007F800000007F800000007F800000007F800 000007F800000007F8000000FFFFC00000FFFFC00000FFFFC00000282E7E9F2E>I<03F03F00FF F07FC0FFF1FFE0FFF3C7F00FF38FF807F70FF807F60FF807FE0FF807FC07F007FC03E007FC0080 07F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F800 0007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F80000FFFFE000FFFF E000FFFFE0001D207E9F22>114 D<00FF870007FFEF001FFFFF003F007F003C001F0078000F00 F8000700F8000700F8000700FC000700FF000000FFF800007FFFC0003FFFF0003FFFFC000FFFFE 0007FFFF0001FFFF80001FFF800000FFC000001FC060000FC0E00007C0E00007C0F00007C0F800 0780F8000F80FE000F00FF803E00FFFFFC00F3FFF800C07FC0001A207D9F21>I<003800003800 00380000380000380000780000780000780000F80000F80001F80003F80007F8001FF800FFFFFE FFFFFEFFFFFE07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80707F80707F80707F80707F80707F80707F80703F80E 03FC0E01FE1C00FFF8007FF0000FE0182E7EAD20>I<01F80003F000FFF801FFF000FFF801FFF0 00FFF801FFF0000FF8001FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8 000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF0 0007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8 000FF00007F8000FF00007F8001FF00007F8001FF00003F8003FF00003F8006FF00001FE03CFF8 0000FFFF8FFF80007FFF0FFF80000FFC0FFF8029207D9F2E>I119 D121 D E /Fp 13 122 df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q 7 122 df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r 13 94 df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s 2 68 df<0000000400000000020000000002000000000100000000008000000000400000000020 FFFFFFFFFCFFFFFFFFFC0000000020000000004000000000800000000100000000020000000002 00000000040026107D922D>33 D<0000FC0007FE001C3E00201E00C01E01801C03801C07003C06 00380E00381C00601C00003C0000380000380000780000780000700000700000F00000F00000F0 0000F00000F00000F00000F80018F800307800707C00C07E00803F83001FFC0007F0001721809F 18>67 D E /Ft 50 122 df<001F83E000F06E3001C078780380F8780300F03007007000070070 000700700007007000070070000700700007007000FFFFFF800700700007007000070070000700 700007007000070070000700700007007000070070000700700007007000070070000700700007 007000070070000700700007007000070070007FE3FF001D20809F1B>11 D<001F81F80000F04F040001C07C06000380F80F000300F00F000700F00F000700700000070070 00000700700000070070000007007000000700700000FFFFFFFF00070070070007007007000700 700700070070070007007007000700700700070070070007007007000700700700070070070007 007007000700700700070070070007007007000700700700070070070007007007000700700700 7FE3FE3FF02420809F26>14 D<7038F87CFC7EFC7E743A04020402040208040804100810082010 40200F0E7E9F17>34 D<0020004000800100020006000C000C0018001800300030003000700060 0060006000E000E000E000E000E000E000E000E000E000E000E000E00060006000600070003000 30003000180018000C000C000600020001000080004000200B2E7DA112>40 D<800040002000100008000C00060006000300030001800180018001C000C000C000C000E000E0 00E000E000E000E000E000E000E000E000E000E000C000C000C001C00180018001800300030006 0006000C00080010002000400080000B2E7DA112>I<70F8FCFC74040404080810102040060E7C 840D>44 DI<70F8F8F87005057C840D>I<03F0000E1C001C0E00180600 380700700380700380700380700380F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0 F003C0F003C0F003C0F003C0F003C07003807003807003807807803807001806001C0E000E1C00 03F000121F7E9D17>48 D<03F0000C1C00100E00200700400780800780F007C0F803C0F803C0F8 03C02007C00007C0000780000780000F00000E00001C0000380000700000600000C00001800003 00000600400C00401800401000803FFF807FFF80FFFF80121E7E9D17>50 D<4000007FFFC07FFF807FFF804001008002008002008004000008000008000010000020000020 0000400000400000C00000C00001C0000180000380000380000380000380000780000780000780 00078000078000078000078000030000121F7D9D17>55 D<03F0000C0C00100600300300200180 6001806001806001807001807803003E03003F06001FC8000FF00003F80007FC000C7E00103F00 300F806003804001C0C001C0C000C0C000C0C000C0C000806001802001001002000C0C0003F000 121F7E9D17>I<03F0000E18001C0C00380600380700700700700380F00380F00380F003C0F003 C0F003C0F003C0F003C07007C07007C03807C0180BC00E13C003E3C00003800003800003800007 00300700780600780E00700C002018001070000FC000121F7E9D17>I<00010000000380000003 8000000380000007C0000007C0000007C0000009E0000009E0000009E0000010F0000010F00000 10F00000207800002078000020780000403C0000403C0000403C0000801E0000801E0000FFFE00 01000F0001000F0001000F00020007800200078002000780040003C00E0003C01F0007E0FFC03F FE1F207F9F22>65 D<000FE0200078186000E004E0038002E0070001E00F0000E01E0000601E00 00603C0000603C0000207C00002078000020F8000000F8000000F8000000F8000000F8000000F8 000000F8000000F8007FFCF80003E0780001E07C0001E03C0001E03C0001E01E0001E01E0001E0 0F0001E0070001E0038002E000E0046000781820000FE0001E217D9F24>71 D73 D75 D77 D<001F800000F0F00001C0380007801E00 0F000F000E0007001E0007803C0003C03C0003C07C0003E0780001E0780001E0F80001F0F80001 F0F80001F0F80001F0F80001F0F80001F0F80001F0F80001F0F80001F0780001E07C0003E07C00 03E03C0003C03C0003C01E0007800E0007000F000F0007801E0001C0380000F0F000001F80001C 217D9F23>79 D82 D<7FFFFFE0780F01E0600F006040 0F0020400F0020C00F0030800F0010800F0010800F0010800F0010000F0000000F0000000F0000 000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000000F0000000F0000000F0000000F0000000F0000001F800007FFFE001C1F7E9E21> 84 D87 D91 D<0804100820102010402040208040804080 40B85CFC7EFC7E7C3E381C0F0E7B9F17>II<1FE0003030007818 00781C00300E00000E00000E00000E0000FE00078E001E0E00380E00780E00F00E10F00E10F00E 10F01E10781E103867200F83C014147E9317>97 D<0E0000FE00000E00000E00000E00000E0000 0E00000E00000E00000E00000E00000E00000E3E000EC3800F01C00F00E00E00E00E00700E0070 0E00780E00780E00780E00780E00780E00780E00700E00700E00E00F00E00D01C00CC300083E00 15207F9F19>I<03F80E0C1C1E381E380C70007000F000F000F000F000F000F000700070003801 38011C020E0C03F010147E9314>I<000380003F80000380000380000380000380000380000380 00038000038000038000038003E380061B801C0780380380380380700380700380F00380F00380 F00380F00380F00380F003807003807003803803803807801C07800E1B8003E3F815207E9F19> I<03F0000E1C001C0E00380700380700700700700380F00380F00380FFFF80F00000F00000F000 007000007000003800801800800C010007060001F80011147F9314>I<007C00C6018F038F0706 0700070007000700070007000700FFF00700070007000700070007000700070007000700070007 000700070007000700070007007FF01020809F0E>I<0000E003E3300E3C301C1C30380E00780F 00780F00780F00780F00780F00380E001C1C001E380033E0002000002000003000003000003FFE 001FFF800FFFC03001E0600070C00030C00030C00030C000306000603000C01C038003FC00141F 7F9417>I<0E0000FE00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E 00000E3E000E43000E81800F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E 01C00E01C00E01C00E01C00E01C00E01C00E01C0FFE7FC16207F9F19>I<1C003E003E003E001C 000000000000000000000000000E007E000E000E000E000E000E000E000E000E000E000E000E00 0E000E000E000E000E000E00FFC00A1F809E0C>I<00E001F001F001F000E00000000000000000 00000000007007F000F00070007000700070007000700070007000700070007000700070007000 700070007000700070007000706070F060F0C061803F000C28829E0E>I<0E0000FE00000E0000 0E00000E00000E00000E00000E00000E00000E00000E00000E00000E0FF00E03C00E03000E0200 0E04000E08000E10000E30000E70000EF8000F38000E1C000E1E000E0E000E07000E07800E0380 0E03C00E03E0FFCFF815207F9F18>I<0E00FE000E000E000E000E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00 FFE00B20809F0C>I<0E1F01F000FE618618000E81C81C000F00F00E000F00F00E000E00E00E00 0E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E 000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E00FFE7FE7FE023147F9326>I<0E 3E00FE43000E81800F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E 01C00E01C00E01C00E01C00E01C00E01C0FFE7FC16147F9319>I<01F800070E001C03803801C0 3801C07000E07000E0F000F0F000F0F000F0F000F0F000F0F000F07000E07000E03801C03801C0 1C0380070E0001F80014147F9317>I<0E3E00FEC3800F01C00F00E00E00E00E00F00E00700E00 780E00780E00780E00780E00780E00780E00700E00F00E00E00F01E00F01C00EC3000E3E000E00 000E00000E00000E00000E00000E00000E00000E0000FFE000151D7F9319>I<03E0800619801C 05803C0780380380780380700380F00380F00380F00380F00380F00380F0038070038078038038 03803807801C0B800E138003E38000038000038000038000038000038000038000038000038000 3FF8151D7E9318>I<0E78FE8C0F1E0F1E0F0C0E000E000E000E000E000E000E000E000E000E00 0E000E000E000E00FFE00F147F9312>I<1F9030704030C010C010C010E00078007F803FE00FF0 0070803880188018C018C018E030D0608F800D147E9312>I<020002000200060006000E000E00 3E00FFF80E000E000E000E000E000E000E000E000E000E000E000E080E080E080E080E08061003 1001E00D1C7F9B12>I<0E01C0FE1FC00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E01C00E01C00E01C00E01C00E01C00E01C00E03C00603C0030DC001F1FC16147F9319>III<7FC3FC 0F01E00701C007018003810001C20000E40000EC00007800003800003C00007C00004E00008700 0107000303800201C00601E01E01E0FF07FE1714809318>II E /Fu 7 117 df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v 86 128 df 0 D<03FFFC00001F8000000F0000000F0000000F0000000F0000003FC00001EF7800078F1E000E 0F07001E0F07803C0F03C07C0F03E0780F01E0F80F01F0F80F01F0F80F01F0F80F01F0F80F01F0 F80F01F0780F01E07C0F03E03C0F03C01E0F07800E0F0700078F1E0001EF7800003FC000000F00 00000F0000000F0000000F0000001F800003FFFC001C227DA123>8 D<001F83E000706E3000C0 7C780180F8780380F0780700700007007000070070000700700007007000070070000700700007 00700007007000FFFFFFC007007000070070000700700007007000070070000700700007007000 070070000700700007007000070070000700700007007000070070000700700007007000070070 0007007000070078007FE3FF801D2380A21C>11 D<001FC0000070200000C01000018038000380 7800070078000700300007000000070000000700000007000000070000000700000007000000FF FFF800070078000700380007003800070038000700380007003800070038000700380007003800 070038000700380007003800070038000700380007003800070038000700380007003800070038 007FE1FF80192380A21B>I<001FD8000070380000C07800018078000380780007003800070038 0007003800070038000700380007003800070038000700380007003800FFFFF800070038000700 380007003800070038000700380007003800070038000700380007003800070038000700380007 003800070038000700380007003800070038000700380007003800070038007FF3FF80192380A2 1B>I<000FC07F00007031C08000E00B004001801E00E003803E01E007003C01E007001C00C007 001C000007001C000007001C000007001C000007001C000007001C000007001C0000FFFFFFFFE0 07001C01E007001C00E007001C00E007001C00E007001C00E007001C00E007001C00E007001C00 E007001C00E007001C00E007001C00E007001C00E007001C00E007001C00E007001C00E007001C 00E007001C00E007001C00E007001C00E07FF1FFCFFE272380A229>I22 D<7038F87CFC7EFC7E743A04020402040204020804080410081008201040200F0F7EA218>34 D<70F8FCFC7404040404080810102040060F7CA20E>39 D<00200040008001000300060004000C 000C00180018003000300030007000600060006000E000E000E000E000E000E000E000E000E000 E000E000E000E000E0006000600060007000300030003000180018000C000C0004000600030001 000080004000200B327CA413>I<800040002000100018000C0004000600060003000300018001 80018001C000C000C000C000E000E000E000E000E000E000E000E000E000E000E000E000E000E0 00C000C000C001C0018001800180030003000600060004000C00180010002000400080000B327D A413>I<0001800000018000000180000001800000018000000180000001800000018000000180 0000018000000180000001800000018000000180000001800000018000FFFFFFFEFFFFFFFE0001 800000018000000180000001800000018000000180000001800000018000000180000001800000 01800000018000000180000001800000018000000180001F227D9C26>43 D<70F8FCFC7404040404080810102040060F7C840E>II<70F8F8F87005 057C840E>I<01F000071C000C06001803003803803803807001C07001C07001C07001C0F001E0 F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0 7001C07001C07001C07803C03803803803801C07000C0600071C0001F00013227EA018>48 D<008003800F80F380038003800380038003800380038003800380038003800380038003800380 03800380038003800380038003800380038003800380038007C0FFFE0F217CA018>I<03F0000C 1C001007002007804003C04003C08003E0F003E0F801E0F801E0F801E02003E00003E00003C000 03C0000780000700000E00001C0000180000300000600000C00001800001000002002004002008 00201800603000403FFFC07FFFC0FFFFC013217EA018>I<03F8000C1E001007002007804007C0 7807C07803C07807C03807C0000780000780000700000F00000E0000380003F000001C00000F00 0007800007800003C00003C00003E02003E07003E0F803E0F803E0F003C04003C0400780200780 100F000C1C0003F00013227EA018>I<000200000600000E00000E00001E00001E00002E00004E 00004E00008E00008E00010E00020E00020E00040E00040E00080E00100E00100E00200E00200E 00400E00800E00FFFFF8000E00000E00000E00000E00000E00000E00000E00001F0001FFF01521 7FA018>I<1000801E07001FFF001FFE001FF80013E00010000010000010000010000010000010 000010F800130E001407001803801003800001C00001C00001E00001E00001E00001E07001E0F0 01E0F001E0E001C08001C04003C04003802007001006000C1C0003F00013227EA018>I<007E00 01C1000300800601C00E03C01C03C0180180380000380000780000700000700000F0F800F30C00 F40600F40300F80380F801C0F001C0F001E0F001E0F001E0F001E0F001E07001E07001E07001E0 3801C03801C01803801C03000C0600070C0001F00013227EA018>I<4000006000007FFFE07FFF C07FFFC0400080C001008001008002008002000004000008000008000010000030000020000060 0000600000600000E00000C00000C00001C00001C00001C00001C00003C00003C00003C00003C0 0003C00003C00003C00003C00001800013237DA118>I<01F800060E0008030010018020018020 00C06000C06000C06000C07000C07801803E01003F02001FC4000FF80003F80003FC00067F0008 3F80100F803007C06001C06000E0C000E0C00060C00060C00060C000606000406000C030008018 03000E0E0003F00013227EA018>I<01F000060C000C0600180700380380700380700380F001C0 F001C0F001C0F001E0F001E0F001E0F001E0F001E07001E07003E03803E01805E00C05E00619E0 03E1E00001C00001C00001C0000380000380300300780700780600700C002018001030000FC000 13227EA018>I<70F8F8F870000000000000000000000070F8F8F87005157C940E>I<70F8F8F870 000000000000000000000070F8F8F87808080808101010204040051F7C940E>I61 D<07E01838201C400E800FF00FF00FF00F000F000E001C003800300060 00C000C000800080018001000100010001000100010000000000000000000000038007C007C007 C0038010237DA217>63 D<000FE00000701C000080020003000180040000400800002008000020 1007C01020183008203008084060040440C0078441C0038481C003828380038283800382838003 828380038283800382838003828380038281C0038241C0038240C007824060078420300B842018 31881007C0F00800000008000000040000000300000E00800078007007C0000FFC001F237DA226 >I<0001800000018000000180000003C0000003C0000003C0000005E0000005E000000DF00000 08F0000008F0000010F800001078000010780000203C0000203C0000203C0000401E0000401E00 00401E0000800F0000800F0000FFFF000100078001000780030007C0020003C0020003C0040003 E0040001E0040001E00C0000F00C0000F03E0001F8FF800FFF20237EA225>II<0007E0100038183000E0063001C00170038000F0 070000F00E0000701E0000701C0000303C0000303C0000307C0000107800001078000010F80000 00F8000000F8000000F8000000F8000000F8000000F8000000F800000078000000780000107C00 00103C0000103C0000101C0000201E0000200E000040070000400380008001C0010000E0020000 381C000007E0001C247DA223>I III<0007F008003C0C1800E0021801C001B8038000F8070000780F 0000381E0000381E0000183C0000183C0000187C0000087800000878000008F8000000F8000000 F8000000F8000000F8000000F8000000F8000000F8001FFF780000F8780000787C0000783C0000 783C0000781E0000781E0000780F00007807000078038000B801C000B800E00318003C0C080007 F00020247DA226>III<03FFF0001F0000 0F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F0000 0F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00700F00F80F00F8 0F00F80E00F01E00401C0020380018700007C00014237EA119>IIIII<000FE00000783C0000E00E0003C00780078003C00F0001E00E0000E01E0000F03C0000 783C0000787C00007C7C00007C7800003C7800003CF800003EF800003EF800003EF800003EF800 003EF800003EF800003EF800003EF800003E7800003C7C00007C7C00007C3C0000783E0000F81E 0000F00F0001E00F0001E0078003C003C0078000E00E0000783C00000FE0001F247DA226>II82 D<03F0200C0C601802603001E07000E0600060E00060E00060E000 20E00020E00020F00000F000007800007F00003FF0001FFE000FFF0003FF80003FC00007E00001 E00000F00000F0000070800070800070800070800070C00060C00060E000C0F000C0C80180C607 0081FC0014247DA21B>I<7FFFFFF87807807860078018400780084007800840078008C007800C 800780048007800480078004800780040007800000078000000780000007800000078000000780 000007800000078000000780000007800000078000000780000007800000078000000780000007 80000007800000078000000780000007800000078000000FC00003FFFF001E227EA123>IIII89 D<7FFFFE7E003E78003C7000786000784000F0C0 00F0C001E08003C08003C0800780000780000F00001F00001E00003C00003C0000780000780000 F00001F00001E00103C00103C0010780010780030F00031E00021E00023C00063C000E78001EF8 007EFFFFFE18227DA11E>II<08041008201020104020 40208040804080408040B85CFC7EFC7E7C3E381C0F0F7AA218>II<04000E001F001B00318060C0C06080200B087AA218>I<0FE0001838003C0C003C0E00 18070000070000070000070000FF0007C7001E07003C0700780700700700F00708F00708F00708 F00F087817083C23900FC1E015157E9418>97 D<0E0000FE00001E00000E00000E00000E00000E 00000E00000E00000E00000E00000E00000E00000E00000E1F000E61C00E80600F00300E00380E 003C0E001C0E001E0E001E0E001E0E001E0E001E0E001E0E001E0E001C0E003C0E00380F00700C 80600C41C0083F0017237FA21B>I<01FE000703000C07801C0780380300780000700000F00000 F00000F00000F00000F00000F00000F000007000007800403800401C00800C010007060001F800 12157E9416>I<0000E0000FE00001E00000E00000E00000E00000E00000E00000E00000E00000 E00000E00000E00000E001F8E00704E00C02E01C01E03800E07800E07000E0F000E0F000E0F000 E0F000E0F000E0F000E0F000E07000E07800E03800E01801E00C02E0070CF001F0FE17237EA21B >I<01FC000707000C03801C01C03801C07801E07000E0F000E0FFFFE0F00000F00000F00000F0 0000F000007000007800203800201C00400E008007030000FC0013157F9416>I<003C00C6018F 038F030F070007000700070007000700070007000700FFF8070007000700070007000700070007 00070007000700070007000700070007000700070007807FF8102380A20F>I<00007001F19807 1E180E0E181C07001C07003C07803C07803C07803C07801C07001C07000E0E000F1C0019F00010 00001000001800001800001FFE000FFFC00FFFE03800F0600030400018C00018C00018C0001860 00306000303800E00E038003FE0015217F9518>I<0E0000FE00001E00000E00000E00000E0000 0E00000E00000E00000E00000E00000E00000E00000E00000E1F800E60C00E80E00F00700F0070 0E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E0070 0E00700E0070FFE7FF18237FA21B>I<1C003E003E003E001C0000000000000000000000000000 0000000E00FE001E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00 0E000E00FFC00A227FA10E>I<01C003E003E003E001C000000000000000000000000000000000 01E00FE001E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000 E000E000E000E000E000E000E060E0F0C0F18061803E000B2C82A10F>I<0E0000FE00001E0000 0E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E03FC0E01F0 0E01C00E01800E02000E04000E08000E10000E38000EF8000F1C000E1E000E0E000E07000E0780 0E03C00E01C00E01E00E00F00E00F8FFE3FE17237FA21A>I<0E00FE001E000E000E000E000E00 0E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E 000E000E000E000E000E000E000E00FFE00B237FA20E>I<0E1FC07F00FE60E183801E807201C0 0F003C00E00F003C00E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800 E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800E00E0038 00E00E003800E0FFE3FF8FFE27157F942A>I<0E1F80FE60C01E80E00F00700F00700E00700E00 700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00 70FFE7FF18157F941B>I<01FC000707000C01801800C03800E0700070700070F00078F00078F0 0078F00078F00078F00078F000787000707800F03800E01C01C00E038007070001FC0015157F94 18>I<0E1F00FE61C00E80600F00700E00380E003C0E001C0E001E0E001E0E001E0E001E0E001E 0E001E0E001E0E003C0E003C0E00380F00700E80E00E41C00E3F000E00000E00000E00000E0000 0E00000E00000E00000E00000E0000FFE000171F7F941B>I<01F8200704600E02601C01603801 E07800E07800E0F000E0F000E0F000E0F000E0F000E0F000E0F000E07000E07800E03801E01C01 E00C02E0070CE001F0E00000E00000E00000E00000E00000E00000E00000E00000E00000E0000F FE171F7E941A>I<0E3CFE461E8F0F0F0F060F000E000E000E000E000E000E000E000E000E000E 000E000E000E000F00FFF010157F9413>I<0F8830786018C018C008C008E008F0007F803FE00F F001F8003C801C800C800CC00CC008E018D0308FC00E157E9413>I<0200020002000200060006 0006000E001E003E00FFF80E000E000E000E000E000E000E000E000E000E000E000E040E040E04 0E040E040E040708030801F00E1F7F9E13>I<0E0070FE07F01E00F00E00700E00700E00700E00 700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00F00E00F00601700382 7800FC7F18157F941B>IIIII<3FFFC0380380300780 200700600E00401C00403C0040380000700000E00001E00001C0000380400700400F00400E00C0 1C0080380080780180700780FFFF8012157F9416>II<7070F8F8F8F8F8F8 70700D057BA118>127 D E /Fw 22 122 df77 D<0000007FFC000000000007FFFFC0000000003FFFFFF800000000 FFE00FFE00000003FF0001FF8000000FFC00007FE000001FF800003FF000003FF000001FF80000 7FE000000FFC0000FFC0000007FE0001FF80000003FF0003FF80000003FF8007FF00000001FFC0 07FF00000001FFC00FFE00000000FFE01FFE00000000FFF01FFC000000007FF01FFC000000007F F03FFC000000007FF83FFC000000007FF87FFC000000007FFC7FF8000000003FFC7FF800000000 3FFC7FF8000000003FFC7FF8000000003FFCFFF8000000003FFEFFF8000000003FFEFFF8000000 003FFEFFF8000000003FFEFFF8000000003FFEFFF8000000003FFEFFF8000000003FFEFFF80000 00003FFEFFF8000000003FFEFFF8000000003FFEFFF8000000003FFEFFF8000000003FFE7FF800 0000003FFC7FFC000000007FFC7FFC000000007FFC7FFC000000007FFC3FFC000000007FF83FFC 000000007FF83FFE00000000FFF81FFE00000000FFF01FFE00000000FFF00FFF00000001FFE00F FF00000001FFE007FF80000003FFC003FF80000003FF8003FFC0000007FF8001FFE000000FFF00 00FFF000001FFE00007FF800003FFC00001FFE0000FFF000000FFF0001FFE0000003FFE00FFF80 000001FFFFFFFF000000003FFFFFF80000000007FFFFC000000000007FFC0000003F3D7BBB4A> 79 D82 D<0003FF800180001FFFF00380007FFFFC078001FFFFFF0F8003FE00FF9F8007F0000F FF800FE00003FF801FC00001FF803F8000007F803F8000007F807F0000003F807F0000001F807F 0000001F80FF0000000F80FF0000000F80FF0000000F80FF8000000780FF8000000780FFC00000 0780FFE000000780FFF8000000007FFE000000007FFFF00000007FFFFF0000003FFFFFF800003F FFFFFF00001FFFFFFFC0000FFFFFFFF00007FFFFFFF80003FFFFFFFC0001FFFFFFFE00007FFFFF FF00003FFFFFFF800007FFFFFF8000007FFFFFC0000007FFFFC00000003FFFE000000003FFE000 000000FFF0000000007FF0000000003FF0700000001FF0F00000001FF0F00000001FF0F0000000 0FF0F00000000FF0F80000000FF0F80000000FE0F80000000FE0FC0000000FE0FC0000001FC0FE 0000001FC0FF0000001F80FFC000003F80FFF000007F00FFFC0001FE00FCFFC007FC00F87FFFFF F800F01FFFFFE000E003FFFF8000C0003FFC00002C3D7BBB37>I<3FFFFFFFFFFFFFC03FFFFFFF FFFFFFC03FFFFFFFFFFFFFC03FFFFFFFFFFFFFC03FF800FFF001FFC07FC000FFF0003FE07F8000 FFF0001FE07F0000FFF0000FE07E0000FFF00007E07C0000FFF00003E07C0000FFF00003E07C00 00FFF00003E0780000FFF00001E0780000FFF00001E0780000FFF00001E0780000FFF00001E0F0 0000FFF00000F0F00000FFF00000F0F00000FFF00000F0F00000FFF00000F0F00000FFF00000F0 000000FFF0000000000000FFF0000000000000FFF0000000000000FFF0000000000000FFF00000 00000000FFF0000000000000FFF0000000000000FFF0000000000000FFF0000000000000FFF000 0000000000FFF0000000000000FFF0000000000000FFF0000000000000FFF0000000000000FFF0 000000000000FFF0000000000000FFF0000000000000FFF0000000000000FFF0000000000000FF F0000000000000FFF0000000000000FFF0000000000000FFF0000000000000FFF0000000000000 FFF0000000000000FFF0000000000000FFF0000000000000FFF0000000000000FFF00000000000 00FFF0000000000000FFF0000000000000FFF0000000000000FFF0000000001FFFFFFFFF800000 1FFFFFFFFF8000001FFFFFFFFF8000001FFFFFFFFF80003C3A7CB945>I<001FFF00000001FFFF F0000003FFFFFC000007F007FE00000FF801FF00001FFC00FF80001FFC007FC0001FFC007FE000 1FFC003FE0000FF8003FF0000FF8003FF00007F0003FF00001C0003FF0000000003FF000000000 3FF0000000003FF0000000FFFFF000000FFFFFF000007FF83FF00001FF803FF00007FE003FF000 0FF8003FF0001FF0003FF0003FE0003FF0007FE0003FF0007FE0003FF000FFC0003FF000FFC000 3FF000FFC0003FF000FFC0003FF000FFC0007FF0007FE0007FF0007FE000DFF0003FF0039FF800 1FFC0F0FFFF007FFFE0FFFF001FFFC07FFF0003FE000FFF02C267DA530>97 D<000000003FC00000003FFFC00000003FFFC00000003FFFC00000003FFFC000000001FFC00000 0000FFC000000000FFC000000000FFC000000000FFC000000000FFC000000000FFC000000000FF C000000000FFC000000000FFC000000000FFC000000000FFC000000000FFC000000000FFC00000 0000FFC000000000FFC000000000FFC00000FFC0FFC0000FFFF8FFC0003FFFFEFFC000FFE03FFF C001FF000FFFC003FE0003FFC007FC0001FFC00FF80000FFC01FF00000FFC01FF00000FFC03FF0 0000FFC03FE00000FFC07FE00000FFC07FE00000FFC07FE00000FFC0FFE00000FFC0FFE00000FF C0FFE00000FFC0FFE00000FFC0FFE00000FFC0FFE00000FFC0FFE00000FFC0FFE00000FFC0FFE0 0000FFC07FE00000FFC07FE00000FFC07FE00000FFC03FF00000FFC03FF00000FFC01FF00000FF C00FF80001FFC007F80003FFC003FC0007FFC001FE000FFFE000FFC07EFFFF003FFFFCFFFF000F FFF0FFFF0001FF80FFFF303C7DBB37>100 D<0001FFC000000FFFF800003FFFFE0000FF80FF00 01FE003F8007FC001FC00FF8000FE00FF8000FF01FF00007F03FF00007F83FF00007F87FE00007 F87FE00003FC7FE00003FC7FE00003FCFFE00003FCFFFFFFFFFCFFFFFFFFFCFFFFFFFFFCFFE000 0000FFE0000000FFE0000000FFE00000007FE00000007FE00000007FE00000003FE00000003FF0 00003C1FF000003C1FF000003C0FF800007807FC0000F803FE0001F001FF0007E000FFC03FC000 3FFFFF000007FFFC000000FFE00026267DA52D>I<00000FF0000000FFFC000003FFFE00000FFC 7F00001FF0FF80003FE1FFC0007FC1FFC000FFC1FFC000FFC1FFC000FF81FFC001FF80FF8001FF 807F0001FF803E0001FF80000001FF80000001FF80000001FF80000001FF80000001FF80000001 FF80000001FF80000001FF800000FFFFFF8000FFFFFF8000FFFFFF8000FFFFFF800001FF800000 01FF80000001FF80000001FF80000001FF80000001FF80000001FF80000001FF80000001FF8000 0001FF80000001FF80000001FF80000001FF80000001FF80000001FF80000001FF80000001FF80 000001FF80000001FF80000001FF80000001FF80000001FF80000001FF80000001FF80000001FF 80000001FF80000001FF80000001FF80000001FF80000001FF8000007FFFFF00007FFFFF00007F FFFF00007FFFFF0000223C7DBB1E>I<000000001F800007FF007FC0003FFFE0FFE000FFFFFBE7 F001FE03FF07F003FC01FE07F007F800FF07F00FF800FF87E01FF0007FC3C01FF0007FC0003FF0 007FE0003FF0007FE0003FF0007FE0003FF0007FE0003FF0007FE0003FF0007FE0003FF0007FE0 001FF0007FC0001FF0007FC0000FF800FF800007F800FF000003FC01FE000001FE03FC000003FF FFF80000073FFFE000000707FF0000000F00000000000F00000000000F00000000000F00000000 000F80000000000FE0000000000FFFFFFC00000FFFFFFF800007FFFFFFF00003FFFFFFF80003FF FFFFFC0001FFFFFFFE0003FFFFFFFF000FFFFFFFFF801FC00007FF803F000000FF807F0000003F C07E0000001FC0FE0000001FC0FE0000001FC0FE0000001FC0FE0000001FC07F0000003F807F00 00003F803F8000007F001FC00000FE000FF00003FC0007FE001FF80001FFFFFFE000003FFFFF00 000003FFF000002C397DA631>I<00FF00000000FFFF00000000FFFF00000000FFFF00000000FF FF0000000007FF0000000003FF0000000003FF0000000003FF0000000003FF0000000003FF0000 000003FF0000000003FF0000000003FF0000000003FF0000000003FF0000000003FF0000000003 FF0000000003FF0000000003FF0000000003FF0000000003FF0000000003FF007FC00003FF01FF F80003FF07FFFC0003FF0F03FE0003FF1C01FF0003FF3001FF8003FF6000FF8003FFE000FFC003 FFC000FFC003FF8000FFC003FF8000FFC003FF8000FFC003FF0000FFC003FF0000FFC003FF0000 FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003 FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000 FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC0FF FFFC3FFFFFFFFFFC3FFFFFFFFFFC3FFFFFFFFFFC3FFFFF303C7CBB37>I<01F00007FC000FFE00 0FFE001FFF001FFF001FFF001FFF001FFF000FFE000FFE0007FC0001F000000000000000000000 00000000000000000000000000000000000000000000FF00FFFF00FFFF00FFFF00FFFF0007FF00 03FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF00 03FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF00 03FF0003FF00FFFFF8FFFFF8FFFFF8FFFFF8153D7DBC1B>I<00FF00FFFF00FFFF00FFFF00FFFF 0007FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF 0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF 0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF 0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF0003FF00FFFF FCFFFFFCFFFFFCFFFFFC163C7DBB1B>108 D<00FE003FE00007FC0000FFFE01FFFC003FFF8000 FFFE07FFFE00FFFFC000FFFE0F81FF01F03FE000FFFE1C00FF83801FF00007FE3800FFC7001FF8 0003FE60007FCC000FF80003FEE0007FFC000FFC0003FEC0007FF8000FFC0003FF80007FF0000F FC0003FF80007FF0000FFC0003FF80007FF0000FFC0003FF00007FE0000FFC0003FF00007FE000 0FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00007FE0 000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00007F E0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF0000 7FE0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00 007FE0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF00007FE0000FFC0003FF 00007FE0000FFC00FFFFFC1FFFFF83FFFFF0FFFFFC1FFFFF83FFFFF0FFFFFC1FFFFF83FFFFF0FF FFFC1FFFFF83FFFFF04C267CA553>I<00FE007FC000FFFE01FFF800FFFE07FFFC00FFFE0F03FE 00FFFE1C01FF0007FE3001FF8003FE6000FF8003FEE000FFC003FEC000FFC003FF8000FFC003FF 8000FFC003FF8000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FF C003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF 0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FF C003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC0FFFFFC3FFFFFFFFFFC3FFFFFFFFF FC3FFFFFFFFFFC3FFFFF30267CA537>I<0000FFC00000000FFFFC0000003FFFFF000000FFC0FF C00001FE001FE00007FC000FF80007F80007F8000FF00003FC001FF00003FE003FF00003FF003F E00001FF007FE00001FF807FE00001FF807FE00001FF807FE00001FF80FFE00001FFC0FFE00001 FFC0FFE00001FFC0FFE00001FFC0FFE00001FFC0FFE00001FFC0FFE00001FFC0FFE00001FFC0FF E00001FFC07FE00001FF807FE00001FF807FE00001FF803FF00003FF003FF00003FF001FF00003 FE000FF80007FC000FF80007FC0007FC000FF80003FE001FF00000FFC0FFC000003FFFFF000000 0FFFFC00000001FFE000002A267DA531>I<00FE03F000FFFE0FFE00FFFE1FFF00FFFE3C3F80FF FE707FC007FE60FFE003FEE0FFE003FEC0FFE003FFC0FFE003FF807FC003FF807FC003FF803F80 03FF800E0003FF00000003FF00000003FF00000003FF00000003FF00000003FF00000003FF0000 0003FF00000003FF00000003FF00000003FF00000003FF00000003FF00000003FF00000003FF00 000003FF00000003FF00000003FF00000003FF00000003FF00000003FF000000FFFFFE0000FFFF FE0000FFFFFE0000FFFFFE000023267DA529>114 D<003FF83801FFFEF807FFFFF80FC00FF81F 0003F83E0000F87C0000F87C000078FC000078FC000078FE000078FF000000FFE00000FFFF8000 7FFFFC007FFFFF003FFFFFC01FFFFFE00FFFFFF007FFFFF801FFFFFC003FFFFE0001FFFE00000F FF000001FF7000007FF000007FF000003FF800003FF800003EFC00003EFE00007EFF00007CFF80 00F8FFF007F0FDFFFFE0F0FFFF80E01FFC0020267DA527>I<0007800000078000000780000007 800000078000000F8000000F8000000F8000000F8000001F8000001F8000003F8000003F800000 7F800000FF800001FF800007FF80001FFFFFF0FFFFFFF0FFFFFFF0FFFFFFF001FF800001FF8000 01FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF80 0001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF803C01FF803C01FF 803C01FF803C01FF803C01FF803C01FF803C01FF803C00FF807800FFC078007FC070003FE0E000 1FFFC00007FF800001FF001E377EB626>I<00FF00003FC0FFFF003FFFC0FFFF003FFFC0FFFF00 3FFFC0FFFF003FFFC007FF0001FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC0 03FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF00 00FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC0 03FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0001FFC003FF00 01FFC003FF0003FFC001FF0003FFC001FF0006FFC000FF800EFFE0007FC03CFFFF003FFFF8FFFF 000FFFE0FFFF0001FF80FFFF30267CA537>I119 D121 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300 TeXDict begin @a4 %%EndSetup %%Page: 1 1 bop 169 36 a Fw(On)33 b(the)h(Mo)s(dularit)m(y)h(of)f(T)-8 b(ermination)34 b(of)436 117 y(T)-8 b(erm)32 b(Rewriting)k(Systems)800 236 y Fv(Enno)17 b(Ohlebusc)o(h)756 296 y(Univ)o(ersit\177)-24 b(at)15 b(Bielefeld)761 357 y(T)l(ec)o(hnisc)o(he)g(F)l(akult\177)-24 b(at)803 417 y(P)o(ostfac)o(h)16 b(100131)773 477 y(W-4800)i(Bielefeld)c(1) 877 537 y(German)o(y)579 597 y(e-mail:)19 b(enno@tec)o(hfak.uni-bielefeld.)o (de)810 717 y(Marc)o(h)d(26,)h(1993)874 870 y Fu(Abstract)190 971 y Ft(It)23 b(is)g(w)o(ell-kno)o(wn)g(that)f(termination)h(is)g(not)f(a)h (mo)q(dular)g(prop)q(ert)o(y)f(of)g(term)g(rewriting)122 1028 y(systems,)14 b(i.e.,)g(it)g(is)h(not)f(preserv)o(ed)h(under)g(disjoin)o(t)g (union.)20 b(The)15 b(ob)s(jectiv)o(e)f(of)g(this)h(pap)q(er)g(is)f(to)122 1084 y(pro)o(vide)k(a)e(\\uniform)i(framew)o(ork")d(for)h(su\016cien)o(t)i (conditions)h(whic)o(h)e(ensure)h(the)f(mo)q(dularit)o(y)122 1141 y(of)i(termination.)34 b(W)l(e)19 b(will)i(pro)o(v)o(e)e(the)h(follo)o (wing)g(result.)34 b(Whenev)o(er)20 b(the)f(disjoin)o(t)i(union)f(of)122 1197 y(t)o(w)o(o)d(terminating)i(term)e(rewriting)i(systems)f(is)h (non-terminating,)g(then)g(one)f(of)g(the)g(systems)122 1253 y(is)f(not)f Fs(C)275 1260 y Fr(E)298 1253 y Ft(-terminating)h(\(i.e.,)f(it)g (lo)q(oses)h(its)f(termination)h(prop)q(ert)o(y)f(when)h(extended)g(with)g (the)122 1310 y(rules)j Fq(C)s(ons)p Ft(\()p Fq(x;)8 b(y)r Ft(\))17 b Fs(!)j Fq(x)f Ft(and)g Fq(C)s(ons)p Ft(\()p Fq(x;)8 b(y)r Ft(\))18 b Fs(!)h Fq(y)r Ft(\))g(and)g(the)g(other)f(is)i(collapsing.) 33 b(This)19 b(result)122 1366 y(has)d(already)h(b)q(een)g(ac)o(hiev)o(ed)g (b)o(y)f(Gramlic)o(h)h([Gra92a)n(])f(for)f Fp(\014nitely)h(br)n(anching)g Ft(term)f(rewriting)122 1423 y(systems.)20 b(A)15 b(completely)i(di\013eren)o (t)e(approac)o(h)g(has)h(to)e(b)q(e)i(tak)o(en,)f(ho)o(w)o(ev)o(er,)f(to)h (pro)o(v)o(e)g(it)g(in)h(full)122 1479 y(generalit)o(y)l(.)33 b(Most)19 b(of)g(the)g(kno)o(wn)h(su\016cien)o(t)g(criteria)g(for)f(the)g (preserv)m(ation)h(of)f(termination)122 1536 y([Rus87,)d(Mid89,)h(K)o(O90,)f (Gra92a)o(])g(follo)o(w)h(as)g(corollaries)h(from)e(our)g(result,)i(and)f (new)g(criteria)122 1592 y(are)j(deriv)o(ed.)36 b(W)l(e)21 b(will)h(moreo)o(v)o(er)d(shed)i(some)f(ligh)o(t)h(on)f(mo)q(dular)h(prop)q (erties)g(of)f(com)o(bined)122 1649 y(systems)e(with)h(shared)g (constructors.)30 b(In)19 b(particular,)h(w)o(e)f(will)h(sho)o(w)e(that)g (the)h(ab)q(o)o(v)o(e)f(result)122 1705 y(do)q(es)d(not)g(carry)g(o)o(v)o(er) f(to)h(constructor)f(sharing)i(systems,)e(and)h(discuss)i(the)e (consequences.)0 1892 y Fo(1)83 b(In)n(tro)r(duction)0 2022 y Fv(During)12 b(the)f(past)i(decade,)e(term)f(rewriting)h(has)i(gained)e (enormous)g(imp)q(ortance)g(in)g(\014elds)g(of)h(computer)0 2083 y(science)e(concerned)h(with)g(sym)o(b)q(olic)e(manipulation.)19 b(Within)10 b(the)h(sub)s(ject)g(of)h(term)e(rewriting)g(mo)q(dular)0 2143 y(asp)q(ects)20 b(ha)o(v)o(e)f(recen)o(tly)e(b)q(een)i(receiving)f (increasing)h(atten)o(tion.)31 b(As)19 b(is)g(w)o(ell-kno)o(wn)f(from)h(soft) o(w)o(are)0 2203 y(engineering,)h(p)q(eople)g(are)g(encouraged)h(to)f(write)g (their)f(programs)i(in)e(a)i(mo)q(dular)f(w)o(a)o(y)f(in)h(order)g(to)0 2263 y(handle)c(large)g(systems.)k(Th)o(us,)c(from)f(a)h(practical)f(p)q(oin) o(t)h(of)h(view,)d(it)i(is)g(w)o(orth)g(kno)o(wing)g(under)g(what)0 2323 y(conditions)23 b(the)g(com)o(bined)e(program)i(inherits)f(prop)q (erties)h(from)f(its)h(constituen)o(t)f(mo)q(dules.)41 b(The)0 2384 y(most)13 b(simple)f(w)o(a)o(y)h(to)i(com)o(bine)c(t)o(w)o(o)j(term)e (rewriting)i(systems)e(\(TRSs\))j(is)e(their)h(disjoin)o(t)f(union.)21 b(This)0 2444 y(means)13 b(that)i(the)f(signatures)h Fn(F)598 2451 y Fm(1)632 2444 y Fv(and)f Fn(F)760 2451 y Fm(2)794 2444 y Fv(of)h(t)o(w)o(o)f(TRSs)h(\()p Fn(F)1125 2451 y Fm(1)1144 2444 y Fl(;)8 b Fn(R)1208 2451 y Fm(1)1228 2444 y Fv(\))14 b(and)h(\()p Fn(F)1409 2451 y Fm(2)1428 2444 y Fl(;)8 b Fn(R)1492 2451 y Fm(2)1512 2444 y Fv(\))14 b(ha)o(v)o(e)g(to)g(b)q(e)h(disjoin)o(t;)0 2504 y(then)j(their)g(disjoin)o(t)g(union)h(is)f(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))18 b(=)g(\()p Fn(F)869 2511 y Fm(1)901 2504 y Fn(])13 b(F)983 2511 y Fm(2)1002 2504 y Fl(;)8 b Fn(R)1066 2511 y Fm(1)1099 2504 y Fn(])13 b(R)1187 2511 y Fm(2)1206 2504 y Fv(\).)29 b(A)18 b(prop)q(ert)o(y)g Fn(P)23 b Fv(of)18 b(TRSs)i(is)e (called)0 2564 y(mo)q(dular)g(if)g(for)g(all)g(disjoin)o(t)g(TRSs)h Fn(R)743 2571 y Fm(1)781 2564 y Fv(and)g Fn(R)920 2571 y Fm(2)958 2564 y Fv(their)f(disjoin)o(t)g(union)g Fn(R)1432 2571 y Fm(1)1465 2564 y Fn(])12 b(R)1552 2571 y Fm(2)1591 2564 y Fv(has)19 b(the)f(prop)q(ert) o(y)0 2624 y Fn(P)23 b Fv(if)c(and)h(only)f(if)g(b)q(oth)i Fn(R)523 2631 y Fm(1)562 2624 y Fv(and)f Fn(R)702 2631 y Fm(2)741 2624 y Fv(ha)o(v)o(e)f(the)g(prop)q(ert)o(y)g Fn(P)t Fv(.)31 b(In)19 b(his)g(pioneering)g(pap)q(er)h([T)l(o)o(y87b)q(],)0 2685 y(T)l(o)o(y)o(ama)13 b(sho)o(w)o(ed)h(that)g(con\015uence)g(is)g(mo)q (dular.)19 b(In)14 b(con)o(trast)g(to)h(this,)f(termination)e(lac)o(ks)h(a)h (mo)q(dular)0 2745 y(b)q(eha)o(vior.)21 b(This)c(is)f(demonstrated)f(b)o(y)h (the)g(follo)o(wing)g(famous)g(example)e(\(cf.)20 b([T)l(o)o(y87a)q(]\).)962 2936 y(1)p eop %%Page: 2 2 bop 0 26 a Fk(Example)16 b(1.1)24 b Fv(The)19 b(term)e(rewriting)h(systems)g (\()p Fn(F)999 33 y Fm(1)1018 26 y Fl(;)8 b Fn(R)1082 33 y Fm(1)1102 26 y Fv(\))18 b(=)h(\()p Fn(f)p Fv(0)p Fl(;)8 b Fv(1)p Fl(;)g(F)f Fn(g)p Fl(;)h Fn(f)p Fl(F)f Fv(\(0)p Fl(;)h Fv(1)p Fl(;)g(x)p Fv(\))p Fn(!)n Fl(F)f Fv(\()p Fl(x;)h(x;)g(x)p Fv(\))p Fn(g)p Fv(\),)0 86 y(and)21 b(\()p Fn(F)154 93 y Fm(2)173 86 y Fl(;)8 b Fn(R)237 93 y Fm(2)257 86 y Fv(\))20 b(=)g(\()p Fn(f)p Fl(g)r Fn(g)p Fl(;)8 b Fn(f)p Fl(g)r Fv(\()p Fl(x;)g(y)r Fv(\))p Fn(!)o Fl(x;)g(g)r Fv(\()p Fl(x;)g(y)r Fv(\))p Fn(!)n Fl(y)r Fn(g)p Fv(\))20 b(are)g(eviden)o(tly)e(terminating)g(but)i(their)f (disjoin)o(t)0 147 y(union)d(is)h(not)f(terminating,)e(for)j(there)f(is)g (the)g(cyclic)e(rewrite)h(deriv)m(ation)157 235 y Fl(t)e Fn(\021)h Fl(F)7 b Fv(\(0)p Fl(;)h Fv(1)p Fl(;)g(g)r Fv(\(0)p Fl(;)g Fv(1\)\))p Fn(!)593 242 y Fr(R)623 247 y Fj(1)643 235 y Fl(F)f Fv(\()p Fl(g)r Fv(\(0)p Fl(;)h Fv(1\))p Fl(;)g(g)r Fv(\(0)p Fl(;)g Fv(1\))p Fl(;)g(g)r Fv(\(0)p Fl(;)g Fv(1\)\))p Fn(!)1214 242 y Fr(R)1244 247 y Fj(2)1263 235 y Fl(F)f Fv(\(0)p Fl(;)h(g)r Fv(\(0)p Fl(;)g Fv(1\))p Fl(;)g(g)r Fv(\(0)p Fl(;)g Fv(1\)\))p Fn(!)1725 242 y Fr(R)1755 247 y Fj(2)1774 235 y Fl(t)73 346 y Fv(In)15 b([Ohl93b])f(an)h(example)e(is)h(giv)o(en)g(whic)o(h)g(sho)o(ws)h (that)h Fn(R)1181 353 y Fm(1)1208 346 y Fn(])8 b(R)1291 353 y Fm(2)1326 346 y Fv(ma)o(y)13 b(b)q(e)i(non-terminating)f(ev)o(en)0 406 y(if)h Fn(R)86 413 y Fm(1)122 406 y Fv(and)h Fn(R)258 413 y Fm(2)294 406 y Fv(are)g(terminating,)e(con\015uen)o(t,)h(irreducible,)e (and)k(v)m(ariable-preserving.)j(Naturally)15 b(the)0 467 y(question)i (arises)g(what)h(restrictions)e(ha)o(v)o(e)h(to)g(b)q(e)g(imp)q(osed)g(on)g (the)g(constituen)o(t)g(TRSs)g(so)h(that)g(their)0 527 y(disjoin)o(t)c(union) h(is)f(again)i(terminating.)j(T)l(o)o(y)o(ama,)13 b(Klop,)i(and)g(Barendregt) f(sho)o(w)o(ed)h(in)f([TKB89])g(that)0 587 y(termination)f(is)i(mo)q(dular)f (for)h(con\015uen)o(t)f(left-linear)g(TRSs.)21 b(But)14 b(the)h(\014rst)g (results)f(w)o(ere)g(obtained)h(b)o(y)0 647 y(in)o(v)o(estigating)f(the)g (distribution)h(of)g(collapsing)g(rules)f(\(a)i(rewrite)e(rule)g(is)g (collapsing)h(if)g(its)f(righ)o(t-hand)0 707 y(side)f(is)g(a)h(v)m(ariable\)) f(and)h(duplicating)e(rules)h(\(a)h(rewrite)e(rule)h Fl(l)h Fn(!)g Fl(r)g Fv(is)g(duplicating)e(if)h Fl(r)i Fv(con)o(tains)e(more)0 768 y(o)q(ccurrences)j(of)i(some)e(v)m(ariable)g(than)i Fl(l)q Fv(\))e(among)h(the)g(TRSs.)24 b(Note)16 b(that)i(in)e(the)h(ab)q(o)o(v)o(e)g (example)e Fn(R)1929 775 y Fm(1)0 828 y Fv(consists)k(of)g(a)g(duplicating)g (rule,)f(whereas)h Fn(R)884 835 y Fm(2)923 828 y Fv(con)o(tains)g(t)o(w)o(o)f (collapsing)h(rules.)29 b(These)18 b(results)h(are)0 888 y(stated)e(in)e(the) i(next)e(theorem.)0 999 y Fk(Theorem)i(1.2)24 b Fv(Let)15 b Fn(R)462 1006 y Fm(1)482 999 y Fl(;)8 b Fn(R)546 1006 y Fm(2)582 999 y Fv(b)q(e)16 b(t)o(w)o(o)f(disjoin)o(t)g(terminating)f(TRSs.)22 b(Their)15 b(disjoin)o(t)g(union)h(is)g(termi-)0 1059 y(nating)h(pro)o(vided) e(that)i(one)f(of)h(the)f(follo)o(wing)g(conditions)g(is)g(satis\014ed:)60 1163 y(1.)24 b(Neither)15 b Fn(R)339 1170 y Fm(1)375 1163 y Fv(nor)i Fn(R)504 1170 y Fm(2)540 1163 y Fv(con)o(tains)f(collapsing)g (rules.)60 1259 y(2.)24 b(Neither)15 b Fn(R)339 1266 y Fm(1)375 1259 y Fv(nor)i Fn(R)504 1266 y Fm(2)540 1259 y Fv(con)o(tains)f(duplicating) g(rules.)60 1355 y(3.)24 b(One)16 b(of)g(the)g(systems)f(con)o(tains)i (neither)e(collapsing)h(nor)h(duplicating)f(rules.)73 1465 y(Statemen)o(ts)e(1)h(and)h(2)g(w)o(ere)e(\014rst)h(pro)o(v)o(ed)g(b)o(y)g (Rusino)o(witc)o(h)f(in)h([Rus87])g(\(Drosten)h(obtained)f(parts)0 1525 y(of)j(these)g(results)g(indep)q(enden)o(tly)f({)h(in)g(2)g(he)g (required)f(righ)o(t-linearit)o(y)l(,)f(cf.)27 b([Dro89)q(]\).)f(The)18 b(pro)q(of)i(of)0 1586 y(the)15 b(last)h(statemen)o(t)d(is)i(due)h(to)f (Middeldorp)g([Mid89].)20 b(An)15 b(equiv)m(alen)o(t)f(form)o(ulation)g(of)h (Theorem)f(1.2)0 1646 y(reads)j(as)g(follo)o(ws.)0 1757 y Fk(Theorem)g(1.3)24 b Fv(Let)17 b Fn(R)464 1764 y Fm(1)484 1757 y Fl(;)8 b Fn(R)548 1764 y Fm(2)585 1757 y Fv(b)q(e)17 b(t)o(w)o(o)g(disjoin)o(t)f(terminating)g (TRSs)h(suc)o(h)g(that)h(their)e(disjoin)o(t)h(union)0 1817 y Fn(R)42 1824 y Fm(1)73 1817 y Fn(])11 b(R)159 1824 y Fm(2)195 1817 y Fv(is)17 b(non-terminating.)j(Then)c Fn(R)793 1824 y Fm(1)829 1817 y Fv(is)g(duplicating)g(and)h Fn(R)1270 1824 y Fm(2)1306 1817 y Fv(is)f(collapsing)g(or)h(vice)e(v)o(ersa.)73 1927 y(A)f(v)o(ery)f(simple)f(in)o(tuitiv)o(e)g(pro)q(of)j(of)f(Theorem)f (1.3)i(can)f(b)q(e)h(found)f(in)g([Ohl93a])g(\(see)g(also)h(the)f(pro)q(of)0 1988 y(of)i(Theorem)f(5.18\).)22 b(Kurihara)16 b(and)g(Oh)o(uc)o(hi)f(w)o (ere)g(the)h(\014rst)g(to)g(observ)o(e)g(that)g(in)f(eac)o(h)h(of)g(the)g (kno)o(wn)0 2048 y(coun)o(terexamples)g(one)i(of)g(the)g(systems)f(w)o(as)i (not)g(simplifyi)o(ng.)25 b(They)18 b(pro)o(v)o(ed)f(in)h([K)o(O90])f(that)i (this)0 2108 y(is)d(essen)o(tial:)0 2219 y Fk(Theorem)h(1.4)24 b Fv(T)l(o)16 b(b)q(e)h(simplifying)c(is)j(a)h(mo)q(dular)f(prop)q(ert)o(y)g (of)g(TRSs.)73 2330 y(W)l(e)e(call)e(a)i(TRS)g(simplifying)d(if)i(its)h (rewrite)e(relation)i(is)f(con)o(tained)g(in)g(some)g(simpli\014cation)e (order-)0 2390 y(ing)17 b(\(as)h(a)g(matter)e(of)h(fact,)g(Kurihara)h(and)g (Oh)o(uc)o(hi)e(used)h(the)g(w)o(ord)h(\\simple)d(termination")h(for)i(this)0 2450 y(case,)g(ho)o(w)o(ev)o(er,)g(\\simple)e(termination")h(in)h(their)g (sense)g(do)q(es)h(not)g(imply)d(termination)h(in)h(general\),)0 2510 y(and)k(w)o(e)e(sa)o(y)h(that)h(a)f(TRS)h(is)e(simply)f(terminating)h (if)g(its)h(rewrite)f(relation)h(is)g(con)o(tained)f(in)h(some)0 2570 y(w)o(ell-founded)f(simpli\014cation)f(ordering.)37 b(Since)20 b(there)g(are)i(v)m(arious)f(di\013eren)o(t)g(de\014nitions)g(of)g(these)0 2630 y(notions)15 b(in)f(the)h(literature,)e(w)o(e)h(feel)f(that)i(it)f(is)g (time)e(to)j(in)o(v)o(estigate)e(their)h(in)o(terdep)q(endences.)19 b(There-)0 2691 y(fore,)f(w)o(e)f(will)g(carefully)f(analyse)i(the)g (di\013erences)f(of)h(these)g(notions)g(and)h(sho)o(w)f(their)g(consequences) 0 2751 y(w.r.t.)28 b(termination)17 b(and)i(mo)q(dularit)o(y)e(asp)q(ects.)29 b(It)18 b(is)h(w)o(ell-kno)o(wn)e(that)j(the)e(simplifying)e(prop)q(ert)o(y)0 2811 y(and)h(simple)d(termination)g(are)j(equiv)m(alen)o(t)d(for)j(\014nite)f (TRSs,)g(hence)f(the)h(ab)q(o)o(v)o(e)g(theorem)f(implies:)962 2936 y(2)p eop %%Page: 3 3 bop 0 26 a Fk(Corollary)18 b(1.5)24 b Fv(Simple)14 b(termination)g(is)i(a)h (mo)q(dular)f(prop)q(ert)o(y)g(of)g Fi(\014nite)i Fv(TRSs.)73 156 y(The)24 b(ab)q(o)o(v)o(e)g(corollary)g(is)f(v)o(ery)g(imp)q(ortan)o(t)g (from)g(a)h(practical)g(p)q(oin)o(t)g(of)g(view)f(b)q(ecause)h(man)o(y)0 216 y(termination)18 b(pro)q(ofs)k(are)e(done)g(semi-mec)n(hanicall)o(y)d (with)j(the)g(aid)g(of)g(theorem)f(pro)o(v)o(ers)g(b)o(y)h(means)0 276 y(of)d(some)f(implem)o(en)n(ted)e(w)o(ell-founded)i(simpli\014cation)e (ordering)j(\(lik)o(e)e(RPO,)i(LPO,)f(RDO,)h(KNS,)f(cf.)0 336 y([Ste89]\).)36 b(Gramlic)o(h)20 b(pro)o(v)o(ed)g(in)i([Gra92a)q(])f(the)g (follo)o(wing)g(abstract)i(result)e(for)g Fi(\014nitely)j(br)n(anching)0 397 y Fv(TRSs,)16 b(i.e.,)e(those)i(TRSs)g Fn(R)h Fv(whic)o(h)e(satisfy)h (that)g(for)g(ev)o(ery)f(rule)g Fl(l)f Fn(!)g Fl(r)h Fn(2)f(R)i Fv(there)g(are)f(only)h(\014nitely)0 457 y(man)o(y)f(di\013eren)o(t)g(rules)h (in)g Fn(R)g Fv(with)g(the)g(same)g(left-hand)g(side)g Fl(l)q Fv(.)0 586 y Fk(Theorem)h(1.6)24 b Fv(Let)d Fn(R)468 593 y Fm(1)488 586 y Fl(;)8 b Fn(R)552 593 y Fm(2)593 586 y Fv(b)q(e)22 b(t)o(w)o(o)f(disjoin)o(t)g Fi(\014nitely)j(br)n(anching)e Fv(terminating)e(TRSs)i(suc)o(h)f(that)0 647 y(their)c(disjoin)o(t)g(union)g Fn(R)471 654 y Fm(1)503 647 y Fn(])12 b(R)590 654 y Fm(2)628 647 y Fv(is)17 b(non-terminating.)24 b(Then)18 b Fn(R)1232 654 y Fm(1)1269 647 y Fv(is)g(not)g Fn(C)1434 654 y Fr(E)1457 647 y Fv(-terminating)e(and)j Fn(R)1879 654 y Fm(2)1916 647 y Fv(is)0 707 y(collapsing)d(or)h(vice)e(v)o(ersa.)73 837 y(A)h(TRS)h Fn(R)g Fv(is)g(called)e Fn(C)514 844 y Fr(E)538 837 y Fv(-terminating)g(if)h (the)h(TRS)g Fn(R)11 b(])h(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)c(y)r Fv(\))13 b Fn(!)h Fl(x;)8 b(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))14 b Fn(!)g Fl(y)r Fn(g)i Fv(is)0 897 y(terminating,)11 b(where)h Fl(C)t(ons)g Fv(is)g(some)g(binary)g(function)g(sym)o(b)q(ol)f(that)i(do)q (es)g(not)g(o)q(ccur)f(in)g(the)g(signature)0 957 y(of)17 b Fn(R)p Fv(.)k(As)16 b(ab)q(o)o(v)o(e)g(this)g(theorem)f(can)h(b)q(e)h (paraphrased)h(as)e(follo)o(ws.)0 1087 y Fk(Theorem)h(1.7)24 b Fv(Let)18 b Fn(R)465 1094 y Fm(1)485 1087 y Fl(;)8 b Fn(R)549 1094 y Fm(2)587 1087 y Fv(b)q(e)18 b(t)o(w)o(o)g(disjoin)o(t)g Fi(\014nitely)j(br)n(anching)e Fv(terminating)d(TRSs.)28 b(Their)18 b(dis-)0 1147 y(join)o(t)d(union)g Fn(R)289 1154 y Fm(1)318 1147 y Fn(])9 b(R)402 1154 y Fm(2)437 1147 y Fv(is)15 b(terminating)e(pro)o (vided)i(that)g(one)h(of)f(the)g(follo)o(wing)g(conditions)g(is)g (satis\014ed:)60 1265 y(1.)24 b(Neither)15 b Fn(R)339 1272 y Fm(1)375 1265 y Fv(nor)i Fn(R)504 1272 y Fm(2)540 1265 y Fv(con)o(tains)f(collapsing)g(rules.)60 1366 y(2.)24 b(Both)16 b Fn(R)285 1373 y Fm(1)321 1366 y Fv(and)h Fn(R)458 1373 y Fm(2)494 1366 y Fv(are)g Fn(C)602 1373 y Fr(E)626 1366 y Fv(-terminating.)60 1466 y(3.)24 b(One)16 b(of)g(the)g(systems)f(is)i Fn(C)621 1473 y Fr(E)644 1466 y Fv(-terminating)e(and)i(do)q(es)g(not)g(con)o(tain)f (collapsing)g(rules.)73 1596 y(In)j([Gra92a)q(])f(it)g(is)h(sho)o(wn)g(that)g (if)f(a)h(terminating)e(TRS)i(is)g(not)g Fn(C)1340 1603 y Fr(E)1364 1596 y Fv(-terminating,)e(then)i(it)f(is)g(du-)0 1656 y(plicating)f(and)h (not)h(simply)c(terminating)h(\(there)i(y)o(et)f(another)h(de\014nition)f(of) h(simple)e(termination)g(is)0 1716 y(used)g(whic)o(h)f(will)f(b)q(e)i(sho)o (wn)g(to)g(b)q(e)g(equiv)m(alen)o(t)e(to)i(ours\).)22 b(Hence)14 b(for)i(the)f(class)h(of)g Fi(\014nitely)i(br)n(anching)0 1776 y Fv(TRSs)g(Theorem)e(1.7)i(con)o(tains)g(Theorem)e(1.2)i(as)g(a)g(sp)q (ecial)f(case.)25 b(Moreo)o(v)o(er,)17 b(it)g(extends)g(Corollary)0 1836 y(1.5)g(to)f(the)g(class)h(of)f Fi(\014nitely)j(br)n(anching)e Fv(TRSs.)73 1917 y(In)c(this)h(pap)q(er,)g(w)o(e)f(generalize)f(the)h(ab)q(o) o(v)o(e)g(results)g(to)h(arbitrary)f(\(i.e.,)f(p)q(ossibly)i(in\014nitely)e (branc)o(h-)0 1978 y(ing\))h(TRSs.)21 b(More)13 b(precisely)e(w)o(e)i(pro)o (v)o(e)g(Theorem)e(1.6)j(for)g(arbitrary)f(disjoin)o(t)g(TRSs)g(solving)g(a)h (recen)o(t)0 2038 y(conjecture)i(of)g(Gramlic)o(h)f([Gra92b)q(].)21 b(It)16 b(should)h(b)q(e)g(p)q(oin)o(ted)f(out)h(that)g(to)f(do)h(this)g(a)g (completely)c(dif-)0 2098 y(feren)o(t)g(approac)o(h)j(than)f(in)f([Gra92a)q (])g(has)i(to)f(b)q(e)f(tak)o(en.)20 b(F)l(rom)14 b(our)h(result,)f(Theorem)f (1.2)i(follo)o(ws)f(as)h(a)0 2158 y(corollary)h(and)h(Corollary)f(1.5)g(can)h (b)q(e)f(pro)o(v)o(ed)f(for)i(arbitrary)f(TRSs.)22 b(This)16 b(settles)f(in)h(particular)g(the)0 2218 y(op)q(en)k(question)f(whether)g (simple)f(termination)f(is)j(a)g(mo)q(dular)f(prop)q(ert)o(y)g(and)h(bridges) f(the)h(existing)0 2279 y(theoretical)15 b(gap.)73 2360 y(One)f(w)o(a)o(y)g (of)h(w)o(eak)o(ening)f(the)g(disjoin)o(tness)g(requiremen)o(t)d(is)k(to)f (allo)o(w)h(shared)g(constructors.)21 b(Con-)0 2420 y(structors)h(are)f(sym)o (b)q(ols)e(that)j(do)f(not)h(o)q(ccur)f(at)g(the)g(ro)q(ot)h(p)q(osition)g (of)f(the)g(left-hand)g(side)g(of)g(an)o(y)0 2480 y(rewrite)15 b(rule.)21 b(W)l(e)16 b(sa)o(y)g(that)h(t)o(w)o(o)f(TRS)g(\()p Fn(F)817 2487 y Fm(1)837 2480 y Fl(;)8 b Fn(R)901 2487 y Fm(1)921 2480 y Fv(\))16 b(and)h(\()p Fn(F)1106 2487 y Fm(2)1125 2480 y Fl(;)8 b Fn(R)1189 2487 y Fm(2)1209 2480 y Fv(\))16 b Fi(shar)n(e)h(c)n (onstructors)f Fv(if)435 2585 y Fn(F)471 2592 y Fm(1)502 2585 y Fn(\\)11 b(F)582 2592 y Fm(2)615 2585 y Fn(\022)j Fv(\()p Fn(F)723 2592 y Fm(1)754 2585 y Fn([)d(F)834 2592 y Fm(2)853 2585 y Fv(\))g Fn(n)g(f)p Fl(r)q(oot)p Fv(\()p Fl(l)q Fv(\))17 b Fn(j)f Fl(l)f Fn(!)e Fl(r)i Fn(2)f(R)1351 2592 y Fm(1)1382 2585 y Fn([)e(R)1469 2592 y Fm(2)1489 2585 y Fn(g)0 2691 y Fv(and)k(call)e(the)h(TRS)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))14 b(=)f(\()p Fn(F)641 2698 y Fm(1)669 2691 y Fn([)c(F)747 2698 y Fm(2)767 2691 y Fl(;)f Fn(R)831 2698 y Fm(1)859 2691 y Fn([)h(R)943 2698 y Fm(2)963 2691 y Fv(\))15 b(their)f(com)o(bined)f(system)h(with)h (shared)g(construc-)0 2751 y(tors.)25 b(A)17 b(prop)q(ert)o(y)g Fn(P)k Fv(is)c(called)f Fi(mo)n(dular)i(for)g(TRSs)g(with)g(shar)n(e)n(d)f(c) n(onstructors)p Fv(,)g(if)g(ev)o(ery)f(com)o(bined)0 2811 y(system)g(has)h (the)g(prop)q(ert)o(y)g Fn(P)k Fv(if)16 b(and)i(only)f(if)f(b)q(oth)i (constituen)o(t)f(TRSs)g(ha)o(v)o(e)g(the)f(prop)q(ert)o(y)h Fn(P)t Fv(.)24 b(Our)962 2936 y(3)p eop %%Page: 4 4 bop 0 26 a Fv(next)13 b(goal)h(is)f(to)g(elucidate)f(what)i(prop)q(erties)f (are)g(also)h(mo)q(dular)f(for)g(constructor)h(sharing)g(TRSs.)20 b(It)13 b(is)0 86 y(kno)o(wn)g(that)h(con\015uence)f(lac)o(ks)f(mo)q(dular)h (b)q(eha)o(vior,)g(whereas)g(lo)q(cal)g(con\015uence)g(is)g(mo)q(dular.)20 b(It)12 b(turns)0 147 y(out)18 b(that)f(w)o(eak)g(termination)f(is)h(also)h (a)g(mo)q(dular)e(prop)q(ert)o(y)i(of)f(constructor)h(sharing)g(TRSs.)25 b(Before)0 207 y(w)o(e)13 b(turn)g(to)g(the)g(question)g(of)g(whether)g(the)g (ab)q(o)o(v)o(e)g(theorems)f(on)h(the)g(mo)q(dularit)o(y)f(of)h(termination)e (also)0 267 y(hold)19 b(in)f(the)g(presence)g(of)h(shared)g(constructors,)h (w)o(e)e(\014rst)h(collect)e(some)h(kno)o(wn)g(results.)28 b(Kurihara)0 327 y(and)17 b(Oh)o(uc)o(hi)e(extended)g(their)h(result)f(in)h ([K)o(O92])g(to)h(com)o(bined)d(systems)h(with)h(shared)h(constructors:)0 460 y Fk(Theorem)g(1.8)24 b Fv(T)l(o)16 b(b)q(e)h(simplifying)c(is)j(a)h(mo)q (dular)f(prop)q(ert)o(y)g(of)g(constructor)h(sharing)g(TRSs.)0 593 y Fk(Corollary)h(1.9)24 b Fv(Simple)9 b(termination)i(is)g(a)h(mo)q (dular)g(prop)q(ert)o(y)g(of)g Fi(\014nite)h Fv(constructor)f(sharing)h (TRSs.)73 726 y(Gramlic)o(h)20 b(sho)o(w)o(ed,)i(in)g([Gra92b)q(],)g(the)f (follo)o(wing)h(extension)f(of)h(Theorem)e(1.6)i(to)g(constructor)0 786 y(sharing)16 b(systems)f(\(a)h(rule)e Fl(l)h Fn(!)f Fl(r)j Fv(is)e(called)f(constructor-lifting)i(if)f Fl(r)q(oot)p Fv(\()p Fl(r)q Fv(\))h(is)g(a)g(shared)g(constructor\).)0 920 y Fk(Theorem)h(1.10)24 b Fv(Let)c Fn(R)495 927 y Fm(1)515 920 y Fl(;)8 b Fn(R)579 927 y Fm(2)619 920 y Fv(b)q(e)21 b(t)o(w)o(o)f Fi(\014nitely)j(br)n(anching)e Fv(terminating)e(TRSs)i(with)f(shared)h(con-)0 980 y(structors)h(suc)o(h)g (that)g(their)f(com)o(bined)e(system)h Fn(R)990 987 y Fm(1)1025 980 y Fn([)15 b(R)1115 987 y Fm(2)1157 980 y Fv(is)21 b(non-terminating.)37 b(Then)22 b Fn(R)1782 987 y Fh(d)1824 980 y Fv(is)f(not)0 1040 y Fn(C)26 1047 y Fr(E)50 1040 y Fv(-terminating)15 b(and)h Fn(R)474 1042 y Fm(\026)468 1051 y Fh(d)505 1040 y Fv(is)g(collapsing)g(or)h (constructor-lifting)f(for)g(some)g Fl(d;)1491 1027 y Fv(\026)1483 1040 y Fl(d)e Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)17 b Fv(with)f Fl(d)e Fn(6)p Fv(=)1916 1027 y(\026)1908 1040 y Fl(d)p Fv(.)73 1173 y(Since)i(non-duplicating)g(TRSs)h(are)f Fn(C)791 1180 y Fr(E)815 1173 y Fv(-terminating,)e(Theorem)h(1.10)i(in)f(particular)g (implies:)0 1306 y Fk(Corollary)i(1.11)24 b Fv(Let)19 b Fn(R)505 1313 y Fm(1)525 1306 y Fl(;)8 b Fn(R)589 1313 y Fm(2)628 1306 y Fv(b)q(e)20 b(t)o(w)o(o)f(terminating)f Fi(\014nitely)j(br)n(anching)g Fv(TRSs)e(with)h(shared)f(con-)0 1366 y(structors.)i(Their)15 b(com)o(bined)d(system)i Fn(R)773 1373 y Fm(1)801 1366 y Fn([)8 b(R)884 1373 y Fm(2)919 1366 y Fv(is)15 b(terminating)e(pro)o(vided)h(that)h (one)g(of)h(the)e(follo)o(wing)0 1426 y(conditions)i(is)g(satis\014ed:)60 1547 y(1.)24 b(Neither)15 b Fn(R)339 1554 y Fm(1)375 1547 y Fv(nor)i Fn(R)504 1554 y Fm(2)540 1547 y Fv(con)o(tains)f(either)f (collapsing)h(or)h(constructor-lifting)f(rules.)60 1649 y(2.)24 b(Neither)15 b Fn(R)339 1656 y Fm(1)375 1649 y Fv(nor)i Fn(R)504 1656 y Fm(2)540 1649 y Fv(con)o(tains)f(duplicating)g(rules.)60 1750 y(3.)24 b(One)17 b(of)g(the)g(systems)f(con)o(tains)i(neither)e (collapsing)h(nor)h(constructor-lifting)f(nor)g(duplicating)122 1810 y(rules.)73 1943 y(Constructor-lifting)h(rules)f(ha)o(v)o(e)g(to)h(b)q (e)f(excluded)g(\(cf.)24 b([Gra92b)q(]\))17 b(b)q(ecause)h(they)f(ma)o(y)f (ha)o(v)o(e)h(the)0 2003 y(same)11 b(impact)g(as)i(collapsing)f(ones)h (\(note)f(that)h(it)f(is)g(easy)g(to)h(obtain)g(an)f(example)f(in)h(whic)o(h) f(b)q(oth)i(TRSs)0 2063 y(are)j(also)h(con\015uen)o(t)f(b)o(y)g(mo)q(difying) f(kno)o(wn)h(coun)o(terexamples)e(to)j(the)f(mo)q(dularit)o(y)e(of)j(con)o(v) o(ergence\).)0 2197 y Fk(Example)f(1.12)24 b Fv(The)14 b(term)f(rewriting)g (systems)g(\()p Fn(F)1008 2204 y Fm(1)1027 2197 y Fl(;)8 b Fn(R)1091 2204 y Fm(1)1111 2197 y Fv(\))14 b(=)g(\()p Fn(f)p Fv(0)p Fl(;)8 b Fv(1)p Fl(;)g(F)f Fn(g)p Fl(;)h Fn(f)p Fl(F)f Fv(\(0)p Fl(;)h Fv(1)p Fl(;)g(x)p Fv(\))p Fn(!)n Fl(F)f Fv(\()p Fl(x;)h(x;)g(x)p Fv(\))p Fn(g)p Fv(\),)0 2257 y(and)18 b(\()p Fn(F)151 2264 y Fm(2)170 2257 y Fl(;)8 b Fn(R)234 2264 y Fm(2)254 2257 y Fv(\))15 b(=)g(\()p Fn(f)p Fl(a;)8 b Fv(0)p Fl(;)g Fv(1)p Fn(g)p Fl(;)g Fn(f)p Fl(a)p Fn(!)o Fv(0)p Fl(;)g(a)p Fn(!)p Fv(1)p Fn(g)p Fv(\))17 b(share)g(the)g(constructors)h(0)f(and)h(1.)23 b(They)17 b(are)g(eviden)o(tly)0 2317 y(terminating)j(but)h(their)g(com)o (bined)e(system)h(is)h(not)h(terminating,)e(for)i(there)e(is)i(the)f(cyclic)e (rewrite)0 2377 y(deriv)m(ation)435 2437 y Fl(F)7 b Fv(\(0)p Fl(;)h Fv(1)p Fl(;)g(a)p Fv(\))p Fn(!)679 2444 y Fr(R)709 2449 y Fj(1)729 2437 y Fl(F)f Fv(\()p Fl(a;)h(a;)g(a)p Fv(\))p Fn(!)975 2444 y Fr(R)1005 2449 y Fj(2)1025 2437 y Fl(F)f Fv(\(0)p Fl(;)h(a;)g(a)p Fv(\))p Fn(!)1270 2444 y Fr(R)1300 2449 y Fj(2)1320 2437 y Fl(F)f Fv(\(0)p Fl(;)h Fv(1)p Fl(;)g(a)p Fv(\))73 2570 y(W)l(e)15 b(will)f(sho)o(w)h(b)o(y)g(coun)o(terexamples)d(that)k(Theorem)d(1.10)j(and)g (Corollary)f(1.9)g(do)h(not)f(extend)f(to)0 2630 y(arbitrary)19 b(TRSs)h(with)f(shared)h(constructors.)32 b(In)19 b(particular,)g Fn(C)1257 2637 y Fr(E)1281 2630 y Fv(-termination)f(and)i(simple)d(termi-)0 2691 y(nation)g(are)g(not)g(mo)q(dular)f(prop)q(erties)h(of)g(TRSs)g(with)g (shared)g(constructors.)23 b(Consequen)o(tly)l(,)16 b(w)o(e)g(are)0 2751 y(in)o(terested)d(in)h(the)h(restrictions)e(under)i(whic)o(h)f(these)g (results)g(do)h(hold.)21 b(W)l(e)14 b(will)g(sho)o(w)h(that)g(Corollary)0 2811 y(1.9)22 b(can)g(b)q(e)h(generalized)d(to)j(the)e(class)i(of)f(TRSs)g (whic)o(h)g(in)o(tro)q(duce)f(only)h(\014nitely)f(man)o(y)f(function)962 2936 y(4)p eop %%Page: 5 5 bop 0 26 a Fv(sym)o(b)q(ols.)20 b(Using)14 b(the)g(metho)q(ds)g(of)h(Gramlic) o(h)d([Gra92b)q(],)i(it)g(is)h(also)g(p)q(ossible)f(to)h(extend)f(this)g (theorem)0 86 y(to)k(the)g(class)g(of)g(\014nitely)f(branc)o(hing)h(TRSs.)27 b(Moreo)o(v)o(er,)16 b(b)o(y)i(a)g(simple)e(pro)q(of)j(similiar)c(to)k(the)e (one)i(of)0 147 y(Theorem)c(1.3)h(giv)o(en)f(in)h([Ohl93a],)f(w)o(e)h(sho)o (w)g(that)h(Corollary)f(1.11)g(can)h(b)q(e)f(generalized)e(to)j(arbitrary)0 207 y(TRSs.)73 288 y(The)22 b(pap)q(er)h(is)f(organized)g(as)h(follo)o(ws:)32 b(Section)22 b(2)g(brie\015y)g(recalls)f(the)g(basic)h(notions)h(of)g(term)0 348 y(rewriting.)h(The)18 b(next)f(section)g(deals)g(with)h(simple)d (termination.)23 b(Since)17 b(there)g(are)g(di\013eren)o(t)g(de\014ni-)0 408 y(tions)g(of)g(simple)d(termination)h(in)h(the)h(literature,)e(w)o(e)h (carefully)f(analyse)i(the)f(relationships)h(b)q(et)o(w)o(een)0 468 y(the)e(di\013eren)o(t)g(notions.)22 b(In)15 b(Section)g(4,)h(w)o(e)f (pro)o(v)o(e)g(the)h(generalization)f(of)h(Theorem)e(1.6)i(for)g(arbitrary)0 529 y(TRSs,)g(from)f(whic)o(h)g(Theorem)g(1.2)h(follo)o(ws)g(as)g(a)h (corollary)e(and)i(whic)o(h)e(also)i(en)o(tails)e(the)g(mo)q(dularit)o(y)0 589 y(of)i Fn(C)82 596 y Fr(E)106 589 y Fv(-termination)e(and)j(simple)d (termination.)21 b(In)c(Section)f(5,)h(w)o(e)g(\014rst)g(in)o(tro)q(duce)f (required)g(notions)0 649 y(of)e(com)o(bined)e(systems)h(with)g(shared)i (constructors)f(and)h(then)f(analyse)g(for)g(whic)o(h)f(cases)h(theorems)f (for)0 709 y(the)g(disjoin)o(t)f(union)i(also)f(hold)g(in)g(the)g(presence)f (of)h(shared)h(constructors.)21 b(Section)12 b(6)i(brie\015y)e(discusses)0 769 y(the)i(consequences)g(of)g(our)h(considerations)g(for)f(conditional)g (term)f(rewriting)g(systems,)g(while)h(the)g(last)0 829 y(section)i(is)g (dedicated)f(to)i(concluding)f(remarks.)0 1017 y Fo(2)83 b(Preliminaries)0 1147 y Fv(In)24 b(this)f(section,)i(w)o(e)e(brie\015y)g(recall)g(the)h(basic) f(notions)i(of)f(term)e(rewriting)h(as)i(surv)o(ey)o(ed)d(in)i(e.g.)0 1207 y(Dersho)o(witz)16 b(and)h(Jouannaud)h([DJ90])e(and)h(Klop)f([Klo92].)73 1288 y(A)k Fi(signatur)n(e)g Fv(is)f(a)h(coun)o(table)g(set)f Fn(F)25 b Fv(of)20 b Fi(function)i(symb)n(ols)e Fv(or)g Fi(op)n(er)n(ators)p Fv(,)e(where)i(ev)o(ery)e Fl(f)25 b Fn(2)20 b(F)0 1348 y Fv(is)f(asso)q (ciated)i(with)f(a)g(natural)g(n)o(um)o(b)q(er)e(denoting)h(its)h(arit)o(y)l (.)30 b Fn(F)1254 1355 y Fh(n)1297 1348 y Fv(denotes)20 b(the)f(set)h(of)g (all)f(function)0 1408 y(sym)o(b)q(ols)j(of)h(arit)o(y)f Fl(n)p Fv(,)i(hence)e Fn(F)29 b Fv(=)715 1375 y Fg(S)750 1419 y Fh(n)p Fr(\025)p Fm(0)827 1408 y Fn(F)863 1415 y Fh(n)886 1408 y Fv(.)41 b(Elemen)o(ts)20 b(of)j Fn(F)1256 1415 y Fm(0)1298 1408 y Fv(are)g(called)f Fi(c)n(onstants)p Fv(.)41 b(The)23 b(set)0 1469 y Fn(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))13 b(of)h Fi(terms)g Fv(built)f(from)g (a)h(signature)h Fn(F)j Fv(and)d(a)f(coun)o(table)g(set)g(of)g Fi(variables)h Fn(V)i Fv(with)d Fn(F)d(\\)6 b(V)18 b Fv(=)c Fn(;)0 1529 y Fv(is)h(the)h(smallest)d(set)j(suc)o(h)f(that)h Fn(V)i(\022)13 b(T)g Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))15 b(and)h(if)f Fl(f)k Fn(2)14 b(F)20 b Fv(has)d(arit)o(y)d Fl(n)i Fv(and)g Fl(t)1543 1536 y Fm(1)1563 1529 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1690 1536 y Fh(n)1727 1529 y Fn(2)14 b(T)f Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))0 1589 y(then)20 b Fl(f)5 b Fv(\()p Fl(t)181 1596 y Fm(1)201 1589 y Fl(;)j(:)g(:)g(:)f(;)h(t)328 1596 y Fh(n)351 1589 y Fv(\))21 b Fn(2)f(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\).)32 b(W)l(e)20 b(write)g Fl(f)26 b Fv(instead)20 b(of)g Fl(f)5 b Fv(\()21 b(\))f(whenev)o(er)f Fl(f)26 b Fv(is)20 b(a)g(constan)o(t.) 34 b(The)0 1649 y(set)17 b(of)f(function)h(sym)o(b)q(ols)e(app)q(earing)j(in) e(a)h(term)e Fl(t)f Fn(2)h(T)e Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))16 b(is)g(denoted)h(b)o(y)f Fn(F)5 b Fl(un)p Fv(\()p Fl(t)p Fv(\))o(,)16 b(and)i(the)e(set)0 1709 y(of)h(v)m(ariables)h(o)q (ccurring)f(in)g Fl(t)f Fv(is)h(denoted)h(b)o(y)e Fn(V)t Fl(ar)q Fv(\()p Fl(t)p Fv(\))o(.)24 b(T)l(erms)16 b(without)i(v)m(ariables)f(are)g (called)f Fi(gr)n(ound)0 1770 y Fv(terms.)34 b(The)22 b(set)f(of)g(all)g (ground)h(terms)e(is)h(denoted)g(b)o(y)g Fn(T)12 b Fv(\()p Fn(F)5 b Fv(\).)36 b(Iden)o(tit)o(y)19 b(of)j(terms)d(and)j(function)0 1830 y(sym)o(b)q(ols)17 b(is)i(denoted)f(b)o(y)g Fn(\021)p Fv(.)28 b(F)l(or)18 b Fl(t)g Fn(2)g(T)12 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))18 b(w)o(e)g(de\014ne)g Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))h(b)o(y:)25 b Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))18 b Fn(\021)g Fl(t)g Fv(if)g Fl(t)f Fn(2)h(V)t Fv(,)g(and)0 1890 y Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))c Fn(\021)g Fl(f)21 b Fv(if)16 b Fl(t)e Fn(\021)f Fl(f)5 b Fv(\()p Fl(t)450 1897 y Fm(1)470 1890 y Fl(;)j(:)g(:)g(:)f(;)h(t)597 1897 y Fh(n)620 1890 y Fv(\).)73 1971 y(A)17 b Fi(substitution)i Fl(\033)g Fv(is)e(a)h(mapping)f (from)g Fn(V)k Fv(to)d Fn(T)12 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))17 b(suc)o(h)g(that)h Fn(f)p Fl(x)e Fn(2)g(V)21 b(j)d Fl(\033)r Fv(\()p Fl(x)p Fv(\))p Fn(6\021)p Fl(x)p Fn(g)e Fv(is)h(\014nite.)0 2031 y(This)f(set)g(is)f(called)g(the)g Fi(domain)h Fv(of)g Fl(\033)i Fv(and)e(will)f(b)q(e)h(denoted)f(b)o(y)h Fn(D)q Fl(om)p Fv(\()p Fl(\033)r Fv(\).)21 b(Occasionally)15 b(w)o(e)g(presen)o(t)0 2091 y(a)k(substitution)h Fl(\033)g Fv(as)g Fn(f)p Fl(x)e Fn(7!)g Fl(\033)r Fv(\()p Fl(x)p Fv(\))g Fn(j)h Fl(x)f Fn(2)h(D)q Fl(om)p Fv(\()p Fl(\033)r Fv(\))p Fn(g)p Fv(.)29 b(The)19 b(substitution)g(with)g (empt)o(y)e(domain)h(will)0 2151 y(b)q(e)h(denoted)h(b)o(y)e Fl(")p Fv(.)30 b(Substitutions)20 b(are)f(extended)f(to)i(morphisms)d(from)h Fn(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))18 b(to)i Fn(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\),)18 b(i.e.)0 2212 y Fl(\033)r Fv(\()p Fl(f)5 b Fv(\()p Fl(t)115 2219 y Fm(1)134 2212 y Fl(;)j(:)g(:)g(:)g(;)g(t)262 2219 y Fh(n)284 2212 y Fv(\)\))14 b Fn(\021)g Fl(f)5 b Fv(\()p Fl(\033)r Fv(\()p Fl(t)504 2219 y Fm(1)523 2212 y Fv(\))p Fl(;)j(:)g(:)g(:)f(;)h(\033)r Fv(\()p Fl(t)718 2219 y Fh(n)741 2212 y Fv(\)\))14 b(for)h(ev)o(ery)e Fl(n)p Fv(-ary)i(function)f(sym)o(b)q(ol)g Fl(f)19 b Fv(and)d(terms)d Fl(t)1766 2219 y Fm(1)1785 2212 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1912 2219 y Fh(n)1935 2212 y Fv(.)0 2272 y(W)l(e)16 b(call)f Fl(\033)r Fv(\()p Fl(t)p Fv(\))h(an)g Fi(instanc)n(e)i Fv(of)e Fl(t)p Fv(.)21 b(W)l(e)16 b(also)h(write)f Fl(t\033)h Fv(instead)f(of)h Fl(\033)r Fv(\()p Fl(t)p Fv(\).)73 2353 y(In)h(order)h(to)f(describ)q(e)g (subterm)f(o)q(ccurrences)h(of)g(a)h(term,)d(w)o(e)i(in)o(tro)q(duce)g(the)g (notationally)g(con-)0 2413 y(v)o(enien)o(t)c(notion)j(\\con)o(text")f (instead)g(of)g(the)g(more)f(precise)g(notion)i(\\p)q(osition")g(\(cf.)k ([Mid90]\).)f(Let)d Ff(2)0 2473 y Fv(b)q(e)i(a)g(sp)q(ecial)g(constan)o(t)g (sym)o(b)q(ol.)28 b(A)19 b Fi(c)n(ontext)h Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)e(;)i Fv(])19 b(is)f(a)i(term)d(in)h Fn(T)13 b Fv(\()p Fn(F)18 b([)13 b(f)p Ff(2)p Fn(g)p Fl(;)8 b Fn(V)t Fv(\).)29 b(If)18 b Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)e(;)i Fv(])0 2533 y(is)16 b(a)g(con)o(text)f(with)h Fl(n)g Fv(o)q(ccurrences)g(of)g Ff(2)g Fv(and)h Fl(t)901 2540 y Fm(1)920 2533 y Fl(;)8 b(:)g(:)g(:)g(;)g(t) 1048 2540 y Fh(n)1087 2533 y Fv(are)16 b(terms,)e(then)i Fl(C)t Fv([)p Fl(t)1499 2540 y Fm(1)1517 2533 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1644 2540 y Fh(n)1667 2533 y Fv(])16 b(is)g(the)f(result)0 2594 y(of)21 b(replacing)f(from)g(left)f(to)i(righ)o(t)g(the)f(o)q(ccurrences)g (of)h Ff(2)g Fv(with)g Fl(t)1275 2601 y Fm(1)1294 2594 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1421 2601 y Fh(n)1444 2594 y Fv(.)35 b(A)20 b(con)o(text)g(con)o(taining)0 2654 y(precisely)15 b(one)i(o)q(ccurrence)e (of)i Ff(2)g Fv(is)g(denoted)f(b)o(y)g Fl(C)t Fv([)g(].)22 b(A)16 b(term)f Fl(t)h Fv(is)h(a)g Fi(subterm)g Fv(of)g(a)g(term)e Fl(s)h Fv(if)g(there)0 2714 y(exists)j(a)g(con)o(text)f Fl(C)t Fv([)h(])f(suc)o(h)h(that)h Fl(s)f Fn(\021)f Fl(C)t Fv([)p Fl(t)p Fv(].)28 b(A)19 b(subterm)f Fl(t)h Fv(of)g Fl(s)g Fv(is)g Fi(pr)n(op)n(er)p Fv(,)f(denoted)h(b)o(y)g Fl(s)p 1822 2716 4 30 v 23 w(>)13 b(t)p Fv(,)19 b(if)0 2774 y Fl(s)14 b Fn(6\021)f Fl(t)p Fv(.)21 b(By)14 b(abuse)h(of)f(notation)i(w)o(e)e(write)g Fn(T)f Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))13 b(for)i Fn(T)e Fv(\()p Fn(F)f([)c(f)p Ff(2)p Fn(g)p Fl(;)g Fn(V)t Fv(\),)14 b(in)o(terpreting)f Ff(2)i Fv(as)g(a)g(sp)q(ecial)962 2936 y(5)p eop %%Page: 6 6 bop 0 26 a Fv(constan)o(t)17 b(whic)o(h)e(is)h(alw)o(a)o(ys)g(a)o(v)m (ailable)g(but)g(used)h(only)f(for)g(the)g(aforemen)o(tioned)f(purp)q(ose.)73 107 y(Let)24 b Fn(!)g Fv(b)q(e)g(a)g(binary)g(relation)f(on)i(terms,)f(i.e.,) f Fn(!)h(\022)j(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))15 b Fn(\002)h(T)d Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))o(.)44 b(The)24 b(re\015exiv)o(e)0 167 y(transitiv)o(e)16 b(closure)h(of)g Fn(!)g Fv(is)g(denoted)g(b)o(y)g Fn(!)859 149 y Fr(\003)879 167 y Fv(.)24 b(If)16 b Fl(s)g Fn(!)1055 149 y Fr(\003)1090 167 y Fl(t)p Fv(,)g(w)o(e)h(sa)o(y)g(that)h Fl(s)f Fi(r)n(e)n(duc)n(es)g Fv(to)g Fl(t)g Fv(and)h(w)o(e)f(call)0 228 y Fl(t)i Fv(a)i Fi(r)n(e)n(duct)e Fv(of)h Fl(s)p Fv(.)32 b(W)l(e)19 b(write)g Fl(s)h Fn( )g Fl(t)f Fv(if)g Fl(t)h Fn(!)f Fl(s)p Fv(;)i(lik)o(ewise)d(for)i Fl(s)1227 209 y Fr(\003)1236 228 y Fn( )p Fl(t)p Fv(.)31 b(The)20 b(transitiv)o(e)f(closure)g(of)h Fn(!)0 288 y Fv(is)f(denoted)g(b)o(y)g Fn(!)360 270 y Fm(+)389 288 y Fv(,)h(and)g Fn($)f Fv(denotes)g(the)g (symmetric)c(closure)k(of)h Fn(!)f Fv(\(i.e.,)f Fn($)h Fv(=)g Fn(!)g([)g( )p Fv(\).)30 b(The)0 348 y(re\015exiv)o(e)14 b(transitiv)o(e)g (closure)i(of)g Fn($)g Fv(is)f(called)g Fi(c)n(onversion)i Fv(and)f(denoted)g(b)o(y)g(=.)21 b(If)15 b Fl(s)f Fv(=)g Fl(t)p Fv(,)h(then)g Fl(s)h Fv(and)0 408 y Fl(t)h Fv(are)h Fi(c)n(onvertible)p Fv(.)28 b(Tw)o(o)18 b(terms)e Fl(t)650 415 y Fm(1)669 408 y Fl(;)8 b(t)709 415 y Fm(2)746 408 y Fv(are)18 b Fi(joinable)p Fv(,)h(denoted)e(b)o(y)h Fl(t)1298 415 y Fm(1)1333 408 y Fn(#)e Fl(t)1392 415 y Fm(2)1412 408 y Fv(,)h(if)g(there)g(exists)g(a)i(term)c Fl(t)1929 415 y Fm(3)0 468 y Fv(suc)o(h)e(that)i Fl(t)229 475 y Fm(1)262 468 y Fn(!)312 450 y Fr(\003)345 468 y Fl(t)363 475 y Fm(3)405 450 y Fr(\003)413 468 y Fn( )f Fl(t)495 475 y Fm(2)514 468 y Fv(.)20 b(Suc)o(h)14 b(a)g(term)e Fl(t)834 475 y Fm(3)867 468 y Fv(is)h(called)g(a)h Fi(c)n(ommon)h(r)n(e)n(duct)e Fv(of)h Fl(t)1491 475 y Fm(1)1525 468 y Fv(and)g Fl(t)1635 475 y Fm(2)1654 468 y Fv(.)21 b(The)13 b(relation)0 529 y Fn(#)18 b Fv(is)g(called)e Fi(joinability)p Fv(.)27 b(A)18 b(term)e Fl(s)i Fv(is)g(a)g Fi(normal)h(form)e Fv(w.r.t.)26 b Fn(!)p Fv(,)17 b(if)h(there)f(is)h(no)g(term)e Fl(t)i Fv(suc)o(h)f(that)0 589 y Fl(s)d Fn(!)g Fl(t)p Fv(.)21 b(A)15 b(term)g Fl(s)h Fv(has)h(a)g (normal)e(form)h(if)f Fl(s)f Fn(!)902 571 y Fr(\003)936 589 y Fl(t)i Fv(for)g(some)f(normal)h(form)f Fl(t)p Fv(.)21 b(The)16 b(set)g(of)h(all)f(normal)0 649 y(forms)e(of)h Fn(!)f Fv(is)h(denoted)f(b)o (y)g Fl(N)5 b(F)i Fv(\()p Fn(!)p Fv(\).)21 b(The)14 b(relation)g Fn(!)h Fv(is)f Fi(we)n(akly)j(terminating)f Fv(if)e(ev)o(ery)f(term)g(has)i (a)0 709 y(normal)g(form;)f(it)h(is)h Fi(terminating)p Fv(,)h(if)e(there)g (is)h(no)g(in\014nite)f(reduction)g(sequence)g Fl(t)1552 716 y Fm(1)1585 709 y Fn(!)e Fl(t)1666 716 y Fm(2)1700 709 y Fn(!)g Fl(t)1781 716 y Fm(3)1814 709 y Fn(!)h Fl(:)8 b(:)g(:)o Fv(.)0 769 y(The)17 b(relation)g Fn(!)g Fv(is)g Fi(c)n(on\015uent)i Fv(if)e(for)g(all)g(terms)f Fl(s;)8 b(t)999 776 y Fm(1)1018 769 y Fl(;)g(t)1058 776 y Fm(2)1094 769 y Fv(with)18 b Fl(t)1225 776 y Fm(1)1252 751 y Fr(\003)1261 769 y Fn( )f Fl(s)f Fn(!)1417 751 y Fr(\003)1452 769 y Fl(t)1470 776 y Fm(2)1506 769 y Fv(w)o(e)h(ha)o(v)o (e)g Fl(t)1711 776 y Fm(1)1746 769 y Fn(#)e Fl(t)1804 776 y Fm(2)1824 769 y Fv(.)24 b(It)17 b(is)0 829 y(w)o(ell-kno)o(wn)c(that)i Fn(!)f Fv(is)g(con\015uen)o(t)g(if)g(and)h(only)f(if)g(ev)o(ery)f(pair)h(of)h (con)o(v)o(ertible)d(terms)g(is)j(joinable.)20 b(The)0 890 y(relation)15 b Fn(!)h Fv(is)f Fi(lo)n(c)n(al)r(ly)j(c)n(on\015uent)f Fv(if)e(for)h(all)f(terms)f Fl(s;)8 b(t)1032 897 y Fm(1)1052 890 y Fl(;)g(t)1092 897 y Fm(2)1126 890 y Fv(with)16 b Fl(t)1255 897 y Fm(1)1288 890 y Fn( )e Fl(s)g Fn(!)f Fl(t)1470 897 y Fm(2)1505 890 y Fv(w)o(e)j(ha)o(v)o(e)e Fl(t)1706 897 y Fm(1)1740 890 y Fn(#)f Fl(t)1796 897 y Fm(2)1816 890 y Fv(.)21 b(If)15 b Fn(!)0 950 y Fv(is)j(con\015uen)o(t)f(and)i(terminating,)d(it)h(is)h (called)f Fi(c)n(onver)n(gent)p Fv(.)27 b(The)18 b(famous)f(Newman's)g(Lemma) e(states)0 1010 y(that)i(termination)d(and)j(lo)q(cal)f(con\015uence)g(of)h Fn(!)f Fv(imply)d(con\015uence.)73 1091 y(A)20 b Fi(term)i(r)n(ewriting)g (system)f Fv(\(TRS)g(for)g(short\))g(is)f(a)i(pair)e(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))21 b(consisting)g(of)g(a)g(signature)g Fn(F)0 1151 y Fv(and)g(a)f(set)g Fn(R)g(\032)g(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))k Fn(\002)i(T)e Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))20 b(of)g Fi(r)n(ewrite)g(rules)h Fv(or)f Fi(r)n(e)n(duction)h(rules)p Fv(.)33 b(Ev)o(ery)19 b(rewrite)g(rule)0 1211 y(\()p Fl(l)q(;)8 b(r)q Fv(\))20 b(m)o(ust)f(satisfy)h(the)h(follo)o (wing)f(t)o(w)o(o)g(constrain)o(ts:)30 b(\(i\))19 b(the)i(left-hand)f(side)g Fl(l)h Fv(is)f(not)h(a)g(v)m(ariable,)0 1272 y(and)c(\(ii\))f(v)m(ariables)h (o)q(ccurring)g(in)f(the)h(righ)o(t-hand)g(side)g Fl(r)h Fv(also)f(o)q(ccur)g (in)g Fl(l)q Fv(.)22 b(Rewrite)16 b(rules)g(\()p Fl(l)q(;)8 b(r)q Fv(\))17 b(will)0 1332 y(b)q(e)i(denoted)g(b)o(y)f Fl(l)h Fn(!)f Fl(r)q Fv(.)29 b(The)18 b(rewrite)g(rules)h(of)g(a)g(TRS)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))18 b(de\014ne)h(a)g Fi(r)n(ewrite)h(r)n(elation) f Fn(!)1847 1339 y Fr(R)1897 1332 y Fv(on)0 1392 y Fn(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))15 b(as)j(follo)o(ws:)j Fl(s)14 b Fn(!)518 1399 y Fr(R)565 1392 y Fl(t)i Fv(if)g(there)f(exists)h(a)h (rewrite)f(rule)g Fl(l)e Fn(!)g Fl(r)k Fv(in)e Fn(R)p Fv(,)h(a)g (substitution)f Fl(\033)i Fv(and)f(a)0 1452 y(con)o(text)f Fl(C)t Fv([)g(])g(suc)o(h)h(that)g Fl(s)d Fn(\021)h Fl(C)t Fv([)p Fl(l)q(\033)r Fv(])f(and)k Fl(t)c Fn(\021)g Fl(C)t Fv([)p Fl(r)q(\033)r Fv(].)21 b(W)l(e)c(call)f Fl(s)f Fn(!)1307 1459 y Fr(R)1353 1452 y Fl(t)i Fv(a)g Fi(r)n(ewrite)h(step)f Fv(or)g Fi(r)n(e)n(duction)0 1512 y(step)p Fv(.)29 b(An)18 b(instance)h(of)g(a)g (left-hand)g(side)f(of)h(a)g(rewrite)f(rule)g(is)g(a)h Fi(r)n(e)n(dex)g Fv(\(reducible)e(expression\).)28 b(A)0 1573 y(TRS)21 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))20 b(has)h(one)f(of)h(the)f(ab)q(o)o(v)o(e)h (prop)q(erties)f(\(e.g.)33 b(termination\))19 b(if)h(its)g(rewrite)f (relation)h(has)0 1633 y(the)c(resp)q(ectiv)o(e)e(prop)q(ert)o(y)l(.)21 b(Moreo)o(v)o(er,)15 b(it)h(is)g(called)f(in\014nite)g(\(\014nite\))h(if)f (the)h(set)g(of)h(rules)e Fn(R)i Fv(is)f(in\014nite)0 1693 y(\(\014nite\).)34 b(A)20 b(TRS)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))20 b(is)h(called)e Fi(\014nitely)k(br)n(anching)f Fv(if)e(for)h(ev)o (ery)e(rule)h Fl(l)i Fn(!)f Fl(r)h Fn(2)g(R)f Fv(there)f(are)0 1753 y(only)14 b(\014nitely)g(man)o(y)f(di\013eren)o(t)h(rules)g(in)g Fn(R)h Fv(with)g(the)f(same)g(left-hand)g(side)h Fl(l)q Fv(.)20 b(W)l(e)14 b(often)h(simply)d(write)0 1813 y Fn(R)19 b Fv(instead)f(of)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))18 b(if)g(there)g(is)g(no)h(am)o(biguit)o(y)d (ab)q(out)k(the)e(underlying)g(signature)g Fn(F)5 b Fv(.)28 b(A)18 b(rewrite)0 1873 y(rule)g Fl(l)h Fn(!)f Fl(r)i Fv(is)e Fi(left-line)n(ar)j Fv(if)d Fl(l)h Fv(do)q(es)h(not)f(con)o(tain)g(m)o (ultiple)c(o)q(ccurrences)k(of)g(the)f(same)g(v)m(ariable.)28 b(A)0 1934 y Fi(left-line)n(ar)18 b Fv(TRS)f(only)f(con)o(tains)h (left-linear)e(rewrite)h(rules.)21 b(A)16 b(rewrite)g(rule)g Fl(l)f Fn(!)f Fl(r)k Fv(is)e Fi(c)n(ol)r(lapsing)i Fv(if)e Fl(r)0 1994 y Fv(is)i(a)h(v)m(ariable)f(and)g Fl(l)g Fn(!)f Fl(r)j Fv(is)e Fi(duplic)n(ating)h Fv(if)f Fl(r)i Fv(con)o(tains)e(more)f(o)q (ccurrences)h(of)g(some)f(v)m(ariable)h(than)0 2054 y Fl(l)q Fv(.)28 b(A)18 b Fi(non-c)n(ol)r(lapsing)k Fv(\()p Fi(non-duplic)n(ating)p Fv(\))f(TRS)e(do)q(es)g(not)h(con)o(tain)e(collapsing)h(\(duplicating\))f (rules.)0 2114 y(A)e(reduction)f(step)h(is)g(called)f Fi(duplic)n(ating)j Fv(if)d(the)h(rewrite)f(rule)h(applied)f(is)h(duplicating.)21 b(The)16 b(Critical)0 2174 y(P)o(air)21 b(Lemma)e(states)i(that)h(a)f(TRS)h (is)f(lo)q(cally)f(con\015uen)o(t)h(if)f(and)i(only)f(if)g(all)f(its)h (critical)e(pairs)j(are)0 2235 y(con)o(v)o(ergen)o(t,)14 b(i.e.,)h(ha)o(v)o (e)g(a)i(common)d(reduct.)73 2316 y(A)g Fi(p)n(artial)i(or)n(dering)e Fv(\()p Fl(A;)8 b(>)p Fv(\))14 b(is)g(a)h(pair)f(consisting)h(of)g(a)g(set)g Fl(A)f Fv(and)h(a)g(binary)f(transitiv)o(e)g(irre\015exiv)o(e)0 2376 y(relation)j Fl(>)g Fv(on)h Fl(A)p Fv(.)24 b(A)17 b(partial)g(ordering)h (is)f(called)f Fi(wel)r(l-founde)o(d)j Fv(if)e(there)g(are)g(no)h(in\014nite) f(sequences)0 2436 y Fl(a)26 2443 y Fm(1)59 2436 y Fl(>)d(a)137 2443 y Fm(2)170 2436 y Fl(>)g(a)248 2443 y Fm(3)281 2436 y Fl(>)g(:)8 b(:)g(:)15 b Fv(of)i(elemen)o(ts)c(from)i Fl(A)p Fv(.)0 2496 y(W)l(e)j(also)h(need)f(the)h(notion)g(of)g(m)o(ultiset)c (ordering.)29 b(A)18 b Fi(multiset)h Fv(is)g(a)g(collection)e(in)h(whic)o(h)g (elemen)o(ts)0 2556 y(are)g(allo)o(w)o(ed)g(to)g(o)q(ccur)h(more)d(than)j (once.)27 b(If)18 b Fl(A)f Fv(is)h(a)h(set,)f(then)g(the)g(set)g(of)h(all)e (\014nite)h(m)o(ultisets)e(o)o(v)o(er)0 2617 y Fl(A)j Fv(is)g(denoted)h(b)o (y)f Fn(M)p Fv(\()p Fl(A)p Fv(\).)30 b(The)20 b Fi(multiset)h(extension)h Fv(of)e(a)f(partial)h(ordering)g(\()p Fl(A;)8 b(>)p Fv(\))18 b(is)i(the)f(partial)0 2677 y(ordering)d(\()p Fn(M)p Fv(\()p Fl(A)p Fv(\))p Fl(;)8 b(>)406 2659 y Fh(mul)470 2677 y Fv(\))16 b(de\014ned)g(as)h(follo)o(ws:)0 2737 y Fl(M)47 2744 y Fm(1)93 2737 y Fl(>)131 2719 y Fh(mul)221 2737 y Fl(M)268 2744 y Fm(2)312 2737 y Fv(if)22 b Fl(M)410 2744 y Fm(2)456 2737 y Fv(=)k(\()p Fl(M)586 2744 y Fm(1)622 2737 y Fn(n)16 b Fl(X)t Fv(\))g Fn(])g Fl(Y)35 b Fv(for)23 b(some)g(m)o(ultisets)e Fl(X)q(;)8 b(Y)37 b Fn(2)26 b(M)p Fv(\()p Fl(A)p Fv(\))d(that)h(satisfy)f(\(i\))0 2797 y Fn(;)d(6)p Fv(=)g Fl(X)25 b Fn(\022)20 b Fl(M)274 2804 y Fm(1)314 2797 y Fv(and)h(\(ii\))e(for)h(all)g Fl(y)h Fn(2)g Fl(Y)31 b Fv(there)20 b(exists)f(an)i Fl(x)f Fn(2)g Fl(X)k Fv(suc)o(h)c(that)h Fl(x)f(>)g(y)r Fv(.)32 b(Dersho)o(witz)962 2936 y(6)p eop %%Page: 7 7 bop 0 26 a Fv(and)21 b(Manna)g(pro)o(v)o(ed)f(in)g([DM79)q(])g(that)h(the)f (m)o(ultiset)e(extension)i(of)h(a)g(w)o(ell-founded)e(ordering)i(is)f(a)0 86 y(w)o(ell-founded)c(ordering.)73 167 y(A)f Fi(r)n(ewrite)i(or)n(dering)e Fv(is)g(a)g(partial)h(ordering)f(\()p Fn(T)e Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))o Fl(;)g(>)p Fv(\))15 b(whic)o(h)g(is)g Fi(close)n(d)i(under)g(c)n(ontexts)g Fv(\(i.e.,)0 228 y(if)h Fl(s)e(>)h(t)h Fv(then)g Fl(C)t Fv([)p Fl(s)p Fv(])d Fl(>)i(C)t Fv([)p Fl(t)p Fv(])f(for)j(all)e(con)o(texts)h Fl(C)t Fv([)f(]\))g(and)i Fi(close)n(d)h(under)f(substitutions)h Fv(\(i.e.,)c(if)h Fl(s)g(>)g(t)0 288 y Fv(then)22 b Fl(s\033)k(>)e(t\033)f Fv(for)f(all)g(substitutions)h Fl(\033)r Fv(\).)38 b(A)22 b Fi(simpli\014c)n(ation)h Fv(ordering)g Fl(>)f Fv(is)g(a)g(rewrite)g(ordering)0 348 y(p)q(ossessing)c(the)e Fi(subterm)i(pr)n(op)n(erty)p Fv(,)c(i.e.,)g Fl(C)t Fv([)p Fl(t)p Fv(])e Fl(>)i(t)h Fv(for)i(all)f(con)o(texts)f Fl(C)t Fv([)h(])d Fn(6\021)h Ff(2)p Fv(.)0 535 y Fo(3)83 b(Simple)26 b(T)-7 b(ermination)0 665 y Fv(The)19 b(notion)g(\\simple)e(termination")h(w) o(as)i(\014rst)f(in)o(tro)q(duced)f(b)o(y)h(Kurihara)g(and)h(Oh)o(uc)o(hi)d (in)i([K)o(O90].)0 726 y(They)c(call)f(\(ha)o(ving)h(\014nite)f(systems)g(in) h(mind\))e(a)i(term)e(rewriting)i(system)e(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))15 b(\\simply)e(terminat-)0 786 y(ing")18 b(if)g(there)f(is)h(a) g(simpli\014cation)e(ordering)i Fl(>)g Fv(on)g(terms)f(suc)o(h)g(that)i Fn(!)1402 793 y Fr(R)1452 786 y Fv(is)e(con)o(tained)h(in)f Fl(>)p Fv(.)27 b(It)17 b(is)0 846 y(w)o(ell-kno)o(wn)f(that)h(in)f(this)g (case)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(terminating)f(if)h(the)g (underlying)g(signature)h Fn(F)k Fv(is)c(\014nite)e(or)0 906 y(the)d(set)h(of)f(rules)h Fn(R)f Fv(is)h(\014nite;)f(this)h(w)o(as)g (\014rst)f(pro)o(v)o(ed)g(b)o(y)g(Dersho)o(witz)g(in)h([Der79].)19 b(Since)12 b(simpli\014cation)0 966 y(orderings)19 b(are)f(a)h(main)e(to)q (ol)i(for)f(pro)o(ving)h(termination)d(of)j(TRSs,)g(Dersho)o(witz's)f (Theorem)f(is)h(often)0 1027 y(cited,)f(where)h(either)g Fn(F)23 b Fv(has)c(to)g(b)q(e)f(\014nite)g(or)h Fn(R)g Fv(has)g(to)f(b)q(e)h (\014nite.)27 b(T)l(o)19 b(the)f(b)q(est)h(of)f(our)h(kno)o(wledge)0 1087 y(there)g(is)f(no)i(criterion)e(men)o(tioned)f(in)i(the)g(literature)f (whic)o(h)g(subsumes)h(b)q(oth)h(cases.)30 b(W)l(e)19 b(will)f(giv)o(e)0 1147 y(suc)o(h)f(a)h(criterion:)k(if)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))17 b(\\in)o(tro)q(duces)h(only)f(\014nitely)f(man)o(y)f (function)i(sym)o(b)q(ols")g(\(i.e.,)e(if)i(the)g(set)0 1207 y(of)g(all)f(those)g(function)g(sym)o(b)q(ols)g(whic)o(h)f(o)q(ccur)i(on)g (the)f(righ)o(t-hand)h(side)f Fl(r)h Fv(but)g(not)g(on)g(the)f(left-hand)0 1267 y(side)21 b Fl(l)h Fv(of)f(some)g(rule)f Fl(l)j Fn(!)f Fl(r)i Fn(2)f(R)e Fv(is)g(\014nite\),)h(then)f(\\simple)e(termination")h (implies)f(termination.)0 1327 y(Hence)14 b(all)h(results)g(dep)q(ending)g (on)h(Dersho)o(witz's)f(Theorem)f(extend)g(to)i(TRSs)g(whic)o(h)e(in)o(tro)q (duce)h(only)0 1388 y(\014nitely)g(man)o(y)g(function)h(sym)o(b)q(ols.)k(Ho)o (w)o(ev)o(er,)14 b(this)i(implication)e(do)q(es)j(not)g(hold)f(in)g(general,) g(that)g(is,)0 1448 y(\\simple)h(termination")h(do)q(es)i(not)f(imply)e (termination.)28 b(Hence)18 b(the)g(notion)i(\\simple)d(termination")0 1508 y(has)e(the)g(wrong)g(connotation,)h(and)f(w)o(e)f(plead)h(for)f (another)i(de\014nition)e(of)h(this)f(notion)h(\(cf.)20 b([Ohl92a)q(]\).)0 1568 y(W)l(e)c(de\014ne)g(instead:)0 1703 y Fk(De\014nition)i(3.1)24 b Fv(A)16 b(binary)h(relation)g Fn(!)g Fv(on)h Fn(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))16 b(is)h(called)g Fi(simplifying)g Fv(if)g(there)g(exists)f(a)i(sim-)0 1763 y(pli\014cation)e(ordering)i Fl(>)f Fv(suc)o(h)g(that)g Fn(!)g(\022)g Fl(>)p Fv(.)24 b(If)16 b Fl(>)h Fv(is)g(also)h(w)o(ell-founded,)e(then)h Fn(!)g Fv(is)g(called)f Fi(simply)0 1824 y(terminating)p Fv(.)35 b(A)20 b(TRS)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))20 b(is)h(called)e Fi(simplifying)i Fv(\(resp.)34 b Fi(simply)21 b(terminating)p Fv(\))h(if)e(its)g(rewrite)0 1884 y(relation)c Fn(!)229 1891 y Fr(R)277 1884 y Fv(is)g(simplifying)e (\(resp.)21 b(simply)14 b(terminating\).)73 2019 y(This)j(is)g(the)g(same)f (de\014nition)h(as)h(in)f(the)f(recen)o(t)g(pap)q(er)i(of)f(Middeldorp)g(and) h(Gramlic)o(h)c([MG92)q(],)0 2079 y(in)k(whic)o(h)g(it)g(is)g(sho)o(wn)h (that)g(simple)d(termination)h(is)h(an)h(undecidable)f(prop)q(ert)o(y)l(,)g (ev)o(en)f(for)i(one)g(rule)0 2139 y(systems.)73 2220 y(It)h(is)h(easy)g(to)g (pro)o(v)o(e)f(that)h(a)g(TRS)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))20 b(is)h(simplifying)d(\(simply)h(terminating\))g(if)h(and)h(only)0 2280 y(if)d(there)h(exists)f(a)i(\(w)o(ell-founded\))e(simpli\014cation)f (ordering)i Fl(>)g Fv(suc)o(h)g(that)g Fl(l)g(>)g(r)h Fv(for)f(ev)o(ery)f (rewrite)0 2340 y(rule)f Fl(l)f Fn(!)f Fl(r)k Fv(in)e Fn(R)p Fv(.)24 b(The)17 b(next)g(lemm)o(ata)e(giv)o(e)i(useful)g(c)o (haracterizations)f(of)i(simplifyi)o(ng)d(and)j(simply)0 2401 y(terminating)d(TRSs,)h(resp)q(ectiv)o(ely)e(\(cf.)20 b([K)o(O90,)c(K)o(O92]) g(and)h([Zan92)q(]\).)0 2535 y Fk(Lemma)f(3.2)24 b Fv(Let)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(b)q(e)g(a)h(TRS,)f(and)h(let)326 2645 y Fn(F)367 2625 y Fh(ar)q(g)437 2645 y Fv(=)c Fn(f)p Fl(f)5 b Fv(\()p Fl(x)589 2652 y Fm(1)609 2645 y Fl(;)j(:)g(:)g(:)g(;)g(x)747 2652 y Fh(n)770 2645 y Fv(\))13 b Fn(!)h Fl(x)894 2652 y Fh(j)928 2645 y Fn(j)i Fl(f)k Fn(2)14 b(F)1085 2652 y Fh(n)1108 2645 y Fl(;)8 b(n)14 b Fn(2)g Fv(I)-9 b(N)p Fl(;)8 b(j)16 b Fn(2)e(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(n)p Fn(gg)p Fl(:)0 2755 y Fv(Then)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(simplifying)e(if)i(and)h(only)f (if)f Fn(!)926 2735 y Fm(+)926 2768 y Fr(R[F)1008 2758 y Fe(ar)q(g)1077 2755 y Fv(is)h(irre\015exiv)o(e.)962 2936 y(7)p eop %%Page: 8 8 bop 0 26 a Fk(Pro)r(of:)23 b Fv(\\if)s(")17 b(:)23 b(Let)17 b Fn(!)447 6 y Fm(+)447 38 y Fr(R[F)529 29 y Fe(ar)q(g)599 26 y Fv(b)q(e)g(irre\015exiv)o(e.)k(By)16 b(de\014nition)g Fn(!)1255 6 y Fm(+)1255 38 y Fr(R[F)1337 29 y Fe(ar)q(g)1407 26 y Fv(is)g(transitiv)o(e.)23 b(An)o(y)16 b(rewrite)0 86 y(relation)c(is)g (closed)h(under)f(con)o(texts)g(and)h(substitutions.)20 b(Hence)12 b(the)g(same)f(holds)i(for)g Fn(!)1668 66 y Fm(+)1668 99 y Fr(R[F)1750 89 y Fe(ar)q(g)1815 86 y Fv(\(b)q(eing)0 147 y(the)22 b(transitiv)o(e)f(closure)h(of)h(a)g(rewrite)e(relation\).)39 b(Finally)l(,)22 b Fn(!)1230 126 y Fm(+)1230 159 y Fr(R[F)1312 149 y Fe(ar)q(g)1387 147 y Fv(has)h(the)f(subterm)f(prop)q(ert)o(y)0 207 y(b)q(ecause)d Fn(!)232 186 y Fm(+)232 219 y Fr(F)260 210 y Fe(ar)q(g)330 207 y Fv(is)g(con)o(tained)f(in)h Fn(!)712 186 y Fm(+)712 219 y Fr(R[F)794 210 y Fe(ar)q(g)846 207 y Fv(.)26 b(Th)o(us,)18 b Fn(!)1075 186 y Fm(+)1075 219 y Fr(R[F)1157 210 y Fe(ar)q(g)1227 207 y Fv(is)f(a)h(simpli\014cation)e(ordering.)26 b(Since)0 267 y Fn(!)50 274 y Fr(R)82 267 y Fn(\022!)171 246 y Fm(+)171 279 y Fr(R[F)253 270 y Fe(ar)q(g)305 267 y Fv(,)16 b(the)g(TRS)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(simplifying.)0 327 y(\\only-if)s(")21 b(:)28 b(Let)20 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))20 b(b)q(e)g(simplifying)e(w.r.t.)31 b(the)20 b(simpli\014cation)e(ordering)i Fl(>)p Fv(.)32 b(W)l(e)20 b(sho)o(w)g(that)0 387 y Fn(!)50 367 y Fm(+)50 400 y Fr(R[F)132 390 y Fe(ar)q(g)202 387 y Fn(\022)d Fl(>)h Fv(\(hence)e Fn(!)519 367 y Fm(+)519 400 y Fr(R[F)601 390 y Fe(ar)q(g)671 387 y Fv(is)i(irre\015exiv)o(e)d(b)q (ecause)i Fl(>)h Fv(is)f(irre\015exiv)o(e\).)23 b(Since)17 b Fl(>)g Fv(is)g(transitiv)o(e,)0 448 y(it)d(su\016ces)g(to)h(sho)o(w)h(that) f Fl(s)f Fn(!)580 455 y Fr(R[F)662 446 y Fe(ar)q(g)728 448 y Fl(t)g Fv(implies)e Fl(s)i(>)f(t)p Fv(.)21 b(So)15 b(let)f Fl(s)g Fn(!)1287 455 y Fr(R[F)1369 446 y Fe(ar)q(g)1435 448 y Fl(t)p Fv(.)20 b(If)14 b Fl(s)p Fn(!)1607 455 y Fr(R)1654 448 y Fl(t)p Fv(,)g(then)g(clearly)0 508 y Fl(s)19 b(>)f(t)p Fv(.)29 b(If)19 b(otherwise)g Fl(s)f Fn(!)522 515 y Fr(F)550 506 y Fe(ar)q(g)622 508 y Fl(t)g Fv(,)i(that)f(is,)g Fl(s)g Fn(\021)f Fl(C)t Fv([)p Fl(f)5 b Fv(\()p Fl(s)1089 515 y Fm(1)1108 508 y Fl(;)j(:)g(:)g(:)g(;)g(s)1241 515 y Fh(n)1264 508 y Fv(\)])18 b(and)i Fl(t)e Fn(\021)h Fl(C)t Fv([)p Fl(s)1583 515 y Fh(j)1600 508 y Fv(])f(where)h Fl(f)24 b Fn(2)19 b(F)1912 515 y Fh(n)1935 508 y Fv(,)0 568 y(then)14 b(the)g(assertion)g(follo)o(ws)g(from)f(the)h (fact)g(that)g Fl(>)g Fv(has)h(the)f(subterm)f(prop)q(ert)o(y)h(and)g(is)g (closed)g(under)0 628 y(con)o(texts.)21 b Ff(2)0 741 y Fk(Lemma)16 b(3.3)24 b Fv(A)12 b(TRS)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))k(is)h(simply)d (terminating)h(if)i(and)g(only)f(if)h(\()p Fn(F)5 b Fl(;)j Fn(R)s([)t(F)1553 723 y Fh(ar)q(g)1609 741 y Fv(\))k(is)h(terminating.)0 822 y Fk(Pro)r(of:)21 b Fv(\\only-if)s(":)h(Let)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(b)q(e)g(simply)d(terminating,)h(i.e.,)g(there)h (is)h(a)h(w)o(ell-founded)e(simpli\014ca-)0 882 y(tion)c(ordering)h Fl(>)f Fv(suc)o(h)h(that)g Fn(!)589 889 y Fr(R)634 882 y Fn(\022)p Fl(>)p Fv(.)20 b(As)11 b(in)g(the)g(pro)q(of)i(of)f(Lemma)d(3.2,)j(it)f (follo)o(ws)h(that)f Fn(!)1724 889 y Fr(R[F)1806 880 y Fe(ar)q(g)1859 882 y Fn(\022)p Fl(>)p Fv(.)0 942 y(Since)k Fl(>)i Fv(is)f(w)o(ell-founded,)f (\()p Fn(F)5 b Fl(;)j Fn(R)j([)g(F)749 924 y Fh(ar)q(g)804 942 y Fv(\))16 b(is)h(terminating.)0 1002 y(\\if)s(":)j(If)12 b(\()p Fn(F)5 b Fl(;)j Fn(R)s([)s(F)363 984 y Fh(ar)q(g)419 1002 y Fv(\))13 b(is)f(terminating,)f(then)i Fn(!)930 982 y Fm(+)930 1015 y Fr(R[F)1012 1005 y Fe(ar)q(g)1076 1002 y Fv(is)g (particularly)e(irre\015exiv)o(e.)17 b(Hence)12 b Fn(!)1815 982 y Fm(+)1815 1015 y Fr(R[F)1897 1005 y Fe(ar)q(g)0 1063 y Fv(is)i(a)g(simpli\014cation)e(ordering)i(\(cf.)20 b(pro)q(of)15 b(of)f(Lemma)d(3.2\).)21 b(Clearly)l(,)13 b(it)h(is)f(w)o(ell-founded)g(and)i (con)o(tains)0 1123 y Fn(!)50 1130 y Fr(R)82 1123 y Fv(.)21 b Ff(2)73 1235 y Fv(Eviden)o(tly)l(,)14 b(\()p Fn(F)5 b Fl(;)j Fn(R)j([)g(F)524 1217 y Fh(ar)q(g)580 1235 y Fv(\))16 b(is)g(terminating)f Fn(,)h Fv(\()p Fn(F)5 b Fl(;)j Fn(R)i([)i(F)1216 1217 y Fh(ar)q(g)1271 1235 y Fv(\))k(is)g(simply)f(terminating.)0 1348 y Fk(Prop)r(osition)j(3.4)24 b Fv(Let)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(b)q(e)h(a)f(TRS.)g(Then) h(the)f(follo)o(wing)g(implicati)o(ons)e(hold:)25 1455 y(1.)50 b(\()p Fn(F)5 b Fl(;)j Fn(R)j([)g(F)333 1436 y Fh(ar)q(g)388 1455 y Fv(\))16 b(is)h(terminating)349 1515 y Fn(m)25 1575 y Fv(2.)50 b Fn(R)16 b Fv(is)g(simply)e(terminating)349 1635 y Fn(+)25 1695 y Fv(3.)50 b Fn(R)16 b Fv(is)g(simplifying)e(and)j (terminating)290 1756 y Fn(+)328 b(+)25 1816 y Fv(4.)50 b Fn(R)16 b Fv(is)g(simplifying)58 b(4'.)21 b Fn(R)16 b Fv(is)g(terminating)290 1876 y Fn(m)25 1936 y Fv(5.)50 b Fn(!)163 1915 y Fm(+)163 1948 y Fr(R[F)245 1939 y Fe(ar)q(g)313 1936 y Fv(is)16 b(irre\015exiv)o(e)0 2014 y Fk(Pro)r(of:)21 b Fv(The)16 b(equiv)m(alences)f(are)h(pro)o(v)o(ed)g (ab)q(o)o(v)o(e)g(and)h(the)f(implications)e(are)i(trivial.)k Ff(2)73 2126 y Fv(The)c(follo)o(wing)g(examples)f(sho)o(w)h(that)h(none)g(of) f(the)g(missing)f(implications)f(holds:)0 2187 y(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(=)g(\()p Fn(f)p Fl(f)281 2194 y Fm(1)300 2187 y Fl(;)8 b(f)346 2194 y Fm(2)366 2187 y Fl(;)g(f)412 2194 y Fm(3)431 2187 y Fl(;)g(f)477 2194 y Fm(4)497 2187 y Fl(;)g(:)g(:)g(:)o Fn(g)p Fl(;)g Fn(f)p Fl(f)672 2194 y Fm(1)708 2187 y Fn(!)15 b Fl(f)797 2194 y Fm(2)833 2187 y Fn(!)h Fl(f)923 2194 y Fm(3)959 2187 y Fn(!)f Fl(f)1048 2194 y Fm(4)1084 2187 y Fn(!)h Fl(:)8 b(:)g(:)o Fn(g)p Fv(\))18 b(is)f(simplifying)e(but)i(not)h (terminat-)0 2247 y(ing,)c(whereas)h(\()p Fn(F)336 2229 y Fr(0)347 2247 y Fl(;)8 b Fn(R)411 2229 y Fr(0)423 2247 y Fv(\))13 b(=)h(\()p Fn(f)p Fl(f)s(;)8 b(g)r Fn(g)p Fl(;)g Fn(f)p Fl(f)d Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\))14 b Fn(!)f Fl(f)5 b Fv(\()p Fl(g)r Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\)\))p Fn(g)p Fv(\))15 b(is)f(terminating)e(but)i(not)h(simplifying.)0 2307 y(Finally)l(,)j(it)g(is) h(sho)o(wn)h(in)e(the)h(next)f(example)f(that)i(in)g(the)g(ab)q(o)o(v)o(e)f (prop)q(osition)j(3)e(do)q(es)h(not)f(imply)d(1)0 2367 y(\(hence)f(it)h(do)q (es)h(not)g(imply)d(2\).)0 2480 y Fk(Example)i(3.5)24 b Fv(Let)17 b Fn(F)h Fv(=)c Fn(f)p Fl(a;)8 b(g)r(;)g(f)665 2487 y Fm(1)684 2480 y Fl(;)g(f)730 2487 y Fm(2)750 2480 y Fl(;)g(f)796 2487 y Fm(3)815 2480 y Fl(;)g(f)861 2487 y Fm(4)881 2480 y Fl(;)g(:)g(:)g(:)o Fn(g)16 b Fv(and)h(let)663 2569 y Fn(R)d Fv(=)g Fn(f)41 b Fl(f)861 2576 y Fm(1)881 2569 y Fv(\()p Fl(a)p Fv(\))13 b Fn(!)h Fl(f)1046 2576 y Fm(2)1066 2569 y Fv(\()p Fl(g)r Fv(\()p Fl(a)p Fv(\)\))p Fl(;)837 2629 y(f)861 2636 y Fm(2)881 2629 y Fv(\()p Fl(a)p Fv(\))f Fn(!)h Fl(f)1046 2636 y Fm(3)1066 2629 y Fv(\()p Fl(g)r Fv(\()p Fl(a)p Fv(\)\))p Fl(;)837 2689 y(f)861 2696 y Fm(3)881 2689 y Fv(\()p Fl(a)p Fv(\))f Fn(!)h Fl(f)1046 2696 y Fm(4)1066 2689 y Fv(\()p Fl(g)r Fv(\()p Fl(a)p Fv(\)\))p Fl(;)837 2749 y(f)861 2756 y Fm(4)881 2749 y Fv(\()p Fl(a)p Fv(\))f Fn(!)h Fl(f)1046 2756 y Fm(5)1066 2749 y Fv(\()p Fl(g)r Fv(\()p Fl(a)p Fv(\)\))p Fl(;)837 2810 y(:)8 b(:)g(:)352 b Fn(g)p Fl(:)962 2936 y Fv(8)p eop %%Page: 9 9 bop 0 26 a Fv(Then)17 b(w)o(e)f(ha)o(v)o(e)f Fn(F)353 8 y Fh(ar)q(g)423 26 y Fv(=)f Fn(f)p Fl(g)r Fv(\()p Fl(x)p Fv(\))g Fn(!)g Fl(x;)8 b(f)743 33 y Fm(1)763 26 y Fv(\()p Fl(x)p Fv(\))14 b Fn(!)g Fl(x;)8 b(f)981 33 y Fm(2)1000 26 y Fv(\()p Fl(x)p Fv(\))14 b Fn(!)g Fl(x;)8 b(f)1218 33 y Fm(3)1237 26 y Fv(\()p Fl(x)p Fv(\))14 b Fn(!)g Fl(x;)8 b(f)1455 33 y Fm(4)1475 26 y Fv(\()p Fl(x)p Fv(\))13 b Fn(!)i Fl(x;)8 b(:)g(:)g(:)n Fn(g)p Fv(.)22 b(Clearly)l(,)0 86 y Fn(R)d Fv(is)f(terminating.)26 b(F)l(urthermore,)17 b Fn(!)752 66 y Fm(+)752 99 y Fr(R[F)834 89 y Fe(ar)q(g)905 86 y Fv(is)h(irre\015exiv)o(e,)e(that)j(is,)f Fn(R)h Fv(is)f(simplifying.)25 b(But)18 b Fn(R)h Fv(is)0 147 y(not)e(simply)d(terminating)g(b)q(ecause)j (there)e(is)i(the)f(in\014nite)f(reduction)h(sequence)85 257 y Fl(f)109 264 y Fm(1)129 257 y Fv(\()p Fl(a)p Fv(\))d Fn(!)256 264 y Fr(R[F)338 255 y Fe(ar)q(g)404 257 y Fl(f)428 264 y Fm(2)448 257 y Fv(\()p Fl(g)r Fv(\()p Fl(a)p Fv(\)\))g Fn(!)638 264 y Fr(R[F)720 255 y Fe(ar)q(g)786 257 y Fl(f)810 264 y Fm(2)830 257 y Fv(\()p Fl(a)p Fv(\))g Fn(!)957 264 y Fr(R[F)1039 255 y Fe(ar)q(g)1105 257 y Fl(f)1129 264 y Fm(3)1149 257 y Fv(\()p Fl(g)r Fv(\()p Fl(a)p Fv(\)\))h Fn(!)1340 264 y Fr(R[F)1422 255 y Fe(ar)q(g)1488 257 y Fl(f)1512 264 y Fm(3)1531 257 y Fv(\()p Fl(a)p Fv(\))g Fn(!)1659 264 y Fr(R[F)1741 255 y Fe(ar)q(g)1807 257 y Fl(:)8 b(:)g(:)0 367 y Fv(The)20 b(p)q(oin)o(t)g(is)f(that)h Fn(!)446 374 y Fr(R)498 367 y Fv(is)g(con)o(tained)f(in)g(a)h(\(non)h(w)o (ell-founded\))e(simpli\014cation)e(ordering)j(but)g(its)0 427 y(termination)15 b(has)i(to)f(b)q(e)h(pro)o(v)o(ed)e(b)o(y)h(other)g (means)f(than)i(simpli\014cation)d(orderings.)73 562 y(Next)h(w)o(e)g(giv)o (e)g(a)h(su\016cien)o(t)f(criterion)g(for)h(the)f(termination)f(of)j(a)f (simplifying)d(relation)i(on)i(terms.)0 622 y(This)24 b(en)o(tails)e(the)h (termination)f(of)i(a)g(simplifying)d(TRS)i(whic)o(h)g(\\in)o(tro)q(duces)h (only)f(\014nitely)f(man)o(y)0 682 y(function)16 b(sym)o(b)q(ols")g(b)q (ecause)g(the)g(asso)q(ciated)h(rewrite)f(relation)f(satis\014es)i(that)g (criterion.)0 817 y Fk(Prop)r(osition)h(3.6)24 b Fv(Let)17 b Fn(!)f Fv(b)q(e)h(a)g(binary)f(relation)h(on)g Fn(T)12 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))16 b(whic)o(h)g(is)h(simplifyi)o(ng.)j (If)c(for)h(ev)o(ery)0 877 y(reduction)d(sequence)g(the)g(set)h(of)g(all)f (function)g(sym)o(b)q(ols)g(o)q(ccurring)h(in)f(the)g(\(terms)f(of)i(that\))g (reduction)0 937 y(sequence)g(is)h(\014nite,)f(then)h Fn(!)h Fv(is)f(terminating.)0 1018 y Fk(Pro)r(of:)21 b Fv(This)c(follo)o(ws)f(from)f (Krusk)m(al's)h(T)l(ree)g(Theorem,)e(see)i([Gal91)q(,)f(Ohl92a)q(].)21 b Ff(2)0 1153 y Fk(De\014nition)d(3.7)24 b Fv(Let)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(b)q(e)g(a)h(TRS.)f(Set)653 1263 y Fn(F)694 1243 y Fr(0)719 1263 y Fv(=)806 1222 y Fg([)771 1314 y Fh(l)p Fr(!)p Fh(r)q Fr(2R)887 1263 y Fv(\()p Fn(F)5 b Fl(un)p Fv(\()p Fl(r)q Fv(\))11 b Fn(n)g(F)5 b Fl(un)p Fv(\()p Fl(l)q Fv(\)\))p Fl(;)0 1411 y Fv(i.e.,)16 b Fn(F)135 1393 y Fr(0)165 1411 y Fv(consists)i(of)g(all)g(those)g(function)g(sym)o(b)q(ols)f (whic)o(h)g(o)q(ccur)i(at)f(the)g(righ)o(t-hand)g(side)g Fl(r)h Fv(but)f(not)0 1471 y(at)g(the)f(left-hand)g(side)g Fl(l)h Fv(of)g(some)e(rule)g Fl(l)h Fn(!)e Fl(r)i Fn(2)f(R)p Fv(.)25 b(W)l(e)17 b(sa)o(y)g(that)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))17 b Fi(intr)n(o)n(duc)n(es)h(only)g(\014nitely)0 1531 y(many)f(function)i(symb) n(ols)d Fv(if)g(the)g(set)g Fn(F)748 1513 y Fr(0)775 1531 y Fv(is)g(\014nite.)73 1666 y(Clearly)l(,)i(if)f(the)h(signature)h Fn(F)k Fv(or)c(the)f(set)g(of)h(rules)f Fn(R)g Fv(of)h(a)f(TRS)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))18 b(is)g(\014nite,)g(then)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))0 1726 y(b)q(elongs)19 b(to)g(the)f(class)g(of)g (TRSs)h(whic)o(h)f(in)o(tro)q(duce)f(only)h(\014nitely)f(man)o(y)g(function)h (sym)o(b)q(ols.)26 b(There-)0 1786 y(fore,)15 b(eac)o(h)g(result)g(stated)h (for)g(TRSs)g(whic)o(h)e(in)o(tro)q(duce)h(only)h(\014nitely)e(man)o(y)g (function)h(sym)o(b)q(ols)f(holds)0 1847 y(particularly)h(for)i(TRSs)f(with)h (\014nite)e(signature)i(and)g(for)f(\014nite)g(TRSs.)0 1982 y Fk(Prop)r(osition)i(3.8)24 b Fv(Let)15 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))15 b(b)q(e)g(a)h(simplifying)c(TRS.)j(If)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))15 b(in)o(tro)q(duces)g(only)f(\014nitely)g(man) o(y)0 2042 y(function)i(sym)o(b)q(ols,)f(then)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(terminating.)0 2123 y Fk(Pro)r(of:)21 b Fv(The)16 b(pro)q(of)i(uses)e(Prop)q(osition)i(3.6)e(and)h(can)g(b)q(e)f (found)h(in)f([Ohl92a].)21 b Ff(2)73 2258 y Fv(No)o(w)15 b(it)g(is)f(easy)h (to)h(sho)o(w)g(that)f(the)g(notions)h(\\simple)d(termination",\\simpl)o (ifyi)o(ng)g(and)i(terminat-)0 2318 y(ing",)20 b(and)g(\\simplifying")e (coincide)g(for)i(the)f(class)h(of)g(TRSs)g(whic)o(h)f(in)o(tro)q(duce)g (only)g(\014nitely)f(man)o(y)0 2378 y(function)e(sym)o(b)q(ols.)0 2513 y Fk(Prop)r(osition)i(3.9)24 b Fv(Let)17 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))17 b(b)q(e)g(a)h(TRS)f(whic)o(h)g(in)o(tro)q(duces)g (only)f(\014nitely)g(man)o(y)g(function)h(sym-)0 2573 y(b)q(ols.)22 b(Then)16 b(the)g(follo)o(wing)g(statemen)o(ts)f(are)h(equiv)m(alen)o(t:)60 2696 y(1.)24 b(\()p Fn(F)5 b Fl(;)j Fn(R)j([)g(F)342 2677 y Fh(ar)q(g)398 2696 y Fv(\))16 b(is)g(terminating.)60 2797 y(2.)24 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(simply)e(terminating.)962 2936 y(9)p eop %%Page: 10 10 bop 60 26 a Fv(3.)24 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(simplifying)e(and)i(terminating.)60 128 y(4.)24 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(simplifying.)60 230 y(5.)24 b Fn(!)172 209 y Fm(+)172 242 y Fr(R[F)254 232 y Fe(ar)q(g)322 230 y Fv(is)16 b(irre\015exiv)o(e.)0 352 y Fk(Pro)r(of:)21 b Fv(By)16 b(Prop)q(osition)h(3.4,)f(it)g(su\016ces)g(to)g(sho)o(w)h(4)g Fn(\))f Fv(1:)0 412 y(If)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))17 b(is)g(simplifying)e(and)j(in)o(tro)q(duces)g(only)f(\014nitely)f(man)o(y)g (function)h(sym)o(b)q(ols,)f(then)i(the)f(same)0 472 y(holds)k(for)g(\()p Fn(F)5 b Fl(;)j Fn(R)13 b([)i(F)437 454 y Fh(ar)q(g)492 472 y Fv(\).)34 b(Therefore,)21 b(it)f(follo)o(ws)g(from)g(Prop)q(osition)h(3.8)g (that)g(\()p Fn(F)5 b Fl(;)j Fn(R)14 b([)g(F)1821 454 y Fh(ar)q(g)1877 472 y Fv(\))20 b(is)0 533 y(terminating.)g Ff(2)0 720 y Fo(4)83 b(On)27 b(the)g(Mo)r(dularit)n(y)g(of)h(T)-7 b(ermination)0 865 y Fd(4.1)70 b(Basic)22 b(Notions)g(of)h(the)g(Disjoin)n(t)e(Union)i(of)g (TRSs)0 978 y Fv(First)17 b(w)o(e)g(giv)o(e)f(a)i(brief)e(o)o(v)o(erview)g (of)i(the)f(basic)g(notions)h(of)g(disjoin)o(t)f(unions)h(of)f(TRSs)h(\(see)f ([T)l(o)o(y87b,)0 1038 y(Mid90]\).)37 b(Let)22 b(\()p Fn(F)366 1045 y Fm(1)385 1038 y Fl(;)8 b Fn(R)449 1045 y Fm(1)469 1038 y Fv(\))22 b(and)g(\()p Fn(F)665 1045 y Fm(2)685 1038 y Fl(;)8 b Fn(R)749 1045 y Fm(2)768 1038 y Fv(\))22 b(b)q(e)g(t)o(w)o(o)f(disjoin)o(t) h(TRSs,)h(i.e.,)e Fn(F)1447 1045 y Fm(1)1481 1038 y Fn(\\)15 b(F)1565 1045 y Fm(2)1607 1038 y Fv(=)23 b Fn(;)p Fv(.)38 b(The)22 b(TRS)0 1098 y(\()p Fn(F)55 1105 y Fm(1)80 1098 y Fn(])6 b(F)155 1105 y Fm(2)174 1098 y Fl(;)i Fn(R)238 1105 y Fm(1)264 1098 y Fn(])e(R)345 1105 y Fm(2)364 1098 y Fv(\))14 b(is)f(called)g(their)f Fi(disjoint)j(union)p Fv(;)g(w)o(e)e(will)g(simply)e(write)i Fn(R)1490 1105 y Fm(1)1515 1098 y Fn(])6 b(R)1596 1105 y Fm(2)1616 1098 y Fv(.)20 b(Other)14 b(authors)0 1158 y(use)i(the)g(notation)h Fn(R)405 1165 y Fm(1)436 1158 y Fn(\010)11 b(R)528 1165 y Fm(2)564 1158 y Fv(and)17 b(call)f(it)f(the)h(direct)g(sum)f(of)h Fn(R)1220 1165 y Fm(1)1257 1158 y Fv(and)g Fn(R)1393 1165 y Fm(2)1413 1158 y Fv(.)73 1239 y(In)i(the)f(sequel)g(let)g Fl(t)g Fn(2)g(T)12 b Fv(\()p Fn(F)619 1246 y Fm(1)651 1239 y Fn(])h(F)733 1246 y Fm(2)752 1239 y Fl(;)8 b Fn(V)t Fv(\).)26 b(Let)18 b Fl(t)e Fn(\021)g Fl(C)t Fv([)p Fl(t)1117 1246 y Fm(1)1136 1239 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1263 1246 y Fh(n)1286 1239 y Fv(])18 b(with)f Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])16 b Fn(6\021)g Ff(2)p Fv(.)27 b(W)l(e)17 b(write)0 1299 y Fl(t)d Fn(\021)f Fl(C)t Fv([)-8 b([)p Fl(t)161 1306 y Fm(1)179 1299 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)306 1306 y Fh(n)329 1299 y Fv(])-8 b(])15 b(if)h Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])13 b Fn(2)h(T)f Fv(\()p Fn(F)740 1306 y Fh(d)760 1299 y Fl(;)8 b Fn(V)t Fv(\))16 b(and)h Fl(r)q(oot)p Fv(\()p Fl(t)1071 1306 y Fm(1)1091 1299 y Fv(\))p Fl(;)8 b(:)g(:)g(:)f(r)q(oot)p Fv(\()p Fl(t)1321 1306 y Fh(n)1345 1299 y Fv(\))14 b Fn(2)g(F)1467 1301 y Fm(\026)1461 1310 y Fh(d)1498 1299 y Fv(for)i(some)g Fl(d;)1750 1286 y Fv(\026)1742 1299 y Fl(d)e Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)0 1360 y Fv(with)16 b Fl(d)e Fn(6)p Fv(=)211 1346 y(\026)202 1360 y Fl(d)p Fv(.)22 b(The)16 b Fl(t)381 1367 y Fh(j)415 1360 y Fv(are)g(the)g Fi(princip)n(al)g Fv(subterms)g(of)g Fl(t)p Fv(.)21 b(Moreo)o(v)o(er,)15 b(w)o(e)g(de\014ne)h(for)h(all)f Fl(t)187 1502 y(r)q(ank)r Fv(\()p Fl(t)p Fv(\))d(=)413 1429 y Fg(\()467 1472 y Fv(1)658 b(,)15 b(if)h Fl(t)e Fn(2)g(T)e Fv(\()p Fn(F)1396 1479 y Fm(1)1416 1472 y Fl(;)c Fn(V)t Fv(\))i Fn([)i(T)g Fv(\()p Fn(F)1641 1479 y Fm(2)1661 1472 y Fl(;)c Fn(V)t Fv(\))467 1532 y(1)k(+)f Fl(max)p Fn(f)p Fl(r)q(ank)r Fv(\()p Fl(t)816 1539 y Fh(j)832 1532 y Fv(\))25 b Fn(j)f Fv(1)15 b Fn(\024)e Fl(j)k Fn(\024)d Fl(n)p Fn(g)p Fv(,)h(if)h Fl(t)e Fn(\021)f Fl(C)t Fv([)-8 b([)p Fl(t)1384 1539 y Fm(1)1402 1532 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1529 1539 y Fh(n)1552 1532 y Fv(])-8 b(])0 1646 y(The)21 b(m)o(ultiset)d Fl(S)s Fv(\()p Fl(t)p Fv(\))j(of)g Fi(sp)n(e)n(cial)g Fv(subterms)f(of)i(a)f(term)e Fl(t)i Fv(is)g(de\014ned)g (b)o(y)f Fl(S)s Fv(\()p Fl(t)p Fv(\))i(=)1575 1613 y Fg(S)1610 1656 y Fh(j)r Fr(\025)p Fm(1)1682 1646 y Fl(S)1715 1628 y Fh(j)1733 1646 y Fv(\()p Fl(t)p Fv(\),)f(where)0 1706 y Fl(S)33 1688 y Fm(1)53 1706 y Fv(\()p Fl(t)p Fv(\))13 b(=)h([)p Fl(t)p Fv(])g(\(in)i (order)g(to)g(distinguish)f(m)o(ultisets)f(from)g(sets,)i(w)o(e)f(use)h(brac) o(k)o(ets)f(instead)h(of)g(braces)g(for)0 1766 y(the)g(former\))f(and)402 1908 y Fl(S)435 1887 y Fh(j)r Fm(+1)499 1908 y Fv(\()p Fl(t)p Fv(\))e(=)620 1835 y Fg(\()674 1877 y Fv([)j(])380 b(,)16 b(if)g Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))d(=)h(1)674 1937 y Fl(S)707 1919 y Fh(j)725 1937 y Fv(\()p Fl(t)762 1944 y Fm(1)781 1937 y Fv(\))d Fn([)h Fl(:)c(:)g(:)i Fn([)h Fl(S)1001 1919 y Fh(j)1019 1937 y Fv(\()p Fl(t)1056 1944 y Fh(n)1080 1937 y Fv(\),)k(if)h Fl(t)d Fn(\021)h Fl(C)t Fv([)-8 b([)p Fl(t)1334 1944 y Fm(1)1352 1937 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1479 1944 y Fh(n)1502 1937 y Fv(])-8 b(])0 2049 y(F)l(urthermore,)14 b(w)o(e)i(de\014ne)g(for)g Fl(d)e Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)p Fv(:)22 b Fl(S)855 2056 y Fh(d)875 2049 y Fv(\()p Fl(t)p Fv(\))14 b(=)g([)p Fl(s)h Fn(j)h Fl(s)e Fn(2)g Fl(S)s Fv(\()p Fl(t)p Fv(\))i Fl(;)24 b(r)q(oot)p Fv(\()p Fl(s)p Fv(\))15 b Fn(2)f(F)1552 2056 y Fh(d)1572 2049 y Fv(].)0 2109 y(The)k(topmost)f(homogeneous)h(part)g(of)g Fl(t)p Fv(,)f(denoted)h(b)o(y)f Fl(top)p Fv(\()p Fl(t)p Fv(\),)h(is)f (obtained)h(from)f Fl(t)g Fv(b)o(y)g(replacing)g(all)0 2169 y(principal)e(subterms)g(with)i Ff(2)p Fv(,)f(i.e.,)524 2307 y Fl(top)p Fv(\()p Fl(t)p Fv(\))e Fn(\021)711 2234 y Fg(\()765 2277 y Fl(t)199 b Fv(if)41 b Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))13 b(=)h(1)765 2337 y Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])41 b(if)g Fl(t)13 b Fn(\021)h Fl(C)t Fv([)-8 b([)p Fl(t)1213 2344 y Fm(1)1230 2337 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1358 2344 y Fh(n)1381 2337 y Fv(])-8 b(])0 2452 y(W)l(e)16 b(further)g(de\014ne)g Fn(T)429 2434 y Fh()19 b Fn(/)f Fl(<)9 b(t)634 1260 y Fm(1)653 1253 y Fl(;)f(:)g(:)g(:)f(;)h(t)780 1260 y Fh(n)813 1253 y Fl(>)18 b Fv(if)g Fl(t)934 1260 y Fh(i)965 1253 y Fn(\021)f Fl(t)1039 1260 y Fh(j)1075 1253 y Fv(whenev)o(er)g Fl(s)1314 1260 y Fh(i)1346 1253 y Fn(\021)g Fl(s)1425 1260 y Fh(j)1443 1253 y Fv(,)i(for)f(all)g(1)g Fn(\024)f Fl(i)g(<)h(j)i Fn(\024)d Fl(n)p Fv(.)0 1313 y(If)f Fl(<)6 b(s)116 1320 y Fm(1)136 1313 y Fl(;)i(:)g(:)g(:)g(;)g(s)269 1320 y Fh(n)299 1313 y Fl(>)16 b Fn(/)h Fl(<)6 b(t)471 1320 y Fm(1)491 1313 y Fl(;)i(:)g(:)g(:)f(;)h(t)618 1320 y Fh(n)647 1313 y Fl(>)17 b Fv(and)h Fl(<)6 b(t)860 1320 y Fm(1)879 1313 y Fl(;)i(:)g(:)g(:)g(;)g(t)1007 1320 y Fh(n)1036 1313 y Fl(>)17 b Fn(/)f Fl(<)6 b(s)1213 1320 y Fm(1)1233 1313 y Fl(;)i(:)g(:)g(:)g(;)g(s)1366 1320 y Fh(n)1396 1313 y Fl(>)p Fv(,)16 b(w)o(e)g(write)h Fl(<)6 b(s)1729 1320 y Fm(1)1749 1313 y Fl(;)i(:)g(:)g(:)f(;)h(s)1881 1320 y Fh(n)1911 1313 y Fl(>)0 1373 y Fn(1)14 b Fl(<)6 b(t)126 1380 y Fm(1)144 1373 y Fl(;)i(:)g(:)g(:)g(;)g(t)272 1380 y Fh(n)300 1373 y Fl(>)p Fv(.)73 1454 y(The)18 b(follo)o(wing)g(facts)g(\(their)f(pro)q(ofs)i(are)f (straigh)o(tforw)o(ard\))h(will)e(b)q(e)h(hea)o(vily)e(used)j(in)e(the)h (sequel)0 1514 y(without)f(b)q(eing)f(explicitly)d(men)o(tioned.)73 1637 y Fn(\017)24 b Fv(If)18 b Fl(s)g Fn(!)264 1619 y Fh(o)301 1637 y Fl(t)p Fv(,)h(then)f Fl(s)h Fn(\021)e Fl(C)t Fn(f)-17 b(f)p Fl(s)658 1644 y Fm(1)678 1637 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)810 1644 y Fh(n)833 1637 y Fn(g)-17 b(g)p Fl(;)8 b(t)18 b Fn(\021)g Fl(C)1020 1619 y Fr(0)1031 1637 y Fn(h)-8 b(h)p Fl(s)1084 1644 y Fh(i)1096 1649 y Fj(1)1116 1637 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1248 1644 y Fh(i)1260 1648 y Fe(m)1292 1637 y Fn(i)-8 b(i)19 b Fv(for)g(some)f (con)o(texts)g Fl(C)t Fn(f)p Fl(;)8 b(:)g(:)g(:)f(;)h Fn(g)p Fv(,)122 1697 y Fl(C)161 1679 y Fr(0)172 1697 y Fn(h)p Fl(;)g(:)g(:)g(:)g(;)g Fn(i)p Fv(,)16 b Fl(i)367 1704 y Fm(1)387 1697 y Fl(;)8 b(:)g(:)g(:)f(;)h(i) 513 1704 y Fh(m)560 1697 y Fn(2)14 b(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(n)p Fn(g)p Fv(,)16 b(and)h(terms)e Fl(s)1104 1704 y Fm(1)1124 1697 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1256 1704 y Fh(n)1280 1697 y Fv(.)22 b(If)16 b Fl(s)e Fn(!)1452 1679 y Fh(o)1485 1697 y Fl(t)i Fv(is)h(not)g (destructiv)o(e)d(at)122 1757 y(lev)o(el)g(1,)i(then)g Fl(t)e Fn(\021)f Fl(C)521 1739 y Fr(0)532 1757 y Fn(f)-17 b(f)p Fl(s)588 1764 y Fh(i)600 1769 y Fj(1)620 1757 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)752 1764 y Fh(i)764 1768 y Fe(m)796 1757 y Fn(g)-17 b(g)p Fv(.)73 1859 y Fn(\017)24 b Fv(If)e Fl(C)t Fn(f)-17 b(f)p Fl(s)272 1866 y Fm(1)292 1859 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)424 1866 y Fh(n)447 1859 y Fn(g)-17 b(g)26 b(!)556 1841 y Fh(o)600 1859 y Fl(C)639 1841 y Fr(0)650 1859 y Fn(h)-8 b(h)p Fl(s)703 1866 y Fh(i)715 1871 y Fj(1)735 1859 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)867 1866 y Fh(i)879 1870 y Fe(m)910 1859 y Fn(i)-8 b(i)p Fv(,)25 b(then)e Fl(C)t Fn(f)p Fl(t)1179 1866 y Fm(1)1197 1859 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1325 1866 y Fh(n)1348 1859 y Fn(g)25 b(!)1448 1841 y Fh(o)1492 1859 y Fl(C)1531 1841 y Fr(0)1542 1859 y Fn(h)p Fl(t)1579 1866 y Fh(i)1591 1871 y Fj(1)1610 1859 y Fl(;)8 b(:)g(:)g(:)g(;)g (t)1738 1866 y Fh(i)1750 1870 y Fe(m)1781 1859 y Fn(i)23 b Fv(b)o(y)f(an)122 1919 y(application)13 b(of)h(the)f(same)f(rule)h(for)h(all) f(terms)e Fl(t)1006 1926 y Fm(1)1026 1919 y Fl(;)d(:)g(:)g(:)f(;)h(t)1153 1926 y Fh(n)1190 1919 y Fv(with)13 b Fl(<)6 b(s)1365 1926 y Fm(1)1384 1919 y Fl(;)i(:)g(:)g(:)f(;)h(s)1516 1926 y Fh(n)1545 1919 y Fl(>)14 b Fn(1)g Fl(<)6 b(t)1723 1926 y Fm(1)1741 1919 y Fl(;)i(:)g(:)g(:)g(;)g(t)1869 1926 y Fh(n)1897 1919 y Fl(>)p Fv(.)73 2021 y Fn(\017)24 b Fv(If)c Fl(s)h Fn(!)269 2003 y Fh(i)304 2021 y Fl(t)p Fv(,)g(then)f Fl(s)h Fn(\021)g Fl(C)t Fv([)-8 b([)p Fl(s)658 2028 y Fm(1)676 2021 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)808 2028 y Fh(j)826 2021 y Fl(;)g(:)g(:)g(:)g(;)g(s)959 2028 y Fh(n)982 2021 y Fv(])-8 b(])20 b(and)h Fl(t)f Fn(\021)h Fl(C)t Fv([)p Fl(s)1295 2028 y Fm(1)1314 2021 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1446 2003 y Fr(0)1446 2033 y Fh(j)1464 2021 y Fl(;)g(:)g(:)g(:)g(;)g(s)1597 2028 y Fh(n)1620 2021 y Fv(])20 b(for)h(some)e(con-)122 2081 y(text)g Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])p Fl(;)g(j)22 b Fn(2)f(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(n)p Fn(g)19 b Fv(and)i(terms)e Fl(s)1015 2088 y Fm(1)1034 2081 y Fl(;)8 b(:)g(:)g(:)g(;)g (s)1167 2088 y Fh(n)1190 2081 y Fl(;)g(s)1235 2063 y Fr(0)1235 2093 y Fh(j)1273 2081 y Fv(with)20 b Fl(s)1411 2088 y Fh(j)1450 2081 y Fn(!)g Fl(s)1543 2063 y Fr(0)1543 2093 y Fh(j)1561 2081 y Fv(.)33 b(If)19 b Fl(s)i Fn(!)1754 2063 y Fh(i)1788 2081 y Fl(t)f Fv(is)f(not)122 2141 y(destructiv)o(e)c(at)h(lev)o(el)e(2,)i(then)h Fl(t)c Fn(\021)h Fl(C)t Fv([)-8 b([)p Fl(s)874 2148 y Fm(1)891 2141 y Fl(;)8 b(:)g(:)g(:)g(;)g(s)1024 2123 y Fr(0)1024 2154 y Fh(j)1042 2141 y Fl(;)g(:)g(:)g(:)f(;)h(s)1174 2148 y Fh(n)1198 2141 y Fv(])-8 b(].)73 2264 y(A)15 b(prop)q(ert)o(y)g Fn(P)20 b Fv(of)15 b(TRSs)h(is)f(called)g Fi(mo)n(dular)f Fv(if)h(for)h(all)e (disjoin)o(t)h(TRSs)h Fn(R)1469 2271 y Fm(1)1504 2264 y Fv(and)g Fn(R)1640 2271 y Fm(2)1676 2264 y Fv(their)e(disjoin)o(t)0 2324 y(union)j Fn(R)178 2331 y Fm(1)209 2324 y Fn(])11 b(R)295 2331 y Fm(2)331 2324 y Fv(has)17 b(the)f(prop)q(ert)o(y)h Fn(P)j Fv(if)c(and)g(only)h(if)e(b)q(oth)j Fn(R)1206 2331 y Fm(1)1242 2324 y Fv(and)f Fn(R)1379 2331 y Fm(2)1415 2324 y Fv(ha)o(v)o(e)f(the)g(prop) q(ert)o(y)g Fn(P)t Fv(.)21 b(Up)0 2384 y(to)c(the)f(end)g(of)g(this)g (section,)g Fn(!)g Fv(denotes)g Fn(!)855 2391 y Fr(R)885 2396 y Fj(1)903 2391 y Fr(]R)957 2396 y Fj(2)976 2384 y Fv(.)0 2549 y Fd(4.2)70 b(Su\016cien)n(t)21 b(Conditions)h(for)i(the)e(Mo)r(dularit)n(y)h (of)g(T)-6 b(ermination)0 2662 y Fv(W)l(e)22 b(next)f(tac)o(kle)f(the)i (question)g(of)g(su\016cien)o(t)e(conditions)i(for)g(the)g(mo)q(dularit)o(y)e (of)i(termination)f(of)0 2723 y(TRSs.)30 b(W)l(e)18 b(b)q(egin)h(b)o(y)g (describing)f(a)h(pro)q(of)h(tec)o(hnique)d(whic)o(h)i(is)f(often)h (applicable)f(and)i(whic)o(h)e(will)0 2783 y(subsequen)o(tly)23 b(b)q(e)g(used)h(in)g(di\013eren)o(t)f(v)m(arian)o(ts.)43 b(Let)24 b Fn(R)1116 2790 y Fm(1)1136 2783 y Fl(;)8 b Fn(R)1200 2790 y Fm(2)1244 2783 y Fv(b)q(e)24 b(t)o(w)o(o)f(disjoin)o(t)g(TRSs)h(that)h(ha)o (v)o(e)950 2936 y(11)p eop %%Page: 12 12 bop 0 26 a Fv(the)18 b(same)g(sp)q(ecial)g(prop)q(ert)o(y)h(\(e.g.)28 b(they)18 b(are)h(non-duplicating\))f(and)i(that)f(are)f(terminating.)27 b(If)18 b(w)o(e)0 86 y(w)o(an)o(t)j(to)g(sho)o(w)h(mo)q(dularit)o(y)d(of)j (termination)d(for)i(TRSs)h(with)f(this)g(sp)q(ecial)f(prop)q(ert)o(y)l(,)i (w)o(e)e(ha)o(v)o(e)g(to)0 147 y(\014rst)f(guaran)o(tee)h(that)f Fn(R)g Fv(=)f Fn(R)597 154 y Fm(1)630 147 y Fn(])13 b(R)718 154 y Fm(2)757 147 y Fv(inherits)18 b(the)h(sp)q(ecial)f(prop)q(ert)o(y)h (from)f(its)h(constituen)o(t)f(TRSs)0 207 y(\(otherwise)23 b(w)o(e)g(only)g(ha)o(v)o(e)g(a)g(su\016cien)o(t)f(condition)i(for)f(the)g (preserv)m(ation)h(of)f(termination)f(under)0 267 y(disjoin)o(t)16 b(union)h(since)f(w)o(e)g(cannot)h(infer)f(the)h(preserv)m(ation)f(of)h (termination)e(if)h(more)g(than)h(t)o(w)o(o)f(TRSs)0 327 y(are)g(com)o (bined\).)i(In)e(order)f(to)h(pro)o(v)o(e)f(that)h(the)f(disjoin)o(t)g(union) h Fn(R)e Fv(=)g Fn(R)1359 334 y Fm(1)1388 327 y Fn(])c(R)1473 334 y Fm(2)1508 327 y Fv(is)16 b(again)g(terminating,)0 387 y(w)o(e)g(pro)q(ceed)g(b)o(y)g(con)o(tradiction.)k(So)d(supp)q(ose)h(that)e (there)g(is)g(an)h(in\014nite)e Fn(R)i Fv(rewrite)e(deriv)m(ation)724 495 y Fl(D)h Fv(:)d Fl(s)830 502 y Fm(1)864 495 y Fn(!)g Fl(s)950 502 y Fm(2)984 495 y Fn(!)g Fl(s)1070 502 y Fm(3)1104 495 y Fn(!)h Fl(:)8 b(:)g(:)0 602 y Fv(W.l.o.g.)23 b(w)o(e)17 b(ma)o(y)f(further)h (assume)f(that)i Fl(D)h Fv(is)e(an)h(in\014nite)e Fn(R)p Fv(-deriv)m(ation)i (of)f(minimal)d(rank)j(\(where)0 663 y Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))e(=)f Fl(r)q(ank)r Fv(\()p Fl(s)398 670 y Fm(1)418 663 y Fv(\)\),)i(i.e.,)e(an)o(y)i Fn(R)p Fv(-deriv)m(ation)h(of)f(smaller)f (rank)h(is)g(\014nite.)21 b(Let)c Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))e(=)f Fl(k)r Fv(.)22 b(By)0 723 y(our)16 b(assumptions,)g(it)g(follo)o (ws)f(that)i(for)f(all)g(indices)f Fl(j)s Fv(,)g Fl(r)q(ank)r Fv(\()p Fl(s)1192 730 y Fh(j)1211 723 y Fv(\))e(=)h Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))j(and)g Fl(r)q(oot)p Fv(\()p Fl(s)1720 730 y Fh(j)1739 723 y Fv(\))c Fn(2)i(F)1855 730 y Fh(d)1891 723 y Fv(for)0 783 y(some)e Fl(d)h Fn(2)h(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)14 b Fv(\(i.e.,)f(the)h(terms)f(all)h(ha)o(v)o(e)f(the)h(same)g (rank)g(and)h(their)f(ro)q(ot)h(sym)o(b)q(ols)e(ha)o(v)o(e)h(the)g(same)0 843 y(color\).)21 b(In)15 b(particular,)g Fn(!)g Fv(is)g(terminating)e(on)j Fn(T)939 825 y Fh()d Fv(1)0 288 y(and)23 b(supp)q(ose)g(that)f(the)g (claim)e(holds)j(for)f(an)o(y)g(term)e Fl(v)k Fv(with)e Fl( )r Fv(\()p Fl(v)r Fv(\))g Fl(<)i(k)r Fv(.)39 b(Let)22 b Fl(t)i Fn(\021)f Fl(C)t Fv([)-8 b([)p Fl(t)1762 295 y Fm(1)1780 288 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1907 295 y Fh(n)1930 288 y Fv(])-8 b(])0 348 y(and)21 b(let)f Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))i Fn(2)g(F)430 355 y Fh(d)471 348 y Fv(for)f(some)e Fl(d)j Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)p Fv(.)35 b(By)20 b(Lemma)f(4.4,)j(there)e (exists)g(a)h Fl(u)g Fn(2)h Fl(S)1752 355 y Fh(d)1772 348 y Fv(\()p Fl(s)p Fv(\))f(with)0 408 y Fl(r)q(ank)r Fv(\()p Fl(u)p Fv(\))14 b(=)h Fl( )r Fv(\()p Fl(t)p Fv(\))h(suc)o(h)g(that)h Fl(u)d Fn(!)652 390 y Fr(\003)652 420 y(R)699 408 y Fl(t)p Fv(.)22 b(Since)16 b Fl(t)899 390 y Fr(0)925 408 y Fn(6\021)e Fl(t)i Fv(there)g(is)h(a)g Fl(j)g Fn(2)e(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)g(n)p Fn(g)16 b Fv(suc)o(h)h(that)g Fl(t)1755 390 y Fr(0)1781 408 y Fn(2)e Fl(S)s Fv(\()p Fl(t)1899 415 y Fh(j)1916 408 y Fv(\).)0 468 y(Let)97 455 y(\026)88 468 y Fl(d)g Fv(=)g Fn(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)k(n)g(f)p Fl(d)p Fn(g)p Fv(.)23 b(Then)17 b(b)o(y)g(Lemma)d(4.2)k(there)e(is)h(a)g Fl(u)1157 450 y Fr(0)1184 468 y Fn(2)e Fl(S)1268 470 y Fm(\026)1262 479 y Fh(d)1282 468 y Fv(\()p Fl(u)p Fv(\))g Fn(\022)f Fl(S)1452 470 y Fm(\026)1446 479 y Fh(d)1467 468 y Fv(\()p Fl(s)p Fv(\))i(suc)o(h)h (that)h Fl(u)1790 450 y Fr(0)1816 468 y Fn(!)1866 450 y Fr(\003)1866 481 y(R)1913 468 y Fl(t)1931 475 y Fh(j)0 529 y Fv(b)q(ecause)f Fl(r)q(oot)p Fv(\()p Fl(t)305 536 y Fh(j)324 529 y Fv(\))e Fn(2)h(F)449 531 y Fm(\026)443 539 y Fh(d)463 529 y Fv(.)24 b(Clearly)l(,)16 b Fl(r)q(ank)r Fv(\()p Fl(u)833 510 y Fr(0)845 529 y Fv(\))f Fl(<)g(r)q(ank)r Fv(\()p Fl(u)p Fv(\),)i(therefore)g Fl( )r Fv(\()p Fl(t)1411 536 y Fh(j)1428 529 y Fv(\))e Fl(<)h( )r Fv(\()p Fl(t)p Fv(\).)23 b(No)o(w)17 b(if)g Fl(t)1819 510 y Fr(0)1845 529 y Fn(\021)e Fl(t)1917 536 y Fh(j)1935 529 y Fv(,)0 589 y(then)h(the)g(lemm)o(a)d(follo)o(ws.)21 b(If)16 b(otherwise)g Fl(t)814 571 y Fr(0)839 589 y Fn(6\021)d Fl(t)909 596 y Fh(j)927 589 y Fv(,)j(then)g(b)o(y)f(the)h(induction)f(h)o(yp)q(othesis)h Fl( )r Fv(\()p Fl(t)1746 571 y Fr(0)1757 589 y Fv(\))e Fl(<)g( )r Fv(\()p Fl(t)1913 596 y Fh(j)1930 589 y Fv(\))0 649 y(and)j(hence)e Fl( )r Fv(\()p Fl(t)301 631 y Fr(0)312 649 y Fv(\))f Fl(<)g( )r Fv(\()p Fl(t)468 656 y Fh(j)485 649 y Fv(\))g Fl(<)g( )r Fv(\()p Fl(t)p Fv(\).)20 b Ff(2)73 778 y Fv(In)f(the)g(rest)f(of)i(this)f(section,)f Fn(C)683 785 y Fr(E)726 778 y Fv(will)g(denote)h(the)g(TRS)g Fn(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)8 b(y)r Fv(\))17 b Fn(!)i Fl(x)p Fv(,)f Fl(C)t(ons)p Fv(\()p Fl(x;)8 b(y)r Fv(\))18 b Fn(!)g Fl(y)r Fn(g)0 838 y Fv(and)f Fn(V)j Fv(will)15 b(denote)h Fn(f)p Fl(z)r Fn(g)g Fv(if)g(not)g(indicated)g(otherwise.)21 b(Recall)15 b Fn(h)p Fl(t)1249 845 y Fm(1)1268 838 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1396 845 y Fh(n)1419 838 y Fn(i)14 b(!)1502 818 y Fm(+)1502 850 y Fr(C)1521 856 y Fb(E)1557 838 y Fl(t)1575 845 y Fh(j)1609 838 y Fv(for)j(an)o(y)f(index)f Fl(j)s Fv(.)73 919 y(The)d(follo)o(wing)f(de\014nition)h(states)g(ho)o(w)g(the)f Fl(d)p Fv(-information)h(con)o(tained)f(in)h(a)g(term)e Fl(s)h Fv(with)h Fl(r)q(oot)p Fv(\()p Fl(s)p Fv(\))j Fn(2)0 979 y(F)36 986 y Fh(d)72 979 y Fv(has)i(to)g(b)q(e)f(rearranged)h(for)g(the)f(purp)q (ose)h(of)f(pro)o(ving)h(Theorem)d(4.12.)0 1109 y Fk(De\014nition)k(4.7)24 b Fv(Let)e Fl(s)j Fn(2)f(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))22 b(with)g Fl(r)q(ank)r Fv(\()p Fl(s)p Fv(\))j(=)f Fl(k)g Fv(and)g Fl(r)q(oot)p Fv(\()p Fl(s)p Fv(\))h Fn(2)f(F)1541 1116 y Fh(d)1561 1109 y Fv(.)40 b(Let)23 b Fn(\037)f Fv(b)q(e)h(some)0 1169 y(arbitrary)15 b(but)h(\014xed)f(total)g(ordering)g(on)h Fn(T)d Fv(\()p Fn(F)h(])9 b(f)p Fl(C)t(ons)p Fn(g)p Fl(;)f Fn(V)t Fv(\).)20 b(W)l(e)15 b(de\014ne)g(\(where)1600 1156 y(\026)1592 1169 y Fl(d)f Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)h(n)g(f)p Fl(d)p Fn(g)p Fv(\):)46 1279 y Fl(L)85 1249 y Fm(\026)79 1258 y Fh(d)79 1291 y Fm(1)100 1279 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)17 b(i)46 1351 y Fl(L)79 1331 y Fh(d)79 1364 y Fm(1)100 1351 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fl(S)s(or)q(t)p Fv(\()p Fn(f)p Fl(t)17 b Fn(j)f Fl(t)d Fn(2)h Fl(S)595 1358 y Fh(d)615 1351 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(r)q(ank)r Fv(\()p Fl(t)p Fv(\))14 b(=)f(1)p Fn(g)p Fv(\))46 1429 y Fl(L)85 1399 y Fm(\026)79 1408 y Fh(d)79 1441 y Fm(2)100 1429 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)p Fl(L)333 1408 y Fh(d)333 1441 y Fm(1)354 1429 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(L)476 1399 y Fm(\026)470 1408 y Fh(d)470 1441 y Fm(1)490 1429 y Fv(\()p Fl(s)p Fv(\))p Fn(i)46 1506 y Fl(L)79 1486 y Fh(d)79 1518 y Fm(2)100 1506 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fl(S)s(or)q(t)p Fv(\()p Fn(f)p Fl(C)t Fv([)p Fl(L)514 1477 y Fm(\026)508 1486 y Fh(d)508 1518 y Fm(1)528 1506 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(:)g(:)g(:)f(;)h(L)737 1477 y Fm(\026)731 1486 y Fh(d)731 1518 y Fm(1)752 1506 y Fv(\()p Fl(s)p Fv(\)])15 b Fn(j)h Fl(t)e Fn(2)g Fl(S)981 1513 y Fh(d)1001 1506 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(r)q(ank)r Fv(\()p Fl(t)p Fv(\))13 b(=)h(2)p Fl(;)8 b(t)14 b Fn(\021)f Fl(C)t Fv([)-8 b([)p Fl(t)1517 1513 y Fm(1)1535 1506 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1662 1513 y Fh(n)1685 1506 y Fv(])-8 b(])o Fn(g)p Fv(\))46 1584 y Fl(L)85 1554 y Fm(\026)79 1563 y Fh(d)79 1596 y Fm(3)100 1584 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)p Fl(L)333 1563 y Fh(d)333 1596 y Fm(2)354 1584 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(L)476 1554 y Fm(\026)470 1563 y Fh(d)470 1596 y Fm(2)490 1584 y Fv(\()p Fl(s)p Fv(\))p Fn(i)46 1661 y Fl(L)79 1640 y Fh(d)79 1673 y Fm(3)100 1661 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fl(S)s(or)q(t)p Fv(\()p Fn(f)p Fl(C)t Fv([)p Fl(L)514 1632 y Fm(\026)508 1640 y Fh(d)508 1673 y(r)q(ank)q Fm(\()p Fh(t)611 1678 y Fj(1)628 1673 y Fm(\))644 1661 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(:)g(:)g(:)f(;)h(L)853 1632 y Fm(\026)847 1640 y Fh(d)847 1673 y(r)q(ank)q Fm(\()p Fh(t)950 1677 y Fe(n)971 1673 y Fm(\))987 1661 y Fv(\()p Fl(s)p Fv(\)])16 b Fn(j)g Fl(t)d Fn(2)h Fl(S)1216 1668 y Fh(d)1236 1661 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(r)q(ank)r Fv(\()p Fl(t)p Fv(\))14 b(=)f(3)p Fl(;)8 b(t)14 b Fn(\021)g Fl(C)t Fv([)-8 b([)p Fl(t)1753 1668 y Fm(1)1770 1661 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1898 1668 y Fh(n)1921 1661 y Fv(])-8 b(])n Fn(g)p Fv(\))214 1728 y(.)214 1745 y(.)214 1761 y(.)0 1834 y Fl(L)39 1805 y Fm(\026)33 1813 y Fh(d)33 1846 y(k)q Fr(\000)p Fm(2)100 1834 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)p Fl(L)333 1813 y Fh(d)333 1846 y(k)q Fr(\000)p Fm(3)400 1834 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(L)522 1805 y Fm(\026)516 1813 y Fh(d)516 1846 y(k)q Fr(\000)p Fm(3)583 1834 y Fv(\()p Fl(s)p Fv(\))p Fn(i)0 1915 y Fl(L)33 1894 y Fh(d)33 1927 y(k)q Fr(\000)p Fm(2)100 1915 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fl(S)s(or)q(t)p Fv(\()p Fn(f)p Fl(C)t Fv([)p Fl(L)514 1885 y Fm(\026)508 1894 y Fh(d)508 1927 y(r)q(ank)q Fm(\()p Fh(t)611 1932 y Fj(1)628 1927 y Fm(\))644 1915 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(:)g(:)g(:)f(;)h(L)853 1885 y Fm(\026)847 1894 y Fh(d)847 1927 y(r)q(ank)q Fm(\()p Fh(t)950 1931 y Fe(n)971 1927 y Fm(\))987 1915 y Fv(\()p Fl(s)p Fv(\)])16 b Fn(j)g Fl(t)d Fn(2)h Fl(S)1216 1922 y Fh(d)1236 1915 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(r)q(ank)r Fv(\()p Fl(t)p Fv(\))14 b(=)f Fl(k)h Fn(\000)c Fv(2)p Fl(;)e(t)14 b Fn(\021)g Fl(C)t Fv([)-8 b([)p Fl(t)1841 1922 y Fm(1)1858 1915 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1986 1922 y Fh(n)2009 1915 y Fv(])-8 b(])n Fn(g)p Fv(\))0 2000 y Fl(L)39 1971 y Fm(\026)33 1980 y Fh(d)33 2013 y(k)q Fr(\000)p Fm(1)100 2000 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)p Fl(L)333 1980 y Fh(d)333 2013 y(k)q Fr(\000)p Fm(2)400 2000 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(L)522 1971 y Fm(\026)516 1980 y Fh(d)516 2013 y(k)q Fr(\000)p Fm(2)583 2000 y Fv(\()p Fl(s)p Fv(\))p Fn(i)0 2073 y Fl(L)33 2052 y Fh(d)33 2085 y(k)q Fr(\000)p Fm(1)100 2073 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)17 b(i)45 2154 y Fl(L)84 2124 y Fm(\026)78 2133 y Fh(d)78 2166 y(k)100 2154 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)17 b(i)45 2231 y Fl(L)78 2210 y Fh(d)78 2243 y(k)100 2231 y Fv(\()p Fl(s)p Fv(\))41 b(=)g Fn(h)p Fl(C)t Fv([)p Fl(L)392 2202 y Fm(\026)386 2210 y Fh(d)386 2243 y(r)q(ank)q Fm(\()p Fh(s)492 2248 y Fj(1)510 2243 y Fm(\))525 2231 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(:)g(:)g(:)g(;)g(L)735 2202 y Fm(\026)729 2210 y Fh(d)729 2243 y(r)q(ank)q Fm(\()p Fh(s)835 2247 y Fe(n)856 2243 y Fm(\))872 2231 y Fv(\()p Fl(s)p Fv(\)])p Fn(i)58 b Fv(if)16 b Fl(s)d Fn(\021)h Fl(C)t Fv([)-8 b([)p Fl(s)1240 2238 y Fm(1)1258 2231 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1390 2238 y Fh(n)1414 2231 y Fv(])-8 b(])0 2336 y(where)20 b Fl(S)s(or)q(t)p Fv(\()p Fn(f)p Fl(t)304 2343 y Fm(1)323 2336 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)451 2343 y Fh(n)474 2336 y Fn(g)p Fv(\))20 b(=)h Fn(h)p Fl(t)634 2344 y Fh(\031)q Fm(\(1\))703 2336 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)830 2344 y Fh(\031)q Fm(\()p Fh(n)p Fm(\))902 2336 y Fn(i)20 b Fv(suc)o(h)g(that)h Fl(t)1183 2344 y Fh(\031)q Fm(\()p Fh(j)r Fm(\))1258 2336 y Fn(\037)8 b Fl(t)1323 2344 y Fh(\031)q Fm(\()p Fh(j)r Fm(+1\))1455 2336 y Fv(for)21 b(1)g Fn(\024)f Fl(j)k(<)d(n)p Fv(.)33 b(As)20 b(w)o(e)0 2396 y(shall)f(see)g(later,)h(the)f(sorting)h(pro)q (cess)g(is)g(necessary)f(in)g(order)h(to)g(cop)q(e)f(with)h(non)g (left-linear)e(rules:)0 2456 y(the)e(succession)g(of)h(the)f(listed)f(elemen) o(ts)e(of)k(a)g(set)f(has)h(to)f(b)q(e)h(uniquely)e(determined.)0 2586 y Fk(Example)h(4.8)24 b Fv(Let)c Fn(R)460 2593 y Fm(1)500 2586 y Fv(=)g Fn(f)p Fl(F)7 b Fv(\()p Fl(x;)h(x;)g(B)778 2593 y Fm(2)796 2586 y Fv(\))19 b Fn(!)h Fl(x;)8 b(G)p Fv(\()p Fl(x)p Fv(\))20 b Fn(!)f Fl(x;)8 b(H)t Fv(\()p Fl(x)p Fv(\))19 b Fn(!)h Fl(A;)8 b(A)19 b Fn(!)g Fl(B)1617 2593 y Fh(j)1655 2586 y Fn(j)h Fl(j)j Fn(2)d Fc(I)-7 b(N)q Fn(g)19 b Fv(and)0 2646 y(let)d Fn(R)113 2653 y Fm(2)147 2646 y Fv(=)e Fn(f)p Fl(f)5 b Fv(\()p Fl(x)p Fv(\))14 b Fn(!)g Fl(x)p Fn(g)p Fv(.)21 b(F)l(or)c(the)f(term)f Fl(s)f Fn(\021)g Fl(F)7 b Fv(\()p Fl(f)e Fv(\()p Fl(G)p Fv(\()p Fl(f)g Fv(\()p Fl(B)1112 2653 y Fm(1)1132 2646 y Fv(\)\)\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\))p Fl(;)j(f)d Fv(\()p Fl(H)t Fv(\()p Fl(a)p Fv(\)\)\))16 b(w)o(e)g(ha)o(v)o(e)g(\(note)g(that)0 2706 y Fl(d)e Fv(=)g(1\))366 2811 y Fl(L)399 2791 y Fm(2)399 2823 y(1)419 2811 y Fv(\()p Fl(s)p Fv(\))g(=)158 b Fn(h)17 b(i)158 b Fl(L)936 2791 y Fm(1)936 2823 y(1)956 2811 y Fv(\()p Fl(s)p Fv(\))14 b(=)f Fn(h)p Fl(A;)8 b(B)1197 2818 y Fm(1)1217 2811 y Fn(i)950 2936 y Fv(15)p eop %%Page: 16 16 bop 366 26 a Fl(L)399 6 y Fm(2)399 39 y(2)419 26 y Fv(\()p Fl(s)p Fv(\))14 b(=)41 b Fn(h)p Fl(L)625 8 y Fm(1)625 39 y(1)646 26 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(L)762 8 y Fm(2)762 39 y(1)781 26 y Fv(\()p Fl(s)p Fv(\))p Fn(i)42 b Fl(L)936 6 y Fm(1)936 39 y(2)956 26 y Fv(\()p Fl(s)p Fv(\))14 b(=)f Fn(h)p Fl(H)t Fv(\()p Fl(L)1197 6 y Fm(2)1197 39 y(1)1218 26 y Fv(\()p Fl(s)p Fv(\)\))p Fn(i)366 99 y Fl(L)399 78 y Fm(2)399 111 y(3)419 99 y Fv(\()p Fl(s)p Fv(\))h(=)41 b Fn(h)p Fl(L)625 81 y Fm(1)625 111 y(2)646 99 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(L)762 81 y Fm(2)762 111 y(2)781 99 y Fv(\()p Fl(s)p Fv(\))p Fn(i)42 b Fl(L)936 78 y Fm(1)936 111 y(3)956 99 y Fv(\()p Fl(s)p Fv(\))14 b(=)f Fn(h)p Fl(G)p Fv(\()p Fl(L)1191 78 y Fm(2)1191 111 y(2)1212 99 y Fv(\()p Fl(s)p Fv(\)\))p Fn(i)366 172 y Fl(L)399 151 y Fm(2)399 184 y(4)419 172 y Fv(\()p Fl(s)p Fv(\))h(=)41 b Fn(h)p Fl(L)625 153 y Fm(1)625 184 y(3)646 172 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(L)762 153 y Fm(2)762 184 y(3)781 172 y Fv(\()p Fl(s)p Fv(\))p Fn(i)42 b Fl(L)936 151 y Fm(1)936 184 y(4)956 172 y Fv(\()p Fl(s)p Fv(\))14 b(=)f Fn(h)k(i)366 244 y Fl(L)399 224 y Fm(2)399 256 y(5)419 244 y Fv(\()p Fl(s)p Fv(\))d(=)158 b Fn(h)17 b(i)158 b Fl(L)936 224 y Fm(1)936 256 y(5)956 244 y Fv(\()p Fl(s)p Fv(\))14 b(=)f Fn(h)p Fl(F)7 b Fv(\()p Fl(L)1192 224 y Fm(2)1192 256 y(4)1212 244 y Fv(\()p Fl(s)p Fv(\))p Fl(;)h(L)1328 224 y Fm(2)1328 256 y(2)1348 244 y Fv(\()p Fl(s)p Fv(\))p Fl(;)g(L)1464 224 y Fm(2)1464 256 y(3)1483 244 y Fv(\()p Fl(s)p Fv(\)\))p Fn(i)0 354 y Fv(where)16 b Fn(\037)g Fv(is)g(some)f(total)i(ordering)f(on)h Fn(T)c Fv(\()p Fn(F)i(])d(f)p Fl(C)t(ons)p Fn(g)p Fl(;)c Fn(V)t Fv(\))15 b(suc)o(h)h(that)h Fl(S)s(or)q(t)p Fv(\()p Fn(f)p Fl(A;)8 b(B)1607 361 y Fm(1)1626 354 y Fn(g)p Fv(\))14 b(=)g Fn(h)p Fl(A;)8 b(B)1851 361 y Fm(1)1870 354 y Fn(i)p Fv(.)0 489 y Fk(Lemma)16 b(4.9)24 b Fv(Let)16 b Fl(s)e Fn(2)g(T)f Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))15 b(with)h Fl(r)q(ank)r Fv(\()p Fl(s)p Fv(\))e(=)g Fl(k)k Fv(and)f Fl(r)q(oot)p Fv(\()p Fl(s)p Fv(\))d Fn(2)g(F)1383 496 y Fh(d)1404 489 y Fv(.)0 549 y(Then)i(w)o(e)g(ha)o(v)o(e)g(for)g(all)g Fl(j;)8 b(l)17 b Fv(with)f(1)e Fn(\024)g Fl(j)j(<)c(l)i(<)f(k)r Fv(:)73 672 y Fn(\017)24 b Fl(L)161 645 y Fm(\026)155 654 y Fh(d)155 684 y(l)175 672 y Fv(\()p Fl(s)p Fv(\))14 b Fn(!)300 651 y Fm(+)300 684 y Fr(C)319 690 y Fb(E)355 672 y Fl(L)394 645 y Fm(\026)388 654 y Fh(d)388 684 y(j)409 672 y Fv(\()p Fl(s)p Fv(\))73 782 y Fn(\017)24 b Fl(L)161 755 y Fm(\026)155 764 y Fh(d)155 794 y(l)175 782 y Fv(\()p Fl(s)p Fv(\))14 b Fn(!)300 761 y Fm(+)300 794 y Fr(C)319 800 y Fb(E)355 782 y Fl(L)388 764 y Fh(d)388 794 y(j)409 782 y Fv(\()p Fl(s)p Fv(\))0 905 y(Moreo)o(v)o(er,)h(for)h Fl(t)e Fn(2)g Fl(S)411 912 y Fh(d)431 905 y Fv(\()p Fl(s)p Fv(\))i(,)g Fl(t)d Fn(\021)h Fl(C)t Fn(f)-17 b(f)p Fl(t)712 912 y Fm(1)731 905 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)858 912 y Fh(n)881 905 y Fn(g)-17 b(g)p Fv(,)16 b(and)h Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))d(=)f Fl(k)1292 886 y Fr(0)1320 905 y Fv(w)o(e)j(ha)o(v)o(e)500 1019 y Fl(L)533 999 y Fh(d)533 1032 y(k)552 1022 y Fb(0)566 1019 y Fv(\()p Fl(s)p Fv(\))e Fn(!)691 999 y Fm(+)691 1032 y Fr(C)710 1038 y Fb(E)746 1019 y Fl(C)t Fn(f)p Fl(L)849 990 y Fm(\026)843 999 y Fh(d)843 1032 y(r)q(ank)q Fm(\()p Fh(t)946 1037 y Fj(1)963 1032 y Fm(\))978 1019 y Fv(\()p Fl(s)p Fv(\))p Fl(;)8 b(:)g(:)g(:)g(;)g(L)1188 990 y Fm(\026)1182 999 y Fh(d)1182 1032 y(r)q(ank)q Fm(\()p Fh(t)1285 1036 y Fe(n)1306 1032 y Fm(\))1322 1019 y Fv(\()p Fl(s)p Fv(\))p Fn(g)27 b Fl(:)0 1150 y Fk(Pro)r(of:)21 b Fv(Straigh)o(tforw)o(ard.)h Ff(2)73 1285 y Fv(With)16 b(the)g(aid)g(of)h(De\014nition)f(4.7)h(w)o(e)e(can)i(de\014ne)f (the)g(required)f(transformation)h(function)g(\010)1861 1292 y Fh(d)1881 1285 y Fv(.)0 1420 y Fk(De\014nition)i(4.10)24 b Fv(Let)16 b Fl(s)e Fn(2)g(T)f Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))15 b(with)h Fl(r)q(oot)p Fv(\()p Fl(s)p Fv(\))f Fn(2)f(F)1105 1427 y Fh(d)1125 1420 y Fv(.)0 1480 y(De\014ne)i(\010)186 1462 y Fh(s)186 1492 y(d)220 1480 y Fv(:)e Fn(f)p Fl(t)f Fn(2)h(T)f Fv(\()p Fn(F)446 1487 y Fm(1)476 1480 y Fn(])f(F)557 1487 y Fm(2)576 1480 y Fl(;)c Fn(V)t Fv(\))16 b Fn(j)g Fl( )r Fv(\()p Fl(s;)8 b(t)p Fv(\))13 b Fn(6)p Fv(=)h Fn(1g)f(!)h(T)f Fv(\()p Fn(F)1145 1487 y Fh(d)1176 1480 y Fn(])e(f)p Fl(C)t(ons)p Fn(g)p Fl(;)d Fn(V)t Fv(\))15 b(b)o(y)73 1603 y Fn(\017)24 b Fv(\010)157 1584 y Fh(s)157 1615 y(d)177 1603 y Fv(\()p Fl(t)p Fv(\))14 b Fn(\021)f Fl(t)p Fv(,)j(if)f Fl(t)f Fn(2)g(T)f Fv(\()p Fn(F)565 1610 y Fh(d)585 1603 y Fl(;)8 b Fn(V)t Fv(\))73 1704 y Fn(\017)24 b Fv(\010)157 1686 y Fh(s)157 1717 y(d)177 1704 y Fv(\()p Fl(t)p Fv(\))14 b Fn(\021)f(h)k(i)p Fv(,)f(if)g Fl(t)d Fn(2)h(T)f Fv(\()p Fn(F)608 1706 y Fm(\026)602 1715 y Fh(d)622 1704 y Fl(;)8 b Fn(V)t Fv(\))73 1806 y Fn(\017)24 b Fv(\010)157 1788 y Fh(s)157 1818 y(d)177 1806 y Fv(\()p Fl(t)p Fv(\))14 b Fn(\021)f Fl(C)t Fv([\010)387 1788 y Fh(s)387 1818 y(d)407 1806 y Fv(\()p Fl(t)444 1813 y Fm(1)463 1806 y Fv(\))p Fl(;)8 b(:)g(:)g(:)g(;)g Fv(\010)627 1788 y Fh(s)627 1818 y(d)647 1806 y Fv(\()p Fl(t)684 1813 y Fh(n)707 1806 y Fv(\)],)15 b(if)h Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))e Fn(2)g(F)1054 1813 y Fh(d)1090 1806 y Fv(and)j Fl(t)d Fn(\021)f Fl(C)t Fv([)-8 b([)p Fl(t)1346 1813 y Fm(1)1364 1806 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1491 1813 y Fh(n)1514 1806 y Fv(])-8 b(])73 1908 y Fn(\017)24 b Fv(\010)157 1890 y Fh(s)157 1920 y(d)177 1908 y Fv(\()p Fl(t)p Fv(\))14 b Fn(\021)f Fl(L)338 1881 y Fm(\026)332 1890 y Fh(d)332 1921 y( )q Fm(\()p Fh(s;t)p Fm(\))425 1908 y Fv(\()p Fl(s)p Fv(\),)j(if)g Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))e Fn(2)g(F)807 1910 y Fm(\026)801 1918 y Fh(d)837 1908 y Fv(and)j Fl(t)c Fn(\021)h Fl(C)t Fv([)-8 b([)p Fl(t)1093 1915 y Fm(1)1111 1908 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1238 1915 y Fh(n)1261 1908 y Fv(])-8 b(])o(.)73 2043 y(Notice)14 b(that)i(\010)365 2024 y Fh(s)365 2055 y(d)401 2043 y Fv(is)f(w)o (ell-de\014ned)f(\(cf.)20 b(Lemma)13 b(4.6\).)21 b(Again)16 b(w)o(e)f(will)f(suppress)i(the)f(sup)q(erscript)g Fl(s)0 2103 y Fv(in)h(\010)92 2085 y Fh(s)92 2115 y(d)129 2103 y Fv(if)f(it)h(is)g(clear) g(from)f(the)h(con)o(text.)0 2238 y Fk(Example)g(4.11)24 b Fv(\(Example)15 b(4.8)h(con)o(tin)o(ued\))g(Consider)g(the)g(reduction)g (sequence)188 2348 y Fl(D)g Fv(:)d Fl(s)h Fn(\021)g Fl(s)384 2355 y Fm(1)417 2348 y Fn(!)467 2355 y Fr(R)497 2360 y Fj(1)530 2348 y Fl(s)553 2355 y Fm(2)587 2348 y Fn(!)637 2355 y Fr(R)667 2360 y Fj(2)700 2348 y Fl(s)723 2355 y Fm(3)756 2348 y Fn(!)806 2355 y Fr(R)836 2360 y Fj(1)870 2348 y Fl(s)893 2355 y Fm(4)926 2348 y Fn(!)976 2355 y Fr(R)1006 2360 y Fj(1)1039 2348 y Fl(s)1062 2355 y Fm(5)1096 2348 y Fn(!)1146 2355 y Fr(R)1176 2360 y Fj(1)1209 2348 y Fl(s)1232 2355 y Fm(6)1265 2348 y Fn(!)1315 2355 y Fr(R)1345 2360 y Fj(2)1379 2348 y Fl(s)1402 2355 y Fm(7)1435 2348 y Fn(!)1485 2355 y Fr(R)1515 2360 y Fj(1)1548 2348 y Fl(s)1571 2355 y Fm(8)1605 2348 y Fn(!)1655 2355 y Fr(R)1685 2360 y Fj(2)1718 2348 y Fl(s)1741 2355 y Fm(9)0 2458 y Fv(where)i Fl(s)p Fv(,)g Fn(R)236 2465 y Fm(1)256 2458 y Fv(,)f(and)i Fn(R)422 2465 y Fm(2)458 2458 y Fv(are)g(as)g(in)f(Example)e(4.8)j(and)262 2563 y Fl(s)285 2570 y Fm(2)318 2563 y Fn(\021)d Fl(F)7 b Fv(\()p Fl(f)e Fv(\()p Fl(G)p Fv(\()p Fl(f)g Fv(\()p Fl(B)619 2570 y Fm(1)639 2563 y Fv(\)\)\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\)\))83 b Fl(s)1073 2570 y Fm(3)1106 2563 y Fn(\021)14 b Fl(F)7 b Fv(\()p Fl(f)e Fv(\()p Fl(G)p Fv(\()p Fl(B)1359 2570 y Fm(1)1379 2563 y Fv(\)\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\)\))262 2624 y Fl(s)285 2631 y Fm(4)318 2624 y Fn(\021)14 b Fl(F)7 b Fv(\()p Fl(f)e Fv(\()p Fl(B)514 2631 y Fm(1)533 2624 y Fv(\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\)\))227 b Fl(s)1073 2631 y Fm(5)1106 2624 y Fn(\021)14 b Fl(F)7 b Fv(\()p Fl(f)e Fv(\()p Fl(B)1302 2631 y Fm(1)1321 2624 y Fv(\))p Fl(;)j(f)d Fv(\()p Fl(B)1447 2631 y Fm(1)1467 2624 y Fv(\))p Fl(;)j(f)d Fv(\()p Fl(A)p Fv(\)\))262 2684 y Fl(s)285 2691 y Fm(6)318 2684 y Fn(\021)14 b Fl(F)7 b Fv(\()p Fl(f)e Fv(\()p Fl(B)514 2691 y Fm(1)533 2684 y Fv(\))p Fl(;)j(f)d Fv(\()p Fl(B)659 2691 y Fm(1)679 2684 y Fv(\))p Fl(;)j(f)d Fv(\()p Fl(B)805 2691 y Fm(2)825 2684 y Fv(\)\))187 b Fl(s)1073 2691 y Fm(7)1106 2684 y Fn(\021)14 b Fl(F)7 b Fv(\()p Fl(f)e Fv(\()p Fl(B)1302 2691 y Fm(1)1321 2684 y Fv(\))p Fl(;)j(f)d Fv(\()p Fl(B)1447 2691 y Fm(1)1467 2684 y Fv(\))p Fl(;)j(B)1545 2691 y Fm(2)1565 2684 y Fv(\))262 2744 y Fl(s)285 2751 y Fm(8)318 2744 y Fn(\021)14 b Fl(f)5 b Fv(\()p Fl(B)456 2751 y Fm(1)476 2744 y Fv(\))555 b Fl(s)1073 2751 y Fm(9)1106 2744 y Fn(\021)14 b Fl(B)1196 2751 y Fm(1)950 2936 y Fv(16)p eop %%Page: 17 17 bop 0 26 a Fv(Applying)15 b(\010)246 33 y Fm(1)283 26 y Fv(to)h Fl(D)i Fv(w)o(e)e(obtain)257 131 y(\010)292 138 y Fm(1)312 131 y Fv(\()p Fl(D)q Fv(\))e(:)g(\010)468 138 y Fm(1)488 131 y Fv(\()p Fl(s)p Fv(\))41 b Fn(\021)14 b Fv(\010)678 138 y Fm(1)698 131 y Fv(\()p Fl(s)740 138 y Fm(1)759 131 y Fv(\))g Fn(!)842 111 y Fm(+)842 144 y Fr(C)861 150 y Fb(E)897 131 y Fv(\010)932 138 y Fm(1)952 131 y Fv(\()p Fl(s)994 138 y Fm(2)1014 131 y Fv(\))g Fn(\021)f Fv(\010)1134 138 y Fm(1)1154 131 y Fv(\()p Fl(s)1196 138 y Fm(3)1216 131 y Fv(\))h Fn(!)1299 111 y Fm(+)1299 144 y Fr(C)1318 150 y Fb(E)1354 131 y Fv(\010)1389 138 y Fm(1)1409 131 y Fv(\()p Fl(s)1451 138 y Fm(4)1471 131 y Fv(\))f Fn(\021)h Fv(\010)1591 138 y Fm(1)1611 131 y Fv(\()p Fl(s)1653 138 y Fm(5)1673 131 y Fv(\))590 192 y Fn(\021)g Fv(\010)678 199 y Fm(1)698 192 y Fv(\()p Fl(s)740 199 y Fm(6)759 192 y Fv(\))g Fn(!)842 171 y Fm(+)842 204 y Fr(R)872 209 y Fj(1)889 204 y Fr(]C)932 210 y Fb(E)968 192 y Fv(\010)1003 199 y Fm(1)1023 192 y Fv(\()p Fl(s)1065 199 y Fm(7)1085 192 y Fv(\))g Fn(!)1168 199 y Fr(R)1198 204 y Fj(1)1231 192 y Fv(\010)1266 199 y Fm(1)1286 192 y Fv(\()p Fl(s)1328 199 y Fm(8)1347 192 y Fv(\))g Fn(!)1430 171 y Fm(+)1430 204 y Fr(R)1460 209 y Fj(1)1477 204 y Fr(]C)1520 210 y Fb(E)1556 192 y Fv(\010)1591 199 y Fm(1)1611 192 y Fv(\()p Fl(s)1653 199 y Fm(9)1673 192 y Fv(\))0 302 y(where)i(\010)176 309 y Fm(1)196 302 y Fv(\()p Fl(s)p Fv(\))e Fn(\021)f Fl(F)7 b Fv(\()p Fl(L)414 284 y Fm(2)414 315 y(4)433 302 y Fv(\()p Fl(s)p Fv(\))p Fl(;)h(L)549 284 y Fm(2)549 315 y(2)569 302 y Fv(\()p Fl(s)p Fv(\))p Fl(;)g(L)685 284 y Fm(2)685 315 y(3)705 302 y Fv(\()p Fl(s)p Fv(\)\))16 b(and)277 410 y(\010)312 417 y Fm(1)332 410 y Fv(\()p Fl(s)374 417 y Fm(2)393 410 y Fv(\))e Fn(\021)g Fl(F)7 b Fv(\()p Fl(L)570 392 y Fm(2)570 423 y(4)589 410 y Fv(\()p Fl(s)p Fv(\))p Fl(;)h(L)705 392 y Fm(2)705 423 y(2)725 410 y Fv(\()p Fl(s)p Fv(\))p Fl(;)g(L)841 392 y Fm(2)841 423 y(2)860 410 y Fv(\()p Fl(s)p Fv(\)\))p Fl(;)41 b Fv(\010)1030 417 y Fm(1)1050 410 y Fv(\()p Fl(s)1092 417 y Fm(4)1112 410 y Fv(\))14 b Fn(\021)f Fl(F)7 b Fv(\()p Fl(L)1288 392 y Fm(2)1288 423 y(2)1307 410 y Fv(\()p Fl(s)p Fv(\))p Fl(;)h(L)1423 392 y Fm(2)1423 423 y(2)1443 410 y Fv(\()p Fl(s)p Fv(\))p Fl(;)g(L)1559 392 y Fm(2)1559 423 y(2)1579 410 y Fv(\()p Fl(s)p Fv(\)\))p Fl(;)277 471 y Fv(\010)312 478 y Fm(1)332 471 y Fv(\()p Fl(s)374 478 y Fm(7)393 471 y Fv(\))14 b Fn(\021)g Fl(F)7 b Fv(\()p Fl(L)570 453 y Fm(2)570 483 y(2)589 471 y Fv(\()p Fl(s)p Fv(\))p Fl(;)h(L)705 453 y Fm(2)705 483 y(2)725 471 y Fv(\()p Fl(s)p Fv(\))p Fl(;)g(B)845 478 y Fm(2)864 471 y Fv(\))p Fl(;)98 b Fv(\010)1030 478 y Fm(1)1050 471 y Fv(\()p Fl(s)1092 478 y Fm(8)1112 471 y Fv(\))14 b Fn(\021)f Fl(L)1230 453 y Fm(2)1230 483 y(2)1250 471 y Fv(\()p Fl(s)p Fv(\))p Fl(;)49 b Fv(\010)1409 478 y Fm(1)1429 471 y Fv(\()p Fl(s)1471 478 y Fm(9)1491 471 y Fv(\))14 b Fn(\021)f Fl(B)1613 478 y Fm(1)73 601 y Fv(The)23 b(next)f(theorem)g(sho)o(ws)h(that)h(a)f Fn(R)841 608 y Fm(1)877 601 y Fn(])15 b(R)967 608 y Fm(2)1010 601 y Fv(deriv)m(ation)23 b(starting)g(from)f(some)g(term)f Fl(s)i Fv(with)0 662 y Fl(r)q(oot)p Fv(\()p Fl(s)p Fv(\))18 b Fn(2)f(F)252 669 y Fh(d)291 662 y Fv(can)h(alw)o(a)o(ys)g(b)q(e)g(transformed)g(in)o(to)g(a)g Fn(R)1068 669 y Fh(d)1101 662 y Fn(])13 b(C)1173 669 y Fr(E)1215 662 y Fv(deriv)m(ation)18 b(starting)h(from)e(\010)1782 669 y Fh(d)1802 662 y Fv(\()p Fl(s)p Fv(\).)27 b(In)0 722 y(fact,)16 b(it)g(expresses)f(a)i(more)e(general)h(statemen)o(t.)0 856 y Fk(Theorem)h(4.12)24 b Fv(Let)d Fl(s;)8 b(t)21 b Fn(2)h(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))20 b(suc)o(h)h(that)g Fl(r)q(oot)p Fv(\()p Fl(s)p Fv(\))i Fn(2)f(F)1276 863 y Fh(d)1317 856 y Fv(and)f Fl( )r Fv(\()p Fl(t)p Fv(\))g Fn(6)p Fv(=)h Fn(1)p Fv(.)35 b(Then)21 b(for)g(all)0 916 y Fl(t)18 898 y Fr(0)46 916 y Fn(2)d(T)13 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))17 b(suc)o(h)h(that)g Fl(t)f Fn(!)594 898 y Fr(\003)594 928 y(R)624 933 y Fj(1)641 928 y Fr(]R)695 933 y Fj(2)732 916 y Fl(t)750 898 y Fr(0)779 916 y Fv(it)h(follo)o(ws)g(\010)1027 923 y Fh(d)1047 916 y Fv(\()p Fl(t)p Fv(\))f Fn(!)1170 898 y Fr(\003)1170 928 y(R)1200 934 y Fe(d)1218 928 y Fr(]C)1261 934 y Fb(E)1300 916 y Fv(\010)1335 923 y Fh(d)1356 916 y Fv(\()p Fl(t)1393 898 y Fr(0)1404 916 y Fv(\).)27 b(In)18 b(particular,)g Fl(t)e Fn(!)1852 898 y Fh(o)1852 928 y Fr(R)1882 934 y Fe(d)1920 916 y Fl(t)1938 898 y Fr(0)0 976 y Fv(implies)e(\010)201 983 y Fh(d)221 976 y Fv(\()p Fl(t)p Fv(\))f Fn(!)340 983 y Fr(R)370 989 y Fe(d)404 976 y Fv(\010)439 983 y Fh(d)460 976 y Fv(\()p Fl(t)497 958 y Fr(0)508 976 y Fv(\).)0 1057 y Fk(Pro)r(of:)19 b Fv(W)l(e)11 b(pro)o(v)o(e)f(the)i(theorem)d(b)o(y)i(\(\014nite\))g (induction)g(on)h Fl( )r Fv(\()p Fl(t)p Fv(\),)f(where)g Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))i Fn(\024)h Fl( )r Fv(\()p Fl(t)p Fv(\))f Fn(\024)g Fl(r)q(ank)r Fv(\()p Fl(s)p Fv(\).)0 1117 y(The)j(base)h(case)f Fl( )r Fv(\()p Fl(t)p Fv(\))d(=)h(1)i(is)h(straigh)o(tforw)o(ard)g(b)q (ecause)f Fl( )r Fv(\()p Fl(t)p Fv(\))d(=)h(1)i(implies)e Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))f(=)h(1.)0 1178 y(So)h(let)f Fl( )r Fv(\()p Fl(t)p Fv(\))e(=)i Fl(k)i(>)e Fv(1)h(and)g(supp)q(ose)g(that)g(the)f(theorem) f(holds)i(for)g(all)f Fl(v)h Fn(2)f(T)f Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))13 b(whith)i Fl( )r Fv(\()p Fl(v)r Fv(\))d Fl(<)i(k)r Fv(.)0 1238 y(W)l(e)i(pro)o(v)o(e)f(the)h(induction)g(step)g(b)o(y)g (induction)g(on)h(the)f(length)g Fl(l)h Fv(of)f(the)g(deriv)m(ation)g Fl(t)e Fn(!)1687 1220 y Fh(l)1687 1250 y Fr(R)1717 1255 y Fj(1)1734 1250 y Fr(]R)1788 1255 y Fj(2)1821 1238 y Fl(t)1839 1220 y Fr(0)1850 1238 y Fv(.)0 1298 y(The)23 b(case)g Fl(l)k Fv(=)e(0)f(is)f (trivial,)g(so)h(consider)f Fl(t)i Fn(!)950 1280 y Fh(l)950 1310 y Fr(R)980 1315 y Fj(1)998 1310 y Fr(]R)1052 1315 y Fj(2)1096 1298 y Fl(t)1114 1280 y Fr(00)1161 1298 y Fn(!)1211 1305 y Fr(R)1241 1310 y Fj(1)1258 1305 y Fr(]R)1312 1310 y Fj(2)1357 1298 y Fl(t)1375 1280 y Fr(0)1386 1298 y Fv(.)42 b(By)23 b(the)g(inner)f (induction)0 1358 y(h)o(yp)q(othesis)c(\(on)g Fl(l)q Fv(\))f(\010)417 1365 y Fh(d)438 1358 y Fv(\()p Fl(t)p Fv(\))f Fn(!)560 1340 y Fr(\003)560 1371 y(R)590 1377 y Fe(d)608 1371 y Fr(]C)651 1377 y Fb(E)689 1358 y Fv(\010)724 1365 y Fh(d)745 1358 y Fv(\()p Fl(t)782 1340 y Fr(00)803 1358 y Fv(\).)25 b(It)18 b(remains)e(to)i(b)q(e)g (sho)o(wn)g(that)h(\010)1520 1365 y Fh(d)1540 1358 y Fv(\()p Fl(t)1577 1340 y Fr(00)1598 1358 y Fv(\))d Fn(!)1683 1340 y Fr(\003)1683 1371 y(R)1713 1377 y Fe(d)1731 1371 y Fr(]C)1774 1377 y Fb(E)1813 1358 y Fv(\010)1848 1365 y Fh(d)1868 1358 y Fv(\()p Fl(t)1905 1340 y Fr(0)1916 1358 y Fv(\).)0 1418 y(The)g(case)g Fl(r)q(ank)r Fv(\()p Fl(t)345 1400 y Fr(00)366 1418 y Fv(\))e(=)g(1)j(is)f (straigh)o(tforw)o(ard,)g(so)h(let)f Fl(r)q(ank)r Fv(\()p Fl(t)1170 1400 y Fr(00)1190 1418 y Fv(\))e Fl(>)g Fv(1.)0 1479 y(W)l(e)i(distinguish)g (b)q(et)o(w)o(een)f(the)h(follo)o(wing)g(cases:)0 1539 y(\(i\))g Fl(r)q(oot)p Fv(\()p Fl(t)192 1521 y Fr(00)213 1539 y Fv(\))e Fn(2)g(F)329 1546 y Fh(d)73 1661 y Fn(\017)24 b Fv(If)17 b Fl(t)190 1643 y Fr(00)228 1661 y Fn(!)278 1643 y Fh(o)278 1673 y Fr(R)308 1679 y Fe(d)345 1661 y Fl(t)363 1643 y Fr(0)374 1661 y Fv(,)h(then)g(w)o(e)g(ma)o(y)e(write)i Fl(t)845 1643 y Fr(00)882 1661 y Fn(\021)f Fl(C)t Fv([)-8 b([)p Fl(t)1015 1668 y Fm(1)1033 1661 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1160 1668 y Fh(n)1183 1661 y Fv(])-8 b(])17 b(and)i Fl(t)1335 1643 y Fr(0)1363 1661 y Fn(\021)d Fl(C)1457 1643 y Fr(0)1468 1661 y Fn(h)-8 b(h)p Fl(t)1516 1668 y Fh(i)1528 1673 y Fj(1)1548 1661 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1675 1668 y Fh(i)1687 1672 y Fe(m)1718 1661 y Fn(i)-8 b(i)19 b Fv(for)f(some)122 1721 y(con)o(texts)f Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)e(;)i Fv(],)p Fl(C)544 1703 y Fr(0)554 1721 y Fv([)p Fl(;)g(:)g(:)g(:)f(;)h Fv(],)p Fl(i)722 1728 y Fm(1)740 1721 y Fl(;)g(:)g(:)g(:)g(;)g(i)867 1728 y Fh(m)915 1721 y Fn(2)16 b(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(n)p Fn(g)p Fv(,)17 b(and)h(terms)e Fl(t)1459 1728 y Fm(1)1478 1721 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1606 1728 y Fh(n)1629 1721 y Fv(.)24 b(Applying)17 b(\010)1915 1728 y Fh(d)1935 1721 y Fv(,)122 1781 y(w)o(e)h(obtain)g(\010)384 1788 y Fh(d)405 1781 y Fv(\()p Fl(t)442 1763 y Fr(00)463 1781 y Fv(\))f Fn(\021)g Fl(C)t Fv([\010)643 1788 y Fh(d)662 1781 y Fv(\()p Fl(t)699 1788 y Fm(1)718 1781 y Fv(\))p Fl(;)8 b(:)g(:)g(:)g(;)g Fv(\010)882 1788 y Fh(d)902 1781 y Fv(\()p Fl(t)939 1788 y Fh(n)962 1781 y Fv(\)])18 b(and)h(\010)1145 1788 y Fh(d)1165 1781 y Fv(\()p Fl(t)1202 1763 y Fr(0)1213 1781 y Fv(\))f Fn(\021)e Fl(C)1344 1763 y Fr(0)1356 1781 y Fn(h)p Fv(\010)1410 1788 y Fh(d)1430 1781 y Fv(\()p Fl(t)1467 1788 y Fh(i)1479 1793 y Fj(1)1498 1781 y Fv(\))p Fl(;)8 b(:)g(:)g(:)g(;)g Fv(\010)1662 1788 y Fh(d)1682 1781 y Fv(\()p Fl(t)1719 1788 y Fh(i)1731 1792 y Fe(m)1762 1781 y Fv(\))p Fn(i)p Fv(.)28 b(Th)o(us)122 1841 y(\010)157 1848 y Fh(d)177 1841 y Fv(\()p Fl(t)214 1823 y Fr(00)235 1841 y Fv(\))19 b Fn(!)323 1848 y Fr(R)353 1854 y Fe(d)391 1841 y Fv(\010)426 1848 y Fh(d)447 1841 y Fv(\()p Fl(t)484 1823 y Fr(0)495 1841 y Fv(\))g(b)o(y)g(the)f(same)g (rule)h(b)q(ecause)g Fl(<)10 b(t)1165 1848 y Fm(1)1184 1841 y Fl(;)e(:)g(:)g(:)g(;)g(t)1312 1848 y Fh(n)1345 1841 y Fl(>)18 b Fn(1)h Fl(<)f Fv(\010)1561 1848 y Fh(d)1582 1841 y Fv(\()p Fl(t)1619 1848 y Fm(1)1638 1841 y Fv(\))p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(\010)1801 1848 y Fh(d)1822 1841 y Fv(\()p Fl(t)1859 1848 y Fh(n)1882 1841 y Fv(\))i Fl(>)122 1902 y Fv(\(i.e.,)k(ev)o(en)h(non)i (left-linear)e(rules)h(do)g(not)h(cause)f(trouble\).)73 2003 y Fn(\017)24 b Fv(If)19 b Fl(t)192 1985 y Fr(00)232 2003 y Fn(!)282 1985 y Fh(i)282 2015 y Fr(R)312 2020 y Fj(1)329 2015 y Fr(]R)383 2020 y Fj(2)421 2003 y Fl(t)439 1985 y Fr(0)450 2003 y Fv(,)h(then)f(w)o(e)g(ha)o(v)o(e)g Fl(t)807 1985 y Fr(00)847 2003 y Fn(\021)f Fl(C)t Fv([)-8 b([)p Fl(t)981 2010 y Fm(1)999 2003 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1126 2010 y Fh(j)1144 2003 y Fl(;)g(:)g(:)g(:)g(;)g(t)1272 2010 y Fh(n)1295 2003 y Fv(])-8 b(])18 b(and)i Fl(t)1449 1985 y Fr(0)1479 2003 y Fn(\021)f Fl(C)t Fv([)p Fl(t)1608 2010 y Fm(1)1626 2003 y Fl(;)8 b(:)g(:)g(:)g(;)g(t) 1754 1985 y Fr(0)1754 2015 y Fh(j)1771 2003 y Fl(;)g(:)g(:)g(:)g(;)g(t)1899 2010 y Fh(n)1922 2003 y Fv(],)122 2063 y(where)20 b Fl(t)285 2070 y Fh(j)324 2063 y Fn(!)374 2070 y Fr(R)404 2075 y Fj(1)421 2070 y Fr(]R)475 2075 y Fj(2)515 2063 y Fl(t)533 2045 y Fr(0)533 2076 y Fh(j)551 2063 y Fv(.)34 b(Since)20 b Fl(t)749 2070 y Fh(j)787 2063 y Fn(2)i Fl(S)s Fv(\()p Fl(t)912 2045 y Fr(00)932 2063 y Fv(\))f(and)g Fl(t)1089 2070 y Fh(j)1128 2063 y Fn(6\021)g Fl(t)1206 2045 y Fr(00)1226 2063 y Fv(,)h(it)e(follo)o(ws)g(from)f(Lemma)g (4.6)h(that)122 2123 y Fl( )r Fv(\()p Fl(t)193 2130 y Fh(j)210 2123 y Fv(\))29 b Fl(<)h( )r Fv(\()p Fl(t)397 2105 y Fr(00)417 2123 y Fv(\))25 b(and)h(th)o(us)f Fl( )r Fv(\()p Fl(t)752 2130 y Fh(j)770 2123 y Fv(\))k Fl(<)g( )r Fv(\()p Fl(t)p Fv(\))24 b(\(recall)h(that)g Fl( )r Fv(\()p Fl(t)1343 2105 y Fr(00)1364 2123 y Fv(\))k Fn(\024)g Fl( )r Fv(\()p Fl(t)p Fv(\))24 b(b)o(y)h(Lemma)e (4.5\).)122 2184 y(The)g(outer)h(induction)e(h)o(yp)q(othesis)i(yields)e (\010)1014 2191 y Fh(d)1035 2184 y Fv(\()p Fl(t)1072 2191 y Fh(j)1089 2184 y Fv(\))k Fn(!)1184 2166 y Fr(\003)1184 2196 y(R)1214 2202 y Fe(d)1232 2196 y Fr(]C)1275 2202 y Fb(E)1323 2184 y Fv(\010)1358 2191 y Fh(d)1379 2184 y Fv(\()p Fl(t)1416 2166 y Fr(0)1416 2196 y Fh(j)1433 2184 y Fv(\).)43 b(Therefore,)24 b(\010)1787 2191 y Fh(d)1807 2184 y Fv(\()p Fl(t)1844 2166 y Fr(00)1865 2184 y Fv(\))i Fn(\021)122 2244 y Fl(C)t Fv([\010)210 2251 y Fh(d)229 2244 y Fv(\()p Fl(t)266 2251 y Fm(1)286 2244 y Fv(\))p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(\010)449 2251 y Fh(d)469 2244 y Fv(\()p Fl(t)506 2251 y Fh(j)524 2244 y Fv(\))p Fl(;)g(:)g(:)g(:)g Fv(\010)666 2251 y Fh(d)686 2244 y Fv(\()p Fl(t)723 2251 y Fh(n)746 2244 y Fv(\)])13 b Fn(!)842 2226 y Fr(\003)842 2256 y(R)872 2262 y Fe(d)890 2256 y Fr(]C)933 2262 y Fb(E)969 2244 y Fl(C)t Fv([\010)1057 2251 y Fh(d)1077 2244 y Fv(\()p Fl(t)1114 2251 y Fm(1)1133 2244 y Fv(\))p Fl(;)8 b(:)g(:)g(:)g(;)g Fv(\010)1297 2251 y Fh(d)1317 2244 y Fv(\()p Fl(t)1354 2226 y Fr(0)1354 2256 y Fh(j)1372 2244 y Fv(\))p Fl(;)g(:)g(:)g(:)f Fv(\010)1513 2251 y Fh(d)1534 2244 y Fv(\()p Fl(t)1571 2251 y Fh(n)1594 2244 y Fv(\)])13 b Fn(\021)h Fv(\010)1728 2251 y Fh(d)1748 2244 y Fv(\()p Fl(t)1785 2226 y Fr(0)1796 2244 y Fv(\).)0 2366 y(\(ii\))h Fl(r)q(oot)p Fv(\()p Fl(t)205 2348 y Fr(00)227 2366 y Fv(\))f Fn(2)g(F)349 2368 y Fm(\026)343 2376 y Fh(d)73 2488 y Fn(\017)24 b Fv(If)16 b Fl(t)189 2470 y Fr(0)214 2488 y Fn(\021)d Fl(z)r Fv(,)j(then)g(the)g(assertion)h(follo)o(ws)f(easily)l(.)73 2589 y Fn(\017)24 b Fv(If)c(also)h Fl(r)q(oot)p Fv(\()p Fl(t)401 2571 y Fr(0)413 2589 y Fv(\))g Fn(2)g(F)549 2591 y Fm(\026)543 2600 y Fh(d)563 2589 y Fv(,)g(then)f(b)o(y)g(Lemma)f(4.9)h(\010)1081 2596 y Fh(d)1102 2589 y Fv(\()p Fl(t)1139 2571 y Fr(00)1160 2589 y Fv(\))h Fn(\021)f Fl(L)1298 2562 y Fm(\026)1292 2571 y Fh(d)1292 2603 y( )q Fm(\()p Fh(t)1343 2593 y Fb(00)1364 2603 y Fm(\))1379 2589 y Fv(\()p Fl(s)p Fv(\))h Fn(!)1511 2571 y Fr(\003)1511 2601 y(C)1530 2607 y Fb(E)1574 2589 y Fl(L)1613 2562 y Fm(\026)1607 2571 y Fh(d)1607 2603 y( )q Fm(\()p Fh(t)1658 2593 y Fb(0)1669 2603 y Fm(\))1685 2589 y Fv(\()p Fl(s)p Fv(\))f Fn(\021)h Fv(\010)1861 2596 y Fh(d)1882 2589 y Fv(\()p Fl(t)1919 2571 y Fr(0)1930 2589 y Fv(\))122 2649 y(b)q(ecause)16 b Fl( )r Fv(\()p Fl(t)373 2631 y Fr(0)384 2649 y Fv(\))e Fn(\024)g Fl( )r Fv(\()p Fl(t)541 2631 y Fr(00)561 2649 y Fv(\).)73 2751 y Fn(\017)24 b Fv(If)e(otherwise)g Fl(r)q(oot)p Fv(\()p Fl(t)524 2733 y Fr(0)536 2751 y Fv(\))j Fn(2)f(F)673 2758 y Fh(d)693 2751 y Fv(,)g(then)e Fl(t)866 2733 y Fr(00)912 2751 y Fn(\021)i Fl(C)t Fv([)-8 b([)p Fl(t)1052 2758 y Fm(1)1070 2751 y Fl(;)8 b(:)g(:)g(:)f(;)h(t) 1197 2758 y Fh(n)1220 2751 y Fv(])-8 b(])23 b Fn(!)1313 2758 y Fr(R)1348 2761 y Fj(\026)1343 2768 y Fe(d)1388 2751 y Fl(t)1406 2733 y Fr(0)1440 2751 y Fv(where)f Fl(t)1605 2733 y Fr(0)1640 2751 y Fn(\021)j Fl(t)1722 2758 y Fh(j)1762 2751 y Fv(for)e(some)122 2811 y Fl(j)d Fn(2)d(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)g(n)p Fn(g)p Fv(.)26 b(Since)18 b Fl( )r Fv(\()p Fl(t)644 2793 y Fr(00)664 2811 y Fv(\))f Fn(\024)g Fl( )r Fv(\()p Fl(t)p Fv(\))f Fn(6)p Fv(=)g Fn(1)p Fv(,)i(there)g(is)g(an)g Fl(s)1268 2793 y Fr(0)1297 2811 y Fn(2)f Fl(S)1383 2813 y Fm(\026)1377 2822 y Fh(d)1397 2811 y Fv(\()p Fl(s)p Fv(\))h(with)g Fl(r)q(ank)r Fv(\()p Fl(s)1736 2793 y Fr(0)1748 2811 y Fv(\))f(=)g Fl( )r Fv(\()p Fl(t)1910 2793 y Fr(00)1930 2811 y Fv(\))950 2936 y(17)p eop %%Page: 18 18 bop 122 26 a Fv(suc)o(h)23 b(that)h Fl(s)375 8 y Fr(0)413 26 y Fn(!)463 8 y Fr(\003)463 39 y(R)493 44 y Fj(1)510 39 y Fr(]R)564 44 y Fj(2)609 26 y Fl(t)627 8 y Fr(00)648 26 y Fv(.)42 b(No)o(w)24 b(\010)858 33 y Fh(d)878 26 y Fv(\()p Fl(s)920 8 y Fr(0)932 26 y Fv(\))i Fn(\021)f Fl(L)1080 -1 y Fm(\026)1074 8 y Fh(d)1074 40 y(r)q(ank)q Fm(\()p Fh(s)1180 30 y Fb(0)1192 40 y Fm(\))1208 26 y Fv(\()p Fl(s)p Fv(\))h Fn(\021)g Fl(L)1399 -1 y Fm(\026)1393 8 y Fh(d)1393 40 y( )q Fm(\()p Fh(t)1444 30 y Fb(00)1464 40 y Fm(\))1480 26 y Fv(\()p Fl(s)p Fv(\))f Fn(\021)h Fv(\010)1666 33 y Fh(d)1687 26 y Fv(\()p Fl(t)1724 8 y Fr(00)1744 26 y Fv(\).)43 b(Hence)122 86 y(it)22 b(remains)g(to)h(sho)o(w)h(\010)595 93 y Fh(d)615 86 y Fv(\()p Fl(s)657 68 y Fr(0)669 86 y Fv(\))h Fn(!)763 68 y Fr(\003)763 99 y(R)793 105 y Fe(d)811 99 y Fr(]C)854 105 y Fb(E)902 86 y Fv(\010)937 93 y Fh(d)957 86 y Fv(\()p Fl(t)994 68 y Fr(0)1005 86 y Fv(\).)42 b(Since)22 b Fl(s)1237 68 y Fr(0)1274 86 y Fn(!)1324 68 y Fr(\003)1324 99 y(R)1354 104 y Fj(1)1371 99 y Fr(]R)1425 104 y Fj(2)1470 86 y Fl(t)1488 68 y Fr(0)1499 86 y Fv(,)i Fl(r)q(oot)p Fv(\()p Fl(t)1661 68 y Fr(0)1673 86 y Fv(\))h Fn(2)h(F)1812 93 y Fh(d)1832 86 y Fv(,)e(and)122 147 y Fl(r)q(oot)p Fv(\()p Fl(s)251 129 y Fr(0)263 147 y Fv(\))c Fn(2)h(F)398 149 y Fm(\026)392 157 y Fh(d)412 147 y Fv(,)f(there)f(exists)g(a)i Fl(u)e Fn(2)i Fl(S)888 154 y Fh(d)908 147 y Fv(\()p Fl(s)950 129 y Fr(0)961 147 y Fv(\))f Fn(\022)g Fl(S)1089 154 y Fh(d)1109 147 y Fv(\()p Fl(s)p Fv(\))g(suc)o(h)g (that)g Fl(u)g Fn(!)1511 129 y Fr(\003)1511 159 y(R)1541 164 y Fj(1)1558 159 y Fr(]R)1612 164 y Fj(2)1651 147 y Fl(t)1669 129 y Fr(0)1680 147 y Fv(.)32 b(Ob)o(viously)l(,)122 207 y Fl( )r Fv(\()p Fl(u)p Fv(\))18 b(=)g Fl(r)q(ank)r Fv(\()p Fl(u)p Fv(\))h Fl(<)f(r)q(ank)r Fv(\()p Fl(s)689 189 y Fr(0)701 207 y Fv(\))g(=)h Fl( )r Fv(\()p Fl(t)866 189 y Fr(00)886 207 y Fv(\))g Fn(\024)f Fl( )r Fv(\()p Fl(t)p Fv(\).)28 b(Consequen)o(tly)19 b(it)f(follo)o(ws)h(from)f(the)h(outer)122 267 y(induction)d(h)o(yp)q (othesis)g(that)h(\010)720 274 y Fh(d)740 267 y Fv(\()p Fl(u)p Fv(\))d Fn(!)870 249 y Fr(\003)870 279 y(R)900 285 y Fe(d)918 279 y Fr(]C)961 285 y Fb(E)997 267 y Fv(\010)1032 274 y Fh(d)1052 267 y Fv(\()p Fl(t)1089 249 y Fr(0)1100 267 y Fv(\).)22 b(Ev)o(en)o(tually)l (,)14 b(w)o(e)h(ha)o(v)o(e)h(\(cf.)21 b(Lemma)14 b(4.9\))122 335 y(\010)157 342 y Fh(d)177 335 y Fv(\()p Fl(t)214 317 y Fr(00)235 335 y Fv(\))g Fn(\021)g Fv(\010)356 342 y Fh(d)376 335 y Fv(\()p Fl(s)418 317 y Fr(0)430 335 y Fv(\))f Fn(\021)h Fl(L)554 308 y Fm(\026)548 317 y Fh(d)548 349 y(r)q(ank)q Fm(\()p Fh(s)654 339 y Fb(0)666 349 y Fm(\))682 335 y Fv(\()p Fl(s)p Fv(\))f Fn(!)806 315 y Fm(+)806 347 y Fr(C)825 353 y Fb(E)862 335 y Fl(L)895 317 y Fh(d)895 349 y(r)q(ank)q Fm(\()p Fh(u)p Fm(\))1021 335 y Fv(\()p Fl(s)p Fv(\))h Fn(!)1146 315 y Fm(+)1146 347 y Fr(C)1165 353 y Fb(E)1201 335 y Fv(\010)1236 342 y Fh(d)1256 335 y Fv(\()p Fl(u)p Fv(\))g Fn(!)1386 317 y Fr(\003)1386 348 y(R)1416 354 y Fe(d)1434 348 y Fr(]C)1477 354 y Fb(E)1513 335 y Fv(\010)1548 342 y Fh(d)1568 335 y Fv(\()p Fl(t)1605 317 y Fr(0)1617 335 y Fv(\))0 452 y Ff(2)73 578 y Fv(After)f(one)i(further)f (de\014nition,)f(w)o(e)h(are)g(able)g(to)g(state)h(and)g(pro)o(v)o(e)e(the)h (main)f(result)g(of)i(the)f(pap)q(er.)0 705 y Fk(De\014nition)k(4.13)24 b Fv(A)12 b(TRS)h Fn(R)g Fv(is)f(called)g Fn(C)807 712 y Fr(E)831 705 y Fv(-)p Fi(terminating)i Fv(if)f(the)f Fi(c)n(ol)r(lapsing)k(extende)n (d)e Fv(term)d(rewriting)0 765 y(system)17 b Fn(R)12 b(])h(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)8 b(y)r Fv(\))16 b Fn(!)h Fl(x;)8 b(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))16 b Fn(!)h Fl(y)r Fn(g)g Fv(is)h(terminating,)f(where)h Fl(C)t(ons)g Fv(is)g(some)f(binary)0 825 y(function)f(sym)o(b)q(ol)f(that)i(do)q(es)g(not)f(o)q(ccur)h(in)f(the)g (signature)g(of)h Fn(R)p Fv(.)73 952 y(Clearly)l(,)e(a)i Fn(C)319 959 y Fr(E)342 952 y Fv(-terminating)e(TRS)i(is)f(terminating.)0 1079 y Fk(Theorem)h(4.14)24 b Fv(Let)14 b Fn(R)489 1086 y Fm(1)509 1079 y Fl(;)8 b Fn(R)573 1086 y Fm(2)608 1079 y Fv(b)q(e)14 b(t)o(w)o(o)h(disjoin)o(t)f(terminating)e(TRSs)j(suc)o(h)g(that)g(their)f (disjoin)o(t)g(union)0 1139 y Fn(R)42 1146 y Fm(1)76 1139 y Fn(])f(R)164 1146 y Fm(2)204 1139 y Fv(is)20 b(non-terminating.)32 b(Then)20 b Fn(R)821 1146 y Fm(1)861 1139 y Fv(is)g(not)h Fn(C)1031 1146 y Fr(E)1055 1139 y Fv(-terminating)d(and)j Fn(R)1481 1146 y Fm(2)1521 1139 y Fv(is)f(collapsing)g(or)g(vice)0 1199 y(v)o(ersa.)0 1280 y Fk(Pro)r(of:)h Fv(Let)679 1340 y Fl(D)16 b Fv(:)d Fl(s)h Fn(\021)g Fl(s)875 1347 y Fm(1)908 1340 y Fn(!)g Fl(s)995 1347 y Fm(2)1028 1340 y Fn(!)g Fl(s)1115 1347 y Fm(3)1149 1340 y Fn(!)f Fl(:)8 b(:)g(:)0 1424 y Fv(b)q(e)19 b(an)g(in\014nite)f Fn(R)349 1431 y Fm(1)382 1424 y Fn(])13 b(R)470 1431 y Fm(2)490 1424 y Fv(-deriv)m(ation)19 b(of)g(minimal)c(rank,)20 b(i.e.,)d(an)o(y)i Fn(R)1350 1431 y Fm(1)1383 1424 y Fn(])13 b(R)1471 1431 y Fm(2)1491 1424 y Fv(-deriv)m(ation)19 b(of)g(smaller)0 1484 y(rank)e(is)f(\014nite.)k (Let)d Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))e(=)f Fl(k)r Fv(.)21 b(Hence)15 b Fl(r)q(ank)r Fv(\()p Fl(s)998 1491 y Fh(j)1016 1484 y Fv(\))f(=)g Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))j(for)g(all)f(indices)f Fl(j)s Fv(.)21 b Fl(r)q(oot)p Fv(\()p Fl(s)1792 1491 y Fm(1)1813 1484 y Fv(\))14 b Fn(2)g(F)1929 1491 y Fh(d)0 1545 y Fv(for)22 b(some)f Fl(d)j Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)p Fv(.)38 b(It)22 b(follo)o(ws)g(that)g Fl(r)q(oot)p Fv(\()p Fl(s)951 1552 y Fh(j)970 1545 y Fv(\))h Fn(2)h(F)1105 1552 y Fh(d)1147 1545 y Fv(for)e(an)o(y)g Fl(j)s Fv(.)38 b(In)22 b(particular,)g(there)g(is)f (no)0 1605 y(reduction)15 b(step)h(whic)o(h)e(is)i(destructiv)o(e)e(at)h(lev) o(el)f(1.)21 b(F)l(rom)15 b(the)g(minim)o(alit)n(y)e(assumption)i(on)h Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))0 1665 y(w)o(e)h(conclude)g(that)h(there)f (is)h(no)g(index)f Fl(l)g Fn(2)f Fc(I)-7 b(N)19 b Fv(suc)o(h)e(that)i(the)e (sub)q(deriv)m(ation)h(of)g Fl(D)i Fv(b)q(eginning)e(at)g Fl(s)1936 1672 y Fh(l)0 1725 y Fv(consists)f(only)f(of)g(inner)g(rewrite)f(steps.)22 b(Th)o(us)16 b(there)g(m)o(ust)f(b)q(e)h(in\014nitely)f(man)o(y)f Fn(!)1596 1707 y Fh(o)1596 1738 y Fr(R)1626 1744 y Fe(d)1647 1725 y Fv(-steps)i(in)g Fl(D)q Fv(.)0 1785 y(No)o(w)k(w)o(e)g(apply)h(the)f (function)g(\010)648 1792 y Fh(d)689 1785 y Fv(to)h(the)f(rewrite)g(deriv)m (ation)g Fl(D)i Fv(and)f(obtain)g(a)g Fn(R)1648 1792 y Fh(d)1682 1785 y Fn(])15 b(C)1756 1792 y Fr(E)1800 1785 y Fv(rewrite)0 1846 y(deriv)m(ation)h(\(note)g(that)h(\010)497 1853 y Fh(d)534 1846 y Fv(is)f(w)o(ell-de\014ned)f(on)h Fl(D)q Fv(\))276 1948 y(\010)311 1955 y Fh(d)331 1948 y Fv(\()p Fl(D)q Fv(\))f(:)e(\010)487 1955 y Fh(d)508 1948 y Fv(\()p Fl(s)p Fv(\))h Fn(\021)f Fv(\010)670 1955 y Fh(d)691 1948 y Fv(\()p Fl(s)733 1955 y Fm(1)752 1948 y Fv(\))h Fn(!)835 1928 y Fr(\003)835 1961 y(R)865 1967 y Fe(d)883 1961 y Fr(]C)926 1967 y Fb(E)962 1948 y Fv(\010)997 1955 y Fh(d)1017 1948 y Fv(\()p Fl(s)1059 1955 y Fm(2)1079 1948 y Fv(\))g Fn(!)1162 1928 y Fr(\003)1162 1961 y(R)1192 1967 y Fe(d)1210 1961 y Fr(]C)1253 1967 y Fb(E)1289 1948 y Fv(\010)1324 1955 y Fh(d)1344 1948 y Fv(\()p Fl(s)1386 1955 y Fm(3)1406 1948 y Fv(\))g Fn(!)1489 1928 y Fr(\003)1489 1961 y(R)1519 1967 y Fe(d)1537 1961 y Fr(]C)1580 1967 y Fb(E)1616 1948 y Fl(:)8 b(:)g(:)0 2051 y Fv(By)15 b(Theorem)g(4.12)i(it)e(follo)o(ws)h(that)g (a)h(reduction)e(step)h(of)g(the)g(form)f Fl(s)1335 2058 y Fh(j)1367 2051 y Fn(!)1417 2033 y Fh(o)1417 2064 y Fr(R)1447 2070 y Fe(d)1481 2051 y Fl(s)1504 2058 y Fh(j)r Fm(+1)1583 2051 y Fv(in)h Fl(D)i Fv(corresp)q(onds)0 2111 y(to)i(a)g(reduction)g(step)f (\010)469 2118 y Fh(d)490 2111 y Fv(\()p Fl(s)532 2118 y Fh(j)550 2111 y Fv(\))h Fn(!)639 2118 y Fr(R)669 2124 y Fe(d)709 2111 y Fv(\010)744 2118 y Fh(d)764 2111 y Fv(\()p Fl(s)806 2118 y Fh(j)r Fm(+1)869 2111 y Fv(\))g(in)g(\010)1004 2118 y Fh(d)1024 2111 y Fv(\()p Fl(D)q Fv(\).)33 b(Since)18 b(there)i(are)f(in\014nitely)f (man)o(y)h(outer)0 2172 y(reduction)12 b(steps)h(in)g Fl(D)q Fv(,)g(the)g(deriv)m(ation)f(\010)795 2179 y Fh(d)816 2172 y Fv(\()p Fl(D)q Fv(\))h(consists)g(of)g(in\014nitely)e(man)o(y)h(reduction)g (steps.)20 b(Hence)0 2232 y Fn(R)42 2239 y Fh(d)79 2232 y Fv(is)c(not)g Fn(C)240 2239 y Fr(E)264 2232 y Fv(-terminating.)0 2292 y(Let)96 2279 y(\026)87 2292 y Fl(d)f Fn(2)f(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)j(n)g(f)p Fl(d)p Fn(g)p Fv(.)21 b(Supp)q(ose)d(that)e Fn(R)796 2294 y Fm(\026)790 2303 y Fh(d)827 2292 y Fv(is)g(non-collapsing.)22 b(Then)17 b(for)f(an)o(y)g Fl(s)1532 2299 y Fh(j)1550 2292 y Fn(!)1600 2299 y Fr(R)1632 2292 y Fl(s)1655 2299 y Fh(j)r Fm(+1)1735 2292 y Fv(w)o(e)g(ha)o(v)o(e)73 2408 y Fn(\017)24 b Fl(s)145 2415 y Fh(j)177 2408 y Fn(!)227 2390 y Fh(o)227 2421 y Fr(R)257 2427 y Fe(d)291 2408 y Fl(s)314 2415 y Fh(j)r Fm(+1)394 2408 y Fv(implies)13 b Fl(top)p Fv(\()p Fl(s)666 2415 y Fh(j)685 2408 y Fv(\))h Fn(!)768 2415 y Fr(R)798 2421 y Fe(d)832 2408 y Fl(top)p Fv(\()p Fl(s)939 2415 y Fh(j)r Fm(+1)1002 2408 y Fv(\),)i(and)73 2508 y Fn(\017)24 b Fl(s)145 2515 y Fh(j)177 2508 y Fn(!)227 2490 y Fh(i)227 2520 y Fr(R)273 2508 y Fl(s)296 2515 y Fh(j)r Fm(+1)375 2508 y Fv(implies)14 b Fl(top)p Fv(\()p Fl(s)648 2515 y Fh(j)667 2508 y Fv(\))g Fn(\021)f Fl(top)p Fv(\()p Fl(s)859 2515 y Fh(j)r Fm(+1)923 2508 y Fv(\).)0 2624 y(Since)i(there)g(are)h(in\014nitely)e(man)o(y)h(outer)h(reduction)f(steps)h (in)g Fl(D)q Fv(,)g(this)g(yields)e(an)j(in\014nite)e Fn(R)1768 2631 y Fh(d)1804 2624 y Fv(deriv)m(a-)0 2684 y(tion,)h(con)o(tradicting)g (the)g(termination)e(of)j Fn(R)858 2691 y Fh(d)878 2684 y Fv(.)k Ff(2)73 2811 y Fv(This)14 b(theorem)e(can)i(b)q(e)f(paraphrased)i(as)g(follo) o(ws)e(\(again)i(w)o(e)e(omit)f(the)h(pro)q(of)i(of)f(the)f(equiv)m (alence\):)950 2936 y(18)p eop %%Page: 19 19 bop 0 26 a Fk(Theorem)17 b(4.15)24 b Fv(Let)13 b Fn(R)488 33 y Fm(1)508 26 y Fl(;)8 b Fn(R)572 33 y Fm(2)606 26 y Fv(b)q(e)14 b(t)o(w)o(o)f(disjoin)o(t)h(terminating)e(TRSs.)20 b(Their)14 b(disjoin)o(t)f(union)h Fn(R)1822 33 y Fm(1)1848 26 y Fn(])6 b(R)1929 33 y Fm(2)0 86 y Fv(is)16 b(terminating)f(pro)o(vided)g(that)i(one)f (of)h(the)f(follo)o(wing)g(conditions)g(is)g(satis\014ed:)60 196 y(1.)24 b(Both)16 b Fn(R)285 203 y Fm(1)321 196 y Fv(and)h Fn(R)458 203 y Fm(2)494 196 y Fv(are)g Fn(C)602 203 y Fr(E)626 196 y Fv(-terminating.)60 293 y(2.)24 b(Both)16 b Fn(R)285 300 y Fm(1)321 293 y Fv(and)h Fn(R)458 300 y Fm(2)494 293 y Fv(are)g(non-collapsing.)60 390 y(3.)24 b(One)16 b(of)g(the)g(systems)f(is)i Fn(C)621 397 y Fr(E)644 390 y Fv(-terminating)e(and)i(non-collapsing.)0 508 y Fk(Lemma)f(4.16)24 b Fv(If)f(a)g(TRS)h(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))23 b(is)h Fn(C)797 515 y Fr(E)820 508 y Fv(-terminating,)g(then)f(the)g(same)g(is)g(true)g(for)h(the)f(system)0 568 y(\()p Fn(F)16 b(])11 b(f)p Fl(C)t(ons)p Fn(g)p Fl(;)d Fn(R)j(])g(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)d(y)r Fv(\))13 b Fn(!)h Fl(x;)8 b(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))k Fn(!)i Fl(y)r Fn(g)p Fv(\).)0 649 y Fk(Pro)r(of:)23 b Fv(Let)17 b Fl(C)t(ons)g Fv(and)g Fl(C)t(ons)603 631 y Fr(0)632 649 y Fv(b)q(e)g(t)o(w)o (o)g(distinct)f(function)h(sym)o(b)q(ols)f(that)h(do)g(not)h(o)q(ccur)f(in)g Fn(F)5 b Fv(.)23 b(Let)0 709 y Fn(C)26 716 y Fr(E)64 709 y Fv(denote)15 b(the)f(TRS)h(\()p Fn(f)p Fl(C)t(ons)p Fn(g)p Fl(;)8 b Fn(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))k Fn(!)i Fl(x;)8 b(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))k Fn(!)i Fl(y)r Fn(g)p Fv(\))g(and)h(let)f Fn(C)1578 691 y Fr(0)1575 721 y(E)1613 709 y Fv(denote)g(the)h(TRS)0 769 y(\()p Fn(f)p Fl(C)t(ons)158 751 y Fr(0)170 769 y Fn(g)p Fl(;)8 b Fn(f)p Fl(C)t(ons)356 751 y Fr(0)367 769 y Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)g Fl(x;)8 b(C)t(ons)721 751 y Fr(0)733 769 y Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)g Fl(y)r Fn(g)p Fv(\).)21 b(Since)15 b Fn(R)h Fv(is)f Fn(C)1287 776 y Fr(E)1311 769 y Fv(-terminating,)f(the)h(TRS)h Fn(R)11 b(])f(C)1925 776 y Fr(E)0 829 y Fv(is)16 b(terminating.)k(Supp)q(ose) d(that)g(there)e(is)h(an)h(in\014nite)e(rewrite)h(deriv)m(ation)445 924 y Fl(D)g Fv(:)d Fl(s)551 931 y Fm(1)585 924 y Fn(!)635 932 y Fm(\()p Fr(R]C)722 938 y Fb(E)742 932 y Fm(\))p Fr(]C)801 921 y Fb(0)799 944 y(E)834 924 y Fl(s)857 931 y Fm(2)891 924 y Fn(!)941 932 y Fm(\()p Fr(R]C)1028 938 y Fb(E)1048 932 y Fm(\))p Fr(]C)1107 921 y Fb(0)1105 944 y(E)1140 924 y Fl(s)1163 931 y Fm(3)1197 924 y Fn(!)1247 932 y Fm(\()p Fr(R]C)1334 938 y Fb(E)1353 932 y Fm(\))p Fr(]C)1412 921 y Fb(0)1410 944 y(E)1446 924 y Fl(:)8 b(:)g(:)0 1019 y Fv(In)13 b(ev)o(ery)f Fl(s)207 1026 y Fh(j)239 1019 y Fv(replace)g(eac)o(h)h Fl(C)t(ons)622 1001 y Fr(0)634 1019 y Fv(\()p Fl(t)671 1026 y Fm(1)690 1019 y Fl(;)8 b(t)730 1026 y Fm(2)749 1019 y Fv(\))14 b(with)f Fl(C)t(ons)p Fv(\()p Fl(t)1041 1026 y Fm(1)1060 1019 y Fl(;)8 b(t)1100 1026 y Fm(2)1119 1019 y Fv(\))14 b(and)g(denote)f(this)g(term)f(b)o(y)h Fl(s)1692 1001 y Fr(0)1692 1031 y Fh(j)1710 1019 y Fv(.)20 b(Note)13 b(that)0 1079 y Fl(C)t(ons)114 1061 y Fr(0)144 1079 y Fv(do)q(es)20 b(not)f(o)q(ccur)g(in)g(an)o(y)f(rule)g(of)i Fn(R)13 b(])g(C)920 1086 y Fr(E)943 1079 y Fv(.)29 b(Then)19 b Fl(D)1157 1061 y Fr(0)1188 1079 y Fv(:)e Fl(s)1242 1061 y Fr(0)1242 1091 y Fm(1)1280 1079 y Fn(!)1330 1086 y Fr(R]C)1403 1092 y Fb(E)1443 1079 y Fl(s)1466 1061 y Fr(0)1466 1091 y Fm(2)1504 1079 y Fn(!)1554 1086 y Fr(R]C)1627 1092 y Fb(E)1668 1079 y Fl(s)1691 1061 y Fr(0)1691 1091 y Fm(3)1728 1079 y Fn(!)1778 1086 y Fr(R]C)1851 1092 y Fb(E)1892 1079 y Fl(:)8 b(:)g(:)0 1139 y Fv(is)19 b(an)h(in\014nite)e(rewrite)h(deriv)m(ation)g(of)h(terms)d Fl(s)912 1121 y Fr(0)912 1151 y Fh(j)950 1139 y Fn(2)i(T)13 b Fv(\()p Fn(F)18 b([)13 b(f)p Fl(C)t(ons)p Fn(g)p Fl(;)8 b Fn(V)t Fv(\),)19 b(where)g Fl(s)1601 1121 y Fr(0)1601 1151 y Fh(j)1639 1139 y Fv(is)g(rewritten)f(to)0 1199 y Fl(s)23 1181 y Fr(0)23 1212 y Fh(j)r Fm(+1)109 1199 y Fv(b)o(y)k Fl(C)t(ons)p Fv(\()p Fl(x;)8 b(y)r Fv(\))23 b Fn(!)h Fl(x)f Fv(\(resp.)39 b Fl(C)t(ons)p Fv(\()p Fl(x;)8 b(y)r Fv(\))24 b Fn(!)g Fl(y)r Fv(\))e(if)g Fl(s)1185 1206 y Fh(j)1226 1199 y Fv(is)g(reduced)g(to)h Fl(s)1558 1206 y Fh(j)r Fm(+1)1643 1199 y Fv(using)g(the)g(rule)0 1260 y Fl(C)t(ons)114 1241 y Fr(0)126 1260 y Fv(\()p Fl(x;)8 b(y)r Fv(\))13 b Fn(!)g Fl(x)j Fv(\(resp.)21 b Fl(C)t(ons)615 1241 y Fr(0)627 1260 y Fv(\()p Fl(x;)8 b(y)r Fv(\))13 b Fn(!)h Fl(y)r Fv(\).)20 b(This)d(con)o(tradicts)f(the)g(termination)e(of)j Fn(R)11 b(])g(C)1788 1267 y Fr(E)1812 1260 y Fv(.)21 b Ff(2)0 1377 y Fk(Corollary)d(4.17)24 b Fv(It)16 b(follo)o(ws)g(that:)73 1486 y Fn(\017)24 b Fv(T)l(ermination)15 b(is)h(a)g(mo)q(dular)g(prop)q(ert)o (y)g(of)h(non-collapsing)g(TRSs.)73 1584 y Fn(\017)24 b(C)148 1591 y Fr(E)172 1584 y Fv(-termination)15 b(is)h(a)g(mo)q(dular)g(prop)q(ert) o(y)g(of)h(TRSs.)0 1693 y Fk(Pro)r(of:)34 b Fv(The)22 b(\014rst)h(part)g(is)f (easy:)33 b(com)o(bining)21 b(t)o(w)o(o)h(non-collapsing)h(terminating)e (TRSs)i(yields)e(a)0 1753 y(non-collapsing)15 b(terminating)e(TRSs.)21 b(In)14 b(the)g(second)h(case)f(w)o(e)g(kno)o(w)h(that)g(the)f(disjoin)o(t)g (union)g(of)h(t)o(w)o(o)0 1813 y Fn(C)26 1820 y Fr(E)50 1813 y Fv(-terminating)f(TRSs)h Fn(R)505 1820 y Fm(1)540 1813 y Fv(and)h Fn(R)676 1820 y Fm(2)711 1813 y Fv(is)f(again)h(terminating)e(but)h (in)g(addition)g(w)o(e)g(ha)o(v)o(e)f(to)i(pro)o(v)o(e)e(that)0 1873 y(it)j(is)h(also)h Fn(C)227 1880 y Fr(E)251 1873 y Fv(-terminating.)25 b(That)18 b(this)g(in)g(fact)g(the)f(case)h(can)g(b)q(e)g(seen)g(as)h(follo)o (ws:)24 b(Let)18 b Fl(C)t(ons)g Fv(b)q(e)g(a)0 1934 y(binary)d(op)q(erator)i (that)f(do)q(es)h(not)f(o)q(ccur)g(in)f Fn(F)872 1941 y Fm(1)901 1934 y Fn(])10 b(F)980 1941 y Fm(2)1000 1934 y Fv(.)21 b(Since)15 b Fn(R)1204 1941 y Fm(2)1239 1934 y Fv(is)h Fn(C)1314 1941 y Fr(E)1338 1934 y Fv(-terminating,)d(the)j(same)e(holds)0 1994 y(for)j Fn(R)117 2001 y Fm(2)148 1994 y Fn(])11 b(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)d(y)r Fv(\))13 b Fn(!)h Fl(x;)8 b(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))k Fn(!)i Fl(y)r Fn(g)p Fv(.)21 b(Hence)15 b(w)o(e)h(ma)o(y)f(conclude)g(b)o(y)h(the)g(ab)q(o)o(v)o (e)g(theorem)0 2054 y(that)471 2114 y Fn(R)513 2121 y Fm(1)544 2114 y Fn(])11 b Fv(\()p Fn(R)649 2121 y Fm(2)680 2114 y Fn(])h(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)c(y)r Fv(\))k Fn(!)i Fl(x;)8 b(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)g Fl(y)r Fn(g)p Fv(\))445 2195 y(=)h(\()p Fn(R)558 2202 y Fm(1)589 2195 y Fn(])d(R)675 2202 y Fm(2)695 2195 y Fv(\))g Fn(])h(f)p Fl(C)t(ons)p Fv(\()p Fl(x;)c(y)r Fv(\))k Fn(!)i Fl(x;)8 b(C)t(ons)p Fv(\()p Fl(x;)g(y)r Fv(\))k Fn(!)i Fl(y)r Fn(g)0 2275 y Fv(is)i(terminating.)k(But)15 b(this)i(amoun)o(ts)e(to)i Fn(C)808 2282 y Fr(E)832 2275 y Fv(-termination)d(of)j Fn(R)1211 2282 y Fm(1)1242 2275 y Fn(])11 b(R)1328 2282 y Fm(2)1348 2275 y Fv(.)21 b Ff(2)73 2393 y Fv(Theorem)d(4.15)j (is)e(a)h(rather)g(abstract)g(result.)31 b(Ho)o(w)20 b(can)g(w)o(e)f(c)o(hec) o(k)f(whether)h(or)h(not)g(a)g(TRS)g(is)0 2453 y Fn(C)26 2460 y Fr(E)50 2453 y Fv(-terminating?)30 b(The)19 b(next)g(prop)q(osition)i (states)f(some)e(su\016cien)o(t)h(conditions)g(that)h(can)g(easier)f(b)q(e)0 2513 y(c)o(hec)o(k)o(ed.)g(But)d(\014rst)g(of)h(all)e(w)o(e)h(need)g(another) h(de\014nition.)0 2630 y Fk(De\014nition)h(4.18)24 b Fv(A)16 b(TRS)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)h(called)e Fi(non-deterministic)n(al)r(ly)20 b(c)n(ol)r(lapsing)f Fv(if)d(there)f(is)i (a)g(term)0 2691 y(that)g(can)g(b)q(e)h(reduced)e(to)h(t)o(w)o(o)g(distinct)f (v)m(ariables.)23 b(Precisely)l(,)15 b(there)h(m)o(ust)g(b)q(e)h(a)g(term)f Fl(s)f Fn(2)g(T)e Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))0 2751 y(with)16 b Fl(s)e Fn(\021)f Fl(C)t Fv([)p Fl(x;)8 b(y)r Fv(])14 b(for)j(some)e(con)o(text)h Fl(C)t Fv([)p Fl(;)8 b(:)g(:)g(:)e(;)i Fv(])16 b(and)h(distinct)e(v)m(ariables)h Fl(x)g Fv(and)h Fl(y)h Fv(suc)o(h)e(that)g Fl(s)e Fn(!)1875 2730 y Fm(+)1875 2763 y Fr(R)1921 2751 y Fl(x)0 2811 y Fv(and)j Fl(s)d Fn(!)182 2790 y Fm(+)182 2823 y Fr(R)227 2811 y Fl(y)r Fv(.)950 2936 y(19)p eop %%Page: 20 20 bop 0 26 a Fk(Prop)r(osition)18 b(4.19)24 b Fv(Let)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(b)q(e)h(a)f(terminating)f(TRS.)60 137 y(1.)24 b(If)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(non-duplicating,)g(then)g(it)g(is)g Fn(C)977 144 y Fr(E)1001 137 y Fv(-terminating.)60 235 y(2.)24 b(If)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(non-deterministically)d(collapsing,)i(then)i (it)e(is)h Fn(C)1317 242 y Fr(E)1341 235 y Fv(-terminating.)60 333 y(3.)24 b(If)16 b(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(is)g(simply)e(terminating,)g(then)i(it)g(is)g Fn(C)1049 340 y Fr(E)1073 333 y Fv(-terminating.)0 444 y Fk(Pro)r(of:)21 b Fv(W)l(e)16 b(ha)o(v)o(e)g(to)g(pro)o(v)o(e)g(that)g Fn(R)c(])f(C)790 451 y Fr(E)830 444 y Fv(is)16 b(terminating.)0 504 y(1.)26 b(Let)18 b Fn(j)p Fl(s)p Fn(j)204 511 y Fh(C)r(ons)305 504 y Fv(denote)g(the)f(n)o(um)o(b)q(er)f(of)i Fl(C)t(ons)g Fv(op)q(erators)h(in) e(a)h(term)e Fl(s)p Fv(.)26 b(Since)17 b(b)q(oth)h Fn(R)g Fv(and)g Fn(C)1842 511 y Fr(E)1884 504 y Fv(are)0 564 y(non-duplicating,)j(it)f(follo) o(ws)g(from)f Fl(s)h Fn(!)797 571 y Fr(R]C)870 577 y Fb(E)913 564 y Fl(t)g Fv(that)h Fn(j)p Fl(s)p Fn(j)1112 571 y Fh(C)r(ons)1216 564 y Fn(\025)f(j)p Fl(t)p Fn(j)1321 571 y Fh(C)r(ons)1405 564 y Fv(.)33 b(W)l(e)20 b(pro)o(v)o(e)f(the)h(claim)e(b)o(y)0 624 y(induction)e(on)g Fl(m)e Fv(=)f Fn(j)p Fl(s)p Fn(j)443 631 y Fh(C)r(ons)527 624 y Fv(.)22 b(The)16 b(case)g Fl(m)d Fv(=)h(0)j(is)e(trivially)g(true.)20 b(The)d(induction)e(h)o(yp)q(othesis)h (states)0 684 y(that)c(an)o(y)g(rewrite)e(deriv)m(ation)i(in)f Fn(R)r(])r(C)731 691 y Fr(E)766 684 y Fv(starting)h(from)f(a)h(term)e Fl(t)h Fv(with)h Fn(j)p Fl(t)p Fn(j)1384 691 y Fh(C)r(ons)1481 684 y Fl(<)i(m)d Fv(is)h(\014nite.)19 b(Supp)q(ose)0 745 y(there)e(exists)g (an)i(in\014nite)e(deriv)m(ation)g Fl(D)h Fv(:)e Fl(s)840 752 y Fm(1)877 745 y Fn(!)927 752 y Fr(R]C)1000 758 y Fb(E)1038 745 y Fl(s)1061 752 y Fm(2)1097 745 y Fn(!)1147 752 y Fr(R]C)1220 758 y Fb(E)1259 745 y Fl(s)1282 752 y Fm(3)1318 745 y Fn(!)1368 752 y Fr(R]C)1441 758 y Fb(E)1479 745 y Fl(:)8 b(:)g(:)17 b Fv(with)h Fn(j)p Fl(s)1704 752 y Fm(1)1724 745 y Fn(j)1738 752 y Fh(C)r(ons)1838 745 y Fv(=)f Fl(m)p Fv(.)0 805 y(Ob)o(viously)l(,)12 b(this)h(implies)d Fn(j)p Fl(s)527 812 y Fh(j)546 805 y Fn(j)560 812 y Fh(C)r(ons)658 805 y Fv(=)j Fl(m)g Fv(for)g(an)o(y)g(index)g Fl(j)s Fv(.)20 b(No)o(w)13 b(if)f Fl(s)1283 812 y Fm(1)1317 805 y Fn(\021)h Fl(C)t(ons)p Fv(\()p Fl(t)1520 812 y Fm(1)1540 805 y Fl(;)8 b(t)1580 812 y Fm(2)1599 805 y Fv(\),)13 b(then)g(w)o(e)f(deriv) o(e)0 865 y(a)k(con)o(tradiction)f(b)q(ecause)g(an)o(y)h(rewrite)e(deriv)m (ation)h(starting)h(from)f Fl(t)1313 872 y Fm(1)1348 865 y Fv(or)g Fl(t)1424 872 y Fm(2)1459 865 y Fv(m)o(ust)f(terminate.)19 b(Hence)0 925 y Fl(r)q(oot)p Fv(\()p Fl(s)129 932 y Fm(1)150 925 y Fv(\))c Fn(2)h(F)5 b Fv(.)24 b(The)17 b(application)g(of)h(a)f(rule)g (from)f Fn(C)1005 932 y Fr(E)1046 925 y Fv(w)o(ould)i(decrease)e(the)h(n)o (um)o(b)q(er)f(of)h(o)q(ccurrences)0 985 y(of)i Fl(C)t(ons)f Fv(op)q(erators,)j(hence)c(there)h(is)h(no)g(application)f(of)h(suc)o(h)f(a)h (rule)f(in)g(the)h(whole)f(deriv)m(ation)g Fl(D)q Fv(.)0 1046 y(No)o(w)g(the)g(claim)e(follo)o(ws)j(b)o(y)e(the)h(same)g(argumen)o(ts)f(as) i(the)f(second)h(part)f(of)h(the)f(pro)q(of)i(of)e(Theorem)0 1106 y(4.14.)0 1166 y(2.)30 b(Let)20 b Fl(C)t Fv([)p Fl(x;)8 b(y)r Fv(])17 b(b)q(e)i(a)h(term)d(with)i Fl(C)t Fv([)p Fl(x;)8 b(y)r Fv(])17 b Fn(!)875 1145 y Fm(+)875 1178 y Fr(R)925 1166 y Fl(x)i Fv(and)h Fl(C)t Fv([)p Fl(x;)8 b(y)r Fv(])17 b Fn(!)1280 1145 y Fm(+)1280 1178 y Fr(R)1330 1166 y Fl(y)r Fv(.)30 b(Supp)q(ose)20 b(that)g(there)e(exists)0 1226 y(an)23 b(in\014nite)g(deriv)m(ation)f Fl(D)27 b Fv(:)e Fl(s)611 1233 y Fm(1)656 1226 y Fn(!)706 1233 y Fr(R]C)779 1239 y Fb(E)827 1226 y Fl(s)850 1233 y Fm(2)895 1226 y Fn(!)945 1233 y Fr(R]C)1018 1239 y Fb(E)1065 1226 y Fl(s)1088 1233 y Fm(3)1133 1226 y Fn(!)1183 1233 y Fr(R]C)1256 1239 y Fb(E)1303 1226 y Fl(:)8 b(:)g(:)p Fv(.)41 b(In)23 b(ev)o(ery)e Fl(s)1642 1233 y Fh(j)1683 1226 y Fv(replace)h(eac)o(h)0 1286 y Fl(C)t(ons)p Fv(\()p Fl(t)151 1293 y Fm(1)170 1286 y Fl(;)8 b(t)210 1293 y Fm(2)229 1286 y Fv(\))20 b(with)g Fl(C)t Fv([)p Fl(t)454 1293 y Fm(1)473 1286 y Fl(;)8 b(t)513 1293 y Fm(2)532 1286 y Fv(])19 b(and)i(denote)f(this)g(term)e(b)o(y)i Fl(s)1139 1268 y Fr(0)1139 1299 y Fh(j)1157 1286 y Fv(.)33 b(Note)19 b(that)i Fl(C)t(ons)f Fv(do)q(es)g(not)h(o)q(ccur)f(in)0 1346 y(an)o(y)d(rule)f(of)i Fn(R)p Fv(.)23 b(Then)18 b Fl(D)497 1328 y Fr(0)524 1346 y Fv(:)d Fl(s)576 1328 y Fr(0)576 1359 y Fm(1)611 1346 y Fn(!)661 1326 y Fm(+)661 1359 y Fr(R)708 1346 y Fl(s)731 1328 y Fr(0)731 1359 y Fm(2)766 1346 y Fn(!)816 1326 y Fm(+)816 1359 y Fr(R)863 1346 y Fl(s)886 1328 y Fr(0)886 1359 y Fm(3)921 1346 y Fn(!)971 1326 y Fm(+)971 1359 y Fr(R)1018 1346 y Fl(:)8 b(:)g(:)17 b Fv(is)f(an)i(in\014nite)e(rewrite)g(deriv)m(ation) h(of)g(terms)0 1407 y Fl(s)23 1389 y Fr(0)23 1419 y Fh(j)60 1407 y Fn(2)i(T)12 b Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\),)19 b(where)g Fl(s)487 1389 y Fr(0)487 1419 y Fh(j)524 1407 y Fv(is)g(rewritten)f (to)h Fl(s)875 1389 y Fr(0)875 1419 y Fh(j)r Fm(+1)958 1407 y Fv(b)o(y)f Fl(C)t Fv([)p Fl(x;)8 b(y)r Fv(])17 b Fn(!)1238 1386 y Fm(+)1238 1419 y Fr(R)1288 1407 y Fl(x)i Fv(\(resp.)30 b Fl(C)t Fv([)p Fl(x;)8 b(y)r Fv(])16 b Fn(!)1694 1386 y Fm(+)1694 1419 y Fr(R)1745 1407 y Fl(y)r Fv(\))i(if)h Fl(s)1879 1414 y Fh(j)1916 1407 y Fv(is)0 1467 y(reduced)d(to)g Fl(s)264 1474 y Fh(j)r Fm(+1)344 1467 y Fv(using)g(the)g(rule)g Fl(C)t(ons)p Fv(\()p Fl(x;)8 b(y)r Fv(\))13 b Fn(!)g Fl(x)j Fv(\(resp.)22 b Fl(C)t(ons)p Fv(\()p Fl(x;)8 b(y)r Fv(\))13 b Fn(!)g Fl(y)r Fv(\).)0 1527 y(3.)22 b(W)l(e)16 b(sho)o(w)g(the)g(stronger)h(statemen)o(t)e (that)i Fn(R)11 b([)g(F)1009 1509 y Fh(ar)q(g)1081 1527 y Fv(is)16 b Fn(C)1156 1534 y Fr(E)1180 1527 y Fv(-terminating.)0 1587 y(Case)23 b(\(i\):)32 b Fn(F)27 b Fv(con)o(tains)22 b(at)g(most)g(constan)o (ts)g(and)h(unary)f(function)g(sym)o(b)q(ols.)38 b(Then)22 b Fn(R)15 b([)h(F)1839 1569 y Fh(ar)q(g)1916 1587 y Fv(is)0 1647 y(non-duplicating)h(and)f(the)g(assertion)h(follo)o(ws)f(from)f(1.)0 1708 y(Case)22 b(\(ii\):)31 b Fn(F)26 b Fv(con)o(tains)c(at)f(least)h(one)f (function)h(sym)o(b)q(ol)e Fl(f)27 b Fv(of)21 b(arit)o(y)g Fl(n)i Fn(\025)f Fv(2.)38 b(Then)21 b Fn(R)15 b([)g(F)1839 1690 y Fh(ar)q(g)1916 1708 y Fv(is)0 1768 y(non-deterministically)e (collapsing)j(and)h(the)f(assertion)h(follo)o(ws)f(from)f(2.)22 b Ff(2)0 1887 y Fk(Corollary)c(4.20)24 b Fv(Let)13 b Fn(R)499 1894 y Fm(1)519 1887 y Fl(;)8 b Fn(R)583 1894 y Fm(2)616 1887 y Fv(b)q(e)13 b(t)o(w)o(o)f(disjoin)o(t)h(terminating)e(TRSs.)20 b(Their)13 b(disjoin)o(t)f(union)h Fn(R)1825 1894 y Fm(1)1849 1887 y Fn(])t(R)1928 1894 y Fm(2)0 1947 y Fv(is)j(terminating)f(pro)o(vided)g (that:)73 2058 y Fn(\017)24 b Fv(One)16 b(of)g(the)g(systems)f(is)i (non-duplicating)f(and)h(non-collapsing.)73 2156 y Fn(\017)24 b Fv(One)16 b(of)g(the)g(systems)f(is)i(simply)d(terminating)g(and)j (non-collapsing.)73 2254 y Fn(\017)24 b Fv(One)16 b(of)g(the)g(systems)f(is)i Fn(C)621 2261 y Fr(E)644 2254 y Fv(-terminating)e(and)i(non-collapsing.)0 2365 y Fk(Pro)r(of:)k Fv(Immedi)o(ate)14 b(consequence)h(of)h(Theorem)f(4.15) i(and)g(Prop)q(ositon)h(4.19.)k Ff(2)0 2484 y Fk(De\014nition)c(4.21)24 b Fv(Let)236 2581 y Fn(A)276 2588 y Fm(0)337 2581 y Fv(=)41 b Fn(fR)17 b(j)f(R)g Fv(is)g Fn(C)663 2588 y Fr(E)687 2581 y Fv(-terminating)f Fn(g)236 2653 y(A)276 2660 y Fm(1)337 2653 y Fv(=)41 b Fn(fR)17 b(j)f(R)g Fv(is)g(non-duplicating)h(and)g(terminating)d Fn(g)236 2726 y(A)276 2733 y Fm(2)337 2726 y Fv(=)41 b Fn(fR)17 b(j)f(R)g Fv(is)g(simply)e(terminating)h Fn(g)236 2799 y(A)276 2806 y Fm(3)337 2799 y Fv(=)41 b Fn(fR)17 b(j)f(R)g Fv(is)g (non-deterministically)d(collapsing)k(and)f(terminating)f Fn(g)950 2936 y Fv(20)p eop %%Page: 21 21 bop 73 26 a Fv(Prop)q(osition)19 b(4.19)f(states)g(that)g Fn(A)728 33 y Fh(j)762 26 y Fn(\022)e(A)857 33 y Fm(0)894 26 y Fv(for)h Fl(j)i Fn(2)e(f)p Fv(1)p Fl(;)8 b Fv(2)p Fl(;)g Fv(3)p Fn(g)p Fv(.)25 b(Th)o(us)18 b(this)f(lemma)e(in)i(conjunction)0 86 y(with)f(the)g(next)g(examples)e(sho)o(ws)j(that)g(w)o(e)f(ha)o(v)o(e)f(the)h (situation)h(depicted)e(in)h(Figure)f(4.1.)0 162 y Fk(Example)h(4.22)598 272 y Fn(R)640 279 y Fm(1)674 272 y Fv(=)e Fn(f)p Fl(h)p Fv(\()p Fl(x;)8 b(y)r Fv(\))13 b Fn(!)g Fl(x;)8 b(h)p Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)g Fl(y)r Fn(g)41 b(2)h(A)1444 279 y Fm(1)1475 272 y Fn(\\)11 b(A)1559 279 y Fm(2)1590 272 y Fn(\\)g(A)1674 279 y Fm(3)987 345 y Fn(R)1029 352 y Fm(2)1062 345 y Fv(=)j Fn(f)p Fl(a)g Fn(!)f Fl(b)p Fn(g)41 b(2)h Fv(\()p Fn(A)1463 352 y Fm(1)1494 345 y Fn(\\)11 b(A)1578 352 y Fm(2)1597 345 y Fv(\))g Fn(n)g(A)1703 352 y Fm(3)111 418 y Fn(R)153 425 y Fm(3)187 418 y Fv(=)j Fn(f)p Fl(f)5 b Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\))14 b Fn(!)f Fl(f)5 b Fv(\()p Fl(g)r Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\)\))p Fl(;)j(h)p Fv(\()p Fl(x;)g(y)r Fv(\))14 b Fn(!)f Fl(x;)8 b(h)p Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)g Fl(y)r Fn(g)41 b(2)h Fv(\()p Fn(A)1463 425 y Fm(1)1494 418 y Fn(\\)11 b(A)1578 425 y Fm(3)1597 418 y Fv(\))g Fn(n)g(A)1703 425 y Fm(2)230 490 y Fn(R)272 497 y Fm(4)306 490 y Fv(=)j Fn(f)p Fl(f)407 497 y Fm(1)427 490 y Fv(\()p Fl(x)p Fv(\))f Fn(!)h Fl(f)594 497 y Fm(2)613 490 y Fv(\()p Fl(x;)8 b(x)p Fv(\))p Fl(;)g(h)p Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)g Fl(x;)8 b(h)p Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)g Fl(y)r Fn(g)41 b(2)h Fv(\()p Fn(A)1463 497 y Fm(2)1494 490 y Fn(\\)11 b(A)1578 497 y Fm(3)1597 490 y Fv(\))g Fn(n)g(A)1703 497 y Fm(1)645 563 y Fn(R)687 570 y Fm(5)721 563 y Fv(=)j Fn(f)p Fl(f)5 b Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\))14 b Fn(!)g Fl(f)5 b Fv(\()p Fl(g)r Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\)\))p Fn(g)41 b(2)h(A)1444 570 y Fm(1)1475 563 y Fn(n)11 b Fv(\()p Fn(A)1570 570 y Fm(2)1600 563 y Fn([)g(A)1684 570 y Fm(3)1704 563 y Fv(\))765 636 y Fn(R)807 643 y Fm(6)840 636 y Fv(=)j Fn(f)p Fl(f)941 643 y Fm(1)961 636 y Fv(\()p Fl(x)p Fv(\))f Fn(!)h Fl(f)1128 643 y Fm(2)1148 636 y Fv(\()p Fl(x;)8 b(x)p Fv(\))p Fn(g)40 b(2)i(A)1444 643 y Fm(2)1475 636 y Fn(n)11 b Fv(\()p Fn(A)1570 643 y Fm(1)1600 636 y Fn([)g(A)1684 643 y Fm(3)1704 636 y Fv(\))981 708 y Fn(R)1023 715 y Fm(7)1057 708 y Fv(=)j Fn(R)1151 715 y Fm(4)1182 708 y Fn([)d(R)1268 715 y Fm(5)1329 708 y Fn(2)42 b(A)1444 715 y Fm(3)1475 708 y Fn(n)11 b Fv(\()p Fn(A)1570 715 y Fm(1)1600 708 y Fn([)g(A)1684 715 y Fm(2)1704 708 y Fv(\))278 781 y Fn(R)320 788 y Fm(8)353 781 y Fv(=)j Fn(f)p Fl(f)5 b Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\))14 b Fn(!)g Fl(f)5 b Fv(\()p Fl(g)r Fv(\()p Fl(f)g Fv(\()p Fl(x)p Fv(\)\)\))p Fl(;)j(f)941 788 y Fm(1)961 781 y Fv(\()p Fl(x)p Fv(\))13 b Fn(!)h Fl(f)1128 788 y Fm(2)1148 781 y Fv(\()p Fl(x;)8 b(x)p Fv(\))p Fn(g)40 b(2)i(A)1444 788 y Fm(0)1475 781 y Fn(n)11 b Fv(\()p Fn(A)1570 788 y Fm(1)1600 781 y Fn([)g(A)1684 788 y Fm(2)1715 781 y Fn([)g(A)1799 788 y Fm(3)1819 781 y Fv(\))912 1123 y Fn(A)952 1130 y Fm(1)881 1187 y Fa(\032\031)881 1071 y(\033)q(\030)854 1217 y Fn(A)894 1224 y Fm(2)840 1252 y Fa(\032\031)840 1136 y(\033\030)969 1217 y Fn(A)1009 1224 y Fm(3)923 1252 y Fa(\032\031)923 1136 y(\033\030)988 1064 y Fn(A)1028 1071 y Fm(0)792 993 y Fa(')p 792 1204 4 130 v 792 1288 a(&)1087 993 y($)p 1087 1204 V 1087 1288 a(\045)p 877 1288 130 4 v 877 993 V 821 1358 a Fv(Figure)16 b(4.1)0 1540 y Fk(Prop)r(osition)i(4.23)24 b Fv(W)l(e)16 b(deriv)o(e:)60 1662 y(1.)24 b(If)16 b Fn(R)213 1669 y Fm(1)246 1662 y Fn(2)f(A)334 1669 y Fh(j)368 1662 y Fv(and)i Fn(R)505 1669 y Fm(2)538 1662 y Fn(2)d(A)625 1669 y Fh(j)660 1662 y Fv(for)i(some)f Fl(j)i Fn(2)d(f)p Fv(0)p Fl(;)8 b Fv(1)p Fl(;)g Fv(2)p Fl(;)g Fv(3)p Fn(g)p Fv(,)17 b(then)f Fn(R)1336 1669 y Fm(1)1367 1662 y Fn(])11 b(R)1453 1669 y Fm(2)1487 1662 y Fn(2)j(A)1574 1669 y Fh(j)1592 1662 y Fv(.)60 1764 y(2.)24 b(If)16 b Fn(R)213 1771 y Fm(1)246 1764 y Fn(2)f(A)334 1771 y Fh(j)368 1764 y Fv(and)i Fn(R)505 1771 y Fm(2)538 1764 y Fn(2)d(A)625 1771 y Fh(k)663 1764 y Fv(for)i(some)g Fl(j;)8 b(k)16 b Fn(2)e(f)p Fv(0)p Fl(;)8 b Fv(1)p Fl(;)g Fv(2)p Fl(;)g Fv(3)p Fn(g)p Fv(,)16 b(then)g Fn(R)1385 1771 y Fm(1)1416 1764 y Fn(])c(R)1503 1771 y Fm(2)1536 1764 y Fn(2)j(A)1624 1771 y Fm(0)1643 1764 y Fv(.)0 1886 y Fk(Pro)r(of:)21 b Fv(1.)h Fl(j)17 b Fv(=)c(0)k(:)k(see)16 b(Corollary)g(4.17) 0 1946 y Fl(j)21 b Fv(=)d(1)h(:)26 b(Clearly)l(,)17 b(the)i(disjoin)o(t)f (union)h(of)g(t)o(w)o(o)f(non-duplicating)h(terminating)e(TRSs)i(is)g(again)g (non-)0 2007 y(duplicating.)i(That)c(it)e(is)h(also)h(terminating)e(follo)o (ws)h(from)f(the)h(ab)q(o)o(v)o(e)g(considerations.)0 2067 y Fl(j)28 b Fv(=)d(2)e(:)34 b(Since)22 b Fn(R)382 2074 y Fh(i)396 2067 y Fv(,)i Fl(i)h Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)p Fv(,)24 b(is)f(simply)e(terminating,)h(the)g(same)g(holds)h(for)g Fn(R)1674 2074 y Fh(i)1704 2067 y Fn([)16 b(F)1794 2043 y Fh(ar)q(g)1789 2078 y(i)1849 2067 y Fv(.)41 b(In)0 2127 y(particular,)16 b Fn(R)281 2134 y Fh(i)306 2127 y Fn([)11 b(F)391 2103 y Fh(ar)q(g)386 2138 y(i)463 2127 y Fv(is)16 b Fn(C)538 2134 y Fr(E)562 2127 y Fv(-terminating.)k(The)c(application)h(of)f(Theorem)f(4.15)i(to)g Fn(R)1682 2134 y Fm(1)1713 2127 y Fn([)11 b(F)1798 2103 y Fh(ar)q(g)1793 2138 y Fm(1)1870 2127 y Fv(and)0 2187 y Fn(R)42 2194 y Fm(2)74 2187 y Fn([)i(F)161 2164 y Fh(ar)q(g)156 2198 y Fm(2)234 2187 y Fv(yields)k(the)h(termination)f(of)h(\()p Fn(R)845 2194 y Fm(1)878 2187 y Fn([)12 b(F)964 2164 y Fh(ar)q(g)959 2198 y Fm(1)1020 2187 y Fv(\))g Fn(])g Fv(\()p Fn(R)1157 2194 y Fm(2)1190 2187 y Fn([)g(F)1276 2164 y Fh(ar)q(g)1271 2198 y Fm(2)1332 2187 y Fv(\),)18 b(or)g(equiv)m(alen)o(tly)l(,)e(the)i(simple)0 2247 y(termination)d(of)h Fn(R)363 2254 y Fm(1)394 2247 y Fn(])11 b(R)480 2254 y Fm(2)500 2247 y Fv(.)0 2308 y Fl(j)26 b Fv(=)c(3)g(:)31 b(Com)o(bining)21 b(a)g(non-deterministically)e(collapsing)i(TRS)g(with)h(an) g(arbitrary)f(other)h(TRS)0 2368 y(yields)15 b(a)i(non-deterministically)c (collapsing)j(TRS.)g(Hence)f(the)h(assertion)h(follo)o(ws)f(as)h(ab)q(o)o(v)o (e.)0 2428 y(2.)22 b(This)16 b(follo)o(ws)g(from)f(Prop)q(osition)j(4.19)f (in)f(conjunction)g(with)g(Corollary)g(4.17.)22 b Ff(2)0 2563 y Fk(Corollary)c(4.24)24 b Fv(According)14 b(to)h(the)g(ab)q(o)o(v)o(e)f (prop)q(osition,)i(w)o(e)e(ha)o(v)o(e)g(in)h(addition)g(to)g(Corollary)g (4.17:)73 2685 y Fn(\017)24 b Fv(T)l(ermination)15 b(is)h(a)g(mo)q(dular)g (prop)q(ert)o(y)g(of)h(non-duplicating)f(TRSs.)73 2787 y Fn(\017)24 b Fv(T)l(ermination)15 b(is)h(a)g(mo)q(dular)g(prop)q(ert)o(y)g(of)h (non-deterministically)c(collapsing)j(TRSs.)950 2936 y(21)p eop %%Page: 22 22 bop 73 26 a Fn(\017)24 b Fv(Simple)14 b(termination)g(is)i(a)h(mo)q(dular)f (prop)q(ert)o(y)g(of)h(TRSs.)73 129 y(W)l(e)j(emphasize)e(that)j(it)f (su\016ces)f(to)i(sho)o(w)g(that)f(eac)o(h)g(constituen)o(t)f(TRS)i(is)f (either)f(simply)f(ter-)0 190 y(minating,)g(non-duplicating,)h (non-deterministically)d(collapsing)j(or)g Fn(C)1373 197 y Fr(E)1397 190 y Fv(-terminating)f(to)h(infer)f(that)0 250 y(their)e(com)o (bination)e(is)i(again)h(terminating.)0 433 y Fo(5)83 b(On)24 b(Mo)r(dular)g(Prop)r(erties)g(of)h(Com)n(bined)d(Systems)g(with)124 524 y(Shared)27 b(Constructors)0 669 y Fd(5.1)70 b(Basic)22 b(Notions)g(of)h(TRSs)g(with)f(Shared)h(Constructors)0 782 y Fv(In)18 b(this)f(section)h(w)o(e)f(w)o(eak)o(en)g(the)h(disjoin)o(tness)f (requiremen)o(t,)e(i.e.,)h(the)i(TRSs)g(are)g(allo)o(w)o(ed)f(to)h(share)0 842 y(sp)q(ecial)e(function)g(sym)o(b)q(ols,)e(so-called)i(constructors.)22 b Fi(Constructors)16 b Fv(are)h(function)f(sym)o(b)q(ols)f(that)h(do)0 902 y(not)g(o)q(ccur)g(at)h(the)e(ro)q(ot)i(p)q(osition)g(of)f(the)g (left-hand)g(side)f(of)h(an)o(y)g(rewrite)f(rule;)g(the)h(others)g(are)g (called)0 963 y Fi(de\014ne)n(d)g(symb)n(ols)p Fv(.)21 b(The)14 b(union)g(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))14 b(=)f(\()p Fn(F)852 970 y Fm(1)878 963 y Fn([)7 b(F)954 970 y Fm(2)974 963 y Fl(;)h Fn(R)1038 970 y Fm(1)1064 963 y Fn([)f(R)1146 970 y Fm(2)1166 963 y Fv(\))14 b(of)h(t)o(w)o(o)e(TRSs)i(\()p Fn(F)1529 970 y Fm(1)1549 963 y Fl(;)8 b Fn(R)1613 970 y Fm(1)1632 963 y Fv(\))14 b(and)h(\()p Fn(F)1813 970 y Fm(2)1833 963 y Fl(;)8 b Fn(R)1897 970 y Fm(2)1916 963 y Fv(\),)0 1023 y(where)388 1083 y Fn(C)17 b Fv(=)d Fn(F)519 1090 y Fm(1)549 1083 y Fn(\\)e(F)630 1090 y Fm(2)663 1083 y Fn(\022)h Fv(\()p Fn(F)770 1090 y Fm(1)801 1083 y Fn([)e(F)881 1090 y Fm(2)901 1083 y Fv(\))g Fn(n)g(f)p Fl(r)q(oot)p Fv(\()p Fl(l)q Fv(\))16 b Fn(j)g Fl(l)f Fn(!)f Fl(r)h Fn(2)f(R)1399 1090 y Fm(1)1430 1083 y Fn([)d(R)1516 1090 y Fm(2)1536 1083 y Fn(g)0 1158 y Fv(is)21 b(called)f(the)h Fi(c)n(ombine)n(d)h(TRS)f(of)h Fv(\()p Fn(F)738 1165 y Fm(1)757 1158 y Fl(;)8 b Fn(R)821 1165 y Fm(1)841 1158 y Fv(\))22 b Fi(and)g Fv(\()p Fn(F)1036 1165 y Fm(2)1055 1158 y Fl(;)8 b Fn(R)1119 1165 y Fm(2)1139 1158 y Fv(\))22 b Fi(with)g(shar)n(e)n(d)f(c)n (onstructors)h Fn(C)s Fv(.)35 b(In)21 b(this)0 1218 y(case)g(w)o(e)f (de\014ne)g Fn(D)367 1225 y Fm(1)409 1218 y Fv(=)h Fn(F)504 1225 y Fm(1)538 1218 y Fn(n)14 b(C)s Fv(,)21 b Fn(D)679 1225 y Fm(2)721 1218 y Fv(=)g Fn(F)816 1225 y Fm(2)850 1218 y Fn(n)14 b(C)s Fv(,)21 b(and)h Fn(D)h Fv(=)e Fn(D)1211 1225 y Fm(1)1245 1218 y Fn(])15 b(D)1331 1225 y Fm(2)1351 1218 y Fv(.)34 b(T)l(o)22 b(b)q(e)e(able)h(to)g(distinguish)0 1279 y(b)q(et)o(w)o(een)15 b(sym)o(b)q(ols)g(from)g(di\013eren)o(t)g(sets,)g(w)o(e)h(use)g(capitals)f Fl(F)q(;)8 b(G;)g(:)g(:)g(:)16 b Fv(for)g(function)g(sym)o(b)q(ols)e(from)h Fn(D)1915 1286 y Fm(1)1935 1279 y Fv(,)0 1339 y(small)h(letters)h Fl(f)s(;)8 b(g)r(;)g(:)g(:)g(:)16 b Fv(for)i(those)g(from)e Fn(D)809 1346 y Fm(2)830 1339 y Fv(,)h(and)h(small)e(capitals)i Fh(C)r Fl(;)8 b Fh(D)q Fl(;)g(:)g(:)g(:)17 b Fv(for)h(shared)g(constructors.) 0 1399 y(As)f(usual)g Fl(x;)8 b(y)r(;)g(z)r(;)g(x)374 1406 y Fm(1)392 1399 y Fl(;)g(y)438 1406 y Fm(1)457 1399 y Fl(;)g(z)502 1406 y Fm(1)522 1399 y Fl(;)g(:)g(:)g(:)16 b Fv(will)f(denote)i(v)m (ariables.)23 b(T)l(o)18 b(emphasize)d(that)i Fn(F)i Fv(=)c Fn(D)f(])d(C)s Fv(,)17 b(w)o(e)f(write)0 1459 y Fn(T)d Fv(\()p Fn(D)q Fl(;)8 b Fn(C)s Fl(;)g Fn(V)t Fv(\))13 b(instead)f(of)h Fn(T)g Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))k(at)h(the)f(appropriate)h (places.)20 b(In)12 b(this)h(section)f Fn(!)g Fv(=)h Fn(!)1670 1466 y Fr(R)1714 1459 y Fv(=)g Fn(!)1815 1466 y Fr(R)1845 1471 y Fj(1)1862 1466 y Fr([R)1916 1471 y Fj(2)1935 1459 y Fv(.)0 1562 y Fk(De\014nition)18 b(5.1)24 b Fv(Let)15 b Fl(s)f Fn(2)g(T)f Fv(\()p Fn(D)625 1569 y Fm(1)656 1562 y Fn(])e(D)738 1569 y Fm(2)758 1562 y Fl(;)d Fn(C)s Fl(;)g Fn(V)t Fv(\).)21 b(Again)15 b(w)o(e)g(color)g(eac)o(h)g(function)g(sym)o(b)q(ol)f(in)h Fl(s)p Fv(.)21 b(F)l(unc-)0 1623 y(tion)e(sym)o(b)q(ols)f(from)g Fn(D)448 1630 y Fm(1)488 1623 y Fv(are)h(colored)g(blac)o(k,)f(those)i(from)e Fn(D)1174 1630 y Fm(2)1213 1623 y Fv(white,)h(and)h(constructors)g(as)f(w)o (ell)f(as)0 1683 y(v)m(ariables)e(are)f(transparen)o(t.)22 b(If)15 b Fl(s)h Fv(do)q(es)g(not)g(con)o(tain)g(white)f(\(blac)o(k\))f (function)i(sym)o(b)q(ols,)e(w)o(e)h(sp)q(eak)i(of)0 1743 y(a)c Fi(black)k(\(white\))e(term)p Fv(.)20 b Fl(s)13 b Fv(is)g(said)g(to)h(b)q(e)f Fi(tr)n(ansp)n(ar)n(ent)f Fv(if)g(it)h(only)g(con)o(tains)g(constructors)h (and)f(v)m(ariables.)0 1803 y(Consequen)o(tly)l(,)j(a)h(transparen)o(t)g (term)e(ma)o(y)h(b)q(e)h(regarded)g(as)g(blac)o(k)f(or)h(white,)f(this)h(is)g (con)o(v)o(enien)o(t)e(for)0 1863 y(later)j(purp)q(oses.)30 b Fl(s)19 b Fv(is)g(called)e Fi(top)j(black)h(\(top)f(white,)h(top)f(tr)n (ansp)n(ar)n(ent\))e Fv(if)g Fl(r)q(oot)p Fv(\()p Fl(s)p Fv(\))h(is)g(blac)o (k)f(\(white,)0 1923 y(transparen)o(t\).)73 2027 y(Sev)o(eral)k (de\014nitions)g(and)h(considerations)g(are)g(symmetric)o(al)d(in)i(the)h (colors)g(blac)o(k)e(and)j(white.)0 2087 y(Therefore,)13 b(w)o(e)g(state)h (the)f(resp)q(ectiv)o(e)f(de\014nitions)i(and)g(considerations)g(only)f(for)h (the)f(color)g(blac)o(k)g(\(the)0 2147 y(same)i(applies)h(m)o(utatis)f(m)o (utandis)g(for)h(the)g(color)h(white\).)0 2250 y Fk(De\014nition)h(5.2)24 b Fv(Let)12 b Fl(s)f Fv(b)q(e)h(a)h(top)f(blac)o(k)f(term)f(suc)o(h)i(that)g Fl(s)i Fn(\021)g Fl(C)1227 2232 y Fh(b)1243 2250 y Fv([)p Fl(s)1280 2257 y Fm(1)1299 2250 y Fl(;)8 b(:)g(:)g(:)g(;)g(s)1432 2257 y Fh(n)1455 2250 y Fv(])k(for)g(some)f(blac)o(k)g(con)o(text)0 2310 y Fl(C)39 2292 y Fh(b)56 2310 y Fv([)p Fl(;)d(:)g(:)g(:)f(;)h Fv(])13 b Fn(6\021)g Ff(2)j Fv(and)g Fl(r)q(oot)p Fv(\()p Fl(s)534 2317 y Fh(j)553 2310 y Fv(\))d Fn(2)h(D)670 2317 y Fm(2)706 2310 y Fv(for)h Fl(j)i Fn(2)d(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)g(n)p Fn(g)p Fv(.)21 b(W)l(e)14 b(denote)i(this)f(b)o(y)f Fl(s)g Fn(\021)g Fl(C)1617 2292 y Fh(b)1633 2310 y Fv([)-8 b([)p Fl(s)1676 2317 y Fm(1)1695 2310 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1827 2317 y Fh(n)1850 2310 y Fv(])-8 b(].)20 b(In)0 2370 y(this)e(case)g(w)o(e)g (de\014ne)f(the)h(m)o(ultiset)d Fl(S)727 2352 y Fh(b)745 2370 y Fv(\()p Fl(s)p Fv(\))i(of)i(all)e Fi(black)k(princip)n(al)d Fv(subterms)f(of)h Fl(s)g Fv(to)h(b)q(e)f Fl(S)1749 2352 y Fh(b)1766 2370 y Fv(\()p Fl(s)p Fv(\))f(=)g([)p Fl(s)p Fv(])0 2431 y(and)e(the)f(set)g(of)h(all)f Fi(white)i(princip)n(al)f Fv(subterms)e(of)i Fl(s)f Fv(to)h(b)q(e)f Fl(S)1148 2413 y Fh(w)1177 2431 y Fv(\()p Fl(s)p Fv(\))f(=)h([)p Fl(s)1340 2438 y Fm(1)1359 2431 y Fl(;)8 b(:)g(:)g(:)g(;)g(s)1492 2438 y Fh(n)1515 2431 y Fv(].)20 b(The)15 b Fi(topmost)g(black)0 2491 y(homo)n(gene)n(ous)g(p) n(art)d Fv(of)h Fl(s)p Fv(,)h(denoted)f(b)o(y)f Fl(top)800 2473 y Fh(b)818 2491 y Fv(\()p Fl(s)p Fv(\),)h(is)g(obtained)g(from)f Fl(s)i Fv(b)o(y)e(replacing)h(all)f(white)h(principal)0 2551 y(subterms)i(with)h Ff(2)p Fv(.)22 b(The)16 b Fi(topmost)i(white)g(homo)n (gene)n(ous)g(p)n(art)d Fv(of)i Fl(s)f Fv(is)g Fl(top)1408 2533 y Fh(w)1437 2551 y Fv(\()p Fl(s)p Fv(\))e Fn(\021)f Ff(2)p Fv(.)22 b(No)o(w)16 b(let)g Fl(s)g Fv(b)q(e)g(a)0 2611 y(top)h(transparen)o (t)f(term)f(suc)o(h)h(that)199 2750 y Fl(s)e Fn(\021)289 2650 y Fg(8)289 2688 y(>)289 2700 y(<)289 2775 y(>)289 2787 y(:)346 2689 y Fl(C)385 2671 y Fh(t)400 2689 y Fv([)p Fl(s)437 2696 y Fm(1)456 2689 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)588 2696 y Fh(l)601 2689 y Fv(])75 b(where)41 b Fl(C)895 2671 y Fh(t)909 2689 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])13 b Fn(2)h(T)f Fv(\()p Fn(C)s Fl(;)8 b Fn(V)t Fv(\))16 b Fl(;)24 b(r)q(oot)p Fv(\()p Fl(s)1453 2696 y Fh(j)1472 2689 y Fv(\))14 b Fn(2)g(D)1590 2696 y Fm(1)1621 2689 y Fn(])e(D)1704 2696 y Fm(2)346 2749 y Fl(C)385 2731 y Fh(b)402 2749 y Fv([)p Fl(t)434 2756 y Fm(1)453 2749 y Fl(;)c(:)g(:)g(:)f(;)h(t)580 2756 y Fh(m)613 2749 y Fv(])63 b(where)41 b Fl(C)895 2731 y Fh(b)912 2749 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])13 b Fn(2)h(T)f Fv(\()p Fn(D)1206 2756 y Fm(1)1226 2749 y Fl(;)8 b Fn(C)s Fl(;)g Fn(V)s Fv(\))16 b Fl(;)25 b(r)q(oot)p Fv(\()p Fl(t)1531 2756 y Fh(j)1549 2749 y Fv(\))14 b Fn(2)g(D)1667 2756 y Fm(2)346 2810 y Fl(C)385 2792 y Fh(w)413 2810 y Fv([)p Fl(u)455 2817 y Fm(1)474 2810 y Fl(;)8 b(:)g(:)g(:)g(;)g(u)612 2817 y Fh(n)635 2810 y Fv(])41 b(where)g Fl(C)895 2792 y Fh(w)923 2810 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])13 b Fn(2)h(T)f Fv(\()p Fn(D)1217 2817 y Fm(2)1237 2810 y Fl(;)8 b Fn(C)s Fl(;)g Fn(V)s Fv(\))17 b Fl(;)24 b(r)q(oot)p Fv(\()p Fl(u)1552 2817 y Fh(j)1570 2810 y Fv(\))14 b Fn(2)g(D)1688 2817 y Fm(1)950 2936 y Fv(22)p eop %%Page: 23 23 bop 0 26 a Fv(F)l(rom)15 b(no)o(w)i(on)f(this)g(will)g(b)q(e)g(denoted)g(b)o (y)732 172 y Fl(s)e Fn(\021)821 72 y Fg(8)821 110 y(>)821 122 y(<)821 197 y(>)821 209 y(:)879 111 y Fl(C)918 93 y Fh(t)932 111 y Fv([)-8 b([)p Fl(s)975 118 y Fm(1)994 111 y Fl(;)8 b(:)g(:)g(:)f(;)h(s) 1126 118 y Fh(l)1139 111 y Fv(])-8 b(])879 171 y Fl(C)918 153 y Fh(b)935 171 y Fv([)g([)p Fl(t)973 178 y Fm(1)991 171 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1118 178 y Fh(m)1151 171 y Fv(])-8 b(])879 232 y Fl(C)918 213 y Fh(w)946 232 y Fv([)g([)p Fl(u)994 239 y Fm(1)1012 232 y Fl(;)8 b(:)g(:)g(:)f(;)h(u)1149 239 y Fh(n)1172 232 y Fv(])-8 b(])0 324 y(In)15 b(this)h(situation,)f(w)o(e)h(de\014ne)f(the) g(m)o(ultiset)e Fl(S)887 306 y Fh(b)904 324 y Fv(\()p Fl(s)p Fv(\))j(\()p Fl(S)1033 306 y Fh(w)1061 324 y Fv(\()p Fl(s)p Fv(\)\))f(of)h Fi(black)j(\(white\))e(princip)n(al)f Fv(subterms)f(of)0 384 y Fl(s)h Fv(to)h(b)q(e)f Fl(S)198 366 y Fh(b)215 384 y Fv(\()p Fl(s)p Fv(\))e(=)g([)p Fl(u)384 391 y Fm(1)403 384 y Fl(;)8 b(:)g(:)g(:)f(;)h(u)540 391 y Fh(n)563 384 y Fv(])16 b(\()p Fl(S)645 366 y Fh(w)673 384 y Fv(\()p Fl(s)p Fv(\))e(=)g([)p Fl(t)832 391 y Fm(1)851 384 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)978 391 y Fh(m)1011 384 y Fv(]\).)0 444 y(The)17 b Fi(topmost)g(black)j (\(white\))e(homo)n(gene)n(ous)g(p)n(art)e Fv(of)h Fl(s)p Fv(,)f(denoted)h(b) o(y)f Fl(top)1397 426 y Fh(b)1415 444 y Fv(\()p Fl(s)p Fv(\))g(\()p Fl(top)1576 426 y Fh(w)1605 444 y Fv(\()p Fl(s)p Fv(\)\),)g(is)h(obtained)0 504 y(from)e Fl(s)h Fv(b)o(y)g(replacing)g(all)g(white)f(\(blac)o(k\))h (principal)f(subterms)g(with)h Ff(2)p Fv(.)0 616 y Fk(Example)g(5.3)24 b Fv(Let)17 b Fn(D)453 623 y Fm(1)487 616 y Fv(=)c Fn(f)p Fl(F)q(;)8 b(A)p Fn(g)p Fv(,)15 b Fn(D)747 623 y Fm(2)781 616 y Fv(=)f Fn(f)p Fl(g)r(;)8 b(b)p Fn(g)p Fv(,)15 b(and)i Fn(C)g Fv(=)d Fn(f)p Fh(C)r Fn(g)p Fv(.)22 b(F)l(or)16 b Fl(s)e Fn(\021)f Fh(C)s Fv(\()p Fl(F)7 b Fv(\()p Fl(b)p Fv(\))p Fl(;)h(g)r Fv(\()p Fl(A)p Fv(\)\))15 b(w)o(e)g(ha)o(v)o(e)447 764 y Fl(s)f Fn(\021)537 664 y Fg(8)537 702 y(>)537 714 y(<)537 789 y(>)537 801 y(:)594 703 y Fl(C)633 685 y Fh(t)648 703 y Fv([)-8 b([)p Fl(F)7 b Fv(\()p Fl(b)p Fv(\))p Fl(;)h(g)r Fv(\()p Fl(A)p Fv(\)])-9 b(])13 b(with)j Fl(C)1070 685 y Fh(t)1084 703 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])14 b Fn(\021)f Fh(C)s Fv(\()p Ff(2)p Fl(;)8 b Ff(2)p Fv(\))594 763 y Fl(C)633 745 y Fh(b)650 763 y Fv([)-8 b([)p Fl(b;)8 b(g)r Fv(\()p Fl(A)p Fv(\)])-8 b(])13 b(with)j Fl(C)996 745 y Fh(b)1013 763 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])13 b Fn(\021)h Fh(C)r Fv(\()p Fl(F)7 b Fv(\()p Ff(2)p Fv(\))p Fl(;)h Ff(2)p Fv(\))594 824 y Fl(C)633 806 y Fh(w)661 824 y Fv([)-8 b([)p Fl(F)7 b Fv(\()p Fl(b)p Fv(\))p Fl(;)h(A)p Fv(])-8 b(])12 b(with)k Fl(C)1020 806 y Fh(w)1048 824 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])13 b Fn(\021)h Fh(C)r Fv(\()p Ff(2)p Fl(;)8 b(g)r Fv(\()p Ff(2)p Fv(\)\))0 916 y(as)17 b(w)o(ell)e(as)i Fl(S)252 898 y Fh(b)269 916 y Fv(\()p Fl(s)p Fv(\))c(=)h([)p Fl(F)7 b Fv(\()p Fl(b)p Fv(\))p Fl(;)h(A)p Fv(])14 b(and)j Fl(S)722 898 y Fh(w)750 916 y Fv(\()p Fl(s)p Fv(\))d(=)g([)p Fl(b;)8 b(g)r Fv(\()p Fl(A)p Fv(\)].)0 1027 y Fk(De\014nition)18 b(5.4)24 b Fv(Let)f Fl(s)g Fv(b)q(e)g(a)h(top)f(blac)o(k)f(term.)41 b(Let)23 b Fl(s)i Fn(\021)h Fl(C)1249 1009 y Fh(b)1265 1027 y Fv([)-8 b([)p Fl(s)1308 1034 y Fm(1)1327 1027 y Fl(;)8 b(:)g(:)g(:)f(;)h(s) 1459 1034 y Fh(n)1482 1027 y Fv(])-8 b(])22 b(and)i Fl(s)h Fn(!)1724 1034 y Fr(R)1782 1027 y Fl(t)e Fv(b)o(y)f(an)0 1087 y(application)16 b(of)h(a)g(rewrite)f(rule)g(of)h Fn(R)d Fv(=)h Fn(R)820 1094 y Fm(1)851 1087 y Fn([)c(R)937 1094 y Fm(2)957 1087 y Fv(.)22 b(W)l(e)17 b(write)f Fl(s)e Fn(!)1290 1069 y Fh(i)1290 1100 y Fr(R)1337 1087 y Fl(t)i Fv(if)g(the)g(rule)g(is)g(applied)g (in)h(one)0 1147 y(of)e(the)f Fl(s)159 1154 y Fh(j)191 1147 y Fv(and)h(w)o(e)f(write)g Fl(s)g Fn(!)564 1129 y Fh(o)564 1160 y Fr(R)610 1147 y Fl(t)g Fv(otherwise.)20 b(The)14 b(relation)g Fn(!)1202 1129 y Fh(i)1202 1160 y Fr(R)1249 1147 y Fv(is)g(called)f Fi(inner)i Fv(reduction)f(and)h Fn(!)1917 1129 y Fh(o)1917 1160 y Fr(R)0 1208 y Fv(is)i(called)e Fi(outer)i Fv(reduction.)23 b(No)o(w)16 b(let)g Fl(s)h Fv(b)q(e)g(a)g(top)h(transparen)o(t)f(term,)e (i.e.,)g Fl(s)f Fn(\021)h Fl(C)1587 1190 y Fh(t)1601 1208 y Fv([)-8 b([)p Fl(s)1644 1215 y Fm(1)1663 1208 y Fl(;)8 b(:)g(:)g(:)f(;)h(s) 1795 1215 y Fh(n)1818 1208 y Fv(])-8 b(])16 b(with)0 1268 y Fl(C)39 1250 y Fh(t)53 1268 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])16 b Fn(6\021)g Ff(2)p Fv(.)26 b(Let)18 b Fl(s)f Fn(!)517 1275 y Fr(R)565 1268 y Fl(t)p Fv(,)g(i.e.,)f Fl(t)g Fn(\021)h Fl(C)837 1250 y Fh(t)851 1268 y Fv([)p Fl(s)888 1275 y Fm(1)907 1268 y Fl(;)8 b(:)g(:)g(:)g(;)g(s)1040 1275 y Fh(j)r Fr(\000)p Fm(1)1103 1268 y Fl(;)g(t)1143 1275 y Fh(j)1160 1268 y Fl(;)g(s)1205 1275 y Fh(j)r Fm(+1)1269 1268 y Fl(;)g(:)g(:)g(:)f(;)h(s)1401 1275 y Fh(n)1424 1268 y Fv(])18 b(for)g(some)e Fl(j)k Fn(2)d(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)f(n)p Fn(g)p Fv(.)0 1328 y(Then)21 b(w)o(e)f(write)g Fl(s)h Fn(!)431 1310 y Fh(i)431 1340 y Fr(R)484 1328 y Fl(t)f Fv(if)g Fl(s)594 1335 y Fh(j)634 1328 y Fn(!)684 1310 y Fh(i)684 1340 y Fr(R)737 1328 y Fl(t)755 1335 y Fh(j)793 1328 y Fv(and)h Fl(s)g Fn(!)986 1310 y Fh(o)986 1340 y Fr(R)1039 1328 y Fl(t)g Fv(if)f Fl(s)1150 1335 y Fh(j)1189 1328 y Fn(!)1239 1310 y Fh(o)1239 1340 y Fr(R)1292 1328 y Fl(t)1310 1335 y Fh(j)1328 1328 y Fv(.)34 b(In)20 b(order)h(to)g(indicate)f(whic)o(h)0 1388 y(TRS)f(the)f(applied)f(rule)h(stems)f(from,)g(w)o(e)h(also)h(use)f(the) g(notation)h Fl(s)f Fn(!)1378 1370 y Fh(o)1378 1401 y Fr(R)1408 1406 y Fj(1)1444 1388 y Fl(t)p Fv(,)p Fl(s)f Fn(!)1566 1370 y Fh(o)1566 1401 y Fr(R)1596 1406 y Fj(2)1632 1388 y Fl(t)p Fv(,)p Fl(s)g Fn(!)1754 1370 y Fh(i)1754 1401 y Fr(R)1784 1406 y Fj(1)1820 1388 y Fl(t)p Fv(,)h(and)0 1448 y Fl(s)c Fn(!)87 1430 y Fh(i)87 1461 y Fr(R)117 1466 y Fj(2)150 1448 y Fl(t)p Fv(.)0 1559 y Fk(De\014nition)k(5.5)24 b Fv(Let)16 b Fl(s)g Fv(b)q(e)h(a)f(top)h(blac)o(k)e(term.)20 b(W)l(e)c(de\014ne)247 1677 y Fl(r)q(ank)r Fv(\()p Fl(s)p Fv(\))e(=)479 1604 y Fg(\()533 1647 y Fv(1)663 b(,)16 b(if)g Fl(s)d Fn(2)h(T)f Fv(\()p Fn(D)1475 1654 y Fm(1)1495 1647 y Fl(;)8 b Fn(C)s Fl(;)g Fn(V)t Fv(\))533 1707 y(1)k(+)f Fl(max)p Fn(f)p Fl(r)q(ank)r Fv(\()p Fl(s)887 1714 y Fh(j)904 1707 y Fv(\))24 b Fn(j)h Fv(1)14 b Fn(\024)g Fl(j)i Fn(\024)e Fl(n)p Fn(g)p Fv(,)i(if)g Fl(s)d Fn(\021)h Fl(C)1423 1689 y Fh(b)1440 1707 y Fv([)-8 b([)p Fl(s)1483 1714 y Fm(1)1501 1707 y Fl(;)8 b(:)g(:)g(:)g(;)g(s)1634 1714 y Fh(n)1657 1707 y Fv(])-8 b(])0 1795 y(No)o(w)16 b(let)g Fl(s)g Fv(b)q(e)g(a)h(top)g (transparen)o(t)f(term.)k(Then)c(w)o(e)g(de\014ne)287 1912 y Fl(r)q(ank)r Fv(\()p Fl(s)p Fv(\))e(=)519 1839 y Fg(\()573 1882 y Fv(0)587 b(,)15 b(if)h Fl(s)e Fn(2)g(T)f Fv(\()p Fn(C)s Fl(;)8 b Fn(V)t Fv(\))573 1942 y Fl(max)p Fn(f)p Fl(r)q(ank)r Fv(\()p Fl(t)837 1949 y Fh(j)854 1942 y Fv(\))24 b Fn(j)h Fv(1)14 b Fn(\024)g Fl(j)i Fn(\024)e Fl(m)p Fn(g)p Fv(,)h(if)h Fl(s)e Fn(\021)f Fl(C)1386 1924 y Fh(t)1401 1942 y Fv([)-8 b([)p Fl(t)1439 1949 y Fm(1)1457 1942 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1584 1949 y Fh(m)1617 1942 y Fv(])-8 b(])73 2054 y(As)19 b(for)g(disjoin)o(t)f(unions,) h(w)o(e)g(ha)o(v)o(e)f Fl(s)g Fn(!)853 2036 y Fr(\003)890 2054 y Fl(t)g Fn(\))g Fl(r)q(ank)r Fv(\()p Fl(s)p Fv(\))g Fn(\025)g Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\).)28 b(W)l(e)19 b(will)e(also)j(use)e(sp)q (ecial)0 2115 y(notations)i(for)g(\\degenerate")g(cases)f(of)h Fl(s)f Fn(\021)g Fl(C)914 2096 y Fh(b)930 2115 y Fv([)-8 b([)p Fl(t)968 2122 y Fm(1)987 2115 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1114 2122 y Fh(m)1147 2115 y Fv(])-8 b(])18 b(and)i Fl(s)f Fn(\021)g Fl(C)1422 2096 y Fh(w)1449 2115 y Fv([)-8 b([)p Fl(u)1497 2122 y Fm(1)1516 2115 y Fl(;)8 b(:)g(:)g(:)f(;)h(u)1653 2122 y Fh(n)1676 2115 y Fv(])-8 b(].)30 b(These)19 b(are)0 2175 y(de\014ned)d(in)g(analogy)h (to)g(those)f(used)h(for)f(the)g(disjoin)o(t)g(union)g(case.)0 2337 y Fd(5.2)70 b(Mo)r(dular)23 b(Prop)r(erties)f(of)h(TRSs)g(with)f(Shared) h(Constructors)0 2450 y Fv(A)c(prop)q(ert)o(y)g Fn(P)k Fv(is)c(called)f Fi(mo)n(dular)h(for)h(TRSs)g(with)g(shar)n(e)n(d)f(c)n(onstructors)g Fv(if)g(for)g(all)g(TRS)h(\()p Fn(F)1827 2457 y Fm(1)1846 2450 y Fl(;)8 b Fn(R)1910 2457 y Fm(1)1930 2450 y Fv(\))0 2510 y(and)16 b(\()p Fn(F)149 2517 y Fm(2)168 2510 y Fl(;)8 b Fn(R)232 2517 y Fm(2)252 2510 y Fv(\))15 b(whic)o(h)g(share)h(at)f(most)g(constructors)h (the)f(com)o(bined)e(system)h(\()p Fn(F)1521 2517 y Fm(1)1549 2510 y Fn([)9 b(F)1627 2517 y Fm(2)1647 2510 y Fl(;)f Fn(R)1711 2517 y Fm(1)1740 2510 y Fn([)h(R)1824 2517 y Fm(2)1844 2510 y Fv(\))15 b(has)0 2570 y(the)h(prop)q(ert)o(y)h Fn(P)j Fv(if)c(and)h(only)f (if)g(b)q(oth)i(\()p Fn(F)802 2577 y Fm(1)821 2570 y Fl(;)8 b Fn(R)885 2577 y Fm(1)905 2570 y Fv(\))16 b(and)h(\()p Fn(F)1090 2577 y Fm(2)1110 2570 y Fl(;)8 b Fn(R)1174 2577 y Fm(2)1194 2570 y Fv(\))16 b(ha)o(v)o(e)g(the)g(prop)q(ert)o(y)h Fn(P)t Fv(.)k(Clearly)l(,)15 b(if)h(a)0 2630 y(prop)q(ert)o(y)h Fn(P)j Fv(is)d(mo)q(dular)f(for)h(TRSs)g(with)g(shared)g(constructors,)g(then)f(it)h (is)f(esp)q(ecially)f(mo)q(dular)i(for)0 2691 y(disjoin)o(t)f(TRSs)i(but)f (not)g(vice)f(v)o(ersa.)23 b(E.g.)h(con\015uence)16 b(is)h(a)g(mo)q(dular)g (prop)q(ert)o(y)g(of)g(\(disjoin)o(t\))f(TRSs)0 2751 y(\(see)e([T)l(o)o(y87b) q(]\))g(but)h(not)g(a)g(mo)q(dular)f(prop)q(ert)o(y)h(of)g(TRSs)g(with)g (shared)g(constructors)g(as)h(is)e(witnessed)0 2811 y(b)o(y)i(the)g(follo)o (wing)g(example)e(\(cf.)20 b([K)o(O92]\):)950 2936 y(23)p eop %%Page: 24 24 bop 0 26 a Fk(Example)16 b(5.6)24 b Fv(The)d(TRSs)h Fn(R)613 33 y Fm(1)655 26 y Fv(=)g Fn(f)p Fl(F)7 b Fv(\()p Fl(x;)h(x)p Fv(\))p Fn(!)n Fl(A;)g(F)f Fv(\()p Fl(x;)h Fh(C)q Fv(\()p Fl(x)p Fv(\)\))p Fn(!)p Fl(B)s Fn(g)20 b Fv(and)i Fn(R)1498 33 y Fm(2)1540 26 y Fv(=)g Fn(f)p Fl(a)p Fn(!)o Fh(C)s Fv(\()p Fl(a)p Fv(\))p Fn(g)e Fv(share)0 86 y(the)15 b(constructor)g Fh(C)j Fv(and)e(are)f(b)q(oth)h (con\015uen)o(t.)21 b(Their)14 b(com)o(bined)f(system)h(is,)h(ho)o(w)o(ev)o (er,)e(not)j(con\015uen)o(t)0 147 y(since)f(the)h(term)f Fl(F)7 b Fv(\()p Fl(a;)h(a)p Fv(\))15 b(has)i(the)f(t)o(w)o(o)g(normal)f(forms)g Fl(A)h Fv(and)h Fl(B)s Fv(.)73 274 y(Note)12 b(that)h(the)f(ab)q(o)o(v)o(e)g (example)e(is)i(also)h(a)g(coun)o(terexample)c(to)k(the)f(mo)q(dularit)o(y)f (of)h(unique)g(normal)0 334 y(forms)19 b(for)i(constructor)f(sharing)h(TRSs.) 34 b(W)l(e)19 b(next)h(presen)o(t)g(some)f(p)q(ositiv)o(e)g(results)h(whic)o (h)f(can)i(b)q(e)0 394 y(pro)o(v)o(ed)16 b(b)o(y)f(the)h(same)f(means)h(as)h (used)f(in)g(Middeldorp)f([Mid90])h(for)g(pro)o(ving)h(the)f(resp)q(ectiv)o (e)e(results)0 455 y(for)j(disjoin)o(t)e(unions.)0 582 y Fk(Prop)r(osition)j (5.7)24 b Fv(Lo)q(cal)14 b(con\015uence)f(is)g(a)g(mo)q(dular)g(prop)q(ert)o (y)g(of)h(TRSs)g(with)f(shared)h(constructors.)0 663 y Fk(Pro)r(of:)21 b Fv(Let)15 b Fn(R)f Fv(=)g Fn(R)409 670 y Fm(1)439 663 y Fn([)c(R)524 670 y Fm(2)559 663 y Fv(b)q(e)16 b(the)f(com)o(bined)e(system)h(of)i(t)o(w)o (o)g(constructor)g(sharing)g(TRSs)g Fn(R)1835 670 y Fm(1)1870 663 y Fv(and)0 723 y Fn(R)42 730 y Fm(2)62 723 y Fv(.)21 b(Let)16 b Fl(C)t(P)7 b Fv(\()p Fn(R)p Fv(\))15 b(denote)h(the)f(set)h(of)g(critical)e (pairs)i(of)g Fn(R)p Fv(.)21 b(Since)15 b Fn(R)1309 730 y Fm(1)1345 723 y Fv(and)h Fn(R)1481 730 y Fm(2)1517 723 y Fv(do)g(not)g(share)g (de\014ned)0 783 y(function)e(sym)o(b)q(ols,)g(rewrite)g(rules)g(of)h Fn(R)763 790 y Fm(1)798 783 y Fv(do)g(not)g(o)o(v)o(erlap)f(with)g(rewrite)g (rules)g(of)h Fn(R)1601 790 y Fm(2)1636 783 y Fv(and)g(vice)f(v)o(ersa.)0 844 y(Hence)i Fl(C)t(P)7 b Fv(\()p Fn(R)p Fv(\))15 b(=)h Fl(C)t(P)7 b Fv(\()p Fn(R)510 851 y Fm(1)530 844 y Fv(\))k Fn([)h Fl(C)t(P)7 b Fv(\()p Fn(R)743 851 y Fm(2)763 844 y Fv(\))17 b(and)h(the)f(assertion)h (follo)o(ws)f(easily)g(from)f(the)h(Critical)f(P)o(air)0 904 y(Lemma.)j Ff(2)73 1031 y Fv(Without)f(c)o(hanging)f(the)g(pro)q(of,)h(the)f (ab)q(o)o(v)o(e)h(prop)q(osition)g(can)f(b)q(e)h(generalized)e(to)h(\(see)g ([Mid90]\):)0 1091 y Fi(The)g(union)g(of)g(two)g(non-interfering)i(\(i.e.,)e (r)n(ewrite)g(rules)g(of)g Fn(R)1221 1098 y Fm(1)1257 1091 y Fi(do)g(not)g(overlap)g(with)h(r)n(ewrite)e(rules)0 1152 y(of)h Fn(R)99 1159 y Fm(2)137 1152 y Fi(and)g(vic)n(e)h(versa\))g(lo)n(c)n (al)r(ly)g(c)n(on\015uent)h(TRSs)e(is)g(lo)n(c)n(al)r(ly)h(c)n(on\015uent.)73 1233 y Fv(In)j(order)g(to)h(sho)o(w)g(mo)q(dularit)o(y)d(of)j(w)o(eak)f (termination,)f(w)o(e)h(mak)o(e)e(use)i(of)h(the)f(follo)o(wing)g(idea:)0 1293 y(Ev)o(ery)13 b(term)e Fl(t)j Fn(2)g(T)f Fv(\()p Fn(D)427 1300 y Fm(1)452 1293 y Fn(])6 b(D)529 1300 y Fm(2)549 1293 y Fl(;)i Fn(C)s Fl(;)g Fn(V)t Fv(\))13 b(can)h(b)q(e)f(reduced)g(to)h(normal) f(form)f(via)h(innermost)g(rewriting;)g(i.e.,)0 1353 y(\014rst)18 b(the)f(b)q(ottom)g(la)o(y)o(er)f(of)i Fl(t)f Fv(is)g(reduced)g(to)h(normal)e (form,)g(then)h(the)h(same)e(is)h(done)h(with)f(the)g(la)o(y)o(er)0 1413 y(ab)q(o)o(v)o(e)g(the)f(b)q(ottom)h(la)o(y)o(er)e(and)j(so)f(on.)24 b(Ev)o(en)o(tually)15 b(the)h(top)h(la)o(y)o(er)f(is)g(reduced)h(to)g(normal) f(form;)f(the)0 1473 y(term)g(obtained)h(is)h(a)g(normal)e(form)g(of)i Fl(t)p Fv(.)22 b(In)16 b(the)g(sequel)f(w)o(e)h(use)h(for)g(an)o(y)f(TRS)h (\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))16 b(the)g(notation)0 1534 y Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)p Fv(\))13 b(=)h Fn(f)p Fl(t)f Fn(2)h(T)f Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))16 b Fn(j)g Fl(t)d Fn(2)h Fl(N)5 b(F)i Fv(\()p Fn(!)846 1541 y Fr(R)878 1534 y Fv(\))p Fn(g)p Fv(.)0 1661 y Fk(Lemma)16 b(5.8)24 b Fv(If)12 b(\()p Fn(F)395 1668 y Fm(1)414 1661 y Fl(;)c Fn(R)478 1668 y Fm(1)498 1661 y Fv(\))k(and)h(\()p Fn(F)675 1668 y Fm(2)695 1661 y Fl(;)8 b Fn(R)759 1668 y Fm(2)779 1661 y Fv(\))k(are)h(arbitrary)f (TRSs,)i(and)f(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))26 b(=)g(\()p Fn(F)1615 1668 y Fm(1)1638 1661 y Fn([)s(F)1710 1668 y Fm(2)1730 1661 y Fl(;)8 b Fn(R)1794 1668 y Fm(1)1818 1661 y Fn([)s(R)1896 1668 y Fm(2)1916 1661 y Fv(\),)0 1721 y(then)16 b Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)p Fv(\))13 b(=)h Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)609 1728 y Fm(1)628 1721 y Fv(\))j Fn(\\)g Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)909 1728 y Fm(2)929 1721 y Fv(\).)0 1802 y Fk(Pro)r(of:)21 b Fv(This)c(follo)o(ws)f(imm)o (ediatel)o(y)d(from)j Fn(!)888 1809 y Fr(R)918 1814 y Fj(1)935 1809 y Fr([R)989 1814 y Fj(2)1008 1802 y Fv(=)p Fn(!)1096 1809 y Fr(R)1126 1814 y Fj(1)1159 1802 y Fn([)e(!)1256 1809 y Fr(R)1286 1814 y Fj(2)1305 1802 y Fv(.)21 b Ff(2)0 1930 y Fk(Prop)r(osition)d(5.9)24 b Fv(W)l(eak)16 b(termination)e(is)i(a)h(mo)q(dular)f(prop)q(ert)o(y)g(of)h (constructor)f(sharing)h(TRSs.)0 2011 y Fk(Pro)r(of:)33 b Fv(Let)22 b(\()p Fn(F)333 2018 y Fm(1)352 2011 y Fl(;)8 b Fn(R)416 2018 y Fm(1)436 2011 y Fv(\))22 b(and)h(\()p Fn(F)633 2018 y Fm(2)652 2011 y Fl(;)8 b Fn(R)716 2018 y Fm(2)736 2011 y Fv(\))22 b(b)q(e)g (constructor)h(sharing)f(TRSs.)39 b(W)l(e)22 b(ha)o(v)o(e)f(to)i(sho)o(w)f (that)0 2071 y(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))14 b(=)f(\()p Fn(F)263 2078 y Fm(1)291 2071 y Fn([)8 b(F)368 2078 y Fm(2)388 2071 y Fl(;)g Fn(R)452 2078 y Fm(1)479 2071 y Fn([)g(R)562 2078 y Fm(2)582 2071 y Fv(\))15 b(is)g(w)o(eakly)e(terminating)g(if)i(and)g (only)f(if)h(b)q(oth)g(\()p Fn(F)1541 2078 y Fm(1)1561 2071 y Fl(;)8 b Fn(R)1625 2078 y Fm(1)1645 2071 y Fv(\))14 b(and)i(\()p Fn(F)1827 2078 y Fm(2)1846 2071 y Fl(;)8 b Fn(R)1910 2078 y Fm(2)1930 2071 y Fv(\))0 2131 y(are)17 b(w)o(eakly)f(terminating.)21 b(The)16 b(\\only-if)s(")i(case)f(b)q(eing)g(trivial,)e(w)o(e)h(turn)h(to)g (pro)o(ving)g(the)f(\\if)s(")i(direc-)0 2191 y(tion.)i(W)l(e)13 b(pro)q(ceed)h(b)o(y)f(induction)f(on)i Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))f(to)h(sho)o(w)g(that)g(ev)o(ery)e(term)f Fl(t)j Fn(2)g(T)e Fv(\()p Fn(F)5 b Fl(;)j Fn(V)t Fv(\))13 b(has)h(a)g(normal)0 2251 y(form)h(w.r.t.)21 b Fn(!)301 2258 y Fr(R)333 2251 y Fv(.)g(If)16 b Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))d(=)h(0,)i(then)g Fl(t)g Fv(is)g(ob)o(viously)g(a)g(normal)g(form.)k(So)d(let)e Fl(r)q(ank)r Fv(\()p Fl(t)p Fv(\))f(=)f Fl(k)j(>)e Fv(0.)0 2312 y(Case)19 b(\(i\):)24 b Fl(t)18 b Fv(is)g(top)g(blac)o(k.)26 b(Then)18 b(w)o(e)g(ma)o(y)f(write)g Fl(t)g Fn(\021)g Fl(C)1103 2293 y Fh(b)1119 2312 y Fn(f)-17 b(f)p Fl(t)1170 2319 y Fm(1)1190 2312 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1317 2319 y Fh(n)1340 2312 y Fn(g)-17 b(g)p Fv(.)27 b(By)18 b(induction)f(h)o(yp)q(othesis,)0 2372 y(for)i(ev)o(ery)f Fl(j)k Fn(2)d(f)p Fv(1)p Fl(;)8 b(:)g(:)g(:)f(;)h(n)p Fn(g)p Fv(,)19 b(w)o(e)g(ha)o(v)o(e)f Fl(t)756 2379 y Fh(j)793 2372 y Fn(!)843 2354 y Fr(\003)843 2384 y(R)894 2372 y Fl(t)912 2354 y Fr(0)912 2384 y Fh(j)949 2372 y Fv(where)g Fl(t)1110 2354 y Fr(0)1110 2384 y Fh(j)1147 2372 y Fn(2)h Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)p Fv(\).)29 b(Then)19 b Fl(C)1637 2354 y Fh(b)1654 2372 y Fn(f)p Fl(t)1697 2354 y Fr(0)1697 2384 y Fm(1)1716 2372 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1844 2354 y Fr(0)1844 2384 y Fh(n)1866 2372 y Fn(g)19 b(\021)11 2429 y Fv(^)0 2442 y Fl(C)39 2424 y Fh(b)56 2442 y Fn(f)-17 b(f)p Fl(s)112 2449 y Fm(1)132 2442 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)264 2449 y Fh(m)297 2442 y Fn(g)-17 b(g)18 b Fv(for)h(some)e(blac)o(k)g(con)o(text)865 2429 y(^)854 2442 y Fl(C)893 2424 y Fh(b)909 2442 y Fn(f)-17 b(f)p Fl(;)8 b(:)g(:)g(:)g(;)g Fn(g)-17 b(g)18 b Fv(and)h(top)g(white)e (normal)h(forms)f Fl(s)1750 2449 y Fm(1)1770 2442 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1902 2449 y Fh(m)1935 2442 y Fv(.)0 2502 y(Cho)q(ose)20 b(fresh)e(v)m(ariables)g Fl(x)524 2509 y Fm(1)543 2502 y Fl(;)8 b(:)g(:)g(:)g(;)g(x)681 2509 y Fh(m)732 2502 y Fv(satisfying)18 b(the)g(prop)q(ert)o(y)g Fl(<)9 b(s)1308 2509 y Fm(1)1328 2502 y Fl(;)f(:)g(:)g(:)f(;)h(s)1460 2509 y Fh(m)1502 2502 y Fl(>)17 b Fn(1)g Fl(<)9 b(x)1699 2509 y Fm(1)1718 2502 y Fl(;)f(:)g(:)g(:)g(;)g(x)1856 2509 y Fh(m)1897 2502 y Fl(>)p Fv(.)0 2562 y(Since)142 2550 y(^)133 2562 y Fn(C)162 2544 y Fh(b)179 2562 y Fn(f)p Fl(x)232 2569 y Fm(1)252 2562 y Fl(;)g(:)g(:)g(:)f(;)h(x)389 2569 y Fh(m)422 2562 y Fn(g)23 b(2)h(T)12 b Fv(\()p Fn(F)621 2569 y Fm(1)641 2562 y Fl(;)c Fn(V)t Fv(\))21 b(and)i(the)e(TRS)h Fn(R)1090 2569 y Fm(1)1132 2562 y Fv(is)f(w)o(eakly)g(terminating,)g(it)h(follo)o(ws)f(that)11 2610 y(^)0 2622 y Fl(C)39 2604 y Fh(b)56 2622 y Fn(f)p Fl(x)109 2629 y Fm(1)128 2622 y Fl(;)8 b(:)g(:)g(:)f(;)h(x)265 2629 y Fh(m)298 2622 y Fn(g)25 b(!)398 2604 y Fr(\003)398 2635 y(R)428 2640 y Fj(1)472 2622 y Fl(C)511 2604 y Fr(\003)530 2622 y Fn(h)p Fl(x)577 2629 y Fh(i)589 2634 y Fj(1)609 2622 y Fl(;)8 b(:)g(:)g(:)f(;)h(x) 746 2629 y Fh(i)758 2635 y Fe(l)772 2622 y Fn(i)23 b Fv(for)g(some)e(normal)h (form)f Fl(C)1355 2604 y Fr(\003)1375 2622 y Fn(h)p Fl(x)1422 2629 y Fh(i)1434 2634 y Fj(1)1453 2622 y Fl(;)8 b(:)g(:)g(:)f(;)h(x)1590 2629 y Fh(i)1602 2635 y Fe(l)1616 2622 y Fn(i)p Fv(.)41 b(By)22 b(the)g(sp)q(e-)0 2691 y(cial)k(c)o(hoice)g(of)h(the)f(v)m(ariables)h Fl(x)657 2698 y Fm(1)676 2691 y Fl(;)8 b(:)g(:)g(:)g(;)g(x)814 2698 y Fh(m)847 2691 y Fv(,)29 b(it)d(follo)o(ws)h(that)g Fl(t)k Fn(!)1335 2673 y Fr(\003)1335 2703 y(R)1365 2708 y Fj(1)1383 2703 y Fr([R)1437 2708 y Fj(2)1499 2678 y Fv(^)1487 2691 y Fl(C)1526 2673 y Fh(b)1543 2691 y Fn(f)-17 b(f)p Fl(s)1599 2698 y Fm(1)1619 2691 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1751 2698 y Fh(m)1784 2691 y Fn(g)-17 b(g)33 b(!)1900 2673 y Fh(o)1900 2703 y Fr(R)1930 2708 y Fj(1)0 2751 y Fl(C)39 2733 y Fr(\003)58 2751 y Fn(h)-8 b(h)p Fl(s)111 2758 y Fh(i)123 2763 y Fj(1)143 2751 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)275 2758 y Fh(i)287 2764 y Fe(l)301 2751 y Fn(i)-8 b(i)19 b(\021)f Fl(t)425 2733 y Fr(0)436 2751 y Fv(.)30 b(Clearly)l(,)18 b Fl(t)680 2733 y Fr(0)709 2751 y Fn(2)h Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)968 2758 y Fm(2)987 2751 y Fv(\).)29 b(By)18 b(construction)h Fl(t)1428 2733 y Fr(0)1458 2751 y Fn(2)f Fl(N)5 b(F)i Fv(\()p Fn(!)1661 2733 y Fh(o)1661 2763 y Fr(R)1691 2768 y Fj(1)1710 2751 y Fv(\))19 b(and)h(since)0 2811 y Fl(s)23 2818 y Fh(i)35 2823 y Fj(1)54 2811 y Fl(;)8 b(:)g(:)g(:)g(;)g(s)187 2818 y Fh(i)199 2824 y Fe(l)232 2811 y Fn(2)20 b Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)o Fv(\),)20 b(w)o(e)f(also)h(ha)o(v)o(e)f Fl(t)854 2793 y Fr(0)884 2811 y Fn(2)h Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)1144 2818 y Fm(1)1163 2811 y Fv(\).)31 b(Hence)18 b(it)h(follo)o(ws)h (from)e(Lemma)f(5.8)950 2936 y(24)p eop %%Page: 25 25 bop 0 26 a Fv(that)17 b Fl(t)124 8 y Fr(0)149 26 y Fn(2)d Fl(N)5 b(F)i Fv(\()p Fn(F)e Fl(;)j Fn(R)o Fv(\).)0 86 y(Case)17 b(\(ii\))e(:)21 b Fl(t)16 b Fv(is)g(top)h(white.)k(Analogous)c(to)g(Case)f(\(i\).)0 147 y(Case)k(\(iii\))e(:)27 b Fl(t)19 b Fv(is)g(top)g(transparen)o(t.)31 b(Let)19 b Fl(t)g Fn(\021)f Fl(C)950 129 y Fh(t)965 147 y Fv([)-8 b([)p Fl(t)1003 154 y Fm(1)1021 147 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1148 154 y Fh(n)1171 147 y Fv(])-8 b(])18 b(with)h Fl(C)1362 129 y Fh(t)1377 147 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h Fv(])18 b Fn(6\021)g Ff(2)p Fv(.)31 b(According)18 b(to)0 207 y(Cases)g(\(i\))f(and)h (\(ii\),)f(ev)o(ery)f Fl(t)547 214 y Fh(j)582 207 y Fv(has)i(a)g(normal)f (form)f Fl(t)1013 189 y Fr(0)1013 219 y Fh(j)1048 207 y Fv(w.r.t.)24 b Fn(!)1237 214 y Fr(R)1269 207 y Fv(.)h(Clearly)l(,)17 b Fl(t)1507 189 y Fr(0)1534 207 y Fn(\021)f Fl(C)1628 189 y Fh(t)1642 207 y Fv([)p Fl(t)1674 189 y Fr(0)1674 219 y Fm(1)1693 207 y Fl(;)8 b(:)g(:)g(:)f(;)h(t)1820 189 y Fr(0)1820 219 y Fh(n)1843 207 y Fv(])17 b(is)g(a)0 267 y(normal)e(form)g(of)i Fl(t)f Fv(w.r.t.)k Fn(!)555 274 y Fr(R)587 267 y Fv(.)i Ff(2)0 432 y Fd(5.3)70 b(Mo)r(dularit)n(y)22 b(of)i(T)-6 b(ermination)21 b(of)i(Constructor)h (Sharing)f(TRSs)0 545 y Fv(W)l(e)e(ha)o(v)o(e)f(seen)h(that,)h(in)e(con)o (trast)i(to)f(termination,)f(simple)f(termination)g(is)i(mo)q(dular.)34 b(But)21 b(do)q(es)0 605 y(this)c(result)f(carry)h(o)o(v)o(er)e(to)j (constructor)f(sharing)g(systems?)22 b(As)17 b(already)g(men)o(tioned,)d (Kurihara)j(and)0 666 y(Oh)o(uc)o(hi)i(sho)o(w)o(ed)h(in)g([K)o(O92])f(that)i (the)f(simplifyi)o(ng)e(prop)q(ert)o(y)i(is)g(mo)q(dular)f(for)i(TRSs)f(with) g(shared)0 726 y(constructors.)i(This)15 b(en)o(tails)f(the)h(mo)q(dularit)o (y)e(of)j(simple)c(termination)i(of)h(\014nite)g(constructor)g(sharing)0 786 y(TRSs.)21 b(Also,)13 b(Gramlic)o(h)f([Gra92b)q(])h(extended)g(his)g (abstract)i(result)e(to)h(\014nitely)e(branc)o(hing)i(TRSs)g(with)0 846 y(shared)i(constructors.)21 b(Ho)o(w)o(ev)o(er,)13 b(the)i(next)g(coun)o (terexamples)e(sho)o(w)j(that)f(Theorem)f(4.14)i(and)g(some)0 906 y(corollaries)g(thereof)g(do)g(not)h(extend)f(to)g(constructor)h(sharing) g(TRSs.)0 1041 y Fk(Example)f(5.10)24 b Fv(Let)19 b Fn(R)487 1048 y Fm(1)525 1041 y Fv(=)f Fn(f)p Fl(F)638 1048 y Fh(j)655 1041 y Fv(\()p Fh(C)702 1048 y(j)720 1041 y Fl(;)8 b(x)p Fv(\))p Fn(!)p Fl(F)871 1048 y Fh(j)r Fm(+1)933 1041 y Fv(\()p Fl(x;)g(x)p Fv(\))p Fl(;)g(F)1103 1048 y Fh(j)1120 1041 y Fv(\()p Fl(x;)g(y)r Fv(\))17 b Fn(!)h Fl(x;)8 b(F)1401 1048 y Fh(j)1418 1041 y Fv(\()p Fl(x;)g(y)r Fv(\))17 b Fn(!)h Fl(y)i Fn(j)f Fl(j)i Fn(2)d Fc(I)-7 b(N)q Fn(g)18 b Fv(and)0 1102 y Fn(R)42 1109 y Fm(2)77 1102 y Fv(=)e Fn(f)p Fl(g)r Fv(\()p Fl(x;)8 b(y)r Fv(\))15 b Fn(!)g Fl(x;)8 b(g)r Fv(\()p Fl(x;)g(y)r Fv(\))14 b Fn(!)i Fl(y)r(;)8 b(a)p Fn(!)o Fh(C)794 1109 y(j)830 1102 y Fn(j)17 b Fl(j)h Fn(2)e Fc(I)-7 b(N)q Fn(g)17 b Fv(whic)o(h)g(share)g(the)g (constructors)h Fn(f)p Fh(C)1716 1109 y(j)1751 1102 y Fn(j)f Fl(j)i Fn(2)d Fc(I)-7 b(N)p Fn(g)p Fv(.)0 1162 y Fn(R)42 1169 y Fm(1)85 1162 y Fv(and)23 b Fn(R)228 1169 y Fm(2)271 1162 y Fv(are)g(b)q(oth)h(simply)d(terminating)g(and)j(non-deterministically)c (collapsing)j(\(hence)f Fn(C)1909 1169 y Fr(E)1933 1162 y Fv(-)0 1222 y(terminating\))15 b(but)h(their)f(com)o(bined)g(system)f(with)i(shared) h(constructors)g Fn(R)g Fv(is)f(not)g(terminating:)431 1332 y Fl(F)463 1339 y Fm(1)483 1332 y Fv(\()p Fh(C)529 1339 y Fm(1)549 1332 y Fl(;)8 b(a)p Fv(\))p Fn(!)665 1339 y Fr(R)697 1332 y Fl(F)729 1339 y Fm(2)749 1332 y Fv(\()p Fl(a;)g(a)p Fv(\))p Fn(!)909 1339 y Fr(R)942 1332 y Fl(F)974 1339 y Fm(2)993 1332 y Fv(\()p Fh(C)1039 1339 y Fm(2)1059 1332 y Fl(;)g(a)p Fv(\))p Fn(!)1175 1339 y Fr(R)1208 1332 y Fl(F)1240 1339 y Fm(3)1259 1332 y Fv(\()p Fl(a;)g(a)p Fv(\))p Fn(!)1420 1339 y Fr(R)1460 1332 y Fl(:)g(:)g(:)73 1467 y Fv(Using)20 b(the)g(results)g(from)f(Section)h (3)h(this)f(can)g(b)q(e)g(in)o(terpreted)f(as)i(follo)o(ws:)29 b(If)20 b Fn(R)1663 1474 y Fm(1)1703 1467 y Fv(and)h Fn(R)1844 1474 y Fm(2)1884 1467 y Fv(are)0 1527 y(simplifying,)13 b(i.e.,)g(their)i (rewrite)g(relations)h(are)g(con)o(tained)f(in)g(some)g(simplication)f (orderings)i Fl(>)1835 1534 y Fm(1)1870 1527 y Fv(and)0 1587 y Fl(>)38 1594 y Fm(2)58 1587 y Fv(,)h(then)h(w)o(e)f(can)h(\014nd)g(a)g (simpli\014cation)e(ordering)i Fl(>)1042 1594 y Fm(1)p Fh(;)p Fm(2)1107 1587 y Fv(whic)o(h)f(con)o(tains)g Fn(!)1490 1594 y Fr(R)1520 1599 y Fj(1)1538 1594 y Fr([R)1592 1599 y Fj(2)1611 1587 y Fv(.)25 b(But)18 b(if)f Fl(>)1833 1594 y Fm(1)1870 1587 y Fv(and)0 1647 y Fl(>)38 1654 y Fm(2)76 1647 y Fv(are)i(additionally)f(w)o (ell-founded)h(\(i.e.,)e Fn(R)876 1654 y Fm(1)915 1647 y Fv(and)i Fn(R)1054 1654 y Fm(2)1093 1647 y Fv(are)g(ev)o(en)e(simply)g(terminating\),) g(then)i Fl(>)1902 1654 y Fm(1)p Fh(;)p Fm(2)0 1708 y Fv(is)e(in)g(general)g (not)g(w)o(ell-founded)g(again.)24 b(In)17 b(the)g(ab)q(o)o(v)o(e)g(example)e (the)i(TRSs)g(share)h(in\014nitely)d(man)o(y)0 1768 y(constructors.)22 b(This)16 b(is)g(not)h(essen)o(tial.)0 1903 y Fk(Example)f(5.11)24 b Fv(Let)d Fn(R)489 1910 y Fm(1)530 1903 y Fv(=)g Fn(f)p Fl(F)646 1910 y Fh(j)664 1903 y Fv(\()p Fh(C)710 1885 y(j)729 1903 y Fv(\()p Fl(x)p Fv(\))p Fl(;)8 b(x)p Fv(\))p Fn(!)o Fl(F)945 1910 y Fh(j)r Fm(+1)1007 1903 y Fv(\()p Fl(x;)g(x)p Fv(\))p Fl(;)g(F)1177 1910 y Fh(j)1194 1903 y Fv(\()p Fl(x;)g(y)r Fv(\))21 b Fn(!)g Fl(x;)8 b(F)1482 1910 y Fh(j)1499 1903 y Fv(\()p Fl(x;)g(y)r Fv(\))20 b Fn(!)h Fl(y)h Fn(j)f Fl(j)j Fn(2)d Fc(I)-7 b(N)q Fn(g)0 1963 y Fv(and)17 b Fn(R)137 1970 y Fm(2)171 1963 y Fv(=)e Fn(f)p Fl(g)r Fv(\()p Fl(x;)8 b(y)r Fv(\))13 b Fn(!)h Fl(x;)8 b(g)r Fv(\()p Fl(x;)g(y)r Fv(\))13 b Fn(!)i Fl(y)r(;)8 b(h)p Fv(\()p Fl(x)p Fv(\))p Fn(!)o Fh(C)950 1945 y(j)968 1963 y Fv(\()p Fl(x)p Fv(\))16 b Fn(j)h Fl(j)g Fn(2)e Fc(I)-7 b(N)q Fn(g)16 b Fv(whic)o(h)g(only)g(share)h(the)f(constructor)0 2023 y Fh(C)r Fv(,)g(where)624 2083 y Fh(C)651 2063 y(j)669 2083 y Fv(\()p Fl(x)p Fv(\))e Fn(\021)f Fh(C)s Fv(\()p Fh(C)r Fv(\()p Fh(C)s Fv(\()p Fl(:)8 b(:)g(:)f Fv(\()p Fh(C)s Fv(\()p Fh(C)r Fv(\()p Fl(x)p Fv(\)\)\))h Fl(:)g(:)g(:)o Fv(\)\)\))801 2113 y Fg(|)p 820 2113 225 5 v 225 w({z)p 1083 2113 V 225 w(})997 2154 y Fh(j)r Fr(\000)p Fh(times)0 2231 y Fn(R)42 2238 y Fm(1)78 2231 y Fv(and)17 b Fn(R)215 2238 y Fm(2)251 2231 y Fv(exhibit)e(the)h(same)f (b)q(eha)o(vior)i(as)f(the)g(TRSs)h(ab)q(o)o(v)o(e.)73 2366 y(Next)e(w)o(e)h(collect)f(some)g(p)q(ositiv)o(e)h(results.)0 2501 y Fk(Prop)r(osition)i(5.12)24 b Fv(Simple)15 b(termination)h(is)h(a)h (mo)q(dular)f(prop)q(ert)o(y)h(of)g(constructor)g(sharing)g(TRSs)0 2561 y(whic)o(h)e(in)o(tro)q(duce)f(only)h(\014nitely)f(man)o(y)g(function)h (sym)o(b)q(ols.)0 2642 y Fk(Pro)r(of:)k Fv(Clearly)l(,)13 b(the)h(union)h(of) f(t)o(w)o(o)g(TRSs)h(whic)o(h)f(in)o(tro)q(duce)f(only)h(\014nitely)f(man)o (y)g(function)h(sym)o(b)q(ols)0 2702 y(also)k(in)o(tro)q(duces)f(only)h (\014nitely)e(man)o(y)g(function)h(sym)o(b)q(ols.)24 b(Th)o(us,)17 b(the)h(prop)q(osition)g(follo)o(ws)g(directly)0 2762 y(from)d(Theorem)g(1.8) i(and)g(Prop)q(osition)g(3.9.)22 b Ff(2)950 2936 y Fv(25)p eop %%Page: 26 26 bop 0 26 a Fk(Prop)r(osition)18 b(5.13)24 b Fv(Simple)19 b(termination)h(is)i (a)g(mo)q(dular)g(prop)q(ert)o(y)g(of)g(\014nitely)e(branc)o(hing)i(TRSs)0 86 y(with)16 b(shared)h(constructors.)0 167 y Fk(Pro)r(of:)i Fv(This)12 b(can)h(b)q(e)f(pro)o(v)o(ed)f(using)i(the)f(metho)q(ds)f(in)h ([Gra92b)q(])g(\(note)g(that)g(the)g(union)h(of)f(t)o(w)o(o)g(\014nitely)0 228 y(branc)o(hing)k(TRSs)h(is)f(again)h(\014nitely)e(branc)o(hing\).)21 b Ff(2)73 362 y Fv(Note)15 b(that)g(Prop)q(osition)h(5.13)f(is)g(not)g(a)g (sp)q(ecial)f(case)h(of)g(Prop)q(osition)h(5.12)g(and)f(vice)e(v)o(ersa.)21 b(Since)0 423 y(simple)d(termination)g(is)h(a)h(mo)q(dular)g(prop)q(ert)o(y)f (of)h(those)g(TRSs)h(b)q(elonging)f(to)g(the)g(aforemen)o(tioned)0 483 y(classes,)c(w)o(e)g(ha)o(v)o(e)f(the)h(follo)o(wing)g(corollary)l(.)0 618 y Fk(Corollary)i(5.14)60 740 y Fv(1.)24 b(Con)o(v)o(ergence)14 b(is)h(a)h(mo)q(dular)f(prop)q(ert)o(y)g(of)g(simply)e(terminating)h(TRSs)i (with)f(shared)g(construc-)122 800 y(tors)i(whic)o(h)e(in)o(tro)q(duce)h (only)g(\014nitely)f(man)o(y)g(function)h(sym)o(b)q(ols.)60 902 y(2.)24 b(Con)o(v)o(ergence)13 b(is)i(a)g(mo)q(dular)f(prop)q(ert)o(y)g (of)h(simply)e(terminating)f(and)k(\014nitely)d(branc)o(hing)h(TRSs)122 962 y(with)i(shared)h(constructors.)0 1085 y Fk(Pro)r(of:)22 b Fv(This)17 b(follo)o(ws)f(from)f(Newman's)g(Lemma)g(in)h(conjunction)h (with)f(Prop)q(osition)i(5.7)f(as)g(w)o(ell)e(as)0 1145 y(Prop)q(osition)i (5.12)g(and)g(Prop)q(osition)h(5.13,)e(resp)q(ectiv)o(ely)l(.)j Ff(2)73 1280 y Fv(T)l(o)f(establish)e(our)i(next)e(result,)g(the)h (generalization)f(of)h(Corollary)g(1.11,)h(w)o(e)e(need)g(a)i(few)e(prereq-)0 1340 y(uisites.)0 1475 y Fk(Lemma)g(5.15)24 b Fv(Let)18 b Fn(T)451 1457 y Fh()p Fv(\))16 b(is)g(a)g(w)o(ell-founded)g (ordering,)g(where)g Fl(>)g Fv(=)e(\()p Fn(!)1396 1542 y Fr(R)1426 1547 y Fj(1)1443 1542 y Fr([R)1497 1547 y Fj(2)1530 1535 y Fn([)p 1578 1537 4 30 v 11 w Fl(>)p Fv(\))1631 1517 y Fm(+)1660 1535 y Fv(.)0 1616 y Fk(Pro)r(of:)21 b Fv(Since)15 b Fl(C)t Fv([)p Fl(s)p Fv(])p 415 1618 V 21 w Fl(>)c(s)j Fn(!)547 1623 y Fr(R)577 1628 y Fj(1)594 1623 y Fr([R)648 1628 y Fj(2)681 1616 y Fl(t)i Fv(implies)d Fl(C)t Fv([)p Fl(s)p Fv(])f Fn(!)1032 1623 y Fr(R)1062 1628 y Fj(1)1080 1623 y Fr([R)1134 1628 y Fj(2)1167 1616 y Fl(C)t Fv([)p Fl(t)p Fv(])p 1275 1618 V 19 w Fl(>)f(t)p Fv(,)16 b(ev)o(ery)f(in\014nite)g(sequence)422 1726 y Fl(s)445 1733 y Fm(1)465 1726 y Fv(\()p Fn(!)534 1733 y Fr(R)564 1738 y Fj(1)581 1733 y Fr([R)635 1738 y Fj(2)668 1726 y Fn([)p 715 1728 V 10 w Fl(>)p Fv(\))p Fl(s)791 1733 y Fm(2)811 1726 y Fv(\()p Fn(!)880 1733 y Fr(R)910 1738 y Fj(1)927 1733 y Fr([R)981 1738 y Fj(2)1014 1726 y Fn([)p 1062 1728 V 11 w Fl(>)p Fv(\))p Fl(s)1138 1733 y Fm(3)1158 1726 y Fv(\()p Fn(!)1227 1733 y Fr(R)1257 1738 y Fj(1)1274 1733 y Fr([R)1328 1738 y Fj(2)1361 1726 y Fn([)p 1408 1728 V 11 w Fl(>)o Fv(\))8 b Fl(:)g(:)g(:)0 1836 y Fv(giv)o(es)15 b(rise)h(to)h(an)g(in\014nite)e Fn(!)552 1843 y Fr(R)582 1848 y Fj(1)599 1843 y Fr([R)653 1848 y Fj(2)688 1836 y Fv(deriv)m(ation,)h(con)o(tradicting)g(the)g(termination)e (of)j Fn(!)1683 1843 y Fr(R)1713 1848 y Fj(1)1730 1843 y Fr([R)1784 1848 y Fj(2)1803 1836 y Fv(.)k Ff(2)0 1971 y Fk(Lemma)16 b(5.16)24 b Fv(Let)16 b Fl(s;)8 b(t)14 b Fn(2)g(T)f Fv(\()p Fn(D)631 1978 y Fm(1)662 1971 y Fn(])e(D)744 1978 y Fm(2)765 1971 y Fl(;)d Fn(C)s Fl(;)g Fn(V)s Fv(\))17 b(suc)o(h)f(that)h Fl(s)d Fn(!)1211 1953 y Fh(o)1211 1983 y Fr(R)1241 1988 y Fj(1)1274 1971 y Fl(t)i Fv(is)h(a)g(non-duplicating)f(reduction)0 2031 y(step.)21 b(Then)c Fl(S)283 2013 y Fh(w)311 2031 y Fv(\()p Fl(t)p Fv(\))c Fn(\022)h Fl(S)466 2013 y Fh(w)494 2031 y Fv(\()p Fl(s)p Fv(\).)0 2112 y Fk(Pro)r(of:)21 b Fv(Case)c(\(i\):)k Fl(s)16 b Fv(is)g(top)h(blac)o(k.)j(W)l(e)c(consider)g(the)g(follo)o(wing)g (sub)q(cases:)60 2235 y(1.)24 b Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))h Fn(2)g(D)386 2242 y Fm(2)406 2235 y Fv(.)41 b(Then)22 b Fl(s)j Fn(\021)f Fl(C)744 2217 y Fh(b)761 2235 y Fv([)-8 b([)p Fl(;)8 b(:)g(:)g(:)e(;)i(t;)g(:)g(:)g(:)f(;)h Fv(])-8 b(])23 b Fn(!)1109 2217 y Fh(o)1109 2247 y Fr(R)1139 2252 y Fj(1)1183 2235 y Fl(t)g Fv(and)g(as)g(a)g(consequence,)g(w)o(e)f(obtain)122 2295 y Fl(S)155 2277 y Fh(w)183 2295 y Fv(\()p Fl(t)p Fv(\))14 b(=)f([)p Fl(t)p Fv(])g Fn(\022)g Fl(S)448 2277 y Fh(w)477 2295 y Fv(\()p Fl(s)p Fv(\).)60 2397 y(2.)24 b Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))d Fn(2)g(C)s Fv(.)33 b(Then)21 b Fl(s)f Fn(\021)h Fl(l)q(\033)h Fn(!)767 2378 y Fh(o)767 2409 y Fr(R)797 2414 y Fj(1)837 2397 y Fl(r)q(\033)g Fn(\021)f Fl(t)f Fv(using)g(a)h(blac)o (k)e(rule)h Fl(l)h Fn(!)g Fl(r)g Fv(with)f Fl(r)q(oot)p Fv(\()p Fl(r)q Fv(\))i Fn(2)f(C)s Fv(.)122 2457 y(Clearly)l(,)13 b Fl(l)i Fn(\021)e Fl(C)t Fv([)p Fl(x)462 2464 y Fm(1)481 2457 y Fl(;)8 b(:)g(:)g(:)f(;)h(x)618 2464 y Fh(n)641 2457 y Fv(])14 b(and)h Fl(r)g Fn(\021)f Fl(C)891 2439 y Fr(0)902 2457 y Fv([)p Fl(x)944 2464 y Fh(i)956 2469 y Fj(1)975 2457 y Fl(;)8 b(:)g(:)g(:)f(;)h(x) 1112 2464 y Fh(i)1124 2468 y Fe(m)1155 2457 y Fv(])14 b(where)g Fn(f)p Fl(x)1375 2464 y Fh(i)1387 2469 y Fj(1)1406 2457 y Fl(;)8 b(:)g(:)g(:)f(;)h(x)1543 2464 y Fh(i)1555 2468 y Fe(m)1586 2457 y Fn(g)14 b(\022)g(f)p Fl(x)1731 2464 y Fm(1)1750 2457 y Fl(;)8 b(:)g(:)g(:)f(;)h(x)1887 2464 y Fh(n)1910 2457 y Fn(g)p Fv(.)122 2517 y(Since)34 b(the)g(rule)f(is)i(non-duplicating,)k(ev)o(en)33 b(the)h(m)o(ultiset)e(inclusion)i([)p Fl(x)1641 2524 y Fh(i)1653 2529 y Fj(1)1671 2517 y Fl(;)8 b(:)g(:)g(:)g(;)g(x)1809 2524 y Fh(i)1821 2528 y Fe(m)1852 2517 y Fv(])44 b Fn(\022)122 2577 y Fv([)p Fl(x)164 2584 y Fm(1)183 2577 y Fl(;)8 b(:)g(:)g(:)f(;)h(x)320 2584 y Fh(n)343 2577 y Fv(])16 b(holds.)22 b(Hence)15 b Fl(S)698 2559 y Fh(w)726 2577 y Fv(\()p Fl(t)p Fv(\))e Fn(\022)h Fl(S)881 2559 y Fh(w)909 2577 y Fv(\()p Fl(s)p Fv(\))i(follo)o(ws.)60 2679 y(3.)24 b Fl(r)q(oot)p Fv(\()p Fl(t)p Fv(\))14 b Fn(2)g(D)364 2686 y Fm(1)384 2679 y Fv(.)22 b(In)16 b(this)g(case)g(the)g(assertion)h (follo)o(ws)f(b)o(y)g(similar)e(argumen)o(ts)h(as)i(in)f(2.)60 2780 y(4.)24 b Fl(t)13 b Fn(2)h(V)t Fv(.)21 b(In)16 b(this)g(case)h(the)f (assertion)g(holds)h(v)m(acuously)l(.)950 2936 y(26)p eop %%Page: 27 27 bop 0 26 a Fv(Case)18 b(\(ii\):)k Fl(s)c Fv(is)f(top)g(transparen)o(t.)25 b(Let)18 b Fl(s)d Fn(\021)h Fl(C)903 8 y Fh(t)917 26 y Fv([)-8 b([)p Fl(s)960 33 y Fm(1)979 26 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1111 33 y Fh(l)1124 26 y Fl(;)g(:)g(:)g(:)f(;)h(s)1256 33 y Fh(n)1280 26 y Fv(])-8 b(])14 b Fn(!)1364 8 y Fh(o)1364 39 y Fr(R)1394 44 y Fj(1)1429 26 y Fl(t)h Fn(\021)h Fl(C)1556 8 y Fh(t)1570 26 y Fv([)p Fl(s)1607 33 y Fm(1)1626 26 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1754 33 y Fh(l)1766 26 y Fl(;)g(:)g(:)g(:)f(;)h(s)1898 33 y Fh(n)1922 26 y Fv(],)0 86 y(i.e.,)23 b Fl(s)124 93 y Fh(l)163 86 y Fn(!)213 68 y Fh(o)213 99 y Fr(R)243 104 y Fj(1)288 86 y Fl(t)306 93 y Fh(l)341 86 y Fv(where)g Fl(s)512 93 y Fh(l)548 86 y Fv(is)g(top)h(blac)o (k.)41 b(No)o(w)23 b(the)g(the)g(assertion)h(follo)o(ws)f(from)f(the)h(fact)g (that)0 147 y Fl(S)33 129 y Fh(w)61 147 y Fv(\()p Fl(t)p Fv(\))14 b(=)f(\()p Fl(S)234 129 y Fh(w)262 147 y Fv(\()p Fl(s)p Fv(\))e Fn(n)g Fl(S)403 129 y Fh(w)432 147 y Fv(\()p Fl(s)474 154 y Fh(l)487 147 y Fv(\)\))g Fn([)g Fl(S)613 129 y Fh(w)641 147 y Fv(\()p Fl(t)678 154 y Fh(l)691 147 y Fv(\))16 b(in)g(conjunction)g(with)g Fl(S)1191 129 y Fh(w)1219 147 y Fv(\()p Fl(t)1256 154 y Fh(l)1269 147 y Fv(\))e Fn(\022)f Fl(S)1387 129 y Fh(w)1415 147 y Fv(\()p Fl(s)1457 154 y Fh(l)1470 147 y Fv(\))k(\(cf.)j(\(i\)\).)h Ff(2)73 282 y Fv(Statemen)o(ts)h(1)i(and)h(2)f(of)g(the)g(next)f(prop)q (osition)i(app)q(eared)f(already)g(in)f(Gramlic)o(h)f([Gra92b)q(].)0 342 y(Statemen)o(t)14 b(3)j(is)f(the)g(new)g(and)h(in)o(teresting)e(part,)h (it)g(is)g(the)g(essence)g(of)g(our)h(pro)q(of.)0 477 y Fk(Prop)r(osition)h (5.17)24 b Fv(Let)13 b Fn(R)553 484 y Fm(1)573 477 y Fl(;)8 b Fn(R)637 484 y Fm(2)671 477 y Fv(b)q(e)13 b(t)o(w)o(o)h(terminating)e(TRSs) i(with)f(shared)h(constructors)g(suc)o(h)f(that)724 587 y Fl(D)j Fv(:)d Fl(s)830 594 y Fm(1)864 587 y Fn(!)g Fl(s)950 594 y Fm(2)984 587 y Fn(!)g Fl(s)1070 594 y Fm(3)1104 587 y Fn(!)h Fl(:)8 b(:)g(:)0 697 y Fv(is)17 b(an)g(in\014nite)f(rewrite)f(deriv)m(ation)i (of)g(minim)o(al)d(rank)j(w.r.t.)22 b Fn(R)1217 704 y Fm(1)1248 697 y Fn([)12 b(R)1335 704 y Fm(2)1355 697 y Fv(,)k(i.e.,)f(an)o(y)h(rewrite) g(deriv)m(ation)0 757 y(w.r.t.)k Fn(R)177 764 y Fm(1)208 757 y Fn([)12 b(R)295 764 y Fm(2)331 757 y Fv(of)17 b(smaller)d(rank)i(is)g (\014nite.)21 b(Then)16 b(w)o(e)g(ha)o(v)o(e)g(for)g(some)f Fl(d;)1424 744 y Fv(\026)1415 757 y Fl(d)g Fn(2)f(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)p Fv(,)16 b Fl(d)e Fn(6)p Fv(=)1751 744 y(\026)1743 757 y Fl(d)p Fv(:)60 879 y(1.)24 b(There)16 b(are)g(in\014nitely) f(man)o(y)f Fn(!)731 861 y Fh(o)731 892 y Fr(R)761 898 y Fe(d)798 879 y Fv(reduction)i(steps)g(in)g(D.)60 981 y(2.)24 b(There)12 b(are)h(in\014nitely)e(man)o(y)g Fn(!)717 988 y Fr(R)752 991 y Fj(\026)747 998 y Fe(d)780 981 y Fv(reduction)h(steps)h(in)g(D)g(whic)o(h)f (are)h(collapsing)f(or)h(constructor)122 1041 y(lifting.)60 1143 y(3.)24 b(There)16 b(are)g(in\014nitely)f(man)o(y)f(duplicating)i Fn(!)986 1125 y Fh(o)986 1155 y Fr(R)1016 1161 y Fe(d)1053 1143 y Fv(reduction)f(steps)i(in)f(D.)0 1265 y Fk(Pro)r(of:)23 b Fv(Let)17 b Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))g(=)e Fl(k)r Fv(.)24 b(Hence)16 b Fl(r)q(ank)r Fv(\()p Fl(s)875 1272 y Fh(j)893 1265 y Fv(\))g(=)f Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))j(for)g(all)f(indices)f Fl(j)s Fv(.)24 b(Moreo)o(v)o(er,)16 b Fn(!)1829 1272 y Fr(R)1859 1277 y Fj(1)1876 1272 y Fr([R)1930 1277 y Fj(2)0 1326 y Fv(is)g(terminating)f (on)i Fn(T)423 1307 y Fh()f Fv(=)k(\()p Fn(!)388 2395 y Fr(R)418 2400 y Fj(1)436 2395 y Fr([R)490 2400 y Fj(2)539 2388 y Fn([)p 587 2390 4 30 v 11 w Fl(>)p Fv(\))640 2370 y Fm(+)670 2388 y Fv(.)51 b(According)25 b(to)i(Lemma)d(5.15,)29 b(\()p Fn(T)1417 2370 y Fh()p Fv(\))26 b(is)g(a)h(w)o(ell-founded)0 2448 y(ordering.)55 b(Let)27 b(\()p Fn(M)p Fv(\()p Fn(T)481 2430 y Fh()609 2430 y Fh(mul)673 2448 y Fv(\))27 b(denote)h(its)f(w)o(ell-founded)g (m)o(ultiset)d(extension.)54 b(Note)27 b(that)0 2509 y Fl(S)33 2491 y Fh(w)61 2509 y Fv(\()p Fl(s)103 2516 y Fh(j)121 2509 y Fv(\))f Fn(\022)f(M)p Fv(\()p Fn(T)348 2491 y Fh()1399 129 y Fh(mul)1478 147 y Fl(S)1511 129 y Fh(w)1539 147 y Fv(\()p Fl(s)1581 154 y Fh(j)r Fm(+1)1644 147 y Fv(\).)73 245 y Fn(\017)24 b Fv(If)16 b Fl(s)194 252 y Fh(j)226 245 y Fn(!)276 252 y Fr(R)306 257 y Fj(2)339 245 y Fl(s)362 252 y Fh(j)r Fm(+1)425 245 y Fv(,)g(then)h(there)e(is)h(a)h(white)f (principal)f(subterm)g Fl(u)f Fn(2)g Fl(S)1429 227 y Fh(w)1457 245 y Fv(\()p Fl(s)1499 252 y Fh(j)1517 245 y Fv(\))j(suc)o(h)f(that)h Fl(u)c Fn(!)1860 252 y Fr(R)1890 257 y Fj(2)1923 245 y Fl(v)122 305 y Fv(for)22 b(some)e Fl(v)r Fv(,)i(i.e.,)e Fl(s)512 312 y Fh(j)553 305 y Fn(\021)j Fl(C)654 287 y Fh(b)670 305 y Fv([)-8 b([)p Fl(;)8 b(:)g(:)g(:)f(;)h(u;)g(:)g(:)g(:)f(;)h Fv(])-8 b(])21 b Fn(!)1027 312 y Fr(R)1057 317 y Fj(2)1099 305 y Fl(C)1138 287 y Fh(b)1155 305 y Fv([)p Fl(;)8 b(:)g(:)g(:)f(;)h(v)r(;)g(:)g(:)g(:)e(;)i Fv(])22 b Fn(\021)h Fl(s)1533 312 y Fh(j)r Fm(+1)1596 305 y Fv(.)37 b(Th)o(us)22 b(w)o(e)f(ha)o(v)o(e)122 366 y Fl(S)155 347 y Fh(w)183 366 y Fv(\()p Fl(s)225 373 y Fh(j)r Fm(+1)288 366 y Fv(\))d(=)f(\()p Fl(S)432 347 y Fh(w)460 366 y Fv(\()p Fl(s)502 373 y Fh(j)520 366 y Fv(\))12 b Fn(n)h Fv([)p Fl(u)p Fv(]\))e Fn([)i Fl(S)754 347 y Fh(w)782 366 y Fv(\()p Fl(v)r Fv(\).)26 b(It)18 b(follo)o(ws)g(from)f Fl(u)g Fn(!)g Fl(v)j Fv(in)e(conjunction)g(with)g Fl(v)p 1866 368 4 30 v 24 w(>)13 b(w)122 426 y Fv(for)23 b(an)o(y)g(principal)f(subterm)g Fl(w)k Fn(2)g Fl(S)862 408 y Fh(w)890 426 y Fv(\()p Fl(v)r Fv(\))c(that)i Fl(u)h(>)g(w)f Fv(for)f(an)o(y)g Fl(w)k Fn(2)e Fl(S)1597 408 y Fh(w)1625 426 y Fv(\()p Fl(v)r Fv(\).)41 b(Therefore)122 486 y Fl(S)155 468 y Fh(w)183 486 y Fv(\()p Fl(s)225 493 y Fh(j)243 486 y Fv(\))14 b Fl(>)314 468 y Fh(mul)393 486 y Fl(S)426 468 y Fh(w)454 486 y Fv(\()p Fl(s)496 493 y Fh(j)r Fm(+1)559 486 y Fv(\).)0 599 y(W)l(e)22 b(conclude)f(from)g(the)h(w)o(ell-foundedness)g (of)g(\()p Fn(M)p Fv(\()p Fn(T)1084 581 y Fh()1211 581 y Fh(mul)1276 599 y Fv(\))22 b(that)g(only)g (\014nitely)f(man)o(y)g Fn(!)1900 581 y Fh(i)1900 611 y Fr(R)1930 616 y Fj(1)0 659 y Fv(and)d Fn(!)146 666 y Fr(R)176 671 y Fj(2)213 659 y Fv(steps)g(can)g(o)q(ccur)g(in)f(the)g(deriv)m(ation)h Fl(D)h Fv(under)f(consideration.)25 b(In)18 b(particular,)f(there)g(are)0 719 y(only)k(\014nitely)g(man)o(y)f Fn(!)470 726 y Fr(R)500 731 y Fj(2)541 719 y Fv(reduction)h(steps)h(whic)o(h)f(are)h(collapsing)f(or) h(constructor-lifting.)38 b(This)0 779 y(con)o(tradicts)16 b(statemen)o(t)e(2.)22 b Ff(2)0 902 y Fk(Theorem)17 b(5.18)24 b Fv(Let)c Fn(R)495 909 y Fm(1)515 902 y Fl(;)8 b Fn(R)579 909 y Fm(2)618 902 y Fv(b)q(e)21 b(t)o(w)o(o)f(terminating)e(TRSs)i(with)g (shared)h(constructors)g(suc)o(h)e(that)0 962 y(their)h(union)h Fn(R)303 969 y Fm(1)337 962 y Fn([)14 b(R)426 969 y Fm(2)467 962 y Fv(is)20 b(non-terminating.)34 b(Then)20 b Fn(R)1086 969 y Fh(d)1128 962 y Fv(is)g(duplicating)g(and)h Fn(R)1587 964 y Fm(\026)1581 972 y Fh(d)1622 962 y Fv(is)g(collapsing)f(or)0 1022 y(constructor-lifting)c(for)g(some)g Fl(d;)652 1009 y Fv(\026)644 1022 y Fl(d)e Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)17 b Fv(with)f Fl(d)e Fn(6)p Fv(=)1077 1009 y(\026)1069 1022 y Fl(d)p Fv(.)0 1103 y Fk(Pro)r(of:)21 b Fv(This)c(is)f(an)g(immedi)o (ate)e(consequence)h(of)i(Prop)q(osition)g(5.17.)22 b Ff(2)73 1225 y Fv(An)16 b(equiv)m(alen)o(t)f(form)o(ulation)g(of)h(Theorem)f(5.18)i (reads)g(as)g(follo)o(ws:)0 1347 y Fk(Theorem)g(5.19)24 b Fv(Let)12 b Fn(R)487 1354 y Fm(1)507 1347 y Fl(;)c Fn(R)571 1354 y Fm(2)604 1347 y Fv(b)q(e)k(t)o(w)o(o)h(terminating)e(TRSs)i(with)g(shared)g (constructors.)20 b(Their)13 b(union)0 1407 y Fn(R)42 1414 y Fm(1)73 1407 y Fn([)e(R)159 1414 y Fm(2)195 1407 y Fv(is)17 b(terminating)d(pro)o(vided)i(that)g(one)h(of)f(the)g(follo)o(wing)g (conditions)g(is)h(satis\014ed:)60 1520 y(1.)24 b(Neither)15 b Fn(R)339 1527 y Fm(1)375 1520 y Fv(nor)i Fn(R)504 1527 y Fm(2)540 1520 y Fv(con)o(tains)f(either)f(collapsing)h(or)h (constructor-lifting)f(rules.)60 1619 y(2.)24 b(Neither)15 b Fn(R)339 1626 y Fm(1)375 1619 y Fv(nor)i Fn(R)504 1626 y Fm(2)540 1619 y Fv(con)o(tains)f(duplicating)g(rules.)60 1717 y(3.)24 b(One)17 b(of)g(the)g(systems)f(con)o(tains)i(neither)e(collapsing)h (nor)h(constructor-lifting)f(nor)g(duplicating)122 1777 y(rules.)0 1963 y Fo(6)83 b(Conditional)27 b(T)-7 b(erm)27 b(Rewriting)g(Systems)0 2093 y Fv(Although)15 b(not)g(all)f(details)g(ha)o(v)o(e)g(b)q(een)g(c)o(hec) o(k)o(ed)f(y)o(et,)g(w)o(e)h(exp)q(ect)g(that)h(Theorem)e(4.14)j(is)e(also)h (true)g(for)0 2153 y(join)f(CTRSs.)21 b(This,)14 b(ho)o(w)o(ev)o(er,)e(do)q (es)j(not)f(seem)e(to)i(lead)g(to)g(practically)e(relev)m(an)o(t)h(results.) 20 b(In)13 b(con)o(trast)0 2214 y(to)k(the)f(unconditional)g(case,)g(the)g (class)h(of)g Fn(C)848 2221 y Fr(E)871 2214 y Fv(-terminating)e(CTRSs)j (comprises)c(neither)i(the)g(class)h(of)0 2274 y(non-duplicating)j (terminating)d(CTRSs)j(nor)g(the)f(class)h(of)f(simply)f(terminating)f(CTRSs) j(\(if)f(simple)0 2334 y(termination)f(of)i(CTRSs)h(is)e(de\014ned)h (according)g(to)g(De\014nition)g(3.1)g({)g(cf.)31 b(also)21 b([Gra92d)q(]\).)31 b(This)20 b(is)0 2394 y(witnessed)c(b)o(y)g(the)g(follo)o (wing)g(non-duplicating)g(CTRS)h(tak)o(en)f(from)f([Mid90].)538 2493 y Fn(R)g Fv(=)e Fn(f)p Fl(F)7 b Fv(\()p Fl(x;)h(y)r Fv(\))k Fn(!)i Fl(F)7 b Fv(\()p Fl(y)r(;)h(x)p Fv(\))k Fn(\()i Fl(x)g Fn(#)f Fl(A;)8 b(y)15 b Fn(#)f Fl(B)s Fn(g)0 2592 y(!)50 2599 y Fr(R)99 2592 y Fv(coincides)h(with)i(the)f(rewrite)g(relation)h(generated)f (b)o(y)g(the)h(TRS)g Fn(R)1379 2574 y Fr(0)1405 2592 y Fv(=)e Fn(f)p Fl(F)7 b Fv(\()p Fl(A;)h(B)s Fv(\))13 b Fn(!)h Fl(F)7 b Fv(\()p Fl(B)s(;)h(A)p Fv(\))p Fn(g)p Fv(.)0 2652 y(Clearly)l(,)j Fn(R)217 2634 y Fr(0)240 2652 y Fv(is)h(simply)d(terminating,)h(th)o(us)i Fn(R)f Fv(is)h(simply)d(terminating)h(and)i(in)f(particular)g(terminating.)0 2712 y(Nev)o(ertheless,)j Fn(R)i Fv(is)g(not)h Fn(C)515 2719 y Fr(E)539 2712 y Fv(-terminating:)290 2811 y Fl(F)7 b Fv(\()p Fl(C)t(ons)p Fv(\()p Fl(A;)h(B)s Fv(\))p Fl(;)g(C)t(ons)p Fv(\()p Fl(A;)g(B)s Fv(\)\))j Fn(!)952 2818 y Fr(R]C)1025 2824 y Fb(E)1061 2811 y Fl(F)c Fv(\()p Fl(C)t(ons)p Fv(\()p Fl(A;)h(B)s Fv(\))p Fl(;)g(C)t(ons)p Fv(\()p Fl(A;)g(B)s Fv(\)\))950 2936 y(28)p eop %%Page: 29 29 bop 0 26 a Fv(b)q(ecause)16 b Fl(C)t(ons)p Fv(\()p Fl(A;)8 b(B)s Fv(\))13 b Fn(!)494 33 y Fr(C)513 39 y Fb(E)549 26 y Fl(A)j Fv(and)h Fl(C)t(ons)p Fv(\()p Fl(A;)8 b(B)s Fv(\))13 b Fn(!)1011 33 y Fr(C)1030 39 y Fb(E)1066 26 y Fl(B)s Fv(.)73 107 y(This)k(example)d(also)k(sho)o(ws)f(that)g(Lemmata)e(3.2)i(and)g(3.3)g (are)g(not)g(true)f(for)h(CTRS:)g(The)f(system)0 167 y(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))21 b(=)g(\()p Fn(f)p Fl(F)q(;)8 b(A;)g(B)s Fn(g)p Fl(;)g Fn(f)p Fl(F)f Fv(\()p Fl(x;)h(y)r Fv(\))18 b Fn(!)i Fl(F)7 b Fv(\()p Fl(y)r(;)h(x)p Fv(\))20 b Fn(\()h Fl(x)f Fn(#)h Fl(A;)8 b(y)22 b Fn(#)f Fl(B)s Fn(g)p Fv(\))f(is)h(simply)d(terminating)h(\(hence)0 228 y(simplifying\),)c(but)k (\()p Fn(F)5 b Fl(;)j Fn(R)k([)h(F)599 209 y Fh(ar)q(g)654 228 y Fv(\))k(=)g(\()p Fn(F)5 b Fl(;)j Fn(R)k([)h(f)p Fl(F)7 b Fv(\()p Fl(x;)h(y)r Fv(\))15 b Fn(!)i Fl(x;)8 b(F)f Fv(\()p Fl(x;)h(y)r Fv(\))15 b Fn(!)i Fl(y)r Fn(g)p Fv(\))h(is)g(not)h(terminating)0 288 y(and)e Fn(!)145 267 y Fm(+)145 300 y Fr(R[F)227 290 y Fe(ar)q(g)295 288 y Fv(is)f(not)h(irre\015exiv)o(e.)73 369 y(A,)f(from)f(a)h(practical)g(p)q(oin)o(t)g(of)h(view,)e(reasonably)h (signi\014can)o(t)g(result)g(for)h(mo)q(dularit)o(y)d(of)j(con)o(v)o(er-)0 429 y(gence)f(for)g(certain)g(\014nite)g(join)g(CTRSs)h(with)f(shared)g (constructors)h(can)g(b)q(e)f(found)h(in)f([Ohl92b].)73 510 y(The)d(ab)q(o)o(v)o(e)g(example)e(also)j(sho)o(ws)g(that)g(the)f(disjoin)o (t)f(union)i(of)f(t)o(w)o(o)g(terminating)f(non-duplicating)0 570 y(CTRSs)17 b(ma)o(y)e(b)q(e)h(non-terminating.)k(Middeldorp)c(pro)o(v)o (ed)f(in)h([Mid90]:)0 688 y Fk(Theorem)h(6.1)24 b Fv(Let)13 b Fn(R)460 695 y Fm(1)494 688 y Fv(and)h Fn(R)628 695 y Fm(2)661 688 y Fv(b)q(e)g(t)o(w)o(o)f(disjoin)o(t)g(terminating)f(join)i(CTRSs.)21 b(Their)13 b(disjoin)o(t)g(union)0 749 y Fn(R)42 756 y Fm(1)73 749 y Fn(])e(R)159 756 y Fm(2)195 749 y Fv(is)17 b(terminating)d(pro)o(vided) i(that)g(one)h(of)f(the)g(follo)o(wing)g(conditions)g(is)h(ful\014lled:)60 859 y(1.)24 b(Both)16 b(systems)f(are)h(non-collapsing.)60 956 y(2.)24 b(Both)16 b(systems)f(are)h(con\015uen)o(t)g(and)h (non-duplicating.)60 1054 y(3.)24 b(Both)16 b(systems)f(are)h(con\015uen)o(t) g(and)h(one)g(of)f(them)f(is)h(non-collapsing)h(and)g(non-duplicating.)73 1172 y(W)l(e)g(p)q(oin)o(t)g(out)h(that)f(a)h(simpler)d(pro)q(of)j(than)g (the)f(one)g(in)g([Mid90)o(])g(can)g(b)q(e)h(ac)o(hiev)o(ed)d(b)o(y)i(a)g (simple)0 1232 y(mo)q(di\014cation)g(of)i(the)f(pro)q(of)h(structure)f (\(resulting)g(in)g(a)h(pro)q(of)g(similiar)d(to)i(that)h(of)g(Theorem)e (5.18\).)0 1293 y(This)f(remark)f(migh)o(t)g(also)h(b)q(e)h(useful)f(for)g (pro)o(ving)g(the)g(conjecture)g(b)q(elo)o(w.)0 1413 y Fk(Pro)r(of)26 b(sk)n(etc)n(h)p Fv(:)33 b(Supp)q(ose)24 b(that)f Fn(R)713 1420 y Fm(1)755 1413 y Fv(and)g Fn(R)898 1420 y Fm(2)940 1413 y Fv(are)g(t)o(w)o(o)f(disjoin)o(t)g(terminating)f(join)h(CTRSs)h(suc)o(h)0 1473 y(that)15 b Fn(R)146 1480 y Fm(1)173 1473 y Fn(])7 b(R)255 1480 y Fm(2)290 1473 y Fv(is)14 b(not)h(terminating.)k(As)14 b(usual)h(assume)f(further)g(that)h Fl(D)h Fv(:)d Fl(s)1441 1480 y Fm(1)1474 1473 y Fn(!)h Fl(s)1561 1480 y Fm(2)1595 1473 y Fn(!)f Fl(s)1681 1480 y Fm(3)1715 1473 y Fn(!)g Fl(:)8 b(:)g(:)14 b Fv(is)g(an)0 1533 y(in\014nite)i Fn(R)208 1540 y Fm(1)239 1533 y Fn(])c(R)326 1540 y Fm(2)362 1533 y Fv(deriv)m(ation)k(of)h(minimal)d (rank.)22 b(Hence)16 b Fl(r)q(ank)r Fv(\()p Fl(s)1263 1540 y Fh(j)1281 1533 y Fv(\))e(=)h Fl(r)q(ank)r Fv(\()p Fl(D)q Fv(\))i(and)h Fl(r)q(oot)p Fv(\()p Fl(s)1793 1540 y Fh(j)1812 1533 y Fv(\))c Fn(2)h(F)1929 1540 y Fh(d)0 1594 y Fv(for)d(some)f Fl(d)j Fn(2)g(f)p Fv(1)p Fl(;)8 b Fv(2)p Fn(g)p Fv(.)20 b(Then)12 b(sho)o(w)h(b)o(y)e(con)o(tradiction)g(that)i Fn(R)1173 1596 y Fm(\026)1167 1604 y Fh(d)1199 1594 y Fv(m)o(ust)d(b)q(e)i(collapsing.)20 b(Finally)11 b(supp)q(ose)0 1654 y(that)20 b(b)q(oth)g(systems)e(are)i (con\015uen)o(t)f(and)h(pro)o(v)o(e)f(b)o(y)g(con)o(tradiction)f(that)i Fn(R)1463 1661 y Fh(d)1503 1654 y Fv(m)o(ust)e(b)q(e)i(duplicating.)0 1714 y(This)12 b(can)g(b)q(e)g(done)g(b)o(y)f(using)h(the)g(pro)q(of)h(ideas) e(for)h(1.)20 b(and)13 b(2.)20 b(in)11 b([Mid90],)h(ho)o(w)o(ev)o(er,)f(w)o (e)g(get)h(3.)20 b(for)12 b(free.)0 1834 y Fk(Conjecture)22 b Fv(\(Middeldorp\):)34 b(The)23 b(disjoin)o(t)f(union)h(of)g(t)o(w)o(o)g (terminating)e(join)i(CTRSs)g(is)g(termi-)0 1895 y(nating)17 b(if)e(one)i(of)f(them)f(is)h(non-collapsing)h(and)g(non-duplicating)g(and)f (the)g(other)h(is)f(con\015uen)o(t.)0 2080 y Fo(7)83 b(Concluding)26 b(Remarks)0 2210 y Fv(While)18 b(\014nishing)h(this)g(pap)q(er,)h(the)e (author)i(receiv)o(ed)d(a)i(cop)o(y)g(of)g([Der93].)29 b(There)19 b(Dersho)o(witz)g(giv)o(es)0 2270 y(a)e(pro)q(of)g(sk)o(etc)o(h)e(for)i (Theorem)e(5.18)i(rev)o(ealing)e(exactly)g(the)h(idea)g(underlying)g(our)h (pro)q(of.)22 b(Moreo)o(v)o(er,)0 2330 y(it)f(con)o(tains)g(some)f (extensions)h(of)h(recen)o(t)e(results)h(of)g(Gramlic)o(h)e([Gra92c)q(])i(to) g(constructor)h(sharing)0 2390 y(systems)11 b(and)i(deals)g(also)g(with)f (hierarc)o(hical)e(TRSs.)21 b(There)12 b(are)g(t)o(w)o(o)h(other)f(recen)o(t) f(pap)q(ers)j(b)o(y)e(Krishna)0 2451 y(Rao)18 b([KR92a)q(,)f(KR92b)q(])g (whic)o(h)g(are)g(also)i(concerned)e(with)g(hierarc)o(hical)f(TRSs.)26 b(These)17 b(are)h(systems)0 2511 y(suc)o(h)e(as)544 2590 y Fn(R)586 2597 y Fm(1)620 2590 y Fv(=)672 2517 y Fg(\()726 2559 y Fl(add)p Fv(\(0)p Fl(;)8 b(x)p Fv(\))114 b Fn(!)41 b Fl(x)726 2619 y(add)p Fv(\()p Fl(S)s Fv(\()p Fl(x)p Fv(\))p Fl(;)8 b(y)r Fv(\))41 b Fn(!)g Fl(S)s Fv(\()p Fl(add)p Fv(\()p Fl(x;)8 b(y)r Fv(\)\))0 2699 y(and)471 2778 y Fn(R)513 2785 y Fm(2)547 2778 y Fv(=)599 2705 y Fg(\()653 2747 y Fl(mul)q(t)p Fv(\(0)p Fl(;)g(x)p Fv(\))112 b Fn(!)41 b Fv(0)653 2808 y Fl(mul)q(t)p Fv(\()p Fl(S)s Fv(\()p Fl(x)p Fv(\))p Fl(;)8 b(y)r Fv(\))39 b Fn(!)i Fl(add)p Fv(\()p Fl(mul)q(t)p Fv(\()p Fl(x;)8 b(y)r Fv(\))p Fl(;)g(y)r Fv(\))950 2936 y(29)p eop %%Page: 30 30 bop 0 26 a Fv(where)17 b(de\014ned)g(sym)o(b)q(ols)f(of)h(the)g(\014rst)g (TRS)h(ma)o(y)d(o)q(ccur)i(as)h(constructors)g(in)f(the)f(second.)24 b(Denoting)0 86 y(the)15 b(set)h(of)g(constructors)g(of)g Fn(R)587 93 y Fh(j)621 86 y Fv(b)o(y)f Fn(C)714 93 y Fh(j)748 86 y Fv(and)h(the)f(set) h(of)f(de\014ned)h(sym)o(b)q(ols)e(b)o(y)h Fn(D)1513 93 y Fh(j)1532 86 y Fv(,)g(the)g(relationship)h(of)0 147 y(the)g(di\013eren)o(t)f(kinds)h (of)h(com)o(binations)e(of)h(TRSs)h(is)f(illustrated)g(in)f([KR92a)q(,)h (KR92b])g(as)h(follo)o(ws:)54 354 y Fn(D)92 361 y Fm(1)23 400 y Fa(\032\031)23 284 y(\033\030)154 436 y Fn(C)180 443 y Fm(1)117 483 y Fa(\032\031)117 367 y(\033)q(\030)284 436 y Fn(C)310 443 y Fm(2)247 483 y Fa(\032\031)247 367 y(\033)q(\030)372 354 y Fn(D)410 361 y Fm(2)342 400 y Fa(\032\031)342 284 y(\033\030)46 589 y Fv(\(i\))f(Disjoin)o(t)g(Union)703 354 y Fn(D)741 361 y Fm(1)672 400 y Fa(\032\031)672 284 y(\033)q(\030)788 436 y Fn(C)814 443 y Fm(1)767 483 y Fa(\032\031)767 367 y(\033\030)903 436 y Fn(C)929 443 y Fm(2)850 483 y Fa(\032\031)850 367 y(\033\030)975 354 y Fn(D)1013 361 y Fm(2)944 400 y Fa(\032\031)944 284 y(\033\030)532 589 y Fv(\(ii\))f(Constructor)i(Sharing)g(TRSs)1439 436 y Fn(D)1477 443 y Fm(1)1417 483 y Fa(\032\031)1417 367 y(\033\030)1591 436 y Fn(C)1617 443 y Fm(1)1547 483 y Fa(\032\031)1547 367 y(\033\030)1518 360 y Fn(C)1544 367 y Fm(2)1482 418 y Fa(\032\031)1482 302 y(\033\030)1512 241 y Fn(D)1550 248 y Fm(2)1482 288 y Fa(\032\031)1482 172 y(\033\030)1264 589 y Fv(\(iii\))e(Hierarc)o(hical)f(TRSs)0 696 y(Although)e(the)f(\014gure)h(suggests)g(that)g(eac)o(h)f(kind)g(of)h (com)o(bination)e(subsumes)g(the)i(kinds)f(to)h(its)f(left,)g(this)0 756 y(ma)o(y)i(not)j(b)q(e)f(the)f(case.)21 b(The)15 b(reason)g(is)g(that)g (hierarc)o(hical)e(TRSs)j(are)e(\(at)i(least)e(in)h([KR92a,)g(KR92b]\))0 816 y(constructor)f(systems)e(b)o(y)h(de\014nition.)20 b(A)14 b(TRS)f(\()p Fn(F)5 b Fl(;)j Fn(R)p Fv(\))14 b(=)g(\()p Fn(D)7 b(])f(C)s Fl(;)i Fn(R)p Fv(\))14 b(is)f(called)g(a)h Fi(c)n(onstructor)h (system)0 876 y Fv(if)e(ev)o(ery)e(left-hand)j(side)e Fl(f)5 b Fv(\()p Fl(t)532 883 y Fm(1)552 876 y Fl(;)j(:)g(:)g(:)f(;)h(t)679 883 y Fh(n)702 876 y Fv(\))14 b(of)f(a)h(rewrite)e(rule)g(of)i Fn(R)f Fv(satis\014es)h Fl(t)1384 883 y Fm(1)1403 876 y Fl(;)8 b(:)g(:)g(:)g(;)g(t)1531 883 y Fh(n)1567 876 y Fn(2)14 b(T)f Fv(\()p Fn(C)s Fl(;)8 b Fn(V)t Fv(\).)20 b(Notice)0 937 y(that)e(the)g(TRSs)g (in)g(the)f(ab)q(o)o(v)o(e)h(example)d(are)j(constructor)g(systems.)25 b(An)18 b(in)o(teresting)e(result)i(in)f(this)0 997 y(regard)i(w)o(as)h (obtained)f(b)o(y)f(Middeldorp)g(and)i(T)l(o)o(y)o(ama)e([MT91];)h(they)f (pro)o(v)o(ed)h(that)g(con)o(v)o(ergence)e(is)0 1057 y(preserv)o(ed)c(under)i (the)f(com)o(bination)f(of)h(constructor)h(systems)e(whic)o(h)h(do)h(not)g (share)g(de\014ned)f(sym)o(b)q(ols)0 1117 y(\(in)24 b(fact,)i(they)e(pro)o(v) o(ed)f(a)i(more)e(general)h(result\).)45 b(Th)o(us)25 b(hierarc)o(hical)d (systems)i(in)g(this)g(limited)0 1177 y(sense)15 b(are)g(more)f(an)h (extension)g(of)g(com)o(binations)f(of)i(constructor)f(systems)f(than)i(a)f (generalization)g(of)0 1238 y(com)o(binations)h(of)i(constructor)f(sharing)h (systems.)23 b(In)17 b(particular,)g(results)g(for)h(hierarc)o(hical)d (systems)0 1298 y(\(pro)o(vided)20 b(that)g(they)g(are)g(de\014ned)g(to)h(b)q (e)f(constructor)h(systems\))e(do)i(not)g(hold)f(automatically)f(for)0 1358 y(disjoin)o(t)d(unions)g(or)h(constructor)g(sharing)g(systems.)0 1439 y Fk(Ac)n(kno)n(wledgemen)n(ts:)j Fv(The)15 b(author)h(is)f(grateful)h (to)f(Rob)q(ert)h(Giegeric)o(h)e(and)i(Aart)f(Middeldorp)g(for)0 1499 y(v)m(aluable)g(commen)o(t)o(s)e(on)i(a)h(previous)e(v)o(ersion)h(of)g (the)g(pap)q(er,)g(to)g(Ank)o(e)f(Bo)q(dzin)h(for)g(t)o(yp)q(esetting)f (parts)0 1559 y(of)j(the)f(man)o(uscript,)e(and)j(to)f(Hugh)h(Osb)q(orne)g (for)f(suggestions)i(for)e(impro)o(ving)e(the)i(English.)0 1746 y Fo(References)0 1877 y Fv([Der79])53 b(N.)11 b(Dersho)o(witz.)j(A)e (Note)f(on)h(Simpli\014cation)e(Orderings.)k Fi(Information)f(Pr)n(o)n(c)n (essing)g(L)n(etters)207 1937 y Fk(9)p Fi(\(5\))p Fv(,)j(pages)h(212{215,)h (1979.)0 2039 y([Der93])53 b(N.)29 b(Dersho)o(witz.)61 b(Hierarc)o(hical)28 b(T)l(ermination.)60 b(Draft,)33 b(Dept.)c(Hebrew)g(Univ)o(ersit)o(y)l(,)207 2099 y(Jerusalem)15 b(91904,)j(Israel,)d(1993.)0 2200 y([DJ90])69 b(N.)16 b(Dersho)o(witz)h(and)g(J.P)l(.)f(Jouannaud.)25 b(Rewrite)16 b(Systems.)21 b(In)c(L.)g(v)m(an)g(Leeu)o(w)o(en,)f(editor,)207 2261 y Fi(Handb)n(o)n(ok)d(of)f(The)n(or)n(etic)n(al)g(Computer)h(Scienc)n (e,)i(V)l(ol.)e(B)p Fv(,)e(c)o(hapter)f(6.)h(North-Holland,)h(1990.)0 2362 y([DM79])49 b(N.)12 b(Dersho)o(witz)g(and)h(Z.)f(Manna.)k(Pro)o(ving)d (T)l(ermination)e(with)h(Multiset)g(Orderings.)j Fi(Com-)207 2423 y(munic)n(ations)j(of)g(the)g(A)o(CM)e Fk(22)p Fi(\(8\))p Fv(,)g(pages)h(465{476,)i(1979.)0 2524 y([Dro89])51 b(K.)18 b(Drosten.)26 b Fi(T)l(ermersetzungssysteme)p Fv(.)h(Informatik-F)l(ac)o(h)o (b)q(eric)o(h)o(te)14 b Fk(210)p Fv(,)k(Springer)f(V)l(er-)207 2584 y(lag,)f(1989.)0 2686 y([Gal91])55 b(J.)20 b(Gallier.)32 b(What's)21 b(so)g(Sp)q(ecial)e(ab)q(out)j(Krusk)m(al's)e(Theorem)f(and)i (the)f(Ordinal)f(\000)1835 2693 y Fm(0)1856 2686 y Fv(?)33 b(A)207 2746 y(Surv)o(ey)18 b(of)h(Some)f(Results)g(in)h(Pro)q(of)h(Theory.) 29 b Fi(A)o(nnals)21 b(of)f(Pur)n(e)f(and)h(Applie)n(d)g(L)n(o)n(gic)f Fk(53)p Fv(,)207 2807 y(pages)e(199{260,)i(1991.)950 2936 y(30)p eop %%Page: 31 31 bop 0 26 a Fv([Gra92a])26 b(B.)11 b(Gramlic)o(h.)g(Generalized)f(Su\016cien)o (t)g(Conditions)i(for)g(Mo)q(dular)g(T)l(ermination)e(of)h(Rewrit-)207 86 y(ing.)24 b(In)17 b Fi(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)i(the)f(Thir)n(d)f (International)j(Confer)n(enc)n(e)f(on)g(A)o(lgebr)n(aic)g(and)g(L)n(o)n(gic) 207 147 y(Pr)n(o)n(gr)n(amming)p Fv(,)12 b(pages)j(53{68.)f(Lecture)g(Notes)f (in)g(Computer)g(Science)e Fk(632)p Fv(,)j(Springer)f(V)l(er-)207 207 y(lag,)j(1992.)0 305 y([Gra92b])25 b(B.)10 b(Gramlic)o(h.)h(Generalized)f (Su\016cien)o(t)g(Conditions)i(for)f(Mo)q(dular)h(T)l(ermination)e(of)h (Rewrit-)207 365 y(ing.)21 b(Extended)16 b(v)o(ersion)g(of)g([Gra92a)r(],)f (submitted,)f(1992.)0 464 y([Gra92c])28 b(B.)20 b(Gramlic)o(h.)33 b(Relating)21 b(Innermost,)f(W)l(eak,)i(Uniform)d(and)j(Mo)q(dular)f(T)l (ermination)f(of)207 524 y(T)l(erm)f(Rewriting)h(Systems.)33 b(In)20 b Fi(Confer)n(enc)n(e)i(on)g(L)n(o)n(gic)e(Pr)n(o)n(gr)n(amming)g (and)i(A)o(utomate)n(d)207 584 y(R)n(e)n(asoning)p Fv(,)f(pages)g(285{296.)h (Lecture)e(Notes)g(in)g(Arti\014cial)f(In)o(telligence)e Fk(624)p Fv(,)k(Springer)207 645 y(V)l(erlag,)16 b(1992.)0 743 y([Gra92d])25 b(B.)15 b(Gramlic)o(h.)k(Su\016cien)o(t)c(Conditions)i(for)g(Mo)q(dular)f(T)l (ermination)f(of)i(Conditional)f(T)l(erm)207 803 y(Rewriting)i(Systems.)26 b(In)18 b Fi(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)h(3r)n(d)e(International)j (Workshop)e(on)g(Condi-)207 863 y(tional)c(T)l(erm)f(R)n(ewriting)g(Systems)p Fv(,)f(pages)g(128{142.)h(Lecture)e(Notes)g(in)g(Computer)f(Science)207 924 y Fk(656)p Fv(,)16 b(Springer)g(V)l(erlag,)g(1992.)0 1022 y([Klo92])55 b(J.W.)31 b(Klop.)68 b(T)l(erm)30 b(Rewriting)h(Systems.)66 b(In)31 b(S.)h(Abramsky)l(,)h(D.)f(Gabba)o(y)l(,)j(and)207 1082 y(T.)12 b(Maibaum,)f(editors,)h Fi(Handb)n(o)n(ok)h(of)h(L)n(o)n(gic)e (in)i(Computer)f(Scienc)n(e,)j(V)l(ol.)e(II)p Fv(,)d(pages)h(1{112.)207 1142 y(Oxford)17 b(Univ)o(ersit)o(y)c(Press,)j(1992.)0 1241 y([K)o(O90])56 b(M.)13 b(Kurihara)h(and)g(A.)f(Oh)o(uc)o(hi.)i(Mo)q(dularit)o (y)e(of)h(Simple)e(T)l(ermination)g(of)h(T)l(erm)g(Rewriting)207 1301 y(Systems.)20 b Fi(Journal)d(of)h(IPS)f(Jap)n(an)g Fk(31)p Fi(\(5\))p Fv(,)f(pages)h(633{642,)h(1990.)0 1400 y([K)o(O92])56 b(M.)13 b(Kurihara)h(and)g(A.)f(Oh)o(uc)o(hi.)i(Mo)q(dularit)o(y)e(of)h (Simple)e(T)l(ermination)g(of)h(T)l(erm)g(Rewriting)207 1460 y(Systems)f(with)h(Shared)h(Constructors.)j Fi(The)n(or)n(etic)n(al)e (Computer)f(Scienc)n(e)j Fk(103)p Fv(,)c(pages)h(273{)207 1520 y(282,)j(1992.)0 1618 y([KR92a])33 b(M.R.K.)16 b(Krishna)i(Rao.)27 b(Mo)q(dular)18 b(Pro)q(ofs)i(for)e(Completeness)e(of)i(Hierarc)o(hical)e (Systems.)207 1679 y(Submitted)f(for)h(Publication,)g(1992.)0 1777 y([KR92b])30 b(M.R.K.)23 b(Krishna)h(Rao.)46 b(Mo)q(dular)25 b(Pro)q(ofs)h(for)f(Simple)d(T)l(ermination)g(of)j(Hierarc)o(hical)207 1837 y(Systems.)20 b(Submitted)15 b(for)h(Publication,)f(1992.)0 1936 y([MG92])48 b(A.)18 b(Middeldorp)g(and)h(B.)e(Gramlic)o(h.)26 b(Simple)16 b(T)l(ermination)h(is)h(Di\016cult.)27 b(Submitted)17 b(for)207 1996 y(Publication,)f(1992.)0 2094 y([Mid89])45 b(A.)18 b(Middeldorp.)26 b(A)18 b(Su\016cien)o(t)e(Condition)j(for)f(the)g(T)l (ermination)f(of)h(the)g(Direct)f(Sum)h(of)207 2155 y(T)l(erm)d(Rewriting)h (Systems.)21 b(In)16 b Fi(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)h(the)g(4th)g(IEEE) g(Symp)n(osium)e(on)i(L)n(o)n(gic)f(in)207 2215 y(Computer)h(Scienc)n(e)p Fv(,)f(pages)g(396{401,)i(1989.)0 2313 y([Mid90])45 b(A.)16 b(Middeldorp.)21 b Fi(Mo)n(dular)16 b(Pr)n(op)n(erties)h(of)g(T)l(erm)g(R)n (ewriting)h(Systems)p Fv(.)k(PhD)17 b(thesis,)f(F)l(ree)207 2373 y(Univ)o(ersit)o(y)e(Amsterdam,)f(1990.)0 2472 y([MT91])51 b(A.)22 b(Middeldorp)f(and)i(Y.)e(T)l(o)o(y)o(ama.)38 b(Completeness)21 b(of)i(Com)o(binations)e(of)i(Constructor)207 2532 y(Systems.)29 b(In)19 b Fi(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)h(4th)f(International)i (Confer)n(enc)n(e)f(on)f(R)n(ewriting)h(T)l(e)n(ch-)207 2592 y(niques)e(and)f(Applic)n(ations)p Fv(,)f(pages)g(188{199.)i(Lecture)c(Notes) i(in)f(Computer)f(Science)g Fk(488)p Fv(,)207 2652 y(Springer)h(V)l(erlag,)g (1991.)0 2751 y([Ohl92a])28 b(E.)15 b(Ohlebusc)o(h.)i(A)d(Note)h(on)g(Simple) d(T)l(ermination)h(of)i(In\014nite)f(T)l(erm)f(Rewriting)h(Systems.)207 2811 y(Rep)q(ort)j(Nr.)e(7,)h(T)l(ec)o(hnisc)o(he)f(F)l(akult\177)-24 b(at,)16 b(Univ)o(ersit\177)-24 b(at)14 b(Bielefeld,)f(1992.)950 2936 y(31)p eop %%Page: 32 32 bop 0 26 a Fv([Ohl92b])25 b(E.)15 b(Ohlebusc)o(h.)k(Com)o(binations)c(of)h (Simplifyi)o(ng)d(Conditional)j(T)l(erm)e(Rewriting)h(Systems.)207 86 y(In)h Fi(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)h(3r)n(d)f(International)i (Workshop)e(on)h(Conditional)g(T)l(erm)g(R)n(ewriting)207 147 y(Systems)p Fv(,)d(pages)g(113{127.)h(Lecture)e(Notes)f(in)h(Computer)f (Science)g Fk(656)p Fv(,)h(Springer)g(V)l(erlag,)207 207 y(1992.)0 309 y([Ohl93a])28 b(E.)16 b(Ohlebusc)o(h.)21 b(A)16 b(Simple)e(Pro)q(of)j(of) g(Su\016cien)o(t)e(Conditions)i(for)f(the)h(T)l(ermination)d(of)j(the)207 369 y(Disjoin)o(t)g(Union)g(of)h(T)l(erm)e(Rewriting)g(Systems.)23 b Fi(Bul)r(letin)e(of)e(the)f(Eur)n(op)n(e)n(an)g(Asso)n(ciation)207 429 y(for)f(The)n(or)n(etic)n(al)g(Computer)g(Scienc)n(e)j Fk(49)p Fv(,)c(pages)h(178{183,)h(F)l(ebruary)e(1993.)0 531 y([Ohl93b])25 b(E.)13 b(Ohlebusc)o(h.)i(T)l(ermination)c(is)i(not)g(Mo)q (dular)g(for)g(Con\015uen)o(t)h(V)l(ariable-Preserving)d(T)l(erm)207 591 y(Rewriting)16 b(Systems.)k(Submitted)14 b(for)j(Publication,)e(1993.)0 692 y([Rus87])49 b(M.)17 b(Rusino)o(witc)o(h.)24 b(On)18 b(T)l(ermination)e (of)i(the)f(Direct)g(Sum)f(of)i(T)l(erm)e(Rewriting)h(Systems.)207 753 y Fi(Information)g(Pr)n(o)n(c)n(essing)g(L)n(etters)h Fk(26)p Fv(,)d(pages)i(65{70,)h(1987.)0 854 y([Ste89])63 b(J.)15 b(Stein)o(bac)o(h.)j (Extensions)d(and)h(Comparison)f(of)h(Simpli\014cation)c(Orderings.)19 b(In)c Fi(Pr)n(o)n(c)n(e)n(e)n(d-)207 915 y(ings)f(of)f(the)h(3r)n(d)e (International)j(Confer)n(enc)n(e)e(on)h(R)n(ewriting)f(T)l(e)n(chniques)i (and)f(Applic)n(ations)p Fv(,)207 975 y(pages)j(434{448.)i(Lecture)d(Notes)g (in)g(Computer)f(Science)g Fk(355)p Fv(,)h(Springer)g(V)l(erlag,)f(1989.)0 1076 y([TKB89])24 b(Y.)17 b(T)l(o)o(y)o(ama,)g(J.W.)g(Klop,)h(and)g(H.P)f (Barendregt.)26 b(T)l(ermination)16 b(for)i(the)g(Direct)f(Sum)g(of)207 1137 y(Left-Linear)e(T)l(erm)e(Rewriting)g(Systems.)k(In)d Fi(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)i(the)g(3r)n(d)e(International)j(Con-)207 1197 y(fer)n(enc)n(e)j(on)g(R)n(ewriting)g(T)l(e)n(chniques)h(and)e(Applic)n (ations)p Fv(,)g(pages)g(477{491.)i(Lecture)d(Notes)207 1257 y(in)e(Computer)g(Science)e Fk(355)p Fv(,)i(Springer)g(V)l(erlag,)f(1989.)0 1359 y([T)l(o)o(y87a])27 b(Y.)11 b(T)l(o)o(y)o(ama.)g(Coun)o(terexamples)f (to)h(T)l(ermination)f(for)h(the)g(Direct)f(Sum)g(of)i(T)l(erm)e(Rewriting) 207 1419 y(Systems.)20 b Fi(Information)d(Pr)n(o)n(c)n(essing)g(L)n(etters)g Fk(25)p Fv(,)f(pages)h(141{143,)i(1987.)0 1521 y([T)l(o)o(y87b])24 b(Y.)13 b(T)l(o)o(y)o(ama.)i(On)f(the)f(Ch)o(urc)o(h-Rosser)g(Prop)q(ert)o(y) h(for)f(the)g(Direct)g(Sum)f(of)i(T)l(erm)e(Rewriting)207 1581 y(Systems.)20 b Fi(Journal)d(of)h(the)g(A)o(CM)f Fk(34)p Fi(\(1\))p Fv(,)e(pages)i(128{143,)i(1987.)0 1683 y([Zan92])50 b(H.)20 b(Zan)o(tema.)34 b(T)l(ermination)20 b(of)h(T)l(erm)f(Rewriting)g(b)o(y)h(In) o(terpretation.)34 b(In)21 b Fi(Pr)n(o)n(c)n(e)n(e)n(dings)207 1743 y(of)c(the)f(3r)n(d)g(International)h(Workshop)f(on)h(Conditional)g(T)l (erm)f(R)n(ewriting)h(Systems)p Fv(,)e(pages)207 1803 y(155{167.)k(Lecture)d (Notes)g(in)g(Computer)f(Science)g Fk(656)p Fv(,)h(Springer)g(V)l(erlag,)f (1992.)950 2936 y(32)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF