%!PS-Adobe-2.0 %%Title: TeX output 1994.09.29:1129 %%Creator: DVILASER/PS, ArborText, Inc. %%BoundingBox: (atend) %%Pages: (atend) %%DocumentFonts: (atend) %%DocumentMedia: Plain 612 792 75 white {} %%EndComments %! % Dvips.pro - included prolog for DviLaser-generated PostScript files. % % Copyright (c) 1986-91, ArborText, Inc. % Permission to copy is granted so long as the PostScript code % is not resold or used in a commercial product. % % $Header: /ati/src/tex/dvips/src/RCS/dvips.pro.dist,v 1.3 1992/04/29 20:00:53 mmb Exp $ % systemdict /setpacking known % use array packing mode if its available {/savepackingmode currentpacking def true setpacking} if /$DviLaser 400 dict def % Begin document /BeginDviLaserDoc { 0 0 moveto vmstatus pop pop 0 eq { $DviLaser begin InitializeState } { /DviLaserJob save def $DviLaser begin InitializeState /DviLaserFonts save def } ifelse } bind def % End document /EndDviLaserDoc { vmstatus pop pop 0 eq { end } { DviLaserFonts restore end DviLaserJob restore } ifelse } bind def $DviLaser begin /tempstr 64 string def /tempint 0 def /tempmatrix matrix def % % Debugging routines % /DebugMode false def /PrintInt { tempstr cvs print } bind def /PrintLn { (\n) print flush } bind def /PrintVMStats { print PrintLn (VM status - ) print vmstatus 3 copy PrintInt (\(total\), ) print PrintInt (\(used\), ) print pop exch sub PrintInt (\(remaining\), ) print PrintInt (\(level\)) print PrintLn } bind def /VMS /PrintVMStats load def /VMSDebug { DebugMode {PrintVMStats} {pop} ifelse } bind def (beginning of common prolog) VMSDebug % Make it easy to bind definitions. /bdef { bind def } bind def /xdef { exch def } bdef % Bind some procedures that do restores and such to avoid InvalRestore problems. /RestoreDviLaserPage { /DviLaserPage where {pop DviLaserPage restore} if } bdef /RestoreDviLaserFonts { DocumentType 0 eq {/DviLaserFonts where {pop DviLaserFonts restore} if } if } bdef /SaveDviLaserFonts { DocumentType 0 eq {/DviLaserFonts save def} if } bdef /SetCurrentPoint { /currentpoint cvx stopped {0 0 moveto currentpoint} if } bdef % Begin page /BP { /Magnification xdef /DviLaserPage save def (beginning of page) VMSDebug } bdef % End page /EP { DviLaserPage restore } bdef % Exit page (temporarily) to add fonts/characters. /XP { % Save current point information so it can be reset later. /Xpos where {pop Xpos} {0} ifelse /Ypos where {pop Ypos} {0} ifelse SetCurrentPoint RestoreDviLaserPage moveto /Ypos xdef /Xpos xdef } bdef % Resume page /RP { /DviLaserPage save def } bdef % Purge all fonts to reclaim memory space. /PF { GlobalMode LocalMode } bdef % Switch to base save/restore level, saving state information. /GlobalMode { /UserSave where {pop UserSave} if % invoke "UserSave" if available PortraitMode PaperWidth PaperHeight PxlResolution Resolution Magnification Ymax RasterScaleFactor SetCurrentPoint RestoreDviLaserPage RestoreDviLaserFonts RecoverState } bdef % Preserve state at the base level. /RecoverState { 10 copy /Ypos xdef /Xpos xdef /RasterScaleFactor xdef /Ymax xdef /Magnification xdef /Resolution xdef /PxlResolution xdef /PaperHeight xdef /PaperWidth xdef /PortraitMode xdef DoInitialScaling PortraitMode not {PaperWidth 0 SetupLandscape} if Xpos Ypos moveto } bdef % Initialize state variables to default values. /InitializeState { /DocumentType 0 def /Resolution 3600.0 def /PxlResolution 300.0 def /RasterScaleFactor PxlResolution Resolution div def /PortraitMode true def 11.0 Resolution mul /PaperHeight xdef 8.5 Resolution mul /PaperWidth xdef /Ymax PaperHeight def /Magnification 1000.0 def /Xpos 0.0 def /Ypos 0.0 def /InitialMatrix matrix currentmatrix def } bdef % Switch from base save/restore level, restoring state information. /LocalMode { /Ypos xdef /Xpos xdef /RasterScaleFactor xdef /Ymax xdef /Magnification xdef /Resolution xdef /PxlResolution xdef /PaperHeight xdef /PaperWidth xdef /PortraitMode xdef DoInitialScaling PortraitMode not {PaperWidth 0 SetupLandscape} if Xpos Ypos moveto /UserRestore where {pop UserRestore} if % invoke "UserRestore" if available SaveDviLaserFonts /DviLaserPage save def } bdef % Abbreviations /S /show load def /SV /save load def /RST /restore load def /Yadjust {Ymax exch sub} bdef % (x,y) position absolute, just set Xpos & Ypos, don't move. /SXY { Yadjust /Ypos xdef /Xpos xdef } bdef % (x,y) position absolute /XY { Yadjust 2 copy /Ypos xdef /Xpos xdef moveto } bdef % (x,0) position absolute /X { currentpoint exch pop 2 copy /Ypos xdef /Xpos xdef moveto } bdef % (0,y) position absolute /Y { currentpoint pop exch Yadjust 2 copy /Ypos xdef /Xpos xdef moveto } bdef % (x,y) position relative /xy { neg rmoveto currentpoint /Ypos xdef /Xpos xdef } bdef % (x,0) position relative /x { 0.0 rmoveto currentpoint /Ypos xdef /Xpos xdef } bdef % (0,y) position relative /y { 0.0 exch neg rmoveto currentpoint /Ypos xdef /Xpos xdef } bdef % % Print a rule. Normally this is done with a filled area defined by rlineto. % Unfortunately, doing it this way introduces a 1 pixel error on some % PostScript printer implementations. In order to get exactly correct % pixel size and positioning, an alternate approach is implemented under % control of boolean "PreciseRulePositioning". With this approach, % we create a temporary font in which the rule is the only character. % Since the alternate mechanism causes documents with lots of rules to % print much more slowly, and since the 1 pixel maximum difference is % rarely noticed, it is not used by default. Enable it by setting % "PreciseRulePositioning" true. % /PreciseRulePositioning false def /R { /ht xdef /wd xdef PreciseRulePositioning ht 1950 le wd 1950 le and PxlResolution 400 le and and {save /tfd 6 dict def tfd begin /FontType 3 def /FontMatrix [1 0 0 1 0 0] def /FontBBox [0 0 wd ht] def /Encoding 256 array dup 97 /a put def /BuildChar { pop % ignore character code pop % ignore font dict, too wd 0 0 0 wd ht setcachedevice wd ht true [1 0 0 -1 0 ht] {} imagemask } def end % tfd /tf tfd definefont setfont (a) show restore } {gsave 0 setgray currentpoint newpath moveto 0.0 ht rlineto wd 0.0 rlineto 0.0 ht neg rlineto wd neg 0.0 rlineto closepath fill grestore } ifelse wd 0.0 rmoveto currentpoint /Ypos xdef /Xpos xdef } bdef /SHADE { /percent xdef /ht xdef /wd xdef /Red xdef /Green xdef /Blue xdef gsave 0.01 percent mul setgray currentpoint newpath moveto 0.0 ht rlineto wd 0.0 rlineto 0.0 ht neg rlineto wd neg 0.0 rlineto closepath fill grestore wd 0.0 rmoveto currentpoint /Ypos xdef /Xpos xdef } bdef % % RES % /RES { /Resolution xdef /PxlResolution xdef /RasterScaleFactor PxlResolution Resolution div def DoInitialScaling } bdef % Document type: 0-document; 1-page; 2-EPS /DT { /DocumentType xdef } bdef % % Do initial scaling. % /DoInitialScaling { InitialMatrix setmatrix 72.0 Resolution div dup scale } bdef % % PM % /PM { XP /PaperWidth xdef /PaperHeight xdef /Ymax PaperHeight def /PortraitMode true def DoInitialScaling RP } bdef % % LM % /LM { XP /PaperWidth xdef /PaperHeight xdef /Ymax PaperWidth def /PortraitMode false def DoInitialScaling PaperWidth 0 SetupLandscape RP } bdef % Change magnification setting /MAG { XP /Magnification xdef RP } bdef % % Switch to landscape mode % /SetupLandscape { translate 90.0 rotate } bdef % % SPB - begin "\special" mode % % This is the PostScript procedure used to transfer from the internal % environment used for the DVI translation code emitted by DVIPS to % a standard PostScript environment. % % Parameters: 0 - Local % 1 - Global % 2 - Inline % /SPB { /spc_mode xdef spc_mode 0 eq spc_mode 2 eq or {XP} {spc_mode 1 eq {GlobalMode} if} ifelse Resolution 72.0 div dup scale % Restore default scaling... Magnification 1000.0 div dup scale % Adjust for any magnification... /Xpos Xpos 72.0 Resolution div mul 1000.0 Magnification div mul def /Ypos Ypos 72.0 Resolution div mul 1000.0 Magnification div mul def } bdef % % SPE - end "\special" mode % % This is the PostScript procedure used to reenter the internal % environment used for the DVI translation code emitted by DVIPS from % the standard PostScript environment provided for processing user-supplied % PostScript code. % % Parameters: 0 - Local % 1 - Global % 2 - Inline % /SPE { /spc_mode xdef 1000.0 Magnification div dup scale % Un-adjust for any magnification... 72.0 Resolution div dup scale % Restore default internal scaling... spc_mode 0 eq spc_mode 2 eq or {RP} {spc_mode 1 eq {LocalMode} if} ifelse } bdef % % PP % /PP { /#copies xdef showpage /#copies 1 def } bdef % % /font-name DMF % /DMF { /psz xdef /nam xdef nam findfont psz scalefont setfont } bdef % % /abcd (xxx) str-concat ==> /abcdxxx % /str-concatstr 64 string def /str-concat { /xxx xdef /nam xdef /namstr nam str-concatstr cvs def /newnam namstr length xxx length add string def newnam 0 namstr putinterval newnam namstr length xxx putinterval newnam cvn } bdef % % /abcdef 2 str-strip ==> /cdef % /str-strip { /num xdef /nam xdef /namstr nam tempstr cvs def /newlen namstr length num sub def namstr num newlen getinterval cvn } bdef % % copydict ==> new-dict on stack % /copydict { dup length 1 add dict /newdict xdef {1 index /FID ne {newdict 3 1 roll put} {pop pop} ifelse } forall newdict } bdef % % DefineCMEncoding % /DefineCMEncoding { /EncodeType xdef /CMEncoding 256 array def /Times-Roman findfont /Encoding get aload pop CMEncoding astore pop EncodeType 11 eq {Do-CM-rm-encoding} if EncodeType 12 eq {Do-CM-it-encoding} if EncodeType 13 eq {Do-CM-tt-encoding} if } bdef % % Do special mappings for the various CM-font types. Characters that % get "covered up" are repositioned in the range (128,128+32). The % range (256-64,255) is filled with the standard ISO Latin/1 mappings. % /Do-standard-CM-encodings { CMEncoding dup 0 /.notdef put dup 1 /.notdef put dup 2 /.notdef put dup 3 /.notdef put dup 4 /.notdef put dup 5 /.notdef put dup 6 /.notdef put dup 7 /.notdef put dup 8 /.notdef put dup 9 /.notdef put dup 10 /.notdef put dup 11 /.notdef put dup 12 /fi put dup 13 /fl put dup 14 /.notdef put dup 15 /.notdef put dup 16 /dotlessi put dup 17 /ogonek put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 127 /dieresis put dup 128 /space put dup 129 /quotedbl put dup 130 /sterling put dup 131 /dollar put dup 132 /less put dup 133 /greater put dup 134 /backslash put dup 135 /asciicircum put dup 136 /underscore put dup 137 /braceleft put dup 138 /bar put dup 139 /braceright put dup 140 /asciitilde put dup 141 /brokenbar put dup 142 /copyright put dup 143 /logicalnot put dup 144 /registered put dup 145 /plusminus put dup 146 /twosuperior put dup 147 /threesuperior put dup 148 /mu put dup 149 /onesuperior put dup 150 /onequarter put dup 151 /onehalf put dup 152 /threequarters put dup 153 /Eth put dup 154 /multiply put dup 155 /Thorn put dup 156 /eth put dup 157 /divide put dup 158 /thorn put dup 159 /ordfeminine put dup 160 /ordmasculine put dup 176 /Lslash put dup 181 /lslash put pop Do-ISO-Latin1-Encoding } bdef % Fill the last 64 positions with ISOLatin1 composite character definitions. /Do-ISO-Latin1-Encoding { CMEncoding dup 192 /Agrave put dup 193 /Aacute put dup 194 /Acircumflex put dup 195 /Atilde put dup 196 /Adieresis put dup 197 /Aring put dup 198 /AE put dup 199 /Ccedilla put dup 200 /Egrave put dup 201 /Eacute put dup 202 /Ecircumflex put dup 203 /Edieresis put dup 204 /Igrave put dup 205 /Iacute put dup 206 /Icircumflex put dup 207 /Idieresis put dup 208 /Eth put dup 209 /Ntilde put dup 210 /Ograve put dup 211 /Oacute put dup 212 /Ocircumflex put dup 213 /Otilde put dup 214 /Odieresis put dup 215 /multiply put dup 216 /Oslash put dup 217 /Ugrave put dup 218 /Uacute put dup 219 /Ucircumflex put dup 220 /Udieresis put dup 221 /Yacute put dup 222 /Thorn put dup 223 /germandbls put dup 224 /agrave put dup 225 /aacute put dup 226 /acircumflex put dup 227 /atilde put dup 228 /adieresis put dup 229 /aring put dup 230 /ae put dup 231 /ccedilla put dup 232 /egrave put dup 233 /eacute put dup 234 /ecircumflex put dup 235 /edieresis put dup 236 /igrave put dup 237 /iacute put dup 238 /icircumflex put dup 239 /idieresis put dup 240 /eth put dup 241 /ntilde put dup 242 /ograve put dup 243 /oacute put dup 244 /ocircumflex put dup 245 /otilde put dup 246 /odieresis put dup 247 /divide put dup 248 /oslash put dup 249 /ugrave put dup 250 /uacute put dup 251 /ucircumflex put dup 252 /udieresis put dup 253 /yacute put dup 254 /thorn put dup 255 /ydieresis put pop } bdef /Do-CM-rm-encoding { Do-standard-CM-encodings CMEncoding dup 32 /.notdef put dup 34 /quotedblright put dup 60 /exclamdown put dup 62 /questiondown put dup 92 /quotedblleft put dup 94 /circumflex put dup 95 /dotaccent put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put pop } bdef /Do-CM-it-encoding { Do-standard-CM-encodings CMEncoding dup 32 /.notdef put dup 34 /quotedblright put dup 36 /sterling put dup 60 /exclamdown put dup 62 /questiondown put dup 92 /quotedblleft put dup 94 /circumflex put dup 95 /dotaccent put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put pop } bdef /Do-CM-tt-encoding { Do-standard-CM-encodings CMEncoding dup 12 /.notdef put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 94 /circumflex put dup 126 /tilde put pop } bdef % % Routines to handle packing/unpacking numbers. % % PackHW --> % /PackHW { /num xdef /pos xdef /target xdef num 16#0000FFFF and 1 pos sub 16 mul bitshift target or } bdef % % PackByte --> % /PackByte { /num xdef /pos xdef /target xdef num 16#000000FF and 3 pos sub 8 mul bitshift target or } bdef % % UnpkHW --> % /UnpkHW { /num xdef /pos xdef num 1 pos sub -16 mul bitshift 16#0000FFFF and dup 16#00007FFF gt {16#00010000 sub} if } bdef % % UnpkByte --> % /UnpkByte { /num xdef /pos xdef num 3 pos sub -8 mul bitshift 16#000000FF and dup 16#0000007F gt {16#00000100 sub} if } bdef % % DefineCMFont % % type 10: "as-is" PostScript font % type 11: CM-mapped PostScript font - roman % type 12: CM-mapped PostScript font - text italic % type 13: CM-mapped PostScript font - typewriter type % /int-dict-name {int (-dict) str-concat} bdef /int-dict {int (-dict) str-concat cvx load} bdef /DF { true % signal that the font is already loaded DefineCMFont } bdef /DNF { false % signal that the font is not already loaded DefineCMFont } bdef /DefineCMFont { /loaded xdef /typ xdef /psz xdef /ext xdef /int xdef typ 10 ne { % font_type = 11, 12, 13 loaded not { /fnam ext 3 str-strip def fnam findfont copydict /newdict xdef typ DefineCMEncoding newdict /Encoding CMEncoding put ext newdict definefont pop } if int-dict-name ext findfont psz scalefont def currentdict int [int-dict /setfont cvx] cvx put } { % font_type = 10 /fnam ext def int-dict-name fnam findfont psz scalefont def currentdict int [int-dict /setfont cvx] cvx put } ifelse } bdef % % % [llx lly urx ury] DefinePXLFont % /PXLF { true % signal that the font is already loaded DefinePXLFont } bdef /PXLNF { false % signal that the font is not already loaded DefinePXLFont } bdef /PXLBuildCharDict 17 dict def /CMEncodingArray 256 array def 0 1 255 {CMEncodingArray exch dup tempstr cvs cvn put} for /RasterConvert {RasterScaleFactor div} bdef /TransformBBox { aload pop /BB-ury xdef /BB-urx xdef /BB-lly xdef /BB-llx xdef [BB-llx RasterConvert BB-lly RasterConvert BB-urx RasterConvert BB-ury RasterConvert] } bdef /DefinePXLFont { /newfont xdef /bb xdef /num xdef /psz xdef /dsz xdef /pxlmag xdef /ext xdef /int xdef /fnam ext (-) str-concat pxlmag tempstr cvs str-concat def newfont not { int-dict-name 13 dict def int-dict begin /FontType 3 def /FontMatrix [1 dsz div 0 0 1 dsz div 0 0] def /FontBBox bb TransformBBox def /Encoding CMEncodingArray def /CharDict 1 dict def CharDict begin /Char-Info num array def end /BuildChar { PXLBuildCharDict begin /char xdef /fontdict xdef fontdict /CharDict get /Char-Info get char get aload pop /rasters xdef /PackedWord1 xdef 0 PackedWord1 UnpkHW 16#7FFF ne { /PackedWord2 xdef /wx 0 PackedWord1 UnpkHW def /rows 2 PackedWord1 UnpkByte def /cols 3 PackedWord1 UnpkByte def /llx 0 PackedWord2 UnpkByte def /lly 1 PackedWord2 UnpkByte def /urx 2 PackedWord2 UnpkByte def /ury 3 PackedWord2 UnpkByte def } { /PackedWord2 xdef /PackedWord3 xdef /PackedWord4 xdef /wx 1 PackedWord1 UnpkHW def /rows 0 PackedWord2 UnpkHW def /cols 1 PackedWord2 UnpkHW def /llx 0 PackedWord3 UnpkHW def /lly 1 PackedWord3 UnpkHW def /urx 0 PackedWord4 UnpkHW def /ury 1 PackedWord4 UnpkHW def } ifelse rows 0 lt { /rows rows neg def /runlength 1 def } { /runlength 0 def } ifelse wx 0 llx RasterConvert lly RasterConvert urx RasterConvert ury RasterConvert setcachedevice rows 0 ne { gsave cols rows true RasterScaleFactor 0 0 RasterScaleFactor neg llx neg .5 add ury .5 add tempmatrix astore {GenerateRasters} imagemask grestore } if end } def end fnam int-dict definefont pop } if int-dict-name fnam findfont psz scalefont def currentdict int [int-dict /setfont cvx] cvx put } bdef % % PXLC % /PXLC { /rasters xdef /runlength xdef /cols xdef /rows xdef /ury xdef /urx xdef /lly xdef /llx xdef /wx xdef /code xdef /int xdef % See if the long or short format is required true cols CKSZ rows CKSZ ury CKSZ urx CKSZ lly CKSZ llx CKSZ TackRunLengthToRows { int-dict /CharDict get /Char-Info get code [0 0 llx PackByte 1 lly PackByte 2 urx PackByte 3 ury PackByte 0 0 wx PackHW 2 rows PackByte 3 cols PackByte rasters] put} { int-dict /CharDict get /Char-Info get code [0 0 urx PackHW 1 ury PackHW 0 0 llx PackHW 1 lly PackHW 0 0 rows PackHW 1 cols PackHW 0 0 16#7FFF PackHW 1 wx PackHW rasters] put} ifelse } bdef /CKSZ {abs 127 le and} bdef /TackRunLengthToRows {runlength 0 ne {/rows rows neg def} if} bdef % % PLOTC % /PLOTC { /rasters xdef /runlength xdef /cols xdef /rows xdef /ury xdef /urx xdef /lly xdef /llx xdef /psz xdef /dsz xdef /wx xdef % "Plot" a character's raster pattern. rows 0 ne { gsave currentpoint translate psz dsz div dup scale cols rows true RasterScaleFactor 0 0 RasterScaleFactor neg llx neg ury tempmatrix astore {GenerateRasters} imagemask grestore } if wx x } bdef % Routine to generate rasters for "imagemask". /GenerateRasters { rasters runlength 1 eq {RunLengthToRasters} if } bdef % Routine to convert from runlength encoding back to rasters. /RunLengthToRasters { % ...not done yet... } bdef % % These procedures handle bitmap processing. % % BMbeg % /BMbeg { /BMmagnification xdef /BMresolution xdef /BMrows xdef /BMcols xdef /BMcurrentrow 0 def gsave 0.0 setgray Resolution BMresolution div dup scale currentpoint translate BMmagnification 1000.0 div dup scale 0.0 BMrows moveto BMrows dup scale currentpoint translate /BMCheckpoint save def } bdef /BMend { BMCheckpoint restore grestore } bdef % % BMswath % /BMswath { /rows xdef /rasters xdef BMcols rows true [BMrows 0 0 BMrows neg 0 BMcurrentrow neg] {rasters} imagemask /BMcurrentrow BMcurrentrow rows add def BMcurrentrow % save this on the stack around a restore... BMCheckpoint restore /BMcurrentrow xdef /BMCheckpoint save def } bdef % % Procedures for implementing the "rotate " special: % ROTB - % - ROTE - /ROTB { XP gsave Xpos Ypos translate rotate % using from the stack Xpos neg Ypos neg translate RP } bdef /ROTE {XP grestore RP} bdef % % Procedures for implementing the "epsfile []" special: % EPSB - % - EPSE - /EPSB { 0 SPB save 4 1 roll % push the savelevel below the parameters /showpage {} def Xpos Ypos translate 1000 div dup scale % using from the stack neg exch neg exch translate % using from the stack } bdef /EPSE {restore 0 SPE} bdef % % Procedure for implementing revision bars: % REVB - % The bar is a line of width barw drawn from (barx,bary1) to (barx,bary2). /REVB { /barw xdef /barx xdef /bary2 xdef /bary1 xdef gsave barw setlinewidth barx bary1 Yadjust moveto barx bary2 Yadjust lineto stroke grestore } bdef % % A small array and two procedures to facilitate The Publisher's % implementation of gray table cells: % GRSP - % GRFB - % % GRSP saves the current DVI location so that it can be retrieved later % by the index . GRFB fills a box whose corners are given by the % indexes and with a halftone gray with the given % level and frequency. The array GRPM holds the coordinates of points % marking the corners of gray table cells. is currently ignored % since modifying it via setscreen is prohibited by Adobe's PostScript % coding conventions. % /GRPM 40 dict def /GRSP {GRPM exch [Xpos Ypos] put} bdef /GRFB { /GRfreq xdef /GRgraylev xdef GRPM exch get aload pop /GRlry xdef /GRlrx xdef GRPM exch get aload pop /GRuly xdef /GRulx xdef gsave % % set the screen frequency if it isn't zero % % (disabled currently...) % GRfreq 0 ne % {currentscreen % 3 -1 roll pop GRfreq 3 1 roll % setscreen} % if % set the gray level GRgraylev setgray % draw and fill the path GRulx GRuly moveto GRlrx GRuly lineto GRlrx GRlry lineto GRulx GRlry lineto closepath fill grestore } bdef % % Procedures for implementing the "paper " option: % SPS - % paper-manual - % etc. The boolean is passed so that a paper source procedure % knows if it is being called at the beginning (false) or end % (true) of a page. /SPS { /eop xdef /name xdef name where {pop eop name cvx exec} if } bdef /paper-manual { {statusdict /manualfeed known {statusdict /manualfeed true put} if} if } bdef /paper-automatic { {statusdict /manualfeed known {statusdict /manualfeed false put} if} if } bdef /paper-top-tray { {} {statusdict /setpapertray known {statusdict begin gsave 0 setpapertray grestore end} if} ifelse } bdef /paper-bottom-tray { {} {statusdict /setpapertray known {statusdict begin gsave 1 setpapertray grestore end} if} ifelse } bdef /paper-both-trays { {} {statusdict /setpapertray known {statusdict begin gsave 2 setpapertray grestore end} if} ifelse } bdef (end of common prolog) VMSDebug end systemdict /setpacking known {savepackingmode setpacking} if % % End of prolog section. % %%EndProlog %%BeginSetup BeginDviLaserDoc 300 300 RES 0 DT %%EndSetup %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 546 0 XY %%EndPageSetup XP /F57 /cmr17 300 71.7 71.7 128 [-4 -19 67 54] PXLNF RP XP /F57 79 52 4 -1 48 50 51 48 0 <00003FC00000 0001FFF80000 0007C03E0000 001F000F8000 003C0003C000 00F80001F000 01F00000F800 03E000007C00 03C000003C00 07C000003E00 0F8000001F00 0F8000001F00 1F0000000F80 1F0000000F80 3E00000007C0 3E00000007C0 7E00000007E0 7E00000007E0 7C00000003E0 7C00000003E0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 FC00000003F0 7E00000007E0 7E00000007E0 7E00000007E0 7E00000007E0 3E00000007C0 3F0000000FC0 1F0000000F80 1F0000000F80 0F8000001F00 0F8000001F00 07C000003E00 03E000007C00 03E000007C00 01F00000F800 00F80001F000 003E0007C000 001F000F8000 0007E07E0000 0001FFF80000 00003FC00000> PXLC RP 546 562 XY F57(O)S XP /F57 114 25 2 0 24 31 31 24 0 <0187F0 3F8FF8 FFBC7C FFB87C 0FF07C 07E07C 07E000 07C000 07C000 07C000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 078000 0FC000 FFFE00 FFFE00> PXLC RP 597 562 XY F57(r)S XP /F57 100 37 3 0 35 50 50 32 0 <00000060 00000FE0 00003FE0 00003FE0 000003E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 000001E0 003F81E0 01FFF1E0 03F079E0 07C01FE0 0F800FE0 1F0007E0 3E0003E0 3C0001E0 7C0001E0 7C0001E0 780001E0 F80001E0 F80001E0 F80001E0 F80001E0 F80001E0 F80001E0 F80001E0 F80001E0 F80001E0 780001E0 7C0001E0 7C0001E0 3C0003E0 3E0003E0 1E0007E0 1F000FE0 0FC03DE0 07E0F9F0 01FFE1FF 007F81FF> PXLC RP 623 562 XY F57(d)S XP /F57 105 18 2 0 15 48 48 16 0 <0F00 1F80 1F80 1F80 1F80 0F00 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0180 1F80 FF80 FF80 0F80 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0FC0 FFF8 FFF8> PXLC RP 659 562 XY F57(i)S XP /F57 110 37 2 0 35 31 31 40 0 <0181FE0000 3F87FF8000 FF9E0FC000 FFB803E000 0FF001E000 07E001F000 07E000F000 07C000F000 07C000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 0FC001F800 FFFC1FFF80 FFFC1FFF80> PXLC RP 677 562 XY F57(n)S XP /F57 97 33 3 0 33 31 31 32 0 <03FF0000 1FFFC000 3E03F000 3F00F800 3F007C00 3F003C00 3F003E00 1E001E00 00001E00 00001E00 00001E00 00001E00 0003FE00 007FFE00 03FF1E00 07F01E00 1FC01E00 3F001E00 7E001E00 7C001E00 FC001E00 F8001E0C F8001E0C F8001E0C F8003E0C FC007E0C 7C007E0C 7E01FF1C 3F87CFB8 1FFF07F0 03FC03C0> PXLC RP 714 562 XY F57(a)S XP /F57 108 18 2 0 16 50 50 16 0 <0180 3F80 FF80 FF80 0F80 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0780 0FC0 FFFC FFFC> PXLC RP 747 562 XY F57(l)S XP /F57 66 47 4 0 43 49 49 40 0 PXLC RP 786 562 XY F57(B)S XP /F57 111 33 2 0 31 31 31 32 0 <001FC000 00FFF800 01E03C00 07800F00 0F000780 1E0003C0 1E0003C0 3C0001E0 3C0001E0 780000F0 780000F0 780000F0 F80000F8 F80000F8 F80000F8 F80000F8 F80000F8 F80000F8 F80000F8 F80000F8 780000F0 7C0001F0 3C0001E0 3C0001E0 1E0003C0 1E0003C0 0F000780 07C01F00 01F07C00 00FFF800 001FC000> PXLC RP 833 562 XY F57(o)S XP /F57 117 37 2 0 35 31 31 40 0 <0180003000 3F8007F000 FF801FF000 FF801FF000 0F8001F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078000F000 078001F000 078001F000 078001F000 078003F000 03C007F000 03C00EF000 01F03CF800 00FFF8FF80 003FC0FF80> PXLC RP 866 562 XY F57(und)S XP /F57 115 26 2 0 24 31 31 24 0 <03FE30 0FFFF0 3E07F0 7801F0 7000F0 E00070 E00070 E00030 E00030 F00030 F80000 7F0000 7FF000 3FFF00 0FFFC0 03FFE0 003FF0 0003F8 0000F8 C0007C C0003C E0001C E0001C E0001C F0001C F80038 F80038 FC00F0 EF03F0 C7FFC0 C1FF00> PXLC RP 976 562 XY F57(s)S XP /F57 102 20 1 0 24 50 50 24 0 <0003F0 001FFC 003E3E 00787E 00F07E 01E07E 01E07E 03E03C 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 FFFFC0 FFFFC0 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 03C000 07E000 7FFF00 7FFF00> PXLC RP 1024 562 XY F57(for)S XP /F57 80 45 4 0 41 49 49 40 0 PXLC RP 1123 562 XY F57(Pro)S XP /F57 103 33 2 -15 32 32 47 32 0 <000001F8 007F07FC 01FFCF3C 07E3FC3C 0F80F83C 0F007818 1F007C00 1E003C00 3E003E00 3E003E00 3E003E00 3E003E00 3E003E00 3E003E00 3E003E00 1E003C00 1F007C00 0F007800 0F80F800 0FE3F000 0DFFC000 1C7F0000 1C000000 1C000000 1C000000 1E000000 1F000000 0FFFF800 0FFFFF00 07FFFFC0 0FFFFFE0 3E000FF0 7C0001F0 780000F0 F0000078 E0000038 E0000038 E0000038 E0000038 F0000078 70000070 780000F0 3C0001E0 1F0007C0 0FE03F80 03FFFE00 007FF000> PXLC RP 1227 562 XY F57(gra)S XP /F57 109 55 2 0 54 31 31 56 0 <0181FE003FC000 3F87FF80FFF000 FF9E0FC3C1F800 FFB803E7007C00 0FF001EE003C00 07E001FC003E00 07E000FC001E00 07C000F8001E00 07C000F8001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 078000F0001E00 0FC001F8003F00 FFFC1FFF83FFF0 FFFC1FFF83FFF0> PXLC RP 1318 562 XY F57(ms)S XP /F43 /cmr12 300 49.8 49.8 128 [-3 -13 48 37] PXLNF RP XP /F43 72 37 2 0 34 34 34 32 0 PXLC RP 461 674 XY F43(H)S XP /F43 101 22 1 0 20 21 21 24 0 <01FC00 07FF00 1F0F80 3E03C0 3C01C0 7801E0 7801E0 FFFFE0 FFFFE0 F00000 F00000 F00000 F00000 F00000 780000 780060 3C00E0 3E01C0 0F83C0 07FF00 01FC00> PXLC RP 498 674 XY F43(e)S XP /F43 108 14 0 0 12 35 35 16 0 <0F00 FF00 FF00 1F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 FFF0 FFF0> PXLC RP 519 674 XY F43(l)S XP /F43 109 41 1 0 39 21 21 40 0 <0F1FC0FE00 FF3FF1FF80 FF71F38F80 1FC0FE07C0 0F807C03C0 0F807C03C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 0F007803C0 FFF3FF9FFC FFF3FF9FFC> PXLC RP 533 674 XY F43(m)S XP /F43 117 27 1 0 25 21 21 24 0 <0F00F0 FF0FF0 FF0FF0 1F01F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F01F0 0F01F0 0F03F0 0787F8 07FEFF 01F8FF> PXLC RP 572 674 XY F43(u)S XP /F43 116 19 1 0 15 31 31 16 0 <0300 0300 0300 0300 0700 0700 0700 0F00 1F00 3F00 FFF8 FFF8 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F0C 0F0C 0F0C 0F0C 0F0C 0F1C 0798 07F8 01F0> PXLC RP 599 674 XY F43(t)S XP /F43 83 27 3 -1 23 35 36 24 0 <03F060 0FFE60 1E1FE0 3C07E0 7803E0 7801E0 F000E0 F000E0 F00060 F00060 F80060 F80000 FC0000 7E0000 7FC000 7FFC00 3FFF00 1FFF80 0FFFC0 01FFE0 001FE0 0007E0 0003F0 0001F0 0001F0 C000F0 C000F0 C000F0 E000F0 E000F0 F001E0 F801E0 FC03C0 FF0780 C7FF00 C1FC00> PXLC RP 635 674 XY F43(S)S XP /F43 99 22 2 0 20 21 21 24 0 <03FE00 0FFF80 1F0F80 3E0F80 3C0F80 780700 780000 F00000 F00000 F00000 F00000 F00000 F00000 F00000 780000 7800C0 3C01C0 3E0180 1F8780 0FFF00 03FC00> PXLC RP 662 674 XY F43(c)S XP /F43 104 27 1 0 25 35 35 24 0 <0F0000 FF0000 FF0000 1F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F1F80 0F3FE0 0F61E0 0FC1F0 0F80F0 0F80F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 FFF3FF FFF3FF> PXLC RP 682 674 XY F43(h)S XP /F43 119 35 1 0 33 21 21 32 0 PXLC RP 708 674 XY F43(w)S XP /F43 105 14 0 0 11 34 34 16 0 <0E00 1F00 1F00 1F00 0E00 0000 0000 0000 0000 0000 0000 0000 0000 0F00 FF00 FF00 1F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 FFE0 FFE0> PXLC RP 743 674 XY F43(ic)S -1 x(h)S -1 x(te)S XP /F43 110 27 1 0 25 21 21 24 0 <0F1F80 FF3FE0 FF61E0 1FC1F0 0F80F0 0F80F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 0F00F0 FFF3FF FFF3FF> PXLC RP 843 674 XY F43(n)S XP /F43 98 27 1 0 24 35 35 24 0 <0F0000 FF0000 FF0000 1F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F0000 0F1F80 0FFFE0 0FE1F0 0F8078 0F003C 0F003C 0F001C 0F001E 0F001E 0F001E 0F001E 0F001E 0F001E 0F001E 0F003C 0F003C 0F807C 0F80F8 0FE1F0 0E7FE0 0C3F80> PXLC RP 869 674 XY F43(b)S 1 x(e)S XP /F43 114 19 1 0 17 21 21 16 0 <0F7C FFFF FFDF 1F9F 0F9F 0F0E 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 0F00 FFF8 FFF8> PXLC RP 919 674 XY F43(r)S XP /F43 103 24 1 -11 22 22 33 24 0 <0000F0 03F9F8 07FFB8 1F1F78 1E0F38 3C0780 3C0780 3C0780 3C0780 3C0780 3C0780 1E0F00 1F1F00 1FFC00 1BF800 180000 180000 1C0000 1FFF00 1FFFC0 0FFFE0 1FFFF0 7C01F8 700078 E00078 E00038 E00038 F00078 F00078 7C01F0 3F07E0 0FFF80 03FE00> PXLC RP 938 674 XY F43(g)S 1084 X(St)S XP /F43 97 24 2 0 23 21 21 24 0 <1FF000 7FFC00 7C3E00 7C1F00 7C0F00 380F00 000F00 000F00 01FF00 0FCF00 3E0F00 7C0F00 780F00 F00F00 F00F18 F00F18 F01F18 F81F18 7C7FB8 3FE7F0 1F83C0> PXLC RP 1130 674 XY F43(anle)S XP /F43 121 26 1 -10 24 21 31 24 0 PXLC RP 1217 674 XY F43(y)S 1259 X(S)S XP /F43 46 14 4 0 9 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 1286 674 XY F43(.)S XP /F43 87 50 1 -1 48 34 35 48 0 PXLC RP 1316 674 XY F43(W)S 1362 X(ainer)S XP /F10 /cmsy7 300 29.1 29.1 128 [-1 -28 37 24] PXLNF RP XP /F10 3 17 2 1 13 14 13 16 0 <0400 0400 0400 C460 E4E0 3F80 0E00 3F80 E4E0 C460 0400 0400 0400> PXLC RP 1484 659 XY F10(\003)S XP /F55 /cmbx10 432 59.8 59.8 128 [-5 -15 69 45] PXLNF RP XP /F55 73 26 1 0 24 41 41 24 0 PXLC RP 300 923 XY F55(I)S XP /F55 110 38 3 0 36 27 27 40 0 PXLC RP 326 923 XY F55(n)S XP /F55 116 27 1 0 22 38 38 24 0 <007000 007000 007000 007000 00F000 00F000 01F000 01F000 03F000 07F000 1FF000 FFFFF0 FFFFF0 FFFFF0 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F038 07F038 07F038 07F038 07F038 07F078 07F878 03FCF0 03FFF0 01FFE0 007FC0> PXLC RP 362 923 XY F55(t)S XP /F55 114 28 2 0 26 27 27 24 0 PXLC RP 389 923 XY F55(r)S XP /F55 111 34 2 0 31 27 27 32 0 <003FE000 01FFFC00 03FFFE00 0FF07F80 1FC01FC0 3F800FE0 3F800FE0 7F0007F0 7F0007F0 7F0007F0 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 7F0007F0 7F0007F0 3F800FE0 3F800FE0 1FC01FC0 0FF07F80 07FFFF00 01FFFC00 003FE000> PXLC RP 417 923 XY F55(o)S XP /F55 100 38 2 0 35 42 42 40 0 <00003FF800 00003FF800 00003FF800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 007FE3F800 03FFFBF800 07FFFFF800 0FF87FF800 1FE01FF800 3FC007F800 7F8003F800 7F8003F800 7F8003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 7F0003F800 7F8003F800 7F8007F800 3FC00FF800 3FE01FF800 1FF87FF800 07FFFFFF80 03FFF3FF80 007FC3FF80> PXLC RP 454 923 XY F55(d)S XP /F55 117 38 3 0 36 27 27 40 0 PXLC RP 492 923 XY F55(u)S XP /F55 99 31 2 0 28 27 27 32 0 <003FF800 01FFFF00 07FFFF80 0FF87F80 1FE07F80 3FC07F80 7FC07F80 7F807F80 7F803F00 FF000000 FF000000 FF000000 FF000000 FF000000 FF000000 FF000000 FF000000 FF800000 7F800000 7F800000 7FC001C0 3FE003C0 1FF007C0 0FFC1F80 07FFFF00 01FFFE00 003FF000> PXLC RP 530 923 XY F55(ct)S XP /F55 105 19 3 0 18 43 43 16 0 <0FC0 1FE0 3FE0 3FF0 3FF0 3FF0 3FF0 3FE0 1FE0 0FC0 0000 0000 0000 0000 0000 0000 FFE0 FFE0 FFE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 FFFE FFFE FFFE> PXLC RP 587 923 XY F55(ion)S XP /F25 /cmr10 300 41.5 41.5 128 [-3 -11 41 31] PXLNF RP XP /F25 73 15 1 0 13 28 28 16 0 PXLC RP 300 1016 XY F25(I)S XP /F25 110 23 1 0 21 18 18 24 0 PXLC RP 315 1016 XY F25(n)S XP /F25 116 16 1 0 13 26 26 16 0 <0C00 0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFE0 FFE0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C30 1C30 1C30 1C30 1C30 1E70 0FE0 07C0> PXLC RP 353 1016 XY F25(t)S XP /F25 104 23 1 0 21 29 29 24 0 PXLC RP 369 1016 XY F25(h)S XP /F25 105 12 1 0 10 29 29 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 392 1016 XY F25(i)S XP /F25 115 16 1 0 14 18 18 16 0 <1FB0 7FF0 F0F0 E070 E030 F030 F800 7FC0 7FE0 1FF0 00F8 C078 C038 E038 F078 F8F0 FFF0 CFC0> PXLC RP 403 1016 XY F25(s)S XP /F25 112 23 1 -8 21 18 26 24 0 PXLC RP 434 1016 XY F25(p)S XP /F25 97 21 1 0 20 18 18 24 0 <1FE000 3FF800 3C3C00 3C1E00 180E00 000E00 000E00 03FE00 1F8E00 7E0E00 780E00 F80E00 F00E60 F00E60 F81E60 7C7E60 7FEFC0 1F8780> PXLC RP 457 1016 XY F25(ap)S XP /F25 101 18 1 0 16 18 18 16 0 <07E0 1FF8 3E7C 781C 781E F01E FFFE FFFE F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 502 1016 XY F25(e)S XP /F25 114 16 1 0 14 18 18 16 0 PXLC RP 521 1016 XY F25(r)S XP /F25 119 30 1 0 28 18 18 32 0 PXLC RP 552 1016 XY F25(w)S -1 x(e)S XP /F25 117 23 1 0 21 18 18 24 0 PXLC RP 613 1016 XY F25(use)S XP /F25 109 35 1 0 34 18 18 40 0 PXLC RP 686 1016 XY F25(meth)S XP /F25 111 21 1 0 19 18 18 24 0 <03F000 0FFC00 1E1E00 380700 780780 700380 F003C0 F003C0 F003C0 F003C0 F003C0 F003C0 700380 780780 380700 1E1E00 0FFC00 03F000> PXLC RP 778 1016 XY F25(o)S XP /F25 100 23 1 0 21 29 29 24 0 <001F80 001F80 000380 000380 000380 000380 000380 000380 000380 000380 000380 07F380 1FFF80 3E1F80 7C0780 780380 F80380 F00380 F00380 F00380 F00380 F00380 F00380 F00380 780780 7C0F80 3E1F80 1FFBF0 07E3F0> PXLC RP 800 1016 XY F25(ds)S 854 X(o)S XP /F25 102 13 0 0 15 29 29 16 0 <00FC 03FE 079E 071E 0F1E 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 875 1016 XY F25(f)S 902 X(pro)S 1 x(of)S 1011 X(theor)S XP /F25 121 22 1 -8 20 18 26 24 0 PXLC RP 1106 1016 XY F25(y)S 1142 X(to)S 1194 X(assi)S XP /F25 103 21 1 -9 19 19 28 24 0 <000380 07E7C0 1FFDC0 3C3DC0 781E00 781E00 781E00 781E00 781E00 781E00 3C3C00 3FF800 37E000 200000 300000 380000 3FFC00 3FFF00 3FFF80 7807C0 F003C0 E001C0 E001C0 F003C0 F807C0 7C0F80 1FFE00 07F800> PXLC RP 1259 1016 XY F25(gn)S 1317 X(ordina)S XP /F25 108 12 1 0 10 29 29 16 0 PXLC RP 1433 1016 XY F25(ls)S 1475 X(to)S XP /F25 99 18 1 0 16 18 18 16 0 <07F8 0FFC 3E3C 3C3C 7818 7800 F000 F000 F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 1527 1016 XY F25(classes)S 49 y 300 X(of)S XP /F25 40 16 3 -11 13 31 42 16 0 <01C0 0380 0380 0700 0E00 0C00 1C00 1800 3800 3800 7000 7000 7000 7000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 7000 7000 7000 7000 3800 3800 1800 1C00 0C00 0E00 0700 0380 0380 01C0> PXLC RP 355 1065 XY F25(\(terminating)S XP /F25 41 16 2 -11 12 31 42 16 0 PXLC RP 583 1065 XY F25(\))S 621 X(programs)S XP /F25 44 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 790 1065 XY F25(,)S 825 X(the)S 904 X(idea)S XP /F25 98 23 1 0 21 29 29 24 0 PXLC RP 1000 1065 XY F25(b)S 1 x(eing)S 1119 X(that)S 1217 X(the)S 1296 X (ordinal)S 1445 X(assignmen)S -1 x(t)S 50 y 300 X(pro)S XP /F25 118 22 1 0 20 18 18 24 0 PXLC RP 359 1115 XY F25(vides)S 469 X(a)S 509 X(uniform)S 670 X(w)S -1 x(a)S -1 x(y)S 760 X(of)S 813 X(measuring)S 1017 X(computational)S 1296 X (comple)S XP /F25 120 22 0 0 21 18 18 24 0 <7F8FF0 7F8FF0 0F0F80 070F00 038E00 01DC00 01D800 00F000 007000 007800 00F800 01DC00 038E00 030E00 070700 1F0780 FF8FF8 FF8FF8> PXLC RP 1423 1115 XY F25(xit)S -1 x(y)S XP /F25 92 21 5 16 19 29 13 16 0 <1818 3C3C 3838 7070 6060 E0E0 C0C0 C0C0 F8F8 FCFC FCFC 7C7C 3838> PXLC RP 1513 1115 XY F25(\\in)S 1587 X(the)S 50 y 300 X(large)S XP /F25 34 21 1 16 15 29 13 16 0 <7070 F8F8 FCFC FCFC 7C7C 0C0C 0C0C 1C1C 1818 3838 7070 F0F0 6060> PXLC RP 388 1165 XY F25(")S XP /F25 46 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 409 1165 XY F25(.)S XP /F25 87 43 1 -1 41 28 29 40 0 PXLC RP 439 1165 XY F25(W)S -3 x(e)S 510 X(are)S 580 X(not)S 653 X (concerned)S 847 X(with)S 942 X(placing)S 1085 X(prior)S 1187 X (\(e.g.)S 1284 X(p)S 1 x(olynomial)S -3 x(\))S 1514 X(b)S 1 x (ounds)S 50 y 300 X(on)S 359 X(computation)S XP /F25 123 21 0 10 20 12 2 24 0 PXLC RP 588 1215 XY F25({length,)S 749 X(but)S 827 X(rather)S 953 X(with)S 1049 X(general)S 1194 X(metho)S 1 x(ds)S 1363 X(of)S 1412 X (assessing)S 1587 X(the)S 50 y 300 X(complexit)S -1 x(y)S 512 X(of)S 561 X(natural)S 708 X(classes)S 842 X(of)S 890 X(programs)S 1075 X (according)S 1263 X(to)S 1315 X(the)S 1388 X(w)S -1 x(a)S -1 x(ys)S 1490 X(in)S 1540 X(whic)S -1 x(h)S 49 y 300 X(they)S 393 X(are)S 461 X(constructe)S 3 x(d.)S 701 X(W)S -3 x(e)S 772 X(b)S 1 x(egin)S 883 X(with)S 977 X(an)S 1034 X(o)S -1 x(v)S -1 x(erview)S 1205 X (of)S 1251 X(the)S 1322 X(metho)S 1 x(d)S 1472 X(in)S 1520 X (section)S XP /F25 50 21 2 0 18 28 28 16 0 <0FE0 1FF8 38FC 703E E03E F01F F81F F81F F80F 700F 001F 001F 001E 003E 007C 0078 00F0 01E0 03C0 0380 0700 0E03 1C03 3803 7006 FFFE FFFE FFFE> PXLC RP 300 1364 XY F25(2,)S 347 X(the)S 419 X(crucial)S 554 X(idea)S 642 X (b)S 1 x(eing)S 754 X(supplied)S 919 X(b)S -1 x(y)S XP /F25 66 29 2 0 26 28 28 24 0 PXLC RP 977 1364 XY F25(Buc)S -1 x(hhol)S XP /F25 122 18 1 0 15 18 18 16 0 <7FFC 7FFC 783C 7078 60F0 61E0 61E0 63C0 0780 078C 0F0C 1E0C 1E1C 3C18 7818 F078 FFF8 FFF8> PXLC RP 1126 1364 XY F25(z)S XP /F25 39 12 3 16 9 29 13 8 0 <70 F8 FC FC 7C 0C 0C 1C 18 38 70 F0 60> PXLC RP 1144 1364 XY F25('s)S XP /F26 /cmmi10 300 41.5 41.5 128 [-1 -10 43 31] PXLNF RP XP /F26 33 26 0 0 23 18 18 24 0 <0C001E 1C001E 18001E 38001E 30000E 701C0E 601C06 601C0E E03C0C C0380C C0381C C03018 E07038 E0F870 FFFFF0 FFFFE0 7FBFC0 3E1F00> PXLC RP 1186 1364 XY F26(!)S XP /F8 /cmr7 300 29.1 29.1 128 [-2 -8 32 22] PXLNF RP XP /F8 43 25 2 -4 22 18 22 24 0 <006000 006000 006000 006000 006000 006000 006000 006000 006000 006000 FFFFF0 FFFFF0 006000 006000 006000 006000 006000 006000 006000 006000 006000 006000> PXLC RP 1214 1349 XY F8(+)S 15 y 2 x F25({rule.)S XP /F25 83 23 2 -1 20 29 30 24 0 <07F180 1FFF80 3C3F80 780F80 700780 E00380 E00380 E00180 E00180 F00000 F80000 7E0000 7FE000 3FFC00 1FFE00 0FFF00 00FF80 000F80 0007C0 0003C0 0003C0 C001C0 C001C0 C001C0 E001C0 F00380 F80780 FE0F00 FFFE00 C3F800> PXLC RP 1363 1364 XY F25(Section)S XP /F25 51 21 2 -1 18 28 29 16 0 <0FE0 1FF8 3C7C 703E 783E 7C3E 783E 783E 003E 003C 007C 00F8 07F0 07E0 0078 003C 003E 001E 001F 001F 701F F81F F81F F81F F03E 703E 787C 3FF8 0FE0> PXLC RP 1509 1364 XY F25(3)S 1545 X(in)S -1 x(tro)S XP /F25 45 14 0 7 11 10 3 16 0 PXLC RP 1631 1364 XY F25(-)S 50 y 300 X(duces)S 414 X(a)S 449 X(large)S 551 X(class)S 648 X(of)S 696 X(higher{order)S 939 X(programs)S 1122 X (based)S 1238 X(on)S XP /F25 80 28 2 0 25 28 28 24 0 PXLC RP 1296 1414 XY F25(Plot)S XP /F25 107 22 1 0 20 29 29 24 0 PXLC RP 1373 1414 XY F25(kin's)S 1472 X(P)S XP /F25 67 30 2 -1 27 29 30 32 0 <003FC180 00FFF180 03F07B80 07C01F80 0F800F80 1F000780 3E000780 3E000380 7C000380 7C000180 78000180 F8000180 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 78000180 7C000180 7C000180 3E000380 3E000300 1F000700 0F800E00 07C01E00 03F87C00 00FFF000 003FC000> PXLC RP 1500 1414 XY F25(C)S XP /F25 70 27 2 0 24 28 28 24 0 PXLC RP 1530 1414 XY F25(F,)S 1583 X(but)S 50 y 300 X(with)S 397 X(\\b)S 1 x(ounded")S XP /F25 12 23 0 0 21 29 29 24 0 <003F00 01FF80 03C3C0 0783C0 0F03C0 0E03C0 0E0000 0E0000 0E0000 0E0000 0E0000 FFFFC0 FFFFC0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 7F87F8 7F87F8> PXLC RP 610 1464 XY F25(\014xed)S 713 X(p)S 1 x(oin)S -1 x(t)S 824 X(op)S 1 x(erators)S 1010 X(con)S -1 x(trolled)S 1205 X(b)S -1 x(y)S 1265 X (giv)S -1 x(en)S 1376 X(w)S -1 x(ell{orderings.)S XP /F25 65 31 1 0 29 29 29 32 0 <00060000 000F0000 000F0000 000F0000 001F8000 001F8000 001F8000 001F8000 0033C000 0033C000 0033C000 0061E000 0061E000 0061E000 00C0F000 00C0F000 00C0F000 01807800 01807800 01FFF800 03FFFC00 03003C00 03003C00 06001E00 06001E00 0E001E00 1F001F00 FFC0FFF0 FFC0FFF0> PXLC RP 300 1514 XY F25(A)S XP /F25 84 30 1 0 28 28 28 32 0 <7FFFFFC0 7FFFFFC0 780F03C0 700F01C0 600F00C0 E00F00E0 C00F0060 C00F0060 C00F0060 C00F0060 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 03FFFC00 03FFFC00> PXLC RP 342 1514 XY F25(T)S -3 x(ait{st)S -1 x(yle)S 531 X(computabilit)S -1 x(y)S 793 X(pro)S 1 x(of)S 898 X(then)S 989 X(ensures)S 1131 X (termination.)S 1372 X(In)S 1421 X(section)S XP /F25 52 21 1 0 19 28 28 24 0 <001C00 001C00 003C00 007C00 007C00 00DC00 01DC00 019C00 039C00 071C00 061C00 0E1C00 0C1C00 181C00 381C00 301C00 601C00 E01C00 FFFFC0 FFFFC0 001C00 001C00 001C00 001C00 001C00 001C00 01FFC0 01FFC0> PXLC RP 1556 1514 XY F25(4)S 1587 X(the)S 50 y 300 X(details)S 430 X(of)S 475 X(the)S 544 X(ordinal)S 683 X(assignmen)S -1 x(t)S 895 X(metho)S 1 x(d)S 1043 X(are)S 1111 X(w)S -1 x(ork)S -1 x(ed)S 1250 X(out)S 1322 X(for)S 1384 X(the)S 1453 X(case)S 1539 X(where)S 49 y 300 X (the)S 374 X(giv)S -1 x(en)S 485 X(w)S -1 x(ell{ordering)S 742 X (is)S XP /F25 106 13 -3 -8 8 29 37 16 0 <01C0 03E0 03E0 03E0 01C0 0000 0000 0000 0000 0000 0000 0FE0 0FE0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 F0E0 F1E0 F3C0 FF80 7E00> PXLC RP 786 1613 XY F25(just)S 871 X(the)S 945 X(ordering)S 1111 X(of)S 1161 X(the)S 1235 X(natural)S 1383 X(n)S -1 x(um)S -1 x(b)S 1 x (ers.)S 1573 X(The)S 50 y 300 X(complexit)S -1 x(y)S 515 X(b)S 1 x (ounds)S 663 X(th)S -1 x(us)S 758 X(obtained)S 932 X(turn)S 1029 X (out)S 1106 X(to)S 1161 X(b)S 1 x(e)S 1221 X(the)S 1296 X(slo)S -1 x (w{gro)S -1 x(wing)S 1554 X(func-)S 50 y 300 X(tions)S XP /F26 71 33 2 -1 31 29 30 32 0 <0003FC18 001FFF38 003F07F8 00FC03F0 01F001F0 03E000F0 07C000F0 0F8000E0 1F000060 1F000060 3E0000E0 3E0000C0 7C000000 7C000000 7C000000 FC000000 F8000000 F800FFF0 F801FFF0 F8000F80 F8000F00 F8000F00 F8001F00 78001F00 7C001E00 3E003E00 1F007E00 0FC1EE00 07FFC400 01FE0000> PXLC RP 400 1713 XY F26(G)S XP /F9 /cmmi7 300 29.1 29.1 128 [-1 -7 32 22] PXLNF RP XP /F9 11 22 1 0 19 13 13 24 0 <03F000 0E38C0 1C1DC0 381D80 701F80 701F80 701F00 F00E00 E00E00 601E00 703E00 38EEC0 1F87C0> PXLC RP 433 1719 XY F9(\013)S -6 y 468 X F25(with)S XP /F26 11 27 1 0 24 18 18 24 0 <00FC00 03FE00 0F8F06 1E078E 3C038C 3C038C 7803DC 7803D8 F801F8 F001F0 F001E0 F001E0 F001C0 F003C0 700FC0 7C3FEE 3FF8FC 0FE078> PXLC RP 561 1713 XY F26(\013)S 600 X F25(b)S 1 x(elo)S -1 x(w)S 716 X(the)S 786 X(Bac)S -1 x(hmann{)S XP /F25 72 31 2 0 28 28 28 32 0 PXLC RP 998 1713 XY F25(Ho)S -1 x(w)S -1 x(ard)S 1150 X(ordinal.)S 1307 X (Th)S -1 x(us)S 1410 X(the)S 1480 X(functions)S 50 y 300 X (computed)S 492 X(b)S -1 x(y)S 549 X(PCF)S XP /F9 60 26 3 -3 21 17 20 24 0 <0001C0 0007C0 000F80 003E00 00FC00 03F000 0FC000 1F0000 7C0000 F80000 F80000 7C0000 1F0000 0FC000 03F000 00FC00 003E00 000F80 0007C0 0001C0> PXLC RP 635 1769 XY F9(<)S -6 y 2 x F25({programs)S 866 X(are)S 935 X(just)S 1018 X(the)S 1089 X(pro)S -1 x(v)S -2 x(ably)S 1259 X(recursiv)S -1 x(e)S 1432 X(functions)S 1612 X(of)S 50 y 300 X(arithmetic.)S XP /F55 49 34 5 0 28 39 39 24 0 <000700 000F00 007F00 07FF00 FFFF00 FFFF00 F8FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 00FF00 7FFFFE 7FFFFE 7FFFFE> PXLC RP 300 1958 XY F55(1)S XP /F55 79 52 4 0 47 41 41 48 0 <0000FFE00000 0007FFFC0000 003FFFFF8000 007FC07FC000 01FF001FF000 03FC0007F800 07F80003FC00 0FF80003FE00 1FF00001FF00 1FF00001FF00 3FE00000FF80 3FE00000FF80 7FE00000FFC0 7FC000007FC0 7FC000007FC0 7FC000007FC0 FFC000007FE0 FFC000007FE0 FFC000007FE0 FFC000007FE0 FFC000007FE0 FFC000007FE0 FFC000007FE0 FFC000007FE0 FFC000007FE0 FFC000007FE0 7FC000007FC0 7FE00000FFC0 7FE00000FFC0 3FE00000FF80 3FE00000FF80 1FF00001FF00 1FF00001FF00 0FF80003FE00 07FC0007FC00 03FE000FF800 01FF001FF000 00FFC07FE000 003FFFFF8000 000FFFFE0000 0000FFE00000> PXLC RP 403 1958 XY F55(Ordin)S XP /F55 97 33 2 0 32 27 27 32 0 <03FFC000 1FFFF800 3FFFFC00 3FC3FE00 3FC0FF00 3FC07F80 3FC07F80 3FC03F80 1F803F80 00003F80 00003F80 007FFF80 03FFFF80 1FFE3F80 3FF03F80 7FC03F80 FF803F80 FF003F80 FE003F80 FE003F80 FE007F80 FF007F80 FF80FF80 7FC3FFC0 3FFFDFFC 1FFF9FFC 03FE07FC> PXLC RP 579 1958 XY F55(a)S XP /F55 108 19 3 0 18 42 42 16 0 PXLC RP 612 1958 XY F55(l)S 654 X(co)S XP /F55 109 57 3 0 56 27 27 56 0 PXLC RP 719 1958 XY F55(m)S XP /F55 112 38 2 -12 35 27 39 40 0 PXLC RP 776 1958 XY F55(pl)S XP /F55 101 32 2 0 29 27 27 32 0 <007FF000 03FFFC00 07FFFF00 0FF8FF80 1FE03F80 3FC01FC0 7F800FC0 7F800FC0 7F000FE0 FF0007E0 FF0007E0 FFFFFFE0 FFFFFFE0 FFFFFFE0 FF000000 FF000000 FF000000 FF000000 7F800000 7F800000 7FC000E0 3FE001E0 1FF003E0 0FFC0FC0 07FFFF80 01FFFF00 003FF800> PXLC RP 833 1958 XY F55(e)S XP /F55 120 36 1 0 34 27 27 40 0 PXLC RP 865 1958 XY F55(xit)S XP /F55 121 36 1 -12 34 27 39 40 0 PXLC RP 945 1958 XY F55(y)S 1004 X(o)S XP /F55 102 21 2 0 27 42 42 32 0 <0007FC00 003FFF00 00FFFF00 01FF7F80 03FC7F80 03F87F80 07F87F80 07F07F80 07F03F00 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 FFFFC000 FFFFC000 FFFFC000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 07F00000 7FFF8000 7FFF8000 7FFF8000> PXLC RP 1039 1958 XY F55(f)S 1083 X(pro)S XP /F55 103 34 2 -13 32 27 40 32 0 <01FFC3F8 07FFFFFC 1FFFFFFC 1FE3FEFC 3F80FE7C 7F80FF7C 7F007F00 7F007F00 7F007F00 7F007F00 7F007F00 7F007F00 7F80FF00 3F80FE00 1FE3FC00 1FFFFC00 1FFFF000 3DFFC000 38000000 3C000000 3C000000 3F000000 3FFFFC00 3FFFFF80 1FFFFFE0 1FFFFFE0 1FFFFFF0 7FFFFFF0 7F001FF8 FE0007F8 FC0003F8 FC0001F8 FC0001F8 FE0003F8 FF0007F8 7F800FF0 3FE03FE0 1FFFFFC0 07FFFF00 00FFF800> PXLC RP 1184 1958 XY F55(gram)S XP /F55 115 27 2 0 24 27 27 24 0 <07FFF0 1FFFF0 3FFFF0 7F03F0 FC01F0 F800F0 F80070 FC0070 FF0000 FFF000 FFFF00 7FFFC0 7FFFF0 3FFFF8 0FFFF8 01FFFC 000FFC 0001FC E000FC F0007C F0007C F800FC FC00F8 FF03F8 FFFFF0 FFFFE0 F3FF80> PXLC RP 1337 1958 XY F55(s)S 2051 Y 300 X F25(Imagine)S 458 X(a)S 493 X (program)S XP /F26 80 27 2 0 30 28 28 32 0 <03FFFF00 03FFFFC0 003E03E0 003C01F0 003C01F0 007C01F0 007C01F0 007801F0 007801F0 00F801E0 00F803E0 00F003C0 00F00780 01F01F00 01FFFE00 01FFF000 01E00000 03E00000 03E00000 03C00000 03C00000 07C00000 07C00000 07800000 07800000 0F800000 FFF80000 FFF80000> PXLC RP 659 2051 XY F26(P)S 705 X F25(as)S 756 X(a)S 791 X(transition)S 980 X(relation)S XP /F27 /cmsy10 300 41.5 41.5 128 [-2 -40 47 33] PXLNF RP XP /F27 0 32 3 9 28 11 2 32 0 PXLC RP 734 2146 XY F27(\000)S XP /F27 33 42 2 3 39 17 14 40 0 <0000000600 0000000700 0000000700 0000000380 00000001C0 00000000E0 FFFFFFFFF8 FFFFFFFFF8 00000000E0 00000001C0 0000000380 0000000700 0000000700 0000000600> PXLC RP 759 2146 XY F27(!)S XP /F9 80 21 1 0 23 20 20 24 0 <07FFE0 07FFF0 01E078 01C038 01C03C 03C03C 03C078 038078 038070 0781E0 07FF80 070000 070000 0F0000 0F0000 0E0000 0E0000 1E0000 FF8000 FF8000> PXLC RP 801 2152 XY F9(P)S XP /F27 18 32 3 -8 28 26 34 32 0 <00FFFF80 03FFFF80 0FC00000 1F000000 3C000000 38000000 70000000 70000000 E0000000 E0000000 C0000000 C0000000 C0000000 C0000000 C0000000 E0000000 E0000000 70000000 70000000 38000000 3C000000 1F000000 0FC00000 03FFFF80 00FFFF80 00000000 00000000 00000000 00000000 00000000 00000000 00000000 7FFFFF80 7FFFFF80> PXLC RP 842 2146 XY F27(\022)S 900 X F25(\(States)S XP /F27 2 32 6 0 25 20 20 24 0 PXLC RP 1041 2146 XY F27(\002)S XP /F25 68 32 2 0 29 28 28 32 0 PXLC RP 1087 2146 XY F25(Data\))S XP /F8 50 17 2 0 14 19 19 16 0 <1F80 7FE0 E3E0 E0F0 E0F0 E070 00F0 00F0 00E0 01C0 0380 0700 0E00 1C30 3830 7070 FFE0 FFE0 FFE0> PXLC RP 1192 2125 XY F8(2)S 2241 Y 300 X F25(so)S 354 X(that)S 447 X(a)S 485 X(terminating)S 714 X(computation)S 960 X(of)S 1010 X F26(P)S 1043 X F25(,)S 1072 X(starting)S 1230 X(in)S 1281 X(initial)S 1404 X (state)S XP /F26 115 19 2 0 17 18 18 16 0 <01F8 07FC 0F1E 0E1E 1E3E 1E1C 1F00 1FE0 0FF8 07F8 007C 203C F038 F038 F078 F0F0 7FE0 3F80> PXLC RP 1509 2241 XY F26(s)S XP /F8 48 17 1 0 15 19 19 16 0 <0FC0 1860 3030 6018 6018 E01C E01C E01C E01C E01C E01C E01C E01C E01C 6018 6018 3030 1860 0FC0> PXLC RP 1529 2247 XY F8(0)S -6 y 1564 X F25(with)S 50 y 300 X(input)S XP /F26 97 22 1 0 21 18 18 24 0 <01FB80 07FF80 0F1F80 1E0F80 3C0780 7C0700 780700 780F00 F80F00 F00E00 F00E00 F01E70 F01E70 F03C60 F07C60 78FEE0 3FDFC0 1F8780> PXLC RP 411 2291 XY F26(a)S 6 y F8(0)S -6 y 1 x F25(,)S 477 X(is)S 518 X(a)S 553 X(se)S XP /F25 113 22 1 -8 21 18 26 24 0 <07F180 1FF980 3F1F80 3C0F80 780780 780380 F00380 F00380 F00380 F00380 F00380 F00380 F80380 780780 7C0F80 3E1F80 1FFB80 07E380 000380 000380 000380 000380 000380 000380 001FF0 001FF0> PXLC RP 588 2291 XY F25(quence)S 2386 Y 576 X(\()S F26(s)S 6 y F8(0)S XP /F26 59 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 630 2386 XY F26(;)S 649 X(a)S 6 y F8(0)S -6 y 1 x F25(\))S 717 X F27 (\000)S 742 X(!)S 6 y F9(P)S -6 y 823 X F25(\()S F26(s)S XP /F8 49 17 3 0 14 19 19 16 0 <0600 1E00 FE00 EE00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0> PXLC RP 859 2392 XY F8(1)S -6 y 1 x F26(;)S 896 X(a)S 6 y F8(1)S -6 y 1 x F25(\))S 964 X F27(\000)S 989 X(!)S 6 y F9(P)S XP /F26 58 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 1070 2386 XY F26(:)S 1089 X(:)S 1108 X(:)S 1130 X F27(\000)S 1155 X (!)S 6 y F9(P)S -6 y 1236 X F25(\()S F26(s)S XP /F9 107 18 1 0 15 20 20 16 0 <3F00 3F00 0F00 0E00 0E00 1E00 1E00 1C78 1CDC 3D9C 3F18 3E00 3C00 7F80 7BC0 71CC 71CC F1DC F1D8 E0F0> PXLC RP 1272 2392 XY F9(k)S -6 y 2 x F26(;)S 1311 X(a)S 6 y F9(k)S -6 y 2 x F25(\))S 2481 Y 300 X(where)S 420 X F26(s)S 7 y F9(k)S -7 y 474 X F25(is)S 515 X(a)S 550 X(terminal)S 716 X(state)S 818 X(and)S 899 X F26(a)S 7 y F9(k)S -7 y 955 X F25(the)S 1026 X(output.)S 50 y 375 X (A)S 423 X(complexit)S -1 x(y{b)S 1 x(ound)S 772 X(for)S 838 X(the)S 913 X(program)S 1082 X(is)S 1127 X(then)S 1225 X(a)S 1263 X (function)S XP /F26 103 20 1 -8 18 18 26 24 0 <00FB80 03FF80 079F80 0F0F80 1E0780 3E0700 3C0700 3C0F00 7C0F00 780E00 780E00 781E00 781E00 783C00 787C00 3CFC00 3FFC00 0FB800 003800 007800 007800 F07000 F0F000 F3E000 FFC000 7F0000> PXLC RP 1428 2531 XY F26(g)S 6 y F9(P)S XP /F25 58 12 3 0 8 18 18 8 0 <70 F8 F8 F8 70 00 00 00 00 00 00 00 00 70 F8 F8 F8 70> PXLC RP 1476 2531 XY F25(:)S XP /F26 78 33 2 0 35 28 28 40 0 <03FE03FF80 03FE03FF80 003F007C00 003F003000 003F803000 003F807000 0077807000 0067C06000 0067C06000 0063C0E000 00E3E0E000 00C3E0C000 00C1E0C000 00C1F1C000 01C1F1C000 0180F18000 0180F98000 01807B8000 03807B8000 03007F0000 03003F0000 03003F0000 07003F0000 06001E0000 06001E0000 1F001E0000 FFE00E0000 FFE00C0000> PXLC RP 1494 2531 XY F26(N)S 1549 X F27(!)S 1607 X F26(N)S 50 y 300 X F25 (suc)S -1 x(h)S 392 X(that)S 481 X(on)S 537 X(inputs)S 663 X F26(a)S 6 y F8(0)S -6 y 716 X F25(of)S 762 X(size)S XP /F27 20 32 3 -8 28 26 34 32 0 <00000380 00000F80 00003F00 0000FC00 0003F000 000FC000 003F0000 00FC0000 03F00000 0FC00000 3F000000 FC000000 F0000000 FC000000 3F000000 0FC00000 03F00000 00FC0000 003F0000 000FC000 0003F000 0000FC00 00003F00 00000F80 00000380 00000000 00000000 00000000 00000000 00000000 00000000 00000000 FFFFFF80 FFFFFF80> PXLC RP 839 2581 XY F27(\024)S XP /F26 110 25 1 0 24 18 18 24 0 <3E1F80 3F3FE0 7771E0 67E0E0 E7C0E0 C780E0 0700E0 0F01E0 0F01C0 0E01C0 0E03C0 1E038E 1E038E 1C078C 1C071C 3C07B8 3C03F0 3801E0> PXLC RP 883 2581 XY F26(n)S 921 X F25(the)S 991 X(length)S XP /F26 107 22 2 0 20 29 29 24 0 <0FC000 1FC000 03C000 03C000 038000 038000 078000 078000 070000 070000 0F0000 0F0F80 0E1F80 0E3980 1E7380 1EE780 1DC780 1F8780 3F0000 3FC000 3BE000 38E000 78E1C0 78E1C0 70E180 70E380 F0F700 F07F00 E03E00> PXLC RP 1116 2581 XY F26(k)S 1152 X F25(of)S 1198 X(the)S 1268 X(ensuing)S 1417 X(computation)S 50 y 300 X(is)S 342 X F27(\024)S 386 X F26(g)S 6 y F9(P)S -6 y 433 X F25(\()S F26(n)S F25(\).)S 300 2675 XY 538 2 R XP /F6 /cmsy6 300 24.9 24.9 128 [0 -24 34 20] PXLNF RP XP /F6 3 16 3 1 13 12 11 16 0 <0C00 0C00 8C40 EDC0 7F80 0C00 7F80 EDC0 8C40 0C00 0C00> PXLC RP 346 2702 XY F6(\003)S XP /F13 /cmr8 300 33.2 33.2 128 [-3 -9 34 25] PXLNF RP XP /F13 66 25 1 0 22 23 23 24 0 PXLC RP 364 2714 XY F13(B)S XP /F13 111 18 1 0 16 14 14 16 0 <07C0 1FF0 3838 701C 701C F01E F01E F01E F01E F01E 701C 3838 1FF0 07C0> PXLC RP 388 2714 XY F13(o)S XP /F13 116 14 1 0 11 20 20 16 0 <1800 1800 1800 1800 3800 7800 FF80 FF80 3800 3800 3800 3800 3800 3800 38C0 38C0 38C0 3DC0 3F80 1F00> PXLC RP 406 2714 XY F13(t)S XP /F13 104 20 0 0 19 23 23 24 0 PXLC RP 420 2714 XY F13(h)S XP /F13 97 18 1 0 17 14 14 16 0 <3FC0 7FF0 78F8 7838 3038 0FF8 3F38 7C38 F838 F03B F03B F8FB 7FFE 3F1C> PXLC RP 453 2714 XY F13(a)S XP /F13 117 20 0 0 19 14 14 24 0 PXLC RP 471 2714 XY F13(utho)S XP /F13 114 14 0 0 13 14 14 16 0 PXLC RP 541 2714 XY F13(r)S XP /F13 115 14 1 0 12 14 14 16 0 <3FC0 7FC0 F1C0 E0C0 F000 FF00 7FC0 3FE0 03E0 C0E0 E0E0 F1E0 FFC0 DF80> PXLC RP 555 2714 XY F13(s)S XP /F13 112 20 0 -6 18 14 20 24 0 PXLC RP 582 2714 XY F13(part)S XP /F13 105 10 0 0 8 23 23 8 0 <3C 3C 3C 3C 00 00 00 00 00 7C 7C 1C 1C 1C 1C 1C 1C 1C 1C 1C 1C FF FF> PXLC RP 647 2714 XY F13(ia)S XP /F13 108 10 0 0 9 23 23 16 0 PXLC RP 675 2714 XY F13(ll)S XP /F13 121 19 1 -6 17 14 20 16 0 PXLC RP 694 2714 XY F13(y)S 726 X(supp)S 1 x(ort)S XP /F13 101 16 1 0 14 14 14 16 0 <0FC0 3FF0 7CF0 7878 FFF8 FFF8 F000 F000 F000 F000 7818 7C38 1FF0 0FE0> PXLC RP 845 2714 XY F13(e)S XP /F13 100 20 1 0 19 23 23 24 0 <007E00 007E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 0FEE00 3FFE00 7C3E00 780E00 F00E00 F00E00 F00E00 F00E00 F00E00 F00E00 781E00 7C3E00 3FFFC0 0FCFC0> PXLC RP 861 2714 XY F13(d)S XP /F13 98 20 0 0 18 23 23 24 0 PXLC RP 894 2714 XY F13(b)S -1 x(y)S 945 X(the)S XP /F13 83 20 2 0 17 23 23 16 0 <0FCC 3FFC 787C 703C E01C E00C E00C F000 FC00 7FC0 7FF0 3FFC 0FFC 007E 001E 001E C00E C00E E00E F01C F83C FFF8 CFE0> PXLC RP 1007 2714 XY F13(S)S XP /F13 99 16 1 0 14 14 14 16 0 <0FF0 1FF8 7C78 7878 F030 F000 F000 F000 F000 F800 7818 7E38 1FF0 0FE0> PXLC RP 1027 2714 XY F13(cie)S XP /F13 110 20 0 0 19 14 14 24 0 PXLC RP 1068 2714 XY F13(nce)S XP /F13 84 25 1 0 23 23 23 24 0 <7FFFF8 7FFFF8 707838 607818 E0781C C0780C C0780C C0780C 007800 007800 007800 007800 007800 007800 007800 007800 007800 007800 007800 007800 007800 0FFFC0 0FFFC0> PXLC RP 1133 2714 XY F13(T)S XP /F13 119 25 1 0 23 14 14 24 0 PXLC RP 1158 2714 XY F13(winnin)S XP /F13 103 18 1 -7 17 14 21 16 0 <1FDF 3FFF 78F7 F078 F078 F078 F078 78F0 3FE0 7FC0 6000 7FF0 7FFC 7FFC F03E E00E E00E F01E 783C 3FF8 0FE0> PXLC RP 1262 2714 XY F13(g)S XP /F13 67 25 2 0 22 23 23 24 0 <00FE30 07FF70 0FC3F0 1F01F0 3C00F0 7C0070 780070 780030 F00030 F00000 F00000 F00000 F00000 F00000 F00030 780030 780030 7C0070 3C0060 1F00E0 0FC3C0 07FF80 00FE00> PXLC RP 1293 2714 XY F13(Con)S -1 x(tract)S 1443 X(SC)S XP /F13 49 18 2 0 14 21 21 16 0 <0300 0F00 FF00 F700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 7FF0 7FF0> PXLC RP 1488 2714 XY F13(1)S XP /F13 42 18 2 11 15 25 14 16 0 <0200 0200 0200 C218 F278 3AE0 0F80 0F80 3AE0 F278 C218 0200 0200 0200> PXLC RP 1506 2714 XY F13(*)S XP /F13 123 18 0 8 17 10 2 24 0 PXLC RP 1523 2714 XY F13({CT)S XP /F13 57 18 1 0 16 21 21 16 0 <0FE0 1FF0 7C78 783C F01C F01E F01E F01E F01E F03E 787E 3FFE 1FDE 021E 001C 783C 783C 7878 78F0 3FE0 1F80> PXLC RP 1592 2714 XY F13(91{)S XP /F13 48 18 1 0 16 21 21 16 0 <07C0 1FF0 3C78 783C 701C 701C F01E F01E F01E F01E F01E F01E F01E F01E F01E 701C 701C 783C 3C78 1FF0 07C0> PXLC RP 300 2753 XY F13(0)S XP /F13 55 18 2 0 17 22 22 16 0 <6000 7FFE 7FFE 7FFE E01C C038 C070 00E0 00C0 01C0 0380 0380 0780 0700 0700 0F00 0F00 0F00 0F00 0F00 0F00 0F00> PXLC RP 318 2753 XY F13(7)S XP /F13 50 18 2 0 15 21 21 16 0 <1F80 7FE0 F1F0 F8F8 F878 F878 7078 0078 00F8 00F0 01F0 01E0 03C0 0780 0E00 1C18 3818 7038 FFF0 FFF0 FFF0> PXLC RP 335 2753 XY F13(2)S XP /F13 52 18 1 0 16 21 21 16 0 <0070 00F0 00F0 01F0 03F0 0770 0670 0E70 1C70 1870 3070 7070 E070 FFFE FFFE 0070 0070 0070 0070 03FE 03FE> PXLC RP 353 2753 XY F13(4)S 382 X(o)S XP /F13 102 11 0 0 13 23 23 16 0 <01F0 07F8 0F78 1E78 1C78 1C00 1C00 1C00 1C00 FF80 FF80 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 400 2753 XY F13(f)S 423 X(the)S XP /F13 69 24 1 0 22 23 23 24 0 PXLC RP 483 2753 XY F13(Europ)S 1 x(ean)S 644 X(Co)S XP /F13 109 29 0 0 29 14 14 32 0 PXLC RP 687 2753 XY F13(mm)S -1 x(unit)S -1 x(y)S XP /F13 46 10 3 0 7 4 4 8 0 PXLC RP 822 2753 XY F13(.)S 2878 Y 941 X F25(3)S XP /F25 55 21 2 -1 19 28 29 24 0 <600000 7FFF80 7FFF80 7FFF80 600700 C00600 C00E00 C01C00 001800 003800 007000 006000 00E000 00E000 01C000 01C000 01C000 03C000 038000 038000 038000 078000 078000 078000 078000 078000 078000 078000 078000> PXLC RP 962 2878 XY F25(7)S XP /F25 57 21 2 -1 18 28 29 16 0 <07E0 1FF0 3E78 381C 781C 700E F00E F00E F00F F00F F00F F00F F00F F01F 701F 781F 3C3F 1FEF 0FCF 010F 000E 001E 781E 781C 783C 7878 78F0 3FE0 1F80> PXLC RP 983 2878 XY F25(9)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 99 395 670 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(3)S XP /F25 56 21 2 -1 18 28 29 16 0 <07E0 1FF8 3C3C 301C 700E 6006 6006 7006 780E 7C0C 3F1C 3FF8 1FF0 0FF0 1FF8 3DFC 787E 703F F01F E00F E007 E007 E007 F007 700E 781E 3C3C 1FF8 07E0> PXLC RP 321 462 XY F25(8)S XP /F25 48 21 2 -1 18 28 29 16 0 <03C0 0FF0 1C38 381C 381C 700E 700E 700E F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F 700E 700E 700E 381C 381C 1C38 0FF0 07E0> PXLC RP 342 462 XY F25(0)S XP /F18 /cmr9 300 37.4 37.4 128 [-3 -10 37 28] PXLNF RP XP /F18 72 29 1 0 27 26 26 32 0 PXLC RP 700 462 XY F18(H)S XP /F18 46 11 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 729 462 XY F18(.)S XP /F18 83 21 2 0 18 26 26 16 0 <0FC6 3FF6 787E 701E E00E E00E E006 E006 F000 FC00 7F80 7FF0 3FFC 0FFE 01FE 003F 000F 000F C007 C007 E007 E00F F00E FC3C DFFC C7F0> PXLC RP 752 462 XY F18(S)S XP /F18 99 17 1 0 15 16 16 16 0 <07F8 1FFC 3E3C 783C 7818 F000 F000 F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 774 462 XY F18(c)S XP /F18 104 21 1 0 19 26 26 24 0 PXLC RP 790 462 XY F18(h)S XP /F18 119 28 1 0 26 16 16 32 0 PXLC RP 810 462 XY F18(w)S XP /F18 105 11 0 0 9 26 26 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 838 462 XY F18(ic)S -1 x(h)S XP /F18 116 15 1 0 12 23 23 16 0 <0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFC0 FFC0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C60 1C60 1C60 1C60 1EE0 0FC0 0780> PXLC RP 885 462 XY F18(t)S XP /F18 101 17 1 0 15 16 16 16 0 <07E0 1FF0 3C78 783C 703C FFFC FFFC F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 900 462 XY F18(e)S XP /F18 110 21 1 0 19 16 16 24 0 PXLC RP 917 462 XY F18(n)S XP /F18 98 21 1 0 19 26 26 24 0 PXLC RP 937 462 XY F18(b)S 1 x(e)S XP /F18 114 15 1 0 13 16 16 16 0 PXLC RP 976 462 XY F18(r)S XP /F18 103 19 1 -8 17 16 24 16 0 <0FCF 1FFF 3CF7 7878 7878 7878 7878 7878 3CF0 3FE0 3FC0 3000 3000 3FF8 3FFC 3FFE 701F E00F E007 E007 F00F 7C3E 3FFC 0FF0> PXLC RP 991 462 XY F18(g)S XP /F18 97 19 1 0 18 16 16 24 0 <3FC000 7FF000 78F800 787800 303800 003800 07F800 3F3800 7C3800 F83800 F03800 F03980 F07980 F8F980 7FFF00 3F1E00> PXLC RP 1023 462 XY F18(an)S XP /F18 100 21 1 0 19 26 26 24 0 <007E00 007E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 0FEE00 1FFE00 3E3E00 781E00 780E00 F00E00 F00E00 F00E00 F00E00 F00E00 F00E00 780E00 781E00 3C3E00 1FFFC0 0FCFC0> PXLC RP 1064 462 XY F18(d)S 1098 X(S.S.)S XP /F18 87 39 1 0 37 26 26 40 0 PXLC RP 1175 462 XY F18(W)S -3 x(ainer)S 553 Y 375 X F25(If)S 422 X(one)S 504 X(wishes)S 639 X(to)S 696 X(compare)S 868 X(and)S 954 X (classify)S 1103 X(the)S 1181 X(complexities)S 1423 X(of)S 1476 X (programs)S 50 y 300 X(in)S 352 X(a)S 391 X(giv)S -1 x(en)S 503 X (language)S XP /F26 76 28 2 0 26 28 28 24 0 <03FFF0 03FFF0 003E00 003C00 003C00 003C00 007C00 007800 007800 007800 00F800 00F000 00F000 00F000 01F000 01E000 01E000 01E007 03E006 03C006 03C00E 03C00C 07C01C 07803C 078078 0F81F8 FFFFF8 FFFFF0> PXLC RP 680 603 XY F26(L)S F25(,)S 738 X(one)S 818 X(w)S -1 x(ould)S 943 X (lik)S -1 x(e)S 1023 X(\(if)S 1081 X(p)S 1 x(ossible\))S 1257 X(to)S 1312 X(arrange)S 1466 X(a)S 1504 X(suitable)S 50 y 300 X (collection)S 482 X(of)S 528 X(b)S 1 x(ounding)S 709 X(functions)S 886 X F26(g)S 6 y F9(P)S -6 y 945 X F25(in)S 991 X(a)S 1024 X (linear,)S 1149 X(w)S -1 x(ell-founded)S 1389 X(scale.)S 1504 X (That)S 1606 X(is,)S 50 y 300 X(one)S 373 X(w)S -1 x(ould)S 492 X (try)S 557 X(to)S 606 X(asso)S 1 x(ciate)S 778 X(with)S 870 X F26 (L)S 909 X F25(a)S 941 X(\\natural")S 1125 X(and)S 1203 X (\\minimal")S 1404 X(w)S -1 x(ell{ordering)S XP /F27 107 21 4 -11 16 31 42 16 0 PXLC RP 300 752 XY F27(k)S F26(L)S F27(k)S F25(,)S 395 X(together)S 558 X (with)S 652 X(a)S 686 X(hierarc)S -1 x(h)S -1 x(y)S 867 X(of)S 914 X (functions)S XP /F27 102 21 2 -10 18 31 41 16 0 <001F 007F 00F8 01E0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 03C0 0F80 FF00 FC00 FF00 0F80 03C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01E0 00F8 007F 001F> PXLC RP 1092 752 XY F27(f)S F26(G)S 7 y F9(\013)S -7 y 1 x F25(:)S 1188 X F26(\013)S XP /F27 50 28 3 -3 24 23 26 24 0 <00FFF8 03FFF8 0FC000 1F0000 3C0000 380000 700000 700000 E00000 E00000 C00000 C00000 FFFFF8 FFFFF8 C00000 C00000 E00000 E00000 700000 700000 380000 3C0000 1F0000 0FC000 03FFF8 00FFF8> PXLC RP 1226 752 XY F27(2)S 1265 X(k)S F26(L)S F27(k)S XP /F27 103 21 2 -10 18 31 41 16 0 PXLC RP 1335 752 XY F27(g)S F25(,)S 1381 X(and)S 1461 X(an)S 1518 X (ordinal)S 50 y 300 X(assignmen)S -1 x(t)S 514 X F26(P)S 558 X F27 (2)S 597 X F26(L)S 637 X F27(\000)S 662 X(!)S 715 X F26(\013)S 7 y F9(P)S -7 y 781 X F27(2)S 820 X(k)S F26(L)S F27(k)S 904 X F25(suc)S -1 x(h)S 997 X(that)S 889 Y 339 X(\(i\))S 404 X F26(G)S 7 y F9 (\013)S XP /F1 /cmmi5 300 20.8 20.8 128 [0 -5 28 16] PXLNF RP XP /F1 80 18 2 0 19 14 14 24 0 <1FFE00 1FFF00 070380 0F0180 0F0380 0E0380 0E0700 1FFC00 1E0000 1C0000 1C0000 3C0000 FF0000 FF0000> PXLC RP 458 900 XY F1(P)S -11 y 498 X F25(is)S 539 X(a)S 574 X(complexit)S -1 x(y)S 785 X(b)S 1 x(ound)S 913 X(for)S 977 X F26(P)S 1010 X F25 (,)S 981 Y 328 X(\(ii\))S 404 X(if)S 442 X F26(\013)S XP /F26 60 32 3 -3 28 23 26 32 0 <00000180 00000780 00001F80 00007E00 0001F800 0007E000 001F8000 007E0000 01F80000 07E00000 1F800000 7E000000 F8000000 F8000000 7E000000 1F800000 07E00000 01F80000 007E0000 001F8000 0007E000 0001F800 00007E00 00001F80 00000780 00000180> PXLC RP 480 981 XY F26(<)S XP /F26 12 23 -1 -8 24 29 37 32 0 <0000FC00 0007FE00 000F8F00 001C0780 00380380 00700380 00E00780 00C00780 01C00700 01800F00 03801E00 038FFC00 030FF800 030FFC00 07001E00 06001E00 06000E00 06000E00 0E001E00 0C001E00 0C001E00 0C001E00 1C003E00 1C003C00 1E007800 1F00F800 3F81F000 33FFC000 31FF8000 30180000 70000000 60000000 60000000 60000000 E0000000 C0000000 C0000000> PXLC RP 524 981 XY F26(\014)S 561 X F27(2)S 600 X(k)S F26(L)S F27(k)S 684 X F25(then)S 778 X F26(G)S XP /F9 12 19 0 -6 17 20 26 24 0 <003F00 00E300 01C380 038380 070380 060780 0E0700 0CFF00 0CCF00 1CFF00 180380 180380 180380 380780 380700 380700 380E00 7C1E00 6E3800 67F000 600000 E00000 C00000 C00000 C00000 C00000> PXLC RP 811 987 XY F9(\014)S -6 y 847 X F25(dominates)S 1046 X F26(G)S 6 y F9(\013)S -6 y 1 x F25(.)S 1068 Y 300 X(The)S 391 X(aim)S 477 X (here)S 573 X(is)S 620 X(to)S 676 X(sho)S -1 x(w)S 785 X(ho)S -1 x (w)S 877 X(this)S 963 X(can)S 1045 X(b)S 1 x(e)S 1107 X(ac)S -1 x (hiev)S -1 x(ed,)S 1292 X(at)S 1349 X(least)S 1451 X(for)S 1520 X (certain)S 50 y 300 X(natural)S 445 X(classes)S 578 X(of)S 625 X (terminating)S 851 X(programs.)S 50 y 375 X(The)S 462 X(metho)S 1 x (ds)S 631 X(are)S 703 X(those)S 813 X(of)S 862 X(pro)S 1 x(of)S 973 X(theory)S 1105 X(and)S 1188 X(the)S 1261 X(origins)S 1397 X (lie)S 1455 X(in)S XP /F25 91 12 4 -10 11 31 41 8 0 PXLC RP 1505 1168 XY F25([)S XP /F25 53 21 2 -1 18 28 29 16 0 <301C 3FFC 3FF8 3FF0 3FC0 3000 3000 3000 3000 3000 33E0 3FF8 3E3C 381E 381E 300E 000F 000F 000F 000F F00F F00F F00F F01E E01E 703C 7878 3FF0 0FC0> PXLC RP 1517 1168 XY F25(5)S -1 x(,)S 1565 X(3)S -1 x(,)S 1613 X(4)S -1 x (,)S XP /F25 49 21 4 0 17 28 28 16 0 <0300 0700 3F00 FF00 C700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 FFF8 FFF8> PXLC RP 300 1217 XY F25(11,)S 367 X(8)S XP /F25 93 12 0 -10 7 31 41 8 0 PXLC RP 388 1217 XY F25(].)S 429 X(First)S 530 X(w)S -1 x(e)S 591 X(need)S 688 X(to)S 739 X(dev)S -1 x(elop)S 889 X(a)S 924 X(little)S 1023 X (\\constructiv)S -1 x(e{ordinal{theory".)S XP /F30 /cmbx10 300 41.5 41.5 128 [-4 -10 47 31] PXLNF RP XP /F30 49 24 3 0 20 27 27 24 0 <006000 01E000 0FE000 FFE000 FFE000 F3E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 7FFF80 7FFF80 7FFF80> PXLC RP 375 1315 XY F30(1)S XP /F30 46 13 3 0 10 7 7 8 0 <7C FE FE FE FE FE 7C> PXLC RP 399 1315 XY F30(.1.)S XP /F30 68 37 2 0 33 28 28 32 0 PXLC RP 465 1315 XY F30(D)S XP /F30 101 22 1 0 20 18 18 24 0 <03FC00 0FFF00 3F0F80 3E07C0 7E03C0 7C03E0 FC03E0 FFFFE0 FFFFE0 FC0000 FC0000 FC0000 7C0000 7E00E0 3F01E0 1FC3C0 0FFF80 03FE00> PXLC RP 502 1315 XY F30(e)S XP /F30 12 27 0 0 25 29 29 32 0 <001FE000 00FFF800 03F87800 07E0FC00 0FC0FC00 0F80FC00 0F80FC00 0F807800 0F800000 0F800000 0F800000 FFFFFC00 FFFFFC00 FFFFFC00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 7FE1FF80 7FE1FF80 7FE1FF80> PXLC RP 523 1315 XY F30(\014)S XP /F30 110 27 1 0 25 18 18 24 0 PXLC RP 550 1315 XY F30(n)S XP /F30 105 13 1 0 12 30 30 16 0 <1E00 3F00 7F00 7F00 7F00 3F00 1E00 0000 0000 0000 0000 0000 FF00 FF00 FF00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 FFE0 FFE0 FFE0> PXLC RP 576 1315 XY F30(i)S XP /F30 116 19 1 0 15 26 26 16 0 <0700 0700 0700 0700 0F00 0F00 0F00 1F00 3FFC FFFC FFFC 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F1C 1F1C 1F1C 1F1C 1FBC 0FF8 07F0> PXLC RP 590 1315 XY F30(ti)S XP /F30 111 24 1 0 22 18 18 24 0 <01FC00 0FFF80 1F07C0 3E03E0 7C01F0 7C01F0 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 7C01F0 7C01F0 3E03E0 1F07C0 0FFF80 01FC00> PXLC RP 622 1315 XY F30(on.)S 706 X F25(The)S 796 X(set)S XP /F25 10 30 2 0 27 29 29 32 0 <00FF8000 03FFE000 0FC1F800 1F007C00 3E003E00 7C001F00 7C001F00 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 7C001F00 7C001F00 3C001E00 3E003E00 1E003C00 1E003C00 0F007800 CF007980 C7007180 C3006180 E380E380 7F80FF00 7F80FF00 7F80FF00 7F80FF00> PXLC RP 865 1315 XY F25(\012)S 6 y F8(1)S -6 y 932 X F25(of)S 984 X(coun)S -1 x(table)S XP /F28 /cmti10 300 41.5 41.5 128 [-1 -11 45 31] PXLNF RP XP /F28 116 14 3 0 15 26 26 16 0 <01C0 01C0 01C0 03C0 03C0 0380 0380 0780 FFE0 FFE0 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C70 3C70 3860 38E0 39C0 3F80 1F00> PXLC RP 1176 1315 XY F28(t)S XP /F28 114 18 3 0 19 18 18 16 0 <1E3E 3F7F 77E7 67C7 E78F C78F 070F 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800> PXLC RP 1190 1315 XY F28(r)S XP /F28 101 19 4 0 18 18 18 16 0 <01F8 07FC 0F1C 1E0C 3C0C 780C 783C 7FF8 FFC0 F000 F000 F000 F000 F00C 701C 7878 3FF0 1FC0> PXLC RP 1205 1315 XY F28(e)S -2 x(e)S XP /F28 123 21 4 10 22 12 2 24 0 PXLC RP 1241 1315 XY F28({)S XP /F28 111 21 4 0 21 18 18 24 0 <01F800 07FE00 0F9F00 1E0F00 3C0700 7C0780 780780 780F80 F80F00 F00F00 F00F00 F01F00 F01E00 F03C00 F07C00 78F800 3FE000 1F8000> PXLC RP 1262 1315 XY F28(or)S XP /F28 100 21 4 0 22 29 29 24 0 <000FC0 001FC0 0003C0 0003C0 000380 000380 000780 000780 000700 000700 000F00 03EF00 07FE00 0F3E00 1E1E00 3C1E00 781C00 781C00 783C00 F83C00 F03800 F03800 F079C0 F079C0 E0F180 F0F180 73F300 7FFF00 3E3E00> PXLC RP 1299 1315 XY F28(d)S XP /F28 105 13 3 0 14 28 28 16 0 <01C0 01E0 01E0 01C0 0000 0000 0000 0000 0000 0000 1F00 3F00 7780 6380 E780 C700 0F00 0E00 0E00 1E00 1C00 1CE0 3CE0 38C0 39C0 3B80 3F80 1E00> PXLC RP 1320 1315 XY F28(i)S XP /F28 110 23 3 0 24 18 18 24 0 <1E3F00 3F7F80 77F380 67C380 E78380 C78380 070380 0F0780 0F0700 0E0700 0E0F00 1E0E38 1E0E38 1C1E30 1C1C70 3C1EE0 3C0FC0 380F80> PXLC RP 1333 1315 XY F28(n)S XP /F28 97 21 4 0 22 18 18 24 0 <03EE00 07FE00 0F3E00 1E1E00 3C1E00 781C00 781C00 783C00 F83C00 F03800 F03800 F079C0 F079C0 E0F180 F0F180 73F300 7FFF00 3E3E00> PXLC RP 1356 1315 XY F28(a)S XP /F28 108 11 4 0 13 29 29 16 0 <1F80 3F80 0780 0780 0700 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800 3800 7800 7800 7000 7000 F700 F700 E600 E600 EE00 FC00 7C00> PXLC RP 1378 1315 XY F28(l)S XP /F28 115 17 3 0 17 18 18 16 0 <03F0 07F8 0F3C 0E3C 1E7C 1E38 1F00 1FE0 0FF0 0FF0 00F8 2078 F070 F070 F0F0 E1E0 7FC0 3F80> PXLC RP 1388 1315 XY F28(s)S 1423 X F25(is)S 1470 X(generated)S 50 y 300 X (inductiv)S -1 x(ely)S 513 X(b)S -1 x(y)S 571 X(three)S 677 X (rules:)S 1458 Y 643 X(0)S 675 X F27(2)S 715 X F25(\012)S 6 y F8(1)S -6 y 1 x F26(;)S 49 y 643 X(\013)S 681 X F27(2)S 721 X F25(\012)S 7 y F8(1)S XP /F25 61 32 2 4 29 16 12 32 0 PXLC RP 792 1507 XY F25(=)S XP /F27 41 42 2 -2 39 22 24 40 0 <0000030000 0000030000 0000038000 000001C000 000000E000 000000E000 FFFFFFF000 FFFFFFFC00 0000001E00 0000000780 00000003E0 00000000F8 00000001F0 0000000780 0000000F00 0000001C00 FFFFFFF800 FFFFFFF000 000000E000 000000C000 000001C000 0000038000 0000030000 0000030000> PXLC RP 818 1507 XY F27(\))S 882 X F26(\013)S XP /F25 43 32 2 -4 29 24 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFE0 FFFFFFE0 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000> PXLC RP 918 1507 XY F25(+)S 960 X(1)S 992 X F27(2)S 1031 X F25(\012)S 7 y F8(1)S -7 y 2 x F26(;)S XP /F27 56 23 0 -1 22 29 30 24 0 PXLC RP 643 1557 XY F27(8)S XP /F26 120 24 2 0 21 18 18 24 0 <0F8F80 3FDFC0 38F9C0 70F1C0 E0F3C0 C0E3C0 00E3C0 01E000 01E000 01C000 01C000 73C0E0 F3C0E0 F380C0 F781C0 E7C780 FFFF00 7C7C00> PXLC RP 666 1557 XY F26(x)S 701 X F27(2)S 741 X F26(N)S 779 X F25(\()S F26 (\013)S XP /F9 120 19 1 0 16 13 13 16 0 <1F7C 33EE 63CE E3CC C380 0380 0780 0700 070C E70C EF1C CF38 FBE0> PXLC RP 821 1564 XY F9(x)S -7 y 854 X F27(2)S 893 X F25(\012)S 7 y F8(1)S -7 y 1 x F25(\))S 981 X(=)S 1006 X F27(\))S 1071 X F26(\013)S 1109 X F25(=)S XP /F27 104 16 4 -11 13 31 42 16 0 <0180 0380 0380 0300 0700 0700 0600 0E00 0E00 0C00 1C00 1C00 1800 3800 3800 3000 7000 7000 6000 E000 E000 E000 E000 6000 7000 7000 3000 3800 3800 1800 1C00 1C00 0C00 0E00 0E00 0600 0700 0700 0300 0380 0380 0180> PXLC RP 1153 1557 XY F27(h)S F26(\013)S 7 y F9(x)S XP /F27 105 16 2 -11 11 31 42 16 0 PXLC RP 1216 1557 XY F27(i)S 1244 X(2)S 1283 X F25(\012)S 7 y F8(1)S -7 y 2 x F26(:)S 1651 Y 375 X F25(Alternativ)S -1 x(ely)S 626 X(w)S -1 x (e)S 688 X(can)S 764 X(de\014ne)S 884 X(\012)S 6 y F8(1)S -6 y 947 X F25(in)S 995 X(terms)S 1111 X(of)S XP /F25 77 38 2 0 35 28 28 40 0 PXLC RP 1159 1651 XY F25(Martin{)S XP /F25 76 26 2 0 23 28 28 24 0 PXLC RP 1305 1651 XY F25(L)S XP /F25 127 21 4 24 16 29 5 16 0 <70E0 F9F0 F9F0 F9F0 70E0> PXLC RP 1331 1651 XY F25(\177)S 1331 X(of)S 3 x('s)S 1410 X(w)S -1 x (ell{founded)S 50 y 300 X(t)S -1 x(yp)S 1 x(e)S 393 X(constructor)S XP /F26 87 39 2 -1 43 28 29 48 0 PXLC RP 613 1701 XY F26(W)S 658 X F25(:)S 1796 Y 691 X(\012)S 6 y F8(1)S XP /F27 17 32 2 0 29 20 20 32 0 PXLC RP 751 1796 XY F27(\021)S 795 X F26(W)S 840 X(x)S 875 X F27(2)S 915 X F26(N)S XP /F8 51 17 1 0 15 19 19 16 0 <1FC0 7FF0 70F8 7078 7078 0078 0078 01F0 0FC0 00F0 0078 003C 003C E03C E03C E07C F0F8 7FF0 1FC0> PXLC RP 948 1802 XY F8(3)S -6 y 2 x F26(:)S XP /F26 82 32 2 -1 31 28 29 32 0 <03FFFE00 03FFFF80 003E03E0 003C01E0 003C01F0 003C01F0 007C01F0 007801F0 007801F0 007803E0 00F803C0 00F00780 00F01F00 00FFFC00 01FFF000 01E07800 01E03C00 01E03C00 03E03C00 03C03C00 03C03C00 03C07C00 07C07C00 07807C00 07807C18 0F807E38 FFF87E70 FFF83FE0 00000FC0> PXLC RP 978 1796 XY F26(R)S 6 y F8(3)S -6 y 1 x F25(\()S F26(x;)S XP /F27 63 32 2 0 30 28 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFF0 FFFFFFF0> PXLC RP 1087 1796 XY F27(?)S F26(;)S XP /F27 62 32 2 0 30 28 28 32 0 PXLC RP 1137 1796 XY F27(>)S F26(;)S 1188 X(N)S 1226 X F25(\))S F26(:)S 1892 Y 375 X F30(1.)S XP /F30 50 24 2 0 21 27 27 24 0 <0FF800 3FFF00 7FFF80 FE7FC0 FE1FC0 FE0FE0 FE0FE0 FE07E0 7C07E0 000FE0 000FE0 001FC0 001F80 003F00 007E00 00FC00 01F000 03E000 07C0E0 0F00E0 1E00E0 3C01E0 7FFFC0 FFFFC0 FFFFC0 FFFFC0 FFFFC0> PXLC RP 412 1892 XY F30(2.)S 465 X(De\014nition.)S 706 X F25(The)S 796 X (sub{tree)S 967 X(partial)S 1105 X(ordering)S 1274 X F26(<)S 1325 X F25(on)S 1387 X(\012)S 6 y F8(1)S -6 y 1454 X F25(is)S 1501 X(just)S 1587 X(the)S 50 y 300 X(transitiv)S -1 x(e)S 485 X(closure)S 624 X (of)S 2037 Y 584 X F26(\013)S 622 X(<)S 666 X(\013)S 702 X F25(+)S 743 X(1)S 847 X(and)S 997 X F27(8)S F26(n)S 1056 X F27(2)S 1095 X F26(N)S 1133 X F25(\()S F26(\013)S XP /F9 110 21 0 0 18 13 13 24 0 <3C7C00 6FCE00 6F8E00 EF0E00 CE0E00 0E0E00 1E1E00 1E1C00 1C1CC0 1C3CC0 3C39C0 3C3B80 381F00> PXLC RP 1176 2043 XY F9(n)S -6 y 1210 X F26(<)S 1254 X F27(h)S F26(\013)S 6 y F9(x)S -6 y 2 x F27(i)S F25(\))S F26(:)S 2133 Y 375 X F30(1.)S XP /F30 51 24 2 0 21 27 27 24 0 <07F800 1FFF00 3FFF80 7E3F80 7E1FC0 7F0FC0 7F0FC0 7E1FC0 3E1FC0 003F80 007F00 03FE00 03FC00 003F00 001F80 000FC0 000FE0 380FE0 7C0FE0 FE0FE0 FE0FE0 FE0FE0 FE1FC0 7C3FC0 7FFF80 3FFE00 07F800> PXLC RP 412 2133 XY F30(3.)S 465 X(De\014nition.)S 706 X F25(F)S -3 x(or)S 778 X(eac)S -1 x(h)S 870 X F26(\013)S 908 X F27(2)S 947 X F25(\012)S 6 y F8(1)S -6 y 1007 X F25(and)S 1086 X(eac)S -1 x(h)S 1177 X F26 (n)S 1214 X F27(2)S 1253 X F26(N)S 1303 X F25(de\014ne)S 1421 X (the)S 1490 X(\014nite)S 1594 X(set)S 49 y 300 X F26(\013)S F25([)S F26(n)S F25(])S 386 X(of)S 430 X(sub{trees)S 610 X(of)S 654 X F26 (\013)S 691 X F25(b)S -1 x(y:)S 764 X(0[)S F26(n)S F25(])S 844 X(=)S XP /F27 59 21 2 -3 18 32 35 16 0 <0018 0018 03F8 0FF8 1C38 383C 307C 707E 6066 6066 60E6 E0E7 E0C7 E0C7 E1C7 E1C7 E187 E187 E187 E387 E387 E307 E307 E707 E707 6606 760E 7E0E 3E0C 3C1C 1C38 1FF0 1FC0 1800 1800> PXLC RP 888 2182 XY F27(;)S F25(,)S 931 X(\()S F26(\013)S 3 x F25(+)S 3 x (1\)[)S F26(n)S F25(])S 1109 X(=)S 1153 X F26(\013)S F25([)S F26(n)S F25(])S XP /F27 91 28 2 -1 25 25 26 24 0 PXLC RP 1231 2182 XY F27([)S 3 x(f)S F26(\013)S F27(g)S F25(,)S 1353 X F27 (h)S F26(\013)S 7 y F9(x)S -7 y 2 x F27(i)S F25([)S F26(n)S F25(])S 1492 X(=)S 1536 X F26(\013)S 7 y F9(n)S -7 y 1 x F25([)S F26(n)S F25 (].)S 2280 Y 375 X(Then)S 483 X F26(\013)S F25([)S F26(n)S F25(])S 572 X(is)S 613 X(easily)S 728 X(seen)S 818 X(to)S 869 X(b)S 1 x(e)S 925 X(linearly)S 1074 X(ordered)S 1224 X(b)S -1 x(y)S 1282 X F26(<)S F25(.)S 2378 Y 375 X F30(1.)S XP /F30 52 24 1 0 22 27 27 24 0 <000780 000780 000F80 001F80 003F80 007F80 00EF80 01EF80 03CF80 038F80 070F80 0E0F80 1C0F80 380F80 700F80 F00F80 FFFFF8 FFFFF8 FFFFF8 000F80 000F80 000F80 000F80 000F80 01FFF8 01FFF8 01FFF8> PXLC RP 412 2378 XY F30(4.)S 465 X(De\014nition.)S 706 X F26(\013)S 748 X F27(2)S 790 X F25(\012)S 6 y F8(1)S -6 y 855 X F25(is)S 898 X(said)S 986 X(to)S 1039 X(b)S 1 x(e)S 1097 X F28(str)S XP /F28 117 22 3 0 23 18 18 24 0 <1F0380 3F8380 738380 638780 E38780 C78700 070700 070F00 0F0F00 0E0E00 0E0E00 0E1E70 1E1E70 1C1C60 1E3C60 0E7CC0 0FFFC0 07CF80> PXLC RP 1146 2378 XY F28(u)S XP /F28 99 19 4 0 19 18 18 16 0 <01F8 07FC 0F9E 1E1E 3C3E 7C1C 7800 7800 F800 F000 F000 F000 F000 F00C F01C 7878 3FF0 1FC0> PXLC RP 1168 2378 XY F28(ctur)S -2 x(e)S -2 x(d)S 1292 X F25(if)S 1332 X (it)S 1376 X(is)S 1420 X(presen)S -1 x(ted)S 1608 X(as)S 49 y 300 X (the)S 371 X(directed)S 531 X(union)S 646 X(of)S 693 X(its)S 751 X (\014nite)S 857 X(subsets)S 1001 X F26(\013)S F25([)S F26(n)S F25 (])S 1089 X(as)S 1140 X(follo)S -1 x(ws:)S 2515 Y 339 X(\(i\))S 404 X F26(\013)S F25([0])S 486 X F27(\022)S 530 X F26(\013)S F25 ([1])S 612 X F27(\022)S 656 X F26(\013)S F25([2])S 738 X F27(\022)S 781 X F26(:)S 800 X(:)S 819 X(:)S 2606 Y 328 X F25(\(ii\))S 404 X F26(\014)S 441 X(<)S 485 X(\013)S XP /F27 40 42 2 -2 39 22 24 40 0 <0006000000 0006000000 000E000000 001C000000 0038000000 0038000000 007FFFFFF8 01FFFFFFF8 03C0000000 0F00000000 3E00000000 F800000000 7C00000000 0F00000000 0780000000 01C0000000 00FFFFFFF8 007FFFFFF8 0038000000 0018000000 001C000000 000E000000 0006000000 0006000000> PXLC RP 535 2606 XY F27(\()S 570 X(\))S XP /F27 57 23 2 0 20 29 29 24 0 PXLC RP 634 2606 XY F27(9)S F26(n)S 693 X F27(2)S 732 X F26(N)S 770 X F25 (\()S F26(\014)S 824 X F27(2)S 863 X F26(\013)S F25([)S F26(n)S F25 (]\).)S 2703 Y 375 X(Th)S -1 x(us)S 481 X(if)S 520 X F26(\013)S 562 X F25(is)S 605 X(structured,)S 819 X F27(f)S F26(\014)S 2 x F25 (:)S 884 X F26(\014)S 923 X(<)S 969 X(\013)S F27(g)S 1032 X F25(is)S XP /F28 119 28 3 0 28 18 18 32 0 <1F01C380 3F81C780 7381C780 6383C780 E3838380 C7838380 07038180 07078380 0F070300 0E070300 0E070300 0E070700 0E0F0600 0E0F0E00 0E1F0C00 0F3F1C00 07FFF800 03F3F000> PXLC RP 1075 2703 XY F28(wel)S 2 x(l{or)S -2 x(der)S -2 x(e)S -2 x(d)S 1311 X F25(with)S 1407 X(least)S 1505 X(elemen)S -1 x(t)S 50 y 300 X(0)S 335 X(and)S 415 X(suc)S -1 x(h)S 509 X(that)S 599 X F26(\014)S 636 X (<)S 680 X(\013)S 730 X F25(=)S 755 X F27(\))S 820 X F26(\014)S 855 X F25(+)S 896 X(1)S 928 X F27(\024)S 972 X F26(\013)S F25(.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup XP /F18 79 30 2 0 27 26 26 32 0 <007F0000 01FFC000 07C1F000 0F007800 1E003C00 3C001E00 38000E00 78000F00 70000700 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 78000F00 78000F00 38000E00 3C001E00 1E003C00 0F007800 07C1F000 01FFC000 007F0000> PXLC RP 692 462 XY F18(Ordina)S XP /F18 108 11 0 0 9 26 26 16 0 PXLC RP 809 462 XY F18(l)S XP /F18 66 27 1 0 24 26 26 24 0 PXLC RP 833 462 XY F18(B)S XP /F18 111 19 1 0 17 16 16 16 0 <07E0 1FF8 3C3C 781E 700E F00F F00F F00F F00F F00F F00F 700E 781E 3C3C 1FF8 07E0> PXLC RP 860 462 XY F18(o)S XP /F18 117 21 1 0 19 16 16 24 0 PXLC RP 879 462 XY F18(und)S XP /F18 115 15 1 0 13 16 16 16 0 <3FE0 7FE0 F0E0 E060 E060 F000 FF80 7FC0 1FE0 01F0 C0F0 E070 E070 F0F0 FFE0 DF80> PXLC RP 943 462 XY F18(s)S XP /F18 102 12 0 0 14 26 26 16 0 <00F8 03FC 07BC 0F3C 0E3C 0E00 0E00 0E00 0E00 0E00 FFC0 FFC0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 971 462 XY F18(for)S XP /F18 80 26 1 0 23 26 26 24 0 PXLC RP 1030 462 XY F18(Progra)S XP /F18 109 32 1 0 30 16 16 32 0 PXLC RP 1144 462 XY F18(ms)S 1583 X F25(381)S 553 Y 375 X F30(1.)S XP /F30 53 24 2 0 21 27 27 24 0 <380380 3FFF80 3FFF80 3FFF00 3FFC00 3FF800 3F8000 380000 380000 380000 3BFC00 3FFF00 3E1F80 3C0FC0 3807C0 0007E0 0007E0 0007E0 7C07E0 FC07E0 FC07E0 FC07E0 FC0FC0 7C1F80 7FFF80 3FFE00 07F800> PXLC RP 412 553 XY F30(5.)S XP /F30 82 36 2 0 35 28 28 40 0 PXLC RP 465 553 XY F30(Re)S XP /F30 109 40 1 0 38 18 18 40 0 PXLC RP 523 553 XY F30(m)S XP /F30 97 23 1 0 22 18 18 24 0 <07FC00 1FFF00 3F1F80 3F0FC0 3F07C0 3F07C0 1E07C0 03FFC0 1FFFC0 7F87C0 7F07C0 FE07C0 FC07C0 FC07C0 FE0FC0 7E3FF8 3FFBF8 0FE1F8> PXLC RP 562 553 XY F30(a)S XP /F30 114 20 1 0 18 18 18 24 0 PXLC RP 586 553 XY F30(r)S XP /F30 107 25 1 0 23 29 29 24 0 PXLC RP 605 553 XY F30(k.)S 664 X F25(Although)S 845 X(w)S -1 x(e)S 904 X (ha)S -1 x(v)S -1 x(e)S 997 X(not)S 1069 X(placed)S 1195 X(an)S -1 x (y)S 1271 X(e)S XP /F25 11 24 0 0 26 29 29 32 0 <003F1F80 01FFFFC0 03C3F3C0 0783E3C0 0F03E3C0 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 FFFFFC00 FFFFFC00 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 7F87FC00 7F87FC00> PXLC RP 1290 553 XY F25(\013ectivit)S -1 x(y)S 1460 X(conditions)S 50 y 300 X(on)S 358 X(the)S 430 X(formation)S 621 X(of)S 669 X (sequences)S 857 X F27(h)S F26(\013)S 6 y F9(x)S -6 y 2 x F27(i)S 949 X(2)S 989 X F25(\012)S 6 y F8(1)S -6 y 2 x F25(,)S 1064 X(the)S 1136 X(particular)S 1328 X(examples)S 1507 X(of)S 1555 X(tree{)S 50 y 300 X(ordinals)S 459 X(that)S 551 X(w)S -1 x(e)S 615 X (construct)S 799 X(here)S 891 X(will)S 972 X(all)S 1032 X(b)S 1 x (e)S 1090 X(recursiv)S -1 x(e,)S 1278 X(and)S 1361 X(also)S 1446 X (structured.)S 50 y 300 X(Th)S -1 x(us)S 406 X(when)S 515 X(w)S -1 x (e)S 577 X(construct)S 761 X(and)S 842 X(\\name")S 996 X(a)S 1031 X (particular)S 1224 X F26(\013)S 1263 X F27(2)S 1304 X F25(\012)S 6 y F8(1)S -6 y 2 x F25(,)S 1379 X(it)S 1422 X(will)S 1501 X (actually)S 49 y 300 X(co)S 1 x(de)S 396 X(a)S 430 X(recursiv)S -1 x (e)S 604 X(w)S -1 x(ell{ordering.)S 845 Y 375 X(F)S -3 x(or)S 452 X (example)S 617 X(an)S -1 x(y)S 698 X(primitiv)S -1 x(e)S PF XP /F25 /cmr10 300 41.5 41.5 128 [-3 -11 41 31] PXLNF RP XP /F25 114 16 1 0 14 18 18 16 0 PXLC RP 878 845 XY F25(r)S XP /F25 101 18 1 0 16 18 18 16 0 <07E0 1FF8 3E7C 781C 781E F01E FFFE FFFE F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 894 845 XY F25(e)S XP /F25 99 18 1 0 16 18 18 16 0 <07F8 0FFC 3E3C 3C3C 7818 7800 F000 F000 F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 913 845 XY F25(c)S XP /F25 117 23 1 0 21 18 18 24 0 PXLC RP 931 845 XY F25(ur)S XP /F25 115 16 1 0 14 18 18 16 0 <1FB0 7FF0 F0F0 E070 E030 F030 F800 7FC0 7FE0 1FF0 00F8 C078 C038 E038 F078 F8F0 FFF0 CFC0> PXLC RP 970 845 XY F25(s)S XP /F25 105 12 1 0 10 29 29 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 987 845 XY F25(i)S XP /F25 118 22 1 0 20 18 18 24 0 PXLC RP 998 845 XY F25(v)S -1 x(e)S XP /F25 100 23 1 0 21 29 29 24 0 <001F80 001F80 000380 000380 000380 000380 000380 000380 000380 000380 000380 07F380 1FFF80 3E1F80 7C0780 780380 F80380 F00380 F00380 F00380 F00380 F00380 F00380 F00380 780780 7C0F80 3E1F80 1FFBF0 07E3F0> PXLC RP 1054 845 XY F25(de)S XP /F25 12 23 0 0 21 29 29 24 0 <003F00 01FF80 03C3C0 0783C0 0F03C0 0E03C0 0E0000 0E0000 0E0000 0E0000 0E0000 FFFFC0 FFFFC0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 7F87F8 7F87F8> PXLC RP 1095 845 XY F25(\014)S XP /F25 110 23 1 0 21 18 18 24 0 PXLC RP 1118 845 XY F25(ni)S XP /F25 116 16 1 0 13 26 26 16 0 <0C00 0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFE0 FFE0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C30 1C30 1C30 1C30 1C30 1E70 0FE0 07C0> PXLC RP 1153 845 XY F25(ti)S XP /F25 111 21 1 0 19 18 18 24 0 <03F000 0FFC00 1E1E00 380700 780780 700380 F003C0 F003C0 F003C0 F003C0 F003C0 F003C0 700380 780780 380700 1E1E00 0FFC00 03F000> PXLC RP 1181 845 XY F25(on)S 1241 X(o)S XP /F25 102 13 0 0 15 29 29 16 0 <00FC 03FE 079E 071E 0F1E 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 1262 845 XY F25(f)S XP /F25 97 21 1 0 20 18 18 24 0 <1FE000 3FF800 3C3C00 3C1E00 180E00 000E00 000E00 03FE00 1F8E00 7E0E00 780E00 F80E00 F00E60 F00E60 F81E60 7C7E60 7FEFC0 1F8780> PXLC RP 1291 845 XY F25(a)S 1328 X(n)S -1 x(u)S XP /F25 109 35 1 0 34 18 18 40 0 PXLC RP 1373 845 XY F25(m)S XP /F25 98 23 1 0 21 29 29 24 0 PXLC RP 1406 845 XY F25(b)S 1 x(er)S XP /F25 123 21 0 10 20 12 2 24 0 PXLC RP 1465 845 XY F25({t)S XP /F25 104 23 1 0 21 29 29 24 0 PXLC RP 1502 845 XY F25(heoretic)S 49 y 300 X(function)S XP /F26 /cmmi10 300 41.5 41.5 128 [-1 -10 43 31] PXLNF RP XP /F26 102 20 2 -8 23 29 37 24 0 <0003F0 0007F0 000EF8 000EF8 001C70 001C00 001C00 001C00 003C00 003800 003800 03FFC0 03FFC0 007800 007000 007000 007000 007000 007000 00F000 00E000 00E000 00E000 00E000 01E000 01C000 01C000 01C000 01C000 03C000 038000 038000 F38000 F70000 FF0000 FE0000 7C0000> PXLC RP 463 894 XY F26(f)S XP /F25 44 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 487 894 XY F25(,)S 513 X(sa)S XP /F25 121 22 1 -8 20 18 26 24 0 PXLC RP 549 894 XY F25(y)S 48 y 788 X F26(f)S XP /F25 40 16 3 -11 13 31 42 16 0 <01C0 0380 0380 0700 0E00 0C00 1C00 1800 3800 3800 7000 7000 7000 7000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 7000 7000 7000 7000 3800 3800 1800 1C00 0C00 0E00 0700 0380 0380 01C0> PXLC RP 813 942 XY F25(\()S XP /F26 97 22 1 0 21 18 18 24 0 <01FB80 07FF80 0F1F80 1E0F80 3C0780 7C0700 780700 780F00 F80F00 F00E00 F00E00 F01E70 F01E70 F03C60 F07C60 78FEE0 3FDFC0 1F8780> PXLC RP 829 942 XY F26(a)S XP /F26 59 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 851 942 XY F26(;)S XP /F25 48 21 2 -1 18 28 29 16 0 <03C0 0FF0 1C38 381C 381C 700E 700E 700E F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F 700E 700E 700E 381C 381C 1C38 0FF0 07E0> PXLC RP 869 942 XY F25(0)S XP /F25 41 16 2 -11 12 31 42 16 0 PXLC RP 890 942 XY F25(\))S XP /F25 61 32 2 4 29 16 12 32 0 PXLC RP 918 942 XY F25(=)S XP /F26 103 20 1 -8 18 18 26 24 0 <00FB80 03FF80 079F80 0F0F80 1E0780 3E0700 3C0700 3C0F00 7C0F00 780E00 780E00 781E00 781E00 783C00 787C00 3CFC00 3FFC00 0FB800 003800 007800 007800 F07000 F0F000 F3E000 FFC000 7F0000> PXLC RP 963 942 XY F26(g)S XP /F8 /cmr7 300 29.1 29.1 128 [-2 -8 32 22] PXLNF RP XP /F8 48 17 1 0 15 19 19 16 0 <0FC0 1860 3030 6018 6018 E01C E01C E01C E01C E01C E01C E01C E01C E01C 6018 6018 3030 1860 0FC0> PXLC RP 983 949 XY F8(0)S -7 y 2 x F25(\()S F26(a)S F25(\))S F26(;)S 50 y 719 X(f)S 743 X F25(\()S F26(a;)S XP /F26 98 18 2 0 17 29 29 16 0 <3F00 7F00 0F00 0F00 0E00 0E00 0E00 1E00 1C00 1C00 1C00 3DF0 3FF8 3F3C 3C1C 781C 701E 701E 703E F03C E03C E03C E07C E078 E0F0 E0F0 73E0 7FC0 3F00> PXLC RP 800 992 XY F26(b)S XP /F25 43 32 2 -4 29 24 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFE0 FFFFFFE0 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000> PXLC RP 828 992 XY F25(+)S XP /F25 49 21 4 0 17 28 28 16 0 <0300 0700 3F00 FF00 C700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 FFF8 FFF8> PXLC RP 869 992 XY F25(1\))S 918 X(=)S 963 X F26(g)S XP /F8 49 17 3 0 14 19 19 16 0 <0600 1E00 FE00 EE00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0> PXLC RP 983 998 XY F8(1)S -6 y 2 x F25(\()S F26(a;)S 1059 X(b;)S 1096 X(f)S 1120 X F25(\()S F26(a;)S 1177 X(b)S F25(\)\))S 74 y 300 X(can)S 376 X(b)S 1 x(e)S XP /F25 108 12 1 0 10 29 29 16 0 PXLC RP 433 1066 XY F25(lifted)S 540 X(in)S 588 X(a)S 623 X(formally)S XP /F25 92 21 5 16 19 29 13 16 0 <1818 3C3C 3838 7070 6060 E0E0 C0C0 C0C0 F8F8 FCFC FCFC 7C7C 3838> PXLC RP 787 1066 XY F25(\\con)S -1 x(tin)S -1 x(uous)S XP /F25 34 21 1 16 15 29 13 16 0 <7070 F8F8 FCFC FCFC 7C7C 0C0C 0C0C 1C1C 1818 3838 7070 F0F0 6060> PXLC RP 1002 1066 XY F25(")S XP /F25 119 30 1 0 28 18 18 32 0 PXLC RP 1036 1066 XY F25(w)S -1 x(a)S -1 x(y)S -3 x(,)S 1129 X(to)S 1179 X (a)S XP /F25 112 23 1 -8 21 18 26 24 0 PXLC RP 1214 1066 XY F25(primitiv)S -1 x(e)S 1392 X(recursion)S 1570 X(o)S -1 x(v)S -1 x(er)S XP /F25 10 30 2 0 27 29 29 32 0 <00FF8000 03FFE000 0FC1F800 1F007C00 3E003E00 7C001F00 7C001F00 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 7C001F00 7C001F00 3C001E00 3E003E00 1E003C00 1E003C00 0F007800 CF007980 C7007180 C3006180 E380E380 7F80FF00 7F80FF00 7F80FF00 7F80FF00> PXLC RP 300 1116 XY F25(\012)S 6 y F8(1)S -6 y 362 X F25(th)S -1 x(us)S XP /F26 39 27 2 -9 25 18 27 24 0 <0C01F0 1C07FC 380FFC 300F1E 701C0E 60380E 603006 60700E E0600C C0600C E0E01C E0C038 70C078 79C0F0 3FC7E0 3FFFC0 0FFF00 03FC00 038000 038000 070000 070000 070000 070000 0E0000 0E0000 0E0000> PXLC RP 765 1162 XY F26(')S F25(\()S XP /F26 11 27 1 0 24 18 18 24 0 <00FC00 03FE00 0F8F06 1E078E 3C038C 3C038C 7803DC 7803D8 F801F8 F001F0 F001E0 F001E0 F001C0 F003C0 700FC0 7C3FEE 3FF8FC 0FE078> PXLC RP 808 1162 XY F26(\013;)S 854 X F25(0\))S 903 X(=)S XP /F26 32 27 1 -8 25 29 37 24 0 <000060 000060 000060 0000E0 0000C0 0000C0 0000C0 0001C0 000180 000180 000180 1F038F 3F830F 73830F 63830F E38707 C78607 070603 070607 0F0E06 0E0C06 0E0C0E 0E0C0C 0E1C1C 0E1838 0F18F0 07D9E0 03FFC0 00FF00 003000 003000 007000 006000 006000 006000 00E000 00C000> PXLC RP 947 1162 XY F26( )S 6 y F8(0)S -6 y 2 x F25(\()S F26(\013)S F25(\))S F26(;)S 50 y 688 X(')S F25(\()S F26(\013;)S XP /F26 12 23 -1 -8 24 29 37 32 0 <0000FC00 0007FE00 000F8F00 001C0780 00380380 00700380 00E00780 00C00780 01C00700 01800F00 03801E00 038FFC00 030FF800 030FFC00 07001E00 06001E00 06000E00 06000E00 0E001E00 0C001E00 0C001E00 0C001E00 1C003E00 1C003C00 1E007800 1F00F800 3F81F000 33FFC000 31FF8000 30180000 70000000 60000000 60000000 60000000 E0000000 C0000000 C0000000> PXLC RP 777 1212 XY F26(\014)S 812 X F25(+)S 853 X(1\))S 903 X(=)S 947 X F26 ( )S 6 y F8(1)S -6 y 2 x F25(\()S F26(\013;)S 1055 X(\014)S 2 x(;)S 1099 X(')S F25(\()S F26(\013;)S 1188 X(\014)S 2 x F25(\)\))S F26(;)S 50 y 709 X(')S F25(\()S F26(\013;)S XP /F27 /cmsy10 300 41.5 41.5 128 [-2 -40 47 33] PXLNF RP XP /F27 104 16 4 -11 13 31 42 16 0 <0180 0380 0380 0300 0700 0700 0600 0E00 0E00 0C00 1C00 1C00 1800 3800 3800 3000 7000 7000 6000 E000 E000 E000 E000 6000 7000 7000 3000 3800 3800 1800 1C00 1C00 0C00 0E00 0E00 0600 0700 0700 0300 0380 0380 0180> PXLC RP 797 1262 XY F27(h)S F26(\014)S XP /F9 /cmmi7 300 29.1 29.1 128 [-1 -7 32 22] PXLNF RP XP /F9 120 19 1 0 16 13 13 16 0 <1F7C 33EE 63CE E3CC C380 0380 0780 0700 070C E70C EF1C CF38 FBE0> PXLC RP 837 1268 XY F9(x)S XP /F27 105 16 2 -11 11 31 42 16 0 PXLC RP 858 1262 XY F27(i)S F25(\))S 903 X(=)S 947 X F27(h)S F26(')S F25 (\()S F26(\013;)S 1052 X(\014)S 6 y F9(x)S -6 y 2 x F25(\))S F27(i)S XP /F26 58 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 1129 1262 XY F26(:)S XP /F25 84 30 1 0 28 28 28 32 0 <7FFFFFC0 7FFFFFC0 780F03C0 700F01C0 600F00C0 E00F00E0 C00F0060 C00F0060 C00F0060 C00F0060 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 03FFFC00 03FFFC00> PXLC RP 300 1335 XY F25(This)S 395 X(automatically)S XP /F25 103 21 1 -9 19 19 28 24 0 <000380 07E7C0 1FFDC0 3C3DC0 781E00 781E00 781E00 781E00 781E00 781E00 3C3C00 3FF800 37E000 200000 300000 380000 3FFC00 3FFF00 3FFF80 7807C0 F003C0 E001C0 E001C0 F003C0 F807C0 7C0F80 1FFE00 07F800> PXLC RP 658 1335 XY F25(giv)S -1 x(es)S 760 X(rise)S 837 X(to)S 888 X(the)S 960 X(ob)S -1 x(vious)S 1110 X(de\014nitions)S 1312 X(of)S 1360 X (addition,)S 1535 X(m)S -1 x(ulti)S XP /F25 45 14 0 7 11 10 3 16 0 PXLC RP 1631 1335 XY F25(-)S 50 y 300 X(plication)S 469 X(and)S 548 X(e)S XP /F25 120 22 0 0 21 18 18 24 0 <7F8FF0 7F8FF0 0F0F80 070F00 038E00 01DC00 01D800 00F000 007000 007800 00F800 01DC00 038E00 030E00 070700 1F0780 FF8FF8 FF8FF8> PXLC RP 566 1385 XY F25(xp)S 1 x(onen)S -1 x(tiation)S 829 X(o)S -1 x(v)S -1 x(er)S 916 X(\012)S 6 y F8(1)S XP /F25 46 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 965 1385 XY F25(.)S XP /F25 83 23 2 -1 20 29 30 24 0 <07F180 1FFF80 3C3F80 780F80 700780 E00380 E00380 E00180 E00180 F00000 F80000 7E0000 7FE000 3FFC00 1FFE00 0FFF00 00FF80 000F80 0007C0 0003C0 0003C0 C001C0 C001C0 C001C0 E001C0 F00380 F80780 FE0F00 FFFE00 C3F800> PXLC RP 994 1385 XY F25(So)S 1050 X(starting)S 1203 X(with)S XP /F26 120 24 2 0 21 18 18 24 0 <0F8F80 3FDFC0 38F9C0 70F1C0 E0F3C0 C0E3C0 00E3C0 01E000 01E000 01C000 01C000 73C0E0 F3C0E0 F380C0 F781C0 E7C780 FFFF00 7C7C00> PXLC RP 1296 1385 XY F26(x)S 1331 X F25(=)S 1375 X(0)S 1402 X(+)S 1440 X(1)S 1467 X(+)S 1505 X(1)S 1533 X F26(:)S 1552 X(:)S 1571 X(:)S 3 x F25 (+)S 1624 X(1)S 50 y 300 X(\()S F26(x)S 354 X F25(times\))S 480 X (w)S -1 x(e)S 541 X(can)S 617 X(de\014ne)S XP /F26 33 26 0 0 23 18 18 24 0 <0C001E 1C001E 18001E 38001E 30000E 701C0E 601C06 601C0E E03C0C C0380C C0381C C03018 E07038 E0F870 FFFFF0 FFFFE0 7FBFC0 3E1F00> PXLC RP 737 1435 XY F26(!)S XP /F27 50 28 3 -3 24 23 26 24 0 <00FFF8 03FFF8 0FC000 1F0000 3C0000 380000 700000 700000 E00000 E00000 C00000 C00000 FFFFF8 FFFFF8 C00000 C00000 E00000 E00000 700000 700000 380000 3C0000 1F0000 0FC000 03FFF8 00FFF8> PXLC RP 776 1435 XY F27(2)S 815 X F25(\012)S 6 y F8(1)S -6 y 877 X F25(b)S -1 x(y)S 935 X(the)S 1006 X(iden)S -1 x(tit)S -1 x(y)S 1159 X(se)S XP /F25 113 22 1 -8 21 18 26 24 0 <07F180 1FF980 3F1F80 3C0F80 780780 780380 F00380 F00380 F00380 F00380 F00380 F00380 F80380 780780 7C0F80 3E1F80 1FFB80 07E380 000380 000380 000380 000380 000380 000380 001FF0 001FF0> PXLC RP 1194 1435 XY F25(quence)S 1331 X F26(!)S 1370 X F25(=)S 1414 X F27 (h)S F26(x)S F27(i)S 1484 X F25(and)S 1564 X(then)S 50 y 300 X (generate)S 466 X(all)S XP /F25 67 30 2 -1 27 29 30 32 0 <003FC180 00FFF180 03F07B80 07C01F80 0F800F80 1F000780 3E000780 3E000380 7C000380 7C000180 78000180 F8000180 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 78000180 7C000180 7C000180 3E000380 3E000300 1F000700 0F800E00 07C01E00 03F87C00 00FFF000 003FC000> PXLC RP 523 1485 XY F25(Can)S -1 x(tor)S 662 X(forms)S 777 X(in)S 825 X (\012)S 6 y F8(1)S -6 y 887 X F25(b)S -1 x(y)S 944 X F26(\013)S 982 X F25(=)S 1026 X F26(!)S XP /F9 11 22 1 0 19 13 13 24 0 <03F000 0E38C0 1C1DC0 381D80 701F80 701F80 701F00 F00E00 E00E00 601E00 703E00 38EEC0 1F87C0> PXLC RP 1054 1470 XY F9(\013)S XP /F0 /cmr5 300 20.8 20.8 128 [-1 -6 26 16] PXLNF RP XP /F0 49 14 3 0 11 13 13 8 0 <18 F8 F8 18 18 18 18 18 18 18 18 FF FF> PXLC RP 1075 1474 XY F0(1)S XP /F27 1 12 3 8 8 13 5 8 0 <70 F8 F8 F8 70> PXLC RP 1102 1485 XY F27(\001)S XP /F26 110 25 1 0 24 18 18 24 0 <3E1F80 3F3FE0 7771E0 67E0E0 E7C0E0 C780E0 0700E0 0F01E0 0F01C0 0E01C0 0E03C0 1E038E 1E038E 1C078C 1C071C 3C07B8 3C03F0 3801E0> PXLC RP 1122 1485 XY F26(n)S 6 y F8(1)S -6 y 1174 X F25(+)S 1215 X F26(!)S -15 y 1 x F9(\013)S XP /F0 50 14 2 0 12 13 13 16 0 <7F00 FF80 F3C0 F1C0 01C0 01C0 0380 0700 1E00 3840 6040 FF80 FF80> PXLC RP 1264 1474 XY F0(2)S 11 y 1291 X F27(\001)S 1311 X F26(n)S XP /F8 50 17 2 0 14 19 19 16 0 <1F80 7FE0 E3E0 E0F0 E0F0 E070 00F0 00F0 00E0 01C0 0380 0700 0E00 1C30 3830 7070 FFE0 FFE0 FFE0> PXLC RP 1336 1491 XY F8(2)S -6 y 1363 X F25(+)S 1404 X F26(:)S 1423 X(:)S 1442 X(:)S 1461 X F25(+)S 1502 X F26(!)S -15 y 1 x F9(\013)S XP /F1 /cmmi5 300 20.8 20.8 128 [0 -5 28 16] PXLNF RP XP /F1 107 16 2 0 13 14 14 16 0 <7800 7800 1800 3800 3800 31E0 33E0 7EE0 78C0 7E00 6720 E360 E340 C3C0> PXLC RP 1551 1474 XY F1(k)S 11 y 1580 X F27(\001)S 1600 X F26(n)S XP /F9 107 18 1 0 15 20 20 16 0 <3F00 3F00 0F00 0E00 0E00 1E00 1E00 1C78 1CDC 3D9C 3F18 3E00 3C00 7F80 7BC0 71CC 71CC F1DC F1D8 E0F0> PXLC RP 1625 1491 XY F9(k)S 43 y 300 X F25(etcetera.)S XP /F30 /cmbx10 300 41.5 41.5 128 [-4 -10 47 31] PXLNF RP XP /F30 49 24 3 0 20 27 27 24 0 <006000 01E000 0FE000 FFE000 FFE000 F3E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 7FFF80 7FFF80 7FFF80> PXLC RP 375 1627 XY F30(1)S XP /F30 46 13 3 0 10 7 7 8 0 <7C FE FE FE FE FE 7C> PXLC RP 399 1627 XY F30(.)S XP /F30 54 24 2 0 21 27 27 24 0 <00FE00 03FF80 0FFF80 1FC7C0 3F0FC0 3E0FC0 7E0FC0 7C0FC0 7C0780 FC2000 FDFE00 FFFF80 FF0FC0 FE07C0 FE07E0 FC07E0 FC07E0 FC07E0 FC07E0 FC07E0 7C07E0 7C07E0 3E07C0 3F0FC0 1FFF80 0FFF00 03FC00> PXLC RP 412 1627 XY F30(6.)S XP /F30 68 37 2 0 33 28 28 32 0 PXLC RP 465 1627 XY F30(D)S XP /F30 101 22 1 0 20 18 18 24 0 <03FC00 0FFF00 3F0F80 3E07C0 7E03C0 7C03E0 FC03E0 FFFFE0 FFFFE0 FC0000 FC0000 FC0000 7C0000 7E00E0 3F01E0 1FC3C0 0FFF80 03FE00> PXLC RP 502 1627 XY F30(e)S XP /F30 12 27 0 0 25 29 29 32 0 <001FE000 00FFF800 03F87800 07E0FC00 0FC0FC00 0F80FC00 0F80FC00 0F807800 0F800000 0F800000 0F800000 FFFFFC00 FFFFFC00 FFFFFC00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 0F807C00 7FE1FF80 7FE1FF80 7FE1FF80> PXLC RP 523 1627 XY F30(\014)S XP /F30 110 27 1 0 25 18 18 24 0 PXLC RP 550 1627 XY F30(n)S XP /F30 105 13 1 0 12 30 30 16 0 <1E00 3F00 7F00 7F00 7F00 3F00 1E00 0000 0000 0000 0000 0000 FF00 FF00 FF00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 FFE0 FFE0 FFE0> PXLC RP 576 1627 XY F30(i)S XP /F30 116 19 1 0 15 26 26 16 0 <0700 0700 0700 0700 0F00 0F00 0F00 1F00 3FFC FFFC FFFC 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F1C 1F1C 1F1C 1F1C 1FBC 0FF8 07F0> PXLC RP 590 1627 XY F30(ti)S XP /F30 111 24 1 0 22 18 18 24 0 <01FC00 0FFF80 1F07C0 3E03E0 7C01F0 7C01F0 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 7C01F0 7C01F0 3E03E0 1F07C0 0FFF80 01FC00> PXLC RP 622 1627 XY F30(on)S 688 X F25(\()S XP /F25 79 32 2 -1 29 29 30 32 0 <003F8000 00FFE000 03E0F800 07803C00 0F001E00 1E000F00 3E000F80 3C000780 7C0007C0 7C0007C0 780003C0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 780003C0 7C0007C0 7C0007C0 3C000780 3E000F80 1E000F00 0F001E00 07803C00 03E0F800 00FFE000 003F8000> PXLC RP 704 1627 XY F25(Ordinal)S XP /F25 65 31 1 0 29 29 29 32 0 <00060000 000F0000 000F0000 000F0000 001F8000 001F8000 001F8000 001F8000 0033C000 0033C000 0033C000 0061E000 0061E000 0061E000 00C0F000 00C0F000 00C0F000 01807800 01807800 01FFF800 03FFFC00 03003C00 03003C00 06001E00 06001E00 0E001E00 1F001F00 FFC0FFF0 FFC0FFF0> PXLC RP 856 1627 XY F25(Assignmen)S -1 x(t\).)S XP /F25 87 43 1 -1 41 28 29 40 0 PXLC RP 1115 1627 XY F25(W)S -3 x(rite)S 1233 X F26(n)S XP /F25 58 12 3 0 8 18 18 8 0 <70 F8 F8 F8 70 00 00 00 00 00 00 00 00 70 F8 F8 F8 70> PXLC RP 1258 1627 XY F25(:)S XP /F26 78 33 2 0 35 28 28 40 0 <03FE03FF80 03FE03FF80 003F007C00 003F003000 003F803000 003F807000 0077807000 0067C06000 0067C06000 0063C0E000 00E3E0E000 00C3E0C000 00C1E0C000 00C1F1C000 01C1F1C000 0180F18000 0180F98000 01807B8000 03807B8000 03007F0000 03003F0000 03003F0000 07003F0000 06001E0000 06001E0000 1F001E0000 FFE00E0000 FFE00C0000> PXLC RP 1277 1627 XY F26(N)S XP /F27 106 12 5 -10 7 31 41 8 0 PXLC RP 1331 1627 XY F27(j)S 1336 X F25(=)S -15 y F9(\013)S XP /F26 80 27 2 0 30 28 28 32 0 <03FFFF00 03FFFFC0 003E03E0 003C01F0 003C01F0 007C01F0 007C01F0 007801F0 007801F0 00F801E0 00F803E0 00F003C0 00F00780 01F01F00 01FFFE00 01FFF000 01E00000 03E00000 03E00000 03C00000 03C00000 07C00000 07C00000 07800000 07800000 0F800000 FFF80000 FFF80000> PXLC RP 1408 1627 XY F26(P)S XP /F27 35 21 3 -8 17 29 37 16 0 <0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 0300 E31C F33C 3B70 1FE0 0FC0 0780 0300 0300> PXLC RP 1457 1627 XY F27(#)S 1494 X F25(to)S 1548 X(mean)S 49 y 300 X(that)S 395 X(for)S 464 X(ev)S -1 x(ery)S 579 X(input)S 695 X F26(a)S 7 y F8(0)S -7 y 754 X F25(of)S 807 X(si)S XP /F25 122 18 1 0 15 18 18 16 0 <7FFC 7FFC 783C 7078 60F0 61E0 61E0 63C0 0780 078C 0F0C 1E0C 1E1C 3C18 7818 F078 FFF8 FFF8> PXLC RP 835 1676 XY F25(ze)S XP /F27 20 32 3 -8 28 26 34 32 0 <00000380 00000F80 00003F00 0000FC00 0003F000 000FC000 003F0000 00FC0000 03F00000 0FC00000 3F000000 FC000000 F0000000 FC000000 3F000000 0FC00000 03F00000 00FC0000 003F0000 000FC000 0003F000 0000FC00 00003F00 00000F80 00000380 00000000 00000000 00000000 00000000 00000000 00000000 00000000 FFFFFF80 FFFFFF80> PXLC RP 891 1676 XY F27(\024)S 943 X F26(n)S 987 X F25(there)S 1099 X(is)S 1146 X(a)S 1185 X(terminating)S 1417 X(computation)S 50 y 300 X (sequence)S 51 y 576 X(\()S XP /F26 115 19 2 0 17 18 18 16 0 <01F8 07FC 0F1E 0E1E 1E3E 1E1C 1F00 1FE0 0FF8 07F8 007C 203C F038 F038 F078 F0F0 7FE0 3F80> PXLC RP 592 1777 XY F26(s)S 6 y F8(0)S -6 y 2 x F26(;)S 649 X(a)S 6 y F8(0)S -6 y 1 x F25(\))S XP /F27 0 32 3 9 28 11 2 32 0 PXLC RP 717 1777 XY F27(\000)S XP /F27 33 42 2 3 39 17 14 40 0 <0000000600 0000000700 0000000700 0000000380 00000001C0 00000000E0 FFFFFFFFF8 FFFFFFFFF8 00000000E0 00000001C0 0000000380 0000000700 0000000700 0000000600> PXLC RP 742 1777 XY F27(!)S XP /F9 80 21 1 0 23 20 20 24 0 <07FFE0 07FFF0 01E078 01C038 01C03C 03C03C 03C078 038078 038070 0781E0 07FF80 070000 070000 0F0000 0F0000 0E0000 0E0000 1E0000 FF8000 FF8000> PXLC RP 784 1783 XY F9(P)S -6 y 823 X F25(\()S F26(s)S 6 y F8(1)S -6 y 2 x F26(;)S 896 X(a)S 6 y F8(1)S -6 y 1 x F25(\))S 964 X F27(\000)S 989 X(!)S 6 y F9(P)S -6 y 1070 X F26(:)S 1089 X(:)S 1108 X(:)S 1130 X F27(\000)S 1155 X(!)S 6 y F9(P)S -6 y 1236 X F25(\()S F26(s)S 6 y F9(k)S -6 y 3 x F26(;)S 1311 X(a)S 6 y F9(k)S -6 y 2 x F25(\))S 1853 Y 300 X(and)S 382 X(an)S 441 X(assignmen)S -1 x(t)S 656 X(of)S 704 X(a)S 740 X(tree{ordinal)S 972 X F26(\014)S XP /F9 105 12 0 0 9 20 20 16 0 <0380 0780 0300 0000 0000 0000 0000 3E00 7600 6F00 EE00 CE00 1E00 1C00 1C00 3D80 3980 3B80 3B00 1E00> PXLC RP 996 1859 XY F9(i)S -6 y 1023 X F27(2)S 1065 X F26(\013)S XP /F25 91 12 4 -10 11 31 41 8 0 PXLC RP 1091 1853 XY F25([)S F26(n)S XP /F25 93 12 0 -10 7 31 41 8 0 PXLC RP 1128 1853 XY F25(])S 1154 X(to)S 1206 X(eac)S -1 x(h)S 1301 X(step)S 1390 X(\()S F26(s)S 6 y F9(i)S -6 y 2 x F26(;)S 1458 X(a)S 6 y F9 (i)S -6 y 2 x F25(\),)S XP /F26 105 14 1 0 13 28 28 16 0 <00E0 01E0 01E0 01C0 0000 0000 0000 0000 0000 0000 1F00 3F80 7380 6380 E380 C780 0700 0700 0F00 0E00 1E00 1C70 1C70 3C60 38E0 3DC0 1F80 0F00> PXLC RP 1537 1853 XY F26(i)S XP /F26 60 32 3 -3 28 23 26 32 0 <00000180 00000780 00001F80 00007E00 0001F800 0007E000 001F8000 007E0000 01F80000 07E00000 1F800000 7E000000 F8000000 F8000000 7E000000 1F800000 07E00000 01F80000 007E0000 001F8000 0007E000 0001F800 00007E00 00001F80 00000780 00000180> PXLC RP 1565 1853 XY F26(<)S XP /F26 107 22 2 0 20 29 29 24 0 <0FC000 1FC000 03C000 03C000 038000 038000 078000 078000 070000 070000 0F0000 0F0F80 0E1F80 0E3980 1E7380 1EE780 1DC780 1F8780 3F0000 3FC000 3BE000 38E000 78E1C0 78E1C0 70E180 70E380 F0F700 F07F00 E03E00> PXLC RP 1611 1853 XY F26(k)S 1 x F25(,)S 49 y 300 X(suc)S -1 x(h)S 394 X (that)S 484 X F26(\014)S 7 y F8(0)S -7 y 537 X F26(<)S 581 X(\014)S 7 y F8(1)S -7 y 635 X F26(<)S 678 X(:)S 697 X(:)S 716 X(:)S 738 X (<)S 782 X(\013)S F25(.)S XP /F25 70 27 2 0 24 28 28 24 0 PXLC RP 375 1995 XY F25(Finally)S 516 X(write)S 622 X F27(j)S 627 X F25(=)S -15 y F9(\013)S 15 y 694 X F26(P)S 738 X F27(#)S 773 X F25(to)S 824 X(mean)S XP /F27 56 23 0 -1 22 29 30 24 0 PXLC RP 934 1995 XY F27(8)S F26(n)S F25(\()S F26(n)S F25(:)S 1042 X F26(N)S 1091 X F27(j)S 1096 X F25(=)S -15 y F9(\013)S 15 y 1163 X F26(P)S 1207 X F27(#)S F25(\).)S XP /F28 /cmti10 300 41.5 41.5 128 [-1 -11 45 31] PXLNF RP XP /F28 78 31 2 0 35 28 28 40 0 <01FE01FF80 01FE01FF80 003F007C00 003F003000 003F803000 003F807000 0077807000 0067C06000 0067C06000 0063C0E000 00E3E0E000 00C3E0C000 00C1E0C000 00C1F1C000 01C1F1C000 0180F18000 0180F98000 01807B8000 03807B8000 03007F0000 03003F0000 03003F0000 07003F0000 06001E0000 06001E0000 1F001E0000 FFC00E0000 FFC00C0000> PXLC RP 375 2044 XY F28(N)S XP /F28 111 21 4 0 21 18 18 24 0 <01F800 07FE00 0F9F00 1E0F00 3C0700 7C0780 780780 780F80 F80F00 F00F00 F00F00 F01F00 F01E00 F03C00 F07C00 78F800 3FE000 1F8000> PXLC RP 406 2044 XY F28(o)S XP /F28 116 14 3 0 15 26 26 16 0 <01C0 01C0 01C0 03C0 03C0 0380 0380 0780 FFE0 FFE0 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C70 3C70 3860 38E0 39C0 3F80 1F00> PXLC RP 427 2044 XY F28(t)S XP /F28 101 19 4 0 18 18 18 16 0 <01F8 07FC 0F1C 1E0C 3C0C 780C 783C 7FF8 FFC0 F000 F000 F000 F000 F00C 701C 7878 3FF0 1FC0> PXLC RP 441 2044 XY F28(e)S 3 x F25(.)S XP /F25 73 15 1 0 13 28 28 16 0 PXLC RP 496 2044 XY F25(In)S 548 X(order)S 658 X(to)S 709 X(ha)S -1 x(v)S -1 x(e)S 806 X F26(n)S F25(:)S 850 X F26(N)S 900 X F27(j)S 905 X F25 (=)S -15 y F9(\013)S 15 y 974 X F26(P)S 1019 X F27(#)S 1055 X F25 (it)S 1097 X(is)S 1140 X(only)S 1232 X(necessary)S 1417 X(that)S 1508 X(the)S 1580 X(size)S 50 y 300 X(of)S 348 X F26(\013)S F25([)S F26(n)S F25(])S 437 X(b)S 1 x(e)S 494 X(at)S 546 X(least)S 643 X (as)S 695 X(big)S 765 X(as)S 816 X(the)S 888 X(maxim)S -1 x(um)S 1083 X(\\computation)S 1346 X(length")S 1495 X(of)S 1543 X F26(P)S 1576 X F25(,)S 1601 X(on)S 50 y 300 X(inputs)S 427 X(of)S 473 X (size)S 552 X F27(\024)S 595 X F26(n)S F25(.)S XP /F25 72 31 2 0 28 28 28 32 0 PXLC RP 650 2144 XY F25(Ho)S -1 x(w)S -1 x(ev)S -1 x(er)S 817 X(in)S 865 X (order)S 973 X(to)S 1023 X(ha)S -1 x(v)S -1 x(e)S 1119 X F27(j)S 1124 X F25(=)S -15 y F9(\013)S 15 y 1191 X F26(P)S 1235 X F27(#)S 1269 X F25(w)S -1 x(e)S 1330 X(need)S 1426 X(to)S 1476 X(construct)S 50 y 300 X F26(\013)S 339 X F25(suc)S -1 x(h)S 431 X(that)S 520 X F26(n)S F25(:)S 564 X F26(N)S 612 X F27(j)S 617 X F25(=)S -15 y F9 (\013)S 15 y 685 X F26(P)S 728 X F27(#)S XP /F28 117 22 3 0 23 18 18 24 0 <1F0380 3F8380 738380 638780 E38780 C78700 070700 070F00 0F0F00 0E0E00 0E0E00 0E1E70 1E1E70 1C1C60 1E3C60 0E7CC0 0FFFC0 07CF80> PXLC RP 762 2194 XY F28(u)S XP /F28 110 23 3 0 24 18 18 24 0 <1E3F00 3F7F80 77F380 67C380 E78380 C78380 070380 0F0780 0F0700 0E0700 0E0F00 1E0E38 1E0E38 1C1E30 1C1C70 3C1EE0 3C0FC0 380F80> PXLC RP 784 2194 XY F28(n)S XP /F28 105 13 3 0 14 28 28 16 0 <01C0 01E0 01E0 01C0 0000 0000 0000 0000 0000 0000 1F00 3F00 7780 6380 E780 C700 0F00 0E00 0E00 1E00 1C00 1CE0 3CE0 38C0 39C0 3B80 3F80 1E00> PXLC RP 807 2194 XY F28(i)S XP /F28 102 13 -1 -8 19 29 37 24 0 <0003E0 000FE0 000FF0 001DF0 001CE0 001C00 003C00 003800 003800 003800 003800 03FF80 07FF80 007000 007000 007000 00F000 00E000 00E000 00E000 00E000 01E000 01C000 01C000 01C000 01C000 03C000 038000 038000 038000 038000 070000 F70000 F70000 FE0000 FC0000 780000> PXLC RP 820 2194 XY F28(fo)S XP /F28 114 18 3 0 19 18 18 16 0 <1E3E 3F7F 77E7 67C7 E78F C78F 070F 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800> PXLC RP 854 2194 XY F28(r)S XP /F28 109 34 3 0 35 18 18 32 0 <1E1F87E0 3F7FDFF0 77F1FC70 67C1F070 E7C1F070 C781E070 0701C070 0F03C0F0 0F03C0E0 0E0380E0 0E0381E0 1E0781C7 1E0781C7 1C0703C6 1C07038E 3C0F03DC 3C0F01F8 380E01F0> PXLC RP 871 2194 XY F28(m)S XP /F28 108 11 4 0 13 29 29 16 0 <1F80 3F80 0780 0780 0700 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800 3800 7800 7800 7000 7000 F700 F700 E600 E600 EE00 FC00 7C00> PXLC RP 905 2194 XY F28(l)S XP /F28 121 20 3 -8 20 18 26 24 0 <1F0380 3F8380 738380 638780 E38700 C78700 070700 070F00 0F0E00 0E0E00 0E0E00 0E1E00 1E1C00 1C1C00 1E3C00 0E7C00 0FF800 07F800 003800 007800 387000 78F000 79E000 73C000 7F8000 3E0000> PXLC RP 916 2194 XY F28(y)S 948 X F25(o)S -1 x(v)S -1 x(er)S 1036 X(all)S 1092 X F26(n)S F25(.)S 1146 X(The)S 1230 X(follo)S -1 x(wing)S 1404 X (is)S 1444 X(immediate.)S 2286 Y 375 X F30(1.)S XP /F30 55 24 3 0 22 28 28 24 0 <700000 7FFFE0 7FFFE0 7FFFE0 7FFFE0 7FFFC0 F00780 E00F00 E01E00 E03C00 003800 007800 00F000 00F000 01F000 01F000 03E000 03E000 03E000 03E000 07E000 07E000 07E000 07E000 07E000 07E000 07E000 03C000> PXLC RP 412 2286 XY F30(7.)S XP /F30 70 30 2 0 27 28 28 32 0 PXLC RP 465 2286 XY F30(Fi)S XP /F30 114 20 1 0 18 18 18 24 0 PXLC RP 508 2286 XY F30(r)S XP /F30 115 19 2 0 16 18 18 16 0 <1FF8 7FF8 7078 E038 E038 F000 FF80 FFF0 7FF8 3FF8 0FFC 007C E03C F01C F01C F878 FFF8 EFE0> PXLC RP 528 2286 XY F30(st)S XP /F30 67 34 3 0 30 28 28 32 0 <003FE060 01FFFCE0 03FFFFE0 0FF81FE0 1FE007E0 3FC003E0 3F8003E0 7F0001E0 7F0001E0 7E0000E0 FE0000E0 FE000000 FE000000 FE000000 FE000000 FE000000 FE000000 FE000000 7E0000E0 7F0000E0 7F0000E0 3F8001E0 3FC003C0 1FE007C0 0FF81F80 03FFFF00 01FFFC00 003FF000> PXLC RP 581 2286 XY F30(Co)S XP /F30 108 13 1 0 12 29 29 16 0 PXLC RP 640 2286 XY F30(ll)S XP /F30 97 23 1 0 22 18 18 24 0 <07FC00 1FFF00 3F1F80 3F0FC0 3F07C0 3F07C0 1E07C0 03FFC0 1FFFC0 7F87C0 7F07C0 FE07C0 FC07C0 FC07C0 FE0FC0 7E3FF8 3FFBF8 0FE1F8> PXLC RP 666 2286 XY F30(a)S XP /F30 112 27 1 -8 24 18 26 24 0 PXLC RP 689 2286 XY F30(psin)S XP /F30 103 24 1 -9 22 18 27 24 0 <07F9F0 1FFFF8 3E1FF8 7C0FF8 7C0FF0 7C0F80 7C0F80 7C0F80 7C0F80 3E1F00 3FFE00 37F800 700000 700000 780000 3FFF80 3FFFE0 1FFFF0 7FFFF0 F801F8 F000F8 F00078 F00078 F800F8 7E03F0 3FFFE0 07FF00> PXLC RP 774 2286 XY F30(g)S XP /F30 76 29 2 0 26 28 28 24 0 PXLC RP 814 2286 XY F30(Le)S XP /F30 109 40 1 0 38 18 18 40 0 PXLC RP 865 2286 XY F30(mma.)S XP /F29 /cmsl10 300 41.5 41.5 128 [-4 -11 46 31] PXLNF RP XP /F29 76 26 2 0 25 28 28 24 0 <0FFFC0 0FFFC0 00F000 00F000 00F000 00F000 00F000 01F000 01E000 01E000 01E000 01E000 01E000 03E000 03C000 03C000 03C000 03C006 03C00C 07C00C 07800C 07801C 078018 078038 078078 0F81F8 FFFFF0 FFFFF0> PXLC RP 1002 2286 XY F29(L)S XP /F29 101 18 2 0 18 18 18 16 0 <01F8 07FE 0F1E 1E0F 3C0F 7C07 7FFF 7FFF 7800 F800 F000 F000 7000 780E 7C0C 3E3C 1FF8 07E0> PXLC RP 1027 2286 XY F29(e)S XP /F29 116 16 4 0 15 26 26 16 0 <0600 0600 0600 0E00 0E00 0C00 1C00 3C00 FFE0 FFE0 1C00 3C00 3800 3800 3800 3800 3800 7800 70C0 70C0 70C0 70C0 71C0 7380 7F00 3E00> PXLC RP 1046 2286 XY F29(t)S XP /F26 71 33 2 -1 31 29 30 32 0 <0003FC18 001FFF38 003F07F8 00FC03F0 01F001F0 03E000F0 07C000F0 0F8000E0 1F000060 1F000060 3E0000E0 3E0000C0 7C000000 7C000000 7C000000 FC000000 F8000000 F800FFF0 F801FFF0 F8000F80 F8000F00 F8000F00 F8001F00 78001F00 7C001E00 3E003E00 1F007E00 0FC1EE00 07FFC400 01FE0000> PXLC RP 1077 2286 XY F26(G)S 6 y F9(\013)S -6 y 1 x F25(\()S F26(n)S F25 (\))S 1204 X(=)S XP /F29 115 16 1 0 16 18 18 16 0 <03F6 0FFE 1E1E 3C0E 380C 3C0C 3E00 3FE0 1FF8 07FC 007C 603C 601C 701C 7038 F878 FFF0 CFC0> PXLC RP 1250 2286 XY F29(s)S XP /F29 105 12 1 0 12 29 29 16 0 <01C0 03E0 03E0 03E0 03C0 0000 0000 0000 0000 0000 0000 1F80 1F80 0780 0780 0700 0700 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 0E00 1E00 FF80 FF80> PXLC RP 1266 2286 XY F29(i)S XP /F29 122 18 1 0 18 18 18 24 0 <0FFF80 0FFF80 0E0F00 1C1E00 183C00 183800 187800 18F000 01E000 03C600 078600 0F0600 0E0E00 1E0C00 3C1C00 783C00 FFF800 FFF800> PXLC RP 1278 2286 XY F29(ze)S XP /F29 111 21 2 0 21 18 18 24 0 <00FC00 03FF00 0F0780 1E03C0 3C01C0 3801C0 7801E0 7801E0 7801E0 F003C0 F003C0 F003C0 700380 700780 780F00 3C1E00 1FF800 07E000> PXLC RP 1329 2286 XY F29(o)S XP /F29 102 13 1 0 19 29 29 24 0 <000FC0 003FC0 007BC0 00F3C0 01E3C0 01C000 01C000 01C000 03C000 038000 038000 3FF800 3FF800 038000 078000 070000 070000 070000 070000 070000 0F0000 0E0000 0E0000 0E0000 0E0000 0E0000 1E0000 FFC000 FFC000> PXLC RP 1349 2286 XY F29(f)S 1377 X F26(\013)S F25([)S F26(n)S F25(])S XP /F29 46 12 3 0 8 5 5 8 0 <78 78 F8 F8 70> PXLC RP 1452 2286 XY F29(.)S XP /F29 84 30 4 0 32 28 28 32 0 <1FFFFFF0 3FFFFFF0 3C0781F0 38078060 70078060 60078060 60078060 600F8060 C00F0060 C00F0060 000F0000 000F0000 000F0000 001F0000 001E0000 001E0000 001E0000 001E0000 001E0000 003E0000 003C0000 003C0000 003C0000 003C0000 003C0000 007C0000 1FFFE000 1FFFE000> PXLC RP 1486 2286 XY F29(T)S XP /F29 104 23 1 0 21 29 29 24 0 <07E000 0FE000 01E000 01C000 01C000 01C000 01C000 01C000 03C000 038000 038000 038FC0 03BFE0 03F1E0 07C0E0 0780E0 0780E0 0700E0 0700E0 0701E0 0F01E0 0E01C0 0E01C0 0E01C0 0E01C0 1E03C0 1E03C0 FF8FF0 FF9FF0> PXLC RP 1516 2286 XY F29(he)S XP /F29 110 23 1 0 21 18 18 24 0 <1F8FC0 1FBFE0 07F1E0 07C0E0 0780E0 0780E0 0700E0 0700E0 0701E0 0F01E0 0E01C0 0E01C0 0E01C0 0E01C0 0E03C0 1E03C0 FF8FF0 FF9FF0> PXLC RP 1557 2286 XY F29(n)S 1595 X(fo)S XP /F29 114 16 1 0 17 18 18 16 0 <1FBE 1FFE 07EF 07CE 078E 0780 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 0E00 1E00 FFC0 FFC0> PXLC RP 1629 2286 XY F29(r)S XP /F29 97 21 2 0 21 18 18 24 0 <07F800 0FFE00 0F1F00 0E0700 0E0700 000700 000700 03FF00 1FCF00 3E0F00 7C0E00 F80E00 F00E60 F01E60 F03E60 F87EC0 7FEFC0 3F8F00> PXLC RP 300 2336 XY F29(a)S XP /F29 108 12 1 0 12 29 29 16 0 <07E0 0FE0 01E0 01C0 01C0 01C0 01C0 01C0 03C0 0380 0380 0380 0380 0380 0780 0700 0700 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 1E00 1E00 FF80 FF80> PXLC RP 321 2336 XY F29(ll)S XP /F29 112 23 0 -8 23 18 26 24 0 <0FCFE0 0FFFF8 03F8F8 03E07C 03C03C 03803E 03801E 03801E 03803E 07803E 07003C 07003C 070078 0700F8 0781F0 0FC3E0 0EFFC0 0E7F00 0E0000 0E0000 1E0000 1E0000 1C0000 1C0000 FF8000 FF8000> PXLC RP 358 2336 XY F29(pro)S XP /F29 103 21 0 -9 22 19 28 24 0 <000038 007EFC 01FFDC 03C79C 0783C0 0F03C0 0F03C0 0F03C0 0F03C0 0F0780 078F00 07FE00 0DF800 080000 180000 1C0000 1FFE00 0FFF80 1FFFC0 3C03C0 7001E0 7001E0 F001C0 F001C0 7807C0 7C1F80 3FFE00 0FF800> PXLC RP 418 2336 XY F29(gra)S XP /F29 109 35 1 0 34 18 18 40 0 <1F8FC0FC00 1FBFF3FF00 07F0F70F00 07C07C0700 0780780700 0780780700 0700700700 0700700700 0700700700 0F00F00F00 0E00E00E00 0E00E00E00 0E00E00E00 0E00E00E00 0E00E00E00 1E01E01E00 FF8FF8FF80 FF8FF8FF80> PXLC RP 475 2336 XY F29(ms)S 540 X F26(P)S XP /F29 44 12 2 -8 8 5 13 8 0 <3C 3C 7C 7C 3C 0C 1C 18 38 38 70 E0 40> PXLC RP 573 2336 XY F29(,)S 2428 Y 555 X F27(j)S 560 X F25(=)S -17 y F9 (\013)S 17 y 627 X F26(P)S 671 X F27(#)S XP /F27 40 42 2 -2 39 22 24 40 0 <0006000000 0006000000 000E000000 001C000000 0038000000 0038000000 007FFFFFF8 01FFFFFFF8 03C0000000 0F00000000 3E00000000 F800000000 7C00000000 0F00000000 0780000000 01C0000000 00FFFFFFF8 007FFFFFF8 0038000000 0018000000 001C000000 000E000000 0006000000 0006000000> PXLC RP 703 2428 XY F27(\()S XP /F27 41 42 2 -2 39 22 24 40 0 <0000030000 0000030000 0000038000 000001C000 000000E000 000000E000 FFFFFFF000 FFFFFFFC00 0000001E00 0000000780 00000003E0 00000000F8 00000001F0 0000000780 0000000F00 0000001C00 FFFFFFF800 FFFFFFF000 000000E000 000000C000 000001C000 0000038000 0000030000 0000030000> PXLC RP 737 2428 XY F27(\))S 802 X F26(G)S 6 y F9(\013)S -6 y 872 X F29(is)S 914 X(a)S XP /F29 99 18 2 0 18 18 18 16 0 <01FE 07FE 0F8F 1E0E 3C0E 3C00 7800 7800 7800 F800 F000 F000 7800 780E 7C0C 3E1C 1FF8 07E0> PXLC RP 949 2428 XY F29(comple)S XP /F29 120 22 1 0 23 18 18 24 0 <1FE3FC 3FE7FC 03C3E0 01C380 01C700 00EE00 00EC00 007800 007000 007800 00F800 01DC00 039E00 070E00 0E0700 1E0780 FF1FF0 FF1FF0> PXLC RP 1075 2428 XY F29(xit)S XP /F29 121 22 0 -8 23 18 26 24 0 <0FF1FE 0FF1FE 03C078 01C060 01C060 01C0C0 01C1C0 01C180 00E380 00E300 00E600 00E600 00EC00 007C00 007800 007800 007000 007000 006000 006000 00C000 70C000 F18000 C70000 FE0000 7C0000> PXLC RP 1124 2428 XY F29(y)S XP /F29 98 23 4 0 23 29 29 24 0 <3F0000 7F0000 0F0000 0E0000 0E0000 0E0000 0E0000 0E0000 1E0000 1C0000 1C0000 1CFE00 1FFF80 1F8780 3E03C0 3C03C0 3801E0 3801E0 3801E0 3803E0 7803E0 7003C0 7003C0 700780 700780 780F00 FC3E00 EFFC00 C7F000> PXLC RP 1160 2428 XY F29(b)S 1 x(o)S XP /F29 117 23 4 0 22 18 18 24 0 PXLC RP 1204 2428 XY F29(un)S XP /F29 100 23 2 0 23 29 29 24 0 <0001F8 0003F8 000078 000070 000070 000070 000070 0000F0 0000F0 0000E0 0000E0 01FCE0 07FFE0 0F87E0 1E03E0 3C01C0 7C01C0 7801C0 7801C0 7803C0 F803C0 F00380 F00380 F00380 780780 780F80 3C3F80 1FFFE0 0FE7E0> PXLC RP 1251 2428 XY F29(d)S 1288 X(for)S 1351 X F26(P)S 1 x(:)S 2520 Y 375 X F28(Note)S 3 x F25(.)S 493 X(It)S 538 X(is)S 579 X(ob)S -1 x (vious)S 730 X(from)S 828 X(the)S 899 X(de\014nition)S 1084 X(of)S 1131 X F26(\013)S F25([)S F26(n)S F25(])S 1220 X(that)S 2612 Y 443 X F26(G)S 6 y F8(0)S -6 y 1 x F25(\()S F26(n)S F25(\))S 562 X(=)S 606 X(0)S F26(;)S 687 X(G)S 6 y F9(\013)S XP /F8 43 25 2 -4 22 18 22 24 0 <006000 006000 006000 006000 006000 006000 006000 006000 006000 006000 FFFFF0 FFFFF0 006000 006000 006000 006000 006000 006000 006000 006000 006000 006000> PXLC RP 741 2618 XY F8(+1)S -6 y 2 x F25(\()S F26(n)S F25(\))S 854 X(=)S 898 X F26(G)S 6 y F9(\013)S -6 y 1 x F25(\()S F26(n)S F25(\))S 1021 X (+)S 1062 X(1)S F26(;)S 1143 X(G)S XP /F10 /cmsy7 300 29.1 29.1 128 [-1 -28 37 24] PXLNF RP XP /F10 104 13 3 -8 9 22 30 8 0 <0C 0C 1C 1C 18 38 38 30 30 70 70 60 E0 E0 C0 C0 E0 E0 60 70 70 30 30 38 38 18 1C 1C 0C 0C> PXLC RP 1175 2619 XY F10(h)S F9(\013)S XP /F1 120 16 2 0 14 9 9 16 0 <3DE0 6F70 CE70 0E60 0C00 EC20 FC60 DCC0 F780> PXLC RP 1211 2623 XY F1(x)S XP /F10 105 13 2 -8 8 22 30 8 0 PXLC RP 1229 2619 XY F10(i)S -7 y 3 x F25(\()S F26(n)S F25(\))S 1313 X(=)S 1357 X F26(G)S 6 y F9(\013)S XP /F1 110 18 2 0 16 9 9 16 0 <7BE0 4E70 DC70 1C60 1860 1864 38EC 38D8 30F0> PXLC RP 1411 2622 XY F1(n)S -10 y 1434 X F25(\()S F26(n)S F25(\))S F26(:)S 2703 Y 300 X(G)S 7 y F9(\013)S -7 y 370 X F25(is)S 411 X(the)S 482 X (so{called)S 657 X(Slo)S -1 x(w)S XP /F25 71 33 2 -1 30 29 30 32 0 <003FC180 00FFF180 03F07B80 07C01F80 0F800F80 1F000780 3E000780 3E000380 7C000380 7C000180 78000180 F8000180 F8000000 F8000000 F8000000 F8000000 F8000000 F800FFF0 F800FFF0 78000780 7C000780 7C000780 3E000780 3E000780 1F000780 0F800F80 07C00F80 03F83F80 00FFF980 003FC000> PXLC RP 754 2703 XY F25(Gro)S -1 x(wing)S 921 X(Hierarc)S -1 x(h)S -1 x(y)S -3 x(.)S XP /F25 78 31 2 0 28 28 28 32 0 PXLC RP 1123 2703 XY F25(Note)S 1223 X(that)S 1312 X(ev)S -1 x(erything)S 1516 X(is)S 1557 X(mea-)S 50 y 300 X(sured)S 410 X(here)S 500 X (\\p)S 1 x(oin)S -1 x(t)S -1 x(wise")S 724 X(with)S 818 X(resp)S 1 x(ect)S 960 X(to)S 1010 X(the)S 1081 X(size)S 1159 X F26(n)S 1197 X F25(of)S 1244 X(inputs.)S 1387 X(Note)S 1486 X(also)S 1569 X(that)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 129 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup XP /F25 51 21 2 -1 18 28 29 16 0 <0FE0 1FF8 3C7C 703E 783E 7C3E 783E 783E 003E 003C 007C 00F8 07F0 07E0 0078 003C 003E 001E 001F 001F 701F F81F F81F F81F F03E 703E 787C 3FF8 0FE0> PXLC RP 300 462 XY F25(3)S XP /F25 56 21 2 -1 18 28 29 16 0 <07E0 1FF8 3C3C 301C 700E 6006 6006 7006 780E 7C0C 3F1C 3FF8 1FF0 0FF0 1FF8 3DFC 787E 703F F01F E00F E007 E007 E007 F007 700E 781E 3C3C 1FF8 07E0> PXLC RP 321 462 XY F25(8)S XP /F25 50 21 2 0 18 28 28 16 0 <0FE0 1FF8 38FC 703E E03E F01F F81F F81F F80F 700F 001F 001F 001E 003E 007C 0078 00F0 01E0 03C0 0380 0700 0E03 1C03 3803 7006 FFFE FFFE FFFE> PXLC RP 342 462 XY F25(2)S XP /F18 /cmr9 300 37.4 37.4 128 [-3 -10 37 28] PXLNF RP XP /F18 72 29 1 0 27 26 26 32 0 PXLC RP 700 462 XY F18(H)S XP /F18 46 11 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 729 462 XY F18(.)S XP /F18 83 21 2 0 18 26 26 16 0 <0FC6 3FF6 787E 701E E00E E00E E006 E006 F000 FC00 7F80 7FF0 3FFC 0FFE 01FE 003F 000F 000F C007 C007 E007 E00F F00E FC3C DFFC C7F0> PXLC RP 752 462 XY F18(S)S XP /F18 99 17 1 0 15 16 16 16 0 <07F8 1FFC 3E3C 783C 7818 F000 F000 F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 774 462 XY F18(c)S XP /F18 104 21 1 0 19 26 26 24 0 PXLC RP 790 462 XY F18(h)S XP /F18 119 28 1 0 26 16 16 32 0 PXLC RP 810 462 XY F18(w)S XP /F18 105 11 0 0 9 26 26 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 838 462 XY F18(ic)S -1 x(h)S XP /F18 116 15 1 0 12 23 23 16 0 <0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFC0 FFC0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C60 1C60 1C60 1C60 1EE0 0FC0 0780> PXLC RP 885 462 XY F18(t)S XP /F18 101 17 1 0 15 16 16 16 0 <07E0 1FF0 3C78 783C 703C FFFC FFFC F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 900 462 XY F18(e)S XP /F18 110 21 1 0 19 16 16 24 0 PXLC RP 917 462 XY F18(n)S XP /F18 98 21 1 0 19 26 26 24 0 PXLC RP 937 462 XY F18(b)S 1 x(e)S XP /F18 114 15 1 0 13 16 16 16 0 PXLC RP 976 462 XY F18(r)S XP /F18 103 19 1 -8 17 16 24 16 0 <0FCF 1FFF 3CF7 7878 7878 7878 7878 7878 3CF0 3FE0 3FC0 3000 3000 3FF8 3FFC 3FFE 701F E00F E007 E007 F00F 7C3E 3FFC 0FF0> PXLC RP 991 462 XY F18(g)S XP /F18 97 19 1 0 18 16 16 24 0 <3FC000 7FF000 78F800 787800 303800 003800 07F800 3F3800 7C3800 F83800 F03800 F03980 F07980 F8F980 7FFF00 3F1E00> PXLC RP 1023 462 XY F18(an)S XP /F18 100 21 1 0 19 26 26 24 0 <007E00 007E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 0FEE00 1FFE00 3E3E00 781E00 780E00 F00E00 F00E00 F00E00 F00E00 F00E00 F00E00 780E00 781E00 3C3E00 1FFFC0 0FCFC0> PXLC RP 1064 462 XY F18(d)S 1098 X(S.S.)S XP /F18 87 39 1 0 37 26 26 40 0 PXLC RP 1175 462 XY F18(W)S -3 x(ainer)S 553 Y 300 X F25(the)S 373 X (function)S 536 X F26(G)S 6 y F9(\013)S -6 y 608 X F25(dep)S 1 x (ends)S 769 X(crucially)S 938 X(on)S 996 X(the)S 1069 X(w)S -1 x(a)S -1 x(y)S 1154 X(in)S 1204 X(whic)S -1 x(h)S 1324 X(a)S 1360 X (structured)S 1562 X(tree-)S 50 y 300 X(ordinal)S 442 X F26(\013)S 484 X F25(is)S 527 X(presen)S -1 x(ted)S 714 X(as)S 766 X(the)S 839 X(directed)S 1000 X(union)S 1116 X(of)S 1165 X(its)S 1224 X (\014nite)S 1331 X(subsets)S 1476 X F26(\013)S F25([)S F26(n)S F25 (],)S 1578 X(and)S 50 y 300 X(not)S XP /F25 106 13 -3 -8 8 29 37 16 0 <01C0 03E0 03E0 03E0 01C0 0000 0000 0000 0000 0000 0000 0FE0 0FE0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 F0E0 F1E0 F3C0 FF80 7E00> PXLC RP 371 653 XY F25(just)S 451 X(on)S 505 X(its)S 561 X(set-theoretic)S 796 X(ordinal)S 934 X(ran)S XP /F25 107 22 1 0 20 29 29 24 0 PXLC RP 994 653 XY F25(k)S 1027 X F27(j)S F26(\013)S F27(j)S F25(.)S 1106 X (Simply)S 1243 X(kno)S -1 x(wing)S 1404 X(that)S 1491 X F27(j)S F26 (\013)S F27(j)S 1552 X(\024)S 1596 X(j)S F26(\014)S 2 x F27(j)S 50 y 300 X F25(giv)S -1 x(es)S 400 X(no)S 457 X(information)S 680 X (whatev)S -1 x(er)S 856 X(ab)S 1 x(out)S 974 X(the)S 1044 X (relativ)S -1 x(e)S 1190 X(sizes)S 1284 X(of)S 1330 X F26(G)S 6 y F9(\013)S -6 y 1398 X F25(and)S 1478 X F26(G)S XP /F9 12 19 0 -6 17 20 26 24 0 <003F00 00E300 01C380 038380 070380 060780 0E0700 0CFF00 0CCF00 1CFF00 180380 180380 180380 380780 380700 380700 380E00 7C1E00 6E3800 67F000 600000 E00000 C00000 C00000 C00000 C00000> PXLC RP 1510 709 XY F9(\014)S -6 y 1533 X F25(,)S 1557 X(since)S 49 y 300 X (for)S 366 X(example,)S 543 X(if)S 584 X F26(f)S 625 X F25(is)S 670 X(an)S 730 X(arbitrary)S 909 X(increasing)S 1106 X(n)S -1 x(um)S -1 x(b)S 1 x(er-theoretic)S 1433 X(function)S 1598 X(w)S -1 x(e)S 50 y 300 X(can)S 381 X(de\014ne)S 506 X(the)S 583 X(structured)S 789 X(tree-ordinal)S 1018 X F26(\013)S XP /F9 102 16 1 -6 15 20 26 16 0 <007C 007C 00FC 00FC 00E0 00E0 01E0 0FFC 0FFC 01C0 01C0 03C0 0380 0380 0380 0380 0380 0780 0700 0700 0700 0F00 EE00 EE00 EC00 7800> PXLC RP 1045 809 XY F9(f)S -7 y 1087 X F25(=)S 1139 X F27(h)S F26(f)S 1179 X F25(\()S F26(x)S F25(\))S F27(i)S 1271 X F25(with)S 1371 X(ordinal)S 1517 X(rank)S 1618 X F26(!)S 50 y 300 X F25(suc)S -1 x(h)S 393 X (that)S 482 X F26(G)S 6 y F9(\013)S XP /F1 102 14 2 -4 13 14 18 16 0 <01E0 03E0 03E0 03E0 0700 1FC0 1FC0 0600 0600 0600 0E00 0C00 0C00 0C00 EC00 F800 D800 F000> PXLC RP 536 863 XY F1(f)S -11 y 568 X F25(=)S 612 X F26(f)S 636 X F25(.)S 667 X(Ho)S -1 x(w)S -1 x(ev)S -1 x(er,)S 845 X(if)S 882 X(instead)S 1025 X(w)S -1 x(e)S 1085 X(require)S 1224 X(the)S 1294 X(existence)S 1470 X(of)S 1517 X(a)S 1550 X("nat-)S 50 y 300 X(ural)S 386 X (transformation")S 689 X(from)S 788 X F26(\013)S 828 X F25(to)S 879 X F26(\014)S 2 x F25(,)S 931 X(that)S 1021 X(is)S 1063 X(an)S 1120 X(order-prese)S 3 x(rving)S 1429 X(map)S XP /F26 25 24 1 0 23 18 18 24 0 <0FFFFC 1FFFFC 3FFFFC 786700 60E600 E0C600 00C600 01CE00 01CE00 018E00 038E00 038E00 070E00 070E00 0F0F00 0E0F00 0E0700 0E0700> PXLC RP 1522 902 XY F26(\031)S 1561 X F25(from)S XP /F27 102 21 2 -10 18 31 41 16 0 <001F 007F 00F8 01E0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 03C0 0F80 FF00 FC00 FF00 0F80 03C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01E0 00F8 007F 001F> PXLC RP 300 952 XY F27(f)S XP /F26 13 21 0 -9 21 18 27 24 0 <07E018 1FF018 3FF838 7FF830 703870 E01C60 C01CE0 C00CC0 000DC0 000D80 000D80 000F00 000F00 000E00 000E00 000E00 000C00 000C00 000C00 000C00 001800 001800 001800 001800 003000 003000 003000> PXLC RP 321 952 XY F26(\015)S 357 X F25(:)S 381 X F26(\015)S 417 X(<)S 462 X (\013)S XP /F27 103 21 2 -10 18 31 41 16 0 PXLC RP 488 952 XY F27(g)S 523 X F25(in)S -1 x(to)S 608 X F27(f)S F26(\015)S 665 X F25(:)S 689 X F26(\015)S 725 X(<)S 769 X(\014)S 2 x F27(g)S 830 X F25(suc)S -1 x(h)S 924 X(that)S 1015 X(for)S 1079 X(ev)S -1 x (ery)S 1189 X F26(n)S F25(,)S 1240 X F26(\031)S 1279 X F25(tak)S -1 x(es)S 1386 X F26(\013)S F25([)S F26(n)S F25(])S 1475 X(in)S -1 x (to)S 1560 X F26(\014)S 2 x F25([)S F26(n)S F25(],)S 50 y 300 X (then)S 392 X(w)S -1 x(e)S 451 X(do)S 506 X(obtain)S 633 X F27(8)S F26(n)S 692 X F25(\()S F26(G)S 6 y F9(\013)S -6 y 2 x F25(\()S F26 (n)S F25(\))S 833 X F27(\024)S 877 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\()S F26(n)S F25(\)\).)S 1035 X(Th)S -1 x(us)S 1138 X(pro)S 1 x (of-theoretic)S 1417 X(ordinal)S 1555 X(anal-)S 49 y 300 X(ysis)S 380 X(is)S 422 X(ab)S 1 x(out)S 540 X(the)S 612 X(w)S -1 x(a)S -1 x (ys)S 712 X(in)S 761 X(whic)S -1 x(h)S 879 X(ordinals)S 1037 X(are)S 1106 X(presen)S -1 x(ted)S 1292 X(\(ordinal)S 1449 X(notations\),)S 50 y 300 X(and)S 378 X(not)S 450 X(ab)S 1 x(out)S 566 X (set-theoretic)S 802 X(ordinals.)S 974 X(What)S 1088 X(w)S -1 x(e)S 1147 X(are)S 1214 X(in)S -1 x(terested)S 1402 X(in)S 1448 X(here,)S 1548 X(is)S 1587 X(the)S 50 y 300 X(dev)S -1 x(elopmen)S -1 x(t)S 542 X(of)S 590 X(a)S 625 X(fairly)S 734 X(general)S 878 X(framew)S -1 x(ork)S 1083 X(for)S 1147 X(assigning)S 1325 X(to)S 1377 X(a)S 1412 X(giv)S -1 x(en)S 1521 X(class)S XP /F25 76 26 2 0 23 28 28 24 0 PXLC RP 1619 1151 XY F25(L)S 50 y 300 X(of)S 348 X(terminating)S 575 X (programs,)S 771 X(a)S 806 X(single)S 923 X(structured)S 1125 X (tree-)S 1223 X(ordinal)S XP /F26 28 18 1 0 21 18 18 24 0 <0FFFF0 1FFFF0 3FFFF0 786000 606000 E0E000 00E000 00C000 01C000 01C000 01C000 01C000 038000 038000 038000 038000 070000 070000> PXLC RP 1364 1201 XY F26(\034)S 1402 X F25(suc)S -1 x(h)S 1496 X(that)S 1587 X(the)S 50 y 300 X(hierarc)S -1 x(h)S -1 x(y)S 483 X(of)S 533 X (functions)S 714 X F27(f)S F26(G)S 6 y F9(\013)S -6 y 806 X F25(:)S 832 X F26(\013)S 874 X(<)S 921 X(\034)S 944 X F27(g)S 980 X F25(is)S 1024 X(a)S 1061 X(w)S -1 x(ell-ordered)S 1297 X(scale)S 1398 X(of)S 1448 X(complexit)S -1 x(y)S 49 y 300 X(b)S 1 x(ounds)S 444 X(for)S 508 X(the)S 579 X(programs)S 762 X(in)S 810 X(L.)S 50 y 375 X(Supp)S 1 x(ose)S 534 X(no)S -1 x(w)S 617 X(that)S 704 X(w)S -1 x(e)S 761 X (ha)S -1 x(v)S -1 x(e)S 854 X(some)S 954 X(language)S XP /F26 76 28 2 0 26 28 28 24 0 <03FFF0 03FFF0 003E00 003C00 003C00 003C00 007C00 007800 007800 007800 00F800 00F000 00F000 00F000 01F000 01E000 01E000 01E007 03E006 03C006 03C00E 03C00C 07C01C 07803C 078078 0F81F8 FFFFF8 FFFFF0> PXLC RP 1124 1350 XY F26(L)S 1163 X F25(within)S 1288 X(whic)S -1 x(h)S 1404 X(w)S -1 x(e)S 1461 X(can)S 1534 X(gener-)S 50 y 300 X(ate)S 367 X(terminating)S 590 X(programs,)S 782 X(b)S -1 x(y)S 837 X (starting)S 989 X(with)S 1081 X(some)S 1183 X(giv)S -1 x(en)S 1288 X (\\basic")S 1431 X(programs)S XP /F26 66 31 2 0 30 28 28 32 0 <03FFFF00 03FFFFC0 003E03E0 003C01F0 003C00F0 003C00F0 007C00F0 007801F0 007801F0 007803E0 00F807C0 00F00F80 00FFFF00 00FFFE00 01F01F00 01E00F80 01E007C0 01E007C0 03E007C0 03C007C0 03C007C0 03C007C0 07C00F80 07801F00 07803E00 0F807C00 FFFFF800 FFFFE000> PXLC RP 1611 1400 XY F26(B)S 50 y 300 X F25(\(for)S 378 X(example)S 539 X (zero,)S 638 X(successor\))S 831 X(and)S 910 X(then)S 1003 X (building)S 1163 X(new)S 1247 X(programs)S 1428 X(b)S -1 x(y)S 1484 X (comp)S 1 x(osi-)S 50 y 300 X(tion)S 382 X(\(sequencing\))S 620 X (and)S 697 X(applications)S 924 X(of)S 968 X(certain)S 1103 X(giv)S -1 x(en)S 1208 X(\\program{constructors")S XP /F26 67 30 2 -1 31 29 30 32 0 <0003FC18 001FFF38 003F07F8 00FC03F0 01F001F0 03E000F0 07C000F0 0F8000E0 1F000060 1F000060 3E0000E0 3E0000C0 7C000000 7C000000 7C000000 FC000000 F8000000 F8000000 F8000000 F8000300 F8000300 F8000700 F8000600 78000E00 7C001C00 3E003800 1F007000 0FC1E000 07FF8000 01FE0000> PXLC RP 300 1549 XY F26(C)S 346 X F25(\(for)S 426 X(example)S 589 X(lo)S 1 x(ops)S 696 X(or)S 747 X(recursors\).)S 50 y 375 X(Our)S 460 X (standing)S 629 X(assumption)S 848 X(on)S 906 X(basic)S 1009 X (programs)S 1192 X(will)S 1270 X(b)S 1 x(e)S 1327 X(that)S 1417 X (they)S 1510 X(op)S 1 x(erate)S 50 y 300 X(in)S 348 X(one)S 425 X (step)S 512 X(and)S 593 X(increase)S 750 X(the)S 822 X(size)S 900 X (of)S 948 X(the)S 1019 X(input)S 1130 X(b)S -1 x(y)S 1188 X(at)S 1238 X(most)S 1340 X(one.)S XP /F25 69 28 2 0 26 28 28 24 0 PXLC RP 375 1699 XY F25(Eac)S -1 x(h)S 474 X(constructor)S 690 X F26(C)S 733 X F25(is)S 771 X(a)S 801 X(uniform)S 953 X(metho)S 1 x(d)S 1101 X (for)S 1160 X(constructing)S 1395 X(new)S 1476 X(programs)S 50 y 300 X F26(C)S 3 x F25(\()S XP /F26 81 33 2 -8 30 29 37 32 0 <0003F800 000FFE00 003E0F80 00F807C0 01E003C0 03C001E0 078001E0 0F8001F0 1F0001F0 1E0001F0 3E0001F0 3E0001F0 7C0001F0 7C0001F0 7C0001F0 F80003E0 F80003E0 F80003E0 F80007C0 F80007C0 F8000780 F8000F80 F83C1F00 787E1E00 78E63C00 3CC37800 3EC3F000 1FC7C000 07FF8100 01FF0100 00038300 00038E00 0003FE00 0003FC00 0003FC00 0003F800 0001F000> PXLC RP 349 1749 XY F26(Q)S F25(\))S 410 X(from)S 506 X(previously)S 703 X (generated)S 890 X(ones)S 980 X F26(Q)S F25(.)S 1043 X(In)S 1093 X (terms)S 1206 X(of)S 1252 X(the)S 1321 X(formalism)S 1512 X (already)S 50 y 300 X(dev)S -1 x(elop)S 1 x(ed,)S 504 X(its)S 562 X (uniformit)S -1 x(y)S 766 X(can)S 842 X(b)S 1 x(e)S 899 X(measured)S 1084 X(b)S -1 x(y)S 1141 X(a)S 1176 X(sequence)S 1348 X F27(h)S F26 (\013)S XP /F9 24 15 1 -6 13 21 27 16 0 <0180 0180 0180 03F0 07B0 1FF0 1C00 3C00 3800 3800 3800 3FC0 1CC0 3FC0 6000 E000 C000 C000 E000 F000 FE00 7F80 3FC0 07C0 01C0 0DC0 0780> PXLC RP 1391 1805 XY F9(\030)S -6 y 3 x F27(i)S 1439 X F25(suc)S -1 x(h)S 1532 X(that)S 1894 Y 681 X F27(8)S XP /F26 24 18 0 -8 17 30 38 24 0 <000800 001800 001800 001800 003F80 00FF80 03FF80 07C000 078000 0F8000 0F0000 0F0000 0F0000 0F0000 0F0000 0FFC00 07FC00 0FFC00 1E0000 1C0000 380000 780000 700000 700000 F00000 F00000 F00000 F80000 7E0000 3F8000 1FF000 07FC00 01FC00 003E00 000E00 031C00 03FC00 01F800> PXLC RP 704 1894 XY F26(\030)S 2 x F27(8)S F26(Q)S F25(\()S F27(j)S 801 X F25(=)S -17 y F9(\030)S 17 y 863 X F26(Q)S 907 X F27(#)S 939 X F25 (=)S 964 X F27(\))S 1018 X(j)S 1023 X F25(=)S -17 y F9(\013)S XP /F1 24 14 2 -4 11 14 18 16 0 <0400 0400 0F80 3F80 7000 6000 6000 7F00 7E00 E000 C000 C000 E000 F800 7F00 0F00 0100 1F00> PXLC RP 1076 1881 XY F1(\030)S 13 y 1106 X F26(C)S 3 x F25(\()S F26(Q)S F25 (\))S 1215 X F27(#)S F25(\))S F26(:)S 1990 Y 300 X F25(Th)S -1 x (us)S 404 X F26(\013)S 442 X F25(=)S 486 X F27(h)S F26(\013)S 6 y F9(\030)S -6 y 3 x F27(i)S 575 X F25(measures)S 752 X(the)S 822 X (complexit)S -1 x(y)S 1031 X(of)S 1077 X F26(C)S 1122 X F25 (itself.)S XP /F25 66 29 2 0 26 28 28 24 0 PXLC RP 1238 1990 XY F25(But)S 1319 X(unfortunately)S 1578 X F26(\013)S 1617 X F25(is)S 49 y 300 X(uncoun)S -1 x(table)S 535 X(and)S 616 X (therefore)S 791 X(no)S 850 X(longer)S 975 X(a)S 1011 X(mem)S -1 x (b)S 1 x(er)S 1171 X(of)S 1219 X(\012)S 7 y F8(1)S -7 y 1 x F25(.)S 1300 X(Therefore)S 1489 X(if)S 1528 X(w)S -1 x(e)S 1589 X(are)S 50 y 300 X(to)S 349 X(mak)S -1 x(e)S 455 X(ordinal)S 594 X(assignmen)S -1 x(ts)S 822 X(to)S 871 X(programs)S 1051 X(in)S 1098 X F26(L)S F25 (,)S 1150 X(it)S 1189 X(is)S 1229 X(quite)S 1332 X(natural)S 1475 X (to)S 1524 X(extend)S 50 y 300 X(\012)S 6 y F8(1)S -6 y 368 X F25 (to)S 424 X(the)S 501 X(next)S 600 X(higher)S 733 X(n)S -1 x(um)S -1 x(b)S 1 x(er)S 889 X(class)S 992 X(\012)S 6 y F8(2)S XP /F27 17 32 2 0 29 20 20 32 0 PXLC RP 1062 2139 XY F27(\021)S XP /F26 87 39 2 -1 43 28 29 48 0 PXLC RP 1115 2139 XY F26(W)S 1160 X(x)S 1204 X F27(2)S 1253 X F26(N)S XP /F8 52 17 1 0 15 19 19 16 0 <0060 00E0 01E0 03E0 06E0 04E0 0CE0 18E0 30E0 60E0 C0E0 FFFC FFFC 00E0 00E0 00E0 00E0 07FC 07FC> PXLC RP 1286 2145 XY F8(4)S -6 y 2 x F26(:)S XP /F26 82 32 2 -1 31 28 29 32 0 <03FFFE00 03FFFF80 003E03E0 003C01E0 003C01F0 003C01F0 007C01F0 007801F0 007801F0 007803E0 00F803C0 00F00780 00F01F00 00FFFC00 01FFF000 01E07800 01E03C00 01E03C00 03E03C00 03C03C00 03C03C00 03C07C00 07C07C00 07807C00 07807C18 0F807E38 FFF87E70 FFF83FE0 00000FC0> PXLC RP 1316 2139 XY F26(R)S 6 y F8(4)S -6 y 1 x F25(\()S F26(x;)S XP /F27 63 32 2 0 30 28 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFF0 FFFFFFF0> PXLC RP 1425 2139 XY F27(?)S F26(;)S XP /F27 62 32 2 0 30 28 28 32 0 PXLC RP 1475 2139 XY F27(>)S F26(;)S 1526 X(N)S 2 x(;)S 1580 X F25(\012)S 6 y F8(1)S -6 y 2 x F25(\))S 50 y 300 X(generated)S 489 X(inductiv)S -1 x(ely)S 703 X(b)S -1 x(y)S 760 X(the)S 832 X(four)S 918 X (rules:)S 2282 Y 642 X(0)S 675 X F27(2)S 714 X F25(\012)S 6 y F8(2)S -6 y 1 x F26(;)S 50 y 642 X(\013)S 681 X F27(2)S 720 X F25(\012)S 6 y F8(2)S -6 y 791 X F25(=)S 816 X F27(\))S 881 X F26(\013)S 917 X F25(+)S 959 X(1)S 991 X F27(2)S 1030 X F25(\012)S 6 y F8(2)S -6 y 2 x F26(;)S 49 y 642 X F27(8)S F26(x)S 701 X F27(2)S 740 X F26(N)S 778 X F25(\()S F26(\013)S 7 y F9(x)S -7 y 853 X F27(2)S 892 X F25 (\012)S 7 y F8(2)S -7 y 2 x F25(\))S 980 X(=)S 1005 X F27(\))S 1070 X F26(\013)S 1108 X F25(=)S 1152 X F27(h)S F26(\013)S 7 y F9(x)S -7 y 1 x F27(i)S 1243 X(2)S 1282 X F25(\012)S 7 y F8(2)S -7 y 2 x F26(;)S 50 y 642 X F27(8)S F26(\030)S 697 X F27(2)S 736 X F25(\012)S 6 y F8 (1)S -6 y 2 x F25(\()S F26(\013)S 6 y F9(\030)S -6 y 857 X F27(2)S 896 X F25(\012)S 6 y F8(2)S -6 y 2 x F25(\))S 984 X(=)S 1009 X F27 (\))S 1074 X F26(\013)S 1112 X F25(=)S 1156 X F27(h)S F26(\013)S 6 y F9(\030)S -6 y 3 x F27(i)S 1245 X(2)S 1284 X F25(\012)S 6 y F8 (2)S -6 y 2 x F26(:)S 2525 Y 375 X F25(Then)S 482 X(w)S -1 x(e)S 542 X(can)S 617 X(deriv)S -1 x(e)S 739 X(eac)S -1 x(h)S 831 X (program)S 997 X(in)S 1044 X F26(L)S 1085 X F25(and)S 1165 X(sim)S -1 x(ultaneously)S 1446 X(assign)S 1568 X(to)S 1617 X(it)S 50 y 300 X(an)S 358 X(ordinal)S 498 X(b)S 1 x(ound)S 626 X(in)S 675 X (\012)S 6 y F8(2)S -6 y 1 x F25(,)S 749 X(according)S 936 X(to)S 986 X(the)S 1058 X(follo)S -1 x(wing)S 1233 X(rules:)S XP /F27 15 21 2 2 18 18 16 16 0 <07E0 1FF8 3FFC 7FFE 7FFE FFFF FFFF FFFF FFFF FFFF FFFF 7FFE 7FFE 3FFC 1FF8 07E0> PXLC RP 362 2662 XY F27(\017)S XP /F28 66 29 2 0 29 28 28 32 0 <01FFFF00 01FFFF80 003E03C0 003C03E0 003C01E0 003C01E0 007C01E0 007803E0 007803E0 007807C0 00F80780 00F01F00 00FFFE00 00FFFC00 01F03E00 01E00F00 01E00F80 01E00F80 03E00F80 03C00F80 03C00F80 03C00F80 07C01F00 07801E00 07803E00 0F80FC00 FFFFF000 FFFFC000> PXLC RP 404 2662 XY F28(B)S XP /F28 97 21 4 0 22 18 18 24 0 <03EE00 07FE00 0F3E00 1E1E00 3C1E00 781C00 781C00 783C00 F83C00 F03800 F03800 F079C0 F079C0 E0F180 F0F180 73F300 7FFF00 3E3E00> PXLC RP 433 2662 XY F28(a)S XP /F28 115 17 3 0 17 18 18 16 0 <03F0 07F8 0F3C 0E3C 1E7C 1E38 1F00 1FE0 0FF0 0FF0 00F8 2078 F070 F070 F0F0 E1E0 7FC0 3F80> PXLC RP 454 2662 XY F28(si)S XP /F28 99 19 4 0 19 18 18 16 0 <01F8 07FC 0F9E 1E1E 3C3E 7C1C 7800 7800 F800 F000 F000 F000 F000 F00C F01C 7878 3FF0 1FC0> PXLC RP 484 2662 XY F28(c)S 2 x F25(.)S XP /F27 96 25 2 0 22 29 29 24 0 PXLC RP 535 2662 XY F27(`)S -15 y F8(1)S 15 y 591 X F26(B)S 2753 Y 362 X F27 (\017)S XP /F28 67 30 6 -1 33 29 30 32 0 <0007F060 001FFCE0 007E0FE0 00F807C0 01F003C0 07E003C0 07C003C0 0F800180 1F000180 1F000180 3E000380 3E000300 7C000000 7C000000 7C000000 FC000000 F8000000 F8000000 F8000000 F8000C00 F8000C00 F8001C00 F8001800 78003800 7C007000 3C00E000 3E01E000 1F878000 0FFF0000 03FC0000> PXLC RP 404 2753 XY F28(Com)S XP /F28 112 21 1 -8 21 18 26 24 0 <078F80 0FDFC0 1DF9E0 19F0E0 39E0F0 31C0F0 01C0F0 03C1F0 03C1E0 0381E0 0381E0 0783E0 0783C0 0783C0 078780 0FCF00 0FFE00 0EF800 0E0000 1E0000 1E0000 1C0000 1C0000 3C0000 FF8000 FF8000> PXLC RP 489 2753 XY F28(p)S -2 x(osition)S 3 x F25(.)S 663 X(If)S 704 X F27 (`)S -15 y F9(\013)S XP /F0 48 14 2 0 12 13 13 16 0 <1E00 6180 4080 C0C0 C0C0 C0C0 C0C0 C0C0 C0C0 C0C0 4080 6180 1E00> PXLC RP 751 2742 XY F0(0)S 11 y 781 X F26(Q)S 828 X F25(and)S 909 X F27(`)S -15 y F9(\013)S 4 y F0(1)S 11 y 985 X F26(P)S 1032 X F25(then)S 1126 X F27(`)S -15 y F9(\013)S 4 y F0(0)S -4 y 2 x F8(+)S F9(\013)S XP /F2 /cmsy5 300 20.8 20.8 128 [0 -20 30 17] PXLNF RP XP /F2 3 15 2 1 12 10 9 16 0 <0C00 0C00 EDC0 7F80 1E00 7F80 EDC0 0C00 0C00> PXLC RP 1236 2726 XY F2(\003)S 21 y 1236 X F0(1)S 6 y 1267 X F26(P)S XP /F27 14 21 2 2 18 18 16 16 0 <07E0 1FF8 3E7C 781E 700E E007 E007 C003 C003 E007 E007 700E 781E 3E7C 1FF8 07E0> PXLC RP 1309 2753 XY F27(\016)S 1339 X F26(Q)S F25(.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup XP /F18 79 30 2 0 27 26 26 32 0 <007F0000 01FFC000 07C1F000 0F007800 1E003C00 3C001E00 38000E00 78000F00 70000700 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 78000F00 78000F00 38000E00 3C001E00 1E003C00 0F007800 07C1F000 01FFC000 007F0000> PXLC RP 692 462 XY F18(Ordina)S XP /F18 108 11 0 0 9 26 26 16 0 PXLC RP 809 462 XY F18(l)S XP /F18 66 27 1 0 24 26 26 24 0 PXLC RP 833 462 XY F18(B)S XP /F18 111 19 1 0 17 16 16 16 0 <07E0 1FF8 3C3C 781E 700E F00F F00F F00F F00F F00F F00F 700E 781E 3C3C 1FF8 07E0> PXLC RP 860 462 XY F18(o)S XP /F18 117 21 1 0 19 16 16 24 0 PXLC RP 879 462 XY F18(und)S XP /F18 115 15 1 0 13 16 16 16 0 <3FE0 7FE0 F0E0 E060 E060 F000 FF80 7FC0 1FE0 01F0 C0F0 E070 E070 F0F0 FFE0 DF80> PXLC RP 943 462 XY F18(s)S XP /F18 102 12 0 0 14 26 26 16 0 <00F8 03FC 07BC 0F3C 0E3C 0E00 0E00 0E00 0E00 0E00 FFC0 FFC0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 971 462 XY F18(for)S XP /F18 80 26 1 0 23 26 26 24 0 PXLC RP 1030 462 XY F18(Progra)S XP /F18 109 32 1 0 30 16 16 32 0 PXLC RP 1144 462 XY F18(ms)S 1583 X F25(383)S 553 Y 362 X F27(\017)S 404 X F28(Constructors)S 3 x F25(.)S 689 X(If)S 737 X F27(8)S F26(\030)S 2 x F27(8)S F26(Q)S F25(\()S F27(j)S 857 X F25(=)S -15 y F9(\030)S 15 y 930 X F26(Q)S 984 X F27(#)S 1027 X F25(=)S 1052 X F27(\))S 1116 X(`)S -15 y F9(\013)S 5 y F1(\030)S 10 y 1203 X F26(C)S 3 x F25 (\()S F26(Q)S F25(\)\))S 1337 X(and)S 1424 X F27(`)S -15 y F9(\013)S 4 y F0(0)S 11 y 1512 X F26(P)S 1564 X F25(then)S 50 y 404 X F27(`)S -15 y F9(\013)S 15 y 464 X F26(C)S 3 x F25(\()S F26(P)S 546 X F25 (\).)S 693 Y 375 X F28(Note)S 3 x F25(.)S 502 X(In)S 557 X(the)S 631 X(comp)S 1 x(osition)S 866 X(rule)S 952 X(the)S 1027 X(ordinal)S 1170 X(b)S 1 x(ound)S 1301 X F26(\013)S 6 y F8(0)S -6 y 1358 X F25 (+)S 1401 X F26(\013)S 6 y F8(1)S -6 y 1463 X F25(w)S -1 x(ould)S 1587 X(ob-)S 50 y 300 X(viously)S 447 X(b)S 1 x(e)S 511 X(the)S 589 X(most)S 697 X(natural)S 849 X(and)S 936 X(desirable.)S 1145 X (Ho)S -1 x(w)S -1 x(ev)S -1 x(er,)S 1332 X(tec)S -1 x(hnical)S 1513 X (reasons)S 50 y 300 X(concerning)S 511 X(the)S 586 X(pro)S 1 x(of)S 697 X(of)S 748 X(the)S 823 X(second)S 960 X(collapsing)S 1155 X (lemma)S 1292 X(b)S 1 x(elo)S -1 x(w)S 1413 X(necessitate)S 1624 X (a)S 50 y 300 X(mo)S 1 x(di\014cation)S 540 X(to)S 593 X F26(\013)S 6 y F8(0)S -6 y 648 X F25(+)S 690 X F26(\013)S XP /F10 3 17 2 1 13 14 13 16 0 <0400 0400 0400 C460 E4E0 3F80 0E00 3F80 E4E0 C460 0400 0400 0400> PXLC RP 717 828 XY F10(\003)S 25 y 717 X F8(1)S -10 y 752 X F25(where)S 873 X F26(\013)S -15 y F10(\003)S 25 y 900 X F8(1)S -10 y 933 X F25 (=)S 980 X F27(h)S F26(\015)S 6 y F9(\030)S -6 y 3 x F27(i)S 1067 X F25(with)S 1163 X F26(\015)S 6 y F8(0)S -6 y 1217 X F25(=)S 1264 X F26(\013)S 6 y F8(1)S -6 y 1324 X F25(and)S 1406 X F26(\015)S 6 y F9(\030)S -6 y 1460 X F25(=)S 1506 X F26(\030)S 1542 X F25(on)S 1601 X(all)S 49 y 300 X F26(\030)S XP /F27 54 0 6 -10 25 30 40 24 0 <000060 0000E0 0000E0 0001C0 0001C0 000180 000380 000380 000700 000700 000E00 000E00 001C00 001C00 001800 003800 003800 007000 007000 00E000 00E000 01C000 01C000 038000 038000 030000 070000 070000 0E0000 0E0000 1C0000 1C0000 380000 380000 300000 700000 700000 E00000 E00000 C00000> PXLC RP 334 892 XY F27(6)S F25(=)S 381 X(0.)S 437 X(Clearly)S 583 X(this)S 666 X(is)S 709 X(a)S 746 X(rather)S 872 X(strange,)S 1031 X (\\non{monotone")S 1354 X(sequence,)S 1539 X(but)S 1617 X(it)S 50 y 300 X(satis\014es)S 455 X(exactly)S 601 X(the)S 675 X(righ)S -1 x (t)S 778 X(iden)S -1 x(tit)S -1 x(y)S 933 X(with)S 1030 X(resp)S 1 x(ec)S 3 x(t)S 1175 X(to)S 1228 X(the)S 1302 X(collapsing)S 1496 X (function)S XP /F26 68 34 2 0 33 28 28 32 0 <03FFFF80 03FFFFE0 003E01F0 003C00F8 003C007C 003C003C 007C003C 0078003E 0078003E 0078003E 00F8003E 00F0003E 00F0003E 00F0003E 01F0007C 01E0007C 01E0007C 01E00078 03E000F8 03C000F0 03C001F0 03C001E0 07C003C0 07800780 07801F00 0F807C00 FFFFF800 FFFFC000> PXLC RP 300 992 XY F26(D)S 349 X F25(de\014ned)S 492 X(b)S 1 x(elo)S -1 x (w.)S 626 X(Other)S 746 X(more)S 850 X(natural)S 995 X(c)S -1 x (hoices)S 1135 X(are)S 1204 X(p)S 1 x(ossible,)S 1372 X(for)S 1435 X (example)S 1598 X(w)S -1 x(e)S 50 y 300 X(could)S 411 X(instead)S 554 X(tak)S -1 x(e)S 644 X F26(\013)S -15 y F10(\003)S 25 y 671 X F8(1)S -10 y 701 X F25(=)S 745 X F26(\013)S 6 y F8(1)S -6 y 799 X F25(+)S 841 X F27(h)S F26(\030)S 2 x F27(i)S F25(.)S 50 y 375 X F28 (Note)S 3 x F25(.)S 492 X(The)S 577 X(uniformit)S -1 x(y)S 780 X (clause)S 901 X(on)S 958 X F26(\013)S 997 X F25(in)S 1044 X(the)S 1115 X(constructor)S 1333 X(rule)S 1415 X(is)S 1456 X(predicated)S 49 y 300 X(up)S 1 x(on)S 406 X(the)S 478 X(assumption)S 698 X F27 (j)S 703 X F25(=)S -15 y F9(\030)S 15 y 766 X F26(Q)S 811 X F27(#)S 846 X F25(and)S 928 X(not)S 1002 X F27(`)S -15 y F9(\030)S 15 y 1058 X F26(Q)S F25(.)S 1123 X(The)S 1209 X(reason)S 1339 X(is)S 1382 X(that)S 1472 X(if)S 1511 X F27(j)S 1516 X F25(=)S -15 y F9 (\030)S 15 y 1579 X F26(Q)S 1624 X F27(#)S 50 y 300 X F25(w)S -1 x (ere)S 394 X(replaced)S 555 X(b)S -1 x(y)S 611 X F27(`)S -15 y F9 (\030)S 15 y 666 X F26(Q)S 711 X F25(the)S 780 X(inductiv)S -1 x(e)S 958 X(de\014nition)S 1140 X(w)S -1 x(ould)S 1259 X(not)S 1331 X(b)S 1 x(e)S 1385 X(\\p)S 1 x(ositiv)S -1 x(e")S 1578 X(and)S 50 y 300 X (w)S -1 x(ould)S 424 X(not)S 501 X(mak)S -1 x(e)S 612 X(go)S 1 x(o)S 1 x(d)S 717 X(sense.)S 849 X(So)S 909 X(w)S -1 x(e)S 973 X(are)S 1046 X(forced)S 1172 X(to)S 1226 X(adopt)S 1347 X(t)S -1 x(w)S -1 x (o)S 1428 X(stages:)S 1573 X(\014rst)S 50 y 300 X(de\014ne)S 420 X F27(j)S 425 X F25(=)S -15 y F9(\030)S 15 y 489 X F25(and)S 570 X (then)S 664 X(de\014ne)S 784 X F27(`)S -15 y F9(\013)S 15 y 2 x F25 (.)S 50 y 375 X F28(Note)S 3 x F25(.)S 492 X(The)S 576 X (constructor{rule)S 885 X(is)S 925 X(nothing)S 1076 X(other)S 1183 X (than)S 1279 X(Buc)S -1 x(hholz)S XP /F25 39 12 3 16 9 29 13 8 0 <70 F8 FC FC 7C 0C 0C 1C 18 38 70 F0 60> PXLC RP 1445 1341 XY F25('s)S 1486 X F26(!)S -15 y 1 x F8(+)S 15 y 3 x F25 ({)S XP /F25 82 31 2 -1 30 28 29 32 0 PXLC RP 1561 1341 XY F25(Rule)S 50 y 300 X(\(cf.)S 370 X([)S XP /F25 53 21 2 -1 18 28 29 16 0 <301C 3FFC 3FF8 3FF0 3FC0 3000 3000 3000 3000 3000 33E0 3FF8 3E3C 381E 381E 300E 000F 000F 000F 000F F00F F00F F00F F01E E01E 703C 7878 3FF0 0FC0> PXLC RP 381 1391 XY F25(5,)S 425 X(p.)S 471 X(1)S XP /F25 57 21 2 -1 18 28 29 16 0 <07E0 1FF0 3E78 381C 781C 700E F00E F00E F00F F00F F00F F00F F00F F01F 701F 781F 3C3F 1FEF 0FCF 010F 000E 001E 781E 781C 783C 7878 78F0 3FE0 1F80> PXLC RP 492 1391 XY F25(95],)S 568 X(where)S 685 X(it)S 724 X(is)S 763 X (called)S 878 X(\012{Rule\),)S 1051 X(dev)S -1 x(elop)S 1 x(ed)S 1241 X(originally)S 1422 X(in)S 1468 X(his)S 1530 X(pro)S 1 x(of{)S 49 y 300 X(theoretic)S 469 X(analysis)S 620 X(of)S 663 X(inductiv)S -1 x(e)S 838 X(de\014nitions,)S 1047 X(but)S 1119 X(it)S 1156 X (app)S 1 x(ears)S 1305 X(here)S 1391 X(in)S 1435 X(a)S 1465 X (somewhat)S 50 y 300 X(simpler)S 446 X(con)S -1 x(text.)S 609 X (What)S 726 X(follo)S -1 x(ws)S 862 X(is)S 904 X(a)S 938 X (fundamen)S -1 x(tal)S 1178 X(idea)S 1266 X(in)S 1314 X(his)S 1379 X (analysis.)S 50 y 375 X(The)S 459 X(p)S 1 x(oin)S -1 x(t)S 566 X (is)S 607 X(that)S 696 X(once)S 789 X(w)S -1 x(e)S 849 X(ha)S -1 x (v)S -1 x(e)S 943 X(deriv)S -1 x(ed)S 1088 X(a)S 1121 X(program)S 1286 X F26(P)S 1331 X F25(with)S 1425 X(uncoun)S -1 x(table)S 50 y 300 X(ordinal)S 441 X(b)S 1 x(ound)S 569 X F26(\013)S 607 X F27(2)S 646 X F25(\012)S 6 y F8(2)S 54 y 926 X F27(`)S -17 y F9(\013)S 17 y 987 X F26(P)S 1736 Y 300 X F25(w)S -1 x(e)S 365 X(kno)S -1 x(w)S 477 X(that)S 570 X(it)S 615 X(ough)S -1 x(t)S 735 X(to)S 789 X(b)S 1 x(e)S 849 X(p)S 1 x(ossible)S 1009 X(to)S 1063 X(collapse)S 1221 X F26(\013)S 1265 X F25(do)S -1 x(wn)S 1378 X(to)S 1433 X(a)S 1471 X (coun)S -1 x(table)S 50 y 300 X(ordinal)S 441 X(b)S 1 x(ound)S 569 X F26(D)S 1 x(\013)S 643 X F27(2)S 682 X F25(\012)S 6 y F8(1)S -6 y 744 X F25(with)S 1884 Y 879 X F27(j)S 884 X F25(=)S XP /F9 68 27 1 0 24 20 20 24 0 <07FFE0 07FFF0 01E078 01C03C 01C01C 03C01E 03C01E 03801E 03801E 07801E 07803C 07003C 07003C 0F0038 0F0070 0E00F0 0E01E0 1E07C0 FFFF00 FFFC00> PXLC RP 915 1867 XY F9(D)S 1 x(\013)S 17 y 979 X F26(P)S 1023 X F27(#)S 1055 X F26(:)S 1982 Y 300 X F25(The)S 386 X(reason)S 517 X(is)S 560 X(simple)S XP /F25 124 42 0 10 41 12 2 48 0 PXLC RP 690 1982 XY F25(|)S 746 X(there)S 853 X(can)S 931 X(only)S 1023 X (b)S 1 x(e)S 1080 X(coun)S -1 x(tably{man)S -1 x(y)S 1392 X(di)S XP /F25 11 24 0 0 26 29 29 32 0 <003F1F80 01FFFFC0 03C3F3C0 0783E3C0 0F03E3C0 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 FFFFFC00 FFFFFC00 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 7F87FC00 7F87FC00> PXLC RP 1427 1982 XY F25(\013eren)S -1 x(t)S 1557 X(com-)S 50 y 300 X (putations)S 485 X(of)S 532 X F26(P)S 565 X F25(,)S 590 X(so)S 641 X (there)S 747 X(has)S 821 X(to)S 872 X(b)S 1 x(e)S 928 X(a)S 963 X (coun)S -1 x(table)S 1151 X(upp)S 1 x(er)S 1270 X(b)S 1 x(ound)S XP /F25 33 12 3 0 8 30 30 8 0 <70 F8 F8 F8 F8 F8 F8 F8 F8 70 70 70 70 70 70 70 70 70 70 70 70 00 00 00 00 70 F8 F8 F8 70> PXLC RP 1384 2032 XY F25(!)S XP /F25 75 32 2 0 30 28 28 32 0 PXLC RP 375 2082 XY F25(Kno)S -1 x(wing)S 551 X(only)S 643 X(that)S 735 X (suc)S -1 x(h)S 831 X(a)S 867 X(collapsing)S 1060 X(map)S 1154 X F26 (D)S 1 x F25(:)S 1208 X(\012)S 6 y F8(2)S -6 y 1271 X F27(!)S 1327 X F25(\012)S 6 y F8(1)S -6 y 1391 X F25(exists,)S 1520 X(w)S -1 x(e)S 1583 X(can)S 50 y 300 X(more)S 405 X(or)S 458 X(less)S 536 X(read)S 630 X(o\013)S 690 X(its)S 750 X(de\014nition)S 936 X(from)S 1035 X (the)S 1108 X(ab)S 1 x(o)S -1 x(v)S -1 x(e)S 1227 X(rules)S 1328 X (for)S 1393 X F27(`)S -15 y F9(\013)S 15 y 1457 X F26(P)S 1490 X F25 (.)S 1524 X(In)S 1577 X(fact)S 50 y 300 X(w)S -1 x(e)S 359 X (de\014ne)S 477 X(\014rst)S 561 X(a)S 594 X(binary)S 723 X(v)S -1 x (ersion)S 862 X(of)S 907 X F26(D)S 1 x F25(:)S 961 X(\012)S 6 y F8 (2)S XP /F27 2 32 6 0 25 20 20 24 0 PXLC RP 1016 2182 XY F27(\002)S 1054 X F25(\012)S 6 y F8(1)S -6 y 1114 X F27 (!)S 1167 X F25(\012)S 6 y F8(1)S -6 y 1 x F25(.)S 1245 X(The)S 1328 X(second)S 1460 X(parameter)S 49 y 300 X F26(\014)S 341 X F25 (in)S 391 X F26(D)S 1 x F25(\()S F26(\013;)S 488 X(\014)S 2 x F25 (\))S 546 X(is)S 589 X(to)S 642 X(b)S 1 x(e)S 700 X(though)S -1 x (t)S 857 X(of)S 906 X(as)S 959 X(a)S 995 X(b)S 1 x(ound)S 1125 X (on)S 1184 X(an)S -1 x(y)S 1264 X(p)S 1 x(oten)S -1 x(tial)S 1441 X (input.)S 1573 X(The)S 50 y 300 X(conditions)S 499 X(on)S 556 X F26 (D)S 606 X F25(are)S 675 X(as)S 726 X(follo)S -1 x(ws:)S 2371 Y 362 X F27(\017)S 404 X F28(Basic)S 2 x F25(.)S 535 X F26(D)S 1 x F25 (\(1)S F26(;)S 626 X(\014)S 2 x F25(\))S 680 X(=)S 723 X F26(\014)S 758 X F25(+)S 800 X(1)S 2469 Y 362 X F27(\017)S 404 X F28(Comp)S -2 x(osition)S XP /F28 123 21 4 10 22 12 2 24 0 PXLC RP 630 2469 XY F28({)S XP /F28 48 21 4 -1 23 28 29 24 0 <003E00 00FF80 01E3C0 0381C0 0701C0 0701C0 0F01E0 0E01E0 1E01C0 1E01C0 1C03C0 3C03C0 3C03C0 3C03C0 780780 780780 780780 780780 700F00 F00F00 F00E00 F01E00 F01E00 F01C00 F03800 707800 78F000 3FC000 1F0000> PXLC RP 651 2469 XY F28(0)S 678 X F25(.)S 708 X F26(D)S 1 x F25(\()S F26 (\013)S 6 y F8(0)S -6 y 814 X F25(+)S 855 X F26(\013)S 6 y F8(1)S -6 y 2 x F26(;)S 920 X(\014)S 2 x F25(\))S 972 X(=)S 1016 X F26(D)S 1 x F25(\()S F26(\013)S 6 y F8(1)S -6 y 2 x F26(;)S 1132 X(D)S 1 x F25(\()S F26(\013)S 6 y F8(0)S -6 y 1 x F26(;)S 1247 X(\014)S 2 x F25(\)\))S 2566 Y 362 X F27(\017)S 404 X F28(Comp)S -2 x(osition{)S XP /F28 49 21 4 0 19 28 28 16 0 <0006 000E 000E 003C 007C 03FC 07FC 0038 0038 0078 0078 0070 0070 00F0 00F0 00E0 00E0 01E0 01E0 01C0 01C0 03C0 03C0 0380 0380 0780 FFF8 FFF8> PXLC RP 651 2566 XY F28(1)S 678 X F25(.)S 708 X F26(D)S 1 x F25(\()S F26 (\013)S 6 y F8(0)S -6 y 814 X F25(+)S 855 X F26(\013)S -15 y F10 (\003)S 25 y 882 X F8(1)S -10 y 2 x F26(;)S 920 X(\014)S 2 x F25 (\))S 973 X(=)S 1017 X F26(D)S 1 x F25(\()S F26(\013)S 6 y F8(0)S -6 y 2 x F26(;)S 1133 X(\014)S 2 x F25(\))S 1183 X(+)S 1225 X F26 (D)S 1 x F25(\()S F26(\013)S 6 y F8(1)S -6 y 1 x F26(;)S 1340 X(D)S 1 x F25(\()S F26(\013)S 6 y F8(0)S -6 y 2 x F26(;)S 1456 X(\014)S 2 x F25(\)\))S 2663 Y 362 X F27(\017)S 404 X F28(Constructors)S 3 x F25(.)S 671 X F26(D)S 1 x F25(\()S F26(\013;)S 768 X(\014)S 2 x F25 (\))S 821 X(=)S 865 X F26(D)S 1 x F25(\()S F26(\013)S 8 y F9(D)S XP /F8 40 13 2 -8 10 22 30 8 0 <03 06 0E 1C 18 38 30 70 70 60 E0 E0 E0 E0 E0 E0 E0 E0 E0 E0 60 70 70 30 38 18 1C 0E 06 03> PXLC RP 971 2671 XY F8(\()S F9(\013)S 4 y F0(0)S XP /F9 59 10 2 -6 6 4 10 8 0 PXLC RP 1022 2671 XY F9(;\014)S XP /F8 41 13 2 -8 10 22 30 8 0 PXLC RP 1052 2671 XY F8(\))S -8 y 2 x F26(;)S 1086 X(\014)S 2 x F25(\))S 2753 Y 300 X(It)S 343 X(is)S 383 X(easy)S 472 X(to)S 521 X(c)S -1 x (hec)S -1 x(k)S 631 X(that)S 719 X(the)S 789 X(follo)S -1 x(wi)S -3 x(ng)S 962 X(recursiv)S -1 x(e)S 1134 X(de\014nition)S 1316 X (of)S 1362 X F26(D)S 1409 X F25(do)S 1 x(es)S 1501 X(actually)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 129 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(38)S XP /F25 52 21 1 0 19 28 28 24 0 <001C00 001C00 003C00 007C00 007C00 00DC00 01DC00 019C00 039C00 071C00 061C00 0E1C00 0C1C00 181C00 381C00 301C00 601C00 E01C00 FFFFC0 FFFFC0 001C00 001C00 001C00 001C00 001C00 001C00 01FFC0 01FFC0> PXLC RP 342 462 XY F25(4)S 700 X F18(H.)S 752 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S 1023 X(and)S 1098 X(S.S.)S 1175 X(W)S -3 x(ainer)S 553 Y 300 X F25(satisfy)S 430 X(these)S 536 X(conditions:)S 642 Y 639 X F26(D)S 1 x F25(\(0)S F26(;)S 730 X (\014)S 2 x F25(\))S 784 X(=)S 829 X F26(\014)S 2 x(;)S 50 y 562 X (D)S 1 x F25(\()S F26(\013)S 649 X F25(+)S 691 X(1)S F26(;)S 731 X (\014)S 2 x F25(\))S 784 X(=)S 829 X F26(D)S 1 x F25(\()S F26 (\013;)S 926 X(\014)S 2 x F25(\))S 977 X(+)S 1018 X(1)S F26(;)S 50 y 633 X(D)S 1 x F25(\()S F26(\013;)S 730 X(\014)S 2 x F25(\))S 784 X (=)S 829 X F27(h)S F26(D)S 1 x F25(\()S F26(\013)S 6 y F9(x)S -6 y 2 x F26(;)S 963 X(\014)S 2 x F25(\))S F27(i)S 1104 X F25(if)S 1142 X F26(\013)S 1180 X F25(=)S 1224 X F27(h)S F26(\013)S 6 y F9(x)S -6 y 1 x F27(i)S F26(;)S 50 y 633 X(D)S 1 x F25(\()S F26(\013;)S 730 X (\014)S 2 x F25(\))S 784 X(=)S 829 X F26(D)S 1 x F25(\()S F26(\013)S 7 y F9(D)S 1 x F8(\()S F9(\013)S 5 y F0(0)S -5 y 2 x F9(;\014)S 1 x F8(\))S -7 y 2 x F26(;)S 1050 X(\014)S 2 x F25(\))S 1175 X(if)S 1213 X F26(\013)S 1251 X F25(=)S 1295 X F27(h)S F26(\013)S 6 y F9 (\030)S -6 y 3 x F27(i)S F26(:)S 884 Y 300 X F25(If)S 344 X(w)S -1 x (e)S 408 X(no)S -1 x(w)S 498 X(put)S 577 X F26(D)S 1 x(\013)S 655 X F25(=)S 704 X F26(D)S 1 x F25(\()S F26(\013;)S 801 X(!)S 1 x F25 (\))S 861 X(then)S 958 X F26(D)S 1 x(\013)S 1037 X F25(is)S 1082 X (exactly)S 1227 X(the)S 1302 X(collapsing)S 1496 X(function)S 50 y 300 X(in)S -1 x(tro)S 1 x(duced)S 514 X(b)S -1 x(y)S 578 X(Buc)S -1 x(hholz)S 764 X(in)S 819 X(c)S -1 x(hapter)S 974 X(I)S XP /F25 86 31 1 -1 29 28 29 32 0 PXLC RP 989 934 XY F25(V)S 1040 X(of)S 1093 X([5],)S 1170 X(page)S 1273 X (195,)S 1368 X(with)S 1469 X(the)S 1547 X(sligh)S -1 x(t)S 50 y 300 X(di\013erence)S 490 X(that)S 583 X(he)S 643 X(starts)S 762 X (with)S 861 X F26(D)S 1 x F25(0)S 935 X(=)S 985 X(0)S 1024 X (rather)S 1152 X(than)S 1253 X(our)S 1331 X F26(D)S 1 x F25(0)S 1405 X(=)S 1456 X F26(!)S 1 x F25(,)S 1513 X(and)S 1598 X(w)S -1 x (e)S 50 y 300 X(obtain)S 1126 Y 375 X F30(1.)S XP /F30 56 24 2 0 21 27 27 24 0 <03FC00 0FFF00 3FFF80 3E0FC0 7807C0 7803C0 7C03C0 7E03C0 7F87C0 7FEF80 3FFF00 3FFE00 1FFF00 0FFF80 3FFFC0 7CFFC0 783FE0 F80FE0 F003E0 F001E0 F001E0 F801E0 F803C0 7E0FC0 3FFF80 1FFF00 07FC00> PXLC RP 412 1126 XY F30(8.)S XP /F30 83 27 3 0 23 28 28 24 0 <07F860 1FFFE0 3FFFE0 7E0FE0 7C03E0 F801E0 F800E0 F800E0 FC0000 FE0000 FFF000 7FFE00 7FFF80 3FFFC0 1FFFE0 07FFE0 007FF0 0007F0 0003F0 0001F0 E001F0 E001F0 F001F0 F803E0 FE07E0 FFFFC0 FFFF80 C3FE00> PXLC RP 465 1126 XY F30(Se)S XP /F30 99 21 2 0 19 18 18 24 0 <03FC00 1FFE00 3F3F00 7E3F00 7E3F00 FC3F00 FC1E00 FC0000 FC0000 FC0000 FC0000 FC0000 FC0000 7E0380 7F0780 3F8F00 1FFE00 03F800> PXLC RP 513 1126 XY F30(con)S XP /F30 100 27 2 0 25 29 29 24 0 <000FF0 000FF0 000FF0 0001F0 0001F0 0001F0 0001F0 0001F0 0001F0 0001F0 0001F0 07F9F0 1FFFF0 3F0FF0 7E03F0 7E01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 7C01F0 7E03F0 3F0FFE 1FFFFE 07F9FE> PXLC RP 585 1126 XY F30(d)S 627 X(Collapsing)S 860 X(Lemma.)S 1218 Y 781 X F27(`)S -17 y F9(\013)S 17 y 842 X F26(P)S 897 X F25(=)S 922 X F27 (\))S 976 X(j)S 981 X F25(=)S -17 y F9(D)S 1 x(\013)S 17 y 1076 X F26(P)S 1120 X F27(#)S 1152 X F26(:)S XP /F30 80 33 2 0 29 28 28 32 0 PXLC RP 375 1310 XY F30(Pro)S 1 x(o)S XP /F30 102 15 0 0 19 29 29 24 0 <003F80 01FFC0 03F7E0 07C7E0 0FC7E0 0F87E0 0F83C0 0F8000 0F8000 0F8000 0F8000 FFF800 FFF800 FFF800 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 7FF800 7FF800 7FF800> PXLC RP 476 1310 XY F30(f.)S 525 X F25(W)S -3 x(e)S 593 X(in)S 638 X(fact)S 717 X(need)S 810 X(to)S 858 X(pro)S -1 x(v)S -1 x(e)S 967 X (something)S 1162 X(rather)S 1284 X(more)S 1384 X(general.)S 1542 X (What)S 50 y 300 X(w)S -1 x(e)S 361 X(shall)S 459 X(sho)S -1 x(w)S 562 X(is)S 604 X(the)S 675 X(follo)S -1 x(wing.)S 867 X(If)S 909 X F27(`)S -15 y F9(\013)S 15 y 970 X F26(P)S 1017 X F25(then)S 1111 X (for)S 1175 X(all)S 1233 X F26(n)S 1272 X F25(and)S 1353 X(for)S 1417 X(all)S 1475 X F26(\014)S XP /F27 21 32 3 -8 28 26 34 32 0 PXLC RP 1512 1360 XY F27(\025)S 1556 X F26(!)S 1598 X F25(w)S -1 x(e)S 50 y 300 X(ha)S -1 x(v)S -1 x(e)S 1493 Y 330 X(\(1\))S 404 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\()S F26(n)S F25(\):)S 535 X F26(N)S 584 X F27(j)S 589 X F25(=)S -15 y F9(D)S 1 x F8(\()S F9(\013;\014)S 1 x F8(\))S 15 y 740 X F26(P)S 784 X F27(#)S 819 X F25(and)S 1578 Y 330 X(\(2\))S 404 X(for)S 463 X(eac)S -1 x(h)S 552 X(input)S 659 X F26(a)S 690 X F25(of)S 734 X(size)S 808 X F27(\024)S 852 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\()S F26(n)S F25(\))S 974 X(the)S 1041 X (output)S 1173 X F26(P)S 1206 X(a)S 1237 X F25(is)S 1274 X(of)S 1318 X(size)S 1392 X F27(\024)S 1436 X F26(G)S 7 y F9(D)S 1 x F8 (\()S F9(\013;\014)S 1 x F8(\))S -7 y 1 x F25(\()S F26(n)S F25(\).)S 1661 Y 300 X(The)S 387 X(result)S 504 X(then)S 600 X(follo)S -1 x (ws)S 738 X(immediately)S 976 X(from)S 1076 X(\(1\))S 1145 X(b)S -1 x(y)S 1204 X(putting)S 1353 X F26(\014)S 1393 X F25(=)S 1440 X F26(!)S 1 x F25(,)S 1494 X(since)S 1598 X(w)S -1 x(e)S 50 y 300 X (tak)S -1 x(e)S 395 X F26(!)S 441 X F25(:=)S 505 X F27(h)S F26(x)S F27(i)S 579 X F25(and)S 665 X(hence)S 785 X F26(G)S XP /F9 33 21 0 0 18 13 13 24 0 <1801C0 1801C0 3801C0 7000C0 60E0C0 60E0C0 60E1C0 E1C180 E1C380 F7EF00 7FFF00 7FFE00 3E7800> PXLC RP 817 1717 XY F9(!)S -6 y 3 x F25(\()S F26(n)S F25(\))S 918 X(=)S 970 X F26(n)S F25(.)S 1039 X(W)S -3 x(e)S 1115 X(pro)S 1 x(ceed)S 1273 X(b)S -1 x(y)S 1336 X(induction)S 1525 X(on)S 1587 X(the)S 50 y 300 X(deriv)S -2 x(ation)S 495 X F27(`)S -15 y F9(\013)S 15 y 556 X F26(P)S 589 X F25(.)S 618 X(There)S 738 X(are)S 807 X(three)S 913 X (cases.)S 50 y 375 X F28(Basic)S 2 x F25(.)S 509 X(Supp)S 1 x(ose)S 673 X F27(`)S -15 y F8(1)S 15 y 730 X F26(B)S 778 X F25(and)S 860 X (let)S 921 X F26(n)S 960 X F25(and)S 1042 X F26(\014)S 1081 X F27 (\025)S 1126 X F26(!)S 1168 X F25(b)S 1 x(e)S 1226 X(giv)S -1 x (en.)S 1353 X(The)S 1439 X(assumption)S 49 y 300 X(on)S 356 X (basic)S 459 X(programs)S 641 X(is)S 681 X(that)S 770 X(giv)S -1 x (en)S 877 X(an)S 933 X(input)S 1043 X(of)S 1089 X(size)S 1166 X F27 (\024)S 1210 X F26(G)S 7 y F9(\014)S -7 y 3 x F25(\()S F26(n)S F25 (\),)S 1346 X(the)S 1417 X(computation)S 50 y 300 X(terminates)S 508 X(in)S 559 X(one)S 638 X(step)S 728 X(with)S 825 X(output)S 963 X(of)S 1013 X(size)S 1094 X F27(\024)S 1142 X F26(G)S 7 y F9 (\014)S -7 y 3 x F25(\()S F26(n)S F25(\))S 1265 X(+)S 1308 X(1.)S 1366 X(Therefore)S 1557 X(since)S 50 y 300 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\()S F26(n)S F25(\))S 422 X(+)S 464 X(1)S 497 X(=)S 542 X F26(G)S 6 y F9(\014)S 1 x F8(+1)S -6 y 2 x F25(\()S F26(n)S F25(\))S 711 X(w)S -1 x(e)S 772 X(ha)S -1 x(v)S -1 x(e)S 869 X F26 (G)S 6 y F9(\014)S -6 y 3 x F25(\()S F26(n)S F25(\):)S 1000 X F26 (N)S 1050 X F27(j)S 1055 X F25(=)S -15 y F9(\014)S 1 x F8(+1)S 15 y 1164 X F26(B)S 1210 X F27(#)S F25(.)S 1262 X(But)S 1345 X F26(D)S 1 x F25(\(1)S F26(;)S 1436 X(\014)S 2 x F25(\))S 1491 X(=)S 1535 X F26(\014)S 1571 X F25(+)S 1613 X(1.)S 50 y 300 X(Hence)S 423 X(the)S 495 X(result.)S 50 y 375 X F28(Comp)S -2 x(osition)S 3 x F25(.)S 633 X(Supp)S 1 x(ose)S 795 X F27(`)S -15 y F9(\013)S 4 y F0(0)S -4 y 2 x F8(+)S F9(\013)S -13 y F2(\003)S 21 y 905 X F0(1)S 7 y 936 X F26 (P)S 975 X F27(\016)S 1002 X F26(Q)S 1047 X F25(arises)S 1159 X (from)S 1255 X F27(`)S -15 y F9(\013)S 4 y F0(0)S 11 y 1332 X F26 (Q)S 1377 X F25(and)S 1457 X F27(`)S -15 y F9(\013)S 4 y F0(1)S 11 y 1533 X F26(P)S 1578 X F25(and)S 50 y 300 X(let)S 360 X F26(n)S 399 X F25(and)S 479 X F26(\014)S 517 X F27(\025)S 560 X F26(!)S 602 X F25 (b)S 1 x(e)S 658 X(giv)S -1 x(en.)S 783 X(By)S 848 X(one)S 924 X (application)S 1138 X(of)S 1186 X(the)S 1257 X(induction)S 1442 X (h)S -1 x(yp)S 1 x(othesis)S 2201 Y 764 X F26(G)S 7 y F9(\014)S -7 y 3 x F25(\()S F26(n)S F25(\):)S 895 X F26(N)S 944 X F27(j)S 949 X F25 (=)S -17 y F9(D)S 1 x F8(\()S F9(\013)S 5 y F0(0)S -5 y 1 x F9(;)S (\014)S 1 x F8(\))S 17 y 1116 X F26(Q)S 1161 X F27(#)S 2293 Y 300 X F25(and)S 379 X(on)S 435 X(input)S 544 X F26(a)S 578 X F25(of)S 624 X(size)S 701 X F27(\024)S 744 X F26(G)S 7 y F9(\014)S -7 y 800 X F25(\()S F26(n)S F25(\))S 869 X(the)S 939 X(output)S 1073 X F26(b)S 1102 X F25(=)S 1146 X F26(Qa)S 1213 X F25(is)S 1253 X(of)S 1299 X (size)S 1376 X F27(\024)S 1420 X F26(G)S 8 y F9(D)S 1 x F8(\()S F9 (\013)S 4 y F0(0)S -4 y 1 x F9(;\014)S 1 x F8(\))S -8 y 2 x F25(\()S F26(n)S F25(\).)S 50 y 300 X(No)S -1 x(w)S 399 X(w)S -1 x(e)S 464 X (can)S 545 X(use)S 621 X(the)S 696 X(induction)S 885 X(h)S -1 x(yp)S 1 x(othesis)S 1094 X(on)S 1156 X F27(`)S -15 y F9(\013)S 4 y F0(1)S 11 y 1240 X F26(P)S 1290 X F25(with)S 1389 X F26(\014)S 1433 X F25 (replaced)S 1601 X(b)S -1 x(y)S 50 y 300 X F26(D)S 1 x F25(\()S F26 (\013)S 6 y F8(0)S -6 y 2 x F26(;)S 416 X(\014)S 2 x F25(\))S 471 X (to)S 522 X(obtain)S 2485 Y 662 X F26(G)S 8 y F9(D)S 1 x F8(\()S F9 (\013)S 4 y F0(0)S -4 y 2 x F9(;\014)S 1 x F8(\))S -8 y 2 x F25(\()S F26(n)S F25(\):)S 895 X F26(N)S 944 X 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F26(:)S 75 y 300 X F25(The)S 385 X(result)S 501 X(then)S 596 X(follo)S -1 x(ws)S 732 X(b)S -1 x(y)S 789 X(prop)S 1 x(ert)S -1 x(y)S 959 X(Comp)S 1 x(osition{1)S 1244 X(of)S 1292 X (the)S 1363 X F26(D)S 1 x F25({function.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 129 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup 692 462 XY F18(Ordinal)S 833 X(Bounds)S 971 X(for)S 1030 X (Programs)S 1583 X F25(385)S 553 Y 375 X F28(Constructors)S 3 x F25 (.)S 643 X(Supp)S 1 x(ose)S 806 X F27(`)S -15 y F9(\013)S 15 y 867 X F26(C)S 3 x F25(\()S F26(P)S 949 X F25(\))S 979 X(arises)S 1093 X (from)S 1191 X F27(`)S -15 y F9(\013)S 4 y F0(0)S 11 y 1269 X F26 (P)S 1316 X F25(and)S 1397 X F27(8)S F26(\030)S 2 x F27(8)S F26(Q)S F25(\()S F27(j)S 1517 X F25(=)S -15 y F9(\030)S 15 y 1579 X F26(Q)S 1624 X F27(#)S 50 y 300 X F25(=)S 325 X F27(\))S 384 X(`)S -15 y F9 (\013)S 4 y F1(\030)S 11 y 467 X F26(C)S 3 x F25(\()S F26(Q)S F25 (\)\),)S 610 X(and)S 694 X(let)S 758 X F26(n)S 800 X F25(and)S 884 X F26(\014)S 927 X F27(\025)S 977 X F26(!)S 1021 X F25(b)S 1 x(e)S 1081 X(giv)S -1 x(en.)S 1216 X(Applying)S 1399 X(the)S 1474 X (induction)S 50 y 300 X(h)S -1 x(yp)S 1 x(othesis)S 505 X(to)S 555 X F27(`)S -15 y F9(\013)S 4 y F0(0)S 11 y 632 X F26(P)S 678 X F25(w)S -1 x(e)S 739 X(obtain)S 733 Y 764 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\()S F26(n)S F25(\):)S 895 X F26(N)S 944 X F27(j)S 949 X F25(=)S -17 y F9(D)S 1 x F8(\()S F9(\013)S 4 y F0(0)S -4 y 2 x F9(;\014)S 1 x F8(\))S 17 y 1116 X F26(P)S 1160 X F27(#)S 813 Y 300 X F25 (where,)S 431 X(on)S 487 X(inputs)S 613 X(of)S 659 X(size)S 736 X F27(\024)S 780 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\()S F26(n)S F25 (\),)S 916 X(the)S 986 X(output)S 1121 X(of)S 1167 X F26(P)S 1212 X F25(is)S 1252 X(of)S 1298 X(size)S 1376 X F27(\024)S 1420 X F26(G)S 8 y F9(D)S 1 x F8(\()S F9(\013)S 4 y F0(0)S -4 y 1 x F9(;\014)S 1 x F8(\))S -8 y 2 x F25(\()S F26(n)S 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F26(N)S 485 X F27(j)S 490 X F25 (=)S -15 y F9(D)S 1 x F8(\()S F9(\013)S 5 y F1(\030)S -5 y 2 x F9 (;)S (\014)S 1 x F8(\))S 15 y 662 X F26(C)S 3 x F25(\()S F26(P)S 744 X F25(\))S 776 X F27(#)S 813 X F25(and)S 897 X(secondly)S -3 x(,)S 1076 X(on)S 1136 X(inputs)S 1266 X(of)S 1316 X(size)S 1398 X F27 (\024)S 1446 X F26(G)S 7 y F9(\014)S -7 y 3 x F25(\()S F26(n)S F25 (\),)S 1587 X(the)S 50 y 300 X(output)S 438 X(giv)S -1 x(en)S 549 X (b)S -1 x(y)S 609 X F26(C)S 3 x F25(\()S F26(P)S 691 X F25(\))S 723 X(is)S 767 X(of)S 817 X(size)S 898 X F27(\024)S 946 X F26(G)S 8 y F9(D)S 1 x F8(\()S F9(\013)S 4 y F1(\030)S -4 y 1 x F9(;\014)S 1 x F8(\))S -8 y 2 x F25(\()S F26(n)S F25(\).)S 1197 X(Since)S 1307 X (with)S 1404 X F26(\030)S 1440 X F25(=)S 1488 X F26(D)S 1 x F25(\()S F26(\013)S 6 y F8(0)S -6 y 2 x F26(;)S 1604 X(\014)S 2 x F25(\))S 50 y 300 X(w)S -1 x(e)S 361 X(ha)S -1 x(v)S -1 x(e)S 457 X F26(D)S 1 x F25(\()S F26(\013)S 6 y F9(\030)S -6 y 3 x F26(;)S 572 X(\014)S 2 x F25(\))S 625 X(=)S 669 X F26(D)S 1 x F25(\()S F26(\013;)S 766 X (\014)S 2 x F25(\),)S 833 X(this)S 914 X(completes)S 1105 X(the)S 1177 X(pro)S 1 x(of.)S 1628 1118 XY 17 17 R 1198 Y 375 X(By)S 441 X (com)S -1 x(bining)S 642 X(the)S 714 X(First)S 817 X(and)S 899 X (Second)S 1040 X(Collapsing)S 1245 X(Lemmas,)S 1422 X(w)S -1 x(e)S 1484 X(therefore)S 50 y 300 X(ha)S -1 x(v)S -1 x(e)S 396 X(a)S 430 X (w)S -1 x(a)S -1 x(y)S 514 X(of)S 562 X(reading)S 709 X(o\013)S 768 X(complexit)S -1 x(y)S 979 X(b)S 1 x(ounds)S 1124 X(for)S 1187 X (programs)S 1370 X(in)S 1418 X F26(L)S F25(:)S 1328 Y 575 X F27(`)S -17 y F9(\013)S 17 y 635 X F26(P)S 691 X F25(=)S 716 X F27(\))S 781 X F26(P)S 827 X F25(has)S 901 X(a)S 935 X(complexit)S -1 x(y)S 1146 X(b)S 1 x(ound)S 1274 X F26(G)S 6 y F9(D)S 1 x(\013)S -6 y 2 x F26(:)S 1408 Y 375 X F25(Of)S 436 X(course)S 566 X(the)S 640 X (tric)S -1 x(k)S 739 X(lies)S 813 X(in)S 864 X(\014nding)S 1005 X (appropriate)S 1231 X(uniform)S 1390 X(measures)S 1570 X F26(\013)S 1613 X F25(=)S 50 y 300 X F27(h)S F26(\013)S 6 y F9(\030)S -6 y 3 x F27(i)S 397 X F25(for)S 467 X F26(L)S F25({constructors.)S 786 X (But)S 875 X(once)S 975 X(this)S 1062 X(is)S 1110 X(done,)S 1228 X (w)S -1 x(e)S 1296 X(ha)S -1 x(v)S -1 x(e)S 1397 X(a)S 1438 X (metho)S 1 x(d)S 1595 X(for)S 50 y 300 X(classifying)S 503 X(the)S 579 X(computational)S 858 X(complexit)S -1 x(y)S 1074 X(of)S 1126 X F26(L)S F25({programs)S 1362 X(in)S 1415 X(terms)S 1535 X(of)S 1587 X (the)S 50 y 300 X F26(G)S 6 y F9(D)S 1 x(\013)S -6 y 1 x F25 ({functions.)S 601 X(These)S 721 X(are)S 791 X(w)S -1 x(ell{kno)S -1 x(wn)S 1014 X(functions)S 1193 X(in)S 1241 X(subrecurs)S 3 x(iv)S -1 x(e)S 1478 X(hierarc)S -1 x(h)S -1 x(y)S 50 y 300 X(theory)S 428 X(|)S 481 X(the)S 550 X(so{called)S 723 X(F)S -3 x(ast)S 812 X (Gro)S -1 x(wing)S 977 X(F)S -3 x(unctions)S 1164 X(|)S 1217 X(and)S 1296 X(they)S 1387 X(pro)S -1 x(vide)S 1532 X(imme-)S 49 y 300 X (diate)S 404 X(access)S 527 X(to)S 577 X(pro)S 1 x(of{theoretic)S 866 X(c)S -1 x(haracterizations)S 1186 X(of)S 1233 X(the)S 1305 X (functions)S 1484 X(de\014nable)S 50 y 300 X(b)S -1 x(y)S 358 X F26 (L)S F25({programs.)S 50 y 375 X(What)S 494 X(w)S -1 x(e)S 558 X (ha)S -1 x(v)S -1 x(e)S 657 X(describ)S 1 x(ed)S 844 X(here)S 937 X (are)S 1009 X(only)S 1103 X(the)S 1178 X(conceptual)S 1388 X (ideas.)S XP /F25 80 28 2 0 25 28 28 24 0 PXLC RP 1517 1757 XY F25(Precise)S 50 y 300 X(results)S 433 X(require)S 573 X(more)S 678 X(detail)S 794 X(concerning)S 1002 X(the)S 1074 X (structure)S 1253 X(of)S 1301 X(the)S 1373 X(language)S 1547 X F26 (L)S F25(.)S 1607 X(In)S 50 y 300 X(the)S 369 X(rest)S 447 X(of)S 492 X(the)S 561 X(pap)S 1 x(er)S 675 X(w)S -1 x(e)S 733 X(w)S -1 x (ork)S 832 X(out)S 903 X(these)S 1007 X(details)S 1136 X(for)S 1197 X (a)S 1229 X(natural)S 1372 X(class)S 1466 X(of)S 1511 X(higher{)S 49 y 300 X(order)S 410 X(programs)S 594 X(based)S 711 X(on)S 771 X (G)S XP /F25 127 21 4 24 16 29 5 16 0 <70E0 F9F0 F9F0 F9F0 70E0> PXLC RP 803 1906 XY F25(\177)S 803 X(odel's)S XP /F25 68 32 2 0 29 28 28 32 0 PXLC RP 920 1906 XY F25(Dialectica)S 1115 X(system)S XP /F26 84 24 1 0 29 28 28 32 0 <1FFFFFF0 3FFFFFF0 3E07C1F0 38078070 70078060 60078060 600F8060 E00F0060 C00F0060 C00F0060 001F0000 001E0000 001E0000 001E0000 003E0000 003C0000 003C0000 003C0000 007C0000 00780000 00780000 00780000 00F80000 00F00000 00F00000 01F00000 7FFF8000 7FFF8000> PXLC RP 1254 1906 XY F26(T)S 1284 X F25(,)S 1311 X(follo)S -1 x(wing)S 1488 X ([11)S -1 x(].)S 1587 X(W)S -3 x(e)S 50 y 300 X(can)S 380 X(ho)S -1 x(w)S -1 x(ev)S -1 x(er)S 543 X(already)S 694 X(see)S 765 X(ho)S -1 x(w)S 855 X(this)S 940 X(analysis)S 1100 X(should)S 1236 X(pro)S 1 x(ceed.)S 1417 X(G\177)S 1450 X(odel's)S 1569 X(T)S 1617 X(is)S 50 y 300 X(based)S 415 X(on)S 472 X(the)S 543 X(primitiv)S -1 x(e)S 719 X(recursion)S 897 X(constructor)S 1116 X(C)S 1159 X(whic)S -1 x (h,)S 1288 X(pro)S -1 x(vided)S 1458 X(with)S 1552 X(a)S 1586 X(n)S -1 x(u-)S 50 y 300 X(merical)S 450 X(input)S XP /F26 109 36 1 0 36 18 18 40 0 <3E0FC1F800 3F3FE3FE00 7778F71E00 67E07E0E00 E7C07C0E00 C780780E00 0780700E00 0F00F01E00 0F00F01C00 0E00E01C00 0E00E03C00 1E01E038E0 1E01E038E0 1C01C078C0 1C01C071C0 3C03C07B80 3C03C03F00 3803801E00> PXLC RP 565 2056 XY F26(m)S F25(,)S 632 X(mak)S -1 x(es)S 762 X(exactly)S 909 X F26(m)S 964 X F25(recursiv)S -1 x(e)S 1142 X(calls)S 1238 X (on)S 1301 X(an)S -1 x(y)S 1383 X(giv)S -1 x(en)S 1496 X(function)S 50 y 300 X(parameter.)S 524 X(Th)S -1 x(us)S 632 X(if)S XP /F25 81 32 2 -8 29 29 37 32 0 <003F8000 00FFE000 03E0F800 07803C00 0F001E00 1E000F00 3E000F80 3C000780 7C0007C0 7C0007C0 780003C0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 780003C0 7C0007C0 7C0007C0 3C1F0780 3E1F8F80 1E398F00 0F30DE00 07B0FC00 03F8F800 00FFE020 003FF020 00007060 000078E0 00003FE0 00003FC0 00003FC0 00001FC0 00000F00> PXLC RP 673 2106 XY F25(Q)S 723 X(computes)S 910 X(the)S 985 X(n)S -1 x (umerical)S 1179 X(input)S 1292 X(with)S 1390 X(ordinal-b)S 1 x (ound)S 50 y 300 X F26(\030)S 339 X F25(then)S 439 X F26(\030)S 478 X F25(should)S 615 X(also)S 704 X(b)S 1 x(e)S 765 X(the)S 842 X (b)S 1 x(ound)S 975 X(on)S 1038 X(the)S 1115 X(deriv)S -2 x(ation)S 1315 X(of)S 1368 X(C\(Q\).)S 1493 X(In)S 1550 X(other)S 49 y 300 X (w)S -1 x(ords)S 425 X(the)S 502 X(ordinal-sequence)S 820 X (measuring)S 1025 X(this)S 1111 X(C)S 1161 X(should)S 1298 X(b)S 1 x(e)S 1360 X(the)S 1437 X(iden)S -1 x(tit)S -1 x(y)S 1596 X(se-)S 50 y 300 X(quence)S 442 X(\012)S 491 X(=)S 542 X F27(h)S F26(\030)S 2 x F27(i)S 613 X F25(in)S 666 X(\012)S 6 y F8(2)S -6 y 2 x F25(.)S 758 X(If)S 804 X F26(k)S 846 X F25(nested)S 979 X(applications)S 1215 X(of)S 1267 X(C)S 1315 X(are)S 1389 X(used)S 1488 X(then)S 1587 X(the)S 50 y 300 X(ordinal-b)S 1 x(ound)S 568 X(will)S 645 X (b)S 1 x(e)S 701 X(\012)S 738 X F27(\001)S 757 X F26(k)S 1 x F25(.)S 809 X(But)S 891 X(w)S -1 x(e)S 951 X(are)S 1019 X(here)S 1108 X(w)S -1 x(orking)S 1264 X(in)S 1312 X(a)S 1346 X(higher-order)S 1580 X (set-)S 50 y 300 X(ting,)S 397 X(so)S 449 X(a)S 484 X (Normalization)S 755 X(Theorem)S 931 X(will)S 1010 X(b)S 1 x(e)S 1066 X(needed)S 1205 X(to)S 1256 X(reduce)S 1388 X(a)S 1423 X (higher-order)S 50 y 300 X(program)S 471 X(with)S 570 X("cut-)S 680 X(rank")S 801 X F26(n)S 845 X F25(to)S 900 X(a)S 939 X (\014rst-order)S 1138 X(one.)S 1243 X(This)S 1343 X(will)S 1426 X (increase)S 1587 X(the)S 49 y 300 X(initial)S 418 X(ordinal)S 558 X (b)S 1 x(ound)S 684 X(\012)S 720 X F27(\001)S 738 X F26(k)S 774 X F25(to)S XP /F26 69 31 2 0 31 28 28 32 0 <03FFFFF8 03FFFFF8 003E00F8 003C0078 003C0030 003C0030 007C0030 00781830 00781830 00783830 00F83000 00F07000 00FFF000 00FFF000 01F0E000 01E06000 01E06000 01E0E0E0 03E0C0C0 03C001C0 03C00180 03C00380 07C00380 07800700 07800F00 0F803E00 FFFFFE00 FFFFFC00> PXLC RP 823 2454 XY F26(E)S XP /F9 110 21 0 0 18 13 13 24 0 <3C7C00 6FCE00 6F8E00 EF0E00 CE0E00 0E0E00 1E1E00 1E1C00 1C1CC0 1C3CC0 3C39C0 3C3B80 381F00> PXLC RP 856 2439 XY F9(n)S 15 y 2 x F25(\(\012)S 931 X F27(\001)S 949 X F26 (k)S 1 x F25(\))S 1001 X(where)S 1119 X F26(E)S 1165 X F25(is)S 1205 X(the)S 1275 X(exp)S 1 x(onen)S -1 x(tial)S 1496 X(function)S 50 y 300 X(on)S 356 X(\012)S 6 y F8(2)S -6 y 1 x F25(.)S 434 X (Collapsing)S 635 X(this)S 714 X(to)S 762 X(the)S 832 X(coun)S -1 x (table)S 1018 X(tree-ordinal)S 1240 X F26(D)S 1 x(E)S -15 y 2 x F9 (n)S 15 y 2 x F25(\(\012)S 1382 X F27(\001)S 1399 X F26(k)S 1 x F25 (\))S 1450 X(then)S 1542 X(yields)S 50 y 300 X(the)S 374 X(desired)S 518 X(ordinal)S 662 X(estimate)S 831 X(for)S 897 X(the)S 971 X(giv)S -1 x(en)S 1082 X(program.)S 1273 X(This)S 1370 X(is)S 1415 X (precisely)S 1587 X(the)S 50 y 300 X(pro)S 1 x(cedure)S 492 X(tak)S -1 x(en)S 604 X(b)S -1 x(y)S 661 X(Buc)S -1 x(hholz)S 841 X(in)S 889 X(his)S 953 X(brief)S 1048 X(analysis)S 1203 X(of)S XP /F26 73 18 1 0 20 28 28 24 0 <03FFE0 03FFE0 003E00 003C00 003C00 003C00 007C00 007800 007800 007800 00F800 00F000 00F000 00F000 01F000 01E000 01E000 01E000 03E000 03C000 03C000 03C000 07C000 078000 078000 0F8000 FFF800 FFF800> PXLC RP 1250 2604 XY F26(I)S 3 x(D)S 6 y F8(1)S -6 y 1337 X F25(on)S 1394 X (pages)S 1507 X(195-19)S XP /F25 54 21 2 -1 18 28 29 16 0 <01F8 07FC 0F8E 1E1E 1C1E 3C1E 381E 7800 7000 7080 F3F0 F7F8 FC3C F81E F80E F00F F00F F00F F00F F00F F00F 700F 700F 780E 381E 3C1C 1E3C 0FF8 07E0> PXLC RP 1624 2604 XY F25(6)S 50 y 300 X(of)S 346 X([5],)S 415 X(but)S 490 X (of)S 537 X(course)S 663 X(the)S 734 X(con)S -1 x(text)S 880 X(and)S 960 X(the)S 1031 X(in)S -1 x(tricate)S 1195 X(details)S 1326 X(are)S 1394 X(di\013eren)S -1 x(t)S 1557 X(here.)S 49 y 300 X(Nev)S -1 x (ertheless)S 537 X(w)S -1 x(e)S 596 X(can)S 670 X(b)S 1 x(egin)S 780 X(to)S 829 X(see)S 894 X(that)S 982 X(the)S 1051 X(upp)S 1 x (er)S 1168 X(b)S 1 x(ound)S 1294 X(on)S 1349 X(the)S 1419 X (ordinal)S 1557 X(mea-)S 50 y 300 X(sures)S 405 X(for)S 469 X (G\177)S 502 X(odel's)S 618 X(T)S 662 X(is)S 705 X(going)S 816 X (to)S 867 X(b)S 1 x(e)S 924 X(the)S 996 X(same)S 1101 X(as)S 1153 X (the)S 1225 X(ordinal)S 1366 X(b)S 1 x(ound)S 1495 X(for)S 1559 X F26(I)S 3 x(D)S 7 y F8(1)S -7 y 2 x F25(,)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 128 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(386)S 700 X F18(H.)S 752 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S 1023 X(and)S 1098 X(S.S.)S 1175 X(W)S -3 x(ainer)S 553 Y 300 X F25(namely)S 447 X(the)S 520 X (Bac)S -1 x(hmann{Ho)S -1 x(w)S -1 x(ard)S 889 X(ordinal.)S 1053 X (Ho)S -1 x(w)S -1 x(ev)S -1 x(er,)S 1235 X(m)S -1 x(uc)S -1 x(h)S 1348 X(w)S -1 x(ork)S 1452 X(is)S 1496 X(required)S 50 y 300 X(in)S 348 X(what)S 452 X(follo)S -1 x(ws,)S 600 X(in)S 648 X(order)S 757 X (to)S 808 X(set)S 873 X(this)S 954 X(up)S 1014 X(precisely)S -3 x (.)S XP /F55 /cmbx10 432 59.8 59.8 128 [-5 -15 69 45] PXLNF RP XP /F55 50 34 3 0 30 39 39 32 0 <01FFC000 0FFFF800 1FFFFE00 3F87FF00 7E01FF80 FF00FFC0 FF807FC0 FF807FE0 FF803FE0 FF803FE0 FF803FE0 7F003FE0 3E003FE0 00003FE0 00007FE0 00007FC0 00007FC0 0000FF80 0000FF80 0001FF00 0003FE00 0007F800 000FF000 000FE000 001FC000 003F00E0 007E00E0 00FC00E0 01F801C0 03F001C0 07C001C0 0FFFFFC0 1FFFFFC0 3FFFFFC0 7FFFFFC0 FFFFFF80 FFFFFF80 FFFFFF80 FFFFFF80> PXLC RP 300 742 XY F55(2)S XP /F55 80 47 3 0 42 41 41 40 0 PXLC RP 403 742 XY F55(P)S XP /F55 67 50 4 0 45 41 41 48 0 <00007FF80380 0007FFFF0F80 001FFFFFDF80 007FF81FFF80 01FFC003FF80 03FF0001FF80 07FE00007F80 0FFC00003F80 1FF800003F80 1FF000001F80 3FF000000F80 3FE000000F80 7FE000000F80 7FE000000780 7FE000000780 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 7FE000000000 7FE000000380 7FE000000380 3FE000000380 3FF000000780 1FF000000780 1FF800000F00 0FFC00001F00 07FE00003E00 03FF00007E00 01FFC001FC00 007FF80FF000 001FFFFFE000 0007FFFF8000 00007FF80000> PXLC RP 450 742 XY F55(C)S XP /F55 70 43 3 0 39 41 41 40 0 PXLC RP 500 742 XY F55(F)S XP /F55 119 50 1 0 48 27 27 48 0 PXLC RP 566 742 XY F55(w)S XP /F55 105 19 3 0 18 43 43 16 0 <0FC0 1FE0 3FE0 3FF0 3FF0 3FF0 3FF0 3FE0 1FE0 0FC0 0000 0000 0000 0000 0000 0000 FFE0 FFE0 FFE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 FFFE FFFE FFFE> PXLC RP 616 742 XY F55(i)S XP /F55 116 27 1 0 22 38 38 24 0 <007000 007000 007000 007000 00F000 00F000 01F000 01F000 03F000 07F000 1FF000 FFFFF0 FFFFF0 FFFFF0 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F038 07F038 07F038 07F038 07F038 07F078 07F878 03FCF0 03FFF0 01FFE0 007FC0> PXLC RP 635 742 XY F55(t)S XP /F55 104 38 3 0 36 42 42 40 0 PXLC RP 661 742 XY F55(h)S XP /F55 98 38 2 0 35 42 42 40 0 PXLC RP 722 742 XY F55(b)S XP /F55 111 34 2 0 31 27 27 32 0 <003FE000 01FFFC00 03FFFE00 0FF07F80 1FC01FC0 3F800FE0 3F800FE0 7F0007F0 7F0007F0 7F0007F0 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 7F0007F0 7F0007F0 3F800FE0 3F800FE0 1FC01FC0 0FF07F80 07FFFF00 01FFFC00 003FE000> PXLC RP 763 742 XY F55(o)S XP /F55 117 38 3 0 36 27 27 40 0 PXLC RP 797 742 XY F55(u)S XP /F55 110 38 3 0 36 27 27 40 0 PXLC RP 835 742 XY F55(n)S XP /F55 100 38 2 0 35 42 42 40 0 <00003FF800 00003FF800 00003FF800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 000003F800 007FE3F800 03FFFBF800 07FFFFF800 0FF87FF800 1FE01FF800 3FC007F800 7F8003F800 7F8003F800 7F8003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 FF0003F800 7F0003F800 7F8003F800 7F8007F800 3FC00FF800 3FE01FF800 1FF87FF800 07FFFFFF80 03FFF3FF80 007FC3FF80> PXLC RP 873 742 XY F55(d)S XP /F55 101 32 2 0 29 27 27 32 0 <007FF000 03FFFC00 07FFFF00 0FF8FF80 1FE03F80 3FC01FC0 7F800FC0 7F800FC0 7F000FE0 FF0007E0 FF0007E0 FFFFFFE0 FFFFFFE0 FFFFFFE0 FF000000 FF000000 FF000000 FF000000 7F800000 7F800000 7FC000E0 3FE001E0 1FF003E0 0FFC0FC0 07FFFF80 01FFFF00 003FF800> PXLC RP 912 742 XY F55(ed)S XP /F55 12 38 1 0 35 42 42 40 0 <0001FFC000 001FFFF000 007FFFF800 01FFC1F800 03FE03FC00 03FC03FC00 07F803FC00 07F803FC00 07F003FC00 07F003FC00 07F001F800 07F0000000 07F0000000 07F0000000 07F0000000 FFFFFFFC00 FFFFFFFC00 FFFFFFFC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 07F001FC00 7FFF1FFFC0 7FFF1FFFC0 7FFF1FFFC0> PXLC RP 1004 742 XY F55(\014)S XP /F55 120 36 1 0 34 27 27 40 0 PXLC RP 1042 742 XY F55(xed)S XP /F55 112 38 2 -12 35 27 39 40 0 PXLC RP 1171 742 XY F55(p)S 2 x(oin)S -2 x(t)S XP /F55 115 27 2 0 24 27 27 24 0 <07FFF0 1FFFF0 3FFFF0 7F03F0 FC01F0 F800F0 F80070 FC0070 FF0000 FFF000 FFFF00 7FFFC0 7FFFF0 3FFFF8 0FFFF8 01FFFC 000FFC 0001FC E000FC F0007C F0007C F800FC FC00F8 FF03F8 FFFFF0 FFFFE0 F3FF80> PXLC RP 1328 742 XY F55(s)S 834 Y 300 X F25(W)S -3 x(e)S 372 X(study)S 488 X (restricted)S 674 X(notions)S 820 X(of)S 868 X(computabilit)S -1 x (y)S 1134 X(in)S 1184 X(the)S 1256 X(con)S -1 x(text)S 1404 X(of)S 1453 X(the)S 1525 X(partial)S 49 y 300 X(con)S -1 x(tin)S -1 x (uous)S 506 X(functionals.)S 733 X(In)S 784 X(particular)S 974 X(w)S -1 x(e)S 1034 X(in)S -1 x(tro)S 1 x(duce)S 1217 X(b)S 1 x(ounded)S 1385 X(v)S -1 x(ersions)S 1541 X(of)S 1587 X(the)S 50 y 300 X (\014xed)S 403 X(p)S 1 x(oin)S -1 x(t)S 514 X(op)S 1 x(erators)S XP /F25 89 31 0 0 30 28 28 32 0 PXLC RP 701 933 XY F25(Y)S XP /F9 26 17 1 -6 16 13 19 16 0 <01F8 039C 070C 0E0E 1C0E 1C0E 3C1E 3C1C 381C 383C 7838 7CF0 77C0 7000 F000 F000 E000 E000 E000> PXLC RP 732 940 XY F9(\032)S -7 y 2 x F25(,)S 780 X(dep)S 1 x(ending)S 982 X (on)S 1043 X(a)S 1080 X(giv)S -1 x(en)S 1191 X(w)S -1 x (ell{ordering.)S 1470 X(The)S 1558 X(\014xed)S 50 y 300 X(p)S 1 x (oin)S -1 x(t)S 413 X(op)S 1 x(erator)S 585 X(Y)S 6 y F9(\032)S -6 y 654 X F25(of)S 706 X(t)S -1 x(yp)S 1 x(e)S 804 X(\()S XP /F26 26 21 1 -9 21 18 27 24 0 <001F00 007FC0 00F1E0 01C1E0 0380E0 0780F0 0700F0 0701F0 0F01E0 0E01E0 0E01E0 1E03E0 1E03C0 1E0780 1E0F80 3F1F00 3FFC00 39F000 380000 780000 780000 700000 700000 F00000 F00000 E00000 E00000> PXLC RP 821 983 XY F26(\032)S 861 X F27(!)S 922 X F26(\032)S F25(\))S 979 X F27(!)S 1040 X F26(\032)S 1080 X F25(satis\014es)S 1238 X(Y)S 6 y F9(\032)S -6 y 2 x F26(G)S 1340 X F25(=)S 1392 X F26(G)S F25(\(Y)S 6 y F9(\032)S -6 y 2 x F26(G)S F25(\).)S 1584 X(F)S -3 x(or)S 50 y 300 X(example,)S 474 X(the)S 546 X(factorial)S 708 X(ma)S -1 x(y)S 798 X(b)S 1 x(e)S 855 X(de\014ned)S 998 X(b)S -1 x(y)S 1125 Y 481 X (Y)S 12 y F30(nat)S XP /F10 33 33 2 2 29 12 10 32 0 <00000E00 00000600 00000700 00000380 FFFFFFE0 FFFFFFE0 00000380 00000700 00000600 00000E00> PXLC RP 580 1137 XY F10(!)S F30(nat)S -12 y 2 x F25(\()S XP /F26 21 24 2 0 21 29 29 24 0 <0F8000 0FE000 01E000 00E000 00F000 00F000 007000 007800 007800 003800 003C00 003C00 001C00 001E00 001E00 001E00 003E00 007F00 00FF00 01E700 03C780 078780 0F0380 1E03C0 1C03C0 3C01C0 7801E0 F001E0 E000E0> PXLC RP 700 1125 XY F26(\025f)S 748 X(\025x:)S F25(cond)S 1 x(\(zerop)S 1018 X F26(x)S F25(\)1\(m)S -1 x(ult)S 1191 X F26(x)S F25(\()S F26 (f)S 1255 X F25(\(pred)S 1364 X F26(x)S F25(\)\)\)\))S F26(:)S 1217 Y 300 X F25(The)S 385 X(in)S -1 x(ten)S -1 x(tion)S 561 X(here)S 651 X (is)S 692 X(that)S 782 X(Y)S 1 x F26(Gx)S 884 X F25(will)S 963 X (only)S 1054 X(use)S 1126 X(its)S 1183 X(functional)S 1378 X (argumen)S -1 x(t)S 1564 X F26(G)S 1610 X F25(in)S 50 y 300 X(suc)S -1 x(h)S 392 X(a)S 425 X(w)S -1 x(a)S -1 x(y)S 507 X(that)S 595 X F26(G)S 640 X F25(itself)S 739 X(calls)S 830 X(its)S 886 X (function)S 1047 X(argumen)S -1 x(t)S 1231 X(at)S 1280 X(n)S -1 x (um)S -1 x(b)S 1 x(ers)S 1446 X(\\less")S 1562 X(than)S 50 y 300 X F26(x)S F25(.)S 358 X(W)S -3 x(e)S 431 X(prop)S 1 x(ose)S 586 X(to)S 639 X(mak)S -1 x(e)S 749 X(this)S 831 X(in)S -1 x(ten)S -1 x(tion)S 1008 X(explicit)S 1156 X(and)S 1238 X(restrict)S 1383 X(Y)S 1430 X (accordingly)S -3 x(.)S 50 y 300 X(So)S 358 X(let)S 418 X(a)S 452 X (w)S -1 x(ell{ordering)S XP /F27 30 32 3 -3 28 22 25 32 0 <00000180 00000180 00000380 00000380 00000300 00000700 00000F00 00001E00 00007C00 0001F800 0007E000 00FF8000 FFFE0000 FFFE0000 00FF8000 0007E000 0001F800 00007C00 00001E00 00000F00 00000700 00000380 00000380 00000180 00000180> PXLC RP 707 1367 XY F27(\036)S 753 X F25(b)S 1 x(e)S 810 X(giv)S -1 x(en.)S 934 X(W)S -3 x(e)S 1006 X(w)S -1 x(an)S -1 x(t)S 1107 X(Y)S 6 y F9 (\032;)S XP /F10 30 26 3 -3 21 17 20 24 0 <0000C0 0000C0 0001C0 0001C0 000380 000780 001F00 007E00 03F800 FFE000 FFE000 03F800 007E00 001F00 000780 000380 0001C0 0001C0 0000C0 0000C0> PXLC RP 1166 1373 XY F10(\036)S -6 y 1207 X F25(to)S 1258 X(satisfy)S 1459 Y 583 X(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26(Gx)S 737 X F25(=)S 781 X F26(G)S F25(\()S F26(\025)S XP /F26 121 20 1 -8 20 18 26 24 0 <1F00E0 3F80E0 7380E0 6381E0 E381C0 C781C0 0701C0 0703C0 0F0380 0E0380 0E0380 0E0780 0E0700 0E0F00 0E0F00 0F3F00 0FFE00 03EE00 000E00 001E00 381C00 783C00 787800 71F000 7FE000 1F8000> PXLC RP 854 1459 XY F26(y)S 1 x(:)S F25(cond\()S F26(y)S 1022 X F27(\036)S 1066 X F26(x)S F25(\)\(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26 (Gy)S 1 x F25(\))S F27(?)S F25(\))S F26(x:)S 1551 Y 300 X F25(This)S 396 X(equalit)S -1 x(y)S -3 x(,)S 564 X(when)S 674 X(view)S -1 x (ed)S 811 X(as)S 863 X(a)S 899 X(con)S -1 x(v)S -1 x(ersion)S 1103 X (rule)S 1187 X(from)S 1286 X(left)S 1360 X(to)S 1412 X(righ)S -1 x (t,)S 1526 X(clearly)S 50 y 300 X(giv)S -1 x(es)S 407 X(rise)S 489 X (to)S 545 X(an)S 608 X(in\014nite)S 754 X(reduction)S 943 X (sequence)S 3 x(:)S 1142 X(just)S 1230 X(rep)S 1 x(eatedly)S 1438 X (expand)S 1587 X(the)S 50 y 300 X(subterm)S 461 X(Y)S 6 y F9(\032;)S F10(\036)S -6 y 3 x F26(Gy)S 616 X F25(b)S -1 x(y)S 673 X(the)S 744 X(same)S 848 X(rule)S 931 X(again.)S 1058 X(Hence)S 1181 X(in)S 1229 X(order)S 1337 X(to)S 1388 X(ac)S -1 x(hiev)S -1 x(e)S 1532 X (strong)S 50 y 300 X(normalization)S 565 X(w)S -1 x(e)S 629 X(ha)S -1 x(v)S -1 x(e)S 727 X(to)S 780 X(mak)S -1 x(e)S 891 X(use)S 966 X (of)S 1015 X(the)S 1089 X(fact)S 1174 X(that)S 1266 X(the)S 1340 X (subterm)S 1504 X(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26(Gy)S 50 y 300 X F25(o)S 1 x(ccurs)S 429 X(in)S 478 X(a)S 513 X (\\guarded")S 715 X(con)S -1 x(text)S 863 X(cond\()S F26(y)S 999 X F27(\036)S 1044 X F26(x)S F25(\)\(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26(Gy)S 1 x F25(\))S F27(?)S 1303 X F25(only)S -3 x(.)S 1409 X (Therefore)S 1598 X(w)S -1 x(e)S 50 y 300 X(restrict)S 440 X(our)S 511 X(con)S -1 x(v)S -1 x(ersion)S 711 X(rules)S 807 X(and)S 885 X (allo)S -1 x(w)S 989 X(con)S -1 x(v)S -1 x(ersion)S 1188 X(of)S 1233 X(a)S 1264 X(term)S 1361 X(cond)S XP /F26 77 40 2 0 42 28 28 40 0 <03FF0001FF 03FF0003FF 003F0003F0 003F0007E0 003F000DE0 003F000FE0 0077801BE0 0067801BC0 00678033C0 00678067C0 00E78067C0 00C780C780 00C780C780 00C7818F80 01C7830F80 0187830F00 0187860F00 0183C61F00 0383CC1F00 0303CC1E00 0303D81E00 0303F03E00 0703F03E00 0603E03C00 0603E03C00 1F03C07C00 FFE387FFC0 FFE387FFC0> PXLC RP 1446 1801 XY F26(M)S 1491 X(N)S 1529 X(L)S 1568 X F25(only)S 56 y 300 X(if)S 338 X F26(M)S 396 X F25(is)S 437 X(a)S 471 X(b)S 1 x(o)S 1 x(olean)S 626 X(constan)S -1 x(t)S 793 X(\(i.e.)S 875 X(t)S 884 X (t)S 1 x(,)S 926 X(f)S 935 X(f)S 964 X(or)S 1014 X F27(?)S XP /F30 98 27 1 0 24 29 29 24 0 PXLC RP 1047 1842 XY F30(b)S 1 x(o)S 1 x(ole)S 15 y 2 x F25(\).)S 1207 X (Then)S 1315 X(clearly)S 1447 X(the)S 1518 X(in\014nite)S 49 y 300 X (reduction)S 489 X(sequence)S 665 X(ab)S 1 x(o)S -1 x(v)S -1 x(e)S 787 X(can)S 867 X(not)S 945 X(b)S 1 x(e)S 1005 X(formed)S 1149 X (an)S -1 x(y)S 1232 X(more,)S 1352 X(but)S 1432 X(still)S 1517 X(w)S -1 x(e)S 1583 X(can)S 50 y 300 X(exp)S 1 x(ect)S 431 X(that)S 521 X (an)S -1 x(y)S 600 X(closed)S 722 X(term)S 822 X(of)S 869 X(ground)S 1010 X(t)S -1 x(yp)S 1 x(e)S 1103 X(reduces)S 1251 X(to)S 1302 X(a)S 1336 X(n)S -1 x(umeral.)S 50 y 375 X(It)S 423 X(is)S 469 X(sho)S -1 x(wn)S 598 X(b)S 1 x(elo)S -1 x(w)S 719 X(that)S 813 X(an)S -1 x (y)S 895 X(term)S 998 X(in)S -1 x(v)S -1 x(olving)S 1179 X(only)S 1273 X(b)S 1 x(ounded)S 1447 X(\014xed)S 1550 X(p)S 1 x(oin)S -1 x (t)S 50 y 300 X(op)S 1 x(erators)S 488 X(is)S 533 X(strongly)S 697 X (normalizable.)S 969 X(Hence)S 1096 X(the)S 1171 X(ob)S 2 x(ject)S 1301 X(\\unde\014ned")S 1535 X(in)S 1587 X(the)S 50 y 300 X(mo)S 1 x(del)S 429 X(can)S 511 X(b)S 1 x(e)S 573 X(view)S -1 x(ed)S 715 X (as)S 771 X(a)S 811 X(\014nite)S 923 X(error.)S 1058 X(This)S 1158 X (is)S 1206 X(in)S 1260 X(con)S -1 x(trast)S 1426 X(to)S 1482 X(the)S 1560 X(term)S 50 y 300 X(language)S 476 X(PCF)S 577 X(of)S 627 X ([9)S -1 x(],)S 700 X(where)S 822 X(the)S 896 X(v)S -2 x(alue)S 1006 X(unde\014ned)S 1198 X(is)S 1242 X(to)S 1295 X(b)S 1 x(e)S 1355 X(in)S -1 x(terpreted)S 1571 X(as)S 1624 X(a)S 49 y 300 X(non)S -1 x(terminating)S 591 X(computation.)S 849 X(The)S 933 X(addition)S 1095 X(of)S 1141 X(constan)S -1 x(ts)S 1324 X(for)S 1386 X (\014nite)S 1491 X(errors)S 1608 X(to)S 50 y 300 X(a)S 332 X(term)S 429 X(language)S 600 X(has)S 672 X(also)S 753 X(b)S 1 x(een)S 848 X (considered)S 1049 X(in)S 1095 X([6].)S 1168 X(Ho)S -1 x(w)S -1 x (ev)S -1 x(er,)S 1345 X(Dosc)S -1 x(h)S 1466 X(only)S 1555 X(deals)S 50 y 300 X(with)S 395 X(strict)S 503 X(functionals.)S 50 y 375 X(F)S -3 x(or)S 451 X(the)S 524 X(particular)S 717 X(case)S 806 X F26(<)S 854 X F25(\(the)S 943 X(standard)S 1118 X(ordering)S 1283 X(of)S 1332 X(the)S 1405 X(natural)S 1552 X(n)S -1 x(um-)S 50 y 300 X(b)S 1 x(ers\))S 405 X(Y)S 6 y F9(\032;)S XP /F9 60 26 3 -3 21 17 20 24 0 <0001C0 0007C0 000F80 003E00 00FC00 03F000 0FC000 1F0000 7C0000 F80000 F80000 7C0000 1F0000 0FC000 03F000 00FC00 003E00 000F80 0007C0 0001C0> PXLC RP 463 2411 XY F9(<)S -6 y 505 X F25(is)S 546 X(de\014nable)S 721 X (from)S 819 X(the)S 890 X(standard)S 1063 X(higher)S 1189 X(order)S 1297 X(recursion)S 1475 X(op)S 1 x(erators)S 49 y 300 X F26(R)S 7 y F9(\032)S -7 y 2 x F25(,)S 376 X(adapted)S 534 X(to)S 585 X(the)S 656 X(partial)S 789 X(con)S -1 x(tin)S -1 x(uous)S 996 X (functionals.)S 1224 X(W)S -3 x(e)S 1295 X(then)S 1389 X(sho)S -1 x (w)S 1491 X(that)S 1580 X(an)S -1 x(y)S 50 y 300 X(closed)S 423 X (term)S 522 X F26(M)S 581 X F25(of)S 629 X(t)S -1 x(yp)S 1 x(e)S 723 X F30(nat)S 803 X F27(!)S 856 X F30(nat)S 939 X F25(in)S -1 x (v)S -1 x(olving)S 1117 X(only)S 1208 X(b)S 1 x(ounded)S 1378 X (\014xed)S 1479 X(p)S 1 x(oin)S -1 x(t)S 1587 X(op-)S 50 y 300 X (erators)S 443 X(Y)S 6 y F9(\032;<)S -6 y 548 X F25(has)S 626 X (the)S 702 X(prop)S 1 x(ert)S -1 x(y)S 876 X(that)S 971 X(v)S -2 x (alues)S 1099 X(of)S 1150 X(the)S 1226 X F26(M)S 1271 X F30(n)S 1316 X F25(for)S 1384 X F26(n)S 1428 X F25(=)S 1479 X(0)S F26(;)S 1519 X F25(1)S F26(;)S 1559 X F25(2)S F26(;)S 1599 X(:)S 1615 X(:)S 1634 X(:)S 50 y 300 X F25(\(if)S 355 X(de\014ned\))S 515 X(are)S 586 X(ev)S -1 x(en)S -1 x(tually)S 785 X(b)S 1 x(ounded)S 955 X(b)S -1 x(y)S 1014 X(a)S 1049 X(function)S 1213 X F26(G)S 6 y F9(\013)S -6 y 1284 X F25(of)S 1332 X(the)S 1405 X(slo)S -1 x(w{gro)S -1 x (wing)S 50 y 300 X(subrecursiv)S -1 x(e)S 538 X(hierarc)S -1 x(h)S -1 x(y)S 722 X(with)S 819 X F26(\013)S 862 X F25(denoting)S 1035 X (an)S 1095 X(ordinal)S 1238 X(b)S 1 x(elo)S -1 x(w)S 1358 X(the)S 1432 X(Bac)S -1 x(hmann{)S 49 y 300 X(Ho)S -1 x(w)S -1 x(ard)S 456 X (ordinal.)S 619 X(F)S -3 x(rom)S 731 X(this)S 814 X(w)S -1 x(e)S 877 X(can)S 956 X(immediately)S 1194 X(b)S 1 x(ound)S 1325 X(the)S 1398 X(complexit)S -1 x(y)S 1612 X(of)S 50 y 300 X(the)S 371 X (program)S 538 X F26(M)S 596 X F25(as)S 647 X(in)S 696 X(the)S 767 X (First)S 869 X(Collapsing)S 1072 X(Lemma.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 129 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup 692 462 XY F18(Ordinal)S 833 X(Bounds)S 971 X(for)S 1030 X (Programs)S 1583 X F25(38)S XP /F25 55 21 2 -1 19 28 29 24 0 <600000 7FFF80 7FFF80 7FFF80 600700 C00600 C00E00 C01C00 001800 003800 007000 006000 00E000 00E000 01C000 01C000 01C000 03C000 038000 038000 038000 078000 078000 078000 078000 078000 078000 078000 078000> PXLC RP 1624 462 XY F25(7)S XP /F48 /cmbx12 300 49.8 49.8 128 [-3 -12 56 37] PXLNF RP XP /F48 50 28 3 0 24 32 32 24 0 <03FC00 1FFF80 3FFFC0 7C3FE0 FC1FF0 FE0FF0 FE07F8 FE07F8 FE03F8 7C03F8 3807F8 0007F8 0007F0 000FF0 000FE0 001FC0 001F80 003F00 007E00 00F800 01F000 01E038 03C038 078038 0F0038 1C0078 3FFFF0 7FFFF0 FFFFF0 FFFFF0 FFFFF0 FFFFF0> PXLC RP 300 553 XY F48(2)S PF XP /F48 /cmbx12 300 49.8 49.8 128 [-3 -12 56 37] PXLNF RP XP /F48 46 16 4 0 12 8 8 8 0 <3C 7E FF FF FF FF 7E 3C> PXLC RP 328 553 XY F48(.)S XP /F48 49 28 4 0 23 32 32 24 0 <001800 007800 03F800 FFF800 FFF800 FDF800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 01F800 7FFFE0 7FFFE0 7FFFE0> PXLC RP 344 553 XY F48(1)S XP /F48 83 31 3 0 27 34 34 24 0 <03FE0C 0FFF9C 1FFFFC 3F03FC 7C00FC 7C007C F8003C F8003C F8001C FC001C FE0000 FF0000 FFF000 7FFF00 7FFFE0 3FFFF0 1FFFF8 0FFFFC 03FFFE 007FFE 0003FF 0000FF 00003F 00003F E0001F E0001F E0001F F0001F F8003E FC007E FF80FC FFFFF8 E7FFF0 C0FFC0> PXLC RP 428 553 XY F48(S)S XP /F48 121 30 1 -10 28 22 32 32 0 PXLC RP 459 553 XY F48(y)S XP /F48 110 31 2 0 28 22 22 32 0 PXLC RP 488 553 XY F48(n)S XP /F48 116 22 1 0 18 32 32 24 0 <01C000 01C000 01C000 01C000 03C000 03C000 03C000 07C000 07C000 0FC000 3FFF00 FFFF00 FFFF00 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC380 0FC380 0FC380 0FC380 0FC380 0FC780 07E700 03FE00 01FC00> PXLC RP 518 553 XY F48(t)S XP /F48 97 27 2 0 26 22 22 24 0 <07FE00 1FFF80 3F0FC0 3F07E0 3F07F0 3F03F0 1E03F0 0003F0 0003F0 01FFF0 0FFFF0 3FE3F0 7F83F0 7F03F0 FF03F0 FE03F0 FE03F0 FE07F0 FF0FF8 7F1FFF 3FFDFF 0FF0FF> PXLC RP 540 553 XY F48(a)S XP /F48 120 30 1 0 28 22 22 32 0 PXLC RP 567 553 XY F48(x)S XP /F48 111 28 2 0 25 22 22 24 0 <00FE00 07FFC0 0F83E0 1F01F0 3E00F8 7E00FC 7E00FC 7E00FC FE00FE FE00FE FE00FE FE00FE FE00FE FE00FE FE00FE 7E00FC 7E00FC 3F01F8 1F01F0 0F83E0 07FFC0 00FE00> PXLC RP 615 553 XY F48(o)S XP /F48 102 17 1 0 20 35 35 24 0 <003F80 01FFC0 03F7E0 07E7E0 07E7E0 0FC7E0 0FC3C0 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 FFFC00 FFFC00 FFFC00 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 0FC000 7FFC00 7FFC00 7FFC00> PXLC RP 643 553 XY F48(f)S XP /F48 80 38 2 0 34 34 34 32 0 PXLC RP 679 553 XY F48(P)S XP /F48 67 40 3 0 36 34 34 40 0 <0007FE0180 003FFFC380 00FFFFEF80 03FF81FF80 07FC007F80 0FF0003F80 1FE0001F80 1FC0000F80 3FC0000780 7F80000780 7F80000780 7F80000380 FF00000380 FF00000000 FF00000000 FF00000000 FF00000000 FF00000000 FF00000000 FF00000000 FF00000000 FF00000000 7F80000380 7F80000380 7F80000380 3FC0000780 1FC0000700 1FE0000F00 0FF0001E00 07FC003C00 03FF81F800 00FFFFF000 003FFFC000 0007FF0000> PXLC RP 717 553 XY F48(C)S XP /F48 70 35 2 0 32 34 34 32 0 PXLC RP 758 553 XY F48(F)S XP /F45 /cmsy10 360 49.8 49.8 128 [-1 -48 54 39] PXLNF RP XP /F45 30 39 4 -2 34 27 29 32 0 <0000000C 0000000C 0000000C 0000001C 0000001C 00000038 00000038 00000070 000000F0 000001E0 000007C0 00001F80 0001FE00 003FF800 FFFF8000 FFFF8000 003FF800 0001FE00 00001F80 000007C0 000001E0 000000F0 00000078 00000038 00000018 0000001C 0000001C 0000000C 0000000C> PXLC RP 793 563 XY F45(\036)S XP /F25 /cmr10 300 41.5 41.5 128 [-3 -11 41 31] PXLNF RP XP /F25 76 26 2 0 23 28 28 24 0 PXLC RP 300 630 XY F25(L)S XP /F25 101 18 1 0 16 18 18 16 0 <07E0 1FF8 3E7C 781C 781E F01E FFFE FFFE F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 326 630 XY F25(e)S XP /F25 116 16 1 0 13 26 26 16 0 <0C00 0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFE0 FFE0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C30 1C30 1C30 1C30 1C30 1E70 0FE0 07C0> PXLC RP 344 630 XY F25(t)S XP /F25 117 23 1 0 21 18 18 24 0 PXLC RP 373 630 XY F25(u)S XP /F25 115 16 1 0 14 18 18 16 0 <1FB0 7FF0 F0F0 E070 E030 F030 F800 7FC0 7FE0 1FF0 00F8 C078 C038 E038 F078 F8F0 FFF0 CFC0> PXLC RP 396 630 XY F25(s)S XP /F25 12 23 0 0 21 29 29 24 0 <003F00 01FF80 03C3C0 0783C0 0F03C0 0E03C0 0E0000 0E0000 0E0000 0E0000 0E0000 FFFFC0 FFFFC0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 7F87F8 7F87F8> PXLC RP 425 630 XY F25(\014)S XP /F25 114 16 1 0 14 18 18 16 0 PXLC RP 448 630 XY F25(rst)S 510 X(re)S XP /F25 118 22 1 0 20 18 18 24 0 PXLC RP 544 630 XY F25(v)S XP /F25 105 12 1 0 10 29 29 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 566 630 XY F25(ie)S XP /F25 119 30 1 0 28 18 18 32 0 PXLC RP 596 630 XY F25(w)S 639 X(s)S XP /F25 111 21 1 0 19 18 18 24 0 <03F000 0FFC00 1E1E00 380700 780780 700380 F003C0 F003C0 F003C0 F003C0 F003C0 F003C0 700380 780780 380700 1E1E00 0FFC00 03F000> PXLC RP 655 630 XY F25(o)S XP /F25 109 35 1 0 34 18 18 40 0 PXLC RP 676 630 XY F25(me)S 741 X(s)S XP /F25 121 22 1 -8 20 18 26 24 0 PXLC RP 758 630 XY F25(y)S XP /F25 110 23 1 0 21 18 18 24 0 PXLC RP 780 630 XY F25(n)S -1 x(t)S XP /F25 97 21 1 0 20 18 18 24 0 <1FE000 3FF800 3C3C00 3C1E00 180E00 000E00 000E00 03FE00 1F8E00 7E0E00 780E00 F80E00 F00E60 F00E60 F81E60 7C7E60 7FEFC0 1F8780> PXLC RP 818 630 XY F25(a)S XP /F25 99 18 1 0 16 18 18 16 0 <07F8 0FFC 3E3C 3C3C 7818 7800 F000 F000 F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 839 630 XY F25(ctica)S XP /F25 108 12 1 0 10 29 29 16 0 PXLC RP 924 630 XY F25(l)S 948 X(issues)S XP /F25 46 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 1050 630 XY F25(.)S XP /F25 83 23 2 -1 20 29 30 24 0 <07F180 1FFF80 3C3F80 780F80 700780 E00380 E00380 E00180 E00180 F00000 F80000 7E0000 7FE000 3FFC00 1FFE00 0FFF00 00FF80 000F80 0007C0 0003C0 0003C0 C001C0 C001C0 C001C0 E001C0 F00380 F80780 FE0F00 FFFE00 C3F800> PXLC RP 1080 630 XY F25(Sim)S XP /F25 112 23 1 -8 21 18 26 24 0 PXLC RP 1149 630 XY F25(ple)S XP /F28 /cmti10 300 41.5 41.5 128 [-1 -11 45 31] PXLNF RP XP /F28 116 14 3 0 15 26 26 16 0 <01C0 01C0 01C0 03C0 03C0 0380 0380 0780 FFE0 FFE0 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C70 3C70 3860 38E0 39C0 3F80 1F00> PXLC RP 1214 630 XY F28(t)S XP /F28 121 20 3 -8 20 18 26 24 0 <1F0380 3F8380 738380 638780 E38700 C78700 070700 070F00 0F0E00 0E0E00 0E0E00 0E1E00 1E1C00 1C1C00 1E3C00 0E7C00 0FF800 07F800 003800 007800 387000 78F000 79E000 73C000 7F8000 3E0000> PXLC RP 1228 630 XY F28(y)S XP /F28 112 21 1 -8 21 18 26 24 0 <078F80 0FDFC0 1DF9E0 19F0E0 39E0F0 31C0F0 01C0F0 03C1F0 03C1E0 0381E0 0381E0 0783E0 0783C0 0783C0 078780 0FCF00 0FFE00 0EF800 0E0000 1E0000 1E0000 1C0000 1C0000 3C0000 FF8000 FF8000> PXLC RP 1248 630 XY F28(p)S XP /F28 101 19 4 0 18 18 18 16 0 <01F8 07FC 0F1C 1E0C 3C0C 780C 783C 7FF8 FFC0 F000 F000 F000 F000 F00C 701C 7878 3FF0 1FC0> PXLC RP 1268 630 XY F28(e)S XP /F28 115 17 3 0 17 18 18 16 0 <03F0 07F8 0F3C 0E3C 1E7C 1E38 1F00 1FE0 0FF0 0FF0 00F8 2078 F070 F070 F0F0 E1E0 7FC0 3F80> PXLC RP 1287 630 XY F28(s)S XP /F26 /cmmi10 300 41.5 41.5 128 [-1 -10 43 31] PXLNF RP XP /F26 26 21 1 -9 21 18 27 24 0 <001F00 007FC0 00F1E0 01C1E0 0380E0 0780F0 0700F0 0701F0 0F01E0 0E01E0 0E01E0 1E03E0 1E03C0 1E0780 1E0F80 3F1F00 3FFC00 39F000 380000 780000 780000 700000 700000 F00000 F00000 E00000 E00000> PXLC RP 1320 630 XY F26(\032)S XP /F26 59 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 1341 630 XY F26(;)S XP /F26 27 24 1 0 23 18 18 24 0 <01FFFC 07FFFC 0FFFFC 1F0F00 3C0700 380700 700380 700780 F00700 E00700 E00700 E00F00 E00E00 E01C00 F03C00 78F800 3FE000 1F8000> PXLC RP 1359 630 XY F26(\033)S -1 x(;)S XP /F26 28 18 1 0 21 18 18 24 0 <0FFFF0 1FFFF0 3FFFF0 786000 606000 E0E000 00E000 00C000 01C000 01C000 01C000 01C000 038000 038000 038000 038000 070000 070000> PXLC RP 1401 630 XY F26(\034)S XP /F26 58 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 1431 630 XY F26(:)S 1450 X(:)S 1469 X(:)S 1492 X F25(are)S XP /F25 98 23 1 0 21 29 29 24 0 PXLC RP 1560 630 XY F25(built)S XP /F25 102 13 0 0 15 29 29 16 0 <00FC 03FE 079E 071E 0F1E 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 300 680 XY F25(from)S XP /F25 103 21 1 -9 19 19 28 24 0 <000380 07E7C0 1FFDC0 3C3DC0 781E00 781E00 781E00 781E00 781E00 781E00 3C3C00 3FF800 37E000 200000 300000 380000 3FFC00 3FFF00 3FFF80 7807C0 F003C0 E001C0 E001C0 F003C0 F807C0 7C0F80 1FFE00 07F800> PXLC RP 394 680 XY F25(groun)S XP /F25 100 23 1 0 21 29 29 24 0 <001F80 001F80 000380 000380 000380 000380 000380 000380 000380 000380 000380 07F380 1FFF80 3E1F80 7C0780 780380 F80380 F00380 F00380 F00380 F00380 F00380 F00380 F00380 780780 7C0F80 3E1F80 1FFBF0 07E3F0> PXLC RP 497 680 XY F25(d)S 530 X(t)S -1 x(yp)S 1 x(es)S XP /F30 /cmbx10 300 41.5 41.5 128 [-4 -10 47 31] PXLNF RP XP /F30 110 27 1 0 25 18 18 24 0 PXLC RP 635 680 XY F30(n)S XP /F30 97 23 1 0 22 18 18 24 0 <07FC00 1FFF00 3F1F80 3F0FC0 3F07C0 3F07C0 1E07C0 03FFC0 1FFFC0 7F87C0 7F07C0 FE07C0 FC07C0 FC07C0 FE0FC0 7E3FF8 3FFBF8 0FE1F8> PXLC RP 662 680 XY F30(a)S XP /F30 116 19 1 0 15 26 26 16 0 <0700 0700 0700 0700 0F00 0F00 0F00 1F00 3FFC FFFC FFFC 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F1C 1F1C 1F1C 1F1C 1FBC 0FF8 07F0> PXLC RP 685 680 XY F30(t)S 713 X F25(and)S XP /F30 98 27 1 0 24 29 29 24 0 PXLC RP 789 680 XY F30(b)S XP /F30 111 24 1 0 22 18 18 24 0 <01FC00 0FFF80 1F07C0 3E03E0 7C01F0 7C01F0 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 7C01F0 7C01F0 3E03E0 1F07C0 0FFF80 01FC00> PXLC RP 817 680 XY F30(o)S 1 x(o)S XP /F30 108 13 1 0 12 29 29 16 0 PXLC RP 866 680 XY F30(l)S XP /F30 101 22 1 0 20 18 18 24 0 <03FC00 0FFF00 3F0F80 3E07C0 7E03C0 7C03E0 FC03E0 FFFFE0 FFFFE0 FC0000 FC0000 FC0000 7C0000 7E00E0 3F01E0 1FC3C0 0FFF80 03FE00> PXLC RP 879 680 XY F30(e)S XP /F25 40 16 3 -11 13 31 42 16 0 <01C0 0380 0380 0700 0E00 0C00 1C00 1800 3800 3800 7000 7000 7000 7000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 7000 7000 7000 7000 3800 3800 1800 1C00 0C00 0E00 0700 0380 0380 01C0> PXLC RP 910 680 XY F25(\(denoted)S 1078 X(b)S -1 x(y)S XP /F26 19 15 2 0 14 18 18 16 0 <0E00 0E00 0E00 1E00 1C00 1C00 1C00 3C00 3800 3800 7800 7070 7070 F060 E0E0 F3C0 7F80 3E00> PXLC RP 1132 680 XY F26(\023)S XP /F25 41 16 2 -11 12 31 42 16 0 PXLC RP 1146 680 XY F25(\))S 1172 X(b)S -1 x(y)S XP /F28 102 13 -1 -8 19 29 37 24 0 <0003E0 000FE0 000FF0 001DF0 001CE0 001C00 003C00 003800 003800 003800 003800 03FF80 07FF80 007000 007000 007000 00F000 00E000 00E000 00E000 00E000 01E000 01C000 01C000 01C000 01C000 03C000 038000 038000 038000 038000 070000 F70000 F70000 FE0000 FC0000 780000> PXLC RP 1225 680 XY F28(f)S XP /F28 117 22 3 0 23 18 18 24 0 <1F0380 3F8380 738380 638780 E38780 C78700 070700 070F00 0F0F00 0E0E00 0E0E00 0E1E70 1E1E70 1C1C60 1E3C60 0E7CC0 0FFFC0 07CF80> PXLC RP 1237 680 XY F28(u)S XP /F28 110 23 3 0 24 18 18 24 0 <1E3F00 3F7F80 77F380 67C380 E78380 C78380 070380 0F0780 0F0700 0E0700 0E0F00 1E0E38 1E0E38 1C1E30 1C1C70 3C1EE0 3C0FC0 380F80> PXLC RP 1260 680 XY F28(n)S XP /F28 99 19 4 0 19 18 18 16 0 <01F8 07FC 0F9E 1E1E 3C3E 7C1C 7800 7800 F800 F000 F000 F000 F000 F00C F01C 7878 3FF0 1FC0> PXLC RP 1283 680 XY F28(ct)S XP /F28 105 13 3 0 14 28 28 16 0 <01C0 01E0 01E0 01C0 0000 0000 0000 0000 0000 0000 1F00 3F00 7780 6380 E780 C700 0F00 0E00 0E00 1E00 1C00 1CE0 3CE0 38C0 39C0 3B80 3F80 1E00> PXLC RP 1316 680 XY F28(i)S XP /F28 111 21 4 0 21 18 18 24 0 <01F800 07FE00 0F9F00 1E0F00 3C0700 7C0780 780780 780F80 F80F00 F00F00 F00F00 F01F00 F01E00 F03C00 F07C00 78F800 3FE000 1F8000> PXLC RP 1329 680 XY F28(on)S 1384 X(typ)S -2 x(e)S 1468 X F25(formation)S 49 y 300 X F26(\032)S XP /F27 /cmsy10 300 41.5 41.5 128 [-2 -40 47 33] PXLNF RP XP /F27 33 42 2 3 39 17 14 40 0 <0000000600 0000000700 0000000700 0000000380 00000001C0 00000000E0 FFFFFFFFF8 FFFFFFFFF8 00000000E0 00000001C0 0000000380 0000000700 0000000700 0000000600> PXLC RP 333 729 XY F27(!)S 386 X F26(\033)S 1 x F25(.)S XP /F25 84 30 1 0 28 28 28 32 0 <7FFFFFC0 7FFFFFC0 780F03C0 700F01C0 600F00C0 E00F00E0 C00F0060 C00F0060 C00F0060 C00F0060 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 03FFFC00 03FFFC00> PXLC RP 441 729 XY F25(T)S XP /F25 104 23 1 0 21 29 29 24 0 PXLC RP 471 729 XY F25(he)S XP /F28 108 11 4 0 13 29 29 16 0 <1F80 3F80 0780 0780 0700 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800 3800 7800 7800 7000 7000 F700 F700 E600 E600 EE00 FC00 7C00> PXLC RP 527 729 XY F28(le)S XP /F28 118 19 3 0 19 18 18 16 0 <1F07 3F8F 738F 638F E387 C787 0703 0707 0F06 0E06 0E06 0E0E 1E0C 1E1C 0E18 0E38 0FF0 07E0> PXLC RP 556 729 XY F28(vel)S 623 X F25(lev)S 1 x(\()S F26(\032)S F25(\))S 743 X(of)S 791 X(a)S 825 X(t)S -1 x(yp)S 1 x(e)S 918 X F26(\032)S 954 X F25(is)S 996 X(de\014ned)S 1139 X(inductiv)S -1 x(ely)S 1352 X (b)S -1 x(y)S 814 Y 683 X(lev)S 1 x(\()S F26(\023)S F25(\))S XP /F25 58 12 3 0 8 18 18 8 0 <70 F8 F8 F8 70 00 00 00 00 00 00 00 00 70 F8 F8 F8 70> PXLC RP 795 814 XY F25(:)S XP /F25 61 32 2 4 29 16 12 32 0 PXLC RP 807 814 XY F25(=)S XP /F25 48 21 2 -1 18 28 29 16 0 <03C0 0FF0 1C38 381C 381C 700E 700E 700E F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F 700E 700E 700E 381C 381C 1C38 0FF0 07E0> PXLC RP 851 814 XY F25(0)S 955 X(for)S 1019 X(an)S -1 x(y)S 1098 X(ground)S 1239 X(t)S -1 x(yp)S 1 x(e)S 1332 X F26(\023)S XP /F25 59 12 3 -8 8 18 26 8 0 <70 F8 F8 F8 70 00 00 00 00 00 00 00 00 70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 1347 814 XY F25(;)S 50 y 587 X(lev\()S F26(\032)S 688 X F27(!)S 741 X F26(\033)S 1 x F25(\))S 795 X(:=)S 851 X(ma)S XP /F25 120 22 0 0 21 18 18 24 0 <7F8FF0 7F8FF0 0F0F80 070F00 038E00 01DC00 01D800 00F000 007000 007800 00F800 01DC00 038E00 030E00 070700 1F0780 FF8FF8 FF8FF8> PXLC RP 907 864 XY F25(x\(lev\()S F26(\032)S F25(\))S XP /F25 43 32 2 -4 29 24 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFE0 FFFFFFE0 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000> PXLC RP 1060 864 XY F25(+)S XP /F25 49 21 4 0 17 28 28 16 0 <0300 0700 3F00 FF00 C700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 FFF8 FFF8> PXLC RP 1102 864 XY F25(1)S F26(;)S 1142 X F25(lev)S -1 x(\()S F26(\033)S 1 x F25(\)\))S F26(:)S 946 Y 300 X F25(The)S 389 X(t)S -1 x(yp)S 1 x(e)S 486 X(constructor)S 710 X F27(!)S 769 X F25(is)S 815 X (understo)S 1 x(o)S 1 x(d)S 1036 X(to)S 1090 X(b)S 1 x(e)S 1150 X (asso)S 1 x(ciativ)S -1 x(e)S 1361 X(to)S 1416 X(the)S 1491 X(righ)S -1 x(t)S XP /F25 44 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 1578 946 XY F25(,)S 1608 X(so)S 50 y 300 X F26(\032)S 338 X F27(!)S 397 X F26(\033)S 439 X F27(!)S 498 X F26(\034)S 538 X F25(means)S 668 X F26(\032)S 706 X F27(!)S 765 X F25(\()S F26(\033)S 823 X F27 (!)S 882 X F26(\034)S 905 X F25(\).)S XP /F25 78 31 2 0 28 28 28 32 0 PXLC RP 961 996 XY F25(Note)S 1064 X(that)S 1157 X(an)S -1 x(y)S 1239 X(t)S -1 x(yp)S 1 x(e)S 1336 X F26(\032)S 1374 X F25(can)S 1454 X(b)S 1 x (e)S 1513 X(written)S 50 y 300 X(uni)S XP /F25 113 22 1 -8 21 18 26 24 0 <07F180 1FF980 3F1F80 3C0F80 780780 780380 F00380 F00380 F00380 F00380 F00380 F00380 F80380 780780 7C0F80 3E1F80 1FFB80 07E380 000380 000380 000380 000380 000380 000380 001FF0 001FF0> PXLC RP 358 1046 XY F25(quely)S 471 X(in)S 522 X(the)S 597 X(form)S XP /F26 126 21 8 22 26 30 8 24 0 <000700 000700 000380 FFFFC0 FFFFC0 000F80 000E00 000C00> PXLC RP 698 1046 XY F26(~)S 698 X(\032)S 735 X F27(!)S 793 X F26(\023)S 824 X F25(:=)S 884 X F26(\032)S XP /F8 /cmr7 300 29.1 29.1 128 [-2 -8 32 22] PXLNF RP XP /F8 49 17 3 0 14 19 19 16 0 <0600 1E00 FE00 EE00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0> PXLC RP 905 1052 XY F8(1)S -6 y 940 X F27(!)S 998 X F26(\032)S XP /F8 50 17 2 0 14 19 19 16 0 <1F80 7FE0 E3E0 E0F0 E0F0 E070 00F0 00F0 00E0 01C0 0380 0700 0E00 1C30 3830 7070 FFE0 FFE0 FFE0> PXLC RP 1019 1052 XY F8(2)S -6 y 1054 X F27(!)S XP /F27 1 12 3 8 8 13 5 8 0 <70 F8 F8 F8 70> PXLC RP 1112 1046 XY F27(\001)S 1131 X(\001)S 1150 X(\001)S 1176 X(!)S 1234 X F26(\032)S XP /F9 /cmmi7 300 29.1 29.1 128 [-1 -7 32 22] PXLNF RP XP /F9 110 21 0 0 18 13 13 24 0 <3C7C00 6FCE00 6F8E00 EF0E00 CE0E00 0E0E00 1E1E00 1E1C00 1C1CC0 1C3CC0 3C39C0 3C3B80 381F00> PXLC RP 1255 1052 XY F9(n)S -6 y 1294 X F27(!)S 1352 X F26(\023)S 1383 X F25 (with)S 1481 X(a)S 1518 X(ground)S 50 y 300 X(t)S -1 x(yp)S 1 x(e)S 393 X F26(\023)S F25(,)S 433 X(and)S 514 X(its)S 572 X(lev)S -1 x (el)S 667 X(is)S 708 X(max\(lev\()S F26(\032)S 6 y F8(1)S -6 y 2 x F25(\))S F26(;)S 945 X F25(lev)S 1 x(\()S F26(\032)S 6 y F8(2)S -6 y 2 x F25(\))S F26(;)S 1089 X(:)S 1108 X(:)S 1127 X(:)S 1144 X(;)S 1163 X F25(lev\()S F26(\032)S 6 y F9(n)S -6 y 2 x F25(\)\))S 1316 X (+)S 1358 X(1.)S 50 y 375 X(Let)S XP /F27 30 32 3 -3 28 22 25 32 0 <00000180 00000180 00000380 00000380 00000300 00000700 00000F00 00001E00 00007C00 0001F800 0007E000 00FF8000 FFFE0000 FFFE0000 00FF8000 0007E000 0001F800 00007C00 00001E00 00000F00 00000700 00000380 00000380 00000180 00000180> PXLC RP 454 1146 XY F27(\036)S 505 X F25(b)S 1 x(e)S 566 X(a)S 606 X(giv)S -1 x(en)S 719 X(w)S -1 x(ell)S XP /F25 123 21 0 10 20 12 2 24 0 PXLC RP 789 1146 XY F25({ordering)S 979 X(of)S 1031 X(the)S 1107 X(natural)S 1257 X(n)S -1 x(um)S -1 x(b)S 1 x(ers.)S XP /F25 65 31 1 0 29 29 29 32 0 <00060000 000F0000 000F0000 000F0000 001F8000 001F8000 001F8000 001F8000 0033C000 0033C000 0033C000 0061E000 0061E000 0061E000 00C0F000 00C0F000 00C0F000 01807800 01807800 01FFF800 03FFFC00 03003C00 03003C00 06001E00 06001E00 0E001E00 1F001F00 FFC0FFF0 FFC0FFF0> PXLC RP 1456 1146 XY F25(A)S 1505 X F28(te)S XP /F28 114 18 3 0 19 18 18 16 0 <1E3E 3F7F 77E7 67C7 E78F C78F 070F 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800> PXLC RP 1538 1146 XY F28(r)S XP /F28 109 34 3 0 35 18 18 32 0 <1E1F87E0 3F7FDFF0 77F1FC70 67C1F070 E7C1F070 C781E070 0701C070 0F03C0F0 0F03C0E0 0E0380E0 0E0381E0 1E0781C7 1E0781C7 1C0703C6 1C07038E 3C0F03DC 3C0F01F8 380E01F0> PXLC RP 1556 1146 XY F28(m)S 1612 X F25(of)S XP /F25 80 28 2 0 25 28 28 24 0 PXLC RP 300 1195 XY F25(P)S XP /F25 67 30 2 -1 27 29 30 32 0 <003FC180 00FFF180 03F07B80 07C01F80 0F800F80 1F000780 3E000780 3E000380 7C000380 7C000180 78000180 F8000180 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 78000180 7C000180 7C000180 3E000380 3E000300 1F000700 0F800E00 07C01E00 03F87C00 00FFF000 003FC000> PXLC RP 328 1195 XY F25(C)S XP /F25 70 27 2 0 24 28 28 24 0 PXLC RP 358 1195 XY F25(F)S XP /F10 /cmsy7 300 29.1 29.1 128 [-1 -28 37 24] PXLNF RP XP /F10 30 26 3 -3 21 17 20 24 0 <0000C0 0000C0 0001C0 0001C0 000380 000780 001F00 007E00 03F800 FFE000 FFE000 03F800 007E00 001F00 000780 000380 0001C0 0001C0 0000C0 0000C0> PXLC RP 385 1202 XY F10(\036)S -7 y 428 X F25(is)S 471 X(built)S 571 X (from)S 670 X(t)S -1 x(yp)S 1 x(ed)S 787 X F28(v)S XP /F28 97 21 4 0 22 18 18 24 0 <03EE00 07FE00 0F3E00 1E1E00 3C1E00 781C00 781C00 783C00 F83C00 F03800 F03800 F079C0 F079C0 E0F180 F0F180 73F300 7FFF00 3E3E00> PXLC RP 806 1195 XY F28(aria)S XP /F28 98 19 5 0 19 29 29 16 0 <3F00 7F00 0F00 0F00 0E00 0E00 0E00 1E00 1C00 1C00 1C00 3FE0 3FF0 3E78 3C38 783C 703C 703C 707C F078 E078 E078 E0F8 E0F0 E0F0 E1E0 F3C0 7F80 3E00> PXLC RP 879 1195 XY F28(bles)S XP /F26 120 24 2 0 21 18 18 24 0 <0F8F80 3FDFC0 38F9C0 70F1C0 E0F3C0 C0E3C0 00E3C0 01E000 01E000 01C000 01C000 73C0E0 F3C0E0 F380C0 F781C0 E7C780 FFFF00 7C7C00> PXLC RP 963 1195 XY F26(x)S XP /F9 26 17 1 -6 16 13 19 16 0 <01F8 039C 070C 0E0E 1C0E 1C0E 3C1E 3C1C 381C 383C 7838 7CF0 77C0 7000 F000 F000 E000 E000 E000> PXLC RP 987 1180 XY F9(\032)S 15 y 2 x F26(;)S XP /F26 121 20 1 -8 20 18 26 24 0 <1F00E0 3F80E0 7380E0 6381E0 E381C0 C781C0 0701C0 0703C0 0F0380 0E0380 0E0380 0E0780 0E0700 0E0F00 0E0F00 0F3F00 0FFE00 03EE00 000E00 001E00 381C00 783C00 787800 71F000 7FE000 1F8000> PXLC RP 1024 1195 XY F26(y)S -15 y 1 x F9(\032)S 15 y 3 x F26(;)S XP /F26 122 19 1 0 18 18 18 24 0 <03E180 07F380 0FFF00 0E3F00 1C0E00 001C00 003800 007000 00E000 01C000 038000 070380 0E0300 1C0700 3F8F00 7FFE00 F3FC00 C0F000> PXLC RP 1084 1195 XY F26(z)S -15 y 2 x F9(\032)S 15 y 2 x F26(;)S 1143 X(:)S 1162 X(:)S 1181 X(:)S 1206 X F25(and)S 1288 X(the)S 1360 X(t)S -1 x (yp)S 1 x(ed)S 1477 X F28(c)S -2 x(onstants)S 50 y 300 X F25 (listed)S 410 X(b)S 1 x(elo)S -1 x(w)S 526 X(b)S -1 x(y)S 583 X F28 (applic)S -2 x(ation)S XP /F26 77 40 2 0 42 28 28 40 0 <03FF0001FF 03FF0003FF 003F0003F0 003F0007E0 003F000DE0 003F000FE0 0077801BE0 0067801BC0 00678033C0 00678067C0 00E78067C0 00C780C780 00C780C780 00C7818F80 01C7830F80 0187830F00 0187860F00 0183C61F00 0383CC1F00 0303CC1E00 0303D81E00 0303F03E00 0703F03E00 0603E03C00 0603E03C00 1F03C07C00 FFE387FFC0 FFE387FFC0> PXLC RP 795 1245 XY F26(M)S -15 y 840 X F9(\032)S XP /F10 33 33 2 2 29 12 10 32 0 <00000E00 00000600 00000700 00000380 FFFFFFE0 FFFFFFE0 00000380 00000700 00000600 00000E00> PXLC RP 857 1230 XY F10(!)S XP /F9 27 19 1 0 18 13 13 24 0 <07FF80 0FFF80 1FFF00 3C7800 783800 703800 F07800 F07000 E07000 E0F000 60E000 73C000 3F0000> PXLC RP 890 1230 XY F9(\033)S XP /F26 78 33 2 0 35 28 28 40 0 <03FE03FF80 03FE03FF80 003F007C00 003F003000 003F803000 003F807000 0077807000 0067C06000 0067C06000 0063C0E000 00E3E0E000 00C3E0C000 00C1E0C000 00C1F1C000 01C1F1C000 0180F18000 0180F98000 01807B8000 03807B8000 03007F0000 03003F0000 03003F0000 07003F0000 06001E0000 06001E0000 1F001E0000 FFE00E0000 FFE00C0000> PXLC RP 912 1245 XY F26(N)S -15 y 950 X F9(\032)S 15 y 982 X F25(and)S XP /F26 21 24 2 0 21 29 29 24 0 <0F8000 0FE000 01E000 00E000 00F000 00F000 007000 007800 007800 003800 003C00 003C00 001C00 001E00 001E00 001E00 003E00 007F00 00FF00 01E700 03C780 078780 0F0380 1E03C0 1C03C0 3C01C0 7801E0 F001E0 E000E0> PXLC RP 1062 1245 XY F26(\025)S F25({)S F28(abstr)S -2 x(action)S 1321 X F26 (\025x)S -15 y F9(\032)S 15 y 2 x F26(M)S -15 y 1433 X F9(\033)S 15 y 3 x F25(.)S 1485 X(The)S 1569 X(con)S XP /F25 45 14 0 7 11 10 3 16 0 PXLC RP 1631 1245 XY F25(-)S 50 y 300 X(stan)S -1 x(ts)S 422 X(are)S 1373 Y 330 X(\(1\))S XP /F27 63 32 2 0 30 28 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFF0 FFFFFFF0> PXLC RP 404 1373 XY F27(?)S -15 y F30(nat)S 15 y 520 X F25(and)S 601 X(0)S 636 X(of)S 683 X(t)S -1 x(yp)S 1 x(e)S 776 X F30(nat)S F25(,)S 1462 Y 330 X(\()S XP /F25 50 21 2 0 18 28 28 16 0 <0FE0 1FF8 38FC 703E E03E F01F F81F F81F F80F 700F 001F 001F 001E 003E 007C 0078 00F0 01E0 03C0 0380 0700 0E03 1C03 3803 7006 FFFE FFFE FFFE> PXLC RP 346 1462 XY F25(2\))S 404 X F27(?)S -15 y F30(b)S 1 x(o)S 1 x(ole)S 15 y 2 x F25(,)S 576 X(t)S 585 X(t)S 615 X(and)S 695 X(f)S 704 X(f)S 734 X(of)S 781 X(t)S -1 x(yp)S 1 x(e)S 875 X F30(b)S 1 x(o)S 1 x (ole)S F25(,)S 1542 Y 330 X(\()S XP /F25 51 21 2 -1 18 28 29 16 0 <0FE0 1FF8 3C7C 703E 783E 7C3E 783E 783E 003E 003C 007C 00F8 07F0 07E0 0078 003C 003E 001E 001F 001F 701F F81F F81F F81F F03E 703E 787C 3FF8 0FE0> PXLC RP 346 1542 XY F25(3\))S 404 X(the)S 474 X(sequen)S -1 x(tial)S 666 X (conditional)S 879 X(cond)S XP /F9 19 12 1 0 10 13 13 16 0 <1C00 1C00 3C00 3800 3800 3800 7800 7000 7180 F180 E300 E700 7C00> PXLC RP 965 1548 XY F9(\023)S -6 y 991 X F25(of)S 1037 X(t)S -1 x(yp)S 1 x (e)S 1129 X F30(b)S 1 x(o)S 1 x(ole)S 1253 X F27(!)S 1306 X F26 (\023)S 1332 X F27(!)S 1385 X F26(\023)S 1411 X F27(!)S 1464 X F26 (\023)S F25(,)S 1503 X(for)S 1565 X(eac)S -1 x(h)S 50 y 404 X (ground)S 545 X(t)S -1 x(yp)S 1 x(e)S 638 X F26(\023)S F25(,)S 1672 Y 330 X(\()S XP /F25 52 21 1 0 19 28 28 24 0 <001C00 001C00 003C00 007C00 007C00 00DC00 01DC00 019C00 039C00 071C00 061C00 0E1C00 0C1C00 181C00 381C00 301C00 601C00 E01C00 FFFFC0 FFFFC0 001C00 001C00 001C00 001C00 001C00 001C00 01FFC0 01FFC0> PXLC RP 346 1672 XY F25(4\))S 404 X(the)S 476 X(parallel)S 624 X (conditional)S 839 X(p)S 1 x(cond)S 10 y 1 x F9(\023)S -10 y 978 X F25(of)S 1025 X(t)S -1 x(yp)S 1 x(e)S 1119 X F30(b)S 1 x(o)S 1 x (ole)S 1244 X F27(!)S 1298 X F26(\023)S 1325 X F27(!)S 1379 X F26 (\023)S 1406 X F27(!)S 1460 X F26(\023)S F25(,)S 1501 X(for)S 1565 X (eac)S -1 x(h)S 50 y 404 X(ground)S 545 X(t)S -1 x(yp)S 1 x(e)S 638 X F26(\023)S F25(,)S 1802 Y 330 X(\()S XP /F25 53 21 2 -1 18 28 29 16 0 <301C 3FFC 3FF8 3FF0 3FC0 3000 3000 3000 3000 3000 33E0 3FF8 3E3C 381E 381E 300E 000F 000F 000F 000F F00F F00F F00F F01E E01E 703C 7878 3FF0 0FC0> PXLC RP 346 1802 XY F25(5\))S XP /F26 99 18 1 0 17 18 18 16 0 <00FC 03FE 0F8F 1E0F 3C1F 3C0E 7800 7800 F800 F000 F000 F000 F000 F006 700E 783C 3FF8 0FE0> PXLC RP 404 1802 XY F26(c)S 7 y F10(\036)S -7 y 462 X F25(and)S 541 X F26 (c)S XP /F8 61 25 2 2 22 12 10 24 0 PXLC RP 559 1809 XY F8(=)S -7 y 599 X F25(of)S 645 X(t)S -1 x(yp)S 1 x(e)S 737 X F30(nat)S 817 X F27(!)S 870 X F30(nat)S 949 X F27(!)S 1002 X F30(b)S 1 x(o)S 1 x(ole)S 1127 X F25(and)S 1206 X(succ)S 1295 X(of)S 1340 X(t)S -1 x(yp)S 1 x(e)S 1432 X F30(nat)S 1512 X F27(!)S 1565 X F30(nat)S -1 x F25(,)S 1883 Y 330 X(\()S XP /F25 54 21 2 -1 18 28 29 16 0 <01F8 07FC 0F8E 1E1E 1C1E 3C1E 381E 7800 7000 7080 F3F0 F7F8 FC3C F81E F80E F00F F00F F00F F00F F00F F00F 700F 700F 780E 381E 3C1C 1E3C 0FF8 07E0> PXLC RP 346 1883 XY F25(6\))S 404 X(for)S 465 X(eac)S -1 x(h)S 555 X(t)S -1 x(yp)S 1 x(e)S 646 X F26(\032)S F25(,)S 690 X(the)S 759 X (constan)S -1 x(t)S XP /F25 89 31 0 0 30 28 28 32 0 PXLC RP 924 1883 XY F25(Y)S 6 y F9(\032)S XP /F9 59 10 2 -6 6 4 10 8 0 PXLC RP 972 1889 XY F9(;)S F10(\036)S -6 y 1021 X F25(of)S 1066 X(t)S -1 x (yp)S 1 x(e)S 1156 X(\()S 1 x F26(~)S 1172 X(\032)S 1206 X F27(!)S 1259 X F30(nat)S -1 x F25(\))S 1355 X F27(!)S 1408 X F25(\()S F26 (\032)S 1457 X F27(!)S 1510 X F26(\032)S F25(\))S 1559 X F27(!)S 1612 X F26(\032)S F25(.)S 1960 Y 300 X(Application)S 529 X(is)S 576 X(understo)S 1 x(o)S 1 x(d)S 798 X(to)S 853 X(b)S 1 x(e)S 914 X (asso)S 1 x(ciativ)S -1 x(e)S 1125 X(to)S 1181 X(the)S 1257 X (left,)S 1347 X(so)S 1402 X F26(M)S 1447 X(N)S XP /F26 76 28 2 0 26 28 28 24 0 <03FFF0 03FFF0 003E00 003C00 003C00 003C00 007C00 007800 007800 007800 00F800 00F000 00F000 00F000 01F000 01E000 01E000 01E007 03E006 03C006 03C00E 03C00C 07C01C 07803C 078078 0F81F8 FFFFF8 FFFFF0> PXLC RP 1485 1960 XY F26(L)S 1532 X F25(means)S 50 y 300 X(\()S F26(M)S 361 X(N)S 399 X F25(\))S F26(L)S F25(.)S 50 y 375 X(Note)S 478 X (that)S 570 X(in)S 621 X(con)S -1 x(trast)S 785 X(to)S XP /F25 91 12 4 -10 11 31 41 8 0 PXLC RP 838 2060 XY F25([)S XP /F25 57 21 2 -1 18 28 29 16 0 <07E0 1FF0 3E78 381C 781C 700E F00E F00E F00F F00F F00F F00F F00F F01F 701F 781F 3C3F 1FEF 0FCF 010F 000E 001E 781E 781C 783C 7878 78F0 3FE0 1F80> PXLC RP 850 2060 XY F25(9)S XP /F25 93 12 0 -10 7 31 41 8 0 PXLC RP 870 2060 XY F25(])S 898 X(w)S -1 x(e)S 962 X(do)S 1022 X(need)S 1122 X(a)S 1159 X(constan)S -1 x(t)S 1329 X(for)S 1396 X(the)S 1470 X (unde)S (\014ned)S 50 y 300 X(ob)S XP /F25 106 13 -3 -8 8 29 37 16 0 <01C0 03E0 03E0 03E0 01C0 0000 0000 0000 0000 0000 0000 0FE0 0FE0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 00E0 F0E0 F1E0 F3C0 FF80 7E00> PXLC RP 346 2110 XY F25(ject)S XP /F27 59 21 2 -3 18 32 35 16 0 <0018 0018 03F8 0FF8 1C38 383C 307C 707E 6066 6066 60E6 E0E7 E0C7 E0C7 E1C7 E1C7 E187 E187 E187 E387 E387 E307 E307 E707 E707 6606 760E 7E0E 3E0C 3C1C 1C38 1FF0 1FC0 1800 1800> PXLC RP 432 2110 XY F27(;)S 472 X F25(in)S 526 X(our)S 606 X(ground)S 753 X (domains.)S 951 X(This)S 1051 X(is)S 1099 X(necessary)S -3 x(,)S 1298 X(since)S 1406 X(w)S -1 x(e)S 1473 X(w)S -1 x(an)S -1 x(t)S 1580 X(an)S -1 x(y)S 50 y 300 X(reduction)S 480 X(sequence)S 647 X (to)S 694 X(terminate,)S 891 X(p)S 1 x(ossibly)S 1046 X(with)S 1136 X (a)S 1166 X(sym)S -1 x(b)S 1 x(ol)S 1303 X(indicating)S 1492 X(an)S 1545 X(error.)S 56 y 300 X(The)S 385 X(constan)S -1 x(ts)S 569 X F27 (?)S -15 y F30(nat)S 15 y 686 X F25(and)S 766 X F27(?)S -15 y F30 (b)S 1 x(o)S 1 x(ole)S 15 y 927 X F25(serv)S -1 x(e)S 1031 X(as)S 1082 X(suc)S -1 x(h)S 1175 X(error)S 1277 X(sym)S -1 x(b)S 1 x (ols.)S 50 y 375 X(The)S 459 X(term)S 557 X(succ)S 1 x(\(succ)S 733 X F26(:)S 752 X(:)S 771 X(:)S 788 X F25(\(succ)S 892 X(0\))S 936 X F26(:)S 955 X(:)S 974 X(:)S -2 x F25(\))S 1013 X(with)S XP /F26 110 25 1 0 24 18 18 24 0 <3E1F80 3F3FE0 7771E0 67E0E0 E7C0E0 C780E0 0700E0 0F01E0 0F01C0 0E01C0 0E03C0 1E038E 1E038E 1C078C 1C071C 3C07B8 3C03F0 3801E0> PXLC RP 1106 2266 XY F26(n)S 1144 X F25(o)S 1 x(ccurrence)S 3 x(s)S 1366 X (of)S 1412 X(succ)S 1501 X(is)S 1541 X(called)S 49 y 300 X(a)S 335 X (n)S -1 x(umeral)S 495 X(and)S 576 X(is)S 617 X(denoted)S 774 X(b)S -1 x(y)S 832 X F30(n)S F25(.)S XP /F48 50 28 3 0 24 32 32 24 0 <03FC00 1FFF80 3FFFC0 7C3FE0 FC1FF0 FE0FF0 FE07F8 FE07F8 FE03F8 7C03F8 3807F8 0007F8 0007F0 000FF0 000FE0 001FC0 001F80 003F00 007E00 00F800 01F000 01E038 03C038 078038 0F0038 1C0078 3FFFF0 7FFFF0 FFFFF0 FFFFF0 FFFFF0 FFFFF0> PXLC RP 300 2430 XY F48(2.2)S XP /F48 68 43 2 0 39 34 34 40 0 PXLC RP 428 2430 XY F48(D)S XP /F48 101 26 2 0 23 22 22 24 0 <01FE00 07FF80 1FC7E0 3F03E0 3F03F0 7E01F0 7E01F8 FE01F8 FE01F8 FFFFF8 FFFFF8 FE0000 FE0000 FE0000 FE0000 7E0000 7F0000 3F0038 1F8078 0FE1F0 07FFE0 00FF80> PXLC RP 471 2430 XY F48(enotat)S XP /F48 105 16 1 0 13 36 36 16 0 <1F00 3F80 7F80 7F80 7F80 7F80 3F80 1F00 0000 0000 0000 0000 0000 0000 FF80 FF80 FF80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 1F80 FFF0 FFF0 FFF0> PXLC RP 626 2430 XY F48(iona)S XP /F48 108 16 1 0 13 35 35 16 0 PXLC RP 728 2430 XY F48(l)S XP /F48 115 22 2 0 19 22 22 24 0 <0FFF00 3FFF00 7C1F00 780F00 F00700 F00700 F80000 FF0000 FFF800 7FFC00 7FFE00 1FFF00 0FFF80 007F80 E00F80 E00F80 F00780 F00780 F80F00 FE1F00 FFFE00 C7F800> PXLC RP 762 2430 XY F48(se)S XP /F48 109 47 2 0 44 22 22 48 0 PXLC RP 810 2430 XY F48(man)S -2 x(ti)S XP /F48 99 25 2 0 22 22 22 24 0 <01FF80 07FFE0 1FC3F0 3F83F0 3F03F0 7F03F0 7E01E0 FE0000 FE0000 FE0000 FE0000 FE0000 FE0000 FE0000 FE0000 7F0000 7F0000 3F8070 3F80F0 1FE1E0 07FFC0 01FF00> PXLC RP 951 2430 XY F48(cs)S 2507 Y 300 X F25(Let)S 377 X(Con)S -1 x(t)S 6 y -1 x F9(\032)S -6 y 501 X F25(b)S 1 x(e)S 560 X(the)S 634 X (domain)S 783 X(of)S 833 X(the)S 907 X(partial)S 1043 X(con)S -1 x (tin)S -1 x(uous)S 1253 X(functionals)S 1467 X(of)S 1516 X(t)S -1 x (yp)S 1 x(e)S 1612 X F26(\032)S F25(,)S 50 y 300 X(in)S 352 X(the)S 427 X(sense)S 536 X(of)S 587 X(Scott)S 699 X(and)S XP /F25 69 28 2 0 26 28 28 24 0 PXLC RP 783 2557 XY F25(Erso)S -1 x(v)S 902 X(\(cf.)S 989 X([13)S -1 x(],)S 1083 X([9])S 1144 X(or)S 1198 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S XP /F25 39 12 3 16 9 29 13 8 0 <70 F8 FC FC 7C 0C 0C 1C 18 38 70 F0 60> PXLC RP 1478 2557 XY F25('s)S 1523 X(lecture)S 50 y 300 X(notes)S 409 X ([12)S -1 x(]\).)S 520 X(F)S -3 x(or)S 595 X(an)S -1 x(y)S 673 X (constan)S -1 x(t)S 841 X F26(c)S -15 y F9(\032)S 15 y 892 X F25 (of)S 940 X(the)S 1011 X(language)S 1185 X(PCF)S 6 y F10(\036)S -6 y 1312 X F25(w)S -1 x(e)S 1373 X(de\014ne)S 1493 X(its)S 1552 X(v)S -2 x(alue)S 49 y 300 X([)S 305 X([)S F26(c)S -15 y F9(\032)S 15 y 1 x F25(])S 358 X(])S XP /F27 50 28 3 -3 24 23 26 24 0 <00FFF8 03FFF8 0FC000 1F0000 3C0000 380000 700000 700000 E00000 E00000 C00000 C00000 FFFFF8 FFFFF8 C00000 C00000 E00000 E00000 700000 700000 380000 3C0000 1F0000 0FC000 03FFF8 00FFF8> PXLC RP 381 2656 XY F27(2)S 420 X F25(Con)S -1 x(t)S 7 y F9(\032)S -7 y 2 x F25(,)S 554 X(as)S 605 X(follo)S -1 x(ws.)S 2753 Y 309 X([)S 314 X ([)S F27(?)S -17 y F30(nat)S 17 y 1 x F25(])S 433 X(])S 455 X(:=)S 511 X([)S 516 X([)S F27(?)S -17 y F30(b)S 1 x(o)S 1 x(ole)S 17 y 1 x F25(])S 678 X(])S 701 X(:=)S 756 X F27(;)S F25(;)S 837 X([)S 842 X([t)S 863 X(t])S 884 X(])S 906 X(:=)S XP /F27 102 21 2 -10 18 31 41 16 0 <001F 007F 00F8 01E0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 03C0 0F80 FF00 FC00 FF00 0F80 03C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01E0 00F8 007F 001F> PXLC RP 962 2753 XY F27(f)S F25(true)S XP /F27 103 21 2 -10 18 31 41 16 0 PXLC RP 1056 2753 XY F27(g)S F25(;)S 1137 X([)S 1142 X([f)S 1163 X(f)S 2 x (])S 1183 X(])S 1206 X(:=)S 1261 X F27(f)S F25(false)S F27(g)S F25 (;)S 1442 X([)S 1447 X([0])S 1485 X(])S 1507 X(:=)S 1562 X F27(f)S F25(0)S F27(g)S F26(:)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(3)S XP /F25 56 21 2 -1 18 28 29 16 0 <07E0 1FF8 3C3C 301C 700E 6006 6006 7006 780E 7C0C 3F1C 3FF8 1FF0 0FF0 1FF8 3DFC 787E 703F F01F E00F E007 E007 E007 F007 700E 781E 3C3C 1FF8 07E0> PXLC RP 321 462 XY F25(88)S XP /F18 /cmr9 300 37.4 37.4 128 [-3 -10 37 28] PXLNF RP XP /F18 72 29 1 0 27 26 26 32 0 PXLC RP 700 462 XY F18(H)S XP /F18 46 11 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 729 462 XY F18(.)S XP /F18 83 21 2 0 18 26 26 16 0 <0FC6 3FF6 787E 701E E00E E00E E006 E006 F000 FC00 7F80 7FF0 3FFC 0FFE 01FE 003F 000F 000F C007 C007 E007 E00F F00E FC3C DFFC C7F0> PXLC RP 752 462 XY F18(S)S XP /F18 99 17 1 0 15 16 16 16 0 <07F8 1FFC 3E3C 783C 7818 F000 F000 F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 774 462 XY F18(c)S XP /F18 104 21 1 0 19 26 26 24 0 PXLC RP 790 462 XY F18(h)S XP /F18 119 28 1 0 26 16 16 32 0 PXLC RP 810 462 XY F18(w)S XP /F18 105 11 0 0 9 26 26 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 838 462 XY F18(ic)S -1 x(h)S XP /F18 116 15 1 0 12 23 23 16 0 <0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFC0 FFC0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C60 1C60 1C60 1C60 1EE0 0FC0 0780> PXLC RP 885 462 XY F18(t)S XP /F18 101 17 1 0 15 16 16 16 0 <07E0 1FF0 3C78 783C 703C FFFC FFFC F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 900 462 XY F18(e)S XP /F18 110 21 1 0 19 16 16 24 0 PXLC RP 917 462 XY F18(n)S XP /F18 98 21 1 0 19 26 26 24 0 PXLC RP 937 462 XY F18(b)S 1 x(e)S XP /F18 114 15 1 0 13 16 16 16 0 PXLC RP 976 462 XY F18(r)S XP /F18 103 19 1 -8 17 16 24 16 0 <0FCF 1FFF 3CF7 7878 7878 7878 7878 7878 3CF0 3FE0 3FC0 3000 3000 3FF8 3FFC 3FFE 701F E00F E007 E007 F00F 7C3E 3FFC 0FF0> PXLC RP 991 462 XY F18(g)S XP /F18 97 19 1 0 18 16 16 24 0 <3FC000 7FF000 78F800 787800 303800 003800 07F800 3F3800 7C3800 F83800 F03800 F03980 F07980 F8F980 7FFF00 3F1E00> PXLC RP 1023 462 XY F18(an)S XP /F18 100 21 1 0 19 26 26 24 0 <007E00 007E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 0FEE00 1FFE00 3E3E00 781E00 780E00 F00E00 F00E00 F00E00 F00E00 F00E00 F00E00 780E00 781E00 3C3E00 1FFFC0 0FCFC0> PXLC RP 1064 462 XY F18(d)S 1098 X(S.S.)S XP /F18 87 39 1 0 37 26 26 40 0 PXLC RP 1175 462 XY F18(W)S -3 x(ainer)S 597 Y 653 X F25([)S 658 X([cond)S 6 y F9(\023)S -6 y 2 x F25(])S 774 X(])S XP /F26 112 21 -1 -8 21 18 26 24 0 <0F87C0 0FDFF0 1DFC78 19F078 39E038 31E03C 01C03C 03C07C 03C078 038078 038078 0780F8 0780F0 0781E0 0783E0 0FC7C0 0FFF00 0E7C00 0E0000 1E0000 1E0000 1C0000 1C0000 3C0000 FF8000 FF8000> PXLC RP 785 597 XY F26(pxy)S 863 X F25(:=)S XP /F67 /cmex10 300 41.5 41.5 128 [-1 -123 60 32] PXLNF RP XP /F67 56 37 16 -38 29 -1 37 16 0 <0038 0078 01F0 03E0 07C0 0F80 0F80 1F00 3E00 3E00 7C00 7C00 7C00 FC00 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800> PXLC RP 919 512 XY F67(8)S XP /F67 60 37 7 -76 21 1 77 16 0 <007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 00F8 00F8 00F8 00F0 01F0 01F0 03E0 03C0 07C0 0F80 1F00 3E00 7C00 F000 E000 F000 7C00 3E00 1F00 0F80 07C0 03C0 03E0 01F0 01F0 00F0 00F8 00F8 00F8 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C> PXLC RP 919 549 XY F67(<)S XP /F67 58 37 16 -36 29 1 37 16 0 PXLC RP 919 624 XY F67(:)S 548 Y 963 X F26(x;)S 1039 X F25(if)S 1077 X F26 (p)S 1110 X F25(=)S 1154 X F27(f)S F25(true)S F27(g)S F25(;)S 50 y 963 X F26(y)S 1 x(;)S 1039 X F25(if)S 1077 X F26(p)S 1110 X F25(=)S 1154 X F27(f)S F25(false)S -1 x F27(g)S F25(;)S 50 y 963 X F27(;)S F26(;)S 1039 X F25(if)S 1077 X F26(p)S 1110 X F25(=)S 1154 X F27(;)S F25(.)S 794 Y 594 X([)S 599 X([p)S 1 x(cond)S 11 y F9(\023)S -11 y 2 x F25(])S 739 X(])S F26(pxy)S 828 X F25(:=)S 697 Y 883 X F67(8)S XP /F67 62 37 16 -13 21 1 14 8 0 PXLC RP 883 734 XY F67(>)S 13 y 883 X(<)S 74 y 883 X(>)S 13 y 883 X(:)S 721 Y 927 X F26(x;)S 1004 X F25(if)S 1042 X F26(p)S 1074 X F25(=)S 1118 X F27(f)S F25(true)S 1 x F27(g)S F25(;)S 50 y 927 X F26(y)S 1 x(;)S 1004 X F25(if)S 1042 X F26(p)S 1074 X F25(=)S 1118 X F27(f)S F25(false)S F27(g)S F25(;)S 50 y 927 X F26(x;)S 1004 X F25(if)S 1042 X F26(p)S 1074 X F25(=)S 1118 X F27(;)S 1153 X F25(and)S 1234 X F26(x)S 1269 X F25(=)S 1313 X F26(y)S 1 x F25(;)S 49 y 927 X F27(;)S F26(;)S 1004 X F25(if)S 1042 X F26(p)S 1074 X F25(=)S 1118 X F27(;)S 1153 X F25(and)S 1234 X F26(x)S XP /F27 54 0 6 -10 25 30 40 24 0 <000060 0000E0 0000E0 0001C0 0001C0 000180 000380 000380 000700 000700 000E00 000E00 001C00 001C00 001800 003800 003800 007000 007000 00E000 00E000 01C000 01C000 038000 038000 030000 070000 070000 0E0000 0E0000 1C0000 1C0000 380000 380000 300000 700000 700000 E00000 E00000 C00000> PXLC RP 1269 870 XY F27(6)S F25(=)S 1313 X F26(y)S 1 x F25(.)S XP /F25 73 15 1 0 13 28 28 16 0 PXLC RP 300 938 XY F25(It)S 345 X(is)S 386 X(w)S -1 x(ell{)S XP /F25 107 22 1 0 20 29 29 24 0 PXLC RP 477 938 XY F25(kno)S -1 x(wn)S 608 X(\(cf.)S 686 X([9)S -1 x(],)S 754 X(p.23)S XP /F25 55 21 2 -1 19 28 29 24 0 <600000 7FFF80 7FFF80 7FFF80 600700 C00600 C00E00 C01C00 001800 003800 007000 006000 00E000 00E000 01C000 01C000 01C000 03C000 038000 038000 038000 078000 078000 078000 078000 078000 078000 078000 078000> PXLC RP 831 938 XY F25(7\))S 881 X(that)S 971 X([)S 976 X([cond)S 6 y -1 x F9(\023)S -6 y 3 x F25(])S 1092 X(])S 1116 X(can)S 1192 X(b)S 1 x (e)S 1248 X(de\014ned)S 1391 X(from)S 1489 X([)S 1494 X([p)S 1 x (cond)S 10 y -1 x F9(\023)S -10 y 3 x F25(])S 1634 X(])S 50 y 300 X (b)S -1 x(y)S 50 y 498 X([)S 503 X([cond)S 6 y F9(\023)S -6 y 2 x F25(])S 619 X(])S F26(pxy)S 708 X F25(=)S 752 X([)S 757 X([p)S 1 x (cond)S 10 y F9(\023)S -10 y 2 x F25(])S 897 X(])S F26(p)S F25(\([)S 951 X([p)S 1 x(cond)S 10 y -1 x F9(\023)S -10 y 2 x F25(])S 1090 X (])S F26(px)S F27(;)S F25(\)\([)S 1205 X([p)S 1 x(cond)S 10 y -1 x F9(\023)S -10 y 2 x F25(])S 1344 X(])S F26(p)S F27(;)S F26(y)S 1 x F25(\))S F26(:)S XP /F25 72 31 2 0 28 28 28 32 0 PXLC RP 300 1112 XY F25(Ho)S -1 x(w)S -1 x(ev)S -1 x(er,)S 482 X(w)S -1 x (e)S 546 X(m)S -1 x(ust)S 652 X(k)S -1 x(eep)S 749 X(the)S 823 X (cond)S 6 y 1 x F9(\023)S -6 y 939 X F25(as)S 993 X(separate)S 1160 X (constan)S -1 x(ts)S 1346 X(in)S 1398 X(our)S 1474 X(language,)S 50 y 300 X(since)S 399 X(they)S 490 X(will)S 566 X(b)S 1 x(e)S 619 X (giv)S -1 x(en)S 725 X(a)S 757 X(sp)S 1 x(ecial)S 890 X(status)S 1010 X(in)S 1056 X(our)S 1127 X(op)S 1 x(erational)S 1342 X(seman)S -1 x(tics,)S 1540 X(whic)S -1 x(h)S 50 y 300 X(mak)S -1 x(es)S 427 X (it)S 471 X(p)S 1 x(ossible)S 630 X(to)S 684 X(use)S 758 X(them)S 867 X(as)S XP /F25 92 21 5 16 19 29 13 16 0 <1818 3C3C 3838 7070 6060 E0E0 C0C0 C0C0 F8F8 FCFC FCFC 7C7C 3838> PXLC RP 920 1212 XY F25(\\guards)S XP /F25 34 21 1 16 15 29 13 16 0 <7070 F8F8 FCFC FCFC 7C7C 0C0C 0C0C 1C1C 1818 3838 7070 F0F0 6060> PXLC RP 1061 1212 XY F25(")S 1098 X(blo)S 1 x(c)S -1 x(king)S 1266 X (in\014nite)S 1409 X(reduction)S 1596 X(se-)S 50 y 300 X(quences.)S 1400 Y 472 X([)S 477 X([)S F26(c)S 6 y F10(\036)S -6 y 1 x F25(])S 539 X(])S F26(xy)S 607 X F25(:=)S 1315 Y 663 X F67(8)S 37 y 663 X (<)S 75 y 663 X(:)S 1351 Y 707 X F27(f)S F25(true)S F27(g)S F26(;)S 881 X F25(if)S 919 X F26(x)S 954 X F25(=)S 998 X F27(f)S F26(n)S F27 (g)S F25(,)S 1090 X F26(y)S 1123 X F25(=)S 1167 X F27(f)S XP /F26 109 36 1 0 36 18 18 40 0 <3E0FC1F800 3F3FE3FE00 7778F71E00 67E07E0E00 E7C07C0E00 C780780E00 0780700E00 0F00F01E00 0F00F01C00 0E00E01C00 0E00E03C00 1E01E038E0 1E01E038E0 1C01C078C0 1C01C071C0 3C03C07B80 3C03C03F00 3803801E00> PXLC RP 1188 1351 XY F26(m)S F27(g)S 1259 X F25(and)S 1340 X F26(n)S 1376 X F27(\036)S 1420 X F26(m)S F25(;)S 50 y 707 X F27(f)S F25(false)S -1 x F27(g)S F26(;)S 881 X F25(if)S 919 X F26(x)S 954 X F25(=)S 998 X F27(f)S F26(n)S F27(g)S F25(,)S 1090 X F26(y)S 1123 X F25(=)S 1167 X F27(f)S F26(m)S F27(g)S 1259 X F25(and)S 1340 X F26(n)S 1376 X F27(6\036)S 1420 X F26(m)S F25(;)S 50 y 707 X F27(;)S F26(;)S 881 X F25(if)S 919 X F26(x)S 954 X F25(=)S 998 X F27(;)S 1033 X F25(or)S 1084 X F26(y)S 1117 X F25(=)S 1161 X F27(;)S F25(,)S 1538 Y 300 X (and)S 381 X(similarly)S 551 X(for)S 614 X F26(c)S 6 y F8(=)S -6 y 673 X F25(and)S 754 X(succ)S 2 x(.)S 50 y 375 X(The)S 458 X(v)S -2 x (alue)S 564 X([)S 569 X([Y)S 6 y F9(\032;)S F10(\036)S -6 y 1 x F25 (])S 671 X(])S 695 X(is)S 735 X(to)S 784 X(b)S 1 x(e)S 839 X(an)S 895 X(ob)S 2 x(ject)S 1019 X(of)S 1065 X(t)S -1 x(yp)S 1 x(e)S 1156 X (\()S 1 x F26(~)S 1172 X(\032)S 1206 X F27(!)S 1259 X F30(nat)S -1 x F25(\))S 1355 X F27(!)S 1408 X F25(\()S F26(\032)S 1457 X F27(!)S 1510 X F26(\032)S F25(\))S 1559 X F27(!)S 1612 X F26(\032)S F25(.)S 49 y 300 X(So)S 354 X(let)S 411 X(ob)S 2 x(jects)S XP /F26 22 25 1 -9 24 18 27 24 0 <038070 038070 038070 0780F0 0780F0 0700E0 0700E0 0F01E0 0F01E0 0E01C0 0E01C0 1E03CE 1E03CE 1E078C 1E0F8C 3F1FDC 3FFBF8 3BF0F0 380000 780000 780000 700000 700000 F00000 F00000 E00000 E00000> PXLC RP 550 1637 XY F26(\026)S 585 X F25(of)S 629 X(t)S -1 x(yp)S 1 x(e)S 720 X F26(~)S 719 X(\032)S 752 X F27(!)S 805 X F30(nat)S F25(,)S XP /F26 71 33 2 -1 31 29 30 32 0 <0003FC18 001FFF38 003F07F8 00FC03F0 01F001F0 03E000F0 07C000F0 0F8000E0 1F000060 1F000060 3E0000E0 3E0000C0 7C000000 7C000000 7C000000 FC000000 F8000000 F800FFF0 F801FFF0 F8000F80 F8000F00 F8000F00 F8001F00 78001F00 7C001E00 3E003E00 1F007E00 0FC1EE00 07FFC400 01FE0000> PXLC RP 896 1637 XY F26(G)S 940 X F25(of)S 984 X(t)S -1 x(yp)S 1 x(e)S 1074 X F26(\032)S 1107 X F27(!)S 1160 X F26(\032)S 1192 X F25(and)S 1268 X F26(~)S 1269 X(x)S 1303 X F25(of)S 1347 X(t)S -1 x(yp)S 1 x(es)S 1454 X F26(~)S 1454 X(\032)S 1486 X F25(b)S 1 x(e)S 1539 X(giv)S -1 x(en.)S XP /F25 87 43 1 -1 41 28 29 40 0 PXLC RP 300 1687 XY F25(W)S -3 x(e)S 372 X(view)S 469 X F26(\026)S 509 X F25 (as)S 561 X(a)S 596 X(measure)S 759 X(function)S 923 X(mapping)S 1094 X(an)S -1 x(y)S 1174 X(sequence)S 1346 X F26(~)S 1347 X(y)S 1383 X F25(of)S 1432 X(ob)S 2 x(jects)S 1575 X(in)S -1 x(to)S 50 y 300 X(the)S 375 X(domain)S 526 X(of)S 576 X(the)S 651 X(w)S -1 x (ell{ordering)S 909 X F27(\036)S F25(,)S 971 X(whic)S -1 x(h)S 1093 X (is)S 1138 X(the)S 1213 X(set)S 1281 X(of)S 1331 X(natural)S 1480 X (n)S -1 x(um)S -1 x(b)S 1 x(ers.)S 50 y 300 X(W)S -3 x(e)S 371 X (no)S -1 x(w)S 457 X(de\014ne)S 576 X([)S 581 X([Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F25(])S 684 X(])S F26(\026G)S -1 x(~)S 754 X(x)S 789 X F25(b)S -1 x(y)S 846 X(trans\014nite)S 1044 X(recursion)S 1222 X(on)S 1279 X F26(\026)S -1 x(~)S 1304 X(x)S F25(,)S 1353 X (as)S 1403 X(follo)S -1 x(ws.)S 1555 X(If)S 1596 X F26(\026)S -1 x (~)S 1621 X(x)S 50 y 300 X F25(is)S 342 X(unde\014ned,)S 542 X (then)S 637 X([)S 642 X([Y)S 6 y F9(\032;)S F10(\036)S -6 y 1 x F25 (])S 744 X(])S F26(\026G)S -1 x(~)S 814 X(x)S 850 X F25(is)S 892 X (unde)S (\014ned.)S XP /F25 79 32 2 -1 29 29 30 32 0 <003F8000 00FFE000 03E0F800 07803C00 0F001E00 1E000F00 3E000F80 3C000780 7C0007C0 7C0007C0 780003C0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 780003C0 7C0007C0 7C0007C0 3C000780 3E000F80 1E000F00 0F001E00 07803C00 03E0F800 00FFE000 003F8000> PXLC RP 1097 1837 XY F25(Otherwise,)S 1305 X(assume)S 1448 X(that)S 1538 X (for)S 1601 X(all)S 50 y 300 X F26(~)S 300 X(y)S 336 X F25(with)S 430 X F26(\026~)S 455 X(y)S 489 X F27(\036)S 532 X F26(\026)S -1 x (~)S 557 X(x)S 595 X F25(the)S 666 X(v)S -2 x(alue)S 774 X([)S 779 X ([Y)S 6 y F9(\032;)S F10(\036)S -6 y 1 x F25(])S 881 X(])S F26 (\026G~)S 951 X(y)S 986 X F25(is)S 1027 X(already)S 1174 X (de\014ned.)S 1333 X(Then)S 1441 X(let)S 1977 Y 799 X([)S 804 X([Y)S 6 y F9(\032;)S F10(\036)S -6 y 1 x F25(])S 906 X(])S F26(\026G)S -1 x(~)S 976 X(x)S 1010 X F25(:=)S 1065 X F26(G)S XP /F26 102 20 2 -8 23 29 37 24 0 <0003F0 0007F0 000EF8 000EF8 001C70 001C00 001C00 001C00 003C00 003800 003800 03FFC0 03FFC0 007800 007000 007000 007000 007000 007000 00F000 00E000 00E000 00E000 00E000 01E000 01C000 01C000 01C000 01C000 03C000 038000 038000 F38000 F70000 FF0000 FE0000 7C0000> PXLC RP 1098 1977 XY F26(f)S 1122 X(~)S 1122 X(x)S 2068 Y 300 X F25(where)S 420 X F26(f)S 459 X F25(is)S 500 X(de\014ned)S 643 X(b)S -1 x(y)S 2181 Y 668 X F26(f)S 692 X(~)S 692 X(y)S 727 X F25(:=)S XP /F67 26 31 5 -97 25 2 99 24 0 <000070 0001F0 0007E0 000F80 001F00 003E00 003C00 007C00 007800 00F800 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 01F000 01E000 03E000 03C000 078000 0F8000 1F0000 3C0000 F80000 E00000 F80000 3C0000 1F0000 0F8000 078000 03C000 03E000 01E000 01F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F000 00F800 007800 007C00 003C00 003E00 001F00 000F80 0007E0 0001F0 000070> PXLC RP 782 2122 XY F67(\032)S 35 y 820 X F25([)S 825 X([Y)S 6 y F9(\032;)S F10(\036)S -6 y 1 x F25(])S 927 X(])S F26(\026G~)S 997 X(y)S 1 x(;)S 1071 X F25(if)S 1109 X F26(\026~)S 1134 X(y)S 1167 X F27(\036)S 1211 X F26(\026)S -1 x(~)S 1236 X(x)S F25(;)S 50 y 820 X F27(;)S F26 (;)S 1071 X F25(otherwise.)S 2294 Y 375 X(Note)S 478 X(that)S 571 X (all)S 632 X(\014nite)S 741 X(functionals)S 955 X(from)S 1056 X (Con)S -1 x(t)S 6 y F9(\032)S -6 y 1181 X F25(are)S 1254 X (de\014nable)S 1432 X(b)S -1 x(y)S 1493 X(terms)S 1612 X(of)S 50 y 300 X(PCF)S 6 y F10(\036)S -6 y 430 X F25(not)S 507 X(in)S -1 x(v)S -1 x(olving)S 688 X(\014xed)S 791 X(p)S 1 x(oin)S -1 x(t)S 902 X (op)S 1 x(erators;)S 1102 X(this)S 1186 X(has)S 1263 X(b)S 1 x(een)S 1364 X(observ)S -1 x(ed)S 1538 X(in)S 1590 X([9)S -1 x(],)S 50 y 300 X(Lemma)S 448 X(4.5.)S 2510 Y 300 X F48(2.)S XP /F48 51 28 2 0 25 32 32 24 0 <01FE00 07FFC0 1FFFE0 1F0FF0 3F07F8 3F07F8 3F83F8 3F03F8 3F03F8 0E07F8 0007F0 0007F0 000FE0 001FC0 01FF00 01FF00 000FE0 0003F8 0003FC 0001FC 0001FE 0001FE 7C01FE FE01FE FE01FE FE01FE FE03FC FE03FC 7E07F8 3FFFF0 1FFFC0 03FF00> PXLC RP 344 2510 XY F48(3)S XP /F48 79 42 3 0 38 34 34 40 0 <0007FC0000 003FFF8000 00FE0FE000 03F803F800 07E000FC00 0FC0007E00 1FC0007F00 1F80003F00 3F80003F80 3F80003F80 7F00001FC0 7F00001FC0 7F00001FC0 FF00001FE0 FF00001FE0 FF00001FE0 FF00001FE0 FF00001FE0 FF00001FE0 FF00001FE0 FF00001FE0 FF00001FE0 7F00001FC0 7F80003FC0 7F80003FC0 3F80003F80 3FC0007F80 1FC0007F00 0FE000FE00 07F001FC00 03F803F800 00FE0FE000 003FFF8000 0007FC0000> PXLC RP 428 2510 XY F48(O)S XP /F48 112 31 2 -10 28 22 32 32 0 PXLC RP 470 2510 XY F48(p)S 2 x(e)S XP /F48 114 23 2 0 21 22 22 24 0 PXLC RP 528 2510 XY F48(rational)S 736 X(seman)S -2 x(tics)S 2587 Y 300 X F25(The)S 385 X(op)S 1 x(erational)S 603 X(seman)S -1 x(tics)S 792 X (is)S 834 X(giv)S -1 x(en)S 942 X(b)S -1 x(y)S 1000 X(the)S 1071 X (follo)S -1 x(wing)S 1246 X(con)S -1 x(v)S -1 x(ersion)S 1448 X (rules.)S 2718 Y 330 X(\(1\))S 404 X(\()S F26(\025x)S -15 y F9 (\032)S 15 y 2 x F26(M)S 532 X F25(\))S F26(N)S 598 X F27(!)S 651 X F26(M)S 696 X F25([)S F26(N)S XP /F26 61 21 2 -10 18 31 41 16 0 <0003 0007 0007 0006 000E 000E 000C 001C 001C 0018 0038 0038 0030 0070 0070 00E0 00E0 00C0 01C0 01C0 0180 0380 0380 0300 0700 0700 0E00 0E00 0C00 1C00 1C00 1800 3800 3800 3000 7000 7000 6000 E000 E000 C000> PXLC RP 741 2718 XY F26(=x)S F25(].)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 130 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup XP /F18 79 30 2 0 27 26 26 32 0 <007F0000 01FFC000 07C1F000 0F007800 1E003C00 3C001E00 38000E00 78000F00 70000700 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 78000F00 78000F00 38000E00 3C001E00 1E003C00 0F007800 07C1F000 01FFC000 007F0000> PXLC RP 692 462 XY F18(Ordina)S XP /F18 108 11 0 0 9 26 26 16 0 PXLC RP 809 462 XY F18(l)S XP /F18 66 27 1 0 24 26 26 24 0 PXLC RP 833 462 XY F18(B)S XP /F18 111 19 1 0 17 16 16 16 0 <07E0 1FF8 3C3C 781E 700E F00F F00F F00F F00F F00F F00F 700E 781E 3C3C 1FF8 07E0> PXLC RP 860 462 XY F18(o)S XP /F18 117 21 1 0 19 16 16 24 0 PXLC RP 879 462 XY F18(und)S XP /F18 115 15 1 0 13 16 16 16 0 <3FE0 7FE0 F0E0 E060 E060 F000 FF80 7FC0 1FE0 01F0 C0F0 E070 E070 F0F0 FFE0 DF80> PXLC RP 943 462 XY F18(s)S XP /F18 102 12 0 0 14 26 26 16 0 <00F8 03FC 07BC 0F3C 0E3C 0E00 0E00 0E00 0E00 0E00 FFC0 FFC0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 971 462 XY F18(for)S XP /F18 80 26 1 0 23 26 26 24 0 PXLC RP 1030 462 XY F18(Progra)S XP /F18 109 32 1 0 30 16 16 32 0 PXLC RP 1144 462 XY F18(ms)S 1583 X F25(389)S 561 Y 330 X(\(2\))S -16 y 447 X F26(M)S 503 X F27(!)S 556 X F26(M)S XP /F10 48 10 1 1 8 16 15 8 0 <1E 1E 1E 3E 3C 3C 38 38 78 70 70 60 E0 E0 C0> PXLC RP 601 528 XY F10(0)S 409 552 XY 242 2 R 35 y 409 X F26(M)S 454 X(N)S 503 X F27(!)S 556 X F26(M)S -17 y 601 X F10(0)S 17 y 1 x F26(N)S -26 y 655 X F25(.)S 687 Y 330 X(\(3\))S 429 X(Y)S 6 y F9(\032;)S F10 (\036)S -6 y 2 x F26(M)S 560 X(N)S -11 y 606 X(~)S 11 y 598 X(N)S 647 X F27(!)S 697 X F25(cond)S 6 y F9(\023)S -6 y 3 x F25(\()S F26 (c)S 6 y F8(=)S -6 y 2 x F25(\()S F26(M)S -11 y 928 X(~)S 11 y 919 X (N)S 957 X F25(\)\()S F26(M)S -11 y 1043 X(~)S 11 y 1034 X(N)S 1072 X F25(\)\))S 55 y 697 X(\()S F26(N)S 751 X F25(\()S F26(\025~)S 791 X (y)S 2 x(:)S F25(cond)S 6 y F9(\023)S -6 y 2 x 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10 y F9 (\023)S -10 y 3 x F25(t)S 562 X(t)S F26(M)S 623 X(N)S 672 X F27(!)S 725 X F26(M)S 770 X F25(;)S 831 X(p)S 1 x(cond)S 10 y 1 x F9(\023)S -10 y 2 x F25(f)S 964 X(f)S 3 x F26(M)S 1025 X(N)S 1074 X F27(!)S 1127 X F26(N)S 1165 X F25(;)S 1227 X(p)S 1 x(cond)S 10 y F9(\023)S -10 y 3 x F26(M)S 1396 X(N)S 1434 X(N)S 1483 X F27(!)S 1536 X F26 (N)S 1574 X F25(;)S 50 y 429 X(p)S 1 x(cond)S 10 y F9(\023)S -10 y 3 x F27(?)S F26(M)S 630 X F27(?)S 673 X(!)S 727 X(?)S F25(;)S 831 X (p)S 1 x(cond)S 10 y 1 x F9(\023)S -10 y 2 x F27(??)S F26(N)S 1069 X F27(!)S 1122 X(?)S F25(;)S 1227 X(p)S 1 x(cond)S 10 y F9(\023)S -10 y 3 x F26(M)S 1396 X F25(t)S 1405 X(tf)S 1430 X(f)S 1457 X F27(!)S 1510 X F26(M)S 1555 X F25(;)S 49 y 429 X(p)S 1 x(cond)S 11 y F9 (\023)S -11 y 3 x F27(?)S F26(c)S XP /F26 100 22 1 0 21 29 29 24 0 <0003F0 0007F0 0000F0 0000F0 0000E0 0000E0 0001E0 0001E0 0001C0 0001C0 0003C0 01FBC0 07FF80 0F1F80 1E0F80 3C0780 7C0700 780700 780F00 F80F00 F00E00 F00E00 F01E70 F01E70 F03C60 F07C60 78FEE0 3FDFC0 1F8780> PXLC RP 603 1194 XY F26(d)S 636 X F27(!)S 689 X(?)S 735 X F25(if)S 773 X F26 (c)S 805 X F25(and)S 886 X F26(d)S 921 X F25(are)S 990 X(di)S XP /F25 11 24 0 0 26 29 29 32 0 <003F1F80 01FFFFC0 03C3F3C0 0783E3C0 0F03E3C0 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 FFFFFC00 FFFFFC00 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 0E01C000 7F87FC00 7F87FC00> PXLC RP 1025 1194 XY F25(\013eren)S -1 x(t)S 1154 X(constan)S -1 x(ts)S 1338 X(or)S 1389 X(n)S -1 x(umerals.)S 1319 Y 330 X(\(7\))S -16 y 558 X F26(M)S 614 X F27(!)S 667 X F26(M)S -17 y 712 X F10(0)S 434 1310 XY 414 2 R 34 y 434 X F25(p)S 1 x(cond)S 11 y F9(\023)S -11 y 3 x F26(M)S 614 X F27(!)S 667 X F25(p)S 1 x(cond)S 11 y F9 (\023)S -11 y 3 x F26(M)S -17 y 836 X F10(0)S -24 y 1208 X F26(N)S 1257 X F27(!)S 1310 X F26(N)S -17 y 1348 X F10(0)S 1039 1310 XY 490 2 R 34 y 1039 X F25(p)S 1 x(cond)S 11 y 1 x F9(\023)S -11 y 2 x F26(M)S 1208 X(N)S 1257 X F27(!)S 1310 X F25(p)S 1 x(cond)S 11 y 1 x F9(\023)S -11 y 2 x F26(M)S 1479 X(N)S -17 y 1517 X F10(0)S 61 y 640 X F26(L)S 680 X F27(!)S 733 X F26(L)S -17 y F10(0)S 434 1395 XY 546 2 R 35 y 434 X F25(p)S 1 x(cond)S 10 y F9(\023)S -10 y 3 x F26(M)S 603 X(N)S 641 X(L)S 680 X F27(!)S 733 X F25(p)S 1 x(cond)S 10 y 1 x F9(\023)S -10 y 2 x F26(M)S 902 X(N)S 940 X(L)S -18 y F10(0)S 1560 Y 330 X F25(\(8\))S 404 X F26(c)S 6 y F10(\036)S XP /F30 109 40 1 0 38 18 18 40 0 PXLC RP 450 1560 XY F30(mn)S 535 X F27(!)S -58 y 596 X F67(\032)S 33 y 634 X F25(t)S 643 X(t)S F26(;)S 712 X F25(if)S 750 X F26(m)S 798 X F27 (\036)S 842 X F26(n)S F25(;)S 50 y 634 X(f)S 643 X(f)S 3 x F26(;)S 712 X F25(otherwise)S -25 y 888 X(;)S 941 X F26(c)S 6 y F10(\036)S -6 y 2 x F27(?)S F26(N)S 1076 X F27(!)S 1137 X(?)S F25(;)S 1222 X F26(c)S 6 y F10(\036)S -6 y 2 x F26(M)S 1313 X F27(?)S 1364 X(!)S 1425 X(?)S F25(;)S 1489 X(similarly)S 72 y 404 X(for)S 467 X F26(c)S 7 y F8(=)S -7 y 3 x F25(.)S 50 y 404 X(succ)S 1 x F27(?)S 524 X(!)S 577 X(?)S F25(.)S 1788 Y 330 X(\(9\))S -16 y 455 X F26(M)S 511 X F27 (!)S 564 X F26(M)S -17 y 609 X F10(0)S 409 1778 XY 258 2 R 35 y 409 X F26(c)S 7 y F10(\036)S -7 y 2 x F26(M)S 511 X F27(!)S 564 X F26(c)S 7 y F10(\036)S -7 y 2 x F26(M)S -17 y 655 X F10(0)S -8 y 672 X F25(;)S -16 y 820 X F26(N)S 870 X F27(!)S 923 X F26(N)S -17 y 961 X F10(0)S 730 1778 XY 334 2 R 35 y 730 X F26(c)S 7 y F10(\036)S -7 y 2 x F26(M)S 821 X(N)S 870 X F27(!)S 923 X F26(c)S 7 y F10 (\036)S -7 y 2 x F26(M)S 1014 X(N)S -17 y 1052 X F10(0)S -8 y 1068 X F25(;)S 1093 X(similarly)S 1263 X(for)S 1327 X F26(c)S 6 y F8(=)S -6 y 1386 X F25(and)S 1467 X(succ)S 1 x(.)S 1933 Y 375 X(It)S 420 X (migh)S -1 x(t)S 538 X(seem)S 640 X(that)S 730 X(the)S 802 X (simpler)S 947 X(rule)S 2014 Y 318 X(\(3'\))S 404 X(Y)S 6 y F9 (\032;)S F10(\036)S -6 y 2 x F26(M)S 535 X(N)S -10 y 582 X(~)S 10 y 573 X(N)S 622 X F27(!)S 675 X F26(N)S 713 X F25(\()S F26(\025~)S 753 X(y)S 2 x(:)S F25(cond)S 6 y F9(\023)S -6 y 2 x F25(\()S F26(c)S 6 y F10(\036)S -6 y 3 x F25(\()S F26(M)S 1009 X(~)S 1010 X(y)S 1 x F25(\)\()S F26(M)S -10 y 1117 X(~)S 10 y 1108 X(N)S 1146 X F25 (\)\)\(Y)S 6 y F9(\032;)S F10(\036)S -6 y 3 x F26(M)S 1326 X(N)S 1363 X(~)S 1364 X(y)S 1 x F25(\))S F27(?)S F25(\))S -10 y 1459 X F26 (~)S 10 y 1449 X(N)S 2101 Y 300 X F25(for)S 364 X(the)S 435 X(Y)S 1 x({op)S 1 x(erator)S 655 X(w)S -1 x(ould)S 776 X(su)S XP /F25 14 35 0 0 33 29 29 40 0 <003F83F000 01FFDFF800 03E1FC3C00 0781F83C00 0F01F03C00 0E01E03C00 0E00E00000 0E00E00000 0E00E00000 0E00E00000 0E00E00000 FFFFFFFC00 FFFFFFFC00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 7FC7FCFF80 7FC7FCFF80> PXLC RP 816 2101 XY F25(\016ce.)S 917 X(Ho)S -1 x(w)S -1 x(ev)S -1 x(er,)S 1096 X(it)S 1138 X(do)S 1 x(es)S 1231 X(not)S 1305 X(in)S 1353 X (case)S 1441 X F26(M)S -10 y 1495 X(~)S 10 y 1486 X(N)S 1538 X F25 (ev)S -2 x(alu-)S 50 y 300 X(ates)S 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457 X F26(M)S 513 X F27(!)S 566 X F26 (M)S -17 y 611 X F10(0)S 409 2555 XY 262 2 R 35 y 409 X F26(\025xM)S 513 X F27(!)S 566 X F26(\025xM)S -18 y 659 X F10(0)S 2703 Y 300 X F25(for)S 367 X(then)S 465 X(follo)S -1 x(wing)S 644 X(an)S 705 X (application)S 923 X(of)S 974 X(rule)S 1061 X(\(3\))S 1131 X(for)S 1198 X(the)S 1273 X(b)S 1 x(ounded)S 1447 X(\014xed)S 1550 X(p)S 1 x(oin)S -1 x(t)S 50 y 300 X(op)S 1 x(erator)S 467 X(w)S -1 x(e)S 529 X(could)S 639 X(easily)S 754 X(construct)S 936 X(an)S 994 X(in)S (\014nite)S 1134 X(reduction)S 1319 X(sequence.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 129 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(390)S 700 X F18(H.)S 752 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S 1023 X(and)S 1098 X(S.S.)S 1175 X(W)S -3 x(ainer)S 553 Y 300 X F48(2.)S XP /F48 52 28 2 0 25 32 32 24 0 <0000E0 0001E0 0003E0 0007E0 000FE0 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X(means)S 1138 X(for)S 1208 X(a)S 1248 X(term)S 1353 X(of)S 1406 X(t)S -1 x(yp)S 1 x(e)S 1505 X F26(\032)S 1546 X F25(to)S 1602 X(b)S 1 x(e)S 50 y 300 X (computable.)S 540 X(The)S 625 X(de\014nition)S 810 X(is)S 851 X(b)S -1 x(y)S 909 X(induction)S 1094 X(on)S 1151 X(t)S -1 x(yp)S 1 x (es.)S 1163 Y 351 X(1.)S 404 X(A)S 450 X(closed)S 574 X(term)S 675 X F26(M)S 736 X F25(of)S 784 X(ground)S 927 X(t)S -1 x(yp)S 1 x(e)S 1022 X(is)S 1065 X(computable)S 1290 X(if)S 1330 X(an)S -1 x(y)S 1410 X(reduction)S 1596 X(se-)S 50 y 404 X(quence)S 541 X(starting)S 696 X(with)S 790 X F26(M)S 849 X F25(terminates.)S 1298 Y 351 X(2.)S 404 X(A)S 448 X(closed)S 571 X(term)S 670 X F26(M)S 728 X F25(of)S 775 X(t)S -1 x(yp)S 1 x(e)S 868 X F26(\032)S 901 X F27(!)S 954 X F26 (\033)S 993 X F25(is)S 1035 X(computable)S 1258 X(if)S 1296 X(for)S 1359 X(an)S -1 x(y)S 1437 X(closed)S 1560 X(term)S 50 y 404 X F26 (N)S 455 X F25(of)S 503 X(t)S -1 x(yp)S 1 x(e)S 596 X F26(\032)S F25 (,)S 643 X(if)S 681 X F26(N)S 733 X 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27 27 24 0 <006000 01E000 0FE000 FFE000 FFE000 F3E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 7FFF80 7FFF80 7FFF80> PXLC RP 412 1627 XY F30(1.)S XP /F30 84 33 2 0 30 28 28 32 0 <7FFFFFE0 7FFFFFE0 7FFFFFE0 7C1F83E0 781F81E0 701F80E0 F01F80F0 E01F8070 E01F8070 E01F8070 E01F8070 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 07FFFE00 07FFFE00 07FFFE00> PXLC RP 465 1627 XY F30(T)S XP /F30 104 27 1 0 25 29 29 24 0 PXLC RP 498 1627 XY F30(heo)S XP /F30 114 20 1 0 18 18 18 24 0 PXLC RP 570 1627 XY F30(rem.)S XP /F29 /cmsl10 300 41.5 41.5 128 [-4 -11 46 31] PXLNF RP XP /F29 65 31 1 0 29 29 29 32 0 <00003000 00007000 00007000 0000F000 0000F800 0001F800 0001F800 00037800 00037800 00067800 00067C00 000C3C00 000C3C00 00183C00 00183C00 00303C00 00303C00 00601E00 00601E00 00FFFE00 00FFFE00 01801E00 01801E00 03001F00 03000F00 07000F00 0F000F00 7FC0FFF0 FFC0FFF0> PXLC RP 686 1627 XY F29(A)S XP /F29 110 23 1 0 21 18 18 24 0 <1F8FC0 1FBFE0 07F1E0 07C0E0 0780E0 0780E0 0700E0 0700E0 0701E0 0F01E0 0E01C0 0E01C0 0E01C0 0E01C0 0E03C0 1E03C0 FF8FF0 FF9FF0> PXLC RP 717 1627 XY F29(n)S XP /F29 121 22 0 -8 23 18 26 24 0 <0FF1FE 0FF1FE 03C078 01C060 01C060 01C0C0 01C1C0 01C180 00E380 00E300 00E600 00E600 00EC00 007C00 007800 007800 007000 007000 006000 006000 00C000 70C000 F18000 C70000 FE0000 7C0000> PXLC RP 739 1627 XY F29(y)S XP /F29 116 16 4 0 15 26 26 16 0 <0600 0600 0600 0E00 0E00 0C00 1C00 3C00 FFE0 FFE0 1C00 3C00 3800 3800 3800 3800 3800 7800 70C0 70C0 70C0 70C0 71C0 7380 7F00 3E00> PXLC RP 774 1627 XY F29(t)S XP /F29 101 18 2 0 18 18 18 16 0 <01F8 07FE 0F1E 1E0F 3C0F 7C07 7FFF 7FFF 7800 F800 F000 F000 7000 780E 7C0C 3E3C 1FF8 07E0> PXLC RP 791 1627 XY F29(e)S XP /F29 114 16 1 0 17 18 18 16 0 <1FBE 1FFE 07EF 07CE 078E 0780 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 0E00 1E00 FFC0 FFC0> PXLC RP 809 1627 XY F29(r)S XP /F29 109 35 1 0 34 18 18 40 0 <1F8FC0FC00 1FBFF3FF00 07F0F70F00 07C07C0700 0780780700 0780780700 0700700700 0700700700 0700700700 0F00F00F00 0E00E00E00 0E00E00E00 0E00E00E00 0E00E00E00 0E00E00E00 1E01E01E00 FF8FF8FF80 FF8FF8FF80> PXLC RP 825 1627 XY F29(m)S 874 X F26(M)S XP /F29 105 12 1 0 12 29 29 16 0 <01C0 03E0 03E0 03E0 03C0 0000 0000 0000 0000 0000 0000 1F80 1F80 0780 0780 0700 0700 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 0E00 1E00 FF80 FF80> PXLC RP 932 1627 XY F29(i)S XP /F29 115 16 1 0 16 18 18 16 0 <03F6 0FFE 1E1E 3C0E 380C 3C0C 3E00 3FE0 1FF8 07FC 007C 603C 601C 701C 7038 F878 FFF0 CFC0> PXLC RP 944 1627 XY F29(s)S XP /F29 99 18 2 0 18 18 18 16 0 <01FE 07FE 0F8F 1E0E 3C0E 3C00 7800 7800 7800 F800 F000 F000 7800 780E 7C0C 3E1C 1FF8 07E0> PXLC RP 974 1627 XY F29(c)S XP /F29 111 21 2 0 21 18 18 24 0 <00FC00 03FF00 0F0780 1E03C0 3C01C0 3801C0 7801E0 7801E0 7801E0 F003C0 F003C0 F003C0 700380 700780 780F00 3C1E00 1FF800 07E000> PXLC RP 993 1627 XY F29(om)S XP /F29 112 23 0 -8 23 18 26 24 0 <0FCFE0 0FFFF8 03F8F8 03E07C 03C03C 03803E 03801E 03801E 03803E 07803E 07003C 07003C 070078 0700F8 0781F0 0FC3E0 0EFFC0 0E7F00 0E0000 0E0000 1E0000 1E0000 1C0000 1C0000 FF8000 FF8000> PXLC RP 1048 1627 XY F29(p)S XP /F29 117 23 4 0 22 18 18 24 0 PXLC RP 1071 1627 XY F29(ut)S XP /F29 97 21 2 0 21 18 18 24 0 <07F800 0FFE00 0F1F00 0E0700 0E0700 000700 000700 03FF00 1FCF00 3E0F00 7C0E00 F80E00 F00E60 F01E60 F03E60 F87EC0 7FEFC0 3F8F00> PXLC RP 1110 1627 XY F29(a)S XP /F29 98 23 4 0 23 29 29 24 0 <3F0000 7F0000 0F0000 0E0000 0E0000 0E0000 0E0000 0E0000 1E0000 1C0000 1C0000 1CFE00 1FFF80 1F8780 3E03C0 3C03C0 3801E0 3801E0 3801E0 3803E0 7803E0 7003C0 7003C0 700780 700780 780F00 FC3E00 EFFC00 C7F000> PXLC RP 1131 1627 XY F29(b)S XP /F29 108 12 1 0 12 29 29 16 0 <07E0 0FE0 01E0 01C0 01C0 01C0 01C0 01C0 03C0 0380 0380 0380 0380 0380 0780 0700 0700 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 1E00 1E00 FF80 FF80> PXLC RP 1154 1627 XY F29(le)S XP /F29 46 12 3 0 8 5 5 8 0 <78 78 F8 F8 70> PXLC RP 1184 1627 XY F29(.)S XP /F30 80 33 2 0 29 28 28 32 0 PXLC RP 375 1720 XY F30(Pro)S 1 x(o)S XP /F30 102 15 0 0 19 29 29 24 0 <003F80 01FFC0 03F7E0 07C7E0 0FC7E0 0F87E0 0F83C0 0F8000 0F8000 0F8000 0F8000 FFF800 FFF800 FFF800 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 7FF800 7FF800 7FF800> PXLC RP 476 1720 XY F30(f.)S XP /F25 66 29 2 0 26 28 28 24 0 PXLC RP 525 1720 XY F25(By)S 590 X(induction)S 774 X(on)S 832 X F26(M)S 877 X F25(.)S 50 y 375 X F26(x)S F25(:)S 428 X(T)S -3 x(rivial.)S 50 y 375 X F26(M)S 420 X(N)S 458 X F25(:)S 494 X(Let)S -11 y 573 X F26(~)S 11 y 571 X(L)S 617 X F25(b)S 1 x(e)S 677 X(a)S 714 X (sequence)S 890 X(of)S 940 X(closed)S 1066 X(computable)S 1293 X (terms.)S 1434 X(W)S -3 x(e)S 1509 X(ha)S -1 x(v)S -1 x(e)S 1608 X (to)S 49 y 300 X(sho)S -1 x(w)S 408 X(that)S 502 X F26(M)S 547 X F25 ([)S -10 y 2 x F26(~)S 10 y 559 X(L)S F25(])S F26(N)S 637 X F25([)S -10 y 2 x F26(~)S 10 y 649 X(L)S -1 x F25(])S 706 X(is)S 753 X 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-3 x(e)S 523 X(m)S -1 x(ust)S 631 X(sho)S -1 x(w)S 739 X (that)S 833 X(\()S F26(\025xM)S 942 X F25([)S -10 y 2 x F26(~)S 10 y 954 X(L)S F25(]\))S F26(N)S -10 y 1060 X(~)S 10 y 1048 X(M)S 1111 X F25(is)S 1158 X(computable,)S 1399 X(i.e.)S 1486 X(that)S 1580 X (an)S -1 x(y)S 50 y 300 X(reduction)S 484 X(sequence)S 655 X (starting)S 808 X(with)S 902 X(that)S 991 X(term)S 1089 X (terminates.)S 1310 X(But)S 1392 X(from)S 1489 X(the)S 1559 X (rules)S 50 y 300 X(of)S 346 X(our)S 418 X(op)S 1 x(erational)S 634 X(seman)S -1 x(tics)S 821 X(it)S 861 X(follo)S -1 x(ws)S 995 X (that)S 1083 X(the)S 1153 X(second)S 1285 X(mem)S -1 x(b)S 1 x(er)S 1443 X(of)S 1488 X(an)S -1 x(y)S 1565 X(suc)S -1 x(h)S 50 y 300 X (sequence)S 468 X(m)S -1 x(ust)S 567 X(b)S 1 x(e)S 619 X F26(M)S 664 X F25([)S F26(N)S 2 x(;)S -11 y 731 X(~)S 11 y 730 X(L)S F25(])S -11 y 782 X F26(~)S 11 y 770 X(M)S 814 X F25(,)S 836 X(whic)S -1 x (h)S 951 X(is)S 989 X(computable)S 1209 X(b)S -1 x(y)S 1262 X (induction)S 1443 X(h)S -1 x(yp)S 1 x(othesis.)S 54 y 375 X(Y)S 7 y F9(\032;)S F10(\036)S -7 y 2 x F25(:)S 502 X(Let)S 582 X F26(M)S 2 x(;)S 643 X(N)S 2 x(;)S -10 y 706 X(~)S 10 y 697 X(N)S 754 X F25 (b)S 1 x(e)S 816 X(closed)S 944 X(computable)S 1173 X(terms.)S 1321 X (W)S -3 x(e)S 1398 X(ha)S -1 x(v)S -1 x(e)S 1500 X(to)S 1556 X(sho)S -1 x(w)S 57 y 300 X(that)S 392 X(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26(M)S 523 X(N)S -11 y 569 X(~)S 11 y 561 X(N)S 614 X F25(is)S 658 X(computable,)S 895 X(i.e.)S 972 X(that)S 1064 X(an)S -1 x(y)S 1144 X(reduction)S 1330 X(sequence)S 1504 X(starting)S 49 y 300 X (with)S 400 X(that)S 496 X(term)S 601 X(terminates.)S 840 X(Since)S 954 X F26(M)S 1019 X F25(is)S 1066 X(computable)S 1296 X(b)S -1 x (y)S 1359 X(assumption,)S 1598 X(w)S -1 x(e)S 50 y 300 X(kno)S -1 x (w)S 409 X(that)S 500 X F26(M)S -10 y 553 X(~)S 10 y 545 X(N)S 597 X F25(reduces)S 746 X(to)S 797 X(a)S 832 X(constan)S -1 x(t)S 1001 X (or)S 1052 X(a)S 1088 X(n)S -1 x(umeral)S 1249 X F26(c)S F25(;)S 1293 X(w)S -1 x(e)S 1355 X(use)S 1427 X(an)S 1486 X(auxiliary)S 50 y 300 X(trans\014nite)S 500 X F27(\036)S F25({induction)S 739 X(on)S 798 X(\(the)S 888 X(v)S -2 x(alue)S 996 X(of)S 3 x(\))S 1065 X F26 (c)S F25(.)S 1117 X(If)S 1160 X F26(c)S 1192 X F25(=)S 1239 X F27 (?)S F25(,)S 1298 X(then)S 1394 X(an)S -1 x(y)S 1474 X(reduction)S 55 y 300 X(sequence)S 475 X(starting)S 632 X(with)S 730 X(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26(M)S 861 X(N)S -11 y 907 X(~)S 11 y 899 X(N)S 953 X F25(m)S -1 x(ust)S 1059 X(after)S 1160 X(its)S 1220 X(\014rst)S 1309 X(t)S -1 x(w)S -1 x(o)S 1390 X(steps)S 1497 X (w)S -1 x(ork)S 1601 X(on)S 56 y 300 X(reducing)S 473 X F26(c)S 6 y F8(=)S -6 y 2 x F25(\()S F26(M)S -11 y 588 X(~)S 11 y 579 X(N)S 617 X F25(\)\()S F26(M)S -11 y 703 X(~)S 11 y 694 X(N)S 732 X F25 (\))S 766 X(un)S -1 x(til)S 869 X(this)S 954 X(term)S 1058 X(has)S 1136 X(b)S 1 x(een)S 1238 X(reduced)S 1397 X(to)S 1452 X(a)S 1491 X (constan)S -1 x(t)S 50 y 300 X(or)S 355 X(a)S 394 X(n)S -1 x (umeral;)S 573 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X(for)S 1030 X (Programs)S 1583 X F25(391)S 553 Y 300 X(as)S 349 X(b)S 1 x(efore)S 473 X(w)S -1 x(e)S 532 X(can)S 607 X(see)S 673 X(that)S 761 X(an)S -1 x(y)S 838 X(reduction)S 1021 X(sequence)S 1192 X(starting)S 1345 X (with)S 1438 X(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26(M)S 1569 X (N)S -10 y 1616 X(~)S 10 y 1607 X(N)S 50 y 300 X F25(m)S -1 x(ust)S 403 X(come)S 509 X(across)S 696 Y 560 X F26(N)S 598 X F25(\()S F26 (\025~)S 638 X(y)S 2 x(:)S F25(cond)S 6 y F9(\023)S -6 y 3 x F25 (\()S F26(c)S 6 y F10(\036)S -6 y 2 x F25(\()S F26(M)S 894 X(~)S 895 X(y)S 2 x F25(\)\()S F26(M)S -11 y 1002 X(~)S 11 y 994 X(N)S 1031 X F25(\)\)\(Y)S 6 y F9(\032;)S F10(\036)S -6 y 3 x F26(M)S 1211 X(N)S 1248 X(~)S 1249 X(y)S 2 x F25(\))S F27(?)S F25(\))S -11 y 1344 X F26(~)S 11 y 1335 X(N)S 1373 X(:)S 788 Y 300 X F25(Since)S 410 X F26(N)S 464 X F25(is)S 508 X(computable)S 734 X(b)S -1 x(y)S 793 X(assumption,)S 1027 X(it)S 1070 X(su\016ces)S 1213 X(to)S 1266 X (sho)S -1 x(w)S 1371 X(that)S 1463 X(the)S 1536 X(closed)S 50 y 300 X(term)S 52 y 620 X F26(\025~)S 644 X(y)S 2 x(:)S F25(cond)S 6 y F9(\023)S -6 y 2 x F25(\()S F26(c)S 6 y F10(\036)S -6 y 3 x F25 (\()S F26(M)S 900 X(~)S 901 X(y)S 1 x F25(\)\()S F26(M)S -11 y 1008 X (~)S 11 y 999 X(N)S 1037 X F25(\)\)\(Y)S 6 y F9(\032;)S F10(\036)S -6 y 3 x F26(M)S 1217 X(N)S 1254 X(~)S 1255 X(y)S 1 x F25(\))S F27 (?)S 973 Y 300 X F25(is)S 345 X(computable.)S 596 X(So)S 657 X(let)S -11 y 723 X F26(~)S 11 y 721 X(L)S 767 X F25(b)S 1 x(e)S 827 X(a)S 865 X(sequence)S 1040 X(of)S 1091 X(closed)S 1217 X(computable)S 1445 X(terms.)S 1587 X(W)S -3 x(e)S 50 y 300 X(m)S -1 x(ust)S 403 X (sho)S -1 x(w)S 506 X(that)S 1115 Y 642 X(cond)S 6 y 1 x F9(\023)S -6 y 2 x F25(\()S F26(c)S 6 y F10(\036)S -6 y 2 x F25(\()S F26(M)S -10 y 866 X(~)S 10 y 865 X(L)S F25(\)\()S F26(M)S -10 y 979 X(~)S 10 y 970 X(N)S 1008 X F25(\)\)\(Y)S 6 y F9(\032;)S F10(\036)S -6 y 3 x F26(M)S 1188 X(N)S -10 y 1227 X(~)S 10 y 1226 X(L)S F25(\))S F27 (?)S 1214 Y 300 X F25(is)S 342 X(computable.)S 585 X(Since)S 694 X F26(M)S 753 X F25(is)S 795 X(computable)S 1020 X(b)S -1 x(y)S 1078 X (assumption,)S 1310 X(w)S -1 x(e)S 1372 X(kno)S -1 x(w)S 1481 X (that)S 1572 X F26(M)S -10 y 1618 X(~)S 10 y 1617 X(L)S 50 y 300 X F25(reduces)S 448 X(to)S 499 X(a)S 534 X(constan)S -1 x(t)S 702 X (or)S 754 X(a)S 789 X(n)S -1 x(umeral)S 949 X F26(d)S F25(.)S 1002 X (If)S 1044 X F26(d)S 1078 X F25(=)S 1122 X F27(?)S F25(,)S 1180 X (then)S 1275 X F26(c)S 6 y F10(\036)S -6 y 2 x F25(\()S F26(M)S -10 y 1383 X(~)S 10 y 1382 X(L)S 1 x F25(\)\()S F26(M)S -10 y 1497 X(~)S 10 y 1488 X(N)S 1526 X F25(\))S 1556 X(m)S -1 x(ust)S 50 y 300 X (reduce)S 428 X(to)S 476 X F27(?)S 518 X F25(and)S 596 X(hence)S 708 X(the)S 776 X(whole)S 890 X(term)S 986 X(m)S -1 x(ust)S 1086 X (reduce)S 1214 X(to)S 1261 X F27(?)S F25(.)S 1322 X(So)S 1377 X (assume)S 1517 X F26(d)S 1550 X F25(=)S 1594 X F30(m)S F25(.)S 50 y 300 X(Then)S 408 X(the)S 480 X(whole)S 597 X(term)S 697 X(m)S -1 x (ust)S 800 X(reduce)S 931 X(to)S 1456 Y 719 X(cond)S 6 y 1 x F9 (\023)S -6 y 2 x F25(\()S F26(c)S 6 y F10(\036)S -6 y 2 x F30(mn)S F25(\)\(Y)S 6 y F9(\032;)S F10(\036)S -6 y 2 x F26(M)S 1111 X(N)S -10 y 1150 X(~)S 10 y 1149 X(L)S F25(\))S F27(?)S 1549 Y 300 X F25 (If)S 344 X F26(m)S 395 X F27(\036)S 442 X F26(n)S 483 X F25 (holds,)S 606 X(then)S 702 X F26(c)S 6 y F10(\036)S -6 y 2 x F30 (mn)S 830 X F27(!)S 886 X F25(t)S 895 X(t)S 928 X(and)S 1010 X (hence)S 1128 X(the)S 1201 X(whole)S 1321 X(term)S 1422 X(m)S -1 x (ust)S 1527 X(reduce)S 50 y 300 X(to)S 354 X(Y)S 6 y F9(\032;)S F10 (\036)S -6 y 2 x F26(M)S 485 X(N)S -11 y 524 X(~)S 11 y 523 X(L)S F25(,)S 580 X(whic)S -1 x(h)S 701 X(b)S -1 x(y)S 762 X(h)S -1 x(yp)S 1 x(othesis)S 969 X(of)S 1020 X(our)S 1096 X(auxiliary)S 1272 X F27 (\036)S F25({induction)S 1513 X(is)S 1557 X(com-)S 50 y 300 X (putable.)S 474 X(If)S 518 X F26(m)S 571 X F27(\036)S 619 X F26(n)S 661 X F25(do)S 1 x(es)S 757 X(not)S 834 X(hold,)S 941 X(then)S 1038 X F26(c)S 6 y F10(\036)S -6 y 2 x F30(mn)S 1166 X F27(!)S 1224 X F25 (f)S 1233 X(f)S 1265 X(and)S 1349 X(hence)S 1467 X(the)S 1541 X (whole)S 49 y 300 X(term)S 399 X(reduces)S 547 X(to)S 598 X F27(?)S F25(.)S 50 y 375 X(cond)S 6 y F9(\023)S -6 y 2 x F25(,)S 507 X(p)S 1 x(cond)S 10 y 1 x F9(\023)S -10 y 2 x F25(:)S 673 X(Let)S 753 X F26(M)S 2 x(;)S 814 X(N)S 2 x(;)S 868 X(L)S 916 X F25(b)S 1 x(e)S 978 X(closed)S 1107 X(computable)S 1336 X(terms.)S 1486 X(W)S -3 x (e)S 1563 X(ha)S -1 x(v)S -1 x(e)S 50 y 300 X(to)S 352 X(sho)S -1 x (w)S 456 X(that)S 548 X(cond)S 6 y F9(\023)S -6 y 2 x F26(M)S 692 X (N)S 730 X(L)S 774 X F25(and)S 856 X(p)S 1 x(cond)S 10 y F9(\023)S -10 y 3 x F26(M)S 1025 X(N)S 1063 X(L)S 1106 X F25(are)S 1176 X (computable,)S 1413 X(i.e.)S 1489 X(that)S 1580 X(an)S -1 x(y)S 50 y 300 X(reduction)S 481 X(sequence)S 649 X(starting)S 800 X(with)S 891 X(either)S 1004 X(of)S 1048 X(them)S 1150 X(terminates.)S 1370 X (But)S 1449 X(this)S 1526 X(clearly)S 50 y 300 X(holds,)S 428 X (since)S 536 X(otherwise)S 727 X(one)S 809 X(of)S 862 X(the)S 940 X (terms)S 1062 X F26(M)S 2 x(;)S 1123 X(N)S 2 x(;)S 1177 X(L)S 1225 X F25(w)S -1 x(ould)S 1352 X(ha)S -1 x(v)S -1 x(e)S 1454 X(an)S 1518 X (in)S (\014nite)S 49 y 300 X(reduction)S 485 X(sequence.)S 50 y 375 X(F)S -3 x(or)S 449 X(the)S 521 X(other)S 629 X(constan)S -1 x(ts)S 813 X (of)S 860 X(PCF)S 6 y F10(\036)S -6 y 987 X F25(the)S 1059 X (argumen)S -1 x(t)S 1245 X(is)S 1286 X(similar.)S 1628 1997 XY 17 17 R XP /F55 /cmbx10 432 59.8 59.8 128 [-5 -15 69 45] PXLNF RP XP /F55 51 34 3 0 30 39 39 32 0 <00FFC000 03FFF800 0FFFFE00 1FC3FF00 1FC1FF00 3FE0FF80 3FE0FF80 3FE0FF80 3FE0FF80 3FE0FF80 3FE0FF80 1FC1FF80 0F81FF00 0003FF00 0003FE00 000FFC00 00FFF000 00FFC000 00FFF800 0003FE00 0001FF00 0000FF80 0000FFC0 00007FC0 00007FE0 3F007FE0 7F807FE0 FFC07FE0 FFC07FE0 FFC07FE0 FFC07FE0 FFC07FE0 FFC0FFC0 FF80FFC0 7F01FF80 3FC3FF00 1FFFFE00 0FFFF800 01FFC000> PXLC RP 300 2137 XY F55(3)S XP /F55 80 47 3 0 42 41 41 40 0 PXLC RP 403 2137 XY F55(P)S XP /F55 114 28 2 0 26 27 27 24 0 PXLC RP 450 2137 XY F55(r)S XP /F55 105 19 3 0 18 43 43 16 0 <0FC0 1FE0 3FE0 3FF0 3FF0 3FF0 3FF0 3FE0 1FE0 0FC0 0000 0000 0000 0000 0000 0000 FFE0 FFE0 FFE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 0FE0 FFFE FFFE FFFE> PXLC RP 478 2137 XY F55(i)S XP /F55 109 57 3 0 56 27 27 56 0 PXLC RP 498 2137 XY F55(mi)S XP /F55 116 27 1 0 22 38 38 24 0 <007000 007000 007000 007000 00F000 00F000 01F000 01F000 03F000 07F000 1FF000 FFFFF0 FFFFF0 FFFFF0 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F000 07F038 07F038 07F038 07F038 07F038 07F078 07F878 03FCF0 03FFF0 01FFE0 007FC0> PXLC RP 574 2137 XY F55(ti)S XP /F55 118 36 1 0 34 27 27 40 0 PXLC RP 620 2137 XY F55(v)S XP /F55 101 32 2 0 29 27 27 32 0 <007FF000 03FFFC00 07FFFF00 0FF8FF80 1FE03F80 3FC01FC0 7F800FC0 7F800FC0 7F000FE0 FF0007E0 FF0007E0 FFFFFFE0 FFFFFFE0 FFFFFFE0 FF000000 FF000000 FF000000 FF000000 7F800000 7F800000 7FC000E0 3FE001E0 1FF003E0 0FFC0FC0 07FFFF80 01FFFF00 003FF800> PXLC RP 654 2137 XY F55(e)S 702 X(re)S XP /F55 99 31 2 0 28 27 27 32 0 <003FF800 01FFFF00 07FFFF80 0FF87F80 1FE07F80 3FC07F80 7FC07F80 7F807F80 7F803F00 FF000000 FF000000 FF000000 FF000000 FF000000 FF000000 FF000000 FF000000 FF800000 7F800000 7F800000 7FC001C0 3FE003C0 1FF007C0 0FFC1F80 07FFFF00 01FFFE00 003FF000> PXLC RP 762 2137 XY F55(c)S XP /F55 117 38 3 0 36 27 27 40 0 PXLC RP 792 2137 XY F55(ur)S XP /F55 115 27 2 0 24 27 27 24 0 <07FFF0 1FFFF0 3FFFF0 7F03F0 FC01F0 F800F0 F80070 FC0070 FF0000 FFF000 FFFF00 7FFFC0 7FFFF0 3FFFF8 0FFFF8 01FFFC 000FFC 0001FC E000FC F0007C F0007C F800FC FC00F8 FF03F8 FFFFF0 FFFFE0 F3FF80> PXLC RP 859 2137 XY F55(si)S XP /F55 111 34 2 0 31 27 27 32 0 <003FE000 01FFFC00 03FFFE00 0FF07F80 1FC01FC0 3F800FE0 3F800FE0 7F0007F0 7F0007F0 7F0007F0 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 FF0007F8 7F0007F0 7F0007F0 3F800FE0 3F800FE0 1FC01FC0 0FF07F80 07FFFF00 01FFFC00 003FE000> PXLC RP 905 2137 XY F55(o)S XP /F55 110 38 3 0 36 27 27 40 0 PXLC RP 940 2137 XY F55(n)S 994 X(in)S XP /F55 104 38 3 0 36 42 42 40 0 PXLC RP 1068 2137 XY F55(hi)S XP /F55 103 34 2 -13 32 27 40 32 0 <01FFC3F8 07FFFFFC 1FFFFFFC 1FE3FEFC 3F80FE7C 7F80FF7C 7F007F00 7F007F00 7F007F00 7F007F00 7F007F00 7F007F00 7F80FF00 3F80FE00 1FE3FC00 1FFFFC00 1FFFF000 3DFFC000 38000000 3C000000 3C000000 3F000000 3FFFFC00 3FFFFF80 1FFFFFE0 1FFFFFE0 1FFFFFF0 7FFFFFF0 7F001FF8 FE0007F8 FC0003F8 FC0001F8 FC0001F8 FE0003F8 FF0007F8 7F800FF0 3FE03FE0 1FFFFFC0 07FFFF00 00FFF800> PXLC RP 1125 2137 XY F55(gher)S 1274 X(t)S XP /F55 121 36 1 -12 34 27 39 40 0 PXLC RP 1299 2137 XY F55(y)S XP /F55 112 38 2 -12 35 27 39 40 0 PXLC RP 1335 2137 XY F55(p)S 2 x(es)S XP /F55 44 19 4 -12 15 10 22 16 0 <3F00 7F80 FFC0 FFC0 FFE0 FFE0 FFE0 FFE0 7FE0 3FE0 00E0 01E0 01E0 01C0 03C0 07C0 0780 0F80 1F00 3E00 3C00 3800> PXLC RP 1434 2137 XY F55(,)S 1471 X(P)S XP /F55 67 50 4 0 45 41 41 48 0 <00007FF80380 0007FFFF0F80 001FFFFFDF80 007FF81FFF80 01FFC003FF80 03FF0001FF80 07FE00007F80 0FFC00003F80 1FF800003F80 1FF000001F80 3FF000000F80 3FE000000F80 7FE000000F80 7FE000000780 7FE000000780 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 FFC000000000 7FE000000000 7FE000000380 7FE000000380 3FE000000380 3FF000000780 1FF000000780 1FF800000F00 0FFC00001F00 07FE00003E00 03FF00007E00 01FFC001FC00 007FF80FF000 001FFFFFE000 0007FFFF8000 00007FF80000> PXLC RP 1518 2137 XY F55(C)S XP /F55 70 43 3 0 39 41 41 40 0 PXLC RP 1567 2137 XY F55(F)S XP /F26 60 32 3 -3 28 23 26 32 0 <00000180 00000780 00001F80 00007E00 0001F800 0007E000 001F8000 007E0000 01F80000 07E00000 1F800000 7E000000 F8000000 F8000000 7E000000 1F800000 07E00000 01F80000 007E0000 001F8000 0007E000 0001F800 00007E00 00001F80 00000780 00000180> PXLC RP 1611 2146 XY F26(<)S 2229 Y 300 X F25(Here)S 396 X(w)S -1 x(e)S 455 X(restrict)S 596 X(ourselv)S -1 x(es)S 769 X(to)S 818 X(the)S 887 X(case)S 972 X(where)S 1090 X(the)S 1159 X(w)S -1 x (ell{ordering)S 1412 X F27(\036)S 1456 X F25(considered)S 50 y 300 X (is)S 338 X(the)S 406 X(standard)S 575 X(ordering)S 735 X F26(<)S 777 X F25(of)S 821 X(the)S 889 X(natural)S 1030 X(n)S -1 x(um)S -1 x (b)S 1 x(ers.)S 1212 X(In)S 1261 X(this)S 1338 X(case)S 1422 X(the)S 1489 X(b)S 1 x(ounded)S 49 y 300 X(\014xed)S 397 X(p)S 1 x(oin)S -1 x(t)S 503 X(op)S 1 x(erator)S 667 X(is)S 706 X(de\014nable)S 878 X(b)S -1 x(y)S 933 X(means)S 1057 X(of)S 1101 X(primitiv)S -1 x (e)S 1276 X(recursion,)S 1463 X(as)S 1511 X(follo)S -1 x(ws.)S XP /F25 68 32 2 0 29 28 28 32 0 PXLC RP 375 2378 XY F25(De\014ne)S 505 X(a)S 541 X(functional)S XP /F26 72 35 2 0 35 28 28 40 0 <03FFE7FF80 03FFE7FF80 003E00F800 003C00F000 003C00F000 003C01F000 007C01F000 007801E000 007801E000 007803E000 00F803E000 00F003C000 00FFFFC000 00FFFFC000 01F007C000 01E0078000 01E0078000 01E00F8000 03E00F8000 03C00F0000 03C00F0000 03C01F0000 07C01F0000 07801E0000 07801E0000 0F803E0000 FFF3FFE000 FFF3FFE000> PXLC RP 737 2378 XY F26(H)S 6 y F9(\032)S -6 y 807 X F25(of)S 855 X(t)S -1 x (yp)S 1 x(e)S 950 X(\()S 1 x F26(~)S 966 X(\032)S 1002 X F27(!)S 1058 X F30(nat)S -1 x F25(\))S 1156 X F27(!)S 1212 X F25(\()S F26 (\032)S 1263 X F27(!)S 1319 X F26(\032)S F25(\))S 1371 X F27(!)S 1426 X F30(nat)S 1509 X F27(!)S 1564 X F26(\032)S 1601 X F25(b)S -1 x(y)S 50 y 300 X(primitiv)S -1 x(e)S 478 X(recursion,)S 667 X (as)S 718 X(in)S XP /F25 71 33 2 -1 30 29 30 32 0 <003FC180 00FFF180 03F07B80 07C01F80 0F800F80 1F000780 3E000780 3E000380 7C000380 7C000180 78000180 F8000180 F8000000 F8000000 F8000000 F8000000 F8000000 F800FFF0 F800FFF0 78000780 7C000780 7C000780 3E000780 3E000780 1F000780 0F800F80 07C00F80 03F83F80 00FFF980 003FC000> PXLC RP 767 2428 XY F25(G)S XP /F25 127 21 4 24 16 29 5 16 0 <70E0 F9F0 F9F0 F9F0 70E0> PXLC RP 799 2428 XY F25(\177)S 799 X(odel's)S 915 X(Dialectica)S 1108 X (system)S XP /F26 84 24 1 0 29 28 28 32 0 <1FFFFFF0 3FFFFFF0 3E07C1F0 38078070 70078060 60078060 600F8060 E00F0060 C00F0060 C00F0060 001F0000 001E0000 001E0000 001E0000 003E0000 003C0000 003C0000 003C0000 007C0000 00780000 00780000 00780000 00F80000 00F00000 00F00000 01F00000 7FFF8000 7FFF8000> PXLC RP 1246 2428 XY F26(T)S 1276 X F25(:)S 2518 Y 689 X F26(H)S 6 y F9 (\032)S -6 y 2 x F26(\026G)S F27(f)S F25(0)S F27(g)S 875 X F25(:=)S 931 X F26(H)S 6 y F9(\032)S -6 y 2 x F26(\026G)S F27(;)S 1075 X F25 (:=)S 1130 X F26(\025)S -1 x(~)S 1154 X(x)S F27(;)S F26(;)S 50 y 613 X(H)S 6 y F9(\032)S -6 y 2 x F26(\026G)S F27(f)S F26(n)S 780 X F25(+)S 821 X(1)S F27(g)S 875 X F25(:=)S XP /F26 75 35 2 0 35 28 28 40 0 <03FFE1FF80 03FFE1FF80 003E007C00 003C00F000 003C01E000 003C038000 007C070000 00780E0000 00781C0000 0078780000 00F8F00000 00F1E00000 00F3F00000 00F7F00000 01FEF80000 01FCF80000 01F8780000 01E07C0000 03E07C0000 03C03C0000 03C03E0000 03C03E0000 07C01F0000 07801F0000 07800F0000 0F801F8000 FFF87FF000 FFF87FF000> PXLC RP 931 2568 XY F26(K)S 6 y F9(\032)S -6 y 3 x F26(\026G)S F27(f)S F26 (n)S F27(g)S F25(\()S F26(H)S 6 y F9(\032)S -6 y 1 x F26(\026G)S F27 (f)S F26(n)S F27(g)S F25(\))S F26(;)S 2661 Y 300 X F25(where)S 420 X F26(K)S 6 y F9(\032)S -6 y 488 X F25(of)S 536 X(t)S -1 x(yp)S 1 x (e)S 629 X(\()S 1 x F26(~)S 645 X(\032)S 678 X F27(!)S 731 X F30 (nat)S F25(\))S 827 X F27(!)S 880 X F25(\()S F26(\032)S 929 X F27 (!)S 982 X F26(\032)S F25(\))S 1032 X F27(!)S 1085 X F30(nat)S 1164 X F27(!)S 1218 X F26(\032)S 1250 X F27(!)S 1304 X F26(\032)S 1339 X F25(is)S 1381 X(de\014ned)S 1524 X(b)S -1 x(y)S 2753 Y 381 X F26(K)S 7 y F9(\032)S -7 y 2 x F26(\026Gxz)S 1 x(~)S 538 X(y)S 571 X F25 (:=)S 626 X([)S 631 X([cond)S 7 y F9(\023)S -7 y 2 x F25(])S 747 X (]\([)S 780 X([)S F26(c)S 7 y F8(=)S -7 y 1 x F25(])S 841 X(]\()S F26(\026~)S 894 X(y)S 1 x F25(\))S F26(x)S F25(\)\()S F26(Gz)S 1 x (~)S 1041 X(y)S 2 x F25(\)\([)S 1100 X([cond)S 7 y F9(\023)S -7 y 2 x F25(])S 1216 X(]\([)S 1249 X([)S F26(c)S XP /F9 60 26 3 -3 21 17 20 24 0 <0001C0 0007C0 000F80 003E00 00FC00 03F000 0FC000 1F0000 7C0000 F80000 F80000 7C0000 1F0000 0FC000 03F000 00FC00 003E00 000F80 0007C0 0001C0> PXLC RP 1278 2760 XY F9(<)S -7 y 2 x F25(])S 1311 X(]\()S F26(\026~)S 1364 X (y)S 1 x F25(\))S F26(x)S F25(\)\()S F26(z)S 1 x(~)S 1478 X(y)S 2 x F25(\))S F27(;)S F25(\))S F26(:)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(392)S 700 X F18(H.)S 752 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S 1023 X(and)S 1098 X(S.S.)S 1175 X(W)S -3 x(ainer)S 553 Y 300 X F25(Then)S 408 X(w)S -1 x(e)S 470 X(ha)S -1 x(v)S -1 x(e,)S 577 X(b)S -1 x(y)S 634 X(induction)S 819 X(on)S 877 X F26(n)S 667 Y 589 X(H)S 7 y F9(\032)S -7 y 2 x F26 (\026G)S F27(f)S F26(n)S F27(g)S F26(~)S 768 X(y)S 800 X F25(=)S -58 y 844 X F67(\032)S 35 y 882 X F25([)S 887 X([Y)S 6 y F9(\032;<)S -6 y 1 x F25(])S 989 X(])S F26(\026G~)S 1059 X(y)S 1 x(;)S 1133 X F25(if)S 1171 X F26(\026~)S 1196 X(y)S 1229 X(<)S 1273 X F27(f)S F26 (n)S F27(g)S F25(;)S 49 y 882 X F27(;)S F26(;)S 1133 X F25 (otherwise.)S 784 Y 300 X(Hence)S 423 X([)S 428 X([Y)S 6 y F9 (\032;<)S -6 y 2 x F25(])S 531 X(])S 556 X(is)S 597 X(explicitely)S 796 X(de\014nable)S 971 X(from)S 1069 X F26(H)S 6 y F9(\032)S -6 y 2 x F25(,)S 1148 X(b)S -1 x(y)S 875 Y 663 X([)S 668 X([Y)S 6 y F9 (\032;<)S -6 y 2 x F25(])S 771 X(])S F26(\026G~)S 841 X(y)S 873 X F25(=)S 917 X F26(H)S 6 y F9(\032)S -6 y 2 x F26(\026G)S F25(\([)S 1050 X([succ])S 1142 X(]\()S F26(\026~)S 1195 X(y)S 1 x F25(\)\))S F26(~)S 1248 X(y)S 2 x(:)S 967 Y 300 X F25(Th)S -1 x(us)S 405 X (PCF)S 6 y F9(<)S -6 y 533 X F25(is)S 575 X(nothing)S 727 X(other)S 836 X(than)S 933 X(a)S 968 X(partial)S 1102 X(v)S -1 x(ersion)S 1244 X(of)S 1291 X(G\177)S 1324 X(odel's)S 1440 X F26(T)S 1470 X F25 (.)S 1501 X(W)S -3 x(e)S 1572 X(no)S -1 x(w)S 49 y 300 X(giv)S -1 x (e)S 381 X(details)S 509 X(of)S 552 X(an)S 605 X(ordinal)S 742 X (analysis)S 894 X(of)S 937 X(this)S 1013 X(system,)S 1159 X(follo)S -1 x(wing)S 1330 X(the)S 1398 X(ideas)S 1497 X(outlined)S 50 y 300 X (in)S 348 X(section)S 487 X(2.)S 50 y 375 X(W)S -3 x(e)S 449 X(sho)S -1 x(w)S 554 X(that)S 647 X(an)S -1 x(y)S 728 X(closed)S 853 X (term)S 955 X F26(M)S 1016 X F25(of)S 1066 X(t)S -1 x(yp)S 1 x(e)S 1161 X F30(nat)S 1246 X F27(!)S 1303 X F30(nat)S 1388 X F25(in)S -1 x(v)S -1 x(olvi)S -3 x(ng)S 1568 X(only)S 50 y 300 X(b)S 1 x (ounded)S 467 X(\014xed)S 564 X(p)S 1 x(oin)S -1 x(t)S 669 X(op)S 1 x(erators)S 850 X(Y)S 6 y F9(\032;<)S -6 y 947 X F25(has)S 1018 X (the)S 1087 X(prop)S 1 x(ert)S -1 x(y)S 1253 X(that)S 1340 X(v)S -2 x(alues)S 1461 X(of)S 1505 X(the)S 1574 X F26(M)S 1619 X F30(n)S 50 y 300 X F25(for)S 364 X F26(n)S 402 X F25(=)S 448 X(0)S F26(;)S 488 X F25(1)S F26(;)S 528 X F25(2)S F26(;)S 565 X(:)S 584 X(:)S 603 X(:)S 628 X F25(\(if)S 683 X(de\014ned\))S 843 X(are)S 914 X (ev)S -1 x(en)S -1 x(tually)S 1113 X(b)S 1 x(ounded)S 1283 X(b)S -1 x(y)S 1342 X(a)S 1377 X(function)S 1541 X F26(G)S XP /F9 11 22 1 0 19 13 13 24 0 <03F000 0E38C0 1C1DC0 381D80 701F80 701F80 701F00 F00E00 E00E00 601E00 703E00 38EEC0 1F87C0> PXLC RP 1573 1222 XY F9(\013)S -6 y 1612 X F25(of)S 50 y 300 X(the)S 371 X (slo)S -1 x(w{gro)S -1 x(wing)S 624 X(subrecurs)S 3 x(iv)S -1 x(e)S 860 X(hierarc)S -1 x(h)S -1 x(y)S 1040 X(with)S XP /F26 11 27 1 0 24 18 18 24 0 <00FC00 03FE00 0F8F06 1E078E 3C038C 3C038C 7803DC 7803D8 F801F8 F001F0 F001E0 F001E0 F001C0 F003C0 700FC0 7C3FEE 3FF8FC 0FE078> PXLC RP 1134 1266 XY F26(\013)S 1174 X F25(denoting)S 1344 X(an)S 1401 X (ordinal)S 1541 X(b)S 1 x(elo)S -1 x(w)S 49 y 300 X(the)S 369 X (Bac)S -1 x(hmann{Ho)S -1 x(w)S -1 x(ard)S 733 X(ordinal.)S 890 X (The)S 973 X(argumen)S -1 x(t)S 1157 X(is)S 1196 X(similar)S 1331 X (to)S 1379 X(the)S 1449 X(one)S 1523 X(in)S 1569 X([11)S -1 x(],)S 50 y 300 X(and)S 381 X(w)S -1 x(e)S 442 X(rely)S 524 X(hea)S -1 x (vily)S 666 X(on)S 723 X(that)S 813 X(pap)S 1 x(er.)S 50 y 375 X (By)S 439 X(the)S 509 X(remark)S 650 X(ab)S 1 x(o)S -1 x(v)S -1 x (e)S 767 X(it)S 807 X(su\016ces)S 947 X(to)S 997 X(deal)S 1084 X (with)S 1177 X(terms)S 1292 X(in)S -1 x(v)S -1 x(olving)S 1468 X (the)S 1539 X(recur-)S 50 y 300 X(sion)S 389 X(op)S 1 x(erators)S XP /F26 82 32 2 -1 31 28 29 32 0 <03FFFE00 03FFFF80 003E03E0 003C01E0 003C01F0 003C01F0 007C01F0 007801F0 007801F0 007803E0 00F803C0 00F00780 00F01F00 00FFFC00 01FFF000 01E07800 01E03C00 01E03C00 03E03C00 03C03C00 03C03C00 03C07C00 07C07C00 07807C00 07807C18 0F807E38 FFF87E70 FFF83FE0 00000FC0> PXLC RP 576 1465 XY F26(R)S 6 y F9(\032)S -6 y 644 X F25(instead)S 791 X (of)S 842 X(the)S XP /F26 89 24 1 0 30 28 28 32 0 PXLC RP 917 1465 XY F26(Y)S 6 y F9(\032;<)S -6 y 2 x F25('s.)S 1064 X F26 (R)S 6 y F9(\032)S -6 y 1132 X F25(has)S 1210 X(t)S -1 x(yp)S 1 x (e)S 1307 X F26(\032)S 1345 X F27(!)S 1404 X F25(\()S F30(nat)S 1506 X F27(!)S 1565 X F26(\032)S 1603 X F27(!)S 50 y 300 X F26 (\032)S F25(\))S 349 X F27(!)S 402 X F30(nat)S 482 X F27(!)S 535 X F26(\032)S F25(,)S 582 X(and)S 663 X(the)S 734 X(con)S -1 x(v)S -1 x (ersion)S 936 X(rules)S 1036 X(asso)S 1 x(ciated)S 1233 X(with)S 1328 X(it)S 1369 X(are)S 1611 Y 768 X F26(R)S 6 y F9(\032)S XP /F26 80 27 2 0 30 28 28 32 0 <03FFFF00 03FFFFC0 003E03E0 003C01F0 003C01F0 007C01F0 007C01F0 007801F0 007801F0 00F801E0 00F803E0 00F003C0 00F00780 01F01F00 01FFFE00 01FFF000 01E00000 03E00000 03E00000 03C00000 03C00000 07C00000 07C00000 07800000 07800000 0F800000 FFF80000 FFF80000> PXLC RP 819 1611 XY F26(P)S XP /F26 81 33 2 -8 30 29 37 32 0 <0003F800 000FFE00 003E0F80 00F807C0 01E003C0 03C001E0 078001E0 0F8001F0 1F0001F0 1E0001F0 3E0001F0 3E0001F0 7C0001F0 7C0001F0 7C0001F0 F80003E0 F80003E0 F80003E0 F80007C0 F80007C0 F8000780 F8000F80 F83C1F00 787E1E00 78E63C00 3CC37800 3EC3F000 1FC7C000 07FF8100 01FF0100 00038300 00038E00 0003FE00 0003FC00 0003FC00 0003F800 0001F000> PXLC RP 851 1611 XY F26(Q)S F25(0)S -11 y 2 x F26(~)S 11 y 905 X(L)S 945 X F27(!)S 999 X F26(P)S -11 y 1033 X(~)S 11 y 1032 X(L;)S 55 y 635 X (R)S 6 y F9(\032)S -6 y 2 x F26(P)S 719 X(Q)S F25(\(succ)S 1 x F26 (M)S 889 X F25(\))S -10 y 2 x F26(~)S 10 y 905 X(L)S 945 X F27(!)S 999 X F26(QM)S 1077 X F25(\()S F26(R)S 6 y F9(\032)S -6 y 2 x F26 (P)S 1177 X(QM)S 1255 X F25(\))S -10 y 2 x F26(~)S 10 y 1271 X(L)S -1 x(:)S 1756 Y 300 X F25(Since)S 407 X(an)S -1 x(y)S 484 X(suc)S -1 x(h)S 576 X(term)S 674 X(has)S 746 X(a)S 779 X(unique)S XP /F26 12 23 -1 -8 24 29 37 32 0 <0000FC00 0007FE00 000F8F00 001C0780 00380380 00700380 00E00780 00C00780 01C00700 01800F00 03801E00 038FFC00 030FF800 030FFC00 07001E00 06001E00 06000E00 06000E00 0E001E00 0C001E00 0C001E00 0C001E00 1C003E00 1C003C00 1E007800 1F00F800 3F81F000 33FFC000 31FF8000 30180000 70000000 60000000 60000000 60000000 E0000000 C0000000 C0000000> PXLC RP 913 1756 XY F26(\014)S XP /F26 17 21 1 -9 20 18 27 24 0 <3E1F80 3F3FE0 7771E0 67E0E0 E7C0E0 C780E0 0700E0 0F01E0 0F01E0 0E01C0 0E01C0 1E03C0 1E03C0 1C0380 1C0380 3C0780 3C0780 380700 000700 000F00 000F00 000E00 000E00 001E00 001E00 001C00 001C00> PXLC RP 938 1756 XY F26(\021)S 1 x(R)S F25({normal)S 1152 X(form,)S 1261 X (w)S -1 x(e)S 1320 X(can)S 1395 X(iden)S -1 x(tify)S 1545 X F26 (\014)S 2 x(\021)S 1 x(R)S F25({)S 49 y 300 X(equiv)S -2 x(alen)S -1 x(t)S 497 X(terms.)S 50 y 375 X(F)S -3 x(or)S 453 X(an)S -1 x(y)S 535 X(term)S 638 X F26(M)S 701 X F25(let)S 765 X(lev\()S F26(M)S 878 X F25(\))S 912 X(denote)S 1050 X(the)S 1125 X(lev)S -1 x(el)S 1223 X(of)S 1274 X(the)S 1350 X(t)S -1 x(yp)S 1 x(e)S 1447 X(of)S 1498 X F26(M)S 1543 X F25(.)S 1584 X(F)S -3 x(or)S 50 y 300 X (terms)S 416 X F26(M)S 475 X F25(with)S 569 X(lev)S 1 x(\()S F26(M)S 683 X F25(\))S 711 X(=)S 755 X(0,)S 801 X(tree)S 884 X(ordinals)S 1042 X F26(\013)S 1080 X F27(2)S XP /F25 10 30 2 0 27 29 29 32 0 <00FF8000 03FFE000 0FC1F800 1F007C00 3E003E00 7C001F00 7C001F00 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 7C001F00 7C001F00 3C001E00 3E003E00 1E003C00 1E003C00 0F007800 CF007980 C7007180 C3006180 E380E380 7F80FF00 7F80FF00 7F80FF00 7F80FF00> PXLC RP 1120 1905 XY F25(\012)S 6 y F8(2)S -6 y 1182 X F25(and)S 1263 X F26 (n)S 1300 X F27(2)S XP /F30 78 37 2 0 34 28 28 32 0 PXLC RP 1340 1905 XY F30(N)S 1391 X F25(w)S -1 x(e)S 1452 X(no)S -1 x(w)S 1539 X(follo)S -1 x(w)S 50 y 300 X(the)S 371 X(ideas)S 475 X(of)S 522 X(section)S 660 X(2)S 695 X(in)S 743 X(de\014ning)S 900 X (inductiv)S -1 x(ely)S 1113 X(the)S 1184 X(relation)S XP /F27 96 25 2 0 22 29 29 24 0 PXLC RP 1336 1955 XY F27(`)S -15 y F9(\013)S 25 y 1361 X(n)S -10 y 1390 X F26(M)S 1435 X F25(,)S 1460 X(to)S 1510 X(b)S 1 x(e)S 1566 X(read)S 50 y 300 X F26(M)S 359 X F25(is)S XP /F28 103 19 2 -8 19 18 26 24 0 <00FB80 01FF80 03CF80 078780 0F0780 1E0700 1E0700 1E0F00 3E0F00 3C0E00 3C0E00 3C1E00 3C1E00 383C00 3C3C00 1CFC00 1FFC00 0FB800 003800 007800 007800 F07000 F0F000 F1E000 FFC000 7F0000> PXLC RP 400 2005 XY F28(gener)S -2 x(ate)S XP /F28 100 21 4 0 22 29 29 24 0 <000FC0 001FC0 0003C0 0003C0 000380 000380 000780 000780 000700 000700 000F00 03EF00 07FE00 0F3E00 1E1E00 3C1E00 781C00 781C00 783C00 F83C00 F03800 F03800 F079C0 F079C0 E0F180 F0F180 73F300 7FFF00 3E3E00> PXLC RP 548 2005 XY F28(d)S 588 X F25(with)S 682 X(size)S 761 X F26(\013)S 801 X F25(and)S 882 X(rank)S 978 X F26(n)S F25(,)S 1028 X(b)S -1 x (y)S 1086 X(the)S 1157 X(rules:)S 2088 Y 362 X F27(\017)S XP /F28 86 31 8 -1 35 28 29 32 0 PXLC RP 404 2088 XY F28(V)S -3 x(ariable)S XP /F28 82 30 2 -1 29 28 29 32 0 <01FFFC00 01FFFF00 003E0F80 003C07C0 003C03C0 003C03C0 007C07C0 007807C0 007807C0 00780780 00F80F80 00F01F00 00F03E00 00FFF800 01FFF000 01E0F000 01E07800 01E07800 03E07800 03C07800 03C07800 03C0F800 07C0F800 0780F800 0780F860 0F80FCE0 FFF0FDC0 FFF07FC0 00001F00> PXLC RP 571 2088 XY F28(R)S -1 x(ule)S XP /F28 46 13 5 0 10 5 5 8 0 <70 F8 F8 F0 F0> PXLC RP 652 2088 XY F28(.)S 693 X F25(If)S 738 X F27(`)S -15 y F9(\013)S 25 y 763 X(n)S -10 y 797 X F26(L)S XP /F9 105 12 0 0 9 20 20 16 0 <0380 0780 0300 0000 0000 0000 0000 3E00 7600 6F00 EE00 CE00 1E00 1C00 1C00 3D80 3980 3B80 3B00 1E00> PXLC RP 826 2094 XY F9(i)S -6 y 843 X F26(~)S 839 X(y)S 6 y F9(i)S -6 y 891 X F25(for)S XP /F26 105 14 1 0 13 28 28 16 0 <00E0 01E0 01E0 01C0 0000 0000 0000 0000 0000 0000 1F00 3F80 7380 6380 E380 C780 0700 0700 0F00 0E00 1E00 1C70 1C70 3C60 38E0 3DC0 1F80 0F00> PXLC RP 958 2088 XY F26(i)S 989 X F25(=)S 1038 X(1)S F26(;)S 1078 X(:)S 1097 X(:)S 1116 X(:)S 1133 X(;)S 1152 X(m)S 1205 X F25(with)S 1303 X F26(m)S XP /F27 21 32 3 -8 28 26 34 32 0 PXLC RP 1356 2088 XY F27(\025)S 1406 X F25(0,)S 1456 X(then)S 1554 X F27(`)S -15 y F9(\013)S XP /F8 43 25 2 -4 22 18 22 24 0 <006000 006000 006000 006000 006000 006000 006000 006000 006000 006000 FFFFF0 FFFFF0 006000 006000 006000 006000 006000 006000 006000 006000 006000 006000> PXLC RP 1601 2073 XY F8(+1)S 25 y 1579 X F9(n)S 40 y 404 X F26(xL)S 6 y F8 (1)S -6 y 481 X F26(:)S 500 X(:)S 519 X(:)S 537 X(L)S XP /F9 109 29 0 0 27 13 13 32 0 <3C7E3E00 6EC7E700 6F87C700 EF078700 CF070700 0E070700 1E0F0F00 1E0F0E00 1C0E0E60 1C0E1E60 3C1E1CE0 3C1E1DC0 381C0F80> PXLC RP 565 2144 XY F9(m)S -6 y 2 x F25(.)S 2226 Y 362 X F27(\017)S XP /F28 67 30 6 -1 33 29 30 32 0 <0007F060 001FFCE0 007E0FE0 00F807C0 01F003C0 07E003C0 07C003C0 0F800180 1F000180 1F000180 3E000380 3E000300 7C000000 7C000000 7C000000 FC000000 F8000000 F8000000 F8000000 F8000C00 F8000C00 F8001C00 F8001800 78003800 7C007000 3C00E000 3E01E000 1F878000 0FFF0000 03FC0000> PXLC RP 404 2226 XY F28(Constant)S 584 X(R)S -1 x(ules.)S 719 X F27(`)S -16 y F9(\013)S F8(+1)S 26 y 744 X F9(n)S -10 y 818 X F27(?)S -16 y F30 (nat)S 16 y 2 x F25(,)S 948 X F27(`)S -16 y F9(\013)S F8(+1)S 26 y 973 X F9(n)S -10 y 1047 X F25(0,)S 1096 X F27(`)S -16 y F9(\013)S F8(+1)S 26 y 1121 X F9(n)S -10 y 1195 X F27(?)S -16 y F30(b)S 1 x (o)S 1 x(ole)S 16 y 2 x F25(,)S 1369 X F27(`)S -16 y F9(\013)S F8 (+1)S 26 y 1394 X F9(n)S -10 y 1468 X F25(t)S 1477 X(t,)S 1521 X F27 (`)S -16 y F9(\013)S F8(+1)S 26 y 1546 X F9(n)S -10 y 1620 X F25(f)S 1629 X(f)S 49 y 404 X(\(an)S -1 x(y)S 499 X F26(\013)S F25(\).)S 574 X(F)S -3 x(or)S 650 X(the)S 722 X(other)S 831 X(constan)S -1 x (ts)S 1016 X(succ)S 1 x(,)S 1119 X(cond)S 7 y F9(\023)S -7 y 3 x F25 (,)S 1245 X(p)S 1 x(cond)S 10 y 1 x F9(\023)S -10 y 2 x F25(,)S 1395 X F26(c)S 7 y F8(=)S -7 y 1455 X F25(and)S 1537 X F26(c)S 7 y F9(<)S -7 y 1598 X F25(w)S -1 x(e)S 50 y 404 X(ha)S -1 x(v)S -1 x (e)S 499 X(rules)S 599 X(similar)S 735 X(to)S 786 X(the)S XP /F25 86 31 1 -1 29 28 29 32 0 PXLC RP 858 2325 XY F25(V)S -3 x(ariable)S XP /F25 82 31 2 -1 30 28 29 32 0 PXLC RP 1021 2325 XY F25(Rule,)S 1130 X(e.g.)S XP /F30 123 24 0 10 23 12 2 24 0 PXLC RP 450 2408 XY F30({)S 495 X F25(If)S 537 X F27(`)S -15 y F9(\013)S 26 y 562 X(n)S -11 y 590 X F26(M)S 635 X F25(,)S 660 X(then)S 755 X F27(`)S -15 y F9(\013)S F8(+1)S 26 y 780 X F9(n)S -11 y 851 X F25 (succ)S 938 X F26(M)S 983 X F25(.)S 67 y 450 X F30({)S 495 X F25 (If)S 537 X F27(`)S -15 y F9(\013)S 25 y 562 X(n)S -10 y 590 X F26 (M)S 635 X F25(,)S 660 X F27(`)S -15 y F9(\013)S 25 y 685 X(n)S -10 y 714 X F26(N)S 766 X F25(and)S 846 X F27(`)S -15 y F9(\013)S 25 y 871 X(n)S -10 y 900 X F26(L)S F25(,)S 954 X(then)S 1048 X F27(`)S -15 y F9(\013)S F8(+1)S 25 y 1073 X F9(n)S -10 y 1144 X F25(cond)S 6 y F9(\023)S -6 y 3 x F26(M)S 1289 X(N)S 1327 X(L)S F25(.)S 2558 Y 362 X F27(\017)S XP /F26 33 26 0 0 23 18 18 24 0 <0C001E 1C001E 18001E 38001E 30000E 701C0E 601C06 601C0E E03C0C C0380C C0381C C03018 E07038 E0F870 FFFFF0 FFFFE0 7FBFC0 3E1F00> PXLC RP 404 2558 XY F26(!)S -15 y 1 x F8(+)S XP /F28 123 21 4 10 22 12 2 24 0 PXLC RP 459 2558 XY F28({R)S -1 x(ule.)S 592 X F25(If)S 634 X F26(\013)S 672 X F25(=)S XP /F27 104 16 4 -11 13 31 42 16 0 <0180 0380 0380 0300 0700 0700 0600 0E00 0E00 0C00 1C00 1C00 1800 3800 3800 3000 7000 7000 6000 E000 E000 E000 E000 6000 7000 7000 3000 3800 3800 1800 1C00 1C00 0C00 0E00 0E00 0600 0700 0700 0300 0380 0380 0180> PXLC RP 716 2558 XY F27(h)S F26(\013)S XP /F9 24 15 1 -6 13 21 27 16 0 <0180 0180 0180 03F0 07B0 1FF0 1C00 3C00 3800 3800 3800 3FC0 1CC0 3FC0 6000 E000 C000 C000 E000 F000 FE00 7F80 3FC0 07C0 01C0 0DC0 0780> PXLC RP 759 2565 XY F9(\030)S XP /F27 105 16 2 -11 11 31 42 16 0 PXLC RP 777 2558 XY F27(i)S F25(,)S 818 X F27(`)S -15 y F9(\013)S XP /F0 /cmr5 300 20.8 20.8 128 [-1 -6 26 16] PXLNF RP XP /F0 48 14 2 0 12 13 13 16 0 <1E00 6180 4080 C0C0 C0C0 C0C0 C0C0 C0C0 C0C0 C0C0 4080 6180 1E00> PXLC RP 865 2547 XY F0(0)S PF XP /F9 /cmmi7 300 29.1 29.1 128 [-1 -7 32 22] PXLNF RP XP /F9 110 21 0 0 18 13 13 24 0 <3C7C00 6FCE00 6F8E00 EF0E00 CE0E00 0E0E00 1E1E00 1E1C00 1C1CC0 1C3CC0 3C39C0 3C3B80 381F00> PXLC RP 844 2569 XY F9(n)S XP /F26 /cmmi10 300 41.5 41.5 128 [-1 -10 43 31] PXLNF RP XP /F26 77 40 2 0 42 28 28 40 0 <03FF0001FF 03FF0003FF 003F0003F0 003F0007E0 003F000DE0 003F000FE0 0077801BE0 0067801BC0 00678033C0 00678067C0 00E78067C0 00C780C780 00C780C780 00C7818F80 01C7830F80 0187830F00 0187860F00 0183C61F00 0383CC1F00 0303CC1E00 0303D81E00 0303F03E00 0703F03E00 0603E03C00 0603E03C00 1F03C07C00 FFE387FFC0 FFE387FFC0> PXLC RP 888 2558 XY F26(M)S XP /F25 /cmr10 300 41.5 41.5 128 [-3 -11 41 31] PXLNF RP XP /F25 97 21 1 0 20 18 18 24 0 <1FE000 3FF800 3C3C00 3C1E00 180E00 000E00 000E00 03FE00 1F8E00 7E0E00 780E00 F80E00 F00E60 F00E60 F81E60 7C7E60 7FEFC0 1F8780> PXLC RP 947 2558 XY F25(a)S XP /F25 110 23 1 0 21 18 18 24 0 PXLC RP 967 2558 XY F25(n)S XP /F25 100 23 1 0 21 29 29 24 0 <001F80 001F80 000380 000380 000380 000380 000380 000380 000380 000380 000380 07F380 1FFF80 3E1F80 7C0780 780380 F80380 F00380 F00380 F00380 F00380 F00380 F00380 F00380 780780 7C0F80 3E1F80 1FFBF0 07E3F0> PXLC RP 990 2558 XY F25(d)S XP /F27 /cmsy10 300 41.5 41.5 128 [-2 -40 47 33] PXLNF RP XP /F27 56 23 0 -1 22 29 30 24 0 PXLC RP 694 2656 XY F27(8)S XP /F26 24 18 0 -8 17 30 38 24 0 <000800 001800 001800 001800 003F80 00FF80 03FF80 07C000 078000 0F8000 0F0000 0F0000 0F0000 0F0000 0F0000 0FFC00 07FC00 0FFC00 1E0000 1C0000 380000 780000 700000 700000 F00000 F00000 F00000 F80000 7E0000 3F8000 1FF000 07FC00 01FC00 003E00 000E00 031C00 03FC00 01F800> PXLC RP 717 2656 XY F26(\030)S XP /F27 50 28 3 -3 24 23 26 24 0 <00FFF8 03FFF8 0FC000 1F0000 3C0000 380000 700000 700000 E00000 E00000 C00000 C00000 FFFFF8 FFFFF8 C00000 C00000 E00000 E00000 700000 700000 380000 3C0000 1F0000 0FC000 03FFF8 00FFF8> PXLC RP 748 2656 XY F27(2)S XP /F25 10 30 2 0 27 29 29 32 0 <00FF8000 03FFE000 0FC1F800 1F007C00 3E003E00 7C001F00 7C001F00 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 F8000F80 7C001F00 7C001F00 3C001E00 3E003E00 1E003C00 1E003C00 0F007800 CF007980 C7007180 C3006180 E380E380 7F80FF00 7F80FF00 7F80FF00 7F80FF00> PXLC RP 788 2656 XY F25(\012)S XP /F8 /cmr7 300 29.1 29.1 128 [-2 -8 32 22] PXLNF RP XP /F8 49 17 3 0 14 19 19 16 0 <0600 1E00 FE00 EE00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0> PXLC RP 817 2662 XY F8(1)S -6 y 2 x F27(8)S XP /F26 109 36 1 0 36 18 18 40 0 <3E0FC1F800 3F3FE3FE00 7778F71E00 67E07E0E00 E7C07C0E00 C780780E00 0780700E00 0F00F01E00 0F00F01C00 0E00E01C00 0E00E03C00 1E01E038E0 1E01E038E0 1C01C078C0 1C01C071C0 3C03C07B80 3C03C03F00 3803801E00> PXLC RP 859 2656 XY F26(m)S XP /F26 58 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 896 2656 XY F26(:)S XP /F27 96 25 2 0 22 29 29 24 0 PXLC RP 919 2656 XY F27(`)S XP /F9 24 15 1 -6 13 21 27 16 0 <0180 0180 0180 03F0 07B0 1FF0 1C00 3C00 3800 3800 3800 3FC0 1CC0 3FC0 6000 E000 C000 C000 E000 F000 FE00 7F80 3FC0 07C0 01C0 0DC0 0780> PXLC RP 944 2636 XY F9(\030)S XP /F8 48 17 1 0 15 19 19 16 0 <0FC0 1860 3030 6018 6018 E01C E01C E01C E01C E01C E01C E01C E01C E01C 6018 6018 3030 1860 0FC0> PXLC RP 944 2667 XY F8(0)S XP /F30 /cmbx10 300 41.5 41.5 128 [-4 -10 47 31] PXLNF RP XP /F30 109 40 1 0 38 18 18 40 0 PXLC RP 967 2656 XY F30(m)S XP /F25 61 32 2 4 29 16 12 32 0 PXLC RP 1030 2656 XY F25(=)S XP /F27 41 42 2 -2 39 22 24 40 0 <0000030000 0000030000 0000038000 000001C000 000000E000 000000E000 FFFFFFF000 FFFFFFFC00 0000001E00 0000000780 00000003E0 00000000F8 00000001F0 0000000780 0000000F00 0000001C00 FFFFFFF800 FFFFFFF000 000000E000 000000C000 000001C000 0000038000 0000030000 0000030000> PXLC RP 1055 2656 XY F27(\))S 1109 X(`)S XP /F9 11 22 1 0 19 13 13 24 0 <03F000 0E38C0 1C1DC0 381D80 701F80 701F80 701F00 F00E00 E00E00 601E00 703E00 38EEC0 1F87C0> PXLC RP 1134 2634 XY F9(\013)S XP /F1 /cmmi5 300 20.8 20.8 128 [0 -5 28 16] PXLNF RP XP /F1 24 14 2 -4 11 14 18 16 0 <0400 0400 0F80 3F80 7000 6000 6000 7F00 7E00 E000 C000 C000 E000 F800 7F00 0F00 0100 1F00> PXLC RP 1155 2638 XY F1(\030)S 23 y 1134 X F9(n)S XP /F26 82 32 2 -1 31 28 29 32 0 <03FFFE00 03FFFF80 003E03E0 003C01E0 003C01F0 003C01F0 007C01F0 007801F0 007801F0 007803E0 00F803C0 00F00780 00F01F00 00FFFC00 01FFF000 01E07800 01E03C00 01E03C00 03E03C00 03C03C00 03C03C00 03C07C00 07C07C00 07807C00 07807C18 0F807E38 FFF87E70 FFF83FE0 00000FC0> PXLC RP 1178 2656 XY F26(R)S XP /F26 80 27 2 0 30 28 28 32 0 <03FFFF00 03FFFFC0 003E03E0 003C01F0 003C01F0 007C01F0 007C01F0 007801F0 007801F0 00F801E0 00F803E0 00F003C0 00F00780 01F01F00 01FFFE00 01FFF000 01E00000 03E00000 03E00000 03C00000 03C00000 07C00000 07C00000 07800000 07800000 0F800000 FFF80000 FFF80000> PXLC RP 1210 2656 XY F26(P)S XP /F26 81 33 2 -8 30 29 37 32 0 <0003F800 000FFE00 003E0F80 00F807C0 01E003C0 03C001E0 078001E0 0F8001F0 1F0001F0 1E0001F0 3E0001F0 3E0001F0 7C0001F0 7C0001F0 7C0001F0 F80003E0 F80003E0 F80003E0 F80007C0 F80007C0 F8000780 F8000F80 F83C1F00 787E1E00 78E63C00 3CC37800 3EC3F000 1FC7C000 07FF8100 01FF0100 00038300 00038E00 0003FE00 0003FC00 0003FC00 0003F800 0001F000> PXLC RP 1243 2656 XY F26(Q)S F30(m)S XP /F26 126 21 8 22 26 30 8 24 0 <000700 000700 000380 FFFFC0 FFFFC0 000F80 000E00 000C00> PXLC RP 1317 2646 XY F26(~)S XP /F26 76 28 2 0 26 28 28 24 0 <03FFF0 03FFF0 003E00 003C00 003C00 003C00 007C00 007800 007800 007800 00F800 00F000 00F000 00F000 01F000 01E000 01E000 01E007 03E006 03C006 03C00E 03C00C 07C01C 07803C 078078 0F81F8 FFFFF8 FFFFF0> PXLC RP 1315 2656 XY F26(L)S XP /F26 59 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 1344 2656 XY F26(;)S XP /F25 116 16 1 0 13 26 26 16 0 <0C00 0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFE0 FFE0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C30 1C30 1C30 1C30 1C30 1E70 0FE0 07C0> PXLC RP 404 2753 XY F25(t)S XP /F25 104 23 1 0 21 29 29 24 0 PXLC RP 420 2753 XY F25(h)S XP /F25 101 18 1 0 16 18 18 16 0 <07E0 1FF8 3E7C 781C 781E F01E FFFE FFFE F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 443 2753 XY F25(en)S 498 X F27(`)S -15 y F9(\013)S 26 y 523 X(n)S -11 y 552 X F26(RP)S 617 X(QM)S -10 y 696 X(~)S 10 y 695 X(L)S XP /F25 46 12 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 722 2753 XY F25(.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup XP /F18 /cmr9 300 37.4 37.4 128 [-3 -10 37 28] PXLNF RP XP /F18 79 30 2 0 27 26 26 32 0 <007F0000 01FFC000 07C1F000 0F007800 1E003C00 3C001E00 38000E00 78000F00 70000700 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 F0000780 78000F00 78000F00 38000E00 3C001E00 1E003C00 0F007800 07C1F000 01FFC000 007F0000> PXLC RP 692 462 XY F18(O)S XP /F18 114 15 1 0 13 16 16 16 0 PXLC RP 722 462 XY F18(r)S XP /F18 100 21 1 0 19 26 26 24 0 <007E00 007E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 000E00 0FEE00 1FFE00 3E3E00 781E00 780E00 F00E00 F00E00 F00E00 F00E00 F00E00 F00E00 780E00 781E00 3C3E00 1FFFC0 0FCFC0> PXLC RP 737 462 XY F18(d)S XP /F18 105 11 0 0 9 26 26 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 758 462 XY F18(i)S XP /F18 110 21 1 0 19 16 16 24 0 PXLC RP 769 462 XY F18(n)S XP /F18 97 19 1 0 18 16 16 24 0 <3FC000 7FF000 78F800 787800 303800 003800 07F800 3F3800 7C3800 F83800 F03800 F03980 F07980 F8F980 7FFF00 3F1E00> PXLC RP 790 462 XY F18(a)S XP /F18 108 11 0 0 9 26 26 16 0 PXLC RP 809 462 XY F18(l)S XP /F18 66 27 1 0 24 26 26 24 0 PXLC RP 833 462 XY F18(B)S XP /F18 111 19 1 0 17 16 16 16 0 <07E0 1FF8 3C3C 781E 700E F00F F00F F00F F00F F00F F00F 700E 781E 3C3C 1FF8 07E0> PXLC RP 860 462 XY F18(o)S XP /F18 117 21 1 0 19 16 16 24 0 PXLC RP 879 462 XY F18(und)S XP /F18 115 15 1 0 13 16 16 16 0 <3FE0 7FE0 F0E0 E060 E060 F000 FF80 7FC0 1FE0 01F0 C0F0 E070 E070 F0F0 FFE0 DF80> PXLC RP 943 462 XY F18(s)S XP /F18 102 12 0 0 14 26 26 16 0 <00F8 03FC 07BC 0F3C 0E3C 0E00 0E00 0E00 0E00 0E00 FFC0 FFC0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 971 462 XY F18(for)S XP /F18 80 26 1 0 23 26 26 24 0 PXLC RP 1030 462 XY F18(Pro)S XP /F18 103 19 1 -8 17 16 24 16 0 <0FCF 1FFF 3CF7 7878 7878 7878 7878 7878 3CF0 3FE0 3FC0 3000 3000 3FF8 3FFC 3FFE 701F E00F E007 E007 F00F 7C3E 3FFC 0FF0> PXLC RP 1090 462 XY F18(gra)S XP /F18 109 32 1 0 30 16 16 32 0 PXLC RP 1144 462 XY F18(ms)S XP /F25 51 21 2 -1 18 28 29 16 0 <0FE0 1FF8 3C7C 703E 783E 7C3E 783E 783E 003E 003C 007C 00F8 07F0 07E0 0078 003C 003E 001E 001F 001F 701F F81F F81F F81F F03E 703E 787C 3FF8 0FE0> PXLC RP 1583 462 XY F25(3)S XP /F25 57 21 2 -1 18 28 29 16 0 <07E0 1FF0 3E78 381C 781C 700E F00E F00E F00F F00F F00F F00F F00F F01F 701F 781F 3C3F 1FEF 0FCF 010F 000E 001E 781E 781C 783C 7878 78F0 3FE0 1F80> PXLC RP 1603 462 XY F25(93)S XP /F27 15 21 2 2 18 18 16 16 0 <07E0 1FF8 3FFC 7FFE 7FFE FFFF FFFF FFFF FFFF FFFF FFFF 7FFE 7FFE 3FFC 1FF8 07E0> PXLC RP 362 553 XY F27(\017)S XP /F28 /cmti10 300 41.5 41.5 128 [-1 -11 45 31] PXLNF RP XP /F28 67 30 6 -1 33 29 30 32 0 <0007F060 001FFCE0 007E0FE0 00F807C0 01F003C0 07E003C0 07C003C0 0F800180 1F000180 1F000180 3E000380 3E000300 7C000000 7C000000 7C000000 FC000000 F8000000 F8000000 F8000000 F8000C00 F8000C00 F8001C00 F8001800 78003800 7C007000 3C00E000 3E01E000 1F878000 0FFF0000 03FC0000> PXLC RP 404 553 XY F28(C)S XP /F28 117 22 3 0 23 18 18 24 0 <1F0380 3F8380 738380 638780 E38780 C78700 070700 070F00 0F0F00 0E0E00 0E0E00 0E1E70 1E1E70 1C1C60 1E3C60 0E7CC0 0FFFC0 07CF80> PXLC RP 433 553 XY F28(u)S XP /F28 116 14 3 0 15 26 26 16 0 <01C0 01C0 01C0 03C0 03C0 0380 0380 0780 FFE0 FFE0 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C70 3C70 3860 38E0 39C0 3F80 1F00> PXLC RP 456 553 XY F28(t)S XP /F28 82 30 2 -1 29 28 29 32 0 <01FFFC00 01FFFF00 003E0F80 003C07C0 003C03C0 003C03C0 007C07C0 007807C0 007807C0 00780780 00F80F80 00F01F00 00F03E00 00FFF800 01FFF000 01E0F000 01E07800 01E07800 03E07800 03C07800 03C07800 03C0F800 07C0F800 0780F800 0780F860 0F80FCE0 FFF0FDC0 FFF07FC0 00001F00> PXLC RP 486 553 XY F28(R)S -1 x(u)S XP /F28 108 11 4 0 13 29 29 16 0 <1F80 3F80 0780 0780 0700 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800 3800 7800 7800 7000 7000 F700 F700 E600 E600 EE00 FC00 7C00> PXLC RP 538 553 XY F28(l)S XP /F28 101 19 4 0 18 18 18 16 0 <01F8 07FC 0F1C 1E0C 3C0C 780C 783C 7FF8 FFC0 F000 F000 F000 F000 F00C 701C 7878 3FF0 1FC0> PXLC RP 549 553 XY F28(e)S XP /F28 46 13 5 0 10 5 5 8 0 <70 F8 F8 F0 F0> PXLC RP 568 553 XY F28(.)S XP /F25 73 15 1 0 13 28 28 16 0 PXLC RP 606 553 XY F25(I)S XP /F25 102 13 0 0 15 29 29 16 0 <00FC 03FE 079E 071E 0F1E 0E00 0E00 0E00 0E00 0E00 0E00 FFE0 FFE0 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 0E00 7FE0 7FE0> PXLC RP 621 553 XY F25(f)S 649 X F27(`)S -15 y F9(\013)S 25 y 674 X(n)S -10 y 707 X F26(M)S 751 X(~)S XP /F26 121 20 1 -8 20 18 26 24 0 <1F00E0 3F80E0 7380E0 6381E0 E381C0 C781C0 0701C0 0703C0 0F0380 0E0380 0E0380 0E0780 0E0700 0E0F00 0E0F00 0F3F00 0FFE00 03EE00 000E00 001E00 381C00 783C00 787800 71F000 7FE000 1F8000> PXLC RP 752 553 XY F26(y)S XP /F25 119 30 1 0 28 18 18 32 0 PXLC RP 790 553 XY F25(w)S XP /F25 105 12 1 0 10 29 29 16 0 <3C00 7C00 7C00 7C00 3C00 0000 0000 0000 0000 0000 0000 FC00 FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 FF80 FF80> PXLC RP 820 553 XY F25(ith)S XP /F25 108 12 1 0 10 29 29 16 0 PXLC RP 886 553 XY F25(le)S XP /F25 118 22 1 0 20 18 18 24 0 PXLC RP 916 553 XY F25(v)S XP /F25 40 16 3 -11 13 31 42 16 0 <01C0 0380 0380 0700 0E00 0C00 1C00 1800 3800 3800 7000 7000 7000 7000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 7000 7000 7000 7000 3800 3800 1800 1C00 0C00 0E00 0700 0380 0380 01C0> PXLC RP 939 553 XY F25(\()S F26(M)S XP /F25 41 16 2 -11 12 31 42 16 0 PXLC RP 1000 553 XY F25(\))S XP /F27 20 32 3 -8 28 26 34 32 0 <00000380 00000F80 00003F00 0000FC00 0003F000 000FC000 003F0000 00FC0000 03F00000 0FC00000 3F000000 FC000000 F0000000 FC000000 3F000000 0FC00000 03F00000 00FC0000 003F0000 000FC000 0003F000 0000FC00 00003F00 00000F80 00000380 00000000 00000000 00000000 00000000 00000000 00000000 00000000 FFFFFF80 FFFFFF80> PXLC RP 1031 553 XY F27(\024)S XP /F26 110 25 1 0 24 18 18 24 0 <3E1F80 3F3FE0 7771E0 67E0E0 E7C0E0 C780E0 0700E0 0F01E0 0F01C0 0E01C0 0E03C0 1E038E 1E038E 1C078C 1C071C 3C07B8 3C03F0 3801E0> PXLC RP 1079 553 XY F26(n)S 1120 X F25(and)S 1203 X F27(`)S -15 y F9(\013)S 25 y 1228 X(n)S -10 y 1260 X F26(L)S XP /F9 105 12 0 0 9 20 20 16 0 <0380 0780 0300 0000 0000 0000 0000 3E00 7600 6F00 EE00 CE00 1E00 1C00 1C00 3D80 3980 3B80 3B00 1E00> PXLC RP 1288 559 XY F9(i)S -6 y 1306 X F26(~)S 1302 X(y)S 6 y F9(i)S -6 y 1353 X F25(f)S XP /F25 111 21 1 0 19 18 18 24 0 <03F000 0FFC00 1E1E00 380700 780780 700380 F003C0 F003C0 F003C0 F003C0 F003C0 F003C0 700380 780780 380700 1E1E00 0FFC00 03F000> PXLC RP 1365 553 XY F25(o)S XP /F25 114 16 1 0 14 18 18 16 0 PXLC RP 1386 553 XY F25(r)S XP /F26 105 14 1 0 13 28 28 16 0 <00E0 01E0 01E0 01C0 0000 0000 0000 0000 0000 0000 1F00 3F80 7380 6380 E380 C780 0700 0700 0F00 0E00 1E00 1C70 1C70 3C60 38E0 3DC0 1F80 0F00> PXLC RP 1418 553 XY F26(i)S 1448 X F25(=)S XP /F25 49 21 4 0 17 28 28 16 0 <0300 0700 3F00 FF00 C700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 FFF8 FFF8> PXLC RP 1496 553 XY F25(1)S F26(;)S 1536 X(:)S 1555 X(:)S 1574 X(:)S 1590 X (;)S 1609 X(m)S 50 y 404 X F25(with)S 498 X F26(m)S XP /F27 21 32 3 -8 28 26 34 32 0 PXLC RP 546 603 XY F27(\025)S 590 X F25(1)S XP /F25 44 12 3 -8 8 5 13 8 0 <70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 611 603 XY F25(,)S 636 X(then)S 731 X F27(`)S -15 y F9(\013)S XP /F8 43 25 2 -4 22 18 22 24 0 <006000 006000 006000 006000 006000 006000 006000 006000 006000 006000 FFFFF0 FFFFF0 006000 006000 006000 006000 006000 006000 006000 006000 006000 006000> PXLC RP 778 588 XY F8(+1)S 25 y 756 X F9(n)S -10 y 826 X F26(M)S 871 X(L)S 6 y F8(1)S -6 y 925 X F26(:)S 944 X(:)S 963 X(:)S 980 X(L)S XP /F9 109 29 0 0 27 13 13 32 0 <3C7E3E00 6EC7E700 6F87C700 EF078700 CF070700 0E070700 1E0F0F00 1E0F0E00 1C0E0E60 1C0E1E60 3C1E1CE0 3C1E1DC0 381C0F80> PXLC RP 1009 609 XY F9(m)S -6 y 2 x F25(.)S 694 Y 362 X F27(\017)S XP /F26 60 32 3 -3 28 23 26 32 0 <00000180 00000780 00001F80 00007E00 0001F800 0007E000 001F8000 007E0000 01F80000 07E00000 1F800000 7E000000 F8000000 F8000000 7E000000 1F800000 07E00000 01F80000 007E0000 001F8000 0007E000 0001F800 00007E00 00001F80 00000780 00000180> PXLC RP 404 694 XY F26(<)S 6 y F8(1)S XP /F28 123 21 4 10 22 12 2 24 0 PXLC RP 455 694 XY F28({R)S -1 x(ule.)S 588 X F25(If)S 630 X F27(`)S XP /F9 12 19 0 -6 17 20 26 24 0 <003F00 00E300 01C380 038380 070380 060780 0E0700 0CFF00 0CCF00 1CFF00 180380 180380 180380 380780 380700 380700 380E00 7C1E00 6E3800 67F000 600000 E00000 C00000 C00000 C00000 C00000> PXLC RP 655 679 XY F9(\014)S 25 y 655 X(n)S -10 y 682 X F26(M)S 741 X F25 (and)S XP /F26 12 23 -1 -8 24 29 37 32 0 <0000FC00 0007FE00 000F8F00 001C0780 00380380 00700380 00E00780 00C00780 01C00700 01800F00 03801E00 038FFC00 030FF800 030FFC00 07001E00 06001E00 06000E00 06000E00 0E001E00 0C001E00 0C001E00 0C001E00 1C003E00 1C003C00 1E007800 1F00F800 3F81F000 33FFC000 31FF8000 30180000 70000000 60000000 60000000 60000000 E0000000 C0000000 C0000000> PXLC RP 822 694 XY F26(\014)S 859 X(<)S 6 y F8(1)S XP /F26 11 27 1 0 24 18 18 24 0 <00FC00 03FE00 0F8F06 1E078E 3C038C 3C038C 7803DC 7803D8 F801F8 F001F0 F001E0 F001E0 F001C0 F003C0 700FC0 7C3FEE 3FF8FC 0FE078> PXLC RP 921 694 XY F26(\013)S F25(,)S 973 X(then)S 1068 X F27(`)S -15 y F9 (\013)S 25 y 1093 X(n)S -10 y 1122 X F26(M)S 1167 X F25(,)S 781 Y 300 X(where)S 420 X F26(\014)S 458 X(<)S 6 y F8(1)S -6 y 520 X F26 (\013)S XP /F25 109 35 1 0 34 18 18 40 0 PXLC RP 561 781 XY F25(mean)S XP /F25 115 16 1 0 14 18 18 16 0 <1FB0 7FF0 F0F0 E070 E030 F030 F800 7FC0 7FE0 1FF0 00F8 C078 C038 E038 F078 F8F0 FFF0 CFC0> PXLC RP 658 781 XY F25(s)S 688 X(that)S 778 X(there)S 885 X(is)S 927 X(a)S XP /F25 12 23 0 0 21 29 29 24 0 <003F00 01FF80 03C3C0 0783C0 0F03C0 0E03C0 0E0000 0E0000 0E0000 0E0000 0E0000 FFFFC0 FFFFC0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 0E01C0 7F87F8 7F87F8> PXLC RP 961 781 XY F25(\014nite)S 1068 X(se)S XP /F25 113 22 1 -8 21 18 26 24 0 <07F180 1FF980 3F1F80 3C0F80 780780 780380 F00380 F00380 F00380 F00380 F00380 F00380 F80380 780780 7C0F80 3E1F80 1FFB80 07E380 000380 000380 000380 000380 000380 000380 001FF0 001FF0> PXLC RP 1102 781 XY F25(q)S XP /F25 117 23 1 0 21 18 18 24 0 PXLC RP 1124 781 XY F25(uen)S XP /F25 99 18 1 0 16 18 18 16 0 <07F8 0FFC 3E3C 3C3C 7818 7800 F000 F000 F000 F000 F000 F000 7800 7806 3C0E 3F1C 0FF8 07F0> PXLC RP 1189 781 XY F25(ce)S 1240 X F26(\014)S 1277 X F25(=)S 1321 X F26 (\014)S 6 y F8(0)S -6 y 2 x F26(;)S 1382 X(\014)S 6 y F8(1)S -6 y 2 x F26(;)S 1443 X(:)S 1462 X(:)S 1481 X(:)S 1498 X(;)S 1517 X (\014)S 6 y F9(n)S -6 y 1574 X F25(=)S 1618 X F26(\013)S 49 y 300 X F25(with)S 395 X F26(n)S XP /F26 62 32 3 -3 28 23 26 32 0 PXLC RP 432 830 XY F26(>)S XP /F25 48 21 2 -1 18 28 29 16 0 <03C0 0FF0 1C38 381C 381C 700E 700E 700E F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F 700E 700E 700E 381C 381C 1C38 0FF0 07E0> PXLC RP 476 830 XY F25(0)S 511 X(suc)S -1 x(h)S 605 X(that)S 695 X(for)S 759 X(all)S 817 X F26(i)S 843 X(<)S 887 X(n)S 926 X F25(either)S 1044 X F26(\014)S 7 y F9(i)S F8(+1)S -7 y 1135 X F25(=)S 1179 X F26 (\014)S 7 y F9(i)S XP /F25 43 32 2 -4 29 24 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFE0 FFFFFFE0 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000> PXLC RP 1226 830 XY F25(+)S 1268 X(1)S 1303 X(or)S 1354 X F26(\014)S 7 y F9 (i)S F8(+1)S -7 y 1445 X F25(=)S XP /F27 104 16 4 -11 13 31 42 16 0 <0180 0380 0380 0300 0700 0700 0600 0E00 0E00 0C00 1C00 1C00 1800 3800 3800 3000 7000 7000 6000 E000 E000 E000 E000 6000 7000 7000 3000 3800 3800 1800 1C00 1C00 0C00 0E00 0E00 0600 0700 0700 0300 0380 0380 0180> PXLC RP 1489 830 XY F27(h)S XP /F26 13 21 0 -9 21 18 27 24 0 <07E018 1FF018 3FF838 7FF830 703870 E01C60 C01CE0 C00CC0 000DC0 000D80 000D80 000F00 000F00 000E00 000E00 000E00 000C00 000C00 000C00 000C00 001800 001800 001800 001800 003000 003000 003000> PXLC RP 1505 830 XY F26(\015)S XP /F9 120 19 1 0 16 13 13 16 0 <1F7C 33EE 63CE E3CC C380 0380 0780 0700 070C E70C EF1C CF38 FBE0> PXLC RP 1527 837 XY F9(x)S XP /F27 105 16 2 -11 11 31 42 16 0 PXLC RP 1548 830 XY F27(i)S 1578 X F25(and)S 50 y 300 X F26(\014)S 7 y F9 (i)S -7 y 349 X F25(=)S 393 X F26(\015)S 7 y F8(0)S -7 y 445 X F25 (or)S 495 X F26(\015)S 7 y F8(1)S -7 y 2 x F25(,)S 559 X(or)S 608 X (else)S 686 X F26(\014)S 7 y F9(i)S F8(+1)S -7 y 776 X F25(=)S 820 X F27(h)S F26(\015)S 7 y F9(\030)S -7 y 876 X F27(i)S 905 X F25(and)S 984 X F26(\014)S 7 y F9(i)S -7 y 1033 X F25(=)S 1077 X F26(\015)S 7 y F8(0)S -7 y 1129 X F25(or)S 1179 X F26(\015)S 7 y F8(1)S -7 y 2 x F25(.)S XP /F25 87 43 1 -1 41 28 29 40 0 PXLC RP 1248 880 XY F25(W)S -3 x(e)S 1318 X(also)S 1400 X(de\014ne)S 1519 X F26(\014)S 1556 X F27(\024)S 7 y F8(1)S -7 y 1618 X F26(\013)S 50 y 300 X F25(to)S 351 X(mean)S 461 X F26(\014)S 499 X(<)S 6 y F8(1)S -6 y 561 X F26(\013)S 602 X F25(or)S 652 X F26(\014)S 690 X F25(=)S 733 X F26(\013)S F25(.)S XP /F25 77 38 2 0 35 28 28 40 0 PXLC RP 375 980 XY F25(More)S XP /F25 112 23 1 -8 21 18 26 24 0 PXLC RP 487 980 XY F25(precisel)S XP /F25 121 22 1 -8 20 18 26 24 0 PXLC RP 621 980 XY F25(y)S -3 x(,)S 671 X(w)S -1 x(e)S 737 X(\014rst)S 827 X (inductiv)S -1 x(ely)S 1046 X(de\014ne)S 1170 X F27(`)S -15 y F9 (\013)S 25 y 1195 X(n)S -10 y 1232 X F26(M)S 1296 X F25(for)S 1364 X F26(\013)S 1410 X F27(2)S 1458 X F25(\012)S 6 y F8(1)S XP /F25 98 23 1 0 21 29 29 24 0 PXLC RP 1525 980 XY F25(b)S -1 x(y)S 1587 X(the)S 50 y 300 X(rules)S XP /F25 103 21 1 -9 19 19 28 24 0 <000380 07E7C0 1FFDC0 3C3DC0 781E00 781E00 781E00 781E00 781E00 781E00 3C3C00 3FF800 37E000 200000 300000 380000 3FFC00 3FFF00 3FFF80 7807C0 F003C0 E001C0 E001C0 F003C0 F807C0 7C0F80 1FFE00 07F800> PXLC RP 405 1030 XY F25(giv)S -1 x(en)S 520 X(e)S XP /F25 120 22 0 0 21 18 18 24 0 <7F8FF0 7F8FF0 0F0F80 070F00 038E00 01DC00 01D800 00F000 007000 007800 00F800 01DC00 038E00 030E00 070700 1F0780 FF8FF8 FF8FF8> PXLC RP 538 1030 XY F25(xcluding)S 711 X(the)S XP /F26 33 26 0 0 23 18 18 24 0 <0C001E 1C001E 18001E 38001E 30000E 701C0E 601C06 601C0E E03C0C C0380C C0381C C03018 E07038 E0F870 FFFFF0 FFFFE0 7FBFC0 3E1F00> PXLC RP 789 1030 XY F26(!)S -15 y 1 x F8(+)S XP /F25 123 21 0 10 20 12 2 24 0 PXLC RP 843 1030 XY F25({)S XP /F25 82 31 2 -1 30 28 29 32 0 PXLC RP 864 1030 XY F25(Rule,)S 981 X(and)S 1067 X(based)S 1189 X(on)S 1252 X (this)S 1339 X(relation)S 1497 X(w)S -1 x(e)S 1564 X(then)S 50 y 300 X(de\014ne)S 420 X F27(`)S -16 y F9(\013)S 26 y 445 X(n)S -10 y 474 X F26(M)S 532 X F25(for)S 596 X F26(\013)S 634 X F27(2)S 673 X F25(\012)S XP /F8 50 17 2 0 14 19 19 16 0 <1F80 7FE0 E3E0 E0F0 E0F0 E070 00F0 00F0 00E0 01C0 0380 0700 0E00 1C30 3830 7070 FFE0 FFE0 FFE0> PXLC RP 703 1086 XY F8(2)S -6 y 736 X F25(b)S -1 x(y)S 793 X(all)S 851 X (the)S 922 X(rules)S 1022 X(giv)S -1 x(en.)S XP /F30 51 24 2 0 21 27 27 24 0 <07F800 1FFF00 3FFF80 7E3F80 7E1FC0 7F0FC0 7F0FC0 7E1FC0 3E1FC0 003F80 007F00 03FE00 03FC00 003F00 001F80 000FC0 000FE0 380FE0 7C0FE0 FE0FE0 FE0FE0 FE0FE0 FE1FC0 7C3FC0 7FFF80 3FFE00 07F800> PXLC RP 375 1177 XY F30(3)S XP /F30 46 13 3 0 10 7 7 8 0 <7C FE FE FE FE FE 7C> PXLC RP 399 1177 XY F30(.)S XP /F30 49 24 3 0 20 27 27 24 0 <006000 01E000 0FE000 FFE000 FFE000 F3E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 03E000 7FFF80 7FFF80 7FFF80> PXLC RP 412 1177 XY F30(1.)S XP /F30 86 36 1 0 34 28 28 40 0 PXLC RP 465 1177 XY F30(V)S XP /F30 97 23 1 0 22 18 18 24 0 <07FC00 1FFF00 3F1F80 3F0FC0 3F07C0 3F07C0 1E07C0 03FFC0 1FFFC0 7F87C0 7F07C0 FE07C0 FC07C0 FC07C0 FE0FC0 7E3FF8 3FFBF8 0FE1F8> PXLC RP 497 1177 XY F30(a)S XP /F30 114 20 1 0 18 18 18 24 0 PXLC RP 520 1177 XY F30(r)S XP /F30 105 13 1 0 12 30 30 16 0 <1E00 3F00 7F00 7F00 7F00 3F00 1E00 0000 0000 0000 0000 0000 FF00 FF00 FF00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 FFE0 FFE0 FFE0> PXLC RP 540 1177 XY F30(ia)S XP /F30 98 27 1 0 24 29 29 24 0 PXLC RP 576 1177 XY F30(b)S XP /F30 108 13 1 0 12 29 29 16 0 PXLC RP 603 1177 XY F30(l)S XP /F30 101 22 1 0 20 18 18 24 0 <03FC00 0FFF00 3F0F80 3E07C0 7E03C0 7C03E0 FC03E0 FFFFE0 FFFFE0 FC0000 FC0000 FC0000 7C0000 7E00E0 3F01E0 1FC3C0 0FFF80 03FE00> PXLC RP 616 1177 XY F30(e)S XP /F30 76 29 2 0 26 28 28 24 0 PXLC RP 654 1177 XY F30(Lemma.)S XP /F29 /cmsl10 300 41.5 41.5 128 [-4 -11 46 31] PXLNF RP XP /F29 73 15 1 0 18 28 28 24 0 <0FFF80 0FFF80 00F000 00F000 00F000 00F000 00F000 01F000 01E000 01E000 01E000 01E000 01E000 03E000 03C000 03C000 03C000 03C000 03C000 07C000 078000 078000 078000 078000 078000 0F8000 FFF800 FFF800> PXLC RP 841 1177 XY F29(I)S XP /F29 102 13 1 0 19 29 29 24 0 <000FC0 003FC0 007BC0 00F3C0 01E3C0 01C000 01C000 01C000 03C000 038000 038000 3FF800 3FF800 038000 078000 070000 070000 070000 070000 070000 0F0000 0E0000 0E0000 0E0000 0E0000 0E0000 1E0000 FFC000 FFC000> PXLC RP 856 1177 XY F29(f)S 883 X F25(1)S 915 X F27(\024)S 6 y F8(1)S -6 y 978 X F26(\015)S XP /F29 97 21 2 0 21 18 18 24 0 <07F800 0FFE00 0F1F00 0E0700 0E0700 000700 000700 03FF00 1FCF00 3E0F00 7C0E00 F80E00 F00E60 F01E60 F03E60 F87EC0 7FEFC0 3F8F00> PXLC RP 1015 1177 XY F29(a)S XP /F29 110 23 1 0 21 18 18 24 0 <1F8FC0 1FBFE0 07F1E0 07C0E0 0780E0 0780E0 0700E0 0700E0 0701E0 0F01E0 0E01C0 0E01C0 0E01C0 0E01C0 0E03C0 1E03C0 FF8FF0 FF9FF0> PXLC RP 1036 1177 XY F29(n)S XP /F29 100 23 2 0 23 29 29 24 0 <0001F8 0003F8 000078 000070 000070 000070 000070 0000F0 0000F0 0000E0 0000E0 01FCE0 07FFE0 0F87E0 1E03E0 3C01C0 7C01C0 7801C0 7801C0 7803C0 F803C0 F00380 F00380 F00380 780780 780F80 3C3F80 1FFFE0 0FE7E0> PXLC RP 1059 1177 XY F29(d)S 1096 X F25(lev\()S XP /F26 120 24 2 0 21 18 18 24 0 <0F8F80 3FDFC0 38F9C0 70F1C0 E0F3C0 C0E3C0 00E3C0 01E000 01E000 01C000 01C000 73C0E0 F3C0E0 F380C0 F781C0 E7C780 FFFF00 7C7C00> PXLC RP 1164 1177 XY F26(x)S F25(\))S 1216 X F26(<)S XP /F26 107 22 2 0 20 29 29 24 0 <0FC000 1FC000 03C000 03C000 038000 038000 078000 078000 070000 070000 0F0000 0F0F80 0E1F80 0E3980 1E7380 1EE780 1DC780 1F8780 3F0000 3FC000 3BE000 38E000 78E1C0 78E1C0 70E180 70E380 F0F700 F07F00 E03E00> PXLC RP 1260 1177 XY F26(k)S XP /F29 44 12 2 -8 8 5 13 8 0 <3C 3C 7C 7C 3C 0C 1C 18 38 38 70 E0 40> PXLC RP 1283 1177 XY F29(,)S XP /F29 116 16 4 0 15 26 26 16 0 <0600 0600 0600 0E00 0E00 0C00 1C00 3C00 FFE0 FFE0 1C00 3C00 3800 3800 3800 3800 3800 7800 70C0 70C0 70C0 70C0 71C0 7380 7F00 3E00> PXLC RP 1308 1177 XY F29(t)S XP /F29 104 23 1 0 21 29 29 24 0 <07E000 0FE000 01E000 01C000 01C000 01C000 01C000 01C000 03C000 038000 038000 038FC0 03BFE0 03F1E0 07C0E0 0780E0 0780E0 0700E0 0700E0 0701E0 0F01E0 0E01C0 0E01C0 0E01C0 0E01C0 1E03C0 1E03C0 FF8FF0 FF9FF0> PXLC RP 1324 1177 XY F29(h)S XP /F29 101 18 2 0 18 18 18 16 0 <01F8 07FE 0F1E 1E0F 3C0F 7C07 7FFF 7FFF 7800 F800 F000 F000 7000 780E 7C0C 3E3C 1FF8 07E0> PXLC RP 1347 1177 XY F29(en)S 1403 X F27(`)S XP /F9 13 17 0 -6 16 13 19 16 0 <0F83 3FC7 7FC6 FBCE E0EC 00EC 0078 0078 0078 0070 0070 0060 0060 0060 0060 00C0 00C0 00C0 00C0> PXLC RP 1428 1162 XY F9(\015)S XP /F10 /cmsy7 300 29.1 29.1 128 [-1 -28 37 24] PXLNF RP XP /F10 1 10 2 5 6 9 4 8 0 PXLC RP 1447 1162 XY F10(\001)S XP /F9 107 18 1 0 15 20 20 16 0 <3F00 3F00 0F00 0E00 0E00 1E00 1E00 1C78 1CDC 3D9C 3F18 3E00 3C00 7F80 7BC0 71CC 71CC F1DC F1D8 E0F0> PXLC RP 1457 1162 XY F9(k)S 25 y 1428 X(n)S -10 y 1482 X F26(x~)S 1506 X(y)S XP /F29 46 12 3 0 8 5 5 8 0 <78 78 F8 F8 70> PXLC RP 1528 1177 XY F29(.)S XP /F30 80 33 2 0 29 28 28 32 0 PXLC RP 375 1282 XY F30(Pr)S XP /F30 111 24 1 0 22 18 18 24 0 <01FC00 0FFF80 1F07C0 3E03E0 7C01F0 7C01F0 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 FC01F8 7C01F0 7C01F0 3E03E0 1F07C0 0FFF80 01FC00> PXLC RP 427 1282 XY F30(o)S 1 x(o)S XP /F30 102 15 0 0 19 29 29 24 0 <003F80 01FFC0 03F7E0 07C7E0 0FC7E0 0F87E0 0F83C0 0F8000 0F8000 0F8000 0F8000 FFF800 FFF800 FFF800 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 0F8000 7FF800 7FF800 7FF800> PXLC RP 476 1282 XY F30(f.)S XP /F25 66 29 2 0 26 28 28 24 0 PXLC RP 525 1282 XY F25(By)S 589 X(induction)S 772 X(on)S 829 X(lev\()S F26 (x)S F25(\).)S 967 X(By)S 1031 X(induction)S 1215 X(h)S -1 x(yp)S 1 x(othesis)S 1418 X F27(`)S -22 y F9(\015)S 2 x F10(\001)S XP /F8 40 13 2 -8 10 22 30 8 0 <03 06 0E 1C 18 38 30 70 70 60 E0 E0 E0 E0 E0 E0 E0 E0 E0 E0 60 70 70 30 38 18 1C 0E 06 03> PXLC RP 1473 1260 XY F8(\()S F9(k)S XP /F10 0 26 3 6 21 8 2 24 0 PXLC RP 1504 1260 XY F10(\000)S F8(1)S XP /F8 41 13 2 -8 10 22 30 8 0 PXLC RP 1546 1260 XY F8(\))S 27 y 1443 X F9(n)S -5 y 1566 X F26(y)S 6 y F9 (i)S -6 y 1603 X F26(~)S XP /F26 122 19 1 0 18 18 18 24 0 <03E180 07F380 0FFF00 0E3F00 1C0E00 001C00 003800 007000 00E000 01C000 038000 070380 0E0300 1C0700 3F8F00 7FFE00 F3FC00 C0F000> PXLC RP 1600 1282 XY F26(z)S 6 y F9(i)S -6 y 2 x F25(,)S 58 y 300 X(hence)S 415 X F27(`)S -22 y F9(\015)S 2 x F10(\001)S F8(\()S F9(k)S 1 x F10 (\000)S F8(1\)+1)S 27 y 440 X F9(n)S -5 y 605 X F26(x~)S 629 X(y)S 663 X F25(b)S -1 x(y)S 720 X(the)S XP /F25 86 31 1 -1 29 28 29 32 0 PXLC RP 791 1340 XY F25(V)S -3 x(ariable)S 954 X(Rule,)S 1063 X(hence)S 1177 X F27(`)S -15 y F9(\015)S 2 x F10(\001)S F9(k)S 25 y 1202 X(n)S -10 y 1257 X F26(x~)S 1281 X(y)S 1316 X F25(b)S -1 x(y)S 1373 X (the)S 1443 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule.)S 1628 1340 XY 17 17 R 1437 Y 375 X F30(3.)S XP /F30 50 24 2 0 21 27 27 24 0 <0FF800 3FFF00 7FFF80 FE7FC0 FE1FC0 FE0FE0 FE0FE0 FE07E0 7C07E0 000FE0 000FE0 001FC0 001F80 003F00 007E00 00FC00 01F000 03E000 07C0E0 0F00E0 1E00E0 3C01E0 7FFFC0 FFFFC0 FFFFC0 FFFFC0 FFFFC0> PXLC RP 412 1437 XY F30(2.)S XP /F30 83 27 3 0 23 28 28 24 0 <07F860 1FFFE0 3FFFE0 7E0FE0 7C03E0 F801E0 F800E0 F800E0 FC0000 FE0000 FFF000 7FFE00 7FFF80 3FFFC0 1FFFE0 07FFE0 007FF0 0007F0 0003F0 0001F0 E001F0 E001F0 F001F0 F803E0 FE07E0 FFFFC0 FFFF80 C3FE00> PXLC RP 465 1437 XY F30(S)S XP /F30 117 27 1 0 25 18 18 24 0 PXLC RP 491 1437 XY F30(ub)S XP /F30 115 19 2 0 16 18 18 16 0 <1FF8 7FF8 7078 E038 E038 F000 FF80 FFF0 7FF8 3FF8 0FFC 007C E03C F01C F01C F878 FFF8 EFE0> PXLC RP 544 1437 XY F30(s)S XP /F30 116 19 1 0 15 26 26 16 0 <0700 0700 0700 0700 0F00 0F00 0F00 1F00 3FFC FFFC FFFC 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F00 1F1C 1F1C 1F1C 1F1C 1FBC 0FF8 07F0> PXLC RP 563 1437 XY F30(titutio)S XP /F30 110 27 1 0 25 18 18 24 0 PXLC RP 696 1437 XY F30(n)S 738 X(Lemma.)S 926 X F29(If)S 969 X F27(`)S -15 y F9(\013)S 25 y 994 X(n)S -10 y 1025 X F26(M)S 1086 X F29(and)S 1168 X F27(`)S -15 y F9(\014)S 25 y 1193 X(n)S XP /F26 78 33 2 0 35 28 28 40 0 <03FE03FF80 03FE03FF80 003F007C00 003F003000 003F803000 003F807000 0077807000 0067C06000 0067C06000 0063C0E000 00E3E0E000 00C3E0C000 00C1E0C000 00C1F1C000 01C1F1C000 0180F18000 0180F98000 01807B8000 03807B8000 03007F0000 03003F0000 03003F0000 07003F0000 06001E0000 06001E0000 1F001E0000 FFE00E0000 FFE00C0000> PXLC RP 1223 1437 XY F26(N)S XP /F9 106 14 -1 -6 12 20 26 16 0 <0038 0038 0030 0000 0000 0000 0000 03C0 06E0 0CF0 1CE0 18E0 00E0 01E0 01E0 01C0 01C0 03C0 03C0 0380 0380 0780 E700 E700 EE00 7800> PXLC RP 1257 1443 XY F9(j)S -6 y 1279 X F26(~)S 1274 X(y)S 6 y F9(j)S XP /F29 119 30 4 0 31 18 18 32 0 PXLC RP 1327 1437 XY F29(w)S XP /F29 105 12 1 0 12 29 29 16 0 <01C0 03E0 03E0 03E0 03C0 0000 0000 0000 0000 0000 0000 1F80 1F80 0780 0780 0700 0700 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 0E00 1E00 FF80 FF80> PXLC RP 1357 1437 XY F29(ith)S 1424 X F25(lev\()S F26(N)S 6 y F9(j)S -6 y 1543 X F25(\))S 1573 X F27(\024)S 1620 X F26(n)S 56 y 300 X F29(f)S XP /F29 111 21 2 0 21 18 18 24 0 <00FC00 03FF00 0F0780 1E03C0 3C01C0 3801C0 7801E0 7801E0 7801E0 F003C0 F003C0 F003C0 700380 700780 780F00 3C1E00 1FF800 07E000> PXLC RP 313 1493 XY F29(o)S XP /F29 114 16 1 0 17 18 18 16 0 <1FBE 1FFE 07EF 07CE 078E 0780 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 0E00 1E00 FFC0 FFC0> PXLC RP 333 1493 XY F29(r)S XP /F26 106 17 0 -8 16 28 36 16 0 <0007 000F 000F 000F 0000 0000 0000 0000 0000 0000 00F8 01FC 039C 071C 0E1C 0C1C 001C 003C 003C 0038 0038 0078 0078 0070 0070 00F0 00F0 00E0 00E0 01E0 01E0 F1C0 F3C0 F780 FF00 7C00> PXLC RP 364 1493 XY F26(j)S 395 X F25(=)S 438 X(1)S F26(;)S 478 X(:)S 497 X (:)S 516 X(:)S 533 X(;)S 552 X(m)S F29(,)S 613 X(then)S 708 X F27 (`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S 25 y 733 X(n)S -10 y 807 X F26(M)S XP /F9 126 17 6 15 19 21 6 16 0 <00C0 00E0 FFF8 FFF8 01E0 01C0> PXLC RP 847 1500 XY F9(~)S 847 X(x)S XP /F25 91 12 4 -10 11 31 41 8 0 PXLC RP 868 1493 XY F25([)S -10 y 889 X F26(~)S 10 y 880 X(N)S XP /F25 93 12 0 -10 7 31 41 8 0 PXLC RP 918 1493 XY F25(])S -1 x F29(.)S 1595 Y 375 X F30(Pro)S 1 x(of.)S 525 X F25(By)S 596 X(induction)S 787 X(on)S 851 X F27(`)S -15 y F9 (\013)S 25 y 876 X(n)S -10 y 915 X F26(M)S 960 X F25(.)S 1009 X(W)S -3 x(e)S 1087 X(write)S 1199 X F26(L)S XP /F10 3 17 2 1 13 14 13 16 0 <0400 0400 0400 C460 E4E0 3F80 0E00 3F80 E4E0 C460 0400 0400 0400> PXLC RP 1228 1580 XY F10(\003)S 15 y 1267 X F25(for)S 1337 X F26(L)S 7 y F9 (~)S 1365 X(x)S -7 y 2 x F25([)S -10 y 1406 X F26(~)S 10 y 1398 X (N)S 1435 X F25(].)S XP /F28 86 31 8 -1 35 28 29 32 0 PXLC RP 1496 1595 XY F28(V)S XP /F28 97 21 4 0 22 18 18 24 0 <03EE00 07FE00 0F3E00 1E1E00 3C1E00 781C00 781C00 783C00 F83C00 F03800 F03800 F079C0 F079C0 E0F180 F0F180 73F300 7FFF00 3E3E00> PXLC RP 1523 1595 XY F28(a)S XP /F28 114 18 3 0 19 18 18 16 0 <1E3E 3F7F 77E7 67C7 E78F C78F 070F 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800> PXLC RP 1545 1595 XY F28(r)S XP /F28 105 13 3 0 14 28 28 16 0 <01C0 01E0 01E0 01C0 0000 0000 0000 0000 0000 0000 1F00 3F00 7780 6380 E780 C700 0F00 0E00 0E00 1E00 1C00 1CE0 3CE0 38C0 39C0 3B80 3F80 1E00> PXLC RP 1562 1595 XY F28(ia)S XP /F28 98 19 5 0 19 29 29 16 0 <3F00 7F00 0F00 0F00 0E00 0E00 0E00 1E00 1C00 1C00 1C00 3FE0 3FF0 3E78 3C38 783C 703C 703C 707C F078 E078 E078 E0F8 E0F0 E0F0 E1E0 F3C0 7F80 3E00> PXLC RP 1596 1595 XY F28(ble)S 50 y 300 X(R)S -1 x(ule)S 3 x F25(.)S 414 X (By)S 478 X(induction)S 661 X(h)S -1 x(yp)S 1 x(othesis)S 864 X F27 (`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S 25 y 889 X(n)S -10 y 964 X F26(L)S -15 y F10(\003)S 26 y 992 X F9(i)S -11 y 1015 X F26(~)S 1011 X(y)S 6 y F9(i)S -6 y 2 x F25(,)S 1070 X(hence)S 1184 X F27(`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S F8(+1)S 25 y 1209 X F9(n)S -10 y 1325 X F26(xL)S -15 y F10(\003)S 25 y 1377 X F8(1)S -10 y 1403 X F26 (:)S 1422 X(:)S 1441 X(:)S 1459 X(L)S -15 y F10(\003)S 25 y 1487 X F9(m)S -10 y 1531 X F25(b)S -1 x(y)S 1587 X(the)S 50 y 300 X(V)S -3 x(ariable)S 463 X(Rule.)S XP /F25 78 31 2 0 28 28 28 32 0 PXLC RP 576 1695 XY F25(No)S -1 x(w)S 670 X(if)S 707 X F26(x)S 744 X F25 (is)S 785 X(one)S 861 X(of)S 907 X(the)S 978 X(v)S -2 x(ariables)S 1149 X F26(x)S 6 y F9(j)S -6 y 1204 X F25(to)S 1253 X(b)S 1 x(e)S 1309 X(substituted)S 1526 X(b)S -1 x(y)S 1583 X F26(N)S 6 y F9(j)S -6 y 3 x F25(,)S 49 y 300 X(w)S -1 x(e)S 364 X(m)S -1 x(ust)S 469 X (use)S 544 X(the)S XP /F25 67 30 2 -1 27 29 30 32 0 <003FC180 00FFF180 03F07B80 07C01F80 0F800F80 1F000780 3E000780 3E000380 7C000380 7C000180 78000180 F8000180 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 F8000000 78000180 7C000180 7C000180 3E000380 3E000300 1F000700 0F800E00 07C01E00 03F87C00 00FFF000 003FC000> PXLC RP 618 1744 XY F25(Cut)S 703 X(Rule)S 803 X(instead)S 949 X(of)S 999 X (the)S 1073 X(V)S -3 x(ariable)S 1240 X(Rule.)S XP /F25 84 30 1 0 28 28 28 32 0 <7FFFFFC0 7FFFFFC0 780F03C0 700F01C0 600F00C0 E00F00E0 C00F0060 C00F0060 C00F0060 C00F0060 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 000F0000 03FFFC00 03FFFC00> PXLC RP 1361 1744 XY F25(This)S 1458 X(is)S 1503 X(p)S 1 x(ossible)S 50 y 300 X(since)S 407 X(lev\()S F26(N)S 6 y F9(j)S -6 y 526 X F25(\))S 562 X F27(\024)S 614 X F26(n)S 657 X F25(b)S -1 x(y)S 720 X(h)S -1 x (yp)S 1 x(othesis)S 929 X(and)S 1015 X F27(`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S 26 y 1040 X(n)S -11 y 1122 X F26(N)S 6 y F9(j)S -6 y 1179 X F26(~)S 1173 X(y)S 6 y F9(j)S -6 y 1230 X F25(b)S -1 x (y)S 1292 X(h)S -1 x(yp)S 1 x(othesis)S 1502 X(and)S 1587 X(the)S 50 y 300 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule.)S 497 X(Then)S 609 X(\(if)S 667 X F26(m)S 721 X(>)S 771 X F25(0\))S 826 X(the)S 901 X(Cut)S 988 X(Rule)S 1089 X(yields)S 1210 X F27(`)S -15 y F9 (\014)S 1 x F8(+)S F9(\013)S F8(+1)S 25 y 1235 X F9(n)S -10 y 1357 X F26(N)S 6 y F9(j)S -6 y 1408 X F26(L)S -15 y F10(\003)S 25 y 1436 X F8(1)S -10 y 1463 X F26(:)S 1482 X(:)S 1501 X(:)S 1518 X(L)S -15 y F10(\003)S 25 y 1546 X F9(m)S -10 y 3 x F25(,)S 1608 X(as)S 50 y 300 X(required.)S 484 X(In)S 537 X(case)S 627 X F26(m)S 677 X F25 (=)S 724 X(0)S 760 X(there)S 868 X(are)S 939 X(no)S 998 X F26(L)S 6 y F9(i)S XP /F25 39 12 3 16 9 29 13 8 0 <70 F8 FC FC 7C 0C 0C 1C 18 38 70 F0 60> PXLC RP 1040 1894 XY F25('s)S 1083 X(and)S 1166 X(w)S -1 x(e)S 1228 X(ha)S -1 x(v)S -1 x(e)S 1325 X(used)S 1422 X(the)S 1495 X(V)S -3 x (ariable)S 50 y 300 X(Rule)S 400 X(to)S 454 X(generate)S 623 X F27 (`)S -15 y F9(\013)S F8(+1)S 25 y 648 X F9(n)S -10 y 723 X F26(x)S 6 y F9(j)S -6 y 3 x F25(.)S 803 X(But)S 888 X(then)S 985 X F27(`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S F8(+1)S 25 y 1010 X F9(n)S -10 y 1131 X F26(N)S 6 y F9(j)S -6 y 1199 X F25(holds)S 1310 X(b)S -1 x (y)S 1371 X(h)S -1 x(yp)S 1 x(othesis)S 1578 X(and)S 50 y 300 X (the)S 369 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule.)S XP /F25 70 27 2 0 24 28 28 24 0 PXLC RP 554 1994 XY F25(F)S -3 x(or)S 626 X(all)S 682 X(other)S 788 X (rules)S 885 X(the)S 954 X(claim)S 1063 X(follo)S -1 x(ws)S 1197 X (easily)S 1309 X(from)S 1405 X(the)S 1474 X(induction)S 49 y 300 X (h)S -1 x(yp)S 1 x(othesis)S 508 X(and)S 593 X(the)S 668 X(same)S 775 X(rule.)S XP /F25 69 28 2 0 26 28 28 24 0 PXLC RP 886 2043 XY F25(E.g.)S 987 X(for)S 1054 X(the)S 1130 X(Constan)S -1 x(t)S 1312 X(Rule)S 1413 X(for)S 1480 X(cond)S XP /F9 19 12 1 0 10 13 13 16 0 <1C00 1C00 3C00 3800 3800 3800 7800 7000 7180 F180 E300 E700 7C00> PXLC RP 1566 2050 XY F9(\023)S -7 y 1598 X F25(w)S -1 x(e)S 50 y 300 X(ha)S -1 x(v)S -1 x(e)S 398 X(b)S -1 x(y)S 458 X(induction)S 644 X(h)S -1 x(yp)S 1 x(othesis)S 851 X F27(`)S -15 y F9(\014)S 1 x F8(+)S F9 (\013)S 25 y 876 X(n)S -10 y 954 X F26(M)S -15 y 999 X F10(\003)S 15 y 2 x F25(,)S 1046 X F27(`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S 25 y 1071 X(n)S -10 y 1148 X F26(N)S -15 y 1186 X F10(\003)S 15 y 1221 X F25(and)S 1304 X F27(`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S 25 y 1329 X(n)S -10 y 1407 X F26(L)S -15 y F10(\003)S 15 y 2 x F25 (.)S 1491 X(Then)S 1601 X(an)S 50 y 300 X(application)S 514 X(of)S 562 X(the)S 633 X(same)S 737 X(rule)S 820 X(yields)S 937 X F27(`)S -15 y F9(\014)S 1 x F8(+)S F9(\013)S F8(+1)S 25 y 962 X F9(n)S -10 y 1079 X F25(cond)S 6 y F9(\023)S -6 y 2 x F26(M)S -15 y 1223 X F10 (\003)S 15 y 2 x F26(N)S -15 y 1280 X F10(\003)S 15 y 2 x F26(L)S -15 y F10(\003)S 15 y 2 x F25(.)S 1628 2143 XY 17 17 R 2240 Y 375 X F30(3.3.)S XP /F30 67 34 3 0 30 28 28 32 0 <003FE060 01FFFCE0 03FFFFE0 0FF81FE0 1FE007E0 3FC003E0 3F8003E0 7F0001E0 7F0001E0 7E0000E0 FE0000E0 FE000000 FE000000 FE000000 FE000000 FE000000 FE000000 FE000000 7E0000E0 7F0000E0 7F0000E0 3F8001E0 3FC003C0 1FE007C0 0FF81F80 03FFFF00 01FFFC00 003FF000> PXLC RP 465 2240 XY F30(Cut)S XP /F30 69 31 2 0 29 28 28 32 0 PXLC RP 560 2240 XY F30(Elimination)S 819 X(Lemma.)S 1007 X F29(If)S 1047 X F27(`)S -15 y F9(\013)S 25 y 1072 X(n)S F8(+1)S -10 y 1141 X F26(M)S 1186 X F29(,)S 1211 X(then)S 1304 X F27(`)S XP /F9 69 24 1 0 23 20 20 24 0 <07FFFC 07FFFC 01E03C 01C01C 01C018 03C018 03C318 038318 038700 07FF00 07FE00 070618 070618 0F0638 0F0030 0E0070 0E00E0 1E03E0 FFFFE0 FFFFC0> PXLC RP 1329 2225 XY F9(E)S 2 x(\013)S 25 y 1329 X(n)S -10 y 1383 X F26(M)S 1428 X F29(,)S 1453 X(where)S XP /F26 69 31 2 0 31 28 28 32 0 <03FFFFF8 03FFFFF8 003E00F8 003C0078 003C0030 003C0030 007C0030 00781830 00781830 00783830 00F83000 00F07000 00FFF000 00FFF000 01F0E000 01E06000 01E06000 01E0E0E0 03E0C0C0 03C001C0 03C00180 03C00380 07C00380 07800700 07800F00 0F803E00 FFFFFE00 FFFFFC00> PXLC RP 1571 2240 XY F26(E)S 1617 X F29(i)S XP /F29 115 16 1 0 16 18 18 16 0 <03F6 0FFE 1E1E 3C0E 380C 3C0C 3E00 3FE0 1FF8 07FC 007C 603C 601C 701C 7038 F878 FFF0 CFC0> PXLC RP 1629 2240 XY F29(s)S 50 y 300 X(the)S 371 X(e)S XP /F29 120 22 1 0 23 18 18 24 0 <1FE3FC 3FE7FC 03C3E0 01C380 01C700 00EE00 00EC00 007800 007000 007800 00F800 01DC00 039E00 070E00 0E0700 1E0780 FF1FF0 FF1FF0> PXLC RP 390 2290 XY F29(x)S XP /F29 112 23 0 -8 23 18 26 24 0 <0FCFE0 0FFFF8 03F8F8 03E07C 03C03C 03803E 03801E 03801E 03803E 07803E 07003C 07003C 070078 0700F8 0781F0 0FC3E0 0EFFC0 0E7F00 0E0000 0E0000 1E0000 1E0000 1C0000 1C0000 FF8000 FF8000> PXLC RP 412 2290 XY F29(p)S 1 x(onen)S -1 x(tiation)S 654 X(f)S XP /F29 117 23 4 0 22 18 18 24 0 PXLC RP 667 2290 XY F29(un)S XP /F29 99 18 2 0 18 18 18 16 0 <01FE 07FE 0F8F 1E0E 3C0E 3C00 7800 7800 7800 F800 F000 F000 7800 780E 7C0C 3E1C 1FF8 07E0> PXLC RP 713 2290 XY F29(ction)S 817 X F26(E)S 2 x(\013)S 888 X F25(=)S XP /F25 50 21 2 0 18 28 28 16 0 <0FE0 1FF8 38FC 703E E03E F01F F81F F81F F80F 700F 001F 001F 001E 003E 007C 0078 00F0 01E0 03C0 0380 0700 0E03 1C03 3803 7006 FFFE FFFE FFFE> PXLC RP 932 2290 XY F25(2)S -15 y F9(\013)S 15 y 990 X F29(de)S XP /F29 12 23 1 0 22 29 29 24 0 <0007F0 001FF8 007C38 00F078 01E078 01C078 01C000 01C000 03C000 038000 038000 3FFFF0 3FFFF0 0380F0 0780F0 0700E0 0700E0 0700E0 0700E0 0701E0 0F01E0 0E01C0 0E01C0 0E01C0 0E01C0 0E03C0 1E03C0 FF8FF0 FF8FF0> PXLC RP 1031 2290 XY F29(\014ned)S 1133 X(on)S 1191 X F25(\012)S 6 y F8(2)S -6 y 1253 X F29(as)S 1304 X(fo)S XP /F29 108 12 1 0 12 29 29 16 0 <07E0 0FE0 01E0 01C0 01C0 01C0 01C0 01C0 03C0 0380 0380 0380 0380 0380 0780 0700 0700 0700 0700 0700 0F00 0E00 0E00 0E00 0E00 1E00 1E00 FF80 FF80> PXLC RP 1337 2290 XY F29(llo)S -1 x(ws)S XP /F29 58 12 3 0 11 18 18 8 0 <0E 1F 1F 1E 1E 00 00 00 00 00 00 00 00 78 78 F8 F8 70> PXLC RP 1426 2290 XY F29(:)S 2385 Y 305 X F26(E)S 2 x F25(0)S 370 X(=)S 414 X(1)S 517 X F26(E)S 2 x F25(\()S F26(\013)S 603 X F25(+)S 644 X (1\))S 692 X(=)S 736 X F26(E)S 2 x(\013)S 805 X F25(+)S 847 X F26 (E)S 2 x(\013)S 990 X(E)S 2 x F27(h)S F26(\013)S 6 y F9(x)S -6 y 1 x F27(i)S 1114 X F25(=)S 1158 X F27(h)S F26(E)S 2 x(\013)S 6 y F9 (x)S -6 y 1 x F27(i)S 1353 X F26(E)S 2 x F27(h)S F26(\013)S 6 y F9 (\030)S -6 y 3 x F27(i)S 1475 X F25(=)S 1519 X F27(h)S F26(E)S 2 x (\013)S 6 y F9(\030)S -6 y 3 x F27(i)S F26(:)S 2480 Y 375 X F30 (Pro)S 1 x(of.)S 525 X F25(By)S 594 X(induction)S 783 X(on)S 845 X F27(`)S -15 y F9(\013)S 26 y 870 X(n)S F8(+1)S -11 y 947 X F26(M)S 992 X F25(.)S 1034 X F28(V)S -3 x(ariable)S 1202 X(R)S -1 x(ule)S 3 x F25(.)S 1329 X(By)S 1399 X(induction)S 1587 X(h)S -1 x(y)S XP /F25 45 14 0 7 11 10 3 16 0 PXLC RP 1631 2480 XY F25(-)S 62 y 300 X(p)S 1 x(othesis)S 463 X F27(`)S -16 y F9(E)S 2 x(\013)S 26 y 488 X(n)S -10 y 547 X F26(L)S 6 y F9(i)S -6 y 593 X F26(~)S 589 X(y)S 6 y F9(i)S -6 y 2 x F25(,)S 652 X(hence)S 770 X F27(`)S -22 y F8(\()S F9(E)S 2 x(\013)S F8(\)+1)S 26 y 795 X F9(n)S -4 y 922 X F26(xL)S 6 y F8(1)S -6 y 999 X F26(:)S 1018 X(:)S 1037 X(:)S 1055 X(L)S 6 y F9(m)S -6 y 1131 X F25(b)S -1 x(y)S 1191 X (the)S 1265 X(V)S -3 x(ariable)S 1431 X(Rule,)S 1543 X(hence)S 57 y 300 X F27(`)S -21 y F9(E)S 2 x F8(\()S F9(\013)S F8(+1\))S 26 y 325 X F9(n)S -5 y 448 X F26(xL)S 7 y F8(1)S -7 y 526 X F26(:)S 545 X (:)S 564 X(:)S 581 X(L)S 7 y F9(m)S -7 y 655 X F25(b)S -1 x(y)S 713 X(the)S 785 X F26(<)S 7 y F8(1)S -7 y 2 x F25({Rule.)S 971 X F28 (C)S XP /F28 111 21 4 0 21 18 18 24 0 <01F800 07FE00 0F9F00 1E0F00 3C0700 7C0780 780780 780F80 F80F00 F00F00 F00F00 F01F00 F01E00 F03C00 F07C00 78F800 3FE000 1F8000> PXLC RP 1001 2599 XY F28(o)S XP /F28 110 23 3 0 24 18 18 24 0 <1E3F00 3F7F80 77F380 67C380 E78380 C78380 070380 0F0780 0F0700 0E0700 0E0F00 1E0E38 1E0E38 1C1E30 1C1C70 3C1EE0 3C0FC0 380F80> PXLC RP 1022 2599 XY F28(n)S XP /F28 115 17 3 0 17 18 18 16 0 <03F0 07F8 0F3C 0E3C 1E7C 1E38 1F00 1FE0 0FF0 0FF0 00F8 2078 F070 F070 F0F0 E1E0 7FC0 3F80> PXLC RP 1046 2599 XY F28(stant)S 1150 X(R)S -1 x(ules)S 3 x F25(.)S XP /F25 83 23 2 -1 20 29 30 24 0 <07F180 1FFF80 3C3F80 780F80 700780 E00380 E00380 E00180 E00180 F00000 F80000 7E0000 7FE000 3FFC00 1FFE00 0FFF00 00FF80 000F80 0007C0 0003C0 0003C0 C001C0 C001C0 C001C0 E001C0 F00380 F80780 FE0F00 FFFE00 C3F800> PXLC RP 1283 2599 XY F25(Similarl)S -3 x(y)S -3 x(.)S 1473 X F26(!)S -15 y 1 x F8(+)S 15 y 3 x F28({R)S -1 x(ule)S 3 x F25(.)S 50 y 300 X (Then)S 408 X F27(`)S -15 y F9(\013)S 25 y 433 X(n)S F8(+1)S -10 y 503 X F26(RP)S 568 X(QM)S -10 y 647 X(~)S 10 y 646 X(L)S 687 X F25 (has)S 761 X(b)S 1 x(een)S 859 X(inferred)S 1013 X(from)S 1111 X F27 (`)S -18 y F9(\013)S XP /F0 /cmr5 300 20.8 20.8 128 [-1 -6 26 16] PXLNF RP XP /F0 48 14 2 0 12 13 13 16 0 <1E00 6180 4080 C0C0 C0C0 C0C0 C0C0 C0C0 C0C0 C0C0 4080 6180 1E00> PXLC RP 1158 2635 XY F0(0)S 25 y 1136 X F9(n)S F8(+1)S -11 y 1205 X F26(M)S 1264 X F25(with)S 1359 X F26(\013)S 1397 X F25(=)S 1441 X F27(h)S F26(\013)S 6 y F9(\030)S -6 y 2 x F27(i)S F25(,)S 1543 X(and)S 2753 Y 629 X F27(8)S F26(\030)S 684 X F27(2)S 723 X F25(\012)S 7 y F8(1)S -7 y 2 x F27(8)S F26(m:)S 854 X F27(`)S -20 y F9(\030)S 31 y 879 X F8(0)S -11 y 903 X F30(m)S 966 X F25(=)S 991 X F27(\))S 1044 X(`)S -22 y F9(\013)S 4 y F1(\030)S 29 y 1069 X F9(n)S F8(+1)S -11 y 1139 X F26(RP)S 1204 X(Q)S F30(m)S -10 y 2 x F26(~)S 10 y 1277 X(L)S -1 x (:)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(39)S XP /F25 52 21 1 0 19 28 28 24 0 <001C00 001C00 003C00 007C00 007C00 00DC00 01DC00 019C00 039C00 071C00 061C00 0E1C00 0C1C00 181C00 381C00 301C00 601C00 E01C00 FFFFC0 FFFFC0 001C00 001C00 001C00 001C00 001C00 001C00 01FFC0 01FFC0> PXLC RP 342 462 XY F25(4)S XP /F18 72 29 1 0 27 26 26 32 0 PXLC RP 700 462 XY F18(H)S XP /F18 46 11 3 0 8 5 5 8 0 <70 F8 F8 F8 70> PXLC RP 729 462 XY F18(.)S XP /F18 83 21 2 0 18 26 26 16 0 <0FC6 3FF6 787E 701E E00E E00E E006 E006 F000 FC00 7F80 7FF0 3FFC 0FFE 01FE 003F 000F 000F C007 C007 E007 E00F F00E FC3C DFFC C7F0> PXLC RP 752 462 XY F18(S)S XP /F18 99 17 1 0 15 16 16 16 0 <07F8 1FFC 3E3C 783C 7818 F000 F000 F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 774 462 XY F18(c)S XP /F18 104 21 1 0 19 26 26 24 0 PXLC RP 790 462 XY F18(h)S XP /F18 119 28 1 0 26 16 16 32 0 PXLC RP 810 462 XY F18(wic)S -1 x(h)S XP /F18 116 15 1 0 12 23 23 16 0 <0C00 0C00 0C00 0C00 1C00 1C00 3C00 FFC0 FFC0 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C60 1C60 1C60 1C60 1EE0 0FC0 0780> PXLC RP 885 462 XY F18(t)S XP /F18 101 17 1 0 15 16 16 16 0 <07E0 1FF0 3C78 783C 703C FFFC FFFC F000 F000 F000 F000 7800 7C0C 3E1C 1FF8 07E0> PXLC RP 900 462 XY F18(en)S XP /F18 98 21 1 0 19 26 26 24 0 PXLC RP 937 462 XY F18(b)S 1 x(erg)S 1023 X(and)S 1098 X(S.S.)S XP /F18 87 39 1 0 37 26 26 40 0 PXLC RP 1175 462 XY F18(W)S -3 x(ainer)S 553 Y 300 X F25(By)S 365 X (induction)S 550 X(h)S -1 x(yp)S 1 x(othesis)S 754 X F27(`)S -15 y F9(E)S 2 x(\013)S 4 y F0(0)S 21 y 779 X F9(n)S -10 y 850 X F26(M)S 909 X F25(and)S 638 Y 629 X F27(8)S F26(\030)S 683 X F27(2)S 723 X F25(\012)S 6 y F8(1)S -6 y 1 x F27(8)S F26(m:)S 854 X F27(`)S -20 y F9(\030)S 31 y 879 X F8(0)S -11 y 902 X F30(m)S 965 X F25(=)S 990 X F27(\))S 1044 X(`)S -23 y F9(E)S 2 x(\013)S 5 y F1(\030)S 23 y 1069 X F9(n)S -5 y 1140 X F26(RP)S 1205 X(Q)S F30(m)S -11 y 2 x F26(~)S 11 y 1278 X(L)S -1 x(:)S 721 Y 300 X F25(No)S -1 x(w)S 396 X F26(E)S 2 x(\013)S 6 y F8(0)S -6 y 489 X F25(=)S 536 X(\()S F26(E)S 2 x (\013)S F25(\))S 6 y F8(0)S -6 y 2 x F25(,)S 675 X(and)S 757 X F26 (E)S 2 x(\013)S 6 y F9(\030)S -6 y 850 X F25(=)S 897 X(\()S F26(E)S 2 x(\013)S F25(\))S 6 y F9(\030)S -6 y 3 x F25(.)S XP /F25 72 31 2 0 28 28 28 32 0 PXLC RP 1043 721 XY F25(Hence)S 1168 X F27(`)S -15 y F9(E)S 2 x(\013)S 25 y 1193 X(n)S -10 y 1251 X F26(RP)S 1316 X(QM)S -11 y 1395 X(~)S 11 y 1394 X(L)S 1436 X F25(b)S -1 x(y)S 1496 X(the)S 1569 X F26(!)S -15 y 1 x F8(+)S 15 y 3 x F25({)S 50 y 300 X(Rule.)S 424 X F28(Cut)S 507 X (R)S -1 x(ule)S 3 x F25(.)S 632 X(By)S 700 X(induction)S 888 X(h)S -1 x(yp)S 1 x(othesis)S 1096 X F27(`)S -15 y F9(E)S 2 x(\013)S 25 y 1121 X(n)S -10 y 1181 X F26(M)S 1225 X(~)S 1226 X(y)S 1265 X F25 (and)S 1349 X F27(`)S -15 y F9(E)S 2 x(\013)S 25 y 1374 X(n)S -10 y 1434 X F26(L)S 6 y F9(i)S -6 y 1480 X F26(~)S 1476 X(y)S 6 y F9(i)S -6 y 2 x F25(.)S 1550 X(Since)S 50 y 300 X(lev\()S F26(M)S 413 X F25 (\))S 441 X F27(\024)S 485 X F26(n)S 514 X F25(+)S 550 X(1,)S 594 X (w)S -1 x(e)S 653 X(ha)S -1 x(v)S -1 x(e)S 746 X(lev\()S F26(L)S 6 y F9(i)S -6 y 2 x F25(\))S 884 X F27(\024)S 928 X F26(n)S 964 X F25(and)S 1042 X(hence)S 1155 X F27(`)S -16 y F9(E)S 2 x(\013)S F8 (+)S F9(E)S 2 x(\013)S 26 y 1180 X(n)S -10 y 1307 X F26(M)S 1352 X (L)S 6 y F8(1)S -6 y 1406 X F26(:)S 1425 X(:)S 1444 X(:)S 1461 X(L)S 6 y F9(m)S -6 y 1532 X F25(b)S -1 x(y)S 1587 X(the)S 57 y 300 X (Substitution)S XP /F25 76 26 2 0 23 28 28 24 0 PXLC RP 540 878 XY F25(Lemma.)S 711 X(But)S 796 X F26(E)S 2 x F25(\()S F26 (\013)S 882 X F25(+)S 925 X(1\))S 978 X(=)S 1025 X F26(E)S 2 x (\013)S 1096 X F25(+)S 1139 X F26(E)S 2 x(\013)S F25(,)S 1227 X(so)S 1280 X F27(`)S -21 y F9(E)S 2 x F8(\()S F9(\013)S F8(+1\))S 26 y 1305 X F9(n)S -5 y 1431 X F26(M)S 1476 X(L)S 7 y F8(1)S -7 y 1530 X F26(:)S 1549 X(:)S 1568 X(:)S 1585 X(L)S 7 y F9(m)S 43 y 300 X F25 (immediately)S -3 x(.)S 560 X F26(<)S 6 y F8(1)S -6 y 2 x F28({R)S -1 x(ule)S 3 x F25(.)S 758 X(By)S 827 X(induction)S 1015 X(h)S -1 x (yp)S 1 x(othesis)S 1224 X F27(`)S -15 y F9(E)S 2 x(\014)S 25 y 1249 X(n)S -10 y 1308 X F26(M)S 1371 X F25(where)S 1494 X F26(\014)S 1538 X(<)S 6 y F8(1)S -6 y 1607 X F26(\013)S F25(.)S 50 y 300 X (Since)S 411 X(from)S 511 X F26(\014)S 552 X(<)S 6 y F8(1)S -6 y 619 X F26(\013)S 662 X F25(w)S -1 x(e)S 725 X(can)S 804 X(easily)S 921 X(infer)S 1019 X F26(E)S 2 x(\014)S 1093 X(<)S 6 y F8(1)S -6 y 1159 X F26(E)S 2 x(\013)S F25(,)S 1248 X(w)S -1 x(e)S 1311 X(get)S 1383 X F27(`)S -15 y F9(E)S 2 x(\013)S 25 y 1408 X(n)S -10 y 1466 X F26(M)S 1527 X F25(b)S -1 x(y)S 1587 X(the)S 50 y 300 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule.)S 1628 1028 XY 17 17 R 1105 Y 375 X F30 (3.)S XP /F30 52 24 1 0 22 27 27 24 0 <000780 000780 000F80 001F80 003F80 007F80 00EF80 01EF80 03CF80 038F80 070F80 0E0F80 1C0F80 380F80 700F80 F00F80 FFFFF8 FFFFF8 FFFFF8 000F80 000F80 000F80 000F80 000F80 01FFF8 01FFF8 01FFF8> PXLC RP 412 1105 XY F30(4.)S 465 X(Se)S XP /F30 99 21 2 0 19 18 18 24 0 <03FC00 1FFE00 3F3F00 7E3F00 7E3F00 FC3F00 FC1E00 FC0000 FC0000 FC0000 FC0000 FC0000 FC0000 7E0380 7F0780 3F8F00 1FFE00 03F800> PXLC RP 513 1105 XY F30(con)S XP /F30 100 27 2 0 25 29 29 24 0 <000FF0 000FF0 000FF0 0001F0 0001F0 0001F0 0001F0 0001F0 0001F0 0001F0 0001F0 07F9F0 1FFFF0 3F0FF0 7E03F0 7E01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 FC01F0 7C01F0 7E03F0 3F0FFE 1FFFFE 07F9FE> PXLC RP 585 1105 XY F30(d)S 627 X(Colla)S XP /F30 112 27 1 -8 24 18 26 24 0 PXLC RP 735 1105 XY F30(psin)S XP /F30 103 24 1 -9 22 18 27 24 0 <07F9F0 1FFFF8 3E1FF8 7C0FF8 7C0FF0 7C0F80 7C0F80 7C0F80 7C0F80 3E1F00 3FFE00 37F800 700000 700000 780000 3FFF80 3FFFE0 1FFFF0 7FFFF0 F801F8 F000F8 F00078 F00078 F800F8 7E03F0 3FFFE0 07FF00> PXLC RP 821 1105 XY F30(g)S 860 X(Lemma.)S XP /F29 82 31 2 -1 31 28 29 32 0 <0FFFF800 0FFFFE00 00F00F80 00F00780 00F007C0 00F003C0 00F003C0 01F007C0 01E007C0 01E007C0 01E00F80 01E00F00 01E03E00 03FFF800 03FFE000 03C0F000 03C07800 03C07800 03C07800 07C07C00 07807C00 07807C00 07807C00 07807C00 07807C38 0F807C30 FFF87E70 FFF03FE0 00000FC0> PXLC RP 1048 1105 XY F29(Recall)S 1179 X(the)S 1257 X(de\014nition)S 1448 X (of)S 1502 X(the)S 1580 X(col)S XP /F29 45 14 1 7 13 10 3 16 0 <7FF0 FFE0 FFE0> PXLC RP 1631 1105 XY F29(-)S 50 y 300 X(lapsin)S XP /F29 103 21 0 -9 22 19 28 24 0 <000038 007EFC 01FFDC 03C79C 0783C0 0F03C0 0F03C0 0F03C0 0F03C0 0F0780 078F00 07FE00 0DF800 080000 180000 1C0000 1FFE00 0FFF80 1FFFC0 3C03C0 7001E0 7001E0 F001C0 F001C0 7807C0 7C1F80 3FFE00 0FF800> PXLC RP 406 1155 XY F29(g)S 446 X(function)S XP /F26 68 34 2 0 33 28 28 32 0 <03FFFF80 03FFFFE0 003E01F0 003C00F8 003C007C 003C003C 007C003C 0078003E 0078003E 0078003E 00F8003E 00F0003E 00F0003E 00F0003E 01F0007C 01E0007C 01E0007C 01E00078 03E000F8 03C000F0 03C001F0 03C001E0 07C003C0 07800780 07801F00 0F807C00 FFFFF800 FFFFC000> PXLC RP 615 1155 XY F26(D)S 1 x(\013)S 698 X F25(=)S 751 X F26(D)S 1 x F25 (\()S F26(\013;)S 848 X(!)S 1 x F25(\))S 910 X F29(in)S 964 X (section)S XP /F29 50 21 2 0 21 28 28 24 0 <007E00 01FF80 0387C0 0703E0 0E03E0 0F01E0 0F01E0 1F01E0 1F01E0 0F01E0 0003E0 0003C0 0007C0 000F80 000F00 001E00 003C00 007800 00F000 01E000 038000 070180 0E0180 1C0300 380300 7FFF00 FFFE00 FFFE00> PXLC RP 1108 1155 XY F29(2.)S XP /F29 83 23 2 -1 24 29 30 24 0 <007F0C 01FFDC 03E1FC 078078 070078 0E0038 0E0038 0E0038 1E0038 1E0000 1F0000 0FC000 0FF800 0FFF00 07FFC0 01FFC0 003FE0 0003E0 0001E0 0000E0 0000E0 6000E0 6000E0 6000E0 7001C0 7001C0 F80380 FE0F00 EFFE00 C3F800> PXLC RP 1176 1155 XY F29(Supp)S 1 x(ose)S 1344 X F26(\013)S 1391 X F27(2)S 1440 X F25(\012)S 6 y F8(2)S -6 y 1508 X F29(has)S 1587 X(the)S 50 y 300 X(prop)S 1 x(ert)S XP /F29 121 22 0 -8 23 18 26 24 0 <0FF1FE 0FF1FE 03C078 01C060 01C060 01C0C0 01C1C0 01C180 00E380 00E300 00E600 00E600 00EC00 007C00 007800 007800 007000 007000 006000 006000 00C000 70C000 F18000 C70000 FE0000 7C0000> PXLC RP 434 1205 XY F29(y)S 471 X(that)S 562 X(e)S XP /F29 118 22 4 0 23 18 18 24 0 PXLC RP 580 1205 XY F29(v)S -1 x(ery)S XP /F29 98 23 4 0 23 29 29 24 0 <3F0000 7F0000 0F0000 0E0000 0E0000 0E0000 0E0000 0E0000 1E0000 1C0000 1C0000 1CFE00 1FFF80 1F8780 3E03C0 3C03C0 3801E0 3801E0 3801E0 3803E0 7803E0 7003C0 7003C0 700780 700780 780F00 FC3E00 EFFC00 C7F000> PXLC RP 673 1205 XY F29(big)S 743 X(li)S XP /F29 109 35 1 0 34 18 18 40 0 <1F8FC0FC00 1FBFF3FF00 07F0F70F00 07C07C0700 0780780700 0780780700 0700700700 0700700700 0700700700 0F00F00F00 0E00E00E00 0E00E00E00 0E00E00E00 0E00E00E00 0E00E00E00 1E01E01E00 FF8FF8FF80 FF8FF8FF80> PXLC RP 766 1205 XY F29(mit)S 844 X F26(\014)S 883 X F25(=)S 929 X F27(h)S F26(\014)S 6 y F9(\030)S -6 y 3 x F27(i)S 1017 X F29(o)S 1 x (ccurring)S 1202 X(in)S 1252 X(its)S 1311 X(construction)S 1550 X (satis-)S 50 y 300 X(\014es)S 372 X F26(\030)S 403 X F27(\024)S 6 y F8(1)S XP /F26 17 21 1 -9 20 18 27 24 0 <3E1F80 3F3FE0 7771E0 67E0E0 E7C0E0 C780E0 0700E0 0F01E0 0F01E0 0E01C0 0E01C0 1E03C0 1E03C0 1C0380 1C0380 3C0780 3C0780 380700 000700 000F00 000F00 000E00 000E00 001E00 001E00 001C00 001C00> PXLC RP 466 1255 XY F26(\021)S 511 X F25(=)S 536 X F27(\))S 601 X F26(\014)S 6 y F9(\030)S -6 y 654 X F27(\024)S 6 y F8(1)S -6 y 716 X F26(\014)S XP /F9 17 17 0 -6 15 13 19 16 0 <3C7C 6FCE 6F8E EF0E CE0E 0E0E 1E1E 1E1E 1C1C 1C1C 3C3C 3C3C 3838 0038 0078 0078 0070 0070 0070> PXLC RP 740 1261 XY F9(\021)S -6 y 3 x F29(.)S XP /F29 84 30 4 0 32 28 28 32 0 <1FFFFFF0 3FFFFFF0 3C0781F0 38078060 70078060 60078060 60078060 600F8060 C00F0060 C00F0060 000F0000 000F0000 000F0000 001F0000 001E0000 001E0000 001E0000 001E0000 001E0000 003E0000 003C0000 003C0000 003C0000 003C0000 003C0000 007C0000 1FFFE000 1FFFE000> PXLC RP 790 1255 XY F29(Then)S 899 X(w)S -1 x(e)S 960 X(ha)S -1 x(v)S -1 x (e)S 1332 Y 809 X F27(`)S -17 y F9(\013)S 27 y 834 X F8(0)S -10 y 862 X F30(m)S 925 X F25(=)S 950 X F27(\))S 1003 X(`)S XP /F9 68 27 1 0 24 20 20 24 0 <07FFE0 07FFF0 01E078 01C03C 01C01C 03C01E 03C01E 03801E 03801E 07801E 07803C 07003C 07003C 0F0038 0F0070 0E00F0 0E01E0 1E07C0 FFFF00 FFFC00> PXLC RP 1029 1315 XY F9(D)S 1 x(\013)S 27 y 1028 X F8(0)S -10 y 1085 X F30 (m)S F26(:)S 1410 Y 375 X F30(Pro)S 1 x(of.)S 525 X F25(By)S 589 X (induction)S 773 X(on)S 830 X F27(`)S -15 y F9(\013)S 25 y 855 X F8 (0)S -10 y 884 X F30(m)S F25(.)S 953 X(Recall)S 1078 X(that)S 1167 X (w)S -1 x(e)S 1228 X(iden)S -1 x(tify)S 1378 X(terms)S 1493 X(with)S 1587 X(the)S 49 y 300 X(same)S 403 X(normal)S 542 X(form.)S 656 X F28(Constant)S 833 X(R)S -1 x(ules)S 3 x F25(.)S 964 X F28(Case)S 1065 X F25(0.)S 1115 X(Clear,)S 1236 X(since)S 1337 X F27(`)S -15 y F9(D)S 1 x F8(1)S 26 y 1362 X(0)S -11 y 1413 X F25(0)S 1447 X(b)S -1 x(y)S 1503 X(the)S 1573 X F26(<)S 7 y F8(1)S -7 y 2 x F25({)S 50 y 300 X(Rule)S 398 X(using)S 507 X(1)S 539 X F26(<)S 7 y F8(1)S -7 y 602 X F26(!)S 639 X F25(+)S 680 X(1.)S 732 X F28(Case)S 834 X F25(succ)S 1 x(.)S 941 X(By)S 1006 X(induction)S 1191 X(h)S -1 x (yp)S 1 x(othesis)S 1396 X F27(`)S -15 y F9(D)S 1 x(\013)S 26 y 1421 X F8(0)S -11 y 1478 X F30(m)S F25(,)S 1543 X(hence)S 58 y 300 X F27(`)S -18 y F9(D)S 1 x(\013)S F8(+1)S 29 y 325 X(0)S -11 y 424 X F25(succ)S 512 X F30(m)S 563 X F25(b)S -1 x(y)S 619 X(the)S 688 X (Constan)S -1 x(t)S 865 X(Rule)S 960 X(for)S 1021 X(succ)S 2 x(,)S 1121 X(hence)S 1234 X F27(`)S -21 y F9(D)S 1 x F8(\()S F9(\013)S F8 (+1\))S 32 y 1259 X(0)S -11 y 1384 X F25(succ)S 1472 X F30(m)S F25 (.)S 1541 X F28(Cases)S 50 y 300 X F25(cond)S 12 y F30(nat)S XP /F28 44 13 3 -8 9 5 13 8 0 <1C 3C 3C 3C 3C 0C 1C 18 38 70 70 E0 C0> PXLC RP 456 1617 XY F28(,)S 487 X F25(p)S 1 x(cond)S 12 y 1 x F30(nat)S -12 y 1 x F25(.)S 708 X(If)S 753 X(cond)S 12 y 1 x F30(nat)S -12 y 1 x F26 (M)S 954 X(N)S 992 X(L)S 1037 X F25(or)S 1092 X(p)S 1 x(cond)S 12 y F30(nat)S -12 y 2 x F26(M)S 1317 X(N)S 1355 X(L)S 1400 X F25(has)S 1478 X(a)S 1516 X(de\014ned)S 50 y 300 X(v)S -2 x(alue)S 408 X (denoted)S 565 X(b)S -1 x(y)S 624 X F30(m)S F25(,)S 690 X(then)S 785 X(this)S 866 X(is)S 909 X(the)S 981 X(v)S -2 x(alue)S 1089 X (of)S 1137 X(either)S 1255 X F26(N)S 1308 X F25(or)S 1359 X F26(L)S F25(.)S 1419 X(Supp)S 1 x(ose)S 1583 X(e.g.)S 50 y 300 X(it)S 340 X (is)S 381 X(the)S 451 X(v)S -2 x(alue)S 557 X(of)S 603 X F26(N)S 641 X F25(.)S 671 X(Then)S 778 X(b)S -1 x(y)S 835 X(induction)S 1018 X(h)S -1 x(yp)S 1 x(othesis)S 1221 X F27(`)S -16 y F9(D)S 1 x (\013)S 26 y 1246 X F8(0)S -10 y 1303 X F30(m)S F25(,)S 1367 X (hence)S 1481 X F27(`)S -18 y F9(D)S 1 x(\013)S F8(+1)S 29 y 1506 X (0)S -11 y 1605 X F30(m)S 49 y 300 X F25(b)S -1 x(y)S 357 X(the)S 428 X(same)S 531 X(Constan)S -1 x(t)S 710 X(Rule)S 807 X(\(since)S 924 X(w)S -1 x(e)S 984 X(iden)S -1 x(tify)S 1135 X(terms)S 1250 X (with)S 1344 X(the)S 1415 X(same)S 1518 X(normal)S 58 y 300 X (form\),)S 429 X(hence)S 546 X F27(`)S -21 y F9(D)S 1 x F8(\()S F9 (\013)S F8(+1\))S 32 y 571 X(0)S -11 y 700 X F30(m)S F25(.)S 777 X F26(!)S -15 y 1 x F8(+)S 15 y 2 x F28({R)S -1 x(ule)S 3 x F25(.)S 974 X(Then)S 1085 X F27(`)S -15 y F9(\013)S 26 y 1110 X F8(0)S -11 y 1142 X F26(RP)S 1207 X(QM)S -10 y 1286 X(~)S 10 y 1285 X(L)S 1329 X F25(has)S 1405 X(b)S 1 x(een)S 1505 X(inferred)S 50 y 300 X(from)S 398 X F27(`)S -18 y F9(\013)S 4 y F0(0)S 25 y 423 X F8(0)S -11 y 468 X F26(M)S 527 X F25(and)S 1957 Y 661 X F27(8)S F26(\030)S 715 X F27(2)S 755 X F25(\012)S 6 y F8(1)S -6 y 1 x F27(8)S F26(n:)S 874 X F27(`)S -20 y F9(\030)S 31 y 899 X F8(0)S -11 y 923 X F30(n)S 972 X F25(=)S 997 X F27(\))S 1051 X(`)S -23 y F9(\013)S 5 y F1(\030)S 29 y 1076 X F8(0)S -11 y 1121 X F26(RP)S 1186 X(Q)S F30(n)S -10 y 2 x F26 (~)S 10 y 1246 X(L)S -1 x(;)S 2040 Y 300 X F25(and)S 380 X F30(m)S 432 X F25(denotes)S 581 X(the)S 652 X(v)S -2 x(alue)S 758 X(of)S 804 X F26(RP)S 869 X(QM)S -10 y 948 X(~)S 10 y 947 X(L)S -1 x F25 (.)S 1004 X(Then)S 1111 X F26(M)S 1168 X F25(m)S -1 x(ust)S 1270 X (ha)S -1 x(v)S -1 x(e)S 1365 X(a)S 1398 X(de\014ned)S 1540 X(v)S -2 x(alue,)S 50 y 300 X(sa)S -1 x(y)S 377 X F26(k)S 1 x F25(.)S 447 X(Hence)S 576 X F30(m)S 636 X F25(also)S 725 X(denotes)S 880 X (the)S 958 X(v)S -2 x(alue)S 1070 X(of)S 1123 X F26(RP)S 1188 X(Q)S XP /F30 107 25 1 0 23 29 29 24 0 PXLC RP 1221 2090 XY F30(k)S -11 y 2 x F26(~)S 11 y 1246 X(L)S F25(.)S 1321 X (W)S -3 x(e)S 1398 X(ha)S -1 x(v)S -1 x(e)S 1500 X(to)S 1556 X(sho)S -1 x(w)S 50 y 300 X F27(`)S -15 y F9(D)S 1 x(\013)S 25 y 325 X F8 (0)S -10 y 386 X F30(m)S F25(.)S 463 X(By)S 531 X(the)S 605 X (induction)S 792 X(h)S -1 x(yp)S 1 x(othesis)S 999 X(w)S -1 x(e)S 1062 X(ha)S -1 x(v)S -1 x(e)S 1161 X F27(`)S -19 y F9(D)S 1 x(\013)S 4 y F0(0)S 26 y 1186 X F8(0)S -11 y 1263 X F30(k)S F25(.)S 1325 X (Then)S 1436 X(with)S 1533 X F26(n)S 1574 X F25(=)S 1622 X F26(k)S 55 y 300 X F25(and)S 385 X F26(\030)S 423 X F25(=)S 474 X F26(D)S 1 x(\013)S 6 y F8(0)S -6 y 573 X F25(w)S -1 x(e)S 638 X(get)S 711 X F27(`)S -25 y F9(\013)S XP /F1 68 23 2 0 21 14 14 24 0 <1FFE00 1FFF80 0703C0 0F00C0 0F00E0 0E00E0 0E00E0 1E00E0 1E01C0 1C01C0 1C0380 3C0F00 FFFC00 FFF000> PXLC RP 758 2175 XY F1(D)S XP /F1 11 19 2 0 16 9 9 16 0 <1F80 39C4 60CC E0E8 C078 C070 C060 61E8 3F38> PXLC RP 782 2175 XY F1(\013)S 6 y F0(0)S 25 y 736 X F8(0)S -11 y 833 X F26 (RP)S 898 X(Q)S F30(k)S -10 y 2 x F26(~)S 10 y 956 X(L)S 1001 X F25 (from)S 1103 X(our)S 1181 X(assumption,)S 1417 X(so)S 1473 X(again)S 1587 X(the)S 50 y 300 X(induction)S 490 X(h)S -1 x(yp)S 1 x (othesis)S 701 X(yields)S 823 X F27(`)S -15 y F9(D)S 1 x(\013)S 25 y 848 X F8(0)S -10 y 914 X F30(m)S 974 X F25(b)S -1 x(y)S 1037 X(de)S (\014nition)S 1227 X(of)S 1280 X F26(D)S 1 x(\013)S F25(.)S 1390 X F26(<)S 6 y F8(1)S -6 y 2 x F28({R)S -1 x(ule)S 3 x F25(.)S 1594 X (By)S 54 y 300 X(induction)S 485 X(h)S -1 x(yp)S 1 x(othesis)S 691 X F27(`)S -19 y F9(D)S 1 x(\014)S 31 y 716 X F8(0)S -12 y 773 X F30 (m)S 828 X F25(where)S 948 X F26(\014)S 987 X(<)S 7 y F8(1)S -7 y 1051 X F26(\013)S F25(,)S 1105 X(hence)S 1221 X F27(`)S -15 y F9(D)S 1 x(\013)S 26 y 1246 X F8(0)S -11 y 1304 X F30(m)S 1359 X F25(b)S -1 x(y)S 1417 X(the)S 1490 X F26(<)S 7 y F8(1)S -7 y 2 x F25({Rule)S 50 y 300 X(using)S 409 X(the)S 480 X(follo)S -1 x(wing)S 655 X (claim)S XP /F25 58 12 3 0 8 18 18 8 0 <70 F8 F8 F8 70 00 00 00 00 00 00 00 00 70 F8 F8 F8 70> PXLC RP 752 2349 XY F25(:)S 2427 Y 748 X F26(\014)S 786 X F27(\024)S 6 y F8 (1)S -6 y 848 X F26(\013)S 898 X F25(=)S 923 X F27(\))S 988 X F26 (D)S 1 x(\014)S 1060 X F27(\024)S 6 y F8(1)S -6 y 1123 X F26(D)S 1 x(\013:)S XP /F25 80 28 2 0 25 28 28 24 0 PXLC RP 300 2504 XY F25(Pro)S 1 x(of)S 416 X(of)S 465 X(the)S 538 X(claim)S 651 X(is)S 694 X(b)S -1 x(y)S 754 X(induction)S 940 X(o)S -1 x(v)S -1 x(er)S 1031 X F26(\013)S 1073 X F25(satisfying)S 1260 X(the)S 1334 X(conditions)S 1534 X(stated)S 50 y 300 X(ab)S 1 x(o)S -1 x(v)S -1 x(e.)S 438 X(The)S XP /F25 122 18 1 0 15 18 18 16 0 <7FFC 7FFC 783C 7078 60F0 61E0 61E0 63C0 0780 078C 0F0C 1E0C 1E1C 3C18 7818 F078 FFF8 FFF8> PXLC RP 524 2554 XY F25(zero,)S 625 X(success)S 3 x(or)S 805 X(and)S 887 X (small)S 997 X(limit)S 1097 X(cases)S 1203 X(are)S 1273 X (immediate.)S 1496 X(Supp)S 1 x(ose)S 50 y 300 X(then)S 394 X(that)S 484 X F26(\013)S 522 X F25(=)S 566 X F27(h)S F26(\013)S 6 y F9 (\030)S -6 y 2 x F27(i)S 656 X F25(where)S 775 X F26(\013)S 6 y F9 (\030)S -6 y 832 X F27(\024)S 6 y F8(1)S -6 y 894 X F26(\013)S 6 y F9(\021)S -6 y 954 X F25(whenev)S -1 x(er)S 1136 X F26(\030)S 1168 X F27(\024)S 6 y F8(1)S -6 y 1230 X F26(\021)S 1 x F25(.)S 1282 X(If)S 1323 X F26(\014)S 1360 X(<)S 6 y F8(1)S -6 y 1423 X F26(\013)S 1463 X F25(then)S 1557 X F26(\014)S 1594 X F27(\024)S 6 y F8(1)S 44 y 300 X F26(\013)S 6 y F9(i)S -6 y 355 X F25(b)S -1 x(y)S 414 X (de\014nition)S 600 X(where)S 721 X F26(i)S 749 X F25(=)S 795 X(0)S 831 X(or)S 883 X(1.)S 937 X(But)S 1021 X(it)S 1064 X(is)S 1107 X (straigh)S -1 x(tforw)S -1 x(ard)S 1400 X(to)S 1452 X(v)S -1 x (erify)S 1569 X(that)S 49 y 300 X F26(i)S 328 X F27(\024)S 7 y F8 (1)S -7 y 393 X F26(!)S 434 X F27(\024)S 7 y F8(1)S -7 y 499 X F26 (D)S 1 x(\015)S 574 X F25(for)S 639 X(ev)S -1 x(ery)S 750 X F26 (\015)S 2 x F25(.)S 808 X(Therefore)S 998 X(with)S 1094 X F26(\015)S 1131 X F25(=)S 1178 X F26(\013)S 7 y F8(0)S -7 y 1238 X F25(w)S -1 x (e)S 1301 X(obtain)S 1431 X F26(\013)S 7 y F9(i)S -7 y 1486 X F27 (\024)S 7 y F8(1)S -7 y 1551 X F26(\013)S 7 y F9(D)S 1 x(\013)S 4 y F0(0)S 39 y 300 X F25(and)S 381 X(hence)S 496 X(b)S -1 x(y)S 554 X (the)S 625 X(induction)S 810 X(h)S -1 x(yp)S 1 x(othesis)S 1014 X F26(D)S 1 x(\014)S 1087 X F27(\024)S 7 y F8(1)S -7 y 1149 X F26(D)S 1 x(\013)S 7 y F9(i)S -7 y 1237 X F27(\024)S 7 y F8(1)S -7 y 1299 X F26(D)S 1 x(\013)S 7 y F9(D)S 1 x(\013)S 4 y F0(0)S -11 y 1441 X F25 (=)S 1485 X F26(D)S 1 x(\013)S F25(.)S 1628 2753 XY 17 17 R %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup 692 462 XY F18(Ordinal)S 833 X(Bounds)S 971 X(for)S 1030 X (Programs)S 1583 X F25(39)S XP /F25 53 21 2 -1 18 28 29 16 0 <301C 3FFC 3FF8 3FF0 3FC0 3000 3000 3000 3000 3000 33E0 3FF8 3E3C 381E 381E 300E 000F 000F 000F 000F F00F F00F F00F F01E E01E 703C 7878 3FF0 0FC0> PXLC RP 1624 462 XY F25(5)S 553 Y 375 X F30(3.)S XP /F30 53 24 2 0 21 27 27 24 0 <380380 3FFF80 3FFF80 3FFF00 3FFC00 3FF800 3F8000 380000 380000 380000 3BFC00 3FFF00 3E1F80 3C0FC0 3807C0 0007E0 0007E0 0007E0 7C07E0 FC07E0 FC07E0 FC07E0 FC0FC0 7C1F80 7FFF80 3FFE00 07F800> PXLC RP 412 553 XY F30(5.)S XP /F30 70 30 2 0 27 28 28 32 0 PXLC RP 465 553 XY F30(First)S 581 X(Collapsing)S 814 X(Lemma.)S XP /F29 76 26 2 0 25 28 28 24 0 <0FFFC0 0FFFC0 00F000 00F000 00F000 00F000 00F000 01F000 01E000 01E000 01E000 01E000 01E000 03E000 03C000 03C000 03C000 03C006 03C00C 07C00C 07800C 07801C 078018 078038 078078 0F81F8 FFFFF0 FFFFF0> PXLC RP 1002 553 XY F29(Let)S XP /F26 71 33 2 -1 31 29 30 32 0 <0003FC18 001FFF38 003F07F8 00FC03F0 01F001F0 03E000F0 07C000F0 0F8000E0 1F000060 1F000060 3E0000E0 3E0000C0 7C000000 7C000000 7C000000 FC000000 F8000000 F800FFF0 F801FFF0 F8000F80 F8000F00 F8000F00 F8001F00 78001F00 7C001E00 3E003E00 1F007E00 0FC1EE00 07FFC400 01FE0000> PXLC RP 1081 553 XY F26(G)S 6 y F9(\013)S -6 y 1 x F29(,)S 1169 X F26(\013)S 1216 X F27(2)S 1264 X F25(\012)S 6 y F8(1)S -6 y 2 x F29(,)S 1345 X (b)S 1 x(e)S 1407 X(the)S 1484 X(follo)S -1 x(wi)S -3 x(ng)S 50 y 300 X(sligh)S -1 x(tly)S 446 X(mo)S 1 x(di\014ed)S 615 X(v)S -1 x (ersion)S 756 X(of)S 803 X(the)S 875 X(slo)S -1 x(w)S XP /F29 123 21 2 10 22 12 2 24 0 PXLC RP 952 603 XY F29({gro)S -1 x(wing)S 1129 X(hierarc)S -1 x(h)S -1 x (y:)S 697 Y 363 X F26(G)S 6 y F8(0)S -6 y 1 x F25(\()S F26(n)S F25 (\))S 483 X(=)S 526 X(0)S F26(;)S 649 X(G)S 6 y F9(\013)S F8(+1)S -6 y 1 x F25(\()S F26(n)S F25(\))S 816 X(=)S 859 X F26(G)S 6 y F9 (\013)S -6 y 2 x F25(\()S F26(n)S F25(\))S 982 X(+)S 1024 X(1)S F26 (;)S 1146 X(G)S XP /F10 104 13 3 -8 9 22 30 8 0 <0C 0C 1C 1C 18 38 38 30 30 70 70 60 E0 E0 C0 C0 E0 E0 60 70 70 30 30 38 38 18 1C 1C 0C 0C> PXLC RP 1179 704 XY F10(h)S F9(\013)S XP /F1 120 16 2 0 14 9 9 16 0 <3DE0 6F70 CE70 0E60 0C00 EC20 FC60 DCC0 F780> PXLC RP 1214 708 XY F1(x)S XP /F10 105 13 2 -8 8 22 30 8 0 PXLC RP 1232 704 XY F10(i)S -7 y 3 x F25(\()S F26(n)S F25(\))S 1316 X(=)S 1360 X(max)S 26 y 1370 X F9(i)S XP /F10 20 26 3 -7 21 20 27 24 0 <0001C0 0007C0 000F80 003E00 00FC00 03F000 0FC000 1F0000 7C0000 F80000 F80000 7C0000 1F0000 0FC000 03F000 00FC00 003E00 000F80 0007C0 0001C0 000000 000000 000000 000000 000000 FFFFC0 FFFFC0> PXLC RP 1381 723 XY F10(\024)S F9(n)S -26 y 1444 X F26(G)S 6 y F9(\013)S XP /F1 105 11 2 0 9 15 15 8 0 <0E 0E 0E 00 00 00 38 6C DC 18 38 32 36 34 3C> PXLC RP 1498 707 XY F1(i)S -10 y 1514 X F25(\()S F26(n)S F25(\))S F26(:)S 810 Y 300 X F29(Then)S 408 X(for)S 472 X(all)S 530 X F26(\013)S 568 X F27(2)S 607 X F25(\012)S 6 y F8(1)S -6 y 669 X F29(w)S -1 x (e)S 731 X(ha)S -1 x(v)S -1 x(e)S 903 Y 802 X F27(`)S -17 y F9 (\013)S 28 y 827 X F8(0)S -11 y 856 X F30(n)S 905 X F25(=)S 930 X F27(\))S 984 X(`)S XP /F9 71 26 2 0 23 20 20 24 0 <007F18 01FFF8 07E1F0 0F8070 1F0070 3E0070 3C0060 7C0000 780000 780000 F80000 F01FF0 F01FF0 F003C0 F00380 780380 7C0380 3E0F80 1FFF80 07F900> PXLC RP 1009 882 XY F9(G)S 4 y F1(\013)S -4 y 2 x F8(\(1\))S 32 y 1009 X(0)S -11 y 1105 X F30(n)S F26(:)S 997 Y 375 X F30(Pro)S 1 x(of.)S 525 X F25(By)S 591 X(induction)S 778 X(on)S 837 X F27(`)S -15 y F9(\013)S 25 y 862 X F8(0)S -10 y 893 X F30(n)S 935 X F25(noting)S 1066 X (that)S 1158 X(the)S 1231 X F26(!)S -15 y 1 x F8(+)S 15 y 3 x F25 ({rule)S 1391 X(cannot)S 1529 X(b)S 1 x(e)S 1587 X(ap-)S 58 y 300 X (plied.)S 433 X F28(Constant)S 616 X(R)S -1 x(ules)S 3 x F25(.)S 763 X F28(Case)S 870 X F25(0.)S 936 X F27(`)S -22 y F9(G)S XP /F0 49 14 3 0 11 13 13 8 0 <18 F8 F8 18 18 18 18 18 18 18 18 FF FF> PXLC RP 987 1037 XY F0(1)S -4 y 2 x F8(\(1\))S 33 y 961 X(0)S -11 y 1061 X F25(0)S 1101 X(since)S 1208 X F26(G)S 6 y F8(1)S -6 y 1 x F25 (\(1\))S 1332 X(=)S 1384 X(1.)S 1451 X F28(Case)S 1557 X F25(succ)S 1 x(.)S 59 y 300 X(By)S 367 X(induction)S 554 X(h)S -1 x(yp)S 1 x (othesis)S 760 X F27(`)S -22 y F9(G)S 4 y F1(\013)S -4 y 2 x F8 (\(1\))S 33 y 785 X(0)S -11 y 885 X F30(n)S F25(,)S 939 X(hence)S 1056 X F27(`)S -22 y F9(G)S 4 y F1(\013)S XP /F0 43 21 2 -4 19 14 18 24 0 <00C000 00C000 00C000 00C000 00C000 00C000 00C000 00C000 FFFF80 FFFF80 00C000 00C000 00C000 00C000 00C000 00C000 00C000 00C000> PXLC RP 1127 1096 XY F0(+1)S -4 y 2 x F8(\(1\))S 33 y 1081 X(0)S -11 y 1217 X F25(succ)S 1304 X F30(n)S 1347 X F25(b)S -1 x(y)S 1406 X(the)S 1480 X (Constan)S -1 x(t)S 49 y 300 X(Rule)S 402 X(for)S 471 X(succ)S 1 x (,)S 579 X(since)S 686 X F26(G)S 7 y F9(\013)S F8(+1)S -7 y 1 x F25 (\(1\))S 857 X(=)S 909 X F26(G)S 7 y F9(\013)S -7 y 1 x F25(\(1\))S 1031 X(+)S 1075 X(1.)S 1141 X F28(Cases)S 1265 X F25(cond)S 13 y F30 (nat)S -13 y 1 x F28(,)S 1454 X F25(p)S 1 x(cond)S 13 y F30(nat)S -13 y 1 x F25(.)S 50 y 300 X(cond)S 12 y F30(nat)S -12 y 2 x F26(M)S 501 X(N)S 539 X(L)S 583 X F25(or)S 636 X(p)S 1 x(cond)S 12 y F30 (nat)S -12 y 1 x F26(M)S 860 X(N)S 898 X(L)S 942 X F25(can)S 1021 X (only)S 1114 X(ha)S -1 x(v)S -1 x(e)S 1212 X F30(n)S 1254 X F25(as)S 1307 X(normal)S 1450 X(form)S 1550 X(when)S 50 y 300 X(either)S 425 X F26(N)S 484 X F25(or)S 542 X F26(L)S 592 X F25(do)S 1 x(es.)S 724 X(Supp)S 1 x(ose)S 894 X(it)S 943 X(is)S 992 X F26(N)S 1030 X F25(.)S 1082 X(Then)S 1197 X(b)S -1 x(y)S 1262 X(induction)S 1454 X (h)S -1 x(yp)S 1 x(othesis)S 57 y 300 X F27(`)S -22 y F9(G)S 4 y F1 (\013)S -4 y 2 x F8(\(1\))S 33 y 325 X(0)S -11 y 427 X F30(n)S F25 (,)S 483 X(hence)S 602 X F27(`)S -22 y F9(G)S 4 y F1(\013)S F0(+1)S -4 y 3 x F8(\(1\))S 33 y 627 X(0)S -11 y 765 X F30(n)S F25(.)S 832 X F26(<)S 6 y F8(1)S -6 y 1 x F28({R)S -1 x(ule)S 3 x F25(.)S 1028 X (By)S 1097 X(induction)S 1285 X(h)S -1 x(yp)S 1 x(othesis)S 1493 X F27(`)S -23 y F9(G)S XP /F1 12 16 2 -4 15 14 18 16 0 <01F0 0738 0C18 0838 1830 17F0 17F0 3030 2030 2030 2030 6060 71C0 5F80 4000 C000 8000 8000> PXLC RP 1544 1301 XY F1(\014)S -4 y 1564 X F8(\(1\))S 34 y 1518 X(0)S -11 y 1618 X F30(n)S 58 y 300 X F25(where)S 419 X F26(\014)S 456 X(<)S 7 y F8(1)S -7 y 519 X F26(\013)S F25(.)S 575 X(W)S -3 x(e)S 646 X (can)S 721 X(then)S 815 X(conclude)S 985 X(that)S 1074 X F27(`)S -21 y F9(G)S 4 y F1(\013)S -4 y 2 x F8(\(1\))S 32 y 1099 X(0)S -11 y 1195 X F30(n)S 1235 X F25(b)S -1 x(y)S 1291 X(the)S 1362 X(follo)S -1 x(wing)S 1537 X(claim:)S 1472 Y 711 X F26(\014)S 748 X F27(\024)S 6 y F8(1)S -6 y 810 X F26(\013)S 860 X F25(=)S 885 X F27(\))S 950 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\(1\))S 1070 X F27(\024)S 1113 X F26(G)S 6 y F9(\013)S -6 y 2 x F25(\(1\))S F26(:)S 1566 Y 300 X F25 (Pro)S 1 x(of)S 415 X(of)S 463 X(claim.)S 594 X(The)S 681 X(cases)S 786 X F26(\013)S 826 X F25(=)S 872 X(0)S 908 X(and)S 990 X F26 (\013)S 1030 X F25(=)S 1076 X F26(\013)S XP /F10 48 10 1 1 8 16 15 8 0 <1E 1E 1E 3E 3C 3C 38 38 78 70 70 60 E0 E0 C0> PXLC RP 1103 1551 XY F10(0)S 15 y 1125 X F25(+)S 1167 X(1)S 1203 X(are)S 1273 X(immediate.)S 1496 X(Supp)S 1 x(ose)S 49 y 300 X(that)S 390 X F26(\013)S 428 X F25(=)S 472 X F27(h)S F26(\013)S 7 y F9(x)S -7 y 1 x F27(i)S F25(.)S 582 X(Then)S 690 X(if)S 728 X F26(\014)S 765 X (<)S 7 y F8(1)S -7 y 827 X F26(\013)S 868 X F25(w)S -1 x(e)S 929 X (ha)S -1 x(v)S -1 x(e)S 1025 X F26(\014)S 1062 X F27(\024)S 7 y F8 (1)S -7 y 1124 X F26(\013)S 7 y F9(i)S -7 y 1178 X F25(where)S 1298 X F26(i)S 1324 X F25(=)S 1368 X(0)S 1402 X(or)S 1453 X F26(i)S 1479 X F25(=)S 1523 X(1.)S 1573 X(The)S 50 y 300 X(induction)S 483 X(h)S -1 x(yp)S 1 x(othesis)S 687 X(then)S 780 X(giv)S -1 x(es)S 880 X F26 (G)S 6 y F9(\014)S -6 y 3 x F25(\(1\))S 1000 X F27(\024)S 1044 X F26 (G)S 6 y F9(\013)S 5 y F1(i)S -11 y 3 x F25(\(1\))S 1179 X(and)S 1258 X(hence)S 1372 X F26(G)S 6 y F9(\014)S -6 y 3 x F25(\(1\))S 1492 X F27(\024)S 1536 X F26(G)S 6 y F9(\013)S -6 y 1 x F25(\(1\))S 50 y 300 X(b)S -1 x(y)S 358 X(the)S 429 X(revised)S 569 X(limit)S 668 X(clause)S 791 X(in)S 839 X(the)S 911 X(de\014nition)S 1095 X (of)S 1142 X F26(G)S 6 y F9(\013)S -6 y 2 x F25(.)S 1628 1715 XY 17 17 R 1810 Y 375 X F30(3.)S XP /F30 54 24 2 0 21 27 27 24 0 <00FE00 03FF80 0FFF80 1FC7C0 3F0FC0 3E0FC0 7E0FC0 7C0FC0 7C0780 FC2000 FDFE00 FFFF80 FF0FC0 FE07C0 FE07E0 FC07E0 FC07E0 FC07E0 FC07E0 FC07E0 7C07E0 7C07E0 3E07C0 3F0FC0 1FFF80 0FFF00 03FC00> PXLC RP 412 1810 XY F30(6.)S 465 X(V)S 497 X(alue)S 598 X(Lemma.)S 785 X F29 (If)S 827 X F27(`)S -15 y F9(k)S 25 y 852 X F8(0)S -10 y 877 X F30 (n)S F29(,)S 929 X(then)S 1024 X F26(n)S 1060 X(<)S 1104 X(k)S 1 x F29(.)S 1904 Y 375 X F30(Pro)S 1 x(of.)S 525 X F25(By)S 595 X (induction)S 785 X(on)S 848 X F27(`)S -15 y F9(k)S 26 y 873 X F8(0)S -11 y 908 X F30(n)S F25(.)S 981 X F28(Constant)S 1164 X(R)S -1 x (ules)S 3 x F25(.)S 1312 X F28(Case)S 1419 X F25(0.)S XP /F25 79 32 2 -1 29 29 30 32 0 <003F8000 00FFE000 03E0F800 07803C00 0F001E00 1E000F00 3E000F80 3C000780 7C0007C0 7C0007C0 780003C0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 F80003E0 780003C0 7C0007C0 7C0007C0 3C000780 3E000F80 1E000F00 0F001E00 07803C00 03E0F800 00FFE000 003F8000> PXLC RP 1486 1904 XY F25(Ob)S -1 x(vious.)S 50 y 300 X F28(Case)S 404 X F25 (succ)S 2 x(.)S 519 X(By)S 587 X(induction)S 774 X(h)S -1 x(yp)S 1 x(othesis)S 981 X F26(n)S 1022 X(<)S 1071 X(k)S 1 x F25(,)S 1122 X (hence)S 1240 X F26(n)S 1276 X F25(+)S 1320 X(1)S 1356 X F26(<)S 1405 X(k)S 1439 X F25(+)S 1482 X(1.)S 1541 X F28(Cases)S 50 y 300 X F25(cond)S 12 y F30(nat)S -12 y 2 x F28(,)S 481 X F25(p)S 1 x(cond)S 12 y 1 x F30(nat)S -12 y 1 x F25(.)S 690 X(cond)S 12 y F30(nat)S -12 y 2 x F26(M)S 891 X(N)S 929 X(L)S 967 X F25(or)S 1015 X(p)S 1 x (cond)S 12 y 1 x F30(nat)S -12 y 1 x F26(M)S 1240 X(N)S 1278 X(L)S 1317 X F25(can)S 1390 X(only)S 1478 X(ha)S -1 x(v)S -1 x(e)S 1571 X F30(n)S 1608 X F25(as)S 50 y 300 X(normal)S 439 X(form)S 535 X (when)S 641 X(either)S 757 X F26(N)S 807 X F25(or)S 856 X F26(L)S 896 X F25(do)S 1 x(es.)S 1005 X(Supp)S 1 x(ose)S 1166 X(it)S 1205 X (is)S 1245 X F26(N)S 1283 X F25(.)S 1312 X(Then)S 1419 X(b)S -1 x (y)S 1474 X(induction)S 50 y 300 X(h)S -1 x(yp)S 1 x(othesis)S 505 X F26(n)S 542 X(<)S 586 X(k)S 1 x F25(,)S 635 X(hence)S 751 X F26(n)S 788 X(<)S 832 X(k)S 865 X F25(+)S 906 X(1.)S 958 X F26(<)S 6 y F8 (1)S -6 y 2 x F28({R)S -1 x(ule)S 3 x F25(.)S 1146 X(Note)S 1246 X (\014rst)S 1332 X(that)S 1423 X(from)S 1521 X F26(\014)S 1559 X(<)S 6 y F8(1)S -6 y 1622 X F26(k)S 49 y 300 X F25(w)S -1 x(e)S 360 X (can)S 435 X(conclude)S 605 X(that)S 694 X F26(\014)S 732 X F25(is)S 773 X(a)S 807 X(n)S -1 x(umeral)S XP /F26 108 12 2 0 11 29 29 16 0 <1F80 3F80 0780 0780 0700 0700 0F00 0F00 0E00 0E00 1E00 1E00 1C00 1C00 3C00 3C00 3800 3800 7800 7800 7000 7000 F700 F700 E600 E600 FE00 FC00 3C00> PXLC RP 966 2153 XY F26(l)S 992 X F25(with)S 1086 X F26(l)S 1110 X(<)S 1154 X (k)S 1 x F25(.)S 1207 X(By)S 1271 X(induction)S 1454 X(h)S -1 x(yp)S 1 x(othesis)S 50 y 300 X F26(n)S 336 X(<)S 380 X(l)S 1 x F25(,)S 419 X(hence)S 534 X F26(n)S 571 X(<)S 614 X(k)S 1 x F25(.o)S 1628 2203 XY 17 17 R 2298 Y 375 X(W)S -3 x(e)S 449 X(no)S -1 x(w)S 538 X(construct)S 724 X(an)S XP /F25 92 21 5 16 19 29 13 16 0 <1818 3C3C 3838 7070 6060 E0E0 C0C0 C0C0 F8F8 FCFC FCFC 7C7C 3838> PXLC RP 784 2298 XY F25(\\initial)S XP /F25 34 21 1 16 15 29 13 16 0 <7070 F8F8 FCFC FCFC 7C7C 0C0C 0C0C 1C1C 1818 3838 7070 F0F0 6060> PXLC RP 911 2298 XY F25(")S 949 X(generation)S 1154 X(tree)S 1240 X(for)S 1307 X(an)S -1 x(y)S 1388 X(term)S 1490 X F26(M)S 1535 X F25(.)S 1573 X(The)S 50 y 300 X(main)S 404 X(p)S 1 x(oin)S -1 x(t)S 512 X (here)S 602 X(is)S 644 X(that)S 734 X(b)S -1 x(y)S 792 X(using)S 900 X(the)S 972 X(Cut)S 1055 X(Rule)S 1152 X(w)S -1 x(e)S 1213 X (can)S 1289 X(\\shortcut")S 1495 X(an)S 1553 X(enor-)S 50 y 300 X (mous)S 409 X(generation)S 612 X(tree.)S 711 X(Consider)S 884 X (for)S 948 X(example)S 1111 X(an)S 1168 X(application)S 1383 X (term)S 1482 X(\()S XP /F26 21 24 2 0 21 29 29 24 0 <0F8000 0FE000 01E000 00E000 00F000 00F000 007000 007800 007800 003800 003C00 003C00 001C00 001E00 001E00 001E00 003E00 007F00 00FF00 01E700 03C780 078780 0F0380 1E03C0 1C03C0 3C01C0 7801E0 F001E0 E000E0> PXLC RP 1498 2398 XY F26(\025xM)S 1591 X F25(\))S F26(N)S 49 y 300 X F25 (where)S 418 X F26(N)S 469 X F25(ma)S -1 x(y)S 557 X(b)S 1 x(e)S 612 X(complex)S 773 X(and)S 852 X F26(M)S 909 X F25(ma)S -1 x(y)S 997 X(con)S -1 x(tain)S 1142 X(man)S -1 x(y)S 1253 X(o)S 1 x (ccurrenc)S 3 x(es)S 1475 X(of)S 1520 X F26(x)S F25(.)S 1573 X(The)S 50 y 300 X(shortcut)S 463 X(is)S 505 X(ac)S -1 x(hiev)S -1 x(ed)S 671 X(b)S -1 x(y)S 728 X(considering)S 945 X(instead)S 1087 X(of)S 1134 X(\()S F26(\025xM)S 1243 X F25(\))S F26(N)S 1297 X F25(,)S 1322 X(whic)S -1 x(h)S 1440 X(ma)S -1 x(y)S 1529 X(ha)S -1 x(v)S -1 x(e)S 1624 X(a)S 50 y 300 X(complex)S 465 X(normal)S 609 X (form,)S 722 X(the)S 796 X(t)S -1 x(w)S -1 x(o)S 878 X(terms)S 996 X (\()S F26(\025xM)S 1105 X F25(\))S F26(y)S 1160 X F25(\(with)S 1273 X (a)S 1310 X(v)S -2 x(ariable)S 1469 X F26(y)S 1 x F25(\))S 1524 X (and)S 1607 X F26(N)S 50 y 300 X F25(separately)S -3 x(.)S 2692 Y 375 X F30(3.)S XP /F30 55 24 3 0 22 28 28 24 0 <700000 7FFFE0 7FFFE0 7FFFE0 7FFFE0 7FFFC0 F00780 E00F00 E01E00 E03C00 003800 007800 00F000 00F000 01F000 01F000 03E000 03E000 03E000 03E000 07E000 07E000 07E000 07E000 07E000 07E000 07E000 03C000> PXLC RP 412 2692 XY F30(7.)S 465 X(Em)S -1 x(b)S 1 x(edding)S 717 X(Lemma.)S XP /F29 65 31 1 0 29 29 29 32 0 <00003000 00007000 00007000 0000F000 0000F800 0001F800 0001F800 00037800 00037800 00067800 00067C00 000C3C00 000C3C00 00183C00 00183C00 00303C00 00303C00 00601E00 00601E00 00FFFE00 00FFFE00 01801E00 01801E00 03001F00 03000F00 07000F00 0F000F00 7FC0FFF0 FFC0FFF0> PXLC RP 904 2692 XY F29(Assume)S 1057 X(that)S 1146 X(all)S 1203 X (subterms)S 1380 X(of)S 1427 X F26(M)S 1484 X F29(ha)S -1 x(v)S -1 x (e)S 1579 X(lev-)S 49 y 300 X(els)S 360 X F27(\024)S 404 X F26(n)S F29(.)S 459 X(Then)S 567 X(w)S -1 x(e)S 628 X(can)S 705 X(\014nd)S 788 X F26(k)S 824 X F29(suc)S -1 x(h)S 918 X(that)S 1008 X F27(`)S XP /F8 10 24 1 0 22 20 20 24 0 <03FE00 0F8F80 1E03C0 3C01E0 7800F0 7800F0 7800F0 7800F0 7800F0 7800F0 3C01E0 3C01E0 1C01C0 0E0380 C60318 C60318 E30638 7F07F0 7F07F0 7F07F0> PXLC RP 1033 2726 XY F8(\012)S F10(\001)S F9(k)S 26 y 1033 X(n)S -11 y 1092 X F26(M)S 1136 X(~)S 1137 X(y)S 1 x F29(,)S 1184 X(where)S 1304 X F25 (\012)S 1345 X(:=)S 1401 X F27(h)S F26(\030)S 2 x F27(i)S 7 y F9 (\030)S XP /F10 50 22 3 -3 17 17 20 16 0 <07FC 1FFC 3E00 7800 7000 6000 E000 E000 C000 FFFC FFFC C000 E000 E000 6000 7000 7800 3E00 1FFC 07FC> PXLC RP 1469 2748 XY F10(2)S F8(\012)S 4 y F0(1)S -11 y 1534 X F29(.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 124 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(39)S XP /F25 54 21 2 -1 18 28 29 16 0 <01F8 07FC 0F8E 1E1E 1C1E 3C1E 381E 7800 7000 7080 F3F0 F7F8 FC3C F81E F80E F00F F00F F00F F00F F00F F00F 700F 700F 780E 381E 3C1C 1E3C 0FF8 07E0> PXLC RP 342 462 XY F25(6)S 700 X F18(H.)S 752 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S 1023 X(and)S 1098 X(S.S.)S 1175 X(W)S -3 x(ainer)S 553 Y 375 X F30(Pro)S 1 x(of.)S 525 X F25 (By)S 597 X(induction)S 788 X(on)S 852 X F26(M)S 897 X F25(.)S 947 X F28(Case)S 1055 X F26(x)S F28(.)S 1130 X F25(The)S 1222 X(claim)S 1339 X(follo)S -1 x(ws)S 1482 X(from)S 1587 X(the)S 56 y 300 X(V)S -3 x(ariable)S 469 X(Lemma)S 622 X(with)S 721 X F26(k)S 763 X F25 (:=)S 827 X(lev\()S F26(x)S F25(\))S 948 X(+)S 992 X(1.)S 1058 X F28 (Cases)S XP /F27 63 32 2 0 30 28 28 32 0 <00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFF0 FFFFFFF0> PXLC RP 1181 609 XY F27(?)S -15 y F30(nat)S 15 y 2 x F28(,)S 1317 X F25(0)S F28(,)S 1371 X F27(?)S -15 y F30(b)S 1 x(o)S 1 x(ole)S 15 y 2 x F28 (,)S 1550 X F25(t)S 1559 X(t)S F28(,)S 1609 X F25(f)S 1618 X(f)S 2 x(.)S 50 y 300 X(Ob)S -1 x(vious.)S 486 X F28(Case)S 590 X F25 (succ)S 2 x(.)S 705 X(By)S 773 X(the)S 847 X(V)S -3 x(ariable)S 1014 X(Rule)S 1114 X F27(`)S -15 y F8(1)S 25 y 1139 X F9(n)S -10 y 1171 X F26(y)S 1 x F25(,)S 1222 X(hence)S 1340 X F27(`)S -15 y F8 (\012)S 25 y 1365 X F9(n)S -10 y 1400 X F26(y)S 1439 X F25(b)S -1 x (y)S 1499 X(the)S 1573 X F26(<)S 6 y F8(1)S -6 y 2 x F25({)S 50 y 300 X(Rule,)S 411 X(hence)S 528 X F27(`)S -15 y F8(\012+1)S 25 y 553 X F9(n)S -10 y 628 X F25(succ)S 716 X F26(y)S 753 X F25(b)S -1 x (y)S 812 X(the)S 885 X(Constan)S -1 x(t)S 1066 X(Rule)S 1165 X(for)S 1230 X(succ)S 1 x(,)S 1334 X(hence)S 1451 X F27(`)S -15 y F8(\012)S F10(\001)S F8(2)S 25 y 1476 X F9(n)S -10 y 1535 X F25(succ)S 1623 X F26(y)S 50 y 300 X F25(again)S 406 X(b)S -1 x(y)S 460 X(the)S 527 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule.)S 710 X F28(Cases)S 825 X F25 (cond)S 6 y 1 x F9(\023)S -6 y 2 x F28(,)S 949 X F25(p)S 1 x(cond)S 10 y 1 x F9(\023)S -10 y 2 x F28(,)S XP /F26 99 18 1 0 17 18 18 16 0 <00FC 03FE 0F8F 1E0F 3C1F 3C0E 7800 7800 F800 F000 F000 F000 F000 F006 700E 783C 3FF8 0FE0> PXLC RP 1098 759 XY F26(c)S XP /F8 61 25 2 2 22 12 10 24 0 PXLC RP 1116 765 XY F8(=)S -6 y 2 x F28(,)S 1167 X F26(c)S XP /F9 60 26 3 -3 21 17 20 24 0 <0001C0 0007C0 000F80 003E00 00FC00 03F000 0FC000 1F0000 7C0000 F80000 F80000 7C0000 1F0000 0FC000 03F000 00FC00 003E00 000F80 0007C0 0001C0> PXLC RP 1185 765 XY F9(<)S -6 y 2 x F25(.)S XP /F25 65 31 1 0 29 29 29 32 0 <00060000 000F0000 000F0000 000F0000 001F8000 001F8000 001F8000 001F8000 0033C000 0033C000 0033C000 0061E000 0061E000 0061E000 00C0F000 00C0F000 00C0F000 01807800 01807800 01FFF800 03FFFC00 03003C00 03003C00 06001E00 06001E00 0E001E00 1F001F00 FFC0FFF0 FFC0FFF0> PXLC RP 1242 759 XY F25(As)S 1299 X(for)S 1358 X(Case)S 1453 X(succ)S 1 x (.)S 1558 X F28(Case)S 57 y 300 X F26(R)S F25(.)S 361 X(Consider)S 530 X F26(Ry)S XP /F26 119 30 1 0 28 18 18 32 0 <1F00E1E0 3F80E1E0 7380E1E0 6381E1E0 E381C0E0 C781C0E0 0701C060 0703C0E0 0F0380C0 0E0380C0 0E0380C0 0E0381C0 0E078180 0E078380 0E0F8300 0F1FCF00 07FFFE00 03F0F800> PXLC RP 584 816 XY F26(w)S 1 x(x~)S 639 X(y)S 1 x F25(.)S 689 X(W)S -3 x(e)S 757 X(w)S -1 x(an)S -1 x(t)S 855 X(to)S 902 X(apply)S 1013 X(the)S 1081 X F26(!)S -15 y 1 x F8(+)S 15 y 3 x F25({Rule)S 1250 X(to)S 1297 X(get)S 1363 X F27(`)S -21 y F8(\012)S F10(\001)S F8(\()S F9 (k)S 1 x F8(+1\))S 26 y 1388 X F9(n)S -5 y 1515 X F26(Ry)S 1 x(w)S 1 x(x~)S 1623 X(y)S 50 y 300 X F25(with)S 398 X F26(k)S 438 X F25 (=)S 487 X(max\(lev\()S F26(y)S 1 x F25(\))S F26(;)S 705 X F25 (1\).)S 783 X(Since)S 894 X(\(\012)S XP /F27 1 12 3 8 8 13 5 8 0 <70 F8 F8 F8 70> PXLC RP 952 866 XY F27(\001)S 975 X F25(\()S F26(k)S 1025 X F25(+)S 1069 X (1\)\))S 6 y F8(0)S -6 y 1158 X F25(=)S 1207 X(\012)S 1249 X F27 (\001)S 1272 X F26(k)S 1312 X F25(and)S 1396 X F27(`)S -15 y F8 (\012)S F10(\001)S F9(k)S 26 y 1421 X F8(0)S -11 y 1485 X F26(x)S 1526 X F25(b)S -1 x(y)S 1587 X(the)S 50 y 300 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule,)S 481 X(w)S -1 x(e)S 542 X(only)S 633 X(ha)S -1 x(v)S -1 x(e)S 729 X(to)S 779 X(sho)S -1 x(w)S 1004 Y 623 X F27 (8)S F26(\030)S 678 X F27(2)S 717 X F25(\012)S 6 y F8(1)S -6 y 2 x F27(8)S F26(m:)S 848 X F27(`)S -20 y F9(\030)S 31 y 873 X F8(0)S -11 y 897 X F30(m)S 960 X F25(=)S 985 X F27(\))S 1038 X(`)S -17 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+)S F9(\030)S 27 y 1063 X(n)S -10 y 1164 X F26(Ry)S 1 x(w)S 1 x F30(m)S F26(~)S 1288 X(y)S 2 x(:)S 1098 Y 300 X F25(This)S 394 X(is)S 434 X(done)S 532 X(b)S -1 x(y)S 589 X(induction)S 772 X(on)S 829 X F27(`)S -20 y F9(\030)S 31 y 854 X F8(0)S -11 y 878 X F30(m)S F25(.)S 947 X(Note)S 1046 X(that)S 1135 X F27(`)S -20 y F9(\030)S 31 y 1160 X F8(0)S -11 y 1184 X F30 (m)S 1236 X F25(can)S 1311 X(only)S 1401 X(b)S 1 x(e)S 1457 X (deriv)S -1 x(ed)S 1601 X(b)S -1 x(y)S 50 y 300 X(a)S 335 X (Constan)S -1 x(t)S 515 X(Rule)S 613 X(or)S 664 X(the)S 736 X F26 (<)S 6 y F8(1)S -6 y 2 x F25({Rule)S 906 X(\(the)S 994 X F26(!)S -15 y 1 x F8(+)S 15 y 3 x F25({Rule)S 1168 X(is)S 1210 X(excluded,)S 1394 X(since)S 1496 X F26(\030)S 1529 X F27(2)S 1569 X F25(\012)S 6 y F8(1)S -6 y 1 x F25(\).)S 49 y 300 X F28(Constant)S 480 X(R)S -1 x(ules)S 3 x F25(.)S 619 X F28(Case)S 723 X F25(0.)S 782 X (Since)S 892 X F26(Ry)S 1 x(w)S 1 x F25(0)S F26(~)S 997 X(y)S 1036 X F25(is)S 1080 X(iden)S -1 x(ti\014ed)S 1264 X(with)S 1361 X F26(y)S 1 x(~)S 1382 X(y)S 1421 X F25(it)S 1464 X(su)S XP /F25 14 35 0 0 33 29 29 40 0 <003F83F000 01FFDFF800 03E1FC3C00 0781F83C00 0F01F03C00 0E01E03C00 0E00E00000 0E00E00000 0E00E00000 0E00E00000 0E00E00000 FFFFFFFC00 FFFFFFFC00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 0E00E01C00 7FC7FCFF80 7FC7FCFF80> PXLC RP 1504 1197 XY F25(\016ces)S 1608 X(to)S 50 y 300 X(sho)S -1 x(w)S 404 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+1)S 26 y 429 X F9(n)S -11 y 533 X F26(y)S 1 x(~)S 554 X(y)S 3 x F25(.)S 612 X (Since)S 722 X(lev\()S F26(y)S 6 y F9(i)S -6 y 3 x F25(\))S 855 X F26(<)S 901 X(k)S 1 x F25(,)S 952 X(this)S 1034 X(follo)S -1 x(ws)S 1172 X(from)S 1272 X(the)S 1345 X(V)S -3 x(ariable)S 1511 X(Lemma)S 50 y 300 X(and)S 377 X(the)S 444 X(V)S -3 x(ariable)S 604 X(Rule.)S 716 X F28(Case)S 815 X F25(succ)S 1 x(.)S 920 X(By)S 981 X (induction)S 1161 X(h)S -1 x(yp)S 1 x(othesis)S 1362 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+)S F9(\030)S 25 y 1387 X(n)S -10 y 1487 X F26(Ry)S 1 x(w)S 1 x F30(m)S F26(~)S 1611 X(y)S 2 x F25 (.)S 50 y 300 X(Since)S 413 X F26(Ry)S 1 x(w)S 1 x F25(\(succ)S 610 X F30(m)S F25(\))S F26(~)S 666 X(y)S 707 X F25(is)S 754 X(iden)S -1 x(ti\014ed)S 940 X(with)S 1039 X F26(w)S 1 x F30(m)S F25(\()S F26 (Ry)S 1 x(w)S 1 x F30(m)S F25(\))S F26(~)S 1266 X(y)S 1307 X F25 (it)S 1354 X(su\016ces)S 1500 X(to)S 1556 X(sho)S -1 x(w)S 50 y 300 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+)S F9(\030)S 1 x F8(+1)S 25 y 325 X F9(n)S -10 y 473 X F26(w)S 1 x F30(m)S F25 (\()S F26(Ry)S 1 x(w)S 1 x F30(m)S F25(\))S F26(~)S 700 X(y)S 2 x F25(.)S 762 X(This)S 860 X(follo)S -1 x(ws)S 999 X(from)S 1101 X (the)S 1176 X(V)S -3 x(ariable)S 1343 X(Rule,)S 1456 X(since)S 1561 X (from)S 54 y 300 X F27(`)S -20 y F9(\030)S 31 y 325 X F8(0)S -11 y 362 X F30(m)S 424 X F25(w)S -1 x(e)S 493 X(get)S 570 X F27(`)S -20 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+)S F9(\030)S 31 y 595 X F8(0)S -11 y 709 X F30(m)S 771 X F25(b)S -1 x(y)S 836 X(the)S 916 X (Substitution)S 1162 X(Lemma.)S 1350 X F28(Cases)S 1477 X F25(cond)S 12 y F30(nat)S -12 y 1 x F28(,)S 50 y 300 X F25(p)S 1 x(cond)S 12 y 1 x F30(nat)S -12 y 1 x F25(.)S 526 X(cond)S 12 y F30(nat)S -12 y 1 x F26(M)S 726 X(N)S 764 X(L)S 811 X F25(or)S 867 X(p)S 1 x(cond)S 12 y 1 x F30(nat)S -12 y 1 x F26(M)S 1092 X(N)S 1130 X(L)S 1177 X F25(can)S 1259 X(only)S 1355 X(ha)S -1 x(v)S -1 x(e)S 1456 X F30(m)S 1515 X F25(as)S 1571 X(nor-)S 50 y 300 X(mal)S 383 X(form)S 483 X (when)S 593 X(either)S 713 X F26(N)S 766 X F25(or)S 819 X F26(L)S 863 X F25(do)S 1 x(es.)S 979 X(Supp)S 1 x(ose)S 1144 X(it)S 1187 X (is)S 1231 X F26(N)S 1269 X F25(.)S 1304 X(Then)S 1415 X(b)S -1 x (y)S 1474 X(induction)S 50 y 300 X(h)S -1 x(yp)S 1 x(othesis)S 506 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+)S F9(\030)S 25 y 531 X(n)S -10 y 633 X F26(Ry)S 1 x(w)S 1 x F30(m)S F26(~)S 757 X(y)S 2 x F25(,)S 806 X(hence)S 922 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+)S F9(\030)S 1 x F8(+1)S 25 y 947 X F9(n)S -10 y 1092 X F26(Ry)S 1 x(w)S 1 x F30(m)S F26(~)S 1216 X(y)S 1253 X F25 (b)S -1 x(y)S 1312 X(the)S 1384 X F26(<)S 6 y F8(1)S -6 y 2 x F25 ({Rule.)S 1573 X F26(<)S 6 y F8(1)S -6 y 2 x F28({)S 50 y 300 X(R)S -1 x(ule)S 3 x F25(.)S 419 X(The)S 507 X(claim)S 619 X(follo)S -1 x (ws)S 757 X(from)S 857 X(the)S 930 X(induction)S 1116 X(h)S -1 x (yp)S 1 x(othesis)S 1323 X(and)S 1405 X(the)S 1478 X F26(<)S 6 y F8 (1)S -6 y 2 x F25({Rule.)S 49 y 300 X F28(Case)S 403 X F26(\025xM)S 496 X F25(.)S 530 X(By)S 596 X(induction)S 782 X(h)S -1 x(yp)S 1 x (othesis)S 988 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 26 y 1013 X(n)S -11 y 1074 X F26(M)S 1118 X(~)S 1119 X(y)S 1 x F25(,)S 1167 X(whic)S -1 x(h)S 1287 X(sa)S -1 x(ys)S 1377 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 26 y 1402 X(n)S -11 y 1463 X F25(\()S F26(\025xM)S 1572 X F25(\))S F26(x~)S 1612 X(y)S 1 x F25(,)S 50 y 300 X(since)S 399 X(b)S 1 x(oth)S 494 X(terms)S 607 X(ha)S -1 x(v)S -1 x(e)S 700 X(the)S 769 X(same)S 871 X(normal)S 1009 X(form)S 1104 X (and)S 1182 X(hence)S 1295 X(are)S 1362 X(iden)S -1 x(ti\014ed.)S 1558 X F28(Case)S 50 y 300 X F26(M)S 345 X(N)S 383 X F25(.)S 411 X (By)S 473 X(induction)S 653 X(h)S -1 x(yp)S 1 x(othesis)S 854 X F27 (`)S -15 y F8(\012)S F10(\001)S F9(k)S 25 y 879 X(n)S -10 y 938 X F26(M)S 983 X(y)S 1 x(~)S 1004 X(y)S 1037 X F25(and)S 1113 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 25 y 1138 X(n)S -10 y 1197 X F26 (N)S 1234 X(~)S 1235 X(x)S F25(.)S 1288 X(Also)S 1377 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 25 y 1402 X(n)S -10 y 1461 X F26(y)S 6 y F9(i)S -6 y 1499 X F26(~)S 1496 X(y)S 6 y F9(i)S -6 y 2 x F25 (.)S 1558 X(Note)S 50 y 300 X(that)S 391 X(b)S -1 x(y)S 449 X(the)S 521 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule)S 691 X(w)S -1 x(e)S 752 X(can)S 829 X(assume)S 973 X(that)S 1064 X(w)S -1 x(e)S 1125 X (ha)S -1 x(v)S -1 x(e)S 1222 X(the)S 1294 X(same)S 1398 X F26(k)S 1436 X F25(in)S 1485 X(all)S 1543 X(cases.)S 55 y 300 X(Then)S 407 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 1 x F8(+1)S 26 y 432 X F9 (n)S -11 y 533 X F26(M)S 578 X(N)S 615 X(~)S 616 X(y)S 651 X F25(b)S -1 x(y)S 707 X(the)S 778 X(Cut)S 860 X(Rule,)S 968 X(hence)S 1082 X F27(`)S -21 y F8(\012)S F10(\001)S F8(\()S F9(k)S 1 x F8(+1\))S 26 y 1107 X F9(n)S -5 y 1234 X F26(M)S 1279 X(N)S 1316 X(~)S 1317 X(y)S 1351 X F25(b)S -1 x(y)S 1408 X(the)S 1478 X F26(<)S 7 y F8(1)S -7 y 2 x F25({Rule.)S 1628 1955 XY 17 17 R 2043 Y 375 X(No)S -1 x(w)S 469 X(w)S -1 x(e)S 529 X(obtain)S 658 X(the)S 729 X(desired)S 869 X (estimate)S 1035 X(in)S 1083 X(terms)S 1198 X(of)S 1244 X(the)S 1315 X(slo)S -1 x(w{gro)S -1 x(wing)S 1569 X(sub-)S 50 y 300 X (recursiv)S -1 x(e)S 473 X(hierarc)S -1 x(h)S -1 x(y)S -3 x(.)S 2181 Y 375 X F30(3.)S XP /F30 56 24 2 0 21 27 27 24 0 <03FC00 0FFF00 3FFF80 3E0FC0 7807C0 7803C0 7C03C0 7E03C0 7F87C0 7FEF80 3FFF00 3FFE00 1FFF00 0FFF80 3FFFC0 7CFFC0 783FE0 F80FE0 F003E0 F001E0 F001E0 F801E0 F803C0 7E0FC0 3FFF80 1FFF00 07FC00> PXLC RP 412 2181 XY F30(8.)S XP /F30 84 33 2 0 30 28 28 32 0 <7FFFFFE0 7FFFFFE0 7FFFFFE0 7C1F83E0 781F81E0 701F80E0 F01F80F0 E01F8070 E01F8070 E01F8070 E01F8070 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 001F8000 07FFFE00 07FFFE00 07FFFE00> PXLC RP 465 2181 XY F30(T)S XP /F30 104 27 1 0 25 29 29 24 0 PXLC RP 498 2181 XY F30(heorem.)S 686 X F29(Let)S 763 X F26(M)S 825 X F29 (b)S 1 x(e)S 885 X(a)S 923 X(closed)S 1049 X(term)S 1151 X(of)S 1202 X(t)S -1 x(yp)S 1 x(e)S 1298 X F30(nat)S XP /F27 33 42 2 3 39 17 14 40 0 <0000000600 0000000700 0000000700 0000000380 00000001C0 00000000E0 FFFFFFFFF8 FFFFFFFFF8 00000000E0 00000001C0 0000000380 0000000700 0000000700 0000000600> PXLC RP 1384 2181 XY F27(!)S 1442 X F30(nat)S F29(.)S 1550 X(Then)S 50 y 300 X(there)S 407 X(is)S 449 X(a)S 484 X F26(k)S 521 X F29(suc)S -1 x(h)S 615 X(that)S 706 X(for)S 770 X(all)S 828 X F26(n)S 865 X F27(\025)S 910 X F25(1)S 945 X F29(the)S 1017 X(v)S -2 x(alue)S 1125 X(of)S 1172 X F26(M)S 1217 X F30(n)S 1258 X F29(is)S 1300 X (either)S 1418 X(unde\014ned)S 1608 X(or)S 50 y 300 X F27(\024)S 344 X F26(G)S 6 y F9(\013)S -6 y 1 x F25(\()S F26(n)S F25(\))S F29 (,)S 483 X(with)S 577 X F26(\013)S 615 X F25(:=)S 671 X F26(D)S 1 x F25(\()S F26(E)S -15 y 2 x F9(k)S 15 y 3 x F25(\(\012)S 831 X F27 (\001)S 852 X F26(k)S 1 x F25(\))S 900 X F27(\001)S 921 X F25(\(1)S 967 X(+)S 1009 X F26(!)S 1 x F25(\)\))S F29(.)S 2369 Y 375 X F30 (Pro)S 1 x(of.)S 525 X F25(Consider)S 702 X F26(M)S 747 X(x)S F25 (.)S 813 X(By)S 882 X(the)S 957 X(Em)S -1 x(b)S 1 x(edding)S 1181 X (Lemma)S 1333 X(w)S -1 x(e)S 1399 X(\014nd)S 1486 X(a)S 1524 X F26 (k)S 1565 X F25(suc)S -1 x(h)S 50 y 300 X(that)S 391 X F27(`)S -15 y F8(\012)S F10(\001)S F9(k)S 27 y 416 X(k)S -12 y 477 X F26(M)S 522 X (x)S F25(.)S 579 X(Hence)S 704 X F27(`)S -20 y F9(\015)S 31 y 729 X F8(0)S -11 y 757 X F26(M)S 802 X(x)S 841 X F25(with)S 936 X F26 (\015)S 974 X F25(:=)S 1031 X F26(E)S -15 y 2 x F9(k)S 15 y 3 x F25 (\()S (\012)S 1141 X F27(\001)S 1162 X F26(k)S 1 x F25(\))S 1216 X(b)S -1 x(y)S 1275 X(the)S 1348 X(Cut)S 1432 X(Elimination)S 50 y 300 X (Lemma.)S 464 X(Note)S 565 X(that)S 655 X F27(`)S -20 y F9(\015)S 2 x F10(\001)S F9(n)S 31 y 680 X F8(0)S -11 y 736 X F30(n)S XP /F25 59 12 3 -8 8 18 26 8 0 <70 F8 F8 F8 70 00 00 00 00 00 00 00 00 70 F8 F8 F8 78 18 18 18 38 30 70 E0 60> PXLC RP 763 2469 XY F25(;)S 788 X(this)S 869 X(follo)S -1 x(ws)S 1005 X (from)S 2557 Y 502 X F26(\015)S 537 X F25(=)S 581 X F26(E)S -17 y 2 x F9(k)S 17 y 3 x F25(\(\012)S 690 X F27(\001)S 711 X F26(k)S 1 x F25(\))S 761 X F26(>)S 6 y F8(1)S -6 y 824 X F26(E)S -17 y 2 x F9 (k)S 17 y 2 x F25(\(\012)S 933 X F27(\001)S 953 X F25(\()S F26(k)S XP /F27 0 32 3 9 28 11 2 32 0 PXLC RP 1002 2557 XY F27(\000)S 1043 X F25(1\))S 1089 X(+)S 1131 X(1\))S 1179 X F27(\025)S 6 y F8(1)S -6 y 1242 X F26(E)S -17 y 2 x F9(k)S 17 y 2 x F25(\(1\))S 1360 X F27(\025)S 6 y F8(1)S -6 y 1422 X F25 (2)S 2645 Y 300 X(b)S -1 x(y)S 360 X(induction)S 547 X(an)S 607 X F26(n)S F25(:)S 666 X(F)S -3 x(or)S 743 X F26(n)S 783 X F25(=)S 831 X(1)S 868 X(w)S -1 x(e)S 931 X(obtain)S 1063 X F27(`)S -20 y F9 (\015)S 31 y 1088 X F8(0)S -11 y 1118 X F25(1)S 1155 X(from)S 1255 X F27(`)S -15 y F8(2)S 25 y 1280 X(0)S -10 y 1308 X F25(1)S 1344 X(b)S -1 x(y)S 1404 X(the)S 1478 X F26(<)S 6 y F8(1)S -6 y 2 x F25 ({Rule,)S 50 y 300 X(and)S 382 X(from)S 480 X F27(`)S -20 y F9 (\015)S 2 x F10(\001)S F9(n)S 31 y 505 X F8(0)S -11 y 563 X F30(n)S 605 X F25(w)S -1 x(e)S 666 X(\014rst)S 753 X(obtain)S 883 X F27(`)S -20 y F9(\015)S 2 x F10(\001)S F9(n)S F8(+1)S 31 y 908 X(0)S -11 y 1008 X F25(\()S F30(n)S 1060 X F25(+)S 1101 X F30(1)S F25(\))S 1156 X (and)S 1237 X(then)S 1333 X(using)S 1442 X(1)S 1476 X F26(<)S 6 y F8(1)S -6 y 1540 X F26(\015)S 1578 X F25(and)S 58 y 300 X(the)S 371 X F26(<)S 7 y F8(1)S -7 y 2 x F25({Rule)S 541 X F27(`)S -21 y F9(\015)S 2 x F10(\001)S F8(\()S F9(n)S F8(+1\))S 32 y 566 X(0)S -11 y 690 X F25(\()S F30(n)S 742 X F25(+)S 784 X F30(1)S -1 x F25 (\))S 1 x(.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 122 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup 692 462 XY F18(Ordinal)S 833 X(Bounds)S 971 X(for)S 1030 X (Programs)S 1583 X F25(39)S XP /F25 55 21 2 -1 19 28 29 24 0 <600000 7FFF80 7FFF80 7FFF80 600700 C00600 C00E00 C01C00 001800 003800 007000 006000 00E000 00E000 01C000 01C000 01C000 03C000 038000 038000 038000 078000 078000 078000 078000 078000 078000 078000 078000> PXLC RP 1624 462 XY F25(7)S 555 Y 375 X(So)S 430 X(w)S -1 x(e)S 488 X(ha)S -1 x(v)S -1 x(e)S 581 X F27(`)S -22 y F9(\015)S 2 x F10(\001)S F8 (\()S F9(n)S F8(+1\))S 33 y 606 X(0)S -11 y 730 X F26(M)S 775 X F30 (n)S 813 X F25(b)S -1 x(y)S 867 X(the)S 936 X(Substitution)S 1171 X (Lemma.)S 1334 X(Let)S 1406 X F30(m)S 1456 X F25(denote)S 1587 X (the)S 50 y 300 X(n)S -1 x(umerical)S 489 X(v)S -2 x(alue)S 595 X (of)S 641 X F26(M)S 686 X F30(n)S 725 X F25(\(assuming)S 920 X(it)S 960 X(is)S 1000 X(de\014ned\).)S 1175 X(Then)S 1282 X(with)S 1376 X F26(\013)S 1414 X F25(:=)S 1469 X F26(D)S 1 x F25(\()S F26(\015)S 1551 X F27(\001)S 1570 X F25(\(1)S 1614 X(+)S 50 y 300 X F26(!)S 1 x F25(\)\))S 371 X(w)S -1 x(e)S 430 X(ha)S -1 x(v)S -1 x(e)S 524 X F26(\013)S 6 y F9(n)S -6 y 585 X F25(=)S 629 X F26(D)S 1 x F25(\()S F26(\015)S 709 X F27(\001)S 726 X F25(\()S F26(n)S 772 X F25(+)S 809 X(1\)\))S 874 X(and)S 953 X(hence)S 1066 X F27(`)S -19 y F9 (\013)S XP /F1 110 18 2 0 16 9 9 16 0 <7BE0 4E70 DC70 1C60 1860 1864 38EC 38D8 30F0> PXLC RP 1113 640 XY F1(n)S 26 y 1091 X F8(0)S -11 y 1140 X F30(m)S 1192 X F25(b)S -1 x(y)S 1248 X(the)S 1317 X(Second)S 1456 X(Collapsing)S 50 y 300 X(Lemma,)S 460 X(since)S 561 X(it)S 602 X(is)S 644 X(easy)S 735 X(to)S 786 X(c)S -1 x(hec)S XP /F25 107 22 1 0 20 29 29 24 0 PXLC RP 862 705 XY F25(k)S 897 X(that)S 987 X(\012,)S 1042 X(\012)S 1081 X F27(\001)S 1102 X F26(k)S 1138 X F25(and)S 1219 X F26(E)S -16 y 2 x F9(k)S 16 y 2 x F25(\(\012)S 1327 X F27(\001)S 1348 X F26(k)S 1 x F25(\))S 1400 X(all)S 1458 X(satisfy)S 1587 X(the)S 49 y 300 X (required)S 463 X(condition)S 645 X(\(the)S 733 X(only)S 824 X(imp)S 1 x(ortan)S -1 x(t)S 1020 X(thing)S 1129 X(to)S 1179 X(notice)S 1302 X(is)S 1344 X(that)S 1434 X F26(E)S 1481 X F25(preserv)S -1 x (es)S 59 y 300 X(the)S 375 X F27(\024)S 6 y F8(1)S -6 y 1 x F25 ({ordering\).)S 651 X(Hence)S 778 X F27(`)S -22 y F9(G)S 4 y F1 (\013)S 4 y(n)S -8 y 870 X F8(\(1\))S 33 y 803 X(0)S -11 y 925 X F30 (m)S 981 X F25(b)S -1 x(y)S 1042 X(the)S 1116 X(First)S 1221 X (Collapsing)S 1427 X(Lemma)S 1578 X(and)S 50 y 300 X F26(m)S 353 X (<)S 401 X(G)S 6 y F9(\013)S 4 y F1(n)S -10 y 478 X F25(\(1\))S 548 X(b)S -1 x(y)S 608 X(the)S 683 X(V)S -3 x(alue)S 801 X(Lemma.)S 974 X(But)S 1059 X(the)S 1134 X F26(G)S 6 y F9(\013)S -6 y 1 x F25 ({functions)S 1393 X(are)S 1465 X(increasing)S 50 y 300 X(and)S 381 X(so)S 50 y 677 X F26(m)S 725 X(<)S 769 X(G)S 7 y F9(\013)S 4 y F1(n)S -11 y 3 x F25(\(1\))S 910 X F27(\024)S 954 X F26(G)S 7 y F9 (\013)S 4 y F1(n)S -11 y 3 x F25(\()S F26(n)S F25(\))S 1099 X F27 (\024)S 1143 X F26(G)S 7 y F9(\013)S -7 y 1 x F25(\()S F26(n)S F25 (\))S F26(:)S 1628 1039 XY 17 17 R 1131 Y 375 X F30(Pro)S 1 x(of.)S 525 X F25(Consider)S 700 X F26(M)S 745 X F25(\(succ)S 851 X F26(x)S F25(\).)S 927 X(By)S 993 X(the)S 1067 X(Em)S -1 x(b)S 1 x(edding)S 1288 X(Lemma)S 1438 X(w)S -1 x(e)S 1501 X(\014nd)S 1586 X(a)S 1622 X F26(k)S 50 y 300 X F25(suc)S -1 x(h)S 398 X(that)S 492 X F27(`)S -16 y F8(\012)S F10(\001)S F9(k)S 27 y 517 X(k)S -11 y 583 X F26(M)S 628 X F25(\(succ)S 731 X F26(x)S F25(\).)S 814 X(Hence)S 941 X F27 (`)S -20 y F9(\015)S 31 y 966 X F8(0)S -11 y 999 X F26(M)S 1044 X F25(\(succ)S 1148 X F26(x)S F25(\))S 1206 X(with)S 1305 X F26(\015)S 1347 X F25(:=)S 1409 X F26(E)S -16 y 2 x F9(k)S 16 y 3 x F25 (\(\012)S 1521 X F27(\001)S 1544 X F26(k)S 1 x F25(\))S 1601 X(b)S -1 x(y)S 49 y 300 X(the)S 374 X(Cut)S 461 X(Eliminati)S -3 x(on)S 690 X(Lemma.)S 863 X(Since)S 975 X F27(`)S -20 y F9(\015)S 2 x F10 (\001)S F9(n)S 31 y 1000 X F8(0)S -11 y 1061 X F25(\()S F30(n)S 1113 X F27(\000)S 1155 X F30(1)S F25(\))S 1212 X(\(b)S -1 x(y)S 1289 X(the)S 1363 X(Constan)S -1 x(t)S 1545 X(Rules)S 59 y 300 X (and)S 384 X(the)S 460 X F26(<)S 6 y F8(1)S -6 y 2 x F25({Rule\))S 649 X(w)S -1 x(e)S 713 X(get)S 786 X F27(`)S -22 y F9(\015)S 2 x F10 (\001)S F8(\()S F9(n)S F8(+1\))S 33 y 811 X(0)S -11 y 942 X F26(M)S 987 X F30(n)S 1031 X F25(b)S -1 x(y)S 1092 X(the)S 1168 X (Substitution)S 1409 X(Lemma.)S 1584 X(Let)S 50 y 300 X F30(m)S 354 X F25(denote)S 489 X(the)S 561 X(n)S -1 x(umerical)S 753 X(v)S -2 x(alue)S 861 X(of)S 909 X F26(M)S 954 X F30(n)S 995 X F25 (\(assuming)S 1192 X(it)S 1235 X(is)S 1277 X(de\014ned\).)S 1455 X (Then)S 1564 X(with)S 50 y 300 X F26(\013)S 346 X F25(:=)S 408 X F26 (D)S 1 x F25(\()S F26(\015)S 496 X F27(\001)S 520 X F25(\(1)S 569 X (+)S 613 X F26(!)S 1 x F25(\)\))S 691 X(w)S -1 x(e)S 757 X(ha)S -1 x (v)S -1 x(e)S 857 X F26(\013)S 6 y F9(n)S -6 y 925 X F25(=)S 976 X F26(D)S 1 x F25(\()S F26(\015)S 1064 X F27(\001)S 1088 X F25(\()S F26(n)S 1141 X F25(+)S 1185 X(1\)\))S 1257 X(and)S 1342 X(hence)S 1462 X F27(`)S -19 y F9(\013)S 4 y F1(n)S 26 y 1487 X F8(0)S -11 y 1543 X F30(m)S 1601 X F25(b)S -1 x(y)S 49 y 300 X(the)S 373 X (Second)S 515 X(Collapsing)S 720 X(Lemma,)S 881 X(since)S 985 X(it)S 1028 X(is)S 1071 X(easy)S 1164 X(to)S 1216 X(c)S -1 x(hec)S -1 x(k)S 1329 X(that)S 1421 X(\012,)S 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21 17 20 24 0 <0000C0 0000C0 0001C0 0001C0 000380 000780 001F00 007E00 03F800 FFE000 FFE000 03F800 007E00 001F00 000780 000380 0001C0 0001C0 0000C0 0000C0> PXLC RP 1353 2660 XY F10(\036)S -6 y 1398 X F25(programs,)S 1595 X(for)S 49 y 300 X(example)S 468 X(that)S 563 X(the)S 639 X(functions)S 824 X(de\014nable)S 1004 X(in)S 1058 X(PCF)S 7 y F9(<)S -7 y 1190 X F25(are)S 1264 X(exactly)S 1413 X(the)S 1489 X(pro)S -1 x(v)S -2 x (ably)S 50 y 300 X(recursiv)S -1 x(e)S 473 X(functions)S 652 X(of)S 700 X(arithmetic.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 129 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY %%EndPageSetup 300 462 XY F25(398)S 700 X F18(H.)S 752 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S 1023 X(and)S 1098 X(S.S.)S 1175 X(W)S -3 x(ainer)S XP /F52 /cmr10 432 59.8 59.8 128 [-3 -16 59 45] PXLNF RP XP /F52 82 44 2 -1 43 41 42 48 0 PXLC RP 300 553 XY F52(R)S XP /F52 69 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00FFF180 03F07B80 07C01F80 0F800F80 1F000780 3E000780 3E000380 7C000380 7C000180 78000180 F8000180 F8000000 F8000000 F8000000 F8000000 F8000000 F800FFF0 F800FFF0 78000780 7C000780 7C000780 3E000780 3E000780 1F000780 0F800F80 07C00F80 03F83F80 00FFF980 003FC000> PXLC RP 453 1703 XY F25(G.)S 510 X(Handley)S 673 X(and)S 752 X(S.)S 800 X (S.)S 848 X(W)S -3 x(ainer.)S 1007 X(Equational)S 1217 X(deriv)S -2 x(ation)S 1411 X(vs.)S 1474 X(computa-)S 50 y 385 X(tion.)S 487 X (T)S -3 x(o)S 548 X(app)S 1 x(ear:)S 701 X(Annals)S 841 X(of)S 888 X (Pure)S 988 X(and)S 1069 X(Applied)S 1225 X(Logic.)S 1843 Y 321 X ([9])S 385 X(G.)S 443 X(D.)S 500 X(Plotkin.)S 663 X(LCF)S 760 X (considered)S 963 X(as)S 1013 X(a)S 1048 X(programming)S 1304 X (language.)S 1493 X F28(The)S -2 x(or)S -2 x(et-)S 50 y 385 X(ic)S -2 x(al)S 462 X(Computer)S 655 X(Scienc)S -2 x(e)S F25(,)S 814 X (5:223{255,)S 1017 X(1977.)S 1984 Y 300 X([10])S 385 X(H.)S 438 X (Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg.)S 746 X(Primitiv)S -1 x(e)S 925 X(recursion)S 1099 X(on)S 1152 X(the)S 1220 X(partial)S 1350 X(con)S -1 x(tin)S -1 x(uous)S 1554 X(func-)S 49 y 385 X(tionals.)S 543 X(In)S 598 X(M.)S 664 X(Bro)S -1 x(y)S -3 x(,)S 776 X(editor,)S 911 X F28(Informati)S XP /F28 107 19 3 0 20 29 29 24 0 <0FC000 1FC000 03C000 03C000 038000 038000 078000 078000 070000 070000 0F0000 0F0F00 0E3F00 0E7B80 1E6780 1ECF00 1D8F00 1F8F00 3F0000 3FC000 3BE000 38E000 78E380 78E380 70E300 70E700 F0E600 F07E00 E07C00> PXLC RP 1084 2033 XY F28(k)S 1120 X(und)S 1204 X(Mathematik)S F25(,)S 1446 X (pages)S 1562 X(251{)S 50 y 385 X(269.)S 473 X(Springer,)S 651 X (Berlin,)S 786 X(1991.)S 2174 Y 300 X([11])S 385 X(H.)S 446 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg.)S 768 X (Pro)S 1 x(ofs)S 902 X(as)S 958 X(programs.)S 1169 X(In)S 1225 X(P)S -3 x(.)S 1280 X(Aczel,)S 1409 X(H.)S 1469 X(Simmons,)S 50 y 385 X (and)S 470 X(S.)S 522 X(S.)S 574 X(W)S -3 x(ainer,)S 733 X (editors,)S 885 X F28(Pr)S -2 x(o)S -2 x(of)S 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(T)S -1 x(yp)S 1 x(e)S 854 X(theory)S -3 x(.)S 997 X(Course)S 1134 X (notes,)S 1253 X(Univ)S -1 x(ersit)S XP /F25 127 21 4 24 16 29 5 16 0 <70E0 F9F0 F9F0 F9F0 70E0> PXLC RP 1418 2414 XY F25(\177)S 1418 X(at)S 1467 X(M)S 1 x(\177)S 1505 X (unc)S -1 x(hen,)S 50 y 385 X(1992.)S 2554 Y 300 X([13])S 385 X(D.)S 449 X(Scott.)S 592 X(Domains)S 771 X(for)S 840 X(denotational)S 1086 X(seman)S -1 x(tics.)S 1309 X(In)S 1368 X(E.)S 1427 X(Nielsen)S 1578 X(and)S 50 y 385 X(E.)S 438 X(M.)S 500 X(Sc)S -1 x(hmidt,)S 673 X(editors,)S 820 X F28(A)S -1 x(utomata,)S 1023 X(L)S -2 x (anguages)S 1223 X(and)S 1303 X(Pr)S -2 x(o)S -2 x(gr)S -2 x (amming)S F25(,)S 1568 X(Lec-)S 50 y 385 X(ture)S 469 X(Notes)S 582 X(in)S 627 X(Computer)S 820 X(Science,)S 974 X(V)S -3 x(olume)S 1120 X(140,)S 1204 X(pages)S 1314 X(577{613.)S 1481 X(Springer,)S 49 y 385 X(Berlin,)S 523 X(1982.)S 641 X(A)S 688 X(corrected)S 870 X (and)S 952 X(expanded)S 1139 X(v)S -1 x(ersion)S 1282 X(of)S 1331 X (a)S 1367 X(pap)S 1 x(er)S 1486 X(prepared)S 50 y 385 X(for)S 449 X (ICALP'82,)S 658 X(Aarh)S -1 x(us,)S 812 X(Denmark.)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 129 395 689 %%PageBoundingBox: (atend) %%BeginPageSetup 1000 BP 3300 2550 PM /paper-automatic false SPS 692 0 XY %%EndPageSetup 692 462 XY F18(Ordinal)S 833 X(Bounds)S 971 X(for)S 1030 X (Programs)S 1583 X F25(399)S 553 Y 300 X([14])S 385 X(A.)S 439 X (S.)S 485 X(T)S -3 x(ro)S 1 x(elstra)S 660 X(and)S 739 X(D.)S 793 X (v)S -2 x(an)S 868 X(Dalen.)S 1002 X F28(Constructivism)S 1288 X (in)S 1336 X(Mathematics.)S 1592 X(A)S -1 x(n)S 50 y 385 X(Intr)S -2 x(o)S -2 x(duction)S F25(,)S 631 X(v)S -1 x(olume)S 772 X(121,)S 858 X(123)S 933 X(of)S 978 X F28(Studies)S 1121 X(in)S 1170 X(L)S -2 x(o)S -2 x(gic)S 1277 X(and)S 1356 X(the)S XP /F28 70 27 2 0 29 28 28 32 0 <01FFFFE0 01FFFFE0 003E03E0 003C01E0 003C00C0 003C00C0 007C00C0 007800C0 007818C0 007838C0 00F83000 00F03000 00F0F000 00FFF000 01FFE000 01E0E000 01E06000 01E0E000 03E0C000 03C0C000 03C00000 03C00000 07C00000 07800000 07800000 0F800000 FFF80000 FFF80000> PXLC RP 1424 603 XY F28(F)S -3 x(oundations)S 50 y 385 X(of)S 434 X (Mathematics)S F25(.)S 694 X(North{Holland,)S 990 X(Amsterdam,)S 1226 X(1988.)S 736 Y 300 X([15])S 385 X(S.)S 435 X(S.)S 484 X(W)S -3 x(ainer.)S 646 X(Slo)S -1 x(w)S 745 X(gro)S -1 x(wing)S 902 X(v)S -1 x(ersus)S 1028 X(fast)S 1109 X(gro)S -1 x(wing.)S XP /F28 74 22 3 -1 25 28 29 24 0 <003FFC 003FFC 0003E0 0003C0 0003C0 0007C0 0007C0 000780 000780 000F80 000F80 000F00 000F00 001F00 001F00 001E00 001E00 003E00 003E00 003C00 003C00 787C00 F87C00 F87800 F8F800 F1F000 F3E000 7FC000 3F0000> PXLC RP 1284 736 XY F28(Journal)S 1437 X(of)S 1487 X(Symb)S -2 x(olic)S 50 y 385 X(L)S -2 x(o)S -2 x(gic)S F25(,)S 505 X(54\(2\):608{6)S -3 x (14,)S XP /F25 74 21 1 -1 17 28 29 16 0 <1FFF 1FFF 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 F878 F878 F878 F8F8 F0F0 71F0 3FE0 1F80> PXLC RP 781 786 XY F25(June)S 881 X(1989.)S 1151 Y 300 X F18(Helm)S -1 x (ut)S 437 X(Sc)S -1 x(h)S -1 x(wic)S -1 x(h)S -1 x(ten)S -1 x(b)S 1 x(erg)S 1049 X(Stanle)S XP /F18 121 20 1 -7 18 16 23 24 0 PXLC RP 1154 1151 XY F18(y)S 1187 X(S.)S 1231 X(W)S -3 x(ainer)S XP /F18 70 25 1 0 22 26 26 24 0 PXLC RP 300 1200 XY F18(F)S -3 x(a)S XP /F18 107 20 1 0 19 26 26 24 0 PXLC RP 341 1200 XY F18(kult)S XP /F18 127 19 3 21 15 26 5 16 0 <79E0 F9F0 F9F0 F9F0 79E0> PXLC RP 408 1200 XY F18(\177)S 408 X(at)S 455 X(f)S 1 x(\177)S 467 X(ur)S XP /F18 77 35 1 0 33 26 26 32 0 PXLC RP 516 1200 XY F18(Mathematik)S 1049 X(Sc)S -1 x(ho)S 1 x(ol)S 1171 X (of)S 1214 X(Mathematics)S XP /F18 85 29 1 0 27 26 26 32 0 PXLC RP 300 1250 XY F18(Uni)S XP /F18 118 20 1 0 18 16 16 24 0 PXLC RP 361 1250 XY F18(v)S -1 x(ersit\177)S 453 X(at)S 500 X(M)S 1 x(\177)S 535 X(unc)S -1 x(hen)S 1049 X(Univ)S -1 x(ersit)S -1 x(y)S 1234 X (of)S XP /F18 76 24 1 0 21 26 26 24 0 PXLC RP 1278 1250 XY F18(Leeds)S XP /F18 84 28 1 0 26 26 26 32 0 <7FFFFF00 7FFFFF00 781E0F00 601E0300 601E0300 E01E0380 C01E0180 C01E0180 C01E0180 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 001E0000 03FFF000 03FFF000> PXLC RP 300 1300 XY F18(Theresienstr)S XP /F18 51 19 2 0 16 24 24 16 0 <0FC0 3FE0 78F0 7878 7C78 7878 3878 0078 00F0 01F0 0FC0 0FC0 00F0 0078 003C 003C 003C F83C F83C F83C F878 70F8 7FF0 1FC0> PXLC RP 520 1300 XY F18(3)S XP /F18 57 19 2 0 16 24 24 16 0 <0FC0 1FE0 3CF0 7878 F038 F038 F03C F03C F03C F03C F03C 707C 787C 3FFC 1FBC 023C 003C 0038 7878 7870 78F0 79E0 3FC0 1F00> PXLC RP 540 1300 XY F18(9)S 1049 X(Leeds)S 1157 X(LS)S XP /F18 50 19 2 0 16 24 24 16 0 <1FC0 3FF0 71F8 F078 F87C F83C F83C 703C 003C 007C 0078 00F8 00F0 01E0 03C0 0780 0700 0E0C 1C0C 380C 7018 FFF8 FFF8 FFF8> PXLC RP 1202 1300 XY F18(2)S 1234 X(9)S XP /F18 74 20 1 0 17 26 26 16 0 <0FFF 0FFF 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 0078 7078 F878 F878 F8F8 F9F0 7FE0 1F80> PXLC RP 1253 1300 XY F18(JT)S XP /F18 68 29 1 0 26 26 26 32 0 PXLC RP 300 1350 XY F18(D)S XP /F18 45 13 0 6 10 9 3 16 0 PXLC RP 329 1350 XY F18(-)S XP /F18 56 19 2 0 16 24 24 16 0 <0FC0 1FE0 3870 7038 6018 6018 7018 7838 7E70 3FF0 3FC0 1FF0 3FF8 79F8 70FC F03C E03C E01C E01C F01C 7038 7C78 3FF0 0FC0> PXLC RP 342 1350 XY F18(8)S XP /F18 48 19 1 0 17 24 24 16 0 <07E0 1FF8 3C3C 381C 700E 700E 700E F00F F00F F00F F00F F00F F00F F00F F00F F00F F00F 700E 700E 700E 381C 3C3C 1FF8 07E0> PXLC RP 361 1350 XY F18(000)S 432 X(M)S 1 x(\177)S 467 X(unc)S -1 x(hen)S 598 X(2)S XP /F18 69 26 1 0 24 26 26 24 0 PXLC RP 1049 1350 XY F18(England)S XP /F18 71 30 2 0 28 26 26 32 0 <007F0600 01FFE600 07E0FE00 0F803E00 1E001E00 3C001E00 3C000E00 78000E00 78000600 F0000600 F0000000 F0000000 F0000000 F0000000 F0000000 F003FFC0 F003FFC0 78001E00 78001E00 3C001E00 3C001E00 1E001E00 0F803E00 07E07E00 01FFFE00 007FC600> PXLC RP 300 1400 XY F18(German)S -1 x(y)S XP /F31 /cmtt10 300 41.5 41.5 128 [0 -10 22 30] PXLNF RP XP /F31 112 22 0 -9 20 18 27 24 0 PXLC RP 1049 1400 XY F31(p)S PF XP /F31 /cmtt10 300 41.5 41.5 128 [0 -10 22 30] PXLNF RP XP /F31 109 22 0 0 21 18 18 24 0 PXLC RP 1071 1400 XY F31(m)S XP /F31 116 22 1 0 18 23 23 24 0 <070000 070000 070000 070000 070000 FFFF00 FFFF00 FFFF00 070000 070000 070000 070000 070000 070000 070000 070100 070380 070380 078780 07CF80 07FF00 03FE00 01FC00> PXLC RP 1093 1400 XY F31(t)S XP /F31 54 22 2 0 19 25 25 24 0 <01FC00 07FE00 0FFF00 1F8F00 3E0F00 7C0F00 780F00 780000 F06000 FFFC00 FFFE00 FFFF00 FC1F00 F80F80 F00780 F00380 F00380 F00380 700780 780780 7C0F80 3E3F00 3FFE00 1FFC00 07F000> PXLC RP 1114 1400 XY F31(6)S XP /F31 115 22 3 0 18 18 18 16 0 <1FFC 7FFC FFFC F87C F01C F01C F800 7FC0 3FF8 0FFC 007E E01E F00E F01E FC3E FFFE FFFC EFF0> PXLC RP 1136 1400 XY F31(ss)S XP /F31 119 22 1 0 20 18 18 24 0 PXLC RP 1180 1400 XY F31(w)S XP /F31 64 22 2 0 20 25 25 24 0 <01FC00 07FE00 0FFF00 1F8F80 3E7F80 7CFF80 79FFC0 79E7C0 F3C3C0 F3C3C0 E381C0 E381C0 E381C0 E381C0 E381C0 F3C3C0 F3C3C0 79E780 79FF80 7CFF00 3E7E00 1F87C0 0FFFC0 07FF80 01FE00> PXLC RP 1202 1400 XY F31(@s)S XP /F31 117 22 0 0 21 18 18 24 0 PXLC RP 1245 1400 XY F31(u)S XP /F31 110 22 0 0 21 18 18 24 0 PXLC RP 1267 1400 XY F31(n)S XP /F31 46 22 8 0 14 6 6 8 0 <78 FC FC FC FC 78> PXLC RP 1289 1400 XY F31(.)S XP /F31 108 22 2 0 19 25 25 24 0 PXLC RP 1311 1400 XY F31(l)S XP /F31 101 22 3 0 19 18 18 16 0 <07E0 1FF8 3FFC 7C3E 781F F00F F007 FFFF FFFF FFFF E000 F000 F007 780F 7E1F 3FFE 0FFC 03F0> PXLC RP 1332 1400 XY F31(ee)S XP /F31 100 22 1 0 21 25 25 24 0 <007F00 007F00 007F00 000700 000700 000700 000700 07F700 1FFF00 3FFF00 7E3F00 7C1F00 F80F00 F00700 E00700 E00700 E00700 E00700 F00F00 F00F00 781F00 7E3F00 3FFFF0 1FFFF0 0FE7F0> PXLC RP 1376 1400 XY F31(ds.)S XP /F31 97 22 2 0 21 18 18 24 0 <3FF000 7FF800 7FFC00 787E00 781E00 000E00 007E00 0FFE00 3FFE00 7FEE00 FC0E00 F00E00 E00E00 F01E00 FC7E00 7FFFE0 7FFFE0 1FE3E0> PXLC RP 1441 1400 XY F31(a)S XP /F31 99 22 3 0 19 18 18 16 0 <07FC 1FFE 3FFE 7E1E 7C1E F800 F000 E000 E000 E000 F000 F000 F807 7C0F 7F1F 3FFE 1FFC 07F8> PXLC RP 1463 1400 XY F31(c.u)S XP /F31 107 22 0 0 20 25 25 24 0 PXLC RP 1529 1400 XY F31(k)S 49 y 300 X(sc)S XP /F31 104 22 0 0 21 25 25 24 0 PXLC RP 344 1449 XY F31(hw)S XP /F31 105 22 3 0 19 26 26 16 0 <0380 07C0 07C0 07C0 0380 0000 0000 0000 7FC0 7FC0 7FC0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 FFFF FFFF FFFF> PXLC RP 387 1449 XY F31(icht@)S XP /F31 114 22 1 0 20 18 18 24 0 PXLC RP 496 1449 XY F31(r)S XP /F31 122 22 1 0 19 18 18 24 0 <7FFFC0 7FFFC0 7FFFC0 700F80 701F00 703E00 007C00 00F800 01F000 03E000 07C000 0F8000 1F01C0 3E01C0 7C01C0 FFFFC0 FFFFC0 FFFFC0> PXLC RP 518 1449 XY F31(z.mathematik)S -3 x(.uni)S XP /F31 45 22 2 11 19 14 3 24 0 PXLC RP 867 1449 XY F31(-muenchen.d)S -3 x(e)S %%PageTrailer EP /paper-automatic true SPS 1 PP %%PageBoundingBox: 72 442 395 689 %%Trailer EndDviLaserDoc %%BoundingBox: 72 99 395 689 %%Pages: 21 %%DocumentFonts: